LMFDB - Embedded newform 354.5.d.a.235.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [354,5,Mod(235,354)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(354, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 5, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("354.235");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 354 = 2 \cdot 3 \cdot 59 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	354.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$36.5929669317$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			235.14
Character	$\chi$	$=$	354.235
Dual form			354.5.d.a.235.13

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.82843i q^{2} +5.19615 q^{3} -8.00000 q^{4} -17.0264 q^{5} +14.6969i q^{6} +47.1245 q^{7} -22.6274i q^{8} +27.0000 q^{9} +O(q^{10})$	\(q+2.82843i q^{2} +5.19615 q^{3} -8.00000 q^{4} -17.0264 q^{5} +14.6969i q^{6} +47.1245 q^{7} -22.6274i q^{8} +27.0000 q^{9} -48.1580i q^{10} -47.3890i q^{11} -41.5692 q^{12} -102.976i q^{13} +133.288i q^{14} -88.4719 q^{15} +64.0000 q^{16} -391.843 q^{17} +76.3675i q^{18} -404.590 q^{19} +136.211 q^{20} +244.866 q^{21} +134.036 q^{22} +655.082i q^{23} -117.576i q^{24} -335.101 q^{25} +291.260 q^{26} +140.296 q^{27} -376.996 q^{28} -363.904 q^{29} -250.236i q^{30} +620.853i q^{31} +181.019i q^{32} -246.240i q^{33} -1108.30i q^{34} -802.362 q^{35} -216.000 q^{36} -1917.15i q^{37} -1144.35i q^{38} -535.079i q^{39} +385.264i q^{40} +393.601 q^{41} +692.586i q^{42} -1612.33i q^{43} +379.112i q^{44} -459.713 q^{45} -1852.85 q^{46} -2351.32i q^{47} +332.554 q^{48} -180.278 q^{49} -947.808i q^{50} -2036.08 q^{51} +823.808i q^{52} +127.124 q^{53} +396.817i q^{54} +806.865i q^{55} -1066.31i q^{56} -2102.31 q^{57} -1029.28i q^{58} +(-3304.16 - 1095.38i) q^{59} +707.775 q^{60} -4318.05i q^{61} -1756.04 q^{62} +1272.36 q^{63} -512.000 q^{64} +1753.31i q^{65} +696.473 q^{66} -3247.13i q^{67} +3134.75 q^{68} +3403.91i q^{69} -2269.42i q^{70} -9153.25 q^{71} -610.940i q^{72} +795.611i q^{73} +5422.52 q^{74} -1741.24 q^{75} +3236.72 q^{76} -2233.18i q^{77} +1513.43 q^{78} +1582.51 q^{79} -1089.69 q^{80} +729.000 q^{81} +1113.27i q^{82} +9271.61i q^{83} -1958.93 q^{84} +6671.69 q^{85} +4560.36 q^{86} -1890.90 q^{87} -1072.29 q^{88} -996.244i q^{89} -1300.27i q^{90} -4852.70i q^{91} -5240.66i q^{92} +3226.05i q^{93} +6650.52 q^{94} +6888.72 q^{95} +940.604i q^{96} +3989.08i q^{97} -509.902i q^{98} -1279.50i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 320 q^{4} - 80 q^{7} + 1080 q^{9}+O(q^{10}) $ 40 * q - 320 * q^4 - 80 * q^7 + 1080 * q^9	$ 40 q - 320 q^{4} - 80 q^{7} + 1080 q^{9} - 144 q^{15} + 2560 q^{16} + 480 q^{17} - 792 q^{19} - 1024 q^{22} + 3400 q^{25} + 768 q^{26} + 640 q^{28} + 1608 q^{29} - 5760 q^{35} - 8640 q^{36} + 6264 q^{41} + 7040 q^{46} + 17912 q^{49} + 1296 q^{51} - 1104 q^{53} + 5040 q^{57} + 13584 q^{59} + 1152 q^{60} - 12288 q^{62} - 2160 q^{63} - 20480 q^{64} + 1152 q^{66} - 3840 q^{68} + 35352 q^{71} + 4608 q^{74} + 3168 q^{75} + 6336 q^{76} - 12672 q^{78} - 15720 q^{79} + 29160 q^{81} - 26872 q^{85} + 18432 q^{86} + 7776 q^{87} + 8192 q^{88} - 18432 q^{94} - 19128 q^{95}+O(q^{100}) $ 40 * q - 320 * q^4 - 80 * q^7 + 1080 * q^9 - 144 * q^15 + 2560 * q^16 + 480 * q^17 - 792 * q^19 - 1024 * q^22 + 3400 * q^25 + 768 * q^26 + 640 * q^28 + 1608 * q^29 - 5760 * q^35 - 8640 * q^36 + 6264 * q^41 + 7040 * q^46 + 17912 * q^49 + 1296 * q^51 - 1104 * q^53 + 5040 * q^57 + 13584 * q^59 + 1152 * q^60 - 12288 * q^62 - 2160 * q^63 - 20480 * q^64 + 1152 * q^66 - 3840 * q^68 + 35352 * q^71 + 4608 * q^74 + 3168 * q^75 + 6336 * q^76 - 12672 * q^78 - 15720 * q^79 + 29160 * q^81 - 26872 * q^85 + 18432 * q^86 + 7776 * q^87 + 8192 * q^88 - 18432 * q^94 - 19128 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/354\mathbb{Z}\right)^\times$.

$n$	$61$	$119$
$\chi(n)$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$			2.82843i			0.707107i
$2$			2.82843i			0.707107i
$3$	5.19615			0.577350
$3$	5.19615			0.577350
$4$	−8.00000			−0.500000
$5$	−17.0264			−0.681057			−0.340528	−	0.940234i	$-0.610606\pi$
$5$	−17.0264			−0.681057			−0.340528	+	0.940234i	$0.610606\pi$
$6$			14.6969i			0.408248i
$7$	47.1245			0.961725			0.480863	−	0.876796i	$-0.340324\pi$
$7$	47.1245			0.961725			0.480863	+	0.876796i	$0.340324\pi$
$8$		−	22.6274i		−	0.353553i
$9$	27.0000			0.333333
$10$		−	48.1580i		−	0.481580i
$11$		−	47.3890i		−	0.391644i	−0.980639	−	0.195822i	$-0.937262\pi$
$11$		−	47.3890i		−	0.391644i	0.980639	−	0.195822i	$-0.0627376\pi$
$12$	−41.5692			−0.288675
$13$		−	102.976i		−	0.609326i	−0.952460	−	0.304663i	$-0.901456\pi$
$13$		−	102.976i		−	0.609326i	0.952460	−	0.304663i	$-0.0985438\pi$
$14$			133.288i			0.680042i
$15$	−88.4719			−0.393208
$16$	64.0000			0.250000
$17$	−391.843			−1.35586			−0.677930	−	0.735127i	$-0.737123\pi$
$17$	−391.843			−1.35586			−0.677930	+	0.735127i	$0.737123\pi$
$18$			76.3675i			0.235702i
$19$	−404.590			−1.12075			−0.560374	−	0.828240i	$-0.689342\pi$
$19$	−404.590			−1.12075			−0.560374	+	0.828240i	$0.689342\pi$
$20$	136.211			0.340528
$21$	244.866			0.555252
$22$	134.036			0.276934
$23$			655.082i			1.23834i	0.785257	+	0.619170i	$0.212531\pi$
$23$			655.082i			1.23834i	−0.785257	+	0.619170i	$0.787469\pi$
$24$		−	117.576i		−	0.204124i
$25$	−335.101			−0.536161
$26$	291.260			0.430858
$27$	140.296			0.192450
$28$	−376.996			−0.480863
$29$	−363.904			−0.432704			−0.216352	−	0.976315i	$-0.569416\pi$
$29$	−363.904			−0.432704			−0.216352	+	0.976315i	$0.569416\pi$
$30$		−	250.236i		−	0.278040i
$31$			620.853i			0.646049i	0.946391	+	0.323025i	$0.104700\pi$
$31$			620.853i			0.646049i	−0.946391	+	0.323025i	$0.895300\pi$
$32$			181.019i			0.176777i
$33$		−	246.240i		−	0.226116i
$34$		−	1108.30i		−	0.958737i
$35$	−802.362			−0.654990
$36$	−216.000			−0.166667
$37$		−	1917.15i		−	1.40040i	−0.713945	−	0.700201i	$-0.753094\pi$
$37$		−	1917.15i		−	1.40040i	0.713945	−	0.700201i	$-0.246906\pi$
$38$		−	1144.35i		−	0.792488i
$39$		−	535.079i		−	0.351794i
$40$			385.264i			0.240790i
$41$	393.601			0.234147			0.117074	−	0.993123i	$-0.462649\pi$
$41$	393.601			0.234147			0.117074	+	0.993123i	$0.462649\pi$
$42$			692.586i			0.392623i
$43$		−	1612.33i		−	0.872001i	−0.899947	−	0.436000i	$-0.856395\pi$
$43$		−	1612.33i		−	0.872001i	0.899947	−	0.436000i	$-0.143605\pi$
$44$			379.112i			0.195822i
$45$	−459.713			−0.227019
$46$	−1852.85			−0.875639
$47$		−	2351.32i		−	1.06443i	−0.846611	−	0.532213i	$-0.821361\pi$
$47$		−	2351.32i		−	1.06443i	0.846611	−	0.532213i	$-0.178639\pi$
$48$	332.554			0.144338
$49$	−180.278			−0.0750844
$50$		−	947.808i		−	0.379123i
$51$	−2036.08			−0.782806
$52$			823.808i			0.304663i
$53$	127.124			0.0452560			0.0226280	−	0.999744i	$-0.492797\pi$
$53$	127.124			0.0452560			0.0226280	+	0.999744i	$0.492797\pi$
$54$			396.817i			0.136083i
$55$			806.865i			0.266732i
$56$		−	1066.31i		−	0.340021i
$57$	−2102.31			−0.647064
$58$		−	1029.28i		−	0.305968i
$59$	−3304.16	−	1095.38i	−0.949200	−	0.314674i
$59$	−3304.16	−	1095.38i	−0.949200	−	0.314674i
$60$	707.775			0.196604
$61$		−	4318.05i		−	1.16045i	−0.814455	−	0.580227i	$-0.802964\pi$
$61$		−	4318.05i		−	1.16045i	0.814455	−	0.580227i	$-0.197036\pi$
$62$	−1756.04			−0.456826
$63$	1272.36			0.320575
$64$	−512.000			−0.125000
$65$			1753.31i			0.414986i
$66$	696.473			0.159888
$67$		−	3247.13i		−	0.723352i	−0.932304	−	0.361676i	$-0.882205\pi$
$67$		−	3247.13i		−	0.723352i	0.932304	−	0.361676i	$-0.117795\pi$
$68$	3134.75			0.677930
$69$			3403.91i			0.714956i
$70$		−	2269.42i		−	0.463148i
$71$	−9153.25			−1.81576			−0.907881	−	0.419229i	$-0.862300\pi$
$71$	−9153.25			−1.81576			−0.907881	+	0.419229i	$0.862300\pi$
$72$		−	610.940i		−	0.117851i
$73$			795.611i			0.149298i	0.997210	+	0.0746492i	$0.0237837\pi$
$73$			795.611i			0.149298i	−0.997210	+	0.0746492i	$0.976216\pi$
$74$	5422.52			0.990234
$75$	−1741.24			−0.309553
$76$	3236.72			0.560374
$77$		−	2233.18i		−	0.376654i
$78$	1513.43			0.248756
$79$	1582.51			0.253567			0.126783	−	0.991930i	$-0.459535\pi$
$79$	1582.51			0.253567			0.126783	+	0.991930i	$0.459535\pi$
$80$	−1089.69			−0.170264
$81$	729.000			0.111111
$82$			1113.27i			0.165567i
$83$			9271.61i			1.34586i	0.739708	+	0.672928i	$0.234964\pi$
$83$			9271.61i			1.34586i	−0.739708	+	0.672928i	$0.765036\pi$
$84$	−1958.93			−0.277626
$85$	6671.69			0.923417
$86$	4560.36			0.616598
$87$	−1890.90			−0.249822
$88$	−1072.29			−0.138467
$89$		−	996.244i		−	0.125773i	−0.998021	−	0.0628863i	$-0.979969\pi$
$89$		−	996.244i		−	0.125773i	0.998021	−	0.0628863i	$-0.0200305\pi$
$90$		−	1300.27i		−	0.160527i
$91$		−	4852.70i		−	0.586004i
$92$		−	5240.66i		−	0.619170i
$93$			3226.05i			0.372997i
$94$	6650.52			0.752662
$95$	6888.72			0.763293
$96$			940.604i			0.102062i
$97$			3989.08i			0.423965i	0.977274	+	0.211982i	$0.0679919\pi$
$97$			3989.08i			0.423965i	−0.977274	+	0.211982i	$0.932008\pi$
$98$		−	509.902i		−	0.0530927i
$99$		−	1279.50i		−	0.130548i
$100$	2680.81			0.268081
$101$		−	16483.2i		−	1.61584i	−0.589292	−	0.807920i	$-0.700593\pi$
$101$		−	16483.2i		−	1.61584i	0.589292	−	0.807920i	$-0.299407\pi$
$102$		−	5758.90i		−	0.553527i
$103$			7094.74i			0.668747i	0.942441	+	0.334374i	$0.108525\pi$
$103$			7094.74i			0.668747i	−0.942441	+	0.334374i	$0.891475\pi$
$104$	−2330.08			−0.215429
$105$	−4169.20			−0.378158
$106$			359.561i			0.0320008i
$107$	−19747.9			−1.72486			−0.862431	−	0.506174i	$-0.831059\pi$
$107$	−19747.9			−1.72486			−0.862431	+	0.506174i	$0.831059\pi$
$108$	−1122.37			−0.0962250
$109$		−	11521.2i		−	0.969717i	−0.874593	−	0.484859i	$-0.838871\pi$
$109$		−	11521.2i		−	0.969717i	0.874593	−	0.484859i	$-0.161129\pi$
$110$	−2282.16			−0.188608
$111$		−	9961.81i		−	0.808523i
$112$	3015.97			0.240431
$113$		−	1694.46i		−	0.132701i	−0.997796	−	0.0663507i	$-0.978864\pi$
$113$		−	1694.46i		−	0.132701i	0.997796	−	0.0663507i	$-0.0211356\pi$
$114$		−	5946.23i		−	0.457543i
$115$		−	11153.7i		−	0.843380i
$116$	2911.23			0.216352
$117$		−	2780.35i		−	0.203109i
$118$	3098.20	−	9345.59i	0.222508	−	0.671186i
$119$	−18465.4			−1.30396
$120$			2001.89i			0.139020i
$121$	12395.3			0.846615
$122$	12213.3			0.820564
$123$	2045.21			0.135185
$124$		−	4966.83i		−	0.323025i
$125$	16347.1			1.04621
$126$			3598.78i			0.226681i
$127$	17115.2			1.06115			0.530574	−	0.847639i	$-0.321977\pi$
$127$	17115.2			1.06115			0.530574	+	0.847639i	$0.321977\pi$
$128$		−	1448.15i		−	0.0883883i
$129$		−	8377.91i		−	0.503450i
$130$	−4959.12			−0.293439
$131$			13988.9i			0.815156i	0.913170	+	0.407578i	$0.133627\pi$
$131$			13988.9i			0.815156i	−0.913170	+	0.407578i	$0.866373\pi$
$132$			1969.92i			0.113058i
$133$	−19066.1			−1.07785
$134$	9184.27			0.511487
$135$	−2388.74			−0.131069
$136$			8866.40i			0.479369i
$137$	−1241.16			−0.0661280			−0.0330640	−	0.999453i	$-0.510527\pi$
$137$	−1241.16			−0.0661280			−0.0330640	+	0.999453i	$0.510527\pi$
$138$	−9627.70			−0.505550
$139$	−33057.5			−1.71096			−0.855482	−	0.517833i	$-0.826739\pi$
$139$	−33057.5			−1.71096			−0.855482	+	0.517833i	$0.826739\pi$
$140$	6418.90			0.327495
$141$		−	12217.8i		−	0.614546i
$142$		−	25889.3i		−	1.28394i
$143$	−4879.93			−0.238639
$144$	1728.00			0.0833333
$145$	6195.99			0.294696
$146$	−2250.33			−0.105570
$147$	−936.750			−0.0433500
$148$			15337.2i			0.700201i
$149$			26856.7i			1.20970i	0.796338	+	0.604852i	$0.206768\pi$
$149$			26856.7i			1.20970i	−0.796338	+	0.604852i	$0.793232\pi$
$150$		−	4924.96i		−	0.218887i
$151$		−	1563.87i		−	0.0685879i	−0.999412	−	0.0342939i	$-0.989082\pi$
$151$		−	1563.87i		−	0.0685879i	0.999412	−	0.0342939i	$-0.0109182\pi$
$152$			9154.82i			0.396244i
$153$	−10579.8			−0.451953
$154$	6316.40			0.266335
$155$		−	10570.9i		−	0.439996i
$156$			4280.63i			0.175897i
$157$		−	22399.4i		−	0.908733i	−0.890815	−	0.454366i	$-0.849866\pi$
$157$		−	22399.4i		−	0.908733i	0.890815	−	0.454366i	$-0.150134\pi$
$158$			4476.01i			0.179299i
$159$	660.556			0.0261286
$160$		−	3082.11i		−	0.120395i
$161$			30870.4i			1.19094i
$162$			2061.92i			0.0785674i
$163$	15919.4			0.599173			0.299586	−	0.954069i	$-0.403151\pi$
$163$	15919.4			0.599173			0.299586	+	0.954069i	$0.403151\pi$
$164$	−3148.81			−0.117074
$165$			4192.59i			0.153998i
$166$	−26224.1			−0.951664
$167$	26147.0			0.937537			0.468768	−	0.883321i	$-0.344698\pi$
$167$	26147.0			0.937537			0.468768	+	0.883321i	$0.344698\pi$
$168$		−	5540.69i		−	0.196311i
$169$	17956.9			0.628722
$170$			18870.4i			0.652955i
$171$	−10923.9			−0.373582
$172$			12898.6i			0.436000i
$173$			34353.7i			1.14784i	0.818912	+	0.573920i	$0.194578\pi$
$173$			34353.7i			1.14784i	−0.818912	+	0.573920i	$0.805422\pi$
$174$		−	5348.28i		−	0.176651i
$175$	−15791.5			−0.515640
$176$		−	3032.89i		−	0.0979111i
$177$	−17168.9	−	5691.76i	−0.548021	−	0.181677i
$178$	2817.80			0.0889346
$179$			41319.4i			1.28958i	0.764360	+	0.644789i	$0.223055\pi$
$179$			41319.4i			1.28958i	−0.764360	+	0.644789i	$0.776945\pi$
$180$	3677.71			0.113509
$181$	4010.42			0.122414			0.0612072	−	0.998125i	$-0.480505\pi$
$181$	4010.42			0.122414			0.0612072	+	0.998125i	$0.480505\pi$
$182$	13725.5			0.414367
$183$		−	22437.2i		−	0.669988i
$184$	14822.8			0.437819
$185$			32642.2i			0.953754i
$186$	−9124.64			−0.263749
$187$			18569.1i			0.531015i
$188$			18810.5i			0.532213i
$189$	6611.39			0.185084
$190$			19484.2i			0.539729i
$191$			63990.2i			1.75407i	0.480426	+	0.877035i	$0.340482\pi$
$191$			63990.2i			1.75407i	−0.480426	+	0.877035i	$0.659518\pi$
$192$	−2660.43			−0.0721688
$193$	−7521.89			−0.201935			−0.100968	−	0.994890i	$-0.532194\pi$
$193$	−7521.89			−0.201935			−0.100968	+	0.994890i	$0.532194\pi$
$194$	−11282.8			−0.299788
$195$			9110.49i			0.239592i
$196$	1442.22			0.0375422
$197$	16306.2			0.420166			0.210083	−	0.977684i	$-0.432627\pi$
$197$	16306.2			0.420166			0.210083	+	0.977684i	$0.432627\pi$
$198$	3618.98			0.0923115
$199$	16277.2			0.411029			0.205514	−	0.978654i	$-0.434113\pi$
$199$	16277.2			0.411029			0.205514	+	0.978654i	$0.434113\pi$
$200$			7582.47i			0.189562i
$201$		−	16872.6i		−	0.417628i
$202$	46621.5			1.14257
$203$	−17148.8			−0.416143
$204$	16288.6			0.391403
$205$	−6701.62			−0.159468
$206$	−20066.9			−0.472876
$207$			17687.2i			0.412780i
$208$		−	6590.47i		−	0.152331i
$209$			19173.1i			0.438934i
$210$		−	11792.3i		−	0.267398i
$211$			31714.8i			0.712357i	0.934418	+	0.356178i	$0.115920\pi$
$211$			31714.8i			0.712357i	−0.934418	+	0.356178i	$0.884080\pi$
$212$	−1016.99			−0.0226280
$213$	−47561.7			−1.04833
$214$		−	55855.6i		−	1.21966i
$215$			27452.2i			0.593882i
$216$		−	3174.54i		−	0.0680414i
$217$			29257.4i			0.621322i
$218$	32586.9			0.685694
$219$			4134.11i			0.0861974i
$220$		−	6454.92i		−	0.133366i
$221$			40350.5i			0.826160i
$222$	28176.3			0.571712
$223$	−51284.3			−1.03127			−0.515637	−	0.856807i	$-0.672445\pi$
$223$	−51284.3			−1.03127			−0.515637	+	0.856807i	$0.672445\pi$
$224$			8530.45i			0.170011i
$225$	−9047.72			−0.178720
$226$	4792.67			0.0938340
$227$		−	55833.5i		−	1.08354i	−0.840528	−	0.541768i	$-0.817755\pi$
$227$		−	55833.5i		−	1.08354i	0.840528	−	0.541768i	$-0.182245\pi$
$228$	16818.5			0.323532
$229$		−	28360.6i		−	0.540810i	−0.962747	−	0.270405i	$-0.912842\pi$
$229$		−	28360.6i		−	0.540810i	0.962747	−	0.270405i	$-0.0871576\pi$
$230$	31547.4			0.596360
$231$		−	11604.0i		−	0.217462i
$232$			8234.21i			0.152984i
$233$		−	29356.2i		−	0.540740i	−0.962757	−	0.270370i	$-0.912854\pi$
$233$		−	29356.2i		−	0.540740i	0.962757	−	0.270370i	$-0.0871460\pi$
$234$	7864.03			0.143619
$235$			40034.5i			0.724934i
$236$	26433.3	+	8763.03i	0.474600	+	0.157337i
$237$	8222.96			0.146397
$238$		−	52228.1i		−	0.922042i
$239$	−62552.5			−1.09509			−0.547544	−	0.836777i	$-0.684437\pi$
$239$	−62552.5			−1.09509			−0.547544	+	0.836777i	$0.684437\pi$
$240$	−5662.20			−0.0983021
$241$	−103865.			−1.78827			−0.894136	−	0.447795i	$-0.852209\pi$
$241$	−103865.			−1.78827			−0.894136	+	0.447795i	$0.852209\pi$
$242$			35059.2i			0.598647i
$243$	3788.00			0.0641500
$244$			34544.4i			0.580227i
$245$	3069.48			0.0511368
$246$			5784.73i			0.0955901i
$247$			41663.1i			0.682900i
$248$	14048.3			0.228413
$249$			48176.7i			0.777031i
$250$			46236.5i			0.739785i
$251$	45415.7			0.720873			0.360436	−	0.932784i	$-0.382628\pi$
$251$	45415.7			0.720873			0.360436	+	0.932784i	$0.382628\pi$
$252$	−10178.9			−0.160288
$253$	31043.7			0.484989
$254$			48409.2i			0.750344i
$255$	34667.1			0.533135
$256$	4096.00			0.0625000
$257$	−13345.3			−0.202051			−0.101026	−	0.994884i	$-0.532212\pi$
$257$	−13345.3			−0.202051			−0.101026	+	0.994884i	$0.532212\pi$
$258$	23696.3			0.355993
$259$		−	90344.9i		−	1.34680i
$260$		−	14026.5i		−	0.207493i
$261$	−9825.42			−0.144235
$262$	−39566.6			−0.576402
$263$	80507.0			1.16392			0.581958	−	0.813219i	$-0.302287\pi$
$263$	80507.0			1.16392			0.581958	+	0.813219i	$0.302287\pi$
$264$	−5571.78			−0.0799441
$265$	−2164.47			−0.0308219
$266$		−	53927.1i		−	0.762156i
$267$		−	5176.64i		−	0.0726148i
$268$			25977.0i			0.361676i
$269$			87611.2i			1.21075i	0.795940	+	0.605376i	$0.206977\pi$
$269$			87611.2i			1.21075i	−0.795940	+	0.605376i	$0.793023\pi$
$270$		−	6756.38i		−	0.0926801i
$271$	−97671.2			−1.32993			−0.664964	−	0.746876i	$-0.731553\pi$
$271$	−97671.2			−1.32993			−0.664964	+	0.746876i	$0.731553\pi$
$272$	−25078.0			−0.338965
$273$		−	25215.4i		−	0.338330i
$274$		−	3510.52i		−	0.0467595i
$275$			15880.1i			0.209985i
$276$		−	27231.2i		−	0.357478i
$277$	50422.2			0.657146			0.328573	−	0.944479i	$-0.393432\pi$
$277$	50422.2			0.657146			0.328573	+	0.944479i	$0.393432\pi$
$278$		−	93500.8i		−	1.20983i
$279$			16763.0i			0.215350i
$280$			18155.4i			0.231574i
$281$	−52292.7			−0.662260			−0.331130	−	0.943585i	$-0.607430\pi$
$281$	−52292.7			−0.662260			−0.331130	+	0.943585i	$0.607430\pi$
$282$	34557.1			0.434550
$283$		−	107656.i		−	1.34420i	−0.740461	−	0.672100i	$-0.765393\pi$
$283$		−	107656.i		−	1.34420i	0.740461	−	0.672100i	$-0.234607\pi$
$284$	73226.0			0.907881
$285$	35794.8			0.440687
$286$		−	13802.5i		−	0.168743i
$287$	18548.3			0.225185
$288$			4887.52i			0.0589256i
$289$	70020.2			0.838354
$290$			17524.9i			0.208382i
$291$			20727.9i			0.244776i
$292$		−	6364.89i		−	0.0746492i
$293$	2791.34			0.0325145			0.0162573	−	0.999868i	$-0.494825\pi$
$293$	2791.34			0.0325145			0.0162573	+	0.999868i	$0.494825\pi$
$294$		−	2649.53i		−	0.0306531i
$295$	56258.1	+	18650.4i	0.646459	+	0.214311i
$296$	−43380.2			−0.495117
$297$		−	6648.49i		−	0.0753720i
$298$	−75962.1			−0.855390
$299$	67457.7			0.754552
$300$	13929.9			0.154776
$301$		−	75980.3i		−	0.838625i
$302$	4423.30			0.0484989
$303$		−	85649.2i		−	0.932906i
$304$	−25893.7			−0.280187
$305$			73520.9i			0.790335i
$306$		−	29924.1i		−	0.319579i
$307$	66093.5			0.701265			0.350633	−	0.936513i	$-0.385967\pi$
$307$	66093.5			0.701265			0.350633	+	0.936513i	$0.385967\pi$
$308$			17865.5i			0.188327i
$309$			36865.3i			0.386101i
$310$	29899.1			0.311124
$311$	31991.4			0.330759			0.165380	−	0.986230i	$-0.447115\pi$
$311$	31991.4			0.330759			0.165380	+	0.986230i	$0.447115\pi$
$312$	−12107.5			−0.124378
$313$		−	54753.5i		−	0.558886i	−0.960162	−	0.279443i	$-0.909850\pi$
$313$		−	54753.5i		−	0.558886i	0.960162	−	0.279443i	$-0.0901499\pi$
$314$	63355.0			0.642571
$315$	−21663.8			−0.218330
$316$	−12660.1			−0.126783
$317$	−170096.			−1.69268			−0.846342	−	0.532640i	$-0.821200\pi$
$317$	−170096.			−1.69268			−0.846342	+	0.532640i	$0.821200\pi$
$318$			1868.33i			0.0184757i
$319$			17245.1i			0.169466i
$320$	8717.53			0.0851321
$321$	−102613.			−0.995850
$322$	−87314.8			−0.842124
$323$	158536.			1.51958
$324$	−5832.00			−0.0555556
$325$			34507.4i			0.326697i
$326$			45026.9i			0.423679i
$327$		−	59866.0i		−	0.559866i
$328$		−	8906.18i		−	0.0827835i
$329$		−	110805.i		−	1.02368i
$330$	−11858.4			−0.108893
$331$	183943.			1.67891			0.839453	−	0.543433i	$-0.182876\pi$
$331$	183943.			1.67891			0.839453	+	0.543433i	$0.182876\pi$
$332$		−	74172.8i		−	0.672928i
$333$		−	51763.1i		−	0.466801i
$334$			73954.8i			0.662938i
$335$			55287.0i			0.492644i
$336$	15671.4			0.138813
$337$		−	65638.0i		−	0.577957i	−0.957336	−	0.288979i	$-0.906684\pi$
$337$		−	65638.0i		−	0.577957i	0.957336	−	0.288979i	$-0.0933156\pi$
$338$			50789.9i			0.444574i
$339$		−	8804.69i		−	0.0766152i
$340$	−53373.5			−0.461709
$341$	29421.6			0.253022
$342$		−	30897.5i		−	0.264163i
$343$	−121642.			−1.03394
$344$	−36482.8			−0.308299
$345$		−	57956.3i		−	0.486926i
$346$	−97166.9			−0.811645
$347$		−	199917.i		−	1.66032i	−0.557527	−	0.830159i	$-0.688250\pi$
$347$		−	199917.i		−	1.66032i	0.557527	−	0.830159i	$-0.311750\pi$
$348$	15127.2			0.124911
$349$		−	58539.3i		−	0.480615i	−0.970697	−	0.240307i	$-0.922752\pi$
$349$		−	58539.3i		−	0.480615i	0.970697	−	0.240307i	$-0.0772482\pi$
$350$		−	44665.0i		−	0.364613i
$351$		−	14447.1i		−	0.117265i
$352$	8578.32			0.0692336
$353$		−	51526.3i		−	0.413504i	−0.978393	−	0.206752i	$-0.933711\pi$
$353$		−	51526.3i		−	0.413504i	0.978393	−	0.206752i	$-0.0662893\pi$
$354$	16098.7	−	48561.1i	0.128465	−	0.387509i
$355$	155847.			1.23664
$356$			7969.96i			0.0628863i
$357$	−95949.2			−0.752844
$358$	−116869.			−0.911870
$359$	76162.3			0.590951			0.295475	−	0.955350i	$-0.404522\pi$
$359$	76162.3			0.590951			0.295475	+	0.955350i	$0.404522\pi$
$360$			10402.1i			0.0802633i
$361$	33371.8			0.256074
$362$			11343.2i			0.0865600i
$363$	64407.8			0.488793
$364$			38821.6i			0.293002i
$365$		−	13546.4i		−	0.101681i
$366$	63462.1			0.473753
$367$			21917.2i			0.162724i	0.996685	+	0.0813621i	$0.0259270\pi$
$367$			21917.2i			0.162724i	−0.996685	+	0.0813621i	$0.974073\pi$
$368$			41925.2i			0.309585i
$369$	10627.2			0.0780490
$370$	−92326.2			−0.674406
$371$	5990.66			0.0435238
$372$		−	25808.4i		−	0.186498i
$373$	−94995.4			−0.682786			−0.341393	−	0.939921i	$-0.610899\pi$
$373$	−94995.4			−0.682786			−0.341393	+	0.939921i	$0.610899\pi$
$374$	−52521.2			−0.375484
$375$	84941.9			0.604032
$376$	−53204.2			−0.376331
$377$			37473.4i			0.263658i
$378$			18699.8i			0.130874i
$379$	49211.2			0.342599			0.171299	−	0.985219i	$-0.445203\pi$
$379$	49211.2			0.342599			0.171299	+	0.985219i	$0.445203\pi$
$380$	−55109.7			−0.381646
$381$	88933.4			0.612654
$382$	−180992.			−1.24032
$383$	−143781.			−0.980179			−0.490089	−	0.871672i	$-0.663036\pi$
$383$	−143781.			−0.980179			−0.490089	+	0.871672i	$0.663036\pi$
$384$		−	7524.83i		−	0.0510310i
$385$			38023.1i			0.256523i
$386$		−	21275.1i		−	0.142790i
$387$		−	43532.9i		−	0.290667i
$388$		−	31912.7i		−	0.211982i
$389$	119384.			0.788948			0.394474	−	0.918907i	$-0.370927\pi$
$389$	119384.			0.788948			0.394474	+	0.918907i	$0.370927\pi$
$390$	−25768.3			−0.169417
$391$		−	256689.i		−	1.67901i
$392$			4079.22i			0.0265463i
$393$			72688.4i			0.470631i
$394$			46121.0i			0.297102i
$395$	−26944.5			−0.172693
$396$			10236.0i			0.0652741i
$397$			289180.i			1.83479i	0.397975	+	0.917396i	$0.369713\pi$
$397$			289180.i			1.83479i	−0.397975	+	0.917396i	$0.630287\pi$
$398$			46038.8i			0.290641i
$399$	−99070.4			−0.622297
$400$	−21446.5			−0.134040
$401$			25378.3i			0.157824i	0.996882	+	0.0789120i	$0.0251446\pi$
$401$			25378.3i			0.157824i	−0.996882	+	0.0789120i	$0.974855\pi$
$402$	47722.9			0.295307
$403$	63933.0			0.393655
$404$			131866.i			0.807920i
$405$	−12412.3			−0.0756730
$406$		−	48504.2i		−	0.294257i
$407$	−90851.8			−0.548460
$408$			46071.2i			0.276764i
$409$			185992.i			1.11185i	0.831231	+	0.555927i	$0.187637\pi$
$409$			185992.i			1.11185i	−0.831231	+	0.555927i	$0.812363\pi$
$410$		−	18955.0i		−	0.112761i
$411$	−6449.24			−0.0381790
$412$		−	56757.9i		−	0.334374i
$413$	−155707.	−	51619.2i	−0.912870	−	0.302630i
$414$	−50027.0			−0.291880
$415$		−	157862.i		−	0.916605i
$416$	18640.7			0.107715
$417$	−171772.			−0.987825
$418$	−54229.7			−0.310373
$419$		−	34104.8i		−	0.194262i	−0.995272	−	0.0971310i	$-0.969033\pi$
$419$		−	34104.8i		−	0.194262i	0.995272	−	0.0971310i	$-0.0309666\pi$
$420$	33353.6			0.189079
$421$		−	130968.i		−	0.738927i	−0.929245	−	0.369464i	$-0.879541\pi$
$421$		−	130968.i		−	0.738927i	0.929245	−	0.369464i	$-0.120459\pi$
$422$	−89703.1			−0.503712
$423$		−	63485.5i		−	0.354808i
$424$		−	2876.49i		−	0.0160004i
$425$	131307.			0.726959
$426$		−	134525.i		−	0.741281i
$427$		−	203486.i		−	1.11604i
$428$	157984.			0.862431
$429$	−25356.9			−0.137778
$430$	−77646.5			−0.419938
$431$			330076.i			1.77688i	0.458988	+	0.888442i	$0.348212\pi$
$431$			330076.i			1.77688i	−0.458988	+	0.888442i	$0.651788\pi$
$432$	8978.95			0.0481125
$433$	113888.			0.607438			0.303719	−	0.952762i	$-0.401772\pi$
$433$	113888.			0.607438			0.303719	+	0.952762i	$0.401772\pi$
$434$	−82752.5			−0.439341
$435$	32195.3			0.170143
$436$			92169.7i			0.484859i
$437$		−	265039.i		−	1.38787i
$438$	−11693.0			−0.0609508
$439$	−56912.3			−0.295309			−0.147655	−	0.989039i	$-0.547172\pi$
$439$	−56912.3			−0.295309			−0.147655	+	0.989039i	$0.547172\pi$
$440$	18257.3			0.0943041
$441$	−4867.50			−0.0250281
$442$	−114128.			−0.584183
$443$			144645.i			0.737048i	0.929618	+	0.368524i	$0.120137\pi$
$443$			144645.i			0.737048i	−0.929618	+	0.368524i	$0.879863\pi$
$444$			79694.5i			0.404261i
$445$			16962.5i			0.0856583i
$446$		−	145054.i		−	0.729222i
$447$			139551.i			0.698423i
$448$	−24127.8			−0.120216
$449$	186744.			0.926307			0.463153	−	0.886278i	$-0.346718\pi$
$449$	186744.			0.926307			0.463153	+	0.886278i	$0.346718\pi$
$450$		−	25590.8i		−	0.126374i
$451$		−	18652.4i		−	0.0917024i
$452$			13555.7i			0.0663507i
$453$		−	8126.12i		−	0.0395992i
$454$	157921.			0.766176
$455$			82624.1i			0.399102i
$456$			47569.8i			0.228772i
$457$		−	170389.i		−	0.815848i	−0.913016	−	0.407924i	$-0.866253\pi$
$457$		−	170389.i		−	0.815848i	0.913016	−	0.407924i	$-0.133747\pi$
$458$	80215.9			0.382410
$459$	−54974.1			−0.260935
$460$			89229.6i			0.421690i
$461$	−44823.5			−0.210913			−0.105457	−	0.994424i	$-0.533630\pi$
$461$	−44823.5			−0.210913			−0.105457	+	0.994424i	$0.533630\pi$
$462$	32821.0			0.153769
$463$		−	38120.7i		−	0.177827i	−0.996039	−	0.0889137i	$-0.971660\pi$
$463$		−	38120.7i		−	0.177827i	0.996039	−	0.0889137i	$-0.0283395\pi$
$464$	−23289.9			−0.108176
$465$		−	54928.1i		−	0.254032i
$466$	83032.0			0.382361
$467$		−	126399.i		−	0.579575i	−0.957091	−	0.289788i	$-0.906415\pi$
$467$		−	126399.i		−	0.579575i	0.957091	−	0.289788i	$-0.0935847\pi$
$468$			22242.8i			0.101554i
$469$		−	153019.i		−	0.695666i
$470$	−113235.			−0.512606
$471$		−	116390.i		−	0.524657i
$472$	−24785.6	+	74764.7i	−0.111254	+	0.335593i
$473$	−76406.6			−0.341514
$474$			23258.1i			0.103518i
$475$	135578.			0.600901
$476$	147723.			0.651982
$477$	3432.35			0.0150853
$478$		−	176925.i		−	0.774344i
$479$	10130.8			0.0441543			0.0220771	−	0.999756i	$-0.492972\pi$
$479$	10130.8			0.0441543			0.0220771	+	0.999756i	$0.492972\pi$
$480$		−	16015.1i		−	0.0695101i
$481$	−197421.			−0.853301
$482$		−	293774.i		−	1.26450i
$483$			160407.i			0.687591i
$484$	−99162.3			−0.423307
$485$		−	67919.8i		−	0.288744i
$486$			10714.1i			0.0453609i
$487$	69063.0			0.291197			0.145599	−	0.989344i	$-0.453489\pi$
$487$	69063.0			0.291197			0.145599	+	0.989344i	$0.453489\pi$
$488$	−97706.2			−0.410282
$489$	82719.7			0.345933
$490$			8681.81i			0.0361592i
$491$	−274527.			−1.13873			−0.569366	−	0.822084i	$-0.692811\pi$
$491$	−274527.			−1.13873			−0.569366	+	0.822084i	$0.692811\pi$
$492$	−16361.7			−0.0675924
$493$	142593.			0.586686
$494$	−117841.			−0.482883
$495$			21785.3i			0.0889107i
$496$			39734.6i			0.161512i
$497$	−431343.			−1.74626
$498$	−136264.			−0.549444
$499$	195379.			0.784652			0.392326	−	0.919826i	$-0.371670\pi$
$499$	195379.			0.784652			0.392326	+	0.919826i	$0.371670\pi$
$500$	−130777.			−0.523107
$501$	135864.			0.541287
$502$			128455.i			0.509734i
$503$		−	167879.i		−	0.663531i	−0.943362	−	0.331765i	$-0.892356\pi$
$503$		−	167879.i		−	0.663531i	0.943362	−	0.331765i	$-0.107644\pi$
$504$		−	28790.3i		−	0.113340i
$505$			280650.i			1.10048i
$506$			87804.7i			0.342939i
$507$	93307.0			0.362993
$508$	−136922.			−0.530574
$509$		−	78739.1i		−	0.303917i	−0.988387	−	0.151958i	$-0.951442\pi$
$509$		−	78739.1i		−	0.303917i	0.988387	−	0.151958i	$-0.0485580\pi$
$510$			98053.4i			0.376984i
$511$			37492.8i			0.143584i
$512$			11585.2i			0.0441942i
$513$	−56762.4			−0.215688
$514$		−	37746.2i		−	0.142872i
$515$		−	120798.i		−	0.455455i
$516$			67023.3i			0.251725i
$517$	−111426.			−0.416876
$518$	255534.			0.952333
$519$			178507.i			0.662705i
$520$	39673.0			0.146720
$521$	−58643.4			−0.216045			−0.108022	−	0.994148i	$-0.534452\pi$
$521$	−58643.4			−0.216045			−0.108022	+	0.994148i	$0.534452\pi$
$522$		−	27790.5i		−	0.101989i
$523$	501319.			1.83278			0.916390	−	0.400286i	$-0.131089\pi$
$523$	501319.			1.83278			0.916390	+	0.400286i	$0.131089\pi$
$524$		−	111911.i		−	0.407578i
$525$	−82054.9			−0.297705
$526$			227708.i			0.823013i
$527$		−	243277.i		−	0.875952i
$528$		−	15759.4i		−	0.0565290i
$529$	−149291.			−0.533486
$530$		−	6122.04i		−	0.0217944i
$531$	−89212.5	−	29575.2i	−0.316400	−	0.104891i
$532$	152529.			0.538925
$533$		−	40531.5i		−	0.142672i
$534$	14641.7			0.0513464
$535$	336237.			1.17473
$536$	−73474.1			−0.255744
$537$			214702.i			0.744538i
$538$	−247802.			−0.856131
$539$			8543.17i			0.0294064i
$540$	19109.9			0.0655347
$541$			82899.7i			0.283242i	0.989921	+	0.141621i	$0.0452315\pi$
$541$			82899.7i			0.283242i	−0.989921	+	0.141621i	$0.954769\pi$
$542$		−	276256.i		−	0.940401i
$543$	20838.7			0.0706760
$544$		−	70931.2i		−	0.239684i
$545$			196165.i			0.660433i
$546$	71319.8			0.239235
$547$	384401.			1.28472			0.642362	−	0.766402i	$-0.277955\pi$
$547$	384401.			1.28472			0.642362	+	0.766402i	$0.277955\pi$
$548$	9929.25			0.0330640
$549$		−	116587.i		−	0.386818i
$550$	−44915.7			−0.148482
$551$	147232.			0.484952
$552$	77021.6			0.252775
$553$	74575.1			0.243862
$554$			142615.i			0.464673i
$555$			169614.i			0.550650i
$556$	264460.			0.855482
$557$	−94376.0			−0.304194			−0.152097	−	0.988366i	$-0.548603\pi$
$557$	−94376.0			−0.304194			−0.152097	+	0.988366i	$0.548603\pi$
$558$	−47413.0			−0.152275
$559$	−166031.			−0.531332
$560$	−51351.2			−0.163747
$561$			96487.6i			0.306581i
$562$		−	147906.i		−	0.468289i
$563$			553574.i			1.74646i	0.487307	+	0.873230i	$0.337979\pi$
$563$			553574.i			1.74646i	−0.487307	+	0.873230i	$0.662021\pi$
$564$			97742.3i			0.307273i
$565$			28850.7i			0.0903772i
$566$	304496.			0.950492
$567$	34353.8			0.106858
$568$			207114.i			0.641969i
$569$		−	561947.i		−	1.73568i	−0.496840	−	0.867842i	$-0.665507\pi$
$569$		−	561947.i		−	1.73568i	0.496840	−	0.867842i	$-0.334493\pi$
$570$			101243.i			0.311613i
$571$			409827.i			1.25698i	0.777817	+	0.628490i	$0.216327\pi$
$571$			409827.i			1.25698i	−0.777817	+	0.628490i	$0.783673\pi$
$572$	39039.4			0.119320
$573$			332503.i			1.01271i
$574$			52462.5i			0.159230i
$575$		−	219519.i		−	0.663950i
$576$	−13824.0			−0.0416667
$577$	−300769.			−0.903402			−0.451701	−	0.892169i	$-0.649183\pi$
$577$	−300769.			−0.903402			−0.451701	+	0.892169i	$0.649183\pi$
$578$			198047.i			0.592806i
$579$	−39084.9			−0.116587
$580$	−49567.9			−0.147348
$581$			436920.i			1.29434i
$582$	−58627.3			−0.173083
$583$		−	6024.28i		−	0.0177243i
$584$	18002.6			0.0527849
$585$			47339.5i			0.138329i
$586$			7895.10i			0.0229912i
$587$		−	198243.i		−	0.575336i	−0.957730	−	0.287668i	$-0.907120\pi$
$587$		−	198243.i		−	0.575336i	0.957730	−	0.287668i	$-0.0928799\pi$
$588$	7494.00			0.0216750
$589$		−	251191.i		−	0.724058i
$590$	−52751.3	+	159122.i	−0.151541	+	0.457116i
$591$	84729.6			0.242583
$592$		−	122698.i		−	0.350101i
$593$	292290.			0.831199			0.415600	−	0.909548i	$-0.363572\pi$
$593$	292290.			0.831199			0.415600	+	0.909548i	$0.363572\pi$
$594$	18804.8			0.0532961
$595$	314400.			0.888074
$596$		−	214853.i		−	0.604852i
$597$	84578.6			0.237308
$598$			190799.i			0.533549i
$599$	536335.			1.49480			0.747399	−	0.664376i	$-0.231302\pi$
$599$	536335.			1.49480			0.747399	+	0.664376i	$0.231302\pi$
$600$			39399.7i			0.109443i
$601$		−	164915.i		−	0.456575i	−0.973594	−	0.228288i	$-0.926687\pi$
$601$		−	164915.i		−	0.456575i	0.973594	−	0.228288i	$-0.0733127\pi$
$602$	214905.			0.592997
$603$		−	87672.5i		−	0.241117i
$604$			12511.0i			0.0342939i
$605$	−211047.			−0.576593
$606$	242252.			0.659664
$607$	528214.			1.43362			0.716808	−	0.697271i	$-0.245602\pi$
$607$	528214.			1.43362			0.716808	+	0.697271i	$0.245602\pi$
$608$		−	73238.6i		−	0.198122i
$609$	−89107.9			−0.240260
$610$	−207948.			−0.558851
$611$	−242129.			−0.648582
$612$	84638.2			0.225977
$613$			622698.i			1.65713i	0.559892	+	0.828566i	$0.310843\pi$
$613$			622698.i			1.65713i	−0.559892	+	0.828566i	$0.689157\pi$
$614$			186941.i			0.495869i
$615$	−34822.6			−0.0920686
$616$	−50531.2			−0.133167
$617$	172256.			0.452485			0.226243	−	0.974071i	$-0.427356\pi$
$617$	172256.			0.452485			0.226243	+	0.974071i	$0.427356\pi$
$618$	−104271.			−0.273015
$619$	533841.			1.39325			0.696627	−	0.717433i	$-0.254683\pi$
$619$	533841.			1.39325			0.696627	+	0.717433i	$0.254683\pi$
$620$			84567.3i			0.219998i
$621$			91905.4i			0.238319i
$622$			90485.3i			0.233882i
$623$		−	46947.6i		−	0.120959i
$624$		−	34245.1i		−	0.0879486i
$625$	−68894.3			−0.176370
$626$	154866.			0.395192
$627$			99626.3i			0.253419i
$628$			179195.i			0.454366i
$629$			751223.i			1.89875i
$630$		−	61274.4i		−	0.154383i
$631$	−160198.			−0.402345			−0.201173	−	0.979556i	$-0.564475\pi$
$631$	−160198.			−0.402345			−0.201173	+	0.979556i	$0.564475\pi$
$632$		−	35808.1i		−	0.0896494i
$633$			164795.i			0.411279i
$634$		−	481104.i		−	1.19691i
$635$	−291411.			−0.722702
$636$	−5284.45			−0.0130643
$637$			18564.3i			0.0457509i
$638$	−48776.4			−0.119831
$639$	−247138.			−0.605254
$640$			24656.9i			0.0601975i
$641$	−70039.6			−0.170462			−0.0852310	−	0.996361i	$-0.527163\pi$
$641$	−70039.6			−0.170462			−0.0852310	+	0.996361i	$0.527163\pi$
$642$		−	290234.i		−	0.704172i
$643$	300889.			0.727753			0.363877	−	0.931447i	$-0.381453\pi$
$643$	300889.			0.727753			0.363877	+	0.931447i	$0.381453\pi$
$644$		−	246963.i		−	0.595471i
$645$			142646.i			0.342878i
$646$			448407.i			1.07450i
$647$	−349066.			−0.833870			−0.416935	−	0.908936i	$-0.636896\pi$
$647$	−349066.			−0.833870			−0.416935	+	0.908936i	$0.636896\pi$
$648$		−	16495.4i		−	0.0392837i
$649$	−51908.9	+	156581.i	−0.123240	+	0.371749i
$650$	−97601.6			−0.231010
$651$			152026.i			0.358720i
$652$	−127355.			−0.299586
$653$	−183235.			−0.429716			−0.214858	−	0.976645i	$-0.568929\pi$
$653$	−183235.			−0.429716			−0.214858	+	0.976645i	$0.568929\pi$
$654$	169326.			0.395885
$655$		−	238181.i		−	0.555168i
$656$	25190.5			0.0585368
$657$			21481.5i			0.0497661i
$658$	313403.			0.723854
$659$		−	193108.i		−	0.444661i	−0.974971	−	0.222331i	$-0.928634\pi$
$659$		−	193108.i		−	0.444661i	0.974971	−	0.222331i	$-0.0713664\pi$
$660$		−	33540.7i		−	0.0769989i
$661$	−530945.			−1.21520			−0.607599	−	0.794244i	$-0.707867\pi$
$661$	−530945.			−1.21520			−0.607599	+	0.794244i	$0.707867\pi$
$662$			520268.i			1.18717i
$663$			209667.i			0.476984i
$664$	209792.			0.475832
$665$	324628.			0.734078
$666$	146408.			0.330078
$667$		−	238387.i		−	0.535835i
$668$	−209176.			−0.468768
$669$	−266481.			−0.595407
$670$	−156375.			−0.348352
$671$	−204628.			−0.454485
$672$			44325.5i			0.0981557i
$673$			784780.i			1.73268i	0.499456	+	0.866339i	$0.333533\pi$
$673$			784780.i			1.73268i	−0.499456	+	0.866339i	$0.666467\pi$
$674$	185652.			0.408677
$675$	−47013.4			−0.103184
$676$	−143655.			−0.314361
$677$	−289714.			−0.632109			−0.316054	−	0.948741i	$-0.602358\pi$
$677$	−289714.			−0.632109			−0.316054	+	0.948741i	$0.602358\pi$
$678$	24903.4			0.0541751
$679$			187984.i			0.407738i
$680$		−	150963.i		−	0.326477i
$681$		−	290120.i		−	0.625580i
$682$			83216.9i			0.178913i
$683$		−	697601.i		−	1.49543i	−0.664021	−	0.747714i	$-0.731151\pi$
$683$		−	697601.i		−	1.49543i	0.664021	−	0.747714i	$-0.268849\pi$
$684$	87391.4			0.186791
$685$	21132.5			0.0450369
$686$		−	344054.i		−	0.731103i
$687$		−	147366.i		−	0.312237i
$688$		−	103189.i		−	0.218000i
$689$		−	13090.7i		−	0.0275756i
$690$	163925.			0.344308
$691$			737684.i			1.54495i	0.635045	+	0.772475i	$0.280982\pi$
$691$			737684.i			1.54495i	−0.635045	+	0.772475i	$0.719018\pi$
$692$		−	274829.i		−	0.573920i
$693$		−	60296.0i		−	0.125551i
$694$	565451.			1.17402
$695$	562851.			1.16526
$696$			42786.2i			0.0883254i
$697$	−154230.			−0.317470
$698$	165574.			0.339846
$699$		−	152539.i		−	0.312196i
$700$	126332.			0.257820
$701$		−	620459.i		−	1.26263i	−0.775526	−	0.631316i	$-0.782515\pi$
$701$		−	620459.i		−	1.26263i	0.775526	−	0.631316i	$-0.217485\pi$
$702$	40862.7			0.0829187
$703$			775660.i			1.56950i
$704$			24263.2i			0.0489556i
$705$			208025.i			0.418541i
$706$	145738.			0.292391
$707$		−	776763.i		−	1.55399i
$708$	137352.	+	45534.0i	0.274010	+	0.0908385i
$709$	70755.6			0.140756			0.0703782	−	0.997520i	$-0.477579\pi$
$709$	70755.6			0.140756			0.0703782	+	0.997520i	$0.477579\pi$
$710$			440802.i			0.874434i
$711$	42727.8			0.0845223
$712$	−22542.4			−0.0444673
$713$	−406710.			−0.800029
$714$		−	271385.i		−	0.532341i
$715$	83087.8			0.162527
$716$		−	330555.i		−	0.644789i
$717$	−325032.			−0.632249
$718$			215420.i			0.417865i
$719$		−	342303.i		−	0.662146i	−0.943605	−	0.331073i	$-0.892589\pi$
$719$		−	342303.i		−	0.662146i	0.943605	−	0.331073i	$-0.107411\pi$
$720$	−29421.7			−0.0567547
$721$			334336.i			0.643151i
$722$			94389.7i			0.181072i
$723$	−539697.			−1.03246
$724$	−32083.3			−0.0612072
$725$	121945.			0.231999
$726$			182173.i			0.345629i
$727$	285752.			0.540655			0.270327	−	0.962768i	$-0.412868\pi$
$727$	285752.			0.540655			0.270327	+	0.962768i	$0.412868\pi$
$728$	−109804.			−0.207184
$729$	19683.0			0.0370370
$730$	38315.0			0.0718991
$731$			631780.i			1.18231i
$732$			179498.i			0.334994i
$733$	−15154.7			−0.0282060			−0.0141030	−	0.999901i	$-0.504489\pi$
$733$	−15154.7			−0.0282060			−0.0141030	+	0.999901i	$0.504489\pi$
$734$	−61991.1			−0.115063
$735$	15949.5			0.0295238
$736$	−118582.			−0.218910
$737$	−153878.			−0.283297
$738$			30058.4i			0.0551890i
$739$		−	325976.i		−	0.596894i	−0.954426	−	0.298447i	$-0.903531\pi$
$739$		−	325976.i		−	0.596894i	0.954426	−	0.298447i	$-0.0964686\pi$
$740$		−	261138.i		−	0.476877i
$741$			216488.i			0.394273i
$742$			16944.2i			0.0307760i
$743$	−639397.			−1.15823			−0.579113	−	0.815247i	$-0.696601\pi$
$743$	−639397.			−1.15823			−0.579113	+	0.815247i	$0.696601\pi$
$744$	72997.2			0.131874
$745$		−	457273.i		−	0.823878i
$746$		−	268688.i		−	0.482803i
$747$			250333.i			0.448619i
$748$		−	148552.i		−	0.265507i
$749$	−930613.			−1.65884
$750$			240252.i			0.427115i
$751$		−	58996.0i		−	0.104603i	−0.998631	−	0.0523013i	$-0.983344\pi$
$751$		−	58996.0i		−	0.104603i	0.998631	−	0.0523013i	$-0.0166556\pi$
$752$		−	150484.i		−	0.266106i
$753$	235987.			0.416196
$754$	−105991.			−0.186434
$755$			26627.1i			0.0467122i
$756$	−52891.1			−0.0925421
$757$	2177.79			0.00380036			0.00190018	−	0.999998i	$-0.499395\pi$
$757$	2177.79			0.00380036			0.00190018	+	0.999998i	$0.499395\pi$
$758$			139190.i			0.242254i
$759$	161308.			0.280009
$760$		−	155874.i		−	0.269865i
$761$	−567896.			−0.980617			−0.490309	−	0.871549i	$-0.663116\pi$
$761$	−567896.			−0.980617			−0.490309	+	0.871549i	$0.663116\pi$
$762$			251542.i			0.433211i
$763$		−	542932.i		−	0.932601i
$764$		−	511922.i		−	0.877035i
$765$	180136.			0.307806
$766$		−	406675.i		−	0.693091i
$767$	−112798.	+	340250.i	−0.191739	+	0.578372i
$768$	21283.4			0.0360844
$769$		−	558745.i		−	0.944846i	−0.881372	−	0.472423i	$-0.843379\pi$
$769$		−	558745.i		−	0.944846i	0.881372	−	0.472423i	$-0.156621\pi$
$770$	−107546.			−0.181389
$771$	−69344.1			−0.116654
$772$	60175.1			0.100968
$773$			393982.i			0.659352i	0.944094	+	0.329676i	$0.106940\pi$
$773$			393982.i			0.659352i	−0.944094	+	0.329676i	$0.893060\pi$
$774$	123130.			0.205533
$775$		−	208049.i		−	0.346387i
$776$	90262.7			0.149894
$777$		−	469446.i		−	0.777577i
$778$			337670.i			0.557871i
$779$	−159247.			−0.262420
$780$		−	72883.9i		−	0.119796i
$781$			433763.i			0.711133i
$782$	726027.			1.18724
$783$	−51054.4			−0.0832740
$784$	−11537.8			−0.0187711
$785$			381381.i			0.618899i
$786$	−205594.			−0.332786
$787$	−579971.			−0.936390			−0.468195	−	0.883625i	$-0.655095\pi$
$787$	−579971.			−0.936390			−0.468195	+	0.883625i	$0.655095\pi$
$788$	−130450.			−0.210083
$789$	418326.			0.671988
$790$		−	76210.5i		−	0.122113i
$791$		−	79850.8i		−	0.127622i
$792$	−28951.8			−0.0461557
$793$	−444655.			−0.707094
$794$	−817924.			−1.29739
$795$	−11246.9			−0.0177950
$796$	−130217.			−0.205514
$797$		−	934407.i		−	1.47102i	−0.677512	−	0.735512i	$-0.736942\pi$
$797$		−	934407.i		−	1.47102i	0.677512	−	0.735512i	$-0.263058\pi$
$798$		−	280213.i		−	0.440031i
$799$			921347.i			1.44321i
$800$		−	60659.7i		−	0.0947808i
$801$		−	26898.6i		−	0.0419242i
$802$	−71780.6			−0.111598
$803$	37703.2			0.0584719
$804$			134981.i			0.208814i
$805$		−	525613.i		−	0.811100i
$806$			180830.i			0.278356i
$807$			455241.i			0.699028i
$808$	−372972.			−0.571286
$809$		−	188625.i		−	0.288205i	−0.989563	−	0.144103i	$-0.953970\pi$
$809$		−	188625.i		−	0.288205i	0.989563	−	0.144103i	$-0.0460295\pi$
$810$		−	35107.2i		−	0.0535089i
$811$		−	1.15002e6i		−	1.74850i	−0.485479	−	0.874248i	$-0.661355\pi$
$811$		−	1.15002e6i		−	1.74850i	0.485479	−	0.874248i	$-0.338645\pi$
$812$	137191.			0.208071
$813$	−507514.			−0.767834
$814$		−	256968.i		−	0.387820i
$815$	−271051.			−0.408071
$816$	−130309.			−0.195701
$817$			652332.i			0.977292i
$818$	−526065.			−0.786200
$819$		−	131023.i		−	0.195335i
$820$	53613.0			0.0797338
$821$			99947.1i			0.148280i	0.997248	+	0.0741402i	$0.0236212\pi$
$821$			99947.1i			0.148280i	−0.997248	+	0.0741402i	$0.976379\pi$
$822$		−	18241.2i		−	0.0269966i
$823$		−	1.20974e6i		−	1.78604i	−0.450017	−	0.893020i	$-0.648582\pi$
$823$		−	1.20974e6i		−	1.78604i	0.450017	−	0.893020i	$-0.351418\pi$
$824$	160536.			0.236438
$825$			82515.4i			0.121235i
$826$	146001.	−	440407.i	0.213991	−	0.645496i
$827$	599251.			0.876189			0.438094	−	0.898929i	$-0.355654\pi$
$827$	599251.			0.876189			0.438094	+	0.898929i	$0.355654\pi$
$828$		−	141498.i		−	0.206390i
$829$	−1.12951e6			−1.64354			−0.821772	−	0.569816i	$-0.807014\pi$
$829$	−1.12951e6			−1.64354			−0.821772	+	0.569816i	$0.807014\pi$
$830$	446502.			0.648138
$831$	262001.			0.379404
$832$			52723.7i			0.0761657i
$833$	70640.6			0.101804
$834$		−	485844.i		−	0.698498i
$835$	−445189.			−0.638516
$836$		−	153385.i		−	0.219467i
$837$			87103.3i			0.124332i
$838$	96463.0			0.137364
$839$		−	806767.i		−	1.14610i	−0.819519	−	0.573052i	$-0.805759\pi$
$839$		−	806767.i		−	1.14610i	0.819519	−	0.573052i	$-0.194241\pi$
$840$			94338.2i			0.133699i
$841$	−574855.			−0.812767
$842$	370434.			0.522501
$843$	−271721.			−0.382356
$844$		−	253719.i		−	0.356178i
$845$	−305742.			−0.428196
$846$	179564.			0.250887
$847$	584122.			0.814211
$848$	8135.94			0.0113140
$849$		−	559395.i		−	0.776074i
$850$			371392.i			0.514038i
$851$	1.25589e6			1.73417
$852$	380494.			0.524165
$853$	209645.			0.288129			0.144065	−	0.989568i	$-0.453983\pi$
$853$	209645.			0.288129			0.144065	+	0.989568i	$0.453983\pi$
$854$	575545.			0.789157
$855$	185995.			0.254431
$856$			446845.i			0.609831i
$857$		−	1.30559e6i		−	1.77765i	−0.458245	−	0.888826i	$-0.651522\pi$
$857$		−	1.30559e6i		−	1.77765i	0.458245	−	0.888826i	$-0.348478\pi$
$858$		−	71720.0i		−	0.0974240i
$859$			484696.i			0.656875i	0.944526	+	0.328438i	$0.106522\pi$
$859$			484696.i			0.656875i	−0.944526	+	0.328438i	$0.893478\pi$
$860$		−	219618.i		−	0.296941i
$861$	96379.7			0.130011
$862$	−933595.			−1.25645
$863$		−	333875.i		−	0.448294i	−0.974555	−	0.224147i	$-0.928040\pi$
$863$		−	333875.i		−	0.448294i	0.974555	−	0.224147i	$-0.0719595\pi$
$864$			25396.3i			0.0340207i
$865$		−	584920.i		−	0.781744i
$866$			322124.i			0.429524i
$867$	363836.			0.484024
$868$		−	234059.i		−	0.310661i
$869$		−	74993.5i		−	0.0993080i
$870$			91062.1i			0.120309i
$871$	−334377.			−0.440757
$872$	−260695.			−0.342847
$873$			107705.i			0.141322i
$874$	749645.			0.981369
$875$	770349.			1.00617
$876$		−	33072.9i		−	0.0430987i
$877$	−1.47646e6			−1.91965			−0.959826	−	0.280595i	$-0.909468\pi$
$877$	−1.47646e6			−1.91965			−0.959826	+	0.280595i	$0.909468\pi$
$878$		−	160972.i		−	0.208815i
$879$	14504.2			0.0187723
$880$			51639.3i			0.0666830i
$881$		−	242199.i		−	0.312047i	−0.987753	−	0.156023i	$-0.950132\pi$
$881$		−	242199.i		−	0.312047i	0.987753	−	0.156023i	$-0.0498675\pi$
$882$		−	13767.4i		−	0.0176976i
$883$	−526071.			−0.674720			−0.337360	−	0.941376i	$-0.609534\pi$
$883$	−526071.			−0.674720			−0.337360	+	0.941376i	$0.609534\pi$
$884$		−	322804.i		−	0.413080i
$885$	292326.	+	96910.2i	0.373233	+	0.123732i
$886$	−409118.			−0.521171
$887$			422441.i			0.536931i	0.963289	+	0.268466i	$0.0865166\pi$
$887$			422441.i			0.536931i	−0.963289	+	0.268466i	$0.913483\pi$
$888$	−225410.			−0.285856
$889$	806548.			1.02053
$890$	−47977.1			−0.0605695
$891$		−	34546.6i		−	0.0435160i
$892$	410274.			0.515637
$893$			951318.i			1.19295i
$894$	−394711.			−0.493860
$895$		−	703521.i		−	0.878276i
$896$		−	68243.6i		−	0.0850053i
$897$	350521.			0.435641
$898$			528193.i			0.654998i
$899$		−	225931.i		−	0.279548i
$900$	72381.8			0.0893602
$901$	−49812.7			−0.0613607
$902$	52756.8			0.0648434
$903$		−	394805.i		−	0.484180i
$904$	−38341.3			−0.0469170
$905$	−68283.0			−0.0833711
$906$	22984.1			0.0280009
$907$	198548.			0.241353			0.120676	−	0.992692i	$-0.461494\pi$
$907$	198548.			0.241353			0.120676	+	0.992692i	$0.461494\pi$
$908$			446668.i			0.541768i
$909$		−	445046.i		−	0.538614i
$910$	−233696.			−0.282208
$911$	205635.			0.247776			0.123888	−	0.992296i	$-0.460464\pi$
$911$	205635.			0.247776			0.123888	+	0.992296i	$0.460464\pi$
$912$	−134548.			−0.161766
$913$	439372.			0.527097
$914$	481933.			0.576891
$915$			382026.i			0.456300i
$916$			226885.i			0.270405i
$917$			659220.i			0.783956i
$918$		−	155490.i		−	0.184509i
$919$			1.60494e6i			1.90033i	0.311748	+	0.950165i	$0.399086\pi$
$919$			1.60494e6i			1.90033i	−0.311748	+	0.950165i	$0.600914\pi$
$920$	−252379.			−0.298180
$921$	343432.			0.404876
$922$		−	126780.i		−	0.149138i
$923$			942566.i			1.10639i
$924$			92831.7i			0.108731i
$925$			642439.i			0.750842i
$926$	107822.			0.125743
$927$			191558.i			0.222916i
$928$		−	65873.7i		−	0.0764920i
$929$			1.41505e6i			1.63961i	0.572640	+	0.819807i	$0.305919\pi$
$929$			1.41505e6i			1.63961i	−0.572640	+	0.819807i	$0.694081\pi$
$930$	155360.			0.179628
$931$	72938.5			0.0841506
$932$			234850.i			0.270370i
$933$	166232.			0.190964
$934$	357510.			0.409822
$935$		−	316165.i		−	0.361651i
$936$	−62912.2			−0.0718097
$937$		−	455249.i		−	0.518525i	−0.965807	−	0.259262i	$-0.916521\pi$
$937$		−	455249.i		−	0.518525i	0.965807	−	0.259262i	$-0.0834794\pi$
$938$	432804.			0.491910
$939$		−	284508.i		−	0.322673i
$940$		−	320276.i		−	0.362467i
$941$		−	1.05591e6i		−	1.19248i	−0.802808	−	0.596238i	$-0.796662\pi$
$941$		−	1.05591e6i		−	1.19248i	0.802808	−	0.596238i	$-0.203338\pi$
$942$	329202.			0.370989
$943$			257841.i			0.289954i
$944$	−211467.	−	70104.3i	−0.237300	−	0.0786684i
$945$	−112568.			−0.126053
$946$		−	216111.i		−	0.241487i
$947$	650000.			0.724792			0.362396	−	0.932024i	$-0.381959\pi$
$947$	650000.			0.724792			0.362396	+	0.932024i	$0.381959\pi$
$948$	−65783.7			−0.0731984
$949$	81928.9			0.0909713
$950$			383474.i			0.424901i
$951$	−883845.			−0.977271
$952$			417825.i			0.461021i
$953$	−304739.			−0.335538			−0.167769	−	0.985826i	$-0.553656\pi$
$953$	−304739.			−0.335538			−0.167769	+	0.985826i	$0.553656\pi$
$954$			9708.15i			0.0106669i
$955$		−	1.08953e6i		−	1.19462i
$956$	500420.			0.547544
$957$			89607.9i			0.0978414i
$958$			28654.2i			0.0312218i
$959$	−58488.9			−0.0635970
$960$	45297.6			0.0491511
$961$	538062.			0.582620
$962$		−	558390.i		−	0.603375i
$963$	−533195.			−0.574954
$964$	830917.			0.894136
$965$	128071.			0.137529
$966$	−453701.			−0.486200
$967$		−	913330.i		−	0.976730i	−0.872639	−	0.488365i	$-0.837593\pi$
$967$		−	913330.i		−	0.976730i	0.872639	−	0.488365i	$-0.162407\pi$
$968$		−	280473.i		−	0.299323i
$969$	823776.			0.877327
$970$	192106.			0.204173
$971$	−53747.2			−0.0570055			−0.0285028	−	0.999594i	$-0.509074\pi$
$971$	−53747.2			−0.0570055			−0.0285028	+	0.999594i	$0.509074\pi$
$972$	−30304.0			−0.0320750
$973$	−1.55782e6			−1.64548
$974$			195340.i			0.205908i
$975$			179306.i			0.188619i
$976$		−	276355.i		−	0.290113i
$977$		−	672926.i		−	0.704982i	−0.935815	−	0.352491i	$-0.885335\pi$
$977$		−	672926.i		−	0.704982i	0.935815	−	0.352491i	$-0.114665\pi$
$978$			233967.i			0.244611i
$979$	−47211.0			−0.0492581
$980$	−24555.9			−0.0255684
$981$		−	311073.i		−	0.323239i
$982$		−	776479.i		−	0.805205i
$983$			1.21590e6i			1.25832i	0.777275	+	0.629161i	$0.216601\pi$
$983$			1.21590e6i			1.25832i	−0.777275	+	0.629161i	$0.783399\pi$
$984$		−	46277.9i		−	0.0477951i
$985$	−277637.			−0.286157
$986$			403315.i			0.414850i
$987$		−	575758.i		−	0.591025i
$988$		−	333304.i		−	0.341450i
$989$	1.05621e6			1.07983
$990$	−61618.3			−0.0628694
$991$		−	234097.i		−	0.238368i	−0.992872	−	0.119184i	$-0.961972\pi$
$991$		−	234097.i		−	0.238368i	0.992872	−	0.119184i	$-0.0380279\pi$
$992$	−112386.			−0.114206
$993$	955793.			0.969316
$994$		−	1.22002e6i		−	1.23479i
$995$	−277142.			−0.279934
$996$		−	385413.i		−	0.388515i
$997$	−1.74383e6			−1.75434			−0.877170	−	0.480180i	$-0.840572\pi$
$997$	−1.74383e6			−1.75434			−0.877170	+	0.480180i	$0.840572\pi$
$998$			552616.i			0.554833i
$999$		−	268969.i		−	0.269508i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	354.5.d.a.235.14	yes	40
3.2	odd	2		1062.5.d.b.235.7		40
59.58	odd	2	inner	354.5.d.a.235.13	✓	40
177.176	even	2		1062.5.d.b.235.8		40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
354.5.d.a.235.13	✓	40	59.58	odd	2	inner
354.5.d.a.235.14	yes	40	1.1	even	1	trivial
1062.5.d.b.235.7		40	3.2	odd	2
1062.5.d.b.235.8		40	177.176	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 354 = 2 \cdot 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	354.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.82843i q^{2} +5.19615 q^{3} -8.00000 q^{4} -17.0264 q^{5} +14.6969i q^{6} +47.1245 q^{7} -22.6274i q^{8} +27.0000 q^{9} +O(q^{10})\)	\(q+2.82843i q^{2} +5.19615 q^{3} -8.00000 q^{4} -17.0264 q^{5} +14.6969i q^{6} +47.1245 q^{7} -22.6274i q^{8} +27.0000 q^{9} -48.1580i q^{10} -47.3890i q^{11} -41.5692 q^{12} -102.976i q^{13} +133.288i q^{14} -88.4719 q^{15} +64.0000 q^{16} -391.843 q^{17} +76.3675i q^{18} -404.590 q^{19} +136.211 q^{20} +244.866 q^{21} +134.036 q^{22} +655.082i q^{23} -117.576i q^{24} -335.101 q^{25} +291.260 q^{26} +140.296 q^{27} -376.996 q^{28} -363.904 q^{29} -250.236i q^{30} +620.853i q^{31} +181.019i q^{32} -246.240i q^{33} -1108.30i q^{34} -802.362 q^{35} -216.000 q^{36} -1917.15i q^{37} -1144.35i q^{38} -535.079i q^{39} +385.264i q^{40} +393.601 q^{41} +692.586i q^{42} -1612.33i q^{43} +379.112i q^{44} -459.713 q^{45} -1852.85 q^{46} -2351.32i q^{47} +332.554 q^{48} -180.278 q^{49} -947.808i q^{50} -2036.08 q^{51} +823.808i q^{52} +127.124 q^{53} +396.817i q^{54} +806.865i q^{55} -1066.31i q^{56} -2102.31 q^{57} -1029.28i q^{58} +(-3304.16 - 1095.38i) q^{59} +707.775 q^{60} -4318.05i q^{61} -1756.04 q^{62} +1272.36 q^{63} -512.000 q^{64} +1753.31i q^{65} +696.473 q^{66} -3247.13i q^{67} +3134.75 q^{68} +3403.91i q^{69} -2269.42i q^{70} -9153.25 q^{71} -610.940i q^{72} +795.611i q^{73} +5422.52 q^{74} -1741.24 q^{75} +3236.72 q^{76} -2233.18i q^{77} +1513.43 q^{78} +1582.51 q^{79} -1089.69 q^{80} +729.000 q^{81} +1113.27i q^{82} +9271.61i q^{83} -1958.93 q^{84} +6671.69 q^{85} +4560.36 q^{86} -1890.90 q^{87} -1072.29 q^{88} -996.244i q^{89} -1300.27i q^{90} -4852.70i q^{91} -5240.66i q^{92} +3226.05i q^{93} +6650.52 q^{94} +6888.72 q^{95} +940.604i q^{96} +3989.08i q^{97} -509.902i q^{98} -1279.50i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 320 q^{4} - 80 q^{7} + 1080 q^{9}+O(q^{10}) \) 40 * q - 320 * q^4 - 80 * q^7 + 1080 * q^9	\( 40 q - 320 q^{4} - 80 q^{7} + 1080 q^{9} - 144 q^{15} + 2560 q^{16} + 480 q^{17} - 792 q^{19} - 1024 q^{22} + 3400 q^{25} + 768 q^{26} + 640 q^{28} + 1608 q^{29} - 5760 q^{35} - 8640 q^{36} + 6264 q^{41} + 7040 q^{46} + 17912 q^{49} + 1296 q^{51} - 1104 q^{53} + 5040 q^{57} + 13584 q^{59} + 1152 q^{60} - 12288 q^{62} - 2160 q^{63} - 20480 q^{64} + 1152 q^{66} - 3840 q^{68} + 35352 q^{71} + 4608 q^{74} + 3168 q^{75} + 6336 q^{76} - 12672 q^{78} - 15720 q^{79} + 29160 q^{81} - 26872 q^{85} + 18432 q^{86} + 7776 q^{87} + 8192 q^{88} - 18432 q^{94} - 19128 q^{95}+O(q^{100}) \) 40 * q - 320 * q^4 - 80 * q^7 + 1080 * q^9 - 144 * q^15 + 2560 * q^16 + 480 * q^17 - 792 * q^19 - 1024 * q^22 + 3400 * q^25 + 768 * q^26 + 640 * q^28 + 1608 * q^29 - 5760 * q^35 - 8640 * q^36 + 6264 * q^41 + 7040 * q^46 + 17912 * q^49 + 1296 * q^51 - 1104 * q^53 + 5040 * q^57 + 13584 * q^59 + 1152 * q^60 - 12288 * q^62 - 2160 * q^63 - 20480 * q^64 + 1152 * q^66 - 3840 * q^68 + 35352 * q^71 + 4608 * q^74 + 3168 * q^75 + 6336 * q^76 - 12672 * q^78 - 15720 * q^79 + 29160 * q^81 - 26872 * q^85 + 18432 * q^86 + 7776 * q^87 + 8192 * q^88 - 18432 * q^94 - 19128 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more