LMFDB - Embedded newform 350.6.c.i.99.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [350,6,Mod(99,350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(350, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("350.99");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 350 = 2 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	350.c (of order $2$, degree $1$, not 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:	$56.1343369345$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{79})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 39x^{2} + 400 $ x^4 - 39*x^2 + 400
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			99.3
Root			$4.44410 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	350.99
Dual form			350.6.c.i.99.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.00000i q^{2} -7.77639i q^{3} -16.0000 q^{4} +31.1056 q^{6} +49.0000i q^{7} -64.0000i q^{8} +182.528 q^{9} +O(q^{10})$	\(q+4.00000i q^{2} -7.77639i q^{3} -16.0000 q^{4} +31.1056 q^{6} +49.0000i q^{7} -64.0000i q^{8} +182.528 q^{9} -588.329 q^{11} +124.422i q^{12} -147.515i q^{13} -196.000 q^{14} +256.000 q^{16} +63.1806i q^{17} +730.111i q^{18} +1612.51 q^{19} +381.043 q^{21} -2353.32i q^{22} +1484.73i q^{23} -497.689 q^{24} +590.061 q^{26} -3309.07i q^{27} -784.000i q^{28} +1691.84 q^{29} -7446.52 q^{31} +1024.00i q^{32} +4575.08i q^{33} -252.722 q^{34} -2920.44 q^{36} +2439.79i q^{37} +6450.03i q^{38} -1147.14 q^{39} +334.413 q^{41} +1524.17i q^{42} -11933.5i q^{43} +9413.27 q^{44} -5938.91 q^{46} -5866.15i q^{47} -1990.76i q^{48} -2401.00 q^{49} +491.317 q^{51} +2360.24i q^{52} -25017.1i q^{53} +13236.3 q^{54} +3136.00 q^{56} -12539.5i q^{57} +6767.38i q^{58} +52348.0 q^{59} +16847.5 q^{61} -29786.1i q^{62} +8943.86i q^{63} -4096.00 q^{64} -18300.3 q^{66} -69080.7i q^{67} -1010.89i q^{68} +11545.8 q^{69} +30528.0 q^{71} -11681.8i q^{72} +46165.7i q^{73} -9759.17 q^{74} -25800.1 q^{76} -28828.1i q^{77} -4588.54i q^{78} +1869.00 q^{79} +18621.6 q^{81} +1337.65i q^{82} -106022. i q^{83} -6096.69 q^{84} +47734.0 q^{86} -13156.4i q^{87} +37653.1i q^{88} +38290.7 q^{89} +7228.25 q^{91} -23755.6i q^{92} +57907.0i q^{93} +23464.6 q^{94} +7963.02 q^{96} -90755.9i q^{97} -9604.00i q^{98} -107386. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 64 q^{4} - 160 q^{6} - 692 q^{9}+O(q^{10}) $ 4 * q - 64 * q^4 - 160 * q^6 - 692 * q^9	$ 4 q - 64 q^{4} - 160 q^{6} - 692 q^{9} - 2140 q^{11} - 784 q^{14} + 1024 q^{16} - 1656 q^{19} - 1960 q^{21} + 2560 q^{24} - 5888 q^{26} + 4492 q^{29} + 576 q^{31} - 15232 q^{34} + 11072 q^{36} - 51376 q^{39} + 22456 q^{41} + 34240 q^{44} - 13232 q^{46} - 9604 q^{49} - 101280 q^{51} + 89920 q^{54} + 12544 q^{56} + 185856 q^{59} + 30344 q^{61} - 16384 q^{64} + 70432 q^{66} + 13688 q^{69} - 41644 q^{71} + 71888 q^{74} + 26496 q^{76} + 144212 q^{79} + 220964 q^{81} + 31360 q^{84} - 27216 q^{86} + 19840 q^{89} - 72128 q^{91} + 124576 q^{94} - 40960 q^{96} + 294380 q^{99}+O(q^{100}) $ 4 * q - 64 * q^4 - 160 * q^6 - 692 * q^9 - 2140 * q^11 - 784 * q^14 + 1024 * q^16 - 1656 * q^19 - 1960 * q^21 + 2560 * q^24 - 5888 * q^26 + 4492 * q^29 + 576 * q^31 - 15232 * q^34 + 11072 * q^36 - 51376 * q^39 + 22456 * q^41 + 34240 * q^44 - 13232 * q^46 - 9604 * q^49 - 101280 * q^51 + 89920 * q^54 + 12544 * q^56 + 185856 * q^59 + 30344 * q^61 - 16384 * q^64 + 70432 * q^66 + 13688 * q^69 - 41644 * q^71 + 71888 * q^74 + 26496 * q^76 + 144212 * q^79 + 220964 * q^81 + 31360 * q^84 - 27216 * q^86 + 19840 * q^89 - 72128 * q^91 + 124576 * q^94 - 40960 * q^96 + 294380 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/350\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$1$	$-1$

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 $

$2$	4.00000i	0.707107i
$2$	4.00000i	0.707107i
$3$	− 7.77639i	− 0.498856i	−0.968393	−	0.249428i	$-0.919757\pi$
$3$	− 7.77639i	− 0.498856i	0.968393	−	0.249428i	$-0.0802425\pi$
$4$	−16.0000	−0.500000
$5$	0	0
$5$	0	0
$6$	31.1056	0.352744
$7$	49.0000i	0.377964i
$7$	49.0000i	0.377964i
$8$	− 64.0000i	− 0.353553i
$9$	182.528	0.751143
$10$	0	0
$11$	−588.329	−1.46602	−0.733008	−	0.680220i	$-0.761884\pi$
$11$	−588.329	−1.46602	−0.733008	+	0.680220i	$0.761884\pi$
$12$	124.422i	0.249428i
$13$	− 147.515i	− 0.242091i	−0.992647	−	0.121045i	$-0.961375\pi$
$13$	− 147.515i	− 0.242091i	0.992647	−	0.121045i	$-0.0386247\pi$
$14$	−196.000	−0.267261
$15$	0	0
$16$	256.000	0.250000
$17$	63.1806i	0.0530226i	0.999649	+	0.0265113i	$0.00843980\pi$
$17$	63.1806i	0.0530226i	−0.999649	+	0.0265113i	$0.991560\pi$
$18$	730.111i	0.531138i
$19$	1612.51	1.02475	0.512375	−	0.858762i	$-0.328766\pi$
$19$	1612.51	1.02475	0.512375	+	0.858762i	$0.328766\pi$
$20$	0	0
$21$	381.043	0.188550
$22$	− 2353.32i	− 1.03663i
$23$	1484.73i	0.585230i	0.956230	+	0.292615i	$0.0945255\pi$
$23$	1484.73i	0.585230i	−0.956230	+	0.292615i	$0.905475\pi$
$24$	−497.689	−0.176372
$25$	0	0
$26$	590.061	0.171184
$27$	− 3309.07i	− 0.873568i
$28$	− 784.000i	− 0.188982i
$29$	1691.84	0.373564	0.186782	−	0.982401i	$-0.440194\pi$
$29$	1691.84	0.373564	0.186782	+	0.982401i	$0.440194\pi$
$30$	0	0
$31$	−7446.52	−1.39171	−0.695855	−	0.718182i	$-0.744975\pi$
$31$	−7446.52	−1.39171	−0.695855	+	0.718182i	$0.744975\pi$
$32$	1024.00i	0.176777i
$33$	4575.08i	0.731330i
$34$	−252.722	−0.0374927
$35$	0	0
$36$	−2920.44	−0.375572
$37$	2439.79i	0.292987i	0.989212	+	0.146494i	$0.0467988\pi$
$37$	2439.79i	0.292987i	−0.989212	+	0.146494i	$0.953201\pi$
$38$	6450.03i	0.724608i
$39$	−1147.14	−0.120768
$40$	0	0
$41$	334.413	0.0310687	0.0155343	−	0.999879i	$-0.495055\pi$
$41$	334.413	0.0310687	0.0155343	+	0.999879i	$0.495055\pi$
$42$	1524.17i	0.133325i
$43$	− 11933.5i	− 0.984229i	−0.870530	−	0.492115i	$-0.836224\pi$
$43$	− 11933.5i	− 0.984229i	0.870530	−	0.492115i	$-0.163776\pi$
$44$	9413.27	0.733008
$45$	0	0
$46$	−5938.91	−0.413820
$47$	− 5866.15i	− 0.387354i	−0.981065	−	0.193677i	$-0.937959\pi$
$47$	− 5866.15i	− 0.387354i	0.981065	−	0.193677i	$-0.0620414\pi$
$48$	− 1990.76i	− 0.124714i
$49$	−2401.00	−0.142857
$50$	0	0
$51$	491.317	0.0264506
$52$	2360.24i	0.121045i
$53$	− 25017.1i	− 1.22334i	−0.791112	−	0.611671i	$-0.790498\pi$
$53$	− 25017.1i	− 1.22334i	0.791112	−	0.611671i	$-0.209502\pi$
$54$	13236.3	0.617706
$55$	0	0
$56$	3136.00	0.133631
$57$	− 12539.5i	− 0.511202i
$58$	6767.38i	0.264150i
$59$	52348.0	1.95781	0.978904	−	0.204322i	$-0.0654991\pi$
$59$	52348.0	1.95781	0.978904	+	0.204322i	$0.0654991\pi$
$60$	0	0
$61$	16847.5	0.579710	0.289855	−	0.957071i	$-0.406393\pi$
$61$	16847.5	0.579710	0.289855	+	0.957071i	$0.406393\pi$
$62$	− 29786.1i	− 0.984088i
$63$	8943.86i	0.283905i
$64$	−4096.00	−0.125000
$65$	0	0
$66$	−18300.3	−0.517129
$67$	− 69080.7i	− 1.88005i	−0.341104	−	0.940026i	$-0.610801\pi$
$67$	− 69080.7i	− 1.88005i	0.341104	−	0.940026i	$-0.389199\pi$
$68$	− 1010.89i	− 0.0265113i
$69$	11545.8	0.291945
$70$	0	0
$71$	30528.0	0.718709	0.359354	−	0.933201i	$-0.382997\pi$
$71$	30528.0	0.718709	0.359354	+	0.933201i	$0.382997\pi$
$72$	− 11681.8i	− 0.265569i
$73$	46165.7i	1.01394i	0.861964	+	0.506970i	$0.169234\pi$
$73$	46165.7i	1.01394i	−0.861964	+	0.506970i	$0.830766\pi$
$74$	−9759.17	−0.207173
$75$	0	0
$76$	−25800.1	−0.512375
$77$	− 28828.1i	− 0.554102i
$78$	− 4588.54i	− 0.0853962i
$79$	1869.00	0.0336932	0.0168466	−	0.999858i	$-0.494637\pi$
$79$	1869.00	0.0336932	0.0168466	+	0.999858i	$0.494637\pi$
$80$	0	0
$81$	18621.6	0.315359
$82$	1337.65i	0.0219689i
$83$	− 106022.i	− 1.68928i	−0.535332	−	0.844642i	$-0.679813\pi$
$83$	− 106022.i	− 1.68928i	0.535332	−	0.844642i	$-0.320187\pi$
$84$	−6096.69	−0.0942748
$85$	0	0
$86$	47734.0	0.695955
$87$	− 13156.4i	− 0.186355i
$88$	37653.1i	0.518315i
$89$	38290.7	0.512411	0.256206	−	0.966622i	$-0.417528\pi$
$89$	38290.7	0.512411	0.256206	+	0.966622i	$0.417528\pi$
$90$	0	0
$91$	7228.25	0.0915018
$92$	− 23755.6i	− 0.292615i
$93$	57907.0i	0.694263i
$94$	23464.6	0.273901
$95$	0	0
$96$	7963.02	0.0881860
$97$	− 90755.9i	− 0.979367i	−0.871900	−	0.489683i	$-0.837112\pi$
$97$	− 90755.9i	− 0.979367i	0.871900	−	0.489683i	$-0.162888\pi$
$98$	− 9604.00i	− 0.101015i
$99$	−107386.	−1.10119
$100$	0	0
$101$	115241.	1.12409	0.562046	−	0.827106i	$-0.310014\pi$
$101$	115241.	1.12409	0.562046	+	0.827106i	$0.310014\pi$
$102$	1965.27i	0.0187034i
$103$	− 8842.67i	− 0.0821278i	−0.999157	−	0.0410639i	$-0.986925\pi$
$103$	− 8842.67i	− 0.0821278i	0.999157	−	0.0410639i	$-0.0130747\pi$
$104$	−9440.98	−0.0855921
$105$	0	0
$106$	100068.	0.865033
$107$	− 131403.i	− 1.10955i	−0.832000	−	0.554775i	$-0.812804\pi$
$107$	− 131403.i	− 1.10955i	0.832000	−	0.554775i	$-0.187196\pi$
$108$	52945.1i	0.436784i
$109$	115386.	0.930223	0.465112	−	0.885252i	$-0.346014\pi$
$109$	115386.	0.930223	0.465112	+	0.885252i	$0.346014\pi$
$110$	0	0
$111$	18972.8	0.146158
$112$	12544.0i	0.0944911i
$113$	114989.i	0.847152i	0.905861	+	0.423576i	$0.139225\pi$
$113$	114989.i	0.847152i	−0.905861	+	0.423576i	$0.860775\pi$
$114$	50158.0	0.361475
$115$	0	0
$116$	−27069.5	−0.186782
$117$	− 26925.6i	− 0.181845i
$118$	209392.i	1.38438i
$119$	−3095.85	−0.0200407
$120$	0	0
$121$	185080.	1.14920
$122$	67390.0i	0.409917i
$123$	− 2600.52i	− 0.0154988i
$124$	119144.	0.695855
$125$	0	0
$126$	−35775.4	−0.200751
$127$	− 205537.i	− 1.13079i	−0.824821	−	0.565394i	$-0.808724\pi$
$127$	− 205537.i	− 1.13079i	0.824821	−	0.565394i	$-0.191276\pi$
$128$	− 16384.0i	− 0.0883883i
$129$	−92799.5	−0.490988
$130$	0	0
$131$	188494.	0.959662	0.479831	−	0.877361i	$-0.340698\pi$
$131$	188494.	0.959662	0.479831	+	0.877361i	$0.340698\pi$
$132$	− 73201.2i	− 0.365665i
$133$	79012.9i	0.387319i
$134$	276323.	1.32940
$135$	0	0
$136$	4043.56	0.0187463
$137$	40844.0i	0.185920i	0.995670	+	0.0929602i	$0.0296329\pi$
$137$	40844.0i	0.185920i	−0.995670	+	0.0929602i	$0.970367\pi$
$138$	46183.2i	0.206437i
$139$	7349.31	0.0322633	0.0161317	−	0.999870i	$-0.494865\pi$
$139$	7349.31	0.0322633	0.0161317	+	0.999870i	$0.494865\pi$
$140$	0	0
$141$	−45617.5	−0.193234
$142$	122112.i	0.508204i
$143$	86787.5i	0.354909i
$144$	46727.1	0.187786
$145$	0	0
$146$	−184663.	−0.716964
$147$	18671.1i	0.0712651i
$148$	− 39036.7i	− 0.146494i
$149$	−277980.	−1.02576	−0.512882	−	0.858459i	$-0.671422\pi$
$149$	−277980.	−1.02576	−0.512882	+	0.858459i	$0.671422\pi$
$150$	0	0
$151$	100921.	0.360197	0.180099	−	0.983649i	$-0.442358\pi$
$151$	100921.	0.360197	0.180099	+	0.983649i	$0.442358\pi$
$152$	− 103201.i	− 0.362304i
$153$	11532.2i	0.0398276i
$154$	115313.	0.391809
$155$	0	0
$156$	18354.2	0.0603842
$157$	− 202146.i	− 0.654511i	−0.944936	−	0.327255i	$-0.893876\pi$
$157$	− 202146.i	− 0.654511i	0.944936	−	0.327255i	$-0.106124\pi$
$158$	7476.02i	0.0238247i
$159$	−194543.	−0.610271
$160$	0	0
$161$	−72751.6	−0.221196
$162$	74486.6i	0.222993i
$163$	120232.i	0.354448i	0.984171	+	0.177224i	$0.0567117\pi$
$163$	120232.i	0.354448i	−0.984171	+	0.177224i	$0.943288\pi$
$164$	−5350.60	−0.0155343
$165$	0	0
$166$	424090.	1.19450
$167$	368050.i	1.02121i	0.859815	+	0.510605i	$0.170579\pi$
$167$	368050.i	1.02121i	−0.859815	+	0.510605i	$0.829421\pi$
$168$	− 24386.8i	− 0.0666624i
$169$	349532.	0.941392
$170$	0	0
$171$	294328.	0.769734
$172$	190936.i	0.492115i
$173$	− 50079.2i	− 0.127216i	−0.997975	−	0.0636080i	$-0.979739\pi$
$173$	− 50079.2i	− 0.127216i	0.997975	−	0.0636080i	$-0.0202607\pi$
$174$	52625.8	0.131773
$175$	0	0
$176$	−150612.	−0.366504
$177$	− 407078.i	− 0.976663i
$178$	153163.i	0.362329i
$179$	366552.	0.855073	0.427536	−	0.903998i	$-0.359382\pi$
$179$	366552.	0.855073	0.427536	+	0.903998i	$0.359382\pi$
$180$	0	0
$181$	237963.	0.539899	0.269949	−	0.962874i	$-0.412993\pi$
$181$	237963.	0.539899	0.269949	+	0.962874i	$0.412993\pi$
$182$	28913.0i	0.0647015i
$183$	− 131013.i	− 0.289192i
$184$	95022.5	0.206910
$185$	0	0
$186$	−231628.	−0.490918
$187$	− 37171.0i	− 0.0777320i
$188$	93858.4i	0.193677i
$189$	162144.	0.330177
$190$	0	0
$191$	180213.	0.357439	0.178720	−	0.983900i	$-0.442804\pi$
$191$	180213.	0.357439	0.178720	+	0.983900i	$0.442804\pi$
$192$	31852.1i	0.0623569i
$193$	− 671065.i	− 1.29679i	−0.761302	−	0.648397i	$-0.775440\pi$
$193$	− 671065.i	− 1.29679i	0.761302	−	0.648397i	$-0.224560\pi$
$194$	363023.	0.692517
$195$	0	0
$196$	38416.0	0.0714286
$197$	− 432242.i	− 0.793527i	−0.917921	−	0.396763i	$-0.870133\pi$
$197$	− 432242.i	− 0.793527i	0.917921	−	0.396763i	$-0.129867\pi$
$198$	− 429546.i	− 0.778657i
$199$	−912639.	−1.63368	−0.816839	−	0.576866i	$-0.804276\pi$
$199$	−912639.	−1.63368	−0.816839	+	0.576866i	$0.804276\pi$
$200$	0	0
$201$	−537198.	−0.937874
$202$	460962.i	0.794853i
$203$	82900.4i	0.141194i
$204$	−7861.07	−0.0132253
$205$	0	0
$206$	35370.7	0.0580731
$207$	271004.i	0.439592i
$208$	− 37763.9i	− 0.0605227i
$209$	−948686.	−1.50230
$210$	0	0
$211$	−984209.	−1.52188	−0.760941	−	0.648821i	$-0.775262\pi$
$211$	−984209.	−1.52188	−0.760941	+	0.648821i	$0.775262\pi$
$212$	400274.i	0.611671i
$213$	− 237398.i	− 0.358532i
$214$	525614.	0.784571
$215$	0	0
$216$	−211780.	−0.308853
$217$	− 364879.i	− 0.526017i
$218$	461544.i	0.657767i
$219$	359002.	0.505809
$220$	0	0
$221$	9320.10	0.0128363
$222$	75891.1i	0.103349i
$223$	− 363329.i	− 0.489257i	−0.969617	−	0.244629i	$-0.921334\pi$
$223$	− 363329.i	− 0.489257i	0.969617	−	0.244629i	$-0.0786661\pi$
$224$	−50176.0	−0.0668153
$225$	0	0
$226$	−459957.	−0.599027
$227$	587124.i	0.756250i	0.925755	+	0.378125i	$0.123431\pi$
$227$	587124.i	0.756250i	−0.925755	+	0.378125i	$0.876569\pi$
$228$	200632.i	0.255601i
$229$	−1.21082e6	−1.52578	−0.762891	−	0.646528i	$-0.776221\pi$
$229$	−1.21082e6	−1.52578	−0.762891	+	0.646528i	$0.776221\pi$
$230$	0	0
$231$	−224179.	−0.276417
$232$	− 108278.i	− 0.132075i
$233$	− 163067.i	− 0.196778i	−0.995148	−	0.0983890i	$-0.968631\pi$
$233$	− 163067.i	− 0.196778i	0.995148	−	0.0983890i	$-0.0313689\pi$
$234$	107703.	0.128584
$235$	0	0
$236$	−837568.	−0.978904
$237$	− 14534.1i	− 0.0168081i
$238$	− 12383.4i	− 0.0141709i
$239$	−1.27658e6	−1.44562	−0.722809	−	0.691048i	$-0.757149\pi$
$239$	−1.27658e6	−1.44562	−0.722809	+	0.691048i	$0.757149\pi$
$240$	0	0
$241$	−506946.	−0.562237	−0.281118	−	0.959673i	$-0.590705\pi$
$241$	−506946.	−0.562237	−0.281118	+	0.959673i	$0.590705\pi$
$242$	740321.i	0.812609i
$243$	− 948913.i	− 1.03089i
$244$	−269560.	−0.289855
$245$	0	0
$246$	10402.1	0.0109593
$247$	− 237870.i	− 0.248083i
$248$	476577.i	0.492044i
$249$	−824472.	−0.842709
$250$	0	0
$251$	−1.72797e6	−1.73122	−0.865611	−	0.500717i	$-0.833070\pi$
$251$	−1.72797e6	−1.73122	−0.865611	+	0.500717i	$0.833070\pi$
$252$	− 143102.i	− 0.141953i
$253$	− 873508.i	− 0.857957i
$254$	822148.	0.799587
$255$	0	0
$256$	65536.0	0.0625000
$257$	− 104724.i	− 0.0989043i	−0.998776	−	0.0494522i	$-0.984252\pi$
$257$	− 104724.i	− 0.0989043i	0.998776	−	0.0494522i	$-0.0157475\pi$
$258$	− 371198.i	− 0.347181i
$259$	−119550.	−0.110739
$260$	0	0
$261$	308809.	0.280600
$262$	753974.i	0.678584i
$263$	− 1.36123e6i	− 1.21351i	−0.794889	−	0.606755i	$-0.792471\pi$
$263$	− 1.36123e6i	− 1.21351i	0.794889	−	0.606755i	$-0.207529\pi$
$264$	292805.	0.258564
$265$	0	0
$266$	−316052.	−0.273876
$267$	− 297764.i	− 0.255619i
$268$	1.10529e6i	0.940026i
$269$	347865.	0.293109	0.146555	−	0.989203i	$-0.453182\pi$
$269$	347865.	0.293109	0.146555	+	0.989203i	$0.453182\pi$
$270$	0	0
$271$	1.18265e6	0.978209	0.489104	−	0.872225i	$-0.337324\pi$
$271$	1.18265e6	0.978209	0.489104	+	0.872225i	$0.337324\pi$
$272$	16174.2i	0.0132557i
$273$	− 56209.7i	− 0.0456462i
$274$	−163376.	−0.131466
$275$	0	0
$276$	−184733.	−0.145973
$277$	628741.i	0.492348i	0.969226	+	0.246174i	$0.0791735\pi$
$277$	628741.i	0.492348i	−0.969226	+	0.246174i	$0.920826\pi$
$278$	29397.2i	0.0228136i
$279$	−1.35920e6	−1.04537
$280$	0	0
$281$	1.51236e6	1.14258	0.571292	−	0.820747i	$-0.306442\pi$
$281$	1.51236e6	1.14258	0.571292	+	0.820747i	$0.306442\pi$
$282$	− 182470.i	− 0.136637i
$283$	1.17843e6i	0.874654i	0.899303	+	0.437327i	$0.144075\pi$
$283$	1.17843e6i	0.874654i	−0.899303	+	0.437327i	$0.855925\pi$
$284$	−488448.	−0.359354
$285$	0	0
$286$	−347150.	−0.250959
$287$	16386.2i	0.0117429i
$288$	186908.i	0.132785i
$289$	1.41587e6	0.997189
$290$	0	0
$291$	−705753.	−0.488563
$292$	− 738651.i	− 0.506970i
$293$	490855.i	0.334029i	0.985954	+	0.167014i	$0.0534127\pi$
$293$	490855.i	0.334029i	−0.985954	+	0.167014i	$0.946587\pi$
$294$	−74684.4	−0.0503920
$295$	0	0
$296$	156147.	0.103587
$297$	1.94682e6i	1.28066i
$298$	− 1.11192e6i	− 0.725325i
$299$	219020.	0.141679
$300$	0	0
$301$	584741.	0.372004
$302$	403685.i	0.254698i
$303$	− 896155.i	− 0.560759i
$304$	412802.	0.256188
$305$	0	0
$306$	−46128.8	−0.0281623
$307$	2.29867e6i	1.39197i	0.718055	+	0.695986i	$0.245033\pi$
$307$	2.29867e6i	1.39197i	−0.718055	+	0.695986i	$0.754967\pi$
$308$	461250.i	0.277051i
$309$	−68764.0	−0.0409699
$310$	0	0
$311$	−1.61778e6	−0.948459	−0.474229	−	0.880401i	$-0.657273\pi$
$311$	−1.61778e6	−0.948459	−0.474229	+	0.880401i	$0.657273\pi$
$312$	73416.7i	0.0426981i
$313$	− 174399.i	− 0.100620i	−0.998734	−	0.0503099i	$-0.983979\pi$
$313$	− 174399.i	− 0.100620i	0.998734	−	0.0503099i	$-0.0160209\pi$
$314$	808586.	0.462809
$315$	0	0
$316$	−29904.1	−0.0168466
$317$	3.28000e6i	1.83327i	0.399730	+	0.916633i	$0.369104\pi$
$317$	3.28000e6i	1.83327i	−0.399730	+	0.916633i	$0.630896\pi$
$318$	− 778171.i	− 0.431527i
$319$	−995361.	−0.547651
$320$	0	0
$321$	−1.02184e6	−0.553506
$322$	− 291006.i	− 0.156409i
$323$	101879.i	0.0543349i
$324$	−297946.	−0.157680
$325$	0	0
$326$	−480929.	−0.250632
$327$	− 897287.i	− 0.464047i
$328$	− 21402.4i	− 0.0109844i
$329$	287441.	0.146406
$330$	0	0
$331$	−82124.6	−0.0412006	−0.0206003	−	0.999788i	$-0.506558\pi$
$331$	−82124.6	−0.0412006	−0.0206003	+	0.999788i	$0.506558\pi$
$332$	1.69636e6i	0.844642i
$333$	445330.i	0.220075i
$334$	−1.47220e6	−0.722105
$335$	0	0
$336$	97547.0	0.0471374
$337$	− 1.84526e6i	− 0.885079i	−0.896749	−	0.442540i	$-0.854078\pi$
$337$	− 1.84526e6i	− 0.885079i	0.896749	−	0.442540i	$-0.145922\pi$
$338$	1.39813e6i	0.665665i
$339$	894201.	0.422606
$340$	0	0
$341$	4.38100e6	2.04027
$342$	1.17731e6i	0.544284i
$343$	− 117649.i	− 0.0539949i
$344$	−763743.	−0.347978
$345$	0	0
$346$	200317.	0.0899553
$347$	2.68081e6i	1.19521i	0.801792	+	0.597603i	$0.203880\pi$
$347$	2.68081e6i	1.19521i	−0.801792	+	0.597603i	$0.796120\pi$
$348$	210503.i	0.0931773i
$349$	−3.46523e6	−1.52289	−0.761444	−	0.648231i	$-0.775509\pi$
$349$	−3.46523e6	−1.52289	−0.761444	+	0.648231i	$0.775509\pi$
$350$	0	0
$351$	−488138.	−0.211483
$352$	− 602449.i	− 0.259157i
$353$	237146.i	0.101293i	0.998717	+	0.0506465i	$0.0161282\pi$
$353$	237146.i	0.101293i	−0.998717	+	0.0506465i	$0.983872\pi$
$354$	1.62831e6	0.690605
$355$	0	0
$356$	−612652.	−0.256206
$357$	24074.5i	0.00999740i
$358$	1.46621e6i	0.604628i
$359$	1.33358e6	0.546115	0.273057	−	0.961998i	$-0.411965\pi$
$359$	1.33358e6	0.546115	0.273057	+	0.961998i	$0.411965\pi$
$360$	0	0
$361$	124084.	0.0501127
$362$	951851.i	0.381766i
$363$	− 1.43926e6i	− 0.573286i
$364$	−115652.	−0.0457509
$365$	0	0
$366$	524051.	0.204489
$367$	− 4.51463e6i	− 1.74967i	−0.484418	−	0.874837i	$-0.660969\pi$
$367$	− 4.51463e6i	− 1.74967i	0.484418	−	0.874837i	$-0.339031\pi$
$368$	380090.i	0.146308i
$369$	61039.6	0.0233370
$370$	0	0
$371$	1.22584e6	0.462380
$372$	− 926512.i	− 0.347131i
$373$	3.81113e6i	1.41834i	0.705035	+	0.709172i	$0.250931\pi$
$373$	3.81113e6i	1.41834i	−0.705035	+	0.709172i	$0.749069\pi$
$374$	148684.	0.0549648
$375$	0	0
$376$	−375434.	−0.136950
$377$	− 249573.i	− 0.0904366i
$378$	648578.i	0.233471i
$379$	896148.	0.320466	0.160233	−	0.987079i	$-0.448775\pi$
$379$	896148.	0.320466	0.160233	+	0.987079i	$0.448775\pi$
$380$	0	0
$381$	−1.59834e6	−0.564100
$382$	720852.i	0.252748i
$383$	− 1.51304e6i	− 0.527051i	−0.964652	−	0.263526i	$-0.915115\pi$
$383$	− 1.51304e6i	− 0.527051i	0.964652	−	0.263526i	$-0.0848854\pi$
$384$	−127408.	−0.0440930
$385$	0	0
$386$	2.68426e6	0.916972
$387$	− 2.17819e6i	− 0.739297i
$388$	1.45209e6i	0.489683i
$389$	5.36445e6	1.79743	0.898714	−	0.438536i	$-0.144503\pi$
$389$	5.36445e6	1.79743	0.898714	+	0.438536i	$0.144503\pi$
$390$	0	0
$391$	−93805.8	−0.0310304
$392$	153664.i	0.0505076i
$393$	− 1.46580e6i	− 0.478733i
$394$	1.72897e6	0.561108
$395$	0	0
$396$	1.71818e6	0.550594
$397$	− 2.84949e6i	− 0.907385i	−0.891158	−	0.453692i	$-0.850107\pi$
$397$	− 2.84949e6i	− 0.907385i	0.891158	−	0.453692i	$-0.149893\pi$
$398$	− 3.65056e6i	− 1.15518i
$399$	614435.	0.193216
$400$	0	0
$401$	2.81190e6	0.873250	0.436625	−	0.899644i	$-0.356174\pi$
$401$	2.81190e6	0.873250	0.436625	+	0.899644i	$0.356174\pi$
$402$	− 2.14879e6i	− 0.663177i
$403$	1.09848e6i	0.336921i
$404$	−1.84385e6	−0.562046
$405$	0	0
$406$	−331602.	−0.0998393
$407$	− 1.43540e6i	− 0.429524i
$408$	− 31444.3i	− 0.00935171i
$409$	−3.76521e6	−1.11296	−0.556482	−	0.830859i	$-0.687849\pi$
$409$	−3.76521e6	−1.11296	−0.556482	+	0.830859i	$0.687849\pi$
$410$	0	0
$411$	317619.	0.0927474
$412$	141483.i	0.0410639i
$413$	2.56505e6i	0.739982i
$414$	−1.08402e6	−0.310838
$415$	0	0
$416$	151056.	0.0427960
$417$	− 57151.1i	− 0.0160948i
$418$	− 3.79474e6i	− 1.06229i
$419$	1.22257e6	0.340202	0.170101	−	0.985427i	$-0.445591\pi$
$419$	1.22257e6	0.340202	0.170101	+	0.985427i	$0.445591\pi$
$420$	0	0
$421$	5.01813e6	1.37987	0.689933	−	0.723874i	$-0.257640\pi$
$421$	5.01813e6	1.37987	0.689933	+	0.723874i	$0.257640\pi$
$422$	− 3.93684e6i	− 1.07613i
$423$	− 1.07074e6i	− 0.290959i
$424$	−1.60110e6	−0.432517
$425$	0	0
$426$	949591.	0.253520
$427$	825527.i	0.219110i
$428$	2.10245e6i	0.554775i
$429$	674894.	0.177048
$430$	0	0
$431$	−4.55140e6	−1.18019	−0.590095	−	0.807334i	$-0.700910\pi$
$431$	−4.55140e6	−1.18019	−0.590095	+	0.807334i	$0.700910\pi$
$432$	− 847122.i	− 0.218392i
$433$	3.75433e6i	0.962306i	0.876637	+	0.481153i	$0.159782\pi$
$433$	3.75433e6i	0.962306i	−0.876637	+	0.481153i	$0.840218\pi$
$434$	1.45952e6	0.371950
$435$	0	0
$436$	−1.84618e6	−0.465112
$437$	2.39413e6i	0.599715i
$438$	1.43601e6i	0.357661i
$439$	−4.57813e6	−1.13378	−0.566888	−	0.823795i	$-0.691853\pi$
$439$	−4.57813e6	−1.13378	−0.566888	+	0.823795i	$0.691853\pi$
$440$	0	0
$441$	−438249.	−0.107306
$442$	37280.4i	0.00907663i
$443$	4.09528e6i	0.991459i	0.868477	+	0.495729i	$0.165099\pi$
$443$	4.09528e6i	0.991459i	−0.868477	+	0.495729i	$0.834901\pi$
$444$	−303564.	−0.0730791
$445$	0	0
$446$	1.45331e6	0.345957
$447$	2.16168e6i	0.511708i
$448$	− 200704.i	− 0.0472456i
$449$	5.87346e6	1.37492	0.687461	−	0.726221i	$-0.258725\pi$
$449$	5.87346e6	1.37492	0.687461	+	0.726221i	$0.258725\pi$
$450$	0	0
$451$	−196745.	−0.0455472
$452$	− 1.83983e6i	− 0.423576i
$453$	− 784804.i	− 0.179687i
$454$	−2.34850e6	−0.534750
$455$	0	0
$456$	−802527.	−0.180737
$457$	− 5.40551e6i	− 1.21073i	−0.795949	−	0.605364i	$-0.793028\pi$
$457$	− 5.40551e6i	− 1.21073i	0.795949	−	0.605364i	$-0.206972\pi$
$458$	− 4.84330e6i	− 1.07889i
$459$	209069.	0.0463188
$460$	0	0
$461$	−2.02999e6	−0.444879	−0.222440	−	0.974946i	$-0.571402\pi$
$461$	−2.02999e6	−0.444879	−0.222440	+	0.974946i	$0.571402\pi$
$462$	− 896715.i	− 0.195456i
$463$	2.20634e6i	0.478321i	0.970980	+	0.239160i	$0.0768721\pi$
$463$	2.20634e6i	0.478321i	−0.970980	+	0.239160i	$0.923128\pi$
$464$	433112.	0.0933911
$465$	0	0
$466$	652268.	0.139143
$467$	− 9.24988e6i	− 1.96266i	−0.192342	−	0.981328i	$-0.561608\pi$
$467$	− 9.24988e6i	− 1.96266i	0.192342	−	0.981328i	$-0.438392\pi$
$468$	430810.i	0.0909225i
$469$	3.38495e6	0.710593
$470$	0	0
$471$	−1.57197e6	−0.326506
$472$	− 3.35027e6i	− 0.692189i
$473$	7.02082e6i	1.44290i
$474$	58136.4	0.0118851
$475$	0	0
$476$	49533.6	0.0100203
$477$	− 4.56632e6i	− 0.918904i
$478$	− 5.10632e6i	− 1.02221i
$479$	8.73710e6	1.73992	0.869959	−	0.493125i	$-0.164145\pi$
$479$	8.73710e6	1.73992	0.869959	+	0.493125i	$0.164145\pi$
$480$	0	0
$481$	359907.	0.0709295
$482$	− 2.02779e6i	− 0.397561i
$483$	565745.i	0.110345i
$484$	−2.96128e6	−0.574601
$485$	0	0
$486$	3.79565e6	0.728947
$487$	5.75797e6i	1.10014i	0.835119	+	0.550069i	$0.185398\pi$
$487$	5.75797e6i	1.10014i	−0.835119	+	0.550069i	$0.814602\pi$
$488$	− 1.07824e6i	− 0.204958i
$489$	934973.	0.176818
$490$	0	0
$491$	21277.9	0.00398314	0.00199157	−	0.999998i	$-0.499366\pi$
$491$	21277.9	0.00398314	0.00199157	+	0.999998i	$0.499366\pi$
$492$	41608.3i	0.00774939i
$493$	106892.i	0.0198074i
$494$	951478.	0.175421
$495$	0	0
$496$	−1.90631e6	−0.347928
$497$	1.49587e6i	0.271646i
$498$	− 3.29789e6i	− 0.595885i
$499$	8.04194e6	1.44580	0.722902	−	0.690950i	$-0.242808\pi$
$499$	8.04194e6	1.44580	0.722902	+	0.690950i	$0.242808\pi$
$500$	0	0
$501$	2.86210e6	0.509437
$502$	− 6.91189e6i	− 1.22416i
$503$	− 5.82427e6i	− 1.02641i	−0.858266	−	0.513206i	$-0.828458\pi$
$503$	− 5.82427e6i	− 1.02641i	0.858266	−	0.513206i	$-0.171542\pi$
$504$	572407.	0.100376
$505$	0	0
$506$	3.49403e6	0.606667
$507$	− 2.71810e6i	− 0.469619i
$508$	3.28859e6i	0.565394i
$509$	−1.43672e6	−0.245798	−0.122899	−	0.992419i	$-0.539219\pi$
$509$	−1.43672e6	−0.245798	−0.122899	+	0.992419i	$0.539219\pi$
$510$	0	0
$511$	−2.26212e6	−0.383233
$512$	262144.i	0.0441942i
$513$	− 5.33590e6i	− 0.895188i
$514$	418898.	0.0699359
$515$	0	0
$516$	1.48479e6	0.245494
$517$	3.45123e6i	0.567868i
$518$	− 478199.i	− 0.0783041i
$519$	−389435.	−0.0634624
$520$	0	0
$521$	−8.10030e6	−1.30739	−0.653697	−	0.756756i	$-0.726783\pi$
$521$	−8.10030e6	−1.30739	−0.653697	+	0.756756i	$0.726783\pi$
$522$	1.23523e6i	0.198414i
$523$	− 375315.i	− 0.0599987i	−0.999550	−	0.0299993i	$-0.990449\pi$
$523$	− 375315.i	− 0.0599987i	0.999550	−	0.0299993i	$-0.00955052\pi$
$524$	−3.01590e6	−0.479831
$525$	0	0
$526$	5.44493e6	0.858081
$527$	− 470475.i	− 0.0737922i
$528$	1.17122e6i	0.182833i
$529$	4.23193e6	0.657505
$530$	0	0
$531$	9.55496e6	1.47059
$532$	− 1.26421e6i	− 0.193660i
$533$	− 49331.0i	− 0.00752145i
$534$	1.19105e6	0.180750
$535$	0	0
$536$	−4.42116e6	−0.664698
$537$	− 2.85045e6i	− 0.426558i
$538$	1.39146e6i	0.207260i
$539$	1.41258e6	0.209431
$540$	0	0
$541$	−8.56631e6	−1.25835	−0.629174	−	0.777265i	$-0.716607\pi$
$541$	−8.56631e6	−1.25835	−0.629174	+	0.777265i	$0.716607\pi$
$542$	4.73058e6i	0.691698i
$543$	− 1.85049e6i	− 0.269332i
$544$	−64696.9	−0.00937316
$545$	0	0
$546$	224839.	0.0322767
$547$	− 5.94955e6i	− 0.850190i	−0.905149	−	0.425095i	$-0.860241\pi$
$547$	− 5.94955e6i	− 0.850190i	0.905149	−	0.425095i	$-0.139759\pi$
$548$	− 653504.i	− 0.0929602i
$549$	3.07514e6	0.435445
$550$	0	0
$551$	2.72811e6	0.382810
$552$	− 738932.i	− 0.103218i
$553$	91581.2i	0.0127348i
$554$	−2.51497e6	−0.348143
$555$	0	0
$556$	−117589.	−0.0161317
$557$	3.48878e6i	0.476470i	0.971208	+	0.238235i	$0.0765689\pi$
$557$	3.48878e6i	0.476470i	−0.971208	+	0.238235i	$0.923431\pi$
$558$	− 5.43679e6i	− 0.739191i
$559$	−1.76037e6	−0.238273
$560$	0	0
$561$	−289056.	−0.0387770
$562$	6.04943e6i	0.807930i
$563$	4.65749e6i	0.619271i	0.950855	+	0.309635i	$0.100207\pi$
$563$	4.65749e6i	0.619271i	−0.950855	+	0.309635i	$0.899793\pi$
$564$	729879.	0.0966170
$565$	0	0
$566$	−4.71371e6	−0.618474
$567$	912460.i	0.119195i
$568$	− 1.95379e6i	− 0.254102i
$569$	5.01530e6	0.649406	0.324703	−	0.945816i	$-0.394736\pi$
$569$	5.01530e6	0.649406	0.324703	+	0.945816i	$0.394736\pi$
$570$	0	0
$571$	−8.50936e6	−1.09221	−0.546106	−	0.837716i	$-0.683890\pi$
$571$	−8.50936e6	−1.09221	−0.546106	+	0.837716i	$0.683890\pi$
$572$	− 1.38860e6i	− 0.177455i
$573$	− 1.40141e6i	− 0.178311i
$574$	−65544.9	−0.00830346
$575$	0	0
$576$	−747634.	−0.0938929
$577$	1.46073e7i	1.82654i	0.407354	+	0.913270i	$0.366452\pi$
$577$	1.46073e7i	1.82654i	−0.407354	+	0.913270i	$0.633548\pi$
$578$	5.66346e6i	0.705119i
$579$	−5.21846e6	−0.646913
$580$	0	0
$581$	5.19510e6	0.638489
$582$	− 2.82301e6i	− 0.345466i
$583$	1.47183e7i	1.79344i
$584$	2.95460e6	0.358482
$585$	0	0
$586$	−1.96342e6	−0.236194
$587$	3.92273e6i	0.469887i	0.972009	+	0.234943i	$0.0754905\pi$
$587$	3.92273e6i	0.469887i	−0.972009	+	0.234943i	$0.924509\pi$
$588$	− 298738.i	− 0.0356325i
$589$	−1.20076e7	−1.42616
$590$	0	0
$591$	−3.36128e6	−0.395855
$592$	624587.i	0.0732468i
$593$	1.98554e6i	0.231869i	0.993257	+	0.115934i	$0.0369862\pi$
$593$	1.98554e6i	0.231869i	−0.993257	+	0.115934i	$0.963014\pi$
$594$	−7.78729e6	−0.905566
$595$	0	0
$596$	4.44768e6	0.512882
$597$	7.09704e6i	0.814969i
$598$	876079.i	0.100182i
$599$	1.21357e7	1.38196	0.690982	−	0.722871i	$-0.257178\pi$
$599$	1.21357e7	1.38196	0.690982	+	0.722871i	$0.257178\pi$
$600$	0	0
$601$	1.45565e7	1.64388	0.821942	−	0.569570i	$-0.192890\pi$
$601$	1.45565e7	1.64388	0.821942	+	0.569570i	$0.192890\pi$
$602$	2.33896e6i	0.263046i
$603$	− 1.26091e7i	− 1.41219i
$604$	−1.61474e6	−0.180099
$605$	0	0
$606$	3.58462e6	0.396517
$607$	− 4.81923e6i	− 0.530892i	−0.964126	−	0.265446i	$-0.914481\pi$
$607$	− 4.81923e6i	− 0.530892i	0.964126	−	0.265446i	$-0.0855192\pi$
$608$	1.65121e6i	0.181152i
$609$	644666.	0.0704354
$610$	0	0
$611$	−865347.	−0.0937750
$612$	− 184515.i	− 0.0199138i
$613$	1.27574e7i	1.37123i	0.727964	+	0.685616i	$0.240467\pi$
$613$	1.27574e7i	1.37123i	−0.727964	+	0.685616i	$0.759533\pi$
$614$	−9.19468e6	−0.984273
$615$	0	0
$616$	−1.84500e6	−0.195905
$617$	− 845244.i	− 0.0893859i	−0.999001	−	0.0446930i	$-0.985769\pi$
$617$	− 845244.i	− 0.0893859i	0.999001	−	0.0446930i	$-0.0142310\pi$
$618$	− 275056.i	− 0.0289701i
$619$	1.64444e7	1.72501	0.862505	−	0.506048i	$-0.168894\pi$
$619$	1.64444e7	1.72501	0.862505	+	0.506048i	$0.168894\pi$
$620$	0	0
$621$	4.91306e6	0.511238
$622$	− 6.47112e6i	− 0.670662i
$623$	1.87625e6i	0.193673i
$624$	−293667.	−0.0301921
$625$	0	0
$626$	697596.	0.0711489
$627$	7.37735e6i	0.749431i
$628$	3.23434e6i	0.327255i
$629$	−154147.	−0.0155349
$630$	0	0
$631$	−1.36092e7	−1.36069	−0.680343	−	0.732894i	$-0.738169\pi$
$631$	−1.36092e7	−1.36069	−0.680343	+	0.732894i	$0.738169\pi$
$632$	− 119616.i	− 0.0119124i
$633$	7.65359e6i	0.759200i
$634$	−1.31200e7	−1.29631
$635$	0	0
$636$	3.11269e6	0.305135
$637$	354184.i	0.0345844i
$638$	− 3.98145e6i	− 0.387248i
$639$	5.57221e6	0.539853
$640$	0	0
$641$	1.20347e6	0.115688	0.0578440	−	0.998326i	$-0.481577\pi$
$641$	1.20347e6	0.115688	0.0578440	+	0.998326i	$0.481577\pi$
$642$	− 4.08738e6i	− 0.391388i
$643$	− 1.31941e7i	− 1.25849i	−0.777205	−	0.629247i	$-0.783363\pi$
$643$	− 1.31941e7i	− 1.25849i	0.777205	−	0.629247i	$-0.216637\pi$
$644$	1.16403e6	0.110598
$645$	0	0
$646$	−407517.	−0.0384206
$647$	− 382498.i	− 0.0359227i	−0.999839	−	0.0179613i	$-0.994282\pi$
$647$	− 382498.i	− 0.0359227i	0.999839	−	0.0179613i	$-0.00571758\pi$
$648$	− 1.19178e6i	− 0.111496i
$649$	−3.07978e7	−2.87018
$650$	0	0
$651$	−2.83744e6	−0.262407
$652$	− 1.92372e6i	− 0.177224i
$653$	1.34972e7i	1.23868i	0.785121	+	0.619342i	$0.212601\pi$
$653$	1.34972e7i	1.23868i	−0.785121	+	0.619342i	$0.787399\pi$
$654$	3.58915e6	0.328131
$655$	0	0
$656$	85609.6	0.00776717
$657$	8.42652e6i	0.761614i
$658$	1.14977e6i	0.103525i
$659$	−1.01824e7	−0.913353	−0.456676	−	0.889633i	$-0.650960\pi$
$659$	−1.01824e7	−0.913353	−0.456676	+	0.889633i	$0.650960\pi$
$660$	0	0
$661$	140948.	0.0125475	0.00627373	−	0.999980i	$-0.498003\pi$
$661$	140948.	0.0125475	0.00627373	+	0.999980i	$0.498003\pi$
$662$	− 328498.i	− 0.0291332i
$663$	− 72476.7i	− 0.00640346i
$664$	−6.78544e6	−0.597252
$665$	0	0
$666$	−1.78132e6	−0.155617
$667$	2.51193e6i	0.218621i
$668$	− 5.88880e6i	− 0.510605i
$669$	−2.82538e6	−0.244069
$670$	0	0
$671$	−9.91187e6	−0.849864
$672$	390188.i	0.0333312i
$673$	− 7.93418e6i	− 0.675250i	−0.941281	−	0.337625i	$-0.890376\pi$
$673$	− 7.93418e6i	− 0.675250i	0.941281	−	0.337625i	$-0.109624\pi$
$674$	7.38103e6	0.625846
$675$	0	0
$676$	−5.59252e6	−0.470696
$677$	1.73979e7i	1.45890i	0.684033	+	0.729451i	$0.260224\pi$
$677$	1.73979e7i	1.45890i	−0.684033	+	0.729451i	$0.739776\pi$
$678$	3.57681e6i	0.298828i
$679$	4.44704e6	0.370166
$680$	0	0
$681$	4.56571e6	0.377260
$682$	1.75240e7i	1.44269i
$683$	− 1.48842e7i	− 1.22088i	−0.792061	−	0.610442i	$-0.790992\pi$
$683$	− 1.48842e7i	− 1.22088i	0.792061	−	0.610442i	$-0.209008\pi$
$684$	−4.70924e6	−0.384867
$685$	0	0
$686$	470596.	0.0381802
$687$	9.41584e6i	0.761145i
$688$	− 3.05497e6i	− 0.246057i
$689$	−3.69041e6	−0.296160
$690$	0	0
$691$	−1.14830e7	−0.914869	−0.457435	−	0.889243i	$-0.651232\pi$
$691$	−1.14830e7	−0.914869	−0.457435	+	0.889243i	$0.651232\pi$
$692$	801267.i	0.0636080i
$693$	− 5.26193e6i	− 0.416210i
$694$	−1.07232e7	−0.845138
$695$	0	0
$696$	−842012.	−0.0658863
$697$	21128.4i	0.00164734i
$698$	− 1.38609e7i	− 1.07684i
$699$	−1.26807e6	−0.0981638
$700$	0	0
$701$	2.06197e7	1.58485	0.792423	−	0.609972i	$-0.208819\pi$
$701$	2.06197e7	1.58485	0.792423	+	0.609972i	$0.208819\pi$
$702$	− 1.95255e6i	− 0.149541i
$703$	3.93418e6i	0.300239i
$704$	2.40980e6	0.183252
$705$	0	0
$706$	−948585.	−0.0716250
$707$	5.64679e6i	0.424867i
$708$	6.51325e6i	0.488332i
$709$	−7.75134e6	−0.579110	−0.289555	−	0.957161i	$-0.593507\pi$
$709$	−7.75134e6	−0.579110	−0.289555	+	0.957161i	$0.593507\pi$
$710$	0	0
$711$	341145.	0.0253084
$712$	− 2.45061e6i	− 0.181165i
$713$	− 1.10560e7i	− 0.814471i
$714$	−96298.1	−0.00706923
$715$	0	0
$716$	−5.86483e6	−0.427536
$717$	9.92719e6i	0.721155i
$718$	5.33433e6i	0.386161i
$719$	−2.39851e7	−1.73029	−0.865146	−	0.501521i	$-0.832774\pi$
$719$	−2.39851e7	−1.73029	−0.865146	+	0.501521i	$0.832774\pi$
$720$	0	0
$721$	433291.	0.0310414
$722$	496336.i	0.0354351i
$723$	3.94221e6i	0.280475i
$724$	−3.80740e6	−0.269949
$725$	0	0
$726$	5.75702e6	0.405374
$727$	− 1.05562e7i	− 0.740750i	−0.928882	−	0.370375i	$-0.879229\pi$
$727$	− 1.05562e7i	− 0.740750i	0.928882	−	0.370375i	$-0.120771\pi$
$728$	− 462608.i	− 0.0323508i
$729$	−2.85406e6	−0.198904
$730$	0	0
$731$	753965.	0.0521864
$732$	2.09620e6i	0.144596i
$733$	− 1.99297e7i	− 1.37007i	−0.728512	−	0.685033i	$-0.759788\pi$
$733$	− 1.99297e7i	− 1.37007i	0.728512	−	0.685033i	$-0.240212\pi$
$734$	1.80585e7	1.23721
$735$	0	0
$736$	−1.52036e6	−0.103455
$737$	4.06422e7i	2.75618i
$738$	244158.i	0.0165018i
$739$	1.38830e7	0.935133	0.467567	−	0.883958i	$-0.345131\pi$
$739$	1.38830e7	0.935133	0.467567	+	0.883958i	$0.345131\pi$
$740$	0	0
$741$	−1.84977e6	−0.123757
$742$	4.90336e6i	0.326952i
$743$	− 1.74009e7i	− 1.15638i	−0.815902	−	0.578190i	$-0.803759\pi$
$743$	− 1.74009e7i	− 1.15638i	0.815902	−	0.578190i	$-0.196241\pi$
$744$	3.70605e6	0.245459
$745$	0	0
$746$	−1.52445e7	−1.00292
$747$	− 1.93520e7i	− 1.26889i
$748$	594735.i	0.0388660i
$749$	6.43877e6	0.419371
$750$	0	0
$751$	−2.22259e7	−1.43800	−0.719001	−	0.695009i	$-0.755400\pi$
$751$	−2.22259e7	−1.43800	−0.719001	+	0.695009i	$0.755400\pi$
$752$	− 1.50173e6i	− 0.0968386i
$753$	1.34374e7i	0.863630i
$754$	998292.	0.0639483
$755$	0	0
$756$	−2.59431e6	−0.165089
$757$	1.78715e6i	0.113350i	0.998393	+	0.0566750i	$0.0180499\pi$
$757$	1.78715e6i	0.113350i	−0.998393	+	0.0566750i	$0.981950\pi$
$758$	3.58459e6i	0.226604i
$759$	−6.79274e6	−0.427997
$760$	0	0
$761$	2.01942e7	1.26406	0.632028	−	0.774946i	$-0.282223\pi$
$761$	2.01942e7	1.26406	0.632028	+	0.774946i	$0.282223\pi$
$762$	− 6.39335e6i	− 0.398879i
$763$	5.65392e6i	0.351591i
$764$	−2.88341e6	−0.178720
$765$	0	0
$766$	6.05215e6	0.372682
$767$	− 7.72213e6i	− 0.473967i
$768$	− 509633.i	− 0.0311785i
$769$	−1.76651e7	−1.07721	−0.538604	−	0.842559i	$-0.681048\pi$
$769$	−1.76651e7	−1.07721	−0.538604	+	0.842559i	$0.681048\pi$
$770$	0	0
$771$	−814378.	−0.0493390
$772$	1.07370e7i	0.648397i
$773$	8.11314e6i	0.488360i	0.969730	+	0.244180i	$0.0785189\pi$
$773$	8.11314e6i	0.488360i	−0.969730	+	0.244180i	$0.921481\pi$
$774$	8.71277e6	0.522762
$775$	0	0
$776$	−5.80838e6	−0.346258
$777$	929666.i	0.0552426i
$778$	2.14578e7i	1.27097i
$779$	539243.	0.0318376
$780$	0	0
$781$	−1.79605e7	−1.05364
$782$	− 375223.i	− 0.0219418i
$783$	− 5.59843e6i	− 0.326334i
$784$	−614656.	−0.0357143
$785$	0	0
$786$	5.86320e6	0.338515
$787$	− 2.26545e7i	− 1.30382i	−0.758297	−	0.651909i	$-0.773969\pi$
$787$	− 2.26545e7i	− 1.30382i	0.758297	−	0.651909i	$-0.226031\pi$
$788$	6.91588e6i	0.396763i
$789$	−1.05855e7	−0.605366
$790$	0	0
$791$	−5.63448e6	−0.320193
$792$	6.87273e6i	0.389329i
$793$	− 2.48526e6i	− 0.140343i
$794$	1.13980e7	0.641618
$795$	0	0
$796$	1.46022e7	0.816839
$797$	− 2.92055e7i	− 1.62862i	−0.580431	−	0.814309i	$-0.697116\pi$
$797$	− 2.92055e7i	− 1.62862i	0.580431	−	0.814309i	$-0.302884\pi$
$798$	2.45774e6i	0.136625i
$799$	370627.	0.0205385
$800$	0	0
$801$	6.98912e6	0.384894
$802$	1.12476e7i	0.617481i
$803$	− 2.71606e7i	− 1.48645i
$804$	8.59517e6	0.468937
$805$	0	0
$806$	−4.39390e6	−0.238239
$807$	− 2.70513e6i	− 0.146219i
$808$	− 7.37539e6i	− 0.397426i
$809$	3.04067e7	1.63342	0.816709	−	0.577050i	$-0.195796\pi$
$809$	3.04067e7	1.63342	0.816709	+	0.577050i	$0.195796\pi$
$810$	0	0
$811$	−1.67648e7	−0.895049	−0.447524	−	0.894272i	$-0.647694\pi$
$811$	−1.67648e7	−0.895049	−0.447524	+	0.894272i	$0.647694\pi$
$812$	− 1.32641e6i	− 0.0705970i
$813$	− 9.19672e6i	− 0.487985i
$814$	5.74160e6	0.303719
$815$	0	0
$816$	125777.	0.00661266
$817$	− 1.92429e7i	− 1.00859i
$818$	− 1.50609e7i	− 0.786985i
$819$	1.31936e6	0.0687309
$820$	0	0
$821$	4.39417e6	0.227520	0.113760	−	0.993508i	$-0.463711\pi$
$821$	4.39417e6	0.227520	0.113760	+	0.993508i	$0.463711\pi$
$822$	1.27048e6i	0.0655823i
$823$	2.59030e7i	1.33306i	0.745477	+	0.666531i	$0.232222\pi$
$823$	2.59030e7i	1.33306i	−0.745477	+	0.666531i	$0.767778\pi$
$824$	−565931.	−0.0290366
$825$	0	0
$826$	−1.02602e7	−0.523246
$827$	2.02511e6i	0.102964i	0.998674	+	0.0514820i	$0.0163945\pi$
$827$	2.02511e6i	0.102964i	−0.998674	+	0.0514820i	$0.983606\pi$
$828$	− 4.33606e6i	− 0.219796i
$829$	−1.62217e7	−0.819802	−0.409901	−	0.912130i	$-0.634437\pi$
$829$	−1.62217e7	−0.819802	−0.409901	+	0.912130i	$0.634437\pi$
$830$	0	0
$831$	4.88934e6	0.245611
$832$	604223.i	0.0302614i
$833$	− 151697.i	− 0.00757466i
$834$	228604.	0.0113807
$835$	0	0
$836$	1.51790e7	0.751150
$837$	2.46410e7i	1.21575i
$838$	4.89026e6i	0.240559i
$839$	2.68249e7	1.31563	0.657815	−	0.753179i	$-0.271481\pi$
$839$	2.68249e7	1.31563	0.657815	+	0.753179i	$0.271481\pi$
$840$	0	0
$841$	−1.76488e7	−0.860450
$842$	2.00725e7i	0.975712i
$843$	− 1.17607e7i	− 0.569985i
$844$	1.57473e7	0.760941
$845$	0	0
$846$	4.28294e6	0.205739
$847$	9.06893e6i	0.434358i
$848$	− 6.40438e6i	− 0.305835i
$849$	9.16390e6	0.436326
$850$	0	0
$851$	−3.62242e6	−0.171465
$852$	3.79836e6i	0.179266i
$853$	3.43612e7i	1.61695i	0.588532	+	0.808474i	$0.299706\pi$
$853$	3.43612e7i	1.61695i	−0.588532	+	0.808474i	$0.700294\pi$
$854$	−3.30211e6	−0.154934
$855$	0	0
$856$	−8.40982e6	−0.392285
$857$	− 1.12064e6i	− 0.0521212i	−0.999660	−	0.0260606i	$-0.991704\pi$
$857$	− 1.12064e6i	− 0.0521212i	0.999660	−	0.0260606i	$-0.00829628\pi$
$858$	2.69957e6i	0.125192i
$859$	−3.04895e7	−1.40983	−0.704916	−	0.709291i	$-0.749015\pi$
$859$	−3.04895e7	−1.40983	−0.704916	+	0.709291i	$0.749015\pi$
$860$	0	0
$861$	127426.	0.00585799
$862$	− 1.82056e7i	− 0.834520i
$863$	2.62444e6i	0.119953i	0.998200	+	0.0599763i	$0.0191025\pi$
$863$	2.62444e6i	0.119953i	−0.998200	+	0.0599763i	$0.980897\pi$
$864$	3.38849e6	0.154426
$865$	0	0
$866$	−1.50173e7	−0.680453
$867$	− 1.10103e7i	− 0.497453i
$868$	5.83807e6i	0.263009i
$869$	−1.09959e6	−0.0493948
$870$	0	0
$871$	−1.01905e7	−0.455143
$872$	− 7.38471e6i	− 0.328884i
$873$	− 1.65655e7i	− 0.735645i
$874$	−9.57653e6	−0.424062
$875$	0	0
$876$	−5.74404e6	−0.252905
$877$	− 5.66401e6i	− 0.248671i	−0.992240	−	0.124335i	$-0.960320\pi$
$877$	− 5.66401e6i	− 0.248671i	0.992240	−	0.124335i	$-0.0396799\pi$
$878$	− 1.83125e7i	− 0.801700i
$879$	3.81708e6	0.166632
$880$	0	0
$881$	2.21548e7	0.961676	0.480838	−	0.876810i	$-0.340333\pi$
$881$	2.21548e7	0.961676	0.480838	+	0.876810i	$0.340333\pi$
$882$	− 1.75300e6i	− 0.0758769i
$883$	− 2.14786e7i	− 0.927050i	−0.886084	−	0.463525i	$-0.846584\pi$
$883$	− 2.14786e7i	− 0.927050i	0.886084	−	0.463525i	$-0.153416\pi$
$884$	−149122.	−0.00641815
$885$	0	0
$886$	−1.63811e7	−0.701067
$887$	− 2.26847e7i	− 0.968106i	−0.875039	−	0.484053i	$-0.839164\pi$
$887$	− 2.26847e7i	− 0.968106i	0.875039	−	0.484053i	$-0.160836\pi$
$888$	− 1.21426e6i	− 0.0516747i
$889$	1.00713e7	0.427397
$890$	0	0
$891$	−1.09557e7	−0.462321
$892$	5.81326e6i	0.244629i
$893$	− 9.45922e6i	− 0.396941i
$894$	−8.64672e6	−0.361832
$895$	0	0
$896$	802816.	0.0334077
$897$	− 1.70318e6i	− 0.0706774i
$898$	2.34938e7i	0.972217i
$899$	−1.25984e7	−0.519894
$900$	0	0
$901$	1.58060e6	0.0648648
$902$	− 786979.i	− 0.0322067i
$903$	− 4.54717e6i	− 0.185576i
$904$	7.35931e6	0.299513
$905$	0	0
$906$	3.13921e6	0.127058
$907$	− 3.98778e7i	− 1.60958i	−0.593560	−	0.804790i	$-0.702278\pi$
$907$	− 3.98778e7i	− 1.60958i	0.593560	−	0.804790i	$-0.297722\pi$
$908$	− 9.39399e6i	− 0.378125i
$909$	2.10346e7	0.844354
$910$	0	0
$911$	1.25677e7	0.501717	0.250858	−	0.968024i	$-0.419287\pi$
$911$	1.25677e7	0.501717	0.250858	+	0.968024i	$0.419287\pi$
$912$	− 3.21011e6i	− 0.127801i
$913$	6.23761e7i	2.47652i
$914$	2.16220e7	0.856113
$915$	0	0
$916$	1.93732e7	0.762891
$917$	9.23619e6i	0.362718i
$918$	836275.i	0.0327524i
$919$	4.18208e6	0.163344	0.0816720	−	0.996659i	$-0.473974\pi$
$919$	4.18208e6	0.163344	0.0816720	+	0.996659i	$0.473974\pi$
$920$	0	0
$921$	1.78753e7	0.694393
$922$	− 8.11997e6i	− 0.314577i
$923$	− 4.50335e6i	− 0.173993i
$924$	3.58686e6	0.138208
$925$	0	0
$926$	−8.82534e6	−0.338224
$927$	− 1.61403e6i	− 0.0616898i
$928$	1.73245e6i	0.0660375i
$929$	−4.65328e7	−1.76897	−0.884484	−	0.466571i	$-0.845489\pi$
$929$	−4.65328e7	−1.76897	−0.884484	+	0.466571i	$0.845489\pi$
$930$	0	0
$931$	−3.87163e6	−0.146393
$932$	2.60907e6i	0.0983890i
$933$	1.25805e7i	0.473144i
$934$	3.69995e7	1.38781
$935$	0	0
$936$	−1.72324e6	−0.0642919
$937$	− 2.69548e7i	− 1.00297i	−0.865167	−	0.501484i	$-0.832787\pi$
$937$	− 2.69548e7i	− 1.00297i	0.865167	−	0.501484i	$-0.167213\pi$
$938$	1.35398e7i	0.502465i
$939$	−1.35620e6	−0.0501947
$940$	0	0
$941$	−6.83412e6	−0.251599	−0.125799	−	0.992056i	$-0.540150\pi$
$941$	−6.83412e6	−0.251599	−0.125799	+	0.992056i	$0.540150\pi$
$942$	− 6.28788e6i	− 0.230875i
$943$	496511.i	0.0181823i
$944$	1.34011e7	0.489452
$945$	0	0
$946$	−2.80833e7	−1.02028
$947$	− 1.70413e7i	− 0.617487i	−0.951145	−	0.308743i	$-0.900092\pi$
$947$	− 1.70413e7i	− 0.617487i	0.951145	−	0.308743i	$-0.0999084\pi$
$948$	232546.i	0.00840403i
$949$	6.81014e6	0.245466
$950$	0	0
$951$	2.55065e7	0.914535
$952$	198134.i	0.00708545i
$953$	− 5.08910e6i	− 0.181513i	−0.995873	−	0.0907567i	$-0.971071\pi$
$953$	− 5.08910e6i	− 0.181513i	0.995873	−	0.0907567i	$-0.0289286\pi$
$954$	1.82653e7	0.649764
$955$	0	0
$956$	2.04253e7	0.722809
$957$	7.74032e6i	0.273199i
$958$	3.49484e7i	1.23031i
$959$	−2.00136e6	−0.0702713
$960$	0	0
$961$	2.68215e7	0.936859
$962$	1.43963e6i	0.0501547i
$963$	− 2.39848e7i	− 0.833431i
$964$	8.11114e6	0.281118
$965$	0	0
$966$	−2.26298e6	−0.0780257
$967$	− 3.03945e7i	− 1.04527i	−0.852556	−	0.522635i	$-0.824949\pi$
$967$	− 3.03945e7i	− 1.04527i	0.852556	−	0.522635i	$-0.175051\pi$
$968$	− 1.18451e7i	− 0.406304i
$969$	792252.	0.0271053
$970$	0	0
$971$	3.40735e7	1.15976	0.579881	−	0.814701i	$-0.303099\pi$
$971$	3.40735e7	1.15976	0.579881	+	0.814701i	$0.303099\pi$
$972$	1.51826e7i	0.515443i
$973$	360116.i	0.0121944i
$974$	−2.30319e7	−0.777915
$975$	0	0
$976$	4.31296e6	0.144928
$977$	1.28684e6i	0.0431307i	0.999767	+	0.0215654i	$0.00686500\pi$
$977$	1.28684e6i	0.0431307i	−0.999767	+	0.0215654i	$0.993135\pi$
$978$	3.73989e6i	0.125029i
$979$	−2.25276e7	−0.751203
$980$	0	0
$981$	2.10612e7	0.698731
$982$	85111.7i	0.00281650i
$983$	4.75992e7i	1.57114i	0.618771	+	0.785571i	$0.287631\pi$
$983$	4.75992e7i	1.57114i	−0.618771	+	0.785571i	$0.712369\pi$
$984$	−166433.	−0.00547965
$985$	0	0
$986$	−427567.	−0.0140059
$987$	− 2.23526e6i	− 0.0730356i
$988$	3.80591e6i	0.124041i
$989$	1.77180e7	0.576001
$990$	0	0
$991$	2.23288e7	0.722239	0.361120	−	0.932520i	$-0.382395\pi$
$991$	2.23288e7	0.722239	0.361120	+	0.932520i	$0.382395\pi$
$992$	− 7.62523e6i	− 0.246022i
$993$	638632.i	0.0205531i
$994$	−5.98349e6	−0.192083
$995$	0	0
$996$	1.31915e7	0.421354
$997$	− 4.81000e7i	− 1.53252i	−0.642528	−	0.766262i	$-0.722114\pi$
$997$	− 4.81000e7i	− 1.53252i	0.642528	−	0.766262i	$-0.277886\pi$
$998$	3.21678e7i	1.02234i
$999$	8.07344e6	0.255944

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.6.c.i.99.3		4
5.2	odd	4		350.6.a.r.1.1	✓	2
5.3	odd	4		350.6.a.s.1.2	yes	2
5.4	even	2	inner	350.6.c.i.99.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.6.a.r.1.1	✓	2	5.2	odd	4
350.6.a.s.1.2	yes	2	5.3	odd	4
350.6.c.i.99.2		4	5.4	even	2	inner
350.6.c.i.99.3		4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	350.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.00000i q^{2} -7.77639i q^{3} -16.0000 q^{4} +31.1056 q^{6} +49.0000i q^{7} -64.0000i q^{8} +182.528 q^{9} +O(q^{10})\)	\(q+4.00000i q^{2} -7.77639i q^{3} -16.0000 q^{4} +31.1056 q^{6} +49.0000i q^{7} -64.0000i q^{8} +182.528 q^{9} -588.329 q^{11} +124.422i q^{12} -147.515i q^{13} -196.000 q^{14} +256.000 q^{16} +63.1806i q^{17} +730.111i q^{18} +1612.51 q^{19} +381.043 q^{21} -2353.32i q^{22} +1484.73i q^{23} -497.689 q^{24} +590.061 q^{26} -3309.07i q^{27} -784.000i q^{28} +1691.84 q^{29} -7446.52 q^{31} +1024.00i q^{32} +4575.08i q^{33} -252.722 q^{34} -2920.44 q^{36} +2439.79i q^{37} +6450.03i q^{38} -1147.14 q^{39} +334.413 q^{41} +1524.17i q^{42} -11933.5i q^{43} +9413.27 q^{44} -5938.91 q^{46} -5866.15i q^{47} -1990.76i q^{48} -2401.00 q^{49} +491.317 q^{51} +2360.24i q^{52} -25017.1i q^{53} +13236.3 q^{54} +3136.00 q^{56} -12539.5i q^{57} +6767.38i q^{58} +52348.0 q^{59} +16847.5 q^{61} -29786.1i q^{62} +8943.86i q^{63} -4096.00 q^{64} -18300.3 q^{66} -69080.7i q^{67} -1010.89i q^{68} +11545.8 q^{69} +30528.0 q^{71} -11681.8i q^{72} +46165.7i q^{73} -9759.17 q^{74} -25800.1 q^{76} -28828.1i q^{77} -4588.54i q^{78} +1869.00 q^{79} +18621.6 q^{81} +1337.65i q^{82} -106022. i q^{83} -6096.69 q^{84} +47734.0 q^{86} -13156.4i q^{87} +37653.1i q^{88} +38290.7 q^{89} +7228.25 q^{91} -23755.6i q^{92} +57907.0i q^{93} +23464.6 q^{94} +7963.02 q^{96} -90755.9i q^{97} -9604.00i q^{98} -107386. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 64 q^{4} - 160 q^{6} - 692 q^{9}+O(q^{10}) \) 4 * q - 64 * q^4 - 160 * q^6 - 692 * q^9	\( 4 q - 64 q^{4} - 160 q^{6} - 692 q^{9} - 2140 q^{11} - 784 q^{14} + 1024 q^{16} - 1656 q^{19} - 1960 q^{21} + 2560 q^{24} - 5888 q^{26} + 4492 q^{29} + 576 q^{31} - 15232 q^{34} + 11072 q^{36} - 51376 q^{39} + 22456 q^{41} + 34240 q^{44} - 13232 q^{46} - 9604 q^{49} - 101280 q^{51} + 89920 q^{54} + 12544 q^{56} + 185856 q^{59} + 30344 q^{61} - 16384 q^{64} + 70432 q^{66} + 13688 q^{69} - 41644 q^{71} + 71888 q^{74} + 26496 q^{76} + 144212 q^{79} + 220964 q^{81} + 31360 q^{84} - 27216 q^{86} + 19840 q^{89} - 72128 q^{91} + 124576 q^{94} - 40960 q^{96} + 294380 q^{99}+O(q^{100}) \) 4 * q - 64 * q^4 - 160 * q^6 - 692 * q^9 - 2140 * q^11 - 784 * q^14 + 1024 * q^16 - 1656 * q^19 - 1960 * q^21 + 2560 * q^24 - 5888 * q^26 + 4492 * q^29 + 576 * q^31 - 15232 * q^34 + 11072 * q^36 - 51376 * q^39 + 22456 * q^41 + 34240 * q^44 - 13232 * q^46 - 9604 * q^49 - 101280 * q^51 + 89920 * q^54 + 12544 * q^56 + 185856 * q^59 + 30344 * q^61 - 16384 * q^64 + 70432 * q^66 + 13688 * q^69 - 41644 * q^71 + 71888 * q^74 + 26496 * q^76 + 144212 * q^79 + 220964 * q^81 + 31360 * q^84 - 27216 * q^86 + 19840 * q^89 - 72128 * q^91 + 124576 * q^94 - 40960 * q^96 + 294380 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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