LMFDB - Embedded newform 1014.4.b.e.337.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1014,4,Mod(337,1014)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1014, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1014.337");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1014 = 2 \cdot 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1014.b (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:	$59.8279367458$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 78)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1014.337
Dual form			1014.4.b.e.337.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{2} -3.00000 q^{3} -4.00000 q^{4} +20.0000i q^{5} +6.00000i q^{6} -32.0000i q^{7} +8.00000i q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-2.00000i q^{2} -3.00000 q^{3} -4.00000 q^{4} +20.0000i q^{5} +6.00000i q^{6} -32.0000i q^{7} +8.00000i q^{8} +9.00000 q^{9} +40.0000 q^{10} +50.0000i q^{11} +12.0000 q^{12} -64.0000 q^{14} -60.0000i q^{15} +16.0000 q^{16} +30.0000 q^{17} -18.0000i q^{18} +120.000i q^{19} -80.0000i q^{20} +96.0000i q^{21} +100.000 q^{22} +20.0000 q^{23} -24.0000i q^{24} -275.000 q^{25} -27.0000 q^{27} +128.000i q^{28} +82.0000 q^{29} -120.000 q^{30} +44.0000i q^{31} -32.0000i q^{32} -150.000i q^{33} -60.0000i q^{34} +640.000 q^{35} -36.0000 q^{36} -306.000i q^{37} +240.000 q^{38} -160.000 q^{40} -108.000i q^{41} +192.000 q^{42} +356.000 q^{43} -200.000i q^{44} +180.000i q^{45} -40.0000i q^{46} -178.000i q^{47} -48.0000 q^{48} -681.000 q^{49} +550.000i q^{50} -90.0000 q^{51} +198.000 q^{53} +54.0000i q^{54} -1000.00 q^{55} +256.000 q^{56} -360.000i q^{57} -164.000i q^{58} +94.0000i q^{59} +240.000i q^{60} -62.0000 q^{61} +88.0000 q^{62} -288.000i q^{63} -64.0000 q^{64} -300.000 q^{66} +140.000i q^{67} -120.000 q^{68} -60.0000 q^{69} -1280.00i q^{70} +778.000i q^{71} +72.0000i q^{72} +62.0000i q^{73} -612.000 q^{74} +825.000 q^{75} -480.000i q^{76} +1600.00 q^{77} -1096.00 q^{79} +320.000i q^{80} +81.0000 q^{81} -216.000 q^{82} +462.000i q^{83} -384.000i q^{84} +600.000i q^{85} -712.000i q^{86} -246.000 q^{87} -400.000 q^{88} +1224.00i q^{89} +360.000 q^{90} -80.0000 q^{92} -132.000i q^{93} -356.000 q^{94} -2400.00 q^{95} +96.0000i q^{96} -614.000i q^{97} +1362.00i q^{98} +450.000i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} - 8 q^{4} + 18 q^{9}+O(q^{10}) $ 2 * q - 6 * q^3 - 8 * q^4 + 18 * q^9	$ 2 q - 6 q^{3} - 8 q^{4} + 18 q^{9} + 80 q^{10} + 24 q^{12} - 128 q^{14} + 32 q^{16} + 60 q^{17} + 200 q^{22} + 40 q^{23} - 550 q^{25} - 54 q^{27} + 164 q^{29} - 240 q^{30} + 1280 q^{35} - 72 q^{36} + 480 q^{38} - 320 q^{40} + 384 q^{42} + 712 q^{43} - 96 q^{48} - 1362 q^{49} - 180 q^{51} + 396 q^{53} - 2000 q^{55} + 512 q^{56} - 124 q^{61} + 176 q^{62} - 128 q^{64} - 600 q^{66} - 240 q^{68} - 120 q^{69} - 1224 q^{74} + 1650 q^{75} + 3200 q^{77} - 2192 q^{79} + 162 q^{81} - 432 q^{82} - 492 q^{87} - 800 q^{88} + 720 q^{90} - 160 q^{92} - 712 q^{94} - 4800 q^{95}+O(q^{100}) $ 2 * q - 6 * q^3 - 8 * q^4 + 18 * q^9 + 80 * q^10 + 24 * q^12 - 128 * q^14 + 32 * q^16 + 60 * q^17 + 200 * q^22 + 40 * q^23 - 550 * q^25 - 54 * q^27 + 164 * q^29 - 240 * q^30 + 1280 * q^35 - 72 * q^36 + 480 * q^38 - 320 * q^40 + 384 * q^42 + 712 * q^43 - 96 * q^48 - 1362 * q^49 - 180 * q^51 + 396 * q^53 - 2000 * q^55 + 512 * q^56 - 124 * q^61 + 176 * q^62 - 128 * q^64 - 600 * q^66 - 240 * q^68 - 120 * q^69 - 1224 * q^74 + 1650 * q^75 + 3200 * q^77 - 2192 * q^79 + 162 * q^81 - 432 * q^82 - 492 * q^87 - 800 * q^88 + 720 * q^90 - 160 * q^92 - 712 * q^94 - 4800 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1014\mathbb{Z}\right)^\times$.

$n$	$677$	$847$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 2.00000i	− 0.707107i
$2$	− 2.00000i	− 0.707107i
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−4.00000	−0.500000
$5$	20.0000i	1.78885i	0.447214	+	0.894427i	$0.352416\pi$
$5$	20.0000i	1.78885i	−0.447214	+	0.894427i	$0.647584\pi$
$6$	6.00000i	0.408248i
$7$	− 32.0000i	− 1.72784i	−0.503631	−	0.863919i	$-0.668003\pi$
$7$	− 32.0000i	− 1.72784i	0.503631	−	0.863919i	$-0.331997\pi$
$8$	8.00000i	0.353553i
$9$	9.00000	0.333333
$10$	40.0000	1.26491
$11$	50.0000i	1.37051i	0.728305	+	0.685253i	$0.240308\pi$
$11$	50.0000i	1.37051i	−0.728305	+	0.685253i	$0.759692\pi$
$12$	12.0000	0.288675
$13$	0	0
$13$	0	0
$14$	−64.0000	−1.22177
$15$	− 60.0000i	− 1.03280i
$16$	16.0000	0.250000
$17$	30.0000	0.428004	0.214002	−	0.976833i	$-0.431350\pi$
$17$	30.0000	0.428004	0.214002	+	0.976833i	$0.431350\pi$
$18$	− 18.0000i	− 0.235702i
$19$	120.000i	1.44894i	0.689306	+	0.724471i	$0.257916\pi$
$19$	120.000i	1.44894i	−0.689306	+	0.724471i	$0.742084\pi$
$20$	− 80.0000i	− 0.894427i
$21$	96.0000i	0.997567i
$22$	100.000	0.969094
$23$	20.0000	0.181317	0.0906584	−	0.995882i	$-0.471103\pi$
$23$	20.0000	0.181317	0.0906584	+	0.995882i	$0.471103\pi$
$24$	− 24.0000i	− 0.204124i
$25$	−275.000	−2.20000
$26$	0	0
$27$	−27.0000	−0.192450
$28$	128.000i	0.863919i
$29$	82.0000	0.525070	0.262535	−	0.964923i	$-0.415442\pi$
$29$	82.0000	0.525070	0.262535	+	0.964923i	$0.415442\pi$
$30$	−120.000	−0.730297
$31$	44.0000i	0.254924i	0.991843	+	0.127462i	$0.0406830\pi$
$31$	44.0000i	0.254924i	−0.991843	+	0.127462i	$0.959317\pi$
$32$	− 32.0000i	− 0.176777i
$33$	− 150.000i	− 0.791262i
$34$	− 60.0000i	− 0.302645i
$35$	640.000	3.09085
$36$	−36.0000	−0.166667
$37$	− 306.000i	− 1.35962i	−0.733386	−	0.679812i	$-0.762061\pi$
$37$	− 306.000i	− 1.35962i	0.733386	−	0.679812i	$-0.237939\pi$
$38$	240.000	1.02456
$39$	0	0
$40$	−160.000	−0.632456
$41$	− 108.000i	− 0.411385i	−0.978617	−	0.205692i	$-0.934055\pi$
$41$	− 108.000i	− 0.411385i	0.978617	−	0.205692i	$-0.0659446\pi$
$42$	192.000	0.705387
$43$	356.000	1.26255	0.631273	−	0.775561i	$-0.282533\pi$
$43$	356.000	1.26255	0.631273	+	0.775561i	$0.282533\pi$
$44$	− 200.000i	− 0.685253i
$45$	180.000i	0.596285i
$46$	− 40.0000i	− 0.128210i
$47$	− 178.000i	− 0.552425i	−0.961097	−	0.276212i	$-0.910921\pi$
$47$	− 178.000i	− 0.552425i	0.961097	−	0.276212i	$-0.0890793\pi$
$48$	−48.0000	−0.144338
$49$	−681.000	−1.98542
$50$	550.000i	1.55563i
$51$	−90.0000	−0.247108
$52$	0	0
$53$	198.000	0.513158	0.256579	−	0.966523i	$-0.417405\pi$
$53$	198.000	0.513158	0.256579	+	0.966523i	$0.417405\pi$
$54$	54.0000i	0.136083i
$55$	−1000.00	−2.45164
$56$	256.000	0.610883
$57$	− 360.000i	− 0.836547i
$58$	− 164.000i	− 0.371280i
$59$	94.0000i	0.207420i	0.994608	+	0.103710i	$0.0330713\pi$
$59$	94.0000i	0.207420i	−0.994608	+	0.103710i	$0.966929\pi$
$60$	240.000i	0.516398i
$61$	−62.0000	−0.130136	−0.0650679	−	0.997881i	$-0.520726\pi$
$61$	−62.0000	−0.130136	−0.0650679	+	0.997881i	$0.520726\pi$
$62$	88.0000	0.180258
$63$	− 288.000i	− 0.575946i
$64$	−64.0000	−0.125000
$65$	0	0
$66$	−300.000	−0.559507
$67$	140.000i	0.255279i	0.991821	+	0.127640i	$0.0407401\pi$
$67$	140.000i	0.255279i	−0.991821	+	0.127640i	$0.959260\pi$
$68$	−120.000	−0.214002
$69$	−60.0000	−0.104683
$70$	− 1280.00i	− 2.18556i
$71$	778.000i	1.30045i	0.759744	+	0.650223i	$0.225324\pi$
$71$	778.000i	1.30045i	−0.759744	+	0.650223i	$0.774676\pi$
$72$	72.0000i	0.117851i
$73$	62.0000i	0.0994048i	0.998764	+	0.0497024i	$0.0158273\pi$
$73$	62.0000i	0.0994048i	−0.998764	+	0.0497024i	$0.984173\pi$
$74$	−612.000	−0.961399
$75$	825.000	1.27017
$76$	− 480.000i	− 0.724471i
$77$	1600.00	2.36801
$78$	0	0
$79$	−1096.00	−1.56088	−0.780441	−	0.625230i	$-0.785005\pi$
$79$	−1096.00	−1.56088	−0.780441	+	0.625230i	$0.785005\pi$
$80$	320.000i	0.447214i
$81$	81.0000	0.111111
$82$	−216.000	−0.290893
$83$	462.000i	0.610977i	0.952196	+	0.305488i	$0.0988197\pi$
$83$	462.000i	0.610977i	−0.952196	+	0.305488i	$0.901180\pi$
$84$	− 384.000i	− 0.498784i
$85$	600.000i	0.765637i
$86$	− 712.000i	− 0.892755i
$87$	−246.000	−0.303149
$88$	−400.000	−0.484547
$89$	1224.00i	1.45779i	0.684623	+	0.728897i	$0.259967\pi$
$89$	1224.00i	1.45779i	−0.684623	+	0.728897i	$0.740033\pi$
$90$	360.000	0.421637
$91$	0	0
$92$	−80.0000	−0.0906584
$93$	− 132.000i	− 0.147180i
$94$	−356.000	−0.390623
$95$	−2400.00	−2.59195
$96$	96.0000i	0.102062i
$97$	− 614.000i	− 0.642704i	−0.946960	−	0.321352i	$-0.895863\pi$
$97$	− 614.000i	− 0.642704i	0.946960	−	0.321352i	$-0.104137\pi$
$98$	1362.00i	1.40391i
$99$	450.000i	0.456835i
$100$	1100.00	1.10000
$101$	−1058.00	−1.04233	−0.521163	−	0.853457i	$-0.674502\pi$
$101$	−1058.00	−1.04233	−0.521163	+	0.853457i	$0.674502\pi$
$102$	180.000i	0.174732i
$103$	−1768.00	−1.69132	−0.845661	−	0.533720i	$-0.820794\pi$
$103$	−1768.00	−1.69132	−0.845661	+	0.533720i	$0.820794\pi$
$104$	0	0
$105$	−1920.00	−1.78450
$106$	− 396.000i	− 0.362858i
$107$	−1808.00	−1.63351	−0.816757	−	0.576982i	$-0.804230\pi$
$107$	−1808.00	−1.63351	−0.816757	+	0.576982i	$0.804230\pi$
$108$	108.000	0.0962250
$109$	1886.00i	1.65730i	0.559765	+	0.828652i	$0.310891\pi$
$109$	1886.00i	1.65730i	−0.559765	+	0.828652i	$0.689109\pi$
$110$	2000.00i	1.73357i
$111$	918.000i	0.784979i
$112$	− 512.000i	− 0.431959i
$113$	1246.00	1.03729	0.518645	−	0.854990i	$-0.326437\pi$
$113$	1246.00	1.03729	0.518645	+	0.854990i	$0.326437\pi$
$114$	−720.000	−0.591528
$115$	400.000i	0.324349i
$116$	−328.000	−0.262535
$117$	0	0
$118$	188.000	0.146668
$119$	− 960.000i	− 0.739521i
$120$	480.000	0.365148
$121$	−1169.00	−0.878287
$122$	124.000i	0.0920199i
$123$	324.000i	0.237513i
$124$	− 176.000i	− 0.127462i
$125$	− 3000.00i	− 2.14663i
$126$	−576.000	−0.407255
$127$	−1624.00	−1.13470	−0.567349	−	0.823477i	$-0.692031\pi$
$127$	−1624.00	−1.13470	−0.567349	+	0.823477i	$0.692031\pi$
$128$	128.000i	0.0883883i
$129$	−1068.00	−0.728931
$130$	0	0
$131$	−2072.00	−1.38192	−0.690960	−	0.722893i	$-0.742812\pi$
$131$	−2072.00	−1.38192	−0.690960	+	0.722893i	$0.742812\pi$
$132$	600.000i	0.395631i
$133$	3840.00	2.50354
$134$	280.000	0.180510
$135$	− 540.000i	− 0.344265i
$136$	240.000i	0.151322i
$137$	− 756.000i	− 0.471456i	−0.971819	−	0.235728i	$-0.924253\pi$
$137$	− 756.000i	− 0.471456i	0.971819	−	0.235728i	$-0.0757474\pi$
$138$	120.000i	0.0740223i
$139$	172.000	0.104956	0.0524779	−	0.998622i	$-0.483288\pi$
$139$	172.000	0.104956	0.0524779	+	0.998622i	$0.483288\pi$
$140$	−2560.00	−1.54542
$141$	534.000i	0.318943i
$142$	1556.00	0.919554
$143$	0	0
$144$	144.000	0.0833333
$145$	1640.00i	0.939273i
$146$	124.000	0.0702898
$147$	2043.00	1.14628
$148$	1224.00i	0.679812i
$149$	− 1272.00i	− 0.699371i	−0.936867	−	0.349686i	$-0.886288\pi$
$149$	− 1272.00i	− 0.699371i	0.936867	−	0.349686i	$-0.113712\pi$
$150$	− 1650.00i	− 0.898146i
$151$	1404.00i	0.756662i	0.925670	+	0.378331i	$0.123502\pi$
$151$	1404.00i	0.756662i	−0.925670	+	0.378331i	$0.876498\pi$
$152$	−960.000	−0.512278
$153$	270.000	0.142668
$154$	− 3200.00i	− 1.67444i
$155$	−880.000	−0.456021
$156$	0	0
$157$	−2170.00	−1.10309	−0.551544	−	0.834146i	$-0.685961\pi$
$157$	−2170.00	−1.10309	−0.551544	+	0.834146i	$0.685961\pi$
$158$	2192.00i	1.10371i
$159$	−594.000	−0.296272
$160$	640.000	0.316228
$161$	− 640.000i	− 0.313286i
$162$	− 162.000i	− 0.0785674i
$163$	248.000i	0.119171i	0.998223	+	0.0595855i	$0.0189779\pi$
$163$	248.000i	0.119171i	−0.998223	+	0.0595855i	$0.981022\pi$
$164$	432.000i	0.205692i
$165$	3000.00	1.41545
$166$	924.000	0.432026
$167$	102.000i	0.0472635i	0.999721	+	0.0236317i	$0.00752291\pi$
$167$	102.000i	0.0472635i	−0.999721	+	0.0236317i	$0.992477\pi$
$168$	−768.000	−0.352693
$169$	0	0
$170$	1200.00	0.541387
$171$	1080.00i	0.482980i
$172$	−1424.00	−0.631273
$173$	−682.000	−0.299720	−0.149860	−	0.988707i	$-0.547882\pi$
$173$	−682.000	−0.299720	−0.149860	+	0.988707i	$0.547882\pi$
$174$	492.000i	0.214359i
$175$	8800.00i	3.80124i
$176$	800.000i	0.342627i
$177$	− 282.000i	− 0.119754i
$178$	2448.00	1.03082
$179$	612.000	0.255548	0.127774	−	0.991803i	$-0.459217\pi$
$179$	612.000	0.255548	0.127774	+	0.991803i	$0.459217\pi$
$180$	− 720.000i	− 0.298142i
$181$	66.0000	0.0271035	0.0135518	−	0.999908i	$-0.495686\pi$
$181$	66.0000	0.0271035	0.0135518	+	0.999908i	$0.495686\pi$
$182$	0	0
$183$	186.000	0.0751340
$184$	160.000i	0.0641052i
$185$	6120.00	2.43217
$186$	−264.000	−0.104072
$187$	1500.00i	0.586582i
$188$	712.000i	0.276212i
$189$	864.000i	0.332522i
$190$	4800.00i	1.83278i
$191$	608.000	0.230332	0.115166	−	0.993346i	$-0.463260\pi$
$191$	608.000	0.230332	0.115166	+	0.993346i	$0.463260\pi$
$192$	192.000	0.0721688
$193$	1370.00i	0.510957i	0.966815	+	0.255479i	$0.0822331\pi$
$193$	1370.00i	0.510957i	−0.966815	+	0.255479i	$0.917767\pi$
$194$	−1228.00	−0.454460
$195$	0	0
$196$	2724.00	0.992711
$197$	4908.00i	1.77503i	0.460781	+	0.887514i	$0.347569\pi$
$197$	4908.00i	1.77503i	−0.460781	+	0.887514i	$0.652431\pi$
$198$	900.000	0.323031
$199$	328.000	0.116841	0.0584204	−	0.998292i	$-0.481394\pi$
$199$	328.000	0.116841	0.0584204	+	0.998292i	$0.481394\pi$
$200$	− 2200.00i	− 0.777817i
$201$	− 420.000i	− 0.147386i
$202$	2116.00i	0.737036i
$203$	− 2624.00i	− 0.907235i
$204$	360.000	0.123554
$205$	2160.00	0.735907
$206$	3536.00i	1.19595i
$207$	180.000	0.0604390
$208$	0	0
$209$	−6000.00	−1.98578
$210$	3840.00i	1.26183i
$211$	1316.00	0.429371	0.214685	−	0.976683i	$-0.431127\pi$
$211$	1316.00	0.429371	0.214685	+	0.976683i	$0.431127\pi$
$212$	−792.000	−0.256579
$213$	− 2334.00i	− 0.750812i
$214$	3616.00i	1.15507i
$215$	7120.00i	2.25851i
$216$	− 216.000i	− 0.0680414i
$217$	1408.00	0.440467
$218$	3772.00	1.17189
$219$	− 186.000i	− 0.0573914i
$220$	4000.00	1.22582
$221$	0	0
$222$	1836.00	0.555064
$223$	1932.00i	0.580163i	0.957002	+	0.290081i	$0.0936824\pi$
$223$	1932.00i	0.580163i	−0.957002	+	0.290081i	$0.906318\pi$
$224$	−1024.00	−0.305441
$225$	−2475.00	−0.733333
$226$	− 2492.00i	− 0.733475i
$227$	− 4998.00i	− 1.46136i	−0.682720	−	0.730680i	$-0.739203\pi$
$227$	− 4998.00i	− 1.46136i	0.682720	−	0.730680i	$-0.260797\pi$
$228$	1440.00i	0.418273i
$229$	− 78.0000i	− 0.0225082i	−0.999937	−	0.0112541i	$-0.996418\pi$
$229$	− 78.0000i	− 0.0225082i	0.999937	−	0.0112541i	$-0.00358237\pi$
$230$	800.000	0.229350
$231$	−4800.00	−1.36717
$232$	656.000i	0.185640i
$233$	1282.00	0.360458	0.180229	−	0.983625i	$-0.442316\pi$
$233$	1282.00	0.360458	0.180229	+	0.983625i	$0.442316\pi$
$234$	0	0
$235$	3560.00	0.988208
$236$	− 376.000i	− 0.103710i
$237$	3288.00	0.901175
$238$	−1920.00	−0.522921
$239$	− 294.000i	− 0.0795702i	−0.999208	−	0.0397851i	$-0.987333\pi$
$239$	− 294.000i	− 0.0795702i	0.999208	−	0.0397851i	$-0.0126673\pi$
$240$	− 960.000i	− 0.258199i
$241$	− 4962.00i	− 1.32627i	−0.748501	−	0.663134i	$-0.769226\pi$
$241$	− 4962.00i	− 1.32627i	0.748501	−	0.663134i	$-0.230774\pi$
$242$	2338.00i	0.621043i
$243$	−243.000	−0.0641500
$244$	248.000	0.0650679
$245$	− 13620.0i	− 3.55163i
$246$	648.000	0.167947
$247$	0	0
$248$	−352.000	−0.0901291
$249$	− 1386.00i	− 0.352748i
$250$	−6000.00	−1.51789
$251$	−744.000	−0.187095	−0.0935475	−	0.995615i	$-0.529821\pi$
$251$	−744.000	−0.187095	−0.0935475	+	0.995615i	$0.529821\pi$
$252$	1152.00i	0.287973i
$253$	1000.00i	0.248496i
$254$	3248.00i	0.802353i
$255$	− 1800.00i	− 0.442041i
$256$	256.000	0.0625000
$257$	1026.00	0.249028	0.124514	−	0.992218i	$-0.460263\pi$
$257$	1026.00	0.249028	0.124514	+	0.992218i	$0.460263\pi$
$258$	2136.00i	0.515432i
$259$	−9792.00	−2.34921
$260$	0	0
$261$	738.000	0.175023
$262$	4144.00i	0.977165i
$263$	−5532.00	−1.29703	−0.648513	−	0.761204i	$-0.724609\pi$
$263$	−5532.00	−1.29703	−0.648513	+	0.761204i	$0.724609\pi$
$264$	1200.00	0.279753
$265$	3960.00i	0.917966i
$266$	− 7680.00i	− 1.77027i
$267$	− 3672.00i	− 0.841658i
$268$	− 560.000i	− 0.127640i
$269$	−3534.00	−0.801010	−0.400505	−	0.916294i	$-0.631165\pi$
$269$	−3534.00	−0.801010	−0.400505	+	0.916294i	$0.631165\pi$
$270$	−1080.00	−0.243432
$271$	2392.00i	0.536176i	0.963394	+	0.268088i	$0.0863918\pi$
$271$	2392.00i	0.536176i	−0.963394	+	0.268088i	$0.913608\pi$
$272$	480.000	0.107001
$273$	0	0
$274$	−1512.00	−0.333370
$275$	− 13750.0i	− 3.01511i
$276$	240.000	0.0523417
$277$	−6102.00	−1.32359	−0.661794	−	0.749686i	$-0.730204\pi$
$277$	−6102.00	−1.32359	−0.661794	+	0.749686i	$0.730204\pi$
$278$	− 344.000i	− 0.0742149i
$279$	396.000i	0.0849746i
$280$	5120.00i	1.09278i
$281$	− 7540.00i	− 1.60071i	−0.599528	−	0.800354i	$-0.704645\pi$
$281$	− 7540.00i	− 1.60071i	0.599528	−	0.800354i	$-0.295355\pi$
$282$	1068.00	0.225527
$283$	2756.00	0.578895	0.289447	−	0.957194i	$-0.406528\pi$
$283$	2756.00	0.578895	0.289447	+	0.957194i	$0.406528\pi$
$284$	− 3112.00i	− 0.650223i
$285$	7200.00	1.49646
$286$	0	0
$287$	−3456.00	−0.710806
$288$	− 288.000i	− 0.0589256i
$289$	−4013.00	−0.816813
$290$	3280.00	0.664166
$291$	1842.00i	0.371065i
$292$	− 248.000i	− 0.0497024i
$293$	968.000i	0.193007i	0.995333	+	0.0965037i	$0.0307660\pi$
$293$	968.000i	0.193007i	−0.995333	+	0.0965037i	$0.969234\pi$
$294$	− 4086.00i	− 0.810545i
$295$	−1880.00	−0.371043
$296$	2448.00	0.480700
$297$	− 1350.00i	− 0.263754i
$298$	−2544.00	−0.494530
$299$	0	0
$300$	−3300.00	−0.635085
$301$	− 11392.0i	− 2.18147i
$302$	2808.00	0.535041
$303$	3174.00	0.601787
$304$	1920.00i	0.362235i
$305$	− 1240.00i	− 0.232794i
$306$	− 540.000i	− 0.100882i
$307$	− 6436.00i	− 1.19649i	−0.801314	−	0.598244i	$-0.795865\pi$
$307$	− 6436.00i	− 1.19649i	0.801314	−	0.598244i	$-0.204135\pi$
$308$	−6400.00	−1.18401
$309$	5304.00	0.976485
$310$	1760.00i	0.322456i
$311$	−7932.00	−1.44625	−0.723123	−	0.690719i	$-0.757294\pi$
$311$	−7932.00	−1.44625	−0.723123	+	0.690719i	$0.757294\pi$
$312$	0	0
$313$	10358.0	1.87051	0.935254	−	0.353978i	$-0.115171\pi$
$313$	10358.0	1.87051	0.935254	+	0.353978i	$0.115171\pi$
$314$	4340.00i	0.780001i
$315$	5760.00	1.03028
$316$	4384.00	0.780441
$317$	2820.00i	0.499643i	0.968292	+	0.249822i	$0.0803720\pi$
$317$	2820.00i	0.499643i	−0.968292	+	0.249822i	$0.919628\pi$
$318$	1188.00i	0.209496i
$319$	4100.00i	0.719611i
$320$	− 1280.00i	− 0.223607i
$321$	5424.00	0.943110
$322$	−1280.00	−0.221527
$323$	3600.00i	0.620153i
$324$	−324.000	−0.0555556
$325$	0	0
$326$	496.000	0.0842666
$327$	− 5658.00i	− 0.956844i
$328$	864.000	0.145446
$329$	−5696.00	−0.954500
$330$	− 6000.00i	− 1.00088i
$331$	4180.00i	0.694120i	0.937843	+	0.347060i	$0.112820\pi$
$331$	4180.00i	0.694120i	−0.937843	+	0.347060i	$0.887180\pi$
$332$	− 1848.00i	− 0.305488i
$333$	− 2754.00i	− 0.453208i
$334$	204.000	0.0334203
$335$	−2800.00	−0.456658
$336$	1536.00i	0.249392i
$337$	5026.00	0.812414	0.406207	−	0.913781i	$-0.366851\pi$
$337$	5026.00	0.812414	0.406207	+	0.913781i	$0.366851\pi$
$338$	0	0
$339$	−3738.00	−0.598880
$340$	− 2400.00i	− 0.382818i
$341$	−2200.00	−0.349374
$342$	2160.00	0.341519
$343$	10816.0i	1.70265i
$344$	2848.00i	0.446378i
$345$	− 1200.00i	− 0.187263i
$346$	1364.00i	0.211934i
$347$	−7332.00	−1.13430	−0.567150	−	0.823614i	$-0.691954\pi$
$347$	−7332.00	−1.13430	−0.567150	+	0.823614i	$0.691954\pi$
$348$	984.000	0.151575
$349$	− 8162.00i	− 1.25187i	−0.779876	−	0.625934i	$-0.784718\pi$
$349$	− 8162.00i	− 1.25187i	0.779876	−	0.625934i	$-0.215282\pi$
$350$	17600.0	2.68788
$351$	0	0
$352$	1600.00	0.242274
$353$	− 1244.00i	− 0.187568i	−0.995593	−	0.0937839i	$-0.970104\pi$
$353$	− 1244.00i	− 0.187568i	0.995593	−	0.0937839i	$-0.0298963\pi$
$354$	−564.000	−0.0846787
$355$	−15560.0	−2.32631
$356$	− 4896.00i	− 0.728897i
$357$	2880.00i	0.426963i
$358$	− 1224.00i	− 0.180699i
$359$	9558.00i	1.40516i	0.711605	+	0.702579i	$0.247968\pi$
$359$	9558.00i	1.40516i	−0.711605	+	0.702579i	$0.752032\pi$
$360$	−1440.00	−0.210819
$361$	−7541.00	−1.09943
$362$	− 132.000i	− 0.0191651i
$363$	3507.00	0.507079
$364$	0	0
$365$	−1240.00	−0.177821
$366$	− 372.000i	− 0.0531277i
$367$	−11032.0	−1.56912	−0.784558	−	0.620055i	$-0.787110\pi$
$367$	−11032.0	−1.56912	−0.784558	+	0.620055i	$0.787110\pi$
$368$	320.000	0.0453292
$369$	− 972.000i	− 0.137128i
$370$	− 12240.0i	− 1.71980i
$371$	− 6336.00i	− 0.886654i
$372$	528.000i	0.0735901i
$373$	5474.00	0.759874	0.379937	−	0.925012i	$-0.375946\pi$
$373$	5474.00	0.759874	0.379937	+	0.925012i	$0.375946\pi$
$374$	3000.00	0.414776
$375$	9000.00i	1.23935i
$376$	1424.00	0.195312
$377$	0	0
$378$	1728.00	0.235129
$379$	7040.00i	0.954144i	0.878864	+	0.477072i	$0.158302\pi$
$379$	7040.00i	0.954144i	−0.878864	+	0.477072i	$0.841698\pi$
$380$	9600.00	1.29597
$381$	4872.00	0.655118
$382$	− 1216.00i	− 0.162869i
$383$	1830.00i	0.244148i	0.992521	+	0.122074i	$0.0389545\pi$
$383$	1830.00i	0.244148i	−0.992521	+	0.122074i	$0.961045\pi$
$384$	− 384.000i	− 0.0510310i
$385$	32000.0i	4.23603i
$386$	2740.00	0.361301
$387$	3204.00	0.420849
$388$	2456.00i	0.321352i
$389$	−10158.0	−1.32399	−0.661994	−	0.749509i	$-0.730289\pi$
$389$	−10158.0	−1.32399	−0.661994	+	0.749509i	$0.730289\pi$
$390$	0	0
$391$	600.000	0.0776044
$392$	− 5448.00i	− 0.701953i
$393$	6216.00	0.797852
$394$	9816.00	1.25513
$395$	− 21920.0i	− 2.79219i
$396$	− 1800.00i	− 0.228418i
$397$	− 12658.0i	− 1.60022i	−0.599854	−	0.800109i	$-0.704775\pi$
$397$	− 12658.0i	− 1.60022i	0.599854	−	0.800109i	$-0.295225\pi$
$398$	− 656.000i	− 0.0826189i
$399$	−11520.0	−1.44542
$400$	−4400.00	−0.550000
$401$	15720.0i	1.95765i	0.204689	+	0.978827i	$0.434382\pi$
$401$	15720.0i	1.95765i	−0.204689	+	0.978827i	$0.565618\pi$
$402$	−840.000	−0.104217
$403$	0	0
$404$	4232.00	0.521163
$405$	1620.00i	0.198762i
$406$	−5248.00	−0.641512
$407$	15300.0	1.86337
$408$	− 720.000i	− 0.0873660i
$409$	− 7654.00i	− 0.925345i	−0.886529	−	0.462672i	$-0.846891\pi$
$409$	− 7654.00i	− 0.925345i	0.886529	−	0.462672i	$-0.153109\pi$
$410$	− 4320.00i	− 0.520365i
$411$	2268.00i	0.272195i
$412$	7072.00	0.845661
$413$	3008.00	0.358387
$414$	− 360.000i	− 0.0427368i
$415$	−9240.00	−1.09295
$416$	0	0
$417$	−516.000	−0.0605962
$418$	12000.0i	1.40416i
$419$	−1848.00	−0.215467	−0.107734	−	0.994180i	$-0.534359\pi$
$419$	−1848.00	−0.215467	−0.107734	+	0.994180i	$0.534359\pi$
$420$	7680.00	0.892251
$421$	12542.0i	1.45192i	0.687735	+	0.725962i	$0.258605\pi$
$421$	12542.0i	1.45192i	−0.687735	+	0.725962i	$0.741395\pi$
$422$	− 2632.00i	− 0.303611i
$423$	− 1602.00i	− 0.184142i
$424$	1584.00i	0.181429i
$425$	−8250.00	−0.941609
$426$	−4668.00	−0.530905
$427$	1984.00i	0.224854i
$428$	7232.00	0.816757
$429$	0	0
$430$	14240.0	1.59701
$431$	5238.00i	0.585396i	0.956205	+	0.292698i	$0.0945530\pi$
$431$	5238.00i	0.585396i	−0.956205	+	0.292698i	$0.905447\pi$
$432$	−432.000	−0.0481125
$433$	8258.00	0.916522	0.458261	−	0.888818i	$-0.348472\pi$
$433$	8258.00	0.916522	0.458261	+	0.888818i	$0.348472\pi$
$434$	− 2816.00i	− 0.311457i
$435$	− 4920.00i	− 0.542290i
$436$	− 7544.00i	− 0.828652i
$437$	2400.00i	0.262718i
$438$	−372.000	−0.0405818
$439$	6304.00	0.685361	0.342681	−	0.939452i	$-0.388665\pi$
$439$	6304.00	0.685361	0.342681	+	0.939452i	$0.388665\pi$
$440$	− 8000.00i	− 0.866784i
$441$	−6129.00	−0.661808
$442$	0	0
$443$	12744.0	1.36678	0.683392	−	0.730051i	$-0.260504\pi$
$443$	12744.0	1.36678	0.683392	+	0.730051i	$0.260504\pi$
$444$	− 3672.00i	− 0.392490i
$445$	−24480.0	−2.60778
$446$	3864.00	0.410237
$447$	3816.00i	0.403782i
$448$	2048.00i	0.215980i
$449$	− 11776.0i	− 1.23774i	−0.785495	−	0.618868i	$-0.787591\pi$
$449$	− 11776.0i	− 1.23774i	0.785495	−	0.618868i	$-0.212409\pi$
$450$	4950.00i	0.518545i
$451$	5400.00	0.563805
$452$	−4984.00	−0.518645
$453$	− 4212.00i	− 0.436859i
$454$	−9996.00	−1.03334
$455$	0	0
$456$	2880.00	0.295764
$457$	− 2134.00i	− 0.218434i	−0.994018	−	0.109217i	$-0.965166\pi$
$457$	− 2134.00i	− 0.218434i	0.994018	−	0.109217i	$-0.0348343\pi$
$458$	−156.000	−0.0159157
$459$	−810.000	−0.0823694
$460$	− 1600.00i	− 0.162175i
$461$	− 2724.00i	− 0.275205i	−0.990488	−	0.137602i	$-0.956060\pi$
$461$	− 2724.00i	− 0.275205i	0.990488	−	0.137602i	$-0.0439396\pi$
$462$	9600.00i	0.966737i
$463$	− 5648.00i	− 0.566922i	−0.958984	−	0.283461i	$-0.908517\pi$
$463$	− 5648.00i	− 0.566922i	0.958984	−	0.283461i	$-0.0914826\pi$
$464$	1312.00	0.131267
$465$	2640.00	0.263284
$466$	− 2564.00i	− 0.254882i
$467$	18224.0	1.80579	0.902897	−	0.429856i	$-0.141436\pi$
$467$	18224.0	1.80579	0.902897	+	0.429856i	$0.141436\pi$
$468$	0	0
$469$	4480.00	0.441081
$470$	− 7120.00i	− 0.698768i
$471$	6510.00	0.636868
$472$	−752.000	−0.0733339
$473$	17800.0i	1.73033i
$474$	− 6576.00i	− 0.637227i
$475$	− 33000.0i	− 3.18767i
$476$	3840.00i	0.369761i
$477$	1782.00	0.171053
$478$	−588.000	−0.0562646
$479$	9066.00i	0.864794i	0.901683	+	0.432397i	$0.142332\pi$
$479$	9066.00i	0.864794i	−0.901683	+	0.432397i	$0.857668\pi$
$480$	−1920.00	−0.182574
$481$	0	0
$482$	−9924.00	−0.937813
$483$	1920.00i	0.180876i
$484$	4676.00	0.439144
$485$	12280.0	1.14970
$486$	486.000i	0.0453609i
$487$	− 8948.00i	− 0.832593i	−0.909229	−	0.416296i	$-0.863328\pi$
$487$	− 8948.00i	− 0.832593i	0.909229	−	0.416296i	$-0.136672\pi$
$488$	− 496.000i	− 0.0460100i
$489$	− 744.000i	− 0.0688034i
$490$	−27240.0	−2.51138
$491$	−8720.00	−0.801483	−0.400741	−	0.916191i	$-0.631247\pi$
$491$	−8720.00	−0.801483	−0.400741	+	0.916191i	$0.631247\pi$
$492$	− 1296.00i	− 0.118756i
$493$	2460.00	0.224732
$494$	0	0
$495$	−9000.00	−0.817212
$496$	704.000i	0.0637309i
$497$	24896.0	2.24696
$498$	−2772.00	−0.249430
$499$	− 6604.00i	− 0.592456i	−0.955117	−	0.296228i	$-0.904271\pi$
$499$	− 6604.00i	− 0.592456i	0.955117	−	0.296228i	$-0.0957289\pi$
$500$	12000.0i	1.07331i
$501$	− 306.000i	− 0.0272876i
$502$	1488.00i	0.132296i
$503$	3404.00	0.301743	0.150872	−	0.988553i	$-0.451792\pi$
$503$	3404.00	0.301743	0.150872	+	0.988553i	$0.451792\pi$
$504$	2304.00	0.203628
$505$	− 21160.0i	− 1.86457i
$506$	2000.00	0.175713
$507$	0	0
$508$	6496.00	0.567349
$509$	76.0000i	0.00661815i	0.999995	+	0.00330908i	$0.00105331\pi$
$509$	76.0000i	0.00661815i	−0.999995	+	0.00330908i	$0.998947\pi$
$510$	−3600.00	−0.312570
$511$	1984.00	0.171755
$512$	− 512.000i	− 0.0441942i
$513$	− 3240.00i	− 0.278849i
$514$	− 2052.00i	− 0.176089i
$515$	− 35360.0i	− 3.02553i
$516$	4272.00	0.364466
$517$	8900.00	0.757102
$518$	19584.0i	1.66114i
$519$	2046.00	0.173043
$520$	0	0
$521$	12054.0	1.01362	0.506809	−	0.862058i	$-0.330825\pi$
$521$	12054.0	1.01362	0.506809	+	0.862058i	$0.330825\pi$
$522$	− 1476.00i	− 0.123760i
$523$	276.000	0.0230758	0.0115379	−	0.999933i	$-0.496327\pi$
$523$	276.000	0.0230758	0.0115379	+	0.999933i	$0.496327\pi$
$524$	8288.00	0.690960
$525$	− 26400.0i	− 2.19465i
$526$	11064.0i	0.917136i
$527$	1320.00i	0.109108i
$528$	− 2400.00i	− 0.197816i
$529$	−11767.0	−0.967124
$530$	7920.00	0.649100
$531$	846.000i	0.0691399i
$532$	−15360.0	−1.25177
$533$	0	0
$534$	−7344.00	−0.595142
$535$	− 36160.0i	− 2.92212i
$536$	−1120.00	−0.0902549
$537$	−1836.00	−0.147540
$538$	7068.00i	0.566400i
$539$	− 34050.0i	− 2.72103i
$540$	2160.00i	0.172133i
$541$	13778.0i	1.09494i	0.836825	+	0.547470i	$0.184409\pi$
$541$	13778.0i	1.09494i	−0.836825	+	0.547470i	$0.815591\pi$
$542$	4784.00	0.379134
$543$	−198.000	−0.0156482
$544$	− 960.000i	− 0.0756611i
$545$	−37720.0	−2.96467
$546$	0	0
$547$	−10844.0	−0.847634	−0.423817	−	0.905748i	$-0.639310\pi$
$547$	−10844.0	−0.847634	−0.423817	+	0.905748i	$0.639310\pi$
$548$	3024.00i	0.235728i
$549$	−558.000	−0.0433786
$550$	−27500.0	−2.13201
$551$	9840.00i	0.760795i
$552$	− 480.000i	− 0.0370112i
$553$	35072.0i	2.69695i
$554$	12204.0i	0.935917i
$555$	−18360.0	−1.40421
$556$	−688.000	−0.0524779
$557$	20544.0i	1.56280i	0.624033	+	0.781398i	$0.285493\pi$
$557$	20544.0i	1.56280i	−0.624033	+	0.781398i	$0.714507\pi$
$558$	792.000	0.0600861
$559$	0	0
$560$	10240.0	0.772712
$561$	− 4500.00i	− 0.338663i
$562$	−15080.0	−1.13187
$563$	−6988.00	−0.523107	−0.261553	−	0.965189i	$-0.584235\pi$
$563$	−6988.00	−0.523107	−0.261553	+	0.965189i	$0.584235\pi$
$564$	− 2136.00i	− 0.159471i
$565$	24920.0i	1.85556i
$566$	− 5512.00i	− 0.409340i
$567$	− 2592.00i	− 0.191982i
$568$	−6224.00	−0.459777
$569$	706.000	0.0520159	0.0260080	−	0.999662i	$-0.491720\pi$
$569$	706.000	0.0520159	0.0260080	+	0.999662i	$0.491720\pi$
$570$	− 14400.0i	− 1.05816i
$571$	17532.0	1.28492	0.642462	−	0.766318i	$-0.277913\pi$
$571$	17532.0	1.28492	0.642462	+	0.766318i	$0.277913\pi$
$572$	0	0
$573$	−1824.00	−0.132982
$574$	6912.00i	0.502616i
$575$	−5500.00	−0.398897
$576$	−576.000	−0.0416667
$577$	14814.0i	1.06883i	0.845222	+	0.534415i	$0.179468\pi$
$577$	14814.0i	1.06883i	−0.845222	+	0.534415i	$0.820532\pi$
$578$	8026.00i	0.577574i
$579$	− 4110.00i	− 0.295001i
$580$	− 6560.00i	− 0.469637i
$581$	14784.0	1.05567
$582$	3684.00	0.262383
$583$	9900.00i	0.703287i
$584$	−496.000	−0.0351449
$585$	0	0
$586$	1936.00	0.136477
$587$	− 14170.0i	− 0.996352i	−0.867076	−	0.498176i	$-0.834003\pi$
$587$	− 14170.0i	− 0.996352i	0.867076	−	0.498176i	$-0.165997\pi$
$588$	−8172.00	−0.573142
$589$	−5280.00	−0.369369
$590$	3760.00i	0.262367i
$591$	− 14724.0i	− 1.02481i
$592$	− 4896.00i	− 0.339906i
$593$	− 11744.0i	− 0.813269i	−0.913591	−	0.406634i	$-0.866702\pi$
$593$	− 11744.0i	− 0.813269i	0.913591	−	0.406634i	$-0.133298\pi$
$594$	−2700.00	−0.186502
$595$	19200.0	1.32290
$596$	5088.00i	0.349686i
$597$	−984.000	−0.0674580
$598$	0	0
$599$	−15076.0	−1.02836	−0.514181	−	0.857682i	$-0.671904\pi$
$599$	−15076.0	−1.02836	−0.514181	+	0.857682i	$0.671904\pi$
$600$	6600.00i	0.449073i
$601$	20230.0	1.37304	0.686522	−	0.727109i	$-0.259137\pi$
$601$	20230.0	1.37304	0.686522	+	0.727109i	$0.259137\pi$
$602$	−22784.0	−1.54254
$603$	1260.00i	0.0850931i
$604$	− 5616.00i	− 0.378331i
$605$	− 23380.0i	− 1.57113i
$606$	− 6348.00i	− 0.425528i
$607$	−28056.0	−1.87604	−0.938021	−	0.346577i	$-0.887344\pi$
$607$	−28056.0	−1.87604	−0.938021	+	0.346577i	$0.887344\pi$
$608$	3840.00	0.256139
$609$	7872.00i	0.523792i
$610$	−2480.00	−0.164610
$611$	0	0
$612$	−1080.00	−0.0713340
$613$	− 27446.0i	− 1.80837i	−0.427136	−	0.904187i	$-0.640478\pi$
$613$	− 27446.0i	− 1.80837i	0.427136	−	0.904187i	$-0.359522\pi$
$614$	−12872.0	−0.846045
$615$	−6480.00	−0.424876
$616$	12800.0i	0.837219i
$617$	− 8804.00i	− 0.574450i	−0.957863	−	0.287225i	$-0.907267\pi$
$617$	− 8804.00i	− 0.574450i	0.957863	−	0.287225i	$-0.0927328\pi$
$618$	− 10608.0i	− 0.690480i
$619$	3508.00i	0.227784i	0.993493	+	0.113892i	$0.0363318\pi$
$619$	3508.00i	0.227784i	−0.993493	+	0.113892i	$0.963668\pi$
$620$	3520.00	0.228011
$621$	−540.000	−0.0348945
$622$	15864.0i	1.02265i
$623$	39168.0	2.51883
$624$	0	0
$625$	25625.0	1.64000
$626$	− 20716.0i	− 1.32265i
$627$	18000.0	1.14649
$628$	8680.00	0.551544
$629$	− 9180.00i	− 0.581925i
$630$	− 11520.0i	− 0.728520i
$631$	22084.0i	1.39326i	0.717428	+	0.696632i	$0.245319\pi$
$631$	22084.0i	1.39326i	−0.717428	+	0.696632i	$0.754681\pi$
$632$	− 8768.00i	− 0.551855i
$633$	−3948.00	−0.247897
$634$	5640.00	0.353301
$635$	− 32480.0i	− 2.02981i
$636$	2376.00	0.148136
$637$	0	0
$638$	8200.00	0.508842
$639$	7002.00i	0.433482i
$640$	−2560.00	−0.158114
$641$	7342.00	0.452405	0.226202	−	0.974080i	$-0.427369\pi$
$641$	7342.00	0.452405	0.226202	+	0.974080i	$0.427369\pi$
$642$	− 10848.0i	− 0.666879i
$643$	− 2996.00i	− 0.183749i	−0.995771	−	0.0918746i	$-0.970714\pi$
$643$	− 2996.00i	− 0.183749i	0.995771	−	0.0918746i	$-0.0292859\pi$
$644$	2560.00i	0.156643i
$645$	− 21360.0i	− 1.30395i
$646$	7200.00	0.438514
$647$	−9344.00	−0.567775	−0.283888	−	0.958858i	$-0.591624\pi$
$647$	−9344.00	−0.567775	−0.283888	+	0.958858i	$0.591624\pi$
$648$	648.000i	0.0392837i
$649$	−4700.00	−0.284270
$650$	0	0
$651$	−4224.00	−0.254304
$652$	− 992.000i	− 0.0595855i
$653$	−16686.0	−0.999960	−0.499980	−	0.866037i	$-0.666659\pi$
$653$	−16686.0	−0.999960	−0.499980	+	0.866037i	$0.666659\pi$
$654$	−11316.0	−0.676591
$655$	− 41440.0i	− 2.47205i
$656$	− 1728.00i	− 0.102846i
$657$	558.000i	0.0331349i
$658$	11392.0i	0.674934i
$659$	31356.0	1.85350	0.926750	−	0.375679i	$-0.122590\pi$
$659$	31356.0	1.85350	0.926750	+	0.375679i	$0.122590\pi$
$660$	−12000.0	−0.707726
$661$	590.000i	0.0347176i	0.999849	+	0.0173588i	$0.00552576\pi$
$661$	590.000i	0.0347176i	−0.999849	+	0.0173588i	$0.994474\pi$
$662$	8360.00	0.490817
$663$	0	0
$664$	−3696.00	−0.216013
$665$	76800.0i	4.47846i
$666$	−5508.00	−0.320466
$667$	1640.00	0.0952040
$668$	− 408.000i	− 0.0236317i
$669$	− 5796.00i	− 0.334957i
$670$	5600.00i	0.322906i
$671$	− 3100.00i	− 0.178352i
$672$	3072.00	0.176347
$673$	−5938.00	−0.340109	−0.170054	−	0.985435i	$-0.554394\pi$
$673$	−5938.00	−0.340109	−0.170054	+	0.985435i	$0.554394\pi$
$674$	− 10052.0i	− 0.574464i
$675$	7425.00	0.423390
$676$	0	0
$677$	9486.00	0.538518	0.269259	−	0.963068i	$-0.413221\pi$
$677$	9486.00	0.538518	0.269259	+	0.963068i	$0.413221\pi$
$678$	7476.00i	0.423472i
$679$	−19648.0	−1.11049
$680$	−4800.00	−0.270694
$681$	14994.0i	0.843717i
$682$	4400.00i	0.247045i
$683$	26162.0i	1.46568i	0.680400	+	0.732841i	$0.261806\pi$
$683$	26162.0i	1.46568i	−0.680400	+	0.732841i	$0.738194\pi$
$684$	− 4320.00i	− 0.241490i
$685$	15120.0	0.843366
$686$	21632.0	1.20396
$687$	234.000i	0.0129951i
$688$	5696.00	0.315637
$689$	0	0
$690$	−2400.00	−0.132415
$691$	17348.0i	0.955064i	0.878614	+	0.477532i	$0.158468\pi$
$691$	17348.0i	0.955064i	−0.878614	+	0.477532i	$0.841532\pi$
$692$	2728.00	0.149860
$693$	14400.0	0.789337
$694$	14664.0i	0.802072i
$695$	3440.00i	0.187751i
$696$	− 1968.00i	− 0.107179i
$697$	− 3240.00i	− 0.176074i
$698$	−16324.0	−0.885204
$699$	−3846.00	−0.208110
$700$	− 35200.0i	− 1.90062i
$701$	−30.0000	−0.00161638	−0.000808191	−	1.00000i	$-0.500257\pi$
$701$	−30.0000	−0.00161638	−0.000808191		1.00000i	$0.500257\pi$
$702$	0	0
$703$	36720.0	1.97002
$704$	− 3200.00i	− 0.171313i
$705$	−10680.0	−0.570542
$706$	−2488.00	−0.132630
$707$	33856.0i	1.80097i
$708$	1128.00i	0.0598769i
$709$	31466.0i	1.66676i	0.552703	+	0.833378i	$0.313596\pi$
$709$	31466.0i	1.66676i	−0.552703	+	0.833378i	$0.686404\pi$
$710$	31120.0i	1.64495i
$711$	−9864.00	−0.520294
$712$	−9792.00	−0.515408
$713$	880.000i	0.0462220i
$714$	5760.00	0.301908
$715$	0	0
$716$	−2448.00	−0.127774
$717$	882.000i	0.0459399i
$718$	19116.0	0.993597
$719$	28892.0	1.49859	0.749297	−	0.662234i	$-0.230391\pi$
$719$	28892.0	1.49859	0.749297	+	0.662234i	$0.230391\pi$
$720$	2880.00i	0.149071i
$721$	56576.0i	2.92233i
$722$	15082.0i	0.777415i
$723$	14886.0i	0.765721i
$724$	−264.000	−0.0135518
$725$	−22550.0	−1.15515
$726$	− 7014.00i	− 0.358559i
$727$	−13384.0	−0.682786	−0.341393	−	0.939921i	$-0.610899\pi$
$727$	−13384.0	−0.682786	−0.341393	+	0.939921i	$0.610899\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	2480.00i	0.125738i
$731$	10680.0	0.540375
$732$	−744.000	−0.0375670
$733$	− 7130.00i	− 0.359280i	−0.983732	−	0.179640i	$-0.942507\pi$
$733$	− 7130.00i	− 0.359280i	0.983732	−	0.179640i	$-0.0574934\pi$
$734$	22064.0i	1.10953i
$735$	40860.0i	2.05054i
$736$	− 640.000i	− 0.0320526i
$737$	−7000.00	−0.349862
$738$	−1944.00	−0.0969643
$739$	− 29268.0i	− 1.45689i	−0.685105	−	0.728444i	$-0.740244\pi$
$739$	− 29268.0i	− 1.45689i	0.685105	−	0.728444i	$-0.259756\pi$
$740$	−24480.0	−1.21608
$741$	0	0
$742$	−12672.0	−0.626959
$743$	9898.00i	0.488725i	0.969684	+	0.244362i	$0.0785786\pi$
$743$	9898.00i	0.488725i	−0.969684	+	0.244362i	$0.921421\pi$
$744$	1056.00	0.0520361
$745$	25440.0	1.25107
$746$	− 10948.0i	− 0.537312i
$747$	4158.00i	0.203659i
$748$	− 6000.00i	− 0.293291i
$749$	57856.0i	2.82245i
$750$	18000.0	0.876356
$751$	15120.0	0.734669	0.367335	−	0.930089i	$-0.380270\pi$
$751$	15120.0	0.734669	0.367335	+	0.930089i	$0.380270\pi$
$752$	− 2848.00i	− 0.138106i
$753$	2232.00	0.108019
$754$	0	0
$755$	−28080.0	−1.35356
$756$	− 3456.00i	− 0.166261i
$757$	−5454.00	−0.261861	−0.130931	−	0.991392i	$-0.541797\pi$
$757$	−5454.00	−0.261861	−0.130931	+	0.991392i	$0.541797\pi$
$758$	14080.0	0.674682
$759$	− 3000.00i	− 0.143469i
$760$	− 19200.0i	− 0.916391i
$761$	− 11988.0i	− 0.571044i	−0.958372	−	0.285522i	$-0.907833\pi$
$761$	− 11988.0i	− 0.571044i	0.958372	−	0.285522i	$-0.0921670\pi$
$762$	− 9744.00i	− 0.463239i
$763$	60352.0	2.86355
$764$	−2432.00	−0.115166
$765$	5400.00i	0.255212i
$766$	3660.00	0.172639
$767$	0	0
$768$	−768.000	−0.0360844
$769$	− 1338.00i	− 0.0627432i	−0.999508	−	0.0313716i	$-0.990012\pi$
$769$	− 1338.00i	− 0.0627432i	0.999508	−	0.0313716i	$-0.00998753\pi$
$770$	64000.0	2.99532
$771$	−3078.00	−0.143776
$772$	− 5480.00i	− 0.255479i
$773$	14408.0i	0.670401i	0.942147	+	0.335200i	$0.108804\pi$
$773$	14408.0i	0.670401i	−0.942147	+	0.335200i	$0.891196\pi$
$774$	− 6408.00i	− 0.297585i
$775$	− 12100.0i	− 0.560832i
$776$	4912.00	0.227230
$777$	29376.0	1.35632
$778$	20316.0i	0.936200i
$779$	12960.0	0.596072
$780$	0	0
$781$	−38900.0	−1.78227
$782$	− 1200.00i	− 0.0548746i
$783$	−2214.00	−0.101050
$784$	−10896.0	−0.496356
$785$	− 43400.0i	− 1.97326i
$786$	− 12432.0i	− 0.564166i
$787$	− 10660.0i	− 0.482831i	−0.970422	−	0.241415i	$-0.922388\pi$
$787$	− 10660.0i	− 0.482831i	0.970422	−	0.241415i	$-0.0776117\pi$
$788$	− 19632.0i	− 0.887514i
$789$	16596.0	0.748838
$790$	−43840.0	−1.97438
$791$	− 39872.0i	− 1.79227i
$792$	−3600.00	−0.161516
$793$	0	0
$794$	−25316.0	−1.13153
$795$	− 11880.0i	− 0.529988i
$796$	−1312.00	−0.0584204
$797$	−1974.00	−0.0877323	−0.0438662	−	0.999037i	$-0.513968\pi$
$797$	−1974.00	−0.0877323	−0.0438662	+	0.999037i	$0.513968\pi$
$798$	23040.0i	1.02206i
$799$	− 5340.00i	− 0.236440i
$800$	8800.00i	0.388909i
$801$	11016.0i	0.485932i
$802$	31440.0	1.38427
$803$	−3100.00	−0.136235
$804$	1680.00i	0.0736928i
$805$	12800.0	0.560423
$806$	0	0
$807$	10602.0	0.462464
$808$	− 8464.00i	− 0.368518i
$809$	31734.0	1.37912	0.689560	−	0.724229i	$-0.257804\pi$
$809$	31734.0	1.37912	0.689560	+	0.724229i	$0.257804\pi$
$810$	3240.00	0.140546
$811$	38824.0i	1.68100i	0.541808	+	0.840502i	$0.317740\pi$
$811$	38824.0i	1.68100i	−0.541808	+	0.840502i	$0.682260\pi$
$812$	10496.0i	0.453617i
$813$	− 7176.00i	− 0.309561i
$814$	− 30600.0i	− 1.31760i
$815$	−4960.00	−0.213179
$816$	−1440.00	−0.0617771
$817$	42720.0i	1.82936i
$818$	−15308.0	−0.654317
$819$	0	0
$820$	−8640.00	−0.367954
$821$	16736.0i	0.711438i	0.934593	+	0.355719i	$0.115764\pi$
$821$	16736.0i	0.711438i	−0.934593	+	0.355719i	$0.884236\pi$
$822$	4536.00	0.192471
$823$	−42096.0	−1.78296	−0.891479	−	0.453062i	$-0.850332\pi$
$823$	−42096.0	−1.78296	−0.891479	+	0.453062i	$0.850332\pi$
$824$	− 14144.0i	− 0.597973i
$825$	41250.0i	1.74078i
$826$	− 6016.00i	− 0.253418i
$827$	24858.0i	1.04522i	0.852572	+	0.522610i	$0.175042\pi$
$827$	24858.0i	1.04522i	−0.852572	+	0.522610i	$0.824958\pi$
$828$	−720.000	−0.0302195
$829$	−922.000	−0.0386277	−0.0193139	−	0.999813i	$-0.506148\pi$
$829$	−922.000	−0.0386277	−0.0193139	+	0.999813i	$0.506148\pi$
$830$	18480.0i	0.772832i
$831$	18306.0	0.764173
$832$	0	0
$833$	−20430.0	−0.849769
$834$	1032.00i	0.0428480i
$835$	−2040.00	−0.0845474
$836$	24000.0	0.992892
$837$	− 1188.00i	− 0.0490601i
$838$	3696.00i	0.152358i
$839$	− 14294.0i	− 0.588181i	−0.955778	−	0.294090i	$-0.904983\pi$
$839$	− 14294.0i	− 0.588181i	0.955778	−	0.294090i	$-0.0950167\pi$
$840$	− 15360.0i	− 0.630917i
$841$	−17665.0	−0.724302
$842$	25084.0	1.02666
$843$	22620.0i	0.924169i
$844$	−5264.00	−0.214685
$845$	0	0
$846$	−3204.00	−0.130208
$847$	37408.0i	1.51754i
$848$	3168.00	0.128290
$849$	−8268.00	−0.334225
$850$	16500.0i	0.665818i
$851$	− 6120.00i	− 0.246523i
$852$	9336.00i	0.375406i
$853$	37966.0i	1.52395i	0.647605	+	0.761976i	$0.275771\pi$
$853$	37966.0i	1.52395i	−0.647605	+	0.761976i	$0.724229\pi$
$854$	3968.00	0.158996
$855$	−21600.0	−0.863982
$856$	− 14464.0i	− 0.577534i
$857$	−39038.0	−1.55602	−0.778012	−	0.628249i	$-0.783772\pi$
$857$	−39038.0	−1.55602	−0.778012	+	0.628249i	$0.783772\pi$
$858$	0	0
$859$	20564.0	0.816804	0.408402	−	0.912802i	$-0.366086\pi$
$859$	20564.0	0.816804	0.408402	+	0.912802i	$0.366086\pi$
$860$	− 28480.0i	− 1.12926i
$861$	10368.0	0.410384
$862$	10476.0	0.413937
$863$	− 39866.0i	− 1.57248i	−0.617918	−	0.786242i	$-0.712024\pi$
$863$	− 39866.0i	− 1.57248i	0.617918	−	0.786242i	$-0.287976\pi$
$864$	864.000i	0.0340207i
$865$	− 13640.0i	− 0.536155i
$866$	− 16516.0i	− 0.648079i
$867$	12039.0	0.471587
$868$	−5632.00	−0.220233
$869$	− 54800.0i	− 2.13920i
$870$	−9840.00	−0.383457
$871$	0	0
$872$	−15088.0	−0.585945
$873$	− 5526.00i	− 0.214235i
$874$	4800.00	0.185769
$875$	−96000.0	−3.70902
$876$	744.000i	0.0286957i
$877$	− 30990.0i	− 1.19322i	−0.802530	−	0.596612i	$-0.796513\pi$
$877$	− 30990.0i	− 1.19322i	0.802530	−	0.596612i	$-0.203487\pi$
$878$	− 12608.0i	− 0.484623i
$879$	− 2904.00i	− 0.111433i
$880$	−16000.0	−0.612909
$881$	4458.00	0.170481	0.0852405	−	0.996360i	$-0.472834\pi$
$881$	4458.00	0.170481	0.0852405	+	0.996360i	$0.472834\pi$
$882$	12258.0i	0.467969i
$883$	3164.00	0.120586	0.0602928	−	0.998181i	$-0.480797\pi$
$883$	3164.00	0.120586	0.0602928	+	0.998181i	$0.480797\pi$
$884$	0	0
$885$	5640.00	0.214222
$886$	− 25488.0i	− 0.966463i
$887$	−32512.0	−1.23072	−0.615359	−	0.788247i	$-0.710989\pi$
$887$	−32512.0	−1.23072	−0.615359	+	0.788247i	$0.710989\pi$
$888$	−7344.00	−0.277532
$889$	51968.0i	1.96057i
$890$	48960.0i	1.84398i
$891$	4050.00i	0.152278i
$892$	− 7728.00i	− 0.290081i
$893$	21360.0	0.800431
$894$	7632.00	0.285517
$895$	12240.0i	0.457138i
$896$	4096.00	0.152721
$897$	0	0
$898$	−23552.0	−0.875212
$899$	3608.00i	0.133853i
$900$	9900.00	0.366667
$901$	5940.00	0.219634
$902$	− 10800.0i	− 0.398670i
$903$	34176.0i	1.25948i
$904$	9968.00i	0.366738i
$905$	1320.00i	0.0484843i
$906$	−8424.00	−0.308906
$907$	−10500.0	−0.384396	−0.192198	−	0.981356i	$-0.561562\pi$
$907$	−10500.0	−0.384396	−0.192198	+	0.981356i	$0.561562\pi$
$908$	19992.0i	0.730680i
$909$	−9522.00	−0.347442
$910$	0	0
$911$	9840.00	0.357864	0.178932	−	0.983861i	$-0.442736\pi$
$911$	9840.00	0.357864	0.178932	+	0.983861i	$0.442736\pi$
$912$	− 5760.00i	− 0.209137i
$913$	−23100.0	−0.837348
$914$	−4268.00	−0.154456
$915$	3720.00i	0.134404i
$916$	312.000i	0.0112541i
$917$	66304.0i	2.38773i
$918$	1620.00i	0.0582440i
$919$	−35040.0	−1.25774	−0.628870	−	0.777511i	$-0.716482\pi$
$919$	−35040.0	−1.25774	−0.628870	+	0.777511i	$0.716482\pi$
$920$	−3200.00	−0.114675
$921$	19308.0i	0.690793i
$922$	−5448.00	−0.194599
$923$	0	0
$924$	19200.0	0.683586
$925$	84150.0i	2.99117i
$926$	−11296.0	−0.400874
$927$	−15912.0	−0.563774
$928$	− 2624.00i	− 0.0928201i
$929$	− 44172.0i	− 1.56000i	−0.625782	−	0.779998i	$-0.715220\pi$
$929$	− 44172.0i	− 1.56000i	0.625782	−	0.779998i	$-0.284780\pi$
$930$	− 5280.00i	− 0.186170i
$931$	− 81720.0i	− 2.87676i
$932$	−5128.00	−0.180229
$933$	23796.0	0.834990
$934$	− 36448.0i	− 1.27689i
$935$	−30000.0	−1.04931
$936$	0	0
$937$	−54018.0	−1.88334	−0.941671	−	0.336535i	$-0.890745\pi$
$937$	−54018.0	−1.88334	−0.941671	+	0.336535i	$0.890745\pi$
$938$	− 8960.00i	− 0.311892i
$939$	−31074.0	−1.07994
$940$	−14240.0	−0.494104
$941$	1672.00i	0.0579231i	0.999581	+	0.0289616i	$0.00922004\pi$
$941$	1672.00i	0.0579231i	−0.999581	+	0.0289616i	$0.990780\pi$
$942$	− 13020.0i	− 0.450334i
$943$	− 2160.00i	− 0.0745910i
$944$	1504.00i	0.0518549i
$945$	−17280.0	−0.594834
$946$	35600.0	1.22353
$947$	5238.00i	0.179738i	0.995954	+	0.0898691i	$0.0286449\pi$
$947$	5238.00i	0.179738i	−0.995954	+	0.0898691i	$0.971355\pi$
$948$	−13152.0	−0.450588
$949$	0	0
$950$	−66000.0	−2.25402
$951$	− 8460.00i	− 0.288469i
$952$	7680.00	0.261460
$953$	50042.0	1.70096	0.850482	−	0.526004i	$-0.176310\pi$
$953$	50042.0	1.70096	0.850482	+	0.526004i	$0.176310\pi$
$954$	− 3564.00i	− 0.120953i
$955$	12160.0i	0.412030i
$956$	1176.00i	0.0397851i
$957$	− 12300.0i	− 0.415468i
$958$	18132.0	0.611501
$959$	−24192.0	−0.814599
$960$	3840.00i	0.129099i
$961$	27855.0	0.935014
$962$	0	0
$963$	−16272.0	−0.544505
$964$	19848.0i	0.663134i
$965$	−27400.0	−0.914028
$966$	3840.00	0.127899
$967$	− 37676.0i	− 1.25293i	−0.779452	−	0.626463i	$-0.784502\pi$
$967$	− 37676.0i	− 1.25293i	0.779452	−	0.626463i	$-0.215498\pi$
$968$	− 9352.00i	− 0.310521i
$969$	− 10800.0i	− 0.358045i
$970$	− 24560.0i	− 0.812963i
$971$	−17364.0	−0.573880	−0.286940	−	0.957949i	$-0.592638\pi$
$971$	−17364.0	−0.573880	−0.286940	+	0.957949i	$0.592638\pi$
$972$	972.000	0.0320750
$973$	− 5504.00i	− 0.181346i
$974$	−17896.0	−0.588732
$975$	0	0
$976$	−992.000	−0.0325340
$977$	14904.0i	0.488046i	0.969769	+	0.244023i	$0.0784673\pi$
$977$	14904.0i	0.488046i	−0.969769	+	0.244023i	$0.921533\pi$
$978$	−1488.00	−0.0486513
$979$	−61200.0	−1.99792
$980$	54480.0i	1.77582i
$981$	16974.0i	0.552434i
$982$	17440.0i	0.566734i
$983$	− 18038.0i	− 0.585272i	−0.956224	−	0.292636i	$-0.905467\pi$
$983$	− 18038.0i	− 0.585272i	0.956224	−	0.292636i	$-0.0945325\pi$
$984$	−2592.00	−0.0839735
$985$	−98160.0	−3.17527
$986$	− 4920.00i	− 0.158909i
$987$	17088.0	0.551081
$988$	0	0
$989$	7120.00	0.228921
$990$	18000.0i	0.577856i
$991$	46176.0	1.48015	0.740075	−	0.672524i	$-0.234790\pi$
$991$	46176.0	1.48015	0.740075	+	0.672524i	$0.234790\pi$
$992$	1408.00	0.0450646
$993$	− 12540.0i	− 0.400750i
$994$	− 49792.0i	− 1.58884i
$995$	6560.00i	0.209011i
$996$	5544.00i	0.176374i
$997$	55838.0	1.77373	0.886864	−	0.462030i	$-0.152879\pi$
$997$	55838.0	1.77373	0.886864	+	0.462030i	$0.152879\pi$
$998$	−13208.0	−0.418930
$999$	8262.00i	0.261660i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1014.4.b.e.337.1		2
13.5	odd	4		1014.4.a.d.1.1		1
13.8	odd	4		78.4.a.d.1.1	✓	1
13.12	even	2	inner	1014.4.b.e.337.2		2
39.8	even	4		234.4.a.f.1.1		1
52.47	even	4		624.4.a.e.1.1		1
65.34	odd	4		1950.4.a.h.1.1		1
104.21	odd	4		2496.4.a.r.1.1		1
104.99	even	4		2496.4.a.i.1.1		1
156.47	odd	4		1872.4.a.r.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
78.4.a.d.1.1	✓	1	13.8	odd	4
234.4.a.f.1.1		1	39.8	even	4
624.4.a.e.1.1		1	52.47	even	4
1014.4.a.d.1.1		1	13.5	odd	4
1014.4.b.e.337.1		2	1.1	even	1	trivial
1014.4.b.e.337.2		2	13.12	even	2	inner
1872.4.a.r.1.1		1	156.47	odd	4
1950.4.a.h.1.1		1	65.34	odd	4
2496.4.a.i.1.1		1	104.99	even	4
2496.4.a.r.1.1		1	104.21	odd	4

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:	\( N \)	\(=\)	\( 1014 = 2 \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1014.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{2} -3.00000 q^{3} -4.00000 q^{4} +20.0000i q^{5} +6.00000i q^{6} -32.0000i q^{7} +8.00000i q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-2.00000i q^{2} -3.00000 q^{3} -4.00000 q^{4} +20.0000i q^{5} +6.00000i q^{6} -32.0000i q^{7} +8.00000i q^{8} +9.00000 q^{9} +40.0000 q^{10} +50.0000i q^{11} +12.0000 q^{12} -64.0000 q^{14} -60.0000i q^{15} +16.0000 q^{16} +30.0000 q^{17} -18.0000i q^{18} +120.000i q^{19} -80.0000i q^{20} +96.0000i q^{21} +100.000 q^{22} +20.0000 q^{23} -24.0000i q^{24} -275.000 q^{25} -27.0000 q^{27} +128.000i q^{28} +82.0000 q^{29} -120.000 q^{30} +44.0000i q^{31} -32.0000i q^{32} -150.000i q^{33} -60.0000i q^{34} +640.000 q^{35} -36.0000 q^{36} -306.000i q^{37} +240.000 q^{38} -160.000 q^{40} -108.000i q^{41} +192.000 q^{42} +356.000 q^{43} -200.000i q^{44} +180.000i q^{45} -40.0000i q^{46} -178.000i q^{47} -48.0000 q^{48} -681.000 q^{49} +550.000i q^{50} -90.0000 q^{51} +198.000 q^{53} +54.0000i q^{54} -1000.00 q^{55} +256.000 q^{56} -360.000i q^{57} -164.000i q^{58} +94.0000i q^{59} +240.000i q^{60} -62.0000 q^{61} +88.0000 q^{62} -288.000i q^{63} -64.0000 q^{64} -300.000 q^{66} +140.000i q^{67} -120.000 q^{68} -60.0000 q^{69} -1280.00i q^{70} +778.000i q^{71} +72.0000i q^{72} +62.0000i q^{73} -612.000 q^{74} +825.000 q^{75} -480.000i q^{76} +1600.00 q^{77} -1096.00 q^{79} +320.000i q^{80} +81.0000 q^{81} -216.000 q^{82} +462.000i q^{83} -384.000i q^{84} +600.000i q^{85} -712.000i q^{86} -246.000 q^{87} -400.000 q^{88} +1224.00i q^{89} +360.000 q^{90} -80.0000 q^{92} -132.000i q^{93} -356.000 q^{94} -2400.00 q^{95} +96.0000i q^{96} -614.000i q^{97} +1362.00i q^{98} +450.000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} - 8 q^{4} + 18 q^{9}+O(q^{10}) \) 2 * q - 6 * q^3 - 8 * q^4 + 18 * q^9	\( 2 q - 6 q^{3} - 8 q^{4} + 18 q^{9} + 80 q^{10} + 24 q^{12} - 128 q^{14} + 32 q^{16} + 60 q^{17} + 200 q^{22} + 40 q^{23} - 550 q^{25} - 54 q^{27} + 164 q^{29} - 240 q^{30} + 1280 q^{35} - 72 q^{36} + 480 q^{38} - 320 q^{40} + 384 q^{42} + 712 q^{43} - 96 q^{48} - 1362 q^{49} - 180 q^{51} + 396 q^{53} - 2000 q^{55} + 512 q^{56} - 124 q^{61} + 176 q^{62} - 128 q^{64} - 600 q^{66} - 240 q^{68} - 120 q^{69} - 1224 q^{74} + 1650 q^{75} + 3200 q^{77} - 2192 q^{79} + 162 q^{81} - 432 q^{82} - 492 q^{87} - 800 q^{88} + 720 q^{90} - 160 q^{92} - 712 q^{94} - 4800 q^{95}+O(q^{100}) \) 2 * q - 6 * q^3 - 8 * q^4 + 18 * q^9 + 80 * q^10 + 24 * q^12 - 128 * q^14 + 32 * q^16 + 60 * q^17 + 200 * q^22 + 40 * q^23 - 550 * q^25 - 54 * q^27 + 164 * q^29 - 240 * q^30 + 1280 * q^35 - 72 * q^36 + 480 * q^38 - 320 * q^40 + 384 * q^42 + 712 * q^43 - 96 * q^48 - 1362 * q^49 - 180 * q^51 + 396 * q^53 - 2000 * q^55 + 512 * q^56 - 124 * q^61 + 176 * q^62 - 128 * q^64 - 600 * q^66 - 240 * q^68 - 120 * q^69 - 1224 * q^74 + 1650 * q^75 + 3200 * q^77 - 2192 * q^79 + 162 * q^81 - 432 * q^82 - 492 * q^87 - 800 * q^88 + 720 * q^90 - 160 * q^92 - 712 * q^94 - 4800 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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