LMFDB - Embedded newform 1617.2.a.x.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1617,2,Mod(1,1617)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1617, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1617.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1617 = 3 \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1617.a (trivial)

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:	yes
Analytic conductor:	$12.9118100068$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.11344.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 4x + 3 $ x^4 - 2x^3 - 4x^2 + 4*x + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 231)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.552409$ of defining polynomial
Character	$\chi$	$=$	1617.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.552409 q^{2} -1.00000 q^{3} -1.69484 q^{4} -1.59002 q^{5} +0.552409 q^{6} +2.04107 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.552409 q^{2} -1.00000 q^{3} -1.69484 q^{4} -1.59002 q^{5} +0.552409 q^{6} +2.04107 q^{8} +1.00000 q^{9} +0.878345 q^{10} -1.00000 q^{11} +1.69484 q^{12} -2.87834 q^{13} +1.59002 q^{15} +2.26218 q^{16} -4.83728 q^{17} -0.552409 q^{18} +1.14589 q^{19} +2.69484 q^{20} +0.552409 q^{22} -3.65187 q^{23} -2.04107 q^{24} -2.47182 q^{25} +1.59002 q^{26} -1.00000 q^{27} -0.325935 q^{29} -0.878345 q^{30} -6.45153 q^{31} -5.33178 q^{32} +1.00000 q^{33} +2.67216 q^{34} -1.69484 q^{36} +1.69484 q^{37} -0.632998 q^{38} +2.87834 q^{39} -3.24535 q^{40} +4.05839 q^{41} +4.62764 q^{43} +1.69484 q^{44} -1.59002 q^{45} +2.01733 q^{46} -0.305156 q^{47} -2.26218 q^{48} +1.36546 q^{50} +4.83728 q^{51} +4.87834 q^{52} +5.71372 q^{53} +0.552409 q^{54} +1.59002 q^{55} -1.14589 q^{57} +0.180050 q^{58} -11.8615 q^{59} -2.69484 q^{60} -1.77353 q^{61} +3.56389 q^{62} -1.57904 q^{64} +4.57664 q^{65} -0.552409 q^{66} +15.2715 q^{67} +8.19843 q^{68} +3.65187 q^{69} +9.16666 q^{71} +2.04107 q^{72} +11.7330 q^{73} -0.936248 q^{74} +2.47182 q^{75} -1.94210 q^{76} -1.59002 q^{78} -8.71027 q^{79} -3.59693 q^{80} +1.00000 q^{81} -2.24190 q^{82} -8.40856 q^{83} +7.69139 q^{85} -2.55635 q^{86} +0.325935 q^{87} -2.04107 q^{88} +5.74979 q^{89} +0.878345 q^{90} +6.18935 q^{92} +6.45153 q^{93} +0.168571 q^{94} -1.82199 q^{95} +5.33178 q^{96} +3.65329 q^{97} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} - 4 q^{3} + 4 q^{4} - 4 q^{5} - 2 q^{6} + 12 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q + 2 * q^2 - 4 * q^3 + 4 * q^4 - 4 * q^5 - 2 * q^6 + 12 * q^8 + 4 * q^9	$ 4 q + 2 q^{2} - 4 q^{3} + 4 q^{4} - 4 q^{5} - 2 q^{6} + 12 q^{8} + 4 q^{9} - 10 q^{10} - 4 q^{11} - 4 q^{12} + 2 q^{13} + 4 q^{15} + 12 q^{16} - 2 q^{17} + 2 q^{18} - 2 q^{22} + 4 q^{23} - 12 q^{24} + 4 q^{25} + 4 q^{26} - 4 q^{27} + 8 q^{29} + 10 q^{30} + 12 q^{31} + 26 q^{32} + 4 q^{33} + 16 q^{34} + 4 q^{36} - 4 q^{37} - 8 q^{38} - 2 q^{39} + 6 q^{40} - 2 q^{41} + 18 q^{43} - 4 q^{44} - 4 q^{45} - 14 q^{46} - 12 q^{47} - 12 q^{48} + 2 q^{50} + 2 q^{51} + 6 q^{52} - 12 q^{53} - 2 q^{54} + 4 q^{55} - 4 q^{58} - 12 q^{59} - 2 q^{61} + 26 q^{62} + 56 q^{64} - 4 q^{65} + 2 q^{66} + 28 q^{67} + 48 q^{68} - 4 q^{69} + 12 q^{71} + 12 q^{72} - 6 q^{73} - 16 q^{74} - 4 q^{75} + 18 q^{76} - 4 q^{78} + 2 q^{79} - 16 q^{80} + 4 q^{81} + 12 q^{82} + 12 q^{83} + 18 q^{85} + 36 q^{86} - 8 q^{87} - 12 q^{88} - 8 q^{89} - 10 q^{90} - 16 q^{92} - 12 q^{93} - 20 q^{94} + 34 q^{95} - 26 q^{96} + 44 q^{97} - 4 q^{99}+O(q^{100}) $ 4 * q + 2 * q^2 - 4 * q^3 + 4 * q^4 - 4 * q^5 - 2 * q^6 + 12 * q^8 + 4 * q^9 - 10 * q^10 - 4 * q^11 - 4 * q^12 + 2 * q^13 + 4 * q^15 + 12 * q^16 - 2 * q^17 + 2 * q^18 - 2 * q^22 + 4 * q^23 - 12 * q^24 + 4 * q^25 + 4 * q^26 - 4 * q^27 + 8 * q^29 + 10 * q^30 + 12 * q^31 + 26 * q^32 + 4 * q^33 + 16 * q^34 + 4 * q^36 - 4 * q^37 - 8 * q^38 - 2 * q^39 + 6 * q^40 - 2 * q^41 + 18 * q^43 - 4 * q^44 - 4 * q^45 - 14 * q^46 - 12 * q^47 - 12 * q^48 + 2 * q^50 + 2 * q^51 + 6 * q^52 - 12 * q^53 - 2 * q^54 + 4 * q^55 - 4 * q^58 - 12 * q^59 - 2 * q^61 + 26 * q^62 + 56 * q^64 - 4 * q^65 + 2 * q^66 + 28 * q^67 + 48 * q^68 - 4 * q^69 + 12 * q^71 + 12 * q^72 - 6 * q^73 - 16 * q^74 - 4 * q^75 + 18 * q^76 - 4 * q^78 + 2 * q^79 - 16 * q^80 + 4 * q^81 + 12 * q^82 + 12 * q^83 + 18 * q^85 + 36 * q^86 - 8 * q^87 - 12 * q^88 - 8 * q^89 - 10 * q^90 - 16 * q^92 - 12 * q^93 - 20 * q^94 + 34 * q^95 - 26 * q^96 + 44 * q^97 - 4 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	−0.552409	−0.390612	−0.195306	−	0.980742i	$-0.562570\pi$
$2$	−0.552409	−0.390612	−0.195306	+	0.980742i	$0.562570\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.69484	−0.847422
$5$	−1.59002	−0.711081	−0.355540	−	0.934661i	$-0.615703\pi$
$5$	−1.59002	−0.711081	−0.355540	+	0.934661i	$0.615703\pi$
$6$	0.552409	0.225520
$7$	0	0
$7$	0	0
$8$	2.04107	0.721626
$9$	1.00000	0.333333
$10$	0.878345	0.277757
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	1.69484	0.489259
$13$	−2.87834	−0.798309	−0.399155	−	0.916884i	$-0.630696\pi$
$13$	−2.87834	−0.798309	−0.399155	+	0.916884i	$0.630696\pi$
$14$	0	0
$15$	1.59002	0.410543
$16$	2.26218	0.565546
$17$	−4.83728	−1.17321	−0.586606	−	0.809872i	$-0.699536\pi$
$17$	−4.83728	−1.17321	−0.586606	+	0.809872i	$0.699536\pi$
$18$	−0.552409	−0.130204
$19$	1.14589	0.262884	0.131442	−	0.991324i	$-0.458039\pi$
$19$	1.14589	0.262884	0.131442	+	0.991324i	$0.458039\pi$
$20$	2.69484	0.602585
$21$	0	0
$22$	0.552409	0.117774
$23$	−3.65187	−0.761468	−0.380734	−	0.924685i	$-0.624329\pi$
$23$	−3.65187	−0.761468	−0.380734	+	0.924685i	$0.624329\pi$
$24$	−2.04107	−0.416631
$25$	−2.47182	−0.494364
$26$	1.59002	0.311830
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−0.325935	−0.0605247	−0.0302623	−	0.999542i	$-0.509634\pi$
$29$	−0.325935	−0.0605247	−0.0302623	+	0.999542i	$0.509634\pi$
$30$	−0.878345	−0.160363
$31$	−6.45153	−1.15873	−0.579365	−	0.815068i	$-0.696699\pi$
$31$	−6.45153	−1.15873	−0.579365	+	0.815068i	$0.696699\pi$
$32$	−5.33178	−0.942535
$33$	1.00000	0.174078
$34$	2.67216	0.458271
$35$	0	0
$36$	−1.69484	−0.282474
$37$	1.69484	0.278631	0.139315	−	0.990248i	$-0.455510\pi$
$37$	1.69484	0.278631	0.139315	+	0.990248i	$0.455510\pi$
$38$	−0.632998	−0.102686
$39$	2.87834	0.460904
$40$	−3.24535	−0.513134
$41$	4.05839	0.633815	0.316907	−	0.948456i	$-0.397356\pi$
$41$	4.05839	0.633815	0.316907	+	0.948456i	$0.397356\pi$
$42$	0	0
$43$	4.62764	0.705709	0.352854	−	0.935678i	$-0.385211\pi$
$43$	4.62764	0.705709	0.352854	+	0.935678i	$0.385211\pi$
$44$	1.69484	0.255507
$45$	−1.59002	−0.237027
$46$	2.01733	0.297439
$47$	−0.305156	−0.0445116	−0.0222558	−	0.999752i	$-0.507085\pi$
$47$	−0.305156	−0.0445116	−0.0222558	+	0.999752i	$0.507085\pi$
$48$	−2.26218	−0.326518
$49$	0	0
$50$	1.36546	0.193105
$51$	4.83728	0.677354
$52$	4.87834	0.676505
$53$	5.71372	0.784839	0.392420	−	0.919786i	$-0.371638\pi$
$53$	5.71372	0.784839	0.392420	+	0.919786i	$0.371638\pi$
$54$	0.552409	0.0751734
$55$	1.59002	0.214399
$56$	0	0
$57$	−1.14589	−0.151776
$58$	0.180050	0.0236417
$59$	−11.8615	−1.54424	−0.772118	−	0.635479i	$-0.780803\pi$
$59$	−11.8615	−1.54424	−0.772118	+	0.635479i	$0.780803\pi$
$60$	−2.69484	−0.347903
$61$	−1.77353	−0.227077	−0.113538	−	0.993534i	$-0.536218\pi$
$61$	−1.77353	−0.227077	−0.113538	+	0.993534i	$0.536218\pi$
$62$	3.56389	0.452614
$63$	0	0
$64$	−1.57904	−0.197380
$65$	4.57664	0.567662
$66$	−0.552409	−0.0679969
$67$	15.2715	1.86571	0.932854	−	0.360254i	$-0.117310\pi$
$67$	15.2715	1.86571	0.932854	+	0.360254i	$0.117310\pi$
$68$	8.19843	0.994206
$69$	3.65187	0.439634
$70$	0	0
$71$	9.16666	1.08788	0.543941	−	0.839123i	$-0.316931\pi$
$71$	9.16666	1.08788	0.543941	+	0.839123i	$0.316931\pi$
$72$	2.04107	0.240542
$73$	11.7330	1.37324	0.686619	−	0.727017i	$-0.259094\pi$
$73$	11.7330	1.37324	0.686619	+	0.727017i	$0.259094\pi$
$74$	−0.936248	−0.108837
$75$	2.47182	0.285421
$76$	−1.94210	−0.222774
$77$	0	0
$78$	−1.59002	−0.180035
$79$	−8.71027	−0.979981	−0.489991	−	0.871728i	$-0.663000\pi$
$79$	−8.71027	−0.979981	−0.489991	+	0.871728i	$0.663000\pi$
$80$	−3.59693	−0.402149
$81$	1.00000	0.111111
$82$	−2.24190	−0.247576
$83$	−8.40856	−0.922959	−0.461480	−	0.887151i	$-0.652681\pi$
$83$	−8.40856	−0.922959	−0.461480	+	0.887151i	$0.652681\pi$
$84$	0	0
$85$	7.69139	0.834249
$86$	−2.55635	−0.275659
$87$	0.325935	0.0349439
$88$	−2.04107	−0.217578
$89$	5.74979	0.609476	0.304738	−	0.952436i	$-0.401431\pi$
$89$	5.74979	0.609476	0.304738	+	0.952436i	$0.401431\pi$
$90$	0.878345	0.0925857
$91$	0	0
$92$	6.18935	0.645284
$93$	6.45153	0.668993
$94$	0.168571	0.0173868
$95$	−1.82199	−0.186932
$96$	5.33178	0.544173
$97$	3.65329	0.370935	0.185467	−	0.982650i	$-0.440620\pi$
$97$	3.65329	0.370935	0.185467	+	0.982650i	$0.440620\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	4.18935	0.418935
$101$	−0.284377	−0.0282966	−0.0141483	−	0.999900i	$-0.504504\pi$
$101$	−0.284377	−0.0282966	−0.0141483	+	0.999900i	$0.504504\pi$
$102$	−2.67216	−0.264583
$103$	16.6934	1.64485	0.822426	−	0.568872i	$-0.192620\pi$
$103$	16.6934	1.64485	0.822426	+	0.568872i	$0.192620\pi$
$104$	−5.87489	−0.576081
$105$	0	0
$106$	−3.15631	−0.306568
$107$	17.9495	1.73524	0.867621	−	0.497225i	$-0.165648\pi$
$107$	17.9495	1.73524	0.867621	+	0.497225i	$0.165648\pi$
$108$	1.69484	0.163086
$109$	11.0880	1.06204	0.531018	−	0.847361i	$-0.321810\pi$
$109$	11.0880	1.06204	0.531018	+	0.847361i	$0.321810\pi$
$110$	−0.878345	−0.0837469
$111$	−1.69484	−0.160867
$112$	0	0
$113$	1.72443	0.162221	0.0811105	−	0.996705i	$-0.474153\pi$
$113$	1.72443	0.162221	0.0811105	+	0.996705i	$0.474153\pi$
$114$	0.632998	0.0592857
$115$	5.80657	0.541465
$116$	0.552409	0.0512899
$117$	−2.87834	−0.266103
$118$	6.55241	0.603198
$119$	0	0
$120$	3.24535	0.296258
$121$	1.00000	0.0909091
$122$	0.979712	0.0886990
$123$	−4.05839	−0.365933
$124$	10.9343	0.981933
$125$	11.8804	1.06261
$126$	0	0
$127$	20.3675	1.80732	0.903661	−	0.428248i	$-0.140869\pi$
$127$	20.3675	1.80732	0.903661	+	0.428248i	$0.140869\pi$
$128$	11.5358	1.01963
$129$	−4.62764	−0.407441
$130$	−2.52818	−0.221736
$131$	−20.1821	−1.76332	−0.881659	−	0.471888i	$-0.843573\pi$
$131$	−20.1821	−1.76332	−0.881659	+	0.471888i	$0.843573\pi$
$132$	−1.69484	−0.147517
$133$	0	0
$134$	−8.43611	−0.728769
$135$	1.59002	0.136848
$136$	−9.87321	−0.846621
$137$	−21.2632	−1.81663	−0.908317	−	0.418282i	$-0.862633\pi$
$137$	−21.2632	−1.81663	−0.908317	+	0.418282i	$0.862633\pi$
$138$	−2.01733	−0.171726
$139$	−5.27951	−0.447802	−0.223901	−	0.974612i	$-0.571879\pi$
$139$	−5.27951	−0.447802	−0.223901	+	0.974612i	$0.571879\pi$
$140$	0	0
$141$	0.305156	0.0256988
$142$	−5.06375	−0.424941
$143$	2.87834	0.240699
$144$	2.26218	0.188515
$145$	0.518245	0.0430379
$146$	−6.48139	−0.536404
$147$	0	0
$148$	−2.87250	−0.236118
$149$	1.36546	0.111863	0.0559313	−	0.998435i	$-0.482187\pi$
$149$	1.36546	0.111863	0.0559313	+	0.998435i	$0.482187\pi$
$150$	−1.36546	−0.111489
$151$	15.9490	1.29791	0.648956	−	0.760826i	$-0.275206\pi$
$151$	15.9490	1.29791	0.648956	+	0.760826i	$0.275206\pi$
$152$	2.33883	0.189704
$153$	−4.83728	−0.391071
$154$	0	0
$155$	10.2581	0.823950
$156$	−4.87834	−0.390580
$157$	8.39659	0.670121	0.335060	−	0.942197i	$-0.391243\pi$
$157$	8.39659	0.670121	0.335060	+	0.942197i	$0.391243\pi$
$158$	4.81163	0.382793
$159$	−5.71372	−0.453127
$160$	8.47767	0.670219
$161$	0	0
$162$	−0.552409	−0.0434014
$163$	−17.3130	−1.35606	−0.678031	−	0.735033i	$-0.737167\pi$
$163$	−17.3130	−1.35606	−0.678031	+	0.735033i	$0.737167\pi$
$164$	−6.87834	−0.537108
$165$	−1.59002	−0.123783
$166$	4.64497	0.360519
$167$	−18.4945	−1.43115	−0.715574	−	0.698537i	$-0.753835\pi$
$167$	−18.4945	−1.43115	−0.715574	+	0.698537i	$0.753835\pi$
$168$	0	0
$169$	−4.71513	−0.362702
$170$	−4.24880	−0.325868
$171$	1.14589	0.0876281
$172$	−7.84313	−0.598033
$173$	−12.2680	−0.932721	−0.466361	−	0.884595i	$-0.654435\pi$
$173$	−12.2680	−0.932721	−0.466361	+	0.884595i	$0.654435\pi$
$174$	−0.180050	−0.0136495
$175$	0	0
$176$	−2.26218	−0.170518
$177$	11.8615	0.891566
$178$	−3.17624	−0.238069
$179$	13.8116	1.03233	0.516165	−	0.856489i	$-0.327359\pi$
$179$	13.8116	1.03233	0.516165	+	0.856489i	$0.327359\pi$
$180$	2.69484	0.200862
$181$	18.7687	1.39506	0.697532	−	0.716554i	$-0.254282\pi$
$181$	18.7687	1.39506	0.697532	+	0.716554i	$0.254282\pi$
$182$	0	0
$183$	1.77353	0.131103
$184$	−7.45371	−0.549495
$185$	−2.69484	−0.198129
$186$	−3.56389	−0.261317
$187$	4.83728	0.353737
$188$	0.517192	0.0377201
$189$	0	0
$190$	1.00648	0.0730179
$191$	8.11679	0.587310	0.293655	−	0.955911i	$-0.405128\pi$
$191$	8.11679	0.587310	0.293655	+	0.955911i	$0.405128\pi$
$192$	1.57904	0.113957
$193$	1.19077	0.0857132	0.0428566	−	0.999081i	$-0.486354\pi$
$193$	1.19077	0.0857132	0.0428566	+	0.999081i	$0.486354\pi$
$194$	−2.01811	−0.144892
$195$	−4.57664	−0.327740
$196$	0	0
$197$	−26.7523	−1.90602	−0.953012	−	0.302933i	$-0.902034\pi$
$197$	−26.7523	−1.90602	−0.953012	+	0.302933i	$0.902034\pi$
$198$	0.552409	0.0392580
$199$	6.03367	0.427716	0.213858	−	0.976865i	$-0.431397\pi$
$199$	6.03367	0.427716	0.213858	+	0.976865i	$0.431397\pi$
$200$	−5.04515	−0.356746
$201$	−15.2715	−1.07717
$202$	0.157093	0.0110530
$203$	0	0
$204$	−8.19843	−0.574005
$205$	−6.45295	−0.450693
$206$	−9.22161	−0.642500
$207$	−3.65187	−0.253823
$208$	−6.51134	−0.451480
$209$	−1.14589	−0.0792626
$210$	0	0
$211$	−20.4601	−1.40853	−0.704264	−	0.709938i	$-0.748723\pi$
$211$	−20.4601	−1.40853	−0.704264	+	0.709938i	$0.748723\pi$
$212$	−9.68386	−0.665090
$213$	−9.16666	−0.628090
$214$	−9.91547	−0.677807
$215$	−7.35806	−0.501816
$216$	−2.04107	−0.138877
$217$	0	0
$218$	−6.12511	−0.414845
$219$	−11.7330	−0.792839
$220$	−2.69484	−0.181686
$221$	13.9234	0.936586
$222$	0.936248	0.0628368
$223$	29.6048	1.98248	0.991242	−	0.132055i	$-0.0421574\pi$
$223$	29.6048	1.98248	0.991242	+	0.132055i	$0.0421574\pi$
$224$	0	0
$225$	−2.47182	−0.164788
$226$	−0.952592	−0.0633655
$227$	24.0831	1.59845	0.799226	−	0.601030i	$-0.205243\pi$
$227$	24.0831	1.59845	0.799226	+	0.601030i	$0.205243\pi$
$228$	1.94210	0.128619
$229$	5.21471	0.344597	0.172299	−	0.985045i	$-0.444881\pi$
$229$	5.21471	0.344597	0.172299	+	0.985045i	$0.444881\pi$
$230$	−3.20760	−0.211503
$231$	0	0
$232$	−0.665256	−0.0436762
$233$	3.14589	0.206094	0.103047	−	0.994676i	$-0.467141\pi$
$233$	3.14589	0.206094	0.103047	+	0.994676i	$0.467141\pi$
$234$	1.59002	0.103943
$235$	0.485206	0.0316513
$236$	20.1034	1.30862
$237$	8.71027	0.565793
$238$	0	0
$239$	2.75465	0.178184	0.0890919	−	0.996023i	$-0.471604\pi$
$239$	2.75465	0.178184	0.0890919	+	0.996023i	$0.471604\pi$
$240$	3.59693	0.232181
$241$	9.02959	0.581647	0.290823	−	0.956777i	$-0.406071\pi$
$241$	9.02959	0.581647	0.290823	+	0.956777i	$0.406071\pi$
$242$	−0.552409	−0.0355102
$243$	−1.00000	−0.0641500
$244$	3.00585	0.192430
$245$	0	0
$246$	2.24190	0.142938
$247$	−3.29825	−0.209863
$248$	−13.1680	−0.836169
$249$	8.40856	0.532871
$250$	−6.56283	−0.415070
$251$	6.73133	0.424878	0.212439	−	0.977174i	$-0.431859\pi$
$251$	6.73133	0.424878	0.212439	+	0.977174i	$0.431859\pi$
$252$	0	0
$253$	3.65187	0.229591
$254$	−11.2512	−0.705963
$255$	−7.69139	−0.481654
$256$	−3.21443	−0.200902
$257$	−27.6893	−1.72721	−0.863607	−	0.504166i	$-0.831800\pi$
$257$	−27.6893	−1.72721	−0.863607	+	0.504166i	$0.831800\pi$
$258$	2.55635	0.159152
$259$	0	0
$260$	−7.75669	−0.481049
$261$	−0.325935	−0.0201749
$262$	11.1488	0.688774
$263$	4.60341	0.283858	0.141929	−	0.989877i	$-0.454669\pi$
$263$	4.60341	0.283858	0.141929	+	0.989877i	$0.454669\pi$
$264$	2.04107	0.125619
$265$	−9.08495	−0.558084
$266$	0	0
$267$	−5.74979	−0.351881
$268$	−25.8828	−1.58104
$269$	−19.1103	−1.16518	−0.582588	−	0.812768i	$-0.697960\pi$
$269$	−19.1103	−1.16518	−0.582588	+	0.812768i	$0.697960\pi$
$270$	−0.878345	−0.0534544
$271$	−15.3729	−0.933834	−0.466917	−	0.884301i	$-0.654635\pi$
$271$	−15.3729	−0.933834	−0.466917	+	0.884301i	$0.654635\pi$
$272$	−10.9428	−0.663505
$273$	0	0
$274$	11.7460	0.709600
$275$	2.47182	0.149056
$276$	−6.18935	−0.372555
$277$	−9.38969	−0.564172	−0.282086	−	0.959389i	$-0.591026\pi$
$277$	−9.38969	−0.564172	−0.282086	+	0.959389i	$0.591026\pi$
$278$	2.91645	0.174917
$279$	−6.45153	−0.386243
$280$	0	0
$281$	26.9788	1.60942	0.804710	−	0.593668i	$-0.202321\pi$
$281$	26.9788	1.60942	0.804710	+	0.593668i	$0.202321\pi$
$282$	−0.168571	−0.0100383
$283$	23.5756	1.40142	0.700712	−	0.713445i	$-0.252866\pi$
$283$	23.5756	1.40142	0.700712	+	0.713445i	$0.252866\pi$
$284$	−15.5361	−0.921896
$285$	1.82199	0.107925
$286$	−1.59002	−0.0940201
$287$	0	0
$288$	−5.33178	−0.314178
$289$	6.39926	0.376427
$290$	−0.286284	−0.0168112
$291$	−3.65329	−0.214159
$292$	−19.8855	−1.16371
$293$	11.8500	0.692286	0.346143	−	0.938182i	$-0.387491\pi$
$293$	11.8500	0.692286	0.346143	+	0.938182i	$0.387491\pi$
$294$	0	0
$295$	18.8601	1.09808
$296$	3.45929	0.201067
$297$	1.00000	0.0580259
$298$	−0.754291	−0.0436949
$299$	10.5113	0.607887
$300$	−4.18935	−0.241872
$301$	0	0
$302$	−8.81038	−0.506980
$303$	0.284377	0.0163370
$304$	2.59220	0.148673
$305$	2.81995	0.161470
$306$	2.67216	0.152757
$307$	−19.5894	−1.11803	−0.559013	−	0.829159i	$-0.688820\pi$
$307$	−19.5894	−1.11803	−0.559013	+	0.829159i	$0.688820\pi$
$308$	0	0
$309$	−16.6934	−0.949656
$310$	−5.66667	−0.321845
$311$	21.4017	1.21358	0.606788	−	0.794863i	$-0.292458\pi$
$311$	21.4017	1.21358	0.606788	+	0.794863i	$0.292458\pi$
$312$	5.87489	0.332600
$313$	13.3278	0.753334	0.376667	−	0.926349i	$-0.377070\pi$
$313$	13.3278	0.753334	0.376667	+	0.926349i	$0.377070\pi$
$314$	−4.63836	−0.261758
$315$	0	0
$316$	14.7625	0.830458
$317$	−7.09552	−0.398524	−0.199262	−	0.979946i	$-0.563854\pi$
$317$	−7.09552	−0.398524	−0.199262	+	0.979946i	$0.563854\pi$
$318$	3.15631	0.176997
$319$	0.325935	0.0182489
$320$	2.51071	0.140353
$321$	−17.9495	−1.00184
$322$	0	0
$323$	−5.54297	−0.308419
$324$	−1.69484	−0.0941580
$325$	7.11475	0.394655
$326$	9.56389	0.529695
$327$	−11.0880	−0.613167
$328$	8.28345	0.457377
$329$	0	0
$330$	0.878345	0.0483513
$331$	20.7350	1.13970	0.569849	−	0.821749i	$-0.307002\pi$
$331$	20.7350	1.13970	0.569849	+	0.821749i	$0.307002\pi$
$332$	14.2512	0.782136
$333$	1.69484	0.0928769
$334$	10.2165	0.559024
$335$	−24.2820	−1.32667
$336$	0	0
$337$	−5.30755	−0.289121	−0.144560	−	0.989496i	$-0.546177\pi$
$337$	−5.30755	−0.289121	−0.144560	+	0.989496i	$0.546177\pi$
$338$	2.60468	0.141676
$339$	−1.72443	−0.0936583
$340$	−13.0357	−0.706961
$341$	6.45153	0.349370
$342$	−0.632998	−0.0342286
$343$	0	0
$344$	9.44532	0.509258
$345$	−5.80657	−0.312615
$346$	6.77698	0.364333
$347$	16.8031	0.902038	0.451019	−	0.892514i	$-0.351061\pi$
$347$	16.8031	0.902038	0.451019	+	0.892514i	$0.351061\pi$
$348$	−0.552409	−0.0296123
$349$	0.870459	0.0465946	0.0232973	−	0.999729i	$-0.492584\pi$
$349$	0.870459	0.0465946	0.0232973	+	0.999729i	$0.492584\pi$
$350$	0	0
$351$	2.87834	0.153635
$352$	5.33178	0.284185
$353$	0.938154	0.0499329	0.0249665	−	0.999688i	$-0.492052\pi$
$353$	0.938154	0.0499329	0.0249665	+	0.999688i	$0.492052\pi$
$354$	−6.55241	−0.348257
$355$	−14.5752	−0.773573
$356$	−9.74499	−0.516483
$357$	0	0
$358$	−7.62968	−0.403241
$359$	−27.0099	−1.42553	−0.712765	−	0.701403i	$-0.752557\pi$
$359$	−27.0099	−1.42553	−0.712765	+	0.701403i	$0.752557\pi$
$360$	−3.24535	−0.171045
$361$	−17.6869	−0.930892
$362$	−10.3680	−0.544929
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	−18.6557	−0.976483
$366$	−0.979712	−0.0512104
$367$	28.4110	1.48304	0.741520	−	0.670931i	$-0.234105\pi$
$367$	28.4110	1.48304	0.741520	+	0.670931i	$0.234105\pi$
$368$	−8.26120	−0.430645
$369$	4.05839	0.211272
$370$	1.48866	0.0773916
$371$	0	0
$372$	−10.9343	−0.566919
$373$	−9.74676	−0.504668	−0.252334	−	0.967640i	$-0.581198\pi$
$373$	−9.74676	−0.504668	−0.252334	+	0.967640i	$0.581198\pi$
$374$	−2.67216	−0.138174
$375$	−11.8804	−0.613500
$376$	−0.622844	−0.0321207
$377$	0.938154	0.0483174
$378$	0	0
$379$	−33.1034	−1.70041	−0.850204	−	0.526454i	$-0.823521\pi$
$379$	−33.1034	−1.70041	−0.850204	+	0.526454i	$0.823521\pi$
$380$	3.08798	0.158410
$381$	−20.3675	−1.04346
$382$	−4.48379	−0.229411
$383$	10.6884	0.546150	0.273075	−	0.961993i	$-0.411959\pi$
$383$	10.6884	0.546150	0.273075	+	0.961993i	$0.411959\pi$
$384$	−11.5358	−0.588686
$385$	0	0
$386$	−0.657790	−0.0334806
$387$	4.62764	0.235236
$388$	−6.19175	−0.314338
$389$	−38.5883	−1.95651	−0.978253	−	0.207414i	$-0.933495\pi$
$389$	−38.5883	−1.95651	−0.978253	+	0.207414i	$0.933495\pi$
$390$	2.52818	0.128019
$391$	17.6651	0.893363
$392$	0	0
$393$	20.1821	1.01805
$394$	14.7782	0.744517
$395$	13.8495	0.696846
$396$	1.69484	0.0851691
$397$	−13.9391	−0.699585	−0.349793	−	0.936827i	$-0.613748\pi$
$397$	−13.9391	−0.699585	−0.349793	+	0.936827i	$0.613748\pi$
$398$	−3.33306	−0.167071
$399$	0	0
$400$	−5.59171	−0.279586
$401$	−13.1371	−0.656034	−0.328017	−	0.944672i	$-0.606380\pi$
$401$	−13.1371	−0.656034	−0.328017	+	0.944672i	$0.606380\pi$
$402$	8.43611	0.420755
$403$	18.5697	0.925025
$404$	0.481975	0.0239791
$405$	−1.59002	−0.0790090
$406$	0	0
$407$	−1.69484	−0.0840103
$408$	9.87321	0.488797
$409$	25.0329	1.23780	0.618898	−	0.785471i	$-0.287579\pi$
$409$	25.0329	1.23780	0.618898	+	0.785471i	$0.287579\pi$
$410$	3.56467	0.176046
$411$	21.2632	1.04883
$412$	−28.2928	−1.39388
$413$	0	0
$414$	2.01733	0.0991463
$415$	13.3698	0.656299
$416$	15.3467	0.752435
$417$	5.27951	0.258539
$418$	0.632998	0.0309609
$419$	−5.04297	−0.246365	−0.123183	−	0.992384i	$-0.539310\pi$
$419$	−5.04297	−0.246365	−0.123183	+	0.992384i	$0.539310\pi$
$420$	0	0
$421$	9.94055	0.484473	0.242236	−	0.970217i	$-0.422119\pi$
$421$	9.94055	0.484473	0.242236	+	0.970217i	$0.422119\pi$
$422$	11.3023	0.550189
$423$	−0.305156	−0.0148372
$424$	11.6621	0.566360
$425$	11.9569	0.579994
$426$	5.06375	0.245340
$427$	0	0
$428$	−30.4216	−1.47048
$429$	−2.87834	−0.138968
$430$	4.06466	0.196015
$431$	13.6876	0.659308	0.329654	−	0.944102i	$-0.393068\pi$
$431$	13.6876	0.659308	0.329654	+	0.944102i	$0.393068\pi$
$432$	−2.26218	−0.108839
$433$	4.26360	0.204895	0.102448	−	0.994738i	$-0.467333\pi$
$433$	4.26360	0.204895	0.102448	+	0.994738i	$0.467333\pi$
$434$	0	0
$435$	−0.518245	−0.0248480
$436$	−18.7924	−0.899993
$437$	−4.18463	−0.200178
$438$	6.48139	0.309693
$439$	23.4032	1.11697	0.558487	−	0.829513i	$-0.311382\pi$
$439$	23.4032	1.11697	0.558487	+	0.829513i	$0.311382\pi$
$440$	3.24535	0.154716
$441$	0	0
$442$	−7.69139	−0.365842
$443$	−14.7192	−0.699330	−0.349665	−	0.936875i	$-0.613705\pi$
$443$	−14.7192	−0.699330	−0.349665	+	0.936875i	$0.613705\pi$
$444$	2.87250	0.136323
$445$	−9.14230	−0.433387
$446$	−16.3540	−0.774383
$447$	−1.36546	−0.0645839
$448$	0	0
$449$	−34.8625	−1.64526	−0.822631	−	0.568575i	$-0.807495\pi$
$449$	−34.8625	−1.64526	−0.822631	+	0.568575i	$0.807495\pi$
$450$	1.36546	0.0643683
$451$	−4.05839	−0.191102
$452$	−2.92264	−0.137470
$453$	−15.9490	−0.749349
$454$	−13.3037	−0.624376
$455$	0	0
$456$	−2.33883	−0.109526
$457$	−10.6014	−0.495911	−0.247955	−	0.968771i	$-0.579759\pi$
$457$	−10.6014	−0.495911	−0.247955	+	0.968771i	$0.579759\pi$
$458$	−2.88065	−0.134604
$459$	4.83728	0.225785
$460$	−9.84122	−0.458849
$461$	6.84418	0.318765	0.159383	−	0.987217i	$-0.449050\pi$
$461$	6.84418	0.318765	0.159383	+	0.987217i	$0.449050\pi$
$462$	0	0
$463$	−11.7299	−0.545136	−0.272568	−	0.962137i	$-0.587873\pi$
$463$	−11.7299	−0.545136	−0.272568	+	0.962137i	$0.587873\pi$
$464$	−0.737325	−0.0342295
$465$	−10.2581	−0.475708
$466$	−1.73782	−0.0805028
$467$	38.2887	1.77179	0.885894	−	0.463887i	$-0.153546\pi$
$467$	38.2887	1.77179	0.885894	+	0.463887i	$0.153546\pi$
$468$	4.87834	0.225502
$469$	0	0
$470$	−0.268032	−0.0123634
$471$	−8.39659	−0.386894
$472$	−24.2101	−1.11436
$473$	−4.62764	−0.212779
$474$	−4.81163	−0.221006
$475$	−2.83242	−0.129961
$476$	0	0
$477$	5.71372	0.261613
$478$	−1.52170	−0.0696008
$479$	−12.6289	−0.577030	−0.288515	−	0.957475i	$-0.593161\pi$
$479$	−12.6289	−0.577030	−0.288515	+	0.957475i	$0.593161\pi$
$480$	−8.47767	−0.386951
$481$	−4.87834	−0.222433
$482$	−4.98803	−0.227199
$483$	0	0
$484$	−1.69484	−0.0770384
$485$	−5.80881	−0.263765
$486$	0.552409	0.0250578
$487$	35.6140	1.61382	0.806911	−	0.590672i	$-0.201137\pi$
$487$	35.6140	1.61382	0.806911	+	0.590672i	$0.201137\pi$
$488$	−3.61988	−0.163864
$489$	17.3130	0.782923
$490$	0	0
$491$	7.59854	0.342917	0.171459	−	0.985191i	$-0.445152\pi$
$491$	7.59854	0.342917	0.171459	+	0.985191i	$0.445152\pi$
$492$	6.87834	0.310100
$493$	1.57664	0.0710083
$494$	1.82199	0.0819751
$495$	1.59002	0.0714663
$496$	−14.5945	−0.655315
$497$	0	0
$498$	−4.64497	−0.208146
$499$	1.60608	0.0718980	0.0359490	−	0.999354i	$-0.488555\pi$
$499$	1.60608	0.0718980	0.0359490	+	0.999354i	$0.488555\pi$
$500$	−20.1354	−0.900482
$501$	18.4945	0.826274
$502$	−3.71845	−0.165963
$503$	38.6291	1.72239	0.861194	−	0.508277i	$-0.169717\pi$
$503$	38.6291	1.72239	0.861194	+	0.508277i	$0.169717\pi$
$504$	0	0
$505$	0.452167	0.0201212
$506$	−2.01733	−0.0896812
$507$	4.71513	0.209406
$508$	−34.5197	−1.53156
$509$	−31.7982	−1.40943	−0.704716	−	0.709489i	$-0.748926\pi$
$509$	−31.7982	−1.40943	−0.704716	+	0.709489i	$0.748926\pi$
$510$	4.24880	0.188140
$511$	0	0
$512$	−21.2960	−0.941159
$513$	−1.14589	−0.0505921
$514$	15.2959	0.674671
$515$	−26.5430	−1.16962
$516$	7.84313	0.345274
$517$	0.305156	0.0134208
$518$	0	0
$519$	12.2680	0.538507
$520$	9.34123	0.409640
$521$	−12.0683	−0.528723	−0.264362	−	0.964424i	$-0.585161\pi$
$521$	−12.0683	−0.528723	−0.264362	+	0.964424i	$0.585161\pi$
$522$	0.180050	0.00788056
$523$	−0.285650	−0.0124906	−0.00624531	−	0.999980i	$-0.501988\pi$
$523$	−0.285650	−0.0124906	−0.00624531	+	0.999980i	$0.501988\pi$
$524$	34.2055	1.49427
$525$	0	0
$526$	−2.54297	−0.110879
$527$	31.2079	1.35944
$528$	2.26218	0.0984489
$529$	−9.66384	−0.420167
$530$	5.01861	0.217995
$531$	−11.8615	−0.514746
$532$	0	0
$533$	−11.6815	−0.505980
$534$	3.17624	0.137449
$535$	−28.5401	−1.23390
$536$	31.1701	1.34634
$537$	−13.8116	−0.596016
$538$	10.5567	0.455132
$539$	0	0
$540$	−2.69484	−0.115968
$541$	2.81002	0.120812	0.0604060	−	0.998174i	$-0.480760\pi$
$541$	2.81002	0.120812	0.0604060	+	0.998174i	$0.480760\pi$
$542$	8.49211	0.364767
$543$	−18.7687	−0.805440
$544$	25.7913	1.10579
$545$	−17.6302	−0.755193
$546$	0	0
$547$	−20.8781	−0.892681	−0.446341	−	0.894863i	$-0.647273\pi$
$547$	−20.8781	−0.892681	−0.446341	+	0.894863i	$0.647273\pi$
$548$	36.0377	1.53946
$549$	−1.77353	−0.0756922
$550$	−1.36546	−0.0582233
$551$	−0.373485	−0.0159110
$552$	7.45371	0.317251
$553$	0	0
$554$	5.18695	0.220372
$555$	2.69484	0.114390
$556$	8.94795	0.379477
$557$	18.2754	0.774355	0.387177	−	0.922005i	$-0.373450\pi$
$557$	18.2754	0.774355	0.387177	+	0.922005i	$0.373450\pi$
$558$	3.56389	0.150871
$559$	−13.3199	−0.563374
$560$	0	0
$561$	−4.83728	−0.204230
$562$	−14.9033	−0.628659
$563$	−16.2989	−0.686916	−0.343458	−	0.939168i	$-0.611598\pi$
$563$	−16.2989	−0.686916	−0.343458	+	0.939168i	$0.611598\pi$
$564$	−0.517192	−0.0217777
$565$	−2.74189	−0.115352
$566$	−13.0234	−0.547413
$567$	0	0
$568$	18.7098	0.785045
$569$	16.7444	0.701963	0.350981	−	0.936382i	$-0.385848\pi$
$569$	16.7444	0.701963	0.350981	+	0.936382i	$0.385848\pi$
$570$	−1.00648	−0.0421569
$571$	24.2923	1.01660	0.508300	−	0.861180i	$-0.330274\pi$
$571$	24.2923	1.01660	0.508300	+	0.861180i	$0.330274\pi$
$572$	−4.87834	−0.203974
$573$	−8.11679	−0.339084
$574$	0	0
$575$	9.02677	0.376442
$576$	−1.57904	−0.0657932
$577$	−22.3173	−0.929080	−0.464540	−	0.885552i	$-0.653780\pi$
$577$	−22.3173	−0.929080	−0.464540	+	0.885552i	$0.653780\pi$
$578$	−3.53501	−0.147037
$579$	−1.19077	−0.0494865
$580$	−0.878345	−0.0364713
$581$	0	0
$582$	2.01811	0.0836533
$583$	−5.71372	−0.236638
$584$	23.9477	0.990964
$585$	4.57664	0.189221
$586$	−6.54607	−0.270416
$587$	26.6944	1.10180	0.550898	−	0.834572i	$-0.314285\pi$
$587$	26.6944	1.10180	0.550898	+	0.834572i	$0.314285\pi$
$588$	0	0
$589$	−7.39272	−0.304612
$590$	−10.4185	−0.428923
$591$	26.7523	1.10044
$592$	3.83405	0.157578
$593$	22.9441	0.942203	0.471101	−	0.882079i	$-0.343857\pi$
$593$	22.9441	0.942203	0.471101	+	0.882079i	$0.343857\pi$
$594$	−0.552409	−0.0226656
$595$	0	0
$596$	−2.31424	−0.0947948
$597$	−6.03367	−0.246942
$598$	−5.80657	−0.237448
$599$	−20.7041	−0.845948	−0.422974	−	0.906142i	$-0.639014\pi$
$599$	−20.7041	−0.845948	−0.422974	+	0.906142i	$0.639014\pi$
$600$	5.04515	0.205967
$601$	11.2354	0.458302	0.229151	−	0.973391i	$-0.426405\pi$
$601$	11.2354	0.458302	0.229151	+	0.973391i	$0.426405\pi$
$602$	0	0
$603$	15.2715	0.621903
$604$	−27.0311	−1.09988
$605$	−1.59002	−0.0646437
$606$	−0.157093	−0.00638145
$607$	7.45400	0.302549	0.151274	−	0.988492i	$-0.451662\pi$
$607$	7.45400	0.302549	0.151274	+	0.988492i	$0.451662\pi$
$608$	−6.10962	−0.247778
$609$	0	0
$610$	−1.55777	−0.0630721
$611$	0.878345	0.0355340
$612$	8.19843	0.331402
$613$	21.1879	0.855773	0.427886	−	0.903833i	$-0.359258\pi$
$613$	21.1879	0.855773	0.427886	+	0.903833i	$0.359258\pi$
$614$	10.8214	0.436715
$615$	6.45295	0.260208
$616$	0	0
$617$	39.7670	1.60096	0.800479	−	0.599361i	$-0.204578\pi$
$617$	39.7670	1.60096	0.800479	+	0.599361i	$0.204578\pi$
$618$	9.22161	0.370947
$619$	−17.2954	−0.695162	−0.347581	−	0.937650i	$-0.612997\pi$
$619$	−17.2954	−0.695162	−0.347581	+	0.937650i	$0.612997\pi$
$620$	−17.3859	−0.698234
$621$	3.65187	0.146545
$622$	−11.8225	−0.474038
$623$	0	0
$624$	6.51134	0.260662
$625$	−6.53100	−0.261240
$626$	−7.36243	−0.294262
$627$	1.14589	0.0457623
$628$	−14.2309	−0.567875
$629$	−8.19843	−0.326893
$630$	0	0
$631$	−22.8639	−0.910198	−0.455099	−	0.890441i	$-0.650396\pi$
$631$	−22.8639	−0.910198	−0.455099	+	0.890441i	$0.650396\pi$
$632$	−17.7782	−0.707180
$633$	20.4601	0.813214
$634$	3.91963	0.155668
$635$	−32.3848	−1.28515
$636$	9.68386	0.383990
$637$	0	0
$638$	−0.180050	−0.00712824
$639$	9.16666	0.362628
$640$	−18.3423	−0.725042
$641$	−6.72569	−0.265649	−0.132824	−	0.991140i	$-0.542405\pi$
$641$	−6.72569	−0.265649	−0.132824	+	0.991140i	$0.542405\pi$
$642$	9.91547	0.391332
$643$	−8.16118	−0.321845	−0.160923	−	0.986967i	$-0.551447\pi$
$643$	−8.16118	−0.321845	−0.160923	+	0.986967i	$0.551447\pi$
$644$	0	0
$645$	7.35806	0.289723
$646$	3.06199	0.120472
$647$	19.6638	0.773065	0.386533	−	0.922276i	$-0.373673\pi$
$647$	19.6638	0.773065	0.386533	+	0.922276i	$0.373673\pi$
$648$	2.04107	0.0801807
$649$	11.8615	0.465605
$650$	−3.93026	−0.154157
$651$	0	0
$652$	29.3429	1.14916
$653$	−22.3035	−0.872802	−0.436401	−	0.899752i	$-0.643747\pi$
$653$	−22.3035	−0.872802	−0.436401	+	0.899752i	$0.643747\pi$
$654$	6.12511	0.239511
$655$	32.0900	1.25386
$656$	9.18083	0.358451
$657$	11.7330	0.457746
$658$	0	0
$659$	19.0497	0.742072	0.371036	−	0.928619i	$-0.379003\pi$
$659$	19.0497	0.742072	0.371036	+	0.928619i	$0.379003\pi$
$660$	2.69484	0.104897
$661$	−24.3223	−0.946029	−0.473015	−	0.881055i	$-0.656834\pi$
$661$	−24.3223	−0.946029	−0.473015	+	0.881055i	$0.656834\pi$
$662$	−11.4542	−0.445180
$663$	−13.9234	−0.540738
$664$	−17.1624	−0.666032
$665$	0	0
$666$	−0.936248	−0.0362789
$667$	1.19027	0.0460876
$668$	31.3453	1.21279
$669$	−29.6048	−1.14459
$670$	13.4136	0.518214
$671$	1.77353	0.0684662
$672$	0	0
$673$	42.6054	1.64232	0.821159	−	0.570699i	$-0.193328\pi$
$673$	42.6054	1.64232	0.821159	+	0.570699i	$0.193328\pi$
$674$	2.93194	0.112934
$675$	2.47182	0.0951404
$676$	7.99141	0.307362
$677$	13.9660	0.536758	0.268379	−	0.963313i	$-0.413512\pi$
$677$	13.9660	0.536758	0.268379	+	0.963313i	$0.413512\pi$
$678$	0.952592	0.0365841
$679$	0	0
$680$	15.6986	0.602016
$681$	−24.0831	−0.922867
$682$	−3.56389	−0.136468
$683$	−5.28345	−0.202166	−0.101083	−	0.994878i	$-0.532231\pi$
$683$	−5.28345	−0.202166	−0.101083	+	0.994878i	$0.532231\pi$
$684$	−1.94210	−0.0742579
$685$	33.8090	1.29177
$686$	0	0
$687$	−5.21471	−0.198953
$688$	10.4686	0.399110
$689$	−16.4460	−0.626544
$690$	3.20760	0.122111
$691$	−28.1316	−1.07018	−0.535088	−	0.844796i	$-0.679721\pi$
$691$	−28.1316	−1.07018	−0.535088	+	0.844796i	$0.679721\pi$
$692$	20.7924	0.790408
$693$	0	0
$694$	−9.28220	−0.352347
$695$	8.39455	0.318424
$696$	0.665256	0.0252165
$697$	−19.6316	−0.743599
$698$	−0.480850	−0.0182004
$699$	−3.14589	−0.118988
$700$	0	0
$701$	29.9421	1.13090	0.565449	−	0.824783i	$-0.308703\pi$
$701$	29.9421	1.13090	0.565449	+	0.824783i	$0.308703\pi$
$702$	−1.59002	−0.0600116
$703$	1.94210	0.0732476
$704$	1.57904	0.0595122
$705$	−0.485206	−0.0182739
$706$	−0.518245	−0.0195044
$707$	0	0
$708$	−20.1034	−0.755532
$709$	−16.7068	−0.627438	−0.313719	−	0.949516i	$-0.601575\pi$
$709$	−16.7068	−0.627438	−0.313719	+	0.949516i	$0.601575\pi$
$710$	8.05149	0.302167
$711$	−8.71027	−0.326660
$712$	11.7357	0.439814
$713$	23.5602	0.882335
$714$	0	0
$715$	−4.57664	−0.171157
$716$	−23.4086	−0.874819
$717$	−2.75465	−0.102874
$718$	14.9205	0.556829
$719$	−23.3495	−0.870791	−0.435395	−	0.900239i	$-0.643391\pi$
$719$	−23.3495	−0.870791	−0.435395	+	0.900239i	$0.643391\pi$
$720$	−3.59693	−0.134050
$721$	0	0
$722$	9.77044	0.363618
$723$	−9.02959	−0.335814
$724$	−31.8099	−1.18221
$725$	0.805654	0.0299212
$726$	0.552409	0.0205018
$727$	−14.4262	−0.535037	−0.267519	−	0.963553i	$-0.586204\pi$
$727$	−14.4262	−0.535037	−0.267519	+	0.963553i	$0.586204\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	10.3056	0.381426
$731$	−22.3852	−0.827946
$732$	−3.00585	−0.111099
$733$	−26.9467	−0.995298	−0.497649	−	0.867378i	$-0.665803\pi$
$733$	−26.9467	−0.995298	−0.497649	+	0.867378i	$0.665803\pi$
$734$	−15.6945	−0.579294
$735$	0	0
$736$	19.4710	0.717710
$737$	−15.2715	−0.562532
$738$	−2.24190	−0.0825253
$739$	−44.5715	−1.63959	−0.819795	−	0.572658i	$-0.805912\pi$
$739$	−44.5715	−1.63959	−0.819795	+	0.572658i	$0.805912\pi$
$740$	4.56734	0.167899
$741$	3.29825	0.121164
$742$	0	0
$743$	44.7041	1.64004	0.820018	−	0.572338i	$-0.193963\pi$
$743$	44.7041	1.64004	0.820018	+	0.572338i	$0.193963\pi$
$744$	13.1680	0.482763
$745$	−2.17111	−0.0795434
$746$	5.38420	0.197130
$747$	−8.40856	−0.307653
$748$	−8.19843	−0.299764
$749$	0	0
$750$	6.56283	0.239641
$751$	44.5924	1.62720	0.813600	−	0.581425i	$-0.197505\pi$
$751$	44.5924	1.62720	0.813600	+	0.581425i	$0.197505\pi$
$752$	−0.690319	−0.0251734
$753$	−6.73133	−0.245303
$754$	−0.518245	−0.0188734
$755$	−25.3593	−0.922920
$756$	0	0
$757$	5.85842	0.212928	0.106464	−	0.994317i	$-0.466047\pi$
$757$	5.85842	0.212928	0.106464	+	0.994317i	$0.466047\pi$
$758$	18.2866	0.664200
$759$	−3.65187	−0.132555
$760$	−3.71880	−0.134895
$761$	38.4034	1.39212	0.696061	−	0.717982i	$-0.254934\pi$
$761$	38.4034	1.39212	0.696061	+	0.717982i	$0.254934\pi$
$762$	11.2512	0.407588
$763$	0	0
$764$	−13.7567	−0.497700
$765$	7.69139	0.278083
$766$	−5.90435	−0.213333
$767$	34.1415	1.23278
$768$	3.21443	0.115991
$769$	1.44505	0.0521099	0.0260549	−	0.999661i	$-0.491706\pi$
$769$	1.44505	0.0521099	0.0260549	+	0.999661i	$0.491706\pi$
$770$	0	0
$771$	27.6893	0.997207
$772$	−2.01816	−0.0726352
$773$	2.11679	0.0761356	0.0380678	−	0.999275i	$-0.487880\pi$
$773$	2.11679	0.0761356	0.0380678	+	0.999275i	$0.487880\pi$
$774$	−2.55635	−0.0918862
$775$	15.9470	0.572834
$776$	7.45660	0.267676
$777$	0	0
$778$	21.3166	0.764236
$779$	4.65046	0.166620
$780$	7.75669	0.277734
$781$	−9.16666	−0.328009
$782$	−9.75838	−0.348959
$783$	0.325935	0.0116480
$784$	0	0
$785$	−13.3508	−0.476510
$786$	−11.1488	−0.397664
$787$	15.2864	0.544902	0.272451	−	0.962170i	$-0.412166\pi$
$787$	15.2864	0.544902	0.272451	+	0.962170i	$0.412166\pi$
$788$	45.3410	1.61521
$789$	−4.60341	−0.163886
$790$	−7.65062	−0.272197
$791$	0	0
$792$	−2.04107	−0.0725261
$793$	5.10482	0.181277
$794$	7.70011	0.273267
$795$	9.08495	0.322210
$796$	−10.2261	−0.362456
$797$	−15.1839	−0.537840	−0.268920	−	0.963163i	$-0.586667\pi$
$797$	−15.1839	−0.537840	−0.268920	+	0.963163i	$0.586667\pi$
$798$	0	0
$799$	1.47613	0.0522216
$800$	13.1792	0.465956
$801$	5.74979	0.203159
$802$	7.25705	0.256255
$803$	−11.7330	−0.414047
$804$	25.8828	0.912815
$805$	0	0
$806$	−10.2581	−0.361326
$807$	19.1103	0.672715
$808$	−0.580433	−0.0204195
$809$	−11.8893	−0.418007	−0.209003	−	0.977915i	$-0.567022\pi$
$809$	−11.8893	−0.418007	−0.209003	+	0.977915i	$0.567022\pi$
$810$	0.878345	0.0308619
$811$	3.83804	0.134772	0.0673859	−	0.997727i	$-0.478534\pi$
$811$	3.83804	0.134772	0.0673859	+	0.997727i	$0.478534\pi$
$812$	0	0
$813$	15.3729	0.539149
$814$	0.936248	0.0328155
$815$	27.5282	0.964270
$816$	10.9428	0.383075
$817$	5.30275	0.185520
$818$	−13.8284	−0.483499
$819$	0	0
$820$	10.9367	0.381928
$821$	31.7745	1.10894	0.554469	−	0.832204i	$-0.312921\pi$
$821$	31.7745	1.10894	0.554469	+	0.832204i	$0.312921\pi$
$822$	−11.7460	−0.409688
$823$	12.3832	0.431651	0.215826	−	0.976432i	$-0.430756\pi$
$823$	12.3832	0.431651	0.215826	+	0.976432i	$0.430756\pi$
$824$	34.0724	1.18697
$825$	−2.47182	−0.0860578
$826$	0	0
$827$	−28.7072	−0.998248	−0.499124	−	0.866530i	$-0.666345\pi$
$827$	−28.7072	−0.998248	−0.499124	+	0.866530i	$0.666345\pi$
$828$	6.18935	0.215095
$829$	−2.71655	−0.0943496	−0.0471748	−	0.998887i	$-0.515022\pi$
$829$	−2.71655	−0.0943496	−0.0471748	+	0.998887i	$0.515022\pi$
$830$	−7.38561	−0.256358
$831$	9.38969	0.325725
$832$	4.54502	0.157570
$833$	0	0
$834$	−2.91645	−0.100988
$835$	29.4067	1.01766
$836$	1.94210	0.0671688
$837$	6.45153	0.222998
$838$	2.78579	0.0962334
$839$	48.9487	1.68990	0.844948	−	0.534848i	$-0.179631\pi$
$839$	48.9487	1.68990	0.844948	+	0.534848i	$0.179631\pi$
$840$	0	0
$841$	−28.8938	−0.996337
$842$	−5.49126	−0.189241
$843$	−26.9788	−0.929199
$844$	34.6766	1.19362
$845$	7.49718	0.257911
$846$	0.168571	0.00579560
$847$	0	0
$848$	12.9255	0.443863
$849$	−23.5756	−0.809112
$850$	−6.60510	−0.226553
$851$	−6.18935	−0.212168
$852$	15.5361	0.532257
$853$	21.5962	0.739441	0.369721	−	0.929143i	$-0.379453\pi$
$853$	21.5962	0.739441	0.369721	+	0.929143i	$0.379453\pi$
$854$	0	0
$855$	−1.82199	−0.0623106
$856$	36.6361	1.25220
$857$	9.44147	0.322514	0.161257	−	0.986912i	$-0.448445\pi$
$857$	9.44147	0.322514	0.161257	+	0.986912i	$0.448445\pi$
$858$	1.59002	0.0542826
$859$	3.50931	0.119736	0.0598680	−	0.998206i	$-0.480932\pi$
$859$	3.50931	0.119736	0.0598680	+	0.998206i	$0.480932\pi$
$860$	12.4708	0.425250
$861$	0	0
$862$	−7.56115	−0.257534
$863$	38.8785	1.32344	0.661721	−	0.749751i	$-0.269827\pi$
$863$	38.8785	1.32344	0.661721	+	0.749751i	$0.269827\pi$
$864$	5.33178	0.181391
$865$	19.5065	0.663240
$866$	−2.35525	−0.0800347
$867$	−6.39926	−0.217330
$868$	0	0
$869$	8.71027	0.295476
$870$	0.286284	0.00970592
$871$	−43.9566	−1.48941
$872$	22.6313	0.766393
$873$	3.65329	0.123645
$874$	2.31163	0.0781919
$875$	0	0
$876$	19.8855	0.671869
$877$	4.39455	0.148394	0.0741968	−	0.997244i	$-0.476361\pi$
$877$	4.39455	0.148394	0.0741968	+	0.997244i	$0.476361\pi$
$878$	−12.9282	−0.436304
$879$	−11.8500	−0.399692
$880$	3.59693	0.121252
$881$	29.9918	1.01045	0.505225	−	0.862988i	$-0.331409\pi$
$881$	29.9918	1.01045	0.505225	+	0.862988i	$0.331409\pi$
$882$	0	0
$883$	16.8653	0.567563	0.283782	−	0.958889i	$-0.408411\pi$
$883$	16.8653	0.567563	0.283782	+	0.958889i	$0.408411\pi$
$884$	−23.5979	−0.793684
$885$	−18.8601	−0.633975
$886$	8.13103	0.273167
$887$	−10.0278	−0.336701	−0.168350	−	0.985727i	$-0.553844\pi$
$887$	−10.0278	−0.336701	−0.168350	+	0.985727i	$0.553844\pi$
$888$	−3.45929	−0.116086
$889$	0	0
$890$	5.05030	0.169286
$891$	−1.00000	−0.0335013
$892$	−50.1755	−1.68000
$893$	−0.349674	−0.0117014
$894$	0.754291	0.0252273
$895$	−21.9608	−0.734070
$896$	0	0
$897$	−10.5113	−0.350964
$898$	19.2584	0.642660
$899$	2.10278	0.0701317
$900$	4.18935	0.139645
$901$	−27.6388	−0.920783
$902$	2.24190	0.0746470
$903$	0	0
$904$	3.51968	0.117063
$905$	−29.8426	−0.992003
$906$	8.81038	0.292705
$907$	−50.5517	−1.67854	−0.839271	−	0.543714i	$-0.817018\pi$
$907$	−50.5517	−1.67854	−0.839271	+	0.543714i	$0.817018\pi$
$908$	−40.8171	−1.35456
$909$	−0.284377	−0.00943219
$910$	0	0
$911$	21.7956	0.722120	0.361060	−	0.932543i	$-0.382415\pi$
$911$	21.7956	0.722120	0.361060	+	0.932543i	$0.382415\pi$
$912$	−2.59220	−0.0858364
$913$	8.40856	0.278283
$914$	5.85629	0.193709
$915$	−2.81995	−0.0932247
$916$	−8.83811	−0.292019
$917$	0	0
$918$	−2.67216	−0.0881944
$919$	22.4978	0.742135	0.371067	−	0.928606i	$-0.378992\pi$
$919$	22.4978	0.742135	0.371067	+	0.928606i	$0.378992\pi$
$920$	11.8516	0.390735
$921$	19.5894	0.645492
$922$	−3.78079	−0.124514
$923$	−26.3848	−0.868467
$924$	0	0
$925$	−4.18935	−0.137745
$926$	6.47972	0.212937
$927$	16.6934	0.548284
$928$	1.73782	0.0570466
$929$	5.67978	0.186348	0.0931738	−	0.995650i	$-0.470299\pi$
$929$	5.67978	0.186348	0.0931738	+	0.995650i	$0.470299\pi$
$930$	5.66667	0.185817
$931$	0	0
$932$	−5.33178	−0.174648
$933$	−21.4017	−0.700659
$934$	−21.1510	−0.692083
$935$	−7.69139	−0.251535
$936$	−5.87489	−0.192027
$937$	−40.6016	−1.32640	−0.663198	−	0.748444i	$-0.730801\pi$
$937$	−40.6016	−1.32640	−0.663198	+	0.748444i	$0.730801\pi$
$938$	0	0
$939$	−13.3278	−0.434938
$940$	−0.822348	−0.0268220
$941$	−17.6473	−0.575285	−0.287643	−	0.957738i	$-0.592872\pi$
$941$	−17.6473	−0.575285	−0.287643	+	0.957738i	$0.592872\pi$
$942$	4.63836	0.151126
$943$	−14.8207	−0.482629
$944$	−26.8329	−0.873337
$945$	0	0
$946$	2.55635	0.0831142
$947$	28.2944	0.919446	0.459723	−	0.888062i	$-0.347949\pi$
$947$	28.2944	0.919446	0.459723	+	0.888062i	$0.347949\pi$
$948$	−14.7625	−0.479465
$949$	−33.7715	−1.09627
$950$	1.56466	0.0507642
$951$	7.09552	0.230088
$952$	0	0
$953$	43.3325	1.40368	0.701839	−	0.712335i	$-0.252362\pi$
$953$	43.3325	1.40368	0.701839	+	0.712335i	$0.252362\pi$
$954$	−3.15631	−0.102189
$955$	−12.9059	−0.417625
$956$	−4.66871	−0.150997
$957$	−0.325935	−0.0105360
$958$	6.97633	0.225395
$959$	0	0
$960$	−2.51071	−0.0810328
$961$	10.6223	0.342654
$962$	2.69484	0.0868853
$963$	17.9495	0.578414
$964$	−15.3037	−0.492900
$965$	−1.89335	−0.0609490
$966$	0	0
$967$	−2.82833	−0.0909529	−0.0454764	−	0.998965i	$-0.514481\pi$
$967$	−2.82833	−0.0909529	−0.0454764	+	0.998965i	$0.514481\pi$
$968$	2.04107	0.0656024
$969$	5.54297	0.178066
$970$	3.20884	0.103030
$971$	28.9329	0.928502	0.464251	−	0.885704i	$-0.346324\pi$
$971$	28.9329	0.928502	0.464251	+	0.885704i	$0.346324\pi$
$972$	1.69484	0.0543621
$973$	0	0
$974$	−19.6735	−0.630379
$975$	−7.11475	−0.227854
$976$	−4.01204	−0.128422
$977$	56.9917	1.82333	0.911663	−	0.410940i	$-0.134799\pi$
$977$	56.9917	1.82333	0.911663	+	0.410940i	$0.134799\pi$
$978$	−9.56389	−0.305819
$979$	−5.74979	−0.183764
$980$	0	0
$981$	11.0880	0.354012
$982$	−4.19751	−0.133948
$983$	−31.1237	−0.992692	−0.496346	−	0.868125i	$-0.665325\pi$
$983$	−31.1237	−0.992692	−0.496346	+	0.868125i	$0.665325\pi$
$984$	−8.28345	−0.264067
$985$	42.5368	1.35534
$986$	−0.870951	−0.0277367
$987$	0	0
$988$	5.59002	0.177842
$989$	−16.8995	−0.537374
$990$	−0.878345	−0.0279156
$991$	8.56875	0.272195	0.136098	−	0.990695i	$-0.456544\pi$
$991$	8.56875	0.272195	0.136098	+	0.990695i	$0.456544\pi$
$992$	34.3982	1.09214
$993$	−20.7350	−0.658005
$994$	0	0
$995$	−9.59369	−0.304140
$996$	−14.2512	−0.451566
$997$	−14.5858	−0.461937	−0.230968	−	0.972961i	$-0.574189\pi$
$997$	−14.5858	−0.461937	−0.230968	+	0.972961i	$0.574189\pi$
$998$	−0.887214	−0.0280843
$999$	−1.69484	−0.0536225

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1617.2.a.x.1.2		4
3.2	odd	2		4851.2.a.bu.1.3		4
7.3	odd	6		231.2.i.e.100.3	yes	8
7.5	odd	6		231.2.i.e.67.3	✓	8
7.6	odd	2		1617.2.a.z.1.2		4
21.5	even	6		693.2.i.i.298.2		8
21.17	even	6		693.2.i.i.100.2		8
21.20	even	2		4851.2.a.bt.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.2.i.e.67.3	✓	8	7.5	odd	6
231.2.i.e.100.3	yes	8	7.3	odd	6
693.2.i.i.100.2		8	21.17	even	6
693.2.i.i.298.2		8	21.5	even	6
1617.2.a.x.1.2		4	1.1	even	1	trivial
1617.2.a.z.1.2		4	7.6	odd	2
4851.2.a.bt.1.3		4	21.20	even	2
4851.2.a.bu.1.3		4	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1617 = 3 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1617.a (trivial)

\(f(q)\)	\(=\)	\(q-0.552409 q^{2} -1.00000 q^{3} -1.69484 q^{4} -1.59002 q^{5} +0.552409 q^{6} +2.04107 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.552409 q^{2} -1.00000 q^{3} -1.69484 q^{4} -1.59002 q^{5} +0.552409 q^{6} +2.04107 q^{8} +1.00000 q^{9} +0.878345 q^{10} -1.00000 q^{11} +1.69484 q^{12} -2.87834 q^{13} +1.59002 q^{15} +2.26218 q^{16} -4.83728 q^{17} -0.552409 q^{18} +1.14589 q^{19} +2.69484 q^{20} +0.552409 q^{22} -3.65187 q^{23} -2.04107 q^{24} -2.47182 q^{25} +1.59002 q^{26} -1.00000 q^{27} -0.325935 q^{29} -0.878345 q^{30} -6.45153 q^{31} -5.33178 q^{32} +1.00000 q^{33} +2.67216 q^{34} -1.69484 q^{36} +1.69484 q^{37} -0.632998 q^{38} +2.87834 q^{39} -3.24535 q^{40} +4.05839 q^{41} +4.62764 q^{43} +1.69484 q^{44} -1.59002 q^{45} +2.01733 q^{46} -0.305156 q^{47} -2.26218 q^{48} +1.36546 q^{50} +4.83728 q^{51} +4.87834 q^{52} +5.71372 q^{53} +0.552409 q^{54} +1.59002 q^{55} -1.14589 q^{57} +0.180050 q^{58} -11.8615 q^{59} -2.69484 q^{60} -1.77353 q^{61} +3.56389 q^{62} -1.57904 q^{64} +4.57664 q^{65} -0.552409 q^{66} +15.2715 q^{67} +8.19843 q^{68} +3.65187 q^{69} +9.16666 q^{71} +2.04107 q^{72} +11.7330 q^{73} -0.936248 q^{74} +2.47182 q^{75} -1.94210 q^{76} -1.59002 q^{78} -8.71027 q^{79} -3.59693 q^{80} +1.00000 q^{81} -2.24190 q^{82} -8.40856 q^{83} +7.69139 q^{85} -2.55635 q^{86} +0.325935 q^{87} -2.04107 q^{88} +5.74979 q^{89} +0.878345 q^{90} +6.18935 q^{92} +6.45153 q^{93} +0.168571 q^{94} -1.82199 q^{95} +5.33178 q^{96} +3.65329 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} - 4 q^{3} + 4 q^{4} - 4 q^{5} - 2 q^{6} + 12 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q + 2 * q^2 - 4 * q^3 + 4 * q^4 - 4 * q^5 - 2 * q^6 + 12 * q^8 + 4 * q^9	\( 4 q + 2 q^{2} - 4 q^{3} + 4 q^{4} - 4 q^{5} - 2 q^{6} + 12 q^{8} + 4 q^{9} - 10 q^{10} - 4 q^{11} - 4 q^{12} + 2 q^{13} + 4 q^{15} + 12 q^{16} - 2 q^{17} + 2 q^{18} - 2 q^{22} + 4 q^{23} - 12 q^{24} + 4 q^{25} + 4 q^{26} - 4 q^{27} + 8 q^{29} + 10 q^{30} + 12 q^{31} + 26 q^{32} + 4 q^{33} + 16 q^{34} + 4 q^{36} - 4 q^{37} - 8 q^{38} - 2 q^{39} + 6 q^{40} - 2 q^{41} + 18 q^{43} - 4 q^{44} - 4 q^{45} - 14 q^{46} - 12 q^{47} - 12 q^{48} + 2 q^{50} + 2 q^{51} + 6 q^{52} - 12 q^{53} - 2 q^{54} + 4 q^{55} - 4 q^{58} - 12 q^{59} - 2 q^{61} + 26 q^{62} + 56 q^{64} - 4 q^{65} + 2 q^{66} + 28 q^{67} + 48 q^{68} - 4 q^{69} + 12 q^{71} + 12 q^{72} - 6 q^{73} - 16 q^{74} - 4 q^{75} + 18 q^{76} - 4 q^{78} + 2 q^{79} - 16 q^{80} + 4 q^{81} + 12 q^{82} + 12 q^{83} + 18 q^{85} + 36 q^{86} - 8 q^{87} - 12 q^{88} - 8 q^{89} - 10 q^{90} - 16 q^{92} - 12 q^{93} - 20 q^{94} + 34 q^{95} - 26 q^{96} + 44 q^{97} - 4 q^{99}+O(q^{100}) \) 4 * q + 2 * q^2 - 4 * q^3 + 4 * q^4 - 4 * q^5 - 2 * q^6 + 12 * q^8 + 4 * q^9 - 10 * q^10 - 4 * q^11 - 4 * q^12 + 2 * q^13 + 4 * q^15 + 12 * q^16 - 2 * q^17 + 2 * q^18 - 2 * q^22 + 4 * q^23 - 12 * q^24 + 4 * q^25 + 4 * q^26 - 4 * q^27 + 8 * q^29 + 10 * q^30 + 12 * q^31 + 26 * q^32 + 4 * q^33 + 16 * q^34 + 4 * q^36 - 4 * q^37 - 8 * q^38 - 2 * q^39 + 6 * q^40 - 2 * q^41 + 18 * q^43 - 4 * q^44 - 4 * q^45 - 14 * q^46 - 12 * q^47 - 12 * q^48 + 2 * q^50 + 2 * q^51 + 6 * q^52 - 12 * q^53 - 2 * q^54 + 4 * q^55 - 4 * q^58 - 12 * q^59 - 2 * q^61 + 26 * q^62 + 56 * q^64 - 4 * q^65 + 2 * q^66 + 28 * q^67 + 48 * q^68 - 4 * q^69 + 12 * q^71 + 12 * q^72 - 6 * q^73 - 16 * q^74 - 4 * q^75 + 18 * q^76 - 4 * q^78 + 2 * q^79 - 16 * q^80 + 4 * q^81 + 12 * q^82 + 12 * q^83 + 18 * q^85 + 36 * q^86 - 8 * q^87 - 12 * q^88 - 8 * q^89 - 10 * q^90 - 16 * q^92 - 12 * q^93 - 20 * q^94 + 34 * q^95 - 26 * q^96 + 44 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more