LMFDB - Embedded newform 1045.2.a.f.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1045,2,Mod(1,1045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1045, 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("1045.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1045 = 5 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1045.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:	$8.34436701122$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.7281497.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 5x^{4} + 7x^{3} + 6x^{2} - 2x - 1 $ x^6 - 2x^5 - 5x^4 + 7x^3 + 6x^2 - 2*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.79049$ of defining polynomial
Character	$\chi$	$=$	1045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.23198 q^{2} -1.34246 q^{3} +2.98176 q^{4} -1.00000 q^{5} +2.99635 q^{6} +4.13295 q^{7} -2.19127 q^{8} -1.19780 q^{9} +O(q^{10})$	\(q-2.23198 q^{2} -1.34246 q^{3} +2.98176 q^{4} -1.00000 q^{5} +2.99635 q^{6} +4.13295 q^{7} -2.19127 q^{8} -1.19780 q^{9} +2.23198 q^{10} -1.00000 q^{11} -4.00289 q^{12} -0.110474 q^{13} -9.22468 q^{14} +1.34246 q^{15} -1.07264 q^{16} -6.99924 q^{17} +2.67348 q^{18} -1.00000 q^{19} -2.98176 q^{20} -5.54832 q^{21} +2.23198 q^{22} +8.36934 q^{23} +2.94168 q^{24} +1.00000 q^{25} +0.246577 q^{26} +5.63538 q^{27} +12.3235 q^{28} -3.28289 q^{29} -2.99635 q^{30} -0.202971 q^{31} +6.77665 q^{32} +1.34246 q^{33} +15.6222 q^{34} -4.13295 q^{35} -3.57156 q^{36} -0.683573 q^{37} +2.23198 q^{38} +0.148307 q^{39} +2.19127 q^{40} -4.97522 q^{41} +12.3838 q^{42} +8.52642 q^{43} -2.98176 q^{44} +1.19780 q^{45} -18.6802 q^{46} +3.85948 q^{47} +1.43998 q^{48} +10.0813 q^{49} -2.23198 q^{50} +9.39619 q^{51} -0.329408 q^{52} -10.1554 q^{53} -12.5781 q^{54} +1.00000 q^{55} -9.05639 q^{56} +1.34246 q^{57} +7.32737 q^{58} -10.4857 q^{59} +4.00289 q^{60} -3.53161 q^{61} +0.453029 q^{62} -4.95046 q^{63} -12.9801 q^{64} +0.110474 q^{65} -2.99635 q^{66} -7.84063 q^{67} -20.8700 q^{68} -11.2355 q^{69} +9.22468 q^{70} -1.58930 q^{71} +2.62471 q^{72} +8.41595 q^{73} +1.52572 q^{74} -1.34246 q^{75} -2.98176 q^{76} -4.13295 q^{77} -0.331020 q^{78} +2.80598 q^{79} +1.07264 q^{80} -3.97186 q^{81} +11.1046 q^{82} -5.52580 q^{83} -16.5437 q^{84} +6.99924 q^{85} -19.0308 q^{86} +4.40715 q^{87} +2.19127 q^{88} +0.654537 q^{89} -2.67348 q^{90} -0.456585 q^{91} +24.9553 q^{92} +0.272481 q^{93} -8.61430 q^{94} +1.00000 q^{95} -9.09738 q^{96} -10.8858 q^{97} -22.5013 q^{98} +1.19780 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{2} - q^{3} + 4 q^{4} - 6 q^{5} + 5 q^{7} - 12 q^{8} + q^{9}+O(q^{10}) $ 6 * q - 2 * q^2 - q^3 + 4 * q^4 - 6 * q^5 + 5 * q^7 - 12 * q^8 + q^9	\( 6 q - 2 q^{2} - q^{3} + 4 q^{4} - 6 q^{5} + 5 q^{7} - 12 q^{8} + q^{9} + 2 q^{10} - 6 q^{11} + q^{12} - 5 q^{13} - 8 q^{14} + q^{15} + 4 q^{16} + q^{17} + 6 q^{18} - 6 q^{19} - 4 q^{20} - 21 q^{21} + 2 q^{22} + 4 q^{23} - q^{24} + 6 q^{25} - 14 q^{26} - 16 q^{27} + 10 q^{28} - 9 q^{29} - 21 q^{31} - q^{32} + q^{33} - 5 q^{35} - 28 q^{36} - 3 q^{37} + 2 q^{38} + 20 q^{39} + 12 q^{40} - 23 q^{41} + q^{42} + 7 q^{43} - 4 q^{44} - q^{45} - 12 q^{46} - 18 q^{47} - 3 q^{49} - 2 q^{50} - 16 q^{51} + 13 q^{52} - 17 q^{53} + q^{54} + 6 q^{55} - 2 q^{56} + q^{57} + 23 q^{58} - 29 q^{59} - q^{60} + 17 q^{61} + 2 q^{62} + 6 q^{63} - 18 q^{64} + 5 q^{65} + 8 q^{67} - q^{68} - 38 q^{69} + 8 q^{70} - 12 q^{71} + 13 q^{72} + 2 q^{73} - 37 q^{74} - q^{75} - 4 q^{76} - 5 q^{77} + q^{78} + 3 q^{79} - 4 q^{80} - 2 q^{81} + 24 q^{82} - 11 q^{83} - 3 q^{84} - q^{85} - 12 q^{86} - 12 q^{87} + 12 q^{88} - 22 q^{89} - 6 q^{90} - 18 q^{91} - 15 q^{92} + 18 q^{93} + 22 q^{94} + 6 q^{95} - 17 q^{96} - 2 q^{97} - q^{98} - q^{99}+O(q^{100}) \) 6 * q - 2 * q^2 - q^3 + 4 * q^4 - 6 * q^5 + 5 * q^7 - 12 * q^8 + q^9 + 2 * q^10 - 6 * q^11 + q^12 - 5 * q^13 - 8 * q^14 + q^15 + 4 * q^16 + q^17 + 6 * q^18 - 6 * q^19 - 4 * q^20 - 21 * q^21 + 2 * q^22 + 4 * q^23 - q^24 + 6 * q^25 - 14 * q^26 - 16 * q^27 + 10 * q^28 - 9 * q^29 - 21 * q^31 - q^32 + q^33 - 5 * q^35 - 28 * q^36 - 3 * q^37 + 2 * q^38 + 20 * q^39 + 12 * q^40 - 23 * q^41 + q^42 + 7 * q^43 - 4 * q^44 - q^45 - 12 * q^46 - 18 * q^47 - 3 * q^49 - 2 * q^50 - 16 * q^51 + 13 * q^52 - 17 * q^53 + q^54 + 6 * q^55 - 2 * q^56 + q^57 + 23 * q^58 - 29 * q^59 - q^60 + 17 * q^61 + 2 * q^62 + 6 * q^63 - 18 * q^64 + 5 * q^65 + 8 * q^67 - q^68 - 38 * q^69 + 8 * q^70 - 12 * q^71 + 13 * q^72 + 2 * q^73 - 37 * q^74 - q^75 - 4 * q^76 - 5 * q^77 + q^78 + 3 * q^79 - 4 * q^80 - 2 * q^81 + 24 * q^82 - 11 * q^83 - 3 * q^84 - q^85 - 12 * q^86 - 12 * q^87 + 12 * q^88 - 22 * q^89 - 6 * q^90 - 18 * q^91 - 15 * q^92 + 18 * q^93 + 22 * q^94 + 6 * q^95 - 17 * q^96 - 2 * q^97 - q^98 - 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 $

$2$	−2.23198	−1.57825	−0.789126	−	0.614232i	$-0.789466\pi$
$2$	−2.23198	−1.57825	−0.789126	+	0.614232i	$0.789466\pi$
$3$	−1.34246	−0.775069	−0.387535	−	0.921855i	$-0.626673\pi$
$3$	−1.34246	−0.775069	−0.387535	+	0.921855i	$0.626673\pi$
$4$	2.98176	1.49088
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	2.99635	1.22325
$7$	4.13295	1.56211	0.781054	−	0.624463i	$-0.214682\pi$
$7$	4.13295	1.56211	0.781054	+	0.624463i	$0.214682\pi$
$8$	−2.19127	−0.774729
$9$	−1.19780	−0.399268
$10$	2.23198	0.705816
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−4.00289	−1.15553
$13$	−0.110474	−0.0306401	−0.0153200	−	0.999883i	$-0.504877\pi$
$13$	−0.110474	−0.0306401	−0.0153200	+	0.999883i	$0.504877\pi$
$14$	−9.22468	−2.46540
$15$	1.34246	0.346621
$16$	−1.07264	−0.268160
$17$	−6.99924	−1.69756	−0.848782	−	0.528743i	$-0.822664\pi$
$17$	−6.99924	−1.69756	−0.848782	+	0.528743i	$0.822664\pi$
$18$	2.67348	0.630145
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−2.98176	−0.666741
$21$	−5.54832	−1.21074
$22$	2.23198	0.475861
$23$	8.36934	1.74513	0.872564	−	0.488500i	$-0.162456\pi$
$23$	8.36934	1.74513	0.872564	+	0.488500i	$0.162456\pi$
$24$	2.94168	0.600469
$25$	1.00000	0.200000
$26$	0.246577	0.0483577
$27$	5.63538	1.08453
$28$	12.3235	2.32891
$29$	−3.28289	−0.609618	−0.304809	−	0.952414i	$-0.598593\pi$
$29$	−3.28289	−0.609618	−0.304809	+	0.952414i	$0.598593\pi$
$30$	−2.99635	−0.547056
$31$	−0.202971	−0.0364547	−0.0182274	−	0.999834i	$-0.505802\pi$
$31$	−0.202971	−0.0364547	−0.0182274	+	0.999834i	$0.505802\pi$
$32$	6.77665	1.19795
$33$	1.34246	0.233692
$34$	15.6222	2.67918
$35$	−4.13295	−0.698596
$36$	−3.57156	−0.595260
$37$	−0.683573	−0.112379	−0.0561893	−	0.998420i	$-0.517895\pi$
$37$	−0.683573	−0.112379	−0.0561893	+	0.998420i	$0.517895\pi$
$38$	2.23198	0.362076
$39$	0.148307	0.0237482
$40$	2.19127	0.346470
$41$	−4.97522	−0.776999	−0.388499	−	0.921449i	$-0.627006\pi$
$41$	−4.97522	−0.776999	−0.388499	+	0.921449i	$0.627006\pi$
$42$	12.3838	1.91086
$43$	8.52642	1.30027	0.650134	−	0.759820i	$-0.274713\pi$
$43$	8.52642	1.30027	0.650134	+	0.759820i	$0.274713\pi$
$44$	−2.98176	−0.449517
$45$	1.19780	0.178558
$46$	−18.6802	−2.75425
$47$	3.85948	0.562963	0.281482	−	0.959567i	$-0.409174\pi$
$47$	3.85948	0.562963	0.281482	+	0.959567i	$0.409174\pi$
$48$	1.43998	0.207843
$49$	10.0813	1.44018
$50$	−2.23198	−0.315650
$51$	9.39619	1.31573
$52$	−0.329408	−0.0456806
$53$	−10.1554	−1.39496	−0.697478	−	0.716607i	$-0.745694\pi$
$53$	−10.1554	−1.39496	−0.697478	+	0.716607i	$0.745694\pi$
$54$	−12.5781	−1.71166
$55$	1.00000	0.134840
$56$	−9.05639	−1.21021
$57$	1.34246	0.177813
$58$	7.32737	0.962131
$59$	−10.4857	−1.36512	−0.682561	−	0.730829i	$-0.739134\pi$
$59$	−10.4857	−1.36512	−0.682561	+	0.730829i	$0.739134\pi$
$60$	4.00289	0.516770
$61$	−3.53161	−0.452177	−0.226088	−	0.974107i	$-0.572594\pi$
$61$	−3.53161	−0.452177	−0.226088	+	0.974107i	$0.572594\pi$
$62$	0.453029	0.0575347
$63$	−4.95046	−0.623699
$64$	−12.9801	−1.62251
$65$	0.110474	0.0137027
$66$	−2.99635	−0.368825
$67$	−7.84063	−0.957886	−0.478943	−	0.877846i	$-0.658980\pi$
$67$	−7.84063	−0.957886	−0.478943	+	0.877846i	$0.658980\pi$
$68$	−20.8700	−2.53086
$69$	−11.2355	−1.35259
$70$	9.22468	1.10256
$71$	−1.58930	−0.188616	−0.0943078	−	0.995543i	$-0.530064\pi$
$71$	−1.58930	−0.188616	−0.0943078	+	0.995543i	$0.530064\pi$
$72$	2.62471	0.309324
$73$	8.41595	0.985012	0.492506	−	0.870309i	$-0.336081\pi$
$73$	8.41595	0.985012	0.492506	+	0.870309i	$0.336081\pi$
$74$	1.52572	0.177362
$75$	−1.34246	−0.155014
$76$	−2.98176	−0.342031
$77$	−4.13295	−0.470993
$78$	−0.331020	−0.0374806
$79$	2.80598	0.315697	0.157848	−	0.987463i	$-0.449544\pi$
$79$	2.80598	0.315697	0.157848	+	0.987463i	$0.449544\pi$
$80$	1.07264	0.119925
$81$	−3.97186	−0.441317
$82$	11.1046	1.22630
$83$	−5.52580	−0.606535	−0.303267	−	0.952905i	$-0.598078\pi$
$83$	−5.52580	−0.606535	−0.303267	+	0.952905i	$0.598078\pi$
$84$	−16.5437	−1.80507
$85$	6.99924	0.759174
$86$	−19.0308	−2.05215
$87$	4.40715	0.472496
$88$	2.19127	0.233590
$89$	0.654537	0.0693808	0.0346904	−	0.999398i	$-0.488955\pi$
$89$	0.654537	0.0693808	0.0346904	+	0.999398i	$0.488955\pi$
$90$	−2.67348	−0.281809
$91$	−0.456585	−0.0478631
$92$	24.9553	2.60177
$93$	0.272481	0.0282549
$94$	−8.61430	−0.888498
$95$	1.00000	0.102598
$96$	−9.09738	−0.928497
$97$	−10.8858	−1.10529	−0.552643	−	0.833418i	$-0.686381\pi$
$97$	−10.8858	−1.10529	−0.552643	+	0.833418i	$0.686381\pi$
$98$	−22.5013	−2.27297
$99$	1.19780	0.120384
$100$	2.98176	0.298176
$101$	9.08470	0.903961	0.451980	−	0.892028i	$-0.350718\pi$
$101$	9.08470	0.903961	0.451980	+	0.892028i	$0.350718\pi$
$102$	−20.9721	−2.07655
$103$	−3.59063	−0.353795	−0.176898	−	0.984229i	$-0.556606\pi$
$103$	−3.59063	−0.353795	−0.176898	+	0.984229i	$0.556606\pi$
$104$	0.242079	0.0237378
$105$	5.54832	0.541460
$106$	22.6668	2.20159
$107$	−7.13069	−0.689350	−0.344675	−	0.938722i	$-0.612011\pi$
$107$	−7.13069	−0.689350	−0.344675	+	0.938722i	$0.612011\pi$
$108$	16.8033	1.61690
$109$	−15.1412	−1.45026	−0.725131	−	0.688611i	$-0.758221\pi$
$109$	−15.1412	−1.45026	−0.725131	+	0.688611i	$0.758221\pi$
$110$	−2.23198	−0.212811
$111$	0.917668	0.0871012
$112$	−4.43317	−0.418896
$113$	−2.67686	−0.251818	−0.125909	−	0.992042i	$-0.540185\pi$
$113$	−2.67686	−0.251818	−0.125909	+	0.992042i	$0.540185\pi$
$114$	−2.99635	−0.280634
$115$	−8.36934	−0.780445
$116$	−9.78879	−0.908866
$117$	0.132327	0.0122336
$118$	23.4039	2.15451
$119$	−28.9275	−2.65178
$120$	−2.94168	−0.268538
$121$	1.00000	0.0909091
$122$	7.88251	0.713649
$123$	6.67903	0.602228
$124$	−0.605211	−0.0543496
$125$	−1.00000	−0.0894427
$126$	11.0494	0.984355
$127$	−13.0799	−1.16065	−0.580326	−	0.814384i	$-0.697075\pi$
$127$	−13.0799	−1.16065	−0.580326	+	0.814384i	$0.697075\pi$
$128$	15.4181	1.36278
$129$	−11.4464	−1.00780
$130$	−0.246577	−0.0216262
$131$	−8.33532	−0.728260	−0.364130	−	0.931348i	$-0.618634\pi$
$131$	−8.33532	−0.728260	−0.364130	+	0.931348i	$0.618634\pi$
$132$	4.00289	0.348407
$133$	−4.13295	−0.358372
$134$	17.5002	1.51179
$135$	−5.63538	−0.485016
$136$	15.3372	1.31515
$137$	−13.0803	−1.11753	−0.558765	−	0.829326i	$-0.688724\pi$
$137$	−13.0803	−1.11753	−0.558765	+	0.829326i	$0.688724\pi$
$138$	25.0775	2.13473
$139$	−18.2803	−1.55052	−0.775259	−	0.631644i	$-0.782381\pi$
$139$	−18.2803	−1.55052	−0.775259	+	0.631644i	$0.782381\pi$
$140$	−12.3235	−1.04152
$141$	−5.18120	−0.436335
$142$	3.54730	0.297683
$143$	0.110474	0.00923833
$144$	1.28481	0.107068
$145$	3.28289	0.272629
$146$	−18.7843	−1.55460
$147$	−13.5337	−1.11624
$148$	−2.03825	−0.167543
$149$	18.3204	1.50087	0.750435	−	0.660945i	$-0.229844\pi$
$149$	18.3204	1.50087	0.750435	+	0.660945i	$0.229844\pi$
$150$	2.99635	0.244651
$151$	8.66581	0.705213	0.352607	−	0.935772i	$-0.385295\pi$
$151$	8.66581	0.705213	0.352607	+	0.935772i	$0.385295\pi$
$152$	2.19127	0.177735
$153$	8.38371	0.677783
$154$	9.22468	0.743346
$155$	0.202971	0.0163030
$156$	0.442216	0.0354056
$157$	−18.7512	−1.49651	−0.748254	−	0.663412i	$-0.769108\pi$
$157$	−18.7512	−1.49651	−0.748254	+	0.663412i	$0.769108\pi$
$158$	−6.26290	−0.498249
$159$	13.6332	1.08119
$160$	−6.77665	−0.535741
$161$	34.5901	2.72608
$162$	8.86513	0.696510
$163$	19.8023	1.55103	0.775517	−	0.631327i	$-0.217489\pi$
$163$	19.8023	1.55103	0.775517	+	0.631327i	$0.217489\pi$
$164$	−14.8349	−1.15841
$165$	−1.34246	−0.104510
$166$	12.3335	0.957265
$167$	1.91597	0.148262	0.0741312	−	0.997248i	$-0.476382\pi$
$167$	1.91597	0.148262	0.0741312	+	0.997248i	$0.476382\pi$
$168$	12.1578	0.937997
$169$	−12.9878	−0.999061
$170$	−15.6222	−1.19817
$171$	1.19780	0.0915983
$172$	25.4237	1.93854
$173$	−3.30818	−0.251516	−0.125758	−	0.992061i	$-0.540136\pi$
$173$	−3.30818	−0.251516	−0.125758	+	0.992061i	$0.540136\pi$
$174$	−9.83669	−0.745718
$175$	4.13295	0.312422
$176$	1.07264	0.0808534
$177$	14.0766	1.05806
$178$	−1.46092	−0.109500
$179$	−21.9873	−1.64341	−0.821703	−	0.569915i	$-0.806976\pi$
$179$	−21.9873	−1.64341	−0.821703	+	0.569915i	$0.806976\pi$
$180$	3.57156	0.266208
$181$	−17.0245	−1.26542	−0.632709	−	0.774389i	$-0.718057\pi$
$181$	−17.0245	−1.26542	−0.632709	+	0.774389i	$0.718057\pi$
$182$	1.01909	0.0755400
$183$	4.74105	0.350468
$184$	−18.3394	−1.35200
$185$	0.683573	0.0502573
$186$	−0.608173	−0.0445934
$187$	6.99924	0.511835
$188$	11.5080	0.839310
$189$	23.2907	1.69415
$190$	−2.23198	−0.161925
$191$	−12.8748	−0.931589	−0.465794	−	0.884893i	$-0.654231\pi$
$191$	−12.8748	−0.931589	−0.465794	+	0.884893i	$0.654231\pi$
$192$	17.4253	1.25756
$193$	2.49248	0.179412	0.0897062	−	0.995968i	$-0.471407\pi$
$193$	2.49248	0.179412	0.0897062	+	0.995968i	$0.471407\pi$
$194$	24.2970	1.74442
$195$	−0.148307	−0.0106205
$196$	30.0599	2.14714
$197$	0.604202	0.0430476	0.0215238	−	0.999768i	$-0.493148\pi$
$197$	0.604202	0.0430476	0.0215238	+	0.999768i	$0.493148\pi$
$198$	−2.67348	−0.189996
$199$	−17.6373	−1.25027	−0.625137	−	0.780515i	$-0.714957\pi$
$199$	−17.6373	−1.25027	−0.625137	+	0.780515i	$0.714957\pi$
$200$	−2.19127	−0.154946
$201$	10.5257	0.742428
$202$	−20.2769	−1.42668
$203$	−13.5680	−0.952289
$204$	28.0171	1.96159
$205$	4.97522	0.347484
$206$	8.01423	0.558378
$207$	−10.0248	−0.696773
$208$	0.118499	0.00821645
$209$	1.00000	0.0691714
$210$	−12.3838	−0.854561
$211$	19.9281	1.37190	0.685952	−	0.727646i	$-0.259386\pi$
$211$	19.9281	1.37190	0.685952	+	0.727646i	$0.259386\pi$
$212$	−30.2810	−2.07971
$213$	2.13357	0.146190
$214$	15.9156	1.08797
$215$	−8.52642	−0.581497
$216$	−12.3486	−0.840217
$217$	−0.838870	−0.0569462
$218$	33.7949	2.28888
$219$	−11.2981	−0.763453
$220$	2.98176	0.201030
$221$	0.773236	0.0520135
$222$	−2.04822	−0.137468
$223$	12.6225	0.845263	0.422631	−	0.906302i	$-0.361106\pi$
$223$	12.6225	0.845263	0.422631	+	0.906302i	$0.361106\pi$
$224$	28.0076	1.87133
$225$	−1.19780	−0.0798536
$226$	5.97472	0.397432
$227$	7.57012	0.502447	0.251223	−	0.967929i	$-0.419167\pi$
$227$	7.57012	0.502447	0.251223	+	0.967929i	$0.419167\pi$
$228$	4.00289	0.265098
$229$	−8.20266	−0.542047	−0.271023	−	0.962573i	$-0.587362\pi$
$229$	−8.20266	−0.542047	−0.271023	+	0.962573i	$0.587362\pi$
$230$	18.6802	1.23174
$231$	5.54832	0.365052
$232$	7.19369	0.472289
$233$	16.9904	1.11308	0.556538	−	0.830822i	$-0.312129\pi$
$233$	16.9904	1.11308	0.556538	+	0.830822i	$0.312129\pi$
$234$	−0.295351	−0.0193077
$235$	−3.85948	−0.251765
$236$	−31.2658	−2.03523
$237$	−3.76691	−0.244687
$238$	64.5657	4.18517
$239$	−15.2540	−0.986699	−0.493350	−	0.869831i	$-0.664228\pi$
$239$	−15.2540	−0.986699	−0.493350	+	0.869831i	$0.664228\pi$
$240$	−1.43998	−0.0929501
$241$	−1.53642	−0.0989698	−0.0494849	−	0.998775i	$-0.515758\pi$
$241$	−1.53642	−0.0989698	−0.0494849	+	0.998775i	$0.515758\pi$
$242$	−2.23198	−0.143477
$243$	−11.5741	−0.742478
$244$	−10.5304	−0.674141
$245$	−10.0813	−0.644069
$246$	−14.9075	−0.950467
$247$	0.110474	0.00702931
$248$	0.444764	0.0282425
$249$	7.41815	0.470106
$250$	2.23198	0.141163
$251$	24.6721	1.55729	0.778646	−	0.627464i	$-0.215907\pi$
$251$	24.6721	1.55729	0.778646	+	0.627464i	$0.215907\pi$
$252$	−14.7611	−0.929860
$253$	−8.36934	−0.526176
$254$	29.1941	1.83180
$255$	−9.39619	−0.588412
$256$	−8.45273	−0.528296
$257$	11.9052	0.742624	0.371312	−	0.928508i	$-0.378908\pi$
$257$	11.9052	0.742624	0.371312	+	0.928508i	$0.378908\pi$
$258$	25.5481	1.59056
$259$	−2.82517	−0.175548
$260$	0.329408	0.0204290
$261$	3.93226	0.243401
$262$	18.6043	1.14938
$263$	−0.849505	−0.0523827	−0.0261914	−	0.999657i	$-0.508338\pi$
$263$	−0.849505	−0.0523827	−0.0261914	+	0.999657i	$0.508338\pi$
$264$	−2.94168	−0.181048
$265$	10.1554	0.623843
$266$	9.22468	0.565602
$267$	−0.878689	−0.0537749
$268$	−23.3789	−1.42809
$269$	−16.6805	−1.01703	−0.508514	−	0.861054i	$-0.669805\pi$
$269$	−16.6805	−1.01703	−0.508514	+	0.861054i	$0.669805\pi$
$270$	12.5781	0.765478
$271$	−13.2403	−0.804291	−0.402146	−	0.915576i	$-0.631735\pi$
$271$	−13.2403	−0.804291	−0.402146	+	0.915576i	$0.631735\pi$
$272$	7.50767	0.455219
$273$	0.612947	0.0370972
$274$	29.1951	1.76374
$275$	−1.00000	−0.0603023
$276$	−33.5015	−2.01655
$277$	−17.2124	−1.03419	−0.517096	−	0.855927i	$-0.672987\pi$
$277$	−17.2124	−1.03419	−0.517096	+	0.855927i	$0.672987\pi$
$278$	40.8014	2.44711
$279$	0.243120	0.0145552
$280$	9.05639	0.541223
$281$	9.32448	0.556252	0.278126	−	0.960545i	$-0.410287\pi$
$281$	9.32448	0.556252	0.278126	+	0.960545i	$0.410287\pi$
$282$	11.5644	0.688647
$283$	12.8642	0.764698	0.382349	−	0.924018i	$-0.375115\pi$
$283$	12.8642	0.764698	0.382349	+	0.924018i	$0.375115\pi$
$284$	−4.73891	−0.281203
$285$	−1.34246	−0.0795204
$286$	−0.246577	−0.0145804
$287$	−20.5623	−1.21376
$288$	−8.11709	−0.478304
$289$	31.9893	1.88172
$290$	−7.32737	−0.430278
$291$	14.6138	0.856673
$292$	25.0943	1.46853
$293$	−14.4092	−0.841796	−0.420898	−	0.907108i	$-0.638285\pi$
$293$	−14.4092	−0.841796	−0.420898	+	0.907108i	$0.638285\pi$
$294$	30.2070	1.76171
$295$	10.4857	0.610501
$296$	1.49789	0.0870630
$297$	−5.63538	−0.326998
$298$	−40.8910	−2.36875
$299$	−0.924597	−0.0534708
$300$	−4.00289	−0.231107
$301$	35.2393	2.03116
$302$	−19.3419	−1.11300
$303$	−12.1958	−0.700632
$304$	1.07264	0.0615202
$305$	3.53161	0.202220
$306$	−18.7123	−1.06971
$307$	−28.2838	−1.61424	−0.807122	−	0.590385i	$-0.798976\pi$
$307$	−28.2838	−1.61424	−0.807122	+	0.590385i	$0.798976\pi$
$308$	−12.3235	−0.702194
$309$	4.82027	0.274216
$310$	−0.453029	−0.0257303
$311$	−12.3942	−0.702809	−0.351404	−	0.936224i	$-0.614296\pi$
$311$	−12.3942	−0.702809	−0.351404	+	0.936224i	$0.614296\pi$
$312$	−0.324981	−0.0183984
$313$	−33.0560	−1.86844	−0.934219	−	0.356701i	$-0.883901\pi$
$313$	−33.0560	−1.86844	−0.934219	+	0.356701i	$0.883901\pi$
$314$	41.8524	2.36187
$315$	4.95046	0.278927
$316$	8.36674	0.470666
$317$	9.95795	0.559294	0.279647	−	0.960103i	$-0.409783\pi$
$317$	9.95795	0.559294	0.279647	+	0.960103i	$0.409783\pi$
$318$	−30.4292	−1.70638
$319$	3.28289	0.183807
$320$	12.9801	0.725610
$321$	9.57266	0.534294
$322$	−77.2045	−4.30244
$323$	6.99924	0.389448
$324$	−11.8431	−0.657951
$325$	−0.110474	−0.00612801
$326$	−44.1984	−2.44792
$327$	20.3264	1.12405
$328$	10.9020	0.601964
$329$	15.9510	0.879409
$330$	2.99635	0.164944
$331$	29.9575	1.64661	0.823307	−	0.567596i	$-0.192126\pi$
$331$	29.9575	1.64661	0.823307	+	0.567596i	$0.192126\pi$
$332$	−16.4766	−0.904270
$333$	0.818785	0.0448692
$334$	−4.27642	−0.233995
$335$	7.84063	0.428380
$336$	5.95135	0.324673
$337$	11.7426	0.639662	0.319831	−	0.947475i	$-0.396374\pi$
$337$	11.7426	0.639662	0.319831	+	0.947475i	$0.396374\pi$
$338$	28.9886	1.57677
$339$	3.59358	0.195177
$340$	20.8700	1.13184
$341$	0.202971	0.0109915
$342$	−2.67348	−0.144565
$343$	12.7348	0.687613
$344$	−18.6837	−1.00736
$345$	11.2355	0.604899
$346$	7.38381	0.396956
$347$	7.01661	0.376671	0.188336	−	0.982105i	$-0.439691\pi$
$347$	7.01661	0.376671	0.188336	+	0.982105i	$0.439691\pi$
$348$	13.1410	0.704434
$349$	4.66034	0.249462	0.124731	−	0.992191i	$-0.460193\pi$
$349$	4.66034	0.249462	0.124731	+	0.992191i	$0.460193\pi$
$350$	−9.22468	−0.493080
$351$	−0.622565	−0.0332301
$352$	−6.77665	−0.361197
$353$	−36.5878	−1.94737	−0.973687	−	0.227889i	$-0.926818\pi$
$353$	−36.5878	−1.94737	−0.973687	+	0.227889i	$0.926818\pi$
$354$	−31.4188	−1.66989
$355$	1.58930	0.0843514
$356$	1.95167	0.103438
$357$	38.8340	2.05531
$358$	49.0753	2.59371
$359$	1.09396	0.0577369	0.0288684	−	0.999583i	$-0.490810\pi$
$359$	1.09396	0.0577369	0.0288684	+	0.999583i	$0.490810\pi$
$360$	−2.62471	−0.138334
$361$	1.00000	0.0526316
$362$	37.9984	1.99715
$363$	−1.34246	−0.0704608
$364$	−1.36143	−0.0713581
$365$	−8.41595	−0.440511
$366$	−10.5819	−0.553127
$367$	−12.3334	−0.643801	−0.321900	−	0.946774i	$-0.604322\pi$
$367$	−12.3334	−0.643801	−0.321900	+	0.946774i	$0.604322\pi$
$368$	−8.97730	−0.467974
$369$	5.95933	0.310231
$370$	−1.52572	−0.0793186
$371$	−41.9719	−2.17907
$372$	0.812471	0.0421247
$373$	−9.20094	−0.476407	−0.238203	−	0.971215i	$-0.576558\pi$
$373$	−9.20094	−0.476407	−0.238203	+	0.971215i	$0.576558\pi$
$374$	−15.6222	−0.807804
$375$	1.34246	0.0693243
$376$	−8.45715	−0.436144
$377$	0.362675	0.0186787
$378$	−51.9846	−2.67380
$379$	36.7806	1.88929	0.944645	−	0.328093i	$-0.106406\pi$
$379$	36.7806	1.88929	0.944645	+	0.328093i	$0.106406\pi$
$380$	2.98176	0.152961
$381$	17.5592	0.899586
$382$	28.7364	1.47028
$383$	23.8111	1.21669	0.608345	−	0.793673i	$-0.291834\pi$
$383$	23.8111	1.21669	0.608345	+	0.793673i	$0.291834\pi$
$384$	−20.6981	−1.05625
$385$	4.13295	0.210635
$386$	−5.56317	−0.283158
$387$	−10.2130	−0.519155
$388$	−32.4588	−1.64785
$389$	3.05680	0.154986	0.0774928	−	0.996993i	$-0.475309\pi$
$389$	3.05680	0.154986	0.0774928	+	0.996993i	$0.475309\pi$
$390$	0.331020	0.0167618
$391$	−58.5790	−2.96247
$392$	−22.0908	−1.11575
$393$	11.1898	0.564452
$394$	−1.34857	−0.0679400
$395$	−2.80598	−0.141184
$396$	3.57156	0.179478
$397$	6.65411	0.333960	0.166980	−	0.985960i	$-0.446598\pi$
$397$	6.65411	0.333960	0.166980	+	0.985960i	$0.446598\pi$
$398$	39.3661	1.97325
$399$	5.54832	0.277763
$400$	−1.07264	−0.0536321
$401$	18.0655	0.902148	0.451074	−	0.892487i	$-0.351041\pi$
$401$	18.0655	0.902148	0.451074	+	0.892487i	$0.351041\pi$
$402$	−23.4933	−1.17174
$403$	0.0224231	0.00111698
$404$	27.0883	1.34770
$405$	3.97186	0.197363
$406$	30.2836	1.50295
$407$	0.683573	0.0338834
$408$	−20.5895	−1.01933
$409$	30.6190	1.51401	0.757005	−	0.653409i	$-0.226662\pi$
$409$	30.6190	1.51401	0.757005	+	0.653409i	$0.226662\pi$
$410$	−11.1046	−0.548418
$411$	17.5598	0.866163
$412$	−10.7064	−0.527465
$413$	−43.3369	−2.13247
$414$	22.3752	1.09968
$415$	5.52580	0.271251
$416$	−0.748646	−0.0367054
$417$	24.5406	1.20176
$418$	−2.23198	−0.109170
$419$	−38.4886	−1.88029	−0.940145	−	0.340774i	$-0.889311\pi$
$419$	−38.4886	−1.88029	−0.940145	+	0.340774i	$0.889311\pi$
$420$	16.5437	0.807251
$421$	−33.2827	−1.62210	−0.811049	−	0.584979i	$-0.801103\pi$
$421$	−33.2827	−1.62210	−0.811049	+	0.584979i	$0.801103\pi$
$422$	−44.4791	−2.16521
$423$	−4.62290	−0.224773
$424$	22.2532	1.08071
$425$	−6.99924	−0.339513
$426$	−4.76210	−0.230725
$427$	−14.5960	−0.706349
$428$	−21.2620	−1.02774
$429$	−0.148307	−0.00716034
$430$	19.0308	0.917749
$431$	4.84543	0.233396	0.116698	−	0.993167i	$-0.462769\pi$
$431$	4.84543	0.233396	0.116698	+	0.993167i	$0.462769\pi$
$432$	−6.04474	−0.290828
$433$	29.2320	1.40480	0.702400	−	0.711783i	$-0.252112\pi$
$433$	29.2320	1.40480	0.702400	+	0.711783i	$0.252112\pi$
$434$	1.87235	0.0898755
$435$	−4.40715	−0.211307
$436$	−45.1473	−2.16216
$437$	−8.36934	−0.400360
$438$	25.2171	1.20492
$439$	−12.2196	−0.583211	−0.291605	−	0.956539i	$-0.594189\pi$
$439$	−12.2196	−0.583211	−0.291605	+	0.956539i	$0.594189\pi$
$440$	−2.19127	−0.104464
$441$	−12.0754	−0.575018
$442$	−1.72585	−0.0820903
$443$	−24.1325	−1.14657	−0.573284	−	0.819357i	$-0.694331\pi$
$443$	−24.1325	−1.14657	−0.573284	+	0.819357i	$0.694331\pi$
$444$	2.73626	0.129857
$445$	−0.654537	−0.0310280
$446$	−28.1732	−1.33404
$447$	−24.5945	−1.16328
$448$	−53.6461	−2.53454
$449$	−8.77225	−0.413988	−0.206994	−	0.978342i	$-0.566368\pi$
$449$	−8.77225	−0.413988	−0.206994	+	0.978342i	$0.566368\pi$
$450$	2.67348	0.126029
$451$	4.97522	0.234274
$452$	−7.98176	−0.375430
$453$	−11.6335	−0.546589
$454$	−16.8964	−0.792987
$455$	0.456585	0.0214050
$456$	−2.94168	−0.137757
$457$	−6.85848	−0.320826	−0.160413	−	0.987050i	$-0.551283\pi$
$457$	−6.85848	−0.320826	−0.160413	+	0.987050i	$0.551283\pi$
$458$	18.3082	0.855487
$459$	−39.4433	−1.84106
$460$	−24.9553	−1.16355
$461$	−24.9007	−1.15974	−0.579870	−	0.814709i	$-0.696896\pi$
$461$	−24.9007	−1.15974	−0.579870	+	0.814709i	$0.696896\pi$
$462$	−12.3838	−0.576145
$463$	−2.79250	−0.129778	−0.0648892	−	0.997892i	$-0.520669\pi$
$463$	−2.79250	−0.129778	−0.0648892	+	0.997892i	$0.520669\pi$
$464$	3.52137	0.163475
$465$	−0.272481	−0.0126360
$466$	−37.9222	−1.75671
$467$	−7.55358	−0.349538	−0.174769	−	0.984609i	$-0.555918\pi$
$467$	−7.55358	−0.349538	−0.174769	+	0.984609i	$0.555918\pi$
$468$	0.394565	0.0182388
$469$	−32.4050	−1.49632
$470$	8.61430	0.397348
$471$	25.1727	1.15990
$472$	22.9770	1.05760
$473$	−8.52642	−0.392045
$474$	8.40768	0.386178
$475$	−1.00000	−0.0458831
$476$	−86.2547	−3.95348
$477$	12.1642	0.556961
$478$	34.0467	1.55726
$479$	28.2961	1.29288	0.646440	−	0.762965i	$-0.276257\pi$
$479$	28.2961	1.29288	0.646440	+	0.762965i	$0.276257\pi$
$480$	9.09738	0.415237
$481$	0.0755172	0.00344329
$482$	3.42928	0.156199
$483$	−46.4357	−2.11290
$484$	2.98176	0.135534
$485$	10.8858	0.494299
$486$	25.8332	1.17182
$487$	−22.0840	−1.00072	−0.500361	−	0.865817i	$-0.666799\pi$
$487$	−22.0840	−1.00072	−0.500361	+	0.865817i	$0.666799\pi$
$488$	7.73870	0.350315
$489$	−26.5837	−1.20216
$490$	22.5013	1.01650
$491$	34.1421	1.54081	0.770405	−	0.637555i	$-0.220054\pi$
$491$	34.1421	1.54081	0.770405	+	0.637555i	$0.220054\pi$
$492$	19.9152	0.897848
$493$	22.9777	1.03487
$494$	−0.246577	−0.0110940
$495$	−1.19780	−0.0538373
$496$	0.217715	0.00977571
$497$	−6.56851	−0.294638
$498$	−16.5572	−0.741946
$499$	−9.33688	−0.417976	−0.208988	−	0.977918i	$-0.567017\pi$
$499$	−9.33688	−0.417976	−0.208988	+	0.977918i	$0.567017\pi$
$500$	−2.98176	−0.133348
$501$	−2.57211	−0.114914
$502$	−55.0678	−2.45780
$503$	16.8794	0.752616	0.376308	−	0.926495i	$-0.377193\pi$
$503$	16.8794	0.752616	0.376308	+	0.926495i	$0.377193\pi$
$504$	10.8478	0.483198
$505$	−9.08470	−0.404264
$506$	18.6802	0.830438
$507$	17.4356	0.774342
$508$	−39.0010	−1.73039
$509$	13.2971	0.589383	0.294691	−	0.955592i	$-0.404783\pi$
$509$	13.2971	0.589383	0.294691	+	0.955592i	$0.404783\pi$
$510$	20.9721	0.928662
$511$	34.7827	1.53870
$512$	−11.9698	−0.528995
$513$	−5.63538	−0.248808
$514$	−26.5722	−1.17205
$515$	3.59063	0.158222
$516$	−34.1303	−1.50250
$517$	−3.85948	−0.169740
$518$	6.30574	0.277058
$519$	4.44110	0.194943
$520$	−0.242079	−0.0106158
$521$	−20.5200	−0.898997	−0.449499	−	0.893281i	$-0.648397\pi$
$521$	−20.5200	−0.898997	−0.449499	+	0.893281i	$0.648397\pi$
$522$	−8.77675	−0.384148
$523$	19.6848	0.860757	0.430378	−	0.902649i	$-0.358380\pi$
$523$	19.6848	0.860757	0.430378	+	0.902649i	$0.358380\pi$
$524$	−24.8539	−1.08575
$525$	−5.54832	−0.242148
$526$	1.89608	0.0826731
$527$	1.42064	0.0618842
$528$	−1.43998	−0.0626670
$529$	47.0458	2.04547
$530$	−22.6668	−0.984581
$531$	12.5598	0.545049
$532$	−12.3235	−0.534289
$533$	0.549634	0.0238073
$534$	1.96122	0.0848703
$535$	7.13069	0.308287
$536$	17.1809	0.742102
$537$	29.5170	1.27375
$538$	37.2306	1.60513
$539$	−10.0813	−0.434231
$540$	−16.8033	−0.723100
$541$	−15.9758	−0.686852	−0.343426	−	0.939180i	$-0.611587\pi$
$541$	−15.9758	−0.686852	−0.343426	+	0.939180i	$0.611587\pi$
$542$	29.5522	1.26937
$543$	22.8547	0.980787
$544$	−47.4314	−2.03360
$545$	15.1412	0.648577
$546$	−1.36809	−0.0585487
$547$	−41.5489	−1.77650	−0.888252	−	0.459357i	$-0.848080\pi$
$547$	−41.5489	−1.77650	−0.888252	+	0.459357i	$0.848080\pi$
$548$	−39.0024	−1.66610
$549$	4.23018	0.180540
$550$	2.23198	0.0951722
$551$	3.28289	0.139856
$552$	24.6200	1.04789
$553$	11.5970	0.493153
$554$	38.4178	1.63222
$555$	−0.917668	−0.0389528
$556$	−54.5075	−2.31163
$557$	30.0766	1.27439	0.637194	−	0.770703i	$-0.280095\pi$
$557$	30.0766	1.27439	0.637194	+	0.770703i	$0.280095\pi$
$558$	−0.542640	−0.0229718
$559$	−0.941951	−0.0398403
$560$	4.43317	0.187336
$561$	−9.39619	−0.396707
$562$	−20.8121	−0.877906
$563$	21.3804	0.901079	0.450539	−	0.892757i	$-0.351232\pi$
$563$	21.3804	0.901079	0.450539	+	0.892757i	$0.351232\pi$
$564$	−15.4491	−0.650523
$565$	2.67686	0.112617
$566$	−28.7127	−1.20689
$567$	−16.4155	−0.689386
$568$	3.48258	0.146126
$569$	11.5873	0.485767	0.242883	−	0.970055i	$-0.421907\pi$
$569$	11.5873	0.485767	0.242883	+	0.970055i	$0.421907\pi$
$570$	2.99635	0.125503
$571$	−25.1618	−1.05299	−0.526494	−	0.850179i	$-0.676494\pi$
$571$	−25.1618	−1.05299	−0.526494	+	0.850179i	$0.676494\pi$
$572$	0.329408	0.0137732
$573$	17.2839	0.722046
$574$	45.8948	1.91561
$575$	8.36934	0.349026
$576$	15.5476	0.647817
$577$	−1.55228	−0.0646222	−0.0323111	−	0.999478i	$-0.510287\pi$
$577$	−1.55228	−0.0646222	−0.0323111	+	0.999478i	$0.510287\pi$
$578$	−71.3996	−2.96983
$579$	−3.34605	−0.139057
$580$	9.78879	0.406457
$581$	−22.8378	−0.947473
$582$	−32.6177	−1.35205
$583$	10.1554	0.420595
$584$	−18.4416	−0.763118
$585$	−0.132327	−0.00547103
$586$	32.1612	1.32857
$587$	−5.32002	−0.219581	−0.109790	−	0.993955i	$-0.535018\pi$
$587$	−5.32002	−0.219581	−0.109790	+	0.993955i	$0.535018\pi$
$588$	−40.3542	−1.66418
$589$	0.202971	0.00836329
$590$	−23.4039	−0.963525
$591$	−0.811117	−0.0333649
$592$	0.733228	0.0301355
$593$	36.2723	1.48952	0.744762	−	0.667331i	$-0.232563\pi$
$593$	36.2723	1.48952	0.744762	+	0.667331i	$0.232563\pi$
$594$	12.5781	0.516085
$595$	28.9275	1.18591
$596$	54.6271	2.23761
$597$	23.6773	0.969048
$598$	2.06369	0.0843904
$599$	−40.0043	−1.63453	−0.817266	−	0.576261i	$-0.804511\pi$
$599$	−40.0043	−1.63453	−0.817266	+	0.576261i	$0.804511\pi$
$600$	2.94168	0.120094
$601$	36.6257	1.49400	0.746998	−	0.664827i	$-0.231495\pi$
$601$	36.6257	1.49400	0.746998	+	0.664827i	$0.231495\pi$
$602$	−78.6535	−3.20568
$603$	9.39154	0.382453
$604$	25.8393	1.05139
$605$	−1.00000	−0.0406558
$606$	27.2209	1.10577
$607$	23.7724	0.964893	0.482446	−	0.875926i	$-0.339748\pi$
$607$	23.7724	0.964893	0.482446	+	0.875926i	$0.339748\pi$
$608$	−6.77665	−0.274829
$609$	18.2145	0.738090
$610$	−7.88251	−0.319153
$611$	−0.426374	−0.0172492
$612$	24.9982	1.01049
$613$	−2.77756	−0.112184	−0.0560922	−	0.998426i	$-0.517864\pi$
$613$	−2.77756	−0.112184	−0.0560922	+	0.998426i	$0.517864\pi$
$614$	63.1291	2.54768
$615$	−6.67903	−0.269324
$616$	9.05639	0.364892
$617$	39.0259	1.57112	0.785562	−	0.618783i	$-0.212374\pi$
$617$	39.0259	1.57112	0.785562	+	0.618783i	$0.212374\pi$
$618$	−10.7588	−0.432781
$619$	45.6890	1.83639	0.918197	−	0.396123i	$-0.129645\pi$
$619$	45.6890	1.83639	0.918197	+	0.396123i	$0.129645\pi$
$620$	0.605211	0.0243059
$621$	47.1644	1.89264
$622$	27.6636	1.10921
$623$	2.70517	0.108380
$624$	−0.159081	−0.00636832
$625$	1.00000	0.0400000
$626$	73.7806	2.94886
$627$	−1.34246	−0.0536127
$628$	−55.9115	−2.23111
$629$	4.78448	0.190770
$630$	−11.0494	−0.440217
$631$	−20.2465	−0.806000	−0.403000	−	0.915200i	$-0.632032\pi$
$631$	−20.2465	−0.806000	−0.403000	+	0.915200i	$0.632032\pi$
$632$	−6.14864	−0.244580
$633$	−26.7526	−1.06332
$634$	−22.2260	−0.882707
$635$	13.0799	0.519059
$636$	40.6510	1.61192
$637$	−1.11372	−0.0441273
$638$	−7.32737	−0.290093
$639$	1.90367	0.0753081
$640$	−15.4181	−0.609453
$641$	22.9945	0.908229	0.454115	−	0.890943i	$-0.349956\pi$
$641$	22.9945	0.908229	0.454115	+	0.890943i	$0.349956\pi$
$642$	−21.3660	−0.843250
$643$	−36.6452	−1.44514	−0.722572	−	0.691295i	$-0.757040\pi$
$643$	−36.6452	−1.44514	−0.722572	+	0.691295i	$0.757040\pi$
$644$	103.139	4.06425
$645$	11.4464	0.450701
$646$	−15.6222	−0.614647
$647$	36.1179	1.41994	0.709970	−	0.704232i	$-0.248709\pi$
$647$	36.1179	1.41994	0.709970	+	0.704232i	$0.248709\pi$
$648$	8.70339	0.341902
$649$	10.4857	0.411600
$650$	0.246577	0.00967155
$651$	1.12615	0.0441373
$652$	59.0455	2.31240
$653$	28.1710	1.10242	0.551208	−	0.834368i	$-0.314167\pi$
$653$	28.1710	1.10242	0.551208	+	0.834368i	$0.314167\pi$
$654$	−45.3683	−1.77404
$655$	8.33532	0.325688
$656$	5.33663	0.208360
$657$	−10.0806	−0.393284
$658$	−35.6025	−1.38793
$659$	−46.4859	−1.81083	−0.905416	−	0.424525i	$-0.860441\pi$
$659$	−46.4859	−1.81083	−0.905416	+	0.424525i	$0.860441\pi$
$660$	−4.00289	−0.155812
$661$	−37.2017	−1.44698	−0.723489	−	0.690336i	$-0.757463\pi$
$661$	−37.2017	−1.44698	−0.723489	+	0.690336i	$0.757463\pi$
$662$	−66.8648	−2.59877
$663$	−1.03804	−0.0403140
$664$	12.1085	0.469900
$665$	4.13295	0.160269
$666$	−1.82752	−0.0708148
$667$	−27.4756	−1.06386
$668$	5.71296	0.221041
$669$	−16.9451	−0.655137
$670$	−17.5002	−0.676091
$671$	3.53161	0.136336
$672$	−37.5990	−1.45041
$673$	31.3183	1.20723	0.603616	−	0.797276i	$-0.293726\pi$
$673$	31.3183	1.20723	0.603616	+	0.797276i	$0.293726\pi$
$674$	−26.2094	−1.00955
$675$	5.63538	0.216906
$676$	−38.7264	−1.48948
$677$	44.6151	1.71470	0.857349	−	0.514736i	$-0.172110\pi$
$677$	44.6151	1.71470	0.857349	+	0.514736i	$0.172110\pi$
$678$	−8.02082	−0.308038
$679$	−44.9905	−1.72658
$680$	−15.3372	−0.588154
$681$	−10.1626	−0.389431
$682$	−0.453029	−0.0173474
$683$	11.0815	0.424024	0.212012	−	0.977267i	$-0.431998\pi$
$683$	11.0815	0.424024	0.212012	+	0.977267i	$0.431998\pi$
$684$	3.57156	0.136562
$685$	13.0803	0.499774
$686$	−28.4238	−1.08523
$687$	11.0117	0.420124
$688$	−9.14579	−0.348680
$689$	1.12191	0.0427415
$690$	−25.0775	−0.954682
$691$	−28.3303	−1.07773	−0.538867	−	0.842391i	$-0.681148\pi$
$691$	−28.3303	−1.07773	−0.538867	+	0.842391i	$0.681148\pi$
$692$	−9.86419	−0.374980
$693$	4.95046	0.188052
$694$	−15.6610	−0.594482
$695$	18.2803	0.693412
$696$	−9.65724	−0.366057
$697$	34.8227	1.31900
$698$	−10.4018	−0.393714
$699$	−22.8089	−0.862711
$700$	12.3235	0.465783
$701$	41.4446	1.56534	0.782671	−	0.622435i	$-0.213857\pi$
$701$	41.4446	1.56534	0.782671	+	0.622435i	$0.213857\pi$
$702$	1.38956	0.0524454
$703$	0.683573	0.0257814
$704$	12.9801	0.489206
$705$	5.18120	0.195135
$706$	81.6635	3.07345
$707$	37.5466	1.41208
$708$	41.9731	1.57744
$709$	−39.1483	−1.47025	−0.735123	−	0.677933i	$-0.762876\pi$
$709$	−39.1483	−1.47025	−0.735123	+	0.677933i	$0.762876\pi$
$710$	−3.54730	−0.133128
$711$	−3.36101	−0.126048
$712$	−1.43426	−0.0537513
$713$	−1.69874	−0.0636182
$714$	−86.6768	−3.24380
$715$	−0.110474	−0.00413151
$716$	−65.5607	−2.45012
$717$	20.4779	0.764760
$718$	−2.44170	−0.0911233
$719$	27.5810	1.02860	0.514299	−	0.857611i	$-0.328052\pi$
$719$	27.5810	1.02860	0.514299	+	0.857611i	$0.328052\pi$
$720$	−1.28481	−0.0478822
$721$	−14.8399	−0.552666
$722$	−2.23198	−0.0830659
$723$	2.06259	0.0767084
$724$	−50.7628	−1.88659
$725$	−3.28289	−0.121924
$726$	2.99635	0.111205
$727$	44.7807	1.66083	0.830413	−	0.557149i	$-0.188105\pi$
$727$	44.7807	1.66083	0.830413	+	0.557149i	$0.188105\pi$
$728$	1.00050	0.0370810
$729$	27.4533	1.01679
$730$	18.7843	0.695237
$731$	−59.6784	−2.20729
$732$	14.1366	0.522506
$733$	−15.0617	−0.556316	−0.278158	−	0.960535i	$-0.589724\pi$
$733$	−15.0617	−0.556316	−0.278158	+	0.960535i	$0.589724\pi$
$734$	27.5281	1.01608
$735$	13.5337	0.499198
$736$	56.7161	2.09058
$737$	7.84063	0.288813
$738$	−13.3011	−0.489622
$739$	−16.7492	−0.616128	−0.308064	−	0.951366i	$-0.599681\pi$
$739$	−16.7492	−0.616128	−0.308064	+	0.951366i	$0.599681\pi$
$740$	2.03825	0.0749274
$741$	−0.148307	−0.00544820
$742$	93.6806	3.43912
$743$	38.4706	1.41135	0.705674	−	0.708537i	$-0.250644\pi$
$743$	38.4706	1.41135	0.705674	+	0.708537i	$0.250644\pi$
$744$	−0.597078	−0.0218899
$745$	−18.3204	−0.671209
$746$	20.5364	0.751890
$747$	6.61882	0.242170
$748$	20.8700	0.763083
$749$	−29.4708	−1.07684
$750$	−2.99635	−0.109411
$751$	−38.7951	−1.41565	−0.707827	−	0.706386i	$-0.750324\pi$
$751$	−38.7951	−1.41565	−0.707827	+	0.706386i	$0.750324\pi$
$752$	−4.13984	−0.150964
$753$	−33.1213	−1.20701
$754$	−0.809486	−0.0294797
$755$	−8.66581	−0.315381
$756$	69.4473	2.52577
$757$	49.3111	1.79224	0.896120	−	0.443811i	$-0.146374\pi$
$757$	49.3111	1.79224	0.896120	+	0.443811i	$0.146374\pi$
$758$	−82.0937	−2.98178
$759$	11.2355	0.407823
$760$	−2.19127	−0.0794856
$761$	−7.21992	−0.261722	−0.130861	−	0.991401i	$-0.541774\pi$
$761$	−7.21992	−0.261722	−0.130861	+	0.991401i	$0.541774\pi$
$762$	−39.1919	−1.41977
$763$	−62.5777	−2.26547
$764$	−38.3896	−1.38889
$765$	−8.38371	−0.303114
$766$	−53.1460	−1.92024
$767$	1.15840	0.0418274
$768$	11.3474	0.409466
$769$	15.7986	0.569714	0.284857	−	0.958570i	$-0.408054\pi$
$769$	15.7986	0.569714	0.284857	+	0.958570i	$0.408054\pi$
$770$	−9.22468	−0.332434
$771$	−15.9822	−0.575585
$772$	7.43196	0.267482
$773$	13.1031	0.471287	0.235643	−	0.971840i	$-0.424280\pi$
$773$	13.1031	0.471287	0.235643	+	0.971840i	$0.424280\pi$
$774$	22.7952	0.819357
$775$	−0.202971	−0.00729095
$776$	23.8537	0.856298
$777$	3.79268	0.136062
$778$	−6.82272	−0.244606
$779$	4.97522	0.178256
$780$	−0.442216	−0.0158339
$781$	1.58930	0.0568697
$782$	130.747	4.67552
$783$	−18.5004	−0.661149
$784$	−10.8136	−0.386200
$785$	18.7512	0.669259
$786$	−24.9755	−0.890847
$787$	−4.60737	−0.164235	−0.0821176	−	0.996623i	$-0.526168\pi$
$787$	−4.60737	−0.164235	−0.0821176	+	0.996623i	$0.526168\pi$
$788$	1.80158	0.0641787
$789$	1.14043	0.0406002
$790$	6.26290	0.222824
$791$	−11.0633	−0.393367
$792$	−2.62471	−0.0932648
$793$	0.390153	0.0138547
$794$	−14.8519	−0.527073
$795$	−13.6332	−0.483521
$796$	−52.5901	−1.86401
$797$	12.6952	0.449687	0.224844	−	0.974395i	$-0.427813\pi$
$797$	12.6952	0.449687	0.224844	+	0.974395i	$0.427813\pi$
$798$	−12.3838	−0.438380
$799$	−27.0134	−0.955666
$800$	6.77665	0.239591
$801$	−0.784006	−0.0277015
$802$	−40.3219	−1.42382
$803$	−8.41595	−0.296992
$804$	31.3852	1.10687
$805$	−34.5901	−1.21914
$806$	−0.0500481	−0.00176287
$807$	22.3929	0.788267
$808$	−19.9070	−0.700325
$809$	−3.22295	−0.113313	−0.0566564	−	0.998394i	$-0.518044\pi$
$809$	−3.22295	−0.113313	−0.0566564	+	0.998394i	$0.518044\pi$
$810$	−8.86513	−0.311489
$811$	18.0724	0.634609	0.317304	−	0.948324i	$-0.397222\pi$
$811$	18.0724	0.634609	0.317304	+	0.948324i	$0.397222\pi$
$812$	−40.4566	−1.41975
$813$	17.7746	0.623381
$814$	−1.52572	−0.0534766
$815$	−19.8023	−0.693643
$816$	−10.0787	−0.352826
$817$	−8.52642	−0.298302
$818$	−68.3410	−2.38949
$819$	0.546899	0.0191102
$820$	14.8349	0.518057
$821$	0.572887	0.0199939	0.00999694	−	0.999950i	$-0.496818\pi$
$821$	0.572887	0.0199939	0.00999694	+	0.999950i	$0.496818\pi$
$822$	−39.1933	−1.36702
$823$	−37.8486	−1.31932	−0.659660	−	0.751564i	$-0.729300\pi$
$823$	−37.8486	−1.31932	−0.659660	+	0.751564i	$0.729300\pi$
$824$	7.86802	0.274095
$825$	1.34246	0.0467384
$826$	96.7273	3.36557
$827$	−10.9219	−0.379792	−0.189896	−	0.981804i	$-0.560815\pi$
$827$	−10.9219	−0.379792	−0.189896	+	0.981804i	$0.560815\pi$
$828$	−29.8916	−1.03880
$829$	49.6425	1.72415	0.862077	−	0.506777i	$-0.169163\pi$
$829$	49.6425	1.72415	0.862077	+	0.506777i	$0.169163\pi$
$830$	−12.3335	−0.428102
$831$	23.1069	0.801571
$832$	1.43397	0.0497139
$833$	−70.5612	−2.44480
$834$	−54.7742	−1.89668
$835$	−1.91597	−0.0663050
$836$	2.98176	0.103126
$837$	−1.14382	−0.0395362
$838$	85.9060	2.96757
$839$	−43.5296	−1.50281	−0.751404	−	0.659842i	$-0.770623\pi$
$839$	−43.5296	−1.50281	−0.751404	+	0.659842i	$0.770623\pi$
$840$	−12.1578	−0.419485
$841$	−18.2226	−0.628366
$842$	74.2864	2.56008
$843$	−12.5177	−0.431134
$844$	59.4206	2.04534
$845$	12.9878	0.446794
$846$	10.3182	0.354748
$847$	4.13295	0.142010
$848$	10.8931	0.374072
$849$	−17.2697	−0.592694
$850$	15.6222	0.535837
$851$	−5.72105	−0.196115
$852$	6.36180	0.217952
$853$	24.4635	0.837616	0.418808	−	0.908075i	$-0.362448\pi$
$853$	24.4635	0.837616	0.418808	+	0.908075i	$0.362448\pi$
$854$	32.5780	1.11480
$855$	−1.19780	−0.0409640
$856$	15.6252	0.534060
$857$	−22.5611	−0.770672	−0.385336	−	0.922776i	$-0.625914\pi$
$857$	−22.5611	−0.770672	−0.385336	+	0.922776i	$0.625914\pi$
$858$	0.331020	0.0113008
$859$	−14.6892	−0.501189	−0.250595	−	0.968092i	$-0.580626\pi$
$859$	−14.6892	−0.501189	−0.250595	+	0.968092i	$0.580626\pi$
$860$	−25.4237	−0.866941
$861$	27.6041	0.940745
$862$	−10.8149	−0.368357
$863$	−41.8413	−1.42429	−0.712147	−	0.702031i	$-0.752277\pi$
$863$	−41.8413	−1.42429	−0.712147	+	0.702031i	$0.752277\pi$
$864$	38.1890	1.29922
$865$	3.30818	0.112482
$866$	−65.2453	−2.21713
$867$	−42.9443	−1.45847
$868$	−2.50131	−0.0848999
$869$	−2.80598	−0.0951862
$870$	9.83669	0.333495
$871$	0.866189	0.0293497
$872$	33.1784	1.12356
$873$	13.0391	0.441305
$874$	18.6802	0.631868
$875$	−4.13295	−0.139719
$876$	−33.6881	−1.13821
$877$	−54.9892	−1.85685	−0.928426	−	0.371516i	$-0.878838\pi$
$877$	−54.9892	−1.85685	−0.928426	+	0.371516i	$0.878838\pi$
$878$	27.2740	0.920453
$879$	19.3438	0.652450
$880$	−1.07264	−0.0361587
$881$	−30.1002	−1.01410	−0.507050	−	0.861917i	$-0.669264\pi$
$881$	−30.1002	−1.01410	−0.507050	+	0.861917i	$0.669264\pi$
$882$	26.9521	0.907524
$883$	54.9963	1.85077	0.925386	−	0.379026i	$-0.123741\pi$
$883$	54.9963	1.85077	0.925386	+	0.379026i	$0.123741\pi$
$884$	2.30560	0.0775458
$885$	−14.0766	−0.473181
$886$	53.8633	1.80957
$887$	32.8266	1.10221	0.551105	−	0.834436i	$-0.314206\pi$
$887$	32.8266	1.10221	0.551105	+	0.834436i	$0.314206\pi$
$888$	−2.01085	−0.0674799
$889$	−54.0585	−1.81306
$890$	1.46092	0.0489700
$891$	3.97186	0.133062
$892$	37.6371	1.26018
$893$	−3.85948	−0.129153
$894$	54.8944	1.83594
$895$	21.9873	0.734954
$896$	63.7222	2.12881
$897$	1.24123	0.0414436
$898$	19.5795	0.653377
$899$	0.666333	0.0222235
$900$	−3.57156	−0.119052
$901$	71.0802	2.36803
$902$	−11.1046	−0.369743
$903$	−47.3073	−1.57429
$904$	5.86572	0.195091
$905$	17.0245	0.565913
$906$	25.9658	0.862655
$907$	14.9631	0.496841	0.248421	−	0.968652i	$-0.420088\pi$
$907$	14.9631	0.496841	0.248421	+	0.968652i	$0.420088\pi$
$908$	22.5723	0.749087
$909$	−10.8817	−0.360922
$910$	−1.01909	−0.0337825
$911$	26.0121	0.861818	0.430909	−	0.902395i	$-0.358193\pi$
$911$	26.0121	0.861818	0.430909	+	0.902395i	$0.358193\pi$
$912$	−1.43998	−0.0476824
$913$	5.52580	0.182877
$914$	15.3080	0.506344
$915$	−4.74105	−0.156734
$916$	−24.4583	−0.808126
$917$	−34.4495	−1.13762
$918$	88.0370	2.90565
$919$	40.8366	1.34707	0.673537	−	0.739153i	$-0.264774\pi$
$919$	40.8366	1.34707	0.673537	+	0.739153i	$0.264774\pi$
$920$	18.3394	0.604633
$921$	37.9699	1.25115
$922$	55.5779	1.83036
$923$	0.175577	0.00577919
$924$	16.5437	0.544249
$925$	−0.683573	−0.0224757
$926$	6.23281	0.204823
$927$	4.30087	0.141259
$928$	−22.2470	−0.730294
$929$	20.3088	0.666309	0.333155	−	0.942872i	$-0.391887\pi$
$929$	20.3088	0.666309	0.333155	+	0.942872i	$0.391887\pi$
$930$	0.608173	0.0199428
$931$	−10.0813	−0.330400
$932$	50.6611	1.65946
$933$	16.6387	0.544725
$934$	16.8595	0.551659
$935$	−6.99924	−0.228899
$936$	−0.289963	−0.00947772
$937$	30.8199	1.00684	0.503422	−	0.864041i	$-0.332074\pi$
$937$	30.8199	1.00684	0.503422	+	0.864041i	$0.332074\pi$
$938$	72.3274	2.36157
$939$	44.3764	1.44817
$940$	−11.5080	−0.375351
$941$	−18.2874	−0.596153	−0.298077	−	0.954542i	$-0.596345\pi$
$941$	−18.2874	−0.596153	−0.298077	+	0.954542i	$0.596345\pi$
$942$	−56.1851	−1.83061
$943$	−41.6393	−1.35596
$944$	11.2474	0.366072
$945$	−23.2907	−0.757648
$946$	19.0308	0.618746
$947$	−53.3021	−1.73209	−0.866043	−	0.499970i	$-0.833344\pi$
$947$	−53.3021	−1.73209	−0.866043	+	0.499970i	$0.833344\pi$
$948$	−11.2320	−0.364799
$949$	−0.929746	−0.0301808
$950$	2.23198	0.0724152
$951$	−13.3681	−0.433492
$952$	63.3878	2.05441
$953$	24.4980	0.793568	0.396784	−	0.917912i	$-0.370126\pi$
$953$	24.4980	0.793568	0.396784	+	0.917912i	$0.370126\pi$
$954$	−27.1503	−0.879024
$955$	12.8748	0.416619
$956$	−45.4837	−1.47105
$957$	−4.40715	−0.142463
$958$	−63.1564	−2.04049
$959$	−54.0604	−1.74570
$960$	−17.4253	−0.562398
$961$	−30.9588	−0.998671
$962$	−0.168553	−0.00543438
$963$	8.54116	0.275235
$964$	−4.58124	−0.147552
$965$	−2.49248	−0.0802357
$966$	103.644	3.33469
$967$	29.9164	0.962047	0.481023	−	0.876708i	$-0.340265\pi$
$967$	29.9164	0.962047	0.481023	+	0.876708i	$0.340265\pi$
$968$	−2.19127	−0.0704299
$969$	−9.39619	−0.301849
$970$	−24.2970	−0.780128
$971$	42.2300	1.35522	0.677612	−	0.735419i	$-0.263015\pi$
$971$	42.2300	1.35522	0.677612	+	0.735419i	$0.263015\pi$
$972$	−34.5111	−1.10694
$973$	−75.5517	−2.42208
$974$	49.2912	1.57939
$975$	0.148307	0.00474963
$976$	3.78816	0.121256
$977$	25.2891	0.809070	0.404535	−	0.914523i	$-0.367433\pi$
$977$	25.2891	0.809070	0.404535	+	0.914523i	$0.367433\pi$
$978$	59.3345	1.89731
$979$	−0.654537	−0.0209191
$980$	−30.0599	−0.960229
$981$	18.1362	0.579043
$982$	−76.2046	−2.43179
$983$	−9.20537	−0.293606	−0.146803	−	0.989166i	$-0.546898\pi$
$983$	−9.20537	−0.293606	−0.146803	+	0.989166i	$0.546898\pi$
$984$	−14.6355	−0.466563
$985$	−0.604202	−0.0192515
$986$	−51.2860	−1.63328
$987$	−21.4136	−0.681603
$988$	0.329408	0.0104799
$989$	71.3605	2.26913
$990$	2.67348	0.0849687
$991$	32.9139	1.04554	0.522772	−	0.852472i	$-0.324898\pi$
$991$	32.9139	1.04554	0.522772	+	0.852472i	$0.324898\pi$
$992$	−1.37547	−0.0436711
$993$	−40.2168	−1.27624
$994$	14.6608	0.465013
$995$	17.6373	0.559139
$996$	22.1191	0.700871
$997$	−6.45140	−0.204318	−0.102159	−	0.994768i	$-0.532575\pi$
$997$	−6.45140	−0.204318	−0.102159	+	0.994768i	$0.532575\pi$
$998$	20.8398	0.659671
$999$	−3.85219	−0.121878

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.2.a.f.1.2	✓	6
3.2	odd	2		9405.2.a.z.1.5		6
5.4	even	2		5225.2.a.l.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.2.a.f.1.2	✓	6	1.1	even	1	trivial
5225.2.a.l.1.5		6	5.4	even	2
9405.2.a.z.1.5		6	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:	\( N \)	\(=\)	\( 1045 = 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1045.a (trivial)

\(f(q)\)	\(=\)	\(q-2.23198 q^{2} -1.34246 q^{3} +2.98176 q^{4} -1.00000 q^{5} +2.99635 q^{6} +4.13295 q^{7} -2.19127 q^{8} -1.19780 q^{9} +O(q^{10})\)	\(q-2.23198 q^{2} -1.34246 q^{3} +2.98176 q^{4} -1.00000 q^{5} +2.99635 q^{6} +4.13295 q^{7} -2.19127 q^{8} -1.19780 q^{9} +2.23198 q^{10} -1.00000 q^{11} -4.00289 q^{12} -0.110474 q^{13} -9.22468 q^{14} +1.34246 q^{15} -1.07264 q^{16} -6.99924 q^{17} +2.67348 q^{18} -1.00000 q^{19} -2.98176 q^{20} -5.54832 q^{21} +2.23198 q^{22} +8.36934 q^{23} +2.94168 q^{24} +1.00000 q^{25} +0.246577 q^{26} +5.63538 q^{27} +12.3235 q^{28} -3.28289 q^{29} -2.99635 q^{30} -0.202971 q^{31} +6.77665 q^{32} +1.34246 q^{33} +15.6222 q^{34} -4.13295 q^{35} -3.57156 q^{36} -0.683573 q^{37} +2.23198 q^{38} +0.148307 q^{39} +2.19127 q^{40} -4.97522 q^{41} +12.3838 q^{42} +8.52642 q^{43} -2.98176 q^{44} +1.19780 q^{45} -18.6802 q^{46} +3.85948 q^{47} +1.43998 q^{48} +10.0813 q^{49} -2.23198 q^{50} +9.39619 q^{51} -0.329408 q^{52} -10.1554 q^{53} -12.5781 q^{54} +1.00000 q^{55} -9.05639 q^{56} +1.34246 q^{57} +7.32737 q^{58} -10.4857 q^{59} +4.00289 q^{60} -3.53161 q^{61} +0.453029 q^{62} -4.95046 q^{63} -12.9801 q^{64} +0.110474 q^{65} -2.99635 q^{66} -7.84063 q^{67} -20.8700 q^{68} -11.2355 q^{69} +9.22468 q^{70} -1.58930 q^{71} +2.62471 q^{72} +8.41595 q^{73} +1.52572 q^{74} -1.34246 q^{75} -2.98176 q^{76} -4.13295 q^{77} -0.331020 q^{78} +2.80598 q^{79} +1.07264 q^{80} -3.97186 q^{81} +11.1046 q^{82} -5.52580 q^{83} -16.5437 q^{84} +6.99924 q^{85} -19.0308 q^{86} +4.40715 q^{87} +2.19127 q^{88} +0.654537 q^{89} -2.67348 q^{90} -0.456585 q^{91} +24.9553 q^{92} +0.272481 q^{93} -8.61430 q^{94} +1.00000 q^{95} -9.09738 q^{96} -10.8858 q^{97} -22.5013 q^{98} +1.19780 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{2} - q^{3} + 4 q^{4} - 6 q^{5} + 5 q^{7} - 12 q^{8} + q^{9}+O(q^{10}) \) 6 * q - 2 * q^2 - q^3 + 4 * q^4 - 6 * q^5 + 5 * q^7 - 12 * q^8 + q^9	\( 6 q - 2 q^{2} - q^{3} + 4 q^{4} - 6 q^{5} + 5 q^{7} - 12 q^{8} + q^{9} + 2 q^{10} - 6 q^{11} + q^{12} - 5 q^{13} - 8 q^{14} + q^{15} + 4 q^{16} + q^{17} + 6 q^{18} - 6 q^{19} - 4 q^{20} - 21 q^{21} + 2 q^{22} + 4 q^{23} - q^{24} + 6 q^{25} - 14 q^{26} - 16 q^{27} + 10 q^{28} - 9 q^{29} - 21 q^{31} - q^{32} + q^{33} - 5 q^{35} - 28 q^{36} - 3 q^{37} + 2 q^{38} + 20 q^{39} + 12 q^{40} - 23 q^{41} + q^{42} + 7 q^{43} - 4 q^{44} - q^{45} - 12 q^{46} - 18 q^{47} - 3 q^{49} - 2 q^{50} - 16 q^{51} + 13 q^{52} - 17 q^{53} + q^{54} + 6 q^{55} - 2 q^{56} + q^{57} + 23 q^{58} - 29 q^{59} - q^{60} + 17 q^{61} + 2 q^{62} + 6 q^{63} - 18 q^{64} + 5 q^{65} + 8 q^{67} - q^{68} - 38 q^{69} + 8 q^{70} - 12 q^{71} + 13 q^{72} + 2 q^{73} - 37 q^{74} - q^{75} - 4 q^{76} - 5 q^{77} + q^{78} + 3 q^{79} - 4 q^{80} - 2 q^{81} + 24 q^{82} - 11 q^{83} - 3 q^{84} - q^{85} - 12 q^{86} - 12 q^{87} + 12 q^{88} - 22 q^{89} - 6 q^{90} - 18 q^{91} - 15 q^{92} + 18 q^{93} + 22 q^{94} + 6 q^{95} - 17 q^{96} - 2 q^{97} - q^{98} - q^{99}+O(q^{100}) \) 6 * q - 2 * q^2 - q^3 + 4 * q^4 - 6 * q^5 + 5 * q^7 - 12 * q^8 + q^9 + 2 * q^10 - 6 * q^11 + q^12 - 5 * q^13 - 8 * q^14 + q^15 + 4 * q^16 + q^17 + 6 * q^18 - 6 * q^19 - 4 * q^20 - 21 * q^21 + 2 * q^22 + 4 * q^23 - q^24 + 6 * q^25 - 14 * q^26 - 16 * q^27 + 10 * q^28 - 9 * q^29 - 21 * q^31 - q^32 + q^33 - 5 * q^35 - 28 * q^36 - 3 * q^37 + 2 * q^38 + 20 * q^39 + 12 * q^40 - 23 * q^41 + q^42 + 7 * q^43 - 4 * q^44 - q^45 - 12 * q^46 - 18 * q^47 - 3 * q^49 - 2 * q^50 - 16 * q^51 + 13 * q^52 - 17 * q^53 + q^54 + 6 * q^55 - 2 * q^56 + q^57 + 23 * q^58 - 29 * q^59 - q^60 + 17 * q^61 + 2 * q^62 + 6 * q^63 - 18 * q^64 + 5 * q^65 + 8 * q^67 - q^68 - 38 * q^69 + 8 * q^70 - 12 * q^71 + 13 * q^72 + 2 * q^73 - 37 * q^74 - q^75 - 4 * q^76 - 5 * q^77 + q^78 + 3 * q^79 - 4 * q^80 - 2 * q^81 + 24 * q^82 - 11 * q^83 - 3 * q^84 - q^85 - 12 * q^86 - 12 * q^87 + 12 * q^88 - 22 * q^89 - 6 * q^90 - 18 * q^91 - 15 * q^92 + 18 * q^93 + 22 * q^94 + 6 * q^95 - 17 * q^96 - 2 * q^97 - q^98 - q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more