LMFDB - Embedded newform 399.2.a.e.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [399,2,Mod(1,399)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("399.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(399, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 399 = 3 \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	399.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,1,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$3.18603104065$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.404.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x - 1 $ x^3 - x^2 - 5*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			1.1
Root			$-0.210756$ of defining polynomial
Character	$\chi$	$=$	399.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.74483 q^{2} +1.00000 q^{3} +5.53407 q^{4} -2.53407 q^{5} -2.74483 q^{6} -1.00000 q^{7} -9.70041 q^{8} +1.00000 q^{9} +6.95558 q^{10} +2.42151 q^{11} +5.53407 q^{12} -0.421512 q^{13} +2.74483 q^{14} -2.53407 q^{15} +15.5578 q^{16} +6.53407 q^{17} -2.74483 q^{18} -1.00000 q^{19} -14.0237 q^{20} -1.00000 q^{21} -6.64663 q^{22} -1.57849 q^{23} -9.70041 q^{24} +1.42151 q^{25} +1.15698 q^{26} +1.00000 q^{27} -5.53407 q^{28} +6.02372 q^{29} +6.95558 q^{30} +3.48965 q^{31} -23.3026 q^{32} +2.42151 q^{33} -17.9349 q^{34} +2.53407 q^{35} +5.53407 q^{36} -2.84302 q^{37} +2.74483 q^{38} -0.421512 q^{39} +24.5815 q^{40} +8.42151 q^{41} +2.74483 q^{42} +11.4897 q^{43} +13.4008 q^{44} -2.53407 q^{45} +4.33268 q^{46} +8.02372 q^{47} +15.5578 q^{48} +1.00000 q^{49} -3.90180 q^{50} +6.53407 q^{51} -2.33268 q^{52} -10.8667 q^{53} -2.74483 q^{54} -6.13628 q^{55} +9.70041 q^{56} -1.00000 q^{57} -16.5341 q^{58} +14.1363 q^{59} -14.0237 q^{60} +6.84302 q^{61} -9.57849 q^{62} -1.00000 q^{63} +32.8461 q^{64} +1.06814 q^{65} -6.64663 q^{66} -9.91116 q^{67} +36.1600 q^{68} -1.57849 q^{69} -6.95558 q^{70} -13.3771 q^{71} -9.70041 q^{72} +7.06814 q^{73} +7.80361 q^{74} +1.42151 q^{75} -5.53407 q^{76} -2.42151 q^{77} +1.15698 q^{78} +14.1363 q^{79} -39.4245 q^{80} +1.00000 q^{81} -23.1156 q^{82} +2.11256 q^{83} -5.53407 q^{84} -16.5578 q^{85} -31.5371 q^{86} +6.02372 q^{87} -23.4897 q^{88} -9.71477 q^{89} +6.95558 q^{90} +0.421512 q^{91} -8.73546 q^{92} +3.48965 q^{93} -22.0237 q^{94} +2.53407 q^{95} -23.3026 q^{96} -4.97930 q^{97} -2.74483 q^{98} +2.42151 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} + 3 q^{3} + 9 q^{4} + q^{6} - 3 q^{7} - 9 q^{8} + 3 q^{9} + 10 q^{10} + 4 q^{11} + 9 q^{12} + 2 q^{13} - q^{14} + 13 q^{16} + 12 q^{17} + q^{18} - 3 q^{19} - 16 q^{20} - 3 q^{21} - 8 q^{22}+ \cdots + 4 q^{99}+O(q^{100}) $ 3 * q + q^2 + 3 * q^3 + 9 * q^4 + q^6 - 3 * q^7 - 9 * q^8 + 3 * q^9 + 10 * q^10 + 4 * q^11 + 9 * q^12 + 2 * q^13 - q^14 + 13 * q^16 + 12 * q^17 + q^18 - 3 * q^19 - 16 * q^20 - 3 * q^21 - 8 * q^22 - 8 * q^23 - 9 * q^24 + q^25 + 10 * q^26 + 3 * q^27 - 9 * q^28 - 8 * q^29 + 10 * q^30 - 8 * q^31 - 27 * q^32 + 4 * q^33 - 6 * q^34 + 9 * q^36 - 2 * q^37 - q^38 + 2 * q^39 + 14 * q^40 + 22 * q^41 - q^42 + 16 * q^43 - 12 * q^46 - 2 * q^47 + 13 * q^48 + 3 * q^49 - 9 * q^50 + 12 * q^51 + 18 * q^52 + q^54 + 12 * q^55 + 9 * q^56 - 3 * q^57 - 42 * q^58 + 12 * q^59 - 16 * q^60 + 14 * q^61 - 32 * q^62 - 3 * q^63 + 29 * q^64 - 12 * q^65 - 8 * q^66 - 8 * q^67 + 52 * q^68 - 8 * q^69 - 10 * q^70 - 26 * q^71 - 9 * q^72 + 6 * q^73 + 18 * q^74 + q^75 - 9 * q^76 - 4 * q^77 + 10 * q^78 + 12 * q^79 - 52 * q^80 + 3 * q^81 - 2 * q^82 + 2 * q^83 - 9 * q^84 - 16 * q^85 - 24 * q^86 - 8 * q^87 - 52 * q^88 - 2 * q^89 + 10 * q^90 - 2 * q^91 - 36 * q^92 - 8 * q^93 - 40 * q^94 - 27 * q^96 + 22 * q^97 + q^98 + 4 * q^99

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 $

$2$	−2.74483	−1.94089	−0.970443	−	0.241332i	$-0.922416\pi$
$2$	−2.74483	−1.94089	−0.970443	+	0.241332i	$0.922416\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	5.53407	2.76704
$5$	−2.53407	−1.13327	−0.566635	−	0.823969i	$-0.691755\pi$
$5$	−2.53407	−1.13327	−0.566635	+	0.823969i	$0.691755\pi$
$6$	−2.74483	−1.12057
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−9.70041	−3.42961
$9$	1.00000	0.333333
$10$	6.95558	2.19955
$11$	2.42151	0.730113	0.365057	−	0.930985i	$-0.381050\pi$
$11$	2.42151	0.730113	0.365057	+	0.930985i	$0.381050\pi$
$12$	5.53407	1.59755
$13$	−0.421512	−0.116906	−0.0584532	−	0.998290i	$-0.518617\pi$
$13$	−0.421512	−0.116906	−0.0584532	+	0.998290i	$0.518617\pi$
$14$	2.74483	0.733586
$15$	−2.53407	−0.654294
$16$	15.5578	3.88945
$17$	6.53407	1.58474	0.792372	−	0.610038i	$-0.208846\pi$
$17$	6.53407	1.58474	0.792372	+	0.610038i	$0.208846\pi$
$18$	−2.74483	−0.646962
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−14.0237	−3.13580
$21$	−1.00000	−0.218218
$22$	−6.64663	−1.41707
$23$	−1.57849	−0.329138	−0.164569	−	0.986366i	$-0.552623\pi$
$23$	−1.57849	−0.329138	−0.164569	+	0.986366i	$0.552623\pi$
$24$	−9.70041	−1.98009
$25$	1.42151	0.284302
$26$	1.15698	0.226902
$27$	1.00000	0.192450
$28$	−5.53407	−1.04584
$29$	6.02372	1.11858	0.559289	−	0.828973i	$-0.311074\pi$
$29$	6.02372	1.11858	0.559289	+	0.828973i	$0.311074\pi$
$30$	6.95558	1.26991
$31$	3.48965	0.626760	0.313380	−	0.949628i	$-0.398539\pi$
$31$	3.48965	0.626760	0.313380	+	0.949628i	$0.398539\pi$
$32$	−23.3026	−4.11936
$33$	2.42151	0.421531
$34$	−17.9349	−3.07581
$35$	2.53407	0.428336
$36$	5.53407	0.922345
$37$	−2.84302	−0.467390	−0.233695	−	0.972310i	$-0.575082\pi$
$37$	−2.84302	−0.467390	−0.233695	+	0.972310i	$0.575082\pi$
$38$	2.74483	0.445270
$39$	−0.421512	−0.0674959
$40$	24.5815	3.88668
$41$	8.42151	1.31522	0.657610	−	0.753359i	$-0.271568\pi$
$41$	8.42151	1.31522	0.657610	+	0.753359i	$0.271568\pi$
$42$	2.74483	0.423536
$43$	11.4897	1.75216	0.876078	−	0.482170i	$-0.160151\pi$
$43$	11.4897	1.75216	0.876078	+	0.482170i	$0.160151\pi$
$44$	13.4008	2.02025
$45$	−2.53407	−0.377757
$46$	4.33268	0.638818
$47$	8.02372	1.17038	0.585190	−	0.810896i	$-0.301020\pi$
$47$	8.02372	1.17038	0.585190	+	0.810896i	$0.301020\pi$
$48$	15.5578	2.24557
$49$	1.00000	0.142857
$50$	−3.90180	−0.551798
$51$	6.53407	0.914953
$52$	−2.33268	−0.323484
$53$	−10.8667	−1.49266	−0.746331	−	0.665575i	$-0.768186\pi$
$53$	−10.8667	−1.49266	−0.746331	+	0.665575i	$0.768186\pi$
$54$	−2.74483	−0.373524
$55$	−6.13628	−0.827416
$56$	9.70041	1.29627
$57$	−1.00000	−0.132453
$58$	−16.5341	−2.17103
$59$	14.1363	1.84039	0.920194	−	0.391464i	$-0.128031\pi$
$59$	14.1363	1.84039	0.920194	+	0.391464i	$0.128031\pi$
$60$	−14.0237	−1.81045
$61$	6.84302	0.876159	0.438080	−	0.898936i	$-0.355659\pi$
$61$	6.84302	0.876159	0.438080	+	0.898936i	$0.355659\pi$
$62$	−9.57849	−1.21647
$63$	−1.00000	−0.125988
$64$	32.8461	4.10576
$65$	1.06814	0.132487
$66$	−6.64663	−0.818143
$67$	−9.91116	−1.21084	−0.605421	−	0.795906i	$-0.706995\pi$
$67$	−9.91116	−1.21084	−0.605421	+	0.795906i	$0.706995\pi$
$68$	36.1600	4.38504
$69$	−1.57849	−0.190028
$70$	−6.95558	−0.831351
$71$	−13.3771	−1.58757	−0.793784	−	0.608199i	$-0.791892\pi$
$71$	−13.3771	−1.58757	−0.793784	+	0.608199i	$0.791892\pi$
$72$	−9.70041	−1.14320
$73$	7.06814	0.827263	0.413632	−	0.910444i	$-0.364260\pi$
$73$	7.06814	0.827263	0.413632	+	0.910444i	$0.364260\pi$
$74$	7.80361	0.907151
$75$	1.42151	0.164142
$76$	−5.53407	−0.634801
$77$	−2.42151	−0.275957
$78$	1.15698	0.131002
$79$	14.1363	1.59046	0.795228	−	0.606311i	$-0.207351\pi$
$79$	14.1363	1.59046	0.795228	+	0.606311i	$0.207351\pi$
$80$	−39.4245	−4.40780
$81$	1.00000	0.111111
$82$	−23.1156	−2.55269
$83$	2.11256	0.231883	0.115942	−	0.993256i	$-0.463011\pi$
$83$	2.11256	0.231883	0.115942	+	0.993256i	$0.463011\pi$
$84$	−5.53407	−0.603817
$85$	−16.5578	−1.79594
$86$	−31.5371	−3.40073
$87$	6.02372	0.645811
$88$	−23.4897	−2.50401
$89$	−9.71477	−1.02976	−0.514882	−	0.857261i	$-0.672164\pi$
$89$	−9.71477	−1.02976	−0.514882	+	0.857261i	$0.672164\pi$
$90$	6.95558	0.733183
$91$	0.421512	0.0441864
$92$	−8.73546	−0.910735
$93$	3.48965	0.361860
$94$	−22.0237	−2.27157
$95$	2.53407	0.259990
$96$	−23.3026	−2.37831
$97$	−4.97930	−0.505572	−0.252786	−	0.967522i	$-0.581347\pi$
$97$	−4.97930	−0.505572	−0.252786	+	0.967522i	$0.581347\pi$
$98$	−2.74483	−0.277269
$99$	2.42151	0.243371
$100$	7.86675	0.786675
$101$	−1.69105	−0.168265	−0.0841327	−	0.996455i	$-0.526812\pi$
$101$	−1.69105	−0.168265	−0.0841327	+	0.996455i	$0.526812\pi$
$102$	−17.9349	−1.77582
$103$	3.15698	0.311066	0.155533	−	0.987831i	$-0.450290\pi$
$103$	3.15698	0.311066	0.155533	+	0.987831i	$0.450290\pi$
$104$	4.08884	0.400943
$105$	2.53407	0.247300
$106$	29.8273	2.89709
$107$	−16.5341	−1.59841	−0.799204	−	0.601059i	$-0.794746\pi$
$107$	−16.5341	−1.59841	−0.799204	+	0.601059i	$0.794746\pi$
$108$	5.53407	0.532516
$109$	1.77488	0.170003	0.0850015	−	0.996381i	$-0.472910\pi$
$109$	1.77488	0.170003	0.0850015	+	0.996381i	$0.472910\pi$
$110$	16.8430	1.60592
$111$	−2.84302	−0.269848
$112$	−15.5578	−1.47007
$113$	10.0237	0.942952	0.471476	−	0.881879i	$-0.343721\pi$
$113$	10.0237	0.942952	0.471476	+	0.881879i	$0.343721\pi$
$114$	2.74483	0.257077
$115$	4.00000	0.373002
$116$	33.3357	3.09514
$117$	−0.421512	−0.0389688
$118$	−38.8016	−3.57198
$119$	−6.53407	−0.598977
$120$	24.5815	2.24398
$121$	−5.13628	−0.466935
$122$	−18.7829	−1.70052
$123$	8.42151	0.759342
$124$	19.3120	1.73427
$125$	9.06814	0.811079
$126$	2.74483	0.244529
$127$	2.08884	0.185354	0.0926771	−	0.995696i	$-0.470458\pi$
$127$	2.08884	0.185354	0.0926771	+	0.995696i	$0.470458\pi$
$128$	−43.5515	−3.84944
$129$	11.4897	1.01161
$130$	−2.93186	−0.257141
$131$	−13.9349	−1.21750	−0.608748	−	0.793363i	$-0.708328\pi$
$131$	−13.9349	−1.21750	−0.608748	+	0.793363i	$0.708328\pi$
$132$	13.4008	1.16639
$133$	1.00000	0.0867110
$134$	27.2044	2.35010
$135$	−2.53407	−0.218098
$136$	−63.3831	−5.43506
$137$	16.9793	1.45064	0.725320	−	0.688412i	$-0.241692\pi$
$137$	16.9793	1.45064	0.725320	+	0.688412i	$0.241692\pi$
$138$	4.33268	0.368822
$139$	−1.06814	−0.0905985	−0.0452992	−	0.998973i	$-0.514424\pi$
$139$	−1.06814	−0.0905985	−0.0452992	+	0.998973i	$0.514424\pi$
$140$	14.0237	1.18522
$141$	8.02372	0.675719
$142$	36.7178	3.08129
$143$	−1.02070	−0.0853549
$144$	15.5578	1.29648
$145$	−15.2645	−1.26765
$146$	−19.4008	−1.60562
$147$	1.00000	0.0824786
$148$	−15.7335	−1.29329
$149$	14.0474	1.15081	0.575406	−	0.817868i	$-0.304844\pi$
$149$	14.0474	1.15081	0.575406	+	0.817868i	$0.304844\pi$
$150$	−3.90180	−0.318581
$151$	−14.7542	−1.20068	−0.600339	−	0.799745i	$-0.704968\pi$
$151$	−14.7542	−1.20068	−0.600339	+	0.799745i	$0.704968\pi$
$152$	9.70041	0.786807
$153$	6.53407	0.528248
$154$	6.64663	0.535601
$155$	−8.84302	−0.710289
$156$	−2.33268	−0.186764
$157$	−16.1363	−1.28782	−0.643908	−	0.765103i	$-0.722688\pi$
$157$	−16.1363	−1.28782	−0.643908	+	0.765103i	$0.722688\pi$
$158$	−38.8016	−3.08689
$159$	−10.8667	−0.861789
$160$	59.0505	4.66835
$161$	1.57849	0.124402
$162$	−2.74483	−0.215654
$163$	−6.64663	−0.520604	−0.260302	−	0.965527i	$-0.583822\pi$
$163$	−6.64663	−0.520604	−0.260302	+	0.965527i	$0.583822\pi$
$164$	46.6052	3.63926
$165$	−6.13628	−0.477709
$166$	−5.79861	−0.450059
$167$	−14.3614	−1.11132	−0.555659	−	0.831410i	$-0.687534\pi$
$167$	−14.3614	−1.11132	−0.555659	+	0.831410i	$0.687534\pi$
$168$	9.70041	0.748403
$169$	−12.8223	−0.986333
$170$	45.4483	3.48572
$171$	−1.00000	−0.0764719
$172$	63.5845	4.84828
$173$	−5.71477	−0.434486	−0.217243	−	0.976118i	$-0.569706\pi$
$173$	−5.71477	−0.434486	−0.217243	+	0.976118i	$0.569706\pi$
$174$	−16.5341	−1.25344
$175$	−1.42151	−0.107456
$176$	37.6734	2.83974
$177$	14.1363	1.06255
$178$	26.6654	1.99865
$179$	−9.60221	−0.717703	−0.358851	−	0.933395i	$-0.616832\pi$
$179$	−9.60221	−0.717703	−0.358851	+	0.933395i	$0.616832\pi$
$180$	−14.0237	−1.04527
$181$	−0.979304	−0.0727911	−0.0363956	−	0.999337i	$-0.511588\pi$
$181$	−0.979304	−0.0727911	−0.0363956	+	0.999337i	$0.511588\pi$
$182$	−1.15698	−0.0857608
$183$	6.84302	0.505851
$184$	15.3120	1.12881
$185$	7.20442	0.529680
$186$	−9.57849	−0.702329
$187$	15.8223	1.15704
$188$	44.4038	3.23848
$189$	−1.00000	−0.0727393
$190$	−6.95558	−0.504611
$191$	−11.7148	−0.847651	−0.423825	−	0.905744i	$-0.639313\pi$
$191$	−11.7148	−0.847651	−0.423825	+	0.905744i	$0.639313\pi$
$192$	32.8461	2.37046
$193$	17.8223	1.28288	0.641440	−	0.767174i	$-0.278337\pi$
$193$	17.8223	1.28288	0.641440	+	0.767174i	$0.278337\pi$
$194$	13.6673	0.981257
$195$	1.06814	0.0764911
$196$	5.53407	0.395291
$197$	−19.1156	−1.36193	−0.680965	−	0.732316i	$-0.738439\pi$
$197$	−19.1156	−1.36193	−0.680965	+	0.732316i	$0.738439\pi$
$198$	−6.64663	−0.472355
$199$	16.0474	1.13757	0.568787	−	0.822485i	$-0.307413\pi$
$199$	16.0474	1.13757	0.568787	+	0.822485i	$0.307413\pi$
$200$	−13.7892	−0.975047
$201$	−9.91116	−0.699080
$202$	4.64163	0.326584
$203$	−6.02372	−0.422782
$204$	36.1600	2.53171
$205$	−21.3407	−1.49050
$206$	−8.66535	−0.603744
$207$	−1.57849	−0.109713
$208$	−6.55779	−0.454701
$209$	−2.42151	−0.167499
$210$	−6.95558	−0.479981
$211$	−9.29326	−0.639774	−0.319887	−	0.947456i	$-0.603645\pi$
$211$	−9.29326	−0.639774	−0.319887	+	0.947456i	$0.603645\pi$
$212$	−60.1373	−4.13025
$213$	−13.3771	−0.916583
$214$	45.3831	3.10233
$215$	−29.1156	−1.98567
$216$	−9.70041	−0.660029
$217$	−3.48965	−0.236893
$218$	−4.87175	−0.329956
$219$	7.06814	0.477621
$220$	−33.9586	−2.28949
$221$	−2.75419	−0.185267
$222$	7.80361	0.523744
$223$	−8.33268	−0.557997	−0.278999	−	0.960292i	$-0.590002\pi$
$223$	−8.33268	−0.557997	−0.278999	+	0.960292i	$0.590002\pi$
$224$	23.3026	1.55697
$225$	1.42151	0.0947675
$226$	−27.5134	−1.83016
$227$	16.8905	1.12106	0.560530	−	0.828134i	$-0.310597\pi$
$227$	16.8905	1.12106	0.560530	+	0.828134i	$0.310597\pi$
$228$	−5.53407	−0.366503
$229$	6.04744	0.399626	0.199813	−	0.979834i	$-0.435966\pi$
$229$	6.04744	0.399626	0.199813	+	0.979834i	$0.435966\pi$
$230$	−10.9793	−0.723954
$231$	−2.42151	−0.159324
$232$	−58.4326	−3.83629
$233$	21.2044	1.38915	0.694574	−	0.719421i	$-0.255593\pi$
$233$	21.2044	1.38915	0.694574	+	0.719421i	$0.255593\pi$
$234$	1.15698	0.0756339
$235$	−20.3327	−1.32636
$236$	78.2312	5.09242
$237$	14.1363	0.918250
$238$	17.9349	1.16255
$239$	−5.80361	−0.375404	−0.187702	−	0.982226i	$-0.560104\pi$
$239$	−5.80361	−0.375404	−0.187702	+	0.982226i	$0.560104\pi$
$240$	−39.4245	−2.54484
$241$	−12.9793	−0.836070	−0.418035	−	0.908431i	$-0.637281\pi$
$241$	−12.9793	−0.836070	−0.418035	+	0.908431i	$0.637281\pi$
$242$	14.0982	0.906266
$243$	1.00000	0.0641500
$244$	37.8698	2.42436
$245$	−2.53407	−0.161896
$246$	−23.1156	−1.47380
$247$	0.421512	0.0268202
$248$	−33.8510	−2.14954
$249$	2.11256	0.133878
$250$	−24.8905	−1.57421
$251$	10.3377	0.652508	0.326254	−	0.945282i	$-0.394213\pi$
$251$	10.3377	0.652508	0.326254	+	0.945282i	$0.394213\pi$
$252$	−5.53407	−0.348614
$253$	−3.82233	−0.240308
$254$	−5.73349	−0.359751
$255$	−16.5578	−1.03689
$256$	53.8491	3.36557
$257$	−15.1757	−0.946634	−0.473317	−	0.880892i	$-0.656944\pi$
$257$	−15.1757	−0.946634	−0.473317	+	0.880892i	$0.656944\pi$
$258$	−31.5371	−1.96341
$259$	2.84302	0.176657
$260$	5.91116	0.366595
$261$	6.02372	0.372859
$262$	38.2488	2.36302
$263$	−8.51035	−0.524771	−0.262385	−	0.964963i	$-0.584509\pi$
$263$	−8.51035	−0.524771	−0.262385	+	0.964963i	$0.584509\pi$
$264$	−23.4897	−1.44569
$265$	27.5371	1.69159
$266$	−2.74483	−0.168296
$267$	−9.71477	−0.594534
$268$	−54.8491	−3.35044
$269$	15.1757	0.925279	0.462639	−	0.886547i	$-0.346902\pi$
$269$	15.1757	0.925279	0.462639	+	0.886547i	$0.346902\pi$
$270$	6.95558	0.423303
$271$	0.843024	0.0512100	0.0256050	−	0.999672i	$-0.491849\pi$
$271$	0.843024	0.0512100	0.0256050	+	0.999672i	$0.491849\pi$
$272$	101.656	6.16378
$273$	0.421512	0.0255111
$274$	−46.6052	−2.81553
$275$	3.44221	0.207573
$276$	−8.73546	−0.525813
$277$	10.5578	0.634356	0.317178	−	0.948366i	$-0.397265\pi$
$277$	10.5578	0.634356	0.317178	+	0.948366i	$0.397265\pi$
$278$	2.93186	0.175841
$279$	3.48965	0.208920
$280$	−24.5815	−1.46903
$281$	15.0444	0.897475	0.448737	−	0.893664i	$-0.351874\pi$
$281$	15.0444	0.897475	0.448737	+	0.893664i	$0.351874\pi$
$282$	−22.0237	−1.31149
$283$	1.24581	0.0740559	0.0370279	−	0.999314i	$-0.488211\pi$
$283$	1.24581	0.0740559	0.0370279	+	0.999314i	$0.488211\pi$
$284$	−74.0298	−4.39286
$285$	2.53407	0.150105
$286$	2.80163	0.165664
$287$	−8.42151	−0.497106
$288$	−23.3026	−1.37312
$289$	25.6941	1.51142
$290$	41.8985	2.46036
$291$	−4.97930	−0.291892
$292$	39.1156	2.28907
$293$	20.4215	1.19304	0.596519	−	0.802599i	$-0.296550\pi$
$293$	20.4215	1.19304	0.596519	+	0.802599i	$0.296550\pi$
$294$	−2.74483	−0.160082
$295$	−35.8223	−2.08566
$296$	27.5785	1.60297
$297$	2.42151	0.140510
$298$	−38.5578	−2.23359
$299$	0.665351	0.0384783
$300$	7.86675	0.454187
$301$	−11.4897	−0.662253
$302$	40.4977	2.33038
$303$	−1.69105	−0.0971481
$304$	−15.5578	−0.892301
$305$	−17.3407	−0.992926
$306$	−17.9349	−1.02527
$307$	−2.64663	−0.151051	−0.0755255	−	0.997144i	$-0.524063\pi$
$307$	−2.64663	−0.151051	−0.0755255	+	0.997144i	$0.524063\pi$
$308$	−13.4008	−0.763582
$309$	3.15698	0.179594
$310$	24.2726	1.37859
$311$	−5.09186	−0.288733	−0.144367	−	0.989524i	$-0.546114\pi$
$311$	−5.09186	−0.288733	−0.144367	+	0.989524i	$0.546114\pi$
$312$	4.08884	0.231485
$313$	19.9586	1.12813	0.564064	−	0.825731i	$-0.309237\pi$
$313$	19.9586	1.12813	0.564064	+	0.825731i	$0.309237\pi$
$314$	44.2913	2.49950
$315$	2.53407	0.142779
$316$	78.2312	4.40085
$317$	−6.81930	−0.383010	−0.191505	−	0.981492i	$-0.561337\pi$
$317$	−6.81930	−0.383010	−0.191505	+	0.981492i	$0.561337\pi$
$318$	29.8273	1.67263
$319$	14.5865	0.816688
$320$	−83.2342	−4.65293
$321$	−16.5341	−0.922842
$322$	−4.33268	−0.241451
$323$	−6.53407	−0.363565
$324$	5.53407	0.307448
$325$	−0.599184	−0.0332367
$326$	18.2438	1.01043
$327$	1.77488	0.0981513
$328$	−81.6921	−4.51069
$329$	−8.02372	−0.442362
$330$	16.8430	0.927178
$331$	−5.91116	−0.324907	−0.162453	−	0.986716i	$-0.551941\pi$
$331$	−5.91116	−0.324907	−0.162453	+	0.986716i	$0.551941\pi$
$332$	11.6910	0.641630
$333$	−2.84302	−0.155797
$334$	39.4195	2.15694
$335$	25.1156	1.37221
$336$	−15.5578	−0.848747
$337$	4.08884	0.222733	0.111367	−	0.993779i	$-0.464477\pi$
$337$	4.08884	0.222733	0.111367	+	0.993779i	$0.464477\pi$
$338$	35.1951	1.91436
$339$	10.0237	0.544414
$340$	−91.6320	−4.96944
$341$	8.45023	0.457606
$342$	2.74483	0.148423
$343$	−1.00000	−0.0539949
$344$	−111.454	−6.00921
$345$	4.00000	0.215353
$346$	15.6860	0.843287
$347$	−17.3534	−0.931578	−0.465789	−	0.884896i	$-0.654229\pi$
$347$	−17.3534	−0.931578	−0.465789	+	0.884896i	$0.654229\pi$
$348$	33.3357	1.78698
$349$	−4.70674	−0.251946	−0.125973	−	0.992034i	$-0.540205\pi$
$349$	−4.70674	−0.251946	−0.125973	+	0.992034i	$0.540205\pi$
$350$	3.90180	0.208560
$351$	−0.421512	−0.0224986
$352$	−56.4276	−3.00760
$353$	20.6704	1.10017	0.550086	−	0.835108i	$-0.314595\pi$
$353$	20.6704	1.10017	0.550086	+	0.835108i	$0.314595\pi$
$354$	−38.8016	−2.06228
$355$	33.8985	1.79915
$356$	−53.7622	−2.84939
$357$	−6.53407	−0.345820
$358$	26.3564	1.39298
$359$	8.10756	0.427901	0.213950	−	0.976845i	$-0.431367\pi$
$359$	8.10756	0.427901	0.213950	+	0.976845i	$0.431367\pi$
$360$	24.5815	1.29556
$361$	1.00000	0.0526316
$362$	2.68802	0.141279
$363$	−5.13628	−0.269585
$364$	2.33268	0.122265
$365$	−17.9112	−0.937513
$366$	−18.7829	−0.981798
$367$	28.8905	1.50807	0.754035	−	0.656834i	$-0.228105\pi$
$367$	28.8905	1.50807	0.754035	+	0.656834i	$0.228105\pi$
$368$	−24.5578	−1.28016
$369$	8.42151	0.438406
$370$	−19.7749	−1.02805
$371$	10.8667	0.564173
$372$	19.3120	1.00128
$373$	11.9586	0.619193	0.309597	−	0.950868i	$-0.399806\pi$
$373$	11.9586	0.619193	0.309597	+	0.950868i	$0.399806\pi$
$374$	−43.4295	−2.24569
$375$	9.06814	0.468277
$376$	−77.8334	−4.01395
$377$	−2.53907	−0.130769
$378$	2.74483	0.141179
$379$	−15.3821	−0.790125	−0.395063	−	0.918654i	$-0.629277\pi$
$379$	−15.3821	−0.790125	−0.395063	+	0.918654i	$0.629277\pi$
$380$	14.0237	0.719402
$381$	2.08884	0.107014
$382$	32.1550	1.64519
$383$	−0.617907	−0.0315736	−0.0157868	−	0.999875i	$-0.505025\pi$
$383$	−0.617907	−0.0315736	−0.0157868	+	0.999875i	$0.505025\pi$
$384$	−43.5515	−2.22248
$385$	6.13628	0.312734
$386$	−48.9192	−2.48992
$387$	11.4897	0.584052
$388$	−27.5558	−1.39893
$389$	−11.0681	−0.561177	−0.280588	−	0.959828i	$-0.590530\pi$
$389$	−11.0681	−0.561177	−0.280588	+	0.959828i	$0.590530\pi$
$390$	−2.93186	−0.148460
$391$	−10.3140	−0.521599
$392$	−9.70041	−0.489945
$393$	−13.9349	−0.702922
$394$	52.4690	2.64335
$395$	−35.8223	−1.80242
$396$	13.4008	0.673416
$397$	−10.4502	−0.524482	−0.262241	−	0.965002i	$-0.584462\pi$
$397$	−10.4502	−0.524482	−0.262241	+	0.965002i	$0.584462\pi$
$398$	−44.0474	−2.20790
$399$	1.00000	0.0500626
$400$	22.1156	1.10578
$401$	3.26953	0.163273	0.0816364	−	0.996662i	$-0.473985\pi$
$401$	3.26953	0.163273	0.0816364	+	0.996662i	$0.473985\pi$
$402$	27.2044	1.35683
$403$	−1.47093	−0.0732722
$404$	−9.35837	−0.465596
$405$	−2.53407	−0.125919
$406$	16.5341	0.820572
$407$	−6.88441	−0.341248
$408$	−63.3831	−3.13793
$409$	7.40082	0.365947	0.182973	−	0.983118i	$-0.441428\pi$
$409$	7.40082	0.365947	0.182973	+	0.983118i	$0.441428\pi$
$410$	58.5765	2.89289
$411$	16.9793	0.837527
$412$	17.4709	0.860731
$413$	−14.1363	−0.695601
$414$	4.33268	0.212939
$415$	−5.35337	−0.262787
$416$	9.82233	0.481579
$417$	−1.06814	−0.0523071
$418$	6.64663	0.325097
$419$	−31.8461	−1.55578	−0.777891	−	0.628400i	$-0.783710\pi$
$419$	−31.8461	−1.55578	−0.777891	+	0.628400i	$0.783710\pi$
$420$	14.0237	0.684288
$421$	40.0061	1.94978	0.974888	−	0.222696i	$-0.0714858\pi$
$421$	40.0061	1.94978	0.974888	+	0.222696i	$0.0714858\pi$
$422$	25.5084	1.24173
$423$	8.02372	0.390127
$424$	105.412	5.11925
$425$	9.28826	0.450547
$426$	36.7178	1.77898
$427$	−6.84302	−0.331157
$428$	−91.5007	−4.42285
$429$	−1.02070	−0.0492797
$430$	79.9172	3.85395
$431$	3.91616	0.188635	0.0943175	−	0.995542i	$-0.469933\pi$
$431$	3.91616	0.188635	0.0943175	+	0.995542i	$0.469933\pi$
$432$	15.5578	0.748525
$433$	18.0000	0.865025	0.432512	−	0.901628i	$-0.357627\pi$
$433$	18.0000	0.865025	0.432512	+	0.901628i	$0.357627\pi$
$434$	9.57849	0.459782
$435$	−15.2645	−0.731878
$436$	9.82233	0.470404
$437$	1.57849	0.0755093
$438$	−19.4008	−0.927007
$439$	17.4483	0.832760	0.416380	−	0.909191i	$-0.363299\pi$
$439$	17.4483	0.832760	0.416380	+	0.909191i	$0.363299\pi$
$440$	59.5244	2.83772
$441$	1.00000	0.0476190
$442$	7.55977	0.359581
$443$	−3.03942	−0.144407	−0.0722036	−	0.997390i	$-0.523003\pi$
$443$	−3.03942	−0.144407	−0.0722036	+	0.997390i	$0.523003\pi$
$444$	−15.7335	−0.746678
$445$	24.6179	1.16700
$446$	22.8717	1.08301
$447$	14.0474	0.664421
$448$	−32.8461	−1.55183
$449$	35.9349	1.69587	0.847936	−	0.530099i	$-0.177845\pi$
$449$	35.9349	1.69587	0.847936	+	0.530099i	$0.177845\pi$
$450$	−3.90180	−0.183933
$451$	20.3928	0.960259
$452$	55.4720	2.60918
$453$	−14.7542	−0.693212
$454$	−46.3614	−2.17585
$455$	−1.06814	−0.0500752
$456$	9.70041	0.454263
$457$	−24.4215	−1.14239	−0.571195	−	0.820814i	$-0.693520\pi$
$457$	−24.4215	−1.14239	−0.571195	+	0.820814i	$0.693520\pi$
$458$	−16.5992	−0.775629
$459$	6.53407	0.304984
$460$	22.1363	1.03211
$461$	10.1313	0.471861	0.235930	−	0.971770i	$-0.424186\pi$
$461$	10.1313	0.471861	0.235930	+	0.971770i	$0.424186\pi$
$462$	6.64663	0.309229
$463$	−5.86372	−0.272510	−0.136255	−	0.990674i	$-0.543507\pi$
$463$	−5.86372	−0.272510	−0.136255	+	0.990674i	$0.543507\pi$
$464$	93.7158	4.35065
$465$	−8.84302	−0.410085
$466$	−58.2024	−2.69618
$467$	−14.9556	−0.692062	−0.346031	−	0.938223i	$-0.612471\pi$
$467$	−14.9556	−0.692062	−0.346031	+	0.938223i	$0.612471\pi$
$468$	−2.33268	−0.107828
$469$	9.91116	0.457655
$470$	55.8097	2.57431
$471$	−16.1363	−0.743521
$472$	−137.128	−6.31181
$473$	27.8223	1.27927
$474$	−38.8016	−1.78222
$475$	−1.42151	−0.0652234
$476$	−36.1600	−1.65739
$477$	−10.8667	−0.497554
$478$	15.9299	0.728616
$479$	−33.7098	−1.54024	−0.770119	−	0.637900i	$-0.779803\pi$
$479$	−33.7098	−1.54024	−0.770119	+	0.637900i	$0.779803\pi$
$480$	59.0505	2.69527
$481$	1.19837	0.0546409
$482$	35.6259	1.62272
$483$	1.57849	0.0718237
$484$	−28.4245	−1.29202
$485$	12.6179	0.572950
$486$	−2.74483	−0.124508
$487$	21.5084	0.974637	0.487319	−	0.873224i	$-0.337975\pi$
$487$	21.5084	0.974637	0.487319	+	0.873224i	$0.337975\pi$
$488$	−66.3801	−3.00489
$489$	−6.64663	−0.300571
$490$	6.95558	0.314221
$491$	−17.3534	−0.783147	−0.391573	−	0.920147i	$-0.628069\pi$
$491$	−17.3534	−0.783147	−0.391573	+	0.920147i	$0.628069\pi$
$492$	46.6052	2.10113
$493$	39.3594	1.77266
$494$	−1.15698	−0.0520548
$495$	−6.13628	−0.275805
$496$	54.2913	2.43775
$497$	13.3771	0.600045
$498$	−5.79861	−0.259842
$499$	−32.9379	−1.47450	−0.737252	−	0.675618i	$-0.763877\pi$
$499$	−32.9379	−1.47450	−0.737252	+	0.675618i	$0.763877\pi$
$500$	50.1837	2.24428
$501$	−14.3614	−0.641620
$502$	−28.3751	−1.26644
$503$	−38.2074	−1.70359	−0.851793	−	0.523879i	$-0.824485\pi$
$503$	−38.2074	−1.70359	−0.851793	+	0.523879i	$0.824485\pi$
$504$	9.70041	0.432091
$505$	4.28523	0.190690
$506$	10.4916	0.466410
$507$	−12.8223	−0.569460
$508$	11.5598	0.512882
$509$	−8.82430	−0.391130	−0.195565	−	0.980691i	$-0.562654\pi$
$509$	−8.82430	−0.391130	−0.195565	+	0.980691i	$0.562654\pi$
$510$	45.4483	2.01248
$511$	−7.06814	−0.312676
$512$	−60.7034	−2.68274
$513$	−1.00000	−0.0441511
$514$	41.6547	1.83731
$515$	−8.00000	−0.352522
$516$	63.5845	2.79915
$517$	19.4295	0.854510
$518$	−7.80361	−0.342871
$519$	−5.71477	−0.250851
$520$	−10.3614	−0.454377
$521$	−24.2913	−1.06422	−0.532110	−	0.846675i	$-0.678601\pi$
$521$	−24.2913	−1.06422	−0.532110	+	0.846675i	$0.678601\pi$
$522$	−16.5341	−0.723677
$523$	19.0969	0.835047	0.417524	−	0.908666i	$-0.362898\pi$
$523$	19.0969	0.835047	0.417524	+	0.908666i	$0.362898\pi$
$524$	−77.1166	−3.36886
$525$	−1.42151	−0.0620399
$526$	23.3594	1.01852
$527$	22.8016	0.993255
$528$	37.6734	1.63952
$529$	−20.5084	−0.891668
$530$	−75.5845	−3.28318
$531$	14.1363	0.613462
$532$	5.53407	0.239932
$533$	−3.54977	−0.153757
$534$	26.6654	1.15392
$535$	41.8985	1.81143
$536$	96.1423	4.15272
$537$	−9.60221	−0.414366
$538$	−41.6547	−1.79586
$539$	2.42151	0.104302
$540$	−14.0237	−0.603485
$541$	−36.2312	−1.55770	−0.778850	−	0.627210i	$-0.784197\pi$
$541$	−36.2312	−1.55770	−0.778850	+	0.627210i	$0.784197\pi$
$542$	−2.31395	−0.0993928
$543$	−0.979304	−0.0420260
$544$	−152.261	−6.52813
$545$	−4.49768	−0.192659
$546$	−1.15698	−0.0495140
$547$	−29.9586	−1.28094	−0.640469	−	0.767984i	$-0.721260\pi$
$547$	−29.9586	−1.28094	−0.640469	+	0.767984i	$0.721260\pi$
$548$	93.9647	4.01397
$549$	6.84302	0.292053
$550$	−9.44826	−0.402875
$551$	−6.02372	−0.256619
$552$	15.3120	0.651721
$553$	−14.1363	−0.601136
$554$	−28.9793	−1.23121
$555$	7.20442	0.305811
$556$	−5.91116	−0.250689
$557$	−14.4402	−0.611852	−0.305926	−	0.952055i	$-0.598966\pi$
$557$	−14.4402	−0.611852	−0.305926	+	0.952055i	$0.598966\pi$
$558$	−9.57849	−0.405490
$559$	−4.84302	−0.204838
$560$	39.4245	1.66599
$561$	15.8223	0.668019
$562$	−41.2943	−1.74190
$563$	−17.9112	−0.754866	−0.377433	−	0.926037i	$-0.623193\pi$
$563$	−17.9112	−0.754866	−0.377433	+	0.926037i	$0.623193\pi$
$564$	44.4038	1.86974
$565$	−25.4008	−1.06862
$566$	−3.41954	−0.143734
$567$	−1.00000	−0.0419961
$568$	129.763	5.44475
$569$	−8.50535	−0.356563	−0.178281	−	0.983980i	$-0.557054\pi$
$569$	−8.50535	−0.356563	−0.178281	+	0.983980i	$0.557054\pi$
$570$	−6.95558	−0.291337
$571$	−6.80163	−0.284639	−0.142320	−	0.989821i	$-0.545456\pi$
$571$	−6.80163	−0.284639	−0.142320	+	0.989821i	$0.545456\pi$
$572$	−5.64860	−0.236180
$573$	−11.7148	−0.489391
$574$	23.1156	0.964826
$575$	−2.24384	−0.0935746
$576$	32.8461	1.36859
$577$	−14.2726	−0.594175	−0.297087	−	0.954850i	$-0.596015\pi$
$577$	−14.2726	−0.594175	−0.297087	+	0.954850i	$0.596015\pi$
$578$	−70.5258	−2.93348
$579$	17.8223	0.740671
$580$	−84.4750	−3.50763
$581$	−2.11256	−0.0876437
$582$	13.6673	0.566529
$583$	−26.3140	−1.08981
$584$	−68.5638	−2.83719
$585$	1.06814	0.0441622
$586$	−56.0535	−2.31555
$587$	40.2963	1.66321	0.831603	−	0.555371i	$-0.187424\pi$
$587$	40.2963	1.66321	0.831603	+	0.555371i	$0.187424\pi$
$588$	5.53407	0.228221
$589$	−3.48965	−0.143789
$590$	98.3261	4.04802
$591$	−19.1156	−0.786310
$592$	−44.2312	−1.81789
$593$	26.1313	1.07308	0.536542	−	0.843874i	$-0.319730\pi$
$593$	26.1313	1.07308	0.536542	+	0.843874i	$0.319730\pi$
$594$	−6.64663	−0.272714
$595$	16.5578	0.678803
$596$	77.7395	3.18434
$597$	16.0474	0.656778
$598$	−1.82627	−0.0746819
$599$	3.06314	0.125157	0.0625783	−	0.998040i	$-0.480068\pi$
$599$	3.06314	0.125157	0.0625783	+	0.998040i	$0.480068\pi$
$600$	−13.7892	−0.562944
$601$	28.1363	1.14770	0.573851	−	0.818959i	$-0.305449\pi$
$601$	28.1363	1.14770	0.573851	+	0.818959i	$0.305449\pi$
$602$	31.5371	1.28536
$603$	−9.91116	−0.403614
$604$	−81.6507	−3.32232
$605$	13.0157	0.529163
$606$	4.64163	0.188553
$607$	28.8430	1.17070	0.585351	−	0.810780i	$-0.300957\pi$
$607$	28.8430	1.17070	0.585351	+	0.810780i	$0.300957\pi$
$608$	23.3026	0.945046
$609$	−6.02372	−0.244094
$610$	47.5972	1.92715
$611$	−3.38209	−0.136825
$612$	36.1600	1.46168
$613$	22.2852	0.900092	0.450046	−	0.893005i	$-0.351408\pi$
$613$	22.2852	0.900092	0.450046	+	0.893005i	$0.351408\pi$
$614$	7.26454	0.293173
$615$	−21.3407	−0.860540
$616$	23.4897	0.946425
$617$	4.13628	0.166520	0.0832602	−	0.996528i	$-0.473467\pi$
$617$	4.13628	0.166520	0.0832602	+	0.996528i	$0.473467\pi$
$618$	−8.66535	−0.348572
$619$	−9.06814	−0.364479	−0.182240	−	0.983254i	$-0.558335\pi$
$619$	−9.06814	−0.364479	−0.182240	+	0.983254i	$0.558335\pi$
$620$	−48.9379	−1.96539
$621$	−1.57849	−0.0633426
$622$	13.9763	0.560398
$623$	9.71477	0.389214
$624$	−6.55779	−0.262522
$625$	−30.0869	−1.20347
$626$	−54.7829	−2.18957
$627$	−2.42151	−0.0967059
$628$	−89.2993	−3.56343
$629$	−18.5765	−0.740694
$630$	−6.95558	−0.277117
$631$	−41.6259	−1.65710	−0.828551	−	0.559913i	$-0.810834\pi$
$631$	−41.6259	−1.65710	−0.828551	+	0.559913i	$0.810834\pi$
$632$	−137.128	−5.45465
$633$	−9.29326	−0.369374
$634$	18.7178	0.743379
$635$	−5.29326	−0.210057
$636$	−60.1373	−2.38460
$637$	−0.421512	−0.0167009
$638$	−40.0374	−1.58510
$639$	−13.3771	−0.529190
$640$	110.362	4.36246
$641$	3.09186	0.122121	0.0610606	−	0.998134i	$-0.480552\pi$
$641$	3.09186	0.122121	0.0610606	+	0.998134i	$0.480552\pi$
$642$	45.3831	1.79113
$643$	2.93186	0.115621	0.0578106	−	0.998328i	$-0.481588\pi$
$643$	2.93186	0.115621	0.0578106	+	0.998328i	$0.481588\pi$
$644$	8.73546	0.344226
$645$	−29.1156	−1.14643
$646$	17.9349	0.705639
$647$	−5.93489	−0.233324	−0.116662	−	0.993172i	$-0.537219\pi$
$647$	−5.93489	−0.233324	−0.116662	+	0.993172i	$0.537219\pi$
$648$	−9.70041	−0.381068
$649$	34.2312	1.34369
$650$	1.64466	0.0645087
$651$	−3.48965	−0.136770
$652$	−36.7829	−1.44053
$653$	1.82233	0.0713132	0.0356566	−	0.999364i	$-0.488648\pi$
$653$	1.82233	0.0713132	0.0356566	+	0.999364i	$0.488648\pi$
$654$	−4.87175	−0.190500
$655$	35.3120	1.37975
$656$	131.020	5.11548
$657$	7.06814	0.275754
$658$	22.0237	0.858574
$659$	10.3978	0.405040	0.202520	−	0.979278i	$-0.435087\pi$
$659$	10.3978	0.405040	0.202520	+	0.979278i	$0.435087\pi$
$660$	−33.9586	−1.32184
$661$	23.2231	0.903276	0.451638	−	0.892201i	$-0.350840\pi$
$661$	23.2231	0.903276	0.451638	+	0.892201i	$0.350840\pi$
$662$	16.2251	0.630607
$663$	−2.75419	−0.106964
$664$	−20.4927	−0.795270
$665$	−2.53407	−0.0982670
$666$	7.80361	0.302384
$667$	−9.50837	−0.368166
$668$	−79.4770	−3.07506
$669$	−8.33268	−0.322160
$670$	−68.9379	−2.66330
$671$	16.5705	0.639696
$672$	23.3026	0.898918
$673$	13.5498	0.522305	0.261153	−	0.965298i	$-0.415897\pi$
$673$	13.5498	0.522305	0.261153	+	0.965298i	$0.415897\pi$
$674$	−11.2231	−0.432299
$675$	1.42151	0.0547140
$676$	−70.9597	−2.72922
$677$	−12.6466	−0.486049	−0.243025	−	0.970020i	$-0.578140\pi$
$677$	−12.6466	−0.486049	−0.243025	+	0.970020i	$0.578140\pi$
$678$	−27.5134	−1.05664
$679$	4.97930	0.191088
$680$	160.617	6.15939
$681$	16.8905	0.647244
$682$	−23.1944	−0.888160
$683$	−13.7799	−0.527273	−0.263636	−	0.964622i	$-0.584922\pi$
$683$	−13.7799	−0.527273	−0.263636	+	0.964622i	$0.584922\pi$
$684$	−5.53407	−0.211600
$685$	−43.0267	−1.64397
$686$	2.74483	0.104798
$687$	6.04744	0.230724
$688$	178.754	6.81492
$689$	4.58046	0.174502
$690$	−10.9793	−0.417975
$691$	1.11559	0.0424389	0.0212194	−	0.999775i	$-0.493245\pi$
$691$	1.11559	0.0424389	0.0212194	+	0.999775i	$0.493245\pi$
$692$	−31.6259	−1.20224
$693$	−2.42151	−0.0919856
$694$	47.6320	1.80809
$695$	2.70674	0.102673
$696$	−58.4326	−2.21488
$697$	55.0267	2.08429
$698$	12.9192	0.488999
$699$	21.2044	0.802025
$700$	−7.86675	−0.297335
$701$	16.7067	0.631005	0.315502	−	0.948925i	$-0.397827\pi$
$701$	16.7067	0.631005	0.315502	+	0.948925i	$0.397827\pi$
$702$	1.15698	0.0436673
$703$	2.84302	0.107227
$704$	79.5371	2.99767
$705$	−20.3327	−0.765773
$706$	−56.7365	−2.13531
$707$	1.69105	0.0635984
$708$	78.2312	2.94011
$709$	−25.1443	−0.944314	−0.472157	−	0.881514i	$-0.656525\pi$
$709$	−25.1443	−0.944314	−0.472157	+	0.881514i	$0.656525\pi$
$710$	−93.0455	−3.49193
$711$	14.1363	0.530152
$712$	94.2372	3.53169
$713$	−5.50837	−0.206290
$714$	17.9349	0.671196
$715$	2.58651	0.0967302
$716$	−53.1393	−1.98591
$717$	−5.80361	−0.216740
$718$	−22.2538	−0.830506
$719$	32.2488	1.20268	0.601339	−	0.798994i	$-0.294634\pi$
$719$	32.2488	1.20268	0.601339	+	0.798994i	$0.294634\pi$
$720$	−39.4245	−1.46927
$721$	−3.15698	−0.117572
$722$	−2.74483	−0.102152
$723$	−12.9793	−0.482706
$724$	−5.41954	−0.201416
$725$	8.56279	0.318014
$726$	14.0982	0.523233
$727$	−30.2312	−1.12121	−0.560606	−	0.828083i	$-0.689432\pi$
$727$	−30.2312	−1.12121	−0.560606	+	0.828083i	$0.689432\pi$
$728$	−4.08884	−0.151542
$729$	1.00000	0.0370370
$730$	49.1630	1.81961
$731$	75.0742	2.77672
$732$	37.8698	1.39971
$733$	25.0267	0.924384	0.462192	−	0.886780i	$-0.347063\pi$
$733$	25.0267	0.924384	0.462192	+	0.886780i	$0.347063\pi$
$734$	−79.2993	−2.92699
$735$	−2.53407	−0.0934706
$736$	36.7829	1.35584
$737$	−24.0000	−0.884051
$738$	−23.1156	−0.850896
$739$	5.53104	0.203463	0.101731	−	0.994812i	$-0.467562\pi$
$739$	5.53104	0.203463	0.101731	+	0.994812i	$0.467562\pi$
$740$	39.8698	1.46564
$741$	0.421512	0.0154846
$742$	−29.8273	−1.09500
$743$	21.6971	0.795989	0.397995	−	0.917388i	$-0.369706\pi$
$743$	21.6971	0.795989	0.397995	+	0.917388i	$0.369706\pi$
$744$	−33.8510	−1.24104
$745$	−35.5972	−1.30418
$746$	−32.8243	−1.20178
$747$	2.11256	0.0772945
$748$	87.5619	3.20158
$749$	16.5341	0.604142
$750$	−24.8905	−0.908871
$751$	−48.9854	−1.78750	−0.893751	−	0.448564i	$-0.851935\pi$
$751$	−48.9854	−1.78750	−0.893751	+	0.448564i	$0.851935\pi$
$752$	124.831	4.55213
$753$	10.3377	0.376726
$754$	6.96931	0.253807
$755$	37.3881	1.36069
$756$	−5.53407	−0.201272
$757$	−4.04139	−0.146887	−0.0734434	−	0.997299i	$-0.523399\pi$
$757$	−4.04139	−0.146887	−0.0734434	+	0.997299i	$0.523399\pi$
$758$	42.2212	1.53354
$759$	−3.82233	−0.138742
$760$	−24.5815	−0.891665
$761$	−36.9429	−1.33918	−0.669590	−	0.742731i	$-0.733530\pi$
$761$	−36.9429	−1.33918	−0.669590	+	0.742731i	$0.733530\pi$
$762$	−5.73349	−0.207702
$763$	−1.77488	−0.0642551
$764$	−64.8304	−2.34548
$765$	−16.5578	−0.598648
$766$	1.69605	0.0612806
$767$	−5.95861	−0.215153
$768$	53.8491	1.94311
$769$	24.1363	0.870377	0.435188	−	0.900339i	$-0.356682\pi$
$769$	24.1363	0.870377	0.435188	+	0.900339i	$0.356682\pi$
$770$	−16.8430	−0.606980
$771$	−15.1757	−0.546539
$772$	98.6300	3.54977
$773$	−25.5371	−0.918506	−0.459253	−	0.888306i	$-0.651883\pi$
$773$	−25.5371	−0.918506	−0.459253	+	0.888306i	$0.651883\pi$
$774$	−31.5371	−1.13358
$775$	4.96058	0.178189
$776$	48.3013	1.73392
$777$	2.84302	0.101993
$778$	30.3801	1.08918
$779$	−8.42151	−0.301732
$780$	5.91116	0.211654
$781$	−32.3928	−1.15911
$782$	28.3100	1.01236
$783$	6.02372	0.215270
$784$	15.5578	0.555635
$785$	40.8905	1.45944
$786$	38.2488	1.36429
$787$	−19.0969	−0.680730	−0.340365	−	0.940293i	$-0.610551\pi$
$787$	−19.0969	−0.680730	−0.340365	+	0.940293i	$0.610551\pi$
$788$	−105.787	−3.76851
$789$	−8.51035	−0.302976
$790$	98.3261	3.49828
$791$	−10.0237	−0.356403
$792$	−23.4897	−0.834668
$793$	−2.88441	−0.102429
$794$	28.6841	1.01796
$795$	27.5371	0.976640
$796$	88.8077	3.14770
$797$	8.02872	0.284392	0.142196	−	0.989839i	$-0.454584\pi$
$797$	8.02872	0.284392	0.142196	+	0.989839i	$0.454584\pi$
$798$	−2.74483	−0.0971658
$799$	52.4276	1.85475
$800$	−33.1249	−1.17114
$801$	−9.71477	−0.343254
$802$	−8.97430	−0.316894
$803$	17.1156	0.603996
$804$	−54.8491	−1.93438
$805$	−4.00000	−0.140981
$806$	4.03745	0.142213
$807$	15.1757	0.534210
$808$	16.4038	0.577085
$809$	−29.0742	−1.02219	−0.511097	−	0.859523i	$-0.670761\pi$
$809$	−29.0742	−1.02219	−0.511097	+	0.859523i	$0.670761\pi$
$810$	6.95558	0.244394
$811$	4.66535	0.163823	0.0819113	−	0.996640i	$-0.473898\pi$
$811$	4.66535	0.163823	0.0819113	+	0.996640i	$0.473898\pi$
$812$	−33.3357	−1.16985
$813$	0.843024	0.0295661
$814$	18.8965	0.662323
$815$	16.8430	0.589985
$816$	101.656	3.55866
$817$	−11.4897	−0.401972
$818$	−20.3140	−0.710261
$819$	0.421512	0.0147288
$820$	−118.101	−4.12426
$821$	−17.5972	−0.614147	−0.307073	−	0.951686i	$-0.599350\pi$
$821$	−17.5972	−0.614147	−0.307073	+	0.951686i	$0.599350\pi$
$822$	−46.6052	−1.62554
$823$	−7.93989	−0.276767	−0.138384	−	0.990379i	$-0.544191\pi$
$823$	−7.93989	−0.276767	−0.138384	+	0.990379i	$0.544191\pi$
$824$	−30.6240	−1.06684
$825$	3.44221	0.119842
$826$	38.8016	1.35008
$827$	−31.7859	−1.10531	−0.552653	−	0.833412i	$-0.686384\pi$
$827$	−31.7859	−1.10531	−0.552653	+	0.833412i	$0.686384\pi$
$828$	−8.73546	−0.303578
$829$	−48.9793	−1.70112	−0.850561	−	0.525877i	$-0.823737\pi$
$829$	−48.9793	−1.70112	−0.850561	+	0.525877i	$0.823737\pi$
$830$	14.6941	0.510039
$831$	10.5578	0.366246
$832$	−13.8450	−0.479989
$833$	6.53407	0.226392
$834$	2.93186	0.101522
$835$	36.3928	1.25942
$836$	−13.4008	−0.463477
$837$	3.48965	0.120620
$838$	87.4119	3.01959
$839$	36.2726	1.25227	0.626134	−	0.779716i	$-0.284636\pi$
$839$	36.2726	1.25227	0.626134	+	0.779716i	$0.284636\pi$
$840$	−24.5815	−0.848143
$841$	7.28523	0.251215
$842$	−109.810	−3.78429
$843$	15.0444	0.518157
$844$	−51.4295	−1.77028
$845$	32.4927	1.11778
$846$	−22.0237	−0.757191
$847$	5.13628	0.176485
$848$	−169.063	−5.80563
$849$	1.24581	0.0427562
$850$	−25.4947	−0.874459
$851$	4.48768	0.153836
$852$	−74.0298	−2.53622
$853$	−34.6654	−1.18692	−0.593460	−	0.804864i	$-0.702238\pi$
$853$	−34.6654	−1.18692	−0.593460	+	0.804864i	$0.702238\pi$
$854$	18.7829	0.642738
$855$	2.53407	0.0866634
$856$	160.387	5.48192
$857$	−4.87175	−0.166416	−0.0832078	−	0.996532i	$-0.526517\pi$
$857$	−4.87175	−0.166416	−0.0832078	+	0.996532i	$0.526517\pi$
$858$	2.80163	0.0956461
$859$	−37.1156	−1.26637	−0.633184	−	0.774002i	$-0.718252\pi$
$859$	−37.1156	−1.26637	−0.633184	+	0.774002i	$0.718252\pi$
$860$	−161.128	−5.49441
$861$	−8.42151	−0.287004
$862$	−10.7492	−0.366119
$863$	−29.1520	−0.992345	−0.496172	−	0.868224i	$-0.665262\pi$
$863$	−29.1520	−0.992345	−0.496172	+	0.868224i	$0.665262\pi$
$864$	−23.3026	−0.792771
$865$	14.4816	0.492390
$866$	−49.4069	−1.67891
$867$	25.6941	0.872616
$868$	−19.3120	−0.655491
$869$	34.2312	1.16121
$870$	41.8985	1.42049
$871$	4.17767	0.141555
$872$	−17.2171	−0.583044
$873$	−4.97930	−0.168524
$874$	−4.33268	−0.146555
$875$	−9.06814	−0.306559
$876$	39.1156	1.32159
$877$	14.4502	0.487950	0.243975	−	0.969782i	$-0.421549\pi$
$877$	14.4502	0.487950	0.243975	+	0.969782i	$0.421549\pi$
$878$	−47.8924	−1.61629
$879$	20.4215	0.688800
$880$	−95.4670	−3.21819
$881$	10.5341	0.354902	0.177451	−	0.984130i	$-0.443215\pi$
$881$	10.5341	0.354902	0.177451	+	0.984130i	$0.443215\pi$
$882$	−2.74483	−0.0924231
$883$	−5.11559	−0.172153	−0.0860766	−	0.996289i	$-0.527433\pi$
$883$	−5.11559	−0.172153	−0.0860766	+	0.996289i	$0.527433\pi$
$884$	−15.2419	−0.512639
$885$	−35.8223	−1.20415
$886$	8.34267	0.280278
$887$	−0.617907	−0.0207473	−0.0103736	−	0.999946i	$-0.503302\pi$
$887$	−0.617907	−0.0207473	−0.0103736	+	0.999946i	$0.503302\pi$
$888$	27.5785	0.925473
$889$	−2.08884	−0.0700573
$890$	−67.5719	−2.26501
$891$	2.42151	0.0811237
$892$	−46.1136	−1.54400
$893$	−8.02372	−0.268504
$894$	−38.5578	−1.28957
$895$	24.3327	0.813352
$896$	43.5515	1.45495
$897$	0.665351	0.0222154
$898$	−98.6350	−3.29149
$899$	21.0207	0.701079
$900$	7.86675	0.262225
$901$	−71.0041	−2.36549
$902$	−55.9747	−1.86375
$903$	−11.4897	−0.382352
$904$	−97.2342	−3.23396
$905$	2.48163	0.0824920
$906$	40.4977	1.34545
$907$	−8.27256	−0.274686	−0.137343	−	0.990524i	$-0.543856\pi$
$907$	−8.27256	−0.274686	−0.137343	+	0.990524i	$0.543856\pi$
$908$	93.4730	3.10201
$909$	−1.69105	−0.0560885
$910$	2.93186	0.0971902
$911$	−5.77988	−0.191496	−0.0957480	−	0.995406i	$-0.530524\pi$
$911$	−5.77988	−0.191496	−0.0957480	+	0.995406i	$0.530524\pi$
$912$	−15.5578	−0.515170
$913$	5.11559	0.169301
$914$	67.0328	2.21725
$915$	−17.3407	−0.573266
$916$	33.4670	1.10578
$917$	13.9349	0.460170
$918$	−17.9349	−0.591939
$919$	4.62791	0.152661	0.0763303	−	0.997083i	$-0.475680\pi$
$919$	4.62791	0.152661	0.0763303	+	0.997083i	$0.475680\pi$
$920$	−38.8016	−1.27925
$921$	−2.64663	−0.0872094
$922$	−27.8086	−0.915828
$923$	5.63860	0.185597
$924$	−13.4008	−0.440854
$925$	−4.04139	−0.132880
$926$	16.0949	0.528911
$927$	3.15698	0.103689
$928$	−140.369	−4.60782
$929$	−29.9536	−0.982746	−0.491373	−	0.870949i	$-0.663505\pi$
$929$	−29.9536	−0.982746	−0.491373	+	0.870949i	$0.663505\pi$
$930$	24.2726	0.795929
$931$	−1.00000	−0.0327737
$932$	117.347	3.84382
$933$	−5.09186	−0.166700
$934$	41.0505	1.34321
$935$	−40.0949	−1.31124
$936$	4.08884	0.133648
$937$	4.58651	0.149835	0.0749174	−	0.997190i	$-0.476131\pi$
$937$	4.58651	0.149835	0.0749174	+	0.997190i	$0.476131\pi$
$938$	−27.2044	−0.888256
$939$	19.9586	0.651325
$940$	−112.522	−3.67008
$941$	7.41082	0.241586	0.120793	−	0.992678i	$-0.461456\pi$
$941$	7.41082	0.241586	0.120793	+	0.992678i	$0.461456\pi$
$942$	44.2913	1.44309
$943$	−13.2933	−0.432888
$944$	219.929	7.15809
$945$	2.53407	0.0824333
$946$	−76.3675	−2.48292
$947$	59.6320	1.93778	0.968890	−	0.247493i	$-0.0796068\pi$
$947$	59.6320	1.93778	0.968890	+	0.247493i	$0.0796068\pi$
$948$	78.2312	2.54083
$949$	−2.97930	−0.0967123
$950$	3.90180	0.126591
$951$	−6.81930	−0.221131
$952$	63.3831	2.05426
$953$	25.7986	0.835699	0.417849	−	0.908516i	$-0.362784\pi$
$953$	25.7986	0.835699	0.417849	+	0.908516i	$0.362784\pi$
$954$	29.8273	0.965695
$955$	29.6860	0.960618
$956$	−32.1176	−1.03876
$957$	14.5865	0.471515
$958$	92.5275	2.98943
$959$	−16.9793	−0.548290
$960$	−83.2342	−2.68637
$961$	−18.8223	−0.607172
$962$	−3.28931	−0.106052
$963$	−16.5341	−0.532803
$964$	−71.8284	−2.31344
$965$	−45.1630	−1.45385
$966$	−4.33268	−0.139402
$967$	55.5845	1.78748	0.893739	−	0.448587i	$-0.148073\pi$
$967$	55.5845	1.78748	0.893739	+	0.448587i	$0.148073\pi$
$968$	49.8240	1.60140
$969$	−6.53407	−0.209905
$970$	−34.6340	−1.11203
$971$	−48.4877	−1.55604	−0.778022	−	0.628237i	$-0.783777\pi$
$971$	−48.4877	−1.55604	−0.778022	+	0.628237i	$0.783777\pi$
$972$	5.53407	0.177505
$973$	1.06814	0.0342430
$974$	−59.0367	−1.89166
$975$	−0.599184	−0.0191892
$976$	106.462	3.40778
$977$	53.0505	1.69723	0.848617	−	0.529007i	$-0.177435\pi$
$977$	53.0505	1.69723	0.848617	+	0.529007i	$0.177435\pi$
$978$	18.2438	0.583374
$979$	−23.5244	−0.751844
$980$	−14.0237	−0.447971
$981$	1.77488	0.0566677
$982$	47.6320	1.52000
$983$	12.1777	0.388407	0.194204	−	0.980961i	$-0.437788\pi$
$983$	12.1777	0.388407	0.194204	+	0.980961i	$0.437788\pi$
$984$	−81.6921	−2.60425
$985$	48.4402	1.54343
$986$	−108.035	−3.44053
$987$	−8.02372	−0.255398
$988$	2.33268	0.0742123
$989$	−18.1363	−0.576700
$990$	16.8430	0.535306
$991$	−2.04139	−0.0648469	−0.0324235	−	0.999474i	$-0.510323\pi$
$991$	−2.04139	−0.0648469	−0.0324235	+	0.999474i	$0.510323\pi$
$992$	−81.3180	−2.58185
$993$	−5.91116	−0.187585
$994$	−36.7178	−1.16462
$995$	−40.6654	−1.28918
$996$	11.6910	0.370445
$997$	−45.6447	−1.44558	−0.722790	−	0.691067i	$-0.757141\pi$
$997$	−45.6447	−1.44558	−0.722790	+	0.691067i	$0.757141\pi$
$998$	90.4088	2.86184
$999$	−2.84302	−0.0899493

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	399.2.a.e.1.1	✓	3
3.2	odd	2		1197.2.a.m.1.3		3
4.3	odd	2		6384.2.a.bu.1.1		3
5.4	even	2		9975.2.a.x.1.3		3
7.6	odd	2		2793.2.a.w.1.1		3
19.18	odd	2		7581.2.a.l.1.3		3
21.20	even	2		8379.2.a.bq.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
399.2.a.e.1.1	✓	3	1.1	even	1	trivial
1197.2.a.m.1.3		3	3.2	odd	2
2793.2.a.w.1.1		3	7.6	odd	2
6384.2.a.bu.1.1		3	4.3	odd	2
7581.2.a.l.1.3		3	19.18	odd	2
8379.2.a.bq.1.3		3	21.20	even	2
9975.2.a.x.1.3		3	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 399 = 3 \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	399.a (trivial)

\(f(q)\)	\(=\)	\(q-2.74483 q^{2} +1.00000 q^{3} +5.53407 q^{4} -2.53407 q^{5} -2.74483 q^{6} -1.00000 q^{7} -9.70041 q^{8} +1.00000 q^{9} +6.95558 q^{10} +2.42151 q^{11} +5.53407 q^{12} -0.421512 q^{13} +2.74483 q^{14} -2.53407 q^{15} +15.5578 q^{16} +6.53407 q^{17} -2.74483 q^{18} -1.00000 q^{19} -14.0237 q^{20} -1.00000 q^{21} -6.64663 q^{22} -1.57849 q^{23} -9.70041 q^{24} +1.42151 q^{25} +1.15698 q^{26} +1.00000 q^{27} -5.53407 q^{28} +6.02372 q^{29} +6.95558 q^{30} +3.48965 q^{31} -23.3026 q^{32} +2.42151 q^{33} -17.9349 q^{34} +2.53407 q^{35} +5.53407 q^{36} -2.84302 q^{37} +2.74483 q^{38} -0.421512 q^{39} +24.5815 q^{40} +8.42151 q^{41} +2.74483 q^{42} +11.4897 q^{43} +13.4008 q^{44} -2.53407 q^{45} +4.33268 q^{46} +8.02372 q^{47} +15.5578 q^{48} +1.00000 q^{49} -3.90180 q^{50} +6.53407 q^{51} -2.33268 q^{52} -10.8667 q^{53} -2.74483 q^{54} -6.13628 q^{55} +9.70041 q^{56} -1.00000 q^{57} -16.5341 q^{58} +14.1363 q^{59} -14.0237 q^{60} +6.84302 q^{61} -9.57849 q^{62} -1.00000 q^{63} +32.8461 q^{64} +1.06814 q^{65} -6.64663 q^{66} -9.91116 q^{67} +36.1600 q^{68} -1.57849 q^{69} -6.95558 q^{70} -13.3771 q^{71} -9.70041 q^{72} +7.06814 q^{73} +7.80361 q^{74} +1.42151 q^{75} -5.53407 q^{76} -2.42151 q^{77} +1.15698 q^{78} +14.1363 q^{79} -39.4245 q^{80} +1.00000 q^{81} -23.1156 q^{82} +2.11256 q^{83} -5.53407 q^{84} -16.5578 q^{85} -31.5371 q^{86} +6.02372 q^{87} -23.4897 q^{88} -9.71477 q^{89} +6.95558 q^{90} +0.421512 q^{91} -8.73546 q^{92} +3.48965 q^{93} -22.0237 q^{94} +2.53407 q^{95} -23.3026 q^{96} -4.97930 q^{97} -2.74483 q^{98} +2.42151 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} + 3 q^{3} + 9 q^{4} + q^{6} - 3 q^{7} - 9 q^{8} + 3 q^{9} + 10 q^{10} + 4 q^{11} + 9 q^{12} + 2 q^{13} - q^{14} + 13 q^{16} + 12 q^{17} + q^{18} - 3 q^{19} - 16 q^{20} - 3 q^{21} - 8 q^{22}+ \cdots + 4 q^{99}+O(q^{100}) \) 3 * q + q^2 + 3 * q^3 + 9 * q^4 + q^6 - 3 * q^7 - 9 * q^8 + 3 * q^9 + 10 * q^10 + 4 * q^11 + 9 * q^12 + 2 * q^13 - q^14 + 13 * q^16 + 12 * q^17 + q^18 - 3 * q^19 - 16 * q^20 - 3 * q^21 - 8 * q^22 - 8 * q^23 - 9 * q^24 + q^25 + 10 * q^26 + 3 * q^27 - 9 * q^28 - 8 * q^29 + 10 * q^30 - 8 * q^31 - 27 * q^32 + 4 * q^33 - 6 * q^34 + 9 * q^36 - 2 * q^37 - q^38 + 2 * q^39 + 14 * q^40 + 22 * q^41 - q^42 + 16 * q^43 - 12 * q^46 - 2 * q^47 + 13 * q^48 + 3 * q^49 - 9 * q^50 + 12 * q^51 + 18 * q^52 + q^54 + 12 * q^55 + 9 * q^56 - 3 * q^57 - 42 * q^58 + 12 * q^59 - 16 * q^60 + 14 * q^61 - 32 * q^62 - 3 * q^63 + 29 * q^64 - 12 * q^65 - 8 * q^66 - 8 * q^67 + 52 * q^68 - 8 * q^69 - 10 * q^70 - 26 * q^71 - 9 * q^72 + 6 * q^73 + 18 * q^74 + q^75 - 9 * q^76 - 4 * q^77 + 10 * q^78 + 12 * q^79 - 52 * q^80 + 3 * q^81 - 2 * q^82 + 2 * q^83 - 9 * q^84 - 16 * q^85 - 24 * q^86 - 8 * q^87 - 52 * q^88 - 2 * q^89 + 10 * q^90 - 2 * q^91 - 36 * q^92 - 8 * q^93 - 40 * q^94 - 27 * q^96 + 22 * q^97 + q^98 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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