LMFDB - Embedded newform 3465.2.a.v.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3465,2,Mod(1,3465)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3465, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3465.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3465 = 3^{2} \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3465.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:	$27.6681643004$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1155)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	3465.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.732051 q^{2} -1.46410 q^{4} +1.00000 q^{5} +1.00000 q^{7} -2.53590 q^{8} +O(q^{10})$	\(q+0.732051 q^{2} -1.46410 q^{4} +1.00000 q^{5} +1.00000 q^{7} -2.53590 q^{8} +0.732051 q^{10} +1.00000 q^{11} -4.19615 q^{13} +0.732051 q^{14} +1.07180 q^{16} +6.46410 q^{17} -1.53590 q^{19} -1.46410 q^{20} +0.732051 q^{22} -8.46410 q^{23} +1.00000 q^{25} -3.07180 q^{26} -1.46410 q^{28} -5.73205 q^{29} -5.26795 q^{31} +5.85641 q^{32} +4.73205 q^{34} +1.00000 q^{35} -4.19615 q^{37} -1.12436 q^{38} -2.53590 q^{40} +7.66025 q^{41} +1.73205 q^{43} -1.46410 q^{44} -6.19615 q^{46} -1.26795 q^{47} +1.00000 q^{49} +0.732051 q^{50} +6.14359 q^{52} -4.46410 q^{53} +1.00000 q^{55} -2.53590 q^{56} -4.19615 q^{58} -13.7321 q^{59} +11.3923 q^{61} -3.85641 q^{62} +2.14359 q^{64} -4.19615 q^{65} +4.92820 q^{67} -9.46410 q^{68} +0.732051 q^{70} -7.66025 q^{71} +8.39230 q^{73} -3.07180 q^{74} +2.24871 q^{76} +1.00000 q^{77} +3.66025 q^{79} +1.07180 q^{80} +5.60770 q^{82} -2.46410 q^{83} +6.46410 q^{85} +1.26795 q^{86} -2.53590 q^{88} -14.6603 q^{89} -4.19615 q^{91} +12.3923 q^{92} -0.928203 q^{94} -1.53590 q^{95} -2.26795 q^{97} +0.732051 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 4 q^{4} + 2 q^{5} + 2 q^{7} - 12 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 4 * q^4 + 2 * q^5 + 2 * q^7 - 12 * q^8	$ 2 q - 2 q^{2} + 4 q^{4} + 2 q^{5} + 2 q^{7} - 12 q^{8} - 2 q^{10} + 2 q^{11} + 2 q^{13} - 2 q^{14} + 16 q^{16} + 6 q^{17} - 10 q^{19} + 4 q^{20} - 2 q^{22} - 10 q^{23} + 2 q^{25} - 20 q^{26} + 4 q^{28} - 8 q^{29} - 14 q^{31} - 16 q^{32} + 6 q^{34} + 2 q^{35} + 2 q^{37} + 22 q^{38} - 12 q^{40} - 2 q^{41} + 4 q^{44} - 2 q^{46} - 6 q^{47} + 2 q^{49} - 2 q^{50} + 40 q^{52} - 2 q^{53} + 2 q^{55} - 12 q^{56} + 2 q^{58} - 24 q^{59} + 2 q^{61} + 20 q^{62} + 32 q^{64} + 2 q^{65} - 4 q^{67} - 12 q^{68} - 2 q^{70} + 2 q^{71} - 4 q^{73} - 20 q^{74} - 44 q^{76} + 2 q^{77} - 10 q^{79} + 16 q^{80} + 32 q^{82} + 2 q^{83} + 6 q^{85} + 6 q^{86} - 12 q^{88} - 12 q^{89} + 2 q^{91} + 4 q^{92} + 12 q^{94} - 10 q^{95} - 8 q^{97} - 2 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 4 * q^4 + 2 * q^5 + 2 * q^7 - 12 * q^8 - 2 * q^10 + 2 * q^11 + 2 * q^13 - 2 * q^14 + 16 * q^16 + 6 * q^17 - 10 * q^19 + 4 * q^20 - 2 * q^22 - 10 * q^23 + 2 * q^25 - 20 * q^26 + 4 * q^28 - 8 * q^29 - 14 * q^31 - 16 * q^32 + 6 * q^34 + 2 * q^35 + 2 * q^37 + 22 * q^38 - 12 * q^40 - 2 * q^41 + 4 * q^44 - 2 * q^46 - 6 * q^47 + 2 * q^49 - 2 * q^50 + 40 * q^52 - 2 * q^53 + 2 * q^55 - 12 * q^56 + 2 * q^58 - 24 * q^59 + 2 * q^61 + 20 * q^62 + 32 * q^64 + 2 * q^65 - 4 * q^67 - 12 * q^68 - 2 * q^70 + 2 * q^71 - 4 * q^73 - 20 * q^74 - 44 * q^76 + 2 * q^77 - 10 * q^79 + 16 * q^80 + 32 * q^82 + 2 * q^83 + 6 * q^85 + 6 * q^86 - 12 * q^88 - 12 * q^89 + 2 * q^91 + 4 * q^92 + 12 * q^94 - 10 * q^95 - 8 * q^97 - 2 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.732051	0.517638	0.258819	−	0.965926i	$-0.416667\pi$
$2$	0.732051	0.517638	0.258819	+	0.965926i	$0.416667\pi$
$3$	0	0
$3$	0	0
$4$	−1.46410	−0.732051
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−2.53590	−0.896575
$9$	0	0
$10$	0.732051	0.231495
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−4.19615	−1.16380	−0.581902	−	0.813259i	$-0.697691\pi$
$13$	−4.19615	−1.16380	−0.581902	+	0.813259i	$0.697691\pi$
$14$	0.732051	0.195649
$15$	0	0
$16$	1.07180	0.267949
$17$	6.46410	1.56777	0.783887	−	0.620903i	$-0.213234\pi$
$17$	6.46410	1.56777	0.783887	+	0.620903i	$0.213234\pi$
$18$	0	0
$19$	−1.53590	−0.352359	−0.176180	−	0.984358i	$-0.556374\pi$
$19$	−1.53590	−0.352359	−0.176180	+	0.984358i	$0.556374\pi$
$20$	−1.46410	−0.327383
$21$	0	0
$22$	0.732051	0.156074
$23$	−8.46410	−1.76489	−0.882444	−	0.470418i	$-0.844103\pi$
$23$	−8.46410	−1.76489	−0.882444	+	0.470418i	$0.844103\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−3.07180	−0.602429
$27$	0	0
$28$	−1.46410	−0.276689
$29$	−5.73205	−1.06442	−0.532208	−	0.846614i	$-0.678637\pi$
$29$	−5.73205	−1.06442	−0.532208	+	0.846614i	$0.678637\pi$
$30$	0	0
$31$	−5.26795	−0.946152	−0.473076	−	0.881022i	$-0.656856\pi$
$31$	−5.26795	−0.946152	−0.473076	+	0.881022i	$0.656856\pi$
$32$	5.85641	1.03528
$33$	0	0
$34$	4.73205	0.811540
$35$	1.00000	0.169031
$36$	0	0
$37$	−4.19615	−0.689843	−0.344922	−	0.938631i	$-0.612095\pi$
$37$	−4.19615	−0.689843	−0.344922	+	0.938631i	$0.612095\pi$
$38$	−1.12436	−0.182395
$39$	0	0
$40$	−2.53590	−0.400961
$41$	7.66025	1.19633	0.598165	−	0.801373i	$-0.295897\pi$
$41$	7.66025	1.19633	0.598165	+	0.801373i	$0.295897\pi$
$42$	0	0
$43$	1.73205	0.264135	0.132068	−	0.991241i	$-0.457838\pi$
$43$	1.73205	0.264135	0.132068	+	0.991241i	$0.457838\pi$
$44$	−1.46410	−0.220722
$45$	0	0
$46$	−6.19615	−0.913573
$47$	−1.26795	−0.184949	−0.0924747	−	0.995715i	$-0.529478\pi$
$47$	−1.26795	−0.184949	−0.0924747	+	0.995715i	$0.529478\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0.732051	0.103528
$51$	0	0
$52$	6.14359	0.851963
$53$	−4.46410	−0.613192	−0.306596	−	0.951840i	$-0.599190\pi$
$53$	−4.46410	−0.613192	−0.306596	+	0.951840i	$0.599190\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	−2.53590	−0.338874
$57$	0	0
$58$	−4.19615	−0.550982
$59$	−13.7321	−1.78776	−0.893880	−	0.448306i	$-0.852028\pi$
$59$	−13.7321	−1.78776	−0.893880	+	0.448306i	$0.852028\pi$
$60$	0	0
$61$	11.3923	1.45864	0.729318	−	0.684175i	$-0.239838\pi$
$61$	11.3923	1.45864	0.729318	+	0.684175i	$0.239838\pi$
$62$	−3.85641	−0.489764
$63$	0	0
$64$	2.14359	0.267949
$65$	−4.19615	−0.520469
$66$	0	0
$67$	4.92820	0.602076	0.301038	−	0.953612i	$-0.402667\pi$
$67$	4.92820	0.602076	0.301038	+	0.953612i	$0.402667\pi$
$68$	−9.46410	−1.14769
$69$	0	0
$70$	0.732051	0.0874968
$71$	−7.66025	−0.909105	−0.454552	−	0.890720i	$-0.650201\pi$
$71$	−7.66025	−0.909105	−0.454552	+	0.890720i	$0.650201\pi$
$72$	0	0
$73$	8.39230	0.982245	0.491122	−	0.871091i	$-0.336587\pi$
$73$	8.39230	0.982245	0.491122	+	0.871091i	$0.336587\pi$
$74$	−3.07180	−0.357089
$75$	0	0
$76$	2.24871	0.257945
$77$	1.00000	0.113961
$78$	0	0
$79$	3.66025	0.411811	0.205905	−	0.978572i	$-0.433986\pi$
$79$	3.66025	0.411811	0.205905	+	0.978572i	$0.433986\pi$
$80$	1.07180	0.119831
$81$	0	0
$82$	5.60770	0.619266
$83$	−2.46410	−0.270470	−0.135235	−	0.990814i	$-0.543179\pi$
$83$	−2.46410	−0.270470	−0.135235	+	0.990814i	$0.543179\pi$
$84$	0	0
$85$	6.46410	0.701130
$86$	1.26795	0.136726
$87$	0	0
$88$	−2.53590	−0.270328
$89$	−14.6603	−1.55398	−0.776992	−	0.629511i	$-0.783255\pi$
$89$	−14.6603	−1.55398	−0.776992	+	0.629511i	$0.783255\pi$
$90$	0	0
$91$	−4.19615	−0.439876
$92$	12.3923	1.29199
$93$	0	0
$94$	−0.928203	−0.0957369
$95$	−1.53590	−0.157580
$96$	0	0
$97$	−2.26795	−0.230275	−0.115138	−	0.993350i	$-0.536731\pi$
$97$	−2.26795	−0.230275	−0.115138	+	0.993350i	$0.536731\pi$
$98$	0.732051	0.0739483
$99$	0	0
$100$	−1.46410	−0.146410
$101$	−16.9282	−1.68442	−0.842210	−	0.539150i	$-0.818745\pi$
$101$	−16.9282	−1.68442	−0.842210	+	0.539150i	$0.818745\pi$
$102$	0	0
$103$	−14.2679	−1.40586	−0.702931	−	0.711258i	$-0.748126\pi$
$103$	−14.2679	−1.40586	−0.702931	+	0.711258i	$0.748126\pi$
$104$	10.6410	1.04344
$105$	0	0
$106$	−3.26795	−0.317411
$107$	−5.26795	−0.509272	−0.254636	−	0.967037i	$-0.581956\pi$
$107$	−5.26795	−0.509272	−0.254636	+	0.967037i	$0.581956\pi$
$108$	0	0
$109$	−5.26795	−0.504578	−0.252289	−	0.967652i	$-0.581183\pi$
$109$	−5.26795	−0.504578	−0.252289	+	0.967652i	$0.581183\pi$
$110$	0.732051	0.0697983
$111$	0	0
$112$	1.07180	0.101275
$113$	−10.8564	−1.02128	−0.510642	−	0.859793i	$-0.670592\pi$
$113$	−10.8564	−1.02128	−0.510642	+	0.859793i	$0.670592\pi$
$114$	0	0
$115$	−8.46410	−0.789282
$116$	8.39230	0.779206
$117$	0	0
$118$	−10.0526	−0.925413
$119$	6.46410	0.592563
$120$	0	0
$121$	1.00000	0.0909091
$122$	8.33975	0.755045
$123$	0	0
$124$	7.71281	0.692631
$125$	1.00000	0.0894427
$126$	0	0
$127$	−22.1244	−1.96322	−0.981610	−	0.190900i	$-0.938859\pi$
$127$	−22.1244	−1.96322	−0.981610	+	0.190900i	$0.938859\pi$
$128$	−10.1436	−0.896575
$129$	0	0
$130$	−3.07180	−0.269414
$131$	−16.1962	−1.41506	−0.707532	−	0.706681i	$-0.750192\pi$
$131$	−16.1962	−1.41506	−0.707532	+	0.706681i	$0.750192\pi$
$132$	0	0
$133$	−1.53590	−0.133179
$134$	3.60770	0.311657
$135$	0	0
$136$	−16.3923	−1.40563
$137$	6.53590	0.558399	0.279200	−	0.960233i	$-0.409931\pi$
$137$	6.53590	0.558399	0.279200	+	0.960233i	$0.409931\pi$
$138$	0	0
$139$	11.4641	0.972372	0.486186	−	0.873855i	$-0.338388\pi$
$139$	11.4641	0.972372	0.486186	+	0.873855i	$0.338388\pi$
$140$	−1.46410	−0.123739
$141$	0	0
$142$	−5.60770	−0.470587
$143$	−4.19615	−0.350900
$144$	0	0
$145$	−5.73205	−0.476021
$146$	6.14359	0.508447
$147$	0	0
$148$	6.14359	0.505000
$149$	−15.4641	−1.26687	−0.633434	−	0.773796i	$-0.718355\pi$
$149$	−15.4641	−1.26687	−0.633434	+	0.773796i	$0.718355\pi$
$150$	0	0
$151$	−11.8564	−0.964861	−0.482430	−	0.875934i	$-0.660246\pi$
$151$	−11.8564	−0.964861	−0.482430	+	0.875934i	$0.660246\pi$
$152$	3.89488	0.315917
$153$	0	0
$154$	0.732051	0.0589903
$155$	−5.26795	−0.423132
$156$	0	0
$157$	−14.1244	−1.12725	−0.563623	−	0.826032i	$-0.690593\pi$
$157$	−14.1244	−1.12725	−0.563623	+	0.826032i	$0.690593\pi$
$158$	2.67949	0.213169
$159$	0	0
$160$	5.85641	0.462990
$161$	−8.46410	−0.667065
$162$	0	0
$163$	19.1244	1.49794	0.748968	−	0.662607i	$-0.230550\pi$
$163$	19.1244	1.49794	0.748968	+	0.662607i	$0.230550\pi$
$164$	−11.2154	−0.875775
$165$	0	0
$166$	−1.80385	−0.140006
$167$	10.3923	0.804181	0.402090	−	0.915600i	$-0.368284\pi$
$167$	10.3923	0.804181	0.402090	+	0.915600i	$0.368284\pi$
$168$	0	0
$169$	4.60770	0.354438
$170$	4.73205	0.362932
$171$	0	0
$172$	−2.53590	−0.193360
$173$	25.4641	1.93600	0.968000	−	0.250951i	$-0.0807432\pi$
$173$	25.4641	1.93600	0.968000	+	0.250951i	$0.0807432\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	1.07180	0.0807897
$177$	0	0
$178$	−10.7321	−0.804401
$179$	20.0526	1.49880	0.749399	−	0.662118i	$-0.230342\pi$
$179$	20.0526	1.49880	0.749399	+	0.662118i	$0.230342\pi$
$180$	0	0
$181$	7.07180	0.525643	0.262821	−	0.964845i	$-0.415347\pi$
$181$	7.07180	0.525643	0.262821	+	0.964845i	$0.415347\pi$
$182$	−3.07180	−0.227697
$183$	0	0
$184$	21.4641	1.58235
$185$	−4.19615	−0.308507
$186$	0	0
$187$	6.46410	0.472702
$188$	1.85641	0.135392
$189$	0	0
$190$	−1.12436	−0.0815693
$191$	1.07180	0.0775525	0.0387762	−	0.999248i	$-0.487654\pi$
$191$	1.07180	0.0775525	0.0387762	+	0.999248i	$0.487654\pi$
$192$	0	0
$193$	−7.46410	−0.537278	−0.268639	−	0.963241i	$-0.586574\pi$
$193$	−7.46410	−0.537278	−0.268639	+	0.963241i	$0.586574\pi$
$194$	−1.66025	−0.119199
$195$	0	0
$196$	−1.46410	−0.104579
$197$	−5.46410	−0.389301	−0.194651	−	0.980873i	$-0.562357\pi$
$197$	−5.46410	−0.389301	−0.194651	+	0.980873i	$0.562357\pi$
$198$	0	0
$199$	−14.9282	−1.05823	−0.529116	−	0.848549i	$-0.677476\pi$
$199$	−14.9282	−1.05823	−0.529116	+	0.848549i	$0.677476\pi$
$200$	−2.53590	−0.179315
$201$	0	0
$202$	−12.3923	−0.871920
$203$	−5.73205	−0.402311
$204$	0	0
$205$	7.66025	0.535015
$206$	−10.4449	−0.727728
$207$	0	0
$208$	−4.49742	−0.311840
$209$	−1.53590	−0.106240
$210$	0	0
$211$	4.53590	0.312264	0.156132	−	0.987736i	$-0.450097\pi$
$211$	4.53590	0.312264	0.156132	+	0.987736i	$0.450097\pi$
$212$	6.53590	0.448887
$213$	0	0
$214$	−3.85641	−0.263619
$215$	1.73205	0.118125
$216$	0	0
$217$	−5.26795	−0.357612
$218$	−3.85641	−0.261189
$219$	0	0
$220$	−1.46410	−0.0987097
$221$	−27.1244	−1.82458
$222$	0	0
$223$	0.803848	0.0538296	0.0269148	−	0.999638i	$-0.491432\pi$
$223$	0.803848	0.0538296	0.0269148	+	0.999638i	$0.491432\pi$
$224$	5.85641	0.391298
$225$	0	0
$226$	−7.94744	−0.528656
$227$	5.39230	0.357900	0.178950	−	0.983858i	$-0.442730\pi$
$227$	5.39230	0.357900	0.178950	+	0.983858i	$0.442730\pi$
$228$	0	0
$229$	28.1962	1.86325	0.931627	−	0.363416i	$-0.118390\pi$
$229$	28.1962	1.86325	0.931627	+	0.363416i	$0.118390\pi$
$230$	−6.19615	−0.408562
$231$	0	0
$232$	14.5359	0.954328
$233$	10.1962	0.667972	0.333986	−	0.942578i	$-0.391606\pi$
$233$	10.1962	0.667972	0.333986	+	0.942578i	$0.391606\pi$
$234$	0	0
$235$	−1.26795	−0.0827119
$236$	20.1051	1.30873
$237$	0	0
$238$	4.73205	0.306733
$239$	−18.1244	−1.17237	−0.586184	−	0.810178i	$-0.699370\pi$
$239$	−18.1244	−1.17237	−0.586184	+	0.810178i	$0.699370\pi$
$240$	0	0
$241$	26.9282	1.73460	0.867299	−	0.497788i	$-0.165854\pi$
$241$	26.9282	1.73460	0.867299	+	0.497788i	$0.165854\pi$
$242$	0.732051	0.0470580
$243$	0	0
$244$	−16.6795	−1.06780
$245$	1.00000	0.0638877
$246$	0	0
$247$	6.44486	0.410077
$248$	13.3590	0.848296
$249$	0	0
$250$	0.732051	0.0462990
$251$	6.92820	0.437304	0.218652	−	0.975803i	$-0.429834\pi$
$251$	6.92820	0.437304	0.218652	+	0.975803i	$0.429834\pi$
$252$	0	0
$253$	−8.46410	−0.532134
$254$	−16.1962	−1.01624
$255$	0	0
$256$	−11.7128	−0.732051
$257$	−13.8038	−0.861060	−0.430530	−	0.902576i	$-0.641673\pi$
$257$	−13.8038	−0.861060	−0.430530	+	0.902576i	$0.641673\pi$
$258$	0	0
$259$	−4.19615	−0.260736
$260$	6.14359	0.381009
$261$	0	0
$262$	−11.8564	−0.732491
$263$	−11.3205	−0.698052	−0.349026	−	0.937113i	$-0.613488\pi$
$263$	−11.3205	−0.698052	−0.349026	+	0.937113i	$0.613488\pi$
$264$	0	0
$265$	−4.46410	−0.274228
$266$	−1.12436	−0.0689387
$267$	0	0
$268$	−7.21539	−0.440750
$269$	−7.58846	−0.462676	−0.231338	−	0.972873i	$-0.574310\pi$
$269$	−7.58846	−0.462676	−0.231338	+	0.972873i	$0.574310\pi$
$270$	0	0
$271$	−17.9282	−1.08906	−0.544530	−	0.838741i	$-0.683292\pi$
$271$	−17.9282	−1.08906	−0.544530	+	0.838741i	$0.683292\pi$
$272$	6.92820	0.420084
$273$	0	0
$274$	4.78461	0.289049
$275$	1.00000	0.0603023
$276$	0	0
$277$	−9.07180	−0.545071	−0.272536	−	0.962146i	$-0.587862\pi$
$277$	−9.07180	−0.545071	−0.272536	+	0.962146i	$0.587862\pi$
$278$	8.39230	0.503337
$279$	0	0
$280$	−2.53590	−0.151549
$281$	−17.0718	−1.01842	−0.509209	−	0.860643i	$-0.670062\pi$
$281$	−17.0718	−1.01842	−0.509209	+	0.860643i	$0.670062\pi$
$282$	0	0
$283$	−0.196152	−0.0116601	−0.00583003	−	0.999983i	$-0.501856\pi$
$283$	−0.196152	−0.0116601	−0.00583003	+	0.999983i	$0.501856\pi$
$284$	11.2154	0.665511
$285$	0	0
$286$	−3.07180	−0.181639
$287$	7.66025	0.452170
$288$	0	0
$289$	24.7846	1.45792
$290$	−4.19615	−0.246407
$291$	0	0
$292$	−12.2872	−0.719053
$293$	−9.00000	−0.525786	−0.262893	−	0.964825i	$-0.584677\pi$
$293$	−9.00000	−0.525786	−0.262893	+	0.964825i	$0.584677\pi$
$294$	0	0
$295$	−13.7321	−0.799511
$296$	10.6410	0.618497
$297$	0	0
$298$	−11.3205	−0.655779
$299$	35.5167	2.05398
$300$	0	0
$301$	1.73205	0.0998337
$302$	−8.67949	−0.499449
$303$	0	0
$304$	−1.64617	−0.0944144
$305$	11.3923	0.652321
$306$	0	0
$307$	−10.3923	−0.593120	−0.296560	−	0.955014i	$-0.595840\pi$
$307$	−10.3923	−0.593120	−0.296560	+	0.955014i	$0.595840\pi$
$308$	−1.46410	−0.0834249
$309$	0	0
$310$	−3.85641	−0.219029
$311$	−7.32051	−0.415108	−0.207554	−	0.978224i	$-0.566550\pi$
$311$	−7.32051	−0.415108	−0.207554	+	0.978224i	$0.566550\pi$
$312$	0	0
$313$	1.73205	0.0979013	0.0489506	−	0.998801i	$-0.484412\pi$
$313$	1.73205	0.0979013	0.0489506	+	0.998801i	$0.484412\pi$
$314$	−10.3397	−0.583506
$315$	0	0
$316$	−5.35898	−0.301466
$317$	−18.9282	−1.06311	−0.531557	−	0.847023i	$-0.678393\pi$
$317$	−18.9282	−1.06311	−0.531557	+	0.847023i	$0.678393\pi$
$318$	0	0
$319$	−5.73205	−0.320933
$320$	2.14359	0.119831
$321$	0	0
$322$	−6.19615	−0.345298
$323$	−9.92820	−0.552420
$324$	0	0
$325$	−4.19615	−0.232761
$326$	14.0000	0.775388
$327$	0	0
$328$	−19.4256	−1.07260
$329$	−1.26795	−0.0699043
$330$	0	0
$331$	8.32051	0.457336	0.228668	−	0.973504i	$-0.426563\pi$
$331$	8.32051	0.457336	0.228668	+	0.973504i	$0.426563\pi$
$332$	3.60770	0.197998
$333$	0	0
$334$	7.60770	0.416275
$335$	4.92820	0.269257
$336$	0	0
$337$	−10.6603	−0.580701	−0.290351	−	0.956920i	$-0.593772\pi$
$337$	−10.6603	−0.580701	−0.290351	+	0.956920i	$0.593772\pi$
$338$	3.37307	0.183471
$339$	0	0
$340$	−9.46410	−0.513263
$341$	−5.26795	−0.285275
$342$	0	0
$343$	1.00000	0.0539949
$344$	−4.39230	−0.236817
$345$	0	0
$346$	18.6410	1.00215
$347$	1.46410	0.0785971	0.0392985	−	0.999228i	$-0.487488\pi$
$347$	1.46410	0.0785971	0.0392985	+	0.999228i	$0.487488\pi$
$348$	0	0
$349$	−15.0000	−0.802932	−0.401466	−	0.915874i	$-0.631499\pi$
$349$	−15.0000	−0.802932	−0.401466	+	0.915874i	$0.631499\pi$
$350$	0.732051	0.0391298
$351$	0	0
$352$	5.85641	0.312148
$353$	−15.4641	−0.823071	−0.411536	−	0.911394i	$-0.635007\pi$
$353$	−15.4641	−0.823071	−0.411536	+	0.911394i	$0.635007\pi$
$354$	0	0
$355$	−7.66025	−0.406564
$356$	21.4641	1.13760
$357$	0	0
$358$	14.6795	0.775835
$359$	34.5167	1.82172	0.910860	−	0.412716i	$-0.135420\pi$
$359$	34.5167	1.82172	0.910860	+	0.412716i	$0.135420\pi$
$360$	0	0
$361$	−16.6410	−0.875843
$362$	5.17691	0.272093
$363$	0	0
$364$	6.14359	0.322012
$365$	8.39230	0.439273
$366$	0	0
$367$	16.8038	0.877154	0.438577	−	0.898694i	$-0.355483\pi$
$367$	16.8038	0.877154	0.438577	+	0.898694i	$0.355483\pi$
$368$	−9.07180	−0.472900
$369$	0	0
$370$	−3.07180	−0.159695
$371$	−4.46410	−0.231765
$372$	0	0
$373$	29.1962	1.51172	0.755860	−	0.654734i	$-0.227219\pi$
$373$	29.1962	1.51172	0.755860	+	0.654734i	$0.227219\pi$
$374$	4.73205	0.244689
$375$	0	0
$376$	3.21539	0.165821
$377$	24.0526	1.23877
$378$	0	0
$379$	20.8564	1.07132	0.535661	−	0.844433i	$-0.320063\pi$
$379$	20.8564	1.07132	0.535661	+	0.844433i	$0.320063\pi$
$380$	2.24871	0.115356
$381$	0	0
$382$	0.784610	0.0401441
$383$	−32.7321	−1.67253	−0.836265	−	0.548326i	$-0.815265\pi$
$383$	−32.7321	−1.67253	−0.836265	+	0.548326i	$0.815265\pi$
$384$	0	0
$385$	1.00000	0.0509647
$386$	−5.46410	−0.278115
$387$	0	0
$388$	3.32051	0.168573
$389$	15.5167	0.786726	0.393363	−	0.919383i	$-0.371312\pi$
$389$	15.5167	0.786726	0.393363	+	0.919383i	$0.371312\pi$
$390$	0	0
$391$	−54.7128	−2.76695
$392$	−2.53590	−0.128082
$393$	0	0
$394$	−4.00000	−0.201517
$395$	3.66025	0.184167
$396$	0	0
$397$	37.8564	1.89996	0.949979	−	0.312313i	$-0.101104\pi$
$397$	37.8564	1.89996	0.949979	+	0.312313i	$0.101104\pi$
$398$	−10.9282	−0.547781
$399$	0	0
$400$	1.07180	0.0535898
$401$	33.1244	1.65415	0.827076	−	0.562091i	$-0.190003\pi$
$401$	33.1244	1.65415	0.827076	+	0.562091i	$0.190003\pi$
$402$	0	0
$403$	22.1051	1.10113
$404$	24.7846	1.23308
$405$	0	0
$406$	−4.19615	−0.208252
$407$	−4.19615	−0.207996
$408$	0	0
$409$	−9.07180	−0.448571	−0.224286	−	0.974523i	$-0.572005\pi$
$409$	−9.07180	−0.448571	−0.224286	+	0.974523i	$0.572005\pi$
$410$	5.60770	0.276944
$411$	0	0
$412$	20.8897	1.02916
$413$	−13.7321	−0.675710
$414$	0	0
$415$	−2.46410	−0.120958
$416$	−24.5744	−1.20486
$417$	0	0
$418$	−1.12436	−0.0549940
$419$	−29.7321	−1.45251	−0.726253	−	0.687428i	$-0.758740\pi$
$419$	−29.7321	−1.45251	−0.726253	+	0.687428i	$0.758740\pi$
$420$	0	0
$421$	24.7128	1.20443	0.602214	−	0.798334i	$-0.294285\pi$
$421$	24.7128	1.20443	0.602214	+	0.798334i	$0.294285\pi$
$422$	3.32051	0.161640
$423$	0	0
$424$	11.3205	0.549772
$425$	6.46410	0.313555
$426$	0	0
$427$	11.3923	0.551312
$428$	7.71281	0.372813
$429$	0	0
$430$	1.26795	0.0611459
$431$	24.3923	1.17494	0.587468	−	0.809247i	$-0.300125\pi$
$431$	24.3923	1.17494	0.587468	+	0.809247i	$0.300125\pi$
$432$	0	0
$433$	9.32051	0.447915	0.223958	−	0.974599i	$-0.428102\pi$
$433$	9.32051	0.447915	0.223958	+	0.974599i	$0.428102\pi$
$434$	−3.85641	−0.185113
$435$	0	0
$436$	7.71281	0.369377
$437$	13.0000	0.621874
$438$	0	0
$439$	−26.7128	−1.27493	−0.637467	−	0.770478i	$-0.720018\pi$
$439$	−26.7128	−1.27493	−0.637467	+	0.770478i	$0.720018\pi$
$440$	−2.53590	−0.120894
$441$	0	0
$442$	−19.8564	−0.944473
$443$	23.4641	1.11481	0.557407	−	0.830240i	$-0.311796\pi$
$443$	23.4641	1.11481	0.557407	+	0.830240i	$0.311796\pi$
$444$	0	0
$445$	−14.6603	−0.694963
$446$	0.588457	0.0278643
$447$	0	0
$448$	2.14359	0.101275
$449$	−16.0526	−0.757567	−0.378784	−	0.925485i	$-0.623658\pi$
$449$	−16.0526	−0.757567	−0.378784	+	0.925485i	$0.623658\pi$
$450$	0	0
$451$	7.66025	0.360707
$452$	15.8949	0.747632
$453$	0	0
$454$	3.94744	0.185263
$455$	−4.19615	−0.196719
$456$	0	0
$457$	−2.66025	−0.124441	−0.0622207	−	0.998062i	$-0.519818\pi$
$457$	−2.66025	−0.124441	−0.0622207	+	0.998062i	$0.519818\pi$
$458$	20.6410	0.964491
$459$	0	0
$460$	12.3923	0.577794
$461$	−29.3205	−1.36559	−0.682796	−	0.730609i	$-0.739236\pi$
$461$	−29.3205	−1.36559	−0.682796	+	0.730609i	$0.739236\pi$
$462$	0	0
$463$	12.5359	0.582593	0.291296	−	0.956633i	$-0.405913\pi$
$463$	12.5359	0.582593	0.291296	+	0.956633i	$0.405913\pi$
$464$	−6.14359	−0.285209
$465$	0	0
$466$	7.46410	0.345768
$467$	27.3205	1.26424	0.632121	−	0.774870i	$-0.282184\pi$
$467$	27.3205	1.26424	0.632121	+	0.774870i	$0.282184\pi$
$468$	0	0
$469$	4.92820	0.227563
$470$	−0.928203	−0.0428148
$471$	0	0
$472$	34.8231	1.60286
$473$	1.73205	0.0796398
$474$	0	0
$475$	−1.53590	−0.0704719
$476$	−9.46410	−0.433786
$477$	0	0
$478$	−13.2679	−0.606862
$479$	6.73205	0.307595	0.153798	−	0.988102i	$-0.450850\pi$
$479$	6.73205	0.307595	0.153798	+	0.988102i	$0.450850\pi$
$480$	0	0
$481$	17.6077	0.802842
$482$	19.7128	0.897894
$483$	0	0
$484$	−1.46410	−0.0665501
$485$	−2.26795	−0.102982
$486$	0	0
$487$	0.535898	0.0242839	0.0121419	−	0.999926i	$-0.496135\pi$
$487$	0.535898	0.0242839	0.0121419	+	0.999926i	$0.496135\pi$
$488$	−28.8897	−1.30778
$489$	0	0
$490$	0.732051	0.0330707
$491$	−3.33975	−0.150721	−0.0753603	−	0.997156i	$-0.524011\pi$
$491$	−3.33975	−0.150721	−0.0753603	+	0.997156i	$0.524011\pi$
$492$	0	0
$493$	−37.0526	−1.66876
$494$	4.71797	0.212271
$495$	0	0
$496$	−5.64617	−0.253521
$497$	−7.66025	−0.343609
$498$	0	0
$499$	29.3923	1.31578	0.657890	−	0.753114i	$-0.271449\pi$
$499$	29.3923	1.31578	0.657890	+	0.753114i	$0.271449\pi$
$500$	−1.46410	−0.0654766
$501$	0	0
$502$	5.07180	0.226365
$503$	23.0000	1.02552	0.512760	−	0.858532i	$-0.328623\pi$
$503$	23.0000	1.02552	0.512760	+	0.858532i	$0.328623\pi$
$504$	0	0
$505$	−16.9282	−0.753295
$506$	−6.19615	−0.275453
$507$	0	0
$508$	32.3923	1.43718
$509$	17.1962	0.762206	0.381103	−	0.924533i	$-0.375544\pi$
$509$	17.1962	0.762206	0.381103	+	0.924533i	$0.375544\pi$
$510$	0	0
$511$	8.39230	0.371254
$512$	11.7128	0.517638
$513$	0	0
$514$	−10.1051	−0.445718
$515$	−14.2679	−0.628721
$516$	0	0
$517$	−1.26795	−0.0557643
$518$	−3.07180	−0.134967
$519$	0	0
$520$	10.6410	0.466639
$521$	9.05256	0.396600	0.198300	−	0.980141i	$-0.436458\pi$
$521$	9.05256	0.396600	0.198300	+	0.980141i	$0.436458\pi$
$522$	0	0
$523$	−27.3205	−1.19464	−0.597321	−	0.802002i	$-0.703768\pi$
$523$	−27.3205	−1.19464	−0.597321	+	0.802002i	$0.703768\pi$
$524$	23.7128	1.03590
$525$	0	0
$526$	−8.28719	−0.361339
$527$	−34.0526	−1.48335
$528$	0	0
$529$	48.6410	2.11483
$530$	−3.26795	−0.141951
$531$	0	0
$532$	2.24871	0.0974940
$533$	−32.1436	−1.39229
$534$	0	0
$535$	−5.26795	−0.227753
$536$	−12.4974	−0.539806
$537$	0	0
$538$	−5.55514	−0.239499
$539$	1.00000	0.0430730
$540$	0	0
$541$	−34.0526	−1.46403	−0.732017	−	0.681286i	$-0.761421\pi$
$541$	−34.0526	−1.46403	−0.732017	+	0.681286i	$0.761421\pi$
$542$	−13.1244	−0.563739
$543$	0	0
$544$	37.8564	1.62308
$545$	−5.26795	−0.225654
$546$	0	0
$547$	23.5885	1.00857	0.504285	−	0.863537i	$-0.331756\pi$
$547$	23.5885	1.00857	0.504285	+	0.863537i	$0.331756\pi$
$548$	−9.56922	−0.408777
$549$	0	0
$550$	0.732051	0.0312148
$551$	8.80385	0.375057
$552$	0	0
$553$	3.66025	0.155650
$554$	−6.64102	−0.282150
$555$	0	0
$556$	−16.7846	−0.711826
$557$	−8.53590	−0.361678	−0.180839	−	0.983513i	$-0.557881\pi$
$557$	−8.53590	−0.361678	−0.180839	+	0.983513i	$0.557881\pi$
$558$	0	0
$559$	−7.26795	−0.307401
$560$	1.07180	0.0452917
$561$	0	0
$562$	−12.4974	−0.527172
$563$	24.9282	1.05060	0.525299	−	0.850918i	$-0.323953\pi$
$563$	24.9282	1.05060	0.525299	+	0.850918i	$0.323953\pi$
$564$	0	0
$565$	−10.8564	−0.456732
$566$	−0.143594	−0.00603569
$567$	0	0
$568$	19.4256	0.815081
$569$	4.66025	0.195368	0.0976840	−	0.995217i	$-0.468857\pi$
$569$	4.66025	0.195368	0.0976840	+	0.995217i	$0.468857\pi$
$570$	0	0
$571$	17.1244	0.716632	0.358316	−	0.933600i	$-0.383351\pi$
$571$	17.1244	0.716632	0.358316	+	0.933600i	$0.383351\pi$
$572$	6.14359	0.256877
$573$	0	0
$574$	5.60770	0.234061
$575$	−8.46410	−0.352977
$576$	0	0
$577$	−7.07180	−0.294403	−0.147201	−	0.989107i	$-0.547027\pi$
$577$	−7.07180	−0.294403	−0.147201	+	0.989107i	$0.547027\pi$
$578$	18.1436	0.754674
$579$	0	0
$580$	8.39230	0.348471
$581$	−2.46410	−0.102228
$582$	0	0
$583$	−4.46410	−0.184884
$584$	−21.2820	−0.880657
$585$	0	0
$586$	−6.58846	−0.272167
$587$	18.5885	0.767228	0.383614	−	0.923494i	$-0.374679\pi$
$587$	18.5885	0.767228	0.383614	+	0.923494i	$0.374679\pi$
$588$	0	0
$589$	8.09103	0.333385
$590$	−10.0526	−0.413857
$591$	0	0
$592$	−4.49742	−0.184843
$593$	−16.0000	−0.657041	−0.328521	−	0.944497i	$-0.606550\pi$
$593$	−16.0000	−0.657041	−0.328521	+	0.944497i	$0.606550\pi$
$594$	0	0
$595$	6.46410	0.265002
$596$	22.6410	0.927412
$597$	0	0
$598$	26.0000	1.06322
$599$	−28.2487	−1.15421	−0.577106	−	0.816670i	$-0.695818\pi$
$599$	−28.2487	−1.15421	−0.577106	+	0.816670i	$0.695818\pi$
$600$	0	0
$601$	17.7846	0.725449	0.362725	−	0.931896i	$-0.381847\pi$
$601$	17.7846	0.725449	0.362725	+	0.931896i	$0.381847\pi$
$602$	1.26795	0.0516778
$603$	0	0
$604$	17.3590	0.706327
$605$	1.00000	0.0406558
$606$	0	0
$607$	23.1769	0.940722	0.470361	−	0.882474i	$-0.344124\pi$
$607$	23.1769	0.940722	0.470361	+	0.882474i	$0.344124\pi$
$608$	−8.99485	−0.364789
$609$	0	0
$610$	8.33975	0.337666
$611$	5.32051	0.215245
$612$	0	0
$613$	34.7846	1.40494	0.702469	−	0.711715i	$-0.252081\pi$
$613$	34.7846	1.40494	0.702469	+	0.711715i	$0.252081\pi$
$614$	−7.60770	−0.307022
$615$	0	0
$616$	−2.53590	−0.102174
$617$	46.6410	1.87770	0.938848	−	0.344331i	$-0.111894\pi$
$617$	46.6410	1.87770	0.938848	+	0.344331i	$0.111894\pi$
$618$	0	0
$619$	−24.9282	−1.00195	−0.500975	−	0.865462i	$-0.667025\pi$
$619$	−24.9282	−1.00195	−0.500975	+	0.865462i	$0.667025\pi$
$620$	7.71281	0.309754
$621$	0	0
$622$	−5.35898	−0.214876
$623$	−14.6603	−0.587351
$624$	0	0
$625$	1.00000	0.0400000
$626$	1.26795	0.0506774
$627$	0	0
$628$	20.6795	0.825202
$629$	−27.1244	−1.08152
$630$	0	0
$631$	−16.7128	−0.665327	−0.332663	−	0.943046i	$-0.607947\pi$
$631$	−16.7128	−0.665327	−0.332663	+	0.943046i	$0.607947\pi$
$632$	−9.28203	−0.369219
$633$	0	0
$634$	−13.8564	−0.550308
$635$	−22.1244	−0.877978
$636$	0	0
$637$	−4.19615	−0.166258
$638$	−4.19615	−0.166127
$639$	0	0
$640$	−10.1436	−0.400961
$641$	28.6410	1.13125	0.565626	−	0.824662i	$-0.308635\pi$
$641$	28.6410	1.13125	0.565626	+	0.824662i	$0.308635\pi$
$642$	0	0
$643$	−6.80385	−0.268318	−0.134159	−	0.990960i	$-0.542833\pi$
$643$	−6.80385	−0.268318	−0.134159	+	0.990960i	$0.542833\pi$
$644$	12.3923	0.488325
$645$	0	0
$646$	−7.26795	−0.285954
$647$	−44.1051	−1.73395	−0.866976	−	0.498351i	$-0.833939\pi$
$647$	−44.1051	−1.73395	−0.866976	+	0.498351i	$0.833939\pi$
$648$	0	0
$649$	−13.7321	−0.539030
$650$	−3.07180	−0.120486
$651$	0	0
$652$	−28.0000	−1.09656
$653$	29.5359	1.15583	0.577915	−	0.816097i	$-0.303867\pi$
$653$	29.5359	1.15583	0.577915	+	0.816097i	$0.303867\pi$
$654$	0	0
$655$	−16.1962	−0.632836
$656$	8.21024	0.320556
$657$	0	0
$658$	−0.928203	−0.0361851
$659$	30.1244	1.17348	0.586739	−	0.809776i	$-0.300411\pi$
$659$	30.1244	1.17348	0.586739	+	0.809776i	$0.300411\pi$
$660$	0	0
$661$	−2.73205	−0.106264	−0.0531322	−	0.998587i	$-0.516920\pi$
$661$	−2.73205	−0.106264	−0.0531322	+	0.998587i	$0.516920\pi$
$662$	6.09103	0.236735
$663$	0	0
$664$	6.24871	0.242497
$665$	−1.53590	−0.0595596
$666$	0	0
$667$	48.5167	1.87857
$668$	−15.2154	−0.588701
$669$	0	0
$670$	3.60770	0.139377
$671$	11.3923	0.439795
$672$	0	0
$673$	−17.5885	−0.677985	−0.338993	−	0.940789i	$-0.610086\pi$
$673$	−17.5885	−0.677985	−0.338993	+	0.940789i	$0.610086\pi$
$674$	−7.80385	−0.300593
$675$	0	0
$676$	−6.74613	−0.259467
$677$	39.2487	1.50845	0.754225	−	0.656616i	$-0.228013\pi$
$677$	39.2487	1.50845	0.754225	+	0.656616i	$0.228013\pi$
$678$	0	0
$679$	−2.26795	−0.0870359
$680$	−16.3923	−0.628616
$681$	0	0
$682$	−3.85641	−0.147669
$683$	46.9282	1.79566	0.897829	−	0.440344i	$-0.145144\pi$
$683$	46.9282	1.79566	0.897829	+	0.440344i	$0.145144\pi$
$684$	0	0
$685$	6.53590	0.249724
$686$	0.732051	0.0279498
$687$	0	0
$688$	1.85641	0.0707748
$689$	18.7321	0.713634
$690$	0	0
$691$	−26.0000	−0.989087	−0.494543	−	0.869153i	$-0.664665\pi$
$691$	−26.0000	−0.989087	−0.494543	+	0.869153i	$0.664665\pi$
$692$	−37.2820	−1.41725
$693$	0	0
$694$	1.07180	0.0406848
$695$	11.4641	0.434858
$696$	0	0
$697$	49.5167	1.87558
$698$	−10.9808	−0.415628
$699$	0	0
$700$	−1.46410	−0.0553378
$701$	−33.1962	−1.25380	−0.626901	−	0.779099i	$-0.715677\pi$
$701$	−33.1962	−1.25380	−0.626901	+	0.779099i	$0.715677\pi$
$702$	0	0
$703$	6.44486	0.243073
$704$	2.14359	0.0807897
$705$	0	0
$706$	−11.3205	−0.426053
$707$	−16.9282	−0.636651
$708$	0	0
$709$	36.4641	1.36944	0.684719	−	0.728807i	$-0.259925\pi$
$709$	36.4641	1.36944	0.684719	+	0.728807i	$0.259925\pi$
$710$	−5.60770	−0.210453
$711$	0	0
$712$	37.1769	1.39326
$713$	44.5885	1.66985
$714$	0	0
$715$	−4.19615	−0.156927
$716$	−29.3590	−1.09720
$717$	0	0
$718$	25.2679	0.942991
$719$	−32.6603	−1.21802	−0.609011	−	0.793162i	$-0.708433\pi$
$719$	−32.6603	−1.21802	−0.609011	+	0.793162i	$0.708433\pi$
$720$	0	0
$721$	−14.2679	−0.531366
$722$	−12.1821	−0.453370
$723$	0	0
$724$	−10.3538	−0.384797
$725$	−5.73205	−0.212883
$726$	0	0
$727$	42.9090	1.59141	0.795703	−	0.605687i	$-0.207102\pi$
$727$	42.9090	1.59141	0.795703	+	0.605687i	$0.207102\pi$
$728$	10.6410	0.394382
$729$	0	0
$730$	6.14359	0.227385
$731$	11.1962	0.414105
$732$	0	0
$733$	51.1769	1.89026	0.945131	−	0.326691i	$-0.105934\pi$
$733$	51.1769	1.89026	0.945131	+	0.326691i	$0.105934\pi$
$734$	12.3013	0.454048
$735$	0	0
$736$	−49.5692	−1.82715
$737$	4.92820	0.181533
$738$	0	0
$739$	8.53590	0.313998	0.156999	−	0.987599i	$-0.449818\pi$
$739$	8.53590	0.313998	0.156999	+	0.987599i	$0.449818\pi$
$740$	6.14359	0.225843
$741$	0	0
$742$	−3.26795	−0.119970
$743$	−19.1769	−0.703533	−0.351766	−	0.936088i	$-0.614419\pi$
$743$	−19.1769	−0.703533	−0.351766	+	0.936088i	$0.614419\pi$
$744$	0	0
$745$	−15.4641	−0.566561
$746$	21.3731	0.782524
$747$	0	0
$748$	−9.46410	−0.346042
$749$	−5.26795	−0.192487
$750$	0	0
$751$	16.3205	0.595544	0.297772	−	0.954637i	$-0.403757\pi$
$751$	16.3205	0.595544	0.297772	+	0.954637i	$0.403757\pi$
$752$	−1.35898	−0.0495570
$753$	0	0
$754$	17.6077	0.641234
$755$	−11.8564	−0.431499
$756$	0	0
$757$	8.53590	0.310243	0.155121	−	0.987895i	$-0.450423\pi$
$757$	8.53590	0.310243	0.155121	+	0.987895i	$0.450423\pi$
$758$	15.2679	0.554557
$759$	0	0
$760$	3.89488	0.141282
$761$	11.0718	0.401352	0.200676	−	0.979658i	$-0.435686\pi$
$761$	11.0718	0.401352	0.200676	+	0.979658i	$0.435686\pi$
$762$	0	0
$763$	−5.26795	−0.190713
$764$	−1.56922	−0.0567724
$765$	0	0
$766$	−23.9615	−0.865765
$767$	57.6218	2.08060
$768$	0	0
$769$	−23.7846	−0.857695	−0.428847	−	0.903377i	$-0.641080\pi$
$769$	−23.7846	−0.857695	−0.428847	+	0.903377i	$0.641080\pi$
$770$	0.732051	0.0263813
$771$	0	0
$772$	10.9282	0.393315
$773$	−36.2487	−1.30378	−0.651888	−	0.758315i	$-0.726023\pi$
$773$	−36.2487	−1.30378	−0.651888	+	0.758315i	$0.726023\pi$
$774$	0	0
$775$	−5.26795	−0.189230
$776$	5.75129	0.206459
$777$	0	0
$778$	11.3590	0.407239
$779$	−11.7654	−0.421538
$780$	0	0
$781$	−7.66025	−0.274105
$782$	−40.0526	−1.43228
$783$	0	0
$784$	1.07180	0.0382785
$785$	−14.1244	−0.504120
$786$	0	0
$787$	−42.6410	−1.51999	−0.759994	−	0.649930i	$-0.774798\pi$
$787$	−42.6410	−1.51999	−0.759994	+	0.649930i	$0.774798\pi$
$788$	8.00000	0.284988
$789$	0	0
$790$	2.67949	0.0953320
$791$	−10.8564	−0.386009
$792$	0	0
$793$	−47.8038	−1.69756
$794$	27.7128	0.983491
$795$	0	0
$796$	21.8564	0.774680
$797$	11.1769	0.395907	0.197953	−	0.980211i	$-0.436571\pi$
$797$	11.1769	0.395907	0.197953	+	0.980211i	$0.436571\pi$
$798$	0	0
$799$	−8.19615	−0.289959
$800$	5.85641	0.207055
$801$	0	0
$802$	24.2487	0.856252
$803$	8.39230	0.296158
$804$	0	0
$805$	−8.46410	−0.298320
$806$	16.1821	0.569989
$807$	0	0
$808$	42.9282	1.51021
$809$	−8.53590	−0.300106	−0.150053	−	0.988678i	$-0.547944\pi$
$809$	−8.53590	−0.300106	−0.150053	+	0.988678i	$0.547944\pi$
$810$	0	0
$811$	4.78461	0.168010	0.0840052	−	0.996465i	$-0.473229\pi$
$811$	4.78461	0.168010	0.0840052	+	0.996465i	$0.473229\pi$
$812$	8.39230	0.294512
$813$	0	0
$814$	−3.07180	−0.107666
$815$	19.1244	0.669897
$816$	0	0
$817$	−2.66025	−0.0930705
$818$	−6.64102	−0.232198
$819$	0	0
$820$	−11.2154	−0.391658
$821$	9.98076	0.348331	0.174165	−	0.984716i	$-0.444277\pi$
$821$	9.98076	0.348331	0.174165	+	0.984716i	$0.444277\pi$
$822$	0	0
$823$	18.4449	0.642948	0.321474	−	0.946918i	$-0.395822\pi$
$823$	18.4449	0.642948	0.321474	+	0.946918i	$0.395822\pi$
$824$	36.1821	1.26046
$825$	0	0
$826$	−10.0526	−0.349773
$827$	−13.4641	−0.468193	−0.234096	−	0.972213i	$-0.575213\pi$
$827$	−13.4641	−0.468193	−0.234096	+	0.972213i	$0.575213\pi$
$828$	0	0
$829$	−35.1769	−1.22174	−0.610872	−	0.791729i	$-0.709181\pi$
$829$	−35.1769	−1.22174	−0.610872	+	0.791729i	$0.709181\pi$
$830$	−1.80385	−0.0626125
$831$	0	0
$832$	−8.99485	−0.311840
$833$	6.46410	0.223968
$834$	0	0
$835$	10.3923	0.359641
$836$	2.24871	0.0777733
$837$	0	0
$838$	−21.7654	−0.751872
$839$	−15.3397	−0.529587	−0.264793	−	0.964305i	$-0.585304\pi$
$839$	−15.3397	−0.529587	−0.264793	+	0.964305i	$0.585304\pi$
$840$	0	0
$841$	3.85641	0.132980
$842$	18.0910	0.623458
$843$	0	0
$844$	−6.64102	−0.228593
$845$	4.60770	0.158510
$846$	0	0
$847$	1.00000	0.0343604
$848$	−4.78461	−0.164304
$849$	0	0
$850$	4.73205	0.162308
$851$	35.5167	1.21750
$852$	0	0
$853$	10.8756	0.372375	0.186187	−	0.982514i	$-0.440387\pi$
$853$	10.8756	0.372375	0.186187	+	0.982514i	$0.440387\pi$
$854$	8.33975	0.285380
$855$	0	0
$856$	13.3590	0.456601
$857$	−49.7128	−1.69816	−0.849079	−	0.528266i	$-0.822842\pi$
$857$	−49.7128	−1.69816	−0.849079	+	0.528266i	$0.822842\pi$
$858$	0	0
$859$	−48.5885	−1.65782	−0.828908	−	0.559384i	$-0.811038\pi$
$859$	−48.5885	−1.65782	−0.828908	+	0.559384i	$0.811038\pi$
$860$	−2.53590	−0.0864734
$861$	0	0
$862$	17.8564	0.608192
$863$	25.5359	0.869252	0.434626	−	0.900611i	$-0.356881\pi$
$863$	25.5359	0.869252	0.434626	+	0.900611i	$0.356881\pi$
$864$	0	0
$865$	25.4641	0.865805
$866$	6.82309	0.231858
$867$	0	0
$868$	7.71281	0.261790
$869$	3.66025	0.124166
$870$	0	0
$871$	−20.6795	−0.700698
$872$	13.3590	0.452392
$873$	0	0
$874$	9.51666	0.321906
$875$	1.00000	0.0338062
$876$	0	0
$877$	18.3731	0.620414	0.310207	−	0.950669i	$-0.399602\pi$
$877$	18.3731	0.620414	0.310207	+	0.950669i	$0.399602\pi$
$878$	−19.5551	−0.659954
$879$	0	0
$880$	1.07180	0.0361303
$881$	−16.9090	−0.569678	−0.284839	−	0.958575i	$-0.591940\pi$
$881$	−16.9090	−0.569678	−0.284839	+	0.958575i	$0.591940\pi$
$882$	0	0
$883$	−37.1769	−1.25110	−0.625551	−	0.780183i	$-0.715126\pi$
$883$	−37.1769	−1.25110	−0.625551	+	0.780183i	$0.715126\pi$
$884$	39.7128	1.33569
$885$	0	0
$886$	17.1769	0.577070
$887$	21.3923	0.718283	0.359142	−	0.933283i	$-0.383069\pi$
$887$	21.3923	0.718283	0.359142	+	0.933283i	$0.383069\pi$
$888$	0	0
$889$	−22.1244	−0.742027
$890$	−10.7321	−0.359739
$891$	0	0
$892$	−1.17691	−0.0394060
$893$	1.94744	0.0651686
$894$	0	0
$895$	20.0526	0.670283
$896$	−10.1436	−0.338874
$897$	0	0
$898$	−11.7513	−0.392146
$899$	30.1962	1.00710
$900$	0	0
$901$	−28.8564	−0.961346
$902$	5.60770	0.186716
$903$	0	0
$904$	27.5307	0.915659
$905$	7.07180	0.235074
$906$	0	0
$907$	18.3397	0.608961	0.304481	−	0.952519i	$-0.401517\pi$
$907$	18.3397	0.608961	0.304481	+	0.952519i	$0.401517\pi$
$908$	−7.89488	−0.262001
$909$	0	0
$910$	−3.07180	−0.101829
$911$	−31.6077	−1.04721	−0.523605	−	0.851961i	$-0.675413\pi$
$911$	−31.6077	−1.04721	−0.523605	+	0.851961i	$0.675413\pi$
$912$	0	0
$913$	−2.46410	−0.0815499
$914$	−1.94744	−0.0644156
$915$	0	0
$916$	−41.2820	−1.36400
$917$	−16.1962	−0.534844
$918$	0	0
$919$	−46.4974	−1.53381	−0.766904	−	0.641762i	$-0.778204\pi$
$919$	−46.4974	−1.53381	−0.766904	+	0.641762i	$0.778204\pi$
$920$	21.4641	0.707650
$921$	0	0
$922$	−21.4641	−0.706883
$923$	32.1436	1.05802
$924$	0	0
$925$	−4.19615	−0.137969
$926$	9.17691	0.301572
$927$	0	0
$928$	−33.5692	−1.10196
$929$	15.7128	0.515521	0.257760	−	0.966209i	$-0.417016\pi$
$929$	15.7128	0.515521	0.257760	+	0.966209i	$0.417016\pi$
$930$	0	0
$931$	−1.53590	−0.0503370
$932$	−14.9282	−0.488990
$933$	0	0
$934$	20.0000	0.654420
$935$	6.46410	0.211399
$936$	0	0
$937$	26.9808	0.881423	0.440712	−	0.897649i	$-0.354726\pi$
$937$	26.9808	0.881423	0.440712	+	0.897649i	$0.354726\pi$
$938$	3.60770	0.117795
$939$	0	0
$940$	1.85641	0.0605493
$941$	−26.8372	−0.874867	−0.437433	−	0.899251i	$-0.644112\pi$
$941$	−26.8372	−0.874867	−0.437433	+	0.899251i	$0.644112\pi$
$942$	0	0
$943$	−64.8372	−2.11139
$944$	−14.7180	−0.479029
$945$	0	0
$946$	1.26795	0.0412246
$947$	12.7128	0.413111	0.206555	−	0.978435i	$-0.433775\pi$
$947$	12.7128	0.413111	0.206555	+	0.978435i	$0.433775\pi$
$948$	0	0
$949$	−35.2154	−1.14314
$950$	−1.12436	−0.0364789
$951$	0	0
$952$	−16.3923	−0.531278
$953$	−8.92820	−0.289213	−0.144606	−	0.989489i	$-0.546192\pi$
$953$	−8.92820	−0.289213	−0.144606	+	0.989489i	$0.546192\pi$
$954$	0	0
$955$	1.07180	0.0346825
$956$	26.5359	0.858232
$957$	0	0
$958$	4.92820	0.159223
$959$	6.53590	0.211055
$960$	0	0
$961$	−3.24871	−0.104797
$962$	12.8897	0.415581
$963$	0	0
$964$	−39.4256	−1.26981
$965$	−7.46410	−0.240278
$966$	0	0
$967$	30.5167	0.981350	0.490675	−	0.871343i	$-0.336750\pi$
$967$	30.5167	0.981350	0.490675	+	0.871343i	$0.336750\pi$
$968$	−2.53590	−0.0815069
$969$	0	0
$970$	−1.66025	−0.0533075
$971$	−42.6603	−1.36903	−0.684516	−	0.728998i	$-0.739987\pi$
$971$	−42.6603	−1.36903	−0.684516	+	0.728998i	$0.739987\pi$
$972$	0	0
$973$	11.4641	0.367522
$974$	0.392305	0.0125703
$975$	0	0
$976$	12.2102	0.390840
$977$	−11.3923	−0.364472	−0.182236	−	0.983255i	$-0.558334\pi$
$977$	−11.3923	−0.364472	−0.182236	+	0.983255i	$0.558334\pi$
$978$	0	0
$979$	−14.6603	−0.468544
$980$	−1.46410	−0.0467690
$981$	0	0
$982$	−2.44486	−0.0780187
$983$	−10.6795	−0.340623	−0.170311	−	0.985390i	$-0.554477\pi$
$983$	−10.6795	−0.340623	−0.170311	+	0.985390i	$0.554477\pi$
$984$	0	0
$985$	−5.46410	−0.174101
$986$	−27.1244	−0.863815
$987$	0	0
$988$	−9.43594	−0.300197
$989$	−14.6603	−0.466169
$990$	0	0
$991$	−30.5692	−0.971063	−0.485532	−	0.874219i	$-0.661374\pi$
$991$	−30.5692	−0.971063	−0.485532	+	0.874219i	$0.661374\pi$
$992$	−30.8513	−0.979528
$993$	0	0
$994$	−5.60770	−0.177865
$995$	−14.9282	−0.473256
$996$	0	0
$997$	−29.7128	−0.941014	−0.470507	−	0.882396i	$-0.655929\pi$
$997$	−29.7128	−0.941014	−0.470507	+	0.882396i	$0.655929\pi$
$998$	21.5167	0.681098
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3465.2.a.v.1.2		2
3.2	odd	2		1155.2.a.r.1.1	✓	2
15.14	odd	2		5775.2.a.bc.1.2		2
21.20	even	2		8085.2.a.bh.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1155.2.a.r.1.1	✓	2	3.2	odd	2
3465.2.a.v.1.2		2	1.1	even	1	trivial
5775.2.a.bc.1.2		2	15.14	odd	2
8085.2.a.bh.1.1		2	21.20	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 3465 = 3^{2} \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3465.a (trivial)

\(f(q)\)	\(=\)	\(q+0.732051 q^{2} -1.46410 q^{4} +1.00000 q^{5} +1.00000 q^{7} -2.53590 q^{8} +O(q^{10})\)	\(q+0.732051 q^{2} -1.46410 q^{4} +1.00000 q^{5} +1.00000 q^{7} -2.53590 q^{8} +0.732051 q^{10} +1.00000 q^{11} -4.19615 q^{13} +0.732051 q^{14} +1.07180 q^{16} +6.46410 q^{17} -1.53590 q^{19} -1.46410 q^{20} +0.732051 q^{22} -8.46410 q^{23} +1.00000 q^{25} -3.07180 q^{26} -1.46410 q^{28} -5.73205 q^{29} -5.26795 q^{31} +5.85641 q^{32} +4.73205 q^{34} +1.00000 q^{35} -4.19615 q^{37} -1.12436 q^{38} -2.53590 q^{40} +7.66025 q^{41} +1.73205 q^{43} -1.46410 q^{44} -6.19615 q^{46} -1.26795 q^{47} +1.00000 q^{49} +0.732051 q^{50} +6.14359 q^{52} -4.46410 q^{53} +1.00000 q^{55} -2.53590 q^{56} -4.19615 q^{58} -13.7321 q^{59} +11.3923 q^{61} -3.85641 q^{62} +2.14359 q^{64} -4.19615 q^{65} +4.92820 q^{67} -9.46410 q^{68} +0.732051 q^{70} -7.66025 q^{71} +8.39230 q^{73} -3.07180 q^{74} +2.24871 q^{76} +1.00000 q^{77} +3.66025 q^{79} +1.07180 q^{80} +5.60770 q^{82} -2.46410 q^{83} +6.46410 q^{85} +1.26795 q^{86} -2.53590 q^{88} -14.6603 q^{89} -4.19615 q^{91} +12.3923 q^{92} -0.928203 q^{94} -1.53590 q^{95} -2.26795 q^{97} +0.732051 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 4 q^{4} + 2 q^{5} + 2 q^{7} - 12 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 4 * q^4 + 2 * q^5 + 2 * q^7 - 12 * q^8	\( 2 q - 2 q^{2} + 4 q^{4} + 2 q^{5} + 2 q^{7} - 12 q^{8} - 2 q^{10} + 2 q^{11} + 2 q^{13} - 2 q^{14} + 16 q^{16} + 6 q^{17} - 10 q^{19} + 4 q^{20} - 2 q^{22} - 10 q^{23} + 2 q^{25} - 20 q^{26} + 4 q^{28} - 8 q^{29} - 14 q^{31} - 16 q^{32} + 6 q^{34} + 2 q^{35} + 2 q^{37} + 22 q^{38} - 12 q^{40} - 2 q^{41} + 4 q^{44} - 2 q^{46} - 6 q^{47} + 2 q^{49} - 2 q^{50} + 40 q^{52} - 2 q^{53} + 2 q^{55} - 12 q^{56} + 2 q^{58} - 24 q^{59} + 2 q^{61} + 20 q^{62} + 32 q^{64} + 2 q^{65} - 4 q^{67} - 12 q^{68} - 2 q^{70} + 2 q^{71} - 4 q^{73} - 20 q^{74} - 44 q^{76} + 2 q^{77} - 10 q^{79} + 16 q^{80} + 32 q^{82} + 2 q^{83} + 6 q^{85} + 6 q^{86} - 12 q^{88} - 12 q^{89} + 2 q^{91} + 4 q^{92} + 12 q^{94} - 10 q^{95} - 8 q^{97} - 2 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 4 * q^4 + 2 * q^5 + 2 * q^7 - 12 * q^8 - 2 * q^10 + 2 * q^11 + 2 * q^13 - 2 * q^14 + 16 * q^16 + 6 * q^17 - 10 * q^19 + 4 * q^20 - 2 * q^22 - 10 * q^23 + 2 * q^25 - 20 * q^26 + 4 * q^28 - 8 * q^29 - 14 * q^31 - 16 * q^32 + 6 * q^34 + 2 * q^35 + 2 * q^37 + 22 * q^38 - 12 * q^40 - 2 * q^41 + 4 * q^44 - 2 * q^46 - 6 * q^47 + 2 * q^49 - 2 * q^50 + 40 * q^52 - 2 * q^53 + 2 * q^55 - 12 * q^56 + 2 * q^58 - 24 * q^59 + 2 * q^61 + 20 * q^62 + 32 * q^64 + 2 * q^65 - 4 * q^67 - 12 * q^68 - 2 * q^70 + 2 * q^71 - 4 * q^73 - 20 * q^74 - 44 * q^76 + 2 * q^77 - 10 * q^79 + 16 * q^80 + 32 * q^82 + 2 * q^83 + 6 * q^85 + 6 * q^86 - 12 * q^88 - 12 * q^89 + 2 * q^91 + 4 * q^92 + 12 * q^94 - 10 * q^95 - 8 * q^97 - 2 * q^98

Introduction

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