LMFDB - Embedded newform 3675.2.a.bn.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3675,2,Mod(1,3675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3675, 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("3675.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3675 = 3 \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3675.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:	$29.3450227428$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.11344.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 4x + 3 $ x^4 - 2x^3 - 4x^2 + 4*x + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.552409$ of defining polynomial
Character	$\chi$	$=$	3675.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.55241 q^{2} -1.00000 q^{3} +0.409975 q^{4} +1.55241 q^{6} +2.46837 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.55241 q^{2} -1.00000 q^{3} +0.409975 q^{4} +1.55241 q^{6} +2.46837 q^{8} +1.00000 q^{9} +4.43075 q^{11} -0.409975 q^{12} -1.73246 q^{13} -4.65187 q^{16} +2.73246 q^{17} -1.55241 q^{18} +0.305156 q^{19} -6.87834 q^{22} -7.02423 q^{23} -2.46837 q^{24} +2.68949 q^{26} -1.00000 q^{27} -7.79430 q^{29} +5.28487 q^{31} +2.28487 q^{32} -4.43075 q^{33} -4.24190 q^{34} +0.409975 q^{36} -3.67406 q^{37} -0.473727 q^{38} +1.73246 q^{39} +6.71562 q^{41} -9.71562 q^{43} +1.81650 q^{44} +10.9045 q^{46} -1.81650 q^{47} +4.65187 q^{48} -2.73246 q^{51} -0.710265 q^{52} -1.71513 q^{53} +1.55241 q^{54} -0.305156 q^{57} +12.1000 q^{58} +1.14243 q^{59} -9.55635 q^{61} -8.20428 q^{62} +5.75669 q^{64} +6.87834 q^{66} -8.38433 q^{67} +1.12024 q^{68} +7.02423 q^{69} +10.2888 q^{71} +2.46837 q^{72} +12.8204 q^{73} +5.70365 q^{74} +0.125106 q^{76} -2.68949 q^{78} +6.71372 q^{79} +1.00000 q^{81} -10.4254 q^{82} -5.09946 q^{83} +15.0826 q^{86} +7.79430 q^{87} +10.9367 q^{88} -4.07065 q^{89} -2.87976 q^{92} -5.28487 q^{93} +2.81995 q^{94} -2.28487 q^{96} +2.87834 q^{97} +4.43075 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} - 4 q^{3} + 4 q^{4} + 2 q^{6} - 6 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q - 2 * q^2 - 4 * q^3 + 4 * q^4 + 2 * q^6 - 6 * q^8 + 4 * q^9	$ 4 q - 2 q^{2} - 4 q^{3} + 4 q^{4} + 2 q^{6} - 6 q^{8} + 4 q^{9} - 4 q^{12} + 2 q^{13} + 2 q^{17} - 2 q^{18} + 12 q^{19} - 14 q^{22} - 10 q^{23} + 6 q^{24} - 6 q^{26} - 4 q^{27} - 6 q^{29} + 8 q^{31} - 4 q^{32} + 4 q^{34} + 4 q^{36} - 24 q^{37} + 8 q^{38} - 2 q^{39} - 4 q^{41} - 8 q^{43} + 10 q^{44} + 16 q^{46} - 10 q^{47} - 2 q^{51} + 34 q^{52} - 20 q^{53} + 2 q^{54} - 12 q^{57} - 10 q^{58} - 2 q^{59} + 8 q^{61} - 10 q^{62} - 4 q^{64} + 14 q^{66} - 6 q^{67} - 30 q^{68} + 10 q^{69} - 14 q^{71} - 6 q^{72} + 12 q^{73} + 20 q^{74} + 16 q^{76} + 6 q^{78} - 8 q^{79} + 4 q^{81} - 18 q^{82} - 6 q^{83} + 24 q^{86} + 6 q^{87} + 12 q^{88} - 8 q^{89} - 46 q^{92} - 8 q^{93} + 16 q^{94} + 4 q^{96} - 2 q^{97}+O(q^{100}) $ 4 * q - 2 * q^2 - 4 * q^3 + 4 * q^4 + 2 * q^6 - 6 * q^8 + 4 * q^9 - 4 * q^12 + 2 * q^13 + 2 * q^17 - 2 * q^18 + 12 * q^19 - 14 * q^22 - 10 * q^23 + 6 * q^24 - 6 * q^26 - 4 * q^27 - 6 * q^29 + 8 * q^31 - 4 * q^32 + 4 * q^34 + 4 * q^36 - 24 * q^37 + 8 * q^38 - 2 * q^39 - 4 * q^41 - 8 * q^43 + 10 * q^44 + 16 * q^46 - 10 * q^47 - 2 * q^51 + 34 * q^52 - 20 * q^53 + 2 * q^54 - 12 * q^57 - 10 * q^58 - 2 * q^59 + 8 * q^61 - 10 * q^62 - 4 * q^64 + 14 * q^66 - 6 * q^67 - 30 * q^68 + 10 * q^69 - 14 * q^71 - 6 * q^72 + 12 * q^73 + 20 * q^74 + 16 * q^76 + 6 * q^78 - 8 * q^79 + 4 * q^81 - 18 * q^82 - 6 * q^83 + 24 * q^86 + 6 * q^87 + 12 * q^88 - 8 * q^89 - 46 * q^92 - 8 * q^93 + 16 * q^94 + 4 * q^96 - 2 * q^97

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 $

$2$	−1.55241	−1.09772	−0.548860	−	0.835915i	$-0.684938\pi$
$2$	−1.55241	−1.09772	−0.548860	+	0.835915i	$0.684938\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0.409975	0.204988
$5$	0	0
$5$	0	0
$6$	1.55241	0.633769
$7$	0	0
$7$	0	0
$8$	2.46837	0.872700
$9$	1.00000	0.333333
$10$	0	0
$11$	4.43075	1.33592	0.667961	−	0.744196i	$-0.267167\pi$
$11$	4.43075	1.33592	0.667961	+	0.744196i	$0.267167\pi$
$12$	−0.409975	−0.118350
$13$	−1.73246	−0.480498	−0.240249	−	0.970711i	$-0.577229\pi$
$13$	−1.73246	−0.480498	−0.240249	+	0.970711i	$0.577229\pi$
$14$	0	0
$15$	0	0
$16$	−4.65187	−1.16297
$17$	2.73246	0.662719	0.331359	−	0.943505i	$-0.392493\pi$
$17$	2.73246	0.662719	0.331359	+	0.943505i	$0.392493\pi$
$18$	−1.55241	−0.365906
$19$	0.305156	0.0700076	0.0350038	−	0.999387i	$-0.488856\pi$
$19$	0.305156	0.0700076	0.0350038	+	0.999387i	$0.488856\pi$
$20$	0	0
$21$	0	0
$22$	−6.87834	−1.46647
$23$	−7.02423	−1.46465	−0.732327	−	0.680954i	$-0.761566\pi$
$23$	−7.02423	−1.46465	−0.732327	+	0.680954i	$0.761566\pi$
$24$	−2.46837	−0.503854
$25$	0	0
$26$	2.68949	0.527452
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−7.79430	−1.44737	−0.723683	−	0.690132i	$-0.757552\pi$
$29$	−7.79430	−1.44737	−0.723683	+	0.690132i	$0.757552\pi$
$30$	0	0
$31$	5.28487	0.949190	0.474595	−	0.880204i	$-0.342594\pi$
$31$	5.28487	0.949190	0.474595	+	0.880204i	$0.342594\pi$
$32$	2.28487	0.403912
$33$	−4.43075	−0.771295
$34$	−4.24190	−0.727479
$35$	0	0
$36$	0.409975	0.0683292
$37$	−3.67406	−0.604013	−0.302006	−	0.953306i	$-0.597656\pi$
$37$	−3.67406	−0.604013	−0.302006	+	0.953306i	$0.597656\pi$
$38$	−0.473727	−0.0768487
$39$	1.73246	0.277415
$40$	0	0
$41$	6.71562	1.04880	0.524402	−	0.851471i	$-0.324289\pi$
$41$	6.71562	1.04880	0.524402	+	0.851471i	$0.324289\pi$
$42$	0	0
$43$	−9.71562	−1.48162	−0.740809	−	0.671715i	$-0.765558\pi$
$43$	−9.71562	−1.48162	−0.740809	+	0.671715i	$0.765558\pi$
$44$	1.81650	0.273848
$45$	0	0
$46$	10.9045	1.60778
$47$	−1.81650	−0.264964	−0.132482	−	0.991185i	$-0.542295\pi$
$47$	−1.81650	−0.264964	−0.132482	+	0.991185i	$0.542295\pi$
$48$	4.65187	0.671440
$49$	0	0
$50$	0	0
$51$	−2.73246	−0.382621
$52$	−0.710265	−0.0984961
$53$	−1.71513	−0.235591	−0.117796	−	0.993038i	$-0.537583\pi$
$53$	−1.71513	−0.235591	−0.117796	+	0.993038i	$0.537583\pi$
$54$	1.55241	0.211256
$55$	0	0
$56$	0	0
$57$	−0.305156	−0.0404189
$58$	12.1000	1.58880
$59$	1.14243	0.148732	0.0743661	−	0.997231i	$-0.476307\pi$
$59$	1.14243	0.148732	0.0743661	+	0.997231i	$0.476307\pi$
$60$	0	0
$61$	−9.55635	−1.22357	−0.611783	−	0.791026i	$-0.709547\pi$
$61$	−9.55635	−1.22357	−0.611783	+	0.791026i	$0.709547\pi$
$62$	−8.20428	−1.04194
$63$	0	0
$64$	5.75669	0.719586
$65$	0	0
$66$	6.87834	0.846666
$67$	−8.38433	−1.02431	−0.512154	−	0.858893i	$-0.671152\pi$
$67$	−8.38433	−1.02431	−0.512154	+	0.858893i	$0.671152\pi$
$68$	1.12024	0.135849
$69$	7.02423	0.845618
$70$	0	0
$71$	10.2888	1.22106	0.610529	−	0.791994i	$-0.290957\pi$
$71$	10.2888	1.22106	0.610529	+	0.791994i	$0.290957\pi$
$72$	2.46837	0.290900
$73$	12.8204	1.50052	0.750260	−	0.661143i	$-0.229928\pi$
$73$	12.8204	1.50052	0.750260	+	0.661143i	$0.229928\pi$
$74$	5.70365	0.663036
$75$	0	0
$76$	0.125106	0.0143507
$77$	0	0
$78$	−2.68949	−0.304524
$79$	6.71372	0.755352	0.377676	−	0.925938i	$-0.376723\pi$
$79$	6.71372	0.755352	0.377676	+	0.925938i	$0.376723\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−10.4254	−1.15129
$83$	−5.09946	−0.559739	−0.279869	−	0.960038i	$-0.590291\pi$
$83$	−5.09946	−0.559739	−0.279869	+	0.960038i	$0.590291\pi$
$84$	0	0
$85$	0	0
$86$	15.0826	1.62640
$87$	7.79430	0.835637
$88$	10.9367	1.16586
$89$	−4.07065	−0.431489	−0.215744	−	0.976450i	$-0.569218\pi$
$89$	−4.07065	−0.431489	−0.215744	+	0.976450i	$0.569218\pi$
$90$	0	0
$91$	0	0
$92$	−2.87976	−0.300236
$93$	−5.28487	−0.548015
$94$	2.81995	0.290856
$95$	0	0
$96$	−2.28487	−0.233198
$97$	2.87834	0.292252	0.146126	−	0.989266i	$-0.453320\pi$
$97$	2.87834	0.292252	0.146126	+	0.989266i	$0.453320\pi$
$98$	0	0
$99$	4.43075	0.445308
$100$	0	0
$101$	−7.21767	−0.718185	−0.359092	−	0.933302i	$-0.616914\pi$
$101$	−7.21767	−0.718185	−0.359092	+	0.933302i	$0.616914\pi$
$102$	4.24190	0.420010
$103$	11.2734	1.11080	0.555400	−	0.831583i	$-0.312565\pi$
$103$	11.2734	1.11080	0.555400	+	0.831583i	$0.312565\pi$
$104$	−4.27635	−0.419331
$105$	0	0
$106$	2.66259	0.258613
$107$	−6.91596	−0.668591	−0.334296	−	0.942468i	$-0.608498\pi$
$107$	−6.91596	−0.668591	−0.334296	+	0.942468i	$0.608498\pi$
$108$	−0.409975	−0.0394499
$109$	4.42195	0.423546	0.211773	−	0.977319i	$-0.432076\pi$
$109$	4.42195	0.423546	0.211773	+	0.977319i	$0.432076\pi$
$110$	0	0
$111$	3.67406	0.348727
$112$	0	0
$113$	2.86151	0.269188	0.134594	−	0.990901i	$-0.457027\pi$
$113$	2.86151	0.269188	0.134594	+	0.990901i	$0.457027\pi$
$114$	0.473727	0.0443686
$115$	0	0
$116$	−3.19547	−0.296692
$117$	−1.73246	−0.160166
$118$	−1.77353	−0.163266
$119$	0	0
$120$	0	0
$121$	8.63158	0.784689
$122$	14.8354	1.34313
$123$	−6.71562	−0.605527
$124$	2.16666	0.194572
$125$	0	0
$126$	0	0
$127$	−13.7325	−1.21856	−0.609279	−	0.792956i	$-0.708541\pi$
$127$	−13.7325	−1.21856	−0.609279	+	0.792956i	$0.708541\pi$
$128$	−13.5065	−1.19382
$129$	9.71562	0.855413
$130$	0	0
$131$	−21.9817	−1.92055	−0.960277	−	0.279048i	$-0.909981\pi$
$131$	−21.9817	−1.92055	−0.960277	+	0.279048i	$0.909981\pi$
$132$	−1.81650	−0.158106
$133$	0	0
$134$	13.0159	1.12440
$135$	0	0
$136$	6.74472	0.578355
$137$	−3.00191	−0.256470	−0.128235	−	0.991744i	$-0.540931\pi$
$137$	−3.00191	−0.256470	−0.128235	+	0.991744i	$0.540931\pi$
$138$	−10.9045	−0.928251
$139$	21.7364	1.84366	0.921829	−	0.387597i	$-0.126695\pi$
$139$	21.7364	1.84366	0.921829	+	0.387597i	$0.126695\pi$
$140$	0	0
$141$	1.81650	0.152977
$142$	−15.9724	−1.34038
$143$	−7.67610	−0.641908
$144$	−4.65187	−0.387656
$145$	0	0
$146$	−19.9026	−1.64715
$147$	0	0
$148$	−1.50628	−0.123815
$149$	−5.21309	−0.427073	−0.213536	−	0.976935i	$-0.568498\pi$
$149$	−5.21309	−0.427073	−0.213536	+	0.976935i	$0.568498\pi$
$150$	0	0
$151$	−9.71372	−0.790491	−0.395246	−	0.918576i	$-0.629340\pi$
$151$	−9.71372	−0.790491	−0.395246	+	0.918576i	$0.629340\pi$
$152$	0.753238	0.0610957
$153$	2.73246	0.220906
$154$	0	0
$155$	0	0
$156$	0.710265	0.0568667
$157$	9.47182	0.755934	0.377967	−	0.925819i	$-0.376623\pi$
$157$	9.47182	0.755934	0.377967	+	0.925819i	$0.376623\pi$
$158$	−10.4224	−0.829165
$159$	1.71513	0.136019
$160$	0	0
$161$	0	0
$162$	−1.55241	−0.121969
$163$	5.39659	0.422694	0.211347	−	0.977411i	$-0.432215\pi$
$163$	5.39659	0.422694	0.211347	+	0.977411i	$0.432215\pi$
$164$	2.75324	0.214992
$165$	0	0
$166$	7.91645	0.614436
$167$	0.312550	0.0241859	0.0120929	−	0.999927i	$-0.496151\pi$
$167$	0.312550	0.0241859	0.0120929	+	0.999927i	$0.496151\pi$
$168$	0	0
$169$	−9.99859	−0.769122
$170$	0	0
$171$	0.305156	0.0233359
$172$	−3.98316	−0.303713
$173$	−9.61475	−0.730996	−0.365498	−	0.930812i	$-0.619101\pi$
$173$	−9.61475	−0.730996	−0.365498	+	0.930812i	$0.619101\pi$
$174$	−12.1000	−0.917295
$175$	0	0
$176$	−20.6113	−1.55363
$177$	−1.14243	−0.0858706
$178$	6.31932	0.473653
$179$	−8.85461	−0.661824	−0.330912	−	0.943662i	$-0.607356\pi$
$179$	−8.85461	−0.661824	−0.330912	+	0.943662i	$0.607356\pi$
$180$	0	0
$181$	16.9234	1.25790	0.628952	−	0.777445i	$-0.283484\pi$
$181$	16.9234	1.25790	0.628952	+	0.777445i	$0.283484\pi$
$182$	0	0
$183$	9.55635	0.706426
$184$	−17.3384	−1.27820
$185$	0	0
$186$	8.20428	0.601567
$187$	12.1069	0.885341
$188$	−0.744719	−0.0543142
$189$	0	0
$190$	0	0
$191$	0.318541	0.0230488	0.0115244	−	0.999934i	$-0.496332\pi$
$191$	0.318541	0.0230488	0.0115244	+	0.999934i	$0.496332\pi$
$192$	−5.75669	−0.415453
$193$	2.09334	0.150682	0.0753410	−	0.997158i	$-0.475995\pi$
$193$	2.09334	0.150682	0.0753410	+	0.997158i	$0.475995\pi$
$194$	−4.46837	−0.320810
$195$	0	0
$196$	0	0
$197$	−22.5798	−1.60874	−0.804372	−	0.594126i	$-0.797498\pi$
$197$	−22.5798	−1.60874	−0.804372	+	0.594126i	$0.797498\pi$
$198$	−6.87834	−0.488823
$199$	7.50408	0.531950	0.265975	−	0.963980i	$-0.414306\pi$
$199$	7.50408	0.531950	0.265975	+	0.963980i	$0.414306\pi$
$200$	0	0
$201$	8.38433	0.591385
$202$	11.2048	0.788365
$203$	0	0
$204$	−1.12024	−0.0784325
$205$	0	0
$206$	−17.5009	−1.21935
$207$	−7.02423	−0.488218
$208$	8.05918	0.558803
$209$	1.35207	0.0935248
$210$	0	0
$211$	7.67216	0.528173	0.264087	−	0.964499i	$-0.414930\pi$
$211$	7.67216	0.528173	0.264087	+	0.964499i	$0.414930\pi$
$212$	−0.703161	−0.0482933
$213$	−10.2888	−0.704978
$214$	10.7364	0.733925
$215$	0	0
$216$	−2.46837	−0.167951
$217$	0	0
$218$	−6.86467	−0.464934
$219$	−12.8204	−0.866325
$220$	0	0
$221$	−4.73387	−0.318435
$222$	−5.70365	−0.382804
$223$	−6.90208	−0.462198	−0.231099	−	0.972930i	$-0.574232\pi$
$223$	−6.90208	−0.462198	−0.231099	+	0.972930i	$0.574232\pi$
$224$	0	0
$225$	0	0
$226$	−4.44223	−0.295493
$227$	−6.34721	−0.421279	−0.210639	−	0.977564i	$-0.567555\pi$
$227$	−6.34721	−0.421279	−0.210639	+	0.977564i	$0.567555\pi$
$228$	−0.125106	−0.00828538
$229$	−6.75669	−0.446495	−0.223247	−	0.974762i	$-0.571666\pi$
$229$	−6.75669	−0.446495	−0.223247	+	0.974762i	$0.571666\pi$
$230$	0	0
$231$	0	0
$232$	−19.2392	−1.26312
$233$	7.41201	0.485577	0.242789	−	0.970079i	$-0.421938\pi$
$233$	7.41201	0.485577	0.242789	+	0.970079i	$0.421938\pi$
$234$	2.68949	0.175817
$235$	0	0
$236$	0.468370	0.0304883
$237$	−6.71372	−0.436103
$238$	0	0
$239$	7.36355	0.476309	0.238154	−	0.971227i	$-0.423458\pi$
$239$	7.36355	0.476309	0.238154	+	0.971227i	$0.423458\pi$
$240$	0	0
$241$	−13.6800	−0.881209	−0.440605	−	0.897701i	$-0.645236\pi$
$241$	−13.6800	−0.881209	−0.440605	+	0.897701i	$0.645236\pi$
$242$	−13.3998	−0.861369
$243$	−1.00000	−0.0641500
$244$	−3.91787	−0.250816
$245$	0	0
$246$	10.4254	0.664699
$247$	−0.528671	−0.0336385
$248$	13.0450	0.828359
$249$	5.09946	0.323165
$250$	0	0
$251$	−23.4843	−1.48231	−0.741157	−	0.671331i	$-0.765723\pi$
$251$	−23.4843	−1.48231	−0.741157	+	0.671331i	$0.765723\pi$
$252$	0	0
$253$	−31.1226	−1.95666
$254$	21.3184	1.33764
$255$	0	0
$256$	9.45420	0.590888
$257$	−14.5974	−0.910562	−0.455281	−	0.890348i	$-0.650461\pi$
$257$	−14.5974	−0.910562	−0.455281	+	0.890348i	$0.650461\pi$
$258$	−15.0826	−0.939003
$259$	0	0
$260$	0	0
$261$	−7.79430	−0.482455
$262$	34.1247	2.10823
$263$	−5.08940	−0.313826	−0.156913	−	0.987612i	$-0.550154\pi$
$263$	−5.08940	−0.313826	−0.156913	+	0.987612i	$0.550154\pi$
$264$	−10.9367	−0.673110
$265$	0	0
$266$	0	0
$267$	4.07065	0.249120
$268$	−3.43737	−0.209971
$269$	−30.5099	−1.86022	−0.930112	−	0.367277i	$-0.880290\pi$
$269$	−30.5099	−1.86022	−0.930112	+	0.367277i	$0.880290\pi$
$270$	0	0
$271$	19.4880	1.18381	0.591907	−	0.806007i	$-0.298375\pi$
$271$	19.4880	1.18381	0.591907	+	0.806007i	$0.298375\pi$
$272$	−12.7110	−0.770720
$273$	0	0
$274$	4.66019	0.281532
$275$	0	0
$276$	2.87976	0.173341
$277$	25.4802	1.53096	0.765478	−	0.643462i	$-0.222502\pi$
$277$	25.4802	1.53096	0.765478	+	0.643462i	$0.222502\pi$
$278$	−33.7438	−2.02382
$279$	5.28487	0.316397
$280$	0	0
$281$	−22.1914	−1.32383	−0.661914	−	0.749580i	$-0.730255\pi$
$281$	−22.1914	−1.32383	−0.661914	+	0.749580i	$0.730255\pi$
$282$	−2.81995	−0.167926
$283$	18.1242	1.07737	0.538685	−	0.842507i	$-0.318921\pi$
$283$	18.1242	1.07737	0.538685	+	0.842507i	$0.318921\pi$
$284$	4.21816	0.250302
$285$	0	0
$286$	11.9165	0.704635
$287$	0	0
$288$	2.28487	0.134637
$289$	−9.53367	−0.560804
$290$	0	0
$291$	−2.87834	−0.168732
$292$	5.25606	0.307588
$293$	13.8958	0.811801	0.405901	−	0.913917i	$-0.366958\pi$
$293$	13.8958	0.811801	0.405901	+	0.913917i	$0.366958\pi$
$294$	0	0
$295$	0	0
$296$	−9.06895	−0.527122
$297$	−4.43075	−0.257098
$298$	8.09285	0.468806
$299$	12.1692	0.703763
$300$	0	0
$301$	0	0
$302$	15.0797	0.867737
$303$	7.21767	0.414644
$304$	−1.41955	−0.0814166
$305$	0	0
$306$	−4.24190	−0.242493
$307$	−29.8332	−1.70267	−0.851335	−	0.524622i	$-0.824207\pi$
$307$	−29.8332	−1.70267	−0.851335	+	0.524622i	$0.824207\pi$
$308$	0	0
$309$	−11.2734	−0.641321
$310$	0	0
$311$	−13.5132	−0.766266	−0.383133	−	0.923693i	$-0.625155\pi$
$311$	−13.5132	−0.766266	−0.383133	+	0.923693i	$0.625155\pi$
$312$	4.27635	0.242101
$313$	28.9857	1.63837	0.819184	−	0.573531i	$-0.194427\pi$
$313$	28.9857	1.63837	0.819184	+	0.573531i	$0.194427\pi$
$314$	−14.7041	−0.829803
$315$	0	0
$316$	2.75246	0.154838
$317$	1.15864	0.0650756	0.0325378	−	0.999471i	$-0.489641\pi$
$317$	1.15864	0.0650756	0.0325378	+	0.999471i	$0.489641\pi$
$318$	−2.66259	−0.149310
$319$	−34.5346	−1.93357
$320$	0	0
$321$	6.91596	0.386011
$322$	0	0
$323$	0.833827	0.0463954
$324$	0.409975	0.0227764
$325$	0	0
$326$	−8.37772	−0.463999
$327$	−4.42195	−0.244534
$328$	16.5766	0.915292
$329$	0	0
$330$	0	0
$331$	−28.3491	−1.55821	−0.779104	−	0.626895i	$-0.784326\pi$
$331$	−28.3491	−1.55821	−0.779104	+	0.626895i	$0.784326\pi$
$332$	−2.09065	−0.114739
$333$	−3.67406	−0.201338
$334$	−0.485206	−0.0265493
$335$	0	0
$336$	0	0
$337$	15.9729	0.870101	0.435051	−	0.900406i	$-0.356730\pi$
$337$	15.9729	0.870101	0.435051	+	0.900406i	$0.356730\pi$
$338$	15.5219	0.844280
$339$	−2.86151	−0.155416
$340$	0	0
$341$	23.4160	1.26805
$342$	−0.473727	−0.0256162
$343$	0	0
$344$	−23.9817	−1.29301
$345$	0	0
$346$	14.9260	0.802428
$347$	13.0120	0.698519	0.349260	−	0.937026i	$-0.386433\pi$
$347$	13.0120	0.698519	0.349260	+	0.937026i	$0.386433\pi$
$348$	3.19547	0.171295
$349$	−32.0724	−1.71680	−0.858398	−	0.512984i	$-0.828540\pi$
$349$	−32.0724	−1.71680	−0.858398	+	0.512984i	$0.828540\pi$
$350$	0	0
$351$	1.73246	0.0924718
$352$	10.1237	0.539595
$353$	−13.3689	−0.711555	−0.355778	−	0.934571i	$-0.615784\pi$
$353$	−13.3689	−0.711555	−0.355778	+	0.934571i	$0.615784\pi$
$354$	1.77353	0.0942618
$355$	0	0
$356$	−1.66887	−0.0884498
$357$	0	0
$358$	13.7460	0.726497
$359$	−13.1195	−0.692420	−0.346210	−	0.938157i	$-0.612531\pi$
$359$	−13.1195	−0.692420	−0.346210	+	0.938157i	$0.612531\pi$
$360$	0	0
$361$	−18.9069	−0.995099
$362$	−26.2720	−1.38082
$363$	−8.63158	−0.453041
$364$	0	0
$365$	0	0
$366$	−14.8354	−0.775457
$367$	−13.1360	−0.685691	−0.342846	−	0.939392i	$-0.611391\pi$
$367$	−13.1360	−0.685691	−0.342846	+	0.939392i	$0.611391\pi$
$368$	32.6758	1.70334
$369$	6.71562	0.349601
$370$	0	0
$371$	0	0
$372$	−2.16666	−0.112336
$373$	−29.6197	−1.53365	−0.766826	−	0.641855i	$-0.778165\pi$
$373$	−29.6197	−1.53365	−0.766826	+	0.641855i	$0.778165\pi$
$374$	−18.7948	−0.971856
$375$	0	0
$376$	−4.48379	−0.231234
$377$	13.5033	0.695456
$378$	0	0
$379$	7.47689	0.384062	0.192031	−	0.981389i	$-0.438493\pi$
$379$	7.47689	0.384062	0.192031	+	0.981389i	$0.438493\pi$
$380$	0	0
$381$	13.7325	0.703535
$382$	−0.494506	−0.0253012
$383$	23.7391	1.21301	0.606505	−	0.795080i	$-0.292571\pi$
$383$	23.7391	1.21301	0.606505	+	0.795080i	$0.292571\pi$
$384$	13.5065	0.689250
$385$	0	0
$386$	−3.24972	−0.165406
$387$	−9.71562	−0.493873
$388$	1.18005	0.0599079
$389$	−14.1305	−0.716443	−0.358221	−	0.933637i	$-0.616617\pi$
$389$	−14.1305	−0.716443	−0.358221	+	0.933637i	$0.616617\pi$
$390$	0	0
$391$	−19.1934	−0.970653
$392$	0	0
$393$	21.9817	1.10883
$394$	35.0531	1.76595
$395$	0	0
$396$	1.81650	0.0912825
$397$	−11.2696	−0.565604	−0.282802	−	0.959178i	$-0.591264\pi$
$397$	−11.2696	−0.565604	−0.282802	+	0.959178i	$0.591264\pi$
$398$	−11.6494	−0.583932
$399$	0	0
$400$	0	0
$401$	23.2968	1.16339	0.581694	−	0.813407i	$-0.302390\pi$
$401$	23.2968	1.16339	0.581694	+	0.813407i	$0.302390\pi$
$402$	−13.0159	−0.649175
$403$	−9.15582	−0.456084
$404$	−2.95906	−0.147219
$405$	0	0
$406$	0	0
$407$	−16.2789	−0.806914
$408$	−6.74472	−0.333913
$409$	−20.6478	−1.02097	−0.510484	−	0.859887i	$-0.670534\pi$
$409$	−20.6478	−1.02097	−0.510484	+	0.859887i	$0.670534\pi$
$410$	0	0
$411$	3.00191	0.148073
$412$	4.62181	0.227700
$413$	0	0
$414$	10.9045	0.535926
$415$	0	0
$416$	−3.95844	−0.194079
$417$	−21.7364	−1.06444
$418$	−2.09897	−0.102664
$419$	5.48254	0.267839	0.133920	−	0.990992i	$-0.457244\pi$
$419$	5.48254	0.267839	0.133920	+	0.990992i	$0.457244\pi$
$420$	0	0
$421$	−3.78079	−0.184264	−0.0921322	−	0.995747i	$-0.529368\pi$
$421$	−3.78079	−0.184264	−0.0921322	+	0.995747i	$0.529368\pi$
$422$	−11.9103	−0.579786
$423$	−1.81650	−0.0883212
$424$	−4.23358	−0.205601
$425$	0	0
$426$	15.9724	0.773868
$427$	0	0
$428$	−2.83537	−0.137053
$429$	7.67610	0.370606
$430$	0	0
$431$	−27.5127	−1.32524	−0.662621	−	0.748955i	$-0.730556\pi$
$431$	−27.5127	−1.32524	−0.662621	+	0.748955i	$0.730556\pi$
$432$	4.65187	0.223813
$433$	−8.75514	−0.420745	−0.210373	−	0.977621i	$-0.567468\pi$
$433$	−8.75514	−0.420745	−0.210373	+	0.977621i	$0.567468\pi$
$434$	0	0
$435$	0	0
$436$	1.81289	0.0868216
$437$	−2.14349	−0.102537
$438$	19.9026	0.950982
$439$	10.6127	0.506517	0.253259	−	0.967399i	$-0.418498\pi$
$439$	10.6127	0.506517	0.253259	+	0.967399i	$0.418498\pi$
$440$	0	0
$441$	0	0
$442$	7.34891	0.349552
$443$	19.9856	0.949543	0.474771	−	0.880109i	$-0.342531\pi$
$443$	19.9856	0.949543	0.474771	+	0.880109i	$0.342531\pi$
$444$	1.50628	0.0714847
$445$	0	0
$446$	10.7149	0.507363
$447$	5.21309	0.246571
$448$	0	0
$449$	3.02578	0.142795	0.0713976	−	0.997448i	$-0.477254\pi$
$449$	3.02578	0.142795	0.0713976	+	0.997448i	$0.477254\pi$
$450$	0	0
$451$	29.7553	1.40112
$452$	1.17315	0.0551802
$453$	9.71372	0.456390
$454$	9.85346	0.462446
$455$	0	0
$456$	−0.753238	−0.0352736
$457$	−2.28135	−0.106717	−0.0533584	−	0.998575i	$-0.516993\pi$
$457$	−2.28135	−0.106717	−0.0533584	+	0.998575i	$0.516993\pi$
$458$	10.4891	0.490126
$459$	−2.73246	−0.127540
$460$	0	0
$461$	24.0678	1.12095	0.560475	−	0.828171i	$-0.310619\pi$
$461$	24.0678	1.12095	0.560475	+	0.828171i	$0.310619\pi$
$462$	0	0
$463$	−11.1290	−0.517211	−0.258605	−	0.965983i	$-0.583263\pi$
$463$	−11.1290	−0.517211	−0.258605	+	0.965983i	$0.583263\pi$
$464$	36.2581	1.68324
$465$	0	0
$466$	−11.5065	−0.533027
$467$	28.5766	1.32237	0.661185	−	0.750223i	$-0.270054\pi$
$467$	28.5766	1.32237	0.661185	+	0.750223i	$0.270054\pi$
$468$	−0.710265	−0.0328320
$469$	0	0
$470$	0	0
$471$	−9.47182	−0.436438
$472$	2.81995	0.129799
$473$	−43.0475	−1.97933
$474$	10.4224	0.478718
$475$	0	0
$476$	0	0
$477$	−1.71513	−0.0785305
$478$	−11.4312	−0.522853
$479$	5.58009	0.254961	0.127480	−	0.991841i	$-0.459311\pi$
$479$	5.58009	0.254961	0.127480	+	0.991841i	$0.459311\pi$
$480$	0	0
$481$	6.36517	0.290227
$482$	21.2370	0.967320
$483$	0	0
$484$	3.53873	0.160852
$485$	0	0
$486$	1.55241	0.0704187
$487$	−8.34684	−0.378232	−0.189116	−	0.981955i	$-0.560562\pi$
$487$	−8.34684	−0.378232	−0.189116	+	0.981955i	$0.560562\pi$
$488$	−23.5886	−1.06781
$489$	−5.39659	−0.244042
$490$	0	0
$491$	0.557405	0.0251554	0.0125777	−	0.999921i	$-0.495996\pi$
$491$	0.557405	0.0251554	0.0125777	+	0.999921i	$0.495996\pi$
$492$	−2.75324	−0.124126
$493$	−21.2976	−0.959197
$494$	0.820713	0.0369256
$495$	0	0
$496$	−24.5845	−1.10388
$497$	0	0
$498$	−7.91645	−0.354745
$499$	−26.0337	−1.16543	−0.582713	−	0.812678i	$-0.698009\pi$
$499$	−26.0337	−1.16543	−0.582713	+	0.812678i	$0.698009\pi$
$500$	0	0
$501$	−0.312550	−0.0139637
$502$	36.4572	1.62717
$503$	5.52409	0.246307	0.123154	−	0.992388i	$-0.460699\pi$
$503$	5.52409	0.246307	0.123154	+	0.992388i	$0.460699\pi$
$504$	0	0
$505$	0	0
$506$	48.3151	2.14787
$507$	9.99859	0.444053
$508$	−5.62997	−0.249789
$509$	−22.8893	−1.01455	−0.507274	−	0.861785i	$-0.669347\pi$
$509$	−22.8893	−1.01455	−0.507274	+	0.861785i	$0.669347\pi$
$510$	0	0
$511$	0	0
$512$	12.3362	0.545186
$513$	−0.305156	−0.0134730
$514$	22.6612	0.999541
$515$	0	0
$516$	3.98316	0.175349
$517$	−8.04846	−0.353971
$518$	0	0
$519$	9.61475	0.422041
$520$	0	0
$521$	−26.1989	−1.14780	−0.573898	−	0.818927i	$-0.694569\pi$
$521$	−26.1989	−1.14780	−0.573898	+	0.818927i	$0.694569\pi$
$522$	12.1000	0.529601
$523$	−26.8760	−1.17521	−0.587603	−	0.809149i	$-0.699928\pi$
$523$	−26.8760	−1.17521	−0.587603	+	0.809149i	$0.699928\pi$
$524$	−9.01197	−0.393690
$525$	0	0
$526$	7.90083	0.344493
$527$	14.4407	0.629046
$528$	20.6113	0.896992
$529$	26.3398	1.14521
$530$	0	0
$531$	1.14243	0.0495774
$532$	0	0
$533$	−11.6345	−0.503948
$534$	−6.31932	−0.273464
$535$	0	0
$536$	−20.6956	−0.893915
$537$	8.85461	0.382104
$538$	47.3639	2.04200
$539$	0	0
$540$	0	0
$541$	−18.7844	−0.807606	−0.403803	−	0.914846i	$-0.632312\pi$
$541$	−18.7844	−0.807606	−0.403803	+	0.914846i	$0.632312\pi$
$542$	−30.2534	−1.29949
$543$	−16.9234	−0.726251
$544$	6.24331	0.267680
$545$	0	0
$546$	0	0
$547$	13.4126	0.573483	0.286742	−	0.958008i	$-0.407428\pi$
$547$	13.4126	0.573483	0.286742	+	0.958008i	$0.407428\pi$
$548$	−1.23071	−0.0525732
$549$	−9.55635	−0.407855
$550$	0	0
$551$	−2.37848	−0.101327
$552$	17.3384	0.737971
$553$	0	0
$554$	−39.5557	−1.68056
$555$	0	0
$556$	8.91138	0.377927
$557$	−40.1845	−1.70267	−0.851336	−	0.524621i	$-0.824207\pi$
$557$	−40.1845	−1.70267	−0.851336	+	0.524621i	$0.824207\pi$
$558$	−8.20428	−0.347315
$559$	16.8319	0.711914
$560$	0	0
$561$	−12.1069	−0.511152
$562$	34.4501	1.45319
$563$	−25.1879	−1.06154	−0.530772	−	0.847514i	$-0.678098\pi$
$563$	−25.1879	−1.06154	−0.530772	+	0.847514i	$0.678098\pi$
$564$	0.744719	0.0313583
$565$	0	0
$566$	−28.1362	−1.18265
$567$	0	0
$568$	25.3966	1.06562
$569$	−11.0351	−0.462617	−0.231309	−	0.972880i	$-0.574301\pi$
$569$	−11.0351	−0.462617	−0.231309	+	0.972880i	$0.574301\pi$
$570$	0	0
$571$	28.7147	1.20167	0.600836	−	0.799372i	$-0.294834\pi$
$571$	28.7147	1.20167	0.600836	+	0.799372i	$0.294834\pi$
$572$	−3.14701	−0.131583
$573$	−0.318541	−0.0133073
$574$	0	0
$575$	0	0
$576$	5.75669	0.239862
$577$	−6.86405	−0.285754	−0.142877	−	0.989740i	$-0.545635\pi$
$577$	−6.86405	−0.285754	−0.142877	+	0.989740i	$0.545635\pi$
$578$	14.8002	0.615605
$579$	−2.09334	−0.0869963
$580$	0	0
$581$	0	0
$582$	4.46837	0.185220
$583$	−7.59933	−0.314732
$584$	31.6456	1.30950
$585$	0	0
$586$	−21.5720	−0.891130
$587$	9.03965	0.373106	0.186553	−	0.982445i	$-0.440268\pi$
$587$	9.03965	0.373106	0.186553	+	0.982445i	$0.440268\pi$
$588$	0	0
$589$	1.61271	0.0664506
$590$	0	0
$591$	22.5798	0.928809
$592$	17.0913	0.702447
$593$	−5.10798	−0.209760	−0.104880	−	0.994485i	$-0.533446\pi$
$593$	−5.10798	−0.209760	−0.104880	+	0.994485i	$0.533446\pi$
$594$	6.87834	0.282222
$595$	0	0
$596$	−2.13724	−0.0875446
$597$	−7.50408	−0.307121
$598$	−18.8916	−0.772534
$599$	−29.9443	−1.22349	−0.611745	−	0.791055i	$-0.709532\pi$
$599$	−29.9443	−1.22349	−0.611745	+	0.791055i	$0.709532\pi$
$600$	0	0
$601$	12.5387	0.511466	0.255733	−	0.966747i	$-0.417683\pi$
$601$	12.5387	0.511466	0.255733	+	0.966747i	$0.417683\pi$
$602$	0	0
$603$	−8.38433	−0.341436
$604$	−3.98238	−0.162041
$605$	0	0
$606$	−11.2048	−0.455163
$607$	9.59805	0.389573	0.194786	−	0.980846i	$-0.437599\pi$
$607$	9.59805	0.389573	0.194786	+	0.980846i	$0.437599\pi$
$608$	0.697242	0.0282769
$609$	0	0
$610$	0	0
$611$	3.14701	0.127314
$612$	1.12024	0.0452830
$613$	−27.6646	−1.11736	−0.558682	−	0.829382i	$-0.688693\pi$
$613$	−27.6646	−1.11736	−0.558682	+	0.829382i	$0.688693\pi$
$614$	46.3133	1.86905
$615$	0	0
$616$	0	0
$617$	20.4155	0.821896	0.410948	−	0.911659i	$-0.365198\pi$
$617$	20.4155	0.821896	0.410948	+	0.911659i	$0.365198\pi$
$618$	17.5009	0.703990
$619$	15.4422	0.620676	0.310338	−	0.950626i	$-0.399558\pi$
$619$	15.4422	0.620676	0.310338	+	0.950626i	$0.399558\pi$
$620$	0	0
$621$	7.02423	0.281873
$622$	20.9781	0.841145
$623$	0	0
$624$	−8.05918	−0.322625
$625$	0	0
$626$	−44.9977	−1.79847
$627$	−1.35207	−0.0539966
$628$	3.88321	0.154957
$629$	−10.0392	−0.400290
$630$	0	0
$631$	−38.1722	−1.51961	−0.759805	−	0.650151i	$-0.774705\pi$
$631$	−38.1722	−1.51961	−0.759805	+	0.650151i	$0.774705\pi$
$632$	16.5719	0.659196
$633$	−7.67216	−0.304941
$634$	−1.79868	−0.0714347
$635$	0	0
$636$	0.703161	0.0278822
$637$	0	0
$638$	53.6119	2.12252
$639$	10.2888	0.407019
$640$	0	0
$641$	−16.7801	−0.662776	−0.331388	−	0.943495i	$-0.607517\pi$
$641$	−16.7801	−0.662776	−0.331388	+	0.943495i	$0.607517\pi$
$642$	−10.7364	−0.423732
$643$	−14.2002	−0.560001	−0.280001	−	0.960000i	$-0.590335\pi$
$643$	−14.2002	−0.560001	−0.280001	+	0.960000i	$0.590335\pi$
$644$	0	0
$645$	0	0
$646$	−1.29444	−0.0509291
$647$	14.4885	0.569602	0.284801	−	0.958587i	$-0.408072\pi$
$647$	14.4885	0.569602	0.284801	+	0.958587i	$0.408072\pi$
$648$	2.46837	0.0969667
$649$	5.06185	0.198695
$650$	0	0
$651$	0	0
$652$	2.21247	0.0866469
$653$	−10.2833	−0.402418	−0.201209	−	0.979548i	$-0.564487\pi$
$653$	−10.2833	−0.402418	−0.201209	+	0.979548i	$0.564487\pi$
$654$	6.86467	0.268430
$655$	0	0
$656$	−31.2402	−1.21973
$657$	12.8204	0.500173
$658$	0	0
$659$	14.9773	0.583433	0.291717	−	0.956505i	$-0.405774\pi$
$659$	14.9773	0.583433	0.291717	+	0.956505i	$0.405774\pi$
$660$	0	0
$661$	−2.44647	−0.0951565	−0.0475782	−	0.998868i	$-0.515150\pi$
$661$	−2.44647	−0.0951565	−0.0475782	+	0.998868i	$0.515150\pi$
$662$	44.0094	1.71048
$663$	4.73387	0.183848
$664$	−12.5874	−0.488484
$665$	0	0
$666$	5.70365	0.221012
$667$	54.7490	2.11989
$668$	0.128138	0.00495780
$669$	6.90208	0.266850
$670$	0	0
$671$	−42.3418	−1.63459
$672$	0	0
$673$	−17.2596	−0.665310	−0.332655	−	0.943049i	$-0.607945\pi$
$673$	−17.2596	−0.665310	−0.332655	+	0.943049i	$0.607945\pi$
$674$	−24.7965	−0.955127
$675$	0	0
$676$	−4.09917	−0.157660
$677$	11.6384	0.447298	0.223649	−	0.974670i	$-0.428203\pi$
$677$	11.6384	0.447298	0.223649	+	0.974670i	$0.428203\pi$
$678$	4.44223	0.170603
$679$	0	0
$680$	0	0
$681$	6.34721	0.243225
$682$	−36.3511	−1.39196
$683$	31.0960	1.18986	0.594928	−	0.803779i	$-0.297181\pi$
$683$	31.0960	1.18986	0.594928	+	0.803779i	$0.297181\pi$
$684$	0.125106	0.00478356
$685$	0	0
$686$	0	0
$687$	6.75669	0.257784
$688$	45.1958	1.72307
$689$	2.97140	0.113201
$690$	0	0
$691$	17.9339	0.682238	0.341119	−	0.940020i	$-0.389194\pi$
$691$	17.9339	0.682238	0.341119	+	0.940020i	$0.389194\pi$
$692$	−3.94181	−0.149845
$693$	0	0
$694$	−20.1999	−0.766778
$695$	0	0
$696$	19.2392	0.729261
$697$	18.3502	0.695062
$698$	49.7895	1.88456
$699$	−7.41201	−0.280348
$700$	0	0
$701$	7.04488	0.266081	0.133041	−	0.991111i	$-0.457526\pi$
$701$	7.04488	0.266081	0.133041	+	0.991111i	$0.457526\pi$
$702$	−2.68949	−0.101508
$703$	−1.12116	−0.0422855
$704$	25.5065	0.961312
$705$	0	0
$706$	20.7540	0.781088
$707$	0	0
$708$	−0.468370	−0.0176024
$709$	−25.9243	−0.973609	−0.486804	−	0.873511i	$-0.661838\pi$
$709$	−25.9243	−0.973609	−0.486804	+	0.873511i	$0.661838\pi$
$710$	0	0
$711$	6.71372	0.251784
$712$	−10.0479	−0.376560
$713$	−37.1221	−1.39023
$714$	0	0
$715$	0	0
$716$	−3.63017	−0.135666
$717$	−7.36355	−0.274997
$718$	20.3668	0.760082
$719$	−31.2855	−1.16675	−0.583376	−	0.812202i	$-0.698269\pi$
$719$	−31.2855	−1.16675	−0.583376	+	0.812202i	$0.698269\pi$
$720$	0	0
$721$	0	0
$722$	29.3512	1.09234
$723$	13.6800	0.508766
$724$	6.93815	0.257854
$725$	0	0
$726$	13.3998	0.497311
$727$	−22.8312	−0.846761	−0.423380	−	0.905952i	$-0.639157\pi$
$727$	−22.8312	−0.846761	−0.423380	+	0.905952i	$0.639157\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−26.5475	−0.981896
$732$	3.91787	0.144809
$733$	5.95434	0.219929	0.109964	−	0.993936i	$-0.464926\pi$
$733$	5.95434	0.219929	0.109964	+	0.993936i	$0.464926\pi$
$734$	20.3924	0.752696
$735$	0	0
$736$	−16.0494	−0.591590
$737$	−37.1489	−1.36840
$738$	−10.4254	−0.383764
$739$	41.5079	1.52689	0.763446	−	0.645872i	$-0.223506\pi$
$739$	41.5079	1.52689	0.763446	+	0.645872i	$0.223506\pi$
$740$	0	0
$741$	0.528671	0.0194212
$742$	0	0
$743$	−6.80015	−0.249473	−0.124737	−	0.992190i	$-0.539809\pi$
$743$	−6.80015	−0.249473	−0.124737	+	0.992190i	$0.539809\pi$
$744$	−13.0450	−0.478253
$745$	0	0
$746$	45.9820	1.68352
$747$	−5.09946	−0.186580
$748$	4.96351	0.181484
$749$	0	0
$750$	0	0
$751$	23.6113	0.861588	0.430794	−	0.902450i	$-0.358233\pi$
$751$	23.6113	0.861588	0.430794	+	0.902450i	$0.358233\pi$
$752$	8.45012	0.308144
$753$	23.4843	0.855815
$754$	−20.9627	−0.763416
$755$	0	0
$756$	0	0
$757$	16.2267	0.589769	0.294884	−	0.955533i	$-0.404719\pi$
$757$	16.2267	0.589769	0.294884	+	0.955533i	$0.404719\pi$
$758$	−11.6072	−0.421592
$759$	31.1226	1.12968
$760$	0	0
$761$	2.55508	0.0926215	0.0463108	−	0.998927i	$-0.485254\pi$
$761$	2.55508	0.0926215	0.0463108	+	0.998927i	$0.485254\pi$
$762$	−21.3184	−0.772284
$763$	0	0
$764$	0.130594	0.00472473
$765$	0	0
$766$	−36.8528	−1.33154
$767$	−1.97922	−0.0714655
$768$	−9.45420	−0.341149
$769$	45.4525	1.63906	0.819530	−	0.573037i	$-0.194235\pi$
$769$	45.4525	1.63906	0.819530	+	0.573037i	$0.194235\pi$
$770$	0	0
$771$	14.5974	0.525713
$772$	0.858217	0.0308879
$773$	51.4891	1.85194	0.925968	−	0.377603i	$-0.123251\pi$
$773$	51.4891	1.85194	0.925968	+	0.377603i	$0.123251\pi$
$774$	15.0826	0.542134
$775$	0	0
$776$	7.10482	0.255048
$777$	0	0
$778$	21.9363	0.786453
$779$	2.04931	0.0734243
$780$	0	0
$781$	45.5872	1.63124
$782$	29.7961	1.06550
$783$	7.79430	0.278546
$784$	0	0
$785$	0	0
$786$	−34.1247	−1.21719
$787$	−20.1514	−0.718321	−0.359161	−	0.933276i	$-0.616937\pi$
$787$	−20.1514	−0.718321	−0.359161	+	0.933276i	$0.616937\pi$
$788$	−9.25716	−0.329773
$789$	5.08940	0.181187
$790$	0	0
$791$	0	0
$792$	10.9367	0.388620
$793$	16.5560	0.587920
$794$	17.4950	0.620874
$795$	0	0
$796$	3.07649	0.109043
$797$	13.0702	0.462971	0.231486	−	0.972838i	$-0.425641\pi$
$797$	13.0702	0.462971	0.231486	+	0.972838i	$0.425641\pi$
$798$	0	0
$799$	−4.96351	−0.175596
$800$	0	0
$801$	−4.07065	−0.143830
$802$	−36.1662	−1.27707
$803$	56.8042	2.00458
$804$	3.43737	0.121227
$805$	0	0
$806$	14.2136	0.500652
$807$	30.5099	1.07400
$808$	−17.8159	−0.626760
$809$	−1.73720	−0.0610765	−0.0305383	−	0.999534i	$-0.509722\pi$
$809$	−1.73720	−0.0610765	−0.0305383	+	0.999534i	$0.509722\pi$
$810$	0	0
$811$	27.9004	0.979715	0.489858	−	0.871802i	$-0.337049\pi$
$811$	27.9004	0.979715	0.489858	+	0.871802i	$0.337049\pi$
$812$	0	0
$813$	−19.4880	−0.683475
$814$	25.2715	0.885765
$815$	0	0
$816$	12.7110	0.444976
$817$	−2.96478	−0.103725
$818$	32.0538	1.12074
$819$	0	0
$820$	0	0
$821$	16.9360	0.591070	0.295535	−	0.955332i	$-0.404502\pi$
$821$	16.9360	0.591070	0.295535	+	0.955332i	$0.404502\pi$
$822$	−4.66019	−0.162543
$823$	9.89031	0.344755	0.172377	−	0.985031i	$-0.444855\pi$
$823$	9.89031	0.344755	0.172377	+	0.985031i	$0.444855\pi$
$824$	27.8269	0.969396
$825$	0	0
$826$	0	0
$827$	35.3201	1.22820	0.614101	−	0.789228i	$-0.289519\pi$
$827$	35.3201	1.22820	0.614101	+	0.789228i	$0.289519\pi$
$828$	−2.87976	−0.100079
$829$	−9.29951	−0.322985	−0.161493	−	0.986874i	$-0.551631\pi$
$829$	−9.29951	−0.322985	−0.161493	+	0.986874i	$0.551631\pi$
$830$	0	0
$831$	−25.4802	−0.883898
$832$	−9.97323	−0.345760
$833$	0	0
$834$	33.7438	1.16845
$835$	0	0
$836$	0.554316	0.0191714
$837$	−5.28487	−0.182672
$838$	−8.51114	−0.294012
$839$	7.93405	0.273914	0.136957	−	0.990577i	$-0.456268\pi$
$839$	7.93405	0.273914	0.136957	+	0.990577i	$0.456268\pi$
$840$	0	0
$841$	31.7512	1.09487
$842$	5.86933	0.202271
$843$	22.1914	0.764312
$844$	3.14539	0.108269
$845$	0	0
$846$	2.81995	0.0969519
$847$	0	0
$848$	7.97857	0.273985
$849$	−18.1242	−0.622020
$850$	0	0
$851$	25.8075	0.884669
$852$	−4.21816	−0.144512
$853$	−30.0757	−1.02977	−0.514887	−	0.857258i	$-0.672166\pi$
$853$	−30.0757	−1.02977	−0.514887	+	0.857258i	$0.672166\pi$
$854$	0	0
$855$	0	0
$856$	−17.0711	−0.583480
$857$	−47.1373	−1.61018	−0.805090	−	0.593153i	$-0.797883\pi$
$857$	−47.1373	−1.61018	−0.805090	+	0.593153i	$0.797883\pi$
$858$	−11.9165	−0.406821
$859$	32.6367	1.11355	0.556774	−	0.830664i	$-0.312039\pi$
$859$	32.6367	1.11355	0.556774	+	0.830664i	$0.312039\pi$
$860$	0	0
$861$	0	0
$862$	42.7110	1.45474
$863$	−37.4102	−1.27346	−0.636728	−	0.771088i	$-0.719713\pi$
$863$	−37.4102	−1.27346	−0.636728	+	0.771088i	$0.719713\pi$
$864$	−2.28487	−0.0777328
$865$	0	0
$866$	13.5916	0.461860
$867$	9.53367	0.323780
$868$	0	0
$869$	29.7468	1.00909
$870$	0	0
$871$	14.5255	0.492178
$872$	10.9150	0.369628
$873$	2.87834	0.0974172
$874$	3.32757	0.112557
$875$	0	0
$876$	−5.25606	−0.177586
$877$	−33.5657	−1.13343	−0.566716	−	0.823913i	$-0.691786\pi$
$877$	−33.5657	−1.13343	−0.566716	+	0.823913i	$0.691786\pi$
$878$	−16.4753	−0.556014
$879$	−13.8958	−0.468694
$880$	0	0
$881$	12.1952	0.410867	0.205433	−	0.978671i	$-0.434140\pi$
$881$	12.1952	0.410867	0.205433	+	0.978671i	$0.434140\pi$
$882$	0	0
$883$	11.0408	0.371552	0.185776	−	0.982592i	$-0.440520\pi$
$883$	11.0408	0.371552	0.185776	+	0.982592i	$0.440520\pi$
$884$	−1.94077	−0.0652752
$885$	0	0
$886$	−31.0258	−1.04233
$887$	1.56093	0.0524108	0.0262054	−	0.999657i	$-0.491658\pi$
$887$	1.56093	0.0524108	0.0262054	+	0.999657i	$0.491658\pi$
$888$	9.06895	0.304334
$889$	0	0
$890$	0	0
$891$	4.43075	0.148436
$892$	−2.82968	−0.0947448
$893$	−0.554316	−0.0185495
$894$	−8.09285	−0.270665
$895$	0	0
$896$	0	0
$897$	−12.1692	−0.406318
$898$	−4.69724	−0.156749
$899$	−41.1919	−1.37383
$900$	0	0
$901$	−4.68653	−0.156131
$902$	−46.1924	−1.53804
$903$	0	0
$904$	7.06326	0.234921
$905$	0	0
$906$	−15.0797	−0.500988
$907$	17.4078	0.578016	0.289008	−	0.957327i	$-0.406675\pi$
$907$	17.4078	0.578016	0.289008	+	0.957327i	$0.406675\pi$
$908$	−2.60220	−0.0863569
$909$	−7.21767	−0.239395
$910$	0	0
$911$	−18.3203	−0.606978	−0.303489	−	0.952835i	$-0.598152\pi$
$911$	−18.3203	−0.606978	−0.303489	+	0.952835i	$0.598152\pi$
$912$	1.41955	0.0470059
$913$	−22.5945	−0.747767
$914$	3.54158	0.117145
$915$	0	0
$916$	−2.77007	−0.0915258
$917$	0	0
$918$	4.24190	0.140003
$919$	22.5927	0.745264	0.372632	−	0.927979i	$-0.378455\pi$
$919$	22.5927	0.745264	0.372632	+	0.927979i	$0.378455\pi$
$920$	0	0
$921$	29.8332	0.983037
$922$	−37.3631	−1.23049
$923$	−17.8249	−0.586715
$924$	0	0
$925$	0	0
$926$	17.2768	0.567752
$927$	11.2734	0.370267
$928$	−17.8090	−0.584608
$929$	−38.6712	−1.26876	−0.634381	−	0.773021i	$-0.718745\pi$
$929$	−38.6712	−1.26876	−0.634381	+	0.773021i	$0.718745\pi$
$930$	0	0
$931$	0	0
$932$	3.03874	0.0995372
$933$	13.5132	0.442404
$934$	−44.3626	−1.45159
$935$	0	0
$936$	−4.27635	−0.139777
$937$	46.7921	1.52863	0.764316	−	0.644842i	$-0.223077\pi$
$937$	46.7921	1.52863	0.764316	+	0.644842i	$0.223077\pi$
$938$	0	0
$939$	−28.9857	−0.945912
$940$	0	0
$941$	41.8922	1.36565	0.682824	−	0.730583i	$-0.260752\pi$
$941$	41.8922	1.36565	0.682824	+	0.730583i	$0.260752\pi$
$942$	14.7041	0.479087
$943$	−47.1721	−1.53613
$944$	−5.31446	−0.172971
$945$	0	0
$946$	66.8274	2.17275
$947$	56.1146	1.82348	0.911740	−	0.410768i	$-0.134739\pi$
$947$	56.1146	1.82348	0.911740	+	0.410768i	$0.134739\pi$
$948$	−2.75246	−0.0893956
$949$	−22.2109	−0.720996
$950$	0	0
$951$	−1.15864	−0.0375714
$952$	0	0
$953$	17.7705	0.575643	0.287821	−	0.957684i	$-0.407069\pi$
$953$	17.7705	0.575643	0.287821	+	0.957684i	$0.407069\pi$
$954$	2.66259	0.0862044
$955$	0	0
$956$	3.01887	0.0976373
$957$	34.5346	1.11635
$958$	−8.66259	−0.279875
$959$	0	0
$960$	0	0
$961$	−3.07016	−0.0990375
$962$	−9.88135	−0.318587
$963$	−6.91596	−0.222864
$964$	−5.60848	−0.180637
$965$	0	0
$966$	0	0
$967$	−22.5942	−0.726579	−0.363290	−	0.931676i	$-0.618346\pi$
$967$	−22.5942	−0.726579	−0.363290	+	0.931676i	$0.618346\pi$
$968$	21.3059	0.684799
$969$	−0.833827	−0.0267864
$970$	0	0
$971$	−41.0488	−1.31732	−0.658660	−	0.752441i	$-0.728876\pi$
$971$	−41.0488	−1.31732	−0.658660	+	0.752441i	$0.728876\pi$
$972$	−0.409975	−0.0131500
$973$	0	0
$974$	12.9577	0.415192
$975$	0	0
$976$	44.4549	1.42297
$977$	−19.4734	−0.623008	−0.311504	−	0.950245i	$-0.600833\pi$
$977$	−19.4734	−0.623008	−0.311504	+	0.950245i	$0.600833\pi$
$978$	8.37772	0.267890
$979$	−18.0361	−0.576435
$980$	0	0
$981$	4.42195	0.141182
$982$	−0.865321	−0.0276135
$983$	−51.7272	−1.64984	−0.824920	−	0.565249i	$-0.808780\pi$
$983$	−51.7272	−1.64984	−0.824920	+	0.565249i	$0.808780\pi$
$984$	−16.5766	−0.528444
$985$	0	0
$986$	33.0626	1.05293
$987$	0	0
$988$	−0.216742	−0.00689548
$989$	68.2448	2.17006
$990$	0	0
$991$	35.9499	1.14198	0.570992	−	0.820955i	$-0.306559\pi$
$991$	35.9499	1.14198	0.570992	+	0.820955i	$0.306559\pi$
$992$	12.0752	0.383389
$993$	28.3491	0.899632
$994$	0	0
$995$	0	0
$996$	2.09065	0.0662449
$997$	−56.0599	−1.77544	−0.887718	−	0.460388i	$-0.847710\pi$
$997$	−56.0599	−1.77544	−0.887718	+	0.460388i	$0.847710\pi$
$998$	40.4149	1.27931
$999$	3.67406	0.116242

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3675.2.a.bn.1.2		4
5.2	odd	4		735.2.d.e.589.3		8
5.3	odd	4		735.2.d.e.589.6		8
5.4	even	2		3675.2.a.cb.1.3		4
7.3	odd	6		525.2.i.k.226.3		8
7.5	odd	6		525.2.i.k.151.3		8
7.6	odd	2		3675.2.a.bp.1.2		4
15.2	even	4		2205.2.d.o.1324.6		8
15.8	even	4		2205.2.d.o.1324.3		8
35.2	odd	12		735.2.q.g.214.3		16
35.3	even	12		105.2.q.a.79.3	yes	16
35.12	even	12		105.2.q.a.4.3	✓	16
35.13	even	4		735.2.d.d.589.6		8
35.17	even	12		105.2.q.a.79.6	yes	16
35.18	odd	12		735.2.q.g.79.3		16
35.19	odd	6		525.2.i.h.151.2		8
35.23	odd	12		735.2.q.g.214.6		16
35.24	odd	6		525.2.i.h.226.2		8
35.27	even	4		735.2.d.d.589.3		8
35.32	odd	12		735.2.q.g.79.6		16
35.33	even	12		105.2.q.a.4.6	yes	16
35.34	odd	2		3675.2.a.bz.1.3		4
105.17	odd	12		315.2.bf.b.289.3		16
105.38	odd	12		315.2.bf.b.289.6		16
105.47	odd	12		315.2.bf.b.109.6		16
105.62	odd	4		2205.2.d.s.1324.6		8
105.68	odd	12		315.2.bf.b.109.3		16
105.83	odd	4		2205.2.d.s.1324.3		8
140.3	odd	12		1680.2.di.d.289.1		16
140.47	odd	12		1680.2.di.d.529.1		16
140.87	odd	12		1680.2.di.d.289.8		16
140.103	odd	12		1680.2.di.d.529.8		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.q.a.4.3	✓	16	35.12	even	12
105.2.q.a.4.6	yes	16	35.33	even	12
105.2.q.a.79.3	yes	16	35.3	even	12
105.2.q.a.79.6	yes	16	35.17	even	12
315.2.bf.b.109.3		16	105.68	odd	12
315.2.bf.b.109.6		16	105.47	odd	12
315.2.bf.b.289.3		16	105.17	odd	12
315.2.bf.b.289.6		16	105.38	odd	12
525.2.i.h.151.2		8	35.19	odd	6
525.2.i.h.226.2		8	35.24	odd	6
525.2.i.k.151.3		8	7.5	odd	6
525.2.i.k.226.3		8	7.3	odd	6
735.2.d.d.589.3		8	35.27	even	4
735.2.d.d.589.6		8	35.13	even	4
735.2.d.e.589.3		8	5.2	odd	4
735.2.d.e.589.6		8	5.3	odd	4
735.2.q.g.79.3		16	35.18	odd	12
735.2.q.g.79.6		16	35.32	odd	12
735.2.q.g.214.3		16	35.2	odd	12
735.2.q.g.214.6		16	35.23	odd	12
1680.2.di.d.289.1		16	140.3	odd	12
1680.2.di.d.289.8		16	140.87	odd	12
1680.2.di.d.529.1		16	140.47	odd	12
1680.2.di.d.529.8		16	140.103	odd	12
2205.2.d.o.1324.3		8	15.8	even	4
2205.2.d.o.1324.6		8	15.2	even	4
2205.2.d.s.1324.3		8	105.83	odd	4
2205.2.d.s.1324.6		8	105.62	odd	4
3675.2.a.bn.1.2		4	1.1	even	1	trivial
3675.2.a.bp.1.2		4	7.6	odd	2
3675.2.a.bz.1.3		4	35.34	odd	2
3675.2.a.cb.1.3		4	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 3675 = 3 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3675.a (trivial)

\(f(q)\)	\(=\)	\(q-1.55241 q^{2} -1.00000 q^{3} +0.409975 q^{4} +1.55241 q^{6} +2.46837 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.55241 q^{2} -1.00000 q^{3} +0.409975 q^{4} +1.55241 q^{6} +2.46837 q^{8} +1.00000 q^{9} +4.43075 q^{11} -0.409975 q^{12} -1.73246 q^{13} -4.65187 q^{16} +2.73246 q^{17} -1.55241 q^{18} +0.305156 q^{19} -6.87834 q^{22} -7.02423 q^{23} -2.46837 q^{24} +2.68949 q^{26} -1.00000 q^{27} -7.79430 q^{29} +5.28487 q^{31} +2.28487 q^{32} -4.43075 q^{33} -4.24190 q^{34} +0.409975 q^{36} -3.67406 q^{37} -0.473727 q^{38} +1.73246 q^{39} +6.71562 q^{41} -9.71562 q^{43} +1.81650 q^{44} +10.9045 q^{46} -1.81650 q^{47} +4.65187 q^{48} -2.73246 q^{51} -0.710265 q^{52} -1.71513 q^{53} +1.55241 q^{54} -0.305156 q^{57} +12.1000 q^{58} +1.14243 q^{59} -9.55635 q^{61} -8.20428 q^{62} +5.75669 q^{64} +6.87834 q^{66} -8.38433 q^{67} +1.12024 q^{68} +7.02423 q^{69} +10.2888 q^{71} +2.46837 q^{72} +12.8204 q^{73} +5.70365 q^{74} +0.125106 q^{76} -2.68949 q^{78} +6.71372 q^{79} +1.00000 q^{81} -10.4254 q^{82} -5.09946 q^{83} +15.0826 q^{86} +7.79430 q^{87} +10.9367 q^{88} -4.07065 q^{89} -2.87976 q^{92} -5.28487 q^{93} +2.81995 q^{94} -2.28487 q^{96} +2.87834 q^{97} +4.43075 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} - 4 q^{3} + 4 q^{4} + 2 q^{6} - 6 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q - 2 * q^2 - 4 * q^3 + 4 * q^4 + 2 * q^6 - 6 * q^8 + 4 * q^9	\( 4 q - 2 q^{2} - 4 q^{3} + 4 q^{4} + 2 q^{6} - 6 q^{8} + 4 q^{9} - 4 q^{12} + 2 q^{13} + 2 q^{17} - 2 q^{18} + 12 q^{19} - 14 q^{22} - 10 q^{23} + 6 q^{24} - 6 q^{26} - 4 q^{27} - 6 q^{29} + 8 q^{31} - 4 q^{32} + 4 q^{34} + 4 q^{36} - 24 q^{37} + 8 q^{38} - 2 q^{39} - 4 q^{41} - 8 q^{43} + 10 q^{44} + 16 q^{46} - 10 q^{47} - 2 q^{51} + 34 q^{52} - 20 q^{53} + 2 q^{54} - 12 q^{57} - 10 q^{58} - 2 q^{59} + 8 q^{61} - 10 q^{62} - 4 q^{64} + 14 q^{66} - 6 q^{67} - 30 q^{68} + 10 q^{69} - 14 q^{71} - 6 q^{72} + 12 q^{73} + 20 q^{74} + 16 q^{76} + 6 q^{78} - 8 q^{79} + 4 q^{81} - 18 q^{82} - 6 q^{83} + 24 q^{86} + 6 q^{87} + 12 q^{88} - 8 q^{89} - 46 q^{92} - 8 q^{93} + 16 q^{94} + 4 q^{96} - 2 q^{97}+O(q^{100}) \) 4 * q - 2 * q^2 - 4 * q^3 + 4 * q^4 + 2 * q^6 - 6 * q^8 + 4 * q^9 - 4 * q^12 + 2 * q^13 + 2 * q^17 - 2 * q^18 + 12 * q^19 - 14 * q^22 - 10 * q^23 + 6 * q^24 - 6 * q^26 - 4 * q^27 - 6 * q^29 + 8 * q^31 - 4 * q^32 + 4 * q^34 + 4 * q^36 - 24 * q^37 + 8 * q^38 - 2 * q^39 - 4 * q^41 - 8 * q^43 + 10 * q^44 + 16 * q^46 - 10 * q^47 - 2 * q^51 + 34 * q^52 - 20 * q^53 + 2 * q^54 - 12 * q^57 - 10 * q^58 - 2 * q^59 + 8 * q^61 - 10 * q^62 - 4 * q^64 + 14 * q^66 - 6 * q^67 - 30 * q^68 + 10 * q^69 - 14 * q^71 - 6 * q^72 + 12 * q^73 + 20 * q^74 + 16 * q^76 + 6 * q^78 - 8 * q^79 + 4 * q^81 - 18 * q^82 - 6 * q^83 + 24 * q^86 + 6 * q^87 + 12 * q^88 - 8 * q^89 - 46 * q^92 - 8 * q^93 + 16 * q^94 + 4 * q^96 - 2 * q^97

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