LMFDB - Embedded newform 3645.2.a.g.1.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3645,2,Mod(1,3645)] 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("3645.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3645, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3645 = 3^{6} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3645.a (trivial)

Newform invariants

comment:select newform

sage:traces = [15,-3,0,9,15] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$29.1054715368$
Analytic rank:	$1$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 3 x^{14} - 15 x^{13} + 47 x^{12} + 84 x^{11} - 279 x^{10} - 219 x^{9} + 783 x^{8} + 279 x^{7} + \cdots + 3 $ x^15 - 3x^14 - 15x^13 + 47x^12 + 84x^11 - 279x^10 - 219x^9 + 783x^8 + 279x^7 - 1054x^6 - 195x^5 + 609x^4 + 88x^3 - 102*x^2 + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3^{3} $
Twist minimal:	no (minimal twist has level 135)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$0.300859$ of defining polynomial
Character	$\chi$	$=$	3645.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.300859 q^{2} -1.90948 q^{4} +1.00000 q^{5} -4.28937 q^{7} +1.17620 q^{8} -0.300859 q^{10} -1.77263 q^{11} +4.04481 q^{13} +1.29050 q^{14} +3.46510 q^{16} +3.90428 q^{17} +1.11369 q^{19} -1.90948 q^{20} +0.533311 q^{22} -8.62555 q^{23} +1.00000 q^{25} -1.21692 q^{26} +8.19048 q^{28} -0.598345 q^{29} +1.62890 q^{31} -3.39491 q^{32} -1.17464 q^{34} -4.28937 q^{35} +5.21715 q^{37} -0.335064 q^{38} +1.17620 q^{40} -0.756985 q^{41} -7.20963 q^{43} +3.38480 q^{44} +2.59507 q^{46} +11.7489 q^{47} +11.3987 q^{49} -0.300859 q^{50} -7.72349 q^{52} -3.97724 q^{53} -1.77263 q^{55} -5.04517 q^{56} +0.180017 q^{58} +13.8195 q^{59} -8.48018 q^{61} -0.490070 q^{62} -5.90880 q^{64} +4.04481 q^{65} +7.83466 q^{67} -7.45517 q^{68} +1.29050 q^{70} +5.18353 q^{71} -4.37528 q^{73} -1.56963 q^{74} -2.12657 q^{76} +7.60345 q^{77} -12.8907 q^{79} +3.46510 q^{80} +0.227746 q^{82} -7.90723 q^{83} +3.90428 q^{85} +2.16908 q^{86} -2.08497 q^{88} +3.25566 q^{89} -17.3497 q^{91} +16.4703 q^{92} -3.53478 q^{94} +1.11369 q^{95} -1.99171 q^{97} -3.42940 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - 3 q^{2} + 9 q^{4} + 15 q^{5} - 12 q^{7} - 9 q^{8} - 3 q^{10} - 12 q^{13} - 3 q^{16} - 12 q^{17} - 24 q^{19} + 9 q^{20} - 18 q^{22} - 18 q^{23} + 15 q^{25} + 9 q^{26} - 30 q^{28} + 3 q^{29} - 24 q^{31}+ \cdots - 15 q^{98}+O(q^{100}) $ 15 * q - 3 * q^2 + 9 * q^4 + 15 * q^5 - 12 * q^7 - 9 * q^8 - 3 * q^10 - 12 * q^13 - 3 * q^16 - 12 * q^17 - 24 * q^19 + 9 * q^20 - 18 * q^22 - 18 * q^23 + 15 * q^25 + 9 * q^26 - 30 * q^28 + 3 * q^29 - 24 * q^31 - 6 * q^32 - 18 * q^34 - 12 * q^35 - 24 * q^37 - 18 * q^38 - 9 * q^40 + 9 * q^41 - 42 * q^43 + 12 * q^44 - 30 * q^46 - 21 * q^47 - 3 * q^49 - 3 * q^50 - 36 * q^52 - 18 * q^53 + 30 * q^56 - 30 * q^58 + 6 * q^59 - 15 * q^61 + 36 * q^62 - 27 * q^64 - 12 * q^65 - 45 * q^67 - 36 * q^68 + 12 * q^71 - 21 * q^73 + 21 * q^74 - 48 * q^76 - 9 * q^77 - 48 * q^79 - 3 * q^80 - 24 * q^82 - 33 * q^83 - 12 * q^85 + 15 * q^86 - 54 * q^88 + 9 * q^89 - 51 * q^91 + 33 * q^92 - 30 * q^94 - 24 * q^95 - 30 * q^97 - 15 * q^98

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 $

$2$	−0.300859	−0.212739	−0.106370	−	0.994327i	$-0.533923\pi$
$2$	−0.300859	−0.212739	−0.106370	+	0.994327i	$0.533923\pi$
$3$	0	0
$3$	0	0
$4$	−1.90948	−0.954742
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−4.28937	−1.62123	−0.810615	−	0.585580i	$-0.800867\pi$
$7$	−4.28937	−1.62123	−0.810615	+	0.585580i	$0.800867\pi$
$8$	1.17620	0.415851
$9$	0	0
$10$	−0.300859	−0.0951400
$11$	−1.77263	−0.534467	−0.267233	−	0.963632i	$-0.586109\pi$
$11$	−1.77263	−0.534467	−0.267233	+	0.963632i	$0.586109\pi$
$12$	0	0
$13$	4.04481	1.12183	0.560914	−	0.827874i	$-0.310450\pi$
$13$	4.04481	1.12183	0.560914	+	0.827874i	$0.310450\pi$
$14$	1.29050	0.344899
$15$	0	0
$16$	3.46510	0.866274
$17$	3.90428	0.946928	0.473464	−	0.880813i	$-0.343003\pi$
$17$	3.90428	0.946928	0.473464	+	0.880813i	$0.343003\pi$
$18$	0	0
$19$	1.11369	0.255498	0.127749	−	0.991807i	$-0.459225\pi$
$19$	1.11369	0.255498	0.127749	+	0.991807i	$0.459225\pi$
$20$	−1.90948	−0.426974
$21$	0	0
$22$	0.533311	0.113702
$23$	−8.62555	−1.79855	−0.899276	−	0.437383i	$-0.855906\pi$
$23$	−8.62555	−1.79855	−0.899276	+	0.437383i	$0.855906\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−1.21692	−0.238657
$27$	0	0
$28$	8.19048	1.54786
$29$	−0.598345	−0.111110	−0.0555549	−	0.998456i	$-0.517693\pi$
$29$	−0.598345	−0.111110	−0.0555549	+	0.998456i	$0.517693\pi$
$30$	0	0
$31$	1.62890	0.292560	0.146280	−	0.989243i	$-0.453270\pi$
$31$	1.62890	0.292560	0.146280	+	0.989243i	$0.453270\pi$
$32$	−3.39491	−0.600141
$33$	0	0
$34$	−1.17464	−0.201449
$35$	−4.28937	−0.725036
$36$	0	0
$37$	5.21715	0.857694	0.428847	−	0.903377i	$-0.358920\pi$
$37$	5.21715	0.857694	0.428847	+	0.903377i	$0.358920\pi$
$38$	−0.335064	−0.0543545
$39$	0	0
$40$	1.17620	0.185974
$41$	−0.756985	−0.118221	−0.0591106	−	0.998251i	$-0.518826\pi$
$41$	−0.756985	−0.118221	−0.0591106	+	0.998251i	$0.518826\pi$
$42$	0	0
$43$	−7.20963	−1.09946	−0.549729	−	0.835343i	$-0.685269\pi$
$43$	−7.20963	−1.09946	−0.549729	+	0.835343i	$0.685269\pi$
$44$	3.38480	0.510278
$45$	0	0
$46$	2.59507	0.382623
$47$	11.7489	1.71376	0.856880	−	0.515516i	$-0.172400\pi$
$47$	11.7489	1.71376	0.856880	+	0.515516i	$0.172400\pi$
$48$	0	0
$49$	11.3987	1.62838
$50$	−0.300859	−0.0425479
$51$	0	0
$52$	−7.72349	−1.07106
$53$	−3.97724	−0.546316	−0.273158	−	0.961969i	$-0.588068\pi$
$53$	−3.97724	−0.546316	−0.273158	+	0.961969i	$0.588068\pi$
$54$	0	0
$55$	−1.77263	−0.239021
$56$	−5.04517	−0.674189
$57$	0	0
$58$	0.180017	0.0236374
$59$	13.8195	1.79915	0.899576	−	0.436764i	$-0.143876\pi$
$59$	13.8195	1.79915	0.899576	+	0.436764i	$0.143876\pi$
$60$	0	0
$61$	−8.48018	−1.08578	−0.542888	−	0.839805i	$-0.682669\pi$
$61$	−8.48018	−1.08578	−0.542888	+	0.839805i	$0.682669\pi$
$62$	−0.490070	−0.0622390
$63$	0	0
$64$	−5.90880	−0.738600
$65$	4.04481	0.501696
$66$	0	0
$67$	7.83466	0.957156	0.478578	−	0.878045i	$-0.341152\pi$
$67$	7.83466	0.957156	0.478578	+	0.878045i	$0.341152\pi$
$68$	−7.45517	−0.904072
$69$	0	0
$70$	1.29050	0.154244
$71$	5.18353	0.615172	0.307586	−	0.951520i	$-0.400479\pi$
$71$	5.18353	0.615172	0.307586	+	0.951520i	$0.400479\pi$
$72$	0	0
$73$	−4.37528	−0.512088	−0.256044	−	0.966665i	$-0.582419\pi$
$73$	−4.37528	−0.512088	−0.256044	+	0.966665i	$0.582419\pi$
$74$	−1.56963	−0.182465
$75$	0	0
$76$	−2.12657	−0.243935
$77$	7.60345	0.866493
$78$	0	0
$79$	−12.8907	−1.45032	−0.725159	−	0.688582i	$-0.758234\pi$
$79$	−12.8907	−1.45032	−0.725159	+	0.688582i	$0.758234\pi$
$80$	3.46510	0.387410
$81$	0	0
$82$	0.227746	0.0251503
$83$	−7.90723	−0.867932	−0.433966	−	0.900929i	$-0.642886\pi$
$83$	−7.90723	−0.867932	−0.433966	+	0.900929i	$0.642886\pi$
$84$	0	0
$85$	3.90428	0.423479
$86$	2.16908	0.233898
$87$	0	0
$88$	−2.08497	−0.222258
$89$	3.25566	0.345100	0.172550	−	0.985001i	$-0.444799\pi$
$89$	3.25566	0.345100	0.172550	+	0.985001i	$0.444799\pi$
$90$	0	0
$91$	−17.3497	−1.81874
$92$	16.4703	1.71715
$93$	0	0
$94$	−3.53478	−0.364584
$95$	1.11369	0.114262
$96$	0	0
$97$	−1.99171	−0.202227	−0.101114	−	0.994875i	$-0.532241\pi$
$97$	−1.99171	−0.202227	−0.101114	+	0.994875i	$0.532241\pi$
$98$	−3.42940	−0.346422
$99$	0	0
$100$	−1.90948	−0.190948
$101$	−9.16732	−0.912183	−0.456091	−	0.889933i	$-0.650751\pi$
$101$	−9.16732	−0.912183	−0.456091	+	0.889933i	$0.650751\pi$
$102$	0	0
$103$	1.81293	0.178633	0.0893165	−	0.996003i	$-0.471532\pi$
$103$	1.81293	0.178633	0.0893165	+	0.996003i	$0.471532\pi$
$104$	4.75751	0.466513
$105$	0	0
$106$	1.19659	0.116223
$107$	−3.71297	−0.358946	−0.179473	−	0.983763i	$-0.557439\pi$
$107$	−3.71297	−0.358946	−0.179473	+	0.983763i	$0.557439\pi$
$108$	0	0
$109$	−3.24138	−0.310468	−0.155234	−	0.987878i	$-0.549613\pi$
$109$	−3.24138	−0.310468	−0.155234	+	0.987878i	$0.549613\pi$
$110$	0.533311	0.0508492
$111$	0	0
$112$	−14.8631	−1.40443
$113$	−12.2993	−1.15702	−0.578509	−	0.815676i	$-0.696365\pi$
$113$	−12.2993	−1.15702	−0.578509	+	0.815676i	$0.696365\pi$
$114$	0	0
$115$	−8.62555	−0.804336
$116$	1.14253	0.106081
$117$	0	0
$118$	−4.15774	−0.382751
$119$	−16.7469	−1.53519
$120$	0	0
$121$	−7.85780	−0.714345
$122$	2.55134	0.230987
$123$	0	0
$124$	−3.11037	−0.279319
$125$	1.00000	0.0894427
$126$	0	0
$127$	−11.0119	−0.977144	−0.488572	−	0.872523i	$-0.662482\pi$
$127$	−11.0119	−0.977144	−0.488572	+	0.872523i	$0.662482\pi$
$128$	8.56754	0.757271
$129$	0	0
$130$	−1.21692	−0.106731
$131$	−17.5454	−1.53295	−0.766475	−	0.642275i	$-0.777991\pi$
$131$	−17.5454	−1.53295	−0.766475	+	0.642275i	$0.777991\pi$
$132$	0	0
$133$	−4.77703	−0.414221
$134$	−2.35713	−0.203625
$135$	0	0
$136$	4.59223	0.393781
$137$	−0.356723	−0.0304769	−0.0152385	−	0.999884i	$-0.504851\pi$
$137$	−0.356723	−0.0304769	−0.0152385	+	0.999884i	$0.504851\pi$
$138$	0	0
$139$	−1.49122	−0.126483	−0.0632416	−	0.997998i	$-0.520144\pi$
$139$	−1.49122	−0.126483	−0.0632416	+	0.997998i	$0.520144\pi$
$140$	8.19048	0.692222
$141$	0	0
$142$	−1.55951	−0.130871
$143$	−7.16993	−0.599579
$144$	0	0
$145$	−0.598345	−0.0496898
$146$	1.31634	0.108941
$147$	0	0
$148$	−9.96206	−0.818876
$149$	18.0992	1.48274	0.741372	−	0.671094i	$-0.234175\pi$
$149$	18.0992	1.48274	0.741372	+	0.671094i	$0.234175\pi$
$150$	0	0
$151$	−21.2099	−1.72603	−0.863017	−	0.505174i	$-0.831428\pi$
$151$	−21.2099	−1.72603	−0.863017	+	0.505174i	$0.831428\pi$
$152$	1.30993	0.106249
$153$	0	0
$154$	−2.28757	−0.184337
$155$	1.62890	0.130837
$156$	0	0
$157$	−5.47578	−0.437015	−0.218507	−	0.975835i	$-0.570119\pi$
$157$	−5.47578	−0.437015	−0.218507	+	0.975835i	$0.570119\pi$
$158$	3.87828	0.308540
$159$	0	0
$160$	−3.39491	−0.268391
$161$	36.9982	2.91586
$162$	0	0
$163$	−17.9869	−1.40884	−0.704420	−	0.709783i	$-0.748793\pi$
$163$	−17.9869	−1.40884	−0.704420	+	0.709783i	$0.748793\pi$
$164$	1.44545	0.112871
$165$	0	0
$166$	2.37896	0.184643
$167$	−19.9377	−1.54283	−0.771415	−	0.636333i	$-0.780450\pi$
$167$	−19.9377	−1.54283	−0.771415	+	0.636333i	$0.780450\pi$
$168$	0	0
$169$	3.36045	0.258496
$170$	−1.17464	−0.0900907
$171$	0	0
$172$	13.7667	1.04970
$173$	−6.76573	−0.514389	−0.257194	−	0.966360i	$-0.582798\pi$
$173$	−6.76573	−0.514389	−0.257194	+	0.966360i	$0.582798\pi$
$174$	0	0
$175$	−4.28937	−0.324246
$176$	−6.14232	−0.462995
$177$	0	0
$178$	−0.979495	−0.0734163
$179$	−9.43971	−0.705557	−0.352778	−	0.935707i	$-0.614763\pi$
$179$	−9.43971	−0.705557	−0.352778	+	0.935707i	$0.614763\pi$
$180$	0	0
$181$	1.13240	0.0841709	0.0420855	−	0.999114i	$-0.486600\pi$
$181$	1.13240	0.0841709	0.0420855	+	0.999114i	$0.486600\pi$
$182$	5.21980	0.386917
$183$	0	0
$184$	−10.1454	−0.747929
$185$	5.21715	0.383572
$186$	0	0
$187$	−6.92084	−0.506102
$188$	−22.4344	−1.63620
$189$	0	0
$190$	−0.335064	−0.0243081
$191$	23.6241	1.70938	0.854689	−	0.519141i	$-0.173748\pi$
$191$	23.6241	1.70938	0.854689	+	0.519141i	$0.173748\pi$
$192$	0	0
$193$	−2.71596	−0.195499	−0.0977497	−	0.995211i	$-0.531164\pi$
$193$	−2.71596	−0.195499	−0.0977497	+	0.995211i	$0.531164\pi$
$194$	0.599223	0.0430217
$195$	0	0
$196$	−21.7656	−1.55469
$197$	14.2743	1.01700	0.508501	−	0.861061i	$-0.330200\pi$
$197$	14.2743	1.01700	0.508501	+	0.861061i	$0.330200\pi$
$198$	0	0
$199$	21.2703	1.50781	0.753905	−	0.656983i	$-0.228168\pi$
$199$	21.2703	1.50781	0.753905	+	0.656983i	$0.228168\pi$
$200$	1.17620	0.0831701
$201$	0	0
$202$	2.75807	0.194057
$203$	2.56652	0.180134
$204$	0	0
$205$	−0.756985	−0.0528701
$206$	−0.545436	−0.0380023
$207$	0	0
$208$	14.0156	0.971810
$209$	−1.97416	−0.136555
$210$	0	0
$211$	−20.1444	−1.38680	−0.693400	−	0.720553i	$-0.743888\pi$
$211$	−20.1444	−1.38680	−0.693400	+	0.720553i	$0.743888\pi$
$212$	7.59447	0.521591
$213$	0	0
$214$	1.11708	0.0763619
$215$	−7.20963	−0.491693
$216$	0	0
$217$	−6.98697	−0.474306
$218$	0.975197	0.0660487
$219$	0	0
$220$	3.38480	0.228203
$221$	15.7921	1.06229
$222$	0	0
$223$	−5.78299	−0.387258	−0.193629	−	0.981075i	$-0.562026\pi$
$223$	−5.78299	−0.387258	−0.193629	+	0.981075i	$0.562026\pi$
$224$	14.5620	0.972967
$225$	0	0
$226$	3.70035	0.246144
$227$	−15.3090	−1.01610	−0.508048	−	0.861329i	$-0.669633\pi$
$227$	−15.3090	−1.01610	−0.508048	+	0.861329i	$0.669633\pi$
$228$	0	0
$229$	2.46710	0.163030	0.0815152	−	0.996672i	$-0.474024\pi$
$229$	2.46710	0.163030	0.0815152	+	0.996672i	$0.474024\pi$
$230$	2.59507	0.171114
$231$	0	0
$232$	−0.703775	−0.0462051
$233$	−8.42853	−0.552171	−0.276086	−	0.961133i	$-0.589037\pi$
$233$	−8.42853	−0.552171	−0.276086	+	0.961133i	$0.589037\pi$
$234$	0	0
$235$	11.7489	0.766417
$236$	−26.3882	−1.71773
$237$	0	0
$238$	5.03846	0.326595
$239$	15.2495	0.986409	0.493204	−	0.869914i	$-0.335826\pi$
$239$	15.2495	0.986409	0.493204	+	0.869914i	$0.335826\pi$
$240$	0	0
$241$	−0.650803	−0.0419219	−0.0209610	−	0.999780i	$-0.506673\pi$
$241$	−0.650803	−0.0419219	−0.0209610	+	0.999780i	$0.506673\pi$
$242$	2.36409	0.151969
$243$	0	0
$244$	16.1928	1.03664
$245$	11.3987	0.728236
$246$	0	0
$247$	4.50466	0.286625
$248$	1.91592	0.121661
$249$	0	0
$250$	−0.300859	−0.0190280
$251$	−6.15859	−0.388727	−0.194364	−	0.980930i	$-0.562264\pi$
$251$	−6.15859	−0.388727	−0.194364	+	0.980930i	$0.562264\pi$
$252$	0	0
$253$	15.2899	0.961266
$254$	3.31302	0.207877
$255$	0	0
$256$	9.23998	0.577499
$257$	−7.01222	−0.437410	−0.218705	−	0.975791i	$-0.570183\pi$
$257$	−7.01222	−0.437410	−0.218705	+	0.975791i	$0.570183\pi$
$258$	0	0
$259$	−22.3783	−1.39052
$260$	−7.72349	−0.478991
$261$	0	0
$262$	5.27869	0.326119
$263$	−19.4528	−1.19951	−0.599756	−	0.800183i	$-0.704736\pi$
$263$	−19.4528	−1.19951	−0.599756	+	0.800183i	$0.704736\pi$
$264$	0	0
$265$	−3.97724	−0.244320
$266$	1.43721	0.0881211
$267$	0	0
$268$	−14.9602	−0.913837
$269$	−10.1856	−0.621026	−0.310513	−	0.950569i	$-0.600501\pi$
$269$	−10.1856	−0.621026	−0.310513	+	0.950569i	$0.600501\pi$
$270$	0	0
$271$	−6.87083	−0.417373	−0.208687	−	0.977983i	$-0.566919\pi$
$271$	−6.87083	−0.417373	−0.208687	+	0.977983i	$0.566919\pi$
$272$	13.5287	0.820299
$273$	0	0
$274$	0.107323	0.00648365
$275$	−1.77263	−0.106893
$276$	0	0
$277$	26.5635	1.59604	0.798022	−	0.602628i	$-0.205880\pi$
$277$	26.5635	1.59604	0.798022	+	0.602628i	$0.205880\pi$
$278$	0.448646	0.0269080
$279$	0	0
$280$	−5.04517	−0.301507
$281$	23.9246	1.42722	0.713611	−	0.700543i	$-0.247059\pi$
$281$	23.9246	1.42722	0.713611	+	0.700543i	$0.247059\pi$
$282$	0	0
$283$	−13.5016	−0.802588	−0.401294	−	0.915949i	$-0.631439\pi$
$283$	−13.5016	−0.802588	−0.401294	+	0.915949i	$0.631439\pi$
$284$	−9.89786	−0.587330
$285$	0	0
$286$	2.15714	0.127554
$287$	3.24699	0.191664
$288$	0	0
$289$	−1.75656	−0.103327
$290$	0.180017	0.0105710
$291$	0	0
$292$	8.35453	0.488912
$293$	5.51414	0.322140	0.161070	−	0.986943i	$-0.448506\pi$
$293$	5.51414	0.322140	0.161070	+	0.986943i	$0.448506\pi$
$294$	0	0
$295$	13.8195	0.804605
$296$	6.13643	0.356673
$297$	0	0
$298$	−5.44531	−0.315438
$299$	−34.8887	−2.01766
$300$	0	0
$301$	30.9248	1.78247
$302$	6.38118	0.367196
$303$	0	0
$304$	3.85904	0.221331
$305$	−8.48018	−0.485574
$306$	0	0
$307$	4.76721	0.272079	0.136039	−	0.990703i	$-0.456563\pi$
$307$	4.76721	0.272079	0.136039	+	0.990703i	$0.456563\pi$
$308$	−14.5187	−0.827278
$309$	0	0
$310$	−0.490070	−0.0278341
$311$	−10.8755	−0.616691	−0.308345	−	0.951274i	$-0.599775\pi$
$311$	−10.8755	−0.616691	−0.308345	+	0.951274i	$0.599775\pi$
$312$	0	0
$313$	−12.4182	−0.701916	−0.350958	−	0.936391i	$-0.614144\pi$
$313$	−12.4182	−0.701916	−0.350958	+	0.936391i	$0.614144\pi$
$314$	1.64744	0.0929703
$315$	0	0
$316$	24.6146	1.38468
$317$	11.0331	0.619678	0.309839	−	0.950789i	$-0.399725\pi$
$317$	11.0331	0.619678	0.309839	+	0.950789i	$0.399725\pi$
$318$	0	0
$319$	1.06064	0.0593845
$320$	−5.90880	−0.330312
$321$	0	0
$322$	−11.1312	−0.620319
$323$	4.34816	0.241938
$324$	0	0
$325$	4.04481	0.224365
$326$	5.41151	0.299716
$327$	0	0
$328$	−0.890368	−0.0491623
$329$	−50.3956	−2.77840
$330$	0	0
$331$	2.02600	0.111359	0.0556796	−	0.998449i	$-0.482267\pi$
$331$	2.02600	0.111359	0.0556796	+	0.998449i	$0.482267\pi$
$332$	15.0987	0.828651
$333$	0	0
$334$	5.99845	0.328221
$335$	7.83466	0.428053
$336$	0	0
$337$	−15.3801	−0.837806	−0.418903	−	0.908031i	$-0.637585\pi$
$337$	−15.3801	−0.837806	−0.418903	+	0.908031i	$0.637585\pi$
$338$	−1.01102	−0.0549923
$339$	0	0
$340$	−7.45517	−0.404313
$341$	−2.88744	−0.156364
$342$	0	0
$343$	−18.8676	−1.01875
$344$	−8.48000	−0.457211
$345$	0	0
$346$	2.03553	0.109431
$347$	−1.15918	−0.0622280	−0.0311140	−	0.999516i	$-0.509906\pi$
$347$	−1.15918	−0.0622280	−0.0311140	+	0.999516i	$0.509906\pi$
$348$	0	0
$349$	−5.16380	−0.276412	−0.138206	−	0.990404i	$-0.544134\pi$
$349$	−5.16380	−0.276412	−0.138206	+	0.990404i	$0.544134\pi$
$350$	1.29050	0.0689799
$351$	0	0
$352$	6.01791	0.320756
$353$	12.1424	0.646277	0.323138	−	0.946352i	$-0.395262\pi$
$353$	12.1424	0.646277	0.323138	+	0.946352i	$0.395262\pi$
$354$	0	0
$355$	5.18353	0.275113
$356$	−6.21663	−0.329481
$357$	0	0
$358$	2.84002	0.150100
$359$	−8.89415	−0.469415	−0.234708	−	0.972066i	$-0.575413\pi$
$359$	−8.89415	−0.469415	−0.234708	+	0.972066i	$0.575413\pi$
$360$	0	0
$361$	−17.7597	−0.934721
$362$	−0.340694	−0.0179065
$363$	0	0
$364$	33.1289	1.73643
$365$	−4.37528	−0.229013
$366$	0	0
$367$	−27.6276	−1.44215	−0.721075	−	0.692857i	$-0.756352\pi$
$367$	−27.6276	−1.44215	−0.721075	+	0.692857i	$0.756352\pi$
$368$	−29.8884	−1.55804
$369$	0	0
$370$	−1.56963	−0.0816010
$371$	17.0598	0.885703
$372$	0	0
$373$	−30.8466	−1.59718	−0.798589	−	0.601877i	$-0.794420\pi$
$373$	−30.8466	−1.59718	−0.798589	+	0.601877i	$0.794420\pi$
$374$	2.08220	0.107668
$375$	0	0
$376$	13.8192	0.712668
$377$	−2.42019	−0.124646
$378$	0	0
$379$	−21.0219	−1.07982	−0.539911	−	0.841722i	$-0.681542\pi$
$379$	−21.0219	−1.07982	−0.539911	+	0.841722i	$0.681542\pi$
$380$	−2.12657	−0.109091
$381$	0	0
$382$	−7.10751	−0.363652
$383$	21.5253	1.09989	0.549945	−	0.835201i	$-0.314649\pi$
$383$	21.5253	1.09989	0.549945	+	0.835201i	$0.314649\pi$
$384$	0	0
$385$	7.60345	0.387508
$386$	0.817122	0.0415904
$387$	0	0
$388$	3.80313	0.193075
$389$	−13.4127	−0.680050	−0.340025	−	0.940416i	$-0.610435\pi$
$389$	−13.4127	−0.680050	−0.340025	+	0.940416i	$0.610435\pi$
$390$	0	0
$391$	−33.6766	−1.70310
$392$	13.4072	0.677165
$393$	0	0
$394$	−4.29455	−0.216357
$395$	−12.8907	−0.648602
$396$	0	0
$397$	−17.0060	−0.853506	−0.426753	−	0.904368i	$-0.640343\pi$
$397$	−17.0060	−0.853506	−0.426753	+	0.904368i	$0.640343\pi$
$398$	−6.39935	−0.320771
$399$	0	0
$400$	3.46510	0.173255
$401$	−6.23973	−0.311597	−0.155799	−	0.987789i	$-0.549795\pi$
$401$	−6.23973	−0.311597	−0.155799	+	0.987789i	$0.549795\pi$
$402$	0	0
$403$	6.58860	0.328201
$404$	17.5049	0.870899
$405$	0	0
$406$	−0.772161	−0.0383217
$407$	−9.24805	−0.458409
$408$	0	0
$409$	9.60027	0.474703	0.237351	−	0.971424i	$-0.423721\pi$
$409$	9.60027	0.474703	0.237351	+	0.971424i	$0.423721\pi$
$410$	0.227746	0.0112476
$411$	0	0
$412$	−3.46176	−0.170548
$413$	−59.2771	−2.91684
$414$	0	0
$415$	−7.90723	−0.388151
$416$	−13.7318	−0.673255
$417$	0	0
$418$	0.593943	0.0290507
$419$	−15.0053	−0.733058	−0.366529	−	0.930407i	$-0.619454\pi$
$419$	−15.0053	−0.733058	−0.366529	+	0.930407i	$0.619454\pi$
$420$	0	0
$421$	37.0390	1.80517	0.902584	−	0.430513i	$-0.141667\pi$
$421$	37.0390	1.80517	0.902584	+	0.430513i	$0.141667\pi$
$422$	6.06064	0.295027
$423$	0	0
$424$	−4.67804	−0.227186
$425$	3.90428	0.189386
$426$	0	0
$427$	36.3746	1.76029
$428$	7.08985	0.342701
$429$	0	0
$430$	2.16908	0.104602
$431$	14.3326	0.690378	0.345189	−	0.938533i	$-0.387815\pi$
$431$	14.3326	0.690378	0.345189	+	0.938533i	$0.387815\pi$
$432$	0	0
$433$	14.4669	0.695233	0.347616	−	0.937637i	$-0.386991\pi$
$433$	14.4669	0.695233	0.347616	+	0.937637i	$0.386991\pi$
$434$	2.10209	0.100904
$435$	0	0
$436$	6.18936	0.296416
$437$	−9.60619	−0.459526
$438$	0	0
$439$	−14.9516	−0.713601	−0.356801	−	0.934181i	$-0.616132\pi$
$439$	−14.9516	−0.713601	−0.356801	+	0.934181i	$0.616132\pi$
$440$	−2.08497	−0.0993970
$441$	0	0
$442$	−4.75119	−0.225991
$443$	34.6483	1.64619	0.823094	−	0.567905i	$-0.192246\pi$
$443$	34.6483	1.64619	0.823094	+	0.567905i	$0.192246\pi$
$444$	0	0
$445$	3.25566	0.154333
$446$	1.73986	0.0823850
$447$	0	0
$448$	25.3450	1.19744
$449$	25.5553	1.20603	0.603015	−	0.797730i	$-0.293966\pi$
$449$	25.5553	1.20603	0.603015	+	0.797730i	$0.293966\pi$
$450$	0	0
$451$	1.34185	0.0631853
$452$	23.4853	1.10465
$453$	0	0
$454$	4.60586	0.216164
$455$	−17.3497	−0.813365
$456$	0	0
$457$	−19.2167	−0.898919	−0.449459	−	0.893301i	$-0.648383\pi$
$457$	−19.2167	−0.898919	−0.449459	+	0.893301i	$0.648383\pi$
$458$	−0.742248	−0.0346830
$459$	0	0
$460$	16.4703	0.767934
$461$	11.2073	0.521976	0.260988	−	0.965342i	$-0.415952\pi$
$461$	11.2073	0.521976	0.260988	+	0.965342i	$0.415952\pi$
$462$	0	0
$463$	−12.7319	−0.591703	−0.295852	−	0.955234i	$-0.595603\pi$
$463$	−12.7319	−0.591703	−0.295852	+	0.955234i	$0.595603\pi$
$464$	−2.07332	−0.0962516
$465$	0	0
$466$	2.53580	0.117469
$467$	32.9377	1.52417	0.762087	−	0.647475i	$-0.224175\pi$
$467$	32.9377	1.52417	0.762087	+	0.647475i	$0.224175\pi$
$468$	0	0
$469$	−33.6057	−1.55177
$470$	−3.53478	−0.163047
$471$	0	0
$472$	16.2546	0.748179
$473$	12.7800	0.587624
$474$	0	0
$475$	1.11369	0.0510996
$476$	31.9780	1.46571
$477$	0	0
$478$	−4.58795	−0.209848
$479$	22.1599	1.01251	0.506255	−	0.862384i	$-0.331029\pi$
$479$	22.1599	1.01251	0.506255	+	0.862384i	$0.331029\pi$
$480$	0	0
$481$	21.1023	0.962184
$482$	0.195800	0.00891845
$483$	0	0
$484$	15.0043	0.682015
$485$	−1.99171	−0.0904388
$486$	0	0
$487$	−15.7072	−0.711761	−0.355881	−	0.934531i	$-0.615819\pi$
$487$	−15.7072	−0.711761	−0.355881	+	0.934531i	$0.615819\pi$
$488$	−9.97442	−0.451521
$489$	0	0
$490$	−3.42940	−0.154924
$491$	26.5847	1.19975	0.599874	−	0.800094i	$-0.295217\pi$
$491$	26.5847	1.19975	0.599874	+	0.800094i	$0.295217\pi$
$492$	0	0
$493$	−2.33611	−0.105213
$494$	−1.35527	−0.0609764
$495$	0	0
$496$	5.64431	0.253437
$497$	−22.2341	−0.997334
$498$	0	0
$499$	−2.11487	−0.0946747	−0.0473374	−	0.998879i	$-0.515074\pi$
$499$	−2.11487	−0.0946747	−0.0473374	+	0.998879i	$0.515074\pi$
$500$	−1.90948	−0.0853947
$501$	0	0
$502$	1.85287	0.0826976
$503$	−9.60052	−0.428066	−0.214033	−	0.976826i	$-0.568660\pi$
$503$	−9.60052	−0.428066	−0.214033	+	0.976826i	$0.568660\pi$
$504$	0	0
$505$	−9.16732	−0.407941
$506$	−4.60010	−0.204499
$507$	0	0
$508$	21.0270	0.932921
$509$	3.94762	0.174975	0.0874876	−	0.996166i	$-0.472116\pi$
$509$	3.94762	0.174975	0.0874876	+	0.996166i	$0.472116\pi$
$510$	0	0
$511$	18.7672	0.830212
$512$	−19.9150	−0.880128
$513$	0	0
$514$	2.10969	0.0930544
$515$	1.81293	0.0798871
$516$	0	0
$517$	−20.8265	−0.915948
$518$	6.73271	0.295818
$519$	0	0
$520$	4.75751	0.208631
$521$	25.3238	1.10946	0.554729	−	0.832031i	$-0.312822\pi$
$521$	25.3238	1.10946	0.554729	+	0.832031i	$0.312822\pi$
$522$	0	0
$523$	−15.5210	−0.678687	−0.339344	−	0.940662i	$-0.610205\pi$
$523$	−15.5210	−0.678687	−0.339344	+	0.940662i	$0.610205\pi$
$524$	33.5027	1.46357
$525$	0	0
$526$	5.85256	0.255184
$527$	6.35970	0.277033
$528$	0	0
$529$	51.4001	2.23479
$530$	1.19659	0.0519765
$531$	0	0
$532$	9.12166	0.395474
$533$	−3.06185	−0.132624
$534$	0	0
$535$	−3.71297	−0.160525
$536$	9.21515	0.398034
$537$	0	0
$538$	3.06443	0.132117
$539$	−20.2056	−0.870317
$540$	0	0
$541$	11.7630	0.505730	0.252865	−	0.967502i	$-0.418627\pi$
$541$	11.7630	0.505730	0.252865	+	0.967502i	$0.418627\pi$
$542$	2.06715	0.0887918
$543$	0	0
$544$	−13.2547	−0.568291
$545$	−3.24138	−0.138845
$546$	0	0
$547$	18.0335	0.771059	0.385529	−	0.922696i	$-0.374019\pi$
$547$	18.0335	0.771059	0.385529	+	0.922696i	$0.374019\pi$
$548$	0.681158	0.0290976
$549$	0	0
$550$	0.533311	0.0227404
$551$	−0.666371	−0.0283883
$552$	0	0
$553$	55.2930	2.35130
$554$	−7.99186	−0.339542
$555$	0	0
$556$	2.84745	0.120759
$557$	−9.90891	−0.419854	−0.209927	−	0.977717i	$-0.567323\pi$
$557$	−9.90891	−0.419854	−0.209927	+	0.977717i	$0.567323\pi$
$558$	0	0
$559$	−29.1616	−1.23340
$560$	−14.8631	−0.628080
$561$	0	0
$562$	−7.19793	−0.303626
$563$	1.16407	0.0490598	0.0245299	−	0.999699i	$-0.492191\pi$
$563$	1.16407	0.0490598	0.0245299	+	0.999699i	$0.492191\pi$
$564$	0	0
$565$	−12.2993	−0.517435
$566$	4.06208	0.170742
$567$	0	0
$568$	6.09688	0.255820
$569$	5.12125	0.214694	0.107347	−	0.994222i	$-0.465764\pi$
$569$	5.12125	0.214694	0.107347	+	0.994222i	$0.465764\pi$
$570$	0	0
$571$	−14.8341	−0.620786	−0.310393	−	0.950608i	$-0.600461\pi$
$571$	−14.8341	−0.620786	−0.310393	+	0.950608i	$0.600461\pi$
$572$	13.6909	0.572444
$573$	0	0
$574$	−0.976885	−0.0407744
$575$	−8.62555	−0.359710
$576$	0	0
$577$	−31.5634	−1.31400	−0.657001	−	0.753890i	$-0.728175\pi$
$577$	−31.5634	−1.31400	−0.657001	+	0.753890i	$0.728175\pi$
$578$	0.528478	0.0219818
$579$	0	0
$580$	1.14253	0.0474410
$581$	33.9170	1.40712
$582$	0	0
$583$	7.05016	0.291988
$584$	−5.14622	−0.212952
$585$	0	0
$586$	−1.65898	−0.0685318
$587$	−17.0065	−0.701935	−0.350967	−	0.936388i	$-0.614147\pi$
$587$	−17.0065	−0.701935	−0.350967	+	0.936388i	$0.614147\pi$
$588$	0	0
$589$	1.81409	0.0747485
$590$	−4.15774	−0.171171
$591$	0	0
$592$	18.0779	0.742998
$593$	−42.7865	−1.75703	−0.878517	−	0.477712i	$-0.841466\pi$
$593$	−42.7865	−1.75703	−0.878517	+	0.477712i	$0.841466\pi$
$594$	0	0
$595$	−16.7469	−0.686557
$596$	−34.5601	−1.41564
$597$	0	0
$598$	10.4966	0.429237
$599$	1.93853	0.0792062	0.0396031	−	0.999215i	$-0.487391\pi$
$599$	1.93853	0.0792062	0.0396031	+	0.999215i	$0.487391\pi$
$600$	0	0
$601$	−19.7131	−0.804113	−0.402056	−	0.915615i	$-0.631704\pi$
$601$	−19.7131	−0.804113	−0.402056	+	0.915615i	$0.631704\pi$
$602$	−9.30400	−0.379203
$603$	0	0
$604$	40.4999	1.64792
$605$	−7.85780	−0.319465
$606$	0	0
$607$	−11.5487	−0.468746	−0.234373	−	0.972147i	$-0.575304\pi$
$607$	−11.5487	−0.468746	−0.234373	+	0.972147i	$0.575304\pi$
$608$	−3.78088	−0.153335
$609$	0	0
$610$	2.55134	0.103301
$611$	47.5222	1.92254
$612$	0	0
$613$	11.9028	0.480748	0.240374	−	0.970680i	$-0.422730\pi$
$613$	11.9028	0.480748	0.240374	+	0.970680i	$0.422730\pi$
$614$	−1.43426	−0.0578819
$615$	0	0
$616$	8.94320	0.360332
$617$	−28.9392	−1.16505	−0.582525	−	0.812813i	$-0.697935\pi$
$617$	−28.9392	−1.16505	−0.582525	+	0.812813i	$0.697935\pi$
$618$	0	0
$619$	−5.87633	−0.236190	−0.118095	−	0.993002i	$-0.537679\pi$
$619$	−5.87633	−0.236190	−0.118095	+	0.993002i	$0.537679\pi$
$620$	−3.11037	−0.124915
$621$	0	0
$622$	3.27198	0.131194
$623$	−13.9647	−0.559485
$624$	0	0
$625$	1.00000	0.0400000
$626$	3.73612	0.149325
$627$	0	0
$628$	10.4559	0.417236
$629$	20.3692	0.812174
$630$	0	0
$631$	20.2847	0.807522	0.403761	−	0.914864i	$-0.367703\pi$
$631$	20.2847	0.807522	0.403761	+	0.914864i	$0.367703\pi$
$632$	−15.1621	−0.603116
$633$	0	0
$634$	−3.31939	−0.131830
$635$	−11.0119	−0.436992
$636$	0	0
$637$	46.1055	1.82677
$638$	−0.319104	−0.0126334
$639$	0	0
$640$	8.56754	0.338662
$641$	−9.62261	−0.380070	−0.190035	−	0.981777i	$-0.560860\pi$
$641$	−9.62261	−0.380070	−0.190035	+	0.981777i	$0.560860\pi$
$642$	0	0
$643$	−6.53427	−0.257686	−0.128843	−	0.991665i	$-0.541126\pi$
$643$	−6.53427	−0.257686	−0.128843	+	0.991665i	$0.541126\pi$
$644$	−70.6474	−2.78390
$645$	0	0
$646$	−1.30818	−0.0514698
$647$	4.88147	0.191910	0.0959551	−	0.995386i	$-0.469409\pi$
$647$	4.88147	0.191910	0.0959551	+	0.995386i	$0.469409\pi$
$648$	0	0
$649$	−24.4969	−0.961587
$650$	−1.21692	−0.0477314
$651$	0	0
$652$	34.3456	1.34508
$653$	−6.27970	−0.245744	−0.122872	−	0.992423i	$-0.539210\pi$
$653$	−6.27970	−0.245744	−0.122872	+	0.992423i	$0.539210\pi$
$654$	0	0
$655$	−17.5454	−0.685556
$656$	−2.62302	−0.102412
$657$	0	0
$658$	15.1620	0.591075
$659$	−7.21880	−0.281205	−0.140602	−	0.990066i	$-0.544904\pi$
$659$	−7.21880	−0.281205	−0.140602	+	0.990066i	$0.544904\pi$
$660$	0	0
$661$	43.6944	1.69951	0.849757	−	0.527175i	$-0.176749\pi$
$661$	43.6944	1.69951	0.849757	+	0.527175i	$0.176749\pi$
$662$	−0.609541	−0.0236905
$663$	0	0
$664$	−9.30052	−0.360930
$665$	−4.77703	−0.185245
$666$	0	0
$667$	5.16105	0.199837
$668$	38.0708	1.47300
$669$	0	0
$670$	−2.35713	−0.0910638
$671$	15.0322	0.580311
$672$	0	0
$673$	35.4872	1.36793	0.683965	−	0.729514i	$-0.260254\pi$
$673$	35.4872	1.36793	0.683965	+	0.729514i	$0.260254\pi$
$674$	4.62723	0.178234
$675$	0	0
$676$	−6.41672	−0.246797
$677$	−14.3532	−0.551637	−0.275818	−	0.961210i	$-0.588949\pi$
$677$	−14.3532	−0.551637	−0.275818	+	0.961210i	$0.588949\pi$
$678$	0	0
$679$	8.54317	0.327857
$680$	4.59223	0.176104
$681$	0	0
$682$	0.868712	0.0332647
$683$	−14.5737	−0.557646	−0.278823	−	0.960343i	$-0.589944\pi$
$683$	−14.5737	−0.557646	−0.278823	+	0.960343i	$0.589944\pi$
$684$	0	0
$685$	−0.356723	−0.0136297
$686$	5.67649	0.216729
$687$	0	0
$688$	−24.9821	−0.952433
$689$	−16.0872	−0.612872
$690$	0	0
$691$	−27.3677	−1.04112	−0.520558	−	0.853826i	$-0.674276\pi$
$691$	−27.3677	−1.04112	−0.520558	+	0.853826i	$0.674276\pi$
$692$	12.9190	0.491108
$693$	0	0
$694$	0.348750	0.0132384
$695$	−1.49122	−0.0565650
$696$	0	0
$697$	−2.95548	−0.111947
$698$	1.55357	0.0588037
$699$	0	0
$700$	8.19048	0.309571
$701$	30.5667	1.15449	0.577243	−	0.816572i	$-0.304128\pi$
$701$	30.5667	1.15449	0.577243	+	0.816572i	$0.304128\pi$
$702$	0	0
$703$	5.81029	0.219139
$704$	10.4741	0.394757
$705$	0	0
$706$	−3.65316	−0.137489
$707$	39.3220	1.47886
$708$	0	0
$709$	14.8107	0.556227	0.278113	−	0.960548i	$-0.410291\pi$
$709$	14.8107	0.556227	0.278113	+	0.960548i	$0.410291\pi$
$710$	−1.55951	−0.0585274
$711$	0	0
$712$	3.82932	0.143510
$713$	−14.0502	−0.526184
$714$	0	0
$715$	−7.16993	−0.268140
$716$	18.0250	0.673625
$717$	0	0
$718$	2.67588	0.0998631
$719$	−5.24852	−0.195737	−0.0978684	−	0.995199i	$-0.531202\pi$
$719$	−5.24852	−0.195737	−0.0978684	+	0.995199i	$0.531202\pi$
$720$	0	0
$721$	−7.77632	−0.289605
$722$	5.34316	0.198852
$723$	0	0
$724$	−2.16231	−0.0803615
$725$	−0.598345	−0.0222220
$726$	0	0
$727$	47.4670	1.76045	0.880226	−	0.474555i	$-0.157391\pi$
$727$	47.4670	1.76045	0.880226	+	0.474555i	$0.157391\pi$
$728$	−20.4067	−0.756324
$729$	0	0
$730$	1.31634	0.0487200
$731$	−28.1485	−1.04111
$732$	0	0
$733$	−38.7927	−1.43284	−0.716421	−	0.697668i	$-0.754221\pi$
$733$	−38.7927	−1.43284	−0.716421	+	0.697668i	$0.754221\pi$
$734$	8.31202	0.306802
$735$	0	0
$736$	29.2830	1.07938
$737$	−13.8879	−0.511568
$738$	0	0
$739$	−15.4247	−0.567407	−0.283704	−	0.958912i	$-0.591563\pi$
$739$	−15.4247	−0.567407	−0.283704	+	0.958912i	$0.591563\pi$
$740$	−9.96206	−0.366213
$741$	0	0
$742$	−5.13261	−0.188424
$743$	41.1249	1.50873	0.754364	−	0.656457i	$-0.227946\pi$
$743$	41.1249	1.50873	0.754364	+	0.656457i	$0.227946\pi$
$744$	0	0
$745$	18.0992	0.663103
$746$	9.28048	0.339783
$747$	0	0
$748$	13.2152	0.483197
$749$	15.9263	0.581934
$750$	0	0
$751$	−51.4762	−1.87839	−0.939196	−	0.343380i	$-0.888428\pi$
$751$	−51.4762	−1.87839	−0.939196	+	0.343380i	$0.888428\pi$
$752$	40.7112	1.48459
$753$	0	0
$754$	0.728135	0.0265171
$755$	−21.2099	−0.771906
$756$	0	0
$757$	36.7855	1.33699	0.668497	−	0.743715i	$-0.266938\pi$
$757$	36.7855	1.33699	0.668497	+	0.743715i	$0.266938\pi$
$758$	6.32463	0.229721
$759$	0	0
$760$	1.30993	0.0475160
$761$	23.7062	0.859351	0.429675	−	0.902983i	$-0.358628\pi$
$761$	23.7062	0.859351	0.429675	+	0.902983i	$0.358628\pi$
$762$	0	0
$763$	13.9035	0.503339
$764$	−45.1098	−1.63201
$765$	0	0
$766$	−6.47607	−0.233990
$767$	55.8974	2.01834
$768$	0	0
$769$	−11.0301	−0.397756	−0.198878	−	0.980024i	$-0.563730\pi$
$769$	−11.0301	−0.397756	−0.198878	+	0.980024i	$0.563730\pi$
$770$	−2.28757	−0.0824382
$771$	0	0
$772$	5.18609	0.186651
$773$	−44.2598	−1.59192	−0.795958	−	0.605352i	$-0.793032\pi$
$773$	−44.2598	−1.59192	−0.795958	+	0.605352i	$0.793032\pi$
$774$	0	0
$775$	1.62890	0.0585120
$776$	−2.34265	−0.0840964
$777$	0	0
$778$	4.03532	0.144673
$779$	−0.843046	−0.0302053
$780$	0	0
$781$	−9.18846	−0.328789
$782$	10.1319	0.362316
$783$	0	0
$784$	39.4976	1.41063
$785$	−5.47578	−0.195439
$786$	0	0
$787$	−41.1437	−1.46662	−0.733308	−	0.679897i	$-0.762024\pi$
$787$	−41.1437	−1.46662	−0.733308	+	0.679897i	$0.762024\pi$
$788$	−27.2566	−0.970975
$789$	0	0
$790$	3.87828	0.137983
$791$	52.7562	1.87579
$792$	0	0
$793$	−34.3007	−1.21805
$794$	5.11640	0.181574
$795$	0	0
$796$	−40.6153	−1.43957
$797$	−23.3683	−0.827748	−0.413874	−	0.910334i	$-0.635825\pi$
$797$	−23.3683	−0.827748	−0.413874	+	0.910334i	$0.635825\pi$
$798$	0	0
$799$	45.8712	1.62281
$800$	−3.39491	−0.120028
$801$	0	0
$802$	1.87728	0.0662890
$803$	7.75574	0.273694
$804$	0	0
$805$	36.9982	1.30401
$806$	−1.98224	−0.0698214
$807$	0	0
$808$	−10.7826	−0.379332
$809$	16.4420	0.578070	0.289035	−	0.957319i	$-0.406666\pi$
$809$	16.4420	0.578070	0.289035	+	0.957319i	$0.406666\pi$
$810$	0	0
$811$	49.2346	1.72886	0.864430	−	0.502753i	$-0.167679\pi$
$811$	49.2346	1.72886	0.864430	+	0.502753i	$0.167679\pi$
$812$	−4.90073	−0.171982
$813$	0	0
$814$	2.78236	0.0975217
$815$	−17.9869	−0.630052
$816$	0	0
$817$	−8.02930	−0.280910
$818$	−2.88833	−0.100988
$819$	0	0
$820$	1.44545	0.0504773
$821$	2.76238	0.0964077	0.0482039	−	0.998838i	$-0.484650\pi$
$821$	2.76238	0.0964077	0.0482039	+	0.998838i	$0.484650\pi$
$822$	0	0
$823$	−32.9929	−1.15006	−0.575029	−	0.818133i	$-0.695009\pi$
$823$	−32.9929	−1.15006	−0.575029	+	0.818133i	$0.695009\pi$
$824$	2.13237	0.0742847
$825$	0	0
$826$	17.8341	0.620526
$827$	54.7826	1.90498	0.952489	−	0.304574i	$-0.0985142\pi$
$827$	54.7826	1.90498	0.952489	+	0.304574i	$0.0985142\pi$
$828$	0	0
$829$	51.0692	1.77370	0.886852	−	0.462053i	$-0.152887\pi$
$829$	51.0692	1.77370	0.886852	+	0.462053i	$0.152887\pi$
$830$	2.37896	0.0825750
$831$	0	0
$832$	−23.9000	−0.828582
$833$	44.5037	1.54196
$834$	0	0
$835$	−19.9377	−0.689974
$836$	3.76962	0.130375
$837$	0	0
$838$	4.51449	0.155950
$839$	41.2173	1.42298	0.711490	−	0.702697i	$-0.248021\pi$
$839$	41.2173	1.42298	0.711490	+	0.702697i	$0.248021\pi$
$840$	0	0
$841$	−28.6420	−0.987655
$842$	−11.1435	−0.384031
$843$	0	0
$844$	38.4655	1.32404
$845$	3.36045	0.115603
$846$	0	0
$847$	33.7050	1.15812
$848$	−13.7815	−0.473259
$849$	0	0
$850$	−1.17464	−0.0402898
$851$	−45.0008	−1.54261
$852$	0	0
$853$	20.6422	0.706776	0.353388	−	0.935477i	$-0.385029\pi$
$853$	20.6422	0.706776	0.353388	+	0.935477i	$0.385029\pi$
$854$	−10.9436	−0.374483
$855$	0	0
$856$	−4.36720	−0.149268
$857$	−36.9400	−1.26185	−0.630923	−	0.775845i	$-0.717324\pi$
$857$	−36.9400	−1.26185	−0.630923	+	0.775845i	$0.717324\pi$
$858$	0	0
$859$	23.6981	0.808570	0.404285	−	0.914633i	$-0.367520\pi$
$859$	23.6981	0.808570	0.404285	+	0.914633i	$0.367520\pi$
$860$	13.7667	0.469440
$861$	0	0
$862$	−4.31210	−0.146871
$863$	−36.6147	−1.24638	−0.623189	−	0.782072i	$-0.714163\pi$
$863$	−36.6147	−1.24638	−0.623189	+	0.782072i	$0.714163\pi$
$864$	0	0
$865$	−6.76573	−0.230042
$866$	−4.35248	−0.147903
$867$	0	0
$868$	13.3415	0.452840
$869$	22.8504	0.775147
$870$	0	0
$871$	31.6897	1.07376
$872$	−3.81252	−0.129108
$873$	0	0
$874$	2.89011	0.0977594
$875$	−4.28937	−0.145007
$876$	0	0
$877$	24.2122	0.817587	0.408793	−	0.912627i	$-0.365950\pi$
$877$	24.2122	0.817587	0.408793	+	0.912627i	$0.365950\pi$
$878$	4.49832	0.151811
$879$	0	0
$880$	−6.14232	−0.207058
$881$	−26.9469	−0.907863	−0.453931	−	0.891037i	$-0.649979\pi$
$881$	−26.9469	−0.907863	−0.453931	+	0.891037i	$0.649979\pi$
$882$	0	0
$883$	0.989181	0.0332886	0.0166443	−	0.999861i	$-0.494702\pi$
$883$	0.989181	0.0332886	0.0166443	+	0.999861i	$0.494702\pi$
$884$	−30.1547	−1.01421
$885$	0	0
$886$	−10.4242	−0.350209
$887$	−24.8187	−0.833331	−0.416665	−	0.909060i	$-0.636801\pi$
$887$	−24.8187	−0.833331	−0.416665	+	0.909060i	$0.636801\pi$
$888$	0	0
$889$	47.2339	1.58418
$890$	−0.979495	−0.0328328
$891$	0	0
$892$	11.0425	0.369731
$893$	13.0847	0.437862
$894$	0	0
$895$	−9.43971	−0.315535
$896$	−36.7493	−1.22771
$897$	0	0
$898$	−7.68854	−0.256570
$899$	−0.974646	−0.0325063
$900$	0	0
$901$	−15.5283	−0.517322
$902$	−0.403708	−0.0134420
$903$	0	0
$904$	−14.4665	−0.481147
$905$	1.13240	0.0376424
$906$	0	0
$907$	3.27950	0.108894	0.0544470	−	0.998517i	$-0.482660\pi$
$907$	3.27950	0.108894	0.0544470	+	0.998517i	$0.482660\pi$
$908$	29.2324	0.970110
$909$	0	0
$910$	5.21980	0.173035
$911$	39.4133	1.30582	0.652910	−	0.757435i	$-0.273548\pi$
$911$	39.4133	1.30582	0.652910	+	0.757435i	$0.273548\pi$
$912$	0	0
$913$	14.0166	0.463881
$914$	5.78151	0.191236
$915$	0	0
$916$	−4.71088	−0.155652
$917$	75.2587	2.48526
$918$	0	0
$919$	−9.02247	−0.297624	−0.148812	−	0.988866i	$-0.547545\pi$
$919$	−9.02247	−0.297624	−0.148812	+	0.988866i	$0.547545\pi$
$920$	−10.1454	−0.334484
$921$	0	0
$922$	−3.37182	−0.111045
$923$	20.9664	0.690116
$924$	0	0
$925$	5.21715	0.171539
$926$	3.83052	0.125879
$927$	0	0
$928$	2.03133	0.0666816
$929$	24.9112	0.817311	0.408655	−	0.912689i	$-0.365998\pi$
$929$	24.9112	0.817311	0.408655	+	0.912689i	$0.365998\pi$
$930$	0	0
$931$	12.6946	0.416049
$932$	16.0941	0.527181
$933$	0	0
$934$	−9.90959	−0.324252
$935$	−6.92084	−0.226336
$936$	0	0
$937$	−25.4029	−0.829876	−0.414938	−	0.909850i	$-0.636197\pi$
$937$	−25.4029	−0.829876	−0.414938	+	0.909850i	$0.636197\pi$
$938$	10.1106	0.330122
$939$	0	0
$940$	−22.4344	−0.731730
$941$	−24.2416	−0.790252	−0.395126	−	0.918627i	$-0.629299\pi$
$941$	−24.2416	−0.790252	−0.395126	+	0.918627i	$0.629299\pi$
$942$	0	0
$943$	6.52941	0.212627
$944$	47.8861	1.55856
$945$	0	0
$946$	−3.84497	−0.125011
$947$	−10.9166	−0.354741	−0.177371	−	0.984144i	$-0.556759\pi$
$947$	−10.9166	−0.354741	−0.177371	+	0.984144i	$0.556759\pi$
$948$	0	0
$949$	−17.6972	−0.574474
$950$	−0.335064	−0.0108709
$951$	0	0
$952$	−19.6978	−0.638409
$953$	14.5735	0.472081	0.236040	−	0.971743i	$-0.424150\pi$
$953$	14.5735	0.472081	0.236040	+	0.971743i	$0.424150\pi$
$954$	0	0
$955$	23.6241	0.764457
$956$	−29.1187	−0.941766
$957$	0	0
$958$	−6.66700	−0.215401
$959$	1.53012	0.0494101
$960$	0	0
$961$	−28.3467	−0.914409
$962$	−6.34883	−0.204695
$963$	0	0
$964$	1.24270	0.0400246
$965$	−2.71596	−0.0874300
$966$	0	0
$967$	−47.9194	−1.54098	−0.770492	−	0.637450i	$-0.779989\pi$
$967$	−47.9194	−1.54098	−0.770492	+	0.637450i	$0.779989\pi$
$968$	−9.24237	−0.297061
$969$	0	0
$970$	0.599223	0.0192399
$971$	−10.0839	−0.323608	−0.161804	−	0.986823i	$-0.551731\pi$
$971$	−10.0839	−0.323608	−0.161804	+	0.986823i	$0.551731\pi$
$972$	0	0
$973$	6.39638	0.205058
$974$	4.72565	0.151420
$975$	0	0
$976$	−29.3846	−0.940579
$977$	−32.2259	−1.03100	−0.515499	−	0.856890i	$-0.672394\pi$
$977$	−32.2259	−1.03100	−0.515499	+	0.856890i	$0.672394\pi$
$978$	0	0
$979$	−5.77107	−0.184444
$980$	−21.7656	−0.695277
$981$	0	0
$982$	−7.99823	−0.255234
$983$	−2.53026	−0.0807029	−0.0403514	−	0.999186i	$-0.512848\pi$
$983$	−2.53026	−0.0807029	−0.0403514	+	0.999186i	$0.512848\pi$
$984$	0	0
$985$	14.2743	0.454817
$986$	0.702839	0.0223830
$987$	0	0
$988$	−8.60158	−0.273653
$989$	62.1870	1.97743
$990$	0	0
$991$	−50.6978	−1.61047	−0.805234	−	0.592957i	$-0.797960\pi$
$991$	−50.6978	−1.61047	−0.805234	+	0.592957i	$0.797960\pi$
$992$	−5.52999	−0.175577
$993$	0	0
$994$	6.68932	0.212172
$995$	21.2703	0.674313
$996$	0	0
$997$	13.9724	0.442509	0.221254	−	0.975216i	$-0.428985\pi$
$997$	13.9724	0.442509	0.221254	+	0.975216i	$0.428985\pi$
$998$	0.636279	0.0201411
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3645.2.a.g.1.7		15
3.2	odd	2		3645.2.a.h.1.9		15
27.2	odd	18		135.2.k.a.31.3	✓	30
27.13	even	9		405.2.k.a.181.3		30
27.14	odd	18		135.2.k.a.61.3	yes	30
27.25	even	9		405.2.k.a.226.3		30
135.2	even	36		675.2.u.c.274.6		60
135.14	odd	18		675.2.l.d.601.3		30
135.29	odd	18		675.2.l.d.301.3		30
135.68	even	36		675.2.u.c.574.6		60
135.83	even	36		675.2.u.c.274.5		60
135.122	even	36		675.2.u.c.574.5		60

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.2.k.a.31.3	✓	30	27.2	odd	18
135.2.k.a.61.3	yes	30	27.14	odd	18
405.2.k.a.181.3		30	27.13	even	9
405.2.k.a.226.3		30	27.25	even	9
675.2.l.d.301.3		30	135.29	odd	18
675.2.l.d.601.3		30	135.14	odd	18
675.2.u.c.274.5		60	135.83	even	36
675.2.u.c.274.6		60	135.2	even	36
675.2.u.c.574.5		60	135.122	even	36
675.2.u.c.574.6		60	135.68	even	36
3645.2.a.g.1.7		15	1.1	even	1	trivial
3645.2.a.h.1.9		15	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3645 = 3^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3645.a (trivial)

\(f(q)\)	\(=\)	\(q-0.300859 q^{2} -1.90948 q^{4} +1.00000 q^{5} -4.28937 q^{7} +1.17620 q^{8} -0.300859 q^{10} -1.77263 q^{11} +4.04481 q^{13} +1.29050 q^{14} +3.46510 q^{16} +3.90428 q^{17} +1.11369 q^{19} -1.90948 q^{20} +0.533311 q^{22} -8.62555 q^{23} +1.00000 q^{25} -1.21692 q^{26} +8.19048 q^{28} -0.598345 q^{29} +1.62890 q^{31} -3.39491 q^{32} -1.17464 q^{34} -4.28937 q^{35} +5.21715 q^{37} -0.335064 q^{38} +1.17620 q^{40} -0.756985 q^{41} -7.20963 q^{43} +3.38480 q^{44} +2.59507 q^{46} +11.7489 q^{47} +11.3987 q^{49} -0.300859 q^{50} -7.72349 q^{52} -3.97724 q^{53} -1.77263 q^{55} -5.04517 q^{56} +0.180017 q^{58} +13.8195 q^{59} -8.48018 q^{61} -0.490070 q^{62} -5.90880 q^{64} +4.04481 q^{65} +7.83466 q^{67} -7.45517 q^{68} +1.29050 q^{70} +5.18353 q^{71} -4.37528 q^{73} -1.56963 q^{74} -2.12657 q^{76} +7.60345 q^{77} -12.8907 q^{79} +3.46510 q^{80} +0.227746 q^{82} -7.90723 q^{83} +3.90428 q^{85} +2.16908 q^{86} -2.08497 q^{88} +3.25566 q^{89} -17.3497 q^{91} +16.4703 q^{92} -3.53478 q^{94} +1.11369 q^{95} -1.99171 q^{97} -3.42940 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - 3 q^{2} + 9 q^{4} + 15 q^{5} - 12 q^{7} - 9 q^{8} - 3 q^{10} - 12 q^{13} - 3 q^{16} - 12 q^{17} - 24 q^{19} + 9 q^{20} - 18 q^{22} - 18 q^{23} + 15 q^{25} + 9 q^{26} - 30 q^{28} + 3 q^{29} - 24 q^{31}+ \cdots - 15 q^{98}+O(q^{100}) \) 15 * q - 3 * q^2 + 9 * q^4 + 15 * q^5 - 12 * q^7 - 9 * q^8 - 3 * q^10 - 12 * q^13 - 3 * q^16 - 12 * q^17 - 24 * q^19 + 9 * q^20 - 18 * q^22 - 18 * q^23 + 15 * q^25 + 9 * q^26 - 30 * q^28 + 3 * q^29 - 24 * q^31 - 6 * q^32 - 18 * q^34 - 12 * q^35 - 24 * q^37 - 18 * q^38 - 9 * q^40 + 9 * q^41 - 42 * q^43 + 12 * q^44 - 30 * q^46 - 21 * q^47 - 3 * q^49 - 3 * q^50 - 36 * q^52 - 18 * q^53 + 30 * q^56 - 30 * q^58 + 6 * q^59 - 15 * q^61 + 36 * q^62 - 27 * q^64 - 12 * q^65 - 45 * q^67 - 36 * q^68 + 12 * q^71 - 21 * q^73 + 21 * q^74 - 48 * q^76 - 9 * q^77 - 48 * q^79 - 3 * q^80 - 24 * q^82 - 33 * q^83 - 12 * q^85 + 15 * q^86 - 54 * q^88 + 9 * q^89 - 51 * q^91 + 33 * q^92 - 30 * q^94 - 24 * q^95 - 30 * q^97 - 15 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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