LMFDB - Embedded newform 3645.2.a.g.1.6

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			1.6
Root			$1.30569$ 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)

$f(q)$	$=$	\(q-1.30569 q^{2} -0.295165 q^{4} +1.00000 q^{5} +0.875317 q^{7} +2.99678 q^{8} -1.30569 q^{10} +2.41809 q^{11} +0.511284 q^{13} -1.14290 q^{14} -3.32255 q^{16} -7.14686 q^{17} +1.42378 q^{19} -0.295165 q^{20} -3.15729 q^{22} -2.21921 q^{23} +1.00000 q^{25} -0.667581 q^{26} -0.258363 q^{28} +5.58162 q^{29} -4.33407 q^{31} -1.65533 q^{32} +9.33161 q^{34} +0.875317 q^{35} -4.34834 q^{37} -1.85902 q^{38} +2.99678 q^{40} -9.25117 q^{41} -10.3472 q^{43} -0.713737 q^{44} +2.89761 q^{46} +10.2588 q^{47} -6.23382 q^{49} -1.30569 q^{50} -0.150913 q^{52} -1.32921 q^{53} +2.41809 q^{55} +2.62314 q^{56} -7.28788 q^{58} +4.76495 q^{59} -13.2046 q^{61} +5.65897 q^{62} +8.80646 q^{64} +0.511284 q^{65} -2.72949 q^{67} +2.10950 q^{68} -1.14290 q^{70} -4.06901 q^{71} -16.3419 q^{73} +5.67760 q^{74} -0.420251 q^{76} +2.11660 q^{77} +6.44868 q^{79} -3.32255 q^{80} +12.0792 q^{82} +10.7958 q^{83} -7.14686 q^{85} +13.5102 q^{86} +7.24650 q^{88} -8.70945 q^{89} +0.447536 q^{91} +0.655033 q^{92} -13.3949 q^{94} +1.42378 q^{95} +15.3410 q^{97} +8.13946 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.30569	−0.923265	−0.461632	−	0.887071i	$-0.652736\pi$
$2$	−1.30569	−0.923265	−0.461632	+	0.887071i	$0.652736\pi$
$3$	0	0
$3$	0	0
$4$	−0.295165	−0.147582
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0.875317	0.330839	0.165419	−	0.986223i	$-0.447102\pi$
$7$	0.875317	0.330839	0.165419	+	0.986223i	$0.447102\pi$
$8$	2.99678	1.05952
$9$	0	0
$10$	−1.30569	−0.412896
$11$	2.41809	0.729083	0.364541	−	0.931187i	$-0.381226\pi$
$11$	2.41809	0.729083	0.364541	+	0.931187i	$0.381226\pi$
$12$	0	0
$13$	0.511284	0.141805	0.0709024	−	0.997483i	$-0.477412\pi$
$13$	0.511284	0.141805	0.0709024	+	0.997483i	$0.477412\pi$
$14$	−1.14290	−0.305452
$15$	0	0
$16$	−3.32255	−0.830637
$17$	−7.14686	−1.73337	−0.866685	−	0.498856i	$-0.833754\pi$
$17$	−7.14686	−1.73337	−0.866685	+	0.498856i	$0.833754\pi$
$18$	0	0
$19$	1.42378	0.326638	0.163319	−	0.986573i	$-0.447780\pi$
$19$	1.42378	0.326638	0.163319	+	0.986573i	$0.447780\pi$
$20$	−0.295165	−0.0660009
$21$	0	0
$22$	−3.15729	−0.673136
$23$	−2.21921	−0.462737	−0.231369	−	0.972866i	$-0.574320\pi$
$23$	−2.21921	−0.462737	−0.231369	+	0.972866i	$0.574320\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−0.667581	−0.130923
$27$	0	0
$28$	−0.258363	−0.0488260
$29$	5.58162	1.03648	0.518240	−	0.855235i	$-0.326587\pi$
$29$	5.58162	1.03648	0.518240	+	0.855235i	$0.326587\pi$
$30$	0	0
$31$	−4.33407	−0.778422	−0.389211	−	0.921149i	$-0.627252\pi$
$31$	−4.33407	−0.778422	−0.389211	+	0.921149i	$0.627252\pi$
$32$	−1.65533	−0.292625
$33$	0	0
$34$	9.33161	1.60036
$35$	0.875317	0.147956
$36$	0	0
$37$	−4.34834	−0.714863	−0.357431	−	0.933939i	$-0.616347\pi$
$37$	−4.34834	−0.714863	−0.357431	+	0.933939i	$0.616347\pi$
$38$	−1.85902	−0.301573
$39$	0	0
$40$	2.99678	0.473833
$41$	−9.25117	−1.44479	−0.722395	−	0.691480i	$-0.756959\pi$
$41$	−9.25117	−1.44479	−0.722395	+	0.691480i	$0.756959\pi$
$42$	0	0
$43$	−10.3472	−1.57793	−0.788965	−	0.614438i	$-0.789383\pi$
$43$	−10.3472	−1.57793	−0.788965	+	0.614438i	$0.789383\pi$
$44$	−0.713737	−0.107600
$45$	0	0
$46$	2.89761	0.427229
$47$	10.2588	1.49640	0.748201	−	0.663472i	$-0.230918\pi$
$47$	10.2588	1.49640	0.748201	+	0.663472i	$0.230918\pi$
$48$	0	0
$49$	−6.23382	−0.890546
$50$	−1.30569	−0.184653
$51$	0	0
$52$	−0.150913	−0.0209279
$53$	−1.32921	−0.182581	−0.0912907	−	0.995824i	$-0.529099\pi$
$53$	−1.32921	−0.182581	−0.0912907	+	0.995824i	$0.529099\pi$
$54$	0	0
$55$	2.41809	0.326056
$56$	2.62314	0.350531
$57$	0	0
$58$	−7.28788	−0.956946
$59$	4.76495	0.620343	0.310172	−	0.950681i	$-0.399614\pi$
$59$	4.76495	0.620343	0.310172	+	0.950681i	$0.399614\pi$
$60$	0	0
$61$	−13.2046	−1.69067	−0.845336	−	0.534234i	$-0.820600\pi$
$61$	−13.2046	−1.69067	−0.845336	+	0.534234i	$0.820600\pi$
$62$	5.65897	0.718690
$63$	0	0
$64$	8.80646	1.10081
$65$	0.511284	0.0634170
$66$	0	0
$67$	−2.72949	−0.333460	−0.166730	−	0.986003i	$-0.553321\pi$
$67$	−2.72949	−0.333460	−0.166730	+	0.986003i	$0.553321\pi$
$68$	2.10950	0.255815
$69$	0	0
$70$	−1.14290	−0.136602
$71$	−4.06901	−0.482902	−0.241451	−	0.970413i	$-0.577623\pi$
$71$	−4.06901	−0.482902	−0.241451	+	0.970413i	$0.577623\pi$
$72$	0	0
$73$	−16.3419	−1.91268	−0.956340	−	0.292257i	$-0.905594\pi$
$73$	−16.3419	−1.91268	−0.956340	+	0.292257i	$0.905594\pi$
$74$	5.67760	0.660008
$75$	0	0
$76$	−0.420251	−0.0482060
$77$	2.11660	0.241209
$78$	0	0
$79$	6.44868	0.725533	0.362766	−	0.931880i	$-0.381832\pi$
$79$	6.44868	0.725533	0.362766	+	0.931880i	$0.381832\pi$
$80$	−3.32255	−0.371472
$81$	0	0
$82$	12.0792	1.33392
$83$	10.7958	1.18499	0.592497	−	0.805573i	$-0.298142\pi$
$83$	10.7958	1.18499	0.592497	+	0.805573i	$0.298142\pi$
$84$	0	0
$85$	−7.14686	−0.775186
$86$	13.5102	1.45685
$87$	0	0
$88$	7.24650	0.772479
$89$	−8.70945	−0.923199	−0.461600	−	0.887088i	$-0.652724\pi$
$89$	−8.70945	−0.923199	−0.461600	+	0.887088i	$0.652724\pi$
$90$	0	0
$91$	0.447536	0.0469145
$92$	0.655033	0.0682919
$93$	0	0
$94$	−13.3949	−1.38157
$95$	1.42378	0.146077
$96$	0	0
$97$	15.3410	1.55764	0.778822	−	0.627244i	$-0.215817\pi$
$97$	15.3410	1.55764	0.778822	+	0.627244i	$0.215817\pi$
$98$	8.13946	0.822209
$99$	0	0
$100$	−0.295165	−0.0295165
$101$	14.0624	1.39926	0.699631	−	0.714505i	$-0.253348\pi$
$101$	14.0624	1.39926	0.699631	+	0.714505i	$0.253348\pi$
$102$	0	0
$103$	−17.3787	−1.71237	−0.856185	−	0.516669i	$-0.827172\pi$
$103$	−17.3787	−1.71237	−0.856185	+	0.516669i	$0.827172\pi$
$104$	1.53221	0.150245
$105$	0	0
$106$	1.73554	0.168571
$107$	12.3424	1.19318	0.596592	−	0.802545i	$-0.296521\pi$
$107$	12.3424	1.19318	0.596592	+	0.802545i	$0.296521\pi$
$108$	0	0
$109$	10.1941	0.976414	0.488207	−	0.872728i	$-0.337651\pi$
$109$	10.1941	0.976414	0.488207	+	0.872728i	$0.337651\pi$
$110$	−3.15729	−0.301036
$111$	0	0
$112$	−2.90828	−0.274807
$113$	3.51805	0.330950	0.165475	−	0.986214i	$-0.447084\pi$
$113$	3.51805	0.330950	0.165475	+	0.986214i	$0.447084\pi$
$114$	0	0
$115$	−2.21921	−0.206942
$116$	−1.64750	−0.152966
$117$	0	0
$118$	−6.22156	−0.572741
$119$	−6.25578	−0.573466
$120$	0	0
$121$	−5.15282	−0.468438
$122$	17.2411	1.56094
$123$	0	0
$124$	1.27927	0.114882
$125$	1.00000	0.0894427
$126$	0	0
$127$	−4.76746	−0.423043	−0.211522	−	0.977373i	$-0.567842\pi$
$127$	−4.76746	−0.423043	−0.211522	+	0.977373i	$0.567842\pi$
$128$	−8.18786	−0.723711
$129$	0	0
$130$	−0.667581	−0.0585507
$131$	5.43969	0.475267	0.237634	−	0.971355i	$-0.423628\pi$
$131$	5.43969	0.475267	0.237634	+	0.971355i	$0.423628\pi$
$132$	0	0
$133$	1.24626	0.108065
$134$	3.56388	0.307872
$135$	0	0
$136$	−21.4176	−1.83654
$137$	5.38051	0.459688	0.229844	−	0.973227i	$-0.426178\pi$
$137$	5.38051	0.459688	0.229844	+	0.973227i	$0.426178\pi$
$138$	0	0
$139$	5.81081	0.492866	0.246433	−	0.969160i	$-0.420741\pi$
$139$	5.81081	0.492866	0.246433	+	0.969160i	$0.420741\pi$
$140$	−0.258363	−0.0218357
$141$	0	0
$142$	5.31288	0.445847
$143$	1.23633	0.103387
$144$	0	0
$145$	5.58162	0.463528
$146$	21.3376	1.76591
$147$	0	0
$148$	1.28348	0.105501
$149$	−2.23916	−0.183439	−0.0917197	−	0.995785i	$-0.529236\pi$
$149$	−2.23916	−0.183439	−0.0917197	+	0.995785i	$0.529236\pi$
$150$	0	0
$151$	−3.91707	−0.318766	−0.159383	−	0.987217i	$-0.550951\pi$
$151$	−3.91707	−0.318766	−0.159383	+	0.987217i	$0.550951\pi$
$152$	4.26676	0.346080
$153$	0	0
$154$	−2.76363	−0.222700
$155$	−4.33407	−0.348121
$156$	0	0
$157$	−4.42110	−0.352842	−0.176421	−	0.984315i	$-0.556452\pi$
$157$	−4.42110	−0.352842	−0.176421	+	0.984315i	$0.556452\pi$
$158$	−8.41999	−0.669859
$159$	0	0
$160$	−1.65533	−0.130866
$161$	−1.94251	−0.153092
$162$	0	0
$163$	−7.00769	−0.548884	−0.274442	−	0.961604i	$-0.588493\pi$
$163$	−7.00769	−0.548884	−0.274442	+	0.961604i	$0.588493\pi$
$164$	2.73062	0.213226
$165$	0	0
$166$	−14.0960	−1.09406
$167$	−15.9979	−1.23795	−0.618976	−	0.785410i	$-0.712452\pi$
$167$	−15.9979	−1.23795	−0.618976	+	0.785410i	$0.712452\pi$
$168$	0	0
$169$	−12.7386	−0.979891
$170$	9.33161	0.715702
$171$	0	0
$172$	3.05412	0.232875
$173$	−17.1530	−1.30412	−0.652059	−	0.758168i	$-0.726095\pi$
$173$	−17.1530	−1.30412	−0.652059	+	0.758168i	$0.726095\pi$
$174$	0	0
$175$	0.875317	0.0661678
$176$	−8.03423	−0.605603
$177$	0	0
$178$	11.3719	0.852357
$179$	−3.59067	−0.268380	−0.134190	−	0.990956i	$-0.542843\pi$
$179$	−3.59067	−0.268380	−0.134190	+	0.990956i	$0.542843\pi$
$180$	0	0
$181$	21.9154	1.62896	0.814479	−	0.580193i	$-0.197023\pi$
$181$	21.9154	1.62896	0.814479	+	0.580193i	$0.197023\pi$
$182$	−0.584345	−0.0433145
$183$	0	0
$184$	−6.65049	−0.490281
$185$	−4.34834	−0.319696
$186$	0	0
$187$	−17.2818	−1.26377
$188$	−3.02804	−0.220843
$189$	0	0
$190$	−1.85902	−0.134868
$191$	−20.4556	−1.48011	−0.740057	−	0.672544i	$-0.765202\pi$
$191$	−20.4556	−1.48011	−0.740057	+	0.672544i	$0.765202\pi$
$192$	0	0
$193$	6.30430	0.453793	0.226897	−	0.973919i	$-0.427142\pi$
$193$	6.30430	0.453793	0.226897	+	0.973919i	$0.427142\pi$
$194$	−20.0307	−1.43812
$195$	0	0
$196$	1.84001	0.131429
$197$	−21.0255	−1.49801	−0.749004	−	0.662565i	$-0.769468\pi$
$197$	−21.0255	−1.49801	−0.749004	+	0.662565i	$0.769468\pi$
$198$	0	0
$199$	0.692612	0.0490980	0.0245490	−	0.999699i	$-0.492185\pi$
$199$	0.692612	0.0490980	0.0245490	+	0.999699i	$0.492185\pi$
$200$	2.99678	0.211904
$201$	0	0
$202$	−18.3612	−1.29189
$203$	4.88569	0.342908
$204$	0	0
$205$	−9.25117	−0.646130
$206$	22.6912	1.58097
$207$	0	0
$208$	−1.69877	−0.117788
$209$	3.44284	0.238146
$210$	0	0
$211$	−10.4254	−0.717713	−0.358857	−	0.933393i	$-0.616833\pi$
$211$	−10.4254	−0.717713	−0.358857	+	0.933393i	$0.616833\pi$
$212$	0.392337	0.0269458
$213$	0	0
$214$	−16.1154	−1.10162
$215$	−10.3472	−0.705672
$216$	0	0
$217$	−3.79369	−0.257532
$218$	−13.3103	−0.901489
$219$	0	0
$220$	−0.713737	−0.0481201
$221$	−3.65408	−0.245800
$222$	0	0
$223$	4.81521	0.322451	0.161225	−	0.986918i	$-0.448455\pi$
$223$	4.81521	0.322451	0.161225	+	0.986918i	$0.448455\pi$
$224$	−1.44894	−0.0968116
$225$	0	0
$226$	−4.59349	−0.305554
$227$	3.57710	0.237421	0.118710	−	0.992929i	$-0.462124\pi$
$227$	3.57710	0.237421	0.118710	+	0.992929i	$0.462124\pi$
$228$	0	0
$229$	−6.87527	−0.454331	−0.227165	−	0.973856i	$-0.572946\pi$
$229$	−6.87527	−0.454331	−0.227165	+	0.973856i	$0.572946\pi$
$230$	2.89761	0.191063
$231$	0	0
$232$	16.7269	1.09817
$233$	−7.54985	−0.494607	−0.247304	−	0.968938i	$-0.579544\pi$
$233$	−7.54985	−0.494607	−0.247304	+	0.968938i	$0.579544\pi$
$234$	0	0
$235$	10.2588	0.669211
$236$	−1.40645	−0.0915518
$237$	0	0
$238$	8.16812	0.529461
$239$	−7.78731	−0.503719	−0.251860	−	0.967764i	$-0.581042\pi$
$239$	−7.78731	−0.503719	−0.251860	+	0.967764i	$0.581042\pi$
$240$	0	0
$241$	12.2989	0.792239	0.396120	−	0.918199i	$-0.370357\pi$
$241$	12.2989	0.792239	0.396120	+	0.918199i	$0.370357\pi$
$242$	6.72800	0.432493
$243$	0	0
$244$	3.89753	0.249514
$245$	−6.23382	−0.398264
$246$	0	0
$247$	0.727957	0.0463188
$248$	−12.9883	−0.824756
$249$	0	0
$250$	−1.30569	−0.0825793
$251$	22.7754	1.43757	0.718784	−	0.695233i	$-0.244699\pi$
$251$	22.7754	1.43757	0.718784	+	0.695233i	$0.244699\pi$
$252$	0	0
$253$	−5.36626	−0.337374
$254$	6.22483	0.390581
$255$	0	0
$256$	−6.92208	−0.432630
$257$	−3.45438	−0.215478	−0.107739	−	0.994179i	$-0.534361\pi$
$257$	−3.45438	−0.215478	−0.107739	+	0.994179i	$0.534361\pi$
$258$	0	0
$259$	−3.80618	−0.236504
$260$	−0.150913	−0.00935924
$261$	0	0
$262$	−7.10256	−0.438798
$263$	−31.7543	−1.95805	−0.979027	−	0.203730i	$-0.934693\pi$
$263$	−31.7543	−1.95805	−0.979027	+	0.203730i	$0.934693\pi$
$264$	0	0
$265$	−1.32921	−0.0816528
$266$	−1.62723	−0.0997722
$267$	0	0
$268$	0.805650	0.0492129
$269$	9.71620	0.592407	0.296204	−	0.955125i	$-0.404279\pi$
$269$	9.71620	0.592407	0.296204	+	0.955125i	$0.404279\pi$
$270$	0	0
$271$	−26.3970	−1.60351	−0.801753	−	0.597655i	$-0.796099\pi$
$271$	−26.3970	−1.60351	−0.801753	+	0.597655i	$0.796099\pi$
$272$	23.7458	1.43980
$273$	0	0
$274$	−7.02530	−0.424414
$275$	2.41809	0.145817
$276$	0	0
$277$	−5.64315	−0.339064	−0.169532	−	0.985525i	$-0.554226\pi$
$277$	−5.64315	−0.339064	−0.169532	+	0.985525i	$0.554226\pi$
$278$	−7.58714	−0.455046
$279$	0	0
$280$	2.62314	0.156762
$281$	29.2188	1.74305	0.871524	−	0.490353i	$-0.163132\pi$
$281$	29.2188	1.74305	0.871524	+	0.490353i	$0.163132\pi$
$282$	0	0
$283$	−12.2896	−0.730543	−0.365272	−	0.930901i	$-0.619024\pi$
$283$	−12.2896	−0.730543	−0.365272	+	0.930901i	$0.619024\pi$
$284$	1.20103	0.0712679
$285$	0	0
$286$	−1.61427	−0.0954539
$287$	−8.09771	−0.477993
$288$	0	0
$289$	34.0777	2.00457
$290$	−7.28788	−0.427959
$291$	0	0
$292$	4.82357	0.282278
$293$	12.6899	0.741350	0.370675	−	0.928763i	$-0.379126\pi$
$293$	12.6899	0.741350	0.370675	+	0.928763i	$0.379126\pi$
$294$	0	0
$295$	4.76495	0.277426
$296$	−13.0310	−0.757413
$297$	0	0
$298$	2.92366	0.169363
$299$	−1.13465	−0.0656184
$300$	0	0
$301$	−9.05707	−0.522041
$302$	5.11449	0.294306
$303$	0	0
$304$	−4.73058	−0.271318
$305$	−13.2046	−0.756092
$306$	0	0
$307$	−14.3449	−0.818705	−0.409352	−	0.912376i	$-0.634245\pi$
$307$	−14.3449	−0.818705	−0.409352	+	0.912376i	$0.634245\pi$
$308$	−0.624746	−0.0355982
$309$	0	0
$310$	5.65897	0.321408
$311$	−3.98915	−0.226204	−0.113102	−	0.993583i	$-0.536079\pi$
$311$	−3.98915	−0.226204	−0.113102	+	0.993583i	$0.536079\pi$
$312$	0	0
$313$	−1.01036	−0.0571089	−0.0285545	−	0.999592i	$-0.509090\pi$
$313$	−1.01036	−0.0571089	−0.0285545	+	0.999592i	$0.509090\pi$
$314$	5.77260	0.325767
$315$	0	0
$316$	−1.90342	−0.107076
$317$	0.329357	0.0184986	0.00924928	−	0.999957i	$-0.497056\pi$
$317$	0.329357	0.0184986	0.00924928	+	0.999957i	$0.497056\pi$
$318$	0	0
$319$	13.4969	0.755680
$320$	8.80646	0.492296
$321$	0	0
$322$	2.53633	0.141344
$323$	−10.1756	−0.566184
$324$	0	0
$325$	0.511284	0.0283610
$326$	9.14989	0.506766
$327$	0	0
$328$	−27.7237	−1.53079
$329$	8.97972	0.495068
$330$	0	0
$331$	−8.73866	−0.480320	−0.240160	−	0.970733i	$-0.577200\pi$
$331$	−8.73866	−0.480320	−0.240160	+	0.970733i	$0.577200\pi$
$332$	−3.18654	−0.174884
$333$	0	0
$334$	20.8883	1.14296
$335$	−2.72949	−0.149128
$336$	0	0
$337$	−29.3357	−1.59802	−0.799009	−	0.601319i	$-0.794642\pi$
$337$	−29.3357	−1.59802	−0.799009	+	0.601319i	$0.794642\pi$
$338$	16.6327	0.904699
$339$	0	0
$340$	2.10950	0.114404
$341$	−10.4802	−0.567534
$342$	0	0
$343$	−11.5838	−0.625466
$344$	−31.0082	−1.67185
$345$	0	0
$346$	22.3966	1.20405
$347$	15.6418	0.839698	0.419849	−	0.907594i	$-0.362083\pi$
$347$	15.6418	0.839698	0.419849	+	0.907594i	$0.362083\pi$
$348$	0	0
$349$	−17.7955	−0.952570	−0.476285	−	0.879291i	$-0.658017\pi$
$349$	−17.7955	−0.952570	−0.476285	+	0.879291i	$0.658017\pi$
$350$	−1.14290	−0.0610904
$351$	0	0
$352$	−4.00276	−0.213348
$353$	−26.9462	−1.43420	−0.717100	−	0.696970i	$-0.754531\pi$
$353$	−26.9462	−1.43420	−0.717100	+	0.696970i	$0.754531\pi$
$354$	0	0
$355$	−4.06901	−0.215961
$356$	2.57072	0.136248
$357$	0	0
$358$	4.68832	0.247785
$359$	5.96933	0.315049	0.157525	−	0.987515i	$-0.449649\pi$
$359$	5.96933	0.315049	0.157525	+	0.987515i	$0.449649\pi$
$360$	0	0
$361$	−16.9728	−0.893308
$362$	−28.6148	−1.50396
$363$	0	0
$364$	−0.132097	−0.00692376
$365$	−16.3419	−0.855376
$366$	0	0
$367$	−25.3254	−1.32198	−0.660988	−	0.750396i	$-0.729863\pi$
$367$	−25.3254	−1.32198	−0.660988	+	0.750396i	$0.729863\pi$
$368$	7.37343	0.384367
$369$	0	0
$370$	5.67760	0.295164
$371$	−1.16348	−0.0604050
$372$	0	0
$373$	7.68704	0.398020	0.199010	−	0.979997i	$-0.436227\pi$
$373$	7.68704	0.398020	0.199010	+	0.979997i	$0.436227\pi$
$374$	22.5647	1.16679
$375$	0	0
$376$	30.7434	1.58547
$377$	2.85380	0.146978
$378$	0	0
$379$	34.2011	1.75679	0.878397	−	0.477932i	$-0.158613\pi$
$379$	34.2011	1.75679	0.878397	+	0.477932i	$0.158613\pi$
$380$	−0.420251	−0.0215584
$381$	0	0
$382$	26.7087	1.36654
$383$	−4.64116	−0.237152	−0.118576	−	0.992945i	$-0.537833\pi$
$383$	−4.64116	−0.237152	−0.118576	+	0.992945i	$0.537833\pi$
$384$	0	0
$385$	2.11660	0.107872
$386$	−8.23148	−0.418971
$387$	0	0
$388$	−4.52813	−0.229881
$389$	−5.29722	−0.268580	−0.134290	−	0.990942i	$-0.542875\pi$
$389$	−5.29722	−0.268580	−0.134290	+	0.990942i	$0.542875\pi$
$390$	0	0
$391$	15.8604	0.802095
$392$	−18.6814	−0.943553
$393$	0	0
$394$	27.4529	1.38306
$395$	6.44868	0.324468
$396$	0	0
$397$	−13.3367	−0.669352	−0.334676	−	0.942333i	$-0.608627\pi$
$397$	−13.3367	−0.669352	−0.334676	+	0.942333i	$0.608627\pi$
$398$	−0.904339	−0.0453304
$399$	0	0
$400$	−3.32255	−0.166127
$401$	0.445017	0.0222231	0.0111115	−	0.999938i	$-0.496463\pi$
$401$	0.445017	0.0222231	0.0111115	+	0.999938i	$0.496463\pi$
$402$	0	0
$403$	−2.21594	−0.110384
$404$	−4.15073	−0.206506
$405$	0	0
$406$	−6.37921	−0.316595
$407$	−10.5147	−0.521194
$408$	0	0
$409$	−9.40699	−0.465146	−0.232573	−	0.972579i	$-0.574714\pi$
$409$	−9.40699	−0.465146	−0.232573	+	0.972579i	$0.574714\pi$
$410$	12.0792	0.596549
$411$	0	0
$412$	5.12957	0.252716
$413$	4.17084	0.205234
$414$	0	0
$415$	10.7958	0.529945
$416$	−0.846347	−0.0414956
$417$	0	0
$418$	−4.49529	−0.219872
$419$	25.6132	1.25129	0.625644	−	0.780109i	$-0.284836\pi$
$419$	25.6132	1.25129	0.625644	+	0.780109i	$0.284836\pi$
$420$	0	0
$421$	8.30798	0.404906	0.202453	−	0.979292i	$-0.435109\pi$
$421$	8.30798	0.404906	0.202453	+	0.979292i	$0.435109\pi$
$422$	13.6124	0.662639
$423$	0	0
$424$	−3.98336	−0.193449
$425$	−7.14686	−0.346674
$426$	0	0
$427$	−11.5582	−0.559340
$428$	−3.64304	−0.176093
$429$	0	0
$430$	13.5102	0.651522
$431$	−30.4309	−1.46580	−0.732902	−	0.680335i	$-0.761834\pi$
$431$	−30.4309	−1.46580	−0.732902	+	0.680335i	$0.761834\pi$
$432$	0	0
$433$	−12.0997	−0.581475	−0.290737	−	0.956803i	$-0.593901\pi$
$433$	−12.0997	−0.581475	−0.290737	+	0.956803i	$0.593901\pi$
$434$	4.95339	0.237771
$435$	0	0
$436$	−3.00893	−0.144102
$437$	−3.15967	−0.151148
$438$	0	0
$439$	10.8885	0.519678	0.259839	−	0.965652i	$-0.416331\pi$
$439$	10.8885	0.519678	0.259839	+	0.965652i	$0.416331\pi$
$440$	7.24650	0.345463
$441$	0	0
$442$	4.77111	0.226938
$443$	−24.0822	−1.14418	−0.572090	−	0.820191i	$-0.693867\pi$
$443$	−24.0822	−1.14418	−0.572090	+	0.820191i	$0.693867\pi$
$444$	0	0
$445$	−8.70945	−0.412867
$446$	−6.28719	−0.297707
$447$	0	0
$448$	7.70844	0.364190
$449$	2.46422	0.116294	0.0581469	−	0.998308i	$-0.481481\pi$
$449$	2.46422	0.116294	0.0581469	+	0.998308i	$0.481481\pi$
$450$	0	0
$451$	−22.3702	−1.05337
$452$	−1.03840	−0.0488424
$453$	0	0
$454$	−4.67060	−0.219202
$455$	0.447536	0.0209808
$456$	0	0
$457$	−16.5775	−0.775461	−0.387731	−	0.921773i	$-0.626741\pi$
$457$	−16.5775	−0.775461	−0.387731	+	0.921773i	$0.626741\pi$
$458$	8.97700	0.419468
$459$	0	0
$460$	0.655033	0.0305411
$461$	22.8463	1.06406	0.532029	−	0.846726i	$-0.321430\pi$
$461$	22.8463	1.06406	0.532029	+	0.846726i	$0.321430\pi$
$462$	0	0
$463$	31.9280	1.48382	0.741909	−	0.670501i	$-0.233921\pi$
$463$	31.9280	1.48382	0.741909	+	0.670501i	$0.233921\pi$
$464$	−18.5452	−0.860939
$465$	0	0
$466$	9.85779	0.456653
$467$	−6.09121	−0.281868	−0.140934	−	0.990019i	$-0.545010\pi$
$467$	−6.09121	−0.281868	−0.140934	+	0.990019i	$0.545010\pi$
$468$	0	0
$469$	−2.38917	−0.110322
$470$	−13.3949	−0.617859
$471$	0	0
$472$	14.2795	0.657267
$473$	−25.0204	−1.15044
$474$	0	0
$475$	1.42378	0.0653276
$476$	1.84649	0.0846335
$477$	0	0
$478$	10.1678	0.465066
$479$	27.7101	1.26610	0.633052	−	0.774109i	$-0.281802\pi$
$479$	27.7101	1.26610	0.633052	+	0.774109i	$0.281802\pi$
$480$	0	0
$481$	−2.22324	−0.101371
$482$	−16.0585	−0.731447
$483$	0	0
$484$	1.52093	0.0691333
$485$	15.3410	0.696600
$486$	0	0
$487$	5.25389	0.238076	0.119038	−	0.992890i	$-0.462019\pi$
$487$	5.25389	0.238076	0.119038	+	0.992890i	$0.462019\pi$
$488$	−39.5712	−1.79131
$489$	0	0
$490$	8.13946	0.367703
$491$	30.0480	1.35605	0.678023	−	0.735041i	$-0.262837\pi$
$491$	30.0480	1.35605	0.678023	+	0.735041i	$0.262837\pi$
$492$	0	0
$493$	−39.8911	−1.79660
$494$	−0.950489	−0.0427645
$495$	0	0
$496$	14.4002	0.646586
$497$	−3.56167	−0.159763
$498$	0	0
$499$	29.9091	1.33892	0.669458	−	0.742850i	$-0.266527\pi$
$499$	29.9091	1.33892	0.669458	+	0.742850i	$0.266527\pi$
$500$	−0.295165	−0.0132002
$501$	0	0
$502$	−29.7376	−1.32726
$503$	−16.7420	−0.746489	−0.373244	−	0.927733i	$-0.621755\pi$
$503$	−16.7420	−0.746489	−0.373244	+	0.927733i	$0.621755\pi$
$504$	0	0
$505$	14.0624	0.625769
$506$	7.00669	0.311485
$507$	0	0
$508$	1.40719	0.0624338
$509$	−24.8195	−1.10011	−0.550053	−	0.835130i	$-0.685392\pi$
$509$	−24.8195	−1.10011	−0.550053	+	0.835130i	$0.685392\pi$
$510$	0	0
$511$	−14.3044	−0.632789
$512$	25.4138	1.12314
$513$	0	0
$514$	4.51036	0.198943
$515$	−17.3787	−0.765795
$516$	0	0
$517$	24.8068	1.09100
$518$	4.96970	0.218356
$519$	0	0
$520$	1.53221	0.0671917
$521$	−13.4986	−0.591383	−0.295692	−	0.955283i	$-0.595550\pi$
$521$	−13.4986	−0.591383	−0.295692	+	0.955283i	$0.595550\pi$
$522$	0	0
$523$	−19.4838	−0.851967	−0.425983	−	0.904731i	$-0.640072\pi$
$523$	−19.4838	−0.851967	−0.425983	+	0.904731i	$0.640072\pi$
$524$	−1.60560	−0.0701412
$525$	0	0
$526$	41.4614	1.80780
$527$	30.9750	1.34929
$528$	0	0
$529$	−18.0751	−0.785874
$530$	1.73554	0.0753872
$531$	0	0
$532$	−0.367853	−0.0159484
$533$	−4.72998	−0.204878
$534$	0	0
$535$	12.3424	0.533608
$536$	−8.17969	−0.353309
$537$	0	0
$538$	−12.6864	−0.546949
$539$	−15.0740	−0.649281
$540$	0	0
$541$	29.9338	1.28695	0.643476	−	0.765466i	$-0.277491\pi$
$541$	29.9338	1.28695	0.643476	+	0.765466i	$0.277491\pi$
$542$	34.4664	1.48046
$543$	0	0
$544$	11.8305	0.507227
$545$	10.1941	0.436666
$546$	0	0
$547$	8.44227	0.360965	0.180483	−	0.983578i	$-0.442234\pi$
$547$	8.44227	0.360965	0.180483	+	0.983578i	$0.442234\pi$
$548$	−1.58814	−0.0678419
$549$	0	0
$550$	−3.15729	−0.134627
$551$	7.94701	0.338554
$552$	0	0
$553$	5.64464	0.240034
$554$	7.36823	0.313046
$555$	0	0
$556$	−1.71515	−0.0727385
$557$	−10.7951	−0.457402	−0.228701	−	0.973497i	$-0.573448\pi$
$557$	−10.7951	−0.457402	−0.228701	+	0.973497i	$0.573448\pi$
$558$	0	0
$559$	−5.29035	−0.223758
$560$	−2.90828	−0.122897
$561$	0	0
$562$	−38.1508	−1.60929
$563$	−15.5336	−0.654661	−0.327331	−	0.944910i	$-0.606149\pi$
$563$	−15.5336	−0.654661	−0.327331	+	0.944910i	$0.606149\pi$
$564$	0	0
$565$	3.51805	0.148005
$566$	16.0465	0.674485
$567$	0	0
$568$	−12.1939	−0.511646
$569$	−36.4539	−1.52823	−0.764113	−	0.645083i	$-0.776823\pi$
$569$	−36.4539	−1.52823	−0.764113	+	0.645083i	$0.776823\pi$
$570$	0	0
$571$	−24.2843	−1.01627	−0.508133	−	0.861279i	$-0.669664\pi$
$571$	−24.2843	−1.01627	−0.508133	+	0.861279i	$0.669664\pi$
$572$	−0.364922	−0.0152582
$573$	0	0
$574$	10.5731	0.441314
$575$	−2.21921	−0.0925475
$576$	0	0
$577$	44.0901	1.83549	0.917747	−	0.397166i	$-0.130006\pi$
$577$	44.0901	1.83549	0.917747	+	0.397166i	$0.130006\pi$
$578$	−44.4950	−1.85075
$579$	0	0
$580$	−1.64750	−0.0684087
$581$	9.44975	0.392042
$582$	0	0
$583$	−3.21416	−0.133117
$584$	−48.9732	−2.02653
$585$	0	0
$586$	−16.5691	−0.684462
$587$	7.39258	0.305124	0.152562	−	0.988294i	$-0.451248\pi$
$587$	7.39258	0.305124	0.152562	+	0.988294i	$0.451248\pi$
$588$	0	0
$589$	−6.17077	−0.254262
$590$	−6.22156	−0.256137
$591$	0	0
$592$	14.4476	0.593791
$593$	33.5458	1.37756	0.688781	−	0.724969i	$-0.258146\pi$
$593$	33.5458	1.37756	0.688781	+	0.724969i	$0.258146\pi$
$594$	0	0
$595$	−6.25578	−0.256462
$596$	0.660923	0.0270725
$597$	0	0
$598$	1.48150	0.0605831
$599$	−6.83459	−0.279254	−0.139627	−	0.990204i	$-0.544590\pi$
$599$	−6.83459	−0.279254	−0.139627	+	0.990204i	$0.544590\pi$
$600$	0	0
$601$	21.9144	0.893905	0.446953	−	0.894558i	$-0.352509\pi$
$601$	21.9144	0.893905	0.446953	+	0.894558i	$0.352509\pi$
$602$	11.8257	0.481982
$603$	0	0
$604$	1.15618	0.0470444
$605$	−5.15282	−0.209492
$606$	0	0
$607$	42.1226	1.70971	0.854853	−	0.518871i	$-0.173648\pi$
$607$	42.1226	1.70971	0.854853	+	0.518871i	$0.173648\pi$
$608$	−2.35684	−0.0955823
$609$	0	0
$610$	17.2411	0.698073
$611$	5.24517	0.212197
$612$	0	0
$613$	−1.28897	−0.0520610	−0.0260305	−	0.999661i	$-0.508287\pi$
$613$	−1.28897	−0.0520610	−0.0260305	+	0.999661i	$0.508287\pi$
$614$	18.7300	0.755881
$615$	0	0
$616$	6.34299	0.255566
$617$	8.24737	0.332027	0.166013	−	0.986124i	$-0.446911\pi$
$617$	8.24737	0.332027	0.166013	+	0.986124i	$0.446911\pi$
$618$	0	0
$619$	35.6864	1.43436	0.717179	−	0.696889i	$-0.245433\pi$
$619$	35.6864	1.43436	0.717179	+	0.696889i	$0.245433\pi$
$620$	1.27927	0.0513766
$621$	0	0
$622$	5.20861	0.208846
$623$	−7.62353	−0.305430
$624$	0	0
$625$	1.00000	0.0400000
$626$	1.31922	0.0527267
$627$	0	0
$628$	1.30495	0.0520733
$629$	31.0770	1.23912
$630$	0	0
$631$	−14.0413	−0.558975	−0.279487	−	0.960149i	$-0.590165\pi$
$631$	−14.0413	−0.558975	−0.279487	+	0.960149i	$0.590165\pi$
$632$	19.3253	0.768718
$633$	0	0
$634$	−0.430040	−0.0170791
$635$	−4.76746	−0.189191
$636$	0	0
$637$	−3.18725	−0.126284
$638$	−17.6228	−0.697693
$639$	0	0
$640$	−8.18786	−0.323654
$641$	13.5576	0.535491	0.267746	−	0.963490i	$-0.413721\pi$
$641$	13.5576	0.535491	0.267746	+	0.963490i	$0.413721\pi$
$642$	0	0
$643$	1.68805	0.0665702	0.0332851	−	0.999446i	$-0.489403\pi$
$643$	1.68805	0.0665702	0.0332851	+	0.999446i	$0.489403\pi$
$644$	0.573362	0.0225936
$645$	0	0
$646$	13.2862	0.522738
$647$	−42.5869	−1.67426	−0.837132	−	0.547001i	$-0.815769\pi$
$647$	−42.5869	−1.67426	−0.837132	+	0.547001i	$0.815769\pi$
$648$	0	0
$649$	11.5221	0.452281
$650$	−0.667581	−0.0261847
$651$	0	0
$652$	2.06842	0.0810057
$653$	14.8606	0.581541	0.290770	−	0.956793i	$-0.406088\pi$
$653$	14.8606	0.581541	0.290770	+	0.956793i	$0.406088\pi$
$654$	0	0
$655$	5.43969	0.212546
$656$	30.7375	1.20010
$657$	0	0
$658$	−11.7248	−0.457079
$659$	−6.78383	−0.264261	−0.132130	−	0.991232i	$-0.542182\pi$
$659$	−6.78383	−0.264261	−0.132130	+	0.991232i	$0.542182\pi$
$660$	0	0
$661$	5.92404	0.230418	0.115209	−	0.993341i	$-0.463246\pi$
$661$	5.92404	0.230418	0.115209	+	0.993341i	$0.463246\pi$
$662$	11.4100	0.443463
$663$	0	0
$664$	32.3527	1.25553
$665$	1.24626	0.0483279
$666$	0	0
$667$	−12.3868	−0.479618
$668$	4.72201	0.182700
$669$	0	0
$670$	3.56388	0.137685
$671$	−31.9299	−1.23264
$672$	0	0
$673$	−29.1472	−1.12354	−0.561771	−	0.827293i	$-0.689880\pi$
$673$	−29.1472	−1.12354	−0.561771	+	0.827293i	$0.689880\pi$
$674$	38.3034	1.47539
$675$	0	0
$676$	3.75999	0.144615
$677$	3.51001	0.134901	0.0674504	−	0.997723i	$-0.478514\pi$
$677$	3.51001	0.134901	0.0674504	+	0.997723i	$0.478514\pi$
$678$	0	0
$679$	13.4283	0.515330
$680$	−21.4176	−0.821327
$681$	0	0
$682$	13.6839	0.523984
$683$	−13.7747	−0.527075	−0.263537	−	0.964649i	$-0.584889\pi$
$683$	−13.7747	−0.527075	−0.263537	+	0.964649i	$0.584889\pi$
$684$	0	0
$685$	5.38051	0.205579
$686$	15.1249	0.577471
$687$	0	0
$688$	34.3790	1.31069
$689$	−0.679605	−0.0258909
$690$	0	0
$691$	−20.2774	−0.771389	−0.385694	−	0.922627i	$-0.626038\pi$
$691$	−20.2774	−0.771389	−0.385694	+	0.922627i	$0.626038\pi$
$692$	5.06296	0.192465
$693$	0	0
$694$	−20.4234	−0.775263
$695$	5.81081	0.220417
$696$	0	0
$697$	66.1169	2.50436
$698$	23.2354	0.879474
$699$	0	0
$700$	−0.258363	−0.00976521
$701$	−6.62140	−0.250087	−0.125043	−	0.992151i	$-0.539907\pi$
$701$	−6.62140	−0.250087	−0.125043	+	0.992151i	$0.539907\pi$
$702$	0	0
$703$	−6.19109	−0.233501
$704$	21.2948	0.802579
$705$	0	0
$706$	35.1834	1.32415
$707$	12.3091	0.462930
$708$	0	0
$709$	−9.75038	−0.366183	−0.183092	−	0.983096i	$-0.558610\pi$
$709$	−9.75038	−0.366183	−0.183092	+	0.983096i	$0.558610\pi$
$710$	5.31288	0.199389
$711$	0	0
$712$	−26.1003	−0.978150
$713$	9.61822	0.360205
$714$	0	0
$715$	1.23633	0.0462363
$716$	1.05984	0.0396081
$717$	0	0
$718$	−7.79412	−0.290874
$719$	35.1862	1.31222	0.656112	−	0.754663i	$-0.272200\pi$
$719$	35.1862	1.31222	0.656112	+	0.754663i	$0.272200\pi$
$720$	0	0
$721$	−15.2118	−0.566519
$722$	22.1613	0.824759
$723$	0	0
$724$	−6.46865	−0.240406
$725$	5.58162	0.207296
$726$	0	0
$727$	−39.0744	−1.44919	−0.724595	−	0.689175i	$-0.757973\pi$
$727$	−39.0744	−1.44919	−0.724595	+	0.689175i	$0.757973\pi$
$728$	1.34117	0.0497070
$729$	0	0
$730$	21.3376	0.789739
$731$	73.9499	2.73514
$732$	0	0
$733$	34.2261	1.26417	0.632084	−	0.774900i	$-0.282200\pi$
$733$	34.2261	1.26417	0.632084	+	0.774900i	$0.282200\pi$
$734$	33.0672	1.22053
$735$	0	0
$736$	3.67354	0.135408
$737$	−6.60017	−0.243120
$738$	0	0
$739$	9.27481	0.341180	0.170590	−	0.985342i	$-0.445433\pi$
$739$	9.27481	0.341180	0.170590	+	0.985342i	$0.445433\pi$
$740$	1.28348	0.0471816
$741$	0	0
$742$	1.51915	0.0557698
$743$	−24.3655	−0.893883	−0.446941	−	0.894563i	$-0.647487\pi$
$743$	−24.3655	−0.893883	−0.446941	+	0.894563i	$0.647487\pi$
$744$	0	0
$745$	−2.23916	−0.0820366
$746$	−10.0369	−0.367478
$747$	0	0
$748$	5.10098	0.186510
$749$	10.8035	0.394752
$750$	0	0
$751$	−41.4309	−1.51184	−0.755918	−	0.654666i	$-0.772809\pi$
$751$	−41.4309	−1.51184	−0.755918	+	0.654666i	$0.772809\pi$
$752$	−34.0854	−1.24297
$753$	0	0
$754$	−3.72618	−0.135700
$755$	−3.91707	−0.142557
$756$	0	0
$757$	−2.02213	−0.0734956	−0.0367478	−	0.999325i	$-0.511700\pi$
$757$	−2.02213	−0.0734956	−0.0367478	+	0.999325i	$0.511700\pi$
$758$	−44.6562	−1.62199
$759$	0	0
$760$	4.26676	0.154772
$761$	−24.0178	−0.870646	−0.435323	−	0.900274i	$-0.643366\pi$
$761$	−24.0178	−0.870646	−0.435323	+	0.900274i	$0.643366\pi$
$762$	0	0
$763$	8.92304	0.323036
$764$	6.03777	0.218439
$765$	0	0
$766$	6.05993	0.218954
$767$	2.43624	0.0879676
$768$	0	0
$769$	3.52388	0.127074	0.0635371	−	0.997979i	$-0.479762\pi$
$769$	3.52388	0.127074	0.0635371	+	0.997979i	$0.479762\pi$
$770$	−2.76363	−0.0995943
$771$	0	0
$772$	−1.86081	−0.0669720
$773$	12.0905	0.434866	0.217433	−	0.976075i	$-0.430232\pi$
$773$	12.0905	0.434866	0.217433	+	0.976075i	$0.430232\pi$
$774$	0	0
$775$	−4.33407	−0.155684
$776$	45.9737	1.65036
$777$	0	0
$778$	6.91654	0.247970
$779$	−13.1717	−0.471923
$780$	0	0
$781$	−9.83924	−0.352076
$782$	−20.7088	−0.740546
$783$	0	0
$784$	20.7122	0.739720
$785$	−4.42110	−0.157796
$786$	0	0
$787$	−29.8291	−1.06329	−0.531646	−	0.846967i	$-0.678426\pi$
$787$	−29.8291	−1.06329	−0.531646	+	0.846967i	$0.678426\pi$
$788$	6.20600	0.221080
$789$	0	0
$790$	−8.41999	−0.299570
$791$	3.07941	0.109491
$792$	0	0
$793$	−6.75129	−0.239745
$794$	17.4137	0.617989
$795$	0	0
$796$	−0.204435	−0.00724600
$797$	−47.2619	−1.67410	−0.837051	−	0.547124i	$-0.815722\pi$
$797$	−47.2619	−1.67410	−0.837051	+	0.547124i	$0.815722\pi$
$798$	0	0
$799$	−73.3183	−2.59382
$800$	−1.65533	−0.0585249
$801$	0	0
$802$	−0.581055	−0.0205178
$803$	−39.5163	−1.39450
$804$	0	0
$805$	−1.94251	−0.0684646
$806$	2.89334	0.101914
$807$	0	0
$808$	42.1419	1.48255
$809$	−13.1706	−0.463052	−0.231526	−	0.972829i	$-0.574372\pi$
$809$	−13.1706	−0.463052	−0.231526	+	0.972829i	$0.574372\pi$
$810$	0	0
$811$	4.32307	0.151803	0.0759017	−	0.997115i	$-0.475816\pi$
$811$	4.32307	0.151803	0.0759017	+	0.997115i	$0.475816\pi$
$812$	−1.44208	−0.0506072
$813$	0	0
$814$	13.7290	0.481200
$815$	−7.00769	−0.245469
$816$	0	0
$817$	−14.7321	−0.515412
$818$	12.2826	0.429453
$819$	0	0
$820$	2.73062	0.0953575
$821$	−4.21823	−0.147217	−0.0736086	−	0.997287i	$-0.523452\pi$
$821$	−4.21823	−0.147217	−0.0736086	+	0.997287i	$0.523452\pi$
$822$	0	0
$823$	−16.8374	−0.586915	−0.293458	−	0.955972i	$-0.594806\pi$
$823$	−16.8374	−0.586915	−0.293458	+	0.955972i	$0.594806\pi$
$824$	−52.0801	−1.81429
$825$	0	0
$826$	−5.44584	−0.189485
$827$	−5.50934	−0.191579	−0.0957893	−	0.995402i	$-0.530537\pi$
$827$	−5.50934	−0.191579	−0.0957893	+	0.995402i	$0.530537\pi$
$828$	0	0
$829$	54.9072	1.90700	0.953502	−	0.301386i	$-0.0974494\pi$
$829$	54.9072	1.90700	0.953502	+	0.301386i	$0.0974494\pi$
$830$	−14.0960	−0.489279
$831$	0	0
$832$	4.50260	0.156100
$833$	44.5523	1.54364
$834$	0	0
$835$	−15.9979	−0.553629
$836$	−1.01621	−0.0351462
$837$	0	0
$838$	−33.4430	−1.15527
$839$	40.7607	1.40722	0.703608	−	0.710588i	$-0.251571\pi$
$839$	40.7607	1.40722	0.703608	+	0.710588i	$0.251571\pi$
$840$	0	0
$841$	2.15449	0.0742927
$842$	−10.8477	−0.373835
$843$	0	0
$844$	3.07721	0.105922
$845$	−12.7386	−0.438221
$846$	0	0
$847$	−4.51035	−0.154978
$848$	4.41637	0.151659
$849$	0	0
$850$	9.33161	0.320072
$851$	9.64988	0.330794
$852$	0	0
$853$	23.8165	0.815461	0.407730	−	0.913102i	$-0.366320\pi$
$853$	23.8165	0.815461	0.407730	+	0.913102i	$0.366320\pi$
$854$	15.0915	0.516419
$855$	0	0
$856$	36.9875	1.26421
$857$	−5.24163	−0.179051	−0.0895254	−	0.995985i	$-0.528535\pi$
$857$	−5.24163	−0.179051	−0.0895254	+	0.995985i	$0.528535\pi$
$858$	0	0
$859$	52.2205	1.78174	0.890870	−	0.454259i	$-0.150096\pi$
$859$	52.2205	1.78174	0.890870	+	0.454259i	$0.150096\pi$
$860$	3.05412	0.104145
$861$	0	0
$862$	39.7334	1.35332
$863$	21.3677	0.727366	0.363683	−	0.931523i	$-0.381519\pi$
$863$	21.3677	0.727366	0.363683	+	0.931523i	$0.381519\pi$
$864$	0	0
$865$	−17.1530	−0.583219
$866$	15.7985	0.536855
$867$	0	0
$868$	1.11976	0.0380073
$869$	15.5935	0.528973
$870$	0	0
$871$	−1.39555	−0.0472863
$872$	30.5494	1.03453
$873$	0	0
$874$	4.12556	0.139549
$875$	0.875317	0.0295911
$876$	0	0
$877$	35.3397	1.19334	0.596668	−	0.802488i	$-0.296491\pi$
$877$	35.3397	1.19334	0.596668	+	0.802488i	$0.296491\pi$
$878$	−14.2170	−0.479800
$879$	0	0
$880$	−8.03423	−0.270834
$881$	−1.26259	−0.0425376	−0.0212688	−	0.999774i	$-0.506771\pi$
$881$	−1.26259	−0.0425376	−0.0212688	+	0.999774i	$0.506771\pi$
$882$	0	0
$883$	−44.4546	−1.49602	−0.748009	−	0.663689i	$-0.768990\pi$
$883$	−44.4546	−1.49602	−0.748009	+	0.663689i	$0.768990\pi$
$884$	1.07856	0.0362758
$885$	0	0
$886$	31.4440	1.05638
$887$	4.80118	0.161208	0.0806040	−	0.996746i	$-0.474315\pi$
$887$	4.80118	0.161208	0.0806040	+	0.996746i	$0.474315\pi$
$888$	0	0
$889$	−4.17304	−0.139959
$890$	11.3719	0.381186
$891$	0	0
$892$	−1.42128	−0.0475881
$893$	14.6063	0.488782
$894$	0	0
$895$	−3.59067	−0.120023
$896$	−7.16698	−0.239432
$897$	0	0
$898$	−3.21752	−0.107370
$899$	−24.1912	−0.806820
$900$	0	0
$901$	9.49970	0.316481
$902$	29.2086	0.972541
$903$	0	0
$904$	10.5428	0.350649
$905$	21.9154	0.728492
$906$	0	0
$907$	22.5939	0.750218	0.375109	−	0.926981i	$-0.377605\pi$
$907$	22.5939	0.750218	0.375109	+	0.926981i	$0.377605\pi$
$908$	−1.05584	−0.0350391
$909$	0	0
$910$	−0.584345	−0.0193708
$911$	−18.2580	−0.604914	−0.302457	−	0.953163i	$-0.597807\pi$
$911$	−18.2580	−0.604914	−0.302457	+	0.953163i	$0.597807\pi$
$912$	0	0
$913$	26.1053	0.863958
$914$	21.6451	0.715956
$915$	0	0
$916$	2.02934	0.0670513
$917$	4.76145	0.157237
$918$	0	0
$919$	−20.5698	−0.678534	−0.339267	−	0.940690i	$-0.610179\pi$
$919$	−20.5698	−0.678534	−0.339267	+	0.940690i	$0.610179\pi$
$920$	−6.65049	−0.219260
$921$	0	0
$922$	−29.8303	−0.982407
$923$	−2.08042	−0.0684779
$924$	0	0
$925$	−4.34834	−0.142973
$926$	−41.6881	−1.36996
$927$	0	0
$928$	−9.23945	−0.303300
$929$	8.10385	0.265879	0.132939	−	0.991124i	$-0.457558\pi$
$929$	8.10385	0.265879	0.132939	+	0.991124i	$0.457558\pi$
$930$	0	0
$931$	−8.87560	−0.290886
$932$	2.22845	0.0729953
$933$	0	0
$934$	7.95325	0.260238
$935$	−17.2818	−0.565175
$936$	0	0
$937$	27.2407	0.889914	0.444957	−	0.895552i	$-0.353219\pi$
$937$	27.2407	0.889914	0.444957	+	0.895552i	$0.353219\pi$
$938$	3.11953	0.101856
$939$	0	0
$940$	−3.02804	−0.0987639
$941$	37.8407	1.23357	0.616786	−	0.787131i	$-0.288434\pi$
$941$	37.8407	1.23357	0.616786	+	0.787131i	$0.288434\pi$
$942$	0	0
$943$	20.5303	0.668559
$944$	−15.8318	−0.515280
$945$	0	0
$946$	32.6690	1.06216
$947$	26.4272	0.858768	0.429384	−	0.903122i	$-0.358731\pi$
$947$	26.4272	0.858768	0.429384	+	0.903122i	$0.358731\pi$
$948$	0	0
$949$	−8.35538	−0.271227
$950$	−1.85902	−0.0603147
$951$	0	0
$952$	−18.7472	−0.607600
$953$	18.1144	0.586782	0.293391	−	0.955993i	$-0.405216\pi$
$953$	18.1144	0.586782	0.293391	+	0.955993i	$0.405216\pi$
$954$	0	0
$955$	−20.4556	−0.661927
$956$	2.29854	0.0743402
$957$	0	0
$958$	−36.1808	−1.16895
$959$	4.70966	0.152083
$960$	0	0
$961$	−12.2158	−0.394058
$962$	2.90287	0.0935922
$963$	0	0
$964$	−3.63019	−0.116921
$965$	6.30430	0.202943
$966$	0	0
$967$	42.7148	1.37362	0.686808	−	0.726839i	$-0.259012\pi$
$967$	42.7148	1.37362	0.686808	+	0.726839i	$0.259012\pi$
$968$	−15.4419	−0.496321
$969$	0	0
$970$	−20.0307	−0.643146
$971$	7.24330	0.232449	0.116224	−	0.993223i	$-0.462921\pi$
$971$	7.24330	0.232449	0.116224	+	0.993223i	$0.462921\pi$
$972$	0	0
$973$	5.08630	0.163059
$974$	−6.85997	−0.219808
$975$	0	0
$976$	43.8728	1.40434
$977$	20.0997	0.643048	0.321524	−	0.946901i	$-0.395805\pi$
$977$	20.0997	0.643048	0.321524	+	0.946901i	$0.395805\pi$
$978$	0	0
$979$	−21.0603	−0.673089
$980$	1.84001	0.0587768
$981$	0	0
$982$	−39.2334	−1.25199
$983$	−33.5146	−1.06895	−0.534475	−	0.845185i	$-0.679490\pi$
$983$	−33.5146	−1.06895	−0.534475	+	0.845185i	$0.679490\pi$
$984$	0	0
$985$	−21.0255	−0.669930
$986$	52.0855	1.65874
$987$	0	0
$988$	−0.214868	−0.00683585
$989$	22.9626	0.730167
$990$	0	0
$991$	−53.2001	−1.68996	−0.844979	−	0.534800i	$-0.820387\pi$
$991$	−53.2001	−1.68996	−0.844979	+	0.534800i	$0.820387\pi$
$992$	7.17434	0.227786
$993$	0	0
$994$	4.65045	0.147503
$995$	0.692612	0.0219573
$996$	0	0
$997$	43.3210	1.37199	0.685995	−	0.727606i	$-0.259367\pi$
$997$	43.3210	1.37199	0.685995	+	0.727606i	$0.259367\pi$
$998$	−39.0521	−1.23617
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3645.2.a.g.1.6		15
3.2	odd	2		3645.2.a.h.1.10		15
27.5	odd	18		135.2.k.a.106.4	✓	30
27.11	odd	18		135.2.k.a.121.4	yes	30
27.16	even	9		405.2.k.a.91.2		30
27.22	even	9		405.2.k.a.316.2		30
135.32	even	36		675.2.u.c.349.8		60
135.38	even	36		675.2.u.c.499.8		60
135.59	odd	18		675.2.l.d.376.2		30
135.92	even	36		675.2.u.c.499.3		60
135.113	even	36		675.2.u.c.349.3		60
135.119	odd	18		675.2.l.d.526.2		30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.2.k.a.106.4	✓	30	27.5	odd	18
135.2.k.a.121.4	yes	30	27.11	odd	18
405.2.k.a.91.2		30	27.16	even	9
405.2.k.a.316.2		30	27.22	even	9
675.2.l.d.376.2		30	135.59	odd	18
675.2.l.d.526.2		30	135.119	odd	18
675.2.u.c.349.3		60	135.113	even	36
675.2.u.c.349.8		60	135.32	even	36
675.2.u.c.499.3		60	135.92	even	36
675.2.u.c.499.8		60	135.38	even	36
3645.2.a.g.1.6		15	1.1	even	1	trivial
3645.2.a.h.1.10		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)

\(f(q)\)	\(=\)	\(q-1.30569 q^{2} -0.295165 q^{4} +1.00000 q^{5} +0.875317 q^{7} +2.99678 q^{8} -1.30569 q^{10} +2.41809 q^{11} +0.511284 q^{13} -1.14290 q^{14} -3.32255 q^{16} -7.14686 q^{17} +1.42378 q^{19} -0.295165 q^{20} -3.15729 q^{22} -2.21921 q^{23} +1.00000 q^{25} -0.667581 q^{26} -0.258363 q^{28} +5.58162 q^{29} -4.33407 q^{31} -1.65533 q^{32} +9.33161 q^{34} +0.875317 q^{35} -4.34834 q^{37} -1.85902 q^{38} +2.99678 q^{40} -9.25117 q^{41} -10.3472 q^{43} -0.713737 q^{44} +2.89761 q^{46} +10.2588 q^{47} -6.23382 q^{49} -1.30569 q^{50} -0.150913 q^{52} -1.32921 q^{53} +2.41809 q^{55} +2.62314 q^{56} -7.28788 q^{58} +4.76495 q^{59} -13.2046 q^{61} +5.65897 q^{62} +8.80646 q^{64} +0.511284 q^{65} -2.72949 q^{67} +2.10950 q^{68} -1.14290 q^{70} -4.06901 q^{71} -16.3419 q^{73} +5.67760 q^{74} -0.420251 q^{76} +2.11660 q^{77} +6.44868 q^{79} -3.32255 q^{80} +12.0792 q^{82} +10.7958 q^{83} -7.14686 q^{85} +13.5102 q^{86} +7.24650 q^{88} -8.70945 q^{89} +0.447536 q^{91} +0.655033 q^{92} -13.3949 q^{94} +1.42378 q^{95} +15.3410 q^{97} +8.13946 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - 3 q^{2} + 9 q^{4} + 15 q^{5} - 12 q^{7} - 9 q^{8} - 3 q^{10} - 12 q^{13} - 3 q^{16} - 12 q^{17} - 24 q^{19} + 9 q^{20} - 18 q^{22} - 18 q^{23} + 15 q^{25} + 9 q^{26} - 30 q^{28} + 3 q^{29} - 24 q^{31}+ \cdots - 15 q^{98}+O(q^{100}) \) 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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