LMFDB - Embedded newform 4232.2.a.bb.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4232,2,Mod(1,4232)] 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("4232.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4232, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4232 = 2^{3} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4232.a (trivial)

Newform invariants

comment:select newform

sage:traces = [15,0,1,0,0,0,10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$33.7926901354$
Analytic rank:	$0$
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} - x^{14} - 30 x^{13} + 28 x^{12} + 354 x^{11} - 302 x^{10} - 2111 x^{9} + 1596 x^{8} + 6777 x^{7} + \cdots - 419 $ x^15 - x^14 - 30x^13 + 28x^12 + 354x^11 - 302x^10 - 2111x^9 + 1596x^8 + 6777x^7 - 4292x^6 - 11506x^5 + 5297x^4 + 9480x^3 - 1825x^2 - 3136*x - 419
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 184)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.67059$ of defining polynomial
Character	$\chi$	$=$	4232.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.67059 q^{3} -3.30085 q^{5} +1.48378 q^{7} -0.209129 q^{9} +2.77429 q^{11} +3.41846 q^{13} +5.51436 q^{15} +3.76631 q^{17} -0.514351 q^{19} -2.47878 q^{21} +5.89559 q^{25} +5.36114 q^{27} +7.25504 q^{29} -7.65695 q^{31} -4.63470 q^{33} -4.89772 q^{35} -3.28450 q^{37} -5.71085 q^{39} -10.7718 q^{41} -6.13046 q^{43} +0.690301 q^{45} -8.93322 q^{47} -4.79840 q^{49} -6.29196 q^{51} -11.3758 q^{53} -9.15751 q^{55} +0.859270 q^{57} -2.03410 q^{59} +10.6262 q^{61} -0.310300 q^{63} -11.2838 q^{65} +5.97360 q^{67} +13.0664 q^{71} +10.0288 q^{73} -9.84911 q^{75} +4.11643 q^{77} +7.58909 q^{79} -8.32888 q^{81} +8.04453 q^{83} -12.4320 q^{85} -12.1202 q^{87} +15.7531 q^{89} +5.07224 q^{91} +12.7916 q^{93} +1.69779 q^{95} +7.14396 q^{97} -0.580184 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + q^{3} + 10 q^{7} + 16 q^{9} + 23 q^{11} + 10 q^{15} + 29 q^{19} + q^{21} + 23 q^{25} + q^{27} - 2 q^{29} + 20 q^{31} + 18 q^{33} - 18 q^{35} + 24 q^{37} - 19 q^{39} + 9 q^{41} + 48 q^{43} + 4 q^{45}+ \cdots + 63 q^{99}+O(q^{100}) $ 15 * q + q^3 + 10 * q^7 + 16 * q^9 + 23 * q^11 + 10 * q^15 + 29 * q^19 + q^21 + 23 * q^25 + q^27 - 2 * q^29 + 20 * q^31 + 18 * q^33 - 18 * q^35 + 24 * q^37 - 19 * q^39 + 9 * q^41 + 48 * q^43 + 4 * q^45 - 36 * q^47 + 25 * q^49 + 35 * q^51 - 5 * q^53 - 10 * q^55 + 23 * q^57 - 22 * q^59 + 12 * q^61 + 35 * q^63 - 26 * q^65 + 58 * q^67 + 2 * q^71 + 5 * q^73 - 17 * q^75 + 26 * q^77 + 26 * q^79 - 21 * q^81 + 68 * q^83 - 72 * q^85 + 19 * q^87 - 6 * q^89 + 71 * q^91 - 55 * q^93 - 12 * q^95 + 40 * q^97 + 63 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.67059	−0.964516	−0.482258	−	0.876029i	$-0.660183\pi$
$3$	−1.67059	−0.964516	−0.482258	+	0.876029i	$0.660183\pi$
$4$	0	0
$5$	−3.30085	−1.47618	−0.738092	−	0.674700i	$-0.764273\pi$
$5$	−3.30085	−1.47618	−0.738092	+	0.674700i	$0.764273\pi$
$6$	0	0
$7$	1.48378	0.560815	0.280408	−	0.959881i	$-0.409530\pi$
$7$	1.48378	0.560815	0.280408	+	0.959881i	$0.409530\pi$
$8$	0	0
$9$	−0.209129	−0.0697095
$10$	0	0
$11$	2.77429	0.836480	0.418240	−	0.908336i	$-0.362647\pi$
$11$	2.77429	0.836480	0.418240	+	0.908336i	$0.362647\pi$
$12$	0	0
$13$	3.41846	0.948112	0.474056	−	0.880495i	$-0.342789\pi$
$13$	3.41846	0.948112	0.474056	+	0.880495i	$0.342789\pi$
$14$	0	0
$15$	5.51436	1.42380
$16$	0	0
$17$	3.76631	0.913464	0.456732	−	0.889604i	$-0.349020\pi$
$17$	3.76631	0.913464	0.456732	+	0.889604i	$0.349020\pi$
$18$	0	0
$19$	−0.514351	−0.118000	−0.0590001	−	0.998258i	$-0.518791\pi$
$19$	−0.514351	−0.118000	−0.0590001	+	0.998258i	$0.518791\pi$
$20$	0	0
$21$	−2.47878	−0.540915
$22$	0	0
$23$	0	0
$23$	0	0
$24$	0	0
$25$	5.89559	1.17912
$26$	0	0
$27$	5.36114	1.03175
$28$	0	0
$29$	7.25504	1.34723	0.673614	−	0.739083i	$-0.264741\pi$
$29$	7.25504	1.34723	0.673614	+	0.739083i	$0.264741\pi$
$30$	0	0
$31$	−7.65695	−1.37523	−0.687614	−	0.726076i	$-0.741342\pi$
$31$	−7.65695	−1.37523	−0.687614	+	0.726076i	$0.741342\pi$
$32$	0	0
$33$	−4.63470	−0.806798
$34$	0	0
$35$	−4.89772	−0.827866
$36$	0	0
$37$	−3.28450	−0.539968	−0.269984	−	0.962865i	$-0.587018\pi$
$37$	−3.28450	−0.539968	−0.269984	+	0.962865i	$0.587018\pi$
$38$	0	0
$39$	−5.71085	−0.914468
$40$	0	0
$41$	−10.7718	−1.68227	−0.841137	−	0.540821i	$-0.818113\pi$
$41$	−10.7718	−1.68227	−0.841137	+	0.540821i	$0.818113\pi$
$42$	0	0
$43$	−6.13046	−0.934887	−0.467443	−	0.884023i	$-0.654825\pi$
$43$	−6.13046	−0.934887	−0.467443	+	0.884023i	$0.654825\pi$
$44$	0	0
$45$	0.690301	0.102904
$46$	0	0
$47$	−8.93322	−1.30304	−0.651522	−	0.758630i	$-0.725869\pi$
$47$	−8.93322	−1.30304	−0.651522	+	0.758630i	$0.725869\pi$
$48$	0	0
$49$	−4.79840	−0.685486
$50$	0	0
$51$	−6.29196	−0.881050
$52$	0	0
$53$	−11.3758	−1.56259	−0.781294	−	0.624164i	$-0.785440\pi$
$53$	−11.3758	−1.56259	−0.781294	+	0.624164i	$0.785440\pi$
$54$	0	0
$55$	−9.15751	−1.23480
$56$	0	0
$57$	0.859270	0.113813
$58$	0	0
$59$	−2.03410	−0.264817	−0.132409	−	0.991195i	$-0.542271\pi$
$59$	−2.03410	−0.264817	−0.132409	+	0.991195i	$0.542271\pi$
$60$	0	0
$61$	10.6262	1.36055	0.680273	−	0.732959i	$-0.261861\pi$
$61$	10.6262	1.36055	0.680273	+	0.732959i	$0.261861\pi$
$62$	0	0
$63$	−0.310300	−0.0390942
$64$	0	0
$65$	−11.2838	−1.39959
$66$	0	0
$67$	5.97360	0.729791	0.364896	−	0.931048i	$-0.381105\pi$
$67$	5.97360	0.729791	0.364896	+	0.931048i	$0.381105\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	13.0664	1.55070	0.775349	−	0.631533i	$-0.217574\pi$
$71$	13.0664	1.55070	0.775349	+	0.631533i	$0.217574\pi$
$72$	0	0
$73$	10.0288	1.17378	0.586892	−	0.809665i	$-0.300351\pi$
$73$	10.0288	1.17378	0.586892	+	0.809665i	$0.300351\pi$
$74$	0	0
$75$	−9.84911	−1.13728
$76$	0	0
$77$	4.11643	0.469111
$78$	0	0
$79$	7.58909	0.853840	0.426920	−	0.904289i	$-0.359599\pi$
$79$	7.58909	0.853840	0.426920	+	0.904289i	$0.359599\pi$
$80$	0	0
$81$	−8.32888	−0.925431
$82$	0	0
$83$	8.04453	0.883002	0.441501	−	0.897261i	$-0.354446\pi$
$83$	8.04453	0.883002	0.441501	+	0.897261i	$0.354446\pi$
$84$	0	0
$85$	−12.4320	−1.34844
$86$	0	0
$87$	−12.1202	−1.29942
$88$	0	0
$89$	15.7531	1.66983	0.834915	−	0.550379i	$-0.185517\pi$
$89$	15.7531	1.66983	0.834915	+	0.550379i	$0.185517\pi$
$90$	0	0
$91$	5.07224	0.531716
$92$	0	0
$93$	12.7916	1.32643
$94$	0	0
$95$	1.69779	0.174190
$96$	0	0
$97$	7.14396	0.725359	0.362679	−	0.931914i	$-0.381862\pi$
$97$	7.14396	0.725359	0.362679	+	0.931914i	$0.381862\pi$
$98$	0	0
$99$	−0.580184	−0.0583107
$100$	0	0
$101$	−6.54788	−0.651538	−0.325769	−	0.945449i	$-0.605623\pi$
$101$	−6.54788	−0.651538	−0.325769	+	0.945449i	$0.605623\pi$
$102$	0	0
$103$	12.7901	1.26025	0.630123	−	0.776495i	$-0.283004\pi$
$103$	12.7901	1.26025	0.630123	+	0.776495i	$0.283004\pi$
$104$	0	0
$105$	8.18209	0.798490
$106$	0	0
$107$	5.69776	0.550823	0.275411	−	0.961326i	$-0.411186\pi$
$107$	5.69776	0.550823	0.275411	+	0.961326i	$0.411186\pi$
$108$	0	0
$109$	−1.49751	−0.143435	−0.0717177	−	0.997425i	$-0.522848\pi$
$109$	−1.49751	−0.143435	−0.0717177	+	0.997425i	$0.522848\pi$
$110$	0	0
$111$	5.48705	0.520808
$112$	0	0
$113$	−10.5976	−0.996936	−0.498468	−	0.866908i	$-0.666104\pi$
$113$	−10.5976	−0.996936	−0.498468	+	0.866908i	$0.666104\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−0.714899	−0.0660924
$118$	0	0
$119$	5.58837	0.512285
$120$	0	0
$121$	−3.30331	−0.300300
$122$	0	0
$123$	17.9953	1.62258
$124$	0	0
$125$	−2.95620	−0.264410
$126$	0	0
$127$	−11.3452	−1.00673	−0.503363	−	0.864075i	$-0.667904\pi$
$127$	−11.3452	−1.00673	−0.503363	+	0.864075i	$0.667904\pi$
$128$	0	0
$129$	10.2415	0.901713
$130$	0	0
$131$	1.56164	0.136441	0.0682206	−	0.997670i	$-0.478268\pi$
$131$	1.56164	0.136441	0.0682206	+	0.997670i	$0.478268\pi$
$132$	0	0
$133$	−0.763183	−0.0661763
$134$	0	0
$135$	−17.6963	−1.52305
$136$	0	0
$137$	−10.3690	−0.885885	−0.442942	−	0.896550i	$-0.646065\pi$
$137$	−10.3690	−0.885885	−0.442942	+	0.896550i	$0.646065\pi$
$138$	0	0
$139$	19.2890	1.63607	0.818034	−	0.575169i	$-0.195064\pi$
$139$	19.2890	1.63607	0.818034	+	0.575169i	$0.195064\pi$
$140$	0	0
$141$	14.9237	1.25681
$142$	0	0
$143$	9.48382	0.793077
$144$	0	0
$145$	−23.9478	−1.98876
$146$	0	0
$147$	8.01616	0.661162
$148$	0	0
$149$	−12.2893	−1.00678	−0.503391	−	0.864059i	$-0.667914\pi$
$149$	−12.2893	−1.00678	−0.503391	+	0.864059i	$0.667914\pi$
$150$	0	0
$151$	0.133912	0.0108976	0.00544881	−	0.999985i	$-0.498266\pi$
$151$	0.133912	0.0108976	0.00544881	+	0.999985i	$0.498266\pi$
$152$	0	0
$153$	−0.787643	−0.0636772
$154$	0	0
$155$	25.2744	2.03009
$156$	0	0
$157$	−0.293633	−0.0234345	−0.0117172	−	0.999931i	$-0.503730\pi$
$157$	−0.293633	−0.0234345	−0.0117172	+	0.999931i	$0.503730\pi$
$158$	0	0
$159$	19.0043	1.50714
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−6.15981	−0.482474	−0.241237	−	0.970466i	$-0.577553\pi$
$163$	−6.15981	−0.482474	−0.241237	+	0.970466i	$0.577553\pi$
$164$	0	0
$165$	15.2984	1.19098
$166$	0	0
$167$	−15.0160	−1.16197	−0.580986	−	0.813913i	$-0.697333\pi$
$167$	−15.0160	−1.16197	−0.580986	+	0.813913i	$0.697333\pi$
$168$	0	0
$169$	−1.31410	−0.101084
$170$	0	0
$171$	0.107565	0.00822574
$172$	0	0
$173$	−1.24346	−0.0945387	−0.0472694	−	0.998882i	$-0.515052\pi$
$173$	−1.24346	−0.0945387	−0.0472694	+	0.998882i	$0.515052\pi$
$174$	0	0
$175$	8.74774	0.661267
$176$	0	0
$177$	3.39815	0.255421
$178$	0	0
$179$	17.3294	1.29526	0.647630	−	0.761955i	$-0.275760\pi$
$179$	17.3294	1.29526	0.647630	+	0.761955i	$0.275760\pi$
$180$	0	0
$181$	20.4880	1.52286	0.761431	−	0.648246i	$-0.224497\pi$
$181$	20.4880	1.52286	0.761431	+	0.648246i	$0.224497\pi$
$182$	0	0
$183$	−17.7520	−1.31227
$184$	0	0
$185$	10.8416	0.797092
$186$	0	0
$187$	10.4488	0.764095
$188$	0	0
$189$	7.95474	0.578622
$190$	0	0
$191$	−7.21168	−0.521819	−0.260909	−	0.965363i	$-0.584022\pi$
$191$	−7.21168	−0.521819	−0.260909	+	0.965363i	$0.584022\pi$
$192$	0	0
$193$	4.65971	0.335413	0.167707	−	0.985837i	$-0.446364\pi$
$193$	4.65971	0.335413	0.167707	+	0.985837i	$0.446364\pi$
$194$	0	0
$195$	18.8507	1.34992
$196$	0	0
$197$	−1.13481	−0.0808521	−0.0404261	−	0.999183i	$-0.512872\pi$
$197$	−1.13481	−0.0808521	−0.0404261	+	0.999183i	$0.512872\pi$
$198$	0	0
$199$	15.8557	1.12398	0.561989	−	0.827145i	$-0.310036\pi$
$199$	15.8557	1.12398	0.561989	+	0.827145i	$0.310036\pi$
$200$	0	0
$201$	−9.97944	−0.703895
$202$	0	0
$203$	10.7649	0.755546
$204$	0	0
$205$	35.5561	2.48335
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−1.42696	−0.0987049
$210$	0	0
$211$	19.5298	1.34449	0.672244	−	0.740329i	$-0.265330\pi$
$211$	19.5298	1.34449	0.672244	+	0.740329i	$0.265330\pi$
$212$	0	0
$213$	−21.8286	−1.49567
$214$	0	0
$215$	20.2357	1.38006
$216$	0	0
$217$	−11.3612	−0.771249
$218$	0	0
$219$	−16.7540	−1.13213
$220$	0	0
$221$	12.8750	0.866066
$222$	0	0
$223$	20.9362	1.40199	0.700994	−	0.713167i	$-0.252740\pi$
$223$	20.9362	1.40199	0.700994	+	0.713167i	$0.252740\pi$
$224$	0	0
$225$	−1.23294	−0.0821957
$226$	0	0
$227$	23.0685	1.53111	0.765555	−	0.643371i	$-0.222465\pi$
$227$	23.0685	1.53111	0.765555	+	0.643371i	$0.222465\pi$
$228$	0	0
$229$	7.93760	0.524532	0.262266	−	0.964996i	$-0.415530\pi$
$229$	7.93760	0.524532	0.262266	+	0.964996i	$0.415530\pi$
$230$	0	0
$231$	−6.87687	−0.452465
$232$	0	0
$233$	−13.7867	−0.903196	−0.451598	−	0.892221i	$-0.649146\pi$
$233$	−13.7867	−0.903196	−0.451598	+	0.892221i	$0.649146\pi$
$234$	0	0
$235$	29.4872	1.92353
$236$	0	0
$237$	−12.6783	−0.823542
$238$	0	0
$239$	−12.6236	−0.816553	−0.408276	−	0.912858i	$-0.633870\pi$
$239$	−12.6236	−0.816553	−0.408276	+	0.912858i	$0.633870\pi$
$240$	0	0
$241$	−9.14683	−0.589199	−0.294599	−	0.955621i	$-0.595186\pi$
$241$	−9.14683	−0.589199	−0.294599	+	0.955621i	$0.595186\pi$
$242$	0	0
$243$	−2.16927	−0.139159
$244$	0	0
$245$	15.8388	1.01190
$246$	0	0
$247$	−1.75829	−0.111877
$248$	0	0
$249$	−13.4391	−0.851670
$250$	0	0
$251$	−9.31248	−0.587799	−0.293899	−	0.955836i	$-0.594953\pi$
$251$	−9.31248	−0.587799	−0.293899	+	0.955836i	$0.594953\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	20.7688	1.30059
$256$	0	0
$257$	−23.0638	−1.43868	−0.719339	−	0.694659i	$-0.755555\pi$
$257$	−23.0638	−1.43868	−0.719339	+	0.694659i	$0.755555\pi$
$258$	0	0
$259$	−4.87347	−0.302823
$260$	0	0
$261$	−1.51724	−0.0939146
$262$	0	0
$263$	26.7098	1.64699	0.823497	−	0.567320i	$-0.192020\pi$
$263$	26.7098	1.64699	0.823497	+	0.567320i	$0.192020\pi$
$264$	0	0
$265$	37.5498	2.30667
$266$	0	0
$267$	−26.3171	−1.61058
$268$	0	0
$269$	−7.85330	−0.478824	−0.239412	−	0.970918i	$-0.576955\pi$
$269$	−7.85330	−0.478824	−0.239412	+	0.970918i	$0.576955\pi$
$270$	0	0
$271$	−17.7118	−1.07591	−0.537957	−	0.842972i	$-0.680804\pi$
$271$	−17.7118	−1.07591	−0.537957	+	0.842972i	$0.680804\pi$
$272$	0	0
$273$	−8.47364	−0.512848
$274$	0	0
$275$	16.3561	0.986309
$276$	0	0
$277$	−2.46981	−0.148396	−0.0741982	−	0.997244i	$-0.523640\pi$
$277$	−2.46981	−0.148396	−0.0741982	+	0.997244i	$0.523640\pi$
$278$	0	0
$279$	1.60129	0.0958666
$280$	0	0
$281$	11.4632	0.683837	0.341918	−	0.939730i	$-0.388923\pi$
$281$	11.4632	0.683837	0.341918	+	0.939730i	$0.388923\pi$
$282$	0	0
$283$	22.9149	1.36215	0.681074	−	0.732215i	$-0.261513\pi$
$283$	22.9149	1.36215	0.681074	+	0.732215i	$0.261513\pi$
$284$	0	0
$285$	−2.83632	−0.168009
$286$	0	0
$287$	−15.9830	−0.943446
$288$	0	0
$289$	−2.81492	−0.165584
$290$	0	0
$291$	−11.9346	−0.699620
$292$	0	0
$293$	30.8559	1.80262	0.901310	−	0.433176i	$-0.142607\pi$
$293$	30.8559	1.80262	0.901310	+	0.433176i	$0.142607\pi$
$294$	0	0
$295$	6.71425	0.390919
$296$	0	0
$297$	14.8734	0.863040
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−9.09625	−0.524299
$302$	0	0
$303$	10.9388	0.628419
$304$	0	0
$305$	−35.0755	−2.00842
$306$	0	0
$307$	8.14371	0.464786	0.232393	−	0.972622i	$-0.425344\pi$
$307$	8.14371	0.464786	0.232393	+	0.972622i	$0.425344\pi$
$308$	0	0
$309$	−21.3670	−1.21553
$310$	0	0
$311$	−14.4868	−0.821471	−0.410736	−	0.911754i	$-0.634728\pi$
$311$	−14.4868	−0.821471	−0.410736	+	0.911754i	$0.634728\pi$
$312$	0	0
$313$	2.71085	0.153226	0.0766132	−	0.997061i	$-0.475589\pi$
$313$	2.71085	0.153226	0.0766132	+	0.997061i	$0.475589\pi$
$314$	0	0
$315$	1.02425	0.0577102
$316$	0	0
$317$	−14.2198	−0.798663	−0.399331	−	0.916807i	$-0.630758\pi$
$317$	−14.2198	−0.798663	−0.399331	+	0.916807i	$0.630758\pi$
$318$	0	0
$319$	20.1276	1.12693
$320$	0	0
$321$	−9.51862	−0.531277
$322$	0	0
$323$	−1.93720	−0.107789
$324$	0	0
$325$	20.1539	1.11793
$326$	0	0
$327$	2.50172	0.138346
$328$	0	0
$329$	−13.2549	−0.730767
$330$	0	0
$331$	29.6231	1.62823	0.814116	−	0.580702i	$-0.197222\pi$
$331$	29.6231	1.62823	0.814116	+	0.580702i	$0.197222\pi$
$332$	0	0
$333$	0.686883	0.0376409
$334$	0	0
$335$	−19.7179	−1.07731
$336$	0	0
$337$	21.9791	1.19728	0.598638	−	0.801019i	$-0.295709\pi$
$337$	21.9791	1.19728	0.598638	+	0.801019i	$0.295709\pi$
$338$	0	0
$339$	17.7042	0.961560
$340$	0	0
$341$	−21.2426	−1.15035
$342$	0	0
$343$	−17.5062	−0.945247
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−8.71737	−0.467973	−0.233986	−	0.972240i	$-0.575177\pi$
$347$	−8.71737	−0.467973	−0.233986	+	0.972240i	$0.575177\pi$
$348$	0	0
$349$	2.36899	0.126809	0.0634047	−	0.997988i	$-0.479804\pi$
$349$	2.36899	0.126809	0.0634047	+	0.997988i	$0.479804\pi$
$350$	0	0
$351$	18.3269	0.978216
$352$	0	0
$353$	6.94446	0.369616	0.184808	−	0.982775i	$-0.440834\pi$
$353$	6.94446	0.369616	0.184808	+	0.982775i	$0.440834\pi$
$354$	0	0
$355$	−43.1302	−2.28911
$356$	0	0
$357$	−9.33587	−0.494107
$358$	0	0
$359$	−12.8278	−0.677028	−0.338514	−	0.940961i	$-0.609924\pi$
$359$	−12.8278	−0.677028	−0.338514	+	0.940961i	$0.609924\pi$
$360$	0	0
$361$	−18.7354	−0.986076
$362$	0	0
$363$	5.51847	0.289645
$364$	0	0
$365$	−33.1036	−1.73272
$366$	0	0
$367$	16.5274	0.862724	0.431362	−	0.902179i	$-0.358033\pi$
$367$	16.5274	0.862724	0.431362	+	0.902179i	$0.358033\pi$
$368$	0	0
$369$	2.25269	0.117271
$370$	0	0
$371$	−16.8792	−0.876323
$372$	0	0
$373$	−10.2405	−0.530231	−0.265115	−	0.964217i	$-0.585410\pi$
$373$	−10.2405	−0.530231	−0.265115	+	0.964217i	$0.585410\pi$
$374$	0	0
$375$	4.93859	0.255028
$376$	0	0
$377$	24.8011	1.27732
$378$	0	0
$379$	23.8829	1.22678	0.613390	−	0.789780i	$-0.289805\pi$
$379$	23.8829	1.22678	0.613390	+	0.789780i	$0.289805\pi$
$380$	0	0
$381$	18.9532	0.971003
$382$	0	0
$383$	13.9381	0.712204	0.356102	−	0.934447i	$-0.384106\pi$
$383$	13.9381	0.712204	0.356102	+	0.934447i	$0.384106\pi$
$384$	0	0
$385$	−13.5877	−0.692494
$386$	0	0
$387$	1.28206	0.0651705
$388$	0	0
$389$	−10.7222	−0.543640	−0.271820	−	0.962348i	$-0.587625\pi$
$389$	−10.7222	−0.543640	−0.271820	+	0.962348i	$0.587625\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−2.60886	−0.131600
$394$	0	0
$395$	−25.0504	−1.26042
$396$	0	0
$397$	12.8056	0.642695	0.321348	−	0.946961i	$-0.395864\pi$
$397$	12.8056	0.642695	0.321348	+	0.946961i	$0.395864\pi$
$398$	0	0
$399$	1.27497	0.0638281
$400$	0	0
$401$	−27.0322	−1.34992	−0.674961	−	0.737853i	$-0.735840\pi$
$401$	−27.0322	−1.34992	−0.674961	+	0.737853i	$0.735840\pi$
$402$	0	0
$403$	−26.1750	−1.30387
$404$	0	0
$405$	27.4924	1.36611
$406$	0	0
$407$	−9.11216	−0.451673
$408$	0	0
$409$	7.55832	0.373735	0.186867	−	0.982385i	$-0.440167\pi$
$409$	7.55832	0.373735	0.186867	+	0.982385i	$0.440167\pi$
$410$	0	0
$411$	17.3224	0.854450
$412$	0	0
$413$	−3.01815	−0.148514
$414$	0	0
$415$	−26.5538	−1.30347
$416$	0	0
$417$	−32.2240	−1.57801
$418$	0	0
$419$	1.18896	0.0580844	0.0290422	−	0.999578i	$-0.490754\pi$
$419$	1.18896	0.0580844	0.0290422	+	0.999578i	$0.490754\pi$
$420$	0	0
$421$	29.5074	1.43810	0.719051	−	0.694957i	$-0.244577\pi$
$421$	29.5074	1.43810	0.719051	+	0.694957i	$0.244577\pi$
$422$	0	0
$423$	1.86819	0.0908346
$424$	0	0
$425$	22.2046	1.07708
$426$	0	0
$427$	15.7669	0.763015
$428$	0	0
$429$	−15.8436	−0.764935
$430$	0	0
$431$	16.7495	0.806795	0.403398	−	0.915025i	$-0.367829\pi$
$431$	16.7495	0.806795	0.403398	+	0.915025i	$0.367829\pi$
$432$	0	0
$433$	30.1801	1.45036	0.725182	−	0.688557i	$-0.241756\pi$
$433$	30.1801	1.45036	0.725182	+	0.688557i	$0.241756\pi$
$434$	0	0
$435$	40.0069	1.91819
$436$	0	0
$437$	0	0
$438$	0	0
$439$	19.0215	0.907848	0.453924	−	0.891040i	$-0.350024\pi$
$439$	19.0215	0.907848	0.453924	+	0.891040i	$0.350024\pi$
$440$	0	0
$441$	1.00348	0.0477849
$442$	0	0
$443$	34.1089	1.62056	0.810281	−	0.586041i	$-0.199314\pi$
$443$	34.1089	1.62056	0.810281	+	0.586041i	$0.199314\pi$
$444$	0	0
$445$	−51.9987	−2.46498
$446$	0	0
$447$	20.5305	0.971057
$448$	0	0
$449$	7.75986	0.366211	0.183105	−	0.983093i	$-0.441385\pi$
$449$	7.75986	0.366211	0.183105	+	0.983093i	$0.441385\pi$
$450$	0	0
$451$	−29.8842	−1.40719
$452$	0	0
$453$	−0.223712	−0.0105109
$454$	0	0
$455$	−16.7427	−0.784910
$456$	0	0
$457$	2.26466	0.105937	0.0529683	−	0.998596i	$-0.483132\pi$
$457$	2.26466	0.105937	0.0529683	+	0.998596i	$0.483132\pi$
$458$	0	0
$459$	20.1917	0.942468
$460$	0	0
$461$	21.3135	0.992668	0.496334	−	0.868132i	$-0.334679\pi$
$461$	21.3135	0.992668	0.496334	+	0.868132i	$0.334679\pi$
$462$	0	0
$463$	−0.493172	−0.0229196	−0.0114598	−	0.999934i	$-0.503648\pi$
$463$	−0.493172	−0.0229196	−0.0114598	+	0.999934i	$0.503648\pi$
$464$	0	0
$465$	−42.2232	−1.95805
$466$	0	0
$467$	20.0616	0.928338	0.464169	−	0.885747i	$-0.346353\pi$
$467$	20.0616	0.928338	0.464169	+	0.885747i	$0.346353\pi$
$468$	0	0
$469$	8.86350	0.409278
$470$	0	0
$471$	0.490540	0.0226029
$472$	0	0
$473$	−17.0077	−0.782015
$474$	0	0
$475$	−3.03240	−0.139136
$476$	0	0
$477$	2.37901	0.108927
$478$	0	0
$479$	−3.71347	−0.169673	−0.0848363	−	0.996395i	$-0.527037\pi$
$479$	−3.71347	−0.169673	−0.0848363	+	0.996395i	$0.527037\pi$
$480$	0	0
$481$	−11.2279	−0.511950
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−23.5811	−1.07076
$486$	0	0
$487$	18.5390	0.840081	0.420041	−	0.907505i	$-0.362016\pi$
$487$	18.5390	0.840081	0.420041	+	0.907505i	$0.362016\pi$
$488$	0	0
$489$	10.2905	0.465354
$490$	0	0
$491$	36.0340	1.62619	0.813096	−	0.582129i	$-0.197780\pi$
$491$	36.0340	1.62619	0.813096	+	0.582129i	$0.197780\pi$
$492$	0	0
$493$	27.3247	1.23064
$494$	0	0
$495$	1.91510	0.0860772
$496$	0	0
$497$	19.3877	0.869655
$498$	0	0
$499$	18.7071	0.837446	0.418723	−	0.908114i	$-0.362478\pi$
$499$	18.7071	0.837446	0.418723	+	0.908114i	$0.362478\pi$
$500$	0	0
$501$	25.0856	1.12074
$502$	0	0
$503$	−33.9153	−1.51221	−0.756104	−	0.654452i	$-0.772899\pi$
$503$	−33.9153	−1.51221	−0.756104	+	0.654452i	$0.772899\pi$
$504$	0	0
$505$	21.6135	0.961790
$506$	0	0
$507$	2.19532	0.0974975
$508$	0	0
$509$	−30.0661	−1.33266	−0.666329	−	0.745658i	$-0.732135\pi$
$509$	−30.0661	−1.33266	−0.666329	+	0.745658i	$0.732135\pi$
$510$	0	0
$511$	14.8805	0.658276
$512$	0	0
$513$	−2.75751	−0.121747
$514$	0	0
$515$	−42.2182	−1.86036
$516$	0	0
$517$	−24.7833	−1.08997
$518$	0	0
$519$	2.07732	0.0911841
$520$	0	0
$521$	11.3843	0.498755	0.249378	−	0.968406i	$-0.419774\pi$
$521$	11.3843	0.498755	0.249378	+	0.968406i	$0.419774\pi$
$522$	0	0
$523$	9.34956	0.408828	0.204414	−	0.978885i	$-0.434471\pi$
$523$	9.34956	0.408828	0.204414	+	0.978885i	$0.434471\pi$
$524$	0	0
$525$	−14.6139	−0.637803
$526$	0	0
$527$	−28.8384	−1.25622
$528$	0	0
$529$	0	0
$530$	0	0
$531$	0.425389	0.0184603
$532$	0	0
$533$	−36.8231	−1.59498
$534$	0	0
$535$	−18.8074	−0.813116
$536$	0	0
$537$	−28.9503	−1.24930
$538$	0	0
$539$	−13.3122	−0.573396
$540$	0	0
$541$	1.01082	0.0434587	0.0217294	−	0.999764i	$-0.493083\pi$
$541$	1.01082	0.0434587	0.0217294	+	0.999764i	$0.493083\pi$
$542$	0	0
$543$	−34.2271	−1.46882
$544$	0	0
$545$	4.94305	0.211737
$546$	0	0
$547$	−10.9526	−0.468300	−0.234150	−	0.972201i	$-0.575231\pi$
$547$	−10.9526	−0.468300	−0.234150	+	0.972201i	$0.575231\pi$
$548$	0	0
$549$	−2.22224	−0.0948431
$550$	0	0
$551$	−3.73164	−0.158973
$552$	0	0
$553$	11.2605	0.478846
$554$	0	0
$555$	−18.1119	−0.768808
$556$	0	0
$557$	2.93193	0.124230	0.0621149	−	0.998069i	$-0.480215\pi$
$557$	2.93193	0.124230	0.0621149	+	0.998069i	$0.480215\pi$
$558$	0	0
$559$	−20.9568	−0.886377
$560$	0	0
$561$	−17.4557	−0.736981
$562$	0	0
$563$	−17.1366	−0.722223	−0.361112	−	0.932523i	$-0.617603\pi$
$563$	−17.1366	−0.722223	−0.361112	+	0.932523i	$0.617603\pi$
$564$	0	0
$565$	34.9810	1.47166
$566$	0	0
$567$	−12.3582	−0.518996
$568$	0	0
$569$	−38.3739	−1.60872	−0.804360	−	0.594143i	$-0.797491\pi$
$569$	−38.3739	−1.60872	−0.804360	+	0.594143i	$0.797491\pi$
$570$	0	0
$571$	16.9152	0.707879	0.353939	−	0.935268i	$-0.384842\pi$
$571$	16.9152	0.707879	0.353939	+	0.935268i	$0.384842\pi$
$572$	0	0
$573$	12.0478	0.503303
$574$	0	0
$575$	0	0
$576$	0	0
$577$	17.2677	0.718866	0.359433	−	0.933171i	$-0.382970\pi$
$577$	17.2677	0.718866	0.359433	+	0.933171i	$0.382970\pi$
$578$	0	0
$579$	−7.78446	−0.323511
$580$	0	0
$581$	11.9363	0.495201
$582$	0	0
$583$	−31.5598	−1.30707
$584$	0	0
$585$	2.35977	0.0975645
$586$	0	0
$587$	32.0512	1.32289	0.661447	−	0.749992i	$-0.269943\pi$
$587$	32.0512	1.32289	0.661447	+	0.749992i	$0.269943\pi$
$588$	0	0
$589$	3.93836	0.162277
$590$	0	0
$591$	1.89581	0.0779831
$592$	0	0
$593$	−3.05933	−0.125632	−0.0628158	−	0.998025i	$-0.520008\pi$
$593$	−3.05933	−0.125632	−0.0628158	+	0.998025i	$0.520008\pi$
$594$	0	0
$595$	−18.4463	−0.756226
$596$	0	0
$597$	−26.4883	−1.08409
$598$	0	0
$599$	−20.8858	−0.853370	−0.426685	−	0.904400i	$-0.640319\pi$
$599$	−20.8858	−0.853370	−0.426685	+	0.904400i	$0.640319\pi$
$600$	0	0
$601$	32.6184	1.33053	0.665267	−	0.746606i	$-0.268318\pi$
$601$	32.6184	1.33053	0.665267	+	0.746606i	$0.268318\pi$
$602$	0	0
$603$	−1.24925	−0.0508734
$604$	0	0
$605$	10.9037	0.443299
$606$	0	0
$607$	−38.7411	−1.57245	−0.786227	−	0.617938i	$-0.787968\pi$
$607$	−38.7411	−1.57245	−0.786227	+	0.617938i	$0.787968\pi$
$608$	0	0
$609$	−17.9837	−0.728736
$610$	0	0
$611$	−30.5379	−1.23543
$612$	0	0
$613$	17.5002	0.706825	0.353412	−	0.935468i	$-0.385021\pi$
$613$	17.5002	0.706825	0.353412	+	0.935468i	$0.385021\pi$
$614$	0	0
$615$	−59.3997	−2.39523
$616$	0	0
$617$	13.5760	0.546547	0.273274	−	0.961936i	$-0.411894\pi$
$617$	13.5760	0.546547	0.273274	+	0.961936i	$0.411894\pi$
$618$	0	0
$619$	−40.9992	−1.64790	−0.823948	−	0.566666i	$-0.808233\pi$
$619$	−40.9992	−1.64790	−0.823948	+	0.566666i	$0.808233\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	23.3742	0.936467
$624$	0	0
$625$	−19.7200	−0.788800
$626$	0	0
$627$	2.38386	0.0952024
$628$	0	0
$629$	−12.3704	−0.493242
$630$	0	0
$631$	5.73600	0.228346	0.114173	−	0.993461i	$-0.463578\pi$
$631$	5.73600	0.228346	0.114173	+	0.993461i	$0.463578\pi$
$632$	0	0
$633$	−32.6263	−1.29678
$634$	0	0
$635$	37.4488	1.48611
$636$	0	0
$637$	−16.4032	−0.649917
$638$	0	0
$639$	−2.73256	−0.108098
$640$	0	0
$641$	18.3019	0.722880	0.361440	−	0.932395i	$-0.382285\pi$
$641$	18.3019	0.722880	0.361440	+	0.932395i	$0.382285\pi$
$642$	0	0
$643$	−14.0965	−0.555913	−0.277957	−	0.960594i	$-0.589657\pi$
$643$	−14.0965	−0.555913	−0.277957	+	0.960594i	$0.589657\pi$
$644$	0	0
$645$	−33.8056	−1.33109
$646$	0	0
$647$	31.2770	1.22963	0.614813	−	0.788673i	$-0.289232\pi$
$647$	31.2770	1.22963	0.614813	+	0.788673i	$0.289232\pi$
$648$	0	0
$649$	−5.64319	−0.221515
$650$	0	0
$651$	18.9799	0.743882
$652$	0	0
$653$	−0.0904533	−0.00353971	−0.00176986	−	0.999998i	$-0.500563\pi$
$653$	−0.0904533	−0.00353971	−0.00176986	+	0.999998i	$0.500563\pi$
$654$	0	0
$655$	−5.15473	−0.201412
$656$	0	0
$657$	−2.09731	−0.0818240
$658$	0	0
$659$	−27.6709	−1.07790	−0.538952	−	0.842336i	$-0.681180\pi$
$659$	−27.6709	−1.07790	−0.538952	+	0.842336i	$0.681180\pi$
$660$	0	0
$661$	−11.7434	−0.456764	−0.228382	−	0.973572i	$-0.573344\pi$
$661$	−11.7434	−0.456764	−0.228382	+	0.973572i	$0.573344\pi$
$662$	0	0
$663$	−21.5088	−0.835334
$664$	0	0
$665$	2.51915	0.0976884
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−34.9757	−1.35224
$670$	0	0
$671$	29.4802	1.13807
$672$	0	0
$673$	28.5751	1.10149	0.550745	−	0.834673i	$-0.314343\pi$
$673$	28.5751	1.10149	0.550745	+	0.834673i	$0.314343\pi$
$674$	0	0
$675$	31.6071	1.21656
$676$	0	0
$677$	28.6106	1.09959	0.549797	−	0.835299i	$-0.314705\pi$
$677$	28.6106	1.09959	0.549797	+	0.835299i	$0.314705\pi$
$678$	0	0
$679$	10.6000	0.406792
$680$	0	0
$681$	−38.5380	−1.47678
$682$	0	0
$683$	11.1726	0.427509	0.213754	−	0.976887i	$-0.431431\pi$
$683$	11.1726	0.427509	0.213754	+	0.976887i	$0.431431\pi$
$684$	0	0
$685$	34.2265	1.30773
$686$	0	0
$687$	−13.2605	−0.505919
$688$	0	0
$689$	−38.8878	−1.48151
$690$	0	0
$691$	−26.1688	−0.995509	−0.497755	−	0.867318i	$-0.665842\pi$
$691$	−26.1688	−0.995509	−0.497755	+	0.867318i	$0.665842\pi$
$692$	0	0
$693$	−0.860864	−0.0327015
$694$	0	0
$695$	−63.6699	−2.41514
$696$	0	0
$697$	−40.5700	−1.53670
$698$	0	0
$699$	23.0319	0.871147
$700$	0	0
$701$	−0.417323	−0.0157621	−0.00788103	−	0.999969i	$-0.502509\pi$
$701$	−0.417323	−0.0157621	−0.00788103	+	0.999969i	$0.502509\pi$
$702$	0	0
$703$	1.68939	0.0637164
$704$	0	0
$705$	−49.2610	−1.85528
$706$	0	0
$707$	−9.71560	−0.365393
$708$	0	0
$709$	−2.56216	−0.0962238	−0.0481119	−	0.998842i	$-0.515320\pi$
$709$	−2.56216	−0.0962238	−0.0481119	+	0.998842i	$0.515320\pi$
$710$	0	0
$711$	−1.58710	−0.0595208
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−31.3046	−1.17073
$716$	0	0
$717$	21.0889	0.787578
$718$	0	0
$719$	29.9255	1.11603	0.558017	−	0.829829i	$-0.311562\pi$
$719$	29.9255	1.11603	0.558017	+	0.829829i	$0.311562\pi$
$720$	0	0
$721$	18.9777	0.706766
$722$	0	0
$723$	15.2806	0.568292
$724$	0	0
$725$	42.7727	1.58854
$726$	0	0
$727$	12.6701	0.469908	0.234954	−	0.972006i	$-0.424506\pi$
$727$	12.6701	0.469908	0.234954	+	0.972006i	$0.424506\pi$
$728$	0	0
$729$	28.6106	1.05965
$730$	0	0
$731$	−23.0892	−0.853986
$732$	0	0
$733$	5.50625	0.203378	0.101689	−	0.994816i	$-0.467575\pi$
$733$	5.50625	0.203378	0.101689	+	0.994816i	$0.467575\pi$
$734$	0	0
$735$	−26.4601	−0.975997
$736$	0	0
$737$	16.5725	0.610456
$738$	0	0
$739$	32.7590	1.20506	0.602529	−	0.798097i	$-0.294160\pi$
$739$	32.7590	1.20506	0.602529	+	0.798097i	$0.294160\pi$
$740$	0	0
$741$	2.93738	0.107907
$742$	0	0
$743$	26.7451	0.981182	0.490591	−	0.871390i	$-0.336781\pi$
$743$	26.7451	0.981182	0.490591	+	0.871390i	$0.336781\pi$
$744$	0	0
$745$	40.5652	1.48620
$746$	0	0
$747$	−1.68234	−0.0615537
$748$	0	0
$749$	8.45421	0.308910
$750$	0	0
$751$	−34.0124	−1.24113	−0.620566	−	0.784154i	$-0.713097\pi$
$751$	−34.0124	−1.24113	−0.620566	+	0.784154i	$0.713097\pi$
$752$	0	0
$753$	15.5573	0.566941
$754$	0	0
$755$	−0.442023	−0.0160869
$756$	0	0
$757$	−44.6288	−1.62206	−0.811031	−	0.585003i	$-0.801093\pi$
$757$	−44.6288	−1.62206	−0.811031	+	0.585003i	$0.801093\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	17.8091	0.645581	0.322791	−	0.946470i	$-0.395379\pi$
$761$	17.8091	0.645581	0.322791	+	0.946470i	$0.395379\pi$
$762$	0	0
$763$	−2.22197	−0.0804407
$764$	0	0
$765$	2.59989	0.0939992
$766$	0	0
$767$	−6.95350	−0.251076
$768$	0	0
$769$	31.9856	1.15343	0.576715	−	0.816946i	$-0.304334\pi$
$769$	31.9856	1.15343	0.576715	+	0.816946i	$0.304334\pi$
$770$	0	0
$771$	38.5301	1.38763
$772$	0	0
$773$	20.2306	0.727644	0.363822	−	0.931468i	$-0.381472\pi$
$773$	20.2306	0.727644	0.363822	+	0.931468i	$0.381472\pi$
$774$	0	0
$775$	−45.1422	−1.62156
$776$	0	0
$777$	8.14157	0.292077
$778$	0	0
$779$	5.54049	0.198509
$780$	0	0
$781$	36.2500	1.29713
$782$	0	0
$783$	38.8953	1.39000
$784$	0	0
$785$	0.969237	0.0345936
$786$	0	0
$787$	27.2923	0.972865	0.486432	−	0.873718i	$-0.338298\pi$
$787$	27.2923	0.972865	0.486432	+	0.873718i	$0.338298\pi$
$788$	0	0
$789$	−44.6211	−1.58855
$790$	0	0
$791$	−15.7244	−0.559097
$792$	0	0
$793$	36.3253	1.28995
$794$	0	0
$795$	−62.7303	−2.22481
$796$	0	0
$797$	−51.1405	−1.81149	−0.905744	−	0.423825i	$-0.860687\pi$
$797$	−51.1405	−1.81149	−0.905744	+	0.423825i	$0.860687\pi$
$798$	0	0
$799$	−33.6452	−1.19028
$800$	0	0
$801$	−3.29443	−0.116403
$802$	0	0
$803$	27.8229	0.981848
$804$	0	0
$805$	0	0
$806$	0	0
$807$	13.1197	0.461834
$808$	0	0
$809$	0.690659	0.0242823	0.0121411	−	0.999926i	$-0.496135\pi$
$809$	0.690659	0.0242823	0.0121411	+	0.999926i	$0.496135\pi$
$810$	0	0
$811$	−32.3309	−1.13529	−0.567645	−	0.823273i	$-0.692145\pi$
$811$	−32.3309	−1.13529	−0.567645	+	0.823273i	$0.692145\pi$
$812$	0	0
$813$	29.5891	1.03774
$814$	0	0
$815$	20.3326	0.712220
$816$	0	0
$817$	3.15321	0.110317
$818$	0	0
$819$	−1.06075	−0.0370656
$820$	0	0
$821$	6.09808	0.212824	0.106412	−	0.994322i	$-0.466064\pi$
$821$	6.09808	0.212824	0.106412	+	0.994322i	$0.466064\pi$
$822$	0	0
$823$	27.6548	0.963986	0.481993	−	0.876175i	$-0.339913\pi$
$823$	27.6548	0.963986	0.481993	+	0.876175i	$0.339913\pi$
$824$	0	0
$825$	−27.3243	−0.951310
$826$	0	0
$827$	17.5955	0.611857	0.305929	−	0.952054i	$-0.401033\pi$
$827$	17.5955	0.611857	0.305929	+	0.952054i	$0.401033\pi$
$828$	0	0
$829$	26.7911	0.930494	0.465247	−	0.885181i	$-0.345965\pi$
$829$	26.7911	0.930494	0.465247	+	0.885181i	$0.345965\pi$
$830$	0	0
$831$	4.12604	0.143131
$832$	0	0
$833$	−18.0723	−0.626167
$834$	0	0
$835$	49.5655	1.71528
$836$	0	0
$837$	−41.0500	−1.41889
$838$	0	0
$839$	−5.62747	−0.194282	−0.0971410	−	0.995271i	$-0.530970\pi$
$839$	−5.62747	−0.194282	−0.0971410	+	0.995271i	$0.530970\pi$
$840$	0	0
$841$	23.6356	0.815022
$842$	0	0
$843$	−19.1503	−0.659571
$844$	0	0
$845$	4.33763	0.149219
$846$	0	0
$847$	−4.90137	−0.168413
$848$	0	0
$849$	−38.2814	−1.31381
$850$	0	0
$851$	0	0
$852$	0	0
$853$	12.9685	0.444032	0.222016	−	0.975043i	$-0.428736\pi$
$853$	12.9685	0.444032	0.222016	+	0.975043i	$0.428736\pi$
$854$	0	0
$855$	−0.355057	−0.0121427
$856$	0	0
$857$	11.6700	0.398641	0.199321	−	0.979934i	$-0.436127\pi$
$857$	11.6700	0.398641	0.199321	+	0.979934i	$0.436127\pi$
$858$	0	0
$859$	−41.2591	−1.40774	−0.703871	−	0.710328i	$-0.748547\pi$
$859$	−41.2591	−1.40774	−0.703871	+	0.710328i	$0.748547\pi$
$860$	0	0
$861$	26.7010	0.909968
$862$	0	0
$863$	5.67249	0.193094	0.0965468	−	0.995328i	$-0.469220\pi$
$863$	5.67249	0.193094	0.0965468	+	0.995328i	$0.469220\pi$
$864$	0	0
$865$	4.10448	0.139557
$866$	0	0
$867$	4.70258	0.159708
$868$	0	0
$869$	21.0544	0.714220
$870$	0	0
$871$	20.4205	0.691924
$872$	0	0
$873$	−1.49401	−0.0505644
$874$	0	0
$875$	−4.38634	−0.148285
$876$	0	0
$877$	21.4440	0.724112	0.362056	−	0.932156i	$-0.382075\pi$
$877$	21.4440	0.724112	0.362056	+	0.932156i	$0.382075\pi$
$878$	0	0
$879$	−51.5475	−1.73865
$880$	0	0
$881$	−8.86932	−0.298815	−0.149407	−	0.988776i	$-0.547737\pi$
$881$	−8.86932	−0.298815	−0.149407	+	0.988776i	$0.547737\pi$
$882$	0	0
$883$	47.1229	1.58581	0.792905	−	0.609345i	$-0.208568\pi$
$883$	47.1229	1.58581	0.792905	+	0.609345i	$0.208568\pi$
$884$	0	0
$885$	−11.2168	−0.377048
$886$	0	0
$887$	6.75358	0.226763	0.113382	−	0.993552i	$-0.463832\pi$
$887$	6.75358	0.226763	0.113382	+	0.993552i	$0.463832\pi$
$888$	0	0
$889$	−16.8338	−0.564587
$890$	0	0
$891$	−23.1067	−0.774105
$892$	0	0
$893$	4.59481	0.153759
$894$	0	0
$895$	−57.2017	−1.91204
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−55.5515	−1.85275
$900$	0	0
$901$	−42.8448	−1.42737
$902$	0	0
$903$	15.1961	0.505695
$904$	0	0
$905$	−67.6278	−2.24802
$906$	0	0
$907$	17.0399	0.565800	0.282900	−	0.959149i	$-0.408704\pi$
$907$	17.0399	0.565800	0.282900	+	0.959149i	$0.408704\pi$
$908$	0	0
$909$	1.36935	0.0454184
$910$	0	0
$911$	−12.1556	−0.402734	−0.201367	−	0.979516i	$-0.564538\pi$
$911$	−12.1556	−0.402734	−0.201367	+	0.979516i	$0.564538\pi$
$912$	0	0
$913$	22.3179	0.738614
$914$	0	0
$915$	58.5968	1.93715
$916$	0	0
$917$	2.31713	0.0765183
$918$	0	0
$919$	1.52112	0.0501772	0.0250886	−	0.999685i	$-0.492013\pi$
$919$	1.52112	0.0501772	0.0250886	+	0.999685i	$0.492013\pi$
$920$	0	0
$921$	−13.6048	−0.448293
$922$	0	0
$923$	44.6671	1.47023
$924$	0	0
$925$	−19.3641	−0.636686
$926$	0	0
$927$	−2.67478	−0.0878512
$928$	0	0
$929$	−4.04569	−0.132735	−0.0663673	−	0.997795i	$-0.521141\pi$
$929$	−4.04569	−0.132735	−0.0663673	+	0.997795i	$0.521141\pi$
$930$	0	0
$931$	2.46806	0.0808875
$932$	0	0
$933$	24.2015	0.792322
$934$	0	0
$935$	−34.4900	−1.12794
$936$	0	0
$937$	−36.7638	−1.20102	−0.600511	−	0.799616i	$-0.705036\pi$
$937$	−36.7638	−1.20102	−0.600511	+	0.799616i	$0.705036\pi$
$938$	0	0
$939$	−4.52872	−0.147789
$940$	0	0
$941$	−10.0727	−0.328360	−0.164180	−	0.986430i	$-0.552498\pi$
$941$	−10.0727	−0.328360	−0.164180	+	0.986430i	$0.552498\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−26.2574	−0.854152
$946$	0	0
$947$	−18.3978	−0.597849	−0.298925	−	0.954277i	$-0.596628\pi$
$947$	−18.3978	−0.597849	−0.298925	+	0.954277i	$0.596628\pi$
$948$	0	0
$949$	34.2832	1.11288
$950$	0	0
$951$	23.7554	0.770323
$952$	0	0
$953$	32.3556	1.04810	0.524050	−	0.851687i	$-0.324420\pi$
$953$	32.3556	1.04810	0.524050	+	0.851687i	$0.324420\pi$
$954$	0	0
$955$	23.8047	0.770300
$956$	0	0
$957$	−33.6250	−1.08694
$958$	0	0
$959$	−15.3853	−0.496818
$960$	0	0
$961$	27.6289	0.891254
$962$	0	0
$963$	−1.19156	−0.0383976
$964$	0	0
$965$	−15.3810	−0.495131
$966$	0	0
$967$	−52.5187	−1.68889	−0.844443	−	0.535645i	$-0.820068\pi$
$967$	−52.5187	−1.68889	−0.844443	+	0.535645i	$0.820068\pi$
$968$	0	0
$969$	3.23627	0.103964
$970$	0	0
$971$	22.2023	0.712505	0.356252	−	0.934390i	$-0.384054\pi$
$971$	22.2023	0.712505	0.356252	+	0.934390i	$0.384054\pi$
$972$	0	0
$973$	28.6205	0.917532
$974$	0	0
$975$	−33.6688	−1.07827
$976$	0	0
$977$	−8.58787	−0.274750	−0.137375	−	0.990519i	$-0.543867\pi$
$977$	−8.58787	−0.274750	−0.137375	+	0.990519i	$0.543867\pi$
$978$	0	0
$979$	43.7038	1.39678
$980$	0	0
$981$	0.313172	0.00999881
$982$	0	0
$983$	18.0356	0.575246	0.287623	−	0.957744i	$-0.407135\pi$
$983$	18.0356	0.575246	0.287623	+	0.957744i	$0.407135\pi$
$984$	0	0
$985$	3.74585	0.119353
$986$	0	0
$987$	22.1435	0.704836
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−32.2791	−1.02538	−0.512689	−	0.858574i	$-0.671351\pi$
$991$	−32.2791	−1.02538	−0.512689	+	0.858574i	$0.671351\pi$
$992$	0	0
$993$	−49.4880	−1.57046
$994$	0	0
$995$	−52.3371	−1.65920
$996$	0	0
$997$	40.1217	1.27067	0.635333	−	0.772238i	$-0.280863\pi$
$997$	40.1217	1.27067	0.635333	+	0.772238i	$0.280863\pi$
$998$	0	0
$999$	−17.6087	−0.557113

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4232.2.a.bb.1.5		15
4.3	odd	2		8464.2.a.cg.1.11		15
23.13	even	11		184.2.i.b.169.1	yes	30
23.16	even	11		184.2.i.b.49.1	✓	30
23.22	odd	2		4232.2.a.ba.1.5		15
92.39	odd	22		368.2.m.e.49.3		30
92.59	odd	22		368.2.m.e.353.3		30
92.91	even	2		8464.2.a.ch.1.11		15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
184.2.i.b.49.1	✓	30	23.16	even	11
184.2.i.b.169.1	yes	30	23.13	even	11
368.2.m.e.49.3		30	92.39	odd	22
368.2.m.e.353.3		30	92.59	odd	22
4232.2.a.ba.1.5		15	23.22	odd	2
4232.2.a.bb.1.5		15	1.1	even	1	trivial
8464.2.a.cg.1.11		15	4.3	odd	2
8464.2.a.ch.1.11		15	92.91	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4232 = 2^{3} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4232.a (trivial)

\(f(q)\)	\(=\)	\(q-1.67059 q^{3} -3.30085 q^{5} +1.48378 q^{7} -0.209129 q^{9} +2.77429 q^{11} +3.41846 q^{13} +5.51436 q^{15} +3.76631 q^{17} -0.514351 q^{19} -2.47878 q^{21} +5.89559 q^{25} +5.36114 q^{27} +7.25504 q^{29} -7.65695 q^{31} -4.63470 q^{33} -4.89772 q^{35} -3.28450 q^{37} -5.71085 q^{39} -10.7718 q^{41} -6.13046 q^{43} +0.690301 q^{45} -8.93322 q^{47} -4.79840 q^{49} -6.29196 q^{51} -11.3758 q^{53} -9.15751 q^{55} +0.859270 q^{57} -2.03410 q^{59} +10.6262 q^{61} -0.310300 q^{63} -11.2838 q^{65} +5.97360 q^{67} +13.0664 q^{71} +10.0288 q^{73} -9.84911 q^{75} +4.11643 q^{77} +7.58909 q^{79} -8.32888 q^{81} +8.04453 q^{83} -12.4320 q^{85} -12.1202 q^{87} +15.7531 q^{89} +5.07224 q^{91} +12.7916 q^{93} +1.69779 q^{95} +7.14396 q^{97} -0.580184 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + q^{3} + 10 q^{7} + 16 q^{9} + 23 q^{11} + 10 q^{15} + 29 q^{19} + q^{21} + 23 q^{25} + q^{27} - 2 q^{29} + 20 q^{31} + 18 q^{33} - 18 q^{35} + 24 q^{37} - 19 q^{39} + 9 q^{41} + 48 q^{43} + 4 q^{45}+ \cdots + 63 q^{99}+O(q^{100}) \) 15 * q + q^3 + 10 * q^7 + 16 * q^9 + 23 * q^11 + 10 * q^15 + 29 * q^19 + q^21 + 23 * q^25 + q^27 - 2 * q^29 + 20 * q^31 + 18 * q^33 - 18 * q^35 + 24 * q^37 - 19 * q^39 + 9 * q^41 + 48 * q^43 + 4 * q^45 - 36 * q^47 + 25 * q^49 + 35 * q^51 - 5 * q^53 - 10 * q^55 + 23 * q^57 - 22 * q^59 + 12 * q^61 + 35 * q^63 - 26 * q^65 + 58 * q^67 + 2 * q^71 + 5 * q^73 - 17 * q^75 + 26 * q^77 + 26 * q^79 - 21 * q^81 + 68 * q^83 - 72 * q^85 + 19 * q^87 - 6 * q^89 + 71 * q^91 - 55 * q^93 - 12 * q^95 + 40 * q^97 + 63 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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