LMFDB - Embedded newform 2312.2.a.w.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2312,2,Mod(1,2312)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2312, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2312.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2312 = 2^{3} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2312.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$18.4614129473$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 32x^{10} + 380x^{8} - 2000x^{6} + 4068x^{4} - 800x^{2} + 32 $ x^12 - 32x^10 + 380x^8 - 2000x^6 + 4068x^4 - 800*x^2 + 32
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 136)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.67543$ of defining polynomial
Character	$\chi$	$=$	2312.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.67543 q^{3} -1.02573 q^{5} -2.33494 q^{7} +4.15793 q^{9} -5.20843 q^{11} -4.35777 q^{13} +2.74428 q^{15} -2.67684 q^{19} +6.24696 q^{21} -4.10630 q^{23} -3.94787 q^{25} -3.09796 q^{27} +3.84029 q^{29} -10.9766 q^{31} +13.9348 q^{33} +2.39502 q^{35} -1.10586 q^{37} +11.6589 q^{39} -2.62484 q^{41} -4.05380 q^{43} -4.26493 q^{45} +10.4047 q^{47} -1.54807 q^{49} -7.77630 q^{53} +5.34245 q^{55} +7.16170 q^{57} -10.8822 q^{59} -7.95894 q^{61} -9.70851 q^{63} +4.46990 q^{65} -6.44882 q^{67} +10.9861 q^{69} -1.87861 q^{71} +13.2020 q^{73} +10.5623 q^{75} +12.1613 q^{77} +2.50172 q^{79} -4.18540 q^{81} -0.0329340 q^{83} -10.2744 q^{87} +7.34936 q^{89} +10.1751 q^{91} +29.3672 q^{93} +2.74572 q^{95} +18.2600 q^{97} -21.6563 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 28 q^{9} + 24 q^{13} + 8 q^{15} - 8 q^{19} + 16 q^{21} + 20 q^{25} + 24 q^{33} - 32 q^{35} + 8 q^{43} + 24 q^{47} + 36 q^{49} + 8 q^{53} + 56 q^{55} - 40 q^{59} + 40 q^{67} + 56 q^{69} + 80 q^{77}+ \cdots + 40 q^{93}+O(q^{100}) $ 12 * q + 28 * q^9 + 24 * q^13 + 8 * q^15 - 8 * q^19 + 16 * q^21 + 20 * q^25 + 24 * q^33 - 32 * q^35 + 8 * q^43 + 24 * q^47 + 36 * q^49 + 8 * q^53 + 56 * q^55 - 40 * q^59 + 40 * q^67 + 56 * q^69 + 80 * q^77 + 60 * q^81 - 24 * q^83 + 24 * q^87 + 48 * q^89 + 40 * q^93

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
$3$	−2.67543	−1.54466	−0.772330	−	0.635221i	$-0.780909\pi$
$3$	−2.67543	−1.54466	−0.772330	+	0.635221i	$0.780909\pi$
$4$	0	0
$5$	−1.02573	−0.458722	−0.229361	−	0.973341i	$-0.573664\pi$
$5$	−1.02573	−0.458722	−0.229361	+	0.973341i	$0.573664\pi$
$6$	0	0
$7$	−2.33494	−0.882523	−0.441262	−	0.897379i	$-0.645469\pi$
$7$	−2.33494	−0.882523	−0.441262	+	0.897379i	$0.645469\pi$
$8$	0	0
$9$	4.15793	1.38598
$10$	0	0
$11$	−5.20843	−1.57040	−0.785200	−	0.619242i	$-0.787440\pi$
$11$	−5.20843	−1.57040	−0.785200	+	0.619242i	$0.787440\pi$
$12$	0	0
$13$	−4.35777	−1.20863	−0.604313	−	0.796747i	$-0.706552\pi$
$13$	−4.35777	−1.20863	−0.604313	+	0.796747i	$0.706552\pi$
$14$	0	0
$15$	2.74428	0.708569
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	−2.67684	−0.614109	−0.307054	−	0.951692i	$-0.599343\pi$
$19$	−2.67684	−0.614109	−0.307054	+	0.951692i	$0.599343\pi$
$20$	0	0
$21$	6.24696	1.36320
$22$	0	0
$23$	−4.10630	−0.856223	−0.428112	−	0.903726i	$-0.640821\pi$
$23$	−4.10630	−0.856223	−0.428112	+	0.903726i	$0.640821\pi$
$24$	0	0
$25$	−3.94787	−0.789575
$26$	0	0
$27$	−3.09796	−0.596204
$28$	0	0
$29$	3.84029	0.713124	0.356562	−	0.934272i	$-0.383949\pi$
$29$	3.84029	0.713124	0.356562	+	0.934272i	$0.383949\pi$
$30$	0	0
$31$	−10.9766	−1.97146	−0.985728	−	0.168344i	$-0.946158\pi$
$31$	−10.9766	−1.97146	−0.985728	+	0.168344i	$0.946158\pi$
$32$	0	0
$33$	13.9348	2.42574
$34$	0	0
$35$	2.39502	0.404832
$36$	0	0
$37$	−1.10586	−0.181802	−0.0909012	−	0.995860i	$-0.528975\pi$
$37$	−1.10586	−0.181802	−0.0909012	+	0.995860i	$0.528975\pi$
$38$	0	0
$39$	11.6589	1.86692
$40$	0	0
$41$	−2.62484	−0.409932	−0.204966	−	0.978769i	$-0.565708\pi$
$41$	−2.62484	−0.409932	−0.204966	+	0.978769i	$0.565708\pi$
$42$	0	0
$43$	−4.05380	−0.618198	−0.309099	−	0.951030i	$-0.600027\pi$
$43$	−4.05380	−0.618198	−0.309099	+	0.951030i	$0.600027\pi$
$44$	0	0
$45$	−4.26493	−0.635778
$46$	0	0
$47$	10.4047	1.51768	0.758838	−	0.651280i	$-0.225767\pi$
$47$	10.4047	1.51768	0.758838	+	0.651280i	$0.225767\pi$
$48$	0	0
$49$	−1.54807	−0.221153
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−7.77630	−1.06816	−0.534078	−	0.845435i	$-0.679341\pi$
$53$	−7.77630	−1.06816	−0.534078	+	0.845435i	$0.679341\pi$
$54$	0	0
$55$	5.34245	0.720376
$56$	0	0
$57$	7.16170	0.948590
$58$	0	0
$59$	−10.8822	−1.41674	−0.708372	−	0.705839i	$-0.750570\pi$
$59$	−10.8822	−1.41674	−0.708372	+	0.705839i	$0.750570\pi$
$60$	0	0
$61$	−7.95894	−1.01904	−0.509519	−	0.860459i	$-0.670177\pi$
$61$	−7.95894	−1.01904	−0.509519	+	0.860459i	$0.670177\pi$
$62$	0	0
$63$	−9.70851	−1.22316
$64$	0	0
$65$	4.46990	0.554423
$66$	0	0
$67$	−6.44882	−0.787848	−0.393924	−	0.919143i	$-0.628883\pi$
$67$	−6.44882	−0.787848	−0.393924	+	0.919143i	$0.628883\pi$
$68$	0	0
$69$	10.9861	1.32257
$70$	0	0
$71$	−1.87861	−0.222950	−0.111475	−	0.993767i	$-0.535558\pi$
$71$	−1.87861	−0.222950	−0.111475	+	0.993767i	$0.535558\pi$
$72$	0	0
$73$	13.2020	1.54517	0.772586	−	0.634910i	$-0.218963\pi$
$73$	13.2020	1.54517	0.772586	+	0.634910i	$0.218963\pi$
$74$	0	0
$75$	10.5623	1.21962
$76$	0	0
$77$	12.1613	1.38591
$78$	0	0
$79$	2.50172	0.281465	0.140733	−	0.990048i	$-0.455054\pi$
$79$	2.50172	0.281465	0.140733	+	0.990048i	$0.455054\pi$
$80$	0	0
$81$	−4.18540	−0.465045
$82$	0	0
$83$	−0.0329340	−0.00361498	−0.00180749	−	0.999998i	$-0.500575\pi$
$83$	−0.0329340	−0.00361498	−0.00180749	+	0.999998i	$0.500575\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−10.2744	−1.10153
$88$	0	0
$89$	7.34936	0.779030	0.389515	−	0.921020i	$-0.372643\pi$
$89$	7.34936	0.779030	0.389515	+	0.921020i	$0.372643\pi$
$90$	0	0
$91$	10.1751	1.06664
$92$	0	0
$93$	29.3672	3.04523
$94$	0	0
$95$	2.74572	0.281705
$96$	0	0
$97$	18.2600	1.85402	0.927009	−	0.375038i	$-0.122370\pi$
$97$	18.2600	1.85402	0.927009	+	0.375038i	$0.122370\pi$
$98$	0	0
$99$	−21.6563	−2.17654
$100$	0	0
$101$	−9.20094	−0.915528	−0.457764	−	0.889074i	$-0.651350\pi$
$101$	−9.20094	−0.915528	−0.457764	+	0.889074i	$0.651350\pi$
$102$	0	0
$103$	7.88599	0.777029	0.388515	−	0.921443i	$-0.372988\pi$
$103$	7.88599	0.777029	0.388515	+	0.921443i	$0.372988\pi$
$104$	0	0
$105$	−6.40771	−0.625329
$106$	0	0
$107$	−4.38509	−0.423923	−0.211961	−	0.977278i	$-0.567985\pi$
$107$	−4.38509	−0.423923	−0.211961	+	0.977278i	$0.567985\pi$
$108$	0	0
$109$	6.08791	0.583116	0.291558	−	0.956553i	$-0.405826\pi$
$109$	6.08791	0.583116	0.291558	+	0.956553i	$0.405826\pi$
$110$	0	0
$111$	2.95866	0.280823
$112$	0	0
$113$	−18.2819	−1.71981	−0.859907	−	0.510451i	$-0.829478\pi$
$113$	−18.2819	−1.71981	−0.859907	+	0.510451i	$0.829478\pi$
$114$	0	0
$115$	4.21197	0.392768
$116$	0	0
$117$	−18.1193	−1.67513
$118$	0	0
$119$	0	0
$120$	0	0
$121$	16.1277	1.46616
$122$	0	0
$123$	7.02259	0.633206
$124$	0	0
$125$	9.17812	0.820916
$126$	0	0
$127$	9.67307	0.858346	0.429173	−	0.903222i	$-0.358805\pi$
$127$	9.67307	0.858346	0.429173	+	0.903222i	$0.358805\pi$
$128$	0	0
$129$	10.8457	0.954907
$130$	0	0
$131$	−9.91665	−0.866422	−0.433211	−	0.901293i	$-0.642619\pi$
$131$	−9.91665	−0.866422	−0.433211	+	0.901293i	$0.642619\pi$
$132$	0	0
$133$	6.25025	0.541965
$134$	0	0
$135$	3.17768	0.273491
$136$	0	0
$137$	2.40322	0.205321	0.102661	−	0.994716i	$-0.467264\pi$
$137$	2.40322	0.205321	0.102661	+	0.994716i	$0.467264\pi$
$138$	0	0
$139$	8.69599	0.737584	0.368792	−	0.929512i	$-0.379771\pi$
$139$	8.69599	0.737584	0.368792	+	0.929512i	$0.379771\pi$
$140$	0	0
$141$	−27.8370	−2.34429
$142$	0	0
$143$	22.6971	1.89803
$144$	0	0
$145$	−3.93911	−0.327125
$146$	0	0
$147$	4.14176	0.341607
$148$	0	0
$149$	−3.90029	−0.319524	−0.159762	−	0.987156i	$-0.551073\pi$
$149$	−3.90029	−0.319524	−0.159762	+	0.987156i	$0.551073\pi$
$150$	0	0
$151$	−3.57881	−0.291240	−0.145620	−	0.989341i	$-0.546518\pi$
$151$	−3.57881	−0.291240	−0.145620	+	0.989341i	$0.546518\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	11.2591	0.904350
$156$	0	0
$157$	13.5186	1.07890	0.539449	−	0.842018i	$-0.318633\pi$
$157$	13.5186	1.07890	0.539449	+	0.842018i	$0.318633\pi$
$158$	0	0
$159$	20.8050	1.64994
$160$	0	0
$161$	9.58795	0.755637
$162$	0	0
$163$	−1.35180	−0.105881	−0.0529407	−	0.998598i	$-0.516859\pi$
$163$	−1.35180	−0.105881	−0.0529407	+	0.998598i	$0.516859\pi$
$164$	0	0
$165$	−14.2934	−1.11274
$166$	0	0
$167$	2.73583	0.211705	0.105852	−	0.994382i	$-0.466243\pi$
$167$	2.73583	0.211705	0.105852	+	0.994382i	$0.466243\pi$
$168$	0	0
$169$	5.99012	0.460778
$170$	0	0
$171$	−11.1301	−0.851141
$172$	0	0
$173$	3.88477	0.295354	0.147677	−	0.989036i	$-0.452820\pi$
$173$	3.88477	0.295354	0.147677	+	0.989036i	$0.452820\pi$
$174$	0	0
$175$	9.21803	0.696818
$176$	0	0
$177$	29.1146	2.18839
$178$	0	0
$179$	−8.36962	−0.625575	−0.312787	−	0.949823i	$-0.601263\pi$
$179$	−8.36962	−0.625575	−0.312787	+	0.949823i	$0.601263\pi$
$180$	0	0
$181$	−7.02160	−0.521911	−0.260956	−	0.965351i	$-0.584038\pi$
$181$	−7.02160	−0.521911	−0.260956	+	0.965351i	$0.584038\pi$
$182$	0	0
$183$	21.2936	1.57407
$184$	0	0
$185$	1.13432	0.0833967
$186$	0	0
$187$	0	0
$188$	0	0
$189$	7.23355	0.526163
$190$	0	0
$191$	12.6238	0.913427	0.456713	−	0.889614i	$-0.349027\pi$
$191$	12.6238	0.913427	0.456713	+	0.889614i	$0.349027\pi$
$192$	0	0
$193$	−18.7679	−1.35094	−0.675472	−	0.737386i	$-0.736060\pi$
$193$	−18.7679	−1.35094	−0.675472	+	0.737386i	$0.736060\pi$
$194$	0	0
$195$	−11.9589	−0.856396
$196$	0	0
$197$	14.4212	1.02747	0.513735	−	0.857949i	$-0.328261\pi$
$197$	14.4212	1.02747	0.513735	+	0.857949i	$0.328261\pi$
$198$	0	0
$199$	9.57318	0.678625	0.339313	−	0.940674i	$-0.389806\pi$
$199$	9.57318	0.678625	0.339313	+	0.940674i	$0.389806\pi$
$200$	0	0
$201$	17.2534	1.21696
$202$	0	0
$203$	−8.96683	−0.629348
$204$	0	0
$205$	2.69239	0.188045
$206$	0	0
$207$	−17.0737	−1.18671
$208$	0	0
$209$	13.9421	0.964397
$210$	0	0
$211$	1.60876	0.110752	0.0553759	−	0.998466i	$-0.482364\pi$
$211$	1.60876	0.110752	0.0553759	+	0.998466i	$0.482364\pi$
$212$	0	0
$213$	5.02609	0.344382
$214$	0	0
$215$	4.15811	0.283581
$216$	0	0
$217$	25.6297	1.73986
$218$	0	0
$219$	−35.3209	−2.38677
$220$	0	0
$221$	0	0
$222$	0	0
$223$	9.48098	0.634893	0.317447	−	0.948276i	$-0.397175\pi$
$223$	9.48098	0.634893	0.317447	+	0.948276i	$0.397175\pi$
$224$	0	0
$225$	−16.4150	−1.09433
$226$	0	0
$227$	−2.61793	−0.173758	−0.0868790	−	0.996219i	$-0.527689\pi$
$227$	−2.61793	−0.173758	−0.0868790	+	0.996219i	$0.527689\pi$
$228$	0	0
$229$	−4.93551	−0.326148	−0.163074	−	0.986614i	$-0.552141\pi$
$229$	−4.93551	−0.326148	−0.163074	+	0.986614i	$0.552141\pi$
$230$	0	0
$231$	−32.5368	−2.14077
$232$	0	0
$233$	2.28244	0.149528	0.0747639	−	0.997201i	$-0.476180\pi$
$233$	2.28244	0.149528	0.0747639	+	0.997201i	$0.476180\pi$
$234$	0	0
$235$	−10.6724	−0.696191
$236$	0	0
$237$	−6.69317	−0.434768
$238$	0	0
$239$	−12.1821	−0.787995	−0.393997	−	0.919112i	$-0.628908\pi$
$239$	−12.1821	−0.787995	−0.393997	+	0.919112i	$0.628908\pi$
$240$	0	0
$241$	14.4072	0.928047	0.464024	−	0.885823i	$-0.346405\pi$
$241$	14.4072	0.928047	0.464024	+	0.885823i	$0.346405\pi$
$242$	0	0
$243$	20.4916	1.31454
$244$	0	0
$245$	1.58791	0.101448
$246$	0	0
$247$	11.6650	0.742228
$248$	0	0
$249$	0.0881127	0.00558392
$250$	0	0
$251$	−17.8445	−1.12634	−0.563168	−	0.826342i	$-0.690418\pi$
$251$	−17.8445	−1.12634	−0.563168	+	0.826342i	$0.690418\pi$
$252$	0	0
$253$	21.3874	1.34461
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−5.40343	−0.337057	−0.168528	−	0.985697i	$-0.553901\pi$
$257$	−5.40343	−0.337057	−0.168528	+	0.985697i	$0.553901\pi$
$258$	0	0
$259$	2.58212	0.160445
$260$	0	0
$261$	15.9677	0.988374
$262$	0	0
$263$	−9.88246	−0.609378	−0.304689	−	0.952452i	$-0.598553\pi$
$263$	−9.88246	−0.609378	−0.304689	+	0.952452i	$0.598553\pi$
$264$	0	0
$265$	7.97640	0.489987
$266$	0	0
$267$	−19.6627	−1.20334
$268$	0	0
$269$	−26.7671	−1.63202	−0.816009	−	0.578039i	$-0.803818\pi$
$269$	−26.7671	−1.63202	−0.816009	+	0.578039i	$0.803818\pi$
$270$	0	0
$271$	−26.6483	−1.61877	−0.809384	−	0.587280i	$-0.800199\pi$
$271$	−26.6483	−1.61877	−0.809384	+	0.587280i	$0.800199\pi$
$272$	0	0
$273$	−27.2228	−1.64760
$274$	0	0
$275$	20.5622	1.23995
$276$	0	0
$277$	−28.0560	−1.68572	−0.842860	−	0.538132i	$-0.819130\pi$
$277$	−28.0560	−1.68572	−0.842860	+	0.538132i	$0.819130\pi$
$278$	0	0
$279$	−45.6400	−2.73239
$280$	0	0
$281$	−21.1636	−1.26251	−0.631256	−	0.775575i	$-0.717460\pi$
$281$	−21.1636	−1.26251	−0.631256	+	0.775575i	$0.717460\pi$
$282$	0	0
$283$	4.38686	0.260772	0.130386	−	0.991463i	$-0.458378\pi$
$283$	4.38686	0.260772	0.130386	+	0.991463i	$0.458378\pi$
$284$	0	0
$285$	−7.34598	−0.435139
$286$	0	0
$287$	6.12885	0.361774
$288$	0	0
$289$	0	0
$290$	0	0
$291$	−48.8533	−2.86383
$292$	0	0
$293$	−11.8892	−0.694573	−0.347286	−	0.937759i	$-0.612897\pi$
$293$	−11.8892	−0.694573	−0.347286	+	0.937759i	$0.612897\pi$
$294$	0	0
$295$	11.1623	0.649891
$296$	0	0
$297$	16.1355	0.936278
$298$	0	0
$299$	17.8943	1.03485
$300$	0	0
$301$	9.46536	0.545574
$302$	0	0
$303$	24.6165	1.41418
$304$	0	0
$305$	8.16375	0.467455
$306$	0	0
$307$	20.9185	1.19388	0.596941	−	0.802285i	$-0.296383\pi$
$307$	20.9185	1.19388	0.596941	+	0.802285i	$0.296383\pi$
$308$	0	0
$309$	−21.0984	−1.20025
$310$	0	0
$311$	6.11686	0.346855	0.173428	−	0.984847i	$-0.444516\pi$
$311$	6.11686	0.346855	0.173428	+	0.984847i	$0.444516\pi$
$312$	0	0
$313$	−28.6754	−1.62083	−0.810414	−	0.585858i	$-0.800758\pi$
$313$	−28.6754	−1.62083	−0.810414	+	0.585858i	$0.800758\pi$
$314$	0	0
$315$	9.95833	0.561088
$316$	0	0
$317$	6.26907	0.352106	0.176053	−	0.984381i	$-0.443667\pi$
$317$	6.26907	0.352106	0.176053	+	0.984381i	$0.443667\pi$
$318$	0	0
$319$	−20.0019	−1.11989
$320$	0	0
$321$	11.7320	0.654817
$322$	0	0
$323$	0	0
$324$	0	0
$325$	17.2039	0.954301
$326$	0	0
$327$	−16.2878	−0.900716
$328$	0	0
$329$	−24.2942	−1.33938
$330$	0	0
$331$	−9.56871	−0.525944	−0.262972	−	0.964803i	$-0.584703\pi$
$331$	−9.56871	−0.525944	−0.262972	+	0.964803i	$0.584703\pi$
$332$	0	0
$333$	−4.59810	−0.251974
$334$	0	0
$335$	6.61476	0.361403
$336$	0	0
$337$	−2.28982	−0.124735	−0.0623673	−	0.998053i	$-0.519865\pi$
$337$	−2.28982	−0.124735	−0.0623673	+	0.998053i	$0.519865\pi$
$338$	0	0
$339$	48.9119	2.65653
$340$	0	0
$341$	57.1709	3.09598
$342$	0	0
$343$	19.9592	1.07770
$344$	0	0
$345$	−11.2688	−0.606693
$346$	0	0
$347$	−3.99933	−0.214695	−0.107348	−	0.994222i	$-0.534236\pi$
$347$	−3.99933	−0.214695	−0.107348	+	0.994222i	$0.534236\pi$
$348$	0	0
$349$	26.5502	1.42120	0.710599	−	0.703597i	$-0.248424\pi$
$349$	26.5502	1.42120	0.710599	+	0.703597i	$0.248424\pi$
$350$	0	0
$351$	13.5002	0.720588
$352$	0	0
$353$	1.75320	0.0933134	0.0466567	−	0.998911i	$-0.485143\pi$
$353$	1.75320	0.0933134	0.0466567	+	0.998911i	$0.485143\pi$
$354$	0	0
$355$	1.92695	0.102272
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−19.8649	−1.04843	−0.524216	−	0.851586i	$-0.675641\pi$
$359$	−19.8649	−1.04843	−0.524216	+	0.851586i	$0.675641\pi$
$360$	0	0
$361$	−11.8345	−0.622870
$362$	0	0
$363$	−43.1486	−2.26471
$364$	0	0
$365$	−13.5417	−0.708804
$366$	0	0
$367$	−17.3289	−0.904560	−0.452280	−	0.891876i	$-0.649389\pi$
$367$	−17.3289	−0.904560	−0.452280	+	0.891876i	$0.649389\pi$
$368$	0	0
$369$	−10.9139	−0.568156
$370$	0	0
$371$	18.1572	0.942673
$372$	0	0
$373$	−0.854460	−0.0442423	−0.0221211	−	0.999755i	$-0.507042\pi$
$373$	−0.854460	−0.0442423	−0.0221211	+	0.999755i	$0.507042\pi$
$374$	0	0
$375$	−24.5554	−1.26804
$376$	0	0
$377$	−16.7351	−0.861901
$378$	0	0
$379$	−29.9048	−1.53611	−0.768053	−	0.640386i	$-0.778774\pi$
$379$	−29.9048	−1.53611	−0.768053	+	0.640386i	$0.778774\pi$
$380$	0	0
$381$	−25.8796	−1.32585
$382$	0	0
$383$	−27.6510	−1.41290	−0.706451	−	0.707762i	$-0.749705\pi$
$383$	−27.6510	−1.41290	−0.706451	+	0.707762i	$0.749705\pi$
$384$	0	0
$385$	−12.4743	−0.635749
$386$	0	0
$387$	−16.8554	−0.856809
$388$	0	0
$389$	29.3110	1.48613	0.743064	−	0.669220i	$-0.233372\pi$
$389$	29.3110	1.48613	0.743064	+	0.669220i	$0.233372\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	26.5313	1.33833
$394$	0	0
$395$	−2.56609	−0.129114
$396$	0	0
$397$	−18.6002	−0.933518	−0.466759	−	0.884384i	$-0.654578\pi$
$397$	−18.6002	−0.933518	−0.466759	+	0.884384i	$0.654578\pi$
$398$	0	0
$399$	−16.7221	−0.837152
$400$	0	0
$401$	−7.93565	−0.396288	−0.198144	−	0.980173i	$-0.563491\pi$
$401$	−7.93565	−0.396288	−0.198144	+	0.980173i	$0.563491\pi$
$402$	0	0
$403$	47.8335	2.38276
$404$	0	0
$405$	4.29310	0.213326
$406$	0	0
$407$	5.75980	0.285503
$408$	0	0
$409$	−6.93345	−0.342837	−0.171418	−	0.985198i	$-0.554835\pi$
$409$	−6.93345	−0.342837	−0.171418	+	0.985198i	$0.554835\pi$
$410$	0	0
$411$	−6.42966	−0.317152
$412$	0	0
$413$	25.4093	1.25031
$414$	0	0
$415$	0.0337815	0.00165827
$416$	0	0
$417$	−23.2655	−1.13932
$418$	0	0
$419$	25.6432	1.25275	0.626376	−	0.779521i	$-0.284537\pi$
$419$	25.6432	1.25275	0.626376	+	0.779521i	$0.284537\pi$
$420$	0	0
$421$	−34.5518	−1.68395	−0.841977	−	0.539513i	$-0.818608\pi$
$421$	−34.5518	−1.68395	−0.841977	+	0.539513i	$0.818608\pi$
$422$	0	0
$423$	43.2619	2.10346
$424$	0	0
$425$	0	0
$426$	0	0
$427$	18.5836	0.899325
$428$	0	0
$429$	−60.7245	−2.93181
$430$	0	0
$431$	2.86931	0.138210	0.0691050	−	0.997609i	$-0.477986\pi$
$431$	2.86931	0.138210	0.0691050	+	0.997609i	$0.477986\pi$
$432$	0	0
$433$	−16.9089	−0.812588	−0.406294	−	0.913742i	$-0.633179\pi$
$433$	−16.9089	−0.812588	−0.406294	+	0.913742i	$0.633179\pi$
$434$	0	0
$435$	10.5388	0.505298
$436$	0	0
$437$	10.9919	0.525814
$438$	0	0
$439$	20.8176	0.993572	0.496786	−	0.867873i	$-0.334513\pi$
$439$	20.8176	0.993572	0.496786	+	0.867873i	$0.334513\pi$
$440$	0	0
$441$	−6.43678	−0.306513
$442$	0	0
$443$	−23.5157	−1.11726	−0.558632	−	0.829416i	$-0.688673\pi$
$443$	−23.5157	−1.11726	−0.558632	+	0.829416i	$0.688673\pi$
$444$	0	0
$445$	−7.53847	−0.357358
$446$	0	0
$447$	10.4349	0.493556
$448$	0	0
$449$	9.60971	0.453510	0.226755	−	0.973952i	$-0.427188\pi$
$449$	9.60971	0.453510	0.226755	+	0.973952i	$0.427188\pi$
$450$	0	0
$451$	13.6713	0.643757
$452$	0	0
$453$	9.57487	0.449867
$454$	0	0
$455$	−10.4369	−0.489291
$456$	0	0
$457$	17.1673	0.803051	0.401526	−	0.915848i	$-0.368480\pi$
$457$	17.1673	0.803051	0.401526	+	0.915848i	$0.368480\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−30.2470	−1.40874	−0.704371	−	0.709832i	$-0.748771\pi$
$461$	−30.2470	−1.40874	−0.704371	+	0.709832i	$0.748771\pi$
$462$	0	0
$463$	33.1695	1.54152	0.770758	−	0.637128i	$-0.219878\pi$
$463$	33.1695	1.54152	0.770758	+	0.637128i	$0.219878\pi$
$464$	0	0
$465$	−30.1228	−1.39691
$466$	0	0
$467$	−5.57527	−0.257993	−0.128996	−	0.991645i	$-0.541176\pi$
$467$	−5.57527	−0.257993	−0.128996	+	0.991645i	$0.541176\pi$
$468$	0	0
$469$	15.0576	0.695294
$470$	0	0
$471$	−36.1680	−1.66653
$472$	0	0
$473$	21.1139	0.970819
$474$	0	0
$475$	10.5678	0.484885
$476$	0	0
$477$	−32.3333	−1.48044
$478$	0	0
$479$	−23.1079	−1.05583	−0.527913	−	0.849298i	$-0.677025\pi$
$479$	−23.1079	−1.05583	−0.527913	+	0.849298i	$0.677025\pi$
$480$	0	0
$481$	4.81908	0.219731
$482$	0	0
$483$	−25.6519	−1.16720
$484$	0	0
$485$	−18.7298	−0.850478
$486$	0	0
$487$	14.8048	0.670868	0.335434	−	0.942064i	$-0.391117\pi$
$487$	14.8048	0.670868	0.335434	+	0.942064i	$0.391117\pi$
$488$	0	0
$489$	3.61665	0.163551
$490$	0	0
$491$	0.524437	0.0236675	0.0118338	−	0.999930i	$-0.496233\pi$
$491$	0.524437	0.0236675	0.0118338	+	0.999930i	$0.496233\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	22.2136	0.998425
$496$	0	0
$497$	4.38644	0.196759
$498$	0	0
$499$	−16.1396	−0.722509	−0.361255	−	0.932467i	$-0.617652\pi$
$499$	−16.1396	−0.722509	−0.361255	+	0.932467i	$0.617652\pi$
$500$	0	0
$501$	−7.31951	−0.327012
$502$	0	0
$503$	−12.8740	−0.574023	−0.287011	−	0.957927i	$-0.592662\pi$
$503$	−12.8740	−0.574023	−0.287011	+	0.957927i	$0.592662\pi$
$504$	0	0
$505$	9.43771	0.419973
$506$	0	0
$507$	−16.0262	−0.711746
$508$	0	0
$509$	−0.684018	−0.0303185	−0.0151593	−	0.999885i	$-0.504826\pi$
$509$	−0.684018	−0.0303185	−0.0151593	+	0.999885i	$0.504826\pi$
$510$	0	0
$511$	−30.8257	−1.36365
$512$	0	0
$513$	8.29275	0.366134
$514$	0	0
$515$	−8.08891	−0.356440
$516$	0	0
$517$	−54.1919	−2.38336
$518$	0	0
$519$	−10.3934	−0.456221
$520$	0	0
$521$	−4.75103	−0.208147	−0.104073	−	0.994570i	$-0.533188\pi$
$521$	−4.75103	−0.208147	−0.104073	+	0.994570i	$0.533188\pi$
$522$	0	0
$523$	40.6707	1.77840	0.889202	−	0.457515i	$-0.151260\pi$
$523$	40.6707	1.77840	0.889202	+	0.457515i	$0.151260\pi$
$524$	0	0
$525$	−24.6622	−1.07635
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−6.13829	−0.266882
$530$	0	0
$531$	−45.2475	−1.96358
$532$	0	0
$533$	11.4385	0.495455
$534$	0	0
$535$	4.49793	0.194463
$536$	0	0
$537$	22.3923	0.966301
$538$	0	0
$539$	8.06302	0.347299
$540$	0	0
$541$	−34.5690	−1.48624	−0.743119	−	0.669159i	$-0.766654\pi$
$541$	−34.5690	−1.48624	−0.743119	+	0.669159i	$0.766654\pi$
$542$	0	0
$543$	18.7858	0.806176
$544$	0	0
$545$	−6.24456	−0.267488
$546$	0	0
$547$	26.4964	1.13290	0.566452	−	0.824094i	$-0.308315\pi$
$547$	26.4964	1.13290	0.566452	+	0.824094i	$0.308315\pi$
$548$	0	0
$549$	−33.0927	−1.41236
$550$	0	0
$551$	−10.2798	−0.437936
$552$	0	0
$553$	−5.84135	−0.248399
$554$	0	0
$555$	−3.03479	−0.128820
$556$	0	0
$557$	−24.1708	−1.02415	−0.512075	−	0.858941i	$-0.671123\pi$
$557$	−24.1708	−1.02415	−0.512075	+	0.858941i	$0.671123\pi$
$558$	0	0
$559$	17.6655	0.747171
$560$	0	0
$561$	0	0
$562$	0	0
$563$	3.53989	0.149189	0.0745943	−	0.997214i	$-0.476234\pi$
$563$	3.53989	0.149189	0.0745943	+	0.997214i	$0.476234\pi$
$564$	0	0
$565$	18.7523	0.788916
$566$	0	0
$567$	9.77265	0.410413
$568$	0	0
$569$	−15.8884	−0.666076	−0.333038	−	0.942913i	$-0.608074\pi$
$569$	−15.8884	−0.666076	−0.333038	+	0.942913i	$0.608074\pi$
$570$	0	0
$571$	17.1570	0.717997	0.358998	−	0.933338i	$-0.383118\pi$
$571$	17.1570	0.717997	0.358998	+	0.933338i	$0.383118\pi$
$572$	0	0
$573$	−33.7741	−1.41093
$574$	0	0
$575$	16.2112	0.676052
$576$	0	0
$577$	−39.0319	−1.62492	−0.812460	−	0.583017i	$-0.801872\pi$
$577$	−39.0319	−1.62492	−0.812460	+	0.583017i	$0.801872\pi$
$578$	0	0
$579$	50.2122	2.08675
$580$	0	0
$581$	0.0768989	0.00319030
$582$	0	0
$583$	40.5023	1.67743
$584$	0	0
$585$	18.5855	0.768418
$586$	0	0
$587$	−15.8714	−0.655081	−0.327540	−	0.944837i	$-0.606220\pi$
$587$	−15.8714	−0.655081	−0.327540	+	0.944837i	$0.606220\pi$
$588$	0	0
$589$	29.3826	1.21069
$590$	0	0
$591$	−38.5830	−1.58709
$592$	0	0
$593$	−10.4046	−0.427264	−0.213632	−	0.976914i	$-0.568529\pi$
$593$	−10.4046	−0.427264	−0.213632	+	0.976914i	$0.568529\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−25.6124	−1.04825
$598$	0	0
$599$	−35.7297	−1.45988	−0.729938	−	0.683513i	$-0.760451\pi$
$599$	−35.7297	−1.45988	−0.729938	+	0.683513i	$0.760451\pi$
$600$	0	0
$601$	27.8337	1.13536	0.567681	−	0.823249i	$-0.307841\pi$
$601$	27.8337	1.13536	0.567681	+	0.823249i	$0.307841\pi$
$602$	0	0
$603$	−26.8137	−1.09194
$604$	0	0
$605$	−16.5427	−0.672558
$606$	0	0
$607$	−17.1987	−0.698073	−0.349036	−	0.937109i	$-0.613491\pi$
$607$	−17.1987	−0.698073	−0.349036	+	0.937109i	$0.613491\pi$
$608$	0	0
$609$	23.9901	0.972130
$610$	0	0
$611$	−45.3411	−1.83430
$612$	0	0
$613$	−44.0031	−1.77727	−0.888634	−	0.458618i	$-0.848345\pi$
$613$	−44.0031	−1.77727	−0.888634	+	0.458618i	$0.848345\pi$
$614$	0	0
$615$	−7.20330	−0.290465
$616$	0	0
$617$	−29.6094	−1.19203	−0.596014	−	0.802974i	$-0.703250\pi$
$617$	−29.6094	−1.19203	−0.596014	+	0.802974i	$0.703250\pi$
$618$	0	0
$619$	26.1735	1.05200	0.526001	−	0.850484i	$-0.323691\pi$
$619$	26.1735	1.05200	0.526001	+	0.850484i	$0.323691\pi$
$620$	0	0
$621$	12.7212	0.510483
$622$	0	0
$623$	−17.1603	−0.687512
$624$	0	0
$625$	10.3251	0.413003
$626$	0	0
$627$	−37.3012	−1.48967
$628$	0	0
$629$	0	0
$630$	0	0
$631$	40.2883	1.60385	0.801925	−	0.597424i	$-0.203809\pi$
$631$	40.2883	1.60385	0.801925	+	0.597424i	$0.203809\pi$
$632$	0	0
$633$	−4.30413	−0.171074
$634$	0	0
$635$	−9.92198	−0.393742
$636$	0	0
$637$	6.74613	0.267292
$638$	0	0
$639$	−7.81114	−0.309004
$640$	0	0
$641$	−7.90788	−0.312342	−0.156171	−	0.987730i	$-0.549915\pi$
$641$	−7.90788	−0.312342	−0.156171	+	0.987730i	$0.549915\pi$
$642$	0	0
$643$	−5.54323	−0.218603	−0.109302	−	0.994009i	$-0.534861\pi$
$643$	−5.54323	−0.218603	−0.109302	+	0.994009i	$0.534861\pi$
$644$	0	0
$645$	−11.1247	−0.438036
$646$	0	0
$647$	18.3835	0.722731	0.361366	−	0.932424i	$-0.382311\pi$
$647$	18.3835	0.722731	0.361366	+	0.932424i	$0.382311\pi$
$648$	0	0
$649$	56.6793	2.22486
$650$	0	0
$651$	−68.5704	−2.68749
$652$	0	0
$653$	35.3400	1.38296	0.691480	−	0.722396i	$-0.256959\pi$
$653$	35.3400	1.38296	0.691480	+	0.722396i	$0.256959\pi$
$654$	0	0
$655$	10.1718	0.397446
$656$	0	0
$657$	54.8928	2.14157
$658$	0	0
$659$	0.950773	0.0370369	0.0185184	−	0.999829i	$-0.494105\pi$
$659$	0.950773	0.0370369	0.0185184	+	0.999829i	$0.494105\pi$
$660$	0	0
$661$	11.5490	0.449203	0.224602	−	0.974451i	$-0.427892\pi$
$661$	11.5490	0.449203	0.224602	+	0.974451i	$0.427892\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−6.41108	−0.248611
$666$	0	0
$667$	−15.7694	−0.610593
$668$	0	0
$669$	−25.3657	−0.980695
$670$	0	0
$671$	41.4536	1.60030
$672$	0	0
$673$	−11.9983	−0.462501	−0.231250	−	0.972894i	$-0.574282\pi$
$673$	−11.9983	−0.462501	−0.231250	+	0.972894i	$0.574282\pi$
$674$	0	0
$675$	12.2304	0.470747
$676$	0	0
$677$	41.6732	1.60163	0.800815	−	0.598912i	$-0.204400\pi$
$677$	41.6732	1.60163	0.800815	+	0.598912i	$0.204400\pi$
$678$	0	0
$679$	−42.6359	−1.63621
$680$	0	0
$681$	7.00409	0.268397
$682$	0	0
$683$	−11.3230	−0.433262	−0.216631	−	0.976254i	$-0.569507\pi$
$683$	−11.3230	−0.433262	−0.216631	+	0.976254i	$0.569507\pi$
$684$	0	0
$685$	−2.46506	−0.0941853
$686$	0	0
$687$	13.2046	0.503787
$688$	0	0
$689$	33.8873	1.29100
$690$	0	0
$691$	24.6925	0.939347	0.469673	−	0.882840i	$-0.344372\pi$
$691$	24.6925	0.939347	0.469673	+	0.882840i	$0.344372\pi$
$692$	0	0
$693$	50.5661	1.92085
$694$	0	0
$695$	−8.91976	−0.338346
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−6.10652	−0.230970
$700$	0	0
$701$	38.0011	1.43528	0.717642	−	0.696413i	$-0.245222\pi$
$701$	38.0011	1.43528	0.717642	+	0.696413i	$0.245222\pi$
$702$	0	0
$703$	2.96021	0.111647
$704$	0	0
$705$	28.5533	1.07538
$706$	0	0
$707$	21.4836	0.807975
$708$	0	0
$709$	19.5644	0.734757	0.367378	−	0.930072i	$-0.380255\pi$
$709$	19.5644	0.734757	0.367378	+	0.930072i	$0.380255\pi$
$710$	0	0
$711$	10.4020	0.390104
$712$	0	0
$713$	45.0733	1.68801
$714$	0	0
$715$	−23.2812	−0.870666
$716$	0	0
$717$	32.5924	1.21718
$718$	0	0
$719$	−39.1974	−1.46182	−0.730909	−	0.682475i	$-0.760904\pi$
$719$	−39.1974	−1.46182	−0.730909	+	0.682475i	$0.760904\pi$
$720$	0	0
$721$	−18.4133	−0.685746
$722$	0	0
$723$	−38.5454	−1.43352
$724$	0	0
$725$	−15.1610	−0.563065
$726$	0	0
$727$	49.8981	1.85062	0.925310	−	0.379212i	$-0.123805\pi$
$727$	49.8981	1.85062	0.925310	+	0.379212i	$0.123805\pi$
$728$	0	0
$729$	−42.2678	−1.56547
$730$	0	0
$731$	0	0
$732$	0	0
$733$	27.9360	1.03184	0.515919	−	0.856637i	$-0.327451\pi$
$733$	27.9360	1.03184	0.515919	+	0.856637i	$0.327451\pi$
$734$	0	0
$735$	−4.24834	−0.156702
$736$	0	0
$737$	33.5882	1.23724
$738$	0	0
$739$	30.1936	1.11069	0.555344	−	0.831621i	$-0.312587\pi$
$739$	30.1936	1.11069	0.555344	+	0.831621i	$0.312587\pi$
$740$	0	0
$741$	−31.2090	−1.14649
$742$	0	0
$743$	5.24624	0.192466	0.0962328	−	0.995359i	$-0.469321\pi$
$743$	5.24624	0.192466	0.0962328	+	0.995359i	$0.469321\pi$
$744$	0	0
$745$	4.00065	0.146572
$746$	0	0
$747$	−0.136937	−0.00501028
$748$	0	0
$749$	10.2389	0.374122
$750$	0	0
$751$	−14.4140	−0.525976	−0.262988	−	0.964799i	$-0.584708\pi$
$751$	−14.4140	−0.525976	−0.262988	+	0.964799i	$0.584708\pi$
$752$	0	0
$753$	47.7418	1.73981
$754$	0	0
$755$	3.67091	0.133598
$756$	0	0
$757$	−6.29402	−0.228760	−0.114380	−	0.993437i	$-0.536488\pi$
$757$	−6.29402	−0.228760	−0.114380	+	0.993437i	$0.536488\pi$
$758$	0	0
$759$	−57.2205	−2.07697
$760$	0	0
$761$	6.45544	0.234010	0.117005	−	0.993131i	$-0.462671\pi$
$761$	6.45544	0.234010	0.117005	+	0.993131i	$0.462671\pi$
$762$	0	0
$763$	−14.2149	−0.514613
$764$	0	0
$765$	0	0
$766$	0	0
$767$	47.4222	1.71232
$768$	0	0
$769$	−0.665285	−0.0239908	−0.0119954	−	0.999928i	$-0.503818\pi$
$769$	−0.665285	−0.0239908	−0.0119954	+	0.999928i	$0.503818\pi$
$770$	0	0
$771$	14.4565	0.520638
$772$	0	0
$773$	−4.37461	−0.157344	−0.0786719	−	0.996901i	$-0.525068\pi$
$773$	−4.37461	−0.157344	−0.0786719	+	0.996901i	$0.525068\pi$
$774$	0	0
$775$	43.3342	1.55661
$776$	0	0
$777$	−6.90827	−0.247833
$778$	0	0
$779$	7.02628	0.251743
$780$	0	0
$781$	9.78461	0.350121
$782$	0	0
$783$	−11.8971	−0.425167
$784$	0	0
$785$	−13.8664	−0.494914
$786$	0	0
$787$	−7.39078	−0.263453	−0.131726	−	0.991286i	$-0.542052\pi$
$787$	−7.39078	−0.263453	−0.131726	+	0.991286i	$0.542052\pi$
$788$	0	0
$789$	26.4398	0.941283
$790$	0	0
$791$	42.6870	1.51778
$792$	0	0
$793$	34.6832	1.23164
$794$	0	0
$795$	−21.3403	−0.756863
$796$	0	0
$797$	−22.5759	−0.799678	−0.399839	−	0.916585i	$-0.630934\pi$
$797$	−22.5759	−0.799678	−0.399839	+	0.916585i	$0.630934\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	30.5581	1.07972
$802$	0	0
$803$	−68.7615	−2.42654
$804$	0	0
$805$	−9.83468	−0.346627
$806$	0	0
$807$	71.6135	2.52091
$808$	0	0
$809$	−4.39402	−0.154486	−0.0772428	−	0.997012i	$-0.524612\pi$
$809$	−4.39402	−0.154486	−0.0772428	+	0.997012i	$0.524612\pi$
$810$	0	0
$811$	−11.3678	−0.399177	−0.199589	−	0.979880i	$-0.563961\pi$
$811$	−11.3678	−0.399177	−0.199589	+	0.979880i	$0.563961\pi$
$812$	0	0
$813$	71.2956	2.50045
$814$	0	0
$815$	1.38659	0.0485700
$816$	0	0
$817$	10.8514	0.379641
$818$	0	0
$819$	42.3074	1.47834
$820$	0	0
$821$	45.0897	1.57364	0.786821	−	0.617181i	$-0.211725\pi$
$821$	45.0897	1.57364	0.786821	+	0.617181i	$0.211725\pi$
$822$	0	0
$823$	10.6982	0.372916	0.186458	−	0.982463i	$-0.440299\pi$
$823$	10.6982	0.372916	0.186458	+	0.982463i	$0.440299\pi$
$824$	0	0
$825$	−55.0128	−1.91530
$826$	0	0
$827$	−50.8622	−1.76865	−0.884326	−	0.466869i	$-0.845382\pi$
$827$	−50.8622	−1.76865	−0.884326	+	0.466869i	$0.845382\pi$
$828$	0	0
$829$	−6.52156	−0.226503	−0.113252	−	0.993566i	$-0.536127\pi$
$829$	−6.52156	−0.226503	−0.113252	+	0.993566i	$0.536127\pi$
$830$	0	0
$831$	75.0619	2.60387
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−2.80623	−0.0971134
$836$	0	0
$837$	34.0051	1.17539
$838$	0	0
$839$	−8.36481	−0.288785	−0.144393	−	0.989520i	$-0.546123\pi$
$839$	−8.36481	−0.288785	−0.144393	+	0.989520i	$0.546123\pi$
$840$	0	0
$841$	−14.2522	−0.491454
$842$	0	0
$843$	56.6216	1.95015
$844$	0	0
$845$	−6.14426	−0.211369
$846$	0	0
$847$	−37.6572	−1.29392
$848$	0	0
$849$	−11.7367	−0.402804
$850$	0	0
$851$	4.54100	0.155663
$852$	0	0
$853$	3.09683	0.106034	0.0530168	−	0.998594i	$-0.483116\pi$
$853$	3.09683	0.106034	0.0530168	+	0.998594i	$0.483116\pi$
$854$	0	0
$855$	11.4165	0.390437
$856$	0	0
$857$	8.20901	0.280415	0.140207	−	0.990122i	$-0.455223\pi$
$857$	8.20901	0.280415	0.140207	+	0.990122i	$0.455223\pi$
$858$	0	0
$859$	−11.2149	−0.382648	−0.191324	−	0.981527i	$-0.561278\pi$
$859$	−11.2149	−0.382648	−0.191324	+	0.981527i	$0.561278\pi$
$860$	0	0
$861$	−16.3973	−0.558819
$862$	0	0
$863$	34.8722	1.18706	0.593532	−	0.804811i	$-0.297733\pi$
$863$	34.8722	1.18706	0.593532	+	0.804811i	$0.297733\pi$
$864$	0	0
$865$	−3.98474	−0.135485
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−13.0300	−0.442013
$870$	0	0
$871$	28.1024	0.952215
$872$	0	0
$873$	75.9237	2.56963
$874$	0	0
$875$	−21.4303	−0.724478
$876$	0	0
$877$	−12.3174	−0.415930	−0.207965	−	0.978136i	$-0.566684\pi$
$877$	−12.3174	−0.415930	−0.207965	+	0.978136i	$0.566684\pi$
$878$	0	0
$879$	31.8087	1.07288
$880$	0	0
$881$	−41.9252	−1.41250	−0.706248	−	0.707965i	$-0.749613\pi$
$881$	−41.9252	−1.41250	−0.706248	+	0.707965i	$0.749613\pi$
$882$	0	0
$883$	−14.6148	−0.491828	−0.245914	−	0.969292i	$-0.579088\pi$
$883$	−14.6148	−0.491828	−0.245914	+	0.969292i	$0.579088\pi$
$884$	0	0
$885$	−29.8638	−1.00386
$886$	0	0
$887$	5.82073	0.195441	0.0977205	−	0.995214i	$-0.468845\pi$
$887$	5.82073	0.195441	0.0977205	+	0.995214i	$0.468845\pi$
$888$	0	0
$889$	−22.5860	−0.757510
$890$	0	0
$891$	21.7994	0.730306
$892$	0	0
$893$	−27.8516	−0.932018
$894$	0	0
$895$	8.58499	0.286965
$896$	0	0
$897$	−47.8750	−1.59850
$898$	0	0
$899$	−42.1534	−1.40589
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−25.3239	−0.842727
$904$	0	0
$905$	7.20228	0.239412
$906$	0	0
$907$	2.41186	0.0800844	0.0400422	−	0.999198i	$-0.487251\pi$
$907$	2.41186	0.0800844	0.0400422	+	0.999198i	$0.487251\pi$
$908$	0	0
$909$	−38.2569	−1.26890
$910$	0	0
$911$	19.8485	0.657611	0.328806	−	0.944398i	$-0.393354\pi$
$911$	19.8485	0.657611	0.328806	+	0.944398i	$0.393354\pi$
$912$	0	0
$913$	0.171534	0.00567696
$914$	0	0
$915$	−21.8415	−0.722059
$916$	0	0
$917$	23.1547	0.764637
$918$	0	0
$919$	16.7830	0.553619	0.276810	−	0.960925i	$-0.410723\pi$
$919$	16.7830	0.553619	0.276810	+	0.960925i	$0.410723\pi$
$920$	0	0
$921$	−55.9660	−1.84414
$922$	0	0
$923$	8.18655	0.269463
$924$	0	0
$925$	4.36580	0.143547
$926$	0	0
$927$	32.7894	1.07694
$928$	0	0
$929$	−23.2751	−0.763632	−0.381816	−	0.924238i	$-0.624701\pi$
$929$	−23.2751	−0.763632	−0.381816	+	0.924238i	$0.624701\pi$
$930$	0	0
$931$	4.14394	0.135812
$932$	0	0
$933$	−16.3652	−0.535774
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−39.7859	−1.29975	−0.649875	−	0.760041i	$-0.725179\pi$
$937$	−39.7859	−1.29975	−0.649875	+	0.760041i	$0.725179\pi$
$938$	0	0
$939$	76.7190	2.50363
$940$	0	0
$941$	−28.5899	−0.932004	−0.466002	−	0.884784i	$-0.654306\pi$
$941$	−28.5899	−0.932004	−0.466002	+	0.884784i	$0.654306\pi$
$942$	0	0
$943$	10.7784	0.350993
$944$	0	0
$945$	−7.41969	−0.241363
$946$	0	0
$947$	−42.7997	−1.39080	−0.695402	−	0.718621i	$-0.744774\pi$
$947$	−42.7997	−1.39080	−0.695402	+	0.718621i	$0.744774\pi$
$948$	0	0
$949$	−57.5310	−1.86754
$950$	0	0
$951$	−16.7725	−0.543885
$952$	0	0
$953$	−41.2093	−1.33490	−0.667451	−	0.744654i	$-0.732614\pi$
$953$	−41.2093	−1.33490	−0.667451	+	0.744654i	$0.732614\pi$
$954$	0	0
$955$	−12.9487	−0.419008
$956$	0	0
$957$	53.5136	1.72985
$958$	0	0
$959$	−5.61137	−0.181201
$960$	0	0
$961$	89.4859	2.88664
$962$	0	0
$963$	−18.2329	−0.587547
$964$	0	0
$965$	19.2509	0.619707
$966$	0	0
$967$	30.2438	0.972574	0.486287	−	0.873799i	$-0.338351\pi$
$967$	30.2438	0.972574	0.486287	+	0.873799i	$0.338351\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−37.4533	−1.20193	−0.600967	−	0.799274i	$-0.705218\pi$
$971$	−37.4533	−1.20193	−0.600967	+	0.799274i	$0.705218\pi$
$972$	0	0
$973$	−20.3046	−0.650935
$974$	0	0
$975$	−46.0279	−1.47407
$976$	0	0
$977$	−38.0539	−1.21745	−0.608726	−	0.793381i	$-0.708319\pi$
$977$	−38.0539	−1.21745	−0.608726	+	0.793381i	$0.708319\pi$
$978$	0	0
$979$	−38.2786	−1.22339
$980$	0	0
$981$	25.3131	0.808185
$982$	0	0
$983$	49.0680	1.56503	0.782513	−	0.622634i	$-0.213938\pi$
$983$	49.0680	1.56503	0.782513	+	0.622634i	$0.213938\pi$
$984$	0	0
$985$	−14.7923	−0.471323
$986$	0	0
$987$	64.9975	2.06889
$988$	0	0
$989$	16.6461	0.529316
$990$	0	0
$991$	−33.6501	−1.06893	−0.534465	−	0.845190i	$-0.679487\pi$
$991$	−33.6501	−1.06893	−0.534465	+	0.845190i	$0.679487\pi$
$992$	0	0
$993$	25.6004	0.812405
$994$	0	0
$995$	−9.81953	−0.311300
$996$	0	0
$997$	38.7913	1.22853	0.614266	−	0.789099i	$-0.289452\pi$
$997$	38.7913	1.22853	0.614266	+	0.789099i	$0.289452\pi$
$998$	0	0
$999$	3.42592	0.108391

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2312.2.a.w.1.3		12
4.3	odd	2		4624.2.a.bt.1.10		12
17.4	even	4		2312.2.b.n.577.10		12
17.11	odd	16		136.2.n.c.121.3	yes	12
17.13	even	4		2312.2.b.n.577.3		12
17.14	odd	16		136.2.n.c.9.3	✓	12
17.16	even	2	inner	2312.2.a.w.1.10		12
51.11	even	16		1224.2.bq.c.937.2		12
51.14	even	16		1224.2.bq.c.145.2		12
68.11	even	16		272.2.v.f.257.1		12
68.31	even	16		272.2.v.f.145.1		12
68.67	odd	2		4624.2.a.bt.1.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.2.n.c.9.3	✓	12	17.14	odd	16
136.2.n.c.121.3	yes	12	17.11	odd	16
272.2.v.f.145.1		12	68.31	even	16
272.2.v.f.257.1		12	68.11	even	16
1224.2.bq.c.145.2		12	51.14	even	16
1224.2.bq.c.937.2		12	51.11	even	16
2312.2.a.w.1.3		12	1.1	even	1	trivial
2312.2.a.w.1.10		12	17.16	even	2	inner
2312.2.b.n.577.3		12	17.13	even	4
2312.2.b.n.577.10		12	17.4	even	4
4624.2.a.bt.1.3		12	68.67	odd	2
4624.2.a.bt.1.10		12	4.3	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2312 = 2^{3} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2312.a (trivial)

\(f(q)\)	\(=\)	\(q-2.67543 q^{3} -1.02573 q^{5} -2.33494 q^{7} +4.15793 q^{9} -5.20843 q^{11} -4.35777 q^{13} +2.74428 q^{15} -2.67684 q^{19} +6.24696 q^{21} -4.10630 q^{23} -3.94787 q^{25} -3.09796 q^{27} +3.84029 q^{29} -10.9766 q^{31} +13.9348 q^{33} +2.39502 q^{35} -1.10586 q^{37} +11.6589 q^{39} -2.62484 q^{41} -4.05380 q^{43} -4.26493 q^{45} +10.4047 q^{47} -1.54807 q^{49} -7.77630 q^{53} +5.34245 q^{55} +7.16170 q^{57} -10.8822 q^{59} -7.95894 q^{61} -9.70851 q^{63} +4.46990 q^{65} -6.44882 q^{67} +10.9861 q^{69} -1.87861 q^{71} +13.2020 q^{73} +10.5623 q^{75} +12.1613 q^{77} +2.50172 q^{79} -4.18540 q^{81} -0.0329340 q^{83} -10.2744 q^{87} +7.34936 q^{89} +10.1751 q^{91} +29.3672 q^{93} +2.74572 q^{95} +18.2600 q^{97} -21.6563 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 28 q^{9} + 24 q^{13} + 8 q^{15} - 8 q^{19} + 16 q^{21} + 20 q^{25} + 24 q^{33} - 32 q^{35} + 8 q^{43} + 24 q^{47} + 36 q^{49} + 8 q^{53} + 56 q^{55} - 40 q^{59} + 40 q^{67} + 56 q^{69} + 80 q^{77}+ \cdots + 40 q^{93}+O(q^{100}) \) 12 * q + 28 * q^9 + 24 * q^13 + 8 * q^15 - 8 * q^19 + 16 * q^21 + 20 * q^25 + 24 * q^33 - 32 * q^35 + 8 * q^43 + 24 * q^47 + 36 * q^49 + 8 * q^53 + 56 * q^55 - 40 * q^59 + 40 * q^67 + 56 * q^69 + 80 * q^77 + 60 * q^81 - 24 * q^83 + 24 * q^87 + 48 * q^89 + 40 * q^93

Introduction

L-functions

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