LMFDB - Embedded newform 1856.4.a.bk.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1856,4,Mod(1,1856)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1856, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1856.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1856 = 2^{6} \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1856.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:	$109.507544971$
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} - 245x^{10} + 21294x^{8} - 755514x^{6} + 8955005x^{4} - 27099393x^{2} + 23710340 $ x^12 - 245x^10 + 21294x^8 - 755514x^6 + 8955005x^4 - 27099393*x^2 + 23710340
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{19} $
Twist minimal:	no (minimal twist has level 928)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-8.30801$ of defining polynomial
Character	$\chi$	$=$	1856.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.30801 q^{3} -11.7163 q^{5} +23.1998 q^{7} +42.0230 q^{9} +O(q^{10})$	$q-8.30801 q^{3} -11.7163 q^{5} +23.1998 q^{7} +42.0230 q^{9} -62.0825 q^{11} +55.0060 q^{13} +97.3390 q^{15} +23.4038 q^{17} +36.9813 q^{19} -192.744 q^{21} +28.7359 q^{23} +12.2713 q^{25} -124.812 q^{27} +29.0000 q^{29} +231.614 q^{31} +515.782 q^{33} -271.816 q^{35} +202.173 q^{37} -456.990 q^{39} -253.327 q^{41} +127.371 q^{43} -492.354 q^{45} -462.422 q^{47} +195.232 q^{49} -194.439 q^{51} +632.724 q^{53} +727.377 q^{55} -307.241 q^{57} -735.673 q^{59} -185.609 q^{61} +974.928 q^{63} -644.466 q^{65} -385.606 q^{67} -238.738 q^{69} -1122.76 q^{71} -63.7142 q^{73} -101.950 q^{75} -1440.30 q^{77} +238.508 q^{79} -97.6858 q^{81} +658.034 q^{83} -274.206 q^{85} -240.932 q^{87} -916.454 q^{89} +1276.13 q^{91} -1924.25 q^{93} -433.283 q^{95} +1078.30 q^{97} -2608.90 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 10 q^{5} + 166 q^{9}+O(q^{10}) $ 12 * q - 10 * q^5 + 166 * q^9	$ 12 q - 10 q^{5} + 166 q^{9} - 30 q^{13} + 336 q^{17} - 56 q^{21} + 294 q^{25} + 348 q^{29} + 214 q^{33} - 196 q^{37} + 940 q^{41} - 1672 q^{45} + 972 q^{49} + 406 q^{53} + 1224 q^{57} - 400 q^{61} + 2998 q^{65} - 1004 q^{69} + 2252 q^{73} - 3032 q^{77} + 3932 q^{81} - 5564 q^{85} + 5212 q^{89} - 3722 q^{93} + 6164 q^{97}+O(q^{100}) $ 12 * q - 10 * q^5 + 166 * q^9 - 30 * q^13 + 336 * q^17 - 56 * q^21 + 294 * q^25 + 348 * q^29 + 214 * q^33 - 196 * q^37 + 940 * q^41 - 1672 * q^45 + 972 * q^49 + 406 * q^53 + 1224 * q^57 - 400 * q^61 + 2998 * q^65 - 1004 * q^69 + 2252 * q^73 - 3032 * q^77 + 3932 * q^81 - 5564 * q^85 + 5212 * q^89 - 3722 * q^93 + 6164 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−8.30801	−1.59888	−0.799439	−	0.600748i	$-0.794870\pi$
$3$	−8.30801	−1.59888	−0.799439	+	0.600748i	$0.794870\pi$
$4$	0	0
$5$	−11.7163	−1.04794	−0.523968	−	0.851738i	$-0.675549\pi$
$5$	−11.7163	−1.04794	−0.523968	+	0.851738i	$0.675549\pi$
$6$	0	0
$7$	23.1998	1.25267	0.626337	−	0.779553i	$-0.284554\pi$
$7$	23.1998	1.25267	0.626337	+	0.779553i	$0.284554\pi$
$8$	0	0
$9$	42.0230	1.55641
$10$	0	0
$11$	−62.0825	−1.70169	−0.850845	−	0.525417i	$-0.823909\pi$
$11$	−62.0825	−1.70169	−0.850845	+	0.525417i	$0.823909\pi$
$12$	0	0
$13$	55.0060	1.17353	0.586766	−	0.809757i	$-0.300401\pi$
$13$	55.0060	1.17353	0.586766	+	0.809757i	$0.300401\pi$
$14$	0	0
$15$	97.3390	1.67552
$16$	0	0
$17$	23.4038	0.333897	0.166949	−	0.985966i	$-0.446609\pi$
$17$	23.4038	0.333897	0.166949	+	0.985966i	$0.446609\pi$
$18$	0	0
$19$	36.9813	0.446531	0.223265	−	0.974758i	$-0.428328\pi$
$19$	36.9813	0.446531	0.223265	+	0.974758i	$0.428328\pi$
$20$	0	0
$21$	−192.744	−2.00287
$22$	0	0
$23$	28.7359	0.260515	0.130257	−	0.991480i	$-0.458420\pi$
$23$	28.7359	0.260515	0.130257	+	0.991480i	$0.458420\pi$
$24$	0	0
$25$	12.2713	0.0981706
$26$	0	0
$27$	−124.812	−0.889630
$28$	0	0
$29$	29.0000	0.185695
$29$	29.0000	0.185695
$30$	0	0
$31$	231.614	1.34190	0.670952	−	0.741500i	$-0.265885\pi$
$31$	231.614	1.34190	0.670952	+	0.741500i	$0.265885\pi$
$32$	0	0
$33$	515.782	2.72079
$34$	0	0
$35$	−271.816	−1.31272
$36$	0	0
$37$	202.173	0.898298	0.449149	−	0.893457i	$-0.351727\pi$
$37$	202.173	0.898298	0.449149	+	0.893457i	$0.351727\pi$
$38$	0	0
$39$	−456.990	−1.87633
$40$	0	0
$41$	−253.327	−0.964953	−0.482477	−	0.875909i	$-0.660263\pi$
$41$	−253.327	−0.964953	−0.482477	+	0.875909i	$0.660263\pi$
$42$	0	0
$43$	127.371	0.451719	0.225859	−	0.974160i	$-0.427481\pi$
$43$	127.371	0.451719	0.225859	+	0.974160i	$0.427481\pi$
$44$	0	0
$45$	−492.354	−1.63102
$46$	0	0
$47$	−462.422	−1.43513	−0.717566	−	0.696491i	$-0.754744\pi$
$47$	−462.422	−1.43513	−0.717566	+	0.696491i	$0.754744\pi$
$48$	0	0
$49$	195.232	0.569191
$50$	0	0
$51$	−194.439	−0.533861
$52$	0	0
$53$	632.724	1.63984	0.819918	−	0.572481i	$-0.194019\pi$
$53$	632.724	1.63984	0.819918	+	0.572481i	$0.194019\pi$
$54$	0	0
$55$	727.377	1.78326
$56$	0	0
$57$	−307.241	−0.713948
$58$	0	0
$59$	−735.673	−1.62333	−0.811665	−	0.584123i	$-0.801439\pi$
$59$	−735.673	−1.62333	−0.811665	+	0.584123i	$0.801439\pi$
$60$	0	0
$61$	−185.609	−0.389586	−0.194793	−	0.980844i	$-0.562404\pi$
$61$	−185.609	−0.389586	−0.194793	+	0.980844i	$0.562404\pi$
$62$	0	0
$63$	974.928	1.94967
$64$	0	0
$65$	−644.466	−1.22979
$66$	0	0
$67$	−385.606	−0.703123	−0.351561	−	0.936165i	$-0.614349\pi$
$67$	−385.606	−0.703123	−0.351561	+	0.936165i	$0.614349\pi$
$68$	0	0
$69$	−238.738	−0.416531
$70$	0	0
$71$	−1122.76	−1.87671	−0.938356	−	0.345670i	$-0.887652\pi$
$71$	−1122.76	−1.87671	−0.938356	+	0.345670i	$0.887652\pi$
$72$	0	0
$73$	−63.7142	−0.102153	−0.0510766	−	0.998695i	$-0.516265\pi$
$73$	−63.7142	−0.102153	−0.0510766	+	0.998695i	$0.516265\pi$
$74$	0	0
$75$	−101.950	−0.156963
$76$	0	0
$77$	−1440.30	−2.13166
$78$	0	0
$79$	238.508	0.339674	0.169837	−	0.985472i	$-0.445676\pi$
$79$	238.508	0.339674	0.169837	+	0.985472i	$0.445676\pi$
$80$	0	0
$81$	−97.6858	−0.134000
$82$	0	0
$83$	658.034	0.870224	0.435112	−	0.900376i	$-0.356709\pi$
$83$	658.034	0.870224	0.435112	+	0.900376i	$0.356709\pi$
$84$	0	0
$85$	−274.206	−0.349903
$86$	0	0
$87$	−240.932	−0.296904
$88$	0	0
$89$	−916.454	−1.09150	−0.545752	−	0.837947i	$-0.683756\pi$
$89$	−916.454	−1.09150	−0.545752	+	0.837947i	$0.683756\pi$
$90$	0	0
$91$	1276.13	1.47005
$92$	0	0
$93$	−1924.25	−2.14554
$94$	0	0
$95$	−433.283	−0.467936
$96$	0	0
$97$	1078.30	1.12871	0.564353	−	0.825533i	$-0.309126\pi$
$97$	1078.30	1.12871	0.564353	+	0.825533i	$0.309126\pi$
$98$	0	0
$99$	−2608.90	−2.64853
$100$	0	0
$101$	1050.27	1.03471	0.517353	−	0.855772i	$-0.326917\pi$
$101$	1050.27	1.03471	0.517353	+	0.855772i	$0.326917\pi$
$102$	0	0
$103$	−1436.51	−1.37421	−0.687107	−	0.726556i	$-0.741120\pi$
$103$	−1436.51	−1.37421	−0.687107	+	0.726556i	$0.741120\pi$
$104$	0	0
$105$	2258.25	2.09888
$106$	0	0
$107$	−607.864	−0.549200	−0.274600	−	0.961559i	$-0.588545\pi$
$107$	−607.864	−0.549200	−0.274600	+	0.961559i	$0.588545\pi$
$108$	0	0
$109$	−1817.10	−1.59675	−0.798377	−	0.602158i	$-0.794308\pi$
$109$	−1817.10	−1.59675	−0.798377	+	0.602158i	$0.794308\pi$
$110$	0	0
$111$	−1679.66	−1.43627
$112$	0	0
$113$	1733.82	1.44340	0.721701	−	0.692204i	$-0.243360\pi$
$113$	1733.82	1.44340	0.721701	+	0.692204i	$0.243360\pi$
$114$	0	0
$115$	−336.678	−0.273003
$116$	0	0
$117$	2311.52	1.82650
$118$	0	0
$119$	542.964	0.418264
$120$	0	0
$121$	2523.24	1.89575
$122$	0	0
$123$	2104.65	1.54284
$124$	0	0
$125$	1320.76	0.945060
$126$	0	0
$127$	−606.984	−0.424103	−0.212052	−	0.977258i	$-0.568015\pi$
$127$	−606.984	−0.424103	−0.212052	+	0.977258i	$0.568015\pi$
$128$	0	0
$129$	−1058.20	−0.722243
$130$	0	0
$131$	481.319	0.321016	0.160508	−	0.987035i	$-0.448687\pi$
$131$	481.319	0.321016	0.160508	+	0.987035i	$0.448687\pi$
$132$	0	0
$133$	857.959	0.559357
$134$	0	0
$135$	1462.33	0.932276
$136$	0	0
$137$	2159.52	1.34672	0.673360	−	0.739315i	$-0.264851\pi$
$137$	2159.52	1.34672	0.673360	+	0.739315i	$0.264851\pi$
$138$	0	0
$139$	−1232.32	−0.751970	−0.375985	−	0.926626i	$-0.622696\pi$
$139$	−1232.32	−0.751970	−0.375985	+	0.926626i	$0.622696\pi$
$140$	0	0
$141$	3841.81	2.29460
$142$	0	0
$143$	−3414.91	−1.99699
$144$	0	0
$145$	−339.772	−0.194597
$146$	0	0
$147$	−1621.99	−0.910066
$148$	0	0
$149$	−1085.43	−0.596789	−0.298395	−	0.954443i	$-0.596451\pi$
$149$	−1085.43	−0.596789	−0.298395	+	0.954443i	$0.596451\pi$
$150$	0	0
$151$	−1852.18	−0.998198	−0.499099	−	0.866545i	$-0.666336\pi$
$151$	−1852.18	−0.998198	−0.499099	+	0.866545i	$0.666336\pi$
$152$	0	0
$153$	983.499	0.519681
$154$	0	0
$155$	−2713.65	−1.40623
$156$	0	0
$157$	1924.87	0.978480	0.489240	−	0.872149i	$-0.337274\pi$
$157$	1924.87	0.978480	0.489240	+	0.872149i	$0.337274\pi$
$158$	0	0
$159$	−5256.68	−2.62190
$160$	0	0
$161$	666.667	0.326340
$162$	0	0
$163$	937.283	0.450391	0.225195	−	0.974314i	$-0.427698\pi$
$163$	937.283	0.450391	0.225195	+	0.974314i	$0.427698\pi$
$164$	0	0
$165$	−6043.05	−2.85122
$166$	0	0
$167$	642.013	0.297488	0.148744	−	0.988876i	$-0.452477\pi$
$167$	642.013	0.297488	0.148744	+	0.988876i	$0.452477\pi$
$168$	0	0
$169$	828.658	0.377177
$170$	0	0
$171$	1554.07	0.694985
$172$	0	0
$173$	−2706.10	−1.18925	−0.594627	−	0.804002i	$-0.702700\pi$
$173$	−2706.10	−1.18925	−0.594627	+	0.804002i	$0.702700\pi$
$174$	0	0
$175$	284.693	0.122976
$176$	0	0
$177$	6111.98	2.59551
$178$	0	0
$179$	611.613	0.255386	0.127693	−	0.991814i	$-0.459243\pi$
$179$	611.613	0.255386	0.127693	+	0.991814i	$0.459243\pi$
$180$	0	0
$181$	457.305	0.187797	0.0938984	−	0.995582i	$-0.470067\pi$
$181$	457.305	0.187797	0.0938984	+	0.995582i	$0.470067\pi$
$182$	0	0
$183$	1542.04	0.622901
$184$	0	0
$185$	−2368.72	−0.941359
$186$	0	0
$187$	−1452.97	−0.568190
$188$	0	0
$189$	−2895.61	−1.11442
$190$	0	0
$191$	−439.501	−0.166498	−0.0832491	−	0.996529i	$-0.526530\pi$
$191$	−439.501	−0.166498	−0.0832491	+	0.996529i	$0.526530\pi$
$192$	0	0
$193$	4902.77	1.82854	0.914271	−	0.405102i	$-0.132764\pi$
$193$	4902.77	1.82854	0.914271	+	0.405102i	$0.132764\pi$
$194$	0	0
$195$	5354.23	1.96628
$196$	0	0
$197$	2525.13	0.913237	0.456619	−	0.889663i	$-0.349060\pi$
$197$	2525.13	0.913237	0.456619	+	0.889663i	$0.349060\pi$
$198$	0	0
$199$	1741.76	0.620452	0.310226	−	0.950663i	$-0.399595\pi$
$199$	1741.76	0.620452	0.310226	+	0.950663i	$0.399595\pi$
$200$	0	0
$201$	3203.62	1.12421
$202$	0	0
$203$	672.795	0.232616
$204$	0	0
$205$	2968.06	1.01121
$206$	0	0
$207$	1207.57	0.405468
$208$	0	0
$209$	−2295.89	−0.759857
$210$	0	0
$211$	3859.94	1.25938	0.629690	−	0.776847i	$-0.283182\pi$
$211$	3859.94	1.25938	0.629690	+	0.776847i	$0.283182\pi$
$212$	0	0
$213$	9327.87	3.00063
$214$	0	0
$215$	−1492.32	−0.473372
$216$	0	0
$217$	5373.40	1.68097
$218$	0	0
$219$	529.338	0.163330
$220$	0	0
$221$	1287.35	0.391839
$222$	0	0
$223$	2151.76	0.646153	0.323077	−	0.946373i	$-0.395283\pi$
$223$	2151.76	0.646153	0.323077	+	0.946373i	$0.395283\pi$
$224$	0	0
$225$	515.679	0.152794
$226$	0	0
$227$	−4030.71	−1.17853	−0.589267	−	0.807938i	$-0.700584\pi$
$227$	−4030.71	−1.17853	−0.589267	+	0.807938i	$0.700584\pi$
$228$	0	0
$229$	−3671.88	−1.05958	−0.529792	−	0.848128i	$-0.677730\pi$
$229$	−3671.88	−1.05958	−0.529792	+	0.848128i	$0.677730\pi$
$230$	0	0
$231$	11966.1	3.40827
$232$	0	0
$233$	2753.83	0.774290	0.387145	−	0.922019i	$-0.373461\pi$
$233$	2753.83	0.774290	0.387145	+	0.922019i	$0.373461\pi$
$234$	0	0
$235$	5417.87	1.50393
$236$	0	0
$237$	−1981.52	−0.543096
$238$	0	0
$239$	5767.48	1.56095	0.780475	−	0.625186i	$-0.214977\pi$
$239$	5767.48	1.56095	0.780475	+	0.625186i	$0.214977\pi$
$240$	0	0
$241$	3993.78	1.06748	0.533738	−	0.845650i	$-0.320787\pi$
$241$	3993.78	1.06748	0.533738	+	0.845650i	$0.320787\pi$
$242$	0	0
$243$	4181.49	1.10388
$244$	0	0
$245$	−2287.40	−0.596476
$246$	0	0
$247$	2034.19	0.524018
$248$	0	0
$249$	−5466.95	−1.39138
$250$	0	0
$251$	−6476.02	−1.62854	−0.814269	−	0.580488i	$-0.802862\pi$
$251$	−6476.02	−1.62854	−0.814269	+	0.580488i	$0.802862\pi$
$252$	0	0
$253$	−1783.99	−0.443315
$254$	0	0
$255$	2278.10	0.559452
$256$	0	0
$257$	7506.57	1.82197	0.910986	−	0.412437i	$-0.135322\pi$
$257$	7506.57	1.82197	0.910986	+	0.412437i	$0.135322\pi$
$258$	0	0
$259$	4690.38	1.12527
$260$	0	0
$261$	1218.67	0.289018
$262$	0	0
$263$	6973.95	1.63510	0.817551	−	0.575856i	$-0.195331\pi$
$263$	6973.95	1.63510	0.817551	+	0.575856i	$0.195331\pi$
$264$	0	0
$265$	−7413.17	−1.71844
$266$	0	0
$267$	7613.91	1.74518
$268$	0	0
$269$	5173.19	1.17255	0.586273	−	0.810113i	$-0.300594\pi$
$269$	5173.19	1.17255	0.586273	+	0.810113i	$0.300594\pi$
$270$	0	0
$271$	−85.7783	−0.0192275	−0.00961376	−	0.999954i	$-0.503060\pi$
$271$	−85.7783	−0.0192275	−0.00961376	+	0.999954i	$0.503060\pi$
$272$	0	0
$273$	−10602.1	−2.35043
$274$	0	0
$275$	−761.835	−0.167056
$276$	0	0
$277$	5893.83	1.27843	0.639216	−	0.769027i	$-0.279259\pi$
$277$	5893.83	1.27843	0.639216	+	0.769027i	$0.279259\pi$
$278$	0	0
$279$	9733.11	2.08855
$280$	0	0
$281$	−7827.39	−1.66172	−0.830859	−	0.556482i	$-0.812151\pi$
$281$	−7827.39	−1.66172	−0.830859	+	0.556482i	$0.812151\pi$
$282$	0	0
$283$	−3389.07	−0.711871	−0.355936	−	0.934511i	$-0.615838\pi$
$283$	−3389.07	−0.711871	−0.355936	+	0.934511i	$0.615838\pi$
$284$	0	0
$285$	3599.72	0.748172
$286$	0	0
$287$	−5877.15	−1.20877
$288$	0	0
$289$	−4365.26	−0.888513
$290$	0	0
$291$	−8958.51	−1.80466
$292$	0	0
$293$	−114.137	−0.0227575	−0.0113788	−	0.999935i	$-0.503622\pi$
$293$	−114.137	−0.0227575	−0.0113788	+	0.999935i	$0.503622\pi$
$294$	0	0
$295$	8619.36	1.70115
$296$	0	0
$297$	7748.62	1.51387
$298$	0	0
$299$	1580.64	0.305722
$300$	0	0
$301$	2954.99	0.565856
$302$	0	0
$303$	−8725.62	−1.65437
$304$	0	0
$305$	2174.64	0.408262
$306$	0	0
$307$	4567.26	0.849078	0.424539	−	0.905410i	$-0.360436\pi$
$307$	4567.26	0.849078	0.424539	+	0.905410i	$0.360436\pi$
$308$	0	0
$309$	11934.6	2.19720
$310$	0	0
$311$	−7139.58	−1.30176	−0.650882	−	0.759179i	$-0.725601\pi$
$311$	−7139.58	−1.30176	−0.650882	+	0.759179i	$0.725601\pi$
$312$	0	0
$313$	−2470.03	−0.446051	−0.223026	−	0.974813i	$-0.571593\pi$
$313$	−2470.03	−0.446051	−0.223026	+	0.974813i	$0.571593\pi$
$314$	0	0
$315$	−11422.5	−2.04313
$316$	0	0
$317$	−6835.81	−1.21116	−0.605579	−	0.795785i	$-0.707059\pi$
$317$	−6835.81	−1.21116	−0.605579	+	0.795785i	$0.707059\pi$
$318$	0	0
$319$	−1800.39	−0.315996
$320$	0	0
$321$	5050.14	0.878104
$322$	0	0
$323$	865.502	0.149095
$324$	0	0
$325$	674.996	0.115206
$326$	0	0
$327$	15096.5	2.55301
$328$	0	0
$329$	−10728.1	−1.79775
$330$	0	0
$331$	5574.77	0.925732	0.462866	−	0.886428i	$-0.346821\pi$
$331$	5574.77	0.925732	0.462866	+	0.886428i	$0.346821\pi$
$332$	0	0
$333$	8495.92	1.39812
$334$	0	0
$335$	4517.87	0.736828
$336$	0	0
$337$	1436.08	0.232131	0.116066	−	0.993242i	$-0.462972\pi$
$337$	1436.08	0.232131	0.116066	+	0.993242i	$0.462972\pi$
$338$	0	0
$339$	−14404.6	−2.30782
$340$	0	0
$341$	−14379.2	−2.28351
$342$	0	0
$343$	−3428.18	−0.539663
$344$	0	0
$345$	2797.12	0.436498
$346$	0	0
$347$	−2288.66	−0.354069	−0.177035	−	0.984205i	$-0.556650\pi$
$347$	−2288.66	−0.354069	−0.177035	+	0.984205i	$0.556650\pi$
$348$	0	0
$349$	3302.23	0.506488	0.253244	−	0.967402i	$-0.418502\pi$
$349$	3302.23	0.506488	0.253244	+	0.967402i	$0.418502\pi$
$350$	0	0
$351$	−6865.39	−1.04401
$352$	0	0
$353$	9008.59	1.35830	0.679148	−	0.734001i	$-0.262349\pi$
$353$	9008.59	1.35830	0.679148	+	0.734001i	$0.262349\pi$
$354$	0	0
$355$	13154.5	1.96668
$356$	0	0
$357$	−4510.95	−0.668753
$358$	0	0
$359$	−628.657	−0.0924212	−0.0462106	−	0.998932i	$-0.514715\pi$
$359$	−628.657	−0.0924212	−0.0462106	+	0.998932i	$0.514715\pi$
$360$	0	0
$361$	−5491.39	−0.800610
$362$	0	0
$363$	−20963.1	−3.03107
$364$	0	0
$365$	746.494	0.107050
$366$	0	0
$367$	−3977.75	−0.565768	−0.282884	−	0.959154i	$-0.591291\pi$
$367$	−3977.75	−0.565768	−0.282884	+	0.959154i	$0.591291\pi$
$368$	0	0
$369$	−10645.6	−1.50186
$370$	0	0
$371$	14679.1	2.05418
$372$	0	0
$373$	−8082.30	−1.12195	−0.560973	−	0.827834i	$-0.689573\pi$
$373$	−8082.30	−1.12195	−0.560973	+	0.827834i	$0.689573\pi$
$374$	0	0
$375$	−10972.9	−1.51103
$376$	0	0
$377$	1595.17	0.217919
$378$	0	0
$379$	5888.22	0.798040	0.399020	−	0.916942i	$-0.369350\pi$
$379$	5888.22	0.798040	0.399020	+	0.916942i	$0.369350\pi$
$380$	0	0
$381$	5042.83	0.678089
$382$	0	0
$383$	−1783.01	−0.237879	−0.118940	−	0.992902i	$-0.537949\pi$
$383$	−1783.01	−0.237879	−0.118940	+	0.992902i	$0.537949\pi$
$384$	0	0
$385$	16875.0	2.23385
$386$	0	0
$387$	5352.52	0.703059
$388$	0	0
$389$	6487.75	0.845609	0.422804	−	0.906221i	$-0.361046\pi$
$389$	6487.75	0.845609	0.422804	+	0.906221i	$0.361046\pi$
$390$	0	0
$391$	672.528	0.0869852
$392$	0	0
$393$	−3998.80	−0.513265
$394$	0	0
$395$	−2794.42	−0.355956
$396$	0	0
$397$	−1235.25	−0.156160	−0.0780800	−	0.996947i	$-0.524879\pi$
$397$	−1235.25	−0.156160	−0.0780800	+	0.996947i	$0.524879\pi$
$398$	0	0
$399$	−7127.94	−0.894344
$400$	0	0
$401$	−11175.6	−1.39172	−0.695862	−	0.718176i	$-0.744977\pi$
$401$	−11175.6	−1.39172	−0.695862	+	0.718176i	$0.744977\pi$
$402$	0	0
$403$	12740.1	1.57477
$404$	0	0
$405$	1144.52	0.140423
$406$	0	0
$407$	−12551.4	−1.52862
$408$	0	0
$409$	1754.32	0.212092	0.106046	−	0.994361i	$-0.466181\pi$
$409$	1754.32	0.212092	0.106046	+	0.994361i	$0.466181\pi$
$410$	0	0
$411$	−17941.3	−2.15324
$412$	0	0
$413$	−17067.5	−2.03350
$414$	0	0
$415$	−7709.71	−0.911939
$416$	0	0
$417$	10238.1	1.20231
$418$	0	0
$419$	12924.5	1.50693	0.753463	−	0.657490i	$-0.228382\pi$
$419$	12924.5	1.50693	0.753463	+	0.657490i	$0.228382\pi$
$420$	0	0
$421$	6837.59	0.791552	0.395776	−	0.918347i	$-0.370476\pi$
$421$	6837.59	0.791552	0.395776	+	0.918347i	$0.370476\pi$
$422$	0	0
$423$	−19432.4	−2.23365
$424$	0	0
$425$	287.196	0.0327789
$426$	0	0
$427$	−4306.09	−0.488024
$428$	0	0
$429$	28371.1	3.19294
$430$	0	0
$431$	1845.32	0.206232	0.103116	−	0.994669i	$-0.467119\pi$
$431$	1845.32	0.206232	0.103116	+	0.994669i	$0.467119\pi$
$432$	0	0
$433$	9935.23	1.10267	0.551336	−	0.834283i	$-0.314118\pi$
$433$	9935.23	1.10267	0.551336	+	0.834283i	$0.314118\pi$
$434$	0	0
$435$	2822.83	0.311137
$436$	0	0
$437$	1062.69	0.116328
$438$	0	0
$439$	6723.58	0.730977	0.365489	−	0.930816i	$-0.380902\pi$
$439$	6723.58	0.730977	0.365489	+	0.930816i	$0.380902\pi$
$440$	0	0
$441$	8204.26	0.885893
$442$	0	0
$443$	−7000.82	−0.750833	−0.375416	−	0.926856i	$-0.622500\pi$
$443$	−7000.82	−0.750833	−0.375416	+	0.926856i	$0.622500\pi$
$444$	0	0
$445$	10737.4	1.14383
$446$	0	0
$447$	9017.73	0.954193
$448$	0	0
$449$	−10213.2	−1.07347	−0.536737	−	0.843750i	$-0.680343\pi$
$449$	−10213.2	−1.07347	−0.536737	+	0.843750i	$0.680343\pi$
$450$	0	0
$451$	15727.2	1.64205
$452$	0	0
$453$	15387.9	1.59600
$454$	0	0
$455$	−14951.5	−1.54052
$456$	0	0
$457$	−16670.1	−1.70633	−0.853165	−	0.521641i	$-0.825320\pi$
$457$	−16670.1	−1.70633	−0.853165	+	0.521641i	$0.825320\pi$
$458$	0	0
$459$	−2921.07	−0.297045
$460$	0	0
$461$	14275.6	1.44226	0.721129	−	0.692800i	$-0.243623\pi$
$461$	14275.6	1.44226	0.721129	+	0.692800i	$0.243623\pi$
$462$	0	0
$463$	−1684.40	−0.169073	−0.0845364	−	0.996420i	$-0.526941\pi$
$463$	−1684.40	−0.169073	−0.0845364	+	0.996420i	$0.526941\pi$
$464$	0	0
$465$	22545.1	2.24839
$466$	0	0
$467$	17086.7	1.69310	0.846552	−	0.532306i	$-0.178674\pi$
$467$	17086.7	1.69310	0.846552	+	0.532306i	$0.178674\pi$
$468$	0	0
$469$	−8945.99	−0.880783
$470$	0	0
$471$	−15991.8	−1.56447
$472$	0	0
$473$	−7907.52	−0.768685
$474$	0	0
$475$	453.809	0.0438362
$476$	0	0
$477$	26589.0	2.55226
$478$	0	0
$479$	−11785.6	−1.12422	−0.562108	−	0.827064i	$-0.690009\pi$
$479$	−11785.6	−1.12422	−0.562108	+	0.827064i	$0.690009\pi$
$480$	0	0
$481$	11120.7	1.05418
$482$	0	0
$483$	−5538.68	−0.521778
$484$	0	0
$485$	−12633.6	−1.18281
$486$	0	0
$487$	12881.8	1.19862	0.599310	−	0.800517i	$-0.295442\pi$
$487$	12881.8	1.19862	0.599310	+	0.800517i	$0.295442\pi$
$488$	0	0
$489$	−7786.96	−0.720120
$490$	0	0
$491$	8183.35	0.752157	0.376079	−	0.926588i	$-0.377272\pi$
$491$	8183.35	0.752157	0.376079	+	0.926588i	$0.377272\pi$
$492$	0	0
$493$	678.710	0.0620032
$494$	0	0
$495$	30566.6	2.77549
$496$	0	0
$497$	−26047.7	−2.35091
$498$	0	0
$499$	−12166.3	−1.09146	−0.545728	−	0.837963i	$-0.683747\pi$
$499$	−12166.3	−1.09146	−0.545728	+	0.837963i	$0.683747\pi$
$500$	0	0
$501$	−5333.85	−0.475646
$502$	0	0
$503$	14543.9	1.28923	0.644614	−	0.764508i	$-0.277018\pi$
$503$	14543.9	1.28923	0.644614	+	0.764508i	$0.277018\pi$
$504$	0	0
$505$	−12305.2	−1.08431
$506$	0	0
$507$	−6884.50	−0.603060
$508$	0	0
$509$	−1535.46	−0.133709	−0.0668546	−	0.997763i	$-0.521296\pi$
$509$	−1535.46	−0.133709	−0.0668546	+	0.997763i	$0.521296\pi$
$510$	0	0
$511$	−1478.16	−0.127965
$512$	0	0
$513$	−4615.69	−0.397247
$514$	0	0
$515$	16830.6	1.44009
$516$	0	0
$517$	28708.3	2.44215
$518$	0	0
$519$	22482.3	1.90147
$520$	0	0
$521$	7692.44	0.646856	0.323428	−	0.946253i	$-0.395165\pi$
$521$	7692.44	0.646856	0.323428	+	0.946253i	$0.395165\pi$
$522$	0	0
$523$	−3378.78	−0.282493	−0.141247	−	0.989974i	$-0.545111\pi$
$523$	−3378.78	−0.282493	−0.141247	+	0.989974i	$0.545111\pi$
$524$	0	0
$525$	−2365.23	−0.196623
$526$	0	0
$527$	5420.64	0.448059
$528$	0	0
$529$	−11341.3	−0.932132
$530$	0	0
$531$	−30915.2	−2.52657
$532$	0	0
$533$	−13934.5	−1.13240
$534$	0	0
$535$	7121.91	0.575527
$536$	0	0
$537$	−5081.29	−0.408331
$538$	0	0
$539$	−12120.5	−0.968586
$540$	0	0
$541$	−7492.69	−0.595446	−0.297723	−	0.954652i	$-0.596227\pi$
$541$	−7492.69	−0.595446	−0.297723	+	0.954652i	$0.596227\pi$
$542$	0	0
$543$	−3799.30	−0.300264
$544$	0	0
$545$	21289.6	1.67330
$546$	0	0
$547$	−5212.11	−0.407411	−0.203706	−	0.979032i	$-0.565299\pi$
$547$	−5212.11	−0.407411	−0.203706	+	0.979032i	$0.565299\pi$
$548$	0	0
$549$	−7799.84	−0.606355
$550$	0	0
$551$	1072.46	0.0829187
$552$	0	0
$553$	5533.34	0.425500
$554$	0	0
$555$	19679.3	1.50512
$556$	0	0
$557$	16670.9	1.26816	0.634082	−	0.773266i	$-0.281378\pi$
$557$	16670.9	1.26816	0.634082	+	0.773266i	$0.281378\pi$
$558$	0	0
$559$	7006.17	0.530106
$560$	0	0
$561$	12071.3	0.908466
$562$	0	0
$563$	9671.46	0.723985	0.361992	−	0.932181i	$-0.382097\pi$
$563$	9671.46	0.723985	0.361992	+	0.932181i	$0.382097\pi$
$564$	0	0
$565$	−20314.0	−1.51259
$566$	0	0
$567$	−2266.30	−0.167858
$568$	0	0
$569$	23335.6	1.71930	0.859648	−	0.510887i	$-0.170683\pi$
$569$	23335.6	1.71930	0.859648	+	0.510887i	$0.170683\pi$
$570$	0	0
$571$	−2026.11	−0.148494	−0.0742469	−	0.997240i	$-0.523655\pi$
$571$	−2026.11	−0.148494	−0.0742469	+	0.997240i	$0.523655\pi$
$572$	0	0
$573$	3651.38	0.266210
$574$	0	0
$575$	352.627	0.0255749
$576$	0	0
$577$	−27023.5	−1.94975	−0.974873	−	0.222760i	$-0.928493\pi$
$577$	−27023.5	−1.94975	−0.974873	+	0.222760i	$0.928493\pi$
$578$	0	0
$579$	−40732.2	−2.92362
$580$	0	0
$581$	15266.3	1.09011
$582$	0	0
$583$	−39281.1	−2.79049
$584$	0	0
$585$	−27082.4	−1.91405
$586$	0	0
$587$	23913.1	1.68143	0.840714	−	0.541479i	$-0.182135\pi$
$587$	23913.1	1.68143	0.840714	+	0.541479i	$0.182135\pi$
$588$	0	0
$589$	8565.37	0.599202
$590$	0	0
$591$	−20978.8	−1.46015
$592$	0	0
$593$	−23939.4	−1.65780	−0.828899	−	0.559399i	$-0.811032\pi$
$593$	−23939.4	−1.65780	−0.828899	+	0.559399i	$0.811032\pi$
$594$	0	0
$595$	−6361.53	−0.438314
$596$	0	0
$597$	−14470.6	−0.992027
$598$	0	0
$599$	−6506.21	−0.443801	−0.221900	−	0.975069i	$-0.571226\pi$
$599$	−6506.21	−0.443801	−0.221900	+	0.975069i	$0.571226\pi$
$600$	0	0
$601$	1426.61	0.0968262	0.0484131	−	0.998827i	$-0.484584\pi$
$601$	1426.61	0.0968262	0.0484131	+	0.998827i	$0.484584\pi$
$602$	0	0
$603$	−16204.3	−1.09435
$604$	0	0
$605$	−29563.0	−1.98662
$606$	0	0
$607$	23679.9	1.58342	0.791711	−	0.610896i	$-0.209191\pi$
$607$	23679.9	1.58342	0.791711	+	0.610896i	$0.209191\pi$
$608$	0	0
$609$	−5589.59	−0.371924
$610$	0	0
$611$	−25436.0	−1.68417
$612$	0	0
$613$	−11376.7	−0.749595	−0.374797	−	0.927107i	$-0.622288\pi$
$613$	−11376.7	−0.749595	−0.374797	+	0.927107i	$0.622288\pi$
$614$	0	0
$615$	−24658.6	−1.61680
$616$	0	0
$617$	20839.6	1.35976	0.679880	−	0.733323i	$-0.262032\pi$
$617$	20839.6	1.35976	0.679880	+	0.733323i	$0.262032\pi$
$618$	0	0
$619$	−1174.67	−0.0762747	−0.0381374	−	0.999273i	$-0.512142\pi$
$619$	−1174.67	−0.0762747	−0.0381374	+	0.999273i	$0.512142\pi$
$620$	0	0
$621$	−3586.57	−0.231762
$622$	0	0
$623$	−21261.6	−1.36730
$624$	0	0
$625$	−17008.3	−1.08853
$626$	0	0
$627$	19074.3	1.21492
$628$	0	0
$629$	4731.62	0.299939
$630$	0	0
$631$	3158.50	0.199268	0.0996338	−	0.995024i	$-0.468233\pi$
$631$	3158.50	0.199268	0.0996338	+	0.995024i	$0.468233\pi$
$632$	0	0
$633$	−32068.4	−2.01359
$634$	0	0
$635$	7111.60	0.444433
$636$	0	0
$637$	10738.9	0.667963
$638$	0	0
$639$	−47181.6	−2.92093
$640$	0	0
$641$	22203.9	1.36818	0.684088	−	0.729400i	$-0.260201\pi$
$641$	22203.9	1.36818	0.684088	+	0.729400i	$0.260201\pi$
$642$	0	0
$643$	−5700.58	−0.349625	−0.174812	−	0.984602i	$-0.555932\pi$
$643$	−5700.58	−0.349625	−0.174812	+	0.984602i	$0.555932\pi$
$644$	0	0
$645$	12398.2	0.756864
$646$	0	0
$647$	17132.2	1.04102	0.520508	−	0.853857i	$-0.325743\pi$
$647$	17132.2	1.04102	0.520508	+	0.853857i	$0.325743\pi$
$648$	0	0
$649$	45672.5	2.76241
$650$	0	0
$651$	−44642.3	−2.68766
$652$	0	0
$653$	−10452.8	−0.626417	−0.313209	−	0.949684i	$-0.601404\pi$
$653$	−10452.8	−0.626417	−0.313209	+	0.949684i	$0.601404\pi$
$654$	0	0
$655$	−5639.27	−0.336404
$656$	0	0
$657$	−2677.46	−0.158992
$658$	0	0
$659$	−9635.72	−0.569582	−0.284791	−	0.958590i	$-0.591924\pi$
$659$	−9635.72	−0.569582	−0.284791	+	0.958590i	$0.591924\pi$
$660$	0	0
$661$	15486.7	0.911289	0.455644	−	0.890162i	$-0.349409\pi$
$661$	15486.7	0.911289	0.455644	+	0.890162i	$0.349409\pi$
$662$	0	0
$663$	−10695.3	−0.626503
$664$	0	0
$665$	−10052.1	−0.586171
$666$	0	0
$667$	833.340	0.0483764
$668$	0	0
$669$	−17876.8	−1.03312
$670$	0	0
$671$	11523.1	0.662955
$672$	0	0
$673$	12254.6	0.701903	0.350951	−	0.936394i	$-0.385858\pi$
$673$	12254.6	0.701903	0.350951	+	0.936394i	$0.385858\pi$
$674$	0	0
$675$	−1531.60	−0.0873355
$676$	0	0
$677$	−7177.70	−0.407476	−0.203738	−	0.979025i	$-0.565309\pi$
$677$	−7177.70	−0.407476	−0.203738	+	0.979025i	$0.565309\pi$
$678$	0	0
$679$	25016.3	1.41390
$680$	0	0
$681$	33487.2	1.88433
$682$	0	0
$683$	13368.3	0.748938	0.374469	−	0.927240i	$-0.377825\pi$
$683$	13368.3	0.748938	0.374469	+	0.927240i	$0.377825\pi$
$684$	0	0
$685$	−25301.6	−1.41128
$686$	0	0
$687$	30506.0	1.69414
$688$	0	0
$689$	34803.6	1.92440
$690$	0	0
$691$	25358.7	1.39608	0.698039	−	0.716059i	$-0.254056\pi$
$691$	25358.7	1.39608	0.698039	+	0.716059i	$0.254056\pi$
$692$	0	0
$693$	−60526.0	−3.31774
$694$	0	0
$695$	14438.2	0.788017
$696$	0	0
$697$	−5928.82	−0.322195
$698$	0	0
$699$	−22878.9	−1.23799
$700$	0	0
$701$	−6638.51	−0.357679	−0.178840	−	0.983878i	$-0.557234\pi$
$701$	−6638.51	−0.357679	−0.178840	+	0.983878i	$0.557234\pi$
$702$	0	0
$703$	7476.61	0.401118
$704$	0	0
$705$	−45011.7	−2.40459
$706$	0	0
$707$	24366.0	1.29615
$708$	0	0
$709$	−10257.3	−0.543329	−0.271664	−	0.962392i	$-0.587574\pi$
$709$	−10257.3	−0.543329	−0.271664	+	0.962392i	$0.587574\pi$
$710$	0	0
$711$	10022.8	0.528671
$712$	0	0
$713$	6655.62	0.349586
$714$	0	0
$715$	40010.1	2.09272
$716$	0	0
$717$	−47916.3	−2.49577
$718$	0	0
$719$	18376.5	0.953167	0.476583	−	0.879129i	$-0.341875\pi$
$719$	18376.5	0.953167	0.476583	+	0.879129i	$0.341875\pi$
$720$	0	0
$721$	−33326.9	−1.72144
$722$	0	0
$723$	−33180.3	−1.70676
$724$	0	0
$725$	355.869	0.0182298
$726$	0	0
$727$	−31217.6	−1.59257	−0.796283	−	0.604924i	$-0.793203\pi$
$727$	−31217.6	−1.59257	−0.796283	+	0.604924i	$0.793203\pi$
$728$	0	0
$729$	−32102.3	−1.63097
$730$	0	0
$731$	2980.97	0.150828
$732$	0	0
$733$	22639.6	1.14081	0.570405	−	0.821364i	$-0.306786\pi$
$733$	22639.6	1.14081	0.570405	+	0.821364i	$0.306786\pi$
$734$	0	0
$735$	19003.7	0.953691
$736$	0	0
$737$	23939.4	1.19650
$738$	0	0
$739$	−36777.9	−1.83071	−0.915356	−	0.402646i	$-0.868090\pi$
$739$	−36777.9	−1.83071	−0.915356	+	0.402646i	$0.868090\pi$
$740$	0	0
$741$	−16900.1	−0.837841
$742$	0	0
$743$	28157.4	1.39030	0.695152	−	0.718863i	$-0.255337\pi$
$743$	28157.4	1.39030	0.695152	+	0.718863i	$0.255337\pi$
$744$	0	0
$745$	12717.2	0.625397
$746$	0	0
$747$	27652.6	1.35442
$748$	0	0
$749$	−14102.3	−0.687968
$750$	0	0
$751$	35664.5	1.73291	0.866457	−	0.499252i	$-0.166392\pi$
$751$	35664.5	1.73291	0.866457	+	0.499252i	$0.166392\pi$
$752$	0	0
$753$	53802.9	2.60383
$754$	0	0
$755$	21700.6	1.04605
$756$	0	0
$757$	26484.4	1.27159	0.635793	−	0.771859i	$-0.280673\pi$
$757$	26484.4	1.27159	0.635793	+	0.771859i	$0.280673\pi$
$758$	0	0
$759$	14821.4	0.708807
$760$	0	0
$761$	−10243.7	−0.487953	−0.243976	−	0.969781i	$-0.578452\pi$
$761$	−10243.7	−0.487953	−0.243976	+	0.969781i	$0.578452\pi$
$762$	0	0
$763$	−42156.3	−2.00021
$764$	0	0
$765$	−11523.0	−0.544593
$766$	0	0
$767$	−40466.4	−1.90503
$768$	0	0
$769$	−15428.6	−0.723498	−0.361749	−	0.932276i	$-0.617820\pi$
$769$	−15428.6	−0.723498	−0.361749	+	0.932276i	$0.617820\pi$
$770$	0	0
$771$	−62364.7	−2.91311
$772$	0	0
$773$	−5436.35	−0.252952	−0.126476	−	0.991970i	$-0.540367\pi$
$773$	−5436.35	−0.252952	−0.126476	+	0.991970i	$0.540367\pi$
$774$	0	0
$775$	2842.21	0.131736
$776$	0	0
$777$	−38967.7	−1.79918
$778$	0	0
$779$	−9368.37	−0.430881
$780$	0	0
$781$	69703.5	3.19358
$782$	0	0
$783$	−3619.54	−0.165200
$784$	0	0
$785$	−22552.3	−1.02538
$786$	0	0
$787$	−12165.7	−0.551030	−0.275515	−	0.961297i	$-0.588848\pi$
$787$	−12165.7	−0.551030	−0.275515	+	0.961297i	$0.588848\pi$
$788$	0	0
$789$	−57939.6	−2.61433
$790$	0	0
$791$	40224.5	1.80811
$792$	0	0
$793$	−10209.6	−0.457192
$794$	0	0
$795$	61588.7	2.74758
$796$	0	0
$797$	−25778.0	−1.14567	−0.572837	−	0.819669i	$-0.694157\pi$
$797$	−25778.0	−1.14567	−0.572837	+	0.819669i	$0.694157\pi$
$798$	0	0
$799$	−10822.4	−0.479187
$800$	0	0
$801$	−38512.2	−1.69883
$802$	0	0
$803$	3955.54	0.173833
$804$	0	0
$805$	−7810.86	−0.341984
$806$	0	0
$807$	−42978.9	−1.87476
$808$	0	0
$809$	−6596.11	−0.286659	−0.143329	−	0.989675i	$-0.545781\pi$
$809$	−6596.11	−0.286659	−0.143329	+	0.989675i	$0.545781\pi$
$810$	0	0
$811$	−44781.9	−1.93897	−0.969486	−	0.245148i	$-0.921164\pi$
$811$	−44781.9	−1.93897	−0.969486	+	0.245148i	$0.921164\pi$
$812$	0	0
$813$	712.647	0.0307425
$814$	0	0
$815$	−10981.5	−0.471981
$816$	0	0
$817$	4710.34	0.201706
$818$	0	0
$819$	53626.9	2.28800
$820$	0	0
$821$	−34370.5	−1.46107	−0.730534	−	0.682876i	$-0.760729\pi$
$821$	−34370.5	−1.46107	−0.730534	+	0.682876i	$0.760729\pi$
$822$	0	0
$823$	6839.47	0.289683	0.144841	−	0.989455i	$-0.453733\pi$
$823$	6839.47	0.289683	0.144841	+	0.989455i	$0.453733\pi$
$824$	0	0
$825$	6329.33	0.267102
$826$	0	0
$827$	44071.5	1.85310	0.926552	−	0.376166i	$-0.122758\pi$
$827$	44071.5	1.85310	0.926552	+	0.376166i	$0.122758\pi$
$828$	0	0
$829$	302.918	0.0126909	0.00634547	−	0.999980i	$-0.497980\pi$
$829$	302.918	0.0126909	0.00634547	+	0.999980i	$0.497980\pi$
$830$	0	0
$831$	−48966.0	−2.04406
$832$	0	0
$833$	4569.18	0.190051
$834$	0	0
$835$	−7522.01	−0.311748
$836$	0	0
$837$	−28908.1	−1.19380
$838$	0	0
$839$	20748.0	0.853753	0.426877	−	0.904310i	$-0.359614\pi$
$839$	20748.0	0.853753	0.426877	+	0.904310i	$0.359614\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	65030.0	2.65688
$844$	0	0
$845$	−9708.79	−0.395257
$846$	0	0
$847$	58538.8	2.37475
$848$	0	0
$849$	28156.5	1.13819
$850$	0	0
$851$	5809.61	0.234020
$852$	0	0
$853$	24903.4	0.999621	0.499810	−	0.866135i	$-0.333403\pi$
$853$	24903.4	0.999621	0.499810	+	0.866135i	$0.333403\pi$
$854$	0	0
$855$	−18207.9	−0.728300
$856$	0	0
$857$	19835.5	0.790628	0.395314	−	0.918546i	$-0.370636\pi$
$857$	19835.5	0.790628	0.395314	+	0.918546i	$0.370636\pi$
$858$	0	0
$859$	15601.2	0.619683	0.309842	−	0.950788i	$-0.399724\pi$
$859$	15601.2	0.619683	0.309842	+	0.950788i	$0.399724\pi$
$860$	0	0
$861$	48827.4	1.93268
$862$	0	0
$863$	19751.2	0.779071	0.389535	−	0.921012i	$-0.372636\pi$
$863$	19751.2	0.779071	0.389535	+	0.921012i	$0.372636\pi$
$864$	0	0
$865$	31705.4	1.24626
$866$	0	0
$867$	36266.6	1.42062
$868$	0	0
$869$	−14807.2	−0.578019
$870$	0	0
$871$	−21210.6	−0.825137
$872$	0	0
$873$	45313.4	1.75673
$874$	0	0
$875$	30641.4	1.18385
$876$	0	0
$877$	−12274.0	−0.472593	−0.236297	−	0.971681i	$-0.575934\pi$
$877$	−12274.0	−0.472593	−0.236297	+	0.971681i	$0.575934\pi$
$878$	0	0
$879$	948.250	0.0363865
$880$	0	0
$881$	27499.6	1.05163	0.525815	−	0.850599i	$-0.323760\pi$
$881$	27499.6	1.05163	0.525815	+	0.850599i	$0.323760\pi$
$882$	0	0
$883$	37153.2	1.41597	0.707987	−	0.706225i	$-0.249603\pi$
$883$	37153.2	1.41597	0.707987	+	0.706225i	$0.249603\pi$
$884$	0	0
$885$	−71609.7	−2.71993
$886$	0	0
$887$	34788.7	1.31690	0.658451	−	0.752624i	$-0.271212\pi$
$887$	34788.7	1.31690	0.658451	+	0.752624i	$0.271212\pi$
$888$	0	0
$889$	−14081.9	−0.531263
$890$	0	0
$891$	6064.58	0.228026
$892$	0	0
$893$	−17100.9	−0.640830
$894$	0	0
$895$	−7165.84	−0.267629
$896$	0	0
$897$	−13132.0	−0.488813
$898$	0	0
$899$	6716.80	0.249185
$900$	0	0
$901$	14808.1	0.547537
$902$	0	0
$903$	−24550.1	−0.904734
$904$	0	0
$905$	−5357.92	−0.196799
$906$	0	0
$907$	1393.51	0.0510152	0.0255076	−	0.999675i	$-0.491880\pi$
$907$	1393.51	0.0510152	0.0255076	+	0.999675i	$0.491880\pi$
$908$	0	0
$909$	44135.4	1.61043
$910$	0	0
$911$	−51070.1	−1.85733	−0.928664	−	0.370921i	$-0.879042\pi$
$911$	−51070.1	−1.85733	−0.928664	+	0.370921i	$0.879042\pi$
$912$	0	0
$913$	−40852.4	−1.48085
$914$	0	0
$915$	−18067.0	−0.652760
$916$	0	0
$917$	11166.5	0.402128
$918$	0	0
$919$	−1729.86	−0.0620923	−0.0310462	−	0.999518i	$-0.509884\pi$
$919$	−1729.86	−0.0620923	−0.0310462	+	0.999518i	$0.509884\pi$
$920$	0	0
$921$	−37944.8	−1.35757
$922$	0	0
$923$	−61758.3	−2.20238
$924$	0	0
$925$	2480.93	0.0881865
$926$	0	0
$927$	−60366.7	−2.13884
$928$	0	0
$929$	22800.4	0.805228	0.402614	−	0.915370i	$-0.368102\pi$
$929$	22800.4	0.805228	0.402614	+	0.915370i	$0.368102\pi$
$930$	0	0
$931$	7219.94	0.254161
$932$	0	0
$933$	59315.7	2.08136
$934$	0	0
$935$	17023.4	0.595427
$936$	0	0
$937$	39841.8	1.38909	0.694543	−	0.719451i	$-0.255606\pi$
$937$	39841.8	1.38909	0.694543	+	0.719451i	$0.255606\pi$
$938$	0	0
$939$	20521.0	0.713182
$940$	0	0
$941$	−4231.91	−0.146606	−0.0733030	−	0.997310i	$-0.523354\pi$
$941$	−4231.91	−0.146606	−0.0733030	+	0.997310i	$0.523354\pi$
$942$	0	0
$943$	−7279.58	−0.251385
$944$	0	0
$945$	33925.8	1.16784
$946$	0	0
$947$	33543.5	1.15102	0.575511	−	0.817794i	$-0.304803\pi$
$947$	33543.5	1.15102	0.575511	+	0.817794i	$0.304803\pi$
$948$	0	0
$949$	−3504.66	−0.119880
$950$	0	0
$951$	56792.0	1.93649
$952$	0	0
$953$	−49082.4	−1.66835	−0.834174	−	0.551502i	$-0.814055\pi$
$953$	−49082.4	−1.66835	−0.834174	+	0.551502i	$0.814055\pi$
$954$	0	0
$955$	5149.31	0.174479
$956$	0	0
$957$	14957.7	0.505239
$958$	0	0
$959$	50100.6	1.68700
$960$	0	0
$961$	23853.9	0.800709
$962$	0	0
$963$	−25544.3	−0.854780
$964$	0	0
$965$	−57442.2	−1.91620
$966$	0	0
$967$	−52198.8	−1.73589	−0.867943	−	0.496665i	$-0.834558\pi$
$967$	−52198.8	−1.73589	−0.867943	+	0.496665i	$0.834558\pi$
$968$	0	0
$969$	−7190.60	−0.238385
$970$	0	0
$971$	−36766.9	−1.21515	−0.607573	−	0.794264i	$-0.707857\pi$
$971$	−36766.9	−1.21515	−0.607573	+	0.794264i	$0.707857\pi$
$972$	0	0
$973$	−28589.6	−0.941973
$974$	0	0
$975$	−5607.88	−0.184201
$976$	0	0
$977$	2047.60	0.0670508	0.0335254	−	0.999438i	$-0.489327\pi$
$977$	2047.60	0.0670508	0.0335254	+	0.999438i	$0.489327\pi$
$978$	0	0
$979$	56895.8	1.85740
$980$	0	0
$981$	−76359.9	−2.48520
$982$	0	0
$983$	4103.30	0.133138	0.0665691	−	0.997782i	$-0.478795\pi$
$983$	4103.30	0.133138	0.0665691	+	0.997782i	$0.478795\pi$
$984$	0	0
$985$	−29585.1	−0.957015
$986$	0	0
$987$	89129.3	2.87438
$988$	0	0
$989$	3660.12	0.117679
$990$	0	0
$991$	−19810.8	−0.635027	−0.317514	−	0.948254i	$-0.602848\pi$
$991$	−19810.8	−0.635027	−0.317514	+	0.948254i	$0.602848\pi$
$992$	0	0
$993$	−46315.3	−1.48013
$994$	0	0
$995$	−20406.9	−0.650195
$996$	0	0
$997$	−39629.9	−1.25887	−0.629434	−	0.777054i	$-0.716713\pi$
$997$	−39629.9	−1.25887	−0.629434	+	0.777054i	$0.716713\pi$
$998$	0	0
$999$	−25233.5	−0.799153

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1856.4.a.bk.1.3		12
4.3	odd	2	inner	1856.4.a.bk.1.10		12
8.3	odd	2		928.4.a.i.1.3	✓	12
8.5	even	2		928.4.a.i.1.10	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
928.4.a.i.1.3	✓	12	8.3	odd	2
928.4.a.i.1.10	yes	12	8.5	even	2
1856.4.a.bk.1.3		12	1.1	even	1	trivial
1856.4.a.bk.1.10		12	4.3	odd	2	inner

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:	\( N \)	\(=\)	\( 1856 = 2^{6} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1856.a (trivial)

\(f(q)\)	\(=\)	\(q-8.30801 q^{3} -11.7163 q^{5} +23.1998 q^{7} +42.0230 q^{9} +O(q^{10})\)	\(q-8.30801 q^{3} -11.7163 q^{5} +23.1998 q^{7} +42.0230 q^{9} -62.0825 q^{11} +55.0060 q^{13} +97.3390 q^{15} +23.4038 q^{17} +36.9813 q^{19} -192.744 q^{21} +28.7359 q^{23} +12.2713 q^{25} -124.812 q^{27} +29.0000 q^{29} +231.614 q^{31} +515.782 q^{33} -271.816 q^{35} +202.173 q^{37} -456.990 q^{39} -253.327 q^{41} +127.371 q^{43} -492.354 q^{45} -462.422 q^{47} +195.232 q^{49} -194.439 q^{51} +632.724 q^{53} +727.377 q^{55} -307.241 q^{57} -735.673 q^{59} -185.609 q^{61} +974.928 q^{63} -644.466 q^{65} -385.606 q^{67} -238.738 q^{69} -1122.76 q^{71} -63.7142 q^{73} -101.950 q^{75} -1440.30 q^{77} +238.508 q^{79} -97.6858 q^{81} +658.034 q^{83} -274.206 q^{85} -240.932 q^{87} -916.454 q^{89} +1276.13 q^{91} -1924.25 q^{93} -433.283 q^{95} +1078.30 q^{97} -2608.90 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 10 q^{5} + 166 q^{9}+O(q^{10}) \) 12 * q - 10 * q^5 + 166 * q^9	\( 12 q - 10 q^{5} + 166 q^{9} - 30 q^{13} + 336 q^{17} - 56 q^{21} + 294 q^{25} + 348 q^{29} + 214 q^{33} - 196 q^{37} + 940 q^{41} - 1672 q^{45} + 972 q^{49} + 406 q^{53} + 1224 q^{57} - 400 q^{61} + 2998 q^{65} - 1004 q^{69} + 2252 q^{73} - 3032 q^{77} + 3932 q^{81} - 5564 q^{85} + 5212 q^{89} - 3722 q^{93} + 6164 q^{97}+O(q^{100}) \) 12 * q - 10 * q^5 + 166 * q^9 - 30 * q^13 + 336 * q^17 - 56 * q^21 + 294 * q^25 + 348 * q^29 + 214 * q^33 - 196 * q^37 + 940 * q^41 - 1672 * q^45 + 972 * q^49 + 406 * q^53 + 1224 * q^57 - 400 * q^61 + 2998 * q^65 - 1004 * q^69 + 2252 * q^73 - 3032 * q^77 + 3932 * q^81 - 5564 * q^85 + 5212 * q^89 - 3722 * q^93 + 6164 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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