LMFDB - Embedded newform 1568.3.c.h.97.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1568,3,Mod(97,1568)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1568, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1568.97");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1568 = 2^{5} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1568.c (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$42.7249054517$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 36x^{14} + 522x^{12} + 3644x^{10} + 12219x^{8} + 15156x^{6} + 15478x^{4} - 10992x^{2} + 11025 $ x^16 + 36x^14 + 522x^12 + 3644x^10 + 12219x^8 + 15156x^6 + 15478x^4 - 10992*x^2 + 11025
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{28} $
Twist minimal:	no (minimal twist has level 224)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			97.11
Root			$-0.707107 - 1.17406i$ of defining polynomial
Character	$\chi$	$=$	1568.97
Dual form			1568.3.c.h.97.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.66037i q^{3} -8.40186i q^{5} +6.24317 q^{9} +O(q^{10})$	$q+1.66037i q^{3} -8.40186i q^{5} +6.24317 q^{9} -5.21586 q^{11} +4.88512i q^{13} +13.9502 q^{15} +7.72228i q^{17} +35.3375i q^{19} -26.6271 q^{23} -45.5912 q^{25} +25.3093i q^{27} +45.1300 q^{29} +40.4376i q^{31} -8.66025i q^{33} +7.95896 q^{37} -8.11110 q^{39} -26.6511i q^{41} -0.403279 q^{43} -52.4542i q^{45} -32.6601i q^{47} -12.8218 q^{51} +81.2235 q^{53} +43.8229i q^{55} -58.6733 q^{57} +8.80248i q^{59} +29.2483i q^{61} +41.0441 q^{65} +87.5584 q^{67} -44.2108i q^{69} +27.5210 q^{71} +87.0045i q^{73} -75.6983i q^{75} +121.693 q^{79} +14.1657 q^{81} +46.6625i q^{83} +64.8815 q^{85} +74.9324i q^{87} +60.8799i q^{89} -67.1414 q^{93} +296.901 q^{95} -66.4681i q^{97} -32.5635 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{9}+O(q^{10}) $ 16 * q - 16 * q^9	$ 16 q - 16 q^{9} - 32 q^{25} + 112 q^{29} - 16 q^{37} + 48 q^{53} - 528 q^{57} + 16 q^{65} - 64 q^{81} + 720 q^{85} + 464 q^{93}+O(q^{100}) $ 16 * q - 16 * q^9 - 32 * q^25 + 112 * q^29 - 16 * q^37 + 48 * q^53 - 528 * q^57 + 16 * q^65 - 64 * q^81 + 720 * q^85 + 464 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1568\mathbb{Z}\right)^\times$.

$n$	$197$	$1471$	$1473$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	1.66037i	0.553457i	0.960948	+	0.276728i	$0.0892502\pi$
$3$	1.66037i	0.553457i	−0.960948	+	0.276728i	$0.910750\pi$
$4$	0	0
$5$	− 8.40186i	− 1.68037i	−0.542298	−	0.840186i	$-0.682446\pi$
$5$	− 8.40186i	− 1.68037i	0.542298	−	0.840186i	$-0.317554\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	6.24317	0.693686
$10$	0	0
$11$	−5.21586	−0.474169	−0.237084	−	0.971489i	$-0.576192\pi$
$11$	−5.21586	−0.474169	−0.237084	+	0.971489i	$0.576192\pi$
$12$	0	0
$13$	4.88512i	0.375778i	0.982190	+	0.187889i	$0.0601646\pi$
$13$	4.88512i	0.375778i	−0.982190	+	0.187889i	$0.939835\pi$
$14$	0	0
$15$	13.9502	0.930013
$16$	0	0
$17$	7.72228i	0.454252i	0.973865	+	0.227126i	$0.0729329\pi$
$17$	7.72228i	0.454252i	−0.973865	+	0.227126i	$0.927067\pi$
$18$	0	0
$19$	35.3375i	1.85987i	0.367727	+	0.929934i	$0.380136\pi$
$19$	35.3375i	1.85987i	−0.367727	+	0.929934i	$0.619864\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−26.6271	−1.15770	−0.578850	−	0.815434i	$-0.696498\pi$
$23$	−26.6271	−1.15770	−0.578850	+	0.815434i	$0.696498\pi$
$24$	0	0
$25$	−45.5912	−1.82365
$26$	0	0
$27$	25.3093i	0.937382i
$28$	0	0
$29$	45.1300	1.55621	0.778103	−	0.628137i	$-0.216182\pi$
$29$	45.1300	1.55621	0.778103	+	0.628137i	$0.216182\pi$
$30$	0	0
$31$	40.4376i	1.30444i	0.758030	+	0.652220i	$0.226162\pi$
$31$	40.4376i	1.30444i	−0.758030	+	0.652220i	$0.773838\pi$
$32$	0	0
$33$	− 8.66025i	− 0.262432i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	7.95896	0.215107	0.107554	−	0.994199i	$-0.465698\pi$
$37$	7.95896	0.215107	0.107554	+	0.994199i	$0.465698\pi$
$38$	0	0
$39$	−8.11110	−0.207977
$40$	0	0
$41$	− 26.6511i	− 0.650027i	−0.945709	−	0.325014i	$-0.894631\pi$
$41$	− 26.6511i	− 0.650027i	0.945709	−	0.325014i	$-0.105369\pi$
$42$	0	0
$43$	−0.403279	−0.00937859	−0.00468930	−	0.999989i	$-0.501493\pi$
$43$	−0.403279	−0.00937859	−0.00468930	+	0.999989i	$0.501493\pi$
$44$	0	0
$45$	− 52.4542i	− 1.16565i
$46$	0	0
$47$	− 32.6601i	− 0.694896i	−0.937699	−	0.347448i	$-0.887048\pi$
$47$	− 32.6601i	− 0.694896i	0.937699	−	0.347448i	$-0.112952\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−12.8218	−0.251409
$52$	0	0
$53$	81.2235	1.53252	0.766259	−	0.642531i	$-0.222116\pi$
$53$	81.2235	1.53252	0.766259	+	0.642531i	$0.222116\pi$
$54$	0	0
$55$	43.8229i	0.796780i
$56$	0	0
$57$	−58.6733	−1.02936
$58$	0	0
$59$	8.80248i	0.149195i	0.997214	+	0.0745973i	$0.0237671\pi$
$59$	8.80248i	0.149195i	−0.997214	+	0.0745973i	$0.976233\pi$
$60$	0	0
$61$	29.2483i	0.479481i	0.970837	+	0.239740i	$0.0770623\pi$
$61$	29.2483i	0.479481i	−0.970837	+	0.239740i	$0.922938\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	41.0441	0.631447
$66$	0	0
$67$	87.5584	1.30684	0.653421	−	0.756995i	$-0.273333\pi$
$67$	87.5584	1.30684	0.653421	+	0.756995i	$0.273333\pi$
$68$	0	0
$69$	− 44.2108i	− 0.640737i
$70$	0	0
$71$	27.5210	0.387620	0.193810	−	0.981039i	$-0.437915\pi$
$71$	27.5210	0.387620	0.193810	+	0.981039i	$0.437915\pi$
$72$	0	0
$73$	87.0045i	1.19184i	0.803043	+	0.595921i	$0.203213\pi$
$73$	87.0045i	1.19184i	−0.803043	+	0.595921i	$0.796787\pi$
$74$	0	0
$75$	− 75.6983i	− 1.00931i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	121.693	1.54042	0.770212	−	0.637788i	$-0.220150\pi$
$79$	121.693	1.54042	0.770212	+	0.637788i	$0.220150\pi$
$80$	0	0
$81$	14.1657	0.174885
$82$	0	0
$83$	46.6625i	0.562198i	0.959679	+	0.281099i	$0.0906990\pi$
$83$	46.6625i	0.562198i	−0.959679	+	0.281099i	$0.909301\pi$
$84$	0	0
$85$	64.8815	0.763312
$86$	0	0
$87$	74.9324i	0.861292i
$88$	0	0
$89$	60.8799i	0.684043i	0.939692	+	0.342022i	$0.111112\pi$
$89$	60.8799i	0.684043i	−0.939692	+	0.342022i	$0.888888\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−67.1414	−0.721951
$94$	0	0
$95$	296.901	3.12527
$96$	0	0
$97$	− 66.4681i	− 0.685238i	−0.939474	−	0.342619i	$-0.888686\pi$
$97$	− 66.4681i	− 0.685238i	0.939474	−	0.342619i	$-0.111314\pi$
$98$	0	0
$99$	−32.5635	−0.328924
$100$	0	0
$101$	− 96.8129i	− 0.958543i	−0.877667	−	0.479272i	$-0.840901\pi$
$101$	− 96.8129i	− 0.958543i	0.877667	−	0.479272i	$-0.159099\pi$
$102$	0	0
$103$	− 172.458i	− 1.67435i	−0.546935	−	0.837175i	$-0.684206\pi$
$103$	− 172.458i	− 1.67435i	0.546935	−	0.837175i	$-0.315794\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	103.181	0.964305	0.482153	−	0.876087i	$-0.339855\pi$
$107$	103.181	0.964305	0.482153	+	0.876087i	$0.339855\pi$
$108$	0	0
$109$	−41.5526	−0.381216	−0.190608	−	0.981666i	$-0.561046\pi$
$109$	−41.5526	−0.381216	−0.190608	+	0.981666i	$0.561046\pi$
$110$	0	0
$111$	13.2148i	0.119052i
$112$	0	0
$113$	−133.885	−1.18482	−0.592409	−	0.805637i	$-0.701823\pi$
$113$	−133.885	−1.18482	−0.592409	+	0.805637i	$0.701823\pi$
$114$	0	0
$115$	223.717i	1.94537i
$116$	0	0
$117$	30.4986i	0.260672i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−93.7948	−0.775164
$122$	0	0
$123$	44.2507	0.359762
$124$	0	0
$125$	173.005i	1.38404i
$126$	0	0
$127$	−132.489	−1.04322	−0.521610	−	0.853184i	$-0.674668\pi$
$127$	−132.489	−1.04322	−0.521610	+	0.853184i	$0.674668\pi$
$128$	0	0
$129$	− 0.669593i	− 0.00519064i
$130$	0	0
$131$	− 74.0380i	− 0.565175i	−0.959241	−	0.282588i	$-0.908807\pi$
$131$	− 74.0380i	− 0.565175i	0.959241	−	0.282588i	$-0.0911928\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	212.645	1.57515
$136$	0	0
$137$	−74.3800	−0.542920	−0.271460	−	0.962450i	$-0.587506\pi$
$137$	−74.3800	−0.542920	−0.271460	+	0.962450i	$0.587506\pi$
$138$	0	0
$139$	123.649i	0.889560i	0.895640	+	0.444780i	$0.146718\pi$
$139$	123.649i	0.889560i	−0.895640	+	0.444780i	$0.853282\pi$
$140$	0	0
$141$	54.2279	0.384595
$142$	0	0
$143$	− 25.4801i	− 0.178182i
$144$	0	0
$145$	− 379.176i	− 2.61500i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	206.591	1.38651	0.693257	−	0.720690i	$-0.256175\pi$
$149$	206.591	1.38651	0.693257	+	0.720690i	$0.256175\pi$
$150$	0	0
$151$	48.4554	0.320897	0.160448	−	0.987044i	$-0.448706\pi$
$151$	48.4554	0.320897	0.160448	+	0.987044i	$0.448706\pi$
$152$	0	0
$153$	48.2115i	0.315108i
$154$	0	0
$155$	339.751	2.19194
$156$	0	0
$157$	177.247i	1.12896i	0.825447	+	0.564480i	$0.190923\pi$
$157$	177.247i	1.12896i	−0.825447	+	0.564480i	$0.809077\pi$
$158$	0	0
$159$	134.861i	0.848183i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−126.953	−0.778853	−0.389427	−	0.921057i	$-0.627327\pi$
$163$	−126.953	−0.778853	−0.389427	+	0.921057i	$0.627327\pi$
$164$	0	0
$165$	−72.7622	−0.440983
$166$	0	0
$167$	191.898i	1.14909i	0.818472	+	0.574546i	$0.194821\pi$
$167$	191.898i	1.14909i	−0.818472	+	0.574546i	$0.805179\pi$
$168$	0	0
$169$	145.136	0.858791
$170$	0	0
$171$	220.618i	1.29016i
$172$	0	0
$173$	188.549i	1.08988i	0.838476	+	0.544939i	$0.183447\pi$
$173$	188.549i	1.08988i	−0.838476	+	0.544939i	$0.816553\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−14.6154	−0.0825728
$178$	0	0
$179$	241.055	1.34667	0.673337	−	0.739336i	$-0.264860\pi$
$179$	241.055	1.34667	0.673337	+	0.739336i	$0.264860\pi$
$180$	0	0
$181$	277.790i	1.53475i	0.641198	+	0.767376i	$0.278438\pi$
$181$	277.790i	1.53475i	−0.641198	+	0.767376i	$0.721562\pi$
$182$	0	0
$183$	−48.5631	−0.265372
$184$	0	0
$185$	− 66.8701i	− 0.361460i
$186$	0	0
$187$	− 40.2783i	− 0.215392i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−264.376	−1.38417	−0.692085	−	0.721816i	$-0.743307\pi$
$191$	−264.376	−1.38417	−0.692085	+	0.721816i	$0.743307\pi$
$192$	0	0
$193$	211.397	1.09532	0.547660	−	0.836701i	$-0.315519\pi$
$193$	211.397	1.09532	0.547660	+	0.836701i	$0.315519\pi$
$194$	0	0
$195$	68.1484i	0.349479i
$196$	0	0
$197$	−79.4949	−0.403527	−0.201764	−	0.979434i	$-0.564667\pi$
$197$	−79.4949	−0.403527	−0.201764	+	0.979434i	$0.564667\pi$
$198$	0	0
$199$	202.956i	1.01988i	0.860210	+	0.509939i	$0.170332\pi$
$199$	202.956i	1.01988i	−0.860210	+	0.509939i	$0.829668\pi$
$200$	0	0
$201$	145.379i	0.723280i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−223.919	−1.09229
$206$	0	0
$207$	−166.237	−0.803079
$208$	0	0
$209$	− 184.315i	− 0.881891i
$210$	0	0
$211$	−166.533	−0.789256	−0.394628	−	0.918841i	$-0.629127\pi$
$211$	−166.533	−0.789256	−0.394628	+	0.918841i	$0.629127\pi$
$212$	0	0
$213$	45.6951i	0.214531i
$214$	0	0
$215$	3.38830i	0.0157595i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−144.460	−0.659633
$220$	0	0
$221$	−37.7242	−0.170698
$222$	0	0
$223$	55.0782i	0.246988i	0.992345	+	0.123494i	$0.0394099\pi$
$223$	55.0782i	0.246988i	−0.992345	+	0.123494i	$0.960590\pi$
$224$	0	0
$225$	−284.634	−1.26504
$226$	0	0
$227$	− 212.603i	− 0.936576i	−0.883576	−	0.468288i	$-0.844871\pi$
$227$	− 212.603i	− 0.936576i	0.883576	−	0.468288i	$-0.155129\pi$
$228$	0	0
$229$	44.4773i	0.194224i	0.995273	+	0.0971120i	$0.0309605\pi$
$229$	44.4773i	0.194224i	−0.995273	+	0.0971120i	$0.969039\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−272.099	−1.16781	−0.583904	−	0.811823i	$-0.698475\pi$
$233$	−272.099	−1.16781	−0.583904	+	0.811823i	$0.698475\pi$
$234$	0	0
$235$	−274.406	−1.16768
$236$	0	0
$237$	202.056i	0.852558i
$238$	0	0
$239$	389.180	1.62837	0.814185	−	0.580605i	$-0.197184\pi$
$239$	389.180	1.62837	0.814185	+	0.580605i	$0.197184\pi$
$240$	0	0
$241$	− 70.3632i	− 0.291964i	−0.989287	−	0.145982i	$-0.953366\pi$
$241$	− 70.3632i	− 0.291964i	0.989287	−	0.145982i	$-0.0466341\pi$
$242$	0	0
$243$	251.304i	1.03417i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−172.628	−0.698898
$248$	0	0
$249$	−77.4770	−0.311153
$250$	0	0
$251$	97.5325i	0.388576i	0.980945	+	0.194288i	$0.0622396\pi$
$251$	97.5325i	0.388576i	−0.980945	+	0.194288i	$0.937760\pi$
$252$	0	0
$253$	138.883	0.548945
$254$	0	0
$255$	107.727i	0.422460i
$256$	0	0
$257$	− 207.148i	− 0.806025i	−0.915194	−	0.403013i	$-0.867963\pi$
$257$	− 207.148i	− 0.806025i	0.915194	−	0.403013i	$-0.132037\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	281.754	1.07952
$262$	0	0
$263$	207.214	0.787887	0.393943	−	0.919135i	$-0.371111\pi$
$263$	207.214	0.787887	0.393943	+	0.919135i	$0.371111\pi$
$264$	0	0
$265$	− 682.428i	− 2.57520i
$266$	0	0
$267$	−101.083	−0.378588
$268$	0	0
$269$	− 35.9747i	− 0.133735i	−0.997762	−	0.0668674i	$-0.978700\pi$
$269$	− 35.9747i	− 0.133735i	0.997762	−	0.0668674i	$-0.0213005\pi$
$270$	0	0
$271$	76.6632i	0.282890i	0.989946	+	0.141445i	$0.0451748\pi$
$271$	76.6632i	0.282890i	−0.989946	+	0.141445i	$0.954825\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	237.797	0.864718
$276$	0	0
$277$	268.003	0.967521	0.483760	−	0.875200i	$-0.339271\pi$
$277$	268.003	0.967521	0.483760	+	0.875200i	$0.339271\pi$
$278$	0	0
$279$	252.459i	0.904871i
$280$	0	0
$281$	−417.336	−1.48518	−0.742591	−	0.669745i	$-0.766403\pi$
$281$	−417.336	−1.48518	−0.742591	+	0.669745i	$0.766403\pi$
$282$	0	0
$283$	143.804i	0.508142i	0.967186	+	0.254071i	$0.0817696\pi$
$283$	143.804i	0.508142i	−0.967186	+	0.254071i	$0.918230\pi$
$284$	0	0
$285$	492.965i	1.72970i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	229.366	0.793655
$290$	0	0
$291$	110.362	0.379250
$292$	0	0
$293$	− 265.694i	− 0.906805i	−0.891306	−	0.453402i	$-0.850210\pi$
$293$	− 265.694i	− 0.906805i	0.891306	−	0.453402i	$-0.149790\pi$
$294$	0	0
$295$	73.9572	0.250702
$296$	0	0
$297$	− 132.010i	− 0.444477i
$298$	0	0
$299$	− 130.076i	− 0.435038i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	160.745	0.530512
$304$	0	0
$305$	245.740	0.805706
$306$	0	0
$307$	− 281.617i	− 0.917318i	−0.888612	−	0.458659i	$-0.848330\pi$
$307$	− 281.617i	− 0.917318i	0.888612	−	0.458659i	$-0.151670\pi$
$308$	0	0
$309$	286.344	0.926680
$310$	0	0
$311$	− 56.3889i	− 0.181315i	−0.995882	−	0.0906574i	$-0.971103\pi$
$311$	− 56.3889i	− 0.181315i	0.995882	−	0.0906574i	$-0.0288968\pi$
$312$	0	0
$313$	− 84.3476i	− 0.269481i	−0.990881	−	0.134741i	$-0.956980\pi$
$313$	− 84.3476i	− 0.269481i	0.990881	−	0.134741i	$-0.0430201\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	159.386	0.502795	0.251398	−	0.967884i	$-0.419110\pi$
$317$	159.386	0.502795	0.251398	+	0.967884i	$0.419110\pi$
$318$	0	0
$319$	−235.391	−0.737904
$320$	0	0
$321$	171.318i	0.533701i
$322$	0	0
$323$	−272.886	−0.844848
$324$	0	0
$325$	− 222.719i	− 0.685288i
$326$	0	0
$327$	− 68.9926i	− 0.210987i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	559.588	1.69060	0.845299	−	0.534293i	$-0.179422\pi$
$331$	559.588	1.69060	0.845299	+	0.534293i	$0.179422\pi$
$332$	0	0
$333$	49.6892	0.149217
$334$	0	0
$335$	− 735.653i	− 2.19598i
$336$	0	0
$337$	140.493	0.416892	0.208446	−	0.978034i	$-0.433159\pi$
$337$	140.493	0.416892	0.208446	+	0.978034i	$0.433159\pi$
$338$	0	0
$339$	− 222.298i	− 0.655746i
$340$	0	0
$341$	− 210.917i	− 0.618525i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−371.453	−1.07668
$346$	0	0
$347$	388.318	1.11907	0.559537	−	0.828806i	$-0.310979\pi$
$347$	388.318	1.11907	0.559537	+	0.828806i	$0.310979\pi$
$348$	0	0
$349$	469.369i	1.34490i	0.740144	+	0.672449i	$0.234757\pi$
$349$	469.369i	1.34490i	−0.740144	+	0.672449i	$0.765243\pi$
$350$	0	0
$351$	−123.639	−0.352248
$352$	0	0
$353$	648.657i	1.83756i	0.394775	+	0.918778i	$0.370822\pi$
$353$	648.657i	1.83756i	−0.394775	+	0.918778i	$0.629178\pi$
$354$	0	0
$355$	− 231.228i	− 0.651346i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−280.601	−0.781619	−0.390810	−	0.920472i	$-0.627805\pi$
$359$	−280.601	−0.781619	−0.390810	+	0.920472i	$0.627805\pi$
$360$	0	0
$361$	−887.738	−2.45911
$362$	0	0
$363$	− 155.734i	− 0.429020i
$364$	0	0
$365$	731.000	2.00274
$366$	0	0
$367$	− 20.0010i	− 0.0544985i	−0.999629	−	0.0272493i	$-0.991325\pi$
$367$	− 20.0010i	− 0.0544985i	0.999629	−	0.0272493i	$-0.00867479\pi$
$368$	0	0
$369$	− 166.388i	− 0.450915i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−160.408	−0.430047	−0.215024	−	0.976609i	$-0.568983\pi$
$373$	−160.408	−0.430047	−0.215024	+	0.976609i	$0.568983\pi$
$374$	0	0
$375$	−287.252	−0.766005
$376$	0	0
$377$	220.465i	0.584788i
$378$	0	0
$379$	−397.426	−1.04862	−0.524308	−	0.851529i	$-0.675676\pi$
$379$	−397.426	−1.04862	−0.524308	+	0.851529i	$0.675676\pi$
$380$	0	0
$381$	− 219.981i	− 0.577377i
$382$	0	0
$383$	− 268.132i	− 0.700084i	−0.936734	−	0.350042i	$-0.886167\pi$
$383$	− 268.132i	− 0.700084i	0.936734	−	0.350042i	$-0.113833\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−2.51774	−0.00650579
$388$	0	0
$389$	−192.682	−0.495326	−0.247663	−	0.968846i	$-0.579663\pi$
$389$	−192.682	−0.495326	−0.247663	+	0.968846i	$0.579663\pi$
$390$	0	0
$391$	− 205.622i	− 0.525887i
$392$	0	0
$393$	122.930	0.312800
$394$	0	0
$395$	− 1022.45i	− 2.58848i
$396$	0	0
$397$	− 138.908i	− 0.349895i	−0.984578	−	0.174948i	$-0.944024\pi$
$397$	− 138.908i	− 0.349895i	0.984578	−	0.174948i	$-0.0559756\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	676.821	1.68783	0.843917	−	0.536474i	$-0.180244\pi$
$401$	676.821	1.68783	0.843917	+	0.536474i	$0.180244\pi$
$402$	0	0
$403$	−197.543	−0.490180
$404$	0	0
$405$	− 119.018i	− 0.293872i
$406$	0	0
$407$	−41.5128	−0.101997
$408$	0	0
$409$	204.573i	0.500177i	0.968223	+	0.250089i	$0.0804598\pi$
$409$	204.573i	0.500177i	−0.968223	+	0.250089i	$0.919540\pi$
$410$	0	0
$411$	− 123.498i	− 0.300483i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	392.052	0.944703
$416$	0	0
$417$	−205.303	−0.492333
$418$	0	0
$419$	516.134i	1.23182i	0.787815	+	0.615911i	$0.211212\pi$
$419$	516.134i	1.23182i	−0.787815	+	0.615911i	$0.788788\pi$
$420$	0	0
$421$	−81.4693	−0.193514	−0.0967569	−	0.995308i	$-0.530847\pi$
$421$	−81.4693	−0.193514	−0.0967569	+	0.995308i	$0.530847\pi$
$422$	0	0
$423$	− 203.903i	− 0.482039i
$424$	0	0
$425$	− 352.068i	− 0.828396i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	42.3064	0.0986162
$430$	0	0
$431$	−97.6451	−0.226555	−0.113277	−	0.993563i	$-0.536135\pi$
$431$	−97.6451	−0.226555	−0.113277	+	0.993563i	$0.536135\pi$
$432$	0	0
$433$	− 476.427i	− 1.10029i	−0.835068	−	0.550146i	$-0.814572\pi$
$433$	− 476.427i	− 1.10029i	0.835068	−	0.550146i	$-0.185428\pi$
$434$	0	0
$435$	629.572	1.44729
$436$	0	0
$437$	− 940.934i	− 2.15317i
$438$	0	0
$439$	269.396i	0.613658i	0.951765	+	0.306829i	$0.0992680\pi$
$439$	269.396i	0.613658i	−0.951765	+	0.306829i	$0.900732\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−341.971	−0.771943	−0.385971	−	0.922511i	$-0.626134\pi$
$443$	−341.971	−0.771943	−0.385971	+	0.922511i	$0.626134\pi$
$444$	0	0
$445$	511.504	1.14945
$446$	0	0
$447$	343.017i	0.767376i
$448$	0	0
$449$	−460.145	−1.02482	−0.512411	−	0.858741i	$-0.671247\pi$
$449$	−460.145	−1.02482	−0.512411	+	0.858741i	$0.671247\pi$
$450$	0	0
$451$	139.008i	0.308223i
$452$	0	0
$453$	80.4539i	0.177602i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	609.373	1.33342	0.666710	−	0.745317i	$-0.267702\pi$
$457$	609.373	1.33342	0.666710	+	0.745317i	$0.267702\pi$
$458$	0	0
$459$	−195.446	−0.425807
$460$	0	0
$461$	102.856i	0.223114i	0.993758	+	0.111557i	$0.0355837\pi$
$461$	102.856i	0.223114i	−0.993758	+	0.111557i	$0.964416\pi$
$462$	0	0
$463$	−541.601	−1.16976	−0.584882	−	0.811118i	$-0.698859\pi$
$463$	−541.601	−1.16976	−0.584882	+	0.811118i	$0.698859\pi$
$464$	0	0
$465$	564.113i	1.21315i
$466$	0	0
$467$	154.809i	0.331497i	0.986168	+	0.165748i	$0.0530040\pi$
$467$	154.809i	0.331497i	−0.986168	+	0.165748i	$0.946996\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−294.295	−0.624830
$472$	0	0
$473$	2.10345	0.00444704
$474$	0	0
$475$	− 1611.08i	− 3.39175i
$476$	0	0
$477$	507.092	1.06309
$478$	0	0
$479$	− 728.668i	− 1.52123i	−0.649205	−	0.760614i	$-0.724898\pi$
$479$	− 728.668i	− 1.52123i	0.649205	−	0.760614i	$-0.275102\pi$
$480$	0	0
$481$	38.8805i	0.0808326i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−558.456	−1.15146
$486$	0	0
$487$	−448.941	−0.921850	−0.460925	−	0.887439i	$-0.652482\pi$
$487$	−448.941	−0.921850	−0.460925	+	0.887439i	$0.652482\pi$
$488$	0	0
$489$	− 210.789i	− 0.431062i
$490$	0	0
$491$	73.5801	0.149858	0.0749288	−	0.997189i	$-0.476127\pi$
$491$	73.5801	0.149858	0.0749288	+	0.997189i	$0.476127\pi$
$492$	0	0
$493$	348.506i	0.706909i
$494$	0	0
$495$	273.594i	0.552715i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−866.413	−1.73630	−0.868149	−	0.496303i	$-0.834691\pi$
$499$	−866.413	−1.73630	−0.868149	+	0.496303i	$0.834691\pi$
$500$	0	0
$501$	−318.622	−0.635973
$502$	0	0
$503$	− 306.742i	− 0.609825i	−0.952380	−	0.304913i	$-0.901373\pi$
$503$	− 306.742i	− 0.609825i	0.952380	−	0.304913i	$-0.0986272\pi$
$504$	0	0
$505$	−813.408	−1.61071
$506$	0	0
$507$	240.979i	0.475303i
$508$	0	0
$509$	− 187.740i	− 0.368841i	−0.982847	−	0.184421i	$-0.940959\pi$
$509$	− 187.740i	− 0.368841i	0.982847	−	0.184421i	$-0.0590409\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−894.367	−1.74341
$514$	0	0
$515$	−1448.97	−2.81353
$516$	0	0
$517$	170.350i	0.329498i
$518$	0	0
$519$	−313.061	−0.603200
$520$	0	0
$521$	− 346.975i	− 0.665978i	−0.942931	−	0.332989i	$-0.891943\pi$
$521$	− 346.975i	− 0.665978i	0.942931	−	0.332989i	$-0.108057\pi$
$522$	0	0
$523$	322.655i	0.616931i	0.951236	+	0.308466i	$0.0998155\pi$
$523$	322.655i	0.616931i	−0.951236	+	0.308466i	$0.900185\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−312.271	−0.592544
$528$	0	0
$529$	180.002	0.340268
$530$	0	0
$531$	54.9554i	0.103494i
$532$	0	0
$533$	130.194	0.244266
$534$	0	0
$535$	− 866.909i	− 1.62039i
$536$	0	0
$537$	400.240i	0.745326i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	494.942	0.914864	0.457432	−	0.889244i	$-0.348769\pi$
$541$	494.942	0.914864	0.457432	+	0.889244i	$0.348769\pi$
$542$	0	0
$543$	−461.234	−0.849418
$544$	0	0
$545$	349.119i	0.640585i
$546$	0	0
$547$	−244.584	−0.447137	−0.223569	−	0.974688i	$-0.571771\pi$
$547$	−244.584	−0.447137	−0.223569	+	0.974688i	$0.571771\pi$
$548$	0	0
$549$	182.602i	0.332609i
$550$	0	0
$551$	1594.78i	2.89434i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	111.029	0.200052
$556$	0	0
$557$	830.165	1.49042	0.745211	−	0.666829i	$-0.232349\pi$
$557$	830.165	1.49042	0.745211	+	0.666829i	$0.232349\pi$
$558$	0	0
$559$	− 1.97007i	− 0.00352427i
$560$	0	0
$561$	66.8769	0.119210
$562$	0	0
$563$	567.336i	1.00770i	0.863791	+	0.503851i	$0.168084\pi$
$563$	567.336i	1.00770i	−0.863791	+	0.503851i	$0.831916\pi$
$564$	0	0
$565$	1124.88i	1.99094i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−648.475	−1.13967	−0.569837	−	0.821757i	$-0.692994\pi$
$569$	−648.475	−1.13967	−0.569837	+	0.821757i	$0.692994\pi$
$570$	0	0
$571$	−509.846	−0.892900	−0.446450	−	0.894809i	$-0.647312\pi$
$571$	−509.846	−0.892900	−0.446450	+	0.894809i	$0.647312\pi$
$572$	0	0
$573$	− 438.963i	− 0.766078i
$574$	0	0
$575$	1213.96	2.11124
$576$	0	0
$577$	− 416.399i	− 0.721662i	−0.932631	−	0.360831i	$-0.882493\pi$
$577$	− 416.399i	− 0.721662i	0.932631	−	0.360831i	$-0.117507\pi$
$578$	0	0
$579$	350.997i	0.606212i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−423.650	−0.726673
$584$	0	0
$585$	256.245	0.438026
$586$	0	0
$587$	581.897i	0.991307i	0.868520	+	0.495654i	$0.165071\pi$
$587$	581.897i	0.991307i	−0.868520	+	0.495654i	$0.834929\pi$
$588$	0	0
$589$	−1428.96	−2.42608
$590$	0	0
$591$	− 131.991i	− 0.223335i
$592$	0	0
$593$	554.846i	0.935659i	0.883819	+	0.467830i	$0.154964\pi$
$593$	554.846i	0.935659i	−0.883819	+	0.467830i	$0.845036\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−336.982	−0.564459
$598$	0	0
$599$	66.7864	0.111496	0.0557482	−	0.998445i	$-0.482246\pi$
$599$	66.7864	0.111496	0.0557482	+	0.998445i	$0.482246\pi$
$600$	0	0
$601$	215.249i	0.358151i	0.983835	+	0.179076i	$0.0573107\pi$
$601$	215.249i	0.358151i	−0.983835	+	0.179076i	$0.942689\pi$
$602$	0	0
$603$	546.642	0.906537
$604$	0	0
$605$	788.051i	1.30256i
$606$	0	0
$607$	− 626.512i	− 1.03215i	−0.856545	−	0.516073i	$-0.827393\pi$
$607$	− 626.512i	− 1.03215i	0.856545	−	0.516073i	$-0.172607\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	159.549	0.261127
$612$	0	0
$613$	120.840	0.197129	0.0985647	−	0.995131i	$-0.468575\pi$
$613$	120.840	0.197129	0.0985647	+	0.995131i	$0.468575\pi$
$614$	0	0
$615$	− 371.788i	− 0.604534i
$616$	0	0
$617$	537.102	0.870506	0.435253	−	0.900308i	$-0.356659\pi$
$617$	537.102	0.870506	0.435253	+	0.900308i	$0.356659\pi$
$618$	0	0
$619$	− 244.175i	− 0.394467i	−0.980357	−	0.197234i	$-0.936804\pi$
$619$	− 244.175i	− 0.394467i	0.980357	−	0.197234i	$-0.0631958\pi$
$620$	0	0
$621$	− 673.913i	− 1.08521i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	313.780	0.502048
$626$	0	0
$627$	306.032	0.488089
$628$	0	0
$629$	61.4613i	0.0977128i
$630$	0	0
$631$	−873.683	−1.38460	−0.692300	−	0.721609i	$-0.743403\pi$
$631$	−873.683	−1.38460	−0.692300	+	0.721609i	$0.743403\pi$
$632$	0	0
$633$	− 276.507i	− 0.436819i
$634$	0	0
$635$	1113.15i	1.75300i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	171.819	0.268887
$640$	0	0
$641$	465.377	0.726017	0.363008	−	0.931786i	$-0.381750\pi$
$641$	465.377	0.726017	0.363008	+	0.931786i	$0.381750\pi$
$642$	0	0
$643$	82.1402i	0.127745i	0.997958	+	0.0638726i	$0.0203451\pi$
$643$	82.1402i	0.127745i	−0.997958	+	0.0638726i	$0.979655\pi$
$644$	0	0
$645$	−5.62583	−0.00872221
$646$	0	0
$647$	179.429i	0.277325i	0.990340	+	0.138663i	$0.0442804\pi$
$647$	179.429i	0.277325i	−0.990340	+	0.138663i	$0.955720\pi$
$648$	0	0
$649$	− 45.9125i	− 0.0707435i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	244.561	0.374520	0.187260	−	0.982310i	$-0.440039\pi$
$653$	244.561	0.374520	0.187260	+	0.982310i	$0.440039\pi$
$654$	0	0
$655$	−622.056	−0.949705
$656$	0	0
$657$	543.184i	0.826764i
$658$	0	0
$659$	−128.978	−0.195718	−0.0978590	−	0.995200i	$-0.531199\pi$
$659$	−128.978	−0.195718	−0.0978590	+	0.995200i	$0.531199\pi$
$660$	0	0
$661$	− 455.457i	− 0.689043i	−0.938779	−	0.344521i	$-0.888041\pi$
$661$	− 455.457i	− 0.689043i	0.938779	−	0.344521i	$-0.111959\pi$
$662$	0	0
$663$	− 62.6362i	− 0.0944739i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1201.68	−1.80162
$668$	0	0
$669$	−91.4503	−0.136697
$670$	0	0
$671$	− 152.555i	− 0.227355i
$672$	0	0
$673$	690.223	1.02559	0.512795	−	0.858511i	$-0.328610\pi$
$673$	690.223	1.02559	0.512795	+	0.858511i	$0.328610\pi$
$674$	0	0
$675$	− 1153.88i	− 1.70946i
$676$	0	0
$677$	− 410.865i	− 0.606891i	−0.952849	−	0.303445i	$-0.901863\pi$
$677$	− 410.865i	− 0.606891i	0.952849	−	0.303445i	$-0.0981370\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	352.999	0.518354
$682$	0	0
$683$	−364.224	−0.533271	−0.266636	−	0.963797i	$-0.585912\pi$
$683$	−364.224	−0.533271	−0.266636	+	0.963797i	$0.585912\pi$
$684$	0	0
$685$	624.930i	0.912307i
$686$	0	0
$687$	−73.8488	−0.107495
$688$	0	0
$689$	396.786i	0.575887i
$690$	0	0
$691$	− 1041.16i	− 1.50674i	−0.657596	−	0.753371i	$-0.728426\pi$
$691$	− 1041.16i	− 1.50674i	0.657596	−	0.753371i	$-0.271574\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1038.88	1.49479
$696$	0	0
$697$	205.807	0.295276
$698$	0	0
$699$	− 451.785i	− 0.646331i
$700$	0	0
$701$	−366.854	−0.523330	−0.261665	−	0.965159i	$-0.584272\pi$
$701$	−366.854	−0.523330	−0.261665	+	0.965159i	$0.584272\pi$
$702$	0	0
$703$	281.250i	0.400071i
$704$	0	0
$705$	− 455.615i	− 0.646262i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−65.3040	−0.0921072	−0.0460536	−	0.998939i	$-0.514665\pi$
$709$	−65.3040	−0.0921072	−0.0460536	+	0.998939i	$0.514665\pi$
$710$	0	0
$711$	759.753	1.06857
$712$	0	0
$713$	− 1076.74i	− 1.51015i
$714$	0	0
$715$	−214.080	−0.299413
$716$	0	0
$717$	646.184i	0.901232i
$718$	0	0
$719$	− 679.230i	− 0.944687i	−0.881415	−	0.472344i	$-0.843408\pi$
$719$	− 679.230i	− 0.944687i	0.881415	−	0.472344i	$-0.156592\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	116.829	0.161589
$724$	0	0
$725$	−2057.53	−2.83797
$726$	0	0
$727$	− 1079.11i	− 1.48433i	−0.670217	−	0.742166i	$-0.733799\pi$
$727$	− 1079.11i	− 1.48433i	0.670217	−	0.742166i	$-0.266201\pi$
$728$	0	0
$729$	−289.766	−0.397485
$730$	0	0
$731$	− 3.11424i	− 0.00426024i
$732$	0	0
$733$	− 348.258i	− 0.475114i	−0.971374	−	0.237557i	$-0.923653\pi$
$733$	− 348.258i	− 0.475114i	0.971374	−	0.237557i	$-0.0763466\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−456.692	−0.619664
$738$	0	0
$739$	541.486	0.732728	0.366364	−	0.930472i	$-0.380602\pi$
$739$	541.486	0.732728	0.366364	+	0.930472i	$0.380602\pi$
$740$	0	0
$741$	− 286.626i	− 0.386810i
$742$	0	0
$743$	−702.388	−0.945340	−0.472670	−	0.881240i	$-0.656710\pi$
$743$	−702.388	−0.945340	−0.472670	+	0.881240i	$0.656710\pi$
$744$	0	0
$745$	− 1735.75i	− 2.32986i
$746$	0	0
$747$	291.322i	0.389989i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	38.0581	0.0506766	0.0253383	−	0.999679i	$-0.491934\pi$
$751$	38.0581	0.0506766	0.0253383	+	0.999679i	$0.491934\pi$
$752$	0	0
$753$	−161.940	−0.215060
$754$	0	0
$755$	− 407.115i	− 0.539226i
$756$	0	0
$757$	−451.849	−0.596895	−0.298447	−	0.954426i	$-0.596469\pi$
$757$	−451.849	−0.596895	−0.298447	+	0.954426i	$0.596469\pi$
$758$	0	0
$759$	230.597i	0.303817i
$760$	0	0
$761$	− 10.4123i	− 0.0136823i	−0.999977	−	0.00684117i	$-0.997822\pi$
$761$	− 10.4123i	− 0.0136823i	0.999977	−	0.00684117i	$-0.00217763\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	405.066	0.529498
$766$	0	0
$767$	−43.0012	−0.0560641
$768$	0	0
$769$	1421.57i	1.84860i	0.381666	+	0.924300i	$0.375351\pi$
$769$	1421.57i	1.84860i	−0.381666	+	0.924300i	$0.624649\pi$
$770$	0	0
$771$	343.943	0.446100
$772$	0	0
$773$	842.410i	1.08979i	0.838503	+	0.544897i	$0.183431\pi$
$773$	842.410i	1.08979i	−0.838503	+	0.544897i	$0.816569\pi$
$774$	0	0
$775$	− 1843.60i	− 2.37884i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	941.784	1.20896
$780$	0	0
$781$	−143.546	−0.183797
$782$	0	0
$783$	1142.21i	1.45876i
$784$	0	0
$785$	1489.20	1.89707
$786$	0	0
$787$	− 483.878i	− 0.614839i	−0.951574	−	0.307419i	$-0.900535\pi$
$787$	− 483.878i	− 0.614839i	0.951574	−	0.307419i	$-0.0994654\pi$
$788$	0	0
$789$	344.052i	0.436061i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−142.882	−0.180178
$794$	0	0
$795$	1133.08	1.42526
$796$	0	0
$797$	10.4020i	0.0130515i	0.999979	+	0.00652574i	$0.00207722\pi$
$797$	10.4020i	0.0130515i	−0.999979	+	0.00652574i	$0.997923\pi$
$798$	0	0
$799$	252.211	0.315658
$800$	0	0
$801$	380.083i	0.474511i
$802$	0	0
$803$	− 453.803i	− 0.565135i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	59.7313	0.0740164
$808$	0	0
$809$	−279.658	−0.345683	−0.172842	−	0.984950i	$-0.555295\pi$
$809$	−279.658	−0.345683	−0.172842	+	0.984950i	$0.555295\pi$
$810$	0	0
$811$	1027.85i	1.26739i	0.773584	+	0.633693i	$0.218462\pi$
$811$	1027.85i	1.26739i	−0.773584	+	0.633693i	$0.781538\pi$
$812$	0	0
$813$	−127.289	−0.156567
$814$	0	0
$815$	1066.64i	1.30876i
$816$	0	0
$817$	− 14.2509i	− 0.0174429i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−476.141	−0.579952	−0.289976	−	0.957034i	$-0.593647\pi$
$821$	−476.141	−0.579952	−0.289976	+	0.957034i	$0.593647\pi$
$822$	0	0
$823$	1110.26	1.34903	0.674517	−	0.738259i	$-0.264352\pi$
$823$	1110.26	1.34903	0.674517	+	0.738259i	$0.264352\pi$
$824$	0	0
$825$	394.832i	0.478584i
$826$	0	0
$827$	−997.533	−1.20621	−0.603103	−	0.797663i	$-0.706069\pi$
$827$	−997.533	−1.20621	−0.603103	+	0.797663i	$0.706069\pi$
$828$	0	0
$829$	731.843i	0.882802i	0.897310	+	0.441401i	$0.145518\pi$
$829$	731.843i	0.882802i	−0.897310	+	0.441401i	$0.854482\pi$
$830$	0	0
$831$	444.985i	0.535481i
$832$	0	0
$833$	0	0
$834$	0	0
$835$	1612.30	1.93090
$836$	0	0
$837$	−1023.45	−1.22276
$838$	0	0
$839$	1200.42i	1.43077i	0.698729	+	0.715386i	$0.253749\pi$
$839$	1200.42i	1.43077i	−0.698729	+	0.715386i	$0.746251\pi$
$840$	0	0
$841$	1195.71	1.42178
$842$	0	0
$843$	− 692.933i	− 0.821984i
$844$	0	0
$845$	− 1219.41i	− 1.44309i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−238.768	−0.281234
$850$	0	0
$851$	−211.924	−0.249029
$852$	0	0
$853$	− 1251.12i	− 1.46673i	−0.679836	−	0.733364i	$-0.737949\pi$
$853$	− 1251.12i	− 1.46673i	0.679836	−	0.733364i	$-0.262051\pi$
$854$	0	0
$855$	1853.60	2.16795
$856$	0	0
$857$	− 227.343i	− 0.265277i	−0.991164	−	0.132639i	$-0.957655\pi$
$857$	− 227.343i	− 0.265277i	0.991164	−	0.132639i	$-0.0423450\pi$
$858$	0	0
$859$	− 398.813i	− 0.464276i	−0.972683	−	0.232138i	$-0.925428\pi$
$859$	− 398.813i	− 0.464276i	0.972683	−	0.232138i	$-0.0745721\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	3.97387	0.00460472	0.00230236	−	0.999997i	$-0.499267\pi$
$863$	3.97387	0.00460472	0.00230236	+	0.999997i	$0.499267\pi$
$864$	0	0
$865$	1584.16	1.83140
$866$	0	0
$867$	380.833i	0.439254i
$868$	0	0
$869$	−634.736	−0.730421
$870$	0	0
$871$	427.733i	0.491083i
$872$	0	0
$873$	− 414.972i	− 0.475340i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	439.044	0.500620	0.250310	−	0.968166i	$-0.419467\pi$
$877$	439.044	0.500620	0.250310	+	0.968166i	$0.419467\pi$
$878$	0	0
$879$	441.150	0.501877
$880$	0	0
$881$	848.178i	0.962744i	0.876516	+	0.481372i	$0.159861\pi$
$881$	848.178i	0.962744i	−0.876516	+	0.481372i	$0.840139\pi$
$882$	0	0
$883$	−1227.48	−1.39012	−0.695062	−	0.718950i	$-0.744623\pi$
$883$	−1227.48	−1.39012	−0.695062	+	0.718950i	$0.744623\pi$
$884$	0	0
$885$	122.796i	0.138753i
$886$	0	0
$887$	− 943.945i	− 1.06420i	−0.846682	−	0.532100i	$-0.821403\pi$
$887$	− 943.945i	− 1.06420i	0.846682	−	0.532100i	$-0.178597\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−73.8863	−0.0829252
$892$	0	0
$893$	1154.13	1.29241
$894$	0	0
$895$	− 2025.31i	− 2.26291i
$896$	0	0
$897$	215.975	0.240775
$898$	0	0
$899$	1824.95i	2.02998i
$900$	0	0
$901$	627.231i	0.696149i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	2333.95	2.57895
$906$	0	0
$907$	95.0939	0.104844	0.0524222	−	0.998625i	$-0.483306\pi$
$907$	95.0939	0.104844	0.0524222	+	0.998625i	$0.483306\pi$
$908$	0	0
$909$	− 604.419i	− 0.664928i
$910$	0	0
$911$	−698.535	−0.766779	−0.383389	−	0.923587i	$-0.625243\pi$
$911$	−698.535	−0.766779	−0.383389	+	0.923587i	$0.625243\pi$
$912$	0	0
$913$	− 243.385i	− 0.266577i
$914$	0	0
$915$	408.020i	0.445923i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−462.459	−0.503220	−0.251610	−	0.967829i	$-0.580960\pi$
$919$	−462.459	−0.503220	−0.251610	+	0.967829i	$0.580960\pi$
$920$	0	0
$921$	467.588	0.507696
$922$	0	0
$923$	134.444i	0.145659i
$924$	0	0
$925$	−362.859	−0.392280
$926$	0	0
$927$	− 1076.69i	− 1.16147i
$928$	0	0
$929$	1806.85i	1.94494i	0.233024	+	0.972471i	$0.425138\pi$
$929$	1806.85i	1.94494i	−0.233024	+	0.972471i	$0.574862\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	93.6265	0.100350
$934$	0	0
$935$	−338.413	−0.361939
$936$	0	0
$937$	− 1569.03i	− 1.67453i	−0.546800	−	0.837263i	$-0.684154\pi$
$937$	− 1569.03i	− 1.67453i	0.546800	−	0.837263i	$-0.315846\pi$
$938$	0	0
$939$	140.048	0.149146
$940$	0	0
$941$	− 1007.11i	− 1.07026i	−0.844770	−	0.535129i	$-0.820263\pi$
$941$	− 1007.11i	− 1.07026i	0.844770	−	0.535129i	$-0.179737\pi$
$942$	0	0
$943$	709.642i	0.752536i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−763.982	−0.806739	−0.403369	−	0.915037i	$-0.632161\pi$
$947$	−763.982	−0.806739	−0.403369	+	0.915037i	$0.632161\pi$
$948$	0	0
$949$	−425.027	−0.447869
$950$	0	0
$951$	264.640i	0.278275i
$952$	0	0
$953$	−698.870	−0.733337	−0.366669	−	0.930352i	$-0.619502\pi$
$953$	−698.870	−0.733337	−0.366669	+	0.930352i	$0.619502\pi$
$954$	0	0
$955$	2221.25i	2.32592i
$956$	0	0
$957$	− 390.837i	− 0.408398i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−674.202	−0.701563
$962$	0	0
$963$	644.174	0.668925
$964$	0	0
$965$	− 1776.13i	− 1.84054i
$966$	0	0
$967$	383.857	0.396956	0.198478	−	0.980105i	$-0.436400\pi$
$967$	383.857	0.396956	0.198478	+	0.980105i	$0.436400\pi$
$968$	0	0
$969$	− 453.092i	− 0.467587i
$970$	0	0
$971$	252.500i	0.260042i	0.991511	+	0.130021i	$0.0415044\pi$
$971$	252.500i	0.260042i	−0.991511	+	0.130021i	$0.958496\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	369.795	0.379277
$976$	0	0
$977$	−67.5646	−0.0691552	−0.0345776	−	0.999402i	$-0.511009\pi$
$977$	−67.5646	−0.0691552	−0.0345776	+	0.999402i	$0.511009\pi$
$978$	0	0
$979$	− 317.541i	− 0.324352i
$980$	0	0
$981$	−259.420	−0.264444
$982$	0	0
$983$	382.208i	0.388818i	0.980921	+	0.194409i	$0.0622789\pi$
$983$	382.208i	0.388818i	−0.980921	+	0.194409i	$0.937721\pi$
$984$	0	0
$985$	667.905i	0.678076i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	10.7382	0.0108576
$990$	0	0
$991$	1838.30	1.85499	0.927496	−	0.373832i	$-0.121956\pi$
$991$	1838.30	1.85499	0.927496	+	0.373832i	$0.121956\pi$
$992$	0	0
$993$	929.123i	0.935673i
$994$	0	0
$995$	1705.21	1.71377
$996$	0	0
$997$	− 1355.84i	− 1.35992i	−0.733250	−	0.679959i	$-0.761998\pi$
$997$	− 1355.84i	− 1.35992i	0.733250	−	0.679959i	$-0.238002\pi$
$998$	0	0
$999$	201.436i	0.201637i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1568.3.c.h.97.11		16
4.3	odd	2	inner	1568.3.c.h.97.5		16
7.4	even	3		224.3.s.a.33.6	yes	16
7.5	odd	6		224.3.s.a.129.6	yes	16
7.6	odd	2	inner	1568.3.c.h.97.6		16
28.11	odd	6		224.3.s.a.33.3	✓	16
28.19	even	6		224.3.s.a.129.3	yes	16
28.27	even	2	inner	1568.3.c.h.97.12		16
56.5	odd	6		448.3.s.g.129.3		16
56.11	odd	6		448.3.s.g.257.6		16
56.19	even	6		448.3.s.g.129.6		16
56.53	even	6		448.3.s.g.257.3		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.3.s.a.33.3	✓	16	28.11	odd	6
224.3.s.a.33.6	yes	16	7.4	even	3
224.3.s.a.129.3	yes	16	28.19	even	6
224.3.s.a.129.6	yes	16	7.5	odd	6
448.3.s.g.129.3		16	56.5	odd	6
448.3.s.g.129.6		16	56.19	even	6
448.3.s.g.257.3		16	56.53	even	6
448.3.s.g.257.6		16	56.11	odd	6
1568.3.c.h.97.5		16	4.3	odd	2	inner
1568.3.c.h.97.6		16	7.6	odd	2	inner
1568.3.c.h.97.11		16	1.1	even	1	trivial
1568.3.c.h.97.12		16	28.27	even	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:	\( N \)	\(=\)	\( 1568 = 2^{5} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1568.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.66037i q^{3} -8.40186i q^{5} +6.24317 q^{9} +O(q^{10})\)	\(q+1.66037i q^{3} -8.40186i q^{5} +6.24317 q^{9} -5.21586 q^{11} +4.88512i q^{13} +13.9502 q^{15} +7.72228i q^{17} +35.3375i q^{19} -26.6271 q^{23} -45.5912 q^{25} +25.3093i q^{27} +45.1300 q^{29} +40.4376i q^{31} -8.66025i q^{33} +7.95896 q^{37} -8.11110 q^{39} -26.6511i q^{41} -0.403279 q^{43} -52.4542i q^{45} -32.6601i q^{47} -12.8218 q^{51} +81.2235 q^{53} +43.8229i q^{55} -58.6733 q^{57} +8.80248i q^{59} +29.2483i q^{61} +41.0441 q^{65} +87.5584 q^{67} -44.2108i q^{69} +27.5210 q^{71} +87.0045i q^{73} -75.6983i q^{75} +121.693 q^{79} +14.1657 q^{81} +46.6625i q^{83} +64.8815 q^{85} +74.9324i q^{87} +60.8799i q^{89} -67.1414 q^{93} +296.901 q^{95} -66.4681i q^{97} -32.5635 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{9}+O(q^{10}) \) 16 * q - 16 * q^9	\( 16 q - 16 q^{9} - 32 q^{25} + 112 q^{29} - 16 q^{37} + 48 q^{53} - 528 q^{57} + 16 q^{65} - 64 q^{81} + 720 q^{85} + 464 q^{93}+O(q^{100}) \) 16 * q - 16 * q^9 - 32 * q^25 + 112 * q^29 - 16 * q^37 + 48 * q^53 - 528 * q^57 + 16 * q^65 - 64 * q^81 + 720 * q^85 + 464 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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