LMFDB - Embedded newform 1568.3.h.a.881.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1568,3,Mod(881,1568)]

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, 1, 1]))

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

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

chi := DirichletCharacter("1568.881");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1568 = 2^{5} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1568.h (of order $2$, degree $1$, not 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:	$28$
Twist minimal:	no (minimal twist has level 56)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.17
Character	$\chi$	$=$	1568.881
Dual form			1568.3.h.a.881.18

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.910863 q^{3} +6.34503 q^{5} -8.17033 q^{9} +O(q^{10})$	$q+0.910863 q^{3} +6.34503 q^{5} -8.17033 q^{9} +13.2146i q^{11} +19.4243 q^{13} +5.77945 q^{15} +15.9268i q^{17} +16.4545 q^{19} -23.9215 q^{23} +15.2593 q^{25} -15.6398 q^{27} -16.6618i q^{29} +12.8588i q^{31} +12.0367i q^{33} +47.5556i q^{37} +17.6929 q^{39} -6.49499i q^{41} +33.2928i q^{43} -51.8409 q^{45} -21.9062i q^{47} +14.5071i q^{51} +37.1846i q^{53} +83.8473i q^{55} +14.9878 q^{57} +54.6855 q^{59} -10.2468 q^{61} +123.248 q^{65} -17.1341i q^{67} -21.7892 q^{69} -32.0568 q^{71} +107.166i q^{73} +13.8992 q^{75} +58.3083 q^{79} +59.2872 q^{81} -36.3441 q^{83} +101.056i q^{85} -15.1766i q^{87} -1.07373i q^{89} +11.7126i q^{93} +104.404 q^{95} +169.517i q^{97} -107.968i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 64 q^{9}+O(q^{10}) $ 28 * q + 64 * q^9	$ 28 q + 64 q^{9} - 28 q^{15} + 60 q^{23} + 64 q^{25} - 40 q^{39} + 124 q^{57} + 104 q^{65} + 136 q^{71} - 324 q^{79} + 36 q^{81} + 580 q^{95}+O(q^{100}) $ 28 * q + 64 * q^9 - 28 * q^15 + 60 * q^23 + 64 * q^25 - 40 * q^39 + 124 * q^57 + 104 * q^65 + 136 * q^71 - 324 * q^79 + 36 * q^81 + 580 * q^95

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
$3$	0.910863	0.303621	0.151810	−	0.988410i	$-0.451490\pi$
$3$	0.910863	0.303621	0.151810	+	0.988410i	$0.451490\pi$
$4$	0	0
$5$	6.34503	1.26901	0.634503	−	0.772921i	$-0.281205\pi$
$5$	6.34503	1.26901	0.634503	+	0.772921i	$0.281205\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−8.17033	−0.907814
$10$	0	0
$11$	13.2146i	1.20133i	0.799500	+	0.600666i	$0.205098\pi$
$11$	13.2146i	1.20133i	−0.799500	+	0.600666i	$0.794902\pi$
$12$	0	0
$13$	19.4243	1.49418	0.747090	−	0.664723i	$-0.231450\pi$
$13$	19.4243	1.49418	0.747090	+	0.664723i	$0.231450\pi$
$14$	0	0
$15$	5.77945	0.385296
$16$	0	0
$17$	15.9268i	0.936869i	0.883498	+	0.468434i	$0.155182\pi$
$17$	15.9268i	0.936869i	−0.883498	+	0.468434i	$0.844818\pi$
$18$	0	0
$19$	16.4545	0.866026	0.433013	−	0.901388i	$-0.357450\pi$
$19$	16.4545	0.866026	0.433013	+	0.901388i	$0.357450\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−23.9215	−1.04006	−0.520032	−	0.854147i	$-0.674080\pi$
$23$	−23.9215	−1.04006	−0.520032	+	0.854147i	$0.674080\pi$
$24$	0	0
$25$	15.2593	0.610374
$26$	0	0
$27$	−15.6398	−0.579252
$28$	0	0
$29$	− 16.6618i	− 0.574544i	−0.957849	−	0.287272i	$-0.907252\pi$
$29$	− 16.6618i	− 0.574544i	0.957849	−	0.287272i	$-0.0927483\pi$
$30$	0	0
$31$	12.8588i	0.414799i	0.978256	+	0.207400i	$0.0665000\pi$
$31$	12.8588i	0.414799i	−0.978256	+	0.207400i	$0.933500\pi$
$32$	0	0
$33$	12.0367i	0.364749i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	47.5556i	1.28529i	0.766165	+	0.642644i	$0.222162\pi$
$37$	47.5556i	1.28529i	−0.766165	+	0.642644i	$0.777838\pi$
$38$	0	0
$39$	17.6929	0.453664
$40$	0	0
$41$	− 6.49499i	− 0.158415i	−0.996858	−	0.0792073i	$-0.974761\pi$
$41$	− 6.49499i	− 0.158415i	0.996858	−	0.0792073i	$-0.0252389\pi$
$42$	0	0
$43$	33.2928i	0.774252i	0.922027	+	0.387126i	$0.126532\pi$
$43$	33.2928i	0.774252i	−0.922027	+	0.387126i	$0.873468\pi$
$44$	0	0
$45$	−51.8409	−1.15202
$46$	0	0
$47$	− 21.9062i	− 0.466089i	−0.972466	−	0.233045i	$-0.925131\pi$
$47$	− 21.9062i	− 0.466089i	0.972466	−	0.233045i	$-0.0748688\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	14.5071i	0.284453i
$52$	0	0
$53$	37.1846i	0.701597i	0.936451	+	0.350798i	$0.114090\pi$
$53$	37.1846i	0.701597i	−0.936451	+	0.350798i	$0.885910\pi$
$54$	0	0
$55$	83.8473i	1.52450i
$56$	0	0
$57$	14.9878	0.262944
$58$	0	0
$59$	54.6855	0.926874	0.463437	−	0.886130i	$-0.346616\pi$
$59$	54.6855	0.926874	0.463437	+	0.886130i	$0.346616\pi$
$60$	0	0
$61$	−10.2468	−0.167980	−0.0839902	−	0.996467i	$-0.526766\pi$
$61$	−10.2468	−0.167980	−0.0839902	+	0.996467i	$0.526766\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	123.248	1.89612
$66$	0	0
$67$	− 17.1341i	− 0.255733i	−0.991791	−	0.127867i	$-0.959187\pi$
$67$	− 17.1341i	− 0.255733i	0.991791	−	0.127867i	$-0.0408129\pi$
$68$	0	0
$69$	−21.7892	−0.315785
$70$	0	0
$71$	−32.0568	−0.451505	−0.225752	−	0.974185i	$-0.572484\pi$
$71$	−32.0568	−0.451505	−0.225752	+	0.974185i	$0.572484\pi$
$72$	0	0
$73$	107.166i	1.46802i	0.679137	+	0.734011i	$0.262354\pi$
$73$	107.166i	1.46802i	−0.679137	+	0.734011i	$0.737646\pi$
$74$	0	0
$75$	13.8992	0.185322
$76$	0	0
$77$	0	0
$78$	0	0
$79$	58.3083	0.738080	0.369040	−	0.929414i	$-0.379687\pi$
$79$	58.3083	0.738080	0.369040	+	0.929414i	$0.379687\pi$
$80$	0	0
$81$	59.2872	0.731941
$82$	0	0
$83$	−36.3441	−0.437880	−0.218940	−	0.975738i	$-0.570260\pi$
$83$	−36.3441	−0.437880	−0.218940	+	0.975738i	$0.570260\pi$
$84$	0	0
$85$	101.056i	1.18889i
$86$	0	0
$87$	− 15.1766i	− 0.174443i
$88$	0	0
$89$	− 1.07373i	− 0.0120644i	−0.999982	−	0.00603222i	$-0.998080\pi$
$89$	− 1.07373i	− 0.0120644i	0.999982	−	0.00603222i	$-0.00192013\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	11.7126i	0.125942i
$94$	0	0
$95$	104.404	1.09899
$96$	0	0
$97$	169.517i	1.74760i	0.486286	+	0.873799i	$0.338351\pi$
$97$	169.517i	1.74760i	−0.486286	+	0.873799i	$0.661649\pi$
$98$	0	0
$99$	− 107.968i	− 1.09059i
$100$	0	0
$101$	−28.1260	−0.278476	−0.139238	−	0.990259i	$-0.544465\pi$
$101$	−28.1260	−0.278476	−0.139238	+	0.990259i	$0.544465\pi$
$102$	0	0
$103$	− 166.310i	− 1.61466i	−0.590098	−	0.807331i	$-0.700911\pi$
$103$	− 166.310i	− 1.61466i	0.590098	−	0.807331i	$-0.299089\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 197.584i	− 1.84658i	−0.384109	−	0.923288i	$-0.625491\pi$
$107$	− 197.584i	− 1.84658i	0.384109	−	0.923288i	$-0.374509\pi$
$108$	0	0
$109$	− 11.5198i	− 0.105686i	−0.998603	−	0.0528431i	$-0.983172\pi$
$109$	− 11.5198i	− 0.105686i	0.998603	−	0.0528431i	$-0.0168283\pi$
$110$	0	0
$111$	43.3167i	0.390240i
$112$	0	0
$113$	−14.7908	−0.130892	−0.0654460	−	0.997856i	$-0.520847\pi$
$113$	−14.7908	−0.130892	−0.0654460	+	0.997856i	$0.520847\pi$
$114$	0	0
$115$	−151.782	−1.31985
$116$	0	0
$117$	−158.703	−1.35644
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−53.6270	−0.443198
$122$	0	0
$123$	− 5.91605i	− 0.0480980i
$124$	0	0
$125$	−61.8047	−0.494438
$126$	0	0
$127$	70.2656	0.553272	0.276636	−	0.960975i	$-0.410780\pi$
$127$	70.2656	0.553272	0.276636	+	0.960975i	$0.410780\pi$
$128$	0	0
$129$	30.3252i	0.235079i
$130$	0	0
$131$	142.129	1.08496	0.542478	−	0.840070i	$-0.317486\pi$
$131$	142.129	1.08496	0.542478	+	0.840070i	$0.317486\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−99.2350	−0.735074
$136$	0	0
$137$	253.074	1.84725	0.923626	−	0.383295i	$-0.125211\pi$
$137$	253.074	1.84725	0.923626	+	0.383295i	$0.125211\pi$
$138$	0	0
$139$	−49.1909	−0.353892	−0.176946	−	0.984221i	$-0.556622\pi$
$139$	−49.1909	−0.353892	−0.176946	+	0.984221i	$0.556622\pi$
$140$	0	0
$141$	− 19.9535i	− 0.141514i
$142$	0	0
$143$	256.686i	1.79501i
$144$	0	0
$145$	− 105.719i	− 0.729099i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 41.6936i	− 0.279823i	−0.990164	−	0.139911i	$-0.955318\pi$
$149$	− 41.6936i	− 0.279823i	0.990164	−	0.139911i	$-0.0446818\pi$
$150$	0	0
$151$	97.6290	0.646549	0.323275	−	0.946305i	$-0.395216\pi$
$151$	97.6290	0.646549	0.323275	+	0.946305i	$0.395216\pi$
$152$	0	0
$153$	− 130.127i	− 0.850503i
$154$	0	0
$155$	81.5892i	0.526382i
$156$	0	0
$157$	−28.1654	−0.179397	−0.0896986	−	0.995969i	$-0.528590\pi$
$157$	−28.1654	−0.179397	−0.0896986	+	0.995969i	$0.528590\pi$
$158$	0	0
$159$	33.8701i	0.213019i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	242.432i	1.48731i	0.668563	+	0.743655i	$0.266910\pi$
$163$	242.432i	1.48731i	−0.668563	+	0.743655i	$0.733090\pi$
$164$	0	0
$165$	76.3734i	0.462869i
$166$	0	0
$167$	− 60.1108i	− 0.359945i	−0.983672	−	0.179972i	$-0.942399\pi$
$167$	− 60.1108i	− 0.359945i	0.983672	−	0.179972i	$-0.0576008\pi$
$168$	0	0
$169$	208.305	1.23257
$170$	0	0
$171$	−134.439	−0.786191
$172$	0	0
$173$	139.364	0.805573	0.402786	−	0.915294i	$-0.368042\pi$
$173$	139.364	0.805573	0.402786	+	0.915294i	$0.368042\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	49.8110	0.281418
$178$	0	0
$179$	291.727i	1.62976i	0.579631	+	0.814879i	$0.303197\pi$
$179$	291.727i	1.62976i	−0.579631	+	0.814879i	$0.696803\pi$
$180$	0	0
$181$	−166.844	−0.921791	−0.460895	−	0.887455i	$-0.652472\pi$
$181$	−166.844	−0.921791	−0.460895	+	0.887455i	$0.652472\pi$
$182$	0	0
$183$	−9.33343	−0.0510024
$184$	0	0
$185$	301.742i	1.63104i
$186$	0	0
$187$	−210.467	−1.12549
$188$	0	0
$189$	0	0
$190$	0	0
$191$	131.356	0.687728	0.343864	−	0.939019i	$-0.388264\pi$
$191$	131.356	0.687728	0.343864	+	0.939019i	$0.388264\pi$
$192$	0	0
$193$	−81.4392	−0.421965	−0.210982	−	0.977490i	$-0.567666\pi$
$193$	−81.4392	−0.421965	−0.210982	+	0.977490i	$0.567666\pi$
$194$	0	0
$195$	112.262	0.575702
$196$	0	0
$197$	− 2.09549i	− 0.0106370i	−0.999986	−	0.00531851i	$-0.998307\pi$
$197$	− 2.09549i	− 0.0106370i	0.999986	−	0.00531851i	$-0.00169294\pi$
$198$	0	0
$199$	126.944i	0.637911i	0.947770	+	0.318955i	$0.103332\pi$
$199$	126.944i	0.637911i	−0.947770	+	0.318955i	$0.896668\pi$
$200$	0	0
$201$	− 15.6068i	− 0.0776459i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	− 41.2109i	− 0.201029i
$206$	0	0
$207$	195.446	0.944184
$208$	0	0
$209$	217.440i	1.04038i
$210$	0	0
$211$	− 7.16822i	− 0.0339726i	−0.999856	−	0.0169863i	$-0.994593\pi$
$211$	− 7.16822i	− 0.0339726i	0.999856	−	0.0169863i	$-0.00540717\pi$
$212$	0	0
$213$	−29.1994	−0.137086
$214$	0	0
$215$	211.244i	0.982530i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	97.6132i	0.445722i
$220$	0	0
$221$	309.367i	1.39985i
$222$	0	0
$223$	279.720i	1.25435i	0.778878	+	0.627175i	$0.215789\pi$
$223$	279.720i	1.25435i	−0.778878	+	0.627175i	$0.784211\pi$
$224$	0	0
$225$	−124.674	−0.554106
$226$	0	0
$227$	304.784	1.34266	0.671330	−	0.741159i	$-0.265724\pi$
$227$	304.784	1.34266	0.671330	+	0.741159i	$0.265724\pi$
$228$	0	0
$229$	414.688	1.81087	0.905433	−	0.424490i	$-0.139547\pi$
$229$	414.688	1.81087	0.905433	+	0.424490i	$0.139547\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−164.926	−0.707836	−0.353918	−	0.935277i	$-0.615151\pi$
$233$	−164.926	−0.707836	−0.353918	+	0.935277i	$0.615151\pi$
$234$	0	0
$235$	− 138.995i	− 0.591469i
$236$	0	0
$237$	53.1109	0.224097
$238$	0	0
$239$	19.1182	0.0799926	0.0399963	−	0.999200i	$-0.487265\pi$
$239$	19.1182	0.0799926	0.0399963	+	0.999200i	$0.487265\pi$
$240$	0	0
$241$	− 350.308i	− 1.45356i	−0.686870	−	0.726780i	$-0.741016\pi$
$241$	− 350.308i	− 1.45356i	0.686870	−	0.726780i	$-0.258984\pi$
$242$	0	0
$243$	194.761	0.801485
$244$	0	0
$245$	0	0
$246$	0	0
$247$	319.618	1.29400
$248$	0	0
$249$	−33.1045	−0.132950
$250$	0	0
$251$	−88.3204	−0.351874	−0.175937	−	0.984401i	$-0.556296\pi$
$251$	−88.3204	−0.351874	−0.175937	+	0.984401i	$0.556296\pi$
$252$	0	0
$253$	− 316.114i	− 1.24946i
$254$	0	0
$255$	92.0479i	0.360972i
$256$	0	0
$257$	− 86.0828i	− 0.334952i	−0.985876	−	0.167476i	$-0.946438\pi$
$257$	− 86.0828i	− 0.334952i	0.985876	−	0.167476i	$-0.0535617\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	136.132i	0.521579i
$262$	0	0
$263$	−319.210	−1.21373	−0.606863	−	0.794806i	$-0.707572\pi$
$263$	−319.210	−1.21373	−0.606863	+	0.794806i	$0.707572\pi$
$264$	0	0
$265$	235.937i	0.890330i
$266$	0	0
$267$	− 0.978025i	− 0.00366302i
$268$	0	0
$269$	57.4680	0.213636	0.106818	−	0.994279i	$-0.465934\pi$
$269$	57.4680	0.213636	0.106818	+	0.994279i	$0.465934\pi$
$270$	0	0
$271$	30.8765i	0.113935i	0.998376	+	0.0569677i	$0.0181432\pi$
$271$	30.8765i	0.113935i	−0.998376	+	0.0569677i	$0.981857\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	201.647i	0.733261i
$276$	0	0
$277$	− 356.185i	− 1.28587i	−0.765923	−	0.642933i	$-0.777718\pi$
$277$	− 356.185i	− 1.28587i	0.765923	−	0.642933i	$-0.222282\pi$
$278$	0	0
$279$	− 105.060i	− 0.376561i
$280$	0	0
$281$	−294.160	−1.04683	−0.523416	−	0.852077i	$-0.675343\pi$
$281$	−294.160	−1.04683	−0.523416	+	0.852077i	$0.675343\pi$
$282$	0	0
$283$	−415.002	−1.46644	−0.733219	−	0.679993i	$-0.761983\pi$
$283$	−415.002	−1.46644	−0.733219	+	0.679993i	$0.761983\pi$
$284$	0	0
$285$	95.0979	0.333677
$286$	0	0
$287$	0	0
$288$	0	0
$289$	35.3381	0.122277
$290$	0	0
$291$	154.407i	0.530608i
$292$	0	0
$293$	370.564	1.26472	0.632362	−	0.774673i	$-0.282085\pi$
$293$	370.564	1.26472	0.632362	+	0.774673i	$0.282085\pi$
$294$	0	0
$295$	346.981	1.17621
$296$	0	0
$297$	− 206.675i	− 0.695874i
$298$	0	0
$299$	−464.658	−1.55404
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−25.6190	−0.0845510
$304$	0	0
$305$	−65.0162	−0.213168
$306$	0	0
$307$	−160.327	−0.522239	−0.261120	−	0.965306i	$-0.584092\pi$
$307$	−160.327	−0.522239	−0.261120	+	0.965306i	$0.584092\pi$
$308$	0	0
$309$	− 151.486i	− 0.490245i
$310$	0	0
$311$	− 472.839i	− 1.52038i	−0.649700	−	0.760191i	$-0.725105\pi$
$311$	− 472.839i	− 1.52038i	0.649700	−	0.760191i	$-0.274895\pi$
$312$	0	0
$313$	− 231.013i	− 0.738060i	−0.929417	−	0.369030i	$-0.879690\pi$
$313$	− 231.013i	− 0.738060i	0.929417	−	0.369030i	$-0.120310\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 225.319i	− 0.710784i	−0.934717	−	0.355392i	$-0.884347\pi$
$317$	− 225.319i	− 0.710784i	0.934717	−	0.355392i	$-0.115653\pi$
$318$	0	0
$319$	220.179	0.690218
$320$	0	0
$321$	− 179.972i	− 0.560659i
$322$	0	0
$323$	262.067i	0.811353i
$324$	0	0
$325$	296.403	0.912008
$326$	0	0
$327$	− 10.4930i	− 0.0320885i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	− 20.6988i	− 0.0625341i	−0.999511	−	0.0312671i	$-0.990046\pi$
$331$	− 20.6988i	− 0.0625341i	0.999511	−	0.0312671i	$-0.00995424\pi$
$332$	0	0
$333$	− 388.545i	− 1.16680i
$334$	0	0
$335$	− 108.716i	− 0.324527i
$336$	0	0
$337$	34.9645	0.103752	0.0518762	−	0.998654i	$-0.483480\pi$
$337$	34.9645	0.103752	0.0518762	+	0.998654i	$0.483480\pi$
$338$	0	0
$339$	−13.4724	−0.0397415
$340$	0	0
$341$	−169.924	−0.498311
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−138.253	−0.400733
$346$	0	0
$347$	− 438.738i	− 1.26438i	−0.774815	−	0.632188i	$-0.782157\pi$
$347$	− 438.738i	− 1.26438i	0.774815	−	0.632188i	$-0.217843\pi$
$348$	0	0
$349$	−435.121	−1.24677	−0.623383	−	0.781917i	$-0.714242\pi$
$349$	−435.121	−1.24677	−0.623383	+	0.781917i	$0.714242\pi$
$350$	0	0
$351$	−303.793	−0.865507
$352$	0	0
$353$	281.109i	0.796343i	0.917311	+	0.398171i	$0.130355\pi$
$353$	281.109i	0.796343i	−0.917311	+	0.398171i	$0.869645\pi$
$354$	0	0
$355$	−203.401	−0.572962
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−263.930	−0.735180	−0.367590	−	0.929988i	$-0.619817\pi$
$359$	−263.930	−0.735180	−0.367590	+	0.929988i	$0.619817\pi$
$360$	0	0
$361$	−90.2495	−0.249999
$362$	0	0
$363$	−48.8468	−0.134564
$364$	0	0
$365$	679.969i	1.86293i
$366$	0	0
$367$	154.939i	0.422176i	0.977467	+	0.211088i	$0.0677007\pi$
$367$	154.939i	0.422176i	−0.977467	+	0.211088i	$0.932299\pi$
$368$	0	0
$369$	53.0662i	0.143811i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 584.862i	− 1.56799i	−0.620765	−	0.783997i	$-0.713178\pi$
$373$	− 584.862i	− 1.56799i	0.620765	−	0.783997i	$-0.286822\pi$
$374$	0	0
$375$	−56.2956	−0.150122
$376$	0	0
$377$	− 323.644i	− 0.858472i
$378$	0	0
$379$	128.176i	0.338195i	0.985599	+	0.169098i	$0.0540853\pi$
$379$	128.176i	0.338195i	−0.985599	+	0.169098i	$0.945915\pi$
$380$	0	0
$381$	64.0023	0.167985
$382$	0	0
$383$	− 249.920i	− 0.652532i	−0.945278	−	0.326266i	$-0.894209\pi$
$383$	− 249.920i	− 0.652532i	0.945278	−	0.326266i	$-0.105791\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 272.013i	− 0.702877i
$388$	0	0
$389$	216.423i	0.556358i	0.960529	+	0.278179i	$0.0897309\pi$
$389$	216.423i	0.556358i	−0.960529	+	0.278179i	$0.910269\pi$
$390$	0	0
$391$	− 380.991i	− 0.974402i
$392$	0	0
$393$	129.460	0.329415
$394$	0	0
$395$	369.968	0.936627
$396$	0	0
$397$	−699.882	−1.76293	−0.881463	−	0.472252i	$-0.843441\pi$
$397$	−699.882	−1.76293	−0.881463	+	0.472252i	$0.843441\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−180.981	−0.451323	−0.225662	−	0.974206i	$-0.572454\pi$
$401$	−180.981	−0.451323	−0.225662	+	0.974206i	$0.572454\pi$
$402$	0	0
$403$	249.773i	0.619785i
$404$	0	0
$405$	376.179	0.928837
$406$	0	0
$407$	−628.431	−1.54406
$408$	0	0
$409$	− 358.843i	− 0.877366i	−0.898642	−	0.438683i	$-0.855445\pi$
$409$	− 358.843i	− 0.877366i	0.898642	−	0.438683i	$-0.144555\pi$
$410$	0	0
$411$	230.515	0.560864
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−230.604	−0.555672
$416$	0	0
$417$	−44.8062	−0.107449
$418$	0	0
$419$	780.890	1.86370	0.931849	−	0.362846i	$-0.118195\pi$
$419$	780.890	1.86370	0.931849	+	0.362846i	$0.118195\pi$
$420$	0	0
$421$	114.961i	0.273068i	0.990635	+	0.136534i	$0.0435962\pi$
$421$	114.961i	0.273068i	−0.990635	+	0.136534i	$0.956404\pi$
$422$	0	0
$423$	178.981i	0.423122i
$424$	0	0
$425$	243.032i	0.571840i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	233.806i	0.545001i
$430$	0	0
$431$	−309.712	−0.718590	−0.359295	−	0.933224i	$-0.616983\pi$
$431$	−309.712	−0.718590	−0.359295	+	0.933224i	$0.616983\pi$
$432$	0	0
$433$	− 595.775i	− 1.37592i	−0.725747	−	0.687962i	$-0.758506\pi$
$433$	− 595.775i	− 1.37592i	0.725747	−	0.687962i	$-0.241494\pi$
$434$	0	0
$435$	− 96.2958i	− 0.221370i
$436$	0	0
$437$	−393.615	−0.900722
$438$	0	0
$439$	806.900i	1.83804i	0.394211	+	0.919020i	$0.371018\pi$
$439$	806.900i	1.83804i	−0.394211	+	0.919020i	$0.628982\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 444.807i	− 1.00408i	−0.864845	−	0.502039i	$-0.832583\pi$
$443$	− 444.807i	− 1.00408i	0.864845	−	0.502039i	$-0.167417\pi$
$444$	0	0
$445$	− 6.81287i	− 0.0153098i
$446$	0	0
$447$	− 37.9772i	− 0.0849601i
$448$	0	0
$449$	262.420	0.584455	0.292228	−	0.956349i	$-0.405604\pi$
$449$	262.420	0.584455	0.292228	+	0.956349i	$0.405604\pi$
$450$	0	0
$451$	85.8291	0.190308
$452$	0	0
$453$	88.9266	0.196306
$454$	0	0
$455$	0	0
$456$	0	0
$457$	389.475	0.852244	0.426122	−	0.904666i	$-0.359879\pi$
$457$	389.475	0.852244	0.426122	+	0.904666i	$0.359879\pi$
$458$	0	0
$459$	− 249.092i	− 0.542683i
$460$	0	0
$461$	−158.714	−0.344283	−0.172141	−	0.985072i	$-0.555069\pi$
$461$	−158.714	−0.344283	−0.172141	+	0.985072i	$0.555069\pi$
$462$	0	0
$463$	528.844	1.14221	0.571106	−	0.820877i	$-0.306515\pi$
$463$	528.844	1.14221	0.571106	+	0.820877i	$0.306515\pi$
$464$	0	0
$465$	74.3166i	0.159821i
$466$	0	0
$467$	436.898	0.935542	0.467771	−	0.883850i	$-0.345057\pi$
$467$	436.898	0.935542	0.467771	+	0.883850i	$0.345057\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−25.6548	−0.0544688
$472$	0	0
$473$	−439.953	−0.930134
$474$	0	0
$475$	251.085	0.528600
$476$	0	0
$477$	− 303.811i	− 0.636920i
$478$	0	0
$479$	− 545.870i	− 1.13960i	−0.821782	−	0.569802i	$-0.807020\pi$
$479$	− 545.870i	− 1.13960i	0.821782	−	0.569802i	$-0.192980\pi$
$480$	0	0
$481$	923.737i	1.92045i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1075.59i	2.21771i
$486$	0	0
$487$	−648.230	−1.33107	−0.665534	−	0.746368i	$-0.731796\pi$
$487$	−648.230	−1.33107	−0.665534	+	0.746368i	$0.731796\pi$
$488$	0	0
$489$	220.822i	0.451579i
$490$	0	0
$491$	− 732.074i	− 1.49098i	−0.666514	−	0.745492i	$-0.732214\pi$
$491$	− 732.074i	− 1.49098i	0.666514	−	0.745492i	$-0.267786\pi$
$492$	0	0
$493$	265.368	0.538272
$494$	0	0
$495$	− 685.060i	− 1.38396i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 26.7040i	− 0.0535151i	−0.999642	−	0.0267575i	$-0.991482\pi$
$499$	− 26.7040i	− 0.0535151i	0.999642	−	0.0267575i	$-0.00851820\pi$
$500$	0	0
$501$	− 54.7527i	− 0.109287i
$502$	0	0
$503$	− 616.414i	− 1.22548i	−0.790286	−	0.612738i	$-0.790068\pi$
$503$	− 616.414i	− 1.22548i	0.790286	−	0.612738i	$-0.209932\pi$
$504$	0	0
$505$	−178.460	−0.353387
$506$	0	0
$507$	189.737	0.374235
$508$	0	0
$509$	−132.753	−0.260811	−0.130405	−	0.991461i	$-0.541628\pi$
$509$	−132.753	−0.260811	−0.130405	+	0.991461i	$0.541628\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−257.345	−0.501648
$514$	0	0
$515$	− 1055.24i	− 2.04902i
$516$	0	0
$517$	289.483	0.559928
$518$	0	0
$519$	126.942	0.244589
$520$	0	0
$521$	− 676.054i	− 1.29761i	−0.760955	−	0.648804i	$-0.775269\pi$
$521$	− 676.054i	− 1.29761i	0.760955	−	0.648804i	$-0.224731\pi$
$522$	0	0
$523$	372.449	0.712139	0.356069	−	0.934460i	$-0.384117\pi$
$523$	372.449	0.712139	0.356069	+	0.934460i	$0.384117\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−204.799	−0.388612
$528$	0	0
$529$	43.2358	0.0817313
$530$	0	0
$531$	−446.799	−0.841429
$532$	0	0
$533$	− 126.161i	− 0.236700i
$534$	0	0
$535$	− 1253.67i	− 2.34331i
$536$	0	0
$537$	265.723i	0.494829i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	− 69.6715i	− 0.128783i	−0.997925	−	0.0643914i	$-0.979489\pi$
$541$	− 69.6715i	− 0.128783i	0.997925	−	0.0643914i	$-0.0205106\pi$
$542$	0	0
$543$	−151.972	−0.279875
$544$	0	0
$545$	− 73.0934i	− 0.134116i
$546$	0	0
$547$	− 466.463i	− 0.852765i	−0.904543	−	0.426383i	$-0.859788\pi$
$547$	− 466.463i	− 0.852765i	0.904543	−	0.426383i	$-0.140212\pi$
$548$	0	0
$549$	83.7197	0.152495
$550$	0	0
$551$	− 274.161i	− 0.497570i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	274.845i	0.495217i
$556$	0	0
$557$	137.219i	0.246353i	0.992385	+	0.123177i	$0.0393082\pi$
$557$	137.219i	0.246353i	−0.992385	+	0.123177i	$0.960692\pi$
$558$	0	0
$559$	646.692i	1.15687i
$560$	0	0
$561$	−191.706	−0.341722
$562$	0	0
$563$	169.126	0.300402	0.150201	−	0.988655i	$-0.452008\pi$
$563$	169.126	0.300402	0.150201	+	0.988655i	$0.452008\pi$
$564$	0	0
$565$	−93.8479	−0.166103
$566$	0	0
$567$	0	0
$568$	0	0
$569$	744.931	1.30919	0.654597	−	0.755978i	$-0.272838\pi$
$569$	744.931	1.30919	0.654597	+	0.755978i	$0.272838\pi$
$570$	0	0
$571$	886.611i	1.55273i	0.630281	+	0.776367i	$0.282940\pi$
$571$	886.611i	1.55273i	−0.630281	+	0.776367i	$0.717060\pi$
$572$	0	0
$573$	119.647	0.208809
$574$	0	0
$575$	−365.026	−0.634827
$576$	0	0
$577$	− 240.062i	− 0.416052i	−0.978123	−	0.208026i	$-0.933296\pi$
$577$	− 240.062i	− 0.416052i	0.978123	−	0.208026i	$-0.0667038\pi$
$578$	0	0
$579$	−74.1799	−0.128117
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−491.382	−0.842850
$584$	0	0
$585$	−1006.98	−1.72133
$586$	0	0
$587$	190.873	0.325168	0.162584	−	0.986695i	$-0.448017\pi$
$587$	190.873	0.325168	0.162584	+	0.986695i	$0.448017\pi$
$588$	0	0
$589$	211.585i	0.359227i
$590$	0	0
$591$	− 1.90871i	− 0.00322962i
$592$	0	0
$593$	− 736.177i	− 1.24145i	−0.784030	−	0.620723i	$-0.786839\pi$
$593$	− 736.177i	− 1.24145i	0.784030	−	0.620723i	$-0.213161\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	115.629i	0.193683i
$598$	0	0
$599$	−1116.66	−1.86421	−0.932104	−	0.362190i	$-0.882029\pi$
$599$	−1116.66	−1.86421	−0.932104	+	0.362190i	$0.882029\pi$
$600$	0	0
$601$	183.100i	0.304659i	0.988330	+	0.152329i	$0.0486774\pi$
$601$	183.100i	0.304659i	−0.988330	+	0.152329i	$0.951323\pi$
$602$	0	0
$603$	139.991i	0.232158i
$604$	0	0
$605$	−340.264	−0.562421
$606$	0	0
$607$	454.982i	0.749558i	0.927114	+	0.374779i	$0.122281\pi$
$607$	454.982i	0.749558i	−0.927114	+	0.374779i	$0.877719\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 425.513i	− 0.696421i
$612$	0	0
$613$	− 268.876i	− 0.438623i	−0.975655	−	0.219312i	$-0.929619\pi$
$613$	− 268.876i	− 0.438623i	0.975655	−	0.219312i	$-0.0703811\pi$
$614$	0	0
$615$	− 37.5375i	− 0.0610366i
$616$	0	0
$617$	−184.934	−0.299731	−0.149866	−	0.988706i	$-0.547884\pi$
$617$	−184.934	−0.299731	−0.149866	+	0.988706i	$0.547884\pi$
$618$	0	0
$619$	993.618	1.60520	0.802599	−	0.596519i	$-0.203450\pi$
$619$	993.618	1.60520	0.802599	+	0.596519i	$0.203450\pi$
$620$	0	0
$621$	374.127	0.602459
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−773.636	−1.23782
$626$	0	0
$627$	198.058i	0.315883i
$628$	0	0
$629$	−757.408	−1.20415
$630$	0	0
$631$	−805.857	−1.27711	−0.638555	−	0.769576i	$-0.720468\pi$
$631$	−805.857	−1.27711	−0.638555	+	0.769576i	$0.720468\pi$
$632$	0	0
$633$	− 6.52926i	− 0.0103148i
$634$	0	0
$635$	445.837	0.702105
$636$	0	0
$637$	0	0
$638$	0	0
$639$	261.915	0.409883
$640$	0	0
$641$	5.50441	0.00858723	0.00429361	−	0.999991i	$-0.498633\pi$
$641$	5.50441	0.00858723	0.00429361	+	0.999991i	$0.498633\pi$
$642$	0	0
$643$	−1024.08	−1.59266	−0.796331	−	0.604861i	$-0.793229\pi$
$643$	−1024.08	−1.59266	−0.796331	+	0.604861i	$0.793229\pi$
$644$	0	0
$645$	192.414i	0.298317i
$646$	0	0
$647$	− 456.573i	− 0.705677i	−0.935684	−	0.352839i	$-0.885216\pi$
$647$	− 456.573i	− 0.705677i	0.935684	−	0.352839i	$-0.114784\pi$
$648$	0	0
$649$	722.650i	1.11348i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 28.2443i	− 0.0432532i	−0.999766	−	0.0216266i	$-0.993116\pi$
$653$	− 28.2443i	− 0.0432532i	0.999766	−	0.0216266i	$-0.00688450\pi$
$654$	0	0
$655$	901.814	1.37681
$656$	0	0
$657$	− 875.579i	− 1.33269i
$658$	0	0
$659$	132.188i	0.200589i	0.994958	+	0.100295i	$0.0319785\pi$
$659$	132.188i	0.200589i	−0.994958	+	0.100295i	$0.968021\pi$
$660$	0	0
$661$	693.847	1.04969	0.524847	−	0.851197i	$-0.324123\pi$
$661$	693.847	1.04969	0.524847	+	0.851197i	$0.324123\pi$
$662$	0	0
$663$	281.791i	0.425024i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	398.574i	0.597562i
$668$	0	0
$669$	254.787i	0.380847i
$670$	0	0
$671$	− 135.408i	− 0.201800i
$672$	0	0
$673$	532.137	0.790694	0.395347	−	0.918532i	$-0.370624\pi$
$673$	532.137	0.790694	0.395347	+	0.918532i	$0.370624\pi$
$674$	0	0
$675$	−238.653	−0.353560
$676$	0	0
$677$	−286.230	−0.422792	−0.211396	−	0.977401i	$-0.567801\pi$
$677$	−286.230	−0.422792	−0.211396	+	0.977401i	$0.567801\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	277.616	0.407659
$682$	0	0
$683$	447.836i	0.655690i	0.944731	+	0.327845i	$0.106322\pi$
$683$	447.836i	0.655690i	−0.944731	+	0.327845i	$0.893678\pi$
$684$	0	0
$685$	1605.76	2.34417
$686$	0	0
$687$	377.724	0.549817
$688$	0	0
$689$	722.287i	1.04831i
$690$	0	0
$691$	−1020.73	−1.47718	−0.738591	−	0.674154i	$-0.764508\pi$
$691$	−1020.73	−1.47718	−0.738591	+	0.674154i	$0.764508\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−312.118	−0.449090
$696$	0	0
$697$	103.444	0.148414
$698$	0	0
$699$	−150.225	−0.214914
$700$	0	0
$701$	− 1311.02i	− 1.87021i	−0.354369	−	0.935106i	$-0.615304\pi$
$701$	− 1311.02i	− 1.87021i	0.354369	−	0.935106i	$-0.384696\pi$
$702$	0	0
$703$	782.504i	1.11309i
$704$	0	0
$705$	− 126.606i	− 0.179582i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	− 537.508i	− 0.758121i	−0.925372	−	0.379061i	$-0.876247\pi$
$709$	− 537.508i	− 0.758121i	0.925372	−	0.379061i	$-0.123753\pi$
$710$	0	0
$711$	−476.398	−0.670040
$712$	0	0
$713$	− 307.601i	− 0.431417i
$714$	0	0
$715$	1628.68i	2.27787i
$716$	0	0
$717$	17.4141	0.0242874
$718$	0	0
$719$	− 269.363i	− 0.374635i	−0.982299	−	0.187318i	$-0.940021\pi$
$719$	− 269.363i	− 0.374635i	0.982299	−	0.187318i	$-0.0599793\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	− 319.083i	− 0.441331i
$724$	0	0
$725$	− 254.248i	− 0.350686i
$726$	0	0
$727$	− 460.316i	− 0.633172i	−0.948564	−	0.316586i	$-0.897463\pi$
$727$	− 460.316i	− 0.633172i	0.948564	−	0.316586i	$-0.102537\pi$
$728$	0	0
$729$	−356.185	−0.488594
$730$	0	0
$731$	−530.247	−0.725372
$732$	0	0
$733$	−66.6821	−0.0909715	−0.0454857	−	0.998965i	$-0.514484\pi$
$733$	−66.6821	−0.0909715	−0.0454857	+	0.998965i	$0.514484\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	226.421	0.307220
$738$	0	0
$739$	− 933.889i	− 1.26372i	−0.775083	−	0.631860i	$-0.782292\pi$
$739$	− 933.889i	− 1.26372i	0.775083	−	0.631860i	$-0.217708\pi$
$740$	0	0
$741$	291.128	0.392885
$742$	0	0
$743$	1198.23	1.61269	0.806345	−	0.591446i	$-0.201443\pi$
$743$	1198.23	1.61269	0.806345	+	0.591446i	$0.201443\pi$
$744$	0	0
$745$	− 264.547i	− 0.355097i
$746$	0	0
$747$	296.943	0.397514
$748$	0	0
$749$	0	0
$750$	0	0
$751$	168.599	0.224499	0.112249	−	0.993680i	$-0.464194\pi$
$751$	168.599	0.224499	0.112249	+	0.993680i	$0.464194\pi$
$752$	0	0
$753$	−80.4478	−0.106836
$754$	0	0
$755$	619.458	0.820474
$756$	0	0
$757$	209.207i	0.276364i	0.990407	+	0.138182i	$0.0441259\pi$
$757$	209.207i	0.276364i	−0.990407	+	0.138182i	$0.955874\pi$
$758$	0	0
$759$	− 287.936i	− 0.379362i
$760$	0	0
$761$	− 553.248i	− 0.727002i	−0.931594	−	0.363501i	$-0.881581\pi$
$761$	− 553.248i	− 0.727002i	0.931594	−	0.363501i	$-0.118419\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	− 825.659i	− 1.07929i
$766$	0	0
$767$	1062.23	1.38492
$768$	0	0
$769$	− 219.524i	− 0.285467i	−0.989761	−	0.142734i	$-0.954411\pi$
$769$	− 219.524i	− 0.285467i	0.989761	−	0.142734i	$-0.0455892\pi$
$770$	0	0
$771$	− 78.4096i	− 0.101699i
$772$	0	0
$773$	−666.674	−0.862451	−0.431225	−	0.902244i	$-0.641919\pi$
$773$	−666.674	−0.862451	−0.431225	+	0.902244i	$0.641919\pi$
$774$	0	0
$775$	196.216i	0.253182i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 106.872i	− 0.137191i
$780$	0	0
$781$	− 423.620i	− 0.542407i
$782$	0	0
$783$	260.587i	0.332806i
$784$	0	0
$785$	−178.710	−0.227656
$786$	0	0
$787$	−919.865	−1.16882	−0.584412	−	0.811457i	$-0.698675\pi$
$787$	−919.865	−1.16882	−0.584412	+	0.811457i	$0.698675\pi$
$788$	0	0
$789$	−290.757	−0.368513
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−199.037	−0.250993
$794$	0	0
$795$	214.907i	0.270323i
$796$	0	0
$797$	−1016.13	−1.27494	−0.637470	−	0.770476i	$-0.720019\pi$
$797$	−1016.13	−1.27494	−0.637470	+	0.770476i	$0.720019\pi$
$798$	0	0
$799$	348.895	0.436664
$800$	0	0
$801$	8.77277i	0.0109523i
$802$	0	0
$803$	−1416.16	−1.76358
$804$	0	0
$805$	0	0
$806$	0	0
$807$	52.3454	0.0648642
$808$	0	0
$809$	1131.90	1.39913	0.699567	−	0.714567i	$-0.253376\pi$
$809$	1131.90	1.39913	0.699567	+	0.714567i	$0.253376\pi$
$810$	0	0
$811$	481.066	0.593176	0.296588	−	0.955006i	$-0.404151\pi$
$811$	481.066	0.593176	0.296588	+	0.955006i	$0.404151\pi$
$812$	0	0
$813$	28.1242i	0.0345931i
$814$	0	0
$815$	1538.23i	1.88740i
$816$	0	0
$817$	547.817i	0.670523i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 727.675i	− 0.886328i	−0.896441	−	0.443164i	$-0.853856\pi$
$821$	− 727.675i	− 0.886328i	0.896441	−	0.443164i	$-0.146144\pi$
$822$	0	0
$823$	626.646	0.761417	0.380709	−	0.924695i	$-0.375680\pi$
$823$	626.646	0.761417	0.380709	+	0.924695i	$0.375680\pi$
$824$	0	0
$825$	183.673i	0.222633i
$826$	0	0
$827$	− 1468.52i	− 1.77572i	−0.460116	−	0.887859i	$-0.652192\pi$
$827$	− 1468.52i	− 1.77572i	0.460116	−	0.887859i	$-0.347808\pi$
$828$	0	0
$829$	−818.704	−0.987580	−0.493790	−	0.869581i	$-0.664389\pi$
$829$	−818.704	−0.987580	−0.493790	+	0.869581i	$0.664389\pi$
$830$	0	0
$831$	− 324.435i	− 0.390416i
$832$	0	0
$833$	0	0
$834$	0	0
$835$	− 381.404i	− 0.456772i
$836$	0	0
$837$	− 201.109i	− 0.240273i
$838$	0	0
$839$	− 1108.84i	− 1.32162i	−0.750555	−	0.660808i	$-0.770214\pi$
$839$	− 1108.84i	− 1.32162i	0.750555	−	0.660808i	$-0.229786\pi$
$840$	0	0
$841$	563.386	0.669900
$842$	0	0
$843$	−267.939	−0.317840
$844$	0	0
$845$	1321.70	1.56414
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−378.010	−0.445241
$850$	0	0
$851$	− 1137.60i	− 1.33678i
$852$	0	0
$853$	−610.400	−0.715592	−0.357796	−	0.933800i	$-0.616472\pi$
$853$	−610.400	−0.715592	−0.357796	+	0.933800i	$0.616472\pi$
$854$	0	0
$855$	−853.017	−0.997680
$856$	0	0
$857$	444.114i	0.518219i	0.965848	+	0.259110i	$0.0834291\pi$
$857$	444.114i	0.518219i	−0.965848	+	0.259110i	$0.916571\pi$
$858$	0	0
$859$	−81.5094	−0.0948887	−0.0474443	−	0.998874i	$-0.515108\pi$
$859$	−81.5094	−0.0948887	−0.0474443	+	0.998874i	$0.515108\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1051.46	1.21838	0.609189	−	0.793025i	$-0.291495\pi$
$863$	1051.46	1.21838	0.609189	+	0.793025i	$0.291495\pi$
$864$	0	0
$865$	884.268	1.02228
$866$	0	0
$867$	32.1882	0.0371259
$868$	0	0
$869$	770.524i	0.886679i
$870$	0	0
$871$	− 332.819i	− 0.382111i
$872$	0	0
$873$	− 1385.01i	− 1.58650i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1559.63i	1.77837i	0.457551	+	0.889183i	$0.348727\pi$
$877$	1559.63i	1.77837i	−0.457551	+	0.889183i	$0.651273\pi$
$878$	0	0
$879$	337.533	0.383997
$880$	0	0
$881$	1515.22i	1.71989i	0.510389	+	0.859944i	$0.329501\pi$
$881$	1515.22i	1.71989i	−0.510389	+	0.859944i	$0.670499\pi$
$882$	0	0
$883$	763.828i	0.865037i	0.901625	+	0.432519i	$0.142375\pi$
$883$	763.828i	0.865037i	−0.901625	+	0.432519i	$0.857625\pi$
$884$	0	0
$885$	316.052	0.357121
$886$	0	0
$887$	573.371i	0.646416i	0.946328	+	0.323208i	$0.104761\pi$
$887$	573.371i	0.646416i	−0.946328	+	0.323208i	$0.895239\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	783.460i	0.879304i
$892$	0	0
$893$	− 360.455i	− 0.403645i
$894$	0	0
$895$	1851.01i	2.06817i
$896$	0	0
$897$	−423.240	−0.471840
$898$	0	0
$899$	214.250	0.238320
$900$	0	0
$901$	−592.231	−0.657304
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1058.63	−1.16976
$906$	0	0
$907$	1021.95i	1.12674i	0.826205	+	0.563369i	$0.190495\pi$
$907$	1021.95i	1.12674i	−0.826205	+	0.563369i	$0.809505\pi$
$908$	0	0
$909$	229.799	0.252804
$910$	0	0
$911$	−630.111	−0.691669	−0.345835	−	0.938295i	$-0.612404\pi$
$911$	−630.111	−0.691669	−0.345835	+	0.938295i	$0.612404\pi$
$912$	0	0
$913$	− 480.274i	− 0.526040i
$914$	0	0
$915$	−59.2208	−0.0647222
$916$	0	0
$917$	0	0
$918$	0	0
$919$	842.979	0.917278	0.458639	−	0.888623i	$-0.348337\pi$
$919$	842.979	0.917278	0.458639	+	0.888623i	$0.348337\pi$
$920$	0	0
$921$	−146.036	−0.158563
$922$	0	0
$923$	−622.683	−0.674630
$924$	0	0
$925$	725.668i	0.784506i
$926$	0	0
$927$	1358.81i	1.46581i
$928$	0	0
$929$	− 774.651i	− 0.833855i	−0.908940	−	0.416927i	$-0.863107\pi$
$929$	− 774.651i	− 0.833855i	0.908940	−	0.416927i	$-0.136893\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	− 430.691i	− 0.461620i
$934$	0	0
$935$	−1335.42	−1.42825
$936$	0	0
$937$	1586.27i	1.69293i	0.532447	+	0.846463i	$0.321273\pi$
$937$	1586.27i	1.69293i	−0.532447	+	0.846463i	$0.678727\pi$
$938$	0	0
$939$	− 210.421i	− 0.224090i
$940$	0	0
$941$	820.046	0.871462	0.435731	−	0.900077i	$-0.356490\pi$
$941$	820.046	0.871462	0.435731	+	0.900077i	$0.356490\pi$
$942$	0	0
$943$	155.370i	0.164761i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	637.335i	0.673005i	0.941683	+	0.336502i	$0.109244\pi$
$947$	637.335i	0.673005i	−0.941683	+	0.336502i	$0.890756\pi$
$948$	0	0
$949$	2081.62i	2.19349i
$950$	0	0
$951$	− 205.234i	− 0.215809i
$952$	0	0
$953$	−350.626	−0.367918	−0.183959	−	0.982934i	$-0.558891\pi$
$953$	−350.626	−0.367918	−0.183959	+	0.982934i	$0.558891\pi$
$954$	0	0
$955$	833.458	0.872731
$956$	0	0
$957$	200.553	0.209564
$958$	0	0
$959$	0	0
$960$	0	0
$961$	795.652	0.827942
$962$	0	0
$963$	1614.32i	1.67635i
$964$	0	0
$965$	−516.734	−0.535475
$966$	0	0
$967$	649.816	0.671992	0.335996	−	0.941863i	$-0.390927\pi$
$967$	649.816	0.671992	0.335996	+	0.941863i	$0.390927\pi$
$968$	0	0
$969$	238.707i	0.246344i
$970$	0	0
$971$	970.610	0.999598	0.499799	−	0.866141i	$-0.333407\pi$
$971$	970.610	0.999598	0.499799	+	0.866141i	$0.333407\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	269.982	0.276905
$976$	0	0
$977$	−600.875	−0.615020	−0.307510	−	0.951545i	$-0.599496\pi$
$977$	−600.875	−0.615020	−0.307510	+	0.951545i	$0.599496\pi$
$978$	0	0
$979$	14.1890	0.0144934
$980$	0	0
$981$	94.1205i	0.0959434i
$982$	0	0
$983$	− 1268.62i	− 1.29056i	−0.763945	−	0.645281i	$-0.776740\pi$
$983$	− 1268.62i	− 1.29056i	0.763945	−	0.645281i	$-0.223260\pi$
$984$	0	0
$985$	− 13.2959i	− 0.0134984i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 796.413i	− 0.805271i
$990$	0	0
$991$	1549.11	1.56318	0.781590	−	0.623793i	$-0.214409\pi$
$991$	1549.11	1.56318	0.781590	+	0.623793i	$0.214409\pi$
$992$	0	0
$993$	− 18.8538i	− 0.0189867i
$994$	0	0
$995$	805.464i	0.809512i
$996$	0	0
$997$	940.938	0.943769	0.471885	−	0.881660i	$-0.343574\pi$
$997$	940.938	0.943769	0.471885	+	0.881660i	$0.343574\pi$
$998$	0	0
$999$	− 743.761i	− 0.744506i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1568.3.h.a.881.17		28
4.3	odd	2		392.3.h.a.293.25		28
7.4	even	3		224.3.n.a.145.6		28
7.5	odd	6		224.3.n.a.17.9		28
7.6	odd	2	inner	1568.3.h.a.881.11		28
8.3	odd	2		392.3.h.a.293.28		28
8.5	even	2	inner	1568.3.h.a.881.12		28
28.3	even	6		392.3.j.e.117.6		28
28.11	odd	6		56.3.j.a.5.6	yes	28
28.19	even	6		56.3.j.a.45.4	yes	28
28.23	odd	6		392.3.j.e.325.4		28
28.27	even	2		392.3.h.a.293.26		28
56.3	even	6		392.3.j.e.117.4		28
56.5	odd	6		224.3.n.a.17.6		28
56.11	odd	6		56.3.j.a.5.4	✓	28
56.13	odd	2	inner	1568.3.h.a.881.18		28
56.19	even	6		56.3.j.a.45.6	yes	28
56.27	even	2		392.3.h.a.293.27		28
56.51	odd	6		392.3.j.e.325.6		28
56.53	even	6		224.3.n.a.145.9		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.3.j.a.5.4	✓	28	56.11	odd	6
56.3.j.a.5.6	yes	28	28.11	odd	6
56.3.j.a.45.4	yes	28	28.19	even	6
56.3.j.a.45.6	yes	28	56.19	even	6
224.3.n.a.17.6		28	56.5	odd	6
224.3.n.a.17.9		28	7.5	odd	6
224.3.n.a.145.6		28	7.4	even	3
224.3.n.a.145.9		28	56.53	even	6
392.3.h.a.293.25		28	4.3	odd	2
392.3.h.a.293.26		28	28.27	even	2
392.3.h.a.293.27		28	56.27	even	2
392.3.h.a.293.28		28	8.3	odd	2
392.3.j.e.117.4		28	56.3	even	6
392.3.j.e.117.6		28	28.3	even	6
392.3.j.e.325.4		28	28.23	odd	6
392.3.j.e.325.6		28	56.51	odd	6
1568.3.h.a.881.11		28	7.6	odd	2	inner
1568.3.h.a.881.12		28	8.5	even	2	inner
1568.3.h.a.881.17		28	1.1	even	1	trivial
1568.3.h.a.881.18		28	56.13	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:	\( N \)	\(=\)	\( 1568 = 2^{5} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1568.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.910863 q^{3} +6.34503 q^{5} -8.17033 q^{9} +O(q^{10})\)	\(q+0.910863 q^{3} +6.34503 q^{5} -8.17033 q^{9} +13.2146i q^{11} +19.4243 q^{13} +5.77945 q^{15} +15.9268i q^{17} +16.4545 q^{19} -23.9215 q^{23} +15.2593 q^{25} -15.6398 q^{27} -16.6618i q^{29} +12.8588i q^{31} +12.0367i q^{33} +47.5556i q^{37} +17.6929 q^{39} -6.49499i q^{41} +33.2928i q^{43} -51.8409 q^{45} -21.9062i q^{47} +14.5071i q^{51} +37.1846i q^{53} +83.8473i q^{55} +14.9878 q^{57} +54.6855 q^{59} -10.2468 q^{61} +123.248 q^{65} -17.1341i q^{67} -21.7892 q^{69} -32.0568 q^{71} +107.166i q^{73} +13.8992 q^{75} +58.3083 q^{79} +59.2872 q^{81} -36.3441 q^{83} +101.056i q^{85} -15.1766i q^{87} -1.07373i q^{89} +11.7126i q^{93} +104.404 q^{95} +169.517i q^{97} -107.968i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 64 q^{9}+O(q^{10}) \) 28 * q + 64 * q^9	\( 28 q + 64 q^{9} - 28 q^{15} + 60 q^{23} + 64 q^{25} - 40 q^{39} + 124 q^{57} + 104 q^{65} + 136 q^{71} - 324 q^{79} + 36 q^{81} + 580 q^{95}+O(q^{100}) \) 28 * q + 64 * q^9 - 28 * q^15 + 60 * q^23 + 64 * q^25 - 40 * q^39 + 124 * q^57 + 104 * q^65 + 136 * q^71 - 324 * q^79 + 36 * q^81 + 580 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more