LMFDB - Embedded newform 1856.2.a.q.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1856,2,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, 2, names="a")

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

chi := DirichletCharacter("1856.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1856 = 2^{6} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
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:	$14.8202346151$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 928)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ 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)

$f(q)$	$=$	$q-2.41421 q^{3} -1.00000 q^{5} +2.00000 q^{7} +2.82843 q^{9} +O(q^{10})$	$q-2.41421 q^{3} -1.00000 q^{5} +2.00000 q^{7} +2.82843 q^{9} +0.414214 q^{11} -3.82843 q^{13} +2.41421 q^{15} -2.82843 q^{17} +3.65685 q^{19} -4.82843 q^{21} +6.82843 q^{23} -4.00000 q^{25} +0.414214 q^{27} -1.00000 q^{29} -0.414214 q^{31} -1.00000 q^{33} -2.00000 q^{35} +5.65685 q^{37} +9.24264 q^{39} +0.828427 q^{41} -5.24264 q^{43} -2.82843 q^{45} +1.58579 q^{47} -3.00000 q^{49} +6.82843 q^{51} +5.48528 q^{53} -0.414214 q^{55} -8.82843 q^{57} -12.4853 q^{59} +4.48528 q^{61} +5.65685 q^{63} +3.82843 q^{65} -4.00000 q^{67} -16.4853 q^{69} +3.17157 q^{71} -8.00000 q^{73} +9.65685 q^{75} +0.828427 q^{77} -8.89949 q^{79} -9.48528 q^{81} +4.48528 q^{83} +2.82843 q^{85} +2.41421 q^{87} -8.82843 q^{89} -7.65685 q^{91} +1.00000 q^{93} -3.65685 q^{95} -8.82843 q^{97} +1.17157 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 2 q^{5} + 4 q^{7}+O(q^{10}) $ 2 * q - 2 * q^3 - 2 * q^5 + 4 * q^7	$ 2 q - 2 q^{3} - 2 q^{5} + 4 q^{7} - 2 q^{11} - 2 q^{13} + 2 q^{15} - 4 q^{19} - 4 q^{21} + 8 q^{23} - 8 q^{25} - 2 q^{27} - 2 q^{29} + 2 q^{31} - 2 q^{33} - 4 q^{35} + 10 q^{39} - 4 q^{41} - 2 q^{43} + 6 q^{47} - 6 q^{49} + 8 q^{51} - 6 q^{53} + 2 q^{55} - 12 q^{57} - 8 q^{59} - 8 q^{61} + 2 q^{65} - 8 q^{67} - 16 q^{69} + 12 q^{71} - 16 q^{73} + 8 q^{75} - 4 q^{77} + 2 q^{79} - 2 q^{81} - 8 q^{83} + 2 q^{87} - 12 q^{89} - 4 q^{91} + 2 q^{93} + 4 q^{95} - 12 q^{97} + 8 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^5 + 4 * q^7 - 2 * q^11 - 2 * q^13 + 2 * q^15 - 4 * q^19 - 4 * q^21 + 8 * q^23 - 8 * q^25 - 2 * q^27 - 2 * q^29 + 2 * q^31 - 2 * q^33 - 4 * q^35 + 10 * q^39 - 4 * q^41 - 2 * q^43 + 6 * q^47 - 6 * q^49 + 8 * q^51 - 6 * q^53 + 2 * q^55 - 12 * q^57 - 8 * q^59 - 8 * q^61 + 2 * q^65 - 8 * q^67 - 16 * q^69 + 12 * q^71 - 16 * q^73 + 8 * q^75 - 4 * q^77 + 2 * q^79 - 2 * q^81 - 8 * q^83 + 2 * q^87 - 12 * q^89 - 4 * q^91 + 2 * q^93 + 4 * q^95 - 12 * q^97 + 8 * q^99

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
$3$	−2.41421	−1.39385	−0.696923	−	0.717146i	$-0.745448\pi$
$3$	−2.41421	−1.39385	−0.696923	+	0.717146i	$0.745448\pi$
$4$	0	0
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	0	0
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	0	0
$9$	2.82843	0.942809
$10$	0	0
$11$	0.414214	0.124890	0.0624450	−	0.998048i	$-0.480110\pi$
$11$	0.414214	0.124890	0.0624450	+	0.998048i	$0.480110\pi$
$12$	0	0
$13$	−3.82843	−1.06181	−0.530907	−	0.847430i	$-0.678149\pi$
$13$	−3.82843	−1.06181	−0.530907	+	0.847430i	$0.678149\pi$
$14$	0	0
$15$	2.41421	0.623347
$16$	0	0
$17$	−2.82843	−0.685994	−0.342997	−	0.939336i	$-0.611442\pi$
$17$	−2.82843	−0.685994	−0.342997	+	0.939336i	$0.611442\pi$
$18$	0	0
$19$	3.65685	0.838940	0.419470	−	0.907769i	$-0.362216\pi$
$19$	3.65685	0.838940	0.419470	+	0.907769i	$0.362216\pi$
$20$	0	0
$21$	−4.82843	−1.05365
$22$	0	0
$23$	6.82843	1.42383	0.711913	−	0.702268i	$-0.247829\pi$
$23$	6.82843	1.42383	0.711913	+	0.702268i	$0.247829\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	0.414214	0.0797154
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−0.414214	−0.0743950	−0.0371975	−	0.999308i	$-0.511843\pi$
$31$	−0.414214	−0.0743950	−0.0371975	+	0.999308i	$0.511843\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	−2.00000	−0.338062
$36$	0	0
$37$	5.65685	0.929981	0.464991	−	0.885316i	$-0.346058\pi$
$37$	5.65685	0.929981	0.464991	+	0.885316i	$0.346058\pi$
$38$	0	0
$39$	9.24264	1.48001
$40$	0	0
$41$	0.828427	0.129379	0.0646893	−	0.997905i	$-0.479394\pi$
$41$	0.828427	0.129379	0.0646893	+	0.997905i	$0.479394\pi$
$42$	0	0
$43$	−5.24264	−0.799495	−0.399748	−	0.916625i	$-0.630902\pi$
$43$	−5.24264	−0.799495	−0.399748	+	0.916625i	$0.630902\pi$
$44$	0	0
$45$	−2.82843	−0.421637
$46$	0	0
$47$	1.58579	0.231311	0.115655	−	0.993289i	$-0.463103\pi$
$47$	1.58579	0.231311	0.115655	+	0.993289i	$0.463103\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	6.82843	0.956171
$52$	0	0
$53$	5.48528	0.753461	0.376731	−	0.926323i	$-0.377048\pi$
$53$	5.48528	0.753461	0.376731	+	0.926323i	$0.377048\pi$
$54$	0	0
$55$	−0.414214	−0.0558525
$56$	0	0
$57$	−8.82843	−1.16935
$58$	0	0
$59$	−12.4853	−1.62545	−0.812723	−	0.582651i	$-0.802016\pi$
$59$	−12.4853	−1.62545	−0.812723	+	0.582651i	$0.802016\pi$
$60$	0	0
$61$	4.48528	0.574281	0.287141	−	0.957888i	$-0.407295\pi$
$61$	4.48528	0.574281	0.287141	+	0.957888i	$0.407295\pi$
$62$	0	0
$63$	5.65685	0.712697
$64$	0	0
$65$	3.82843	0.474858
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	0	0
$69$	−16.4853	−1.98459
$70$	0	0
$71$	3.17157	0.376396	0.188198	−	0.982131i	$-0.439735\pi$
$71$	3.17157	0.376396	0.188198	+	0.982131i	$0.439735\pi$
$72$	0	0
$73$	−8.00000	−0.936329	−0.468165	−	0.883641i	$-0.655085\pi$
$73$	−8.00000	−0.936329	−0.468165	+	0.883641i	$0.655085\pi$
$74$	0	0
$75$	9.65685	1.11508
$76$	0	0
$77$	0.828427	0.0944080
$78$	0	0
$79$	−8.89949	−1.00127	−0.500636	−	0.865658i	$-0.666900\pi$
$79$	−8.89949	−1.00127	−0.500636	+	0.865658i	$0.666900\pi$
$80$	0	0
$81$	−9.48528	−1.05392
$82$	0	0
$83$	4.48528	0.492324	0.246162	−	0.969229i	$-0.420831\pi$
$83$	4.48528	0.492324	0.246162	+	0.969229i	$0.420831\pi$
$84$	0	0
$85$	2.82843	0.306786
$86$	0	0
$87$	2.41421	0.258831
$88$	0	0
$89$	−8.82843	−0.935811	−0.467906	−	0.883778i	$-0.654991\pi$
$89$	−8.82843	−0.935811	−0.467906	+	0.883778i	$0.654991\pi$
$90$	0	0
$91$	−7.65685	−0.802656
$92$	0	0
$93$	1.00000	0.103695
$94$	0	0
$95$	−3.65685	−0.375185
$96$	0	0
$97$	−8.82843	−0.896391	−0.448195	−	0.893936i	$-0.647933\pi$
$97$	−8.82843	−0.896391	−0.448195	+	0.893936i	$0.647933\pi$
$98$	0	0
$99$	1.17157	0.117748
$100$	0	0
$101$	−11.3137	−1.12576	−0.562878	−	0.826540i	$-0.690306\pi$
$101$	−11.3137	−1.12576	−0.562878	+	0.826540i	$0.690306\pi$
$102$	0	0
$103$	8.14214	0.802268	0.401134	−	0.916019i	$-0.368616\pi$
$103$	8.14214	0.802268	0.401134	+	0.916019i	$0.368616\pi$
$104$	0	0
$105$	4.82843	0.471206
$106$	0	0
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	0	0
$109$	−10.6569	−1.02074	−0.510371	−	0.859954i	$-0.670492\pi$
$109$	−10.6569	−1.02074	−0.510371	+	0.859954i	$0.670492\pi$
$110$	0	0
$111$	−13.6569	−1.29625
$112$	0	0
$113$	7.65685	0.720296	0.360148	−	0.932895i	$-0.382726\pi$
$113$	7.65685	0.720296	0.360148	+	0.932895i	$0.382726\pi$
$114$	0	0
$115$	−6.82843	−0.636754
$116$	0	0
$117$	−10.8284	−1.00109
$118$	0	0
$119$	−5.65685	−0.518563
$120$	0	0
$121$	−10.8284	−0.984402
$122$	0	0
$123$	−2.00000	−0.180334
$124$	0	0
$125$	9.00000	0.804984
$126$	0	0
$127$	−19.6569	−1.74426	−0.872132	−	0.489271i	$-0.837263\pi$
$127$	−19.6569	−1.74426	−0.872132	+	0.489271i	$0.837263\pi$
$128$	0	0
$129$	12.6569	1.11437
$130$	0	0
$131$	−5.31371	−0.464261	−0.232130	−	0.972685i	$-0.574570\pi$
$131$	−5.31371	−0.464261	−0.232130	+	0.972685i	$0.574570\pi$
$132$	0	0
$133$	7.31371	0.634179
$134$	0	0
$135$	−0.414214	−0.0356498
$136$	0	0
$137$	−15.3137	−1.30834	−0.654169	−	0.756348i	$-0.726982\pi$
$137$	−15.3137	−1.30834	−0.654169	+	0.756348i	$0.726982\pi$
$138$	0	0
$139$	−12.4853	−1.05899	−0.529494	−	0.848314i	$-0.677618\pi$
$139$	−12.4853	−1.05899	−0.529494	+	0.848314i	$0.677618\pi$
$140$	0	0
$141$	−3.82843	−0.322412
$142$	0	0
$143$	−1.58579	−0.132610
$144$	0	0
$145$	1.00000	0.0830455
$146$	0	0
$147$	7.24264	0.597363
$148$	0	0
$149$	−11.8284	−0.969023	−0.484511	−	0.874785i	$-0.661003\pi$
$149$	−11.8284	−0.969023	−0.484511	+	0.874785i	$0.661003\pi$
$150$	0	0
$151$	−0.343146	−0.0279248	−0.0139624	−	0.999903i	$-0.504445\pi$
$151$	−0.343146	−0.0279248	−0.0139624	+	0.999903i	$0.504445\pi$
$152$	0	0
$153$	−8.00000	−0.646762
$154$	0	0
$155$	0.414214	0.0332704
$156$	0	0
$157$	−10.4853	−0.836817	−0.418408	−	0.908259i	$-0.637412\pi$
$157$	−10.4853	−0.836817	−0.418408	+	0.908259i	$0.637412\pi$
$158$	0	0
$159$	−13.2426	−1.05021
$160$	0	0
$161$	13.6569	1.07631
$162$	0	0
$163$	10.4142	0.815704	0.407852	−	0.913048i	$-0.366278\pi$
$163$	10.4142	0.815704	0.407852	+	0.913048i	$0.366278\pi$
$164$	0	0
$165$	1.00000	0.0778499
$166$	0	0
$167$	−10.4853	−0.811375	−0.405688	−	0.914012i	$-0.632968\pi$
$167$	−10.4853	−0.811375	−0.405688	+	0.914012i	$0.632968\pi$
$168$	0	0
$169$	1.65685	0.127450
$170$	0	0
$171$	10.3431	0.790960
$172$	0	0
$173$	15.6569	1.19037	0.595184	−	0.803589i	$-0.297079\pi$
$173$	15.6569	1.19037	0.595184	+	0.803589i	$0.297079\pi$
$174$	0	0
$175$	−8.00000	−0.604743
$176$	0	0
$177$	30.1421	2.26562
$178$	0	0
$179$	14.4853	1.08268	0.541340	−	0.840804i	$-0.317917\pi$
$179$	14.4853	1.08268	0.541340	+	0.840804i	$0.317917\pi$
$180$	0	0
$181$	−26.3137	−1.95588	−0.977941	−	0.208880i	$-0.933018\pi$
$181$	−26.3137	−1.95588	−0.977941	+	0.208880i	$0.933018\pi$
$182$	0	0
$183$	−10.8284	−0.800460
$184$	0	0
$185$	−5.65685	−0.415900
$186$	0	0
$187$	−1.17157	−0.0856739
$188$	0	0
$189$	0.828427	0.0602592
$190$	0	0
$191$	5.31371	0.384486	0.192243	−	0.981347i	$-0.438424\pi$
$191$	5.31371	0.384486	0.192243	+	0.981347i	$0.438424\pi$
$192$	0	0
$193$	23.4558	1.68839	0.844194	−	0.536037i	$-0.180079\pi$
$193$	23.4558	1.68839	0.844194	+	0.536037i	$0.180079\pi$
$194$	0	0
$195$	−9.24264	−0.661879
$196$	0	0
$197$	17.3137	1.23355	0.616775	−	0.787139i	$-0.288439\pi$
$197$	17.3137	1.23355	0.616775	+	0.787139i	$0.288439\pi$
$198$	0	0
$199$	−15.6569	−1.10988	−0.554942	−	0.831889i	$-0.687260\pi$
$199$	−15.6569	−1.10988	−0.554942	+	0.831889i	$0.687260\pi$
$200$	0	0
$201$	9.65685	0.681142
$202$	0	0
$203$	−2.00000	−0.140372
$204$	0	0
$205$	−0.828427	−0.0578599
$206$	0	0
$207$	19.3137	1.34240
$208$	0	0
$209$	1.51472	0.104775
$210$	0	0
$211$	22.4142	1.54306	0.771529	−	0.636194i	$-0.219492\pi$
$211$	22.4142	1.54306	0.771529	+	0.636194i	$0.219492\pi$
$212$	0	0
$213$	−7.65685	−0.524639
$214$	0	0
$215$	5.24264	0.357545
$216$	0	0
$217$	−0.828427	−0.0562373
$218$	0	0
$219$	19.3137	1.30510
$220$	0	0
$221$	10.8284	0.728399
$222$	0	0
$223$	−0.828427	−0.0554756	−0.0277378	−	0.999615i	$-0.508830\pi$
$223$	−0.828427	−0.0554756	−0.0277378	+	0.999615i	$0.508830\pi$
$224$	0	0
$225$	−11.3137	−0.754247
$226$	0	0
$227$	−25.7990	−1.71234	−0.856170	−	0.516695i	$-0.827162\pi$
$227$	−25.7990	−1.71234	−0.856170	+	0.516695i	$0.827162\pi$
$228$	0	0
$229$	6.48528	0.428559	0.214280	−	0.976772i	$-0.431260\pi$
$229$	6.48528	0.428559	0.214280	+	0.976772i	$0.431260\pi$
$230$	0	0
$231$	−2.00000	−0.131590
$232$	0	0
$233$	18.3137	1.19977	0.599885	−	0.800086i	$-0.295213\pi$
$233$	18.3137	1.19977	0.599885	+	0.800086i	$0.295213\pi$
$234$	0	0
$235$	−1.58579	−0.103445
$236$	0	0
$237$	21.4853	1.39562
$238$	0	0
$239$	13.1716	0.851998	0.425999	−	0.904724i	$-0.359923\pi$
$239$	13.1716	0.851998	0.425999	+	0.904724i	$0.359923\pi$
$240$	0	0
$241$	−21.6274	−1.39314	−0.696572	−	0.717487i	$-0.745292\pi$
$241$	−21.6274	−1.39314	−0.696572	+	0.717487i	$0.745292\pi$
$242$	0	0
$243$	21.6569	1.38929
$244$	0	0
$245$	3.00000	0.191663
$246$	0	0
$247$	−14.0000	−0.890799
$248$	0	0
$249$	−10.8284	−0.686224
$250$	0	0
$251$	−0.757359	−0.0478041	−0.0239020	−	0.999714i	$-0.507609\pi$
$251$	−0.757359	−0.0478041	−0.0239020	+	0.999714i	$0.507609\pi$
$252$	0	0
$253$	2.82843	0.177822
$254$	0	0
$255$	−6.82843	−0.427613
$256$	0	0
$257$	−24.7990	−1.54692	−0.773459	−	0.633846i	$-0.781475\pi$
$257$	−24.7990	−1.54692	−0.773459	+	0.633846i	$0.781475\pi$
$258$	0	0
$259$	11.3137	0.703000
$260$	0	0
$261$	−2.82843	−0.175075
$262$	0	0
$263$	0.272078	0.0167770	0.00838852	−	0.999965i	$-0.497330\pi$
$263$	0.272078	0.0167770	0.00838852	+	0.999965i	$0.497330\pi$
$264$	0	0
$265$	−5.48528	−0.336958
$266$	0	0
$267$	21.3137	1.30438
$268$	0	0
$269$	28.4853	1.73678	0.868389	−	0.495883i	$-0.165156\pi$
$269$	28.4853	1.73678	0.868389	+	0.495883i	$0.165156\pi$
$270$	0	0
$271$	−5.24264	−0.318468	−0.159234	−	0.987241i	$-0.550902\pi$
$271$	−5.24264	−0.318468	−0.159234	+	0.987241i	$0.550902\pi$
$272$	0	0
$273$	18.4853	1.11878
$274$	0	0
$275$	−1.65685	−0.0999121
$276$	0	0
$277$	6.00000	0.360505	0.180253	−	0.983620i	$-0.442309\pi$
$277$	6.00000	0.360505	0.180253	+	0.983620i	$0.442309\pi$
$278$	0	0
$279$	−1.17157	−0.0701402
$280$	0	0
$281$	11.9706	0.714104	0.357052	−	0.934085i	$-0.383782\pi$
$281$	11.9706	0.714104	0.357052	+	0.934085i	$0.383782\pi$
$282$	0	0
$283$	3.51472	0.208928	0.104464	−	0.994529i	$-0.466687\pi$
$283$	3.51472	0.208928	0.104464	+	0.994529i	$0.466687\pi$
$284$	0	0
$285$	8.82843	0.522951
$286$	0	0
$287$	1.65685	0.0978010
$288$	0	0
$289$	−9.00000	−0.529412
$290$	0	0
$291$	21.3137	1.24943
$292$	0	0
$293$	−13.3137	−0.777795	−0.388898	−	0.921281i	$-0.627144\pi$
$293$	−13.3137	−0.777795	−0.388898	+	0.921281i	$0.627144\pi$
$294$	0	0
$295$	12.4853	0.726921
$296$	0	0
$297$	0.171573	0.00995567
$298$	0	0
$299$	−26.1421	−1.51184
$300$	0	0
$301$	−10.4853	−0.604362
$302$	0	0
$303$	27.3137	1.56913
$304$	0	0
$305$	−4.48528	−0.256826
$306$	0	0
$307$	18.0711	1.03137	0.515685	−	0.856778i	$-0.327537\pi$
$307$	18.0711	1.03137	0.515685	+	0.856778i	$0.327537\pi$
$308$	0	0
$309$	−19.6569	−1.11824
$310$	0	0
$311$	29.3137	1.66223	0.831114	−	0.556102i	$-0.187704\pi$
$311$	29.3137	1.66223	0.831114	+	0.556102i	$0.187704\pi$
$312$	0	0
$313$	−33.4853	−1.89270	−0.946350	−	0.323143i	$-0.895260\pi$
$313$	−33.4853	−1.89270	−0.946350	+	0.323143i	$0.895260\pi$
$314$	0	0
$315$	−5.65685	−0.318728
$316$	0	0
$317$	17.1716	0.964452	0.482226	−	0.876047i	$-0.339828\pi$
$317$	17.1716	0.964452	0.482226	+	0.876047i	$0.339828\pi$
$318$	0	0
$319$	−0.414214	−0.0231915
$320$	0	0
$321$	14.4853	0.808490
$322$	0	0
$323$	−10.3431	−0.575508
$324$	0	0
$325$	15.3137	0.849452
$326$	0	0
$327$	25.7279	1.42276
$328$	0	0
$329$	3.17157	0.174854
$330$	0	0
$331$	18.4142	1.01214	0.506068	−	0.862493i	$-0.331098\pi$
$331$	18.4142	1.01214	0.506068	+	0.862493i	$0.331098\pi$
$332$	0	0
$333$	16.0000	0.876795
$334$	0	0
$335$	4.00000	0.218543
$336$	0	0
$337$	−4.48528	−0.244329	−0.122164	−	0.992510i	$-0.538984\pi$
$337$	−4.48528	−0.244329	−0.122164	+	0.992510i	$0.538984\pi$
$338$	0	0
$339$	−18.4853	−1.00398
$340$	0	0
$341$	−0.171573	−0.00929119
$342$	0	0
$343$	−20.0000	−1.07990
$344$	0	0
$345$	16.4853	0.887538
$346$	0	0
$347$	−27.1716	−1.45865	−0.729323	−	0.684169i	$-0.760165\pi$
$347$	−27.1716	−1.45865	−0.729323	+	0.684169i	$0.760165\pi$
$348$	0	0
$349$	−21.1421	−1.13171	−0.565856	−	0.824504i	$-0.691454\pi$
$349$	−21.1421	−1.13171	−0.565856	+	0.824504i	$0.691454\pi$
$350$	0	0
$351$	−1.58579	−0.0846430
$352$	0	0
$353$	−3.65685	−0.194635	−0.0973174	−	0.995253i	$-0.531026\pi$
$353$	−3.65685	−0.194635	−0.0973174	+	0.995253i	$0.531026\pi$
$354$	0	0
$355$	−3.17157	−0.168330
$356$	0	0
$357$	13.6569	0.722797
$358$	0	0
$359$	−2.27208	−0.119916	−0.0599578	−	0.998201i	$-0.519097\pi$
$359$	−2.27208	−0.119916	−0.0599578	+	0.998201i	$0.519097\pi$
$360$	0	0
$361$	−5.62742	−0.296180
$362$	0	0
$363$	26.1421	1.37211
$364$	0	0
$365$	8.00000	0.418739
$366$	0	0
$367$	34.9706	1.82545	0.912724	−	0.408576i	$-0.133975\pi$
$367$	34.9706	1.82545	0.912724	+	0.408576i	$0.133975\pi$
$368$	0	0
$369$	2.34315	0.121979
$370$	0	0
$371$	10.9706	0.569563
$372$	0	0
$373$	−11.0000	−0.569558	−0.284779	−	0.958593i	$-0.591920\pi$
$373$	−11.0000	−0.569558	−0.284779	+	0.958593i	$0.591920\pi$
$374$	0	0
$375$	−21.7279	−1.12203
$376$	0	0
$377$	3.82843	0.197174
$378$	0	0
$379$	−28.6274	−1.47049	−0.735246	−	0.677801i	$-0.762933\pi$
$379$	−28.6274	−1.47049	−0.735246	+	0.677801i	$0.762933\pi$
$380$	0	0
$381$	47.4558	2.43124
$382$	0	0
$383$	−6.97056	−0.356179	−0.178090	−	0.984014i	$-0.556992\pi$
$383$	−6.97056	−0.356179	−0.178090	+	0.984014i	$0.556992\pi$
$384$	0	0
$385$	−0.828427	−0.0422206
$386$	0	0
$387$	−14.8284	−0.753771
$388$	0	0
$389$	−32.9706	−1.67167	−0.835837	−	0.548978i	$-0.815017\pi$
$389$	−32.9706	−1.67167	−0.835837	+	0.548978i	$0.815017\pi$
$390$	0	0
$391$	−19.3137	−0.976736
$392$	0	0
$393$	12.8284	0.647109
$394$	0	0
$395$	8.89949	0.447782
$396$	0	0
$397$	6.65685	0.334098	0.167049	−	0.985949i	$-0.446576\pi$
$397$	6.65685	0.334098	0.167049	+	0.985949i	$0.446576\pi$
$398$	0	0
$399$	−17.6569	−0.883948
$400$	0	0
$401$	19.9706	0.997282	0.498641	−	0.866809i	$-0.333833\pi$
$401$	19.9706	0.997282	0.498641	+	0.866809i	$0.333833\pi$
$402$	0	0
$403$	1.58579	0.0789936
$404$	0	0
$405$	9.48528	0.471327
$406$	0	0
$407$	2.34315	0.116145
$408$	0	0
$409$	16.6274	0.822173	0.411086	−	0.911596i	$-0.365149\pi$
$409$	16.6274	0.822173	0.411086	+	0.911596i	$0.365149\pi$
$410$	0	0
$411$	36.9706	1.82362
$412$	0	0
$413$	−24.9706	−1.22872
$414$	0	0
$415$	−4.48528	−0.220174
$416$	0	0
$417$	30.1421	1.47607
$418$	0	0
$419$	−31.1716	−1.52283	−0.761415	−	0.648264i	$-0.775495\pi$
$419$	−31.1716	−1.52283	−0.761415	+	0.648264i	$0.775495\pi$
$420$	0	0
$421$	−18.1421	−0.884194	−0.442097	−	0.896967i	$-0.645765\pi$
$421$	−18.1421	−0.884194	−0.442097	+	0.896967i	$0.645765\pi$
$422$	0	0
$423$	4.48528	0.218082
$424$	0	0
$425$	11.3137	0.548795
$426$	0	0
$427$	8.97056	0.434116
$428$	0	0
$429$	3.82843	0.184838
$430$	0	0
$431$	23.7990	1.14636	0.573179	−	0.819431i	$-0.305710\pi$
$431$	23.7990	1.14636	0.573179	+	0.819431i	$0.305710\pi$
$432$	0	0
$433$	28.0000	1.34559	0.672797	−	0.739827i	$-0.265093\pi$
$433$	28.0000	1.34559	0.672797	+	0.739827i	$0.265093\pi$
$434$	0	0
$435$	−2.41421	−0.115753
$436$	0	0
$437$	24.9706	1.19450
$438$	0	0
$439$	−2.14214	−0.102239	−0.0511193	−	0.998693i	$-0.516279\pi$
$439$	−2.14214	−0.102239	−0.0511193	+	0.998693i	$0.516279\pi$
$440$	0	0
$441$	−8.48528	−0.404061
$442$	0	0
$443$	−4.34315	−0.206349	−0.103175	−	0.994663i	$-0.532900\pi$
$443$	−4.34315	−0.206349	−0.103175	+	0.994663i	$0.532900\pi$
$444$	0	0
$445$	8.82843	0.418508
$446$	0	0
$447$	28.5563	1.35067
$448$	0	0
$449$	37.3137	1.76094	0.880471	−	0.474099i	$-0.157226\pi$
$449$	37.3137	1.76094	0.880471	+	0.474099i	$0.157226\pi$
$450$	0	0
$451$	0.343146	0.0161581
$452$	0	0
$453$	0.828427	0.0389229
$454$	0	0
$455$	7.65685	0.358959
$456$	0	0
$457$	30.2843	1.41664	0.708319	−	0.705892i	$-0.249454\pi$
$457$	30.2843	1.41664	0.708319	+	0.705892i	$0.249454\pi$
$458$	0	0
$459$	−1.17157	−0.0546843
$460$	0	0
$461$	−5.31371	−0.247484	−0.123742	−	0.992314i	$-0.539490\pi$
$461$	−5.31371	−0.247484	−0.123742	+	0.992314i	$0.539490\pi$
$462$	0	0
$463$	−15.7990	−0.734241	−0.367121	−	0.930173i	$-0.619656\pi$
$463$	−15.7990	−0.734241	−0.367121	+	0.930173i	$0.619656\pi$
$464$	0	0
$465$	−1.00000	−0.0463739
$466$	0	0
$467$	−29.2426	−1.35319	−0.676594	−	0.736356i	$-0.736545\pi$
$467$	−29.2426	−1.35319	−0.676594	+	0.736356i	$0.736545\pi$
$468$	0	0
$469$	−8.00000	−0.369406
$470$	0	0
$471$	25.3137	1.16639
$472$	0	0
$473$	−2.17157	−0.0998490
$474$	0	0
$475$	−14.6274	−0.671152
$476$	0	0
$477$	15.5147	0.710370
$478$	0	0
$479$	−16.5563	−0.756479	−0.378239	−	0.925708i	$-0.623470\pi$
$479$	−16.5563	−0.756479	−0.378239	+	0.925708i	$0.623470\pi$
$480$	0	0
$481$	−21.6569	−0.987468
$482$	0	0
$483$	−32.9706	−1.50021
$484$	0	0
$485$	8.82843	0.400878
$486$	0	0
$487$	14.6863	0.665499	0.332750	−	0.943015i	$-0.392024\pi$
$487$	14.6863	0.665499	0.332750	+	0.943015i	$0.392024\pi$
$488$	0	0
$489$	−25.1421	−1.13697
$490$	0	0
$491$	0.272078	0.0122787	0.00613935	−	0.999981i	$-0.498046\pi$
$491$	0.272078	0.0122787	0.00613935	+	0.999981i	$0.498046\pi$
$492$	0	0
$493$	2.82843	0.127386
$494$	0	0
$495$	−1.17157	−0.0526583
$496$	0	0
$497$	6.34315	0.284529
$498$	0	0
$499$	−23.1127	−1.03467	−0.517333	−	0.855784i	$-0.673075\pi$
$499$	−23.1127	−1.03467	−0.517333	+	0.855784i	$0.673075\pi$
$500$	0	0
$501$	25.3137	1.13093
$502$	0	0
$503$	30.0711	1.34080	0.670401	−	0.741999i	$-0.266122\pi$
$503$	30.0711	1.34080	0.670401	+	0.741999i	$0.266122\pi$
$504$	0	0
$505$	11.3137	0.503453
$506$	0	0
$507$	−4.00000	−0.177646
$508$	0	0
$509$	−32.4558	−1.43858	−0.719290	−	0.694710i	$-0.755533\pi$
$509$	−32.4558	−1.43858	−0.719290	+	0.694710i	$0.755533\pi$
$510$	0	0
$511$	−16.0000	−0.707798
$512$	0	0
$513$	1.51472	0.0668765
$514$	0	0
$515$	−8.14214	−0.358785
$516$	0	0
$517$	0.656854	0.0288884
$518$	0	0
$519$	−37.7990	−1.65919
$520$	0	0
$521$	−1.14214	−0.0500379	−0.0250189	−	0.999687i	$-0.507965\pi$
$521$	−1.14214	−0.0500379	−0.0250189	+	0.999687i	$0.507965\pi$
$522$	0	0
$523$	4.97056	0.217348	0.108674	−	0.994077i	$-0.465340\pi$
$523$	4.97056	0.217348	0.108674	+	0.994077i	$0.465340\pi$
$524$	0	0
$525$	19.3137	0.842919
$526$	0	0
$527$	1.17157	0.0510345
$528$	0	0
$529$	23.6274	1.02728
$530$	0	0
$531$	−35.3137	−1.53248
$532$	0	0
$533$	−3.17157	−0.137376
$534$	0	0
$535$	6.00000	0.259403
$536$	0	0
$537$	−34.9706	−1.50909
$538$	0	0
$539$	−1.24264	−0.0535243
$540$	0	0
$541$	−42.6274	−1.83270	−0.916348	−	0.400383i	$-0.868877\pi$
$541$	−42.6274	−1.83270	−0.916348	+	0.400383i	$0.868877\pi$
$542$	0	0
$543$	63.5269	2.72620
$544$	0	0
$545$	10.6569	0.456489
$546$	0	0
$547$	−21.3137	−0.911308	−0.455654	−	0.890157i	$-0.650595\pi$
$547$	−21.3137	−0.911308	−0.455654	+	0.890157i	$0.650595\pi$
$548$	0	0
$549$	12.6863	0.541438
$550$	0	0
$551$	−3.65685	−0.155787
$552$	0	0
$553$	−17.7990	−0.756890
$554$	0	0
$555$	13.6569	0.579701
$556$	0	0
$557$	22.6863	0.961249	0.480625	−	0.876926i	$-0.340410\pi$
$557$	22.6863	0.961249	0.480625	+	0.876926i	$0.340410\pi$
$558$	0	0
$559$	20.0711	0.848916
$560$	0	0
$561$	2.82843	0.119416
$562$	0	0
$563$	42.6985	1.79953	0.899763	−	0.436378i	$-0.143739\pi$
$563$	42.6985	1.79953	0.899763	+	0.436378i	$0.143739\pi$
$564$	0	0
$565$	−7.65685	−0.322126
$566$	0	0
$567$	−18.9706	−0.796689
$568$	0	0
$569$	7.65685	0.320992	0.160496	−	0.987036i	$-0.448691\pi$
$569$	7.65685	0.320992	0.160496	+	0.987036i	$0.448691\pi$
$570$	0	0
$571$	36.2843	1.51845	0.759225	−	0.650829i	$-0.225578\pi$
$571$	36.2843	1.51845	0.759225	+	0.650829i	$0.225578\pi$
$572$	0	0
$573$	−12.8284	−0.535915
$574$	0	0
$575$	−27.3137	−1.13906
$576$	0	0
$577$	−33.4558	−1.39279	−0.696393	−	0.717661i	$-0.745213\pi$
$577$	−33.4558	−1.39279	−0.696393	+	0.717661i	$0.745213\pi$
$578$	0	0
$579$	−56.6274	−2.35336
$580$	0	0
$581$	8.97056	0.372162
$582$	0	0
$583$	2.27208	0.0940999
$584$	0	0
$585$	10.8284	0.447700
$586$	0	0
$587$	−29.1716	−1.20404	−0.602020	−	0.798481i	$-0.705637\pi$
$587$	−29.1716	−1.20404	−0.602020	+	0.798481i	$0.705637\pi$
$588$	0	0
$589$	−1.51472	−0.0624129
$590$	0	0
$591$	−41.7990	−1.71938
$592$	0	0
$593$	42.4558	1.74345	0.871726	−	0.489993i	$-0.163001\pi$
$593$	42.4558	1.74345	0.871726	+	0.489993i	$0.163001\pi$
$594$	0	0
$595$	5.65685	0.231908
$596$	0	0
$597$	37.7990	1.54701
$598$	0	0
$599$	19.1005	0.780425	0.390213	−	0.920725i	$-0.372401\pi$
$599$	19.1005	0.780425	0.390213	+	0.920725i	$0.372401\pi$
$600$	0	0
$601$	−15.8579	−0.646856	−0.323428	−	0.946253i	$-0.604835\pi$
$601$	−15.8579	−0.646856	−0.323428	+	0.946253i	$0.604835\pi$
$602$	0	0
$603$	−11.3137	−0.460730
$604$	0	0
$605$	10.8284	0.440238
$606$	0	0
$607$	40.0122	1.62404	0.812022	−	0.583626i	$-0.198367\pi$
$607$	40.0122	1.62404	0.812022	+	0.583626i	$0.198367\pi$
$608$	0	0
$609$	4.82843	0.195658
$610$	0	0
$611$	−6.07107	−0.245609
$612$	0	0
$613$	−15.6274	−0.631185	−0.315593	−	0.948895i	$-0.602203\pi$
$613$	−15.6274	−0.631185	−0.315593	+	0.948895i	$0.602203\pi$
$614$	0	0
$615$	2.00000	0.0806478
$616$	0	0
$617$	8.68629	0.349697	0.174848	−	0.984595i	$-0.444056\pi$
$617$	8.68629	0.349697	0.174848	+	0.984595i	$0.444056\pi$
$618$	0	0
$619$	9.72792	0.390998	0.195499	−	0.980704i	$-0.437367\pi$
$619$	9.72792	0.390998	0.195499	+	0.980704i	$0.437367\pi$
$620$	0	0
$621$	2.82843	0.113501
$622$	0	0
$623$	−17.6569	−0.707407
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	−3.65685	−0.146041
$628$	0	0
$629$	−16.0000	−0.637962
$630$	0	0
$631$	−39.1716	−1.55940	−0.779698	−	0.626156i	$-0.784627\pi$
$631$	−39.1716	−1.55940	−0.779698	+	0.626156i	$0.784627\pi$
$632$	0	0
$633$	−54.1127	−2.15079
$634$	0	0
$635$	19.6569	0.780058
$636$	0	0
$637$	11.4853	0.455063
$638$	0	0
$639$	8.97056	0.354870
$640$	0	0
$641$	−40.7696	−1.61030	−0.805150	−	0.593071i	$-0.797915\pi$
$641$	−40.7696	−1.61030	−0.805150	+	0.593071i	$0.797915\pi$
$642$	0	0
$643$	8.34315	0.329022	0.164511	−	0.986375i	$-0.447395\pi$
$643$	8.34315	0.329022	0.164511	+	0.986375i	$0.447395\pi$
$644$	0	0
$645$	−12.6569	−0.498363
$646$	0	0
$647$	−46.4264	−1.82521	−0.912605	−	0.408842i	$-0.865933\pi$
$647$	−46.4264	−1.82521	−0.912605	+	0.408842i	$0.865933\pi$
$648$	0	0
$649$	−5.17157	−0.203002
$650$	0	0
$651$	2.00000	0.0783862
$652$	0	0
$653$	−9.51472	−0.372340	−0.186170	−	0.982518i	$-0.559607\pi$
$653$	−9.51472	−0.372340	−0.186170	+	0.982518i	$0.559607\pi$
$654$	0	0
$655$	5.31371	0.207624
$656$	0	0
$657$	−22.6274	−0.882780
$658$	0	0
$659$	−20.7574	−0.808592	−0.404296	−	0.914628i	$-0.632483\pi$
$659$	−20.7574	−0.808592	−0.404296	+	0.914628i	$0.632483\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	0	0
$663$	−26.1421	−1.01528
$664$	0	0
$665$	−7.31371	−0.283613
$666$	0	0
$667$	−6.82843	−0.264398
$668$	0	0
$669$	2.00000	0.0773245
$670$	0	0
$671$	1.85786	0.0717221
$672$	0	0
$673$	−26.3137	−1.01432	−0.507159	−	0.861852i	$-0.669304\pi$
$673$	−26.3137	−1.01432	−0.507159	+	0.861852i	$0.669304\pi$
$674$	0	0
$675$	−1.65685	−0.0637723
$676$	0	0
$677$	−6.00000	−0.230599	−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000	−0.230599	−0.115299	+	0.993331i	$0.536783\pi$
$678$	0	0
$679$	−17.6569	−0.677608
$680$	0	0
$681$	62.2843	2.38674
$682$	0	0
$683$	24.9706	0.955472	0.477736	−	0.878503i	$-0.341458\pi$
$683$	24.9706	0.955472	0.477736	+	0.878503i	$0.341458\pi$
$684$	0	0
$685$	15.3137	0.585107
$686$	0	0
$687$	−15.6569	−0.597346
$688$	0	0
$689$	−21.0000	−0.800036
$690$	0	0
$691$	45.6569	1.73687	0.868434	−	0.495804i	$-0.165127\pi$
$691$	45.6569	1.73687	0.868434	+	0.495804i	$0.165127\pi$
$692$	0	0
$693$	2.34315	0.0890087
$694$	0	0
$695$	12.4853	0.473594
$696$	0	0
$697$	−2.34315	−0.0887530
$698$	0	0
$699$	−44.2132	−1.67230
$700$	0	0
$701$	39.7696	1.50208	0.751038	−	0.660259i	$-0.229554\pi$
$701$	39.7696	1.50208	0.751038	+	0.660259i	$0.229554\pi$
$702$	0	0
$703$	20.6863	0.780198
$704$	0	0
$705$	3.82843	0.144187
$706$	0	0
$707$	−22.6274	−0.850992
$708$	0	0
$709$	9.82843	0.369114	0.184557	−	0.982822i	$-0.440915\pi$
$709$	9.82843	0.369114	0.184557	+	0.982822i	$0.440915\pi$
$710$	0	0
$711$	−25.1716	−0.944008
$712$	0	0
$713$	−2.82843	−0.105925
$714$	0	0
$715$	1.58579	0.0593051
$716$	0	0
$717$	−31.7990	−1.18756
$718$	0	0
$719$	4.14214	0.154476	0.0772378	−	0.997013i	$-0.475390\pi$
$719$	4.14214	0.154476	0.0772378	+	0.997013i	$0.475390\pi$
$720$	0	0
$721$	16.2843	0.606458
$722$	0	0
$723$	52.2132	1.94183
$724$	0	0
$725$	4.00000	0.148556
$726$	0	0
$727$	21.3137	0.790482	0.395241	−	0.918578i	$-0.370661\pi$
$727$	21.3137	0.790482	0.395241	+	0.918578i	$0.370661\pi$
$728$	0	0
$729$	−23.8284	−0.882534
$730$	0	0
$731$	14.8284	0.548449
$732$	0	0
$733$	0.970563	0.0358486	0.0179243	−	0.999839i	$-0.494294\pi$
$733$	0.970563	0.0358486	0.0179243	+	0.999839i	$0.494294\pi$
$734$	0	0
$735$	−7.24264	−0.267149
$736$	0	0
$737$	−1.65685	−0.0610310
$738$	0	0
$739$	−20.0711	−0.738326	−0.369163	−	0.929365i	$-0.620356\pi$
$739$	−20.0711	−0.738326	−0.369163	+	0.929365i	$0.620356\pi$
$740$	0	0
$741$	33.7990	1.24164
$742$	0	0
$743$	39.2548	1.44012	0.720060	−	0.693912i	$-0.244114\pi$
$743$	39.2548	1.44012	0.720060	+	0.693912i	$0.244114\pi$
$744$	0	0
$745$	11.8284	0.433360
$746$	0	0
$747$	12.6863	0.464167
$748$	0	0
$749$	−12.0000	−0.438470
$750$	0	0
$751$	−31.6569	−1.15518	−0.577588	−	0.816329i	$-0.696006\pi$
$751$	−31.6569	−1.15518	−0.577588	+	0.816329i	$0.696006\pi$
$752$	0	0
$753$	1.82843	0.0666316
$754$	0	0
$755$	0.343146	0.0124884
$756$	0	0
$757$	−7.79899	−0.283459	−0.141730	−	0.989905i	$-0.545266\pi$
$757$	−7.79899	−0.283459	−0.141730	+	0.989905i	$0.545266\pi$
$758$	0	0
$759$	−6.82843	−0.247856
$760$	0	0
$761$	−9.02944	−0.327317	−0.163658	−	0.986517i	$-0.552329\pi$
$761$	−9.02944	−0.327317	−0.163658	+	0.986517i	$0.552329\pi$
$762$	0	0
$763$	−21.3137	−0.771608
$764$	0	0
$765$	8.00000	0.289241
$766$	0	0
$767$	47.7990	1.72592
$768$	0	0
$769$	5.45584	0.196743	0.0983714	−	0.995150i	$-0.468637\pi$
$769$	5.45584	0.196743	0.0983714	+	0.995150i	$0.468637\pi$
$770$	0	0
$771$	59.8701	2.15617
$772$	0	0
$773$	−39.1716	−1.40890	−0.704452	−	0.709752i	$-0.748807\pi$
$773$	−39.1716	−1.40890	−0.704452	+	0.709752i	$0.748807\pi$
$774$	0	0
$775$	1.65685	0.0595160
$776$	0	0
$777$	−27.3137	−0.979874
$778$	0	0
$779$	3.02944	0.108541
$780$	0	0
$781$	1.31371	0.0470082
$782$	0	0
$783$	−0.414214	−0.0148028
$784$	0	0
$785$	10.4853	0.374236
$786$	0	0
$787$	−2.20101	−0.0784575	−0.0392288	−	0.999230i	$-0.512490\pi$
$787$	−2.20101	−0.0784575	−0.0392288	+	0.999230i	$0.512490\pi$
$788$	0	0
$789$	−0.656854	−0.0233846
$790$	0	0
$791$	15.3137	0.544493
$792$	0	0
$793$	−17.1716	−0.609780
$794$	0	0
$795$	13.2426	0.469668
$796$	0	0
$797$	−10.1421	−0.359253	−0.179626	−	0.983735i	$-0.557489\pi$
$797$	−10.1421	−0.359253	−0.179626	+	0.983735i	$0.557489\pi$
$798$	0	0
$799$	−4.48528	−0.158678
$800$	0	0
$801$	−24.9706	−0.882291
$802$	0	0
$803$	−3.31371	−0.116938
$804$	0	0
$805$	−13.6569	−0.481341
$806$	0	0
$807$	−68.7696	−2.42080
$808$	0	0
$809$	−3.31371	−0.116504	−0.0582519	−	0.998302i	$-0.518553\pi$
$809$	−3.31371	−0.116504	−0.0582519	+	0.998302i	$0.518553\pi$
$810$	0	0
$811$	−35.6569	−1.25208	−0.626041	−	0.779790i	$-0.715326\pi$
$811$	−35.6569	−1.25208	−0.626041	+	0.779790i	$0.715326\pi$
$812$	0	0
$813$	12.6569	0.443895
$814$	0	0
$815$	−10.4142	−0.364794
$816$	0	0
$817$	−19.1716	−0.670728
$818$	0	0
$819$	−21.6569	−0.756752
$820$	0	0
$821$	−4.79899	−0.167486	−0.0837430	−	0.996487i	$-0.526687\pi$
$821$	−4.79899	−0.167486	−0.0837430	+	0.996487i	$0.526687\pi$
$822$	0	0
$823$	−24.3431	−0.848549	−0.424274	−	0.905534i	$-0.639471\pi$
$823$	−24.3431	−0.848549	−0.424274	+	0.905534i	$0.639471\pi$
$824$	0	0
$825$	4.00000	0.139262
$826$	0	0
$827$	16.7574	0.582710	0.291355	−	0.956615i	$-0.405894\pi$
$827$	16.7574	0.582710	0.291355	+	0.956615i	$0.405894\pi$
$828$	0	0
$829$	−11.5147	−0.399923	−0.199961	−	0.979804i	$-0.564082\pi$
$829$	−11.5147	−0.399923	−0.199961	+	0.979804i	$0.564082\pi$
$830$	0	0
$831$	−14.4853	−0.502489
$832$	0	0
$833$	8.48528	0.293998
$834$	0	0
$835$	10.4853	0.362858
$836$	0	0
$837$	−0.171573	−0.00593043
$838$	0	0
$839$	8.21320	0.283551	0.141776	−	0.989899i	$-0.454719\pi$
$839$	8.21320	0.283551	0.141776	+	0.989899i	$0.454719\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−28.8995	−0.995351
$844$	0	0
$845$	−1.65685	−0.0569975
$846$	0	0
$847$	−21.6569	−0.744138
$848$	0	0
$849$	−8.48528	−0.291214
$850$	0	0
$851$	38.6274	1.32413
$852$	0	0
$853$	−22.0000	−0.753266	−0.376633	−	0.926363i	$-0.622918\pi$
$853$	−22.0000	−0.753266	−0.376633	+	0.926363i	$0.622918\pi$
$854$	0	0
$855$	−10.3431	−0.353728
$856$	0	0
$857$	28.4558	0.972033	0.486017	−	0.873950i	$-0.338449\pi$
$857$	28.4558	0.972033	0.486017	+	0.873950i	$0.338449\pi$
$858$	0	0
$859$	52.5563	1.79320	0.896600	−	0.442842i	$-0.146030\pi$
$859$	52.5563	1.79320	0.896600	+	0.442842i	$0.146030\pi$
$860$	0	0
$861$	−4.00000	−0.136320
$862$	0	0
$863$	−17.5147	−0.596208	−0.298104	−	0.954533i	$-0.596354\pi$
$863$	−17.5147	−0.596208	−0.298104	+	0.954533i	$0.596354\pi$
$864$	0	0
$865$	−15.6569	−0.532349
$866$	0	0
$867$	21.7279	0.737919
$868$	0	0
$869$	−3.68629	−0.125049
$870$	0	0
$871$	15.3137	0.518885
$872$	0	0
$873$	−24.9706	−0.845126
$874$	0	0
$875$	18.0000	0.608511
$876$	0	0
$877$	−18.5147	−0.625198	−0.312599	−	0.949885i	$-0.601200\pi$
$877$	−18.5147	−0.625198	−0.312599	+	0.949885i	$0.601200\pi$
$878$	0	0
$879$	32.1421	1.08413
$880$	0	0
$881$	26.2843	0.885540	0.442770	−	0.896635i	$-0.353996\pi$
$881$	26.2843	0.885540	0.442770	+	0.896635i	$0.353996\pi$
$882$	0	0
$883$	32.6274	1.09800	0.549000	−	0.835822i	$-0.315009\pi$
$883$	32.6274	1.09800	0.549000	+	0.835822i	$0.315009\pi$
$884$	0	0
$885$	−30.1421	−1.01322
$886$	0	0
$887$	−11.0416	−0.370742	−0.185371	−	0.982669i	$-0.559349\pi$
$887$	−11.0416	−0.370742	−0.185371	+	0.982669i	$0.559349\pi$
$888$	0	0
$889$	−39.3137	−1.31854
$890$	0	0
$891$	−3.92893	−0.131624
$892$	0	0
$893$	5.79899	0.194056
$894$	0	0
$895$	−14.4853	−0.484190
$896$	0	0
$897$	63.1127	2.10727
$898$	0	0
$899$	0.414214	0.0138148
$900$	0	0
$901$	−15.5147	−0.516870
$902$	0	0
$903$	25.3137	0.842387
$904$	0	0
$905$	26.3137	0.874697
$906$	0	0
$907$	10.2843	0.341484	0.170742	−	0.985316i	$-0.445384\pi$
$907$	10.2843	0.341484	0.170742	+	0.985316i	$0.445384\pi$
$908$	0	0
$909$	−32.0000	−1.06137
$910$	0	0
$911$	−17.9289	−0.594012	−0.297006	−	0.954876i	$-0.595988\pi$
$911$	−17.9289	−0.594012	−0.297006	+	0.954876i	$0.595988\pi$
$912$	0	0
$913$	1.85786	0.0614863
$914$	0	0
$915$	10.8284	0.357977
$916$	0	0
$917$	−10.6274	−0.350948
$918$	0	0
$919$	53.7990	1.77466	0.887332	−	0.461130i	$-0.152556\pi$
$919$	53.7990	1.77466	0.887332	+	0.461130i	$0.152556\pi$
$920$	0	0
$921$	−43.6274	−1.43757
$922$	0	0
$923$	−12.1421	−0.399663
$924$	0	0
$925$	−22.6274	−0.743985
$926$	0	0
$927$	23.0294	0.756386
$928$	0	0
$929$	−34.0000	−1.11550	−0.557752	−	0.830008i	$-0.688336\pi$
$929$	−34.0000	−1.11550	−0.557752	+	0.830008i	$0.688336\pi$
$930$	0	0
$931$	−10.9706	−0.359546
$932$	0	0
$933$	−70.7696	−2.31689
$934$	0	0
$935$	1.17157	0.0383145
$936$	0	0
$937$	53.3137	1.74168	0.870841	−	0.491564i	$-0.163575\pi$
$937$	53.3137	1.74168	0.870841	+	0.491564i	$0.163575\pi$
$938$	0	0
$939$	80.8406	2.63813
$940$	0	0
$941$	13.3431	0.434974	0.217487	−	0.976063i	$-0.430214\pi$
$941$	13.3431	0.434974	0.217487	+	0.976063i	$0.430214\pi$
$942$	0	0
$943$	5.65685	0.184213
$944$	0	0
$945$	−0.828427	−0.0269487
$946$	0	0
$947$	10.0711	0.327266	0.163633	−	0.986521i	$-0.447679\pi$
$947$	10.0711	0.327266	0.163633	+	0.986521i	$0.447679\pi$
$948$	0	0
$949$	30.6274	0.994208
$950$	0	0
$951$	−41.4558	−1.34430
$952$	0	0
$953$	−5.68629	−0.184197	−0.0920985	−	0.995750i	$-0.529357\pi$
$953$	−5.68629	−0.184197	−0.0920985	+	0.995750i	$0.529357\pi$
$954$	0	0
$955$	−5.31371	−0.171948
$956$	0	0
$957$	1.00000	0.0323254
$958$	0	0
$959$	−30.6274	−0.989011
$960$	0	0
$961$	−30.8284	−0.994465
$962$	0	0
$963$	−16.9706	−0.546869
$964$	0	0
$965$	−23.4558	−0.755070
$966$	0	0
$967$	35.5269	1.14247	0.571234	−	0.820787i	$-0.306465\pi$
$967$	35.5269	1.14247	0.571234	+	0.820787i	$0.306465\pi$
$968$	0	0
$969$	24.9706	0.802170
$970$	0	0
$971$	9.02944	0.289768	0.144884	−	0.989449i	$-0.453719\pi$
$971$	9.02944	0.289768	0.144884	+	0.989449i	$0.453719\pi$
$972$	0	0
$973$	−24.9706	−0.800519
$974$	0	0
$975$	−36.9706	−1.18401
$976$	0	0
$977$	−14.4558	−0.462483	−0.231242	−	0.972896i	$-0.574279\pi$
$977$	−14.4558	−0.462483	−0.231242	+	0.972896i	$0.574279\pi$
$978$	0	0
$979$	−3.65685	−0.116874
$980$	0	0
$981$	−30.1421	−0.962364
$982$	0	0
$983$	4.21320	0.134380	0.0671902	−	0.997740i	$-0.478597\pi$
$983$	4.21320	0.134380	0.0671902	+	0.997740i	$0.478597\pi$
$984$	0	0
$985$	−17.3137	−0.551661
$986$	0	0
$987$	−7.65685	−0.243720
$988$	0	0
$989$	−35.7990	−1.13834
$990$	0	0
$991$	54.7696	1.73981	0.869906	−	0.493217i	$-0.164179\pi$
$991$	54.7696	1.73981	0.869906	+	0.493217i	$0.164179\pi$
$992$	0	0
$993$	−44.4558	−1.41076
$994$	0	0
$995$	15.6569	0.496356
$996$	0	0
$997$	23.3137	0.738353	0.369176	−	0.929359i	$-0.379640\pi$
$997$	23.3137	0.738353	0.369176	+	0.929359i	$0.379640\pi$
$998$	0	0
$999$	2.34315	0.0741339

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1856.2.a.q.1.1		2
4.3	odd	2		1856.2.a.u.1.2		2
8.3	odd	2		928.2.a.c.1.1	✓	2
8.5	even	2		928.2.a.e.1.2	yes	2
24.5	odd	2		8352.2.a.o.1.2		2
24.11	even	2		8352.2.a.l.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
928.2.a.c.1.1	✓	2	8.3	odd	2
928.2.a.e.1.2	yes	2	8.5	even	2
1856.2.a.q.1.1		2	1.1	even	1	trivial
1856.2.a.u.1.2		2	4.3	odd	2
8352.2.a.l.1.1		2	24.11	even	2
8352.2.a.o.1.2		2	24.5	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-2.41421 q^{3} -1.00000 q^{5} +2.00000 q^{7} +2.82843 q^{9} +O(q^{10})\)	\(q-2.41421 q^{3} -1.00000 q^{5} +2.00000 q^{7} +2.82843 q^{9} +0.414214 q^{11} -3.82843 q^{13} +2.41421 q^{15} -2.82843 q^{17} +3.65685 q^{19} -4.82843 q^{21} +6.82843 q^{23} -4.00000 q^{25} +0.414214 q^{27} -1.00000 q^{29} -0.414214 q^{31} -1.00000 q^{33} -2.00000 q^{35} +5.65685 q^{37} +9.24264 q^{39} +0.828427 q^{41} -5.24264 q^{43} -2.82843 q^{45} +1.58579 q^{47} -3.00000 q^{49} +6.82843 q^{51} +5.48528 q^{53} -0.414214 q^{55} -8.82843 q^{57} -12.4853 q^{59} +4.48528 q^{61} +5.65685 q^{63} +3.82843 q^{65} -4.00000 q^{67} -16.4853 q^{69} +3.17157 q^{71} -8.00000 q^{73} +9.65685 q^{75} +0.828427 q^{77} -8.89949 q^{79} -9.48528 q^{81} +4.48528 q^{83} +2.82843 q^{85} +2.41421 q^{87} -8.82843 q^{89} -7.65685 q^{91} +1.00000 q^{93} -3.65685 q^{95} -8.82843 q^{97} +1.17157 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{5} + 4 q^{7}+O(q^{10}) \) 2 * q - 2 * q^3 - 2 * q^5 + 4 * q^7	\( 2 q - 2 q^{3} - 2 q^{5} + 4 q^{7} - 2 q^{11} - 2 q^{13} + 2 q^{15} - 4 q^{19} - 4 q^{21} + 8 q^{23} - 8 q^{25} - 2 q^{27} - 2 q^{29} + 2 q^{31} - 2 q^{33} - 4 q^{35} + 10 q^{39} - 4 q^{41} - 2 q^{43} + 6 q^{47} - 6 q^{49} + 8 q^{51} - 6 q^{53} + 2 q^{55} - 12 q^{57} - 8 q^{59} - 8 q^{61} + 2 q^{65} - 8 q^{67} - 16 q^{69} + 12 q^{71} - 16 q^{73} + 8 q^{75} - 4 q^{77} + 2 q^{79} - 2 q^{81} - 8 q^{83} + 2 q^{87} - 12 q^{89} - 4 q^{91} + 2 q^{93} + 4 q^{95} - 12 q^{97} + 8 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^5 + 4 * q^7 - 2 * q^11 - 2 * q^13 + 2 * q^15 - 4 * q^19 - 4 * q^21 + 8 * q^23 - 8 * q^25 - 2 * q^27 - 2 * q^29 + 2 * q^31 - 2 * q^33 - 4 * q^35 + 10 * q^39 - 4 * q^41 - 2 * q^43 + 6 * q^47 - 6 * q^49 + 8 * q^51 - 6 * q^53 + 2 * q^55 - 12 * q^57 - 8 * q^59 - 8 * q^61 + 2 * q^65 - 8 * q^67 - 16 * q^69 + 12 * q^71 - 16 * q^73 + 8 * q^75 - 4 * q^77 + 2 * q^79 - 2 * q^81 - 8 * q^83 + 2 * q^87 - 12 * q^89 - 4 * q^91 + 2 * q^93 + 4 * q^95 - 12 * q^97 + 8 * q^99

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