LMFDB - Embedded newform 1568.2.b.g.785.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1568,2,Mod(785,1568)]

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, 0]))

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

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

chi := DirichletCharacter("1568.785");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1568 = 2^{5} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1568.b (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:	$12.5205430369$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 8x^{10} + 27x^{8} + 14x^{6} + 25x^{4} - 42x^{2} + 16 $ x^12 + 8x^10 + 27x^8 + 14x^6 + 25x^4 - 42*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 392)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			785.12
Root			$-0.707107 - 0.164821i$ of defining polynomial
Character	$\chi$	$=$	1568.785
Dual form			1568.2.b.g.785.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.86877i q^{3} -3.19841i q^{5} -5.22982 q^{9} +O(q^{10})$	$q+2.86877i q^{3} -3.19841i q^{5} -5.22982 q^{9} +1.45212i q^{11} +1.14479i q^{13} +9.17548 q^{15} +5.58002 q^{17} +1.47444i q^{19} -3.28415 q^{23} -5.22982 q^{25} -6.39682i q^{27} +3.59086i q^{29} +1.01237 q^{31} -4.16580 q^{33} +6.49511i q^{37} -3.28415 q^{39} +7.39608 q^{41} +7.59434i q^{43} +16.7271i q^{45} +11.9637 q^{47} +16.0078i q^{51} +11.0183i q^{53} +4.64449 q^{55} -4.22982 q^{57} +6.31671i q^{59} -3.19841i q^{61} +3.66152 q^{65} +2.60492i q^{67} -9.42145i q^{69} -12.4596 q^{71} -9.21213 q^{73} -15.0031i q^{75} +13.8913 q^{79} +2.66152 q^{81} +7.87125i q^{83} -17.8472i q^{85} -10.3013 q^{87} +3.23027 q^{89} +2.90425i q^{93} +4.71585 q^{95} +0.401845 q^{97} -7.59434i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{9}+O(q^{10}) $ 12 * q - 12 * q^9	$ 12 q - 12 q^{9} + 16 q^{15} - 32 q^{23} - 12 q^{25} - 32 q^{39} + 8 q^{65} - 48 q^{71} + 80 q^{79} - 4 q^{81} + 64 q^{95}+O(q^{100}) $ 12 * q - 12 * q^9 + 16 * q^15 - 32 * q^23 - 12 * q^25 - 32 * q^39 + 8 * q^65 - 48 * q^71 + 80 * q^79 - 4 * q^81 + 64 * 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
$3$	2.86877i	1.65628i	0.560519	+	0.828141i	$0.310601\pi$
$3$	2.86877i	1.65628i	−0.560519	+	0.828141i	$0.689399\pi$
$4$	0	0
$5$	− 3.19841i	− 1.43037i	−0.698934	−	0.715186i	$-0.746342\pi$
$5$	− 3.19841i	− 1.43037i	0.698934	−	0.715186i	$-0.253658\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−5.22982	−1.74327
$10$	0	0
$11$	1.45212i	0.437832i	0.975744	+	0.218916i	$0.0702521\pi$
$11$	1.45212i	0.437832i	−0.975744	+	0.218916i	$0.929748\pi$
$12$	0	0
$13$	1.14479i	0.317509i	0.987318	+	0.158754i	$0.0507478\pi$
$13$	1.14479i	0.317509i	−0.987318	+	0.158754i	$0.949252\pi$
$14$	0	0
$15$	9.17548	2.36910
$16$	0	0
$17$	5.58002	1.35335	0.676676	−	0.736281i	$-0.263420\pi$
$17$	5.58002	1.35335	0.676676	+	0.736281i	$0.263420\pi$
$18$	0	0
$19$	1.47444i	0.338259i	0.985594	+	0.169130i	$0.0540956\pi$
$19$	1.47444i	0.338259i	−0.985594	+	0.169130i	$0.945904\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3.28415	−0.684792	−0.342396	−	0.939556i	$-0.611238\pi$
$23$	−3.28415	−0.684792	−0.342396	+	0.939556i	$0.611238\pi$
$24$	0	0
$25$	−5.22982	−1.04596
$26$	0	0
$27$	− 6.39682i	− 1.23107i
$28$	0	0
$29$	3.59086i	0.666806i	0.942784	+	0.333403i	$0.108197\pi$
$29$	3.59086i	0.666806i	−0.942784	+	0.333403i	$0.891803\pi$
$30$	0	0
$31$	1.01237	0.181827	0.0909134	−	0.995859i	$-0.471021\pi$
$31$	1.01237	0.181827	0.0909134	+	0.995859i	$0.471021\pi$
$32$	0	0
$33$	−4.16580	−0.725173
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6.49511i	1.06779i	0.845551	+	0.533895i	$0.179272\pi$
$37$	6.49511i	1.06779i	−0.845551	+	0.533895i	$0.820728\pi$
$38$	0	0
$39$	−3.28415	−0.525884
$40$	0	0
$41$	7.39608	1.15507	0.577536	−	0.816365i	$-0.304014\pi$
$41$	7.39608	1.15507	0.577536	+	0.816365i	$0.304014\pi$
$42$	0	0
$43$	7.59434i	1.15813i	0.815283	+	0.579063i	$0.196581\pi$
$43$	7.59434i	1.15813i	−0.815283	+	0.579063i	$0.803419\pi$
$44$	0	0
$45$	16.7271i	2.49353i
$46$	0	0
$47$	11.9637	1.74509	0.872544	−	0.488535i	$-0.162469\pi$
$47$	11.9637	1.74509	0.872544	+	0.488535i	$0.162469\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	16.0078i	2.24153i
$52$	0	0
$53$	11.0183i	1.51349i	0.653713	+	0.756743i	$0.273210\pi$
$53$	11.0183i	1.51349i	−0.653713	+	0.756743i	$0.726790\pi$
$54$	0	0
$55$	4.64449	0.626262
$56$	0	0
$57$	−4.22982	−0.560253
$58$	0	0
$59$	6.31671i	0.822365i	0.911553	+	0.411183i	$0.134884\pi$
$59$	6.31671i	0.822365i	−0.911553	+	0.411183i	$0.865116\pi$
$60$	0	0
$61$	− 3.19841i	− 0.409514i	−0.978813	−	0.204757i	$-0.934360\pi$
$61$	− 3.19841i	− 0.409514i	0.978813	−	0.204757i	$-0.0656404\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	3.66152	0.454156
$66$	0	0
$67$	2.60492i	0.318242i	0.987259	+	0.159121i	$0.0508660\pi$
$67$	2.60492i	0.318242i	−0.987259	+	0.159121i	$0.949134\pi$
$68$	0	0
$69$	− 9.42145i	− 1.13421i
$70$	0	0
$71$	−12.4596	−1.47869	−0.739343	−	0.673329i	$-0.764864\pi$
$71$	−12.4596	−1.47869	−0.739343	+	0.673329i	$0.764864\pi$
$72$	0	0
$73$	−9.21213	−1.07820	−0.539099	−	0.842242i	$-0.681235\pi$
$73$	−9.21213	−1.07820	−0.539099	+	0.842242i	$0.681235\pi$
$74$	0	0
$75$	− 15.0031i	− 1.73241i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	13.8913	1.56290	0.781449	−	0.623970i	$-0.214481\pi$
$79$	13.8913	1.56290	0.781449	+	0.623970i	$0.214481\pi$
$80$	0	0
$81$	2.66152	0.295725
$82$	0	0
$83$	7.87125i	0.863982i	0.901878	+	0.431991i	$0.142189\pi$
$83$	7.87125i	0.863982i	−0.901878	+	0.431991i	$0.857811\pi$
$84$	0	0
$85$	− 17.8472i	− 1.93580i
$86$	0	0
$87$	−10.3013	−1.10442
$88$	0	0
$89$	3.23027	0.342408	0.171204	−	0.985236i	$-0.445234\pi$
$89$	3.23027	0.342408	0.171204	+	0.985236i	$0.445234\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	2.90425i	0.301157i
$94$	0	0
$95$	4.71585	0.483836
$96$	0	0
$97$	0.401845	0.0408012	0.0204006	−	0.999792i	$-0.493506\pi$
$97$	0.401845	0.0408012	0.0204006	+	0.999792i	$0.493506\pi$
$98$	0	0
$99$	− 7.59434i	− 0.763260i
$100$	0	0
$101$	− 6.22304i	− 0.619216i	−0.950864	−	0.309608i	$-0.899802\pi$
$101$	− 6.22304i	− 0.619216i	0.950864	−	0.309608i	$-0.100198\pi$
$102$	0	0
$103$	−7.31924	−0.721186	−0.360593	−	0.932723i	$-0.617426\pi$
$103$	−7.31924	−0.721186	−0.360593	+	0.932723i	$0.617426\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 4.46966i	− 0.432099i	−0.976382	−	0.216049i	$-0.930683\pi$
$107$	− 4.46966i	− 0.432099i	0.976382	−	0.216049i	$-0.0693172\pi$
$108$	0	0
$109$	− 12.6373i	− 1.21044i	−0.796060	−	0.605218i	$-0.793086\pi$
$109$	− 12.6373i	− 1.21044i	0.796060	−	0.605218i	$-0.206914\pi$
$110$	0	0
$111$	−18.6329	−1.76856
$112$	0	0
$113$	−10.2298	−0.962340	−0.481170	−	0.876627i	$-0.659788\pi$
$113$	−10.2298	−0.962340	−0.481170	+	0.876627i	$0.659788\pi$
$114$	0	0
$115$	10.5040i	0.979507i
$116$	0	0
$117$	− 5.98706i	− 0.553504i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	8.89134	0.808303
$122$	0	0
$123$	21.2176i	1.91313i
$124$	0	0
$125$	0.735042i	0.0657442i
$126$	0	0
$127$	−15.0668	−1.33696	−0.668482	−	0.743728i	$-0.733056\pi$
$127$	−15.0668	−1.33696	−0.668482	+	0.743728i	$0.733056\pi$
$128$	0	0
$129$	−21.7864	−1.91818
$130$	0	0
$131$	4.92238i	0.430070i	0.976606	+	0.215035i	$0.0689866\pi$
$131$	4.92238i	0.430070i	−0.976606	+	0.215035i	$0.931013\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−20.4596	−1.76088
$136$	0	0
$137$	18.3510	1.56783	0.783914	−	0.620869i	$-0.213220\pi$
$137$	18.3510	1.56783	0.783914	+	0.620869i	$0.213220\pi$
$138$	0	0
$139$	− 17.0567i	− 1.44673i	−0.690464	−	0.723366i	$-0.742594\pi$
$139$	− 17.0567i	− 1.44673i	0.690464	−	0.723366i	$-0.257406\pi$
$140$	0	0
$141$	34.3211i	2.89036i
$142$	0	0
$143$	−1.66238	−0.139016
$144$	0	0
$145$	11.4850	0.953781
$146$	0	0
$147$	0	0
$148$	0	0
$149$	20.3985i	1.67111i	0.549405	+	0.835556i	$0.314854\pi$
$149$	20.3985i	1.67111i	−0.549405	+	0.835556i	$0.685146\pi$
$150$	0	0
$151$	−4.71585	−0.383771	−0.191885	−	0.981417i	$-0.561460\pi$
$151$	−4.71585	−0.383771	−0.191885	+	0.981417i	$0.561460\pi$
$152$	0	0
$153$	−29.1825	−2.35926
$154$	0	0
$155$	− 3.23797i	− 0.260080i
$156$	0	0
$157$	9.01170i	0.719212i	0.933104	+	0.359606i	$0.117089\pi$
$157$	9.01170i	0.719212i	−0.933104	+	0.359606i	$0.882911\pi$
$158$	0	0
$159$	−31.6090	−2.50676
$160$	0	0
$161$	0	0
$162$	0	0
$163$	4.69009i	0.367356i	0.982986	+	0.183678i	$0.0588004\pi$
$163$	4.69009i	0.367356i	−0.982986	+	0.183678i	$0.941200\pi$
$164$	0	0
$165$	13.3239i	1.03727i
$166$	0	0
$167$	13.9885	1.08246	0.541230	−	0.840875i	$-0.317959\pi$
$167$	13.9885	1.08246	0.541230	+	0.840875i	$0.317959\pi$
$168$	0	0
$169$	11.6894	0.899188
$170$	0	0
$171$	− 7.71104i	− 0.589678i
$172$	0	0
$173$	7.54161i	0.573378i	0.958024	+	0.286689i	$0.0925546\pi$
$173$	7.54161i	0.573378i	−0.958024	+	0.286689i	$0.907445\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−18.1212	−1.36207
$178$	0	0
$179$	− 14.8894i	− 1.11288i	−0.830887	−	0.556441i	$-0.812166\pi$
$179$	− 14.8894i	− 1.11288i	0.830887	−	0.556441i	$-0.187834\pi$
$180$	0	0
$181$	− 16.4911i	− 1.22577i	−0.790170	−	0.612887i	$-0.790008\pi$
$181$	− 16.4911i	− 1.22577i	0.790170	−	0.612887i	$-0.209992\pi$
$182$	0	0
$183$	9.17548	0.678271
$184$	0	0
$185$	20.7740	1.52734
$186$	0	0
$187$	8.10288i	0.592541i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−2.35097	−0.170110	−0.0850550	−	0.996376i	$-0.527107\pi$
$191$	−2.35097	−0.170110	−0.0850550	+	0.996376i	$0.527107\pi$
$192$	0	0
$193$	−9.55286	−0.687630	−0.343815	−	0.939037i	$-0.611719\pi$
$193$	−9.55286	−0.687630	−0.343815	+	0.939037i	$0.611719\pi$
$194$	0	0
$195$	10.5040i	0.752210i
$196$	0	0
$197$	− 12.0578i	− 0.859086i	−0.903046	−	0.429543i	$-0.858675\pi$
$197$	− 12.0578i	− 0.859086i	0.903046	−	0.429543i	$-0.141325\pi$
$198$	0	0
$199$	−3.99447	−0.283160	−0.141580	−	0.989927i	$-0.545218\pi$
$199$	−3.99447	−0.283160	−0.141580	+	0.989927i	$0.545218\pi$
$200$	0	0
$201$	−7.47291	−0.527099
$202$	0	0
$203$	0	0
$204$	0	0
$205$	− 23.6557i	− 1.65218i
$206$	0	0
$207$	17.1755	1.19378
$208$	0	0
$209$	−2.14107	−0.148101
$210$	0	0
$211$	− 6.44154i	− 0.443454i	−0.975109	−	0.221727i	$-0.928831\pi$
$211$	− 6.44154i	− 0.443454i	0.975109	−	0.221727i	$-0.0711694\pi$
$212$	0	0
$213$	− 35.7438i	− 2.44912i
$214$	0	0
$215$	24.2898	1.65655
$216$	0	0
$217$	0	0
$218$	0	0
$219$	− 26.4275i	− 1.78580i
$220$	0	0
$221$	6.38797i	0.429702i
$222$	0	0
$223$	19.2830	1.29128	0.645641	−	0.763641i	$-0.276590\pi$
$223$	19.2830	1.29128	0.645641	+	0.763641i	$0.276590\pi$
$224$	0	0
$225$	27.3510	1.82340
$226$	0	0
$227$	− 9.68889i	− 0.643074i	−0.946897	−	0.321537i	$-0.895800\pi$
$227$	− 9.68889i	− 0.643074i	0.946897	−	0.321537i	$-0.104200\pi$
$228$	0	0
$229$	9.35925i	0.618477i	0.950985	+	0.309238i	$0.100074\pi$
$229$	9.35925i	0.618477i	−0.950985	+	0.309238i	$0.899926\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	16.9193	1.10842	0.554209	−	0.832378i	$-0.313021\pi$
$233$	16.9193	1.10842	0.554209	+	0.832378i	$0.313021\pi$
$234$	0	0
$235$	− 38.2649i	− 2.49613i
$236$	0	0
$237$	39.8510i	2.58860i
$238$	0	0
$239$	4.71585	0.305043	0.152522	−	0.988300i	$-0.451261\pi$
$239$	4.71585	0.305043	0.152522	+	0.988300i	$0.451261\pi$
$240$	0	0
$241$	8.88713	0.572470	0.286235	−	0.958159i	$-0.407596\pi$
$241$	8.88713	0.572470	0.286235	+	0.958159i	$0.407596\pi$
$242$	0	0
$243$	− 11.5552i	− 0.741265i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1.68793	−0.107400
$248$	0	0
$249$	−22.5808	−1.43100
$250$	0	0
$251$	16.3974i	1.03500i	0.855684	+	0.517499i	$0.173137\pi$
$251$	16.3974i	1.03500i	−0.855684	+	0.517499i	$0.826863\pi$
$252$	0	0
$253$	− 4.76899i	− 0.299824i
$254$	0	0
$255$	51.1994	3.20623
$256$	0	0
$257$	12.7279	0.793946	0.396973	−	0.917830i	$-0.370061\pi$
$257$	12.7279	0.793946	0.396973	+	0.917830i	$0.370061\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	− 18.7795i	− 1.16242i
$262$	0	0
$263$	−21.8913	−1.34988	−0.674939	−	0.737874i	$-0.735830\pi$
$263$	−21.8913	−1.34988	−0.674939	+	0.737874i	$0.735830\pi$
$264$	0	0
$265$	35.2412	2.16485
$266$	0	0
$267$	9.26689i	0.567125i
$268$	0	0
$269$	18.5176i	1.12904i	0.825420	+	0.564519i	$0.190938\pi$
$269$	18.5176i	1.12904i	−0.825420	+	0.564519i	$0.809062\pi$
$270$	0	0
$271$	−13.9335	−0.846397	−0.423199	−	0.906037i	$-0.639093\pi$
$271$	−13.9335	−0.846397	−0.423199	+	0.906037i	$0.639093\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 7.59434i	− 0.457956i
$276$	0	0
$277$	− 13.9226i	− 0.836527i	−0.908326	−	0.418264i	$-0.862639\pi$
$277$	− 13.9226i	− 0.836527i	0.908326	−	0.418264i	$-0.137361\pi$
$278$	0	0
$279$	−5.29450	−0.316973
$280$	0	0
$281$	1.43171	0.0854084	0.0427042	−	0.999088i	$-0.486403\pi$
$281$	1.43171	0.0854084	0.0427042	+	0.999088i	$0.486403\pi$
$282$	0	0
$283$	4.49907i	0.267442i	0.991019	+	0.133721i	$0.0426926\pi$
$283$	4.49907i	0.267442i	−0.991019	+	0.133721i	$0.957307\pi$
$284$	0	0
$285$	13.5287i	0.801370i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	14.1366	0.831564
$290$	0	0
$291$	1.15280i	0.0675783i
$292$	0	0
$293$	8.04068i	0.469741i	0.972027	+	0.234871i	$0.0754667\pi$
$293$	8.04068i	0.469741i	−0.972027	+	0.234871i	$0.924533\pi$
$294$	0	0
$295$	20.2034	1.17629
$296$	0	0
$297$	9.28897	0.539001
$298$	0	0
$299$	− 3.75967i	− 0.217428i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	17.8524	1.02560
$304$	0	0
$305$	−10.2298	−0.585758
$306$	0	0
$307$	− 11.9785i	− 0.683648i	−0.939764	−	0.341824i	$-0.888955\pi$
$307$	− 11.9785i	− 0.683648i	0.939764	−	0.341824i	$-0.111045\pi$
$308$	0	0
$309$	− 20.9972i	− 1.19449i
$310$	0	0
$311$	−21.6701	−1.22880	−0.614398	−	0.788996i	$-0.710601\pi$
$311$	−21.6701	−1.22880	−0.614398	+	0.788996i	$0.710601\pi$
$312$	0	0
$313$	9.42081	0.532496	0.266248	−	0.963905i	$-0.414216\pi$
$313$	9.42081	0.532496	0.266248	+	0.963905i	$0.414216\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 9.15360i	− 0.514117i	−0.966396	−	0.257059i	$-0.917247\pi$
$317$	− 9.15360i	− 0.514117i	0.966396	−	0.257059i	$-0.0827534\pi$
$318$	0	0
$319$	−5.21438	−0.291949
$320$	0	0
$321$	12.8224	0.715678
$322$	0	0
$323$	8.22739i	0.457784i
$324$	0	0
$325$	− 5.98706i	− 0.332103i
$326$	0	0
$327$	36.2535	2.00482
$328$	0	0
$329$	0	0
$330$	0	0
$331$	19.8788i	1.09264i	0.837578	+	0.546318i	$0.183971\pi$
$331$	19.8788i	1.09264i	−0.837578	+	0.546318i	$0.816029\pi$
$332$	0	0
$333$	− 33.9682i	− 1.86145i
$334$	0	0
$335$	8.33161	0.455204
$336$	0	0
$337$	−19.5529	−1.06511	−0.532556	−	0.846395i	$-0.678768\pi$
$337$	−19.5529	−1.06511	−0.532556	+	0.846395i	$0.678768\pi$
$338$	0	0
$339$	− 29.3469i	− 1.59391i
$340$	0	0
$341$	1.47008i	0.0796096i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−30.1336	−1.62234
$346$	0	0
$347$	31.0043i	1.66440i	0.554479	+	0.832198i	$0.312918\pi$
$347$	31.0043i	1.66440i	−0.554479	+	0.832198i	$0.687082\pi$
$348$	0	0
$349$	− 14.6735i	− 0.785453i	−0.919655	−	0.392726i	$-0.871532\pi$
$349$	− 14.6735i	− 0.785453i	0.919655	−	0.392726i	$-0.128468\pi$
$350$	0	0
$351$	7.32304	0.390875
$352$	0	0
$353$	25.6667	1.36610	0.683049	−	0.730372i	$-0.260653\pi$
$353$	25.6667	1.36610	0.683049	+	0.730372i	$0.260653\pi$
$354$	0	0
$355$	39.8510i	2.11507i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	4.20341	0.221847	0.110924	−	0.993829i	$-0.464619\pi$
$359$	4.20341	0.221847	0.110924	+	0.993829i	$0.464619\pi$
$360$	0	0
$361$	16.8260	0.885581
$362$	0	0
$363$	25.5072i	1.33878i
$364$	0	0
$365$	29.4642i	1.54222i
$366$	0	0
$367$	35.6035	1.85849	0.929244	−	0.369467i	$-0.120460\pi$
$367$	35.6035	1.85849	0.929244	+	0.369467i	$0.120460\pi$
$368$	0	0
$369$	−38.6801	−2.01361
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 9.97884i	− 0.516684i	−0.966053	−	0.258342i	$-0.916824\pi$
$373$	− 9.97884i	− 0.516684i	0.966053	−	0.258342i	$-0.0831762\pi$
$374$	0	0
$375$	−2.10866	−0.108891
$376$	0	0
$377$	−4.11080	−0.211717
$378$	0	0
$379$	− 24.5283i	− 1.25993i	−0.776622	−	0.629967i	$-0.783068\pi$
$379$	− 24.5283i	− 1.25993i	0.776622	−	0.629967i	$-0.216932\pi$
$380$	0	0
$381$	− 43.2232i	− 2.21439i
$382$	0	0
$383$	−8.63896	−0.441430	−0.220715	−	0.975338i	$-0.570839\pi$
$383$	−8.63896	−0.441430	−0.220715	+	0.975338i	$0.570839\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 39.7170i	− 2.01893i
$388$	0	0
$389$	− 12.6373i	− 0.640738i	−0.947293	−	0.320369i	$-0.896193\pi$
$389$	− 12.6373i	− 0.640738i	0.947293	−	0.320369i	$-0.103807\pi$
$390$	0	0
$391$	−18.3256	−0.926765
$392$	0	0
$393$	−14.1212	−0.712318
$394$	0	0
$395$	− 44.4302i	− 2.23552i
$396$	0	0
$397$	1.87984i	0.0943463i	0.998887	+	0.0471732i	$0.0150213\pi$
$397$	1.87984i	0.0943463i	−0.998887	+	0.0471732i	$0.984979\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	14.5808	0.728129	0.364065	−	0.931374i	$-0.381389\pi$
$401$	14.5808	0.728129	0.364065	+	0.931374i	$0.381389\pi$
$402$	0	0
$403$	1.15895i	0.0577316i
$404$	0	0
$405$	− 8.51263i	− 0.422996i
$406$	0	0
$407$	−9.43171	−0.467512
$408$	0	0
$409$	−30.1948	−1.49304	−0.746519	−	0.665364i	$-0.768276\pi$
$409$	−30.1948	−1.49304	−0.746519	+	0.665364i	$0.768276\pi$
$410$	0	0
$411$	52.6446i	2.59677i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	25.1755	1.23582
$416$	0	0
$417$	48.9317	2.39620
$418$	0	0
$419$	7.28773i	0.356029i	0.984028	+	0.178014i	$0.0569673\pi$
$419$	7.28773i	0.356029i	−0.984028	+	0.178014i	$0.943033\pi$
$420$	0	0
$421$	− 2.30560i	− 0.112368i	−0.998420	−	0.0561840i	$-0.982107\pi$
$421$	− 2.30560i	− 0.112368i	0.998420	−	0.0561840i	$-0.0178933\pi$
$422$	0	0
$423$	−62.5681	−3.04216
$424$	0	0
$425$	−29.1825	−1.41556
$426$	0	0
$427$	0	0
$428$	0	0
$429$	− 4.76899i	− 0.230249i
$430$	0	0
$431$	2.36489	0.113913	0.0569563	−	0.998377i	$-0.481860\pi$
$431$	2.36489	0.113913	0.0569563	+	0.998377i	$0.481860\pi$
$432$	0	0
$433$	−11.0832	−0.532624	−0.266312	−	0.963887i	$-0.585805\pi$
$433$	−11.0832	−0.532624	−0.266312	+	0.963887i	$0.585805\pi$
$434$	0	0
$435$	32.9479i	1.57973i
$436$	0	0
$437$	− 4.84227i	− 0.231637i
$438$	0	0
$439$	6.66922	0.318305	0.159152	−	0.987254i	$-0.449124\pi$
$439$	6.66922	0.318305	0.159152	+	0.987254i	$0.449124\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 1.33883i	− 0.0636098i	−0.999494	−	0.0318049i	$-0.989874\pi$
$443$	− 1.33883i	− 0.0636098i	0.999494	−	0.0318049i	$-0.0101255\pi$
$444$	0	0
$445$	− 10.3317i	− 0.489771i
$446$	0	0
$447$	−58.5186	−2.76783
$448$	0	0
$449$	−4.35097	−0.205335	−0.102667	−	0.994716i	$-0.532738\pi$
$449$	−4.35097	−0.205335	−0.102667	+	0.994716i	$0.532738\pi$
$450$	0	0
$451$	10.7400i	0.505728i
$452$	0	0
$453$	− 13.5287i	− 0.635633i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−22.0125	−1.02970	−0.514850	−	0.857280i	$-0.672153\pi$
$457$	−22.0125	−1.02970	−0.514850	+	0.857280i	$0.672153\pi$
$458$	0	0
$459$	− 35.6943i	− 1.66607i
$460$	0	0
$461$	6.80657i	0.317014i	0.987358	+	0.158507i	$0.0506680\pi$
$461$	6.80657i	0.317014i	−0.987358	+	0.158507i	$0.949332\pi$
$462$	0	0
$463$	−1.43171	−0.0665370	−0.0332685	−	0.999446i	$-0.510592\pi$
$463$	−1.43171	−0.0665370	−0.0332685	+	0.999446i	$0.510592\pi$
$464$	0	0
$465$	9.28897	0.430766
$466$	0	0
$467$	− 33.7974i	− 1.56396i	−0.623306	−	0.781978i	$-0.714211\pi$
$467$	− 33.7974i	− 1.56396i	0.623306	−	0.781978i	$-0.285789\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−25.8524	−1.19122
$472$	0	0
$473$	−11.0279	−0.507065
$474$	0	0
$475$	− 7.71104i	− 0.353807i
$476$	0	0
$477$	− 57.6239i	− 2.63842i
$478$	0	0
$479$	−21.9577	−1.00327	−0.501637	−	0.865078i	$-0.667269\pi$
$479$	−21.9577	−1.00327	−0.501637	+	0.865078i	$0.667269\pi$
$480$	0	0
$481$	−7.43557	−0.339033
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 1.28526i	− 0.0583608i
$486$	0	0
$487$	37.1227	1.68219	0.841094	−	0.540888i	$-0.181912\pi$
$487$	37.1227	1.68219	0.841094	+	0.540888i	$0.181912\pi$
$488$	0	0
$489$	−13.4548	−0.608446
$490$	0	0
$491$	3.31071i	0.149410i	0.997206	+	0.0747051i	$0.0238016\pi$
$491$	3.31071i	0.149410i	−0.997206	+	0.0747051i	$0.976198\pi$
$492$	0	0
$493$	20.0371i	0.902424i
$494$	0	0
$495$	−24.2898	−1.09175
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 24.5345i	− 1.09831i	−0.835719	−	0.549157i	$-0.814949\pi$
$499$	− 24.5345i	− 1.09831i	0.835719	−	0.549157i	$-0.185051\pi$
$500$	0	0
$501$	40.1296i	1.79286i
$502$	0	0
$503$	3.68712	0.164401	0.0822003	−	0.996616i	$-0.473805\pi$
$503$	3.68712	0.164401	0.0822003	+	0.996616i	$0.473805\pi$
$504$	0	0
$505$	−19.9038	−0.885708
$506$	0	0
$507$	33.5343i	1.48931i
$508$	0	0
$509$	− 29.7567i	− 1.31894i	−0.751730	−	0.659471i	$-0.770780\pi$
$509$	− 29.7567i	− 1.31894i	0.751730	−	0.659471i	$-0.229220\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	9.43171	0.416420
$514$	0	0
$515$	23.4099i	1.03156i
$516$	0	0
$517$	17.3728i	0.764055i
$518$	0	0
$519$	−21.6351	−0.949676
$520$	0	0
$521$	−30.1948	−1.32286	−0.661430	−	0.750007i	$-0.730050\pi$
$521$	−30.1948	−1.32286	−0.661430	+	0.750007i	$0.730050\pi$
$522$	0	0
$523$	− 42.5324i	− 1.85981i	−0.367796	−	0.929906i	$-0.619888\pi$
$523$	− 42.5324i	− 1.85981i	0.367796	−	0.929906i	$-0.380112\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5.64903	0.246076
$528$	0	0
$529$	−12.2144	−0.531060
$530$	0	0
$531$	− 33.0352i	− 1.43361i
$532$	0	0
$533$	8.46699i	0.366746i
$534$	0	0
$535$	−14.2958	−0.618062
$536$	0	0
$537$	42.7141	1.84325
$538$	0	0
$539$	0	0
$540$	0	0
$541$	5.20985i	0.223989i	0.993709	+	0.111994i	$0.0357239\pi$
$541$	5.20985i	0.223989i	−0.993709	+	0.111994i	$0.964276\pi$
$542$	0	0
$543$	47.3091	2.03023
$544$	0	0
$545$	−40.4193	−1.73137
$546$	0	0
$547$	− 37.1465i	− 1.58827i	−0.607743	−	0.794134i	$-0.707925\pi$
$547$	− 37.1465i	− 1.58827i	0.607743	−	0.794134i	$-0.292075\pi$
$548$	0	0
$549$	16.7271i	0.713895i
$550$	0	0
$551$	−5.29450	−0.225553
$552$	0	0
$553$	0	0
$554$	0	0
$555$	59.5958i	2.52970i
$556$	0	0
$557$	30.8182i	1.30581i	0.757440	+	0.652905i	$0.226450\pi$
$557$	30.8182i	1.30581i	−0.757440	+	0.652905i	$0.773550\pi$
$558$	0	0
$559$	−8.69396	−0.367715
$560$	0	0
$561$	−23.2453	−0.981415
$562$	0	0
$563$	− 20.4289i	− 0.860976i	−0.902596	−	0.430488i	$-0.858341\pi$
$563$	− 20.4289i	− 0.860976i	0.902596	−	0.430488i	$-0.141659\pi$
$564$	0	0
$565$	32.7191i	1.37650i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	28.0125	1.17434	0.587172	−	0.809462i	$-0.300241\pi$
$569$	28.0125	1.17434	0.587172	+	0.809462i	$0.300241\pi$
$570$	0	0
$571$	− 14.5618i	− 0.609392i	−0.952450	−	0.304696i	$-0.901445\pi$
$571$	− 14.5618i	− 0.609392i	0.952450	−	0.304696i	$-0.0985549\pi$
$572$	0	0
$573$	− 6.74437i	− 0.281750i
$574$	0	0
$575$	17.1755	0.716267
$576$	0	0
$577$	−20.3172	−0.845815	−0.422907	−	0.906173i	$-0.638990\pi$
$577$	−20.3172	−0.845815	−0.422907	+	0.906173i	$0.638990\pi$
$578$	0	0
$579$	− 27.4049i	− 1.13891i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−16.0000	−0.662652
$584$	0	0
$585$	−19.1491	−0.791717
$586$	0	0
$587$	8.05859i	0.332614i	0.986074	+	0.166307i	$0.0531842\pi$
$587$	8.05859i	0.332614i	−0.986074	+	0.166307i	$0.946816\pi$
$588$	0	0
$589$	1.49267i	0.0615046i
$590$	0	0
$591$	34.5911	1.42289
$592$	0	0
$593$	−26.0664	−1.07042	−0.535209	−	0.844720i	$-0.679767\pi$
$593$	−26.0664	−1.07042	−0.535209	+	0.844720i	$0.679767\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 11.4592i	− 0.468994i
$598$	0	0
$599$	−47.4876	−1.94029	−0.970144	−	0.242528i	$-0.922023\pi$
$599$	−47.4876	−1.94029	−0.970144	+	0.242528i	$0.922023\pi$
$600$	0	0
$601$	2.00922	0.0819580	0.0409790	−	0.999160i	$-0.486952\pi$
$601$	2.00922	0.0819580	0.0409790	+	0.999160i	$0.486952\pi$
$602$	0	0
$603$	− 13.6233i	− 0.554782i
$604$	0	0
$605$	− 28.4381i	− 1.15617i
$606$	0	0
$607$	41.5480	1.68638	0.843191	−	0.537614i	$-0.180674\pi$
$607$	41.5480	1.68638	0.843191	+	0.537614i	$0.180674\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	13.6960i	0.554081i
$612$	0	0
$613$	− 35.0077i	− 1.41395i	−0.707239	−	0.706974i	$-0.750060\pi$
$613$	− 35.0077i	− 1.41395i	0.707239	−	0.706974i	$-0.249940\pi$
$614$	0	0
$615$	67.8626	2.73648
$616$	0	0
$617$	6.58078	0.264932	0.132466	−	0.991188i	$-0.457710\pi$
$617$	6.58078	0.264932	0.132466	+	0.991188i	$0.457710\pi$
$618$	0	0
$619$	− 0.391842i	− 0.0157495i	−0.999969	−	0.00787473i	$-0.997493\pi$
$619$	− 0.391842i	− 0.0157495i	0.999969	−	0.00787473i	$-0.00250663\pi$
$620$	0	0
$621$	21.0081i	0.843025i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−23.7981	−0.951924
$626$	0	0
$627$	− 6.14222i	− 0.245296i
$628$	0	0
$629$	36.2428i	1.44510i
$630$	0	0
$631$	−23.4876	−0.935025	−0.467512	−	0.883987i	$-0.654850\pi$
$631$	−23.4876	−0.935025	−0.467512	+	0.883987i	$0.654850\pi$
$632$	0	0
$633$	18.4793	0.734485
$634$	0	0
$635$	48.1898i	1.91236i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	65.1616	2.57775
$640$	0	0
$641$	−15.9347	−0.629383	−0.314691	−	0.949194i	$-0.601901\pi$
$641$	−15.9347	−0.629383	−0.314691	+	0.949194i	$0.601901\pi$
$642$	0	0
$643$	− 44.8491i	− 1.76868i	−0.466847	−	0.884338i	$-0.654610\pi$
$643$	− 44.8491i	− 1.76868i	0.466847	−	0.884338i	$-0.345390\pi$
$644$	0	0
$645$	69.6817i	2.74372i
$646$	0	0
$647$	23.9825	0.942847	0.471424	−	0.881907i	$-0.343740\pi$
$647$	23.9825	0.942847	0.471424	+	0.881907i	$0.343740\pi$
$648$	0	0
$649$	−9.17264	−0.360058
$650$	0	0
$651$	0	0
$652$	0	0
$653$	9.06564i	0.354766i	0.984142	+	0.177383i	$0.0567631\pi$
$653$	9.06564i	0.354766i	−0.984142	+	0.177383i	$0.943237\pi$
$654$	0	0
$655$	15.7438	0.615160
$656$	0	0
$657$	48.1778	1.87959
$658$	0	0
$659$	21.9578i	0.855354i	0.903932	+	0.427677i	$0.140668\pi$
$659$	21.9578i	0.855354i	−0.903932	+	0.427677i	$0.859332\pi$
$660$	0	0
$661$	26.0641i	1.01377i	0.862012	+	0.506887i	$0.169204\pi$
$661$	26.0641i	1.01377i	−0.862012	+	0.506887i	$0.830796\pi$
$662$	0	0
$663$	−18.3256	−0.711707
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 11.7929i	− 0.456624i
$668$	0	0
$669$	55.3183i	2.13873i
$670$	0	0
$671$	4.64449	0.179298
$672$	0	0
$673$	30.5683	1.17832	0.589161	−	0.808016i	$-0.299459\pi$
$673$	30.5683	1.17832	0.589161	+	0.808016i	$0.299459\pi$
$674$	0	0
$675$	33.4542i	1.28765i
$676$	0	0
$677$	− 11.9119i	− 0.457813i	−0.973448	−	0.228906i	$-0.926485\pi$
$677$	− 11.9119i	− 0.457813i	0.973448	−	0.228906i	$-0.0735149\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	27.7952	1.06511
$682$	0	0
$683$	− 18.9402i	− 0.724728i	−0.932037	−	0.362364i	$-0.881970\pi$
$683$	− 18.9402i	− 0.724728i	0.932037	−	0.362364i	$-0.118030\pi$
$684$	0	0
$685$	− 58.6939i	− 2.24258i
$686$	0	0
$687$	−26.8495	−1.02437
$688$	0	0
$689$	−12.6137	−0.480545
$690$	0	0
$691$	8.55765i	0.325549i	0.986663	+	0.162774i	$0.0520442\pi$
$691$	8.55765i	0.325549i	−0.986663	+	0.162774i	$0.947956\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−54.5544	−2.06937
$696$	0	0
$697$	41.2702	1.56322
$698$	0	0
$699$	48.5374i	1.83585i
$700$	0	0
$701$	− 3.59086i	− 0.135625i	−0.997698	−	0.0678125i	$-0.978398\pi$
$701$	− 3.59086i	− 0.135625i	0.997698	−	0.0678125i	$-0.0216020\pi$
$702$	0	0
$703$	−9.57663	−0.361190
$704$	0	0
$705$	109.773	4.13429
$706$	0	0
$707$	0	0
$708$	0	0
$709$	36.8725i	1.38477i	0.721526	+	0.692387i	$0.243441\pi$
$709$	36.8725i	1.38477i	−0.721526	+	0.692387i	$0.756559\pi$
$710$	0	0
$711$	−72.6491	−2.72455
$712$	0	0
$713$	−3.32477	−0.124514
$714$	0	0
$715$	5.31698i	0.198844i
$716$	0	0
$717$	13.5287i	0.505238i
$718$	0	0
$719$	−43.9901	−1.64055	−0.820277	−	0.571966i	$-0.806181\pi$
$719$	−43.9901	−1.64055	−0.820277	+	0.571966i	$0.806181\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	25.4951i	0.948172i
$724$	0	0
$725$	− 18.7795i	− 0.697455i
$726$	0	0
$727$	50.1870	1.86133	0.930666	−	0.365870i	$-0.119228\pi$
$727$	50.1870	1.86133	0.930666	+	0.365870i	$0.119228\pi$
$728$	0	0
$729$	41.1336	1.52347
$730$	0	0
$731$	42.3765i	1.56735i
$732$	0	0
$733$	6.30750i	0.232973i	0.993192	+	0.116486i	$0.0371632\pi$
$733$	6.30750i	0.232973i	−0.993192	+	0.116486i	$0.962837\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−3.78267	−0.139336
$738$	0	0
$739$	34.4000i	1.26542i	0.774387	+	0.632712i	$0.218058\pi$
$739$	34.4000i	1.26542i	−0.774387	+	0.632712i	$0.781942\pi$
$740$	0	0
$741$	− 4.84227i	− 0.177885i
$742$	0	0
$743$	9.93023	0.364305	0.182152	−	0.983270i	$-0.441694\pi$
$743$	9.93023	0.364305	0.182152	+	0.983270i	$0.441694\pi$
$744$	0	0
$745$	65.2428	2.39031
$746$	0	0
$747$	− 41.1652i	− 1.50616i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−10.7717	−0.393065	−0.196532	−	0.980497i	$-0.562968\pi$
$751$	−10.7717	−0.393065	−0.196532	+	0.980497i	$0.562968\pi$
$752$	0	0
$753$	−47.0404	−1.71425
$754$	0	0
$755$	15.0832i	0.548935i
$756$	0	0
$757$	16.2090i	0.589127i	0.955632	+	0.294563i	$0.0951742\pi$
$757$	16.2090i	0.589127i	−0.955632	+	0.294563i	$0.904826\pi$
$758$	0	0
$759$	13.6811	0.496593
$760$	0	0
$761$	−5.37134	−0.194711	−0.0973554	−	0.995250i	$-0.531038\pi$
$761$	−5.37134	−0.194711	−0.0973554	+	0.995250i	$0.531038\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	93.3374i	3.37462i
$766$	0	0
$767$	−7.23133	−0.261108
$768$	0	0
$769$	10.1872	0.367358	0.183679	−	0.982986i	$-0.441199\pi$
$769$	10.1872	0.367358	0.183679	+	0.982986i	$0.441199\pi$
$770$	0	0
$771$	36.5134i	1.31500i
$772$	0	0
$773$	51.3882i	1.84831i	0.382022	+	0.924153i	$0.375228\pi$
$773$	51.3882i	1.84831i	−0.382022	+	0.924153i	$0.624772\pi$
$774$	0	0
$775$	−5.29450	−0.190184
$776$	0	0
$777$	0	0
$778$	0	0
$779$	10.9051i	0.390714i
$780$	0	0
$781$	− 18.0929i	− 0.647416i
$782$	0	0
$783$	22.9701	0.820884
$784$	0	0
$785$	28.8231	1.02874
$786$	0	0
$787$	45.4326i	1.61950i	0.586776	+	0.809749i	$0.300397\pi$
$787$	45.4326i	1.61950i	−0.586776	+	0.809749i	$0.699603\pi$
$788$	0	0
$789$	− 62.8011i	− 2.23578i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	3.66152	0.130024
$794$	0	0
$795$	101.099i	3.58560i
$796$	0	0
$797$	− 30.4917i	− 1.08007i	−0.841642	−	0.540036i	$-0.818410\pi$
$797$	− 30.4917i	− 1.08007i	0.841642	−	0.540036i	$-0.181590\pi$
$798$	0	0
$799$	66.7578	2.36172
$800$	0	0
$801$	−16.8937	−0.596910
$802$	0	0
$803$	− 13.3772i	− 0.472070i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−53.1227	−1.87001
$808$	0	0
$809$	−18.9068	−0.664727	−0.332363	−	0.943151i	$-0.607846\pi$
$809$	−18.9068	−0.664727	−0.332363	+	0.943151i	$0.607846\pi$
$810$	0	0
$811$	− 34.8800i	− 1.22480i	−0.790548	−	0.612401i	$-0.790204\pi$
$811$	− 34.8800i	− 1.22480i	0.790548	−	0.612401i	$-0.209796\pi$
$812$	0	0
$813$	− 39.9718i	− 1.40187i
$814$	0	0
$815$	15.0008	0.525456
$816$	0	0
$817$	−11.1974	−0.391747
$818$	0	0
$819$	0	0
$820$	0	0
$821$	10.3509i	0.361249i	0.983552	+	0.180624i	$0.0578119\pi$
$821$	10.3509i	0.361249i	−0.983552	+	0.180624i	$0.942188\pi$
$822$	0	0
$823$	−7.32304	−0.255265	−0.127633	−	0.991822i	$-0.540738\pi$
$823$	−7.32304	−0.255265	−0.127633	+	0.991822i	$0.540738\pi$
$824$	0	0
$825$	21.7864	0.758504
$826$	0	0
$827$	− 33.6881i	− 1.17145i	−0.810510	−	0.585725i	$-0.800810\pi$
$827$	− 33.6881i	− 1.17145i	0.810510	−	0.585725i	$-0.199190\pi$
$828$	0	0
$829$	− 11.8848i	− 0.412777i	−0.978470	−	0.206388i	$-0.933829\pi$
$829$	− 11.8848i	− 0.412777i	0.978470	−	0.206388i	$-0.0661710\pi$
$830$	0	0
$831$	39.9406	1.38553
$832$	0	0
$833$	0	0
$834$	0	0
$835$	− 44.7408i	− 1.54832i
$836$	0	0
$837$	− 6.47594i	− 0.223841i
$838$	0	0
$839$	−53.8741	−1.85994	−0.929970	−	0.367635i	$-0.880168\pi$
$839$	−53.8741	−1.85994	−0.929970	+	0.367635i	$0.880168\pi$
$840$	0	0
$841$	16.1057	0.555369
$842$	0	0
$843$	4.10723i	0.141460i
$844$	0	0
$845$	− 37.3876i	− 1.28617i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−12.9068	−0.442959
$850$	0	0
$851$	− 21.3309i	− 0.731214i
$852$	0	0
$853$	− 31.6588i	− 1.08398i	−0.840386	−	0.541988i	$-0.817672\pi$
$853$	− 31.6588i	− 1.08398i	0.840386	−	0.541988i	$-0.182328\pi$
$854$	0	0
$855$	−24.6630	−0.843458
$856$	0	0
$857$	−3.02159	−0.103216	−0.0516078	−	0.998667i	$-0.516435\pi$
$857$	−3.02159	−0.103216	−0.0516078	+	0.998667i	$0.516435\pi$
$858$	0	0
$859$	− 14.9274i	− 0.509315i	−0.967031	−	0.254657i	$-0.918037\pi$
$859$	− 14.9274i	− 0.509315i	0.967031	−	0.254657i	$-0.0819627\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	2.86341	0.0974716	0.0487358	−	0.998812i	$-0.484481\pi$
$863$	2.86341	0.0974716	0.0487358	+	0.998812i	$0.484481\pi$
$864$	0	0
$865$	24.1212	0.820144
$866$	0	0
$867$	40.5546i	1.37730i
$868$	0	0
$869$	20.1719i	0.684286i
$870$	0	0
$871$	−2.98210	−0.101045
$872$	0	0
$873$	−2.10157	−0.0711275
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 25.6275i	− 0.865381i	−0.901543	−	0.432690i	$-0.857564\pi$
$877$	− 25.6275i	− 0.865381i	0.901543	−	0.432690i	$-0.142436\pi$
$878$	0	0
$879$	−23.0668	−0.778024
$880$	0	0
$881$	22.1882	0.747540	0.373770	−	0.927521i	$-0.378065\pi$
$881$	22.1882	0.747540	0.373770	+	0.927521i	$0.378065\pi$
$882$	0	0
$883$	− 26.5063i	− 0.892010i	−0.895031	−	0.446005i	$-0.852846\pi$
$883$	− 26.5063i	− 0.892010i	0.895031	−	0.446005i	$-0.147154\pi$
$884$	0	0
$885$	57.9588i	1.94827i
$886$	0	0
$887$	−11.0064	−0.369557	−0.184779	−	0.982780i	$-0.559157\pi$
$887$	−11.0064	−0.369557	−0.184779	+	0.982780i	$0.559157\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	3.86486i	0.129478i
$892$	0	0
$893$	17.6398i	0.590292i
$894$	0	0
$895$	−47.6222	−1.59184
$896$	0	0
$897$	10.7856	0.360121
$898$	0	0
$899$	3.63528i	0.121243i
$900$	0	0
$901$	61.4825i	2.04828i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−52.7453	−1.75331
$906$	0	0
$907$	− 57.5895i	− 1.91223i	−0.292995	−	0.956114i	$-0.594652\pi$
$907$	− 57.5895i	− 1.91223i	0.292995	−	0.956114i	$-0.405348\pi$
$908$	0	0
$909$	32.5453i	1.07946i
$910$	0	0
$911$	−2.77170	−0.0918306	−0.0459153	−	0.998945i	$-0.514620\pi$
$911$	−2.77170	−0.0918306	−0.0459153	+	0.998945i	$0.514620\pi$
$912$	0	0
$913$	−11.4300	−0.378279
$914$	0	0
$915$	− 29.3469i	− 0.970180i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−20.4596	−0.674901	−0.337450	−	0.941343i	$-0.609565\pi$
$919$	−20.4596	−0.674901	−0.337450	+	0.941343i	$0.609565\pi$
$920$	0	0
$921$	34.3635	1.13231
$922$	0	0
$923$	− 14.2637i	− 0.469496i
$924$	0	0
$925$	− 33.9682i	− 1.11687i
$926$	0	0
$927$	38.2783	1.25722
$928$	0	0
$929$	11.5442	0.378754	0.189377	−	0.981904i	$-0.439353\pi$
$929$	11.5442	0.378754	0.189377	+	0.981904i	$0.439353\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	− 62.1663i	− 2.03523i
$934$	0	0
$935$	25.9163	0.847554
$936$	0	0
$937$	−46.4780	−1.51837	−0.759185	−	0.650874i	$-0.774402\pi$
$937$	−46.4780	−1.51837	−0.759185	+	0.650874i	$0.774402\pi$
$938$	0	0
$939$	27.0261i	0.881964i
$940$	0	0
$941$	− 39.6772i	− 1.29344i	−0.762728	−	0.646720i	$-0.776140\pi$
$941$	− 39.6772i	− 1.29344i	0.762728	−	0.646720i	$-0.223860\pi$
$942$	0	0
$943$	−24.2898	−0.790985
$944$	0	0
$945$	0	0
$946$	0	0
$947$	50.2561i	1.63310i	0.577271	+	0.816552i	$0.304118\pi$
$947$	50.2561i	1.63310i	−0.577271	+	0.816552i	$0.695882\pi$
$948$	0	0
$949$	− 10.5460i	− 0.342338i
$950$	0	0
$951$	26.2595	0.851524
$952$	0	0
$953$	26.4596	0.857111	0.428556	−	0.903515i	$-0.359023\pi$
$953$	26.4596	0.857111	0.428556	+	0.903515i	$0.359023\pi$
$954$	0	0
$955$	7.51935i	0.243320i
$956$	0	0
$957$	− 14.9588i	− 0.483550i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−29.9751	−0.966939
$962$	0	0
$963$	23.3755i	0.753265i
$964$	0	0
$965$	30.5539i	0.983566i
$966$	0	0
$967$	6.98903	0.224752	0.112376	−	0.993666i	$-0.464154\pi$
$967$	6.98903	0.224752	0.112376	+	0.993666i	$0.464154\pi$
$968$	0	0
$969$	−23.6024	−0.758220
$970$	0	0
$971$	− 26.3180i	− 0.844583i	−0.906460	−	0.422292i	$-0.861226\pi$
$971$	− 26.3180i	− 0.844583i	0.906460	−	0.422292i	$-0.138774\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	17.1755	0.550056
$976$	0	0
$977$	32.9193	1.05318	0.526590	−	0.850119i	$-0.323470\pi$
$977$	32.9193	1.05318	0.526590	+	0.850119i	$0.323470\pi$
$978$	0	0
$979$	4.69076i	0.149917i
$980$	0	0
$981$	66.0909i	2.11012i
$982$	0	0
$983$	−26.9646	−0.860036	−0.430018	−	0.902820i	$-0.641493\pi$
$983$	−26.9646	−0.860036	−0.430018	+	0.902820i	$0.641493\pi$
$984$	0	0
$985$	−38.5659	−1.22881
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 24.9409i	− 0.793076i
$990$	0	0
$991$	12.4596	0.395793	0.197897	−	0.980223i	$-0.436589\pi$
$991$	12.4596	0.395793	0.197897	+	0.980223i	$0.436589\pi$
$992$	0	0
$993$	−57.0275	−1.80971
$994$	0	0
$995$	12.7759i	0.405025i
$996$	0	0
$997$	44.8670i	1.42095i	0.703721	+	0.710477i	$0.251521\pi$
$997$	44.8670i	1.42095i	−0.703721	+	0.710477i	$0.748479\pi$
$998$	0	0
$999$	41.5480	1.31452

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1568.2.b.g.785.12		12
4.3	odd	2		392.2.b.g.197.5	✓	12
7.2	even	3		1568.2.t.h.753.1		24
7.3	odd	6		1568.2.t.h.177.2		24
7.4	even	3		1568.2.t.h.177.12		24
7.5	odd	6		1568.2.t.h.753.11		24
7.6	odd	2	inner	1568.2.b.g.785.2		12
8.3	odd	2		392.2.b.g.197.8	yes	12
8.5	even	2	inner	1568.2.b.g.785.1		12
28.3	even	6		392.2.p.h.373.12		24
28.11	odd	6		392.2.p.h.373.11		24
28.19	even	6		392.2.p.h.165.3		24
28.23	odd	6		392.2.p.h.165.4		24
28.27	even	2		392.2.b.g.197.6	yes	12
56.3	even	6		392.2.p.h.373.3		24
56.5	odd	6		1568.2.t.h.753.2		24
56.11	odd	6		392.2.p.h.373.4		24
56.13	odd	2	inner	1568.2.b.g.785.11		12
56.19	even	6		392.2.p.h.165.12		24
56.27	even	2		392.2.b.g.197.7	yes	12
56.37	even	6		1568.2.t.h.753.12		24
56.45	odd	6		1568.2.t.h.177.11		24
56.51	odd	6		392.2.p.h.165.11		24
56.53	even	6		1568.2.t.h.177.1		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.2.b.g.197.5	✓	12	4.3	odd	2
392.2.b.g.197.6	yes	12	28.27	even	2
392.2.b.g.197.7	yes	12	56.27	even	2
392.2.b.g.197.8	yes	12	8.3	odd	2
392.2.p.h.165.3		24	28.19	even	6
392.2.p.h.165.4		24	28.23	odd	6
392.2.p.h.165.11		24	56.51	odd	6
392.2.p.h.165.12		24	56.19	even	6
392.2.p.h.373.3		24	56.3	even	6
392.2.p.h.373.4		24	56.11	odd	6
392.2.p.h.373.11		24	28.11	odd	6
392.2.p.h.373.12		24	28.3	even	6
1568.2.b.g.785.1		12	8.5	even	2	inner
1568.2.b.g.785.2		12	7.6	odd	2	inner
1568.2.b.g.785.11		12	56.13	odd	2	inner
1568.2.b.g.785.12		12	1.1	even	1	trivial
1568.2.t.h.177.1		24	56.53	even	6
1568.2.t.h.177.2		24	7.3	odd	6
1568.2.t.h.177.11		24	56.45	odd	6
1568.2.t.h.177.12		24	7.4	even	3
1568.2.t.h.753.1		24	7.2	even	3
1568.2.t.h.753.2		24	56.5	odd	6
1568.2.t.h.753.11		24	7.5	odd	6
1568.2.t.h.753.12		24	56.37	even	6

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:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1568.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.86877i q^{3} -3.19841i q^{5} -5.22982 q^{9} +O(q^{10})\)	\(q+2.86877i q^{3} -3.19841i q^{5} -5.22982 q^{9} +1.45212i q^{11} +1.14479i q^{13} +9.17548 q^{15} +5.58002 q^{17} +1.47444i q^{19} -3.28415 q^{23} -5.22982 q^{25} -6.39682i q^{27} +3.59086i q^{29} +1.01237 q^{31} -4.16580 q^{33} +6.49511i q^{37} -3.28415 q^{39} +7.39608 q^{41} +7.59434i q^{43} +16.7271i q^{45} +11.9637 q^{47} +16.0078i q^{51} +11.0183i q^{53} +4.64449 q^{55} -4.22982 q^{57} +6.31671i q^{59} -3.19841i q^{61} +3.66152 q^{65} +2.60492i q^{67} -9.42145i q^{69} -12.4596 q^{71} -9.21213 q^{73} -15.0031i q^{75} +13.8913 q^{79} +2.66152 q^{81} +7.87125i q^{83} -17.8472i q^{85} -10.3013 q^{87} +3.23027 q^{89} +2.90425i q^{93} +4.71585 q^{95} +0.401845 q^{97} -7.59434i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{9}+O(q^{10}) \) 12 * q - 12 * q^9	\( 12 q - 12 q^{9} + 16 q^{15} - 32 q^{23} - 12 q^{25} - 32 q^{39} + 8 q^{65} - 48 q^{71} + 80 q^{79} - 4 q^{81} + 64 q^{95}+O(q^{100}) \) 12 * q - 12 * q^9 + 16 * q^15 - 32 * q^23 - 12 * q^25 - 32 * q^39 + 8 * q^65 - 48 * q^71 + 80 * q^79 - 4 * q^81 + 64 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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