LMFDB - Embedded newform 1568.2.a.s.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1568,2,Mod(1,1568)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("1568.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1568 = 2^{5} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1568.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:	$12.5205430369$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 224)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	1568.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205 q^{3} +1.00000 q^{5} +O(q^{10})$	$q+1.73205 q^{3} +1.00000 q^{5} +5.19615 q^{11} +1.73205 q^{15} +5.00000 q^{17} +1.73205 q^{19} -1.73205 q^{23} -4.00000 q^{25} -5.19615 q^{27} +8.00000 q^{29} -8.66025 q^{31} +9.00000 q^{33} -5.00000 q^{37} +4.00000 q^{41} +6.92820 q^{43} +8.66025 q^{47} +8.66025 q^{51} -1.00000 q^{53} +5.19615 q^{55} +3.00000 q^{57} +1.73205 q^{59} +11.0000 q^{61} -12.1244 q^{67} -3.00000 q^{69} -13.8564 q^{71} +15.0000 q^{73} -6.92820 q^{75} -1.73205 q^{79} -9.00000 q^{81} +6.92820 q^{83} +5.00000 q^{85} +13.8564 q^{87} +7.00000 q^{89} -15.0000 q^{93} +1.73205 q^{95} +12.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5}+O(q^{10}) $ 2 * q + 2 * q^5	$ 2 q + 2 q^{5} + 10 q^{17} - 8 q^{25} + 16 q^{29} + 18 q^{33} - 10 q^{37} + 8 q^{41} - 2 q^{53} + 6 q^{57} + 22 q^{61} - 6 q^{69} + 30 q^{73} - 18 q^{81} + 10 q^{85} + 14 q^{89} - 30 q^{93} + 24 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 + 10 * q^17 - 8 * q^25 + 16 * q^29 + 18 * q^33 - 10 * q^37 + 8 * q^41 - 2 * q^53 + 6 * q^57 + 22 * q^61 - 6 * q^69 + 30 * q^73 - 18 * q^81 + 10 * q^85 + 14 * q^89 - 30 * q^93 + 24 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.73205	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$3$	1.73205	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$4$	0	0
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.19615	1.56670	0.783349	−	0.621582i	$-0.213510\pi$
$11$	5.19615	1.56670	0.783349	+	0.621582i	$0.213510\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	1.73205	0.447214
$16$	0	0
$17$	5.00000	1.21268	0.606339	−	0.795206i	$-0.292637\pi$
$17$	5.00000	1.21268	0.606339	+	0.795206i	$0.292637\pi$
$18$	0	0
$19$	1.73205	0.397360	0.198680	−	0.980064i	$-0.436335\pi$
$19$	1.73205	0.397360	0.198680	+	0.980064i	$0.436335\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.73205	−0.361158	−0.180579	−	0.983561i	$-0.557797\pi$
$23$	−1.73205	−0.361158	−0.180579	+	0.983561i	$0.557797\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	−5.19615	−1.00000
$28$	0	0
$29$	8.00000	1.48556	0.742781	−	0.669534i	$-0.233506\pi$
$29$	8.00000	1.48556	0.742781	+	0.669534i	$0.233506\pi$
$30$	0	0
$31$	−8.66025	−1.55543	−0.777714	−	0.628619i	$-0.783621\pi$
$31$	−8.66025	−1.55543	−0.777714	+	0.628619i	$0.783621\pi$
$32$	0	0
$33$	9.00000	1.56670
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−5.00000	−0.821995	−0.410997	−	0.911636i	$-0.634819\pi$
$37$	−5.00000	−0.821995	−0.410997	+	0.911636i	$0.634819\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	4.00000	0.624695	0.312348	−	0.949968i	$-0.398885\pi$
$41$	4.00000	0.624695	0.312348	+	0.949968i	$0.398885\pi$
$42$	0	0
$43$	6.92820	1.05654	0.528271	−	0.849076i	$-0.322841\pi$
$43$	6.92820	1.05654	0.528271	+	0.849076i	$0.322841\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.66025	1.26323	0.631614	−	0.775283i	$-0.282393\pi$
$47$	8.66025	1.26323	0.631614	+	0.775283i	$0.282393\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	8.66025	1.21268
$52$	0	0
$53$	−1.00000	−0.137361	−0.0686803	−	0.997639i	$-0.521879\pi$
$53$	−1.00000	−0.137361	−0.0686803	+	0.997639i	$0.521879\pi$
$54$	0	0
$55$	5.19615	0.700649
$56$	0	0
$57$	3.00000	0.397360
$58$	0	0
$59$	1.73205	0.225494	0.112747	−	0.993624i	$-0.464035\pi$
$59$	1.73205	0.225494	0.112747	+	0.993624i	$0.464035\pi$
$60$	0	0
$61$	11.0000	1.40841	0.704203	−	0.709999i	$-0.251305\pi$
$61$	11.0000	1.40841	0.704203	+	0.709999i	$0.251305\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−12.1244	−1.48123	−0.740613	−	0.671932i	$-0.765465\pi$
$67$	−12.1244	−1.48123	−0.740613	+	0.671932i	$0.765465\pi$
$68$	0	0
$69$	−3.00000	−0.361158
$70$	0	0
$71$	−13.8564	−1.64445	−0.822226	−	0.569160i	$-0.807268\pi$
$71$	−13.8564	−1.64445	−0.822226	+	0.569160i	$0.807268\pi$
$72$	0	0
$73$	15.0000	1.75562	0.877809	−	0.479012i	$-0.159005\pi$
$73$	15.0000	1.75562	0.877809	+	0.479012i	$0.159005\pi$
$74$	0	0
$75$	−6.92820	−0.800000
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−1.73205	−0.194871	−0.0974355	−	0.995242i	$-0.531064\pi$
$79$	−1.73205	−0.194871	−0.0974355	+	0.995242i	$0.531064\pi$
$80$	0	0
$81$	−9.00000	−1.00000
$82$	0	0
$83$	6.92820	0.760469	0.380235	−	0.924890i	$-0.375843\pi$
$83$	6.92820	0.760469	0.380235	+	0.924890i	$0.375843\pi$
$84$	0	0
$85$	5.00000	0.542326
$86$	0	0
$87$	13.8564	1.48556
$88$	0	0
$89$	7.00000	0.741999	0.370999	−	0.928633i	$-0.379015\pi$
$89$	7.00000	0.741999	0.370999	+	0.928633i	$0.379015\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−15.0000	−1.55543
$94$	0	0
$95$	1.73205	0.177705
$96$	0	0
$97$	12.0000	1.21842	0.609208	−	0.793011i	$-0.291488\pi$
$97$	12.0000	1.21842	0.609208	+	0.793011i	$0.291488\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−11.0000	−1.09454	−0.547270	−	0.836956i	$-0.684333\pi$
$101$	−11.0000	−1.09454	−0.547270	+	0.836956i	$0.684333\pi$
$102$	0	0
$103$	−15.5885	−1.53598	−0.767988	−	0.640464i	$-0.778742\pi$
$103$	−15.5885	−1.53598	−0.767988	+	0.640464i	$0.778742\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−12.1244	−1.17211	−0.586053	−	0.810273i	$-0.699319\pi$
$107$	−12.1244	−1.17211	−0.586053	+	0.810273i	$0.699319\pi$
$108$	0	0
$109$	−3.00000	−0.287348	−0.143674	−	0.989625i	$-0.545892\pi$
$109$	−3.00000	−0.287348	−0.143674	+	0.989625i	$0.545892\pi$
$110$	0	0
$111$	−8.66025	−0.821995
$112$	0	0
$113$	4.00000	0.376288	0.188144	−	0.982141i	$-0.439753\pi$
$113$	4.00000	0.376288	0.188144	+	0.982141i	$0.439753\pi$
$114$	0	0
$115$	−1.73205	−0.161515
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	16.0000	1.45455
$122$	0	0
$123$	6.92820	0.624695
$124$	0	0
$125$	−9.00000	−0.804984
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	0	0
$129$	12.0000	1.05654
$130$	0	0
$131$	−8.66025	−0.756650	−0.378325	−	0.925673i	$-0.623500\pi$
$131$	−8.66025	−0.756650	−0.378325	+	0.925673i	$0.623500\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−5.19615	−0.447214
$136$	0	0
$137$	5.00000	0.427179	0.213589	−	0.976924i	$-0.431485\pi$
$137$	5.00000	0.427179	0.213589	+	0.976924i	$0.431485\pi$
$138$	0	0
$139$	6.92820	0.587643	0.293821	−	0.955860i	$-0.405073\pi$
$139$	6.92820	0.587643	0.293821	+	0.955860i	$0.405073\pi$
$140$	0	0
$141$	15.0000	1.26323
$142$	0	0
$143$	0	0
$144$	0	0
$145$	8.00000	0.664364
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−7.00000	−0.573462	−0.286731	−	0.958011i	$-0.592569\pi$
$149$	−7.00000	−0.573462	−0.286731	+	0.958011i	$0.592569\pi$
$150$	0	0
$151$	−8.66025	−0.704761	−0.352381	−	0.935857i	$-0.614628\pi$
$151$	−8.66025	−0.704761	−0.352381	+	0.935857i	$0.614628\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−8.66025	−0.695608
$156$	0	0
$157$	7.00000	0.558661	0.279330	−	0.960195i	$-0.409888\pi$
$157$	7.00000	0.558661	0.279330	+	0.960195i	$0.409888\pi$
$158$	0	0
$159$	−1.73205	−0.137361
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−8.66025	−0.678323	−0.339162	−	0.940728i	$-0.610143\pi$
$163$	−8.66025	−0.678323	−0.339162	+	0.940728i	$0.610143\pi$
$164$	0	0
$165$	9.00000	0.700649
$166$	0	0
$167$	−3.46410	−0.268060	−0.134030	−	0.990977i	$-0.542792\pi$
$167$	−3.46410	−0.268060	−0.134030	+	0.990977i	$0.542792\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−5.00000	−0.380143	−0.190071	−	0.981770i	$-0.560872\pi$
$173$	−5.00000	−0.380143	−0.190071	+	0.981770i	$0.560872\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	3.00000	0.225494
$178$	0	0
$179$	−8.66025	−0.647298	−0.323649	−	0.946177i	$-0.604910\pi$
$179$	−8.66025	−0.647298	−0.323649	+	0.946177i	$0.604910\pi$
$180$	0	0
$181$	−6.00000	−0.445976	−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000	−0.445976	−0.222988	+	0.974821i	$0.571581\pi$
$182$	0	0
$183$	19.0526	1.40841
$184$	0	0
$185$	−5.00000	−0.367607
$186$	0	0
$187$	25.9808	1.89990
$188$	0	0
$189$	0	0
$190$	0	0
$191$	22.5167	1.62925	0.814624	−	0.579989i	$-0.196943\pi$
$191$	22.5167	1.62925	0.814624	+	0.579989i	$0.196943\pi$
$192$	0	0
$193$	−11.0000	−0.791797	−0.395899	−	0.918294i	$-0.629567\pi$
$193$	−11.0000	−0.791797	−0.395899	+	0.918294i	$0.629567\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−8.00000	−0.569976	−0.284988	−	0.958531i	$-0.591990\pi$
$197$	−8.00000	−0.569976	−0.284988	+	0.958531i	$0.591990\pi$
$198$	0	0
$199$	−8.66025	−0.613909	−0.306955	−	0.951724i	$-0.599310\pi$
$199$	−8.66025	−0.613909	−0.306955	+	0.951724i	$0.599310\pi$
$200$	0	0
$201$	−21.0000	−1.48123
$202$	0	0
$203$	0	0
$204$	0	0
$205$	4.00000	0.279372
$206$	0	0
$207$	0	0
$208$	0	0
$209$	9.00000	0.622543
$210$	0	0
$211$	−20.7846	−1.43087	−0.715436	−	0.698679i	$-0.753772\pi$
$211$	−20.7846	−1.43087	−0.715436	+	0.698679i	$0.753772\pi$
$212$	0	0
$213$	−24.0000	−1.64445
$214$	0	0
$215$	6.92820	0.472500
$216$	0	0
$217$	0	0
$218$	0	0
$219$	25.9808	1.75562
$220$	0	0
$221$	0	0
$222$	0	0
$223$	13.8564	0.927894	0.463947	−	0.885863i	$-0.346433\pi$
$223$	13.8564	0.927894	0.463947	+	0.885863i	$0.346433\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−8.66025	−0.574801	−0.287401	−	0.957810i	$-0.592791\pi$
$227$	−8.66025	−0.574801	−0.287401	+	0.957810i	$0.592791\pi$
$228$	0	0
$229$	−15.0000	−0.991228	−0.495614	−	0.868543i	$-0.665057\pi$
$229$	−15.0000	−0.991228	−0.495614	+	0.868543i	$0.665057\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	5.00000	0.327561	0.163780	−	0.986497i	$-0.447631\pi$
$233$	5.00000	0.327561	0.163780	+	0.986497i	$0.447631\pi$
$234$	0	0
$235$	8.66025	0.564933
$236$	0	0
$237$	−3.00000	−0.194871
$238$	0	0
$239$	17.3205	1.12037	0.560185	−	0.828367i	$-0.310730\pi$
$239$	17.3205	1.12037	0.560185	+	0.828367i	$0.310730\pi$
$240$	0	0
$241$	15.0000	0.966235	0.483117	−	0.875556i	$-0.339504\pi$
$241$	15.0000	0.966235	0.483117	+	0.875556i	$0.339504\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	12.0000	0.760469
$250$	0	0
$251$	−17.3205	−1.09326	−0.546630	−	0.837374i	$-0.684090\pi$
$251$	−17.3205	−1.09326	−0.546630	+	0.837374i	$0.684090\pi$
$252$	0	0
$253$	−9.00000	−0.565825
$254$	0	0
$255$	8.66025	0.542326
$256$	0	0
$257$	5.00000	0.311891	0.155946	−	0.987766i	$-0.450158\pi$
$257$	5.00000	0.311891	0.155946	+	0.987766i	$0.450158\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	1.73205	0.106803	0.0534014	−	0.998573i	$-0.482994\pi$
$263$	1.73205	0.106803	0.0534014	+	0.998573i	$0.482994\pi$
$264$	0	0
$265$	−1.00000	−0.0614295
$266$	0	0
$267$	12.1244	0.741999
$268$	0	0
$269$	−17.0000	−1.03651	−0.518254	−	0.855227i	$-0.673418\pi$
$269$	−17.0000	−1.03651	−0.518254	+	0.855227i	$0.673418\pi$
$270$	0	0
$271$	12.1244	0.736502	0.368251	−	0.929726i	$-0.379957\pi$
$271$	12.1244	0.736502	0.368251	+	0.929726i	$0.379957\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−20.7846	−1.25336
$276$	0	0
$277$	15.0000	0.901263	0.450631	−	0.892710i	$-0.351199\pi$
$277$	15.0000	0.901263	0.450631	+	0.892710i	$0.351199\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−20.0000	−1.19310	−0.596550	−	0.802576i	$-0.703462\pi$
$281$	−20.0000	−1.19310	−0.596550	+	0.802576i	$0.703462\pi$
$282$	0	0
$283$	−8.66025	−0.514799	−0.257399	−	0.966305i	$-0.582866\pi$
$283$	−8.66025	−0.514799	−0.257399	+	0.966305i	$0.582866\pi$
$284$	0	0
$285$	3.00000	0.177705
$286$	0	0
$287$	0	0
$288$	0	0
$289$	8.00000	0.470588
$290$	0	0
$291$	20.7846	1.21842
$292$	0	0
$293$	−10.0000	−0.584206	−0.292103	−	0.956387i	$-0.594355\pi$
$293$	−10.0000	−0.584206	−0.292103	+	0.956387i	$0.594355\pi$
$294$	0	0
$295$	1.73205	0.100844
$296$	0	0
$297$	−27.0000	−1.56670
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−19.0526	−1.09454
$304$	0	0
$305$	11.0000	0.629858
$306$	0	0
$307$	−20.7846	−1.18624	−0.593120	−	0.805114i	$-0.702104\pi$
$307$	−20.7846	−1.18624	−0.593120	+	0.805114i	$0.702104\pi$
$308$	0	0
$309$	−27.0000	−1.53598
$310$	0	0
$311$	−12.1244	−0.687509	−0.343755	−	0.939060i	$-0.611699\pi$
$311$	−12.1244	−0.687509	−0.343755	+	0.939060i	$0.611699\pi$
$312$	0	0
$313$	−25.0000	−1.41308	−0.706542	−	0.707671i	$-0.749746\pi$
$313$	−25.0000	−1.41308	−0.706542	+	0.707671i	$0.749746\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−5.00000	−0.280828	−0.140414	−	0.990093i	$-0.544843\pi$
$317$	−5.00000	−0.280828	−0.140414	+	0.990093i	$0.544843\pi$
$318$	0	0
$319$	41.5692	2.32743
$320$	0	0
$321$	−21.0000	−1.17211
$322$	0	0
$323$	8.66025	0.481869
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−5.19615	−0.287348
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−8.66025	−0.476011	−0.238005	−	0.971264i	$-0.576494\pi$
$331$	−8.66025	−0.476011	−0.238005	+	0.971264i	$0.576494\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−12.1244	−0.662424
$336$	0	0
$337$	−12.0000	−0.653682	−0.326841	−	0.945079i	$-0.605984\pi$
$337$	−12.0000	−0.653682	−0.326841	+	0.945079i	$0.605984\pi$
$338$	0	0
$339$	6.92820	0.376288
$340$	0	0
$341$	−45.0000	−2.43689
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−3.00000	−0.161515
$346$	0	0
$347$	−12.1244	−0.650870	−0.325435	−	0.945564i	$-0.605511\pi$
$347$	−12.1244	−0.650870	−0.325435	+	0.945564i	$0.605511\pi$
$348$	0	0
$349$	32.0000	1.71292	0.856460	−	0.516213i	$-0.172659\pi$
$349$	32.0000	1.71292	0.856460	+	0.516213i	$0.172659\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−11.0000	−0.585471	−0.292735	−	0.956193i	$-0.594566\pi$
$353$	−11.0000	−0.585471	−0.292735	+	0.956193i	$0.594566\pi$
$354$	0	0
$355$	−13.8564	−0.735422
$356$	0	0
$357$	0	0
$358$	0	0
$359$	8.66025	0.457071	0.228535	−	0.973536i	$-0.426606\pi$
$359$	8.66025	0.457071	0.228535	+	0.973536i	$0.426606\pi$
$360$	0	0
$361$	−16.0000	−0.842105
$362$	0	0
$363$	27.7128	1.45455
$364$	0	0
$365$	15.0000	0.785136
$366$	0	0
$367$	29.4449	1.53701	0.768505	−	0.639844i	$-0.221001\pi$
$367$	29.4449	1.53701	0.768505	+	0.639844i	$0.221001\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	35.0000	1.81223	0.906116	−	0.423030i	$-0.139034\pi$
$373$	35.0000	1.81223	0.906116	+	0.423030i	$0.139034\pi$
$374$	0	0
$375$	−15.5885	−0.804984
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−10.3923	−0.533817	−0.266908	−	0.963722i	$-0.586002\pi$
$379$	−10.3923	−0.533817	−0.266908	+	0.963722i	$0.586002\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−19.0526	−0.973540	−0.486770	−	0.873530i	$-0.661825\pi$
$383$	−19.0526	−0.973540	−0.486770	+	0.873530i	$0.661825\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	5.00000	0.253510	0.126755	−	0.991934i	$-0.459544\pi$
$389$	5.00000	0.253510	0.126755	+	0.991934i	$0.459544\pi$
$390$	0	0
$391$	−8.66025	−0.437968
$392$	0	0
$393$	−15.0000	−0.756650
$394$	0	0
$395$	−1.73205	−0.0871489
$396$	0	0
$397$	17.0000	0.853206	0.426603	−	0.904439i	$-0.359710\pi$
$397$	17.0000	0.853206	0.426603	+	0.904439i	$0.359710\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−19.0000	−0.948815	−0.474407	−	0.880305i	$-0.657338\pi$
$401$	−19.0000	−0.948815	−0.474407	+	0.880305i	$0.657338\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−9.00000	−0.447214
$406$	0	0
$407$	−25.9808	−1.28782
$408$	0	0
$409$	−3.00000	−0.148340	−0.0741702	−	0.997246i	$-0.523631\pi$
$409$	−3.00000	−0.148340	−0.0741702	+	0.997246i	$0.523631\pi$
$410$	0	0
$411$	8.66025	0.427179
$412$	0	0
$413$	0	0
$414$	0	0
$415$	6.92820	0.340092
$416$	0	0
$417$	12.0000	0.587643
$418$	0	0
$419$	34.6410	1.69232	0.846162	−	0.532925i	$-0.178907\pi$
$419$	34.6410	1.69232	0.846162	+	0.532925i	$0.178907\pi$
$420$	0	0
$421$	−24.0000	−1.16969	−0.584844	−	0.811146i	$-0.698844\pi$
$421$	−24.0000	−1.16969	−0.584844	+	0.811146i	$0.698844\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−20.0000	−0.970143
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−22.5167	−1.08459	−0.542295	−	0.840188i	$-0.682444\pi$
$431$	−22.5167	−1.08459	−0.542295	+	0.840188i	$0.682444\pi$
$432$	0	0
$433$	4.00000	0.192228	0.0961139	−	0.995370i	$-0.469359\pi$
$433$	4.00000	0.192228	0.0961139	+	0.995370i	$0.469359\pi$
$434$	0	0
$435$	13.8564	0.664364
$436$	0	0
$437$	−3.00000	−0.143509
$438$	0	0
$439$	8.66025	0.413331	0.206666	−	0.978412i	$-0.433739\pi$
$439$	8.66025	0.413331	0.206666	+	0.978412i	$0.433739\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	15.5885	0.740630	0.370315	−	0.928906i	$-0.379250\pi$
$443$	15.5885	0.740630	0.370315	+	0.928906i	$0.379250\pi$
$444$	0	0
$445$	7.00000	0.331832
$446$	0	0
$447$	−12.1244	−0.573462
$448$	0	0
$449$	28.0000	1.32140	0.660701	−	0.750649i	$-0.270259\pi$
$449$	28.0000	1.32140	0.660701	+	0.750649i	$0.270259\pi$
$450$	0	0
$451$	20.7846	0.978709
$452$	0	0
$453$	−15.0000	−0.704761
$454$	0	0
$455$	0	0
$456$	0	0
$457$	15.0000	0.701670	0.350835	−	0.936437i	$-0.385898\pi$
$457$	15.0000	0.701670	0.350835	+	0.936437i	$0.385898\pi$
$458$	0	0
$459$	−25.9808	−1.21268
$460$	0	0
$461$	−40.0000	−1.86299	−0.931493	−	0.363760i	$-0.881493\pi$
$461$	−40.0000	−1.86299	−0.931493	+	0.363760i	$0.881493\pi$
$462$	0	0
$463$	−27.7128	−1.28792	−0.643962	−	0.765058i	$-0.722710\pi$
$463$	−27.7128	−1.28792	−0.643962	+	0.765058i	$0.722710\pi$
$464$	0	0
$465$	−15.0000	−0.695608
$466$	0	0
$467$	5.19615	0.240449	0.120225	−	0.992747i	$-0.461639\pi$
$467$	5.19615	0.240449	0.120225	+	0.992747i	$0.461639\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	12.1244	0.558661
$472$	0	0
$473$	36.0000	1.65528
$474$	0	0
$475$	−6.92820	−0.317888
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−25.9808	−1.18709	−0.593546	−	0.804800i	$-0.702272\pi$
$479$	−25.9808	−1.18709	−0.593546	+	0.804800i	$0.702272\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	12.0000	0.544892
$486$	0	0
$487$	−25.9808	−1.17730	−0.588650	−	0.808388i	$-0.700341\pi$
$487$	−25.9808	−1.17730	−0.588650	+	0.808388i	$0.700341\pi$
$488$	0	0
$489$	−15.0000	−0.678323
$490$	0	0
$491$	17.3205	0.781664	0.390832	−	0.920462i	$-0.372187\pi$
$491$	17.3205	0.781664	0.390832	+	0.920462i	$0.372187\pi$
$492$	0	0
$493$	40.0000	1.80151
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	19.0526	0.852910	0.426455	−	0.904509i	$-0.359762\pi$
$499$	19.0526	0.852910	0.426455	+	0.904509i	$0.359762\pi$
$500$	0	0
$501$	−6.00000	−0.268060
$502$	0	0
$503$	27.7128	1.23565	0.617827	−	0.786314i	$-0.288013\pi$
$503$	27.7128	1.23565	0.617827	+	0.786314i	$0.288013\pi$
$504$	0	0
$505$	−11.0000	−0.489494
$506$	0	0
$507$	−22.5167	−1.00000
$508$	0	0
$509$	17.0000	0.753512	0.376756	−	0.926313i	$-0.377040\pi$
$509$	17.0000	0.753512	0.376756	+	0.926313i	$0.377040\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−9.00000	−0.397360
$514$	0	0
$515$	−15.5885	−0.686909
$516$	0	0
$517$	45.0000	1.97910
$518$	0	0
$519$	−8.66025	−0.380143
$520$	0	0
$521$	−25.0000	−1.09527	−0.547635	−	0.836717i	$-0.684472\pi$
$521$	−25.0000	−1.09527	−0.547635	+	0.836717i	$0.684472\pi$
$522$	0	0
$523$	−25.9808	−1.13606	−0.568030	−	0.823008i	$-0.692294\pi$
$523$	−25.9808	−1.13606	−0.568030	+	0.823008i	$0.692294\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−43.3013	−1.88623
$528$	0	0
$529$	−20.0000	−0.869565
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−12.1244	−0.524182
$536$	0	0
$537$	−15.0000	−0.647298
$538$	0	0
$539$	0	0
$540$	0	0
$541$	9.00000	0.386940	0.193470	−	0.981106i	$-0.438026\pi$
$541$	9.00000	0.386940	0.193470	+	0.981106i	$0.438026\pi$
$542$	0	0
$543$	−10.3923	−0.445976
$544$	0	0
$545$	−3.00000	−0.128506
$546$	0	0
$547$	−3.46410	−0.148114	−0.0740571	−	0.997254i	$-0.523595\pi$
$547$	−3.46410	−0.148114	−0.0740571	+	0.997254i	$0.523595\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	13.8564	0.590303
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−8.66025	−0.367607
$556$	0	0
$557$	−43.0000	−1.82197	−0.910984	−	0.412441i	$-0.864676\pi$
$557$	−43.0000	−1.82197	−0.910984	+	0.412441i	$0.864676\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	45.0000	1.89990
$562$	0	0
$563$	15.5885	0.656975	0.328488	−	0.944508i	$-0.393461\pi$
$563$	15.5885	0.656975	0.328488	+	0.944508i	$0.393461\pi$
$564$	0	0
$565$	4.00000	0.168281
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−25.0000	−1.04805	−0.524027	−	0.851701i	$-0.675571\pi$
$569$	−25.0000	−1.04805	−0.524027	+	0.851701i	$0.675571\pi$
$570$	0	0
$571$	5.19615	0.217452	0.108726	−	0.994072i	$-0.465323\pi$
$571$	5.19615	0.217452	0.108726	+	0.994072i	$0.465323\pi$
$572$	0	0
$573$	39.0000	1.62925
$574$	0	0
$575$	6.92820	0.288926
$576$	0	0
$577$	23.0000	0.957503	0.478751	−	0.877951i	$-0.341090\pi$
$577$	23.0000	0.957503	0.478751	+	0.877951i	$0.341090\pi$
$578$	0	0
$579$	−19.0526	−0.791797
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−5.19615	−0.215203
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−34.6410	−1.42979	−0.714894	−	0.699233i	$-0.753525\pi$
$587$	−34.6410	−1.42979	−0.714894	+	0.699233i	$0.753525\pi$
$588$	0	0
$589$	−15.0000	−0.618064
$590$	0	0
$591$	−13.8564	−0.569976
$592$	0	0
$593$	29.0000	1.19089	0.595444	−	0.803397i	$-0.296976\pi$
$593$	29.0000	1.19089	0.595444	+	0.803397i	$0.296976\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−15.0000	−0.613909
$598$	0	0
$599$	43.3013	1.76924	0.884621	−	0.466311i	$-0.154417\pi$
$599$	43.3013	1.76924	0.884621	+	0.466311i	$0.154417\pi$
$600$	0	0
$601$	−20.0000	−0.815817	−0.407909	−	0.913023i	$-0.633742\pi$
$601$	−20.0000	−0.815817	−0.407909	+	0.913023i	$0.633742\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	16.0000	0.650493
$606$	0	0
$607$	−5.19615	−0.210905	−0.105453	−	0.994424i	$-0.533629\pi$
$607$	−5.19615	−0.210905	−0.105453	+	0.994424i	$0.533629\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	5.00000	0.201948	0.100974	−	0.994889i	$-0.467804\pi$
$613$	5.00000	0.201948	0.100974	+	0.994889i	$0.467804\pi$
$614$	0	0
$615$	6.92820	0.279372
$616$	0	0
$617$	−20.0000	−0.805170	−0.402585	−	0.915383i	$-0.631888\pi$
$617$	−20.0000	−0.805170	−0.402585	+	0.915383i	$0.631888\pi$
$618$	0	0
$619$	15.5885	0.626553	0.313276	−	0.949662i	$-0.398573\pi$
$619$	15.5885	0.626553	0.313276	+	0.949662i	$0.398573\pi$
$620$	0	0
$621$	9.00000	0.361158
$622$	0	0
$623$	0	0
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	15.5885	0.622543
$628$	0	0
$629$	−25.0000	−0.996815
$630$	0	0
$631$	13.8564	0.551615	0.275807	−	0.961213i	$-0.411055\pi$
$631$	13.8564	0.551615	0.275807	+	0.961213i	$0.411055\pi$
$632$	0	0
$633$	−36.0000	−1.43087
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−1.00000	−0.0394976	−0.0197488	−	0.999805i	$-0.506287\pi$
$641$	−1.00000	−0.0394976	−0.0197488	+	0.999805i	$0.506287\pi$
$642$	0	0
$643$	−34.6410	−1.36611	−0.683054	−	0.730368i	$-0.739349\pi$
$643$	−34.6410	−1.36611	−0.683054	+	0.730368i	$0.739349\pi$
$644$	0	0
$645$	12.0000	0.472500
$646$	0	0
$647$	29.4449	1.15760	0.578799	−	0.815471i	$-0.303522\pi$
$647$	29.4449	1.15760	0.578799	+	0.815471i	$0.303522\pi$
$648$	0	0
$649$	9.00000	0.353281
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−5.00000	−0.195665	−0.0978326	−	0.995203i	$-0.531191\pi$
$653$	−5.00000	−0.195665	−0.0978326	+	0.995203i	$0.531191\pi$
$654$	0	0
$655$	−8.66025	−0.338384
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−6.92820	−0.269884	−0.134942	−	0.990853i	$-0.543085\pi$
$659$	−6.92820	−0.269884	−0.134942	+	0.990853i	$0.543085\pi$
$660$	0	0
$661$	31.0000	1.20576	0.602880	−	0.797832i	$-0.294020\pi$
$661$	31.0000	1.20576	0.602880	+	0.797832i	$0.294020\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−13.8564	−0.536522
$668$	0	0
$669$	24.0000	0.927894
$670$	0	0
$671$	57.1577	2.20655
$672$	0	0
$673$	20.0000	0.770943	0.385472	−	0.922720i	$-0.374039\pi$
$673$	20.0000	0.770943	0.385472	+	0.922720i	$0.374039\pi$
$674$	0	0
$675$	20.7846	0.800000
$676$	0	0
$677$	25.0000	0.960828	0.480414	−	0.877042i	$-0.340486\pi$
$677$	25.0000	0.960828	0.480414	+	0.877042i	$0.340486\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−15.0000	−0.574801
$682$	0	0
$683$	32.9090	1.25923	0.629613	−	0.776909i	$-0.283213\pi$
$683$	32.9090	1.25923	0.629613	+	0.776909i	$0.283213\pi$
$684$	0	0
$685$	5.00000	0.191040
$686$	0	0
$687$	−25.9808	−0.991228
$688$	0	0
$689$	0	0
$690$	0	0
$691$	5.19615	0.197671	0.0988355	−	0.995104i	$-0.468488\pi$
$691$	5.19615	0.197671	0.0988355	+	0.995104i	$0.468488\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	6.92820	0.262802
$696$	0	0
$697$	20.0000	0.757554
$698$	0	0
$699$	8.66025	0.327561
$700$	0	0
$701$	14.0000	0.528773	0.264386	−	0.964417i	$-0.414831\pi$
$701$	14.0000	0.528773	0.264386	+	0.964417i	$0.414831\pi$
$702$	0	0
$703$	−8.66025	−0.326628
$704$	0	0
$705$	15.0000	0.564933
$706$	0	0
$707$	0	0
$708$	0	0
$709$	27.0000	1.01401	0.507003	−	0.861944i	$-0.330753\pi$
$709$	27.0000	1.01401	0.507003	+	0.861944i	$0.330753\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	15.0000	0.561754
$714$	0	0
$715$	0	0
$716$	0	0
$717$	30.0000	1.12037
$718$	0	0
$719$	−15.5885	−0.581351	−0.290676	−	0.956822i	$-0.593880\pi$
$719$	−15.5885	−0.581351	−0.290676	+	0.956822i	$0.593880\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	25.9808	0.966235
$724$	0	0
$725$	−32.0000	−1.18845
$726$	0	0
$727$	−13.8564	−0.513906	−0.256953	−	0.966424i	$-0.582719\pi$
$727$	−13.8564	−0.513906	−0.256953	+	0.966424i	$0.582719\pi$
$728$	0	0
$729$	27.0000	1.00000
$730$	0	0
$731$	34.6410	1.28124
$732$	0	0
$733$	51.0000	1.88373	0.941864	−	0.335994i	$-0.109072\pi$
$733$	51.0000	1.88373	0.941864	+	0.335994i	$0.109072\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−63.0000	−2.32063
$738$	0	0
$739$	19.0526	0.700860	0.350430	−	0.936589i	$-0.386036\pi$
$739$	19.0526	0.700860	0.350430	+	0.936589i	$0.386036\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	27.7128	1.01668	0.508342	−	0.861155i	$-0.330258\pi$
$743$	27.7128	1.01668	0.508342	+	0.861155i	$0.330258\pi$
$744$	0	0
$745$	−7.00000	−0.256460
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	8.66025	0.316017	0.158009	−	0.987438i	$-0.449493\pi$
$751$	8.66025	0.316017	0.158009	+	0.987438i	$0.449493\pi$
$752$	0	0
$753$	−30.0000	−1.09326
$754$	0	0
$755$	−8.66025	−0.315179
$756$	0	0
$757$	0	0		−	1.00000i	$-0.5\pi$
$757$	0	0			1.00000i	$0.5\pi$
$758$	0	0
$759$	−15.5885	−0.565825
$760$	0	0
$761$	13.0000	0.471250	0.235625	−	0.971844i	$-0.424286\pi$
$761$	13.0000	0.471250	0.235625	+	0.971844i	$0.424286\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	20.0000	0.721218	0.360609	−	0.932717i	$-0.382569\pi$
$769$	20.0000	0.721218	0.360609	+	0.932717i	$0.382569\pi$
$770$	0	0
$771$	8.66025	0.311891
$772$	0	0
$773$	5.00000	0.179838	0.0899188	−	0.995949i	$-0.471339\pi$
$773$	5.00000	0.179838	0.0899188	+	0.995949i	$0.471339\pi$
$774$	0	0
$775$	34.6410	1.24434
$776$	0	0
$777$	0	0
$778$	0	0
$779$	6.92820	0.248229
$780$	0	0
$781$	−72.0000	−2.57636
$782$	0	0
$783$	−41.5692	−1.48556
$784$	0	0
$785$	7.00000	0.249841
$786$	0	0
$787$	−39.8372	−1.42004	−0.710021	−	0.704181i	$-0.751315\pi$
$787$	−39.8372	−1.42004	−0.710021	+	0.704181i	$0.751315\pi$
$788$	0	0
$789$	3.00000	0.106803
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−1.73205	−0.0614295
$796$	0	0
$797$	40.0000	1.41687	0.708436	−	0.705775i	$-0.249401\pi$
$797$	40.0000	1.41687	0.708436	+	0.705775i	$0.249401\pi$
$798$	0	0
$799$	43.3013	1.53189
$800$	0	0
$801$	0	0
$802$	0	0
$803$	77.9423	2.75052
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−29.4449	−1.03651
$808$	0	0
$809$	7.00000	0.246107	0.123053	−	0.992400i	$-0.460731\pi$
$809$	7.00000	0.246107	0.123053	+	0.992400i	$0.460731\pi$
$810$	0	0
$811$	34.6410	1.21641	0.608205	−	0.793780i	$-0.291890\pi$
$811$	34.6410	1.21641	0.608205	+	0.793780i	$0.291890\pi$
$812$	0	0
$813$	21.0000	0.736502
$814$	0	0
$815$	−8.66025	−0.303355
$816$	0	0
$817$	12.0000	0.419827
$818$	0	0
$819$	0	0
$820$	0	0
$821$	25.0000	0.872506	0.436253	−	0.899824i	$-0.356305\pi$
$821$	25.0000	0.872506	0.436253	+	0.899824i	$0.356305\pi$
$822$	0	0
$823$	15.5885	0.543379	0.271690	−	0.962385i	$-0.412418\pi$
$823$	15.5885	0.543379	0.271690	+	0.962385i	$0.412418\pi$
$824$	0	0
$825$	−36.0000	−1.25336
$826$	0	0
$827$	−20.7846	−0.722752	−0.361376	−	0.932420i	$-0.617693\pi$
$827$	−20.7846	−0.722752	−0.361376	+	0.932420i	$0.617693\pi$
$828$	0	0
$829$	45.0000	1.56291	0.781457	−	0.623959i	$-0.214477\pi$
$829$	45.0000	1.56291	0.781457	+	0.623959i	$0.214477\pi$
$830$	0	0
$831$	25.9808	0.901263
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−3.46410	−0.119880
$836$	0	0
$837$	45.0000	1.55543
$838$	0	0
$839$	−41.5692	−1.43513	−0.717564	−	0.696492i	$-0.754743\pi$
$839$	−41.5692	−1.43513	−0.717564	+	0.696492i	$0.754743\pi$
$840$	0	0
$841$	35.0000	1.20690
$842$	0	0
$843$	−34.6410	−1.19310
$844$	0	0
$845$	−13.0000	−0.447214
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−15.0000	−0.514799
$850$	0	0
$851$	8.66025	0.296870
$852$	0	0
$853$	16.0000	0.547830	0.273915	−	0.961754i	$-0.411681\pi$
$853$	16.0000	0.547830	0.273915	+	0.961754i	$0.411681\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	7.00000	0.239115	0.119558	−	0.992827i	$-0.461852\pi$
$857$	7.00000	0.239115	0.119558	+	0.992827i	$0.461852\pi$
$858$	0	0
$859$	−25.9808	−0.886452	−0.443226	−	0.896410i	$-0.646166\pi$
$859$	−25.9808	−0.886452	−0.443226	+	0.896410i	$0.646166\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	53.6936	1.82775	0.913875	−	0.405995i	$-0.133075\pi$
$863$	53.6936	1.82775	0.913875	+	0.405995i	$0.133075\pi$
$864$	0	0
$865$	−5.00000	−0.170005
$866$	0	0
$867$	13.8564	0.470588
$868$	0	0
$869$	−9.00000	−0.305304
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−7.00000	−0.236373	−0.118187	−	0.992991i	$-0.537708\pi$
$877$	−7.00000	−0.236373	−0.118187	+	0.992991i	$0.537708\pi$
$878$	0	0
$879$	−17.3205	−0.584206
$880$	0	0
$881$	−34.0000	−1.14549	−0.572745	−	0.819734i	$-0.694121\pi$
$881$	−34.0000	−1.14549	−0.572745	+	0.819734i	$0.694121\pi$
$882$	0	0
$883$	34.6410	1.16576	0.582882	−	0.812557i	$-0.301925\pi$
$883$	34.6410	1.16576	0.582882	+	0.812557i	$0.301925\pi$
$884$	0	0
$885$	3.00000	0.100844
$886$	0	0
$887$	−46.7654	−1.57023	−0.785114	−	0.619352i	$-0.787396\pi$
$887$	−46.7654	−1.57023	−0.785114	+	0.619352i	$0.787396\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−46.7654	−1.56670
$892$	0	0
$893$	15.0000	0.501956
$894$	0	0
$895$	−8.66025	−0.289480
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−69.2820	−2.31069
$900$	0	0
$901$	−5.00000	−0.166574
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−6.00000	−0.199447
$906$	0	0
$907$	−39.8372	−1.32277	−0.661386	−	0.750046i	$-0.730031\pi$
$907$	−39.8372	−1.32277	−0.661386	+	0.750046i	$0.730031\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	13.8564	0.459083	0.229542	−	0.973299i	$-0.426277\pi$
$911$	13.8564	0.459083	0.229542	+	0.973299i	$0.426277\pi$
$912$	0	0
$913$	36.0000	1.19143
$914$	0	0
$915$	19.0526	0.629858
$916$	0	0
$917$	0	0
$918$	0	0
$919$	36.3731	1.19984	0.599918	−	0.800061i	$-0.295200\pi$
$919$	36.3731	1.19984	0.599918	+	0.800061i	$0.295200\pi$
$920$	0	0
$921$	−36.0000	−1.18624
$922$	0	0
$923$	0	0
$924$	0	0
$925$	20.0000	0.657596
$926$	0	0
$927$	0	0
$928$	0	0
$929$	47.0000	1.54202	0.771010	−	0.636823i	$-0.219752\pi$
$929$	47.0000	1.54202	0.771010	+	0.636823i	$0.219752\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−21.0000	−0.687509
$934$	0	0
$935$	25.9808	0.849662
$936$	0	0
$937$	10.0000	0.326686	0.163343	−	0.986569i	$-0.447772\pi$
$937$	10.0000	0.326686	0.163343	+	0.986569i	$0.447772\pi$
$938$	0	0
$939$	−43.3013	−1.41308
$940$	0	0
$941$	−25.0000	−0.814977	−0.407488	−	0.913210i	$-0.633595\pi$
$941$	−25.0000	−0.814977	−0.407488	+	0.913210i	$0.633595\pi$
$942$	0	0
$943$	−6.92820	−0.225613
$944$	0	0
$945$	0	0
$946$	0	0
$947$	5.19615	0.168852	0.0844261	−	0.996430i	$-0.473094\pi$
$947$	5.19615	0.168852	0.0844261	+	0.996430i	$0.473094\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−8.66025	−0.280828
$952$	0	0
$953$	20.0000	0.647864	0.323932	−	0.946080i	$-0.394995\pi$
$953$	20.0000	0.647864	0.323932	+	0.946080i	$0.394995\pi$
$954$	0	0
$955$	22.5167	0.728622
$956$	0	0
$957$	72.0000	2.32743
$958$	0	0
$959$	0	0
$960$	0	0
$961$	44.0000	1.41935
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−11.0000	−0.354103
$966$	0	0
$967$	13.8564	0.445592	0.222796	−	0.974865i	$-0.428482\pi$
$967$	13.8564	0.445592	0.222796	+	0.974865i	$0.428482\pi$
$968$	0	0
$969$	15.0000	0.481869
$970$	0	0
$971$	−8.66025	−0.277921	−0.138960	−	0.990298i	$-0.544376\pi$
$971$	−8.66025	−0.277921	−0.138960	+	0.990298i	$0.544376\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	37.0000	1.18373	0.591867	−	0.806035i	$-0.298391\pi$
$977$	37.0000	1.18373	0.591867	+	0.806035i	$0.298391\pi$
$978$	0	0
$979$	36.3731	1.16249
$980$	0	0
$981$	0	0
$982$	0	0
$983$	32.9090	1.04963	0.524816	−	0.851215i	$-0.324134\pi$
$983$	32.9090	1.04963	0.524816	+	0.851215i	$0.324134\pi$
$984$	0	0
$985$	−8.00000	−0.254901
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−12.0000	−0.381578
$990$	0	0
$991$	−25.9808	−0.825306	−0.412653	−	0.910888i	$-0.635398\pi$
$991$	−25.9808	−0.825306	−0.412653	+	0.910888i	$0.635398\pi$
$992$	0	0
$993$	−15.0000	−0.476011
$994$	0	0
$995$	−8.66025	−0.274549
$996$	0	0
$997$	23.0000	0.728417	0.364209	−	0.931317i	$-0.381339\pi$
$997$	23.0000	0.728417	0.364209	+	0.931317i	$0.381339\pi$
$998$	0	0
$999$	25.9808	0.821995

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1568.2.a.s.1.2		2
4.3	odd	2	inner	1568.2.a.s.1.1		2
7.2	even	3		224.2.i.b.193.1	yes	4
7.3	odd	6		1568.2.i.u.961.2		4
7.4	even	3		224.2.i.b.65.1	✓	4
7.5	odd	6		1568.2.i.u.1537.2		4
7.6	odd	2		1568.2.a.n.1.1		2
8.3	odd	2		3136.2.a.bh.1.2		2
8.5	even	2		3136.2.a.bh.1.1		2
21.2	odd	6		2016.2.s.r.865.2		4
21.11	odd	6		2016.2.s.r.289.2		4
28.3	even	6		1568.2.i.u.961.1		4
28.11	odd	6		224.2.i.b.65.2	yes	4
28.19	even	6		1568.2.i.u.1537.1		4
28.23	odd	6		224.2.i.b.193.2	yes	4
28.27	even	2		1568.2.a.n.1.2		2
56.11	odd	6		448.2.i.i.65.1		4
56.13	odd	2		3136.2.a.bu.1.2		2
56.27	even	2		3136.2.a.bu.1.1		2
56.37	even	6		448.2.i.i.193.2		4
56.51	odd	6		448.2.i.i.193.1		4
56.53	even	6		448.2.i.i.65.2		4
84.11	even	6		2016.2.s.r.289.1		4
84.23	even	6		2016.2.s.r.865.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.2.i.b.65.1	✓	4	7.4	even	3
224.2.i.b.65.2	yes	4	28.11	odd	6
224.2.i.b.193.1	yes	4	7.2	even	3
224.2.i.b.193.2	yes	4	28.23	odd	6
448.2.i.i.65.1		4	56.11	odd	6
448.2.i.i.65.2		4	56.53	even	6
448.2.i.i.193.1		4	56.51	odd	6
448.2.i.i.193.2		4	56.37	even	6
1568.2.a.n.1.1		2	7.6	odd	2
1568.2.a.n.1.2		2	28.27	even	2
1568.2.a.s.1.1		2	4.3	odd	2	inner
1568.2.a.s.1.2		2	1.1	even	1	trivial
1568.2.i.u.961.1		4	28.3	even	6
1568.2.i.u.961.2		4	7.3	odd	6
1568.2.i.u.1537.1		4	28.19	even	6
1568.2.i.u.1537.2		4	7.5	odd	6
2016.2.s.r.289.1		4	84.11	even	6
2016.2.s.r.289.2		4	21.11	odd	6
2016.2.s.r.865.1		4	84.23	even	6
2016.2.s.r.865.2		4	21.2	odd	6
3136.2.a.bh.1.1		2	8.5	even	2
3136.2.a.bh.1.2		2	8.3	odd	2
3136.2.a.bu.1.1		2	56.27	even	2
3136.2.a.bu.1.2		2	56.13	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:	\( N \)	\(=\)	\( 1568 = 2^{5} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1568.a (trivial)

\(f(q)\)	\(=\)	\(q+1.73205 q^{3} +1.00000 q^{5} +O(q^{10})\)	\(q+1.73205 q^{3} +1.00000 q^{5} +5.19615 q^{11} +1.73205 q^{15} +5.00000 q^{17} +1.73205 q^{19} -1.73205 q^{23} -4.00000 q^{25} -5.19615 q^{27} +8.00000 q^{29} -8.66025 q^{31} +9.00000 q^{33} -5.00000 q^{37} +4.00000 q^{41} +6.92820 q^{43} +8.66025 q^{47} +8.66025 q^{51} -1.00000 q^{53} +5.19615 q^{55} +3.00000 q^{57} +1.73205 q^{59} +11.0000 q^{61} -12.1244 q^{67} -3.00000 q^{69} -13.8564 q^{71} +15.0000 q^{73} -6.92820 q^{75} -1.73205 q^{79} -9.00000 q^{81} +6.92820 q^{83} +5.00000 q^{85} +13.8564 q^{87} +7.00000 q^{89} -15.0000 q^{93} +1.73205 q^{95} +12.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5}+O(q^{10}) \) 2 * q + 2 * q^5	\( 2 q + 2 q^{5} + 10 q^{17} - 8 q^{25} + 16 q^{29} + 18 q^{33} - 10 q^{37} + 8 q^{41} - 2 q^{53} + 6 q^{57} + 22 q^{61} - 6 q^{69} + 30 q^{73} - 18 q^{81} + 10 q^{85} + 14 q^{89} - 30 q^{93} + 24 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 + 10 * q^17 - 8 * q^25 + 16 * q^29 + 18 * q^33 - 10 * q^37 + 8 * q^41 - 2 * q^53 + 6 * q^57 + 22 * q^61 - 6 * q^69 + 30 * q^73 - 18 * q^81 + 10 * q^85 + 14 * q^89 - 30 * q^93 + 24 * q^97

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