LMFDB - Embedded newform 7840.2.a.bq.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7840,2,Mod(1,7840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7840, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("7840.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7840 = 2^{5} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7840.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:	$62.6027151847$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{24})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 1 $ x^4 - 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.517638$ of defining polynomial
Character	$\chi$	$=$	7840.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205 q^{3} -1.00000 q^{5} +O(q^{10})$	$q+1.73205 q^{3} -1.00000 q^{5} +0.717439 q^{11} +4.41421 q^{13} -1.73205 q^{15} -1.24264 q^{17} +4.89898 q^{19} +5.91359 q^{23} +1.00000 q^{25} -5.19615 q^{27} +1.00000 q^{29} -1.01461 q^{31} +1.24264 q^{33} -4.24264 q^{37} +7.64564 q^{39} +1.41421 q^{41} +6.92820 q^{43} -0.297173 q^{47} -2.15232 q^{51} -0.242641 q^{53} -0.717439 q^{55} +8.48528 q^{57} -12.2474 q^{59} +0.343146 q^{61} -4.41421 q^{65} -10.8126 q^{67} +10.2426 q^{69} +10.3923 q^{71} +2.82843 q^{73} +1.73205 q^{75} +2.74666 q^{79} -9.00000 q^{81} +6.92820 q^{83} +1.24264 q^{85} +1.73205 q^{87} +8.82843 q^{89} -1.75736 q^{93} -4.89898 q^{95} -10.0711 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{5}+O(q^{10}) $ 4 * q - 4 * q^5	$ 4 q - 4 q^{5} + 12 q^{13} + 12 q^{17} + 4 q^{25} + 4 q^{29} - 12 q^{33} + 16 q^{53} + 24 q^{61} - 12 q^{65} + 24 q^{69} - 36 q^{81} - 12 q^{85} + 24 q^{89} - 24 q^{93} - 12 q^{97}+O(q^{100}) $ 4 * q - 4 * q^5 + 12 * q^13 + 12 * q^17 + 4 * q^25 + 4 * q^29 - 12 * q^33 + 16 * q^53 + 24 * q^61 - 12 * q^65 + 24 * q^69 - 36 * q^81 - 12 * q^85 + 24 * q^89 - 24 * q^93 - 12 * 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
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0.717439	0.216316	0.108158	−	0.994134i	$-0.465505\pi$
$11$	0.717439	0.216316	0.108158	+	0.994134i	$0.465505\pi$
$12$	0	0
$13$	4.41421	1.22428	0.612141	−	0.790748i	$-0.290308\pi$
$13$	4.41421	1.22428	0.612141	+	0.790748i	$0.290308\pi$
$14$	0	0
$15$	−1.73205	−0.447214
$16$	0	0
$17$	−1.24264	−0.301385	−0.150692	−	0.988581i	$-0.548150\pi$
$17$	−1.24264	−0.301385	−0.150692	+	0.988581i	$0.548150\pi$
$18$	0	0
$19$	4.89898	1.12390	0.561951	−	0.827170i	$-0.310051\pi$
$19$	4.89898	1.12390	0.561951	+	0.827170i	$0.310051\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.91359	1.23307	0.616535	−	0.787328i	$-0.288536\pi$
$23$	5.91359	1.23307	0.616535	+	0.787328i	$0.288536\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.19615	−1.00000
$28$	0	0
$29$	1.00000	0.185695	0.0928477	−	0.995680i	$-0.470403\pi$
$29$	1.00000	0.185695	0.0928477	+	0.995680i	$0.470403\pi$
$30$	0	0
$31$	−1.01461	−0.182230	−0.0911148	−	0.995840i	$-0.529043\pi$
$31$	−1.01461	−0.182230	−0.0911148	+	0.995840i	$0.529043\pi$
$32$	0	0
$33$	1.24264	0.216316
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−4.24264	−0.697486	−0.348743	−	0.937218i	$-0.613391\pi$
$37$	−4.24264	−0.697486	−0.348743	+	0.937218i	$0.613391\pi$
$38$	0	0
$39$	7.64564	1.22428
$40$	0	0
$41$	1.41421	0.220863	0.110432	−	0.993884i	$-0.464777\pi$
$41$	1.41421	0.220863	0.110432	+	0.993884i	$0.464777\pi$
$42$	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
$47$	−0.297173	−0.0433471	−0.0216736	−	0.999765i	$-0.506899\pi$
$47$	−0.297173	−0.0433471	−0.0216736	+	0.999765i	$0.506899\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−2.15232	−0.301385
$52$	0	0
$53$	−0.242641	−0.0333293	−0.0166646	−	0.999861i	$-0.505305\pi$
$53$	−0.242641	−0.0333293	−0.0166646	+	0.999861i	$0.505305\pi$
$54$	0	0
$55$	−0.717439	−0.0967394
$56$	0	0
$57$	8.48528	1.12390
$58$	0	0
$59$	−12.2474	−1.59448	−0.797241	−	0.603661i	$-0.793708\pi$
$59$	−12.2474	−1.59448	−0.797241	+	0.603661i	$0.793708\pi$
$60$	0	0
$61$	0.343146	0.0439353	0.0219677	−	0.999759i	$-0.493007\pi$
$61$	0.343146	0.0439353	0.0219677	+	0.999759i	$0.493007\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−4.41421	−0.547516
$66$	0	0
$67$	−10.8126	−1.32097	−0.660483	−	0.750841i	$-0.729648\pi$
$67$	−10.8126	−1.32097	−0.660483	+	0.750841i	$0.729648\pi$
$68$	0	0
$69$	10.2426	1.23307
$70$	0	0
$71$	10.3923	1.23334	0.616670	−	0.787222i	$-0.288481\pi$
$71$	10.3923	1.23334	0.616670	+	0.787222i	$0.288481\pi$
$72$	0	0
$73$	2.82843	0.331042	0.165521	−	0.986206i	$-0.447069\pi$
$73$	2.82843	0.331042	0.165521	+	0.986206i	$0.447069\pi$
$74$	0	0
$75$	1.73205	0.200000
$76$	0	0
$77$	0	0
$78$	0	0
$79$	2.74666	0.309024	0.154512	−	0.987991i	$-0.450620\pi$
$79$	2.74666	0.309024	0.154512	+	0.987991i	$0.450620\pi$
$80$	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
$85$	1.24264	0.134783
$86$	0	0
$87$	1.73205	0.185695
$88$	0	0
$89$	8.82843	0.935811	0.467906	−	0.883778i	$-0.345009\pi$
$89$	8.82843	0.935811	0.467906	+	0.883778i	$0.345009\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−1.75736	−0.182230
$94$	0	0
$95$	−4.89898	−0.502625
$96$	0	0
$97$	−10.0711	−1.02256	−0.511281	−	0.859414i	$-0.670829\pi$
$97$	−10.0711	−1.02256	−0.511281	+	0.859414i	$0.670829\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	6.34315	0.631167	0.315583	−	0.948898i	$-0.397800\pi$
$101$	6.34315	0.631167	0.315583	+	0.948898i	$0.397800\pi$
$102$	0	0
$103$	14.9941	1.47741	0.738707	−	0.674027i	$-0.235437\pi$
$103$	14.9941	1.47741	0.738707	+	0.674027i	$0.235437\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6.33386	0.612317	0.306159	−	0.951981i	$-0.400956\pi$
$107$	6.33386	0.612317	0.306159	+	0.951981i	$0.400956\pi$
$108$	0	0
$109$	−15.4853	−1.48322	−0.741610	−	0.670831i	$-0.765938\pi$
$109$	−15.4853	−1.48322	−0.741610	+	0.670831i	$0.765938\pi$
$110$	0	0
$111$	−7.34847	−0.697486
$112$	0	0
$113$	8.72792	0.821054	0.410527	−	0.911848i	$-0.365345\pi$
$113$	8.72792	0.821054	0.410527	+	0.911848i	$0.365345\pi$
$114$	0	0
$115$	−5.91359	−0.551445
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.4853	−0.953207
$122$	0	0
$123$	2.44949	0.220863
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	10.8126	0.959461	0.479730	−	0.877416i	$-0.340735\pi$
$127$	10.8126	0.959461	0.479730	+	0.877416i	$0.340735\pi$
$128$	0	0
$129$	12.0000	1.05654
$130$	0	0
$131$	14.2767	1.24736	0.623679	−	0.781680i	$-0.285637\pi$
$131$	14.2767	1.24736	0.623679	+	0.781680i	$0.285637\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	5.19615	0.447214
$136$	0	0
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	0	0
$139$	−3.88437	−0.329468	−0.164734	−	0.986338i	$-0.552677\pi$
$139$	−3.88437	−0.329468	−0.164734	+	0.986338i	$0.552677\pi$
$140$	0	0
$141$	−0.514719	−0.0433471
$142$	0	0
$143$	3.16693	0.264832
$144$	0	0
$145$	−1.00000	−0.0830455
$146$	0	0
$147$	0	0
$148$	0	0
$149$	12.4853	1.02283	0.511417	−	0.859333i	$-0.329121\pi$
$149$	12.4853	1.02283	0.511417	+	0.859333i	$0.329121\pi$
$150$	0	0
$151$	4.77589	0.388656	0.194328	−	0.980937i	$-0.437747\pi$
$151$	4.77589	0.388656	0.194328	+	0.980937i	$0.437747\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1.01461	0.0814956
$156$	0	0
$157$	8.82843	0.704585	0.352293	−	0.935890i	$-0.385402\pi$
$157$	8.82843	0.704585	0.352293	+	0.935890i	$0.385402\pi$
$158$	0	0
$159$	−0.420266	−0.0333293
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−16.3059	−1.27718	−0.638588	−	0.769549i	$-0.720481\pi$
$163$	−16.3059	−1.27718	−0.638588	+	0.769549i	$0.720481\pi$
$164$	0	0
$165$	−1.24264	−0.0967394
$166$	0	0
$167$	−12.9649	−1.00325	−0.501627	−	0.865084i	$-0.667265\pi$
$167$	−12.9649	−1.00325	−0.501627	+	0.865084i	$0.667265\pi$
$168$	0	0
$169$	6.48528	0.498868
$170$	0	0
$171$	0	0
$172$	0	0
$173$	18.8995	1.43690	0.718451	−	0.695578i	$-0.244851\pi$
$173$	18.8995	1.43690	0.718451	+	0.695578i	$0.244851\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−21.2132	−1.59448
$178$	0	0
$179$	16.1318	1.20575	0.602874	−	0.797836i	$-0.294022\pi$
$179$	16.1318	1.20575	0.602874	+	0.797836i	$0.294022\pi$
$180$	0	0
$181$	19.4142	1.44305	0.721524	−	0.692390i	$-0.243442\pi$
$181$	19.4142	1.44305	0.721524	+	0.692390i	$0.243442\pi$
$182$	0	0
$183$	0.594346	0.0439353
$184$	0	0
$185$	4.24264	0.311925
$186$	0	0
$187$	−0.891519	−0.0651943
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2.74666	0.198742	0.0993708	−	0.995050i	$-0.468317\pi$
$191$	2.74666	0.198742	0.0993708	+	0.995050i	$0.468317\pi$
$192$	0	0
$193$	−14.9706	−1.07760	−0.538802	−	0.842432i	$-0.681123\pi$
$193$	−14.9706	−1.07760	−0.538802	+	0.842432i	$0.681123\pi$
$194$	0	0
$195$	−7.64564	−0.547516
$196$	0	0
$197$	26.2426	1.86971	0.934855	−	0.355029i	$-0.115529\pi$
$197$	26.2426	1.86971	0.934855	+	0.355029i	$0.115529\pi$
$198$	0	0
$199$	23.2341	1.64702	0.823511	−	0.567301i	$-0.192012\pi$
$199$	23.2341	1.64702	0.823511	+	0.567301i	$0.192012\pi$
$200$	0	0
$201$	−18.7279	−1.32097
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−1.41421	−0.0987730
$206$	0	0
$207$	0	0
$208$	0	0
$209$	3.51472	0.243118
$210$	0	0
$211$	−4.18154	−0.287869	−0.143935	−	0.989587i	$-0.545975\pi$
$211$	−4.18154	−0.287869	−0.143935	+	0.989587i	$0.545975\pi$
$212$	0	0
$213$	18.0000	1.23334
$214$	0	0
$215$	−6.92820	−0.472500
$216$	0	0
$217$	0	0
$218$	0	0
$219$	4.89898	0.331042
$220$	0	0
$221$	−5.48528	−0.368980
$222$	0	0
$223$	−7.22538	−0.483847	−0.241923	−	0.970295i	$-0.577778\pi$
$223$	−7.22538	−0.483847	−0.241923	+	0.970295i	$0.577778\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−21.3280	−1.41559	−0.707794	−	0.706419i	$-0.750309\pi$
$227$	−21.3280	−1.41559	−0.707794	+	0.706419i	$0.750309\pi$
$228$	0	0
$229$	7.07107	0.467269	0.233635	−	0.972324i	$-0.424938\pi$
$229$	7.07107	0.467269	0.233635	+	0.972324i	$0.424938\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−0.485281	−0.0317918	−0.0158959	−	0.999874i	$-0.505060\pi$
$233$	−0.485281	−0.0317918	−0.0158959	+	0.999874i	$0.505060\pi$
$234$	0	0
$235$	0.297173	0.0193854
$236$	0	0
$237$	4.75736	0.309024
$238$	0	0
$239$	−5.61642	−0.363296	−0.181648	−	0.983364i	$-0.558143\pi$
$239$	−5.61642	−0.363296	−0.181648	+	0.983364i	$0.558143\pi$
$240$	0	0
$241$	21.5563	1.38857	0.694283	−	0.719702i	$-0.255722\pi$
$241$	21.5563	1.38857	0.694283	+	0.719702i	$0.255722\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	21.6251	1.37597
$248$	0	0
$249$	12.0000	0.760469
$250$	0	0
$251$	17.7408	1.11979	0.559894	−	0.828564i	$-0.310842\pi$
$251$	17.7408	1.11979	0.559894	+	0.828564i	$0.310842\pi$
$252$	0	0
$253$	4.24264	0.266733
$254$	0	0
$255$	2.15232	0.134783
$256$	0	0
$257$	5.31371	0.331460	0.165730	−	0.986171i	$-0.447002\pi$
$257$	5.31371	0.331460	0.165730	+	0.986171i	$0.447002\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	11.2328	0.692646	0.346323	−	0.938115i	$-0.387430\pi$
$263$	11.2328	0.692646	0.346323	+	0.938115i	$0.387430\pi$
$264$	0	0
$265$	0.242641	0.0149053
$266$	0	0
$267$	15.2913	0.935811
$268$	0	0
$269$	−0.727922	−0.0443822	−0.0221911	−	0.999754i	$-0.507064\pi$
$269$	−0.727922	−0.0443822	−0.0221911	+	0.999754i	$0.507064\pi$
$270$	0	0
$271$	−6.33386	−0.384754	−0.192377	−	0.981321i	$-0.561620\pi$
$271$	−6.33386	−0.384754	−0.192377	+	0.981321i	$0.561620\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0.717439	0.0432632
$276$	0	0
$277$	1.75736	0.105589	0.0527947	−	0.998605i	$-0.483187\pi$
$277$	1.75736	0.105589	0.0527947	+	0.998605i	$0.483187\pi$
$278$	0	0
$279$	0	0
$280$	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
$283$	−6.03668	−0.358844	−0.179422	−	0.983772i	$-0.557423\pi$
$283$	−6.03668	−0.358844	−0.179422	+	0.983772i	$0.557423\pi$
$284$	0	0
$285$	−8.48528	−0.502625
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−15.4558	−0.909167
$290$	0	0
$291$	−17.4436	−1.02256
$292$	0	0
$293$	22.4142	1.30945	0.654726	−	0.755866i	$-0.272784\pi$
$293$	22.4142	1.30945	0.654726	+	0.755866i	$0.272784\pi$
$294$	0	0
$295$	12.2474	0.713074
$296$	0	0
$297$	−3.72792	−0.216316
$298$	0	0
$299$	26.1039	1.50962
$300$	0	0
$301$	0	0
$302$	0	0
$303$	10.9867	0.631167
$304$	0	0
$305$	−0.343146	−0.0196485
$306$	0	0
$307$	19.2987	1.10144	0.550719	−	0.834691i	$-0.314354\pi$
$307$	19.2987	1.10144	0.550719	+	0.834691i	$0.314354\pi$
$308$	0	0
$309$	25.9706	1.47741
$310$	0	0
$311$	−8.95743	−0.507929	−0.253965	−	0.967214i	$-0.581735\pi$
$311$	−8.95743	−0.507929	−0.253965	+	0.967214i	$0.581735\pi$
$312$	0	0
$313$	−24.5563	−1.38801	−0.694003	−	0.719972i	$-0.744155\pi$
$313$	−24.5563	−1.38801	−0.694003	+	0.719972i	$0.744155\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	26.9706	1.51482	0.757409	−	0.652941i	$-0.226465\pi$
$317$	26.9706	1.51482	0.757409	+	0.652941i	$0.226465\pi$
$318$	0	0
$319$	0.717439	0.0401689
$320$	0	0
$321$	10.9706	0.612317
$322$	0	0
$323$	−6.08767	−0.338727
$324$	0	0
$325$	4.41421	0.244857
$326$	0	0
$327$	−26.8213	−1.48322
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−5.49333	−0.301940	−0.150970	−	0.988538i	$-0.548240\pi$
$331$	−5.49333	−0.301940	−0.150970	+	0.988538i	$0.548240\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	10.8126	0.590754
$336$	0	0
$337$	31.6985	1.72673	0.863363	−	0.504583i	$-0.168354\pi$
$337$	31.6985	1.72673	0.863363	+	0.504583i	$0.168354\pi$
$338$	0	0
$339$	15.1172	0.821054
$340$	0	0
$341$	−0.727922	−0.0394192
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−10.2426	−0.551445
$346$	0	0
$347$	−23.4803	−1.26049	−0.630244	−	0.776397i	$-0.717045\pi$
$347$	−23.4803	−1.26049	−0.630244	+	0.776397i	$0.717045\pi$
$348$	0	0
$349$	9.17157	0.490943	0.245472	−	0.969404i	$-0.421057\pi$
$349$	9.17157	0.490943	0.245472	+	0.969404i	$0.421057\pi$
$350$	0	0
$351$	−22.9369	−1.22428
$352$	0	0
$353$	10.7574	0.572556	0.286278	−	0.958147i	$-0.407582\pi$
$353$	10.7574	0.572556	0.286278	+	0.958147i	$0.407582\pi$
$354$	0	0
$355$	−10.3923	−0.551566
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−31.4231	−1.65845	−0.829224	−	0.558917i	$-0.811217\pi$
$359$	−31.4231	−1.65845	−0.829224	+	0.558917i	$0.811217\pi$
$360$	0	0
$361$	5.00000	0.263158
$362$	0	0
$363$	−18.1610	−0.953207
$364$	0	0
$365$	−2.82843	−0.148047
$366$	0	0
$367$	−12.7187	−0.663911	−0.331955	−	0.943295i	$-0.607708\pi$
$367$	−12.7187	−0.663911	−0.331955	+	0.943295i	$0.607708\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	1.51472	0.0784292	0.0392146	−	0.999231i	$-0.487514\pi$
$373$	1.51472	0.0784292	0.0392146	+	0.999231i	$0.487514\pi$
$374$	0	0
$375$	−1.73205	−0.0894427
$376$	0	0
$377$	4.41421	0.227344
$378$	0	0
$379$	−23.0600	−1.18451	−0.592257	−	0.805749i	$-0.701763\pi$
$379$	−23.0600	−1.18451	−0.592257	+	0.805749i	$0.701763\pi$
$380$	0	0
$381$	18.7279	0.959461
$382$	0	0
$383$	−19.5959	−1.00130	−0.500652	−	0.865648i	$-0.666906\pi$
$383$	−19.5959	−1.00130	−0.500652	+	0.865648i	$0.666906\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−18.4558	−0.935748	−0.467874	−	0.883795i	$-0.654980\pi$
$389$	−18.4558	−0.935748	−0.467874	+	0.883795i	$0.654980\pi$
$390$	0	0
$391$	−7.34847	−0.371628
$392$	0	0
$393$	24.7279	1.24736
$394$	0	0
$395$	−2.74666	−0.138200
$396$	0	0
$397$	30.5563	1.53358	0.766790	−	0.641899i	$-0.221853\pi$
$397$	30.5563	1.53358	0.766790	+	0.641899i	$0.221853\pi$
$398$	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
$403$	−4.47871	−0.223101
$404$	0	0
$405$	9.00000	0.447214
$406$	0	0
$407$	−3.04384	−0.150877
$408$	0	0
$409$	20.1421	0.995965	0.497982	−	0.867187i	$-0.334074\pi$
$409$	20.1421	0.995965	0.497982	+	0.867187i	$0.334074\pi$
$410$	0	0
$411$	−3.46410	−0.170872
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−6.92820	−0.340092
$416$	0	0
$417$	−6.72792	−0.329468
$418$	0	0
$419$	3.29002	0.160728	0.0803640	−	0.996766i	$-0.474392\pi$
$419$	3.29002	0.160728	0.0803640	+	0.996766i	$0.474392\pi$
$420$	0	0
$421$	−18.5147	−0.902352	−0.451176	−	0.892435i	$-0.648995\pi$
$421$	−18.5147	−0.902352	−0.451176	+	0.892435i	$0.648995\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−1.24264	−0.0602769
$426$	0	0
$427$	0	0
$428$	0	0
$429$	5.48528	0.264832
$430$	0	0
$431$	9.92105	0.477880	0.238940	−	0.971034i	$-0.423200\pi$
$431$	9.92105	0.477880	0.238940	+	0.971034i	$0.423200\pi$
$432$	0	0
$433$	−17.3137	−0.832044	−0.416022	−	0.909355i	$-0.636576\pi$
$433$	−17.3137	−0.832044	−0.416022	+	0.909355i	$0.636576\pi$
$434$	0	0
$435$	−1.73205	−0.0830455
$436$	0	0
$437$	28.9706	1.38585
$438$	0	0
$439$	31.5972	1.50805	0.754026	−	0.656845i	$-0.228109\pi$
$439$	31.5972	1.50805	0.754026	+	0.656845i	$0.228109\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−24.4949	−1.16379	−0.581894	−	0.813265i	$-0.697688\pi$
$443$	−24.4949	−1.16379	−0.581894	+	0.813265i	$0.697688\pi$
$444$	0	0
$445$	−8.82843	−0.418508
$446$	0	0
$447$	21.6251	1.02283
$448$	0	0
$449$	30.4558	1.43730	0.718650	−	0.695372i	$-0.244760\pi$
$449$	30.4558	1.43730	0.718650	+	0.695372i	$0.244760\pi$
$450$	0	0
$451$	1.01461	0.0477762
$452$	0	0
$453$	8.27208	0.388656
$454$	0	0
$455$	0	0
$456$	0	0
$457$	1.75736	0.0822058	0.0411029	−	0.999155i	$-0.486913\pi$
$457$	1.75736	0.0822058	0.0411029	+	0.999155i	$0.486913\pi$
$458$	0	0
$459$	6.45695	0.301385
$460$	0	0
$461$	23.3137	1.08583	0.542914	−	0.839788i	$-0.317321\pi$
$461$	23.3137	1.08583	0.542914	+	0.839788i	$0.317321\pi$
$462$	0	0
$463$	−34.0467	−1.58228	−0.791141	−	0.611633i	$-0.790513\pi$
$463$	−34.0467	−1.58228	−0.791141	+	0.611633i	$0.790513\pi$
$464$	0	0
$465$	1.75736	0.0814956
$466$	0	0
$467$	−38.0541	−1.76094	−0.880468	−	0.474106i	$-0.842771\pi$
$467$	−38.0541	−1.76094	−0.880468	+	0.474106i	$0.842771\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	15.2913	0.704585
$472$	0	0
$473$	4.97056	0.228547
$474$	0	0
$475$	4.89898	0.224781
$476$	0	0
$477$	0	0
$478$	0	0
$479$	30.1623	1.37815	0.689075	−	0.724690i	$-0.258017\pi$
$479$	30.1623	1.37815	0.689075	+	0.724690i	$0.258017\pi$
$480$	0	0
$481$	−18.7279	−0.853920
$482$	0	0
$483$	0	0
$484$	0	0
$485$	10.0711	0.457304
$486$	0	0
$487$	−22.0454	−0.998973	−0.499486	−	0.866322i	$-0.666478\pi$
$487$	−22.0454	−0.998973	−0.499486	+	0.866322i	$0.666478\pi$
$488$	0	0
$489$	−28.2426	−1.27718
$490$	0	0
$491$	7.05130	0.318221	0.159110	−	0.987261i	$-0.449137\pi$
$491$	7.05130	0.318221	0.159110	+	0.987261i	$0.449137\pi$
$492$	0	0
$493$	−1.24264	−0.0559657
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−31.3000	−1.40118	−0.700590	−	0.713564i	$-0.747080\pi$
$499$	−31.3000	−1.40118	−0.700590	+	0.713564i	$0.747080\pi$
$500$	0	0
$501$	−22.4558	−1.00325
$502$	0	0
$503$	0.297173	0.0132503	0.00662514	−	0.999978i	$-0.497891\pi$
$503$	0.297173	0.0132503	0.00662514	+	0.999978i	$0.497891\pi$
$504$	0	0
$505$	−6.34315	−0.282266
$506$	0	0
$507$	11.2328	0.498868
$508$	0	0
$509$	16.2426	0.719942	0.359971	−	0.932963i	$-0.382787\pi$
$509$	16.2426	0.719942	0.359971	+	0.932963i	$0.382787\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−25.4558	−1.12390
$514$	0	0
$515$	−14.9941	−0.660719
$516$	0	0
$517$	−0.213203	−0.00937668
$518$	0	0
$519$	32.7349	1.43690
$520$	0	0
$521$	8.48528	0.371747	0.185873	−	0.982574i	$-0.440489\pi$
$521$	8.48528	0.371747	0.185873	+	0.982574i	$0.440489\pi$
$522$	0	0
$523$	25.6836	1.12306	0.561532	−	0.827455i	$-0.310212\pi$
$523$	25.6836	1.12306	0.561532	+	0.827455i	$0.310212\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.26080	0.0549212
$528$	0	0
$529$	11.9706	0.520459
$530$	0	0
$531$	0	0
$532$	0	0
$533$	6.24264	0.270399
$534$	0	0
$535$	−6.33386	−0.273837
$536$	0	0
$537$	27.9411	1.20575
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−33.0000	−1.41878	−0.709390	−	0.704816i	$-0.751030\pi$
$541$	−33.0000	−1.41878	−0.709390	+	0.704816i	$0.751030\pi$
$542$	0	0
$543$	33.6264	1.44305
$544$	0	0
$545$	15.4853	0.663317
$546$	0	0
$547$	−21.3790	−0.914098	−0.457049	−	0.889441i	$-0.651094\pi$
$547$	−21.3790	−0.914098	−0.457049	+	0.889441i	$0.651094\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	4.89898	0.208704
$552$	0	0
$553$	0	0
$554$	0	0
$555$	7.34847	0.311925
$556$	0	0
$557$	−17.4558	−0.739628	−0.369814	−	0.929106i	$-0.620579\pi$
$557$	−17.4558	−0.739628	−0.369814	+	0.929106i	$0.620579\pi$
$558$	0	0
$559$	30.5826	1.29350
$560$	0	0
$561$	−1.54416	−0.0651943
$562$	0	0
$563$	34.0467	1.43490	0.717448	−	0.696612i	$-0.245310\pi$
$563$	34.0467	1.43490	0.717448	+	0.696612i	$0.245310\pi$
$564$	0	0
$565$	−8.72792	−0.367186
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−0.970563	−0.0406881	−0.0203441	−	0.999793i	$-0.506476\pi$
$569$	−0.970563	−0.0406881	−0.0203441	+	0.999793i	$0.506476\pi$
$570$	0	0
$571$	29.9882	1.25497	0.627484	−	0.778629i	$-0.284085\pi$
$571$	29.9882	1.25497	0.627484	+	0.778629i	$0.284085\pi$
$572$	0	0
$573$	4.75736	0.198742
$574$	0	0
$575$	5.91359	0.246614
$576$	0	0
$577$	−11.8701	−0.494157	−0.247078	−	0.968995i	$-0.579471\pi$
$577$	−11.8701	−0.494157	−0.247078	+	0.968995i	$0.579471\pi$
$578$	0	0
$579$	−25.9298	−1.07760
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−0.174080	−0.00720965
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−16.7262	−0.690363	−0.345181	−	0.938536i	$-0.612183\pi$
$587$	−16.7262	−0.690363	−0.345181	+	0.938536i	$0.612183\pi$
$588$	0	0
$589$	−4.97056	−0.204808
$590$	0	0
$591$	45.4536	1.86971
$592$	0	0
$593$	7.24264	0.297420	0.148710	−	0.988881i	$-0.452488\pi$
$593$	7.24264	0.297420	0.148710	+	0.988881i	$0.452488\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	40.2426	1.64702
$598$	0	0
$599$	−16.0087	−0.654099	−0.327049	−	0.945007i	$-0.606054\pi$
$599$	−16.0087	−0.654099	−0.327049	+	0.945007i	$0.606054\pi$
$600$	0	0
$601$	−23.6569	−0.964983	−0.482492	−	0.875901i	$-0.660268\pi$
$601$	−23.6569	−0.964983	−0.482492	+	0.875901i	$0.660268\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	10.4853	0.426287
$606$	0	0
$607$	−44.7361	−1.81578	−0.907892	−	0.419204i	$-0.862309\pi$
$607$	−44.7361	−1.81578	−0.907892	+	0.419204i	$0.862309\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1.31178	−0.0530691
$612$	0	0
$613$	−30.0000	−1.21169	−0.605844	−	0.795583i	$-0.707165\pi$
$613$	−30.0000	−1.21169	−0.605844	+	0.795583i	$0.707165\pi$
$614$	0	0
$615$	−2.44949	−0.0987730
$616$	0	0
$617$	4.78680	0.192709	0.0963546	−	0.995347i	$-0.469282\pi$
$617$	4.78680	0.192709	0.0963546	+	0.995347i	$0.469282\pi$
$618$	0	0
$619$	−16.3059	−0.655389	−0.327695	−	0.944784i	$-0.606272\pi$
$619$	−16.3059	−0.655389	−0.327695	+	0.944784i	$0.606272\pi$
$620$	0	0
$621$	−30.7279	−1.23307
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	6.08767	0.243118
$628$	0	0
$629$	5.27208	0.210212
$630$	0	0
$631$	−15.4144	−0.613637	−0.306818	−	0.951768i	$-0.599264\pi$
$631$	−15.4144	−0.613637	−0.306818	+	0.951768i	$0.599264\pi$
$632$	0	0
$633$	−7.24264	−0.287869
$634$	0	0
$635$	−10.8126	−0.429084
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−17.4558	−0.689464	−0.344732	−	0.938701i	$-0.612030\pi$
$641$	−17.4558	−0.689464	−0.344732	+	0.938701i	$0.612030\pi$
$642$	0	0
$643$	38.6485	1.52415	0.762074	−	0.647490i	$-0.224181\pi$
$643$	38.6485	1.52415	0.762074	+	0.647490i	$0.224181\pi$
$644$	0	0
$645$	−12.0000	−0.472500
$646$	0	0
$647$	−24.8431	−0.976681	−0.488341	−	0.872653i	$-0.662398\pi$
$647$	−24.8431	−0.976681	−0.488341	+	0.872653i	$0.662398\pi$
$648$	0	0
$649$	−8.78680	−0.344912
$650$	0	0
$651$	0	0
$652$	0	0
$653$	12.9706	0.507577	0.253789	−	0.967260i	$-0.418323\pi$
$653$	12.9706	0.507577	0.253789	+	0.967260i	$0.418323\pi$
$654$	0	0
$655$	−14.2767	−0.557836
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−39.9093	−1.55464	−0.777322	−	0.629103i	$-0.783423\pi$
$659$	−39.9093	−1.55464	−0.777322	+	0.629103i	$0.783423\pi$
$660$	0	0
$661$	43.7990	1.70358	0.851792	−	0.523881i	$-0.175516\pi$
$661$	43.7990	1.70358	0.851792	+	0.523881i	$0.175516\pi$
$662$	0	0
$663$	−9.50079	−0.368980
$664$	0	0
$665$	0	0
$666$	0	0
$667$	5.91359	0.228975
$668$	0	0
$669$	−12.5147	−0.483847
$670$	0	0
$671$	0.246186	0.00950391
$672$	0	0
$673$	43.4558	1.67510	0.837550	−	0.546361i	$-0.183987\pi$
$673$	43.4558	1.67510	0.837550	+	0.546361i	$0.183987\pi$
$674$	0	0
$675$	−5.19615	−0.200000
$676$	0	0
$677$	33.3848	1.28308	0.641541	−	0.767089i	$-0.278296\pi$
$677$	33.3848	1.28308	0.641541	+	0.767089i	$0.278296\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−36.9411	−1.41559
$682$	0	0
$683$	−3.46410	−0.132550	−0.0662751	−	0.997801i	$-0.521111\pi$
$683$	−3.46410	−0.132550	−0.0662751	+	0.997801i	$0.521111\pi$
$684$	0	0
$685$	2.00000	0.0764161
$686$	0	0
$687$	12.2474	0.467269
$688$	0	0
$689$	−1.07107	−0.0408044
$690$	0	0
$691$	−41.2211	−1.56812	−0.784062	−	0.620683i	$-0.786855\pi$
$691$	−41.2211	−1.56812	−0.784062	+	0.620683i	$0.786855\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	3.88437	0.147342
$696$	0	0
$697$	−1.75736	−0.0665647
$698$	0	0
$699$	−0.840532	−0.0317918
$700$	0	0
$701$	−44.4558	−1.67907	−0.839537	−	0.543302i	$-0.817174\pi$
$701$	−44.4558	−1.67907	−0.839537	+	0.543302i	$0.817174\pi$
$702$	0	0
$703$	−20.7846	−0.783906
$704$	0	0
$705$	0.514719	0.0193854
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−43.0000	−1.61490	−0.807449	−	0.589937i	$-0.799153\pi$
$709$	−43.0000	−1.61490	−0.807449	+	0.589937i	$0.799153\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−6.00000	−0.224702
$714$	0	0
$715$	−3.16693	−0.118436
$716$	0	0
$717$	−9.72792	−0.363296
$718$	0	0
$719$	3.63818	0.135681	0.0678406	−	0.997696i	$-0.478389\pi$
$719$	3.63818	0.135681	0.0678406	+	0.997696i	$0.478389\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	37.3367	1.38857
$724$	0	0
$725$	1.00000	0.0371391
$726$	0	0
$727$	−20.1903	−0.748815	−0.374408	−	0.927264i	$-0.622154\pi$
$727$	−20.1903	−0.748815	−0.374408	+	0.927264i	$0.622154\pi$
$728$	0	0
$729$	27.0000	1.00000
$730$	0	0
$731$	−8.60927	−0.318425
$732$	0	0
$733$	3.38478	0.125020	0.0625098	−	0.998044i	$-0.480090\pi$
$733$	3.38478	0.125020	0.0625098	+	0.998044i	$0.480090\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−7.75736	−0.285746
$738$	0	0
$739$	39.9093	1.46809	0.734043	−	0.679103i	$-0.237631\pi$
$739$	39.9093	1.46809	0.734043	+	0.679103i	$0.237631\pi$
$740$	0	0
$741$	37.4558	1.37597
$742$	0	0
$743$	−35.3075	−1.29531	−0.647653	−	0.761936i	$-0.724249\pi$
$743$	−35.3075	−1.29531	−0.647653	+	0.761936i	$0.724249\pi$
$744$	0	0
$745$	−12.4853	−0.457425
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−50.6497	−1.84824	−0.924118	−	0.382108i	$-0.875198\pi$
$751$	−50.6497	−1.84824	−0.924118	+	0.382108i	$0.875198\pi$
$752$	0	0
$753$	30.7279	1.11979
$754$	0	0
$755$	−4.77589	−0.173812
$756$	0	0
$757$	12.4853	0.453785	0.226893	−	0.973920i	$-0.427143\pi$
$757$	12.4853	0.453785	0.226893	+	0.973920i	$0.427143\pi$
$758$	0	0
$759$	7.34847	0.266733
$760$	0	0
$761$	−41.3553	−1.49913	−0.749565	−	0.661931i	$-0.769737\pi$
$761$	−41.3553	−1.49913	−0.749565	+	0.661931i	$0.769737\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−54.0629	−1.95210
$768$	0	0
$769$	−27.1716	−0.979832	−0.489916	−	0.871770i	$-0.662973\pi$
$769$	−27.1716	−0.979832	−0.489916	+	0.871770i	$0.662973\pi$
$770$	0	0
$771$	9.20361	0.331460
$772$	0	0
$773$	−3.38478	−0.121742	−0.0608710	−	0.998146i	$-0.519388\pi$
$773$	−3.38478	−0.121742	−0.0608710	+	0.998146i	$0.519388\pi$
$774$	0	0
$775$	−1.01461	−0.0364459
$776$	0	0
$777$	0	0
$778$	0	0
$779$	6.92820	0.248229
$780$	0	0
$781$	7.45584	0.266791
$782$	0	0
$783$	−5.19615	−0.185695
$784$	0	0
$785$	−8.82843	−0.315100
$786$	0	0
$787$	−9.25460	−0.329891	−0.164945	−	0.986303i	$-0.552745\pi$
$787$	−9.25460	−0.329891	−0.164945	+	0.986303i	$0.552745\pi$
$788$	0	0
$789$	19.4558	0.692646
$790$	0	0
$791$	0	0
$792$	0	0
$793$	1.51472	0.0537892
$794$	0	0
$795$	0.420266	0.0149053
$796$	0	0
$797$	23.1005	0.818262	0.409131	−	0.912476i	$-0.365832\pi$
$797$	23.1005	0.818262	0.409131	+	0.912476i	$0.365832\pi$
$798$	0	0
$799$	0.369279	0.0130642
$800$	0	0
$801$	0	0
$802$	0	0
$803$	2.02922	0.0716098
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−1.26080	−0.0443822
$808$	0	0
$809$	22.5147	0.791575	0.395788	−	0.918342i	$-0.370472\pi$
$809$	22.5147	0.791575	0.395788	+	0.918342i	$0.370472\pi$
$810$	0	0
$811$	−24.6690	−0.866245	−0.433122	−	0.901335i	$-0.642588\pi$
$811$	−24.6690	−0.866245	−0.433122	+	0.901335i	$0.642588\pi$
$812$	0	0
$813$	−10.9706	−0.384754
$814$	0	0
$815$	16.3059	0.571171
$816$	0	0
$817$	33.9411	1.18745
$818$	0	0
$819$	0	0
$820$	0	0
$821$	31.4264	1.09679	0.548395	−	0.836220i	$-0.315239\pi$
$821$	31.4264	1.09679	0.548395	+	0.836220i	$0.315239\pi$
$822$	0	0
$823$	−3.63818	−0.126819	−0.0634095	−	0.997988i	$-0.520197\pi$
$823$	−3.63818	−0.126819	−0.0634095	+	0.997988i	$0.520197\pi$
$824$	0	0
$825$	1.24264	0.0432632
$826$	0	0
$827$	44.8592	1.55991	0.779954	−	0.625836i	$-0.215242\pi$
$827$	44.8592	1.55991	0.779954	+	0.625836i	$0.215242\pi$
$828$	0	0
$829$	−41.0122	−1.42441	−0.712206	−	0.701970i	$-0.752304\pi$
$829$	−41.0122	−1.42441	−0.712206	+	0.701970i	$0.752304\pi$
$830$	0	0
$831$	3.04384	0.105589
$832$	0	0
$833$	0	0
$834$	0	0
$835$	12.9649	0.448668
$836$	0	0
$837$	5.27208	0.182230
$838$	0	0
$839$	2.44949	0.0845658	0.0422829	−	0.999106i	$-0.486537\pi$
$839$	2.44949	0.0845658	0.0422829	+	0.999106i	$0.486537\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	0	0
$843$	20.7336	0.714104
$844$	0	0
$845$	−6.48528	−0.223100
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−10.4558	−0.358844
$850$	0	0
$851$	−25.0892	−0.860048
$852$	0	0
$853$	−3.17157	−0.108593	−0.0542963	−	0.998525i	$-0.517292\pi$
$853$	−3.17157	−0.108593	−0.0542963	+	0.998525i	$0.517292\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−44.1421	−1.50787	−0.753933	−	0.656951i	$-0.771846\pi$
$857$	−44.1421	−1.50787	−0.753933	+	0.656951i	$0.771846\pi$
$858$	0	0
$859$	−4.89898	−0.167151	−0.0835755	−	0.996501i	$-0.526634\pi$
$859$	−4.89898	−0.167151	−0.0835755	+	0.996501i	$0.526634\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−50.9469	−1.73425	−0.867127	−	0.498088i	$-0.834036\pi$
$863$	−50.9469	−1.73425	−0.867127	+	0.498088i	$0.834036\pi$
$864$	0	0
$865$	−18.8995	−0.642602
$866$	0	0
$867$	−26.7703	−0.909167
$868$	0	0
$869$	1.97056	0.0668468
$870$	0	0
$871$	−47.7290	−1.61724
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	52.9706	1.78869	0.894344	−	0.447379i	$-0.147643\pi$
$877$	52.9706	1.78869	0.894344	+	0.447379i	$0.147643\pi$
$878$	0	0
$879$	38.8226	1.30945
$880$	0	0
$881$	57.9411	1.95209	0.976043	−	0.217577i	$-0.0698155\pi$
$881$	57.9411	1.95209	0.976043	+	0.217577i	$0.0698155\pi$
$882$	0	0
$883$	40.9749	1.37891	0.689457	−	0.724326i	$-0.257849\pi$
$883$	40.9749	1.37891	0.689457	+	0.724326i	$0.257849\pi$
$884$	0	0
$885$	21.2132	0.713074
$886$	0	0
$887$	45.5257	1.52860	0.764302	−	0.644859i	$-0.223084\pi$
$887$	45.5257	1.52860	0.764302	+	0.644859i	$0.223084\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−6.45695	−0.216316
$892$	0	0
$893$	−1.45584	−0.0487180
$894$	0	0
$895$	−16.1318	−0.539227
$896$	0	0
$897$	45.2132	1.50962
$898$	0	0
$899$	−1.01461	−0.0338392
$900$	0	0
$901$	0.301515	0.0100449
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−19.4142	−0.645350
$906$	0	0
$907$	−51.1931	−1.69984	−0.849919	−	0.526913i	$-0.823350\pi$
$907$	−51.1931	−1.69984	−0.849919	+	0.526913i	$0.823350\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−25.6836	−0.850935	−0.425468	−	0.904974i	$-0.639890\pi$
$911$	−25.6836	−0.850935	−0.425468	+	0.904974i	$0.639890\pi$
$912$	0	0
$913$	4.97056	0.164502
$914$	0	0
$915$	−0.594346	−0.0196485
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−11.3559	−0.374598	−0.187299	−	0.982303i	$-0.559973\pi$
$919$	−11.3559	−0.374598	−0.187299	+	0.982303i	$0.559973\pi$
$920$	0	0
$921$	33.4264	1.10144
$922$	0	0
$923$	45.8739	1.50996
$924$	0	0
$925$	−4.24264	−0.139497
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−26.5269	−0.870320	−0.435160	−	0.900353i	$-0.643308\pi$
$929$	−26.5269	−0.870320	−0.435160	+	0.900353i	$0.643308\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−15.5147	−0.507929
$934$	0	0
$935$	0.891519	0.0291558
$936$	0	0
$937$	−22.4142	−0.732240	−0.366120	−	0.930568i	$-0.619314\pi$
$937$	−22.4142	−0.732240	−0.366120	+	0.930568i	$0.619314\pi$
$938$	0	0
$939$	−42.5328	−1.38801
$940$	0	0
$941$	−33.5147	−1.09255	−0.546274	−	0.837606i	$-0.683954\pi$
$941$	−33.5147	−1.09255	−0.546274	+	0.837606i	$0.683954\pi$
$942$	0	0
$943$	8.36308	0.272339
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−2.44949	−0.0795977	−0.0397989	−	0.999208i	$-0.512672\pi$
$947$	−2.44949	−0.0795977	−0.0397989	+	0.999208i	$0.512672\pi$
$948$	0	0
$949$	12.4853	0.405289
$950$	0	0
$951$	46.7144	1.51482
$952$	0	0
$953$	−51.6985	−1.67468	−0.837339	−	0.546684i	$-0.815890\pi$
$953$	−51.6985	−1.67468	−0.837339	+	0.546684i	$0.815890\pi$
$954$	0	0
$955$	−2.74666	−0.0888799
$956$	0	0
$957$	1.24264	0.0401689
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−29.9706	−0.966792
$962$	0	0
$963$	0	0
$964$	0	0
$965$	14.9706	0.481919
$966$	0	0
$967$	−23.0600	−0.741560	−0.370780	−	0.928721i	$-0.620910\pi$
$967$	−23.0600	−0.741560	−0.370780	+	0.928721i	$0.620910\pi$
$968$	0	0
$969$	−10.5442	−0.338727
$970$	0	0
$971$	33.2782	1.06795	0.533975	−	0.845500i	$-0.320698\pi$
$971$	33.2782	1.06795	0.533975	+	0.845500i	$0.320698\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	7.64564	0.244857
$976$	0	0
$977$	−2.54416	−0.0813948	−0.0406974	−	0.999172i	$-0.512958\pi$
$977$	−2.54416	−0.0813948	−0.0406974	+	0.999172i	$0.512958\pi$
$978$	0	0
$979$	6.33386	0.202431
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−19.2987	−0.615534	−0.307767	−	0.951462i	$-0.599582\pi$
$983$	−19.2987	−0.615534	−0.307767	+	0.951462i	$0.599582\pi$
$984$	0	0
$985$	−26.2426	−0.836160
$986$	0	0
$987$	0	0
$988$	0	0
$989$	40.9706	1.30279
$990$	0	0
$991$	5.14517	0.163442	0.0817208	−	0.996655i	$-0.473958\pi$
$991$	5.14517	0.163442	0.0817208	+	0.996655i	$0.473958\pi$
$992$	0	0
$993$	−9.51472	−0.301940
$994$	0	0
$995$	−23.2341	−0.736570
$996$	0	0
$997$	7.92893	0.251112	0.125556	−	0.992087i	$-0.459929\pi$
$997$	7.92893	0.251112	0.125556	+	0.992087i	$0.459929\pi$
$998$	0	0
$999$	22.0454	0.697486

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7840.2.a.bq.1.4	yes	4
4.3	odd	2	inner	7840.2.a.bq.1.1	✓	4
7.6	odd	2		7840.2.a.bt.1.2	yes	4
28.27	even	2		7840.2.a.bt.1.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7840.2.a.bq.1.1	✓	4	4.3	odd	2	inner
7840.2.a.bq.1.4	yes	4	1.1	even	1	trivial
7840.2.a.bt.1.2	yes	4	7.6	odd	2
7840.2.a.bt.1.3	yes	4	28.27	even	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:	\( N \)	\(=\)	\( 7840 = 2^{5} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7840.a (trivial)

\(f(q)\)	\(=\)	\(q+1.73205 q^{3} -1.00000 q^{5} +O(q^{10})\)	\(q+1.73205 q^{3} -1.00000 q^{5} +0.717439 q^{11} +4.41421 q^{13} -1.73205 q^{15} -1.24264 q^{17} +4.89898 q^{19} +5.91359 q^{23} +1.00000 q^{25} -5.19615 q^{27} +1.00000 q^{29} -1.01461 q^{31} +1.24264 q^{33} -4.24264 q^{37} +7.64564 q^{39} +1.41421 q^{41} +6.92820 q^{43} -0.297173 q^{47} -2.15232 q^{51} -0.242641 q^{53} -0.717439 q^{55} +8.48528 q^{57} -12.2474 q^{59} +0.343146 q^{61} -4.41421 q^{65} -10.8126 q^{67} +10.2426 q^{69} +10.3923 q^{71} +2.82843 q^{73} +1.73205 q^{75} +2.74666 q^{79} -9.00000 q^{81} +6.92820 q^{83} +1.24264 q^{85} +1.73205 q^{87} +8.82843 q^{89} -1.75736 q^{93} -4.89898 q^{95} -10.0711 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{5}+O(q^{10}) \) 4 * q - 4 * q^5	\( 4 q - 4 q^{5} + 12 q^{13} + 12 q^{17} + 4 q^{25} + 4 q^{29} - 12 q^{33} + 16 q^{53} + 24 q^{61} - 12 q^{65} + 24 q^{69} - 36 q^{81} - 12 q^{85} + 24 q^{89} - 24 q^{93} - 12 q^{97}+O(q^{100}) \) 4 * q - 4 * q^5 + 12 * q^13 + 12 * q^17 + 4 * q^25 + 4 * q^29 - 12 * q^33 + 16 * q^53 + 24 * q^61 - 12 * q^65 + 24 * q^69 - 36 * q^81 - 12 * q^85 + 24 * q^89 - 24 * q^93 - 12 * q^97

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