LMFDB - Embedded newform 7728.2.a.bj.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7728,2,Mod(1,7728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7728, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("7728.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7728.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:	$61.7083906820$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{33}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 8 $ x^2 - x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3864)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.37228$ of defining polynomial
Character	$\chi$	$=$	7728.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -2.37228 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -2.37228 q^{5} -1.00000 q^{7} +1.00000 q^{9} -3.37228 q^{11} -0.372281 q^{13} -2.37228 q^{15} +4.00000 q^{17} -7.37228 q^{19} -1.00000 q^{21} -1.00000 q^{23} +0.627719 q^{25} +1.00000 q^{27} -0.372281 q^{29} +6.00000 q^{31} -3.37228 q^{33} +2.37228 q^{35} -8.37228 q^{37} -0.372281 q^{39} +5.74456 q^{41} +2.37228 q^{43} -2.37228 q^{45} +7.74456 q^{47} +1.00000 q^{49} +4.00000 q^{51} -10.1168 q^{53} +8.00000 q^{55} -7.37228 q^{57} -9.37228 q^{59} -2.62772 q^{61} -1.00000 q^{63} +0.883156 q^{65} +4.00000 q^{67} -1.00000 q^{69} -12.7446 q^{71} +12.0000 q^{73} +0.627719 q^{75} +3.37228 q^{77} +2.74456 q^{79} +1.00000 q^{81} -2.74456 q^{83} -9.48913 q^{85} -0.372281 q^{87} +15.4891 q^{89} +0.372281 q^{91} +6.00000 q^{93} +17.4891 q^{95} +5.11684 q^{97} -3.37228 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + q^5 - 2 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} + q^{5} - 2 q^{7} + 2 q^{9} - q^{11} + 5 q^{13} + q^{15} + 8 q^{17} - 9 q^{19} - 2 q^{21} - 2 q^{23} + 7 q^{25} + 2 q^{27} + 5 q^{29} + 12 q^{31} - q^{33} - q^{35} - 11 q^{37} + 5 q^{39} - q^{43} + q^{45} + 4 q^{47} + 2 q^{49} + 8 q^{51} - 3 q^{53} + 16 q^{55} - 9 q^{57} - 13 q^{59} - 11 q^{61} - 2 q^{63} + 19 q^{65} + 8 q^{67} - 2 q^{69} - 14 q^{71} + 24 q^{73} + 7 q^{75} + q^{77} - 6 q^{79} + 2 q^{81} + 6 q^{83} + 4 q^{85} + 5 q^{87} + 8 q^{89} - 5 q^{91} + 12 q^{93} + 12 q^{95} - 7 q^{97} - q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + q^5 - 2 * q^7 + 2 * q^9 - q^11 + 5 * q^13 + q^15 + 8 * q^17 - 9 * q^19 - 2 * q^21 - 2 * q^23 + 7 * q^25 + 2 * q^27 + 5 * q^29 + 12 * q^31 - q^33 - q^35 - 11 * q^37 + 5 * q^39 - q^43 + q^45 + 4 * q^47 + 2 * q^49 + 8 * q^51 - 3 * q^53 + 16 * q^55 - 9 * q^57 - 13 * q^59 - 11 * q^61 - 2 * q^63 + 19 * q^65 + 8 * q^67 - 2 * q^69 - 14 * q^71 + 24 * q^73 + 7 * q^75 + q^77 - 6 * q^79 + 2 * q^81 + 6 * q^83 + 4 * q^85 + 5 * q^87 + 8 * q^89 - 5 * q^91 + 12 * q^93 + 12 * q^95 - 7 * q^97 - q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−2.37228	−1.06092	−0.530458	−	0.847711i	$-0.677980\pi$
$5$	−2.37228	−1.06092	−0.530458	+	0.847711i	$0.677980\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.37228	−1.01678	−0.508391	−	0.861127i	$-0.669759\pi$
$11$	−3.37228	−1.01678	−0.508391	+	0.861127i	$0.669759\pi$
$12$	0	0
$13$	−0.372281	−0.103252	−0.0516261	−	0.998666i	$-0.516440\pi$
$13$	−0.372281	−0.103252	−0.0516261	+	0.998666i	$0.516440\pi$
$14$	0	0
$15$	−2.37228	−0.612520
$16$	0	0
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	0	0
$19$	−7.37228	−1.69132	−0.845659	−	0.533724i	$-0.820792\pi$
$19$	−7.37228	−1.69132	−0.845659	+	0.533724i	$0.820792\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	0.627719	0.125544
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−0.372281	−0.0691309	−0.0345655	−	0.999402i	$-0.511005\pi$
$29$	−0.372281	−0.0691309	−0.0345655	+	0.999402i	$0.511005\pi$
$30$	0	0
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	0	0
$33$	−3.37228	−0.587039
$34$	0	0
$35$	2.37228	0.400989
$36$	0	0
$37$	−8.37228	−1.37639	−0.688197	−	0.725524i	$-0.741598\pi$
$37$	−8.37228	−1.37639	−0.688197	+	0.725524i	$0.741598\pi$
$38$	0	0
$39$	−0.372281	−0.0596127
$40$	0	0
$41$	5.74456	0.897150	0.448575	−	0.893745i	$-0.351932\pi$
$41$	5.74456	0.897150	0.448575	+	0.893745i	$0.351932\pi$
$42$	0	0
$43$	2.37228	0.361770	0.180885	−	0.983504i	$-0.442104\pi$
$43$	2.37228	0.361770	0.180885	+	0.983504i	$0.442104\pi$
$44$	0	0
$45$	−2.37228	−0.353639
$46$	0	0
$47$	7.74456	1.12966	0.564830	−	0.825207i	$-0.308942\pi$
$47$	7.74456	1.12966	0.564830	+	0.825207i	$0.308942\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	4.00000	0.560112
$52$	0	0
$53$	−10.1168	−1.38966	−0.694828	−	0.719176i	$-0.744519\pi$
$53$	−10.1168	−1.38966	−0.694828	+	0.719176i	$0.744519\pi$
$54$	0	0
$55$	8.00000	1.07872
$56$	0	0
$57$	−7.37228	−0.976483
$58$	0	0
$59$	−9.37228	−1.22017	−0.610084	−	0.792337i	$-0.708864\pi$
$59$	−9.37228	−1.22017	−0.610084	+	0.792337i	$0.708864\pi$
$60$	0	0
$61$	−2.62772	−0.336445	−0.168222	−	0.985749i	$-0.553803\pi$
$61$	−2.62772	−0.336445	−0.168222	+	0.985749i	$0.553803\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	0.883156	0.109542
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−12.7446	−1.51250	−0.756251	−	0.654282i	$-0.772971\pi$
$71$	−12.7446	−1.51250	−0.756251	+	0.654282i	$0.772971\pi$
$72$	0	0
$73$	12.0000	1.40449	0.702247	−	0.711934i	$-0.252180\pi$
$73$	12.0000	1.40449	0.702247	+	0.711934i	$0.252180\pi$
$74$	0	0
$75$	0.627719	0.0724827
$76$	0	0
$77$	3.37228	0.384307
$78$	0	0
$79$	2.74456	0.308787	0.154394	−	0.988009i	$-0.450658\pi$
$79$	2.74456	0.308787	0.154394	+	0.988009i	$0.450658\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−2.74456	−0.301255	−0.150627	−	0.988591i	$-0.548129\pi$
$83$	−2.74456	−0.301255	−0.150627	+	0.988591i	$0.548129\pi$
$84$	0	0
$85$	−9.48913	−1.02924
$86$	0	0
$87$	−0.372281	−0.0399127
$88$	0	0
$89$	15.4891	1.64184	0.820922	−	0.571040i	$-0.193460\pi$
$89$	15.4891	1.64184	0.820922	+	0.571040i	$0.193460\pi$
$90$	0	0
$91$	0.372281	0.0390257
$92$	0	0
$93$	6.00000	0.622171
$94$	0	0
$95$	17.4891	1.79435
$96$	0	0
$97$	5.11684	0.519537	0.259768	−	0.965671i	$-0.416354\pi$
$97$	5.11684	0.519537	0.259768	+	0.965671i	$0.416354\pi$
$98$	0	0
$99$	−3.37228	−0.338927
$100$	0	0
$101$	12.1168	1.20567	0.602836	−	0.797865i	$-0.294038\pi$
$101$	12.1168	1.20567	0.602836	+	0.797865i	$0.294038\pi$
$102$	0	0
$103$	−7.00000	−0.689730	−0.344865	−	0.938652i	$-0.612075\pi$
$103$	−7.00000	−0.689730	−0.344865	+	0.938652i	$0.612075\pi$
$104$	0	0
$105$	2.37228	0.231511
$106$	0	0
$107$	5.48913	0.530654	0.265327	−	0.964159i	$-0.414520\pi$
$107$	5.48913	0.530654	0.265327	+	0.964159i	$0.414520\pi$
$108$	0	0
$109$	10.3723	0.993484	0.496742	−	0.867898i	$-0.334529\pi$
$109$	10.3723	0.993484	0.496742	+	0.867898i	$0.334529\pi$
$110$	0	0
$111$	−8.37228	−0.794662
$112$	0	0
$113$	−7.62772	−0.717555	−0.358778	−	0.933423i	$-0.616806\pi$
$113$	−7.62772	−0.717555	−0.358778	+	0.933423i	$0.616806\pi$
$114$	0	0
$115$	2.37228	0.221216
$116$	0	0
$117$	−0.372281	−0.0344174
$118$	0	0
$119$	−4.00000	−0.366679
$120$	0	0
$121$	0.372281	0.0338438
$122$	0	0
$123$	5.74456	0.517970
$124$	0	0
$125$	10.3723	0.927725
$126$	0	0
$127$	1.00000	0.0887357	0.0443678	−	0.999015i	$-0.485873\pi$
$127$	1.00000	0.0887357	0.0443678	+	0.999015i	$0.485873\pi$
$128$	0	0
$129$	2.37228	0.208868
$130$	0	0
$131$	−4.11684	−0.359690	−0.179845	−	0.983695i	$-0.557560\pi$
$131$	−4.11684	−0.359690	−0.179845	+	0.983695i	$0.557560\pi$
$132$	0	0
$133$	7.37228	0.639258
$134$	0	0
$135$	−2.37228	−0.204173
$136$	0	0
$137$	18.4891	1.57963	0.789816	−	0.613343i	$-0.210176\pi$
$137$	18.4891	1.57963	0.789816	+	0.613343i	$0.210176\pi$
$138$	0	0
$139$	5.62772	0.477337	0.238668	−	0.971101i	$-0.423289\pi$
$139$	5.62772	0.477337	0.238668	+	0.971101i	$0.423289\pi$
$140$	0	0
$141$	7.74456	0.652210
$142$	0	0
$143$	1.25544	0.104985
$144$	0	0
$145$	0.883156	0.0733421
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	22.1168	1.81188	0.905941	−	0.423403i	$-0.139165\pi$
$149$	22.1168	1.81188	0.905941	+	0.423403i	$0.139165\pi$
$150$	0	0
$151$	−4.48913	−0.365320	−0.182660	−	0.983176i	$-0.558471\pi$
$151$	−4.48913	−0.365320	−0.182660	+	0.983176i	$0.558471\pi$
$152$	0	0
$153$	4.00000	0.323381
$154$	0	0
$155$	−14.2337	−1.14328
$156$	0	0
$157$	−12.8614	−1.02645	−0.513226	−	0.858254i	$-0.671550\pi$
$157$	−12.8614	−1.02645	−0.513226	+	0.858254i	$0.671550\pi$
$158$	0	0
$159$	−10.1168	−0.802318
$160$	0	0
$161$	1.00000	0.0788110
$162$	0	0
$163$	3.37228	0.264137	0.132069	−	0.991241i	$-0.457838\pi$
$163$	3.37228	0.264137	0.132069	+	0.991241i	$0.457838\pi$
$164$	0	0
$165$	8.00000	0.622799
$166$	0	0
$167$	−1.88316	−0.145723	−0.0728615	−	0.997342i	$-0.523213\pi$
$167$	−1.88316	−0.145723	−0.0728615	+	0.997342i	$0.523213\pi$
$168$	0	0
$169$	−12.8614	−0.989339
$170$	0	0
$171$	−7.37228	−0.563772
$172$	0	0
$173$	−7.48913	−0.569388	−0.284694	−	0.958618i	$-0.591892\pi$
$173$	−7.48913	−0.569388	−0.284694	+	0.958618i	$0.591892\pi$
$174$	0	0
$175$	−0.627719	−0.0474511
$176$	0	0
$177$	−9.37228	−0.704464
$178$	0	0
$179$	15.1168	1.12989	0.564943	−	0.825130i	$-0.308898\pi$
$179$	15.1168	1.12989	0.564943	+	0.825130i	$0.308898\pi$
$180$	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
$183$	−2.62772	−0.194247
$184$	0	0
$185$	19.8614	1.46024
$186$	0	0
$187$	−13.4891	−0.986423
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	12.8614	0.930619	0.465309	−	0.885148i	$-0.345943\pi$
$191$	12.8614	0.930619	0.465309	+	0.885148i	$0.345943\pi$
$192$	0	0
$193$	27.2337	1.96032	0.980162	−	0.198199i	$-0.0635091\pi$
$193$	27.2337	1.96032	0.980162	+	0.198199i	$0.0635091\pi$
$194$	0	0
$195$	0.883156	0.0632441
$196$	0	0
$197$	9.86141	0.702596	0.351298	−	0.936264i	$-0.385740\pi$
$197$	9.86141	0.702596	0.351298	+	0.936264i	$0.385740\pi$
$198$	0	0
$199$	−1.00000	−0.0708881	−0.0354441	−	0.999372i	$-0.511285\pi$
$199$	−1.00000	−0.0708881	−0.0354441	+	0.999372i	$0.511285\pi$
$200$	0	0
$201$	4.00000	0.282138
$202$	0	0
$203$	0.372281	0.0261290
$204$	0	0
$205$	−13.6277	−0.951801
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	24.8614	1.71970
$210$	0	0
$211$	11.3723	0.782900	0.391450	−	0.920199i	$-0.371974\pi$
$211$	11.3723	0.782900	0.391450	+	0.920199i	$0.371974\pi$
$212$	0	0
$213$	−12.7446	−0.873243
$214$	0	0
$215$	−5.62772	−0.383807
$216$	0	0
$217$	−6.00000	−0.407307
$218$	0	0
$219$	12.0000	0.810885
$220$	0	0
$221$	−1.48913	−0.100169
$222$	0	0
$223$	−14.0000	−0.937509	−0.468755	−	0.883328i	$-0.655297\pi$
$223$	−14.0000	−0.937509	−0.468755	+	0.883328i	$0.655297\pi$
$224$	0	0
$225$	0.627719	0.0418479
$226$	0	0
$227$	9.11684	0.605106	0.302553	−	0.953133i	$-0.402161\pi$
$227$	9.11684	0.605106	0.302553	+	0.953133i	$0.402161\pi$
$228$	0	0
$229$	−26.3505	−1.74129	−0.870646	−	0.491910i	$-0.836299\pi$
$229$	−26.3505	−1.74129	−0.870646	+	0.491910i	$0.836299\pi$
$230$	0	0
$231$	3.37228	0.221880
$232$	0	0
$233$	3.25544	0.213271	0.106635	−	0.994298i	$-0.465992\pi$
$233$	3.25544	0.213271	0.106635	+	0.994298i	$0.465992\pi$
$234$	0	0
$235$	−18.3723	−1.19848
$236$	0	0
$237$	2.74456	0.178279
$238$	0	0
$239$	9.48913	0.613800	0.306900	−	0.951742i	$-0.400708\pi$
$239$	9.48913	0.613800	0.306900	+	0.951742i	$0.400708\pi$
$240$	0	0
$241$	22.4891	1.44865	0.724326	−	0.689458i	$-0.242151\pi$
$241$	22.4891	1.44865	0.724326	+	0.689458i	$0.242151\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−2.37228	−0.151559
$246$	0	0
$247$	2.74456	0.174632
$248$	0	0
$249$	−2.74456	−0.173930
$250$	0	0
$251$	−9.86141	−0.622446	−0.311223	−	0.950337i	$-0.600739\pi$
$251$	−9.86141	−0.622446	−0.311223	+	0.950337i	$0.600739\pi$
$252$	0	0
$253$	3.37228	0.212014
$254$	0	0
$255$	−9.48913	−0.594232
$256$	0	0
$257$	28.1168	1.75388	0.876940	−	0.480599i	$-0.159581\pi$
$257$	28.1168	1.75388	0.876940	+	0.480599i	$0.159581\pi$
$258$	0	0
$259$	8.37228	0.520228
$260$	0	0
$261$	−0.372281	−0.0230436
$262$	0	0
$263$	−1.00000	−0.0616626	−0.0308313	−	0.999525i	$-0.509815\pi$
$263$	−1.00000	−0.0616626	−0.0308313	+	0.999525i	$0.509815\pi$
$264$	0	0
$265$	24.0000	1.47431
$266$	0	0
$267$	15.4891	0.947919
$268$	0	0
$269$	23.4891	1.43216	0.716079	−	0.698020i	$-0.245935\pi$
$269$	23.4891	1.43216	0.716079	+	0.698020i	$0.245935\pi$
$270$	0	0
$271$	−21.4891	−1.30537	−0.652686	−	0.757629i	$-0.726358\pi$
$271$	−21.4891	−1.30537	−0.652686	+	0.757629i	$0.726358\pi$
$272$	0	0
$273$	0.372281	0.0225315
$274$	0	0
$275$	−2.11684	−0.127650
$276$	0	0
$277$	18.1168	1.08854	0.544268	−	0.838912i	$-0.316808\pi$
$277$	18.1168	1.08854	0.544268	+	0.838912i	$0.316808\pi$
$278$	0	0
$279$	6.00000	0.359211
$280$	0	0
$281$	7.62772	0.455032	0.227516	−	0.973774i	$-0.426940\pi$
$281$	7.62772	0.455032	0.227516	+	0.973774i	$0.426940\pi$
$282$	0	0
$283$	−16.7446	−0.995361	−0.497680	−	0.867360i	$-0.665815\pi$
$283$	−16.7446	−0.995361	−0.497680	+	0.867360i	$0.665815\pi$
$284$	0	0
$285$	17.4891	1.03597
$286$	0	0
$287$	−5.74456	−0.339091
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	5.11684	0.299955
$292$	0	0
$293$	22.2337	1.29891	0.649453	−	0.760402i	$-0.274998\pi$
$293$	22.2337	1.29891	0.649453	+	0.760402i	$0.274998\pi$
$294$	0	0
$295$	22.2337	1.29450
$296$	0	0
$297$	−3.37228	−0.195680
$298$	0	0
$299$	0.372281	0.0215296
$300$	0	0
$301$	−2.37228	−0.136736
$302$	0	0
$303$	12.1168	0.696094
$304$	0	0
$305$	6.23369	0.356940
$306$	0	0
$307$	26.3723	1.50515	0.752573	−	0.658509i	$-0.228813\pi$
$307$	26.3723	1.50515	0.752573	+	0.658509i	$0.228813\pi$
$308$	0	0
$309$	−7.00000	−0.398216
$310$	0	0
$311$	−16.6277	−0.942871	−0.471436	−	0.881900i	$-0.656264\pi$
$311$	−16.6277	−0.942871	−0.471436	+	0.881900i	$0.656264\pi$
$312$	0	0
$313$	−31.0951	−1.75760	−0.878799	−	0.477192i	$-0.841655\pi$
$313$	−31.0951	−1.75760	−0.878799	+	0.477192i	$0.841655\pi$
$314$	0	0
$315$	2.37228	0.133663
$316$	0	0
$317$	10.8832	0.611259	0.305629	−	0.952151i	$-0.401133\pi$
$317$	10.8832	0.611259	0.305629	+	0.952151i	$0.401133\pi$
$318$	0	0
$319$	1.25544	0.0702910
$320$	0	0
$321$	5.48913	0.306373
$322$	0	0
$323$	−29.4891	−1.64082
$324$	0	0
$325$	−0.233688	−0.0129627
$326$	0	0
$327$	10.3723	0.573588
$328$	0	0
$329$	−7.74456	−0.426972
$330$	0	0
$331$	3.37228	0.185357	0.0926787	−	0.995696i	$-0.470457\pi$
$331$	3.37228	0.185357	0.0926787	+	0.995696i	$0.470457\pi$
$332$	0	0
$333$	−8.37228	−0.458798
$334$	0	0
$335$	−9.48913	−0.518446
$336$	0	0
$337$	−30.7446	−1.67476	−0.837382	−	0.546619i	$-0.815915\pi$
$337$	−30.7446	−1.67476	−0.837382	+	0.546619i	$0.815915\pi$
$338$	0	0
$339$	−7.62772	−0.414281
$340$	0	0
$341$	−20.2337	−1.09572
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	2.37228	0.127719
$346$	0	0
$347$	12.3723	0.664179	0.332089	−	0.943248i	$-0.392246\pi$
$347$	12.3723	0.664179	0.332089	+	0.943248i	$0.392246\pi$
$348$	0	0
$349$	−8.51087	−0.455577	−0.227788	−	0.973711i	$-0.573149\pi$
$349$	−8.51087	−0.455577	−0.227788	+	0.973711i	$0.573149\pi$
$350$	0	0
$351$	−0.372281	−0.0198709
$352$	0	0
$353$	10.8832	0.579252	0.289626	−	0.957140i	$-0.406469\pi$
$353$	10.8832	0.579252	0.289626	+	0.957140i	$0.406469\pi$
$354$	0	0
$355$	30.2337	1.60464
$356$	0	0
$357$	−4.00000	−0.211702
$358$	0	0
$359$	−3.86141	−0.203797	−0.101899	−	0.994795i	$-0.532492\pi$
$359$	−3.86141	−0.203797	−0.101899	+	0.994795i	$0.532492\pi$
$360$	0	0
$361$	35.3505	1.86055
$362$	0	0
$363$	0.372281	0.0195397
$364$	0	0
$365$	−28.4674	−1.49005
$366$	0	0
$367$	−26.4891	−1.38272	−0.691361	−	0.722510i	$-0.742988\pi$
$367$	−26.4891	−1.38272	−0.691361	+	0.722510i	$0.742988\pi$
$368$	0	0
$369$	5.74456	0.299050
$370$	0	0
$371$	10.1168	0.525240
$372$	0	0
$373$	3.48913	0.180660	0.0903300	−	0.995912i	$-0.471208\pi$
$373$	3.48913	0.180660	0.0903300	+	0.995912i	$0.471208\pi$
$374$	0	0
$375$	10.3723	0.535622
$376$	0	0
$377$	0.138593	0.00713792
$378$	0	0
$379$	13.8614	0.712013	0.356006	−	0.934484i	$-0.384138\pi$
$379$	13.8614	0.712013	0.356006	+	0.934484i	$0.384138\pi$
$380$	0	0
$381$	1.00000	0.0512316
$382$	0	0
$383$	8.23369	0.420722	0.210361	−	0.977624i	$-0.432536\pi$
$383$	8.23369	0.420722	0.210361	+	0.977624i	$0.432536\pi$
$384$	0	0
$385$	−8.00000	−0.407718
$386$	0	0
$387$	2.37228	0.120590
$388$	0	0
$389$	−3.48913	−0.176906	−0.0884528	−	0.996080i	$-0.528192\pi$
$389$	−3.48913	−0.176906	−0.0884528	+	0.996080i	$0.528192\pi$
$390$	0	0
$391$	−4.00000	−0.202289
$392$	0	0
$393$	−4.11684	−0.207667
$394$	0	0
$395$	−6.51087	−0.327598
$396$	0	0
$397$	14.0000	0.702640	0.351320	−	0.936255i	$-0.385733\pi$
$397$	14.0000	0.702640	0.351320	+	0.936255i	$0.385733\pi$
$398$	0	0
$399$	7.37228	0.369076
$400$	0	0
$401$	−4.11684	−0.205585	−0.102793	−	0.994703i	$-0.532778\pi$
$401$	−4.11684	−0.205585	−0.102793	+	0.994703i	$0.532778\pi$
$402$	0	0
$403$	−2.23369	−0.111268
$404$	0	0
$405$	−2.37228	−0.117880
$406$	0	0
$407$	28.2337	1.39949
$408$	0	0
$409$	16.2337	0.802704	0.401352	−	0.915924i	$-0.368540\pi$
$409$	16.2337	0.802704	0.401352	+	0.915924i	$0.368540\pi$
$410$	0	0
$411$	18.4891	0.912001
$412$	0	0
$413$	9.37228	0.461180
$414$	0	0
$415$	6.51087	0.319606
$416$	0	0
$417$	5.62772	0.275591
$418$	0	0
$419$	34.7446	1.69738	0.848691	−	0.528888i	$-0.177391\pi$
$419$	34.7446	1.69738	0.848691	+	0.528888i	$0.177391\pi$
$420$	0	0
$421$	26.3723	1.28531	0.642653	−	0.766157i	$-0.277834\pi$
$421$	26.3723	1.28531	0.642653	+	0.766157i	$0.277834\pi$
$422$	0	0
$423$	7.74456	0.376554
$424$	0	0
$425$	2.51087	0.121795
$426$	0	0
$427$	2.62772	0.127164
$428$	0	0
$429$	1.25544	0.0606131
$430$	0	0
$431$	−38.4891	−1.85396	−0.926978	−	0.375116i	$-0.877603\pi$
$431$	−38.4891	−1.85396	−0.926978	+	0.375116i	$0.877603\pi$
$432$	0	0
$433$	−5.51087	−0.264836	−0.132418	−	0.991194i	$-0.542274\pi$
$433$	−5.51087	−0.264836	−0.132418	+	0.991194i	$0.542274\pi$
$434$	0	0
$435$	0.883156	0.0423441
$436$	0	0
$437$	7.37228	0.352664
$438$	0	0
$439$	−7.25544	−0.346283	−0.173142	−	0.984897i	$-0.555392\pi$
$439$	−7.25544	−0.346283	−0.173142	+	0.984897i	$0.555392\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	29.3505	1.39449	0.697243	−	0.716835i	$-0.254410\pi$
$443$	29.3505	1.39449	0.697243	+	0.716835i	$0.254410\pi$
$444$	0	0
$445$	−36.7446	−1.74186
$446$	0	0
$447$	22.1168	1.04609
$448$	0	0
$449$	30.4674	1.43784	0.718922	−	0.695091i	$-0.244636\pi$
$449$	30.4674	1.43784	0.718922	+	0.695091i	$0.244636\pi$
$450$	0	0
$451$	−19.3723	−0.912205
$452$	0	0
$453$	−4.48913	−0.210918
$454$	0	0
$455$	−0.883156	−0.0414030
$456$	0	0
$457$	12.2337	0.572268	0.286134	−	0.958190i	$-0.407630\pi$
$457$	12.2337	0.572268	0.286134	+	0.958190i	$0.407630\pi$
$458$	0	0
$459$	4.00000	0.186704
$460$	0	0
$461$	−32.9783	−1.53595	−0.767975	−	0.640480i	$-0.778736\pi$
$461$	−32.9783	−1.53595	−0.767975	+	0.640480i	$0.778736\pi$
$462$	0	0
$463$	−9.23369	−0.429126	−0.214563	−	0.976710i	$-0.568833\pi$
$463$	−9.23369	−0.429126	−0.214563	+	0.976710i	$0.568833\pi$
$464$	0	0
$465$	−14.2337	−0.660071
$466$	0	0
$467$	−41.1168	−1.90266	−0.951330	−	0.308173i	$-0.900282\pi$
$467$	−41.1168	−1.90266	−0.951330	+	0.308173i	$0.900282\pi$
$468$	0	0
$469$	−4.00000	−0.184703
$470$	0	0
$471$	−12.8614	−0.592622
$472$	0	0
$473$	−8.00000	−0.367840
$474$	0	0
$475$	−4.62772	−0.212334
$476$	0	0
$477$	−10.1168	−0.463218
$478$	0	0
$479$	31.7228	1.44945	0.724726	−	0.689037i	$-0.241966\pi$
$479$	31.7228	1.44945	0.724726	+	0.689037i	$0.241966\pi$
$480$	0	0
$481$	3.11684	0.142116
$482$	0	0
$483$	1.00000	0.0455016
$484$	0	0
$485$	−12.1386	−0.551185
$486$	0	0
$487$	6.37228	0.288756	0.144378	−	0.989523i	$-0.453882\pi$
$487$	6.37228	0.288756	0.144378	+	0.989523i	$0.453882\pi$
$488$	0	0
$489$	3.37228	0.152500
$490$	0	0
$491$	40.2337	1.81572	0.907860	−	0.419272i	$-0.137715\pi$
$491$	40.2337	1.81572	0.907860	+	0.419272i	$0.137715\pi$
$492$	0	0
$493$	−1.48913	−0.0670668
$494$	0	0
$495$	8.00000	0.359573
$496$	0	0
$497$	12.7446	0.571672
$498$	0	0
$499$	7.25544	0.324798	0.162399	−	0.986725i	$-0.448077\pi$
$499$	7.25544	0.324798	0.162399	+	0.986725i	$0.448077\pi$
$500$	0	0
$501$	−1.88316	−0.0841332
$502$	0	0
$503$	1.76631	0.0787560	0.0393780	−	0.999224i	$-0.487462\pi$
$503$	1.76631	0.0787560	0.0393780	+	0.999224i	$0.487462\pi$
$504$	0	0
$505$	−28.7446	−1.27912
$506$	0	0
$507$	−12.8614	−0.571195
$508$	0	0
$509$	13.8832	0.615360	0.307680	−	0.951490i	$-0.400447\pi$
$509$	13.8832	0.615360	0.307680	+	0.951490i	$0.400447\pi$
$510$	0	0
$511$	−12.0000	−0.530849
$512$	0	0
$513$	−7.37228	−0.325494
$514$	0	0
$515$	16.6060	0.731746
$516$	0	0
$517$	−26.1168	−1.14862
$518$	0	0
$519$	−7.48913	−0.328736
$520$	0	0
$521$	11.4891	0.503348	0.251674	−	0.967812i	$-0.419019\pi$
$521$	11.4891	0.503348	0.251674	+	0.967812i	$0.419019\pi$
$522$	0	0
$523$	2.39403	0.104684	0.0523418	−	0.998629i	$-0.483331\pi$
$523$	2.39403	0.104684	0.0523418	+	0.998629i	$0.483331\pi$
$524$	0	0
$525$	−0.627719	−0.0273959
$526$	0	0
$527$	24.0000	1.04546
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−9.37228	−0.406722
$532$	0	0
$533$	−2.13859	−0.0926328
$534$	0	0
$535$	−13.0217	−0.562979
$536$	0	0
$537$	15.1168	0.652340
$538$	0	0
$539$	−3.37228	−0.145254
$540$	0	0
$541$	33.0951	1.42287	0.711435	−	0.702752i	$-0.248046\pi$
$541$	33.0951	1.42287	0.711435	+	0.702752i	$0.248046\pi$
$542$	0	0
$543$	−6.00000	−0.257485
$544$	0	0
$545$	−24.6060	−1.05400
$546$	0	0
$547$	−20.7446	−0.886973	−0.443487	−	0.896281i	$-0.646259\pi$
$547$	−20.7446	−0.886973	−0.443487	+	0.896281i	$0.646259\pi$
$548$	0	0
$549$	−2.62772	−0.112148
$550$	0	0
$551$	2.74456	0.116922
$552$	0	0
$553$	−2.74456	−0.116711
$554$	0	0
$555$	19.8614	0.843070
$556$	0	0
$557$	−3.76631	−0.159584	−0.0797919	−	0.996812i	$-0.525426\pi$
$557$	−3.76631	−0.159584	−0.0797919	+	0.996812i	$0.525426\pi$
$558$	0	0
$559$	−0.883156	−0.0373535
$560$	0	0
$561$	−13.4891	−0.569511
$562$	0	0
$563$	22.3723	0.942879	0.471440	−	0.881898i	$-0.343735\pi$
$563$	22.3723	0.942879	0.471440	+	0.881898i	$0.343735\pi$
$564$	0	0
$565$	18.0951	0.761266
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−1.00000	−0.0419222	−0.0209611	−	0.999780i	$-0.506673\pi$
$569$	−1.00000	−0.0419222	−0.0209611	+	0.999780i	$0.506673\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	0	0
$573$	12.8614	0.537293
$574$	0	0
$575$	−0.627719	−0.0261777
$576$	0	0
$577$	−12.5109	−0.520835	−0.260417	−	0.965496i	$-0.583860\pi$
$577$	−12.5109	−0.520835	−0.260417	+	0.965496i	$0.583860\pi$
$578$	0	0
$579$	27.2337	1.13179
$580$	0	0
$581$	2.74456	0.113864
$582$	0	0
$583$	34.1168	1.41298
$584$	0	0
$585$	0.883156	0.0365140
$586$	0	0
$587$	−14.3505	−0.592310	−0.296155	−	0.955140i	$-0.595704\pi$
$587$	−14.3505	−0.592310	−0.296155	+	0.955140i	$0.595704\pi$
$588$	0	0
$589$	−44.2337	−1.82262
$590$	0	0
$591$	9.86141	0.405644
$592$	0	0
$593$	29.1168	1.19569	0.597843	−	0.801613i	$-0.296025\pi$
$593$	29.1168	1.19569	0.597843	+	0.801613i	$0.296025\pi$
$594$	0	0
$595$	9.48913	0.389016
$596$	0	0
$597$	−1.00000	−0.0409273
$598$	0	0
$599$	10.7446	0.439011	0.219505	−	0.975611i	$-0.429556\pi$
$599$	10.7446	0.439011	0.219505	+	0.975611i	$0.429556\pi$
$600$	0	0
$601$	−14.9783	−0.610976	−0.305488	−	0.952196i	$-0.598820\pi$
$601$	−14.9783	−0.610976	−0.305488	+	0.952196i	$0.598820\pi$
$602$	0	0
$603$	4.00000	0.162893
$604$	0	0
$605$	−0.883156	−0.0359054
$606$	0	0
$607$	−16.7446	−0.679641	−0.339820	−	0.940490i	$-0.610366\pi$
$607$	−16.7446	−0.679641	−0.339820	+	0.940490i	$0.610366\pi$
$608$	0	0
$609$	0.372281	0.0150856
$610$	0	0
$611$	−2.88316	−0.116640
$612$	0	0
$613$	19.3505	0.781561	0.390780	−	0.920484i	$-0.372205\pi$
$613$	19.3505	0.781561	0.390780	+	0.920484i	$0.372205\pi$
$614$	0	0
$615$	−13.6277	−0.549523
$616$	0	0
$617$	−16.9783	−0.683519	−0.341759	−	0.939788i	$-0.611023\pi$
$617$	−16.9783	−0.683519	−0.341759	+	0.939788i	$0.611023\pi$
$618$	0	0
$619$	−35.7228	−1.43582	−0.717911	−	0.696135i	$-0.754901\pi$
$619$	−35.7228	−1.43582	−0.717911	+	0.696135i	$0.754901\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	−15.4891	−0.620559
$624$	0	0
$625$	−27.7446	−1.10978
$626$	0	0
$627$	24.8614	0.992869
$628$	0	0
$629$	−33.4891	−1.33530
$630$	0	0
$631$	−29.7228	−1.18325	−0.591623	−	0.806215i	$-0.701513\pi$
$631$	−29.7228	−1.18325	−0.591623	+	0.806215i	$0.701513\pi$
$632$	0	0
$633$	11.3723	0.452008
$634$	0	0
$635$	−2.37228	−0.0941411
$636$	0	0
$637$	−0.372281	−0.0147503
$638$	0	0
$639$	−12.7446	−0.504167
$640$	0	0
$641$	32.4891	1.28324	0.641622	−	0.767021i	$-0.278262\pi$
$641$	32.4891	1.28324	0.641622	+	0.767021i	$0.278262\pi$
$642$	0	0
$643$	0.116844	0.00460788	0.00230394	−	0.999997i	$-0.499267\pi$
$643$	0.116844	0.00460788	0.00230394	+	0.999997i	$0.499267\pi$
$644$	0	0
$645$	−5.62772	−0.221591
$646$	0	0
$647$	−48.4674	−1.90545	−0.952725	−	0.303835i	$-0.901733\pi$
$647$	−48.4674	−1.90545	−0.952725	+	0.303835i	$0.901733\pi$
$648$	0	0
$649$	31.6060	1.24064
$650$	0	0
$651$	−6.00000	−0.235159
$652$	0	0
$653$	−43.1168	−1.68729	−0.843646	−	0.536899i	$-0.819595\pi$
$653$	−43.1168	−1.68729	−0.843646	+	0.536899i	$0.819595\pi$
$654$	0	0
$655$	9.76631	0.381601
$656$	0	0
$657$	12.0000	0.468165
$658$	0	0
$659$	−16.0000	−0.623272	−0.311636	−	0.950202i	$-0.600877\pi$
$659$	−16.0000	−0.623272	−0.311636	+	0.950202i	$0.600877\pi$
$660$	0	0
$661$	−36.1168	−1.40478	−0.702391	−	0.711791i	$-0.747884\pi$
$661$	−36.1168	−1.40478	−0.702391	+	0.711791i	$0.747884\pi$
$662$	0	0
$663$	−1.48913	−0.0578328
$664$	0	0
$665$	−17.4891	−0.678199
$666$	0	0
$667$	0.372281	0.0144148
$668$	0	0
$669$	−14.0000	−0.541271
$670$	0	0
$671$	8.86141	0.342091
$672$	0	0
$673$	4.25544	0.164035	0.0820175	−	0.996631i	$-0.473864\pi$
$673$	4.25544	0.164035	0.0820175	+	0.996631i	$0.473864\pi$
$674$	0	0
$675$	0.627719	0.0241609
$676$	0	0
$677$	11.2554	0.432582	0.216291	−	0.976329i	$-0.430604\pi$
$677$	11.2554	0.432582	0.216291	+	0.976329i	$0.430604\pi$
$678$	0	0
$679$	−5.11684	−0.196366
$680$	0	0
$681$	9.11684	0.349358
$682$	0	0
$683$	−30.9783	−1.18535	−0.592675	−	0.805442i	$-0.701928\pi$
$683$	−30.9783	−1.18535	−0.592675	+	0.805442i	$0.701928\pi$
$684$	0	0
$685$	−43.8614	−1.67586
$686$	0	0
$687$	−26.3505	−1.00534
$688$	0	0
$689$	3.76631	0.143485
$690$	0	0
$691$	3.39403	0.129115	0.0645575	−	0.997914i	$-0.479436\pi$
$691$	3.39403	0.129115	0.0645575	+	0.997914i	$0.479436\pi$
$692$	0	0
$693$	3.37228	0.128102
$694$	0	0
$695$	−13.3505	−0.506415
$696$	0	0
$697$	22.9783	0.870363
$698$	0	0
$699$	3.25544	0.123132
$700$	0	0
$701$	42.3505	1.59956	0.799779	−	0.600295i	$-0.204950\pi$
$701$	42.3505	1.59956	0.799779	+	0.600295i	$0.204950\pi$
$702$	0	0
$703$	61.7228	2.32792
$704$	0	0
$705$	−18.3723	−0.691940
$706$	0	0
$707$	−12.1168	−0.455701
$708$	0	0
$709$	3.48913	0.131037	0.0655184	−	0.997851i	$-0.479130\pi$
$709$	3.48913	0.131037	0.0655184	+	0.997851i	$0.479130\pi$
$710$	0	0
$711$	2.74456	0.102929
$712$	0	0
$713$	−6.00000	−0.224702
$714$	0	0
$715$	−2.97825	−0.111380
$716$	0	0
$717$	9.48913	0.354378
$718$	0	0
$719$	−2.37228	−0.0884712	−0.0442356	−	0.999021i	$-0.514085\pi$
$719$	−2.37228	−0.0884712	−0.0442356	+	0.999021i	$0.514085\pi$
$720$	0	0
$721$	7.00000	0.260694
$722$	0	0
$723$	22.4891	0.836380
$724$	0	0
$725$	−0.233688	−0.00867895
$726$	0	0
$727$	18.3505	0.680584	0.340292	−	0.940320i	$-0.389474\pi$
$727$	18.3505	0.680584	0.340292	+	0.940320i	$0.389474\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	9.48913	0.350968
$732$	0	0
$733$	35.2119	1.30058	0.650291	−	0.759685i	$-0.274647\pi$
$733$	35.2119	1.30058	0.650291	+	0.759685i	$0.274647\pi$
$734$	0	0
$735$	−2.37228	−0.0875029
$736$	0	0
$737$	−13.4891	−0.496878
$738$	0	0
$739$	24.7446	0.910243	0.455122	−	0.890429i	$-0.349596\pi$
$739$	24.7446	0.910243	0.455122	+	0.890429i	$0.349596\pi$
$740$	0	0
$741$	2.74456	0.100824
$742$	0	0
$743$	−19.3723	−0.710700	−0.355350	−	0.934733i	$-0.615638\pi$
$743$	−19.3723	−0.710700	−0.355350	+	0.934733i	$0.615638\pi$
$744$	0	0
$745$	−52.4674	−1.92226
$746$	0	0
$747$	−2.74456	−0.100418
$748$	0	0
$749$	−5.48913	−0.200568
$750$	0	0
$751$	−22.9783	−0.838488	−0.419244	−	0.907874i	$-0.637705\pi$
$751$	−22.9783	−0.838488	−0.419244	+	0.907874i	$0.637705\pi$
$752$	0	0
$753$	−9.86141	−0.359370
$754$	0	0
$755$	10.6495	0.387574
$756$	0	0
$757$	14.2337	0.517332	0.258666	−	0.965967i	$-0.416717\pi$
$757$	14.2337	0.517332	0.258666	+	0.965967i	$0.416717\pi$
$758$	0	0
$759$	3.37228	0.122406
$760$	0	0
$761$	−35.0951	−1.27220	−0.636098	−	0.771608i	$-0.719453\pi$
$761$	−35.0951	−1.27220	−0.636098	+	0.771608i	$0.719453\pi$
$762$	0	0
$763$	−10.3723	−0.375502
$764$	0	0
$765$	−9.48913	−0.343080
$766$	0	0
$767$	3.48913	0.125985
$768$	0	0
$769$	−31.3505	−1.13053	−0.565265	−	0.824910i	$-0.691226\pi$
$769$	−31.3505	−1.13053	−0.565265	+	0.824910i	$0.691226\pi$
$770$	0	0
$771$	28.1168	1.01260
$772$	0	0
$773$	−34.3723	−1.23629	−0.618143	−	0.786066i	$-0.712115\pi$
$773$	−34.3723	−1.23629	−0.618143	+	0.786066i	$0.712115\pi$
$774$	0	0
$775$	3.76631	0.135290
$776$	0	0
$777$	8.37228	0.300354
$778$	0	0
$779$	−42.3505	−1.51737
$780$	0	0
$781$	42.9783	1.53788
$782$	0	0
$783$	−0.372281	−0.0133042
$784$	0	0
$785$	30.5109	1.08898
$786$	0	0
$787$	47.8397	1.70530	0.852650	−	0.522483i	$-0.174994\pi$
$787$	47.8397	1.70530	0.852650	+	0.522483i	$0.174994\pi$
$788$	0	0
$789$	−1.00000	−0.0356009
$790$	0	0
$791$	7.62772	0.271210
$792$	0	0
$793$	0.978251	0.0347387
$794$	0	0
$795$	24.0000	0.851192
$796$	0	0
$797$	−10.6060	−0.375683	−0.187841	−	0.982199i	$-0.560149\pi$
$797$	−10.6060	−0.375683	−0.187841	+	0.982199i	$0.560149\pi$
$798$	0	0
$799$	30.9783	1.09593
$800$	0	0
$801$	15.4891	0.547281
$802$	0	0
$803$	−40.4674	−1.42806
$804$	0	0
$805$	−2.37228	−0.0836119
$806$	0	0
$807$	23.4891	0.826856
$808$	0	0
$809$	34.7446	1.22155	0.610777	−	0.791803i	$-0.290857\pi$
$809$	34.7446	1.22155	0.610777	+	0.791803i	$0.290857\pi$
$810$	0	0
$811$	17.8614	0.627199	0.313599	−	0.949555i	$-0.398465\pi$
$811$	17.8614	0.627199	0.313599	+	0.949555i	$0.398465\pi$
$812$	0	0
$813$	−21.4891	−0.753657
$814$	0	0
$815$	−8.00000	−0.280228
$816$	0	0
$817$	−17.4891	−0.611867
$818$	0	0
$819$	0.372281	0.0130086
$820$	0	0
$821$	−9.76631	−0.340847	−0.170423	−	0.985371i	$-0.554514\pi$
$821$	−9.76631	−0.340847	−0.170423	+	0.985371i	$0.554514\pi$
$822$	0	0
$823$	20.7663	0.723868	0.361934	−	0.932204i	$-0.382117\pi$
$823$	20.7663	0.723868	0.361934	+	0.932204i	$0.382117\pi$
$824$	0	0
$825$	−2.11684	−0.0736990
$826$	0	0
$827$	−9.60597	−0.334032	−0.167016	−	0.985954i	$-0.553413\pi$
$827$	−9.60597	−0.334032	−0.167016	+	0.985954i	$0.553413\pi$
$828$	0	0
$829$	42.4674	1.47495	0.737476	−	0.675373i	$-0.236017\pi$
$829$	42.4674	1.47495	0.737476	+	0.675373i	$0.236017\pi$
$830$	0	0
$831$	18.1168	0.628466
$832$	0	0
$833$	4.00000	0.138592
$834$	0	0
$835$	4.46738	0.154600
$836$	0	0
$837$	6.00000	0.207390
$838$	0	0
$839$	42.0000	1.45000	0.725001	−	0.688748i	$-0.241839\pi$
$839$	42.0000	1.45000	0.725001	+	0.688748i	$0.241839\pi$
$840$	0	0
$841$	−28.8614	−0.995221
$842$	0	0
$843$	7.62772	0.262713
$844$	0	0
$845$	30.5109	1.04961
$846$	0	0
$847$	−0.372281	−0.0127917
$848$	0	0
$849$	−16.7446	−0.574672
$850$	0	0
$851$	8.37228	0.286998
$852$	0	0
$853$	−37.3505	−1.27886	−0.639429	−	0.768850i	$-0.720829\pi$
$853$	−37.3505	−1.27886	−0.639429	+	0.768850i	$0.720829\pi$
$854$	0	0
$855$	17.4891	0.598115
$856$	0	0
$857$	−15.7446	−0.537824	−0.268912	−	0.963165i	$-0.586664\pi$
$857$	−15.7446	−0.537824	−0.268912	+	0.963165i	$0.586664\pi$
$858$	0	0
$859$	−0.883156	−0.0301329	−0.0150664	−	0.999886i	$-0.504796\pi$
$859$	−0.883156	−0.0301329	−0.0150664	+	0.999886i	$0.504796\pi$
$860$	0	0
$861$	−5.74456	−0.195774
$862$	0	0
$863$	25.7228	0.875615	0.437807	−	0.899069i	$-0.355755\pi$
$863$	25.7228	0.875615	0.437807	+	0.899069i	$0.355755\pi$
$864$	0	0
$865$	17.7663	0.604073
$866$	0	0
$867$	−1.00000	−0.0339618
$868$	0	0
$869$	−9.25544	−0.313969
$870$	0	0
$871$	−1.48913	−0.0504571
$872$	0	0
$873$	5.11684	0.173179
$874$	0	0
$875$	−10.3723	−0.350647
$876$	0	0
$877$	−50.3505	−1.70022	−0.850108	−	0.526608i	$-0.823464\pi$
$877$	−50.3505	−1.70022	−0.850108	+	0.526608i	$0.823464\pi$
$878$	0	0
$879$	22.2337	0.749924
$880$	0	0
$881$	−0.510875	−0.0172118	−0.00860590	−	0.999963i	$-0.502739\pi$
$881$	−0.510875	−0.0172118	−0.00860590	+	0.999963i	$0.502739\pi$
$882$	0	0
$883$	31.7228	1.06756	0.533779	−	0.845624i	$-0.320771\pi$
$883$	31.7228	1.06756	0.533779	+	0.845624i	$0.320771\pi$
$884$	0	0
$885$	22.2337	0.747377
$886$	0	0
$887$	−45.9565	−1.54307	−0.771534	−	0.636188i	$-0.780510\pi$
$887$	−45.9565	−1.54307	−0.771534	+	0.636188i	$0.780510\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	0	0
$891$	−3.37228	−0.112976
$892$	0	0
$893$	−57.0951	−1.91061
$894$	0	0
$895$	−35.8614	−1.19871
$896$	0	0
$897$	0.372281	0.0124301
$898$	0	0
$899$	−2.23369	−0.0744977
$900$	0	0
$901$	−40.4674	−1.34816
$902$	0	0
$903$	−2.37228	−0.0789446
$904$	0	0
$905$	14.2337	0.473144
$906$	0	0
$907$	−23.8614	−0.792305	−0.396153	−	0.918185i	$-0.629655\pi$
$907$	−23.8614	−0.792305	−0.396153	+	0.918185i	$0.629655\pi$
$908$	0	0
$909$	12.1168	0.401890
$910$	0	0
$911$	−16.8832	−0.559364	−0.279682	−	0.960093i	$-0.590229\pi$
$911$	−16.8832	−0.559364	−0.279682	+	0.960093i	$0.590229\pi$
$912$	0	0
$913$	9.25544	0.306310
$914$	0	0
$915$	6.23369	0.206079
$916$	0	0
$917$	4.11684	0.135950
$918$	0	0
$919$	11.7663	0.388135	0.194067	−	0.980988i	$-0.437832\pi$
$919$	11.7663	0.388135	0.194067	+	0.980988i	$0.437832\pi$
$920$	0	0
$921$	26.3723	0.868996
$922$	0	0
$923$	4.74456	0.156169
$924$	0	0
$925$	−5.25544	−0.172798
$926$	0	0
$927$	−7.00000	−0.229910
$928$	0	0
$929$	9.11684	0.299114	0.149557	−	0.988753i	$-0.452215\pi$
$929$	9.11684	0.299114	0.149557	+	0.988753i	$0.452215\pi$
$930$	0	0
$931$	−7.37228	−0.241617
$932$	0	0
$933$	−16.6277	−0.544367
$934$	0	0
$935$	32.0000	1.04651
$936$	0	0
$937$	−43.7446	−1.42907	−0.714536	−	0.699598i	$-0.753362\pi$
$937$	−43.7446	−1.42907	−0.714536	+	0.699598i	$0.753362\pi$
$938$	0	0
$939$	−31.0951	−1.01475
$940$	0	0
$941$	54.0951	1.76345	0.881725	−	0.471764i	$-0.156383\pi$
$941$	54.0951	1.76345	0.881725	+	0.471764i	$0.156383\pi$
$942$	0	0
$943$	−5.74456	−0.187069
$944$	0	0
$945$	2.37228	0.0771703
$946$	0	0
$947$	−22.0951	−0.717994	−0.358997	−	0.933339i	$-0.616881\pi$
$947$	−22.0951	−0.717994	−0.358997	+	0.933339i	$0.616881\pi$
$948$	0	0
$949$	−4.46738	−0.145017
$950$	0	0
$951$	10.8832	0.352911
$952$	0	0
$953$	−9.37228	−0.303598	−0.151799	−	0.988411i	$-0.548507\pi$
$953$	−9.37228	−0.303598	−0.151799	+	0.988411i	$0.548507\pi$
$954$	0	0
$955$	−30.5109	−0.987309
$956$	0	0
$957$	1.25544	0.0405825
$958$	0	0
$959$	−18.4891	−0.597045
$960$	0	0
$961$	5.00000	0.161290
$962$	0	0
$963$	5.48913	0.176885
$964$	0	0
$965$	−64.6060	−2.07974
$966$	0	0
$967$	−4.00000	−0.128631	−0.0643157	−	0.997930i	$-0.520486\pi$
$967$	−4.00000	−0.128631	−0.0643157	+	0.997930i	$0.520486\pi$
$968$	0	0
$969$	−29.4891	−0.947327
$970$	0	0
$971$	27.7663	0.891063	0.445532	−	0.895266i	$-0.353015\pi$
$971$	27.7663	0.891063	0.445532	+	0.895266i	$0.353015\pi$
$972$	0	0
$973$	−5.62772	−0.180416
$974$	0	0
$975$	−0.233688	−0.00748400
$976$	0	0
$977$	38.4891	1.23138	0.615688	−	0.787990i	$-0.288878\pi$
$977$	38.4891	1.23138	0.615688	+	0.787990i	$0.288878\pi$
$978$	0	0
$979$	−52.2337	−1.66940
$980$	0	0
$981$	10.3723	0.331161
$982$	0	0
$983$	3.48913	0.111286	0.0556429	−	0.998451i	$-0.482279\pi$
$983$	3.48913	0.111286	0.0556429	+	0.998451i	$0.482279\pi$
$984$	0	0
$985$	−23.3940	−0.745396
$986$	0	0
$987$	−7.74456	−0.246512
$988$	0	0
$989$	−2.37228	−0.0754342
$990$	0	0
$991$	−5.64947	−0.179461	−0.0897306	−	0.995966i	$-0.528601\pi$
$991$	−5.64947	−0.179461	−0.0897306	+	0.995966i	$0.528601\pi$
$992$	0	0
$993$	3.37228	0.107016
$994$	0	0
$995$	2.37228	0.0752064
$996$	0	0
$997$	−30.2337	−0.957511	−0.478755	−	0.877948i	$-0.658912\pi$
$997$	−30.2337	−0.957511	−0.478755	+	0.877948i	$0.658912\pi$
$998$	0	0
$999$	−8.37228	−0.264887

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7728.2.a.bj.1.1		2
4.3	odd	2		3864.2.a.i.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3864.2.a.i.1.1	✓	2	4.3	odd	2
7728.2.a.bj.1.1		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7728.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -2.37228 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -2.37228 q^{5} -1.00000 q^{7} +1.00000 q^{9} -3.37228 q^{11} -0.372281 q^{13} -2.37228 q^{15} +4.00000 q^{17} -7.37228 q^{19} -1.00000 q^{21} -1.00000 q^{23} +0.627719 q^{25} +1.00000 q^{27} -0.372281 q^{29} +6.00000 q^{31} -3.37228 q^{33} +2.37228 q^{35} -8.37228 q^{37} -0.372281 q^{39} +5.74456 q^{41} +2.37228 q^{43} -2.37228 q^{45} +7.74456 q^{47} +1.00000 q^{49} +4.00000 q^{51} -10.1168 q^{53} +8.00000 q^{55} -7.37228 q^{57} -9.37228 q^{59} -2.62772 q^{61} -1.00000 q^{63} +0.883156 q^{65} +4.00000 q^{67} -1.00000 q^{69} -12.7446 q^{71} +12.0000 q^{73} +0.627719 q^{75} +3.37228 q^{77} +2.74456 q^{79} +1.00000 q^{81} -2.74456 q^{83} -9.48913 q^{85} -0.372281 q^{87} +15.4891 q^{89} +0.372281 q^{91} +6.00000 q^{93} +17.4891 q^{95} +5.11684 q^{97} -3.37228 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + q^5 - 2 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} + q^{5} - 2 q^{7} + 2 q^{9} - q^{11} + 5 q^{13} + q^{15} + 8 q^{17} - 9 q^{19} - 2 q^{21} - 2 q^{23} + 7 q^{25} + 2 q^{27} + 5 q^{29} + 12 q^{31} - q^{33} - q^{35} - 11 q^{37} + 5 q^{39} - q^{43} + q^{45} + 4 q^{47} + 2 q^{49} + 8 q^{51} - 3 q^{53} + 16 q^{55} - 9 q^{57} - 13 q^{59} - 11 q^{61} - 2 q^{63} + 19 q^{65} + 8 q^{67} - 2 q^{69} - 14 q^{71} + 24 q^{73} + 7 q^{75} + q^{77} - 6 q^{79} + 2 q^{81} + 6 q^{83} + 4 q^{85} + 5 q^{87} + 8 q^{89} - 5 q^{91} + 12 q^{93} + 12 q^{95} - 7 q^{97} - q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + q^5 - 2 * q^7 + 2 * q^9 - q^11 + 5 * q^13 + q^15 + 8 * q^17 - 9 * q^19 - 2 * q^21 - 2 * q^23 + 7 * q^25 + 2 * q^27 + 5 * q^29 + 12 * q^31 - q^33 - q^35 - 11 * q^37 + 5 * q^39 - q^43 + q^45 + 4 * q^47 + 2 * q^49 + 8 * q^51 - 3 * q^53 + 16 * q^55 - 9 * q^57 - 13 * q^59 - 11 * q^61 - 2 * q^63 + 19 * q^65 + 8 * q^67 - 2 * q^69 - 14 * q^71 + 24 * q^73 + 7 * q^75 + q^77 - 6 * q^79 + 2 * q^81 + 6 * q^83 + 4 * q^85 + 5 * q^87 + 8 * q^89 - 5 * q^91 + 12 * q^93 + 12 * q^95 - 7 * q^97 - q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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