LMFDB - Embedded newform 7728.2.a.ch.1.5

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:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 11x^{4} + 23x^{3} + 9x^{2} - 23x + 2 $ x^6 - 3x^5 - 11x^4 + 23x^3 + 9x^2 - 23*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 3864)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.23871$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +3.34309 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +3.34309 q^{5} +1.00000 q^{7} +1.00000 q^{9} +4.36801 q^{11} +1.50992 q^{13} +3.34309 q^{15} +5.56451 q^{17} +4.36801 q^{19} +1.00000 q^{21} +1.00000 q^{23} +6.17627 q^{25} +1.00000 q^{27} -5.73398 q^{29} -10.9233 q^{31} +4.36801 q^{33} +3.34309 q^{35} -0.336292 q^{37} +1.50992 q^{39} +7.82598 q^{41} +9.58018 q^{43} +3.34309 q^{45} -9.39049 q^{47} +1.00000 q^{49} +5.56451 q^{51} +1.31138 q^{53} +14.6027 q^{55} +4.36801 q^{57} +2.02287 q^{59} +4.87588 q^{61} +1.00000 q^{63} +5.04779 q^{65} -12.1987 q^{67} +1.00000 q^{69} -12.4154 q^{71} +9.23086 q^{73} +6.17627 q^{75} +4.36801 q^{77} -0.878321 q^{79} +1.00000 q^{81} -12.9622 q^{83} +18.6027 q^{85} -5.73398 q^{87} -13.1016 q^{89} +1.50992 q^{91} -10.9233 q^{93} +14.6027 q^{95} +8.31098 q^{97} +4.36801 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{3} + 2 q^{5} + 6 q^{7} + 6 q^{9}+O(q^{10}) $ 6 * q + 6 * q^3 + 2 * q^5 + 6 * q^7 + 6 * q^9	$ 6 q + 6 q^{3} + 2 q^{5} + 6 q^{7} + 6 q^{9} + 3 q^{11} + 2 q^{15} + 6 q^{17} + 3 q^{19} + 6 q^{21} + 6 q^{23} + 10 q^{25} + 6 q^{27} + 5 q^{29} + 4 q^{31} + 3 q^{33} + 2 q^{35} + q^{37} + 12 q^{41} + 6 q^{43} + 2 q^{45} + 6 q^{47} + 6 q^{49} + 6 q^{51} + 10 q^{53} - 3 q^{55} + 3 q^{57} + 14 q^{59} + 4 q^{61} + 6 q^{63} + 27 q^{65} - 7 q^{67} + 6 q^{69} - 7 q^{71} + 10 q^{73} + 10 q^{75} + 3 q^{77} - 14 q^{79} + 6 q^{81} - 14 q^{83} + 21 q^{85} + 5 q^{87} + 25 q^{89} + 4 q^{93} - 3 q^{95} + 11 q^{97} + 3 q^{99}+O(q^{100}) $ 6 * q + 6 * q^3 + 2 * q^5 + 6 * q^7 + 6 * q^9 + 3 * q^11 + 2 * q^15 + 6 * q^17 + 3 * q^19 + 6 * q^21 + 6 * q^23 + 10 * q^25 + 6 * q^27 + 5 * q^29 + 4 * q^31 + 3 * q^33 + 2 * q^35 + q^37 + 12 * q^41 + 6 * q^43 + 2 * q^45 + 6 * q^47 + 6 * q^49 + 6 * q^51 + 10 * q^53 - 3 * q^55 + 3 * q^57 + 14 * q^59 + 4 * q^61 + 6 * q^63 + 27 * q^65 - 7 * q^67 + 6 * q^69 - 7 * q^71 + 10 * q^73 + 10 * q^75 + 3 * q^77 - 14 * q^79 + 6 * q^81 - 14 * q^83 + 21 * q^85 + 5 * q^87 + 25 * q^89 + 4 * q^93 - 3 * q^95 + 11 * q^97 + 3 * 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
$5$	3.34309	1.49508	0.747538	−	0.664219i	$-0.231236\pi$
$5$	3.34309	1.49508	0.747538	+	0.664219i	$0.231236\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.36801	1.31700	0.658502	−	0.752579i	$-0.271190\pi$
$11$	4.36801	1.31700	0.658502	+	0.752579i	$0.271190\pi$
$12$	0	0
$13$	1.50992	0.418775	0.209388	−	0.977833i	$-0.432853\pi$
$13$	1.50992	0.418775	0.209388	+	0.977833i	$0.432853\pi$
$14$	0	0
$15$	3.34309	0.863183
$16$	0	0
$17$	5.56451	1.34959	0.674796	−	0.738005i	$-0.264232\pi$
$17$	5.56451	1.34959	0.674796	+	0.738005i	$0.264232\pi$
$18$	0	0
$19$	4.36801	1.00209	0.501045	−	0.865421i	$-0.332949\pi$
$19$	4.36801	1.00209	0.501045	+	0.865421i	$0.332949\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	6.17627	1.23525
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−5.73398	−1.06477	−0.532386	−	0.846502i	$-0.678705\pi$
$29$	−5.73398	−1.06477	−0.532386	+	0.846502i	$0.678705\pi$
$30$	0	0
$31$	−10.9233	−1.96188	−0.980939	−	0.194315i	$-0.937752\pi$
$31$	−10.9233	−1.96188	−0.980939	+	0.194315i	$0.937752\pi$
$32$	0	0
$33$	4.36801	0.760373
$34$	0	0
$35$	3.34309	0.565086
$36$	0	0
$37$	−0.336292	−0.0552861	−0.0276430	−	0.999618i	$-0.508800\pi$
$37$	−0.336292	−0.0552861	−0.0276430	+	0.999618i	$0.508800\pi$
$38$	0	0
$39$	1.50992	0.241780
$40$	0	0
$41$	7.82598	1.22221	0.611106	−	0.791548i	$-0.290725\pi$
$41$	7.82598	1.22221	0.611106	+	0.791548i	$0.290725\pi$
$42$	0	0
$43$	9.58018	1.46096	0.730482	−	0.682932i	$-0.239295\pi$
$43$	9.58018	1.46096	0.730482	+	0.682932i	$0.239295\pi$
$44$	0	0
$45$	3.34309	0.498359
$46$	0	0
$47$	−9.39049	−1.36974	−0.684872	−	0.728664i	$-0.740142\pi$
$47$	−9.39049	−1.36974	−0.684872	+	0.728664i	$0.740142\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	5.56451	0.779187
$52$	0	0
$53$	1.31138	0.180131	0.0900656	−	0.995936i	$-0.471292\pi$
$53$	1.31138	0.180131	0.0900656	+	0.995936i	$0.471292\pi$
$54$	0	0
$55$	14.6027	1.96902
$56$	0	0
$57$	4.36801	0.578557
$58$	0	0
$59$	2.02287	0.263356	0.131678	−	0.991293i	$-0.457964\pi$
$59$	2.02287	0.263356	0.131678	+	0.991293i	$0.457964\pi$
$60$	0	0
$61$	4.87588	0.624293	0.312146	−	0.950034i	$-0.398952\pi$
$61$	4.87588	0.624293	0.312146	+	0.950034i	$0.398952\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	5.04779	0.626101
$66$	0	0
$67$	−12.1987	−1.49031	−0.745157	−	0.666889i	$-0.767626\pi$
$67$	−12.1987	−1.49031	−0.745157	+	0.666889i	$0.767626\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−12.4154	−1.47344	−0.736719	−	0.676199i	$-0.763626\pi$
$71$	−12.4154	−1.47344	−0.736719	+	0.676199i	$0.763626\pi$
$72$	0	0
$73$	9.23086	1.08039	0.540195	−	0.841540i	$-0.318350\pi$
$73$	9.23086	1.08039	0.540195	+	0.841540i	$0.318350\pi$
$74$	0	0
$75$	6.17627	0.713174
$76$	0	0
$77$	4.36801	0.497781
$78$	0	0
$79$	−0.878321	−0.0988188	−0.0494094	−	0.998779i	$-0.515734\pi$
$79$	−0.878321	−0.0988188	−0.0494094	+	0.998779i	$0.515734\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−12.9622	−1.42279	−0.711393	−	0.702795i	$-0.751935\pi$
$83$	−12.9622	−1.42279	−0.711393	+	0.702795i	$0.751935\pi$
$84$	0	0
$85$	18.6027	2.01774
$86$	0	0
$87$	−5.73398	−0.614747
$88$	0	0
$89$	−13.1016	−1.38877	−0.694383	−	0.719606i	$-0.744323\pi$
$89$	−13.1016	−1.38877	−0.694383	+	0.719606i	$0.744323\pi$
$90$	0	0
$91$	1.50992	0.158282
$92$	0	0
$93$	−10.9233	−1.13269
$94$	0	0
$95$	14.6027	1.49820
$96$	0	0
$97$	8.31098	0.843852	0.421926	−	0.906630i	$-0.361354\pi$
$97$	8.31098	0.843852	0.421926	+	0.906630i	$0.361354\pi$
$98$	0	0
$99$	4.36801	0.439001
$100$	0	0
$101$	−2.70906	−0.269561	−0.134781	−	0.990875i	$-0.543033\pi$
$101$	−2.70906	−0.269561	−0.134781	+	0.990875i	$0.543033\pi$
$102$	0	0
$103$	−8.41032	−0.828693	−0.414347	−	0.910119i	$-0.635990\pi$
$103$	−8.41032	−0.828693	−0.414347	+	0.910119i	$0.635990\pi$
$104$	0	0
$105$	3.34309	0.326252
$106$	0	0
$107$	12.6977	1.22753	0.613765	−	0.789489i	$-0.289654\pi$
$107$	12.6977	1.22753	0.613765	+	0.789489i	$0.289654\pi$
$108$	0	0
$109$	−8.45850	−0.810178	−0.405089	−	0.914277i	$-0.632759\pi$
$109$	−8.45850	−0.810178	−0.405089	+	0.914277i	$0.632759\pi$
$110$	0	0
$111$	−0.336292	−0.0319194
$112$	0	0
$113$	−16.8356	−1.58376	−0.791878	−	0.610679i	$-0.790897\pi$
$113$	−16.8356	−1.58376	−0.791878	+	0.610679i	$0.790897\pi$
$114$	0	0
$115$	3.34309	0.311745
$116$	0	0
$117$	1.50992	0.139592
$118$	0	0
$119$	5.56451	0.510097
$120$	0	0
$121$	8.07951	0.734501
$122$	0	0
$123$	7.82598	0.705645
$124$	0	0
$125$	3.93238	0.351723
$126$	0	0
$127$	−10.3837	−0.921403	−0.460702	−	0.887555i	$-0.652402\pi$
$127$	−10.3837	−0.921403	−0.460702	+	0.887555i	$0.652402\pi$
$128$	0	0
$129$	9.58018	0.843488
$130$	0	0
$131$	−9.76569	−0.853233	−0.426616	−	0.904433i	$-0.640295\pi$
$131$	−9.76569	−0.853233	−0.426616	+	0.904433i	$0.640295\pi$
$132$	0	0
$133$	4.36801	0.378754
$134$	0	0
$135$	3.34309	0.287728
$136$	0	0
$137$	−6.51217	−0.556372	−0.278186	−	0.960527i	$-0.589733\pi$
$137$	−6.51217	−0.556372	−0.278186	+	0.960527i	$0.589733\pi$
$138$	0	0
$139$	−9.69717	−0.822504	−0.411252	−	0.911522i	$-0.634908\pi$
$139$	−9.69717	−0.822504	−0.411252	+	0.911522i	$0.634908\pi$
$140$	0	0
$141$	−9.39049	−0.790822
$142$	0	0
$143$	6.59533	0.551529
$144$	0	0
$145$	−19.1692	−1.59192
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	10.1623	0.832526	0.416263	−	0.909244i	$-0.363340\pi$
$149$	10.1623	0.832526	0.416263	+	0.909244i	$0.363340\pi$
$150$	0	0
$151$	−4.98811	−0.405927	−0.202964	−	0.979186i	$-0.565057\pi$
$151$	−4.98811	−0.405927	−0.202964	+	0.979186i	$0.565057\pi$
$152$	0	0
$153$	5.56451	0.449864
$154$	0	0
$155$	−36.5175	−2.93316
$156$	0	0
$157$	−19.6843	−1.57098	−0.785488	−	0.618877i	$-0.787588\pi$
$157$	−19.6843	−1.57098	−0.785488	+	0.618877i	$0.787588\pi$
$158$	0	0
$159$	1.31138	0.103999
$160$	0	0
$161$	1.00000	0.0788110
$162$	0	0
$163$	−6.31606	−0.494712	−0.247356	−	0.968925i	$-0.579562\pi$
$163$	−6.31606	−0.494712	−0.247356	+	0.968925i	$0.579562\pi$
$164$	0	0
$165$	14.6027	1.13682
$166$	0	0
$167$	−13.0153	−1.00715	−0.503576	−	0.863951i	$-0.667983\pi$
$167$	−13.0153	−1.00715	−0.503576	+	0.863951i	$0.667983\pi$
$168$	0	0
$169$	−10.7202	−0.824627
$170$	0	0
$171$	4.36801	0.334030
$172$	0	0
$173$	7.61434	0.578908	0.289454	−	0.957192i	$-0.406526\pi$
$173$	7.61434	0.578908	0.289454	+	0.957192i	$0.406526\pi$
$174$	0	0
$175$	6.17627	0.466882
$176$	0	0
$177$	2.02287	0.152048
$178$	0	0
$179$	19.9196	1.48886	0.744431	−	0.667699i	$-0.232721\pi$
$179$	19.9196	1.48886	0.744431	+	0.667699i	$0.232721\pi$
$180$	0	0
$181$	6.22617	0.462788	0.231394	−	0.972860i	$-0.425671\pi$
$181$	6.22617	0.462788	0.231394	+	0.972860i	$0.425671\pi$
$182$	0	0
$183$	4.87588	0.360436
$184$	0	0
$185$	−1.12426	−0.0826569
$186$	0	0
$187$	24.3058	1.77742
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	−1.46517	−0.106016	−0.0530079	−	0.998594i	$-0.516881\pi$
$191$	−1.46517	−0.106016	−0.0530079	+	0.998594i	$0.516881\pi$
$192$	0	0
$193$	4.65856	0.335330	0.167665	−	0.985844i	$-0.446377\pi$
$193$	4.65856	0.335330	0.167665	+	0.985844i	$0.446377\pi$
$194$	0	0
$195$	5.04779	0.361480
$196$	0	0
$197$	21.3115	1.51838	0.759191	−	0.650868i	$-0.225595\pi$
$197$	21.3115	1.51838	0.759191	+	0.650868i	$0.225595\pi$
$198$	0	0
$199$	15.2715	1.08257	0.541285	−	0.840839i	$-0.317938\pi$
$199$	15.2715	1.08257	0.541285	+	0.840839i	$0.317938\pi$
$200$	0	0
$201$	−12.1987	−0.860433
$202$	0	0
$203$	−5.73398	−0.402446
$204$	0	0
$205$	26.1630	1.82730
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	19.0795	1.31976
$210$	0	0
$211$	12.4779	0.859017	0.429508	−	0.903063i	$-0.358687\pi$
$211$	12.4779	0.859017	0.429508	+	0.903063i	$0.358687\pi$
$212$	0	0
$213$	−12.4154	−0.850689
$214$	0	0
$215$	32.0274	2.18425
$216$	0	0
$217$	−10.9233	−0.741520
$218$	0	0
$219$	9.23086	0.623764
$220$	0	0
$221$	8.40194	0.565175
$222$	0	0
$223$	−5.93495	−0.397434	−0.198717	−	0.980057i	$-0.563677\pi$
$223$	−5.93495	−0.397434	−0.198717	+	0.980057i	$0.563677\pi$
$224$	0	0
$225$	6.17627	0.411751
$226$	0	0
$227$	−15.0544	−0.999196	−0.499598	−	0.866257i	$-0.666519\pi$
$227$	−15.0544	−0.999196	−0.499598	+	0.866257i	$0.666519\pi$
$228$	0	0
$229$	9.40616	0.621577	0.310788	−	0.950479i	$-0.399407\pi$
$229$	9.40616	0.621577	0.310788	+	0.950479i	$0.399407\pi$
$230$	0	0
$231$	4.36801	0.287394
$232$	0	0
$233$	4.83730	0.316902	0.158451	−	0.987367i	$-0.449350\pi$
$233$	4.83730	0.316902	0.158451	+	0.987367i	$0.449350\pi$
$234$	0	0
$235$	−31.3933	−2.04787
$236$	0	0
$237$	−0.878321	−0.0570531
$238$	0	0
$239$	−11.7812	−0.762065	−0.381032	−	0.924562i	$-0.624431\pi$
$239$	−11.7812	−0.762065	−0.381032	+	0.924562i	$0.624431\pi$
$240$	0	0
$241$	−0.910038	−0.0586207	−0.0293104	−	0.999570i	$-0.509331\pi$
$241$	−0.910038	−0.0586207	−0.0293104	+	0.999570i	$0.509331\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	3.34309	0.213582
$246$	0	0
$247$	6.59533	0.419651
$248$	0	0
$249$	−12.9622	−0.821445
$250$	0	0
$251$	−23.5808	−1.48840	−0.744202	−	0.667955i	$-0.767170\pi$
$251$	−23.5808	−1.48840	−0.744202	+	0.667955i	$0.767170\pi$
$252$	0	0
$253$	4.36801	0.274614
$254$	0	0
$255$	18.6027	1.16494
$256$	0	0
$257$	−1.17119	−0.0730565	−0.0365283	−	0.999333i	$-0.511630\pi$
$257$	−1.17119	−0.0730565	−0.0365283	+	0.999333i	$0.511630\pi$
$258$	0	0
$259$	−0.336292	−0.0208962
$260$	0	0
$261$	−5.73398	−0.354924
$262$	0	0
$263$	−29.2939	−1.80634	−0.903171	−	0.429282i	$-0.858767\pi$
$263$	−29.2939	−1.80634	−0.903171	+	0.429282i	$0.858767\pi$
$264$	0	0
$265$	4.38405	0.269310
$266$	0	0
$267$	−13.1016	−0.801804
$268$	0	0
$269$	17.4092	1.06146	0.530728	−	0.847542i	$-0.321919\pi$
$269$	17.4092	1.06146	0.530728	+	0.847542i	$0.321919\pi$
$270$	0	0
$271$	13.3068	0.808328	0.404164	−	0.914687i	$-0.367563\pi$
$271$	13.3068	0.808328	0.404164	+	0.914687i	$0.367563\pi$
$272$	0	0
$273$	1.50992	0.0913843
$274$	0	0
$275$	26.9780	1.62683
$276$	0	0
$277$	13.5866	0.816339	0.408170	−	0.912906i	$-0.366167\pi$
$277$	13.5866	0.816339	0.408170	+	0.912906i	$0.366167\pi$
$278$	0	0
$279$	−10.9233	−0.653959
$280$	0	0
$281$	1.09451	0.0652931	0.0326466	−	0.999467i	$-0.489606\pi$
$281$	1.09451	0.0652931	0.0326466	+	0.999467i	$0.489606\pi$
$282$	0	0
$283$	32.0003	1.90222	0.951112	−	0.308848i	$-0.0999433\pi$
$283$	32.0003	1.90222	0.951112	+	0.308848i	$0.0999433\pi$
$284$	0	0
$285$	14.6027	0.864987
$286$	0	0
$287$	7.82598	0.461953
$288$	0	0
$289$	13.9637	0.821396
$290$	0	0
$291$	8.31098	0.487198
$292$	0	0
$293$	15.6909	0.916671	0.458335	−	0.888779i	$-0.348446\pi$
$293$	15.6909	0.916671	0.458335	+	0.888779i	$0.348446\pi$
$294$	0	0
$295$	6.76265	0.393737
$296$	0	0
$297$	4.36801	0.253458
$298$	0	0
$299$	1.50992	0.0873207
$300$	0	0
$301$	9.58018	0.552193
$302$	0	0
$303$	−2.70906	−0.155631
$304$	0	0
$305$	16.3005	0.933365
$306$	0	0
$307$	−29.4453	−1.68053	−0.840266	−	0.542175i	$-0.817601\pi$
$307$	−29.4453	−1.68053	−0.840266	+	0.542175i	$0.817601\pi$
$308$	0	0
$309$	−8.41032	−0.478446
$310$	0	0
$311$	−7.85680	−0.445518	−0.222759	−	0.974874i	$-0.571506\pi$
$311$	−7.85680	−0.445518	−0.222759	+	0.974874i	$0.571506\pi$
$312$	0	0
$313$	−27.3253	−1.54452	−0.772259	−	0.635308i	$-0.780873\pi$
$313$	−27.3253	−1.54452	−0.772259	+	0.635308i	$0.780873\pi$
$314$	0	0
$315$	3.34309	0.188362
$316$	0	0
$317$	−14.0716	−0.790337	−0.395169	−	0.918609i	$-0.629314\pi$
$317$	−14.0716	−0.790337	−0.395169	+	0.918609i	$0.629314\pi$
$318$	0	0
$319$	−25.0461	−1.40231
$320$	0	0
$321$	12.6977	0.708715
$322$	0	0
$323$	24.3058	1.35241
$324$	0	0
$325$	9.32565	0.517294
$326$	0	0
$327$	−8.45850	−0.467756
$328$	0	0
$329$	−9.39049	−0.517714
$330$	0	0
$331$	−8.18758	−0.450030	−0.225015	−	0.974355i	$-0.572243\pi$
$331$	−8.18758	−0.450030	−0.225015	+	0.974355i	$0.572243\pi$
$332$	0	0
$333$	−0.336292	−0.0184287
$334$	0	0
$335$	−40.7815	−2.22813
$336$	0	0
$337$	−17.3409	−0.944617	−0.472308	−	0.881433i	$-0.656579\pi$
$337$	−17.3409	−0.944617	−0.472308	+	0.881433i	$0.656579\pi$
$338$	0	0
$339$	−16.8356	−0.914382
$340$	0	0
$341$	−47.7130	−2.58380
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	3.34309	0.179986
$346$	0	0
$347$	−1.77852	−0.0954758	−0.0477379	−	0.998860i	$-0.515201\pi$
$347$	−1.77852	−0.0954758	−0.0477379	+	0.998860i	$0.515201\pi$
$348$	0	0
$349$	14.9499	0.800250	0.400125	−	0.916461i	$-0.368967\pi$
$349$	14.9499	0.800250	0.400125	+	0.916461i	$0.368967\pi$
$350$	0	0
$351$	1.50992	0.0805934
$352$	0	0
$353$	21.1188	1.12404	0.562020	−	0.827123i	$-0.310024\pi$
$353$	21.1188	1.12404	0.562020	+	0.827123i	$0.310024\pi$
$354$	0	0
$355$	−41.5058	−2.20290
$356$	0	0
$357$	5.56451	0.294505
$358$	0	0
$359$	24.2586	1.28032	0.640159	−	0.768243i	$-0.278869\pi$
$359$	24.2586	1.28032	0.640159	+	0.768243i	$0.278869\pi$
$360$	0	0
$361$	0.0795069	0.00418458
$362$	0	0
$363$	8.07951	0.424064
$364$	0	0
$365$	30.8596	1.61527
$366$	0	0
$367$	15.2920	0.798238	0.399119	−	0.916899i	$-0.369316\pi$
$367$	15.2920	0.798238	0.399119	+	0.916899i	$0.369316\pi$
$368$	0	0
$369$	7.82598	0.407404
$370$	0	0
$371$	1.31138	0.0680832
$372$	0	0
$373$	−10.2219	−0.529272	−0.264636	−	0.964348i	$-0.585252\pi$
$373$	−10.2219	−0.529272	−0.264636	+	0.964348i	$0.585252\pi$
$374$	0	0
$375$	3.93238	0.203067
$376$	0	0
$377$	−8.65782	−0.445900
$378$	0	0
$379$	9.82297	0.504572	0.252286	−	0.967653i	$-0.418818\pi$
$379$	9.82297	0.504572	0.252286	+	0.967653i	$0.418818\pi$
$380$	0	0
$381$	−10.3837	−0.531972
$382$	0	0
$383$	−28.3009	−1.44611	−0.723055	−	0.690790i	$-0.757263\pi$
$383$	−28.3009	−1.44611	−0.723055	+	0.690790i	$0.757263\pi$
$384$	0	0
$385$	14.6027	0.744221
$386$	0	0
$387$	9.58018	0.486988
$388$	0	0
$389$	−7.23769	−0.366966	−0.183483	−	0.983023i	$-0.558737\pi$
$389$	−7.23769	−0.366966	−0.183483	+	0.983023i	$0.558737\pi$
$390$	0	0
$391$	5.56451	0.281409
$392$	0	0
$393$	−9.76569	−0.492614
$394$	0	0
$395$	−2.93631	−0.147742
$396$	0	0
$397$	−12.3472	−0.619691	−0.309845	−	0.950787i	$-0.600277\pi$
$397$	−12.3472	−0.619691	−0.309845	+	0.950787i	$0.600277\pi$
$398$	0	0
$399$	4.36801	0.218674
$400$	0	0
$401$	−2.14429	−0.107081	−0.0535405	−	0.998566i	$-0.517051\pi$
$401$	−2.14429	−0.107081	−0.0535405	+	0.998566i	$0.517051\pi$
$402$	0	0
$403$	−16.4932	−0.821586
$404$	0	0
$405$	3.34309	0.166120
$406$	0	0
$407$	−1.46893	−0.0728120
$408$	0	0
$409$	11.9027	0.588548	0.294274	−	0.955721i	$-0.404922\pi$
$409$	11.9027	0.588548	0.294274	+	0.955721i	$0.404922\pi$
$410$	0	0
$411$	−6.51217	−0.321221
$412$	0	0
$413$	2.02287	0.0995391
$414$	0	0
$415$	−43.3338	−2.12717
$416$	0	0
$417$	−9.69717	−0.474873
$418$	0	0
$419$	−10.1436	−0.495548	−0.247774	−	0.968818i	$-0.579699\pi$
$419$	−10.1436	−0.495548	−0.247774	+	0.968818i	$0.579699\pi$
$420$	0	0
$421$	14.6197	0.712521	0.356260	−	0.934387i	$-0.384052\pi$
$421$	14.6197	0.712521	0.356260	+	0.934387i	$0.384052\pi$
$422$	0	0
$423$	−9.39049	−0.456581
$424$	0	0
$425$	34.3679	1.66709
$426$	0	0
$427$	4.87588	0.235960
$428$	0	0
$429$	6.59533	0.318425
$430$	0	0
$431$	−13.9585	−0.672360	−0.336180	−	0.941798i	$-0.609135\pi$
$431$	−13.9585	−0.672360	−0.336180	+	0.941798i	$0.609135\pi$
$432$	0	0
$433$	−0.975653	−0.0468869	−0.0234435	−	0.999725i	$-0.507463\pi$
$433$	−0.975653	−0.0468869	−0.0234435	+	0.999725i	$0.507463\pi$
$434$	0	0
$435$	−19.1692	−0.919093
$436$	0	0
$437$	4.36801	0.208950
$438$	0	0
$439$	−10.5178	−0.501988	−0.250994	−	0.967989i	$-0.580757\pi$
$439$	−10.5178	−0.501988	−0.250994	+	0.967989i	$0.580757\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	21.8350	1.03741	0.518707	−	0.854952i	$-0.326414\pi$
$443$	21.8350	1.03741	0.518707	+	0.854952i	$0.326414\pi$
$444$	0	0
$445$	−43.7998	−2.07631
$446$	0	0
$447$	10.1623	0.480659
$448$	0	0
$449$	3.43821	0.162259	0.0811295	−	0.996704i	$-0.474147\pi$
$449$	3.43821	0.162259	0.0811295	+	0.996704i	$0.474147\pi$
$450$	0	0
$451$	34.1840	1.60966
$452$	0	0
$453$	−4.98811	−0.234362
$454$	0	0
$455$	5.04779	0.236644
$456$	0	0
$457$	0.249057	0.0116504	0.00582519	−	0.999983i	$-0.498146\pi$
$457$	0.249057	0.0116504	0.00582519	+	0.999983i	$0.498146\pi$
$458$	0	0
$459$	5.56451	0.259729
$460$	0	0
$461$	28.3729	1.32146	0.660729	−	0.750625i	$-0.270247\pi$
$461$	28.3729	1.32146	0.660729	+	0.750625i	$0.270247\pi$
$462$	0	0
$463$	−9.97074	−0.463379	−0.231690	−	0.972790i	$-0.574425\pi$
$463$	−9.97074	−0.463379	−0.231690	+	0.972790i	$0.574425\pi$
$464$	0	0
$465$	−36.5175	−1.69346
$466$	0	0
$467$	20.0525	0.927918	0.463959	−	0.885857i	$-0.346428\pi$
$467$	20.0525	0.927918	0.463959	+	0.885857i	$0.346428\pi$
$468$	0	0
$469$	−12.1987	−0.563286
$470$	0	0
$471$	−19.6843	−0.907004
$472$	0	0
$473$	41.8463	1.92410
$474$	0	0
$475$	26.9780	1.23784
$476$	0	0
$477$	1.31138	0.0600438
$478$	0	0
$479$	5.17151	0.236292	0.118146	−	0.992996i	$-0.462305\pi$
$479$	5.17151	0.236292	0.118146	+	0.992996i	$0.462305\pi$
$480$	0	0
$481$	−0.507773	−0.0231525
$482$	0	0
$483$	1.00000	0.0455016
$484$	0	0
$485$	27.7844	1.26162
$486$	0	0
$487$	33.3508	1.51127	0.755634	−	0.654994i	$-0.227329\pi$
$487$	33.3508	1.51127	0.755634	+	0.654994i	$0.227329\pi$
$488$	0	0
$489$	−6.31606	−0.285622
$490$	0	0
$491$	39.5448	1.78463	0.892317	−	0.451410i	$-0.149079\pi$
$491$	39.5448	1.78463	0.892317	+	0.451410i	$0.149079\pi$
$492$	0	0
$493$	−31.9067	−1.43701
$494$	0	0
$495$	14.6027	0.656341
$496$	0	0
$497$	−12.4154	−0.556907
$498$	0	0
$499$	7.48316	0.334992	0.167496	−	0.985873i	$-0.446432\pi$
$499$	7.48316	0.334992	0.167496	+	0.985873i	$0.446432\pi$
$500$	0	0
$501$	−13.0153	−0.581480
$502$	0	0
$503$	−35.0864	−1.56442	−0.782212	−	0.623013i	$-0.785908\pi$
$503$	−35.0864	−1.56442	−0.782212	+	0.623013i	$0.785908\pi$
$504$	0	0
$505$	−9.05663	−0.403015
$506$	0	0
$507$	−10.7202	−0.476099
$508$	0	0
$509$	−9.36959	−0.415300	−0.207650	−	0.978203i	$-0.566581\pi$
$509$	−9.36959	−0.415300	−0.207650	+	0.978203i	$0.566581\pi$
$510$	0	0
$511$	9.23086	0.408349
$512$	0	0
$513$	4.36801	0.192852
$514$	0	0
$515$	−28.1165	−1.23896
$516$	0	0
$517$	−41.0177	−1.80396
$518$	0	0
$519$	7.61434	0.334232
$520$	0	0
$521$	−18.1405	−0.794751	−0.397376	−	0.917656i	$-0.630079\pi$
$521$	−18.1405	−0.794751	−0.397376	+	0.917656i	$0.630079\pi$
$522$	0	0
$523$	24.6685	1.07868	0.539340	−	0.842088i	$-0.318674\pi$
$523$	24.6685	1.07868	0.539340	+	0.842088i	$0.318674\pi$
$524$	0	0
$525$	6.17627	0.269555
$526$	0	0
$527$	−60.7826	−2.64773
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	2.02287	0.0877852
$532$	0	0
$533$	11.8166	0.511833
$534$	0	0
$535$	42.4495	1.83525
$536$	0	0
$537$	19.9196	0.859595
$538$	0	0
$539$	4.36801	0.188143
$540$	0	0
$541$	22.9043	0.984734	0.492367	−	0.870388i	$-0.336132\pi$
$541$	22.9043	0.984734	0.492367	+	0.870388i	$0.336132\pi$
$542$	0	0
$543$	6.22617	0.267191
$544$	0	0
$545$	−28.2776	−1.21128
$546$	0	0
$547$	1.40263	0.0599721	0.0299861	−	0.999550i	$-0.490454\pi$
$547$	1.40263	0.0599721	0.0299861	+	0.999550i	$0.490454\pi$
$548$	0	0
$549$	4.87588	0.208098
$550$	0	0
$551$	−25.0461	−1.06700
$552$	0	0
$553$	−0.878321	−0.0373500
$554$	0	0
$555$	−1.12426	−0.0477220
$556$	0	0
$557$	32.8328	1.39117	0.695584	−	0.718444i	$-0.255146\pi$
$557$	32.8328	1.39117	0.695584	+	0.718444i	$0.255146\pi$
$558$	0	0
$559$	14.4653	0.611816
$560$	0	0
$561$	24.3058	1.02619
$562$	0	0
$563$	32.0271	1.34978	0.674891	−	0.737918i	$-0.264191\pi$
$563$	32.0271	1.34978	0.674891	+	0.737918i	$0.264191\pi$
$564$	0	0
$565$	−56.2829	−2.36784
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	26.8483	1.12554	0.562769	−	0.826614i	$-0.309736\pi$
$569$	26.8483	1.12554	0.562769	+	0.826614i	$0.309736\pi$
$570$	0	0
$571$	−10.4695	−0.438136	−0.219068	−	0.975710i	$-0.570302\pi$
$571$	−10.4695	−0.438136	−0.219068	+	0.975710i	$0.570302\pi$
$572$	0	0
$573$	−1.46517	−0.0612082
$574$	0	0
$575$	6.17627	0.257568
$576$	0	0
$577$	−25.9984	−1.08233	−0.541164	−	0.840917i	$-0.682016\pi$
$577$	−25.9984	−1.08233	−0.541164	+	0.840917i	$0.682016\pi$
$578$	0	0
$579$	4.65856	0.193603
$580$	0	0
$581$	−12.9622	−0.537762
$582$	0	0
$583$	5.72810	0.237234
$584$	0	0
$585$	5.04779	0.208700
$586$	0	0
$587$	23.1003	0.953451	0.476725	−	0.879052i	$-0.341824\pi$
$587$	23.1003	0.953451	0.476725	+	0.879052i	$0.341824\pi$
$588$	0	0
$589$	−47.7130	−1.96598
$590$	0	0
$591$	21.3115	0.876639
$592$	0	0
$593$	26.5087	1.08858	0.544291	−	0.838897i	$-0.316799\pi$
$593$	26.5087	1.08858	0.544291	+	0.838897i	$0.316799\pi$
$594$	0	0
$595$	18.6027	0.762635
$596$	0	0
$597$	15.2715	0.625022
$598$	0	0
$599$	42.2215	1.72512	0.862562	−	0.505952i	$-0.168859\pi$
$599$	42.2215	1.72512	0.862562	+	0.505952i	$0.168859\pi$
$600$	0	0
$601$	−28.4707	−1.16134	−0.580672	−	0.814137i	$-0.697210\pi$
$601$	−28.4707	−1.16134	−0.580672	+	0.814137i	$0.697210\pi$
$602$	0	0
$603$	−12.1987	−0.496771
$604$	0	0
$605$	27.0105	1.09813
$606$	0	0
$607$	37.2462	1.51178	0.755889	−	0.654700i	$-0.227205\pi$
$607$	37.2462	1.51178	0.755889	+	0.654700i	$0.227205\pi$
$608$	0	0
$609$	−5.73398	−0.232352
$610$	0	0
$611$	−14.1788	−0.573615
$612$	0	0
$613$	1.95309	0.0788845	0.0394422	−	0.999222i	$-0.487442\pi$
$613$	1.95309	0.0788845	0.0394422	+	0.999222i	$0.487442\pi$
$614$	0	0
$615$	26.1630	1.05499
$616$	0	0
$617$	39.6809	1.59749	0.798746	−	0.601669i	$-0.205497\pi$
$617$	39.6809	1.59749	0.798746	+	0.601669i	$0.205497\pi$
$618$	0	0
$619$	−43.3165	−1.74104	−0.870518	−	0.492136i	$-0.836216\pi$
$619$	−43.3165	−1.74104	−0.870518	+	0.492136i	$0.836216\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	−13.1016	−0.524904
$624$	0	0
$625$	−17.7350	−0.709402
$626$	0	0
$627$	19.0795	0.761962
$628$	0	0
$629$	−1.87130	−0.0746136
$630$	0	0
$631$	49.8769	1.98557	0.992783	−	0.119922i	$-0.0382644\pi$
$631$	49.8769	1.98557	0.992783	+	0.119922i	$0.0382644\pi$
$632$	0	0
$633$	12.4779	0.495954
$634$	0	0
$635$	−34.7136	−1.37757
$636$	0	0
$637$	1.50992	0.0598251
$638$	0	0
$639$	−12.4154	−0.491146
$640$	0	0
$641$	20.7445	0.819360	0.409680	−	0.912229i	$-0.365640\pi$
$641$	20.7445	0.819360	0.409680	+	0.912229i	$0.365640\pi$
$642$	0	0
$643$	−11.9002	−0.469299	−0.234649	−	0.972080i	$-0.575394\pi$
$643$	−11.9002	−0.469299	−0.234649	+	0.972080i	$0.575394\pi$
$644$	0	0
$645$	32.0274	1.26108
$646$	0	0
$647$	20.5754	0.808902	0.404451	−	0.914560i	$-0.367463\pi$
$647$	20.5754	0.808902	0.404451	+	0.914560i	$0.367463\pi$
$648$	0	0
$649$	8.83593	0.346841
$650$	0	0
$651$	−10.9233	−0.428117
$652$	0	0
$653$	13.8467	0.541862	0.270931	−	0.962599i	$-0.412668\pi$
$653$	13.8467	0.541862	0.270931	+	0.962599i	$0.412668\pi$
$654$	0	0
$655$	−32.6476	−1.27565
$656$	0	0
$657$	9.23086	0.360130
$658$	0	0
$659$	−8.77396	−0.341785	−0.170892	−	0.985290i	$-0.554665\pi$
$659$	−8.77396	−0.341785	−0.170892	+	0.985290i	$0.554665\pi$
$660$	0	0
$661$	−45.3931	−1.76559	−0.882794	−	0.469760i	$-0.844340\pi$
$661$	−45.3931	−1.76559	−0.882794	+	0.469760i	$0.844340\pi$
$662$	0	0
$663$	8.40194	0.326304
$664$	0	0
$665$	14.6027	0.566267
$666$	0	0
$667$	−5.73398	−0.222020
$668$	0	0
$669$	−5.93495	−0.229459
$670$	0	0
$671$	21.2979	0.822196
$672$	0	0
$673$	−21.7908	−0.839974	−0.419987	−	0.907530i	$-0.637965\pi$
$673$	−21.7908	−0.839974	−0.419987	+	0.907530i	$0.637965\pi$
$674$	0	0
$675$	6.17627	0.237725
$676$	0	0
$677$	−44.3602	−1.70490	−0.852451	−	0.522807i	$-0.824885\pi$
$677$	−44.3602	−1.70490	−0.852451	+	0.522807i	$0.824885\pi$
$678$	0	0
$679$	8.31098	0.318946
$680$	0	0
$681$	−15.0544	−0.576886
$682$	0	0
$683$	10.1053	0.386668	0.193334	−	0.981133i	$-0.438070\pi$
$683$	10.1053	0.386668	0.193334	+	0.981133i	$0.438070\pi$
$684$	0	0
$685$	−21.7708	−0.831819
$686$	0	0
$687$	9.40616	0.358868
$688$	0	0
$689$	1.98007	0.0754345
$690$	0	0
$691$	26.7112	1.01614	0.508070	−	0.861316i	$-0.330359\pi$
$691$	26.7112	1.01614	0.508070	+	0.861316i	$0.330359\pi$
$692$	0	0
$693$	4.36801	0.165927
$694$	0	0
$695$	−32.4186	−1.22971
$696$	0	0
$697$	43.5477	1.64949
$698$	0	0
$699$	4.83730	0.182963
$700$	0	0
$701$	5.61896	0.212225	0.106113	−	0.994354i	$-0.466160\pi$
$701$	5.61896	0.212225	0.106113	+	0.994354i	$0.466160\pi$
$702$	0	0
$703$	−1.46893	−0.0554016
$704$	0	0
$705$	−31.3933	−1.18234
$706$	0	0
$707$	−2.70906	−0.101885
$708$	0	0
$709$	−20.2873	−0.761904	−0.380952	−	0.924595i	$-0.624404\pi$
$709$	−20.2873	−0.761904	−0.380952	+	0.924595i	$0.624404\pi$
$710$	0	0
$711$	−0.878321	−0.0329396
$712$	0	0
$713$	−10.9233	−0.409080
$714$	0	0
$715$	22.0488	0.824578
$716$	0	0
$717$	−11.7812	−0.439978
$718$	0	0
$719$	45.4985	1.69681	0.848405	−	0.529348i	$-0.177563\pi$
$719$	45.4985	1.69681	0.848405	+	0.529348i	$0.177563\pi$
$720$	0	0
$721$	−8.41032	−0.313217
$722$	0	0
$723$	−0.910038	−0.0338447
$724$	0	0
$725$	−35.4146	−1.31526
$726$	0	0
$727$	38.7942	1.43880	0.719400	−	0.694597i	$-0.244417\pi$
$727$	38.7942	1.43880	0.719400	+	0.694597i	$0.244417\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	53.3090	1.97170
$732$	0	0
$733$	−33.5370	−1.23872	−0.619359	−	0.785108i	$-0.712607\pi$
$733$	−33.5370	−1.23872	−0.619359	+	0.785108i	$0.712607\pi$
$734$	0	0
$735$	3.34309	0.123312
$736$	0	0
$737$	−53.2842	−1.96275
$738$	0	0
$739$	−20.3756	−0.749530	−0.374765	−	0.927120i	$-0.622277\pi$
$739$	−20.3756	−0.749530	−0.374765	+	0.927120i	$0.622277\pi$
$740$	0	0
$741$	6.59533	0.242285
$742$	0	0
$743$	37.6960	1.38293	0.691466	−	0.722409i	$-0.256965\pi$
$743$	37.6960	1.38293	0.691466	+	0.722409i	$0.256965\pi$
$744$	0	0
$745$	33.9734	1.24469
$746$	0	0
$747$	−12.9622	−0.474262
$748$	0	0
$749$	12.6977	0.463963
$750$	0	0
$751$	−11.0991	−0.405013	−0.202507	−	0.979281i	$-0.564909\pi$
$751$	−11.0991	−0.405013	−0.202507	+	0.979281i	$0.564909\pi$
$752$	0	0
$753$	−23.5808	−0.859330
$754$	0	0
$755$	−16.6757	−0.606892
$756$	0	0
$757$	−16.7983	−0.610545	−0.305272	−	0.952265i	$-0.598747\pi$
$757$	−16.7983	−0.610545	−0.305272	+	0.952265i	$0.598747\pi$
$758$	0	0
$759$	4.36801	0.158549
$760$	0	0
$761$	28.6568	1.03881	0.519405	−	0.854528i	$-0.326154\pi$
$761$	28.6568	1.03881	0.519405	+	0.854528i	$0.326154\pi$
$762$	0	0
$763$	−8.45850	−0.306218
$764$	0	0
$765$	18.6027	0.672581
$766$	0	0
$767$	3.05437	0.110287
$768$	0	0
$769$	46.3103	1.66999	0.834996	−	0.550256i	$-0.185470\pi$
$769$	46.3103	1.66999	0.834996	+	0.550256i	$0.185470\pi$
$770$	0	0
$771$	−1.17119	−0.0421792
$772$	0	0
$773$	−13.7491	−0.494521	−0.247261	−	0.968949i	$-0.579530\pi$
$773$	−13.7491	−0.494521	−0.247261	+	0.968949i	$0.579530\pi$
$774$	0	0
$775$	−67.4651	−2.42342
$776$	0	0
$777$	−0.336292	−0.0120644
$778$	0	0
$779$	34.1840	1.22477
$780$	0	0
$781$	−54.2306	−1.94052
$782$	0	0
$783$	−5.73398	−0.204916
$784$	0	0
$785$	−65.8064	−2.34873
$786$	0	0
$787$	35.9759	1.28240	0.641200	−	0.767373i	$-0.278437\pi$
$787$	35.9759	1.28240	0.641200	+	0.767373i	$0.278437\pi$
$788$	0	0
$789$	−29.2939	−1.04289
$790$	0	0
$791$	−16.8356	−0.598604
$792$	0	0
$793$	7.36217	0.261438
$794$	0	0
$795$	4.38405	0.155486
$796$	0	0
$797$	47.8653	1.69547	0.847737	−	0.530417i	$-0.177964\pi$
$797$	47.8653	1.69547	0.847737	+	0.530417i	$0.177964\pi$
$798$	0	0
$799$	−52.2534	−1.84859
$800$	0	0
$801$	−13.1016	−0.462922
$802$	0	0
$803$	40.3205	1.42288
$804$	0	0
$805$	3.34309	0.117829
$806$	0	0
$807$	17.4092	0.612832
$808$	0	0
$809$	−35.4802	−1.24742	−0.623709	−	0.781657i	$-0.714375\pi$
$809$	−35.4802	−1.24742	−0.623709	+	0.781657i	$0.714375\pi$
$810$	0	0
$811$	19.0909	0.670372	0.335186	−	0.942152i	$-0.391201\pi$
$811$	19.0909	0.670372	0.335186	+	0.942152i	$0.391201\pi$
$812$	0	0
$813$	13.3068	0.466688
$814$	0	0
$815$	−21.1152	−0.739633
$816$	0	0
$817$	41.8463	1.46402
$818$	0	0
$819$	1.50992	0.0527607
$820$	0	0
$821$	33.5983	1.17259	0.586295	−	0.810098i	$-0.300586\pi$
$821$	33.5983	1.17259	0.586295	+	0.810098i	$0.300586\pi$
$822$	0	0
$823$	−19.4558	−0.678187	−0.339093	−	0.940753i	$-0.610120\pi$
$823$	−19.4558	−0.678187	−0.339093	+	0.940753i	$0.610120\pi$
$824$	0	0
$825$	26.9780	0.939254
$826$	0	0
$827$	−12.7785	−0.444353	−0.222177	−	0.975006i	$-0.571316\pi$
$827$	−12.7785	−0.444353	−0.222177	+	0.975006i	$0.571316\pi$
$828$	0	0
$829$	−23.4640	−0.814940	−0.407470	−	0.913219i	$-0.633589\pi$
$829$	−23.4640	−0.814940	−0.407470	+	0.913219i	$0.633589\pi$
$830$	0	0
$831$	13.5866	0.471314
$832$	0	0
$833$	5.56451	0.192799
$834$	0	0
$835$	−43.5113	−1.50577
$836$	0	0
$837$	−10.9233	−0.377564
$838$	0	0
$839$	49.5145	1.70943	0.854716	−	0.519096i	$-0.173731\pi$
$839$	49.5145	1.70943	0.854716	+	0.519096i	$0.173731\pi$
$840$	0	0
$841$	3.87847	0.133740
$842$	0	0
$843$	1.09451	0.0376970
$844$	0	0
$845$	−35.8385	−1.23288
$846$	0	0
$847$	8.07951	0.277615
$848$	0	0
$849$	32.0003	1.09825
$850$	0	0
$851$	−0.336292	−0.0115279
$852$	0	0
$853$	16.8514	0.576982	0.288491	−	0.957483i	$-0.406846\pi$
$853$	16.8514	0.576982	0.288491	+	0.957483i	$0.406846\pi$
$854$	0	0
$855$	14.6027	0.499400
$856$	0	0
$857$	5.93126	0.202608	0.101304	−	0.994856i	$-0.467699\pi$
$857$	5.93126	0.202608	0.101304	+	0.994856i	$0.467699\pi$
$858$	0	0
$859$	−52.5264	−1.79218	−0.896088	−	0.443876i	$-0.853603\pi$
$859$	−52.5264	−1.79218	−0.896088	+	0.443876i	$0.853603\pi$
$860$	0	0
$861$	7.82598	0.266709
$862$	0	0
$863$	−26.3038	−0.895391	−0.447695	−	0.894186i	$-0.647755\pi$
$863$	−26.3038	−0.895391	−0.447695	+	0.894186i	$0.647755\pi$
$864$	0	0
$865$	25.4554	0.865511
$866$	0	0
$867$	13.9637	0.474233
$868$	0	0
$869$	−3.83651	−0.130145
$870$	0	0
$871$	−18.4191	−0.624107
$872$	0	0
$873$	8.31098	0.281284
$874$	0	0
$875$	3.93238	0.132939
$876$	0	0
$877$	−41.9167	−1.41543	−0.707713	−	0.706500i	$-0.750273\pi$
$877$	−41.9167	−1.41543	−0.707713	+	0.706500i	$0.750273\pi$
$878$	0	0
$879$	15.6909	0.529240
$880$	0	0
$881$	−12.8183	−0.431861	−0.215931	−	0.976409i	$-0.569278\pi$
$881$	−12.8183	−0.431861	−0.215931	+	0.976409i	$0.569278\pi$
$882$	0	0
$883$	−54.4558	−1.83258	−0.916291	−	0.400513i	$-0.868832\pi$
$883$	−54.4558	−1.83258	−0.916291	+	0.400513i	$0.868832\pi$
$884$	0	0
$885$	6.76265	0.227324
$886$	0	0
$887$	−47.1571	−1.58338	−0.791691	−	0.610922i	$-0.790799\pi$
$887$	−47.1571	−1.58338	−0.791691	+	0.610922i	$0.790799\pi$
$888$	0	0
$889$	−10.3837	−0.348258
$890$	0	0
$891$	4.36801	0.146334
$892$	0	0
$893$	−41.0177	−1.37261
$894$	0	0
$895$	66.5931	2.22596
$896$	0	0
$897$	1.50992	0.0504146
$898$	0	0
$899$	62.6338	2.08895
$900$	0	0
$901$	7.29716	0.243104
$902$	0	0
$903$	9.58018	0.318809
$904$	0	0
$905$	20.8147	0.691903
$906$	0	0
$907$	56.9065	1.88955	0.944775	−	0.327720i	$-0.106280\pi$
$907$	56.9065	1.88955	0.944775	+	0.327720i	$0.106280\pi$
$908$	0	0
$909$	−2.70906	−0.0898538
$910$	0	0
$911$	−26.9942	−0.894358	−0.447179	−	0.894445i	$-0.647571\pi$
$911$	−26.9942	−0.894358	−0.447179	+	0.894445i	$0.647571\pi$
$912$	0	0
$913$	−56.6190	−1.87381
$914$	0	0
$915$	16.3005	0.538879
$916$	0	0
$917$	−9.76569	−0.322492
$918$	0	0
$919$	39.7594	1.31154	0.655770	−	0.754960i	$-0.272344\pi$
$919$	39.7594	1.31154	0.655770	+	0.754960i	$0.272344\pi$
$920$	0	0
$921$	−29.4453	−0.970255
$922$	0	0
$923$	−18.7462	−0.617039
$924$	0	0
$925$	−2.07703	−0.0682924
$926$	0	0
$927$	−8.41032	−0.276231
$928$	0	0
$929$	−27.7938	−0.911884	−0.455942	−	0.890010i	$-0.650697\pi$
$929$	−27.7938	−0.911884	−0.455942	+	0.890010i	$0.650697\pi$
$930$	0	0
$931$	4.36801	0.143156
$932$	0	0
$933$	−7.85680	−0.257220
$934$	0	0
$935$	81.2566	2.65737
$936$	0	0
$937$	−58.2369	−1.90252	−0.951259	−	0.308393i	$-0.900209\pi$
$937$	−58.2369	−1.90252	−0.951259	+	0.308393i	$0.900209\pi$
$938$	0	0
$939$	−27.3253	−0.891728
$940$	0	0
$941$	33.8951	1.10495	0.552475	−	0.833530i	$-0.313684\pi$
$941$	33.8951	1.10495	0.552475	+	0.833530i	$0.313684\pi$
$942$	0	0
$943$	7.82598	0.254849
$944$	0	0
$945$	3.34309	0.108751
$946$	0	0
$947$	49.0637	1.59436	0.797178	−	0.603744i	$-0.206325\pi$
$947$	49.0637	1.59436	0.797178	+	0.603744i	$0.206325\pi$
$948$	0	0
$949$	13.9378	0.452441
$950$	0	0
$951$	−14.0716	−0.456301
$952$	0	0
$953$	18.7705	0.608035	0.304017	−	0.952667i	$-0.401672\pi$
$953$	18.7705	0.608035	0.304017	+	0.952667i	$0.401672\pi$
$954$	0	0
$955$	−4.89819	−0.158502
$956$	0	0
$957$	−25.0461	−0.809624
$958$	0	0
$959$	−6.51217	−0.210289
$960$	0	0
$961$	88.3180	2.84897
$962$	0	0
$963$	12.6977	0.409177
$964$	0	0
$965$	15.5740	0.501344
$966$	0	0
$967$	15.2423	0.490161	0.245080	−	0.969503i	$-0.421186\pi$
$967$	15.2423	0.490161	0.245080	+	0.969503i	$0.421186\pi$
$968$	0	0
$969$	24.3058	0.780815
$970$	0	0
$971$	−10.7150	−0.343861	−0.171931	−	0.985109i	$-0.555000\pi$
$971$	−10.7150	−0.343861	−0.171931	+	0.985109i	$0.555000\pi$
$972$	0	0
$973$	−9.69717	−0.310877
$974$	0	0
$975$	9.32565	0.298660
$976$	0	0
$977$	−10.8224	−0.346240	−0.173120	−	0.984901i	$-0.555385\pi$
$977$	−10.8224	−0.346240	−0.173120	+	0.984901i	$0.555385\pi$
$978$	0	0
$979$	−57.2279	−1.82901
$980$	0	0
$981$	−8.45850	−0.270059
$982$	0	0
$983$	49.0256	1.56368	0.781838	−	0.623482i	$-0.214283\pi$
$983$	49.0256	1.56368	0.781838	+	0.623482i	$0.214283\pi$
$984$	0	0
$985$	71.2464	2.27010
$986$	0	0
$987$	−9.39049	−0.298903
$988$	0	0
$989$	9.58018	0.304632
$990$	0	0
$991$	21.1406	0.671553	0.335777	−	0.941942i	$-0.391001\pi$
$991$	21.1406	0.671553	0.335777	+	0.941942i	$0.391001\pi$
$992$	0	0
$993$	−8.18758	−0.259825
$994$	0	0
$995$	51.0541	1.61852
$996$	0	0
$997$	0.223489	0.00707797	0.00353899	−	0.999994i	$-0.498874\pi$
$997$	0.223489	0.00707797	0.00353899	+	0.999994i	$0.498874\pi$
$998$	0	0
$999$	−0.336292	−0.0106398

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7728.2.a.ch.1.5		6
4.3	odd	2		3864.2.a.w.1.5	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3864.2.a.w.1.5	✓	6	4.3	odd	2
7728.2.a.ch.1.5		6	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.34309 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +3.34309 q^{5} +1.00000 q^{7} +1.00000 q^{9} +4.36801 q^{11} +1.50992 q^{13} +3.34309 q^{15} +5.56451 q^{17} +4.36801 q^{19} +1.00000 q^{21} +1.00000 q^{23} +6.17627 q^{25} +1.00000 q^{27} -5.73398 q^{29} -10.9233 q^{31} +4.36801 q^{33} +3.34309 q^{35} -0.336292 q^{37} +1.50992 q^{39} +7.82598 q^{41} +9.58018 q^{43} +3.34309 q^{45} -9.39049 q^{47} +1.00000 q^{49} +5.56451 q^{51} +1.31138 q^{53} +14.6027 q^{55} +4.36801 q^{57} +2.02287 q^{59} +4.87588 q^{61} +1.00000 q^{63} +5.04779 q^{65} -12.1987 q^{67} +1.00000 q^{69} -12.4154 q^{71} +9.23086 q^{73} +6.17627 q^{75} +4.36801 q^{77} -0.878321 q^{79} +1.00000 q^{81} -12.9622 q^{83} +18.6027 q^{85} -5.73398 q^{87} -13.1016 q^{89} +1.50992 q^{91} -10.9233 q^{93} +14.6027 q^{95} +8.31098 q^{97} +4.36801 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{3} + 2 q^{5} + 6 q^{7} + 6 q^{9}+O(q^{10}) \) 6 * q + 6 * q^3 + 2 * q^5 + 6 * q^7 + 6 * q^9	\( 6 q + 6 q^{3} + 2 q^{5} + 6 q^{7} + 6 q^{9} + 3 q^{11} + 2 q^{15} + 6 q^{17} + 3 q^{19} + 6 q^{21} + 6 q^{23} + 10 q^{25} + 6 q^{27} + 5 q^{29} + 4 q^{31} + 3 q^{33} + 2 q^{35} + q^{37} + 12 q^{41} + 6 q^{43} + 2 q^{45} + 6 q^{47} + 6 q^{49} + 6 q^{51} + 10 q^{53} - 3 q^{55} + 3 q^{57} + 14 q^{59} + 4 q^{61} + 6 q^{63} + 27 q^{65} - 7 q^{67} + 6 q^{69} - 7 q^{71} + 10 q^{73} + 10 q^{75} + 3 q^{77} - 14 q^{79} + 6 q^{81} - 14 q^{83} + 21 q^{85} + 5 q^{87} + 25 q^{89} + 4 q^{93} - 3 q^{95} + 11 q^{97} + 3 q^{99}+O(q^{100}) \) 6 * q + 6 * q^3 + 2 * q^5 + 6 * q^7 + 6 * q^9 + 3 * q^11 + 2 * q^15 + 6 * q^17 + 3 * q^19 + 6 * q^21 + 6 * q^23 + 10 * q^25 + 6 * q^27 + 5 * q^29 + 4 * q^31 + 3 * q^33 + 2 * q^35 + q^37 + 12 * q^41 + 6 * q^43 + 2 * q^45 + 6 * q^47 + 6 * q^49 + 6 * q^51 + 10 * q^53 - 3 * q^55 + 3 * q^57 + 14 * q^59 + 4 * q^61 + 6 * q^63 + 27 * q^65 - 7 * q^67 + 6 * q^69 - 7 * q^71 + 10 * q^73 + 10 * q^75 + 3 * q^77 - 14 * q^79 + 6 * q^81 - 14 * q^83 + 21 * q^85 + 5 * q^87 + 25 * q^89 + 4 * q^93 - 3 * q^95 + 11 * q^97 + 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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