LMFDB - Embedded newform 7728.2.a.cb.1.2

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:	$1$
Dimension:	$4$
Coefficient field:	4.4.2225.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 5x^{2} + 2x + 4 $ x^4 - x^3 - 5x^2 + 2x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
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			1.2
Root			$1.13856$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -1.08564 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.08564 q^{5} -1.00000 q^{7} +1.00000 q^{9} +4.07830 q^{11} +5.16394 q^{13} -1.08564 q^{15} -2.56511 q^{17} -4.07830 q^{19} -1.00000 q^{21} +1.00000 q^{23} -3.82139 q^{25} +1.00000 q^{27} -10.6326 q^{29} -6.55044 q^{31} +4.07830 q^{33} +1.08564 q^{35} -3.90703 q^{37} +5.16394 q^{39} -1.43489 q^{41} -12.8484 q^{43} -1.08564 q^{45} +9.98533 q^{47} +1.00000 q^{49} -2.56511 q^{51} +9.15309 q^{53} -4.42756 q^{55} -4.07830 q^{57} -3.63989 q^{59} +1.90414 q^{61} -1.00000 q^{63} -5.60617 q^{65} +8.28331 q^{67} +1.00000 q^{69} -14.1903 q^{71} -4.64723 q^{73} -3.82139 q^{75} -4.07830 q^{77} +6.55044 q^{79} +1.00000 q^{81} -8.86597 q^{83} +2.78478 q^{85} -10.6326 q^{87} -10.6501 q^{89} -5.16394 q^{91} -6.55044 q^{93} +4.42756 q^{95} -4.64723 q^{97} +4.07830 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} - 3 q^{5} - 4 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 - 3 * q^5 - 4 * q^7 + 4 * q^9	$ 4 q + 4 q^{3} - 3 q^{5} - 4 q^{7} + 4 q^{9} - 2 q^{11} + q^{13} - 3 q^{15} - 8 q^{17} + 2 q^{19} - 4 q^{21} + 4 q^{23} - q^{25} + 4 q^{27} - 10 q^{29} + 10 q^{31} - 2 q^{33} + 3 q^{35} + q^{39} - 8 q^{41} - 13 q^{43} - 3 q^{45} + 6 q^{47} + 4 q^{49} - 8 q^{51} + 5 q^{53} - 3 q^{55} + 2 q^{57} + q^{59} + 5 q^{61} - 4 q^{63} - 22 q^{65} - 3 q^{67} + 4 q^{69} - 5 q^{71} - 20 q^{73} - q^{75} + 2 q^{77} - 10 q^{79} + 4 q^{81} - 18 q^{83} + 25 q^{85} - 10 q^{87} - 31 q^{89} - q^{91} + 10 q^{93} + 3 q^{95} - 20 q^{97} - 2 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 - 3 * q^5 - 4 * q^7 + 4 * q^9 - 2 * q^11 + q^13 - 3 * q^15 - 8 * q^17 + 2 * q^19 - 4 * q^21 + 4 * q^23 - q^25 + 4 * q^27 - 10 * q^29 + 10 * q^31 - 2 * q^33 + 3 * q^35 + q^39 - 8 * q^41 - 13 * q^43 - 3 * q^45 + 6 * q^47 + 4 * q^49 - 8 * q^51 + 5 * q^53 - 3 * q^55 + 2 * q^57 + q^59 + 5 * q^61 - 4 * q^63 - 22 * q^65 - 3 * q^67 + 4 * q^69 - 5 * q^71 - 20 * q^73 - q^75 + 2 * q^77 - 10 * q^79 + 4 * q^81 - 18 * q^83 + 25 * q^85 - 10 * q^87 - 31 * q^89 - q^91 + 10 * q^93 + 3 * q^95 - 20 * q^97 - 2 * 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
$5$	−1.08564	−0.485512	−0.242756	−	0.970087i	$-0.578051\pi$
$5$	−1.08564	−0.485512	−0.242756	+	0.970087i	$0.578051\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.07830	1.22965	0.614827	−	0.788662i	$-0.289226\pi$
$11$	4.07830	1.22965	0.614827	+	0.788662i	$0.289226\pi$
$12$	0	0
$13$	5.16394	1.43222	0.716110	−	0.697988i	$-0.245921\pi$
$13$	5.16394	1.43222	0.716110	+	0.697988i	$0.245921\pi$
$14$	0	0
$15$	−1.08564	−0.280310
$16$	0	0
$17$	−2.56511	−0.622130	−0.311065	−	0.950389i	$-0.600686\pi$
$17$	−2.56511	−0.622130	−0.311065	+	0.950389i	$0.600686\pi$
$18$	0	0
$19$	−4.07830	−0.935627	−0.467814	−	0.883827i	$-0.654958\pi$
$19$	−4.07830	−0.935627	−0.467814	+	0.883827i	$0.654958\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−3.82139	−0.764278
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−10.6326	−1.97442	−0.987208	−	0.159435i	$-0.949033\pi$
$29$	−10.6326	−1.97442	−0.987208	+	0.159435i	$0.949033\pi$
$30$	0	0
$31$	−6.55044	−1.17649	−0.588247	−	0.808681i	$-0.700182\pi$
$31$	−6.55044	−1.17649	−0.588247	+	0.808681i	$0.700182\pi$
$32$	0	0
$33$	4.07830	0.709942
$34$	0	0
$35$	1.08564	0.183506
$36$	0	0
$37$	−3.90703	−0.642312	−0.321156	−	0.947026i	$-0.604071\pi$
$37$	−3.90703	−0.642312	−0.321156	+	0.947026i	$0.604071\pi$
$38$	0	0
$39$	5.16394	0.826892
$40$	0	0
$41$	−1.43489	−0.224093	−0.112046	−	0.993703i	$-0.535740\pi$
$41$	−1.43489	−0.224093	−0.112046	+	0.993703i	$0.535740\pi$
$42$	0	0
$43$	−12.8484	−1.95936	−0.979682	−	0.200556i	$-0.935725\pi$
$43$	−12.8484	−1.95936	−0.979682	+	0.200556i	$0.935725\pi$
$44$	0	0
$45$	−1.08564	−0.161837
$46$	0	0
$47$	9.98533	1.45651	0.728255	−	0.685306i	$-0.240332\pi$
$47$	9.98533	1.45651	0.728255	+	0.685306i	$0.240332\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−2.56511	−0.359187
$52$	0	0
$53$	9.15309	1.25727	0.628637	−	0.777699i	$-0.283613\pi$
$53$	9.15309	1.25727	0.628637	+	0.777699i	$0.283613\pi$
$54$	0	0
$55$	−4.42756	−0.597012
$56$	0	0
$57$	−4.07830	−0.540185
$58$	0	0
$59$	−3.63989	−0.473874	−0.236937	−	0.971525i	$-0.576143\pi$
$59$	−3.63989	−0.473874	−0.236937	+	0.971525i	$0.576143\pi$
$60$	0	0
$61$	1.90414	0.243800	0.121900	−	0.992542i	$-0.461101\pi$
$61$	1.90414	0.243800	0.121900	+	0.992542i	$0.461101\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	−5.60617	−0.695360
$66$	0	0
$67$	8.28331	1.01197	0.505983	−	0.862543i	$-0.331130\pi$
$67$	8.28331	1.01197	0.505983	+	0.862543i	$0.331130\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−14.1903	−1.68408	−0.842041	−	0.539413i	$-0.818646\pi$
$71$	−14.1903	−1.68408	−0.842041	+	0.539413i	$0.818646\pi$
$72$	0	0
$73$	−4.64723	−0.543917	−0.271958	−	0.962309i	$-0.587671\pi$
$73$	−4.64723	−0.543917	−0.271958	+	0.962309i	$0.587671\pi$
$74$	0	0
$75$	−3.82139	−0.441256
$76$	0	0
$77$	−4.07830	−0.464766
$78$	0	0
$79$	6.55044	0.736982	0.368491	−	0.929631i	$-0.379875\pi$
$79$	6.55044	0.736982	0.368491	+	0.929631i	$0.379875\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−8.86597	−0.973166	−0.486583	−	0.873634i	$-0.661757\pi$
$83$	−8.86597	−0.973166	−0.486583	+	0.873634i	$0.661757\pi$
$84$	0	0
$85$	2.78478	0.302051
$86$	0	0
$87$	−10.6326	−1.13993
$88$	0	0
$89$	−10.6501	−1.12891	−0.564455	−	0.825464i	$-0.690914\pi$
$89$	−10.6501	−1.12891	−0.564455	+	0.825464i	$0.690914\pi$
$90$	0	0
$91$	−5.16394	−0.541328
$92$	0	0
$93$	−6.55044	−0.679249
$94$	0	0
$95$	4.42756	0.454258
$96$	0	0
$97$	−4.64723	−0.471855	−0.235927	−	0.971771i	$-0.575813\pi$
$97$	−4.64723	−0.471855	−0.235927	+	0.971771i	$0.575813\pi$
$98$	0	0
$99$	4.07830	0.409885
$100$	0	0
$101$	−11.9003	−1.18413	−0.592063	−	0.805892i	$-0.701686\pi$
$101$	−11.9003	−1.18413	−0.592063	+	0.805892i	$0.701686\pi$
$102$	0	0
$103$	11.3689	1.12022	0.560108	−	0.828420i	$-0.310760\pi$
$103$	11.3689	1.12022	0.560108	+	0.828420i	$0.310760\pi$
$104$	0	0
$105$	1.08564	0.105947
$106$	0	0
$107$	15.7857	1.52606	0.763028	−	0.646365i	$-0.223712\pi$
$107$	15.7857	1.52606	0.763028	+	0.646365i	$0.223712\pi$
$108$	0	0
$109$	−1.33903	−0.128256	−0.0641281	−	0.997942i	$-0.520427\pi$
$109$	−1.33903	−0.128256	−0.0641281	+	0.997942i	$0.520427\pi$
$110$	0	0
$111$	−3.90703	−0.370839
$112$	0	0
$113$	6.58417	0.619386	0.309693	−	0.950837i	$-0.399774\pi$
$113$	6.58417	0.619386	0.309693	+	0.950837i	$0.399774\pi$
$114$	0	0
$115$	−1.08564	−0.101236
$116$	0	0
$117$	5.16394	0.477407
$118$	0	0
$119$	2.56511	0.235143
$120$	0	0
$121$	5.63256	0.512051
$122$	0	0
$123$	−1.43489	−0.129380
$124$	0	0
$125$	9.57683	0.856578
$126$	0	0
$127$	12.3616	1.09692	0.548458	−	0.836178i	$-0.315215\pi$
$127$	12.3616	1.09692	0.548458	+	0.836178i	$0.315215\pi$
$128$	0	0
$129$	−12.8484	−1.13124
$130$	0	0
$131$	−7.59150	−0.663272	−0.331636	−	0.943407i	$-0.607601\pi$
$131$	−7.59150	−0.663272	−0.331636	+	0.943407i	$0.607601\pi$
$132$	0	0
$133$	4.07830	0.353634
$134$	0	0
$135$	−1.08564	−0.0934368
$136$	0	0
$137$	−10.8185	−0.924287	−0.462144	−	0.886805i	$-0.652920\pi$
$137$	−10.8185	−0.924287	−0.462144	+	0.886805i	$0.652920\pi$
$138$	0	0
$139$	12.6733	1.07494	0.537468	−	0.843284i	$-0.319381\pi$
$139$	12.6733	1.07494	0.537468	+	0.843284i	$0.319381\pi$
$140$	0	0
$141$	9.98533	0.840917
$142$	0	0
$143$	21.0601	1.76114
$144$	0	0
$145$	11.5431	0.958603
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	−12.5613	−1.02906	−0.514530	−	0.857472i	$-0.672034\pi$
$149$	−12.5613	−1.02906	−0.514530	+	0.857472i	$0.672034\pi$
$150$	0	0
$151$	21.3543	1.73779	0.868893	−	0.495000i	$-0.164832\pi$
$151$	21.3543	1.73779	0.868893	+	0.495000i	$0.164832\pi$
$152$	0	0
$153$	−2.56511	−0.207377
$154$	0	0
$155$	7.11140	0.571202
$156$	0	0
$157$	−11.4311	−0.912299	−0.456150	−	0.889903i	$-0.650772\pi$
$157$	−11.4311	−0.912299	−0.456150	+	0.889903i	$0.650772\pi$
$158$	0	0
$159$	9.15309	0.725887
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	−16.8376	−1.31882	−0.659410	−	0.751784i	$-0.729194\pi$
$163$	−16.8376	−1.31882	−0.659410	+	0.751784i	$0.729194\pi$
$164$	0	0
$165$	−4.42756	−0.344685
$166$	0	0
$167$	6.82554	0.528176	0.264088	−	0.964499i	$-0.414929\pi$
$167$	6.82554	0.528176	0.264088	+	0.964499i	$0.414929\pi$
$168$	0	0
$169$	13.6663	1.05125
$170$	0	0
$171$	−4.07830	−0.311876
$172$	0	0
$173$	−24.2753	−1.84562	−0.922810	−	0.385255i	$-0.874113\pi$
$173$	−24.2753	−1.84562	−0.922810	+	0.385255i	$0.874113\pi$
$174$	0	0
$175$	3.82139	0.288870
$176$	0	0
$177$	−3.63989	−0.273591
$178$	0	0
$179$	−22.4803	−1.68026	−0.840130	−	0.542385i	$-0.817521\pi$
$179$	−22.4803	−1.68026	−0.840130	+	0.542385i	$0.817521\pi$
$180$	0	0
$181$	9.19385	0.683374	0.341687	−	0.939814i	$-0.389002\pi$
$181$	9.19385	0.683374	0.341687	+	0.939814i	$0.389002\pi$
$182$	0	0
$183$	1.90414	0.140758
$184$	0	0
$185$	4.24162	0.311850
$186$	0	0
$187$	−10.4613	−0.765005
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−21.3543	−1.54514	−0.772571	−	0.634929i	$-0.781029\pi$
$191$	−21.3543	−1.54514	−0.772571	+	0.634929i	$0.781029\pi$
$192$	0	0
$193$	−18.2458	−1.31336	−0.656679	−	0.754170i	$-0.728039\pi$
$193$	−18.2458	−1.31336	−0.656679	+	0.754170i	$0.728039\pi$
$194$	0	0
$195$	−5.60617	−0.401466
$196$	0	0
$197$	15.1493	1.07934	0.539671	−	0.841876i	$-0.318549\pi$
$197$	15.1493	1.07934	0.539671	+	0.841876i	$0.318549\pi$
$198$	0	0
$199$	6.53138	0.462997	0.231499	−	0.972835i	$-0.425637\pi$
$199$	6.53138	0.462997	0.231499	+	0.972835i	$0.425637\pi$
$200$	0	0
$201$	8.28331	0.584259
$202$	0	0
$203$	10.6326	0.746259
$204$	0	0
$205$	1.55777	0.108800
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	−16.6326	−1.15050
$210$	0	0
$211$	2.64723	0.182243	0.0911214	−	0.995840i	$-0.470955\pi$
$211$	2.64723	0.182243	0.0911214	+	0.995840i	$0.470955\pi$
$212$	0	0
$213$	−14.1903	−0.972306
$214$	0	0
$215$	13.9487	0.951295
$216$	0	0
$217$	6.55044	0.444673
$218$	0	0
$219$	−4.64723	−0.314031
$220$	0	0
$221$	−13.2461	−0.891027
$222$	0	0
$223$	17.7818	1.19076	0.595380	−	0.803444i	$-0.297002\pi$
$223$	17.7818	1.19076	0.595380	+	0.803444i	$0.297002\pi$
$224$	0	0
$225$	−3.82139	−0.254759
$226$	0	0
$227$	13.7695	0.913913	0.456956	−	0.889489i	$-0.348940\pi$
$227$	13.7695	0.913913	0.456956	+	0.889489i	$0.348940\pi$
$228$	0	0
$229$	−20.6112	−1.36203	−0.681013	−	0.732272i	$-0.738460\pi$
$229$	−20.6112	−1.36203	−0.681013	+	0.732272i	$0.738460\pi$
$230$	0	0
$231$	−4.07830	−0.268333
$232$	0	0
$233$	3.37183	0.220896	0.110448	−	0.993882i	$-0.464771\pi$
$233$	3.37183	0.220896	0.110448	+	0.993882i	$0.464771\pi$
$234$	0	0
$235$	−10.8405	−0.707153
$236$	0	0
$237$	6.55044	0.425497
$238$	0	0
$239$	−17.7818	−1.15021	−0.575106	−	0.818079i	$-0.695039\pi$
$239$	−17.7818	−1.15021	−0.575106	+	0.818079i	$0.695039\pi$
$240$	0	0
$241$	−25.0630	−1.61445	−0.807225	−	0.590244i	$-0.799032\pi$
$241$	−25.0630	−1.61445	−0.807225	+	0.590244i	$0.799032\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−1.08564	−0.0693588
$246$	0	0
$247$	−21.0601	−1.34002
$248$	0	0
$249$	−8.86597	−0.561858
$250$	0	0
$251$	−0.465103	−0.0293571	−0.0146785	−	0.999892i	$-0.504672\pi$
$251$	−0.465103	−0.0293571	−0.0146785	+	0.999892i	$0.504672\pi$
$252$	0	0
$253$	4.07830	0.256401
$254$	0	0
$255$	2.78478	0.174389
$256$	0	0
$257$	17.5930	1.09742	0.548711	−	0.836012i	$-0.315119\pi$
$257$	17.5930	1.09742	0.548711	+	0.836012i	$0.315119\pi$
$258$	0	0
$259$	3.90703	0.242771
$260$	0	0
$261$	−10.6326	−0.658139
$262$	0	0
$263$	−5.69532	−0.351189	−0.175594	−	0.984463i	$-0.556185\pi$
$263$	−5.69532	−0.351189	−0.175594	+	0.984463i	$0.556185\pi$
$264$	0	0
$265$	−9.93694	−0.610421
$266$	0	0
$267$	−10.6501	−0.651777
$268$	0	0
$269$	8.27949	0.504809	0.252405	−	0.967622i	$-0.418779\pi$
$269$	8.27949	0.504809	0.252405	+	0.967622i	$0.418779\pi$
$270$	0	0
$271$	12.0366	0.731172	0.365586	−	0.930778i	$-0.380869\pi$
$271$	12.0366	0.731172	0.365586	+	0.930778i	$0.380869\pi$
$272$	0	0
$273$	−5.16394	−0.312536
$274$	0	0
$275$	−15.5848	−0.939798
$276$	0	0
$277$	−12.5329	−0.753028	−0.376514	−	0.926411i	$-0.622877\pi$
$277$	−12.5329	−0.753028	−0.376514	+	0.926411i	$0.622877\pi$
$278$	0	0
$279$	−6.55044	−0.392165
$280$	0	0
$281$	−13.2181	−0.788526	−0.394263	−	0.918998i	$-0.629000\pi$
$281$	−13.2181	−0.788526	−0.394263	+	0.918998i	$0.629000\pi$
$282$	0	0
$283$	−15.3314	−0.911357	−0.455679	−	0.890144i	$-0.650603\pi$
$283$	−15.3314	−0.911357	−0.455679	+	0.890144i	$0.650603\pi$
$284$	0	0
$285$	4.42756	0.262266
$286$	0	0
$287$	1.43489	0.0846990
$288$	0	0
$289$	−10.4202	−0.612954
$290$	0	0
$291$	−4.64723	−0.272425
$292$	0	0
$293$	−11.9977	−0.700912	−0.350456	−	0.936579i	$-0.613973\pi$
$293$	−11.9977	−0.700912	−0.350456	+	0.936579i	$0.613973\pi$
$294$	0	0
$295$	3.95161	0.230071
$296$	0	0
$297$	4.07830	0.236647
$298$	0	0
$299$	5.16394	0.298638
$300$	0	0
$301$	12.8484	0.740570
$302$	0	0
$303$	−11.9003	−0.683656
$304$	0	0
$305$	−2.06721	−0.118368
$306$	0	0
$307$	14.7147	0.839811	0.419906	−	0.907568i	$-0.362063\pi$
$307$	14.7147	0.839811	0.419906	+	0.907568i	$0.362063\pi$
$308$	0	0
$309$	11.3689	0.646757
$310$	0	0
$311$	31.0455	1.76043	0.880213	−	0.474579i	$-0.157400\pi$
$311$	31.0455	1.76043	0.880213	+	0.474579i	$0.157400\pi$
$312$	0	0
$313$	−23.9502	−1.35375	−0.676873	−	0.736100i	$-0.736665\pi$
$313$	−23.9502	−1.35375	−0.676873	+	0.736100i	$0.736665\pi$
$314$	0	0
$315$	1.08564	0.0611687
$316$	0	0
$317$	−9.79650	−0.550226	−0.275113	−	0.961412i	$-0.588715\pi$
$317$	−9.79650	−0.550226	−0.275113	+	0.961412i	$0.588715\pi$
$318$	0	0
$319$	−43.3628	−2.42785
$320$	0	0
$321$	15.7857	0.881069
$322$	0	0
$323$	10.4613	0.582082
$324$	0	0
$325$	−19.7334	−1.09461
$326$	0	0
$327$	−1.33903	−0.0740487
$328$	0	0
$329$	−9.98533	−0.550509
$330$	0	0
$331$	3.37658	0.185593	0.0927967	−	0.995685i	$-0.470419\pi$
$331$	3.37658	0.185593	0.0927967	+	0.995685i	$0.470419\pi$
$332$	0	0
$333$	−3.90703	−0.214104
$334$	0	0
$335$	−8.99267	−0.491322
$336$	0	0
$337$	−23.1142	−1.25911	−0.629554	−	0.776956i	$-0.716762\pi$
$337$	−23.1142	−1.25911	−0.629554	+	0.776956i	$0.716762\pi$
$338$	0	0
$339$	6.58417	0.357603
$340$	0	0
$341$	−26.7147	−1.44668
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−1.08564	−0.0584488
$346$	0	0
$347$	21.6443	1.16193	0.580963	−	0.813930i	$-0.302676\pi$
$347$	21.6443	1.16193	0.580963	+	0.813930i	$0.302676\pi$
$348$	0	0
$349$	2.42756	0.129944	0.0649721	−	0.997887i	$-0.479304\pi$
$349$	2.42756	0.129944	0.0649721	+	0.997887i	$0.479304\pi$
$350$	0	0
$351$	5.16394	0.275631
$352$	0	0
$353$	−15.3528	−0.817146	−0.408573	−	0.912726i	$-0.633973\pi$
$353$	−15.3528	−0.817146	−0.408573	+	0.912726i	$0.633973\pi$
$354$	0	0
$355$	15.4056	0.817642
$356$	0	0
$357$	2.56511	0.135760
$358$	0	0
$359$	−14.0646	−0.742299	−0.371150	−	0.928573i	$-0.621036\pi$
$359$	−14.0646	−0.742299	−0.371150	+	0.928573i	$0.621036\pi$
$360$	0	0
$361$	−2.36744	−0.124602
$362$	0	0
$363$	5.63256	0.295633
$364$	0	0
$365$	5.04520	0.264078
$366$	0	0
$367$	−21.9018	−1.14327	−0.571633	−	0.820509i	$-0.693690\pi$
$367$	−21.9018	−1.14327	−0.571633	+	0.820509i	$0.693690\pi$
$368$	0	0
$369$	−1.43489	−0.0746975
$370$	0	0
$371$	−9.15309	−0.475205
$372$	0	0
$373$	26.1458	1.35378	0.676888	−	0.736086i	$-0.263328\pi$
$373$	26.1458	1.35378	0.676888	+	0.736086i	$0.263328\pi$
$374$	0	0
$375$	9.57683	0.494546
$376$	0	0
$377$	−54.9059	−2.82780
$378$	0	0
$379$	−20.2657	−1.04098	−0.520491	−	0.853867i	$-0.674251\pi$
$379$	−20.2657	−1.04098	−0.520491	+	0.853867i	$0.674251\pi$
$380$	0	0
$381$	12.3616	0.633304
$382$	0	0
$383$	−7.05723	−0.360608	−0.180304	−	0.983611i	$-0.557708\pi$
$383$	−7.05723	−0.360608	−0.180304	+	0.983611i	$0.557708\pi$
$384$	0	0
$385$	4.42756	0.225649
$386$	0	0
$387$	−12.8484	−0.653122
$388$	0	0
$389$	−29.7956	−1.51070	−0.755348	−	0.655324i	$-0.772532\pi$
$389$	−29.7956	−1.51070	−0.755348	+	0.655324i	$0.772532\pi$
$390$	0	0
$391$	−2.56511	−0.129723
$392$	0	0
$393$	−7.59150	−0.382941
$394$	0	0
$395$	−7.11140	−0.357813
$396$	0	0
$397$	33.9356	1.70318	0.851588	−	0.524211i	$-0.175640\pi$
$397$	33.9356	1.70318	0.851588	+	0.524211i	$0.175640\pi$
$398$	0	0
$399$	4.07830	0.204171
$400$	0	0
$401$	−1.27829	−0.0638345	−0.0319173	−	0.999491i	$-0.510161\pi$
$401$	−1.27829	−0.0638345	−0.0319173	+	0.999491i	$0.510161\pi$
$402$	0	0
$403$	−33.8261	−1.68500
$404$	0	0
$405$	−1.08564	−0.0539458
$406$	0	0
$407$	−15.9340	−0.789822
$408$	0	0
$409$	−2.53000	−0.125100	−0.0625501	−	0.998042i	$-0.519923\pi$
$409$	−2.53000	−0.125100	−0.0625501	+	0.998042i	$0.519923\pi$
$410$	0	0
$411$	−10.8185	−0.533637
$412$	0	0
$413$	3.63989	0.179107
$414$	0	0
$415$	9.62523	0.472484
$416$	0	0
$417$	12.6733	0.620615
$418$	0	0
$419$	8.79010	0.429424	0.214712	−	0.976677i	$-0.431119\pi$
$419$	8.79010	0.429424	0.214712	+	0.976677i	$0.431119\pi$
$420$	0	0
$421$	−3.70353	−0.180499	−0.0902495	−	0.995919i	$-0.528766\pi$
$421$	−3.70353	−0.180499	−0.0902495	+	0.995919i	$0.528766\pi$
$422$	0	0
$423$	9.98533	0.485503
$424$	0	0
$425$	9.80228	0.475480
$426$	0	0
$427$	−1.90414	−0.0921478
$428$	0	0
$429$	21.0601	1.01679
$430$	0	0
$431$	−18.8080	−0.905949	−0.452974	−	0.891524i	$-0.649637\pi$
$431$	−18.8080	−0.905949	−0.452974	+	0.891524i	$0.649637\pi$
$432$	0	0
$433$	3.80702	0.182954	0.0914770	−	0.995807i	$-0.470841\pi$
$433$	3.80702	0.182954	0.0914770	+	0.995807i	$0.470841\pi$
$434$	0	0
$435$	11.5431	0.553450
$436$	0	0
$437$	−4.07830	−0.195092
$438$	0	0
$439$	37.6385	1.79639	0.898194	−	0.439599i	$-0.144880\pi$
$439$	37.6385	1.79639	0.898194	+	0.439599i	$0.144880\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	1.62937	0.0774138	0.0387069	−	0.999251i	$-0.487676\pi$
$443$	1.62937	0.0774138	0.0387069	+	0.999251i	$0.487676\pi$
$444$	0	0
$445$	11.5622	0.548099
$446$	0	0
$447$	−12.5613	−0.594129
$448$	0	0
$449$	−22.4816	−1.06097	−0.530486	−	0.847694i	$-0.677991\pi$
$449$	−22.4816	−1.06097	−0.530486	+	0.847694i	$0.677991\pi$
$450$	0	0
$451$	−5.85193	−0.275557
$452$	0	0
$453$	21.3543	1.00331
$454$	0	0
$455$	5.60617	0.262821
$456$	0	0
$457$	20.7554	0.970899	0.485449	−	0.874265i	$-0.338656\pi$
$457$	20.7554	0.970899	0.485449	+	0.874265i	$0.338656\pi$
$458$	0	0
$459$	−2.56511	−0.119729
$460$	0	0
$461$	35.4393	1.65057	0.825286	−	0.564715i	$-0.191014\pi$
$461$	35.4393	1.65057	0.825286	+	0.564715i	$0.191014\pi$
$462$	0	0
$463$	37.6015	1.74749	0.873746	−	0.486383i	$-0.161684\pi$
$463$	37.6015	1.74749	0.873746	+	0.486383i	$0.161684\pi$
$464$	0	0
$465$	7.11140	0.329783
$466$	0	0
$467$	−32.0490	−1.48305	−0.741525	−	0.670926i	$-0.765897\pi$
$467$	−32.0490	−1.48305	−0.741525	+	0.670926i	$0.765897\pi$
$468$	0	0
$469$	−8.28331	−0.382488
$470$	0	0
$471$	−11.4311	−0.526716
$472$	0	0
$473$	−52.3997	−2.40934
$474$	0	0
$475$	15.5848	0.715079
$476$	0	0
$477$	9.15309	0.419091
$478$	0	0
$479$	−29.0132	−1.32565	−0.662824	−	0.748775i	$-0.730642\pi$
$479$	−29.0132	−1.32565	−0.662824	+	0.748775i	$0.730642\pi$
$480$	0	0
$481$	−20.1757	−0.919931
$482$	0	0
$483$	−1.00000	−0.0455016
$484$	0	0
$485$	5.04520	0.229091
$486$	0	0
$487$	−27.0279	−1.22475	−0.612375	−	0.790567i	$-0.709786\pi$
$487$	−27.0279	−1.22475	−0.612375	+	0.790567i	$0.709786\pi$
$488$	0	0
$489$	−16.8376	−0.761421
$490$	0	0
$491$	4.85776	0.219228	0.109614	−	0.993974i	$-0.465039\pi$
$491$	4.85776	0.219228	0.109614	+	0.993974i	$0.465039\pi$
$492$	0	0
$493$	27.2737	1.22834
$494$	0	0
$495$	−4.42756	−0.199004
$496$	0	0
$497$	14.1903	0.636523
$498$	0	0
$499$	−5.49032	−0.245780	−0.122890	−	0.992420i	$-0.539216\pi$
$499$	−5.49032	−0.245780	−0.122890	+	0.992420i	$0.539216\pi$
$500$	0	0
$501$	6.82554	0.304942
$502$	0	0
$503$	−4.70266	−0.209681	−0.104841	−	0.994489i	$-0.533433\pi$
$503$	−4.70266	−0.209681	−0.104841	+	0.994489i	$0.533433\pi$
$504$	0	0
$505$	12.9194	0.574907
$506$	0	0
$507$	13.6663	0.606941
$508$	0	0
$509$	−13.1155	−0.581336	−0.290668	−	0.956824i	$-0.593878\pi$
$509$	−13.1155	−0.581336	−0.290668	+	0.956824i	$0.593878\pi$
$510$	0	0
$511$	4.64723	0.205581
$512$	0	0
$513$	−4.07830	−0.180062
$514$	0	0
$515$	−12.3425	−0.543878
$516$	0	0
$517$	40.7232	1.79101
$518$	0	0
$519$	−24.2753	−1.06557
$520$	0	0
$521$	−22.5068	−0.986040	−0.493020	−	0.870018i	$-0.664107\pi$
$521$	−22.5068	−0.986040	−0.493020	+	0.870018i	$0.664107\pi$
$522$	0	0
$523$	1.34192	0.0586781	0.0293391	−	0.999570i	$-0.490660\pi$
$523$	1.34192	0.0586781	0.0293391	+	0.999570i	$0.490660\pi$
$524$	0	0
$525$	3.82139	0.166779
$526$	0	0
$527$	16.8026	0.731932
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−3.63989	−0.157958
$532$	0	0
$533$	−7.40970	−0.320950
$534$	0	0
$535$	−17.1375	−0.740918
$536$	0	0
$537$	−22.4803	−0.970098
$538$	0	0
$539$	4.07830	0.175665
$540$	0	0
$541$	44.2909	1.90422	0.952108	−	0.305763i	$-0.0989117\pi$
$541$	44.2909	1.90422	0.952108	+	0.305763i	$0.0989117\pi$
$542$	0	0
$543$	9.19385	0.394546
$544$	0	0
$545$	1.45370	0.0622699
$546$	0	0
$547$	8.00414	0.342233	0.171116	−	0.985251i	$-0.445263\pi$
$547$	8.00414	0.342233	0.171116	+	0.985251i	$0.445263\pi$
$548$	0	0
$549$	1.90414	0.0812667
$550$	0	0
$551$	43.3628	1.84732
$552$	0	0
$553$	−6.55044	−0.278553
$554$	0	0
$555$	4.24162	0.180047
$556$	0	0
$557$	32.9362	1.39555	0.697775	−	0.716317i	$-0.254173\pi$
$557$	32.9362	1.39555	0.697775	+	0.716317i	$0.254173\pi$
$558$	0	0
$559$	−66.3484	−2.80624
$560$	0	0
$561$	−10.4613	−0.441676
$562$	0	0
$563$	−20.4033	−0.859896	−0.429948	−	0.902854i	$-0.641468\pi$
$563$	−20.4033	−0.859896	−0.429948	+	0.902854i	$0.641468\pi$
$564$	0	0
$565$	−7.14802	−0.300719
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	5.50084	0.230607	0.115304	−	0.993330i	$-0.463216\pi$
$569$	5.50084	0.230607	0.115304	+	0.993330i	$0.463216\pi$
$570$	0	0
$571$	12.4285	0.520116	0.260058	−	0.965593i	$-0.416258\pi$
$571$	12.4285	0.520116	0.260058	+	0.965593i	$0.416258\pi$
$572$	0	0
$573$	−21.3543	−0.892088
$574$	0	0
$575$	−3.82139	−0.159363
$576$	0	0
$577$	−9.83576	−0.409468	−0.204734	−	0.978818i	$-0.565633\pi$
$577$	−9.83576	−0.409468	−0.204734	+	0.978818i	$0.565633\pi$
$578$	0	0
$579$	−18.2458	−0.758268
$580$	0	0
$581$	8.86597	0.367822
$582$	0	0
$583$	37.3291	1.54601
$584$	0	0
$585$	−5.60617	−0.231787
$586$	0	0
$587$	9.20332	0.379861	0.189931	−	0.981797i	$-0.439174\pi$
$587$	9.20332	0.379861	0.189931	+	0.981797i	$0.439174\pi$
$588$	0	0
$589$	26.7147	1.10076
$590$	0	0
$591$	15.1493	0.623158
$592$	0	0
$593$	13.1772	0.541124	0.270562	−	0.962703i	$-0.412791\pi$
$593$	13.1772	0.541124	0.270562	+	0.962703i	$0.412791\pi$
$594$	0	0
$595$	−2.78478	−0.114165
$596$	0	0
$597$	6.53138	0.267312
$598$	0	0
$599$	−28.9499	−1.18286	−0.591431	−	0.806356i	$-0.701437\pi$
$599$	−28.9499	−1.18286	−0.591431	+	0.806356i	$0.701437\pi$
$600$	0	0
$601$	38.7542	1.58082	0.790408	−	0.612581i	$-0.209869\pi$
$601$	38.7542	1.58082	0.790408	+	0.612581i	$0.209869\pi$
$602$	0	0
$603$	8.28331	0.337322
$604$	0	0
$605$	−6.11492	−0.248607
$606$	0	0
$607$	−2.54630	−0.103351	−0.0516755	−	0.998664i	$-0.516456\pi$
$607$	−2.54630	−0.103351	−0.0516755	+	0.998664i	$0.516456\pi$
$608$	0	0
$609$	10.6326	0.430853
$610$	0	0
$611$	51.5637	2.08604
$612$	0	0
$613$	15.1251	0.610899	0.305449	−	0.952208i	$-0.401193\pi$
$613$	15.1251	0.610899	0.305449	+	0.952208i	$0.401193\pi$
$614$	0	0
$615$	1.55777	0.0628155
$616$	0	0
$617$	3.59439	0.144705	0.0723523	−	0.997379i	$-0.476949\pi$
$617$	3.59439	0.144705	0.0723523	+	0.997379i	$0.476949\pi$
$618$	0	0
$619$	28.2231	1.13438	0.567192	−	0.823586i	$-0.308030\pi$
$619$	28.2231	1.13438	0.567192	+	0.823586i	$0.308030\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	10.6501	0.426688
$624$	0	0
$625$	8.70999	0.348400
$626$	0	0
$627$	−16.6326	−0.664240
$628$	0	0
$629$	10.0219	0.399601
$630$	0	0
$631$	35.4596	1.41162	0.705812	−	0.708399i	$-0.250582\pi$
$631$	35.4596	1.41162	0.705812	+	0.708399i	$0.250582\pi$
$632$	0	0
$633$	2.64723	0.105218
$634$	0	0
$635$	−13.4202	−0.532565
$636$	0	0
$637$	5.16394	0.204603
$638$	0	0
$639$	−14.1903	−0.561361
$640$	0	0
$641$	28.5710	1.12849	0.564243	−	0.825609i	$-0.309168\pi$
$641$	28.5710	1.12849	0.564243	+	0.825609i	$0.309168\pi$
$642$	0	0
$643$	−26.1429	−1.03097	−0.515487	−	0.856897i	$-0.672389\pi$
$643$	−26.1429	−1.03097	−0.515487	+	0.856897i	$0.672389\pi$
$644$	0	0
$645$	13.9487	0.549230
$646$	0	0
$647$	42.6235	1.67570	0.837852	−	0.545897i	$-0.183811\pi$
$647$	42.6235	1.67570	0.837852	+	0.545897i	$0.183811\pi$
$648$	0	0
$649$	−14.8446	−0.582701
$650$	0	0
$651$	6.55044	0.256732
$652$	0	0
$653$	−13.8857	−0.543388	−0.271694	−	0.962384i	$-0.587584\pi$
$653$	−13.8857	−0.543388	−0.271694	+	0.962384i	$0.587584\pi$
$654$	0	0
$655$	8.24162	0.322027
$656$	0	0
$657$	−4.64723	−0.181306
$658$	0	0
$659$	−37.1498	−1.44715	−0.723576	−	0.690244i	$-0.757503\pi$
$659$	−37.1498	−1.44715	−0.723576	+	0.690244i	$0.757503\pi$
$660$	0	0
$661$	16.5088	0.642116	0.321058	−	0.947060i	$-0.395962\pi$
$661$	16.5088	0.642116	0.321058	+	0.947060i	$0.395962\pi$
$662$	0	0
$663$	−13.2461	−0.514434
$664$	0	0
$665$	−4.42756	−0.171693
$666$	0	0
$667$	−10.6326	−0.411694
$668$	0	0
$669$	17.7818	0.687485
$670$	0	0
$671$	7.76566	0.299790
$672$	0	0
$673$	−0.536374	−0.0206757	−0.0103379	−	0.999947i	$-0.503291\pi$
$673$	−0.536374	−0.0206757	−0.0103379	+	0.999947i	$0.503291\pi$
$674$	0	0
$675$	−3.82139	−0.147085
$676$	0	0
$677$	10.9376	0.420365	0.210182	−	0.977662i	$-0.432594\pi$
$677$	10.9376	0.420365	0.210182	+	0.977662i	$0.432594\pi$
$678$	0	0
$679$	4.64723	0.178344
$680$	0	0
$681$	13.7695	0.527648
$682$	0	0
$683$	−18.6472	−0.713516	−0.356758	−	0.934197i	$-0.616118\pi$
$683$	−18.6472	−0.713516	−0.356758	+	0.934197i	$0.616118\pi$
$684$	0	0
$685$	11.7450	0.448752
$686$	0	0
$687$	−20.6112	−0.786366
$688$	0	0
$689$	47.2660	1.80069
$690$	0	0
$691$	−42.6385	−1.62204	−0.811022	−	0.585016i	$-0.801088\pi$
$691$	−42.6385	−1.62204	−0.811022	+	0.585016i	$0.801088\pi$
$692$	0	0
$693$	−4.07830	−0.154922
$694$	0	0
$695$	−13.7586	−0.521895
$696$	0	0
$697$	3.68065	0.139415
$698$	0	0
$699$	3.37183	0.127534
$700$	0	0
$701$	−43.8910	−1.65774	−0.828870	−	0.559442i	$-0.811016\pi$
$701$	−43.8910	−1.65774	−0.828870	+	0.559442i	$0.811016\pi$
$702$	0	0
$703$	15.9340	0.600964
$704$	0	0
$705$	−10.8405	−0.408275
$706$	0	0
$707$	11.9003	0.447558
$708$	0	0
$709$	0.123938	0.00465461	0.00232730	−	0.999997i	$-0.499259\pi$
$709$	0.123938	0.00465461	0.00232730	+	0.999997i	$0.499259\pi$
$710$	0	0
$711$	6.55044	0.245661
$712$	0	0
$713$	−6.55044	−0.245316
$714$	0	0
$715$	−22.8637	−0.855052
$716$	0	0
$717$	−17.7818	−0.664075
$718$	0	0
$719$	44.8215	1.67156	0.835780	−	0.549064i	$-0.185016\pi$
$719$	44.8215	1.67156	0.835780	+	0.549064i	$0.185016\pi$
$720$	0	0
$721$	−11.3689	−0.423402
$722$	0	0
$723$	−25.0630	−0.932103
$724$	0	0
$725$	40.6312	1.50900
$726$	0	0
$727$	−15.5023	−0.574950	−0.287475	−	0.957788i	$-0.592816\pi$
$727$	−15.5023	−0.574950	−0.287475	+	0.957788i	$0.592816\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	32.9576	1.21898
$732$	0	0
$733$	−0.458725	−0.0169434	−0.00847169	−	0.999964i	$-0.502697\pi$
$733$	−0.458725	−0.0169434	−0.00847169	+	0.999964i	$0.502697\pi$
$734$	0	0
$735$	−1.08564	−0.0400443
$736$	0	0
$737$	33.7818	1.24437
$738$	0	0
$739$	46.2707	1.70210	0.851048	−	0.525088i	$-0.175968\pi$
$739$	46.2707	1.70210	0.851048	+	0.525088i	$0.175968\pi$
$740$	0	0
$741$	−21.0601	−0.773663
$742$	0	0
$743$	13.9461	0.511632	0.255816	−	0.966726i	$-0.417656\pi$
$743$	13.9461	0.511632	0.255816	+	0.966726i	$0.417656\pi$
$744$	0	0
$745$	13.6370	0.499621
$746$	0	0
$747$	−8.86597	−0.324389
$748$	0	0
$749$	−15.7857	−0.576795
$750$	0	0
$751$	−35.3355	−1.28941	−0.644705	−	0.764432i	$-0.723020\pi$
$751$	−35.3355	−1.28941	−0.644705	+	0.764432i	$0.723020\pi$
$752$	0	0
$753$	−0.465103	−0.0169493
$754$	0	0
$755$	−23.1830	−0.843716
$756$	0	0
$757$	21.8249	0.793240	0.396620	−	0.917983i	$-0.370183\pi$
$757$	21.8249	0.793240	0.396620	+	0.917983i	$0.370183\pi$
$758$	0	0
$759$	4.07830	0.148033
$760$	0	0
$761$	1.68979	0.0612549	0.0306275	−	0.999531i	$-0.490249\pi$
$761$	1.68979	0.0612549	0.0306275	+	0.999531i	$0.490249\pi$
$762$	0	0
$763$	1.33903	0.0484763
$764$	0	0
$765$	2.78478	0.100684
$766$	0	0
$767$	−18.7962	−0.678691
$768$	0	0
$769$	−5.72490	−0.206445	−0.103223	−	0.994658i	$-0.532915\pi$
$769$	−5.72490	−0.206445	−0.103223	+	0.994658i	$0.532915\pi$
$770$	0	0
$771$	17.5930	0.633597
$772$	0	0
$773$	15.9451	0.573507	0.286754	−	0.958004i	$-0.407424\pi$
$773$	15.9451	0.573507	0.286754	+	0.958004i	$0.407424\pi$
$774$	0	0
$775$	25.0318	0.899169
$776$	0	0
$777$	3.90703	0.140164
$778$	0	0
$779$	5.85193	0.209667
$780$	0	0
$781$	−57.8725	−2.07084
$782$	0	0
$783$	−10.6326	−0.379977
$784$	0	0
$785$	12.4100	0.442932
$786$	0	0
$787$	32.2082	1.14810	0.574049	−	0.818821i	$-0.305372\pi$
$787$	32.2082	1.14810	0.574049	+	0.818821i	$0.305372\pi$
$788$	0	0
$789$	−5.69532	−0.202759
$790$	0	0
$791$	−6.58417	−0.234106
$792$	0	0
$793$	9.83287	0.349175
$794$	0	0
$795$	−9.93694	−0.352427
$796$	0	0
$797$	20.1238	0.712822	0.356411	−	0.934329i	$-0.384000\pi$
$797$	20.1238	0.712822	0.356411	+	0.934329i	$0.384000\pi$
$798$	0	0
$799$	−25.6134	−0.906139
$800$	0	0
$801$	−10.6501	−0.376303
$802$	0	0
$803$	−18.9528	−0.668830
$804$	0	0
$805$	1.08564	0.0382637
$806$	0	0
$807$	8.27949	0.291452
$808$	0	0
$809$	−36.4437	−1.28129	−0.640647	−	0.767836i	$-0.721334\pi$
$809$	−36.4437	−1.28129	−0.640647	+	0.767836i	$0.721334\pi$
$810$	0	0
$811$	47.7264	1.67590	0.837950	−	0.545746i	$-0.183754\pi$
$811$	47.7264	1.67590	0.837950	+	0.545746i	$0.183754\pi$
$812$	0	0
$813$	12.0366	0.422143
$814$	0	0
$815$	18.2795	0.640303
$816$	0	0
$817$	52.3997	1.83323
$818$	0	0
$819$	−5.16394	−0.180443
$820$	0	0
$821$	14.3883	0.502155	0.251078	−	0.967967i	$-0.419215\pi$
$821$	14.3883	0.502155	0.251078	+	0.967967i	$0.419215\pi$
$822$	0	0
$823$	−32.9484	−1.14851	−0.574255	−	0.818677i	$-0.694708\pi$
$823$	−32.9484	−1.14851	−0.574255	+	0.818677i	$0.694708\pi$
$824$	0	0
$825$	−15.5848	−0.542593
$826$	0	0
$827$	−7.61162	−0.264682	−0.132341	−	0.991204i	$-0.542249\pi$
$827$	−7.61162	−0.264682	−0.132341	+	0.991204i	$0.542249\pi$
$828$	0	0
$829$	−40.2033	−1.39632	−0.698159	−	0.715943i	$-0.745997\pi$
$829$	−40.2033	−1.39632	−0.698159	+	0.715943i	$0.745997\pi$
$830$	0	0
$831$	−12.5329	−0.434761
$832$	0	0
$833$	−2.56511	−0.0888757
$834$	0	0
$835$	−7.41006	−0.256436
$836$	0	0
$837$	−6.55044	−0.226416
$838$	0	0
$839$	4.18330	0.144424	0.0722118	−	0.997389i	$-0.476994\pi$
$839$	4.18330	0.144424	0.0722118	+	0.997389i	$0.476994\pi$
$840$	0	0
$841$	84.0513	2.89832
$842$	0	0
$843$	−13.2181	−0.455256
$844$	0	0
$845$	−14.8366	−0.510396
$846$	0	0
$847$	−5.63256	−0.193537
$848$	0	0
$849$	−15.3314	−0.526172
$850$	0	0
$851$	−3.90703	−0.133931
$852$	0	0
$853$	−1.72490	−0.0590596	−0.0295298	−	0.999564i	$-0.509401\pi$
$853$	−1.72490	−0.0590596	−0.0295298	+	0.999564i	$0.509401\pi$
$854$	0	0
$855$	4.42756	0.151419
$856$	0	0
$857$	−37.8445	−1.29274	−0.646372	−	0.763022i	$-0.723715\pi$
$857$	−37.8445	−1.29274	−0.646372	+	0.763022i	$0.723715\pi$
$858$	0	0
$859$	29.2651	0.998513	0.499257	−	0.866454i	$-0.333607\pi$
$859$	29.2651	0.998513	0.499257	+	0.866454i	$0.333607\pi$
$860$	0	0
$861$	1.43489	0.0489010
$862$	0	0
$863$	−51.5637	−1.75525	−0.877624	−	0.479350i	$-0.840872\pi$
$863$	−51.5637	−1.75525	−0.877624	+	0.479350i	$0.840872\pi$
$864$	0	0
$865$	26.3542	0.896070
$866$	0	0
$867$	−10.4202	−0.353889
$868$	0	0
$869$	26.7147	0.906234
$870$	0	0
$871$	42.7745	1.44936
$872$	0	0
$873$	−4.64723	−0.157285
$874$	0	0
$875$	−9.57683	−0.323756
$876$	0	0
$877$	21.1583	0.714465	0.357232	−	0.934016i	$-0.383720\pi$
$877$	21.1583	0.714465	0.357232	+	0.934016i	$0.383720\pi$
$878$	0	0
$879$	−11.9977	−0.404672
$880$	0	0
$881$	31.0258	1.04529	0.522643	−	0.852552i	$-0.324946\pi$
$881$	31.0258	1.04529	0.522643	+	0.852552i	$0.324946\pi$
$882$	0	0
$883$	48.1597	1.62070	0.810352	−	0.585943i	$-0.199276\pi$
$883$	48.1597	1.62070	0.810352	+	0.585943i	$0.199276\pi$
$884$	0	0
$885$	3.95161	0.132832
$886$	0	0
$887$	−26.3176	−0.883659	−0.441829	−	0.897099i	$-0.645670\pi$
$887$	−26.3176	−0.883659	−0.441829	+	0.897099i	$0.645670\pi$
$888$	0	0
$889$	−12.3616	−0.414595
$890$	0	0
$891$	4.07830	0.136628
$892$	0	0
$893$	−40.7232	−1.36275
$894$	0	0
$895$	24.4055	0.815786
$896$	0	0
$897$	5.16394	0.172419
$898$	0	0
$899$	69.6479	2.32289
$900$	0	0
$901$	−23.4787	−0.782188
$902$	0	0
$903$	12.8484	0.427568
$904$	0	0
$905$	−9.98119	−0.331786
$906$	0	0
$907$	18.5303	0.615289	0.307645	−	0.951501i	$-0.400459\pi$
$907$	18.5303	0.615289	0.307645	+	0.951501i	$0.400459\pi$
$908$	0	0
$909$	−11.9003	−0.394709
$910$	0	0
$911$	25.0117	0.828675	0.414338	−	0.910123i	$-0.364013\pi$
$911$	25.0117	0.828675	0.414338	+	0.910123i	$0.364013\pi$
$912$	0	0
$913$	−36.1581	−1.19666
$914$	0	0
$915$	−2.06721	−0.0683397
$916$	0	0
$917$	7.59150	0.250693
$918$	0	0
$919$	−36.7434	−1.21205	−0.606027	−	0.795444i	$-0.707238\pi$
$919$	−36.7434	−1.21205	−0.606027	+	0.795444i	$0.707238\pi$
$920$	0	0
$921$	14.7147	0.484865
$922$	0	0
$923$	−73.2780	−2.41198
$924$	0	0
$925$	14.9303	0.490905
$926$	0	0
$927$	11.3689	0.373405
$928$	0	0
$929$	−57.9256	−1.90048	−0.950239	−	0.311521i	$-0.899161\pi$
$929$	−57.9256	−1.90048	−0.950239	+	0.311521i	$0.899161\pi$
$930$	0	0
$931$	−4.07830	−0.133661
$932$	0	0
$933$	31.0455	1.01638
$934$	0	0
$935$	11.3572	0.371419
$936$	0	0
$937$	1.71762	0.0561123	0.0280562	−	0.999606i	$-0.491068\pi$
$937$	1.71762	0.0561123	0.0280562	+	0.999606i	$0.491068\pi$
$938$	0	0
$939$	−23.9502	−0.781586
$940$	0	0
$941$	29.2552	0.953691	0.476846	−	0.878987i	$-0.341780\pi$
$941$	29.2552	0.953691	0.476846	+	0.878987i	$0.341780\pi$
$942$	0	0
$943$	−1.43489	−0.0467265
$944$	0	0
$945$	1.08564	0.0353158
$946$	0	0
$947$	11.2779	0.366484	0.183242	−	0.983068i	$-0.441341\pi$
$947$	11.2779	0.366484	0.183242	+	0.983068i	$0.441341\pi$
$948$	0	0
$949$	−23.9980	−0.779008
$950$	0	0
$951$	−9.79650	−0.317673
$952$	0	0
$953$	−35.0449	−1.13521	−0.567607	−	0.823300i	$-0.692131\pi$
$953$	−35.0449	−1.13521	−0.567607	+	0.823300i	$0.692131\pi$
$954$	0	0
$955$	23.1830	0.750184
$956$	0	0
$957$	−43.3628	−1.40172
$958$	0	0
$959$	10.8185	0.349348
$960$	0	0
$961$	11.9083	0.384137
$962$	0	0
$963$	15.7857	0.508686
$964$	0	0
$965$	19.8083	0.637651
$966$	0	0
$967$	−43.4153	−1.39614	−0.698072	−	0.716028i	$-0.745958\pi$
$967$	−43.4153	−1.39614	−0.698072	+	0.716028i	$0.745958\pi$
$968$	0	0
$969$	10.4613	0.336065
$970$	0	0
$971$	19.2948	0.619199	0.309600	−	0.950867i	$-0.399805\pi$
$971$	19.2948	0.619199	0.309600	+	0.950867i	$0.399805\pi$
$972$	0	0
$973$	−12.6733	−0.406288
$974$	0	0
$975$	−19.7334	−0.631976
$976$	0	0
$977$	−43.1065	−1.37910	−0.689550	−	0.724238i	$-0.742192\pi$
$977$	−43.1065	−1.37910	−0.689550	+	0.724238i	$0.742192\pi$
$978$	0	0
$979$	−43.4344	−1.38817
$980$	0	0
$981$	−1.33903	−0.0427520
$982$	0	0
$983$	−30.7687	−0.981370	−0.490685	−	0.871337i	$-0.663253\pi$
$983$	−30.7687	−0.981370	−0.490685	+	0.871337i	$0.663253\pi$
$984$	0	0
$985$	−16.4466	−0.524033
$986$	0	0
$987$	−9.98533	−0.317837
$988$	0	0
$989$	−12.8484	−0.408556
$990$	0	0
$991$	−19.8948	−0.631979	−0.315989	−	0.948763i	$-0.602336\pi$
$991$	−19.8948	−0.631979	−0.315989	+	0.948763i	$0.602336\pi$
$992$	0	0
$993$	3.37658	0.107152
$994$	0	0
$995$	−7.09071	−0.224791
$996$	0	0
$997$	18.2530	0.578080	0.289040	−	0.957317i	$-0.406664\pi$
$997$	18.2530	0.578080	0.289040	+	0.957317i	$0.406664\pi$
$998$	0	0
$999$	−3.90703	−0.123613

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7728.2.a.cb.1.2		4
4.3	odd	2		3864.2.a.r.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3864.2.a.r.1.2	✓	4	4.3	odd	2
7728.2.a.cb.1.2		4	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.08564 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.08564 q^{5} -1.00000 q^{7} +1.00000 q^{9} +4.07830 q^{11} +5.16394 q^{13} -1.08564 q^{15} -2.56511 q^{17} -4.07830 q^{19} -1.00000 q^{21} +1.00000 q^{23} -3.82139 q^{25} +1.00000 q^{27} -10.6326 q^{29} -6.55044 q^{31} +4.07830 q^{33} +1.08564 q^{35} -3.90703 q^{37} +5.16394 q^{39} -1.43489 q^{41} -12.8484 q^{43} -1.08564 q^{45} +9.98533 q^{47} +1.00000 q^{49} -2.56511 q^{51} +9.15309 q^{53} -4.42756 q^{55} -4.07830 q^{57} -3.63989 q^{59} +1.90414 q^{61} -1.00000 q^{63} -5.60617 q^{65} +8.28331 q^{67} +1.00000 q^{69} -14.1903 q^{71} -4.64723 q^{73} -3.82139 q^{75} -4.07830 q^{77} +6.55044 q^{79} +1.00000 q^{81} -8.86597 q^{83} +2.78478 q^{85} -10.6326 q^{87} -10.6501 q^{89} -5.16394 q^{91} -6.55044 q^{93} +4.42756 q^{95} -4.64723 q^{97} +4.07830 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} - 3 q^{5} - 4 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 - 3 * q^5 - 4 * q^7 + 4 * q^9	\( 4 q + 4 q^{3} - 3 q^{5} - 4 q^{7} + 4 q^{9} - 2 q^{11} + q^{13} - 3 q^{15} - 8 q^{17} + 2 q^{19} - 4 q^{21} + 4 q^{23} - q^{25} + 4 q^{27} - 10 q^{29} + 10 q^{31} - 2 q^{33} + 3 q^{35} + q^{39} - 8 q^{41} - 13 q^{43} - 3 q^{45} + 6 q^{47} + 4 q^{49} - 8 q^{51} + 5 q^{53} - 3 q^{55} + 2 q^{57} + q^{59} + 5 q^{61} - 4 q^{63} - 22 q^{65} - 3 q^{67} + 4 q^{69} - 5 q^{71} - 20 q^{73} - q^{75} + 2 q^{77} - 10 q^{79} + 4 q^{81} - 18 q^{83} + 25 q^{85} - 10 q^{87} - 31 q^{89} - q^{91} + 10 q^{93} + 3 q^{95} - 20 q^{97} - 2 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 - 3 * q^5 - 4 * q^7 + 4 * q^9 - 2 * q^11 + q^13 - 3 * q^15 - 8 * q^17 + 2 * q^19 - 4 * q^21 + 4 * q^23 - q^25 + 4 * q^27 - 10 * q^29 + 10 * q^31 - 2 * q^33 + 3 * q^35 + q^39 - 8 * q^41 - 13 * q^43 - 3 * q^45 + 6 * q^47 + 4 * q^49 - 8 * q^51 + 5 * q^53 - 3 * q^55 + 2 * q^57 + q^59 + 5 * q^61 - 4 * q^63 - 22 * q^65 - 3 * q^67 + 4 * q^69 - 5 * q^71 - 20 * q^73 - q^75 + 2 * q^77 - 10 * q^79 + 4 * q^81 - 18 * q^83 + 25 * q^85 - 10 * q^87 - 31 * q^89 - q^91 + 10 * q^93 + 3 * q^95 - 20 * q^97 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

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