LMFDB - Embedded newform 7728.2.a.ce.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:	$0$
Dimension:	$4$
Coefficient field:	4.4.15317.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 5x + 2 $ x^4 - 2x^3 - 4x^2 + 5*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 483)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.69353$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +1.04900 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.04900 q^{5} +1.00000 q^{7} +1.00000 q^{9} -0.180969 q^{11} -4.89961 q^{13} +1.04900 q^{15} +2.82550 q^{17} +0.180969 q^{19} +1.00000 q^{21} -1.00000 q^{23} -3.89961 q^{25} +1.00000 q^{27} -5.68466 q^{29} +0.825498 q^{31} -0.180969 q^{33} +1.04900 q^{35} +6.46356 q^{37} -4.89961 q^{39} -3.47003 q^{41} -0.125506 q^{43} +1.04900 q^{45} +12.0316 q^{47} +1.00000 q^{49} +2.82550 q^{51} +8.55269 q^{53} -0.189835 q^{55} +0.180969 q^{57} +4.30647 q^{59} +9.01625 q^{61} +1.00000 q^{63} -5.13967 q^{65} +15.7074 q^{67} -1.00000 q^{69} +2.22350 q^{71} +13.0717 q^{73} -3.89961 q^{75} -0.180969 q^{77} +16.1948 q^{79} +1.00000 q^{81} -10.8757 q^{83} +2.96394 q^{85} -5.68466 q^{87} -14.7466 q^{89} -4.89961 q^{91} +0.825498 q^{93} +0.189835 q^{95} +10.4636 q^{97} -0.180969 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 5 q^{5} + 4 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 + 5 * q^5 + 4 * q^7 + 4 * q^9	$ 4 q + 4 q^{3} + 5 q^{5} + 4 q^{7} + 4 q^{9} - q^{11} + 7 q^{13} + 5 q^{15} + 2 q^{17} + q^{19} + 4 q^{21} - 4 q^{23} + 11 q^{25} + 4 q^{27} + 2 q^{29} - 6 q^{31} - q^{33} + 5 q^{35} + 16 q^{37} + 7 q^{39} + 5 q^{41} - 9 q^{43} + 5 q^{45} + 21 q^{47} + 4 q^{49} + 2 q^{51} + 10 q^{53} - 17 q^{55} + q^{57} + 26 q^{59} + 2 q^{61} + 4 q^{63} + 26 q^{65} - 5 q^{67} - 4 q^{69} + 19 q^{71} + 10 q^{73} + 11 q^{75} - q^{77} + 6 q^{79} + 4 q^{81} + 2 q^{83} - 7 q^{85} + 2 q^{87} - 17 q^{89} + 7 q^{91} - 6 q^{93} + 17 q^{95} + 32 q^{97} - q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 + 5 * q^5 + 4 * q^7 + 4 * q^9 - q^11 + 7 * q^13 + 5 * q^15 + 2 * q^17 + q^19 + 4 * q^21 - 4 * q^23 + 11 * q^25 + 4 * q^27 + 2 * q^29 - 6 * q^31 - q^33 + 5 * q^35 + 16 * q^37 + 7 * q^39 + 5 * q^41 - 9 * q^43 + 5 * q^45 + 21 * q^47 + 4 * q^49 + 2 * q^51 + 10 * q^53 - 17 * q^55 + q^57 + 26 * q^59 + 2 * q^61 + 4 * q^63 + 26 * q^65 - 5 * q^67 - 4 * q^69 + 19 * q^71 + 10 * q^73 + 11 * q^75 - q^77 + 6 * q^79 + 4 * q^81 + 2 * q^83 - 7 * q^85 + 2 * q^87 - 17 * q^89 + 7 * q^91 - 6 * q^93 + 17 * q^95 + 32 * 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
$5$	1.04900	0.469125	0.234563	−	0.972101i	$-0.424634\pi$
$5$	1.04900	0.469125	0.234563	+	0.972101i	$0.424634\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.180969	−0.0545641	−0.0272820	−	0.999628i	$-0.508685\pi$
$11$	−0.180969	−0.0545641	−0.0272820	+	0.999628i	$0.508685\pi$
$12$	0	0
$13$	−4.89961	−1.35891	−0.679453	−	0.733719i	$-0.737783\pi$
$13$	−4.89961	−1.35891	−0.679453	+	0.733719i	$0.737783\pi$
$14$	0	0
$15$	1.04900	0.270850
$16$	0	0
$17$	2.82550	0.685284	0.342642	−	0.939466i	$-0.388678\pi$
$17$	2.82550	0.685284	0.342642	+	0.939466i	$0.388678\pi$
$18$	0	0
$19$	0.180969	0.0415170	0.0207585	−	0.999785i	$-0.493392\pi$
$19$	0.180969	0.0415170	0.0207585	+	0.999785i	$0.493392\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−3.89961	−0.779921
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−5.68466	−1.05561	−0.527807	−	0.849364i	$-0.676986\pi$
$29$	−5.68466	−1.05561	−0.527807	+	0.849364i	$0.676986\pi$
$30$	0	0
$31$	0.825498	0.148264	0.0741319	−	0.997248i	$-0.476381\pi$
$31$	0.825498	0.148264	0.0741319	+	0.997248i	$0.476381\pi$
$32$	0	0
$33$	−0.180969	−0.0315026
$34$	0	0
$35$	1.04900	0.177313
$36$	0	0
$37$	6.46356	1.06260	0.531301	−	0.847183i	$-0.321703\pi$
$37$	6.46356	1.06260	0.531301	+	0.847183i	$0.321703\pi$
$38$	0	0
$39$	−4.89961	−0.784565
$40$	0	0
$41$	−3.47003	−0.541927	−0.270964	−	0.962590i	$-0.587342\pi$
$41$	−3.47003	−0.541927	−0.270964	+	0.962590i	$0.587342\pi$
$42$	0	0
$43$	−0.125506	−0.0191395	−0.00956976	−	0.999954i	$-0.503046\pi$
$43$	−0.125506	−0.0191395	−0.00956976	+	0.999954i	$0.503046\pi$
$44$	0	0
$45$	1.04900	0.156375
$46$	0	0
$47$	12.0316	1.75499	0.877493	−	0.479589i	$-0.159214\pi$
$47$	12.0316	1.75499	0.877493	+	0.479589i	$0.159214\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	2.82550	0.395649
$52$	0	0
$53$	8.55269	1.17480	0.587401	−	0.809296i	$-0.300151\pi$
$53$	8.55269	1.17480	0.587401	+	0.809296i	$0.300151\pi$
$54$	0	0
$55$	−0.189835	−0.0255974
$56$	0	0
$57$	0.180969	0.0239699
$58$	0	0
$59$	4.30647	0.560655	0.280328	−	0.959904i	$-0.409557\pi$
$59$	4.30647	0.560655	0.280328	+	0.959904i	$0.409557\pi$
$60$	0	0
$61$	9.01625	1.15441	0.577206	−	0.816599i	$-0.304143\pi$
$61$	9.01625	1.15441	0.577206	+	0.816599i	$0.304143\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	−5.13967	−0.637497
$66$	0	0
$67$	15.7074	1.91896	0.959480	−	0.281775i	$-0.0909233\pi$
$67$	15.7074	1.91896	0.959480	+	0.281775i	$0.0909233\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	2.22350	0.263881	0.131940	−	0.991258i	$-0.457879\pi$
$71$	2.22350	0.263881	0.131940	+	0.991258i	$0.457879\pi$
$72$	0	0
$73$	13.0717	1.52993	0.764964	−	0.644073i	$-0.222757\pi$
$73$	13.0717	1.52993	0.764964	+	0.644073i	$0.222757\pi$
$74$	0	0
$75$	−3.89961	−0.450288
$76$	0	0
$77$	−0.180969	−0.0206233
$78$	0	0
$79$	16.1948	1.82206	0.911029	−	0.412341i	$-0.135289\pi$
$79$	16.1948	1.82206	0.911029	+	0.412341i	$0.135289\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−10.8757	−1.19377	−0.596883	−	0.802328i	$-0.703594\pi$
$83$	−10.8757	−1.19377	−0.596883	+	0.802328i	$0.703594\pi$
$84$	0	0
$85$	2.96394	0.321484
$86$	0	0
$87$	−5.68466	−0.609459
$88$	0	0
$89$	−14.7466	−1.56314	−0.781568	−	0.623821i	$-0.785580\pi$
$89$	−14.7466	−1.56314	−0.781568	+	0.623821i	$0.785580\pi$
$90$	0	0
$91$	−4.89961	−0.513618
$92$	0	0
$93$	0.825498	0.0856001
$94$	0	0
$95$	0.189835	0.0194767
$96$	0	0
$97$	10.4636	1.06241	0.531207	−	0.847242i	$-0.321739\pi$
$97$	10.4636	1.06241	0.531207	+	0.847242i	$0.321739\pi$
$98$	0	0
$99$	−0.180969	−0.0181880
$100$	0	0
$101$	14.7316	1.46585	0.732923	−	0.680312i	$-0.238156\pi$
$101$	14.7316	1.46585	0.732923	+	0.680312i	$0.238156\pi$
$102$	0	0
$103$	−5.96362	−0.587613	−0.293806	−	0.955865i	$-0.594922\pi$
$103$	−5.96362	−0.587613	−0.293806	+	0.955865i	$0.594922\pi$
$104$	0	0
$105$	1.04900	0.102372
$106$	0	0
$107$	8.55060	0.826618	0.413309	−	0.910591i	$-0.364373\pi$
$107$	8.55060	0.826618	0.413309	+	0.910591i	$0.364373\pi$
$108$	0	0
$109$	9.00123	0.862162	0.431081	−	0.902313i	$-0.358132\pi$
$109$	9.00123	0.862162	0.431081	+	0.902313i	$0.358132\pi$
$110$	0	0
$111$	6.46356	0.613494
$112$	0	0
$113$	9.81016	0.922863	0.461431	−	0.887176i	$-0.347336\pi$
$113$	9.81016	0.922863	0.461431	+	0.887176i	$0.347336\pi$
$114$	0	0
$115$	−1.04900	−0.0978194
$116$	0	0
$117$	−4.89961	−0.452969
$118$	0	0
$119$	2.82550	0.259013
$120$	0	0
$121$	−10.9673	−0.997023
$122$	0	0
$123$	−3.47003	−0.312882
$124$	0	0
$125$	−9.33565	−0.835006
$126$	0	0
$127$	−13.6421	−1.21054	−0.605272	−	0.796019i	$-0.706935\pi$
$127$	−13.6421	−1.21054	−0.605272	+	0.796019i	$0.706935\pi$
$128$	0	0
$129$	−0.125506	−0.0110502
$130$	0	0
$131$	9.72104	0.849331	0.424666	−	0.905350i	$-0.360392\pi$
$131$	9.72104	0.849331	0.424666	+	0.905350i	$0.360392\pi$
$132$	0	0
$133$	0.180969	0.0156920
$134$	0	0
$135$	1.04900	0.0902832
$136$	0	0
$137$	−5.96725	−0.509817	−0.254908	−	0.966965i	$-0.582045\pi$
$137$	−5.96725	−0.509817	−0.254908	+	0.966965i	$0.582045\pi$
$138$	0	0
$139$	−7.36317	−0.624536	−0.312268	−	0.949994i	$-0.601089\pi$
$139$	−7.36317	−0.624536	−0.312268	+	0.949994i	$0.601089\pi$
$140$	0	0
$141$	12.0316	1.01324
$142$	0	0
$143$	0.886675	0.0741475
$144$	0	0
$145$	−5.96318	−0.495216
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	21.9288	1.79648	0.898238	−	0.439509i	$-0.144848\pi$
$149$	21.9288	1.79648	0.898238	+	0.439509i	$0.144848\pi$
$150$	0	0
$151$	−16.7927	−1.36657	−0.683287	−	0.730150i	$-0.739450\pi$
$151$	−16.7927	−1.36657	−0.683287	+	0.730150i	$0.739450\pi$
$152$	0	0
$153$	2.82550	0.228428
$154$	0	0
$155$	0.865944	0.0695543
$156$	0	0
$157$	16.7425	1.33620	0.668099	−	0.744072i	$-0.267108\pi$
$157$	16.7425	1.33620	0.668099	+	0.744072i	$0.267108\pi$
$158$	0	0
$159$	8.55269	0.678272
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	−25.1308	−1.96840	−0.984198	−	0.177070i	$-0.943338\pi$
$163$	−25.1308	−1.96840	−0.984198	+	0.177070i	$0.943338\pi$
$164$	0	0
$165$	−0.189835	−0.0147787
$166$	0	0
$167$	−9.98018	−0.772290	−0.386145	−	0.922438i	$-0.626194\pi$
$167$	−9.98018	−0.772290	−0.386145	+	0.922438i	$0.626194\pi$
$168$	0	0
$169$	11.0061	0.846627
$170$	0	0
$171$	0.180969	0.0138390
$172$	0	0
$173$	−12.5065	−0.950853	−0.475427	−	0.879755i	$-0.657706\pi$
$173$	−12.5065	−0.950853	−0.475427	+	0.879755i	$0.657706\pi$
$174$	0	0
$175$	−3.89961	−0.294783
$176$	0	0
$177$	4.30647	0.323694
$178$	0	0
$179$	19.3081	1.44316	0.721579	−	0.692332i	$-0.243417\pi$
$179$	19.3081	1.44316	0.721579	+	0.692332i	$0.243417\pi$
$180$	0	0
$181$	17.7656	1.32050	0.660251	−	0.751045i	$-0.270450\pi$
$181$	17.7656	1.32050	0.660251	+	0.751045i	$0.270450\pi$
$182$	0	0
$183$	9.01625	0.666500
$184$	0	0
$185$	6.78025	0.498494
$186$	0	0
$187$	−0.511326	−0.0373919
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	8.79275	0.636221	0.318110	−	0.948054i	$-0.396952\pi$
$191$	8.79275	0.636221	0.318110	+	0.948054i	$0.396952\pi$
$192$	0	0
$193$	−6.79275	−0.488953	−0.244476	−	0.969655i	$-0.578616\pi$
$193$	−6.79275	−0.488953	−0.244476	+	0.969655i	$0.578616\pi$
$194$	0	0
$195$	−5.13967	−0.368059
$196$	0	0
$197$	14.6534	1.04401	0.522006	−	0.852942i	$-0.325184\pi$
$197$	14.6534	1.04401	0.522006	+	0.852942i	$0.325184\pi$
$198$	0	0
$199$	4.45469	0.315785	0.157892	−	0.987456i	$-0.449530\pi$
$199$	4.45469	0.315785	0.157892	+	0.987456i	$0.449530\pi$
$200$	0	0
$201$	15.7074	1.10791
$202$	0	0
$203$	−5.68466	−0.398985
$204$	0	0
$205$	−3.64004	−0.254232
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	−0.0327496	−0.00226534
$210$	0	0
$211$	4.59793	0.316535	0.158267	−	0.987396i	$-0.449409\pi$
$211$	4.59793	0.316535	0.158267	+	0.987396i	$0.449409\pi$
$212$	0	0
$213$	2.22350	0.152352
$214$	0	0
$215$	−0.131656	−0.00897884
$216$	0	0
$217$	0.825498	0.0560384
$218$	0	0
$219$	13.0717	0.883304
$220$	0	0
$221$	−13.8438	−0.931237
$222$	0	0
$223$	−13.0442	−0.873504	−0.436752	−	0.899582i	$-0.643871\pi$
$223$	−13.0442	−0.873504	−0.436752	+	0.899582i	$0.643871\pi$
$224$	0	0
$225$	−3.89961	−0.259974
$226$	0	0
$227$	0.320321	0.0212604	0.0106302	−	0.999943i	$-0.496616\pi$
$227$	0.320321	0.0212604	0.0106302	+	0.999943i	$0.496616\pi$
$228$	0	0
$229$	8.88459	0.587110	0.293555	−	0.955942i	$-0.405162\pi$
$229$	8.88459	0.587110	0.293555	+	0.955942i	$0.405162\pi$
$230$	0	0
$231$	−0.180969	−0.0119069
$232$	0	0
$233$	−0.418313	−0.0274046	−0.0137023	−	0.999906i	$-0.504362\pi$
$233$	−0.418313	−0.0274046	−0.0137023	+	0.999906i	$0.504362\pi$
$234$	0	0
$235$	12.6211	0.823309
$236$	0	0
$237$	16.1948	1.05197
$238$	0	0
$239$	15.0571	0.973965	0.486982	−	0.873412i	$-0.338098\pi$
$239$	15.0571	0.973965	0.486982	+	0.873412i	$0.338098\pi$
$240$	0	0
$241$	5.31701	0.342498	0.171249	−	0.985228i	$-0.445220\pi$
$241$	5.31701	0.342498	0.171249	+	0.985228i	$0.445220\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	1.04900	0.0670179
$246$	0	0
$247$	−0.886675	−0.0564178
$248$	0	0
$249$	−10.8757	−0.689221
$250$	0	0
$251$	7.19107	0.453896	0.226948	−	0.973907i	$-0.427125\pi$
$251$	7.19107	0.453896	0.226948	+	0.973907i	$0.427125\pi$
$252$	0	0
$253$	0.180969	0.0113774
$254$	0	0
$255$	2.96394	0.185609
$256$	0	0
$257$	−8.77255	−0.547217	−0.273608	−	0.961841i	$-0.588217\pi$
$257$	−8.77255	−0.547217	−0.273608	+	0.961841i	$0.588217\pi$
$258$	0	0
$259$	6.46356	0.401626
$260$	0	0
$261$	−5.68466	−0.351872
$262$	0	0
$263$	−16.0304	−0.988477	−0.494239	−	0.869326i	$-0.664553\pi$
$263$	−16.0304	−0.988477	−0.494239	+	0.869326i	$0.664553\pi$
$264$	0	0
$265$	8.97173	0.551129
$266$	0	0
$267$	−14.7466	−0.902476
$268$	0	0
$269$	6.27373	0.382516	0.191258	−	0.981540i	$-0.438743\pi$
$269$	6.27373	0.382516	0.191258	+	0.981540i	$0.438743\pi$
$270$	0	0
$271$	17.1397	1.04116	0.520580	−	0.853813i	$-0.325716\pi$
$271$	17.1397	1.04116	0.520580	+	0.853813i	$0.325716\pi$
$272$	0	0
$273$	−4.89961	−0.296538
$274$	0	0
$275$	0.705706	0.0425557
$276$	0	0
$277$	−23.9563	−1.43939	−0.719697	−	0.694288i	$-0.755719\pi$
$277$	−23.9563	−1.43939	−0.719697	+	0.694288i	$0.755719\pi$
$278$	0	0
$279$	0.825498	0.0494212
$280$	0	0
$281$	−7.27132	−0.433771	−0.216885	−	0.976197i	$-0.569590\pi$
$281$	−7.27132	−0.433771	−0.216885	+	0.976197i	$0.569590\pi$
$282$	0	0
$283$	−26.5365	−1.57743	−0.788716	−	0.614758i	$-0.789254\pi$
$283$	−26.5365	−1.57743	−0.788716	+	0.614758i	$0.789254\pi$
$284$	0	0
$285$	0.189835	0.0112449
$286$	0	0
$287$	−3.47003	−0.204829
$288$	0	0
$289$	−9.01656	−0.530386
$290$	0	0
$291$	10.4636	0.613385
$292$	0	0
$293$	−32.7086	−1.91086	−0.955428	−	0.295223i	$-0.904606\pi$
$293$	−32.7086	−1.91086	−0.955428	+	0.295223i	$0.904606\pi$
$294$	0	0
$295$	4.51748	0.263018
$296$	0	0
$297$	−0.180969	−0.0105009
$298$	0	0
$299$	4.89961	0.283352
$300$	0	0
$301$	−0.125506	−0.00723406
$302$	0	0
$303$	14.7316	0.846307
$304$	0	0
$305$	9.45801	0.541564
$306$	0	0
$307$	20.4281	1.16589	0.582946	−	0.812511i	$-0.301900\pi$
$307$	20.4281	1.16589	0.582946	+	0.812511i	$0.301900\pi$
$308$	0	0
$309$	−5.96362	−0.339258
$310$	0	0
$311$	−15.9515	−0.904526	−0.452263	−	0.891884i	$-0.649383\pi$
$311$	−15.9515	−0.904526	−0.452263	+	0.891884i	$0.649383\pi$
$312$	0	0
$313$	−2.91328	−0.164668	−0.0823340	−	0.996605i	$-0.526237\pi$
$313$	−2.91328	−0.164668	−0.0823340	+	0.996605i	$0.526237\pi$
$314$	0	0
$315$	1.04900	0.0591042
$316$	0	0
$317$	0.295090	0.0165739	0.00828695	−	0.999966i	$-0.497362\pi$
$317$	0.295090	0.0165739	0.00828695	+	0.999966i	$0.497362\pi$
$318$	0	0
$319$	1.02874	0.0575986
$320$	0	0
$321$	8.55060	0.477248
$322$	0	0
$323$	0.511326	0.0284510
$324$	0	0
$325$	19.1065	1.05984
$326$	0	0
$327$	9.00123	0.497769
$328$	0	0
$329$	12.0316	0.663322
$330$	0	0
$331$	−10.8907	−0.598609	−0.299305	−	0.954158i	$-0.596755\pi$
$331$	−10.8907	−0.598609	−0.299305	+	0.954158i	$0.596755\pi$
$332$	0	0
$333$	6.46356	0.354201
$334$	0	0
$335$	16.4770	0.900233
$336$	0	0
$337$	1.29884	0.0707522	0.0353761	−	0.999374i	$-0.488737\pi$
$337$	1.29884	0.0707522	0.0353761	+	0.999374i	$0.488737\pi$
$338$	0	0
$339$	9.81016	0.532815
$340$	0	0
$341$	−0.149389	−0.00808987
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−1.04900	−0.0564761
$346$	0	0
$347$	−18.2928	−0.982009	−0.491005	−	0.871157i	$-0.663370\pi$
$347$	−18.2928	−0.982009	−0.491005	+	0.871157i	$0.663370\pi$
$348$	0	0
$349$	−22.9333	−1.22759	−0.613795	−	0.789466i	$-0.710358\pi$
$349$	−22.9333	−1.22759	−0.613795	+	0.789466i	$0.710358\pi$
$350$	0	0
$351$	−4.89961	−0.261522
$352$	0	0
$353$	15.5867	0.829595	0.414797	−	0.909914i	$-0.363853\pi$
$353$	15.5867	0.829595	0.414797	+	0.909914i	$0.363853\pi$
$354$	0	0
$355$	2.33244	0.123793
$356$	0	0
$357$	2.82550	0.149541
$358$	0	0
$359$	21.6094	1.14050	0.570250	−	0.821471i	$-0.306846\pi$
$359$	21.6094	1.14050	0.570250	+	0.821471i	$0.306846\pi$
$360$	0	0
$361$	−18.9673	−0.998276
$362$	0	0
$363$	−10.9673	−0.575631
$364$	0	0
$365$	13.7122	0.717728
$366$	0	0
$367$	4.46399	0.233019	0.116509	−	0.993190i	$-0.462830\pi$
$367$	4.46399	0.233019	0.116509	+	0.993190i	$0.462830\pi$
$368$	0	0
$369$	−3.47003	−0.180642
$370$	0	0
$371$	8.55269	0.444033
$372$	0	0
$373$	−30.7527	−1.59232	−0.796158	−	0.605089i	$-0.793138\pi$
$373$	−30.7527	−1.59232	−0.796158	+	0.605089i	$0.793138\pi$
$374$	0	0
$375$	−9.33565	−0.482091
$376$	0	0
$377$	27.8526	1.43448
$378$	0	0
$379$	−22.7013	−1.16609	−0.583045	−	0.812440i	$-0.698139\pi$
$379$	−22.7013	−1.16609	−0.583045	+	0.812440i	$0.698139\pi$
$380$	0	0
$381$	−13.6421	−0.698907
$382$	0	0
$383$	18.3478	0.937531	0.468765	−	0.883323i	$-0.344699\pi$
$383$	18.3478	0.937531	0.468765	+	0.883323i	$0.344699\pi$
$384$	0	0
$385$	−0.189835	−0.00967490
$386$	0	0
$387$	−0.125506	−0.00637984
$388$	0	0
$389$	−22.0620	−1.11859	−0.559294	−	0.828970i	$-0.688928\pi$
$389$	−22.0620	−1.11859	−0.559294	+	0.828970i	$0.688928\pi$
$390$	0	0
$391$	−2.82550	−0.142892
$392$	0	0
$393$	9.72104	0.490362
$394$	0	0
$395$	16.9883	0.854774
$396$	0	0
$397$	23.3045	1.16962	0.584808	−	0.811171i	$-0.301170\pi$
$397$	23.3045	1.16962	0.584808	+	0.811171i	$0.301170\pi$
$398$	0	0
$399$	0.180969	0.00905976
$400$	0	0
$401$	14.2087	0.709547	0.354773	−	0.934952i	$-0.384558\pi$
$401$	14.2087	0.709547	0.354773	+	0.934952i	$0.384558\pi$
$402$	0	0
$403$	−4.04461	−0.201477
$404$	0	0
$405$	1.04900	0.0521250
$406$	0	0
$407$	−1.16970	−0.0579799
$408$	0	0
$409$	−33.2758	−1.64538	−0.822692	−	0.568488i	$-0.807529\pi$
$409$	−33.2758	−1.64538	−0.822692	+	0.568488i	$0.807529\pi$
$410$	0	0
$411$	−5.96725	−0.294343
$412$	0	0
$413$	4.30647	0.211908
$414$	0	0
$415$	−11.4086	−0.560026
$416$	0	0
$417$	−7.36317	−0.360576
$418$	0	0
$419$	−1.93819	−0.0946867	−0.0473434	−	0.998879i	$-0.515075\pi$
$419$	−1.93819	−0.0946867	−0.0473434	+	0.998879i	$0.515075\pi$
$420$	0	0
$421$	−3.10852	−0.151500	−0.0757501	−	0.997127i	$-0.524135\pi$
$421$	−3.10852	−0.151500	−0.0757501	+	0.997127i	$0.524135\pi$
$422$	0	0
$423$	12.0316	0.584995
$424$	0	0
$425$	−11.0183	−0.534468
$426$	0	0
$427$	9.01625	0.436327
$428$	0	0
$429$	0.886675	0.0428091
$430$	0	0
$431$	27.2204	1.31116	0.655579	−	0.755126i	$-0.272424\pi$
$431$	27.2204	1.31116	0.655579	+	0.755126i	$0.272424\pi$
$432$	0	0
$433$	−12.2146	−0.586998	−0.293499	−	0.955959i	$-0.594820\pi$
$433$	−12.2146	−0.586998	−0.293499	+	0.955959i	$0.594820\pi$
$434$	0	0
$435$	−5.96318	−0.285913
$436$	0	0
$437$	−0.180969	−0.00865690
$438$	0	0
$439$	−28.7859	−1.37387	−0.686937	−	0.726717i	$-0.741045\pi$
$439$	−28.7859	−1.37387	−0.686937	+	0.726717i	$0.741045\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	32.1005	1.52514	0.762569	−	0.646907i	$-0.223938\pi$
$443$	32.1005	1.52514	0.762569	+	0.646907i	$0.223938\pi$
$444$	0	0
$445$	−15.4691	−0.733306
$446$	0	0
$447$	21.9288	1.03720
$448$	0	0
$449$	−25.6596	−1.21095	−0.605476	−	0.795864i	$-0.707017\pi$
$449$	−25.6596	−1.21095	−0.605476	+	0.795864i	$0.707017\pi$
$450$	0	0
$451$	0.627966	0.0295697
$452$	0	0
$453$	−16.7927	−0.788992
$454$	0	0
$455$	−5.13967	−0.240951
$456$	0	0
$457$	16.8061	0.786157	0.393078	−	0.919505i	$-0.371410\pi$
$457$	16.8061	0.786157	0.393078	+	0.919505i	$0.371410\pi$
$458$	0	0
$459$	2.82550	0.131883
$460$	0	0
$461$	8.73248	0.406712	0.203356	−	0.979105i	$-0.434815\pi$
$461$	8.73248	0.406712	0.203356	+	0.979105i	$0.434815\pi$
$462$	0	0
$463$	−18.2312	−0.847275	−0.423638	−	0.905832i	$-0.639247\pi$
$463$	−18.2312	−0.847275	−0.423638	+	0.905832i	$0.639247\pi$
$464$	0	0
$465$	0.865944	0.0401572
$466$	0	0
$467$	−13.8127	−0.639175	−0.319587	−	0.947557i	$-0.603544\pi$
$467$	−13.8127	−0.639175	−0.319587	+	0.947557i	$0.603544\pi$
$468$	0	0
$469$	15.7074	0.725299
$470$	0	0
$471$	16.7425	0.771455
$472$	0	0
$473$	0.0227127	0.00104433
$474$	0	0
$475$	−0.705706	−0.0323800
$476$	0	0
$477$	8.55269	0.391601
$478$	0	0
$479$	19.4814	0.890128	0.445064	−	0.895499i	$-0.353181\pi$
$479$	19.4814	0.890128	0.445064	+	0.895499i	$0.353181\pi$
$480$	0	0
$481$	−31.6689	−1.44398
$482$	0	0
$483$	−1.00000	−0.0455016
$484$	0	0
$485$	10.9762	0.498405
$486$	0	0
$487$	−27.5268	−1.24736	−0.623680	−	0.781680i	$-0.714363\pi$
$487$	−27.5268	−1.24736	−0.623680	+	0.781680i	$0.714363\pi$
$488$	0	0
$489$	−25.1308	−1.13645
$490$	0	0
$491$	−15.3547	−0.692950	−0.346475	−	0.938059i	$-0.612621\pi$
$491$	−15.3547	−0.692950	−0.346475	+	0.938059i	$0.612621\pi$
$492$	0	0
$493$	−16.0620	−0.723396
$494$	0	0
$495$	−0.189835	−0.00853246
$496$	0	0
$497$	2.22350	0.0997375
$498$	0	0
$499$	8.11621	0.363331	0.181666	−	0.983360i	$-0.441851\pi$
$499$	8.11621	0.363331	0.181666	+	0.983360i	$0.441851\pi$
$500$	0	0
$501$	−9.98018	−0.445882
$502$	0	0
$503$	19.1576	0.854194	0.427097	−	0.904206i	$-0.359536\pi$
$503$	19.1576	0.854194	0.427097	+	0.904206i	$0.359536\pi$
$504$	0	0
$505$	15.4534	0.687666
$506$	0	0
$507$	11.0061	0.488800
$508$	0	0
$509$	−3.87856	−0.171914	−0.0859571	−	0.996299i	$-0.527395\pi$
$509$	−3.87856	−0.171914	−0.0859571	+	0.996299i	$0.527395\pi$
$510$	0	0
$511$	13.0717	0.578258
$512$	0	0
$513$	0.180969	0.00798996
$514$	0	0
$515$	−6.25581	−0.275664
$516$	0	0
$517$	−2.17734	−0.0957592
$518$	0	0
$519$	−12.5065	−0.548976
$520$	0	0
$521$	35.3522	1.54881	0.774404	−	0.632691i	$-0.218050\pi$
$521$	35.3522	1.54881	0.774404	+	0.632691i	$0.218050\pi$
$522$	0	0
$523$	39.3231	1.71948	0.859740	−	0.510733i	$-0.170626\pi$
$523$	39.3231	1.71948	0.859740	+	0.510733i	$0.170626\pi$
$524$	0	0
$525$	−3.89961	−0.170193
$526$	0	0
$527$	2.33244	0.101603
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	4.30647	0.186885
$532$	0	0
$533$	17.0018	0.736428
$534$	0	0
$535$	8.96955	0.387787
$536$	0	0
$537$	19.3081	0.833208
$538$	0	0
$539$	−0.180969	−0.00779487
$540$	0	0
$541$	−2.43081	−0.104509	−0.0522544	−	0.998634i	$-0.516641\pi$
$541$	−2.43081	−0.104509	−0.0522544	+	0.998634i	$0.516641\pi$
$542$	0	0
$543$	17.7656	0.762393
$544$	0	0
$545$	9.44226	0.404462
$546$	0	0
$547$	20.0320	0.856507	0.428254	−	0.903659i	$-0.359129\pi$
$547$	20.0320	0.856507	0.428254	+	0.903659i	$0.359129\pi$
$548$	0	0
$549$	9.01625	0.384804
$550$	0	0
$551$	−1.02874	−0.0438260
$552$	0	0
$553$	16.1948	0.688674
$554$	0	0
$555$	6.78025	0.287806
$556$	0	0
$557$	34.5733	1.46492	0.732459	−	0.680811i	$-0.238372\pi$
$557$	34.5733	1.46492	0.732459	+	0.680811i	$0.238372\pi$
$558$	0	0
$559$	0.614931	0.0260088
$560$	0	0
$561$	−0.511326	−0.0215882
$562$	0	0
$563$	−11.6307	−0.490177	−0.245089	−	0.969501i	$-0.578817\pi$
$563$	−11.6307	−0.490177	−0.245089	+	0.969501i	$0.578817\pi$
$564$	0	0
$565$	10.2908	0.432938
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	20.0340	0.839871	0.419935	−	0.907554i	$-0.362053\pi$
$569$	20.0340	0.839871	0.419935	+	0.907554i	$0.362053\pi$
$570$	0	0
$571$	30.6328	1.28194	0.640972	−	0.767564i	$-0.278531\pi$
$571$	30.6328	1.28194	0.640972	+	0.767564i	$0.278531\pi$
$572$	0	0
$573$	8.79275	0.367322
$574$	0	0
$575$	3.89961	0.162625
$576$	0	0
$577$	6.02586	0.250860	0.125430	−	0.992102i	$-0.459969\pi$
$577$	6.02586	0.250860	0.125430	+	0.992102i	$0.459969\pi$
$578$	0	0
$579$	−6.79275	−0.282297
$580$	0	0
$581$	−10.8757	−0.451201
$582$	0	0
$583$	−1.54777	−0.0641020
$584$	0	0
$585$	−5.13967	−0.212499
$586$	0	0
$587$	0.00886676	0.000365971 0	0.000182985	−	1.00000i	$-0.499942\pi$
$587$	0.00886676	0.000365971 0	0.000182985		1.00000i	$0.499942\pi$
$588$	0	0
$589$	0.149389	0.00615547
$590$	0	0
$591$	14.6534	0.602760
$592$	0	0
$593$	−46.0454	−1.89086	−0.945429	−	0.325827i	$-0.894357\pi$
$593$	−46.0454	−1.89086	−0.945429	+	0.325827i	$0.894357\pi$
$594$	0	0
$595$	2.96394	0.121510
$596$	0	0
$597$	4.45469	0.182318
$598$	0	0
$599$	24.6357	1.00659	0.503293	−	0.864116i	$-0.332122\pi$
$599$	24.6357	1.00659	0.503293	+	0.864116i	$0.332122\pi$
$600$	0	0
$601$	−32.4037	−1.32177	−0.660887	−	0.750486i	$-0.729820\pi$
$601$	−32.4037	−1.32177	−0.660887	+	0.750486i	$0.729820\pi$
$602$	0	0
$603$	15.7074	0.639654
$604$	0	0
$605$	−11.5046	−0.467729
$606$	0	0
$607$	−10.5554	−0.428431	−0.214215	−	0.976786i	$-0.568719\pi$
$607$	−10.5554	−0.428431	−0.214215	+	0.976786i	$0.568719\pi$
$608$	0	0
$609$	−5.68466	−0.230354
$610$	0	0
$611$	−58.9500	−2.38486
$612$	0	0
$613$	4.26511	0.172266	0.0861332	−	0.996284i	$-0.472549\pi$
$613$	4.26511	0.172266	0.0861332	+	0.996284i	$0.472549\pi$
$614$	0	0
$615$	−3.64004	−0.146781
$616$	0	0
$617$	−9.32470	−0.375398	−0.187699	−	0.982227i	$-0.560103\pi$
$617$	−9.32470	−0.375398	−0.187699	+	0.982227i	$0.560103\pi$
$618$	0	0
$619$	−3.64538	−0.146520	−0.0732601	−	0.997313i	$-0.523340\pi$
$619$	−3.64538	−0.146520	−0.0732601	+	0.997313i	$0.523340\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	−14.7466	−0.590810
$624$	0	0
$625$	9.70497	0.388199
$626$	0	0
$627$	−0.0327496	−0.00130789
$628$	0	0
$629$	18.2628	0.728185
$630$	0	0
$631$	−15.3155	−0.609699	−0.304849	−	0.952401i	$-0.598606\pi$
$631$	−15.3155	−0.609699	−0.304849	+	0.952401i	$0.598606\pi$
$632$	0	0
$633$	4.59793	0.182751
$634$	0	0
$635$	−14.3105	−0.567896
$636$	0	0
$637$	−4.89961	−0.194129
$638$	0	0
$639$	2.22350	0.0879602
$640$	0	0
$641$	17.8252	0.704052	0.352026	−	0.935990i	$-0.385493\pi$
$641$	17.8252	0.704052	0.352026	+	0.935990i	$0.385493\pi$
$642$	0	0
$643$	10.2196	0.403022	0.201511	−	0.979486i	$-0.435415\pi$
$643$	10.2196	0.403022	0.201511	+	0.979486i	$0.435415\pi$
$644$	0	0
$645$	−0.131656	−0.00518393
$646$	0	0
$647$	1.71217	0.0673124	0.0336562	−	0.999433i	$-0.489285\pi$
$647$	1.71217	0.0673124	0.0336562	+	0.999433i	$0.489285\pi$
$648$	0	0
$649$	−0.779336	−0.0305916
$650$	0	0
$651$	0.825498	0.0323538
$652$	0	0
$653$	−2.80044	−0.109590	−0.0547949	−	0.998498i	$-0.517451\pi$
$653$	−2.80044	−0.109590	−0.0547949	+	0.998498i	$0.517451\pi$
$654$	0	0
$655$	10.1973	0.398443
$656$	0	0
$657$	13.0717	0.509976
$658$	0	0
$659$	−25.7153	−1.00173	−0.500863	−	0.865526i	$-0.666984\pi$
$659$	−25.7153	−1.00173	−0.500863	+	0.865526i	$0.666984\pi$
$660$	0	0
$661$	10.5129	0.408903	0.204452	−	0.978877i	$-0.434459\pi$
$661$	10.5129	0.408903	0.204452	+	0.978877i	$0.434459\pi$
$662$	0	0
$663$	−13.8438	−0.537650
$664$	0	0
$665$	0.189835	0.00736150
$666$	0	0
$667$	5.68466	0.220111
$668$	0	0
$669$	−13.0442	−0.504318
$670$	0	0
$671$	−1.63166	−0.0629894
$672$	0	0
$673$	23.1235	0.891345	0.445672	−	0.895196i	$-0.352965\pi$
$673$	23.1235	0.891345	0.445672	+	0.895196i	$0.352965\pi$
$674$	0	0
$675$	−3.89961	−0.150096
$676$	0	0
$677$	−44.2843	−1.70198	−0.850992	−	0.525178i	$-0.823999\pi$
$677$	−44.2843	−1.70198	−0.850992	+	0.525178i	$0.823999\pi$
$678$	0	0
$679$	10.4636	0.401555
$680$	0	0
$681$	0.320321	0.0122747
$682$	0	0
$683$	−25.9663	−0.993574	−0.496787	−	0.867872i	$-0.665487\pi$
$683$	−25.9663	−0.993574	−0.496787	+	0.867872i	$0.665487\pi$
$684$	0	0
$685$	−6.25962	−0.239168
$686$	0	0
$687$	8.88459	0.338968
$688$	0	0
$689$	−41.9048	−1.59645
$690$	0	0
$691$	11.2697	0.428719	0.214359	−	0.976755i	$-0.431234\pi$
$691$	11.2697	0.428719	0.214359	+	0.976755i	$0.431234\pi$
$692$	0	0
$693$	−0.180969	−0.00687443
$694$	0	0
$695$	−7.72393	−0.292986
$696$	0	0
$697$	−9.80455	−0.371374
$698$	0	0
$699$	−0.418313	−0.0158221
$700$	0	0
$701$	41.4152	1.56423	0.782115	−	0.623134i	$-0.214141\pi$
$701$	41.4152	1.56423	0.782115	+	0.623134i	$0.214141\pi$
$702$	0	0
$703$	1.16970	0.0441161
$704$	0	0
$705$	12.6211	0.475337
$706$	0	0
$707$	14.7316	0.554038
$708$	0	0
$709$	27.9629	1.05017	0.525084	−	0.851050i	$-0.324034\pi$
$709$	27.9629	1.05017	0.525084	+	0.851050i	$0.324034\pi$
$710$	0	0
$711$	16.1948	0.607353
$712$	0	0
$713$	−0.825498	−0.0309151
$714$	0	0
$715$	0.930118	0.0347845
$716$	0	0
$717$	15.0571	0.562319
$718$	0	0
$719$	39.1607	1.46045	0.730224	−	0.683207i	$-0.239415\pi$
$719$	39.1607	1.46045	0.730224	+	0.683207i	$0.239415\pi$
$720$	0	0
$721$	−5.96362	−0.222097
$722$	0	0
$723$	5.31701	0.197742
$724$	0	0
$725$	22.1679	0.823296
$726$	0	0
$727$	39.7139	1.47291	0.736453	−	0.676488i	$-0.236499\pi$
$727$	39.7139	1.47291	0.736453	+	0.676488i	$0.236499\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−0.354618	−0.0131160
$732$	0	0
$733$	−19.6633	−0.726280	−0.363140	−	0.931735i	$-0.618295\pi$
$733$	−19.6633	−0.726280	−0.363140	+	0.931735i	$0.618295\pi$
$734$	0	0
$735$	1.04900	0.0386928
$736$	0	0
$737$	−2.84254	−0.104706
$738$	0	0
$739$	7.05398	0.259485	0.129742	−	0.991548i	$-0.458585\pi$
$739$	7.05398	0.259485	0.129742	+	0.991548i	$0.458585\pi$
$740$	0	0
$741$	−0.886675	−0.0325728
$742$	0	0
$743$	33.3545	1.22366	0.611829	−	0.790990i	$-0.290434\pi$
$743$	33.3545	1.22366	0.611829	+	0.790990i	$0.290434\pi$
$744$	0	0
$745$	23.0032	0.842772
$746$	0	0
$747$	−10.8757	−0.397922
$748$	0	0
$749$	8.55060	0.312432
$750$	0	0
$751$	−5.86039	−0.213849	−0.106924	−	0.994267i	$-0.534100\pi$
$751$	−5.86039	−0.213849	−0.106924	+	0.994267i	$0.534100\pi$
$752$	0	0
$753$	7.19107	0.262057
$754$	0	0
$755$	−17.6155	−0.641095
$756$	0	0
$757$	−4.25464	−0.154638	−0.0773188	−	0.997006i	$-0.524636\pi$
$757$	−4.25464	−0.154638	−0.0773188	+	0.997006i	$0.524636\pi$
$758$	0	0
$759$	0.180969	0.00656874
$760$	0	0
$761$	16.2300	0.588338	0.294169	−	0.955753i	$-0.404957\pi$
$761$	16.2300	0.588338	0.294169	+	0.955753i	$0.404957\pi$
$762$	0	0
$763$	9.00123	0.325866
$764$	0	0
$765$	2.96394	0.107161
$766$	0	0
$767$	−21.1000	−0.761878
$768$	0	0
$769$	−34.4633	−1.24278	−0.621388	−	0.783503i	$-0.713431\pi$
$769$	−34.4633	−1.24278	−0.621388	+	0.783503i	$0.713431\pi$
$770$	0	0
$771$	−8.77255	−0.315936
$772$	0	0
$773$	40.0968	1.44218	0.721091	−	0.692840i	$-0.243641\pi$
$773$	40.0968	1.44218	0.721091	+	0.692840i	$0.243641\pi$
$774$	0	0
$775$	−3.21912	−0.115634
$776$	0	0
$777$	6.46356	0.231879
$778$	0	0
$779$	−0.627966	−0.0224992
$780$	0	0
$781$	−0.402383	−0.0143984
$782$	0	0
$783$	−5.68466	−0.203153
$784$	0	0
$785$	17.5628	0.626845
$786$	0	0
$787$	10.7641	0.383697	0.191849	−	0.981425i	$-0.438552\pi$
$787$	10.7641	0.383697	0.191849	+	0.981425i	$0.438552\pi$
$788$	0	0
$789$	−16.0304	−0.570698
$790$	0	0
$791$	9.81016	0.348809
$792$	0	0
$793$	−44.1761	−1.56874
$794$	0	0
$795$	8.97173	0.318195
$796$	0	0
$797$	28.7263	1.01754	0.508769	−	0.860903i	$-0.330101\pi$
$797$	28.7263	1.01754	0.508769	+	0.860903i	$0.330101\pi$
$798$	0	0
$799$	33.9952	1.20266
$800$	0	0
$801$	−14.7466	−0.521045
$802$	0	0
$803$	−2.36557	−0.0834791
$804$	0	0
$805$	−1.04900	−0.0369723
$806$	0	0
$807$	6.27373	0.220846
$808$	0	0
$809$	−7.02676	−0.247048	−0.123524	−	0.992342i	$-0.539420\pi$
$809$	−7.02676	−0.247048	−0.123524	+	0.992342i	$0.539420\pi$
$810$	0	0
$811$	−29.3729	−1.03142	−0.515712	−	0.856762i	$-0.672472\pi$
$811$	−29.3729	−1.03142	−0.515712	+	0.856762i	$0.672472\pi$
$812$	0	0
$813$	17.1397	0.601114
$814$	0	0
$815$	−26.3621	−0.923425
$816$	0	0
$817$	−0.0227127	−0.000794616 0
$818$	0	0
$819$	−4.89961	−0.171206
$820$	0	0
$821$	−20.8592	−0.727990	−0.363995	−	0.931401i	$-0.618587\pi$
$821$	−20.8592	−0.727990	−0.363995	+	0.931401i	$0.618587\pi$
$822$	0	0
$823$	−8.32421	−0.290164	−0.145082	−	0.989420i	$-0.546345\pi$
$823$	−8.32421	−0.290164	−0.145082	+	0.989420i	$0.546345\pi$
$824$	0	0
$825$	0.705706	0.0245695
$826$	0	0
$827$	−26.0102	−0.904462	−0.452231	−	0.891901i	$-0.649372\pi$
$827$	−26.0102	−0.904462	−0.452231	+	0.891901i	$0.649372\pi$
$828$	0	0
$829$	20.5616	0.714132	0.357066	−	0.934079i	$-0.383777\pi$
$829$	20.5616	0.714132	0.357066	+	0.934079i	$0.383777\pi$
$830$	0	0
$831$	−23.9563	−0.831035
$832$	0	0
$833$	2.82550	0.0978977
$834$	0	0
$835$	−10.4692	−0.362301
$836$	0	0
$837$	0.825498	0.0285334
$838$	0	0
$839$	−11.8568	−0.409341	−0.204670	−	0.978831i	$-0.565612\pi$
$839$	−11.8568	−0.409341	−0.204670	+	0.978831i	$0.565612\pi$
$840$	0	0
$841$	3.31534	0.114322
$842$	0	0
$843$	−7.27132	−0.250438
$844$	0	0
$845$	11.5454	0.397174
$846$	0	0
$847$	−10.9673	−0.376839
$848$	0	0
$849$	−26.5365	−0.910730
$850$	0	0
$851$	−6.46356	−0.221568
$852$	0	0
$853$	30.2041	1.03417	0.517085	−	0.855934i	$-0.327017\pi$
$853$	30.2041	1.03417	0.517085	+	0.855934i	$0.327017\pi$
$854$	0	0
$855$	0.189835	0.00649223
$856$	0	0
$857$	−50.5499	−1.72675	−0.863375	−	0.504562i	$-0.831654\pi$
$857$	−50.5499	−1.72675	−0.863375	+	0.504562i	$0.831654\pi$
$858$	0	0
$859$	−10.0834	−0.344040	−0.172020	−	0.985093i	$-0.555029\pi$
$859$	−10.0834	−0.344040	−0.172020	+	0.985093i	$0.555029\pi$
$860$	0	0
$861$	−3.47003	−0.118258
$862$	0	0
$863$	−39.8098	−1.35514	−0.677571	−	0.735458i	$-0.736967\pi$
$863$	−39.8098	−1.35514	−0.677571	+	0.735458i	$0.736967\pi$
$864$	0	0
$865$	−13.1193	−0.446069
$866$	0	0
$867$	−9.01656	−0.306219
$868$	0	0
$869$	−2.93075	−0.0994190
$870$	0	0
$871$	−76.9599	−2.60769
$872$	0	0
$873$	10.4636	0.354138
$874$	0	0
$875$	−9.33565	−0.315603
$876$	0	0
$877$	−30.8941	−1.04322	−0.521609	−	0.853184i	$-0.674668\pi$
$877$	−30.8941	−1.04322	−0.521609	+	0.853184i	$0.674668\pi$
$878$	0	0
$879$	−32.7086	−1.10323
$880$	0	0
$881$	15.2972	0.515375	0.257688	−	0.966228i	$-0.417039\pi$
$881$	15.2972	0.515375	0.257688	+	0.966228i	$0.417039\pi$
$882$	0	0
$883$	−3.35890	−0.113036	−0.0565180	−	0.998402i	$-0.518000\pi$
$883$	−3.35890	−0.113036	−0.0565180	+	0.998402i	$0.518000\pi$
$884$	0	0
$885$	4.51748	0.151853
$886$	0	0
$887$	51.2123	1.71954	0.859770	−	0.510681i	$-0.170607\pi$
$887$	51.2123	1.71954	0.859770	+	0.510681i	$0.170607\pi$
$888$	0	0
$889$	−13.6421	−0.457542
$890$	0	0
$891$	−0.180969	−0.00606267
$892$	0	0
$893$	2.17734	0.0728618
$894$	0	0
$895$	20.2542	0.677022
$896$	0	0
$897$	4.89961	0.163593
$898$	0	0
$899$	−4.69267	−0.156509
$900$	0	0
$901$	24.1656	0.805073
$902$	0	0
$903$	−0.125506	−0.00417659
$904$	0	0
$905$	18.6360	0.619481
$906$	0	0
$907$	−27.1380	−0.901103	−0.450552	−	0.892750i	$-0.648773\pi$
$907$	−27.1380	−0.901103	−0.450552	+	0.892750i	$0.648773\pi$
$908$	0	0
$909$	14.7316	0.488615
$910$	0	0
$911$	0.642982	0.0213029	0.0106515	−	0.999943i	$-0.496609\pi$
$911$	0.642982	0.0213029	0.0106515	+	0.999943i	$0.496609\pi$
$912$	0	0
$913$	1.96816	0.0651367
$914$	0	0
$915$	9.45801	0.312672
$916$	0	0
$917$	9.72104	0.321017
$918$	0	0
$919$	−51.8741	−1.71117	−0.855585	−	0.517662i	$-0.826802\pi$
$919$	−51.8741	−1.71117	−0.855585	+	0.517662i	$0.826802\pi$
$920$	0	0
$921$	20.4281	0.673129
$922$	0	0
$923$	−10.8943	−0.358589
$924$	0	0
$925$	−25.2053	−0.828747
$926$	0	0
$927$	−5.96362	−0.195871
$928$	0	0
$929$	−2.08704	−0.0684736	−0.0342368	−	0.999414i	$-0.510900\pi$
$929$	−2.08704	−0.0684736	−0.0342368	+	0.999414i	$0.510900\pi$
$930$	0	0
$931$	0.180969	0.00593100
$932$	0	0
$933$	−15.9515	−0.522229
$934$	0	0
$935$	−0.536379	−0.0175415
$936$	0	0
$937$	−35.6141	−1.16346	−0.581731	−	0.813382i	$-0.697624\pi$
$937$	−35.6141	−1.16346	−0.581731	+	0.813382i	$0.697624\pi$
$938$	0	0
$939$	−2.91328	−0.0950711
$940$	0	0
$941$	−19.5524	−0.637389	−0.318695	−	0.947857i	$-0.603244\pi$
$941$	−19.5524	−0.637389	−0.318695	+	0.947857i	$0.603244\pi$
$942$	0	0
$943$	3.47003	0.113000
$944$	0	0
$945$	1.04900	0.0341238
$946$	0	0
$947$	14.2793	0.464016	0.232008	−	0.972714i	$-0.425470\pi$
$947$	14.2793	0.464016	0.232008	+	0.972714i	$0.425470\pi$
$948$	0	0
$949$	−64.0462	−2.07903
$950$	0	0
$951$	0.295090	0.00956894
$952$	0	0
$953$	36.6697	1.18785	0.593924	−	0.804521i	$-0.297578\pi$
$953$	36.6697	1.18785	0.593924	+	0.804521i	$0.297578\pi$
$954$	0	0
$955$	9.22356	0.298467
$956$	0	0
$957$	1.02874	0.0332546
$958$	0	0
$959$	−5.96725	−0.192693
$960$	0	0
$961$	−30.3186	−0.978018
$962$	0	0
$963$	8.55060	0.275539
$964$	0	0
$965$	−7.12557	−0.229380
$966$	0	0
$967$	−3.23340	−0.103979	−0.0519895	−	0.998648i	$-0.516556\pi$
$967$	−3.23340	−0.103979	−0.0519895	+	0.998648i	$0.516556\pi$
$968$	0	0
$969$	0.511326	0.0164262
$970$	0	0
$971$	−35.0009	−1.12323	−0.561617	−	0.827398i	$-0.689820\pi$
$971$	−35.0009	−1.12323	−0.561617	+	0.827398i	$0.689820\pi$
$972$	0	0
$973$	−7.36317	−0.236052
$974$	0	0
$975$	19.1065	0.611899
$976$	0	0
$977$	11.1142	0.355576	0.177788	−	0.984069i	$-0.443106\pi$
$977$	11.1142	0.355576	0.177788	+	0.984069i	$0.443106\pi$
$978$	0	0
$979$	2.66867	0.0852910
$980$	0	0
$981$	9.00123	0.287387
$982$	0	0
$983$	−12.0343	−0.383834	−0.191917	−	0.981411i	$-0.561471\pi$
$983$	−12.0343	−0.383834	−0.191917	+	0.981411i	$0.561471\pi$
$984$	0	0
$985$	15.3714	0.489772
$986$	0	0
$987$	12.0316	0.382969
$988$	0	0
$989$	0.125506	0.00399087
$990$	0	0
$991$	−59.7732	−1.89876	−0.949380	−	0.314130i	$-0.898287\pi$
$991$	−59.7732	−1.89876	−0.949380	+	0.314130i	$0.898287\pi$
$992$	0	0
$993$	−10.8907	−0.345607
$994$	0	0
$995$	4.67296	0.148143
$996$	0	0
$997$	24.4287	0.773666	0.386833	−	0.922150i	$-0.373569\pi$
$997$	24.4287	0.773666	0.386833	+	0.922150i	$0.373569\pi$
$998$	0	0
$999$	6.46356	0.204498

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7728.2.a.ce.1.2		4
4.3	odd	2		483.2.a.j.1.4	✓	4
12.11	even	2		1449.2.a.o.1.1		4
28.27	even	2		3381.2.a.x.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.a.j.1.4	✓	4	4.3	odd	2
1449.2.a.o.1.1		4	12.11	even	2
3381.2.a.x.1.4		4	28.27	even	2
7728.2.a.ce.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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.04900 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.04900 q^{5} +1.00000 q^{7} +1.00000 q^{9} -0.180969 q^{11} -4.89961 q^{13} +1.04900 q^{15} +2.82550 q^{17} +0.180969 q^{19} +1.00000 q^{21} -1.00000 q^{23} -3.89961 q^{25} +1.00000 q^{27} -5.68466 q^{29} +0.825498 q^{31} -0.180969 q^{33} +1.04900 q^{35} +6.46356 q^{37} -4.89961 q^{39} -3.47003 q^{41} -0.125506 q^{43} +1.04900 q^{45} +12.0316 q^{47} +1.00000 q^{49} +2.82550 q^{51} +8.55269 q^{53} -0.189835 q^{55} +0.180969 q^{57} +4.30647 q^{59} +9.01625 q^{61} +1.00000 q^{63} -5.13967 q^{65} +15.7074 q^{67} -1.00000 q^{69} +2.22350 q^{71} +13.0717 q^{73} -3.89961 q^{75} -0.180969 q^{77} +16.1948 q^{79} +1.00000 q^{81} -10.8757 q^{83} +2.96394 q^{85} -5.68466 q^{87} -14.7466 q^{89} -4.89961 q^{91} +0.825498 q^{93} +0.189835 q^{95} +10.4636 q^{97} -0.180969 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 5 q^{5} + 4 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + 5 * q^5 + 4 * q^7 + 4 * q^9	\( 4 q + 4 q^{3} + 5 q^{5} + 4 q^{7} + 4 q^{9} - q^{11} + 7 q^{13} + 5 q^{15} + 2 q^{17} + q^{19} + 4 q^{21} - 4 q^{23} + 11 q^{25} + 4 q^{27} + 2 q^{29} - 6 q^{31} - q^{33} + 5 q^{35} + 16 q^{37} + 7 q^{39} + 5 q^{41} - 9 q^{43} + 5 q^{45} + 21 q^{47} + 4 q^{49} + 2 q^{51} + 10 q^{53} - 17 q^{55} + q^{57} + 26 q^{59} + 2 q^{61} + 4 q^{63} + 26 q^{65} - 5 q^{67} - 4 q^{69} + 19 q^{71} + 10 q^{73} + 11 q^{75} - q^{77} + 6 q^{79} + 4 q^{81} + 2 q^{83} - 7 q^{85} + 2 q^{87} - 17 q^{89} + 7 q^{91} - 6 q^{93} + 17 q^{95} + 32 q^{97} - q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 + 5 * q^5 + 4 * q^7 + 4 * q^9 - q^11 + 7 * q^13 + 5 * q^15 + 2 * q^17 + q^19 + 4 * q^21 - 4 * q^23 + 11 * q^25 + 4 * q^27 + 2 * q^29 - 6 * q^31 - q^33 + 5 * q^35 + 16 * q^37 + 7 * q^39 + 5 * q^41 - 9 * q^43 + 5 * q^45 + 21 * q^47 + 4 * q^49 + 2 * q^51 + 10 * q^53 - 17 * q^55 + q^57 + 26 * q^59 + 2 * q^61 + 4 * q^63 + 26 * q^65 - 5 * q^67 - 4 * q^69 + 19 * q^71 + 10 * q^73 + 11 * q^75 - q^77 + 6 * q^79 + 4 * q^81 + 2 * q^83 - 7 * q^85 + 2 * q^87 - 17 * q^89 + 7 * q^91 - 6 * q^93 + 17 * q^95 + 32 * q^97 - q^99

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