LMFDB - Embedded newform 7728.2.a.x.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("7728.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$61.7083906820$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
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			1.1
Root			$2.30278$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -4.30278 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -4.30278 q^{5} +1.00000 q^{7} +1.00000 q^{9} +5.00000 q^{11} -1.30278 q^{13} +4.30278 q^{15} +1.60555 q^{17} -5.60555 q^{19} -1.00000 q^{21} -1.00000 q^{23} +13.5139 q^{25} -1.00000 q^{27} -8.21110 q^{29} -3.00000 q^{31} -5.00000 q^{33} -4.30278 q^{35} -9.00000 q^{37} +1.30278 q^{39} +2.21110 q^{41} +12.5139 q^{43} -4.30278 q^{45} +1.39445 q^{47} +1.00000 q^{49} -1.60555 q^{51} +5.51388 q^{53} -21.5139 q^{55} +5.60555 q^{57} +6.90833 q^{59} -11.9083 q^{61} +1.00000 q^{63} +5.60555 q^{65} +1.09167 q^{67} +1.00000 q^{69} +9.90833 q^{71} +12.2111 q^{73} -13.5139 q^{75} +5.00000 q^{77} +1.00000 q^{79} +1.00000 q^{81} +1.60555 q^{83} -6.90833 q^{85} +8.21110 q^{87} +0.0916731 q^{89} -1.30278 q^{91} +3.00000 q^{93} +24.1194 q^{95} -10.3944 q^{97} +5.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 5 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - 5 * q^5 + 2 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} - 5 q^{5} + 2 q^{7} + 2 q^{9} + 10 q^{11} + q^{13} + 5 q^{15} - 4 q^{17} - 4 q^{19} - 2 q^{21} - 2 q^{23} + 9 q^{25} - 2 q^{27} - 2 q^{29} - 6 q^{31} - 10 q^{33} - 5 q^{35} - 18 q^{37} - q^{39} - 10 q^{41} + 7 q^{43} - 5 q^{45} + 10 q^{47} + 2 q^{49} + 4 q^{51} - 7 q^{53} - 25 q^{55} + 4 q^{57} + 3 q^{59} - 13 q^{61} + 2 q^{63} + 4 q^{65} + 13 q^{67} + 2 q^{69} + 9 q^{71} + 10 q^{73} - 9 q^{75} + 10 q^{77} + 2 q^{79} + 2 q^{81} - 4 q^{83} - 3 q^{85} + 2 q^{87} + 11 q^{89} + q^{91} + 6 q^{93} + 23 q^{95} - 28 q^{97} + 10 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 5 * q^5 + 2 * q^7 + 2 * q^9 + 10 * q^11 + q^13 + 5 * q^15 - 4 * q^17 - 4 * q^19 - 2 * q^21 - 2 * q^23 + 9 * q^25 - 2 * q^27 - 2 * q^29 - 6 * q^31 - 10 * q^33 - 5 * q^35 - 18 * q^37 - q^39 - 10 * q^41 + 7 * q^43 - 5 * q^45 + 10 * q^47 + 2 * q^49 + 4 * q^51 - 7 * q^53 - 25 * q^55 + 4 * q^57 + 3 * q^59 - 13 * q^61 + 2 * q^63 + 4 * q^65 + 13 * q^67 + 2 * q^69 + 9 * q^71 + 10 * q^73 - 9 * q^75 + 10 * q^77 + 2 * q^79 + 2 * q^81 - 4 * q^83 - 3 * q^85 + 2 * q^87 + 11 * q^89 + q^91 + 6 * q^93 + 23 * q^95 - 28 * q^97 + 10 * 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
$5$	−4.30278	−1.92426	−0.962130	−	0.272591i	$-0.912119\pi$
$5$	−4.30278	−1.92426	−0.962130	+	0.272591i	$0.912119\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.00000	1.50756	0.753778	−	0.657129i	$-0.228229\pi$
$11$	5.00000	1.50756	0.753778	+	0.657129i	$0.228229\pi$
$12$	0	0
$13$	−1.30278	−0.361325	−0.180662	−	0.983545i	$-0.557824\pi$
$13$	−1.30278	−0.361325	−0.180662	+	0.983545i	$0.557824\pi$
$14$	0	0
$15$	4.30278	1.11097
$16$	0	0
$17$	1.60555	0.389403	0.194702	−	0.980863i	$-0.437626\pi$
$17$	1.60555	0.389403	0.194702	+	0.980863i	$0.437626\pi$
$18$	0	0
$19$	−5.60555	−1.28600	−0.643001	−	0.765865i	$-0.722311\pi$
$19$	−5.60555	−1.28600	−0.643001	+	0.765865i	$0.722311\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	13.5139	2.70278
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−8.21110	−1.52476	−0.762382	−	0.647128i	$-0.775970\pi$
$29$	−8.21110	−1.52476	−0.762382	+	0.647128i	$0.775970\pi$
$30$	0	0
$31$	−3.00000	−0.538816	−0.269408	−	0.963026i	$-0.586828\pi$
$31$	−3.00000	−0.538816	−0.269408	+	0.963026i	$0.586828\pi$
$32$	0	0
$33$	−5.00000	−0.870388
$34$	0	0
$35$	−4.30278	−0.727302
$36$	0	0
$37$	−9.00000	−1.47959	−0.739795	−	0.672832i	$-0.765078\pi$
$37$	−9.00000	−1.47959	−0.739795	+	0.672832i	$0.765078\pi$
$38$	0	0
$39$	1.30278	0.208611
$40$	0	0
$41$	2.21110	0.345316	0.172658	−	0.984982i	$-0.444764\pi$
$41$	2.21110	0.345316	0.172658	+	0.984982i	$0.444764\pi$
$42$	0	0
$43$	12.5139	1.90835	0.954174	−	0.299252i	$-0.0967370\pi$
$43$	12.5139	1.90835	0.954174	+	0.299252i	$0.0967370\pi$
$44$	0	0
$45$	−4.30278	−0.641420
$46$	0	0
$47$	1.39445	0.203401	0.101701	−	0.994815i	$-0.467572\pi$
$47$	1.39445	0.203401	0.101701	+	0.994815i	$0.467572\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−1.60555	−0.224822
$52$	0	0
$53$	5.51388	0.757389	0.378695	−	0.925522i	$-0.376373\pi$
$53$	5.51388	0.757389	0.378695	+	0.925522i	$0.376373\pi$
$54$	0	0
$55$	−21.5139	−2.90093
$56$	0	0
$57$	5.60555	0.742473
$58$	0	0
$59$	6.90833	0.899388	0.449694	−	0.893183i	$-0.351533\pi$
$59$	6.90833	0.899388	0.449694	+	0.893183i	$0.351533\pi$
$60$	0	0
$61$	−11.9083	−1.52471	−0.762353	−	0.647162i	$-0.775956\pi$
$61$	−11.9083	−1.52471	−0.762353	+	0.647162i	$0.775956\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	5.60555	0.695283
$66$	0	0
$67$	1.09167	0.133369	0.0666845	−	0.997774i	$-0.478758\pi$
$67$	1.09167	0.133369	0.0666845	+	0.997774i	$0.478758\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	9.90833	1.17590	0.587951	−	0.808897i	$-0.299935\pi$
$71$	9.90833	1.17590	0.587951	+	0.808897i	$0.299935\pi$
$72$	0	0
$73$	12.2111	1.42920	0.714601	−	0.699533i	$-0.246608\pi$
$73$	12.2111	1.42920	0.714601	+	0.699533i	$0.246608\pi$
$74$	0	0
$75$	−13.5139	−1.56045
$76$	0	0
$77$	5.00000	0.569803
$78$	0	0
$79$	1.00000	0.112509	0.0562544	−	0.998416i	$-0.482084\pi$
$79$	1.00000	0.112509	0.0562544	+	0.998416i	$0.482084\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	1.60555	0.176232	0.0881161	−	0.996110i	$-0.471915\pi$
$83$	1.60555	0.176232	0.0881161	+	0.996110i	$0.471915\pi$
$84$	0	0
$85$	−6.90833	−0.749313
$86$	0	0
$87$	8.21110	0.880323
$88$	0	0
$89$	0.0916731	0.00971733	0.00485866	−	0.999988i	$-0.498453\pi$
$89$	0.0916731	0.00971733	0.00485866	+	0.999988i	$0.498453\pi$
$90$	0	0
$91$	−1.30278	−0.136568
$92$	0	0
$93$	3.00000	0.311086
$94$	0	0
$95$	24.1194	2.47460
$96$	0	0
$97$	−10.3944	−1.05540	−0.527698	−	0.849432i	$-0.676945\pi$
$97$	−10.3944	−1.05540	−0.527698	+	0.849432i	$0.676945\pi$
$98$	0	0
$99$	5.00000	0.502519
$100$	0	0
$101$	5.69722	0.566895	0.283448	−	0.958988i	$-0.408522\pi$
$101$	5.69722	0.566895	0.283448	+	0.958988i	$0.408522\pi$
$102$	0	0
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	0	0
$105$	4.30278	0.419908
$106$	0	0
$107$	10.5139	1.01641	0.508207	−	0.861235i	$-0.330308\pi$
$107$	10.5139	1.01641	0.508207	+	0.861235i	$0.330308\pi$
$108$	0	0
$109$	6.90833	0.661698	0.330849	−	0.943684i	$-0.392665\pi$
$109$	6.90833	0.661698	0.330849	+	0.943684i	$0.392665\pi$
$110$	0	0
$111$	9.00000	0.854242
$112$	0	0
$113$	1.69722	0.159661	0.0798307	−	0.996808i	$-0.474562\pi$
$113$	1.69722	0.159661	0.0798307	+	0.996808i	$0.474562\pi$
$114$	0	0
$115$	4.30278	0.401236
$116$	0	0
$117$	−1.30278	−0.120442
$118$	0	0
$119$	1.60555	0.147181
$120$	0	0
$121$	14.0000	1.27273
$122$	0	0
$123$	−2.21110	−0.199368
$124$	0	0
$125$	−36.6333	−3.27658
$126$	0	0
$127$	−5.30278	−0.470545	−0.235273	−	0.971929i	$-0.575598\pi$
$127$	−5.30278	−0.470545	−0.235273	+	0.971929i	$0.575598\pi$
$128$	0	0
$129$	−12.5139	−1.10179
$130$	0	0
$131$	−17.6056	−1.53820	−0.769102	−	0.639126i	$-0.779296\pi$
$131$	−17.6056	−1.53820	−0.769102	+	0.639126i	$0.779296\pi$
$132$	0	0
$133$	−5.60555	−0.486063
$134$	0	0
$135$	4.30278	0.370324
$136$	0	0
$137$	−4.81665	−0.411515	−0.205757	−	0.978603i	$-0.565966\pi$
$137$	−4.81665	−0.411515	−0.205757	+	0.978603i	$0.565966\pi$
$138$	0	0
$139$	−5.09167	−0.431870	−0.215935	−	0.976408i	$-0.569280\pi$
$139$	−5.09167	−0.431870	−0.215935	+	0.976408i	$0.569280\pi$
$140$	0	0
$141$	−1.39445	−0.117434
$142$	0	0
$143$	−6.51388	−0.544718
$144$	0	0
$145$	35.3305	2.93404
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	1.39445	0.114238	0.0571188	−	0.998367i	$-0.481809\pi$
$149$	1.39445	0.114238	0.0571188	+	0.998367i	$0.481809\pi$
$150$	0	0
$151$	−9.39445	−0.764509	−0.382255	−	0.924057i	$-0.624852\pi$
$151$	−9.39445	−0.764509	−0.382255	+	0.924057i	$0.624852\pi$
$152$	0	0
$153$	1.60555	0.129801
$154$	0	0
$155$	12.9083	1.03682
$156$	0	0
$157$	17.8167	1.42192	0.710962	−	0.703231i	$-0.248260\pi$
$157$	17.8167	1.42192	0.710962	+	0.703231i	$0.248260\pi$
$158$	0	0
$159$	−5.51388	−0.437279
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	−18.7250	−1.46665	−0.733327	−	0.679876i	$-0.762033\pi$
$163$	−18.7250	−1.46665	−0.733327	+	0.679876i	$0.762033\pi$
$164$	0	0
$165$	21.5139	1.67485
$166$	0	0
$167$	18.8167	1.45608	0.728038	−	0.685537i	$-0.240432\pi$
$167$	18.8167	1.45608	0.728038	+	0.685537i	$0.240432\pi$
$168$	0	0
$169$	−11.3028	−0.869444
$170$	0	0
$171$	−5.60555	−0.428667
$172$	0	0
$173$	−3.78890	−0.288065	−0.144032	−	0.989573i	$-0.546007\pi$
$173$	−3.78890	−0.288065	−0.144032	+	0.989573i	$0.546007\pi$
$174$	0	0
$175$	13.5139	1.02155
$176$	0	0
$177$	−6.90833	−0.519262
$178$	0	0
$179$	6.30278	0.471092	0.235546	−	0.971863i	$-0.424312\pi$
$179$	6.30278	0.471092	0.235546	+	0.971863i	$0.424312\pi$
$180$	0	0
$181$	14.8167	1.10131	0.550657	−	0.834732i	$-0.314377\pi$
$181$	14.8167	1.10131	0.550657	+	0.834732i	$0.314377\pi$
$182$	0	0
$183$	11.9083	0.880289
$184$	0	0
$185$	38.7250	2.84712
$186$	0	0
$187$	8.02776	0.587048
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−2.60555	−0.188531	−0.0942655	−	0.995547i	$-0.530050\pi$
$191$	−2.60555	−0.188531	−0.0942655	+	0.995547i	$0.530050\pi$
$192$	0	0
$193$	−27.0278	−1.94550	−0.972750	−	0.231856i	$-0.925520\pi$
$193$	−27.0278	−1.94550	−0.972750	+	0.231856i	$0.925520\pi$
$194$	0	0
$195$	−5.60555	−0.401422
$196$	0	0
$197$	−17.9083	−1.27592	−0.637958	−	0.770071i	$-0.720221\pi$
$197$	−17.9083	−1.27592	−0.637958	+	0.770071i	$0.720221\pi$
$198$	0	0
$199$	5.51388	0.390868	0.195434	−	0.980717i	$-0.437388\pi$
$199$	5.51388	0.390868	0.195434	+	0.980717i	$0.437388\pi$
$200$	0	0
$201$	−1.09167	−0.0770007
$202$	0	0
$203$	−8.21110	−0.576306
$204$	0	0
$205$	−9.51388	−0.664478
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	−28.0278	−1.93872
$210$	0	0
$211$	−1.42221	−0.0979086	−0.0489543	−	0.998801i	$-0.515589\pi$
$211$	−1.42221	−0.0979086	−0.0489543	+	0.998801i	$0.515589\pi$
$212$	0	0
$213$	−9.90833	−0.678907
$214$	0	0
$215$	−53.8444	−3.67216
$216$	0	0
$217$	−3.00000	−0.203653
$218$	0	0
$219$	−12.2111	−0.825150
$220$	0	0
$221$	−2.09167	−0.140701
$222$	0	0
$223$	−9.09167	−0.608823	−0.304412	−	0.952541i	$-0.598460\pi$
$223$	−9.09167	−0.608823	−0.304412	+	0.952541i	$0.598460\pi$
$224$	0	0
$225$	13.5139	0.900925
$226$	0	0
$227$	−19.3305	−1.28301	−0.641506	−	0.767118i	$-0.721690\pi$
$227$	−19.3305	−1.28301	−0.641506	+	0.767118i	$0.721690\pi$
$228$	0	0
$229$	−15.5139	−1.02519	−0.512593	−	0.858632i	$-0.671315\pi$
$229$	−15.5139	−1.02519	−0.512593	+	0.858632i	$0.671315\pi$
$230$	0	0
$231$	−5.00000	−0.328976
$232$	0	0
$233$	25.3305	1.65946	0.829729	−	0.558166i	$-0.188495\pi$
$233$	25.3305	1.65946	0.829729	+	0.558166i	$0.188495\pi$
$234$	0	0
$235$	−6.00000	−0.391397
$236$	0	0
$237$	−1.00000	−0.0649570
$238$	0	0
$239$	24.9083	1.61119	0.805593	−	0.592470i	$-0.201847\pi$
$239$	24.9083	1.61119	0.805593	+	0.592470i	$0.201847\pi$
$240$	0	0
$241$	−24.0278	−1.54776	−0.773882	−	0.633330i	$-0.781688\pi$
$241$	−24.0278	−1.54776	−0.773882	+	0.633330i	$0.781688\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−4.30278	−0.274894
$246$	0	0
$247$	7.30278	0.464664
$248$	0	0
$249$	−1.60555	−0.101748
$250$	0	0
$251$	−2.18335	−0.137812	−0.0689058	−	0.997623i	$-0.521951\pi$
$251$	−2.18335	−0.137812	−0.0689058	+	0.997623i	$0.521951\pi$
$252$	0	0
$253$	−5.00000	−0.314347
$254$	0	0
$255$	6.90833	0.432616
$256$	0	0
$257$	−9.02776	−0.563136	−0.281568	−	0.959541i	$-0.590854\pi$
$257$	−9.02776	−0.563136	−0.281568	+	0.959541i	$0.590854\pi$
$258$	0	0
$259$	−9.00000	−0.559233
$260$	0	0
$261$	−8.21110	−0.508254
$262$	0	0
$263$	−17.6056	−1.08560	−0.542802	−	0.839860i	$-0.682637\pi$
$263$	−17.6056	−1.08560	−0.542802	+	0.839860i	$0.682637\pi$
$264$	0	0
$265$	−23.7250	−1.45741
$266$	0	0
$267$	−0.0916731	−0.00561030
$268$	0	0
$269$	−9.90833	−0.604121	−0.302061	−	0.953289i	$-0.597675\pi$
$269$	−9.90833	−0.604121	−0.302061	+	0.953289i	$0.597675\pi$
$270$	0	0
$271$	3.60555	0.219022	0.109511	−	0.993986i	$-0.465072\pi$
$271$	3.60555	0.219022	0.109511	+	0.993986i	$0.465072\pi$
$272$	0	0
$273$	1.30278	0.0788476
$274$	0	0
$275$	67.5694	4.07459
$276$	0	0
$277$	−8.69722	−0.522566	−0.261283	−	0.965262i	$-0.584146\pi$
$277$	−8.69722	−0.522566	−0.261283	+	0.965262i	$0.584146\pi$
$278$	0	0
$279$	−3.00000	−0.179605
$280$	0	0
$281$	6.18335	0.368868	0.184434	−	0.982845i	$-0.440955\pi$
$281$	6.18335	0.368868	0.184434	+	0.982845i	$0.440955\pi$
$282$	0	0
$283$	−13.6972	−0.814215	−0.407108	−	0.913380i	$-0.633463\pi$
$283$	−13.6972	−0.814215	−0.407108	+	0.913380i	$0.633463\pi$
$284$	0	0
$285$	−24.1194	−1.42871
$286$	0	0
$287$	2.21110	0.130517
$288$	0	0
$289$	−14.4222	−0.848365
$290$	0	0
$291$	10.3944	0.609333
$292$	0	0
$293$	32.8444	1.91879	0.959395	−	0.282064i	$-0.0910192\pi$
$293$	32.8444	1.91879	0.959395	+	0.282064i	$0.0910192\pi$
$294$	0	0
$295$	−29.7250	−1.73066
$296$	0	0
$297$	−5.00000	−0.290129
$298$	0	0
$299$	1.30278	0.0753415
$300$	0	0
$301$	12.5139	0.721288
$302$	0	0
$303$	−5.69722	−0.327297
$304$	0	0
$305$	51.2389	2.93393
$306$	0	0
$307$	−3.78890	−0.216244	−0.108122	−	0.994138i	$-0.534484\pi$
$307$	−3.78890	−0.216244	−0.108122	+	0.994138i	$0.534484\pi$
$308$	0	0
$309$	−4.00000	−0.227552
$310$	0	0
$311$	−8.51388	−0.482778	−0.241389	−	0.970428i	$-0.577603\pi$
$311$	−8.51388	−0.482778	−0.241389	+	0.970428i	$0.577603\pi$
$312$	0	0
$313$	−13.2111	−0.746736	−0.373368	−	0.927683i	$-0.621797\pi$
$313$	−13.2111	−0.746736	−0.373368	+	0.927683i	$0.621797\pi$
$314$	0	0
$315$	−4.30278	−0.242434
$316$	0	0
$317$	−12.4861	−0.701290	−0.350645	−	0.936508i	$-0.614038\pi$
$317$	−12.4861	−0.701290	−0.350645	+	0.936508i	$0.614038\pi$
$318$	0	0
$319$	−41.0555	−2.29867
$320$	0	0
$321$	−10.5139	−0.586827
$322$	0	0
$323$	−9.00000	−0.500773
$324$	0	0
$325$	−17.6056	−0.976580
$326$	0	0
$327$	−6.90833	−0.382031
$328$	0	0
$329$	1.39445	0.0768784
$330$	0	0
$331$	25.6333	1.40893	0.704467	−	0.709737i	$-0.251186\pi$
$331$	25.6333	1.40893	0.704467	+	0.709737i	$0.251186\pi$
$332$	0	0
$333$	−9.00000	−0.493197
$334$	0	0
$335$	−4.69722	−0.256637
$336$	0	0
$337$	−19.3028	−1.05149	−0.525745	−	0.850642i	$-0.676213\pi$
$337$	−19.3028	−1.05149	−0.525745	+	0.850642i	$0.676213\pi$
$338$	0	0
$339$	−1.69722	−0.0921806
$340$	0	0
$341$	−15.0000	−0.812296
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−4.30278	−0.231654
$346$	0	0
$347$	28.6333	1.53712	0.768558	−	0.639780i	$-0.220974\pi$
$347$	28.6333	1.53712	0.768558	+	0.639780i	$0.220974\pi$
$348$	0	0
$349$	−20.9083	−1.11920	−0.559599	−	0.828764i	$-0.689045\pi$
$349$	−20.9083	−1.11920	−0.559599	+	0.828764i	$0.689045\pi$
$350$	0	0
$351$	1.30278	0.0695370
$352$	0	0
$353$	−11.0000	−0.585471	−0.292735	−	0.956193i	$-0.594566\pi$
$353$	−11.0000	−0.585471	−0.292735	+	0.956193i	$0.594566\pi$
$354$	0	0
$355$	−42.6333	−2.26274
$356$	0	0
$357$	−1.60555	−0.0849748
$358$	0	0
$359$	−18.5416	−0.978590	−0.489295	−	0.872118i	$-0.662746\pi$
$359$	−18.5416	−0.978590	−0.489295	+	0.872118i	$0.662746\pi$
$360$	0	0
$361$	12.4222	0.653800
$362$	0	0
$363$	−14.0000	−0.734809
$364$	0	0
$365$	−52.5416	−2.75015
$366$	0	0
$367$	−3.48612	−0.181974	−0.0909870	−	0.995852i	$-0.529002\pi$
$367$	−3.48612	−0.181974	−0.0909870	+	0.995852i	$0.529002\pi$
$368$	0	0
$369$	2.21110	0.115105
$370$	0	0
$371$	5.51388	0.286266
$372$	0	0
$373$	1.78890	0.0926256	0.0463128	−	0.998927i	$-0.485253\pi$
$373$	1.78890	0.0926256	0.0463128	+	0.998927i	$0.485253\pi$
$374$	0	0
$375$	36.6333	1.89174
$376$	0	0
$377$	10.6972	0.550935
$378$	0	0
$379$	−6.42221	−0.329887	−0.164943	−	0.986303i	$-0.552744\pi$
$379$	−6.42221	−0.329887	−0.164943	+	0.986303i	$0.552744\pi$
$380$	0	0
$381$	5.30278	0.271669
$382$	0	0
$383$	16.8167	0.859291	0.429645	−	0.902998i	$-0.358639\pi$
$383$	16.8167	0.859291	0.429645	+	0.902998i	$0.358639\pi$
$384$	0	0
$385$	−21.5139	−1.09645
$386$	0	0
$387$	12.5139	0.636116
$388$	0	0
$389$	6.63331	0.336322	0.168161	−	0.985760i	$-0.446217\pi$
$389$	6.63331	0.336322	0.168161	+	0.985760i	$0.446217\pi$
$390$	0	0
$391$	−1.60555	−0.0811962
$392$	0	0
$393$	17.6056	0.888083
$394$	0	0
$395$	−4.30278	−0.216496
$396$	0	0
$397$	−34.6056	−1.73680	−0.868401	−	0.495862i	$-0.834852\pi$
$397$	−34.6056	−1.73680	−0.868401	+	0.495862i	$0.834852\pi$
$398$	0	0
$399$	5.60555	0.280629
$400$	0	0
$401$	15.4222	0.770148	0.385074	−	0.922886i	$-0.374176\pi$
$401$	15.4222	0.770148	0.385074	+	0.922886i	$0.374176\pi$
$402$	0	0
$403$	3.90833	0.194688
$404$	0	0
$405$	−4.30278	−0.213807
$406$	0	0
$407$	−45.0000	−2.23057
$408$	0	0
$409$	−12.0278	−0.594734	−0.297367	−	0.954763i	$-0.596109\pi$
$409$	−12.0278	−0.594734	−0.297367	+	0.954763i	$0.596109\pi$
$410$	0	0
$411$	4.81665	0.237588
$412$	0	0
$413$	6.90833	0.339937
$414$	0	0
$415$	−6.90833	−0.339116
$416$	0	0
$417$	5.09167	0.249340
$418$	0	0
$419$	10.1194	0.494366	0.247183	−	0.968969i	$-0.420495\pi$
$419$	10.1194	0.494366	0.247183	+	0.968969i	$0.420495\pi$
$420$	0	0
$421$	13.3028	0.648338	0.324169	−	0.945999i	$-0.394915\pi$
$421$	13.3028	0.648338	0.324169	+	0.945999i	$0.394915\pi$
$422$	0	0
$423$	1.39445	0.0678004
$424$	0	0
$425$	21.6972	1.05247
$426$	0	0
$427$	−11.9083	−0.576284
$428$	0	0
$429$	6.51388	0.314493
$430$	0	0
$431$	−23.7250	−1.14279	−0.571396	−	0.820674i	$-0.693598\pi$
$431$	−23.7250	−1.14279	−0.571396	+	0.820674i	$0.693598\pi$
$432$	0	0
$433$	15.0278	0.722188	0.361094	−	0.932529i	$-0.382403\pi$
$433$	15.0278	0.722188	0.361094	+	0.932529i	$0.382403\pi$
$434$	0	0
$435$	−35.3305	−1.69397
$436$	0	0
$437$	5.60555	0.268150
$438$	0	0
$439$	−5.78890	−0.276289	−0.138145	−	0.990412i	$-0.544114\pi$
$439$	−5.78890	−0.276289	−0.138145	+	0.990412i	$0.544114\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−24.8444	−1.18039	−0.590197	−	0.807259i	$-0.700950\pi$
$443$	−24.8444	−1.18039	−0.590197	+	0.807259i	$0.700950\pi$
$444$	0	0
$445$	−0.394449	−0.0186987
$446$	0	0
$447$	−1.39445	−0.0659552
$448$	0	0
$449$	−3.27502	−0.154558	−0.0772789	−	0.997010i	$-0.524623\pi$
$449$	−3.27502	−0.154558	−0.0772789	+	0.997010i	$0.524623\pi$
$450$	0	0
$451$	11.0555	0.520584
$452$	0	0
$453$	9.39445	0.441390
$454$	0	0
$455$	5.60555	0.262792
$456$	0	0
$457$	−36.1194	−1.68960	−0.844798	−	0.535086i	$-0.820279\pi$
$457$	−36.1194	−1.68960	−0.844798	+	0.535086i	$0.820279\pi$
$458$	0	0
$459$	−1.60555	−0.0749407
$460$	0	0
$461$	−24.7250	−1.15156	−0.575779	−	0.817606i	$-0.695301\pi$
$461$	−24.7250	−1.15156	−0.575779	+	0.817606i	$0.695301\pi$
$462$	0	0
$463$	−2.81665	−0.130901	−0.0654505	−	0.997856i	$-0.520848\pi$
$463$	−2.81665	−0.130901	−0.0654505	+	0.997856i	$0.520848\pi$
$464$	0	0
$465$	−12.9083	−0.598609
$466$	0	0
$467$	−16.3944	−0.758645	−0.379322	−	0.925265i	$-0.623843\pi$
$467$	−16.3944	−0.758645	−0.379322	+	0.925265i	$0.623843\pi$
$468$	0	0
$469$	1.09167	0.0504088
$470$	0	0
$471$	−17.8167	−0.820948
$472$	0	0
$473$	62.5694	2.87694
$474$	0	0
$475$	−75.7527	−3.47577
$476$	0	0
$477$	5.51388	0.252463
$478$	0	0
$479$	3.78890	0.173119	0.0865596	−	0.996247i	$-0.472413\pi$
$479$	3.78890	0.173119	0.0865596	+	0.996247i	$0.472413\pi$
$480$	0	0
$481$	11.7250	0.534613
$482$	0	0
$483$	1.00000	0.0455016
$484$	0	0
$485$	44.7250	2.03086
$486$	0	0
$487$	−20.8167	−0.943293	−0.471646	−	0.881788i	$-0.656340\pi$
$487$	−20.8167	−0.943293	−0.471646	+	0.881788i	$0.656340\pi$
$488$	0	0
$489$	18.7250	0.846773
$490$	0	0
$491$	−28.9361	−1.30587	−0.652934	−	0.757415i	$-0.726462\pi$
$491$	−28.9361	−1.30587	−0.652934	+	0.757415i	$0.726462\pi$
$492$	0	0
$493$	−13.1833	−0.593748
$494$	0	0
$495$	−21.5139	−0.966977
$496$	0	0
$497$	9.90833	0.444449
$498$	0	0
$499$	−21.0917	−0.944193	−0.472096	−	0.881547i	$-0.656503\pi$
$499$	−21.0917	−0.944193	−0.472096	+	0.881547i	$0.656503\pi$
$500$	0	0
$501$	−18.8167	−0.840666
$502$	0	0
$503$	14.7250	0.656554	0.328277	−	0.944581i	$-0.393532\pi$
$503$	14.7250	0.656554	0.328277	+	0.944581i	$0.393532\pi$
$504$	0	0
$505$	−24.5139	−1.09085
$506$	0	0
$507$	11.3028	0.501974
$508$	0	0
$509$	31.4500	1.39400	0.696998	−	0.717074i	$-0.254519\pi$
$509$	31.4500	1.39400	0.696998	+	0.717074i	$0.254519\pi$
$510$	0	0
$511$	12.2111	0.540187
$512$	0	0
$513$	5.60555	0.247491
$514$	0	0
$515$	−17.2111	−0.758412
$516$	0	0
$517$	6.97224	0.306639
$518$	0	0
$519$	3.78890	0.166314
$520$	0	0
$521$	−21.6333	−0.947772	−0.473886	−	0.880586i	$-0.657149\pi$
$521$	−21.6333	−0.947772	−0.473886	+	0.880586i	$0.657149\pi$
$522$	0	0
$523$	−28.4222	−1.24282	−0.621408	−	0.783487i	$-0.713439\pi$
$523$	−28.4222	−1.24282	−0.621408	+	0.783487i	$0.713439\pi$
$524$	0	0
$525$	−13.5139	−0.589794
$526$	0	0
$527$	−4.81665	−0.209817
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	6.90833	0.299796
$532$	0	0
$533$	−2.88057	−0.124771
$534$	0	0
$535$	−45.2389	−1.95585
$536$	0	0
$537$	−6.30278	−0.271985
$538$	0	0
$539$	5.00000	0.215365
$540$	0	0
$541$	15.8167	0.680011	0.340006	−	0.940423i	$-0.389571\pi$
$541$	15.8167	0.680011	0.340006	+	0.940423i	$0.389571\pi$
$542$	0	0
$543$	−14.8167	−0.635843
$544$	0	0
$545$	−29.7250	−1.27328
$546$	0	0
$547$	−24.7250	−1.05716	−0.528582	−	0.848882i	$-0.677276\pi$
$547$	−24.7250	−1.05716	−0.528582	+	0.848882i	$0.677276\pi$
$548$	0	0
$549$	−11.9083	−0.508235
$550$	0	0
$551$	46.0278	1.96085
$552$	0	0
$553$	1.00000	0.0425243
$554$	0	0
$555$	−38.7250	−1.64378
$556$	0	0
$557$	−31.0278	−1.31469	−0.657344	−	0.753591i	$-0.728320\pi$
$557$	−31.0278	−1.31469	−0.657344	+	0.753591i	$0.728320\pi$
$558$	0	0
$559$	−16.3028	−0.689534
$560$	0	0
$561$	−8.02776	−0.338932
$562$	0	0
$563$	3.66947	0.154650	0.0773248	−	0.997006i	$-0.475362\pi$
$563$	3.66947	0.154650	0.0773248	+	0.997006i	$0.475362\pi$
$564$	0	0
$565$	−7.30278	−0.307230
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	25.2111	1.05690	0.528452	−	0.848963i	$-0.322773\pi$
$569$	25.2111	1.05690	0.528452	+	0.848963i	$0.322773\pi$
$570$	0	0
$571$	4.21110	0.176229	0.0881146	−	0.996110i	$-0.471916\pi$
$571$	4.21110	0.176229	0.0881146	+	0.996110i	$0.471916\pi$
$572$	0	0
$573$	2.60555	0.108848
$574$	0	0
$575$	−13.5139	−0.563568
$576$	0	0
$577$	−7.57779	−0.315468	−0.157734	−	0.987482i	$-0.550419\pi$
$577$	−7.57779	−0.315468	−0.157734	+	0.987482i	$0.550419\pi$
$578$	0	0
$579$	27.0278	1.12324
$580$	0	0
$581$	1.60555	0.0666095
$582$	0	0
$583$	27.5694	1.14181
$584$	0	0
$585$	5.60555	0.231761
$586$	0	0
$587$	−22.1472	−0.914112	−0.457056	−	0.889438i	$-0.651096\pi$
$587$	−22.1472	−0.914112	−0.457056	+	0.889438i	$0.651096\pi$
$588$	0	0
$589$	16.8167	0.692918
$590$	0	0
$591$	17.9083	0.736650
$592$	0	0
$593$	0.972244	0.0399253	0.0199626	−	0.999801i	$-0.493645\pi$
$593$	0.972244	0.0399253	0.0199626	+	0.999801i	$0.493645\pi$
$594$	0	0
$595$	−6.90833	−0.283214
$596$	0	0
$597$	−5.51388	−0.225668
$598$	0	0
$599$	21.4861	0.877899	0.438950	−	0.898512i	$-0.355351\pi$
$599$	21.4861	0.877899	0.438950	+	0.898512i	$0.355351\pi$
$600$	0	0
$601$	−5.93608	−0.242138	−0.121069	−	0.992644i	$-0.538632\pi$
$601$	−5.93608	−0.242138	−0.121069	+	0.992644i	$0.538632\pi$
$602$	0	0
$603$	1.09167	0.0444564
$604$	0	0
$605$	−60.2389	−2.44906
$606$	0	0
$607$	42.5139	1.72559	0.862793	−	0.505558i	$-0.168713\pi$
$607$	42.5139	1.72559	0.862793	+	0.505558i	$0.168713\pi$
$608$	0	0
$609$	8.21110	0.332731
$610$	0	0
$611$	−1.81665	−0.0734939
$612$	0	0
$613$	−18.8167	−0.759997	−0.379999	−	0.924987i	$-0.624076\pi$
$613$	−18.8167	−0.759997	−0.379999	+	0.924987i	$0.624076\pi$
$614$	0	0
$615$	9.51388	0.383637
$616$	0	0
$617$	−17.7250	−0.713581	−0.356790	−	0.934184i	$-0.616129\pi$
$617$	−17.7250	−0.713581	−0.356790	+	0.934184i	$0.616129\pi$
$618$	0	0
$619$	−35.1194	−1.41157	−0.705785	−	0.708427i	$-0.749405\pi$
$619$	−35.1194	−1.41157	−0.705785	+	0.708427i	$0.749405\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	0.0916731	0.00367280
$624$	0	0
$625$	90.0555	3.60222
$626$	0	0
$627$	28.0278	1.11932
$628$	0	0
$629$	−14.4500	−0.576158
$630$	0	0
$631$	48.0278	1.91195	0.955977	−	0.293440i	$-0.0948002\pi$
$631$	48.0278	1.91195	0.955977	+	0.293440i	$0.0948002\pi$
$632$	0	0
$633$	1.42221	0.0565276
$634$	0	0
$635$	22.8167	0.905451
$636$	0	0
$637$	−1.30278	−0.0516179
$638$	0	0
$639$	9.90833	0.391967
$640$	0	0
$641$	43.5416	1.71979	0.859896	−	0.510470i	$-0.170529\pi$
$641$	43.5416	1.71979	0.859896	+	0.510470i	$0.170529\pi$
$642$	0	0
$643$	46.5694	1.83652	0.918259	−	0.395981i	$-0.129595\pi$
$643$	46.5694	1.83652	0.918259	+	0.395981i	$0.129595\pi$
$644$	0	0
$645$	53.8444	2.12012
$646$	0	0
$647$	23.3305	0.917218	0.458609	−	0.888638i	$-0.348348\pi$
$647$	23.3305	0.917218	0.458609	+	0.888638i	$0.348348\pi$
$648$	0	0
$649$	34.5416	1.35588
$650$	0	0
$651$	3.00000	0.117579
$652$	0	0
$653$	12.4861	0.488620	0.244310	−	0.969697i	$-0.421439\pi$
$653$	12.4861	0.488620	0.244310	+	0.969697i	$0.421439\pi$
$654$	0	0
$655$	75.7527	2.95990
$656$	0	0
$657$	12.2111	0.476400
$658$	0	0
$659$	0.633308	0.0246702	0.0123351	−	0.999924i	$-0.496074\pi$
$659$	0.633308	0.0246702	0.0123351	+	0.999924i	$0.496074\pi$
$660$	0	0
$661$	−0.816654	−0.0317642	−0.0158821	−	0.999874i	$-0.505056\pi$
$661$	−0.816654	−0.0317642	−0.0158821	+	0.999874i	$0.505056\pi$
$662$	0	0
$663$	2.09167	0.0812339
$664$	0	0
$665$	24.1194	0.935311
$666$	0	0
$667$	8.21110	0.317935
$668$	0	0
$669$	9.09167	0.351504
$670$	0	0
$671$	−59.5416	−2.29858
$672$	0	0
$673$	16.6333	0.641167	0.320583	−	0.947220i	$-0.396121\pi$
$673$	16.6333	0.641167	0.320583	+	0.947220i	$0.396121\pi$
$674$	0	0
$675$	−13.5139	−0.520149
$676$	0	0
$677$	−24.1472	−0.928052	−0.464026	−	0.885822i	$-0.653596\pi$
$677$	−24.1472	−0.928052	−0.464026	+	0.885822i	$0.653596\pi$
$678$	0	0
$679$	−10.3944	−0.398902
$680$	0	0
$681$	19.3305	0.740748
$682$	0	0
$683$	−27.4222	−1.04928	−0.524641	−	0.851324i	$-0.675800\pi$
$683$	−27.4222	−1.04928	−0.524641	+	0.851324i	$0.675800\pi$
$684$	0	0
$685$	20.7250	0.791861
$686$	0	0
$687$	15.5139	0.591891
$688$	0	0
$689$	−7.18335	−0.273664
$690$	0	0
$691$	13.4861	0.513036	0.256518	−	0.966539i	$-0.417425\pi$
$691$	13.4861	0.513036	0.256518	+	0.966539i	$0.417425\pi$
$692$	0	0
$693$	5.00000	0.189934
$694$	0	0
$695$	21.9083	0.831030
$696$	0	0
$697$	3.55004	0.134467
$698$	0	0
$699$	−25.3305	−0.958089
$700$	0	0
$701$	28.5416	1.07800	0.539001	−	0.842305i	$-0.318802\pi$
$701$	28.5416	1.07800	0.539001	+	0.842305i	$0.318802\pi$
$702$	0	0
$703$	50.4500	1.90276
$704$	0	0
$705$	6.00000	0.225973
$706$	0	0
$707$	5.69722	0.214266
$708$	0	0
$709$	−7.66947	−0.288033	−0.144016	−	0.989575i	$-0.546002\pi$
$709$	−7.66947	−0.288033	−0.144016	+	0.989575i	$0.546002\pi$
$710$	0	0
$711$	1.00000	0.0375029
$712$	0	0
$713$	3.00000	0.112351
$714$	0	0
$715$	28.0278	1.04818
$716$	0	0
$717$	−24.9083	−0.930219
$718$	0	0
$719$	0.211103	0.00787280	0.00393640	−	0.999992i	$-0.498747\pi$
$719$	0.211103	0.00787280	0.00393640	+	0.999992i	$0.498747\pi$
$720$	0	0
$721$	4.00000	0.148968
$722$	0	0
$723$	24.0278	0.893602
$724$	0	0
$725$	−110.964	−4.12109
$726$	0	0
$727$	−46.8722	−1.73839	−0.869196	−	0.494467i	$-0.835363\pi$
$727$	−46.8722	−1.73839	−0.869196	+	0.494467i	$0.835363\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	20.0917	0.743117
$732$	0	0
$733$	−19.3944	−0.716350	−0.358175	−	0.933654i	$-0.616601\pi$
$733$	−19.3944	−0.716350	−0.358175	+	0.933654i	$0.616601\pi$
$734$	0	0
$735$	4.30278	0.158710
$736$	0	0
$737$	5.45837	0.201061
$738$	0	0
$739$	14.5778	0.536253	0.268126	−	0.963384i	$-0.413596\pi$
$739$	14.5778	0.536253	0.268126	+	0.963384i	$0.413596\pi$
$740$	0	0
$741$	−7.30278	−0.268274
$742$	0	0
$743$	−12.4861	−0.458071	−0.229036	−	0.973418i	$-0.573557\pi$
$743$	−12.4861	−0.458071	−0.229036	+	0.973418i	$0.573557\pi$
$744$	0	0
$745$	−6.00000	−0.219823
$746$	0	0
$747$	1.60555	0.0587440
$748$	0	0
$749$	10.5139	0.384169
$750$	0	0
$751$	−12.3028	−0.448935	−0.224467	−	0.974482i	$-0.572064\pi$
$751$	−12.3028	−0.448935	−0.224467	+	0.974482i	$0.572064\pi$
$752$	0	0
$753$	2.18335	0.0795656
$754$	0	0
$755$	40.4222	1.47111
$756$	0	0
$757$	−46.6333	−1.69492	−0.847458	−	0.530862i	$-0.821868\pi$
$757$	−46.6333	−1.69492	−0.847458	+	0.530862i	$0.821868\pi$
$758$	0	0
$759$	5.00000	0.181489
$760$	0	0
$761$	12.0000	0.435000	0.217500	−	0.976060i	$-0.430210\pi$
$761$	12.0000	0.435000	0.217500	+	0.976060i	$0.430210\pi$
$762$	0	0
$763$	6.90833	0.250098
$764$	0	0
$765$	−6.90833	−0.249771
$766$	0	0
$767$	−9.00000	−0.324971
$768$	0	0
$769$	37.4500	1.35048	0.675240	−	0.737598i	$-0.264040\pi$
$769$	37.4500	1.35048	0.675240	+	0.737598i	$0.264040\pi$
$770$	0	0
$771$	9.02776	0.325127
$772$	0	0
$773$	−25.8444	−0.929559	−0.464779	−	0.885427i	$-0.653866\pi$
$773$	−25.8444	−0.929559	−0.464779	+	0.885427i	$0.653866\pi$
$774$	0	0
$775$	−40.5416	−1.45630
$776$	0	0
$777$	9.00000	0.322873
$778$	0	0
$779$	−12.3944	−0.444077
$780$	0	0
$781$	49.5416	1.77274
$782$	0	0
$783$	8.21110	0.293441
$784$	0	0
$785$	−76.6611	−2.73615
$786$	0	0
$787$	29.1472	1.03898	0.519492	−	0.854475i	$-0.326121\pi$
$787$	29.1472	1.03898	0.519492	+	0.854475i	$0.326121\pi$
$788$	0	0
$789$	17.6056	0.626774
$790$	0	0
$791$	1.69722	0.0603464
$792$	0	0
$793$	15.5139	0.550914
$794$	0	0
$795$	23.7250	0.841438
$796$	0	0
$797$	−2.97224	−0.105282	−0.0526411	−	0.998613i	$-0.516764\pi$
$797$	−2.97224	−0.105282	−0.0526411	+	0.998613i	$0.516764\pi$
$798$	0	0
$799$	2.23886	0.0792051
$800$	0	0
$801$	0.0916731	0.00323911
$802$	0	0
$803$	61.0555	2.15460
$804$	0	0
$805$	4.30278	0.151653
$806$	0	0
$807$	9.90833	0.348790
$808$	0	0
$809$	−20.9361	−0.736073	−0.368037	−	0.929811i	$-0.619970\pi$
$809$	−20.9361	−0.736073	−0.368037	+	0.929811i	$0.619970\pi$
$810$	0	0
$811$	17.6056	0.618215	0.309107	−	0.951027i	$-0.399970\pi$
$811$	17.6056	0.618215	0.309107	+	0.951027i	$0.399970\pi$
$812$	0	0
$813$	−3.60555	−0.126452
$814$	0	0
$815$	80.5694	2.82222
$816$	0	0
$817$	−70.1472	−2.45414
$818$	0	0
$819$	−1.30278	−0.0455227
$820$	0	0
$821$	−26.6056	−0.928540	−0.464270	−	0.885694i	$-0.653683\pi$
$821$	−26.6056	−0.928540	−0.464270	+	0.885694i	$0.653683\pi$
$822$	0	0
$823$	11.5139	0.401349	0.200674	−	0.979658i	$-0.435687\pi$
$823$	11.5139	0.401349	0.200674	+	0.979658i	$0.435687\pi$
$824$	0	0
$825$	−67.5694	−2.35246
$826$	0	0
$827$	−0.908327	−0.0315856	−0.0157928	−	0.999875i	$-0.505027\pi$
$827$	−0.908327	−0.0315856	−0.0157928	+	0.999875i	$0.505027\pi$
$828$	0	0
$829$	34.4500	1.19650	0.598248	−	0.801311i	$-0.295864\pi$
$829$	34.4500	1.19650	0.598248	+	0.801311i	$0.295864\pi$
$830$	0	0
$831$	8.69722	0.301703
$832$	0	0
$833$	1.60555	0.0556291
$834$	0	0
$835$	−80.9638	−2.80187
$836$	0	0
$837$	3.00000	0.103695
$838$	0	0
$839$	−34.6972	−1.19788	−0.598941	−	0.800793i	$-0.704411\pi$
$839$	−34.6972	−1.19788	−0.598941	+	0.800793i	$0.704411\pi$
$840$	0	0
$841$	38.4222	1.32490
$842$	0	0
$843$	−6.18335	−0.212966
$844$	0	0
$845$	48.6333	1.67304
$846$	0	0
$847$	14.0000	0.481046
$848$	0	0
$849$	13.6972	0.470088
$850$	0	0
$851$	9.00000	0.308516
$852$	0	0
$853$	−7.76114	−0.265736	−0.132868	−	0.991134i	$-0.542419\pi$
$853$	−7.76114	−0.265736	−0.132868	+	0.991134i	$0.542419\pi$
$854$	0	0
$855$	24.1194	0.824867
$856$	0	0
$857$	0.238859	0.00815927	0.00407963	−	0.999992i	$-0.498701\pi$
$857$	0.238859	0.00815927	0.00407963	+	0.999992i	$0.498701\pi$
$858$	0	0
$859$	−28.7889	−0.982265	−0.491132	−	0.871085i	$-0.663417\pi$
$859$	−28.7889	−0.982265	−0.491132	+	0.871085i	$0.663417\pi$
$860$	0	0
$861$	−2.21110	−0.0753542
$862$	0	0
$863$	56.2389	1.91439	0.957197	−	0.289439i	$-0.0934687\pi$
$863$	56.2389	1.91439	0.957197	+	0.289439i	$0.0934687\pi$
$864$	0	0
$865$	16.3028	0.554311
$866$	0	0
$867$	14.4222	0.489804
$868$	0	0
$869$	5.00000	0.169613
$870$	0	0
$871$	−1.42221	−0.0481896
$872$	0	0
$873$	−10.3944	−0.351799
$874$	0	0
$875$	−36.6333	−1.23843
$876$	0	0
$877$	−29.2111	−0.986389	−0.493194	−	0.869919i	$-0.664171\pi$
$877$	−29.2111	−0.986389	−0.493194	+	0.869919i	$0.664171\pi$
$878$	0	0
$879$	−32.8444	−1.10781
$880$	0	0
$881$	−22.1833	−0.747376	−0.373688	−	0.927554i	$-0.621907\pi$
$881$	−22.1833	−0.747376	−0.373688	+	0.927554i	$0.621907\pi$
$882$	0	0
$883$	36.5139	1.22879	0.614395	−	0.788999i	$-0.289400\pi$
$883$	36.5139	1.22879	0.614395	+	0.788999i	$0.289400\pi$
$884$	0	0
$885$	29.7250	0.999194
$886$	0	0
$887$	−34.9361	−1.17304	−0.586519	−	0.809935i	$-0.699502\pi$
$887$	−34.9361	−1.17304	−0.586519	+	0.809935i	$0.699502\pi$
$888$	0	0
$889$	−5.30278	−0.177849
$890$	0	0
$891$	5.00000	0.167506
$892$	0	0
$893$	−7.81665	−0.261574
$894$	0	0
$895$	−27.1194	−0.906503
$896$	0	0
$897$	−1.30278	−0.0434984
$898$	0	0
$899$	24.6333	0.821567
$900$	0	0
$901$	8.85281	0.294930
$902$	0	0
$903$	−12.5139	−0.416436
$904$	0	0
$905$	−63.7527	−2.11921
$906$	0	0
$907$	−53.1472	−1.76472	−0.882362	−	0.470572i	$-0.844048\pi$
$907$	−53.1472	−1.76472	−0.882362	+	0.470572i	$0.844048\pi$
$908$	0	0
$909$	5.69722	0.188965
$910$	0	0
$911$	−26.2389	−0.869332	−0.434666	−	0.900592i	$-0.643134\pi$
$911$	−26.2389	−0.869332	−0.434666	+	0.900592i	$0.643134\pi$
$912$	0	0
$913$	8.02776	0.265680
$914$	0	0
$915$	−51.2389	−1.69390
$916$	0	0
$917$	−17.6056	−0.581387
$918$	0	0
$919$	17.6056	0.580754	0.290377	−	0.956912i	$-0.406219\pi$
$919$	17.6056	0.580754	0.290377	+	0.956912i	$0.406219\pi$
$920$	0	0
$921$	3.78890	0.124848
$922$	0	0
$923$	−12.9083	−0.424883
$924$	0	0
$925$	−121.625	−3.99900
$926$	0	0
$927$	4.00000	0.131377
$928$	0	0
$929$	0.669468	0.0219645	0.0109823	−	0.999940i	$-0.496504\pi$
$929$	0.669468	0.0219645	0.0109823	+	0.999940i	$0.496504\pi$
$930$	0	0
$931$	−5.60555	−0.183715
$932$	0	0
$933$	8.51388	0.278732
$934$	0	0
$935$	−34.5416	−1.12963
$936$	0	0
$937$	−33.4222	−1.09186	−0.545928	−	0.837832i	$-0.683823\pi$
$937$	−33.4222	−1.09186	−0.545928	+	0.837832i	$0.683823\pi$
$938$	0	0
$939$	13.2111	0.431128
$940$	0	0
$941$	56.8444	1.85307	0.926537	−	0.376203i	$-0.122770\pi$
$941$	56.8444	1.85307	0.926537	+	0.376203i	$0.122770\pi$
$942$	0	0
$943$	−2.21110	−0.0720034
$944$	0	0
$945$	4.30278	0.139969
$946$	0	0
$947$	−29.6333	−0.962953	−0.481477	−	0.876459i	$-0.659899\pi$
$947$	−29.6333	−0.962953	−0.481477	+	0.876459i	$0.659899\pi$
$948$	0	0
$949$	−15.9083	−0.516406
$950$	0	0
$951$	12.4861	0.404890
$952$	0	0
$953$	−11.6695	−0.378011	−0.189006	−	0.981976i	$-0.560526\pi$
$953$	−11.6695	−0.378011	−0.189006	+	0.981976i	$0.560526\pi$
$954$	0	0
$955$	11.2111	0.362783
$956$	0	0
$957$	41.0555	1.32714
$958$	0	0
$959$	−4.81665	−0.155538
$960$	0	0
$961$	−22.0000	−0.709677
$962$	0	0
$963$	10.5139	0.338805
$964$	0	0
$965$	116.294	3.74365
$966$	0	0
$967$	45.2666	1.45568	0.727838	−	0.685749i	$-0.240525\pi$
$967$	45.2666	1.45568	0.727838	+	0.685749i	$0.240525\pi$
$968$	0	0
$969$	9.00000	0.289122
$970$	0	0
$971$	45.9083	1.47327	0.736634	−	0.676291i	$-0.236414\pi$
$971$	45.9083	1.47327	0.736634	+	0.676291i	$0.236414\pi$
$972$	0	0
$973$	−5.09167	−0.163232
$974$	0	0
$975$	17.6056	0.563829
$976$	0	0
$977$	−30.1472	−0.964494	−0.482247	−	0.876035i	$-0.660179\pi$
$977$	−30.1472	−0.964494	−0.482247	+	0.876035i	$0.660179\pi$
$978$	0	0
$979$	0.458365	0.0146494
$980$	0	0
$981$	6.90833	0.220566
$982$	0	0
$983$	1.18335	0.0377429	0.0188714	−	0.999822i	$-0.493993\pi$
$983$	1.18335	0.0377429	0.0188714	+	0.999822i	$0.493993\pi$
$984$	0	0
$985$	77.0555	2.45519
$986$	0	0
$987$	−1.39445	−0.0443858
$988$	0	0
$989$	−12.5139	−0.397918
$990$	0	0
$991$	−17.4861	−0.555465	−0.277732	−	0.960658i	$-0.589583\pi$
$991$	−17.4861	−0.555465	−0.277732	+	0.960658i	$0.589583\pi$
$992$	0	0
$993$	−25.6333	−0.813448
$994$	0	0
$995$	−23.7250	−0.752132
$996$	0	0
$997$	36.0278	1.14101	0.570505	−	0.821294i	$-0.306747\pi$
$997$	36.0278	1.14101	0.570505	+	0.821294i	$0.306747\pi$
$998$	0	0
$999$	9.00000	0.284747

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.a.f.1.2	✓	2	4.3	odd	2
1449.2.a.j.1.1		2	12.11	even	2
3381.2.a.p.1.2		2	28.27	even	2
7728.2.a.x.1.1		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -4.30278 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -4.30278 q^{5} +1.00000 q^{7} +1.00000 q^{9} +5.00000 q^{11} -1.30278 q^{13} +4.30278 q^{15} +1.60555 q^{17} -5.60555 q^{19} -1.00000 q^{21} -1.00000 q^{23} +13.5139 q^{25} -1.00000 q^{27} -8.21110 q^{29} -3.00000 q^{31} -5.00000 q^{33} -4.30278 q^{35} -9.00000 q^{37} +1.30278 q^{39} +2.21110 q^{41} +12.5139 q^{43} -4.30278 q^{45} +1.39445 q^{47} +1.00000 q^{49} -1.60555 q^{51} +5.51388 q^{53} -21.5139 q^{55} +5.60555 q^{57} +6.90833 q^{59} -11.9083 q^{61} +1.00000 q^{63} +5.60555 q^{65} +1.09167 q^{67} +1.00000 q^{69} +9.90833 q^{71} +12.2111 q^{73} -13.5139 q^{75} +5.00000 q^{77} +1.00000 q^{79} +1.00000 q^{81} +1.60555 q^{83} -6.90833 q^{85} +8.21110 q^{87} +0.0916731 q^{89} -1.30278 q^{91} +3.00000 q^{93} +24.1194 q^{95} -10.3944 q^{97} +5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 5 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 5 * q^5 + 2 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} - 5 q^{5} + 2 q^{7} + 2 q^{9} + 10 q^{11} + q^{13} + 5 q^{15} - 4 q^{17} - 4 q^{19} - 2 q^{21} - 2 q^{23} + 9 q^{25} - 2 q^{27} - 2 q^{29} - 6 q^{31} - 10 q^{33} - 5 q^{35} - 18 q^{37} - q^{39} - 10 q^{41} + 7 q^{43} - 5 q^{45} + 10 q^{47} + 2 q^{49} + 4 q^{51} - 7 q^{53} - 25 q^{55} + 4 q^{57} + 3 q^{59} - 13 q^{61} + 2 q^{63} + 4 q^{65} + 13 q^{67} + 2 q^{69} + 9 q^{71} + 10 q^{73} - 9 q^{75} + 10 q^{77} + 2 q^{79} + 2 q^{81} - 4 q^{83} - 3 q^{85} + 2 q^{87} + 11 q^{89} + q^{91} + 6 q^{93} + 23 q^{95} - 28 q^{97} + 10 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 5 * q^5 + 2 * q^7 + 2 * q^9 + 10 * q^11 + q^13 + 5 * q^15 - 4 * q^17 - 4 * q^19 - 2 * q^21 - 2 * q^23 + 9 * q^25 - 2 * q^27 - 2 * q^29 - 6 * q^31 - 10 * q^33 - 5 * q^35 - 18 * q^37 - q^39 - 10 * q^41 + 7 * q^43 - 5 * q^45 + 10 * q^47 + 2 * q^49 + 4 * q^51 - 7 * q^53 - 25 * q^55 + 4 * q^57 + 3 * q^59 - 13 * q^61 + 2 * q^63 + 4 * q^65 + 13 * q^67 + 2 * q^69 + 9 * q^71 + 10 * q^73 - 9 * q^75 + 10 * q^77 + 2 * q^79 + 2 * q^81 - 4 * q^83 - 3 * q^85 + 2 * q^87 + 11 * q^89 + q^91 + 6 * q^93 + 23 * q^95 - 28 * q^97 + 10 * q^99

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