LMFDB - Embedded newform 7728.2.a.ce.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("7728.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$61.7083906820$
Analytic rank:	$0$
Dimension:	$4$
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			1.1
Root			$1.32973$ of defining polynomial
Character	$\chi$	$=$	7728.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -2.73589 q^{5} +1.00000 q^{7} +1.00000 q^{9} +2.50407 q^{11} +1.48511 q^{13} -2.73589 q^{15} +0.902098 q^{17} -2.50407 q^{19} +1.00000 q^{21} -1.00000 q^{23} +2.48511 q^{25} +1.00000 q^{27} +6.68466 q^{29} -1.09790 q^{31} +2.50407 q^{33} -2.73589 q^{35} +9.91023 q^{37} +1.48511 q^{39} -2.30826 q^{41} -5.83380 q^{43} -2.73589 q^{45} +6.74671 q^{47} +1.00000 q^{49} +0.902098 q^{51} -4.91648 q^{53} -6.85086 q^{55} -2.50407 q^{57} +7.32973 q^{59} -1.00625 q^{61} +1.00000 q^{63} -4.06311 q^{65} -11.2929 q^{67} -1.00000 q^{69} +0.362008 q^{71} -5.34411 q^{73} +2.48511 q^{75} +2.50407 q^{77} -10.4672 q^{79} +1.00000 q^{81} -7.59946 q^{83} -2.46805 q^{85} +6.68466 q^{87} +10.6245 q^{89} +1.48511 q^{91} -1.09790 q^{93} +6.85086 q^{95} +13.9102 q^{97} +2.50407 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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}+ \cdots - 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

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−2.73589	−1.22353	−0.611764	−	0.791040i	$-0.709540\pi$
$5$	−2.73589	−1.22353	−0.611764	+	0.791040i	$0.709540\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.50407	0.755005	0.377502	−	0.926009i	$-0.376783\pi$
$11$	2.50407	0.755005	0.377502	+	0.926009i	$0.376783\pi$
$12$	0	0
$13$	1.48511	0.411896	0.205948	−	0.978563i	$-0.433972\pi$
$13$	1.48511	0.411896	0.205948	+	0.978563i	$0.433972\pi$
$14$	0	0
$15$	−2.73589	−0.706405
$16$	0	0
$17$	0.902098	0.218791	0.109396	−	0.993998i	$-0.465108\pi$
$17$	0.902098	0.218791	0.109396	+	0.993998i	$0.465108\pi$
$18$	0	0
$19$	−2.50407	−0.574473	−0.287236	−	0.957860i	$-0.592736\pi$
$19$	−2.50407	−0.574473	−0.287236	+	0.957860i	$0.592736\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	2.48511	0.497023
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	6.68466	1.24131	0.620655	−	0.784084i	$-0.286867\pi$
$29$	6.68466	1.24131	0.620655	+	0.784084i	$0.286867\pi$
$30$	0	0
$31$	−1.09790	−0.197189	−0.0985945	−	0.995128i	$-0.531435\pi$
$31$	−1.09790	−0.197189	−0.0985945	+	0.995128i	$0.531435\pi$
$32$	0	0
$33$	2.50407	0.435902
$34$	0	0
$35$	−2.73589	−0.462450
$36$	0	0
$37$	9.91023	1.62923	0.814616	−	0.580000i	$-0.196948\pi$
$37$	9.91023	1.62923	0.814616	+	0.580000i	$0.196948\pi$
$38$	0	0
$39$	1.48511	0.237808
$40$	0	0
$41$	−2.30826	−0.360490	−0.180245	−	0.983622i	$-0.557689\pi$
$41$	−2.30826	−0.360490	−0.180245	+	0.983622i	$0.557689\pi$
$42$	0	0
$43$	−5.83380	−0.889645	−0.444823	−	0.895619i	$-0.646733\pi$
$43$	−5.83380	−0.889645	−0.444823	+	0.895619i	$0.646733\pi$
$44$	0	0
$45$	−2.73589	−0.407843
$46$	0	0
$47$	6.74671	0.984109	0.492055	−	0.870564i	$-0.336246\pi$
$47$	6.74671	0.984109	0.492055	+	0.870564i	$0.336246\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0.902098	0.126319
$52$	0	0
$53$	−4.91648	−0.675331	−0.337666	−	0.941266i	$-0.609637\pi$
$53$	−4.91648	−0.675331	−0.337666	+	0.941266i	$0.609637\pi$
$54$	0	0
$55$	−6.85086	−0.923770
$56$	0	0
$57$	−2.50407	−0.331672
$58$	0	0
$59$	7.32973	0.954249	0.477125	−	0.878836i	$-0.341679\pi$
$59$	7.32973	0.954249	0.477125	+	0.878836i	$0.341679\pi$
$60$	0	0
$61$	−1.00625	−0.128837	−0.0644185	−	0.997923i	$-0.520519\pi$
$61$	−1.00625	−0.128837	−0.0644185	+	0.997923i	$0.520519\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	−4.06311	−0.503967
$66$	0	0
$67$	−11.2929	−1.37964	−0.689822	−	0.723979i	$-0.742311\pi$
$67$	−11.2929	−1.37964	−0.689822	+	0.723979i	$0.742311\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	0.362008	0.0429624	0.0214812	−	0.999769i	$-0.493162\pi$
$71$	0.362008	0.0429624	0.0214812	+	0.999769i	$0.493162\pi$
$72$	0	0
$73$	−5.34411	−0.625481	−0.312741	−	0.949839i	$-0.601247\pi$
$73$	−5.34411	−0.625481	−0.312741	+	0.949839i	$0.601247\pi$
$74$	0	0
$75$	2.48511	0.286956
$76$	0	0
$77$	2.50407	0.285365
$78$	0	0
$79$	−10.4672	−1.17765	−0.588827	−	0.808259i	$-0.700410\pi$
$79$	−10.4672	−1.17765	−0.588827	+	0.808259i	$0.700410\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−7.59946	−0.834149	−0.417075	−	0.908872i	$-0.636945\pi$
$83$	−7.59946	−0.834149	−0.417075	+	0.908872i	$0.636945\pi$
$84$	0	0
$85$	−2.46805	−0.267697
$86$	0	0
$87$	6.68466	0.716671
$88$	0	0
$89$	10.6245	1.12619	0.563097	−	0.826391i	$-0.309610\pi$
$89$	10.6245	1.12619	0.563097	+	0.826391i	$0.309610\pi$
$90$	0	0
$91$	1.48511	0.155682
$92$	0	0
$93$	−1.09790	−0.113847
$94$	0	0
$95$	6.85086	0.702884
$96$	0	0
$97$	13.9102	1.41237	0.706185	−	0.708027i	$-0.250415\pi$
$97$	13.9102	1.41237	0.706185	+	0.708027i	$0.250415\pi$
$98$	0	0
$99$	2.50407	0.251668
$100$	0	0
$101$	1.81502	0.180601	0.0903004	−	0.995915i	$-0.471217\pi$
$101$	1.81502	0.180601	0.0903004	+	0.995915i	$0.471217\pi$
$102$	0	0
$103$	16.6605	1.64161	0.820805	−	0.571209i	$-0.193525\pi$
$103$	16.6605	1.64161	0.820805	+	0.571209i	$0.193525\pi$
$104$	0	0
$105$	−2.73589	−0.266996
$106$	0	0
$107$	−1.68092	−0.162500	−0.0812502	−	0.996694i	$-0.525891\pi$
$107$	−1.68092	−0.162500	−0.0812502	+	0.996694i	$0.525891\pi$
$108$	0	0
$109$	11.4333	1.09511	0.547554	−	0.836771i	$-0.315559\pi$
$109$	11.4333	1.09511	0.547554	+	0.836771i	$0.315559\pi$
$110$	0	0
$111$	9.91023	0.940638
$112$	0	0
$113$	3.14914	0.296246	0.148123	−	0.988969i	$-0.452677\pi$
$113$	3.14914	0.296246	0.148123	+	0.988969i	$0.452677\pi$
$114$	0	0
$115$	2.73589	0.255123
$116$	0	0
$117$	1.48511	0.137299
$118$	0	0
$119$	0.902098	0.0826952
$120$	0	0
$121$	−4.72964	−0.429968
$122$	0	0
$123$	−2.30826	−0.208129
$124$	0	0
$125$	6.88046	0.615407
$126$	0	0
$127$	−0.449266	−0.0398659	−0.0199329	−	0.999801i	$-0.506345\pi$
$127$	−0.449266	−0.0398659	−0.0199329	+	0.999801i	$0.506345\pi$
$128$	0	0
$129$	−5.83380	−0.513637
$130$	0	0
$131$	19.9759	1.74530	0.872649	−	0.488347i	$-0.162400\pi$
$131$	19.9759	1.74530	0.872649	+	0.488347i	$0.162400\pi$
$132$	0	0
$133$	−2.50407	−0.217130
$134$	0	0
$135$	−2.73589	−0.235468
$136$	0	0
$137$	0.270356	0.0230981	0.0115490	−	0.999933i	$-0.496324\pi$
$137$	0.270356	0.0230981	0.0115490	+	0.999933i	$0.496324\pi$
$138$	0	0
$139$	−4.42512	−0.375334	−0.187667	−	0.982233i	$-0.560093\pi$
$139$	−4.42512	−0.375334	−0.187667	+	0.982233i	$0.560093\pi$
$140$	0	0
$141$	6.74671	0.568176
$142$	0	0
$143$	3.71883	0.310984
$144$	0	0
$145$	−18.2885	−1.51878
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	−3.69530	−0.302731	−0.151365	−	0.988478i	$-0.548367\pi$
$149$	−3.69530	−0.302731	−0.151365	+	0.988478i	$0.548367\pi$
$150$	0	0
$151$	−8.63174	−0.702441	−0.351221	−	0.936293i	$-0.614233\pi$
$151$	−8.63174	−0.702441	−0.351221	+	0.936293i	$0.614233\pi$
$152$	0	0
$153$	0.902098	0.0729303
$154$	0	0
$155$	3.00374	0.241266
$156$	0	0
$157$	9.93438	0.792850	0.396425	−	0.918067i	$-0.370251\pi$
$157$	9.93438	0.792850	0.396425	+	0.918067i	$0.370251\pi$
$158$	0	0
$159$	−4.91648	−0.389903
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	−14.7082	−1.15203	−0.576017	−	0.817438i	$-0.695394\pi$
$163$	−14.7082	−1.15203	−0.576017	+	0.817438i	$0.695394\pi$
$164$	0	0
$165$	−6.85086	−0.533339
$166$	0	0
$167$	5.47430	0.423614	0.211807	−	0.977312i	$-0.432065\pi$
$167$	5.47430	0.423614	0.211807	+	0.977312i	$0.432065\pi$
$168$	0	0
$169$	−10.7944	−0.830341
$170$	0	0
$171$	−2.50407	−0.191491
$172$	0	0
$173$	18.1727	1.38165	0.690823	−	0.723024i	$-0.257248\pi$
$173$	18.1727	1.38165	0.690823	+	0.723024i	$0.257248\pi$
$174$	0	0
$175$	2.48511	0.187857
$176$	0	0
$177$	7.32973	0.550936
$178$	0	0
$179$	−10.1860	−0.761341	−0.380670	−	0.924711i	$-0.624307\pi$
$179$	−10.1860	−0.761341	−0.380670	+	0.924711i	$0.624307\pi$
$180$	0	0
$181$	13.5186	1.00483	0.502416	−	0.864626i	$-0.332445\pi$
$181$	13.5186	1.00483	0.502416	+	0.864626i	$0.332445\pi$
$182$	0	0
$183$	−1.00625	−0.0743841
$184$	0	0
$185$	−27.1133	−1.99341
$186$	0	0
$187$	2.25892	0.165188
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	0.631742	0.0457113	0.0228556	−	0.999739i	$-0.492724\pi$
$191$	0.631742	0.0457113	0.0228556	+	0.999739i	$0.492724\pi$
$192$	0	0
$193$	1.36826	0.0984893	0.0492447	−	0.998787i	$-0.484319\pi$
$193$	1.36826	0.0984893	0.0492447	+	0.998787i	$0.484319\pi$
$194$	0	0
$195$	−4.06311	−0.290966
$196$	0	0
$197$	24.7611	1.76416	0.882078	−	0.471104i	$-0.156144\pi$
$197$	24.7611	1.76416	0.882078	+	0.471104i	$0.156144\pi$
$198$	0	0
$199$	−1.44470	−0.102412	−0.0512059	−	0.998688i	$-0.516306\pi$
$199$	−1.44470	−0.102412	−0.0512059	+	0.998688i	$0.516306\pi$
$200$	0	0
$201$	−11.2929	−0.796538
$202$	0	0
$203$	6.68466	0.469171
$204$	0	0
$205$	6.31517	0.441070
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	−6.27036	−0.433730
$210$	0	0
$211$	23.0990	1.59020	0.795099	−	0.606480i	$-0.207419\pi$
$211$	23.0990	1.59020	0.795099	+	0.606480i	$0.207419\pi$
$212$	0	0
$213$	0.362008	0.0248044
$214$	0	0
$215$	15.9606	1.08851
$216$	0	0
$217$	−1.09790	−0.0745304
$218$	0	0
$219$	−5.34411	−0.361122
$220$	0	0
$221$	1.33972	0.0901192
$222$	0	0
$223$	18.6497	1.24888	0.624438	−	0.781074i	$-0.285328\pi$
$223$	18.6497	1.24888	0.624438	+	0.781074i	$0.285328\pi$
$224$	0	0
$225$	2.48511	0.165674
$226$	0	0
$227$	−20.6334	−1.36949	−0.684744	−	0.728783i	$-0.740086\pi$
$227$	−20.6334	−1.36949	−0.684744	+	0.728783i	$0.740086\pi$
$228$	0	0
$229$	14.9544	0.988214	0.494107	−	0.869401i	$-0.335495\pi$
$229$	14.9544	0.988214	0.494107	+	0.869401i	$0.335495\pi$
$230$	0	0
$231$	2.50407	0.164756
$232$	0	0
$233$	28.1052	1.84123	0.920617	−	0.390467i	$-0.127687\pi$
$233$	28.1052	1.84123	0.920617	+	0.390467i	$0.127687\pi$
$234$	0	0
$235$	−18.4583	−1.20409
$236$	0	0
$237$	−10.4672	−0.679919
$238$	0	0
$239$	−25.8536	−1.67233	−0.836166	−	0.548476i	$-0.815208\pi$
$239$	−25.8536	−1.67233	−0.836166	+	0.548476i	$0.815208\pi$
$240$	0	0
$241$	−14.8311	−0.955356	−0.477678	−	0.878535i	$-0.658521\pi$
$241$	−14.8311	−0.955356	−0.477678	+	0.878535i	$0.658521\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−2.73589	−0.174790
$246$	0	0
$247$	−3.71883	−0.236623
$248$	0	0
$249$	−7.59946	−0.481596
$250$	0	0
$251$	16.2841	1.02784	0.513922	−	0.857837i	$-0.328192\pi$
$251$	16.2841	1.02784	0.513922	+	0.857837i	$0.328192\pi$
$252$	0	0
$253$	−2.50407	−0.157429
$254$	0	0
$255$	−2.46805	−0.154555
$256$	0	0
$257$	22.9446	1.43125	0.715623	−	0.698486i	$-0.246143\pi$
$257$	22.9446	1.43125	0.715623	+	0.698486i	$0.246143\pi$
$258$	0	0
$259$	9.91023	0.615792
$260$	0	0
$261$	6.68466	0.413770
$262$	0	0
$263$	0.776932	0.0479077	0.0239538	−	0.999713i	$-0.492375\pi$
$263$	0.776932	0.0479077	0.0239538	+	0.999713i	$0.492375\pi$
$264$	0	0
$265$	13.4510	0.826287
$266$	0	0
$267$	10.6245	0.650208
$268$	0	0
$269$	3.05937	0.186533	0.0932666	−	0.995641i	$-0.470269\pi$
$269$	3.05937	0.186533	0.0932666	+	0.995641i	$0.470269\pi$
$270$	0	0
$271$	16.0631	0.975765	0.487882	−	0.872909i	$-0.337770\pi$
$271$	16.0631	0.975765	0.487882	+	0.872909i	$0.337770\pi$
$272$	0	0
$273$	1.48511	0.0898832
$274$	0	0
$275$	6.22289	0.375255
$276$	0	0
$277$	−11.6103	−0.697594	−0.348797	−	0.937198i	$-0.613410\pi$
$277$	−11.6103	−0.697594	−0.348797	+	0.937198i	$0.613410\pi$
$278$	0	0
$279$	−1.09790	−0.0657296
$280$	0	0
$281$	9.89753	0.590437	0.295219	−	0.955430i	$-0.404608\pi$
$281$	9.89753	0.590437	0.295219	+	0.955430i	$0.404608\pi$
$282$	0	0
$283$	−13.9994	−0.832177	−0.416088	−	0.909324i	$-0.636599\pi$
$283$	−13.9994	−0.832177	−0.416088	+	0.909324i	$0.636599\pi$
$284$	0	0
$285$	6.85086	0.405810
$286$	0	0
$287$	−2.30826	−0.136253
$288$	0	0
$289$	−16.1862	−0.952130
$290$	0	0
$291$	13.9102	0.815432
$292$	0	0
$293$	−8.14038	−0.475566	−0.237783	−	0.971318i	$-0.576421\pi$
$293$	−8.14038	−0.475566	−0.237783	+	0.971318i	$0.576421\pi$
$294$	0	0
$295$	−20.0534	−1.16755
$296$	0	0
$297$	2.50407	0.145301
$298$	0	0
$299$	−1.48511	−0.0858863
$300$	0	0
$301$	−5.83380	−0.336254
$302$	0	0
$303$	1.81502	0.104270
$304$	0	0
$305$	2.75299	0.157636
$306$	0	0
$307$	−13.5095	−0.771027	−0.385514	−	0.922702i	$-0.625976\pi$
$307$	−13.5095	−0.771027	−0.385514	+	0.922702i	$0.625976\pi$
$308$	0	0
$309$	16.6605	0.947783
$310$	0	0
$311$	24.3035	1.37813	0.689063	−	0.724701i	$-0.258022\pi$
$311$	24.3035	1.37813	0.689063	+	0.724701i	$0.258022\pi$
$312$	0	0
$313$	−33.7836	−1.90956	−0.954782	−	0.297308i	$-0.903911\pi$
$313$	−33.7836	−1.90956	−0.954782	+	0.297308i	$0.903911\pi$
$314$	0	0
$315$	−2.73589	−0.154150
$316$	0	0
$317$	32.1594	1.80625	0.903126	−	0.429376i	$-0.141267\pi$
$317$	32.1594	1.80625	0.903126	+	0.429376i	$0.141267\pi$
$318$	0	0
$319$	16.7388	0.937195
$320$	0	0
$321$	−1.68092	−0.0938196
$322$	0	0
$323$	−2.25892	−0.125689
$324$	0	0
$325$	3.69068	0.204722
$326$	0	0
$327$	11.4333	0.632261
$328$	0	0
$329$	6.74671	0.371958
$330$	0	0
$331$	4.84004	0.266033	0.133016	−	0.991114i	$-0.457534\pi$
$331$	4.84004	0.266033	0.133016	+	0.991114i	$0.457534\pi$
$332$	0	0
$333$	9.91023	0.543077
$334$	0	0
$335$	30.8961	1.68803
$336$	0	0
$337$	−2.59195	−0.141192	−0.0705962	−	0.997505i	$-0.522490\pi$
$337$	−2.59195	−0.141192	−0.0705962	+	0.997505i	$0.522490\pi$
$338$	0	0
$339$	3.14914	0.171038
$340$	0	0
$341$	−2.74922	−0.148879
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	2.73589	0.147296
$346$	0	0
$347$	15.9390	0.855651	0.427825	−	0.903861i	$-0.359280\pi$
$347$	15.9390	0.855651	0.427825	+	0.903861i	$0.359280\pi$
$348$	0	0
$349$	−8.02603	−0.429624	−0.214812	−	0.976655i	$-0.568914\pi$
$349$	−8.02603	−0.429624	−0.214812	+	0.976655i	$0.568914\pi$
$350$	0	0
$351$	1.48511	0.0792695
$352$	0	0
$353$	10.7871	0.574141	0.287070	−	0.957909i	$-0.407319\pi$
$353$	10.7871	0.574141	0.287070	+	0.957909i	$0.407319\pi$
$354$	0	0
$355$	−0.990415	−0.0525658
$356$	0	0
$357$	0.902098	0.0477441
$358$	0	0
$359$	2.17891	0.114998	0.0574992	−	0.998346i	$-0.481687\pi$
$359$	2.17891	0.114998	0.0574992	+	0.998346i	$0.481687\pi$
$360$	0	0
$361$	−12.7296	−0.669981
$362$	0	0
$363$	−4.72964	−0.248242
$364$	0	0
$365$	14.6209	0.765294
$366$	0	0
$367$	−27.0388	−1.41141	−0.705707	−	0.708504i	$-0.749370\pi$
$367$	−27.0388	−1.41141	−0.705707	+	0.708504i	$0.749370\pi$
$368$	0	0
$369$	−2.30826	−0.120163
$370$	0	0
$371$	−4.91648	−0.255251
$372$	0	0
$373$	16.4189	0.850140	0.425070	−	0.905161i	$-0.360249\pi$
$373$	16.4189	0.850140	0.425070	+	0.905161i	$0.360249\pi$
$374$	0	0
$375$	6.88046	0.355306
$376$	0	0
$377$	9.92748	0.511291
$378$	0	0
$379$	34.6399	1.77933	0.889667	−	0.456610i	$-0.150936\pi$
$379$	34.6399	1.77933	0.889667	+	0.456610i	$0.150936\pi$
$380$	0	0
$381$	−0.449266	−0.0230166
$382$	0	0
$383$	10.6722	0.545322	0.272661	−	0.962110i	$-0.412096\pi$
$383$	10.6722	0.545322	0.272661	+	0.962110i	$0.412096\pi$
$384$	0	0
$385$	−6.85086	−0.349152
$386$	0	0
$387$	−5.83380	−0.296548
$388$	0	0
$389$	0.0302201	0.00153222	0.000766110	−	1.00000i	$-0.499756\pi$
$389$	0.0302201	0.00153222	0.000766110		1.00000i	$0.499756\pi$
$390$	0	0
$391$	−0.902098	−0.0456211
$392$	0	0
$393$	19.9759	1.00765
$394$	0	0
$395$	28.6372	1.44089
$396$	0	0
$397$	20.4749	1.02761	0.513803	−	0.857908i	$-0.328236\pi$
$397$	20.4749	1.02761	0.513803	+	0.857908i	$0.328236\pi$
$398$	0	0
$399$	−2.50407	−0.125360
$400$	0	0
$401$	−36.4304	−1.81925	−0.909623	−	0.415435i	$-0.863629\pi$
$401$	−36.4304	−1.81925	−0.909623	+	0.415435i	$0.863629\pi$
$402$	0	0
$403$	−1.63051	−0.0812214
$404$	0	0
$405$	−2.73589	−0.135948
$406$	0	0
$407$	24.8159	1.23008
$408$	0	0
$409$	37.4054	1.84958	0.924790	−	0.380479i	$-0.124241\pi$
$409$	37.4054	1.84958	0.924790	+	0.380479i	$0.124241\pi$
$410$	0	0
$411$	0.270356	0.0133357
$412$	0	0
$413$	7.32973	0.360672
$414$	0	0
$415$	20.7913	1.02061
$416$	0	0
$417$	−4.42512	−0.216699
$418$	0	0
$419$	37.2017	1.81742	0.908710	−	0.417428i	$-0.137068\pi$
$419$	37.2017	1.81742	0.908710	+	0.417428i	$0.137068\pi$
$420$	0	0
$421$	27.6326	1.34673	0.673366	−	0.739309i	$-0.264848\pi$
$421$	27.6326	1.34673	0.673366	+	0.739309i	$0.264848\pi$
$422$	0	0
$423$	6.74671	0.328036
$424$	0	0
$425$	2.24182	0.108744
$426$	0	0
$427$	−1.00625	−0.0486958
$428$	0	0
$429$	3.71883	0.179547
$430$	0	0
$431$	−35.0676	−1.68915	−0.844573	−	0.535441i	$-0.820145\pi$
$431$	−35.0676	−1.68915	−0.844573	+	0.535441i	$0.820145\pi$
$432$	0	0
$433$	−1.00708	−0.0483970	−0.0241985	−	0.999707i	$-0.507703\pi$
$433$	−1.00708	−0.0483970	−0.0241985	+	0.999707i	$0.507703\pi$
$434$	0	0
$435$	−18.2885	−0.876867
$436$	0	0
$437$	2.50407	0.119786
$438$	0	0
$439$	4.04649	0.193129	0.0965643	−	0.995327i	$-0.469215\pi$
$439$	4.04649	0.193129	0.0965643	+	0.995327i	$0.469215\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	29.3947	1.39659	0.698293	−	0.715812i	$-0.253943\pi$
$443$	29.3947	1.39659	0.698293	+	0.715812i	$0.253943\pi$
$444$	0	0
$445$	−29.0675	−1.37793
$446$	0	0
$447$	−3.69530	−0.174782
$448$	0	0
$449$	−4.87627	−0.230126	−0.115063	−	0.993358i	$-0.536707\pi$
$449$	−4.87627	−0.230126	−0.115063	+	0.993358i	$0.536707\pi$
$450$	0	0
$451$	−5.78005	−0.272172
$452$	0	0
$453$	−8.63174	−0.405555
$454$	0	0
$455$	−4.06311	−0.190482
$456$	0	0
$457$	−35.5212	−1.66161	−0.830806	−	0.556562i	$-0.812120\pi$
$457$	−35.5212	−1.66161	−0.830806	+	0.556562i	$0.812120\pi$
$458$	0	0
$459$	0.902098	0.0421064
$460$	0	0
$461$	−18.9442	−0.882319	−0.441160	−	0.897429i	$-0.645433\pi$
$461$	−18.9442	−0.882319	−0.441160	+	0.897429i	$0.645433\pi$
$462$	0	0
$463$	−14.1933	−0.659618	−0.329809	−	0.944048i	$-0.606984\pi$
$463$	−14.1933	−0.659618	−0.329809	+	0.944048i	$0.606984\pi$
$464$	0	0
$465$	3.00374	0.139295
$466$	0	0
$467$	31.0355	1.43615	0.718075	−	0.695966i	$-0.245024\pi$
$467$	31.0355	1.43615	0.718075	+	0.695966i	$0.245024\pi$
$468$	0	0
$469$	−11.2929	−0.521457
$470$	0	0
$471$	9.93438	0.457752
$472$	0	0
$473$	−14.6082	−0.671687
$474$	0	0
$475$	−6.22289	−0.285526
$476$	0	0
$477$	−4.91648	−0.225110
$478$	0	0
$479$	−10.5214	−0.480735	−0.240367	−	0.970682i	$-0.577268\pi$
$479$	−10.5214	−0.480735	−0.240367	+	0.970682i	$0.577268\pi$
$480$	0	0
$481$	14.7178	0.671075
$482$	0	0
$483$	−1.00000	−0.0455016
$484$	0	0
$485$	−38.0569	−1.72808
$486$	0	0
$487$	31.7378	1.43818	0.719089	−	0.694918i	$-0.244559\pi$
$487$	31.7378	1.43818	0.719089	+	0.694918i	$0.244559\pi$
$488$	0	0
$489$	−14.7082	−0.665127
$490$	0	0
$491$	31.8788	1.43867	0.719336	−	0.694662i	$-0.244446\pi$
$491$	31.8788	1.43867	0.719336	+	0.694662i	$0.244446\pi$
$492$	0	0
$493$	6.03022	0.271587
$494$	0	0
$495$	−6.85086	−0.307923
$496$	0	0
$497$	0.362008	0.0162383
$498$	0	0
$499$	39.4279	1.76503	0.882517	−	0.470280i	$-0.155847\pi$
$499$	39.4279	1.76503	0.882517	+	0.470280i	$0.155847\pi$
$500$	0	0
$501$	5.47430	0.244573
$502$	0	0
$503$	−24.4589	−1.09057	−0.545284	−	0.838251i	$-0.683578\pi$
$503$	−24.4589	−1.09057	−0.545284	+	0.838251i	$0.683578\pi$
$504$	0	0
$505$	−4.96569	−0.220970
$506$	0	0
$507$	−10.7944	−0.479398
$508$	0	0
$509$	20.3927	0.903889	0.451944	−	0.892046i	$-0.350731\pi$
$509$	20.3927	0.903889	0.451944	+	0.892046i	$0.350731\pi$
$510$	0	0
$511$	−5.34411	−0.236410
$512$	0	0
$513$	−2.50407	−0.110557
$514$	0	0
$515$	−45.5814	−2.00856
$516$	0	0
$517$	16.8942	0.743007
$518$	0	0
$519$	18.1727	0.797694
$520$	0	0
$521$	26.3058	1.15248	0.576238	−	0.817282i	$-0.304520\pi$
$521$	26.3058	1.15248	0.576238	+	0.817282i	$0.304520\pi$
$522$	0	0
$523$	40.4256	1.76769	0.883843	−	0.467783i	$-0.154947\pi$
$523$	40.4256	1.76769	0.883843	+	0.467783i	$0.154947\pi$
$524$	0	0
$525$	2.48511	0.108459
$526$	0	0
$527$	−0.990415	−0.0431432
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	7.32973	0.318083
$532$	0	0
$533$	−3.42804	−0.148485
$534$	0	0
$535$	4.59881	0.198824
$536$	0	0
$537$	−10.1860	−0.439560
$538$	0	0
$539$	2.50407	0.107858
$540$	0	0
$541$	0.360122	0.0154828	0.00774142	−	0.999970i	$-0.497536\pi$
$541$	0.360122	0.0154828	0.00774142	+	0.999970i	$0.497536\pi$
$542$	0	0
$543$	13.5186	0.580140
$544$	0	0
$545$	−31.2802	−1.33990
$546$	0	0
$547$	−20.2023	−0.863789	−0.431894	−	0.901924i	$-0.642155\pi$
$547$	−20.2023	−0.863789	−0.431894	+	0.901924i	$0.642155\pi$
$548$	0	0
$549$	−1.00625	−0.0429457
$550$	0	0
$551$	−16.7388	−0.713099
$552$	0	0
$553$	−10.4672	−0.445111
$554$	0	0
$555$	−27.1133	−1.15090
$556$	0	0
$557$	9.71086	0.411463	0.205731	−	0.978609i	$-0.434043\pi$
$557$	9.71086	0.411463	0.205731	+	0.978609i	$0.434043\pi$
$558$	0	0
$559$	−8.66385	−0.366442
$560$	0	0
$561$	2.25892	0.0953715
$562$	0	0
$563$	−27.2789	−1.14967	−0.574835	−	0.818269i	$-0.694934\pi$
$563$	−27.2789	−1.14967	−0.574835	+	0.818269i	$0.694934\pi$
$564$	0	0
$565$	−8.61570	−0.362465
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	19.6132	0.822229	0.411115	−	0.911584i	$-0.365140\pi$
$569$	19.6132	0.822229	0.411115	+	0.911584i	$0.365140\pi$
$570$	0	0
$571$	−21.1859	−0.886601	−0.443301	−	0.896373i	$-0.646193\pi$
$571$	−21.1859	−0.886601	−0.443301	+	0.896373i	$0.646193\pi$
$572$	0	0
$573$	0.631742	0.0263914
$574$	0	0
$575$	−2.48511	−0.103636
$576$	0	0
$577$	−12.4079	−0.516547	−0.258273	−	0.966072i	$-0.583154\pi$
$577$	−12.4079	−0.516547	−0.258273	+	0.966072i	$0.583154\pi$
$578$	0	0
$579$	1.36826	0.0568629
$580$	0	0
$581$	−7.59946	−0.315279
$582$	0	0
$583$	−12.3112	−0.509878
$584$	0	0
$585$	−4.06311	−0.167989
$586$	0	0
$587$	9.35493	0.386119	0.193060	−	0.981187i	$-0.438159\pi$
$587$	9.35493	0.386119	0.193060	+	0.981187i	$0.438159\pi$
$588$	0	0
$589$	2.74922	0.113280
$590$	0	0
$591$	24.7611	1.01854
$592$	0	0
$593$	−16.7836	−0.689218	−0.344609	−	0.938746i	$-0.611989\pi$
$593$	−16.7836	−0.689218	−0.344609	+	0.938746i	$0.611989\pi$
$594$	0	0
$595$	−2.46805	−0.101180
$596$	0	0
$597$	−1.44470	−0.0591275
$598$	0	0
$599$	16.0512	0.655836	0.327918	−	0.944706i	$-0.393653\pi$
$599$	16.0512	0.655836	0.327918	+	0.944706i	$0.393653\pi$
$600$	0	0
$601$	−24.4364	−0.996781	−0.498390	−	0.866953i	$-0.666075\pi$
$601$	−24.4364	−0.996781	−0.498390	+	0.866953i	$0.666075\pi$
$602$	0	0
$603$	−11.2929	−0.459882
$604$	0	0
$605$	12.9398	0.526078
$606$	0	0
$607$	−28.2329	−1.14594	−0.572969	−	0.819577i	$-0.694208\pi$
$607$	−28.2329	−1.14594	−0.572969	+	0.819577i	$0.694208\pi$
$608$	0	0
$609$	6.68466	0.270876
$610$	0	0
$611$	10.0196	0.405351
$612$	0	0
$613$	17.9873	0.726500	0.363250	−	0.931692i	$-0.381667\pi$
$613$	17.9873	0.726500	0.363250	+	0.931692i	$0.381667\pi$
$614$	0	0
$615$	6.31517	0.254652
$616$	0	0
$617$	12.9998	0.523353	0.261677	−	0.965156i	$-0.415725\pi$
$617$	12.9998	0.523353	0.261677	+	0.965156i	$0.415725\pi$
$618$	0	0
$619$	1.26266	0.0507505	0.0253752	−	0.999678i	$-0.491922\pi$
$619$	1.26266	0.0507505	0.0253752	+	0.999678i	$0.491922\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	10.6245	0.425661
$624$	0	0
$625$	−31.2498	−1.24999
$626$	0	0
$627$	−6.27036	−0.250414
$628$	0	0
$629$	8.94001	0.356461
$630$	0	0
$631$	24.4568	0.973611	0.486806	−	0.873510i	$-0.338162\pi$
$631$	24.4568	0.973611	0.486806	+	0.873510i	$0.338162\pi$
$632$	0	0
$633$	23.0990	0.918101
$634$	0	0
$635$	1.22914	0.0487771
$636$	0	0
$637$	1.48511	0.0588423
$638$	0	0
$639$	0.362008	0.0143208
$640$	0	0
$641$	−1.29037	−0.0509665	−0.0254833	−	0.999675i	$-0.508112\pi$
$641$	−1.29037	−0.0509665	−0.0254833	+	0.999675i	$0.508112\pi$
$642$	0	0
$643$	−34.3110	−1.35309	−0.676547	−	0.736399i	$-0.736524\pi$
$643$	−34.3110	−1.35309	−0.676547	+	0.736399i	$0.736524\pi$
$644$	0	0
$645$	15.9606	0.628450
$646$	0	0
$647$	2.62092	0.103039	0.0515196	−	0.998672i	$-0.483594\pi$
$647$	2.62092	0.103039	0.0515196	+	0.998672i	$0.483594\pi$
$648$	0	0
$649$	18.3541	0.720463
$650$	0	0
$651$	−1.09790	−0.0430302
$652$	0	0
$653$	7.53697	0.294944	0.147472	−	0.989066i	$-0.452886\pi$
$653$	7.53697	0.294944	0.147472	+	0.989066i	$0.452886\pi$
$654$	0	0
$655$	−54.6518	−2.13542
$656$	0	0
$657$	−5.34411	−0.208494
$658$	0	0
$659$	−22.8213	−0.888990	−0.444495	−	0.895781i	$-0.646617\pi$
$659$	−22.8213	−0.888990	−0.444495	+	0.895781i	$0.646617\pi$
$660$	0	0
$661$	27.3668	1.06445	0.532223	−	0.846604i	$-0.321357\pi$
$661$	27.3668	1.06445	0.532223	+	0.846604i	$0.321357\pi$
$662$	0	0
$663$	1.33972	0.0520304
$664$	0	0
$665$	6.85086	0.265665
$666$	0	0
$667$	−6.68466	−0.258831
$668$	0	0
$669$	18.6497	0.721039
$670$	0	0
$671$	−2.51972	−0.0972726
$672$	0	0
$673$	22.9790	0.885774	0.442887	−	0.896577i	$-0.353954\pi$
$673$	22.9790	0.885774	0.442887	+	0.896577i	$0.353954\pi$
$674$	0	0
$675$	2.48511	0.0956521
$676$	0	0
$677$	−8.80814	−0.338524	−0.169262	−	0.985571i	$-0.554138\pi$
$677$	−8.80814	−0.338524	−0.169262	+	0.985571i	$0.554138\pi$
$678$	0	0
$679$	13.9102	0.533826
$680$	0	0
$681$	−20.6334	−0.790674
$682$	0	0
$683$	−34.4889	−1.31968	−0.659840	−	0.751406i	$-0.729376\pi$
$683$	−34.4889	−1.31968	−0.659840	+	0.751406i	$0.729376\pi$
$684$	0	0
$685$	−0.739666	−0.0282612
$686$	0	0
$687$	14.9544	0.570546
$688$	0	0
$689$	−7.30154	−0.278166
$690$	0	0
$691$	−37.6110	−1.43079	−0.715395	−	0.698721i	$-0.753753\pi$
$691$	−37.6110	−1.43079	−0.715395	+	0.698721i	$0.753753\pi$
$692$	0	0
$693$	2.50407	0.0951217
$694$	0	0
$695$	12.1067	0.459232
$696$	0	0
$697$	−2.08228	−0.0788721
$698$	0	0
$699$	28.1052	1.06304
$700$	0	0
$701$	−47.5348	−1.79536	−0.897682	−	0.440644i	$-0.854750\pi$
$701$	−47.5348	−1.79536	−0.897682	+	0.440644i	$0.854750\pi$
$702$	0	0
$703$	−24.8159	−0.935949
$704$	0	0
$705$	−18.4583	−0.695179
$706$	0	0
$707$	1.81502	0.0682607
$708$	0	0
$709$	−41.1332	−1.54479	−0.772395	−	0.635143i	$-0.780941\pi$
$709$	−41.1332	−1.54479	−0.772395	+	0.635143i	$0.780941\pi$
$710$	0	0
$711$	−10.4672	−0.392551
$712$	0	0
$713$	1.09790	0.0411167
$714$	0	0
$715$	−10.1743	−0.380498
$716$	0	0
$717$	−25.8536	−0.965522
$718$	0	0
$719$	−8.25857	−0.307993	−0.153996	−	0.988071i	$-0.549214\pi$
$719$	−8.25857	−0.307993	−0.153996	+	0.988071i	$0.549214\pi$
$720$	0	0
$721$	16.6605	0.620470
$722$	0	0
$723$	−14.8311	−0.551575
$724$	0	0
$725$	16.6121	0.616959
$726$	0	0
$727$	−0.985235	−0.0365403	−0.0182702	−	0.999833i	$-0.505816\pi$
$727$	−0.985235	−0.0365403	−0.0182702	+	0.999833i	$0.505816\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−5.26266	−0.194646
$732$	0	0
$733$	27.7847	1.02625	0.513125	−	0.858314i	$-0.328488\pi$
$733$	27.7847	1.02625	0.513125	+	0.858314i	$0.328488\pi$
$734$	0	0
$735$	−2.73589	−0.100915
$736$	0	0
$737$	−28.2781	−1.04164
$738$	0	0
$739$	−30.0540	−1.10555	−0.552777	−	0.833329i	$-0.686432\pi$
$739$	−30.0540	−1.10555	−0.552777	+	0.833329i	$0.686432\pi$
$740$	0	0
$741$	−3.71883	−0.136614
$742$	0	0
$743$	−40.1617	−1.47339	−0.736695	−	0.676225i	$-0.763615\pi$
$743$	−40.1617	−1.47339	−0.736695	+	0.676225i	$0.763615\pi$
$744$	0	0
$745$	10.1100	0.370400
$746$	0	0
$747$	−7.59946	−0.278050
$748$	0	0
$749$	−1.68092	−0.0614194
$750$	0	0
$751$	2.15350	0.0785823	0.0392912	−	0.999228i	$-0.487490\pi$
$751$	2.15350	0.0785823	0.0392912	+	0.999228i	$0.487490\pi$
$752$	0	0
$753$	16.2841	0.593426
$754$	0	0
$755$	23.6155	0.859457
$756$	0	0
$757$	−32.0577	−1.16516	−0.582579	−	0.812774i	$-0.697956\pi$
$757$	−32.0577	−1.16516	−0.582579	+	0.812774i	$0.697956\pi$
$758$	0	0
$759$	−2.50407	−0.0908919
$760$	0	0
$761$	0.669651	0.0242748	0.0121374	−	0.999926i	$-0.496136\pi$
$761$	0.669651	0.0242748	0.0121374	+	0.999926i	$0.496136\pi$
$762$	0	0
$763$	11.4333	0.413912
$764$	0	0
$765$	−2.46805	−0.0892324
$766$	0	0
$767$	10.8855	0.393052
$768$	0	0
$769$	43.5115	1.56906	0.784532	−	0.620089i	$-0.212903\pi$
$769$	43.5115	1.56906	0.784532	+	0.620089i	$0.212903\pi$
$770$	0	0
$771$	22.9446	0.826331
$772$	0	0
$773$	21.0046	0.755482	0.377741	−	0.925911i	$-0.376701\pi$
$773$	21.0046	0.755482	0.377741	+	0.925911i	$0.376701\pi$
$774$	0	0
$775$	−2.72841	−0.0980074
$776$	0	0
$777$	9.91023	0.355528
$778$	0	0
$779$	5.78005	0.207092
$780$	0	0
$781$	0.906493	0.0324369
$782$	0	0
$783$	6.68466	0.238890
$784$	0	0
$785$	−27.1794	−0.970075
$786$	0	0
$787$	−22.1975	−0.791255	−0.395627	−	0.918411i	$-0.629473\pi$
$787$	−22.1975	−0.791255	−0.395627	+	0.918411i	$0.629473\pi$
$788$	0	0
$789$	0.776932	0.0276595
$790$	0	0
$791$	3.14914	0.111970
$792$	0	0
$793$	−1.49440	−0.0530675
$794$	0	0
$795$	13.4510	0.477057
$796$	0	0
$797$	22.8502	0.809397	0.404699	−	0.914450i	$-0.367376\pi$
$797$	22.8502	0.809397	0.404699	+	0.914450i	$0.367376\pi$
$798$	0	0
$799$	6.08620	0.215314
$800$	0	0
$801$	10.6245	0.375398
$802$	0	0
$803$	−13.3820	−0.472241
$804$	0	0
$805$	2.73589	0.0964276
$806$	0	0
$807$	3.05937	0.107695
$808$	0	0
$809$	−38.0621	−1.33819	−0.669097	−	0.743175i	$-0.733319\pi$
$809$	−38.0621	−1.33819	−0.669097	+	0.743175i	$0.733319\pi$
$810$	0	0
$811$	−21.0208	−0.738142	−0.369071	−	0.929401i	$-0.620324\pi$
$811$	−21.0208	−0.738142	−0.369071	+	0.929401i	$0.620324\pi$
$812$	0	0
$813$	16.0631	0.563358
$814$	0	0
$815$	40.2400	1.40955
$816$	0	0
$817$	14.6082	0.511077
$818$	0	0
$819$	1.48511	0.0518941
$820$	0	0
$821$	−10.4132	−0.363425	−0.181712	−	0.983352i	$-0.558164\pi$
$821$	−10.4132	−0.363425	−0.181712	+	0.983352i	$0.558164\pi$
$822$	0	0
$823$	−30.0396	−1.04711	−0.523557	−	0.851991i	$-0.675395\pi$
$823$	−30.0396	−1.04711	−0.523557	+	0.851991i	$0.675395\pi$
$824$	0	0
$825$	6.22289	0.216653
$826$	0	0
$827$	−28.6978	−0.997920	−0.498960	−	0.866625i	$-0.666285\pi$
$827$	−28.6978	−0.997920	−0.498960	+	0.866625i	$0.666285\pi$
$828$	0	0
$829$	16.4384	0.570931	0.285465	−	0.958389i	$-0.407852\pi$
$829$	16.4384	0.570931	0.285465	+	0.958389i	$0.407852\pi$
$830$	0	0
$831$	−11.6103	−0.402756
$832$	0	0
$833$	0.902098	0.0312559
$834$	0	0
$835$	−14.9771	−0.518304
$836$	0	0
$837$	−1.09790	−0.0379490
$838$	0	0
$839$	12.5437	0.433055	0.216528	−	0.976277i	$-0.430527\pi$
$839$	12.5437	0.433055	0.216528	+	0.976277i	$0.430527\pi$
$840$	0	0
$841$	15.6847	0.540850
$842$	0	0
$843$	9.89753	0.340889
$844$	0	0
$845$	29.5324	1.01595
$846$	0	0
$847$	−4.72964	−0.162512
$848$	0	0
$849$	−13.9994	−0.480457
$850$	0	0
$851$	−9.91023	−0.339718
$852$	0	0
$853$	−22.0613	−0.755365	−0.377683	−	0.925935i	$-0.623279\pi$
$853$	−22.0613	−0.755365	−0.377683	+	0.925935i	$0.623279\pi$
$854$	0	0
$855$	6.85086	0.234295
$856$	0	0
$857$	6.15359	0.210203	0.105101	−	0.994462i	$-0.466483\pi$
$857$	6.15359	0.210203	0.105101	+	0.994462i	$0.466483\pi$
$858$	0	0
$859$	−23.0698	−0.787131	−0.393566	−	0.919296i	$-0.628759\pi$
$859$	−23.0698	−0.787131	−0.393566	+	0.919296i	$0.628759\pi$
$860$	0	0
$861$	−2.30826	−0.0786655
$862$	0	0
$863$	5.22145	0.177740	0.0888702	−	0.996043i	$-0.471674\pi$
$863$	5.22145	0.177740	0.0888702	+	0.996043i	$0.471674\pi$
$864$	0	0
$865$	−49.7186	−1.69048
$866$	0	0
$867$	−16.1862	−0.549713
$868$	0	0
$869$	−26.2106	−0.889135
$870$	0	0
$871$	−16.7712	−0.568271
$872$	0	0
$873$	13.9102	0.470790
$874$	0	0
$875$	6.88046	0.232602
$876$	0	0
$877$	49.8716	1.68404	0.842022	−	0.539443i	$-0.181365\pi$
$877$	49.8716	1.68404	0.842022	+	0.539443i	$0.181365\pi$
$878$	0	0
$879$	−8.14038	−0.274568
$880$	0	0
$881$	−20.3054	−0.684107	−0.342053	−	0.939681i	$-0.611122\pi$
$881$	−20.3054	−0.684107	−0.342053	+	0.939681i	$0.611122\pi$
$882$	0	0
$883$	50.3500	1.69441	0.847206	−	0.531265i	$-0.178283\pi$
$883$	50.3500	1.69441	0.847206	+	0.531265i	$0.178283\pi$
$884$	0	0
$885$	−20.0534	−0.674086
$886$	0	0
$887$	−47.2694	−1.58715	−0.793576	−	0.608471i	$-0.791783\pi$
$887$	−47.2694	−1.58715	−0.793576	+	0.608471i	$0.791783\pi$
$888$	0	0
$889$	−0.449266	−0.0150679
$890$	0	0
$891$	2.50407	0.0838894
$892$	0	0
$893$	−16.8942	−0.565344
$894$	0	0
$895$	27.8679	0.931522
$896$	0	0
$897$	−1.48511	−0.0495865
$898$	0	0
$899$	−7.33910	−0.244773
$900$	0	0
$901$	−4.43515	−0.147756
$902$	0	0
$903$	−5.83380	−0.194137
$904$	0	0
$905$	−36.9855	−1.22944
$906$	0	0
$907$	5.65035	0.187617	0.0938084	−	0.995590i	$-0.470096\pi$
$907$	5.65035	0.187617	0.0938084	+	0.995590i	$0.470096\pi$
$908$	0	0
$909$	1.81502	0.0602003
$910$	0	0
$911$	−18.2196	−0.603641	−0.301820	−	0.953365i	$-0.597594\pi$
$911$	−18.2196	−0.603641	−0.301820	+	0.953365i	$0.597594\pi$
$912$	0	0
$913$	−19.0296	−0.629787
$914$	0	0
$915$	2.75299	0.0910111
$916$	0	0
$917$	19.9759	0.659661
$918$	0	0
$919$	−7.79561	−0.257154	−0.128577	−	0.991700i	$-0.541041\pi$
$919$	−7.79561	−0.257154	−0.128577	+	0.991700i	$0.541041\pi$
$920$	0	0
$921$	−13.5095	−0.445153
$922$	0	0
$923$	0.537623	0.0176961
$924$	0	0
$925$	24.6281	0.809766
$926$	0	0
$927$	16.6605	0.547203
$928$	0	0
$929$	11.5912	0.380293	0.190147	−	0.981756i	$-0.439104\pi$
$929$	11.5912	0.380293	0.190147	+	0.981756i	$0.439104\pi$
$930$	0	0
$931$	−2.50407	−0.0820675
$932$	0	0
$933$	24.3035	0.795662
$934$	0	0
$935$	−6.18015	−0.202113
$936$	0	0
$937$	−32.0050	−1.04556	−0.522778	−	0.852469i	$-0.675105\pi$
$937$	−32.0050	−1.04556	−0.522778	+	0.852469i	$0.675105\pi$
$938$	0	0
$939$	−33.7836	−1.10249
$940$	0	0
$941$	11.1090	0.362141	0.181071	−	0.983470i	$-0.442044\pi$
$941$	11.1090	0.362141	0.181071	+	0.983470i	$0.442044\pi$
$942$	0	0
$943$	2.30826	0.0751674
$944$	0	0
$945$	−2.73589	−0.0889986
$946$	0	0
$947$	12.1262	0.394049	0.197025	−	0.980399i	$-0.436872\pi$
$947$	12.1262	0.394049	0.197025	+	0.980399i	$0.436872\pi$
$948$	0	0
$949$	−7.93661	−0.257633
$950$	0	0
$951$	32.1594	1.04284
$952$	0	0
$953$	27.6645	0.896140	0.448070	−	0.893999i	$-0.352112\pi$
$953$	27.6645	0.896140	0.448070	+	0.893999i	$0.352112\pi$
$954$	0	0
$955$	−1.72838	−0.0559291
$956$	0	0
$957$	16.7388	0.541090
$958$	0	0
$959$	0.270356	0.00873026
$960$	0	0
$961$	−29.7946	−0.961117
$962$	0	0
$963$	−1.68092	−0.0541668
$964$	0	0
$965$	−3.74341	−0.120505
$966$	0	0
$967$	56.1838	1.80675	0.903374	−	0.428853i	$-0.141082\pi$
$967$	56.1838	1.80675	0.903374	+	0.428853i	$0.141082\pi$
$968$	0	0
$969$	−2.25892	−0.0725668
$970$	0	0
$971$	43.9884	1.41166	0.705828	−	0.708383i	$-0.250575\pi$
$971$	43.9884	1.41166	0.705828	+	0.708383i	$0.250575\pi$
$972$	0	0
$973$	−4.42512	−0.141863
$974$	0	0
$975$	3.69068	0.118196
$976$	0	0
$977$	−6.47804	−0.207251	−0.103625	−	0.994616i	$-0.533044\pi$
$977$	−6.47804	−0.207251	−0.103625	+	0.994616i	$0.533044\pi$
$978$	0	0
$979$	26.6044	0.850282
$980$	0	0
$981$	11.4333	0.365036
$982$	0	0
$983$	−37.8961	−1.20870	−0.604349	−	0.796720i	$-0.706567\pi$
$983$	−37.8961	−1.20870	−0.604349	+	0.796720i	$0.706567\pi$
$984$	0	0
$985$	−67.7437	−2.15849
$986$	0	0
$987$	6.74671	0.214750
$988$	0	0
$989$	5.83380	0.185504
$990$	0	0
$991$	−53.3499	−1.69472	−0.847358	−	0.531022i	$-0.821808\pi$
$991$	−53.3499	−1.69472	−0.847358	+	0.531022i	$0.821808\pi$
$992$	0	0
$993$	4.84004	0.153594
$994$	0	0
$995$	3.95254	0.125304
$996$	0	0
$997$	24.8754	0.787813	0.393907	−	0.919150i	$-0.371123\pi$
$997$	24.8754	0.787813	0.393907	+	0.919150i	$0.371123\pi$
$998$	0	0
$999$	9.91023	0.313546

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7728.2.a.ce.1.1		4
4.3	odd	2		483.2.a.j.1.2	✓	4
12.11	even	2		1449.2.a.o.1.3		4
28.27	even	2		3381.2.a.x.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.a.j.1.2	✓	4	4.3	odd	2
1449.2.a.o.1.3		4	12.11	even	2
3381.2.a.x.1.2		4	28.27	even	2
7728.2.a.ce.1.1		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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -2.73589 q^{5} +1.00000 q^{7} +1.00000 q^{9} +2.50407 q^{11} +1.48511 q^{13} -2.73589 q^{15} +0.902098 q^{17} -2.50407 q^{19} +1.00000 q^{21} -1.00000 q^{23} +2.48511 q^{25} +1.00000 q^{27} +6.68466 q^{29} -1.09790 q^{31} +2.50407 q^{33} -2.73589 q^{35} +9.91023 q^{37} +1.48511 q^{39} -2.30826 q^{41} -5.83380 q^{43} -2.73589 q^{45} +6.74671 q^{47} +1.00000 q^{49} +0.902098 q^{51} -4.91648 q^{53} -6.85086 q^{55} -2.50407 q^{57} +7.32973 q^{59} -1.00625 q^{61} +1.00000 q^{63} -4.06311 q^{65} -11.2929 q^{67} -1.00000 q^{69} +0.362008 q^{71} -5.34411 q^{73} +2.48511 q^{75} +2.50407 q^{77} -10.4672 q^{79} +1.00000 q^{81} -7.59946 q^{83} -2.46805 q^{85} +6.68466 q^{87} +10.6245 q^{89} +1.48511 q^{91} -1.09790 q^{93} +6.85086 q^{95} +13.9102 q^{97} +2.50407 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 5 q^{5} + 4 q^{7} + 4 q^{9} - q^{11} + 7 q^{13} + 5 q^{15} + 2 q^{17} + q^{19} + 4 q^{21} - 4 q^{23} + 11 q^{25} + 4 q^{27} + 2 q^{29} - 6 q^{31} - q^{33} + 5 q^{35} + 16 q^{37} + 7 q^{39}+ \cdots - q^{99}+O(q^{100}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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