LMFDB - Embedded newform 7728.2.a.cd.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:	4.4.24197.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} - x + 2 $ x^4 - 6*x^2 - 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			$-0.700017$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -1.15706 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.15706 q^{5} -1.00000 q^{7} +1.00000 q^{9} -3.66704 q^{11} +2.34710 q^{13} -1.15706 q^{15} +4.80996 q^{17} -7.06707 q^{19} -1.00000 q^{21} +1.00000 q^{23} -3.66122 q^{25} +1.00000 q^{27} -6.52411 q^{29} -2.80996 q^{31} -3.66704 q^{33} +1.15706 q^{35} +5.40993 q^{37} +2.34710 q^{39} +0.647082 q^{41} +4.76709 q^{43} -1.15706 q^{45} -4.85708 q^{47} +1.00000 q^{49} +4.80996 q^{51} +7.63405 q^{53} +4.24297 q^{55} -7.06707 q^{57} +10.2142 q^{59} +9.91001 q^{61} -1.00000 q^{63} -2.71573 q^{65} -6.07114 q^{67} +1.00000 q^{69} +3.65290 q^{71} +11.8183 q^{73} -3.66122 q^{75} +3.66704 q^{77} +9.12408 q^{79} +1.00000 q^{81} +15.8582 q^{83} -5.56541 q^{85} -6.52411 q^{87} +4.66122 q^{89} -2.34710 q^{91} -2.80996 q^{93} +8.17701 q^{95} -4.90419 q^{97} -3.66704 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 5 q^{11} + 7 q^{13} + 5 q^{15} + 12 q^{17} - 3 q^{19} - 4 q^{21} + 4 q^{23} + 7 q^{25} + 4 q^{27} + 6 q^{29} - 4 q^{31} + 5 q^{33} - 5 q^{35} + 20 q^{37} + 7 q^{39} + 3 q^{41} - 9 q^{43} + 5 q^{45} - 7 q^{47} + 4 q^{49} + 12 q^{51} - 6 q^{53} + 21 q^{55} - 3 q^{57} + 2 q^{59} + 24 q^{61} - 4 q^{63} - 14 q^{65} - q^{67} + 4 q^{69} + 17 q^{71} + 16 q^{73} + 7 q^{75} - 5 q^{77} + 10 q^{79} + 4 q^{81} - 8 q^{83} + 17 q^{85} + 6 q^{87} - 3 q^{89} - 7 q^{91} - 4 q^{93} + 3 q^{95} - 2 q^{97} + 5 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 + 5 * q^5 - 4 * q^7 + 4 * q^9 + 5 * q^11 + 7 * q^13 + 5 * q^15 + 12 * q^17 - 3 * q^19 - 4 * q^21 + 4 * q^23 + 7 * q^25 + 4 * q^27 + 6 * q^29 - 4 * q^31 + 5 * q^33 - 5 * q^35 + 20 * q^37 + 7 * q^39 + 3 * q^41 - 9 * q^43 + 5 * q^45 - 7 * q^47 + 4 * q^49 + 12 * q^51 - 6 * q^53 + 21 * q^55 - 3 * q^57 + 2 * q^59 + 24 * q^61 - 4 * q^63 - 14 * q^65 - q^67 + 4 * q^69 + 17 * q^71 + 16 * q^73 + 7 * q^75 - 5 * q^77 + 10 * q^79 + 4 * q^81 - 8 * q^83 + 17 * q^85 + 6 * q^87 - 3 * q^89 - 7 * q^91 - 4 * q^93 + 3 * q^95 - 2 * q^97 + 5 * 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
$5$	−1.15706	−0.517452	−0.258726	−	0.965951i	$-0.583303\pi$
$5$	−1.15706	−0.517452	−0.258726	+	0.965951i	$0.583303\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.66704	−1.10565	−0.552826	−	0.833296i	$-0.686451\pi$
$11$	−3.66704	−1.10565	−0.552826	+	0.833296i	$0.686451\pi$
$12$	0	0
$13$	2.34710	0.650968	0.325484	−	0.945548i	$-0.394473\pi$
$13$	2.34710	0.650968	0.325484	+	0.945548i	$0.394473\pi$
$14$	0	0
$15$	−1.15706	−0.298751
$16$	0	0
$17$	4.80996	1.16659	0.583293	−	0.812262i	$-0.301764\pi$
$17$	4.80996	1.16659	0.583293	+	0.812262i	$0.301764\pi$
$18$	0	0
$19$	−7.06707	−1.62130	−0.810648	−	0.585533i	$-0.800885\pi$
$19$	−7.06707	−1.62130	−0.810648	+	0.585533i	$0.800885\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−3.66122	−0.732243
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−6.52411	−1.21150	−0.605748	−	0.795656i	$-0.707126\pi$
$29$	−6.52411	−1.21150	−0.605748	+	0.795656i	$0.707126\pi$
$30$	0	0
$31$	−2.80996	−0.504684	−0.252342	−	0.967638i	$-0.581201\pi$
$31$	−2.80996	−0.504684	−0.252342	+	0.967638i	$0.581201\pi$
$32$	0	0
$33$	−3.66704	−0.638349
$34$	0	0
$35$	1.15706	0.195579
$36$	0	0
$37$	5.40993	0.889387	0.444693	−	0.895683i	$-0.353313\pi$
$37$	5.40993	0.889387	0.444693	+	0.895683i	$0.353313\pi$
$38$	0	0
$39$	2.34710	0.375837
$40$	0	0
$41$	0.647082	0.101057	0.0505286	−	0.998723i	$-0.483909\pi$
$41$	0.647082	0.101057	0.0505286	+	0.998723i	$0.483909\pi$
$42$	0	0
$43$	4.76709	0.726974	0.363487	−	0.931599i	$-0.381586\pi$
$43$	4.76709	0.726974	0.363487	+	0.931599i	$0.381586\pi$
$44$	0	0
$45$	−1.15706	−0.172484
$46$	0	0
$47$	−4.85708	−0.708477	−0.354239	−	0.935155i	$-0.615260\pi$
$47$	−4.85708	−0.708477	−0.354239	+	0.935155i	$0.615260\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	4.80996	0.673529
$52$	0	0
$53$	7.63405	1.04862	0.524309	−	0.851528i	$-0.324324\pi$
$53$	7.63405	1.04862	0.524309	+	0.851528i	$0.324324\pi$
$54$	0	0
$55$	4.24297	0.572123
$56$	0	0
$57$	−7.06707	−0.936056
$58$	0	0
$59$	10.2142	1.32978	0.664890	−	0.746941i	$-0.268478\pi$
$59$	10.2142	1.32978	0.664890	+	0.746941i	$0.268478\pi$
$60$	0	0
$61$	9.91001	1.26885	0.634423	−	0.772986i	$-0.281238\pi$
$61$	9.91001	1.26885	0.634423	+	0.772986i	$0.281238\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	−2.71573	−0.336845
$66$	0	0
$67$	−6.07114	−0.741708	−0.370854	−	0.928691i	$-0.620935\pi$
$67$	−6.07114	−0.741708	−0.370854	+	0.928691i	$0.620935\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	3.65290	0.433520	0.216760	−	0.976225i	$-0.430451\pi$
$71$	3.65290	0.433520	0.216760	+	0.976225i	$0.430451\pi$
$72$	0	0
$73$	11.8183	1.38322	0.691612	−	0.722269i	$-0.256901\pi$
$73$	11.8183	1.38322	0.691612	+	0.722269i	$0.256901\pi$
$74$	0	0
$75$	−3.66122	−0.422761
$76$	0	0
$77$	3.66704	0.417897
$78$	0	0
$79$	9.12408	1.02654	0.513269	−	0.858228i	$-0.328434\pi$
$79$	9.12408	1.02654	0.513269	+	0.858228i	$0.328434\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	15.8582	1.74066	0.870331	−	0.492468i	$-0.163905\pi$
$83$	15.8582	1.74066	0.870331	+	0.492468i	$0.163905\pi$
$84$	0	0
$85$	−5.56541	−0.603653
$86$	0	0
$87$	−6.52411	−0.699458
$88$	0	0
$89$	4.66122	0.494088	0.247044	−	0.969004i	$-0.420541\pi$
$89$	4.66122	0.494088	0.247044	+	0.969004i	$0.420541\pi$
$90$	0	0
$91$	−2.34710	−0.246043
$92$	0	0
$93$	−2.80996	−0.291379
$94$	0	0
$95$	8.17701	0.838944
$96$	0	0
$97$	−4.90419	−0.497945	−0.248973	−	0.968511i	$-0.580093\pi$
$97$	−4.90419	−0.497945	−0.248973	+	0.968511i	$0.580093\pi$
$98$	0	0
$99$	−3.66704	−0.368551
$100$	0	0
$101$	−5.80589	−0.577707	−0.288854	−	0.957373i	$-0.593274\pi$
$101$	−5.80589	−0.577707	−0.288854	+	0.957373i	$0.593274\pi$
$102$	0	0
$103$	−17.1169	−1.68658	−0.843288	−	0.537463i	$-0.819383\pi$
$103$	−17.1169	−1.68658	−0.843288	+	0.537463i	$0.819383\pi$
$104$	0	0
$105$	1.15706	0.112917
$106$	0	0
$107$	8.34710	0.806944	0.403472	−	0.914992i	$-0.367803\pi$
$107$	8.34710	0.806944	0.403472	+	0.914992i	$0.367803\pi$
$108$	0	0
$109$	16.3771	1.56864	0.784321	−	0.620355i	$-0.213011\pi$
$109$	16.3771	1.56864	0.784321	+	0.620355i	$0.213011\pi$
$110$	0	0
$111$	5.40993	0.513488
$112$	0	0
$113$	−14.1770	−1.33366	−0.666831	−	0.745209i	$-0.732350\pi$
$113$	−14.1770	−1.33366	−0.666831	+	0.745209i	$0.732350\pi$
$114$	0	0
$115$	−1.15706	−0.107896
$116$	0	0
$117$	2.34710	0.216989
$118$	0	0
$119$	−4.80996	−0.440928
$120$	0	0
$121$	2.44715	0.222468
$122$	0	0
$123$	0.647082	0.0583454
$124$	0	0
$125$	10.0215	0.896353
$126$	0	0
$127$	−13.1184	−1.16407	−0.582036	−	0.813163i	$-0.697744\pi$
$127$	−13.1184	−1.16407	−0.582036	+	0.813163i	$0.697744\pi$
$128$	0	0
$129$	4.76709	0.419718
$130$	0	0
$131$	1.28696	0.112442	0.0562209	−	0.998418i	$-0.482095\pi$
$131$	1.28696	0.112442	0.0562209	+	0.998418i	$0.482095\pi$
$132$	0	0
$133$	7.06707	0.612793
$134$	0	0
$135$	−1.15706	−0.0995837
$136$	0	0
$137$	−11.3152	−0.966725	−0.483362	−	0.875420i	$-0.660585\pi$
$137$	−11.3152	−0.966725	−0.483362	+	0.875420i	$0.660585\pi$
$138$	0	0
$139$	−8.93717	−0.758041	−0.379020	−	0.925388i	$-0.623739\pi$
$139$	−8.93717	−0.758041	−0.379020	+	0.925388i	$0.623739\pi$
$140$	0	0
$141$	−4.85708	−0.409040
$142$	0	0
$143$	−8.60689	−0.719745
$144$	0	0
$145$	7.54878	0.626892
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	1.67693	0.137379	0.0686897	−	0.997638i	$-0.478118\pi$
$149$	1.67693	0.137379	0.0686897	+	0.997638i	$0.478118\pi$
$150$	0	0
$151$	19.0110	1.54709	0.773547	−	0.633739i	$-0.218481\pi$
$151$	19.0110	1.54709	0.773547	+	0.633739i	$0.218481\pi$
$152$	0	0
$153$	4.80996	0.388862
$154$	0	0
$155$	3.25129	0.261150
$156$	0	0
$157$	12.5712	1.00329	0.501647	−	0.865073i	$-0.332728\pi$
$157$	12.5712	1.00329	0.501647	+	0.865073i	$0.332728\pi$
$158$	0	0
$159$	7.63405	0.605420
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	15.6740	1.22768	0.613840	−	0.789431i	$-0.289624\pi$
$163$	15.6740	1.22768	0.613840	+	0.789431i	$0.289624\pi$
$164$	0	0
$165$	4.24297	0.330315
$166$	0	0
$167$	−4.96120	−0.383909	−0.191955	−	0.981404i	$-0.561483\pi$
$167$	−4.96120	−0.383909	−0.191955	+	0.981404i	$0.561483\pi$
$168$	0	0
$169$	−7.49113	−0.576241
$170$	0	0
$171$	−7.06707	−0.540432
$172$	0	0
$173$	12.9042	0.981087	0.490544	−	0.871417i	$-0.336798\pi$
$173$	12.9042	0.981087	0.490544	+	0.871417i	$0.336798\pi$
$174$	0	0
$175$	3.66122	0.276762
$176$	0	0
$177$	10.2142	0.767749
$178$	0	0
$179$	0.262762	0.0196398	0.00981989	−	0.999952i	$-0.496874\pi$
$179$	0.262762	0.0196398	0.00981989	+	0.999952i	$0.496874\pi$
$180$	0	0
$181$	−12.1440	−0.902659	−0.451329	−	0.892357i	$-0.649050\pi$
$181$	−12.1440	−0.902659	−0.451329	+	0.892357i	$0.649050\pi$
$182$	0	0
$183$	9.91001	0.732569
$184$	0	0
$185$	−6.25960	−0.460215
$186$	0	0
$187$	−17.6383	−1.28984
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−2.68524	−0.194297	−0.0971487	−	0.995270i	$-0.530972\pi$
$191$	−2.68524	−0.194297	−0.0971487	+	0.995270i	$0.530972\pi$
$192$	0	0
$193$	−18.7252	−1.34787	−0.673933	−	0.738793i	$-0.735396\pi$
$193$	−18.7252	−1.34787	−0.673933	+	0.738793i	$0.735396\pi$
$194$	0	0
$195$	−2.71573	−0.194477
$196$	0	0
$197$	19.3953	1.38186	0.690930	−	0.722922i	$-0.257201\pi$
$197$	19.3953	1.38186	0.690930	+	0.722922i	$0.257201\pi$
$198$	0	0
$199$	−1.10005	−0.0779805	−0.0389902	−	0.999240i	$-0.512414\pi$
$199$	−1.10005	−0.0779805	−0.0389902	+	0.999240i	$0.512414\pi$
$200$	0	0
$201$	−6.07114	−0.428225
$202$	0	0
$203$	6.52411	0.457903
$204$	0	0
$205$	−0.748712	−0.0522923
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	25.9152	1.79259
$210$	0	0
$211$	−25.5235	−1.75711	−0.878554	−	0.477643i	$-0.841491\pi$
$211$	−25.5235	−1.75711	−0.878554	+	0.477643i	$0.841491\pi$
$212$	0	0
$213$	3.65290	0.250293
$214$	0	0
$215$	−5.51580	−0.376174
$216$	0	0
$217$	2.80996	0.190753
$218$	0	0
$219$	11.8183	0.798605
$220$	0	0
$221$	11.2894	0.759411
$222$	0	0
$223$	17.1772	1.15027	0.575134	−	0.818059i	$-0.304950\pi$
$223$	17.1772	1.15027	0.575134	+	0.818059i	$0.304950\pi$
$224$	0	0
$225$	−3.66122	−0.244081
$226$	0	0
$227$	26.0334	1.72790	0.863950	−	0.503577i	$-0.167983\pi$
$227$	26.0334	1.72790	0.863950	+	0.503577i	$0.167983\pi$
$228$	0	0
$229$	−7.01430	−0.463518	−0.231759	−	0.972773i	$-0.574448\pi$
$229$	−7.01430	−0.463518	−0.231759	+	0.972773i	$0.574448\pi$
$230$	0	0
$231$	3.66704	0.241273
$232$	0	0
$233$	0.671109	0.0439658	0.0219829	−	0.999758i	$-0.493002\pi$
$233$	0.671109	0.0439658	0.0219829	+	0.999758i	$0.493002\pi$
$234$	0	0
$235$	5.61992	0.366603
$236$	0	0
$237$	9.12408	0.592673
$238$	0	0
$239$	26.5571	1.71784	0.858918	−	0.512114i	$-0.171137\pi$
$239$	26.5571	1.71784	0.858918	+	0.512114i	$0.171137\pi$
$240$	0	0
$241$	−4.97284	−0.320329	−0.160164	−	0.987090i	$-0.551202\pi$
$241$	−4.97284	−0.320329	−0.160164	+	0.987090i	$0.551202\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−1.15706	−0.0739218
$246$	0	0
$247$	−16.5871	−1.05541
$248$	0	0
$249$	15.8582	1.00497
$250$	0	0
$251$	14.4281	0.910696	0.455348	−	0.890314i	$-0.349515\pi$
$251$	14.4281	0.910696	0.455348	+	0.890314i	$0.349515\pi$
$252$	0	0
$253$	−3.66704	−0.230545
$254$	0	0
$255$	−5.56541	−0.348519
$256$	0	0
$257$	5.44889	0.339893	0.169946	−	0.985453i	$-0.445641\pi$
$257$	5.44889	0.339893	0.169946	+	0.985453i	$0.445641\pi$
$258$	0	0
$259$	−5.40993	−0.336157
$260$	0	0
$261$	−6.52411	−0.403832
$262$	0	0
$263$	−15.8128	−0.975060	−0.487530	−	0.873106i	$-0.662102\pi$
$263$	−15.8128	−0.975060	−0.487530	+	0.873106i	$0.662102\pi$
$264$	0	0
$265$	−8.83305	−0.542610
$266$	0	0
$267$	4.66122	0.285262
$268$	0	0
$269$	11.9011	0.725620	0.362810	−	0.931863i	$-0.381817\pi$
$269$	11.9011	0.725620	0.362810	+	0.931863i	$0.381817\pi$
$270$	0	0
$271$	16.6664	1.01241	0.506206	−	0.862413i	$-0.331048\pi$
$271$	16.6664	1.01241	0.506206	+	0.862413i	$0.331048\pi$
$272$	0	0
$273$	−2.34710	−0.142053
$274$	0	0
$275$	13.4258	0.809607
$276$	0	0
$277$	20.1299	1.20949	0.604744	−	0.796420i	$-0.293275\pi$
$277$	20.1299	1.20949	0.604744	+	0.796420i	$0.293275\pi$
$278$	0	0
$279$	−2.80996	−0.168228
$280$	0	0
$281$	19.4022	1.15744	0.578720	−	0.815526i	$-0.303552\pi$
$281$	19.4022	1.15744	0.578720	+	0.815526i	$0.303552\pi$
$282$	0	0
$283$	10.8313	0.643854	0.321927	−	0.946764i	$-0.395669\pi$
$283$	10.8313	0.643854	0.321927	+	0.946764i	$0.395669\pi$
$284$	0	0
$285$	8.17701	0.484364
$286$	0	0
$287$	−0.647082	−0.0381960
$288$	0	0
$289$	6.13572	0.360925
$290$	0	0
$291$	−4.90419	−0.287489
$292$	0	0
$293$	8.89430	0.519610	0.259805	−	0.965661i	$-0.416342\pi$
$293$	8.89430	0.519610	0.259805	+	0.965661i	$0.416342\pi$
$294$	0	0
$295$	−11.8185	−0.688098
$296$	0	0
$297$	−3.66704	−0.212783
$298$	0	0
$299$	2.34710	0.135736
$300$	0	0
$301$	−4.76709	−0.274770
$302$	0	0
$303$	−5.80589	−0.333539
$304$	0	0
$305$	−11.4665	−0.656568
$306$	0	0
$307$	−19.7442	−1.12686	−0.563429	−	0.826164i	$-0.690518\pi$
$307$	−19.7442	−1.12686	−0.563429	+	0.826164i	$0.690518\pi$
$308$	0	0
$309$	−17.1169	−0.973745
$310$	0	0
$311$	28.5898	1.62118	0.810589	−	0.585615i	$-0.199147\pi$
$311$	28.5898	1.62118	0.810589	+	0.585615i	$0.199147\pi$
$312$	0	0
$313$	12.6570	0.715415	0.357707	−	0.933834i	$-0.383559\pi$
$313$	12.6570	0.715415	0.357707	+	0.933834i	$0.383559\pi$
$314$	0	0
$315$	1.15706	0.0651929
$316$	0	0
$317$	4.83130	0.271353	0.135676	−	0.990753i	$-0.456679\pi$
$317$	4.83130	0.271353	0.135676	+	0.990753i	$0.456679\pi$
$318$	0	0
$319$	23.9241	1.33949
$320$	0	0
$321$	8.34710	0.465890
$322$	0	0
$323$	−33.9923	−1.89138
$324$	0	0
$325$	−8.59323	−0.476667
$326$	0	0
$327$	16.3771	0.905656
$328$	0	0
$329$	4.85708	0.267779
$330$	0	0
$331$	6.84544	0.376259	0.188130	−	0.982144i	$-0.439758\pi$
$331$	6.84544	0.376259	0.188130	+	0.982144i	$0.439758\pi$
$332$	0	0
$333$	5.40993	0.296462
$334$	0	0
$335$	7.02467	0.383799
$336$	0	0
$337$	−8.17527	−0.445335	−0.222668	−	0.974894i	$-0.571476\pi$
$337$	−8.17527	−0.445335	−0.222668	+	0.974894i	$0.571476\pi$
$338$	0	0
$339$	−14.1770	−0.769990
$340$	0	0
$341$	10.3042	0.558005
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−1.15706	−0.0622939
$346$	0	0
$347$	−21.2699	−1.14183	−0.570913	−	0.821011i	$-0.693411\pi$
$347$	−21.2699	−1.14183	−0.570913	+	0.821011i	$0.693411\pi$
$348$	0	0
$349$	1.96527	0.105199	0.0525993	−	0.998616i	$-0.483249\pi$
$349$	1.96527	0.105199	0.0525993	+	0.998616i	$0.483249\pi$
$350$	0	0
$351$	2.34710	0.125279
$352$	0	0
$353$	10.3124	0.548872	0.274436	−	0.961605i	$-0.411509\pi$
$353$	10.3124	0.548872	0.274436	+	0.961605i	$0.411509\pi$
$354$	0	0
$355$	−4.22662	−0.224326
$356$	0	0
$357$	−4.80996	−0.254570
$358$	0	0
$359$	26.3856	1.39258	0.696289	−	0.717761i	$-0.254833\pi$
$359$	26.3856	1.39258	0.696289	+	0.717761i	$0.254833\pi$
$360$	0	0
$361$	30.9435	1.62860
$362$	0	0
$363$	2.44715	0.128442
$364$	0	0
$365$	−13.6744	−0.715753
$366$	0	0
$367$	33.5667	1.75217	0.876083	−	0.482160i	$-0.160148\pi$
$367$	33.5667	1.75217	0.876083	+	0.482160i	$0.160148\pi$
$368$	0	0
$369$	0.647082	0.0336857
$370$	0	0
$371$	−7.63405	−0.396340
$372$	0	0
$373$	−17.0698	−0.883838	−0.441919	−	0.897055i	$-0.645702\pi$
$373$	−17.0698	−0.883838	−0.441919	+	0.897055i	$0.645702\pi$
$374$	0	0
$375$	10.0215	0.517510
$376$	0	0
$377$	−15.3127	−0.788646
$378$	0	0
$379$	15.7541	0.809232	0.404616	−	0.914487i	$-0.367405\pi$
$379$	15.7541	0.809232	0.404616	+	0.914487i	$0.367405\pi$
$380$	0	0
$381$	−13.1184	−0.672077
$382$	0	0
$383$	−3.85818	−0.197144	−0.0985719	−	0.995130i	$-0.531427\pi$
$383$	−3.85818	−0.197144	−0.0985719	+	0.995130i	$0.531427\pi$
$384$	0	0
$385$	−4.24297	−0.216242
$386$	0	0
$387$	4.76709	0.242325
$388$	0	0
$389$	35.0666	1.77795	0.888973	−	0.457959i	$-0.151419\pi$
$389$	35.0666	1.77795	0.888973	+	0.457959i	$0.151419\pi$
$390$	0	0
$391$	4.80996	0.243250
$392$	0	0
$393$	1.28696	0.0649183
$394$	0	0
$395$	−10.5571	−0.531185
$396$	0	0
$397$	−15.3422	−0.770004	−0.385002	−	0.922916i	$-0.625799\pi$
$397$	−15.3422	−0.770004	−0.385002	+	0.922916i	$0.625799\pi$
$398$	0	0
$399$	7.06707	0.353796
$400$	0	0
$401$	−9.24705	−0.461776	−0.230888	−	0.972980i	$-0.574163\pi$
$401$	−9.24705	−0.461776	−0.230888	+	0.972980i	$0.574163\pi$
$402$	0	0
$403$	−6.59525	−0.328533
$404$	0	0
$405$	−1.15706	−0.0574947
$406$	0	0
$407$	−19.8384	−0.983353
$408$	0	0
$409$	−16.6498	−0.823278	−0.411639	−	0.911347i	$-0.635044\pi$
$409$	−16.6498	−0.823278	−0.411639	+	0.911347i	$0.635044\pi$
$410$	0	0
$411$	−11.3152	−0.558139
$412$	0	0
$413$	−10.2142	−0.502610
$414$	0	0
$415$	−18.3488	−0.900709
$416$	0	0
$417$	−8.93717	−0.437655
$418$	0	0
$419$	6.86306	0.335282	0.167641	−	0.985848i	$-0.446385\pi$
$419$	6.86306	0.335282	0.167641	+	0.985848i	$0.446385\pi$
$420$	0	0
$421$	−13.1754	−0.642131	−0.321066	−	0.947057i	$-0.604041\pi$
$421$	−13.1754	−0.642131	−0.321066	+	0.947057i	$0.604041\pi$
$422$	0	0
$423$	−4.85708	−0.236159
$424$	0	0
$425$	−17.6103	−0.854225
$426$	0	0
$427$	−9.91001	−0.479579
$428$	0	0
$429$	−8.60689	−0.415545
$430$	0	0
$431$	26.0837	1.25641	0.628204	−	0.778048i	$-0.283790\pi$
$431$	26.0837	1.25641	0.628204	+	0.778048i	$0.283790\pi$
$432$	0	0
$433$	−39.8676	−1.91591	−0.957957	−	0.286911i	$-0.907372\pi$
$433$	−39.8676	−1.91591	−0.957957	+	0.286911i	$0.907372\pi$
$434$	0	0
$435$	7.54878	0.361936
$436$	0	0
$437$	−7.06707	−0.338064
$438$	0	0
$439$	17.9160	0.855084	0.427542	−	0.903996i	$-0.359380\pi$
$439$	17.9160	0.855084	0.427542	+	0.903996i	$0.359380\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	3.02810	0.143869	0.0719347	−	0.997409i	$-0.477083\pi$
$443$	3.02810	0.143869	0.0719347	+	0.997409i	$0.477083\pi$
$444$	0	0
$445$	−5.39330	−0.255667
$446$	0	0
$447$	1.67693	0.0793160
$448$	0	0
$449$	−30.0253	−1.41698	−0.708491	−	0.705720i	$-0.750624\pi$
$449$	−30.0253	−1.41698	−0.708491	+	0.705720i	$0.750624\pi$
$450$	0	0
$451$	−2.37287	−0.111734
$452$	0	0
$453$	19.0110	0.893215
$454$	0	0
$455$	2.71573	0.127315
$456$	0	0
$457$	12.2169	0.571483	0.285742	−	0.958307i	$-0.407760\pi$
$457$	12.2169	0.571483	0.285742	+	0.958307i	$0.407760\pi$
$458$	0	0
$459$	4.80996	0.224510
$460$	0	0
$461$	33.7511	1.57194	0.785972	−	0.618261i	$-0.212163\pi$
$461$	33.7511	1.57194	0.785972	+	0.618261i	$0.212163\pi$
$462$	0	0
$463$	21.9187	1.01865	0.509324	−	0.860575i	$-0.329896\pi$
$463$	21.9187	1.01865	0.509324	+	0.860575i	$0.329896\pi$
$464$	0	0
$465$	3.25129	0.150775
$466$	0	0
$467$	15.6499	0.724193	0.362096	−	0.932141i	$-0.382061\pi$
$467$	15.6499	0.724193	0.362096	+	0.932141i	$0.382061\pi$
$468$	0	0
$469$	6.07114	0.280339
$470$	0	0
$471$	12.5712	0.579251
$472$	0	0
$473$	−17.4811	−0.803780
$474$	0	0
$475$	25.8741	1.18718
$476$	0	0
$477$	7.63405	0.349539
$478$	0	0
$479$	−6.20999	−0.283742	−0.141871	−	0.989885i	$-0.545312\pi$
$479$	−6.20999	−0.283742	−0.141871	+	0.989885i	$0.545312\pi$
$480$	0	0
$481$	12.6976	0.578962
$482$	0	0
$483$	−1.00000	−0.0455016
$484$	0	0
$485$	5.67444	0.257663
$486$	0	0
$487$	7.82642	0.354649	0.177325	−	0.984152i	$-0.443256\pi$
$487$	7.82642	0.354649	0.177325	+	0.984152i	$0.443256\pi$
$488$	0	0
$489$	15.6740	0.708801
$490$	0	0
$491$	32.0550	1.44662	0.723311	−	0.690523i	$-0.242620\pi$
$491$	32.0550	1.44662	0.723311	+	0.690523i	$0.242620\pi$
$492$	0	0
$493$	−31.3807	−1.41332
$494$	0	0
$495$	4.24297	0.190708
$496$	0	0
$497$	−3.65290	−0.163855
$498$	0	0
$499$	−42.3592	−1.89626	−0.948129	−	0.317885i	$-0.897027\pi$
$499$	−42.3592	−1.89626	−0.948129	+	0.317885i	$0.897027\pi$
$500$	0	0
$501$	−4.96120	−0.221650
$502$	0	0
$503$	−11.6053	−0.517455	−0.258728	−	0.965950i	$-0.583303\pi$
$503$	−11.6053	−0.517455	−0.258728	+	0.965950i	$0.583303\pi$
$504$	0	0
$505$	6.71775	0.298936
$506$	0	0
$507$	−7.49113	−0.332693
$508$	0	0
$509$	−12.1512	−0.538594	−0.269297	−	0.963057i	$-0.586791\pi$
$509$	−12.1512	−0.538594	−0.269297	+	0.963057i	$0.586791\pi$
$510$	0	0
$511$	−11.8183	−0.522810
$512$	0	0
$513$	−7.06707	−0.312019
$514$	0	0
$515$	19.8052	0.872722
$516$	0	0
$517$	17.8111	0.783330
$518$	0	0
$519$	12.9042	0.566431
$520$	0	0
$521$	−15.3884	−0.674178	−0.337089	−	0.941473i	$-0.609442\pi$
$521$	−15.3884	−0.674178	−0.337089	+	0.941473i	$0.609442\pi$
$522$	0	0
$523$	14.3536	0.627637	0.313818	−	0.949483i	$-0.398392\pi$
$523$	14.3536	0.627637	0.313818	+	0.949483i	$0.398392\pi$
$524$	0	0
$525$	3.66122	0.159789
$526$	0	0
$527$	−13.5158	−0.588757
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	10.2142	0.443260
$532$	0	0
$533$	1.51876	0.0657850
$534$	0	0
$535$	−9.65808	−0.417555
$536$	0	0
$537$	0.262762	0.0113390
$538$	0	0
$539$	−3.66704	−0.157950
$540$	0	0
$541$	31.1451	1.33903	0.669517	−	0.742797i	$-0.266501\pi$
$541$	31.1451	1.33903	0.669517	+	0.742797i	$0.266501\pi$
$542$	0	0
$543$	−12.1440	−0.521150
$544$	0	0
$545$	−18.9493	−0.811698
$546$	0	0
$547$	−12.6630	−0.541429	−0.270715	−	0.962660i	$-0.587260\pi$
$547$	−12.6630	−0.541429	−0.270715	+	0.962660i	$0.587260\pi$
$548$	0	0
$549$	9.91001	0.422949
$550$	0	0
$551$	46.1063	1.96420
$552$	0	0
$553$	−9.12408	−0.387995
$554$	0	0
$555$	−6.25960	−0.265705
$556$	0	0
$557$	−30.7740	−1.30394	−0.651968	−	0.758246i	$-0.726057\pi$
$557$	−30.7740	−1.30394	−0.651968	+	0.758246i	$0.726057\pi$
$558$	0	0
$559$	11.1888	0.473237
$560$	0	0
$561$	−17.6383	−0.744689
$562$	0	0
$563$	−35.4352	−1.49342	−0.746708	−	0.665152i	$-0.768367\pi$
$563$	−35.4352	−1.49342	−0.746708	+	0.665152i	$0.768367\pi$
$564$	0	0
$565$	16.4036	0.690106
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	2.15939	0.0905262	0.0452631	−	0.998975i	$-0.485587\pi$
$569$	2.15939	0.0905262	0.0452631	+	0.998975i	$0.485587\pi$
$570$	0	0
$571$	26.4182	1.10557	0.552784	−	0.833324i	$-0.313565\pi$
$571$	26.4182	1.10557	0.552784	+	0.833324i	$0.313565\pi$
$572$	0	0
$573$	−2.68524	−0.112178
$574$	0	0
$575$	−3.66122	−0.152683
$576$	0	0
$577$	−12.9885	−0.540719	−0.270360	−	0.962759i	$-0.587143\pi$
$577$	−12.9885	−0.540719	−0.270360	+	0.962759i	$0.587143\pi$
$578$	0	0
$579$	−18.7252	−0.778191
$580$	0	0
$581$	−15.8582	−0.657908
$582$	0	0
$583$	−27.9943	−1.15941
$584$	0	0
$585$	−2.71573	−0.112282
$586$	0	0
$587$	−42.1437	−1.73946	−0.869729	−	0.493529i	$-0.835707\pi$
$587$	−42.1437	−1.73946	−0.869729	+	0.493529i	$0.835707\pi$
$588$	0	0
$589$	19.8582	0.818242
$590$	0	0
$591$	19.3953	0.797817
$592$	0	0
$593$	10.2777	0.422055	0.211027	−	0.977480i	$-0.432319\pi$
$593$	10.2777	0.422055	0.211027	+	0.977480i	$0.432319\pi$
$594$	0	0
$595$	5.56541	0.228159
$596$	0	0
$597$	−1.10005	−0.0450220
$598$	0	0
$599$	6.73566	0.275212	0.137606	−	0.990487i	$-0.456059\pi$
$599$	6.73566	0.275212	0.137606	+	0.990487i	$0.456059\pi$
$600$	0	0
$601$	−45.6995	−1.86412	−0.932062	−	0.362300i	$-0.881992\pi$
$601$	−45.6995	−1.86412	−0.932062	+	0.362300i	$0.881992\pi$
$602$	0	0
$603$	−6.07114	−0.247236
$604$	0	0
$605$	−2.83149	−0.115117
$606$	0	0
$607$	−46.6518	−1.89354	−0.946769	−	0.321914i	$-0.895674\pi$
$607$	−46.6518	−1.89354	−0.946769	+	0.321914i	$0.895674\pi$
$608$	0	0
$609$	6.52411	0.264370
$610$	0	0
$611$	−11.4000	−0.461196
$612$	0	0
$613$	40.2204	1.62448	0.812242	−	0.583320i	$-0.198247\pi$
$613$	40.2204	1.62448	0.812242	+	0.583320i	$0.198247\pi$
$614$	0	0
$615$	−0.748712	−0.0301910
$616$	0	0
$617$	11.1211	0.447719	0.223860	−	0.974621i	$-0.428134\pi$
$617$	11.1211	0.447719	0.223860	+	0.974621i	$0.428134\pi$
$618$	0	0
$619$	45.1974	1.81664	0.908319	−	0.418278i	$-0.137366\pi$
$619$	45.1974	1.81664	0.908319	+	0.418278i	$0.137366\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	−4.66122	−0.186748
$624$	0	0
$625$	6.71058	0.268423
$626$	0	0
$627$	25.9152	1.03495
$628$	0	0
$629$	26.0215	1.03755
$630$	0	0
$631$	22.9463	0.913478	0.456739	−	0.889601i	$-0.349017\pi$
$631$	22.9463	0.913478	0.456739	+	0.889601i	$0.349017\pi$
$632$	0	0
$633$	−25.5235	−1.01447
$634$	0	0
$635$	15.1788	0.602352
$636$	0	0
$637$	2.34710	0.0929954
$638$	0	0
$639$	3.65290	0.144507
$640$	0	0
$641$	−12.5876	−0.497179	−0.248590	−	0.968609i	$-0.579967\pi$
$641$	−12.5876	−0.497179	−0.248590	+	0.968609i	$0.579967\pi$
$642$	0	0
$643$	−0.547672	−0.0215981	−0.0107990	−	0.999942i	$-0.503438\pi$
$643$	−0.547672	−0.0215981	−0.0107990	+	0.999942i	$0.503438\pi$
$644$	0	0
$645$	−5.51580	−0.217184
$646$	0	0
$647$	16.7393	0.658089	0.329045	−	0.944314i	$-0.393273\pi$
$647$	16.7393	0.658089	0.329045	+	0.944314i	$0.393273\pi$
$648$	0	0
$649$	−37.4560	−1.47027
$650$	0	0
$651$	2.80996	0.110131
$652$	0	0
$653$	2.63120	0.102967	0.0514834	−	0.998674i	$-0.483605\pi$
$653$	2.63120	0.102967	0.0514834	+	0.998674i	$0.483605\pi$
$654$	0	0
$655$	−1.48908	−0.0581833
$656$	0	0
$657$	11.8183	0.461075
$658$	0	0
$659$	28.7983	1.12182	0.560912	−	0.827876i	$-0.310451\pi$
$659$	28.7983	1.12182	0.560912	+	0.827876i	$0.310451\pi$
$660$	0	0
$661$	2.76143	0.107407	0.0537036	−	0.998557i	$-0.482897\pi$
$661$	2.76143	0.107407	0.0537036	+	0.998557i	$0.482897\pi$
$662$	0	0
$663$	11.2894	0.438446
$664$	0	0
$665$	−8.17701	−0.317091
$666$	0	0
$667$	−6.52411	−0.252615
$668$	0	0
$669$	17.1772	0.664108
$670$	0	0
$671$	−36.3404	−1.40290
$672$	0	0
$673$	32.8777	1.26734	0.633670	−	0.773603i	$-0.281548\pi$
$673$	32.8777	1.26734	0.633670	+	0.773603i	$0.281548\pi$
$674$	0	0
$675$	−3.66122	−0.140920
$676$	0	0
$677$	34.5014	1.32599	0.662997	−	0.748622i	$-0.269284\pi$
$677$	34.5014	1.32599	0.662997	+	0.748622i	$0.269284\pi$
$678$	0	0
$679$	4.90419	0.188206
$680$	0	0
$681$	26.0334	0.997604
$682$	0	0
$683$	−34.9805	−1.33849	−0.669246	−	0.743041i	$-0.733383\pi$
$683$	−34.9805	−1.33849	−0.669246	+	0.743041i	$0.733383\pi$
$684$	0	0
$685$	13.0924	0.500234
$686$	0	0
$687$	−7.01430	−0.267612
$688$	0	0
$689$	17.9179	0.682617
$690$	0	0
$691$	−34.0895	−1.29683	−0.648413	−	0.761289i	$-0.724567\pi$
$691$	−34.0895	−1.29683	−0.648413	+	0.761289i	$0.724567\pi$
$692$	0	0
$693$	3.66704	0.139299
$694$	0	0
$695$	10.3408	0.392250
$696$	0	0
$697$	3.11244	0.117892
$698$	0	0
$699$	0.671109	0.0253837
$700$	0	0
$701$	−43.6564	−1.64888	−0.824439	−	0.565950i	$-0.808509\pi$
$701$	−43.6564	−1.64888	−0.824439	+	0.565950i	$0.808509\pi$
$702$	0	0
$703$	−38.2323	−1.44196
$704$	0	0
$705$	5.61992	0.211658
$706$	0	0
$707$	5.80589	0.218353
$708$	0	0
$709$	11.8247	0.444087	0.222044	−	0.975037i	$-0.428727\pi$
$709$	11.8247	0.444087	0.222044	+	0.975037i	$0.428727\pi$
$710$	0	0
$711$	9.12408	0.342180
$712$	0	0
$713$	−2.80996	−0.105234
$714$	0	0
$715$	9.95868	0.372433
$716$	0	0
$717$	26.5571	0.991793
$718$	0	0
$719$	−33.8406	−1.26204	−0.631021	−	0.775766i	$-0.717364\pi$
$719$	−33.8406	−1.26204	−0.631021	+	0.775766i	$0.717364\pi$
$720$	0	0
$721$	17.1169	0.637466
$722$	0	0
$723$	−4.97284	−0.184942
$724$	0	0
$725$	23.8862	0.887110
$726$	0	0
$727$	−43.3730	−1.60862	−0.804308	−	0.594212i	$-0.797464\pi$
$727$	−43.3730	−1.60862	−0.804308	+	0.594212i	$0.797464\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	22.9295	0.848078
$732$	0	0
$733$	−35.3627	−1.30615	−0.653075	−	0.757293i	$-0.726521\pi$
$733$	−35.3627	−1.30615	−0.653075	+	0.757293i	$0.726521\pi$
$734$	0	0
$735$	−1.15706	−0.0426787
$736$	0	0
$737$	22.2631	0.820072
$738$	0	0
$739$	20.9639	0.771169	0.385584	−	0.922673i	$-0.374000\pi$
$739$	20.9639	0.771169	0.385584	+	0.922673i	$0.374000\pi$
$740$	0	0
$741$	−16.5871	−0.609343
$742$	0	0
$743$	3.66859	0.134587	0.0672937	−	0.997733i	$-0.478564\pi$
$743$	3.66859	0.134587	0.0672937	+	0.997733i	$0.478564\pi$
$744$	0	0
$745$	−1.94031	−0.0710873
$746$	0	0
$747$	15.8582	0.580221
$748$	0	0
$749$	−8.34710	−0.304996
$750$	0	0
$751$	−26.4450	−0.964990	−0.482495	−	0.875899i	$-0.660269\pi$
$751$	−26.4450	−0.964990	−0.482495	+	0.875899i	$0.660269\pi$
$752$	0	0
$753$	14.4281	0.525790
$754$	0	0
$755$	−21.9968	−0.800547
$756$	0	0
$757$	31.6239	1.14939	0.574695	−	0.818368i	$-0.305121\pi$
$757$	31.6239	1.14939	0.574695	+	0.818368i	$0.305121\pi$
$758$	0	0
$759$	−3.66704	−0.133105
$760$	0	0
$761$	13.8173	0.500878	0.250439	−	0.968132i	$-0.419425\pi$
$761$	13.8173	0.500878	0.250439	+	0.968132i	$0.419425\pi$
$762$	0	0
$763$	−16.3771	−0.592891
$764$	0	0
$765$	−5.56541	−0.201218
$766$	0	0
$767$	23.9738	0.865644
$768$	0	0
$769$	24.1624	0.871319	0.435659	−	0.900112i	$-0.356515\pi$
$769$	24.1624	0.871319	0.435659	+	0.900112i	$0.356515\pi$
$770$	0	0
$771$	5.44889	0.196237
$772$	0	0
$773$	24.9557	0.897595	0.448798	−	0.893633i	$-0.351852\pi$
$773$	24.9557	0.897595	0.448798	+	0.893633i	$0.351852\pi$
$774$	0	0
$775$	10.2879	0.369551
$776$	0	0
$777$	−5.40993	−0.194080
$778$	0	0
$779$	−4.57297	−0.163844
$780$	0	0
$781$	−13.3953	−0.479322
$782$	0	0
$783$	−6.52411	−0.233153
$784$	0	0
$785$	−14.5456	−0.519156
$786$	0	0
$787$	−27.6309	−0.984935	−0.492468	−	0.870331i	$-0.663905\pi$
$787$	−27.6309	−0.984935	−0.492468	+	0.870331i	$0.663905\pi$
$788$	0	0
$789$	−15.8128	−0.562951
$790$	0	0
$791$	14.1770	0.504077
$792$	0	0
$793$	23.2598	0.825979
$794$	0	0
$795$	−8.83305	−0.313276
$796$	0	0
$797$	−47.8222	−1.69395	−0.846975	−	0.531632i	$-0.821579\pi$
$797$	−47.8222	−1.69395	−0.846975	+	0.531632i	$0.821579\pi$
$798$	0	0
$799$	−23.3623	−0.826500
$800$	0	0
$801$	4.66122	0.164696
$802$	0	0
$803$	−43.3380	−1.52937
$804$	0	0
$805$	1.15706	0.0407810
$806$	0	0
$807$	11.9011	0.418937
$808$	0	0
$809$	−53.9350	−1.89625	−0.948127	−	0.317891i	$-0.897025\pi$
$809$	−53.9350	−1.89625	−0.948127	+	0.317891i	$0.897025\pi$
$810$	0	0
$811$	−29.8321	−1.04755	−0.523774	−	0.851857i	$-0.675476\pi$
$811$	−29.8321	−1.04755	−0.523774	+	0.851857i	$0.675476\pi$
$812$	0	0
$813$	16.6664	0.584516
$814$	0	0
$815$	−18.1357	−0.635266
$816$	0	0
$817$	−33.6893	−1.17864
$818$	0	0
$819$	−2.34710	−0.0820143
$820$	0	0
$821$	38.4517	1.34198	0.670988	−	0.741469i	$-0.265870\pi$
$821$	38.4517	1.34198	0.670988	+	0.741469i	$0.265870\pi$
$822$	0	0
$823$	−22.4705	−0.783274	−0.391637	−	0.920120i	$-0.628091\pi$
$823$	−22.4705	−0.783274	−0.391637	+	0.920120i	$0.628091\pi$
$824$	0	0
$825$	13.4258	0.467427
$826$	0	0
$827$	−15.5282	−0.539968	−0.269984	−	0.962865i	$-0.587018\pi$
$827$	−15.5282	−0.539968	−0.269984	+	0.962865i	$0.587018\pi$
$828$	0	0
$829$	−35.6581	−1.23846	−0.619228	−	0.785211i	$-0.712554\pi$
$829$	−35.6581	−1.23846	−0.619228	+	0.785211i	$0.712554\pi$
$830$	0	0
$831$	20.1299	0.698298
$832$	0	0
$833$	4.80996	0.166655
$834$	0	0
$835$	5.74040	0.198655
$836$	0	0
$837$	−2.80996	−0.0971264
$838$	0	0
$839$	−16.9411	−0.584873	−0.292436	−	0.956285i	$-0.594466\pi$
$839$	−16.9411	−0.584873	−0.292436	+	0.956285i	$0.594466\pi$
$840$	0	0
$841$	13.5640	0.467725
$842$	0	0
$843$	19.4022	0.668249
$844$	0	0
$845$	8.66768	0.298177
$846$	0	0
$847$	−2.44715	−0.0840850
$848$	0	0
$849$	10.8313	0.371729
$850$	0	0
$851$	5.40993	0.185450
$852$	0	0
$853$	−2.83149	−0.0969485	−0.0484742	−	0.998824i	$-0.515436\pi$
$853$	−2.83149	−0.0969485	−0.0484742	+	0.998824i	$0.515436\pi$
$854$	0	0
$855$	8.17701	0.279648
$856$	0	0
$857$	−3.73125	−0.127457	−0.0637286	−	0.997967i	$-0.520299\pi$
$857$	−3.73125	−0.127457	−0.0637286	+	0.997967i	$0.520299\pi$
$858$	0	0
$859$	−3.83133	−0.130723	−0.0653616	−	0.997862i	$-0.520820\pi$
$859$	−3.83133	−0.130723	−0.0653616	+	0.997862i	$0.520820\pi$
$860$	0	0
$861$	−0.647082	−0.0220525
$862$	0	0
$863$	21.3080	0.725333	0.362667	−	0.931919i	$-0.381866\pi$
$863$	21.3080	0.725333	0.362667	+	0.931919i	$0.381866\pi$
$864$	0	0
$865$	−14.9309	−0.507666
$866$	0	0
$867$	6.13572	0.208380
$868$	0	0
$869$	−33.4583	−1.13500
$870$	0	0
$871$	−14.2496	−0.482828
$872$	0	0
$873$	−4.90419	−0.165982
$874$	0	0
$875$	−10.0215	−0.338790
$876$	0	0
$877$	−34.4456	−1.16314	−0.581572	−	0.813495i	$-0.697562\pi$
$877$	−34.4456	−1.16314	−0.581572	+	0.813495i	$0.697562\pi$
$878$	0	0
$879$	8.89430	0.299997
$880$	0	0
$881$	5.70600	0.192240	0.0961201	−	0.995370i	$-0.469357\pi$
$881$	5.70600	0.192240	0.0961201	+	0.995370i	$0.469357\pi$
$882$	0	0
$883$	−5.89756	−0.198469	−0.0992344	−	0.995064i	$-0.531639\pi$
$883$	−5.89756	−0.198469	−0.0992344	+	0.995064i	$0.531639\pi$
$884$	0	0
$885$	−11.8185	−0.397273
$886$	0	0
$887$	41.8153	1.40402	0.702010	−	0.712167i	$-0.252286\pi$
$887$	41.8153	1.40402	0.702010	+	0.712167i	$0.252286\pi$
$888$	0	0
$889$	13.1184	0.439978
$890$	0	0
$891$	−3.66704	−0.122850
$892$	0	0
$893$	34.3253	1.14865
$894$	0	0
$895$	−0.304031	−0.0101626
$896$	0	0
$897$	2.34710	0.0783673
$898$	0	0
$899$	18.3325	0.611423
$900$	0	0
$901$	36.7195	1.22330
$902$	0	0
$903$	−4.76709	−0.158639
$904$	0	0
$905$	14.0514	0.467083
$906$	0	0
$907$	39.9593	1.32683	0.663414	−	0.748253i	$-0.269107\pi$
$907$	39.9593	1.32683	0.663414	+	0.748253i	$0.269107\pi$
$908$	0	0
$909$	−5.80589	−0.192569
$910$	0	0
$911$	25.1107	0.831954	0.415977	−	0.909375i	$-0.363440\pi$
$911$	25.1107	0.831954	0.415977	+	0.909375i	$0.363440\pi$
$912$	0	0
$913$	−58.1525	−1.92457
$914$	0	0
$915$	−11.4665	−0.379070
$916$	0	0
$917$	−1.28696	−0.0424990
$918$	0	0
$919$	−37.6701	−1.24262	−0.621310	−	0.783565i	$-0.713399\pi$
$919$	−37.6701	−1.24262	−0.621310	+	0.783565i	$0.713399\pi$
$920$	0	0
$921$	−19.7442	−0.650592
$922$	0	0
$923$	8.57372	0.282207
$924$	0	0
$925$	−19.8069	−0.651247
$926$	0	0
$927$	−17.1169	−0.562192
$928$	0	0
$929$	−15.5425	−0.509933	−0.254967	−	0.966950i	$-0.582064\pi$
$929$	−15.5425	−0.509933	−0.254967	+	0.966950i	$0.582064\pi$
$930$	0	0
$931$	−7.06707	−0.231614
$932$	0	0
$933$	28.5898	0.935988
$934$	0	0
$935$	20.4085	0.667431
$936$	0	0
$937$	18.2576	0.596449	0.298224	−	0.954496i	$-0.403606\pi$
$937$	18.2576	0.596449	0.298224	+	0.954496i	$0.403606\pi$
$938$	0	0
$939$	12.6570	0.413045
$940$	0	0
$941$	−0.0776022	−0.00252976	−0.00126488	−	0.999999i	$-0.500403\pi$
$941$	−0.0776022	−0.00252976	−0.00126488	+	0.999999i	$0.500403\pi$
$942$	0	0
$943$	0.647082	0.0210719
$944$	0	0
$945$	1.15706	0.0376391
$946$	0	0
$947$	−10.0364	−0.326140	−0.163070	−	0.986615i	$-0.552140\pi$
$947$	−10.0364	−0.326140	−0.163070	+	0.986615i	$0.552140\pi$
$948$	0	0
$949$	27.7387	0.900435
$950$	0	0
$951$	4.83130	0.156666
$952$	0	0
$953$	51.2265	1.65939	0.829695	−	0.558218i	$-0.188515\pi$
$953$	51.2265	1.65939	0.829695	+	0.558218i	$0.188515\pi$
$954$	0	0
$955$	3.10698	0.100540
$956$	0	0
$957$	23.9241	0.773358
$958$	0	0
$959$	11.3152	0.365388
$960$	0	0
$961$	−23.1041	−0.745294
$962$	0	0
$963$	8.34710	0.268981
$964$	0	0
$965$	21.6661	0.697456
$966$	0	0
$967$	−12.2315	−0.393339	−0.196670	−	0.980470i	$-0.563013\pi$
$967$	−12.2315	−0.393339	−0.196670	+	0.980470i	$0.563013\pi$
$968$	0	0
$969$	−33.9923	−1.09199
$970$	0	0
$971$	−33.5753	−1.07748	−0.538741	−	0.842471i	$-0.681100\pi$
$971$	−33.5753	−1.07748	−0.538741	+	0.842471i	$0.681100\pi$
$972$	0	0
$973$	8.93717	0.286513
$974$	0	0
$975$	−8.59323	−0.275204
$976$	0	0
$977$	23.7149	0.758707	0.379353	−	0.925252i	$-0.376146\pi$
$977$	23.7149	0.758707	0.379353	+	0.925252i	$0.376146\pi$
$978$	0	0
$979$	−17.0928	−0.546290
$980$	0	0
$981$	16.3771	0.522881
$982$	0	0
$983$	1.47810	0.0471441	0.0235721	−	0.999722i	$-0.492496\pi$
$983$	1.47810	0.0471441	0.0235721	+	0.999722i	$0.492496\pi$
$984$	0	0
$985$	−22.4415	−0.715046
$986$	0	0
$987$	4.85708	0.154602
$988$	0	0
$989$	4.76709	0.151584
$990$	0	0
$991$	−20.5203	−0.651850	−0.325925	−	0.945396i	$-0.605676\pi$
$991$	−20.5203	−0.651850	−0.325925	+	0.945396i	$0.605676\pi$
$992$	0	0
$993$	6.84544	0.217233
$994$	0	0
$995$	1.27282	0.0403512
$996$	0	0
$997$	−22.7211	−0.719583	−0.359791	−	0.933033i	$-0.617152\pi$
$997$	−22.7211	−0.719583	−0.359791	+	0.933033i	$0.617152\pi$
$998$	0	0
$999$	5.40993	0.171163

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7728.2.a.cd.1.1		4
4.3	odd	2		483.2.a.i.1.3	✓	4
12.11	even	2		1449.2.a.p.1.2		4
28.27	even	2		3381.2.a.w.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.a.i.1.3	✓	4	4.3	odd	2
1449.2.a.p.1.2		4	12.11	even	2
3381.2.a.w.1.3		4	28.27	even	2
7728.2.a.cd.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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.15706 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.15706 q^{5} -1.00000 q^{7} +1.00000 q^{9} -3.66704 q^{11} +2.34710 q^{13} -1.15706 q^{15} +4.80996 q^{17} -7.06707 q^{19} -1.00000 q^{21} +1.00000 q^{23} -3.66122 q^{25} +1.00000 q^{27} -6.52411 q^{29} -2.80996 q^{31} -3.66704 q^{33} +1.15706 q^{35} +5.40993 q^{37} +2.34710 q^{39} +0.647082 q^{41} +4.76709 q^{43} -1.15706 q^{45} -4.85708 q^{47} +1.00000 q^{49} +4.80996 q^{51} +7.63405 q^{53} +4.24297 q^{55} -7.06707 q^{57} +10.2142 q^{59} +9.91001 q^{61} -1.00000 q^{63} -2.71573 q^{65} -6.07114 q^{67} +1.00000 q^{69} +3.65290 q^{71} +11.8183 q^{73} -3.66122 q^{75} +3.66704 q^{77} +9.12408 q^{79} +1.00000 q^{81} +15.8582 q^{83} -5.56541 q^{85} -6.52411 q^{87} +4.66122 q^{89} -2.34710 q^{91} -2.80996 q^{93} +8.17701 q^{95} -4.90419 q^{97} -3.66704 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 5 q^{5} - 4 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + 5 * q^5 - 4 * q^7 + 4 * q^9	\( 4 q + 4 q^{3} + 5 q^{5} - 4 q^{7} + 4 q^{9} + 5 q^{11} + 7 q^{13} + 5 q^{15} + 12 q^{17} - 3 q^{19} - 4 q^{21} + 4 q^{23} + 7 q^{25} + 4 q^{27} + 6 q^{29} - 4 q^{31} + 5 q^{33} - 5 q^{35} + 20 q^{37} + 7 q^{39} + 3 q^{41} - 9 q^{43} + 5 q^{45} - 7 q^{47} + 4 q^{49} + 12 q^{51} - 6 q^{53} + 21 q^{55} - 3 q^{57} + 2 q^{59} + 24 q^{61} - 4 q^{63} - 14 q^{65} - q^{67} + 4 q^{69} + 17 q^{71} + 16 q^{73} + 7 q^{75} - 5 q^{77} + 10 q^{79} + 4 q^{81} - 8 q^{83} + 17 q^{85} + 6 q^{87} - 3 q^{89} - 7 q^{91} - 4 q^{93} + 3 q^{95} - 2 q^{97} + 5 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 + 5 * q^5 - 4 * q^7 + 4 * q^9 + 5 * q^11 + 7 * q^13 + 5 * q^15 + 12 * q^17 - 3 * q^19 - 4 * q^21 + 4 * q^23 + 7 * q^25 + 4 * q^27 + 6 * q^29 - 4 * q^31 + 5 * q^33 - 5 * q^35 + 20 * q^37 + 7 * q^39 + 3 * q^41 - 9 * q^43 + 5 * q^45 - 7 * q^47 + 4 * q^49 + 12 * q^51 - 6 * q^53 + 21 * q^55 - 3 * q^57 + 2 * q^59 + 24 * q^61 - 4 * q^63 - 14 * q^65 - q^67 + 4 * q^69 + 17 * q^71 + 16 * q^73 + 7 * q^75 - 5 * q^77 + 10 * q^79 + 4 * q^81 - 8 * q^83 + 17 * q^85 + 6 * q^87 - 3 * q^89 - 7 * q^91 - 4 * q^93 + 3 * q^95 - 2 * q^97 + 5 * q^99

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