LMFDB - Embedded newform 7728.2.a.by.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:	$3$
Coefficient field:	3.3.1229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 7x + 6 $ x^3 - x^2 - 7*x + 6
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3864)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.59261$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -2.59261 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -2.59261 q^{5} -1.00000 q^{7} +1.00000 q^{9} +1.87096 q^{11} +0.721651 q^{13} -2.59261 q^{15} +5.31427 q^{17} +7.31427 q^{19} -1.00000 q^{21} -1.00000 q^{23} +1.72165 q^{25} +1.00000 q^{27} -5.31427 q^{29} -2.12904 q^{31} +1.87096 q^{33} +2.59261 q^{35} -6.49950 q^{37} +0.721651 q^{39} +4.12904 q^{41} +1.27835 q^{43} -2.59261 q^{45} -12.9272 q^{47} +1.00000 q^{49} +5.31427 q^{51} +10.3345 q^{53} -4.85069 q^{55} +7.31427 q^{57} +10.0359 q^{59} -7.09211 q^{61} -1.00000 q^{63} -1.87096 q^{65} -2.03592 q^{67} -1.00000 q^{69} +11.9069 q^{71} -5.57234 q^{73} +1.72165 q^{75} -1.87096 q^{77} +11.3143 q^{79} +1.00000 q^{81} -1.87096 q^{83} -13.7778 q^{85} -5.31427 q^{87} -1.90688 q^{89} -0.721651 q^{91} -2.12904 q^{93} -18.9631 q^{95} +7.87096 q^{97} +1.87096 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} + q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 + q^5 - 3 * q^7 + 3 * q^9	$ 3 q + 3 q^{3} + q^{5} - 3 q^{7} + 3 q^{9} + 2 q^{11} - 3 q^{13} + q^{15} + 2 q^{17} + 8 q^{19} - 3 q^{21} - 3 q^{23} + 3 q^{27} - 2 q^{29} - 10 q^{31} + 2 q^{33} - q^{35} + 12 q^{37} - 3 q^{39} + 16 q^{41} + 9 q^{43} + q^{45} - 14 q^{47} + 3 q^{49} + 2 q^{51} + 15 q^{53} - 13 q^{55} + 8 q^{57} + 11 q^{59} + 19 q^{61} - 3 q^{63} - 2 q^{65} + 13 q^{67} - 3 q^{69} + 13 q^{71} - 10 q^{73} - 2 q^{77} + 20 q^{79} + 3 q^{81} - 2 q^{83} - 15 q^{85} - 2 q^{87} + 17 q^{89} + 3 q^{91} - 10 q^{93} - 13 q^{95} + 20 q^{97} + 2 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 + q^5 - 3 * q^7 + 3 * q^9 + 2 * q^11 - 3 * q^13 + q^15 + 2 * q^17 + 8 * q^19 - 3 * q^21 - 3 * q^23 + 3 * q^27 - 2 * q^29 - 10 * q^31 + 2 * q^33 - q^35 + 12 * q^37 - 3 * q^39 + 16 * q^41 + 9 * q^43 + q^45 - 14 * q^47 + 3 * q^49 + 2 * q^51 + 15 * q^53 - 13 * q^55 + 8 * q^57 + 11 * q^59 + 19 * q^61 - 3 * q^63 - 2 * q^65 + 13 * q^67 - 3 * q^69 + 13 * q^71 - 10 * q^73 - 2 * q^77 + 20 * q^79 + 3 * q^81 - 2 * q^83 - 15 * q^85 - 2 * q^87 + 17 * q^89 + 3 * q^91 - 10 * q^93 - 13 * q^95 + 20 * q^97 + 2 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−2.59261	−1.15945	−0.579726	−	0.814811i	$-0.696841\pi$
$5$	−2.59261	−1.15945	−0.579726	+	0.814811i	$0.696841\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.87096	0.564117	0.282058	−	0.959397i	$-0.408983\pi$
$11$	1.87096	0.564117	0.282058	+	0.959397i	$0.408983\pi$
$12$	0	0
$13$	0.721651	0.200150	0.100075	−	0.994980i	$-0.468092\pi$
$13$	0.721651	0.200150	0.100075	+	0.994980i	$0.468092\pi$
$14$	0	0
$15$	−2.59261	−0.669410
$16$	0	0
$17$	5.31427	1.28890	0.644449	−	0.764647i	$-0.277087\pi$
$17$	5.31427	1.28890	0.644449	+	0.764647i	$0.277087\pi$
$18$	0	0
$19$	7.31427	1.67801	0.839004	−	0.544125i	$-0.183138\pi$
$19$	7.31427	1.67801	0.839004	+	0.544125i	$0.183138\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	1.72165	0.344330
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−5.31427	−0.986834	−0.493417	−	0.869793i	$-0.664252\pi$
$29$	−5.31427	−0.986834	−0.493417	+	0.869793i	$0.664252\pi$
$30$	0	0
$31$	−2.12904	−0.382386	−0.191193	−	0.981552i	$-0.561236\pi$
$31$	−2.12904	−0.382386	−0.191193	+	0.981552i	$0.561236\pi$
$32$	0	0
$33$	1.87096	0.325693
$34$	0	0
$35$	2.59261	0.438232
$36$	0	0
$37$	−6.49950	−1.06851	−0.534255	−	0.845323i	$-0.679408\pi$
$37$	−6.49950	−1.06851	−0.534255	+	0.845323i	$0.679408\pi$
$38$	0	0
$39$	0.721651	0.115557
$40$	0	0
$41$	4.12904	0.644847	0.322424	−	0.946595i	$-0.395502\pi$
$41$	4.12904	0.644847	0.322424	+	0.946595i	$0.395502\pi$
$42$	0	0
$43$	1.27835	0.194946	0.0974732	−	0.995238i	$-0.468924\pi$
$43$	1.27835	0.194946	0.0974732	+	0.995238i	$0.468924\pi$
$44$	0	0
$45$	−2.59261	−0.386484
$46$	0	0
$47$	−12.9272	−1.88562	−0.942810	−	0.333331i	$-0.891827\pi$
$47$	−12.9272	−1.88562	−0.942810	+	0.333331i	$0.891827\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	5.31427	0.744146
$52$	0	0
$53$	10.3345	1.41956	0.709779	−	0.704424i	$-0.248795\pi$
$53$	10.3345	1.41956	0.709779	+	0.704424i	$0.248795\pi$
$54$	0	0
$55$	−4.85069	−0.654067
$56$	0	0
$57$	7.31427	0.968798
$58$	0	0
$59$	10.0359	1.30657	0.653283	−	0.757114i	$-0.273391\pi$
$59$	10.0359	1.30657	0.653283	+	0.757114i	$0.273391\pi$
$60$	0	0
$61$	−7.09211	−0.908052	−0.454026	−	0.890989i	$-0.650013\pi$
$61$	−7.09211	−0.908052	−0.454026	+	0.890989i	$0.650013\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	−1.87096	−0.232064
$66$	0	0
$67$	−2.03592	−0.248727	−0.124363	−	0.992237i	$-0.539689\pi$
$67$	−2.03592	−0.248727	−0.124363	+	0.992237i	$0.539689\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	11.9069	1.41309	0.706543	−	0.707670i	$-0.250254\pi$
$71$	11.9069	1.41309	0.706543	+	0.707670i	$0.250254\pi$
$72$	0	0
$73$	−5.57234	−0.652193	−0.326096	−	0.945336i	$-0.605733\pi$
$73$	−5.57234	−0.652193	−0.326096	+	0.945336i	$0.605733\pi$
$74$	0	0
$75$	1.72165	0.198799
$76$	0	0
$77$	−1.87096	−0.213216
$78$	0	0
$79$	11.3143	1.27295	0.636477	−	0.771296i	$-0.280391\pi$
$79$	11.3143	1.27295	0.636477	+	0.771296i	$0.280391\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−1.87096	−0.205365	−0.102682	−	0.994714i	$-0.532743\pi$
$83$	−1.87096	−0.205365	−0.102682	+	0.994714i	$0.532743\pi$
$84$	0	0
$85$	−13.7778	−1.49442
$86$	0	0
$87$	−5.31427	−0.569749
$88$	0	0
$89$	−1.90688	−0.202129	−0.101064	−	0.994880i	$-0.532225\pi$
$89$	−1.90688	−0.202129	−0.101064	+	0.994880i	$0.532225\pi$
$90$	0	0
$91$	−0.721651	−0.0756496
$92$	0	0
$93$	−2.12904	−0.220771
$94$	0	0
$95$	−18.9631	−1.94557
$96$	0	0
$97$	7.87096	0.799175	0.399588	−	0.916695i	$-0.369153\pi$
$97$	7.87096	0.799175	0.399588	+	0.916695i	$0.369153\pi$
$98$	0	0
$99$	1.87096	0.188039
$100$	0	0
$101$	12.0359	1.19762	0.598809	−	0.800892i	$-0.295641\pi$
$101$	12.0359	1.19762	0.598809	+	0.800892i	$0.295641\pi$
$102$	0	0
$103$	−10.3705	−1.02183	−0.510916	−	0.859631i	$-0.670694\pi$
$103$	−10.3705	−1.02183	−0.510916	+	0.859631i	$0.670694\pi$
$104$	0	0
$105$	2.59261	0.253013
$106$	0	0
$107$	−14.5354	−1.40519	−0.702596	−	0.711589i	$-0.747976\pi$
$107$	−14.5354	−1.40519	−0.702596	+	0.711589i	$0.747976\pi$
$108$	0	0
$109$	1.14931	0.110084	0.0550421	−	0.998484i	$-0.482471\pi$
$109$	1.14931	0.110084	0.0550421	+	0.998484i	$0.482471\pi$
$110$	0	0
$111$	−6.49950	−0.616905
$112$	0	0
$113$	19.7778	1.86054	0.930272	−	0.366872i	$-0.119571\pi$
$113$	19.7778	1.86054	0.930272	+	0.366872i	$0.119571\pi$
$114$	0	0
$115$	2.59261	0.241763
$116$	0	0
$117$	0.721651	0.0667167
$118$	0	0
$119$	−5.31427	−0.487158
$120$	0	0
$121$	−7.49950	−0.681772
$122$	0	0
$123$	4.12904	0.372303
$124$	0	0
$125$	8.49950	0.760218
$126$	0	0
$127$	6.46358	0.573550	0.286775	−	0.957998i	$-0.407417\pi$
$127$	6.46358	0.573550	0.286775	+	0.957998i	$0.407417\pi$
$128$	0	0
$129$	1.27835	0.112552
$130$	0	0
$131$	−6.12904	−0.535496	−0.267748	−	0.963489i	$-0.586280\pi$
$131$	−6.12904	−0.535496	−0.267748	+	0.963489i	$0.586280\pi$
$132$	0	0
$133$	−7.31427	−0.634227
$134$	0	0
$135$	−2.59261	−0.223137
$136$	0	0
$137$	7.87096	0.672462	0.336231	−	0.941780i	$-0.390848\pi$
$137$	7.87096	0.672462	0.336231	+	0.941780i	$0.390848\pi$
$138$	0	0
$139$	−14.9631	−1.26915	−0.634576	−	0.772861i	$-0.718825\pi$
$139$	−14.9631	−1.26915	−0.634576	+	0.772861i	$0.718825\pi$
$140$	0	0
$141$	−12.9272	−1.08866
$142$	0	0
$143$	1.35018	0.112908
$144$	0	0
$145$	13.7778	1.14419
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	−11.1852	−0.916330	−0.458165	−	0.888867i	$-0.651493\pi$
$149$	−11.1852	−0.916330	−0.458165	+	0.888867i	$0.651493\pi$
$150$	0	0
$151$	−12.9272	−1.05200	−0.525999	−	0.850485i	$-0.676308\pi$
$151$	−12.9272	−1.05200	−0.525999	+	0.850485i	$0.676308\pi$
$152$	0	0
$153$	5.31427	0.429633
$154$	0	0
$155$	5.51977	0.443359
$156$	0	0
$157$	14.3299	1.14365	0.571826	−	0.820375i	$-0.306235\pi$
$157$	14.3299	1.14365	0.571826	+	0.820375i	$0.306235\pi$
$158$	0	0
$159$	10.3345	0.819582
$160$	0	0
$161$	1.00000	0.0788110
$162$	0	0
$163$	13.7778	1.07916	0.539582	−	0.841933i	$-0.318582\pi$
$163$	13.7778	1.07916	0.539582	+	0.841933i	$0.318582\pi$
$164$	0	0
$165$	−4.85069	−0.377626
$166$	0	0
$167$	20.4995	1.58630	0.793149	−	0.609027i	$-0.208440\pi$
$167$	20.4995	1.58630	0.793149	+	0.609027i	$0.208440\pi$
$168$	0	0
$169$	−12.4792	−0.959940
$170$	0	0
$171$	7.31427	0.559336
$172$	0	0
$173$	22.7576	1.73023	0.865113	−	0.501577i	$-0.167247\pi$
$173$	22.7576	1.73023	0.865113	+	0.501577i	$0.167247\pi$
$174$	0	0
$175$	−1.72165	−0.130145
$176$	0	0
$177$	10.0359	0.754346
$178$	0	0
$179$	−16.6644	−1.24556	−0.622780	−	0.782397i	$-0.713997\pi$
$179$	−16.6644	−1.24556	−0.622780	+	0.782397i	$0.713997\pi$
$180$	0	0
$181$	25.1280	1.86775	0.933876	−	0.357598i	$-0.116404\pi$
$181$	25.1280	1.86775	0.933876	+	0.357598i	$0.116404\pi$
$182$	0	0
$183$	−7.09211	−0.524264
$184$	0	0
$185$	16.8507	1.23889
$186$	0	0
$187$	9.94280	0.727089
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−10.5567	−0.763856	−0.381928	−	0.924192i	$-0.624740\pi$
$191$	−10.5567	−0.763856	−0.381928	+	0.924192i	$0.624740\pi$
$192$	0	0
$193$	7.44330	0.535781	0.267890	−	0.963449i	$-0.413674\pi$
$193$	7.44330	0.535781	0.267890	+	0.963449i	$0.413674\pi$
$194$	0	0
$195$	−1.87096	−0.133982
$196$	0	0
$197$	7.02028	0.500174	0.250087	−	0.968223i	$-0.419541\pi$
$197$	7.02028	0.500174	0.250087	+	0.968223i	$0.419541\pi$
$198$	0	0
$199$	−3.66546	−0.259837	−0.129919	−	0.991525i	$-0.541472\pi$
$199$	−3.66546	−0.259837	−0.129919	+	0.991525i	$0.541472\pi$
$200$	0	0
$201$	−2.03592	−0.143603
$202$	0	0
$203$	5.31427	0.372988
$204$	0	0
$205$	−10.7050	−0.747670
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	13.6847	0.946592
$210$	0	0
$211$	−0.685734	−0.0472079	−0.0236039	−	0.999721i	$-0.507514\pi$
$211$	−0.685734	−0.0472079	−0.0236039	+	0.999721i	$0.507514\pi$
$212$	0	0
$213$	11.9069	0.815846
$214$	0	0
$215$	−3.31427	−0.226031
$216$	0	0
$217$	2.12904	0.144528
$218$	0	0
$219$	−5.57234	−0.376544
$220$	0	0
$221$	3.83505	0.257973
$222$	0	0
$223$	28.6644	1.91951	0.959757	−	0.280833i	$-0.0906106\pi$
$223$	28.6644	1.91951	0.959757	+	0.280833i	$0.0906106\pi$
$224$	0	0
$225$	1.72165	0.114777
$226$	0	0
$227$	−24.1483	−1.60278	−0.801390	−	0.598143i	$-0.795906\pi$
$227$	−24.1483	−1.60278	−0.801390	+	0.598143i	$0.795906\pi$
$228$	0	0
$229$	15.7778	1.04263	0.521315	−	0.853365i	$-0.325442\pi$
$229$	15.7778	1.04263	0.521315	+	0.853365i	$0.325442\pi$
$230$	0	0
$231$	−1.87096	−0.123100
$232$	0	0
$233$	19.7778	1.29569	0.647845	−	0.761772i	$-0.275671\pi$
$233$	19.7778	1.29569	0.647845	+	0.761772i	$0.275671\pi$
$234$	0	0
$235$	33.5151	2.18629
$236$	0	0
$237$	11.3143	0.734941
$238$	0	0
$239$	16.4064	1.06124	0.530620	−	0.847610i	$-0.321959\pi$
$239$	16.4064	1.06124	0.530620	+	0.847610i	$0.321959\pi$
$240$	0	0
$241$	14.4995	0.933995	0.466997	−	0.884259i	$-0.345336\pi$
$241$	14.4995	0.933995	0.466997	+	0.884259i	$0.345336\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−2.59261	−0.165636
$246$	0	0
$247$	5.27835	0.335853
$248$	0	0
$249$	−1.87096	−0.118567
$250$	0	0
$251$	−7.55569	−0.476911	−0.238455	−	0.971153i	$-0.576641\pi$
$251$	−7.55569	−0.476911	−0.238455	+	0.971153i	$0.576641\pi$
$252$	0	0
$253$	−1.87096	−0.117626
$254$	0	0
$255$	−13.7778	−0.862802
$256$	0	0
$257$	−6.66908	−0.416006	−0.208003	−	0.978128i	$-0.566696\pi$
$257$	−6.66908	−0.416006	−0.208003	+	0.978128i	$0.566696\pi$
$258$	0	0
$259$	6.49950	0.403859
$260$	0	0
$261$	−5.31427	−0.328945
$262$	0	0
$263$	24.0552	1.48331	0.741653	−	0.670784i	$-0.234042\pi$
$263$	24.0552	1.48331	0.741653	+	0.670784i	$0.234042\pi$
$264$	0	0
$265$	−26.7935	−1.64591
$266$	0	0
$267$	−1.90688	−0.116699
$268$	0	0
$269$	8.72165	0.531768	0.265884	−	0.964005i	$-0.414336\pi$
$269$	8.72165	0.531768	0.265884	+	0.964005i	$0.414336\pi$
$270$	0	0
$271$	−22.5400	−1.36921	−0.684605	−	0.728914i	$-0.740025\pi$
$271$	−22.5400	−1.36921	−0.684605	+	0.728914i	$0.740025\pi$
$272$	0	0
$273$	−0.721651	−0.0436763
$274$	0	0
$275$	3.22115	0.194242
$276$	0	0
$277$	21.9787	1.32057	0.660286	−	0.751014i	$-0.270435\pi$
$277$	21.9787	1.32057	0.660286	+	0.751014i	$0.270435\pi$
$278$	0	0
$279$	−2.12904	−0.127462
$280$	0	0
$281$	14.9272	0.890480	0.445240	−	0.895411i	$-0.353118\pi$
$281$	14.9272	0.890480	0.445240	+	0.895411i	$0.353118\pi$
$282$	0	0
$283$	−32.6644	−1.94170	−0.970850	−	0.239688i	$-0.922955\pi$
$283$	−32.6644	−1.94170	−0.970850	+	0.239688i	$0.922955\pi$
$284$	0	0
$285$	−18.9631	−1.12328
$286$	0	0
$287$	−4.12904	−0.243729
$288$	0	0
$289$	11.2414	0.661260
$290$	0	0
$291$	7.87096	0.461404
$292$	0	0
$293$	−11.5151	−0.672721	−0.336361	−	0.941733i	$-0.609196\pi$
$293$	−11.5151	−0.672721	−0.336361	+	0.941733i	$0.609196\pi$
$294$	0	0
$295$	−26.0193	−1.51490
$296$	0	0
$297$	1.87096	0.108564
$298$	0	0
$299$	−0.721651	−0.0417342
$300$	0	0
$301$	−1.27835	−0.0736828
$302$	0	0
$303$	12.0359	0.691445
$304$	0	0
$305$	18.3871	1.05284
$306$	0	0
$307$	29.7566	1.69830	0.849148	−	0.528155i	$-0.177116\pi$
$307$	29.7566	1.69830	0.849148	+	0.528155i	$0.177116\pi$
$308$	0	0
$309$	−10.3705	−0.589955
$310$	0	0
$311$	15.4626	0.876802	0.438401	−	0.898780i	$-0.355545\pi$
$311$	15.4626	0.876802	0.438401	+	0.898780i	$0.355545\pi$
$312$	0	0
$313$	16.6285	0.939900	0.469950	−	0.882693i	$-0.344272\pi$
$313$	16.6285	0.939900	0.469950	+	0.882693i	$0.344272\pi$
$314$	0	0
$315$	2.59261	0.146077
$316$	0	0
$317$	16.9631	0.952741	0.476371	−	0.879245i	$-0.341952\pi$
$317$	16.9631	0.952741	0.476371	+	0.879245i	$0.341952\pi$
$318$	0	0
$319$	−9.94280	−0.556690
$320$	0	0
$321$	−14.5354	−0.811288
$322$	0	0
$323$	38.8700	2.16278
$324$	0	0
$325$	1.24243	0.0689177
$326$	0	0
$327$	1.14931	0.0635571
$328$	0	0
$329$	12.9272	0.712697
$330$	0	0
$331$	7.74193	0.425535	0.212767	−	0.977103i	$-0.431752\pi$
$331$	7.74193	0.425535	0.212767	+	0.977103i	$0.431752\pi$
$332$	0	0
$333$	−6.49950	−0.356170
$334$	0	0
$335$	5.27835	0.288387
$336$	0	0
$337$	−13.9787	−0.761469	−0.380735	−	0.924684i	$-0.624329\pi$
$337$	−13.9787	−0.761469	−0.380735	+	0.924684i	$0.624329\pi$
$338$	0	0
$339$	19.7778	1.07419
$340$	0	0
$341$	−3.98335	−0.215710
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	2.59261	0.139582
$346$	0	0
$347$	9.35482	0.502193	0.251096	−	0.967962i	$-0.419209\pi$
$347$	9.35482	0.502193	0.251096	+	0.967962i	$0.419209\pi$
$348$	0	0
$349$	−0.0359171	−0.00192260	−0.000961300	−	1.00000i	$-0.500306\pi$
$349$	−0.0359171	−0.00192260	−0.000961300		1.00000i	$0.500306\pi$
$350$	0	0
$351$	0.721651	0.0385189
$352$	0	0
$353$	11.0967	0.590620	0.295310	−	0.955401i	$-0.404577\pi$
$353$	11.0967	0.590620	0.295310	+	0.955401i	$0.404577\pi$
$354$	0	0
$355$	−30.8700	−1.63841
$356$	0	0
$357$	−5.31427	−0.281261
$358$	0	0
$359$	13.8497	0.730958	0.365479	−	0.930820i	$-0.380905\pi$
$359$	13.8497	0.730958	0.365479	+	0.930820i	$0.380905\pi$
$360$	0	0
$361$	34.4985	1.81571
$362$	0	0
$363$	−7.49950	−0.393621
$364$	0	0
$365$	14.4469	0.756187
$366$	0	0
$367$	−1.53642	−0.0802006	−0.0401003	−	0.999196i	$-0.512768\pi$
$367$	−1.53642	−0.0802006	−0.0401003	+	0.999196i	$0.512768\pi$
$368$	0	0
$369$	4.12904	0.214949
$370$	0	0
$371$	−10.3345	−0.536543
$372$	0	0
$373$	16.6119	0.860131	0.430065	−	0.902798i	$-0.358491\pi$
$373$	16.6119	0.860131	0.430065	+	0.902798i	$0.358491\pi$
$374$	0	0
$375$	8.49950	0.438912
$376$	0	0
$377$	−3.83505	−0.197515
$378$	0	0
$379$	−27.6275	−1.41913	−0.709565	−	0.704640i	$-0.751109\pi$
$379$	−27.6275	−1.41913	−0.709565	+	0.704640i	$0.751109\pi$
$380$	0	0
$381$	6.46358	0.331139
$382$	0	0
$383$	8.24142	0.421117	0.210559	−	0.977581i	$-0.432472\pi$
$383$	8.24142	0.421117	0.210559	+	0.977581i	$0.432472\pi$
$384$	0	0
$385$	4.85069	0.247214
$386$	0	0
$387$	1.27835	0.0649821
$388$	0	0
$389$	−20.4589	−1.03731	−0.518655	−	0.854984i	$-0.673567\pi$
$389$	−20.4589	−1.03731	−0.518655	+	0.854984i	$0.673567\pi$
$390$	0	0
$391$	−5.31427	−0.268754
$392$	0	0
$393$	−6.12904	−0.309169
$394$	0	0
$395$	−29.3335	−1.47593
$396$	0	0
$397$	−11.7014	−0.587275	−0.293638	−	0.955917i	$-0.594866\pi$
$397$	−11.7014	−0.587275	−0.293638	+	0.955917i	$0.594866\pi$
$398$	0	0
$399$	−7.31427	−0.366171
$400$	0	0
$401$	4.45894	0.222669	0.111335	−	0.993783i	$-0.464488\pi$
$401$	4.45894	0.222669	0.111335	+	0.993783i	$0.464488\pi$
$402$	0	0
$403$	−1.53642	−0.0765346
$404$	0	0
$405$	−2.59261	−0.128828
$406$	0	0
$407$	−12.1603	−0.602765
$408$	0	0
$409$	−6.75757	−0.334140	−0.167070	−	0.985945i	$-0.553431\pi$
$409$	−6.75757	−0.334140	−0.167070	+	0.985945i	$0.553431\pi$
$410$	0	0
$411$	7.87096	0.388246
$412$	0	0
$413$	−10.0359	−0.493835
$414$	0	0
$415$	4.85069	0.238111
$416$	0	0
$417$	−14.9631	−0.732745
$418$	0	0
$419$	11.4074	0.557287	0.278644	−	0.960395i	$-0.410115\pi$
$419$	11.4074	0.557287	0.278644	+	0.960395i	$0.410115\pi$
$420$	0	0
$421$	−37.3907	−1.82231	−0.911156	−	0.412061i	$-0.864809\pi$
$421$	−37.3907	−1.82231	−0.911156	+	0.412061i	$0.864809\pi$
$422$	0	0
$423$	−12.9272	−0.628540
$424$	0	0
$425$	9.14931	0.443807
$426$	0	0
$427$	7.09211	0.343211
$428$	0	0
$429$	1.35018	0.0651875
$430$	0	0
$431$	34.0193	1.63865	0.819325	−	0.573329i	$-0.194348\pi$
$431$	34.0193	1.63865	0.819325	+	0.573329i	$0.194348\pi$
$432$	0	0
$433$	−0.0405518	−0.00194879	−0.000974397	−	1.00000i	$-0.500310\pi$
$433$	−0.0405518	−0.00194879	−0.000974397		1.00000i	$0.500310\pi$
$434$	0	0
$435$	13.7778	0.660597
$436$	0	0
$437$	−7.31427	−0.349889
$438$	0	0
$439$	−14.8700	−0.709704	−0.354852	−	0.934922i	$-0.615469\pi$
$439$	−14.8700	−0.709704	−0.354852	+	0.934922i	$0.615469\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	14.3705	0.682761	0.341381	−	0.939925i	$-0.389106\pi$
$443$	14.3705	0.682761	0.341381	+	0.939925i	$0.389106\pi$
$444$	0	0
$445$	4.94381	0.234359
$446$	0	0
$447$	−11.1852	−0.529043
$448$	0	0
$449$	21.4792	1.01367	0.506834	−	0.862044i	$-0.330816\pi$
$449$	21.4792	1.01367	0.506834	+	0.862044i	$0.330816\pi$
$450$	0	0
$451$	7.72528	0.363769
$452$	0	0
$453$	−12.9272	−0.607371
$454$	0	0
$455$	1.87096	0.0877121
$456$	0	0
$457$	0.705001	0.0329785	0.0164893	−	0.999864i	$-0.494751\pi$
$457$	0.705001	0.0329785	0.0164893	+	0.999864i	$0.494751\pi$
$458$	0	0
$459$	5.31427	0.248049
$460$	0	0
$461$	30.9225	1.44021	0.720103	−	0.693867i	$-0.244095\pi$
$461$	30.9225	1.44021	0.720103	+	0.693867i	$0.244095\pi$
$462$	0	0
$463$	−5.53179	−0.257084	−0.128542	−	0.991704i	$-0.541030\pi$
$463$	−5.53179	−0.257084	−0.128542	+	0.991704i	$0.541030\pi$
$464$	0	0
$465$	5.51977	0.255973
$466$	0	0
$467$	28.2414	1.30686	0.653429	−	0.756988i	$-0.273330\pi$
$467$	28.2414	1.30686	0.653429	+	0.756988i	$0.273330\pi$
$468$	0	0
$469$	2.03592	0.0940099
$470$	0	0
$471$	14.3299	0.660287
$472$	0	0
$473$	2.39174	0.109973
$474$	0	0
$475$	12.5926	0.577789
$476$	0	0
$477$	10.3345	0.473186
$478$	0	0
$479$	26.9418	1.23100	0.615501	−	0.788136i	$-0.288954\pi$
$479$	26.9418	1.23100	0.615501	+	0.788136i	$0.288954\pi$
$480$	0	0
$481$	−4.69037	−0.213862
$482$	0	0
$483$	1.00000	0.0455016
$484$	0	0
$485$	−20.4064	−0.926606
$486$	0	0
$487$	−0.757568	−0.0343287	−0.0171643	−	0.999853i	$-0.505464\pi$
$487$	−0.757568	−0.0343287	−0.0171643	+	0.999853i	$0.505464\pi$
$488$	0	0
$489$	13.7778	0.623056
$490$	0	0
$491$	−32.5760	−1.47013	−0.735066	−	0.677995i	$-0.762849\pi$
$491$	−32.5760	−1.47013	−0.735066	+	0.677995i	$0.762849\pi$
$492$	0	0
$493$	−28.2414	−1.27193
$494$	0	0
$495$	−4.85069	−0.218022
$496$	0	0
$497$	−11.9069	−0.534097
$498$	0	0
$499$	27.8902	1.24854	0.624269	−	0.781209i	$-0.285397\pi$
$499$	27.8902	1.24854	0.624269	+	0.781209i	$0.285397\pi$
$500$	0	0
$501$	20.4995	0.915850
$502$	0	0
$503$	−15.0349	−0.670373	−0.335187	−	0.942152i	$-0.608799\pi$
$503$	−15.0349	−0.670373	−0.335187	+	0.942152i	$0.608799\pi$
$504$	0	0
$505$	−31.2045	−1.38858
$506$	0	0
$507$	−12.4792	−0.554222
$508$	0	0
$509$	−34.2966	−1.52017	−0.760085	−	0.649823i	$-0.774843\pi$
$509$	−34.2966	−1.52017	−0.760085	+	0.649823i	$0.774843\pi$
$510$	0	0
$511$	5.57234	0.246506
$512$	0	0
$513$	7.31427	0.322933
$514$	0	0
$515$	26.8866	1.18477
$516$	0	0
$517$	−24.1862	−1.06371
$518$	0	0
$519$	22.7576	0.998946
$520$	0	0
$521$	7.11340	0.311644	0.155822	−	0.987785i	$-0.450197\pi$
$521$	7.11340	0.311644	0.155822	+	0.987785i	$0.450197\pi$
$522$	0	0
$523$	−23.7419	−1.03816	−0.519081	−	0.854725i	$-0.673726\pi$
$523$	−23.7419	−1.03816	−0.519081	+	0.854725i	$0.673726\pi$
$524$	0	0
$525$	−1.72165	−0.0751390
$526$	0	0
$527$	−11.3143	−0.492857
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	10.0359	0.435522
$532$	0	0
$533$	2.97972	0.129066
$534$	0	0
$535$	37.6847	1.62925
$536$	0	0
$537$	−16.6644	−0.719124
$538$	0	0
$539$	1.87096	0.0805881
$540$	0	0
$541$	1.55569	0.0668843	0.0334421	−	0.999441i	$-0.489353\pi$
$541$	1.55569	0.0668843	0.0334421	+	0.999441i	$0.489353\pi$
$542$	0	0
$543$	25.1280	1.07835
$544$	0	0
$545$	−2.97972	−0.127637
$546$	0	0
$547$	10.5521	0.451174	0.225587	−	0.974223i	$-0.427570\pi$
$547$	10.5521	0.451174	0.225587	+	0.974223i	$0.427570\pi$
$548$	0	0
$549$	−7.09211	−0.302684
$550$	0	0
$551$	−38.8700	−1.65592
$552$	0	0
$553$	−11.3143	−0.481132
$554$	0	0
$555$	16.8507	0.715272
$556$	0	0
$557$	32.7004	1.38556	0.692779	−	0.721149i	$-0.256386\pi$
$557$	32.7004	1.38556	0.692779	+	0.721149i	$0.256386\pi$
$558$	0	0
$559$	0.922522	0.0390185
$560$	0	0
$561$	9.94280	0.419785
$562$	0	0
$563$	−33.0921	−1.39467	−0.697333	−	0.716747i	$-0.745630\pi$
$563$	−33.0921	−1.39467	−0.697333	+	0.716747i	$0.745630\pi$
$564$	0	0
$565$	−51.2763	−2.15721
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	17.1447	0.718742	0.359371	−	0.933195i	$-0.382991\pi$
$569$	17.1447	0.718742	0.359371	+	0.933195i	$0.382991\pi$
$570$	0	0
$571$	12.0885	0.505887	0.252944	−	0.967481i	$-0.418601\pi$
$571$	12.0885	0.505887	0.252944	+	0.967481i	$0.418601\pi$
$572$	0	0
$573$	−10.5567	−0.441012
$574$	0	0
$575$	−1.72165	−0.0717978
$576$	0	0
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	0	0
$579$	7.44330	0.309333
$580$	0	0
$581$	1.87096	0.0776206
$582$	0	0
$583$	19.3356	0.800797
$584$	0	0
$585$	−1.87096	−0.0773548
$586$	0	0
$587$	−36.1317	−1.49131	−0.745656	−	0.666331i	$-0.767864\pi$
$587$	−36.1317	−1.49131	−0.745656	+	0.666331i	$0.767864\pi$
$588$	0	0
$589$	−15.5723	−0.641647
$590$	0	0
$591$	7.02028	0.288776
$592$	0	0
$593$	34.3299	1.40976	0.704880	−	0.709326i	$-0.251001\pi$
$593$	34.3299	1.40976	0.704880	+	0.709326i	$0.251001\pi$
$594$	0	0
$595$	13.7778	0.564837
$596$	0	0
$597$	−3.66546	−0.150017
$598$	0	0
$599$	18.5354	0.757336	0.378668	−	0.925532i	$-0.376382\pi$
$599$	18.5354	0.757336	0.378668	+	0.925532i	$0.376382\pi$
$600$	0	0
$601$	−11.8497	−0.483358	−0.241679	−	0.970356i	$-0.577698\pi$
$601$	−11.8497	−0.483358	−0.241679	+	0.970356i	$0.577698\pi$
$602$	0	0
$603$	−2.03592	−0.0829090
$604$	0	0
$605$	19.4433	0.790483
$606$	0	0
$607$	36.5760	1.48457	0.742286	−	0.670083i	$-0.233741\pi$
$607$	36.5760	1.48457	0.742286	+	0.670083i	$0.233741\pi$
$608$	0	0
$609$	5.31427	0.215345
$610$	0	0
$611$	−9.32890	−0.377407
$612$	0	0
$613$	4.54005	0.183371	0.0916854	−	0.995788i	$-0.470775\pi$
$613$	4.54005	0.183371	0.0916854	+	0.995788i	$0.470775\pi$
$614$	0	0
$615$	−10.7050	−0.431667
$616$	0	0
$617$	−28.7935	−1.15918	−0.579591	−	0.814907i	$-0.696788\pi$
$617$	−28.7935	−1.15918	−0.579591	+	0.814907i	$0.696788\pi$
$618$	0	0
$619$	−44.1317	−1.77380	−0.886900	−	0.461961i	$-0.847146\pi$
$619$	−44.1317	−1.77380	−0.886900	+	0.461961i	$0.847146\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	1.90688	0.0763976
$624$	0	0
$625$	−30.6442	−1.22577
$626$	0	0
$627$	13.6847	0.546515
$628$	0	0
$629$	−34.5400	−1.37720
$630$	0	0
$631$	15.3861	0.612511	0.306255	−	0.951949i	$-0.400924\pi$
$631$	15.3861	0.612511	0.306255	+	0.951949i	$0.400924\pi$
$632$	0	0
$633$	−0.685734	−0.0272555
$634$	0	0
$635$	−16.7576	−0.665004
$636$	0	0
$637$	0.721651	0.0285929
$638$	0	0
$639$	11.9069	0.471029
$640$	0	0
$641$	22.5760	0.891697	0.445848	−	0.895108i	$-0.352902\pi$
$641$	22.5760	0.891697	0.445848	+	0.895108i	$0.352902\pi$
$642$	0	0
$643$	−31.0183	−1.22324	−0.611620	−	0.791151i	$-0.709482\pi$
$643$	−31.0183	−1.22324	−0.611620	+	0.791151i	$0.709482\pi$
$644$	0	0
$645$	−3.31427	−0.130499
$646$	0	0
$647$	20.1483	0.792112	0.396056	−	0.918226i	$-0.370379\pi$
$647$	20.1483	0.792112	0.396056	+	0.918226i	$0.370379\pi$
$648$	0	0
$649$	18.7768	0.737055
$650$	0	0
$651$	2.12904	0.0834435
$652$	0	0
$653$	−23.3335	−0.913112	−0.456556	−	0.889695i	$-0.650917\pi$
$653$	−23.3335	−0.913112	−0.456556	+	0.889695i	$0.650917\pi$
$654$	0	0
$655$	15.8902	0.620883
$656$	0	0
$657$	−5.57234	−0.217398
$658$	0	0
$659$	34.0146	1.32502	0.662511	−	0.749052i	$-0.269491\pi$
$659$	34.0146	1.32502	0.662511	+	0.749052i	$0.269491\pi$
$660$	0	0
$661$	9.50050	0.369527	0.184763	−	0.982783i	$-0.440848\pi$
$661$	9.50050	0.369527	0.184763	+	0.982783i	$0.440848\pi$
$662$	0	0
$663$	3.83505	0.148941
$664$	0	0
$665$	18.9631	0.735356
$666$	0	0
$667$	5.31427	0.205769
$668$	0	0
$669$	28.6644	1.10823
$670$	0	0
$671$	−13.2691	−0.512247
$672$	0	0
$673$	−6.61390	−0.254947	−0.127474	−	0.991842i	$-0.540687\pi$
$673$	−6.61390	−0.254947	−0.127474	+	0.991842i	$0.540687\pi$
$674$	0	0
$675$	1.72165	0.0662664
$676$	0	0
$677$	2.24606	0.0863230	0.0431615	−	0.999068i	$-0.486257\pi$
$677$	2.24606	0.0863230	0.0431615	+	0.999068i	$0.486257\pi$
$678$	0	0
$679$	−7.87096	−0.302060
$680$	0	0
$681$	−24.1483	−0.925365
$682$	0	0
$683$	−9.49849	−0.363449	−0.181725	−	0.983349i	$-0.558168\pi$
$683$	−9.49849	−0.363449	−0.181725	+	0.983349i	$0.558168\pi$
$684$	0	0
$685$	−20.4064	−0.779688
$686$	0	0
$687$	15.7778	0.601962
$688$	0	0
$689$	7.45793	0.284125
$690$	0	0
$691$	−29.6894	−1.12944	−0.564718	−	0.825284i	$-0.691015\pi$
$691$	−29.6894	−1.12944	−0.564718	+	0.825284i	$0.691015\pi$
$692$	0	0
$693$	−1.87096	−0.0710720
$694$	0	0
$695$	38.7935	1.47152
$696$	0	0
$697$	21.9428	0.831143
$698$	0	0
$699$	19.7778	0.748067
$700$	0	0
$701$	−3.96408	−0.149721	−0.0748607	−	0.997194i	$-0.523851\pi$
$701$	−3.96408	−0.149721	−0.0748607	+	0.997194i	$0.523851\pi$
$702$	0	0
$703$	−47.5390	−1.79297
$704$	0	0
$705$	33.5151	1.26225
$706$	0	0
$707$	−12.0359	−0.452657
$708$	0	0
$709$	−15.4646	−0.580785	−0.290392	−	0.956908i	$-0.593786\pi$
$709$	−15.4646	−0.580785	−0.290392	+	0.956908i	$0.593786\pi$
$710$	0	0
$711$	11.3143	0.424318
$712$	0	0
$713$	2.12904	0.0797330
$714$	0	0
$715$	−3.50050	−0.130911
$716$	0	0
$717$	16.4064	0.612707
$718$	0	0
$719$	−48.0552	−1.79216	−0.896078	−	0.443897i	$-0.853596\pi$
$719$	−48.0552	−1.79216	−0.896078	+	0.443897i	$0.853596\pi$
$720$	0	0
$721$	10.3705	0.386216
$722$	0	0
$723$	14.4995	0.539242
$724$	0	0
$725$	−9.14931	−0.339797
$726$	0	0
$727$	−45.8284	−1.69968	−0.849841	−	0.527040i	$-0.823302\pi$
$727$	−45.8284	−1.69968	−0.849841	+	0.527040i	$0.823302\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	6.79349	0.251266
$732$	0	0
$733$	−29.2165	−1.07914	−0.539568	−	0.841942i	$-0.681413\pi$
$733$	−29.2165	−1.07914	−0.539568	+	0.841942i	$0.681413\pi$
$734$	0	0
$735$	−2.59261	−0.0956300
$736$	0	0
$737$	−3.80913	−0.140311
$738$	0	0
$739$	−0.685734	−0.0252251	−0.0126126	−	0.999920i	$-0.504015\pi$
$739$	−0.685734	−0.0252251	−0.0126126	+	0.999920i	$0.504015\pi$
$740$	0	0
$741$	5.27835	0.193905
$742$	0	0
$743$	−32.7456	−1.20132	−0.600659	−	0.799505i	$-0.705095\pi$
$743$	−32.7456	−1.20132	−0.600659	+	0.799505i	$0.705095\pi$
$744$	0	0
$745$	28.9990	1.06244
$746$	0	0
$747$	−1.87096	−0.0684550
$748$	0	0
$749$	14.5354	0.531112
$750$	0	0
$751$	22.3751	0.816479	0.408239	−	0.912875i	$-0.366143\pi$
$751$	22.3751	0.816479	0.408239	+	0.912875i	$0.366143\pi$
$752$	0	0
$753$	−7.55569	−0.275345
$754$	0	0
$755$	33.5151	1.21974
$756$	0	0
$757$	−42.7576	−1.55405	−0.777025	−	0.629470i	$-0.783272\pi$
$757$	−42.7576	−1.55405	−0.777025	+	0.629470i	$0.783272\pi$
$758$	0	0
$759$	−1.87096	−0.0679117
$760$	0	0
$761$	−33.3694	−1.20964	−0.604821	−	0.796362i	$-0.706755\pi$
$761$	−33.3694	−1.20964	−0.604821	+	0.796362i	$0.706755\pi$
$762$	0	0
$763$	−1.14931	−0.0416079
$764$	0	0
$765$	−13.7778	−0.498139
$766$	0	0
$767$	7.24243	0.261509
$768$	0	0
$769$	−22.8461	−0.823850	−0.411925	−	0.911218i	$-0.635143\pi$
$769$	−22.8461	−0.823850	−0.411925	+	0.911218i	$0.635143\pi$
$770$	0	0
$771$	−6.66908	−0.240181
$772$	0	0
$773$	45.2091	1.62606	0.813030	−	0.582222i	$-0.197817\pi$
$773$	45.2091	1.62606	0.813030	+	0.582222i	$0.197817\pi$
$774$	0	0
$775$	−3.66546	−0.131667
$776$	0	0
$777$	6.49950	0.233168
$778$	0	0
$779$	30.2009	1.08206
$780$	0	0
$781$	22.2773	0.797146
$782$	0	0
$783$	−5.31427	−0.189916
$784$	0	0
$785$	−37.1519	−1.32601
$786$	0	0
$787$	29.5916	1.05483	0.527413	−	0.849609i	$-0.323162\pi$
$787$	29.5916	1.05483	0.527413	+	0.849609i	$0.323162\pi$
$788$	0	0
$789$	24.0552	0.856387
$790$	0	0
$791$	−19.7778	−0.703219
$792$	0	0
$793$	−5.11803	−0.181747
$794$	0	0
$795$	−26.7935	−0.950267
$796$	0	0
$797$	−27.7014	−0.981233	−0.490617	−	0.871376i	$-0.663228\pi$
$797$	−27.7014	−0.981233	−0.490617	+	0.871376i	$0.663228\pi$
$798$	0	0
$799$	−68.6983	−2.43037
$800$	0	0
$801$	−1.90688	−0.0673763
$802$	0	0
$803$	−10.4256	−0.367913
$804$	0	0
$805$	−2.59261	−0.0913777
$806$	0	0
$807$	8.72165	0.307017
$808$	0	0
$809$	36.8653	1.29612	0.648058	−	0.761591i	$-0.275582\pi$
$809$	36.8653	1.29612	0.648058	+	0.761591i	$0.275582\pi$
$810$	0	0
$811$	−12.9438	−0.454519	−0.227259	−	0.973834i	$-0.572976\pi$
$811$	−12.9438	−0.454519	−0.227259	+	0.973834i	$0.572976\pi$
$812$	0	0
$813$	−22.5400	−0.790514
$814$	0	0
$815$	−35.7206	−1.25124
$816$	0	0
$817$	9.35018	0.327121
$818$	0	0
$819$	−0.721651	−0.0252165
$820$	0	0
$821$	6.66908	0.232753	0.116376	−	0.993205i	$-0.462872\pi$
$821$	6.66908	0.232753	0.116376	+	0.993205i	$0.462872\pi$
$822$	0	0
$823$	−8.92252	−0.311020	−0.155510	−	0.987834i	$-0.549702\pi$
$823$	−8.92252	−0.311020	−0.155510	+	0.987834i	$0.549702\pi$
$824$	0	0
$825$	3.22115	0.112146
$826$	0	0
$827$	−25.5198	−0.887409	−0.443705	−	0.896173i	$-0.646336\pi$
$827$	−25.5198	−0.887409	−0.443705	+	0.896173i	$0.646336\pi$
$828$	0	0
$829$	−37.6109	−1.30628	−0.653140	−	0.757237i	$-0.726549\pi$
$829$	−37.6109	−1.30628	−0.653140	+	0.757237i	$0.726549\pi$
$830$	0	0
$831$	21.9787	0.762433
$832$	0	0
$833$	5.31427	0.184128
$834$	0	0
$835$	−53.1473	−1.83924
$836$	0	0
$837$	−2.12904	−0.0735903
$838$	0	0
$839$	4.42303	0.152700	0.0763499	−	0.997081i	$-0.475673\pi$
$839$	4.42303	0.152700	0.0763499	+	0.997081i	$0.475673\pi$
$840$	0	0
$841$	−0.758577	−0.0261578
$842$	0	0
$843$	14.9272	0.514119
$844$	0	0
$845$	32.3538	1.11300
$846$	0	0
$847$	7.49950	0.257686
$848$	0	0
$849$	−32.6644	−1.12104
$850$	0	0
$851$	6.49950	0.222800
$852$	0	0
$853$	−28.0406	−0.960090	−0.480045	−	0.877244i	$-0.659380\pi$
$853$	−28.0406	−0.960090	−0.480045	+	0.877244i	$0.659380\pi$
$854$	0	0
$855$	−18.9631	−0.648523
$856$	0	0
$857$	−23.6681	−0.808486	−0.404243	−	0.914652i	$-0.632465\pi$
$857$	−23.6681	−0.808486	−0.404243	+	0.914652i	$0.632465\pi$
$858$	0	0
$859$	49.5962	1.69220	0.846101	−	0.533023i	$-0.178944\pi$
$859$	49.5962	1.69220	0.846101	+	0.533023i	$0.178944\pi$
$860$	0	0
$861$	−4.12904	−0.140717
$862$	0	0
$863$	−5.92615	−0.201728	−0.100864	−	0.994900i	$-0.532161\pi$
$863$	−5.92615	−0.201728	−0.100864	+	0.994900i	$0.532161\pi$
$864$	0	0
$865$	−59.0016	−2.00611
$866$	0	0
$867$	11.2414	0.381779
$868$	0	0
$869$	21.1686	0.718095
$870$	0	0
$871$	−1.46922	−0.0497827
$872$	0	0
$873$	7.87096	0.266392
$874$	0	0
$875$	−8.49950	−0.287335
$876$	0	0
$877$	43.0801	1.45471	0.727356	−	0.686261i	$-0.240749\pi$
$877$	43.0801	1.45471	0.727356	+	0.686261i	$0.240749\pi$
$878$	0	0
$879$	−11.5151	−0.388396
$880$	0	0
$881$	−19.5870	−0.659902	−0.329951	−	0.943998i	$-0.607032\pi$
$881$	−19.5870	−0.659902	−0.329951	+	0.943998i	$0.607032\pi$
$882$	0	0
$883$	17.9382	0.603667	0.301834	−	0.953361i	$-0.402401\pi$
$883$	17.9382	0.603667	0.301834	+	0.953361i	$0.402401\pi$
$884$	0	0
$885$	−26.0193	−0.874628
$886$	0	0
$887$	−2.53541	−0.0851308	−0.0425654	−	0.999094i	$-0.513553\pi$
$887$	−2.53541	−0.0851308	−0.0425654	+	0.999094i	$0.513553\pi$
$888$	0	0
$889$	−6.46358	−0.216781
$890$	0	0
$891$	1.87096	0.0626796
$892$	0	0
$893$	−94.5527	−3.16408
$894$	0	0
$895$	43.2045	1.44417
$896$	0	0
$897$	−0.721651	−0.0240952
$898$	0	0
$899$	11.3143	0.377352
$900$	0	0
$901$	54.9205	1.82967
$902$	0	0
$903$	−1.27835	−0.0425408
$904$	0	0
$905$	−65.1473	−2.16557
$906$	0	0
$907$	−28.7363	−0.954173	−0.477086	−	0.878856i	$-0.658307\pi$
$907$	−28.7363	−0.954173	−0.477086	+	0.878856i	$0.658307\pi$
$908$	0	0
$909$	12.0359	0.399206
$910$	0	0
$911$	−38.7815	−1.28489	−0.642444	−	0.766333i	$-0.722079\pi$
$911$	−38.7815	−1.28489	−0.642444	+	0.766333i	$0.722079\pi$
$912$	0	0
$913$	−3.50050	−0.115850
$914$	0	0
$915$	18.3871	0.607859
$916$	0	0
$917$	6.12904	0.202399
$918$	0	0
$919$	−23.7253	−0.782625	−0.391312	−	0.920258i	$-0.627979\pi$
$919$	−23.7253	−0.782625	−0.391312	+	0.920258i	$0.627979\pi$
$920$	0	0
$921$	29.7566	0.980512
$922$	0	0
$923$	8.59261	0.282829
$924$	0	0
$925$	−11.1899	−0.367920
$926$	0	0
$927$	−10.3705	−0.340611
$928$	0	0
$929$	−55.0329	−1.80557	−0.902785	−	0.430092i	$-0.858481\pi$
$929$	−55.0329	−1.80557	−0.902785	+	0.430092i	$0.858481\pi$
$930$	0	0
$931$	7.31427	0.239715
$932$	0	0
$933$	15.4626	0.506222
$934$	0	0
$935$	−25.7778	−0.843026
$936$	0	0
$937$	12.3153	0.402323	0.201161	−	0.979558i	$-0.435528\pi$
$937$	12.3153	0.402323	0.201161	+	0.979558i	$0.435528\pi$
$938$	0	0
$939$	16.6285	0.542652
$940$	0	0
$941$	−28.7096	−0.935907	−0.467954	−	0.883753i	$-0.655009\pi$
$941$	−28.7096	−0.935907	−0.467954	+	0.883753i	$0.655009\pi$
$942$	0	0
$943$	−4.12904	−0.134460
$944$	0	0
$945$	2.59261	0.0843378
$946$	0	0
$947$	53.1427	1.72690	0.863452	−	0.504431i	$-0.168298\pi$
$947$	53.1427	1.72690	0.863452	+	0.504431i	$0.168298\pi$
$948$	0	0
$949$	−4.02129	−0.130536
$950$	0	0
$951$	16.9631	0.550065
$952$	0	0
$953$	35.4939	1.14976	0.574879	−	0.818238i	$-0.305049\pi$
$953$	35.4939	1.14976	0.574879	+	0.818238i	$0.305049\pi$
$954$	0	0
$955$	27.3694	0.885655
$956$	0	0
$957$	−9.94280	−0.321405
$958$	0	0
$959$	−7.87096	−0.254167
$960$	0	0
$961$	−26.4672	−0.853781
$962$	0	0
$963$	−14.5354	−0.468397
$964$	0	0
$965$	−19.2976	−0.621212
$966$	0	0
$967$	7.00101	0.225137	0.112569	−	0.993644i	$-0.464092\pi$
$967$	7.00101	0.225137	0.112569	+	0.993644i	$0.464092\pi$
$968$	0	0
$969$	38.8700	1.24868
$970$	0	0
$971$	13.6894	0.439312	0.219656	−	0.975577i	$-0.429506\pi$
$971$	13.6894	0.439312	0.219656	+	0.975577i	$0.429506\pi$
$972$	0	0
$973$	14.9631	0.479694
$974$	0	0
$975$	1.24243	0.0397897
$976$	0	0
$977$	−18.9059	−0.604852	−0.302426	−	0.953173i	$-0.597797\pi$
$977$	−18.9059	−0.604852	−0.302426	+	0.953173i	$0.597797\pi$
$978$	0	0
$979$	−3.56770	−0.114024
$980$	0	0
$981$	1.14931	0.0366947
$982$	0	0
$983$	−36.1270	−1.15227	−0.576136	−	0.817354i	$-0.695440\pi$
$983$	−36.1270	−1.15227	−0.576136	+	0.817354i	$0.695440\pi$
$984$	0	0
$985$	−18.2009	−0.579928
$986$	0	0
$987$	12.9272	0.411476
$988$	0	0
$989$	−1.27835	−0.0406491
$990$	0	0
$991$	−0.317891	−0.0100982	−0.00504908	−	0.999987i	$-0.501607\pi$
$991$	−0.317891	−0.0100982	−0.00504908	+	0.999987i	$0.501607\pi$
$992$	0	0
$993$	7.74193	0.245683
$994$	0	0
$995$	9.50312	0.301269
$996$	0	0
$997$	−51.7566	−1.63915	−0.819573	−	0.572974i	$-0.805789\pi$
$997$	−51.7566	−1.63915	−0.819573	+	0.572974i	$0.805789\pi$
$998$	0	0
$999$	−6.49950	−0.205635

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7728.2.a.by.1.1		3
4.3	odd	2		3864.2.a.n.1.1	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3864.2.a.n.1.1	✓	3	4.3	odd	2
7728.2.a.by.1.1		3	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -2.59261 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -2.59261 q^{5} -1.00000 q^{7} +1.00000 q^{9} +1.87096 q^{11} +0.721651 q^{13} -2.59261 q^{15} +5.31427 q^{17} +7.31427 q^{19} -1.00000 q^{21} -1.00000 q^{23} +1.72165 q^{25} +1.00000 q^{27} -5.31427 q^{29} -2.12904 q^{31} +1.87096 q^{33} +2.59261 q^{35} -6.49950 q^{37} +0.721651 q^{39} +4.12904 q^{41} +1.27835 q^{43} -2.59261 q^{45} -12.9272 q^{47} +1.00000 q^{49} +5.31427 q^{51} +10.3345 q^{53} -4.85069 q^{55} +7.31427 q^{57} +10.0359 q^{59} -7.09211 q^{61} -1.00000 q^{63} -1.87096 q^{65} -2.03592 q^{67} -1.00000 q^{69} +11.9069 q^{71} -5.57234 q^{73} +1.72165 q^{75} -1.87096 q^{77} +11.3143 q^{79} +1.00000 q^{81} -1.87096 q^{83} -13.7778 q^{85} -5.31427 q^{87} -1.90688 q^{89} -0.721651 q^{91} -2.12904 q^{93} -18.9631 q^{95} +7.87096 q^{97} +1.87096 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} + q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 + q^5 - 3 * q^7 + 3 * q^9	\( 3 q + 3 q^{3} + q^{5} - 3 q^{7} + 3 q^{9} + 2 q^{11} - 3 q^{13} + q^{15} + 2 q^{17} + 8 q^{19} - 3 q^{21} - 3 q^{23} + 3 q^{27} - 2 q^{29} - 10 q^{31} + 2 q^{33} - q^{35} + 12 q^{37} - 3 q^{39} + 16 q^{41} + 9 q^{43} + q^{45} - 14 q^{47} + 3 q^{49} + 2 q^{51} + 15 q^{53} - 13 q^{55} + 8 q^{57} + 11 q^{59} + 19 q^{61} - 3 q^{63} - 2 q^{65} + 13 q^{67} - 3 q^{69} + 13 q^{71} - 10 q^{73} - 2 q^{77} + 20 q^{79} + 3 q^{81} - 2 q^{83} - 15 q^{85} - 2 q^{87} + 17 q^{89} + 3 q^{91} - 10 q^{93} - 13 q^{95} + 20 q^{97} + 2 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 + q^5 - 3 * q^7 + 3 * q^9 + 2 * q^11 - 3 * q^13 + q^15 + 2 * q^17 + 8 * q^19 - 3 * q^21 - 3 * q^23 + 3 * q^27 - 2 * q^29 - 10 * q^31 + 2 * q^33 - q^35 + 12 * q^37 - 3 * q^39 + 16 * q^41 + 9 * q^43 + q^45 - 14 * q^47 + 3 * q^49 + 2 * q^51 + 15 * q^53 - 13 * q^55 + 8 * q^57 + 11 * q^59 + 19 * q^61 - 3 * q^63 - 2 * q^65 + 13 * q^67 - 3 * q^69 + 13 * q^71 - 10 * q^73 - 2 * q^77 + 20 * q^79 + 3 * q^81 - 2 * q^83 - 15 * q^85 - 2 * q^87 + 17 * q^89 + 3 * q^91 - 10 * q^93 - 13 * q^95 + 20 * q^97 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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