LMFDB - Embedded newform 7728.2.a.bw.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("7728.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$61.7083906820$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
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			1.3
Root			$2.11491$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +1.47283 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.47283 q^{5} +1.00000 q^{7} +1.00000 q^{9} -4.58774 q^{11} +3.11491 q^{13} +1.47283 q^{15} -6.58774 q^{17} +6.81756 q^{19} +1.00000 q^{21} -1.00000 q^{23} -2.83076 q^{25} +1.00000 q^{27} -7.30359 q^{29} -1.64207 q^{31} -4.58774 q^{33} +1.47283 q^{35} -8.10170 q^{37} +3.11491 q^{39} -9.87189 q^{41} -1.11491 q^{43} +1.47283 q^{45} -10.2298 q^{47} +1.00000 q^{49} -6.58774 q^{51} -5.47283 q^{53} -6.75698 q^{55} +6.81756 q^{57} -12.9868 q^{59} -7.49228 q^{61} +1.00000 q^{63} +4.58774 q^{65} +8.64832 q^{67} -1.00000 q^{69} +6.96735 q^{71} +15.0474 q^{73} -2.83076 q^{75} -4.58774 q^{77} +2.69641 q^{79} +1.00000 q^{81} -10.1017 q^{83} -9.70265 q^{85} -7.30359 q^{87} +11.9519 q^{89} +3.11491 q^{91} -1.64207 q^{93} +10.0411 q^{95} +7.04737 q^{97} -4.58774 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 8 q^{17} - 4 q^{19} + 3 q^{21} - 3 q^{23} - 4 q^{25} + 3 q^{27} - 12 q^{29} - 4 q^{31} - 2 q^{33} - q^{35} + 2 q^{37} + 3 q^{39} - 16 q^{41} + 3 q^{43} - q^{45} - 18 q^{47} + 3 q^{49} - 8 q^{51} - 11 q^{53} - 13 q^{55} - 4 q^{57} - 19 q^{59} - 9 q^{61} + 3 q^{63} + 2 q^{65} - 3 q^{67} - 3 q^{69} + 9 q^{71} + 8 q^{73} - 4 q^{75} - 2 q^{77} + 18 q^{79} + 3 q^{81} - 4 q^{83} - 11 q^{85} - 12 q^{87} - 3 q^{89} + 3 q^{91} - 4 q^{93} + 21 q^{95} - 16 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 - 8 * q^17 - 4 * q^19 + 3 * q^21 - 3 * q^23 - 4 * q^25 + 3 * q^27 - 12 * q^29 - 4 * q^31 - 2 * q^33 - q^35 + 2 * q^37 + 3 * q^39 - 16 * q^41 + 3 * q^43 - q^45 - 18 * q^47 + 3 * q^49 - 8 * q^51 - 11 * q^53 - 13 * q^55 - 4 * q^57 - 19 * q^59 - 9 * q^61 + 3 * q^63 + 2 * q^65 - 3 * q^67 - 3 * q^69 + 9 * q^71 + 8 * q^73 - 4 * q^75 - 2 * q^77 + 18 * q^79 + 3 * q^81 - 4 * q^83 - 11 * q^85 - 12 * q^87 - 3 * q^89 + 3 * q^91 - 4 * q^93 + 21 * q^95 - 16 * 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
$5$	1.47283	0.658671	0.329336	−	0.944213i	$-0.393175\pi$
$5$	1.47283	0.658671	0.329336	+	0.944213i	$0.393175\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.58774	−1.38326	−0.691628	−	0.722254i	$-0.743106\pi$
$11$	−4.58774	−1.38326	−0.691628	+	0.722254i	$0.743106\pi$
$12$	0	0
$13$	3.11491	0.863920	0.431960	−	0.901893i	$-0.357822\pi$
$13$	3.11491	0.863920	0.431960	+	0.901893i	$0.357822\pi$
$14$	0	0
$15$	1.47283	0.380284
$16$	0	0
$17$	−6.58774	−1.59776	−0.798881	−	0.601489i	$-0.794574\pi$
$17$	−6.58774	−1.59776	−0.798881	+	0.601489i	$0.794574\pi$
$18$	0	0
$19$	6.81756	1.56405	0.782027	−	0.623244i	$-0.214186\pi$
$19$	6.81756	1.56405	0.782027	+	0.623244i	$0.214186\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−2.83076	−0.566152
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−7.30359	−1.35624	−0.678122	−	0.734950i	$-0.737206\pi$
$29$	−7.30359	−1.35624	−0.678122	+	0.734950i	$0.737206\pi$
$30$	0	0
$31$	−1.64207	−0.294925	−0.147463	−	0.989068i	$-0.547111\pi$
$31$	−1.64207	−0.294925	−0.147463	+	0.989068i	$0.547111\pi$
$32$	0	0
$33$	−4.58774	−0.798623
$34$	0	0
$35$	1.47283	0.248954
$36$	0	0
$37$	−8.10170	−1.33191	−0.665956	−	0.745991i	$-0.731976\pi$
$37$	−8.10170	−1.33191	−0.665956	+	0.745991i	$0.731976\pi$
$38$	0	0
$39$	3.11491	0.498784
$40$	0	0
$41$	−9.87189	−1.54173	−0.770865	−	0.636999i	$-0.780176\pi$
$41$	−9.87189	−1.54173	−0.770865	+	0.636999i	$0.780176\pi$
$42$	0	0
$43$	−1.11491	−0.170022	−0.0850109	−	0.996380i	$-0.527093\pi$
$43$	−1.11491	−0.170022	−0.0850109	+	0.996380i	$0.527093\pi$
$44$	0	0
$45$	1.47283	0.219557
$46$	0	0
$47$	−10.2298	−1.49217	−0.746086	−	0.665850i	$-0.768069\pi$
$47$	−10.2298	−1.49217	−0.746086	+	0.665850i	$0.768069\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−6.58774	−0.922468
$52$	0	0
$53$	−5.47283	−0.751752	−0.375876	−	0.926670i	$-0.622658\pi$
$53$	−5.47283	−0.751752	−0.375876	+	0.926670i	$0.622658\pi$
$54$	0	0
$55$	−6.75698	−0.911111
$56$	0	0
$57$	6.81756	0.903007
$58$	0	0
$59$	−12.9868	−1.69074	−0.845368	−	0.534184i	$-0.820619\pi$
$59$	−12.9868	−1.69074	−0.845368	+	0.534184i	$0.820619\pi$
$60$	0	0
$61$	−7.49228	−0.959288	−0.479644	−	0.877463i	$-0.659234\pi$
$61$	−7.49228	−0.959288	−0.479644	+	0.877463i	$0.659234\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	4.58774	0.569039
$66$	0	0
$67$	8.64832	1.05656	0.528280	−	0.849070i	$-0.322837\pi$
$67$	8.64832	1.05656	0.528280	+	0.849070i	$0.322837\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	6.96735	0.826872	0.413436	−	0.910533i	$-0.364328\pi$
$71$	6.96735	0.826872	0.413436	+	0.910533i	$0.364328\pi$
$72$	0	0
$73$	15.0474	1.76116	0.880581	−	0.473896i	$-0.157153\pi$
$73$	15.0474	1.76116	0.880581	+	0.473896i	$0.157153\pi$
$74$	0	0
$75$	−2.83076	−0.326868
$76$	0	0
$77$	−4.58774	−0.522822
$78$	0	0
$79$	2.69641	0.303369	0.151685	−	0.988429i	$-0.451530\pi$
$79$	2.69641	0.303369	0.151685	+	0.988429i	$0.451530\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−10.1017	−1.10881	−0.554403	−	0.832248i	$-0.687053\pi$
$83$	−10.1017	−1.10881	−0.554403	+	0.832248i	$0.687053\pi$
$84$	0	0
$85$	−9.70265	−1.05240
$86$	0	0
$87$	−7.30359	−0.783027
$88$	0	0
$89$	11.9519	1.26690	0.633450	−	0.773784i	$-0.281638\pi$
$89$	11.9519	1.26690	0.633450	+	0.773784i	$0.281638\pi$
$90$	0	0
$91$	3.11491	0.326531
$92$	0	0
$93$	−1.64207	−0.170275
$94$	0	0
$95$	10.0411	1.03020
$96$	0	0
$97$	7.04737	0.715552	0.357776	−	0.933807i	$-0.383535\pi$
$97$	7.04737	0.715552	0.357776	+	0.933807i	$0.383535\pi$
$98$	0	0
$99$	−4.58774	−0.461085
$100$	0	0
$101$	0.0411284	0.00409243	0.00204622	−	0.999998i	$-0.499349\pi$
$101$	0.0411284	0.00409243	0.00204622	+	0.999998i	$0.499349\pi$
$102$	0	0
$103$	16.9193	1.66710	0.833552	−	0.552441i	$-0.186303\pi$
$103$	16.9193	1.66710	0.833552	+	0.552441i	$0.186303\pi$
$104$	0	0
$105$	1.47283	0.143734
$106$	0	0
$107$	4.39905	0.425273	0.212636	−	0.977131i	$-0.431795\pi$
$107$	4.39905	0.425273	0.212636	+	0.977131i	$0.431795\pi$
$108$	0	0
$109$	−14.6483	−1.40305	−0.701527	−	0.712643i	$-0.747498\pi$
$109$	−14.6483	−1.40305	−0.701527	+	0.712643i	$0.747498\pi$
$110$	0	0
$111$	−8.10170	−0.768980
$112$	0	0
$113$	−18.2709	−1.71879	−0.859393	−	0.511316i	$-0.829158\pi$
$113$	−18.2709	−1.71879	−0.859393	+	0.511316i	$0.829158\pi$
$114$	0	0
$115$	−1.47283	−0.137342
$116$	0	0
$117$	3.11491	0.287973
$118$	0	0
$119$	−6.58774	−0.603897
$120$	0	0
$121$	10.0474	0.913397
$122$	0	0
$123$	−9.87189	−0.890118
$124$	0	0
$125$	−11.5334	−1.03158
$126$	0	0
$127$	−3.68320	−0.326831	−0.163416	−	0.986557i	$-0.552251\pi$
$127$	−3.68320	−0.326831	−0.163416	+	0.986557i	$0.552251\pi$
$128$	0	0
$129$	−1.11491	−0.0981621
$130$	0	0
$131$	15.1949	1.32759	0.663794	−	0.747916i	$-0.268945\pi$
$131$	15.1949	1.32759	0.663794	+	0.747916i	$0.268945\pi$
$132$	0	0
$133$	6.81756	0.591157
$134$	0	0
$135$	1.47283	0.126761
$136$	0	0
$137$	−8.01945	−0.685148	−0.342574	−	0.939491i	$-0.611299\pi$
$137$	−8.01945	−0.685148	−0.342574	+	0.939491i	$0.611299\pi$
$138$	0	0
$139$	11.5551	0.980090	0.490045	−	0.871697i	$-0.336980\pi$
$139$	11.5551	0.980090	0.490045	+	0.871697i	$0.336980\pi$
$140$	0	0
$141$	−10.2298	−0.861506
$142$	0	0
$143$	−14.2904	−1.19502
$144$	0	0
$145$	−10.7570	−0.893319
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	1.40530	0.115126	0.0575632	−	0.998342i	$-0.481667\pi$
$149$	1.40530	0.115126	0.0575632	+	0.998342i	$0.481667\pi$
$150$	0	0
$151$	−14.6894	−1.19541	−0.597705	−	0.801716i	$-0.703921\pi$
$151$	−14.6894	−1.19541	−0.597705	+	0.801716i	$0.703921\pi$
$152$	0	0
$153$	−6.58774	−0.532587
$154$	0	0
$155$	−2.41850	−0.194259
$156$	0	0
$157$	−0.689445	−0.0550237	−0.0275119	−	0.999621i	$-0.508758\pi$
$157$	−0.689445	−0.0550237	−0.0275119	+	0.999621i	$0.508758\pi$
$158$	0	0
$159$	−5.47283	−0.434024
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	−19.6762	−1.54116	−0.770581	−	0.637342i	$-0.780034\pi$
$163$	−19.6762	−1.54116	−0.770581	+	0.637342i	$0.780034\pi$
$164$	0	0
$165$	−6.75698	−0.526030
$166$	0	0
$167$	15.9108	1.23121	0.615607	−	0.788054i	$-0.288911\pi$
$167$	15.9108	1.23121	0.615607	+	0.788054i	$0.288911\pi$
$168$	0	0
$169$	−3.29735	−0.253642
$170$	0	0
$171$	6.81756	0.521352
$172$	0	0
$173$	10.6266	0.807928	0.403964	−	0.914775i	$-0.367632\pi$
$173$	10.6266	0.807928	0.403964	+	0.914775i	$0.367632\pi$
$174$	0	0
$175$	−2.83076	−0.213985
$176$	0	0
$177$	−12.9868	−0.976147
$178$	0	0
$179$	1.95887	0.146413	0.0732065	−	0.997317i	$-0.476677\pi$
$179$	1.95887	0.146413	0.0732065	+	0.997317i	$0.476677\pi$
$180$	0	0
$181$	6.58774	0.489663	0.244831	−	0.969566i	$-0.421267\pi$
$181$	6.58774	0.489663	0.244831	+	0.969566i	$0.421267\pi$
$182$	0	0
$183$	−7.49228	−0.553845
$184$	0	0
$185$	−11.9325	−0.877292
$186$	0	0
$187$	30.2229	2.21011
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	5.77018	0.417516	0.208758	−	0.977967i	$-0.433058\pi$
$191$	5.77018	0.417516	0.208758	+	0.977967i	$0.433058\pi$
$192$	0	0
$193$	−17.1491	−1.23442	−0.617209	−	0.786799i	$-0.711737\pi$
$193$	−17.1491	−1.23442	−0.617209	+	0.786799i	$0.711737\pi$
$194$	0	0
$195$	4.58774	0.328535
$196$	0	0
$197$	−6.73753	−0.480029	−0.240015	−	0.970769i	$-0.577152\pi$
$197$	−6.73753	−0.480029	−0.240015	+	0.970769i	$0.577152\pi$
$198$	0	0
$199$	−3.85021	−0.272934	−0.136467	−	0.990645i	$-0.543575\pi$
$199$	−3.85021	−0.272934	−0.136467	+	0.990645i	$0.543575\pi$
$200$	0	0
$201$	8.64832	0.610005
$202$	0	0
$203$	−7.30359	−0.512612
$204$	0	0
$205$	−14.5397	−1.01549
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	−31.2772	−2.16349
$210$	0	0
$211$	−3.53341	−0.243250	−0.121625	−	0.992576i	$-0.538811\pi$
$211$	−3.53341	−0.243250	−0.121625	+	0.992576i	$0.538811\pi$
$212$	0	0
$213$	6.96735	0.477395
$214$	0	0
$215$	−1.64207	−0.111988
$216$	0	0
$217$	−1.64207	−0.111471
$218$	0	0
$219$	15.0474	1.01681
$220$	0	0
$221$	−20.5202	−1.38034
$222$	0	0
$223$	10.4185	0.697674	0.348837	−	0.937183i	$-0.386577\pi$
$223$	10.4185	0.697674	0.348837	+	0.937183i	$0.386577\pi$
$224$	0	0
$225$	−2.83076	−0.188717
$226$	0	0
$227$	24.0147	1.59391	0.796957	−	0.604037i	$-0.206442\pi$
$227$	24.0147	1.59391	0.796957	+	0.604037i	$0.206442\pi$
$228$	0	0
$229$	5.13435	0.339288	0.169644	−	0.985505i	$-0.445738\pi$
$229$	5.13435	0.339288	0.169644	+	0.985505i	$0.445738\pi$
$230$	0	0
$231$	−4.58774	−0.301851
$232$	0	0
$233$	0.500759	0.0328058	0.0164029	−	0.999865i	$-0.494779\pi$
$233$	0.500759	0.0328058	0.0164029	+	0.999865i	$0.494779\pi$
$234$	0	0
$235$	−15.0668	−0.982851
$236$	0	0
$237$	2.69641	0.175150
$238$	0	0
$239$	19.9714	1.29184	0.645920	−	0.763405i	$-0.276474\pi$
$239$	19.9714	1.29184	0.645920	+	0.763405i	$0.276474\pi$
$240$	0	0
$241$	−19.0474	−1.22695	−0.613475	−	0.789715i	$-0.710229\pi$
$241$	−19.0474	−1.22695	−0.613475	+	0.789715i	$0.710229\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	1.47283	0.0940959
$246$	0	0
$247$	21.2361	1.35122
$248$	0	0
$249$	−10.1017	−0.640169
$250$	0	0
$251$	13.9736	0.882005	0.441003	−	0.897506i	$-0.354623\pi$
$251$	13.9736	0.882005	0.441003	+	0.897506i	$0.354623\pi$
$252$	0	0
$253$	4.58774	0.288429
$254$	0	0
$255$	−9.70265	−0.607603
$256$	0	0
$257$	22.3246	1.39257	0.696284	−	0.717767i	$-0.254835\pi$
$257$	22.3246	1.39257	0.696284	+	0.717767i	$0.254835\pi$
$258$	0	0
$259$	−8.10170	−0.503415
$260$	0	0
$261$	−7.30359	−0.452081
$262$	0	0
$263$	−23.2772	−1.43533	−0.717666	−	0.696387i	$-0.754790\pi$
$263$	−23.2772	−1.43533	−0.717666	+	0.696387i	$0.754790\pi$
$264$	0	0
$265$	−8.06058	−0.495157
$266$	0	0
$267$	11.9519	0.731445
$268$	0	0
$269$	−1.60095	−0.0976114	−0.0488057	−	0.998808i	$-0.515542\pi$
$269$	−1.60095	−0.0976114	−0.0488057	+	0.998808i	$0.515542\pi$
$270$	0	0
$271$	−27.3594	−1.66197	−0.830984	−	0.556296i	$-0.812222\pi$
$271$	−27.3594	−1.66197	−0.830984	+	0.556296i	$0.812222\pi$
$272$	0	0
$273$	3.11491	0.188523
$274$	0	0
$275$	12.9868	0.783133
$276$	0	0
$277$	10.7375	0.645156	0.322578	−	0.946543i	$-0.395451\pi$
$277$	10.7375	0.645156	0.322578	+	0.946543i	$0.395451\pi$
$278$	0	0
$279$	−1.64207	−0.0983084
$280$	0	0
$281$	0.689445	0.0411289	0.0205644	−	0.999789i	$-0.493454\pi$
$281$	0.689445	0.0411289	0.0205644	+	0.999789i	$0.493454\pi$
$282$	0	0
$283$	−33.1902	−1.97295	−0.986476	−	0.163903i	$-0.947592\pi$
$283$	−33.1902	−1.97295	−0.986476	+	0.163903i	$0.947592\pi$
$284$	0	0
$285$	10.0411	0.594785
$286$	0	0
$287$	−9.87189	−0.582719
$288$	0	0
$289$	26.3983	1.55284
$290$	0	0
$291$	7.04737	0.413124
$292$	0	0
$293$	16.0947	0.940265	0.470132	−	0.882596i	$-0.344206\pi$
$293$	16.0947	0.940265	0.470132	+	0.882596i	$0.344206\pi$
$294$	0	0
$295$	−19.1274	−1.11364
$296$	0	0
$297$	−4.58774	−0.266208
$298$	0	0
$299$	−3.11491	−0.180140
$300$	0	0
$301$	−1.11491	−0.0642622
$302$	0	0
$303$	0.0411284	0.00236277
$304$	0	0
$305$	−11.0349	−0.631856
$306$	0	0
$307$	6.56133	0.374475	0.187238	−	0.982315i	$-0.440047\pi$
$307$	6.56133	0.374475	0.187238	+	0.982315i	$0.440047\pi$
$308$	0	0
$309$	16.9193	0.962503
$310$	0	0
$311$	23.2097	1.31610	0.658049	−	0.752975i	$-0.271382\pi$
$311$	23.2097	1.31610	0.658049	+	0.752975i	$0.271382\pi$
$312$	0	0
$313$	11.1755	0.631676	0.315838	−	0.948813i	$-0.397714\pi$
$313$	11.1755	0.631676	0.315838	+	0.948813i	$0.397714\pi$
$314$	0	0
$315$	1.47283	0.0829848
$316$	0	0
$317$	−12.0411	−0.676297	−0.338149	−	0.941093i	$-0.609801\pi$
$317$	−12.0411	−0.676297	−0.338149	+	0.941093i	$0.609801\pi$
$318$	0	0
$319$	33.5070	1.87603
$320$	0	0
$321$	4.39905	0.245531
$322$	0	0
$323$	−44.9123	−2.49899
$324$	0	0
$325$	−8.81756	−0.489110
$326$	0	0
$327$	−14.6483	−0.810054
$328$	0	0
$329$	−10.2298	−0.563988
$330$	0	0
$331$	−19.0668	−1.04801	−0.524004	−	0.851716i	$-0.675562\pi$
$331$	−19.0668	−1.04801	−0.524004	+	0.851716i	$0.675562\pi$
$332$	0	0
$333$	−8.10170	−0.443971
$334$	0	0
$335$	12.7375	0.695926
$336$	0	0
$337$	−19.1149	−1.04126	−0.520628	−	0.853784i	$-0.674302\pi$
$337$	−19.1149	−1.04126	−0.520628	+	0.853784i	$0.674302\pi$
$338$	0	0
$339$	−18.2709	−0.992341
$340$	0	0
$341$	7.53341	0.407957
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−1.47283	−0.0792947
$346$	0	0
$347$	8.50548	0.456598	0.228299	−	0.973591i	$-0.426684\pi$
$347$	8.50548	0.456598	0.228299	+	0.973591i	$0.426684\pi$
$348$	0	0
$349$	−3.24302	−0.173595	−0.0867974	−	0.996226i	$-0.527663\pi$
$349$	−3.24302	−0.173595	−0.0867974	+	0.996226i	$0.527663\pi$
$350$	0	0
$351$	3.11491	0.166261
$352$	0	0
$353$	2.58774	0.137732	0.0688658	−	0.997626i	$-0.478062\pi$
$353$	2.58774	0.137732	0.0688658	+	0.997626i	$0.478062\pi$
$354$	0	0
$355$	10.2617	0.544637
$356$	0	0
$357$	−6.58774	−0.348660
$358$	0	0
$359$	27.5940	1.45635	0.728177	−	0.685389i	$-0.240368\pi$
$359$	27.5940	1.45635	0.728177	+	0.685389i	$0.240368\pi$
$360$	0	0
$361$	27.4791	1.44627
$362$	0	0
$363$	10.0474	0.527350
$364$	0	0
$365$	22.1623	1.16003
$366$	0	0
$367$	−23.4659	−1.22491	−0.612454	−	0.790506i	$-0.709818\pi$
$367$	−23.4659	−1.22491	−0.612454	+	0.790506i	$0.709818\pi$
$368$	0	0
$369$	−9.87189	−0.513910
$370$	0	0
$371$	−5.47283	−0.284135
$372$	0	0
$373$	1.33000	0.0688649	0.0344324	−	0.999407i	$-0.489038\pi$
$373$	1.33000	0.0688649	0.0344324	+	0.999407i	$0.489038\pi$
$374$	0	0
$375$	−11.5334	−0.595583
$376$	0	0
$377$	−22.7500	−1.17169
$378$	0	0
$379$	4.00000	0.205466	0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000	0.205466	0.102733	+	0.994709i	$0.467241\pi$
$380$	0	0
$381$	−3.68320	−0.188696
$382$	0	0
$383$	5.42474	0.277192	0.138596	−	0.990349i	$-0.455741\pi$
$383$	5.42474	0.277192	0.138596	+	0.990349i	$0.455741\pi$
$384$	0	0
$385$	−6.75698	−0.344368
$386$	0	0
$387$	−1.11491	−0.0566739
$388$	0	0
$389$	33.4806	1.69753	0.848767	−	0.528767i	$-0.177346\pi$
$389$	33.4806	1.69753	0.848767	+	0.528767i	$0.177346\pi$
$390$	0	0
$391$	6.58774	0.333156
$392$	0	0
$393$	15.1949	0.766483
$394$	0	0
$395$	3.97136	0.199821
$396$	0	0
$397$	20.9457	1.05123	0.525616	−	0.850722i	$-0.323835\pi$
$397$	20.9457	1.05123	0.525616	+	0.850722i	$0.323835\pi$
$398$	0	0
$399$	6.81756	0.341305
$400$	0	0
$401$	−20.1017	−1.00383	−0.501916	−	0.864917i	$-0.667371\pi$
$401$	−20.1017	−1.00383	−0.501916	+	0.864917i	$0.667371\pi$
$402$	0	0
$403$	−5.11491	−0.254792
$404$	0	0
$405$	1.47283	0.0731857
$406$	0	0
$407$	37.1685	1.84238
$408$	0	0
$409$	1.53341	0.0758222	0.0379111	−	0.999281i	$-0.487930\pi$
$409$	1.53341	0.0758222	0.0379111	+	0.999281i	$0.487930\pi$
$410$	0	0
$411$	−8.01945	−0.395570
$412$	0	0
$413$	−12.9868	−0.639038
$414$	0	0
$415$	−14.8781	−0.730339
$416$	0	0
$417$	11.5551	0.565855
$418$	0	0
$419$	25.1468	1.22850	0.614252	−	0.789110i	$-0.289458\pi$
$419$	25.1468	1.22850	0.614252	+	0.789110i	$0.289458\pi$
$420$	0	0
$421$	−28.1164	−1.37031	−0.685155	−	0.728397i	$-0.740266\pi$
$421$	−28.1164	−1.37031	−0.685155	+	0.728397i	$0.740266\pi$
$422$	0	0
$423$	−10.2298	−0.497391
$424$	0	0
$425$	18.6483	0.904576
$426$	0	0
$427$	−7.49228	−0.362577
$428$	0	0
$429$	−14.2904	−0.689947
$430$	0	0
$431$	32.5202	1.56644	0.783222	−	0.621743i	$-0.213575\pi$
$431$	32.5202	1.56644	0.783222	+	0.621743i	$0.213575\pi$
$432$	0	0
$433$	−24.4721	−1.17605	−0.588027	−	0.808841i	$-0.700095\pi$
$433$	−24.4721	−1.17605	−0.588027	+	0.808841i	$0.700095\pi$
$434$	0	0
$435$	−10.7570	−0.515758
$436$	0	0
$437$	−6.81756	−0.326128
$438$	0	0
$439$	−16.9651	−0.809701	−0.404850	−	0.914383i	$-0.632676\pi$
$439$	−16.9651	−0.809701	−0.404850	+	0.914383i	$0.632676\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	7.02792	0.333907	0.166953	−	0.985965i	$-0.446607\pi$
$443$	7.02792	0.333907	0.166953	+	0.985965i	$0.446607\pi$
$444$	0	0
$445$	17.6032	0.834471
$446$	0	0
$447$	1.40530	0.0664683
$448$	0	0
$449$	−9.47283	−0.447051	−0.223525	−	0.974698i	$-0.571757\pi$
$449$	−9.47283	−0.447051	−0.223525	+	0.974698i	$0.571757\pi$
$450$	0	0
$451$	45.2897	2.13261
$452$	0	0
$453$	−14.6894	−0.690170
$454$	0	0
$455$	4.58774	0.215077
$456$	0	0
$457$	4.53965	0.212356	0.106178	−	0.994347i	$-0.466139\pi$
$457$	4.53965	0.212356	0.106178	+	0.994347i	$0.466139\pi$
$458$	0	0
$459$	−6.58774	−0.307489
$460$	0	0
$461$	−26.3098	−1.22537	−0.612686	−	0.790327i	$-0.709911\pi$
$461$	−26.3098	−1.22537	−0.612686	+	0.790327i	$0.709911\pi$
$462$	0	0
$463$	−25.0085	−1.16224	−0.581121	−	0.813817i	$-0.697386\pi$
$463$	−25.0085	−1.16224	−0.581121	+	0.813817i	$0.697386\pi$
$464$	0	0
$465$	−2.41850	−0.112155
$466$	0	0
$467$	21.8066	1.00909	0.504544	−	0.863386i	$-0.331661\pi$
$467$	21.8066	1.00909	0.504544	+	0.863386i	$0.331661\pi$
$468$	0	0
$469$	8.64832	0.399342
$470$	0	0
$471$	−0.689445	−0.0317680
$472$	0	0
$473$	5.11491	0.235184
$474$	0	0
$475$	−19.2989	−0.885493
$476$	0	0
$477$	−5.47283	−0.250584
$478$	0	0
$479$	−4.80507	−0.219549	−0.109775	−	0.993957i	$-0.535013\pi$
$479$	−4.80507	−0.219549	−0.109775	+	0.993957i	$0.535013\pi$
$480$	0	0
$481$	−25.2361	−1.15067
$482$	0	0
$483$	−1.00000	−0.0455016
$484$	0	0
$485$	10.3796	0.471314
$486$	0	0
$487$	26.6825	1.20910	0.604549	−	0.796568i	$-0.293353\pi$
$487$	26.6825	1.20910	0.604549	+	0.796568i	$0.293353\pi$
$488$	0	0
$489$	−19.6762	−0.889790
$490$	0	0
$491$	−28.7764	−1.29866	−0.649331	−	0.760506i	$-0.724951\pi$
$491$	−28.7764	−1.29866	−0.649331	+	0.760506i	$0.724951\pi$
$492$	0	0
$493$	48.1142	2.16695
$494$	0	0
$495$	−6.75698	−0.303704
$496$	0	0
$497$	6.96735	0.312528
$498$	0	0
$499$	3.55509	0.159148	0.0795739	−	0.996829i	$-0.474644\pi$
$499$	3.55509	0.159148	0.0795739	+	0.996829i	$0.474644\pi$
$500$	0	0
$501$	15.9108	0.710841
$502$	0	0
$503$	3.60791	0.160869	0.0804343	−	0.996760i	$-0.474369\pi$
$503$	3.60791	0.160869	0.0804343	+	0.996760i	$0.474369\pi$
$504$	0	0
$505$	0.0605754	0.00269557
$506$	0	0
$507$	−3.29735	−0.146440
$508$	0	0
$509$	−10.3774	−0.459969	−0.229984	−	0.973194i	$-0.573868\pi$
$509$	−10.3774	−0.459969	−0.229984	+	0.973194i	$0.573868\pi$
$510$	0	0
$511$	15.0474	0.665657
$512$	0	0
$513$	6.81756	0.301002
$514$	0	0
$515$	24.9193	1.09807
$516$	0	0
$517$	46.9317	2.06406
$518$	0	0
$519$	10.6266	0.466458
$520$	0	0
$521$	−43.3789	−1.90046	−0.950232	−	0.311544i	$-0.899154\pi$
$521$	−43.3789	−1.90046	−0.950232	+	0.311544i	$0.899154\pi$
$522$	0	0
$523$	0.715853	0.0313021	0.0156510	−	0.999878i	$-0.495018\pi$
$523$	0.715853	0.0313021	0.0156510	+	0.999878i	$0.495018\pi$
$524$	0	0
$525$	−2.83076	−0.123545
$526$	0	0
$527$	10.8176	0.471220
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−12.9868	−0.563579
$532$	0	0
$533$	−30.7500	−1.33193
$534$	0	0
$535$	6.47908	0.280115
$536$	0	0
$537$	1.95887	0.0845315
$538$	0	0
$539$	−4.58774	−0.197608
$540$	0	0
$541$	−13.9177	−0.598371	−0.299185	−	0.954195i	$-0.596715\pi$
$541$	−13.9177	−0.598371	−0.299185	+	0.954195i	$0.596715\pi$
$542$	0	0
$543$	6.58774	0.282707
$544$	0	0
$545$	−21.5745	−0.924152
$546$	0	0
$547$	−12.6872	−0.542466	−0.271233	−	0.962514i	$-0.587431\pi$
$547$	−12.6872	−0.542466	−0.271233	+	0.962514i	$0.587431\pi$
$548$	0	0
$549$	−7.49228	−0.319763
$550$	0	0
$551$	−49.7927	−2.12124
$552$	0	0
$553$	2.69641	0.114663
$554$	0	0
$555$	−11.9325	−0.506505
$556$	0	0
$557$	0.0653013	0.00276691	0.00138345	−	0.999999i	$-0.499560\pi$
$557$	0.0653013	0.00276691	0.00138345	+	0.999999i	$0.499560\pi$
$558$	0	0
$559$	−3.47283	−0.146885
$560$	0	0
$561$	30.2229	1.27601
$562$	0	0
$563$	−18.6678	−0.786752	−0.393376	−	0.919378i	$-0.628693\pi$
$563$	−18.6678	−0.786752	−0.393376	+	0.919378i	$0.628693\pi$
$564$	0	0
$565$	−26.9101	−1.13211
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−37.7438	−1.58230	−0.791151	−	0.611621i	$-0.790518\pi$
$569$	−37.7438	−1.58230	−0.791151	+	0.611621i	$0.790518\pi$
$570$	0	0
$571$	−14.1840	−0.593580	−0.296790	−	0.954943i	$-0.595916\pi$
$571$	−14.1840	−0.593580	−0.296790	+	0.954943i	$0.595916\pi$
$572$	0	0
$573$	5.77018	0.241053
$574$	0	0
$575$	2.83076	0.118051
$576$	0	0
$577$	−14.2423	−0.592915	−0.296457	−	0.955046i	$-0.595805\pi$
$577$	−14.2423	−0.592915	−0.296457	+	0.955046i	$0.595805\pi$
$578$	0	0
$579$	−17.1491	−0.712691
$580$	0	0
$581$	−10.1017	−0.419089
$582$	0	0
$583$	25.1079	1.03986
$584$	0	0
$585$	4.58774	0.189680
$586$	0	0
$587$	−34.8323	−1.43768	−0.718841	−	0.695175i	$-0.755327\pi$
$587$	−34.8323	−1.43768	−0.718841	+	0.695175i	$0.755327\pi$
$588$	0	0
$589$	−11.1949	−0.461279
$590$	0	0
$591$	−6.73753	−0.277145
$592$	0	0
$593$	−12.6894	−0.521093	−0.260547	−	0.965461i	$-0.583903\pi$
$593$	−12.6894	−0.521093	−0.260547	+	0.965461i	$0.583903\pi$
$594$	0	0
$595$	−9.70265	−0.397770
$596$	0	0
$597$	−3.85021	−0.157578
$598$	0	0
$599$	−26.4504	−1.08074	−0.540368	−	0.841429i	$-0.681715\pi$
$599$	−26.4504	−1.08074	−0.540368	+	0.841429i	$0.681715\pi$
$600$	0	0
$601$	−8.04113	−0.328004	−0.164002	−	0.986460i	$-0.552440\pi$
$601$	−8.04113	−0.328004	−0.164002	+	0.986460i	$0.552440\pi$
$602$	0	0
$603$	8.64832	0.352187
$604$	0	0
$605$	14.7981	0.601629
$606$	0	0
$607$	−19.6398	−0.797156	−0.398578	−	0.917134i	$-0.630496\pi$
$607$	−19.6398	−0.797156	−0.398578	+	0.917134i	$0.630496\pi$
$608$	0	0
$609$	−7.30359	−0.295957
$610$	0	0
$611$	−31.8649	−1.28912
$612$	0	0
$613$	30.3315	1.22508	0.612539	−	0.790440i	$-0.290148\pi$
$613$	30.3315	1.22508	0.612539	+	0.790440i	$0.290148\pi$
$614$	0	0
$615$	−14.5397	−0.586295
$616$	0	0
$617$	−41.8821	−1.68611	−0.843056	−	0.537826i	$-0.819246\pi$
$617$	−41.8821	−1.68611	−0.843056	+	0.537826i	$0.819246\pi$
$618$	0	0
$619$	−11.3014	−0.454240	−0.227120	−	0.973867i	$-0.572931\pi$
$619$	−11.3014	−0.454240	−0.227120	+	0.973867i	$0.572931\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	11.9519	0.478843
$624$	0	0
$625$	−2.83299	−0.113320
$626$	0	0
$627$	−31.2772	−1.24909
$628$	0	0
$629$	53.3719	2.12808
$630$	0	0
$631$	17.1296	0.681920	0.340960	−	0.940078i	$-0.389248\pi$
$631$	17.1296	0.681920	0.340960	+	0.940078i	$0.389248\pi$
$632$	0	0
$633$	−3.53341	−0.140440
$634$	0	0
$635$	−5.42474	−0.215274
$636$	0	0
$637$	3.11491	0.123417
$638$	0	0
$639$	6.96735	0.275624
$640$	0	0
$641$	39.6134	1.56464	0.782318	−	0.622879i	$-0.214037\pi$
$641$	39.6134	1.56464	0.782318	+	0.622879i	$0.214037\pi$
$642$	0	0
$643$	13.4534	0.530550	0.265275	−	0.964173i	$-0.414537\pi$
$643$	13.4534	0.530550	0.265275	+	0.964173i	$0.414537\pi$
$644$	0	0
$645$	−1.64207	−0.0646566
$646$	0	0
$647$	−41.4853	−1.63096	−0.815478	−	0.578788i	$-0.803526\pi$
$647$	−41.4853	−1.63096	−0.815478	+	0.578788i	$0.803526\pi$
$648$	0	0
$649$	59.5801	2.33872
$650$	0	0
$651$	−1.64207	−0.0643579
$652$	0	0
$653$	19.5815	0.766283	0.383142	−	0.923690i	$-0.374842\pi$
$653$	19.5815	0.766283	0.383142	+	0.923690i	$0.374842\pi$
$654$	0	0
$655$	22.3796	0.874444
$656$	0	0
$657$	15.0474	0.587054
$658$	0	0
$659$	−15.4512	−0.601891	−0.300946	−	0.953641i	$-0.597302\pi$
$659$	−15.4512	−0.601891	−0.300946	+	0.953641i	$0.597302\pi$
$660$	0	0
$661$	40.9387	1.59233	0.796166	−	0.605079i	$-0.206858\pi$
$661$	40.9387	1.59233	0.796166	+	0.605079i	$0.206858\pi$
$662$	0	0
$663$	−20.5202	−0.796939
$664$	0	0
$665$	10.0411	0.389378
$666$	0	0
$667$	7.30359	0.282796
$668$	0	0
$669$	10.4185	0.402803
$670$	0	0
$671$	34.3726	1.32694
$672$	0	0
$673$	−30.0753	−1.15932	−0.579659	−	0.814859i	$-0.696814\pi$
$673$	−30.0753	−1.15932	−0.579659	+	0.814859i	$0.696814\pi$
$674$	0	0
$675$	−2.83076	−0.108956
$676$	0	0
$677$	27.8697	1.07112	0.535559	−	0.844498i	$-0.320101\pi$
$677$	27.8697	1.07112	0.535559	+	0.844498i	$0.320101\pi$
$678$	0	0
$679$	7.04737	0.270453
$680$	0	0
$681$	24.0147	0.920246
$682$	0	0
$683$	−12.2882	−0.470193	−0.235097	−	0.971972i	$-0.575541\pi$
$683$	−12.2882	−0.470193	−0.235097	+	0.971972i	$0.575541\pi$
$684$	0	0
$685$	−11.8113	−0.451287
$686$	0	0
$687$	5.13435	0.195888
$688$	0	0
$689$	−17.0474	−0.649453
$690$	0	0
$691$	−7.68320	−0.292283	−0.146141	−	0.989264i	$-0.546685\pi$
$691$	−7.68320	−0.292283	−0.146141	+	0.989264i	$0.546685\pi$
$692$	0	0
$693$	−4.58774	−0.174274
$694$	0	0
$695$	17.0187	0.645557
$696$	0	0
$697$	65.0335	2.46332
$698$	0	0
$699$	0.500759	0.0189404
$700$	0	0
$701$	−11.7415	−0.443472	−0.221736	−	0.975107i	$-0.571172\pi$
$701$	−11.7415	−0.443472	−0.221736	+	0.975107i	$0.571172\pi$
$702$	0	0
$703$	−55.2338	−2.08318
$704$	0	0
$705$	−15.0668	−0.567449
$706$	0	0
$707$	0.0411284	0.00154679
$708$	0	0
$709$	−4.32928	−0.162590	−0.0812948	−	0.996690i	$-0.525906\pi$
$709$	−4.32928	−0.162590	−0.0812948	+	0.996690i	$0.525906\pi$
$710$	0	0
$711$	2.69641	0.101123
$712$	0	0
$713$	1.64207	0.0614961
$714$	0	0
$715$	−21.0474	−0.787127
$716$	0	0
$717$	19.9714	0.745844
$718$	0	0
$719$	24.9512	0.930523	0.465261	−	0.885173i	$-0.345960\pi$
$719$	24.9512	0.930523	0.465261	+	0.885173i	$0.345960\pi$
$720$	0	0
$721$	16.9193	0.630106
$722$	0	0
$723$	−19.0474	−0.708379
$724$	0	0
$725$	20.6747	0.767840
$726$	0	0
$727$	−16.7089	−0.619699	−0.309849	−	0.950786i	$-0.600279\pi$
$727$	−16.7089	−0.619699	−0.309849	+	0.950786i	$0.600279\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	7.34472	0.271654
$732$	0	0
$733$	20.5249	0.758106	0.379053	−	0.925375i	$-0.376250\pi$
$733$	20.5249	0.758106	0.379053	+	0.925375i	$0.376250\pi$
$734$	0	0
$735$	1.47283	0.0543263
$736$	0	0
$737$	−39.6762	−1.46149
$738$	0	0
$739$	25.0040	0.919787	0.459894	−	0.887974i	$-0.347888\pi$
$739$	25.0040	0.919787	0.459894	+	0.887974i	$0.347888\pi$
$740$	0	0
$741$	21.2361	0.780126
$742$	0	0
$743$	30.4574	1.11737	0.558687	−	0.829379i	$-0.311305\pi$
$743$	30.4574	1.11737	0.558687	+	0.829379i	$0.311305\pi$
$744$	0	0
$745$	2.06977	0.0758305
$746$	0	0
$747$	−10.1017	−0.369602
$748$	0	0
$749$	4.39905	0.160738
$750$	0	0
$751$	−15.4200	−0.562684	−0.281342	−	0.959607i	$-0.590780\pi$
$751$	−15.4200	−0.562684	−0.281342	+	0.959607i	$0.590780\pi$
$752$	0	0
$753$	13.9736	0.509226
$754$	0	0
$755$	−21.6351	−0.787382
$756$	0	0
$757$	43.4247	1.57830	0.789150	−	0.614201i	$-0.210522\pi$
$757$	43.4247	1.57830	0.789150	+	0.614201i	$0.210522\pi$
$758$	0	0
$759$	4.58774	0.166524
$760$	0	0
$761$	−2.20341	−0.0798735	−0.0399367	−	0.999202i	$-0.512716\pi$
$761$	−2.20341	−0.0798735	−0.0399367	+	0.999202i	$0.512716\pi$
$762$	0	0
$763$	−14.6483	−0.530305
$764$	0	0
$765$	−9.70265	−0.350800
$766$	0	0
$767$	−40.4527	−1.46066
$768$	0	0
$769$	23.2577	0.838696	0.419348	−	0.907826i	$-0.362259\pi$
$769$	23.2577	0.838696	0.419348	+	0.907826i	$0.362259\pi$
$770$	0	0
$771$	22.3246	0.803999
$772$	0	0
$773$	38.4527	1.38305	0.691523	−	0.722354i	$-0.256940\pi$
$773$	38.4527	1.38305	0.691523	+	0.722354i	$0.256940\pi$
$774$	0	0
$775$	4.64832	0.166972
$776$	0	0
$777$	−8.10170	−0.290647
$778$	0	0
$779$	−67.3022	−2.41135
$780$	0	0
$781$	−31.9644	−1.14378
$782$	0	0
$783$	−7.30359	−0.261009
$784$	0	0
$785$	−1.01544	−0.0362425
$786$	0	0
$787$	−41.6109	−1.48327	−0.741635	−	0.670804i	$-0.765949\pi$
$787$	−41.6109	−1.48327	−0.741635	+	0.670804i	$0.765949\pi$
$788$	0	0
$789$	−23.2772	−0.828690
$790$	0	0
$791$	−18.2709	−0.649640
$792$	0	0
$793$	−23.3378	−0.828748
$794$	0	0
$795$	−8.06058	−0.285879
$796$	0	0
$797$	−4.24374	−0.150321	−0.0751604	−	0.997171i	$-0.523947\pi$
$797$	−4.24374	−0.150321	−0.0751604	+	0.997171i	$0.523947\pi$
$798$	0	0
$799$	67.3914	2.38414
$800$	0	0
$801$	11.9519	0.422300
$802$	0	0
$803$	−69.0335	−2.43614
$804$	0	0
$805$	−1.47283	−0.0519106
$806$	0	0
$807$	−1.60095	−0.0563559
$808$	0	0
$809$	−34.6803	−1.21929	−0.609646	−	0.792674i	$-0.708689\pi$
$809$	−34.6803	−1.21929	−0.609646	+	0.792674i	$0.708689\pi$
$810$	0	0
$811$	55.1949	1.93816	0.969078	−	0.246754i	$-0.0793641\pi$
$811$	55.1949	1.93816	0.969078	+	0.246754i	$0.0793641\pi$
$812$	0	0
$813$	−27.3594	−0.959538
$814$	0	0
$815$	−28.9798	−1.01512
$816$	0	0
$817$	−7.60095	−0.265923
$818$	0	0
$819$	3.11491	0.108844
$820$	0	0
$821$	−12.6506	−0.441507	−0.220754	−	0.975330i	$-0.570852\pi$
$821$	−12.6506	−0.441507	−0.220754	+	0.975330i	$0.570852\pi$
$822$	0	0
$823$	−48.5521	−1.69242	−0.846211	−	0.532849i	$-0.821122\pi$
$823$	−48.5521	−1.69242	−0.846211	+	0.532849i	$0.821122\pi$
$824$	0	0
$825$	12.9868	0.452142
$826$	0	0
$827$	−29.6065	−1.02952	−0.514759	−	0.857335i	$-0.672119\pi$
$827$	−29.6065	−1.02952	−0.514759	+	0.857335i	$0.672119\pi$
$828$	0	0
$829$	53.3550	1.85309	0.926547	−	0.376178i	$-0.122762\pi$
$829$	53.3550	1.85309	0.926547	+	0.376178i	$0.122762\pi$
$830$	0	0
$831$	10.7375	0.372481
$832$	0	0
$833$	−6.58774	−0.228252
$834$	0	0
$835$	23.4339	0.810965
$836$	0	0
$837$	−1.64207	−0.0567584
$838$	0	0
$839$	43.4270	1.49927	0.749633	−	0.661854i	$-0.230230\pi$
$839$	43.4270	1.49927	0.749633	+	0.661854i	$0.230230\pi$
$840$	0	0
$841$	24.3425	0.839396
$842$	0	0
$843$	0.689445	0.0237458
$844$	0	0
$845$	−4.85645	−0.167067
$846$	0	0
$847$	10.0474	0.345232
$848$	0	0
$849$	−33.1902	−1.13908
$850$	0	0
$851$	8.10170	0.277723
$852$	0	0
$853$	−49.0963	−1.68102	−0.840512	−	0.541793i	$-0.817746\pi$
$853$	−49.0963	−1.68102	−0.840512	+	0.541793i	$0.817746\pi$
$854$	0	0
$855$	10.0411	0.343399
$856$	0	0
$857$	24.1770	0.825871	0.412935	−	0.910760i	$-0.364504\pi$
$857$	24.1770	0.825871	0.412935	+	0.910760i	$0.364504\pi$
$858$	0	0
$859$	−48.2034	−1.64468	−0.822340	−	0.568997i	$-0.807332\pi$
$859$	−48.2034	−1.64468	−0.822340	+	0.568997i	$0.807332\pi$
$860$	0	0
$861$	−9.87189	−0.336433
$862$	0	0
$863$	−42.8929	−1.46009	−0.730045	−	0.683399i	$-0.760501\pi$
$863$	−42.8929	−1.46009	−0.730045	+	0.683399i	$0.760501\pi$
$864$	0	0
$865$	15.6513	0.532159
$866$	0	0
$867$	26.3983	0.896535
$868$	0	0
$869$	−12.3704	−0.419638
$870$	0	0
$871$	26.9387	0.912783
$872$	0	0
$873$	7.04737	0.238517
$874$	0	0
$875$	−11.5334	−0.389900
$876$	0	0
$877$	−49.3091	−1.66505	−0.832525	−	0.553987i	$-0.813106\pi$
$877$	−49.3091	−1.66505	−0.832525	+	0.553987i	$0.813106\pi$
$878$	0	0
$879$	16.0947	0.542862
$880$	0	0
$881$	11.0404	0.371961	0.185980	−	0.982553i	$-0.440454\pi$
$881$	11.0404	0.371961	0.185980	+	0.982553i	$0.440454\pi$
$882$	0	0
$883$	55.9116	1.88157	0.940787	−	0.338997i	$-0.110088\pi$
$883$	55.9116	1.88157	0.940787	+	0.338997i	$0.110088\pi$
$884$	0	0
$885$	−19.1274	−0.642960
$886$	0	0
$887$	46.0202	1.54521	0.772604	−	0.634888i	$-0.218954\pi$
$887$	46.0202	1.54521	0.772604	+	0.634888i	$0.218954\pi$
$888$	0	0
$889$	−3.68320	−0.123531
$890$	0	0
$891$	−4.58774	−0.153695
$892$	0	0
$893$	−69.7423	−2.33384
$894$	0	0
$895$	2.88509	0.0964380
$896$	0	0
$897$	−3.11491	−0.104004
$898$	0	0
$899$	11.9930	0.399990
$900$	0	0
$901$	36.0536	1.20112
$902$	0	0
$903$	−1.11491	−0.0371018
$904$	0	0
$905$	9.70265	0.322527
$906$	0	0
$907$	17.4464	0.579299	0.289650	−	0.957133i	$-0.406461\pi$
$907$	17.4464	0.579299	0.289650	+	0.957133i	$0.406461\pi$
$908$	0	0
$909$	0.0411284	0.00136414
$910$	0	0
$911$	−58.7144	−1.94530	−0.972648	−	0.232285i	$-0.925380\pi$
$911$	−58.7144	−1.94530	−0.972648	+	0.232285i	$0.925380\pi$
$912$	0	0
$913$	46.3440	1.53376
$914$	0	0
$915$	−11.0349	−0.364802
$916$	0	0
$917$	15.1949	0.501781
$918$	0	0
$919$	1.64207	0.0541670	0.0270835	−	0.999633i	$-0.491378\pi$
$919$	1.64207	0.0541670	0.0270835	+	0.999633i	$0.491378\pi$
$920$	0	0
$921$	6.56133	0.216203
$922$	0	0
$923$	21.7026	0.714351
$924$	0	0
$925$	22.9340	0.754065
$926$	0	0
$927$	16.9193	0.555701
$928$	0	0
$929$	−1.45891	−0.0478654	−0.0239327	−	0.999714i	$-0.507619\pi$
$929$	−1.45891	−0.0478654	−0.0239327	+	0.999714i	$0.507619\pi$
$930$	0	0
$931$	6.81756	0.223436
$932$	0	0
$933$	23.2097	0.759850
$934$	0	0
$935$	44.5132	1.45574
$936$	0	0
$937$	−8.98207	−0.293431	−0.146716	−	0.989179i	$-0.546870\pi$
$937$	−8.98207	−0.293431	−0.146716	+	0.989179i	$0.546870\pi$
$938$	0	0
$939$	11.1755	0.364698
$940$	0	0
$941$	37.9861	1.23831	0.619155	−	0.785268i	$-0.287475\pi$
$941$	37.9861	1.23831	0.619155	+	0.785268i	$0.287475\pi$
$942$	0	0
$943$	9.87189	0.321473
$944$	0	0
$945$	1.47283	0.0479113
$946$	0	0
$947$	12.4985	0.406147	0.203074	−	0.979163i	$-0.434907\pi$
$947$	12.4985	0.406147	0.203074	+	0.979163i	$0.434907\pi$
$948$	0	0
$949$	46.8712	1.52150
$950$	0	0
$951$	−12.0411	−0.390460
$952$	0	0
$953$	9.22357	0.298781	0.149390	−	0.988778i	$-0.452269\pi$
$953$	9.22357	0.298781	0.149390	+	0.988778i	$0.452269\pi$
$954$	0	0
$955$	8.49852	0.275006
$956$	0	0
$957$	33.5070	1.08313
$958$	0	0
$959$	−8.01945	−0.258961
$960$	0	0
$961$	−28.3036	−0.913019
$962$	0	0
$963$	4.39905	0.141758
$964$	0	0
$965$	−25.2577	−0.813075
$966$	0	0
$967$	−7.74378	−0.249023	−0.124512	−	0.992218i	$-0.539736\pi$
$967$	−7.74378	−0.249023	−0.124512	+	0.992218i	$0.539736\pi$
$968$	0	0
$969$	−44.9123	−1.44279
$970$	0	0
$971$	31.8044	1.02065	0.510325	−	0.859982i	$-0.329525\pi$
$971$	31.8044	1.02065	0.510325	+	0.859982i	$0.329525\pi$
$972$	0	0
$973$	11.5551	0.370439
$974$	0	0
$975$	−8.81756	−0.282388
$976$	0	0
$977$	47.7779	1.52855	0.764276	−	0.644889i	$-0.223097\pi$
$977$	47.7779	1.52855	0.764276	+	0.644889i	$0.223097\pi$
$978$	0	0
$979$	−54.8323	−1.75245
$980$	0	0
$981$	−14.6483	−0.467685
$982$	0	0
$983$	−16.7089	−0.532931	−0.266465	−	0.963844i	$-0.585856\pi$
$983$	−16.7089	−0.532931	−0.266465	+	0.963844i	$0.585856\pi$
$984$	0	0
$985$	−9.92327	−0.316182
$986$	0	0
$987$	−10.2298	−0.325619
$988$	0	0
$989$	1.11491	0.0354520
$990$	0	0
$991$	21.7919	0.692241	0.346121	−	0.938190i	$-0.387499\pi$
$991$	21.7919	0.692241	0.346121	+	0.938190i	$0.387499\pi$
$992$	0	0
$993$	−19.0668	−0.605067
$994$	0	0
$995$	−5.67072	−0.179774
$996$	0	0
$997$	26.0070	0.823649	0.411824	−	0.911263i	$-0.364892\pi$
$997$	26.0070	0.823649	0.411824	+	0.911263i	$0.364892\pi$
$998$	0	0
$999$	−8.10170	−0.256327

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7728.2.a.bw.1.3		3
4.3	odd	2		3864.2.a.l.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3864.2.a.l.1.3	✓	3	4.3	odd	2
7728.2.a.bw.1.3		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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.47283 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.47283 q^{5} +1.00000 q^{7} +1.00000 q^{9} -4.58774 q^{11} +3.11491 q^{13} +1.47283 q^{15} -6.58774 q^{17} +6.81756 q^{19} +1.00000 q^{21} -1.00000 q^{23} -2.83076 q^{25} +1.00000 q^{27} -7.30359 q^{29} -1.64207 q^{31} -4.58774 q^{33} +1.47283 q^{35} -8.10170 q^{37} +3.11491 q^{39} -9.87189 q^{41} -1.11491 q^{43} +1.47283 q^{45} -10.2298 q^{47} +1.00000 q^{49} -6.58774 q^{51} -5.47283 q^{53} -6.75698 q^{55} +6.81756 q^{57} -12.9868 q^{59} -7.49228 q^{61} +1.00000 q^{63} +4.58774 q^{65} +8.64832 q^{67} -1.00000 q^{69} +6.96735 q^{71} +15.0474 q^{73} -2.83076 q^{75} -4.58774 q^{77} +2.69641 q^{79} +1.00000 q^{81} -10.1017 q^{83} -9.70265 q^{85} -7.30359 q^{87} +11.9519 q^{89} +3.11491 q^{91} -1.64207 q^{93} +10.0411 q^{95} +7.04737 q^{97} -4.58774 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} - q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 - q^5 + 3 * q^7 + 3 * q^9	\( 3 q + 3 q^{3} - q^{5} + 3 q^{7} + 3 q^{9} - 2 q^{11} + 3 q^{13} - q^{15} - 8 q^{17} - 4 q^{19} + 3 q^{21} - 3 q^{23} - 4 q^{25} + 3 q^{27} - 12 q^{29} - 4 q^{31} - 2 q^{33} - q^{35} + 2 q^{37} + 3 q^{39} - 16 q^{41} + 3 q^{43} - q^{45} - 18 q^{47} + 3 q^{49} - 8 q^{51} - 11 q^{53} - 13 q^{55} - 4 q^{57} - 19 q^{59} - 9 q^{61} + 3 q^{63} + 2 q^{65} - 3 q^{67} - 3 q^{69} + 9 q^{71} + 8 q^{73} - 4 q^{75} - 2 q^{77} + 18 q^{79} + 3 q^{81} - 4 q^{83} - 11 q^{85} - 12 q^{87} - 3 q^{89} + 3 q^{91} - 4 q^{93} + 21 q^{95} - 16 q^{97} - 2 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 - q^5 + 3 * q^7 + 3 * q^9 - 2 * q^11 + 3 * q^13 - q^15 - 8 * q^17 - 4 * q^19 + 3 * q^21 - 3 * q^23 - 4 * q^25 + 3 * q^27 - 12 * q^29 - 4 * q^31 - 2 * q^33 - q^35 + 2 * q^37 + 3 * q^39 - 16 * q^41 + 3 * q^43 - q^45 - 18 * q^47 + 3 * q^49 - 8 * q^51 - 11 * q^53 - 13 * q^55 - 4 * q^57 - 19 * q^59 - 9 * q^61 + 3 * q^63 + 2 * q^65 - 3 * q^67 - 3 * q^69 + 9 * q^71 + 8 * q^73 - 4 * q^75 - 2 * q^77 + 18 * q^79 + 3 * q^81 - 4 * q^83 - 11 * q^85 - 12 * q^87 - 3 * q^89 + 3 * q^91 - 4 * q^93 + 21 * q^95 - 16 * q^97 - 2 * q^99

Introduction

L-functions

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