LMFDB - Embedded newform 4864.2.a.be.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4864,2,Mod(1,4864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4864, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4864.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4864 = 2^{8} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4864.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:	$38.8392355432$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.892.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 8x + 10 $ x^3 - x^2 - 8*x + 10
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2432)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.91729$ of defining polynomial
Character	$\chi$	$=$	4864.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.91729 q^{3} -3.91729 q^{5} +0.593272 q^{7} +5.51056 q^{9} +O(q^{10})$	$q-2.91729 q^{3} -3.91729 q^{5} +0.593272 q^{7} +5.51056 q^{9} +3.51056 q^{11} +2.51056 q^{13} +11.4278 q^{15} +3.00000 q^{17} -1.00000 q^{19} -1.73074 q^{21} +7.32401 q^{23} +10.3451 q^{25} -7.32401 q^{27} +5.73074 q^{29} +4.40673 q^{31} -10.2413 q^{33} -2.32401 q^{35} -3.42784 q^{37} -7.32401 q^{39} -10.2413 q^{41} -4.69710 q^{43} -21.5864 q^{45} +2.08271 q^{47} -6.64803 q^{49} -8.75186 q^{51} +1.08271 q^{53} -13.7519 q^{55} +2.91729 q^{57} -8.51056 q^{59} +7.10383 q^{61} +3.26926 q^{63} -9.83457 q^{65} +12.9173 q^{67} -21.3662 q^{69} +12.8557 q^{71} -16.2288 q^{73} -30.1797 q^{75} +2.08271 q^{77} +4.40673 q^{79} +4.83457 q^{81} +13.4615 q^{83} -11.7519 q^{85} -16.7182 q^{87} +0.241300 q^{89} +1.48944 q^{91} -12.8557 q^{93} +3.91729 q^{95} +15.4615 q^{97} +19.3451 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{3} - 2 q^{5} + 3 q^{7} + 8 q^{9}+O(q^{10}) $ 3 * q + q^3 - 2 * q^5 + 3 * q^7 + 8 * q^9	$ 3 q + q^{3} - 2 q^{5} + 3 q^{7} + 8 q^{9} + 2 q^{11} - q^{13} + 16 q^{15} + 9 q^{17} - 3 q^{19} + 7 q^{21} + 11 q^{23} + 3 q^{25} - 11 q^{27} + 5 q^{29} + 12 q^{31} - 10 q^{33} + 4 q^{35} + 8 q^{37} - 11 q^{39} - 10 q^{41} - 8 q^{43} - 16 q^{45} + 16 q^{47} + 2 q^{49} + 3 q^{51} + 13 q^{53} - 12 q^{55} - q^{57} - 17 q^{59} + 14 q^{61} + 22 q^{63} - 10 q^{65} + 29 q^{67} - 19 q^{69} + 2 q^{71} - 17 q^{73} - 43 q^{75} + 16 q^{77} + 12 q^{79} - 5 q^{81} + 16 q^{83} - 6 q^{85} - 27 q^{87} - 20 q^{89} + 13 q^{91} - 2 q^{93} + 2 q^{95} + 22 q^{97} + 30 q^{99}+O(q^{100}) $ 3 * q + q^3 - 2 * q^5 + 3 * q^7 + 8 * q^9 + 2 * q^11 - q^13 + 16 * q^15 + 9 * q^17 - 3 * q^19 + 7 * q^21 + 11 * q^23 + 3 * q^25 - 11 * q^27 + 5 * q^29 + 12 * q^31 - 10 * q^33 + 4 * q^35 + 8 * q^37 - 11 * q^39 - 10 * q^41 - 8 * q^43 - 16 * q^45 + 16 * q^47 + 2 * q^49 + 3 * q^51 + 13 * q^53 - 12 * q^55 - q^57 - 17 * q^59 + 14 * q^61 + 22 * q^63 - 10 * q^65 + 29 * q^67 - 19 * q^69 + 2 * q^71 - 17 * q^73 - 43 * q^75 + 16 * q^77 + 12 * q^79 - 5 * q^81 + 16 * q^83 - 6 * q^85 - 27 * q^87 - 20 * q^89 + 13 * q^91 - 2 * q^93 + 2 * q^95 + 22 * q^97 + 30 * 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
$3$	−2.91729	−1.68430	−0.842148	−	0.539247i	$-0.818709\pi$
$3$	−2.91729	−1.68430	−0.842148	+	0.539247i	$0.818709\pi$
$4$	0	0
$5$	−3.91729	−1.75186	−0.875932	−	0.482435i	$-0.839752\pi$
$5$	−3.91729	−1.75186	−0.875932	+	0.482435i	$0.839752\pi$
$6$	0	0
$7$	0.593272	0.224236	0.112118	−	0.993695i	$-0.464237\pi$
$7$	0.593272	0.224236	0.112118	+	0.993695i	$0.464237\pi$
$8$	0	0
$9$	5.51056	1.83685
$10$	0	0
$11$	3.51056	1.05847	0.529236	−	0.848474i	$-0.322478\pi$
$11$	3.51056	1.05847	0.529236	+	0.848474i	$0.322478\pi$
$12$	0	0
$13$	2.51056	0.696303	0.348152	−	0.937438i	$-0.386809\pi$
$13$	2.51056	0.696303	0.348152	+	0.937438i	$0.386809\pi$
$14$	0	0
$15$	11.4278	2.95066
$16$	0	0
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−1.73074	−0.377679
$22$	0	0
$23$	7.32401	1.52716	0.763581	−	0.645712i	$-0.223439\pi$
$23$	7.32401	1.52716	0.763581	+	0.645712i	$0.223439\pi$
$24$	0	0
$25$	10.3451	2.06903
$26$	0	0
$27$	−7.32401	−1.40951
$28$	0	0
$29$	5.73074	1.06417	0.532086	−	0.846690i	$-0.321408\pi$
$29$	5.73074	1.06417	0.532086	+	0.846690i	$0.321408\pi$
$30$	0	0
$31$	4.40673	0.791472	0.395736	−	0.918364i	$-0.370490\pi$
$31$	4.40673	0.791472	0.395736	+	0.918364i	$0.370490\pi$
$32$	0	0
$33$	−10.2413	−1.78278
$34$	0	0
$35$	−2.32401	−0.392830
$36$	0	0
$37$	−3.42784	−0.563534	−0.281767	−	0.959483i	$-0.590921\pi$
$37$	−3.42784	−0.563534	−0.281767	+	0.959483i	$0.590921\pi$
$38$	0	0
$39$	−7.32401	−1.17278
$40$	0	0
$41$	−10.2413	−1.59942	−0.799711	−	0.600385i	$-0.795014\pi$
$41$	−10.2413	−1.59942	−0.799711	+	0.600385i	$0.795014\pi$
$42$	0	0
$43$	−4.69710	−0.716301	−0.358151	−	0.933664i	$-0.616593\pi$
$43$	−4.69710	−0.716301	−0.358151	+	0.933664i	$0.616593\pi$
$44$	0	0
$45$	−21.5864	−3.21791
$46$	0	0
$47$	2.08271	0.303795	0.151898	−	0.988396i	$-0.451462\pi$
$47$	2.08271	0.303795	0.151898	+	0.988396i	$0.451462\pi$
$48$	0	0
$49$	−6.64803	−0.949718
$50$	0	0
$51$	−8.75186	−1.22551
$52$	0	0
$53$	1.08271	0.148722	0.0743611	−	0.997231i	$-0.476308\pi$
$53$	1.08271	0.148722	0.0743611	+	0.997231i	$0.476308\pi$
$54$	0	0
$55$	−13.7519	−1.85430
$56$	0	0
$57$	2.91729	0.386404
$58$	0	0
$59$	−8.51056	−1.10798	−0.553990	−	0.832523i	$-0.686895\pi$
$59$	−8.51056	−1.10798	−0.553990	+	0.832523i	$0.686895\pi$
$60$	0	0
$61$	7.10383	0.909552	0.454776	−	0.890606i	$-0.349719\pi$
$61$	7.10383	0.909552	0.454776	+	0.890606i	$0.349719\pi$
$62$	0	0
$63$	3.26926	0.411888
$64$	0	0
$65$	−9.83457	−1.21983
$66$	0	0
$67$	12.9173	1.57810	0.789049	−	0.614330i	$-0.210574\pi$
$67$	12.9173	1.57810	0.789049	+	0.614330i	$0.210574\pi$
$68$	0	0
$69$	−21.3662	−2.57219
$70$	0	0
$71$	12.8557	1.52569	0.762845	−	0.646582i	$-0.223802\pi$
$71$	12.8557	1.52569	0.762845	+	0.646582i	$0.223802\pi$
$72$	0	0
$73$	−16.2288	−1.89943	−0.949717	−	0.313109i	$-0.898629\pi$
$73$	−16.2288	−1.89943	−0.949717	+	0.313109i	$0.898629\pi$
$74$	0	0
$75$	−30.1797	−3.48485
$76$	0	0
$77$	2.08271	0.237347
$78$	0	0
$79$	4.40673	0.495796	0.247898	−	0.968786i	$-0.420260\pi$
$79$	4.40673	0.495796	0.247898	+	0.968786i	$0.420260\pi$
$80$	0	0
$81$	4.83457	0.537175
$82$	0	0
$83$	13.4615	1.47759	0.738795	−	0.673930i	$-0.235395\pi$
$83$	13.4615	1.47759	0.738795	+	0.673930i	$0.235395\pi$
$84$	0	0
$85$	−11.7519	−1.27467
$86$	0	0
$87$	−16.7182	−1.79238
$88$	0	0
$89$	0.241300	0.0255778	0.0127889	−	0.999918i	$-0.495929\pi$
$89$	0.241300	0.0255778	0.0127889	+	0.999918i	$0.495929\pi$
$90$	0	0
$91$	1.48944	0.156136
$92$	0	0
$93$	−12.8557	−1.33307
$94$	0	0
$95$	3.91729	0.401905
$96$	0	0
$97$	15.4615	1.56988	0.784938	−	0.619574i	$-0.212695\pi$
$97$	15.4615	1.56988	0.784938	+	0.619574i	$0.212695\pi$
$98$	0	0
$99$	19.3451	1.94426
$100$	0	0
$101$	−15.8346	−1.57560	−0.787799	−	0.615932i	$-0.788780\pi$
$101$	−15.8346	−1.57560	−0.787799	+	0.615932i	$0.788780\pi$
$102$	0	0
$103$	13.4615	1.32640	0.663200	−	0.748442i	$-0.269198\pi$
$103$	13.4615	1.32640	0.663200	+	0.748442i	$0.269198\pi$
$104$	0	0
$105$	6.77981	0.661642
$106$	0	0
$107$	−0.302899	−0.0292824	−0.0146412	−	0.999893i	$-0.504661\pi$
$107$	−0.302899	−0.0292824	−0.0146412	+	0.999893i	$0.504661\pi$
$108$	0	0
$109$	5.29037	0.506726	0.253363	−	0.967371i	$-0.418463\pi$
$109$	5.29037	0.506726	0.253363	+	0.967371i	$0.418463\pi$
$110$	0	0
$111$	10.0000	0.949158
$112$	0	0
$113$	2.77981	0.261503	0.130751	−	0.991415i	$-0.458261\pi$
$113$	2.77981	0.261503	0.130751	+	0.991415i	$0.458261\pi$
$114$	0	0
$115$	−28.6903	−2.67538
$116$	0	0
$117$	13.8346	1.27901
$118$	0	0
$119$	1.77981	0.163155
$120$	0	0
$121$	1.32401	0.120365
$122$	0	0
$123$	29.8768	2.69390
$124$	0	0
$125$	−20.9384	−1.87279
$126$	0	0
$127$	−14.2835	−1.26746	−0.633729	−	0.773555i	$-0.718477\pi$
$127$	−14.2835	−1.26746	−0.633729	+	0.773555i	$0.718477\pi$
$128$	0	0
$129$	13.7028	1.20646
$130$	0	0
$131$	16.3662	1.42993	0.714963	−	0.699163i	$-0.246444\pi$
$131$	16.3662	1.42993	0.714963	+	0.699163i	$0.246444\pi$
$132$	0	0
$133$	−0.593272	−0.0514432
$134$	0	0
$135$	28.6903	2.46926
$136$	0	0
$137$	−4.66914	−0.398912	−0.199456	−	0.979907i	$-0.563917\pi$
$137$	−4.66914	−0.398912	−0.199456	+	0.979907i	$0.563917\pi$
$138$	0	0
$139$	−6.48944	−0.550427	−0.275214	−	0.961383i	$-0.588749\pi$
$139$	−6.48944	−0.550427	−0.275214	+	0.961383i	$0.588749\pi$
$140$	0	0
$141$	−6.07587	−0.511681
$142$	0	0
$143$	8.81346	0.737018
$144$	0	0
$145$	−22.4490	−1.86428
$146$	0	0
$147$	19.3942	1.59961
$148$	0	0
$149$	6.73074	0.551404	0.275702	−	0.961243i	$-0.411090\pi$
$149$	6.73074	0.551404	0.275702	+	0.961243i	$0.411090\pi$
$150$	0	0
$151$	18.2077	1.48172	0.740859	−	0.671660i	$-0.234419\pi$
$151$	18.2077	1.48172	0.740859	+	0.671660i	$0.234419\pi$
$152$	0	0
$153$	16.5317	1.33651
$154$	0	0
$155$	−17.2624	−1.38655
$156$	0	0
$157$	−16.0422	−1.28031	−0.640155	−	0.768246i	$-0.721130\pi$
$157$	−16.0422	−1.28031	−0.640155	+	0.768246i	$0.721130\pi$
$158$	0	0
$159$	−3.15859	−0.250492
$160$	0	0
$161$	4.34513	0.342444
$162$	0	0
$163$	−24.0422	−1.88313	−0.941566	−	0.336827i	$-0.890646\pi$
$163$	−24.0422	−1.88313	−0.941566	+	0.336827i	$0.890646\pi$
$164$	0	0
$165$	40.1181	3.12319
$166$	0	0
$167$	−19.6355	−1.51944	−0.759720	−	0.650250i	$-0.774664\pi$
$167$	−19.6355	−1.51944	−0.759720	+	0.650250i	$0.774664\pi$
$168$	0	0
$169$	−6.69710	−0.515162
$170$	0	0
$171$	−5.51056	−0.421403
$172$	0	0
$173$	−8.37309	−0.636594	−0.318297	−	0.947991i	$-0.603111\pi$
$173$	−8.37309	−0.636594	−0.318297	+	0.947991i	$0.603111\pi$
$174$	0	0
$175$	6.13747	0.463949
$176$	0	0
$177$	24.8277	1.86617
$178$	0	0
$179$	10.2413	0.765471	0.382735	−	0.923858i	$-0.374982\pi$
$179$	10.2413	0.765471	0.382735	+	0.923858i	$0.374982\pi$
$180$	0	0
$181$	24.5248	1.82292	0.911458	−	0.411393i	$-0.134958\pi$
$181$	24.5248	1.82292	0.911458	+	0.411393i	$0.134958\pi$
$182$	0	0
$183$	−20.7239	−1.53195
$184$	0	0
$185$	13.4278	0.987235
$186$	0	0
$187$	10.5317	0.770152
$188$	0	0
$189$	−4.34513	−0.316062
$190$	0	0
$191$	−9.28353	−0.671733	−0.335866	−	0.941910i	$-0.609029\pi$
$191$	−9.28353	−0.671733	−0.335866	+	0.941910i	$0.609029\pi$
$192$	0	0
$193$	−8.03364	−0.578274	−0.289137	−	0.957288i	$-0.593368\pi$
$193$	−8.03364	−0.578274	−0.289137	+	0.957288i	$0.593368\pi$
$194$	0	0
$195$	28.6903	2.05455
$196$	0	0
$197$	−5.62691	−0.400901	−0.200451	−	0.979704i	$-0.564241\pi$
$197$	−5.62691	−0.400901	−0.200451	+	0.979704i	$0.564241\pi$
$198$	0	0
$199$	−18.0970	−1.28286	−0.641431	−	0.767181i	$-0.721659\pi$
$199$	−18.0970	−1.28286	−0.641431	+	0.767181i	$0.721659\pi$
$200$	0	0
$201$	−37.6834	−2.65798
$202$	0	0
$203$	3.39989	0.238625
$204$	0	0
$205$	40.1181	2.80197
$206$	0	0
$207$	40.3594	2.80517
$208$	0	0
$209$	−3.51056	−0.242830
$210$	0	0
$211$	0.302899	0.0208524	0.0104262	−	0.999946i	$-0.496681\pi$
$211$	0.302899	0.0208524	0.0104262	+	0.999946i	$0.496681\pi$
$212$	0	0
$213$	−37.5037	−2.56971
$214$	0	0
$215$	18.3999	1.25486
$216$	0	0
$217$	2.61439	0.177476
$218$	0	0
$219$	47.3440	3.19921
$220$	0	0
$221$	7.53167	0.506635
$222$	0	0
$223$	−0.847099	−0.0567259	−0.0283630	−	0.999598i	$-0.509029\pi$
$223$	−0.847099	−0.0567259	−0.0283630	+	0.999598i	$0.509029\pi$
$224$	0	0
$225$	57.0074	3.80050
$226$	0	0
$227$	12.5528	0.833158	0.416579	−	0.909100i	$-0.363229\pi$
$227$	12.5528	0.833158	0.416579	+	0.909100i	$0.363229\pi$
$228$	0	0
$229$	14.7730	0.976226	0.488113	−	0.872781i	$-0.337685\pi$
$229$	14.7730	0.976226	0.488113	+	0.872781i	$0.337685\pi$
$230$	0	0
$231$	−6.07587	−0.399763
$232$	0	0
$233$	7.71822	0.505637	0.252819	−	0.967514i	$-0.418642\pi$
$233$	7.71822	0.505637	0.252819	+	0.967514i	$0.418642\pi$
$234$	0	0
$235$	−8.15859	−0.532207
$236$	0	0
$237$	−12.8557	−0.835067
$238$	0	0
$239$	27.6144	1.78623	0.893113	−	0.449832i	$-0.148516\pi$
$239$	27.6144	1.78623	0.893113	+	0.449832i	$0.148516\pi$
$240$	0	0
$241$	−1.42784	−0.0919755	−0.0459877	−	0.998942i	$-0.514644\pi$
$241$	−1.42784	−0.0919755	−0.0459877	+	0.998942i	$0.514644\pi$
$242$	0	0
$243$	7.86821	0.504746
$244$	0	0
$245$	26.0422	1.66378
$246$	0	0
$247$	−2.51056	−0.159743
$248$	0	0
$249$	−39.2710	−2.48870
$250$	0	0
$251$	20.5317	1.29595	0.647974	−	0.761663i	$-0.275617\pi$
$251$	20.5317	1.29595	0.647974	+	0.761663i	$0.275617\pi$
$252$	0	0
$253$	25.7114	1.61646
$254$	0	0
$255$	34.2835	2.14692
$256$	0	0
$257$	−25.5037	−1.59088	−0.795439	−	0.606034i	$-0.792760\pi$
$257$	−25.5037	−1.59088	−0.795439	+	0.606034i	$0.792760\pi$
$258$	0	0
$259$	−2.03364	−0.126364
$260$	0	0
$261$	31.5796	1.95473
$262$	0	0
$263$	13.7519	0.847976	0.423988	−	0.905668i	$-0.360630\pi$
$263$	13.7519	0.847976	0.423988	+	0.905668i	$0.360630\pi$
$264$	0	0
$265$	−4.24130	−0.260541
$266$	0	0
$267$	−0.703942	−0.0430806
$268$	0	0
$269$	20.8220	1.26954	0.634771	−	0.772700i	$-0.281094\pi$
$269$	20.8220	1.26954	0.634771	+	0.772700i	$0.281094\pi$
$270$	0	0
$271$	14.8277	0.900720	0.450360	−	0.892847i	$-0.351296\pi$
$271$	14.8277	0.900720	0.450360	+	0.892847i	$0.351296\pi$
$272$	0	0
$273$	−4.34513	−0.262979
$274$	0	0
$275$	36.3172	2.19001
$276$	0	0
$277$	−15.9173	−0.956377	−0.478189	−	0.878257i	$-0.658706\pi$
$277$	−15.9173	−0.956377	−0.478189	+	0.878257i	$0.658706\pi$
$278$	0	0
$279$	24.2835	1.45382
$280$	0	0
$281$	4.64803	0.277278	0.138639	−	0.990343i	$-0.455727\pi$
$281$	4.64803	0.277278	0.138639	+	0.990343i	$0.455727\pi$
$282$	0	0
$283$	−3.99316	−0.237369	−0.118684	−	0.992932i	$-0.537868\pi$
$283$	−3.99316	−0.237369	−0.118684	+	0.992932i	$0.537868\pi$
$284$	0	0
$285$	−11.4278	−0.676927
$286$	0	0
$287$	−6.07587	−0.358647
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	−45.1056	−2.64414
$292$	0	0
$293$	3.11636	0.182059	0.0910297	−	0.995848i	$-0.470984\pi$
$293$	3.11636	0.182059	0.0910297	+	0.995848i	$0.470984\pi$
$294$	0	0
$295$	33.3383	1.94103
$296$	0	0
$297$	−25.7114	−1.49193
$298$	0	0
$299$	18.3874	1.06337
$300$	0	0
$301$	−2.78666	−0.160620
$302$	0	0
$303$	46.1940	2.65377
$304$	0	0
$305$	−27.8277	−1.59341
$306$	0	0
$307$	−23.7028	−1.35279	−0.676395	−	0.736539i	$-0.736459\pi$
$307$	−23.7028	−1.35279	−0.676395	+	0.736539i	$0.736459\pi$
$308$	0	0
$309$	−39.2710	−2.23405
$310$	0	0
$311$	1.90301	0.107910	0.0539550	−	0.998543i	$-0.482817\pi$
$311$	1.90301	0.107910	0.0539550	+	0.998543i	$0.482817\pi$
$312$	0	0
$313$	31.6834	1.79085	0.895426	−	0.445210i	$-0.146871\pi$
$313$	31.6834	1.79085	0.895426	+	0.445210i	$0.146871\pi$
$314$	0	0
$315$	−12.8066	−0.721571
$316$	0	0
$317$	12.1797	0.684080	0.342040	−	0.939685i	$-0.388882\pi$
$317$	12.1797	0.684080	0.342040	+	0.939685i	$0.388882\pi$
$318$	0	0
$319$	20.1181	1.12640
$320$	0	0
$321$	0.883644	0.0493202
$322$	0	0
$323$	−3.00000	−0.166924
$324$	0	0
$325$	25.9720	1.44067
$326$	0	0
$327$	−15.4335	−0.853476
$328$	0	0
$329$	1.23562	0.0681217
$330$	0	0
$331$	7.08271	0.389301	0.194651	−	0.980873i	$-0.437643\pi$
$331$	7.08271	0.389301	0.194651	+	0.980873i	$0.437643\pi$
$332$	0	0
$333$	−18.8893	−1.03513
$334$	0	0
$335$	−50.6007	−2.76461
$336$	0	0
$337$	−9.02112	−0.491411	−0.245706	−	0.969344i	$-0.579020\pi$
$337$	−9.02112	−0.491411	−0.245706	+	0.969344i	$0.579020\pi$
$338$	0	0
$339$	−8.10951	−0.440448
$340$	0	0
$341$	15.4701	0.837751
$342$	0	0
$343$	−8.09699	−0.437196
$344$	0	0
$345$	83.6977	4.50613
$346$	0	0
$347$	−28.0354	−1.50502	−0.752509	−	0.658582i	$-0.771157\pi$
$347$	−28.0354	−1.50502	−0.752509	+	0.658582i	$0.771157\pi$
$348$	0	0
$349$	4.39989	0.235521	0.117760	−	0.993042i	$-0.462429\pi$
$349$	4.39989	0.235521	0.117760	+	0.993042i	$0.462429\pi$
$350$	0	0
$351$	−18.3874	−0.981445
$352$	0	0
$353$	−0.950928	−0.0506128	−0.0253064	−	0.999680i	$-0.508056\pi$
$353$	−0.950928	−0.0506128	−0.0253064	+	0.999680i	$0.508056\pi$
$354$	0	0
$355$	−50.3594	−2.67280
$356$	0	0
$357$	−5.19223	−0.274802
$358$	0	0
$359$	12.3856	0.653688	0.326844	−	0.945078i	$-0.394015\pi$
$359$	12.3856	0.653688	0.326844	+	0.945078i	$0.394015\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−3.86253	−0.202730
$364$	0	0
$365$	63.5727	3.32755
$366$	0	0
$367$	−20.4826	−1.06918	−0.534592	−	0.845111i	$-0.679535\pi$
$367$	−20.4826	−1.06918	−0.534592	+	0.845111i	$0.679535\pi$
$368$	0	0
$369$	−56.4353	−2.93790
$370$	0	0
$371$	0.642343	0.0333488
$372$	0	0
$373$	−4.88364	−0.252865	−0.126433	−	0.991975i	$-0.540353\pi$
$373$	−4.88364	−0.252865	−0.126433	+	0.991975i	$0.540353\pi$
$374$	0	0
$375$	61.0833	3.15433
$376$	0	0
$377$	14.3874	0.740987
$378$	0	0
$379$	−0.179702	−0.00923065	−0.00461532	−	0.999989i	$-0.501469\pi$
$379$	−0.179702	−0.00923065	−0.00461532	+	0.999989i	$0.501469\pi$
$380$	0	0
$381$	41.6691	2.13477
$382$	0	0
$383$	7.86821	0.402047	0.201023	−	0.979586i	$-0.435573\pi$
$383$	7.86821	0.402047	0.201023	+	0.979586i	$0.435573\pi$
$384$	0	0
$385$	−8.15859	−0.415800
$386$	0	0
$387$	−25.8836	−1.31574
$388$	0	0
$389$	31.8614	1.61544	0.807718	−	0.589569i	$-0.200702\pi$
$389$	31.8614	1.61544	0.807718	+	0.589569i	$0.200702\pi$
$390$	0	0
$391$	21.9720	1.11117
$392$	0	0
$393$	−47.7450	−2.40842
$394$	0	0
$395$	−17.2624	−0.868566
$396$	0	0
$397$	2.93840	0.147474	0.0737371	−	0.997278i	$-0.476507\pi$
$397$	2.93840	0.147474	0.0737371	+	0.997278i	$0.476507\pi$
$398$	0	0
$399$	1.73074	0.0866455
$400$	0	0
$401$	4.77981	0.238693	0.119346	−	0.992853i	$-0.461920\pi$
$401$	4.77981	0.238693	0.119346	+	0.992853i	$0.461920\pi$
$402$	0	0
$403$	11.0633	0.551104
$404$	0	0
$405$	−18.9384	−0.941057
$406$	0	0
$407$	−12.0336	−0.596485
$408$	0	0
$409$	−17.8009	−0.880199	−0.440100	−	0.897949i	$-0.645057\pi$
$409$	−17.8009	−0.880199	−0.440100	+	0.897949i	$0.645057\pi$
$410$	0	0
$411$	13.6212	0.671886
$412$	0	0
$413$	−5.04907	−0.248449
$414$	0	0
$415$	−52.7325	−2.58854
$416$	0	0
$417$	18.9316	0.927082
$418$	0	0
$419$	16.9230	0.826741	0.413371	−	0.910563i	$-0.364351\pi$
$419$	16.9230	0.826741	0.413371	+	0.910563i	$0.364351\pi$
$420$	0	0
$421$	11.9384	0.581842	0.290921	−	0.956747i	$-0.406038\pi$
$421$	11.9384	0.581842	0.290921	+	0.956747i	$0.406038\pi$
$422$	0	0
$423$	11.4769	0.558027
$424$	0	0
$425$	31.0354	1.50544
$426$	0	0
$427$	4.21450	0.203954
$428$	0	0
$429$	−25.7114	−1.24136
$430$	0	0
$431$	−29.1729	−1.40521	−0.702604	−	0.711581i	$-0.747979\pi$
$431$	−29.1729	−1.40521	−0.702604	+	0.711581i	$0.747979\pi$
$432$	0	0
$433$	−5.91044	−0.284038	−0.142019	−	0.989864i	$-0.545359\pi$
$433$	−5.91044	−0.284038	−0.142019	+	0.989864i	$0.545359\pi$
$434$	0	0
$435$	65.4900	3.14001
$436$	0	0
$437$	−7.32401	−0.350355
$438$	0	0
$439$	−37.7450	−1.80147	−0.900736	−	0.434368i	$-0.856972\pi$
$439$	−37.7450	−1.80147	−0.900736	+	0.434368i	$0.856972\pi$
$440$	0	0
$441$	−36.6343	−1.74449
$442$	0	0
$443$	9.95093	0.472783	0.236391	−	0.971658i	$-0.424035\pi$
$443$	9.95093	0.472783	0.236391	+	0.971658i	$0.424035\pi$
$444$	0	0
$445$	−0.945243	−0.0448088
$446$	0	0
$447$	−19.6355	−0.928727
$448$	0	0
$449$	21.6269	1.02064	0.510319	−	0.859985i	$-0.329527\pi$
$449$	21.6269	1.02064	0.510319	+	0.859985i	$0.329527\pi$
$450$	0	0
$451$	−35.9527	−1.69295
$452$	0	0
$453$	−53.1169	−2.49565
$454$	0	0
$455$	−5.83457	−0.273529
$456$	0	0
$457$	5.49628	0.257105	0.128553	−	0.991703i	$-0.458967\pi$
$457$	5.49628	0.257105	0.128553	+	0.991703i	$0.458967\pi$
$458$	0	0
$459$	−21.9720	−1.02557
$460$	0	0
$461$	5.42100	0.252481	0.126241	−	0.992000i	$-0.459709\pi$
$461$	5.42100	0.252481	0.126241	+	0.992000i	$0.459709\pi$
$462$	0	0
$463$	7.31149	0.339794	0.169897	−	0.985462i	$-0.445657\pi$
$463$	7.31149	0.339794	0.169897	+	0.985462i	$0.445657\pi$
$464$	0	0
$465$	50.3594	2.33536
$466$	0	0
$467$	−6.97204	−0.322628	−0.161314	−	0.986903i	$-0.551573\pi$
$467$	−6.97204	−0.322628	−0.161314	+	0.986903i	$0.551573\pi$
$468$	0	0
$469$	7.66346	0.353866
$470$	0	0
$471$	46.7998	2.15642
$472$	0	0
$473$	−16.4894	−0.758185
$474$	0	0
$475$	−10.3451	−0.474667
$476$	0	0
$477$	5.96636	0.273181
$478$	0	0
$479$	25.2288	1.15273	0.576366	−	0.817192i	$-0.304470\pi$
$479$	25.2288	1.15273	0.576366	+	0.817192i	$0.304470\pi$
$480$	0	0
$481$	−8.60580	−0.392391
$482$	0	0
$483$	−12.6760	−0.576777
$484$	0	0
$485$	−60.5671	−2.75021
$486$	0	0
$487$	−1.06335	−0.0481848	−0.0240924	−	0.999710i	$-0.507670\pi$
$487$	−1.06335	−0.0481848	−0.0240924	+	0.999710i	$0.507670\pi$
$488$	0	0
$489$	70.1381	3.17175
$490$	0	0
$491$	18.6903	0.843480	0.421740	−	0.906717i	$-0.361420\pi$
$491$	18.6903	0.843480	0.421740	+	0.906717i	$0.361420\pi$
$492$	0	0
$493$	17.1922	0.774299
$494$	0	0
$495$	−75.7804	−3.40608
$496$	0	0
$497$	7.62691	0.342114
$498$	0	0
$499$	−15.8836	−0.711050	−0.355525	−	0.934667i	$-0.615698\pi$
$499$	−15.8836	−0.711050	−0.355525	+	0.934667i	$0.615698\pi$
$500$	0	0
$501$	57.2824	2.55919
$502$	0	0
$503$	−21.2430	−0.947181	−0.473590	−	0.880745i	$-0.657042\pi$
$503$	−21.2430	−0.947181	−0.473590	+	0.880745i	$0.657042\pi$
$504$	0	0
$505$	62.0285	2.76023
$506$	0	0
$507$	19.5374	0.867685
$508$	0	0
$509$	20.6230	0.914097	0.457049	−	0.889442i	$-0.348907\pi$
$509$	20.6230	0.914097	0.457049	+	0.889442i	$0.348907\pi$
$510$	0	0
$511$	−9.62807	−0.425921
$512$	0	0
$513$	7.32401	0.323363
$514$	0	0
$515$	−52.7325	−2.32367
$516$	0	0
$517$	7.31149	0.321559
$518$	0	0
$519$	24.4267	1.07221
$520$	0	0
$521$	−5.26242	−0.230551	−0.115275	−	0.993334i	$-0.536775\pi$
$521$	−5.26242	−0.230551	−0.115275	+	0.993334i	$0.536775\pi$
$522$	0	0
$523$	−2.67599	−0.117013	−0.0585063	−	0.998287i	$-0.518634\pi$
$523$	−2.67599	−0.117013	−0.0585063	+	0.998287i	$0.518634\pi$
$524$	0	0
$525$	−17.9048	−0.781428
$526$	0	0
$527$	13.2202	0.575880
$528$	0	0
$529$	30.6412	1.33223
$530$	0	0
$531$	−46.8979	−2.03520
$532$	0	0
$533$	−25.7114	−1.11368
$534$	0	0
$535$	1.18654	0.0512987
$536$	0	0
$537$	−29.8768	−1.28928
$538$	0	0
$539$	−23.3383	−1.00525
$540$	0	0
$541$	−22.5653	−0.970159	−0.485079	−	0.874470i	$-0.661209\pi$
$541$	−22.5653	−0.970159	−0.485079	+	0.874470i	$0.661209\pi$
$542$	0	0
$543$	−71.5459	−3.07033
$544$	0	0
$545$	−20.7239	−0.887714
$546$	0	0
$547$	8.08446	0.345667	0.172833	−	0.984951i	$-0.444708\pi$
$547$	8.08446	0.345667	0.172833	+	0.984951i	$0.444708\pi$
$548$	0	0
$549$	39.1461	1.67071
$550$	0	0
$551$	−5.73074	−0.244138
$552$	0	0
$553$	2.61439	0.111175
$554$	0	0
$555$	−39.1729	−1.66280
$556$	0	0
$557$	1.12888	0.0478323	0.0239162	−	0.999714i	$-0.492387\pi$
$557$	1.12888	0.0478323	0.0239162	+	0.999714i	$0.492387\pi$
$558$	0	0
$559$	−11.7923	−0.498763
$560$	0	0
$561$	−30.7239	−1.29716
$562$	0	0
$563$	35.4701	1.49489	0.747443	−	0.664326i	$-0.231281\pi$
$563$	35.4701	1.49489	0.747443	+	0.664326i	$0.231281\pi$
$564$	0	0
$565$	−10.8893	−0.458118
$566$	0	0
$567$	2.86821	0.120454
$568$	0	0
$569$	24.0000	1.00613	0.503066	−	0.864248i	$-0.332205\pi$
$569$	24.0000	1.00613	0.503066	+	0.864248i	$0.332205\pi$
$570$	0	0
$571$	−12.2499	−0.512642	−0.256321	−	0.966592i	$-0.582510\pi$
$571$	−12.2499	−0.512642	−0.256321	+	0.966592i	$0.582510\pi$
$572$	0	0
$573$	27.0827	1.13140
$574$	0	0
$575$	75.7679	3.15974
$576$	0	0
$577$	−22.8768	−0.952374	−0.476187	−	0.879344i	$-0.657981\pi$
$577$	−22.8768	−0.952374	−0.476187	+	0.879344i	$0.657981\pi$
$578$	0	0
$579$	23.4364	0.973985
$580$	0	0
$581$	7.98632	0.331328
$582$	0	0
$583$	3.80093	0.157418
$584$	0	0
$585$	−54.1940	−2.24065
$586$	0	0
$587$	−29.2219	−1.20612	−0.603059	−	0.797697i	$-0.706052\pi$
$587$	−29.2219	−1.20612	−0.603059	+	0.797697i	$0.706052\pi$
$588$	0	0
$589$	−4.40673	−0.181576
$590$	0	0
$591$	16.4153	0.675236
$592$	0	0
$593$	−1.35197	−0.0555188	−0.0277594	−	0.999615i	$-0.508837\pi$
$593$	−1.35197	−0.0555188	−0.0277594	+	0.999615i	$0.508837\pi$
$594$	0	0
$595$	−6.97204	−0.285826
$596$	0	0
$597$	52.7941	2.16072
$598$	0	0
$599$	41.1814	1.68263	0.841314	−	0.540546i	$-0.181782\pi$
$599$	41.1814	1.68263	0.841314	+	0.540546i	$0.181782\pi$
$600$	0	0
$601$	−5.25383	−0.214308	−0.107154	−	0.994242i	$-0.534174\pi$
$601$	−5.25383	−0.214308	−0.107154	+	0.994242i	$0.534174\pi$
$602$	0	0
$603$	71.1814	2.89873
$604$	0	0
$605$	−5.18654	−0.210863
$606$	0	0
$607$	9.17286	0.372315	0.186157	−	0.982520i	$-0.440397\pi$
$607$	9.17286	0.372315	0.186157	+	0.982520i	$0.440397\pi$
$608$	0	0
$609$	−9.91844	−0.401916
$610$	0	0
$611$	5.22877	0.211534
$612$	0	0
$613$	−28.3577	−1.14535	−0.572677	−	0.819781i	$-0.694095\pi$
$613$	−28.3577	−1.14535	−0.572677	+	0.819781i	$0.694095\pi$
$614$	0	0
$615$	−117.036	−4.71935
$616$	0	0
$617$	19.4546	0.783214	0.391607	−	0.920132i	$-0.371919\pi$
$617$	19.4546	0.783214	0.391607	+	0.920132i	$0.371919\pi$
$618$	0	0
$619$	−25.2538	−1.01504	−0.507519	−	0.861641i	$-0.669437\pi$
$619$	−25.2538	−1.01504	−0.507519	+	0.861641i	$0.669437\pi$
$620$	0	0
$621$	−53.6412	−2.15255
$622$	0	0
$623$	0.143157	0.00573545
$624$	0	0
$625$	30.2961	1.21184
$626$	0	0
$627$	10.2413	0.408998
$628$	0	0
$629$	−10.2835	−0.410031
$630$	0	0
$631$	−23.6287	−0.940642	−0.470321	−	0.882495i	$-0.655862\pi$
$631$	−23.6287	−0.940642	−0.470321	+	0.882495i	$0.655862\pi$
$632$	0	0
$633$	−0.883644	−0.0351217
$634$	0	0
$635$	55.9527	2.22041
$636$	0	0
$637$	−16.6903	−0.661292
$638$	0	0
$639$	70.8420	2.80247
$640$	0	0
$641$	26.9230	1.06339	0.531697	−	0.846935i	$-0.321555\pi$
$641$	26.9230	1.06339	0.531697	+	0.846935i	$0.321555\pi$
$642$	0	0
$643$	−13.4124	−0.528934	−0.264467	−	0.964395i	$-0.585196\pi$
$643$	−13.4124	−0.528934	−0.264467	+	0.964395i	$0.585196\pi$
$644$	0	0
$645$	−53.6777	−2.11356
$646$	0	0
$647$	6.55104	0.257548	0.128774	−	0.991674i	$-0.458896\pi$
$647$	6.55104	0.257548	0.128774	+	0.991674i	$0.458896\pi$
$648$	0	0
$649$	−29.8768	−1.17277
$650$	0	0
$651$	−7.62691	−0.298922
$652$	0	0
$653$	−2.52308	−0.0987359	−0.0493680	−	0.998781i	$-0.515721\pi$
$653$	−2.52308	−0.0987359	−0.0493680	+	0.998781i	$0.515721\pi$
$654$	0	0
$655$	−64.1113	−2.50503
$656$	0	0
$657$	−89.4296	−3.48898
$658$	0	0
$659$	5.17227	0.201483	0.100742	−	0.994913i	$-0.467878\pi$
$659$	5.17227	0.201483	0.100742	+	0.994913i	$0.467878\pi$
$660$	0	0
$661$	41.6412	1.61965	0.809827	−	0.586668i	$-0.199561\pi$
$661$	41.6412	1.61965	0.809827	+	0.586668i	$0.199561\pi$
$662$	0	0
$663$	−21.9720	−0.853323
$664$	0	0
$665$	2.32401	0.0901214
$666$	0	0
$667$	41.9720	1.62516
$668$	0	0
$669$	2.47123	0.0955433
$670$	0	0
$671$	24.9384	0.962736
$672$	0	0
$673$	−16.7325	−0.644990	−0.322495	−	0.946571i	$-0.604522\pi$
$673$	−16.7325	−0.644990	−0.322495	+	0.946571i	$0.604522\pi$
$674$	0	0
$675$	−75.7679	−2.91631
$676$	0	0
$677$	1.45580	0.0559510	0.0279755	−	0.999609i	$-0.491094\pi$
$677$	1.45580	0.0559510	0.0279755	+	0.999609i	$0.491094\pi$
$678$	0	0
$679$	9.17286	0.352022
$680$	0	0
$681$	−36.6201	−1.40328
$682$	0	0
$683$	8.81346	0.337238	0.168619	−	0.985681i	$-0.446069\pi$
$683$	8.81346	0.337238	0.168619	+	0.985681i	$0.446069\pi$
$684$	0	0
$685$	18.2904	0.698839
$686$	0	0
$687$	−43.0970	−1.64425
$688$	0	0
$689$	2.71822	0.103556
$690$	0	0
$691$	12.6834	0.482500	0.241250	−	0.970463i	$-0.422443\pi$
$691$	12.6834	0.482500	0.241250	+	0.970463i	$0.422443\pi$
$692$	0	0
$693$	11.4769	0.435972
$694$	0	0
$695$	25.4210	0.964274
$696$	0	0
$697$	−30.7239	−1.16375
$698$	0	0
$699$	−22.5162	−0.851643
$700$	0	0
$701$	−1.46149	−0.0551996	−0.0275998	−	0.999619i	$-0.508786\pi$
$701$	−1.46149	−0.0551996	−0.0275998	+	0.999619i	$0.508786\pi$
$702$	0	0
$703$	3.42784	0.129284
$704$	0	0
$705$	23.8009	0.896395
$706$	0	0
$707$	−9.39420	−0.353305
$708$	0	0
$709$	−19.7114	−0.740276	−0.370138	−	0.928977i	$-0.620690\pi$
$709$	−19.7114	−0.740276	−0.370138	+	0.928977i	$0.620690\pi$
$710$	0	0
$711$	24.2835	0.910704
$712$	0	0
$713$	32.2749	1.20871
$714$	0	0
$715$	−34.5248	−1.29116
$716$	0	0
$717$	−80.5591	−3.00853
$718$	0	0
$719$	20.5933	0.767999	0.384000	−	0.923333i	$-0.374546\pi$
$719$	20.5933	0.767999	0.384000	+	0.923333i	$0.374546\pi$
$720$	0	0
$721$	7.98632	0.297426
$722$	0	0
$723$	4.16543	0.154914
$724$	0	0
$725$	59.2853	2.20180
$726$	0	0
$727$	4.66056	0.172850	0.0864252	−	0.996258i	$-0.472456\pi$
$727$	4.66056	0.172850	0.0864252	+	0.996258i	$0.472456\pi$
$728$	0	0
$729$	−37.4575	−1.38732
$730$	0	0
$731$	−14.0913	−0.521186
$732$	0	0
$733$	34.4826	1.27364	0.636822	−	0.771011i	$-0.280249\pi$
$733$	34.4826	1.27364	0.636822	+	0.771011i	$0.280249\pi$
$734$	0	0
$735$	−75.9726	−2.80229
$736$	0	0
$737$	45.3469	1.67037
$738$	0	0
$739$	−12.4222	−0.456956	−0.228478	−	0.973549i	$-0.573375\pi$
$739$	−12.4222	−0.456956	−0.228478	+	0.973549i	$0.573375\pi$
$740$	0	0
$741$	7.32401	0.269054
$742$	0	0
$743$	6.77981	0.248727	0.124364	−	0.992237i	$-0.460311\pi$
$743$	6.77981	0.248727	0.124364	+	0.992237i	$0.460311\pi$
$744$	0	0
$745$	−26.3662	−0.965984
$746$	0	0
$747$	74.1803	2.71411
$748$	0	0
$749$	−0.179702	−0.00656615
$750$	0	0
$751$	−40.4826	−1.47723	−0.738616	−	0.674127i	$-0.764520\pi$
$751$	−40.4826	−1.47723	−0.738616	+	0.674127i	$0.764520\pi$
$752$	0	0
$753$	−59.8968	−2.18276
$754$	0	0
$755$	−71.3246	−2.59577
$756$	0	0
$757$	−23.5191	−0.854818	−0.427409	−	0.904058i	$-0.640574\pi$
$757$	−23.5191	−0.854818	−0.427409	+	0.904058i	$0.640574\pi$
$758$	0	0
$759$	−75.0074	−2.72260
$760$	0	0
$761$	−2.32692	−0.0843507	−0.0421753	−	0.999110i	$-0.513429\pi$
$761$	−2.32692	−0.0843507	−0.0421753	+	0.999110i	$0.513429\pi$
$762$	0	0
$763$	3.13863	0.113626
$764$	0	0
$765$	−64.7593	−2.34138
$766$	0	0
$767$	−21.3662	−0.771490
$768$	0	0
$769$	34.8768	1.25769	0.628845	−	0.777531i	$-0.283528\pi$
$769$	34.8768	1.25769	0.628845	+	0.777531i	$0.283528\pi$
$770$	0	0
$771$	74.4016	2.67951
$772$	0	0
$773$	−5.94699	−0.213898	−0.106949	−	0.994264i	$-0.534108\pi$
$773$	−5.94699	−0.213898	−0.106949	+	0.994264i	$0.534108\pi$
$774$	0	0
$775$	45.5882	1.63758
$776$	0	0
$777$	5.93272	0.212835
$778$	0	0
$779$	10.2413	0.366933
$780$	0	0
$781$	45.1306	1.61490
$782$	0	0
$783$	−41.9720	−1.49996
$784$	0	0
$785$	62.8420	2.24293
$786$	0	0
$787$	−0.277845	−0.00990411	−0.00495206	−	0.999988i	$-0.501576\pi$
$787$	−0.277845	−0.00990411	−0.00495206	+	0.999988i	$0.501576\pi$
$788$	0	0
$789$	−40.1181	−1.42824
$790$	0	0
$791$	1.64919	0.0586383
$792$	0	0
$793$	17.8346	0.633324
$794$	0	0
$795$	12.3731	0.438828
$796$	0	0
$797$	5.01543	0.177656	0.0888278	−	0.996047i	$-0.471688\pi$
$797$	5.01543	0.177656	0.0888278	+	0.996047i	$0.471688\pi$
$798$	0	0
$799$	6.24814	0.221043
$800$	0	0
$801$	1.32970	0.0469826
$802$	0	0
$803$	−56.9720	−2.01050
$804$	0	0
$805$	−17.0211	−0.599915
$806$	0	0
$807$	−60.7439	−2.13829
$808$	0	0
$809$	−18.9190	−0.665158	−0.332579	−	0.943075i	$-0.607919\pi$
$809$	−18.9190	−0.665158	−0.332579	+	0.943075i	$0.607919\pi$
$810$	0	0
$811$	45.3103	1.59106	0.795530	−	0.605914i	$-0.207192\pi$
$811$	45.3103	1.59106	0.795530	+	0.605914i	$0.207192\pi$
$812$	0	0
$813$	−43.2567	−1.51708
$814$	0	0
$815$	94.1803	3.29899
$816$	0	0
$817$	4.69710	0.164331
$818$	0	0
$819$	8.20766	0.286799
$820$	0	0
$821$	12.6498	0.441480	0.220740	−	0.975333i	$-0.429153\pi$
$821$	12.6498	0.441480	0.220740	+	0.975333i	$0.429153\pi$
$822$	0	0
$823$	43.9065	1.53048	0.765242	−	0.643742i	$-0.222619\pi$
$823$	43.9065	1.53048	0.765242	+	0.643742i	$0.222619\pi$
$824$	0	0
$825$	−105.948	−3.68862
$826$	0	0
$827$	−13.0405	−0.453462	−0.226731	−	0.973957i	$-0.572804\pi$
$827$	−13.0405	−0.453462	−0.226731	+	0.973957i	$0.572804\pi$
$828$	0	0
$829$	−49.4507	−1.71749	−0.858747	−	0.512400i	$-0.828757\pi$
$829$	−49.4507	−1.71749	−0.858747	+	0.512400i	$0.828757\pi$
$830$	0	0
$831$	46.4353	1.61082
$832$	0	0
$833$	−19.9441	−0.691022
$834$	0	0
$835$	76.9179	2.66185
$836$	0	0
$837$	−32.2749	−1.11559
$838$	0	0
$839$	−15.2288	−0.525756	−0.262878	−	0.964829i	$-0.584672\pi$
$839$	−15.2288	−0.525756	−0.262878	+	0.964829i	$0.584672\pi$
$840$	0	0
$841$	3.84141	0.132463
$842$	0	0
$843$	−13.5596	−0.467018
$844$	0	0
$845$	26.2345	0.902493
$846$	0	0
$847$	0.785500	0.0269901
$848$	0	0
$849$	11.6492	0.399799
$850$	0	0
$851$	−25.1056	−0.860608
$852$	0	0
$853$	15.4193	0.527945	0.263973	−	0.964530i	$-0.414967\pi$
$853$	15.4193	0.527945	0.263973	+	0.964530i	$0.414967\pi$
$854$	0	0
$855$	21.5864	0.738240
$856$	0	0
$857$	−14.3422	−0.489921	−0.244961	−	0.969533i	$-0.578775\pi$
$857$	−14.3422	−0.489921	−0.244961	+	0.969533i	$0.578775\pi$
$858$	0	0
$859$	46.2681	1.57865	0.789324	−	0.613977i	$-0.210431\pi$
$859$	46.2681	1.57865	0.789324	+	0.613977i	$0.210431\pi$
$860$	0	0
$861$	17.7251	0.604068
$862$	0	0
$863$	25.9527	0.883439	0.441720	−	0.897153i	$-0.354369\pi$
$863$	25.9527	0.883439	0.441720	+	0.897153i	$0.354369\pi$
$864$	0	0
$865$	32.7998	1.11523
$866$	0	0
$867$	23.3383	0.792610
$868$	0	0
$869$	15.4701	0.524786
$870$	0	0
$871$	32.4296	1.09883
$872$	0	0
$873$	85.2014	2.88363
$874$	0	0
$875$	−12.4222	−0.419946
$876$	0	0
$877$	−21.3526	−0.721025	−0.360512	−	0.932754i	$-0.617398\pi$
$877$	−21.3526	−0.721025	−0.360512	+	0.932754i	$0.617398\pi$
$878$	0	0
$879$	−9.09130	−0.306642
$880$	0	0
$881$	53.2642	1.79452	0.897258	−	0.441507i	$-0.145556\pi$
$881$	53.2642	1.79452	0.897258	+	0.441507i	$0.145556\pi$
$882$	0	0
$883$	−38.1586	−1.28414	−0.642069	−	0.766647i	$-0.721924\pi$
$883$	−38.1586	−1.28414	−0.642069	+	0.766647i	$0.721924\pi$
$884$	0	0
$885$	−97.2573	−3.26927
$886$	0	0
$887$	−11.1615	−0.374766	−0.187383	−	0.982287i	$-0.560001\pi$
$887$	−11.1615	−0.374766	−0.187383	+	0.982287i	$0.560001\pi$
$888$	0	0
$889$	−8.47401	−0.284209
$890$	0	0
$891$	16.9720	0.568585
$892$	0	0
$893$	−2.08271	−0.0696954
$894$	0	0
$895$	−40.1181	−1.34100
$896$	0	0
$897$	−53.6412	−1.79103
$898$	0	0
$899$	25.2538	0.842262
$900$	0	0
$901$	3.24814	0.108211
$902$	0	0
$903$	8.12947	0.270532
$904$	0	0
$905$	−96.0708	−3.19350
$906$	0	0
$907$	32.7941	1.08891	0.544455	−	0.838790i	$-0.316737\pi$
$907$	32.7941	1.08891	0.544455	+	0.838790i	$0.316737\pi$
$908$	0	0
$909$	−87.2573	−2.89414
$910$	0	0
$911$	46.8979	1.55380	0.776899	−	0.629626i	$-0.216792\pi$
$911$	46.8979	1.55380	0.776899	+	0.629626i	$0.216792\pi$
$912$	0	0
$913$	47.2573	1.56399
$914$	0	0
$915$	81.1814	2.68378
$916$	0	0
$917$	9.70963	0.320640
$918$	0	0
$919$	−45.1871	−1.49059	−0.745293	−	0.666737i	$-0.767690\pi$
$919$	−45.1871	−1.49059	−0.745293	+	0.666737i	$0.767690\pi$
$920$	0	0
$921$	69.1478	2.27850
$922$	0	0
$923$	32.2749	1.06234
$924$	0	0
$925$	−35.4615	−1.16597
$926$	0	0
$927$	74.1803	2.43640
$928$	0	0
$929$	−51.0776	−1.67580	−0.837901	−	0.545822i	$-0.816217\pi$
$929$	−51.0776	−1.67580	−0.837901	+	0.545822i	$0.816217\pi$
$930$	0	0
$931$	6.64803	0.217880
$932$	0	0
$933$	−5.55163	−0.181752
$934$	0	0
$935$	−41.2556	−1.34920
$936$	0	0
$937$	24.5037	0.800502	0.400251	−	0.916406i	$-0.368923\pi$
$937$	24.5037	0.800502	0.400251	+	0.916406i	$0.368923\pi$
$938$	0	0
$939$	−92.4296	−3.01633
$940$	0	0
$941$	−55.1871	−1.79905	−0.899525	−	0.436870i	$-0.856087\pi$
$941$	−55.1871	−1.79905	−0.899525	+	0.436870i	$0.856087\pi$
$942$	0	0
$943$	−75.0074	−2.44258
$944$	0	0
$945$	17.0211	0.553697
$946$	0	0
$947$	29.9019	0.971680	0.485840	−	0.874048i	$-0.338514\pi$
$947$	29.9019	0.971680	0.485840	+	0.874048i	$0.338514\pi$
$948$	0	0
$949$	−40.7433	−1.32258
$950$	0	0
$951$	−35.5317	−1.15219
$952$	0	0
$953$	26.0086	0.842501	0.421250	−	0.906944i	$-0.361591\pi$
$953$	26.0086	0.842501	0.421250	+	0.906944i	$0.361591\pi$
$954$	0	0
$955$	36.3662	1.17678
$956$	0	0
$957$	−58.6903	−1.89719
$958$	0	0
$959$	−2.77007	−0.0894502
$960$	0	0
$961$	−11.5807	−0.373572
$962$	0	0
$963$	−1.66914	−0.0537874
$964$	0	0
$965$	31.4701	1.01306
$966$	0	0
$967$	50.8420	1.63497	0.817484	−	0.575951i	$-0.195368\pi$
$967$	50.8420	1.63497	0.817484	+	0.575951i	$0.195368\pi$
$968$	0	0
$969$	8.75186	0.281150
$970$	0	0
$971$	22.1267	0.710079	0.355040	−	0.934851i	$-0.384467\pi$
$971$	22.1267	0.710079	0.355040	+	0.934851i	$0.384467\pi$
$972$	0	0
$973$	−3.85000	−0.123425
$974$	0	0
$975$	−75.7679	−2.42651
$976$	0	0
$977$	43.1814	1.38150	0.690748	−	0.723095i	$-0.257281\pi$
$977$	43.1814	1.38150	0.690748	+	0.723095i	$0.257281\pi$
$978$	0	0
$979$	0.847099	0.0270734
$980$	0	0
$981$	29.1529	0.930780
$982$	0	0
$983$	34.8842	1.11263	0.556317	−	0.830970i	$-0.312214\pi$
$983$	34.8842	1.11263	0.556317	+	0.830970i	$0.312214\pi$
$984$	0	0
$985$	22.0422	0.702324
$986$	0	0
$987$	−3.60464	−0.114737
$988$	0	0
$989$	−34.4016	−1.09391
$990$	0	0
$991$	7.36056	0.233816	0.116908	−	0.993143i	$-0.462702\pi$
$991$	7.36056	0.233816	0.116908	+	0.993143i	$0.462702\pi$
$992$	0	0
$993$	−20.6623	−0.655698
$994$	0	0
$995$	70.8911	2.24740
$996$	0	0
$997$	−37.2978	−1.18123	−0.590617	−	0.806952i	$-0.701115\pi$
$997$	−37.2978	−1.18123	−0.590617	+	0.806952i	$0.701115\pi$
$998$	0	0
$999$	25.1056	0.794305

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4864.2.a.be.1.1		3
4.3	odd	2		4864.2.a.bc.1.3		3
8.3	odd	2		4864.2.a.bf.1.1		3
8.5	even	2		4864.2.a.bd.1.3		3
16.3	odd	4		2432.2.c.g.1217.6	yes	6
16.5	even	4		2432.2.c.f.1217.6	yes	6
16.11	odd	4		2432.2.c.g.1217.1	yes	6
16.13	even	4		2432.2.c.f.1217.1	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2432.2.c.f.1217.1	✓	6	16.13	even	4
2432.2.c.f.1217.6	yes	6	16.5	even	4
2432.2.c.g.1217.1	yes	6	16.11	odd	4
2432.2.c.g.1217.6	yes	6	16.3	odd	4
4864.2.a.bc.1.3		3	4.3	odd	2
4864.2.a.bd.1.3		3	8.5	even	2
4864.2.a.be.1.1		3	1.1	even	1	trivial
4864.2.a.bf.1.1		3	8.3	odd	2

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:	\( N \)	\(=\)	\( 4864 = 2^{8} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4864.a (trivial)

\(f(q)\)	\(=\)	\(q-2.91729 q^{3} -3.91729 q^{5} +0.593272 q^{7} +5.51056 q^{9} +O(q^{10})\)	\(q-2.91729 q^{3} -3.91729 q^{5} +0.593272 q^{7} +5.51056 q^{9} +3.51056 q^{11} +2.51056 q^{13} +11.4278 q^{15} +3.00000 q^{17} -1.00000 q^{19} -1.73074 q^{21} +7.32401 q^{23} +10.3451 q^{25} -7.32401 q^{27} +5.73074 q^{29} +4.40673 q^{31} -10.2413 q^{33} -2.32401 q^{35} -3.42784 q^{37} -7.32401 q^{39} -10.2413 q^{41} -4.69710 q^{43} -21.5864 q^{45} +2.08271 q^{47} -6.64803 q^{49} -8.75186 q^{51} +1.08271 q^{53} -13.7519 q^{55} +2.91729 q^{57} -8.51056 q^{59} +7.10383 q^{61} +3.26926 q^{63} -9.83457 q^{65} +12.9173 q^{67} -21.3662 q^{69} +12.8557 q^{71} -16.2288 q^{73} -30.1797 q^{75} +2.08271 q^{77} +4.40673 q^{79} +4.83457 q^{81} +13.4615 q^{83} -11.7519 q^{85} -16.7182 q^{87} +0.241300 q^{89} +1.48944 q^{91} -12.8557 q^{93} +3.91729 q^{95} +15.4615 q^{97} +19.3451 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{3} - 2 q^{5} + 3 q^{7} + 8 q^{9}+O(q^{10}) \) 3 * q + q^3 - 2 * q^5 + 3 * q^7 + 8 * q^9	\( 3 q + q^{3} - 2 q^{5} + 3 q^{7} + 8 q^{9} + 2 q^{11} - q^{13} + 16 q^{15} + 9 q^{17} - 3 q^{19} + 7 q^{21} + 11 q^{23} + 3 q^{25} - 11 q^{27} + 5 q^{29} + 12 q^{31} - 10 q^{33} + 4 q^{35} + 8 q^{37} - 11 q^{39} - 10 q^{41} - 8 q^{43} - 16 q^{45} + 16 q^{47} + 2 q^{49} + 3 q^{51} + 13 q^{53} - 12 q^{55} - q^{57} - 17 q^{59} + 14 q^{61} + 22 q^{63} - 10 q^{65} + 29 q^{67} - 19 q^{69} + 2 q^{71} - 17 q^{73} - 43 q^{75} + 16 q^{77} + 12 q^{79} - 5 q^{81} + 16 q^{83} - 6 q^{85} - 27 q^{87} - 20 q^{89} + 13 q^{91} - 2 q^{93} + 2 q^{95} + 22 q^{97} + 30 q^{99}+O(q^{100}) \) 3 * q + q^3 - 2 * q^5 + 3 * q^7 + 8 * q^9 + 2 * q^11 - q^13 + 16 * q^15 + 9 * q^17 - 3 * q^19 + 7 * q^21 + 11 * q^23 + 3 * q^25 - 11 * q^27 + 5 * q^29 + 12 * q^31 - 10 * q^33 + 4 * q^35 + 8 * q^37 - 11 * q^39 - 10 * q^41 - 8 * q^43 - 16 * q^45 + 16 * q^47 + 2 * q^49 + 3 * q^51 + 13 * q^53 - 12 * q^55 - q^57 - 17 * q^59 + 14 * q^61 + 22 * q^63 - 10 * q^65 + 29 * q^67 - 19 * q^69 + 2 * q^71 - 17 * q^73 - 43 * q^75 + 16 * q^77 + 12 * q^79 - 5 * q^81 + 16 * q^83 - 6 * q^85 - 27 * q^87 - 20 * q^89 + 13 * q^91 - 2 * q^93 + 2 * q^95 + 22 * q^97 + 30 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more