LMFDB - Embedded newform 6384.2.a.cb.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6384,2,Mod(1,6384)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6384, 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("6384.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6384 = 2^{4} \cdot 3 \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6384.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:	$50.9764966504$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.9248.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 2 $ x^4 - 5*x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 3192)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.13578$ of defining polynomial
Character	$\chi$	$=$	6384.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +3.33513 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +3.33513 q^{5} +1.00000 q^{7} +1.00000 q^{9} +4.27156 q^{11} +3.33513 q^{15} -4.18668 q^{17} +1.00000 q^{19} +1.00000 q^{21} -6.99596 q^{23} +6.12311 q^{25} +1.00000 q^{27} +1.78797 q^{29} +6.27156 q^{31} +4.27156 q^{33} +3.33513 q^{35} -5.52181 q^{37} -3.87285 q^{41} +5.25025 q^{43} +3.33513 q^{45} +11.4795 q^{47} +1.00000 q^{49} -4.18668 q^{51} -4.45824 q^{53} +14.2462 q^{55} +1.00000 q^{57} +4.79741 q^{59} +10.5431 q^{61} +1.00000 q^{63} -1.97465 q^{67} -6.99596 q^{69} +12.3311 q^{71} -3.87285 q^{73} +6.12311 q^{75} +4.27156 q^{77} +4.32569 q^{79} +1.00000 q^{81} -4.33109 q^{83} -13.9631 q^{85} +1.78797 q^{87} +14.9165 q^{89} +6.27156 q^{93} +3.33513 q^{95} +0.398706 q^{97} +4.27156 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 4 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 + 4 * q^7 + 4 * q^9	$ 4 q + 4 q^{3} + 4 q^{7} + 4 q^{9} - 4 q^{17} + 4 q^{19} + 4 q^{21} - 4 q^{23} + 8 q^{25} + 4 q^{27} + 4 q^{29} + 8 q^{31} + 4 q^{37} - 8 q^{41} + 12 q^{43} + 8 q^{47} + 4 q^{49} - 4 q^{51} + 12 q^{53} + 24 q^{55} + 4 q^{57} + 8 q^{61} + 4 q^{63} + 8 q^{67} - 4 q^{69} + 12 q^{71} - 8 q^{73} + 8 q^{75} + 20 q^{79} + 4 q^{81} + 20 q^{83} - 4 q^{85} + 4 q^{87} + 8 q^{93} - 8 q^{97}+O(q^{100}) $ 4 * q + 4 * q^3 + 4 * q^7 + 4 * q^9 - 4 * q^17 + 4 * q^19 + 4 * q^21 - 4 * q^23 + 8 * q^25 + 4 * q^27 + 4 * q^29 + 8 * q^31 + 4 * q^37 - 8 * q^41 + 12 * q^43 + 8 * q^47 + 4 * q^49 - 4 * q^51 + 12 * q^53 + 24 * q^55 + 4 * q^57 + 8 * q^61 + 4 * q^63 + 8 * q^67 - 4 * q^69 + 12 * q^71 - 8 * q^73 + 8 * q^75 + 20 * q^79 + 4 * q^81 + 20 * q^83 - 4 * q^85 + 4 * q^87 + 8 * q^93 - 8 * q^97

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
$5$	3.33513	1.49152	0.745758	−	0.666217i	$-0.232087\pi$
$5$	3.33513	1.49152	0.745758	+	0.666217i	$0.232087\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.27156	1.28792	0.643962	−	0.765058i	$-0.277290\pi$
$11$	4.27156	1.28792	0.643962	+	0.765058i	$0.277290\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	3.33513	0.861127
$16$	0	0
$17$	−4.18668	−1.01542	−0.507709	−	0.861528i	$-0.669508\pi$
$17$	−4.18668	−1.01542	−0.507709	+	0.861528i	$0.669508\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	−6.99596	−1.45876	−0.729379	−	0.684110i	$-0.760191\pi$
$23$	−6.99596	−1.45876	−0.729379	+	0.684110i	$0.760191\pi$
$24$	0	0
$25$	6.12311	1.22462
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	1.78797	0.332018	0.166009	−	0.986124i	$-0.446912\pi$
$29$	1.78797	0.332018	0.166009	+	0.986124i	$0.446912\pi$
$30$	0	0
$31$	6.27156	1.12641	0.563203	−	0.826319i	$-0.309569\pi$
$31$	6.27156	1.12641	0.563203	+	0.826319i	$0.309569\pi$
$32$	0	0
$33$	4.27156	0.743583
$34$	0	0
$35$	3.33513	0.563740
$36$	0	0
$37$	−5.52181	−0.907780	−0.453890	−	0.891058i	$-0.649964\pi$
$37$	−5.52181	−0.907780	−0.453890	+	0.891058i	$0.649964\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−3.87285	−0.604838	−0.302419	−	0.953175i	$-0.597794\pi$
$41$	−3.87285	−0.604838	−0.302419	+	0.953175i	$0.597794\pi$
$42$	0	0
$43$	5.25025	0.800656	0.400328	−	0.916372i	$-0.368896\pi$
$43$	5.25025	0.800656	0.400328	+	0.916372i	$0.368896\pi$
$44$	0	0
$45$	3.33513	0.497172
$46$	0	0
$47$	11.4795	1.67446	0.837232	−	0.546848i	$-0.184173\pi$
$47$	11.4795	1.67446	0.837232	+	0.546848i	$0.184173\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−4.18668	−0.586252
$52$	0	0
$53$	−4.45824	−0.612386	−0.306193	−	0.951969i	$-0.599055\pi$
$53$	−4.45824	−0.612386	−0.306193	+	0.951969i	$0.599055\pi$
$54$	0	0
$55$	14.2462	1.92096
$56$	0	0
$57$	1.00000	0.132453
$58$	0	0
$59$	4.79741	0.624570	0.312285	−	0.949989i	$-0.398906\pi$
$59$	4.79741	0.624570	0.312285	+	0.949989i	$0.398906\pi$
$60$	0	0
$61$	10.5431	1.34991	0.674954	−	0.737860i	$-0.264164\pi$
$61$	10.5431	1.34991	0.674954	+	0.737860i	$0.264164\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−1.97465	−0.241242	−0.120621	−	0.992699i	$-0.538489\pi$
$67$	−1.97465	−0.241242	−0.120621	+	0.992699i	$0.538489\pi$
$68$	0	0
$69$	−6.99596	−0.842214
$70$	0	0
$71$	12.3311	1.46343	0.731716	−	0.681610i	$-0.238720\pi$
$71$	12.3311	1.46343	0.731716	+	0.681610i	$0.238720\pi$
$72$	0	0
$73$	−3.87285	−0.453283	−0.226642	−	0.973978i	$-0.572775\pi$
$73$	−3.87285	−0.453283	−0.226642	+	0.973978i	$0.572775\pi$
$74$	0	0
$75$	6.12311	0.707035
$76$	0	0
$77$	4.27156	0.486789
$78$	0	0
$79$	4.32569	0.486679	0.243339	−	0.969941i	$-0.421757\pi$
$79$	4.32569	0.486679	0.243339	+	0.969941i	$0.421757\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−4.33109	−0.475399	−0.237699	−	0.971339i	$-0.576393\pi$
$83$	−4.33109	−0.475399	−0.237699	+	0.971339i	$0.576393\pi$
$84$	0	0
$85$	−13.9631	−1.51451
$86$	0	0
$87$	1.78797	0.191691
$88$	0	0
$89$	14.9165	1.58114	0.790572	−	0.612370i	$-0.209784\pi$
$89$	14.9165	1.58114	0.790572	+	0.612370i	$0.209784\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	6.27156	0.650330
$94$	0	0
$95$	3.33513	0.342177
$96$	0	0
$97$	0.398706	0.0404824	0.0202412	−	0.999795i	$-0.493557\pi$
$97$	0.398706	0.0404824	0.0202412	+	0.999795i	$0.493557\pi$
$98$	0	0
$99$	4.27156	0.429308
$100$	0	0
$101$	0.410574	0.0408536	0.0204268	−	0.999791i	$-0.493497\pi$
$101$	0.410574	0.0408536	0.0204268	+	0.999791i	$0.493497\pi$
$102$	0	0
$103$	−8.06493	−0.794661	−0.397330	−	0.917676i	$-0.630063\pi$
$103$	−8.06493	−0.794661	−0.397330	+	0.917676i	$0.630063\pi$
$104$	0	0
$105$	3.33513	0.325476
$106$	0	0
$107$	1.91512	0.185142	0.0925709	−	0.995706i	$-0.470492\pi$
$107$	1.91512	0.185142	0.0925709	+	0.995706i	$0.470492\pi$
$108$	0	0
$109$	−0.894158	−0.0856448	−0.0428224	−	0.999083i	$-0.513635\pi$
$109$	−0.894158	−0.0856448	−0.0428224	+	0.999083i	$0.513635\pi$
$110$	0	0
$111$	−5.52181	−0.524107
$112$	0	0
$113$	6.33109	0.595579	0.297789	−	0.954632i	$-0.403751\pi$
$113$	6.33109	0.595579	0.297789	+	0.954632i	$0.403751\pi$
$114$	0	0
$115$	−23.3324	−2.17576
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−4.18668	−0.383792
$120$	0	0
$121$	7.24621	0.658746
$122$	0	0
$123$	−3.87285	−0.349203
$124$	0	0
$125$	3.74571	0.335026
$126$	0	0
$127$	11.2503	0.998299	0.499149	−	0.866516i	$-0.333646\pi$
$127$	11.2503	0.998299	0.499149	+	0.866516i	$0.333646\pi$
$128$	0	0
$129$	5.25025	0.462259
$130$	0	0
$131$	3.78797	0.330957	0.165478	−	0.986213i	$-0.447083\pi$
$131$	3.78797	0.330957	0.165478	+	0.986213i	$0.447083\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	3.33513	0.287042
$136$	0	0
$137$	−11.9919	−1.02454	−0.512269	−	0.858825i	$-0.671195\pi$
$137$	−11.9919	−1.02454	−0.512269	+	0.858825i	$0.671195\pi$
$138$	0	0
$139$	−3.46768	−0.294124	−0.147062	−	0.989127i	$-0.546982\pi$
$139$	−3.46768	−0.294124	−0.147062	+	0.989127i	$0.546982\pi$
$140$	0	0
$141$	11.4795	0.966752
$142$	0	0
$143$	0	0
$144$	0	0
$145$	5.96313	0.495211
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	1.40275	0.114918	0.0574588	−	0.998348i	$-0.481700\pi$
$149$	1.40275	0.114918	0.0574588	+	0.998348i	$0.481700\pi$
$150$	0	0
$151$	1.00404	0.0817077	0.0408539	−	0.999165i	$-0.486992\pi$
$151$	1.00404	0.0817077	0.0408539	+	0.999165i	$0.486992\pi$
$152$	0	0
$153$	−4.18668	−0.338473
$154$	0	0
$155$	20.9165	1.68005
$156$	0	0
$157$	−12.2462	−0.977354	−0.488677	−	0.872465i	$-0.662520\pi$
$157$	−12.2462	−0.977354	−0.488677	+	0.872465i	$0.662520\pi$
$158$	0	0
$159$	−4.45824	−0.353561
$160$	0	0
$161$	−6.99596	−0.551359
$162$	0	0
$163$	−10.5908	−0.829534	−0.414767	−	0.909928i	$-0.636137\pi$
$163$	−10.5908	−0.829534	−0.414767	+	0.909928i	$0.636137\pi$
$164$	0	0
$165$	14.2462	1.10907
$166$	0	0
$167$	−4.11906	−0.318743	−0.159371	−	0.987219i	$-0.550947\pi$
$167$	−4.11906	−0.318743	−0.159371	+	0.987219i	$0.550947\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	1.00000	0.0764719
$172$	0	0
$173$	−8.67026	−0.659188	−0.329594	−	0.944123i	$-0.606912\pi$
$173$	−8.67026	−0.659188	−0.329594	+	0.944123i	$0.606912\pi$
$174$	0	0
$175$	6.12311	0.462863
$176$	0	0
$177$	4.79741	0.360596
$178$	0	0
$179$	14.8316	1.10857	0.554283	−	0.832328i	$-0.312993\pi$
$179$	14.8316	1.10857	0.554283	+	0.832328i	$0.312993\pi$
$180$	0	0
$181$	−16.7179	−1.24263	−0.621317	−	0.783559i	$-0.713402\pi$
$181$	−16.7179	−1.24263	−0.621317	+	0.783559i	$0.713402\pi$
$182$	0	0
$183$	10.5431	0.779370
$184$	0	0
$185$	−18.4160	−1.35397
$186$	0	0
$187$	−17.8836	−1.30778
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	−10.3446	−0.748507	−0.374253	−	0.927326i	$-0.622101\pi$
$191$	−10.3446	−0.748507	−0.374253	+	0.927326i	$0.622101\pi$
$192$	0	0
$193$	4.04261	0.290994	0.145497	−	0.989359i	$-0.453522\pi$
$193$	4.04261	0.290994	0.145497	+	0.989359i	$0.453522\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−13.2675	−0.945271	−0.472636	−	0.881258i	$-0.656697\pi$
$197$	−13.2675	−0.945271	−0.472636	+	0.881258i	$0.656697\pi$
$198$	0	0
$199$	17.1708	1.21720	0.608602	−	0.793476i	$-0.291731\pi$
$199$	17.1708	1.21720	0.608602	+	0.793476i	$0.291731\pi$
$200$	0	0
$201$	−1.97465	−0.139281
$202$	0	0
$203$	1.78797	0.125491
$204$	0	0
$205$	−12.9165	−0.902126
$206$	0	0
$207$	−6.99596	−0.486253
$208$	0	0
$209$	4.27156	0.295470
$210$	0	0
$211$	17.2387	1.18676	0.593381	−	0.804921i	$-0.297793\pi$
$211$	17.2387	1.18676	0.593381	+	0.804921i	$0.297793\pi$
$212$	0	0
$213$	12.3311	0.844912
$214$	0	0
$215$	17.5103	1.19419
$216$	0	0
$217$	6.27156	0.425741
$218$	0	0
$219$	−3.87285	−0.261703
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−0.517770	−0.0346724	−0.0173362	−	0.999850i	$-0.505519\pi$
$223$	−0.517770	−0.0346724	−0.0173362	+	0.999850i	$0.505519\pi$
$224$	0	0
$225$	6.12311	0.408207
$226$	0	0
$227$	18.9591	1.25836	0.629179	−	0.777260i	$-0.283391\pi$
$227$	18.9591	1.25836	0.629179	+	0.777260i	$0.283391\pi$
$228$	0	0
$229$	−29.5022	−1.94956	−0.974780	−	0.223167i	$-0.928360\pi$
$229$	−29.5022	−1.94956	−0.974780	+	0.223167i	$0.928360\pi$
$230$	0	0
$231$	4.27156	0.281048
$232$	0	0
$233$	13.8648	0.908311	0.454156	−	0.890922i	$-0.349941\pi$
$233$	13.8648	0.908311	0.454156	+	0.890922i	$0.349941\pi$
$234$	0	0
$235$	38.2858	2.49749
$236$	0	0
$237$	4.32569	0.280984
$238$	0	0
$239$	−29.6155	−1.91567	−0.957835	−	0.287320i	$-0.907236\pi$
$239$	−29.6155	−1.91567	−0.957835	+	0.287320i	$0.907236\pi$
$240$	0	0
$241$	15.1880	0.978347	0.489174	−	0.872186i	$-0.337298\pi$
$241$	15.1880	0.978347	0.489174	+	0.872186i	$0.337298\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	3.33513	0.213074
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−4.33109	−0.274472
$250$	0	0
$251$	−26.7233	−1.68676	−0.843381	−	0.537317i	$-0.819438\pi$
$251$	−26.7233	−1.68676	−0.843381	+	0.537317i	$0.819438\pi$
$252$	0	0
$253$	−29.8836	−1.87877
$254$	0	0
$255$	−13.9631	−0.874405
$256$	0	0
$257$	−26.2888	−1.63985	−0.819926	−	0.572470i	$-0.805985\pi$
$257$	−26.2888	−1.63985	−0.819926	+	0.572470i	$0.805985\pi$
$258$	0	0
$259$	−5.52181	−0.343109
$260$	0	0
$261$	1.78797	0.110673
$262$	0	0
$263$	−21.0148	−1.29583	−0.647915	−	0.761713i	$-0.724359\pi$
$263$	−21.0148	−1.29583	−0.647915	+	0.761713i	$0.724359\pi$
$264$	0	0
$265$	−14.8688	−0.913384
$266$	0	0
$267$	14.9165	0.912873
$268$	0	0
$269$	13.9493	0.850504	0.425252	−	0.905075i	$-0.360185\pi$
$269$	13.9493	0.850504	0.425252	+	0.905075i	$0.360185\pi$
$270$	0	0
$271$	25.0862	1.52388	0.761940	−	0.647648i	$-0.224247\pi$
$271$	25.0862	1.52388	0.761940	+	0.647648i	$0.224247\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	26.1552	1.57722
$276$	0	0
$277$	6.62260	0.397913	0.198957	−	0.980008i	$-0.436245\pi$
$277$	6.62260	0.397913	0.198957	+	0.980008i	$0.436245\pi$
$278$	0	0
$279$	6.27156	0.375468
$280$	0	0
$281$	4.25464	0.253810	0.126905	−	0.991915i	$-0.459496\pi$
$281$	4.25464	0.253810	0.126905	+	0.991915i	$0.459496\pi$
$282$	0	0
$283$	−24.8658	−1.47812	−0.739059	−	0.673641i	$-0.764729\pi$
$283$	−24.8658	−1.47812	−0.739059	+	0.673641i	$0.764729\pi$
$284$	0	0
$285$	3.33513	0.197556
$286$	0	0
$287$	−3.87285	−0.228607
$288$	0	0
$289$	0.528283	0.0310755
$290$	0	0
$291$	0.398706	0.0233725
$292$	0	0
$293$	2.62765	0.153509	0.0767546	−	0.997050i	$-0.475544\pi$
$293$	2.62765	0.153509	0.0767546	+	0.997050i	$0.475544\pi$
$294$	0	0
$295$	16.0000	0.931556
$296$	0	0
$297$	4.27156	0.247861
$298$	0	0
$299$	0	0
$300$	0	0
$301$	5.25025	0.302620
$302$	0	0
$303$	0.410574	0.0235868
$304$	0	0
$305$	35.1627	2.01341
$306$	0	0
$307$	11.5472	0.659031	0.329516	−	0.944150i	$-0.393115\pi$
$307$	11.5472	0.659031	0.329516	+	0.944150i	$0.393115\pi$
$308$	0	0
$309$	−8.06493	−0.458798
$310$	0	0
$311$	25.4477	1.44301	0.721504	−	0.692410i	$-0.243451\pi$
$311$	25.4477	1.44301	0.721504	+	0.692410i	$0.243451\pi$
$312$	0	0
$313$	4.04261	0.228502	0.114251	−	0.993452i	$-0.463553\pi$
$313$	4.04261	0.228502	0.114251	+	0.993452i	$0.463553\pi$
$314$	0	0
$315$	3.33513	0.187913
$316$	0	0
$317$	20.3737	1.14430	0.572151	−	0.820149i	$-0.306109\pi$
$317$	20.3737	1.14430	0.572151	+	0.820149i	$0.306109\pi$
$318$	0	0
$319$	7.63743	0.427614
$320$	0	0
$321$	1.91512	0.106892
$322$	0	0
$323$	−4.18668	−0.232953
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−0.894158	−0.0494471
$328$	0	0
$329$	11.4795	0.632888
$330$	0	0
$331$	−27.2069	−1.49543	−0.747713	−	0.664021i	$-0.768848\pi$
$331$	−27.2069	−1.49543	−0.747713	+	0.664021i	$0.768848\pi$
$332$	0	0
$333$	−5.52181	−0.302593
$334$	0	0
$335$	−6.58573	−0.359817
$336$	0	0
$337$	−14.6277	−0.796819	−0.398410	−	0.917208i	$-0.630438\pi$
$337$	−14.6277	−0.796819	−0.398410	+	0.917208i	$0.630438\pi$
$338$	0	0
$339$	6.33109	0.343858
$340$	0	0
$341$	26.7893	1.45072
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−23.3324	−1.25618
$346$	0	0
$347$	−22.6875	−1.21793	−0.608965	−	0.793197i	$-0.708415\pi$
$347$	−22.6875	−1.21793	−0.608965	+	0.793197i	$0.708415\pi$
$348$	0	0
$349$	22.4924	1.20399	0.601996	−	0.798499i	$-0.294372\pi$
$349$	22.4924	1.20399	0.601996	+	0.798499i	$0.294372\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−34.2401	−1.82242	−0.911208	−	0.411947i	$-0.864849\pi$
$353$	−34.2401	−1.82242	−0.911208	+	0.411947i	$0.864849\pi$
$354$	0	0
$355$	41.1258	2.18273
$356$	0	0
$357$	−4.18668	−0.221583
$358$	0	0
$359$	3.19956	0.168866	0.0844331	−	0.996429i	$-0.473092\pi$
$359$	3.19956	0.168866	0.0844331	+	0.996429i	$0.473092\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	7.24621	0.380327
$364$	0	0
$365$	−12.9165	−0.676079
$366$	0	0
$367$	29.3136	1.53016	0.765078	−	0.643938i	$-0.222700\pi$
$367$	29.3136	1.53016	0.765078	+	0.643938i	$0.222700\pi$
$368$	0	0
$369$	−3.87285	−0.201613
$370$	0	0
$371$	−4.45824	−0.231460
$372$	0	0
$373$	−11.5644	−0.598783	−0.299392	−	0.954130i	$-0.596784\pi$
$373$	−11.5644	−0.598783	−0.299392	+	0.954130i	$0.596784\pi$
$374$	0	0
$375$	3.74571	0.193427
$376$	0	0
$377$	0	0
$378$	0	0
$379$	17.9747	0.923296	0.461648	−	0.887063i	$-0.347258\pi$
$379$	17.9747	0.923296	0.461648	+	0.887063i	$0.347258\pi$
$380$	0	0
$381$	11.2503	0.576368
$382$	0	0
$383$	−19.5022	−0.996516	−0.498258	−	0.867029i	$-0.666027\pi$
$383$	−19.5022	−0.996516	−0.498258	+	0.867029i	$0.666027\pi$
$384$	0	0
$385$	14.2462	0.726054
$386$	0	0
$387$	5.25025	0.266885
$388$	0	0
$389$	18.0541	0.915381	0.457691	−	0.889112i	$-0.348677\pi$
$389$	18.0541	0.915381	0.457691	+	0.889112i	$0.348677\pi$
$390$	0	0
$391$	29.2898	1.48125
$392$	0	0
$393$	3.78797	0.191078
$394$	0	0
$395$	14.4268	0.725889
$396$	0	0
$397$	11.8836	0.596423	0.298212	−	0.954500i	$-0.403610\pi$
$397$	11.8836	0.596423	0.298212	+	0.954500i	$0.403610\pi$
$398$	0	0
$399$	1.00000	0.0500626
$400$	0	0
$401$	−22.9933	−1.14823	−0.574115	−	0.818775i	$-0.694654\pi$
$401$	−22.9933	−1.14823	−0.574115	+	0.818775i	$0.694654\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	3.33513	0.165724
$406$	0	0
$407$	−23.5867	−1.16915
$408$	0	0
$409$	−26.5654	−1.31358	−0.656788	−	0.754075i	$-0.728085\pi$
$409$	−26.5654	−1.31358	−0.656788	+	0.754075i	$0.728085\pi$
$410$	0	0
$411$	−11.9919	−0.591518
$412$	0	0
$413$	4.79741	0.236065
$414$	0	0
$415$	−14.4448	−0.709065
$416$	0	0
$417$	−3.46768	−0.169813
$418$	0	0
$419$	−28.3419	−1.38459	−0.692296	−	0.721614i	$-0.743401\pi$
$419$	−28.3419	−1.38459	−0.692296	+	0.721614i	$0.743401\pi$
$420$	0	0
$421$	−24.4809	−1.19313	−0.596563	−	0.802566i	$-0.703467\pi$
$421$	−24.4809	−1.19313	−0.596563	+	0.802566i	$0.703467\pi$
$422$	0	0
$423$	11.4795	0.558154
$424$	0	0
$425$	−25.6355	−1.24350
$426$	0	0
$427$	10.5431	0.510217
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−29.9259	−1.44148	−0.720740	−	0.693205i	$-0.756198\pi$
$431$	−29.9259	−1.44148	−0.720740	+	0.693205i	$0.756198\pi$
$432$	0	0
$433$	−6.18702	−0.297329	−0.148665	−	0.988888i	$-0.547497\pi$
$433$	−6.18702	−0.297329	−0.148665	+	0.988888i	$0.547497\pi$
$434$	0	0
$435$	5.96313	0.285910
$436$	0	0
$437$	−6.99596	−0.334662
$438$	0	0
$439$	9.62017	0.459146	0.229573	−	0.973291i	$-0.426267\pi$
$439$	9.62017	0.459146	0.229573	+	0.973291i	$0.426267\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−39.9774	−1.89938	−0.949691	−	0.313189i	$-0.898603\pi$
$443$	−39.9774	−1.89938	−0.949691	+	0.313189i	$0.898603\pi$
$444$	0	0
$445$	49.7484	2.35830
$446$	0	0
$447$	1.40275	0.0663477
$448$	0	0
$449$	25.9178	1.22314	0.611569	−	0.791191i	$-0.290539\pi$
$449$	25.9178	1.22314	0.611569	+	0.791191i	$0.290539\pi$
$450$	0	0
$451$	−16.5431	−0.778985
$452$	0	0
$453$	1.00404	0.0471740
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−10.4636	−0.489468	−0.244734	−	0.969590i	$-0.578701\pi$
$457$	−10.4636	−0.489468	−0.244734	+	0.969590i	$0.578701\pi$
$458$	0	0
$459$	−4.18668	−0.195417
$460$	0	0
$461$	−8.78393	−0.409108	−0.204554	−	0.978855i	$-0.565574\pi$
$461$	−8.78393	−0.409108	−0.204554	+	0.978855i	$0.565574\pi$
$462$	0	0
$463$	6.91377	0.321310	0.160655	−	0.987011i	$-0.448639\pi$
$463$	6.91377	0.321310	0.160655	+	0.987011i	$0.448639\pi$
$464$	0	0
$465$	20.9165	0.969978
$466$	0	0
$467$	−16.5008	−0.763568	−0.381784	−	0.924252i	$-0.624690\pi$
$467$	−16.5008	−0.763568	−0.381784	+	0.924252i	$0.624690\pi$
$468$	0	0
$469$	−1.97465	−0.0911810
$470$	0	0
$471$	−12.2462	−0.564276
$472$	0	0
$473$	22.4268	1.03118
$474$	0	0
$475$	6.12311	0.280947
$476$	0	0
$477$	−4.45824	−0.204129
$478$	0	0
$479$	−23.1907	−1.05961	−0.529806	−	0.848119i	$-0.677735\pi$
$479$	−23.1907	−1.05961	−0.529806	+	0.848119i	$0.677735\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−6.99596	−0.318327
$484$	0	0
$485$	1.32974	0.0603802
$486$	0	0
$487$	21.5580	0.976884	0.488442	−	0.872596i	$-0.337565\pi$
$487$	21.5580	0.976884	0.488442	+	0.872596i	$0.337565\pi$
$488$	0	0
$489$	−10.5908	−0.478932
$490$	0	0
$491$	−29.7312	−1.34175	−0.670874	−	0.741571i	$-0.734081\pi$
$491$	−29.7312	−1.34175	−0.670874	+	0.741571i	$0.734081\pi$
$492$	0	0
$493$	−7.48567	−0.337138
$494$	0	0
$495$	14.2462	0.640320
$496$	0	0
$497$	12.3311	0.553125
$498$	0	0
$499$	9.03554	0.404486	0.202243	−	0.979335i	$-0.435177\pi$
$499$	9.03554	0.404486	0.202243	+	0.979335i	$0.435177\pi$
$500$	0	0
$501$	−4.11906	−0.184026
$502$	0	0
$503$	40.0910	1.78757	0.893785	−	0.448495i	$-0.148040\pi$
$503$	40.0910	1.78757	0.893785	+	0.448495i	$0.148040\pi$
$504$	0	0
$505$	1.36932	0.0609338
$506$	0	0
$507$	−13.0000	−0.577350
$508$	0	0
$509$	−23.2898	−1.03230	−0.516152	−	0.856497i	$-0.672636\pi$
$509$	−23.2898	−1.03230	−0.516152	+	0.856497i	$0.672636\pi$
$510$	0	0
$511$	−3.87285	−0.171325
$512$	0	0
$513$	1.00000	0.0441511
$514$	0	0
$515$	−26.8976	−1.18525
$516$	0	0
$517$	49.0355	2.15658
$518$	0	0
$519$	−8.67026	−0.380582
$520$	0	0
$521$	−14.7467	−0.646065	−0.323033	−	0.946388i	$-0.604702\pi$
$521$	−14.7467	−0.646065	−0.323033	+	0.946388i	$0.604702\pi$
$522$	0	0
$523$	11.5472	0.504922	0.252461	−	0.967607i	$-0.418760\pi$
$523$	11.5472	0.504922	0.252461	+	0.967607i	$0.418760\pi$
$524$	0	0
$525$	6.12311	0.267234
$526$	0	0
$527$	−26.2570	−1.14377
$528$	0	0
$529$	25.9434	1.12798
$530$	0	0
$531$	4.79741	0.208190
$532$	0	0
$533$	0	0
$534$	0	0
$535$	6.38718	0.276142
$536$	0	0
$537$	14.8316	0.640031
$538$	0	0
$539$	4.27156	0.183989
$540$	0	0
$541$	30.4924	1.31097	0.655486	−	0.755207i	$-0.272464\pi$
$541$	30.4924	1.31097	0.655486	+	0.755207i	$0.272464\pi$
$542$	0	0
$543$	−16.7179	−0.717435
$544$	0	0
$545$	−2.98214	−0.127741
$546$	0	0
$547$	−26.6368	−1.13891	−0.569454	−	0.822023i	$-0.692845\pi$
$547$	−26.6368	−1.13891	−0.569454	+	0.822023i	$0.692845\pi$
$548$	0	0
$549$	10.5431	0.449969
$550$	0	0
$551$	1.78797	0.0761702
$552$	0	0
$553$	4.32569	0.183947
$554$	0	0
$555$	−18.4160	−0.781714
$556$	0	0
$557$	19.9033	0.843328	0.421664	−	0.906752i	$-0.361446\pi$
$557$	19.9033	0.843328	0.421664	+	0.906752i	$0.361446\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−17.8836	−0.755048
$562$	0	0
$563$	13.4596	0.567254	0.283627	−	0.958935i	$-0.408462\pi$
$563$	13.4596	0.567254	0.283627	+	0.958935i	$0.408462\pi$
$564$	0	0
$565$	21.1150	0.888316
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	14.1613	0.593674	0.296837	−	0.954928i	$-0.404068\pi$
$569$	14.1613	0.593674	0.296837	+	0.954928i	$0.404068\pi$
$570$	0	0
$571$	−4.84811	−0.202887	−0.101443	−	0.994841i	$-0.532346\pi$
$571$	−4.84811	−0.202887	−0.101443	+	0.994841i	$0.532346\pi$
$572$	0	0
$573$	−10.3446	−0.432151
$574$	0	0
$575$	−42.8370	−1.78643
$576$	0	0
$577$	−21.4914	−0.894699	−0.447350	−	0.894359i	$-0.647632\pi$
$577$	−21.4914	−0.894699	−0.447350	+	0.894359i	$0.647632\pi$
$578$	0	0
$579$	4.04261	0.168005
$580$	0	0
$581$	−4.33109	−0.179684
$582$	0	0
$583$	−19.0436	−0.788706
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−2.79707	−0.115447	−0.0577237	−	0.998333i	$-0.518384\pi$
$587$	−2.79707	−0.115447	−0.0577237	+	0.998333i	$0.518384\pi$
$588$	0	0
$589$	6.27156	0.258415
$590$	0	0
$591$	−13.2675	−0.545753
$592$	0	0
$593$	−15.6781	−0.643822	−0.321911	−	0.946770i	$-0.604325\pi$
$593$	−15.6781	−0.643822	−0.321911	+	0.946770i	$0.604325\pi$
$594$	0	0
$595$	−13.9631	−0.572432
$596$	0	0
$597$	17.1708	0.702753
$598$	0	0
$599$	−5.26373	−0.215070	−0.107535	−	0.994201i	$-0.534296\pi$
$599$	−5.26373	−0.215070	−0.107535	+	0.994201i	$0.534296\pi$
$600$	0	0
$601$	36.2743	1.47966	0.739829	−	0.672795i	$-0.234906\pi$
$601$	36.2743	1.47966	0.739829	+	0.672795i	$0.234906\pi$
$602$	0	0
$603$	−1.97465	−0.0804141
$604$	0	0
$605$	24.1671	0.982531
$606$	0	0
$607$	12.2239	0.496153	0.248076	−	0.968741i	$-0.420202\pi$
$607$	12.2239	0.496153	0.248076	+	0.968741i	$0.420202\pi$
$608$	0	0
$609$	1.78797	0.0724523
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−21.2530	−0.858399	−0.429200	−	0.903210i	$-0.641204\pi$
$613$	−21.2530	−0.858399	−0.429200	+	0.903210i	$0.641204\pi$
$614$	0	0
$615$	−12.9165	−0.520843
$616$	0	0
$617$	30.1299	1.21298	0.606491	−	0.795090i	$-0.292577\pi$
$617$	30.1299	1.21298	0.606491	+	0.795090i	$0.292577\pi$
$618$	0	0
$619$	10.6196	0.426837	0.213418	−	0.976961i	$-0.431540\pi$
$619$	10.6196	0.426837	0.213418	+	0.976961i	$0.431540\pi$
$620$	0	0
$621$	−6.99596	−0.280738
$622$	0	0
$623$	14.9165	0.597616
$624$	0	0
$625$	−18.1231	−0.724924
$626$	0	0
$627$	4.27156	0.170590
$628$	0	0
$629$	23.1181	0.921777
$630$	0	0
$631$	2.95335	0.117571	0.0587854	−	0.998271i	$-0.481277\pi$
$631$	2.95335	0.117571	0.0587854	+	0.998271i	$0.481277\pi$
$632$	0	0
$633$	17.2387	0.685178
$634$	0	0
$635$	37.5211	1.48898
$636$	0	0
$637$	0	0
$638$	0	0
$639$	12.3311	0.487810
$640$	0	0
$641$	8.90603	0.351767	0.175883	−	0.984411i	$-0.443722\pi$
$641$	8.90603	0.351767	0.175883	+	0.984411i	$0.443722\pi$
$642$	0	0
$643$	19.4170	0.765731	0.382865	−	0.923804i	$-0.374937\pi$
$643$	19.4170	0.765731	0.382865	+	0.923804i	$0.374937\pi$
$644$	0	0
$645$	17.5103	0.689467
$646$	0	0
$647$	−39.8529	−1.56678	−0.783390	−	0.621531i	$-0.786511\pi$
$647$	−39.8529	−1.56678	−0.783390	+	0.621531i	$0.786511\pi$
$648$	0	0
$649$	20.4924	0.804398
$650$	0	0
$651$	6.27156	0.245802
$652$	0	0
$653$	−5.57251	−0.218069	−0.109034	−	0.994038i	$-0.534776\pi$
$653$	−5.57251	−0.218069	−0.109034	+	0.994038i	$0.534776\pi$
$654$	0	0
$655$	12.6334	0.493627
$656$	0	0
$657$	−3.87285	−0.151094
$658$	0	0
$659$	2.66184	0.103690	0.0518452	−	0.998655i	$-0.483490\pi$
$659$	2.66184	0.103690	0.0518452	+	0.998655i	$0.483490\pi$
$660$	0	0
$661$	39.8756	1.55098	0.775490	−	0.631360i	$-0.217503\pi$
$661$	39.8756	1.55098	0.775490	+	0.631360i	$0.217503\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	3.33513	0.129331
$666$	0	0
$667$	−12.5086	−0.484334
$668$	0	0
$669$	−0.517770	−0.0200181
$670$	0	0
$671$	45.0355	1.73858
$672$	0	0
$673$	−30.8895	−1.19070	−0.595352	−	0.803465i	$-0.702987\pi$
$673$	−30.8895	−1.19070	−0.595352	+	0.803465i	$0.702987\pi$
$674$	0	0
$675$	6.12311	0.235678
$676$	0	0
$677$	1.84103	0.0707567	0.0353783	−	0.999374i	$-0.488736\pi$
$677$	1.84103	0.0707567	0.0353783	+	0.999374i	$0.488736\pi$
$678$	0	0
$679$	0.398706	0.0153009
$680$	0	0
$681$	18.9591	0.726514
$682$	0	0
$683$	−31.7988	−1.21675	−0.608373	−	0.793651i	$-0.708178\pi$
$683$	−31.7988	−1.21675	−0.608373	+	0.793651i	$0.708178\pi$
$684$	0	0
$685$	−39.9946	−1.52812
$686$	0	0
$687$	−29.5022	−1.12558
$688$	0	0
$689$	0	0
$690$	0	0
$691$	22.8976	0.871066	0.435533	−	0.900173i	$-0.356560\pi$
$691$	22.8976	0.871066	0.435533	+	0.900173i	$0.356560\pi$
$692$	0	0
$693$	4.27156	0.162263
$694$	0	0
$695$	−11.5652	−0.438691
$696$	0	0
$697$	16.2144	0.614164
$698$	0	0
$699$	13.8648	0.524414
$700$	0	0
$701$	22.7541	0.859409	0.429705	−	0.902970i	$-0.358618\pi$
$701$	22.7541	0.859409	0.429705	+	0.902970i	$0.358618\pi$
$702$	0	0
$703$	−5.52181	−0.208259
$704$	0	0
$705$	38.2858	1.44193
$706$	0	0
$707$	0.410574	0.0154412
$708$	0	0
$709$	26.5143	0.995766	0.497883	−	0.867244i	$-0.334111\pi$
$709$	26.5143	0.995766	0.497883	+	0.867244i	$0.334111\pi$
$710$	0	0
$711$	4.32569	0.162226
$712$	0	0
$713$	−43.8756	−1.64315
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−29.6155	−1.10601
$718$	0	0
$719$	4.53125	0.168987	0.0844935	−	0.996424i	$-0.473073\pi$
$719$	4.53125	0.168987	0.0844935	+	0.996424i	$0.473073\pi$
$720$	0	0
$721$	−8.06493	−0.300354
$722$	0	0
$723$	15.1880	0.564849
$724$	0	0
$725$	10.9480	0.406597
$726$	0	0
$727$	−21.8459	−0.810219	−0.405110	−	0.914268i	$-0.632767\pi$
$727$	−21.8459	−0.810219	−0.405110	+	0.914268i	$0.632767\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−21.9811	−0.813001
$732$	0	0
$733$	−28.1617	−1.04018	−0.520088	−	0.854113i	$-0.674101\pi$
$733$	−28.1617	−1.04018	−0.520088	+	0.854113i	$0.674101\pi$
$734$	0	0
$735$	3.33513	0.123018
$736$	0	0
$737$	−8.43484	−0.310702
$738$	0	0
$739$	16.6334	0.611869	0.305935	−	0.952053i	$-0.401031\pi$
$739$	16.6334	0.611869	0.305935	+	0.952053i	$0.401031\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	51.7319	1.89786	0.948930	−	0.315487i	$-0.102168\pi$
$743$	51.7319	1.89786	0.948930	+	0.315487i	$0.102168\pi$
$744$	0	0
$745$	4.67835	0.171401
$746$	0	0
$747$	−4.33109	−0.158466
$748$	0	0
$749$	1.91512	0.0699770
$750$	0	0
$751$	22.0396	0.804236	0.402118	−	0.915588i	$-0.368274\pi$
$751$	22.0396	0.804236	0.402118	+	0.915588i	$0.368274\pi$
$752$	0	0
$753$	−26.7233	−0.973852
$754$	0	0
$755$	3.34861	0.121868
$756$	0	0
$757$	38.5270	1.40029	0.700143	−	0.714003i	$-0.253120\pi$
$757$	38.5270	1.40029	0.700143	+	0.714003i	$0.253120\pi$
$758$	0	0
$759$	−29.8836	−1.08471
$760$	0	0
$761$	32.1813	1.16657	0.583286	−	0.812267i	$-0.301767\pi$
$761$	32.1813	1.16657	0.583286	+	0.812267i	$0.301767\pi$
$762$	0	0
$763$	−0.894158	−0.0323707
$764$	0	0
$765$	−13.9631	−0.504838
$766$	0	0
$767$	0	0
$768$	0	0
$769$	21.6374	0.780266	0.390133	−	0.920759i	$-0.372429\pi$
$769$	21.6374	0.780266	0.390133	+	0.920759i	$0.372429\pi$
$770$	0	0
$771$	−26.2888	−0.946769
$772$	0	0
$773$	0.816287	0.0293598	0.0146799	−	0.999892i	$-0.495327\pi$
$773$	0.816287	0.0293598	0.0146799	+	0.999892i	$0.495327\pi$
$774$	0	0
$775$	38.4014	1.37942
$776$	0	0
$777$	−5.52181	−0.198094
$778$	0	0
$779$	−3.87285	−0.138759
$780$	0	0
$781$	52.6730	1.88479
$782$	0	0
$783$	1.78797	0.0638970
$784$	0	0
$785$	−40.8427	−1.45774
$786$	0	0
$787$	−44.9798	−1.60336	−0.801678	−	0.597756i	$-0.796059\pi$
$787$	−44.9798	−1.60336	−0.801678	+	0.597756i	$0.796059\pi$
$788$	0	0
$789$	−21.0148	−0.748148
$790$	0	0
$791$	6.33109	0.225108
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−14.8688	−0.527342
$796$	0	0
$797$	−2.42405	−0.0858644	−0.0429322	−	0.999078i	$-0.513670\pi$
$797$	−2.42405	−0.0858644	−0.0429322	+	0.999078i	$0.513670\pi$
$798$	0	0
$799$	−48.0612	−1.70028
$800$	0	0
$801$	14.9165	0.527048
$802$	0	0
$803$	−16.5431	−0.583794
$804$	0	0
$805$	−23.3324	−0.822361
$806$	0	0
$807$	13.9493	0.491039
$808$	0	0
$809$	13.6950	0.481491	0.240745	−	0.970588i	$-0.422608\pi$
$809$	13.6950	0.481491	0.240745	+	0.970588i	$0.422608\pi$
$810$	0	0
$811$	−36.8895	−1.29537	−0.647683	−	0.761910i	$-0.724262\pi$
$811$	−36.8895	−1.29537	−0.647683	+	0.761910i	$0.724262\pi$
$812$	0	0
$813$	25.0862	0.879813
$814$	0	0
$815$	−35.3217	−1.23726
$816$	0	0
$817$	5.25025	0.183683
$818$	0	0
$819$	0	0
$820$	0	0
$821$	17.5110	0.611139	0.305569	−	0.952170i	$-0.401153\pi$
$821$	17.5110	0.611139	0.305569	+	0.952170i	$0.401153\pi$
$822$	0	0
$823$	−31.8705	−1.11094	−0.555468	−	0.831538i	$-0.687461\pi$
$823$	−31.8705	−1.11094	−0.555468	+	0.831538i	$0.687461\pi$
$824$	0	0
$825$	26.1552	0.910607
$826$	0	0
$827$	26.5773	0.924183	0.462092	−	0.886832i	$-0.347099\pi$
$827$	26.5773	0.924183	0.462092	+	0.886832i	$0.347099\pi$
$828$	0	0
$829$	34.3473	1.19293	0.596465	−	0.802639i	$-0.296571\pi$
$829$	34.3473	1.19293	0.596465	+	0.802639i	$0.296571\pi$
$830$	0	0
$831$	6.62260	0.229735
$832$	0	0
$833$	−4.18668	−0.145060
$834$	0	0
$835$	−13.7376	−0.475410
$836$	0	0
$837$	6.27156	0.216777
$838$	0	0
$839$	−39.9709	−1.37995	−0.689974	−	0.723834i	$-0.742378\pi$
$839$	−39.9709	−1.37995	−0.689974	+	0.723834i	$0.742378\pi$
$840$	0	0
$841$	−25.8032	−0.889764
$842$	0	0
$843$	4.25464	0.146537
$844$	0	0
$845$	−43.3567	−1.49152
$846$	0	0
$847$	7.24621	0.248983
$848$	0	0
$849$	−24.8658	−0.853391
$850$	0	0
$851$	38.6304	1.32423
$852$	0	0
$853$	10.5431	0.360989	0.180495	−	0.983576i	$-0.442230\pi$
$853$	10.5431	0.360989	0.180495	+	0.983576i	$0.442230\pi$
$854$	0	0
$855$	3.33513	0.114059
$856$	0	0
$857$	−9.26408	−0.316455	−0.158227	−	0.987403i	$-0.550578\pi$
$857$	−9.26408	−0.316455	−0.158227	+	0.987403i	$0.550578\pi$
$858$	0	0
$859$	−51.2242	−1.74775	−0.873873	−	0.486154i	$-0.838399\pi$
$859$	−51.2242	−1.74775	−0.873873	+	0.486154i	$0.838399\pi$
$860$	0	0
$861$	−3.87285	−0.131986
$862$	0	0
$863$	45.0041	1.53196	0.765978	−	0.642867i	$-0.222255\pi$
$863$	45.0041	1.53196	0.765978	+	0.642867i	$0.222255\pi$
$864$	0	0
$865$	−28.9165	−0.983190
$866$	0	0
$867$	0.528283	0.0179414
$868$	0	0
$869$	18.4775	0.626805
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0.398706	0.0134941
$874$	0	0
$875$	3.74571	0.126628
$876$	0	0
$877$	−20.5147	−0.692734	−0.346367	−	0.938099i	$-0.612585\pi$
$877$	−20.5147	−0.692734	−0.346367	+	0.938099i	$0.612585\pi$
$878$	0	0
$879$	2.62765	0.0886285
$880$	0	0
$881$	−23.0030	−0.774990	−0.387495	−	0.921872i	$-0.626659\pi$
$881$	−23.0030	−0.774990	−0.387495	+	0.921872i	$0.626659\pi$
$882$	0	0
$883$	−5.28983	−0.178017	−0.0890085	−	0.996031i	$-0.528370\pi$
$883$	−5.28983	−0.178017	−0.0890085	+	0.996031i	$0.528370\pi$
$884$	0	0
$885$	16.0000	0.537834
$886$	0	0
$887$	53.8944	1.80960	0.904799	−	0.425839i	$-0.140021\pi$
$887$	53.8944	1.80960	0.904799	+	0.425839i	$0.140021\pi$
$888$	0	0
$889$	11.2503	0.377321
$890$	0	0
$891$	4.27156	0.143103
$892$	0	0
$893$	11.4795	0.384148
$894$	0	0
$895$	49.4653	1.65344
$896$	0	0
$897$	0	0
$898$	0	0
$899$	11.2134	0.373987
$900$	0	0
$901$	18.6652	0.621828
$902$	0	0
$903$	5.25025	0.174717
$904$	0	0
$905$	−55.7565	−1.85341
$906$	0	0
$907$	33.2226	1.10314	0.551569	−	0.834130i	$-0.314030\pi$
$907$	33.2226	1.10314	0.551569	+	0.834130i	$0.314030\pi$
$908$	0	0
$909$	0.410574	0.0136179
$910$	0	0
$911$	−44.3527	−1.46947	−0.734735	−	0.678354i	$-0.762693\pi$
$911$	−44.3527	−1.46947	−0.734735	+	0.678354i	$0.762693\pi$
$912$	0	0
$913$	−18.5005	−0.612277
$914$	0	0
$915$	35.1627	1.16244
$916$	0	0
$917$	3.78797	0.125090
$918$	0	0
$919$	−56.4763	−1.86298	−0.931490	−	0.363767i	$-0.881490\pi$
$919$	−56.4763	−1.86298	−0.931490	+	0.363767i	$0.881490\pi$
$920$	0	0
$921$	11.5472	0.380492
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−33.8106	−1.11169
$926$	0	0
$927$	−8.06493	−0.264887
$928$	0	0
$929$	−2.35373	−0.0772233	−0.0386117	−	0.999254i	$-0.512294\pi$
$929$	−2.35373	−0.0772233	−0.0386117	+	0.999254i	$0.512294\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	25.4477	0.833121
$934$	0	0
$935$	−59.6443	−1.95058
$936$	0	0
$937$	10.9909	0.359057	0.179529	−	0.983753i	$-0.442543\pi$
$937$	10.9909	0.359057	0.179529	+	0.983753i	$0.442543\pi$
$938$	0	0
$939$	4.04261	0.131926
$940$	0	0
$941$	−0.0587775	−0.00191609	−0.000958046	−	1.00000i	$-0.500305\pi$
$941$	−0.0587775	−0.00191609	−0.000958046		1.00000i	$0.500305\pi$
$942$	0	0
$943$	27.0943	0.882312
$944$	0	0
$945$	3.33513	0.108492
$946$	0	0
$947$	22.9149	0.744633	0.372317	−	0.928106i	$-0.378564\pi$
$947$	22.9149	0.744633	0.372317	+	0.928106i	$0.378564\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	20.3737	0.660663
$952$	0	0
$953$	−7.40653	−0.239921	−0.119961	−	0.992779i	$-0.538277\pi$
$953$	−7.40653	−0.239921	−0.119961	+	0.992779i	$0.538277\pi$
$954$	0	0
$955$	−34.5005	−1.11641
$956$	0	0
$957$	7.63743	0.246883
$958$	0	0
$959$	−11.9919	−0.387239
$960$	0	0
$961$	8.33244	0.268789
$962$	0	0
$963$	1.91512	0.0617139
$964$	0	0
$965$	13.4826	0.434022
$966$	0	0
$967$	41.0305	1.31945	0.659726	−	0.751506i	$-0.270672\pi$
$967$	41.0305	1.31945	0.659726	+	0.751506i	$0.270672\pi$
$968$	0	0
$969$	−4.18668	−0.134496
$970$	0	0
$971$	−24.5431	−0.787626	−0.393813	−	0.919191i	$-0.628844\pi$
$971$	−24.5431	−0.787626	−0.393813	+	0.919191i	$0.628844\pi$
$972$	0	0
$973$	−3.46768	−0.111169
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−39.8414	−1.27464	−0.637319	−	0.770600i	$-0.719957\pi$
$977$	−39.8414	−1.27464	−0.637319	+	0.770600i	$0.719957\pi$
$978$	0	0
$979$	63.7166	2.03639
$980$	0	0
$981$	−0.894158	−0.0285483
$982$	0	0
$983$	−46.7663	−1.49161	−0.745806	−	0.666163i	$-0.767936\pi$
$983$	−46.7663	−1.49161	−0.745806	+	0.666163i	$0.767936\pi$
$984$	0	0
$985$	−44.2489	−1.40989
$986$	0	0
$987$	11.4795	0.365398
$988$	0	0
$989$	−36.7306	−1.16796
$990$	0	0
$991$	13.4626	0.427654	0.213827	−	0.976872i	$-0.431407\pi$
$991$	13.4626	0.427654	0.213827	+	0.976872i	$0.431407\pi$
$992$	0	0
$993$	−27.2069	−0.863385
$994$	0	0
$995$	57.2668	1.81548
$996$	0	0
$997$	22.4348	0.710519	0.355259	−	0.934768i	$-0.384393\pi$
$997$	22.4348	0.710519	0.355259	+	0.934768i	$0.384393\pi$
$998$	0	0
$999$	−5.52181	−0.174702

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6384.2.a.cb.1.4		4
4.3	odd	2		3192.2.a.x.1.4	✓	4
12.11	even	2		9576.2.a.cj.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3192.2.a.x.1.4	✓	4	4.3	odd	2
6384.2.a.cb.1.4		4	1.1	even	1	trivial
9576.2.a.cj.1.1		4	12.11	even	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:	\( N \)	\(=\)	\( 6384 = 2^{4} \cdot 3 \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6384.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.33513 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +3.33513 q^{5} +1.00000 q^{7} +1.00000 q^{9} +4.27156 q^{11} +3.33513 q^{15} -4.18668 q^{17} +1.00000 q^{19} +1.00000 q^{21} -6.99596 q^{23} +6.12311 q^{25} +1.00000 q^{27} +1.78797 q^{29} +6.27156 q^{31} +4.27156 q^{33} +3.33513 q^{35} -5.52181 q^{37} -3.87285 q^{41} +5.25025 q^{43} +3.33513 q^{45} +11.4795 q^{47} +1.00000 q^{49} -4.18668 q^{51} -4.45824 q^{53} +14.2462 q^{55} +1.00000 q^{57} +4.79741 q^{59} +10.5431 q^{61} +1.00000 q^{63} -1.97465 q^{67} -6.99596 q^{69} +12.3311 q^{71} -3.87285 q^{73} +6.12311 q^{75} +4.27156 q^{77} +4.32569 q^{79} +1.00000 q^{81} -4.33109 q^{83} -13.9631 q^{85} +1.78797 q^{87} +14.9165 q^{89} +6.27156 q^{93} +3.33513 q^{95} +0.398706 q^{97} +4.27156 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 4 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + 4 * q^7 + 4 * q^9	\( 4 q + 4 q^{3} + 4 q^{7} + 4 q^{9} - 4 q^{17} + 4 q^{19} + 4 q^{21} - 4 q^{23} + 8 q^{25} + 4 q^{27} + 4 q^{29} + 8 q^{31} + 4 q^{37} - 8 q^{41} + 12 q^{43} + 8 q^{47} + 4 q^{49} - 4 q^{51} + 12 q^{53} + 24 q^{55} + 4 q^{57} + 8 q^{61} + 4 q^{63} + 8 q^{67} - 4 q^{69} + 12 q^{71} - 8 q^{73} + 8 q^{75} + 20 q^{79} + 4 q^{81} + 20 q^{83} - 4 q^{85} + 4 q^{87} + 8 q^{93} - 8 q^{97}+O(q^{100}) \) 4 * q + 4 * q^3 + 4 * q^7 + 4 * q^9 - 4 * q^17 + 4 * q^19 + 4 * q^21 - 4 * q^23 + 8 * q^25 + 4 * q^27 + 4 * q^29 + 8 * q^31 + 4 * q^37 - 8 * q^41 + 12 * q^43 + 8 * q^47 + 4 * q^49 - 4 * q^51 + 12 * q^53 + 24 * q^55 + 4 * q^57 + 8 * q^61 + 4 * q^63 + 8 * q^67 - 4 * q^69 + 12 * q^71 - 8 * q^73 + 8 * q^75 + 20 * q^79 + 4 * q^81 + 20 * q^83 - 4 * q^85 + 4 * q^87 + 8 * q^93 - 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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