LMFDB - Embedded newform 8001.2.a.ba.1.24

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8001,2,Mod(1,8001)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8001, 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("8001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8001 = 3^{2} \cdot 7 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8001.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:	$63.8883066572$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.24
Character	$\chi$	$=$	8001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.476800 q^{2} -1.77266 q^{4} +3.87912 q^{5} +1.00000 q^{7} -1.79880 q^{8} +O(q^{10})$	\(q+0.476800 q^{2} -1.77266 q^{4} +3.87912 q^{5} +1.00000 q^{7} -1.79880 q^{8} +1.84956 q^{10} -5.00409 q^{11} -0.407559 q^{13} +0.476800 q^{14} +2.68765 q^{16} -0.710255 q^{17} +8.42156 q^{19} -6.87638 q^{20} -2.38595 q^{22} +5.39892 q^{23} +10.0476 q^{25} -0.194324 q^{26} -1.77266 q^{28} -8.15929 q^{29} -0.794645 q^{31} +4.87908 q^{32} -0.338649 q^{34} +3.87912 q^{35} +6.57907 q^{37} +4.01540 q^{38} -6.97778 q^{40} +2.29873 q^{41} +4.17556 q^{43} +8.87055 q^{44} +2.57420 q^{46} -9.25600 q^{47} +1.00000 q^{49} +4.79069 q^{50} +0.722464 q^{52} -1.76295 q^{53} -19.4115 q^{55} -1.79880 q^{56} -3.89035 q^{58} +1.08670 q^{59} +1.11237 q^{61} -0.378886 q^{62} -3.04897 q^{64} -1.58097 q^{65} +0.810556 q^{67} +1.25904 q^{68} +1.84956 q^{70} -9.28378 q^{71} -11.8007 q^{73} +3.13690 q^{74} -14.9286 q^{76} -5.00409 q^{77} +3.16006 q^{79} +10.4257 q^{80} +1.09603 q^{82} -1.77907 q^{83} -2.75517 q^{85} +1.99091 q^{86} +9.00137 q^{88} +7.40408 q^{89} -0.407559 q^{91} -9.57046 q^{92} -4.41326 q^{94} +32.6683 q^{95} +1.89293 q^{97} +0.476800 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 54 q^{4} + 40 q^{7}+O(q^{10}) $ 40 * q + 54 * q^4 + 40 * q^7	$ 40 q + 54 q^{4} + 40 q^{7} + 20 q^{10} + 10 q^{13} + 90 q^{16} + 38 q^{19} + 14 q^{22} + 84 q^{25} + 54 q^{28} + 66 q^{31} + 22 q^{34} + 40 q^{37} + 26 q^{40} + 38 q^{43} + 28 q^{46} + 40 q^{49} + 28 q^{52} + 60 q^{55} + 42 q^{58} + 54 q^{61} + 124 q^{64} + 48 q^{67} + 20 q^{70} + 16 q^{76} + 102 q^{79} + 48 q^{82} + 104 q^{85} + 48 q^{88} + 10 q^{91} - 10 q^{94} + 8 q^{97}+O(q^{100}) $ 40 * q + 54 * q^4 + 40 * q^7 + 20 * q^10 + 10 * q^13 + 90 * q^16 + 38 * q^19 + 14 * q^22 + 84 * q^25 + 54 * q^28 + 66 * q^31 + 22 * q^34 + 40 * q^37 + 26 * q^40 + 38 * q^43 + 28 * q^46 + 40 * q^49 + 28 * q^52 + 60 * q^55 + 42 * q^58 + 54 * q^61 + 124 * q^64 + 48 * q^67 + 20 * q^70 + 16 * q^76 + 102 * q^79 + 48 * q^82 + 104 * q^85 + 48 * q^88 + 10 * q^91 - 10 * q^94 + 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 $

$2$	0.476800	0.337148	0.168574	−	0.985689i	$-0.446084\pi$
$2$	0.476800	0.337148	0.168574	+	0.985689i	$0.446084\pi$
$3$	0	0
$3$	0	0
$4$	−1.77266	−0.886331
$5$	3.87912	1.73480	0.867398	−	0.497614i	$-0.165791\pi$
$5$	3.87912	1.73480	0.867398	+	0.497614i	$0.165791\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.79880	−0.635973
$9$	0	0
$10$	1.84956	0.584884
$11$	−5.00409	−1.50879	−0.754394	−	0.656422i	$-0.772069\pi$
$11$	−5.00409	−1.50879	−0.754394	+	0.656422i	$0.772069\pi$
$12$	0	0
$13$	−0.407559	−0.113037	−0.0565183	−	0.998402i	$-0.518000\pi$
$13$	−0.407559	−0.113037	−0.0565183	+	0.998402i	$0.518000\pi$
$14$	0.476800	0.127430
$15$	0	0
$16$	2.68765	0.671914
$17$	−0.710255	−0.172262	−0.0861310	−	0.996284i	$-0.527450\pi$
$17$	−0.710255	−0.172262	−0.0861310	+	0.996284i	$0.527450\pi$
$18$	0	0
$19$	8.42156	1.93204	0.966020	−	0.258469i	$-0.0832178\pi$
$19$	8.42156	1.93204	0.966020	+	0.258469i	$0.0832178\pi$
$20$	−6.87638	−1.53760
$21$	0	0
$22$	−2.38595	−0.508685
$23$	5.39892	1.12575	0.562876	−	0.826541i	$-0.309695\pi$
$23$	5.39892	1.12575	0.562876	+	0.826541i	$0.309695\pi$
$24$	0	0
$25$	10.0476	2.00952
$26$	−0.194324	−0.0381101
$27$	0	0
$28$	−1.77266	−0.335002
$29$	−8.15929	−1.51514	−0.757571	−	0.652753i	$-0.773614\pi$
$29$	−8.15929	−1.51514	−0.757571	+	0.652753i	$0.773614\pi$
$30$	0	0
$31$	−0.794645	−0.142722	−0.0713612	−	0.997451i	$-0.522734\pi$
$31$	−0.794645	−0.142722	−0.0713612	+	0.997451i	$0.522734\pi$
$32$	4.87908	0.862508
$33$	0	0
$34$	−0.338649	−0.0580779
$35$	3.87912	0.655692
$36$	0	0
$37$	6.57907	1.08159	0.540796	−	0.841154i	$-0.318123\pi$
$37$	6.57907	1.08159	0.540796	+	0.841154i	$0.318123\pi$
$38$	4.01540	0.651384
$39$	0	0
$40$	−6.97778	−1.10328
$41$	2.29873	0.359001	0.179500	−	0.983758i	$-0.442552\pi$
$41$	2.29873	0.359001	0.179500	+	0.983758i	$0.442552\pi$
$42$	0	0
$43$	4.17556	0.636767	0.318384	−	0.947962i	$-0.396860\pi$
$43$	4.17556	0.636767	0.318384	+	0.947962i	$0.396860\pi$
$44$	8.87055	1.33729
$45$	0	0
$46$	2.57420	0.379545
$47$	−9.25600	−1.35013	−0.675063	−	0.737760i	$-0.735884\pi$
$47$	−9.25600	−1.35013	−0.675063	+	0.737760i	$0.735884\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	4.79069	0.677506
$51$	0	0
$52$	0.722464	0.100188
$53$	−1.76295	−0.242160	−0.121080	−	0.992643i	$-0.538636\pi$
$53$	−1.76295	−0.242160	−0.121080	+	0.992643i	$0.538636\pi$
$54$	0	0
$55$	−19.4115	−2.61744
$56$	−1.79880	−0.240375
$57$	0	0
$58$	−3.89035	−0.510827
$59$	1.08670	0.141477	0.0707384	−	0.997495i	$-0.477464\pi$
$59$	1.08670	0.141477	0.0707384	+	0.997495i	$0.477464\pi$
$60$	0	0
$61$	1.11237	0.142424	0.0712120	−	0.997461i	$-0.477313\pi$
$61$	1.11237	0.142424	0.0712120	+	0.997461i	$0.477313\pi$
$62$	−0.378886	−0.0481186
$63$	0	0
$64$	−3.04897	−0.381121
$65$	−1.58097	−0.196095
$66$	0	0
$67$	0.810556	0.0990251	0.0495126	−	0.998774i	$-0.484233\pi$
$67$	0.810556	0.0990251	0.0495126	+	0.998774i	$0.484233\pi$
$68$	1.25904	0.152681
$69$	0	0
$70$	1.84956	0.221065
$71$	−9.28378	−1.10178	−0.550891	−	0.834577i	$-0.685712\pi$
$71$	−9.28378	−1.10178	−0.550891	+	0.834577i	$0.685712\pi$
$72$	0	0
$73$	−11.8007	−1.38117	−0.690586	−	0.723250i	$-0.742647\pi$
$73$	−11.8007	−1.38117	−0.690586	+	0.723250i	$0.742647\pi$
$74$	3.13690	0.364657
$75$	0	0
$76$	−14.9286	−1.71243
$77$	−5.00409	−0.570268
$78$	0	0
$79$	3.16006	0.355535	0.177767	−	0.984073i	$-0.443113\pi$
$79$	3.16006	0.355535	0.177767	+	0.984073i	$0.443113\pi$
$80$	10.4257	1.16563
$81$	0	0
$82$	1.09603	0.121037
$83$	−1.77907	−0.195278	−0.0976391	−	0.995222i	$-0.531129\pi$
$83$	−1.77907	−0.195278	−0.0976391	+	0.995222i	$0.531129\pi$
$84$	0	0
$85$	−2.75517	−0.298840
$86$	1.99091	0.214685
$87$	0	0
$88$	9.00137	0.959549
$89$	7.40408	0.784831	0.392415	−	0.919788i	$-0.371640\pi$
$89$	7.40408	0.784831	0.392415	+	0.919788i	$0.371640\pi$
$90$	0	0
$91$	−0.407559	−0.0427238
$92$	−9.57046	−0.997789
$93$	0	0
$94$	−4.41326	−0.455193
$95$	32.6683	3.35170
$96$	0	0
$97$	1.89293	0.192198	0.0960990	−	0.995372i	$-0.469363\pi$
$97$	1.89293	0.192198	0.0960990	+	0.995372i	$0.469363\pi$
$98$	0.476800	0.0481640
$99$	0	0
$100$	−17.8110	−1.78110
$101$	19.4439	1.93474	0.967368	−	0.253376i	$-0.0815411\pi$
$101$	19.4439	1.93474	0.967368	+	0.253376i	$0.0815411\pi$
$102$	0	0
$103$	13.9767	1.37717	0.688585	−	0.725156i	$-0.258232\pi$
$103$	13.9767	1.37717	0.688585	+	0.725156i	$0.258232\pi$
$104$	0.733119	0.0718882
$105$	0	0
$106$	−0.840576	−0.0816440
$107$	6.69599	0.647326	0.323663	−	0.946172i	$-0.395086\pi$
$107$	6.69599	0.647326	0.323663	+	0.946172i	$0.395086\pi$
$108$	0	0
$109$	0.827402	0.0792507	0.0396254	−	0.999215i	$-0.487384\pi$
$109$	0.827402	0.0792507	0.0396254	+	0.999215i	$0.487384\pi$
$110$	−9.25538	−0.882466
$111$	0	0
$112$	2.68765	0.253960
$113$	8.28680	0.779557	0.389778	−	0.920909i	$-0.372552\pi$
$113$	8.28680	0.779557	0.389778	+	0.920909i	$0.372552\pi$
$114$	0	0
$115$	20.9431	1.95295
$116$	14.4637	1.34292
$117$	0	0
$118$	0.518140	0.0476986
$119$	−0.710255	−0.0651089
$120$	0	0
$121$	14.0409	1.27644
$122$	0.530376	0.0480180
$123$	0	0
$124$	1.40864	0.126499
$125$	19.5803	1.75131
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	−11.2119	−0.991002
$129$	0	0
$130$	−0.753807	−0.0661132
$131$	21.3079	1.86168	0.930838	−	0.365432i	$-0.119079\pi$
$131$	21.3079	1.86168	0.930838	+	0.365432i	$0.119079\pi$
$132$	0	0
$133$	8.42156	0.730242
$134$	0.386473	0.0333862
$135$	0	0
$136$	1.27761	0.109554
$137$	12.9060	1.10263	0.551316	−	0.834296i	$-0.314126\pi$
$137$	12.9060	1.10263	0.551316	+	0.834296i	$0.314126\pi$
$138$	0	0
$139$	14.3363	1.21599	0.607993	−	0.793943i	$-0.291975\pi$
$139$	14.3363	1.21599	0.607993	+	0.793943i	$0.291975\pi$
$140$	−6.87638	−0.581160
$141$	0	0
$142$	−4.42650	−0.371464
$143$	2.03946	0.170548
$144$	0	0
$145$	−31.6509	−2.62846
$146$	−5.62659	−0.465660
$147$	0	0
$148$	−11.6625	−0.958649
$149$	−20.1914	−1.65414	−0.827070	−	0.562099i	$-0.809994\pi$
$149$	−20.1914	−1.65414	−0.827070	+	0.562099i	$0.809994\pi$
$150$	0	0
$151$	−9.64467	−0.784872	−0.392436	−	0.919779i	$-0.628368\pi$
$151$	−9.64467	−0.784872	−0.392436	+	0.919779i	$0.628368\pi$
$152$	−15.1487	−1.22873
$153$	0	0
$154$	−2.38595	−0.192265
$155$	−3.08253	−0.247594
$156$	0	0
$157$	−4.48469	−0.357917	−0.178959	−	0.983857i	$-0.557273\pi$
$157$	−4.48469	−0.357917	−0.178959	+	0.983857i	$0.557273\pi$
$158$	1.50672	0.119868
$159$	0	0
$160$	18.9266	1.49628
$161$	5.39892	0.425494
$162$	0	0
$163$	−0.315541	−0.0247151	−0.0123575	−	0.999924i	$-0.503934\pi$
$163$	−0.315541	−0.0247151	−0.0123575	+	0.999924i	$0.503934\pi$
$164$	−4.07487	−0.318194
$165$	0	0
$166$	−0.848259	−0.0658377
$167$	9.99045	0.773084	0.386542	−	0.922272i	$-0.373669\pi$
$167$	9.99045	0.773084	0.386542	+	0.922272i	$0.373669\pi$
$168$	0	0
$169$	−12.8339	−0.987223
$170$	−1.31366	−0.100753
$171$	0	0
$172$	−7.40186	−0.564387
$173$	−3.81632	−0.290149	−0.145075	−	0.989421i	$-0.546342\pi$
$173$	−3.81632	−0.290149	−0.145075	+	0.989421i	$0.546342\pi$
$174$	0	0
$175$	10.0476	0.759527
$176$	−13.4493	−1.01378
$177$	0	0
$178$	3.53026	0.264604
$179$	−2.61980	−0.195813	−0.0979064	−	0.995196i	$-0.531215\pi$
$179$	−2.61980	−0.195813	−0.0979064	+	0.995196i	$0.531215\pi$
$180$	0	0
$181$	23.6108	1.75498	0.877490	−	0.479594i	$-0.159216\pi$
$181$	23.6108	1.75498	0.877490	+	0.479594i	$0.159216\pi$
$182$	−0.194324	−0.0144043
$183$	0	0
$184$	−9.71160	−0.715948
$185$	25.5210	1.87634
$186$	0	0
$187$	3.55417	0.259907
$188$	16.4078	1.19666
$189$	0	0
$190$	15.5762	1.13002
$191$	−3.07391	−0.222420	−0.111210	−	0.993797i	$-0.535473\pi$
$191$	−3.07391	−0.222420	−0.111210	+	0.993797i	$0.535473\pi$
$192$	0	0
$193$	10.0691	0.724788	0.362394	−	0.932025i	$-0.381959\pi$
$193$	10.0691	0.724788	0.362394	+	0.932025i	$0.381959\pi$
$194$	0.902549	0.0647993
$195$	0	0
$196$	−1.77266	−0.126619
$197$	−11.4982	−0.819213	−0.409607	−	0.912262i	$-0.634334\pi$
$197$	−11.4982	−0.819213	−0.409607	+	0.912262i	$0.634334\pi$
$198$	0	0
$199$	5.67839	0.402531	0.201265	−	0.979537i	$-0.435495\pi$
$199$	5.67839	0.402531	0.201265	+	0.979537i	$0.435495\pi$
$200$	−18.0737	−1.27800
$201$	0	0
$202$	9.27082	0.652293
$203$	−8.15929	−0.572670
$204$	0	0
$205$	8.91705	0.622793
$206$	6.66411	0.464311
$207$	0	0
$208$	−1.09538	−0.0759508
$209$	−42.1422	−2.91504
$210$	0	0
$211$	11.0929	0.763664	0.381832	−	0.924232i	$-0.375293\pi$
$211$	11.0929	0.763664	0.381832	+	0.924232i	$0.375293\pi$
$212$	3.12512	0.214634
$213$	0	0
$214$	3.19264	0.218245
$215$	16.1975	1.10466
$216$	0	0
$217$	−0.794645	−0.0539440
$218$	0.394505	0.0267192
$219$	0	0
$220$	34.4100	2.31992
$221$	0.289471	0.0194719
$222$	0	0
$223$	−2.19491	−0.146982	−0.0734911	−	0.997296i	$-0.523414\pi$
$223$	−2.19491	−0.146982	−0.0734911	+	0.997296i	$0.523414\pi$
$224$	4.87908	0.325997
$225$	0	0
$226$	3.95114	0.262826
$227$	10.2975	0.683466	0.341733	−	0.939797i	$-0.388986\pi$
$227$	10.2975	0.683466	0.341733	+	0.939797i	$0.388986\pi$
$228$	0	0
$229$	7.13681	0.471614	0.235807	−	0.971800i	$-0.424227\pi$
$229$	7.13681	0.471614	0.235807	+	0.971800i	$0.424227\pi$
$230$	9.98565	0.658434
$231$	0	0
$232$	14.6770	0.963590
$233$	−8.96887	−0.587570	−0.293785	−	0.955871i	$-0.594915\pi$
$233$	−8.96887	−0.587570	−0.293785	+	0.955871i	$0.594915\pi$
$234$	0	0
$235$	−35.9052	−2.34219
$236$	−1.92636	−0.125395
$237$	0	0
$238$	−0.338649	−0.0219514
$239$	12.4853	0.807606	0.403803	−	0.914846i	$-0.367688\pi$
$239$	12.4853	0.807606	0.403803	+	0.914846i	$0.367688\pi$
$240$	0	0
$241$	−20.4848	−1.31954	−0.659771	−	0.751467i	$-0.729347\pi$
$241$	−20.4848	−1.31954	−0.659771	+	0.751467i	$0.729347\pi$
$242$	6.69468	0.430350
$243$	0	0
$244$	−1.97185	−0.126235
$245$	3.87912	0.247828
$246$	0	0
$247$	−3.43228	−0.218391
$248$	1.42941	0.0907676
$249$	0	0
$250$	9.33586	0.590452
$251$	−13.5065	−0.852526	−0.426263	−	0.904599i	$-0.640170\pi$
$251$	−13.5065	−0.852526	−0.426263	+	0.904599i	$0.640170\pi$
$252$	0	0
$253$	−27.0166	−1.69852
$254$	−0.476800	−0.0299171
$255$	0	0
$256$	0.752097	0.0470061
$257$	25.1877	1.57116	0.785582	−	0.618757i	$-0.212364\pi$
$257$	25.1877	1.57116	0.785582	+	0.618757i	$0.212364\pi$
$258$	0	0
$259$	6.57907	0.408804
$260$	2.80253	0.173805
$261$	0	0
$262$	10.1596	0.627661
$263$	−13.6271	−0.840280	−0.420140	−	0.907459i	$-0.638019\pi$
$263$	−13.6271	−0.840280	−0.420140	+	0.907459i	$0.638019\pi$
$264$	0	0
$265$	−6.83872	−0.420099
$266$	4.01540	0.246200
$267$	0	0
$268$	−1.43684	−0.0877690
$269$	−9.37573	−0.571648	−0.285824	−	0.958282i	$-0.592267\pi$
$269$	−9.37573	−0.571648	−0.285824	+	0.958282i	$0.592267\pi$
$270$	0	0
$271$	21.4873	1.30526	0.652629	−	0.757677i	$-0.273666\pi$
$271$	21.4873	1.30526	0.652629	+	0.757677i	$0.273666\pi$
$272$	−1.90892	−0.115745
$273$	0	0
$274$	6.15357	0.371751
$275$	−50.2790	−3.03194
$276$	0	0
$277$	32.0287	1.92442	0.962209	−	0.272311i	$-0.0877879\pi$
$277$	32.0287	1.92442	0.962209	+	0.272311i	$0.0877879\pi$
$278$	6.83552	0.409967
$279$	0	0
$280$	−6.97778	−0.417002
$281$	3.17026	0.189122	0.0945610	−	0.995519i	$-0.469855\pi$
$281$	3.17026	0.189122	0.0945610	+	0.995519i	$0.469855\pi$
$282$	0	0
$283$	3.00390	0.178563	0.0892816	−	0.996006i	$-0.471543\pi$
$283$	3.00390	0.178563	0.0892816	+	0.996006i	$0.471543\pi$
$284$	16.4570	0.976543
$285$	0	0
$286$	0.972414	0.0575000
$287$	2.29873	0.135690
$288$	0	0
$289$	−16.4955	−0.970326
$290$	−15.0911	−0.886182
$291$	0	0
$292$	20.9187	1.22418
$293$	−23.1537	−1.35265	−0.676327	−	0.736602i	$-0.736429\pi$
$293$	−23.1537	−1.35265	−0.676327	+	0.736602i	$0.736429\pi$
$294$	0	0
$295$	4.21546	0.245433
$296$	−11.8345	−0.687864
$297$	0	0
$298$	−9.62723	−0.557691
$299$	−2.20038	−0.127251
$300$	0	0
$301$	4.17556	0.240675
$302$	−4.59858	−0.264618
$303$	0	0
$304$	22.6343	1.29816
$305$	4.31501	0.247077
$306$	0	0
$307$	22.6103	1.29044	0.645219	−	0.763998i	$-0.276766\pi$
$307$	22.6103	1.29044	0.645219	+	0.763998i	$0.276766\pi$
$308$	8.87055	0.505447
$309$	0	0
$310$	−1.46975	−0.0834760
$311$	17.3980	0.986550	0.493275	−	0.869873i	$-0.335800\pi$
$311$	17.3980	0.986550	0.493275	+	0.869873i	$0.335800\pi$
$312$	0	0
$313$	10.0319	0.567036	0.283518	−	0.958967i	$-0.408498\pi$
$313$	10.0319	0.567036	0.283518	+	0.958967i	$0.408498\pi$
$314$	−2.13830	−0.120671
$315$	0	0
$316$	−5.60173	−0.315122
$317$	18.7272	1.05182	0.525911	−	0.850540i	$-0.323725\pi$
$317$	18.7272	1.05182	0.525911	+	0.850540i	$0.323725\pi$
$318$	0	0
$319$	40.8298	2.28603
$320$	−11.8273	−0.661167
$321$	0	0
$322$	2.57420	0.143455
$323$	−5.98145	−0.332817
$324$	0	0
$325$	−4.09499	−0.227149
$326$	−0.150450	−0.00833264
$327$	0	0
$328$	−4.13496	−0.228315
$329$	−9.25600	−0.510300
$330$	0	0
$331$	−21.6572	−1.19039	−0.595193	−	0.803582i	$-0.702925\pi$
$331$	−21.6572	−1.19039	−0.595193	+	0.803582i	$0.702925\pi$
$332$	3.15369	0.173081
$333$	0	0
$334$	4.76344	0.260644
$335$	3.14425	0.171788
$336$	0	0
$337$	−32.9302	−1.79382	−0.896912	−	0.442209i	$-0.854195\pi$
$337$	−32.9302	−1.79382	−0.896912	+	0.442209i	$0.854195\pi$
$338$	−6.11920	−0.332840
$339$	0	0
$340$	4.88398	0.264871
$341$	3.97647	0.215338
$342$	0	0
$343$	1.00000	0.0539949
$344$	−7.51102	−0.404967
$345$	0	0
$346$	−1.81962	−0.0978233
$347$	11.2049	0.601508	0.300754	−	0.953702i	$-0.402762\pi$
$347$	11.2049	0.601508	0.300754	+	0.953702i	$0.402762\pi$
$348$	0	0
$349$	33.7169	1.80482	0.902412	−	0.430874i	$-0.141795\pi$
$349$	33.7169	1.80482	0.902412	+	0.430874i	$0.141795\pi$
$350$	4.79069	0.256073
$351$	0	0
$352$	−24.4153	−1.30134
$353$	−11.0999	−0.590786	−0.295393	−	0.955376i	$-0.595451\pi$
$353$	−11.0999	−0.590786	−0.295393	+	0.955376i	$0.595451\pi$
$354$	0	0
$355$	−36.0129	−1.91137
$356$	−13.1249	−0.695620
$357$	0	0
$358$	−1.24912	−0.0660179
$359$	0.385888	0.0203664	0.0101832	−	0.999948i	$-0.496759\pi$
$359$	0.385888	0.0203664	0.0101832	+	0.999948i	$0.496759\pi$
$360$	0	0
$361$	51.9227	2.73278
$362$	11.2576	0.591689
$363$	0	0
$364$	0.722464	0.0378674
$365$	−45.7765	−2.39605
$366$	0	0
$367$	−28.4165	−1.48333	−0.741664	−	0.670772i	$-0.765963\pi$
$367$	−28.4165	−1.48333	−0.741664	+	0.670772i	$0.765963\pi$
$368$	14.5104	0.756409
$369$	0	0
$370$	12.1684	0.632606
$371$	−1.76295	−0.0915280
$372$	0	0
$373$	13.0502	0.675712	0.337856	−	0.941198i	$-0.390298\pi$
$373$	13.0502	0.675712	0.337856	+	0.941198i	$0.390298\pi$
$374$	1.69463	0.0876272
$375$	0	0
$376$	16.6497	0.858644
$377$	3.32539	0.171266
$378$	0	0
$379$	26.1620	1.34385	0.671926	−	0.740618i	$-0.265467\pi$
$379$	26.1620	1.34385	0.671926	+	0.740618i	$0.265467\pi$
$380$	−57.9098	−2.97071
$381$	0	0
$382$	−1.46564	−0.0749885
$383$	17.7855	0.908798	0.454399	−	0.890798i	$-0.349854\pi$
$383$	17.7855	0.908798	0.454399	+	0.890798i	$0.349854\pi$
$384$	0	0
$385$	−19.4115	−0.989300
$386$	4.80093	0.244361
$387$	0	0
$388$	−3.35553	−0.170351
$389$	−4.85940	−0.246382	−0.123191	−	0.992383i	$-0.539313\pi$
$389$	−4.85940	−0.246382	−0.123191	+	0.992383i	$0.539313\pi$
$390$	0	0
$391$	−3.83461	−0.193924
$392$	−1.79880	−0.0908533
$393$	0	0
$394$	−5.48234	−0.276196
$395$	12.2583	0.616781
$396$	0	0
$397$	14.9858	0.752116	0.376058	−	0.926596i	$-0.377279\pi$
$397$	14.9858	0.752116	0.376058	+	0.926596i	$0.377279\pi$
$398$	2.70746	0.135713
$399$	0	0
$400$	27.0045	1.35022
$401$	−36.5532	−1.82538	−0.912689	−	0.408655i	$-0.865998\pi$
$401$	−36.5532	−1.82538	−0.912689	+	0.408655i	$0.865998\pi$
$402$	0	0
$403$	0.323865	0.0161328
$404$	−34.4674	−1.71482
$405$	0	0
$406$	−3.89035	−0.193075
$407$	−32.9222	−1.63189
$408$	0	0
$409$	−24.9043	−1.23144	−0.615720	−	0.787965i	$-0.711135\pi$
$409$	−24.9043	−1.23144	−0.615720	+	0.787965i	$0.711135\pi$
$410$	4.25164	0.209974
$411$	0	0
$412$	−24.7761	−1.22063
$413$	1.08670	0.0534732
$414$	0	0
$415$	−6.90123	−0.338768
$416$	−1.98851	−0.0974949
$417$	0	0
$418$	−20.0934	−0.982800
$419$	0.748228	0.0365533	0.0182767	−	0.999833i	$-0.494182\pi$
$419$	0.748228	0.0365533	0.0182767	+	0.999833i	$0.494182\pi$
$420$	0	0
$421$	−13.3895	−0.652562	−0.326281	−	0.945273i	$-0.605796\pi$
$421$	−13.3895	−0.652562	−0.326281	+	0.945273i	$0.605796\pi$
$422$	5.28908	0.257468
$423$	0	0
$424$	3.17121	0.154008
$425$	−7.13635	−0.346164
$426$	0	0
$427$	1.11237	0.0538312
$428$	−11.8697	−0.573745
$429$	0	0
$430$	7.72298	0.372435
$431$	−29.1853	−1.40581	−0.702904	−	0.711285i	$-0.748114\pi$
$431$	−29.1853	−1.40581	−0.702904	+	0.711285i	$0.748114\pi$
$432$	0	0
$433$	−0.0171978	−0.000826473 0	−0.000413236	−	1.00000i	$-0.500132\pi$
$433$	−0.0171978	−0.000826473 0	−0.000413236		1.00000i	$0.500132\pi$
$434$	−0.378886	−0.0181871
$435$	0	0
$436$	−1.46670	−0.0702424
$437$	45.4673	2.17500
$438$	0	0
$439$	32.5422	1.55316	0.776578	−	0.630021i	$-0.216954\pi$
$439$	32.5422	1.55316	0.776578	+	0.630021i	$0.216954\pi$
$440$	34.9174	1.66462
$441$	0	0
$442$	0.138020	0.00656492
$443$	18.6179	0.884561	0.442280	−	0.896877i	$-0.354170\pi$
$443$	18.6179	0.884561	0.442280	+	0.896877i	$0.354170\pi$
$444$	0	0
$445$	28.7213	1.36152
$446$	−1.04653	−0.0495548
$447$	0	0
$448$	−3.04897	−0.144050
$449$	4.41081	0.208159	0.104080	−	0.994569i	$-0.466810\pi$
$449$	4.41081	0.208159	0.104080	+	0.994569i	$0.466810\pi$
$450$	0	0
$451$	−11.5030	−0.541656
$452$	−14.6897	−0.690945
$453$	0	0
$454$	4.90982	0.230429
$455$	−1.58097	−0.0741171
$456$	0	0
$457$	31.9405	1.49411	0.747057	−	0.664760i	$-0.231466\pi$
$457$	31.9405	1.49411	0.747057	+	0.664760i	$0.231466\pi$
$458$	3.40283	0.159004
$459$	0	0
$460$	−37.1250	−1.73096
$461$	22.3641	1.04160	0.520799	−	0.853679i	$-0.325634\pi$
$461$	22.3641	1.04160	0.520799	+	0.853679i	$0.325634\pi$
$462$	0	0
$463$	22.7564	1.05758	0.528790	−	0.848753i	$-0.322646\pi$
$463$	22.7564	1.05758	0.528790	+	0.848753i	$0.322646\pi$
$464$	−21.9294	−1.01804
$465$	0	0
$466$	−4.27636	−0.198098
$467$	13.5933	0.629023	0.314512	−	0.949254i	$-0.398159\pi$
$467$	13.5933	0.629023	0.314512	+	0.949254i	$0.398159\pi$
$468$	0	0
$469$	0.810556	0.0374280
$470$	−17.1196	−0.789667
$471$	0	0
$472$	−1.95477	−0.0899754
$473$	−20.8949	−0.960747
$474$	0	0
$475$	84.6165	3.88247
$476$	1.25904	0.0577081
$477$	0	0
$478$	5.95298	0.272283
$479$	0.420777	0.0192258	0.00961289	−	0.999954i	$-0.496940\pi$
$479$	0.420777	0.0192258	0.00961289	+	0.999954i	$0.496940\pi$
$480$	0	0
$481$	−2.68136	−0.122259
$482$	−9.76714	−0.444881
$483$	0	0
$484$	−24.8897	−1.13135
$485$	7.34292	0.333425
$486$	0	0
$487$	7.04291	0.319145	0.159572	−	0.987186i	$-0.448988\pi$
$487$	7.04291	0.319145	0.159572	+	0.987186i	$0.448988\pi$
$488$	−2.00093	−0.0905779
$489$	0	0
$490$	1.84956	0.0835548
$491$	−3.92441	−0.177106	−0.0885530	−	0.996071i	$-0.528224\pi$
$491$	−3.92441	−0.177106	−0.0885530	+	0.996071i	$0.528224\pi$
$492$	0	0
$493$	5.79517	0.261001
$494$	−1.63651	−0.0736302
$495$	0	0
$496$	−2.13573	−0.0958972
$497$	−9.28378	−0.416434
$498$	0	0
$499$	−13.5608	−0.607065	−0.303532	−	0.952821i	$-0.598166\pi$
$499$	−13.5608	−0.607065	−0.303532	+	0.952821i	$0.598166\pi$
$500$	−34.7092	−1.55224
$501$	0	0
$502$	−6.43992	−0.287428
$503$	−1.31965	−0.0588401	−0.0294201	−	0.999567i	$-0.509366\pi$
$503$	−1.31965	−0.0588401	−0.0294201	+	0.999567i	$0.509366\pi$
$504$	0	0
$505$	75.4251	3.35637
$506$	−12.8815	−0.572654
$507$	0	0
$508$	1.77266	0.0786492
$509$	2.94653	0.130602	0.0653012	−	0.997866i	$-0.479199\pi$
$509$	2.94653	0.130602	0.0653012	+	0.997866i	$0.479199\pi$
$510$	0	0
$511$	−11.8007	−0.522034
$512$	22.7824	1.00685
$513$	0	0
$514$	12.0095	0.529715
$515$	54.2175	2.38911
$516$	0	0
$517$	46.3178	2.03705
$518$	3.13690	0.137827
$519$	0	0
$520$	2.84386	0.124711
$521$	−24.2412	−1.06203	−0.531014	−	0.847363i	$-0.678189\pi$
$521$	−24.2412	−1.06203	−0.531014	+	0.847363i	$0.678189\pi$
$522$	0	0
$523$	−13.6601	−0.597315	−0.298658	−	0.954360i	$-0.596539\pi$
$523$	−13.6601	−0.597315	−0.298658	+	0.954360i	$0.596539\pi$
$524$	−37.7716	−1.65006
$525$	0	0
$526$	−6.49738	−0.283299
$527$	0.564400	0.0245857
$528$	0	0
$529$	6.14832	0.267318
$530$	−3.26070	−0.141636
$531$	0	0
$532$	−14.9286	−0.647236
$533$	−0.936867	−0.0405802
$534$	0	0
$535$	25.9746	1.12298
$536$	−1.45803	−0.0629773
$537$	0	0
$538$	−4.47034	−0.192730
$539$	−5.00409	−0.215541
$540$	0	0
$541$	34.6037	1.48773	0.743865	−	0.668330i	$-0.232991\pi$
$541$	34.6037	1.48773	0.743865	+	0.668330i	$0.232991\pi$
$542$	10.2451	0.440066
$543$	0	0
$544$	−3.46539	−0.148577
$545$	3.20959	0.137484
$546$	0	0
$547$	−32.3891	−1.38486	−0.692428	−	0.721487i	$-0.743459\pi$
$547$	−32.3891	−1.38486	−0.692428	+	0.721487i	$0.743459\pi$
$548$	−22.8780	−0.977298
$549$	0	0
$550$	−23.9730	−1.02221
$551$	−68.7140	−2.92731
$552$	0	0
$553$	3.16006	0.134380
$554$	15.2713	0.648814
$555$	0	0
$556$	−25.4133	−1.07777
$557$	−36.2257	−1.53493	−0.767467	−	0.641089i	$-0.778483\pi$
$557$	−36.2257	−1.53493	−0.767467	+	0.641089i	$0.778483\pi$
$558$	0	0
$559$	−1.70179	−0.0719780
$560$	10.4257	0.440568
$561$	0	0
$562$	1.51158	0.0637622
$563$	−1.20547	−0.0508046	−0.0254023	−	0.999677i	$-0.508087\pi$
$563$	−1.20547	−0.0508046	−0.0254023	+	0.999677i	$0.508087\pi$
$564$	0	0
$565$	32.1455	1.35237
$566$	1.43226	0.0602023
$567$	0	0
$568$	16.6997	0.700704
$569$	−30.1557	−1.26419	−0.632095	−	0.774891i	$-0.717805\pi$
$569$	−30.1557	−1.26419	−0.632095	+	0.774891i	$0.717805\pi$
$570$	0	0
$571$	−0.362635	−0.0151758	−0.00758791	−	0.999971i	$-0.502415\pi$
$571$	−0.362635	−0.0151758	−0.00758791	+	0.999971i	$0.502415\pi$
$572$	−3.61527	−0.151162
$573$	0	0
$574$	1.09603	0.0457475
$575$	54.2462	2.26222
$576$	0	0
$577$	−34.0853	−1.41899	−0.709496	−	0.704710i	$-0.751077\pi$
$577$	−34.0853	−1.41899	−0.709496	+	0.704710i	$0.751077\pi$
$578$	−7.86507	−0.327144
$579$	0	0
$580$	56.1063	2.32969
$581$	−1.77907	−0.0738082
$582$	0	0
$583$	8.82197	0.365369
$584$	21.2272	0.878389
$585$	0	0
$586$	−11.0397	−0.456045
$587$	35.1456	1.45062	0.725308	−	0.688425i	$-0.241698\pi$
$587$	35.1456	1.45062	0.725308	+	0.688425i	$0.241698\pi$
$588$	0	0
$589$	−6.69215	−0.275745
$590$	2.00993	0.0827474
$591$	0	0
$592$	17.6823	0.726737
$593$	22.7973	0.936172	0.468086	−	0.883683i	$-0.344944\pi$
$593$	22.7973	0.936172	0.468086	+	0.883683i	$0.344944\pi$
$594$	0	0
$595$	−2.75517	−0.112951
$596$	35.7924	1.46612
$597$	0	0
$598$	−1.04914	−0.0429025
$599$	24.3667	0.995595	0.497797	−	0.867293i	$-0.334142\pi$
$599$	24.3667	0.995595	0.497797	+	0.867293i	$0.334142\pi$
$600$	0	0
$601$	12.8415	0.523818	0.261909	−	0.965093i	$-0.415648\pi$
$601$	12.8415	0.523818	0.261909	+	0.965093i	$0.415648\pi$
$602$	1.99091	0.0811433
$603$	0	0
$604$	17.0967	0.695657
$605$	54.4663	2.21437
$606$	0	0
$607$	−21.5300	−0.873875	−0.436938	−	0.899492i	$-0.643937\pi$
$607$	−21.5300	−0.873875	−0.436938	+	0.899492i	$0.643937\pi$
$608$	41.0895	1.66640
$609$	0	0
$610$	2.05740	0.0833015
$611$	3.77237	0.152614
$612$	0	0
$613$	−10.1695	−0.410743	−0.205372	−	0.978684i	$-0.565840\pi$
$613$	−10.1695	−0.410743	−0.205372	+	0.978684i	$0.565840\pi$
$614$	10.7806	0.435069
$615$	0	0
$616$	9.00137	0.362675
$617$	−31.1448	−1.25384	−0.626921	−	0.779083i	$-0.715685\pi$
$617$	−31.1448	−1.25384	−0.626921	+	0.779083i	$0.715685\pi$
$618$	0	0
$619$	−13.7319	−0.551933	−0.275967	−	0.961167i	$-0.588998\pi$
$619$	−13.7319	−0.551933	−0.275967	+	0.961167i	$0.588998\pi$
$620$	5.46428	0.219451
$621$	0	0
$622$	8.29536	0.332614
$623$	7.40408	0.296638
$624$	0	0
$625$	25.7162	1.02865
$626$	4.78320	0.191175
$627$	0	0
$628$	7.94984	0.317233
$629$	−4.67282	−0.186317
$630$	0	0
$631$	−20.6765	−0.823118	−0.411559	−	0.911383i	$-0.635016\pi$
$631$	−20.6765	−0.823118	−0.411559	+	0.911383i	$0.635016\pi$
$632$	−5.68434	−0.226111
$633$	0	0
$634$	8.92910	0.354620
$635$	−3.87912	−0.153938
$636$	0	0
$637$	−0.407559	−0.0161481
$638$	19.4676	0.770731
$639$	0	0
$640$	−43.4924	−1.71919
$641$	−11.4317	−0.451525	−0.225763	−	0.974182i	$-0.572487\pi$
$641$	−11.4317	−0.451525	−0.225763	+	0.974182i	$0.572487\pi$
$642$	0	0
$643$	10.0657	0.396951	0.198475	−	0.980106i	$-0.436401\pi$
$643$	10.0657	0.396951	0.198475	+	0.980106i	$0.436401\pi$
$644$	−9.57046	−0.377129
$645$	0	0
$646$	−2.85196	−0.112209
$647$	−3.59533	−0.141347	−0.0706735	−	0.997500i	$-0.522515\pi$
$647$	−3.59533	−0.141347	−0.0706735	+	0.997500i	$0.522515\pi$
$648$	0	0
$649$	−5.43796	−0.213458
$650$	−1.95249	−0.0765830
$651$	0	0
$652$	0.559347	0.0219057
$653$	−34.9356	−1.36714	−0.683569	−	0.729886i	$-0.739573\pi$
$653$	−34.9356	−1.36714	−0.683569	+	0.729886i	$0.739573\pi$
$654$	0	0
$655$	82.6558	3.22963
$656$	6.17818	0.241218
$657$	0	0
$658$	−4.41326	−0.172047
$659$	11.3713	0.442965	0.221482	−	0.975164i	$-0.428910\pi$
$659$	11.3713	0.442965	0.221482	+	0.975164i	$0.428910\pi$
$660$	0	0
$661$	10.4734	0.407366	0.203683	−	0.979037i	$-0.434709\pi$
$661$	10.4734	0.407366	0.203683	+	0.979037i	$0.434709\pi$
$662$	−10.3261	−0.401337
$663$	0	0
$664$	3.20020	0.124192
$665$	32.6683	1.26682
$666$	0	0
$667$	−44.0513	−1.70567
$668$	−17.7097	−0.685209
$669$	0	0
$670$	1.49918	0.0579182
$671$	−5.56638	−0.214888
$672$	0	0
$673$	−7.72784	−0.297886	−0.148943	−	0.988846i	$-0.547587\pi$
$673$	−7.72784	−0.297886	−0.148943	+	0.988846i	$0.547587\pi$
$674$	−15.7011	−0.604785
$675$	0	0
$676$	22.7502	0.875006
$677$	47.4205	1.82252	0.911260	−	0.411832i	$-0.135111\pi$
$677$	47.4205	1.82252	0.911260	+	0.411832i	$0.135111\pi$
$678$	0	0
$679$	1.89293	0.0726440
$680$	4.95600	0.190054
$681$	0	0
$682$	1.89598	0.0726008
$683$	25.1455	0.962167	0.481084	−	0.876675i	$-0.340243\pi$
$683$	25.1455	0.962167	0.481084	+	0.876675i	$0.340243\pi$
$684$	0	0
$685$	50.0639	1.91284
$686$	0.476800	0.0182043
$687$	0	0
$688$	11.2225	0.427853
$689$	0.718508	0.0273730
$690$	0	0
$691$	−38.2318	−1.45441	−0.727203	−	0.686422i	$-0.759180\pi$
$691$	−38.2318	−1.45441	−0.727203	+	0.686422i	$0.759180\pi$
$692$	6.76504	0.257168
$693$	0	0
$694$	5.34247	0.202798
$695$	55.6121	2.10949
$696$	0	0
$697$	−1.63268	−0.0618422
$698$	16.0762	0.608493
$699$	0	0
$700$	−17.8110	−0.673192
$701$	−39.4939	−1.49166	−0.745832	−	0.666134i	$-0.767948\pi$
$701$	−39.4939	−1.49166	−0.745832	+	0.666134i	$0.767948\pi$
$702$	0	0
$703$	55.4061	2.08968
$704$	15.2573	0.575031
$705$	0	0
$706$	−5.29241	−0.199182
$707$	19.4439	0.731261
$708$	0	0
$709$	13.2635	0.498120	0.249060	−	0.968488i	$-0.419878\pi$
$709$	13.2635	0.498120	0.249060	+	0.968488i	$0.419878\pi$
$710$	−17.1709	−0.644414
$711$	0	0
$712$	−13.3185	−0.499131
$713$	−4.29022	−0.160670
$714$	0	0
$715$	7.91132	0.295867
$716$	4.64401	0.173555
$717$	0	0
$718$	0.183991	0.00686650
$719$	4.92349	0.183615	0.0918075	−	0.995777i	$-0.470736\pi$
$719$	4.92349	0.183615	0.0918075	+	0.995777i	$0.470736\pi$
$720$	0	0
$721$	13.9767	0.520521
$722$	24.7567	0.921351
$723$	0	0
$724$	−41.8541	−1.55549
$725$	−81.9813	−3.04471
$726$	0	0
$727$	−26.1568	−0.970101	−0.485050	−	0.874486i	$-0.661199\pi$
$727$	−26.1568	−0.970101	−0.485050	+	0.874486i	$0.661199\pi$
$728$	0.733119	0.0271712
$729$	0	0
$730$	−21.8262	−0.807825
$731$	−2.96571	−0.109691
$732$	0	0
$733$	−20.2501	−0.747955	−0.373977	−	0.927438i	$-0.622006\pi$
$733$	−20.2501	−0.747955	−0.373977	+	0.927438i	$0.622006\pi$
$734$	−13.5490	−0.500101
$735$	0	0
$736$	26.3418	0.970970
$737$	−4.05609	−0.149408
$738$	0	0
$739$	−16.4553	−0.605316	−0.302658	−	0.953099i	$-0.597874\pi$
$739$	−16.4553	−0.605316	−0.302658	+	0.953099i	$0.597874\pi$
$740$	−45.2402	−1.66306
$741$	0	0
$742$	−0.840576	−0.0308585
$743$	33.0398	1.21211	0.606056	−	0.795422i	$-0.292751\pi$
$743$	33.0398	1.21211	0.606056	+	0.795422i	$0.292751\pi$
$744$	0	0
$745$	−78.3248	−2.86960
$746$	6.22232	0.227815
$747$	0	0
$748$	−6.30035	−0.230364
$749$	6.69599	0.244666
$750$	0	0
$751$	−9.64889	−0.352093	−0.176047	−	0.984382i	$-0.556331\pi$
$751$	−9.64889	−0.352093	−0.176047	+	0.984382i	$0.556331\pi$
$752$	−24.8769	−0.907168
$753$	0	0
$754$	1.58555	0.0577422
$755$	−37.4129	−1.36159
$756$	0	0
$757$	−24.8217	−0.902159	−0.451079	−	0.892484i	$-0.648961\pi$
$757$	−24.8217	−0.902159	−0.451079	+	0.892484i	$0.648961\pi$
$758$	12.4740	0.453077
$759$	0	0
$760$	−58.7638	−2.13159
$761$	−6.25290	−0.226668	−0.113334	−	0.993557i	$-0.536153\pi$
$761$	−6.25290	−0.226668	−0.113334	+	0.993557i	$0.536153\pi$
$762$	0	0
$763$	0.827402	0.0299540
$764$	5.44900	0.197138
$765$	0	0
$766$	8.48014	0.306400
$767$	−0.442896	−0.0159920
$768$	0	0
$769$	23.5097	0.847782	0.423891	−	0.905713i	$-0.360664\pi$
$769$	23.5097	0.847782	0.423891	+	0.905713i	$0.360664\pi$
$770$	−9.25538	−0.333541
$771$	0	0
$772$	−17.8491	−0.642402
$773$	17.0244	0.612326	0.306163	−	0.951979i	$-0.400955\pi$
$773$	17.0244	0.612326	0.306163	+	0.951979i	$0.400955\pi$
$774$	0	0
$775$	−7.98427	−0.286804
$776$	−3.40501	−0.122233
$777$	0	0
$778$	−2.31696	−0.0830671
$779$	19.3589	0.693604
$780$	0	0
$781$	46.4568	1.66236
$782$	−1.82834	−0.0653813
$783$	0	0
$784$	2.68765	0.0959877
$785$	−17.3967	−0.620914
$786$	0	0
$787$	−34.5601	−1.23193	−0.615966	−	0.787772i	$-0.711234\pi$
$787$	−34.5601	−1.23193	−0.615966	+	0.787772i	$0.711234\pi$
$788$	20.3824	0.726094
$789$	0	0
$790$	5.84474	0.207947
$791$	8.28680	0.294645
$792$	0	0
$793$	−0.453355	−0.0160991
$794$	7.14523	0.253575
$795$	0	0
$796$	−10.0659	−0.356775
$797$	−9.18832	−0.325467	−0.162734	−	0.986670i	$-0.552031\pi$
$797$	−9.18832	−0.325467	−0.162734	+	0.986670i	$0.552031\pi$
$798$	0	0
$799$	6.57411	0.232575
$800$	49.0230	1.73323
$801$	0	0
$802$	−17.4285	−0.615423
$803$	59.0519	2.08390
$804$	0	0
$805$	20.9431	0.738146
$806$	0.154419	0.00543916
$807$	0	0
$808$	−34.9757	−1.23044
$809$	52.9283	1.86086	0.930429	−	0.366472i	$-0.119434\pi$
$809$	52.9283	1.86086	0.930429	+	0.366472i	$0.119434\pi$
$810$	0	0
$811$	55.1436	1.93635	0.968177	−	0.250268i	$-0.0805188\pi$
$811$	55.1436	1.93635	0.968177	+	0.250268i	$0.0805188\pi$
$812$	14.4637	0.507575
$813$	0	0
$814$	−15.6973	−0.550190
$815$	−1.22402	−0.0428756
$816$	0	0
$817$	35.1648	1.23026
$818$	−11.8744	−0.415178
$819$	0	0
$820$	−15.8069	−0.552001
$821$	−0.850062	−0.0296674	−0.0148337	−	0.999890i	$-0.504722\pi$
$821$	−0.850062	−0.0296674	−0.0148337	+	0.999890i	$0.504722\pi$
$822$	0	0
$823$	−27.0436	−0.942680	−0.471340	−	0.881952i	$-0.656230\pi$
$823$	−27.0436	−0.942680	−0.471340	+	0.881952i	$0.656230\pi$
$824$	−25.1414	−0.875843
$825$	0	0
$826$	0.518140	0.0180284
$827$	−46.5194	−1.61764	−0.808819	−	0.588058i	$-0.799893\pi$
$827$	−46.5194	−1.61764	−0.808819	+	0.588058i	$0.799893\pi$
$828$	0	0
$829$	6.39637	0.222155	0.111078	−	0.993812i	$-0.464570\pi$
$829$	6.39637	0.222155	0.111078	+	0.993812i	$0.464570\pi$
$830$	−3.29050	−0.114215
$831$	0	0
$832$	1.24263	0.0430806
$833$	−0.710255	−0.0246089
$834$	0	0
$835$	38.7542	1.34114
$836$	74.7039	2.58369
$837$	0	0
$838$	0.356755	0.0123239
$839$	−13.0104	−0.449169	−0.224584	−	0.974455i	$-0.572102\pi$
$839$	−13.0104	−0.449169	−0.224584	+	0.974455i	$0.572102\pi$
$840$	0	0
$841$	37.5740	1.29565
$842$	−6.38409	−0.220010
$843$	0	0
$844$	−19.6639	−0.676859
$845$	−49.7843	−1.71263
$846$	0	0
$847$	14.0409	0.482450
$848$	−4.73821	−0.162711
$849$	0	0
$850$	−3.40261	−0.116709
$851$	35.5199	1.21761
$852$	0	0
$853$	−48.1861	−1.64986	−0.824930	−	0.565234i	$-0.808786\pi$
$853$	−48.1861	−1.64986	−0.824930	+	0.565234i	$0.808786\pi$
$854$	0.530376	0.0181491
$855$	0	0
$856$	−12.0448	−0.411682
$857$	24.7884	0.846756	0.423378	−	0.905953i	$-0.360844\pi$
$857$	24.7884	0.846756	0.423378	+	0.905953i	$0.360844\pi$
$858$	0	0
$859$	−3.20536	−0.109365	−0.0546827	−	0.998504i	$-0.517415\pi$
$859$	−3.20536	−0.109365	−0.0546827	+	0.998504i	$0.517415\pi$
$860$	−28.7127	−0.979096
$861$	0	0
$862$	−13.9156	−0.473966
$863$	30.4040	1.03496	0.517482	−	0.855694i	$-0.326869\pi$
$863$	30.4040	1.03496	0.517482	+	0.855694i	$0.326869\pi$
$864$	0	0
$865$	−14.8040	−0.503350
$866$	−0.00819990	−0.000278644 0
$867$	0	0
$868$	1.40864	0.0478122
$869$	−15.8132	−0.536427
$870$	0	0
$871$	−0.330349	−0.0111935
$872$	−1.48833	−0.0504013
$873$	0	0
$874$	21.6788	0.733297
$875$	19.5803	0.661934
$876$	0	0
$877$	−31.0743	−1.04930	−0.524652	−	0.851317i	$-0.675805\pi$
$877$	−31.0743	−1.04930	−0.524652	+	0.851317i	$0.675805\pi$
$878$	15.5161	0.523644
$879$	0	0
$880$	−52.1713	−1.75869
$881$	−35.7896	−1.20578	−0.602890	−	0.797824i	$-0.705984\pi$
$881$	−35.7896	−1.20578	−0.602890	+	0.797824i	$0.705984\pi$
$882$	0	0
$883$	47.5463	1.60006	0.800030	−	0.599959i	$-0.204817\pi$
$883$	47.5463	1.60006	0.800030	+	0.599959i	$0.204817\pi$
$884$	−0.513134	−0.0172586
$885$	0	0
$886$	8.87699	0.298228
$887$	52.6046	1.76629	0.883146	−	0.469099i	$-0.155421\pi$
$887$	52.6046	1.76629	0.883146	+	0.469099i	$0.155421\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	13.6943	0.459035
$891$	0	0
$892$	3.89084	0.130275
$893$	−77.9500	−2.60850
$894$	0	0
$895$	−10.1625	−0.339695
$896$	−11.2119	−0.374564
$897$	0	0
$898$	2.10307	0.0701805
$899$	6.48374	0.216245
$900$	0	0
$901$	1.25215	0.0417150
$902$	−5.48464	−0.182618
$903$	0	0
$904$	−14.9063	−0.495777
$905$	91.5894	3.04453
$906$	0	0
$907$	−43.8180	−1.45495	−0.727477	−	0.686132i	$-0.759307\pi$
$907$	−43.8180	−1.45495	−0.727477	+	0.686132i	$0.759307\pi$
$908$	−18.2539	−0.605777
$909$	0	0
$910$	−0.753807	−0.0249885
$911$	−32.7123	−1.08381	−0.541903	−	0.840441i	$-0.682296\pi$
$911$	−32.7123	−1.08381	−0.541903	+	0.840441i	$0.682296\pi$
$912$	0	0
$913$	8.90261	0.294633
$914$	15.2292	0.503738
$915$	0	0
$916$	−12.6512	−0.418006
$917$	21.3079	0.703648
$918$	0	0
$919$	−48.6840	−1.60594	−0.802969	−	0.596021i	$-0.796748\pi$
$919$	−48.6840	−1.60594	−0.802969	+	0.596021i	$0.796748\pi$
$920$	−37.6725	−1.24202
$921$	0	0
$922$	10.6632	0.351173
$923$	3.78369	0.124542
$924$	0	0
$925$	66.1039	2.17348
$926$	10.8502	0.356561
$927$	0	0
$928$	−39.8098	−1.30682
$929$	37.8583	1.24209	0.621045	−	0.783775i	$-0.286708\pi$
$929$	37.8583	1.24209	0.621045	+	0.783775i	$0.286708\pi$
$930$	0	0
$931$	8.42156	0.276006
$932$	15.8988	0.520782
$933$	0	0
$934$	6.48128	0.212074
$935$	13.7871	0.450886
$936$	0	0
$937$	−11.8499	−0.387121	−0.193560	−	0.981088i	$-0.562004\pi$
$937$	−11.8499	−0.387121	−0.193560	+	0.981088i	$0.562004\pi$
$938$	0.386473	0.0126188
$939$	0	0
$940$	63.6477	2.07596
$941$	8.12653	0.264917	0.132459	−	0.991189i	$-0.457713\pi$
$941$	8.12653	0.264917	0.132459	+	0.991189i	$0.457713\pi$
$942$	0	0
$943$	12.4106	0.404146
$944$	2.92068	0.0950602
$945$	0	0
$946$	−9.96267	−0.323914
$947$	−15.4114	−0.500802	−0.250401	−	0.968142i	$-0.580562\pi$
$947$	−15.4114	−0.500802	−0.250401	+	0.968142i	$0.580562\pi$
$948$	0	0
$949$	4.80950	0.156123
$950$	40.3451	1.30897
$951$	0	0
$952$	1.27761	0.0414075
$953$	−42.9768	−1.39216	−0.696078	−	0.717966i	$-0.745073\pi$
$953$	−42.9768	−1.39216	−0.696078	+	0.717966i	$0.745073\pi$
$954$	0	0
$955$	−11.9241	−0.385854
$956$	−22.1322	−0.715807
$957$	0	0
$958$	0.200626	0.00648194
$959$	12.9060	0.416756
$960$	0	0
$961$	−30.3685	−0.979630
$962$	−1.27847	−0.0412196
$963$	0	0
$964$	36.3126	1.16955
$965$	39.0592	1.25736
$966$	0	0
$967$	34.6077	1.11291	0.556454	−	0.830879i	$-0.312162\pi$
$967$	34.6077	1.11291	0.556454	+	0.830879i	$0.312162\pi$
$968$	−25.2568	−0.811783
$969$	0	0
$970$	3.50110	0.112414
$971$	2.44650	0.0785118	0.0392559	−	0.999229i	$-0.487501\pi$
$971$	2.44650	0.0785118	0.0392559	+	0.999229i	$0.487501\pi$
$972$	0	0
$973$	14.3363	0.459599
$974$	3.35806	0.107599
$975$	0	0
$976$	2.98966	0.0956967
$977$	−46.9898	−1.50334	−0.751668	−	0.659542i	$-0.770750\pi$
$977$	−46.9898	−1.50334	−0.751668	+	0.659542i	$0.770750\pi$
$978$	0	0
$979$	−37.0506	−1.18414
$980$	−6.87638	−0.219658
$981$	0	0
$982$	−1.87116	−0.0597110
$983$	3.47547	0.110850	0.0554251	−	0.998463i	$-0.482349\pi$
$983$	3.47547	0.110850	0.0554251	+	0.998463i	$0.482349\pi$
$984$	0	0
$985$	−44.6030	−1.42117
$986$	2.76314	0.0879962
$987$	0	0
$988$	6.08428	0.193567
$989$	22.5435	0.716842
$990$	0	0
$991$	−3.72105	−0.118203	−0.0591015	−	0.998252i	$-0.518824\pi$
$991$	−3.72105	−0.118203	−0.0591015	+	0.998252i	$0.518824\pi$
$992$	−3.87714	−0.123099
$993$	0	0
$994$	−4.42650	−0.140400
$995$	22.0272	0.698309
$996$	0	0
$997$	26.8809	0.851327	0.425664	−	0.904881i	$-0.360041\pi$
$997$	26.8809	0.851327	0.425664	+	0.904881i	$0.360041\pi$
$998$	−6.46579	−0.204671
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.ba.1.24	yes	40
3.2	odd	2	inner	8001.2.a.ba.1.17	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8001.2.a.ba.1.17	✓	40	3.2	odd	2	inner
8001.2.a.ba.1.24	yes	40	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:	\( N \)	\(=\)	\( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8001.a (trivial)

\(f(q)\)	\(=\)	\(q+0.476800 q^{2} -1.77266 q^{4} +3.87912 q^{5} +1.00000 q^{7} -1.79880 q^{8} +O(q^{10})\)	\(q+0.476800 q^{2} -1.77266 q^{4} +3.87912 q^{5} +1.00000 q^{7} -1.79880 q^{8} +1.84956 q^{10} -5.00409 q^{11} -0.407559 q^{13} +0.476800 q^{14} +2.68765 q^{16} -0.710255 q^{17} +8.42156 q^{19} -6.87638 q^{20} -2.38595 q^{22} +5.39892 q^{23} +10.0476 q^{25} -0.194324 q^{26} -1.77266 q^{28} -8.15929 q^{29} -0.794645 q^{31} +4.87908 q^{32} -0.338649 q^{34} +3.87912 q^{35} +6.57907 q^{37} +4.01540 q^{38} -6.97778 q^{40} +2.29873 q^{41} +4.17556 q^{43} +8.87055 q^{44} +2.57420 q^{46} -9.25600 q^{47} +1.00000 q^{49} +4.79069 q^{50} +0.722464 q^{52} -1.76295 q^{53} -19.4115 q^{55} -1.79880 q^{56} -3.89035 q^{58} +1.08670 q^{59} +1.11237 q^{61} -0.378886 q^{62} -3.04897 q^{64} -1.58097 q^{65} +0.810556 q^{67} +1.25904 q^{68} +1.84956 q^{70} -9.28378 q^{71} -11.8007 q^{73} +3.13690 q^{74} -14.9286 q^{76} -5.00409 q^{77} +3.16006 q^{79} +10.4257 q^{80} +1.09603 q^{82} -1.77907 q^{83} -2.75517 q^{85} +1.99091 q^{86} +9.00137 q^{88} +7.40408 q^{89} -0.407559 q^{91} -9.57046 q^{92} -4.41326 q^{94} +32.6683 q^{95} +1.89293 q^{97} +0.476800 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 54 q^{4} + 40 q^{7}+O(q^{10}) \) 40 * q + 54 * q^4 + 40 * q^7	\( 40 q + 54 q^{4} + 40 q^{7} + 20 q^{10} + 10 q^{13} + 90 q^{16} + 38 q^{19} + 14 q^{22} + 84 q^{25} + 54 q^{28} + 66 q^{31} + 22 q^{34} + 40 q^{37} + 26 q^{40} + 38 q^{43} + 28 q^{46} + 40 q^{49} + 28 q^{52} + 60 q^{55} + 42 q^{58} + 54 q^{61} + 124 q^{64} + 48 q^{67} + 20 q^{70} + 16 q^{76} + 102 q^{79} + 48 q^{82} + 104 q^{85} + 48 q^{88} + 10 q^{91} - 10 q^{94} + 8 q^{97}+O(q^{100}) \) 40 * q + 54 * q^4 + 40 * q^7 + 20 * q^10 + 10 * q^13 + 90 * q^16 + 38 * q^19 + 14 * q^22 + 84 * q^25 + 54 * q^28 + 66 * q^31 + 22 * q^34 + 40 * q^37 + 26 * q^40 + 38 * q^43 + 28 * q^46 + 40 * q^49 + 28 * q^52 + 60 * q^55 + 42 * q^58 + 54 * q^61 + 124 * q^64 + 48 * q^67 + 20 * q^70 + 16 * q^76 + 102 * q^79 + 48 * q^82 + 104 * q^85 + 48 * q^88 + 10 * q^91 - 10 * q^94 + 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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