LMFDB - Embedded newform 8001.2.a.y.1.12

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:	$28$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
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)

$f(q)$	$=$	$q-0.512239 q^{2} -1.73761 q^{4} +1.76990 q^{5} -1.00000 q^{7} +1.91455 q^{8} +O(q^{10})$	\(q-0.512239 q^{2} -1.73761 q^{4} +1.76990 q^{5} -1.00000 q^{7} +1.91455 q^{8} -0.906610 q^{10} -4.52139 q^{11} +4.50554 q^{13} +0.512239 q^{14} +2.49452 q^{16} +5.81738 q^{17} +6.77727 q^{19} -3.07539 q^{20} +2.31603 q^{22} -0.708155 q^{23} -1.86746 q^{25} -2.30791 q^{26} +1.73761 q^{28} +8.00119 q^{29} +8.44906 q^{31} -5.10689 q^{32} -2.97989 q^{34} -1.76990 q^{35} +1.40518 q^{37} -3.47158 q^{38} +3.38856 q^{40} -3.06668 q^{41} -3.92263 q^{43} +7.85641 q^{44} +0.362744 q^{46} +9.77893 q^{47} +1.00000 q^{49} +0.956586 q^{50} -7.82888 q^{52} -10.7565 q^{53} -8.00239 q^{55} -1.91455 q^{56} -4.09852 q^{58} +10.1623 q^{59} -9.39913 q^{61} -4.32794 q^{62} -2.37309 q^{64} +7.97434 q^{65} -9.67794 q^{67} -10.1083 q^{68} +0.906610 q^{70} +7.71069 q^{71} -14.8339 q^{73} -0.719789 q^{74} -11.7763 q^{76} +4.52139 q^{77} +13.3576 q^{79} +4.41504 q^{80} +1.57087 q^{82} -15.4133 q^{83} +10.2962 q^{85} +2.00932 q^{86} -8.65641 q^{88} -4.77308 q^{89} -4.50554 q^{91} +1.23050 q^{92} -5.00914 q^{94} +11.9951 q^{95} +10.7708 q^{97} -0.512239 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 30 q^{4} - 28 q^{7}+O(q^{10}) $ 28 * q + 30 * q^4 - 28 * q^7	$ 28 q + 30 q^{4} - 28 q^{7} + 4 q^{10} + 8 q^{13} + 42 q^{16} + 34 q^{19} - 10 q^{22} + 14 q^{25} - 30 q^{28} + 56 q^{31} - 6 q^{37} + 38 q^{40} + 18 q^{43} + 16 q^{46} + 28 q^{49} + 18 q^{52} + 48 q^{55} + 2 q^{58} + 36 q^{61} + 76 q^{64} - 4 q^{70} + 50 q^{73} + 132 q^{76} + 66 q^{79} - 36 q^{82} + 20 q^{85} + 6 q^{88} - 8 q^{91} + 54 q^{94} - 8 q^{97}+O(q^{100}) $ 28 * q + 30 * q^4 - 28 * q^7 + 4 * q^10 + 8 * q^13 + 42 * q^16 + 34 * q^19 - 10 * q^22 + 14 * q^25 - 30 * q^28 + 56 * q^31 - 6 * q^37 + 38 * q^40 + 18 * q^43 + 16 * q^46 + 28 * q^49 + 18 * q^52 + 48 * q^55 + 2 * q^58 + 36 * q^61 + 76 * q^64 - 4 * q^70 + 50 * q^73 + 132 * q^76 + 66 * q^79 - 36 * q^82 + 20 * q^85 + 6 * q^88 - 8 * q^91 + 54 * 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 $

$2$	−0.512239	−0.362207	−0.181104	−	0.983464i	$-0.557967\pi$
$2$	−0.512239	−0.362207	−0.181104	+	0.983464i	$0.557967\pi$
$3$	0	0
$3$	0	0
$4$	−1.73761	−0.868806
$5$	1.76990	0.791522	0.395761	−	0.918353i	$-0.370481\pi$
$5$	1.76990	0.791522	0.395761	+	0.918353i	$0.370481\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.91455	0.676895
$9$	0	0
$10$	−0.906610	−0.286695
$11$	−4.52139	−1.36325	−0.681624	−	0.731702i	$-0.738726\pi$
$11$	−4.52139	−1.36325	−0.681624	+	0.731702i	$0.738726\pi$
$12$	0	0
$13$	4.50554	1.24961	0.624806	−	0.780780i	$-0.285178\pi$
$13$	4.50554	1.24961	0.624806	+	0.780780i	$0.285178\pi$
$14$	0.512239	0.136902
$15$	0	0
$16$	2.49452	0.623629
$17$	5.81738	1.41092	0.705461	−	0.708749i	$-0.250740\pi$
$17$	5.81738	1.41092	0.705461	+	0.708749i	$0.250740\pi$
$18$	0	0
$19$	6.77727	1.55481	0.777406	−	0.628999i	$-0.216535\pi$
$19$	6.77727	1.55481	0.777406	+	0.628999i	$0.216535\pi$
$20$	−3.07539	−0.687679
$21$	0	0
$22$	2.31603	0.493779
$23$	−0.708155	−0.147660	−0.0738302	−	0.997271i	$-0.523522\pi$
$23$	−0.708155	−0.147660	−0.0738302	+	0.997271i	$0.523522\pi$
$24$	0	0
$25$	−1.86746	−0.373492
$26$	−2.30791	−0.452619
$27$	0	0
$28$	1.73761	0.328378
$29$	8.00119	1.48578	0.742892	−	0.669412i	$-0.233454\pi$
$29$	8.00119	1.48578	0.742892	+	0.669412i	$0.233454\pi$
$30$	0	0
$31$	8.44906	1.51750	0.758748	−	0.651384i	$-0.225811\pi$
$31$	8.44906	1.51750	0.758748	+	0.651384i	$0.225811\pi$
$32$	−5.10689	−0.902778
$33$	0	0
$34$	−2.97989	−0.511046
$35$	−1.76990	−0.299167
$36$	0	0
$37$	1.40518	0.231011	0.115505	−	0.993307i	$-0.463151\pi$
$37$	1.40518	0.231011	0.115505	+	0.993307i	$0.463151\pi$
$38$	−3.47158	−0.563165
$39$	0	0
$40$	3.38856	0.535778
$41$	−3.06668	−0.478936	−0.239468	−	0.970904i	$-0.576973\pi$
$41$	−3.06668	−0.478936	−0.239468	+	0.970904i	$0.576973\pi$
$42$	0	0
$43$	−3.92263	−0.598196	−0.299098	−	0.954222i	$-0.596686\pi$
$43$	−3.92263	−0.598196	−0.299098	+	0.954222i	$0.596686\pi$
$44$	7.85641	1.18440
$45$	0	0
$46$	0.362744	0.0534837
$47$	9.77893	1.42640	0.713201	−	0.700959i	$-0.247244\pi$
$47$	9.77893	1.42640	0.713201	+	0.700959i	$0.247244\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0.956586	0.135282
$51$	0	0
$52$	−7.82888	−1.08567
$53$	−10.7565	−1.47752	−0.738758	−	0.673971i	$-0.764587\pi$
$53$	−10.7565	−1.47752	−0.738758	+	0.673971i	$0.764587\pi$
$54$	0	0
$55$	−8.00239	−1.07904
$56$	−1.91455	−0.255842
$57$	0	0
$58$	−4.09852	−0.538162
$59$	10.1623	1.32302	0.661508	−	0.749938i	$-0.269917\pi$
$59$	10.1623	1.32302	0.661508	+	0.749938i	$0.269917\pi$
$60$	0	0
$61$	−9.39913	−1.20344	−0.601718	−	0.798709i	$-0.705517\pi$
$61$	−9.39913	−1.20344	−0.601718	+	0.798709i	$0.705517\pi$
$62$	−4.32794	−0.549649
$63$	0	0
$64$	−2.37309	−0.296636
$65$	7.97434	0.989095
$66$	0	0
$67$	−9.67794	−1.18235	−0.591174	−	0.806544i	$-0.701335\pi$
$67$	−9.67794	−1.18235	−0.591174	+	0.806544i	$0.701335\pi$
$68$	−10.1083	−1.22582
$69$	0	0
$70$	0.906610	0.108361
$71$	7.71069	0.915091	0.457545	−	0.889186i	$-0.348729\pi$
$71$	7.71069	0.915091	0.457545	+	0.889186i	$0.348729\pi$
$72$	0	0
$73$	−14.8339	−1.73617	−0.868087	−	0.496411i	$-0.834651\pi$
$73$	−14.8339	−1.73617	−0.868087	+	0.496411i	$0.834651\pi$
$74$	−0.719789	−0.0836738
$75$	0	0
$76$	−11.7763	−1.35083
$77$	4.52139	0.515260
$78$	0	0
$79$	13.3576	1.50284	0.751421	−	0.659823i	$-0.229369\pi$
$79$	13.3576	1.50284	0.751421	+	0.659823i	$0.229369\pi$
$80$	4.41504	0.493616
$81$	0	0
$82$	1.57087	0.173474
$83$	−15.4133	−1.69183	−0.845916	−	0.533316i	$-0.820946\pi$
$83$	−15.4133	−1.69183	−0.845916	+	0.533316i	$0.820946\pi$
$84$	0	0
$85$	10.2962	1.11678
$86$	2.00932	0.216671
$87$	0	0
$88$	−8.65641	−0.922777
$89$	−4.77308	−0.505946	−0.252973	−	0.967473i	$-0.581408\pi$
$89$	−4.77308	−0.505946	−0.252973	+	0.967473i	$0.581408\pi$
$90$	0	0
$91$	−4.50554	−0.472309
$92$	1.23050	0.128288
$93$	0	0
$94$	−5.00914	−0.516654
$95$	11.9951	1.23067
$96$	0	0
$97$	10.7708	1.09360	0.546802	−	0.837262i	$-0.315845\pi$
$97$	10.7708	1.09360	0.546802	+	0.837262i	$0.315845\pi$
$98$	−0.512239	−0.0517439
$99$	0	0
$100$	3.24492	0.324492
$101$	5.91868	0.588931	0.294465	−	0.955662i	$-0.404858\pi$
$101$	5.91868	0.588931	0.294465	+	0.955662i	$0.404858\pi$
$102$	0	0
$103$	−2.82386	−0.278243	−0.139122	−	0.990275i	$-0.544428\pi$
$103$	−2.82386	−0.278243	−0.139122	+	0.990275i	$0.544428\pi$
$104$	8.62608	0.845856
$105$	0	0
$106$	5.50988	0.535167
$107$	−7.39664	−0.715060	−0.357530	−	0.933902i	$-0.616381\pi$
$107$	−7.39664	−0.715060	−0.357530	+	0.933902i	$0.616381\pi$
$108$	0	0
$109$	6.62604	0.634660	0.317330	−	0.948315i	$-0.397214\pi$
$109$	6.62604	0.634660	0.317330	+	0.948315i	$0.397214\pi$
$110$	4.09913	0.390837
$111$	0	0
$112$	−2.49452	−0.235710
$113$	−10.0353	−0.944043	−0.472022	−	0.881587i	$-0.656476\pi$
$113$	−10.0353	−0.944043	−0.472022	+	0.881587i	$0.656476\pi$
$114$	0	0
$115$	−1.25336	−0.116877
$116$	−13.9030	−1.29086
$117$	0	0
$118$	−5.20551	−0.479206
$119$	−5.81738	−0.533278
$120$	0	0
$121$	9.44292	0.858448
$122$	4.81460	0.435893
$123$	0	0
$124$	−14.6812	−1.31841
$125$	−12.1547	−1.08715
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	11.4294	1.01022
$129$	0	0
$130$	−4.08477	−0.358258
$131$	8.12188	0.709612	0.354806	−	0.934940i	$-0.384547\pi$
$131$	8.12188	0.709612	0.354806	+	0.934940i	$0.384547\pi$
$132$	0	0
$133$	−6.77727	−0.587664
$134$	4.95741	0.428255
$135$	0	0
$136$	11.1377	0.955046
$137$	−14.7897	−1.26357	−0.631783	−	0.775145i	$-0.717677\pi$
$137$	−14.7897	−1.26357	−0.631783	+	0.775145i	$0.717677\pi$
$138$	0	0
$139$	21.6135	1.83323	0.916615	−	0.399772i	$-0.130911\pi$
$139$	21.6135	1.83323	0.916615	+	0.399772i	$0.130911\pi$
$140$	3.07539	0.259918
$141$	0	0
$142$	−3.94971	−0.331453
$143$	−20.3713	−1.70353
$144$	0	0
$145$	14.1613	1.17603
$146$	7.59849	0.628855
$147$	0	0
$148$	−2.44166	−0.200703
$149$	4.37072	0.358063	0.179032	−	0.983843i	$-0.442704\pi$
$149$	4.37072	0.358063	0.179032	+	0.983843i	$0.442704\pi$
$150$	0	0
$151$	10.2661	0.835447	0.417723	−	0.908574i	$-0.362828\pi$
$151$	10.2661	0.835447	0.417723	+	0.908574i	$0.362828\pi$
$152$	12.9754	1.05245
$153$	0	0
$154$	−2.31603	−0.186631
$155$	14.9540	1.20113
$156$	0	0
$157$	10.6227	0.847784	0.423892	−	0.905713i	$-0.360664\pi$
$157$	10.6227	0.847784	0.423892	+	0.905713i	$0.360664\pi$
$158$	−6.84226	−0.544341
$159$	0	0
$160$	−9.03867	−0.714569
$161$	0.708155	0.0558104
$162$	0	0
$163$	12.3229	0.965201	0.482601	−	0.875840i	$-0.339692\pi$
$163$	12.3229	0.965201	0.482601	+	0.875840i	$0.339692\pi$
$164$	5.32870	0.416102
$165$	0	0
$166$	7.89530	0.612794
$167$	12.8824	0.996867	0.498434	−	0.866928i	$-0.333909\pi$
$167$	12.8824	0.996867	0.498434	+	0.866928i	$0.333909\pi$
$168$	0	0
$169$	7.29988	0.561529
$170$	−5.27409	−0.404505
$171$	0	0
$172$	6.81601	0.519716
$173$	−24.6030	−1.87053	−0.935267	−	0.353944i	$-0.884840\pi$
$173$	−24.6030	−1.87053	−0.935267	+	0.353944i	$0.884840\pi$
$174$	0	0
$175$	1.86746	0.141167
$176$	−11.2787	−0.850162
$177$	0	0
$178$	2.44496	0.183257
$179$	18.6149	1.39135	0.695673	−	0.718359i	$-0.255106\pi$
$179$	18.6149	1.39135	0.695673	+	0.718359i	$0.255106\pi$
$180$	0	0
$181$	−17.2959	−1.28559	−0.642797	−	0.766037i	$-0.722226\pi$
$181$	−17.2959	−1.28559	−0.642797	+	0.766037i	$0.722226\pi$
$182$	2.30791	0.171074
$183$	0	0
$184$	−1.35580	−0.0999507
$185$	2.48703	0.182850
$186$	0	0
$187$	−26.3026	−1.92344
$188$	−16.9920	−1.23927
$189$	0	0
$190$	−6.14434	−0.445757
$191$	5.42727	0.392704	0.196352	−	0.980534i	$-0.437091\pi$
$191$	5.42727	0.392704	0.196352	+	0.980534i	$0.437091\pi$
$192$	0	0
$193$	25.7520	1.85367	0.926836	−	0.375466i	$-0.122517\pi$
$193$	25.7520	1.85367	0.926836	+	0.375466i	$0.122517\pi$
$194$	−5.51720	−0.396112
$195$	0	0
$196$	−1.73761	−0.124115
$197$	−5.11687	−0.364562	−0.182281	−	0.983247i	$-0.558348\pi$
$197$	−5.11687	−0.364562	−0.182281	+	0.983247i	$0.558348\pi$
$198$	0	0
$199$	−19.8835	−1.40950	−0.704751	−	0.709455i	$-0.748941\pi$
$199$	−19.8835	−1.40950	−0.704751	+	0.709455i	$0.748941\pi$
$200$	−3.57535	−0.252815
$201$	0	0
$202$	−3.03178	−0.213315
$203$	−8.00119	−0.561573
$204$	0	0
$205$	−5.42772	−0.379088
$206$	1.44649	0.100782
$207$	0	0
$208$	11.2391	0.779294
$209$	−30.6427	−2.11960
$210$	0	0
$211$	−1.94855	−0.134144	−0.0670718	−	0.997748i	$-0.521366\pi$
$211$	−1.94855	−0.134144	−0.0670718	+	0.997748i	$0.521366\pi$
$212$	18.6906	1.28367
$213$	0	0
$214$	3.78884	0.259000
$215$	−6.94266	−0.473485
$216$	0	0
$217$	−8.44906	−0.573560
$218$	−3.39412	−0.229879
$219$	0	0
$220$	13.9050	0.937478
$221$	26.2104	1.76310
$222$	0	0
$223$	−9.65436	−0.646504	−0.323252	−	0.946313i	$-0.604776\pi$
$223$	−9.65436	−0.646504	−0.323252	+	0.946313i	$0.604776\pi$
$224$	5.10689	0.341218
$225$	0	0
$226$	5.14048	0.341940
$227$	−1.02714	−0.0681735	−0.0340867	−	0.999419i	$-0.510852\pi$
$227$	−1.02714	−0.0681735	−0.0340867	+	0.999419i	$0.510852\pi$
$228$	0	0
$229$	27.5983	1.82375	0.911875	−	0.410468i	$-0.134635\pi$
$229$	27.5983	1.82375	0.911875	+	0.410468i	$0.134635\pi$
$230$	0.642020	0.0423335
$231$	0	0
$232$	15.3187	1.00572
$233$	20.8738	1.36749	0.683745	−	0.729721i	$-0.260350\pi$
$233$	20.8738	1.36749	0.683745	+	0.729721i	$0.260350\pi$
$234$	0	0
$235$	17.3077	1.12903
$236$	−17.6581	−1.14944
$237$	0	0
$238$	2.97989	0.193157
$239$	1.60427	0.103772	0.0518859	−	0.998653i	$-0.483477\pi$
$239$	1.60427	0.103772	0.0518859	+	0.998653i	$0.483477\pi$
$240$	0	0
$241$	2.27009	0.146229	0.0731147	−	0.997324i	$-0.476706\pi$
$241$	2.27009	0.146229	0.0731147	+	0.997324i	$0.476706\pi$
$242$	−4.83703	−0.310936
$243$	0	0
$244$	16.3320	1.04555
$245$	1.76990	0.113075
$246$	0	0
$247$	30.5353	1.94291
$248$	16.1761	1.02719
$249$	0	0
$250$	6.22611	0.393774
$251$	1.10118	0.0695059	0.0347529	−	0.999396i	$-0.488936\pi$
$251$	1.10118	0.0695059	0.0347529	+	0.999396i	$0.488936\pi$
$252$	0	0
$253$	3.20184	0.201298
$254$	−0.512239	−0.0321407
$255$	0	0
$256$	−1.10838	−0.0692739
$257$	12.7463	0.795095	0.397547	−	0.917582i	$-0.369861\pi$
$257$	12.7463	0.795095	0.397547	+	0.917582i	$0.369861\pi$
$258$	0	0
$259$	−1.40518	−0.0873138
$260$	−13.8563	−0.859332
$261$	0	0
$262$	−4.16034	−0.257027
$263$	3.52690	0.217478	0.108739	−	0.994070i	$-0.465319\pi$
$263$	3.52690	0.217478	0.108739	+	0.994070i	$0.465319\pi$
$264$	0	0
$265$	−19.0379	−1.16949
$266$	3.47158	0.212856
$267$	0	0
$268$	16.8165	1.02723
$269$	−26.5855	−1.62095	−0.810474	−	0.585775i	$-0.800790\pi$
$269$	−26.5855	−1.62095	−0.810474	+	0.585775i	$0.800790\pi$
$270$	0	0
$271$	27.6486	1.67953	0.839767	−	0.542948i	$-0.182692\pi$
$271$	27.6486	1.67953	0.839767	+	0.542948i	$0.182692\pi$
$272$	14.5115	0.879892
$273$	0	0
$274$	7.57584	0.457673
$275$	8.44352	0.509163
$276$	0	0
$277$	−6.46296	−0.388321	−0.194161	−	0.980970i	$-0.562198\pi$
$277$	−6.46296	−0.388321	−0.194161	+	0.980970i	$0.562198\pi$
$278$	−11.0712	−0.664009
$279$	0	0
$280$	−3.38856	−0.202505
$281$	10.9292	0.651983	0.325991	−	0.945373i	$-0.394302\pi$
$281$	10.9292	0.651983	0.325991	+	0.945373i	$0.394302\pi$
$282$	0	0
$283$	5.49884	0.326872	0.163436	−	0.986554i	$-0.447742\pi$
$283$	5.49884	0.326872	0.163436	+	0.986554i	$0.447742\pi$
$284$	−13.3982	−0.795036
$285$	0	0
$286$	10.4350	0.617032
$287$	3.06668	0.181021
$288$	0	0
$289$	16.8419	0.990699
$290$	−7.25396	−0.425967
$291$	0	0
$292$	25.7755	1.50840
$293$	−15.2212	−0.889234	−0.444617	−	0.895721i	$-0.646660\pi$
$293$	−15.2212	−0.889234	−0.444617	+	0.895721i	$0.646660\pi$
$294$	0	0
$295$	17.9862	1.04720
$296$	2.69029	0.156370
$297$	0	0
$298$	−2.23885	−0.129693
$299$	−3.19062	−0.184518
$300$	0	0
$301$	3.92263	0.226097
$302$	−5.25872	−0.302605
$303$	0	0
$304$	16.9060	0.969627
$305$	−16.6355	−0.952546
$306$	0	0
$307$	−17.3444	−0.989897	−0.494948	−	0.868922i	$-0.664813\pi$
$307$	−17.3444	−0.989897	−0.494948	+	0.868922i	$0.664813\pi$
$308$	−7.85641	−0.447661
$309$	0	0
$310$	−7.66001	−0.435059
$311$	−9.58784	−0.543676	−0.271838	−	0.962343i	$-0.587632\pi$
$311$	−9.58784	−0.543676	−0.271838	+	0.962343i	$0.587632\pi$
$312$	0	0
$313$	33.3451	1.88478	0.942388	−	0.334523i	$-0.108575\pi$
$313$	33.3451	1.88478	0.942388	+	0.334523i	$0.108575\pi$
$314$	−5.44136	−0.307074
$315$	0	0
$316$	−23.2102	−1.30568
$317$	−25.9435	−1.45713	−0.728566	−	0.684976i	$-0.759813\pi$
$317$	−25.9435	−1.45713	−0.728566	+	0.684976i	$0.759813\pi$
$318$	0	0
$319$	−36.1764	−2.02549
$320$	−4.20013	−0.234794
$321$	0	0
$322$	−0.362744	−0.0202149
$323$	39.4260	2.19372
$324$	0	0
$325$	−8.41392	−0.466721
$326$	−6.31225	−0.349603
$327$	0	0
$328$	−5.87132	−0.324189
$329$	−9.77893	−0.539130
$330$	0	0
$331$	26.2177	1.44105	0.720527	−	0.693427i	$-0.243900\pi$
$331$	26.2177	1.44105	0.720527	+	0.693427i	$0.243900\pi$
$332$	26.7824	1.46987
$333$	0	0
$334$	−6.59885	−0.361073
$335$	−17.1290	−0.935855
$336$	0	0
$337$	−22.2457	−1.21180	−0.605901	−	0.795540i	$-0.707187\pi$
$337$	−22.2457	−1.21180	−0.605901	+	0.795540i	$0.707187\pi$
$338$	−3.73928	−0.203390
$339$	0	0
$340$	−17.8907	−0.970261
$341$	−38.2015	−2.06873
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−7.51007	−0.404916
$345$	0	0
$346$	12.6026	0.677521
$347$	25.4988	1.36885	0.684423	−	0.729085i	$-0.260054\pi$
$347$	25.4988	1.36885	0.684423	+	0.729085i	$0.260054\pi$
$348$	0	0
$349$	18.4861	0.989536	0.494768	−	0.869025i	$-0.335253\pi$
$349$	18.4861	0.989536	0.494768	+	0.869025i	$0.335253\pi$
$350$	−0.956586	−0.0511317
$351$	0	0
$352$	23.0902	1.23071
$353$	−13.1753	−0.701251	−0.350626	−	0.936516i	$-0.614031\pi$
$353$	−13.1753	−0.701251	−0.350626	+	0.936516i	$0.614031\pi$
$354$	0	0
$355$	13.6471	0.724315
$356$	8.29377	0.439569
$357$	0	0
$358$	−9.53529	−0.503956
$359$	−22.4620	−1.18550	−0.592750	−	0.805387i	$-0.701958\pi$
$359$	−22.4620	−1.18550	−0.592750	+	0.805387i	$0.701958\pi$
$360$	0	0
$361$	26.9314	1.41744
$362$	8.85962	0.465651
$363$	0	0
$364$	7.82888	0.410345
$365$	−26.2545	−1.37422
$366$	0	0
$367$	20.0265	1.04537	0.522687	−	0.852524i	$-0.324929\pi$
$367$	20.0265	1.04537	0.522687	+	0.852524i	$0.324929\pi$
$368$	−1.76650	−0.0920854
$369$	0	0
$370$	−1.27395	−0.0662297
$371$	10.7565	0.558448
$372$	0	0
$373$	−6.24279	−0.323239	−0.161620	−	0.986853i	$-0.551672\pi$
$373$	−6.24279	−0.323239	−0.161620	+	0.986853i	$0.551672\pi$
$374$	13.4732	0.696683
$375$	0	0
$376$	18.7222	0.965526
$377$	36.0497	1.85665
$378$	0	0
$379$	3.99359	0.205137	0.102568	−	0.994726i	$-0.467294\pi$
$379$	3.99359	0.205137	0.102568	+	0.994726i	$0.467294\pi$
$380$	−20.8428	−1.06921
$381$	0	0
$382$	−2.78006	−0.142240
$383$	20.9510	1.07055	0.535274	−	0.844679i	$-0.320208\pi$
$383$	20.9510	1.07055	0.535274	+	0.844679i	$0.320208\pi$
$384$	0	0
$385$	8.00239	0.407840
$386$	−13.1912	−0.671414
$387$	0	0
$388$	−18.7154	−0.950130
$389$	−26.6789	−1.35267	−0.676337	−	0.736592i	$-0.736434\pi$
$389$	−26.6789	−1.35267	−0.676337	+	0.736592i	$0.736434\pi$
$390$	0	0
$391$	−4.11960	−0.208337
$392$	1.91455	0.0966993
$393$	0	0
$394$	2.62106	0.132047
$395$	23.6415	1.18953
$396$	0	0
$397$	−25.2639	−1.26796	−0.633980	−	0.773350i	$-0.718580\pi$
$397$	−25.2639	−1.26796	−0.633980	+	0.773350i	$0.718580\pi$
$398$	10.1851	0.510532
$399$	0	0
$400$	−4.65842	−0.232921
$401$	−4.62871	−0.231147	−0.115573	−	0.993299i	$-0.536871\pi$
$401$	−4.62871	−0.231147	−0.115573	+	0.993299i	$0.536871\pi$
$402$	0	0
$403$	38.0676	1.89628
$404$	−10.2844	−0.511666
$405$	0	0
$406$	4.09852	0.203406
$407$	−6.35337	−0.314925
$408$	0	0
$409$	−8.67492	−0.428947	−0.214474	−	0.976730i	$-0.568804\pi$
$409$	−8.67492	−0.428947	−0.214474	+	0.976730i	$0.568804\pi$
$410$	2.78029	0.137309
$411$	0	0
$412$	4.90678	0.241740
$413$	−10.1623	−0.500053
$414$	0	0
$415$	−27.2800	−1.33912
$416$	−23.0093	−1.12812
$417$	0	0
$418$	15.6964	0.767734
$419$	−6.05288	−0.295703	−0.147851	−	0.989010i	$-0.547236\pi$
$419$	−6.05288	−0.295703	−0.147851	+	0.989010i	$0.547236\pi$
$420$	0	0
$421$	−6.79082	−0.330964	−0.165482	−	0.986213i	$-0.552918\pi$
$421$	−6.79082	−0.330964	−0.165482	+	0.986213i	$0.552918\pi$
$422$	0.998121	0.0485878
$423$	0	0
$424$	−20.5938	−1.00012
$425$	−10.8637	−0.526969
$426$	0	0
$427$	9.39913	0.454856
$428$	12.8525	0.621248
$429$	0	0
$430$	3.55630	0.171500
$431$	0.560045	0.0269764	0.0134882	−	0.999909i	$-0.495706\pi$
$431$	0.560045	0.0269764	0.0134882	+	0.999909i	$0.495706\pi$
$432$	0	0
$433$	25.6991	1.23502	0.617509	−	0.786564i	$-0.288142\pi$
$433$	25.6991	1.23502	0.617509	+	0.786564i	$0.288142\pi$
$434$	4.32794	0.207748
$435$	0	0
$436$	−11.5135	−0.551396
$437$	−4.79936	−0.229584
$438$	0	0
$439$	−3.51782	−0.167896	−0.0839482	−	0.996470i	$-0.526753\pi$
$439$	−3.51782	−0.167896	−0.0839482	+	0.996470i	$0.526753\pi$
$440$	−15.3210	−0.730398
$441$	0	0
$442$	−13.4260	−0.638609
$443$	−14.9106	−0.708426	−0.354213	−	0.935165i	$-0.615251\pi$
$443$	−14.9106	−0.708426	−0.354213	+	0.935165i	$0.615251\pi$
$444$	0	0
$445$	−8.44787	−0.400468
$446$	4.94533	0.234168
$447$	0	0
$448$	2.37309	0.112118
$449$	33.5903	1.58522	0.792612	−	0.609727i	$-0.208721\pi$
$449$	33.5903	1.58522	0.792612	+	0.609727i	$0.208721\pi$
$450$	0	0
$451$	13.8657	0.652908
$452$	17.4375	0.820190
$453$	0	0
$454$	0.526139	0.0246929
$455$	−7.97434	−0.373843
$456$	0	0
$457$	−13.2864	−0.621511	−0.310756	−	0.950490i	$-0.600582\pi$
$457$	−13.2864	−0.621511	−0.310756	+	0.950490i	$0.600582\pi$
$458$	−14.1369	−0.660576
$459$	0	0
$460$	2.17785	0.101543
$461$	−20.5088	−0.955191	−0.477595	−	0.878580i	$-0.658491\pi$
$461$	−20.5088	−0.955191	−0.477595	+	0.878580i	$0.658491\pi$
$462$	0	0
$463$	−19.6108	−0.911392	−0.455696	−	0.890135i	$-0.650610\pi$
$463$	−19.6108	−0.911392	−0.455696	+	0.890135i	$0.650610\pi$
$464$	19.9591	0.926578
$465$	0	0
$466$	−10.6924	−0.495315
$467$	−19.6880	−0.911053	−0.455526	−	0.890222i	$-0.650549\pi$
$467$	−19.6880	−0.911053	−0.455526	+	0.890222i	$0.650549\pi$
$468$	0	0
$469$	9.67794	0.446886
$470$	−8.86567	−0.408943
$471$	0	0
$472$	19.4562	0.895543
$473$	17.7357	0.815490
$474$	0	0
$475$	−12.6563	−0.580711
$476$	10.1083	0.463315
$477$	0	0
$478$	−0.821770	−0.0375869
$479$	−20.8553	−0.952904	−0.476452	−	0.879200i	$-0.658077\pi$
$479$	−20.8553	−0.952904	−0.476452	+	0.879200i	$0.658077\pi$
$480$	0	0
$481$	6.33111	0.288674
$482$	−1.16283	−0.0529654
$483$	0	0
$484$	−16.4081	−0.745824
$485$	19.0631	0.865612
$486$	0	0
$487$	−8.76655	−0.397250	−0.198625	−	0.980076i	$-0.563648\pi$
$487$	−8.76655	−0.397250	−0.198625	+	0.980076i	$0.563648\pi$
$488$	−17.9951	−0.814600
$489$	0	0
$490$	−0.906610	−0.0409565
$491$	14.8972	0.672299	0.336150	−	0.941809i	$-0.390875\pi$
$491$	14.8972	0.672299	0.336150	+	0.941809i	$0.390875\pi$
$492$	0	0
$493$	46.5459	2.09632
$494$	−15.6413	−0.703737
$495$	0	0
$496$	21.0763	0.946355
$497$	−7.71069	−0.345872
$498$	0	0
$499$	29.1802	1.30629	0.653143	−	0.757235i	$-0.273450\pi$
$499$	29.1802	1.30629	0.653143	+	0.757235i	$0.273450\pi$
$500$	21.1202	0.944522
$501$	0	0
$502$	−0.564067	−0.0251756
$503$	23.4436	1.04530	0.522650	−	0.852547i	$-0.324943\pi$
$503$	23.4436	1.04530	0.522650	+	0.852547i	$0.324943\pi$
$504$	0	0
$505$	10.4755	0.466152
$506$	−1.64011	−0.0729116
$507$	0	0
$508$	−1.73761	−0.0770940
$509$	−7.08841	−0.314188	−0.157094	−	0.987584i	$-0.550213\pi$
$509$	−7.08841	−0.314188	−0.157094	+	0.987584i	$0.550213\pi$
$510$	0	0
$511$	14.8339	0.656212
$512$	−22.2910	−0.985131
$513$	0	0
$514$	−6.52917	−0.287989
$515$	−4.99795	−0.220236
$516$	0	0
$517$	−44.2143	−1.94454
$518$	0.719789	0.0316257
$519$	0	0
$520$	15.2673	0.669514
$521$	9.75971	0.427581	0.213790	−	0.976880i	$-0.431419\pi$
$521$	9.75971	0.427581	0.213790	+	0.976880i	$0.431419\pi$
$522$	0	0
$523$	32.8898	1.43817	0.719086	−	0.694921i	$-0.244561\pi$
$523$	32.8898	1.43817	0.719086	+	0.694921i	$0.244561\pi$
$524$	−14.1127	−0.616515
$525$	0	0
$526$	−1.80662	−0.0787722
$527$	49.1514	2.14107
$528$	0	0
$529$	−22.4985	−0.978196
$530$	9.75193	0.423597
$531$	0	0
$532$	11.7763	0.510566
$533$	−13.8171	−0.598483
$534$	0	0
$535$	−13.0913	−0.565986
$536$	−18.5289	−0.800326
$537$	0	0
$538$	13.6181	0.587119
$539$	−4.52139	−0.194750
$540$	0	0
$541$	−20.9664	−0.901414	−0.450707	−	0.892672i	$-0.648828\pi$
$541$	−20.9664	−0.901414	−0.450707	+	0.892672i	$0.648828\pi$
$542$	−14.1627	−0.608339
$543$	0	0
$544$	−29.7087	−1.27375
$545$	11.7274	0.502347
$546$	0	0
$547$	28.7390	1.22879	0.614395	−	0.788998i	$-0.289400\pi$
$547$	28.7390	1.22879	0.614395	+	0.788998i	$0.289400\pi$
$548$	25.6987	1.09779
$549$	0	0
$550$	−4.32510	−0.184423
$551$	54.2262	2.31011
$552$	0	0
$553$	−13.3576	−0.568021
$554$	3.31058	0.140653
$555$	0	0
$556$	−37.5558	−1.59272
$557$	−31.6142	−1.33954	−0.669769	−	0.742570i	$-0.733607\pi$
$557$	−31.6142	−1.33954	−0.669769	+	0.742570i	$0.733607\pi$
$558$	0	0
$559$	−17.6736	−0.747512
$560$	−4.41504	−0.186569
$561$	0	0
$562$	−5.59837	−0.236153
$563$	−8.46335	−0.356688	−0.178344	−	0.983968i	$-0.557074\pi$
$563$	−8.46335	−0.356688	−0.178344	+	0.983968i	$0.557074\pi$
$564$	0	0
$565$	−17.7615	−0.747231
$566$	−2.81672	−0.118396
$567$	0	0
$568$	14.7625	0.619421
$569$	21.5578	0.903751	0.451875	−	0.892081i	$-0.350755\pi$
$569$	21.5578	0.903751	0.451875	+	0.892081i	$0.350755\pi$
$570$	0	0
$571$	6.14040	0.256968	0.128484	−	0.991712i	$-0.458989\pi$
$571$	6.14040	0.256968	0.128484	+	0.991712i	$0.458989\pi$
$572$	35.3974	1.48004
$573$	0	0
$574$	−1.57087	−0.0655670
$575$	1.32245	0.0551501
$576$	0	0
$577$	−2.78498	−0.115940	−0.0579701	−	0.998318i	$-0.518463\pi$
$577$	−2.78498	−0.115940	−0.0579701	+	0.998318i	$0.518463\pi$
$578$	−8.62707	−0.358839
$579$	0	0
$580$	−24.6068	−1.02174
$581$	15.4133	0.639452
$582$	0	0
$583$	48.6342	2.01422
$584$	−28.4002	−1.17521
$585$	0	0
$586$	7.79690	0.322087
$587$	8.46781	0.349504	0.174752	−	0.984612i	$-0.444088\pi$
$587$	8.46781	0.349504	0.174752	+	0.984612i	$0.444088\pi$
$588$	0	0
$589$	57.2616	2.35942
$590$	−9.21322	−0.379302
$591$	0	0
$592$	3.50525	0.144065
$593$	12.5095	0.513704	0.256852	−	0.966451i	$-0.417315\pi$
$593$	12.5095	0.513704	0.256852	+	0.966451i	$0.417315\pi$
$594$	0	0
$595$	−10.2962	−0.422102
$596$	−7.59461	−0.311087
$597$	0	0
$598$	1.63436	0.0668339
$599$	46.3136	1.89232	0.946161	−	0.323695i	$-0.104925\pi$
$599$	46.3136	1.89232	0.946161	+	0.323695i	$0.104925\pi$
$600$	0	0
$601$	24.5205	1.00021	0.500106	−	0.865964i	$-0.333294\pi$
$601$	24.5205	1.00021	0.500106	+	0.865964i	$0.333294\pi$
$602$	−2.00932	−0.0818939
$603$	0	0
$604$	−17.8386	−0.725841
$605$	16.7130	0.679480
$606$	0	0
$607$	11.7627	0.477434	0.238717	−	0.971089i	$-0.423273\pi$
$607$	11.7627	0.477434	0.238717	+	0.971089i	$0.423273\pi$
$608$	−34.6108	−1.40365
$609$	0	0
$610$	8.52135	0.345019
$611$	44.0593	1.78245
$612$	0	0
$613$	30.9049	1.24824	0.624119	−	0.781329i	$-0.285458\pi$
$613$	30.9049	1.24824	0.624119	+	0.781329i	$0.285458\pi$
$614$	8.88447	0.358548
$615$	0	0
$616$	8.65641	0.348777
$617$	−13.1853	−0.530819	−0.265409	−	0.964136i	$-0.585507\pi$
$617$	−13.1853	−0.530819	−0.265409	+	0.964136i	$0.585507\pi$
$618$	0	0
$619$	9.45213	0.379913	0.189957	−	0.981792i	$-0.439165\pi$
$619$	9.45213	0.379913	0.189957	+	0.981792i	$0.439165\pi$
$620$	−25.9842	−1.04355
$621$	0	0
$622$	4.91126	0.196924
$623$	4.77308	0.191230
$624$	0	0
$625$	−12.1753	−0.487011
$626$	−17.0806	−0.682680
$627$	0	0
$628$	−18.4581	−0.736559
$629$	8.17448	0.325938
$630$	0	0
$631$	−4.85549	−0.193294	−0.0966471	−	0.995319i	$-0.530812\pi$
$631$	−4.85549	−0.193294	−0.0966471	+	0.995319i	$0.530812\pi$
$632$	25.5737	1.01727
$633$	0	0
$634$	13.2893	0.527784
$635$	1.76990	0.0702362
$636$	0	0
$637$	4.50554	0.178516
$638$	18.5310	0.733648
$639$	0	0
$640$	20.2288	0.799613
$641$	15.7405	0.621714	0.310857	−	0.950457i	$-0.399384\pi$
$641$	15.7405	0.621714	0.310857	+	0.950457i	$0.399384\pi$
$642$	0	0
$643$	−15.8560	−0.625301	−0.312650	−	0.949868i	$-0.601217\pi$
$643$	−15.8560	−0.625301	−0.312650	+	0.949868i	$0.601217\pi$
$644$	−1.23050	−0.0484884
$645$	0	0
$646$	−20.1955	−0.794581
$647$	−15.1236	−0.594568	−0.297284	−	0.954789i	$-0.596081\pi$
$647$	−15.1236	−0.594568	−0.297284	+	0.954789i	$0.596081\pi$
$648$	0	0
$649$	−45.9476	−1.80360
$650$	4.30994	0.169050
$651$	0	0
$652$	−21.4124	−0.838573
$653$	20.8010	0.814005	0.407002	−	0.913427i	$-0.366574\pi$
$653$	20.8010	0.814005	0.407002	+	0.913427i	$0.366574\pi$
$654$	0	0
$655$	14.3749	0.561674
$656$	−7.64989	−0.298678
$657$	0	0
$658$	5.00914	0.195277
$659$	−31.6785	−1.23402	−0.617009	−	0.786956i	$-0.711656\pi$
$659$	−31.6785	−1.23402	−0.617009	+	0.786956i	$0.711656\pi$
$660$	0	0
$661$	43.8992	1.70748	0.853741	−	0.520699i	$-0.174328\pi$
$661$	43.8992	1.70748	0.853741	+	0.520699i	$0.174328\pi$
$662$	−13.4297	−0.521961
$663$	0	0
$664$	−29.5096	−1.14519
$665$	−11.9951	−0.465149
$666$	0	0
$667$	−5.66608	−0.219391
$668$	−22.3845	−0.866084
$669$	0	0
$670$	8.77411	0.338974
$671$	42.4971	1.64058
$672$	0	0
$673$	−17.7216	−0.683119	−0.341559	−	0.939860i	$-0.610955\pi$
$673$	−17.7216	−0.683119	−0.341559	+	0.939860i	$0.610955\pi$
$674$	11.3951	0.438923
$675$	0	0
$676$	−12.6844	−0.487860
$677$	12.4005	0.476588	0.238294	−	0.971193i	$-0.423412\pi$
$677$	12.4005	0.476588	0.238294	+	0.971193i	$0.423412\pi$
$678$	0	0
$679$	−10.7708	−0.413344
$680$	19.7125	0.755940
$681$	0	0
$682$	19.5683	0.749308
$683$	1.68638	0.0645277	0.0322638	−	0.999479i	$-0.489728\pi$
$683$	1.68638	0.0645277	0.0322638	+	0.999479i	$0.489728\pi$
$684$	0	0
$685$	−26.1762	−1.00014
$686$	0.512239	0.0195574
$687$	0	0
$688$	−9.78507	−0.373052
$689$	−48.4637	−1.84632
$690$	0	0
$691$	25.0625	0.953421	0.476711	−	0.879060i	$-0.341829\pi$
$691$	25.0625	0.953421	0.476711	+	0.879060i	$0.341829\pi$
$692$	42.7505	1.62513
$693$	0	0
$694$	−13.0615	−0.495807
$695$	38.2536	1.45104
$696$	0	0
$697$	−17.8401	−0.675740
$698$	−9.46927	−0.358417
$699$	0	0
$700$	−3.24492	−0.122647
$701$	10.7055	0.404341	0.202170	−	0.979350i	$-0.435201\pi$
$701$	10.7055	0.404341	0.202170	+	0.979350i	$0.435201\pi$
$702$	0	0
$703$	9.52331	0.359178
$704$	10.7297	0.404389
$705$	0	0
$706$	6.74891	0.253998
$707$	−5.91868	−0.222595
$708$	0	0
$709$	36.9834	1.38894	0.694471	−	0.719521i	$-0.255639\pi$
$709$	36.9834	1.38894	0.694471	+	0.719521i	$0.255639\pi$
$710$	−6.99059	−0.262352
$711$	0	0
$712$	−9.13831	−0.342472
$713$	−5.98324	−0.224074
$714$	0	0
$715$	−36.0551	−1.34838
$716$	−32.3455	−1.20881
$717$	0	0
$718$	11.5059	0.429397
$719$	−19.8493	−0.740253	−0.370127	−	0.928981i	$-0.620686\pi$
$719$	−19.8493	−0.740253	−0.370127	+	0.928981i	$0.620686\pi$
$720$	0	0
$721$	2.82386	0.105166
$722$	−13.7953	−0.513408
$723$	0	0
$724$	30.0535	1.11693
$725$	−14.9419	−0.554929
$726$	0	0
$727$	24.7556	0.918133	0.459066	−	0.888402i	$-0.348184\pi$
$727$	24.7556	0.918133	0.459066	+	0.888402i	$0.348184\pi$
$728$	−8.62608	−0.319704
$729$	0	0
$730$	13.4485	0.497753
$731$	−22.8194	−0.844007
$732$	0	0
$733$	45.8614	1.69393	0.846964	−	0.531650i	$-0.178428\pi$
$733$	45.8614	1.69393	0.846964	+	0.531650i	$0.178428\pi$
$734$	−10.2583	−0.378643
$735$	0	0
$736$	3.61647	0.133305
$737$	43.7577	1.61183
$738$	0	0
$739$	7.00988	0.257863	0.128931	−	0.991654i	$-0.458845\pi$
$739$	7.00988	0.257863	0.128931	+	0.991654i	$0.458845\pi$
$740$	−4.32149	−0.158861
$741$	0	0
$742$	−5.50988	−0.202274
$743$	39.8673	1.46259	0.731295	−	0.682062i	$-0.238916\pi$
$743$	39.8673	1.46259	0.731295	+	0.682062i	$0.238916\pi$
$744$	0	0
$745$	7.73573	0.283415
$746$	3.19780	0.117080
$747$	0	0
$748$	45.7037	1.67109
$749$	7.39664	0.270267
$750$	0	0
$751$	38.8090	1.41616	0.708081	−	0.706131i	$-0.249561\pi$
$751$	38.8090	1.41616	0.708081	+	0.706131i	$0.249561\pi$
$752$	24.3937	0.889547
$753$	0	0
$754$	−18.4660	−0.672493
$755$	18.1700	0.661275
$756$	0	0
$757$	−21.5550	−0.783431	−0.391715	−	0.920086i	$-0.628118\pi$
$757$	−21.5550	−0.783431	−0.391715	+	0.920086i	$0.628118\pi$
$758$	−2.04567	−0.0743021
$759$	0	0
$760$	22.9652	0.833034
$761$	−52.8809	−1.91693	−0.958466	−	0.285208i	$-0.907937\pi$
$761$	−52.8809	−1.91693	−0.958466	+	0.285208i	$0.907937\pi$
$762$	0	0
$763$	−6.62604	−0.239879
$764$	−9.43049	−0.341183
$765$	0	0
$766$	−10.7319	−0.387760
$767$	45.7865	1.65326
$768$	0	0
$769$	7.78240	0.280641	0.140320	−	0.990106i	$-0.455187\pi$
$769$	7.78240	0.280641	0.140320	+	0.990106i	$0.455187\pi$
$770$	−4.09913	−0.147723
$771$	0	0
$772$	−44.7470	−1.61048
$773$	1.15634	0.0415908	0.0207954	−	0.999784i	$-0.493380\pi$
$773$	1.15634	0.0415908	0.0207954	+	0.999784i	$0.493380\pi$
$774$	0	0
$775$	−15.7783	−0.566774
$776$	20.6211	0.740256
$777$	0	0
$778$	13.6660	0.489949
$779$	−20.7837	−0.744655
$780$	0	0
$781$	−34.8630	−1.24750
$782$	2.11022	0.0754613
$783$	0	0
$784$	2.49452	0.0890899
$785$	18.8011	0.671040
$786$	0	0
$787$	−13.3058	−0.474301	−0.237150	−	0.971473i	$-0.576213\pi$
$787$	−13.3058	−0.474301	−0.237150	+	0.971473i	$0.576213\pi$
$788$	8.89113	0.316733
$789$	0	0
$790$	−12.1101	−0.430858
$791$	10.0353	0.356815
$792$	0	0
$793$	−42.3482	−1.50383
$794$	12.9412	0.459264
$795$	0	0
$796$	34.5498	1.22458
$797$	30.0527	1.06452	0.532260	−	0.846581i	$-0.321343\pi$
$797$	30.0527	1.06452	0.532260	+	0.846581i	$0.321343\pi$
$798$	0	0
$799$	56.8877	2.01254
$800$	9.53692	0.337181
$801$	0	0
$802$	2.37100	0.0837230
$803$	67.0697	2.36684
$804$	0	0
$805$	1.25336	0.0441752
$806$	−19.4997	−0.686847
$807$	0	0
$808$	11.3316	0.398644
$809$	2.96928	0.104394	0.0521972	−	0.998637i	$-0.483378\pi$
$809$	2.96928	0.104394	0.0521972	+	0.998637i	$0.483378\pi$
$810$	0	0
$811$	−42.1036	−1.47846	−0.739229	−	0.673454i	$-0.764810\pi$
$811$	−42.1036	−1.47846	−0.739229	+	0.673454i	$0.764810\pi$
$812$	13.9030	0.487898
$813$	0	0
$814$	3.25444	0.114068
$815$	21.8102	0.763978
$816$	0	0
$817$	−26.5847	−0.930082
$818$	4.44363	0.155368
$819$	0	0
$820$	9.43126	0.329354
$821$	−33.9197	−1.18381	−0.591903	−	0.806009i	$-0.701623\pi$
$821$	−33.9197	−1.18381	−0.591903	+	0.806009i	$0.701623\pi$
$822$	0	0
$823$	−19.8587	−0.692230	−0.346115	−	0.938192i	$-0.612499\pi$
$823$	−19.8587	−0.692230	−0.346115	+	0.938192i	$0.612499\pi$
$824$	−5.40642	−0.188342
$825$	0	0
$826$	5.20551	0.181123
$827$	28.0532	0.975507	0.487753	−	0.872981i	$-0.337817\pi$
$827$	28.0532	0.975507	0.487753	+	0.872981i	$0.337817\pi$
$828$	0	0
$829$	29.5414	1.02602	0.513008	−	0.858384i	$-0.328531\pi$
$829$	29.5414	1.02602	0.513008	+	0.858384i	$0.328531\pi$
$830$	13.9739	0.485040
$831$	0	0
$832$	−10.6920	−0.370680
$833$	5.81738	0.201560
$834$	0	0
$835$	22.8005	0.789043
$836$	53.2450	1.84152
$837$	0	0
$838$	3.10052	0.107106
$839$	3.38509	0.116866	0.0584331	−	0.998291i	$-0.481390\pi$
$839$	3.38509	0.116866	0.0584331	+	0.998291i	$0.481390\pi$
$840$	0	0
$841$	35.0190	1.20755
$842$	3.47852	0.119878
$843$	0	0
$844$	3.38582	0.116545
$845$	12.9200	0.444463
$846$	0	0
$847$	−9.44292	−0.324463
$848$	−26.8322	−0.921422
$849$	0	0
$850$	5.56483	0.190872
$851$	−0.995087	−0.0341111
$852$	0	0
$853$	−44.5454	−1.52520	−0.762602	−	0.646868i	$-0.776079\pi$
$853$	−44.5454	−1.52520	−0.762602	+	0.646868i	$0.776079\pi$
$854$	−4.81460	−0.164752
$855$	0	0
$856$	−14.1612	−0.484021
$857$	3.18608	0.108834	0.0544172	−	0.998518i	$-0.482670\pi$
$857$	3.18608	0.108834	0.0544172	+	0.998518i	$0.482670\pi$
$858$	0	0
$859$	−2.23145	−0.0761361	−0.0380681	−	0.999275i	$-0.512120\pi$
$859$	−2.23145	−0.0761361	−0.0380681	+	0.999275i	$0.512120\pi$
$860$	12.0636	0.411367
$861$	0	0
$862$	−0.286877	−0.00977105
$863$	31.3862	1.06840	0.534199	−	0.845359i	$-0.320613\pi$
$863$	31.3862	1.06840	0.534199	+	0.845359i	$0.320613\pi$
$864$	0	0
$865$	−43.5448	−1.48057
$866$	−13.1640	−0.447333
$867$	0	0
$868$	14.6812	0.498312
$869$	−60.3947	−2.04875
$870$	0	0
$871$	−43.6043	−1.47748
$872$	12.6859	0.429598
$873$	0	0
$874$	2.45842	0.0831572
$875$	12.1547	0.410904
$876$	0	0
$877$	−19.7012	−0.665262	−0.332631	−	0.943057i	$-0.607936\pi$
$877$	−19.7012	−0.665262	−0.332631	+	0.943057i	$0.607936\pi$
$878$	1.80196	0.0608133
$879$	0	0
$880$	−19.9621	−0.672922
$881$	31.4991	1.06123	0.530615	−	0.847613i	$-0.321961\pi$
$881$	31.4991	1.06123	0.530615	+	0.847613i	$0.321961\pi$
$882$	0	0
$883$	35.4081	1.19158	0.595789	−	0.803141i	$-0.296839\pi$
$883$	35.4081	1.19158	0.595789	+	0.803141i	$0.296839\pi$
$884$	−45.5435	−1.53179
$885$	0	0
$886$	7.63781	0.256597
$887$	−2.81610	−0.0945554	−0.0472777	−	0.998882i	$-0.515055\pi$
$887$	−2.81610	−0.0945554	−0.0472777	+	0.998882i	$0.515055\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	4.32733	0.145052
$891$	0	0
$892$	16.7755	0.561686
$893$	66.2744	2.21779
$894$	0	0
$895$	32.9465	1.10128
$896$	−11.4294	−0.381828
$897$	0	0
$898$	−17.2062	−0.574180
$899$	67.6025	2.25467
$900$	0	0
$901$	−62.5745	−2.08466
$902$	−7.10253	−0.236488
$903$	0	0
$904$	−19.2131	−0.639019
$905$	−30.6120	−1.01758
$906$	0	0
$907$	20.9748	0.696456	0.348228	−	0.937410i	$-0.386783\pi$
$907$	20.9748	0.696456	0.348228	+	0.937410i	$0.386783\pi$
$908$	1.78477	0.0592295
$909$	0	0
$910$	4.08477	0.135409
$911$	−23.5213	−0.779296	−0.389648	−	0.920964i	$-0.627403\pi$
$911$	−23.5213	−0.779296	−0.389648	+	0.920964i	$0.627403\pi$
$912$	0	0
$913$	69.6896	2.30639
$914$	6.80581	0.225116
$915$	0	0
$916$	−47.9552	−1.58448
$917$	−8.12188	−0.268208
$918$	0	0
$919$	53.3169	1.75876	0.879381	−	0.476119i	$-0.157957\pi$
$919$	53.3169	1.75876	0.879381	+	0.476119i	$0.157957\pi$
$920$	−2.39962	−0.0791132
$921$	0	0
$922$	10.5054	0.345977
$923$	34.7408	1.14351
$924$	0	0
$925$	−2.62413	−0.0862808
$926$	10.0454	0.330113
$927$	0	0
$928$	−40.8612	−1.34133
$929$	−24.8127	−0.814078	−0.407039	−	0.913411i	$-0.633439\pi$
$929$	−24.8127	−0.814078	−0.407039	+	0.913411i	$0.633439\pi$
$930$	0	0
$931$	6.77727	0.222116
$932$	−36.2706	−1.18808
$933$	0	0
$934$	10.0850	0.329990
$935$	−46.5529	−1.52244
$936$	0	0
$937$	−23.9476	−0.782335	−0.391168	−	0.920319i	$-0.627929\pi$
$937$	−23.9476	−0.782335	−0.391168	+	0.920319i	$0.627929\pi$
$938$	−4.95741	−0.161865
$939$	0	0
$940$	−30.0741	−0.980908
$941$	19.1449	0.624108	0.312054	−	0.950064i	$-0.398983\pi$
$941$	19.1449	0.624108	0.312054	+	0.950064i	$0.398983\pi$
$942$	0	0
$943$	2.17169	0.0707198
$944$	25.3500	0.825071
$945$	0	0
$946$	−9.08493	−0.295376
$947$	−7.26965	−0.236232	−0.118116	−	0.993000i	$-0.537685\pi$
$947$	−7.26965	−0.236232	−0.118116	+	0.993000i	$0.537685\pi$
$948$	0	0
$949$	−66.8346	−2.16954
$950$	6.48305	0.210338
$951$	0	0
$952$	−11.1377	−0.360974
$953$	−32.7433	−1.06066	−0.530330	−	0.847791i	$-0.677932\pi$
$953$	−32.7433	−1.06066	−0.530330	+	0.847791i	$0.677932\pi$
$954$	0	0
$955$	9.60571	0.310834
$956$	−2.78760	−0.0901575
$957$	0	0
$958$	10.6829	0.345149
$959$	14.7897	0.477583
$960$	0	0
$961$	40.3867	1.30280
$962$	−3.24304	−0.104560
$963$	0	0
$964$	−3.94454	−0.127045
$965$	45.5785	1.46722
$966$	0	0
$967$	−48.1172	−1.54735	−0.773673	−	0.633585i	$-0.781583\pi$
$967$	−48.1172	−1.54735	−0.773673	+	0.633585i	$0.781583\pi$
$968$	18.0789	0.581079
$969$	0	0
$970$	−9.76487	−0.313531
$971$	−30.7999	−0.988415	−0.494207	−	0.869344i	$-0.664542\pi$
$971$	−30.7999	−0.988415	−0.494207	+	0.869344i	$0.664542\pi$
$972$	0	0
$973$	−21.6135	−0.692896
$974$	4.49057	0.143887
$975$	0	0
$976$	−23.4463	−0.750498
$977$	−27.6351	−0.884124	−0.442062	−	0.896985i	$-0.645753\pi$
$977$	−27.6351	−0.884124	−0.442062	+	0.896985i	$0.645753\pi$
$978$	0	0
$979$	21.5810	0.689730
$980$	−3.07539	−0.0982399
$981$	0	0
$982$	−7.63090	−0.243512
$983$	−9.55349	−0.304709	−0.152354	−	0.988326i	$-0.548686\pi$
$983$	−9.55349	−0.304709	−0.152354	+	0.988326i	$0.548686\pi$
$984$	0	0
$985$	−9.05633	−0.288559
$986$	−23.8426	−0.759304
$987$	0	0
$988$	−53.0584	−1.68801
$989$	2.77783	0.0883298
$990$	0	0
$991$	−15.7123	−0.499118	−0.249559	−	0.968360i	$-0.580286\pi$
$991$	−15.7123	−0.499118	−0.249559	+	0.968360i	$0.580286\pi$
$992$	−43.1484	−1.36996
$993$	0	0
$994$	3.94971	0.125277
$995$	−35.1917	−1.11565
$996$	0	0
$997$	38.0575	1.20529	0.602647	−	0.798008i	$-0.294113\pi$
$997$	38.0575	1.20529	0.602647	+	0.798008i	$0.294113\pi$
$998$	−14.9472	−0.473146
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.y.1.12	✓	28
3.2	odd	2	inner	8001.2.a.y.1.17	yes	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8001.2.a.y.1.12	✓	28	1.1	even	1	trivial
8001.2.a.y.1.17	yes	28	3.2	odd	2	inner

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)

\(f(q)\)	\(=\)	\(q-0.512239 q^{2} -1.73761 q^{4} +1.76990 q^{5} -1.00000 q^{7} +1.91455 q^{8} +O(q^{10})\)	\(q-0.512239 q^{2} -1.73761 q^{4} +1.76990 q^{5} -1.00000 q^{7} +1.91455 q^{8} -0.906610 q^{10} -4.52139 q^{11} +4.50554 q^{13} +0.512239 q^{14} +2.49452 q^{16} +5.81738 q^{17} +6.77727 q^{19} -3.07539 q^{20} +2.31603 q^{22} -0.708155 q^{23} -1.86746 q^{25} -2.30791 q^{26} +1.73761 q^{28} +8.00119 q^{29} +8.44906 q^{31} -5.10689 q^{32} -2.97989 q^{34} -1.76990 q^{35} +1.40518 q^{37} -3.47158 q^{38} +3.38856 q^{40} -3.06668 q^{41} -3.92263 q^{43} +7.85641 q^{44} +0.362744 q^{46} +9.77893 q^{47} +1.00000 q^{49} +0.956586 q^{50} -7.82888 q^{52} -10.7565 q^{53} -8.00239 q^{55} -1.91455 q^{56} -4.09852 q^{58} +10.1623 q^{59} -9.39913 q^{61} -4.32794 q^{62} -2.37309 q^{64} +7.97434 q^{65} -9.67794 q^{67} -10.1083 q^{68} +0.906610 q^{70} +7.71069 q^{71} -14.8339 q^{73} -0.719789 q^{74} -11.7763 q^{76} +4.52139 q^{77} +13.3576 q^{79} +4.41504 q^{80} +1.57087 q^{82} -15.4133 q^{83} +10.2962 q^{85} +2.00932 q^{86} -8.65641 q^{88} -4.77308 q^{89} -4.50554 q^{91} +1.23050 q^{92} -5.00914 q^{94} +11.9951 q^{95} +10.7708 q^{97} -0.512239 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 30 q^{4} - 28 q^{7}+O(q^{10}) \) 28 * q + 30 * q^4 - 28 * q^7	\( 28 q + 30 q^{4} - 28 q^{7} + 4 q^{10} + 8 q^{13} + 42 q^{16} + 34 q^{19} - 10 q^{22} + 14 q^{25} - 30 q^{28} + 56 q^{31} - 6 q^{37} + 38 q^{40} + 18 q^{43} + 16 q^{46} + 28 q^{49} + 18 q^{52} + 48 q^{55} + 2 q^{58} + 36 q^{61} + 76 q^{64} - 4 q^{70} + 50 q^{73} + 132 q^{76} + 66 q^{79} - 36 q^{82} + 20 q^{85} + 6 q^{88} - 8 q^{91} + 54 q^{94} - 8 q^{97}+O(q^{100}) \) 28 * q + 30 * q^4 - 28 * q^7 + 4 * q^10 + 8 * q^13 + 42 * q^16 + 34 * q^19 - 10 * q^22 + 14 * q^25 - 30 * q^28 + 56 * q^31 - 6 * q^37 + 38 * q^40 + 18 * q^43 + 16 * q^46 + 28 * q^49 + 18 * q^52 + 48 * q^55 + 2 * q^58 + 36 * q^61 + 76 * q^64 - 4 * q^70 + 50 * q^73 + 132 * q^76 + 66 * q^79 - 36 * q^82 + 20 * q^85 + 6 * q^88 - 8 * q^91 + 54 * q^94 - 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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