LMFDB - Embedded newform 8001.2.a.z.1.4

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:	$1$
Dimension:	$32$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
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)

$f(q)$	$=$	$q-2.41110 q^{2} +3.81340 q^{4} -2.91376 q^{5} -1.00000 q^{7} -4.37229 q^{8} +O(q^{10})$	\(q-2.41110 q^{2} +3.81340 q^{4} -2.91376 q^{5} -1.00000 q^{7} -4.37229 q^{8} +7.02536 q^{10} +4.84187 q^{11} -5.92231 q^{13} +2.41110 q^{14} +2.91522 q^{16} +7.70060 q^{17} -7.16283 q^{19} -11.1113 q^{20} -11.6742 q^{22} -6.74473 q^{23} +3.48998 q^{25} +14.2793 q^{26} -3.81340 q^{28} +1.16404 q^{29} -3.19532 q^{31} +1.71568 q^{32} -18.5669 q^{34} +2.91376 q^{35} +6.23173 q^{37} +17.2703 q^{38} +12.7398 q^{40} -1.69054 q^{41} +10.5590 q^{43} +18.4640 q^{44} +16.2622 q^{46} +4.01302 q^{47} +1.00000 q^{49} -8.41468 q^{50} -22.5841 q^{52} +4.73182 q^{53} -14.1080 q^{55} +4.37229 q^{56} -2.80661 q^{58} +1.42000 q^{59} +5.30923 q^{61} +7.70423 q^{62} -9.96713 q^{64} +17.2562 q^{65} +0.727254 q^{67} +29.3655 q^{68} -7.02536 q^{70} -6.03872 q^{71} -7.68043 q^{73} -15.0253 q^{74} -27.3148 q^{76} -4.84187 q^{77} -4.20331 q^{79} -8.49425 q^{80} +4.07607 q^{82} -14.0855 q^{83} -22.4377 q^{85} -25.4588 q^{86} -21.1701 q^{88} +13.2348 q^{89} +5.92231 q^{91} -25.7204 q^{92} -9.67580 q^{94} +20.8708 q^{95} -0.0256030 q^{97} -2.41110 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 30 q^{4} - 32 q^{7}+O(q^{10}) $ 32 * q + 30 * q^4 - 32 * q^7	$ 32 q + 30 q^{4} - 32 q^{7} - 16 q^{10} - 14 q^{13} + 18 q^{16} - 30 q^{19} - 10 q^{22} + 36 q^{25} - 30 q^{28} - 58 q^{31} - 34 q^{34} + 8 q^{37} - 34 q^{40} + 6 q^{43} - 36 q^{46} + 32 q^{49} - 56 q^{52} - 88 q^{55} - 22 q^{58} - 46 q^{61} + 20 q^{64} - 8 q^{67} + 16 q^{70} - 60 q^{73} - 128 q^{76} - 74 q^{79} - 52 q^{82} - 16 q^{85} - 64 q^{88} + 14 q^{91} - 58 q^{94} - 44 q^{97}+O(q^{100}) $ 32 * q + 30 * q^4 - 32 * q^7 - 16 * q^10 - 14 * q^13 + 18 * q^16 - 30 * q^19 - 10 * q^22 + 36 * q^25 - 30 * q^28 - 58 * q^31 - 34 * q^34 + 8 * q^37 - 34 * q^40 + 6 * q^43 - 36 * q^46 + 32 * q^49 - 56 * q^52 - 88 * q^55 - 22 * q^58 - 46 * q^61 + 20 * q^64 - 8 * q^67 + 16 * q^70 - 60 * q^73 - 128 * q^76 - 74 * q^79 - 52 * q^82 - 16 * q^85 - 64 * q^88 + 14 * q^91 - 58 * q^94 - 44 * 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 $

$2$	−2.41110	−1.70490	−0.852452	−	0.522805i	$-0.824886\pi$
$2$	−2.41110	−1.70490	−0.852452	+	0.522805i	$0.824886\pi$
$3$	0	0
$3$	0	0
$4$	3.81340	1.90670
$5$	−2.91376	−1.30307	−0.651536	−	0.758618i	$-0.725875\pi$
$5$	−2.91376	−1.30307	−0.651536	+	0.758618i	$0.725875\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−4.37229	−1.54584
$9$	0	0
$10$	7.02536	2.22161
$11$	4.84187	1.45988	0.729940	−	0.683512i	$-0.239548\pi$
$11$	4.84187	1.45988	0.729940	+	0.683512i	$0.239548\pi$
$12$	0	0
$13$	−5.92231	−1.64255	−0.821277	−	0.570530i	$-0.806738\pi$
$13$	−5.92231	−1.64255	−0.821277	+	0.570530i	$0.806738\pi$
$14$	2.41110	0.644393
$15$	0	0
$16$	2.91522	0.728806
$17$	7.70060	1.86767	0.933835	−	0.357705i	$-0.116441\pi$
$17$	7.70060	1.86767	0.933835	+	0.357705i	$0.116441\pi$
$18$	0	0
$19$	−7.16283	−1.64327	−0.821633	−	0.570016i	$-0.806937\pi$
$19$	−7.16283	−1.64327	−0.821633	+	0.570016i	$0.806937\pi$
$20$	−11.1113	−2.48457
$21$	0	0
$22$	−11.6742	−2.48896
$23$	−6.74473	−1.40637	−0.703187	−	0.711005i	$-0.748240\pi$
$23$	−6.74473	−1.40637	−0.703187	+	0.711005i	$0.748240\pi$
$24$	0	0
$25$	3.48998	0.697995
$26$	14.2793	2.80040
$27$	0	0
$28$	−3.81340	−0.720665
$29$	1.16404	0.216156	0.108078	−	0.994142i	$-0.465530\pi$
$29$	1.16404	0.216156	0.108078	+	0.994142i	$0.465530\pi$
$30$	0	0
$31$	−3.19532	−0.573896	−0.286948	−	0.957946i	$-0.592641\pi$
$31$	−3.19532	−0.573896	−0.286948	+	0.957946i	$0.592641\pi$
$32$	1.71568	0.303293
$33$	0	0
$34$	−18.5669	−3.18420
$35$	2.91376	0.492515
$36$	0	0
$37$	6.23173	1.02449	0.512245	−	0.858840i	$-0.328814\pi$
$37$	6.23173	1.02449	0.512245	+	0.858840i	$0.328814\pi$
$38$	17.2703	2.80161
$39$	0	0
$40$	12.7398	2.01434
$41$	−1.69054	−0.264019	−0.132009	−	0.991248i	$-0.542143\pi$
$41$	−1.69054	−0.264019	−0.132009	+	0.991248i	$0.542143\pi$
$42$	0	0
$43$	10.5590	1.61023	0.805115	−	0.593119i	$-0.202104\pi$
$43$	10.5590	1.61023	0.805115	+	0.593119i	$0.202104\pi$
$44$	18.4640	2.78355
$45$	0	0
$46$	16.2622	2.39773
$47$	4.01302	0.585359	0.292680	−	0.956211i	$-0.405453\pi$
$47$	4.01302	0.585359	0.292680	+	0.956211i	$0.405453\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−8.41468	−1.19002
$51$	0	0
$52$	−22.5841	−3.13186
$53$	4.73182	0.649965	0.324982	−	0.945720i	$-0.394642\pi$
$53$	4.73182	0.649965	0.324982	+	0.945720i	$0.394642\pi$
$54$	0	0
$55$	−14.1080	−1.90233
$56$	4.37229	0.584272
$57$	0	0
$58$	−2.80661	−0.368526
$59$	1.42000	0.184868	0.0924338	−	0.995719i	$-0.470535\pi$
$59$	1.42000	0.184868	0.0924338	+	0.995719i	$0.470535\pi$
$60$	0	0
$61$	5.30923	0.679777	0.339888	−	0.940466i	$-0.389611\pi$
$61$	5.30923	0.679777	0.339888	+	0.940466i	$0.389611\pi$
$62$	7.70423	0.978439
$63$	0	0
$64$	−9.96713	−1.24589
$65$	17.2562	2.14036
$66$	0	0
$67$	0.727254	0.0888482	0.0444241	−	0.999013i	$-0.485855\pi$
$67$	0.727254	0.0888482	0.0444241	+	0.999013i	$0.485855\pi$
$68$	29.3655	3.56109
$69$	0	0
$70$	−7.02536	−0.839691
$71$	−6.03872	−0.716664	−0.358332	−	0.933594i	$-0.616654\pi$
$71$	−6.03872	−0.716664	−0.358332	+	0.933594i	$0.616654\pi$
$72$	0	0
$73$	−7.68043	−0.898926	−0.449463	−	0.893299i	$-0.648385\pi$
$73$	−7.68043	−0.898926	−0.449463	+	0.893299i	$0.648385\pi$
$74$	−15.0253	−1.74666
$75$	0	0
$76$	−27.3148	−3.13322
$77$	−4.84187	−0.551782
$78$	0	0
$79$	−4.20331	−0.472909	−0.236455	−	0.971643i	$-0.575985\pi$
$79$	−4.20331	−0.472909	−0.236455	+	0.971643i	$0.575985\pi$
$80$	−8.49425	−0.949686
$81$	0	0
$82$	4.07607	0.450127
$83$	−14.0855	−1.54608	−0.773040	−	0.634358i	$-0.781265\pi$
$83$	−14.0855	−1.54608	−0.773040	+	0.634358i	$0.781265\pi$
$84$	0	0
$85$	−22.4377	−2.43371
$86$	−25.4588	−2.74529
$87$	0	0
$88$	−21.1701	−2.25674
$89$	13.2348	1.40289	0.701444	−	0.712725i	$-0.252539\pi$
$89$	13.2348	1.40289	0.701444	+	0.712725i	$0.252539\pi$
$90$	0	0
$91$	5.92231	0.620827
$92$	−25.7204	−2.68153
$93$	0	0
$94$	−9.67580	−0.997982
$95$	20.8708	2.14129
$96$	0	0
$97$	−0.0256030	−0.00259960	−0.00129980	−	0.999999i	$-0.500414\pi$
$97$	−0.0256030	−0.00259960	−0.00129980	+	0.999999i	$0.500414\pi$
$98$	−2.41110	−0.243558
$99$	0	0
$100$	13.3087	1.33087
$101$	−8.54546	−0.850305	−0.425152	−	0.905122i	$-0.639780\pi$
$101$	−8.54546	−0.850305	−0.425152	+	0.905122i	$0.639780\pi$
$102$	0	0
$103$	−12.8304	−1.26421	−0.632107	−	0.774881i	$-0.717810\pi$
$103$	−12.8304	−1.26421	−0.632107	+	0.774881i	$0.717810\pi$
$104$	25.8941	2.53912
$105$	0	0
$106$	−11.4089	−1.10813
$107$	14.8632	1.43688	0.718441	−	0.695587i	$-0.244856\pi$
$107$	14.8632	1.43688	0.718441	+	0.695587i	$0.244856\pi$
$108$	0	0
$109$	2.94497	0.282077	0.141039	−	0.990004i	$-0.454956\pi$
$109$	2.94497	0.282077	0.141039	+	0.990004i	$0.454956\pi$
$110$	34.0159	3.24329
$111$	0	0
$112$	−2.91522	−0.275463
$113$	10.5406	0.991577	0.495789	−	0.868443i	$-0.334879\pi$
$113$	10.5406	0.991577	0.495789	+	0.868443i	$0.334879\pi$
$114$	0	0
$115$	19.6525	1.83260
$116$	4.43894	0.412145
$117$	0	0
$118$	−3.42375	−0.315182
$119$	−7.70060	−0.705913
$120$	0	0
$121$	12.4437	1.13125
$122$	−12.8011	−1.15895
$123$	0	0
$124$	−12.1850	−1.09425
$125$	4.39984	0.393534
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	20.6004	1.82083
$129$	0	0
$130$	−41.6064	−3.64912
$131$	−19.9073	−1.73931	−0.869655	−	0.493659i	$-0.835659\pi$
$131$	−19.9073	−1.73931	−0.869655	+	0.493659i	$0.835659\pi$
$132$	0	0
$133$	7.16283	0.621096
$134$	−1.75348	−0.151478
$135$	0	0
$136$	−33.6692	−2.88711
$137$	−1.07424	−0.0917787	−0.0458894	−	0.998947i	$-0.514612\pi$
$137$	−1.07424	−0.0917787	−0.0458894	+	0.998947i	$0.514612\pi$
$138$	0	0
$139$	−5.35399	−0.454120	−0.227060	−	0.973881i	$-0.572911\pi$
$139$	−5.35399	−0.454120	−0.227060	+	0.973881i	$0.572911\pi$
$140$	11.1113	0.939078
$141$	0	0
$142$	14.5600	1.22184
$143$	−28.6751	−2.39793
$144$	0	0
$145$	−3.39172	−0.281667
$146$	18.5183	1.53258
$147$	0	0
$148$	23.7641	1.95339
$149$	3.75043	0.307247	0.153623	−	0.988129i	$-0.450906\pi$
$149$	3.75043	0.307247	0.153623	+	0.988129i	$0.450906\pi$
$150$	0	0
$151$	−3.64734	−0.296817	−0.148408	−	0.988926i	$-0.547415\pi$
$151$	−3.64734	−0.296817	−0.148408	+	0.988926i	$0.547415\pi$
$152$	31.3180	2.54022
$153$	0	0
$154$	11.6742	0.940737
$155$	9.31038	0.747828
$156$	0	0
$157$	17.4878	1.39568	0.697841	−	0.716253i	$-0.254144\pi$
$157$	17.4878	1.39568	0.697841	+	0.716253i	$0.254144\pi$
$158$	10.1346	0.806265
$159$	0	0
$160$	−4.99909	−0.395213
$161$	6.74473	0.531559
$162$	0	0
$163$	−5.22206	−0.409023	−0.204512	−	0.978864i	$-0.565561\pi$
$163$	−5.22206	−0.409023	−0.204512	+	0.978864i	$0.565561\pi$
$164$	−6.44672	−0.503405
$165$	0	0
$166$	33.9614	2.63592
$167$	−10.5060	−0.812979	−0.406489	−	0.913655i	$-0.633247\pi$
$167$	−10.5060	−0.812979	−0.406489	+	0.913655i	$0.633247\pi$
$168$	0	0
$169$	22.0738	1.69798
$170$	54.0994	4.14924
$171$	0	0
$172$	40.2656	3.07023
$173$	1.45272	0.110448	0.0552240	−	0.998474i	$-0.482413\pi$
$173$	1.45272	0.110448	0.0552240	+	0.998474i	$0.482413\pi$
$174$	0	0
$175$	−3.48998	−0.263817
$176$	14.1151	1.06397
$177$	0	0
$178$	−31.9105	−2.39179
$179$	15.7102	1.17423	0.587117	−	0.809502i	$-0.300263\pi$
$179$	15.7102	1.17423	0.587117	+	0.809502i	$0.300263\pi$
$180$	0	0
$181$	17.3252	1.28777	0.643886	−	0.765121i	$-0.277321\pi$
$181$	17.3252	1.28777	0.643886	+	0.765121i	$0.277321\pi$
$182$	−14.2793	−1.05845
$183$	0	0
$184$	29.4899	2.17403
$185$	−18.1577	−1.33498
$186$	0	0
$187$	37.2853	2.72657
$188$	15.3033	1.11611
$189$	0	0
$190$	−50.3215	−3.65070
$191$	−10.6002	−0.767002	−0.383501	−	0.923541i	$-0.625282\pi$
$191$	−10.6002	−0.767002	−0.383501	+	0.923541i	$0.625282\pi$
$192$	0	0
$193$	25.0481	1.80300	0.901500	−	0.432779i	$-0.142467\pi$
$193$	25.0481	1.80300	0.901500	+	0.432779i	$0.142467\pi$
$194$	0.0617315	0.00443206
$195$	0	0
$196$	3.81340	0.272386
$197$	14.5870	1.03928	0.519641	−	0.854385i	$-0.326066\pi$
$197$	14.5870	1.03928	0.519641	+	0.854385i	$0.326066\pi$
$198$	0	0
$199$	19.8687	1.40846	0.704228	−	0.709974i	$-0.251293\pi$
$199$	19.8687	1.40846	0.704228	+	0.709974i	$0.251293\pi$
$200$	−15.2592	−1.07899
$201$	0	0
$202$	20.6039	1.44969
$203$	−1.16404	−0.0816994
$204$	0	0
$205$	4.92584	0.344035
$206$	30.9353	2.15537
$207$	0	0
$208$	−17.2649	−1.19710
$209$	−34.6815	−2.39897
$210$	0	0
$211$	−16.8972	−1.16325	−0.581624	−	0.813458i	$-0.697582\pi$
$211$	−16.8972	−1.16325	−0.581624	+	0.813458i	$0.697582\pi$
$212$	18.0443	1.23929
$213$	0	0
$214$	−35.8367	−2.44975
$215$	−30.7663	−2.09824
$216$	0	0
$217$	3.19532	0.216912
$218$	−7.10062	−0.480914
$219$	0	0
$220$	−53.7996	−3.62717
$221$	−45.6053	−3.06775
$222$	0	0
$223$	−3.68685	−0.246890	−0.123445	−	0.992351i	$-0.539394\pi$
$223$	−3.68685	−0.246890	−0.123445	+	0.992351i	$0.539394\pi$
$224$	−1.71568	−0.114634
$225$	0	0
$226$	−25.4145	−1.69054
$227$	23.4616	1.55720	0.778602	−	0.627518i	$-0.215929\pi$
$227$	23.4616	1.55720	0.778602	+	0.627518i	$0.215929\pi$
$228$	0	0
$229$	−25.9869	−1.71726	−0.858631	−	0.512595i	$-0.828684\pi$
$229$	−25.9869	−1.71726	−0.858631	+	0.512595i	$0.828684\pi$
$230$	−47.3841	−3.12442
$231$	0	0
$232$	−5.08951	−0.334143
$233$	−24.6677	−1.61604	−0.808018	−	0.589158i	$-0.799459\pi$
$233$	−24.6677	−1.61604	−0.808018	+	0.589158i	$0.799459\pi$
$234$	0	0
$235$	−11.6930	−0.762765
$236$	5.41501	0.352487
$237$	0	0
$238$	18.5669	1.20351
$239$	20.9419	1.35462	0.677310	−	0.735698i	$-0.263146\pi$
$239$	20.9419	1.35462	0.677310	+	0.735698i	$0.263146\pi$
$240$	0	0
$241$	−2.75038	−0.177167	−0.0885837	−	0.996069i	$-0.528234\pi$
$241$	−2.75038	−0.177167	−0.0885837	+	0.996069i	$0.528234\pi$
$242$	−30.0031	−1.92867
$243$	0	0
$244$	20.2462	1.29613
$245$	−2.91376	−0.186153
$246$	0	0
$247$	42.4205	2.69915
$248$	13.9709	0.887151
$249$	0	0
$250$	−10.6085	−0.670937
$251$	12.4711	0.787171	0.393586	−	0.919288i	$-0.371234\pi$
$251$	12.4711	0.787171	0.393586	+	0.919288i	$0.371234\pi$
$252$	0	0
$253$	−32.6571	−2.05314
$254$	2.41110	0.151286
$255$	0	0
$256$	−29.7353	−1.85846
$257$	27.8640	1.73811	0.869053	−	0.494718i	$-0.164729\pi$
$257$	27.8640	1.73811	0.869053	+	0.494718i	$0.164729\pi$
$258$	0	0
$259$	−6.23173	−0.387221
$260$	65.8047	4.08103
$261$	0	0
$262$	47.9985	2.96536
$263$	−9.19650	−0.567081	−0.283540	−	0.958960i	$-0.591509\pi$
$263$	−9.19650	−0.567081	−0.283540	+	0.958960i	$0.591509\pi$
$264$	0	0
$265$	−13.7874	−0.846951
$266$	−17.2703	−1.05891
$267$	0	0
$268$	2.77331	0.169407
$269$	−16.8197	−1.02552	−0.512759	−	0.858533i	$-0.671376\pi$
$269$	−16.8197	−1.02552	−0.512759	+	0.858533i	$0.671376\pi$
$270$	0	0
$271$	−2.27897	−0.138438	−0.0692188	−	0.997602i	$-0.522051\pi$
$271$	−2.27897	−0.138438	−0.0692188	+	0.997602i	$0.522051\pi$
$272$	22.4490	1.36117
$273$	0	0
$274$	2.59011	0.156474
$275$	16.8980	1.01899
$276$	0	0
$277$	2.00040	0.120192	0.0600961	−	0.998193i	$-0.480859\pi$
$277$	2.00040	0.120192	0.0600961	+	0.998193i	$0.480859\pi$
$278$	12.9090	0.774231
$279$	0	0
$280$	−12.7398	−0.761348
$281$	14.6564	0.874325	0.437162	−	0.899383i	$-0.355984\pi$
$281$	14.6564	0.874325	0.437162	+	0.899383i	$0.355984\pi$
$282$	0	0
$283$	−21.9212	−1.30308	−0.651539	−	0.758615i	$-0.725876\pi$
$283$	−21.9212	−1.30308	−0.651539	+	0.758615i	$0.725876\pi$
$284$	−23.0281	−1.36646
$285$	0	0
$286$	69.1385	4.08824
$287$	1.69054	0.0997897
$288$	0	0
$289$	42.2992	2.48819
$290$	8.17778	0.480216
$291$	0	0
$292$	−29.2885	−1.71398
$293$	15.8863	0.928089	0.464044	−	0.885812i	$-0.346398\pi$
$293$	15.8863	0.928089	0.464044	+	0.885812i	$0.346398\pi$
$294$	0	0
$295$	−4.13752	−0.240896
$296$	−27.2469	−1.58369
$297$	0	0
$298$	−9.04265	−0.523827
$299$	39.9444	2.31004
$300$	0	0
$301$	−10.5590	−0.608610
$302$	8.79411	0.506044
$303$	0	0
$304$	−20.8813	−1.19762
$305$	−15.4698	−0.885798
$306$	0	0
$307$	−16.9610	−0.968014	−0.484007	−	0.875064i	$-0.660819\pi$
$307$	−16.9610	−0.968014	−0.484007	+	0.875064i	$0.660819\pi$
$308$	−18.4640	−1.05208
$309$	0	0
$310$	−22.4483	−1.27498
$311$	10.4649	0.593408	0.296704	−	0.954969i	$-0.404112\pi$
$311$	10.4649	0.593408	0.296704	+	0.954969i	$0.404112\pi$
$312$	0	0
$313$	12.9768	0.733494	0.366747	−	0.930321i	$-0.380471\pi$
$313$	12.9768	0.733494	0.366747	+	0.930321i	$0.380471\pi$
$314$	−42.1649	−2.37950
$315$	0	0
$316$	−16.0289	−0.901696
$317$	−33.4039	−1.87615	−0.938076	−	0.346430i	$-0.887394\pi$
$317$	−33.4039	−1.87615	−0.938076	+	0.346430i	$0.887394\pi$
$318$	0	0
$319$	5.63612	0.315562
$320$	29.0418	1.62349
$321$	0	0
$322$	−16.2622	−0.906258
$323$	−55.1581	−3.06908
$324$	0	0
$325$	−20.6687	−1.14649
$326$	12.5909	0.697346
$327$	0	0
$328$	7.39155	0.408130
$329$	−4.01302	−0.221245
$330$	0	0
$331$	−16.8998	−0.928896	−0.464448	−	0.885600i	$-0.653747\pi$
$331$	−16.8998	−0.928896	−0.464448	+	0.885600i	$0.653747\pi$
$332$	−53.7135	−2.94791
$333$	0	0
$334$	25.3310	1.38605
$335$	−2.11904	−0.115776
$336$	0	0
$337$	21.1038	1.14960	0.574799	−	0.818295i	$-0.305080\pi$
$337$	21.1038	1.14960	0.574799	+	0.818295i	$0.305080\pi$
$338$	−53.2221	−2.89490
$339$	0	0
$340$	−85.5638	−4.64035
$341$	−15.4713	−0.837819
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−46.1669	−2.48915
$345$	0	0
$346$	−3.50264	−0.188303
$347$	−14.5910	−0.783287	−0.391644	−	0.920117i	$-0.628093\pi$
$347$	−14.5910	−0.783287	−0.391644	+	0.920117i	$0.628093\pi$
$348$	0	0
$349$	2.54477	0.136218	0.0681091	−	0.997678i	$-0.478303\pi$
$349$	2.54477	0.136218	0.0681091	+	0.997678i	$0.478303\pi$
$350$	8.41468	0.449784
$351$	0	0
$352$	8.30712	0.442771
$353$	34.5501	1.83892	0.919458	−	0.393188i	$-0.128628\pi$
$353$	34.5501	1.83892	0.919458	+	0.393188i	$0.128628\pi$
$354$	0	0
$355$	17.5954	0.933864
$356$	50.4697	2.67489
$357$	0	0
$358$	−37.8788	−2.00196
$359$	11.9986	0.633260	0.316630	−	0.948549i	$-0.397449\pi$
$359$	11.9986	0.633260	0.316630	+	0.948549i	$0.397449\pi$
$360$	0	0
$361$	32.3062	1.70033
$362$	−41.7728	−2.19553
$363$	0	0
$364$	22.5841	1.18373
$365$	22.3789	1.17136
$366$	0	0
$367$	−3.64367	−0.190198	−0.0950989	−	0.995468i	$-0.530317\pi$
$367$	−3.64367	−0.190198	−0.0950989	+	0.995468i	$0.530317\pi$
$368$	−19.6624	−1.02497
$369$	0	0
$370$	43.7801	2.27602
$371$	−4.73182	−0.245664
$372$	0	0
$373$	20.2311	1.04753	0.523765	−	0.851863i	$-0.324527\pi$
$373$	20.2311	1.04753	0.523765	+	0.851863i	$0.324527\pi$
$374$	−89.8986	−4.64854
$375$	0	0
$376$	−17.5461	−0.904871
$377$	−6.89379	−0.355048
$378$	0	0
$379$	−19.3071	−0.991739	−0.495870	−	0.868397i	$-0.665151\pi$
$379$	−19.3071	−0.991739	−0.495870	+	0.868397i	$0.665151\pi$
$380$	79.5886	4.08281
$381$	0	0
$382$	25.5581	1.30767
$383$	−17.7304	−0.905981	−0.452991	−	0.891515i	$-0.649643\pi$
$383$	−17.7304	−0.905981	−0.452991	+	0.891515i	$0.649643\pi$
$384$	0	0
$385$	14.1080	0.719012
$386$	−60.3934	−3.07394
$387$	0	0
$388$	−0.0976347	−0.00495665
$389$	−31.0388	−1.57373	−0.786865	−	0.617125i	$-0.788297\pi$
$389$	−31.0388	−1.57373	−0.786865	+	0.617125i	$0.788297\pi$
$390$	0	0
$391$	−51.9384	−2.62664
$392$	−4.37229	−0.220834
$393$	0	0
$394$	−35.1708	−1.77188
$395$	12.2474	0.616234
$396$	0	0
$397$	−6.66235	−0.334374	−0.167187	−	0.985925i	$-0.553468\pi$
$397$	−6.66235	−0.334374	−0.167187	+	0.985925i	$0.553468\pi$
$398$	−47.9054	−2.40128
$399$	0	0
$400$	10.1741	0.508703
$401$	−28.5013	−1.42329	−0.711643	−	0.702541i	$-0.752049\pi$
$401$	−28.5013	−1.42329	−0.711643	+	0.702541i	$0.752049\pi$
$402$	0	0
$403$	18.9237	0.942655
$404$	−32.5873	−1.62128
$405$	0	0
$406$	2.80661	0.139290
$407$	30.1732	1.49563
$408$	0	0
$409$	−26.0751	−1.28933	−0.644665	−	0.764465i	$-0.723003\pi$
$409$	−26.0751	−1.28933	−0.644665	+	0.764465i	$0.723003\pi$
$410$	−11.8767	−0.586547
$411$	0	0
$412$	−48.9274	−2.41048
$413$	−1.42000	−0.0698734
$414$	0	0
$415$	41.0416	2.01465
$416$	−10.1608	−0.498175
$417$	0	0
$418$	83.6206	4.09002
$419$	−15.1006	−0.737714	−0.368857	−	0.929486i	$-0.620251\pi$
$419$	−15.1006	−0.737714	−0.368857	+	0.929486i	$0.620251\pi$
$420$	0	0
$421$	−7.28830	−0.355210	−0.177605	−	0.984102i	$-0.556835\pi$
$421$	−7.28830	−0.355210	−0.177605	+	0.984102i	$0.556835\pi$
$422$	40.7407	1.98323
$423$	0	0
$424$	−20.6889	−1.00474
$425$	26.8749	1.30362
$426$	0	0
$427$	−5.30923	−0.256931
$428$	56.6795	2.73971
$429$	0	0
$430$	74.1806	3.57731
$431$	−18.7701	−0.904123	−0.452062	−	0.891987i	$-0.649311\pi$
$431$	−18.7701	−0.904123	−0.452062	+	0.891987i	$0.649311\pi$
$432$	0	0
$433$	−31.2440	−1.50149	−0.750747	−	0.660590i	$-0.770306\pi$
$433$	−31.2440	−1.50149	−0.750747	+	0.660590i	$0.770306\pi$
$434$	−7.70423	−0.369815
$435$	0	0
$436$	11.2304	0.537836
$437$	48.3114	2.31105
$438$	0	0
$439$	−13.7246	−0.655039	−0.327520	−	0.944844i	$-0.606213\pi$
$439$	−13.7246	−0.655039	−0.327520	+	0.944844i	$0.606213\pi$
$440$	61.6844	2.94069
$441$	0	0
$442$	109.959	5.23022
$443$	30.9731	1.47158	0.735789	−	0.677211i	$-0.236811\pi$
$443$	30.9731	1.47158	0.735789	+	0.677211i	$0.236811\pi$
$444$	0	0
$445$	−38.5630	−1.82806
$446$	8.88936	0.420923
$447$	0	0
$448$	9.96713	0.470903
$449$	−18.9463	−0.894130	−0.447065	−	0.894502i	$-0.647531\pi$
$449$	−18.9463	−0.894130	−0.447065	+	0.894502i	$0.647531\pi$
$450$	0	0
$451$	−8.18540	−0.385435
$452$	40.1956	1.89064
$453$	0	0
$454$	−56.5684	−2.65488
$455$	−17.2562	−0.808982
$456$	0	0
$457$	−6.49341	−0.303749	−0.151874	−	0.988400i	$-0.548531\pi$
$457$	−6.49341	−0.303749	−0.151874	+	0.988400i	$0.548531\pi$
$458$	62.6570	2.92777
$459$	0	0
$460$	74.9429	3.49423
$461$	28.4651	1.32575	0.662876	−	0.748729i	$-0.269336\pi$
$461$	28.4651	1.32575	0.662876	+	0.748729i	$0.269336\pi$
$462$	0	0
$463$	−4.49350	−0.208831	−0.104415	−	0.994534i	$-0.533297\pi$
$463$	−4.49350	−0.208831	−0.104415	+	0.994534i	$0.533297\pi$
$464$	3.39343	0.157536
$465$	0	0
$466$	59.4763	2.75519
$467$	−1.30117	−0.0602111	−0.0301055	−	0.999547i	$-0.509584\pi$
$467$	−1.30117	−0.0602111	−0.0301055	+	0.999547i	$0.509584\pi$
$468$	0	0
$469$	−0.727254	−0.0335815
$470$	28.1929	1.30044
$471$	0	0
$472$	−6.20863	−0.285775
$473$	51.1252	2.35074
$474$	0	0
$475$	−24.9981	−1.14699
$476$	−29.3655	−1.34596
$477$	0	0
$478$	−50.4930	−2.30950
$479$	24.2021	1.10582	0.552910	−	0.833241i	$-0.313517\pi$
$479$	24.2021	1.10582	0.552910	+	0.833241i	$0.313517\pi$
$480$	0	0
$481$	−36.9062	−1.68278
$482$	6.63143	0.302054
$483$	0	0
$484$	47.4529	2.15695
$485$	0.0746010	0.00338746
$486$	0	0
$487$	−16.8759	−0.764722	−0.382361	−	0.924013i	$-0.624889\pi$
$487$	−16.8759	−0.764722	−0.382361	+	0.924013i	$0.624889\pi$
$488$	−23.2135	−1.05082
$489$	0	0
$490$	7.02536	0.317373
$491$	5.48154	0.247379	0.123689	−	0.992321i	$-0.460527\pi$
$491$	5.48154	0.247379	0.123689	+	0.992321i	$0.460527\pi$
$492$	0	0
$493$	8.96378	0.403709
$494$	−102.280	−4.60180
$495$	0	0
$496$	−9.31507	−0.418259
$497$	6.03872	0.270874
$498$	0	0
$499$	−18.1726	−0.813517	−0.406759	−	0.913536i	$-0.633341\pi$
$499$	−18.1726	−0.813517	−0.406759	+	0.913536i	$0.633341\pi$
$500$	16.7784	0.750351
$501$	0	0
$502$	−30.0692	−1.34205
$503$	12.4515	0.555183	0.277592	−	0.960699i	$-0.410464\pi$
$503$	12.4515	0.555183	0.277592	+	0.960699i	$0.410464\pi$
$504$	0	0
$505$	24.8994	1.10801
$506$	78.7396	3.50040
$507$	0	0
$508$	−3.81340	−0.169192
$509$	−20.7279	−0.918749	−0.459375	−	0.888243i	$-0.651926\pi$
$509$	−20.7279	−0.918749	−0.459375	+	0.888243i	$0.651926\pi$
$510$	0	0
$511$	7.68043	0.339762
$512$	30.4940	1.34766
$513$	0	0
$514$	−67.1828	−2.96331
$515$	37.3846	1.64736
$516$	0	0
$517$	19.4305	0.854554
$518$	15.0253	0.660174
$519$	0	0
$520$	−75.4490	−3.30866
$521$	−31.8644	−1.39600	−0.698002	−	0.716096i	$-0.745927\pi$
$521$	−31.8644	−1.39600	−0.698002	+	0.716096i	$0.745927\pi$
$522$	0	0
$523$	31.9579	1.39742	0.698710	−	0.715405i	$-0.253758\pi$
$523$	31.9579	1.39742	0.698710	+	0.715405i	$0.253758\pi$
$524$	−75.9146	−3.31634
$525$	0	0
$526$	22.1737	0.966819
$527$	−24.6059	−1.07185
$528$	0	0
$529$	22.4914	0.977886
$530$	33.2427	1.44397
$531$	0	0
$532$	27.3148	1.18424
$533$	10.0119	0.433665
$534$	0	0
$535$	−43.3078	−1.87236
$536$	−3.17977	−0.137345
$537$	0	0
$538$	40.5541	1.74841
$539$	4.84187	0.208554
$540$	0	0
$541$	−43.1995	−1.85729	−0.928646	−	0.370967i	$-0.879026\pi$
$541$	−43.1995	−1.85729	−0.928646	+	0.370967i	$0.879026\pi$
$542$	5.49482	0.236023
$543$	0	0
$544$	13.2118	0.566451
$545$	−8.58093	−0.367567
$546$	0	0
$547$	6.00358	0.256695	0.128347	−	0.991729i	$-0.459033\pi$
$547$	6.00358	0.256695	0.128347	+	0.991729i	$0.459033\pi$
$548$	−4.09652	−0.174995
$549$	0	0
$550$	−40.7428	−1.73728
$551$	−8.33781	−0.355203
$552$	0	0
$553$	4.20331	0.178743
$554$	−4.82316	−0.204916
$555$	0	0
$556$	−20.4169	−0.865870
$557$	11.8692	0.502914	0.251457	−	0.967868i	$-0.419090\pi$
$557$	11.8692	0.502914	0.251457	+	0.967868i	$0.419090\pi$
$558$	0	0
$559$	−62.5336	−2.64489
$560$	8.49425	0.358948
$561$	0	0
$562$	−35.3379	−1.49064
$563$	24.3997	1.02832	0.514162	−	0.857693i	$-0.328103\pi$
$563$	24.3997	1.02832	0.514162	+	0.857693i	$0.328103\pi$
$564$	0	0
$565$	−30.7128	−1.29210
$566$	52.8541	2.22162
$567$	0	0
$568$	26.4030	1.10785
$569$	−6.37571	−0.267284	−0.133642	−	0.991030i	$-0.542667\pi$
$569$	−6.37571	−0.267284	−0.133642	+	0.991030i	$0.542667\pi$
$570$	0	0
$571$	4.76600	0.199451	0.0997255	−	0.995015i	$-0.468204\pi$
$571$	4.76600	0.199451	0.0997255	+	0.995015i	$0.468204\pi$
$572$	−109.350	−4.57213
$573$	0	0
$574$	−4.07607	−0.170132
$575$	−23.5389	−0.981642
$576$	0	0
$577$	10.7666	0.448220	0.224110	−	0.974564i	$-0.428053\pi$
$577$	10.7666	0.448220	0.224110	+	0.974564i	$0.428053\pi$
$578$	−101.988	−4.24212
$579$	0	0
$580$	−12.9340	−0.537055
$581$	14.0855	0.584363
$582$	0	0
$583$	22.9108	0.948870
$584$	33.5810	1.38959
$585$	0	0
$586$	−38.3035	−1.58230
$587$	0.0116818	0.000482159 0	0.000241079	−	1.00000i	$-0.499923\pi$
$587$	0.0116818	0.000482159 0	0.000241079		1.00000i	$0.499923\pi$
$588$	0	0
$589$	22.8875	0.943065
$590$	9.97597	0.410704
$591$	0	0
$592$	18.1669	0.746654
$593$	−37.5294	−1.54115	−0.770573	−	0.637351i	$-0.780030\pi$
$593$	−37.5294	−1.54115	−0.770573	+	0.637351i	$0.780030\pi$
$594$	0	0
$595$	22.4377	0.919855
$596$	14.3019	0.585828
$597$	0	0
$598$	−96.3099	−3.93840
$599$	−28.8340	−1.17812	−0.589062	−	0.808088i	$-0.700503\pi$
$599$	−28.8340	−1.17812	−0.589062	+	0.808088i	$0.700503\pi$
$600$	0	0
$601$	32.9469	1.34393	0.671966	−	0.740582i	$-0.265450\pi$
$601$	32.9469	1.34393	0.671966	+	0.740582i	$0.265450\pi$
$602$	25.4588	1.03762
$603$	0	0
$604$	−13.9088	−0.565940
$605$	−36.2580	−1.47410
$606$	0	0
$607$	−37.9893	−1.54194	−0.770970	−	0.636872i	$-0.780228\pi$
$607$	−37.9893	−1.54194	−0.770970	+	0.636872i	$0.780228\pi$
$608$	−12.2892	−0.498391
$609$	0	0
$610$	37.2992	1.51020
$611$	−23.7664	−0.961484
$612$	0	0
$613$	−17.9743	−0.725975	−0.362987	−	0.931794i	$-0.618243\pi$
$613$	−17.9743	−0.725975	−0.362987	+	0.931794i	$0.618243\pi$
$614$	40.8946	1.65037
$615$	0	0
$616$	21.1701	0.852966
$617$	25.2975	1.01844	0.509220	−	0.860637i	$-0.329934\pi$
$617$	25.2975	1.01844	0.509220	+	0.860637i	$0.329934\pi$
$618$	0	0
$619$	20.1324	0.809190	0.404595	−	0.914496i	$-0.367412\pi$
$619$	20.1324	0.809190	0.404595	+	0.914496i	$0.367412\pi$
$620$	35.5042	1.42588
$621$	0	0
$622$	−25.2318	−1.01170
$623$	−13.2348	−0.530242
$624$	0	0
$625$	−30.2699	−1.21080
$626$	−31.2884	−1.25054
$627$	0	0
$628$	66.6882	2.66115
$629$	47.9880	1.91341
$630$	0	0
$631$	−11.4163	−0.454476	−0.227238	−	0.973839i	$-0.572969\pi$
$631$	−11.4163	−0.454476	−0.227238	+	0.973839i	$0.572969\pi$
$632$	18.3781	0.731041
$633$	0	0
$634$	80.5402	3.19866
$635$	2.91376	0.115629
$636$	0	0
$637$	−5.92231	−0.234651
$638$	−13.5892	−0.538003
$639$	0	0
$640$	−60.0245	−2.37268
$641$	19.5013	0.770256	0.385128	−	0.922863i	$-0.374157\pi$
$641$	19.5013	0.770256	0.385128	+	0.922863i	$0.374157\pi$
$642$	0	0
$643$	−49.5234	−1.95301	−0.976506	−	0.215492i	$-0.930864\pi$
$643$	−49.5234	−1.95301	−0.976506	+	0.215492i	$0.930864\pi$
$644$	25.7204	1.01352
$645$	0	0
$646$	132.992	5.23249
$647$	13.2292	0.520095	0.260047	−	0.965596i	$-0.416262\pi$
$647$	13.2292	0.520095	0.260047	+	0.965596i	$0.416262\pi$
$648$	0	0
$649$	6.87544	0.269884
$650$	49.8344	1.95466
$651$	0	0
$652$	−19.9138	−0.779885
$653$	5.45442	0.213448	0.106724	−	0.994289i	$-0.465964\pi$
$653$	5.45442	0.213448	0.106724	+	0.994289i	$0.465964\pi$
$654$	0	0
$655$	58.0051	2.26645
$656$	−4.92832	−0.192418
$657$	0	0
$658$	9.67580	0.377202
$659$	12.0550	0.469596	0.234798	−	0.972044i	$-0.424557\pi$
$659$	12.0550	0.469596	0.234798	+	0.972044i	$0.424557\pi$
$660$	0	0
$661$	−1.99457	−0.0775799	−0.0387900	−	0.999247i	$-0.512350\pi$
$661$	−1.99457	−0.0775799	−0.0387900	+	0.999247i	$0.512350\pi$
$662$	40.7471	1.58368
$663$	0	0
$664$	61.5857	2.38999
$665$	−20.8708	−0.809333
$666$	0	0
$667$	−7.85112	−0.303997
$668$	−40.0636	−1.55011
$669$	0	0
$670$	5.10922	0.197386
$671$	25.7066	0.992392
$672$	0	0
$673$	−10.7995	−0.416289	−0.208144	−	0.978098i	$-0.566742\pi$
$673$	−10.7995	−0.416289	−0.208144	+	0.978098i	$0.566742\pi$
$674$	−50.8834	−1.95995
$675$	0	0
$676$	84.1762	3.23754
$677$	−15.9165	−0.611722	−0.305861	−	0.952076i	$-0.598944\pi$
$677$	−15.9165	−0.611722	−0.305861	+	0.952076i	$0.598944\pi$
$678$	0	0
$679$	0.0256030	0.000982555 0
$680$	98.1040	3.76212
$681$	0	0
$682$	37.3029	1.42840
$683$	−2.33667	−0.0894101	−0.0447050	−	0.999000i	$-0.514235\pi$
$683$	−2.33667	−0.0894101	−0.0447050	+	0.999000i	$0.514235\pi$
$684$	0	0
$685$	3.13008	0.119594
$686$	2.41110	0.0920562
$687$	0	0
$688$	30.7818	1.17355
$689$	−28.0233	−1.06760
$690$	0	0
$691$	−13.2180	−0.502837	−0.251419	−	0.967878i	$-0.580897\pi$
$691$	−13.2180	−0.502837	−0.251419	+	0.967878i	$0.580897\pi$
$692$	5.53979	0.210591
$693$	0	0
$694$	35.1804	1.33543
$695$	15.6002	0.591750
$696$	0	0
$697$	−13.0182	−0.493100
$698$	−6.13568	−0.232239
$699$	0	0
$700$	−13.3087	−0.503021
$701$	−11.0311	−0.416639	−0.208320	−	0.978061i	$-0.566799\pi$
$701$	−11.0311	−0.416639	−0.208320	+	0.978061i	$0.566799\pi$
$702$	0	0
$703$	−44.6368	−1.68351
$704$	−48.2596	−1.81885
$705$	0	0
$706$	−83.3037	−3.13518
$707$	8.54546	0.321385
$708$	0	0
$709$	−44.7628	−1.68110	−0.840550	−	0.541733i	$-0.817768\pi$
$709$	−44.7628	−1.68110	−0.840550	+	0.541733i	$0.817768\pi$
$710$	−42.4242	−1.59215
$711$	0	0
$712$	−57.8665	−2.16864
$713$	21.5516	0.807112
$714$	0	0
$715$	83.5522	3.12467
$716$	59.9092	2.23891
$717$	0	0
$718$	−28.9297	−1.07965
$719$	−22.5152	−0.839674	−0.419837	−	0.907599i	$-0.637913\pi$
$719$	−22.5152	−0.839674	−0.419837	+	0.907599i	$0.637913\pi$
$720$	0	0
$721$	12.8304	0.477828
$722$	−77.8934	−2.89889
$723$	0	0
$724$	66.0680	2.45540
$725$	4.06246	0.150876
$726$	0	0
$727$	45.2526	1.67832	0.839162	−	0.543881i	$-0.183046\pi$
$727$	45.2526	1.67832	0.839162	+	0.543881i	$0.183046\pi$
$728$	−25.8941	−0.959698
$729$	0	0
$730$	−53.9577	−1.99707
$731$	81.3105	3.00738
$732$	0	0
$733$	−0.823994	−0.0304349	−0.0152175	−	0.999884i	$-0.504844\pi$
$733$	−0.823994	−0.0304349	−0.0152175	+	0.999884i	$0.504844\pi$
$734$	8.78524	0.324269
$735$	0	0
$736$	−11.5718	−0.426543
$737$	3.52127	0.129708
$738$	0	0
$739$	−33.7283	−1.24071	−0.620357	−	0.784320i	$-0.713012\pi$
$739$	−33.7283	−1.24071	−0.620357	+	0.784320i	$0.713012\pi$
$740$	−69.2427	−2.54541
$741$	0	0
$742$	11.4089	0.418833
$743$	42.3307	1.55296	0.776481	−	0.630141i	$-0.217003\pi$
$743$	42.3307	1.55296	0.776481	+	0.630141i	$0.217003\pi$
$744$	0	0
$745$	−10.9278	−0.400365
$746$	−48.7793	−1.78594
$747$	0	0
$748$	142.184	5.19875
$749$	−14.8632	−0.543091
$750$	0	0
$751$	24.1432	0.880997	0.440498	−	0.897753i	$-0.354802\pi$
$751$	24.1432	0.880997	0.440498	+	0.897753i	$0.354802\pi$
$752$	11.6989	0.426613
$753$	0	0
$754$	16.6216	0.605324
$755$	10.6275	0.386773
$756$	0	0
$757$	−34.4792	−1.25317	−0.626585	−	0.779353i	$-0.715548\pi$
$757$	−34.4792	−1.25317	−0.626585	+	0.779353i	$0.715548\pi$
$758$	46.5513	1.69082
$759$	0	0
$760$	−91.2530	−3.31009
$761$	21.9434	0.795447	0.397724	−	0.917505i	$-0.369800\pi$
$761$	21.9434	0.795447	0.397724	+	0.917505i	$0.369800\pi$
$762$	0	0
$763$	−2.94497	−0.106615
$764$	−40.4227	−1.46244
$765$	0	0
$766$	42.7498	1.54461
$767$	−8.40966	−0.303655
$768$	0	0
$769$	−28.9123	−1.04260	−0.521302	−	0.853372i	$-0.674554\pi$
$769$	−28.9123	−1.04260	−0.521302	+	0.853372i	$0.674554\pi$
$770$	−34.0159	−1.22585
$771$	0	0
$772$	95.5184	3.43778
$773$	−37.5638	−1.35108	−0.675538	−	0.737325i	$-0.736089\pi$
$773$	−37.5638	−1.35108	−0.675538	+	0.737325i	$0.736089\pi$
$774$	0	0
$775$	−11.1516	−0.400577
$776$	0.111944	0.00401855
$777$	0	0
$778$	74.8377	2.68306
$779$	12.1091	0.433853
$780$	0	0
$781$	−29.2387	−1.04624
$782$	125.229	4.47817
$783$	0	0
$784$	2.91522	0.104115
$785$	−50.9553	−1.81867
$786$	0	0
$787$	−6.93126	−0.247073	−0.123536	−	0.992340i	$-0.539424\pi$
$787$	−6.93126	−0.247073	−0.123536	+	0.992340i	$0.539424\pi$
$788$	55.6261	1.98160
$789$	0	0
$790$	−29.5297	−1.05062
$791$	−10.5406	−0.374781
$792$	0	0
$793$	−31.4429	−1.11657
$794$	16.0636	0.570075
$795$	0	0
$796$	75.7674	2.68550
$797$	−19.9833	−0.707847	−0.353923	−	0.935274i	$-0.615153\pi$
$797$	−19.9833	−0.707847	−0.353923	+	0.935274i	$0.615153\pi$
$798$	0	0
$799$	30.9027	1.09326
$800$	5.98770	0.211697
$801$	0	0
$802$	68.7194	2.42657
$803$	−37.1876	−1.31232
$804$	0	0
$805$	−19.6525	−0.692660
$806$	−45.6269	−1.60714
$807$	0	0
$808$	37.3632	1.31443
$809$	−36.8912	−1.29702	−0.648512	−	0.761204i	$-0.724609\pi$
$809$	−36.8912	−1.29702	−0.648512	+	0.761204i	$0.724609\pi$
$810$	0	0
$811$	−50.0117	−1.75615	−0.878075	−	0.478524i	$-0.841172\pi$
$811$	−50.0117	−1.75615	−0.878075	+	0.478524i	$0.841172\pi$
$812$	−4.43894	−0.155776
$813$	0	0
$814$	−72.7506	−2.54991
$815$	15.2158	0.532987
$816$	0	0
$817$	−75.6323	−2.64604
$818$	62.8696	2.19818
$819$	0	0
$820$	18.7842	0.655972
$821$	−41.7480	−1.45702	−0.728508	−	0.685037i	$-0.759786\pi$
$821$	−41.7480	−1.45702	−0.728508	+	0.685037i	$0.759786\pi$
$822$	0	0
$823$	32.5077	1.13315	0.566574	−	0.824011i	$-0.308269\pi$
$823$	32.5077	1.13315	0.566574	+	0.824011i	$0.308269\pi$
$824$	56.0981	1.95427
$825$	0	0
$826$	3.42375	0.119128
$827$	41.7566	1.45202	0.726010	−	0.687685i	$-0.241373\pi$
$827$	41.7566	1.45202	0.726010	+	0.687685i	$0.241373\pi$
$828$	0	0
$829$	40.4492	1.40486	0.702430	−	0.711753i	$-0.252098\pi$
$829$	40.4492	1.40486	0.702430	+	0.711753i	$0.252098\pi$
$830$	−98.9553	−3.43479
$831$	0	0
$832$	59.0285	2.04644
$833$	7.70060	0.266810
$834$	0	0
$835$	30.6119	1.05937
$836$	−132.255	−4.57412
$837$	0	0
$838$	36.4091	1.25773
$839$	−28.2930	−0.976782	−0.488391	−	0.872625i	$-0.662416\pi$
$839$	−28.2930	−0.976782	−0.488391	+	0.872625i	$0.662416\pi$
$840$	0	0
$841$	−27.6450	−0.953276
$842$	17.5728	0.605600
$843$	0	0
$844$	−64.4356	−2.21797
$845$	−64.3176	−2.21259
$846$	0	0
$847$	−12.4437	−0.427571
$848$	13.7943	0.473698
$849$	0	0
$850$	−64.7981	−2.22256
$851$	−42.0313	−1.44081
$852$	0	0
$853$	−10.1811	−0.348594	−0.174297	−	0.984693i	$-0.555765\pi$
$853$	−10.1811	−0.348594	−0.174297	+	0.984693i	$0.555765\pi$
$854$	12.8011	0.438044
$855$	0	0
$856$	−64.9864	−2.22119
$857$	−36.0529	−1.23154	−0.615772	−	0.787925i	$-0.711156\pi$
$857$	−36.0529	−1.23154	−0.615772	+	0.787925i	$0.711156\pi$
$858$	0	0
$859$	27.2611	0.930138	0.465069	−	0.885274i	$-0.346029\pi$
$859$	27.2611	0.930138	0.465069	+	0.885274i	$0.346029\pi$
$860$	−117.324	−4.00072
$861$	0	0
$862$	45.2566	1.54144
$863$	−47.4525	−1.61530	−0.807651	−	0.589661i	$-0.799261\pi$
$863$	−47.4525	−1.61530	−0.807651	+	0.589661i	$0.799261\pi$
$864$	0	0
$865$	−4.23286	−0.143922
$866$	75.3325	2.55990
$867$	0	0
$868$	12.1850	0.413587
$869$	−20.3519	−0.690390
$870$	0	0
$871$	−4.30703	−0.145938
$872$	−12.8763	−0.436045
$873$	0	0
$874$	−116.484	−3.94011
$875$	−4.39984	−0.148742
$876$	0	0
$877$	11.7097	0.395410	0.197705	−	0.980262i	$-0.436651\pi$
$877$	11.7097	0.395410	0.197705	+	0.980262i	$0.436651\pi$
$878$	33.0914	1.11678
$879$	0	0
$880$	−41.1281	−1.38643
$881$	−34.8353	−1.17363	−0.586815	−	0.809721i	$-0.699618\pi$
$881$	−34.8353	−1.17363	−0.586815	+	0.809721i	$0.699618\pi$
$882$	0	0
$883$	−30.0139	−1.01005	−0.505025	−	0.863105i	$-0.668517\pi$
$883$	−30.0139	−1.01005	−0.505025	+	0.863105i	$0.668517\pi$
$884$	−173.911	−5.84927
$885$	0	0
$886$	−74.6793	−2.50890
$887$	17.5113	0.587971	0.293986	−	0.955810i	$-0.405018\pi$
$887$	17.5113	0.587971	0.293986	+	0.955810i	$0.405018\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	92.9793	3.11667
$891$	0	0
$892$	−14.0594	−0.470745
$893$	−28.7446	−0.961902
$894$	0	0
$895$	−45.7756	−1.53011
$896$	−20.6004	−0.688210
$897$	0	0
$898$	45.6813	1.52441
$899$	−3.71947	−0.124051
$900$	0	0
$901$	36.4378	1.21392
$902$	19.7358	0.657131
$903$	0	0
$904$	−46.0866	−1.53282
$905$	−50.4814	−1.67806
$906$	0	0
$907$	5.82321	0.193356	0.0966782	−	0.995316i	$-0.469178\pi$
$907$	5.82321	0.193356	0.0966782	+	0.995316i	$0.469178\pi$
$908$	89.4687	2.96912
$909$	0	0
$910$	41.6064	1.37924
$911$	6.00000	0.198789	0.0993944	−	0.995048i	$-0.468309\pi$
$911$	6.00000	0.198789	0.0993944	+	0.995048i	$0.468309\pi$
$912$	0	0
$913$	−68.1999	−2.25709
$914$	15.6563	0.517863
$915$	0	0
$916$	−99.0984	−3.27430
$917$	19.9073	0.657398
$918$	0	0
$919$	−56.7463	−1.87189	−0.935944	−	0.352148i	$-0.885451\pi$
$919$	−56.7463	−1.87189	−0.935944	+	0.352148i	$0.885451\pi$
$920$	−85.9264	−2.83291
$921$	0	0
$922$	−68.6322	−2.26028
$923$	35.7632	1.17716
$924$	0	0
$925$	21.7486	0.715089
$926$	10.8343	0.356037
$927$	0	0
$928$	1.99712	0.0655587
$929$	−17.9985	−0.590512	−0.295256	−	0.955418i	$-0.595405\pi$
$929$	−17.9985	−0.590512	−0.295256	+	0.955418i	$0.595405\pi$
$930$	0	0
$931$	−7.16283	−0.234752
$932$	−94.0679	−3.08130
$933$	0	0
$934$	3.13726	0.102654
$935$	−108.640	−3.55292
$936$	0	0
$937$	−52.5134	−1.71554	−0.857768	−	0.514036i	$-0.828150\pi$
$937$	−52.5134	−1.71554	−0.857768	+	0.514036i	$0.828150\pi$
$938$	1.75348	0.0572532
$939$	0	0
$940$	−44.5900	−1.45436
$941$	−10.8023	−0.352144	−0.176072	−	0.984377i	$-0.556339\pi$
$941$	−10.8023	−0.352144	−0.176072	+	0.984377i	$0.556339\pi$
$942$	0	0
$943$	11.4023	0.371309
$944$	4.13960	0.134733
$945$	0	0
$946$	−123.268	−4.00779
$947$	9.48649	0.308269	0.154135	−	0.988050i	$-0.450741\pi$
$947$	9.48649	0.308269	0.154135	+	0.988050i	$0.450741\pi$
$948$	0	0
$949$	45.4859	1.47653
$950$	60.2730	1.95551
$951$	0	0
$952$	33.6692	1.09123
$953$	43.7726	1.41793	0.708966	−	0.705242i	$-0.249162\pi$
$953$	43.7726	1.41793	0.708966	+	0.705242i	$0.249162\pi$
$954$	0	0
$955$	30.8863	0.999458
$956$	79.8599	2.58285
$957$	0	0
$958$	−58.3536	−1.88532
$959$	1.07424	0.0346891
$960$	0	0
$961$	−20.7899	−0.670643
$962$	88.9846	2.86898
$963$	0	0
$964$	−10.4883	−0.337805
$965$	−72.9840	−2.34944
$966$	0	0
$967$	−16.9435	−0.544865	−0.272432	−	0.962175i	$-0.587828\pi$
$967$	−16.9435	−0.544865	−0.272432	+	0.962175i	$0.587828\pi$
$968$	−54.4076	−1.74873
$969$	0	0
$970$	−0.179871	−0.00577529
$971$	−32.4646	−1.04184	−0.520919	−	0.853606i	$-0.674411\pi$
$971$	−32.4646	−1.04184	−0.520919	+	0.853606i	$0.674411\pi$
$972$	0	0
$973$	5.35399	0.171641
$974$	40.6896	1.30378
$975$	0	0
$976$	15.4776	0.495425
$977$	−30.0860	−0.962535	−0.481268	−	0.876574i	$-0.659823\pi$
$977$	−30.0860	−0.962535	−0.481268	+	0.876574i	$0.659823\pi$
$978$	0	0
$979$	64.0813	2.04805
$980$	−11.1113	−0.354938
$981$	0	0
$982$	−13.2165	−0.421757
$983$	−36.4707	−1.16323	−0.581617	−	0.813462i	$-0.697580\pi$
$983$	−36.4707	−1.16323	−0.581617	+	0.813462i	$0.697580\pi$
$984$	0	0
$985$	−42.5030	−1.35426
$986$	−21.6126	−0.688285
$987$	0	0
$988$	161.767	5.14648
$989$	−71.2175	−2.26458
$990$	0	0
$991$	36.7882	1.16862	0.584308	−	0.811532i	$-0.301366\pi$
$991$	36.7882	1.16862	0.584308	+	0.811532i	$0.301366\pi$
$992$	−5.48216	−0.174059
$993$	0	0
$994$	−14.5600	−0.461814
$995$	−57.8926	−1.83532
$996$	0	0
$997$	−6.17533	−0.195575	−0.0977873	−	0.995207i	$-0.531176\pi$
$997$	−6.17533	−0.195575	−0.0977873	+	0.995207i	$0.531176\pi$
$998$	43.8159	1.38697
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.z.1.4	✓	32
3.2	odd	2	inner	8001.2.a.z.1.29	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8001.2.a.z.1.4	✓	32	1.1	even	1	trivial
8001.2.a.z.1.29	yes	32	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)

\(f(q)\)	\(=\)	\(q-2.41110 q^{2} +3.81340 q^{4} -2.91376 q^{5} -1.00000 q^{7} -4.37229 q^{8} +O(q^{10})\)	\(q-2.41110 q^{2} +3.81340 q^{4} -2.91376 q^{5} -1.00000 q^{7} -4.37229 q^{8} +7.02536 q^{10} +4.84187 q^{11} -5.92231 q^{13} +2.41110 q^{14} +2.91522 q^{16} +7.70060 q^{17} -7.16283 q^{19} -11.1113 q^{20} -11.6742 q^{22} -6.74473 q^{23} +3.48998 q^{25} +14.2793 q^{26} -3.81340 q^{28} +1.16404 q^{29} -3.19532 q^{31} +1.71568 q^{32} -18.5669 q^{34} +2.91376 q^{35} +6.23173 q^{37} +17.2703 q^{38} +12.7398 q^{40} -1.69054 q^{41} +10.5590 q^{43} +18.4640 q^{44} +16.2622 q^{46} +4.01302 q^{47} +1.00000 q^{49} -8.41468 q^{50} -22.5841 q^{52} +4.73182 q^{53} -14.1080 q^{55} +4.37229 q^{56} -2.80661 q^{58} +1.42000 q^{59} +5.30923 q^{61} +7.70423 q^{62} -9.96713 q^{64} +17.2562 q^{65} +0.727254 q^{67} +29.3655 q^{68} -7.02536 q^{70} -6.03872 q^{71} -7.68043 q^{73} -15.0253 q^{74} -27.3148 q^{76} -4.84187 q^{77} -4.20331 q^{79} -8.49425 q^{80} +4.07607 q^{82} -14.0855 q^{83} -22.4377 q^{85} -25.4588 q^{86} -21.1701 q^{88} +13.2348 q^{89} +5.92231 q^{91} -25.7204 q^{92} -9.67580 q^{94} +20.8708 q^{95} -0.0256030 q^{97} -2.41110 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 30 q^{4} - 32 q^{7}+O(q^{10}) \) 32 * q + 30 * q^4 - 32 * q^7	\( 32 q + 30 q^{4} - 32 q^{7} - 16 q^{10} - 14 q^{13} + 18 q^{16} - 30 q^{19} - 10 q^{22} + 36 q^{25} - 30 q^{28} - 58 q^{31} - 34 q^{34} + 8 q^{37} - 34 q^{40} + 6 q^{43} - 36 q^{46} + 32 q^{49} - 56 q^{52} - 88 q^{55} - 22 q^{58} - 46 q^{61} + 20 q^{64} - 8 q^{67} + 16 q^{70} - 60 q^{73} - 128 q^{76} - 74 q^{79} - 52 q^{82} - 16 q^{85} - 64 q^{88} + 14 q^{91} - 58 q^{94} - 44 q^{97}+O(q^{100}) \) 32 * q + 30 * q^4 - 32 * q^7 - 16 * q^10 - 14 * q^13 + 18 * q^16 - 30 * q^19 - 10 * q^22 + 36 * q^25 - 30 * q^28 - 58 * q^31 - 34 * q^34 + 8 * q^37 - 34 * q^40 + 6 * q^43 - 36 * q^46 + 32 * q^49 - 56 * q^52 - 88 * q^55 - 22 * q^58 - 46 * q^61 + 20 * q^64 - 8 * q^67 + 16 * q^70 - 60 * q^73 - 128 * q^76 - 74 * q^79 - 52 * q^82 - 16 * q^85 - 64 * q^88 + 14 * q^91 - 58 * q^94 - 44 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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