LMFDB - Embedded newform 8001.2.a.y.1.14

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			1.14
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)

$f(q)$	$=$	$q-0.0958522 q^{2} -1.99081 q^{4} +1.26637 q^{5} -1.00000 q^{7} +0.382528 q^{8} +O(q^{10})$	\(q-0.0958522 q^{2} -1.99081 q^{4} +1.26637 q^{5} -1.00000 q^{7} +0.382528 q^{8} -0.121384 q^{10} -3.81636 q^{11} +4.33409 q^{13} +0.0958522 q^{14} +3.94496 q^{16} -6.65918 q^{17} +0.635612 q^{19} -2.52110 q^{20} +0.365807 q^{22} -3.34860 q^{23} -3.39631 q^{25} -0.415432 q^{26} +1.99081 q^{28} -5.89589 q^{29} -3.22259 q^{31} -1.14319 q^{32} +0.638297 q^{34} -1.26637 q^{35} +1.02231 q^{37} -0.0609248 q^{38} +0.484422 q^{40} +7.10867 q^{41} +4.51017 q^{43} +7.59765 q^{44} +0.320970 q^{46} -5.62563 q^{47} +1.00000 q^{49} +0.325544 q^{50} -8.62836 q^{52} +1.17123 q^{53} -4.83292 q^{55} -0.382528 q^{56} +0.565134 q^{58} +6.02243 q^{59} +5.62912 q^{61} +0.308893 q^{62} -7.78034 q^{64} +5.48855 q^{65} +5.74306 q^{67} +13.2572 q^{68} +0.121384 q^{70} +11.1586 q^{71} +7.30292 q^{73} -0.0979904 q^{74} -1.26538 q^{76} +3.81636 q^{77} -6.82806 q^{79} +4.99577 q^{80} -0.681382 q^{82} -7.82200 q^{83} -8.43298 q^{85} -0.432310 q^{86} -1.45987 q^{88} +1.47470 q^{89} -4.33409 q^{91} +6.66643 q^{92} +0.539229 q^{94} +0.804918 q^{95} -14.1081 q^{97} -0.0958522 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.0958522	−0.0677778	−0.0338889	−	0.999426i	$-0.510789\pi$
$2$	−0.0958522	−0.0677778	−0.0338889	+	0.999426i	$0.510789\pi$
$3$	0	0
$3$	0	0
$4$	−1.99081	−0.995406
$5$	1.26637	0.566337	0.283168	−	0.959070i	$-0.408614\pi$
$5$	1.26637	0.566337	0.283168	+	0.959070i	$0.408614\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0.382528	0.135244
$9$	0	0
$10$	−0.121384	−0.0383851
$11$	−3.81636	−1.15068	−0.575338	−	0.817916i	$-0.695129\pi$
$11$	−3.81636	−1.15068	−0.575338	+	0.817916i	$0.695129\pi$
$12$	0	0
$13$	4.33409	1.20206	0.601030	−	0.799226i	$-0.294757\pi$
$13$	4.33409	1.20206	0.601030	+	0.799226i	$0.294757\pi$
$14$	0.0958522	0.0256176
$15$	0	0
$16$	3.94496	0.986240
$17$	−6.65918	−1.61509	−0.807544	−	0.589807i	$-0.799204\pi$
$17$	−6.65918	−1.61509	−0.807544	+	0.589807i	$0.799204\pi$
$18$	0	0
$19$	0.635612	0.145819	0.0729097	−	0.997339i	$-0.476772\pi$
$19$	0.635612	0.145819	0.0729097	+	0.997339i	$0.476772\pi$
$20$	−2.52110	−0.563735
$21$	0	0
$22$	0.365807	0.0779902
$23$	−3.34860	−0.698230	−0.349115	−	0.937080i	$-0.613518\pi$
$23$	−3.34860	−0.698230	−0.349115	+	0.937080i	$0.613518\pi$
$24$	0	0
$25$	−3.39631	−0.679262
$26$	−0.415432	−0.0814730
$27$	0	0
$28$	1.99081	0.376228
$29$	−5.89589	−1.09484	−0.547420	−	0.836858i	$-0.684390\pi$
$29$	−5.89589	−1.09484	−0.547420	+	0.836858i	$0.684390\pi$
$30$	0	0
$31$	−3.22259	−0.578795	−0.289397	−	0.957209i	$-0.593455\pi$
$31$	−3.22259	−0.578795	−0.289397	+	0.957209i	$0.593455\pi$
$32$	−1.14319	−0.202089
$33$	0	0
$34$	0.638297	0.109467
$35$	−1.26637	−0.214055
$36$	0	0
$37$	1.02231	0.168066	0.0840332	−	0.996463i	$-0.473220\pi$
$37$	1.02231	0.168066	0.0840332	+	0.996463i	$0.473220\pi$
$38$	−0.0609248	−0.00988331
$39$	0	0
$40$	0.484422	0.0765938
$41$	7.10867	1.11019	0.555094	−	0.831788i	$-0.312682\pi$
$41$	7.10867	1.11019	0.555094	+	0.831788i	$0.312682\pi$
$42$	0	0
$43$	4.51017	0.687794	0.343897	−	0.939007i	$-0.388253\pi$
$43$	4.51017	0.687794	0.343897	+	0.939007i	$0.388253\pi$
$44$	7.59765	1.14539
$45$	0	0
$46$	0.320970	0.0473245
$47$	−5.62563	−0.820582	−0.410291	−	0.911955i	$-0.634573\pi$
$47$	−5.62563	−0.820582	−0.410291	+	0.911955i	$0.634573\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0.325544	0.0460389
$51$	0	0
$52$	−8.62836	−1.19654
$53$	1.17123	0.160881	0.0804407	−	0.996759i	$-0.474367\pi$
$53$	1.17123	0.160881	0.0804407	+	0.996759i	$0.474367\pi$
$54$	0	0
$55$	−4.83292	−0.651670
$56$	−0.382528	−0.0511175
$57$	0	0
$58$	0.565134	0.0742058
$59$	6.02243	0.784054	0.392027	−	0.919954i	$-0.371774\pi$
$59$	6.02243	0.784054	0.392027	+	0.919954i	$0.371774\pi$
$60$	0	0
$61$	5.62912	0.720735	0.360367	−	0.932810i	$-0.382651\pi$
$61$	5.62912	0.720735	0.360367	+	0.932810i	$0.382651\pi$
$62$	0.308893	0.0392294
$63$	0	0
$64$	−7.78034	−0.972542
$65$	5.48855	0.680771
$66$	0	0
$67$	5.74306	0.701626	0.350813	−	0.936446i	$-0.385905\pi$
$67$	5.74306	0.701626	0.350813	+	0.936446i	$0.385905\pi$
$68$	13.2572	1.60767
$69$	0	0
$70$	0.121384	0.0145082
$71$	11.1586	1.32428	0.662140	−	0.749380i	$-0.269648\pi$
$71$	11.1586	1.32428	0.662140	+	0.749380i	$0.269648\pi$
$72$	0	0
$73$	7.30292	0.854742	0.427371	−	0.904076i	$-0.359440\pi$
$73$	7.30292	0.854742	0.427371	+	0.904076i	$0.359440\pi$
$74$	−0.0979904	−0.0113912
$75$	0	0
$76$	−1.26538	−0.145149
$77$	3.81636	0.434914
$78$	0	0
$79$	−6.82806	−0.768217	−0.384109	−	0.923288i	$-0.625491\pi$
$79$	−6.82806	−0.768217	−0.384109	+	0.923288i	$0.625491\pi$
$80$	4.99577	0.558544
$81$	0	0
$82$	−0.681382	−0.0752461
$83$	−7.82200	−0.858576	−0.429288	−	0.903168i	$-0.641236\pi$
$83$	−7.82200	−0.858576	−0.429288	+	0.903168i	$0.641236\pi$
$84$	0	0
$85$	−8.43298	−0.914685
$86$	−0.432310	−0.0466172
$87$	0	0
$88$	−1.45987	−0.155622
$89$	1.47470	0.156318	0.0781590	−	0.996941i	$-0.475096\pi$
$89$	1.47470	0.156318	0.0781590	+	0.996941i	$0.475096\pi$
$90$	0	0
$91$	−4.33409	−0.454336
$92$	6.66643	0.695023
$93$	0	0
$94$	0.539229	0.0556172
$95$	0.804918	0.0825829
$96$	0	0
$97$	−14.1081	−1.43246	−0.716231	−	0.697863i	$-0.754134\pi$
$97$	−14.1081	−1.43246	−0.716231	+	0.697863i	$0.754134\pi$
$98$	−0.0958522	−0.00968254
$99$	0	0
$100$	6.76142	0.676142
$101$	−6.15691	−0.612635	−0.306318	−	0.951929i	$-0.599097\pi$
$101$	−6.15691	−0.612635	−0.306318	+	0.951929i	$0.599097\pi$
$102$	0	0
$103$	17.7177	1.74578	0.872888	−	0.487920i	$-0.162244\pi$
$103$	17.7177	1.74578	0.872888	+	0.487920i	$0.162244\pi$
$104$	1.65791	0.162572
$105$	0	0
$106$	−0.112265	−0.0109042
$107$	0.266921	0.0258043	0.0129021	−	0.999917i	$-0.495893\pi$
$107$	0.266921	0.0258043	0.0129021	+	0.999917i	$0.495893\pi$
$108$	0	0
$109$	−5.58786	−0.535220	−0.267610	−	0.963527i	$-0.586234\pi$
$109$	−5.58786	−0.535220	−0.267610	+	0.963527i	$0.586234\pi$
$110$	0.463246	0.0441687
$111$	0	0
$112$	−3.94496	−0.372764
$113$	−7.43321	−0.699257	−0.349629	−	0.936888i	$-0.613692\pi$
$113$	−7.43321	−0.699257	−0.349629	+	0.936888i	$0.613692\pi$
$114$	0	0
$115$	−4.24055	−0.395434
$116$	11.7376	1.08981
$117$	0	0
$118$	−0.577264	−0.0531414
$119$	6.65918	0.610446
$120$	0	0
$121$	3.56460	0.324054
$122$	−0.539564	−0.0488498
$123$	0	0
$124$	6.41558	0.576136
$125$	−10.6328	−0.951028
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	3.03214	0.268006
$129$	0	0
$130$	−0.526090	−0.0461411
$131$	10.0704	0.879852	0.439926	−	0.898034i	$-0.355005\pi$
$131$	10.0704	0.879852	0.439926	+	0.898034i	$0.355005\pi$
$132$	0	0
$133$	−0.635612	−0.0551145
$134$	−0.550485	−0.0475546
$135$	0	0
$136$	−2.54733	−0.218431
$137$	−1.08742	−0.0929046	−0.0464523	−	0.998921i	$-0.514792\pi$
$137$	−1.08742	−0.0929046	−0.0464523	+	0.998921i	$0.514792\pi$
$138$	0	0
$139$	−16.8187	−1.42655	−0.713274	−	0.700886i	$-0.752788\pi$
$139$	−16.8187	−1.42655	−0.713274	+	0.700886i	$0.752788\pi$
$140$	2.52110	0.213072
$141$	0	0
$142$	−1.06958	−0.0897568
$143$	−16.5404	−1.38318
$144$	0	0
$145$	−7.46637	−0.620048
$146$	−0.700001	−0.0579325
$147$	0	0
$148$	−2.03522	−0.167294
$149$	7.42345	0.608153	0.304076	−	0.952648i	$-0.401652\pi$
$149$	7.42345	0.608153	0.304076	+	0.952648i	$0.401652\pi$
$150$	0	0
$151$	14.2503	1.15968	0.579838	−	0.814732i	$-0.303116\pi$
$151$	14.2503	1.15968	0.579838	+	0.814732i	$0.303116\pi$
$152$	0.243139	0.0197212
$153$	0	0
$154$	−0.365807	−0.0294775
$155$	−4.08099	−0.327793
$156$	0	0
$157$	−3.75230	−0.299466	−0.149733	−	0.988726i	$-0.547841\pi$
$157$	−3.75230	−0.299466	−0.149733	+	0.988726i	$0.547841\pi$
$158$	0.654485	0.0520680
$159$	0	0
$160$	−1.44770	−0.114451
$161$	3.34860	0.263906
$162$	0	0
$163$	16.2486	1.27269	0.636344	−	0.771406i	$-0.280446\pi$
$163$	16.2486	1.27269	0.636344	+	0.771406i	$0.280446\pi$
$164$	−14.1520	−1.10509
$165$	0	0
$166$	0.749756	0.0581924
$167$	9.85761	0.762804	0.381402	−	0.924409i	$-0.375441\pi$
$167$	9.85761	0.762804	0.381402	+	0.924409i	$0.375441\pi$
$168$	0	0
$169$	5.78434	0.444949
$170$	0.808319	0.0619953
$171$	0	0
$172$	−8.97890	−0.684635
$173$	−5.01326	−0.381151	−0.190575	−	0.981673i	$-0.561035\pi$
$173$	−5.01326	−0.381151	−0.190575	+	0.981673i	$0.561035\pi$
$174$	0	0
$175$	3.39631	0.256737
$176$	−15.0554	−1.13484
$177$	0	0
$178$	−0.141353	−0.0105949
$179$	−6.91717	−0.517014	−0.258507	−	0.966009i	$-0.583230\pi$
$179$	−6.91717	−0.517014	−0.258507	+	0.966009i	$0.583230\pi$
$180$	0	0
$181$	−6.55394	−0.487151	−0.243575	−	0.969882i	$-0.578320\pi$
$181$	−6.55394	−0.487151	−0.243575	+	0.969882i	$0.578320\pi$
$182$	0.415432	0.0307939
$183$	0	0
$184$	−1.28093	−0.0944316
$185$	1.29462	0.0951822
$186$	0	0
$187$	25.4138	1.85844
$188$	11.1996	0.816813
$189$	0	0
$190$	−0.0771532	−0.00559728
$191$	17.3623	1.25629	0.628147	−	0.778095i	$-0.283814\pi$
$191$	17.3623	1.25629	0.628147	+	0.778095i	$0.283814\pi$
$192$	0	0
$193$	−13.1367	−0.945601	−0.472801	−	0.881169i	$-0.656757\pi$
$193$	−13.1367	−0.945601	−0.472801	+	0.881169i	$0.656757\pi$
$194$	1.35229	0.0970891
$195$	0	0
$196$	−1.99081	−0.142201
$197$	−0.622155	−0.0443267	−0.0221634	−	0.999754i	$-0.507055\pi$
$197$	−0.622155	−0.0443267	−0.0221634	+	0.999754i	$0.507055\pi$
$198$	0	0
$199$	−1.27384	−0.0903001	−0.0451501	−	0.998980i	$-0.514377\pi$
$199$	−1.27384	−0.0903001	−0.0451501	+	0.998980i	$0.514377\pi$
$200$	−1.29919	−0.0918663
$201$	0	0
$202$	0.590153	0.0415231
$203$	5.89589	0.413810
$204$	0	0
$205$	9.00220	0.628741
$206$	−1.69828	−0.118325
$207$	0	0
$208$	17.0978	1.18552
$209$	−2.42572	−0.167791
$210$	0	0
$211$	15.3752	1.05847	0.529237	−	0.848474i	$-0.322478\pi$
$211$	15.3752	1.05847	0.529237	+	0.848474i	$0.322478\pi$
$212$	−2.33171	−0.160142
$213$	0	0
$214$	−0.0255850	−0.00174895
$215$	5.71153	0.389523
$216$	0	0
$217$	3.22259	0.218764
$218$	0.535609	0.0362760
$219$	0	0
$220$	9.62143	0.648676
$221$	−28.8615	−1.94143
$222$	0	0
$223$	11.7223	0.784983	0.392492	−	0.919756i	$-0.371613\pi$
$223$	11.7223	0.784983	0.392492	+	0.919756i	$0.371613\pi$
$224$	1.14319	0.0763826
$225$	0	0
$226$	0.712489	0.0473941
$227$	−15.0463	−0.998655	−0.499327	−	0.866413i	$-0.666420\pi$
$227$	−15.0463	−0.998655	−0.499327	+	0.866413i	$0.666420\pi$
$228$	0	0
$229$	10.5521	0.697301	0.348651	−	0.937253i	$-0.386640\pi$
$229$	10.5521	0.697301	0.348651	+	0.937253i	$0.386640\pi$
$230$	0.406467	0.0268016
$231$	0	0
$232$	−2.25535	−0.148071
$233$	−2.55680	−0.167501	−0.0837507	−	0.996487i	$-0.526690\pi$
$233$	−2.55680	−0.167501	−0.0837507	+	0.996487i	$0.526690\pi$
$234$	0	0
$235$	−7.12412	−0.464726
$236$	−11.9895	−0.780452
$237$	0	0
$238$	−0.638297	−0.0413747
$239$	−11.8858	−0.768830	−0.384415	−	0.923160i	$-0.625597\pi$
$239$	−11.8858	−0.768830	−0.384415	+	0.923160i	$0.625597\pi$
$240$	0	0
$241$	12.0407	0.775607	0.387804	−	0.921742i	$-0.373234\pi$
$241$	12.0407	0.775607	0.387804	+	0.921742i	$0.373234\pi$
$242$	−0.341675	−0.0219637
$243$	0	0
$244$	−11.2065	−0.717424
$245$	1.26637	0.0809053
$246$	0	0
$247$	2.75480	0.175284
$248$	−1.23273	−0.0782786
$249$	0	0
$250$	1.01918	0.0644586
$251$	14.0925	0.889513	0.444756	−	0.895652i	$-0.353290\pi$
$251$	14.0925	0.889513	0.444756	+	0.895652i	$0.353290\pi$
$252$	0	0
$253$	12.7794	0.803437
$254$	−0.0958522	−0.00601430
$255$	0	0
$256$	15.2700	0.954378
$257$	−21.1606	−1.31996	−0.659981	−	0.751283i	$-0.729436\pi$
$257$	−21.1606	−1.31996	−0.659981	+	0.751283i	$0.729436\pi$
$258$	0	0
$259$	−1.02231	−0.0635231
$260$	−10.9267	−0.677644
$261$	0	0
$262$	−0.965267	−0.0596344
$263$	5.22017	0.321890	0.160945	−	0.986963i	$-0.448546\pi$
$263$	5.22017	0.321890	0.160945	+	0.986963i	$0.448546\pi$
$264$	0	0
$265$	1.48321	0.0911131
$266$	0.0609248	0.00373554
$267$	0	0
$268$	−11.4333	−0.698403
$269$	18.2060	1.11004	0.555019	−	0.831838i	$-0.312711\pi$
$269$	18.2060	1.11004	0.555019	+	0.831838i	$0.312711\pi$
$270$	0	0
$271$	8.95597	0.544036	0.272018	−	0.962292i	$-0.412309\pi$
$271$	8.95597	0.544036	0.272018	+	0.962292i	$0.412309\pi$
$272$	−26.2702	−1.59286
$273$	0	0
$274$	0.104232	0.00629687
$275$	12.9615	0.781611
$276$	0	0
$277$	28.7235	1.72583	0.862915	−	0.505349i	$-0.168636\pi$
$277$	28.7235	1.72583	0.862915	+	0.505349i	$0.168636\pi$
$278$	1.61211	0.0966882
$279$	0	0
$280$	−0.484422	−0.0289497
$281$	20.5362	1.22509	0.612545	−	0.790436i	$-0.290146\pi$
$281$	20.5362	1.22509	0.612545	+	0.790436i	$0.290146\pi$
$282$	0	0
$283$	25.9163	1.54056	0.770281	−	0.637704i	$-0.220116\pi$
$283$	25.9163	1.54056	0.770281	+	0.637704i	$0.220116\pi$
$284$	−22.2147	−1.31820
$285$	0	0
$286$	1.58544	0.0937489
$287$	−7.10867	−0.419612
$288$	0	0
$289$	27.3447	1.60851
$290$	0.715668	0.0420255
$291$	0	0
$292$	−14.5387	−0.850815
$293$	6.15259	0.359438	0.179719	−	0.983718i	$-0.442481\pi$
$293$	6.15259	0.359438	0.179719	+	0.983718i	$0.442481\pi$
$294$	0	0
$295$	7.62662	0.444039
$296$	0.391061	0.0227300
$297$	0	0
$298$	−0.711554	−0.0412192
$299$	−14.5131	−0.839315
$300$	0	0
$301$	−4.51017	−0.259962
$302$	−1.36593	−0.0786003
$303$	0	0
$304$	2.50746	0.143813
$305$	7.12854	0.408179
$306$	0	0
$307$	11.4262	0.652126	0.326063	−	0.945348i	$-0.394278\pi$
$307$	11.4262	0.652126	0.326063	+	0.945348i	$0.394278\pi$
$308$	−7.59765	−0.432917
$309$	0	0
$310$	0.391172	0.0222171
$311$	28.7019	1.62753	0.813767	−	0.581191i	$-0.197413\pi$
$311$	28.7019	1.62753	0.813767	+	0.581191i	$0.197413\pi$
$312$	0	0
$313$	2.05815	0.116334	0.0581668	−	0.998307i	$-0.481474\pi$
$313$	2.05815	0.116334	0.0581668	+	0.998307i	$0.481474\pi$
$314$	0.359666	0.0202971
$315$	0	0
$316$	13.5934	0.764688
$317$	15.6084	0.876654	0.438327	−	0.898816i	$-0.355571\pi$
$317$	15.6084	0.876654	0.438327	+	0.898816i	$0.355571\pi$
$318$	0	0
$319$	22.5008	1.25981
$320$	−9.85277	−0.550787
$321$	0	0
$322$	−0.320970	−0.0178870
$323$	−4.23265	−0.235511
$324$	0	0
$325$	−14.7199	−0.816514
$326$	−1.55746	−0.0862599
$327$	0	0
$328$	2.71927	0.150146
$329$	5.62563	0.310151
$330$	0	0
$331$	15.7809	0.867394	0.433697	−	0.901059i	$-0.357209\pi$
$331$	15.7809	0.867394	0.433697	+	0.901059i	$0.357209\pi$
$332$	15.5721	0.854632
$333$	0	0
$334$	−0.944874	−0.0517012
$335$	7.27282	0.397357
$336$	0	0
$337$	24.8367	1.35294	0.676472	−	0.736468i	$-0.263508\pi$
$337$	24.8367	1.35294	0.676472	+	0.736468i	$0.263508\pi$
$338$	−0.554442	−0.0301576
$339$	0	0
$340$	16.7885	0.910483
$341$	12.2986	0.666005
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	1.72527	0.0930202
$345$	0	0
$346$	0.480532	0.0258336
$347$	−5.35114	−0.287265	−0.143632	−	0.989631i	$-0.545878\pi$
$347$	−5.35114	−0.287265	−0.143632	+	0.989631i	$0.545878\pi$
$348$	0	0
$349$	−4.47824	−0.239714	−0.119857	−	0.992791i	$-0.538244\pi$
$349$	−4.47824	−0.239714	−0.119857	+	0.992791i	$0.538244\pi$
$350$	−0.325544	−0.0174011
$351$	0	0
$352$	4.36282	0.232539
$353$	−11.9927	−0.638306	−0.319153	−	0.947703i	$-0.603398\pi$
$353$	−11.9927	−0.638306	−0.319153	+	0.947703i	$0.603398\pi$
$354$	0	0
$355$	14.1309	0.749989
$356$	−2.93585	−0.155600
$357$	0	0
$358$	0.663027	0.0350420
$359$	−1.85710	−0.0980142	−0.0490071	−	0.998798i	$-0.515606\pi$
$359$	−1.85710	−0.0980142	−0.0490071	+	0.998798i	$0.515606\pi$
$360$	0	0
$361$	−18.5960	−0.978737
$362$	0.628210	0.0330180
$363$	0	0
$364$	8.62836	0.452249
$365$	9.24818	0.484072
$366$	0	0
$367$	2.96824	0.154941	0.0774705	−	0.996995i	$-0.475316\pi$
$367$	2.96824	0.154941	0.0774705	+	0.996995i	$0.475316\pi$
$368$	−13.2101	−0.688623
$369$	0	0
$370$	−0.124092	−0.00645123
$371$	−1.17123	−0.0608075
$372$	0	0
$373$	24.5776	1.27258	0.636290	−	0.771450i	$-0.280468\pi$
$373$	24.5776	1.27258	0.636290	+	0.771450i	$0.280468\pi$
$374$	−2.43597	−0.125961
$375$	0	0
$376$	−2.15196	−0.110979
$377$	−25.5533	−1.31606
$378$	0	0
$379$	33.4482	1.71812	0.859058	−	0.511878i	$-0.171050\pi$
$379$	33.4482	1.71812	0.859058	+	0.511878i	$0.171050\pi$
$380$	−1.60244	−0.0822035
$381$	0	0
$382$	−1.66422	−0.0851488
$383$	−17.8344	−0.911296	−0.455648	−	0.890160i	$-0.650592\pi$
$383$	−17.8344	−0.911296	−0.455648	+	0.890160i	$0.650592\pi$
$384$	0	0
$385$	4.83292	0.246308
$386$	1.25918	0.0640907
$387$	0	0
$388$	28.0866	1.42588
$389$	−20.0265	−1.01539	−0.507693	−	0.861538i	$-0.669502\pi$
$389$	−20.0265	−1.01539	−0.507693	+	0.861538i	$0.669502\pi$
$390$	0	0
$391$	22.2989	1.12770
$392$	0.382528	0.0193206
$393$	0	0
$394$	0.0596350	0.00300437
$395$	−8.64684	−0.435070
$396$	0	0
$397$	−12.3422	−0.619439	−0.309719	−	0.950828i	$-0.600235\pi$
$397$	−12.3422	−0.619439	−0.309719	+	0.950828i	$0.600235\pi$
$398$	0.122100	0.00612034
$399$	0	0
$400$	−13.3983	−0.669916
$401$	31.8047	1.58825	0.794126	−	0.607754i	$-0.207929\pi$
$401$	31.8047	1.58825	0.794126	+	0.607754i	$0.207929\pi$
$402$	0	0
$403$	−13.9670	−0.695746
$404$	12.2573	0.609821
$405$	0	0
$406$	−0.565134	−0.0280471
$407$	−3.90149	−0.193390
$408$	0	0
$409$	−3.64012	−0.179992	−0.0899961	−	0.995942i	$-0.528685\pi$
$409$	−3.64012	−0.179992	−0.0899961	+	0.995942i	$0.528685\pi$
$410$	−0.862881	−0.0426146
$411$	0	0
$412$	−35.2726	−1.73776
$413$	−6.02243	−0.296345
$414$	0	0
$415$	−9.90553	−0.486243
$416$	−4.95469	−0.242924
$417$	0	0
$418$	0.232511	0.0113725
$419$	−9.23283	−0.451053	−0.225527	−	0.974237i	$-0.572410\pi$
$419$	−9.23283	−0.451053	−0.225527	+	0.974237i	$0.572410\pi$
$420$	0	0
$421$	5.42277	0.264289	0.132145	−	0.991230i	$-0.457814\pi$
$421$	5.42277	0.264289	0.132145	+	0.991230i	$0.457814\pi$
$422$	−1.47375	−0.0717410
$423$	0	0
$424$	0.448030	0.0217583
$425$	22.6167	1.09707
$426$	0	0
$427$	−5.62912	−0.272412
$428$	−0.531390	−0.0256857
$429$	0	0
$430$	−0.547463	−0.0264010
$431$	−13.8921	−0.669160	−0.334580	−	0.942367i	$-0.608594\pi$
$431$	−13.8921	−0.669160	−0.334580	+	0.942367i	$0.608594\pi$
$432$	0	0
$433$	−20.8389	−1.00145	−0.500727	−	0.865605i	$-0.666934\pi$
$433$	−20.8389	−1.00145	−0.500727	+	0.865605i	$0.666934\pi$
$434$	−0.308893	−0.0148273
$435$	0	0
$436$	11.1244	0.532762
$437$	−2.12841	−0.101815
$438$	0	0
$439$	−13.5145	−0.645011	−0.322505	−	0.946568i	$-0.604525\pi$
$439$	−13.5145	−0.645011	−0.322505	+	0.946568i	$0.604525\pi$
$440$	−1.84873	−0.0881346
$441$	0	0
$442$	2.76644	0.131586
$443$	29.3679	1.39531	0.697656	−	0.716433i	$-0.254227\pi$
$443$	29.3679	1.39531	0.697656	+	0.716433i	$0.254227\pi$
$444$	0	0
$445$	1.86751	0.0885287
$446$	−1.12361	−0.0532044
$447$	0	0
$448$	7.78034	0.367587
$449$	3.66521	0.172972	0.0864859	−	0.996253i	$-0.472436\pi$
$449$	3.66521	0.172972	0.0864859	+	0.996253i	$0.472436\pi$
$450$	0	0
$451$	−27.1292	−1.27747
$452$	14.7981	0.696045
$453$	0	0
$454$	1.44222	0.0676866
$455$	−5.48855	−0.257307
$456$	0	0
$457$	4.44008	0.207698	0.103849	−	0.994593i	$-0.466884\pi$
$457$	4.44008	0.207698	0.103849	+	0.994593i	$0.466884\pi$
$458$	−1.01144	−0.0472615
$459$	0	0
$460$	8.44215	0.393617
$461$	−25.7832	−1.20084	−0.600421	−	0.799684i	$-0.705000\pi$
$461$	−25.7832	−1.20084	−0.600421	+	0.799684i	$0.705000\pi$
$462$	0	0
$463$	−37.5798	−1.74648	−0.873240	−	0.487291i	$-0.837985\pi$
$463$	−37.5798	−1.74648	−0.873240	+	0.487291i	$0.837985\pi$
$464$	−23.2590	−1.07977
$465$	0	0
$466$	0.245075	0.0113529
$467$	−19.4322	−0.899216	−0.449608	−	0.893226i	$-0.648436\pi$
$467$	−19.4322	−0.899216	−0.449608	+	0.893226i	$0.648436\pi$
$468$	0	0
$469$	−5.74306	−0.265190
$470$	0.682862	0.0314981
$471$	0	0
$472$	2.30375	0.106039
$473$	−17.2124	−0.791428
$474$	0	0
$475$	−2.15874	−0.0990496
$476$	−13.2572	−0.607642
$477$	0	0
$478$	1.13928	0.0521096
$479$	−26.0260	−1.18916	−0.594579	−	0.804038i	$-0.702681\pi$
$479$	−26.0260	−1.18916	−0.594579	+	0.804038i	$0.702681\pi$
$480$	0	0
$481$	4.43077	0.202026
$482$	−1.15412	−0.0525689
$483$	0	0
$484$	−7.09644	−0.322566
$485$	−17.8661	−0.811256
$486$	0	0
$487$	10.2081	0.462573	0.231287	−	0.972886i	$-0.425706\pi$
$487$	10.2081	0.462573	0.231287	+	0.972886i	$0.425706\pi$
$488$	2.15330	0.0974752
$489$	0	0
$490$	−0.121384	−0.00548358
$491$	23.4197	1.05692	0.528458	−	0.848959i	$-0.322770\pi$
$491$	23.4197	1.05692	0.528458	+	0.848959i	$0.322770\pi$
$492$	0	0
$493$	39.2618	1.76826
$494$	−0.264054	−0.0118803
$495$	0	0
$496$	−12.7130	−0.570830
$497$	−11.1586	−0.500531
$498$	0	0
$499$	10.8342	0.485006	0.242503	−	0.970151i	$-0.422032\pi$
$499$	10.8342	0.485006	0.242503	+	0.970151i	$0.422032\pi$
$500$	21.1680	0.946660
$501$	0	0
$502$	−1.35080	−0.0602892
$503$	31.1992	1.39110	0.695552	−	0.718475i	$-0.255160\pi$
$503$	31.1992	1.39110	0.695552	+	0.718475i	$0.255160\pi$
$504$	0	0
$505$	−7.79691	−0.346958
$506$	−1.22494	−0.0544551
$507$	0	0
$508$	−1.99081	−0.0883280
$509$	5.38386	0.238635	0.119318	−	0.992856i	$-0.461929\pi$
$509$	5.38386	0.238635	0.119318	+	0.992856i	$0.461929\pi$
$510$	0	0
$511$	−7.30292	−0.323062
$512$	−7.52795	−0.332692
$513$	0	0
$514$	2.02829	0.0894640
$515$	22.4371	0.988698
$516$	0	0
$517$	21.4694	0.944224
$518$	0.0979904	0.00430545
$519$	0	0
$520$	2.09953	0.0920703
$521$	20.8338	0.912746	0.456373	−	0.889788i	$-0.349148\pi$
$521$	20.8338	0.912746	0.456373	+	0.889788i	$0.349148\pi$
$522$	0	0
$523$	32.6149	1.42615	0.713075	−	0.701088i	$-0.247302\pi$
$523$	32.6149	1.42615	0.713075	+	0.701088i	$0.247302\pi$
$524$	−20.0482	−0.875810
$525$	0	0
$526$	−0.500365	−0.0218170
$527$	21.4598	0.934805
$528$	0	0
$529$	−11.7869	−0.512474
$530$	−0.142169	−0.00617544
$531$	0	0
$532$	1.26538	0.0548613
$533$	30.8096	1.33451
$534$	0	0
$535$	0.338021	0.0146139
$536$	2.19688	0.0948908
$537$	0	0
$538$	−1.74508	−0.0752359
$539$	−3.81636	−0.164382
$540$	0	0
$541$	0.0614040	0.00263996	0.00131998	−	0.999999i	$-0.499580\pi$
$541$	0.0614040	0.00263996	0.00131998	+	0.999999i	$0.499580\pi$
$542$	−0.858449	−0.0368736
$543$	0	0
$544$	7.61271	0.326392
$545$	−7.07629	−0.303115
$546$	0	0
$547$	−3.50340	−0.149795	−0.0748973	−	0.997191i	$-0.523863\pi$
$547$	−3.50340	−0.149795	−0.0748973	+	0.997191i	$0.523863\pi$
$548$	2.16485	0.0924778
$549$	0	0
$550$	−1.24239	−0.0529758
$551$	−3.74750	−0.159649
$552$	0	0
$553$	6.82806	0.290359
$554$	−2.75322	−0.116973
$555$	0	0
$556$	33.4830	1.41999
$557$	32.3766	1.37184	0.685920	−	0.727677i	$-0.259400\pi$
$557$	32.3766	1.37184	0.685920	+	0.727677i	$0.259400\pi$
$558$	0	0
$559$	19.5475	0.826770
$560$	−4.99577	−0.211110
$561$	0	0
$562$	−1.96844	−0.0830338
$563$	−7.69977	−0.324507	−0.162253	−	0.986749i	$-0.551876\pi$
$563$	−7.69977	−0.324507	−0.162253	+	0.986749i	$0.551876\pi$
$564$	0	0
$565$	−9.41318	−0.396015
$566$	−2.48413	−0.104416
$567$	0	0
$568$	4.26847	0.179101
$569$	−24.3873	−1.02237	−0.511183	−	0.859472i	$-0.670793\pi$
$569$	−24.3873	−1.02237	−0.511183	+	0.859472i	$0.670793\pi$
$570$	0	0
$571$	−23.5247	−0.984479	−0.492240	−	0.870460i	$-0.663822\pi$
$571$	−23.5247	−0.984479	−0.492240	+	0.870460i	$0.663822\pi$
$572$	32.9289	1.37683
$573$	0	0
$574$	0.681382	0.0284403
$575$	11.3729	0.474282
$576$	0	0
$577$	−6.11550	−0.254591	−0.127296	−	0.991865i	$-0.540630\pi$
$577$	−6.11550	−0.254591	−0.127296	+	0.991865i	$0.540630\pi$
$578$	−2.62105	−0.109021
$579$	0	0
$580$	14.8641	0.617200
$581$	7.82200	0.324511
$582$	0	0
$583$	−4.46985	−0.185122
$584$	2.79357	0.115599
$585$	0	0
$586$	−0.589739	−0.0243619
$587$	41.9754	1.73251	0.866256	−	0.499600i	$-0.166520\pi$
$587$	41.9754	1.73251	0.866256	+	0.499600i	$0.166520\pi$
$588$	0	0
$589$	−2.04832	−0.0843995
$590$	−0.731028	−0.0300960
$591$	0	0
$592$	4.03296	0.165754
$593$	12.0895	0.496456	0.248228	−	0.968702i	$-0.420152\pi$
$593$	12.0895	0.496456	0.248228	+	0.968702i	$0.420152\pi$
$594$	0	0
$595$	8.43298	0.345718
$596$	−14.7787	−0.605359
$597$	0	0
$598$	1.39111	0.0568869
$599$	25.1578	1.02792	0.513959	−	0.857815i	$-0.328178\pi$
$599$	25.1578	1.02792	0.513959	+	0.857815i	$0.328178\pi$
$600$	0	0
$601$	−5.79385	−0.236336	−0.118168	−	0.992994i	$-0.537702\pi$
$601$	−5.79385	−0.236336	−0.118168	+	0.992994i	$0.537702\pi$
$602$	0.432310	0.0176196
$603$	0	0
$604$	−28.3698	−1.15435
$605$	4.51409	0.183524
$606$	0	0
$607$	3.32817	0.135086	0.0675431	−	0.997716i	$-0.478484\pi$
$607$	3.32817	0.135086	0.0675431	+	0.997716i	$0.478484\pi$
$608$	−0.726625	−0.0294685
$609$	0	0
$610$	−0.683286	−0.0276654
$611$	−24.3820	−0.986389
$612$	0	0
$613$	8.89568	0.359293	0.179647	−	0.983731i	$-0.442505\pi$
$613$	8.89568	0.359293	0.179647	+	0.983731i	$0.442505\pi$
$614$	−1.09522	−0.0441996
$615$	0	0
$616$	1.45987	0.0588196
$617$	3.16076	0.127248	0.0636238	−	0.997974i	$-0.479734\pi$
$617$	3.16076	0.127248	0.0636238	+	0.997974i	$0.479734\pi$
$618$	0	0
$619$	36.8371	1.48061	0.740304	−	0.672273i	$-0.234682\pi$
$619$	36.8371	1.48061	0.740304	+	0.672273i	$0.234682\pi$
$620$	8.12448	0.326287
$621$	0	0
$622$	−2.75114	−0.110311
$623$	−1.47470	−0.0590827
$624$	0	0
$625$	3.51650	0.140660
$626$	−0.197278	−0.00788483
$627$	0	0
$628$	7.47012	0.298090
$629$	−6.80773	−0.271442
$630$	0	0
$631$	−17.0398	−0.678345	−0.339172	−	0.940724i	$-0.610147\pi$
$631$	−17.0398	−0.678345	−0.339172	+	0.940724i	$0.610147\pi$
$632$	−2.61193	−0.103897
$633$	0	0
$634$	−1.49610	−0.0594176
$635$	1.26637	0.0502543
$636$	0	0
$637$	4.33409	0.171723
$638$	−2.15676	−0.0853868
$639$	0	0
$640$	3.83981	0.151782
$641$	−32.9956	−1.30325	−0.651623	−	0.758543i	$-0.725912\pi$
$641$	−32.9956	−1.30325	−0.651623	+	0.758543i	$0.725912\pi$
$642$	0	0
$643$	1.55797	0.0614405	0.0307203	−	0.999528i	$-0.490220\pi$
$643$	1.55797	0.0614405	0.0307203	+	0.999528i	$0.490220\pi$
$644$	−6.66643	−0.262694
$645$	0	0
$646$	0.405709	0.0159624
$647$	14.7373	0.579381	0.289691	−	0.957120i	$-0.406448\pi$
$647$	14.7373	0.579381	0.289691	+	0.957120i	$0.406448\pi$
$648$	0	0
$649$	−22.9838	−0.902192
$650$	1.41094	0.0553415
$651$	0	0
$652$	−32.3479	−1.26684
$653$	14.5714	0.570223	0.285111	−	0.958494i	$-0.407969\pi$
$653$	14.5714	0.570223	0.285111	+	0.958494i	$0.407969\pi$
$654$	0	0
$655$	12.7528	0.498293
$656$	28.0434	1.09491
$657$	0	0
$658$	−0.539229	−0.0210213
$659$	16.4172	0.639523	0.319762	−	0.947498i	$-0.396397\pi$
$659$	16.4172	0.639523	0.319762	+	0.947498i	$0.396397\pi$
$660$	0	0
$661$	13.8785	0.539809	0.269905	−	0.962887i	$-0.413008\pi$
$661$	13.8785	0.539809	0.269905	+	0.962887i	$0.413008\pi$
$662$	−1.51263	−0.0587900
$663$	0	0
$664$	−2.99214	−0.116117
$665$	−0.804918	−0.0312134
$666$	0	0
$667$	19.7430	0.764450
$668$	−19.6246	−0.759300
$669$	0	0
$670$	−0.697116	−0.0269319
$671$	−21.4827	−0.829332
$672$	0	0
$673$	16.9690	0.654106	0.327053	−	0.945006i	$-0.393945\pi$
$673$	16.9690	0.654106	0.327053	+	0.945006i	$0.393945\pi$
$674$	−2.38066	−0.0916995
$675$	0	0
$676$	−11.5155	−0.442905
$677$	−41.6406	−1.60038	−0.800190	−	0.599747i	$-0.795268\pi$
$677$	−41.6406	−1.60038	−0.800190	+	0.599747i	$0.795268\pi$
$678$	0	0
$679$	14.1081	0.541420
$680$	−3.22585	−0.123706
$681$	0	0
$682$	−1.17885	−0.0451403
$683$	−12.8820	−0.492916	−0.246458	−	0.969153i	$-0.579267\pi$
$683$	−12.8820	−0.492916	−0.246458	+	0.969153i	$0.579267\pi$
$684$	0	0
$685$	−1.37707	−0.0526153
$686$	0.0958522	0.00365966
$687$	0	0
$688$	17.7924	0.678330
$689$	5.07624	0.193389
$690$	0	0
$691$	23.6851	0.901023	0.450512	−	0.892771i	$-0.351242\pi$
$691$	23.6851	0.901023	0.450512	+	0.892771i	$0.351242\pi$
$692$	9.98046	0.379400
$693$	0	0
$694$	0.512919	0.0194701
$695$	−21.2987	−0.807906
$696$	0	0
$697$	−47.3379	−1.79305
$698$	0.429249	0.0162473
$699$	0	0
$700$	−6.76142	−0.255558
$701$	51.4264	1.94235	0.971175	−	0.238369i	$-0.0766127\pi$
$701$	51.4264	1.94235	0.971175	+	0.238369i	$0.0766127\pi$
$702$	0	0
$703$	0.649791	0.0245073
$704$	29.6926	1.11908
$705$	0	0
$706$	1.14952	0.0432629
$707$	6.15691	0.231554
$708$	0	0
$709$	33.3449	1.25229	0.626147	−	0.779705i	$-0.284631\pi$
$709$	33.3449	1.25229	0.626147	+	0.779705i	$0.284631\pi$
$710$	−1.35448	−0.0508326
$711$	0	0
$712$	0.564115	0.0211411
$713$	10.7912	0.404132
$714$	0	0
$715$	−20.9463	−0.783347
$716$	13.7708	0.514639
$717$	0	0
$718$	0.178007	0.00664318
$719$	−25.4167	−0.947883	−0.473941	−	0.880556i	$-0.657169\pi$
$719$	−25.4167	−0.947883	−0.473941	+	0.880556i	$0.657169\pi$
$720$	0	0
$721$	−17.7177	−0.659842
$722$	1.78247	0.0663366
$723$	0	0
$724$	13.0477	0.484913
$725$	20.0243	0.743683
$726$	0	0
$727$	−33.7669	−1.25235	−0.626173	−	0.779684i	$-0.715380\pi$
$727$	−33.7669	−1.25235	−0.626173	+	0.779684i	$0.715380\pi$
$728$	−1.65791	−0.0614463
$729$	0	0
$730$	−0.886459	−0.0328093
$731$	−30.0340	−1.11085
$732$	0	0
$733$	−25.1390	−0.928531	−0.464266	−	0.885696i	$-0.653682\pi$
$733$	−25.1390	−0.928531	−0.464266	+	0.885696i	$0.653682\pi$
$734$	−0.284513	−0.0105016
$735$	0	0
$736$	3.82808	0.141105
$737$	−21.9176	−0.807344
$738$	0	0
$739$	−39.8845	−1.46718	−0.733588	−	0.679594i	$-0.762156\pi$
$739$	−39.8845	−1.46718	−0.733588	+	0.679594i	$0.762156\pi$
$740$	−2.57734	−0.0947449
$741$	0	0
$742$	0.112265	0.00412139
$743$	−34.5585	−1.26783	−0.633914	−	0.773404i	$-0.718553\pi$
$743$	−34.5585	−1.26783	−0.633914	+	0.773404i	$0.718553\pi$
$744$	0	0
$745$	9.40082	0.344419
$746$	−2.35582	−0.0862527
$747$	0	0
$748$	−50.5942	−1.84991
$749$	−0.266921	−0.00975309
$750$	0	0
$751$	39.4560	1.43977	0.719885	−	0.694094i	$-0.244195\pi$
$751$	39.4560	1.43977	0.719885	+	0.694094i	$0.244195\pi$
$752$	−22.1929	−0.809291
$753$	0	0
$754$	2.44934	0.0891998
$755$	18.0462	0.656767
$756$	0	0
$757$	−1.66834	−0.0606370	−0.0303185	−	0.999540i	$-0.509652\pi$
$757$	−1.66834	−0.0606370	−0.0303185	+	0.999540i	$0.509652\pi$
$758$	−3.20608	−0.116450
$759$	0	0
$760$	0.307904	0.0111689
$761$	22.3197	0.809087	0.404543	−	0.914519i	$-0.367430\pi$
$761$	22.3197	0.809087	0.404543	+	0.914519i	$0.367430\pi$
$762$	0	0
$763$	5.58786	0.202294
$764$	−34.5651	−1.25052
$765$	0	0
$766$	1.70947	0.0617656
$767$	26.1018	0.942480
$768$	0	0
$769$	−41.1244	−1.48298	−0.741492	−	0.670962i	$-0.765881\pi$
$769$	−41.1244	−1.48298	−0.741492	+	0.670962i	$0.765881\pi$
$770$	−0.463246	−0.0166942
$771$	0	0
$772$	26.1527	0.941258
$773$	−20.9238	−0.752578	−0.376289	−	0.926502i	$-0.622800\pi$
$773$	−20.9238	−0.752578	−0.376289	+	0.926502i	$0.622800\pi$
$774$	0	0
$775$	10.9449	0.393154
$776$	−5.39675	−0.193732
$777$	0	0
$778$	1.91959	0.0688206
$779$	4.51836	0.161887
$780$	0	0
$781$	−42.5852	−1.52382
$782$	−2.13740	−0.0764333
$783$	0	0
$784$	3.94496	0.140891
$785$	−4.75179	−0.169599
$786$	0	0
$787$	19.9400	0.710784	0.355392	−	0.934717i	$-0.384347\pi$
$787$	19.9400	0.710784	0.355392	+	0.934717i	$0.384347\pi$
$788$	1.23859	0.0441231
$789$	0	0
$790$	0.828819	0.0294881
$791$	7.43321	0.264294
$792$	0	0
$793$	24.3971	0.866367
$794$	1.18303	0.0419842
$795$	0	0
$796$	2.53598	0.0898853
$797$	−17.7408	−0.628411	−0.314205	−	0.949355i	$-0.601738\pi$
$797$	−17.7408	−0.628411	−0.314205	+	0.949355i	$0.601738\pi$
$798$	0	0
$799$	37.4621	1.32531
$800$	3.88263	0.137272
$801$	0	0
$802$	−3.04855	−0.107648
$803$	−27.8705	−0.983530
$804$	0	0
$805$	4.24055	0.149460
$806$	1.33877	0.0471561
$807$	0	0
$808$	−2.35519	−0.0828554
$809$	41.6391	1.46395	0.731976	−	0.681331i	$-0.238598\pi$
$809$	41.6391	1.46395	0.731976	+	0.681331i	$0.238598\pi$
$810$	0	0
$811$	44.0807	1.54788	0.773942	−	0.633256i	$-0.218282\pi$
$811$	44.0807	1.54788	0.773942	+	0.633256i	$0.218282\pi$
$812$	−11.7376	−0.411910
$813$	0	0
$814$	0.373967	0.0131075
$815$	20.5767	0.720770
$816$	0	0
$817$	2.86672	0.100294
$818$	0.348913	0.0121995
$819$	0	0
$820$	−17.9217	−0.625852
$821$	−35.8438	−1.25096	−0.625478	−	0.780242i	$-0.715096\pi$
$821$	−35.8438	−1.25096	−0.625478	+	0.780242i	$0.715096\pi$
$822$	0	0
$823$	10.7907	0.376141	0.188070	−	0.982156i	$-0.439777\pi$
$823$	10.7907	0.376141	0.188070	+	0.982156i	$0.439777\pi$
$824$	6.77752	0.236106
$825$	0	0
$826$	0.577264	0.0200856
$827$	13.7809	0.479207	0.239604	−	0.970871i	$-0.422982\pi$
$827$	13.7809	0.479207	0.239604	+	0.970871i	$0.422982\pi$
$828$	0	0
$829$	−25.9498	−0.901273	−0.450636	−	0.892708i	$-0.648803\pi$
$829$	−25.9498	−0.901273	−0.450636	+	0.892708i	$0.648803\pi$
$830$	0.949467	0.0329565
$831$	0	0
$832$	−33.7207	−1.16905
$833$	−6.65918	−0.230727
$834$	0	0
$835$	12.4834	0.432004
$836$	4.82916	0.167020
$837$	0	0
$838$	0.884987	0.0305714
$839$	29.2765	1.01074	0.505368	−	0.862904i	$-0.331357\pi$
$839$	29.2765	1.01074	0.505368	+	0.862904i	$0.331357\pi$
$840$	0	0
$841$	5.76154	0.198674
$842$	−0.519784	−0.0179129
$843$	0	0
$844$	−30.6092	−1.05361
$845$	7.32510	0.251991
$846$	0	0
$847$	−3.56460	−0.122481
$848$	4.62047	0.158668
$849$	0	0
$850$	−2.16786	−0.0743569
$851$	−3.42329	−0.117349
$852$	0	0
$853$	49.8635	1.70729	0.853646	−	0.520853i	$-0.174386\pi$
$853$	49.8635	1.70729	0.853646	+	0.520853i	$0.174386\pi$
$854$	0.539564	0.0184635
$855$	0	0
$856$	0.102105	0.00348988
$857$	48.9517	1.67216	0.836080	−	0.548608i	$-0.184842\pi$
$857$	48.9517	1.67216	0.836080	+	0.548608i	$0.184842\pi$
$858$	0	0
$859$	52.6366	1.79594	0.897968	−	0.440060i	$-0.145043\pi$
$859$	52.6366	1.79594	0.897968	+	0.440060i	$0.145043\pi$
$860$	−11.3706	−0.387734
$861$	0	0
$862$	1.33159	0.0453541
$863$	−20.4155	−0.694952	−0.347476	−	0.937689i	$-0.612961\pi$
$863$	−20.4155	−0.694952	−0.347476	+	0.937689i	$0.612961\pi$
$864$	0	0
$865$	−6.34863	−0.215860
$866$	1.99746	0.0678764
$867$	0	0
$868$	−6.41558	−0.217759
$869$	26.0583	0.883969
$870$	0	0
$871$	24.8909	0.843397
$872$	−2.13752	−0.0723854
$873$	0	0
$874$	0.204013	0.00690083
$875$	10.6328	0.359455
$876$	0	0
$877$	18.0957	0.611047	0.305524	−	0.952185i	$-0.401169\pi$
$877$	18.0957	0.611047	0.305524	+	0.952185i	$0.401169\pi$
$878$	1.29539	0.0437174
$879$	0	0
$880$	−19.0656	−0.642703
$881$	−53.9666	−1.81818	−0.909090	−	0.416600i	$-0.863222\pi$
$881$	−53.9666	−1.81818	−0.909090	+	0.416600i	$0.863222\pi$
$882$	0	0
$883$	−2.01437	−0.0677889	−0.0338945	−	0.999425i	$-0.510791\pi$
$883$	−2.01437	−0.0677889	−0.0338945	+	0.999425i	$0.510791\pi$
$884$	57.4578	1.93252
$885$	0	0
$886$	−2.81498	−0.0945711
$887$	−12.7082	−0.426700	−0.213350	−	0.976976i	$-0.568437\pi$
$887$	−12.7082	−0.426700	−0.213350	+	0.976976i	$0.568437\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	−0.179005	−0.00600028
$891$	0	0
$892$	−23.3369	−0.781377
$893$	−3.57572	−0.119657
$894$	0	0
$895$	−8.75969	−0.292804
$896$	−3.03214	−0.101297
$897$	0	0
$898$	−0.351318	−0.0117236
$899$	19.0001	0.633688
$900$	0	0
$901$	−7.79946	−0.259838
$902$	2.60040	0.0865838
$903$	0	0
$904$	−2.84341	−0.0945705
$905$	−8.29970	−0.275891
$906$	0	0
$907$	−13.7859	−0.457753	−0.228877	−	0.973455i	$-0.573505\pi$
$907$	−13.7859	−0.457753	−0.228877	+	0.973455i	$0.573505\pi$
$908$	29.9543	0.994067
$909$	0	0
$910$	0.526090	0.0174397
$911$	47.4797	1.57307	0.786536	−	0.617544i	$-0.211872\pi$
$911$	47.4797	1.57307	0.786536	+	0.617544i	$0.211872\pi$
$912$	0	0
$913$	29.8516	0.987943
$914$	−0.425591	−0.0140773
$915$	0	0
$916$	−21.0072	−0.694098
$917$	−10.0704	−0.332553
$918$	0	0
$919$	−0.951786	−0.0313965	−0.0156983	−	0.999877i	$-0.504997\pi$
$919$	−0.951786	−0.0313965	−0.0156983	+	0.999877i	$0.504997\pi$
$920$	−1.62213	−0.0534801
$921$	0	0
$922$	2.47138	0.0813904
$923$	48.3623	1.59187
$924$	0	0
$925$	−3.47208	−0.114161
$926$	3.60210	0.118372
$927$	0	0
$928$	6.74012	0.221255
$929$	−20.0992	−0.659435	−0.329717	−	0.944080i	$-0.606953\pi$
$929$	−20.0992	−0.659435	−0.329717	+	0.944080i	$0.606953\pi$
$930$	0	0
$931$	0.635612	0.0208313
$932$	5.09011	0.166732
$933$	0	0
$934$	1.86262	0.0609468
$935$	32.1833	1.05251
$936$	0	0
$937$	−32.8606	−1.07351	−0.536755	−	0.843738i	$-0.680350\pi$
$937$	−32.8606	−1.07351	−0.536755	+	0.843738i	$0.680350\pi$
$938$	0.550485	0.0179740
$939$	0	0
$940$	14.1828	0.462591
$941$	−2.15918	−0.0703871	−0.0351936	−	0.999381i	$-0.511205\pi$
$941$	−2.15918	−0.0703871	−0.0351936	+	0.999381i	$0.511205\pi$
$942$	0	0
$943$	−23.8041	−0.775167
$944$	23.7582	0.773265
$945$	0	0
$946$	1.64985	0.0536412
$947$	7.88837	0.256338	0.128169	−	0.991752i	$-0.459090\pi$
$947$	7.88837	0.256338	0.128169	+	0.991752i	$0.459090\pi$
$948$	0	0
$949$	31.6515	1.02745
$950$	0.206920	0.00671336
$951$	0	0
$952$	2.54733	0.0825593
$953$	−4.91726	−0.159286	−0.0796428	−	0.996823i	$-0.525378\pi$
$953$	−4.91726	−0.159286	−0.0796428	+	0.996823i	$0.525378\pi$
$954$	0	0
$955$	21.9871	0.711486
$956$	23.6625	0.765299
$957$	0	0
$958$	2.49465	0.0805984
$959$	1.08742	0.0351146
$960$	0	0
$961$	−20.6149	−0.664997
$962$	−0.424699	−0.0136929
$963$	0	0
$964$	−23.9707	−0.772044
$965$	−16.6359	−0.535529
$966$	0	0
$967$	6.23416	0.200477	0.100238	−	0.994963i	$-0.468039\pi$
$967$	6.23416	0.200477	0.100238	+	0.994963i	$0.468039\pi$
$968$	1.36356	0.0438265
$969$	0	0
$970$	1.71250	0.0549851
$971$	−46.9401	−1.50638	−0.753190	−	0.657803i	$-0.771486\pi$
$971$	−46.9401	−1.50638	−0.753190	+	0.657803i	$0.771486\pi$
$972$	0	0
$973$	16.8187	0.539184
$974$	−0.978469	−0.0313522
$975$	0	0
$976$	22.2066	0.710817
$977$	29.2796	0.936737	0.468368	−	0.883533i	$-0.344842\pi$
$977$	29.2796	0.936737	0.468368	+	0.883533i	$0.344842\pi$
$978$	0	0
$979$	−5.62799	−0.179871
$980$	−2.52110	−0.0805336
$981$	0	0
$982$	−2.24483	−0.0716355
$983$	−33.1814	−1.05832	−0.529160	−	0.848522i	$-0.677493\pi$
$983$	−33.1814	−1.05832	−0.529160	+	0.848522i	$0.677493\pi$
$984$	0	0
$985$	−0.787877	−0.0251039
$986$	−3.76333	−0.119849
$987$	0	0
$988$	−5.48429	−0.174478
$989$	−15.1027	−0.480239
$990$	0	0
$991$	−44.8275	−1.42399	−0.711996	−	0.702184i	$-0.752208\pi$
$991$	−44.8275	−1.42399	−0.711996	+	0.702184i	$0.752208\pi$
$992$	3.68404	0.116968
$993$	0	0
$994$	1.06958	0.0339249
$995$	−1.61315	−0.0511403
$996$	0	0
$997$	−4.58468	−0.145198	−0.0725991	−	0.997361i	$-0.523129\pi$
$997$	−4.58468	−0.145198	−0.0725991	+	0.997361i	$0.523129\pi$
$998$	−1.03848	−0.0328726
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.y.1.14	✓	28
3.2	odd	2	inner	8001.2.a.y.1.15	yes	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8001.2.a.y.1.14	✓	28	1.1	even	1	trivial
8001.2.a.y.1.15	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)

\(f(q)\)	\(=\)	\(q-0.0958522 q^{2} -1.99081 q^{4} +1.26637 q^{5} -1.00000 q^{7} +0.382528 q^{8} +O(q^{10})\)	\(q-0.0958522 q^{2} -1.99081 q^{4} +1.26637 q^{5} -1.00000 q^{7} +0.382528 q^{8} -0.121384 q^{10} -3.81636 q^{11} +4.33409 q^{13} +0.0958522 q^{14} +3.94496 q^{16} -6.65918 q^{17} +0.635612 q^{19} -2.52110 q^{20} +0.365807 q^{22} -3.34860 q^{23} -3.39631 q^{25} -0.415432 q^{26} +1.99081 q^{28} -5.89589 q^{29} -3.22259 q^{31} -1.14319 q^{32} +0.638297 q^{34} -1.26637 q^{35} +1.02231 q^{37} -0.0609248 q^{38} +0.484422 q^{40} +7.10867 q^{41} +4.51017 q^{43} +7.59765 q^{44} +0.320970 q^{46} -5.62563 q^{47} +1.00000 q^{49} +0.325544 q^{50} -8.62836 q^{52} +1.17123 q^{53} -4.83292 q^{55} -0.382528 q^{56} +0.565134 q^{58} +6.02243 q^{59} +5.62912 q^{61} +0.308893 q^{62} -7.78034 q^{64} +5.48855 q^{65} +5.74306 q^{67} +13.2572 q^{68} +0.121384 q^{70} +11.1586 q^{71} +7.30292 q^{73} -0.0979904 q^{74} -1.26538 q^{76} +3.81636 q^{77} -6.82806 q^{79} +4.99577 q^{80} -0.681382 q^{82} -7.82200 q^{83} -8.43298 q^{85} -0.432310 q^{86} -1.45987 q^{88} +1.47470 q^{89} -4.33409 q^{91} +6.66643 q^{92} +0.539229 q^{94} +0.804918 q^{95} -14.1081 q^{97} -0.0958522 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 30 q^{4} - 28 q^{7}+O(q^{10}) \) 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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