LMFDB - Embedded newform 8001.2.a.z.1.5

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			1.5
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)

$f(q)$	$=$	$q-2.23843 q^{2} +3.01059 q^{4} -0.715002 q^{5} -1.00000 q^{7} -2.26213 q^{8} +O(q^{10})$	\(q-2.23843 q^{2} +3.01059 q^{4} -0.715002 q^{5} -1.00000 q^{7} -2.26213 q^{8} +1.60049 q^{10} -3.48743 q^{11} -3.88025 q^{13} +2.23843 q^{14} -0.957541 q^{16} +2.02085 q^{17} +1.32595 q^{19} -2.15258 q^{20} +7.80638 q^{22} +4.77444 q^{23} -4.48877 q^{25} +8.68568 q^{26} -3.01059 q^{28} +8.26004 q^{29} -4.73204 q^{31} +6.66766 q^{32} -4.52353 q^{34} +0.715002 q^{35} -10.4983 q^{37} -2.96805 q^{38} +1.61743 q^{40} -0.827289 q^{41} +2.69972 q^{43} -10.4992 q^{44} -10.6873 q^{46} +9.44282 q^{47} +1.00000 q^{49} +10.0478 q^{50} -11.6818 q^{52} -7.12745 q^{53} +2.49352 q^{55} +2.26213 q^{56} -18.4895 q^{58} +6.64479 q^{59} +9.79645 q^{61} +10.5924 q^{62} -13.0100 q^{64} +2.77439 q^{65} +10.4839 q^{67} +6.08393 q^{68} -1.60049 q^{70} -8.80029 q^{71} +6.52998 q^{73} +23.4997 q^{74} +3.99189 q^{76} +3.48743 q^{77} -1.16978 q^{79} +0.684644 q^{80} +1.85183 q^{82} +4.68620 q^{83} -1.44491 q^{85} -6.04313 q^{86} +7.88902 q^{88} +3.95955 q^{89} +3.88025 q^{91} +14.3739 q^{92} -21.1371 q^{94} -0.948058 q^{95} +6.83283 q^{97} -2.23843 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.23843	−1.58281	−0.791406	−	0.611291i	$-0.790651\pi$
$2$	−2.23843	−1.58281	−0.791406	+	0.611291i	$0.790651\pi$
$3$	0	0
$3$	0	0
$4$	3.01059	1.50529
$5$	−0.715002	−0.319759	−0.159879	−	0.987137i	$-0.551111\pi$
$5$	−0.715002	−0.319759	−0.159879	+	0.987137i	$0.551111\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−2.26213	−0.799784
$9$	0	0
$10$	1.60049	0.506118
$11$	−3.48743	−1.05150	−0.525750	−	0.850639i	$-0.676215\pi$
$11$	−3.48743	−1.05150	−0.525750	+	0.850639i	$0.676215\pi$
$12$	0	0
$13$	−3.88025	−1.07619	−0.538093	−	0.842885i	$-0.680855\pi$
$13$	−3.88025	−1.07619	−0.538093	+	0.842885i	$0.680855\pi$
$14$	2.23843	0.598247
$15$	0	0
$16$	−0.957541	−0.239385
$17$	2.02085	0.490127	0.245064	−	0.969507i	$-0.421191\pi$
$17$	2.02085	0.490127	0.245064	+	0.969507i	$0.421191\pi$
$18$	0	0
$19$	1.32595	0.304194	0.152097	−	0.988366i	$-0.451397\pi$
$19$	1.32595	0.304194	0.152097	+	0.988366i	$0.451397\pi$
$20$	−2.15258	−0.481331
$21$	0	0
$22$	7.80638	1.66433
$23$	4.77444	0.995540	0.497770	−	0.867309i	$-0.334152\pi$
$23$	4.77444	0.995540	0.497770	+	0.867309i	$0.334152\pi$
$24$	0	0
$25$	−4.48877	−0.897754
$26$	8.68568	1.70340
$27$	0	0
$28$	−3.01059	−0.568947
$29$	8.26004	1.53385	0.766925	−	0.641737i	$-0.221786\pi$
$29$	8.26004	1.53385	0.766925	+	0.641737i	$0.221786\pi$
$30$	0	0
$31$	−4.73204	−0.849900	−0.424950	−	0.905217i	$-0.639708\pi$
$31$	−4.73204	−0.849900	−0.424950	+	0.905217i	$0.639708\pi$
$32$	6.66766	1.17869
$33$	0	0
$34$	−4.52353	−0.775779
$35$	0.715002	0.120857
$36$	0	0
$37$	−10.4983	−1.72591	−0.862954	−	0.505283i	$-0.831388\pi$
$37$	−10.4983	−1.72591	−0.862954	+	0.505283i	$0.831388\pi$
$38$	−2.96805	−0.481482
$39$	0	0
$40$	1.61743	0.255738
$41$	−0.827289	−0.129201	−0.0646004	−	0.997911i	$-0.520577\pi$
$41$	−0.827289	−0.129201	−0.0646004	+	0.997911i	$0.520577\pi$
$42$	0	0
$43$	2.69972	0.411703	0.205851	−	0.978583i	$-0.434004\pi$
$43$	2.69972	0.411703	0.205851	+	0.978583i	$0.434004\pi$
$44$	−10.4992	−1.58281
$45$	0	0
$46$	−10.6873	−1.57575
$47$	9.44282	1.37738	0.688688	−	0.725058i	$-0.258187\pi$
$47$	9.44282	1.37738	0.688688	+	0.725058i	$0.258187\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	10.0478	1.42098
$51$	0	0
$52$	−11.6818	−1.61998
$53$	−7.12745	−0.979030	−0.489515	−	0.871995i	$-0.662826\pi$
$53$	−7.12745	−0.979030	−0.489515	+	0.871995i	$0.662826\pi$
$54$	0	0
$55$	2.49352	0.336226
$56$	2.26213	0.302290
$57$	0	0
$58$	−18.4895	−2.42780
$59$	6.64479	0.865078	0.432539	−	0.901615i	$-0.357618\pi$
$59$	6.64479	0.865078	0.432539	+	0.901615i	$0.357618\pi$
$60$	0	0
$61$	9.79645	1.25431	0.627153	−	0.778896i	$-0.284220\pi$
$61$	9.79645	1.25431	0.627153	+	0.778896i	$0.284220\pi$
$62$	10.5924	1.34523
$63$	0	0
$64$	−13.0100	−1.62625
$65$	2.77439	0.344120
$66$	0	0
$67$	10.4839	1.28081	0.640407	−	0.768036i	$-0.278766\pi$
$67$	10.4839	1.28081	0.640407	+	0.768036i	$0.278766\pi$
$68$	6.08393	0.737785
$69$	0	0
$70$	−1.60049	−0.191295
$71$	−8.80029	−1.04440	−0.522201	−	0.852823i	$-0.674889\pi$
$71$	−8.80029	−1.04440	−0.522201	+	0.852823i	$0.674889\pi$
$72$	0	0
$73$	6.52998	0.764276	0.382138	−	0.924105i	$-0.375188\pi$
$73$	6.52998	0.764276	0.382138	+	0.924105i	$0.375188\pi$
$74$	23.4997	2.73179
$75$	0	0
$76$	3.99189	0.457901
$77$	3.48743	0.397429
$78$	0	0
$79$	−1.16978	−0.131611	−0.0658053	−	0.997832i	$-0.520962\pi$
$79$	−1.16978	−0.131611	−0.0658053	+	0.997832i	$0.520962\pi$
$80$	0.684644	0.0765455
$81$	0	0
$82$	1.85183	0.204501
$83$	4.68620	0.514377	0.257189	−	0.966361i	$-0.417204\pi$
$83$	4.68620	0.514377	0.257189	+	0.966361i	$0.417204\pi$
$84$	0	0
$85$	−1.44491	−0.156722
$86$	−6.04313	−0.651648
$87$	0	0
$88$	7.88902	0.840973
$89$	3.95955	0.419712	0.209856	−	0.977732i	$-0.432701\pi$
$89$	3.95955	0.419712	0.209856	+	0.977732i	$0.432701\pi$
$90$	0	0
$91$	3.88025	0.406760
$92$	14.3739	1.49858
$93$	0	0
$94$	−21.1371	−2.18013
$95$	−0.948058	−0.0972687
$96$	0	0
$97$	6.83283	0.693768	0.346884	−	0.937908i	$-0.387240\pi$
$97$	6.83283	0.693768	0.346884	+	0.937908i	$0.387240\pi$
$98$	−2.23843	−0.226116
$99$	0	0
$100$	−13.5138	−1.35138
$101$	14.4018	1.43303	0.716517	−	0.697569i	$-0.245735\pi$
$101$	14.4018	1.43303	0.716517	+	0.697569i	$0.245735\pi$
$102$	0	0
$103$	4.78657	0.471635	0.235817	−	0.971797i	$-0.424223\pi$
$103$	4.78657	0.471635	0.235817	+	0.971797i	$0.424223\pi$
$104$	8.77763	0.860717
$105$	0	0
$106$	15.9543	1.54962
$107$	−8.11171	−0.784189	−0.392094	−	0.919925i	$-0.628249\pi$
$107$	−8.11171	−0.784189	−0.392094	+	0.919925i	$0.628249\pi$
$108$	0	0
$109$	0.982270	0.0940844	0.0470422	−	0.998893i	$-0.485020\pi$
$109$	0.982270	0.0940844	0.0470422	+	0.998893i	$0.485020\pi$
$110$	−5.58158	−0.532183
$111$	0	0
$112$	0.957541	0.0904791
$113$	−5.77508	−0.543273	−0.271637	−	0.962400i	$-0.587565\pi$
$113$	−5.77508	−0.543273	−0.271637	+	0.962400i	$0.587565\pi$
$114$	0	0
$115$	−3.41374	−0.318333
$116$	24.8676	2.30889
$117$	0	0
$118$	−14.8739	−1.36926
$119$	−2.02085	−0.185251
$120$	0	0
$121$	1.16215	0.105650
$122$	−21.9287	−1.98533
$123$	0	0
$124$	−14.2462	−1.27935
$125$	6.78450	0.606824
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	15.7868	1.39537
$129$	0	0
$130$	−6.21028	−0.544678
$131$	1.06804	0.0933152	0.0466576	−	0.998911i	$-0.485143\pi$
$131$	1.06804	0.0933152	0.0466576	+	0.998911i	$0.485143\pi$
$132$	0	0
$133$	−1.32595	−0.114975
$134$	−23.4675	−2.02729
$135$	0	0
$136$	−4.57142	−0.391996
$137$	−12.8058	−1.09408	−0.547038	−	0.837108i	$-0.684245\pi$
$137$	−12.8058	−1.09408	−0.547038	+	0.837108i	$0.684245\pi$
$138$	0	0
$139$	−4.13929	−0.351090	−0.175545	−	0.984471i	$-0.556169\pi$
$139$	−4.13929	−0.351090	−0.175545	+	0.984471i	$0.556169\pi$
$140$	2.15258	0.181926
$141$	0	0
$142$	19.6989	1.65309
$143$	13.5321	1.13161
$144$	0	0
$145$	−5.90595	−0.490462
$146$	−14.6169	−1.20971
$147$	0	0
$148$	−31.6060	−2.59800
$149$	−13.5650	−1.11129	−0.555646	−	0.831419i	$-0.687529\pi$
$149$	−13.5650	−1.11129	−0.555646	+	0.831419i	$0.687529\pi$
$150$	0	0
$151$	3.40613	0.277187	0.138594	−	0.990349i	$-0.455742\pi$
$151$	3.40613	0.277187	0.138594	+	0.990349i	$0.455742\pi$
$152$	−2.99948	−0.243290
$153$	0	0
$154$	−7.80638	−0.629056
$155$	3.38342	0.271763
$156$	0	0
$157$	−20.5876	−1.64307	−0.821536	−	0.570157i	$-0.806882\pi$
$157$	−20.5876	−1.64307	−0.821536	+	0.570157i	$0.806882\pi$
$158$	2.61848	0.208315
$159$	0	0
$160$	−4.76739	−0.376895
$161$	−4.77444	−0.376279
$162$	0	0
$163$	14.3948	1.12749	0.563744	−	0.825950i	$-0.309360\pi$
$163$	14.3948	1.12749	0.563744	+	0.825950i	$0.309360\pi$
$164$	−2.49063	−0.194485
$165$	0	0
$166$	−10.4898	−0.814163
$167$	−14.7266	−1.13958	−0.569788	−	0.821792i	$-0.692975\pi$
$167$	−14.7266	−1.13958	−0.569788	+	0.821792i	$0.692975\pi$
$168$	0	0
$169$	2.05631	0.158178
$170$	3.23433	0.248062
$171$	0	0
$172$	8.12773	0.619733
$173$	12.1096	0.920678	0.460339	−	0.887743i	$-0.347728\pi$
$173$	12.1096	0.920678	0.460339	+	0.887743i	$0.347728\pi$
$174$	0	0
$175$	4.48877	0.339319
$176$	3.33935	0.251713
$177$	0	0
$178$	−8.86320	−0.664325
$179$	−10.2816	−0.768484	−0.384242	−	0.923232i	$-0.625537\pi$
$179$	−10.2816	−0.768484	−0.384242	+	0.923232i	$0.625537\pi$
$180$	0	0
$181$	−6.97609	−0.518529	−0.259264	−	0.965806i	$-0.583480\pi$
$181$	−6.97609	−0.518529	−0.259264	+	0.965806i	$0.583480\pi$
$182$	−8.68568	−0.643825
$183$	0	0
$184$	−10.8004	−0.796217
$185$	7.50630	0.551874
$186$	0	0
$187$	−7.04755	−0.515368
$188$	28.4284	2.07336
$189$	0	0
$190$	2.12217	0.153958
$191$	16.8925	1.22230	0.611150	−	0.791515i	$-0.290707\pi$
$191$	16.8925	1.22230	0.611150	+	0.791515i	$0.290707\pi$
$192$	0	0
$193$	5.01576	0.361043	0.180521	−	0.983571i	$-0.442222\pi$
$193$	5.01576	0.361043	0.180521	+	0.983571i	$0.442222\pi$
$194$	−15.2948	−1.09810
$195$	0	0
$196$	3.01059	0.215042
$197$	−8.79497	−0.626616	−0.313308	−	0.949652i	$-0.601437\pi$
$197$	−8.79497	−0.626616	−0.313308	+	0.949652i	$0.601437\pi$
$198$	0	0
$199$	−21.9618	−1.55683	−0.778417	−	0.627747i	$-0.783977\pi$
$199$	−21.9618	−1.55683	−0.778417	+	0.627747i	$0.783977\pi$
$200$	10.1542	0.718010
$201$	0	0
$202$	−32.2375	−2.26822
$203$	−8.26004	−0.579741
$204$	0	0
$205$	0.591514	0.0413131
$206$	−10.7144	−0.746509
$207$	0	0
$208$	3.71549	0.257623
$209$	−4.62416	−0.319860
$210$	0	0
$211$	10.3601	0.713221	0.356610	−	0.934253i	$-0.383932\pi$
$211$	10.3601	0.713221	0.356610	+	0.934253i	$0.383932\pi$
$212$	−21.4578	−1.47373
$213$	0	0
$214$	18.1575	1.24122
$215$	−1.93030	−0.131646
$216$	0	0
$217$	4.73204	0.321232
$218$	−2.19875	−0.148918
$219$	0	0
$220$	7.50696	0.506119
$221$	−7.84138	−0.527468
$222$	0	0
$223$	−6.35678	−0.425681	−0.212841	−	0.977087i	$-0.568272\pi$
$223$	−6.35678	−0.425681	−0.212841	+	0.977087i	$0.568272\pi$
$224$	−6.66766	−0.445501
$225$	0	0
$226$	12.9271	0.859900
$227$	7.68768	0.510249	0.255124	−	0.966908i	$-0.417884\pi$
$227$	7.68768	0.510249	0.255124	+	0.966908i	$0.417884\pi$
$228$	0	0
$229$	−10.5663	−0.698244	−0.349122	−	0.937077i	$-0.613520\pi$
$229$	−10.5663	−0.698244	−0.349122	+	0.937077i	$0.613520\pi$
$230$	7.64142	0.503861
$231$	0	0
$232$	−18.6853	−1.22675
$233$	25.1211	1.64574	0.822869	−	0.568231i	$-0.192372\pi$
$233$	25.1211	1.64574	0.822869	+	0.568231i	$0.192372\pi$
$234$	0	0
$235$	−6.75164	−0.440428
$236$	20.0047	1.30220
$237$	0	0
$238$	4.52353	0.293217
$239$	−10.8412	−0.701257	−0.350629	−	0.936515i	$-0.614032\pi$
$239$	−10.8412	−0.701257	−0.350629	+	0.936515i	$0.614032\pi$
$240$	0	0
$241$	−17.8749	−1.15142	−0.575712	−	0.817652i	$-0.695275\pi$
$241$	−17.8749	−1.15142	−0.575712	+	0.817652i	$0.695275\pi$
$242$	−2.60141	−0.167225
$243$	0	0
$244$	29.4931	1.88810
$245$	−0.715002	−0.0456798
$246$	0	0
$247$	−5.14502	−0.327369
$248$	10.7045	0.679737
$249$	0	0
$250$	−15.1866	−0.960488
$251$	4.79178	0.302455	0.151227	−	0.988499i	$-0.451677\pi$
$251$	4.79178	0.302455	0.151227	+	0.988499i	$0.451677\pi$
$252$	0	0
$253$	−16.6505	−1.04681
$254$	2.23843	0.140452
$255$	0	0
$256$	−9.31760	−0.582350
$257$	6.08661	0.379672	0.189836	−	0.981816i	$-0.439204\pi$
$257$	6.08661	0.379672	0.189836	+	0.981816i	$0.439204\pi$
$258$	0	0
$259$	10.4983	0.652332
$260$	8.35253	0.518002
$261$	0	0
$262$	−2.39074	−0.147700
$263$	25.1166	1.54876	0.774378	−	0.632723i	$-0.218063\pi$
$263$	25.1166	1.54876	0.774378	+	0.632723i	$0.218063\pi$
$264$	0	0
$265$	5.09614	0.313053
$266$	2.96805	0.181983
$267$	0	0
$268$	31.5627	1.92800
$269$	12.2844	0.748995	0.374498	−	0.927228i	$-0.377815\pi$
$269$	12.2844	0.748995	0.374498	+	0.927228i	$0.377815\pi$
$270$	0	0
$271$	−0.354378	−0.0215269	−0.0107635	−	0.999942i	$-0.503426\pi$
$271$	−0.354378	−0.0215269	−0.0107635	+	0.999942i	$0.503426\pi$
$272$	−1.93504	−0.117329
$273$	0	0
$274$	28.6650	1.73172
$275$	15.6543	0.943988
$276$	0	0
$277$	−18.8044	−1.12984	−0.564922	−	0.825144i	$-0.691094\pi$
$277$	−18.8044	−1.12984	−0.564922	+	0.825144i	$0.691094\pi$
$278$	9.26552	0.555709
$279$	0	0
$280$	−1.61743	−0.0966599
$281$	22.7302	1.35597	0.677985	−	0.735076i	$-0.262854\pi$
$281$	22.7302	1.35597	0.677985	+	0.735076i	$0.262854\pi$
$282$	0	0
$283$	−9.78820	−0.581848	−0.290924	−	0.956746i	$-0.593963\pi$
$283$	−9.78820	−0.581848	−0.290924	+	0.956746i	$0.593963\pi$
$284$	−26.4940	−1.57213
$285$	0	0
$286$	−30.2907	−1.79112
$287$	0.827289	0.0488333
$288$	0	0
$289$	−12.9162	−0.759775
$290$	13.2201	0.776309
$291$	0	0
$292$	19.6591	1.15046
$293$	−6.16187	−0.359981	−0.179990	−	0.983668i	$-0.557607\pi$
$293$	−6.16187	−0.359981	−0.179990	+	0.983668i	$0.557607\pi$
$294$	0	0
$295$	−4.75104	−0.276616
$296$	23.7485	1.38035
$297$	0	0
$298$	30.3645	1.75897
$299$	−18.5260	−1.07139
$300$	0	0
$301$	−2.69972	−0.155609
$302$	−7.62440	−0.438735
$303$	0	0
$304$	−1.26965	−0.0728195
$305$	−7.00449	−0.401076
$306$	0	0
$307$	−13.7893	−0.786999	−0.393500	−	0.919325i	$-0.628736\pi$
$307$	−13.7893	−0.786999	−0.393500	+	0.919325i	$0.628736\pi$
$308$	10.4992	0.598248
$309$	0	0
$310$	−7.57357	−0.430150
$311$	−7.99152	−0.453158	−0.226579	−	0.973993i	$-0.572754\pi$
$311$	−7.99152	−0.453158	−0.226579	+	0.973993i	$0.572754\pi$
$312$	0	0
$313$	8.19550	0.463237	0.231618	−	0.972807i	$-0.425598\pi$
$313$	8.19550	0.463237	0.231618	+	0.972807i	$0.425598\pi$
$314$	46.0840	2.60067
$315$	0	0
$316$	−3.52173	−0.198113
$317$	9.71596	0.545703	0.272851	−	0.962056i	$-0.412033\pi$
$317$	9.71596	0.545703	0.272851	+	0.962056i	$0.412033\pi$
$318$	0	0
$319$	−28.8063	−1.61284
$320$	9.30220	0.520009
$321$	0	0
$322$	10.6873	0.595578
$323$	2.67954	0.149094
$324$	0	0
$325$	17.4175	0.966151
$326$	−32.2218	−1.78460
$327$	0	0
$328$	1.87144	0.103333
$329$	−9.44282	−0.520599
$330$	0	0
$331$	26.7460	1.47009	0.735047	−	0.678017i	$-0.237160\pi$
$331$	26.7460	1.47009	0.735047	+	0.678017i	$0.237160\pi$
$332$	14.1082	0.774289
$333$	0	0
$334$	32.9644	1.80373
$335$	−7.49602	−0.409551
$336$	0	0
$337$	7.80788	0.425322	0.212661	−	0.977126i	$-0.431787\pi$
$337$	7.80788	0.425322	0.212661	+	0.977126i	$0.431787\pi$
$338$	−4.60292	−0.250366
$339$	0	0
$340$	−4.35003	−0.235913
$341$	16.5027	0.893669
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−6.10711	−0.329273
$345$	0	0
$346$	−27.1066	−1.45726
$347$	−16.8929	−0.906859	−0.453429	−	0.891292i	$-0.649800\pi$
$347$	−16.8929	−0.906859	−0.453429	+	0.891292i	$0.649800\pi$
$348$	0	0
$349$	−14.8133	−0.792937	−0.396468	−	0.918048i	$-0.629764\pi$
$349$	−14.8133	−0.792937	−0.396468	+	0.918048i	$0.629764\pi$
$350$	−10.0478	−0.537078
$351$	0	0
$352$	−23.2530	−1.23939
$353$	−32.0204	−1.70427	−0.852137	−	0.523319i	$-0.824694\pi$
$353$	−32.0204	−1.70427	−0.852137	+	0.523319i	$0.824694\pi$
$354$	0	0
$355$	6.29223	0.333957
$356$	11.9206	0.631790
$357$	0	0
$358$	23.0147	1.21637
$359$	−7.14129	−0.376903	−0.188451	−	0.982083i	$-0.560347\pi$
$359$	−7.14129	−0.376903	−0.188451	+	0.982083i	$0.560347\pi$
$360$	0	0
$361$	−17.2419	−0.907466
$362$	15.6155	0.820733
$363$	0	0
$364$	11.6818	0.612294
$365$	−4.66895	−0.244384
$366$	0	0
$367$	19.0108	0.992356	0.496178	−	0.868221i	$-0.334736\pi$
$367$	19.0108	0.992356	0.496178	+	0.868221i	$0.334736\pi$
$368$	−4.57172	−0.238317
$369$	0	0
$370$	−16.8024	−0.873513
$371$	7.12745	0.370039
$372$	0	0
$373$	32.6219	1.68910	0.844550	−	0.535476i	$-0.179868\pi$
$373$	32.6219	1.68910	0.844550	+	0.535476i	$0.179868\pi$
$374$	15.7755	0.815731
$375$	0	0
$376$	−21.3609	−1.10160
$377$	−32.0510	−1.65071
$378$	0	0
$379$	9.55764	0.490943	0.245471	−	0.969404i	$-0.421057\pi$
$379$	9.55764	0.490943	0.245471	+	0.969404i	$0.421057\pi$
$380$	−2.85421	−0.146418
$381$	0	0
$382$	−37.8128	−1.93467
$383$	−19.9990	−1.02190	−0.510950	−	0.859610i	$-0.670706\pi$
$383$	−19.9990	−1.02190	−0.510950	+	0.859610i	$0.670706\pi$
$384$	0	0
$385$	−2.49352	−0.127082
$386$	−11.2275	−0.571462
$387$	0	0
$388$	20.5708	1.04432
$389$	−4.07409	−0.206565	−0.103282	−	0.994652i	$-0.532934\pi$
$389$	−4.07409	−0.206565	−0.103282	+	0.994652i	$0.532934\pi$
$390$	0	0
$391$	9.64841	0.487941
$392$	−2.26213	−0.114255
$393$	0	0
$394$	19.6870	0.991814
$395$	0.836396	0.0420837
$396$	0	0
$397$	21.7833	1.09327	0.546636	−	0.837370i	$-0.315908\pi$
$397$	21.7833	1.09327	0.546636	+	0.837370i	$0.315908\pi$
$398$	49.1602	2.46418
$399$	0	0
$400$	4.29818	0.214909
$401$	17.3330	0.865568	0.432784	−	0.901498i	$-0.357531\pi$
$401$	17.3330	0.865568	0.432784	+	0.901498i	$0.357531\pi$
$402$	0	0
$403$	18.3615	0.914651
$404$	43.3579	2.15714
$405$	0	0
$406$	18.4895	0.917621
$407$	36.6120	1.81479
$408$	0	0
$409$	−12.4744	−0.616821	−0.308410	−	0.951253i	$-0.599797\pi$
$409$	−12.4744	−0.616821	−0.308410	+	0.951253i	$0.599797\pi$
$410$	−1.32406	−0.0653909
$411$	0	0
$412$	14.4104	0.709948
$413$	−6.64479	−0.326969
$414$	0	0
$415$	−3.35065	−0.164477
$416$	−25.8721	−1.26849
$417$	0	0
$418$	10.3509	0.506278
$419$	−10.7981	−0.527524	−0.263762	−	0.964588i	$-0.584963\pi$
$419$	−10.7981	−0.527524	−0.263762	+	0.964588i	$0.584963\pi$
$420$	0	0
$421$	−7.59997	−0.370400	−0.185200	−	0.982701i	$-0.559293\pi$
$421$	−7.59997	−0.370400	−0.185200	+	0.982701i	$0.559293\pi$
$422$	−23.1905	−1.12889
$423$	0	0
$424$	16.1232	0.783013
$425$	−9.07111	−0.440014
$426$	0	0
$427$	−9.79645	−0.474083
$428$	−24.4210	−1.18043
$429$	0	0
$430$	4.32086	0.208370
$431$	−34.8208	−1.67726	−0.838630	−	0.544701i	$-0.816643\pi$
$431$	−34.8208	−1.67726	−0.838630	+	0.544701i	$0.816643\pi$
$432$	0	0
$433$	−33.9006	−1.62916	−0.814578	−	0.580054i	$-0.803032\pi$
$433$	−33.9006	−1.62916	−0.814578	+	0.580054i	$0.803032\pi$
$434$	−10.5924	−0.508450
$435$	0	0
$436$	2.95721	0.141625
$437$	6.33067	0.302837
$438$	0	0
$439$	6.04032	0.288289	0.144144	−	0.989557i	$-0.453957\pi$
$439$	6.04032	0.288289	0.144144	+	0.989557i	$0.453957\pi$
$440$	−5.64067	−0.268908
$441$	0	0
$442$	17.5524	0.834883
$443$	−32.0832	−1.52432	−0.762159	−	0.647390i	$-0.775860\pi$
$443$	−32.0832	−1.52432	−0.762159	+	0.647390i	$0.775860\pi$
$444$	0	0
$445$	−2.83109	−0.134207
$446$	14.2292	0.673774
$447$	0	0
$448$	13.0100	0.614666
$449$	−7.01644	−0.331126	−0.165563	−	0.986199i	$-0.552944\pi$
$449$	−7.01644	−0.331126	−0.165563	+	0.986199i	$0.552944\pi$
$450$	0	0
$451$	2.88511	0.135855
$452$	−17.3864	−0.817786
$453$	0	0
$454$	−17.2084	−0.807628
$455$	−2.77439	−0.130065
$456$	0	0
$457$	16.8482	0.788126	0.394063	−	0.919083i	$-0.371069\pi$
$457$	16.8482	0.788126	0.394063	+	0.919083i	$0.371069\pi$
$458$	23.6521	1.10519
$459$	0	0
$460$	−10.2774	−0.479184
$461$	−17.7603	−0.827181	−0.413591	−	0.910463i	$-0.635726\pi$
$461$	−17.7603	−0.827181	−0.413591	+	0.910463i	$0.635726\pi$
$462$	0	0
$463$	10.1828	0.473236	0.236618	−	0.971603i	$-0.423961\pi$
$463$	10.1828	0.473236	0.236618	+	0.971603i	$0.423961\pi$
$464$	−7.90932	−0.367181
$465$	0	0
$466$	−56.2319	−2.60489
$467$	−1.65415	−0.0765449	−0.0382724	−	0.999267i	$-0.512185\pi$
$467$	−1.65415	−0.0765449	−0.0382724	+	0.999267i	$0.512185\pi$
$468$	0	0
$469$	−10.4839	−0.484102
$470$	15.1131	0.697115
$471$	0	0
$472$	−15.0314	−0.691876
$473$	−9.41506	−0.432905
$474$	0	0
$475$	−5.95189	−0.273091
$476$	−6.08393	−0.278857
$477$	0	0
$478$	24.2672	1.10996
$479$	−12.7961	−0.584670	−0.292335	−	0.956316i	$-0.594432\pi$
$479$	−12.7961	−0.584670	−0.292335	+	0.956316i	$0.594432\pi$
$480$	0	0
$481$	40.7359	1.85740
$482$	40.0118	1.82249
$483$	0	0
$484$	3.49877	0.159035
$485$	−4.88549	−0.221839
$486$	0	0
$487$	5.81280	0.263403	0.131702	−	0.991289i	$-0.457956\pi$
$487$	5.81280	0.263403	0.131702	+	0.991289i	$0.457956\pi$
$488$	−22.1609	−1.00317
$489$	0	0
$490$	1.60049	0.0723026
$491$	5.28049	0.238305	0.119153	−	0.992876i	$-0.461982\pi$
$491$	5.28049	0.238305	0.119153	+	0.992876i	$0.461982\pi$
$492$	0	0
$493$	16.6923	0.751782
$494$	11.5168	0.518164
$495$	0	0
$496$	4.53113	0.203454
$497$	8.80029	0.394747
$498$	0	0
$499$	−20.0549	−0.897778	−0.448889	−	0.893587i	$-0.648180\pi$
$499$	−20.0549	−0.897778	−0.448889	+	0.893587i	$0.648180\pi$
$500$	20.4253	0.913448
$501$	0	0
$502$	−10.7261	−0.478729
$503$	14.1949	0.632920	0.316460	−	0.948606i	$-0.397506\pi$
$503$	14.1949	0.632920	0.316460	+	0.948606i	$0.397506\pi$
$504$	0	0
$505$	−10.2973	−0.458225
$506$	37.2711	1.65690
$507$	0	0
$508$	−3.01059	−0.133573
$509$	34.0499	1.50923	0.754617	−	0.656166i	$-0.227823\pi$
$509$	34.0499	1.50923	0.754617	+	0.656166i	$0.227823\pi$
$510$	0	0
$511$	−6.52998	−0.288869
$512$	−10.7167	−0.473617
$513$	0	0
$514$	−13.6245	−0.600950
$515$	−3.42241	−0.150809
$516$	0	0
$517$	−32.9311	−1.44831
$518$	−23.4997	−1.03252
$519$	0	0
$520$	−6.27603	−0.275222
$521$	4.82296	0.211298	0.105649	−	0.994403i	$-0.466308\pi$
$521$	4.82296	0.211298	0.105649	+	0.994403i	$0.466308\pi$
$522$	0	0
$523$	−2.29591	−0.100393	−0.0501965	−	0.998739i	$-0.515985\pi$
$523$	−2.29591	−0.100393	−0.0501965	+	0.998739i	$0.515985\pi$
$524$	3.21543	0.140467
$525$	0	0
$526$	−56.2219	−2.45139
$527$	−9.56273	−0.416559
$528$	0	0
$529$	−0.204717	−0.00890074
$530$	−11.4074	−0.495505
$531$	0	0
$532$	−3.99189	−0.173070
$533$	3.21009	0.139044
$534$	0	0
$535$	5.79989	0.250751
$536$	−23.7160	−1.02437
$537$	0	0
$538$	−27.4979	−1.18552
$539$	−3.48743	−0.150214
$540$	0	0
$541$	−13.1848	−0.566857	−0.283429	−	0.958993i	$-0.591472\pi$
$541$	−13.1848	−0.566857	−0.283429	+	0.958993i	$0.591472\pi$
$542$	0.793252	0.0340731
$543$	0	0
$544$	13.4743	0.577706
$545$	−0.702325	−0.0300843
$546$	0	0
$547$	−0.144810	−0.00619164	−0.00309582	−	0.999995i	$-0.500985\pi$
$547$	−0.144810	−0.00619164	−0.00309582	+	0.999995i	$0.500985\pi$
$548$	−38.5531	−1.64691
$549$	0	0
$550$	−35.0410	−1.49416
$551$	10.9524	0.466588
$552$	0	0
$553$	1.16978	0.0497441
$554$	42.0923	1.78833
$555$	0	0
$556$	−12.4617	−0.528493
$557$	−6.35824	−0.269407	−0.134704	−	0.990886i	$-0.543008\pi$
$557$	−6.35824	−0.269407	−0.134704	+	0.990886i	$0.543008\pi$
$558$	0	0
$559$	−10.4756	−0.443069
$560$	−0.684644	−0.0289315
$561$	0	0
$562$	−50.8800	−2.14624
$563$	−27.7532	−1.16966	−0.584829	−	0.811157i	$-0.698838\pi$
$563$	−27.7532	−1.16966	−0.584829	+	0.811157i	$0.698838\pi$
$564$	0	0
$565$	4.12919	0.173716
$566$	21.9102	0.920956
$567$	0	0
$568$	19.9074	0.835296
$569$	6.89351	0.288991	0.144495	−	0.989505i	$-0.453844\pi$
$569$	6.89351	0.288991	0.144495	+	0.989505i	$0.453844\pi$
$570$	0	0
$571$	15.4924	0.648337	0.324168	−	0.945999i	$-0.394916\pi$
$571$	15.4924	0.648337	0.324168	+	0.945999i	$0.394916\pi$
$572$	40.7395	1.70340
$573$	0	0
$574$	−1.85183	−0.0772940
$575$	−21.4314	−0.893750
$576$	0	0
$577$	26.1702	1.08948	0.544741	−	0.838605i	$-0.316628\pi$
$577$	26.1702	1.08948	0.544741	+	0.838605i	$0.316628\pi$
$578$	28.9120	1.20258
$579$	0	0
$580$	−17.7804	−0.738289
$581$	−4.68620	−0.194416
$582$	0	0
$583$	24.8565	1.02945
$584$	−14.7717	−0.611256
$585$	0	0
$586$	13.7929	0.569781
$587$	34.8064	1.43662	0.718308	−	0.695725i	$-0.244917\pi$
$587$	34.8064	1.43662	0.718308	+	0.695725i	$0.244917\pi$
$588$	0	0
$589$	−6.27446	−0.258534
$590$	10.6349	0.437831
$591$	0	0
$592$	10.0525	0.413157
$593$	−24.0643	−0.988203	−0.494101	−	0.869404i	$-0.664503\pi$
$593$	−24.0643	−0.988203	−0.494101	+	0.869404i	$0.664503\pi$
$594$	0	0
$595$	1.44491	0.0592355
$596$	−40.8387	−1.67282
$597$	0	0
$598$	41.4692	1.69580
$599$	−0.508407	−0.0207729	−0.0103865	−	0.999946i	$-0.503306\pi$
$599$	−0.508407	−0.0207729	−0.0103865	+	0.999946i	$0.503306\pi$
$600$	0	0
$601$	23.1016	0.942333	0.471166	−	0.882044i	$-0.343833\pi$
$601$	23.1016	0.942333	0.471166	+	0.882044i	$0.343833\pi$
$602$	6.04313	0.246300
$603$	0	0
$604$	10.2545	0.417248
$605$	−0.830943	−0.0337826
$606$	0	0
$607$	−35.7095	−1.44940	−0.724701	−	0.689063i	$-0.758022\pi$
$607$	−35.7095	−1.44940	−0.724701	+	0.689063i	$0.758022\pi$
$608$	8.84098	0.358549
$609$	0	0
$610$	15.6791	0.634827
$611$	−36.6405	−1.48231
$612$	0	0
$613$	−9.75584	−0.394035	−0.197017	−	0.980400i	$-0.563126\pi$
$613$	−9.75584	−0.394035	−0.197017	+	0.980400i	$0.563126\pi$
$614$	30.8665	1.24567
$615$	0	0
$616$	−7.88902	−0.317858
$617$	4.62163	0.186060	0.0930299	−	0.995663i	$-0.470345\pi$
$617$	4.62163	0.186060	0.0930299	+	0.995663i	$0.470345\pi$
$618$	0	0
$619$	−46.6480	−1.87494	−0.937471	−	0.348062i	$-0.886840\pi$
$619$	−46.6480	−1.87494	−0.937471	+	0.348062i	$0.886840\pi$
$620$	10.1861	0.409083
$621$	0	0
$622$	17.8885	0.717263
$623$	−3.95955	−0.158636
$624$	0	0
$625$	17.5929	0.703717
$626$	−18.3451	−0.733217
$627$	0	0
$628$	−61.9808	−2.47330
$629$	−21.2154	−0.845914
$630$	0	0
$631$	−40.7101	−1.62064	−0.810322	−	0.585984i	$-0.800708\pi$
$631$	−40.7101	−1.62064	−0.810322	+	0.585984i	$0.800708\pi$
$632$	2.64620	0.105260
$633$	0	0
$634$	−21.7485	−0.863745
$635$	0.715002	0.0283740
$636$	0	0
$637$	−3.88025	−0.153741
$638$	64.4810	2.55283
$639$	0	0
$640$	−11.2876	−0.446181
$641$	16.0725	0.634826	0.317413	−	0.948287i	$-0.397186\pi$
$641$	16.0725	0.634826	0.317413	+	0.948287i	$0.397186\pi$
$642$	0	0
$643$	6.75968	0.266576	0.133288	−	0.991077i	$-0.457446\pi$
$643$	6.75968	0.266576	0.133288	+	0.991077i	$0.457446\pi$
$644$	−14.3739	−0.566410
$645$	0	0
$646$	−5.99798	−0.235987
$647$	26.3074	1.03425	0.517125	−	0.855910i	$-0.327002\pi$
$647$	26.3074	1.03425	0.517125	+	0.855910i	$0.327002\pi$
$648$	0	0
$649$	−23.1732	−0.909628
$650$	−38.9880	−1.52924
$651$	0	0
$652$	43.3368	1.69720
$653$	15.4078	0.602954	0.301477	−	0.953473i	$-0.402520\pi$
$653$	15.4078	0.602954	0.301477	+	0.953473i	$0.402520\pi$
$654$	0	0
$655$	−0.763652	−0.0298384
$656$	0.792163	0.0309288
$657$	0	0
$658$	21.1371	0.824011
$659$	−11.1639	−0.434885	−0.217442	−	0.976073i	$-0.569771\pi$
$659$	−11.1639	−0.434885	−0.217442	+	0.976073i	$0.569771\pi$
$660$	0	0
$661$	16.7973	0.653338	0.326669	−	0.945139i	$-0.394074\pi$
$661$	16.7973	0.653338	0.326669	+	0.945139i	$0.394074\pi$
$662$	−59.8692	−2.32688
$663$	0	0
$664$	−10.6008	−0.411391
$665$	0.948058	0.0367641
$666$	0	0
$667$	39.4371	1.52701
$668$	−44.3356	−1.71539
$669$	0	0
$670$	16.7794	0.648243
$671$	−34.1644	−1.31890
$672$	0	0
$673$	36.1418	1.39317	0.696583	−	0.717476i	$-0.254703\pi$
$673$	36.1418	1.39317	0.696583	+	0.717476i	$0.254703\pi$
$674$	−17.4774	−0.673205
$675$	0	0
$676$	6.19071	0.238104
$677$	36.2462	1.39306	0.696528	−	0.717530i	$-0.254727\pi$
$677$	36.2462	1.39306	0.696528	+	0.717530i	$0.254727\pi$
$678$	0	0
$679$	−6.83283	−0.262220
$680$	3.26858	0.125344
$681$	0	0
$682$	−36.9401	−1.41451
$683$	−21.5032	−0.822796	−0.411398	−	0.911456i	$-0.634959\pi$
$683$	−21.5032	−0.822796	−0.411398	+	0.911456i	$0.634959\pi$
$684$	0	0
$685$	9.15620	0.349840
$686$	2.23843	0.0854638
$687$	0	0
$688$	−2.58509	−0.0985555
$689$	27.6562	1.05362
$690$	0	0
$691$	−16.8476	−0.640913	−0.320456	−	0.947263i	$-0.603836\pi$
$691$	−16.8476	−0.640913	−0.320456	+	0.947263i	$0.603836\pi$
$692$	36.4571	1.38589
$693$	0	0
$694$	37.8137	1.43539
$695$	2.95960	0.112264
$696$	0	0
$697$	−1.67182	−0.0633248
$698$	33.1585	1.25507
$699$	0	0
$700$	13.5138	0.510775
$701$	8.39278	0.316991	0.158495	−	0.987360i	$-0.449336\pi$
$701$	8.39278	0.316991	0.158495	+	0.987360i	$0.449336\pi$
$702$	0	0
$703$	−13.9202	−0.525011
$704$	45.3715	1.71000
$705$	0	0
$706$	71.6756	2.69755
$707$	−14.4018	−0.541636
$708$	0	0
$709$	−22.8508	−0.858181	−0.429091	−	0.903261i	$-0.641166\pi$
$709$	−22.8508	−0.858181	−0.429091	+	0.903261i	$0.641166\pi$
$710$	−14.0847	−0.528591
$711$	0	0
$712$	−8.95703	−0.335679
$713$	−22.5929	−0.846109
$714$	0	0
$715$	−9.67547	−0.361842
$716$	−30.9537	−1.15679
$717$	0	0
$718$	15.9853	0.596566
$719$	−29.0142	−1.08205	−0.541023	−	0.841008i	$-0.681963\pi$
$719$	−29.0142	−1.08205	−0.541023	+	0.841008i	$0.681963\pi$
$720$	0	0
$721$	−4.78657	−0.178261
$722$	38.5948	1.43635
$723$	0	0
$724$	−21.0021	−0.780538
$725$	−37.0774	−1.37702
$726$	0	0
$727$	1.71536	0.0636190	0.0318095	−	0.999494i	$-0.489873\pi$
$727$	1.71536	0.0636190	0.0318095	+	0.999494i	$0.489873\pi$
$728$	−8.77763	−0.325321
$729$	0	0
$730$	10.4511	0.386814
$731$	5.45571	0.201787
$732$	0	0
$733$	32.0548	1.18397	0.591986	−	0.805948i	$-0.298344\pi$
$733$	32.0548	1.18397	0.591986	+	0.805948i	$0.298344\pi$
$734$	−42.5544	−1.57071
$735$	0	0
$736$	31.8343	1.17343
$737$	−36.5619	−1.34677
$738$	0	0
$739$	15.0497	0.553613	0.276807	−	0.960926i	$-0.410724\pi$
$739$	15.0497	0.553613	0.276807	+	0.960926i	$0.410724\pi$
$740$	22.5984	0.830732
$741$	0	0
$742$	−15.9543	−0.585701
$743$	1.95125	0.0715845	0.0357922	−	0.999359i	$-0.488605\pi$
$743$	1.95125	0.0715845	0.0357922	+	0.999359i	$0.488605\pi$
$744$	0	0
$745$	9.69904	0.355345
$746$	−73.0221	−2.67353
$747$	0	0
$748$	−21.2173	−0.775780
$749$	8.11171	0.296395
$750$	0	0
$751$	−13.9594	−0.509385	−0.254692	−	0.967022i	$-0.581974\pi$
$751$	−13.9594	−0.509385	−0.254692	+	0.967022i	$0.581974\pi$
$752$	−9.04188	−0.329724
$753$	0	0
$754$	71.7440	2.61276
$755$	−2.43539	−0.0886330
$756$	0	0
$757$	−23.9434	−0.870238	−0.435119	−	0.900373i	$-0.643294\pi$
$757$	−23.9434	−0.870238	−0.435119	+	0.900373i	$0.643294\pi$
$758$	−21.3941	−0.777070
$759$	0	0
$760$	2.14463	0.0777940
$761$	−17.6709	−0.640569	−0.320285	−	0.947321i	$-0.603779\pi$
$761$	−17.6709	−0.640569	−0.320285	+	0.947321i	$0.603779\pi$
$762$	0	0
$763$	−0.982270	−0.0355606
$764$	50.8564	1.83992
$765$	0	0
$766$	44.7664	1.61748
$767$	−25.7834	−0.930985
$768$	0	0
$769$	−3.34345	−0.120568	−0.0602839	−	0.998181i	$-0.519201\pi$
$769$	−3.34345	−0.120568	−0.0602839	+	0.998181i	$0.519201\pi$
$770$	5.58158	0.201146
$771$	0	0
$772$	15.1004	0.543475
$773$	−35.6247	−1.28133	−0.640665	−	0.767821i	$-0.721341\pi$
$773$	−35.6247	−1.28133	−0.640665	+	0.767821i	$0.721341\pi$
$774$	0	0
$775$	21.2411	0.763002
$776$	−15.4568	−0.554865
$777$	0	0
$778$	9.11958	0.326953
$779$	−1.09694	−0.0393021
$780$	0	0
$781$	30.6904	1.09819
$782$	−21.5973	−0.772319
$783$	0	0
$784$	−0.957541	−0.0341979
$785$	14.7202	0.525386
$786$	0	0
$787$	−49.3147	−1.75788	−0.878939	−	0.476935i	$-0.841748\pi$
$787$	−49.3147	−1.75788	−0.878939	+	0.476935i	$0.841748\pi$
$788$	−26.4780	−0.943240
$789$	0	0
$790$	−1.87222	−0.0666105
$791$	5.77508	0.205338
$792$	0	0
$793$	−38.0126	−1.34987
$794$	−48.7605	−1.73045
$795$	0	0
$796$	−66.1181	−2.34349
$797$	18.8764	0.668637	0.334318	−	0.942460i	$-0.391494\pi$
$797$	18.8764	0.668637	0.334318	+	0.942460i	$0.391494\pi$
$798$	0	0
$799$	19.0825	0.675089
$800$	−29.9296	−1.05817
$801$	0	0
$802$	−38.7988	−1.37003
$803$	−22.7728	−0.803636
$804$	0	0
$805$	3.41374	0.120318
$806$	−41.1010	−1.44772
$807$	0	0
$808$	−32.5788	−1.14612
$809$	27.6464	0.971994	0.485997	−	0.873960i	$-0.338457\pi$
$809$	27.6464	0.971994	0.485997	+	0.873960i	$0.338457\pi$
$810$	0	0
$811$	−40.0155	−1.40514	−0.702568	−	0.711617i	$-0.747963\pi$
$811$	−40.0155	−1.40514	−0.702568	+	0.711617i	$0.747963\pi$
$812$	−24.8676	−0.872680
$813$	0	0
$814$	−81.9536	−2.87247
$815$	−10.2923	−0.360524
$816$	0	0
$817$	3.57969	0.125237
$818$	27.9232	0.976311
$819$	0	0
$820$	1.78080	0.0621884
$821$	−7.00349	−0.244423	−0.122212	−	0.992504i	$-0.538999\pi$
$821$	−7.00349	−0.244423	−0.122212	+	0.992504i	$0.538999\pi$
$822$	0	0
$823$	7.48955	0.261069	0.130535	−	0.991444i	$-0.458331\pi$
$823$	7.48955	0.261069	0.130535	+	0.991444i	$0.458331\pi$
$824$	−10.8278	−0.377206
$825$	0	0
$826$	14.8739	0.517530
$827$	5.69107	0.197898	0.0989490	−	0.995093i	$-0.468452\pi$
$827$	5.69107	0.197898	0.0989490	+	0.995093i	$0.468452\pi$
$828$	0	0
$829$	−46.9867	−1.63192	−0.815958	−	0.578111i	$-0.803790\pi$
$829$	−46.9867	−1.63192	−0.815958	+	0.578111i	$0.803790\pi$
$830$	7.50020	0.260336
$831$	0	0
$832$	50.4821	1.75015
$833$	2.02085	0.0700182
$834$	0	0
$835$	10.5295	0.364389
$836$	−13.9214	−0.481483
$837$	0	0
$838$	24.1709	0.834971
$839$	−23.2141	−0.801441	−0.400721	−	0.916200i	$-0.631240\pi$
$839$	−23.2141	−0.801441	−0.400721	+	0.916200i	$0.631240\pi$
$840$	0	0
$841$	39.2282	1.35270
$842$	17.0120	0.586273
$843$	0	0
$844$	31.1901	1.07361
$845$	−1.47027	−0.0505788
$846$	0	0
$847$	−1.16215	−0.0399321
$848$	6.82482	0.234365
$849$	0	0
$850$	20.3051	0.696459
$851$	−50.1234	−1.71821
$852$	0	0
$853$	−30.3267	−1.03837	−0.519183	−	0.854663i	$-0.673764\pi$
$853$	−30.3267	−1.03837	−0.519183	+	0.854663i	$0.673764\pi$
$854$	21.9287	0.750385
$855$	0	0
$856$	18.3498	0.627182
$857$	−22.4832	−0.768011	−0.384005	−	0.923331i	$-0.625456\pi$
$857$	−22.4832	−0.768011	−0.384005	+	0.923331i	$0.625456\pi$
$858$	0	0
$859$	−36.3023	−1.23862	−0.619309	−	0.785147i	$-0.712587\pi$
$859$	−36.3023	−1.23862	−0.619309	+	0.785147i	$0.712587\pi$
$860$	−5.81134	−0.198165
$861$	0	0
$862$	77.9441	2.65479
$863$	−32.5087	−1.10661	−0.553305	−	0.832979i	$-0.686633\pi$
$863$	−32.5087	−1.10661	−0.553305	+	0.832979i	$0.686633\pi$
$864$	0	0
$865$	−8.65842	−0.294395
$866$	75.8841	2.57865
$867$	0	0
$868$	14.2462	0.483549
$869$	4.07953	0.138388
$870$	0	0
$871$	−40.6802	−1.37839
$872$	−2.22202	−0.0752472
$873$	0	0
$874$	−14.1708	−0.479334
$875$	−6.78450	−0.229358
$876$	0	0
$877$	−6.43887	−0.217425	−0.108713	−	0.994073i	$-0.534673\pi$
$877$	−6.43887	−0.217425	−0.108713	+	0.994073i	$0.534673\pi$
$878$	−13.5209	−0.456307
$879$	0	0
$880$	−2.38765	−0.0804876
$881$	−50.9542	−1.71669	−0.858346	−	0.513072i	$-0.828507\pi$
$881$	−50.9542	−1.71669	−0.858346	+	0.513072i	$0.828507\pi$
$882$	0	0
$883$	−1.41232	−0.0475284	−0.0237642	−	0.999718i	$-0.507565\pi$
$883$	−1.41232	−0.0475284	−0.0237642	+	0.999718i	$0.507565\pi$
$884$	−23.6072	−0.793994
$885$	0	0
$886$	71.8161	2.41271
$887$	−9.49697	−0.318877	−0.159438	−	0.987208i	$-0.550968\pi$
$887$	−9.49697	−0.318877	−0.159438	+	0.987208i	$0.550968\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	6.33721	0.212424
$891$	0	0
$892$	−19.1376	−0.640775
$893$	12.5207	0.418990
$894$	0	0
$895$	7.35138	0.245730
$896$	−15.7868	−0.527399
$897$	0	0
$898$	15.7058	0.524110
$899$	−39.0869	−1.30362
$900$	0	0
$901$	−14.4035	−0.479849
$902$	−6.45813	−0.215032
$903$	0	0
$904$	13.0640	0.434502
$905$	4.98792	0.165804
$906$	0	0
$907$	−48.9510	−1.62539	−0.812695	−	0.582689i	$-0.802000\pi$
$907$	−48.9510	−1.62539	−0.812695	+	0.582689i	$0.802000\pi$
$908$	23.1444	0.768074
$909$	0	0
$910$	6.21028	0.205869
$911$	−5.99796	−0.198721	−0.0993607	−	0.995051i	$-0.531680\pi$
$911$	−5.99796	−0.198721	−0.0993607	+	0.995051i	$0.531680\pi$
$912$	0	0
$913$	−16.3428	−0.540867
$914$	−37.7136	−1.24746
$915$	0	0
$916$	−31.8109	−1.05106
$917$	−1.06804	−0.0352698
$918$	0	0
$919$	−0.999658	−0.0329757	−0.0164878	−	0.999864i	$-0.505248\pi$
$919$	−0.999658	−0.0329757	−0.0164878	+	0.999864i	$0.505248\pi$
$920$	7.72232	0.254597
$921$	0	0
$922$	39.7553	1.30927
$923$	34.1473	1.12397
$924$	0	0
$925$	47.1244	1.54944
$926$	−22.7936	−0.749044
$927$	0	0
$928$	55.0751	1.80793
$929$	−55.3468	−1.81587	−0.907935	−	0.419110i	$-0.862342\pi$
$929$	−55.3468	−1.81587	−0.907935	+	0.419110i	$0.862342\pi$
$930$	0	0
$931$	1.32595	0.0434563
$932$	75.6293	2.47732
$933$	0	0
$934$	3.70270	0.121156
$935$	5.03902	0.164794
$936$	0	0
$937$	−1.27880	−0.0417766	−0.0208883	−	0.999782i	$-0.506649\pi$
$937$	−1.27880	−0.0417766	−0.0208883	+	0.999782i	$0.506649\pi$
$938$	23.4675	0.766242
$939$	0	0
$940$	−20.3264	−0.662974
$941$	−22.7871	−0.742837	−0.371419	−	0.928466i	$-0.621128\pi$
$941$	−22.7871	−0.742837	−0.371419	+	0.928466i	$0.621128\pi$
$942$	0	0
$943$	−3.94984	−0.128625
$944$	−6.36265	−0.207087
$945$	0	0
$946$	21.0750	0.685207
$947$	−8.91484	−0.289693	−0.144847	−	0.989454i	$-0.546269\pi$
$947$	−8.91484	−0.289693	−0.144847	+	0.989454i	$0.546269\pi$
$948$	0	0
$949$	−25.3379	−0.822504
$950$	13.3229	0.432252
$951$	0	0
$952$	4.57142	0.148161
$953$	−8.13381	−0.263480	−0.131740	−	0.991284i	$-0.542056\pi$
$953$	−8.13381	−0.263480	−0.131740	+	0.991284i	$0.542056\pi$
$954$	0	0
$955$	−12.0782	−0.390841
$956$	−32.6383	−1.05560
$957$	0	0
$958$	28.6433	0.925423
$959$	12.8058	0.413522
$960$	0	0
$961$	−8.60776	−0.277670
$962$	−91.1847	−2.93991
$963$	0	0
$964$	−53.8140	−1.73323
$965$	−3.58628	−0.115447
$966$	0	0
$967$	−15.9426	−0.512679	−0.256339	−	0.966587i	$-0.582516\pi$
$967$	−15.9426	−0.512679	−0.256339	+	0.966587i	$0.582516\pi$
$968$	−2.62895	−0.0844975
$969$	0	0
$970$	10.9358	0.351129
$971$	43.2242	1.38713	0.693566	−	0.720393i	$-0.256039\pi$
$971$	43.2242	1.38713	0.693566	+	0.720393i	$0.256039\pi$
$972$	0	0
$973$	4.13929	0.132699
$974$	−13.0116	−0.416918
$975$	0	0
$976$	−9.38050	−0.300262
$977$	11.4772	0.367190	0.183595	−	0.983002i	$-0.441227\pi$
$977$	11.4772	0.367190	0.183595	+	0.983002i	$0.441227\pi$
$978$	0	0
$979$	−13.8087	−0.441327
$980$	−2.15258	−0.0687616
$981$	0	0
$982$	−11.8200	−0.377192
$983$	32.2991	1.03018	0.515090	−	0.857136i	$-0.327758\pi$
$983$	32.2991	1.03018	0.515090	+	0.857136i	$0.327758\pi$
$984$	0	0
$985$	6.28842	0.200366
$986$	−37.3645	−1.18993
$987$	0	0
$988$	−15.4895	−0.492787
$989$	12.8896	0.409866
$990$	0	0
$991$	−46.4069	−1.47416	−0.737081	−	0.675804i	$-0.763796\pi$
$991$	−46.4069	−1.47416	−0.737081	+	0.675804i	$0.763796\pi$
$992$	−31.5516	−1.00177
$993$	0	0
$994$	−19.6989	−0.624810
$995$	15.7028	0.497812
$996$	0	0
$997$	36.8281	1.16636	0.583179	−	0.812344i	$-0.301809\pi$
$997$	36.8281	1.16636	0.583179	+	0.812344i	$0.301809\pi$
$998$	44.8915	1.42101
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.z.1.5	✓	32
3.2	odd	2	inner	8001.2.a.z.1.28	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8001.2.a.z.1.5	✓	32	1.1	even	1	trivial
8001.2.a.z.1.28	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)

\(f(q)\)	\(=\)	\(q-2.23843 q^{2} +3.01059 q^{4} -0.715002 q^{5} -1.00000 q^{7} -2.26213 q^{8} +O(q^{10})\)	\(q-2.23843 q^{2} +3.01059 q^{4} -0.715002 q^{5} -1.00000 q^{7} -2.26213 q^{8} +1.60049 q^{10} -3.48743 q^{11} -3.88025 q^{13} +2.23843 q^{14} -0.957541 q^{16} +2.02085 q^{17} +1.32595 q^{19} -2.15258 q^{20} +7.80638 q^{22} +4.77444 q^{23} -4.48877 q^{25} +8.68568 q^{26} -3.01059 q^{28} +8.26004 q^{29} -4.73204 q^{31} +6.66766 q^{32} -4.52353 q^{34} +0.715002 q^{35} -10.4983 q^{37} -2.96805 q^{38} +1.61743 q^{40} -0.827289 q^{41} +2.69972 q^{43} -10.4992 q^{44} -10.6873 q^{46} +9.44282 q^{47} +1.00000 q^{49} +10.0478 q^{50} -11.6818 q^{52} -7.12745 q^{53} +2.49352 q^{55} +2.26213 q^{56} -18.4895 q^{58} +6.64479 q^{59} +9.79645 q^{61} +10.5924 q^{62} -13.0100 q^{64} +2.77439 q^{65} +10.4839 q^{67} +6.08393 q^{68} -1.60049 q^{70} -8.80029 q^{71} +6.52998 q^{73} +23.4997 q^{74} +3.99189 q^{76} +3.48743 q^{77} -1.16978 q^{79} +0.684644 q^{80} +1.85183 q^{82} +4.68620 q^{83} -1.44491 q^{85} -6.04313 q^{86} +7.88902 q^{88} +3.95955 q^{89} +3.88025 q^{91} +14.3739 q^{92} -21.1371 q^{94} -0.948058 q^{95} +6.83283 q^{97} -2.23843 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 30 q^{4} - 32 q^{7}+O(q^{10}) \) 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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