LMFDB - Embedded newform 8001.2.a.x.1.12

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8001,2,Mod(1,8001)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage: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")

magma://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:traces = [22,0,0,10,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$63.8883066572$
Analytic rank:	$1$
Dimension:	$22$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Character	$\chi$	$=$	8001.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.384540 q^{2} -1.85213 q^{4} -3.33517 q^{5} +1.00000 q^{7} -1.48130 q^{8} -1.28251 q^{10} -0.515399 q^{11} +0.383569 q^{13} +0.384540 q^{14} +3.13464 q^{16} -5.06965 q^{17} -3.66209 q^{19} +6.17717 q^{20} -0.198192 q^{22} +5.75988 q^{23} +6.12339 q^{25} +0.147498 q^{26} -1.85213 q^{28} +5.53042 q^{29} -1.26331 q^{31} +4.16799 q^{32} -1.94948 q^{34} -3.33517 q^{35} -1.25412 q^{37} -1.40822 q^{38} +4.94039 q^{40} +3.87363 q^{41} -10.3608 q^{43} +0.954585 q^{44} +2.21491 q^{46} +11.4758 q^{47} +1.00000 q^{49} +2.35469 q^{50} -0.710420 q^{52} +7.61321 q^{53} +1.71894 q^{55} -1.48130 q^{56} +2.12667 q^{58} +10.0119 q^{59} +4.93654 q^{61} -0.485793 q^{62} -4.66651 q^{64} -1.27927 q^{65} -7.85043 q^{67} +9.38964 q^{68} -1.28251 q^{70} -10.4327 q^{71} -7.05406 q^{73} -0.482260 q^{74} +6.78267 q^{76} -0.515399 q^{77} -1.78677 q^{79} -10.4546 q^{80} +1.48957 q^{82} +1.60224 q^{83} +16.9082 q^{85} -3.98415 q^{86} +0.763460 q^{88} +8.42939 q^{89} +0.383569 q^{91} -10.6680 q^{92} +4.41291 q^{94} +12.2137 q^{95} +11.4989 q^{97} +0.384540 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$22 q + 10 q^{4} + 22 q^{7} - 4 q^{10} - 10 q^{13} - 6 q^{16} - 18 q^{19} - 34 q^{22} - 26 q^{25} + 10 q^{28} - 42 q^{31} - 16 q^{34} - 36 q^{37} - 46 q^{40} - 54 q^{43} - 12 q^{46} + 22 q^{49} - 22 q^{52}+ \cdots - 56 q^{97}+O(q^{100})$ 22 * q + 10 * q^4 + 22 * q^7 - 4 * q^10 - 10 * q^13 - 6 * q^16 - 18 * q^19 - 34 * q^22 - 26 * q^25 + 10 * q^28 - 42 * q^31 - 16 * q^34 - 36 * q^37 - 46 * q^40 - 54 * q^43 - 12 * q^46 + 22 * q^49 - 22 * q^52 - 28 * q^55 - 26 * q^58 - 22 * q^61 - 76 * q^64 - 48 * q^67 - 4 * q^70 - 48 * q^73 - 20 * q^76 - 94 * q^79 - 8 * q^82 - 40 * q^85 - 26 * q^88 - 10 * q^91 - 26 * q^94 - 56 * q^97

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

)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$\alpha_n$			$\theta_n$
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$

$2$	0.384540	0.271911	0.135956	−	0.990715i	$-0.456590\pi$
$2$	0.384540	0.271911	0.135956	+	0.990715i	$0.456590\pi$
$3$	0	0
$3$	0	0
$4$	−1.85213	−0.926064
$5$	−3.33517	−1.49154	−0.745768	−	0.666206i	$-0.767917\pi$
$5$	−3.33517	−1.49154	−0.745768	+	0.666206i	$0.767917\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.48130	−0.523718
$9$	0	0
$10$	−1.28251	−0.405565
$11$	−0.515399	−0.155399	−0.0776993	−	0.996977i	$-0.524757\pi$
$11$	−0.515399	−0.155399	−0.0776993	+	0.996977i	$0.524757\pi$
$12$	0	0
$13$	0.383569	0.106383	0.0531915	−	0.998584i	$-0.483061\pi$
$13$	0.383569	0.106383	0.0531915	+	0.998584i	$0.483061\pi$
$14$	0.384540	0.102773
$15$	0	0
$16$	3.13464	0.783660
$17$	−5.06965	−1.22957	−0.614785	−	0.788695i	$-0.710757\pi$
$17$	−5.06965	−1.22957	−0.614785	+	0.788695i	$0.710757\pi$
$18$	0	0
$19$	−3.66209	−0.840142	−0.420071	−	0.907491i	$-0.637995\pi$
$19$	−3.66209	−0.840142	−0.420071	+	0.907491i	$0.637995\pi$
$20$	6.17717	1.38126
$21$	0	0
$22$	−0.198192	−0.0422546
$23$	5.75988	1.20102	0.600509	−	0.799618i	$-0.294965\pi$
$23$	5.75988	1.20102	0.600509	+	0.799618i	$0.294965\pi$
$24$	0	0
$25$	6.12339	1.22468
$26$	0.147498	0.0289267
$27$	0	0
$28$	−1.85213	−0.350019
$29$	5.53042	1.02697	0.513487	−	0.858097i	$-0.328353\pi$
$29$	5.53042	1.02697	0.513487	+	0.858097i	$0.328353\pi$
$30$	0	0
$31$	−1.26331	−0.226897	−0.113449	−	0.993544i	$-0.536190\pi$
$31$	−1.26331	−0.226897	−0.113449	+	0.993544i	$0.536190\pi$
$32$	4.16799	0.736804
$33$	0	0
$34$	−1.94948	−0.334334
$35$	−3.33517	−0.563747
$36$	0	0
$37$	−1.25412	−0.206176	−0.103088	−	0.994672i	$-0.532872\pi$
$37$	−1.25412	−0.206176	−0.103088	+	0.994672i	$0.532872\pi$
$38$	−1.40822	−0.228444
$39$	0	0
$40$	4.94039	0.781144
$41$	3.87363	0.604960	0.302480	−	0.953156i	$-0.402185\pi$
$41$	3.87363	0.604960	0.302480	+	0.953156i	$0.402185\pi$
$42$	0	0
$43$	−10.3608	−1.58001	−0.790004	−	0.613101i	$-0.789922\pi$
$43$	−10.3608	−1.58001	−0.790004	+	0.613101i	$0.789922\pi$
$44$	0.954585	0.143909
$45$	0	0
$46$	2.21491	0.326570
$47$	11.4758	1.67392	0.836959	−	0.547266i	$-0.184331\pi$
$47$	11.4758	1.67392	0.836959	+	0.547266i	$0.184331\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	2.35469	0.333003
$51$	0	0
$52$	−0.710420	−0.0985175
$53$	7.61321	1.04575	0.522877	−	0.852408i	$-0.324859\pi$
$53$	7.61321	1.04575	0.522877	+	0.852408i	$0.324859\pi$
$54$	0	0
$55$	1.71894	0.231782
$56$	−1.48130	−0.197947
$57$	0	0
$58$	2.12667	0.279246
$59$	10.0119	1.30344	0.651718	−	0.758461i	$-0.274048\pi$
$59$	10.0119	1.30344	0.651718	+	0.758461i	$0.274048\pi$
$60$	0	0
$61$	4.93654	0.632059	0.316030	−	0.948749i	$-0.397650\pi$
$61$	4.93654	0.632059	0.316030	+	0.948749i	$0.397650\pi$
$62$	−0.485793	−0.0616958
$63$	0	0
$64$	−4.66651	−0.583314
$65$	−1.27927	−0.158674
$66$	0	0
$67$	−7.85043	−0.959083	−0.479542	−	0.877519i	$-0.659197\pi$
$67$	−7.85043	−0.959083	−0.479542	+	0.877519i	$0.659197\pi$
$68$	9.38964	1.13866
$69$	0	0
$70$	−1.28251	−0.153289
$71$	−10.4327	−1.23813	−0.619065	−	0.785339i	$-0.712488\pi$
$71$	−10.4327	−1.23813	−0.619065	+	0.785339i	$0.712488\pi$
$72$	0	0
$73$	−7.05406	−0.825616	−0.412808	−	0.910818i	$-0.635452\pi$
$73$	−7.05406	−0.825616	−0.412808	+	0.910818i	$0.635452\pi$
$74$	−0.482260	−0.0560616
$75$	0	0
$76$	6.78267	0.778026
$77$	−0.515399	−0.0587351
$78$	0	0
$79$	−1.78677	−0.201027	−0.100514	−	0.994936i	$-0.532049\pi$
$79$	−1.78677	−0.201027	−0.100514	+	0.994936i	$0.532049\pi$
$80$	−10.4546	−1.16886
$81$	0	0
$82$	1.48957	0.164495
$83$	1.60224	0.175868	0.0879342	−	0.996126i	$-0.471973\pi$
$83$	1.60224	0.175868	0.0879342	+	0.996126i	$0.471973\pi$
$84$	0	0
$85$	16.9082	1.83395
$86$	−3.98415	−0.429622
$87$	0	0
$88$	0.763460	0.0813851
$89$	8.42939	0.893513	0.446757	−	0.894656i	$-0.352579\pi$
$89$	8.42939	0.893513	0.446757	+	0.894656i	$0.352579\pi$
$90$	0	0
$91$	0.383569	0.0402090
$92$	−10.6680	−1.11222
$93$	0	0
$94$	4.41291	0.455157
$95$	12.2137	1.25310
$96$	0	0
$97$	11.4989	1.16753	0.583766	−	0.811922i	$-0.301578\pi$
$97$	11.4989	1.16753	0.583766	+	0.811922i	$0.301578\pi$
$98$	0.384540	0.0388444
$99$	0	0
$100$	−11.3413	−1.13413
$101$	6.51118	0.647886	0.323943	−	0.946077i	$-0.394991\pi$
$101$	6.51118	0.647886	0.323943	+	0.946077i	$0.394991\pi$
$102$	0	0
$103$	7.35590	0.724798	0.362399	−	0.932023i	$-0.381958\pi$
$103$	7.35590	0.724798	0.362399	+	0.932023i	$0.381958\pi$
$104$	−0.568181	−0.0557147
$105$	0	0
$106$	2.92759	0.284352
$107$	0.745709	0.0720904	0.0360452	−	0.999350i	$-0.488524\pi$
$107$	0.745709	0.0720904	0.0360452	+	0.999350i	$0.488524\pi$
$108$	0	0
$109$	−6.40003	−0.613011	−0.306506	−	0.951869i	$-0.599160\pi$
$109$	−6.40003	−0.613011	−0.306506	+	0.951869i	$0.599160\pi$
$110$	0.661004	0.0630242
$111$	0	0
$112$	3.13464	0.296195
$113$	−0.288023	−0.0270949	−0.0135475	−	0.999908i	$-0.504312\pi$
$113$	−0.288023	−0.0270949	−0.0135475	+	0.999908i	$0.504312\pi$
$114$	0	0
$115$	−19.2102	−1.79136
$116$	−10.2431	−0.951044
$117$	0	0
$118$	3.84998	0.354419
$119$	−5.06965	−0.464734
$120$	0	0
$121$	−10.7344	−0.975851
$122$	1.89830	0.171864
$123$	0	0
$124$	2.33981	0.210121
$125$	−3.74670	−0.335115
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	−10.1304	−0.895414
$129$	0	0
$130$	−0.491931	−0.0431452
$131$	15.3614	1.34213	0.671066	−	0.741398i	$-0.265837\pi$
$131$	15.3614	1.34213	0.671066	+	0.741398i	$0.265837\pi$
$132$	0	0
$133$	−3.66209	−0.317544
$134$	−3.01881	−0.260785
$135$	0	0
$136$	7.50967	0.643949
$137$	−1.99537	−0.170476	−0.0852380	−	0.996361i	$-0.527165\pi$
$137$	−1.99537	−0.170476	−0.0852380	+	0.996361i	$0.527165\pi$
$138$	0	0
$139$	−16.8068	−1.42553	−0.712767	−	0.701401i	$-0.752558\pi$
$139$	−16.8068	−1.42553	−0.712767	+	0.701401i	$0.752558\pi$
$140$	6.17717	0.522066
$141$	0	0
$142$	−4.01179	−0.336662
$143$	−0.197691	−0.0165318
$144$	0	0
$145$	−18.4449	−1.53177
$146$	−2.71257	−0.224494
$147$	0	0
$148$	2.32279	0.190933
$149$	−8.33107	−0.682508	−0.341254	−	0.939971i	$-0.610852\pi$
$149$	−8.33107	−0.682508	−0.341254	+	0.939971i	$0.610852\pi$
$150$	0	0
$151$	−5.65737	−0.460390	−0.230195	−	0.973144i	$-0.573936\pi$
$151$	−5.65737	−0.460390	−0.230195	+	0.973144i	$0.573936\pi$
$152$	5.42466	0.439998
$153$	0	0
$154$	−0.198192	−0.0159707
$155$	4.21336	0.338425
$156$	0	0
$157$	17.5110	1.39753	0.698764	−	0.715352i	$-0.253734\pi$
$157$	17.5110	1.39753	0.698764	+	0.715352i	$0.253734\pi$
$158$	−0.687085	−0.0546616
$159$	0	0
$160$	−13.9010	−1.09897
$161$	5.75988	0.453942
$162$	0	0
$163$	5.81050	0.455114	0.227557	−	0.973765i	$-0.426926\pi$
$163$	5.81050	0.455114	0.227557	+	0.973765i	$0.426926\pi$
$164$	−7.17447	−0.560232
$165$	0	0
$166$	0.616125	0.0478206
$167$	−8.15458	−0.631020	−0.315510	−	0.948922i	$-0.602176\pi$
$167$	−8.15458	−0.631020	−0.315510	+	0.948922i	$0.602176\pi$
$168$	0	0
$169$	−12.8529	−0.988683
$170$	6.50187	0.498671
$171$	0	0
$172$	19.1896	1.46319
$173$	−5.26477	−0.400273	−0.200137	−	0.979768i	$-0.564139\pi$
$173$	−5.26477	−0.400273	−0.200137	+	0.979768i	$0.564139\pi$
$174$	0	0
$175$	6.12339	0.462885
$176$	−1.61559	−0.121780
$177$	0	0
$178$	3.24144	0.242956
$179$	−5.90241	−0.441167	−0.220583	−	0.975368i	$-0.570796\pi$
$179$	−5.90241	−0.441167	−0.220583	+	0.975368i	$0.570796\pi$
$180$	0	0
$181$	16.3682	1.21664	0.608321	−	0.793691i	$-0.291843\pi$
$181$	16.3682	1.21664	0.608321	+	0.793691i	$0.291843\pi$
$182$	0.147498	0.0109333
$183$	0	0
$184$	−8.53211	−0.628995
$185$	4.18271	0.307519
$186$	0	0
$187$	2.61289	0.191074
$188$	−21.2547	−1.55016
$189$	0	0
$190$	4.69667	0.340732
$191$	−8.17284	−0.591366	−0.295683	−	0.955286i	$-0.595547\pi$
$191$	−8.17284	−0.591366	−0.295683	+	0.955286i	$0.595547\pi$
$192$	0	0
$193$	−19.2374	−1.38474	−0.692370	−	0.721542i	$-0.743433\pi$
$193$	−19.2374	−1.38474	−0.692370	+	0.721542i	$0.743433\pi$
$194$	4.42178	0.317465
$195$	0	0
$196$	−1.85213	−0.132295
$197$	−17.2148	−1.22651	−0.613253	−	0.789886i	$-0.710139\pi$
$197$	−17.2148	−1.22651	−0.613253	+	0.789886i	$0.710139\pi$
$198$	0	0
$199$	11.1830	0.792744	0.396372	−	0.918090i	$-0.370269\pi$
$199$	11.1830	0.792744	0.396372	+	0.918090i	$0.370269\pi$
$200$	−9.07057	−0.641386
$201$	0	0
$202$	2.50381	0.176167
$203$	5.53042	0.388160
$204$	0	0
$205$	−12.9192	−0.902319
$206$	2.82864	0.197081
$207$	0	0
$208$	1.20235	0.0833680
$209$	1.88744	0.130557
$210$	0	0
$211$	−18.2779	−1.25831	−0.629153	−	0.777282i	$-0.716598\pi$
$211$	−18.2779	−1.25831	−0.629153	+	0.777282i	$0.716598\pi$
$212$	−14.1006	−0.968436
$213$	0	0
$214$	0.286755	0.0196022
$215$	34.5551	2.35664
$216$	0	0
$217$	−1.26331	−0.0857590
$218$	−2.46107	−0.166685
$219$	0	0
$220$	−3.18371	−0.214645
$221$	−1.94456	−0.130805
$222$	0	0
$223$	−28.2048	−1.88873	−0.944365	−	0.328899i	$-0.893322\pi$
$223$	−28.2048	−1.88873	−0.944365	+	0.328899i	$0.893322\pi$
$224$	4.16799	0.278486
$225$	0	0
$226$	−0.110757	−0.00736742
$227$	−21.6717	−1.43840	−0.719202	−	0.694801i	$-0.755492\pi$
$227$	−21.6717	−1.43840	−0.719202	+	0.694801i	$0.755492\pi$
$228$	0	0
$229$	25.5347	1.68738	0.843689	−	0.536832i	$-0.180379\pi$
$229$	25.5347	1.68738	0.843689	+	0.536832i	$0.180379\pi$
$230$	−7.38710	−0.487091
$231$	0	0
$232$	−8.19221	−0.537845
$233$	−23.6494	−1.54932	−0.774661	−	0.632376i	$-0.782080\pi$
$233$	−23.6494	−1.54932	−0.774661	+	0.632376i	$0.782080\pi$
$234$	0	0
$235$	−38.2738	−2.49671
$236$	−18.5433	−1.20707
$237$	0	0
$238$	−1.94948	−0.126366
$239$	−8.23753	−0.532841	−0.266421	−	0.963857i	$-0.585841\pi$
$239$	−8.23753	−0.532841	−0.266421	+	0.963857i	$0.585841\pi$
$240$	0	0
$241$	0.447585	0.0288315	0.0144158	−	0.999896i	$-0.495411\pi$
$241$	0.447585	0.0288315	0.0144158	+	0.999896i	$0.495411\pi$
$242$	−4.12780	−0.265345
$243$	0	0
$244$	−9.14311	−0.585327
$245$	−3.33517	−0.213076
$246$	0	0
$247$	−1.40467	−0.0893768
$248$	1.87134	0.118830
$249$	0	0
$250$	−1.44076	−0.0911214
$251$	−8.12028	−0.512547	−0.256274	−	0.966604i	$-0.582495\pi$
$251$	−8.12028	−0.512547	−0.256274	+	0.966604i	$0.582495\pi$
$252$	0	0
$253$	−2.96864	−0.186637
$254$	0.384540	0.0241282
$255$	0	0
$256$	5.43746	0.339841
$257$	0.667162	0.0416164	0.0208082	−	0.999783i	$-0.493376\pi$
$257$	0.667162	0.0416164	0.0208082	+	0.999783i	$0.493376\pi$
$258$	0	0
$259$	−1.25412	−0.0779273
$260$	2.36937	0.146942
$261$	0	0
$262$	5.90708	0.364940
$263$	24.6687	1.52114	0.760570	−	0.649256i	$-0.224920\pi$
$263$	24.6687	1.52114	0.760570	+	0.649256i	$0.224920\pi$
$264$	0	0
$265$	−25.3914	−1.55978
$266$	−1.40822	−0.0863437
$267$	0	0
$268$	14.5400	0.888173
$269$	20.2132	1.23242	0.616210	−	0.787582i	$-0.288667\pi$
$269$	20.2132	1.23242	0.616210	+	0.787582i	$0.288667\pi$
$270$	0	0
$271$	−4.68284	−0.284462	−0.142231	−	0.989833i	$-0.545428\pi$
$271$	−4.68284	−0.284462	−0.142231	+	0.989833i	$0.545428\pi$
$272$	−15.8915	−0.963565
$273$	0	0
$274$	−0.767301	−0.0463543
$275$	−3.15599	−0.190313
$276$	0	0
$277$	10.2489	0.615796	0.307898	−	0.951419i	$-0.400374\pi$
$277$	10.2489	0.615796	0.307898	+	0.951419i	$0.400374\pi$
$278$	−6.46289	−0.387618
$279$	0	0
$280$	4.94039	0.295245
$281$	−9.49428	−0.566381	−0.283191	−	0.959064i	$-0.591393\pi$
$281$	−9.49428	−0.566381	−0.283191	+	0.959064i	$0.591393\pi$
$282$	0	0
$283$	2.56660	0.152569	0.0762844	−	0.997086i	$-0.475694\pi$
$283$	2.56660	0.152569	0.0762844	+	0.997086i	$0.475694\pi$
$284$	19.3227	1.14659
$285$	0	0
$286$	−0.0760202	−0.00449517
$287$	3.87363	0.228653
$288$	0	0
$289$	8.70134	0.511844
$290$	−7.09282	−0.416505
$291$	0	0
$292$	13.0650	0.764573
$293$	11.6103	0.678279	0.339140	−	0.940736i	$-0.389864\pi$
$293$	11.6103	0.678279	0.339140	+	0.940736i	$0.389864\pi$
$294$	0	0
$295$	−33.3914	−1.94412
$296$	1.85773	0.107978
$297$	0	0
$298$	−3.20363	−0.185582
$299$	2.20931	0.127768
$300$	0	0
$301$	−10.3608	−0.597187
$302$	−2.17549	−0.125185
$303$	0	0
$304$	−11.4793	−0.658385
$305$	−16.4642	−0.942738
$306$	0	0
$307$	−9.00122	−0.513727	−0.256863	−	0.966448i	$-0.582689\pi$
$307$	−9.00122	−0.513727	−0.256863	+	0.966448i	$0.582689\pi$
$308$	0.954585	0.0543925
$309$	0	0
$310$	1.62021	0.0920215
$311$	−4.00207	−0.226937	−0.113468	−	0.993542i	$-0.536196\pi$
$311$	−4.00207	−0.226937	−0.113468	+	0.993542i	$0.536196\pi$
$312$	0	0
$313$	−5.90161	−0.333578	−0.166789	−	0.985993i	$-0.553340\pi$
$313$	−5.90161	−0.333578	−0.166789	+	0.985993i	$0.553340\pi$
$314$	6.73368	0.380004
$315$	0	0
$316$	3.30933	0.186164
$317$	−4.98975	−0.280252	−0.140126	−	0.990134i	$-0.544751\pi$
$317$	−4.98975	−0.280252	−0.140126	+	0.990134i	$0.544751\pi$
$318$	0	0
$319$	−2.85037	−0.159590
$320$	15.5636	0.870034
$321$	0	0
$322$	2.21491	0.123432
$323$	18.5655	1.03301
$324$	0	0
$325$	2.34874	0.130285
$326$	2.23437	0.123750
$327$	0	0
$328$	−5.73801	−0.316829
$329$	11.4758	0.632681
$330$	0	0
$331$	32.1337	1.76623	0.883114	−	0.469159i	$-0.155443\pi$
$331$	32.1337	1.76623	0.883114	+	0.469159i	$0.155443\pi$
$332$	−2.96755	−0.162865
$333$	0	0
$334$	−3.13577	−0.171581
$335$	26.1826	1.43051
$336$	0	0
$337$	−0.613743	−0.0334327	−0.0167163	−	0.999860i	$-0.505321\pi$
$337$	−0.613743	−0.0334327	−0.0167163	+	0.999860i	$0.505321\pi$
$338$	−4.94245	−0.268834
$339$	0	0
$340$	−31.3161	−1.69835
$341$	0.651108	0.0352595
$342$	0	0
$343$	1.00000	0.0539949
$344$	15.3475	0.827480
$345$	0	0
$346$	−2.02452	−0.108839
$347$	18.8505	1.01195	0.505975	−	0.862548i	$-0.331133\pi$
$347$	18.8505	1.01195	0.505975	+	0.862548i	$0.331133\pi$
$348$	0	0
$349$	−16.9302	−0.906255	−0.453128	−	0.891446i	$-0.649692\pi$
$349$	−16.9302	−0.906255	−0.453128	+	0.891446i	$0.649692\pi$
$350$	2.35469	0.125863
$351$	0	0
$352$	−2.14818	−0.114498
$353$	−25.1141	−1.33669	−0.668343	−	0.743853i	$-0.732996\pi$
$353$	−25.1141	−1.33669	−0.668343	+	0.743853i	$0.732996\pi$
$354$	0	0
$355$	34.7948	1.84672
$356$	−15.6123	−0.827451
$357$	0	0
$358$	−2.26971	−0.119958
$359$	−31.2366	−1.64860	−0.824301	−	0.566151i	$-0.808432\pi$
$359$	−31.2366	−1.64860	−0.824301	+	0.566151i	$0.808432\pi$
$360$	0	0
$361$	−5.58906	−0.294161
$362$	6.29425	0.330819
$363$	0	0
$364$	−0.710420	−0.0372361
$365$	23.5265	1.23144
$366$	0	0
$367$	10.1840	0.531601	0.265800	−	0.964028i	$-0.414364\pi$
$367$	10.1840	0.531601	0.265800	+	0.964028i	$0.414364\pi$
$368$	18.0551	0.941190
$369$	0	0
$370$	1.60842	0.0836179
$371$	7.61321	0.395258
$372$	0	0
$373$	37.5615	1.94486	0.972431	−	0.233192i	$-0.0749170\pi$
$373$	37.5615	1.94486	0.972431	+	0.233192i	$0.0749170\pi$
$374$	1.00476	0.0519550
$375$	0	0
$376$	−16.9991	−0.876661
$377$	2.12130	0.109253
$378$	0	0
$379$	−15.0663	−0.773903	−0.386951	−	0.922100i	$-0.626472\pi$
$379$	−15.0663	−0.773903	−0.386951	+	0.922100i	$0.626472\pi$
$380$	−22.6214	−1.16045
$381$	0	0
$382$	−3.14279	−0.160799
$383$	−30.0842	−1.53723	−0.768615	−	0.639711i	$-0.779054\pi$
$383$	−30.0842	−1.53723	−0.768615	+	0.639711i	$0.779054\pi$
$384$	0	0
$385$	1.71894	0.0876055
$386$	−7.39757	−0.376526
$387$	0	0
$388$	−21.2974	−1.08121
$389$	−24.7463	−1.25469	−0.627343	−	0.778743i	$-0.715858\pi$
$389$	−24.7463	−1.25469	−0.627343	+	0.778743i	$0.715858\pi$
$390$	0	0
$391$	−29.2006	−1.47674
$392$	−1.48130	−0.0748169
$393$	0	0
$394$	−6.61980	−0.333501
$395$	5.95919	0.299839
$396$	0	0
$397$	−15.2008	−0.762907	−0.381453	−	0.924388i	$-0.624576\pi$
$397$	−15.2008	−0.762907	−0.381453	+	0.924388i	$0.624576\pi$
$398$	4.30032	0.215556
$399$	0	0
$400$	19.1946	0.959730
$401$	−16.0162	−0.799811	−0.399906	−	0.916556i	$-0.630957\pi$
$401$	−16.0162	−0.799811	−0.399906	+	0.916556i	$0.630957\pi$
$402$	0	0
$403$	−0.484567	−0.0241380
$404$	−12.0595	−0.599984
$405$	0	0
$406$	2.12667	0.105545
$407$	0.646373	0.0320395
$408$	0	0
$409$	−18.5824	−0.918842	−0.459421	−	0.888219i	$-0.651943\pi$
$409$	−18.5824	−0.918842	−0.459421	+	0.888219i	$0.651943\pi$
$410$	−4.96797	−0.245351
$411$	0	0
$412$	−13.6241	−0.671210
$413$	10.0119	0.492653
$414$	0	0
$415$	−5.34374	−0.262314
$416$	1.59871	0.0783834
$417$	0	0
$418$	0.725797	0.0354999
$419$	−27.6284	−1.34974	−0.674869	−	0.737938i	$-0.735800\pi$
$419$	−27.6284	−1.34974	−0.674869	+	0.737938i	$0.735800\pi$
$420$	0	0
$421$	−3.81539	−0.185951	−0.0929753	−	0.995668i	$-0.529638\pi$
$421$	−3.81539	−0.185951	−0.0929753	+	0.995668i	$0.529638\pi$
$422$	−7.02861	−0.342147
$423$	0	0
$424$	−11.2774	−0.547681
$425$	−31.0434	−1.50583
$426$	0	0
$427$	4.93654	0.238896
$428$	−1.38115	−0.0667603
$429$	0	0
$430$	13.2878	0.640796
$431$	−5.14646	−0.247896	−0.123948	−	0.992289i	$-0.539556\pi$
$431$	−5.14646	−0.247896	−0.123948	+	0.992289i	$0.539556\pi$
$432$	0	0
$433$	−27.3715	−1.31539	−0.657695	−	0.753285i	$-0.728468\pi$
$433$	−27.3715	−1.31539	−0.657695	+	0.753285i	$0.728468\pi$
$434$	−0.485793	−0.0233188
$435$	0	0
$436$	11.8537	0.567688
$437$	−21.0932	−1.00903
$438$	0	0
$439$	−28.0636	−1.33940	−0.669700	−	0.742631i	$-0.733577\pi$
$439$	−28.0636	−1.33940	−0.669700	+	0.742631i	$0.733577\pi$
$440$	−2.54627	−0.121389
$441$	0	0
$442$	−0.747763	−0.0355674
$443$	26.9119	1.27862	0.639311	−	0.768948i	$-0.279220\pi$
$443$	26.9119	1.27862	0.639311	+	0.768948i	$0.279220\pi$
$444$	0	0
$445$	−28.1135	−1.33271
$446$	−10.8459	−0.513567
$447$	0	0
$448$	−4.66651	−0.220472
$449$	−5.82666	−0.274977	−0.137488	−	0.990503i	$-0.543903\pi$
$449$	−5.82666	−0.274977	−0.137488	+	0.990503i	$0.543903\pi$
$450$	0	0
$451$	−1.99647	−0.0940099
$452$	0.533456	0.0250917
$453$	0	0
$454$	−8.33366	−0.391118
$455$	−1.27927	−0.0599731
$456$	0	0
$457$	−22.6300	−1.05859	−0.529293	−	0.848439i	$-0.677543\pi$
$457$	−22.6300	−1.05859	−0.529293	+	0.848439i	$0.677543\pi$
$458$	9.81911	0.458817
$459$	0	0
$460$	35.5798	1.65892
$461$	−2.86588	−0.133478	−0.0667388	−	0.997770i	$-0.521259\pi$
$461$	−2.86588	−0.133478	−0.0667388	+	0.997770i	$0.521259\pi$
$462$	0	0
$463$	−1.76337	−0.0819508	−0.0409754	−	0.999160i	$-0.513047\pi$
$463$	−1.76337	−0.0819508	−0.0409754	+	0.999160i	$0.513047\pi$
$464$	17.3359	0.804798
$465$	0	0
$466$	−9.09415	−0.421278
$467$	36.8967	1.70738	0.853688	−	0.520785i	$-0.174361\pi$
$467$	36.8967	1.70738	0.853688	+	0.520785i	$0.174361\pi$
$468$	0	0
$469$	−7.85043	−0.362499
$470$	−14.7178	−0.678882
$471$	0	0
$472$	−14.8306	−0.682634
$473$	5.33995	0.245531
$474$	0	0
$475$	−22.4244	−1.02890
$476$	9.38964	0.430374
$477$	0	0
$478$	−3.16766	−0.144885
$479$	21.8419	0.997984	0.498992	−	0.866607i	$-0.333704\pi$
$479$	21.8419	0.997984	0.498992	+	0.866607i	$0.333704\pi$
$480$	0	0
$481$	−0.481043	−0.0219337
$482$	0.172115	0.00783961
$483$	0	0
$484$	19.8814	0.903701
$485$	−38.3507	−1.74142
$486$	0	0
$487$	20.5914	0.933084	0.466542	−	0.884499i	$-0.345500\pi$
$487$	20.5914	0.933084	0.466542	+	0.884499i	$0.345500\pi$
$488$	−7.31249	−0.331021
$489$	0	0
$490$	−1.28251	−0.0579379
$491$	−15.3723	−0.693742	−0.346871	−	0.937913i	$-0.612756\pi$
$491$	−15.3723	−0.693742	−0.346871	+	0.937913i	$0.612756\pi$
$492$	0	0
$493$	−28.0373	−1.26274
$494$	−0.540151	−0.0243026
$495$	0	0
$496$	−3.96002	−0.177810
$497$	−10.4327	−0.467969
$498$	0	0
$499$	−3.82462	−0.171214	−0.0856068	−	0.996329i	$-0.527283\pi$
$499$	−3.82462	−0.171214	−0.0856068	+	0.996329i	$0.527283\pi$
$500$	6.93937	0.310338
$501$	0	0
$502$	−3.12258	−0.139367
$503$	7.01316	0.312701	0.156351	−	0.987702i	$-0.450027\pi$
$503$	7.01316	0.312701	0.156351	+	0.987702i	$0.450027\pi$
$504$	0	0
$505$	−21.7159	−0.966345
$506$	−1.14156	−0.0507486
$507$	0	0
$508$	−1.85213	−0.0821749
$509$	0.628790	0.0278706	0.0139353	−	0.999903i	$-0.495564\pi$
$509$	0.628790	0.0278706	0.0139353	+	0.999903i	$0.495564\pi$
$510$	0	0
$511$	−7.05406	−0.312053
$512$	22.3518	0.987820
$513$	0	0
$514$	0.256551	0.0113160
$515$	−24.5332	−1.08106
$516$	0	0
$517$	−5.91461	−0.260124
$518$	−0.482260	−0.0211893
$519$	0	0
$520$	1.89498	0.0831005
$521$	8.64677	0.378822	0.189411	−	0.981898i	$-0.439342\pi$
$521$	8.64677	0.378822	0.189411	+	0.981898i	$0.439342\pi$
$522$	0	0
$523$	14.1272	0.617741	0.308870	−	0.951104i	$-0.400049\pi$
$523$	14.1272	0.617741	0.308870	+	0.951104i	$0.400049\pi$
$524$	−28.4513	−1.24290
$525$	0	0
$526$	9.48613	0.413615
$527$	6.40453	0.278986
$528$	0	0
$529$	10.1763	0.442446
$530$	−9.76401	−0.424122
$531$	0	0
$532$	6.78267	0.294066
$533$	1.48581	0.0643575
$534$	0	0
$535$	−2.48707	−0.107525
$536$	11.6288	0.502289
$537$	0	0
$538$	7.77279	0.335109
$539$	−0.515399	−0.0221998
$540$	0	0
$541$	7.29682	0.313715	0.156857	−	0.987621i	$-0.449864\pi$
$541$	7.29682	0.313715	0.156857	+	0.987621i	$0.449864\pi$
$542$	−1.80074	−0.0773484
$543$	0	0
$544$	−21.1303	−0.905953
$545$	21.3452	0.914328
$546$	0	0
$547$	22.6540	0.968614	0.484307	−	0.874898i	$-0.339072\pi$
$547$	22.6540	0.968614	0.484307	+	0.874898i	$0.339072\pi$
$548$	3.69568	0.157872
$549$	0	0
$550$	−1.21360	−0.0517483
$551$	−20.2529	−0.862804
$552$	0	0
$553$	−1.78677	−0.0759812
$554$	3.94111	0.167442
$555$	0	0
$556$	31.1283	1.32014
$557$	−4.61204	−0.195418	−0.0977092	−	0.995215i	$-0.531151\pi$
$557$	−4.61204	−0.195418	−0.0977092	+	0.995215i	$0.531151\pi$
$558$	0	0
$559$	−3.97409	−0.168086
$560$	−10.4546	−0.441786
$561$	0	0
$562$	−3.65093	−0.154005
$563$	−7.01174	−0.295509	−0.147755	−	0.989024i	$-0.547205\pi$
$563$	−7.01174	−0.295509	−0.147755	+	0.989024i	$0.547205\pi$
$564$	0	0
$565$	0.960608	0.0404131
$566$	0.986963	0.0414852
$567$	0	0
$568$	15.4539	0.648432
$569$	−26.5515	−1.11310	−0.556548	−	0.830815i	$-0.687875\pi$
$569$	−26.5515	−1.11310	−0.556548	+	0.830815i	$0.687875\pi$
$570$	0	0
$571$	−39.3223	−1.64559	−0.822794	−	0.568340i	$-0.807586\pi$
$571$	−39.3223	−1.64559	−0.822794	+	0.568340i	$0.807586\pi$
$572$	0.366149	0.0153095
$573$	0	0
$574$	1.48957	0.0621734
$575$	35.2700	1.47086
$576$	0	0
$577$	−12.1039	−0.503894	−0.251947	−	0.967741i	$-0.581071\pi$
$577$	−12.1039	−0.503894	−0.251947	+	0.967741i	$0.581071\pi$
$578$	3.34602	0.139176
$579$	0	0
$580$	34.1624	1.41852
$581$	1.60224	0.0664720
$582$	0	0
$583$	−3.92384	−0.162509
$584$	10.4492	0.432390
$585$	0	0
$586$	4.46462	0.184432
$587$	8.91072	0.367785	0.183892	−	0.982946i	$-0.441130\pi$
$587$	8.91072	0.367785	0.183892	+	0.982946i	$0.441130\pi$
$588$	0	0
$589$	4.62636	0.190626
$590$	−12.8403	−0.528628
$591$	0	0
$592$	−3.93122	−0.161572
$593$	17.2992	0.710392	0.355196	−	0.934792i	$-0.384414\pi$
$593$	17.2992	0.710392	0.355196	+	0.934792i	$0.384414\pi$
$594$	0	0
$595$	16.9082	0.693167
$596$	15.4302	0.632047
$597$	0	0
$598$	0.849571	0.0347415
$599$	22.9576	0.938024	0.469012	−	0.883192i	$-0.344610\pi$
$599$	22.9576	0.938024	0.469012	+	0.883192i	$0.344610\pi$
$600$	0	0
$601$	25.7556	1.05059	0.525297	−	0.850919i	$-0.323954\pi$
$601$	25.7556	1.05059	0.525297	+	0.850919i	$0.323954\pi$
$602$	−3.98415	−0.162382
$603$	0	0
$604$	10.4782	0.426351
$605$	35.8010	1.45552
$606$	0	0
$607$	−23.7127	−0.962470	−0.481235	−	0.876592i	$-0.659812\pi$
$607$	−23.7127	−0.962470	−0.481235	+	0.876592i	$0.659812\pi$
$608$	−15.2636	−0.619020
$609$	0	0
$610$	−6.33116	−0.256341
$611$	4.40177	0.178076
$612$	0	0
$613$	−4.91662	−0.198580	−0.0992902	−	0.995059i	$-0.531657\pi$
$613$	−4.91662	−0.198580	−0.0992902	+	0.995059i	$0.531657\pi$
$614$	−3.46133	−0.139688
$615$	0	0
$616$	0.763460	0.0307607
$617$	32.2646	1.29892	0.649461	−	0.760395i	$-0.274994\pi$
$617$	32.2646	1.29892	0.649461	+	0.760395i	$0.274994\pi$
$618$	0	0
$619$	−16.7286	−0.672379	−0.336189	−	0.941794i	$-0.609138\pi$
$619$	−16.7286	−0.672379	−0.336189	+	0.941794i	$0.609138\pi$
$620$	−7.80368	−0.313403
$621$	0	0
$622$	−1.53896	−0.0617066
$623$	8.42939	0.337716
$624$	0	0
$625$	−18.1211	−0.724842
$626$	−2.26941	−0.0907037
$627$	0	0
$628$	−32.4326	−1.29420
$629$	6.35796	0.253508
$630$	0	0
$631$	6.40569	0.255007	0.127503	−	0.991838i	$-0.459304\pi$
$631$	6.40569	0.255007	0.127503	+	0.991838i	$0.459304\pi$
$632$	2.64674	0.105282
$633$	0	0
$634$	−1.91876	−0.0762037
$635$	−3.33517	−0.132352
$636$	0	0
$637$	0.383569	0.0151976
$638$	−1.09608	−0.0433944
$639$	0	0
$640$	33.7868	1.33554
$641$	5.69431	0.224912	0.112456	−	0.993657i	$-0.464128\pi$
$641$	5.69431	0.224912	0.112456	+	0.993657i	$0.464128\pi$
$642$	0	0
$643$	−37.9909	−1.49821	−0.749107	−	0.662449i	$-0.769517\pi$
$643$	−37.9909	−1.49821	−0.749107	+	0.662449i	$0.769517\pi$
$644$	−10.6680	−0.420380
$645$	0	0
$646$	7.13920	0.280888
$647$	26.7420	1.05133	0.525667	−	0.850690i	$-0.323816\pi$
$647$	26.7420	1.05133	0.525667	+	0.850690i	$0.323816\pi$
$648$	0	0
$649$	−5.16011	−0.202552
$650$	0.903187	0.0354259
$651$	0	0
$652$	−10.7618	−0.421465
$653$	−45.1353	−1.76628	−0.883140	−	0.469109i	$-0.844575\pi$
$653$	−45.1353	−1.76628	−0.883140	+	0.469109i	$0.844575\pi$
$654$	0	0
$655$	−51.2329	−2.00184
$656$	12.1424	0.474083
$657$	0	0
$658$	4.41291	0.172033
$659$	−18.8007	−0.732372	−0.366186	−	0.930542i	$-0.619337\pi$
$659$	−18.8007	−0.732372	−0.366186	+	0.930542i	$0.619337\pi$
$660$	0	0
$661$	−11.7878	−0.458493	−0.229246	−	0.973368i	$-0.573626\pi$
$661$	−11.7878	−0.458493	−0.229246	+	0.973368i	$0.573626\pi$
$662$	12.3567	0.480257
$663$	0	0
$664$	−2.37339	−0.0921055
$665$	12.2137	0.473628
$666$	0	0
$667$	31.8546	1.23342
$668$	15.1033	0.584365
$669$	0	0
$670$	10.0683	0.388971
$671$	−2.54429	−0.0982211
$672$	0	0
$673$	−20.9627	−0.808054	−0.404027	−	0.914747i	$-0.632390\pi$
$673$	−20.9627	−0.808054	−0.404027	+	0.914747i	$0.632390\pi$
$674$	−0.236009	−0.00909072
$675$	0	0
$676$	23.8052	0.915584
$677$	5.81638	0.223542	0.111771	−	0.993734i	$-0.464348\pi$
$677$	5.81638	0.223542	0.111771	+	0.993734i	$0.464348\pi$
$678$	0	0
$679$	11.4989	0.441286
$680$	−25.0460	−0.960472
$681$	0	0
$682$	0.250377	0.00958744
$683$	−31.0134	−1.18670	−0.593348	−	0.804946i	$-0.702194\pi$
$683$	−31.0134	−1.18670	−0.593348	+	0.804946i	$0.702194\pi$
$684$	0	0
$685$	6.65491	0.254271
$686$	0.384540	0.0146818
$687$	0	0
$688$	−32.4774	−1.23819
$689$	2.92019	0.111251
$690$	0	0
$691$	3.18940	0.121330	0.0606652	−	0.998158i	$-0.480678\pi$
$691$	3.18940	0.121330	0.0606652	+	0.998158i	$0.480678\pi$
$692$	9.75103	0.370679
$693$	0	0
$694$	7.24879	0.275160
$695$	56.0536	2.12623
$696$	0	0
$697$	−19.6380	−0.743841
$698$	−6.51036	−0.246421
$699$	0	0
$700$	−11.3413	−0.428661
$701$	29.7208	1.12254	0.561269	−	0.827633i	$-0.310313\pi$
$701$	29.7208	1.12254	0.561269	+	0.827633i	$0.310313\pi$
$702$	0	0
$703$	4.59271	0.173217
$704$	2.40512	0.0906462
$705$	0	0
$706$	−9.65737	−0.363460
$707$	6.51118	0.244878
$708$	0	0
$709$	38.3780	1.44132	0.720659	−	0.693290i	$-0.243840\pi$
$709$	38.3780	1.44132	0.720659	+	0.693290i	$0.243840\pi$
$710$	13.3800	0.502143
$711$	0	0
$712$	−12.4864	−0.467949
$713$	−7.27651	−0.272508
$714$	0	0
$715$	0.659335	0.0246577
$716$	10.9320	0.408549
$717$	0	0
$718$	−12.0117	−0.448273
$719$	1.89887	0.0708159	0.0354079	−	0.999373i	$-0.488727\pi$
$719$	1.89887	0.0708159	0.0354079	+	0.999373i	$0.488727\pi$
$720$	0	0
$721$	7.35590	0.273948
$722$	−2.14922	−0.0799857
$723$	0	0
$724$	−30.3161	−1.12669
$725$	33.8649	1.25771
$726$	0	0
$727$	−53.3979	−1.98042	−0.990210	−	0.139588i	$-0.955422\pi$
$727$	−53.3979	−1.98042	−0.990210	+	0.139588i	$0.955422\pi$
$728$	−0.568181	−0.0210582
$729$	0	0
$730$	9.04690	0.334841
$731$	52.5257	1.94273
$732$	0	0
$733$	−7.30289	−0.269738	−0.134869	−	0.990863i	$-0.543061\pi$
$733$	−7.30289	−0.269738	−0.134869	+	0.990863i	$0.543061\pi$
$734$	3.91616	0.144548
$735$	0	0
$736$	24.0072	0.884915
$737$	4.04610	0.149040
$738$	0	0
$739$	45.1757	1.66182	0.830908	−	0.556410i	$-0.187822\pi$
$739$	45.1757	1.66182	0.830908	+	0.556410i	$0.187822\pi$
$740$	−7.74692	−0.284783
$741$	0	0
$742$	2.92759	0.107475
$743$	16.4803	0.604605	0.302302	−	0.953212i	$-0.402245\pi$
$743$	16.4803	0.604605	0.302302	+	0.953212i	$0.402245\pi$
$744$	0	0
$745$	27.7856	1.01799
$746$	14.4439	0.528829
$747$	0	0
$748$	−4.83941	−0.176946
$749$	0.745709	0.0272476
$750$	0	0
$751$	52.4513	1.91398	0.956988	−	0.290126i	$-0.0936974\pi$
$751$	52.4513	1.91398	0.956988	+	0.290126i	$0.0936974\pi$
$752$	35.9725	1.31178
$753$	0	0
$754$	0.815726	0.0297070
$755$	18.8683	0.686689
$756$	0	0
$757$	−8.28967	−0.301293	−0.150647	−	0.988588i	$-0.548136\pi$
$757$	−8.28967	−0.301293	−0.150647	+	0.988588i	$0.548136\pi$
$758$	−5.79359	−0.210433
$759$	0	0
$760$	−18.0922	−0.656272
$761$	22.2851	0.807833	0.403917	−	0.914796i	$-0.367649\pi$
$761$	22.2851	0.807833	0.403917	+	0.914796i	$0.367649\pi$
$762$	0	0
$763$	−6.40003	−0.231696
$764$	15.1371	0.547643
$765$	0	0
$766$	−11.5686	−0.417990
$767$	3.84025	0.138664
$768$	0	0
$769$	4.88525	0.176167	0.0880833	−	0.996113i	$-0.471926\pi$
$769$	4.88525	0.176167	0.0880833	+	0.996113i	$0.471926\pi$
$770$	0.661004	0.0238209
$771$	0	0
$772$	35.6302	1.28236
$773$	0.798838	0.0287322	0.0143661	−	0.999897i	$-0.495427\pi$
$773$	0.798838	0.0287322	0.0143661	+	0.999897i	$0.495427\pi$
$774$	0	0
$775$	−7.73573	−0.277876
$776$	−17.0333	−0.611458
$777$	0	0
$778$	−9.51594	−0.341163
$779$	−14.1856	−0.508252
$780$	0	0
$781$	5.37699	0.192404
$782$	−11.2288	−0.401541
$783$	0	0
$784$	3.13464	0.111951
$785$	−58.4022	−2.08446
$786$	0	0
$787$	1.43503	0.0511534	0.0255767	−	0.999673i	$-0.491858\pi$
$787$	1.43503	0.0511534	0.0255767	+	0.999673i	$0.491858\pi$
$788$	31.8841	1.13582
$789$	0	0
$790$	2.29155	0.0815297
$791$	−0.288023	−0.0102409
$792$	0	0
$793$	1.89351	0.0672403
$794$	−5.84532	−0.207443
$795$	0	0
$796$	−20.7124	−0.734132
$797$	17.4604	0.618480	0.309240	−	0.950984i	$-0.399925\pi$
$797$	17.4604	0.618480	0.309240	+	0.950984i	$0.399925\pi$
$798$	0	0
$799$	−58.1783	−2.05820
$800$	25.5222	0.902348
$801$	0	0
$802$	−6.15888	−0.217478
$803$	3.63566	0.128300
$804$	0	0
$805$	−19.2102	−0.677071
$806$	−0.186335	−0.00656339
$807$	0	0
$808$	−9.64500	−0.339310
$809$	−21.1752	−0.744481	−0.372241	−	0.928136i	$-0.621410\pi$
$809$	−21.1752	−0.744481	−0.372241	+	0.928136i	$0.621410\pi$
$810$	0	0
$811$	52.3113	1.83690	0.918448	−	0.395541i	$-0.129443\pi$
$811$	52.3113	1.83690	0.918448	+	0.395541i	$0.129443\pi$
$812$	−10.2431	−0.359461
$813$	0	0
$814$	0.248556	0.00871190
$815$	−19.3790	−0.678818
$816$	0	0
$817$	37.9423	1.32743
$818$	−7.14569	−0.249843
$819$	0	0
$820$	23.9281	0.835606
$821$	−41.4667	−1.44720	−0.723599	−	0.690220i	$-0.757514\pi$
$821$	−41.4667	−1.44720	−0.723599	+	0.690220i	$0.757514\pi$
$822$	0	0
$823$	−49.0058	−1.70824	−0.854118	−	0.520079i	$-0.825902\pi$
$823$	−49.0058	−1.70824	−0.854118	+	0.520079i	$0.825902\pi$
$824$	−10.8963	−0.379590
$825$	0	0
$826$	3.84998	0.133958
$827$	7.16803	0.249257	0.124628	−	0.992203i	$-0.460226\pi$
$827$	7.16803	0.249257	0.124628	+	0.992203i	$0.460226\pi$
$828$	0	0
$829$	27.7807	0.964865	0.482433	−	0.875933i	$-0.339753\pi$
$829$	27.7807	0.964865	0.482433	+	0.875933i	$0.339753\pi$
$830$	−2.05488	−0.0713261
$831$	0	0
$832$	−1.78993	−0.0620547
$833$	−5.06965	−0.175653
$834$	0	0
$835$	27.1969	0.941189
$836$	−3.49578	−0.120904
$837$	0	0
$838$	−10.6243	−0.367009
$839$	20.0605	0.692565	0.346283	−	0.938130i	$-0.387444\pi$
$839$	20.0605	0.692565	0.346283	+	0.938130i	$0.387444\pi$
$840$	0	0
$841$	1.58560	0.0546758
$842$	−1.46717	−0.0505620
$843$	0	0
$844$	33.8531	1.16527
$845$	42.8666	1.47466
$846$	0	0
$847$	−10.7344	−0.368837
$848$	23.8647	0.819516
$849$	0	0
$850$	−11.9375	−0.409451
$851$	−7.22359	−0.247622
$852$	0	0
$853$	−24.9589	−0.854577	−0.427288	−	0.904115i	$-0.640531\pi$
$853$	−24.9589	−0.854577	−0.427288	+	0.904115i	$0.640531\pi$
$854$	1.89830	0.0649584
$855$	0	0
$856$	−1.10462	−0.0377550
$857$	−29.4234	−1.00508	−0.502541	−	0.864553i	$-0.667602\pi$
$857$	−29.4234	−1.00508	−0.502541	+	0.864553i	$0.667602\pi$
$858$	0	0
$859$	−16.2218	−0.553479	−0.276740	−	0.960945i	$-0.589254\pi$
$859$	−16.2218	−0.553479	−0.276740	+	0.960945i	$0.589254\pi$
$860$	−64.0005	−2.18240
$861$	0	0
$862$	−1.97902	−0.0674057
$863$	51.4795	1.75238	0.876191	−	0.481963i	$-0.160076\pi$
$863$	51.4795	1.75238	0.876191	+	0.481963i	$0.160076\pi$
$864$	0	0
$865$	17.5589	0.597022
$866$	−10.5254	−0.357669
$867$	0	0
$868$	2.33981	0.0794184
$869$	0.920899	0.0312394
$870$	0	0
$871$	−3.01119	−0.102030
$872$	9.48035	0.321045
$873$	0	0
$874$	−8.11120	−0.274366
$875$	−3.74670	−0.126661
$876$	0	0
$877$	17.3199	0.584851	0.292426	−	0.956288i	$-0.405538\pi$
$877$	17.3199	0.584851	0.292426	+	0.956288i	$0.405538\pi$
$878$	−10.7916	−0.364198
$879$	0	0
$880$	5.38827	0.181639
$881$	30.6844	1.03379	0.516893	−	0.856050i	$-0.327089\pi$
$881$	30.6844	1.03379	0.516893	+	0.856050i	$0.327089\pi$
$882$	0	0
$883$	−10.9184	−0.367433	−0.183717	−	0.982979i	$-0.558813\pi$
$883$	−10.9184	−0.367433	−0.183717	+	0.982979i	$0.558813\pi$
$884$	3.60158	0.121134
$885$	0	0
$886$	10.3487	0.347671
$887$	29.2574	0.982369	0.491184	−	0.871056i	$-0.336564\pi$
$887$	29.2574	0.982369	0.491184	+	0.871056i	$0.336564\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	−10.8108	−0.362378
$891$	0	0
$892$	52.2388	1.74909
$893$	−42.0255	−1.40633
$894$	0	0
$895$	19.6856	0.658016
$896$	−10.1304	−0.338435
$897$	0	0
$898$	−2.24058	−0.0747693
$899$	−6.98664	−0.233017
$900$	0	0
$901$	−38.5963	−1.28583
$902$	−0.767722	−0.0255623
$903$	0	0
$904$	0.426649	0.0141901
$905$	−54.5910	−1.81466
$906$	0	0
$907$	26.4871	0.879489	0.439745	−	0.898123i	$-0.355069\pi$
$907$	26.4871	0.879489	0.439745	+	0.898123i	$0.355069\pi$
$908$	40.1388	1.33205
$909$	0	0
$910$	−0.491931	−0.0163074
$911$	19.5456	0.647574	0.323787	−	0.946130i	$-0.395044\pi$
$911$	19.5456	0.647574	0.323787	+	0.946130i	$0.395044\pi$
$912$	0	0
$913$	−0.825791	−0.0273297
$914$	−8.70214	−0.287841
$915$	0	0
$916$	−47.2935	−1.56262
$917$	15.3614	0.507278
$918$	0	0
$919$	−20.5054	−0.676410	−0.338205	−	0.941073i	$-0.609820\pi$
$919$	−20.5054	−0.676410	−0.338205	+	0.941073i	$0.609820\pi$
$920$	28.4561	0.938169
$921$	0	0
$922$	−1.10205	−0.0362940
$923$	−4.00165	−0.131716
$924$	0	0
$925$	−7.67947	−0.252500
$926$	−0.678087	−0.0222833
$927$	0	0
$928$	23.0508	0.756679
$929$	9.40768	0.308656	0.154328	−	0.988020i	$-0.450679\pi$
$929$	9.40768	0.308656	0.154328	+	0.988020i	$0.450679\pi$
$930$	0	0
$931$	−3.66209	−0.120020
$932$	43.8017	1.43477
$933$	0	0
$934$	14.1883	0.464254
$935$	−8.71445	−0.284993
$936$	0	0
$937$	−13.5377	−0.442258	−0.221129	−	0.975245i	$-0.570974\pi$
$937$	−13.5377	−0.442258	−0.221129	+	0.975245i	$0.570974\pi$
$938$	−3.01881	−0.0985676
$939$	0	0
$940$	70.8880	2.31211
$941$	27.9975	0.912694	0.456347	−	0.889802i	$-0.349158\pi$
$941$	27.9975	0.912694	0.456347	+	0.889802i	$0.349158\pi$
$942$	0	0
$943$	22.3117	0.726568
$944$	31.3836	1.02145
$945$	0	0
$946$	2.05343	0.0667626
$947$	−25.3791	−0.824711	−0.412356	−	0.911023i	$-0.635294\pi$
$947$	−25.3791	−0.824711	−0.412356	+	0.911023i	$0.635294\pi$
$948$	0	0
$949$	−2.70572	−0.0878315
$950$	−8.62310	−0.279770
$951$	0	0
$952$	7.50967	0.243390
$953$	−34.6314	−1.12182	−0.560911	−	0.827876i	$-0.689549\pi$
$953$	−34.6314	−1.12182	−0.560911	+	0.827876i	$0.689549\pi$
$954$	0	0
$955$	27.2578	0.882043
$956$	15.2570	0.493445
$957$	0	0
$958$	8.39911	0.271363
$959$	−1.99537	−0.0644339
$960$	0	0
$961$	−29.4040	−0.948518
$962$	−0.184980	−0.00596400
$963$	0	0
$964$	−0.828986	−0.0266998
$965$	64.1602	2.06539
$966$	0	0
$967$	−39.7563	−1.27848	−0.639239	−	0.769008i	$-0.720750\pi$
$967$	−39.7563	−1.27848	−0.639239	+	0.769008i	$0.720750\pi$
$968$	15.9008	0.511071
$969$	0	0
$970$	−14.7474	−0.473510
$971$	−6.25687	−0.200793	−0.100396	−	0.994948i	$-0.532011\pi$
$971$	−6.25687	−0.200793	−0.100396	+	0.994948i	$0.532011\pi$
$972$	0	0
$973$	−16.8068	−0.538801
$974$	7.91822	0.253716
$975$	0	0
$976$	15.4743	0.495319
$977$	31.5360	1.00893	0.504463	−	0.863433i	$-0.331690\pi$
$977$	31.5360	1.00893	0.504463	+	0.863433i	$0.331690\pi$
$978$	0	0
$979$	−4.34450	−0.138851
$980$	6.17717	0.197323
$981$	0	0
$982$	−5.91127	−0.188636
$983$	−28.0672	−0.895204	−0.447602	−	0.894233i	$-0.647722\pi$
$983$	−28.0672	−0.895204	−0.447602	+	0.894233i	$0.647722\pi$
$984$	0	0
$985$	57.4145	1.82938
$986$	−10.7815	−0.343352
$987$	0	0
$988$	2.60162	0.0827687
$989$	−59.6771	−1.89762
$990$	0	0
$991$	13.0091	0.413246	0.206623	−	0.978421i	$-0.433753\pi$
$991$	13.0091	0.413246	0.206623	+	0.978421i	$0.433753\pi$
$992$	−5.26546	−0.167179
$993$	0	0
$994$	−4.01179	−0.127246
$995$	−37.2973	−1.18241
$996$	0	0
$997$	13.4874	0.427152	0.213576	−	0.976926i	$-0.431489\pi$
$997$	13.4874	0.427152	0.213576	+	0.976926i	$0.431489\pi$
$998$	−1.47072	−0.0465549
$999$	0	0

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Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.x.1.12	yes	22
3.2	odd	2	inner	8001.2.a.x.1.11	✓	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8001.2.a.x.1.11	✓	22	3.2	odd	2	inner
8001.2.a.x.1.12	yes	22	1.1	even	1	trivial

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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

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Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	8001.2.a.x.1.12

Level	$8001$
Weight	$2$
Character	8001.1
Self dual	yes
Analytic conductor	$63.888$
Analytic rank	$1$
Dimension	$22$
Inner twists	$2$