LMFDB - Embedded newform 8001.2.a.x.1.11

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:	$22$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
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)

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

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 $

$2$	−0.384540	−0.271911	−0.135956	−	0.990715i	$-0.543410\pi$
$2$	−0.384540	−0.271911	−0.135956	+	0.990715i	$0.543410\pi$
$3$	0	0
$3$	0	0
$4$	−1.85213	−0.926064
$5$	3.33517	1.49154	0.745768	−	0.666206i	$-0.232083\pi$
$5$	3.33517	1.49154	0.745768	+	0.666206i	$0.232083\pi$
$6$	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	$-0.475243\pi$
$11$	0.515399	0.155399	0.0776993	+	0.996977i	$0.475243\pi$
$12$	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	$-0.289243\pi$
$17$	5.06965	1.22957	0.614785	+	0.788695i	$0.289243\pi$
$18$	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	$-0.705035\pi$
$23$	−5.75988	−1.20102	−0.600509	+	0.799618i	$0.705035\pi$
$24$	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	$-0.671647\pi$
$29$	−5.53042	−1.02697	−0.513487	+	0.858097i	$0.671647\pi$
$30$	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	$-0.597815\pi$
$41$	−3.87363	−0.604960	−0.302480	+	0.953156i	$0.597815\pi$
$42$	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	$-0.815669\pi$
$47$	−11.4758	−1.67392	−0.836959	+	0.547266i	$0.815669\pi$
$48$	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	$-0.675141\pi$
$53$	−7.61321	−1.04575	−0.522877	+	0.852408i	$0.675141\pi$
$54$	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	$-0.725952\pi$
$59$	−10.0119	−1.30344	−0.651718	+	0.758461i	$0.725952\pi$
$60$	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	$-0.287512\pi$
$71$	10.4327	1.23813	0.619065	+	0.785339i	$0.287512\pi$
$72$	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	$-0.528027\pi$
$83$	−1.60224	−0.175868	−0.0879342	+	0.996126i	$0.528027\pi$
$84$	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	$-0.647421\pi$
$89$	−8.42939	−0.893513	−0.446757	+	0.894656i	$0.647421\pi$
$90$	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	$-0.605009\pi$
$101$	−6.51118	−0.647886	−0.323943	+	0.946077i	$0.605009\pi$
$102$	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	$-0.511476\pi$
$107$	−0.745709	−0.0720904	−0.0360452	+	0.999350i	$0.511476\pi$
$108$	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	$-0.495688\pi$
$113$	0.288023	0.0270949	0.0135475	+	0.999908i	$0.495688\pi$
$114$	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	$-0.734163\pi$
$131$	−15.3614	−1.34213	−0.671066	+	0.741398i	$0.734163\pi$
$132$	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	$-0.472835\pi$
$137$	1.99537	0.170476	0.0852380	+	0.996361i	$0.472835\pi$
$138$	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	$-0.389148\pi$
$149$	8.33107	0.682508	0.341254	+	0.939971i	$0.389148\pi$
$150$	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	$-0.397824\pi$
$167$	8.15458	0.631020	0.315510	+	0.948922i	$0.397824\pi$
$168$	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	$-0.435861\pi$
$173$	5.26477	0.400273	0.200137	+	0.979768i	$0.435861\pi$
$174$	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	$-0.429204\pi$
$179$	5.90241	0.441167	0.220583	+	0.975368i	$0.429204\pi$
$180$	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	$-0.404453\pi$
$191$	8.17284	0.591366	0.295683	+	0.955286i	$0.404453\pi$
$192$	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	$-0.289861\pi$
$197$	17.2148	1.22651	0.613253	+	0.789886i	$0.289861\pi$
$198$	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	$-0.244508\pi$
$227$	21.6717	1.43840	0.719202	+	0.694801i	$0.244508\pi$
$228$	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	$-0.217920\pi$
$233$	23.6494	1.54932	0.774661	+	0.632376i	$0.217920\pi$
$234$	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	$-0.414159\pi$
$239$	8.23753	0.532841	0.266421	+	0.963857i	$0.414159\pi$
$240$	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	$-0.417505\pi$
$251$	8.12028	0.512547	0.256274	+	0.966604i	$0.417505\pi$
$252$	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	$-0.506624\pi$
$257$	−0.667162	−0.0416164	−0.0208082	+	0.999783i	$0.506624\pi$
$258$	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	$-0.775080\pi$
$263$	−24.6687	−1.52114	−0.760570	+	0.649256i	$0.775080\pi$
$264$	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	$-0.711333\pi$
$269$	−20.2132	−1.23242	−0.616210	+	0.787582i	$0.711333\pi$
$270$	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	$-0.408607\pi$
$281$	9.49428	0.566381	0.283191	+	0.959064i	$0.408607\pi$
$282$	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	$-0.610136\pi$
$293$	−11.6103	−0.678279	−0.339140	+	0.940736i	$0.610136\pi$
$294$	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	$-0.463804\pi$
$311$	4.00207	0.226937	0.113468	+	0.993542i	$0.463804\pi$
$312$	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	$-0.455249\pi$
$317$	4.98975	0.280252	0.140126	+	0.990134i	$0.455249\pi$
$318$	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	$-0.668867\pi$
$347$	−18.8505	−1.01195	−0.505975	+	0.862548i	$0.668867\pi$
$348$	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	$-0.267004\pi$
$353$	25.1141	1.33669	0.668343	+	0.743853i	$0.267004\pi$
$354$	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	$-0.191568\pi$
$359$	31.2366	1.64860	0.824301	+	0.566151i	$0.191568\pi$
$360$	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	$-0.220946\pi$
$383$	30.0842	1.53723	0.768615	+	0.639711i	$0.220946\pi$
$384$	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	$-0.284142\pi$
$389$	24.7463	1.25469	0.627343	+	0.778743i	$0.284142\pi$
$390$	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	$-0.369043\pi$
$401$	16.0162	0.799811	0.399906	+	0.916556i	$0.369043\pi$
$402$	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	$-0.264200\pi$
$419$	27.6284	1.34974	0.674869	+	0.737938i	$0.264200\pi$
$420$	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	$-0.460444\pi$
$431$	5.14646	0.247896	0.123948	+	0.992289i	$0.460444\pi$
$432$	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	$-0.720780\pi$
$443$	−26.9119	−1.27862	−0.639311	+	0.768948i	$0.720780\pi$
$444$	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	$-0.456097\pi$
$449$	5.82666	0.274977	0.137488	+	0.990503i	$0.456097\pi$
$450$	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	$-0.478741\pi$
$461$	2.86588	0.133478	0.0667388	+	0.997770i	$0.478741\pi$
$462$	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	$-0.825639\pi$
$467$	−36.8967	−1.70738	−0.853688	+	0.520785i	$0.825639\pi$
$468$	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	$-0.666296\pi$
$479$	−21.8419	−0.997984	−0.498992	+	0.866607i	$0.666296\pi$
$480$	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	$-0.387244\pi$
$491$	15.3723	0.693742	0.346871	+	0.937913i	$0.387244\pi$
$492$	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	$-0.549973\pi$
$503$	−7.01316	−0.312701	−0.156351	+	0.987702i	$0.549973\pi$
$504$	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	$-0.504436\pi$
$509$	−0.628790	−0.0278706	−0.0139353	+	0.999903i	$0.504436\pi$
$510$	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	$-0.560658\pi$
$521$	−8.64677	−0.378822	−0.189411	+	0.981898i	$0.560658\pi$
$522$	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	$-0.468849\pi$
$557$	4.61204	0.195418	0.0977092	+	0.995215i	$0.468849\pi$
$558$	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	$-0.452795\pi$
$563$	7.01174	0.295509	0.147755	+	0.989024i	$0.452795\pi$
$564$	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	$-0.312125\pi$
$569$	26.5515	1.11310	0.556548	+	0.830815i	$0.312125\pi$
$570$	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	$-0.558870\pi$
$587$	−8.91072	−0.367785	−0.183892	+	0.982946i	$0.558870\pi$
$588$	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	$-0.615586\pi$
$593$	−17.2992	−0.710392	−0.355196	+	0.934792i	$0.615586\pi$
$594$	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	$-0.655390\pi$
$599$	−22.9576	−0.938024	−0.469012	+	0.883192i	$0.655390\pi$
$600$	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	$-0.725006\pi$
$617$	−32.2646	−1.29892	−0.649461	+	0.760395i	$0.725006\pi$
$618$	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	$-0.535872\pi$
$641$	−5.69431	−0.224912	−0.112456	+	0.993657i	$0.535872\pi$
$642$	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	$-0.676184\pi$
$647$	−26.7420	−1.05133	−0.525667	+	0.850690i	$0.676184\pi$
$648$	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	$-0.155425\pi$
$653$	45.1353	1.76628	0.883140	+	0.469109i	$0.155425\pi$
$654$	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	$-0.380663\pi$
$659$	18.8007	0.732372	0.366186	+	0.930542i	$0.380663\pi$
$660$	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	$-0.535652\pi$
$677$	−5.81638	−0.223542	−0.111771	+	0.993734i	$0.535652\pi$
$678$	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	$-0.297806\pi$
$683$	31.0134	1.18670	0.593348	+	0.804946i	$0.297806\pi$
$684$	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	$-0.689687\pi$
$701$	−29.7208	−1.12254	−0.561269	+	0.827633i	$0.689687\pi$
$702$	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	$-0.511273\pi$
$719$	−1.89887	−0.0708159	−0.0354079	+	0.999373i	$0.511273\pi$
$720$	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	$-0.597755\pi$
$743$	−16.4803	−0.604605	−0.302302	+	0.953212i	$0.597755\pi$
$744$	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	$-0.632351\pi$
$761$	−22.2851	−0.807833	−0.403917	+	0.914796i	$0.632351\pi$
$762$	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	$-0.504573\pi$
$773$	−0.798838	−0.0287322	−0.0143661	+	0.999897i	$0.504573\pi$
$774$	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	$-0.600075\pi$
$797$	−17.4604	−0.618480	−0.309240	+	0.950984i	$0.600075\pi$
$798$	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	$-0.378590\pi$
$809$	21.1752	0.744481	0.372241	+	0.928136i	$0.378590\pi$
$810$	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	$-0.242486\pi$
$821$	41.4667	1.44720	0.723599	+	0.690220i	$0.242486\pi$
$822$	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	$-0.539774\pi$
$827$	−7.16803	−0.249257	−0.124628	+	0.992203i	$0.539774\pi$
$828$	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	$-0.612556\pi$
$839$	−20.0605	−0.692565	−0.346283	+	0.938130i	$0.612556\pi$
$840$	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	$-0.332398\pi$
$857$	29.4234	1.00508	0.502541	+	0.864553i	$0.332398\pi$
$858$	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	$-0.839924\pi$
$863$	−51.4795	−1.75238	−0.876191	+	0.481963i	$0.839924\pi$
$864$	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	$-0.672911\pi$
$881$	−30.6844	−1.03379	−0.516893	+	0.856050i	$0.672911\pi$
$882$	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	$-0.663436\pi$
$887$	−29.2574	−0.982369	−0.491184	+	0.871056i	$0.663436\pi$
$888$	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	$-0.604956\pi$
$911$	−19.5456	−0.647574	−0.323787	+	0.946130i	$0.604956\pi$
$912$	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	$-0.549321\pi$
$929$	−9.40768	−0.308656	−0.154328	+	0.988020i	$0.549321\pi$
$930$	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	$-0.650842\pi$
$941$	−27.9975	−0.912694	−0.456347	+	0.889802i	$0.650842\pi$
$942$	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	$-0.364706\pi$
$947$	25.3791	0.824711	0.412356	+	0.911023i	$0.364706\pi$
$948$	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	$-0.310451\pi$
$953$	34.6314	1.12182	0.560911	+	0.827876i	$0.310451\pi$
$954$	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	$-0.467989\pi$
$971$	6.25687	0.200793	0.100396	+	0.994948i	$0.467989\pi$
$972$	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	$-0.668310\pi$
$977$	−31.5360	−1.00893	−0.504463	+	0.863433i	$0.668310\pi$
$978$	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	$-0.352278\pi$
$983$	28.0672	0.895204	0.447602	+	0.894233i	$0.352278\pi$
$984$	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

Twists

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

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

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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