LMFDB - Embedded newform 8001.2.a.y.1.22

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8001,2,Mod(1,8001)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8001, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8001 = 3^{2} \cdot 7 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8001.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$63.8883066572$
Analytic rank:	$0$
Dimension:	$28$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.22
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)

$f(q)$	$=$	$q+1.55816 q^{2} +0.427866 q^{4} +3.44623 q^{5} -1.00000 q^{7} -2.44964 q^{8} +O(q^{10})$	\(q+1.55816 q^{2} +0.427866 q^{4} +3.44623 q^{5} -1.00000 q^{7} -2.44964 q^{8} +5.36979 q^{10} -2.27859 q^{11} +3.98958 q^{13} -1.55816 q^{14} -4.67266 q^{16} -0.0529312 q^{17} +1.72011 q^{19} +1.47453 q^{20} -3.55040 q^{22} -1.87769 q^{23} +6.87652 q^{25} +6.21641 q^{26} -0.427866 q^{28} -7.03231 q^{29} +9.09736 q^{31} -2.38149 q^{32} -0.0824753 q^{34} -3.44623 q^{35} +6.34320 q^{37} +2.68021 q^{38} -8.44202 q^{40} -4.26175 q^{41} +7.77408 q^{43} -0.974929 q^{44} -2.92575 q^{46} +2.35651 q^{47} +1.00000 q^{49} +10.7147 q^{50} +1.70701 q^{52} +5.55020 q^{53} -7.85254 q^{55} +2.44964 q^{56} -10.9575 q^{58} -1.27476 q^{59} -0.634867 q^{61} +14.1751 q^{62} +5.63459 q^{64} +13.7490 q^{65} +13.4623 q^{67} -0.0226474 q^{68} -5.36979 q^{70} +10.9262 q^{71} +2.51705 q^{73} +9.88372 q^{74} +0.735976 q^{76} +2.27859 q^{77} +3.37139 q^{79} -16.1031 q^{80} -6.64049 q^{82} +12.0668 q^{83} -0.182413 q^{85} +12.1133 q^{86} +5.58171 q^{88} -7.60952 q^{89} -3.98958 q^{91} -0.803401 q^{92} +3.67182 q^{94} +5.92790 q^{95} -1.35845 q^{97} +1.55816 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 30 q^{4} - 28 q^{7}+O(q^{10}) $ 28 * q + 30 * q^4 - 28 * q^7	$ 28 q + 30 q^{4} - 28 q^{7} + 4 q^{10} + 8 q^{13} + 42 q^{16} + 34 q^{19} - 10 q^{22} + 14 q^{25} - 30 q^{28} + 56 q^{31} - 6 q^{37} + 38 q^{40} + 18 q^{43} + 16 q^{46} + 28 q^{49} + 18 q^{52} + 48 q^{55} + 2 q^{58} + 36 q^{61} + 76 q^{64} - 4 q^{70} + 50 q^{73} + 132 q^{76} + 66 q^{79} - 36 q^{82} + 20 q^{85} + 6 q^{88} - 8 q^{91} + 54 q^{94} - 8 q^{97}+O(q^{100}) $ 28 * q + 30 * q^4 - 28 * q^7 + 4 * q^10 + 8 * q^13 + 42 * q^16 + 34 * q^19 - 10 * q^22 + 14 * q^25 - 30 * q^28 + 56 * q^31 - 6 * q^37 + 38 * q^40 + 18 * q^43 + 16 * q^46 + 28 * q^49 + 18 * q^52 + 48 * q^55 + 2 * q^58 + 36 * q^61 + 76 * q^64 - 4 * q^70 + 50 * q^73 + 132 * q^76 + 66 * q^79 - 36 * q^82 + 20 * q^85 + 6 * q^88 - 8 * q^91 + 54 * q^94 - 8 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$	1.55816	1.10179	0.550893	−	0.834576i	$-0.314287\pi$
$2$	1.55816	1.10179	0.550893	+	0.834576i	$0.314287\pi$
$3$	0	0
$3$	0	0
$4$	0.427866	0.213933
$5$	3.44623	1.54120	0.770601	−	0.637318i	$-0.219956\pi$
$5$	3.44623	1.54120	0.770601	+	0.637318i	$0.219956\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−2.44964	−0.866078
$9$	0	0
$10$	5.36979	1.69808
$11$	−2.27859	−0.687019	−0.343510	−	0.939149i	$-0.611616\pi$
$11$	−2.27859	−0.687019	−0.343510	+	0.939149i	$0.611616\pi$
$12$	0	0
$13$	3.98958	1.10651	0.553255	−	0.833012i	$-0.313385\pi$
$13$	3.98958	1.10651	0.553255	+	0.833012i	$0.313385\pi$
$14$	−1.55816	−0.416436
$15$	0	0
$16$	−4.67266	−1.16817
$17$	−0.0529312	−0.0128377	−0.00641884	−	0.999979i	$-0.502043\pi$
$17$	−0.0529312	−0.0128377	−0.00641884	+	0.999979i	$0.502043\pi$
$18$	0	0
$19$	1.72011	0.394620	0.197310	−	0.980341i	$-0.436779\pi$
$19$	1.72011	0.394620	0.197310	+	0.980341i	$0.436779\pi$
$20$	1.47453	0.329714
$21$	0	0
$22$	−3.55040	−0.756948
$23$	−1.87769	−0.391526	−0.195763	−	0.980651i	$-0.562718\pi$
$23$	−1.87769	−0.391526	−0.195763	+	0.980651i	$0.562718\pi$
$24$	0	0
$25$	6.87652	1.37530
$26$	6.21641	1.21914
$27$	0	0
$28$	−0.427866	−0.0808591
$29$	−7.03231	−1.30587	−0.652933	−	0.757415i	$-0.726462\pi$
$29$	−7.03231	−1.30587	−0.652933	+	0.757415i	$0.726462\pi$
$30$	0	0
$31$	9.09736	1.63393	0.816967	−	0.576685i	$-0.195654\pi$
$31$	9.09736	1.63393	0.816967	+	0.576685i	$0.195654\pi$
$32$	−2.38149	−0.420991
$33$	0	0
$34$	−0.0824753	−0.0141444
$35$	−3.44623	−0.582520
$36$	0	0
$37$	6.34320	1.04282	0.521408	−	0.853308i	$-0.325407\pi$
$37$	6.34320	1.04282	0.521408	+	0.853308i	$0.325407\pi$
$38$	2.68021	0.434787
$39$	0	0
$40$	−8.44202	−1.33480
$41$	−4.26175	−0.665573	−0.332786	−	0.943002i	$-0.607989\pi$
$41$	−4.26175	−0.665573	−0.332786	+	0.943002i	$0.607989\pi$
$42$	0	0
$43$	7.77408	1.18554	0.592768	−	0.805373i	$-0.298035\pi$
$43$	7.77408	1.18554	0.592768	+	0.805373i	$0.298035\pi$
$44$	−0.974929	−0.146976
$45$	0	0
$46$	−2.92575	−0.431378
$47$	2.35651	0.343732	0.171866	−	0.985120i	$-0.445020\pi$
$47$	2.35651	0.343732	0.171866	+	0.985120i	$0.445020\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	10.7147	1.51529
$51$	0	0
$52$	1.70701	0.236719
$53$	5.55020	0.762378	0.381189	−	0.924497i	$-0.375515\pi$
$53$	5.55020	0.762378	0.381189	+	0.924497i	$0.375515\pi$
$54$	0	0
$55$	−7.85254	−1.05884
$56$	2.44964	0.327347
$57$	0	0
$58$	−10.9575	−1.43879
$59$	−1.27476	−0.165960	−0.0829799	−	0.996551i	$-0.526444\pi$
$59$	−1.27476	−0.165960	−0.0829799	+	0.996551i	$0.526444\pi$
$60$	0	0
$61$	−0.634867	−0.0812864	−0.0406432	−	0.999174i	$-0.512941\pi$
$61$	−0.634867	−0.0812864	−0.0406432	+	0.999174i	$0.512941\pi$
$62$	14.1751	1.80025
$63$	0	0
$64$	5.63459	0.704323
$65$	13.7490	1.70536
$66$	0	0
$67$	13.4623	1.64468	0.822341	−	0.568995i	$-0.192668\pi$
$67$	13.4623	1.64468	0.822341	+	0.568995i	$0.192668\pi$
$68$	−0.0226474	−0.00274641
$69$	0	0
$70$	−5.36979	−0.641812
$71$	10.9262	1.29670	0.648350	−	0.761342i	$-0.275459\pi$
$71$	10.9262	1.29670	0.648350	+	0.761342i	$0.275459\pi$
$72$	0	0
$73$	2.51705	0.294599	0.147299	−	0.989092i	$-0.452942\pi$
$73$	2.51705	0.294599	0.147299	+	0.989092i	$0.452942\pi$
$74$	9.88372	1.14896
$75$	0	0
$76$	0.735976	0.0844223
$77$	2.27859	0.259669
$78$	0	0
$79$	3.37139	0.379311	0.189655	−	0.981851i	$-0.439263\pi$
$79$	3.37139	0.379311	0.189655	+	0.981851i	$0.439263\pi$
$80$	−16.1031	−1.80038
$81$	0	0
$82$	−6.64049	−0.733319
$83$	12.0668	1.32451	0.662254	−	0.749279i	$-0.269600\pi$
$83$	12.0668	1.32451	0.662254	+	0.749279i	$0.269600\pi$
$84$	0	0
$85$	−0.182413	−0.0197855
$86$	12.1133	1.30621
$87$	0	0
$88$	5.58171	0.595012
$89$	−7.60952	−0.806608	−0.403304	−	0.915066i	$-0.632138\pi$
$89$	−7.60952	−0.806608	−0.403304	+	0.915066i	$0.632138\pi$
$90$	0	0
$91$	−3.98958	−0.418222
$92$	−0.803401	−0.0837604
$93$	0	0
$94$	3.67182	0.378719
$95$	5.92790	0.608189
$96$	0	0
$97$	−1.35845	−0.137929	−0.0689646	−	0.997619i	$-0.521970\pi$
$97$	−1.35845	−0.137929	−0.0689646	+	0.997619i	$0.521970\pi$
$98$	1.55816	0.157398
$99$	0	0
$100$	2.94223	0.294223
$101$	−13.5619	−1.34946	−0.674728	−	0.738066i	$-0.735739\pi$
$101$	−13.5619	−1.34946	−0.674728	+	0.738066i	$0.735739\pi$
$102$	0	0
$103$	3.89661	0.383945	0.191972	−	0.981400i	$-0.438512\pi$
$103$	3.89661	0.383945	0.191972	+	0.981400i	$0.438512\pi$
$104$	−9.77303	−0.958324
$105$	0	0
$106$	8.64810	0.839978
$107$	1.08713	0.105097	0.0525484	−	0.998618i	$-0.483266\pi$
$107$	1.08713	0.105097	0.0525484	+	0.998618i	$0.483266\pi$
$108$	0	0
$109$	−4.53804	−0.434665	−0.217333	−	0.976098i	$-0.569736\pi$
$109$	−4.53804	−0.434665	−0.217333	+	0.976098i	$0.569736\pi$
$110$	−12.2355	−1.16661
$111$	0	0
$112$	4.67266	0.441525
$113$	16.5468	1.55659	0.778294	−	0.627900i	$-0.216085\pi$
$113$	16.5468	1.55659	0.778294	+	0.627900i	$0.216085\pi$
$114$	0	0
$115$	−6.47097	−0.603421
$116$	−3.00889	−0.279368
$117$	0	0
$118$	−1.98628	−0.182852
$119$	0.0529312	0.00485219
$120$	0	0
$121$	−5.80805	−0.528005
$122$	−0.989225	−0.0895603
$123$	0	0
$124$	3.89245	0.349552
$125$	6.46694	0.578421
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	13.5426	1.19701
$129$	0	0
$130$	21.4232	1.87894
$131$	−19.8956	−1.73829	−0.869145	−	0.494557i	$-0.835330\pi$
$131$	−19.8956	−1.73829	−0.869145	+	0.494557i	$0.835330\pi$
$132$	0	0
$133$	−1.72011	−0.149152
$134$	20.9764	1.81209
$135$	0	0
$136$	0.129662	0.0111184
$137$	−13.5891	−1.16099	−0.580496	−	0.814263i	$-0.697141\pi$
$137$	−13.5891	−1.16099	−0.580496	+	0.814263i	$0.697141\pi$
$138$	0	0
$139$	6.79716	0.576528	0.288264	−	0.957551i	$-0.406922\pi$
$139$	6.79716	0.576528	0.288264	+	0.957551i	$0.406922\pi$
$140$	−1.47453	−0.124620
$141$	0	0
$142$	17.0248	1.42869
$143$	−9.09060	−0.760194
$144$	0	0
$145$	−24.2350	−2.01261
$146$	3.92197	0.324585
$147$	0	0
$148$	2.71404	0.223093
$149$	15.5718	1.27569	0.637845	−	0.770165i	$-0.279826\pi$
$149$	15.5718	1.27569	0.637845	+	0.770165i	$0.279826\pi$
$150$	0	0
$151$	−2.61015	−0.212411	−0.106205	−	0.994344i	$-0.533870\pi$
$151$	−2.61015	−0.212411	−0.106205	+	0.994344i	$0.533870\pi$
$152$	−4.21364	−0.341772
$153$	0	0
$154$	3.55040	0.286100
$155$	31.3516	2.51822
$156$	0	0
$157$	20.7507	1.65609	0.828044	−	0.560663i	$-0.189454\pi$
$157$	20.7507	1.65609	0.828044	+	0.560663i	$0.189454\pi$
$158$	5.25317	0.417920
$159$	0	0
$160$	−8.20716	−0.648833
$161$	1.87769	0.147983
$162$	0	0
$163$	−13.4533	−1.05374	−0.526872	−	0.849944i	$-0.676635\pi$
$163$	−13.4533	−1.05374	−0.526872	+	0.849944i	$0.676635\pi$
$164$	−1.82346	−0.142388
$165$	0	0
$166$	18.8021	1.45932
$167$	6.88448	0.532737	0.266369	−	0.963871i	$-0.414176\pi$
$167$	6.88448	0.532737	0.266369	+	0.963871i	$0.414176\pi$
$168$	0	0
$169$	2.91676	0.224366
$170$	−0.284229	−0.0217994
$171$	0	0
$172$	3.32626	0.253625
$173$	13.8817	1.05540	0.527702	−	0.849429i	$-0.323054\pi$
$173$	13.8817	1.05540	0.527702	+	0.849429i	$0.323054\pi$
$174$	0	0
$175$	−6.87652	−0.519816
$176$	10.6471	0.802552
$177$	0	0
$178$	−11.8569	−0.888709
$179$	18.3169	1.36907	0.684535	−	0.728980i	$-0.260005\pi$
$179$	18.3169	1.36907	0.684535	+	0.728980i	$0.260005\pi$
$180$	0	0
$181$	2.68529	0.199596	0.0997981	−	0.995008i	$-0.468180\pi$
$181$	2.68529	0.199596	0.0997981	+	0.995008i	$0.468180\pi$
$182$	−6.21641	−0.460791
$183$	0	0
$184$	4.59967	0.339092
$185$	21.8601	1.60719
$186$	0	0
$187$	0.120608	0.00881974
$188$	1.00827	0.0735356
$189$	0	0
$190$	9.23662	0.670095
$191$	−2.49756	−0.180717	−0.0903585	−	0.995909i	$-0.528801\pi$
$191$	−2.49756	−0.180717	−0.0903585	+	0.995909i	$0.528801\pi$
$192$	0	0
$193$	11.8223	0.850985	0.425493	−	0.904962i	$-0.360101\pi$
$193$	11.8223	0.850985	0.425493	+	0.904962i	$0.360101\pi$
$194$	−2.11668	−0.151969
$195$	0	0
$196$	0.427866	0.0305619
$197$	−10.9652	−0.781238	−0.390619	−	0.920552i	$-0.627739\pi$
$197$	−10.9652	−0.781238	−0.390619	+	0.920552i	$0.627739\pi$
$198$	0	0
$199$	−1.13883	−0.0807294	−0.0403647	−	0.999185i	$-0.512852\pi$
$199$	−1.13883	−0.0807294	−0.0403647	+	0.999185i	$0.512852\pi$
$200$	−16.8450	−1.19112
$201$	0	0
$202$	−21.1316	−1.48681
$203$	7.03231	0.493571
$204$	0	0
$205$	−14.6870	−1.02578
$206$	6.07155	0.423025
$207$	0	0
$208$	−18.6420	−1.29259
$209$	−3.91941	−0.271112
$210$	0	0
$211$	−7.20279	−0.495860	−0.247930	−	0.968778i	$-0.579750\pi$
$211$	−7.20279	−0.495860	−0.247930	+	0.968778i	$0.579750\pi$
$212$	2.37474	0.163098
$213$	0	0
$214$	1.69392	0.115794
$215$	26.7913	1.82715
$216$	0	0
$217$	−9.09736	−0.617569
$218$	−7.07100	−0.478908
$219$	0	0
$220$	−3.35983	−0.226520
$221$	−0.211173	−0.0142050
$222$	0	0
$223$	−0.637280	−0.0426754	−0.0213377	−	0.999772i	$-0.506793\pi$
$223$	−0.637280	−0.0426754	−0.0213377	+	0.999772i	$0.506793\pi$
$224$	2.38149	0.159120
$225$	0	0
$226$	25.7825	1.71503
$227$	12.2197	0.811053	0.405527	−	0.914083i	$-0.367088\pi$
$227$	12.2197	0.811053	0.405527	+	0.914083i	$0.367088\pi$
$228$	0	0
$229$	−21.3523	−1.41100	−0.705499	−	0.708711i	$-0.749277\pi$
$229$	−21.3523	−1.41100	−0.705499	+	0.708711i	$0.749277\pi$
$230$	−10.0828	−0.664841
$231$	0	0
$232$	17.2266	1.13098
$233$	12.6328	0.827602	0.413801	−	0.910367i	$-0.364201\pi$
$233$	12.6328	0.827602	0.413801	+	0.910367i	$0.364201\pi$
$234$	0	0
$235$	8.12107	0.529760
$236$	−0.545427	−0.0355043
$237$	0	0
$238$	0.0824753	0.00534608
$239$	6.51318	0.421302	0.210651	−	0.977561i	$-0.432442\pi$
$239$	6.51318	0.421302	0.210651	+	0.977561i	$0.432442\pi$
$240$	0	0
$241$	−4.32504	−0.278601	−0.139300	−	0.990250i	$-0.544485\pi$
$241$	−4.32504	−0.278601	−0.139300	+	0.990250i	$0.544485\pi$
$242$	−9.04988	−0.581748
$243$	0	0
$244$	−0.271638	−0.0173899
$245$	3.44623	0.220172
$246$	0	0
$247$	6.86251	0.436651
$248$	−22.2852	−1.41511
$249$	0	0
$250$	10.0765	0.637296
$251$	−2.72975	−0.172301	−0.0861503	−	0.996282i	$-0.527457\pi$
$251$	−2.72975	−0.172301	−0.0861503	+	0.996282i	$0.527457\pi$
$252$	0	0
$253$	4.27848	0.268986
$254$	1.55816	0.0977677
$255$	0	0
$256$	9.83233	0.614520
$257$	22.5804	1.40853	0.704263	−	0.709939i	$-0.251278\pi$
$257$	22.5804	1.40853	0.704263	+	0.709939i	$0.251278\pi$
$258$	0	0
$259$	−6.34320	−0.394147
$260$	5.88274	0.364832
$261$	0	0
$262$	−31.0006	−1.91522
$263$	5.61193	0.346047	0.173023	−	0.984918i	$-0.444646\pi$
$263$	5.61193	0.346047	0.173023	+	0.984918i	$0.444646\pi$
$264$	0	0
$265$	19.1273	1.17498
$266$	−2.68021	−0.164334
$267$	0	0
$268$	5.76006	0.351852
$269$	−12.9633	−0.790387	−0.395194	−	0.918598i	$-0.629323\pi$
$269$	−12.9633	−0.790387	−0.395194	+	0.918598i	$0.629323\pi$
$270$	0	0
$271$	17.1284	1.04048	0.520239	−	0.854021i	$-0.325843\pi$
$271$	17.1284	1.04048	0.520239	+	0.854021i	$0.325843\pi$
$272$	0.247329	0.0149965
$273$	0	0
$274$	−21.1740	−1.27917
$275$	−15.6687	−0.944861
$276$	0	0
$277$	−15.5162	−0.932277	−0.466139	−	0.884712i	$-0.654355\pi$
$277$	−15.5162	−0.932277	−0.466139	+	0.884712i	$0.654355\pi$
$278$	10.5911	0.635210
$279$	0	0
$280$	8.44202	0.504507
$281$	−25.5289	−1.52293	−0.761465	−	0.648206i	$-0.775519\pi$
$281$	−25.5289	−1.52293	−0.761465	+	0.648206i	$0.775519\pi$
$282$	0	0
$283$	29.9408	1.77980	0.889899	−	0.456157i	$-0.150774\pi$
$283$	29.9408	1.77980	0.889899	+	0.456157i	$0.150774\pi$
$284$	4.67495	0.277407
$285$	0	0
$286$	−14.1646	−0.837572
$287$	4.26175	0.251563
$288$	0	0
$289$	−16.9972	−0.999835
$290$	−37.7620	−2.21746
$291$	0	0
$292$	1.07696	0.0630244
$293$	−12.6506	−0.739058	−0.369529	−	0.929219i	$-0.620481\pi$
$293$	−12.6506	−0.739058	−0.369529	+	0.929219i	$0.620481\pi$
$294$	0	0
$295$	−4.39312	−0.255778
$296$	−15.5385	−0.903159
$297$	0	0
$298$	24.2633	1.40554
$299$	−7.49121	−0.433228
$300$	0	0
$301$	−7.77408	−0.448090
$302$	−4.06703	−0.234032
$303$	0	0
$304$	−8.03749	−0.460982
$305$	−2.18790	−0.125279
$306$	0	0
$307$	4.70432	0.268490	0.134245	−	0.990948i	$-0.457139\pi$
$307$	4.70432	0.268490	0.134245	+	0.990948i	$0.457139\pi$
$308$	0.974929	0.0555518
$309$	0	0
$310$	48.8509	2.77454
$311$	−33.3195	−1.88938	−0.944688	−	0.327971i	$-0.893635\pi$
$311$	−33.3195	−1.88938	−0.944688	+	0.327971i	$0.893635\pi$
$312$	0	0
$313$	−13.5589	−0.766392	−0.383196	−	0.923667i	$-0.625177\pi$
$313$	−13.5589	−0.766392	−0.383196	+	0.923667i	$0.625177\pi$
$314$	32.3330	1.82465
$315$	0	0
$316$	1.44250	0.0811472
$317$	7.75829	0.435749	0.217875	−	0.975977i	$-0.430088\pi$
$317$	7.75829	0.435749	0.217875	+	0.975977i	$0.430088\pi$
$318$	0	0
$319$	16.0237	0.897156
$320$	19.4181	1.08550
$321$	0	0
$322$	2.92575	0.163046
$323$	−0.0910473	−0.00506601
$324$	0	0
$325$	27.4345	1.52179
$326$	−20.9624	−1.16100
$327$	0	0
$328$	10.4397	0.576438
$329$	−2.35651	−0.129918
$330$	0	0
$331$	−12.3472	−0.678666	−0.339333	−	0.940666i	$-0.610201\pi$
$331$	−12.3472	−0.678666	−0.339333	+	0.940666i	$0.610201\pi$
$332$	5.16299	0.283356
$333$	0	0
$334$	10.7271	0.586963
$335$	46.3942	2.53479
$336$	0	0
$337$	25.1681	1.37099	0.685497	−	0.728076i	$-0.259585\pi$
$337$	25.1681	1.37099	0.685497	+	0.728076i	$0.259585\pi$
$338$	4.54478	0.247204
$339$	0	0
$340$	−0.0780484	−0.00423277
$341$	−20.7291	−1.12254
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−19.0437	−1.02677
$345$	0	0
$346$	21.6299	1.16283
$347$	−22.4956	−1.20763	−0.603813	−	0.797126i	$-0.706352\pi$
$347$	−22.4956	−1.20763	−0.603813	+	0.797126i	$0.706352\pi$
$348$	0	0
$349$	−6.77129	−0.362459	−0.181229	−	0.983441i	$-0.558008\pi$
$349$	−6.77129	−0.362459	−0.181229	+	0.983441i	$0.558008\pi$
$350$	−10.7147	−0.572727
$351$	0	0
$352$	5.42642	0.289229
$353$	−13.0742	−0.695870	−0.347935	−	0.937519i	$-0.613117\pi$
$353$	−13.0742	−0.695870	−0.347935	+	0.937519i	$0.613117\pi$
$354$	0	0
$355$	37.6542	1.99848
$356$	−3.25586	−0.172560
$357$	0	0
$358$	28.5407	1.50842
$359$	2.11758	0.111762	0.0558809	−	0.998437i	$-0.482203\pi$
$359$	2.11758	0.111762	0.0558809	+	0.998437i	$0.482203\pi$
$360$	0	0
$361$	−16.0412	−0.844275
$362$	4.18412	0.219912
$363$	0	0
$364$	−1.70701	−0.0894715
$365$	8.67435	0.454036
$366$	0	0
$367$	−12.0952	−0.631363	−0.315682	−	0.948865i	$-0.602233\pi$
$367$	−12.0952	−0.631363	−0.315682	+	0.948865i	$0.602233\pi$
$368$	8.77383	0.457367
$369$	0	0
$370$	34.0616	1.77078
$371$	−5.55020	−0.288152
$372$	0	0
$373$	14.7769	0.765120	0.382560	−	0.923931i	$-0.375043\pi$
$373$	14.7769	0.765120	0.382560	+	0.923931i	$0.375043\pi$
$374$	0.187927	0.00971747
$375$	0	0
$376$	−5.77259	−0.297698
$377$	−28.0560	−1.44496
$378$	0	0
$379$	−30.3485	−1.55890	−0.779449	−	0.626466i	$-0.784501\pi$
$379$	−30.3485	−1.55890	−0.779449	+	0.626466i	$0.784501\pi$
$380$	2.53635	0.130112
$381$	0	0
$382$	−3.89160	−0.199111
$383$	−14.0435	−0.717591	−0.358796	−	0.933416i	$-0.616812\pi$
$383$	−14.0435	−0.717591	−0.358796	+	0.933416i	$0.616812\pi$
$384$	0	0
$385$	7.85254	0.400202
$386$	18.4210	0.937604
$387$	0	0
$388$	−0.581233	−0.0295076
$389$	−16.0956	−0.816081	−0.408041	−	0.912964i	$-0.633788\pi$
$389$	−16.0956	−0.816081	−0.408041	+	0.912964i	$0.633788\pi$
$390$	0	0
$391$	0.0993885	0.00502629
$392$	−2.44964	−0.123725
$393$	0	0
$394$	−17.0855	−0.860757
$395$	11.6186	0.584595
$396$	0	0
$397$	−2.22502	−0.111671	−0.0558353	−	0.998440i	$-0.517782\pi$
$397$	−2.22502	−0.111671	−0.0558353	+	0.998440i	$0.517782\pi$
$398$	−1.77448	−0.0889465
$399$	0	0
$400$	−32.1317	−1.60658
$401$	−23.2854	−1.16282	−0.581409	−	0.813612i	$-0.697498\pi$
$401$	−23.2854	−1.16282	−0.581409	+	0.813612i	$0.697498\pi$
$402$	0	0
$403$	36.2946	1.80796
$404$	−5.80266	−0.288693
$405$	0	0
$406$	10.9575	0.543810
$407$	−14.4535	−0.716434
$408$	0	0
$409$	22.4639	1.11077	0.555384	−	0.831594i	$-0.312571\pi$
$409$	22.4639	1.11077	0.555384	+	0.831594i	$0.312571\pi$
$410$	−22.8847	−1.13019
$411$	0	0
$412$	1.66723	0.0821384
$413$	1.27476	0.0627269
$414$	0	0
$415$	41.5852	2.04133
$416$	−9.50113	−0.465831
$417$	0	0
$418$	−6.10708	−0.298707
$419$	−3.22903	−0.157748	−0.0788742	−	0.996885i	$-0.525133\pi$
$419$	−3.22903	−0.157748	−0.0788742	+	0.996885i	$0.525133\pi$
$420$	0	0
$421$	25.3101	1.23354	0.616768	−	0.787145i	$-0.288442\pi$
$421$	25.3101	1.23354	0.616768	+	0.787145i	$0.288442\pi$
$422$	−11.2231	−0.546332
$423$	0	0
$424$	−13.5960	−0.660279
$425$	−0.363982	−0.0176557
$426$	0	0
$427$	0.634867	0.0307234
$428$	0.465146	0.0224837
$429$	0	0
$430$	41.7451	2.01313
$431$	26.4606	1.27456	0.637282	−	0.770631i	$-0.280059\pi$
$431$	26.4606	1.27456	0.637282	+	0.770631i	$0.280059\pi$
$432$	0	0
$433$	−26.4649	−1.27182	−0.635911	−	0.771762i	$-0.719376\pi$
$433$	−26.4649	−1.27182	−0.635911	+	0.771762i	$0.719376\pi$
$434$	−14.1751	−0.680429
$435$	0	0
$436$	−1.94167	−0.0929893
$437$	−3.22984	−0.154504
$438$	0	0
$439$	−2.47368	−0.118062	−0.0590311	−	0.998256i	$-0.518801\pi$
$439$	−2.47368	−0.118062	−0.0590311	+	0.998256i	$0.518801\pi$
$440$	19.2359	0.917034
$441$	0	0
$442$	−0.329042	−0.0156509
$443$	−12.4369	−0.590893	−0.295446	−	0.955359i	$-0.595468\pi$
$443$	−12.4369	−0.590893	−0.295446	+	0.955359i	$0.595468\pi$
$444$	0	0
$445$	−26.2242	−1.24315
$446$	−0.992986	−0.0470192
$447$	0	0
$448$	−5.63459	−0.266209
$449$	22.3958	1.05692	0.528461	−	0.848957i	$-0.322769\pi$
$449$	22.3958	1.05692	0.528461	+	0.848957i	$0.322769\pi$
$450$	0	0
$451$	9.71075	0.457261
$452$	7.07979	0.333006
$453$	0	0
$454$	19.0403	0.893607
$455$	−13.7490	−0.644564
$456$	0	0
$457$	−10.9978	−0.514456	−0.257228	−	0.966351i	$-0.582809\pi$
$457$	−10.9978	−0.514456	−0.257228	+	0.966351i	$0.582809\pi$
$458$	−33.2703	−1.55462
$459$	0	0
$460$	−2.76871	−0.129092
$461$	−18.5307	−0.863063	−0.431531	−	0.902098i	$-0.642027\pi$
$461$	−18.5307	−0.863063	−0.431531	+	0.902098i	$0.642027\pi$
$462$	0	0
$463$	19.1725	0.891021	0.445511	−	0.895277i	$-0.353022\pi$
$463$	19.1725	0.891021	0.445511	+	0.895277i	$0.353022\pi$
$464$	32.8596	1.52547
$465$	0	0
$466$	19.6839	0.911841
$467$	22.1986	1.02723	0.513614	−	0.858021i	$-0.328306\pi$
$467$	22.1986	1.02723	0.513614	+	0.858021i	$0.328306\pi$
$468$	0	0
$469$	−13.4623	−0.621631
$470$	12.6539	0.583682
$471$	0	0
$472$	3.12270	0.143734
$473$	−17.7139	−0.814486
$474$	0	0
$475$	11.8284	0.542723
$476$	0.0226474	0.00103804
$477$	0	0
$478$	10.1486	0.464185
$479$	−10.2477	−0.468228	−0.234114	−	0.972209i	$-0.575219\pi$
$479$	−10.2477	−0.468228	−0.234114	+	0.972209i	$0.575219\pi$
$480$	0	0
$481$	25.3067	1.15389
$482$	−6.73911	−0.306958
$483$	0	0
$484$	−2.48507	−0.112958
$485$	−4.68152	−0.212577
$486$	0	0
$487$	−5.41244	−0.245261	−0.122630	−	0.992452i	$-0.539133\pi$
$487$	−5.41244	−0.245261	−0.122630	+	0.992452i	$0.539133\pi$
$488$	1.55519	0.0704004
$489$	0	0
$490$	5.36979	0.242582
$491$	6.33853	0.286054	0.143027	−	0.989719i	$-0.454316\pi$
$491$	6.33853	0.286054	0.143027	+	0.989719i	$0.454316\pi$
$492$	0	0
$493$	0.372228	0.0167643
$494$	10.6929	0.481096
$495$	0	0
$496$	−42.5089	−1.90871
$497$	−10.9262	−0.490106
$498$	0	0
$499$	33.6332	1.50563	0.752815	−	0.658232i	$-0.228696\pi$
$499$	33.6332	1.50563	0.752815	+	0.658232i	$0.228696\pi$
$500$	2.76699	0.123743
$501$	0	0
$502$	−4.25340	−0.189838
$503$	10.4483	0.465868	0.232934	−	0.972493i	$-0.425167\pi$
$503$	10.4483	0.465868	0.232934	+	0.972493i	$0.425167\pi$
$504$	0	0
$505$	−46.7374	−2.07979
$506$	6.66657	0.296365
$507$	0	0
$508$	0.427866	0.0189835
$509$	11.0671	0.490539	0.245269	−	0.969455i	$-0.421124\pi$
$509$	11.0671	0.490539	0.245269	+	0.969455i	$0.421124\pi$
$510$	0	0
$511$	−2.51705	−0.111348
$512$	−11.7648	−0.519935
$513$	0	0
$514$	35.1839	1.55189
$515$	13.4286	0.591736
$516$	0	0
$517$	−5.36950	−0.236150
$518$	−9.88372	−0.434266
$519$	0	0
$520$	−33.6801	−1.47697
$521$	−0.993866	−0.0435421	−0.0217710	−	0.999763i	$-0.506930\pi$
$521$	−0.993866	−0.0435421	−0.0217710	+	0.999763i	$0.506930\pi$
$522$	0	0
$523$	−35.2011	−1.53924	−0.769618	−	0.638504i	$-0.779553\pi$
$523$	−35.2011	−1.53924	−0.769618	+	0.638504i	$0.779553\pi$
$524$	−8.51267	−0.371878
$525$	0	0
$526$	8.74429	0.381269
$527$	−0.481534	−0.0209759
$528$	0	0
$529$	−19.4743	−0.846707
$530$	29.8034	1.29458
$531$	0	0
$532$	−0.735976	−0.0319086
$533$	−17.0026	−0.736464
$534$	0	0
$535$	3.74650	0.161975
$536$	−32.9778	−1.42442
$537$	0	0
$538$	−20.1989	−0.870838
$539$	−2.27859	−0.0981456
$540$	0	0
$541$	45.0516	1.93692	0.968460	−	0.249168i	$-0.0801573\pi$
$541$	45.0516	1.93692	0.968460	+	0.249168i	$0.0801573\pi$
$542$	26.6888	1.14638
$543$	0	0
$544$	0.126055	0.00540455
$545$	−15.6391	−0.669907
$546$	0	0
$547$	34.8448	1.48986	0.744928	−	0.667145i	$-0.232484\pi$
$547$	34.8448	1.48986	0.744928	+	0.667145i	$0.232484\pi$
$548$	−5.81430	−0.248375
$549$	0	0
$550$	−24.4144	−1.04103
$551$	−12.0963	−0.515321
$552$	0	0
$553$	−3.37139	−0.143366
$554$	−24.1767	−1.02717
$555$	0	0
$556$	2.90827	0.123338
$557$	−3.54550	−0.150228	−0.0751138	−	0.997175i	$-0.523932\pi$
$557$	−3.54550	−0.150228	−0.0751138	+	0.997175i	$0.523932\pi$
$558$	0	0
$559$	31.0153	1.31181
$560$	16.1031	0.680480
$561$	0	0
$562$	−39.7782	−1.67794
$563$	30.1967	1.27264	0.636320	−	0.771425i	$-0.280456\pi$
$563$	30.1967	1.27264	0.636320	+	0.771425i	$0.280456\pi$
$564$	0	0
$565$	57.0240	2.39902
$566$	46.6527	1.96096
$567$	0	0
$568$	−26.7652	−1.12304
$569$	26.5115	1.11142	0.555709	−	0.831377i	$-0.312447\pi$
$569$	26.5115	1.11142	0.555709	+	0.831377i	$0.312447\pi$
$570$	0	0
$571$	12.1819	0.509797	0.254899	−	0.966968i	$-0.417958\pi$
$571$	12.1819	0.509797	0.254899	+	0.966968i	$0.417958\pi$
$572$	−3.88956	−0.162631
$573$	0	0
$574$	6.64049	0.277169
$575$	−12.9120	−0.538468
$576$	0	0
$577$	10.0676	0.419118	0.209559	−	0.977796i	$-0.432797\pi$
$577$	10.0676	0.419118	0.209559	+	0.977796i	$0.432797\pi$
$578$	−26.4844	−1.10160
$579$	0	0
$580$	−10.3693	−0.430563
$581$	−12.0668	−0.500617
$582$	0	0
$583$	−12.6466	−0.523769
$584$	−6.16587	−0.255145
$585$	0	0
$586$	−19.7117	−0.814284
$587$	−21.8759	−0.902916	−0.451458	−	0.892292i	$-0.649096\pi$
$587$	−21.8759	−0.902916	−0.451458	+	0.892292i	$0.649096\pi$
$588$	0	0
$589$	15.6484	0.644783
$590$	−6.84520	−0.281812
$591$	0	0
$592$	−29.6396	−1.21818
$593$	−5.04605	−0.207217	−0.103608	−	0.994618i	$-0.533039\pi$
$593$	−5.04605	−0.207217	−0.103608	+	0.994618i	$0.533039\pi$
$594$	0	0
$595$	0.182413	0.00747821
$596$	6.66264	0.272912
$597$	0	0
$598$	−11.6725	−0.477325
$599$	−25.7301	−1.05130	−0.525652	−	0.850700i	$-0.676179\pi$
$599$	−25.7301	−1.05130	−0.525652	+	0.850700i	$0.676179\pi$
$600$	0	0
$601$	29.0768	1.18607	0.593034	−	0.805177i	$-0.297930\pi$
$601$	29.0768	1.18607	0.593034	+	0.805177i	$0.297930\pi$
$602$	−12.1133	−0.493700
$603$	0	0
$604$	−1.11679	−0.0454417
$605$	−20.0159	−0.813762
$606$	0	0
$607$	−4.37198	−0.177453	−0.0887265	−	0.996056i	$-0.528280\pi$
$607$	−4.37198	−0.177453	−0.0887265	+	0.996056i	$0.528280\pi$
$608$	−4.09641	−0.166132
$609$	0	0
$610$	−3.40910	−0.138030
$611$	9.40147	0.380343
$612$	0	0
$613$	−8.17849	−0.330326	−0.165163	−	0.986266i	$-0.552815\pi$
$613$	−8.17849	−0.330326	−0.165163	+	0.986266i	$0.552815\pi$
$614$	7.33009	0.295818
$615$	0	0
$616$	−5.58171	−0.224893
$617$	11.6087	0.467351	0.233675	−	0.972315i	$-0.424925\pi$
$617$	11.6087	0.467351	0.233675	+	0.972315i	$0.424925\pi$
$618$	0	0
$619$	7.08338	0.284705	0.142353	−	0.989816i	$-0.454533\pi$
$619$	7.08338	0.284705	0.142353	+	0.989816i	$0.454533\pi$
$620$	13.4143	0.538731
$621$	0	0
$622$	−51.9171	−2.08169
$623$	7.60952	0.304869
$624$	0	0
$625$	−12.0960	−0.483841
$626$	−21.1269	−0.844400
$627$	0	0
$628$	8.87853	0.354292
$629$	−0.335753	−0.0133873
$630$	0	0
$631$	18.7221	0.745315	0.372658	−	0.927969i	$-0.378447\pi$
$631$	18.7221	0.745315	0.372658	+	0.927969i	$0.378447\pi$
$632$	−8.25868	−0.328513
$633$	0	0
$634$	12.0887	0.480102
$635$	3.44623	0.136760
$636$	0	0
$637$	3.98958	0.158073
$638$	24.9675	0.988474
$639$	0	0
$640$	46.6708	1.84483
$641$	−9.63790	−0.380674	−0.190337	−	0.981719i	$-0.560958\pi$
$641$	−9.63790	−0.380674	−0.190337	+	0.981719i	$0.560958\pi$
$642$	0	0
$643$	43.6291	1.72056	0.860282	−	0.509819i	$-0.170288\pi$
$643$	43.6291	1.72056	0.860282	+	0.509819i	$0.170288\pi$
$644$	0.803401	0.0316585
$645$	0	0
$646$	−0.141866	−0.00558166
$647$	6.96274	0.273734	0.136867	−	0.990589i	$-0.456297\pi$
$647$	6.96274	0.273734	0.136867	+	0.990589i	$0.456297\pi$
$648$	0	0
$649$	2.90465	0.114018
$650$	42.7473	1.67669
$651$	0	0
$652$	−5.75622	−0.225431
$653$	−32.7190	−1.28039	−0.640197	−	0.768211i	$-0.721147\pi$
$653$	−32.7190	−1.28039	−0.640197	+	0.768211i	$0.721147\pi$
$654$	0	0
$655$	−68.5650	−2.67906
$656$	19.9137	0.777499
$657$	0	0
$658$	−3.67182	−0.143142
$659$	17.1823	0.669328	0.334664	−	0.942338i	$-0.391377\pi$
$659$	17.1823	0.669328	0.334664	+	0.942338i	$0.391377\pi$
$660$	0	0
$661$	−35.2327	−1.37039	−0.685197	−	0.728358i	$-0.740284\pi$
$661$	−35.2327	−1.37039	−0.685197	+	0.728358i	$0.740284\pi$
$662$	−19.2390	−0.747745
$663$	0	0
$664$	−29.5594	−1.14713
$665$	−5.92790	−0.229874
$666$	0	0
$667$	13.2045	0.511281
$668$	2.94564	0.113970
$669$	0	0
$670$	72.2897	2.79279
$671$	1.44660	0.0558453
$672$	0	0
$673$	−50.4438	−1.94447	−0.972233	−	0.234014i	$-0.924814\pi$
$673$	−50.4438	−1.94447	−0.972233	+	0.234014i	$0.924814\pi$
$674$	39.2160	1.51054
$675$	0	0
$676$	1.24798	0.0479993
$677$	3.26125	0.125340	0.0626700	−	0.998034i	$-0.480038\pi$
$677$	3.26125	0.125340	0.0626700	+	0.998034i	$0.480038\pi$
$678$	0	0
$679$	1.35845	0.0521323
$680$	0.446846	0.0171358
$681$	0	0
$682$	−32.2993	−1.23680
$683$	−26.8083	−1.02579	−0.512895	−	0.858451i	$-0.671427\pi$
$683$	−26.8083	−1.02579	−0.512895	+	0.858451i	$0.671427\pi$
$684$	0	0
$685$	−46.8311	−1.78932
$686$	−1.55816	−0.0594909
$687$	0	0
$688$	−36.3256	−1.38490
$689$	22.1430	0.843580
$690$	0	0
$691$	−50.2889	−1.91308	−0.956539	−	0.291604i	$-0.905811\pi$
$691$	−50.2889	−1.91308	−0.956539	+	0.291604i	$0.905811\pi$
$692$	5.93950	0.225786
$693$	0	0
$694$	−35.0517	−1.33054
$695$	23.4246	0.888546
$696$	0	0
$697$	0.225579	0.00854442
$698$	−10.5508	−0.399352
$699$	0	0
$700$	−2.94223	−0.111206
$701$	−1.21875	−0.0460317	−0.0230159	−	0.999735i	$-0.507327\pi$
$701$	−1.21875	−0.0460317	−0.0230159	+	0.999735i	$0.507327\pi$
$702$	0	0
$703$	10.9110	0.411516
$704$	−12.8389	−0.483884
$705$	0	0
$706$	−20.3717	−0.766700
$707$	13.5619	0.510047
$708$	0	0
$709$	−11.5445	−0.433563	−0.216781	−	0.976220i	$-0.569556\pi$
$709$	−11.5445	−0.433563	−0.216781	+	0.976220i	$0.569556\pi$
$710$	58.6713	2.20189
$711$	0	0
$712$	18.6406	0.698585
$713$	−17.0820	−0.639728
$714$	0	0
$715$	−31.3283	−1.17161
$716$	7.83718	0.292889
$717$	0	0
$718$	3.29954	0.123138
$719$	−38.2527	−1.42658	−0.713292	−	0.700867i	$-0.752797\pi$
$719$	−38.2527	−1.42658	−0.713292	+	0.700867i	$0.752797\pi$
$720$	0	0
$721$	−3.89661	−0.145117
$722$	−24.9948	−0.930211
$723$	0	0
$724$	1.14895	0.0427002
$725$	−48.3578	−1.79597
$726$	0	0
$727$	17.2829	0.640989	0.320495	−	0.947250i	$-0.396151\pi$
$727$	17.2829	0.640989	0.320495	+	0.947250i	$0.396151\pi$
$728$	9.77303	0.362213
$729$	0	0
$730$	13.5160	0.500251
$731$	−0.411491	−0.0152195
$732$	0	0
$733$	49.8471	1.84114	0.920572	−	0.390574i	$-0.127723\pi$
$733$	49.8471	1.84114	0.920572	+	0.390574i	$0.127723\pi$
$734$	−18.8462	−0.695627
$735$	0	0
$736$	4.47170	0.164829
$737$	−30.6750	−1.12993
$738$	0	0
$739$	−34.5099	−1.26947	−0.634733	−	0.772731i	$-0.718890\pi$
$739$	−34.5099	−1.26947	−0.634733	+	0.772731i	$0.718890\pi$
$740$	9.35321	0.343831
$741$	0	0
$742$	−8.64810	−0.317482
$743$	53.6257	1.96734	0.983669	−	0.179988i	$-0.0576059\pi$
$743$	53.6257	1.96734	0.983669	+	0.179988i	$0.0576059\pi$
$744$	0	0
$745$	53.6640	1.96610
$746$	23.0248	0.842998
$747$	0	0
$748$	0.0516041	0.00188683
$749$	−1.08713	−0.0397228
$750$	0	0
$751$	3.22553	0.117701	0.0588507	−	0.998267i	$-0.481256\pi$
$751$	3.22553	0.117701	0.0588507	+	0.998267i	$0.481256\pi$
$752$	−11.0112	−0.401536
$753$	0	0
$754$	−43.7157	−1.59203
$755$	−8.99519	−0.327368
$756$	0	0
$757$	45.8223	1.66544	0.832720	−	0.553695i	$-0.186783\pi$
$757$	45.8223	1.66544	0.832720	+	0.553695i	$0.186783\pi$
$758$	−47.2879	−1.71757
$759$	0	0
$760$	−14.5212	−0.526739
$761$	−20.3134	−0.736359	−0.368179	−	0.929755i	$-0.620019\pi$
$761$	−20.3134	−0.736359	−0.368179	+	0.929755i	$0.620019\pi$
$762$	0	0
$763$	4.53804	0.164288
$764$	−1.06862	−0.0386613
$765$	0	0
$766$	−21.8821	−0.790632
$767$	−5.08576	−0.183636
$768$	0	0
$769$	−15.8488	−0.571524	−0.285762	−	0.958301i	$-0.592247\pi$
$769$	−15.8488	−0.571524	−0.285762	+	0.958301i	$0.592247\pi$
$770$	12.2355	0.440937
$771$	0	0
$772$	5.05835	0.182054
$773$	−46.3290	−1.66634	−0.833169	−	0.553018i	$-0.813476\pi$
$773$	−46.3290	−1.66634	−0.833169	+	0.553018i	$0.813476\pi$
$774$	0	0
$775$	62.5582	2.24716
$776$	3.32770	0.119457
$777$	0	0
$778$	−25.0796	−0.899147
$779$	−7.33067	−0.262648
$780$	0	0
$781$	−24.8962	−0.890858
$782$	0.154863	0.00553790
$783$	0	0
$784$	−4.67266	−0.166881
$785$	71.5118	2.55237
$786$	0	0
$787$	3.98888	0.142188	0.0710941	−	0.997470i	$-0.477351\pi$
$787$	3.98888	0.142188	0.0710941	+	0.997470i	$0.477351\pi$
$788$	−4.69164	−0.167133
$789$	0	0
$790$	18.1036	0.644099
$791$	−16.5468	−0.588335
$792$	0	0
$793$	−2.53285	−0.0899443
$794$	−3.46694	−0.123037
$795$	0	0
$796$	−0.487266	−0.0172707
$797$	2.17009	0.0768685	0.0384343	−	0.999261i	$-0.487763\pi$
$797$	2.17009	0.0768685	0.0384343	+	0.999261i	$0.487763\pi$
$798$	0	0
$799$	−0.124733	−0.00441272
$800$	−16.3763	−0.578991
$801$	0	0
$802$	−36.2824	−1.28118
$803$	−5.73532	−0.202395
$804$	0	0
$805$	6.47097	0.228072
$806$	56.5529	1.99199
$807$	0	0
$808$	33.2217	1.16873
$809$	−25.1917	−0.885691	−0.442846	−	0.896598i	$-0.646031\pi$
$809$	−25.1917	−0.885691	−0.442846	+	0.896598i	$0.646031\pi$
$810$	0	0
$811$	1.18218	0.0415120	0.0207560	−	0.999785i	$-0.493393\pi$
$811$	1.18218	0.0415120	0.0207560	+	0.999785i	$0.493393\pi$
$812$	3.00889	0.105591
$813$	0	0
$814$	−22.5209	−0.789357
$815$	−46.3633	−1.62403
$816$	0	0
$817$	13.3723	0.467836
$818$	35.0024	1.22383
$819$	0	0
$820$	−6.28406	−0.219449
$821$	0.626182	0.0218539	0.0109270	−	0.999940i	$-0.496522\pi$
$821$	0.626182	0.0218539	0.0109270	+	0.999940i	$0.496522\pi$
$822$	0	0
$823$	−23.4237	−0.816499	−0.408250	−	0.912870i	$-0.633861\pi$
$823$	−23.4237	−0.816499	−0.408250	+	0.912870i	$0.633861\pi$
$824$	−9.54529	−0.332526
$825$	0	0
$826$	1.98628	0.0691116
$827$	50.6591	1.76159	0.880795	−	0.473499i	$-0.157009\pi$
$827$	50.6591	1.76159	0.880795	+	0.473499i	$0.157009\pi$
$828$	0	0
$829$	−7.39849	−0.256960	−0.128480	−	0.991712i	$-0.541010\pi$
$829$	−7.39849	−0.256960	−0.128480	+	0.991712i	$0.541010\pi$
$830$	64.7964	2.24911
$831$	0	0
$832$	22.4796	0.779341
$833$	−0.0529312	−0.00183396
$834$	0	0
$835$	23.7255	0.821056
$836$	−1.67698	−0.0579997
$837$	0	0
$838$	−5.03135	−0.173805
$839$	1.66468	0.0574712	0.0287356	−	0.999587i	$-0.490852\pi$
$839$	1.66468	0.0574712	0.0287356	+	0.999587i	$0.490852\pi$
$840$	0	0
$841$	20.4534	0.705289
$842$	39.4371	1.35909
$843$	0	0
$844$	−3.08183	−0.106081
$845$	10.0518	0.345794
$846$	0	0
$847$	5.80805	0.199567
$848$	−25.9342	−0.890584
$849$	0	0
$850$	−0.567143	−0.0194529
$851$	−11.9106	−0.408289
$852$	0	0
$853$	5.00598	0.171402	0.0857008	−	0.996321i	$-0.472687\pi$
$853$	5.00598	0.171402	0.0857008	+	0.996321i	$0.472687\pi$
$854$	0.989225	0.0338506
$855$	0	0
$856$	−2.66307	−0.0910219
$857$	−11.7641	−0.401854	−0.200927	−	0.979606i	$-0.564395\pi$
$857$	−11.7641	−0.401854	−0.200927	+	0.979606i	$0.564395\pi$
$858$	0	0
$859$	6.07366	0.207231	0.103615	−	0.994617i	$-0.466959\pi$
$859$	6.07366	0.207231	0.103615	+	0.994617i	$0.466959\pi$
$860$	11.4631	0.390888
$861$	0	0
$862$	41.2299	1.40430
$863$	−1.15208	−0.0392172	−0.0196086	−	0.999808i	$-0.506242\pi$
$863$	−1.15208	−0.0392172	−0.0196086	+	0.999808i	$0.506242\pi$
$864$	0	0
$865$	47.8395	1.62659
$866$	−41.2366	−1.40128
$867$	0	0
$868$	−3.89245	−0.132118
$869$	−7.68200	−0.260594
$870$	0	0
$871$	53.7089	1.81986
$872$	11.1166	0.376454
$873$	0	0
$874$	−5.03261	−0.170230
$875$	−6.46694	−0.218623
$876$	0	0
$877$	−32.8595	−1.10959	−0.554793	−	0.831989i	$-0.687202\pi$
$877$	−32.8595	−1.10959	−0.554793	+	0.831989i	$0.687202\pi$
$878$	−3.85439	−0.130079
$879$	0	0
$880$	36.6923	1.23690
$881$	25.5268	0.860019	0.430010	−	0.902824i	$-0.358510\pi$
$881$	25.5268	0.860019	0.430010	+	0.902824i	$0.358510\pi$
$882$	0	0
$883$	−14.7396	−0.496029	−0.248014	−	0.968756i	$-0.579778\pi$
$883$	−14.7396	−0.496029	−0.248014	+	0.968756i	$0.579778\pi$
$884$	−0.0903538	−0.00303893
$885$	0	0
$886$	−19.3786	−0.651038
$887$	10.9986	0.369296	0.184648	−	0.982805i	$-0.440886\pi$
$887$	10.9986	0.369296	0.184648	+	0.982805i	$0.440886\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	−40.8615	−1.36968
$891$	0	0
$892$	−0.272671	−0.00912969
$893$	4.05345	0.135643
$894$	0	0
$895$	63.1243	2.11001
$896$	−13.5426	−0.452425
$897$	0	0
$898$	34.8963	1.16450
$899$	−63.9754	−2.13370
$900$	0	0
$901$	−0.293778	−0.00978718
$902$	15.1309	0.503804
$903$	0	0
$904$	−40.5335	−1.34813
$905$	9.25414	0.307618
$906$	0	0
$907$	−27.1314	−0.900882	−0.450441	−	0.892806i	$-0.648733\pi$
$907$	−27.1314	−0.900882	−0.450441	+	0.892806i	$0.648733\pi$
$908$	5.22842	0.173511
$909$	0	0
$910$	−21.4232	−0.710172
$911$	−22.6674	−0.751004	−0.375502	−	0.926822i	$-0.622530\pi$
$911$	−22.6674	−0.751004	−0.375502	+	0.926822i	$0.622530\pi$
$912$	0	0
$913$	−27.4953	−0.909963
$914$	−17.1364	−0.566820
$915$	0	0
$916$	−9.13592	−0.301859
$917$	19.8956	0.657012
$918$	0	0
$919$	−43.5972	−1.43814	−0.719070	−	0.694937i	$-0.755432\pi$
$919$	−43.5972	−1.43814	−0.719070	+	0.694937i	$0.755432\pi$
$920$	15.8515	0.522610
$921$	0	0
$922$	−28.8739	−0.950910
$923$	43.5909	1.43481
$924$	0	0
$925$	43.6191	1.43419
$926$	29.8738	0.981715
$927$	0	0
$928$	16.7473	0.549759
$929$	48.1771	1.58064	0.790320	−	0.612695i	$-0.209914\pi$
$929$	48.1771	1.58064	0.790320	+	0.612695i	$0.209914\pi$
$930$	0	0
$931$	1.72011	0.0563743
$932$	5.40515	0.177051
$933$	0	0
$934$	34.5890	1.13179
$935$	0.415644	0.0135930
$936$	0	0
$937$	−5.64674	−0.184471	−0.0922355	−	0.995737i	$-0.529401\pi$
$937$	−5.64674	−0.184471	−0.0922355	+	0.995737i	$0.529401\pi$
$938$	−20.9764	−0.684905
$939$	0	0
$940$	3.47473	0.113333
$941$	−20.2918	−0.661492	−0.330746	−	0.943720i	$-0.607300\pi$
$941$	−20.2918	−0.661492	−0.330746	+	0.943720i	$0.607300\pi$
$942$	0	0
$943$	8.00225	0.260589
$944$	5.95653	0.193869
$945$	0	0
$946$	−27.6011	−0.897389
$947$	−8.31493	−0.270199	−0.135100	−	0.990832i	$-0.543135\pi$
$947$	−8.31493	−0.270199	−0.135100	+	0.990832i	$0.543135\pi$
$948$	0	0
$949$	10.0420	0.325977
$950$	18.4305	0.597965
$951$	0	0
$952$	−0.129662	−0.00420237
$953$	−30.0421	−0.973158	−0.486579	−	0.873637i	$-0.661755\pi$
$953$	−30.0421	−0.973158	−0.486579	+	0.873637i	$0.661755\pi$
$954$	0	0
$955$	−8.60717	−0.278521
$956$	2.78677	0.0901305
$957$	0	0
$958$	−15.9675	−0.515887
$959$	13.5891	0.438814
$960$	0	0
$961$	51.7619	1.66974
$962$	39.4319	1.27134
$963$	0	0
$964$	−1.85054	−0.0596019
$965$	40.7423	1.31154
$966$	0	0
$967$	21.2031	0.681845	0.340922	−	0.940091i	$-0.389261\pi$
$967$	21.2031	0.681845	0.340922	+	0.940091i	$0.389261\pi$
$968$	14.2276	0.457293
$969$	0	0
$970$	−7.29456	−0.234214
$971$	17.1885	0.551605	0.275802	−	0.961214i	$-0.411056\pi$
$971$	17.1885	0.551605	0.275802	+	0.961214i	$0.411056\pi$
$972$	0	0
$973$	−6.79716	−0.217907
$974$	−8.43345	−0.270225
$975$	0	0
$976$	2.96652	0.0949560
$977$	3.06434	0.0980368	0.0490184	−	0.998798i	$-0.484391\pi$
$977$	3.06434	0.0980368	0.0490184	+	0.998798i	$0.484391\pi$
$978$	0	0
$979$	17.3389	0.554155
$980$	1.47453	0.0471020
$981$	0	0
$982$	9.87644	0.315170
$983$	10.5337	0.335973	0.167987	−	0.985789i	$-0.446273\pi$
$983$	10.5337	0.335973	0.167987	+	0.985789i	$0.446273\pi$
$984$	0	0
$985$	−37.7886	−1.20405
$986$	0.579992	0.0184707
$987$	0	0
$988$	2.93624	0.0934142
$989$	−14.5973	−0.464168
$990$	0	0
$991$	4.19699	0.133322	0.0666608	−	0.997776i	$-0.478765\pi$
$991$	4.19699	0.133322	0.0666608	+	0.997776i	$0.478765\pi$
$992$	−21.6652	−0.687872
$993$	0	0
$994$	−17.0248	−0.539993
$995$	−3.92467	−0.124420
$996$	0	0
$997$	−28.8412	−0.913410	−0.456705	−	0.889618i	$-0.650970\pi$
$997$	−28.8412	−0.913410	−0.456705	+	0.889618i	$0.650970\pi$
$998$	52.4060	1.65888
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.y.1.22	yes	28
3.2	odd	2	inner	8001.2.a.y.1.7	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8001.2.a.y.1.7	✓	28	3.2	odd	2	inner
8001.2.a.y.1.22	yes	28	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+1.55816 q^{2} +0.427866 q^{4} +3.44623 q^{5} -1.00000 q^{7} -2.44964 q^{8} +O(q^{10})\)	\(q+1.55816 q^{2} +0.427866 q^{4} +3.44623 q^{5} -1.00000 q^{7} -2.44964 q^{8} +5.36979 q^{10} -2.27859 q^{11} +3.98958 q^{13} -1.55816 q^{14} -4.67266 q^{16} -0.0529312 q^{17} +1.72011 q^{19} +1.47453 q^{20} -3.55040 q^{22} -1.87769 q^{23} +6.87652 q^{25} +6.21641 q^{26} -0.427866 q^{28} -7.03231 q^{29} +9.09736 q^{31} -2.38149 q^{32} -0.0824753 q^{34} -3.44623 q^{35} +6.34320 q^{37} +2.68021 q^{38} -8.44202 q^{40} -4.26175 q^{41} +7.77408 q^{43} -0.974929 q^{44} -2.92575 q^{46} +2.35651 q^{47} +1.00000 q^{49} +10.7147 q^{50} +1.70701 q^{52} +5.55020 q^{53} -7.85254 q^{55} +2.44964 q^{56} -10.9575 q^{58} -1.27476 q^{59} -0.634867 q^{61} +14.1751 q^{62} +5.63459 q^{64} +13.7490 q^{65} +13.4623 q^{67} -0.0226474 q^{68} -5.36979 q^{70} +10.9262 q^{71} +2.51705 q^{73} +9.88372 q^{74} +0.735976 q^{76} +2.27859 q^{77} +3.37139 q^{79} -16.1031 q^{80} -6.64049 q^{82} +12.0668 q^{83} -0.182413 q^{85} +12.1133 q^{86} +5.58171 q^{88} -7.60952 q^{89} -3.98958 q^{91} -0.803401 q^{92} +3.67182 q^{94} +5.92790 q^{95} -1.35845 q^{97} +1.55816 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 30 q^{4} - 28 q^{7}+O(q^{10}) \) 28 * q + 30 * q^4 - 28 * q^7	\( 28 q + 30 q^{4} - 28 q^{7} + 4 q^{10} + 8 q^{13} + 42 q^{16} + 34 q^{19} - 10 q^{22} + 14 q^{25} - 30 q^{28} + 56 q^{31} - 6 q^{37} + 38 q^{40} + 18 q^{43} + 16 q^{46} + 28 q^{49} + 18 q^{52} + 48 q^{55} + 2 q^{58} + 36 q^{61} + 76 q^{64} - 4 q^{70} + 50 q^{73} + 132 q^{76} + 66 q^{79} - 36 q^{82} + 20 q^{85} + 6 q^{88} - 8 q^{91} + 54 q^{94} - 8 q^{97}+O(q^{100}) \) 28 * q + 30 * q^4 - 28 * q^7 + 4 * q^10 + 8 * q^13 + 42 * q^16 + 34 * q^19 - 10 * q^22 + 14 * q^25 - 30 * q^28 + 56 * q^31 - 6 * q^37 + 38 * q^40 + 18 * q^43 + 16 * q^46 + 28 * q^49 + 18 * q^52 + 48 * q^55 + 2 * q^58 + 36 * q^61 + 76 * q^64 - 4 * q^70 + 50 * q^73 + 132 * q^76 + 66 * q^79 - 36 * q^82 + 20 * q^85 + 6 * q^88 - 8 * q^91 + 54 * q^94 - 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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