LMFDB - Embedded newform 8001.2.a.x.1.1

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			1.1
Character	$\chi$	$=$	8001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.37730 q^{2} +3.65154 q^{4} +2.39021 q^{5} +1.00000 q^{7} -3.92621 q^{8} +O(q^{10})$	\(q-2.37730 q^{2} +3.65154 q^{4} +2.39021 q^{5} +1.00000 q^{7} -3.92621 q^{8} -5.68223 q^{10} +1.77927 q^{11} -3.89190 q^{13} -2.37730 q^{14} +2.03069 q^{16} -5.42289 q^{17} +2.10743 q^{19} +8.72794 q^{20} -4.22985 q^{22} +4.43391 q^{23} +0.713082 q^{25} +9.25221 q^{26} +3.65154 q^{28} -5.14774 q^{29} -3.27115 q^{31} +3.02488 q^{32} +12.8918 q^{34} +2.39021 q^{35} -6.54578 q^{37} -5.01000 q^{38} -9.38445 q^{40} -5.69237 q^{41} -1.32983 q^{43} +6.49708 q^{44} -10.5407 q^{46} +1.09568 q^{47} +1.00000 q^{49} -1.69521 q^{50} -14.2114 q^{52} +9.83411 q^{53} +4.25282 q^{55} -3.92621 q^{56} +12.2377 q^{58} +5.41776 q^{59} +7.48664 q^{61} +7.77649 q^{62} -11.2524 q^{64} -9.30244 q^{65} +9.52331 q^{67} -19.8019 q^{68} -5.68223 q^{70} +9.03485 q^{71} -15.3815 q^{73} +15.5613 q^{74} +7.69538 q^{76} +1.77927 q^{77} -10.0332 q^{79} +4.85376 q^{80} +13.5325 q^{82} +13.7503 q^{83} -12.9618 q^{85} +3.16141 q^{86} -6.98579 q^{88} +11.1847 q^{89} -3.89190 q^{91} +16.1906 q^{92} -2.60475 q^{94} +5.03720 q^{95} -15.9249 q^{97} -2.37730 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.37730	−1.68100	−0.840502	−	0.541809i	$-0.817740\pi$
$2$	−2.37730	−1.68100	−0.840502	+	0.541809i	$0.817740\pi$
$3$	0	0
$3$	0	0
$4$	3.65154	1.82577
$5$	2.39021	1.06893	0.534466	−	0.845190i	$-0.320513\pi$
$5$	2.39021	1.06893	0.534466	+	0.845190i	$0.320513\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−3.92621	−1.38813
$9$	0	0
$10$	−5.68223	−1.79688
$11$	1.77927	0.536470	0.268235	−	0.963354i	$-0.413560\pi$
$11$	1.77927	0.536470	0.268235	+	0.963354i	$0.413560\pi$
$12$	0	0
$13$	−3.89190	−1.07942	−0.539710	−	0.841851i	$-0.681466\pi$
$13$	−3.89190	−1.07942	−0.539710	+	0.841851i	$0.681466\pi$
$14$	−2.37730	−0.635360
$15$	0	0
$16$	2.03069	0.507671
$17$	−5.42289	−1.31524	−0.657622	−	0.753348i	$-0.728438\pi$
$17$	−5.42289	−1.31524	−0.657622	+	0.753348i	$0.728438\pi$
$18$	0	0
$19$	2.10743	0.483478	0.241739	−	0.970341i	$-0.422282\pi$
$19$	2.10743	0.483478	0.241739	+	0.970341i	$0.422282\pi$
$20$	8.72794	1.95163
$21$	0	0
$22$	−4.22985	−0.901807
$23$	4.43391	0.924533	0.462267	−	0.886741i	$-0.347036\pi$
$23$	4.43391	0.924533	0.462267	+	0.886741i	$0.347036\pi$
$24$	0	0
$25$	0.713082	0.142616
$26$	9.25221	1.81451
$27$	0	0
$28$	3.65154	0.690077
$29$	−5.14774	−0.955911	−0.477956	−	0.878384i	$-0.658622\pi$
$29$	−5.14774	−0.955911	−0.477956	+	0.878384i	$0.658622\pi$
$30$	0	0
$31$	−3.27115	−0.587515	−0.293758	−	0.955880i	$-0.594906\pi$
$31$	−3.27115	−0.587515	−0.293758	+	0.955880i	$0.594906\pi$
$32$	3.02488	0.534728
$33$	0	0
$34$	12.8918	2.21093
$35$	2.39021	0.404018
$36$	0	0
$37$	−6.54578	−1.07612	−0.538060	−	0.842906i	$-0.680843\pi$
$37$	−6.54578	−1.07612	−0.538060	+	0.842906i	$0.680843\pi$
$38$	−5.01000	−0.812729
$39$	0	0
$40$	−9.38445	−1.48381
$41$	−5.69237	−0.888999	−0.444500	−	0.895779i	$-0.646619\pi$
$41$	−5.69237	−0.888999	−0.444500	+	0.895779i	$0.646619\pi$
$42$	0	0
$43$	−1.32983	−0.202798	−0.101399	−	0.994846i	$-0.532332\pi$
$43$	−1.32983	−0.202798	−0.101399	+	0.994846i	$0.532332\pi$
$44$	6.49708	0.979471
$45$	0	0
$46$	−10.5407	−1.55414
$47$	1.09568	0.159821	0.0799104	−	0.996802i	$-0.474537\pi$
$47$	1.09568	0.159821	0.0799104	+	0.996802i	$0.474537\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−1.69521	−0.239739
$51$	0	0
$52$	−14.2114	−1.97077
$53$	9.83411	1.35082	0.675409	−	0.737443i	$-0.263967\pi$
$53$	9.83411	1.35082	0.675409	+	0.737443i	$0.263967\pi$
$54$	0	0
$55$	4.25282	0.573450
$56$	−3.92621	−0.524662
$57$	0	0
$58$	12.2377	1.60689
$59$	5.41776	0.705332	0.352666	−	0.935749i	$-0.385275\pi$
$59$	5.41776	0.705332	0.352666	+	0.935749i	$0.385275\pi$
$60$	0	0
$61$	7.48664	0.958566	0.479283	−	0.877660i	$-0.340897\pi$
$61$	7.48664	0.958566	0.479283	+	0.877660i	$0.340897\pi$
$62$	7.77649	0.987615
$63$	0	0
$64$	−11.2524	−1.40655
$65$	−9.30244	−1.15383
$66$	0	0
$67$	9.52331	1.16346	0.581729	−	0.813383i	$-0.302377\pi$
$67$	9.52331	1.16346	0.581729	+	0.813383i	$0.302377\pi$
$68$	−19.8019	−2.40134
$69$	0	0
$70$	−5.68223	−0.679156
$71$	9.03485	1.07224	0.536119	−	0.844142i	$-0.319890\pi$
$71$	9.03485	1.07224	0.536119	+	0.844142i	$0.319890\pi$
$72$	0	0
$73$	−15.3815	−1.80027	−0.900133	−	0.435615i	$-0.856531\pi$
$73$	−15.3815	−1.80027	−0.900133	+	0.435615i	$0.856531\pi$
$74$	15.5613	1.80896
$75$	0	0
$76$	7.69538	0.882721
$77$	1.77927	0.202766
$78$	0	0
$79$	−10.0332	−1.12882	−0.564409	−	0.825495i	$-0.690896\pi$
$79$	−10.0332	−1.12882	−0.564409	+	0.825495i	$0.690896\pi$
$80$	4.85376	0.542666
$81$	0	0
$82$	13.5325	1.49441
$83$	13.7503	1.50929	0.754646	−	0.656132i	$-0.227809\pi$
$83$	13.7503	1.50929	0.754646	+	0.656132i	$0.227809\pi$
$84$	0	0
$85$	−12.9618	−1.40591
$86$	3.16141	0.340903
$87$	0	0
$88$	−6.98579	−0.744687
$89$	11.1847	1.18558	0.592791	−	0.805357i	$-0.298026\pi$
$89$	11.1847	1.18558	0.592791	+	0.805357i	$0.298026\pi$
$90$	0	0
$91$	−3.89190	−0.407982
$92$	16.1906	1.68799
$93$	0	0
$94$	−2.60475	−0.268659
$95$	5.03720	0.516806
$96$	0	0
$97$	−15.9249	−1.61693	−0.808465	−	0.588544i	$-0.799701\pi$
$97$	−15.9249	−1.61693	−0.808465	+	0.588544i	$0.799701\pi$
$98$	−2.37730	−0.240143
$99$	0	0
$100$	2.60385	0.260385
$101$	−13.6181	−1.35506	−0.677528	−	0.735497i	$-0.736949\pi$
$101$	−13.6181	−1.35506	−0.677528	+	0.735497i	$0.736949\pi$
$102$	0	0
$103$	17.9168	1.76540	0.882699	−	0.469938i	$-0.155724\pi$
$103$	17.9168	1.76540	0.882699	+	0.469938i	$0.155724\pi$
$104$	15.2804	1.49837
$105$	0	0
$106$	−23.3786	−2.27073
$107$	13.0629	1.26284	0.631418	−	0.775443i	$-0.282473\pi$
$107$	13.0629	1.26284	0.631418	+	0.775443i	$0.282473\pi$
$108$	0	0
$109$	−16.4832	−1.57881	−0.789404	−	0.613874i	$-0.789610\pi$
$109$	−16.4832	−1.57881	−0.789404	+	0.613874i	$0.789610\pi$
$110$	−10.1102	−0.963971
$111$	0	0
$112$	2.03069	0.191882
$113$	9.03932	0.850348	0.425174	−	0.905112i	$-0.360213\pi$
$113$	9.03932	0.850348	0.425174	+	0.905112i	$0.360213\pi$
$114$	0	0
$115$	10.5979	0.988263
$116$	−18.7972	−1.74528
$117$	0	0
$118$	−12.8796	−1.18567
$119$	−5.42289	−0.497116
$120$	0	0
$121$	−7.83420	−0.712200
$122$	−17.7980	−1.61135
$123$	0	0
$124$	−11.9447	−1.07267
$125$	−10.2466	−0.916485
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	20.7006	1.82969
$129$	0	0
$130$	22.1147	1.93959
$131$	−5.99318	−0.523626	−0.261813	−	0.965119i	$-0.584320\pi$
$131$	−5.99318	−0.523626	−0.261813	+	0.965119i	$0.584320\pi$
$132$	0	0
$133$	2.10743	0.182738
$134$	−22.6397	−1.95578
$135$	0	0
$136$	21.2914	1.82572
$137$	−11.2954	−0.965028	−0.482514	−	0.875888i	$-0.660276\pi$
$137$	−11.2954	−0.965028	−0.482514	+	0.875888i	$0.660276\pi$
$138$	0	0
$139$	−4.25689	−0.361065	−0.180532	−	0.983569i	$-0.557782\pi$
$139$	−4.25689	−0.361065	−0.180532	+	0.983569i	$0.557782\pi$
$140$	8.72794	0.737646
$141$	0	0
$142$	−21.4785	−1.80244
$143$	−6.92474	−0.579076
$144$	0	0
$145$	−12.3042	−1.02180
$146$	36.5664	3.02625
$147$	0	0
$148$	−23.9022	−1.96475
$149$	−19.2370	−1.57596	−0.787978	−	0.615703i	$-0.788872\pi$
$149$	−19.2370	−1.57596	−0.787978	+	0.615703i	$0.788872\pi$
$150$	0	0
$151$	−22.7475	−1.85116	−0.925581	−	0.378549i	$-0.876423\pi$
$151$	−22.7475	−1.85116	−0.925581	+	0.378549i	$0.876423\pi$
$152$	−8.27423	−0.671129
$153$	0	0
$154$	−4.22985	−0.340851
$155$	−7.81871	−0.628014
$156$	0	0
$157$	3.85447	0.307620	0.153810	−	0.988100i	$-0.450846\pi$
$157$	3.85447	0.307620	0.153810	+	0.988100i	$0.450846\pi$
$158$	23.8518	1.89755
$159$	0	0
$160$	7.23008	0.571588
$161$	4.43391	0.349441
$162$	0	0
$163$	0.845269	0.0662066	0.0331033	−	0.999452i	$-0.489461\pi$
$163$	0.845269	0.0662066	0.0331033	+	0.999452i	$0.489461\pi$
$164$	−20.7860	−1.62311
$165$	0	0
$166$	−32.6886	−2.53712
$167$	−5.67362	−0.439038	−0.219519	−	0.975608i	$-0.570449\pi$
$167$	−5.67362	−0.439038	−0.219519	+	0.975608i	$0.570449\pi$
$168$	0	0
$169$	2.14689	0.165146
$170$	30.8141	2.36333
$171$	0	0
$172$	−4.85594	−0.370262
$173$	−9.82536	−0.747008	−0.373504	−	0.927629i	$-0.621844\pi$
$173$	−9.82536	−0.747008	−0.373504	+	0.927629i	$0.621844\pi$
$174$	0	0
$175$	0.713082	0.0539039
$176$	3.61314	0.272350
$177$	0	0
$178$	−26.5895	−1.99297
$179$	−8.10246	−0.605606	−0.302803	−	0.953053i	$-0.597922\pi$
$179$	−8.10246	−0.605606	−0.302803	+	0.953053i	$0.597922\pi$
$180$	0	0
$181$	−3.47991	−0.258660	−0.129330	−	0.991602i	$-0.541283\pi$
$181$	−3.47991	−0.258660	−0.129330	+	0.991602i	$0.541283\pi$
$182$	9.25221	0.685819
$183$	0	0
$184$	−17.4085	−1.28337
$185$	−15.6458	−1.15030
$186$	0	0
$187$	−9.64878	−0.705589
$188$	4.00091	0.291796
$189$	0	0
$190$	−11.9749	−0.868752
$191$	−3.14356	−0.227460	−0.113730	−	0.993512i	$-0.536280\pi$
$191$	−3.14356	−0.227460	−0.113730	+	0.993512i	$0.536280\pi$
$192$	0	0
$193$	15.0395	1.08257	0.541283	−	0.840840i	$-0.317939\pi$
$193$	15.0395	1.08257	0.541283	+	0.840840i	$0.317939\pi$
$194$	37.8583	2.71806
$195$	0	0
$196$	3.65154	0.260825
$197$	−10.8281	−0.771470	−0.385735	−	0.922610i	$-0.626052\pi$
$197$	−10.8281	−0.771470	−0.385735	+	0.922610i	$0.626052\pi$
$198$	0	0
$199$	5.16261	0.365968	0.182984	−	0.983116i	$-0.441424\pi$
$199$	5.16261	0.365968	0.182984	+	0.983116i	$0.441424\pi$
$200$	−2.79971	−0.197969
$201$	0	0
$202$	32.3744	2.27785
$203$	−5.14774	−0.361301
$204$	0	0
$205$	−13.6059	−0.950280
$206$	−42.5937	−2.96764
$207$	0	0
$208$	−7.90323	−0.547990
$209$	3.74969	0.259371
$210$	0	0
$211$	−11.1113	−0.764933	−0.382467	−	0.923969i	$-0.624925\pi$
$211$	−11.1113	−0.764933	−0.382467	+	0.923969i	$0.624925\pi$
$212$	35.9097	2.46629
$213$	0	0
$214$	−31.0543	−2.12283
$215$	−3.17857	−0.216777
$216$	0	0
$217$	−3.27115	−0.222060
$218$	39.1856	2.65398
$219$	0	0
$220$	15.5294	1.04699
$221$	21.1054	1.41970
$222$	0	0
$223$	−14.3331	−0.959818	−0.479909	−	0.877318i	$-0.659330\pi$
$223$	−14.3331	−0.959818	−0.479909	+	0.877318i	$0.659330\pi$
$224$	3.02488	0.202108
$225$	0	0
$226$	−21.4892	−1.42944
$227$	12.4571	0.826810	0.413405	−	0.910547i	$-0.364339\pi$
$227$	12.4571	0.826810	0.413405	+	0.910547i	$0.364339\pi$
$228$	0	0
$229$	5.89683	0.389673	0.194837	−	0.980836i	$-0.437582\pi$
$229$	5.89683	0.389673	0.194837	+	0.980836i	$0.437582\pi$
$230$	−25.1945	−1.66127
$231$	0	0
$232$	20.2111	1.32692
$233$	−18.8741	−1.23648	−0.618241	−	0.785989i	$-0.712154\pi$
$233$	−18.8741	−1.23648	−0.618241	+	0.785989i	$0.712154\pi$
$234$	0	0
$235$	2.61889	0.170838
$236$	19.7832	1.28778
$237$	0	0
$238$	12.8918	0.835653
$239$	−0.470676	−0.0304455	−0.0152228	−	0.999884i	$-0.504846\pi$
$239$	−0.470676	−0.0304455	−0.0152228	+	0.999884i	$0.504846\pi$
$240$	0	0
$241$	18.4826	1.19057	0.595283	−	0.803516i	$-0.297040\pi$
$241$	18.4826	1.19057	0.595283	+	0.803516i	$0.297040\pi$
$242$	18.6242	1.19721
$243$	0	0
$244$	27.3378	1.75012
$245$	2.39021	0.152705
$246$	0	0
$247$	−8.20192	−0.521876
$248$	12.8432	0.815545
$249$	0	0
$250$	24.3593	1.54061
$251$	−17.2370	−1.08799	−0.543995	−	0.839088i	$-0.683089\pi$
$251$	−17.2370	−1.08799	−0.543995	+	0.839088i	$0.683089\pi$
$252$	0	0
$253$	7.88911	0.495984
$254$	−2.37730	−0.149165
$255$	0	0
$256$	−26.7066	−1.66916
$257$	−18.1864	−1.13444	−0.567218	−	0.823568i	$-0.691980\pi$
$257$	−18.1864	−1.13444	−0.567218	+	0.823568i	$0.691980\pi$
$258$	0	0
$259$	−6.54578	−0.406735
$260$	−33.9683	−2.10662
$261$	0	0
$262$	14.2476	0.880218
$263$	−1.66315	−0.102554	−0.0512772	−	0.998684i	$-0.516329\pi$
$263$	−1.66315	−0.102554	−0.0512772	+	0.998684i	$0.516329\pi$
$264$	0	0
$265$	23.5055	1.44393
$266$	−5.01000	−0.307183
$267$	0	0
$268$	34.7748	2.12421
$269$	−25.9370	−1.58140	−0.790702	−	0.612201i	$-0.790284\pi$
$269$	−25.9370	−1.58140	−0.790702	+	0.612201i	$0.790284\pi$
$270$	0	0
$271$	−23.5372	−1.42979	−0.714893	−	0.699234i	$-0.753525\pi$
$271$	−23.5372	−1.42979	−0.714893	+	0.699234i	$0.753525\pi$
$272$	−11.0122	−0.667712
$273$	0	0
$274$	26.8524	1.62222
$275$	1.26876	0.0765093
$276$	0	0
$277$	−17.7524	−1.06664	−0.533318	−	0.845915i	$-0.679055\pi$
$277$	−17.7524	−1.06664	−0.533318	+	0.845915i	$0.679055\pi$
$278$	10.1199	0.606951
$279$	0	0
$280$	−9.38445	−0.560828
$281$	6.25116	0.372913	0.186456	−	0.982463i	$-0.440300\pi$
$281$	6.25116	0.372913	0.186456	+	0.982463i	$0.440300\pi$
$282$	0	0
$283$	−28.0366	−1.66660	−0.833301	−	0.552819i	$-0.813552\pi$
$283$	−28.0366	−1.66660	−0.833301	+	0.552819i	$0.813552\pi$
$284$	32.9911	1.95766
$285$	0	0
$286$	16.4622	0.973428
$287$	−5.69237	−0.336010
$288$	0	0
$289$	12.4077	0.729868
$290$	29.2506	1.71766
$291$	0	0
$292$	−56.1662	−3.28688
$293$	10.7284	0.626758	0.313379	−	0.949628i	$-0.398539\pi$
$293$	10.7284	0.626758	0.313379	+	0.949628i	$0.398539\pi$
$294$	0	0
$295$	12.9496	0.753952
$296$	25.7001	1.49379
$297$	0	0
$298$	45.7321	2.64919
$299$	−17.2563	−0.997959
$300$	0	0
$301$	−1.32983	−0.0766503
$302$	54.0775	3.11181
$303$	0	0
$304$	4.27953	0.245448
$305$	17.8946	1.02464
$306$	0	0
$307$	27.1249	1.54810	0.774050	−	0.633125i	$-0.218228\pi$
$307$	27.1249	1.54810	0.774050	+	0.633125i	$0.218228\pi$
$308$	6.49708	0.370205
$309$	0	0
$310$	18.5874	1.05569
$311$	34.1691	1.93755	0.968775	−	0.247942i	$-0.0797542\pi$
$311$	34.1691	1.93755	0.968775	+	0.247942i	$0.0797542\pi$
$312$	0	0
$313$	28.2787	1.59841	0.799204	−	0.601060i	$-0.205255\pi$
$313$	28.2787	1.59841	0.799204	+	0.601060i	$0.205255\pi$
$314$	−9.16322	−0.517111
$315$	0	0
$316$	−36.6365	−2.06097
$317$	16.7315	0.939737	0.469869	−	0.882736i	$-0.344301\pi$
$317$	16.7315	0.939737	0.469869	+	0.882736i	$0.344301\pi$
$318$	0	0
$319$	−9.15921	−0.512817
$320$	−26.8956	−1.50351
$321$	0	0
$322$	−10.5407	−0.587411
$323$	−11.4284	−0.635892
$324$	0	0
$325$	−2.77524	−0.153943
$326$	−2.00946	−0.111293
$327$	0	0
$328$	22.3495	1.23404
$329$	1.09568	0.0604066
$330$	0	0
$331$	−25.7958	−1.41787	−0.708933	−	0.705276i	$-0.750823\pi$
$331$	−25.7958	−1.41787	−0.708933	+	0.705276i	$0.750823\pi$
$332$	50.2098	2.75562
$333$	0	0
$334$	13.4879	0.738024
$335$	22.7627	1.24366
$336$	0	0
$337$	−27.9470	−1.52237	−0.761185	−	0.648535i	$-0.775382\pi$
$337$	−27.9470	−1.52237	−0.761185	+	0.648535i	$0.775382\pi$
$338$	−5.10380	−0.277610
$339$	0	0
$340$	−47.3307	−2.56687
$341$	−5.82025	−0.315184
$342$	0	0
$343$	1.00000	0.0539949
$344$	5.22121	0.281509
$345$	0	0
$346$	23.3578	1.25572
$347$	−10.6375	−0.571050	−0.285525	−	0.958371i	$-0.592168\pi$
$347$	−10.6375	−0.571050	−0.285525	+	0.958371i	$0.592168\pi$
$348$	0	0
$349$	6.67075	0.357077	0.178539	−	0.983933i	$-0.442863\pi$
$349$	6.67075	0.357077	0.178539	+	0.983933i	$0.442863\pi$
$350$	−1.69521	−0.0906126
$351$	0	0
$352$	5.38207	0.286865
$353$	18.3993	0.979295	0.489648	−	0.871920i	$-0.337125\pi$
$353$	18.3993	0.979295	0.489648	+	0.871920i	$0.337125\pi$
$354$	0	0
$355$	21.5951	1.14615
$356$	40.8416	2.16460
$357$	0	0
$358$	19.2620	1.01803
$359$	22.0277	1.16258	0.581289	−	0.813697i	$-0.302549\pi$
$359$	22.0277	1.16258	0.581289	+	0.813697i	$0.302549\pi$
$360$	0	0
$361$	−14.5587	−0.766249
$362$	8.27278	0.434808
$363$	0	0
$364$	−14.2114	−0.744882
$365$	−36.7649	−1.92436
$366$	0	0
$367$	25.6671	1.33981	0.669907	−	0.742445i	$-0.266334\pi$
$367$	25.6671	1.33981	0.669907	+	0.742445i	$0.266334\pi$
$368$	9.00387	0.469359
$369$	0	0
$370$	37.1946	1.93366
$371$	9.83411	0.510561
$372$	0	0
$373$	21.3357	1.10472	0.552360	−	0.833605i	$-0.313727\pi$
$373$	21.3357	1.10472	0.552360	+	0.833605i	$0.313727\pi$
$374$	22.9380	1.18610
$375$	0	0
$376$	−4.30185	−0.221851
$377$	20.0345	1.03183
$378$	0	0
$379$	−24.8485	−1.27638	−0.638191	−	0.769878i	$-0.720317\pi$
$379$	−24.8485	−1.27638	−0.638191	+	0.769878i	$0.720317\pi$
$380$	18.3935	0.943569
$381$	0	0
$382$	7.47317	0.382361
$383$	−24.8695	−1.27077	−0.635386	−	0.772195i	$-0.719159\pi$
$383$	−24.8695	−1.27077	−0.635386	+	0.772195i	$0.719159\pi$
$384$	0	0
$385$	4.25282	0.216744
$386$	−35.7534	−1.81980
$387$	0	0
$388$	−58.1505	−2.95215
$389$	−21.5250	−1.09136	−0.545679	−	0.837994i	$-0.683728\pi$
$389$	−21.5250	−1.09136	−0.545679	+	0.837994i	$0.683728\pi$
$390$	0	0
$391$	−24.0446	−1.21599
$392$	−3.92621	−0.198304
$393$	0	0
$394$	25.7416	1.29684
$395$	−23.9813	−1.20663
$396$	0	0
$397$	−28.0977	−1.41018	−0.705091	−	0.709116i	$-0.749094\pi$
$397$	−28.0977	−1.41018	−0.705091	+	0.709116i	$0.749094\pi$
$398$	−12.2731	−0.615193
$399$	0	0
$400$	1.44804	0.0724022
$401$	−10.4875	−0.523723	−0.261862	−	0.965105i	$-0.584336\pi$
$401$	−10.4875	−0.523723	−0.261862	+	0.965105i	$0.584336\pi$
$402$	0	0
$403$	12.7310	0.634175
$404$	−49.7272	−2.47402
$405$	0	0
$406$	12.2377	0.607347
$407$	−11.6467	−0.577306
$408$	0	0
$409$	16.1874	0.800415	0.400207	−	0.916425i	$-0.368938\pi$
$409$	16.1874	0.800415	0.400207	+	0.916425i	$0.368938\pi$
$410$	32.3454	1.59742
$411$	0	0
$412$	65.4241	3.22322
$413$	5.41776	0.266590
$414$	0	0
$415$	32.8661	1.61333
$416$	−11.7725	−0.577196
$417$	0	0
$418$	−8.91413	−0.436004
$419$	8.77126	0.428504	0.214252	−	0.976778i	$-0.431269\pi$
$419$	8.77126	0.428504	0.214252	+	0.976778i	$0.431269\pi$
$420$	0	0
$421$	−14.9972	−0.730921	−0.365460	−	0.930827i	$-0.619088\pi$
$421$	−14.9972	−0.730921	−0.365460	+	0.930827i	$0.619088\pi$
$422$	26.4149	1.28585
$423$	0	0
$424$	−38.6108	−1.87511
$425$	−3.86696	−0.187575
$426$	0	0
$427$	7.48664	0.362304
$428$	47.6997	2.30565
$429$	0	0
$430$	7.55642	0.364403
$431$	4.98369	0.240056	0.120028	−	0.992771i	$-0.461702\pi$
$431$	4.98369	0.240056	0.120028	+	0.992771i	$0.461702\pi$
$432$	0	0
$433$	7.45889	0.358451	0.179226	−	0.983808i	$-0.442641\pi$
$433$	7.45889	0.358451	0.179226	+	0.983808i	$0.442641\pi$
$434$	7.77649	0.373283
$435$	0	0
$436$	−60.1893	−2.88254
$437$	9.34416	0.446992
$438$	0	0
$439$	5.71363	0.272696	0.136348	−	0.990661i	$-0.456463\pi$
$439$	5.71363	0.272696	0.136348	+	0.990661i	$0.456463\pi$
$440$	−16.6975	−0.796020
$441$	0	0
$442$	−50.1737	−2.38652
$443$	−25.5248	−1.21272	−0.606359	−	0.795191i	$-0.707371\pi$
$443$	−25.5248	−1.21272	−0.606359	+	0.795191i	$0.707371\pi$
$444$	0	0
$445$	26.7338	1.26731
$446$	34.0741	1.61346
$447$	0	0
$448$	−11.2524	−0.531626
$449$	−21.5723	−1.01806	−0.509031	−	0.860748i	$-0.669996\pi$
$449$	−21.5723	−1.01806	−0.509031	+	0.860748i	$0.669996\pi$
$450$	0	0
$451$	−10.1283	−0.476921
$452$	33.0075	1.55254
$453$	0	0
$454$	−29.6143	−1.38987
$455$	−9.30244	−0.436105
$456$	0	0
$457$	−14.7294	−0.689011	−0.344506	−	0.938784i	$-0.611953\pi$
$457$	−14.7294	−0.689011	−0.344506	+	0.938784i	$0.611953\pi$
$458$	−14.0185	−0.655042
$459$	0	0
$460$	38.6989	1.80434
$461$	−11.1473	−0.519183	−0.259592	−	0.965718i	$-0.583588\pi$
$461$	−11.1473	−0.519183	−0.259592	+	0.965718i	$0.583588\pi$
$462$	0	0
$463$	22.6768	1.05388	0.526939	−	0.849903i	$-0.323340\pi$
$463$	22.6768	1.05388	0.526939	+	0.849903i	$0.323340\pi$
$464$	−10.4534	−0.485289
$465$	0	0
$466$	44.8693	2.07853
$467$	−41.6989	−1.92960	−0.964798	−	0.262990i	$-0.915291\pi$
$467$	−41.6989	−1.92960	−0.964798	+	0.262990i	$0.915291\pi$
$468$	0	0
$469$	9.52331	0.439746
$470$	−6.22588	−0.287178
$471$	0	0
$472$	−21.2713	−0.979089
$473$	−2.36613	−0.108795
$474$	0	0
$475$	1.50277	0.0689519
$476$	−19.8019	−0.907620
$477$	0	0
$478$	1.11894	0.0511790
$479$	−1.44457	−0.0660041	−0.0330021	−	0.999455i	$-0.510507\pi$
$479$	−1.44457	−0.0660041	−0.0330021	+	0.999455i	$0.510507\pi$
$480$	0	0
$481$	25.4755	1.16158
$482$	−43.9386	−2.00135
$483$	0	0
$484$	−28.6069	−1.30032
$485$	−38.0638	−1.72839
$486$	0	0
$487$	19.1111	0.866009	0.433004	−	0.901392i	$-0.357453\pi$
$487$	19.1111	0.866009	0.433004	+	0.901392i	$0.357453\pi$
$488$	−29.3941	−1.33061
$489$	0	0
$490$	−5.68223	−0.256697
$491$	−11.9270	−0.538257	−0.269128	−	0.963104i	$-0.586736\pi$
$491$	−11.9270	−0.538257	−0.269128	+	0.963104i	$0.586736\pi$
$492$	0	0
$493$	27.9156	1.25726
$494$	19.4984	0.877275
$495$	0	0
$496$	−6.64267	−0.298265
$497$	9.03485	0.405268
$498$	0	0
$499$	7.37097	0.329970	0.164985	−	0.986296i	$-0.447242\pi$
$499$	7.37097	0.329970	0.164985	+	0.986296i	$0.447242\pi$
$500$	−37.4160	−1.67329
$501$	0	0
$502$	40.9775	1.82892
$503$	−11.6390	−0.518959	−0.259479	−	0.965749i	$-0.583551\pi$
$503$	−11.6390	−0.518959	−0.259479	+	0.965749i	$0.583551\pi$
$504$	0	0
$505$	−32.5501	−1.44846
$506$	−18.7548	−0.833751
$507$	0	0
$508$	3.65154	0.162011
$509$	−2.73858	−0.121385	−0.0606927	−	0.998156i	$-0.519331\pi$
$509$	−2.73858	−0.121385	−0.0606927	+	0.998156i	$0.519331\pi$
$510$	0	0
$511$	−15.3815	−0.680437
$512$	22.0884	0.976178
$513$	0	0
$514$	43.2345	1.90699
$515$	42.8249	1.88709
$516$	0	0
$517$	1.94950	0.0857390
$518$	15.5613	0.683723
$519$	0	0
$520$	36.5234	1.60166
$521$	34.8802	1.52813	0.764065	−	0.645139i	$-0.223201\pi$
$521$	34.8802	1.52813	0.764065	+	0.645139i	$0.223201\pi$
$522$	0	0
$523$	10.0250	0.438363	0.219181	−	0.975684i	$-0.429661\pi$
$523$	10.0250	0.438363	0.219181	+	0.975684i	$0.429661\pi$
$524$	−21.8843	−0.956022
$525$	0	0
$526$	3.95381	0.172394
$527$	17.7391	0.772726
$528$	0	0
$529$	−3.34048	−0.145238
$530$	−55.8796	−2.42726
$531$	0	0
$532$	7.69538	0.333637
$533$	22.1542	0.959603
$534$	0	0
$535$	31.2230	1.34989
$536$	−37.3905	−1.61503
$537$	0	0
$538$	61.6598	2.65834
$539$	1.77927	0.0766385
$540$	0	0
$541$	−40.2719	−1.73143	−0.865713	−	0.500541i	$-0.833134\pi$
$541$	−40.2719	−1.73143	−0.865713	+	0.500541i	$0.833134\pi$
$542$	55.9550	2.40347
$543$	0	0
$544$	−16.4036	−0.703298
$545$	−39.3983	−1.68764
$546$	0	0
$547$	20.6434	0.882646	0.441323	−	0.897348i	$-0.354509\pi$
$547$	20.6434	0.882646	0.441323	+	0.897348i	$0.354509\pi$
$548$	−41.2455	−1.76192
$549$	0	0
$550$	−3.01623	−0.128612
$551$	−10.8485	−0.462162
$552$	0	0
$553$	−10.0332	−0.426653
$554$	42.2026	1.79302
$555$	0	0
$556$	−15.5442	−0.659222
$557$	21.7984	0.923627	0.461813	−	0.886977i	$-0.347199\pi$
$557$	21.7984	0.923627	0.461813	+	0.886977i	$0.347199\pi$
$558$	0	0
$559$	5.17558	0.218904
$560$	4.85376	0.205109
$561$	0	0
$562$	−14.8609	−0.626868
$563$	−6.59466	−0.277932	−0.138966	−	0.990297i	$-0.544378\pi$
$563$	−6.59466	−0.277932	−0.138966	+	0.990297i	$0.544378\pi$
$564$	0	0
$565$	21.6058	0.908964
$566$	66.6513	2.80156
$567$	0	0
$568$	−35.4727	−1.48840
$569$	34.0392	1.42700	0.713500	−	0.700656i	$-0.247109\pi$
$569$	34.0392	1.42700	0.713500	+	0.700656i	$0.247109\pi$
$570$	0	0
$571$	33.8094	1.41488	0.707439	−	0.706774i	$-0.249850\pi$
$571$	33.8094	1.41488	0.707439	+	0.706774i	$0.249850\pi$
$572$	−25.2860	−1.05726
$573$	0	0
$574$	13.5325	0.564834
$575$	3.16174	0.131854
$576$	0	0
$577$	−21.4795	−0.894202	−0.447101	−	0.894484i	$-0.647544\pi$
$577$	−21.4795	−0.894202	−0.447101	+	0.894484i	$0.647544\pi$
$578$	−29.4969	−1.22691
$579$	0	0
$580$	−44.9292	−1.86558
$581$	13.7503	0.570459
$582$	0	0
$583$	17.4975	0.724673
$584$	60.3910	2.49900
$585$	0	0
$586$	−25.5045	−1.05358
$587$	33.3158	1.37509	0.687545	−	0.726142i	$-0.258689\pi$
$587$	33.3158	1.37509	0.687545	+	0.726142i	$0.258689\pi$
$588$	0	0
$589$	−6.89372	−0.284051
$590$	−30.7849	−1.26740
$591$	0	0
$592$	−13.2924	−0.546315
$593$	0.745797	0.0306262	0.0153131	−	0.999883i	$-0.495125\pi$
$593$	0.745797	0.0306262	0.0153131	+	0.999883i	$0.495125\pi$
$594$	0	0
$595$	−12.9618	−0.531383
$596$	−70.2447	−2.87734
$597$	0	0
$598$	41.0234	1.67757
$599$	−37.8832	−1.54787	−0.773933	−	0.633267i	$-0.781713\pi$
$599$	−37.8832	−1.54787	−0.773933	+	0.633267i	$0.781713\pi$
$600$	0	0
$601$	42.7271	1.74287	0.871437	−	0.490508i	$-0.163189\pi$
$601$	42.7271	1.74287	0.871437	+	0.490508i	$0.163189\pi$
$602$	3.16141	0.128849
$603$	0	0
$604$	−83.0634	−3.37980
$605$	−18.7254	−0.761294
$606$	0	0
$607$	12.4704	0.506158	0.253079	−	0.967446i	$-0.418557\pi$
$607$	12.4704	0.506158	0.253079	+	0.967446i	$0.418557\pi$
$608$	6.37473	0.258529
$609$	0	0
$610$	−42.5408	−1.72243
$611$	−4.26426	−0.172514
$612$	0	0
$613$	17.6588	0.713232	0.356616	−	0.934251i	$-0.383930\pi$
$613$	17.6588	0.713232	0.356616	+	0.934251i	$0.383930\pi$
$614$	−64.4839	−2.60236
$615$	0	0
$616$	−6.98579	−0.281465
$617$	−42.3218	−1.70381	−0.851907	−	0.523694i	$-0.824554\pi$
$617$	−42.3218	−1.70381	−0.851907	+	0.523694i	$0.824554\pi$
$618$	0	0
$619$	32.8929	1.32208	0.661039	−	0.750352i	$-0.270116\pi$
$619$	32.8929	1.32208	0.661039	+	0.750352i	$0.270116\pi$
$620$	−28.5504	−1.14661
$621$	0	0
$622$	−81.2300	−3.25703
$623$	11.1847	0.448108
$624$	0	0
$625$	−28.0569	−1.12228
$626$	−67.2269	−2.68693
$627$	0	0
$628$	14.0748	0.561644
$629$	35.4971	1.41536
$630$	0	0
$631$	−14.9475	−0.595049	−0.297524	−	0.954714i	$-0.596161\pi$
$631$	−14.9475	−0.595049	−0.297524	+	0.954714i	$0.596161\pi$
$632$	39.3923	1.56694
$633$	0	0
$634$	−39.7759	−1.57970
$635$	2.39021	0.0948524
$636$	0	0
$637$	−3.89190	−0.154203
$638$	21.7742	0.862048
$639$	0	0
$640$	49.4786	1.95581
$641$	19.9028	0.786115	0.393057	−	0.919514i	$-0.371417\pi$
$641$	19.9028	0.786115	0.393057	+	0.919514i	$0.371417\pi$
$642$	0	0
$643$	23.6127	0.931195	0.465597	−	0.884997i	$-0.345839\pi$
$643$	23.6127	0.931195	0.465597	+	0.884997i	$0.345839\pi$
$644$	16.1906	0.637999
$645$	0	0
$646$	27.1687	1.06894
$647$	47.1230	1.85260	0.926298	−	0.376792i	$-0.122973\pi$
$647$	47.1230	1.85260	0.926298	+	0.376792i	$0.122973\pi$
$648$	0	0
$649$	9.63965	0.378389
$650$	6.59758	0.258778
$651$	0	0
$652$	3.08654	0.120878
$653$	−46.5868	−1.82308	−0.911541	−	0.411208i	$-0.865107\pi$
$653$	−46.5868	−1.82308	−0.911541	+	0.411208i	$0.865107\pi$
$654$	0	0
$655$	−14.3249	−0.559721
$656$	−11.5594	−0.451320
$657$	0	0
$658$	−2.60475	−0.101544
$659$	10.1687	0.396116	0.198058	−	0.980190i	$-0.436537\pi$
$659$	10.1687	0.396116	0.198058	+	0.980190i	$0.436537\pi$
$660$	0	0
$661$	−4.13865	−0.160975	−0.0804874	−	0.996756i	$-0.525648\pi$
$661$	−4.13865	−0.160975	−0.0804874	+	0.996756i	$0.525648\pi$
$662$	61.3243	2.38344
$663$	0	0
$664$	−53.9866	−2.09509
$665$	5.03720	0.195334
$666$	0	0
$667$	−22.8246	−0.883772
$668$	−20.7175	−0.801583
$669$	0	0
$670$	−54.1136	−2.09059
$671$	13.3207	0.514241
$672$	0	0
$673$	−30.3075	−1.16827	−0.584133	−	0.811658i	$-0.698565\pi$
$673$	−30.3075	−1.16827	−0.584133	+	0.811658i	$0.698565\pi$
$674$	66.4383	2.55911
$675$	0	0
$676$	7.83947	0.301518
$677$	24.2290	0.931195	0.465598	−	0.884997i	$-0.345839\pi$
$677$	24.2290	0.931195	0.465598	+	0.884997i	$0.345839\pi$
$678$	0	0
$679$	−15.9249	−0.611142
$680$	50.8909	1.95158
$681$	0	0
$682$	13.8365	0.529825
$683$	−30.2938	−1.15916	−0.579579	−	0.814916i	$-0.696783\pi$
$683$	−30.2938	−1.15916	−0.579579	+	0.814916i	$0.696783\pi$
$684$	0	0
$685$	−26.9982	−1.03155
$686$	−2.37730	−0.0907656
$687$	0	0
$688$	−2.70047	−0.102955
$689$	−38.2734	−1.45810
$690$	0	0
$691$	−6.21641	−0.236483	−0.118242	−	0.992985i	$-0.537726\pi$
$691$	−6.21641	−0.236483	−0.118242	+	0.992985i	$0.537726\pi$
$692$	−35.8777	−1.36387
$693$	0	0
$694$	25.2885	0.959937
$695$	−10.1748	−0.385954
$696$	0	0
$697$	30.8691	1.16925
$698$	−15.8584	−0.600248
$699$	0	0
$700$	2.60385	0.0984162
$701$	16.0761	0.607185	0.303593	−	0.952802i	$-0.401814\pi$
$701$	16.0761	0.607185	0.303593	+	0.952802i	$0.401814\pi$
$702$	0	0
$703$	−13.7948	−0.520281
$704$	−20.0211	−0.754572
$705$	0	0
$706$	−43.7406	−1.64620
$707$	−13.6181	−0.512163
$708$	0	0
$709$	−37.2668	−1.39958	−0.699791	−	0.714347i	$-0.746724\pi$
$709$	−37.2668	−1.39958	−0.699791	+	0.714347i	$0.746724\pi$
$710$	−51.3381	−1.92668
$711$	0	0
$712$	−43.9137	−1.64574
$713$	−14.5039	−0.543177
$714$	0	0
$715$	−16.5515	−0.618993
$716$	−29.5865	−1.10570
$717$	0	0
$718$	−52.3664	−1.95430
$719$	−10.3482	−0.385924	−0.192962	−	0.981206i	$-0.561809\pi$
$719$	−10.3482	−0.385924	−0.192962	+	0.981206i	$0.561809\pi$
$720$	0	0
$721$	17.9168	0.667258
$722$	34.6104	1.28807
$723$	0	0
$724$	−12.7070	−0.472254
$725$	−3.67076	−0.136329
$726$	0	0
$727$	−19.5143	−0.723744	−0.361872	−	0.932228i	$-0.617862\pi$
$727$	−19.5143	−0.723744	−0.361872	+	0.932228i	$0.617862\pi$
$728$	15.2804	0.566330
$729$	0	0
$730$	87.4011	3.23486
$731$	7.21154	0.266728
$732$	0	0
$733$	−6.21475	−0.229547	−0.114774	−	0.993392i	$-0.536614\pi$
$733$	−6.21475	−0.229547	−0.114774	+	0.993392i	$0.536614\pi$
$734$	−61.0184	−2.25223
$735$	0	0
$736$	13.4120	0.494374
$737$	16.9445	0.624160
$738$	0	0
$739$	38.5175	1.41689	0.708444	−	0.705767i	$-0.249398\pi$
$739$	38.5175	1.41689	0.708444	+	0.705767i	$0.249398\pi$
$740$	−57.1312	−2.10019
$741$	0	0
$742$	−23.3786	−0.858255
$743$	−12.1792	−0.446813	−0.223406	−	0.974725i	$-0.571718\pi$
$743$	−12.1792	−0.446813	−0.223406	+	0.974725i	$0.571718\pi$
$744$	0	0
$745$	−45.9804	−1.68459
$746$	−50.7213	−1.85704
$747$	0	0
$748$	−35.2329	−1.28824
$749$	13.0629	0.477307
$750$	0	0
$751$	−52.9831	−1.93338	−0.966691	−	0.255947i	$-0.917613\pi$
$751$	−52.9831	−1.93338	−0.966691	+	0.255947i	$0.917613\pi$
$752$	2.22497	0.0811364
$753$	0	0
$754$	−47.6280	−1.73451
$755$	−54.3711	−1.97877
$756$	0	0
$757$	−18.0503	−0.656048	−0.328024	−	0.944669i	$-0.606383\pi$
$757$	−18.0503	−0.656048	−0.328024	+	0.944669i	$0.606383\pi$
$758$	59.0723	2.14560
$759$	0	0
$760$	−19.7771	−0.717391
$761$	2.93317	0.106327	0.0531637	−	0.998586i	$-0.483069\pi$
$761$	2.93317	0.106327	0.0531637	+	0.998586i	$0.483069\pi$
$762$	0	0
$763$	−16.4832	−0.596733
$764$	−11.4788	−0.415290
$765$	0	0
$766$	59.1222	2.13617
$767$	−21.0854	−0.761349
$768$	0	0
$769$	−42.9825	−1.54999	−0.774994	−	0.631969i	$-0.782247\pi$
$769$	−42.9825	−1.54999	−0.774994	+	0.631969i	$0.782247\pi$
$770$	−10.1102	−0.364347
$771$	0	0
$772$	54.9174	1.97652
$773$	−38.2233	−1.37480	−0.687399	−	0.726280i	$-0.741247\pi$
$773$	−38.2233	−1.37480	−0.687399	+	0.726280i	$0.741247\pi$
$774$	0	0
$775$	−2.33259	−0.0837892
$776$	62.5246	2.24450
$777$	0	0
$778$	51.1712	1.83458
$779$	−11.9963	−0.429812
$780$	0	0
$781$	16.0754	0.575224
$782$	57.1611	2.04408
$783$	0	0
$784$	2.03069	0.0725245
$785$	9.21298	0.328825
$786$	0	0
$787$	−19.1811	−0.683734	−0.341867	−	0.939748i	$-0.611059\pi$
$787$	−19.1811	−0.683734	−0.341867	+	0.939748i	$0.611059\pi$
$788$	−39.5393	−1.40853
$789$	0	0
$790$	57.0107	2.02835
$791$	9.03932	0.321401
$792$	0	0
$793$	−29.1373	−1.03469
$794$	66.7966	2.37052
$795$	0	0
$796$	18.8515	0.668174
$797$	35.0781	1.24253	0.621265	−	0.783600i	$-0.286619\pi$
$797$	35.0781	1.24253	0.621265	+	0.783600i	$0.286619\pi$
$798$	0	0
$799$	−5.94173	−0.210203
$800$	2.15699	0.0762610
$801$	0	0
$802$	24.9320	0.880380
$803$	−27.3678	−0.965788
$804$	0	0
$805$	10.5979	0.373528
$806$	−30.2653	−1.06605
$807$	0	0
$808$	53.4677	1.88099
$809$	−25.3341	−0.890700	−0.445350	−	0.895357i	$-0.646921\pi$
$809$	−25.3341	−0.890700	−0.445350	+	0.895357i	$0.646921\pi$
$810$	0	0
$811$	1.49295	0.0524247	0.0262124	−	0.999656i	$-0.491655\pi$
$811$	1.49295	0.0524247	0.0262124	+	0.999656i	$0.491655\pi$
$812$	−18.7972	−0.659652
$813$	0	0
$814$	27.6877	0.970453
$815$	2.02037	0.0707703
$816$	0	0
$817$	−2.80253	−0.0980483
$818$	−38.4822	−1.34550
$819$	0	0
$820$	−49.6827	−1.73500
$821$	41.6801	1.45465	0.727323	−	0.686295i	$-0.240764\pi$
$821$	41.6801	1.45465	0.727323	+	0.686295i	$0.240764\pi$
$822$	0	0
$823$	29.2314	1.01894	0.509471	−	0.860488i	$-0.329841\pi$
$823$	29.2314	1.01894	0.509471	+	0.860488i	$0.329841\pi$
$824$	−70.3453	−2.45059
$825$	0	0
$826$	−12.8796	−0.448139
$827$	−19.7972	−0.688416	−0.344208	−	0.938893i	$-0.611852\pi$
$827$	−19.7972	−0.688416	−0.344208	+	0.938893i	$0.611852\pi$
$828$	0	0
$829$	−18.1482	−0.630312	−0.315156	−	0.949040i	$-0.602057\pi$
$829$	−18.1482	−0.630312	−0.315156	+	0.949040i	$0.602057\pi$
$830$	−78.1324	−2.71201
$831$	0	0
$832$	43.7933	1.51826
$833$	−5.42289	−0.187892
$834$	0	0
$835$	−13.5611	−0.469302
$836$	13.6922	0.473553
$837$	0	0
$838$	−20.8519	−0.720317
$839$	42.2384	1.45823	0.729116	−	0.684390i	$-0.239931\pi$
$839$	42.2384	1.45823	0.729116	+	0.684390i	$0.239931\pi$
$840$	0	0
$841$	−2.50077	−0.0862334
$842$	35.6529	1.22868
$843$	0	0
$844$	−40.5734	−1.39659
$845$	5.13151	0.176529
$846$	0	0
$847$	−7.83420	−0.269186
$848$	19.9700	0.685772
$849$	0	0
$850$	9.19292	0.315315
$851$	−29.0234	−0.994909
$852$	0	0
$853$	−34.6625	−1.18682	−0.593411	−	0.804899i	$-0.702219\pi$
$853$	−34.6625	−1.18682	−0.593411	+	0.804899i	$0.702219\pi$
$854$	−17.7980	−0.609034
$855$	0	0
$856$	−51.2876	−1.75297
$857$	−15.1342	−0.516973	−0.258487	−	0.966015i	$-0.583224\pi$
$857$	−15.1342	−0.516973	−0.258487	+	0.966015i	$0.583224\pi$
$858$	0	0
$859$	−30.2820	−1.03321	−0.516605	−	0.856224i	$-0.672804\pi$
$859$	−30.2820	−1.03321	−0.516605	+	0.856224i	$0.672804\pi$
$860$	−11.6067	−0.395785
$861$	0	0
$862$	−11.8477	−0.403535
$863$	12.8721	0.438170	0.219085	−	0.975706i	$-0.429693\pi$
$863$	12.8721	0.438170	0.219085	+	0.975706i	$0.429693\pi$
$864$	0	0
$865$	−23.4846	−0.798501
$866$	−17.7320	−0.602558
$867$	0	0
$868$	−11.9447	−0.405431
$869$	−17.8517	−0.605577
$870$	0	0
$871$	−37.0638	−1.25586
$872$	64.7167	2.19158
$873$	0	0
$874$	−22.2138	−0.751395
$875$	−10.2466	−0.346399
$876$	0	0
$877$	38.5065	1.30027	0.650136	−	0.759818i	$-0.274712\pi$
$877$	38.5065	1.30027	0.650136	+	0.759818i	$0.274712\pi$
$878$	−13.5830	−0.458404
$879$	0	0
$880$	8.63613	0.291124
$881$	−28.5535	−0.961990	−0.480995	−	0.876723i	$-0.659725\pi$
$881$	−28.5535	−0.961990	−0.480995	+	0.876723i	$0.659725\pi$
$882$	0	0
$883$	30.1231	1.01372	0.506861	−	0.862028i	$-0.330806\pi$
$883$	30.1231	1.01372	0.506861	+	0.862028i	$0.330806\pi$
$884$	77.0671	2.59205
$885$	0	0
$886$	60.6800	2.03858
$887$	−33.6858	−1.13106	−0.565529	−	0.824728i	$-0.691328\pi$
$887$	−33.6858	−1.13106	−0.565529	+	0.824728i	$0.691328\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	−63.5543	−2.13035
$891$	0	0
$892$	−52.3381	−1.75241
$893$	2.30906	0.0772699
$894$	0	0
$895$	−19.3665	−0.647352
$896$	20.7006	0.691557
$897$	0	0
$898$	51.2839	1.71137
$899$	16.8390	0.561612
$900$	0	0
$901$	−53.3293	−1.77666
$902$	24.0779	0.801706
$903$	0	0
$904$	−35.4903	−1.18039
$905$	−8.31770	−0.276490
$906$	0	0
$907$	22.2244	0.737948	0.368974	−	0.929440i	$-0.379709\pi$
$907$	22.2244	0.737948	0.368974	+	0.929440i	$0.379709\pi$
$908$	45.4878	1.50957
$909$	0	0
$910$	22.1147	0.733094
$911$	39.5309	1.30972	0.654859	−	0.755751i	$-0.272728\pi$
$911$	39.5309	1.30972	0.654859	+	0.755751i	$0.272728\pi$
$912$	0	0
$913$	24.4655	0.809689
$914$	35.0161	1.15823
$915$	0	0
$916$	21.5325	0.711455
$917$	−5.99318	−0.197912
$918$	0	0
$919$	−38.4543	−1.26849	−0.634246	−	0.773131i	$-0.718689\pi$
$919$	−38.4543	−1.26849	−0.634246	+	0.773131i	$0.718689\pi$
$920$	−41.6098	−1.37183
$921$	0	0
$922$	26.5005	0.872749
$923$	−35.1627	−1.15740
$924$	0	0
$925$	−4.66768	−0.153472
$926$	−53.9094	−1.77157
$927$	0	0
$928$	−15.5713	−0.511153
$929$	−36.5743	−1.19996	−0.599981	−	0.800014i	$-0.704825\pi$
$929$	−36.5743	−1.19996	−0.599981	+	0.800014i	$0.704825\pi$
$930$	0	0
$931$	2.10743	0.0690683
$932$	−68.9195	−2.25753
$933$	0	0
$934$	99.1308	3.24366
$935$	−23.0626	−0.754227
$936$	0	0
$937$	28.5437	0.932481	0.466240	−	0.884658i	$-0.345608\pi$
$937$	28.5437	0.932481	0.466240	+	0.884658i	$0.345608\pi$
$938$	−22.6397	−0.739214
$939$	0	0
$940$	9.56299	0.311910
$941$	40.8541	1.33181	0.665903	−	0.746039i	$-0.268047\pi$
$941$	40.8541	1.33181	0.665903	+	0.746039i	$0.268047\pi$
$942$	0	0
$943$	−25.2394	−0.821910
$944$	11.0018	0.358077
$945$	0	0
$946$	5.62500	0.182884
$947$	−48.5027	−1.57612	−0.788062	−	0.615596i	$-0.788915\pi$
$947$	−48.5027	−1.57612	−0.788062	+	0.615596i	$0.788915\pi$
$948$	0	0
$949$	59.8632	1.94324
$950$	−3.57254	−0.115908
$951$	0	0
$952$	21.2914	0.690059
$953$	8.20109	0.265659	0.132830	−	0.991139i	$-0.457594\pi$
$953$	8.20109	0.265659	0.132830	+	0.991139i	$0.457594\pi$
$954$	0	0
$955$	−7.51375	−0.243139
$956$	−1.71869	−0.0555865
$957$	0	0
$958$	3.43418	0.110953
$959$	−11.2954	−0.364746
$960$	0	0
$961$	−20.2996	−0.654826
$962$	−60.5629	−1.95263
$963$	0	0
$964$	67.4899	2.17370
$965$	35.9475	1.15719
$966$	0	0
$967$	−36.8790	−1.18595	−0.592974	−	0.805222i	$-0.702046\pi$
$967$	−36.8790	−1.18595	−0.592974	+	0.805222i	$0.702046\pi$
$968$	30.7587	0.988623
$969$	0	0
$970$	90.4890	2.90543
$971$	−30.7349	−0.986330	−0.493165	−	0.869936i	$-0.664160\pi$
$971$	−30.7349	−0.986330	−0.493165	+	0.869936i	$0.664160\pi$
$972$	0	0
$973$	−4.25689	−0.136470
$974$	−45.4329	−1.45576
$975$	0	0
$976$	15.2030	0.486636
$977$	−0.363388	−0.0116258	−0.00581291	−	0.999983i	$-0.501850\pi$
$977$	−0.363388	−0.0116258	−0.00581291	+	0.999983i	$0.501850\pi$
$978$	0	0
$979$	19.9007	0.636028
$980$	8.72794	0.278804
$981$	0	0
$982$	28.3540	0.904811
$983$	33.2316	1.05992	0.529962	−	0.848022i	$-0.322206\pi$
$983$	33.2316	1.05992	0.529962	+	0.848022i	$0.322206\pi$
$984$	0	0
$985$	−25.8814	−0.824649
$986$	−66.3638	−2.11345
$987$	0	0
$988$	−29.9497	−0.952826
$989$	−5.89635	−0.187493
$990$	0	0
$991$	−47.1118	−1.49656	−0.748278	−	0.663385i	$-0.769119\pi$
$991$	−47.1118	−1.49656	−0.748278	+	0.663385i	$0.769119\pi$
$992$	−9.89482	−0.314161
$993$	0	0
$994$	−21.4785	−0.681257
$995$	12.3397	0.391195
$996$	0	0
$997$	−58.5951	−1.85573	−0.927863	−	0.372920i	$-0.878357\pi$
$997$	−58.5951	−1.85573	−0.927863	+	0.372920i	$0.878357\pi$
$998$	−17.5230	−0.554681
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.x.1.1	✓	22
3.2	odd	2	inner	8001.2.a.x.1.22	yes	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8001.2.a.x.1.1	✓	22	1.1	even	1	trivial
8001.2.a.x.1.22	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)

\(f(q)\)	\(=\)	\(q-2.37730 q^{2} +3.65154 q^{4} +2.39021 q^{5} +1.00000 q^{7} -3.92621 q^{8} +O(q^{10})\)	\(q-2.37730 q^{2} +3.65154 q^{4} +2.39021 q^{5} +1.00000 q^{7} -3.92621 q^{8} -5.68223 q^{10} +1.77927 q^{11} -3.89190 q^{13} -2.37730 q^{14} +2.03069 q^{16} -5.42289 q^{17} +2.10743 q^{19} +8.72794 q^{20} -4.22985 q^{22} +4.43391 q^{23} +0.713082 q^{25} +9.25221 q^{26} +3.65154 q^{28} -5.14774 q^{29} -3.27115 q^{31} +3.02488 q^{32} +12.8918 q^{34} +2.39021 q^{35} -6.54578 q^{37} -5.01000 q^{38} -9.38445 q^{40} -5.69237 q^{41} -1.32983 q^{43} +6.49708 q^{44} -10.5407 q^{46} +1.09568 q^{47} +1.00000 q^{49} -1.69521 q^{50} -14.2114 q^{52} +9.83411 q^{53} +4.25282 q^{55} -3.92621 q^{56} +12.2377 q^{58} +5.41776 q^{59} +7.48664 q^{61} +7.77649 q^{62} -11.2524 q^{64} -9.30244 q^{65} +9.52331 q^{67} -19.8019 q^{68} -5.68223 q^{70} +9.03485 q^{71} -15.3815 q^{73} +15.5613 q^{74} +7.69538 q^{76} +1.77927 q^{77} -10.0332 q^{79} +4.85376 q^{80} +13.5325 q^{82} +13.7503 q^{83} -12.9618 q^{85} +3.16141 q^{86} -6.98579 q^{88} +11.1847 q^{89} -3.89190 q^{91} +16.1906 q^{92} -2.60475 q^{94} +5.03720 q^{95} -15.9249 q^{97} -2.37730 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 10 q^{4} + 22 q^{7}+O(q^{10}) \) 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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