LMFDB - Embedded newform 4024.2.a.e.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4024,2,Mod(1,4024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4024, 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("4024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4024 = 2^{3} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4024.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:	$32.1318017734$
Analytic rank:	$1$
Dimension:	$29$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Character	$\chi$	$=$	4024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.70977 q^{3} +2.99505 q^{5} -2.81033 q^{7} +4.34283 q^{9} +O(q^{10})$	$q-2.70977 q^{3} +2.99505 q^{5} -2.81033 q^{7} +4.34283 q^{9} -3.12276 q^{11} +1.74024 q^{13} -8.11588 q^{15} +0.354534 q^{17} -0.768079 q^{19} +7.61534 q^{21} +2.62736 q^{23} +3.97031 q^{25} -3.63877 q^{27} -2.66360 q^{29} +7.34612 q^{31} +8.46195 q^{33} -8.41708 q^{35} -5.78314 q^{37} -4.71564 q^{39} -6.53130 q^{41} +11.2103 q^{43} +13.0070 q^{45} -11.5729 q^{47} +0.897962 q^{49} -0.960705 q^{51} +5.59194 q^{53} -9.35281 q^{55} +2.08131 q^{57} -5.94669 q^{59} +13.2099 q^{61} -12.2048 q^{63} +5.21209 q^{65} -14.9803 q^{67} -7.11953 q^{69} -1.16535 q^{71} +7.40908 q^{73} -10.7586 q^{75} +8.77599 q^{77} +1.11929 q^{79} -3.16829 q^{81} +12.0106 q^{83} +1.06185 q^{85} +7.21773 q^{87} +8.47565 q^{89} -4.89064 q^{91} -19.9063 q^{93} -2.30043 q^{95} -11.3290 q^{97} -13.5616 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 29 q - 7 q^{3} - 4 q^{5} - 13 q^{7} + 20 q^{9}+O(q^{10}) $ 29 * q - 7 * q^3 - 4 * q^5 - 13 * q^7 + 20 * q^9	$ 29 q - 7 q^{3} - 4 q^{5} - 13 q^{7} + 20 q^{9} - 27 q^{11} + 16 q^{13} - 14 q^{15} - 15 q^{17} - 14 q^{19} + q^{21} - 25 q^{23} + 21 q^{25} - 25 q^{27} - 13 q^{29} - 27 q^{31} - 9 q^{33} - 29 q^{35} + 35 q^{37} - 38 q^{39} - 30 q^{41} - 38 q^{43} + q^{45} - 35 q^{47} + 14 q^{49} - 21 q^{51} + 2 q^{53} - 25 q^{55} - 25 q^{57} - 40 q^{59} + 10 q^{61} - 56 q^{63} - 50 q^{65} - 31 q^{67} + 11 q^{69} - 65 q^{71} - 23 q^{73} - 32 q^{75} + 13 q^{77} - 44 q^{79} - 7 q^{81} - 41 q^{83} + 26 q^{85} - 25 q^{87} - 48 q^{89} - 44 q^{91} + 25 q^{93} - 75 q^{95} - 18 q^{97} - 80 q^{99}+O(q^{100}) $ 29 * q - 7 * q^3 - 4 * q^5 - 13 * q^7 + 20 * q^9 - 27 * q^11 + 16 * q^13 - 14 * q^15 - 15 * q^17 - 14 * q^19 + q^21 - 25 * q^23 + 21 * q^25 - 25 * q^27 - 13 * q^29 - 27 * q^31 - 9 * q^33 - 29 * q^35 + 35 * q^37 - 38 * q^39 - 30 * q^41 - 38 * q^43 + q^45 - 35 * q^47 + 14 * q^49 - 21 * q^51 + 2 * q^53 - 25 * q^55 - 25 * q^57 - 40 * q^59 + 10 * q^61 - 56 * q^63 - 50 * q^65 - 31 * q^67 + 11 * q^69 - 65 * q^71 - 23 * q^73 - 32 * q^75 + 13 * q^77 - 44 * q^79 - 7 * q^81 - 41 * q^83 + 26 * q^85 - 25 * q^87 - 48 * q^89 - 44 * q^91 + 25 * q^93 - 75 * q^95 - 18 * q^97 - 80 * q^99

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	0
$2$	0	0
$3$	−2.70977	−1.56448	−0.782242	−	0.622974i	$-0.785924\pi$
$3$	−2.70977	−1.56448	−0.782242	+	0.622974i	$0.785924\pi$
$4$	0	0
$5$	2.99505	1.33943	0.669713	−	0.742620i	$-0.266417\pi$
$5$	2.99505	1.33943	0.669713	+	0.742620i	$0.266417\pi$
$6$	0	0
$7$	−2.81033	−1.06221	−0.531103	−	0.847307i	$-0.678222\pi$
$7$	−2.81033	−1.06221	−0.531103	+	0.847307i	$0.678222\pi$
$8$	0	0
$9$	4.34283	1.44761
$10$	0	0
$11$	−3.12276	−0.941547	−0.470774	−	0.882254i	$-0.656025\pi$
$11$	−3.12276	−0.941547	−0.470774	+	0.882254i	$0.656025\pi$
$12$	0	0
$13$	1.74024	0.482655	0.241328	−	0.970444i	$-0.422417\pi$
$13$	1.74024	0.482655	0.241328	+	0.970444i	$0.422417\pi$
$14$	0	0
$15$	−8.11588	−2.09551
$16$	0	0
$17$	0.354534	0.0859872	0.0429936	−	0.999075i	$-0.486310\pi$
$17$	0.354534	0.0859872	0.0429936	+	0.999075i	$0.486310\pi$
$18$	0	0
$19$	−0.768079	−0.176209	−0.0881047	−	0.996111i	$-0.528081\pi$
$19$	−0.768079	−0.176209	−0.0881047	+	0.996111i	$0.528081\pi$
$20$	0	0
$21$	7.61534	1.66180
$22$	0	0
$23$	2.62736	0.547842	0.273921	−	0.961752i	$-0.411679\pi$
$23$	2.62736	0.547842	0.273921	+	0.961752i	$0.411679\pi$
$24$	0	0
$25$	3.97031	0.794062
$26$	0	0
$27$	−3.63877	−0.700281
$28$	0	0
$29$	−2.66360	−0.494618	−0.247309	−	0.968937i	$-0.579546\pi$
$29$	−2.66360	−0.494618	−0.247309	+	0.968937i	$0.579546\pi$
$30$	0	0
$31$	7.34612	1.31940	0.659701	−	0.751528i	$-0.270683\pi$
$31$	7.34612	1.31940	0.659701	+	0.751528i	$0.270683\pi$
$32$	0	0
$33$	8.46195	1.47304
$34$	0	0
$35$	−8.41708	−1.42275
$36$	0	0
$37$	−5.78314	−0.950742	−0.475371	−	0.879785i	$-0.657686\pi$
$37$	−5.78314	−0.950742	−0.475371	+	0.879785i	$0.657686\pi$
$38$	0	0
$39$	−4.71564	−0.755106
$40$	0	0
$41$	−6.53130	−1.02002	−0.510009	−	0.860169i	$-0.670358\pi$
$41$	−6.53130	−1.02002	−0.510009	+	0.860169i	$0.670358\pi$
$42$	0	0
$43$	11.2103	1.70955	0.854776	−	0.518998i	$-0.173695\pi$
$43$	11.2103	1.70955	0.854776	+	0.518998i	$0.173695\pi$
$44$	0	0
$45$	13.0070	1.93897
$46$	0	0
$47$	−11.5729	−1.68808	−0.844042	−	0.536278i	$-0.819830\pi$
$47$	−11.5729	−1.68808	−0.844042	+	0.536278i	$0.819830\pi$
$48$	0	0
$49$	0.897962	0.128280
$50$	0	0
$51$	−0.960705	−0.134526
$52$	0	0
$53$	5.59194	0.768112	0.384056	−	0.923310i	$-0.374527\pi$
$53$	5.59194	0.768112	0.384056	+	0.923310i	$0.374527\pi$
$54$	0	0
$55$	−9.35281	−1.26113
$56$	0	0
$57$	2.08131	0.275677
$58$	0	0
$59$	−5.94669	−0.774193	−0.387097	−	0.922039i	$-0.626522\pi$
$59$	−5.94669	−0.774193	−0.387097	+	0.922039i	$0.626522\pi$
$60$	0	0
$61$	13.2099	1.69136	0.845679	−	0.533691i	$-0.179196\pi$
$61$	13.2099	1.69136	0.845679	+	0.533691i	$0.179196\pi$
$62$	0	0
$63$	−12.2048	−1.53766
$64$	0	0
$65$	5.21209	0.646481
$66$	0	0
$67$	−14.9803	−1.83014	−0.915068	−	0.403300i	$-0.867863\pi$
$67$	−14.9803	−1.83014	−0.915068	+	0.403300i	$0.867863\pi$
$68$	0	0
$69$	−7.11953	−0.857090
$70$	0	0
$71$	−1.16535	−0.138302	−0.0691509	−	0.997606i	$-0.522029\pi$
$71$	−1.16535	−0.138302	−0.0691509	+	0.997606i	$0.522029\pi$
$72$	0	0
$73$	7.40908	0.867168	0.433584	−	0.901113i	$-0.357249\pi$
$73$	7.40908	0.867168	0.433584	+	0.901113i	$0.357249\pi$
$74$	0	0
$75$	−10.7586	−1.24230
$76$	0	0
$77$	8.77599	1.00012
$78$	0	0
$79$	1.11929	0.125930	0.0629649	−	0.998016i	$-0.479944\pi$
$79$	1.11929	0.125930	0.0629649	+	0.998016i	$0.479944\pi$
$80$	0	0
$81$	−3.16829	−0.352033
$82$	0	0
$83$	12.0106	1.31833	0.659165	−	0.751998i	$-0.270910\pi$
$83$	12.0106	1.31833	0.659165	+	0.751998i	$0.270910\pi$
$84$	0	0
$85$	1.06185	0.115173
$86$	0	0
$87$	7.21773	0.773821
$88$	0	0
$89$	8.47565	0.898417	0.449208	−	0.893427i	$-0.351706\pi$
$89$	8.47565	0.898417	0.449208	+	0.893427i	$0.351706\pi$
$90$	0	0
$91$	−4.89064	−0.512679
$92$	0	0
$93$	−19.9063	−2.06418
$94$	0	0
$95$	−2.30043	−0.236019
$96$	0	0
$97$	−11.3290	−1.15029	−0.575144	−	0.818052i	$-0.695054\pi$
$97$	−11.3290	−1.15029	−0.575144	+	0.818052i	$0.695054\pi$
$98$	0	0
$99$	−13.5616	−1.36299
$100$	0	0
$101$	−19.4663	−1.93697	−0.968484	−	0.249077i	$-0.919873\pi$
$101$	−19.4663	−1.93697	−0.968484	+	0.249077i	$0.919873\pi$
$102$	0	0
$103$	−1.87783	−0.185028	−0.0925140	−	0.995711i	$-0.529490\pi$
$103$	−1.87783	−0.185028	−0.0925140	+	0.995711i	$0.529490\pi$
$104$	0	0
$105$	22.8083	2.22586
$106$	0	0
$107$	−2.62843	−0.254100	−0.127050	−	0.991896i	$-0.540551\pi$
$107$	−2.62843	−0.254100	−0.127050	+	0.991896i	$0.540551\pi$
$108$	0	0
$109$	−5.16595	−0.494808	−0.247404	−	0.968912i	$-0.579578\pi$
$109$	−5.16595	−0.494808	−0.247404	+	0.968912i	$0.579578\pi$
$110$	0	0
$111$	15.6710	1.48742
$112$	0	0
$113$	11.0836	1.04265	0.521326	−	0.853357i	$-0.325437\pi$
$113$	11.0836	1.04265	0.521326	+	0.853357i	$0.325437\pi$
$114$	0	0
$115$	7.86906	0.733794
$116$	0	0
$117$	7.55756	0.698697
$118$	0	0
$119$	−0.996359	−0.0913361
$120$	0	0
$121$	−1.24837	−0.113489
$122$	0	0
$123$	17.6983	1.59580
$124$	0	0
$125$	−3.08397	−0.275839
$126$	0	0
$127$	9.57655	0.849781	0.424890	−	0.905245i	$-0.360313\pi$
$127$	9.57655	0.849781	0.424890	+	0.905245i	$0.360313\pi$
$128$	0	0
$129$	−30.3772	−2.67457
$130$	0	0
$131$	−9.48401	−0.828622	−0.414311	−	0.910135i	$-0.635977\pi$
$131$	−9.48401	−0.828622	−0.414311	+	0.910135i	$0.635977\pi$
$132$	0	0
$133$	2.15856	0.187170
$134$	0	0
$135$	−10.8983	−0.937975
$136$	0	0
$137$	−5.20247	−0.444477	−0.222238	−	0.974992i	$-0.571336\pi$
$137$	−5.20247	−0.444477	−0.222238	+	0.974992i	$0.571336\pi$
$138$	0	0
$139$	1.07215	0.0909387	0.0454693	−	0.998966i	$-0.485522\pi$
$139$	1.07215	0.0909387	0.0454693	+	0.998966i	$0.485522\pi$
$140$	0	0
$141$	31.3599	2.64098
$142$	0	0
$143$	−5.43434	−0.454443
$144$	0	0
$145$	−7.97760	−0.662504
$146$	0	0
$147$	−2.43327	−0.200692
$148$	0	0
$149$	1.75537	0.143806	0.0719029	−	0.997412i	$-0.477093\pi$
$149$	1.75537	0.143806	0.0719029	+	0.997412i	$0.477093\pi$
$150$	0	0
$151$	7.31043	0.594915	0.297457	−	0.954735i	$-0.403861\pi$
$151$	7.31043	0.594915	0.297457	+	0.954735i	$0.403861\pi$
$152$	0	0
$153$	1.53968	0.124476
$154$	0	0
$155$	22.0020	1.76724
$156$	0	0
$157$	−13.6037	−1.08570	−0.542849	−	0.839831i	$-0.682654\pi$
$157$	−13.6037	−1.08570	−0.542849	+	0.839831i	$0.682654\pi$
$158$	0	0
$159$	−15.1529	−1.20170
$160$	0	0
$161$	−7.38375	−0.581921
$162$	0	0
$163$	13.7790	1.07925	0.539626	−	0.841905i	$-0.318566\pi$
$163$	13.7790	1.07925	0.539626	+	0.841905i	$0.318566\pi$
$164$	0	0
$165$	25.3439	1.97302
$166$	0	0
$167$	0.215463	0.0166730	0.00833651	−	0.999965i	$-0.497346\pi$
$167$	0.215463	0.0166730	0.00833651	+	0.999965i	$0.497346\pi$
$168$	0	0
$169$	−9.97157	−0.767044
$170$	0	0
$171$	−3.33564	−0.255083
$172$	0	0
$173$	−1.40112	−0.106525	−0.0532626	−	0.998581i	$-0.516962\pi$
$173$	−1.40112	−0.106525	−0.0532626	+	0.998581i	$0.516962\pi$
$174$	0	0
$175$	−11.1579	−0.843457
$176$	0	0
$177$	16.1141	1.21121
$178$	0	0
$179$	11.3646	0.849433	0.424716	−	0.905326i	$-0.360374\pi$
$179$	11.3646	0.849433	0.424716	+	0.905326i	$0.360374\pi$
$180$	0	0
$181$	7.09891	0.527658	0.263829	−	0.964569i	$-0.415015\pi$
$181$	7.09891	0.527658	0.263829	+	0.964569i	$0.415015\pi$
$182$	0	0
$183$	−35.7958	−2.64610
$184$	0	0
$185$	−17.3208	−1.27345
$186$	0	0
$187$	−1.10713	−0.0809610
$188$	0	0
$189$	10.2261	0.743842
$190$	0	0
$191$	0.692976	0.0501420	0.0250710	−	0.999686i	$-0.492019\pi$
$191$	0.692976	0.0501420	0.0250710	+	0.999686i	$0.492019\pi$
$192$	0	0
$193$	−5.18433	−0.373176	−0.186588	−	0.982438i	$-0.559743\pi$
$193$	−5.18433	−0.373176	−0.186588	+	0.982438i	$0.559743\pi$
$194$	0	0
$195$	−14.1236	−1.01141
$196$	0	0
$197$	7.96963	0.567813	0.283906	−	0.958852i	$-0.408370\pi$
$197$	7.96963	0.567813	0.283906	+	0.958852i	$0.408370\pi$
$198$	0	0
$199$	−16.9943	−1.20469	−0.602346	−	0.798235i	$-0.705767\pi$
$199$	−16.9943	−1.20469	−0.602346	+	0.798235i	$0.705767\pi$
$200$	0	0
$201$	40.5931	2.86322
$202$	0	0
$203$	7.48559	0.525385
$204$	0	0
$205$	−19.5615	−1.36624
$206$	0	0
$207$	11.4102	0.793063
$208$	0	0
$209$	2.39852	0.165909
$210$	0	0
$211$	−21.9438	−1.51068	−0.755338	−	0.655335i	$-0.772527\pi$
$211$	−21.9438	−1.51068	−0.755338	+	0.655335i	$0.772527\pi$
$212$	0	0
$213$	3.15783	0.216371
$214$	0	0
$215$	33.5753	2.28982
$216$	0	0
$217$	−20.6450	−1.40148
$218$	0	0
$219$	−20.0769	−1.35667
$220$	0	0
$221$	0.616974	0.0415022
$222$	0	0
$223$	−0.575659	−0.0385490	−0.0192745	−	0.999814i	$-0.506136\pi$
$223$	−0.575659	−0.0385490	−0.0192745	+	0.999814i	$0.506136\pi$
$224$	0	0
$225$	17.2424	1.14949
$226$	0	0
$227$	3.95462	0.262478	0.131239	−	0.991351i	$-0.458105\pi$
$227$	3.95462	0.262478	0.131239	+	0.991351i	$0.458105\pi$
$228$	0	0
$229$	−11.2286	−0.742008	−0.371004	−	0.928631i	$-0.620986\pi$
$229$	−11.2286	−0.742008	−0.371004	+	0.928631i	$0.620986\pi$
$230$	0	0
$231$	−23.7809	−1.56467
$232$	0	0
$233$	−16.5598	−1.08487	−0.542435	−	0.840098i	$-0.682497\pi$
$233$	−16.5598	−1.08487	−0.542435	+	0.840098i	$0.682497\pi$
$234$	0	0
$235$	−34.6614	−2.26106
$236$	0	0
$237$	−3.03301	−0.197015
$238$	0	0
$239$	−15.0555	−0.973861	−0.486930	−	0.873441i	$-0.661883\pi$
$239$	−15.0555	−0.973861	−0.486930	+	0.873441i	$0.661883\pi$
$240$	0	0
$241$	−21.9393	−1.41323	−0.706616	−	0.707597i	$-0.749779\pi$
$241$	−21.9393	−1.41323	−0.706616	+	0.707597i	$0.749779\pi$
$242$	0	0
$243$	19.5016	1.25103
$244$	0	0
$245$	2.68944	0.171822
$246$	0	0
$247$	−1.33664	−0.0850483
$248$	0	0
$249$	−32.5458	−2.06251
$250$	0	0
$251$	−19.5993	−1.23710	−0.618549	−	0.785746i	$-0.712279\pi$
$251$	−19.5993	−1.23710	−0.618549	+	0.785746i	$0.712279\pi$
$252$	0	0
$253$	−8.20461	−0.515819
$254$	0	0
$255$	−2.87736	−0.180187
$256$	0	0
$257$	−8.04127	−0.501601	−0.250800	−	0.968039i	$-0.580694\pi$
$257$	−8.04127	−0.501601	−0.250800	+	0.968039i	$0.580694\pi$
$258$	0	0
$259$	16.2525	1.00988
$260$	0	0
$261$	−11.5676	−0.716014
$262$	0	0
$263$	−22.8002	−1.40592	−0.702959	−	0.711230i	$-0.748138\pi$
$263$	−22.8002	−1.40592	−0.702959	+	0.711230i	$0.748138\pi$
$264$	0	0
$265$	16.7481	1.02883
$266$	0	0
$267$	−22.9670	−1.40556
$268$	0	0
$269$	−20.0914	−1.22499	−0.612497	−	0.790473i	$-0.709835\pi$
$269$	−20.0914	−1.22499	−0.612497	+	0.790473i	$0.709835\pi$
$270$	0	0
$271$	−32.4263	−1.96976	−0.984880	−	0.173238i	$-0.944577\pi$
$271$	−32.4263	−1.96976	−0.984880	+	0.173238i	$0.944577\pi$
$272$	0	0
$273$	13.2525	0.802078
$274$	0	0
$275$	−12.3983	−0.747647
$276$	0	0
$277$	−28.1639	−1.69220	−0.846101	−	0.533022i	$-0.821056\pi$
$277$	−28.1639	−1.69220	−0.846101	+	0.533022i	$0.821056\pi$
$278$	0	0
$279$	31.9030	1.90998
$280$	0	0
$281$	−11.2374	−0.670369	−0.335184	−	0.942153i	$-0.608799\pi$
$281$	−11.2374	−0.670369	−0.335184	+	0.942153i	$0.608799\pi$
$282$	0	0
$283$	−7.23234	−0.429918	−0.214959	−	0.976623i	$-0.568962\pi$
$283$	−7.23234	−0.429918	−0.214959	+	0.976623i	$0.568962\pi$
$284$	0	0
$285$	6.23363	0.369249
$286$	0	0
$287$	18.3551	1.08347
$288$	0	0
$289$	−16.8743	−0.992606
$290$	0	0
$291$	30.6990	1.79961
$292$	0	0
$293$	−13.7687	−0.804377	−0.402188	−	0.915557i	$-0.631750\pi$
$293$	−13.7687	−0.804377	−0.402188	+	0.915557i	$0.631750\pi$
$294$	0	0
$295$	−17.8106	−1.03697
$296$	0	0
$297$	11.3630	0.659348
$298$	0	0
$299$	4.57223	0.264419
$300$	0	0
$301$	−31.5046	−1.81589
$302$	0	0
$303$	52.7491	3.03036
$304$	0	0
$305$	39.5644	2.26545
$306$	0	0
$307$	−2.52117	−0.143891	−0.0719455	−	0.997409i	$-0.522921\pi$
$307$	−2.52117	−0.143891	−0.0719455	+	0.997409i	$0.522921\pi$
$308$	0	0
$309$	5.08848	0.289473
$310$	0	0
$311$	7.39558	0.419365	0.209682	−	0.977770i	$-0.432757\pi$
$311$	7.39558	0.419365	0.209682	+	0.977770i	$0.432757\pi$
$312$	0	0
$313$	22.4338	1.26803	0.634017	−	0.773319i	$-0.281405\pi$
$313$	22.4338	1.26803	0.634017	+	0.773319i	$0.281405\pi$
$314$	0	0
$315$	−36.5540	−2.05958
$316$	0	0
$317$	14.2967	0.802983	0.401492	−	0.915863i	$-0.368492\pi$
$317$	14.2967	0.802983	0.401492	+	0.915863i	$0.368492\pi$
$318$	0	0
$319$	8.31777	0.465706
$320$	0	0
$321$	7.12243	0.397535
$322$	0	0
$323$	−0.272310	−0.0151517
$324$	0	0
$325$	6.90928	0.383258
$326$	0	0
$327$	13.9985	0.774120
$328$	0	0
$329$	32.5237	1.79309
$330$	0	0
$331$	3.15872	0.173619	0.0868095	−	0.996225i	$-0.472333\pi$
$331$	3.15872	0.173619	0.0868095	+	0.996225i	$0.472333\pi$
$332$	0	0
$333$	−25.1152	−1.37631
$334$	0	0
$335$	−44.8667	−2.45133
$336$	0	0
$337$	−1.98100	−0.107912	−0.0539559	−	0.998543i	$-0.517183\pi$
$337$	−1.98100	−0.107912	−0.0539559	+	0.998543i	$0.517183\pi$
$338$	0	0
$339$	−30.0338	−1.63121
$340$	0	0
$341$	−22.9402	−1.24228
$342$	0	0
$343$	17.1487	0.925945
$344$	0	0
$345$	−21.3233	−1.14801
$346$	0	0
$347$	−0.146376	−0.00785785	−0.00392893	−	0.999992i	$-0.501251\pi$
$347$	−0.146376	−0.00785785	−0.00392893	+	0.999992i	$0.501251\pi$
$348$	0	0
$349$	22.8643	1.22390	0.611949	−	0.790897i	$-0.290386\pi$
$349$	22.8643	1.22390	0.611949	+	0.790897i	$0.290386\pi$
$350$	0	0
$351$	−6.33232	−0.337994
$352$	0	0
$353$	−22.3723	−1.19076	−0.595379	−	0.803445i	$-0.702998\pi$
$353$	−22.3723	−1.19076	−0.595379	+	0.803445i	$0.702998\pi$
$354$	0	0
$355$	−3.49028	−0.185245
$356$	0	0
$357$	2.69990	0.142894
$358$	0	0
$359$	7.61613	0.401964	0.200982	−	0.979595i	$-0.435587\pi$
$359$	7.61613	0.401964	0.200982	+	0.979595i	$0.435587\pi$
$360$	0	0
$361$	−18.4101	−0.968950
$362$	0	0
$363$	3.38280	0.177551
$364$	0	0
$365$	22.1906	1.16151
$366$	0	0
$367$	−30.3532	−1.58443	−0.792213	−	0.610245i	$-0.791071\pi$
$367$	−30.3532	−1.58443	−0.792213	+	0.610245i	$0.791071\pi$
$368$	0	0
$369$	−28.3643	−1.47659
$370$	0	0
$371$	−15.7152	−0.815893
$372$	0	0
$373$	7.61473	0.394276	0.197138	−	0.980376i	$-0.436835\pi$
$373$	7.61473	0.394276	0.197138	+	0.980376i	$0.436835\pi$
$374$	0	0
$375$	8.35684	0.431545
$376$	0	0
$377$	−4.63529	−0.238730
$378$	0	0
$379$	−2.42539	−0.124584	−0.0622920	−	0.998058i	$-0.519841\pi$
$379$	−2.42539	−0.124584	−0.0622920	+	0.998058i	$0.519841\pi$
$380$	0	0
$381$	−25.9502	−1.32947
$382$	0	0
$383$	−33.9212	−1.73329	−0.866645	−	0.498925i	$-0.833728\pi$
$383$	−33.9212	−1.73329	−0.866645	+	0.498925i	$0.833728\pi$
$384$	0	0
$385$	26.2845	1.33958
$386$	0	0
$387$	48.6844	2.47477
$388$	0	0
$389$	−15.2462	−0.773012	−0.386506	−	0.922287i	$-0.626318\pi$
$389$	−15.2462	−0.773012	−0.386506	+	0.922287i	$0.626318\pi$
$390$	0	0
$391$	0.931489	0.0471074
$392$	0	0
$393$	25.6995	1.29637
$394$	0	0
$395$	3.35232	0.168674
$396$	0	0
$397$	10.9380	0.548961	0.274480	−	0.961593i	$-0.411494\pi$
$397$	10.9380	0.548961	0.274480	+	0.961593i	$0.411494\pi$
$398$	0	0
$399$	−5.84918	−0.292825
$400$	0	0
$401$	−5.36987	−0.268159	−0.134079	−	0.990971i	$-0.542808\pi$
$401$	−5.36987	−0.268159	−0.134079	+	0.990971i	$0.542808\pi$
$402$	0	0
$403$	12.7840	0.636816
$404$	0	0
$405$	−9.48919	−0.471522
$406$	0	0
$407$	18.0594	0.895169
$408$	0	0
$409$	13.7255	0.678681	0.339340	−	0.940664i	$-0.389796\pi$
$409$	13.7255	0.678681	0.339340	+	0.940664i	$0.389796\pi$
$410$	0	0
$411$	14.0975	0.695377
$412$	0	0
$413$	16.7122	0.822352
$414$	0	0
$415$	35.9722	1.76581
$416$	0	0
$417$	−2.90528	−0.142272
$418$	0	0
$419$	−13.4151	−0.655370	−0.327685	−	0.944787i	$-0.606268\pi$
$419$	−13.4151	−0.655370	−0.327685	+	0.944787i	$0.606268\pi$
$420$	0	0
$421$	24.3875	1.18857	0.594287	−	0.804253i	$-0.297434\pi$
$421$	24.3875	1.18857	0.594287	+	0.804253i	$0.297434\pi$
$422$	0	0
$423$	−50.2592	−2.44369
$424$	0	0
$425$	1.40761	0.0682792
$426$	0	0
$427$	−37.1243	−1.79657
$428$	0	0
$429$	14.7258	0.710968
$430$	0	0
$431$	−16.4221	−0.791024	−0.395512	−	0.918461i	$-0.629433\pi$
$431$	−16.4221	−0.791024	−0.395512	+	0.918461i	$0.629433\pi$
$432$	0	0
$433$	11.2692	0.541565	0.270782	−	0.962641i	$-0.412718\pi$
$433$	11.2692	0.541565	0.270782	+	0.962641i	$0.412718\pi$
$434$	0	0
$435$	21.6174	1.03648
$436$	0	0
$437$	−2.01802	−0.0965349
$438$	0	0
$439$	19.8673	0.948214	0.474107	−	0.880467i	$-0.342771\pi$
$439$	19.8673	0.948214	0.474107	+	0.880467i	$0.342771\pi$
$440$	0	0
$441$	3.89970	0.185700
$442$	0	0
$443$	28.6536	1.36138	0.680688	−	0.732574i	$-0.261681\pi$
$443$	28.6536	1.36138	0.680688	+	0.732574i	$0.261681\pi$
$444$	0	0
$445$	25.3850	1.20336
$446$	0	0
$447$	−4.75665	−0.224982
$448$	0	0
$449$	1.09278	0.0515713	0.0257856	−	0.999667i	$-0.491791\pi$
$449$	1.09278	0.0515713	0.0257856	+	0.999667i	$0.491791\pi$
$450$	0	0
$451$	20.3957	0.960394
$452$	0	0
$453$	−19.8096	−0.930735
$454$	0	0
$455$	−14.6477	−0.686695
$456$	0	0
$457$	23.9445	1.12008	0.560039	−	0.828466i	$-0.310786\pi$
$457$	23.9445	1.12008	0.560039	+	0.828466i	$0.310786\pi$
$458$	0	0
$459$	−1.29007	−0.0602152
$460$	0	0
$461$	−9.32724	−0.434413	−0.217207	−	0.976126i	$-0.569695\pi$
$461$	−9.32724	−0.434413	−0.217207	+	0.976126i	$0.569695\pi$
$462$	0	0
$463$	−35.5833	−1.65370	−0.826848	−	0.562426i	$-0.809868\pi$
$463$	−35.5833	−1.65370	−0.826848	+	0.562426i	$0.809868\pi$
$464$	0	0
$465$	−59.6202	−2.76482
$466$	0	0
$467$	28.1915	1.30455	0.652274	−	0.757983i	$-0.273815\pi$
$467$	28.1915	1.30455	0.652274	+	0.757983i	$0.273815\pi$
$468$	0	0
$469$	42.0996	1.94398
$470$	0	0
$471$	36.8630	1.69856
$472$	0	0
$473$	−35.0070	−1.60962
$474$	0	0
$475$	−3.04951	−0.139921
$476$	0	0
$477$	24.2849	1.11193
$478$	0	0
$479$	−23.3939	−1.06889	−0.534447	−	0.845202i	$-0.679480\pi$
$479$	−23.3939	−1.06889	−0.534447	+	0.845202i	$0.679480\pi$
$480$	0	0
$481$	−10.0640	−0.458881
$482$	0	0
$483$	20.0082	0.910406
$484$	0	0
$485$	−33.9310	−1.54073
$486$	0	0
$487$	2.27433	0.103060	0.0515299	−	0.998671i	$-0.483590\pi$
$487$	2.27433	0.103060	0.0515299	+	0.998671i	$0.483590\pi$
$488$	0	0
$489$	−37.3378	−1.68847
$490$	0	0
$491$	43.1549	1.94755	0.973777	−	0.227503i	$-0.0730563\pi$
$491$	43.1549	1.94755	0.973777	+	0.227503i	$0.0730563\pi$
$492$	0	0
$493$	−0.944336	−0.0425308
$494$	0	0
$495$	−40.6177	−1.82563
$496$	0	0
$497$	3.27502	0.146905
$498$	0	0
$499$	−19.8690	−0.889457	−0.444728	−	0.895665i	$-0.646700\pi$
$499$	−19.8690	−0.889457	−0.444728	+	0.895665i	$0.646700\pi$
$500$	0	0
$501$	−0.583854	−0.0260847
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	−58.3024	−2.59442
$506$	0	0
$507$	27.0206	1.20003
$508$	0	0
$509$	−3.89159	−0.172492	−0.0862459	−	0.996274i	$-0.527487\pi$
$509$	−3.89159	−0.172492	−0.0862459	+	0.996274i	$0.527487\pi$
$510$	0	0
$511$	−20.8220	−0.921110
$512$	0	0
$513$	2.79486	0.123396
$514$	0	0
$515$	−5.62419	−0.247831
$516$	0	0
$517$	36.1394	1.58941
$518$	0	0
$519$	3.79671	0.166657
$520$	0	0
$521$	24.2976	1.06450	0.532248	−	0.846589i	$-0.321348\pi$
$521$	24.2976	1.06450	0.532248	+	0.846589i	$0.321348\pi$
$522$	0	0
$523$	2.81521	0.123101	0.0615503	−	0.998104i	$-0.480396\pi$
$523$	2.81521	0.123101	0.0615503	+	0.998104i	$0.480396\pi$
$524$	0	0
$525$	30.2353	1.31958
$526$	0	0
$527$	2.60445	0.113452
$528$	0	0
$529$	−16.0970	−0.699869
$530$	0	0
$531$	−25.8255	−1.12073
$532$	0	0
$533$	−11.3660	−0.492316
$534$	0	0
$535$	−7.87227	−0.340348
$536$	0	0
$537$	−30.7955	−1.32892
$538$	0	0
$539$	−2.80412	−0.120782
$540$	0	0
$541$	−7.52225	−0.323407	−0.161703	−	0.986839i	$-0.551699\pi$
$541$	−7.52225	−0.323407	−0.161703	+	0.986839i	$0.551699\pi$
$542$	0	0
$543$	−19.2364	−0.825513
$544$	0	0
$545$	−15.4723	−0.662759
$546$	0	0
$547$	9.91432	0.423906	0.211953	−	0.977280i	$-0.432018\pi$
$547$	9.91432	0.423906	0.211953	+	0.977280i	$0.432018\pi$
$548$	0	0
$549$	57.3685	2.44843
$550$	0	0
$551$	2.04585	0.0871562
$552$	0	0
$553$	−3.14557	−0.133763
$554$	0	0
$555$	46.9353	1.99229
$556$	0	0
$557$	19.5890	0.830014	0.415007	−	0.909818i	$-0.363779\pi$
$557$	19.5890	0.830014	0.415007	+	0.909818i	$0.363779\pi$
$558$	0	0
$559$	19.5085	0.825124
$560$	0	0
$561$	3.00005	0.126662
$562$	0	0
$563$	31.5090	1.32794	0.663972	−	0.747757i	$-0.268869\pi$
$563$	31.5090	1.32794	0.663972	+	0.747757i	$0.268869\pi$
$564$	0	0
$565$	33.1958	1.39656
$566$	0	0
$567$	8.90396	0.373931
$568$	0	0
$569$	23.1860	0.972009	0.486004	−	0.873956i	$-0.338454\pi$
$569$	23.1860	0.972009	0.486004	+	0.873956i	$0.338454\pi$
$570$	0	0
$571$	−6.68722	−0.279851	−0.139926	−	0.990162i	$-0.544686\pi$
$571$	−6.68722	−0.279851	−0.139926	+	0.990162i	$0.544686\pi$
$572$	0	0
$573$	−1.87780	−0.0784463
$574$	0	0
$575$	10.4314	0.435021
$576$	0	0
$577$	15.1082	0.628961	0.314481	−	0.949264i	$-0.398170\pi$
$577$	15.1082	0.628961	0.314481	+	0.949264i	$0.398170\pi$
$578$	0	0
$579$	14.0483	0.583828
$580$	0	0
$581$	−33.7536	−1.40034
$582$	0	0
$583$	−17.4623	−0.723214
$584$	0	0
$585$	22.6353	0.935853
$586$	0	0
$587$	21.0236	0.867737	0.433868	−	0.900976i	$-0.357148\pi$
$587$	21.0236	0.867737	0.433868	+	0.900976i	$0.357148\pi$
$588$	0	0
$589$	−5.64240	−0.232491
$590$	0	0
$591$	−21.5958	−0.888334
$592$	0	0
$593$	30.6148	1.25720	0.628600	−	0.777729i	$-0.283628\pi$
$593$	30.6148	1.25720	0.628600	+	0.777729i	$0.283628\pi$
$594$	0	0
$595$	−2.98414	−0.122338
$596$	0	0
$597$	46.0505	1.88472
$598$	0	0
$599$	−35.6585	−1.45697	−0.728484	−	0.685062i	$-0.759775\pi$
$599$	−35.6585	−1.45697	−0.728484	+	0.685062i	$0.759775\pi$
$600$	0	0
$601$	−15.9128	−0.649095	−0.324547	−	0.945869i	$-0.605212\pi$
$601$	−15.9128	−0.649095	−0.324547	+	0.945869i	$0.605212\pi$
$602$	0	0
$603$	−65.0570	−2.64933
$604$	0	0
$605$	−3.73894	−0.152010
$606$	0	0
$607$	−28.1539	−1.14273	−0.571367	−	0.820695i	$-0.693587\pi$
$607$	−28.1539	−1.14273	−0.571367	+	0.820695i	$0.693587\pi$
$608$	0	0
$609$	−20.2842	−0.821957
$610$	0	0
$611$	−20.1396	−0.814762
$612$	0	0
$613$	−16.5677	−0.669163	−0.334582	−	0.942367i	$-0.608595\pi$
$613$	−16.5677	−0.669163	−0.334582	+	0.942367i	$0.608595\pi$
$614$	0	0
$615$	53.0072	2.13746
$616$	0	0
$617$	2.53831	0.102188	0.0510942	−	0.998694i	$-0.483729\pi$
$617$	2.53831	0.102188	0.0510942	+	0.998694i	$0.483729\pi$
$618$	0	0
$619$	27.1828	1.09257	0.546284	−	0.837600i	$-0.316042\pi$
$619$	27.1828	1.09257	0.546284	+	0.837600i	$0.316042\pi$
$620$	0	0
$621$	−9.56034	−0.383643
$622$	0	0
$623$	−23.8194	−0.954303
$624$	0	0
$625$	−29.0882	−1.16353
$626$	0	0
$627$	−6.49944	−0.259563
$628$	0	0
$629$	−2.05032	−0.0817517
$630$	0	0
$631$	35.3939	1.40901	0.704504	−	0.709700i	$-0.251169\pi$
$631$	35.3939	1.40901	0.704504	+	0.709700i	$0.251169\pi$
$632$	0	0
$633$	59.4627	2.36343
$634$	0	0
$635$	28.6822	1.13822
$636$	0	0
$637$	1.56267	0.0619151
$638$	0	0
$639$	−5.06093	−0.200207
$640$	0	0
$641$	−30.6442	−1.21037	−0.605187	−	0.796084i	$-0.706902\pi$
$641$	−30.6442	−1.21037	−0.605187	+	0.796084i	$0.706902\pi$
$642$	0	0
$643$	1.62411	0.0640487	0.0320243	−	0.999487i	$-0.489805\pi$
$643$	1.62411	0.0640487	0.0320243	+	0.999487i	$0.489805\pi$
$644$	0	0
$645$	−90.9813	−3.58238
$646$	0	0
$647$	5.74932	0.226029	0.113015	−	0.993593i	$-0.463949\pi$
$647$	5.74932	0.226029	0.113015	+	0.993593i	$0.463949\pi$
$648$	0	0
$649$	18.5701	0.728939
$650$	0	0
$651$	55.9432	2.19259
$652$	0	0
$653$	−18.1171	−0.708978	−0.354489	−	0.935060i	$-0.615345\pi$
$653$	−18.1171	−0.708978	−0.354489	+	0.935060i	$0.615345\pi$
$654$	0	0
$655$	−28.4051	−1.10988
$656$	0	0
$657$	32.1764	1.25532
$658$	0	0
$659$	−36.4989	−1.42180	−0.710899	−	0.703295i	$-0.751711\pi$
$659$	−36.4989	−1.42180	−0.710899	+	0.703295i	$0.751711\pi$
$660$	0	0
$661$	12.2644	0.477029	0.238515	−	0.971139i	$-0.423340\pi$
$661$	12.2644	0.477029	0.238515	+	0.971139i	$0.423340\pi$
$662$	0	0
$663$	−1.67186	−0.0649295
$664$	0	0
$665$	6.46498	0.250701
$666$	0	0
$667$	−6.99822	−0.270972
$668$	0	0
$669$	1.55990	0.0603092
$670$	0	0
$671$	−41.2514	−1.59249
$672$	0	0
$673$	−16.7774	−0.646723	−0.323361	−	0.946276i	$-0.604813\pi$
$673$	−16.7774	−0.646723	−0.323361	+	0.946276i	$0.604813\pi$
$674$	0	0
$675$	−14.4470	−0.556066
$676$	0	0
$677$	−8.46021	−0.325152	−0.162576	−	0.986696i	$-0.551980\pi$
$677$	−8.46021	−0.325152	−0.162576	+	0.986696i	$0.551980\pi$
$678$	0	0
$679$	31.8383	1.22184
$680$	0	0
$681$	−10.7161	−0.410642
$682$	0	0
$683$	18.0048	0.688935	0.344467	−	0.938798i	$-0.388060\pi$
$683$	18.0048	0.688935	0.344467	+	0.938798i	$0.388060\pi$
$684$	0	0
$685$	−15.5816	−0.595344
$686$	0	0
$687$	30.4269	1.16086
$688$	0	0
$689$	9.73131	0.370733
$690$	0	0
$691$	5.44316	0.207068	0.103534	−	0.994626i	$-0.466985\pi$
$691$	5.44316	0.207068	0.103534	+	0.994626i	$0.466985\pi$
$692$	0	0
$693$	38.1127	1.44778
$694$	0	0
$695$	3.21114	0.121806
$696$	0	0
$697$	−2.31557	−0.0877084
$698$	0	0
$699$	44.8732	1.69726
$700$	0	0
$701$	−3.41090	−0.128828	−0.0644140	−	0.997923i	$-0.520518\pi$
$701$	−3.41090	−0.128828	−0.0644140	+	0.997923i	$0.520518\pi$
$702$	0	0
$703$	4.44191	0.167530
$704$	0	0
$705$	93.9244	3.53740
$706$	0	0
$707$	54.7067	2.05746
$708$	0	0
$709$	13.8789	0.521234	0.260617	−	0.965442i	$-0.416074\pi$
$709$	13.8789	0.521234	0.260617	+	0.965442i	$0.416074\pi$
$710$	0	0
$711$	4.86089	0.182298
$712$	0	0
$713$	19.3009	0.722824
$714$	0	0
$715$	−16.2761	−0.608692
$716$	0	0
$717$	40.7969	1.52359
$718$	0	0
$719$	−22.1395	−0.825663	−0.412831	−	0.910808i	$-0.635460\pi$
$719$	−22.1395	−0.825663	−0.412831	+	0.910808i	$0.635460\pi$
$720$	0	0
$721$	5.27732	0.196538
$722$	0	0
$723$	59.4503	2.21098
$724$	0	0
$725$	−10.5753	−0.392757
$726$	0	0
$727$	−16.3001	−0.604539	−0.302269	−	0.953223i	$-0.597744\pi$
$727$	−16.3001	−0.604539	−0.302269	+	0.953223i	$0.597744\pi$
$728$	0	0
$729$	−43.3400	−1.60519
$730$	0	0
$731$	3.97443	0.147000
$732$	0	0
$733$	−39.8299	−1.47115	−0.735575	−	0.677443i	$-0.763088\pi$
$733$	−39.8299	−1.47115	−0.735575	+	0.677443i	$0.763088\pi$
$734$	0	0
$735$	−7.28775	−0.268813
$736$	0	0
$737$	46.7799	1.72316
$738$	0	0
$739$	40.6147	1.49404	0.747019	−	0.664803i	$-0.231485\pi$
$739$	40.6147	1.49404	0.747019	+	0.664803i	$0.231485\pi$
$740$	0	0
$741$	3.62198	0.133057
$742$	0	0
$743$	−27.4295	−1.00629	−0.503145	−	0.864202i	$-0.667824\pi$
$743$	−27.4295	−1.00629	−0.503145	+	0.864202i	$0.667824\pi$
$744$	0	0
$745$	5.25743	0.192617
$746$	0	0
$747$	52.1599	1.90843
$748$	0	0
$749$	7.38676	0.269906
$750$	0	0
$751$	−10.8572	−0.396186	−0.198093	−	0.980183i	$-0.563475\pi$
$751$	−10.8572	−0.396186	−0.198093	+	0.980183i	$0.563475\pi$
$752$	0	0
$753$	53.1096	1.93542
$754$	0	0
$755$	21.8951	0.796844
$756$	0	0
$757$	44.1358	1.60414	0.802071	−	0.597229i	$-0.203732\pi$
$757$	44.1358	1.60414	0.802071	+	0.597229i	$0.203732\pi$
$758$	0	0
$759$	22.2326	0.806991
$760$	0	0
$761$	1.93822	0.0702603	0.0351302	−	0.999383i	$-0.488815\pi$
$761$	1.93822	0.0702603	0.0351302	+	0.999383i	$0.488815\pi$
$762$	0	0
$763$	14.5180	0.525588
$764$	0	0
$765$	4.61143	0.166726
$766$	0	0
$767$	−10.3487	−0.373668
$768$	0	0
$769$	21.2432	0.766051	0.383025	−	0.923738i	$-0.374882\pi$
$769$	21.2432	0.766051	0.383025	+	0.923738i	$0.374882\pi$
$770$	0	0
$771$	21.7900	0.784746
$772$	0	0
$773$	24.0879	0.866382	0.433191	−	0.901302i	$-0.357388\pi$
$773$	24.0879	0.866382	0.433191	+	0.901302i	$0.357388\pi$
$774$	0	0
$775$	29.1664	1.04769
$776$	0	0
$777$	−44.0406	−1.57995
$778$	0	0
$779$	5.01655	0.179737
$780$	0	0
$781$	3.63911	0.130218
$782$	0	0
$783$	9.69221	0.346371
$784$	0	0
$785$	−40.7439	−1.45421
$786$	0	0
$787$	37.1361	1.32376	0.661880	−	0.749610i	$-0.269759\pi$
$787$	37.1361	1.32376	0.661880	+	0.749610i	$0.269759\pi$
$788$	0	0
$789$	61.7831	2.19954
$790$	0	0
$791$	−31.1485	−1.10751
$792$	0	0
$793$	22.9884	0.816343
$794$	0	0
$795$	−45.3835	−1.60959
$796$	0	0
$797$	48.0147	1.70077	0.850385	−	0.526161i	$-0.176369\pi$
$797$	48.0147	1.70077	0.850385	+	0.526161i	$0.176369\pi$
$798$	0	0
$799$	−4.10300	−0.145154
$800$	0	0
$801$	36.8083	1.30056
$802$	0	0
$803$	−23.1368	−0.816479
$804$	0	0
$805$	−22.1147	−0.779440
$806$	0	0
$807$	54.4429	1.91648
$808$	0	0
$809$	−3.38115	−0.118875	−0.0594375	−	0.998232i	$-0.518931\pi$
$809$	−3.38115	−0.118875	−0.0594375	+	0.998232i	$0.518931\pi$
$810$	0	0
$811$	−56.4925	−1.98372	−0.991860	−	0.127335i	$-0.959358\pi$
$811$	−56.4925	−1.98372	−0.991860	+	0.127335i	$0.959358\pi$
$812$	0	0
$813$	87.8678	3.08166
$814$	0	0
$815$	41.2687	1.44558
$816$	0	0
$817$	−8.61037	−0.301239
$818$	0	0
$819$	−21.2393	−0.742160
$820$	0	0
$821$	43.8042	1.52878	0.764389	−	0.644756i	$-0.223041\pi$
$821$	43.8042	1.52878	0.764389	+	0.644756i	$0.223041\pi$
$822$	0	0
$823$	−9.62813	−0.335616	−0.167808	−	0.985820i	$-0.553669\pi$
$823$	−9.62813	−0.335616	−0.167808	+	0.985820i	$0.553669\pi$
$824$	0	0
$825$	33.5966	1.16968
$826$	0	0
$827$	−8.45617	−0.294050	−0.147025	−	0.989133i	$-0.546970\pi$
$827$	−8.45617	−0.294050	−0.147025	+	0.989133i	$0.546970\pi$
$828$	0	0
$829$	32.5042	1.12892	0.564458	−	0.825462i	$-0.309085\pi$
$829$	32.5042	1.12892	0.564458	+	0.825462i	$0.309085\pi$
$830$	0	0
$831$	76.3175	2.64742
$832$	0	0
$833$	0.318358	0.0110305
$834$	0	0
$835$	0.645321	0.0223323
$836$	0	0
$837$	−26.7308	−0.923952
$838$	0	0
$839$	−22.5760	−0.779409	−0.389705	−	0.920940i	$-0.627423\pi$
$839$	−22.5760	−0.779409	−0.389705	+	0.920940i	$0.627423\pi$
$840$	0	0
$841$	−21.9053	−0.755353
$842$	0	0
$843$	30.4508	1.04878
$844$	0	0
$845$	−29.8653	−1.02740
$846$	0	0
$847$	3.50835	0.120548
$848$	0	0
$849$	19.5980	0.672600
$850$	0	0
$851$	−15.1944	−0.520857
$852$	0	0
$853$	−15.9550	−0.546288	−0.273144	−	0.961973i	$-0.588064\pi$
$853$	−15.9550	−0.546288	−0.273144	+	0.961973i	$0.588064\pi$
$854$	0	0
$855$	−9.99039	−0.341664
$856$	0	0
$857$	41.3618	1.41289	0.706447	−	0.707766i	$-0.250297\pi$
$857$	41.3618	1.41289	0.706447	+	0.707766i	$0.250297\pi$
$858$	0	0
$859$	32.3343	1.10323	0.551616	−	0.834098i	$-0.314011\pi$
$859$	32.3343	1.10323	0.551616	+	0.834098i	$0.314011\pi$
$860$	0	0
$861$	−49.7380	−1.69507
$862$	0	0
$863$	−7.96393	−0.271095	−0.135548	−	0.990771i	$-0.543279\pi$
$863$	−7.96393	−0.271095	−0.135548	+	0.990771i	$0.543279\pi$
$864$	0	0
$865$	−4.19643	−0.142683
$866$	0	0
$867$	45.7254	1.55292
$868$	0	0
$869$	−3.49527	−0.118569
$870$	0	0
$871$	−26.0693	−0.883324
$872$	0	0
$873$	−49.2001	−1.66517
$874$	0	0
$875$	8.66698	0.292997
$876$	0	0
$877$	−23.1312	−0.781084	−0.390542	−	0.920585i	$-0.627712\pi$
$877$	−23.1312	−0.781084	−0.390542	+	0.920585i	$0.627712\pi$
$878$	0	0
$879$	37.3100	1.25843
$880$	0	0
$881$	−8.58314	−0.289173	−0.144587	−	0.989492i	$-0.546185\pi$
$881$	−8.58314	−0.289173	−0.144587	+	0.989492i	$0.546185\pi$
$882$	0	0
$883$	−8.75584	−0.294657	−0.147329	−	0.989088i	$-0.547068\pi$
$883$	−8.75584	−0.294657	−0.147329	+	0.989088i	$0.547068\pi$
$884$	0	0
$885$	48.2626	1.62233
$886$	0	0
$887$	−0.196253	−0.00658952	−0.00329476	−	0.999995i	$-0.501049\pi$
$887$	−0.196253	−0.00658952	−0.00329476	+	0.999995i	$0.501049\pi$
$888$	0	0
$889$	−26.9133	−0.902642
$890$	0	0
$891$	9.89382	0.331455
$892$	0	0
$893$	8.88891	0.297456
$894$	0	0
$895$	34.0376	1.13775
$896$	0	0
$897$	−12.3897	−0.413679
$898$	0	0
$899$	−19.5671	−0.652599
$900$	0	0
$901$	1.98253	0.0660478
$902$	0	0
$903$	85.3701	2.84094
$904$	0	0
$905$	21.2616	0.706759
$906$	0	0
$907$	40.9980	1.36132	0.680658	−	0.732601i	$-0.261694\pi$
$907$	40.9980	1.36132	0.680658	+	0.732601i	$0.261694\pi$
$908$	0	0
$909$	−84.5388	−2.80398
$910$	0	0
$911$	3.20171	0.106078	0.0530388	−	0.998592i	$-0.483109\pi$
$911$	3.20171	0.106078	0.0530388	+	0.998592i	$0.483109\pi$
$912$	0	0
$913$	−37.5061	−1.24127
$914$	0	0
$915$	−107.210	−3.54426
$916$	0	0
$917$	26.6532	0.880167
$918$	0	0
$919$	14.1175	0.465693	0.232847	−	0.972513i	$-0.425196\pi$
$919$	14.1175	0.465693	0.232847	+	0.972513i	$0.425196\pi$
$920$	0	0
$921$	6.83180	0.225115
$922$	0	0
$923$	−2.02799	−0.0667521
$924$	0	0
$925$	−22.9609	−0.754948
$926$	0	0
$927$	−8.15510	−0.267849
$928$	0	0
$929$	11.5018	0.377362	0.188681	−	0.982038i	$-0.439579\pi$
$929$	11.5018	0.377362	0.188681	+	0.982038i	$0.439579\pi$
$930$	0	0
$931$	−0.689705	−0.0226042
$932$	0	0
$933$	−20.0403	−0.656090
$934$	0	0
$935$	−3.31589	−0.108441
$936$	0	0
$937$	9.71496	0.317374	0.158687	−	0.987329i	$-0.449274\pi$
$937$	9.71496	0.317374	0.158687	+	0.987329i	$0.449274\pi$
$938$	0	0
$939$	−60.7904	−1.98382
$940$	0	0
$941$	−38.7861	−1.26439	−0.632196	−	0.774808i	$-0.717846\pi$
$941$	−38.7861	−1.26439	−0.632196	+	0.774808i	$0.717846\pi$
$942$	0	0
$943$	−17.1601	−0.558808
$944$	0	0
$945$	30.6278	0.996322
$946$	0	0
$947$	22.6534	0.736137	0.368069	−	0.929799i	$-0.380019\pi$
$947$	22.6534	0.736137	0.368069	+	0.929799i	$0.380019\pi$
$948$	0	0
$949$	12.8936	0.418543
$950$	0	0
$951$	−38.7407	−1.25625
$952$	0	0
$953$	−1.43476	−0.0464763	−0.0232381	−	0.999730i	$-0.507398\pi$
$953$	−1.43476	−0.0464763	−0.0232381	+	0.999730i	$0.507398\pi$
$954$	0	0
$955$	2.07550	0.0671615
$956$	0	0
$957$	−22.5392	−0.728589
$958$	0	0
$959$	14.6207	0.472126
$960$	0	0
$961$	22.9655	0.740821
$962$	0	0
$963$	−11.4148	−0.367838
$964$	0	0
$965$	−15.5273	−0.499842
$966$	0	0
$967$	41.9663	1.34954	0.674772	−	0.738026i	$-0.264242\pi$
$967$	41.9663	1.34954	0.674772	+	0.738026i	$0.264242\pi$
$968$	0	0
$969$	0.737897	0.0237047
$970$	0	0
$971$	−25.8547	−0.829715	−0.414858	−	0.909886i	$-0.636169\pi$
$971$	−25.8547	−0.829715	−0.414858	+	0.909886i	$0.636169\pi$
$972$	0	0
$973$	−3.01310	−0.0965955
$974$	0	0
$975$	−18.7225	−0.599601
$976$	0	0
$977$	−50.1775	−1.60532	−0.802660	−	0.596437i	$-0.796583\pi$
$977$	−50.1775	−1.60532	−0.802660	+	0.596437i	$0.796583\pi$
$978$	0	0
$979$	−26.4674	−0.845902
$980$	0	0
$981$	−22.4349	−0.716290
$982$	0	0
$983$	33.2745	1.06129	0.530646	−	0.847593i	$-0.321949\pi$
$983$	33.2745	1.06129	0.530646	+	0.847593i	$0.321949\pi$
$984$	0	0
$985$	23.8694	0.760543
$986$	0	0
$987$	−88.1317	−2.80526
$988$	0	0
$989$	29.4534	0.936564
$990$	0	0
$991$	49.8364	1.58311	0.791553	−	0.611101i	$-0.209273\pi$
$991$	49.8364	1.58311	0.791553	+	0.611101i	$0.209273\pi$
$992$	0	0
$993$	−8.55940	−0.271624
$994$	0	0
$995$	−50.8986	−1.61359
$996$	0	0
$997$	2.54632	0.0806426	0.0403213	−	0.999187i	$-0.487162\pi$
$997$	2.54632	0.0806426	0.0403213	+	0.999187i	$0.487162\pi$
$998$	0	0
$999$	21.0435	0.665787

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4024.2.a.e.1.4	✓	29
4.3	odd	2		8048.2.a.w.1.26		29

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.e.1.4	✓	29	1.1	even	1	trivial
8048.2.a.w.1.26		29	4.3	odd	2

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:	\( N \)	\(=\)	\( 4024 = 2^{3} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4024.a (trivial)

\(f(q)\)	\(=\)	\(q-2.70977 q^{3} +2.99505 q^{5} -2.81033 q^{7} +4.34283 q^{9} +O(q^{10})\)	\(q-2.70977 q^{3} +2.99505 q^{5} -2.81033 q^{7} +4.34283 q^{9} -3.12276 q^{11} +1.74024 q^{13} -8.11588 q^{15} +0.354534 q^{17} -0.768079 q^{19} +7.61534 q^{21} +2.62736 q^{23} +3.97031 q^{25} -3.63877 q^{27} -2.66360 q^{29} +7.34612 q^{31} +8.46195 q^{33} -8.41708 q^{35} -5.78314 q^{37} -4.71564 q^{39} -6.53130 q^{41} +11.2103 q^{43} +13.0070 q^{45} -11.5729 q^{47} +0.897962 q^{49} -0.960705 q^{51} +5.59194 q^{53} -9.35281 q^{55} +2.08131 q^{57} -5.94669 q^{59} +13.2099 q^{61} -12.2048 q^{63} +5.21209 q^{65} -14.9803 q^{67} -7.11953 q^{69} -1.16535 q^{71} +7.40908 q^{73} -10.7586 q^{75} +8.77599 q^{77} +1.11929 q^{79} -3.16829 q^{81} +12.0106 q^{83} +1.06185 q^{85} +7.21773 q^{87} +8.47565 q^{89} -4.89064 q^{91} -19.9063 q^{93} -2.30043 q^{95} -11.3290 q^{97} -13.5616 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 29 q - 7 q^{3} - 4 q^{5} - 13 q^{7} + 20 q^{9}+O(q^{10}) \) 29 * q - 7 * q^3 - 4 * q^5 - 13 * q^7 + 20 * q^9	\( 29 q - 7 q^{3} - 4 q^{5} - 13 q^{7} + 20 q^{9} - 27 q^{11} + 16 q^{13} - 14 q^{15} - 15 q^{17} - 14 q^{19} + q^{21} - 25 q^{23} + 21 q^{25} - 25 q^{27} - 13 q^{29} - 27 q^{31} - 9 q^{33} - 29 q^{35} + 35 q^{37} - 38 q^{39} - 30 q^{41} - 38 q^{43} + q^{45} - 35 q^{47} + 14 q^{49} - 21 q^{51} + 2 q^{53} - 25 q^{55} - 25 q^{57} - 40 q^{59} + 10 q^{61} - 56 q^{63} - 50 q^{65} - 31 q^{67} + 11 q^{69} - 65 q^{71} - 23 q^{73} - 32 q^{75} + 13 q^{77} - 44 q^{79} - 7 q^{81} - 41 q^{83} + 26 q^{85} - 25 q^{87} - 48 q^{89} - 44 q^{91} + 25 q^{93} - 75 q^{95} - 18 q^{97} - 80 q^{99}+O(q^{100}) \) 29 * q - 7 * q^3 - 4 * q^5 - 13 * q^7 + 20 * q^9 - 27 * q^11 + 16 * q^13 - 14 * q^15 - 15 * q^17 - 14 * q^19 + q^21 - 25 * q^23 + 21 * q^25 - 25 * q^27 - 13 * q^29 - 27 * q^31 - 9 * q^33 - 29 * q^35 + 35 * q^37 - 38 * q^39 - 30 * q^41 - 38 * q^43 + q^45 - 35 * q^47 + 14 * q^49 - 21 * q^51 + 2 * q^53 - 25 * q^55 - 25 * q^57 - 40 * q^59 + 10 * q^61 - 56 * q^63 - 50 * q^65 - 31 * q^67 + 11 * q^69 - 65 * q^71 - 23 * q^73 - 32 * q^75 + 13 * q^77 - 44 * q^79 - 7 * q^81 - 41 * q^83 + 26 * q^85 - 25 * q^87 - 48 * q^89 - 44 * q^91 + 25 * q^93 - 75 * q^95 - 18 * q^97 - 80 * q^99

Introduction

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Modular forms

Varieties

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