LMFDB - Embedded newform 4024.2.a.e.1.11

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			1.11
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)

$f(q)$	$=$	$q-1.10456 q^{3} -2.70098 q^{5} +3.17910 q^{7} -1.77994 q^{9} +O(q^{10})$	$q-1.10456 q^{3} -2.70098 q^{5} +3.17910 q^{7} -1.77994 q^{9} +1.34124 q^{11} -4.41191 q^{13} +2.98341 q^{15} -0.889369 q^{17} +7.80979 q^{19} -3.51153 q^{21} -6.37871 q^{23} +2.29528 q^{25} +5.27975 q^{27} +4.35006 q^{29} +7.10696 q^{31} -1.48148 q^{33} -8.58669 q^{35} +1.46251 q^{37} +4.87324 q^{39} -9.53740 q^{41} +0.885660 q^{43} +4.80757 q^{45} +6.15515 q^{47} +3.10670 q^{49} +0.982366 q^{51} -6.04722 q^{53} -3.62265 q^{55} -8.62643 q^{57} -7.23783 q^{59} +0.532473 q^{61} -5.65860 q^{63} +11.9165 q^{65} +11.1711 q^{67} +7.04570 q^{69} -6.84254 q^{71} -9.94461 q^{73} -2.53529 q^{75} +4.26393 q^{77} +8.29396 q^{79} -0.492018 q^{81} +11.7595 q^{83} +2.40217 q^{85} -4.80492 q^{87} -18.1492 q^{89} -14.0259 q^{91} -7.85010 q^{93} -21.0941 q^{95} +3.51041 q^{97} -2.38732 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.10456	−0.637721	−0.318860	−	0.947802i	$-0.603300\pi$
$3$	−1.10456	−0.637721	−0.318860	+	0.947802i	$0.603300\pi$
$4$	0	0
$5$	−2.70098	−1.20791	−0.603957	−	0.797017i	$-0.706410\pi$
$5$	−2.70098	−1.20791	−0.603957	+	0.797017i	$0.706410\pi$
$6$	0	0
$7$	3.17910	1.20159	0.600794	−	0.799404i	$-0.294851\pi$
$7$	3.17910	1.20159	0.600794	+	0.799404i	$0.294851\pi$
$8$	0	0
$9$	−1.77994	−0.593312
$10$	0	0
$11$	1.34124	0.404398	0.202199	−	0.979344i	$-0.435191\pi$
$11$	1.34124	0.404398	0.202199	+	0.979344i	$0.435191\pi$
$12$	0	0
$13$	−4.41191	−1.22364	−0.611821	−	0.790996i	$-0.709563\pi$
$13$	−4.41191	−1.22364	−0.611821	+	0.790996i	$0.709563\pi$
$14$	0	0
$15$	2.98341	0.770312
$16$	0	0
$17$	−0.889369	−0.215704	−0.107852	−	0.994167i	$-0.534397\pi$
$17$	−0.889369	−0.215704	−0.107852	+	0.994167i	$0.534397\pi$
$18$	0	0
$19$	7.80979	1.79169	0.895845	−	0.444367i	$-0.146571\pi$
$19$	7.80979	1.79169	0.895845	+	0.444367i	$0.146571\pi$
$20$	0	0
$21$	−3.51153	−0.766278
$22$	0	0
$23$	−6.37871	−1.33005	−0.665027	−	0.746819i	$-0.731580\pi$
$23$	−6.37871	−1.33005	−0.665027	+	0.746819i	$0.731580\pi$
$24$	0	0
$25$	2.29528	0.459057
$26$	0	0
$27$	5.27975	1.01609
$28$	0	0
$29$	4.35006	0.807785	0.403893	−	0.914806i	$-0.367657\pi$
$29$	4.35006	0.807785	0.403893	+	0.914806i	$0.367657\pi$
$30$	0	0
$31$	7.10696	1.27645	0.638224	−	0.769851i	$-0.279669\pi$
$31$	7.10696	1.27645	0.638224	+	0.769851i	$0.279669\pi$
$32$	0	0
$33$	−1.48148	−0.257893
$34$	0	0
$35$	−8.58669	−1.45142
$36$	0	0
$37$	1.46251	0.240436	0.120218	−	0.992748i	$-0.461641\pi$
$37$	1.46251	0.240436	0.120218	+	0.992748i	$0.461641\pi$
$38$	0	0
$39$	4.87324	0.780343
$40$	0	0
$41$	−9.53740	−1.48949	−0.744746	−	0.667348i	$-0.767429\pi$
$41$	−9.53740	−1.48949	−0.744746	+	0.667348i	$0.767429\pi$
$42$	0	0
$43$	0.885660	0.135062	0.0675309	−	0.997717i	$-0.478488\pi$
$43$	0.885660	0.135062	0.0675309	+	0.997717i	$0.478488\pi$
$44$	0	0
$45$	4.80757	0.716670
$46$	0	0
$47$	6.15515	0.897821	0.448910	−	0.893577i	$-0.351812\pi$
$47$	6.15515	0.897821	0.448910	+	0.893577i	$0.351812\pi$
$48$	0	0
$49$	3.10670	0.443815
$50$	0	0
$51$	0.982366	0.137559
$52$	0	0
$53$	−6.04722	−0.830649	−0.415325	−	0.909673i	$-0.636332\pi$
$53$	−6.04722	−0.830649	−0.415325	+	0.909673i	$0.636332\pi$
$54$	0	0
$55$	−3.62265	−0.488478
$56$	0	0
$57$	−8.62643	−1.14260
$58$	0	0
$59$	−7.23783	−0.942285	−0.471143	−	0.882057i	$-0.656158\pi$
$59$	−7.23783	−0.942285	−0.471143	+	0.882057i	$0.656158\pi$
$60$	0	0
$61$	0.532473	0.0681762	0.0340881	−	0.999419i	$-0.489147\pi$
$61$	0.532473	0.0681762	0.0340881	+	0.999419i	$0.489147\pi$
$62$	0	0
$63$	−5.65860	−0.712917
$64$	0	0
$65$	11.9165	1.47806
$66$	0	0
$67$	11.1711	1.36476	0.682381	−	0.730997i	$-0.260945\pi$
$67$	11.1711	1.36476	0.682381	+	0.730997i	$0.260945\pi$
$68$	0	0
$69$	7.04570	0.848203
$70$	0	0
$71$	−6.84254	−0.812060	−0.406030	−	0.913860i	$-0.633087\pi$
$71$	−6.84254	−0.812060	−0.406030	+	0.913860i	$0.633087\pi$
$72$	0	0
$73$	−9.94461	−1.16393	−0.581964	−	0.813214i	$-0.697716\pi$
$73$	−9.94461	−1.16393	−0.581964	+	0.813214i	$0.697716\pi$
$74$	0	0
$75$	−2.53529	−0.292750
$76$	0	0
$77$	4.26393	0.485920
$78$	0	0
$79$	8.29396	0.933143	0.466572	−	0.884483i	$-0.345489\pi$
$79$	8.29396	0.933143	0.466572	+	0.884483i	$0.345489\pi$
$80$	0	0
$81$	−0.492018	−0.0546686
$82$	0	0
$83$	11.7595	1.29078	0.645388	−	0.763855i	$-0.276696\pi$
$83$	11.7595	1.29078	0.645388	+	0.763855i	$0.276696\pi$
$84$	0	0
$85$	2.40217	0.260552
$86$	0	0
$87$	−4.80492	−0.515141
$88$	0	0
$89$	−18.1492	−1.92381	−0.961905	−	0.273384i	$-0.911857\pi$
$89$	−18.1492	−1.92381	−0.961905	+	0.273384i	$0.911857\pi$
$90$	0	0
$91$	−14.0259	−1.47032
$92$	0	0
$93$	−7.85010	−0.814018
$94$	0	0
$95$	−21.0941	−2.16421
$96$	0	0
$97$	3.51041	0.356428	0.178214	−	0.983992i	$-0.442968\pi$
$97$	3.51041	0.356428	0.178214	+	0.983992i	$0.442968\pi$
$98$	0	0
$99$	−2.38732	−0.239934
$100$	0	0
$101$	5.92728	0.589787	0.294893	−	0.955530i	$-0.404716\pi$
$101$	5.92728	0.589787	0.294893	+	0.955530i	$0.404716\pi$
$102$	0	0
$103$	−13.4142	−1.32174	−0.660870	−	0.750501i	$-0.729812\pi$
$103$	−13.4142	−1.32174	−0.660870	+	0.750501i	$0.729812\pi$
$104$	0	0
$105$	9.48456	0.925598
$106$	0	0
$107$	−9.94886	−0.961793	−0.480896	−	0.876777i	$-0.659689\pi$
$107$	−9.94886	−0.961793	−0.480896	+	0.876777i	$0.659689\pi$
$108$	0	0
$109$	2.34714	0.224816	0.112408	−	0.993662i	$-0.464144\pi$
$109$	2.34714	0.224816	0.112408	+	0.993662i	$0.464144\pi$
$110$	0	0
$111$	−1.61544	−0.153331
$112$	0	0
$113$	8.38102	0.788420	0.394210	−	0.919020i	$-0.371018\pi$
$113$	8.38102	0.788420	0.394210	+	0.919020i	$0.371018\pi$
$114$	0	0
$115$	17.2288	1.60659
$116$	0	0
$117$	7.85291	0.726002
$118$	0	0
$119$	−2.82740	−0.259187
$120$	0	0
$121$	−9.20109	−0.836462
$122$	0	0
$123$	10.5347	0.949880
$124$	0	0
$125$	7.30538	0.653413
$126$	0	0
$127$	14.2469	1.26421	0.632104	−	0.774884i	$-0.282192\pi$
$127$	14.2469	1.26421	0.632104	+	0.774884i	$0.282192\pi$
$128$	0	0
$129$	−0.978269	−0.0861317
$130$	0	0
$131$	−10.5500	−0.921757	−0.460879	−	0.887463i	$-0.652466\pi$
$131$	−10.5500	−0.921757	−0.460879	+	0.887463i	$0.652466\pi$
$132$	0	0
$133$	24.8282	2.15287
$134$	0	0
$135$	−14.2605	−1.22735
$136$	0	0
$137$	9.04935	0.773138	0.386569	−	0.922260i	$-0.373660\pi$
$137$	9.04935	0.773138	0.386569	+	0.922260i	$0.373660\pi$
$138$	0	0
$139$	−9.71190	−0.823752	−0.411876	−	0.911240i	$-0.635126\pi$
$139$	−9.71190	−0.823752	−0.411876	+	0.911240i	$0.635126\pi$
$140$	0	0
$141$	−6.79876	−0.572559
$142$	0	0
$143$	−5.91741	−0.494839
$144$	0	0
$145$	−11.7494	−0.975735
$146$	0	0
$147$	−3.43155	−0.283030
$148$	0	0
$149$	−11.5236	−0.944054	−0.472027	−	0.881584i	$-0.656478\pi$
$149$	−11.5236	−0.944054	−0.472027	+	0.881584i	$0.656478\pi$
$150$	0	0
$151$	−11.4885	−0.934923	−0.467461	−	0.884014i	$-0.654831\pi$
$151$	−11.4885	−0.934923	−0.467461	+	0.884014i	$0.654831\pi$
$152$	0	0
$153$	1.58302	0.127980
$154$	0	0
$155$	−19.1958	−1.54184
$156$	0	0
$157$	2.91858	0.232928	0.116464	−	0.993195i	$-0.462844\pi$
$157$	2.91858	0.232928	0.116464	+	0.993195i	$0.462844\pi$
$158$	0	0
$159$	6.67954	0.529722
$160$	0	0
$161$	−20.2786	−1.59818
$162$	0	0
$163$	−13.6579	−1.06977	−0.534884	−	0.844926i	$-0.679645\pi$
$163$	−13.6579	−1.06977	−0.534884	+	0.844926i	$0.679645\pi$
$164$	0	0
$165$	4.00145	0.311513
$166$	0	0
$167$	7.00489	0.542055	0.271027	−	0.962572i	$-0.412637\pi$
$167$	7.00489	0.542055	0.271027	+	0.962572i	$0.412637\pi$
$168$	0	0
$169$	6.46493	0.497302
$170$	0	0
$171$	−13.9009	−1.06303
$172$	0	0
$173$	14.0549	1.06858	0.534288	−	0.845302i	$-0.320580\pi$
$173$	14.0549	1.06858	0.534288	+	0.845302i	$0.320580\pi$
$174$	0	0
$175$	7.29695	0.551597
$176$	0	0
$177$	7.99465	0.600915
$178$	0	0
$179$	−26.1006	−1.95085	−0.975425	−	0.220330i	$-0.929287\pi$
$179$	−26.1006	−1.95085	−0.975425	+	0.220330i	$0.929287\pi$
$180$	0	0
$181$	−14.5864	−1.08420	−0.542100	−	0.840314i	$-0.682371\pi$
$181$	−14.5864	−1.08420	−0.542100	+	0.840314i	$0.682371\pi$
$182$	0	0
$183$	−0.588151	−0.0434774
$184$	0	0
$185$	−3.95022	−0.290426
$186$	0	0
$187$	−1.19285	−0.0872301
$188$	0	0
$189$	16.7849	1.22092
$190$	0	0
$191$	−5.04805	−0.365264	−0.182632	−	0.983181i	$-0.558462\pi$
$191$	−5.04805	−0.365264	−0.182632	+	0.983181i	$0.558462\pi$
$192$	0	0
$193$	−19.2569	−1.38614	−0.693070	−	0.720871i	$-0.743742\pi$
$193$	−19.2569	−1.38614	−0.693070	+	0.720871i	$0.743742\pi$
$194$	0	0
$195$	−13.1625	−0.942587
$196$	0	0
$197$	−1.73951	−0.123935	−0.0619674	−	0.998078i	$-0.519737\pi$
$197$	−1.73951	−0.123935	−0.0619674	+	0.998078i	$0.519737\pi$
$198$	0	0
$199$	−7.90361	−0.560272	−0.280136	−	0.959960i	$-0.590380\pi$
$199$	−7.90361	−0.560272	−0.280136	+	0.959960i	$0.590380\pi$
$200$	0	0
$201$	−12.3392	−0.870337
$202$	0	0
$203$	13.8293	0.970625
$204$	0	0
$205$	25.7603	1.79918
$206$	0	0
$207$	11.3537	0.789137
$208$	0	0
$209$	10.4748	0.724556
$210$	0	0
$211$	−9.18065	−0.632022	−0.316011	−	0.948756i	$-0.602344\pi$
$211$	−9.18065	−0.632022	−0.316011	+	0.948756i	$0.602344\pi$
$212$	0	0
$213$	7.55803	0.517867
$214$	0	0
$215$	−2.39215	−0.163143
$216$	0	0
$217$	22.5938	1.53377
$218$	0	0
$219$	10.9845	0.742262
$220$	0	0
$221$	3.92381	0.263944
$222$	0	0
$223$	−26.9362	−1.80378	−0.901890	−	0.431965i	$-0.857820\pi$
$223$	−26.9362	−1.80378	−0.901890	+	0.431965i	$0.857820\pi$
$224$	0	0
$225$	−4.08546	−0.272364
$226$	0	0
$227$	24.5406	1.62881	0.814407	−	0.580294i	$-0.197062\pi$
$227$	24.5406	1.62881	0.814407	+	0.580294i	$0.197062\pi$
$228$	0	0
$229$	−0.429634	−0.0283910	−0.0141955	−	0.999899i	$-0.504519\pi$
$229$	−0.429634	−0.0283910	−0.0141955	+	0.999899i	$0.504519\pi$
$230$	0	0
$231$	−4.70979	−0.309881
$232$	0	0
$233$	−5.52478	−0.361941	−0.180970	−	0.983489i	$-0.557924\pi$
$233$	−5.52478	−0.361941	−0.180970	+	0.983489i	$0.557924\pi$
$234$	0	0
$235$	−16.6249	−1.08449
$236$	0	0
$237$	−9.16122	−0.595085
$238$	0	0
$239$	−5.12741	−0.331665	−0.165832	−	0.986154i	$-0.553031\pi$
$239$	−5.12741	−0.331665	−0.165832	+	0.986154i	$0.553031\pi$
$240$	0	0
$241$	−16.6988	−1.07566	−0.537831	−	0.843053i	$-0.680756\pi$
$241$	−16.6988	−1.07566	−0.537831	+	0.843053i	$0.680756\pi$
$242$	0	0
$243$	−15.2958	−0.981225
$244$	0	0
$245$	−8.39114	−0.536090
$246$	0	0
$247$	−34.4561	−2.19239
$248$	0	0
$249$	−12.9892	−0.823155
$250$	0	0
$251$	−8.77558	−0.553910	−0.276955	−	0.960883i	$-0.589325\pi$
$251$	−8.77558	−0.553910	−0.276955	+	0.960883i	$0.589325\pi$
$252$	0	0
$253$	−8.55536	−0.537871
$254$	0	0
$255$	−2.65335	−0.166159
$256$	0	0
$257$	−12.0030	−0.748724	−0.374362	−	0.927283i	$-0.622138\pi$
$257$	−12.0030	−0.748724	−0.374362	+	0.927283i	$0.622138\pi$
$258$	0	0
$259$	4.64948	0.288905
$260$	0	0
$261$	−7.74282	−0.479269
$262$	0	0
$263$	−28.6541	−1.76689	−0.883444	−	0.468537i	$-0.844781\pi$
$263$	−28.6541	−1.76689	−0.883444	+	0.468537i	$0.844781\pi$
$264$	0	0
$265$	16.3334	1.00335
$266$	0	0
$267$	20.0470	1.22685
$268$	0	0
$269$	−14.2025	−0.865940	−0.432970	−	0.901408i	$-0.642534\pi$
$269$	−14.2025	−0.865940	−0.432970	+	0.901408i	$0.642534\pi$
$270$	0	0
$271$	10.3041	0.625929	0.312964	−	0.949765i	$-0.398678\pi$
$271$	10.3041	0.625929	0.312964	+	0.949765i	$0.398678\pi$
$272$	0	0
$273$	15.4925	0.937651
$274$	0	0
$275$	3.07852	0.185642
$276$	0	0
$277$	−17.9985	−1.08143	−0.540713	−	0.841207i	$-0.681846\pi$
$277$	−17.9985	−1.08143	−0.540713	+	0.841207i	$0.681846\pi$
$278$	0	0
$279$	−12.6499	−0.757332
$280$	0	0
$281$	−1.67335	−0.0998235	−0.0499117	−	0.998754i	$-0.515894\pi$
$281$	−1.67335	−0.0998235	−0.0499117	+	0.998754i	$0.515894\pi$
$282$	0	0
$283$	18.1408	1.07836	0.539178	−	0.842192i	$-0.318735\pi$
$283$	18.1408	1.07836	0.539178	+	0.842192i	$0.318735\pi$
$284$	0	0
$285$	23.2998	1.38016
$286$	0	0
$287$	−30.3204	−1.78976
$288$	0	0
$289$	−16.2090	−0.953472
$290$	0	0
$291$	−3.87748	−0.227302
$292$	0	0
$293$	−9.89751	−0.578218	−0.289109	−	0.957296i	$-0.593359\pi$
$293$	−9.89751	−0.578218	−0.289109	+	0.957296i	$0.593359\pi$
$294$	0	0
$295$	19.5492	1.13820
$296$	0	0
$297$	7.08139	0.410904
$298$	0	0
$299$	28.1423	1.62751
$300$	0	0
$301$	2.81560	0.162289
$302$	0	0
$303$	−6.54707	−0.376119
$304$	0	0
$305$	−1.43820	−0.0823510
$306$	0	0
$307$	−1.33888	−0.0764140	−0.0382070	−	0.999270i	$-0.512165\pi$
$307$	−1.33888	−0.0764140	−0.0382070	+	0.999270i	$0.512165\pi$
$308$	0	0
$309$	14.8168	0.842901
$310$	0	0
$311$	6.58308	0.373292	0.186646	−	0.982427i	$-0.440238\pi$
$311$	6.58308	0.373292	0.186646	+	0.982427i	$0.440238\pi$
$312$	0	0
$313$	−29.2468	−1.65313	−0.826564	−	0.562843i	$-0.809708\pi$
$313$	−29.2468	−1.65313	−0.826564	+	0.562843i	$0.809708\pi$
$314$	0	0
$315$	15.2838	0.861143
$316$	0	0
$317$	24.9788	1.40295	0.701473	−	0.712696i	$-0.252526\pi$
$317$	24.9788	1.40295	0.701473	+	0.712696i	$0.252526\pi$
$318$	0	0
$319$	5.83445	0.326667
$320$	0	0
$321$	10.9892	0.613355
$322$	0	0
$323$	−6.94579	−0.386474
$324$	0	0
$325$	−10.1266	−0.561722
$326$	0	0
$327$	−2.59257	−0.143370
$328$	0	0
$329$	19.5679	1.07881
$330$	0	0
$331$	13.2722	0.729506	0.364753	−	0.931104i	$-0.381153\pi$
$331$	13.2722	0.729506	0.364753	+	0.931104i	$0.381153\pi$
$332$	0	0
$333$	−2.60318	−0.142653
$334$	0	0
$335$	−30.1728	−1.64851
$336$	0	0
$337$	−32.6876	−1.78061	−0.890304	−	0.455366i	$-0.849508\pi$
$337$	−32.6876	−1.78061	−0.890304	+	0.455366i	$0.849508\pi$
$338$	0	0
$339$	−9.25738	−0.502792
$340$	0	0
$341$	9.53212	0.516193
$342$	0	0
$343$	−12.3772	−0.668306
$344$	0	0
$345$	−19.0303	−1.02456
$346$	0	0
$347$	−17.2247	−0.924673	−0.462336	−	0.886705i	$-0.652989\pi$
$347$	−17.2247	−0.924673	−0.462336	+	0.886705i	$0.652989\pi$
$348$	0	0
$349$	−8.62312	−0.461585	−0.230793	−	0.973003i	$-0.574132\pi$
$349$	−8.62312	−0.461585	−0.230793	+	0.973003i	$0.574132\pi$
$350$	0	0
$351$	−23.2938	−1.24333
$352$	0	0
$353$	14.9819	0.797407	0.398704	−	0.917080i	$-0.369460\pi$
$353$	14.9819	0.797407	0.398704	+	0.917080i	$0.369460\pi$
$354$	0	0
$355$	18.4815	0.980898
$356$	0	0
$357$	3.12304	0.165289
$358$	0	0
$359$	−12.1093	−0.639102	−0.319551	−	0.947569i	$-0.603532\pi$
$359$	−12.1093	−0.639102	−0.319551	+	0.947569i	$0.603532\pi$
$360$	0	0
$361$	41.9929	2.21015
$362$	0	0
$363$	10.1632	0.533429
$364$	0	0
$365$	26.8602	1.40593
$366$	0	0
$367$	6.97389	0.364034	0.182017	−	0.983295i	$-0.441737\pi$
$367$	6.97389	0.364034	0.182017	+	0.983295i	$0.441737\pi$
$368$	0	0
$369$	16.9760	0.883733
$370$	0	0
$371$	−19.2247	−0.998098
$372$	0	0
$373$	−1.75944	−0.0911005	−0.0455502	−	0.998962i	$-0.514504\pi$
$373$	−1.75944	−0.0911005	−0.0455502	+	0.998962i	$0.514504\pi$
$374$	0	0
$375$	−8.06927	−0.416695
$376$	0	0
$377$	−19.1920	−0.988440
$378$	0	0
$379$	30.6072	1.57218	0.786092	−	0.618109i	$-0.212101\pi$
$379$	30.6072	1.57218	0.786092	+	0.618109i	$0.212101\pi$
$380$	0	0
$381$	−15.7366	−0.806212
$382$	0	0
$383$	3.97507	0.203117	0.101558	−	0.994830i	$-0.467617\pi$
$383$	3.97507	0.203117	0.101558	+	0.994830i	$0.467617\pi$
$384$	0	0
$385$	−11.5168	−0.586950
$386$	0	0
$387$	−1.57642	−0.0801338
$388$	0	0
$389$	4.57204	0.231811	0.115906	−	0.993260i	$-0.463023\pi$
$389$	4.57204	0.231811	0.115906	+	0.993260i	$0.463023\pi$
$390$	0	0
$391$	5.67303	0.286897
$392$	0	0
$393$	11.6532	0.587824
$394$	0	0
$395$	−22.4018	−1.12716
$396$	0	0
$397$	26.1852	1.31420	0.657098	−	0.753805i	$-0.271784\pi$
$397$	26.1852	1.31420	0.657098	+	0.753805i	$0.271784\pi$
$398$	0	0
$399$	−27.4243	−1.37293
$400$	0	0
$401$	−22.0947	−1.10336	−0.551678	−	0.834057i	$-0.686012\pi$
$401$	−22.0947	−1.10336	−0.551678	+	0.834057i	$0.686012\pi$
$402$	0	0
$403$	−31.3553	−1.56192
$404$	0	0
$405$	1.32893	0.0660350
$406$	0	0
$407$	1.96158	0.0972317
$408$	0	0
$409$	10.1127	0.500043	0.250021	−	0.968240i	$-0.419562\pi$
$409$	10.1127	0.500043	0.250021	+	0.968240i	$0.419562\pi$
$410$	0	0
$411$	−9.99560	−0.493046
$412$	0	0
$413$	−23.0098	−1.13224
$414$	0	0
$415$	−31.7622	−1.55915
$416$	0	0
$417$	10.7274	0.525324
$418$	0	0
$419$	−9.67002	−0.472412	−0.236206	−	0.971703i	$-0.575904\pi$
$419$	−9.67002	−0.472412	−0.236206	+	0.971703i	$0.575904\pi$
$420$	0	0
$421$	20.0191	0.975671	0.487836	−	0.872935i	$-0.337787\pi$
$421$	20.0191	0.975671	0.487836	+	0.872935i	$0.337787\pi$
$422$	0	0
$423$	−10.9558	−0.532688
$424$	0	0
$425$	−2.04135	−0.0990202
$426$	0	0
$427$	1.69279	0.0819197
$428$	0	0
$429$	6.53616	0.315569
$430$	0	0
$431$	−10.9741	−0.528605	−0.264303	−	0.964440i	$-0.585142\pi$
$431$	−10.9741	−0.528605	−0.264303	+	0.964440i	$0.585142\pi$
$432$	0	0
$433$	18.4519	0.886741	0.443370	−	0.896339i	$-0.353783\pi$
$433$	18.4519	0.886741	0.443370	+	0.896339i	$0.353783\pi$
$434$	0	0
$435$	12.9780	0.622247
$436$	0	0
$437$	−49.8164	−2.38304
$438$	0	0
$439$	−12.7669	−0.609332	−0.304666	−	0.952459i	$-0.598545\pi$
$439$	−12.7669	−0.609332	−0.304666	+	0.952459i	$0.598545\pi$
$440$	0	0
$441$	−5.52973	−0.263321
$442$	0	0
$443$	24.1635	1.14804	0.574022	−	0.818840i	$-0.305382\pi$
$443$	24.1635	1.14804	0.574022	+	0.818840i	$0.305382\pi$
$444$	0	0
$445$	49.0206	2.32380
$446$	0	0
$447$	12.7286	0.602043
$448$	0	0
$449$	−10.6864	−0.504321	−0.252161	−	0.967685i	$-0.581141\pi$
$449$	−10.6864	−0.504321	−0.252161	+	0.967685i	$0.581141\pi$
$450$	0	0
$451$	−12.7919	−0.602347
$452$	0	0
$453$	12.6898	0.596220
$454$	0	0
$455$	37.8837	1.77601
$456$	0	0
$457$	−6.07758	−0.284297	−0.142149	−	0.989845i	$-0.545401\pi$
$457$	−6.07758	−0.284297	−0.142149	+	0.989845i	$0.545401\pi$
$458$	0	0
$459$	−4.69565	−0.219174
$460$	0	0
$461$	−23.7488	−1.10609	−0.553045	−	0.833151i	$-0.686534\pi$
$461$	−23.7488	−1.10609	−0.553045	+	0.833151i	$0.686534\pi$
$462$	0	0
$463$	16.4729	0.765562	0.382781	−	0.923839i	$-0.374966\pi$
$463$	16.4729	0.765562	0.382781	+	0.923839i	$0.374966\pi$
$464$	0	0
$465$	21.2030	0.983264
$466$	0	0
$467$	24.7890	1.14710	0.573550	−	0.819171i	$-0.305566\pi$
$467$	24.7890	1.14710	0.573550	+	0.819171i	$0.305566\pi$
$468$	0	0
$469$	35.5139	1.63988
$470$	0	0
$471$	−3.22376	−0.148543
$472$	0	0
$473$	1.18788	0.0546187
$474$	0	0
$475$	17.9257	0.822487
$476$	0	0
$477$	10.7637	0.492834
$478$	0	0
$479$	29.5396	1.34970	0.674849	−	0.737956i	$-0.264209\pi$
$479$	29.5396	1.34970	0.674849	+	0.737956i	$0.264209\pi$
$480$	0	0
$481$	−6.45247	−0.294207
$482$	0	0
$483$	22.3990	1.01919
$484$	0	0
$485$	−9.48155	−0.430535
$486$	0	0
$487$	9.74979	0.441805	0.220903	−	0.975296i	$-0.429100\pi$
$487$	9.74979	0.441805	0.220903	+	0.975296i	$0.429100\pi$
$488$	0	0
$489$	15.0860	0.682213
$490$	0	0
$491$	14.5972	0.658765	0.329382	−	0.944197i	$-0.393160\pi$
$491$	14.5972	0.658765	0.329382	+	0.944197i	$0.393160\pi$
$492$	0	0
$493$	−3.86880	−0.174242
$494$	0	0
$495$	6.44809	0.289820
$496$	0	0
$497$	−21.7531	−0.975761
$498$	0	0
$499$	−8.74641	−0.391543	−0.195772	−	0.980650i	$-0.562721\pi$
$499$	−8.74641	−0.391543	−0.195772	+	0.980650i	$0.562721\pi$
$500$	0	0
$501$	−7.73736	−0.345680
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	−16.0095	−0.712412
$506$	0	0
$507$	−7.14093	−0.317140
$508$	0	0
$509$	−20.1710	−0.894065	−0.447032	−	0.894518i	$-0.647519\pi$
$509$	−20.1710	−0.894065	−0.447032	+	0.894518i	$0.647519\pi$
$510$	0	0
$511$	−31.6150	−1.39856
$512$	0	0
$513$	41.2338	1.82052
$514$	0	0
$515$	36.2314	1.59655
$516$	0	0
$517$	8.25551	0.363077
$518$	0	0
$519$	−15.5246	−0.681453
$520$	0	0
$521$	21.4209	0.938466	0.469233	−	0.883074i	$-0.344530\pi$
$521$	21.4209	0.938466	0.469233	+	0.883074i	$0.344530\pi$
$522$	0	0
$523$	25.9170	1.13327	0.566635	−	0.823969i	$-0.308245\pi$
$523$	25.9170	1.13327	0.566635	+	0.823969i	$0.308245\pi$
$524$	0	0
$525$	−8.05995	−0.351765
$526$	0	0
$527$	−6.32071	−0.275335
$528$	0	0
$529$	17.6880	0.769043
$530$	0	0
$531$	12.8829	0.559069
$532$	0	0
$533$	42.0781	1.82261
$534$	0	0
$535$	26.8717	1.16176
$536$	0	0
$537$	28.8298	1.24410
$538$	0	0
$539$	4.16682	0.179478
$540$	0	0
$541$	34.7371	1.49347	0.746733	−	0.665124i	$-0.231621\pi$
$541$	34.7371	1.49347	0.746733	+	0.665124i	$0.231621\pi$
$542$	0	0
$543$	16.1117	0.691417
$544$	0	0
$545$	−6.33958	−0.271558
$546$	0	0
$547$	32.3136	1.38163	0.690814	−	0.723032i	$-0.257252\pi$
$547$	32.3136	1.38163	0.690814	+	0.723032i	$0.257252\pi$
$548$	0	0
$549$	−0.947768	−0.0404498
$550$	0	0
$551$	33.9730	1.44730
$552$	0	0
$553$	26.3674	1.12125
$554$	0	0
$555$	4.36327	0.185210
$556$	0	0
$557$	−45.1232	−1.91193	−0.955967	−	0.293475i	$-0.905188\pi$
$557$	−45.1232	−1.91193	−0.955967	+	0.293475i	$0.905188\pi$
$558$	0	0
$559$	−3.90745	−0.165267
$560$	0	0
$561$	1.31758	0.0556285
$562$	0	0
$563$	25.3419	1.06803	0.534017	−	0.845474i	$-0.320682\pi$
$563$	25.3419	1.06803	0.534017	+	0.845474i	$0.320682\pi$
$564$	0	0
$565$	−22.6370	−0.952344
$566$	0	0
$567$	−1.56418	−0.0656892
$568$	0	0
$569$	−8.03786	−0.336965	−0.168482	−	0.985705i	$-0.553887\pi$
$569$	−8.03786	−0.336965	−0.168482	+	0.985705i	$0.553887\pi$
$570$	0	0
$571$	−26.1754	−1.09540	−0.547702	−	0.836673i	$-0.684497\pi$
$571$	−26.1754	−1.09540	−0.547702	+	0.836673i	$0.684497\pi$
$572$	0	0
$573$	5.57590	0.232936
$574$	0	0
$575$	−14.6410	−0.610570
$576$	0	0
$577$	−32.4673	−1.35163	−0.675816	−	0.737070i	$-0.736209\pi$
$577$	−32.4673	−1.35163	−0.675816	+	0.737070i	$0.736209\pi$
$578$	0	0
$579$	21.2705	0.883970
$580$	0	0
$581$	37.3848	1.55098
$582$	0	0
$583$	−8.11075	−0.335913
$584$	0	0
$585$	−21.2106	−0.876948
$586$	0	0
$587$	−14.0340	−0.579244	−0.289622	−	0.957141i	$-0.593530\pi$
$587$	−14.0340	−0.579244	−0.289622	+	0.957141i	$0.593530\pi$
$588$	0	0
$589$	55.5039	2.28700
$590$	0	0
$591$	1.92140	0.0790359
$592$	0	0
$593$	26.9545	1.10689	0.553445	−	0.832886i	$-0.313313\pi$
$593$	26.9545	1.10689	0.553445	+	0.832886i	$0.313313\pi$
$594$	0	0
$595$	7.63674	0.313076
$596$	0	0
$597$	8.73005	0.357297
$598$	0	0
$599$	3.60982	0.147493	0.0737466	−	0.997277i	$-0.476504\pi$
$599$	3.60982	0.147493	0.0737466	+	0.997277i	$0.476504\pi$
$600$	0	0
$601$	14.6192	0.596329	0.298164	−	0.954515i	$-0.403626\pi$
$601$	14.6192	0.596329	0.298164	+	0.954515i	$0.403626\pi$
$602$	0	0
$603$	−19.8838	−0.809729
$604$	0	0
$605$	24.8519	1.01037
$606$	0	0
$607$	39.4136	1.59975	0.799875	−	0.600167i	$-0.204899\pi$
$607$	39.4136	1.59975	0.799875	+	0.600167i	$0.204899\pi$
$608$	0	0
$609$	−15.2753	−0.618988
$610$	0	0
$611$	−27.1560	−1.09861
$612$	0	0
$613$	32.5036	1.31281	0.656403	−	0.754410i	$-0.272077\pi$
$613$	32.5036	1.31281	0.656403	+	0.754410i	$0.272077\pi$
$614$	0	0
$615$	−28.4539	−1.14737
$616$	0	0
$617$	12.0277	0.484215	0.242108	−	0.970249i	$-0.422161\pi$
$617$	12.0277	0.484215	0.242108	+	0.970249i	$0.422161\pi$
$618$	0	0
$619$	39.5932	1.59139	0.795694	−	0.605699i	$-0.207107\pi$
$619$	39.5932	1.59139	0.795694	+	0.605699i	$0.207107\pi$
$620$	0	0
$621$	−33.6780	−1.35145
$622$	0	0
$623$	−57.6981	−2.31163
$624$	0	0
$625$	−31.2081	−1.24832
$626$	0	0
$627$	−11.5701	−0.462064
$628$	0	0
$629$	−1.30071	−0.0518628
$630$	0	0
$631$	35.3609	1.40769	0.703847	−	0.710351i	$-0.251464\pi$
$631$	35.3609	1.40769	0.703847	+	0.710351i	$0.251464\pi$
$632$	0	0
$633$	10.1406	0.403054
$634$	0	0
$635$	−38.4806	−1.52705
$636$	0	0
$637$	−13.7065	−0.543071
$638$	0	0
$639$	12.1793	0.481805
$640$	0	0
$641$	−44.5729	−1.76052	−0.880262	−	0.474489i	$-0.842633\pi$
$641$	−44.5729	−1.76052	−0.880262	+	0.474489i	$0.842633\pi$
$642$	0	0
$643$	−24.8747	−0.980961	−0.490480	−	0.871452i	$-0.663179\pi$
$643$	−24.8747	−0.980961	−0.490480	+	0.871452i	$0.663179\pi$
$644$	0	0
$645$	2.64228	0.104040
$646$	0	0
$647$	17.4991	0.687959	0.343980	−	0.938977i	$-0.388225\pi$
$647$	17.4991	0.687959	0.343980	+	0.938977i	$0.388225\pi$
$648$	0	0
$649$	−9.70764	−0.381058
$650$	0	0
$651$	−24.9563	−0.978114
$652$	0	0
$653$	−31.1479	−1.21891	−0.609455	−	0.792821i	$-0.708612\pi$
$653$	−31.1479	−1.21891	−0.609455	+	0.792821i	$0.708612\pi$
$654$	0	0
$655$	28.4953	1.11340
$656$	0	0
$657$	17.7008	0.690573
$658$	0	0
$659$	−44.6807	−1.74051	−0.870256	−	0.492600i	$-0.836046\pi$
$659$	−44.6807	−1.74051	−0.870256	+	0.492600i	$0.836046\pi$
$660$	0	0
$661$	18.5941	0.723227	0.361614	−	0.932328i	$-0.382226\pi$
$661$	18.5941	0.723227	0.361614	+	0.932328i	$0.382226\pi$
$662$	0	0
$663$	−4.33411	−0.168323
$664$	0	0
$665$	−67.0603	−2.60049
$666$	0	0
$667$	−27.7478	−1.07440
$668$	0	0
$669$	29.7528	1.15031
$670$	0	0
$671$	0.714172	0.0275703
$672$	0	0
$673$	−23.6364	−0.911116	−0.455558	−	0.890206i	$-0.650560\pi$
$673$	−23.6364	−0.911116	−0.455558	+	0.890206i	$0.650560\pi$
$674$	0	0
$675$	12.1185	0.466442
$676$	0	0
$677$	−40.2742	−1.54786	−0.773931	−	0.633269i	$-0.781713\pi$
$677$	−40.2742	−1.54786	−0.773931	+	0.633269i	$0.781713\pi$
$678$	0	0
$679$	11.1600	0.428280
$680$	0	0
$681$	−27.1066	−1.03873
$682$	0	0
$683$	18.8839	0.722571	0.361285	−	0.932455i	$-0.382338\pi$
$683$	18.8839	0.722571	0.361285	+	0.932455i	$0.382338\pi$
$684$	0	0
$685$	−24.4421	−0.933885
$686$	0	0
$687$	0.474559	0.0181056
$688$	0	0
$689$	26.6798	1.01642
$690$	0	0
$691$	−9.08244	−0.345512	−0.172756	−	0.984965i	$-0.555267\pi$
$691$	−9.08244	−0.345512	−0.172756	+	0.984965i	$0.555267\pi$
$692$	0	0
$693$	−7.58952	−0.288302
$694$	0	0
$695$	26.2316	0.995022
$696$	0	0
$697$	8.48227	0.321289
$698$	0	0
$699$	6.10248	0.230817
$700$	0	0
$701$	−23.7936	−0.898674	−0.449337	−	0.893362i	$-0.648340\pi$
$701$	−23.7936	−0.898674	−0.449337	+	0.893362i	$0.648340\pi$
$702$	0	0
$703$	11.4219	0.430786
$704$	0	0
$705$	18.3633	0.691602
$706$	0	0
$707$	18.8434	0.708681
$708$	0	0
$709$	−24.2463	−0.910590	−0.455295	−	0.890341i	$-0.650466\pi$
$709$	−24.2463	−0.910590	−0.455295	+	0.890341i	$0.650466\pi$
$710$	0	0
$711$	−14.7627	−0.553645
$712$	0	0
$713$	−45.3333	−1.69774
$714$	0	0
$715$	15.9828	0.597723
$716$	0	0
$717$	5.66356	0.211510
$718$	0	0
$719$	42.5172	1.58562	0.792812	−	0.609467i	$-0.208616\pi$
$719$	42.5172	1.58562	0.792812	+	0.609467i	$0.208616\pi$
$720$	0	0
$721$	−42.6451	−1.58819
$722$	0	0
$723$	18.4449	0.685972
$724$	0	0
$725$	9.98461	0.370819
$726$	0	0
$727$	−26.7548	−0.992282	−0.496141	−	0.868242i	$-0.665250\pi$
$727$	−26.7548	−0.992282	−0.496141	+	0.868242i	$0.665250\pi$
$728$	0	0
$729$	18.3712	0.680416
$730$	0	0
$731$	−0.787678	−0.0291333
$732$	0	0
$733$	−20.8166	−0.768880	−0.384440	−	0.923150i	$-0.625606\pi$
$733$	−20.8166	−0.768880	−0.384440	+	0.923150i	$0.625606\pi$
$734$	0	0
$735$	9.26855	0.341876
$736$	0	0
$737$	14.9830	0.551907
$738$	0	0
$739$	9.28965	0.341725	0.170863	−	0.985295i	$-0.445345\pi$
$739$	9.28965	0.341725	0.170863	+	0.985295i	$0.445345\pi$
$740$	0	0
$741$	38.0590	1.39813
$742$	0	0
$743$	40.6571	1.49157	0.745783	−	0.666189i	$-0.232076\pi$
$743$	40.6571	1.49157	0.745783	+	0.666189i	$0.232076\pi$
$744$	0	0
$745$	31.1251	1.14034
$746$	0	0
$747$	−20.9312	−0.765833
$748$	0	0
$749$	−31.6285	−1.15568
$750$	0	0
$751$	20.9085	0.762961	0.381480	−	0.924377i	$-0.375414\pi$
$751$	20.9085	0.762961	0.381480	+	0.924377i	$0.375414\pi$
$752$	0	0
$753$	9.69320	0.353240
$754$	0	0
$755$	31.0302	1.12931
$756$	0	0
$757$	25.6088	0.930766	0.465383	−	0.885109i	$-0.345917\pi$
$757$	25.6088	0.930766	0.465383	+	0.885109i	$0.345917\pi$
$758$	0	0
$759$	9.44995	0.343012
$760$	0	0
$761$	−46.4880	−1.68519	−0.842595	−	0.538548i	$-0.818973\pi$
$761$	−46.4880	−1.68519	−0.842595	+	0.538548i	$0.818973\pi$
$762$	0	0
$763$	7.46181	0.270136
$764$	0	0
$765$	−4.27570	−0.154588
$766$	0	0
$767$	31.9326	1.15302
$768$	0	0
$769$	22.9773	0.828582	0.414291	−	0.910144i	$-0.364030\pi$
$769$	22.9773	0.828582	0.414291	+	0.910144i	$0.364030\pi$
$770$	0	0
$771$	13.2581	0.477477
$772$	0	0
$773$	2.48858	0.0895078	0.0447539	−	0.998998i	$-0.485750\pi$
$773$	2.48858	0.0895078	0.0447539	+	0.998998i	$0.485750\pi$
$774$	0	0
$775$	16.3125	0.585962
$776$	0	0
$777$	−5.13565	−0.184241
$778$	0	0
$779$	−74.4851	−2.66871
$780$	0	0
$781$	−9.17746	−0.328395
$782$	0	0
$783$	22.9672	0.820781
$784$	0	0
$785$	−7.88302	−0.281357
$786$	0	0
$787$	−11.1015	−0.395726	−0.197863	−	0.980230i	$-0.563400\pi$
$787$	−11.1015	−0.395726	−0.197863	+	0.980230i	$0.563400\pi$
$788$	0	0
$789$	31.6503	1.12678
$790$	0	0
$791$	26.6441	0.947357
$792$	0	0
$793$	−2.34922	−0.0834233
$794$	0	0
$795$	−18.0413	−0.639859
$796$	0	0
$797$	3.98641	0.141206	0.0706030	−	0.997504i	$-0.477508\pi$
$797$	3.98641	0.141206	0.0706030	+	0.997504i	$0.477508\pi$
$798$	0	0
$799$	−5.47420	−0.193663
$800$	0	0
$801$	32.3044	1.14142
$802$	0	0
$803$	−13.3381	−0.470691
$804$	0	0
$805$	54.7720	1.93046
$806$	0	0
$807$	15.6875	0.552228
$808$	0	0
$809$	0.724971	0.0254886	0.0127443	−	0.999919i	$-0.495943\pi$
$809$	0.724971	0.0254886	0.0127443	+	0.999919i	$0.495943\pi$
$810$	0	0
$811$	−35.4483	−1.24476	−0.622378	−	0.782716i	$-0.713834\pi$
$811$	−35.4483	−1.24476	−0.622378	+	0.782716i	$0.713834\pi$
$812$	0	0
$813$	−11.3815	−0.399168
$814$	0	0
$815$	36.8896	1.29219
$816$	0	0
$817$	6.91682	0.241989
$818$	0	0
$819$	24.9652	0.872356
$820$	0	0
$821$	9.19425	0.320881	0.160441	−	0.987045i	$-0.448708\pi$
$821$	9.19425	0.320881	0.160441	+	0.987045i	$0.448708\pi$
$822$	0	0
$823$	−29.2341	−1.01904	−0.509519	−	0.860459i	$-0.670177\pi$
$823$	−29.2341	−1.01904	−0.509519	+	0.860459i	$0.670177\pi$
$824$	0	0
$825$	−3.40042	−0.118388
$826$	0	0
$827$	6.30999	0.219420	0.109710	−	0.993964i	$-0.465008\pi$
$827$	6.30999	0.219420	0.109710	+	0.993964i	$0.465008\pi$
$828$	0	0
$829$	11.2994	0.392445	0.196223	−	0.980559i	$-0.437132\pi$
$829$	11.2994	0.392445	0.196223	+	0.980559i	$0.437132\pi$
$830$	0	0
$831$	19.8805	0.689648
$832$	0	0
$833$	−2.76300	−0.0957324
$834$	0	0
$835$	−18.9201	−0.654756
$836$	0	0
$837$	37.5230	1.29698
$838$	0	0
$839$	4.06672	0.140399	0.0701994	−	0.997533i	$-0.477636\pi$
$839$	4.06672	0.140399	0.0701994	+	0.997533i	$0.477636\pi$
$840$	0	0
$841$	−10.0770	−0.347483
$842$	0	0
$843$	1.84832	0.0636595
$844$	0	0
$845$	−17.4616	−0.600698
$846$	0	0
$847$	−29.2512	−1.00508
$848$	0	0
$849$	−20.0376	−0.687690
$850$	0	0
$851$	−9.32895	−0.319792
$852$	0	0
$853$	−4.80455	−0.164505	−0.0822523	−	0.996612i	$-0.526211\pi$
$853$	−4.80455	−0.164505	−0.0822523	+	0.996612i	$0.526211\pi$
$854$	0	0
$855$	37.5461	1.28405
$856$	0	0
$857$	−30.9687	−1.05787	−0.528936	−	0.848662i	$-0.677409\pi$
$857$	−30.9687	−1.05787	−0.528936	+	0.848662i	$0.677409\pi$
$858$	0	0
$859$	−52.9741	−1.80745	−0.903727	−	0.428110i	$-0.859180\pi$
$859$	−52.9741	−1.80745	−0.903727	+	0.428110i	$0.859180\pi$
$860$	0	0
$861$	33.4908	1.14136
$862$	0	0
$863$	−24.3010	−0.827215	−0.413607	−	0.910455i	$-0.635731\pi$
$863$	−24.3010	−0.827215	−0.413607	+	0.910455i	$0.635731\pi$
$864$	0	0
$865$	−37.9621	−1.29075
$866$	0	0
$867$	17.9039	0.608049
$868$	0	0
$869$	11.1242	0.377361
$870$	0	0
$871$	−49.2856	−1.66998
$872$	0	0
$873$	−6.24831	−0.211473
$874$	0	0
$875$	23.2246	0.785133
$876$	0	0
$877$	28.5563	0.964279	0.482139	−	0.876095i	$-0.339860\pi$
$877$	28.5563	0.964279	0.482139	+	0.876095i	$0.339860\pi$
$878$	0	0
$879$	10.9324	0.368742
$880$	0	0
$881$	−52.2466	−1.76023	−0.880116	−	0.474758i	$-0.842536\pi$
$881$	−52.2466	−1.76023	−0.880116	+	0.474758i	$0.842536\pi$
$882$	0	0
$883$	−0.565523	−0.0190314	−0.00951568	−	0.999955i	$-0.503029\pi$
$883$	−0.565523	−0.0190314	−0.00951568	+	0.999955i	$0.503029\pi$
$884$	0	0
$885$	−21.5934	−0.725854
$886$	0	0
$887$	−32.8527	−1.10308	−0.551542	−	0.834147i	$-0.685960\pi$
$887$	−32.8527	−1.10308	−0.551542	+	0.834147i	$0.685960\pi$
$888$	0	0
$889$	45.2924	1.51906
$890$	0	0
$891$	−0.659912	−0.0221079
$892$	0	0
$893$	48.0705	1.60862
$894$	0	0
$895$	70.4972	2.35646
$896$	0	0
$897$	−31.0850	−1.03790
$898$	0	0
$899$	30.9157	1.03110
$900$	0	0
$901$	5.37821	0.179174
$902$	0	0
$903$	−3.11002	−0.103495
$904$	0	0
$905$	39.3976	1.30962
$906$	0	0
$907$	−26.2254	−0.870801	−0.435401	−	0.900237i	$-0.643393\pi$
$907$	−26.2254	−0.870801	−0.435401	+	0.900237i	$0.643393\pi$
$908$	0	0
$909$	−10.5502	−0.349928
$910$	0	0
$911$	−7.52518	−0.249320	−0.124660	−	0.992199i	$-0.539784\pi$
$911$	−7.52518	−0.249320	−0.124660	+	0.992199i	$0.539784\pi$
$912$	0	0
$913$	15.7723	0.521987
$914$	0	0
$915$	1.58858	0.0525169
$916$	0	0
$917$	−33.5395	−1.10757
$918$	0	0
$919$	−53.0436	−1.74975	−0.874874	−	0.484351i	$-0.839056\pi$
$919$	−53.0436	−1.74975	−0.874874	+	0.484351i	$0.839056\pi$
$920$	0	0
$921$	1.47888	0.0487308
$922$	0	0
$923$	30.1886	0.993671
$924$	0	0
$925$	3.35688	0.110374
$926$	0	0
$927$	23.8764	0.784204
$928$	0	0
$929$	0.767535	0.0251820	0.0125910	−	0.999921i	$-0.495992\pi$
$929$	0.767535	0.0251820	0.0125910	+	0.999921i	$0.495992\pi$
$930$	0	0
$931$	24.2627	0.795178
$932$	0	0
$933$	−7.27144	−0.238056
$934$	0	0
$935$	3.22187	0.105366
$936$	0	0
$937$	−13.9673	−0.456292	−0.228146	−	0.973627i	$-0.573266\pi$
$937$	−13.9673	−0.456292	−0.228146	+	0.973627i	$0.573266\pi$
$938$	0	0
$939$	32.3050	1.05423
$940$	0	0
$941$	2.35475	0.0767626	0.0383813	−	0.999263i	$-0.487780\pi$
$941$	2.35475	0.0767626	0.0383813	+	0.999263i	$0.487780\pi$
$942$	0	0
$943$	60.8363	1.98110
$944$	0	0
$945$	−45.3356	−1.47477
$946$	0	0
$947$	−46.7850	−1.52031	−0.760155	−	0.649742i	$-0.774877\pi$
$947$	−46.7850	−1.52031	−0.760155	+	0.649742i	$0.774877\pi$
$948$	0	0
$949$	43.8747	1.42423
$950$	0	0
$951$	−27.5907	−0.894688
$952$	0	0
$953$	−10.7637	−0.348670	−0.174335	−	0.984686i	$-0.555778\pi$
$953$	−10.7637	−0.348670	−0.174335	+	0.984686i	$0.555778\pi$
$954$	0	0
$955$	13.6347	0.441208
$956$	0	0
$957$	−6.44453	−0.208322
$958$	0	0
$959$	28.7688	0.928994
$960$	0	0
$961$	19.5089	0.629320
$962$	0	0
$963$	17.7083	0.570643
$964$	0	0
$965$	52.0124	1.67434
$966$	0	0
$967$	34.6320	1.11369	0.556845	−	0.830616i	$-0.312012\pi$
$967$	34.6320	1.11369	0.556845	+	0.830616i	$0.312012\pi$
$968$	0	0
$969$	7.67207	0.246463
$970$	0	0
$971$	34.4582	1.10582	0.552909	−	0.833242i	$-0.313518\pi$
$971$	34.4582	1.10582	0.552909	+	0.833242i	$0.313518\pi$
$972$	0	0
$973$	−30.8751	−0.989811
$974$	0	0
$975$	11.1855	0.358222
$976$	0	0
$977$	−17.1515	−0.548727	−0.274363	−	0.961626i	$-0.588467\pi$
$977$	−17.1515	−0.548727	−0.274363	+	0.961626i	$0.588467\pi$
$978$	0	0
$979$	−24.3423	−0.777985
$980$	0	0
$981$	−4.17777	−0.133386
$982$	0	0
$983$	45.9299	1.46494	0.732469	−	0.680801i	$-0.238368\pi$
$983$	45.9299	1.46494	0.732469	+	0.680801i	$0.238368\pi$
$984$	0	0
$985$	4.69838	0.149703
$986$	0	0
$987$	−21.6140	−0.687980
$988$	0	0
$989$	−5.64937	−0.179640
$990$	0	0
$991$	36.2831	1.15257	0.576285	−	0.817249i	$-0.304502\pi$
$991$	36.2831	1.15257	0.576285	+	0.817249i	$0.304502\pi$
$992$	0	0
$993$	−14.6600	−0.465221
$994$	0	0
$995$	21.3475	0.676761
$996$	0	0
$997$	58.0242	1.83764	0.918822	−	0.394672i	$-0.129142\pi$
$997$	58.0242	1.83764	0.918822	+	0.394672i	$0.129142\pi$
$998$	0	0
$999$	7.72170	0.244304

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4024.2.a.e.1.11	✓	29
4.3	odd	2		8048.2.a.w.1.19		29

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.e.1.11	✓	29	1.1	even	1	trivial
8048.2.a.w.1.19		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)

\(f(q)\)	\(=\)	\(q-1.10456 q^{3} -2.70098 q^{5} +3.17910 q^{7} -1.77994 q^{9} +O(q^{10})\)	\(q-1.10456 q^{3} -2.70098 q^{5} +3.17910 q^{7} -1.77994 q^{9} +1.34124 q^{11} -4.41191 q^{13} +2.98341 q^{15} -0.889369 q^{17} +7.80979 q^{19} -3.51153 q^{21} -6.37871 q^{23} +2.29528 q^{25} +5.27975 q^{27} +4.35006 q^{29} +7.10696 q^{31} -1.48148 q^{33} -8.58669 q^{35} +1.46251 q^{37} +4.87324 q^{39} -9.53740 q^{41} +0.885660 q^{43} +4.80757 q^{45} +6.15515 q^{47} +3.10670 q^{49} +0.982366 q^{51} -6.04722 q^{53} -3.62265 q^{55} -8.62643 q^{57} -7.23783 q^{59} +0.532473 q^{61} -5.65860 q^{63} +11.9165 q^{65} +11.1711 q^{67} +7.04570 q^{69} -6.84254 q^{71} -9.94461 q^{73} -2.53529 q^{75} +4.26393 q^{77} +8.29396 q^{79} -0.492018 q^{81} +11.7595 q^{83} +2.40217 q^{85} -4.80492 q^{87} -18.1492 q^{89} -14.0259 q^{91} -7.85010 q^{93} -21.0941 q^{95} +3.51041 q^{97} -2.38732 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 29 q - 7 q^{3} - 4 q^{5} - 13 q^{7} + 20 q^{9}+O(q^{10}) \) 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

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