LMFDB - Embedded newform 1444.4.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1444,4,Mod(1,1444)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1444, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1444.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1444 = 2^{2} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1444.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:	$85.1987580483$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 270 x^{13} - 73 x^{12} + 27762 x^{11} + 12723 x^{10} - 1362566 x^{9} - 774753 x^{8} + \cdots + 1913127803 $ x^15 - 270x^13 - 73x^12 + 27762x^11 + 12723x^10 - 1362566x^9 - 774753x^8 + 33065742x^7 + 19540307x^6 - 381558678x^5 - 144249837x^4 + 1975989754x^3 - 108686793x^2 - 3642813399*x + 1913127803
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{2}\cdot 19 $
Twist minimal:	no (minimal twist has level 76)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-8.83535$ of defining polynomial
Character	$\chi$	$=$	1444.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.83535 q^{3} +15.4758 q^{5} -26.4755 q^{7} +51.0635 q^{9} +O(q^{10})$	$q-8.83535 q^{3} +15.4758 q^{5} -26.4755 q^{7} +51.0635 q^{9} -35.6203 q^{11} +87.0829 q^{13} -136.735 q^{15} -79.1436 q^{17} +233.920 q^{21} -3.56888 q^{23} +114.502 q^{25} -212.609 q^{27} +18.4631 q^{29} +70.8145 q^{31} +314.718 q^{33} -409.730 q^{35} -218.640 q^{37} -769.408 q^{39} +249.357 q^{41} -14.7973 q^{43} +790.251 q^{45} -21.5106 q^{47} +357.951 q^{49} +699.262 q^{51} -339.695 q^{53} -551.254 q^{55} +280.821 q^{59} -494.864 q^{61} -1351.93 q^{63} +1347.68 q^{65} +46.7348 q^{67} +31.5323 q^{69} +447.872 q^{71} -649.958 q^{73} -1011.66 q^{75} +943.063 q^{77} -1007.34 q^{79} +499.766 q^{81} +272.991 q^{83} -1224.81 q^{85} -163.128 q^{87} -948.525 q^{89} -2305.56 q^{91} -625.672 q^{93} +804.827 q^{97} -1818.89 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 12 q^{5} - 6 q^{7} + 135 q^{9}+O(q^{10}) $ 15 * q + 12 * q^5 - 6 * q^7 + 135 * q^9	$ 15 q + 12 q^{5} - 6 q^{7} + 135 q^{9} - 42 q^{11} + 150 q^{13} - 153 q^{17} + 216 q^{21} + 72 q^{23} + 207 q^{25} + 219 q^{27} + 462 q^{29} + 30 q^{31} + 309 q^{33} - 84 q^{35} - 30 q^{37} - 1086 q^{39} + 1368 q^{41} + 345 q^{43} + 882 q^{45} - 1134 q^{47} + 525 q^{49} + 1212 q^{51} + 612 q^{53} + 1536 q^{55} + 2190 q^{59} - 1032 q^{61} - 1770 q^{63} + 1530 q^{65} + 618 q^{67} + 756 q^{69} - 804 q^{71} + 996 q^{73} - 1758 q^{77} + 630 q^{79} + 1947 q^{81} + 2382 q^{83} - 4290 q^{85} + 1110 q^{87} + 4965 q^{89} + 864 q^{91} + 2736 q^{93} + 4410 q^{97} + 2385 q^{99}+O(q^{100}) $ 15 * q + 12 * q^5 - 6 * q^7 + 135 * q^9 - 42 * q^11 + 150 * q^13 - 153 * q^17 + 216 * q^21 + 72 * q^23 + 207 * q^25 + 219 * q^27 + 462 * q^29 + 30 * q^31 + 309 * q^33 - 84 * q^35 - 30 * q^37 - 1086 * q^39 + 1368 * q^41 + 345 * q^43 + 882 * q^45 - 1134 * q^47 + 525 * q^49 + 1212 * q^51 + 612 * q^53 + 1536 * q^55 + 2190 * q^59 - 1032 * q^61 - 1770 * q^63 + 1530 * q^65 + 618 * q^67 + 756 * q^69 - 804 * q^71 + 996 * q^73 - 1758 * q^77 + 630 * q^79 + 1947 * q^81 + 2382 * q^83 - 4290 * q^85 + 1110 * q^87 + 4965 * q^89 + 864 * q^91 + 2736 * q^93 + 4410 * q^97 + 2385 * 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
$3$	−8.83535	−1.70036	−0.850182	−	0.526488i	$-0.823508\pi$
$3$	−8.83535	−1.70036	−0.850182	+	0.526488i	$0.823508\pi$
$4$	0	0
$5$	15.4758	1.38420	0.692101	−	0.721801i	$-0.256685\pi$
$5$	15.4758	1.38420	0.692101	+	0.721801i	$0.256685\pi$
$6$	0	0
$7$	−26.4755	−1.42954	−0.714771	−	0.699359i	$-0.753469\pi$
$7$	−26.4755	−1.42954	−0.714771	+	0.699359i	$0.753469\pi$
$8$	0	0
$9$	51.0635	1.89124
$10$	0	0
$11$	−35.6203	−0.976356	−0.488178	−	0.872744i	$-0.662338\pi$
$11$	−35.6203	−0.976356	−0.488178	+	0.872744i	$0.662338\pi$
$12$	0	0
$13$	87.0829	1.85788	0.928940	−	0.370230i	$-0.120721\pi$
$13$	87.0829	1.85788	0.928940	+	0.370230i	$0.120721\pi$
$14$	0	0
$15$	−136.735	−2.35365
$16$	0	0
$17$	−79.1436	−1.12913	−0.564563	−	0.825390i	$-0.690955\pi$
$17$	−79.1436	−1.12913	−0.564563	+	0.825390i	$0.690955\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	233.920	2.43074
$22$	0	0
$23$	−3.56888	−0.0323549	−0.0161774	−	0.999869i	$-0.505150\pi$
$23$	−3.56888	−0.0323549	−0.0161774	+	0.999869i	$0.505150\pi$
$24$	0	0
$25$	114.502	0.916015
$26$	0	0
$27$	−212.609	−1.51543
$28$	0	0
$29$	18.4631	0.118225	0.0591123	−	0.998251i	$-0.481173\pi$
$29$	18.4631	0.118225	0.0591123	+	0.998251i	$0.481173\pi$
$30$	0	0
$31$	70.8145	0.410280	0.205140	−	0.978733i	$-0.434235\pi$
$31$	70.8145	0.410280	0.205140	+	0.978733i	$0.434235\pi$
$32$	0	0
$33$	314.718	1.66016
$34$	0	0
$35$	−409.730	−1.97877
$36$	0	0
$37$	−218.640	−0.971464	−0.485732	−	0.874108i	$-0.661447\pi$
$37$	−218.640	−0.971464	−0.485732	+	0.874108i	$0.661447\pi$
$38$	0	0
$39$	−769.408	−3.15907
$40$	0	0
$41$	249.357	0.949829	0.474915	−	0.880032i	$-0.342479\pi$
$41$	249.357	0.949829	0.474915	+	0.880032i	$0.342479\pi$
$42$	0	0
$43$	−14.7973	−0.0524784	−0.0262392	−	0.999656i	$-0.508353\pi$
$43$	−14.7973	−0.0524784	−0.0262392	+	0.999656i	$0.508353\pi$
$44$	0	0
$45$	790.251	2.61786
$46$	0	0
$47$	−21.5106	−0.0667583	−0.0333792	−	0.999443i	$-0.510627\pi$
$47$	−21.5106	−0.0667583	−0.0333792	+	0.999443i	$0.510627\pi$
$48$	0	0
$49$	357.951	1.04359
$50$	0	0
$51$	699.262	1.91993
$52$	0	0
$53$	−339.695	−0.880392	−0.440196	−	0.897902i	$-0.645091\pi$
$53$	−339.695	−0.880392	−0.440196	+	0.897902i	$0.645091\pi$
$54$	0	0
$55$	−551.254	−1.35147
$56$	0	0
$57$	0	0
$58$	0	0
$59$	280.821	0.619657	0.309829	−	0.950792i	$-0.399728\pi$
$59$	280.821	0.619657	0.309829	+	0.950792i	$0.399728\pi$
$60$	0	0
$61$	−494.864	−1.03870	−0.519351	−	0.854561i	$-0.673826\pi$
$61$	−494.864	−1.03870	−0.519351	+	0.854561i	$0.673826\pi$
$62$	0	0
$63$	−1351.93	−2.70361
$64$	0	0
$65$	1347.68	2.57168
$66$	0	0
$67$	46.7348	0.0852174	0.0426087	−	0.999092i	$-0.486433\pi$
$67$	46.7348	0.0852174	0.0426087	+	0.999092i	$0.486433\pi$
$68$	0	0
$69$	31.5323	0.0550151
$70$	0	0
$71$	447.872	0.748629	0.374315	−	0.927302i	$-0.377878\pi$
$71$	447.872	0.748629	0.374315	+	0.927302i	$0.377878\pi$
$72$	0	0
$73$	−649.958	−1.04208	−0.521040	−	0.853532i	$-0.674456\pi$
$73$	−649.958	−1.04208	−0.521040	+	0.853532i	$0.674456\pi$
$74$	0	0
$75$	−1011.66	−1.55756
$76$	0	0
$77$	943.063	1.39574
$78$	0	0
$79$	−1007.34	−1.43461	−0.717307	−	0.696757i	$-0.754625\pi$
$79$	−1007.34	−1.43461	−0.717307	+	0.696757i	$0.754625\pi$
$80$	0	0
$81$	499.766	0.685550
$82$	0	0
$83$	272.991	0.361020	0.180510	−	0.983573i	$-0.442225\pi$
$83$	272.991	0.361020	0.180510	+	0.983573i	$0.442225\pi$
$84$	0	0
$85$	−1224.81	−1.56294
$86$	0	0
$87$	−163.128	−0.201025
$88$	0	0
$89$	−948.525	−1.12970	−0.564851	−	0.825193i	$-0.691066\pi$
$89$	−948.525	−1.12970	−0.564851	+	0.825193i	$0.691066\pi$
$90$	0	0
$91$	−2305.56	−2.65592
$92$	0	0
$93$	−625.672	−0.697625
$94$	0	0
$95$	0	0
$96$	0	0
$97$	804.827	0.842451	0.421226	−	0.906956i	$-0.361600\pi$
$97$	804.827	0.842451	0.421226	+	0.906956i	$0.361600\pi$
$98$	0	0
$99$	−1818.89	−1.84652
$100$	0	0
$101$	−1416.98	−1.39598	−0.697992	−	0.716105i	$-0.745923\pi$
$101$	−1416.98	−1.39598	−0.697992	+	0.716105i	$0.745923\pi$
$102$	0	0
$103$	232.359	0.222281	0.111141	−	0.993805i	$-0.464550\pi$
$103$	232.359	0.222281	0.111141	+	0.993805i	$0.464550\pi$
$104$	0	0
$105$	3620.11	3.36464
$106$	0	0
$107$	−266.394	−0.240685	−0.120342	−	0.992732i	$-0.538399\pi$
$107$	−266.394	−0.240685	−0.120342	+	0.992732i	$0.538399\pi$
$108$	0	0
$109$	680.025	0.597565	0.298783	−	0.954321i	$-0.403419\pi$
$109$	680.025	0.597565	0.298783	+	0.954321i	$0.403419\pi$
$110$	0	0
$111$	1931.76	1.65184
$112$	0	0
$113$	1157.41	0.963536	0.481768	−	0.876299i	$-0.339995\pi$
$113$	1157.41	0.963536	0.481768	+	0.876299i	$0.339995\pi$
$114$	0	0
$115$	−55.2314	−0.0447857
$116$	0	0
$117$	4446.76	3.51370
$118$	0	0
$119$	2095.36	1.61413
$120$	0	0
$121$	−62.1970	−0.0467295
$122$	0	0
$123$	−2203.16	−1.61506
$124$	0	0
$125$	−162.467	−0.116252
$126$	0	0
$127$	900.063	0.628880	0.314440	−	0.949277i	$-0.398183\pi$
$127$	900.063	0.628880	0.314440	+	0.949277i	$0.398183\pi$
$128$	0	0
$129$	130.740	0.0892324
$130$	0	0
$131$	−224.870	−0.149977	−0.0749885	−	0.997184i	$-0.523892\pi$
$131$	−224.870	−0.149977	−0.0749885	+	0.997184i	$0.523892\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−3290.31	−2.09767
$136$	0	0
$137$	2669.65	1.66484	0.832421	−	0.554144i	$-0.186954\pi$
$137$	2669.65	1.66484	0.832421	+	0.554144i	$0.186954\pi$
$138$	0	0
$139$	2721.37	1.66060	0.830301	−	0.557316i	$-0.188169\pi$
$139$	2721.37	1.66060	0.830301	+	0.557316i	$0.188169\pi$
$140$	0	0
$141$	190.054	0.113514
$142$	0	0
$143$	−3101.91	−1.81395
$144$	0	0
$145$	285.732	0.163647
$146$	0	0
$147$	−3162.62	−1.77448
$148$	0	0
$149$	2380.58	1.30889	0.654444	−	0.756110i	$-0.272903\pi$
$149$	2380.58	1.30889	0.654444	+	0.756110i	$0.272903\pi$
$150$	0	0
$151$	2211.66	1.19194	0.595969	−	0.803007i	$-0.296768\pi$
$151$	2211.66	1.19194	0.595969	+	0.803007i	$0.296768\pi$
$152$	0	0
$153$	−4041.35	−2.13545
$154$	0	0
$155$	1095.92	0.567910
$156$	0	0
$157$	361.359	0.183692	0.0918459	−	0.995773i	$-0.470723\pi$
$157$	361.359	0.183692	0.0918459	+	0.995773i	$0.470723\pi$
$158$	0	0
$159$	3001.33	1.49699
$160$	0	0
$161$	94.4877	0.0462527
$162$	0	0
$163$	−1933.69	−0.929193	−0.464596	−	0.885523i	$-0.653801\pi$
$163$	−1933.69	−0.929193	−0.464596	+	0.885523i	$0.653801\pi$
$164$	0	0
$165$	4870.52	2.29800
$166$	0	0
$167$	−3364.97	−1.55922	−0.779608	−	0.626268i	$-0.784582\pi$
$167$	−3364.97	−1.55922	−0.779608	+	0.626268i	$0.784582\pi$
$168$	0	0
$169$	5386.43	2.45172
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−1348.26	−0.592522	−0.296261	−	0.955107i	$-0.595740\pi$
$173$	−1348.26	−0.592522	−0.296261	+	0.955107i	$0.595740\pi$
$174$	0	0
$175$	−3031.49	−1.30948
$176$	0	0
$177$	−2481.15	−1.05364
$178$	0	0
$179$	1487.55	0.621144	0.310572	−	0.950550i	$-0.399479\pi$
$179$	1487.55	0.621144	0.310572	+	0.950550i	$0.399479\pi$
$180$	0	0
$181$	672.639	0.276226	0.138113	−	0.990417i	$-0.455896\pi$
$181$	672.639	0.276226	0.138113	+	0.990417i	$0.455896\pi$
$182$	0	0
$183$	4372.30	1.76617
$184$	0	0
$185$	−3383.64	−1.34470
$186$	0	0
$187$	2819.12	1.10243
$188$	0	0
$189$	5628.94	2.16638
$190$	0	0
$191$	5207.47	1.97277	0.986386	−	0.164449i	$-0.0525847\pi$
$191$	5207.47	1.97277	0.986386	+	0.164449i	$0.0525847\pi$
$192$	0	0
$193$	−826.437	−0.308229	−0.154115	−	0.988053i	$-0.549253\pi$
$193$	−826.437	−0.308229	−0.154115	+	0.988053i	$0.549253\pi$
$194$	0	0
$195$	−11907.2	−4.37280
$196$	0	0
$197$	2441.77	0.883090	0.441545	−	0.897239i	$-0.354430\pi$
$197$	2441.77	0.883090	0.441545	+	0.897239i	$0.354430\pi$
$198$	0	0
$199$	2651.28	0.944442	0.472221	−	0.881480i	$-0.343452\pi$
$199$	2651.28	0.944442	0.472221	+	0.881480i	$0.343452\pi$
$200$	0	0
$201$	−412.918	−0.144901
$202$	0	0
$203$	−488.820	−0.169007
$204$	0	0
$205$	3859.01	1.31476
$206$	0	0
$207$	−182.239	−0.0611909
$208$	0	0
$209$	0	0
$210$	0	0
$211$	1866.97	0.609136	0.304568	−	0.952491i	$-0.401488\pi$
$211$	1866.97	0.609136	0.304568	+	0.952491i	$0.401488\pi$
$212$	0	0
$213$	−3957.11	−1.27294
$214$	0	0
$215$	−229.001	−0.0726407
$216$	0	0
$217$	−1874.85	−0.586512
$218$	0	0
$219$	5742.61	1.77192
$220$	0	0
$221$	−6892.05	−2.09778
$222$	0	0
$223$	920.479	0.276412	0.138206	−	0.990404i	$-0.455866\pi$
$223$	920.479	0.276412	0.138206	+	0.990404i	$0.455866\pi$
$224$	0	0
$225$	5846.87	1.73240
$226$	0	0
$227$	1036.03	0.302925	0.151463	−	0.988463i	$-0.451602\pi$
$227$	1036.03	0.302925	0.151463	+	0.988463i	$0.451602\pi$
$228$	0	0
$229$	−2109.77	−0.608810	−0.304405	−	0.952543i	$-0.598458\pi$
$229$	−2109.77	−0.608810	−0.304405	+	0.952543i	$0.598458\pi$
$230$	0	0
$231$	−8332.30	−2.37327
$232$	0	0
$233$	−5110.85	−1.43701	−0.718504	−	0.695523i	$-0.755173\pi$
$233$	−5110.85	−1.43701	−0.718504	+	0.695523i	$0.755173\pi$
$234$	0	0
$235$	−332.895	−0.0924070
$236$	0	0
$237$	8900.20	2.43937
$238$	0	0
$239$	6263.00	1.69506	0.847531	−	0.530746i	$-0.178088\pi$
$239$	6263.00	1.69506	0.847531	+	0.530746i	$0.178088\pi$
$240$	0	0
$241$	3433.47	0.917714	0.458857	−	0.888510i	$-0.348259\pi$
$241$	3433.47	0.917714	0.458857	+	0.888510i	$0.348259\pi$
$242$	0	0
$243$	1324.85	0.349749
$244$	0	0
$245$	5539.59	1.44454
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−2411.98	−0.613866
$250$	0	0
$251$	2057.11	0.517304	0.258652	−	0.965971i	$-0.416722\pi$
$251$	2057.11	0.517304	0.258652	+	0.965971i	$0.416722\pi$
$252$	0	0
$253$	127.124	0.0315899
$254$	0	0
$255$	10821.7	2.65757
$256$	0	0
$257$	7717.54	1.87318	0.936590	−	0.350428i	$-0.113964\pi$
$257$	7717.54	1.87318	0.936590	+	0.350428i	$0.113964\pi$
$258$	0	0
$259$	5788.59	1.38875
$260$	0	0
$261$	942.791	0.223591
$262$	0	0
$263$	4949.09	1.16036	0.580179	−	0.814489i	$-0.302983\pi$
$263$	4949.09	1.16036	0.580179	+	0.814489i	$0.302983\pi$
$264$	0	0
$265$	−5257.08	−1.21864
$266$	0	0
$267$	8380.55	1.92090
$268$	0	0
$269$	6436.44	1.45887	0.729436	−	0.684049i	$-0.239783\pi$
$269$	6436.44	1.45887	0.729436	+	0.684049i	$0.239783\pi$
$270$	0	0
$271$	7364.29	1.65073	0.825367	−	0.564597i	$-0.190968\pi$
$271$	7364.29	1.65073	0.825367	+	0.564597i	$0.190968\pi$
$272$	0	0
$273$	20370.4	4.51603
$274$	0	0
$275$	−4078.59	−0.894357
$276$	0	0
$277$	2286.38	0.495940	0.247970	−	0.968768i	$-0.420236\pi$
$277$	2286.38	0.495940	0.247970	+	0.968768i	$0.420236\pi$
$278$	0	0
$279$	3616.04	0.775937
$280$	0	0
$281$	5446.50	1.15627	0.578133	−	0.815942i	$-0.303781\pi$
$281$	5446.50	1.15627	0.578133	+	0.815942i	$0.303781\pi$
$282$	0	0
$283$	−6285.65	−1.32029	−0.660147	−	0.751136i	$-0.729506\pi$
$283$	−6285.65	−1.32029	−0.660147	+	0.751136i	$0.729506\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−6601.84	−1.35782
$288$	0	0
$289$	1350.71	0.274926
$290$	0	0
$291$	−7110.93	−1.43247
$292$	0	0
$293$	−5267.33	−1.05024	−0.525121	−	0.851028i	$-0.675980\pi$
$293$	−5267.33	−1.05024	−0.525121	+	0.851028i	$0.675980\pi$
$294$	0	0
$295$	4345.94	0.857731
$296$	0	0
$297$	7573.21	1.47960
$298$	0	0
$299$	−310.788	−0.0601115
$300$	0	0
$301$	391.766	0.0750200
$302$	0	0
$303$	12519.5	2.37368
$304$	0	0
$305$	−7658.44	−1.43777
$306$	0	0
$307$	6295.49	1.17037	0.585183	−	0.810901i	$-0.301023\pi$
$307$	6295.49	1.17037	0.585183	+	0.810901i	$0.301023\pi$
$308$	0	0
$309$	−2052.97	−0.377960
$310$	0	0
$311$	−9438.92	−1.72100	−0.860502	−	0.509448i	$-0.829850\pi$
$311$	−9438.92	−1.72100	−0.860502	+	0.509448i	$0.829850\pi$
$312$	0	0
$313$	998.681	0.180347	0.0901737	−	0.995926i	$-0.471258\pi$
$313$	998.681	0.180347	0.0901737	+	0.995926i	$0.471258\pi$
$314$	0	0
$315$	−20922.3	−3.74234
$316$	0	0
$317$	−6150.37	−1.08971	−0.544857	−	0.838529i	$-0.683416\pi$
$317$	−6150.37	−1.08971	−0.544857	+	0.838529i	$0.683416\pi$
$318$	0	0
$319$	−657.661	−0.115429
$320$	0	0
$321$	2353.68	0.409252
$322$	0	0
$323$	0	0
$324$	0	0
$325$	9971.15	1.70185
$326$	0	0
$327$	−6008.26	−1.01608
$328$	0	0
$329$	569.503	0.0954338
$330$	0	0
$331$	−6545.77	−1.08697	−0.543487	−	0.839418i	$-0.682896\pi$
$331$	−6545.77	−1.08697	−0.543487	+	0.839418i	$0.682896\pi$
$332$	0	0
$333$	−11164.5	−1.83727
$334$	0	0
$335$	723.260	0.117958
$336$	0	0
$337$	6619.75	1.07003	0.535016	−	0.844842i	$-0.320306\pi$
$337$	6619.75	1.07003	0.535016	+	0.844842i	$0.320306\pi$
$338$	0	0
$339$	−10226.1	−1.63836
$340$	0	0
$341$	−2522.43	−0.400579
$342$	0	0
$343$	−395.830	−0.0623115
$344$	0	0
$345$	487.989	0.0761520
$346$	0	0
$347$	4931.59	0.762944	0.381472	−	0.924380i	$-0.375417\pi$
$347$	4931.59	0.762944	0.381472	+	0.924380i	$0.375417\pi$
$348$	0	0
$349$	−3308.75	−0.507487	−0.253744	−	0.967271i	$-0.581662\pi$
$349$	−3308.75	−0.507487	−0.253744	+	0.967271i	$0.581662\pi$
$350$	0	0
$351$	−18514.6	−2.81550
$352$	0	0
$353$	7081.85	1.06779	0.533893	−	0.845552i	$-0.320728\pi$
$353$	7081.85	1.06779	0.533893	+	0.845552i	$0.320728\pi$
$354$	0	0
$355$	6931.21	1.03625
$356$	0	0
$357$	−18513.3	−2.74461
$358$	0	0
$359$	−3193.16	−0.469439	−0.234719	−	0.972063i	$-0.575417\pi$
$359$	−3193.16	−0.469439	−0.234719	+	0.972063i	$0.575417\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	549.532	0.0794572
$364$	0	0
$365$	−10058.7	−1.44245
$366$	0	0
$367$	−205.330	−0.0292047	−0.0146023	−	0.999893i	$-0.504648\pi$
$367$	−205.330	−0.0292047	−0.0146023	+	0.999893i	$0.504648\pi$
$368$	0	0
$369$	12733.0	1.79636
$370$	0	0
$371$	8993.60	1.25856
$372$	0	0
$373$	7790.07	1.08138	0.540689	−	0.841222i	$-0.318163\pi$
$373$	7790.07	1.08138	0.540689	+	0.841222i	$0.318163\pi$
$374$	0	0
$375$	1435.45	0.197671
$376$	0	0
$377$	1607.82	0.219647
$378$	0	0
$379$	11635.5	1.57698	0.788491	−	0.615046i	$-0.210862\pi$
$379$	11635.5	1.57698	0.788491	+	0.615046i	$0.210862\pi$
$380$	0	0
$381$	−7952.38	−1.06932
$382$	0	0
$383$	−2557.76	−0.341241	−0.170621	−	0.985337i	$-0.554577\pi$
$383$	−2557.76	−0.341241	−0.170621	+	0.985337i	$0.554577\pi$
$384$	0	0
$385$	14594.7	1.93199
$386$	0	0
$387$	−755.603	−0.0992492
$388$	0	0
$389$	−8756.81	−1.14136	−0.570679	−	0.821174i	$-0.693320\pi$
$389$	−8756.81	−1.14136	−0.570679	+	0.821174i	$0.693320\pi$
$390$	0	0
$391$	282.454	0.0365327
$392$	0	0
$393$	1986.81	0.255016
$394$	0	0
$395$	−15589.4	−1.98580
$396$	0	0
$397$	−9071.91	−1.14687	−0.573434	−	0.819252i	$-0.694389\pi$
$397$	−9071.91	−1.14687	−0.573434	+	0.819252i	$0.694389\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	145.237	0.0180867	0.00904336	−	0.999959i	$-0.497121\pi$
$401$	145.237	0.0180867	0.00904336	+	0.999959i	$0.497121\pi$
$402$	0	0
$403$	6166.74	0.762250
$404$	0	0
$405$	7734.30	0.948940
$406$	0	0
$407$	7788.01	0.948494
$408$	0	0
$409$	−3775.39	−0.456433	−0.228216	−	0.973610i	$-0.573289\pi$
$409$	−3775.39	−0.456433	−0.228216	+	0.973610i	$0.573289\pi$
$410$	0	0
$411$	−23587.3	−2.83084
$412$	0	0
$413$	−7434.87	−0.885826
$414$	0	0
$415$	4224.77	0.499725
$416$	0	0
$417$	−24044.3	−2.82363
$418$	0	0
$419$	5805.97	0.676945	0.338473	−	0.940976i	$-0.390090\pi$
$419$	5805.97	0.676945	0.338473	+	0.940976i	$0.390090\pi$
$420$	0	0
$421$	6534.90	0.756511	0.378256	−	0.925701i	$-0.376524\pi$
$421$	6534.90	0.756511	0.378256	+	0.925701i	$0.376524\pi$
$422$	0	0
$423$	−1098.41	−0.126256
$424$	0	0
$425$	−9062.09	−1.03430
$426$	0	0
$427$	13101.8	1.48487
$428$	0	0
$429$	27406.5	3.08438
$430$	0	0
$431$	4575.57	0.511362	0.255681	−	0.966761i	$-0.417700\pi$
$431$	4575.57	0.511362	0.255681	+	0.966761i	$0.417700\pi$
$432$	0	0
$433$	−2373.95	−0.263475	−0.131738	−	0.991285i	$-0.542056\pi$
$433$	−2373.95	−0.263475	−0.131738	+	0.991285i	$0.542056\pi$
$434$	0	0
$435$	−2524.55	−0.278259
$436$	0	0
$437$	0	0
$438$	0	0
$439$	9703.87	1.05499	0.527495	−	0.849558i	$-0.323131\pi$
$439$	9703.87	1.05499	0.527495	+	0.849558i	$0.323131\pi$
$440$	0	0
$441$	18278.2	1.97368
$442$	0	0
$443$	−10234.8	−1.09767	−0.548836	−	0.835930i	$-0.684929\pi$
$443$	−10234.8	−1.09767	−0.548836	+	0.835930i	$0.684929\pi$
$444$	0	0
$445$	−14679.2	−1.56373
$446$	0	0
$447$	−21033.2	−2.22559
$448$	0	0
$449$	−18289.1	−1.92230	−0.961152	−	0.276019i	$-0.910985\pi$
$449$	−18289.1	−1.92230	−0.961152	+	0.276019i	$0.910985\pi$
$450$	0	0
$451$	−8882.16	−0.927371
$452$	0	0
$453$	−19540.8	−2.02673
$454$	0	0
$455$	−35680.5	−3.67633
$456$	0	0
$457$	−4252.19	−0.435249	−0.217625	−	0.976033i	$-0.569831\pi$
$457$	−4252.19	−0.435249	−0.217625	+	0.976033i	$0.569831\pi$
$458$	0	0
$459$	16826.7	1.71112
$460$	0	0
$461$	3738.48	0.377697	0.188848	−	0.982006i	$-0.439525\pi$
$461$	3738.48	0.377697	0.188848	+	0.982006i	$0.439525\pi$
$462$	0	0
$463$	−8644.00	−0.867647	−0.433824	−	0.900998i	$-0.642836\pi$
$463$	−8644.00	−0.867647	−0.433824	+	0.900998i	$0.642836\pi$
$464$	0	0
$465$	−9682.80	−0.965654
$466$	0	0
$467$	−776.052	−0.0768981	−0.0384490	−	0.999261i	$-0.512242\pi$
$467$	−776.052	−0.0768981	−0.0384490	+	0.999261i	$0.512242\pi$
$468$	0	0
$469$	−1237.33	−0.121822
$470$	0	0
$471$	−3192.74	−0.312343
$472$	0	0
$473$	527.084	0.0512376
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−17346.0	−1.66503
$478$	0	0
$479$	−17654.0	−1.68399	−0.841993	−	0.539488i	$-0.818618\pi$
$479$	−17654.0	−1.68399	−0.841993	+	0.539488i	$0.818618\pi$
$480$	0	0
$481$	−19039.8	−1.80486
$482$	0	0
$483$	−834.833	−0.0786464
$484$	0	0
$485$	12455.4	1.16612
$486$	0	0
$487$	−9655.64	−0.898437	−0.449219	−	0.893422i	$-0.648298\pi$
$487$	−9655.64	−0.898437	−0.449219	+	0.893422i	$0.648298\pi$
$488$	0	0
$489$	17084.8	1.57997
$490$	0	0
$491$	−249.662	−0.0229472	−0.0114736	−	0.999934i	$-0.503652\pi$
$491$	−249.662	−0.0229472	−0.0114736	+	0.999934i	$0.503652\pi$
$492$	0	0
$493$	−1461.24	−0.133490
$494$	0	0
$495$	−28148.9	−2.55596
$496$	0	0
$497$	−11857.6	−1.07020
$498$	0	0
$499$	−2109.11	−0.189211	−0.0946057	−	0.995515i	$-0.530159\pi$
$499$	−2109.11	−0.189211	−0.0946057	+	0.995515i	$0.530159\pi$
$500$	0	0
$501$	29730.7	2.65124
$502$	0	0
$503$	4975.66	0.441061	0.220530	−	0.975380i	$-0.429221\pi$
$503$	4975.66	0.441061	0.220530	+	0.975380i	$0.429221\pi$
$504$	0	0
$505$	−21928.9	−1.93232
$506$	0	0
$507$	−47591.0	−4.16882
$508$	0	0
$509$	15383.8	1.33964	0.669820	−	0.742523i	$-0.266371\pi$
$509$	15383.8	1.33964	0.669820	+	0.742523i	$0.266371\pi$
$510$	0	0
$511$	17208.0	1.48970
$512$	0	0
$513$	0	0
$514$	0	0
$515$	3595.95	0.307682
$516$	0	0
$517$	766.213	0.0651799
$518$	0	0
$519$	11912.4	1.00750
$520$	0	0
$521$	−206.935	−0.0174011	−0.00870055	−	0.999962i	$-0.502770\pi$
$521$	−206.935	−0.0174011	−0.00870055	+	0.999962i	$0.502770\pi$
$522$	0	0
$523$	3314.04	0.277080	0.138540	−	0.990357i	$-0.455759\pi$
$523$	3314.04	0.277080	0.138540	+	0.990357i	$0.455759\pi$
$524$	0	0
$525$	26784.3	2.22660
$526$	0	0
$527$	−5604.52	−0.463257
$528$	0	0
$529$	−12154.3	−0.998953
$530$	0	0
$531$	14339.7	1.17192
$532$	0	0
$533$	21714.7	1.76467
$534$	0	0
$535$	−4122.67	−0.333156
$536$	0	0
$537$	−13143.0	−1.05617
$538$	0	0
$539$	−12750.3	−1.01891
$540$	0	0
$541$	265.676	0.0211133	0.0105567	−	0.999944i	$-0.496640\pi$
$541$	265.676	0.0211133	0.0105567	+	0.999944i	$0.496640\pi$
$542$	0	0
$543$	−5943.00	−0.469684
$544$	0	0
$545$	10524.0	0.827151
$546$	0	0
$547$	−22576.0	−1.76468	−0.882339	−	0.470615i	$-0.844032\pi$
$547$	−22576.0	−1.76468	−0.882339	+	0.470615i	$0.844032\pi$
$548$	0	0
$549$	−25269.5	−1.96444
$550$	0	0
$551$	0	0
$552$	0	0
$553$	26669.8	2.05084
$554$	0	0
$555$	29895.6	2.28648
$556$	0	0
$557$	19463.2	1.48058	0.740289	−	0.672289i	$-0.234689\pi$
$557$	19463.2	1.48058	0.740289	+	0.672289i	$0.234689\pi$
$558$	0	0
$559$	−1288.59	−0.0974985
$560$	0	0
$561$	−24907.9	−1.87453
$562$	0	0
$563$	5945.34	0.445056	0.222528	−	0.974926i	$-0.428569\pi$
$563$	5945.34	0.445056	0.222528	+	0.974926i	$0.428569\pi$
$564$	0	0
$565$	17911.8	1.33373
$566$	0	0
$567$	−13231.5	−0.980022
$568$	0	0
$569$	10615.3	0.782103	0.391051	−	0.920369i	$-0.372112\pi$
$569$	10615.3	0.782103	0.391051	+	0.920369i	$0.372112\pi$
$570$	0	0
$571$	18071.4	1.32446	0.662230	−	0.749301i	$-0.269610\pi$
$571$	18071.4	1.32446	0.662230	+	0.749301i	$0.269610\pi$
$572$	0	0
$573$	−46009.8	−3.35443
$574$	0	0
$575$	−408.643	−0.0296376
$576$	0	0
$577$	−24495.1	−1.76732	−0.883662	−	0.468125i	$-0.844930\pi$
$577$	−24495.1	−1.76732	−0.883662	+	0.468125i	$0.844930\pi$
$578$	0	0
$579$	7301.87	0.524102
$580$	0	0
$581$	−7227.58	−0.516094
$582$	0	0
$583$	12100.0	0.859575
$584$	0	0
$585$	68817.3	4.86367
$586$	0	0
$587$	−14853.6	−1.04442	−0.522208	−	0.852818i	$-0.674891\pi$
$587$	−14853.6	−1.04442	−0.522208	+	0.852818i	$0.674891\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−21573.9	−1.50158
$592$	0	0
$593$	−5733.38	−0.397035	−0.198518	−	0.980097i	$-0.563613\pi$
$593$	−5733.38	−0.397035	−0.198518	+	0.980097i	$0.563613\pi$
$594$	0	0
$595$	32427.5	2.23429
$596$	0	0
$597$	−23425.0	−1.60590
$598$	0	0
$599$	−14948.0	−1.01963	−0.509816	−	0.860283i	$-0.670287\pi$
$599$	−14948.0	−1.01963	−0.509816	+	0.860283i	$0.670287\pi$
$600$	0	0
$601$	7596.70	0.515600	0.257800	−	0.966198i	$-0.417002\pi$
$601$	7596.70	0.515600	0.257800	+	0.966198i	$0.417002\pi$
$602$	0	0
$603$	2386.44	0.161167
$604$	0	0
$605$	−962.551	−0.0646831
$606$	0	0
$607$	24917.7	1.66619	0.833095	−	0.553130i	$-0.186567\pi$
$607$	24917.7	1.66619	0.833095	+	0.553130i	$0.186567\pi$
$608$	0	0
$609$	4318.89	0.287373
$610$	0	0
$611$	−1873.20	−0.124029
$612$	0	0
$613$	20641.8	1.36006	0.680029	−	0.733185i	$-0.261967\pi$
$613$	20641.8	1.36006	0.680029	+	0.733185i	$0.261967\pi$
$614$	0	0
$615$	−34095.7	−2.23556
$616$	0	0
$617$	28225.4	1.84167	0.920836	−	0.389950i	$-0.127508\pi$
$617$	28225.4	1.84167	0.920836	+	0.389950i	$0.127508\pi$
$618$	0	0
$619$	27010.3	1.75385	0.876927	−	0.480624i	$-0.159590\pi$
$619$	27010.3	1.75385	0.876927	+	0.480624i	$0.159590\pi$
$620$	0	0
$621$	758.777	0.0490317
$622$	0	0
$623$	25112.6	1.61495
$624$	0	0
$625$	−16827.1	−1.07693
$626$	0	0
$627$	0	0
$628$	0	0
$629$	17303.9	1.09690
$630$	0	0
$631$	−3463.69	−0.218522	−0.109261	−	0.994013i	$-0.534848\pi$
$631$	−3463.69	−0.218522	−0.109261	+	0.994013i	$0.534848\pi$
$632$	0	0
$633$	−16495.4	−1.03575
$634$	0	0
$635$	13929.2	0.870496
$636$	0	0
$637$	31171.4	1.93886
$638$	0	0
$639$	22869.9	1.41584
$640$	0	0
$641$	23527.2	1.44972	0.724860	−	0.688896i	$-0.241905\pi$
$641$	23527.2	1.44972	0.724860	+	0.688896i	$0.241905\pi$
$642$	0	0
$643$	16858.0	1.03392	0.516962	−	0.856009i	$-0.327063\pi$
$643$	16858.0	1.03392	0.516962	+	0.856009i	$0.327063\pi$
$644$	0	0
$645$	2023.31	0.123516
$646$	0	0
$647$	5338.90	0.324411	0.162205	−	0.986757i	$-0.448139\pi$
$647$	5338.90	0.324411	0.162205	+	0.986757i	$0.448139\pi$
$648$	0	0
$649$	−10002.9	−0.605006
$650$	0	0
$651$	16565.0	0.997284
$652$	0	0
$653$	−933.585	−0.0559479	−0.0279740	−	0.999609i	$-0.508906\pi$
$653$	−933.585	−0.0559479	−0.0279740	+	0.999609i	$0.508906\pi$
$654$	0	0
$655$	−3480.05	−0.207598
$656$	0	0
$657$	−33189.1	−1.97082
$658$	0	0
$659$	2894.56	0.171102	0.0855510	−	0.996334i	$-0.472735\pi$
$659$	2894.56	0.171102	0.0855510	+	0.996334i	$0.472735\pi$
$660$	0	0
$661$	12358.6	0.727224	0.363612	−	0.931550i	$-0.381543\pi$
$661$	12358.6	0.727224	0.363612	+	0.931550i	$0.381543\pi$
$662$	0	0
$663$	60893.7	3.56699
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−65.8926	−0.00382514
$668$	0	0
$669$	−8132.76	−0.470001
$670$	0	0
$671$	17627.2	1.01414
$672$	0	0
$673$	−22847.5	−1.30863	−0.654313	−	0.756224i	$-0.727042\pi$
$673$	−22847.5	−1.30863	−0.654313	+	0.756224i	$0.727042\pi$
$674$	0	0
$675$	−24344.2	−1.38816
$676$	0	0
$677$	6843.30	0.388492	0.194246	−	0.980953i	$-0.437774\pi$
$677$	6843.30	0.388492	0.194246	+	0.980953i	$0.437774\pi$
$678$	0	0
$679$	−21308.2	−1.20432
$680$	0	0
$681$	−9153.73	−0.515083
$682$	0	0
$683$	−26766.1	−1.49952	−0.749762	−	0.661707i	$-0.769832\pi$
$683$	−26766.1	−1.49952	−0.749762	+	0.661707i	$0.769832\pi$
$684$	0	0
$685$	41315.1	2.30448
$686$	0	0
$687$	18640.5	1.03520
$688$	0	0
$689$	−29581.7	−1.63566
$690$	0	0
$691$	−1732.37	−0.0953725	−0.0476862	−	0.998862i	$-0.515185\pi$
$691$	−1732.37	−0.0953725	−0.0476862	+	0.998862i	$0.515185\pi$
$692$	0	0
$693$	48156.1	2.63968
$694$	0	0
$695$	42115.5	2.29861
$696$	0	0
$697$	−19735.0	−1.07248
$698$	0	0
$699$	45156.1	2.44344
$700$	0	0
$701$	29850.5	1.60833	0.804163	−	0.594409i	$-0.202614\pi$
$701$	29850.5	1.60833	0.804163	+	0.594409i	$0.202614\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	2941.24	0.157126
$706$	0	0
$707$	37515.1	1.99562
$708$	0	0
$709$	11733.2	0.621509	0.310755	−	0.950490i	$-0.399418\pi$
$709$	11733.2	0.621509	0.310755	+	0.950490i	$0.399418\pi$
$710$	0	0
$711$	−51438.3	−2.71320
$712$	0	0
$713$	−252.728	−0.0132746
$714$	0	0
$715$	−48004.8	−2.51088
$716$	0	0
$717$	−55335.8	−2.88222
$718$	0	0
$719$	−9696.40	−0.502941	−0.251471	−	0.967865i	$-0.580914\pi$
$719$	−9696.40	−0.502941	−0.251471	+	0.967865i	$0.580914\pi$
$720$	0	0
$721$	−6151.81	−0.317761
$722$	0	0
$723$	−30335.9	−1.56045
$724$	0	0
$725$	2114.06	0.108296
$726$	0	0
$727$	−6461.61	−0.329639	−0.164820	−	0.986324i	$-0.552704\pi$
$727$	−6461.61	−0.329639	−0.164820	+	0.986324i	$0.552704\pi$
$728$	0	0
$729$	−25199.2	−1.28025
$730$	0	0
$731$	1171.11	0.0592547
$732$	0	0
$733$	22660.2	1.14185	0.570924	−	0.821003i	$-0.306585\pi$
$733$	22660.2	1.14185	0.570924	+	0.821003i	$0.306585\pi$
$734$	0	0
$735$	−48944.3	−2.45624
$736$	0	0
$737$	−1664.71	−0.0832024
$738$	0	0
$739$	32081.2	1.59693	0.798463	−	0.602044i	$-0.205647\pi$
$739$	32081.2	1.59693	0.798463	+	0.602044i	$0.205647\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	7340.85	0.362463	0.181231	−	0.983440i	$-0.441992\pi$
$743$	7340.85	0.362463	0.181231	+	0.983440i	$0.441992\pi$
$744$	0	0
$745$	36841.4	1.81177
$746$	0	0
$747$	13939.9	0.682776
$748$	0	0
$749$	7052.90	0.344069
$750$	0	0
$751$	−30423.7	−1.47826	−0.739132	−	0.673561i	$-0.764764\pi$
$751$	−30423.7	−1.47826	−0.739132	+	0.673561i	$0.764764\pi$
$752$	0	0
$753$	−18175.3	−0.879606
$754$	0	0
$755$	34227.4	1.64988
$756$	0	0
$757$	11922.5	0.572430	0.286215	−	0.958165i	$-0.407603\pi$
$757$	11922.5	0.572430	0.286215	+	0.958165i	$0.407603\pi$
$758$	0	0
$759$	−1123.19	−0.0537143
$760$	0	0
$761$	−5576.45	−0.265632	−0.132816	−	0.991141i	$-0.542402\pi$
$761$	−5576.45	−0.265632	−0.132816	+	0.991141i	$0.542402\pi$
$762$	0	0
$763$	−18004.0	−0.854244
$764$	0	0
$765$	−62543.3	−2.95589
$766$	0	0
$767$	24454.7	1.15125
$768$	0	0
$769$	−10898.1	−0.511048	−0.255524	−	0.966803i	$-0.582248\pi$
$769$	−10898.1	−0.511048	−0.255524	+	0.966803i	$0.582248\pi$
$770$	0	0
$771$	−68187.2	−3.18509
$772$	0	0
$773$	−13198.2	−0.614109	−0.307055	−	0.951692i	$-0.599343\pi$
$773$	−13198.2	−0.614109	−0.307055	+	0.951692i	$0.599343\pi$
$774$	0	0
$775$	8108.40	0.375822
$776$	0	0
$777$	−51144.3	−2.36138
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−15953.3	−0.730929
$782$	0	0
$783$	−3925.43	−0.179162
$784$	0	0
$785$	5592.34	0.254266
$786$	0	0
$787$	13747.8	0.622687	0.311344	−	0.950297i	$-0.399221\pi$
$787$	13747.8	0.622687	0.311344	+	0.950297i	$0.399221\pi$
$788$	0	0
$789$	−43727.0	−1.97303
$790$	0	0
$791$	−30642.9	−1.37741
$792$	0	0
$793$	−43094.2	−1.92978
$794$	0	0
$795$	46448.1	2.07213
$796$	0	0
$797$	21384.3	0.950403	0.475201	−	0.879877i	$-0.342375\pi$
$797$	21384.3	0.950403	0.475201	+	0.879877i	$0.342375\pi$
$798$	0	0
$799$	1702.43	0.0753786
$800$	0	0
$801$	−48435.0	−2.13654
$802$	0	0
$803$	23151.7	1.01744
$804$	0	0
$805$	1462.28	0.0640230
$806$	0	0
$807$	−56868.2	−2.48061
$808$	0	0
$809$	25679.5	1.11600	0.557999	−	0.829842i	$-0.311569\pi$
$809$	25679.5	1.11600	0.557999	+	0.829842i	$0.311569\pi$
$810$	0	0
$811$	−7885.04	−0.341407	−0.170704	−	0.985322i	$-0.554604\pi$
$811$	−7885.04	−0.341407	−0.170704	+	0.985322i	$0.554604\pi$
$812$	0	0
$813$	−65066.1	−2.80685
$814$	0	0
$815$	−29925.5	−1.28619
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−117730.	−5.02298
$820$	0	0
$821$	−5300.28	−0.225312	−0.112656	−	0.993634i	$-0.535936\pi$
$821$	−5300.28	−0.225312	−0.112656	+	0.993634i	$0.535936\pi$
$822$	0	0
$823$	−26642.2	−1.12842	−0.564210	−	0.825631i	$-0.690819\pi$
$823$	−26642.2	−1.12842	−0.564210	+	0.825631i	$0.690819\pi$
$824$	0	0
$825$	36035.8	1.52073
$826$	0	0
$827$	31825.7	1.33820	0.669098	−	0.743174i	$-0.266681\pi$
$827$	31825.7	1.33820	0.669098	+	0.743174i	$0.266681\pi$
$828$	0	0
$829$	19200.6	0.804420	0.402210	−	0.915547i	$-0.368242\pi$
$829$	19200.6	0.804420	0.402210	+	0.915547i	$0.368242\pi$
$830$	0	0
$831$	−20201.0	−0.843280
$832$	0	0
$833$	−28329.5	−1.17834
$834$	0	0
$835$	−52075.7	−2.15827
$836$	0	0
$837$	−15055.8	−0.621752
$838$	0	0
$839$	20050.8	0.825067	0.412534	−	0.910942i	$-0.364644\pi$
$839$	20050.8	0.825067	0.412534	+	0.910942i	$0.364644\pi$
$840$	0	0
$841$	−24048.1	−0.986023
$842$	0	0
$843$	−48121.7	−1.96607
$844$	0	0
$845$	83359.5	3.39368
$846$	0	0
$847$	1646.69	0.0668018
$848$	0	0
$849$	55536.0	2.24498
$850$	0	0
$851$	780.299	0.0314316
$852$	0	0
$853$	16263.0	0.652793	0.326397	−	0.945233i	$-0.394165\pi$
$853$	16263.0	0.652793	0.326397	+	0.945233i	$0.394165\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	18273.0	0.728348	0.364174	−	0.931331i	$-0.381351\pi$
$857$	18273.0	0.728348	0.364174	+	0.931331i	$0.381351\pi$
$858$	0	0
$859$	−21126.3	−0.839139	−0.419570	−	0.907723i	$-0.637819\pi$
$859$	−21126.3	−0.839139	−0.419570	+	0.907723i	$0.637819\pi$
$860$	0	0
$861$	58329.6	2.30879
$862$	0	0
$863$	−15852.6	−0.625293	−0.312646	−	0.949870i	$-0.601215\pi$
$863$	−15852.6	−0.625293	−0.312646	+	0.949870i	$0.601215\pi$
$864$	0	0
$865$	−20865.5	−0.820170
$866$	0	0
$867$	−11934.0	−0.467474
$868$	0	0
$869$	35881.7	1.40069
$870$	0	0
$871$	4069.80	0.158324
$872$	0	0
$873$	41097.3	1.59328
$874$	0	0
$875$	4301.39	0.166187
$876$	0	0
$877$	10144.0	0.390579	0.195290	−	0.980746i	$-0.437435\pi$
$877$	10144.0	0.390579	0.195290	+	0.980746i	$0.437435\pi$
$878$	0	0
$879$	46538.7	1.78579
$880$	0	0
$881$	−19183.9	−0.733624	−0.366812	−	0.930295i	$-0.619551\pi$
$881$	−19183.9	−0.733624	−0.366812	+	0.930295i	$0.619551\pi$
$882$	0	0
$883$	−29878.6	−1.13873	−0.569363	−	0.822086i	$-0.692810\pi$
$883$	−29878.6	−1.13873	−0.569363	+	0.822086i	$0.692810\pi$
$884$	0	0
$885$	−38398.0	−1.45846
$886$	0	0
$887$	−16378.9	−0.620011	−0.310006	−	0.950735i	$-0.600331\pi$
$887$	−16378.9	−0.620011	−0.310006	+	0.950735i	$0.600331\pi$
$888$	0	0
$889$	−23829.6	−0.899009
$890$	0	0
$891$	−17801.8	−0.669341
$892$	0	0
$893$	0	0
$894$	0	0
$895$	23021.1	0.859788
$896$	0	0
$897$	2745.92	0.102212
$898$	0	0
$899$	1307.46	0.0485051
$900$	0	0
$901$	26884.7	0.994073
$902$	0	0
$903$	−3461.39	−0.127561
$904$	0	0
$905$	10409.7	0.382352
$906$	0	0
$907$	−36323.6	−1.32977	−0.664887	−	0.746944i	$-0.731520\pi$
$907$	−36323.6	−1.32977	−0.664887	+	0.746944i	$0.731520\pi$
$908$	0	0
$909$	−72355.8	−2.64014
$910$	0	0
$911$	9813.73	0.356908	0.178454	−	0.983948i	$-0.442890\pi$
$911$	9813.73	0.356908	0.178454	+	0.983948i	$0.442890\pi$
$912$	0	0
$913$	−9724.02	−0.352484
$914$	0	0
$915$	67665.0	2.44474
$916$	0	0
$917$	5953.54	0.214398
$918$	0	0
$919$	−2321.90	−0.0833434	−0.0416717	−	0.999131i	$-0.513268\pi$
$919$	−2321.90	−0.0833434	−0.0416717	+	0.999131i	$0.513268\pi$
$920$	0	0
$921$	−55622.9	−1.99005
$922$	0	0
$923$	39002.0	1.39086
$924$	0	0
$925$	−25034.7	−0.889875
$926$	0	0
$927$	11865.1	0.420388
$928$	0	0
$929$	15178.6	0.536055	0.268028	−	0.963411i	$-0.413628\pi$
$929$	15178.6	0.536055	0.268028	+	0.963411i	$0.413628\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	83396.2	2.92633
$934$	0	0
$935$	43628.2	1.52598
$936$	0	0
$937$	−40627.6	−1.41648	−0.708242	−	0.705970i	$-0.750511\pi$
$937$	−40627.6	−1.41648	−0.708242	+	0.705970i	$0.750511\pi$
$938$	0	0
$939$	−8823.70	−0.306657
$940$	0	0
$941$	6250.98	0.216553	0.108276	−	0.994121i	$-0.465467\pi$
$941$	6250.98	0.216553	0.108276	+	0.994121i	$0.465467\pi$
$942$	0	0
$943$	−889.924	−0.0307316
$944$	0	0
$945$	87112.6	2.99870
$946$	0	0
$947$	−20648.0	−0.708522	−0.354261	−	0.935147i	$-0.615268\pi$
$947$	−20648.0	−0.708522	−0.354261	+	0.935147i	$0.615268\pi$
$948$	0	0
$949$	−56600.2	−1.93606
$950$	0	0
$951$	54340.7	1.85291
$952$	0	0
$953$	11314.9	0.384601	0.192301	−	0.981336i	$-0.438405\pi$
$953$	11314.9	0.384601	0.192301	+	0.981336i	$0.438405\pi$
$954$	0	0
$955$	80590.0	2.73071
$956$	0	0
$957$	5810.67	0.196272
$958$	0	0
$959$	−70680.2	−2.37996
$960$	0	0
$961$	−24776.3	−0.831671
$962$	0	0
$963$	−13603.0	−0.455193
$964$	0	0
$965$	−12789.8	−0.426652
$966$	0	0
$967$	7018.25	0.233394	0.116697	−	0.993168i	$-0.462769\pi$
$967$	7018.25	0.233394	0.116697	+	0.993168i	$0.462769\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	33418.4	1.10448	0.552239	−	0.833686i	$-0.313773\pi$
$971$	33418.4	1.10448	0.552239	+	0.833686i	$0.313773\pi$
$972$	0	0
$973$	−72049.6	−2.37390
$974$	0	0
$975$	−88098.7	−2.89376
$976$	0	0
$977$	−36451.3	−1.19363	−0.596817	−	0.802378i	$-0.703568\pi$
$977$	−36451.3	−1.19363	−0.596817	+	0.802378i	$0.703568\pi$
$978$	0	0
$979$	33786.7	1.10299
$980$	0	0
$981$	34724.5	1.13014
$982$	0	0
$983$	3952.35	0.128240	0.0641202	−	0.997942i	$-0.479576\pi$
$983$	3952.35	0.128240	0.0641202	+	0.997942i	$0.479576\pi$
$984$	0	0
$985$	37788.4	1.22238
$986$	0	0
$987$	−5031.76	−0.162272
$988$	0	0
$989$	52.8098	0.00169793
$990$	0	0
$991$	27428.6	0.879211	0.439606	−	0.898191i	$-0.355118\pi$
$991$	27428.6	0.879211	0.439606	+	0.898191i	$0.355118\pi$
$992$	0	0
$993$	57834.2	1.84825
$994$	0	0
$995$	41030.8	1.30730
$996$	0	0
$997$	12827.4	0.407472	0.203736	−	0.979026i	$-0.434692\pi$
$997$	12827.4	0.407472	0.203736	+	0.979026i	$0.434692\pi$
$998$	0	0
$999$	46484.9	1.47219

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1444.4.a.k.1.1		15
19.14	odd	18		76.4.i.a.25.1	✓	30
19.15	odd	18		76.4.i.a.73.1	yes	30
19.18	odd	2		1444.4.a.j.1.15		15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.4.i.a.25.1	✓	30	19.14	odd	18
76.4.i.a.73.1	yes	30	19.15	odd	18
1444.4.a.j.1.15		15	19.18	odd	2
1444.4.a.k.1.1		15	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1444 = 2^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1444.a (trivial)

\(f(q)\)	\(=\)	\(q-8.83535 q^{3} +15.4758 q^{5} -26.4755 q^{7} +51.0635 q^{9} +O(q^{10})\)	\(q-8.83535 q^{3} +15.4758 q^{5} -26.4755 q^{7} +51.0635 q^{9} -35.6203 q^{11} +87.0829 q^{13} -136.735 q^{15} -79.1436 q^{17} +233.920 q^{21} -3.56888 q^{23} +114.502 q^{25} -212.609 q^{27} +18.4631 q^{29} +70.8145 q^{31} +314.718 q^{33} -409.730 q^{35} -218.640 q^{37} -769.408 q^{39} +249.357 q^{41} -14.7973 q^{43} +790.251 q^{45} -21.5106 q^{47} +357.951 q^{49} +699.262 q^{51} -339.695 q^{53} -551.254 q^{55} +280.821 q^{59} -494.864 q^{61} -1351.93 q^{63} +1347.68 q^{65} +46.7348 q^{67} +31.5323 q^{69} +447.872 q^{71} -649.958 q^{73} -1011.66 q^{75} +943.063 q^{77} -1007.34 q^{79} +499.766 q^{81} +272.991 q^{83} -1224.81 q^{85} -163.128 q^{87} -948.525 q^{89} -2305.56 q^{91} -625.672 q^{93} +804.827 q^{97} -1818.89 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 12 q^{5} - 6 q^{7} + 135 q^{9}+O(q^{10}) \) 15 * q + 12 * q^5 - 6 * q^7 + 135 * q^9	\( 15 q + 12 q^{5} - 6 q^{7} + 135 q^{9} - 42 q^{11} + 150 q^{13} - 153 q^{17} + 216 q^{21} + 72 q^{23} + 207 q^{25} + 219 q^{27} + 462 q^{29} + 30 q^{31} + 309 q^{33} - 84 q^{35} - 30 q^{37} - 1086 q^{39} + 1368 q^{41} + 345 q^{43} + 882 q^{45} - 1134 q^{47} + 525 q^{49} + 1212 q^{51} + 612 q^{53} + 1536 q^{55} + 2190 q^{59} - 1032 q^{61} - 1770 q^{63} + 1530 q^{65} + 618 q^{67} + 756 q^{69} - 804 q^{71} + 996 q^{73} - 1758 q^{77} + 630 q^{79} + 1947 q^{81} + 2382 q^{83} - 4290 q^{85} + 1110 q^{87} + 4965 q^{89} + 864 q^{91} + 2736 q^{93} + 4410 q^{97} + 2385 q^{99}+O(q^{100}) \) 15 * q + 12 * q^5 - 6 * q^7 + 135 * q^9 - 42 * q^11 + 150 * q^13 - 153 * q^17 + 216 * q^21 + 72 * q^23 + 207 * q^25 + 219 * q^27 + 462 * q^29 + 30 * q^31 + 309 * q^33 - 84 * q^35 - 30 * q^37 - 1086 * q^39 + 1368 * q^41 + 345 * q^43 + 882 * q^45 - 1134 * q^47 + 525 * q^49 + 1212 * q^51 + 612 * q^53 + 1536 * q^55 + 2190 * q^59 - 1032 * q^61 - 1770 * q^63 + 1530 * q^65 + 618 * q^67 + 756 * q^69 - 804 * q^71 + 996 * q^73 - 1758 * q^77 + 630 * q^79 + 1947 * q^81 + 2382 * q^83 - 4290 * q^85 + 1110 * q^87 + 4965 * q^89 + 864 * q^91 + 2736 * q^93 + 4410 * q^97 + 2385 * q^99

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