LMFDB - Embedded newform 6036.2.a.i.1.22

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6036,2,Mod(1,6036)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6036, 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("6036.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6036 = 2^{2} \cdot 3 \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6036.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:	$48.1977026600$
Analytic rank:	$0$
Dimension:	$26$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.22
Character	$\chi$	$=$	6036.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +3.20667 q^{5} -2.04552 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +3.20667 q^{5} -2.04552 q^{7} +1.00000 q^{9} +0.355494 q^{11} -1.61287 q^{13} -3.20667 q^{15} -1.55291 q^{17} +4.81312 q^{19} +2.04552 q^{21} -1.33494 q^{23} +5.28275 q^{25} -1.00000 q^{27} +7.93111 q^{29} +0.483432 q^{31} -0.355494 q^{33} -6.55932 q^{35} +1.13976 q^{37} +1.61287 q^{39} +2.30777 q^{41} -8.90360 q^{43} +3.20667 q^{45} -8.25441 q^{47} -2.81584 q^{49} +1.55291 q^{51} +5.66316 q^{53} +1.13995 q^{55} -4.81312 q^{57} +8.23682 q^{59} +9.44976 q^{61} -2.04552 q^{63} -5.17194 q^{65} +3.12891 q^{67} +1.33494 q^{69} +2.32186 q^{71} -5.59535 q^{73} -5.28275 q^{75} -0.727172 q^{77} +1.77428 q^{79} +1.00000 q^{81} +1.56097 q^{83} -4.97966 q^{85} -7.93111 q^{87} +8.47054 q^{89} +3.29916 q^{91} -0.483432 q^{93} +15.4341 q^{95} +19.3315 q^{97} +0.355494 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 26 q - 26 q^{3} + 6 q^{5} + 5 q^{7} + 26 q^{9}+O(q^{10}) $ 26 * q - 26 * q^3 + 6 * q^5 + 5 * q^7 + 26 * q^9	$ 26 q - 26 q^{3} + 6 q^{5} + 5 q^{7} + 26 q^{9} - 11 q^{11} + 13 q^{13} - 6 q^{15} + 12 q^{17} - q^{19} - 5 q^{21} - 22 q^{23} + 48 q^{25} - 26 q^{27} + 6 q^{29} + 19 q^{31} + 11 q^{33} - 21 q^{35} + 20 q^{37} - 13 q^{39} + 25 q^{41} + 4 q^{43} + 6 q^{45} + 8 q^{47} + 67 q^{49} - 12 q^{51} - 5 q^{53} + 20 q^{55} + q^{57} - 18 q^{59} + 43 q^{61} + 5 q^{63} + 41 q^{65} + 5 q^{67} + 22 q^{69} - q^{71} + 22 q^{73} - 48 q^{75} + 23 q^{77} + 16 q^{79} + 26 q^{81} - 19 q^{83} + 29 q^{85} - 6 q^{87} + 49 q^{89} - 13 q^{91} - 19 q^{93} - 26 q^{95} + 25 q^{97} - 11 q^{99}+O(q^{100}) $ 26 * q - 26 * q^3 + 6 * q^5 + 5 * q^7 + 26 * q^9 - 11 * q^11 + 13 * q^13 - 6 * q^15 + 12 * q^17 - q^19 - 5 * q^21 - 22 * q^23 + 48 * q^25 - 26 * q^27 + 6 * q^29 + 19 * q^31 + 11 * q^33 - 21 * q^35 + 20 * q^37 - 13 * q^39 + 25 * q^41 + 4 * q^43 + 6 * q^45 + 8 * q^47 + 67 * q^49 - 12 * q^51 - 5 * q^53 + 20 * q^55 + q^57 - 18 * q^59 + 43 * q^61 + 5 * q^63 + 41 * q^65 + 5 * q^67 + 22 * q^69 - q^71 + 22 * q^73 - 48 * q^75 + 23 * q^77 + 16 * q^79 + 26 * q^81 - 19 * q^83 + 29 * q^85 - 6 * q^87 + 49 * q^89 - 13 * q^91 - 19 * q^93 - 26 * q^95 + 25 * q^97 - 11 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	3.20667	1.43407	0.717034	−	0.697038i	$-0.245499\pi$
$5$	3.20667	1.43407	0.717034	+	0.697038i	$0.245499\pi$
$6$	0	0
$7$	−2.04552	−0.773135	−0.386568	−	0.922261i	$-0.626339\pi$
$7$	−2.04552	−0.773135	−0.386568	+	0.922261i	$0.626339\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0.355494	0.107186	0.0535928	−	0.998563i	$-0.482933\pi$
$11$	0.355494	0.107186	0.0535928	+	0.998563i	$0.482933\pi$
$12$	0	0
$13$	−1.61287	−0.447329	−0.223664	−	0.974666i	$-0.571802\pi$
$13$	−1.61287	−0.447329	−0.223664	+	0.974666i	$0.571802\pi$
$14$	0	0
$15$	−3.20667	−0.827959
$16$	0	0
$17$	−1.55291	−0.376635	−0.188318	−	0.982108i	$-0.560303\pi$
$17$	−1.55291	−0.376635	−0.188318	+	0.982108i	$0.560303\pi$
$18$	0	0
$19$	4.81312	1.10421	0.552103	−	0.833776i	$-0.313825\pi$
$19$	4.81312	1.10421	0.552103	+	0.833776i	$0.313825\pi$
$20$	0	0
$21$	2.04552	0.446370
$22$	0	0
$23$	−1.33494	−0.278355	−0.139178	−	0.990267i	$-0.544446\pi$
$23$	−1.33494	−0.278355	−0.139178	+	0.990267i	$0.544446\pi$
$24$	0	0
$25$	5.28275	1.05655
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	7.93111	1.47277	0.736385	−	0.676562i	$-0.236531\pi$
$29$	7.93111	1.47277	0.736385	+	0.676562i	$0.236531\pi$
$30$	0	0
$31$	0.483432	0.0868270	0.0434135	−	0.999057i	$-0.486177\pi$
$31$	0.483432	0.0868270	0.0434135	+	0.999057i	$0.486177\pi$
$32$	0	0
$33$	−0.355494	−0.0618836
$34$	0	0
$35$	−6.55932	−1.10873
$36$	0	0
$37$	1.13976	0.187375	0.0936876	−	0.995602i	$-0.470135\pi$
$37$	1.13976	0.187375	0.0936876	+	0.995602i	$0.470135\pi$
$38$	0	0
$39$	1.61287	0.258266
$40$	0	0
$41$	2.30777	0.360413	0.180206	−	0.983629i	$-0.442323\pi$
$41$	2.30777	0.360413	0.180206	+	0.983629i	$0.442323\pi$
$42$	0	0
$43$	−8.90360	−1.35779	−0.678893	−	0.734237i	$-0.737540\pi$
$43$	−8.90360	−1.35779	−0.678893	+	0.734237i	$0.737540\pi$
$44$	0	0
$45$	3.20667	0.478023
$46$	0	0
$47$	−8.25441	−1.20403	−0.602014	−	0.798485i	$-0.705635\pi$
$47$	−8.25441	−1.20403	−0.602014	+	0.798485i	$0.705635\pi$
$48$	0	0
$49$	−2.81584	−0.402262
$50$	0	0
$51$	1.55291	0.217450
$52$	0	0
$53$	5.66316	0.777894	0.388947	−	0.921260i	$-0.372839\pi$
$53$	5.66316	0.777894	0.388947	+	0.921260i	$0.372839\pi$
$54$	0	0
$55$	1.13995	0.153711
$56$	0	0
$57$	−4.81312	−0.637514
$58$	0	0
$59$	8.23682	1.07234	0.536171	−	0.844109i	$-0.319870\pi$
$59$	8.23682	1.07234	0.536171	+	0.844109i	$0.319870\pi$
$60$	0	0
$61$	9.44976	1.20992	0.604959	−	0.796257i	$-0.293189\pi$
$61$	9.44976	1.20992	0.604959	+	0.796257i	$0.293189\pi$
$62$	0	0
$63$	−2.04552	−0.257712
$64$	0	0
$65$	−5.17194	−0.641500
$66$	0	0
$67$	3.12891	0.382257	0.191128	−	0.981565i	$-0.438785\pi$
$67$	3.12891	0.382257	0.191128	+	0.981565i	$0.438785\pi$
$68$	0	0
$69$	1.33494	0.160708
$70$	0	0
$71$	2.32186	0.275555	0.137777	−	0.990463i	$-0.456004\pi$
$71$	2.32186	0.275555	0.137777	+	0.990463i	$0.456004\pi$
$72$	0	0
$73$	−5.59535	−0.654886	−0.327443	−	0.944871i	$-0.606187\pi$
$73$	−5.59535	−0.654886	−0.327443	+	0.944871i	$0.606187\pi$
$74$	0	0
$75$	−5.28275	−0.610000
$76$	0	0
$77$	−0.727172	−0.0828689
$78$	0	0
$79$	1.77428	0.199622	0.0998109	−	0.995006i	$-0.468176\pi$
$79$	1.77428	0.199622	0.0998109	+	0.995006i	$0.468176\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	1.56097	0.171339	0.0856694	−	0.996324i	$-0.472697\pi$
$83$	1.56097	0.171339	0.0856694	+	0.996324i	$0.472697\pi$
$84$	0	0
$85$	−4.97966	−0.540120
$86$	0	0
$87$	−7.93111	−0.850304
$88$	0	0
$89$	8.47054	0.897875	0.448938	−	0.893563i	$-0.351803\pi$
$89$	8.47054	0.897875	0.448938	+	0.893563i	$0.351803\pi$
$90$	0	0
$91$	3.29916	0.345846
$92$	0	0
$93$	−0.483432	−0.0501296
$94$	0	0
$95$	15.4341	1.58351
$96$	0	0
$97$	19.3315	1.96282	0.981410	−	0.191925i	$-0.0614730\pi$
$97$	19.3315	1.96282	0.981410	+	0.191925i	$0.0614730\pi$
$98$	0	0
$99$	0.355494	0.0357285
$100$	0	0
$101$	−0.364219	−0.0362412	−0.0181206	−	0.999836i	$-0.505768\pi$
$101$	−0.364219	−0.0362412	−0.0181206	+	0.999836i	$0.505768\pi$
$102$	0	0
$103$	−18.5620	−1.82897	−0.914486	−	0.404617i	$-0.867405\pi$
$103$	−18.5620	−1.82897	−0.914486	+	0.404617i	$0.867405\pi$
$104$	0	0
$105$	6.55932	0.640124
$106$	0	0
$107$	13.4838	1.30353	0.651765	−	0.758421i	$-0.274029\pi$
$107$	13.4838	1.30353	0.651765	+	0.758421i	$0.274029\pi$
$108$	0	0
$109$	14.1562	1.35592	0.677959	−	0.735099i	$-0.262865\pi$
$109$	14.1562	1.35592	0.677959	+	0.735099i	$0.262865\pi$
$110$	0	0
$111$	−1.13976	−0.108181
$112$	0	0
$113$	17.2333	1.62117	0.810587	−	0.585619i	$-0.199148\pi$
$113$	17.2333	1.62117	0.810587	+	0.585619i	$0.199148\pi$
$114$	0	0
$115$	−4.28073	−0.399180
$116$	0	0
$117$	−1.61287	−0.149110
$118$	0	0
$119$	3.17651	0.291190
$120$	0	0
$121$	−10.8736	−0.988511
$122$	0	0
$123$	−2.30777	−0.208085
$124$	0	0
$125$	0.906694	0.0810972
$126$	0	0
$127$	−15.0181	−1.33264	−0.666318	−	0.745667i	$-0.732131\pi$
$127$	−15.0181	−1.33264	−0.666318	+	0.745667i	$0.732131\pi$
$128$	0	0
$129$	8.90360	0.783919
$130$	0	0
$131$	9.51739	0.831539	0.415769	−	0.909470i	$-0.363512\pi$
$131$	9.51739	0.831539	0.415769	+	0.909470i	$0.363512\pi$
$132$	0	0
$133$	−9.84536	−0.853701
$134$	0	0
$135$	−3.20667	−0.275986
$136$	0	0
$137$	−16.3519	−1.39703	−0.698517	−	0.715594i	$-0.746156\pi$
$137$	−16.3519	−1.39703	−0.698517	+	0.715594i	$0.746156\pi$
$138$	0	0
$139$	4.46275	0.378526	0.189263	−	0.981926i	$-0.439390\pi$
$139$	4.46275	0.378526	0.189263	+	0.981926i	$0.439390\pi$
$140$	0	0
$141$	8.25441	0.695146
$142$	0	0
$143$	−0.573365	−0.0479472
$144$	0	0
$145$	25.4325	2.11205
$146$	0	0
$147$	2.81584	0.232246
$148$	0	0
$149$	−9.53461	−0.781106	−0.390553	−	0.920581i	$-0.627716\pi$
$149$	−9.53461	−0.781106	−0.390553	+	0.920581i	$0.627716\pi$
$150$	0	0
$151$	23.3435	1.89967	0.949834	−	0.312754i	$-0.101251\pi$
$151$	23.3435	1.89967	0.949834	+	0.312754i	$0.101251\pi$
$152$	0	0
$153$	−1.55291	−0.125545
$154$	0	0
$155$	1.55021	0.124516
$156$	0	0
$157$	22.9112	1.82851	0.914254	−	0.405140i	$-0.132777\pi$
$157$	22.9112	1.82851	0.914254	+	0.405140i	$0.132777\pi$
$158$	0	0
$159$	−5.66316	−0.449118
$160$	0	0
$161$	2.73066	0.215206
$162$	0	0
$163$	−13.0075	−1.01882	−0.509412	−	0.860523i	$-0.670137\pi$
$163$	−13.0075	−1.01882	−0.509412	+	0.860523i	$0.670137\pi$
$164$	0	0
$165$	−1.13995	−0.0887453
$166$	0	0
$167$	8.21488	0.635687	0.317843	−	0.948143i	$-0.397041\pi$
$167$	8.21488	0.635687	0.317843	+	0.948143i	$0.397041\pi$
$168$	0	0
$169$	−10.3987	−0.799897
$170$	0	0
$171$	4.81312	0.368069
$172$	0	0
$173$	0.106417	0.00809073	0.00404536	−	0.999992i	$-0.498712\pi$
$173$	0.106417	0.00809073	0.00404536	+	0.999992i	$0.498712\pi$
$174$	0	0
$175$	−10.8060	−0.816856
$176$	0	0
$177$	−8.23682	−0.619118
$178$	0	0
$179$	16.3832	1.22454	0.612268	−	0.790651i	$-0.290257\pi$
$179$	16.3832	1.22454	0.612268	+	0.790651i	$0.290257\pi$
$180$	0	0
$181$	3.95316	0.293836	0.146918	−	0.989149i	$-0.453065\pi$
$181$	3.95316	0.293836	0.146918	+	0.989149i	$0.453065\pi$
$182$	0	0
$183$	−9.44976	−0.698546
$184$	0	0
$185$	3.65483	0.268709
$186$	0	0
$187$	−0.552049	−0.0403698
$188$	0	0
$189$	2.04552	0.148790
$190$	0	0
$191$	19.2065	1.38974	0.694869	−	0.719137i	$-0.255463\pi$
$191$	19.2065	1.38974	0.694869	+	0.719137i	$0.255463\pi$
$192$	0	0
$193$	−16.8901	−1.21578	−0.607889	−	0.794022i	$-0.707984\pi$
$193$	−16.8901	−1.21578	−0.607889	+	0.794022i	$0.707984\pi$
$194$	0	0
$195$	5.17194	0.370370
$196$	0	0
$197$	−4.28196	−0.305077	−0.152539	−	0.988298i	$-0.548745\pi$
$197$	−4.28196	−0.305077	−0.152539	+	0.988298i	$0.548745\pi$
$198$	0	0
$199$	−9.80097	−0.694772	−0.347386	−	0.937722i	$-0.612931\pi$
$199$	−9.80097	−0.694772	−0.347386	+	0.937722i	$0.612931\pi$
$200$	0	0
$201$	−3.12891	−0.220696
$202$	0	0
$203$	−16.2233	−1.13865
$204$	0	0
$205$	7.40026	0.516857
$206$	0	0
$207$	−1.33494	−0.0927851
$208$	0	0
$209$	1.71104	0.118355
$210$	0	0
$211$	14.6970	1.01178	0.505891	−	0.862597i	$-0.331164\pi$
$211$	14.6970	1.01178	0.505891	+	0.862597i	$0.331164\pi$
$212$	0	0
$213$	−2.32186	−0.159091
$214$	0	0
$215$	−28.5509	−1.94716
$216$	0	0
$217$	−0.988872	−0.0671290
$218$	0	0
$219$	5.59535	0.378098
$220$	0	0
$221$	2.50463	0.168480
$222$	0	0
$223$	−1.91766	−0.128416	−0.0642080	−	0.997937i	$-0.520452\pi$
$223$	−1.91766	−0.128416	−0.0642080	+	0.997937i	$0.520452\pi$
$224$	0	0
$225$	5.28275	0.352183
$226$	0	0
$227$	−11.7977	−0.783039	−0.391519	−	0.920170i	$-0.628050\pi$
$227$	−11.7977	−0.783039	−0.391519	+	0.920170i	$0.628050\pi$
$228$	0	0
$229$	−18.0395	−1.19209	−0.596043	−	0.802952i	$-0.703261\pi$
$229$	−18.0395	−1.19209	−0.596043	+	0.802952i	$0.703261\pi$
$230$	0	0
$231$	0.727172	0.0478444
$232$	0	0
$233$	24.6146	1.61256	0.806278	−	0.591536i	$-0.201478\pi$
$233$	24.6146	1.61256	0.806278	+	0.591536i	$0.201478\pi$
$234$	0	0
$235$	−26.4692	−1.72666
$236$	0	0
$237$	−1.77428	−0.115252
$238$	0	0
$239$	−15.2348	−0.985456	−0.492728	−	0.870183i	$-0.664000\pi$
$239$	−15.2348	−0.985456	−0.492728	+	0.870183i	$0.664000\pi$
$240$	0	0
$241$	−24.9578	−1.60767	−0.803837	−	0.594850i	$-0.797211\pi$
$241$	−24.9578	−1.60767	−0.803837	+	0.594850i	$0.797211\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−9.02946	−0.576871
$246$	0	0
$247$	−7.76293	−0.493944
$248$	0	0
$249$	−1.56097	−0.0989225
$250$	0	0
$251$	8.01656	0.506001	0.253000	−	0.967466i	$-0.418583\pi$
$251$	8.01656	0.506001	0.253000	+	0.967466i	$0.418583\pi$
$252$	0	0
$253$	−0.474565	−0.0298357
$254$	0	0
$255$	4.97966	0.311839
$256$	0	0
$257$	−7.64905	−0.477135	−0.238567	−	0.971126i	$-0.576678\pi$
$257$	−7.64905	−0.477135	−0.238567	+	0.971126i	$0.576678\pi$
$258$	0	0
$259$	−2.33140	−0.144866
$260$	0	0
$261$	7.93111	0.490924
$262$	0	0
$263$	−1.93205	−0.119136	−0.0595678	−	0.998224i	$-0.518972\pi$
$263$	−1.93205	−0.119136	−0.0595678	+	0.998224i	$0.518972\pi$
$264$	0	0
$265$	18.1599	1.11555
$266$	0	0
$267$	−8.47054	−0.518388
$268$	0	0
$269$	16.3658	0.997842	0.498921	−	0.866648i	$-0.333730\pi$
$269$	16.3658	0.997842	0.498921	+	0.866648i	$0.333730\pi$
$270$	0	0
$271$	−3.81117	−0.231512	−0.115756	−	0.993278i	$-0.536929\pi$
$271$	−3.81117	−0.231512	−0.115756	+	0.993278i	$0.536929\pi$
$272$	0	0
$273$	−3.29916	−0.199674
$274$	0	0
$275$	1.87799	0.113247
$276$	0	0
$277$	30.6315	1.84047	0.920233	−	0.391372i	$-0.127999\pi$
$277$	30.6315	1.84047	0.920233	+	0.391372i	$0.127999\pi$
$278$	0	0
$279$	0.483432	0.0289423
$280$	0	0
$281$	23.9820	1.43065	0.715325	−	0.698792i	$-0.246279\pi$
$281$	23.9820	1.43065	0.715325	+	0.698792i	$0.246279\pi$
$282$	0	0
$283$	12.4704	0.741291	0.370645	−	0.928774i	$-0.379137\pi$
$283$	12.4704	0.741291	0.370645	+	0.928774i	$0.379137\pi$
$284$	0	0
$285$	−15.4341	−0.914238
$286$	0	0
$287$	−4.72059	−0.278648
$288$	0	0
$289$	−14.5885	−0.858146
$290$	0	0
$291$	−19.3315	−1.13323
$292$	0	0
$293$	28.9893	1.69357	0.846787	−	0.531932i	$-0.178534\pi$
$293$	28.9893	1.69357	0.846787	+	0.531932i	$0.178534\pi$
$294$	0	0
$295$	26.4128	1.53781
$296$	0	0
$297$	−0.355494	−0.0206279
$298$	0	0
$299$	2.15309	0.124516
$300$	0	0
$301$	18.2125	1.04975
$302$	0	0
$303$	0.364219	0.0209239
$304$	0	0
$305$	30.3023	1.73510
$306$	0	0
$307$	3.71327	0.211928	0.105964	−	0.994370i	$-0.466207\pi$
$307$	3.71327	0.211928	0.105964	+	0.994370i	$0.466207\pi$
$308$	0	0
$309$	18.5620	1.05596
$310$	0	0
$311$	−9.94769	−0.564082	−0.282041	−	0.959402i	$-0.591011\pi$
$311$	−9.94769	−0.564082	−0.282041	+	0.959402i	$0.591011\pi$
$312$	0	0
$313$	15.5920	0.881309	0.440655	−	0.897677i	$-0.354746\pi$
$313$	15.5920	0.881309	0.440655	+	0.897677i	$0.354746\pi$
$314$	0	0
$315$	−6.55932	−0.369576
$316$	0	0
$317$	11.5736	0.650036	0.325018	−	0.945708i	$-0.394630\pi$
$317$	11.5736	0.650036	0.325018	+	0.945708i	$0.394630\pi$
$318$	0	0
$319$	2.81946	0.157860
$320$	0	0
$321$	−13.4838	−0.752593
$322$	0	0
$323$	−7.47433	−0.415883
$324$	0	0
$325$	−8.52038	−0.472626
$326$	0	0
$327$	−14.1562	−0.782840
$328$	0	0
$329$	16.8846	0.930877
$330$	0	0
$331$	1.66646	0.0915967	0.0457984	−	0.998951i	$-0.485417\pi$
$331$	1.66646	0.0915967	0.0457984	+	0.998951i	$0.485417\pi$
$332$	0	0
$333$	1.13976	0.0624584
$334$	0	0
$335$	10.0334	0.548182
$336$	0	0
$337$	26.2350	1.42911	0.714556	−	0.699578i	$-0.246629\pi$
$337$	26.2350	1.42911	0.714556	+	0.699578i	$0.246629\pi$
$338$	0	0
$339$	−17.2333	−0.935985
$340$	0	0
$341$	0.171857	0.00930660
$342$	0	0
$343$	20.0785	1.08414
$344$	0	0
$345$	4.28073	0.230467
$346$	0	0
$347$	−34.9097	−1.87405	−0.937025	−	0.349262i	$-0.886432\pi$
$347$	−34.9097	−1.87405	−0.937025	+	0.349262i	$0.886432\pi$
$348$	0	0
$349$	19.7788	1.05873	0.529367	−	0.848393i	$-0.322429\pi$
$349$	19.7788	1.05873	0.529367	+	0.848393i	$0.322429\pi$
$350$	0	0
$351$	1.61287	0.0860885
$352$	0	0
$353$	−23.8489	−1.26935	−0.634674	−	0.772780i	$-0.718866\pi$
$353$	−23.8489	−1.26935	−0.634674	+	0.772780i	$0.718866\pi$
$354$	0	0
$355$	7.44546	0.395164
$356$	0	0
$357$	−3.17651	−0.168118
$358$	0	0
$359$	−28.5291	−1.50571	−0.752854	−	0.658188i	$-0.771323\pi$
$359$	−28.5291	−1.50571	−0.752854	+	0.658188i	$0.771323\pi$
$360$	0	0
$361$	4.16616	0.219272
$362$	0	0
$363$	10.8736	0.570717
$364$	0	0
$365$	−17.9424	−0.939150
$366$	0	0
$367$	−3.81132	−0.198949	−0.0994746	−	0.995040i	$-0.531716\pi$
$367$	−3.81132	−0.198949	−0.0994746	+	0.995040i	$0.531716\pi$
$368$	0	0
$369$	2.30777	0.120138
$370$	0	0
$371$	−11.5841	−0.601417
$372$	0	0
$373$	27.9275	1.44603	0.723015	−	0.690832i	$-0.242756\pi$
$373$	27.9275	1.44603	0.723015	+	0.690832i	$0.242756\pi$
$374$	0	0
$375$	−0.906694	−0.0468215
$376$	0	0
$377$	−12.7918	−0.658813
$378$	0	0
$379$	−3.66848	−0.188437	−0.0942185	−	0.995552i	$-0.530035\pi$
$379$	−3.66848	−0.188437	−0.0942185	+	0.995552i	$0.530035\pi$
$380$	0	0
$381$	15.0181	0.769398
$382$	0	0
$383$	−6.84066	−0.349541	−0.174771	−	0.984609i	$-0.555918\pi$
$383$	−6.84066	−0.349541	−0.174771	+	0.984609i	$0.555918\pi$
$384$	0	0
$385$	−2.33180	−0.118840
$386$	0	0
$387$	−8.90360	−0.452596
$388$	0	0
$389$	8.25768	0.418681	0.209340	−	0.977843i	$-0.432868\pi$
$389$	8.25768	0.418681	0.209340	+	0.977843i	$0.432868\pi$
$390$	0	0
$391$	2.07304	0.104838
$392$	0	0
$393$	−9.51739	−0.480089
$394$	0	0
$395$	5.68953	0.286271
$396$	0	0
$397$	18.2537	0.916125	0.458062	−	0.888920i	$-0.348544\pi$
$397$	18.2537	0.916125	0.458062	+	0.888920i	$0.348544\pi$
$398$	0	0
$399$	9.84536	0.492884
$400$	0	0
$401$	15.6976	0.783902	0.391951	−	0.919986i	$-0.371800\pi$
$401$	15.6976	0.783902	0.391951	+	0.919986i	$0.371800\pi$
$402$	0	0
$403$	−0.779712	−0.0388402
$404$	0	0
$405$	3.20667	0.159341
$406$	0	0
$407$	0.405178	0.0200839
$408$	0	0
$409$	25.6545	1.26853	0.634267	−	0.773114i	$-0.281302\pi$
$409$	25.6545	1.26853	0.634267	+	0.773114i	$0.281302\pi$
$410$	0	0
$411$	16.3519	0.806578
$412$	0	0
$413$	−16.8486	−0.829066
$414$	0	0
$415$	5.00552	0.245711
$416$	0	0
$417$	−4.46275	−0.218542
$418$	0	0
$419$	5.81293	0.283980	0.141990	−	0.989868i	$-0.454650\pi$
$419$	5.81293	0.283980	0.141990	+	0.989868i	$0.454650\pi$
$420$	0	0
$421$	36.9695	1.80178	0.900891	−	0.434046i	$-0.142914\pi$
$421$	36.9695	1.80178	0.900891	+	0.434046i	$0.142914\pi$
$422$	0	0
$423$	−8.25441	−0.401343
$424$	0	0
$425$	−8.20362	−0.397934
$426$	0	0
$427$	−19.3297	−0.935430
$428$	0	0
$429$	0.573365	0.0276823
$430$	0	0
$431$	17.2190	0.829408	0.414704	−	0.909956i	$-0.363885\pi$
$431$	17.2190	0.829408	0.414704	+	0.909956i	$0.363885\pi$
$432$	0	0
$433$	−2.44881	−0.117682	−0.0588412	−	0.998267i	$-0.518741\pi$
$433$	−2.44881	−0.117682	−0.0588412	+	0.998267i	$0.518741\pi$
$434$	0	0
$435$	−25.4325	−1.21939
$436$	0	0
$437$	−6.42525	−0.307362
$438$	0	0
$439$	29.9521	1.42954	0.714768	−	0.699361i	$-0.246532\pi$
$439$	29.9521	1.42954	0.714768	+	0.699361i	$0.246532\pi$
$440$	0	0
$441$	−2.81584	−0.134087
$442$	0	0
$443$	6.47518	0.307645	0.153822	−	0.988099i	$-0.450842\pi$
$443$	6.47518	0.307645	0.153822	+	0.988099i	$0.450842\pi$
$444$	0	0
$445$	27.1622	1.28761
$446$	0	0
$447$	9.53461	0.450972
$448$	0	0
$449$	19.2838	0.910059	0.455029	−	0.890476i	$-0.349629\pi$
$449$	19.2838	0.910059	0.455029	+	0.890476i	$0.349629\pi$
$450$	0	0
$451$	0.820399	0.0386311
$452$	0	0
$453$	−23.3435	−1.09677
$454$	0	0
$455$	10.5793	0.495966
$456$	0	0
$457$	−10.8764	−0.508776	−0.254388	−	0.967102i	$-0.581874\pi$
$457$	−10.8764	−0.508776	−0.254388	+	0.967102i	$0.581874\pi$
$458$	0	0
$459$	1.55291	0.0724834
$460$	0	0
$461$	−30.8443	−1.43656	−0.718280	−	0.695754i	$-0.755070\pi$
$461$	−30.8443	−1.43656	−0.718280	+	0.695754i	$0.755070\pi$
$462$	0	0
$463$	−31.5182	−1.46477	−0.732387	−	0.680888i	$-0.761594\pi$
$463$	−31.5182	−1.46477	−0.732387	+	0.680888i	$0.761594\pi$
$464$	0	0
$465$	−1.55021	−0.0718892
$466$	0	0
$467$	18.2421	0.844145	0.422072	−	0.906562i	$-0.361303\pi$
$467$	18.2421	0.844145	0.422072	+	0.906562i	$0.361303\pi$
$468$	0	0
$469$	−6.40025	−0.295536
$470$	0	0
$471$	−22.9112	−1.05569
$472$	0	0
$473$	−3.16518	−0.145535
$474$	0	0
$475$	25.4265	1.16665
$476$	0	0
$477$	5.66316	0.259298
$478$	0	0
$479$	14.4284	0.659253	0.329626	−	0.944111i	$-0.393077\pi$
$479$	14.4284	0.659253	0.329626	+	0.944111i	$0.393077\pi$
$480$	0	0
$481$	−1.83828	−0.0838184
$482$	0	0
$483$	−2.73066	−0.124249
$484$	0	0
$485$	61.9899	2.81482
$486$	0	0
$487$	22.6553	1.02661	0.513304	−	0.858207i	$-0.328421\pi$
$487$	22.6553	1.02661	0.513304	+	0.858207i	$0.328421\pi$
$488$	0	0
$489$	13.0075	0.588219
$490$	0	0
$491$	−29.7201	−1.34125	−0.670624	−	0.741797i	$-0.733974\pi$
$491$	−29.7201	−1.34125	−0.670624	+	0.741797i	$0.733974\pi$
$492$	0	0
$493$	−12.3163	−0.554697
$494$	0	0
$495$	1.13995	0.0512371
$496$	0	0
$497$	−4.74943	−0.213041
$498$	0	0
$499$	20.2190	0.905127	0.452564	−	0.891732i	$-0.350510\pi$
$499$	20.2190	0.905127	0.452564	+	0.891732i	$0.350510\pi$
$500$	0	0
$501$	−8.21488	−0.367014
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	−1.16793	−0.0519723
$506$	0	0
$507$	10.3987	0.461821
$508$	0	0
$509$	−12.8314	−0.568744	−0.284372	−	0.958714i	$-0.591785\pi$
$509$	−12.8314	−0.568744	−0.284372	+	0.958714i	$0.591785\pi$
$510$	0	0
$511$	11.4454	0.506315
$512$	0	0
$513$	−4.81312	−0.212505
$514$	0	0
$515$	−59.5224	−2.62287
$516$	0	0
$517$	−2.93439	−0.129054
$518$	0	0
$519$	−0.106417	−0.00467118
$520$	0	0
$521$	4.05437	0.177625	0.0888126	−	0.996048i	$-0.471693\pi$
$521$	4.05437	0.177625	0.0888126	+	0.996048i	$0.471693\pi$
$522$	0	0
$523$	17.4966	0.765072	0.382536	−	0.923941i	$-0.375051\pi$
$523$	17.4966	0.765072	0.382536	+	0.923941i	$0.375051\pi$
$524$	0	0
$525$	10.8060	0.471612
$526$	0	0
$527$	−0.750725	−0.0327021
$528$	0	0
$529$	−21.2179	−0.922518
$530$	0	0
$531$	8.23682	0.357448
$532$	0	0
$533$	−3.72213	−0.161223
$534$	0	0
$535$	43.2382	1.86935
$536$	0	0
$537$	−16.3832	−0.706986
$538$	0	0
$539$	−1.00101	−0.0431167
$540$	0	0
$541$	−5.39908	−0.232125	−0.116062	−	0.993242i	$-0.537027\pi$
$541$	−5.39908	−0.232125	−0.116062	+	0.993242i	$0.537027\pi$
$542$	0	0
$543$	−3.95316	−0.169646
$544$	0	0
$545$	45.3943	1.94448
$546$	0	0
$547$	−34.0206	−1.45462	−0.727309	−	0.686311i	$-0.759229\pi$
$547$	−34.0206	−1.45462	−0.727309	+	0.686311i	$0.759229\pi$
$548$	0	0
$549$	9.44976	0.403306
$550$	0	0
$551$	38.1734	1.62624
$552$	0	0
$553$	−3.62933	−0.154335
$554$	0	0
$555$	−3.65483	−0.155139
$556$	0	0
$557$	−28.8278	−1.22147	−0.610736	−	0.791834i	$-0.709126\pi$
$557$	−28.8278	−1.22147	−0.610736	+	0.791834i	$0.709126\pi$
$558$	0	0
$559$	14.3603	0.607377
$560$	0	0
$561$	0.552049	0.0233075
$562$	0	0
$563$	−9.86817	−0.415894	−0.207947	−	0.978140i	$-0.566678\pi$
$563$	−9.86817	−0.415894	−0.207947	+	0.978140i	$0.566678\pi$
$564$	0	0
$565$	55.2616	2.32487
$566$	0	0
$567$	−2.04552	−0.0859039
$568$	0	0
$569$	−32.6392	−1.36830	−0.684152	−	0.729339i	$-0.739828\pi$
$569$	−32.6392	−1.36830	−0.684152	+	0.729339i	$0.739828\pi$
$570$	0	0
$571$	−4.49123	−0.187952	−0.0939760	−	0.995574i	$-0.529958\pi$
$571$	−4.49123	−0.187952	−0.0939760	+	0.995574i	$0.529958\pi$
$572$	0	0
$573$	−19.2065	−0.802365
$574$	0	0
$575$	−7.05218	−0.294096
$576$	0	0
$577$	19.4551	0.809927	0.404964	−	0.914333i	$-0.367284\pi$
$577$	19.4551	0.809927	0.404964	+	0.914333i	$0.367284\pi$
$578$	0	0
$579$	16.8901	0.701930
$580$	0	0
$581$	−3.19300	−0.132468
$582$	0	0
$583$	2.01322	0.0833790
$584$	0	0
$585$	−5.17194	−0.213833
$586$	0	0
$587$	−6.30301	−0.260153	−0.130077	−	0.991504i	$-0.541522\pi$
$587$	−6.30301	−0.260153	−0.130077	+	0.991504i	$0.541522\pi$
$588$	0	0
$589$	2.32682	0.0958749
$590$	0	0
$591$	4.28196	0.176136
$592$	0	0
$593$	4.30462	0.176770	0.0883849	−	0.996086i	$-0.471829\pi$
$593$	4.30462	0.176770	0.0883849	+	0.996086i	$0.471829\pi$
$594$	0	0
$595$	10.1860	0.417586
$596$	0	0
$597$	9.80097	0.401127
$598$	0	0
$599$	−21.0971	−0.862005	−0.431002	−	0.902351i	$-0.641840\pi$
$599$	−21.0971	−0.862005	−0.431002	+	0.902351i	$0.641840\pi$
$600$	0	0
$601$	−9.51242	−0.388020	−0.194010	−	0.981000i	$-0.562149\pi$
$601$	−9.51242	−0.388020	−0.194010	+	0.981000i	$0.562149\pi$
$602$	0	0
$603$	3.12891	0.127419
$604$	0	0
$605$	−34.8682	−1.41759
$606$	0	0
$607$	6.50083	0.263861	0.131930	−	0.991259i	$-0.457882\pi$
$607$	6.50083	0.263861	0.131930	+	0.991259i	$0.457882\pi$
$608$	0	0
$609$	16.2233	0.657400
$610$	0	0
$611$	13.3133	0.538597
$612$	0	0
$613$	14.0793	0.568659	0.284330	−	0.958727i	$-0.408229\pi$
$613$	14.0793	0.568659	0.284330	+	0.958727i	$0.408229\pi$
$614$	0	0
$615$	−7.40026	−0.298407
$616$	0	0
$617$	0.0474085	0.00190860	0.000954298	−	1.00000i	$-0.499696\pi$
$617$	0.0474085	0.00190860	0.000954298		1.00000i	$0.499696\pi$
$618$	0	0
$619$	47.8205	1.92207	0.961033	−	0.276432i	$-0.0891522\pi$
$619$	47.8205	1.92207	0.961033	+	0.276432i	$0.0891522\pi$
$620$	0	0
$621$	1.33494	0.0535695
$622$	0	0
$623$	−17.3267	−0.694179
$624$	0	0
$625$	−23.5063	−0.940252
$626$	0	0
$627$	−1.71104	−0.0683323
$628$	0	0
$629$	−1.76994	−0.0705721
$630$	0	0
$631$	−43.8287	−1.74479	−0.872397	−	0.488797i	$-0.837436\pi$
$631$	−43.8287	−1.74479	−0.872397	+	0.488797i	$0.837436\pi$
$632$	0	0
$633$	−14.6970	−0.584153
$634$	0	0
$635$	−48.1580	−1.91109
$636$	0	0
$637$	4.54157	0.179944
$638$	0	0
$639$	2.32186	0.0918515
$640$	0	0
$641$	−32.3484	−1.27769	−0.638843	−	0.769337i	$-0.720587\pi$
$641$	−32.3484	−1.27769	−0.638843	+	0.769337i	$0.720587\pi$
$642$	0	0
$643$	19.4469	0.766911	0.383456	−	0.923559i	$-0.374734\pi$
$643$	19.4469	0.766911	0.383456	+	0.923559i	$0.374734\pi$
$644$	0	0
$645$	28.5509	1.12419
$646$	0	0
$647$	7.48638	0.294320	0.147160	−	0.989113i	$-0.452987\pi$
$647$	7.48638	0.294320	0.147160	+	0.989113i	$0.452987\pi$
$648$	0	0
$649$	2.92814	0.114940
$650$	0	0
$651$	0.988872	0.0387569
$652$	0	0
$653$	11.4388	0.447636	0.223818	−	0.974631i	$-0.428148\pi$
$653$	11.4388	0.447636	0.223818	+	0.974631i	$0.428148\pi$
$654$	0	0
$655$	30.5192	1.19248
$656$	0	0
$657$	−5.59535	−0.218295
$658$	0	0
$659$	0.937416	0.0365165	0.0182583	−	0.999833i	$-0.494188\pi$
$659$	0.937416	0.0365165	0.0182583	+	0.999833i	$0.494188\pi$
$660$	0	0
$661$	16.7766	0.652536	0.326268	−	0.945277i	$-0.394209\pi$
$661$	16.7766	0.652536	0.326268	+	0.945277i	$0.394209\pi$
$662$	0	0
$663$	−2.50463	−0.0972718
$664$	0	0
$665$	−31.5708	−1.22426
$666$	0	0
$667$	−10.5876	−0.409953
$668$	0	0
$669$	1.91766	0.0741410
$670$	0	0
$671$	3.35934	0.129686
$672$	0	0
$673$	32.7901	1.26396	0.631982	−	0.774983i	$-0.282242\pi$
$673$	32.7901	1.26396	0.631982	+	0.774983i	$0.282242\pi$
$674$	0	0
$675$	−5.28275	−0.203333
$676$	0	0
$677$	−3.36683	−0.129398	−0.0646990	−	0.997905i	$-0.520609\pi$
$677$	−3.36683	−0.129398	−0.0646990	+	0.997905i	$0.520609\pi$
$678$	0	0
$679$	−39.5431	−1.51752
$680$	0	0
$681$	11.7977	0.452088
$682$	0	0
$683$	16.3978	0.627446	0.313723	−	0.949515i	$-0.398424\pi$
$683$	16.3978	0.627446	0.313723	+	0.949515i	$0.398424\pi$
$684$	0	0
$685$	−52.4351	−2.00344
$686$	0	0
$687$	18.0395	0.688252
$688$	0	0
$689$	−9.13392	−0.347975
$690$	0	0
$691$	11.4656	0.436173	0.218087	−	0.975929i	$-0.430018\pi$
$691$	11.4656	0.436173	0.218087	+	0.975929i	$0.430018\pi$
$692$	0	0
$693$	−0.727172	−0.0276230
$694$	0	0
$695$	14.3106	0.542831
$696$	0	0
$697$	−3.58375	−0.135744
$698$	0	0
$699$	−24.6146	−0.931010
$700$	0	0
$701$	−24.8694	−0.939304	−0.469652	−	0.882852i	$-0.655621\pi$
$701$	−24.8694	−0.939304	−0.469652	+	0.882852i	$0.655621\pi$
$702$	0	0
$703$	5.48580	0.206901
$704$	0	0
$705$	26.4692	0.996887
$706$	0	0
$707$	0.745019	0.0280193
$708$	0	0
$709$	22.1078	0.830278	0.415139	−	0.909758i	$-0.363733\pi$
$709$	22.1078	0.830278	0.415139	+	0.909758i	$0.363733\pi$
$710$	0	0
$711$	1.77428	0.0665406
$712$	0	0
$713$	−0.645355	−0.0241687
$714$	0	0
$715$	−1.83859	−0.0687595
$716$	0	0
$717$	15.2348	0.568953
$718$	0	0
$719$	−32.0669	−1.19589	−0.597947	−	0.801536i	$-0.704017\pi$
$719$	−32.0669	−1.19589	−0.597947	+	0.801536i	$0.704017\pi$
$720$	0	0
$721$	37.9691	1.41404
$722$	0	0
$723$	24.9578	0.928190
$724$	0	0
$725$	41.8981	1.55606
$726$	0	0
$727$	15.8303	0.587113	0.293557	−	0.955942i	$-0.405161\pi$
$727$	15.8303	0.587113	0.293557	+	0.955942i	$0.405161\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	13.8265	0.511390
$732$	0	0
$733$	−34.8048	−1.28554	−0.642772	−	0.766057i	$-0.722216\pi$
$733$	−34.8048	−1.28554	−0.642772	+	0.766057i	$0.722216\pi$
$734$	0	0
$735$	9.02946	0.333057
$736$	0	0
$737$	1.11231	0.0409724
$738$	0	0
$739$	3.45086	0.126942	0.0634709	−	0.997984i	$-0.479783\pi$
$739$	3.45086	0.126942	0.0634709	+	0.997984i	$0.479783\pi$
$740$	0	0
$741$	7.76293	0.285178
$742$	0	0
$743$	0.726179	0.0266409	0.0133205	−	0.999911i	$-0.495760\pi$
$743$	0.726179	0.0266409	0.0133205	+	0.999911i	$0.495760\pi$
$744$	0	0
$745$	−30.5744	−1.12016
$746$	0	0
$747$	1.56097	0.0571129
$748$	0	0
$749$	−27.5815	−1.00780
$750$	0	0
$751$	−0.588927	−0.0214902	−0.0107451	−	0.999942i	$-0.503420\pi$
$751$	−0.588927	−0.0214902	−0.0107451	+	0.999942i	$0.503420\pi$
$752$	0	0
$753$	−8.01656	−0.292140
$754$	0	0
$755$	74.8550	2.72425
$756$	0	0
$757$	49.3023	1.79192	0.895961	−	0.444133i	$-0.146488\pi$
$757$	49.3023	1.79192	0.895961	+	0.444133i	$0.146488\pi$
$758$	0	0
$759$	0.474565	0.0172256
$760$	0	0
$761$	25.4987	0.924327	0.462164	−	0.886795i	$-0.347073\pi$
$761$	25.4987	0.924327	0.462164	+	0.886795i	$0.347073\pi$
$762$	0	0
$763$	−28.9568	−1.04831
$764$	0	0
$765$	−4.97966	−0.180040
$766$	0	0
$767$	−13.2849	−0.479690
$768$	0	0
$769$	−42.5951	−1.53602	−0.768009	−	0.640439i	$-0.778752\pi$
$769$	−42.5951	−1.53602	−0.768009	+	0.640439i	$0.778752\pi$
$770$	0	0
$771$	7.64905	0.275474
$772$	0	0
$773$	−27.7911	−0.999575	−0.499787	−	0.866148i	$-0.666589\pi$
$773$	−27.7911	−0.999575	−0.499787	+	0.866148i	$0.666589\pi$
$774$	0	0
$775$	2.55385	0.0917371
$776$	0	0
$777$	2.33140	0.0836386
$778$	0	0
$779$	11.1076	0.397970
$780$	0	0
$781$	0.825409	0.0295355
$782$	0	0
$783$	−7.93111	−0.283435
$784$	0	0
$785$	73.4686	2.62221
$786$	0	0
$787$	−30.3298	−1.08114	−0.540571	−	0.841299i	$-0.681792\pi$
$787$	−30.3298	−1.08114	−0.540571	+	0.841299i	$0.681792\pi$
$788$	0	0
$789$	1.93205	0.0687829
$790$	0	0
$791$	−35.2511	−1.25339
$792$	0	0
$793$	−15.2412	−0.541231
$794$	0	0
$795$	−18.1599	−0.644065
$796$	0	0
$797$	34.7822	1.23205	0.616024	−	0.787727i	$-0.288742\pi$
$797$	34.7822	1.23205	0.616024	+	0.787727i	$0.288742\pi$
$798$	0	0
$799$	12.8183	0.453479
$800$	0	0
$801$	8.47054	0.299292
$802$	0	0
$803$	−1.98911	−0.0701943
$804$	0	0
$805$	8.75633	0.308620
$806$	0	0
$807$	−16.3658	−0.576104
$808$	0	0
$809$	−44.5203	−1.56525	−0.782626	−	0.622492i	$-0.786120\pi$
$809$	−44.5203	−1.56525	−0.782626	+	0.622492i	$0.786120\pi$
$810$	0	0
$811$	4.01832	0.141102	0.0705512	−	0.997508i	$-0.477524\pi$
$811$	4.01832	0.141102	0.0705512	+	0.997508i	$0.477524\pi$
$812$	0	0
$813$	3.81117	0.133664
$814$	0	0
$815$	−41.7107	−1.46106
$816$	0	0
$817$	−42.8541	−1.49928
$818$	0	0
$819$	3.29916	0.115282
$820$	0	0
$821$	−29.4349	−1.02729	−0.513643	−	0.858004i	$-0.671704\pi$
$821$	−29.4349	−1.02729	−0.513643	+	0.858004i	$0.671704\pi$
$822$	0	0
$823$	26.8766	0.936861	0.468430	−	0.883500i	$-0.344820\pi$
$823$	26.8766	0.936861	0.468430	+	0.883500i	$0.344820\pi$
$824$	0	0
$825$	−1.87799	−0.0653831
$826$	0	0
$827$	39.3556	1.36853	0.684265	−	0.729234i	$-0.260123\pi$
$827$	39.3556	1.36853	0.684265	+	0.729234i	$0.260123\pi$
$828$	0	0
$829$	−4.01885	−0.139581	−0.0697903	−	0.997562i	$-0.522233\pi$
$829$	−4.01885	−0.139581	−0.0697903	+	0.997562i	$0.522233\pi$
$830$	0	0
$831$	−30.6315	−1.06259
$832$	0	0
$833$	4.37273	0.151506
$834$	0	0
$835$	26.3424	0.911618
$836$	0	0
$837$	−0.483432	−0.0167099
$838$	0	0
$839$	−55.9413	−1.93131	−0.965655	−	0.259829i	$-0.916334\pi$
$839$	−55.9413	−1.93131	−0.965655	+	0.259829i	$0.916334\pi$
$840$	0	0
$841$	33.9025	1.16905
$842$	0	0
$843$	−23.9820	−0.825986
$844$	0	0
$845$	−33.3451	−1.14711
$846$	0	0
$847$	22.2422	0.764253
$848$	0	0
$849$	−12.4704	−0.427984
$850$	0	0
$851$	−1.52152	−0.0521569
$852$	0	0
$853$	−4.25427	−0.145663	−0.0728317	−	0.997344i	$-0.523204\pi$
$853$	−4.25427	−0.145663	−0.0728317	+	0.997344i	$0.523204\pi$
$854$	0	0
$855$	15.4341	0.527836
$856$	0	0
$857$	−0.924108	−0.0315669	−0.0157835	−	0.999875i	$-0.505024\pi$
$857$	−0.924108	−0.0315669	−0.0157835	+	0.999875i	$0.505024\pi$
$858$	0	0
$859$	9.68928	0.330594	0.165297	−	0.986244i	$-0.447142\pi$
$859$	9.68928	0.330594	0.165297	+	0.986244i	$0.447142\pi$
$860$	0	0
$861$	4.72059	0.160877
$862$	0	0
$863$	37.7072	1.28357	0.641784	−	0.766885i	$-0.278195\pi$
$863$	37.7072	1.28357	0.641784	+	0.766885i	$0.278195\pi$
$864$	0	0
$865$	0.341244	0.0116026
$866$	0	0
$867$	14.5885	0.495451
$868$	0	0
$869$	0.630745	0.0213966
$870$	0	0
$871$	−5.04651	−0.170995
$872$	0	0
$873$	19.3315	0.654273
$874$	0	0
$875$	−1.85466	−0.0626991
$876$	0	0
$877$	−12.1216	−0.409319	−0.204659	−	0.978833i	$-0.565609\pi$
$877$	−12.1216	−0.409319	−0.204659	+	0.978833i	$0.565609\pi$
$878$	0	0
$879$	−28.9893	−0.977786
$880$	0	0
$881$	21.5912	0.727424	0.363712	−	0.931511i	$-0.381509\pi$
$881$	21.5912	0.727424	0.363712	+	0.931511i	$0.381509\pi$
$882$	0	0
$883$	18.0573	0.607677	0.303839	−	0.952723i	$-0.401732\pi$
$883$	18.0573	0.607677	0.303839	+	0.952723i	$0.401732\pi$
$884$	0	0
$885$	−26.4128	−0.887856
$886$	0	0
$887$	−35.3028	−1.18535	−0.592675	−	0.805442i	$-0.701928\pi$
$887$	−35.3028	−1.18535	−0.592675	+	0.805442i	$0.701928\pi$
$888$	0	0
$889$	30.7198	1.03031
$890$	0	0
$891$	0.355494	0.0119095
$892$	0	0
$893$	−39.7295	−1.32950
$894$	0	0
$895$	52.5355	1.75607
$896$	0	0
$897$	−2.15309	−0.0718895
$898$	0	0
$899$	3.83416	0.127876
$900$	0	0
$901$	−8.79435	−0.292982
$902$	0	0
$903$	−18.2125	−0.606075
$904$	0	0
$905$	12.6765	0.421381
$906$	0	0
$907$	33.6263	1.11654	0.558272	−	0.829658i	$-0.311464\pi$
$907$	33.6263	1.11654	0.558272	+	0.829658i	$0.311464\pi$
$908$	0	0
$909$	−0.364219	−0.0120804
$910$	0	0
$911$	−22.0973	−0.732115	−0.366057	−	0.930592i	$-0.619293\pi$
$911$	−22.0973	−0.732115	−0.366057	+	0.930592i	$0.619293\pi$
$912$	0	0
$913$	0.554916	0.0183650
$914$	0	0
$915$	−30.3023	−1.00176
$916$	0	0
$917$	−19.4681	−0.642892
$918$	0	0
$919$	49.6894	1.63910	0.819551	−	0.573006i	$-0.194223\pi$
$919$	49.6894	1.63910	0.819551	+	0.573006i	$0.194223\pi$
$920$	0	0
$921$	−3.71327	−0.122356
$922$	0	0
$923$	−3.74486	−0.123264
$924$	0	0
$925$	6.02106	0.197971
$926$	0	0
$927$	−18.5620	−0.609657
$928$	0	0
$929$	17.0783	0.560320	0.280160	−	0.959953i	$-0.409613\pi$
$929$	17.0783	0.560320	0.280160	+	0.959953i	$0.409613\pi$
$930$	0	0
$931$	−13.5530	−0.444180
$932$	0	0
$933$	9.94769	0.325673
$934$	0	0
$935$	−1.77024	−0.0578931
$936$	0	0
$937$	23.6264	0.771840	0.385920	−	0.922532i	$-0.373884\pi$
$937$	23.6264	0.771840	0.385920	+	0.922532i	$0.373884\pi$
$938$	0	0
$939$	−15.5920	−0.508824
$940$	0	0
$941$	−33.5726	−1.09444	−0.547218	−	0.836990i	$-0.684313\pi$
$941$	−33.5726	−1.09444	−0.547218	+	0.836990i	$0.684313\pi$
$942$	0	0
$943$	−3.08074	−0.100323
$944$	0	0
$945$	6.55932	0.213375
$946$	0	0
$947$	37.8914	1.23131	0.615653	−	0.788017i	$-0.288892\pi$
$947$	37.8914	1.23131	0.615653	+	0.788017i	$0.288892\pi$
$948$	0	0
$949$	9.02455	0.292949
$950$	0	0
$951$	−11.5736	−0.375298
$952$	0	0
$953$	−48.8903	−1.58371	−0.791856	−	0.610707i	$-0.790885\pi$
$953$	−48.8903	−1.58371	−0.791856	+	0.610707i	$0.790885\pi$
$954$	0	0
$955$	61.5891	1.99298
$956$	0	0
$957$	−2.81946	−0.0911403
$958$	0	0
$959$	33.4481	1.08010
$960$	0	0
$961$	−30.7663	−0.992461
$962$	0	0
$963$	13.4838	0.434510
$964$	0	0
$965$	−54.1612	−1.74351
$966$	0	0
$967$	−47.9705	−1.54263	−0.771314	−	0.636454i	$-0.780400\pi$
$967$	−47.9705	−1.54263	−0.771314	+	0.636454i	$0.780400\pi$
$968$	0	0
$969$	7.47433	0.240110
$970$	0	0
$971$	−20.1765	−0.647494	−0.323747	−	0.946144i	$-0.604943\pi$
$971$	−20.1765	−0.647494	−0.323747	+	0.946144i	$0.604943\pi$
$972$	0	0
$973$	−9.12866	−0.292651
$974$	0	0
$975$	8.52038	0.272871
$976$	0	0
$977$	−59.9685	−1.91856	−0.959281	−	0.282452i	$-0.908852\pi$
$977$	−59.9685	−1.91856	−0.959281	+	0.282452i	$0.908852\pi$
$978$	0	0
$979$	3.01123	0.0962392
$980$	0	0
$981$	14.1562	0.451973
$982$	0	0
$983$	−9.05772	−0.288896	−0.144448	−	0.989512i	$-0.546141\pi$
$983$	−9.05772	−0.288896	−0.144448	+	0.989512i	$0.546141\pi$
$984$	0	0
$985$	−13.7309	−0.437501
$986$	0	0
$987$	−16.8846	−0.537442
$988$	0	0
$989$	11.8858	0.377947
$990$	0	0
$991$	−29.0948	−0.924228	−0.462114	−	0.886821i	$-0.652909\pi$
$991$	−29.0948	−0.924228	−0.462114	+	0.886821i	$0.652909\pi$
$992$	0	0
$993$	−1.66646	−0.0528834
$994$	0	0
$995$	−31.4285	−0.996350
$996$	0	0
$997$	−24.9206	−0.789243	−0.394621	−	0.918844i	$-0.629124\pi$
$997$	−24.9206	−0.789243	−0.394621	+	0.918844i	$0.629124\pi$
$998$	0	0
$999$	−1.13976	−0.0360604

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6036.2.a.i.1.22	✓	26

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6036.2.a.i.1.22	✓	26	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:	\( N \)	\(=\)	\( 6036 = 2^{2} \cdot 3 \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6036.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +3.20667 q^{5} -2.04552 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +3.20667 q^{5} -2.04552 q^{7} +1.00000 q^{9} +0.355494 q^{11} -1.61287 q^{13} -3.20667 q^{15} -1.55291 q^{17} +4.81312 q^{19} +2.04552 q^{21} -1.33494 q^{23} +5.28275 q^{25} -1.00000 q^{27} +7.93111 q^{29} +0.483432 q^{31} -0.355494 q^{33} -6.55932 q^{35} +1.13976 q^{37} +1.61287 q^{39} +2.30777 q^{41} -8.90360 q^{43} +3.20667 q^{45} -8.25441 q^{47} -2.81584 q^{49} +1.55291 q^{51} +5.66316 q^{53} +1.13995 q^{55} -4.81312 q^{57} +8.23682 q^{59} +9.44976 q^{61} -2.04552 q^{63} -5.17194 q^{65} +3.12891 q^{67} +1.33494 q^{69} +2.32186 q^{71} -5.59535 q^{73} -5.28275 q^{75} -0.727172 q^{77} +1.77428 q^{79} +1.00000 q^{81} +1.56097 q^{83} -4.97966 q^{85} -7.93111 q^{87} +8.47054 q^{89} +3.29916 q^{91} -0.483432 q^{93} +15.4341 q^{95} +19.3315 q^{97} +0.355494 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 26 q - 26 q^{3} + 6 q^{5} + 5 q^{7} + 26 q^{9}+O(q^{10}) \) 26 * q - 26 * q^3 + 6 * q^5 + 5 * q^7 + 26 * q^9	\( 26 q - 26 q^{3} + 6 q^{5} + 5 q^{7} + 26 q^{9} - 11 q^{11} + 13 q^{13} - 6 q^{15} + 12 q^{17} - q^{19} - 5 q^{21} - 22 q^{23} + 48 q^{25} - 26 q^{27} + 6 q^{29} + 19 q^{31} + 11 q^{33} - 21 q^{35} + 20 q^{37} - 13 q^{39} + 25 q^{41} + 4 q^{43} + 6 q^{45} + 8 q^{47} + 67 q^{49} - 12 q^{51} - 5 q^{53} + 20 q^{55} + q^{57} - 18 q^{59} + 43 q^{61} + 5 q^{63} + 41 q^{65} + 5 q^{67} + 22 q^{69} - q^{71} + 22 q^{73} - 48 q^{75} + 23 q^{77} + 16 q^{79} + 26 q^{81} - 19 q^{83} + 29 q^{85} - 6 q^{87} + 49 q^{89} - 13 q^{91} - 19 q^{93} - 26 q^{95} + 25 q^{97} - 11 q^{99}+O(q^{100}) \) 26 * q - 26 * q^3 + 6 * q^5 + 5 * q^7 + 26 * q^9 - 11 * q^11 + 13 * q^13 - 6 * q^15 + 12 * q^17 - q^19 - 5 * q^21 - 22 * q^23 + 48 * q^25 - 26 * q^27 + 6 * q^29 + 19 * q^31 + 11 * q^33 - 21 * q^35 + 20 * q^37 - 13 * q^39 + 25 * q^41 + 4 * q^43 + 6 * q^45 + 8 * q^47 + 67 * q^49 - 12 * q^51 - 5 * q^53 + 20 * q^55 + q^57 - 18 * q^59 + 43 * q^61 + 5 * q^63 + 41 * q^65 + 5 * q^67 + 22 * q^69 - q^71 + 22 * q^73 - 48 * q^75 + 23 * q^77 + 16 * q^79 + 26 * q^81 - 19 * q^83 + 29 * q^85 - 6 * q^87 + 49 * q^89 - 13 * q^91 - 19 * q^93 - 26 * q^95 + 25 * q^97 - 11 * q^99

Introduction

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