LMFDB - Embedded newform 1296.5.e.g.161.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1296,5,Mod(161,1296)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1296, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("1296.161");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 1296 = 2^{4} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	1296.e (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$133.967472157$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} + 6x^{6} + 121x^{5} + 1104x^{4} - 1647x^{3} + 6529x^{2} + 85254x + 440076 $ x^8 - 3x^7 + 6x^6 + 121x^5 + 1104x^4 - 1647x^3 + 6529x^2 + 85254*x + 440076
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{20} $
Twist minimal:	no (minimal twist has level 36)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.7
Root			$-3.05006 + 3.25531i$ of defining polynomial
Character	$\chi$	$=$	1296.161
Dual form			1296.5.e.g.161.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+31.6564i q^{5} +75.3660 q^{7} +O(q^{10})$	$q+31.6564i q^{5} +75.3660 q^{7} -142.336i q^{11} -192.616 q^{13} -325.855i q^{17} -314.164 q^{19} +512.516i q^{23} -377.128 q^{25} +157.748i q^{29} +367.450 q^{31} +2385.82i q^{35} +1737.04 q^{37} -395.273i q^{41} -720.762 q^{43} +2486.89i q^{47} +3279.03 q^{49} +3986.04i q^{53} +4505.86 q^{55} +2467.30i q^{59} +2480.98 q^{61} -6097.53i q^{65} +6596.38 q^{67} +5828.07i q^{71} -8790.44 q^{73} -10727.3i q^{77} +3869.04 q^{79} +12034.9i q^{83} +10315.4 q^{85} -7637.03i q^{89} -14516.7 q^{91} -9945.30i q^{95} -2911.50 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 26 q^{7}+O(q^{10}) $ 8 * q + 26 * q^7	$ 8 q + 26 q^{7} + 10 q^{13} - 562 q^{19} - 706 q^{25} + 374 q^{31} + 16 q^{37} - 136 q^{43} + 654 q^{49} + 1818 q^{55} + 3874 q^{61} + 308 q^{67} - 7802 q^{73} - 4390 q^{79} + 6084 q^{85} - 15830 q^{91} - 14564 q^{97}+O(q^{100}) $ 8 * q + 26 * q^7 + 10 * q^13 - 562 * q^19 - 706 * q^25 + 374 * q^31 + 16 * q^37 - 136 * q^43 + 654 * q^49 + 1818 * q^55 + 3874 * q^61 + 308 * q^67 - 7802 * q^73 - 4390 * q^79 + 6084 * q^85 - 15830 * q^91 - 14564 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times$.

$n$	$325$	$1135$	$1217$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	31.6564i	1.26626i	0.774047	+	0.633128i	$0.218229\pi$
$5$	31.6564i	1.26626i	−0.774047	+	0.633128i	$0.781771\pi$
$6$	0	0
$7$	75.3660	1.53808	0.769041	−	0.639200i	$-0.220734\pi$
$7$	75.3660	1.53808	0.769041	+	0.639200i	$0.220734\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 142.336i	− 1.17633i	−0.808740	−	0.588167i	$-0.799850\pi$
$11$	− 142.336i	− 1.17633i	0.808740	−	0.588167i	$-0.200150\pi$
$12$	0	0
$13$	−192.616	−1.13974	−0.569870	−	0.821735i	$-0.693006\pi$
$13$	−192.616	−1.13974	−0.569870	+	0.821735i	$0.693006\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 325.855i	− 1.12753i	−0.825937	−	0.563763i	$-0.809353\pi$
$17$	− 325.855i	− 1.12753i	0.825937	−	0.563763i	$-0.190647\pi$
$18$	0	0
$19$	−314.164	−0.870260	−0.435130	−	0.900368i	$-0.643298\pi$
$19$	−314.164	−0.870260	−0.435130	+	0.900368i	$0.643298\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	512.516i	0.968840i	0.874836	+	0.484420i	$0.160969\pi$
$23$	512.516i	0.968840i	−0.874836	+	0.484420i	$0.839031\pi$
$24$	0	0
$25$	−377.128	−0.603405
$26$	0	0
$27$	0	0
$28$	0	0
$29$	157.748i	0.187572i	0.995592	+	0.0937859i	$0.0298969\pi$
$29$	157.748i	0.187572i	−0.995592	+	0.0937859i	$0.970103\pi$
$30$	0	0
$31$	367.450	0.382362	0.191181	−	0.981555i	$-0.438768\pi$
$31$	367.450	0.382362	0.191181	+	0.981555i	$0.438768\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2385.82i	1.94760i
$36$	0	0
$37$	1737.04	1.26884	0.634419	−	0.772989i	$-0.281239\pi$
$37$	1737.04	1.26884	0.634419	+	0.772989i	$0.281239\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 395.273i	− 0.235142i	−0.993064	−	0.117571i	$-0.962489\pi$
$41$	− 395.273i	− 0.235142i	0.993064	−	0.117571i	$-0.0375107\pi$
$42$	0	0
$43$	−720.762	−0.389812	−0.194906	−	0.980822i	$-0.562440\pi$
$43$	−720.762	−0.389812	−0.194906	+	0.980822i	$0.562440\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2486.89i	1.12580i	0.826525	+	0.562900i	$0.190314\pi$
$47$	2486.89i	1.12580i	−0.826525	+	0.562900i	$0.809686\pi$
$48$	0	0
$49$	3279.03	1.36569
$50$	0	0
$51$	0	0
$52$	0	0
$53$	3986.04i	1.41902i	0.704694	+	0.709512i	$0.251084\pi$
$53$	3986.04i	1.41902i	−0.704694	+	0.709512i	$0.748916\pi$
$54$	0	0
$55$	4505.86	1.48954
$56$	0	0
$57$	0	0
$58$	0	0
$59$	2467.30i	0.708791i	0.935096	+	0.354396i	$0.115313\pi$
$59$	2467.30i	0.708791i	−0.935096	+	0.354396i	$0.884687\pi$
$60$	0	0
$61$	2480.98	0.666750	0.333375	−	0.942794i	$-0.391813\pi$
$61$	2480.98	0.666750	0.333375	+	0.942794i	$0.391813\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 6097.53i	− 1.44320i
$66$	0	0
$67$	6596.38	1.46946	0.734728	−	0.678362i	$-0.237310\pi$
$67$	6596.38	1.46946	0.734728	+	0.678362i	$0.237310\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	5828.07i	1.15613i	0.815990	+	0.578066i	$0.196193\pi$
$71$	5828.07i	1.15613i	−0.815990	+	0.578066i	$0.803807\pi$
$72$	0	0
$73$	−8790.44	−1.64955	−0.824774	−	0.565463i	$-0.808697\pi$
$73$	−8790.44	−1.64955	−0.824774	+	0.565463i	$0.808697\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 10727.3i	− 1.80930i
$78$	0	0
$79$	3869.04	0.619939	0.309970	−	0.950746i	$-0.399681\pi$
$79$	3869.04	0.619939	0.309970	+	0.950746i	$0.399681\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	12034.9i	1.74697i	0.486854	+	0.873483i	$0.338144\pi$
$83$	12034.9i	1.74697i	−0.486854	+	0.873483i	$0.661856\pi$
$84$	0	0
$85$	10315.4	1.42774
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 7637.03i	− 0.964150i	−0.876130	−	0.482075i	$-0.839883\pi$
$89$	− 7637.03i	− 0.964150i	0.876130	−	0.482075i	$-0.160117\pi$
$90$	0	0
$91$	−14516.7	−1.75301
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 9945.30i	− 1.10197i
$96$	0	0
$97$	−2911.50	−0.309438	−0.154719	−	0.987959i	$-0.549447\pi$
$97$	−2911.50	−0.309438	−0.154719	+	0.987959i	$0.549447\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	10903.7i	1.06889i	0.845204	+	0.534444i	$0.179479\pi$
$101$	10903.7i	1.06889i	−0.845204	+	0.534444i	$0.820521\pi$
$102$	0	0
$103$	7277.11	0.685938	0.342969	−	0.939347i	$-0.388567\pi$
$103$	7277.11	0.685938	0.342969	+	0.939347i	$0.388567\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5525.48i	0.482617i	0.970449	+	0.241308i	$0.0775765\pi$
$107$	5525.48i	0.482617i	−0.970449	+	0.241308i	$0.922424\pi$
$108$	0	0
$109$	7186.35	0.604861	0.302431	−	0.953171i	$-0.402202\pi$
$109$	7186.35	0.604861	0.302431	+	0.953171i	$0.402202\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 12403.9i	− 0.971403i	−0.874125	−	0.485702i	$-0.838564\pi$
$113$	− 12403.9i	− 0.971403i	0.874125	−	0.485702i	$-0.161436\pi$
$114$	0	0
$115$	−16224.4	−1.22680
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 24558.4i	− 1.73423i
$120$	0	0
$121$	−5618.63	−0.383760
$122$	0	0
$123$	0	0
$124$	0	0
$125$	7846.74i	0.502191i
$126$	0	0
$127$	−9013.28	−0.558824	−0.279412	−	0.960171i	$-0.590140\pi$
$127$	−9013.28	−0.558824	−0.279412	+	0.960171i	$0.590140\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 357.266i	− 0.0208185i	−0.999946	−	0.0104092i	$-0.996687\pi$
$131$	− 357.266i	− 0.0208185i	0.999946	−	0.0104092i	$-0.00331342\pi$
$132$	0	0
$133$	−23677.3	−1.33853
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 14793.1i	− 0.788166i	−0.919075	−	0.394083i	$-0.871062\pi$
$137$	− 14793.1i	− 0.788166i	0.919075	−	0.394083i	$-0.128938\pi$
$138$	0	0
$139$	−2159.23	−0.111756	−0.0558779	−	0.998438i	$-0.517796\pi$
$139$	−2159.23	−0.111756	−0.0558779	+	0.998438i	$0.517796\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	27416.2i	1.34071i
$144$	0	0
$145$	−4993.73	−0.237514
$146$	0	0
$147$	0	0
$148$	0	0
$149$	28846.9i	1.29935i	0.760212	+	0.649675i	$0.225095\pi$
$149$	28846.9i	1.29935i	−0.760212	+	0.649675i	$0.774905\pi$
$150$	0	0
$151$	9505.29	0.416880	0.208440	−	0.978035i	$-0.433161\pi$
$151$	9505.29	0.416880	0.208440	+	0.978035i	$0.433161\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	11632.1i	0.484168i
$156$	0	0
$157$	−9875.36	−0.400639	−0.200320	−	0.979731i	$-0.564198\pi$
$157$	−9875.36	−0.400639	−0.200320	+	0.979731i	$0.564198\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	38626.3i	1.49015i
$162$	0	0
$163$	22464.8	0.845525	0.422763	−	0.906240i	$-0.361060\pi$
$163$	22464.8	0.845525	0.422763	+	0.906240i	$0.361060\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	16968.0i	0.608413i	0.952606	+	0.304206i	$0.0983912\pi$
$167$	16968.0i	0.608413i	−0.952606	+	0.304206i	$0.901609\pi$
$168$	0	0
$169$	8539.88	0.299005
$170$	0	0
$171$	0	0
$172$	0	0
$173$	37460.6i	1.25165i	0.779964	+	0.625825i	$0.215237\pi$
$173$	37460.6i	1.25165i	−0.779964	+	0.625825i	$0.784763\pi$
$174$	0	0
$175$	−28422.6	−0.928085
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 44760.9i	− 1.39699i	−0.715615	−	0.698495i	$-0.753853\pi$
$179$	− 44760.9i	− 1.39699i	0.715615	−	0.698495i	$-0.246147\pi$
$180$	0	0
$181$	−29208.0	−0.891548	−0.445774	−	0.895146i	$-0.647071\pi$
$181$	−29208.0	−0.891548	−0.445774	+	0.895146i	$0.647071\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	54988.4i	1.60667i
$186$	0	0
$187$	−46381.0	−1.32635
$188$	0	0
$189$	0	0
$190$	0	0
$191$	30008.9i	0.822589i	0.911503	+	0.411294i	$0.134923\pi$
$191$	30008.9i	0.822589i	−0.911503	+	0.411294i	$0.865077\pi$
$192$	0	0
$193$	−17231.7	−0.462609	−0.231305	−	0.972881i	$-0.574299\pi$
$193$	−17231.7	−0.462609	−0.231305	+	0.972881i	$0.574299\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	67892.1i	1.74939i	0.484673	+	0.874695i	$0.338938\pi$
$197$	67892.1i	1.74939i	−0.484673	+	0.874695i	$0.661062\pi$
$198$	0	0
$199$	26996.5	0.681713	0.340857	−	0.940115i	$-0.389283\pi$
$199$	26996.5	0.681713	0.340857	+	0.940115i	$0.389283\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	11888.8i	0.288501i
$204$	0	0
$205$	12512.9	0.297750
$206$	0	0
$207$	0	0
$208$	0	0
$209$	44716.9i	1.02372i
$210$	0	0
$211$	−23639.1	−0.530965	−0.265482	−	0.964116i	$-0.585531\pi$
$211$	−23639.1	−0.530965	−0.265482	+	0.964116i	$0.585531\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 22816.7i	− 0.493601i
$216$	0	0
$217$	27693.2	0.588103
$218$	0	0
$219$	0	0
$220$	0	0
$221$	62764.8i	1.28509i
$222$	0	0
$223$	57956.0	1.16544	0.582718	−	0.812675i	$-0.301989\pi$
$223$	57956.0	1.16544	0.582718	+	0.812675i	$0.301989\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	10060.3i	0.195236i	0.995224	+	0.0976178i	$0.0311223\pi$
$227$	10060.3i	0.195236i	−0.995224	+	0.0976178i	$0.968878\pi$
$228$	0	0
$229$	57799.7	1.10219	0.551093	−	0.834444i	$-0.314211\pi$
$229$	57799.7	1.10219	0.551093	+	0.834444i	$0.314211\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 68323.8i	− 1.25852i	−0.777195	−	0.629260i	$-0.783358\pi$
$233$	− 68323.8i	− 1.25852i	0.777195	−	0.629260i	$-0.216642\pi$
$234$	0	0
$235$	−78726.0	−1.42555
$236$	0	0
$237$	0	0
$238$	0	0
$239$	34782.1i	0.608920i	0.952525	+	0.304460i	$0.0984760\pi$
$239$	34782.1i	0.608920i	−0.952525	+	0.304460i	$0.901524\pi$
$240$	0	0
$241$	91965.5	1.58340	0.791701	−	0.610909i	$-0.209196\pi$
$241$	91965.5	1.58340	0.791701	+	0.610909i	$0.209196\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	103802.i	1.72932i
$246$	0	0
$247$	60512.9	0.991869
$248$	0	0
$249$	0	0
$250$	0	0
$251$	69806.6i	1.10802i	0.832509	+	0.554012i	$0.186904\pi$
$251$	69806.6i	1.10802i	−0.832509	+	0.554012i	$0.813096\pi$
$252$	0	0
$253$	72949.7	1.13968
$254$	0	0
$255$	0	0
$256$	0	0
$257$	17563.4i	0.265915i	0.991122	+	0.132957i	$0.0424473\pi$
$257$	17563.4i	0.265915i	−0.991122	+	0.132957i	$0.957553\pi$
$258$	0	0
$259$	130914.	1.95158
$260$	0	0
$261$	0	0
$262$	0	0
$263$	30590.8i	0.442261i	0.975244	+	0.221131i	$0.0709747\pi$
$263$	30590.8i	0.442261i	−0.975244	+	0.221131i	$0.929025\pi$
$264$	0	0
$265$	−126184.	−1.79685
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 59574.8i	− 0.823299i	−0.911342	−	0.411650i	$-0.864953\pi$
$269$	− 59574.8i	− 0.823299i	0.911342	−	0.411650i	$-0.135047\pi$
$270$	0	0
$271$	94290.1	1.28389	0.641945	−	0.766751i	$-0.278128\pi$
$271$	94290.1	1.28389	0.641945	+	0.766751i	$0.278128\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	53679.0i	0.709805i
$276$	0	0
$277$	−99375.5	−1.29515	−0.647574	−	0.762002i	$-0.724217\pi$
$277$	−99375.5	−1.29515	−0.647574	+	0.762002i	$0.724217\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 79375.1i	− 1.00524i	−0.864506	−	0.502622i	$-0.832369\pi$
$281$	− 79375.1i	− 1.00524i	0.864506	−	0.502622i	$-0.167631\pi$
$282$	0	0
$283$	−66919.9	−0.835570	−0.417785	−	0.908546i	$-0.637193\pi$
$283$	−66919.9	−0.835570	−0.417785	+	0.908546i	$0.637193\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 29790.2i	− 0.361667i
$288$	0	0
$289$	−22660.4	−0.271314
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 55964.9i	− 0.651900i	−0.945387	−	0.325950i	$-0.894316\pi$
$293$	− 55964.9i	− 0.651900i	0.945387	−	0.325950i	$-0.105684\pi$
$294$	0	0
$295$	−78105.9	−0.897511
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 98718.8i	− 1.10422i
$300$	0	0
$301$	−54320.9	−0.599562
$302$	0	0
$303$	0	0
$304$	0	0
$305$	78538.8i	0.844276i
$306$	0	0
$307$	−8707.80	−0.0923914	−0.0461957	−	0.998932i	$-0.514710\pi$
$307$	−8707.80	−0.0923914	−0.0461957	+	0.998932i	$0.514710\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	55817.4i	0.577097i	0.957465	+	0.288549i	$0.0931727\pi$
$311$	55817.4i	0.577097i	−0.957465	+	0.288549i	$0.906827\pi$
$312$	0	0
$313$	59039.3	0.602633	0.301316	−	0.953524i	$-0.402574\pi$
$313$	59039.3	0.602633	0.301316	+	0.953524i	$0.402574\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	30342.3i	0.301947i	0.988538	+	0.150973i	$0.0482408\pi$
$317$	30342.3i	0.301947i	−0.988538	+	0.150973i	$0.951759\pi$
$318$	0	0
$319$	22453.3	0.220647
$320$	0	0
$321$	0	0
$322$	0	0
$323$	102372.i	0.981240i
$324$	0	0
$325$	72640.8	0.687724
$326$	0	0
$327$	0	0
$328$	0	0
$329$	187427.i	1.73157i
$330$	0	0
$331$	−126921.	−1.15845	−0.579225	−	0.815168i	$-0.696645\pi$
$331$	−126921.	−1.15845	−0.579225	+	0.815168i	$0.696645\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	208818.i	1.86071i
$336$	0	0
$337$	178396.	1.57082	0.785410	−	0.618976i	$-0.212452\pi$
$337$	178396.	1.57082	0.785410	+	0.618976i	$0.212452\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 52301.4i	− 0.449785i
$342$	0	0
$343$	66173.6	0.562466
$344$	0	0
$345$	0	0
$346$	0	0
$347$	112862.i	0.937322i	0.883378	+	0.468661i	$0.155263\pi$
$347$	112862.i	0.937322i	−0.883378	+	0.468661i	$0.844737\pi$
$348$	0	0
$349$	−73508.3	−0.603511	−0.301756	−	0.953385i	$-0.597573\pi$
$349$	−73508.3	−0.603511	−0.301756	+	0.953385i	$0.597573\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 211360.i	− 1.69618i	−0.529850	−	0.848091i	$-0.677752\pi$
$353$	− 211360.i	− 1.69618i	0.529850	−	0.848091i	$-0.322248\pi$
$354$	0	0
$355$	−184496.	−1.46396
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 36755.1i	− 0.285186i	−0.989781	−	0.142593i	$-0.954456\pi$
$359$	− 36755.1i	− 0.285186i	0.989781	−	0.142593i	$-0.0455440\pi$
$360$	0	0
$361$	−31622.1	−0.242648
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 278274.i	− 2.08875i
$366$	0	0
$367$	147154.	1.09255	0.546273	−	0.837607i	$-0.316046\pi$
$367$	147154.	1.09255	0.546273	+	0.837607i	$0.316046\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	300411.i	2.18257i
$372$	0	0
$373$	−43598.1	−0.313365	−0.156682	−	0.987649i	$-0.550080\pi$
$373$	−43598.1	−0.313365	−0.156682	+	0.987649i	$0.550080\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 30384.7i	− 0.213783i
$378$	0	0
$379$	169833.	1.18234	0.591170	−	0.806547i	$-0.298666\pi$
$379$	169833.	1.18234	0.591170	+	0.806547i	$0.298666\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	122661.i	0.836198i	0.908401	+	0.418099i	$0.137304\pi$
$383$	122661.i	0.836198i	−0.908401	+	0.418099i	$0.862696\pi$
$384$	0	0
$385$	339588.	2.29103
$386$	0	0
$387$	0	0
$388$	0	0
$389$	58306.9i	0.385319i	0.981266	+	0.192660i	$0.0617113\pi$
$389$	58306.9i	0.385319i	−0.981266	+	0.192660i	$0.938289\pi$
$390$	0	0
$391$	167006.	1.09239
$392$	0	0
$393$	0	0
$394$	0	0
$395$	122480.i	0.785002i
$396$	0	0
$397$	−134572.	−0.853834	−0.426917	−	0.904291i	$-0.640400\pi$
$397$	−134572.	−0.853834	−0.426917	+	0.904291i	$0.640400\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	214375.i	1.33317i	0.745430	+	0.666584i	$0.232244\pi$
$401$	214375.i	1.33317i	−0.745430	+	0.666584i	$0.767756\pi$
$402$	0	0
$403$	−70776.6	−0.435793
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 247244.i	− 1.49258i
$408$	0	0
$409$	−48739.0	−0.291360	−0.145680	−	0.989332i	$-0.546537\pi$
$409$	−48739.0	−0.291360	−0.145680	+	0.989332i	$0.546537\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	185951.i	1.09018i
$414$	0	0
$415$	−380980.	−2.21211
$416$	0	0
$417$	0	0
$418$	0	0
$419$	68878.3i	0.392332i	0.980571	+	0.196166i	$0.0628492\pi$
$419$	68878.3i	0.392332i	−0.980571	+	0.196166i	$0.937151\pi$
$420$	0	0
$421$	−315354.	−1.77924	−0.889620	−	0.456701i	$-0.849031\pi$
$421$	−315354.	−1.77924	−0.889620	+	0.456701i	$0.849031\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	122889.i	0.680354i
$426$	0	0
$427$	186981.	1.02552
$428$	0	0
$429$	0	0
$430$	0	0
$431$	151597.i	0.816088i	0.912962	+	0.408044i	$0.133789\pi$
$431$	151597.i	0.816088i	−0.912962	+	0.408044i	$0.866211\pi$
$432$	0	0
$433$	−237835.	−1.26853	−0.634265	−	0.773116i	$-0.718697\pi$
$433$	−237835.	−1.26853	−0.634265	+	0.773116i	$0.718697\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 161014.i	− 0.843143i
$438$	0	0
$439$	−64700.8	−0.335723	−0.167861	−	0.985811i	$-0.553686\pi$
$439$	−64700.8	−0.335723	−0.167861	+	0.985811i	$0.553686\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 5160.92i	− 0.0262978i	−0.999914	−	0.0131489i	$-0.995814\pi$
$443$	− 5160.92i	− 0.0262978i	0.999914	−	0.0131489i	$-0.00418555\pi$
$444$	0	0
$445$	241761.	1.22086
$446$	0	0
$447$	0	0
$448$	0	0
$449$	165679.i	0.821816i	0.911677	+	0.410908i	$0.134788\pi$
$449$	165679.i	0.821816i	−0.911677	+	0.410908i	$0.865212\pi$
$450$	0	0
$451$	−56261.8	−0.276605
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 459546.i	− 2.21976i
$456$	0	0
$457$	−408977.	−1.95824	−0.979121	−	0.203276i	$-0.934841\pi$
$457$	−408977.	−1.95824	−0.979121	+	0.203276i	$0.934841\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	87163.6i	0.410141i	0.978747	+	0.205071i	$0.0657424\pi$
$461$	87163.6i	0.410141i	−0.978747	+	0.205071i	$0.934258\pi$
$462$	0	0
$463$	−16806.5	−0.0783997	−0.0391999	−	0.999231i	$-0.512481\pi$
$463$	−16806.5	−0.0783997	−0.0391999	+	0.999231i	$0.512481\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	394500.i	1.80889i	0.426585	+	0.904447i	$0.359716\pi$
$467$	394500.i	1.80889i	−0.426585	+	0.904447i	$0.640284\pi$
$468$	0	0
$469$	497143.	2.26014
$470$	0	0
$471$	0	0
$472$	0	0
$473$	102591.i	0.458548i
$474$	0	0
$475$	118480.	0.525119
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 104393.i	− 0.454987i	−0.973780	−	0.227493i	$-0.926947\pi$
$479$	− 104393.i	− 0.454987i	0.973780	−	0.227493i	$-0.0730530\pi$
$480$	0	0
$481$	−334581.	−1.44614
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 92167.7i	− 0.391828i
$486$	0	0
$487$	−106236.	−0.447936	−0.223968	−	0.974597i	$-0.571901\pi$
$487$	−106236.	−0.447936	−0.223968	+	0.974597i	$0.571901\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 91028.1i	− 0.377583i	−0.982017	−	0.188792i	$-0.939543\pi$
$491$	− 91028.1i	− 0.377583i	0.982017	−	0.188792i	$-0.0604570\pi$
$492$	0	0
$493$	51402.9	0.211492
$494$	0	0
$495$	0	0
$496$	0	0
$497$	439238.i	1.77823i
$498$	0	0
$499$	138352.	0.555628	0.277814	−	0.960635i	$-0.410390\pi$
$499$	138352.	0.555628	0.277814	+	0.960635i	$0.410390\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 63792.4i	− 0.252135i	−0.992022	−	0.126067i	$-0.959764\pi$
$503$	− 63792.4i	− 0.252135i	0.992022	−	0.126067i	$-0.0402356\pi$
$504$	0	0
$505$	−345172.	−1.35348
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 489170.i	− 1.88810i	−0.329805	−	0.944049i	$-0.606983\pi$
$509$	− 489170.i	− 1.88810i	0.329805	−	0.944049i	$-0.393017\pi$
$510$	0	0
$511$	−662500.	−2.53714
$512$	0	0
$513$	0	0
$514$	0	0
$515$	230367.i	0.868573i
$516$	0	0
$517$	353975.	1.32432
$518$	0	0
$519$	0	0
$520$	0	0
$521$	339386.i	1.25031i	0.780500	+	0.625156i	$0.214965\pi$
$521$	339386.i	1.25031i	−0.780500	+	0.625156i	$0.785035\pi$
$522$	0	0
$523$	−379243.	−1.38648	−0.693242	−	0.720705i	$-0.743818\pi$
$523$	−379243.	−1.38648	−0.693242	+	0.720705i	$0.743818\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 119735.i	− 0.431123i
$528$	0	0
$529$	17167.9	0.0613487
$530$	0	0
$531$	0	0
$532$	0	0
$533$	76135.9i	0.268000i
$534$	0	0
$535$	−174917.	−0.611116
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 466725.i	− 1.60651i
$540$	0	0
$541$	−494913.	−1.69096	−0.845481	−	0.534005i	$-0.820686\pi$
$541$	−494913.	−1.69096	−0.845481	+	0.534005i	$0.820686\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	227494.i	0.765909i
$546$	0	0
$547$	186091.	0.621942	0.310971	−	0.950419i	$-0.399346\pi$
$547$	186091.	0.621942	0.310971	+	0.950419i	$0.399346\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 49558.7i	− 0.163236i
$552$	0	0
$553$	291594.	0.953517
$554$	0	0
$555$	0	0
$556$	0	0
$557$	119662.i	0.385698i	0.981228	+	0.192849i	$0.0617728\pi$
$557$	119662.i	0.385698i	−0.981228	+	0.192849i	$0.938227\pi$
$558$	0	0
$559$	138830.	0.444283
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 26425.9i	− 0.0833707i	−0.999131	−	0.0416854i	$-0.986727\pi$
$563$	− 26425.9i	− 0.0833707i	0.999131	−	0.0416854i	$-0.0132727\pi$
$564$	0	0
$565$	392661.	1.23005
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 523729.i	− 1.61764i	−0.588057	−	0.808820i	$-0.700107\pi$
$569$	− 523729.i	− 1.61764i	0.588057	−	0.808820i	$-0.299893\pi$
$570$	0	0
$571$	323172.	0.991201	0.495600	−	0.868551i	$-0.334948\pi$
$571$	323172.	0.991201	0.495600	+	0.868551i	$0.334948\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 193284.i	− 0.584603i
$576$	0	0
$577$	350692.	1.05335	0.526676	−	0.850066i	$-0.323438\pi$
$577$	350692.	1.05335	0.526676	+	0.850066i	$0.323438\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	907018.i	2.68698i
$582$	0	0
$583$	567358.	1.66924
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 479659.i	− 1.39205i	−0.718016	−	0.696027i	$-0.754949\pi$
$587$	− 479659.i	− 1.39205i	0.718016	−	0.696027i	$-0.245051\pi$
$588$	0	0
$589$	−115439.	−0.332754
$590$	0	0
$591$	0	0
$592$	0	0
$593$	177843.i	0.505740i	0.967500	+	0.252870i	$0.0813745\pi$
$593$	177843.i	0.505740i	−0.967500	+	0.252870i	$0.918625\pi$
$594$	0	0
$595$	777430.	2.19597
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 19604.2i	− 0.0546382i	−0.999627	−	0.0273191i	$-0.991303\pi$
$599$	− 19604.2i	− 0.0546382i	0.999627	−	0.0273191i	$-0.00869701\pi$
$600$	0	0
$601$	394807.	1.09304	0.546520	−	0.837446i	$-0.315952\pi$
$601$	394807.	1.09304	0.546520	+	0.837446i	$0.315952\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 177866.i	− 0.485938i
$606$	0	0
$607$	−703458.	−1.90924	−0.954620	−	0.297826i	$-0.903739\pi$
$607$	−703458.	−1.90924	−0.954620	+	0.297826i	$0.903739\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 479015.i	− 1.28312i
$612$	0	0
$613$	96392.2	0.256520	0.128260	−	0.991741i	$-0.459061\pi$
$613$	96392.2	0.256520	0.128260	+	0.991741i	$0.459061\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 507805.i	− 1.33391i	−0.745097	−	0.666956i	$-0.767597\pi$
$617$	− 507805.i	− 1.33391i	0.745097	−	0.666956i	$-0.232403\pi$
$618$	0	0
$619$	86907.3	0.226817	0.113408	−	0.993548i	$-0.463823\pi$
$619$	86907.3	0.226817	0.113408	+	0.993548i	$0.463823\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 575572.i	− 1.48294i
$624$	0	0
$625$	−484104.	−1.23931
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 566023.i	− 1.43065i
$630$	0	0
$631$	571591.	1.43558	0.717788	−	0.696261i	$-0.245154\pi$
$631$	571591.	1.43558	0.717788	+	0.696261i	$0.245154\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 285328.i	− 0.707615i
$636$	0	0
$637$	−631593.	−1.55653
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 383711.i	− 0.933874i	−0.884291	−	0.466937i	$-0.845357\pi$
$641$	− 383711.i	− 0.933874i	0.884291	−	0.466937i	$-0.154643\pi$
$642$	0	0
$643$	810239.	1.95971	0.979853	−	0.199719i	$-0.0640030\pi$
$643$	810239.	1.95971	0.979853	+	0.199719i	$0.0640030\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 687173.i	− 1.64156i	−0.571243	−	0.820781i	$-0.693539\pi$
$647$	− 687173.i	− 1.64156i	0.571243	−	0.820781i	$-0.306461\pi$
$648$	0	0
$649$	351187.	0.833775
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 218781.i	− 0.513079i	−0.966534	−	0.256539i	$-0.917418\pi$
$653$	− 218781.i	− 0.513079i	0.966534	−	0.256539i	$-0.0825823\pi$
$654$	0	0
$655$	11309.7	0.0263615
$656$	0	0
$657$	0	0
$658$	0	0
$659$	82328.1i	0.189573i	0.995498	+	0.0947867i	$0.0302169\pi$
$659$	82328.1i	0.189573i	−0.995498	+	0.0947867i	$0.969783\pi$
$660$	0	0
$661$	410034.	0.938462	0.469231	−	0.883075i	$-0.344531\pi$
$661$	410034.	0.938462	0.469231	+	0.883075i	$0.344531\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 749537.i	− 1.69492i
$666$	0	0
$667$	−80848.4	−0.181727
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 353133.i	− 0.784320i
$672$	0	0
$673$	531728.	1.17398	0.586989	−	0.809595i	$-0.300313\pi$
$673$	531728.	1.17398	0.586989	+	0.809595i	$0.300313\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 172721.i	− 0.376849i	−0.982088	−	0.188424i	$-0.939662\pi$
$677$	− 172721.i	− 0.376849i	0.982088	−	0.188424i	$-0.0603380\pi$
$678$	0	0
$679$	−219428.	−0.475941
$680$	0	0
$681$	0	0
$682$	0	0
$683$	452646.i	0.970324i	0.874424	+	0.485162i	$0.161239\pi$
$683$	452646.i	0.970324i	−0.874424	+	0.485162i	$0.838761\pi$
$684$	0	0
$685$	468296.	0.998019
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 767774.i	− 1.61732i
$690$	0	0
$691$	−235590.	−0.493401	−0.246701	−	0.969092i	$-0.579346\pi$
$691$	−235590.	−0.493401	−0.246701	+	0.969092i	$0.579346\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 68353.6i	− 0.141512i
$696$	0	0
$697$	−128802.	−0.265128
$698$	0	0
$699$	0	0
$700$	0	0
$701$	41298.8i	0.0840430i	0.999117	+	0.0420215i	$0.0133798\pi$
$701$	41298.8i	0.0840430i	−0.999117	+	0.0420215i	$0.986620\pi$
$702$	0	0
$703$	−545715.	−1.10422
$704$	0	0
$705$	0	0
$706$	0	0
$707$	821769.i	1.64404i
$708$	0	0
$709$	65829.7	0.130957	0.0654786	−	0.997854i	$-0.479143\pi$
$709$	65829.7	0.130957	0.0654786	+	0.997854i	$0.479143\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	188324.i	0.370447i
$714$	0	0
$715$	−867900.	−1.69769
$716$	0	0
$717$	0	0
$718$	0	0
$719$	551850.i	1.06749i	0.845646	+	0.533745i	$0.179216\pi$
$719$	551850.i	1.06749i	−0.845646	+	0.533745i	$0.820784\pi$
$720$	0	0
$721$	548447.	1.05503
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 59491.1i	− 0.113182i
$726$	0	0
$727$	−139892.	−0.264682	−0.132341	−	0.991204i	$-0.542249\pi$
$727$	−139892.	−0.264682	−0.132341	+	0.991204i	$0.542249\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	234864.i	0.439523i
$732$	0	0
$733$	−696073.	−1.29553	−0.647764	−	0.761841i	$-0.724296\pi$
$733$	−696073.	−1.29553	−0.647764	+	0.761841i	$0.724296\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 938905.i	− 1.72857i
$738$	0	0
$739$	−360220.	−0.659598	−0.329799	−	0.944051i	$-0.606981\pi$
$739$	−360220.	−0.659598	−0.329799	+	0.944051i	$0.606981\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 63663.4i	− 0.115322i	−0.998336	−	0.0576610i	$-0.981636\pi$
$743$	− 63663.4i	− 0.115322i	0.998336	−	0.0576610i	$-0.0183643\pi$
$744$	0	0
$745$	−913188.	−1.64531
$746$	0	0
$747$	0	0
$748$	0	0
$749$	416433.i	0.742303i
$750$	0	0
$751$	739103.	1.31046	0.655232	−	0.755427i	$-0.272571\pi$
$751$	739103.	1.31046	0.655232	+	0.755427i	$0.272571\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	300903.i	0.527877i
$756$	0	0
$757$	−315416.	−0.550416	−0.275208	−	0.961385i	$-0.588747\pi$
$757$	−315416.	−0.550416	−0.275208	+	0.961385i	$0.588747\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	939940.i	1.62305i	0.584320	+	0.811523i	$0.301361\pi$
$761$	939940.i	1.62305i	−0.584320	+	0.811523i	$0.698639\pi$
$762$	0	0
$763$	541607.	0.930325
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 475242.i	− 0.807837i
$768$	0	0
$769$	347007.	0.586794	0.293397	−	0.955991i	$-0.405214\pi$
$769$	347007.	0.586794	0.293397	+	0.955991i	$0.405214\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 4630.27i	− 0.00774903i	−0.999992	−	0.00387451i	$-0.998767\pi$
$773$	− 4630.27i	− 0.00774903i	0.999992	−	0.00387451i	$-0.00123330\pi$
$774$	0	0
$775$	−138576.	−0.230719
$776$	0	0
$777$	0	0
$778$	0	0
$779$	124181.i	0.204634i
$780$	0	0
$781$	829545.	1.36000
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 312618.i	− 0.507312i
$786$	0	0
$787$	−502954.	−0.812043	−0.406021	−	0.913864i	$-0.633084\pi$
$787$	−502954.	−0.812043	−0.406021	+	0.913864i	$0.633084\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 934828.i	− 1.49410i
$792$	0	0
$793$	−477875.	−0.759921
$794$	0	0
$795$	0	0
$796$	0	0
$797$	243111.i	0.382727i	0.981519	+	0.191363i	$0.0612909\pi$
$797$	243111.i	0.382727i	−0.981519	+	0.191363i	$0.938709\pi$
$798$	0	0
$799$	810366.	1.26937
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1.25120e6i	1.94042i
$804$	0	0
$805$	−1.22277e6	−1.88692
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 600348.i	− 0.917289i	−0.888620	−	0.458644i	$-0.848335\pi$
$809$	− 600348.i	− 0.917289i	0.888620	−	0.458644i	$-0.151665\pi$
$810$	0	0
$811$	168852.	0.256722	0.128361	−	0.991727i	$-0.459028\pi$
$811$	168852.	0.256722	0.128361	+	0.991727i	$0.459028\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	711153.i	1.07065i
$816$	0	0
$817$	226437.	0.339237
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 514466.i	− 0.763256i	−0.924316	−	0.381628i	$-0.875363\pi$
$821$	− 514466.i	− 0.763256i	0.924316	−	0.381628i	$-0.124637\pi$
$822$	0	0
$823$	−1.20424e6	−1.77793	−0.888963	−	0.457979i	$-0.848574\pi$
$823$	−1.20424e6	−1.77793	−0.888963	+	0.457979i	$0.848574\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 682316.i	− 0.997641i	−0.866705	−	0.498821i	$-0.833767\pi$
$827$	− 682316.i	− 0.997641i	0.866705	−	0.498821i	$-0.166233\pi$
$828$	0	0
$829$	−1.14603e6	−1.66758	−0.833791	−	0.552081i	$-0.813834\pi$
$829$	−1.14603e6	−1.66758	−0.833791	+	0.552081i	$0.813834\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 1.06849e6i	− 1.53985i
$834$	0	0
$835$	−537146.	−0.770406
$836$	0	0
$837$	0	0
$838$	0	0
$839$	328739.i	0.467011i	0.972355	+	0.233506i	$0.0750198\pi$
$839$	328739.i	0.467011i	−0.972355	+	0.233506i	$0.924980\pi$
$840$	0	0
$841$	682397.	0.964817
$842$	0	0
$843$	0	0
$844$	0	0
$845$	270342.i	0.378617i
$846$	0	0
$847$	−423453.	−0.590254
$848$	0	0
$849$	0	0
$850$	0	0
$851$	890261.i	1.22930i
$852$	0	0
$853$	377527.	0.518860	0.259430	−	0.965762i	$-0.416465\pi$
$853$	377527.	0.518860	0.259430	+	0.965762i	$0.416465\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 430415.i	− 0.586038i	−0.956107	−	0.293019i	$-0.905340\pi$
$857$	− 430415.i	− 0.586038i	0.956107	−	0.293019i	$-0.0946599\pi$
$858$	0	0
$859$	168732.	0.228671	0.114335	−	0.993442i	$-0.463526\pi$
$859$	168732.	0.228671	0.114335	+	0.993442i	$0.463526\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	641651.i	0.861543i	0.902461	+	0.430772i	$0.141759\pi$
$863$	641651.i	0.861543i	−0.902461	+	0.430772i	$0.858241\pi$
$864$	0	0
$865$	−1.18587e6	−1.58491
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 550705.i	− 0.729255i
$870$	0	0
$871$	−1.27057e6	−1.67480
$872$	0	0
$873$	0	0
$874$	0	0
$875$	591377.i	0.772411i
$876$	0	0
$877$	−1.11973e6	−1.45584	−0.727921	−	0.685661i	$-0.759513\pi$
$877$	−1.11973e6	−1.45584	−0.727921	+	0.685661i	$0.759513\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 293548.i	− 0.378205i	−0.981957	−	0.189103i	$-0.939442\pi$
$881$	− 293548.i	− 0.378205i	0.981957	−	0.189103i	$-0.0605579\pi$
$882$	0	0
$883$	1.07664e6	1.38085	0.690426	−	0.723403i	$-0.257423\pi$
$883$	1.07664e6	1.38085	0.690426	+	0.723403i	$0.257423\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1.26144e6i	− 1.60331i	−0.597784	−	0.801657i	$-0.703952\pi$
$887$	− 1.26144e6i	− 1.60331i	0.597784	−	0.801657i	$-0.296048\pi$
$888$	0	0
$889$	−679295.	−0.859517
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 781291.i	− 0.979738i
$894$	0	0
$895$	1.41697e6	1.76895
$896$	0	0
$897$	0	0
$898$	0	0
$899$	57964.4i	0.0717203i
$900$	0	0
$901$	1.29887e6	1.59999
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 924620.i	− 1.12893i
$906$	0	0
$907$	213403.	0.259409	0.129705	−	0.991553i	$-0.458597\pi$
$907$	213403.	0.259409	0.129705	+	0.991553i	$0.458597\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 848823.i	− 1.02278i	−0.859350	−	0.511388i	$-0.829132\pi$
$911$	− 848823.i	− 1.02278i	0.859350	−	0.511388i	$-0.170868\pi$
$912$	0	0
$913$	1.71300e6	2.05501
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 26925.7i	− 0.0320205i
$918$	0	0
$919$	650728.	0.770493	0.385246	−	0.922814i	$-0.374116\pi$
$919$	650728.	0.770493	0.385246	+	0.922814i	$0.374116\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 1.12258e6i	− 1.31769i
$924$	0	0
$925$	−655086.	−0.765623
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 273367.i	− 0.316748i	−0.987379	−	0.158374i	$-0.949375\pi$
$929$	− 273367.i	− 0.316748i	0.987379	−	0.158374i	$-0.0506252\pi$
$930$	0	0
$931$	−1.03015e6	−1.18851
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 1.46826e6i	− 1.67949i
$936$	0	0
$937$	1.19093e6	1.35646	0.678232	−	0.734848i	$-0.262746\pi$
$937$	1.19093e6	1.35646	0.678232	+	0.734848i	$0.262746\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1.40642e6i	1.58831i	0.607717	+	0.794153i	$0.292085\pi$
$941$	1.40642e6i	1.58831i	−0.607717	+	0.794153i	$0.707915\pi$
$942$	0	0
$943$	202584.	0.227815
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 1.31602e6i	− 1.46745i	−0.679449	−	0.733723i	$-0.737781\pi$
$947$	− 1.31602e6i	− 1.46745i	0.679449	−	0.733723i	$-0.262219\pi$
$948$	0	0
$949$	1.69318e6	1.88005
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1.09774e6i	1.20868i	0.796725	+	0.604342i	$0.206564\pi$
$953$	1.09774e6i	1.20868i	−0.796725	+	0.604342i	$0.793436\pi$
$954$	0	0
$955$	−949973.	−1.04161
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 1.11489e6i	− 1.21226i
$960$	0	0
$961$	−788502.	−0.853799
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 545495.i	− 0.585782i
$966$	0	0
$967$	49000.8	0.0524023	0.0262012	−	0.999657i	$-0.491659\pi$
$967$	49000.8	0.0524023	0.0262012	+	0.999657i	$0.491659\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1.83854e6i	1.95000i	0.222198	+	0.975001i	$0.428677\pi$
$971$	1.83854e6i	1.95000i	−0.222198	+	0.975001i	$0.571323\pi$
$972$	0	0
$973$	−162733.	−0.171890
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 634276.i	− 0.664492i	−0.943193	−	0.332246i	$-0.892194\pi$
$977$	− 634276.i	− 0.664492i	0.943193	−	0.332246i	$-0.107806\pi$
$978$	0	0
$979$	−1.08703e6	−1.13416
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 914165.i	− 0.946057i	−0.881047	−	0.473029i	$-0.843161\pi$
$983$	− 914165.i	− 0.946057i	0.881047	−	0.473029i	$-0.156839\pi$
$984$	0	0
$985$	−2.14922e6	−2.21518
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 369402.i	− 0.377665i
$990$	0	0
$991$	119.608	0.000121790 0	6.08950e−5	−	1.00000i	$-0.499981\pi$
$991$	119.608	0.000121790 0	6.08950e−5		1.00000i	$0.499981\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	854613.i	0.863224i
$996$	0	0
$997$	1.13918e6	1.14605	0.573025	−	0.819538i	$-0.305770\pi$
$997$	1.13918e6	1.14605	0.573025	+	0.819538i	$0.305770\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1296.5.e.g.161.7		8
3.2	odd	2	inner	1296.5.e.g.161.2		8
4.3	odd	2		324.5.c.a.161.7		8
9.2	odd	6		144.5.q.c.113.4		8
9.4	even	3		144.5.q.c.65.4		8
9.5	odd	6		432.5.q.c.305.1		8
9.7	even	3		432.5.q.c.17.1		8
12.11	even	2		324.5.c.a.161.2		8
36.7	odd	6		108.5.g.a.17.1		8
36.11	even	6		36.5.g.a.5.1	✓	8
36.23	even	6		108.5.g.a.89.1		8
36.31	odd	6		36.5.g.a.29.1	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
36.5.g.a.5.1	✓	8	36.11	even	6
36.5.g.a.29.1	yes	8	36.31	odd	6
108.5.g.a.17.1		8	36.7	odd	6
108.5.g.a.89.1		8	36.23	even	6
144.5.q.c.65.4		8	9.4	even	3
144.5.q.c.113.4		8	9.2	odd	6
324.5.c.a.161.2		8	12.11	even	2
324.5.c.a.161.7		8	4.3	odd	2
432.5.q.c.17.1		8	9.7	even	3
432.5.q.c.305.1		8	9.5	odd	6
1296.5.e.g.161.2		8	3.2	odd	2	inner
1296.5.e.g.161.7		8	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:	\( N \)	\(=\)	\( 1296 = 2^{4} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	1296.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+31.6564i q^{5} +75.3660 q^{7} +O(q^{10})\)	\(q+31.6564i q^{5} +75.3660 q^{7} -142.336i q^{11} -192.616 q^{13} -325.855i q^{17} -314.164 q^{19} +512.516i q^{23} -377.128 q^{25} +157.748i q^{29} +367.450 q^{31} +2385.82i q^{35} +1737.04 q^{37} -395.273i q^{41} -720.762 q^{43} +2486.89i q^{47} +3279.03 q^{49} +3986.04i q^{53} +4505.86 q^{55} +2467.30i q^{59} +2480.98 q^{61} -6097.53i q^{65} +6596.38 q^{67} +5828.07i q^{71} -8790.44 q^{73} -10727.3i q^{77} +3869.04 q^{79} +12034.9i q^{83} +10315.4 q^{85} -7637.03i q^{89} -14516.7 q^{91} -9945.30i q^{95} -2911.50 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 26 q^{7}+O(q^{10}) \) 8 * q + 26 * q^7	\( 8 q + 26 q^{7} + 10 q^{13} - 562 q^{19} - 706 q^{25} + 374 q^{31} + 16 q^{37} - 136 q^{43} + 654 q^{49} + 1818 q^{55} + 3874 q^{61} + 308 q^{67} - 7802 q^{73} - 4390 q^{79} + 6084 q^{85} - 15830 q^{91} - 14564 q^{97}+O(q^{100}) \) 8 * q + 26 * q^7 + 10 * q^13 - 562 * q^19 - 706 * q^25 + 374 * q^31 + 16 * q^37 - 136 * q^43 + 654 * q^49 + 1818 * q^55 + 3874 * q^61 + 308 * q^67 - 7802 * q^73 - 4390 * q^79 + 6084 * q^85 - 15830 * q^91 - 14564 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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