LMFDB - Embedded newform 1296.5.e.c.161.3

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:	$6$
Coefficient field:	6.0.39400128.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} + 11x^{4} + 14x^{3} + 98x^{2} + 20x + 4 $ x^6 - x^5 + 11x^4 + 14x^3 + 98x^2 + 20x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{9} $
Twist minimal:	no (minimal twist has level 9)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.3
Root			$1.89154 - 3.27625i$ of defining polynomial
Character	$\chi$	$=$	1296.161
Dual form			1296.5.e.c.161.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-11.7830i q^{5} +53.2728 q^{7} -124.971i q^{11} -74.6104 q^{13} -7.70989i q^{17} +54.1307 q^{19} +399.954i q^{23} +486.161 q^{25} -540.653i q^{29} +1532.17 q^{31} -627.714i q^{35} -1719.10 q^{37} -1259.63i q^{41} +2607.79 q^{43} +800.034i q^{47} +436.996 q^{49} -4229.81i q^{53} -1472.54 q^{55} -3333.19i q^{59} -15.0169 q^{61} +879.134i q^{65} -5182.39 q^{67} +1924.16i q^{71} +949.554 q^{73} -6657.57i q^{77} +237.980 q^{79} +13165.9i q^{83} -90.8456 q^{85} -575.925i q^{89} -3974.71 q^{91} -637.822i q^{95} +15123.4 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 24 q^{7} + 12 q^{13} + 258 q^{19} + 546 q^{25} + 2580 q^{31} + 12 q^{37} - 570 q^{43} + 3726 q^{49} - 2016 q^{55} - 7260 q^{61} - 10110 q^{67} - 14622 q^{73} + 9528 q^{79} - 24732 q^{85} - 34836 q^{91}+ \cdots + 57918 q^{97}+O(q^{100}) $ 6 * q + 24 * q^7 + 12 * q^13 + 258 * q^19 + 546 * q^25 + 2580 * q^31 + 12 * q^37 - 570 * q^43 + 3726 * q^49 - 2016 * q^55 - 7260 * q^61 - 10110 * q^67 - 14622 * q^73 + 9528 * q^79 - 24732 * q^85 - 34836 * q^91 + 57918 * q^97

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
$5$	− 11.7830i	− 0.471320i	−0.971836	−	0.235660i	$-0.924275\pi$
$5$	− 11.7830i	− 0.471320i	0.971836	−	0.235660i	$-0.0757252\pi$
$6$	0	0
$7$	53.2728	1.08720	0.543600	−	0.839344i	$-0.317061\pi$
$7$	53.2728	1.08720	0.543600	+	0.839344i	$0.317061\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 124.971i	− 1.03282i	−0.856341	−	0.516410i	$-0.827268\pi$
$11$	− 124.971i	− 1.03282i	0.856341	−	0.516410i	$-0.172732\pi$
$12$	0	0
$13$	−74.6104	−0.441482	−0.220741	−	0.975332i	$-0.570848\pi$
$13$	−74.6104	−0.441482	−0.220741	+	0.975332i	$0.570848\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 7.70989i	− 0.0266778i	−0.999911	−	0.0133389i	$-0.995754\pi$
$17$	− 7.70989i	− 0.0266778i	0.999911	−	0.0133389i	$-0.00424603\pi$
$18$	0	0
$19$	54.1307	0.149946	0.0749732	−	0.997186i	$-0.476113\pi$
$19$	54.1307	0.149946	0.0749732	+	0.997186i	$0.476113\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	399.954i	0.756057i	0.925794	+	0.378029i	$0.123398\pi$
$23$	399.954i	0.756057i	−0.925794	+	0.378029i	$0.876602\pi$
$24$	0	0
$25$	486.161	0.777858
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 540.653i	− 0.642869i	−0.946932	−	0.321435i	$-0.895835\pi$
$29$	− 540.653i	− 0.642869i	0.946932	−	0.321435i	$-0.104165\pi$
$30$	0	0
$31$	1532.17	1.59435	0.797173	−	0.603751i	$-0.206328\pi$
$31$	1532.17	1.59435	0.797173	+	0.603751i	$0.206328\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 627.714i	− 0.512419i
$36$	0	0
$37$	−1719.10	−1.25574	−0.627868	−	0.778320i	$-0.716072\pi$
$37$	−1719.10	−1.25574	−0.627868	+	0.778320i	$0.716072\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 1259.63i	− 0.749335i	−0.927159	−	0.374668i	$-0.877757\pi$
$41$	− 1259.63i	− 0.749335i	0.927159	−	0.374668i	$-0.122243\pi$
$42$	0	0
$43$	2607.79	1.41038	0.705189	−	0.709020i	$-0.250862\pi$
$43$	2607.79	1.41038	0.705189	+	0.709020i	$0.250862\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	800.034i	0.362170i	0.983467	+	0.181085i	$0.0579609\pi$
$47$	800.034i	0.362170i	−0.983467	+	0.181085i	$0.942039\pi$
$48$	0	0
$49$	436.996	0.182006
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 4229.81i	− 1.50581i	−0.658131	−	0.752904i	$-0.728653\pi$
$53$	− 4229.81i	− 1.50581i	0.658131	−	0.752904i	$-0.271347\pi$
$54$	0	0
$55$	−1472.54	−0.486789
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 3333.19i	− 0.957538i	−0.877941	−	0.478769i	$-0.841083\pi$
$59$	− 3333.19i	− 0.957538i	0.877941	−	0.478769i	$-0.158917\pi$
$60$	0	0
$61$	−15.0169	−0.00403570	−0.00201785	−	0.999998i	$-0.500642\pi$
$61$	−15.0169	−0.00403570	−0.00201785	+	0.999998i	$0.500642\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	879.134i	0.208079i
$66$	0	0
$67$	−5182.39	−1.15447	−0.577233	−	0.816580i	$-0.695867\pi$
$67$	−5182.39	−1.15447	−0.577233	+	0.816580i	$0.695867\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	1924.16i	0.381701i	0.981619	+	0.190851i	$0.0611246\pi$
$71$	1924.16i	0.381701i	−0.981619	+	0.190851i	$0.938875\pi$
$72$	0	0
$73$	949.554	0.178186	0.0890931	−	0.996023i	$-0.471603\pi$
$73$	949.554	0.178186	0.0890931	+	0.996023i	$0.471603\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 6657.57i	− 1.12288i
$78$	0	0
$79$	237.980	0.0381318	0.0190659	−	0.999818i	$-0.493931\pi$
$79$	237.980	0.0381318	0.0190659	+	0.999818i	$0.493931\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	13165.9i	1.91115i	0.294750	+	0.955574i	$0.404764\pi$
$83$	13165.9i	1.91115i	−0.294750	+	0.955574i	$0.595236\pi$
$84$	0	0
$85$	−90.8456	−0.0125738
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 575.925i	− 0.0727086i	−0.999339	−	0.0363543i	$-0.988426\pi$
$89$	− 575.925i	− 0.0727086i	0.999339	−	0.0363543i	$-0.0115745\pi$
$90$	0	0
$91$	−3974.71	−0.479979
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 637.822i	− 0.0706728i
$96$	0	0
$97$	15123.4	1.60733	0.803667	−	0.595079i	$-0.202879\pi$
$97$	15123.4	1.60733	0.803667	+	0.595079i	$0.202879\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	11060.8i	1.08428i	0.840287	+	0.542141i	$0.182386\pi$
$101$	11060.8i	1.08428i	−0.840287	+	0.542141i	$0.817614\pi$
$102$	0	0
$103$	3562.92	0.335839	0.167920	−	0.985801i	$-0.446295\pi$
$103$	3562.92	0.335839	0.167920	+	0.985801i	$0.446295\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 11432.7i	− 0.998576i	−0.866436	−	0.499288i	$-0.833595\pi$
$107$	− 11432.7i	− 0.998576i	0.866436	−	0.499288i	$-0.166405\pi$
$108$	0	0
$109$	−4780.43	−0.402359	−0.201180	−	0.979554i	$-0.564478\pi$
$109$	−4780.43	−0.402359	−0.201180	+	0.979554i	$0.564478\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 4491.39i	− 0.351742i	−0.984413	−	0.175871i	$-0.943726\pi$
$113$	− 4491.39i	− 0.351742i	0.984413	−	0.175871i	$-0.0562741\pi$
$114$	0	0
$115$	4712.66	0.356345
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 410.728i	− 0.0290041i
$120$	0	0
$121$	−976.812	−0.0667176
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 13092.8i	− 0.837940i
$126$	0	0
$127$	−13521.3	−0.838322	−0.419161	−	0.907912i	$-0.637676\pi$
$127$	−13521.3	−0.838322	−0.419161	+	0.907912i	$0.637676\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	3396.97i	0.197947i	0.995090	+	0.0989735i	$0.0315559\pi$
$131$	3396.97i	0.197947i	−0.995090	+	0.0989735i	$0.968444\pi$
$132$	0	0
$133$	2883.69	0.163022
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3823.19i	0.203697i	0.994800	+	0.101849i	$0.0324757\pi$
$137$	3823.19i	0.203697i	−0.994800	+	0.101849i	$0.967524\pi$
$138$	0	0
$139$	−3476.63	−0.179940	−0.0899702	−	0.995944i	$-0.528677\pi$
$139$	−3476.63	−0.179940	−0.0899702	+	0.995944i	$0.528677\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	9324.16i	0.455971i
$144$	0	0
$145$	−6370.51	−0.302997
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 8910.94i	− 0.401376i	−0.979655	−	0.200688i	$-0.935682\pi$
$149$	− 8910.94i	− 0.401376i	0.979655	−	0.200688i	$-0.0643177\pi$
$150$	0	0
$151$	9915.33	0.434864	0.217432	−	0.976076i	$-0.430232\pi$
$151$	9915.33	0.434864	0.217432	+	0.976076i	$0.430232\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 18053.5i	− 0.751447i
$156$	0	0
$157$	5646.94	0.229094	0.114547	−	0.993418i	$-0.463458\pi$
$157$	5646.94	0.229094	0.114547	+	0.993418i	$0.463458\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	21306.7i	0.821986i
$162$	0	0
$163$	2153.55	0.0810552	0.0405276	−	0.999178i	$-0.487096\pi$
$163$	2153.55	0.0810552	0.0405276	+	0.999178i	$0.487096\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 37020.2i	− 1.32741i	−0.747994	−	0.663706i	$-0.768983\pi$
$167$	− 37020.2i	− 1.32741i	0.747994	−	0.663706i	$-0.231017\pi$
$168$	0	0
$169$	−22994.3	−0.805094
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 48585.4i	− 1.62336i	−0.584105	−	0.811678i	$-0.698554\pi$
$173$	− 48585.4i	− 1.62336i	0.584105	−	0.811678i	$-0.301446\pi$
$174$	0	0
$175$	25899.2	0.845687
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 47717.9i	− 1.48928i	−0.667468	−	0.744639i	$-0.732622\pi$
$179$	− 47717.9i	− 1.48928i	0.667468	−	0.744639i	$-0.267378\pi$
$180$	0	0
$181$	−45767.2	−1.39700	−0.698502	−	0.715608i	$-0.746150\pi$
$181$	−45767.2	−1.39700	−0.698502	+	0.715608i	$0.746150\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	20256.2i	0.591854i
$186$	0	0
$187$	−963.514	−0.0275534
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 46719.5i	− 1.28065i	−0.768103	−	0.640326i	$-0.778799\pi$
$191$	− 46719.5i	− 1.28065i	0.768103	−	0.640326i	$-0.221201\pi$
$192$	0	0
$193$	−27207.8	−0.730430	−0.365215	−	0.930923i	$-0.619005\pi$
$193$	−27207.8	−0.730430	−0.365215	+	0.930923i	$0.619005\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 64665.2i	− 1.66624i	−0.553090	−	0.833122i	$-0.686551\pi$
$197$	− 64665.2i	− 1.66624i	0.553090	−	0.833122i	$-0.313449\pi$
$198$	0	0
$199$	25841.3	0.652542	0.326271	−	0.945276i	$-0.394208\pi$
$199$	25841.3	0.652542	0.326271	+	0.945276i	$0.394208\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 28802.1i	− 0.698928i
$204$	0	0
$205$	−14842.3	−0.353177
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 6764.78i	− 0.154868i
$210$	0	0
$211$	−46058.7	−1.03454	−0.517269	−	0.855823i	$-0.673051\pi$
$211$	−46058.7	−1.03454	−0.517269	+	0.855823i	$0.673051\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 30727.6i	− 0.664739i
$216$	0	0
$217$	81622.8	1.73337
$218$	0	0
$219$	0	0
$220$	0	0
$221$	575.238i	0.0117778i
$222$	0	0
$223$	−21911.1	−0.440610	−0.220305	−	0.975431i	$-0.570705\pi$
$223$	−21911.1	−0.440610	−0.220305	+	0.975431i	$0.570705\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 49785.7i	− 0.966168i	−0.875574	−	0.483084i	$-0.839517\pi$
$227$	− 49785.7i	− 0.966168i	0.875574	−	0.483084i	$-0.160483\pi$
$228$	0	0
$229$	28521.0	0.543868	0.271934	−	0.962316i	$-0.412337\pi$
$229$	28521.0	0.543868	0.271934	+	0.962316i	$0.412337\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	44050.2i	0.811403i	0.914006	+	0.405701i	$0.132973\pi$
$233$	44050.2i	0.811403i	−0.914006	+	0.405701i	$0.867027\pi$
$234$	0	0
$235$	9426.80	0.170698
$236$	0	0
$237$	0	0
$238$	0	0
$239$	20575.2i	0.360203i	0.983648	+	0.180102i	$0.0576427\pi$
$239$	20575.2i	0.360203i	−0.983648	+	0.180102i	$0.942357\pi$
$240$	0	0
$241$	−57917.5	−0.997185	−0.498593	−	0.866836i	$-0.666150\pi$
$241$	−57917.5	−0.997185	−0.498593	+	0.866836i	$0.666150\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 5149.12i	− 0.0857829i
$246$	0	0
$247$	−4038.71	−0.0661986
$248$	0	0
$249$	0	0
$250$	0	0
$251$	54140.4i	0.859357i	0.902982	+	0.429679i	$0.141373\pi$
$251$	54140.4i	0.859357i	−0.902982	+	0.429679i	$0.858627\pi$
$252$	0	0
$253$	49982.8	0.780871
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 113764.i	− 1.72242i	−0.508248	−	0.861211i	$-0.669707\pi$
$257$	− 113764.i	− 1.72242i	0.508248	−	0.861211i	$-0.330293\pi$
$258$	0	0
$259$	−91581.5	−1.36524
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 44363.7i	− 0.641381i	−0.947184	−	0.320691i	$-0.896085\pi$
$263$	− 44363.7i	− 0.641381i	0.947184	−	0.320691i	$-0.103915\pi$
$264$	0	0
$265$	−49839.9	−0.709717
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 4429.77i	− 0.0612176i	−0.999531	−	0.0306088i	$-0.990255\pi$
$269$	− 4429.77i	− 0.0612176i	0.999531	−	0.0306088i	$-0.00974461\pi$
$270$	0	0
$271$	86111.9	1.17253	0.586266	−	0.810119i	$-0.300597\pi$
$271$	86111.9	1.17253	0.586266	+	0.810119i	$0.300597\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 60756.1i	− 0.803387i
$276$	0	0
$277$	70613.4	0.920297	0.460148	−	0.887842i	$-0.347796\pi$
$277$	70613.4	0.920297	0.460148	+	0.887842i	$0.347796\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 149254.i	− 1.89023i	−0.326740	−	0.945114i	$-0.605950\pi$
$281$	− 149254.i	− 1.89023i	0.326740	−	0.945114i	$-0.394050\pi$
$282$	0	0
$283$	83926.7	1.04792	0.523959	−	0.851743i	$-0.324454\pi$
$283$	83926.7	1.04792	0.523959	+	0.851743i	$0.324454\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 67104.2i	− 0.814678i
$288$	0	0
$289$	83461.6	0.999288
$290$	0	0
$291$	0	0
$292$	0	0
$293$	35903.0i	0.418211i	0.977893	+	0.209106i	$0.0670553\pi$
$293$	35903.0i	0.418211i	−0.977893	+	0.209106i	$0.932945\pi$
$294$	0	0
$295$	−39275.0	−0.451307
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 29840.7i	− 0.333785i
$300$	0	0
$301$	138924.	1.53336
$302$	0	0
$303$	0	0
$304$	0	0
$305$	176.944i	0.00190211i
$306$	0	0
$307$	−7054.30	−0.0748475	−0.0374238	−	0.999299i	$-0.511915\pi$
$307$	−7054.30	−0.0748475	−0.0374238	+	0.999299i	$0.511915\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	85661.4i	0.885655i	0.896607	+	0.442827i	$0.146025\pi$
$311$	85661.4i	0.885655i	−0.896607	+	0.442827i	$0.853975\pi$
$312$	0	0
$313$	73442.5	0.749650	0.374825	−	0.927096i	$-0.377703\pi$
$313$	73442.5	0.749650	0.374825	+	0.927096i	$0.377703\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	49239.6i	0.490000i	0.969523	+	0.245000i	$0.0787880\pi$
$317$	49239.6i	0.490000i	−0.969523	+	0.245000i	$0.921212\pi$
$318$	0	0
$319$	−67566.1	−0.663968
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 417.341i	− 0.00400024i
$324$	0	0
$325$	−36272.7	−0.343410
$326$	0	0
$327$	0	0
$328$	0	0
$329$	42620.1i	0.393752i
$330$	0	0
$331$	82011.1	0.748543	0.374271	−	0.927319i	$-0.377893\pi$
$331$	82011.1	0.748543	0.374271	+	0.927319i	$0.377893\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	61064.1i	0.544122i
$336$	0	0
$337$	−46982.9	−0.413694	−0.206847	−	0.978373i	$-0.566320\pi$
$337$	−46982.9	−0.413694	−0.206847	+	0.978373i	$0.566320\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 191477.i	− 1.64667i
$342$	0	0
$343$	−104628.	−0.889324
$344$	0	0
$345$	0	0
$346$	0	0
$347$	44319.7i	0.368076i	0.982919	+	0.184038i	$0.0589170\pi$
$347$	44319.7i	0.368076i	−0.982919	+	0.184038i	$0.941083\pi$
$348$	0	0
$349$	−186463.	−1.53088	−0.765442	−	0.643504i	$-0.777480\pi$
$349$	−186463.	−1.53088	−0.765442	+	0.643504i	$0.777480\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 169694.i	− 1.36181i	−0.732370	−	0.680907i	$-0.761586\pi$
$353$	− 169694.i	− 1.36181i	0.732370	−	0.680907i	$-0.238414\pi$
$354$	0	0
$355$	22672.3	0.179903
$356$	0	0
$357$	0	0
$358$	0	0
$359$	69656.7i	0.540473i	0.962794	+	0.270236i	$0.0871019\pi$
$359$	69656.7i	0.540473i	−0.962794	+	0.270236i	$0.912898\pi$
$360$	0	0
$361$	−127391.	−0.977516
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 11188.6i	− 0.0839827i
$366$	0	0
$367$	176168.	1.30796	0.653982	−	0.756510i	$-0.273097\pi$
$367$	176168.	1.30796	0.653982	+	0.756510i	$0.273097\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 225334.i	− 1.63712i
$372$	0	0
$373$	67187.1	0.482912	0.241456	−	0.970412i	$-0.422375\pi$
$373$	67187.1	0.482912	0.241456	+	0.970412i	$0.422375\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	40338.3i	0.283815i
$378$	0	0
$379$	−43976.9	−0.306158	−0.153079	−	0.988214i	$-0.548919\pi$
$379$	−43976.9	−0.306158	−0.153079	+	0.988214i	$0.548919\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 185140.i	− 1.26212i	−0.775733	−	0.631062i	$-0.782619\pi$
$383$	− 185140.i	− 1.26212i	0.775733	−	0.631062i	$-0.217381\pi$
$384$	0	0
$385$	−78446.2	−0.529237
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 30442.6i	− 0.201179i	−0.994928	−	0.100589i	$-0.967927\pi$
$389$	− 30442.6i	− 0.201179i	0.994928	−	0.100589i	$-0.0320728\pi$
$390$	0	0
$391$	3083.60	0.0201699
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 2804.12i	− 0.0179723i
$396$	0	0
$397$	−235724.	−1.49563	−0.747813	−	0.663910i	$-0.768896\pi$
$397$	−235724.	−1.49563	−0.747813	+	0.663910i	$0.768896\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 234208.i	− 1.45651i	−0.685308	−	0.728254i	$-0.740332\pi$
$401$	− 234208.i	− 1.45651i	0.685308	−	0.728254i	$-0.259668\pi$
$402$	0	0
$403$	−114316.	−0.703874
$404$	0	0
$405$	0	0
$406$	0	0
$407$	214839.i	1.29695i
$408$	0	0
$409$	−297786.	−1.78015	−0.890076	−	0.455812i	$-0.849349\pi$
$409$	−297786.	−1.78015	−0.890076	+	0.455812i	$0.849349\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 177569.i	− 1.04104i
$414$	0	0
$415$	155134.	0.900763
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 36874.2i	− 0.210036i	−0.994470	−	0.105018i	$-0.966510\pi$
$419$	− 36874.2i	− 0.210036i	0.994470	−	0.105018i	$-0.0334901\pi$
$420$	0	0
$421$	240610.	1.35753	0.678764	−	0.734356i	$-0.262516\pi$
$421$	240610.	1.35753	0.678764	+	0.734356i	$0.262516\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 3748.25i	− 0.0207515i
$426$	0	0
$427$	−799.990	−0.00438762
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 162758.i	− 0.876169i	−0.898934	−	0.438084i	$-0.855657\pi$
$431$	− 162758.i	− 0.876169i	0.898934	−	0.438084i	$-0.144343\pi$
$432$	0	0
$433$	266637.	1.42215	0.711073	−	0.703118i	$-0.248209\pi$
$433$	266637.	1.42215	0.711073	+	0.703118i	$0.248209\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	21649.8i	0.113368i
$438$	0	0
$439$	36817.5	0.191040	0.0955201	−	0.995428i	$-0.469549\pi$
$439$	36817.5	0.191040	0.0955201	+	0.995428i	$0.469549\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	137544.i	0.700865i	0.936588	+	0.350432i	$0.113965\pi$
$443$	137544.i	0.700865i	−0.936588	+	0.350432i	$0.886035\pi$
$444$	0	0
$445$	−6786.12	−0.0342690
$446$	0	0
$447$	0	0
$448$	0	0
$449$	174405.i	0.865101i	0.901610	+	0.432550i	$0.142386\pi$
$449$	174405.i	0.865101i	−0.901610	+	0.432550i	$0.857614\pi$
$450$	0	0
$451$	−157418.	−0.773929
$452$	0	0
$453$	0	0
$454$	0	0
$455$	46834.0i	0.226224i
$456$	0	0
$457$	115309.	0.552119	0.276059	−	0.961141i	$-0.410971\pi$
$457$	115309.	0.552119	0.276059	+	0.961141i	$0.410971\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	405463.i	1.90787i	0.300011	+	0.953936i	$0.403010\pi$
$461$	405463.i	1.90787i	−0.300011	+	0.953936i	$0.596990\pi$
$462$	0	0
$463$	−72586.7	−0.338606	−0.169303	−	0.985564i	$-0.554152\pi$
$463$	−72586.7	−0.338606	−0.169303	+	0.985564i	$0.554152\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 358806.i	− 1.64523i	−0.568602	−	0.822613i	$-0.692515\pi$
$467$	− 358806.i	− 1.64523i	0.568602	−	0.822613i	$-0.307485\pi$
$468$	0	0
$469$	−276081.	−1.25514
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 325899.i	− 1.45667i
$474$	0	0
$475$	26316.2	0.116637
$476$	0	0
$477$	0	0
$478$	0	0
$479$	25599.5i	0.111573i	0.998443	+	0.0557867i	$0.0177667\pi$
$479$	25599.5i	0.111573i	−0.998443	+	0.0557867i	$0.982233\pi$
$480$	0	0
$481$	128263.	0.554385
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 178199.i	− 0.757569i
$486$	0	0
$487$	−247261.	−1.04255	−0.521277	−	0.853388i	$-0.674544\pi$
$487$	−247261.	−1.04255	−0.521277	+	0.853388i	$0.674544\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 342256.i	− 1.41967i	−0.704366	−	0.709837i	$-0.748769\pi$
$491$	− 342256.i	− 1.41967i	0.704366	−	0.709837i	$-0.251231\pi$
$492$	0	0
$493$	−4168.37	−0.0171503
$494$	0	0
$495$	0	0
$496$	0	0
$497$	102505.i	0.414986i
$498$	0	0
$499$	197838.	0.794527	0.397264	−	0.917704i	$-0.369960\pi$
$499$	197838.	0.794527	0.397264	+	0.917704i	$0.369960\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	187051.i	0.739305i	0.929170	+	0.369652i	$0.120523\pi$
$503$	187051.i	0.739305i	−0.929170	+	0.369652i	$0.879477\pi$
$504$	0	0
$505$	130329.	0.511044
$506$	0	0
$507$	0	0
$508$	0	0
$509$	28880.8i	0.111474i	0.998445	+	0.0557370i	$0.0177508\pi$
$509$	28880.8i	0.111474i	−0.998445	+	0.0557370i	$0.982249\pi$
$510$	0	0
$511$	50585.4	0.193724
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 41981.9i	− 0.158288i
$516$	0	0
$517$	99981.2	0.374057
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 269661.i	− 0.993443i	−0.867910	−	0.496721i	$-0.834537\pi$
$521$	− 269661.i	− 0.993443i	0.867910	−	0.496721i	$-0.165463\pi$
$522$	0	0
$523$	187282.	0.684689	0.342344	−	0.939575i	$-0.388779\pi$
$523$	187282.	0.684689	0.342344	+	0.939575i	$0.388779\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 11812.8i	− 0.0425336i
$528$	0	0
$529$	119878.	0.428378
$530$	0	0
$531$	0	0
$532$	0	0
$533$	93981.7i	0.330818i
$534$	0	0
$535$	−134711.	−0.470649
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 54611.9i	− 0.187979i
$540$	0	0
$541$	−25587.9	−0.0874259	−0.0437129	−	0.999044i	$-0.513919\pi$
$541$	−25587.9	−0.0874259	−0.0437129	+	0.999044i	$0.513919\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	56327.8i	0.189640i
$546$	0	0
$547$	276849.	0.925269	0.462634	−	0.886549i	$-0.346904\pi$
$547$	276849.	0.925269	0.462634	+	0.886549i	$0.346904\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 29265.9i	− 0.0963959i
$552$	0	0
$553$	12677.9	0.0414569
$554$	0	0
$555$	0	0
$556$	0	0
$557$	461786.i	1.48844i	0.667936	+	0.744219i	$0.267178\pi$
$557$	461786.i	1.48844i	−0.667936	+	0.744219i	$0.732822\pi$
$558$	0	0
$559$	−194568.	−0.622656
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 177944.i	− 0.561393i	−0.959797	−	0.280697i	$-0.909435\pi$
$563$	− 177944.i	− 0.561393i	0.959797	−	0.280697i	$-0.0905654\pi$
$564$	0	0
$565$	−52922.0	−0.165783
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 5902.63i	− 0.0182314i	−0.999958	−	0.00911572i	$-0.997098\pi$
$569$	− 5902.63i	− 0.0182314i	0.999958	−	0.00911572i	$-0.00290166\pi$
$570$	0	0
$571$	−476692.	−1.46206	−0.731031	−	0.682344i	$-0.760961\pi$
$571$	−476692.	−1.46206	−0.731031	+	0.682344i	$0.760961\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	194442.i	0.588105i
$576$	0	0
$577$	209901.	0.630468	0.315234	−	0.949014i	$-0.397917\pi$
$577$	209901.	0.630468	0.315234	+	0.949014i	$0.397917\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	701385.i	2.07780i
$582$	0	0
$583$	−528605.	−1.55523
$584$	0	0
$585$	0	0
$586$	0	0
$587$	180770.i	0.524627i	0.964983	+	0.262313i	$0.0844854\pi$
$587$	180770.i	0.524627i	−0.964983	+	0.262313i	$0.915515\pi$
$588$	0	0
$589$	82937.2	0.239066
$590$	0	0
$591$	0	0
$592$	0	0
$593$	468097.i	1.33115i	0.746332	+	0.665574i	$0.231813\pi$
$593$	468097.i	1.33115i	−0.746332	+	0.665574i	$0.768187\pi$
$594$	0	0
$595$	−4839.60	−0.0136702
$596$	0	0
$597$	0	0
$598$	0	0
$599$	644967.i	1.79756i	0.438399	+	0.898780i	$0.355546\pi$
$599$	644967.i	1.79756i	−0.438399	+	0.898780i	$0.644454\pi$
$600$	0	0
$601$	255033.	0.706070	0.353035	−	0.935610i	$-0.385150\pi$
$601$	255033.	0.706070	0.353035	+	0.935610i	$0.385150\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	11509.8i	0.0314453i
$606$	0	0
$607$	352702.	0.957261	0.478631	−	0.878016i	$-0.341133\pi$
$607$	352702.	0.957261	0.478631	+	0.878016i	$0.341133\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 59690.9i	− 0.159892i
$612$	0	0
$613$	−37013.8	−0.0985015	−0.0492508	−	0.998786i	$-0.515683\pi$
$613$	−37013.8	−0.0985015	−0.0492508	+	0.998786i	$0.515683\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	570187.i	1.49778i	0.662696	+	0.748888i	$0.269412\pi$
$617$	570187.i	1.49778i	−0.662696	+	0.748888i	$0.730588\pi$
$618$	0	0
$619$	440814.	1.15047	0.575233	−	0.817990i	$-0.304911\pi$
$619$	440814.	1.15047	0.575233	+	0.817990i	$0.304911\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 30681.2i	− 0.0790489i
$624$	0	0
$625$	149578.	0.382920
$626$	0	0
$627$	0	0
$628$	0	0
$629$	13254.1i	0.0335003i
$630$	0	0
$631$	80926.3	0.203250	0.101625	−	0.994823i	$-0.467596\pi$
$631$	80926.3	0.203250	0.101625	+	0.994823i	$0.467596\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	159321.i	0.395118i
$636$	0	0
$637$	−32604.4	−0.0803522
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 152324.i	− 0.370726i	−0.982670	−	0.185363i	$-0.940654\pi$
$641$	− 152324.i	− 0.370726i	0.982670	−	0.185363i	$-0.0593461\pi$
$642$	0	0
$643$	−116874.	−0.282681	−0.141340	−	0.989961i	$-0.545141\pi$
$643$	−116874.	−0.282681	−0.141340	+	0.989961i	$0.545141\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	481407.i	1.15002i	0.818148	+	0.575008i	$0.195001\pi$
$647$	481407.i	1.15002i	−0.818148	+	0.575008i	$0.804999\pi$
$648$	0	0
$649$	−416553.	−0.988965
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 460831.i	− 1.08073i	−0.841432	−	0.540363i	$-0.818287\pi$
$653$	− 460831.i	− 1.08073i	0.841432	−	0.540363i	$-0.181713\pi$
$654$	0	0
$655$	40026.5	0.0932964
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 165076.i	− 0.380114i	−0.981773	−	0.190057i	$-0.939133\pi$
$659$	− 165076.i	− 0.380114i	0.981773	−	0.190057i	$-0.0608674\pi$
$660$	0	0
$661$	−281756.	−0.644868	−0.322434	−	0.946592i	$-0.604501\pi$
$661$	−281756.	−0.644868	−0.322434	+	0.946592i	$0.604501\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 33978.6i	− 0.0768355i
$666$	0	0
$667$	216236.	0.486046
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1876.67i	0.00416816i
$672$	0	0
$673$	−756775.	−1.67085	−0.835423	−	0.549608i	$-0.814777\pi$
$673$	−756775.	−1.67085	−0.835423	+	0.549608i	$0.814777\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 68188.8i	− 0.148777i	−0.997229	−	0.0743885i	$-0.976300\pi$
$677$	− 68188.8i	− 0.148777i	0.997229	−	0.0743885i	$-0.0237005\pi$
$678$	0	0
$679$	805667.	1.74749
$680$	0	0
$681$	0	0
$682$	0	0
$683$	31351.1i	0.0672066i	0.999435	+	0.0336033i	$0.0106983\pi$
$683$	31351.1i	0.0672066i	−0.999435	+	0.0336033i	$0.989302\pi$
$684$	0	0
$685$	45048.7	0.0960065
$686$	0	0
$687$	0	0
$688$	0	0
$689$	315588.i	0.664786i
$690$	0	0
$691$	−374884.	−0.785129	−0.392565	−	0.919724i	$-0.628412\pi$
$691$	−374884.	−0.785129	−0.392565	+	0.919724i	$0.628412\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	40965.1i	0.0848095i
$696$	0	0
$697$	−9711.63	−0.0199906
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 97448.5i	− 0.198308i	−0.995072	−	0.0991538i	$-0.968386\pi$
$701$	− 97448.5i	− 0.198308i	0.995072	−	0.0991538i	$-0.0316136\pi$
$702$	0	0
$703$	−93056.2	−0.188293
$704$	0	0
$705$	0	0
$706$	0	0
$707$	589238.i	1.17883i
$708$	0	0
$709$	−292525.	−0.581931	−0.290965	−	0.956734i	$-0.593976\pi$
$709$	−292525.	−0.581931	−0.290965	+	0.956734i	$0.593976\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	612796.i	1.20542i
$714$	0	0
$715$	109867.	0.214908
$716$	0	0
$717$	0	0
$718$	0	0
$719$	655579.i	1.26814i	0.773276	+	0.634070i	$0.218617\pi$
$719$	655579.i	1.26814i	−0.773276	+	0.634070i	$0.781383\pi$
$720$	0	0
$721$	189807.	0.365125
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 262844.i	− 0.500061i
$726$	0	0
$727$	−259261.	−0.490533	−0.245266	−	0.969456i	$-0.578875\pi$
$727$	−259261.	−0.490533	−0.245266	+	0.969456i	$0.578875\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 20105.8i	− 0.0376258i
$732$	0	0
$733$	82508.9	0.153565	0.0767826	−	0.997048i	$-0.475535\pi$
$733$	82508.9	0.153565	0.0767826	+	0.997048i	$0.475535\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	647650.i	1.19235i
$738$	0	0
$739$	730583.	1.33777	0.668884	−	0.743367i	$-0.266772\pi$
$739$	730583.	1.33777	0.668884	+	0.743367i	$0.266772\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	451784.i	0.818377i	0.912450	+	0.409189i	$0.134188\pi$
$743$	451784.i	0.818377i	−0.912450	+	0.409189i	$0.865812\pi$
$744$	0	0
$745$	−104998.	−0.189176
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 609052.i	− 1.08565i
$750$	0	0
$751$	−580162.	−1.02865	−0.514327	−	0.857594i	$-0.671958\pi$
$751$	−580162.	−1.02865	−0.514327	+	0.857594i	$0.671958\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 116832.i	− 0.204960i
$756$	0	0
$757$	1.00242e6	1.74928	0.874641	−	0.484771i	$-0.161097\pi$
$757$	1.00242e6	1.74928	0.874641	+	0.484771i	$0.161097\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	84717.1i	0.146286i	0.997321	+	0.0731429i	$0.0233029\pi$
$761$	84717.1i	0.146286i	−0.997321	+	0.0731429i	$0.976697\pi$
$762$	0	0
$763$	−254667.	−0.437445
$764$	0	0
$765$	0	0
$766$	0	0
$767$	248691.i	0.422736i
$768$	0	0
$769$	−683525.	−1.15585	−0.577926	−	0.816089i	$-0.696138\pi$
$769$	−683525.	−1.15585	−0.577926	+	0.816089i	$0.696138\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	235888.i	0.394772i	0.980326	+	0.197386i	$0.0632453\pi$
$773$	235888.i	0.394772i	−0.980326	+	0.197386i	$0.936755\pi$
$774$	0	0
$775$	744879.	1.24017
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 68184.8i	− 0.112360i
$780$	0	0
$781$	240464.	0.394229
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 66537.8i	− 0.107977i
$786$	0	0
$787$	457456.	0.738585	0.369292	−	0.929313i	$-0.379600\pi$
$787$	457456.	0.738585	0.369292	+	0.929313i	$0.379600\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 239269.i	− 0.382414i
$792$	0	0
$793$	1120.41	0.00178169
$794$	0	0
$795$	0	0
$796$	0	0
$797$	170922.i	0.269080i	0.990908	+	0.134540i	$0.0429556\pi$
$797$	170922.i	0.269080i	−0.990908	+	0.134540i	$0.957044\pi$
$798$	0	0
$799$	6168.17	0.00966191
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 118667.i	− 0.184034i
$804$	0	0
$805$	251057.	0.387418
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 252280.i	− 0.385465i	−0.981251	−	0.192733i	$-0.938265\pi$
$809$	− 252280.i	− 0.385465i	0.981251	−	0.192733i	$-0.0617350\pi$
$810$	0	0
$811$	−166784.	−0.253579	−0.126789	−	0.991930i	$-0.540467\pi$
$811$	−166784.	−0.253579	−0.126789	+	0.991930i	$0.540467\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 25375.3i	− 0.0382029i
$816$	0	0
$817$	141161.	0.211481
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 272312.i	− 0.403999i	−0.979386	−	0.201999i	$-0.935256\pi$
$821$	− 272312.i	− 0.403999i	0.979386	−	0.201999i	$-0.0647439\pi$
$822$	0	0
$823$	−964628.	−1.42416	−0.712082	−	0.702096i	$-0.752248\pi$
$823$	−964628.	−1.42416	−0.712082	+	0.702096i	$0.752248\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	430905.i	0.630043i	0.949085	+	0.315021i	$0.102012\pi$
$827$	430905.i	0.630043i	−0.949085	+	0.315021i	$0.897988\pi$
$828$	0	0
$829$	462870.	0.673520	0.336760	−	0.941591i	$-0.390669\pi$
$829$	462870.	0.673520	0.336760	+	0.941591i	$0.390669\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 3369.19i	− 0.00485551i
$834$	0	0
$835$	−436209.	−0.625635
$836$	0	0
$837$	0	0
$838$	0	0
$839$	804698.i	1.14317i	0.820545	+	0.571583i	$0.193670\pi$
$839$	804698.i	1.14317i	−0.820545	+	0.571583i	$0.806330\pi$
$840$	0	0
$841$	414975.	0.586719
$842$	0	0
$843$	0	0
$844$	0	0
$845$	270942.i	0.379457i
$846$	0	0
$847$	−52037.6	−0.0725354
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 687563.i	− 0.949409i
$852$	0	0
$853$	211580.	0.290788	0.145394	−	0.989374i	$-0.453555\pi$
$853$	211580.	0.290788	0.145394	+	0.989374i	$0.453555\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 283303.i	− 0.385736i	−0.981225	−	0.192868i	$-0.938221\pi$
$857$	− 283303.i	− 0.385736i	0.981225	−	0.192868i	$-0.0617788\pi$
$858$	0	0
$859$	30997.2	0.0420083	0.0210042	−	0.999779i	$-0.493314\pi$
$859$	30997.2	0.0420083	0.0210042	+	0.999779i	$0.493314\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1.35611e6i	1.82085i	0.413677	+	0.910424i	$0.364244\pi$
$863$	1.35611e6i	1.82085i	−0.413677	+	0.910424i	$0.635756\pi$
$864$	0	0
$865$	−572482.	−0.765120
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 29740.7i	− 0.0393833i
$870$	0	0
$871$	386661.	0.509675
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 697491.i	− 0.911009i
$876$	0	0
$877$	−452930.	−0.588887	−0.294443	−	0.955669i	$-0.595134\pi$
$877$	−452930.	−0.588887	−0.294443	+	0.955669i	$0.595134\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 1.13788e6i	− 1.46603i	−0.680211	−	0.733016i	$-0.738112\pi$
$881$	− 1.13788e6i	− 1.46603i	0.680211	−	0.733016i	$-0.261888\pi$
$882$	0	0
$883$	−970367.	−1.24456	−0.622279	−	0.782796i	$-0.713793\pi$
$883$	−970367.	−1.24456	−0.622279	+	0.782796i	$0.713793\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1.29729e6i	1.64889i	0.565944	+	0.824444i	$0.308512\pi$
$887$	1.29729e6i	1.64889i	−0.565944	+	0.824444i	$0.691488\pi$
$888$	0	0
$889$	−720318.	−0.911425
$890$	0	0
$891$	0	0
$892$	0	0
$893$	43306.4i	0.0543061i
$894$	0	0
$895$	−562261.	−0.701926
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 828370.i	− 1.02496i
$900$	0	0
$901$	−32611.4	−0.0401716
$902$	0	0
$903$	0	0
$904$	0	0
$905$	539275.i	0.658436i
$906$	0	0
$907$	−153968.	−0.187161	−0.0935807	−	0.995612i	$-0.529831\pi$
$907$	−153968.	−0.187161	−0.0935807	+	0.995612i	$0.529831\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1.07095e6i	1.29042i	0.764006	+	0.645209i	$0.223230\pi$
$911$	1.07095e6i	1.29042i	−0.764006	+	0.645209i	$0.776770\pi$
$912$	0	0
$913$	1.64536e6	1.97387
$914$	0	0
$915$	0	0
$916$	0	0
$917$	180966.i	0.215208i
$918$	0	0
$919$	−747688.	−0.885298	−0.442649	−	0.896695i	$-0.645961\pi$
$919$	−747688.	−0.885298	−0.442649	+	0.896695i	$0.645961\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 143562.i	− 0.168514i
$924$	0	0
$925$	−835761.	−0.976784
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1.48404e6i	1.71955i	0.510675	+	0.859774i	$0.329396\pi$
$929$	1.48404e6i	1.71955i	−0.510675	+	0.859774i	$0.670604\pi$
$930$	0	0
$931$	23654.9	0.0272911
$932$	0	0
$933$	0	0
$934$	0	0
$935$	11353.1i	0.0129865i
$936$	0	0
$937$	−650560.	−0.740983	−0.370491	−	0.928836i	$-0.620811\pi$
$937$	−650560.	−0.740983	−0.370491	+	0.928836i	$0.620811\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	406958.i	0.459590i	0.973239	+	0.229795i	$0.0738056\pi$
$941$	406958.i	0.459590i	−0.973239	+	0.229795i	$0.926194\pi$
$942$	0	0
$943$	503795.	0.566540
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 830009.i	− 0.925514i	−0.886485	−	0.462757i	$-0.846860\pi$
$947$	− 830009.i	− 0.925514i	0.886485	−	0.462757i	$-0.153140\pi$
$948$	0	0
$949$	−70846.6	−0.0786659
$950$	0	0
$951$	0	0
$952$	0	0
$953$	927863.i	1.02164i	0.859688	+	0.510820i	$0.170658\pi$
$953$	927863.i	1.02164i	−0.859688	+	0.510820i	$0.829342\pi$
$954$	0	0
$955$	−550496.	−0.603597
$956$	0	0
$957$	0	0
$958$	0	0
$959$	203672.i	0.221460i
$960$	0	0
$961$	1.42401e6	1.54194
$962$	0	0
$963$	0	0
$964$	0	0
$965$	320589.i	0.344266i
$966$	0	0
$967$	1.11248e6	1.18970	0.594851	−	0.803836i	$-0.297211\pi$
$967$	1.11248e6	1.18970	0.594851	+	0.803836i	$0.297211\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	68160.6i	0.0722927i	0.999347	+	0.0361464i	$0.0115083\pi$
$971$	68160.6i	0.0722927i	−0.999347	+	0.0361464i	$0.988492\pi$
$972$	0	0
$973$	−185210.	−0.195631
$974$	0	0
$975$	0	0
$976$	0	0
$977$	791113.i	0.828799i	0.910095	+	0.414399i	$0.136008\pi$
$977$	791113.i	0.828799i	−0.910095	+	0.414399i	$0.863992\pi$
$978$	0	0
$979$	−71974.0	−0.0750949
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1.09523e6i	1.13344i	0.823912	+	0.566718i	$0.191787\pi$
$983$	1.09523e6i	1.13344i	−0.823912	+	0.566718i	$0.808213\pi$
$984$	0	0
$985$	−761950.	−0.785334
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1.04300e6i	1.06633i
$990$	0	0
$991$	−278492.	−0.283573	−0.141787	−	0.989897i	$-0.545285\pi$
$991$	−278492.	−0.283573	−0.141787	+	0.989897i	$0.545285\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 304488.i	− 0.307556i
$996$	0	0
$997$	1.33145e6	1.33947	0.669736	−	0.742599i	$-0.266407\pi$
$997$	1.33145e6	1.33947	0.669736	+	0.742599i	$0.266407\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1296.5.e.c.161.3		6
3.2	odd	2	inner	1296.5.e.c.161.4		6
4.3	odd	2		81.5.b.a.80.1		6
9.2	odd	6		432.5.q.a.17.2		6
9.4	even	3		432.5.q.a.305.2		6
9.5	odd	6		144.5.q.a.65.2		6
9.7	even	3		144.5.q.a.113.2		6
12.11	even	2		81.5.b.a.80.6		6
36.7	odd	6		9.5.d.a.5.1	yes	6
36.11	even	6		27.5.d.a.17.3		6
36.23	even	6		9.5.d.a.2.1	✓	6
36.31	odd	6		27.5.d.a.8.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9.5.d.a.2.1	✓	6	36.23	even	6
9.5.d.a.5.1	yes	6	36.7	odd	6
27.5.d.a.8.3		6	36.31	odd	6
27.5.d.a.17.3		6	36.11	even	6
81.5.b.a.80.1		6	4.3	odd	2
81.5.b.a.80.6		6	12.11	even	2
144.5.q.a.65.2		6	9.5	odd	6
144.5.q.a.113.2		6	9.7	even	3
432.5.q.a.17.2		6	9.2	odd	6
432.5.q.a.305.2		6	9.4	even	3
1296.5.e.c.161.3		6	1.1	even	1	trivial
1296.5.e.c.161.4		6	3.2	odd	2	inner

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)

\(f(q)\)	\(=\)	\(q-11.7830i q^{5} +53.2728 q^{7} -124.971i q^{11} -74.6104 q^{13} -7.70989i q^{17} +54.1307 q^{19} +399.954i q^{23} +486.161 q^{25} -540.653i q^{29} +1532.17 q^{31} -627.714i q^{35} -1719.10 q^{37} -1259.63i q^{41} +2607.79 q^{43} +800.034i q^{47} +436.996 q^{49} -4229.81i q^{53} -1472.54 q^{55} -3333.19i q^{59} -15.0169 q^{61} +879.134i q^{65} -5182.39 q^{67} +1924.16i q^{71} +949.554 q^{73} -6657.57i q^{77} +237.980 q^{79} +13165.9i q^{83} -90.8456 q^{85} -575.925i q^{89} -3974.71 q^{91} -637.822i q^{95} +15123.4 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 24 q^{7} + 12 q^{13} + 258 q^{19} + 546 q^{25} + 2580 q^{31} + 12 q^{37} - 570 q^{43} + 3726 q^{49} - 2016 q^{55} - 7260 q^{61} - 10110 q^{67} - 14622 q^{73} + 9528 q^{79} - 24732 q^{85} - 34836 q^{91}+ \cdots + 57918 q^{97}+O(q^{100}) \) 6 * q + 24 * q^7 + 12 * q^13 + 258 * q^19 + 546 * q^25 + 2580 * q^31 + 12 * q^37 - 570 * q^43 + 3726 * q^49 - 2016 * q^55 - 7260 * q^61 - 10110 * q^67 - 14622 * q^73 + 9528 * q^79 - 24732 * q^85 - 34836 * q^91 + 57918 * q^97

Introduction

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