LMFDB - Embedded newform 1296.5.e.c.161.1

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			161.1
Root			$-0.102534 - 0.177594i$ of defining polynomial
Character	$\chi$	$=$	1296.161
Dual form			1296.5.e.c.161.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-34.7338i q^{5} +31.2109 q^{7} +57.7314i q^{11} -73.2956 q^{13} -386.985i q^{17} -115.791 q^{19} +548.312i q^{23} -581.437 q^{25} +785.291i q^{29} -544.734 q^{31} -1084.07i q^{35} +898.827 q^{37} +2588.85i q^{41} -2000.11 q^{43} -811.345i q^{47} -1426.88 q^{49} +2221.00i q^{53} +2005.23 q^{55} -1512.26i q^{59} -1902.56 q^{61} +2545.83i q^{65} -4507.09 q^{67} -3993.54i q^{71} -3436.70 q^{73} +1801.85i q^{77} -1202.78 q^{79} +9256.34i q^{83} -13441.4 q^{85} -8929.99i q^{89} -2287.62 q^{91} +4021.86i q^{95} +6670.29 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	− 34.7338i	− 1.38935i	−0.719323	−	0.694676i	$-0.755548\pi$
$5$	− 34.7338i	− 1.38935i	0.719323	−	0.694676i	$-0.244452\pi$
$6$	0	0
$7$	31.2109	0.636956	0.318478	−	0.947930i	$-0.396828\pi$
$7$	31.2109	0.636956	0.318478	+	0.947930i	$0.396828\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	57.7314i	0.477119i	0.971128	+	0.238559i	$0.0766752\pi$
$11$	57.7314i	0.477119i	−0.971128	+	0.238559i	$0.923325\pi$
$12$	0	0
$13$	−73.2956	−0.433702	−0.216851	−	0.976205i	$-0.569578\pi$
$13$	−73.2956	−0.433702	−0.216851	+	0.976205i	$0.569578\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 386.985i	− 1.33905i	−0.742791	−	0.669524i	$-0.766498\pi$
$17$	− 386.985i	− 1.33905i	0.742791	−	0.669524i	$-0.233502\pi$
$18$	0	0
$19$	−115.791	−0.320750	−0.160375	−	0.987056i	$-0.551270\pi$
$19$	−115.791	−0.320750	−0.160375	+	0.987056i	$0.551270\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	548.312i	1.03651i	0.855227	+	0.518253i	$0.173417\pi$
$23$	548.312i	1.03651i	−0.855227	+	0.518253i	$0.826583\pi$
$24$	0	0
$25$	−581.437	−0.930299
$26$	0	0
$27$	0	0
$28$	0	0
$29$	785.291i	0.933758i	0.884321	+	0.466879i	$0.154622\pi$
$29$	785.291i	0.933758i	−0.884321	+	0.466879i	$0.845378\pi$
$30$	0	0
$31$	−544.734	−0.566840	−0.283420	−	0.958996i	$-0.591469\pi$
$31$	−544.734	−0.566840	−0.283420	+	0.958996i	$0.591469\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 1084.07i	− 0.884957i
$36$	0	0
$37$	898.827	0.656557	0.328279	−	0.944581i	$-0.393532\pi$
$37$	898.827	0.656557	0.328279	+	0.944581i	$0.393532\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2588.85i	1.54007i	0.638003	+	0.770034i	$0.279761\pi$
$41$	2588.85i	1.54007i	−0.638003	+	0.770034i	$0.720239\pi$
$42$	0	0
$43$	−2000.11	−1.08172	−0.540862	−	0.841111i	$-0.681902\pi$
$43$	−2000.11	−1.08172	−0.540862	+	0.841111i	$0.681902\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 811.345i	− 0.367291i	−0.982993	−	0.183645i	$-0.941210\pi$
$47$	− 811.345i	− 0.367291i	0.982993	−	0.183645i	$-0.0587898\pi$
$48$	0	0
$49$	−1426.88	−0.594287
$50$	0	0
$51$	0	0
$52$	0	0
$53$	2221.00i	0.790672i	0.918537	+	0.395336i	$0.129372\pi$
$53$	2221.00i	0.790672i	−0.918537	+	0.395336i	$0.870628\pi$
$54$	0	0
$55$	2005.23	0.662886
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 1512.26i	− 0.434432i	−0.976124	−	0.217216i	$-0.930302\pi$
$59$	− 1512.26i	− 0.434432i	0.976124	−	0.217216i	$-0.0696976\pi$
$60$	0	0
$61$	−1902.56	−0.511304	−0.255652	−	0.966769i	$-0.582290\pi$
$61$	−1902.56	−0.511304	−0.255652	+	0.966769i	$0.582290\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2545.83i	0.602564i
$66$	0	0
$67$	−4507.09	−1.00403	−0.502015	−	0.864859i	$-0.667408\pi$
$67$	−4507.09	−1.00403	−0.502015	+	0.864859i	$0.667408\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 3993.54i	− 0.792213i	−0.918205	−	0.396106i	$-0.870361\pi$
$71$	− 3993.54i	− 0.792213i	0.918205	−	0.396106i	$-0.129639\pi$
$72$	0	0
$73$	−3436.70	−0.644905	−0.322452	−	0.946586i	$-0.604507\pi$
$73$	−3436.70	−0.644905	−0.322452	+	0.946586i	$0.604507\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1801.85i	0.303904i
$78$	0	0
$79$	−1202.78	−0.192722	−0.0963608	−	0.995346i	$-0.530720\pi$
$79$	−1202.78	−0.192722	−0.0963608	+	0.995346i	$0.530720\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	9256.34i	1.34364i	0.740714	+	0.671820i	$0.234487\pi$
$83$	9256.34i	1.34364i	−0.740714	+	0.671820i	$0.765513\pi$
$84$	0	0
$85$	−13441.4	−1.86041
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 8929.99i	− 1.12738i	−0.825986	−	0.563691i	$-0.809381\pi$
$89$	− 8929.99i	− 1.12738i	0.825986	−	0.563691i	$-0.190619\pi$
$90$	0	0
$91$	−2287.62	−0.276249
$92$	0	0
$93$	0	0
$94$	0	0
$95$	4021.86i	0.445635i
$96$	0	0
$97$	6670.29	0.708926	0.354463	−	0.935070i	$-0.384664\pi$
$97$	6670.29	0.708926	0.354463	+	0.935070i	$0.384664\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	9150.06i	0.896977i	0.893789	+	0.448489i	$0.148038\pi$
$101$	9150.06i	0.896977i	−0.893789	+	0.448489i	$0.851962\pi$
$102$	0	0
$103$	−15312.4	−1.44334	−0.721670	−	0.692237i	$-0.756625\pi$
$103$	−15312.4	−1.44334	−0.721670	+	0.692237i	$0.756625\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 6099.28i	− 0.532735i	−0.963872	−	0.266367i	$-0.914177\pi$
$107$	− 6099.28i	− 0.532735i	0.963872	−	0.266367i	$-0.0858234\pi$
$108$	0	0
$109$	15169.5	1.27679	0.638393	−	0.769710i	$-0.279599\pi$
$109$	15169.5	1.27679	0.638393	+	0.769710i	$0.279599\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 1373.06i	− 0.107531i	−0.998554	−	0.0537655i	$-0.982878\pi$
$113$	− 1373.06i	− 0.107531i	0.998554	−	0.0537655i	$-0.0171224\pi$
$114$	0	0
$115$	19045.0	1.44007
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 12078.1i	− 0.852914i
$120$	0	0
$121$	11308.1	0.772358
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 1513.10i	− 0.0968386i
$126$	0	0
$127$	19152.4	1.18745	0.593726	−	0.804667i	$-0.297656\pi$
$127$	19152.4	1.18745	0.593726	+	0.804667i	$0.297656\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	3020.00i	0.175980i	0.996121	+	0.0879902i	$0.0280444\pi$
$131$	3020.00i	0.175980i	−0.996121	+	0.0879902i	$0.971956\pi$
$132$	0	0
$133$	−3613.93	−0.204304
$134$	0	0
$135$	0	0
$136$	0	0
$137$	10147.9i	0.540671i	0.962766	+	0.270336i	$0.0871346\pi$
$137$	10147.9i	0.540671i	−0.962766	+	0.270336i	$0.912865\pi$
$138$	0	0
$139$	35126.4	1.81804	0.909021	−	0.416751i	$-0.136831\pi$
$139$	35126.4	1.81804	0.909021	+	0.416751i	$0.136831\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 4231.45i	− 0.206927i
$144$	0	0
$145$	27276.1	1.29732
$146$	0	0
$147$	0	0
$148$	0	0
$149$	37382.0i	1.68380i	0.539636	+	0.841899i	$0.318562\pi$
$149$	37382.0i	1.68380i	−0.539636	+	0.841899i	$0.681438\pi$
$150$	0	0
$151$	−33270.0	−1.45915	−0.729573	−	0.683903i	$-0.760281\pi$
$151$	−33270.0	−1.45915	−0.729573	+	0.683903i	$0.760281\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	18920.7i	0.787541i
$156$	0	0
$157$	8080.33	0.327816	0.163908	−	0.986476i	$-0.447590\pi$
$157$	8080.33	0.327816	0.163908	+	0.986476i	$0.447590\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	17113.3i	0.660209i
$162$	0	0
$163$	25427.1	0.957022	0.478511	−	0.878081i	$-0.341177\pi$
$163$	25427.1	0.957022	0.478511	+	0.878081i	$0.341177\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	35052.8i	1.25687i	0.777863	+	0.628434i	$0.216304\pi$
$167$	35052.8i	1.25687i	−0.777863	+	0.628434i	$0.783696\pi$
$168$	0	0
$169$	−23188.8	−0.811903
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 14700.5i	− 0.491177i	−0.969374	−	0.245589i	$-0.921019\pi$
$173$	− 14700.5i	− 0.491177i	0.969374	−	0.245589i	$-0.0789813\pi$
$174$	0	0
$175$	−18147.2	−0.592560
$176$	0	0
$177$	0	0
$178$	0	0
$179$	37052.5i	1.15641i	0.815892	+	0.578205i	$0.196247\pi$
$179$	37052.5i	1.15641i	−0.815892	+	0.578205i	$0.803753\pi$
$180$	0	0
$181$	−39664.7	−1.21073	−0.605365	−	0.795948i	$-0.706973\pi$
$181$	−39664.7	−1.21073	−0.605365	+	0.795948i	$0.706973\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 31219.7i	− 0.912189i
$186$	0	0
$187$	22341.2	0.638885
$188$	0	0
$189$	0	0
$190$	0	0
$191$	57637.0i	1.57992i	0.613160	+	0.789959i	$0.289898\pi$
$191$	57637.0i	1.57992i	−0.613160	+	0.789959i	$0.710102\pi$
$192$	0	0
$193$	4179.63	0.112208	0.0561039	−	0.998425i	$-0.482132\pi$
$193$	4179.63	0.112208	0.0561039	+	0.998425i	$0.482132\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 22191.5i	− 0.571812i	−0.958258	−	0.285906i	$-0.907705\pi$
$197$	− 22191.5i	− 0.571812i	0.958258	−	0.285906i	$-0.0922945\pi$
$198$	0	0
$199$	−50608.7	−1.27797	−0.638983	−	0.769221i	$-0.720645\pi$
$199$	−50608.7	−1.27797	−0.638983	+	0.769221i	$0.720645\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	24509.6i	0.594763i
$204$	0	0
$205$	89920.7	2.13970
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 6684.77i	− 0.153036i
$210$	0	0
$211$	8918.05	0.200311	0.100156	−	0.994972i	$-0.468066\pi$
$211$	8918.05	0.200311	0.100156	+	0.994972i	$0.468066\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	69471.4i	1.50290i
$216$	0	0
$217$	−17001.6	−0.361052
$218$	0	0
$219$	0	0
$220$	0	0
$221$	28364.3i	0.580747i
$222$	0	0
$223$	8497.53	0.170877	0.0854384	−	0.996343i	$-0.472771\pi$
$223$	8497.53	0.170877	0.0854384	+	0.996343i	$0.472771\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	19102.8i	0.370719i	0.982671	+	0.185360i	$0.0593450\pi$
$227$	19102.8i	0.370719i	−0.982671	+	0.185360i	$0.940655\pi$
$228$	0	0
$229$	−10723.0	−0.204478	−0.102239	−	0.994760i	$-0.532601\pi$
$229$	−10723.0	−0.204478	−0.102239	+	0.994760i	$0.532601\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	102200.i	1.88251i	0.337694	+	0.941256i	$0.390353\pi$
$233$	102200.i	1.88251i	−0.337694	+	0.941256i	$0.609647\pi$
$234$	0	0
$235$	−28181.1	−0.510296
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 15520.2i	− 0.271707i	−0.990729	−	0.135854i	$-0.956622\pi$
$239$	− 15520.2i	− 0.271707i	0.990729	−	0.135854i	$-0.0433777\pi$
$240$	0	0
$241$	−71957.8	−1.23892	−0.619461	−	0.785028i	$-0.712649\pi$
$241$	−71957.8	−1.23892	−0.619461	+	0.785028i	$0.712649\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	49561.1i	0.825674i
$246$	0	0
$247$	8486.96	0.139110
$248$	0	0
$249$	0	0
$250$	0	0
$251$	41487.8i	0.658526i	0.944238	+	0.329263i	$0.106800\pi$
$251$	41487.8i	0.658526i	−0.944238	+	0.329263i	$0.893200\pi$
$252$	0	0
$253$	−31654.8	−0.494537
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 72529.3i	− 1.09811i	−0.835785	−	0.549057i	$-0.814987\pi$
$257$	− 72529.3i	− 1.09811i	0.835785	−	0.549057i	$-0.185013\pi$
$258$	0	0
$259$	28053.1	0.418198
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 85890.2i	− 1.24174i	−0.783912	−	0.620872i	$-0.786779\pi$
$263$	− 85890.2i	− 1.24174i	0.783912	−	0.620872i	$-0.213221\pi$
$264$	0	0
$265$	77143.7	1.09852
$266$	0	0
$267$	0	0
$268$	0	0
$269$	88967.6i	1.22950i	0.788724	+	0.614748i	$0.210742\pi$
$269$	88967.6i	1.22950i	−0.788724	+	0.614748i	$0.789258\pi$
$270$	0	0
$271$	−96541.6	−1.31455	−0.657273	−	0.753652i	$-0.728290\pi$
$271$	−96541.6	−1.31455	−0.657273	+	0.753652i	$0.728290\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 33567.2i	− 0.443863i
$276$	0	0
$277$	−23541.6	−0.306815	−0.153407	−	0.988163i	$-0.549025\pi$
$277$	−23541.6	−0.306815	−0.153407	+	0.988163i	$0.549025\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	58409.9i	0.739731i	0.929085	+	0.369865i	$0.120596\pi$
$281$	58409.9i	0.739731i	−0.929085	+	0.369865i	$0.879404\pi$
$282$	0	0
$283$	76117.3	0.950408	0.475204	−	0.879876i	$-0.342374\pi$
$283$	76117.3	0.950408	0.475204	+	0.879876i	$0.342374\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	80800.3i	0.980956i
$288$	0	0
$289$	−66236.1	−0.793047
$290$	0	0
$291$	0	0
$292$	0	0
$293$	139911.i	1.62973i	0.579649	+	0.814866i	$0.303190\pi$
$293$	139911.i	1.62973i	−0.579649	+	0.814866i	$0.696810\pi$
$294$	0	0
$295$	−52526.4	−0.603579
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 40188.8i	− 0.449534i
$300$	0	0
$301$	−62425.1	−0.689011
$302$	0	0
$303$	0	0
$304$	0	0
$305$	66083.2i	0.710381i
$306$	0	0
$307$	81796.8	0.867880	0.433940	−	0.900942i	$-0.357123\pi$
$307$	81796.8	0.867880	0.433940	+	0.900942i	$0.357123\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	1702.71i	0.0176043i	0.999961	+	0.00880216i	$0.00280185\pi$
$311$	1702.71i	0.0176043i	−0.999961	+	0.00880216i	$0.997198\pi$
$312$	0	0
$313$	−69960.1	−0.714105	−0.357052	−	0.934084i	$-0.616218\pi$
$313$	−69960.1	−0.714105	−0.357052	+	0.934084i	$0.616218\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 116629.i	− 1.16062i	−0.814397	−	0.580309i	$-0.802932\pi$
$317$	− 116629.i	− 1.16062i	0.814397	−	0.580309i	$-0.197068\pi$
$318$	0	0
$319$	−45335.9	−0.445514
$320$	0	0
$321$	0	0
$322$	0	0
$323$	44809.3i	0.429500i
$324$	0	0
$325$	42616.8	0.403472
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 25322.8i	− 0.233948i
$330$	0	0
$331$	−100836.	−0.920366	−0.460183	−	0.887824i	$-0.652216\pi$
$331$	−100836.	−0.920366	−0.460183	+	0.887824i	$0.652216\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	156549.i	1.39495i
$336$	0	0
$337$	40094.5	0.353041	0.176520	−	0.984297i	$-0.443516\pi$
$337$	40094.5	0.353041	0.176520	+	0.984297i	$0.443516\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 31448.2i	− 0.270450i
$342$	0	0
$343$	−119471.	−1.01549
$344$	0	0
$345$	0	0
$346$	0	0
$347$	226362.i	1.87994i	0.341254	+	0.939971i	$0.389148\pi$
$347$	226362.i	1.87994i	−0.341254	+	0.939971i	$0.610852\pi$
$348$	0	0
$349$	79598.3	0.653511	0.326755	−	0.945109i	$-0.394045\pi$
$349$	79598.3	0.653511	0.326755	+	0.945109i	$0.394045\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	139919.i	1.12286i	0.827523	+	0.561431i	$0.189749\pi$
$353$	139919.i	1.12286i	−0.827523	+	0.561431i	$0.810251\pi$
$354$	0	0
$355$	−138711.	−1.10066
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 211616.i	− 1.64195i	−0.570963	−	0.820976i	$-0.693430\pi$
$359$	− 211616.i	− 1.64195i	0.570963	−	0.820976i	$-0.306570\pi$
$360$	0	0
$361$	−116913.	−0.897119
$362$	0	0
$363$	0	0
$364$	0	0
$365$	119370.i	0.896000i
$366$	0	0
$367$	31993.6	0.237537	0.118769	−	0.992922i	$-0.462105\pi$
$367$	31993.6	0.237537	0.118769	+	0.992922i	$0.462105\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	69319.2i	0.503623i
$372$	0	0
$373$	−123148.	−0.885133	−0.442567	−	0.896736i	$-0.645932\pi$
$373$	−123148.	−0.885133	−0.442567	+	0.896736i	$0.645932\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 57558.3i	− 0.404972i
$378$	0	0
$379$	116524.	0.811218	0.405609	−	0.914047i	$-0.367059\pi$
$379$	116524.	0.811218	0.405609	+	0.914047i	$0.367059\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 181112.i	− 1.23467i	−0.786701	−	0.617334i	$-0.788213\pi$
$383$	− 181112.i	− 1.23467i	0.786701	−	0.617334i	$-0.211787\pi$
$384$	0	0
$385$	62585.0	0.422229
$386$	0	0
$387$	0	0
$388$	0	0
$389$	50827.4i	0.335892i	0.985796	+	0.167946i	$0.0537134\pi$
$389$	50827.4i	0.335892i	−0.985796	+	0.167946i	$0.946287\pi$
$390$	0	0
$391$	212188.	1.38793
$392$	0	0
$393$	0	0
$394$	0	0
$395$	41777.0i	0.267758i
$396$	0	0
$397$	228710.	1.45112	0.725561	−	0.688158i	$-0.241580\pi$
$397$	228710.	1.45112	0.725561	+	0.688158i	$0.241580\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	188930.i	1.17493i	0.809250	+	0.587465i	$0.199874\pi$
$401$	188930.i	1.17493i	−0.809250	+	0.587465i	$0.800126\pi$
$402$	0	0
$403$	39926.5	0.245840
$404$	0	0
$405$	0	0
$406$	0	0
$407$	51890.5i	0.313256i
$408$	0	0
$409$	277427.	1.65845	0.829223	−	0.558918i	$-0.188783\pi$
$409$	277427.	1.65845	0.829223	+	0.558918i	$0.188783\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 47198.8i	− 0.276714i
$414$	0	0
$415$	321508.	1.86679
$416$	0	0
$417$	0	0
$418$	0	0
$419$	102422.i	0.583397i	0.956510	+	0.291699i	$0.0942205\pi$
$419$	102422.i	0.583397i	−0.956510	+	0.291699i	$0.905780\pi$
$420$	0	0
$421$	47135.8	0.265942	0.132971	−	0.991120i	$-0.457548\pi$
$421$	47135.8	0.265942	0.132971	+	0.991120i	$0.457548\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	225007.i	1.24571i
$426$	0	0
$427$	−59380.6	−0.325678
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 45556.1i	− 0.245240i	−0.992454	−	0.122620i	$-0.960870\pi$
$431$	− 45556.1i	− 0.245240i	0.992454	−	0.122620i	$-0.0391297\pi$
$432$	0	0
$433$	209599.	1.11793	0.558965	−	0.829192i	$-0.311199\pi$
$433$	209599.	1.11793	0.558965	+	0.829192i	$0.311199\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 63489.5i	− 0.332460i
$438$	0	0
$439$	183684.	0.953110	0.476555	−	0.879145i	$-0.341885\pi$
$439$	183684.	0.953110	0.476555	+	0.879145i	$0.341885\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 115943.i	− 0.590795i	−0.955374	−	0.295398i	$-0.904548\pi$
$443$	− 115943.i	− 0.590795i	0.955374	−	0.295398i	$-0.0954521\pi$
$444$	0	0
$445$	−310172.	−1.56633
$446$	0	0
$447$	0	0
$448$	0	0
$449$	328940.i	1.63164i	0.578305	+	0.815820i	$0.303714\pi$
$449$	328940.i	1.63164i	−0.578305	+	0.815820i	$0.696286\pi$
$450$	0	0
$451$	−149458.	−0.734795
$452$	0	0
$453$	0	0
$454$	0	0
$455$	79457.6i	0.383807i
$456$	0	0
$457$	−212737.	−1.01862	−0.509308	−	0.860584i	$-0.670099\pi$
$457$	−212737.	−1.01862	−0.509308	+	0.860584i	$0.670099\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	48448.1i	0.227968i	0.993483	+	0.113984i	$0.0363613\pi$
$461$	48448.1i	0.227968i	−0.993483	+	0.113984i	$0.963639\pi$
$462$	0	0
$463$	−169101.	−0.788832	−0.394416	−	0.918932i	$-0.629053\pi$
$463$	−169101.	−0.788832	−0.394416	+	0.918932i	$0.629053\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	54646.4i	0.250569i	0.992121	+	0.125285i	$0.0399844\pi$
$467$	54646.4i	0.250569i	−0.992121	+	0.125285i	$0.960016\pi$
$468$	0	0
$469$	−140670.	−0.639524
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 115469.i	− 0.516111i
$474$	0	0
$475$	67325.1	0.298394
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 138577.i	− 0.603978i	−0.953311	−	0.301989i	$-0.902349\pi$
$479$	− 138577.i	− 0.603978i	0.953311	−	0.301989i	$-0.0976506\pi$
$480$	0	0
$481$	−65880.0	−0.284750
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 231684.i	− 0.984948i
$486$	0	0
$487$	−23464.1	−0.0989340	−0.0494670	−	0.998776i	$-0.515752\pi$
$487$	−23464.1	−0.0989340	−0.0494670	+	0.998776i	$0.515752\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 74326.7i	− 0.308306i	−0.988047	−	0.154153i	$-0.950735\pi$
$491$	− 74326.7i	− 0.308306i	0.988047	−	0.154153i	$-0.0492649\pi$
$492$	0	0
$493$	303895.	1.25035
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 124642.i	− 0.504605i
$498$	0	0
$499$	18972.0	0.0761925	0.0380963	−	0.999274i	$-0.487871\pi$
$499$	18972.0	0.0761925	0.0380963	+	0.999274i	$0.487871\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	117856.i	0.465818i	0.972498	+	0.232909i	$0.0748245\pi$
$503$	117856.i	0.465818i	−0.972498	+	0.232909i	$0.925175\pi$
$504$	0	0
$505$	317816.	1.24622
$506$	0	0
$507$	0	0
$508$	0	0
$509$	183460.i	0.708120i	0.935223	+	0.354060i	$0.115199\pi$
$509$	183460.i	0.708120i	−0.935223	+	0.354060i	$0.884801\pi$
$510$	0	0
$511$	−107262.	−0.410776
$512$	0	0
$513$	0	0
$514$	0	0
$515$	531858.i	2.00531i
$516$	0	0
$517$	46840.1	0.175241
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 409498.i	− 1.50861i	−0.656526	−	0.754303i	$-0.727975\pi$
$521$	− 409498.i	− 1.50861i	0.656526	−	0.754303i	$-0.272025\pi$
$522$	0	0
$523$	211852.	0.774513	0.387256	−	0.921972i	$-0.373423\pi$
$523$	211852.	0.774513	0.387256	+	0.921972i	$0.373423\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	210804.i	0.759026i
$528$	0	0
$529$	−20804.7	−0.0743448
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 189751.i	− 0.667930i
$534$	0	0
$535$	−211851.	−0.740156
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 82375.9i	− 0.283545i
$540$	0	0
$541$	44016.5	0.150391	0.0751954	−	0.997169i	$-0.476042\pi$
$541$	44016.5	0.150391	0.0751954	+	0.997169i	$0.476042\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 526894.i	− 1.77391i
$546$	0	0
$547$	42859.1	0.143241	0.0716207	−	0.997432i	$-0.477183\pi$
$547$	42859.1	0.143241	0.0716207	+	0.997432i	$0.477183\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 90929.5i	− 0.299503i
$552$	0	0
$553$	−37539.6	−0.122755
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 311007.i	− 1.00244i	−0.865319	−	0.501222i	$-0.832884\pi$
$557$	− 311007.i	− 1.00244i	0.865319	−	0.501222i	$-0.167116\pi$
$558$	0	0
$559$	146599.	0.469145
$560$	0	0
$561$	0	0
$562$	0	0
$563$	285768.i	0.901563i	0.892634	+	0.450782i	$0.148855\pi$
$563$	285768.i	0.901563i	−0.892634	+	0.450782i	$0.851145\pi$
$564$	0	0
$565$	−47691.7	−0.149398
$566$	0	0
$567$	0	0
$568$	0	0
$569$	136683.i	0.422171i	0.977468	+	0.211086i	$0.0676999\pi$
$569$	136683.i	0.422171i	−0.977468	+	0.211086i	$0.932300\pi$
$570$	0	0
$571$	28510.0	0.0874431	0.0437215	−	0.999044i	$-0.486079\pi$
$571$	28510.0	0.0874431	0.0437215	+	0.999044i	$0.486079\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 318809.i	− 0.964261i
$576$	0	0
$577$	−293742.	−0.882297	−0.441149	−	0.897434i	$-0.645429\pi$
$577$	−293742.	−0.882297	−0.441149	+	0.897434i	$0.645429\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	288898.i	0.855840i
$582$	0	0
$583$	−128221.	−0.377244
$584$	0	0
$585$	0	0
$586$	0	0
$587$	386055.i	1.12040i	0.828357	+	0.560200i	$0.189276\pi$
$587$	386055.i	1.12040i	−0.828357	+	0.560200i	$0.810724\pi$
$588$	0	0
$589$	63075.2	0.181814
$590$	0	0
$591$	0	0
$592$	0	0
$593$	305581.i	0.868995i	0.900673	+	0.434497i	$0.143074\pi$
$593$	305581.i	0.868995i	−0.900673	+	0.434497i	$0.856926\pi$
$594$	0	0
$595$	−419519.	−1.18500
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 264738.i	− 0.737840i	−0.929461	−	0.368920i	$-0.879728\pi$
$599$	− 264738.i	− 0.737840i	0.929461	−	0.368920i	$-0.120272\pi$
$600$	0	0
$601$	−132203.	−0.366009	−0.183005	−	0.983112i	$-0.558582\pi$
$601$	−132203.	−0.366009	−0.183005	+	0.983112i	$0.558582\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 392773.i	− 1.07308i
$606$	0	0
$607$	254146.	0.689772	0.344886	−	0.938645i	$-0.387918\pi$
$607$	254146.	0.689772	0.344886	+	0.938645i	$0.387918\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	59468.0i	0.159295i
$612$	0	0
$613$	−492878.	−1.31165	−0.655826	−	0.754912i	$-0.727680\pi$
$613$	−492878.	−1.31165	−0.655826	+	0.754912i	$0.727680\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 121278.i	− 0.318574i	−0.987232	−	0.159287i	$-0.949081\pi$
$617$	− 121278.i	− 0.318574i	0.987232	−	0.159287i	$-0.0509195\pi$
$618$	0	0
$619$	−307364.	−0.802180	−0.401090	−	0.916039i	$-0.631369\pi$
$619$	−307364.	−0.802180	−0.401090	+	0.916039i	$0.631369\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 278713.i	− 0.718092i
$624$	0	0
$625$	−415954.	−1.06484
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 347832.i	− 0.879161i
$630$	0	0
$631$	254196.	0.638425	0.319212	−	0.947683i	$-0.396582\pi$
$631$	254196.	0.638425	0.319212	+	0.947683i	$0.396582\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 665236.i	− 1.64979i
$636$	0	0
$637$	104584.	0.257743
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 365987.i	− 0.890736i	−0.895348	−	0.445368i	$-0.853073\pi$
$641$	− 365987.i	− 0.890736i	0.895348	−	0.445368i	$-0.146927\pi$
$642$	0	0
$643$	−230783.	−0.558189	−0.279094	−	0.960264i	$-0.590034\pi$
$643$	−230783.	−0.558189	−0.279094	+	0.960264i	$0.590034\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 278596.i	− 0.665529i	−0.943010	−	0.332764i	$-0.892019\pi$
$647$	− 278596.i	− 0.665529i	0.943010	−	0.332764i	$-0.107981\pi$
$648$	0	0
$649$	87304.7	0.207276
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 78506.8i	− 0.184111i	−0.995754	−	0.0920557i	$-0.970656\pi$
$653$	− 78506.8i	− 0.184111i	0.995754	−	0.0920557i	$-0.0293438\pi$
$654$	0	0
$655$	104896.	0.244499
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 857273.i	− 1.97401i	−0.160703	−	0.987003i	$-0.551376\pi$
$659$	− 857273.i	− 1.97401i	0.160703	−	0.987003i	$-0.448624\pi$
$660$	0	0
$661$	22344.0	0.0511396	0.0255698	−	0.999673i	$-0.491860\pi$
$661$	22344.0	0.0511396	0.0255698	+	0.999673i	$0.491860\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	125526.i	0.283850i
$666$	0	0
$667$	−430584.	−0.967846
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 109838.i	− 0.243953i
$672$	0	0
$673$	−663478.	−1.46486	−0.732430	−	0.680842i	$-0.761614\pi$
$673$	−663478.	−1.46486	−0.732430	+	0.680842i	$0.761614\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	228121.i	0.497722i	0.968539	+	0.248861i	$0.0800563\pi$
$677$	228121.i	0.497722i	−0.968539	+	0.248861i	$0.919944\pi$
$678$	0	0
$679$	208185.	0.451555
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 53606.6i	− 0.114915i	−0.998348	−	0.0574575i	$-0.981701\pi$
$683$	− 53606.6i	− 0.114915i	0.998348	−	0.0574575i	$-0.0182994\pi$
$684$	0	0
$685$	352474.	0.751182
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 162789.i	− 0.342915i
$690$	0	0
$691$	−757326.	−1.58609	−0.793043	−	0.609166i	$-0.791504\pi$
$691$	−757326.	−1.58609	−0.793043	+	0.609166i	$0.791504\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 1.22007e6i	− 2.52590i
$696$	0	0
$697$	1.00185e6	2.06222
$698$	0	0
$699$	0	0
$700$	0	0
$701$	506359.i	1.03044i	0.857058	+	0.515220i	$0.172290\pi$
$701$	506359.i	1.03044i	−0.857058	+	0.515220i	$0.827710\pi$
$702$	0	0
$703$	−104076.	−0.210591
$704$	0	0
$705$	0	0
$706$	0	0
$707$	285581.i	0.571335i
$708$	0	0
$709$	−651243.	−1.29554	−0.647770	−	0.761836i	$-0.724298\pi$
$709$	−651243.	−1.29554	−0.647770	+	0.761836i	$0.724298\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 298684.i	− 0.587533i
$714$	0	0
$715$	−146974.	−0.287495
$716$	0	0
$717$	0	0
$718$	0	0
$719$	69724.6i	0.134874i	0.997724	+	0.0674370i	$0.0214822\pi$
$719$	69724.6i	0.134874i	−0.997724	+	0.0674370i	$0.978518\pi$
$720$	0	0
$721$	−477913.	−0.919345
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 456597.i	− 0.868675i
$726$	0	0
$727$	747728.	1.41473	0.707367	−	0.706846i	$-0.249883\pi$
$727$	747728.	1.41473	0.707367	+	0.706846i	$0.249883\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	774011.i	1.44848i
$732$	0	0
$733$	65738.5	0.122352	0.0611761	−	0.998127i	$-0.480515\pi$
$733$	65738.5	0.122352	0.0611761	+	0.998127i	$0.480515\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 260201.i	− 0.479042i
$738$	0	0
$739$	−977921.	−1.79067	−0.895334	−	0.445395i	$-0.853063\pi$
$739$	−977921.	−1.79067	−0.895334	+	0.445395i	$0.853063\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 504874.i	− 0.914545i	−0.889327	−	0.457272i	$-0.848826\pi$
$743$	− 504874.i	− 0.914545i	0.889327	−	0.457272i	$-0.151174\pi$
$744$	0	0
$745$	1.29842e6	2.33939
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 190364.i	− 0.339329i
$750$	0	0
$751$	532791.	0.944663	0.472332	−	0.881421i	$-0.343412\pi$
$751$	532791.	0.944663	0.472332	+	0.881421i	$0.343412\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1.15559e6i	2.02727i
$756$	0	0
$757$	−293571.	−0.512297	−0.256148	−	0.966637i	$-0.582454\pi$
$757$	−293571.	−0.512297	−0.256148	+	0.966637i	$0.582454\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 25691.2i	− 0.0443624i	−0.999754	−	0.0221812i	$-0.992939\pi$
$761$	− 25691.2i	− 0.0443624i	0.999754	−	0.0221812i	$-0.00706108\pi$
$762$	0	0
$763$	473453.	0.813257
$764$	0	0
$765$	0	0
$766$	0	0
$767$	110842.i	0.188414i
$768$	0	0
$769$	−647144.	−1.09433	−0.547165	−	0.837025i	$-0.684293\pi$
$769$	−647144.	−1.09433	−0.547165	+	0.837025i	$0.684293\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	610238.i	1.02127i	0.859798	+	0.510634i	$0.170589\pi$
$773$	610238.i	1.02127i	−0.859798	+	0.510634i	$0.829411\pi$
$774$	0	0
$775$	316728.	0.527331
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 299766.i	− 0.493977i
$780$	0	0
$781$	230553.	0.377980
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 280661.i	− 0.455451i
$786$	0	0
$787$	580321.	0.936955	0.468477	−	0.883475i	$-0.344803\pi$
$787$	580321.	0.936955	0.468477	+	0.883475i	$0.344803\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 42854.5i	− 0.0684926i
$792$	0	0
$793$	139449.	0.221753
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 1.05714e6i	− 1.66424i	−0.554598	−	0.832118i	$-0.687128\pi$
$797$	− 1.05714e6i	− 1.66424i	0.554598	−	0.832118i	$-0.312872\pi$
$798$	0	0
$799$	−313978.	−0.491820
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 198405.i	− 0.307696i
$804$	0	0
$805$	594409.	0.917263
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 355969.i	− 0.543895i	−0.962312	−	0.271947i	$-0.912332\pi$
$809$	− 355969.i	− 0.543895i	0.962312	−	0.271947i	$-0.0876677\pi$
$810$	0	0
$811$	136417.	0.207409	0.103704	−	0.994608i	$-0.466930\pi$
$811$	136417.	0.207409	0.103704	+	0.994608i	$0.466930\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 883181.i	− 1.32964i
$816$	0	0
$817$	231594.	0.346964
$818$	0	0
$819$	0	0
$820$	0	0
$821$	455979.i	0.676485i	0.941059	+	0.338243i	$0.109832\pi$
$821$	455979.i	0.676485i	−0.941059	+	0.338243i	$0.890168\pi$
$822$	0	0
$823$	−503590.	−0.743494	−0.371747	−	0.928334i	$-0.621241\pi$
$823$	−503590.	−0.743494	−0.371747	+	0.928334i	$0.621241\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 551861.i	− 0.806898i	−0.915002	−	0.403449i	$-0.867811\pi$
$827$	− 551861.i	− 0.806898i	0.915002	−	0.403449i	$-0.132189\pi$
$828$	0	0
$829$	−182308.	−0.265275	−0.132638	−	0.991165i	$-0.542345\pi$
$829$	−182308.	−0.265275	−0.132638	+	0.991165i	$0.542345\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	552182.i	0.795778i
$834$	0	0
$835$	1.21752e6	1.74623
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 415684.i	− 0.590526i	−0.955416	−	0.295263i	$-0.904593\pi$
$839$	− 415684.i	− 0.590526i	0.955416	−	0.295263i	$-0.0954073\pi$
$840$	0	0
$841$	90599.4	0.128095
$842$	0	0
$843$	0	0
$844$	0	0
$845$	805434.i	1.12802i
$846$	0	0
$847$	352935.	0.491958
$848$	0	0
$849$	0	0
$850$	0	0
$851$	492837.i	0.680525i
$852$	0	0
$853$	150271.	0.206527	0.103263	−	0.994654i	$-0.467072\pi$
$853$	150271.	0.206527	0.103263	+	0.994654i	$0.467072\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 144768.i	− 0.197111i	−0.995132	−	0.0985554i	$-0.968578\pi$
$857$	− 144768.i	− 0.197111i	0.995132	−	0.0985554i	$-0.0314222\pi$
$858$	0	0
$859$	−333897.	−0.452508	−0.226254	−	0.974068i	$-0.572648\pi$
$859$	−333897.	−0.452508	−0.226254	+	0.974068i	$0.572648\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 1.13280e6i	− 1.52101i	−0.649330	−	0.760507i	$-0.724951\pi$
$863$	− 1.13280e6i	− 1.52101i	0.649330	−	0.760507i	$-0.275049\pi$
$864$	0	0
$865$	−510603.	−0.682418
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 69437.9i	− 0.0919511i
$870$	0	0
$871$	330350.	0.435450
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 47225.2i	− 0.0616820i
$876$	0	0
$877$	496541.	0.645589	0.322795	−	0.946469i	$-0.395378\pi$
$877$	496541.	0.645589	0.322795	+	0.946469i	$0.395378\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 1.21533e6i	− 1.56583i	−0.622131	−	0.782913i	$-0.713733\pi$
$881$	− 1.21533e6i	− 1.56583i	0.622131	−	0.782913i	$-0.286267\pi$
$882$	0	0
$883$	999070.	1.28137	0.640685	−	0.767804i	$-0.278651\pi$
$883$	999070.	1.28137	0.640685	+	0.767804i	$0.278651\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1.08316e6i	− 1.37672i	−0.725367	−	0.688362i	$-0.758330\pi$
$887$	− 1.08316e6i	− 1.37672i	0.725367	−	0.688362i	$-0.241670\pi$
$888$	0	0
$889$	597763.	0.756355
$890$	0	0
$891$	0	0
$892$	0	0
$893$	93946.4i	0.117809i
$894$	0	0
$895$	1.28698e6	1.60666
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 427774.i	− 0.529292i
$900$	0	0
$901$	859491.	1.05875
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1.37771e6i	1.68213i
$906$	0	0
$907$	−1.27685e6	−1.55212	−0.776061	−	0.630658i	$-0.782785\pi$
$907$	−1.27685e6	−1.55212	−0.776061	+	0.630658i	$0.782785\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	273486.i	0.329532i	0.986333	+	0.164766i	$0.0526869\pi$
$911$	273486.i	0.329532i	−0.986333	+	0.164766i	$0.947313\pi$
$912$	0	0
$913$	−534381.	−0.641076
$914$	0	0
$915$	0	0
$916$	0	0
$917$	94256.7i	0.112092i
$918$	0	0
$919$	−1.15882e6	−1.37210	−0.686051	−	0.727553i	$-0.740657\pi$
$919$	−1.15882e6	−1.37210	−0.686051	+	0.727553i	$0.740657\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	292709.i	0.343584i
$924$	0	0
$925$	−522611.	−0.610795
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 852706.i	− 0.988025i	−0.869455	−	0.494012i	$-0.835530\pi$
$929$	− 852706.i	− 0.988025i	0.869455	−	0.494012i	$-0.164470\pi$
$930$	0	0
$931$	165220.	0.190618
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 775993.i	− 0.887636i
$936$	0	0
$937$	629989.	0.717552	0.358776	−	0.933424i	$-0.383194\pi$
$937$	629989.	0.717552	0.358776	+	0.933424i	$0.383194\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 423252.i	− 0.477991i	−0.971021	−	0.238996i	$-0.923182\pi$
$941$	− 423252.i	− 0.477991i	0.971021	−	0.238996i	$-0.0768182\pi$
$942$	0	0
$943$	−1.41950e6	−1.59629
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 1.39444e6i	− 1.55489i	−0.628952	−	0.777444i	$-0.716516\pi$
$947$	− 1.39444e6i	− 1.55489i	0.628952	−	0.777444i	$-0.283484\pi$
$948$	0	0
$949$	251895.	0.279696
$950$	0	0
$951$	0	0
$952$	0	0
$953$	24001.0i	0.0264267i	0.999913	+	0.0132134i	$0.00420607\pi$
$953$	24001.0i	0.0264267i	−0.999913	+	0.0132134i	$0.995794\pi$
$954$	0	0
$955$	2.00195e6	2.19506
$956$	0	0
$957$	0	0
$958$	0	0
$959$	316723.i	0.344384i
$960$	0	0
$961$	−626786.	−0.678692
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 145174.i	− 0.155896i
$966$	0	0
$967$	−1.04406e6	−1.11654	−0.558268	−	0.829660i	$-0.688534\pi$
$967$	−1.04406e6	−1.11654	−0.558268	+	0.829660i	$0.688534\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	605213.i	0.641904i	0.947096	+	0.320952i	$0.104003\pi$
$971$	605213.i	0.641904i	−0.947096	+	0.320952i	$0.895997\pi$
$972$	0	0
$973$	1.09632e6	1.15801
$974$	0	0
$975$	0	0
$976$	0	0
$977$	48105.1i	0.0503967i	0.999682	+	0.0251983i	$0.00802173\pi$
$977$	48105.1i	0.0503967i	−0.999682	+	0.0251983i	$0.991978\pi$
$978$	0	0
$979$	515540.	0.537895
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 123883.i	− 0.128205i	−0.997943	−	0.0641024i	$-0.979582\pi$
$983$	− 123883.i	− 0.128205i	0.997943	−	0.0641024i	$-0.0204184\pi$
$984$	0	0
$985$	−770794.	−0.794448
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 1.09668e6i	− 1.12121i
$990$	0	0
$991$	−851418.	−0.866953	−0.433477	−	0.901165i	$-0.642713\pi$
$991$	−851418.	−0.866953	−0.433477	+	0.901165i	$0.642713\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	1.75783e6i	1.77555i
$996$	0	0
$997$	−1.60305e6	−1.61271	−0.806355	−	0.591432i	$-0.798563\pi$
$997$	−1.60305e6	−1.61271	−0.806355	+	0.591432i	$0.798563\pi$
$998$	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.161.1		6
3.2	odd	2	inner	1296.5.e.c.161.6		6
4.3	odd	2		81.5.b.a.80.3		6
9.2	odd	6		144.5.q.a.113.1		6
9.4	even	3		144.5.q.a.65.1		6
9.5	odd	6		432.5.q.a.305.3		6
9.7	even	3		432.5.q.a.17.3		6
12.11	even	2		81.5.b.a.80.4		6
36.7	odd	6		27.5.d.a.17.2		6
36.11	even	6		9.5.d.a.5.2	yes	6
36.23	even	6		27.5.d.a.8.2		6
36.31	odd	6		9.5.d.a.2.2	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9.5.d.a.2.2	✓	6	36.31	odd	6
9.5.d.a.5.2	yes	6	36.11	even	6
27.5.d.a.8.2		6	36.23	even	6
27.5.d.a.17.2		6	36.7	odd	6
81.5.b.a.80.3		6	4.3	odd	2
81.5.b.a.80.4		6	12.11	even	2
144.5.q.a.65.1		6	9.4	even	3
144.5.q.a.113.1		6	9.2	odd	6
432.5.q.a.17.3		6	9.7	even	3
432.5.q.a.305.3		6	9.5	odd	6
1296.5.e.c.161.1		6	1.1	even	1	trivial
1296.5.e.c.161.6		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)

\(f(q)\)	\(=\)	\(q-34.7338i q^{5} +31.2109 q^{7} +57.7314i q^{11} -73.2956 q^{13} -386.985i q^{17} -115.791 q^{19} +548.312i q^{23} -581.437 q^{25} +785.291i q^{29} -544.734 q^{31} -1084.07i q^{35} +898.827 q^{37} +2588.85i q^{41} -2000.11 q^{43} -811.345i q^{47} -1426.88 q^{49} +2221.00i q^{53} +2005.23 q^{55} -1512.26i q^{59} -1902.56 q^{61} +2545.83i q^{65} -4507.09 q^{67} -3993.54i q^{71} -3436.70 q^{73} +1801.85i q^{77} -1202.78 q^{79} +9256.34i q^{83} -13441.4 q^{85} -8929.99i q^{89} -2287.62 q^{91} +4021.86i q^{95} +6670.29 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 24 q^{7} + 12 q^{13} + 258 q^{19} + 546 q^{25} + 2580 q^{31} + 12 q^{37} - 570 q^{43} + 3726 q^{49} - 2016 q^{55} - 7260 q^{61} - 10110 q^{67} - 14622 q^{73} + 9528 q^{79} - 24732 q^{85} - 34836 q^{91}+ \cdots + 57918 q^{97}+O(q^{100}) \) 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

L-functions

Modular forms

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Database

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