LMFDB - Embedded newform 9036.2.a.l.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9036,2,Mod(1,9036)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9036, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("9036.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9036 = 2^{2} \cdot 3^{2} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9036.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$72.1528232664$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 5 x^{11} - 26 x^{10} + 136 x^{9} + 267 x^{8} - 1337 x^{7} - 1553 x^{6} + 5791 x^{5} + \cdots + 4088 $ x^12 - 5x^11 - 26x^10 + 136x^9 + 267x^8 - 1337x^7 - 1553x^6 + 5791x^5 + 5621x^4 - 10299x^3 - 9634x^2 + 4990*x + 4088
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 3012)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.63471$ of defining polynomial
Character	$\chi$	$=$	9036.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.63471 q^{5} +3.32358 q^{7} +O(q^{10})$	$q-3.63471 q^{5} +3.32358 q^{7} -2.80544 q^{11} -5.06285 q^{13} -0.258688 q^{17} -0.0440954 q^{19} -3.41394 q^{23} +8.21115 q^{25} -6.64269 q^{29} +2.18438 q^{31} -12.0803 q^{35} -0.445739 q^{37} -3.55134 q^{41} -5.90031 q^{43} +3.67346 q^{47} +4.04617 q^{49} -9.20577 q^{53} +10.1970 q^{55} +4.88626 q^{59} -0.241305 q^{61} +18.4020 q^{65} +5.53993 q^{67} +12.4938 q^{71} +12.2342 q^{73} -9.32408 q^{77} -8.12857 q^{79} -11.1495 q^{83} +0.940257 q^{85} +12.6547 q^{89} -16.8268 q^{91} +0.160274 q^{95} +1.59098 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 7 q^{5}+O(q^{10}) $ 12 * q - 7 * q^5	$ 12 q - 7 q^{5} + q^{11} + 17 q^{13} - 17 q^{17} + 3 q^{19} + 8 q^{23} + 19 q^{25} - 15 q^{29} - 2 q^{31} + 7 q^{35} + 14 q^{37} - 5 q^{41} + 9 q^{43} + 5 q^{47} + 22 q^{49} - 20 q^{53} - 4 q^{55} + q^{59} + 21 q^{61} - 8 q^{65} + 21 q^{67} + 17 q^{71} + 45 q^{73} - 40 q^{77} + 6 q^{79} + q^{83} + 31 q^{85} - 22 q^{89} + 32 q^{91} + 13 q^{95} + 29 q^{97}+O(q^{100}) $ 12 * q - 7 * q^5 + q^11 + 17 * q^13 - 17 * q^17 + 3 * q^19 + 8 * q^23 + 19 * q^25 - 15 * q^29 - 2 * q^31 + 7 * q^35 + 14 * q^37 - 5 * q^41 + 9 * q^43 + 5 * q^47 + 22 * q^49 - 20 * q^53 - 4 * q^55 + q^59 + 21 * q^61 - 8 * q^65 + 21 * q^67 + 17 * q^71 + 45 * q^73 - 40 * q^77 + 6 * q^79 + q^83 + 31 * q^85 - 22 * q^89 + 32 * q^91 + 13 * q^95 + 29 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−3.63471	−1.62549	−0.812747	−	0.582617i	$-0.802029\pi$
$5$	−3.63471	−1.62549	−0.812747	+	0.582617i	$0.802029\pi$
$6$	0	0
$7$	3.32358	1.25619	0.628097	−	0.778135i	$-0.283834\pi$
$7$	3.32358	1.25619	0.628097	+	0.778135i	$0.283834\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2.80544	−0.845871	−0.422935	−	0.906160i	$-0.639000\pi$
$11$	−2.80544	−0.845871	−0.422935	+	0.906160i	$0.639000\pi$
$12$	0	0
$13$	−5.06285	−1.40418	−0.702092	−	0.712087i	$-0.747750\pi$
$13$	−5.06285	−1.40418	−0.702092	+	0.712087i	$0.747750\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−0.258688	−0.0627411	−0.0313705	−	0.999508i	$-0.509987\pi$
$17$	−0.258688	−0.0627411	−0.0313705	+	0.999508i	$0.509987\pi$
$18$	0	0
$19$	−0.0440954	−0.0101162	−0.00505809	−	0.999987i	$-0.501610\pi$
$19$	−0.0440954	−0.0101162	−0.00505809	+	0.999987i	$0.501610\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3.41394	−0.711856	−0.355928	−	0.934513i	$-0.615835\pi$
$23$	−3.41394	−0.711856	−0.355928	+	0.934513i	$0.615835\pi$
$24$	0	0
$25$	8.21115	1.64223
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−6.64269	−1.23352	−0.616758	−	0.787152i	$-0.711554\pi$
$29$	−6.64269	−1.23352	−0.616758	+	0.787152i	$0.711554\pi$
$30$	0	0
$31$	2.18438	0.392327	0.196163	−	0.980571i	$-0.437152\pi$
$31$	2.18438	0.392327	0.196163	+	0.980571i	$0.437152\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−12.0803	−2.04194
$36$	0	0
$37$	−0.445739	−0.0732790	−0.0366395	−	0.999329i	$-0.511665\pi$
$37$	−0.445739	−0.0732790	−0.0366395	+	0.999329i	$0.511665\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−3.55134	−0.554626	−0.277313	−	0.960780i	$-0.589444\pi$
$41$	−3.55134	−0.554626	−0.277313	+	0.960780i	$0.589444\pi$
$42$	0	0
$43$	−5.90031	−0.899789	−0.449894	−	0.893082i	$-0.648538\pi$
$43$	−5.90031	−0.899789	−0.449894	+	0.893082i	$0.648538\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3.67346	0.535829	0.267915	−	0.963443i	$-0.413665\pi$
$47$	3.67346	0.535829	0.267915	+	0.963443i	$0.413665\pi$
$48$	0	0
$49$	4.04617	0.578025
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−9.20577	−1.26451	−0.632255	−	0.774761i	$-0.717870\pi$
$53$	−9.20577	−1.26451	−0.632255	+	0.774761i	$0.717870\pi$
$54$	0	0
$55$	10.1970	1.37496
$56$	0	0
$57$	0	0
$58$	0	0
$59$	4.88626	0.636136	0.318068	−	0.948068i	$-0.396966\pi$
$59$	4.88626	0.636136	0.318068	+	0.948068i	$0.396966\pi$
$60$	0	0
$61$	−0.241305	−0.0308960	−0.0154480	−	0.999881i	$-0.504917\pi$
$61$	−0.241305	−0.0308960	−0.0154480	+	0.999881i	$0.504917\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	18.4020	2.28249
$66$	0	0
$67$	5.53993	0.676810	0.338405	−	0.941001i	$-0.390113\pi$
$67$	5.53993	0.676810	0.338405	+	0.941001i	$0.390113\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	12.4938	1.48274	0.741369	−	0.671098i	$-0.234177\pi$
$71$	12.4938	1.48274	0.741369	+	0.671098i	$0.234177\pi$
$72$	0	0
$73$	12.2342	1.43191	0.715953	−	0.698148i	$-0.245992\pi$
$73$	12.2342	1.43191	0.715953	+	0.698148i	$0.245992\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−9.32408	−1.06258
$78$	0	0
$79$	−8.12857	−0.914536	−0.457268	−	0.889329i	$-0.651172\pi$
$79$	−8.12857	−0.914536	−0.457268	+	0.889329i	$0.651172\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−11.1495	−1.22382	−0.611909	−	0.790928i	$-0.709598\pi$
$83$	−11.1495	−1.22382	−0.611909	+	0.790928i	$0.709598\pi$
$84$	0	0
$85$	0.940257	0.101985
$86$	0	0
$87$	0	0
$88$	0	0
$89$	12.6547	1.34140	0.670700	−	0.741728i	$-0.265994\pi$
$89$	12.6547	1.34140	0.670700	+	0.741728i	$0.265994\pi$
$90$	0	0
$91$	−16.8268	−1.76393
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0.160274	0.0164438
$96$	0	0
$97$	1.59098	0.161540	0.0807700	−	0.996733i	$-0.474262\pi$
$97$	1.59098	0.161540	0.0807700	+	0.996733i	$0.474262\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	5.61555	0.558768	0.279384	−	0.960179i	$-0.409870\pi$
$101$	5.61555	0.558768	0.279384	+	0.960179i	$0.409870\pi$
$102$	0	0
$103$	−11.4759	−1.13075	−0.565376	−	0.824833i	$-0.691269\pi$
$103$	−11.4759	−1.13075	−0.565376	+	0.824833i	$0.691269\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.64285	0.448841	0.224421	−	0.974492i	$-0.427951\pi$
$107$	4.64285	0.448841	0.224421	+	0.974492i	$0.427951\pi$
$108$	0	0
$109$	7.83988	0.750924	0.375462	−	0.926838i	$-0.377484\pi$
$109$	7.83988	0.750924	0.375462	+	0.926838i	$0.377484\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1.43293	−0.134799	−0.0673994	−	0.997726i	$-0.521470\pi$
$113$	−1.43293	−0.134799	−0.0673994	+	0.997726i	$0.521470\pi$
$114$	0	0
$115$	12.4087	1.15712
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−0.859770	−0.0788150
$120$	0	0
$121$	−3.12953	−0.284503
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−11.6716	−1.04394
$126$	0	0
$127$	−16.4563	−1.46026	−0.730131	−	0.683308i	$-0.760541\pi$
$127$	−16.4563	−1.46026	−0.730131	+	0.683308i	$0.760541\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−10.2875	−0.898825	−0.449412	−	0.893324i	$-0.648367\pi$
$131$	−10.2875	−0.898825	−0.449412	+	0.893324i	$0.648367\pi$
$132$	0	0
$133$	−0.146555	−0.0127079
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−3.27103	−0.279463	−0.139731	−	0.990189i	$-0.544624\pi$
$137$	−3.27103	−0.279463	−0.139731	+	0.990189i	$0.544624\pi$
$138$	0	0
$139$	−5.15378	−0.437138	−0.218569	−	0.975821i	$-0.570139\pi$
$139$	−5.15378	−0.437138	−0.218569	+	0.975821i	$0.570139\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	14.2035	1.18776
$144$	0	0
$145$	24.1443	2.00507
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−3.04479	−0.249439	−0.124719	−	0.992192i	$-0.539803\pi$
$149$	−3.04479	−0.249439	−0.124719	+	0.992192i	$0.539803\pi$
$150$	0	0
$151$	−4.16544	−0.338979	−0.169489	−	0.985532i	$-0.554212\pi$
$151$	−4.16544	−0.338979	−0.169489	+	0.985532i	$0.554212\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−7.93960	−0.637724
$156$	0	0
$157$	6.13942	0.489979	0.244990	−	0.969526i	$-0.421215\pi$
$157$	6.13942	0.489979	0.244990	+	0.969526i	$0.421215\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−11.3465	−0.894230
$162$	0	0
$163$	15.7788	1.23589	0.617947	−	0.786220i	$-0.287965\pi$
$163$	15.7788	1.23589	0.617947	+	0.786220i	$0.287965\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−9.00996	−0.697212	−0.348606	−	0.937269i	$-0.613345\pi$
$167$	−9.00996	−0.697212	−0.348606	+	0.937269i	$0.613345\pi$
$168$	0	0
$169$	12.6325	0.971731
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−19.2387	−1.46269	−0.731345	−	0.682008i	$-0.761107\pi$
$173$	−19.2387	−1.46269	−0.731345	+	0.682008i	$0.761107\pi$
$174$	0	0
$175$	27.2904	2.06296
$176$	0	0
$177$	0	0
$178$	0	0
$179$	4.77206	0.356681	0.178340	−	0.983969i	$-0.442927\pi$
$179$	4.77206	0.356681	0.178340	+	0.983969i	$0.442927\pi$
$180$	0	0
$181$	−1.26671	−0.0941539	−0.0470769	−	0.998891i	$-0.514991\pi$
$181$	−1.26671	−0.0941539	−0.0470769	+	0.998891i	$0.514991\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1.62013	0.119115
$186$	0	0
$187$	0.725733	0.0530708
$188$	0	0
$189$	0	0
$190$	0	0
$191$	16.1861	1.17118	0.585592	−	0.810606i	$-0.300862\pi$
$191$	16.1861	1.17118	0.585592	+	0.810606i	$0.300862\pi$
$192$	0	0
$193$	17.2472	1.24148	0.620741	−	0.784016i	$-0.286832\pi$
$193$	17.2472	1.24148	0.620741	+	0.784016i	$0.286832\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3.57399	0.254637	0.127318	−	0.991862i	$-0.459363\pi$
$197$	3.57399	0.254637	0.127318	+	0.991862i	$0.459363\pi$
$198$	0	0
$199$	25.6040	1.81502	0.907510	−	0.420032i	$-0.137981\pi$
$199$	25.6040	1.81502	0.907510	+	0.420032i	$0.137981\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−22.0775	−1.54954
$204$	0	0
$205$	12.9081	0.901542
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0.123707	0.00855698
$210$	0	0
$211$	−11.3965	−0.784570	−0.392285	−	0.919844i	$-0.628315\pi$
$211$	−11.3965	−0.784570	−0.392285	+	0.919844i	$0.628315\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	21.4459	1.46260
$216$	0	0
$217$	7.25996	0.492838
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1.30970	0.0881000
$222$	0	0
$223$	7.73374	0.517889	0.258945	−	0.965892i	$-0.416625\pi$
$223$	7.73374	0.517889	0.258945	+	0.965892i	$0.416625\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	24.7940	1.64564	0.822819	−	0.568303i	$-0.192400\pi$
$227$	24.7940	1.64564	0.822819	+	0.568303i	$0.192400\pi$
$228$	0	0
$229$	6.37077	0.420993	0.210496	−	0.977595i	$-0.432492\pi$
$229$	6.37077	0.420993	0.210496	+	0.977595i	$0.432492\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	8.72907	0.571860	0.285930	−	0.958250i	$-0.407697\pi$
$233$	8.72907	0.571860	0.285930	+	0.958250i	$0.407697\pi$
$234$	0	0
$235$	−13.3520	−0.870987
$236$	0	0
$237$	0	0
$238$	0	0
$239$	7.12233	0.460705	0.230353	−	0.973107i	$-0.426012\pi$
$239$	7.12233	0.460705	0.230353	+	0.973107i	$0.426012\pi$
$240$	0	0
$241$	28.0836	1.80902	0.904512	−	0.426449i	$-0.140236\pi$
$241$	28.0836	1.80902	0.904512	+	0.426449i	$0.140236\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−14.7067	−0.939576
$246$	0	0
$247$	0.223249	0.0142050
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	9.57759	0.602138
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−9.02619	−0.563038	−0.281519	−	0.959556i	$-0.590838\pi$
$257$	−9.02619	−0.563038	−0.281519	+	0.959556i	$0.590838\pi$
$258$	0	0
$259$	−1.48145	−0.0920527
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−11.1135	−0.685286	−0.342643	−	0.939466i	$-0.611322\pi$
$263$	−11.1135	−0.685286	−0.342643	+	0.939466i	$0.611322\pi$
$264$	0	0
$265$	33.4603	2.05545
$266$	0	0
$267$	0	0
$268$	0	0
$269$	3.94261	0.240385	0.120193	−	0.992751i	$-0.461649\pi$
$269$	3.94261	0.240385	0.120193	+	0.992751i	$0.461649\pi$
$270$	0	0
$271$	−18.8123	−1.14277	−0.571383	−	0.820683i	$-0.693593\pi$
$271$	−18.8123	−1.14277	−0.571383	+	0.820683i	$0.693593\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−23.0358	−1.38911
$276$	0	0
$277$	14.2169	0.854212	0.427106	−	0.904202i	$-0.359533\pi$
$277$	14.2169	0.854212	0.427106	+	0.904202i	$0.359533\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	30.2462	1.80433	0.902167	−	0.431387i	$-0.141975\pi$
$281$	30.2462	1.80433	0.902167	+	0.431387i	$0.141975\pi$
$282$	0	0
$283$	−8.72473	−0.518632	−0.259316	−	0.965793i	$-0.583497\pi$
$283$	−8.72473	−0.518632	−0.259316	+	0.965793i	$0.583497\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−11.8032	−0.696719
$288$	0	0
$289$	−16.9331	−0.996064
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−17.4612	−1.02009	−0.510047	−	0.860147i	$-0.670372\pi$
$293$	−17.4612	−1.02009	−0.510047	+	0.860147i	$0.670372\pi$
$294$	0	0
$295$	−17.7601	−1.03404
$296$	0	0
$297$	0	0
$298$	0	0
$299$	17.2843	0.999576
$300$	0	0
$301$	−19.6101	−1.13031
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0.877076	0.0502212
$306$	0	0
$307$	−7.08036	−0.404097	−0.202049	−	0.979375i	$-0.564760\pi$
$307$	−7.08036	−0.404097	−0.202049	+	0.979375i	$0.564760\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−18.2946	−1.03739	−0.518695	−	0.854959i	$-0.673582\pi$
$311$	−18.2946	−1.03739	−0.518695	+	0.854959i	$0.673582\pi$
$312$	0	0
$313$	31.4714	1.77887	0.889434	−	0.457064i	$-0.151099\pi$
$313$	31.4714	1.77887	0.889434	+	0.457064i	$0.151099\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	21.7841	1.22352	0.611758	−	0.791045i	$-0.290463\pi$
$317$	21.7841	1.22352	0.611758	+	0.791045i	$0.290463\pi$
$318$	0	0
$319$	18.6356	1.04340
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0.0114070	0.000634700 0
$324$	0	0
$325$	−41.5719	−2.30599
$326$	0	0
$327$	0	0
$328$	0	0
$329$	12.2090	0.673106
$330$	0	0
$331$	−20.0693	−1.10311	−0.551555	−	0.834138i	$-0.685965\pi$
$331$	−20.0693	−1.10311	−0.551555	+	0.834138i	$0.685965\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−20.1361	−1.10015
$336$	0	0
$337$	−16.0040	−0.871793	−0.435896	−	0.899997i	$-0.643569\pi$
$337$	−16.0040	−0.871793	−0.435896	+	0.899997i	$0.643569\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−6.12814	−0.331857
$342$	0	0
$343$	−9.81727	−0.530083
$344$	0	0
$345$	0	0
$346$	0	0
$347$	23.4926	1.26115	0.630574	−	0.776129i	$-0.282820\pi$
$347$	23.4926	1.26115	0.630574	+	0.776129i	$0.282820\pi$
$348$	0	0
$349$	20.1475	1.07847	0.539236	−	0.842155i	$-0.318713\pi$
$349$	20.1475	1.07847	0.539236	+	0.842155i	$0.318713\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	5.42735	0.288868	0.144434	−	0.989514i	$-0.453864\pi$
$353$	5.42735	0.288868	0.144434	+	0.989514i	$0.453864\pi$
$354$	0	0
$355$	−45.4113	−2.41018
$356$	0	0
$357$	0	0
$358$	0	0
$359$	23.4563	1.23798	0.618989	−	0.785400i	$-0.287543\pi$
$359$	23.4563	1.23798	0.618989	+	0.785400i	$0.287543\pi$
$360$	0	0
$361$	−18.9981	−0.999898
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−44.4679	−2.32756
$366$	0	0
$367$	0.996464	0.0520150	0.0260075	−	0.999662i	$-0.491721\pi$
$367$	0.996464	0.0520150	0.0260075	+	0.999662i	$0.491721\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−30.5961	−1.58847
$372$	0	0
$373$	25.0389	1.29646	0.648232	−	0.761443i	$-0.275509\pi$
$373$	25.0389	1.29646	0.648232	+	0.761443i	$0.275509\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	33.6310	1.73208
$378$	0	0
$379$	14.0123	0.719766	0.359883	−	0.932997i	$-0.382817\pi$
$379$	14.0123	0.719766	0.359883	+	0.932997i	$0.382817\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−9.23009	−0.471635	−0.235818	−	0.971797i	$-0.575777\pi$
$383$	−9.23009	−0.471635	−0.235818	+	0.971797i	$0.575777\pi$
$384$	0	0
$385$	33.8904	1.72721
$386$	0	0
$387$	0	0
$388$	0	0
$389$	25.5712	1.29651	0.648256	−	0.761422i	$-0.275499\pi$
$389$	25.5712	1.29651	0.648256	+	0.761422i	$0.275499\pi$
$390$	0	0
$391$	0.883146	0.0446626
$392$	0	0
$393$	0	0
$394$	0	0
$395$	29.5450	1.48657
$396$	0	0
$397$	19.3174	0.969515	0.484757	−	0.874649i	$-0.338908\pi$
$397$	19.3174	0.969515	0.484757	+	0.874649i	$0.338908\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−15.1825	−0.758178	−0.379089	−	0.925360i	$-0.623763\pi$
$401$	−15.1825	−0.758178	−0.379089	+	0.925360i	$0.623763\pi$
$402$	0	0
$403$	−11.0592	−0.550898
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.25049	0.0619846
$408$	0	0
$409$	−7.52976	−0.372323	−0.186161	−	0.982519i	$-0.559605\pi$
$409$	−7.52976	−0.372323	−0.186161	+	0.982519i	$0.559605\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	16.2399	0.799111
$414$	0	0
$415$	40.5253	1.98931
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−17.4288	−0.851452	−0.425726	−	0.904852i	$-0.639981\pi$
$419$	−17.4288	−0.851452	−0.425726	+	0.904852i	$0.639981\pi$
$420$	0	0
$421$	21.7434	1.05971	0.529853	−	0.848089i	$-0.322247\pi$
$421$	21.7434	1.05971	0.529853	+	0.848089i	$0.322247\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−2.12413	−0.103035
$426$	0	0
$427$	−0.801997	−0.0388114
$428$	0	0
$429$	0	0
$430$	0	0
$431$	31.9469	1.53883	0.769413	−	0.638751i	$-0.220549\pi$
$431$	31.9469	1.53883	0.769413	+	0.638751i	$0.220549\pi$
$432$	0	0
$433$	−15.1069	−0.725990	−0.362995	−	0.931791i	$-0.618246\pi$
$433$	−15.1069	−0.725990	−0.362995	+	0.931791i	$0.618246\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0.150539	0.00720127
$438$	0	0
$439$	9.37578	0.447482	0.223741	−	0.974649i	$-0.428173\pi$
$439$	9.37578	0.447482	0.223741	+	0.974649i	$0.428173\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−2.17324	−0.103254	−0.0516270	−	0.998666i	$-0.516441\pi$
$443$	−2.17324	−0.103254	−0.0516270	+	0.998666i	$0.516441\pi$
$444$	0	0
$445$	−45.9964	−2.18044
$446$	0	0
$447$	0	0
$448$	0	0
$449$	6.41035	0.302523	0.151262	−	0.988494i	$-0.451666\pi$
$449$	6.41035	0.302523	0.151262	+	0.988494i	$0.451666\pi$
$450$	0	0
$451$	9.96306	0.469142
$452$	0	0
$453$	0	0
$454$	0	0
$455$	61.1606	2.86725
$456$	0	0
$457$	−17.1374	−0.801655	−0.400827	−	0.916154i	$-0.631277\pi$
$457$	−17.1374	−0.801655	−0.400827	+	0.916154i	$0.631277\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	21.4546	0.999238	0.499619	−	0.866245i	$-0.333473\pi$
$461$	21.4546	0.999238	0.499619	+	0.866245i	$0.333473\pi$
$462$	0	0
$463$	11.2917	0.524771	0.262386	−	0.964963i	$-0.415491\pi$
$463$	11.2917	0.524771	0.262386	+	0.964963i	$0.415491\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	9.00630	0.416762	0.208381	−	0.978048i	$-0.433181\pi$
$467$	9.00630	0.416762	0.208381	+	0.978048i	$0.433181\pi$
$468$	0	0
$469$	18.4124	0.850205
$470$	0	0
$471$	0	0
$472$	0	0
$473$	16.5529	0.761105
$474$	0	0
$475$	−0.362074	−0.0166131
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−23.2772	−1.06356	−0.531782	−	0.846881i	$-0.678478\pi$
$479$	−23.2772	−1.06356	−0.531782	+	0.846881i	$0.678478\pi$
$480$	0	0
$481$	2.25671	0.102897
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−5.78277	−0.262582
$486$	0	0
$487$	−44.0899	−1.99790	−0.998952	−	0.0457652i	$-0.985427\pi$
$487$	−44.0899	−1.99790	−0.998952	+	0.0457652i	$0.985427\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−21.6179	−0.975600	−0.487800	−	0.872955i	$-0.662200\pi$
$491$	−21.6179	−0.975600	−0.487800	+	0.872955i	$0.662200\pi$
$492$	0	0
$493$	1.71838	0.0773922
$494$	0	0
$495$	0	0
$496$	0	0
$497$	41.5240	1.86261
$498$	0	0
$499$	20.3820	0.912422	0.456211	−	0.889872i	$-0.349206\pi$
$499$	20.3820	0.912422	0.456211	+	0.889872i	$0.349206\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−28.6622	−1.27798	−0.638992	−	0.769213i	$-0.720648\pi$
$503$	−28.6622	−1.27798	−0.638992	+	0.769213i	$0.720648\pi$
$504$	0	0
$505$	−20.4109	−0.908274
$506$	0	0
$507$	0	0
$508$	0	0
$509$	22.8523	1.01291	0.506454	−	0.862267i	$-0.330956\pi$
$509$	22.8523	1.01291	0.506454	+	0.862267i	$0.330956\pi$
$510$	0	0
$511$	40.6614	1.79875
$512$	0	0
$513$	0	0
$514$	0	0
$515$	41.7116	1.83803
$516$	0	0
$517$	−10.3057	−0.453242
$518$	0	0
$519$	0	0
$520$	0	0
$521$	2.67869	0.117356	0.0586778	−	0.998277i	$-0.481312\pi$
$521$	2.67869	0.117356	0.0586778	+	0.998277i	$0.481312\pi$
$522$	0	0
$523$	0.762025	0.0333210	0.0166605	−	0.999861i	$-0.494697\pi$
$523$	0.762025	0.0333210	0.0166605	+	0.999861i	$0.494697\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−0.565073	−0.0246150
$528$	0	0
$529$	−11.3450	−0.493261
$530$	0	0
$531$	0	0
$532$	0	0
$533$	17.9799	0.778797
$534$	0	0
$535$	−16.8754	−0.729589
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−11.3513	−0.488934
$540$	0	0
$541$	−14.3905	−0.618698	−0.309349	−	0.950949i	$-0.600111\pi$
$541$	−14.3905	−0.618698	−0.309349	+	0.950949i	$0.600111\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−28.4957	−1.22062
$546$	0	0
$547$	38.6531	1.65269	0.826343	−	0.563167i	$-0.190417\pi$
$547$	38.6531	1.65269	0.826343	+	0.563167i	$0.190417\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0.292912	0.0124785
$552$	0	0
$553$	−27.0159	−1.14884
$554$	0	0
$555$	0	0
$556$	0	0
$557$	26.7894	1.13510	0.567551	−	0.823338i	$-0.307891\pi$
$557$	26.7894	1.13510	0.567551	+	0.823338i	$0.307891\pi$
$558$	0	0
$559$	29.8724	1.26347
$560$	0	0
$561$	0	0
$562$	0	0
$563$	37.4053	1.57644	0.788222	−	0.615391i	$-0.211002\pi$
$563$	37.4053	1.57644	0.788222	+	0.615391i	$0.211002\pi$
$564$	0	0
$565$	5.20830	0.219115
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−21.6612	−0.908086	−0.454043	−	0.890980i	$-0.650019\pi$
$569$	−21.6612	−0.908086	−0.454043	+	0.890980i	$0.650019\pi$
$570$	0	0
$571$	−4.23099	−0.177062	−0.0885308	−	0.996073i	$-0.528217\pi$
$571$	−4.23099	−0.177062	−0.0885308	+	0.996073i	$0.528217\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−28.0324	−1.16903
$576$	0	0
$577$	−7.35179	−0.306059	−0.153030	−	0.988222i	$-0.548903\pi$
$577$	−7.35179	−0.306059	−0.153030	+	0.988222i	$0.548903\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−37.0563	−1.53735
$582$	0	0
$583$	25.8262	1.06961
$584$	0	0
$585$	0	0
$586$	0	0
$587$	17.7520	0.732703	0.366352	−	0.930477i	$-0.380607\pi$
$587$	17.7520	0.732703	0.366352	+	0.930477i	$0.380607\pi$
$588$	0	0
$589$	−0.0963212	−0.00396885
$590$	0	0
$591$	0	0
$592$	0	0
$593$	28.1381	1.15549	0.577747	−	0.816216i	$-0.303932\pi$
$593$	28.1381	1.15549	0.577747	+	0.816216i	$0.303932\pi$
$594$	0	0
$595$	3.12502	0.128113
$596$	0	0
$597$	0	0
$598$	0	0
$599$	14.1223	0.577023	0.288511	−	0.957476i	$-0.406840\pi$
$599$	14.1223	0.577023	0.288511	+	0.957476i	$0.406840\pi$
$600$	0	0
$601$	35.2433	1.43760	0.718802	−	0.695215i	$-0.244691\pi$
$601$	35.2433	1.43760	0.718802	+	0.695215i	$0.244691\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	11.3750	0.462458
$606$	0	0
$607$	−0.917545	−0.0372420	−0.0186210	−	0.999827i	$-0.505928\pi$
$607$	−0.917545	−0.0372420	−0.0186210	+	0.999827i	$0.505928\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−18.5982	−0.752403
$612$	0	0
$613$	1.66049	0.0670666	0.0335333	−	0.999438i	$-0.489324\pi$
$613$	1.66049	0.0670666	0.0335333	+	0.999438i	$0.489324\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	46.7747	1.88308	0.941539	−	0.336904i	$-0.109380\pi$
$617$	46.7747	1.88308	0.941539	+	0.336904i	$0.109380\pi$
$618$	0	0
$619$	11.8250	0.475286	0.237643	−	0.971353i	$-0.423625\pi$
$619$	11.8250	0.475286	0.237643	+	0.971353i	$0.423625\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	42.0590	1.68506
$624$	0	0
$625$	1.36723	0.0546892
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0.115307	0.00459760
$630$	0	0
$631$	5.64771	0.224832	0.112416	−	0.993661i	$-0.464141\pi$
$631$	5.64771	0.224832	0.112416	+	0.993661i	$0.464141\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	59.8140	2.37365
$636$	0	0
$637$	−20.4852	−0.811653
$638$	0	0
$639$	0	0
$640$	0	0
$641$	48.3565	1.90997	0.954984	−	0.296658i	$-0.0958722\pi$
$641$	48.3565	1.90997	0.954984	+	0.296658i	$0.0958722\pi$
$642$	0	0
$643$	7.75384	0.305782	0.152891	−	0.988243i	$-0.451142\pi$
$643$	7.75384	0.305782	0.152891	+	0.988243i	$0.451142\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−15.4277	−0.606525	−0.303263	−	0.952907i	$-0.598076\pi$
$647$	−15.4277	−0.606525	−0.303263	+	0.952907i	$0.598076\pi$
$648$	0	0
$649$	−13.7081	−0.538089
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−16.7419	−0.655161	−0.327580	−	0.944823i	$-0.606233\pi$
$653$	−16.7419	−0.655161	−0.327580	+	0.944823i	$0.606233\pi$
$654$	0	0
$655$	37.3922	1.46103
$656$	0	0
$657$	0	0
$658$	0	0
$659$	48.0655	1.87237	0.936183	−	0.351514i	$-0.114333\pi$
$659$	48.0655	1.87237	0.936183	+	0.351514i	$0.114333\pi$
$660$	0	0
$661$	−22.4868	−0.874635	−0.437318	−	0.899307i	$-0.644071\pi$
$661$	−22.4868	−0.874635	−0.437318	+	0.899307i	$0.644071\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0.532684	0.0206566
$666$	0	0
$667$	22.6778	0.878086
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0.676966	0.0261340
$672$	0	0
$673$	−6.25651	−0.241171	−0.120585	−	0.992703i	$-0.538477\pi$
$673$	−6.25651	−0.241171	−0.120585	+	0.992703i	$0.538477\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−27.1443	−1.04324	−0.521620	−	0.853178i	$-0.674672\pi$
$677$	−27.1443	−1.04324	−0.521620	+	0.853178i	$0.674672\pi$
$678$	0	0
$679$	5.28776	0.202926
$680$	0	0
$681$	0	0
$682$	0	0
$683$	4.88376	0.186872	0.0934359	−	0.995625i	$-0.470215\pi$
$683$	4.88376	0.186872	0.0934359	+	0.995625i	$0.470215\pi$
$684$	0	0
$685$	11.8892	0.454265
$686$	0	0
$687$	0	0
$688$	0	0
$689$	46.6075	1.77560
$690$	0	0
$691$	−12.0722	−0.459250	−0.229625	−	0.973279i	$-0.573750\pi$
$691$	−12.0722	−0.459250	−0.229625	+	0.973279i	$0.573750\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	18.7325	0.710566
$696$	0	0
$697$	0.918690	0.0347978
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−11.9878	−0.452773	−0.226387	−	0.974038i	$-0.572691\pi$
$701$	−11.9878	−0.452773	−0.226387	+	0.974038i	$0.572691\pi$
$702$	0	0
$703$	0.0196550	0.000741304 0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	18.6637	0.701922
$708$	0	0
$709$	−36.1642	−1.35817	−0.679087	−	0.734058i	$-0.737624\pi$
$709$	−36.1642	−1.35817	−0.679087	+	0.734058i	$0.737624\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−7.45735	−0.279280
$714$	0	0
$715$	−51.6257	−1.93069
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−0.438462	−0.0163519	−0.00817594	−	0.999967i	$-0.502603\pi$
$719$	−0.438462	−0.0163519	−0.00817594	+	0.999967i	$0.502603\pi$
$720$	0	0
$721$	−38.1410	−1.42045
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−54.5441	−2.02572
$726$	0	0
$727$	−37.1691	−1.37852	−0.689262	−	0.724512i	$-0.742065\pi$
$727$	−37.1691	−1.37852	−0.689262	+	0.724512i	$0.742065\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1.52634	0.0564537
$732$	0	0
$733$	18.3596	0.678125	0.339063	−	0.940764i	$-0.389890\pi$
$733$	18.3596	0.678125	0.339063	+	0.940764i	$0.389890\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−15.5419	−0.572494
$738$	0	0
$739$	47.3893	1.74324	0.871622	−	0.490179i	$-0.163069\pi$
$739$	47.3893	1.74324	0.871622	+	0.490179i	$0.163069\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	30.9010	1.13365	0.566824	−	0.823839i	$-0.308172\pi$
$743$	30.9010	1.13365	0.566824	+	0.823839i	$0.308172\pi$
$744$	0	0
$745$	11.0669	0.405461
$746$	0	0
$747$	0	0
$748$	0	0
$749$	15.4309	0.563832
$750$	0	0
$751$	−8.56019	−0.312366	−0.156183	−	0.987728i	$-0.549919\pi$
$751$	−8.56019	−0.312366	−0.156183	+	0.987728i	$0.549919\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	15.1402	0.551008
$756$	0	0
$757$	1.71015	0.0621564	0.0310782	−	0.999517i	$-0.490106\pi$
$757$	1.71015	0.0621564	0.0310782	+	0.999517i	$0.490106\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−27.1887	−0.985590	−0.492795	−	0.870145i	$-0.664025\pi$
$761$	−27.1887	−0.985590	−0.492795	+	0.870145i	$0.664025\pi$
$762$	0	0
$763$	26.0564	0.943307
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−24.7384	−0.893252
$768$	0	0
$769$	−0.596153	−0.0214978	−0.0107489	−	0.999942i	$-0.503422\pi$
$769$	−0.596153	−0.0214978	−0.0107489	+	0.999942i	$0.503422\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−47.3507	−1.70309	−0.851543	−	0.524284i	$-0.824333\pi$
$773$	−47.3507	−1.70309	−0.851543	+	0.524284i	$0.824333\pi$
$774$	0	0
$775$	17.9363	0.644290
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0.156598	0.00561070
$780$	0	0
$781$	−35.0505	−1.25420
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−22.3150	−0.796458
$786$	0	0
$787$	−37.5617	−1.33893	−0.669464	−	0.742844i	$-0.733476\pi$
$787$	−37.5617	−1.33893	−0.669464	+	0.742844i	$0.733476\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−4.76246	−0.169334
$792$	0	0
$793$	1.22169	0.0433836
$794$	0	0
$795$	0	0
$796$	0	0
$797$	31.9517	1.13179	0.565893	−	0.824479i	$-0.308532\pi$
$797$	31.9517	1.13179	0.565893	+	0.824479i	$0.308532\pi$
$798$	0	0
$799$	−0.950281	−0.0336185
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−34.3223	−1.21121
$804$	0	0
$805$	41.2413	1.45356
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−40.6511	−1.42922	−0.714608	−	0.699526i	$-0.753395\pi$
$809$	−40.6511	−1.42922	−0.714608	+	0.699526i	$0.753395\pi$
$810$	0	0
$811$	54.2842	1.90618	0.953088	−	0.302695i	$-0.0978862\pi$
$811$	54.2842	1.90618	0.953088	+	0.302695i	$0.0978862\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−57.3516	−2.00894
$816$	0	0
$817$	0.260177	0.00910243
$818$	0	0
$819$	0	0
$820$	0	0
$821$	2.06835	0.0721861	0.0360930	−	0.999348i	$-0.488509\pi$
$821$	2.06835	0.0721861	0.0360930	+	0.999348i	$0.488509\pi$
$822$	0	0
$823$	−18.0500	−0.629182	−0.314591	−	0.949227i	$-0.601867\pi$
$823$	−18.0500	−0.629182	−0.314591	+	0.949227i	$0.601867\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	9.25888	0.321963	0.160981	−	0.986957i	$-0.448534\pi$
$827$	9.25888	0.321963	0.160981	+	0.986957i	$0.448534\pi$
$828$	0	0
$829$	−1.93169	−0.0670902	−0.0335451	−	0.999437i	$-0.510680\pi$
$829$	−1.93169	−0.0670902	−0.0335451	+	0.999437i	$0.510680\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−1.04670	−0.0362659
$834$	0	0
$835$	32.7486	1.13331
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−29.3391	−1.01290	−0.506449	−	0.862270i	$-0.669042\pi$
$839$	−29.3391	−1.01290	−0.506449	+	0.862270i	$0.669042\pi$
$840$	0	0
$841$	15.1254	0.521564
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−45.9155	−1.57954
$846$	0	0
$847$	−10.4012	−0.357391
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1.52173	0.0521641
$852$	0	0
$853$	−26.5439	−0.908846	−0.454423	−	0.890786i	$-0.650155\pi$
$853$	−26.5439	−0.908846	−0.454423	+	0.890786i	$0.650155\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	15.6509	0.534624	0.267312	−	0.963610i	$-0.413865\pi$
$857$	15.6509	0.534624	0.267312	+	0.963610i	$0.413865\pi$
$858$	0	0
$859$	−40.8957	−1.39534	−0.697671	−	0.716418i	$-0.745780\pi$
$859$	−40.8957	−1.39534	−0.697671	+	0.716418i	$0.745780\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−22.8364	−0.777359	−0.388679	−	0.921373i	$-0.627069\pi$
$863$	−22.8364	−0.777359	−0.388679	+	0.921373i	$0.627069\pi$
$864$	0	0
$865$	69.9271	2.37759
$866$	0	0
$867$	0	0
$868$	0	0
$869$	22.8042	0.773579
$870$	0	0
$871$	−28.0479	−0.950365
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−38.7915	−1.31139
$876$	0	0
$877$	39.6353	1.33839	0.669194	−	0.743087i	$-0.266639\pi$
$877$	39.6353	1.33839	0.669194	+	0.743087i	$0.266639\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	16.0538	0.540868	0.270434	−	0.962739i	$-0.412833\pi$
$881$	16.0538	0.540868	0.270434	+	0.962739i	$0.412833\pi$
$882$	0	0
$883$	50.5426	1.70089	0.850447	−	0.526061i	$-0.176332\pi$
$883$	50.5426	1.70089	0.850447	+	0.526061i	$0.176332\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	45.6278	1.53203	0.766016	−	0.642821i	$-0.222236\pi$
$887$	45.6278	1.53203	0.766016	+	0.642821i	$0.222236\pi$
$888$	0	0
$889$	−54.6938	−1.83437
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−0.161983	−0.00542055
$894$	0	0
$895$	−17.3451	−0.579782
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−14.5102	−0.483941
$900$	0	0
$901$	2.38142	0.0793367
$902$	0	0
$903$	0	0
$904$	0	0
$905$	4.60413	0.153047
$906$	0	0
$907$	33.4334	1.11014	0.555069	−	0.831804i	$-0.312692\pi$
$907$	33.4334	1.11014	0.555069	+	0.831804i	$0.312692\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	19.8240	0.656798	0.328399	−	0.944539i	$-0.393491\pi$
$911$	19.8240	0.656798	0.328399	+	0.944539i	$0.393491\pi$
$912$	0	0
$913$	31.2792	1.03519
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−34.1914	−1.12910
$918$	0	0
$919$	30.0663	0.991796	0.495898	−	0.868381i	$-0.334839\pi$
$919$	30.0663	0.991796	0.495898	+	0.868381i	$0.334839\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−63.2541	−2.08204
$924$	0	0
$925$	−3.66003	−0.120341
$926$	0	0
$927$	0	0
$928$	0	0
$929$	29.4975	0.967781	0.483891	−	0.875129i	$-0.339223\pi$
$929$	29.4975	0.967781	0.483891	+	0.875129i	$0.339223\pi$
$930$	0	0
$931$	−0.178418	−0.00584740
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−2.63783	−0.0862663
$936$	0	0
$937$	−6.63418	−0.216729	−0.108365	−	0.994111i	$-0.534561\pi$
$937$	−6.63418	−0.216729	−0.108365	+	0.994111i	$0.534561\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	36.1025	1.17691	0.588454	−	0.808531i	$-0.299737\pi$
$941$	36.1025	1.17691	0.588454	+	0.808531i	$0.299737\pi$
$942$	0	0
$943$	12.1241	0.394814
$944$	0	0
$945$	0	0
$946$	0	0
$947$	23.6464	0.768404	0.384202	−	0.923249i	$-0.374477\pi$
$947$	23.6464	0.768404	0.384202	+	0.923249i	$0.374477\pi$
$948$	0	0
$949$	−61.9401	−2.01066
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−42.6546	−1.38172	−0.690859	−	0.722990i	$-0.742767\pi$
$953$	−42.6546	−1.38172	−0.690859	+	0.722990i	$0.742767\pi$
$954$	0	0
$955$	−58.8318	−1.90375
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−10.8715	−0.351059
$960$	0	0
$961$	−26.2285	−0.846080
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−62.6887	−2.01802
$966$	0	0
$967$	37.1538	1.19478	0.597392	−	0.801949i	$-0.296204\pi$
$967$	37.1538	1.19478	0.597392	+	0.801949i	$0.296204\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	41.5675	1.33396	0.666982	−	0.745074i	$-0.267586\pi$
$971$	41.5675	1.33396	0.666982	+	0.745074i	$0.267586\pi$
$972$	0	0
$973$	−17.1290	−0.549131
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−17.0478	−0.545407	−0.272704	−	0.962098i	$-0.587918\pi$
$977$	−17.0478	−0.545407	−0.272704	+	0.962098i	$0.587918\pi$
$978$	0	0
$979$	−35.5021	−1.13465
$980$	0	0
$981$	0	0
$982$	0	0
$983$	21.2149	0.676651	0.338325	−	0.941029i	$-0.390140\pi$
$983$	21.2149	0.676651	0.338325	+	0.941029i	$0.390140\pi$
$984$	0	0
$985$	−12.9904	−0.413910
$986$	0	0
$987$	0	0
$988$	0	0
$989$	20.1433	0.640520
$990$	0	0
$991$	−23.0424	−0.731967	−0.365984	−	0.930621i	$-0.619267\pi$
$991$	−23.0424	−0.731967	−0.365984	+	0.930621i	$0.619267\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−93.0632	−2.95030
$996$	0	0
$997$	11.7371	0.371717	0.185859	−	0.982576i	$-0.440493\pi$
$997$	11.7371	0.371717	0.185859	+	0.982576i	$0.440493\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9036.2.a.l.1.2		12
3.2	odd	2		3012.2.a.i.1.11	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3012.2.a.i.1.11	✓	12	3.2	odd	2
9036.2.a.l.1.2		12	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:	\( N \)	\(=\)	\( 9036 = 2^{2} \cdot 3^{2} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9036.a (trivial)

\(f(q)\)	\(=\)	\(q-3.63471 q^{5} +3.32358 q^{7} +O(q^{10})\)	\(q-3.63471 q^{5} +3.32358 q^{7} -2.80544 q^{11} -5.06285 q^{13} -0.258688 q^{17} -0.0440954 q^{19} -3.41394 q^{23} +8.21115 q^{25} -6.64269 q^{29} +2.18438 q^{31} -12.0803 q^{35} -0.445739 q^{37} -3.55134 q^{41} -5.90031 q^{43} +3.67346 q^{47} +4.04617 q^{49} -9.20577 q^{53} +10.1970 q^{55} +4.88626 q^{59} -0.241305 q^{61} +18.4020 q^{65} +5.53993 q^{67} +12.4938 q^{71} +12.2342 q^{73} -9.32408 q^{77} -8.12857 q^{79} -11.1495 q^{83} +0.940257 q^{85} +12.6547 q^{89} -16.8268 q^{91} +0.160274 q^{95} +1.59098 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 7 q^{5}+O(q^{10}) \) 12 * q - 7 * q^5	\( 12 q - 7 q^{5} + q^{11} + 17 q^{13} - 17 q^{17} + 3 q^{19} + 8 q^{23} + 19 q^{25} - 15 q^{29} - 2 q^{31} + 7 q^{35} + 14 q^{37} - 5 q^{41} + 9 q^{43} + 5 q^{47} + 22 q^{49} - 20 q^{53} - 4 q^{55} + q^{59} + 21 q^{61} - 8 q^{65} + 21 q^{67} + 17 q^{71} + 45 q^{73} - 40 q^{77} + 6 q^{79} + q^{83} + 31 q^{85} - 22 q^{89} + 32 q^{91} + 13 q^{95} + 29 q^{97}+O(q^{100}) \) 12 * q - 7 * q^5 + q^11 + 17 * q^13 - 17 * q^17 + 3 * q^19 + 8 * q^23 + 19 * q^25 - 15 * q^29 - 2 * q^31 + 7 * q^35 + 14 * q^37 - 5 * q^41 + 9 * q^43 + 5 * q^47 + 22 * q^49 - 20 * q^53 - 4 * q^55 + q^59 + 21 * q^61 - 8 * q^65 + 21 * q^67 + 17 * q^71 + 45 * q^73 - 40 * q^77 + 6 * q^79 + q^83 + 31 * q^85 - 22 * q^89 + 32 * q^91 + 13 * q^95 + 29 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more