LMFDB - Embedded newform 324.5.c.a.161.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [324,5,Mod(161,324)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(324, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("324.161");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 324 = 2^{2} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	324.c (of order $2$, degree $1$, 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:	$33.4918680392$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} + 6x^{6} + 121x^{5} + 1104x^{4} - 1647x^{3} + 6529x^{2} + 85254x + 440076 $ x^8 - 3x^7 + 6x^6 + 121x^5 + 1104x^4 - 1647x^3 + 6529x^2 + 85254*x + 440076
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{20} $
Twist minimal:	no (minimal twist has level 36)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.4
Root			$3.72537 + 4.42407i$ of defining polynomial
Character	$\chi$	$=$	324.161
Dual form			324.5.c.a.161.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.86801i q^{5} +61.8763 q^{7} +O(q^{10})$	$q-8.86801i q^{5} +61.8763 q^{7} -109.425i q^{11} -155.726 q^{13} -395.955i q^{17} +140.350 q^{19} +927.015i q^{23} +546.358 q^{25} -373.953i q^{29} -1043.66 q^{31} -548.719i q^{35} -194.990 q^{37} -2706.78i q^{41} +335.925 q^{43} -2850.67i q^{47} +1427.67 q^{49} -2765.43i q^{53} -970.381 q^{55} -5028.20i q^{59} -7047.18 q^{61} +1380.98i q^{65} +6879.93 q^{67} +821.812i q^{71} +4091.53 q^{73} -6770.81i q^{77} +7567.12 q^{79} -7826.31i q^{83} -3511.33 q^{85} -1283.28i q^{89} -9635.72 q^{91} -1244.63i q^{95} -3780.65 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 26 q^{7}+O(q^{10}) $ 8 * q - 26 * q^7	$ 8 q - 26 q^{7} + 10 q^{13} + 562 q^{19} - 706 q^{25} - 374 q^{31} + 16 q^{37} + 136 q^{43} + 654 q^{49} - 1818 q^{55} + 3874 q^{61} - 308 q^{67} - 7802 q^{73} + 4390 q^{79} + 6084 q^{85} + 15830 q^{91} - 14564 q^{97}+O(q^{100}) $ 8 * q - 26 * q^7 + 10 * q^13 + 562 * q^19 - 706 * q^25 - 374 * q^31 + 16 * q^37 + 136 * q^43 + 654 * q^49 - 1818 * q^55 + 3874 * q^61 - 308 * q^67 - 7802 * q^73 + 4390 * q^79 + 6084 * q^85 + 15830 * q^91 - 14564 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/324\mathbb{Z}\right)^\times$.

$n$	$163$	$245$
$\chi(n)$	$1$	$-1$

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
$5$	− 8.86801i	− 0.354720i	−0.984146	−	0.177360i	$-0.943244\pi$
$5$	− 8.86801i	− 0.354720i	0.984146	−	0.177360i	$-0.0567557\pi$
$6$	0	0
$7$	61.8763	1.26278	0.631390	−	0.775465i	$-0.282485\pi$
$7$	61.8763	1.26278	0.631390	+	0.775465i	$0.282485\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 109.425i	− 0.904338i	−0.891932	−	0.452169i	$-0.850650\pi$
$11$	− 109.425i	− 0.904338i	0.891932	−	0.452169i	$-0.149350\pi$
$12$	0	0
$13$	−155.726	−0.921453	−0.460727	−	0.887542i	$-0.652411\pi$
$13$	−155.726	−0.921453	−0.460727	+	0.887542i	$0.652411\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 395.955i	− 1.37009i	−0.728502	−	0.685044i	$-0.759783\pi$
$17$	− 395.955i	− 1.37009i	0.728502	−	0.685044i	$-0.240217\pi$
$18$	0	0
$19$	140.350	0.388782	0.194391	−	0.980924i	$-0.437727\pi$
$19$	140.350	0.388782	0.194391	+	0.980924i	$0.437727\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	927.015i	1.75239i	0.481956	+	0.876195i	$0.339926\pi$
$23$	927.015i	1.75239i	−0.481956	+	0.876195i	$0.660074\pi$
$24$	0	0
$25$	546.358	0.874174
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 373.953i	− 0.444653i	−0.974972	−	0.222326i	$-0.928635\pi$
$29$	− 373.953i	− 0.444653i	0.974972	−	0.222326i	$-0.0713650\pi$
$30$	0	0
$31$	−1043.66	−1.08602	−0.543008	−	0.839728i	$-0.682715\pi$
$31$	−1043.66	−1.08602	−0.543008	+	0.839728i	$0.682715\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 548.719i	− 0.447934i
$36$	0	0
$37$	−194.990	−0.142432	−0.0712162	−	0.997461i	$-0.522688\pi$
$37$	−194.990	−0.142432	−0.0712162	+	0.997461i	$0.522688\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 2706.78i	− 1.61022i	−0.593126	−	0.805110i	$-0.702106\pi$
$41$	− 2706.78i	− 1.61022i	0.593126	−	0.805110i	$-0.297894\pi$
$42$	0	0
$43$	335.925	0.181679	0.0908397	−	0.995866i	$-0.471045\pi$
$43$	335.925	0.181679	0.0908397	+	0.995866i	$0.471045\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 2850.67i	− 1.29048i	−0.763979	−	0.645241i	$-0.776757\pi$
$47$	− 2850.67i	− 1.29048i	0.763979	−	0.645241i	$-0.223243\pi$
$48$	0	0
$49$	1427.67	0.594615
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 2765.43i	− 0.984490i	−0.870457	−	0.492245i	$-0.836176\pi$
$53$	− 2765.43i	− 0.984490i	0.870457	−	0.492245i	$-0.163824\pi$
$54$	0	0
$55$	−970.381	−0.320787
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 5028.20i	− 1.44447i	−0.691647	−	0.722236i	$-0.743115\pi$
$59$	− 5028.20i	− 1.44447i	0.691647	−	0.722236i	$-0.256885\pi$
$60$	0	0
$61$	−7047.18	−1.89389	−0.946947	−	0.321391i	$-0.895850\pi$
$61$	−7047.18	−1.89389	−0.946947	+	0.321391i	$0.895850\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1380.98i	0.326858i
$66$	0	0
$67$	6879.93	1.53262	0.766309	−	0.642472i	$-0.222091\pi$
$67$	6879.93	1.53262	0.766309	+	0.642472i	$0.222091\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	821.812i	0.163026i	0.996672	+	0.0815128i	$0.0259752\pi$
$71$	821.812i	0.163026i	−0.996672	+	0.0815128i	$0.974025\pi$
$72$	0	0
$73$	4091.53	0.767786	0.383893	−	0.923378i	$-0.374583\pi$
$73$	4091.53	0.767786	0.383893	+	0.923378i	$0.374583\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 6770.81i	− 1.14198i
$78$	0	0
$79$	7567.12	1.21249	0.606243	−	0.795280i	$-0.292676\pi$
$79$	7567.12	1.21249	0.606243	+	0.795280i	$0.292676\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 7826.31i	− 1.13606i	−0.823008	−	0.568029i	$-0.807706\pi$
$83$	− 7826.31i	− 1.13606i	0.823008	−	0.568029i	$-0.192294\pi$
$84$	0	0
$85$	−3511.33	−0.485998
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 1283.28i	− 0.162010i	−0.996714	−	0.0810051i	$-0.974187\pi$
$89$	− 1283.28i	− 0.162010i	0.996714	−	0.0810051i	$-0.0258130\pi$
$90$	0	0
$91$	−9635.72	−1.16359
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 1244.63i	− 0.137909i
$96$	0	0
$97$	−3780.65	−0.401812	−0.200906	−	0.979611i	$-0.564389\pi$
$97$	−3780.65	−0.401812	−0.200906	+	0.979611i	$0.564389\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1195.11i	0.117156i	0.998283	+	0.0585779i	$0.0186566\pi$
$101$	1195.11i	0.117156i	−0.998283	+	0.0585779i	$0.981343\pi$
$102$	0	0
$103$	−13535.0	−1.27580	−0.637901	−	0.770118i	$-0.720197\pi$
$103$	−13535.0	−1.27580	−0.637901	+	0.770118i	$0.720197\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 18095.0i	− 1.58049i	−0.612792	−	0.790244i	$-0.709954\pi$
$107$	− 18095.0i	− 1.58049i	0.612792	−	0.790244i	$-0.290046\pi$
$108$	0	0
$109$	10676.5	0.898621	0.449311	−	0.893376i	$-0.351670\pi$
$109$	10676.5	0.898621	0.449311	+	0.893376i	$0.351670\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	12798.4i	1.00230i	0.865359	+	0.501152i	$0.167090\pi$
$113$	12798.4i	1.00230i	−0.865359	+	0.501152i	$0.832910\pi$
$114$	0	0
$115$	8220.77	0.621608
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 24500.2i	− 1.73012i
$120$	0	0
$121$	2667.18	0.182172
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 10387.6i	− 0.664807i
$126$	0	0
$127$	449.363	0.0278606	0.0139303	−	0.999903i	$-0.495566\pi$
$127$	449.363	0.0278606	0.0139303	+	0.999903i	$0.495566\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	11634.1i	0.677939i	0.940797	+	0.338970i	$0.110078\pi$
$131$	11634.1i	0.677939i	−0.940797	+	0.338970i	$0.889922\pi$
$132$	0	0
$133$	8684.35	0.490946
$134$	0	0
$135$	0	0
$136$	0	0
$137$	28175.4i	1.50117i	0.660775	+	0.750584i	$0.270228\pi$
$137$	28175.4i	1.50117i	−0.660775	+	0.750584i	$0.729772\pi$
$138$	0	0
$139$	−9495.16	−0.491442	−0.245721	−	0.969341i	$-0.579025\pi$
$139$	−9495.16	−0.491442	−0.245721	+	0.969341i	$0.579025\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	17040.3i	0.833305i
$144$	0	0
$145$	−3316.22	−0.157727
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1998.09i	0.0899998i	0.998987	+	0.0449999i	$0.0143288\pi$
$149$	1998.09i	0.0899998i	−0.998987	+	0.0449999i	$0.985671\pi$
$150$	0	0
$151$	−3711.23	−0.162766	−0.0813830	−	0.996683i	$-0.525934\pi$
$151$	−3711.23	−0.162766	−0.0813830	+	0.996683i	$0.525934\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	9255.19i	0.385232i
$156$	0	0
$157$	2916.86	0.118336	0.0591679	−	0.998248i	$-0.481155\pi$
$157$	2916.86	0.118336	0.0591679	+	0.998248i	$0.481155\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	57360.2i	2.21289i
$162$	0	0
$163$	−5975.21	−0.224894	−0.112447	−	0.993658i	$-0.535869\pi$
$163$	−5975.21	−0.224894	−0.112447	+	0.993658i	$0.535869\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 4169.57i	− 0.149506i	−0.997202	−	0.0747529i	$-0.976183\pi$
$167$	− 4169.57i	− 0.149506i	0.997202	−	0.0747529i	$-0.0238168\pi$
$168$	0	0
$169$	−4310.55	−0.150924
$170$	0	0
$171$	0	0
$172$	0	0
$173$	30842.4i	1.03052i	0.857035	+	0.515259i	$0.172304\pi$
$173$	30842.4i	1.03052i	−0.857035	+	0.515259i	$0.827696\pi$
$174$	0	0
$175$	33806.6	1.10389
$176$	0	0
$177$	0	0
$178$	0	0
$179$	5977.90i	0.186570i	0.995639	+	0.0932852i	$0.0297368\pi$
$179$	5977.90i	0.186570i	−0.995639	+	0.0932852i	$0.970263\pi$
$180$	0	0
$181$	−6456.49	−0.197079	−0.0985393	−	0.995133i	$-0.531417\pi$
$181$	−6456.49	−0.197079	−0.0985393	+	0.995133i	$0.531417\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1729.17i	0.0505237i
$186$	0	0
$187$	−43327.4	−1.23902
$188$	0	0
$189$	0	0
$190$	0	0
$191$	23836.9i	0.653405i	0.945127	+	0.326703i	$0.105938\pi$
$191$	23836.9i	0.653405i	−0.945127	+	0.326703i	$0.894062\pi$
$192$	0	0
$193$	40163.4	1.07824	0.539121	−	0.842229i	$-0.318757\pi$
$193$	40163.4	1.07824	0.539121	+	0.842229i	$0.318757\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 3622.11i	− 0.0933318i	−0.998911	−	0.0466659i	$-0.985140\pi$
$197$	− 3622.11i	− 0.0933318i	0.998911	−	0.0466659i	$-0.0148596\pi$
$198$	0	0
$199$	46416.3	1.17210	0.586049	−	0.810276i	$-0.300683\pi$
$199$	46416.3	1.17210	0.586049	+	0.810276i	$0.300683\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 23138.8i	− 0.561499i
$204$	0	0
$205$	−24003.7	−0.571177
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 15357.8i	− 0.351590i
$210$	0	0
$211$	−1528.19	−0.0343251	−0.0171625	−	0.999853i	$-0.505463\pi$
$211$	−1528.19	−0.0343251	−0.0171625	+	0.999853i	$0.505463\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 2978.99i	− 0.0644453i
$216$	0	0
$217$	−64577.8	−1.37140
$218$	0	0
$219$	0	0
$220$	0	0
$221$	61660.3i	1.26247i
$222$	0	0
$223$	27482.8	0.552651	0.276326	−	0.961064i	$-0.410883\pi$
$223$	27482.8	0.552651	0.276326	+	0.961064i	$0.410883\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	45239.4i	0.877940i	0.898502	+	0.438970i	$0.144657\pi$
$227$	45239.4i	0.877940i	−0.898502	+	0.438970i	$0.855343\pi$
$228$	0	0
$229$	−80148.6	−1.52836	−0.764178	−	0.645005i	$-0.776855\pi$
$229$	−80148.6	−1.52836	−0.764178	+	0.645005i	$0.776855\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	90756.0i	1.67172i	0.548942	+	0.835860i	$0.315031\pi$
$233$	90756.0i	1.67172i	−0.548942	+	0.835860i	$0.684969\pi$
$234$	0	0
$235$	−25279.8	−0.457760
$236$	0	0
$237$	0	0
$238$	0	0
$239$	817.902i	0.0143188i	0.999974	+	0.00715938i	$0.00227892\pi$
$239$	817.902i	0.0143188i	−0.999974	+	0.00715938i	$0.997721\pi$
$240$	0	0
$241$	74316.2	1.27953	0.639764	−	0.768572i	$-0.279032\pi$
$241$	74316.2	1.27953	0.639764	+	0.768572i	$0.279032\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 12660.6i	− 0.210922i
$246$	0	0
$247$	−21856.1	−0.358244
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 16304.4i	− 0.258796i	−0.991593	−	0.129398i	$-0.958695\pi$
$251$	− 16304.4i	− 0.258796i	0.991593	−	0.129398i	$-0.0413045\pi$
$252$	0	0
$253$	101439.	1.58475
$254$	0	0
$255$	0	0
$256$	0	0
$257$	125552.i	1.90090i	0.310880	+	0.950449i	$0.399376\pi$
$257$	125552.i	1.90090i	−0.310880	+	0.950449i	$0.600624\pi$
$258$	0	0
$259$	−12065.3	−0.179861
$260$	0	0
$261$	0	0
$262$	0	0
$263$	16133.3i	0.233245i	0.993176	+	0.116623i	$0.0372068\pi$
$263$	16133.3i	0.233245i	−0.993176	+	0.116623i	$0.962793\pi$
$264$	0	0
$265$	−24523.9	−0.349219
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 75561.0i	− 1.04422i	−0.852877	−	0.522112i	$-0.825144\pi$
$269$	− 75561.0i	− 1.04422i	0.852877	−	0.522112i	$-0.174856\pi$
$270$	0	0
$271$	−14842.2	−0.202097	−0.101049	−	0.994881i	$-0.532220\pi$
$271$	−14842.2	−0.202097	−0.101049	+	0.994881i	$0.532220\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 59785.2i	− 0.790549i
$276$	0	0
$277$	80762.3	1.05257	0.526283	−	0.850310i	$-0.323585\pi$
$277$	80762.3	1.05257	0.526283	+	0.850310i	$0.323585\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 66262.3i	− 0.839177i	−0.907714	−	0.419589i	$-0.862174\pi$
$281$	− 66262.3i	− 0.839177i	0.907714	−	0.419589i	$-0.137826\pi$
$282$	0	0
$283$	14577.9	0.182021	0.0910104	−	0.995850i	$-0.470990\pi$
$283$	14577.9	0.182021	0.0910104	+	0.995850i	$0.470990\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 167485.i	− 2.03335i
$288$	0	0
$289$	−73259.5	−0.877139
$290$	0	0
$291$	0	0
$292$	0	0
$293$	75087.6i	0.874648i	0.899304	+	0.437324i	$0.144074\pi$
$293$	75087.6i	0.874648i	−0.899304	+	0.437324i	$0.855926\pi$
$294$	0	0
$295$	−44590.2	−0.512383
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 144360.i	− 1.61475i
$300$	0	0
$301$	20785.8	0.229421
$302$	0	0
$303$	0	0
$304$	0	0
$305$	62494.4i	0.671802i
$306$	0	0
$307$	146994.	1.55963	0.779817	−	0.626007i	$-0.215312\pi$
$307$	146994.	1.55963	0.779817	+	0.626007i	$0.215312\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 145765.i	− 1.50707i	−0.657407	−	0.753536i	$-0.728347\pi$
$311$	− 145765.i	− 1.50707i	0.657407	−	0.753536i	$-0.271653\pi$
$312$	0	0
$313$	−75733.7	−0.773037	−0.386519	−	0.922282i	$-0.626323\pi$
$313$	−75733.7	−0.773037	−0.386519	+	0.922282i	$0.626323\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	92732.5i	0.922813i	0.887189	+	0.461406i	$0.152655\pi$
$317$	92732.5i	0.922813i	−0.887189	+	0.461406i	$0.847345\pi$
$318$	0	0
$319$	−40919.8	−0.402116
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 55572.4i	− 0.532665i
$324$	0	0
$325$	−85082.0	−0.805510
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 176389.i	− 1.62960i
$330$	0	0
$331$	10809.8	0.0986647	0.0493323	−	0.998782i	$-0.484291\pi$
$331$	10809.8	0.0986647	0.0493323	+	0.998782i	$0.484291\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 61011.2i	− 0.543651i
$336$	0	0
$337$	130647.	1.15038	0.575190	−	0.818020i	$-0.304928\pi$
$337$	130647.	1.15038	0.575190	+	0.818020i	$0.304928\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	114203.i	0.982125i
$342$	0	0
$343$	−60225.9	−0.511912
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 163561.i	− 1.35838i	−0.733961	−	0.679191i	$-0.762331\pi$
$347$	− 163561.i	− 1.35838i	0.733961	−	0.679191i	$-0.237669\pi$
$348$	0	0
$349$	14465.8	0.118766	0.0593831	−	0.998235i	$-0.481087\pi$
$349$	14465.8	0.118766	0.0593831	+	0.998235i	$0.481087\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	9639.63i	0.0773591i	0.999252	+	0.0386795i	$0.0123151\pi$
$353$	9639.63i	0.0773591i	−0.999252	+	0.0386795i	$0.987685\pi$
$354$	0	0
$355$	7287.84	0.0578285
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 56543.7i	− 0.438728i	−0.975643	−	0.219364i	$-0.929602\pi$
$359$	− 56543.7i	− 0.438728i	0.975643	−	0.219364i	$-0.0703982\pi$
$360$	0	0
$361$	−110623.	−0.848849
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 36283.7i	− 0.272349i
$366$	0	0
$367$	−159963.	−1.18765	−0.593824	−	0.804595i	$-0.702382\pi$
$367$	−159963.	−1.18765	−0.593824	+	0.804595i	$0.702382\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 171115.i	− 1.24320i
$372$	0	0
$373$	−104973.	−0.754502	−0.377251	−	0.926111i	$-0.623131\pi$
$373$	−104973.	−0.754502	−0.377251	+	0.926111i	$0.623131\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	58234.0i	0.409727i
$378$	0	0
$379$	25050.7	0.174398	0.0871989	−	0.996191i	$-0.472208\pi$
$379$	25050.7	0.174398	0.0871989	+	0.996191i	$0.472208\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	52543.4i	0.358196i	0.983831	+	0.179098i	$0.0573179\pi$
$383$	52543.4i	0.358196i	−0.983831	+	0.179098i	$0.942682\pi$
$384$	0	0
$385$	−60043.6	−0.405084
$386$	0	0
$387$	0	0
$388$	0	0
$389$	202167.i	1.33602i	0.744154	+	0.668008i	$0.232853\pi$
$389$	202167.i	1.33602i	−0.744154	+	0.668008i	$0.767147\pi$
$390$	0	0
$391$	367056.	2.40093
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 67105.3i	− 0.430093i
$396$	0	0
$397$	303603.	1.92631	0.963154	−	0.268951i	$-0.0866770\pi$
$397$	303603.	1.92631	0.963154	+	0.268951i	$0.0866770\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 118278.i	− 0.735553i	−0.929914	−	0.367776i	$-0.880119\pi$
$401$	− 118278.i	− 0.735553i	0.929914	−	0.367776i	$-0.119881\pi$
$402$	0	0
$403$	162525.	1.00071
$404$	0	0
$405$	0	0
$406$	0	0
$407$	21336.8i	0.128807i
$408$	0	0
$409$	41066.4	0.245494	0.122747	−	0.992438i	$-0.460830\pi$
$409$	41066.4	0.245494	0.122747	+	0.992438i	$0.460830\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 311127.i	− 1.82405i
$414$	0	0
$415$	−69403.7	−0.402983
$416$	0	0
$417$	0	0
$418$	0	0
$419$	100062.i	0.569953i	0.958535	+	0.284976i	$0.0919858\pi$
$419$	100062.i	0.569953i	−0.958535	+	0.284976i	$0.908014\pi$
$420$	0	0
$421$	−220514.	−1.24415	−0.622074	−	0.782958i	$-0.713710\pi$
$421$	−220514.	−1.24415	−0.622074	+	0.782958i	$0.713710\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 216333.i	− 1.19769i
$426$	0	0
$427$	−436053.	−2.39157
$428$	0	0
$429$	0	0
$430$	0	0
$431$	211457.i	1.13833i	0.822224	+	0.569163i	$0.192733\pi$
$431$	211457.i	1.13833i	−0.822224	+	0.569163i	$0.807267\pi$
$432$	0	0
$433$	−106655.	−0.568859	−0.284430	−	0.958697i	$-0.591804\pi$
$433$	−106655.	−0.568859	−0.284430	+	0.958697i	$0.591804\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	130107.i	0.681298i
$438$	0	0
$439$	214390.	1.11244	0.556218	−	0.831037i	$-0.312252\pi$
$439$	214390.	1.11244	0.556218	+	0.831037i	$0.312252\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 201627.i	− 1.02741i	−0.857968	−	0.513703i	$-0.828273\pi$
$443$	− 201627.i	− 1.02741i	0.857968	−	0.513703i	$-0.171727\pi$
$444$	0	0
$445$	−11380.2	−0.0574683
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 136895.i	− 0.679039i	−0.940599	−	0.339519i	$-0.889736\pi$
$449$	− 136895.i	− 0.679039i	0.940599	−	0.339519i	$-0.110264\pi$
$450$	0	0
$451$	−296189.	−1.45618
$452$	0	0
$453$	0	0
$454$	0	0
$455$	85449.6i	0.412750i
$456$	0	0
$457$	366676.	1.75570	0.877850	−	0.478936i	$-0.158977\pi$
$457$	366676.	1.75570	0.877850	+	0.478936i	$0.158977\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	48936.0i	0.230264i	0.993350	+	0.115132i	$0.0367291\pi$
$461$	48936.0i	0.230264i	−0.993350	+	0.115132i	$0.963271\pi$
$462$	0	0
$463$	−287028.	−1.33894	−0.669472	−	0.742837i	$-0.733479\pi$
$463$	−287028.	−1.33894	−0.669472	+	0.742837i	$0.733479\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 43613.3i	− 0.199979i	−0.994988	−	0.0999896i	$-0.968119\pi$
$467$	− 43613.3i	− 0.199979i	0.994988	−	0.0999896i	$-0.0318809\pi$
$468$	0	0
$469$	425704.	1.93536
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 36758.6i	− 0.164300i
$474$	0	0
$475$	76681.5	0.339863
$476$	0	0
$477$	0	0
$478$	0	0
$479$	229869.i	1.00187i	0.865486	+	0.500933i	$0.167010\pi$
$479$	229869.i	1.00187i	−0.865486	+	0.500933i	$0.832990\pi$
$480$	0	0
$481$	30364.9	0.131245
$482$	0	0
$483$	0	0
$484$	0	0
$485$	33526.8i	0.142531i
$486$	0	0
$487$	−15921.9	−0.0671330	−0.0335665	−	0.999436i	$-0.510687\pi$
$487$	−15921.9	−0.0671330	−0.0335665	+	0.999436i	$0.510687\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	168056.i	0.697092i	0.937292	+	0.348546i	$0.113324\pi$
$491$	168056.i	0.697092i	−0.937292	+	0.348546i	$0.886676\pi$
$492$	0	0
$493$	−148069.	−0.609213
$494$	0	0
$495$	0	0
$496$	0	0
$497$	50850.7i	0.205866i
$498$	0	0
$499$	356057.	1.42994	0.714971	−	0.699154i	$-0.246440\pi$
$499$	356057.	1.42994	0.714971	+	0.699154i	$0.246440\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	180472.i	0.713301i	0.934238	+	0.356651i	$0.116081\pi$
$503$	180472.i	0.713301i	−0.934238	+	0.356651i	$0.883919\pi$
$504$	0	0
$505$	10598.2	0.0415575
$506$	0	0
$507$	0	0
$508$	0	0
$509$	398746.i	1.53908i	0.638599	+	0.769540i	$0.279514\pi$
$509$	398746.i	1.53908i	−0.638599	+	0.769540i	$0.720486\pi$
$510$	0	0
$511$	253169.	0.969545
$512$	0	0
$513$	0	0
$514$	0	0
$515$	120028.i	0.452553i
$516$	0	0
$517$	−311935.	−1.16703
$518$	0	0
$519$	0	0
$520$	0	0
$521$	485387.i	1.78819i	0.447880	+	0.894094i	$0.352179\pi$
$521$	485387.i	1.78819i	−0.447880	+	0.894094i	$0.647821\pi$
$522$	0	0
$523$	80589.4	0.294628	0.147314	−	0.989090i	$-0.452937\pi$
$523$	80589.4	0.294628	0.147314	+	0.989090i	$0.452937\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	413243.i	1.48794i
$528$	0	0
$529$	−579515.	−2.07087
$530$	0	0
$531$	0	0
$532$	0	0
$533$	421515.i	1.48374i
$534$	0	0
$535$	−160467.	−0.560631
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 156223.i	− 0.537734i
$540$	0	0
$541$	−446700.	−1.52623	−0.763117	−	0.646260i	$-0.776332\pi$
$541$	−446700.	−1.52623	−0.763117	+	0.646260i	$0.776332\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 94679.4i	− 0.318759i
$546$	0	0
$547$	−253916.	−0.848625	−0.424312	−	0.905516i	$-0.639484\pi$
$547$	−253916.	−0.848625	−0.424312	+	0.905516i	$0.639484\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 52484.4i	− 0.172873i
$552$	0	0
$553$	468225.	1.53110
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 49265.2i	− 0.158792i	−0.996843	−	0.0793962i	$-0.974701\pi$
$557$	− 49265.2i	− 0.158792i	0.996843	−	0.0793962i	$-0.0252992\pi$
$558$	0	0
$559$	−52312.1	−0.167409
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 395329.i	− 1.24722i	−0.781737	−	0.623608i	$-0.785666\pi$
$563$	− 395329.i	− 1.24722i	0.781737	−	0.623608i	$-0.214334\pi$
$564$	0	0
$565$	113497.	0.355538
$566$	0	0
$567$	0	0
$568$	0	0
$569$	259200.i	0.800591i	0.916386	+	0.400296i	$0.131093\pi$
$569$	259200.i	0.800591i	−0.916386	+	0.400296i	$0.868907\pi$
$570$	0	0
$571$	452457.	1.38773	0.693864	−	0.720106i	$-0.255907\pi$
$571$	452457.	1.38773	0.693864	+	0.720106i	$0.255907\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	506482.i	1.53189i
$576$	0	0
$577$	499076.	1.49905	0.749523	−	0.661978i	$-0.230283\pi$
$577$	499076.	1.49905	0.749523	+	0.661978i	$0.230283\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 484263.i	− 1.43459i
$582$	0	0
$583$	−302607.	−0.890312
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 275.865i	0 0.000800608i	−1.00000		0.000400304i	$-0.999873\pi$
$587$	− 275.865i	0 0.000800608i	1.00000		0.000400304i	$-0.000127421\pi$
$588$	0	0
$589$	−146478.	−0.422223
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 10939.7i	− 0.0311096i	−0.999879	−	0.0155548i	$-0.995049\pi$
$593$	− 10939.7i	− 0.0311096i	0.999879	−	0.0155548i	$-0.00495145\pi$
$594$	0	0
$595$	−217268.	−0.613708
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 16053.4i	− 0.0447418i	−0.999750	−	0.0223709i	$-0.992879\pi$
$599$	− 16053.4i	− 0.0447418i	0.999750	−	0.0223709i	$-0.00712147\pi$
$600$	0	0
$601$	535133.	1.48154	0.740769	−	0.671760i	$-0.234461\pi$
$601$	535133.	1.48154	0.740769	+	0.671760i	$0.234461\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 23652.6i	− 0.0646201i
$606$	0	0
$607$	−75795.5	−0.205715	−0.102858	−	0.994696i	$-0.532799\pi$
$607$	−75795.5	−0.205715	−0.102858	+	0.994696i	$0.532799\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	443923.i	1.18912i
$612$	0	0
$613$	−279865.	−0.744778	−0.372389	−	0.928077i	$-0.621461\pi$
$613$	−279865.	−0.744778	−0.372389	+	0.928077i	$0.621461\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	2172.10i	0.00570570i	0.999996	+	0.00285285i	$0.000908091\pi$
$617$	2172.10i	0.00570570i	−0.999996	+	0.00285285i	$0.999092\pi$
$618$	0	0
$619$	7929.74	0.0206956	0.0103478	−	0.999946i	$-0.496706\pi$
$619$	7929.74	0.0206956	0.0103478	+	0.999946i	$0.496706\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 79404.7i	− 0.204583i
$624$	0	0
$625$	249357.	0.638353
$626$	0	0
$627$	0	0
$628$	0	0
$629$	77207.3i	0.195145i
$630$	0	0
$631$	546257.	1.37195	0.685975	−	0.727625i	$-0.259376\pi$
$631$	546257.	1.37195	0.685975	+	0.727625i	$0.259376\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 3984.95i	− 0.00988271i
$636$	0	0
$637$	−222325.	−0.547910
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 210026.i	− 0.511160i	−0.966788	−	0.255580i	$-0.917734\pi$
$641$	− 210026.i	− 0.511160i	0.966788	−	0.255580i	$-0.0822664\pi$
$642$	0	0
$643$	−79969.4	−0.193420	−0.0967101	−	0.995313i	$-0.530832\pi$
$643$	−79969.4	−0.193420	−0.0967101	+	0.995313i	$0.530832\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 247483.i	− 0.591203i	−0.955311	−	0.295601i	$-0.904480\pi$
$647$	− 247483.i	− 0.591203i	0.955311	−	0.295601i	$-0.0955200\pi$
$648$	0	0
$649$	−550211.	−1.30629
$650$	0	0
$651$	0	0
$652$	0	0
$653$	41230.6i	0.0966926i	0.998831	+	0.0483463i	$0.0153951\pi$
$653$	41230.6i	0.0966926i	−0.998831	+	0.0483463i	$0.984605\pi$
$654$	0	0
$655$	103171.	0.240479
$656$	0	0
$657$	0	0
$658$	0	0
$659$	819626.i	1.88732i	0.330921	+	0.943659i	$0.392641\pi$
$659$	819626.i	1.88732i	−0.330921	+	0.943659i	$0.607359\pi$
$660$	0	0
$661$	240147.	0.549636	0.274818	−	0.961496i	$-0.411382\pi$
$661$	240147.	0.549636	0.274818	+	0.961496i	$0.411382\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 77012.8i	− 0.174149i
$666$	0	0
$667$	346660.	0.779205
$668$	0	0
$669$	0	0
$670$	0	0
$671$	771137.i	1.71272i
$672$	0	0
$673$	−203278.	−0.448807	−0.224404	−	0.974496i	$-0.572043\pi$
$673$	−203278.	−0.448807	−0.224404	+	0.974496i	$0.572043\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 341062.i	− 0.744142i	−0.928204	−	0.372071i	$-0.878648\pi$
$677$	− 341062.i	− 0.744142i	0.928204	−	0.372071i	$-0.121352\pi$
$678$	0	0
$679$	−233932.	−0.507401
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 753067.i	− 1.61433i	−0.590326	−	0.807165i	$-0.701001\pi$
$683$	− 753067.i	− 1.61433i	0.590326	−	0.807165i	$-0.298999\pi$
$684$	0	0
$685$	249860.	0.532495
$686$	0	0
$687$	0	0
$688$	0	0
$689$	430649.i	0.907161i
$690$	0	0
$691$	−404888.	−0.847967	−0.423983	−	0.905670i	$-0.639369\pi$
$691$	−404888.	−0.847967	−0.423983	+	0.905670i	$0.639369\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	84203.1i	0.174325i
$696$	0	0
$697$	−1.07176e6	−2.20614
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 62736.6i	− 0.127669i	−0.997961	−	0.0638344i	$-0.979667\pi$
$701$	− 62736.6i	− 0.127669i	0.997961	−	0.0638344i	$-0.0203329\pi$
$702$	0	0
$703$	−27366.9	−0.0553751
$704$	0	0
$705$	0	0
$706$	0	0
$707$	73948.7i	0.147942i
$708$	0	0
$709$	957035.	1.90386	0.951931	−	0.306312i	$-0.0990951\pi$
$709$	957035.	1.90386	0.951931	+	0.306312i	$0.0990951\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 967489.i	− 1.90312i
$714$	0	0
$715$	151113.	0.295590
$716$	0	0
$717$	0	0
$718$	0	0
$719$	109853.i	0.212498i	0.994340	+	0.106249i	$0.0338841\pi$
$719$	109853.i	0.212498i	−0.994340	+	0.106249i	$0.966116\pi$
$720$	0	0
$721$	−837495.	−1.61106
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 204312.i	− 0.388704i
$726$	0	0
$727$	224454.	0.424677	0.212339	−	0.977196i	$-0.431892\pi$
$727$	224454.	0.424677	0.212339	+	0.977196i	$0.431892\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 133011.i	− 0.248917i
$732$	0	0
$733$	52793.6	0.0982592	0.0491296	−	0.998792i	$-0.484355\pi$
$733$	52793.6	0.0982592	0.0491296	+	0.998792i	$0.484355\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 752836.i	− 1.38601i
$738$	0	0
$739$	639035.	1.17013	0.585067	−	0.810985i	$-0.301068\pi$
$739$	639035.	1.17013	0.585067	+	0.810985i	$0.301068\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 748058.i	− 1.35506i	−0.735496	−	0.677529i	$-0.763051\pi$
$743$	− 748058.i	− 1.35506i	0.735496	−	0.677529i	$-0.236949\pi$
$744$	0	0
$745$	17719.0	0.0319248
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 1.11965e6i	− 1.99581i
$750$	0	0
$751$	−355572.	−0.630446	−0.315223	−	0.949018i	$-0.602079\pi$
$751$	−355572.	−0.630446	−0.315223	+	0.949018i	$0.602079\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	32911.2i	0.0577364i
$756$	0	0
$757$	173106.	0.302079	0.151040	−	0.988528i	$-0.451738\pi$
$757$	173106.	0.302079	0.151040	+	0.988528i	$0.451738\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 200363.i	− 0.345978i	−0.984924	−	0.172989i	$-0.944658\pi$
$761$	− 200363.i	− 0.345978i	0.984924	−	0.172989i	$-0.0553425\pi$
$762$	0	0
$763$	660623.	1.13476
$764$	0	0
$765$	0	0
$766$	0	0
$767$	783020.i	1.33101i
$768$	0	0
$769$	−401839.	−0.679516	−0.339758	−	0.940513i	$-0.610345\pi$
$769$	−401839.	−0.679516	−0.339758	+	0.940513i	$0.610345\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 817430.i	− 1.36802i	−0.729474	−	0.684009i	$-0.760235\pi$
$773$	− 817430.i	− 1.36802i	0.729474	−	0.684009i	$-0.239765\pi$
$774$	0	0
$775$	−570213.	−0.949366
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 379897.i	− 0.626024i
$780$	0	0
$781$	89926.8	0.147430
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 25866.7i	− 0.0419761i
$786$	0	0
$787$	98405.6	0.158880	0.0794402	−	0.996840i	$-0.474687\pi$
$787$	98405.6	0.158880	0.0794402	+	0.996840i	$0.474687\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	791919.i	1.26569i
$792$	0	0
$793$	1.09743e6	1.74513
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 331751.i	− 0.522271i	−0.965302	−	0.261136i	$-0.915903\pi$
$797$	− 331751.i	− 0.522271i	0.965302	−	0.261136i	$-0.0840970\pi$
$798$	0	0
$799$	−1.12874e6	−1.76807
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 447715.i	− 0.694338i
$804$	0	0
$805$	508671.	0.784955
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 317676.i	− 0.485387i	−0.970103	−	0.242693i	$-0.921969\pi$
$809$	− 317676.i	− 0.485387i	0.970103	−	0.242693i	$-0.0780309\pi$
$810$	0	0
$811$	−1.02772e6	−1.56255	−0.781274	−	0.624188i	$-0.785430\pi$
$811$	−1.02772e6	−1.56255	−0.781274	+	0.624188i	$0.785430\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	52988.2i	0.0797745i
$816$	0	0
$817$	47147.2	0.0706336
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 1.15868e6i	− 1.71900i	−0.511134	−	0.859501i	$-0.670774\pi$
$821$	− 1.15868e6i	− 1.71900i	0.511134	−	0.859501i	$-0.329226\pi$
$822$	0	0
$823$	884143.	1.30534	0.652669	−	0.757644i	$-0.273649\pi$
$823$	884143.	1.30534	0.652669	+	0.757644i	$0.273649\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 772574.i	− 1.12961i	−0.825224	−	0.564806i	$-0.808951\pi$
$827$	− 772574.i	− 1.12961i	0.825224	−	0.564806i	$-0.191049\pi$
$828$	0	0
$829$	−83501.8	−0.121503	−0.0607515	−	0.998153i	$-0.519350\pi$
$829$	−83501.8	−0.121503	−0.0607515	+	0.998153i	$0.519350\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 565294.i	− 0.814675i
$834$	0	0
$835$	−36975.7	−0.0530327
$836$	0	0
$837$	0	0
$838$	0	0
$839$	753803.i	1.07086i	0.844579	+	0.535432i	$0.179851\pi$
$839$	753803.i	1.07086i	−0.844579	+	0.535432i	$0.820149\pi$
$840$	0	0
$841$	567440.	0.802284
$842$	0	0
$843$	0	0
$844$	0	0
$845$	38226.0i	0.0535359i
$846$	0	0
$847$	165035.	0.230043
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 180759.i	− 0.249597i
$852$	0	0
$853$	−388249.	−0.533596	−0.266798	−	0.963752i	$-0.585966\pi$
$853$	−388249.	−0.533596	−0.266798	+	0.963752i	$0.585966\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1.17381e6i	1.59821i	0.601190	+	0.799106i	$0.294694\pi$
$857$	1.17381e6i	1.59821i	−0.601190	+	0.799106i	$0.705306\pi$
$858$	0	0
$859$	−520391.	−0.705250	−0.352625	−	0.935765i	$-0.614711\pi$
$859$	−520391.	−0.705250	−0.352625	+	0.935765i	$0.614711\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	387934.i	0.520878i	0.965490	+	0.260439i	$0.0838673\pi$
$863$	387934.i	0.520878i	−0.965490	+	0.260439i	$0.916133\pi$
$864$	0	0
$865$	273510.	0.365546
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 828032.i	− 1.09650i
$870$	0	0
$871$	−1.07138e6	−1.41224
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 642747.i	− 0.839506i
$876$	0	0
$877$	1.19270e6	1.55071	0.775355	−	0.631526i	$-0.217571\pi$
$877$	1.19270e6	1.55071	0.775355	+	0.631526i	$0.217571\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	563606.i	0.726146i	0.931761	+	0.363073i	$0.118272\pi$
$881$	563606.i	0.726146i	−0.931761	+	0.363073i	$0.881728\pi$
$882$	0	0
$883$	886399.	1.13686	0.568431	−	0.822731i	$-0.307551\pi$
$883$	886399.	1.13686	0.568431	+	0.822731i	$0.307551\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	220599.i	0.280386i	0.990124	+	0.140193i	$0.0447724\pi$
$887$	220599.i	0.280386i	−0.990124	+	0.140193i	$0.955228\pi$
$888$	0	0
$889$	27804.9	0.0351818
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 400093.i	− 0.501716i
$894$	0	0
$895$	53012.1	0.0661803
$896$	0	0
$897$	0	0
$898$	0	0
$899$	390280.i	0.482900i
$900$	0	0
$901$	−1.09499e6	−1.34884
$902$	0	0
$903$	0	0
$904$	0	0
$905$	57256.2i	0.0699078i
$906$	0	0
$907$	−285722.	−0.347320	−0.173660	−	0.984806i	$-0.555559\pi$
$907$	−285722.	−0.347320	−0.173660	+	0.984806i	$0.555559\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1.35700e6i	1.63509i	0.575861	+	0.817547i	$0.304667\pi$
$911$	1.35700e6i	1.63509i	−0.575861	+	0.817547i	$0.695333\pi$
$912$	0	0
$913$	−856393.	−1.02738
$914$	0	0
$915$	0	0
$916$	0	0
$917$	719876.i	0.856089i
$918$	0	0
$919$	−185850.	−0.220055	−0.110028	−	0.993929i	$-0.535094\pi$
$919$	−185850.	−0.220055	−0.110028	+	0.993929i	$0.535094\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 127977.i	− 0.150221i
$924$	0	0
$925$	−106534.	−0.124511
$926$	0	0
$927$	0	0
$928$	0	0
$929$	400366.i	0.463902i	0.972727	+	0.231951i	$0.0745109\pi$
$929$	400366.i	0.463902i	−0.972727	+	0.231951i	$0.925489\pi$
$930$	0	0
$931$	200374.	0.231176
$932$	0	0
$933$	0	0
$934$	0	0
$935$	384227.i	0.439506i
$936$	0	0
$937$	−351423.	−0.400268	−0.200134	−	0.979769i	$-0.564138\pi$
$937$	−351423.	−0.400268	−0.200134	+	0.979769i	$0.564138\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 1.04261e6i	− 1.17745i	−0.808333	−	0.588726i	$-0.799630\pi$
$941$	− 1.04261e6i	− 1.17745i	0.808333	−	0.588726i	$-0.200370\pi$
$942$	0	0
$943$	2.50922e6	2.82173
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 1.12321e6i	− 1.25245i	−0.779640	−	0.626227i	$-0.784598\pi$
$947$	− 1.12321e6i	− 1.25245i	0.779640	−	0.626227i	$-0.215402\pi$
$948$	0	0
$949$	−637156.	−0.707478
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 1.20759e6i	− 1.32964i	−0.747005	−	0.664819i	$-0.768509\pi$
$953$	− 1.20759e6i	− 1.32964i	0.747005	−	0.664819i	$-0.231491\pi$
$954$	0	0
$955$	211385.	0.231776
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1.74339e6i	1.89565i
$960$	0	0
$961$	165707.	0.179429
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 356169.i	− 0.382474i
$966$	0	0
$967$	453786.	0.485286	0.242643	−	0.970116i	$-0.421986\pi$
$967$	453786.	0.485286	0.242643	+	0.970116i	$0.421986\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	35561.3i	0.0377171i	0.999822	+	0.0188586i	$0.00600322\pi$
$971$	35561.3i	0.0377171i	−0.999822	+	0.0188586i	$0.993997\pi$
$972$	0	0
$973$	−587525.	−0.620584
$974$	0	0
$975$	0	0
$976$	0	0
$977$	660569.i	0.692036i	0.938228	+	0.346018i	$0.112466\pi$
$977$	660569.i	0.692036i	−0.938228	+	0.346018i	$0.887534\pi$
$978$	0	0
$979$	−140423.	−0.146512
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 397775.i	− 0.411653i	−0.978589	−	0.205826i	$-0.934012\pi$
$983$	− 397775.i	− 0.411653i	0.978589	−	0.205826i	$-0.0659882\pi$
$984$	0	0
$985$	−32120.9	−0.0331067
$986$	0	0
$987$	0	0
$988$	0	0
$989$	311407.i	0.318373i
$990$	0	0
$991$	910142.	0.926748	0.463374	−	0.886163i	$-0.346639\pi$
$991$	910142.	0.926748	0.463374	+	0.886163i	$0.346639\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 411620.i	− 0.415767i
$996$	0	0
$997$	−956404.	−0.962168	−0.481084	−	0.876675i	$-0.659757\pi$
$997$	−956404.	−0.962168	−0.481084	+	0.876675i	$0.659757\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	324.5.c.a.161.4		8
3.2	odd	2	inner	324.5.c.a.161.5		8
4.3	odd	2		1296.5.e.g.161.4		8
9.2	odd	6		36.5.g.a.5.3	✓	8
9.4	even	3		36.5.g.a.29.3	yes	8
9.5	odd	6		108.5.g.a.89.3		8
9.7	even	3		108.5.g.a.17.3		8
12.11	even	2		1296.5.e.g.161.5		8
36.7	odd	6		432.5.q.c.17.3		8
36.11	even	6		144.5.q.c.113.2		8
36.23	even	6		432.5.q.c.305.3		8
36.31	odd	6		144.5.q.c.65.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
36.5.g.a.5.3	✓	8	9.2	odd	6
36.5.g.a.29.3	yes	8	9.4	even	3
108.5.g.a.17.3		8	9.7	even	3
108.5.g.a.89.3		8	9.5	odd	6
144.5.q.c.65.2		8	36.31	odd	6
144.5.q.c.113.2		8	36.11	even	6
324.5.c.a.161.4		8	1.1	even	1	trivial
324.5.c.a.161.5		8	3.2	odd	2	inner
432.5.q.c.17.3		8	36.7	odd	6
432.5.q.c.305.3		8	36.23	even	6
1296.5.e.g.161.4		8	4.3	odd	2
1296.5.e.g.161.5		8	12.11	even	2

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:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	324.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-8.86801i q^{5} +61.8763 q^{7} +O(q^{10})\)	\(q-8.86801i q^{5} +61.8763 q^{7} -109.425i q^{11} -155.726 q^{13} -395.955i q^{17} +140.350 q^{19} +927.015i q^{23} +546.358 q^{25} -373.953i q^{29} -1043.66 q^{31} -548.719i q^{35} -194.990 q^{37} -2706.78i q^{41} +335.925 q^{43} -2850.67i q^{47} +1427.67 q^{49} -2765.43i q^{53} -970.381 q^{55} -5028.20i q^{59} -7047.18 q^{61} +1380.98i q^{65} +6879.93 q^{67} +821.812i q^{71} +4091.53 q^{73} -6770.81i q^{77} +7567.12 q^{79} -7826.31i q^{83} -3511.33 q^{85} -1283.28i q^{89} -9635.72 q^{91} -1244.63i q^{95} -3780.65 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 26 q^{7}+O(q^{10}) \) 8 * q - 26 * q^7	\( 8 q - 26 q^{7} + 10 q^{13} + 562 q^{19} - 706 q^{25} - 374 q^{31} + 16 q^{37} + 136 q^{43} + 654 q^{49} - 1818 q^{55} + 3874 q^{61} - 308 q^{67} - 7802 q^{73} + 4390 q^{79} + 6084 q^{85} + 15830 q^{91} - 14564 q^{97}+O(q^{100}) \) 8 * q - 26 * q^7 + 10 * q^13 + 562 * q^19 - 706 * q^25 - 374 * q^31 + 16 * q^37 + 136 * q^43 + 654 * q^49 - 1818 * q^55 + 3874 * q^61 - 308 * q^67 - 7802 * q^73 + 4390 * q^79 + 6084 * q^85 + 15830 * q^91 - 14564 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more