LMFDB - Embedded newform 9072.2.a.cm.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9072,2,Mod(1,9072)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9072, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("9072.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9072 = 2^{4} \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9072.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:	$72.4402847137$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.1425384.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 12x^{3} - 3x^{2} + 21x - 9 $ x^5 - 12x^3 - 3x^2 + 21*x - 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 504)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.96807$ of defining polynomial
Character	$\chi$	$=$	9072.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.52433 q^{5} +1.00000 q^{7} +O(q^{10})$	$q-3.52433 q^{5} +1.00000 q^{7} -4.16771 q^{11} +3.04103 q^{13} -3.79546 q^{17} -5.40544 q^{19} +8.97080 q^{23} +7.42092 q^{25} +2.92214 q^{29} +6.17535 q^{31} -3.52433 q^{35} -4.66649 q^{37} +3.67657 q^{41} +2.63554 q^{43} +1.72104 q^{47} +1.00000 q^{49} -11.4209 q^{53} +14.6884 q^{55} -1.12896 q^{59} +12.0719 q^{61} -10.7176 q^{65} -2.25565 q^{67} -9.92978 q^{71} +2.50106 q^{73} -4.16771 q^{77} +6.33527 q^{79} -11.0464 q^{83} +13.3765 q^{85} +13.4697 q^{89} +3.04103 q^{91} +19.0506 q^{95} +18.9374 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 3 q^{5} + 5 q^{7}+O(q^{10}) $ 5 * q - 3 * q^5 + 5 * q^7	$ 5 q - 3 q^{5} + 5 q^{7} - 4 q^{11} + 3 q^{13} - q^{19} - 8 q^{23} + 10 q^{25} - 9 q^{29} - 3 q^{31} - 3 q^{35} - 3 q^{37} - 12 q^{41} - 5 q^{43} - 3 q^{47} + 5 q^{49} - 30 q^{53} - 22 q^{55} - 7 q^{59} + 14 q^{61} - 11 q^{65} - 8 q^{67} - 9 q^{71} + 15 q^{73} - 4 q^{77} - 3 q^{79} - 20 q^{83} + 21 q^{85} + 12 q^{89} + 3 q^{91} + 12 q^{95} + 37 q^{97}+O(q^{100}) $ 5 * q - 3 * q^5 + 5 * q^7 - 4 * q^11 + 3 * q^13 - q^19 - 8 * q^23 + 10 * q^25 - 9 * q^29 - 3 * q^31 - 3 * q^35 - 3 * q^37 - 12 * q^41 - 5 * q^43 - 3 * q^47 + 5 * q^49 - 30 * q^53 - 22 * q^55 - 7 * q^59 + 14 * q^61 - 11 * q^65 - 8 * q^67 - 9 * q^71 + 15 * q^73 - 4 * q^77 - 3 * q^79 - 20 * q^83 + 21 * q^85 + 12 * q^89 + 3 * q^91 + 12 * q^95 + 37 * 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
$5$	−3.52433	−1.57613	−0.788065	−	0.615592i	$-0.788917\pi$
$5$	−3.52433	−1.57613	−0.788065	+	0.615592i	$0.788917\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−4.16771	−1.25661	−0.628306	−	0.777966i	$-0.716251\pi$
$11$	−4.16771	−1.25661	−0.628306	+	0.777966i	$0.716251\pi$
$12$	0	0
$13$	3.04103	0.843430	0.421715	−	0.906729i	$-0.361428\pi$
$13$	3.04103	0.843430	0.421715	+	0.906729i	$0.361428\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.79546	−0.920533	−0.460267	−	0.887781i	$-0.652246\pi$
$17$	−3.79546	−0.920533	−0.460267	+	0.887781i	$0.652246\pi$
$18$	0	0
$19$	−5.40544	−1.24009	−0.620047	−	0.784565i	$-0.712886\pi$
$19$	−5.40544	−1.24009	−0.620047	+	0.784565i	$0.712886\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	8.97080	1.87054	0.935271	−	0.353932i	$-0.115156\pi$
$23$	8.97080	1.87054	0.935271	+	0.353932i	$0.115156\pi$
$24$	0	0
$25$	7.42092	1.48418
$26$	0	0
$27$	0	0
$28$	0	0
$29$	2.92214	0.542628	0.271314	−	0.962491i	$-0.412542\pi$
$29$	2.92214	0.542628	0.271314	+	0.962491i	$0.412542\pi$
$30$	0	0
$31$	6.17535	1.10913	0.554563	−	0.832142i	$-0.312886\pi$
$31$	6.17535	1.10913	0.554563	+	0.832142i	$0.312886\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.52433	−0.595721
$36$	0	0
$37$	−4.66649	−0.767166	−0.383583	−	0.923506i	$-0.625310\pi$
$37$	−4.66649	−0.767166	−0.383583	+	0.923506i	$0.625310\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	3.67657	0.574183	0.287092	−	0.957903i	$-0.407312\pi$
$41$	3.67657	0.574183	0.287092	+	0.957903i	$0.407312\pi$
$42$	0	0
$43$	2.63554	0.401916	0.200958	−	0.979600i	$-0.435595\pi$
$43$	2.63554	0.401916	0.200958	+	0.979600i	$0.435595\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.72104	0.251039	0.125519	−	0.992091i	$-0.459940\pi$
$47$	1.72104	0.251039	0.125519	+	0.992091i	$0.459940\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−11.4209	−1.56878	−0.784392	−	0.620265i	$-0.787025\pi$
$53$	−11.4209	−1.56878	−0.784392	+	0.620265i	$0.787025\pi$
$54$	0	0
$55$	14.6884	1.98058
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−1.12896	−0.146979	−0.0734894	−	0.997296i	$-0.523413\pi$
$59$	−1.12896	−0.146979	−0.0734894	+	0.997296i	$0.523413\pi$
$60$	0	0
$61$	12.0719	1.54565	0.772826	−	0.634617i	$-0.218842\pi$
$61$	12.0719	1.54565	0.772826	+	0.634617i	$0.218842\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−10.7176	−1.32935
$66$	0	0
$67$	−2.25565	−0.275571	−0.137786	−	0.990462i	$-0.543998\pi$
$67$	−2.25565	−0.275571	−0.137786	+	0.990462i	$0.543998\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−9.92978	−1.17845	−0.589224	−	0.807970i	$-0.700566\pi$
$71$	−9.92978	−1.17845	−0.589224	+	0.807970i	$0.700566\pi$
$72$	0	0
$73$	2.50106	0.292727	0.146364	−	0.989231i	$-0.453243\pi$
$73$	2.50106	0.292727	0.146364	+	0.989231i	$0.453243\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−4.16771	−0.474955
$78$	0	0
$79$	6.33527	0.712773	0.356387	−	0.934339i	$-0.384009\pi$
$79$	6.33527	0.712773	0.356387	+	0.934339i	$0.384009\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−11.0464	−1.21250	−0.606249	−	0.795275i	$-0.707327\pi$
$83$	−11.0464	−1.21250	−0.606249	+	0.795275i	$0.707327\pi$
$84$	0	0
$85$	13.3765	1.45088
$86$	0	0
$87$	0	0
$88$	0	0
$89$	13.4697	1.42779	0.713895	−	0.700253i	$-0.246929\pi$
$89$	13.4697	1.42779	0.713895	+	0.700253i	$0.246929\pi$
$90$	0	0
$91$	3.04103	0.318786
$92$	0	0
$93$	0	0
$94$	0	0
$95$	19.0506	1.95455
$96$	0	0
$97$	18.9374	1.92280	0.961401	−	0.275150i	$-0.0887274\pi$
$97$	18.9374	1.92280	0.961401	+	0.275150i	$0.0887274\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−9.65881	−0.961087	−0.480544	−	0.876971i	$-0.659561\pi$
$101$	−9.65881	−0.961087	−0.480544	+	0.876971i	$0.659561\pi$
$102$	0	0
$103$	1.96317	0.193437	0.0967183	−	0.995312i	$-0.469165\pi$
$103$	1.96317	0.193437	0.0967183	+	0.995312i	$0.469165\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	18.5072	1.78916	0.894578	−	0.446912i	$-0.147476\pi$
$107$	18.5072	1.78916	0.894578	+	0.446912i	$0.147476\pi$
$108$	0	0
$109$	2.67237	0.255967	0.127983	−	0.991776i	$-0.459150\pi$
$109$	2.67237	0.255967	0.127983	+	0.991776i	$0.459150\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−9.31211	−0.876009	−0.438005	−	0.898973i	$-0.644315\pi$
$113$	−9.31211	−0.876009	−0.438005	+	0.898973i	$0.644315\pi$
$114$	0	0
$115$	−31.6161	−2.94822
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−3.79546	−0.347929
$120$	0	0
$121$	6.36982	0.579074
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−8.53213	−0.763137
$126$	0	0
$127$	−16.7663	−1.48777	−0.743883	−	0.668310i	$-0.767018\pi$
$127$	−16.7663	−1.48777	−0.743883	+	0.668310i	$0.767018\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−2.73131	−0.238636	−0.119318	−	0.992856i	$-0.538071\pi$
$131$	−2.73131	−0.238636	−0.119318	+	0.992856i	$0.538071\pi$
$132$	0	0
$133$	−5.40544	−0.468711
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−16.2574	−1.38896	−0.694482	−	0.719510i	$-0.744366\pi$
$137$	−16.2574	−1.38896	−0.694482	+	0.719510i	$0.744366\pi$
$138$	0	0
$139$	−1.30811	−0.110953	−0.0554764	−	0.998460i	$-0.517668\pi$
$139$	−1.30811	−0.110953	−0.0554764	+	0.998460i	$0.517668\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−12.6741	−1.05986
$144$	0	0
$145$	−10.2986	−0.855251
$146$	0	0
$147$	0	0
$148$	0	0
$149$	13.0916	1.07251	0.536253	−	0.844057i	$-0.319839\pi$
$149$	13.0916	1.07251	0.536253	+	0.844057i	$0.319839\pi$
$150$	0	0
$151$	−14.2187	−1.15710	−0.578549	−	0.815648i	$-0.696381\pi$
$151$	−14.2187	−1.15710	−0.578549	+	0.815648i	$0.696381\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−21.7640	−1.74812
$156$	0	0
$157$	−9.74087	−0.777406	−0.388703	−	0.921363i	$-0.627077\pi$
$157$	−9.74087	−0.777406	−0.388703	+	0.921363i	$0.627077\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	8.97080	0.706998
$162$	0	0
$163$	−18.2129	−1.42655	−0.713274	−	0.700886i	$-0.752788\pi$
$163$	−18.2129	−1.42655	−0.713274	+	0.700886i	$0.752788\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−15.6295	−1.20945	−0.604724	−	0.796435i	$-0.706716\pi$
$167$	−15.6295	−1.20945	−0.604724	+	0.796435i	$0.706716\pi$
$168$	0	0
$169$	−3.75215	−0.288627
$170$	0	0
$171$	0	0
$172$	0	0
$173$	10.4798	0.796766	0.398383	−	0.917219i	$-0.369572\pi$
$173$	10.4798	0.796766	0.398383	+	0.917219i	$0.369572\pi$
$174$	0	0
$175$	7.42092	0.560969
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−18.0874	−1.35192	−0.675958	−	0.736940i	$-0.736270\pi$
$179$	−18.0874	−1.35192	−0.675958	+	0.736940i	$0.736270\pi$
$180$	0	0
$181$	22.4020	1.66513	0.832564	−	0.553929i	$-0.186872\pi$
$181$	22.4020	1.66513	0.832564	+	0.553929i	$0.186872\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	16.4463	1.20915
$186$	0	0
$187$	15.8184	1.15675
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−3.38981	−0.245278	−0.122639	−	0.992451i	$-0.539136\pi$
$191$	−3.38981	−0.245278	−0.122639	+	0.992451i	$0.539136\pi$
$192$	0	0
$193$	15.9966	1.15146	0.575729	−	0.817641i	$-0.304718\pi$
$193$	15.9966	1.15146	0.575729	+	0.817641i	$0.304718\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	6.75083	0.480977	0.240488	−	0.970652i	$-0.422692\pi$
$197$	6.75083	0.480977	0.240488	+	0.970652i	$0.422692\pi$
$198$	0	0
$199$	−11.4209	−0.809608	−0.404804	−	0.914404i	$-0.632660\pi$
$199$	−11.4209	−0.809608	−0.404804	+	0.914404i	$0.632660\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2.92214	0.205094
$204$	0	0
$205$	−12.9574	−0.904987
$206$	0	0
$207$	0	0
$208$	0	0
$209$	22.5283	1.55832
$210$	0	0
$211$	−12.8029	−0.881390	−0.440695	−	0.897657i	$-0.645268\pi$
$211$	−12.8029	−0.881390	−0.440695	+	0.897657i	$0.645268\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−9.28851	−0.633471
$216$	0	0
$217$	6.17535	0.419210
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−11.5421	−0.776405
$222$	0	0
$223$	11.7271	0.785308	0.392654	−	0.919686i	$-0.371557\pi$
$223$	11.7271	0.785308	0.392654	+	0.919686i	$0.371557\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−12.6062	−0.836705	−0.418353	−	0.908285i	$-0.637392\pi$
$227$	−12.6062	−0.836705	−0.418353	+	0.908285i	$0.637392\pi$
$228$	0	0
$229$	−8.68960	−0.574225	−0.287113	−	0.957897i	$-0.592695\pi$
$229$	−8.68960	−0.574225	−0.287113	+	0.957897i	$0.592695\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−12.9251	−0.846748	−0.423374	−	0.905955i	$-0.639154\pi$
$233$	−12.9251	−0.846748	−0.423374	+	0.905955i	$0.639154\pi$
$234$	0	0
$235$	−6.06550	−0.395670
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−21.6041	−1.39745	−0.698725	−	0.715390i	$-0.746249\pi$
$239$	−21.6041	−1.39745	−0.698725	+	0.715390i	$0.746249\pi$
$240$	0	0
$241$	3.32359	0.214091	0.107046	−	0.994254i	$-0.465861\pi$
$241$	3.32359	0.214091	0.107046	+	0.994254i	$0.465861\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−3.52433	−0.225161
$246$	0	0
$247$	−16.4381	−1.04593
$248$	0	0
$249$	0	0
$250$	0	0
$251$	2.67969	0.169140	0.0845701	−	0.996418i	$-0.473048\pi$
$251$	2.67969	0.169140	0.0845701	+	0.996418i	$0.473048\pi$
$252$	0	0
$253$	−37.3877	−2.35055
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−7.50886	−0.468390	−0.234195	−	0.972190i	$-0.575245\pi$
$257$	−7.50886	−0.468390	−0.234195	+	0.972190i	$0.575245\pi$
$258$	0	0
$259$	−4.66649	−0.289962
$260$	0	0
$261$	0	0
$262$	0	0
$263$	18.7012	1.15317	0.576584	−	0.817038i	$-0.304385\pi$
$263$	18.7012	1.15317	0.576584	+	0.817038i	$0.304385\pi$
$264$	0	0
$265$	40.2511	2.47261
$266$	0	0
$267$	0	0
$268$	0	0
$269$	17.0035	1.03672	0.518360	−	0.855163i	$-0.326543\pi$
$269$	17.0035	1.03672	0.518360	+	0.855163i	$0.326543\pi$
$270$	0	0
$271$	−23.1427	−1.40582	−0.702910	−	0.711279i	$-0.748116\pi$
$271$	−23.1427	−1.40582	−0.702910	+	0.711279i	$0.748116\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−30.9282	−1.86504
$276$	0	0
$277$	−13.8618	−0.832877	−0.416438	−	0.909164i	$-0.636722\pi$
$277$	−13.8618	−0.832877	−0.416438	+	0.909164i	$0.636722\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−9.13852	−0.545158	−0.272579	−	0.962133i	$-0.587877\pi$
$281$	−9.13852	−0.545158	−0.272579	+	0.962133i	$0.587877\pi$
$282$	0	0
$283$	6.90842	0.410663	0.205331	−	0.978692i	$-0.434173\pi$
$283$	6.90842	0.410663	0.205331	+	0.978692i	$0.434173\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3.67657	0.217021
$288$	0	0
$289$	−2.59451	−0.152618
$290$	0	0
$291$	0	0
$292$	0	0
$293$	18.6017	1.08672	0.543361	−	0.839499i	$-0.317152\pi$
$293$	18.6017	1.08672	0.543361	+	0.839499i	$0.317152\pi$
$294$	0	0
$295$	3.97885	0.231657
$296$	0	0
$297$	0	0
$298$	0	0
$299$	27.2805	1.57767
$300$	0	0
$301$	2.63554	0.151910
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−42.5455	−2.43615
$306$	0	0
$307$	−13.9553	−0.796473	−0.398236	−	0.917283i	$-0.630378\pi$
$307$	−13.9553	−0.796473	−0.398236	+	0.917283i	$0.630378\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−17.7228	−1.00497	−0.502484	−	0.864587i	$-0.667580\pi$
$311$	−17.7228	−1.00497	−0.502484	+	0.864587i	$0.667580\pi$
$312$	0	0
$313$	14.7661	0.834630	0.417315	−	0.908762i	$-0.362971\pi$
$313$	14.7661	0.834630	0.417315	+	0.908762i	$0.362971\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2.57639	0.144705	0.0723523	−	0.997379i	$-0.476949\pi$
$317$	2.57639	0.144705	0.0723523	+	0.997379i	$0.476949\pi$
$318$	0	0
$319$	−12.1786	−0.681872
$320$	0	0
$321$	0	0
$322$	0	0
$323$	20.5161	1.14155
$324$	0	0
$325$	22.5672	1.25180
$326$	0	0
$327$	0	0
$328$	0	0
$329$	1.72104	0.0948838
$330$	0	0
$331$	0.317554	0.0174543	0.00872716	−	0.999962i	$-0.497222\pi$
$331$	0.317554	0.0174543	0.00872716	+	0.999962i	$0.497222\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	7.94965	0.434336
$336$	0	0
$337$	−13.9552	−0.760189	−0.380095	−	0.924948i	$-0.624109\pi$
$337$	−13.9552	−0.760189	−0.380095	+	0.924948i	$0.624109\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−25.7371	−1.39374
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	24.0801	1.29269	0.646344	−	0.763046i	$-0.276297\pi$
$347$	24.0801	1.29269	0.646344	+	0.763046i	$0.276297\pi$
$348$	0	0
$349$	−26.3233	−1.40905	−0.704526	−	0.709678i	$-0.748840\pi$
$349$	−26.3233	−1.40905	−0.704526	+	0.709678i	$0.748840\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−2.61003	−0.138918	−0.0694590	−	0.997585i	$-0.522127\pi$
$353$	−2.61003	−0.138918	−0.0694590	+	0.997585i	$0.522127\pi$
$354$	0	0
$355$	34.9958	1.85739
$356$	0	0
$357$	0	0
$358$	0	0
$359$	3.48082	0.183711	0.0918553	−	0.995772i	$-0.470720\pi$
$359$	3.48082	0.183711	0.0918553	+	0.995772i	$0.470720\pi$
$360$	0	0
$361$	10.2188	0.537832
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−8.81458	−0.461376
$366$	0	0
$367$	−1.92198	−0.100327	−0.0501633	−	0.998741i	$-0.515974\pi$
$367$	−1.92198	−0.100327	−0.0501633	+	0.998741i	$0.515974\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−11.4209	−0.592945
$372$	0	0
$373$	32.2456	1.66961	0.834806	−	0.550545i	$-0.185580\pi$
$373$	32.2456	1.66961	0.834806	+	0.550545i	$0.185580\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	8.88631	0.457668
$378$	0	0
$379$	4.98892	0.256264	0.128132	−	0.991757i	$-0.459102\pi$
$379$	4.98892	0.256264	0.128132	+	0.991757i	$0.459102\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−0.568755	−0.0290620	−0.0145310	−	0.999894i	$-0.504626\pi$
$383$	−0.568755	−0.0290620	−0.0145310	+	0.999894i	$0.504626\pi$
$384$	0	0
$385$	14.6884	0.748590
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−24.8518	−1.26003	−0.630017	−	0.776581i	$-0.716952\pi$
$389$	−24.8518	−1.26003	−0.630017	+	0.776581i	$0.716952\pi$
$390$	0	0
$391$	−34.0483	−1.72190
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−22.3276	−1.12342
$396$	0	0
$397$	−3.87576	−0.194519	−0.0972593	−	0.995259i	$-0.531008\pi$
$397$	−3.87576	−0.194519	−0.0972593	+	0.995259i	$0.531008\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	14.8640	0.742273	0.371136	−	0.928578i	$-0.378968\pi$
$401$	14.8640	0.742273	0.371136	+	0.928578i	$0.378968\pi$
$402$	0	0
$403$	18.7794	0.935469
$404$	0	0
$405$	0	0
$406$	0	0
$407$	19.4486	0.964031
$408$	0	0
$409$	−23.3414	−1.15416	−0.577078	−	0.816689i	$-0.695807\pi$
$409$	−23.3414	−1.15416	−0.577078	+	0.816689i	$0.695807\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−1.12896	−0.0555527
$414$	0	0
$415$	38.9311	1.91105
$416$	0	0
$417$	0	0
$418$	0	0
$419$	21.5498	1.05278	0.526389	−	0.850244i	$-0.323545\pi$
$419$	21.5498	1.05278	0.526389	+	0.850244i	$0.323545\pi$
$420$	0	0
$421$	−19.6780	−0.959048	−0.479524	−	0.877529i	$-0.659191\pi$
$421$	−19.6780	−0.959048	−0.479524	+	0.877529i	$0.659191\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−28.1658	−1.36624
$426$	0	0
$427$	12.0719	0.584202
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−19.3862	−0.933798	−0.466899	−	0.884311i	$-0.654629\pi$
$431$	−19.3862	−0.933798	−0.466899	+	0.884311i	$0.654629\pi$
$432$	0	0
$433$	34.9983	1.68191	0.840956	−	0.541104i	$-0.181993\pi$
$433$	34.9983	1.68191	0.840956	+	0.541104i	$0.181993\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−48.4912	−2.31965
$438$	0	0
$439$	3.88875	0.185600	0.0927999	−	0.995685i	$-0.470418\pi$
$439$	3.88875	0.185600	0.0927999	+	0.995685i	$0.470418\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−21.7264	−1.03225	−0.516126	−	0.856513i	$-0.672626\pi$
$443$	−21.7264	−1.03225	−0.516126	+	0.856513i	$0.672626\pi$
$444$	0	0
$445$	−47.4718	−2.25038
$446$	0	0
$447$	0	0
$448$	0	0
$449$	19.3098	0.911287	0.455643	−	0.890162i	$-0.349409\pi$
$449$	19.3098	0.911287	0.455643	+	0.890162i	$0.349409\pi$
$450$	0	0
$451$	−15.3229	−0.721526
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−10.7176	−0.502449
$456$	0	0
$457$	−16.0773	−0.752066	−0.376033	−	0.926606i	$-0.622712\pi$
$457$	−16.0773	−0.752066	−0.376033	+	0.926606i	$0.622712\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−33.3776	−1.55455	−0.777274	−	0.629162i	$-0.783398\pi$
$461$	−33.3776	−1.55455	−0.777274	+	0.629162i	$0.783398\pi$
$462$	0	0
$463$	−13.3525	−0.620541	−0.310271	−	0.950648i	$-0.600420\pi$
$463$	−13.3525	−0.620541	−0.310271	+	0.950648i	$0.600420\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	11.4976	0.532047	0.266023	−	0.963967i	$-0.414290\pi$
$467$	11.4976	0.532047	0.266023	+	0.963967i	$0.414290\pi$
$468$	0	0
$469$	−2.25565	−0.104156
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−10.9842	−0.505052
$474$	0	0
$475$	−40.1134	−1.84053
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−4.96157	−0.226700	−0.113350	−	0.993555i	$-0.536158\pi$
$479$	−4.96157	−0.226700	−0.113350	+	0.993555i	$0.536158\pi$
$480$	0	0
$481$	−14.1909	−0.647051
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−66.7417	−3.03059
$486$	0	0
$487$	−26.4801	−1.19993	−0.599964	−	0.800027i	$-0.704818\pi$
$487$	−26.4801	−1.19993	−0.599964	+	0.800027i	$0.704818\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−21.4038	−0.965940	−0.482970	−	0.875637i	$-0.660442\pi$
$491$	−21.4038	−0.965940	−0.482970	+	0.875637i	$0.660442\pi$
$492$	0	0
$493$	−11.0909	−0.499507
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−9.92978	−0.445411
$498$	0	0
$499$	13.2711	0.594095	0.297048	−	0.954863i	$-0.403998\pi$
$499$	13.2711	0.594095	0.297048	+	0.954863i	$0.403998\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−32.3009	−1.44022	−0.720112	−	0.693857i	$-0.755910\pi$
$503$	−32.3009	−1.44022	−0.720112	+	0.693857i	$0.755910\pi$
$504$	0	0
$505$	34.0409	1.51480
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−22.6802	−1.00528	−0.502642	−	0.864495i	$-0.667639\pi$
$509$	−22.6802	−1.00528	−0.502642	+	0.864495i	$0.667639\pi$
$510$	0	0
$511$	2.50106	0.110641
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−6.91886	−0.304881
$516$	0	0
$517$	−7.17278	−0.315459
$518$	0	0
$519$	0	0
$520$	0	0
$521$	23.2159	1.01710	0.508552	−	0.861031i	$-0.330181\pi$
$521$	23.2159	1.01710	0.508552	+	0.861031i	$0.330181\pi$
$522$	0	0
$523$	20.4196	0.892885	0.446442	−	0.894812i	$-0.352691\pi$
$523$	20.4196	0.892885	0.446442	+	0.894812i	$0.352691\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−23.4383	−1.02099
$528$	0	0
$529$	57.4753	2.49893
$530$	0	0
$531$	0	0
$532$	0	0
$533$	11.1805	0.484283
$534$	0	0
$535$	−65.2254	−2.81994
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−4.16771	−0.179516
$540$	0	0
$541$	−39.4226	−1.69491	−0.847455	−	0.530868i	$-0.821866\pi$
$541$	−39.4226	−1.69491	−0.847455	+	0.530868i	$0.821866\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−9.41832	−0.403437
$546$	0	0
$547$	−24.5822	−1.05106	−0.525529	−	0.850776i	$-0.676133\pi$
$547$	−24.5822	−1.05106	−0.525529	+	0.850776i	$0.676133\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−15.7955	−0.672909
$552$	0	0
$553$	6.33527	0.269403
$554$	0	0
$555$	0	0
$556$	0	0
$557$	12.4619	0.528026	0.264013	−	0.964519i	$-0.414954\pi$
$557$	12.4619	0.528026	0.264013	+	0.964519i	$0.414954\pi$
$558$	0	0
$559$	8.01475	0.338988
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−10.4990	−0.442479	−0.221240	−	0.975219i	$-0.571010\pi$
$563$	−10.4990	−0.442479	−0.221240	+	0.975219i	$0.571010\pi$
$564$	0	0
$565$	32.8190	1.38070
$566$	0	0
$567$	0	0
$568$	0	0
$569$	10.2340	0.429030	0.214515	−	0.976721i	$-0.431183\pi$
$569$	10.2340	0.429030	0.214515	+	0.976721i	$0.431183\pi$
$570$	0	0
$571$	4.69973	0.196677	0.0983387	−	0.995153i	$-0.468647\pi$
$571$	4.69973	0.196677	0.0983387	+	0.995153i	$0.468647\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	66.5716	2.77623
$576$	0	0
$577$	−27.3980	−1.14060	−0.570298	−	0.821438i	$-0.693172\pi$
$577$	−27.3980	−1.14060	−0.570298	+	0.821438i	$0.693172\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−11.0464	−0.458281
$582$	0	0
$583$	47.5991	1.97135
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−0.669885	−0.0276491	−0.0138246	−	0.999904i	$-0.504401\pi$
$587$	−0.669885	−0.0276491	−0.0138246	+	0.999904i	$0.504401\pi$
$588$	0	0
$589$	−33.3805	−1.37542
$590$	0	0
$591$	0	0
$592$	0	0
$593$	9.83861	0.404023	0.202012	−	0.979383i	$-0.435252\pi$
$593$	9.83861	0.404023	0.202012	+	0.979383i	$0.435252\pi$
$594$	0	0
$595$	13.3765	0.548381
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−0.0270318	−0.00110449	−0.000552245	−	1.00000i	$-0.500176\pi$
$599$	−0.0270318	−0.00110449	−0.000552245		1.00000i	$0.500176\pi$
$600$	0	0
$601$	−36.3352	−1.48215	−0.741073	−	0.671425i	$-0.765683\pi$
$601$	−36.3352	−1.48215	−0.741073	+	0.671425i	$0.765683\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−22.4493	−0.912696
$606$	0	0
$607$	−21.3181	−0.865273	−0.432637	−	0.901568i	$-0.642417\pi$
$607$	−21.3181	−0.865273	−0.432637	+	0.901568i	$0.642417\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	5.23372	0.211734
$612$	0	0
$613$	−12.5229	−0.505793	−0.252897	−	0.967493i	$-0.581383\pi$
$613$	−12.5229	−0.505793	−0.252897	+	0.967493i	$0.581383\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	44.5382	1.79304	0.896521	−	0.443001i	$-0.146086\pi$
$617$	44.5382	1.79304	0.896521	+	0.443001i	$0.146086\pi$
$618$	0	0
$619$	−28.8200	−1.15837	−0.579186	−	0.815195i	$-0.696629\pi$
$619$	−28.8200	−1.15837	−0.579186	+	0.815195i	$0.696629\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	13.4697	0.539654
$624$	0	0
$625$	−7.03455	−0.281382
$626$	0	0
$627$	0	0
$628$	0	0
$629$	17.7115	0.706202
$630$	0	0
$631$	35.3192	1.40604	0.703018	−	0.711172i	$-0.251835\pi$
$631$	35.3192	1.40604	0.703018	+	0.711172i	$0.251835\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	59.0899	2.34491
$636$	0	0
$637$	3.04103	0.120490
$638$	0	0
$639$	0	0
$640$	0	0
$641$	36.4522	1.43978	0.719888	−	0.694090i	$-0.244193\pi$
$641$	36.4522	1.43978	0.719888	+	0.694090i	$0.244193\pi$
$642$	0	0
$643$	−39.7258	−1.56663	−0.783317	−	0.621623i	$-0.786474\pi$
$643$	−39.7258	−1.56663	−0.783317	+	0.621623i	$0.786474\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−0.649904	−0.0255504	−0.0127752	−	0.999918i	$-0.504067\pi$
$647$	−0.649904	−0.0255504	−0.0127752	+	0.999918i	$0.504067\pi$
$648$	0	0
$649$	4.70520	0.184695
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−14.8048	−0.579356	−0.289678	−	0.957124i	$-0.593548\pi$
$653$	−14.8048	−0.579356	−0.289678	+	0.957124i	$0.593548\pi$
$654$	0	0
$655$	9.62606	0.376121
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−7.07426	−0.275574	−0.137787	−	0.990462i	$-0.543999\pi$
$659$	−7.07426	−0.275574	−0.137787	+	0.990462i	$0.543999\pi$
$660$	0	0
$661$	−24.6837	−0.960086	−0.480043	−	0.877245i	$-0.659379\pi$
$661$	−24.6837	−0.960086	−0.480043	+	0.877245i	$0.659379\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	19.0506	0.738750
$666$	0	0
$667$	26.2139	1.01501
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−50.3123	−1.94229
$672$	0	0
$673$	−33.1017	−1.27598	−0.637988	−	0.770046i	$-0.720233\pi$
$673$	−33.1017	−1.27598	−0.637988	+	0.770046i	$0.720233\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−11.1801	−0.429685	−0.214842	−	0.976649i	$-0.568924\pi$
$677$	−11.1801	−0.429685	−0.214842	+	0.976649i	$0.568924\pi$
$678$	0	0
$679$	18.9374	0.726751
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−33.1925	−1.27008	−0.635039	−	0.772480i	$-0.719016\pi$
$683$	−33.1925	−1.27008	−0.635039	+	0.772480i	$0.719016\pi$
$684$	0	0
$685$	57.2965	2.18919
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−34.7313	−1.32316
$690$	0	0
$691$	−17.9292	−0.682059	−0.341030	−	0.940053i	$-0.610776\pi$
$691$	−17.9292	−0.682059	−0.341030	+	0.940053i	$0.610776\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	4.61023	0.174876
$696$	0	0
$697$	−13.9543	−0.528555
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−22.6754	−0.856438	−0.428219	−	0.903675i	$-0.640859\pi$
$701$	−22.6754	−0.856438	−0.428219	+	0.903675i	$0.640859\pi$
$702$	0	0
$703$	25.2245	0.951358
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−9.65881	−0.363257
$708$	0	0
$709$	8.95444	0.336291	0.168146	−	0.985762i	$-0.446222\pi$
$709$	8.95444	0.336291	0.168146	+	0.985762i	$0.446222\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	55.3978	2.07467
$714$	0	0
$715$	44.6678	1.67048
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−33.9896	−1.26760	−0.633799	−	0.773498i	$-0.718505\pi$
$719$	−33.9896	−1.26760	−0.633799	+	0.773498i	$0.718505\pi$
$720$	0	0
$721$	1.96317	0.0731122
$722$	0	0
$723$	0	0
$724$	0	0
$725$	21.6850	0.805359
$726$	0	0
$727$	−13.9021	−0.515599	−0.257799	−	0.966198i	$-0.582997\pi$
$727$	−13.9021	−0.515599	−0.257799	+	0.966198i	$0.582997\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−10.0031	−0.369977
$732$	0	0
$733$	28.6361	1.05770	0.528849	−	0.848716i	$-0.322624\pi$
$733$	28.6361	1.05770	0.528849	+	0.848716i	$0.322624\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	9.40089	0.346286
$738$	0	0
$739$	−16.2398	−0.597389	−0.298695	−	0.954349i	$-0.596551\pi$
$739$	−16.2398	−0.597389	−0.298695	+	0.954349i	$0.596551\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	10.2045	0.374368	0.187184	−	0.982325i	$-0.440064\pi$
$743$	10.2045	0.374368	0.187184	+	0.982325i	$0.440064\pi$
$744$	0	0
$745$	−46.1392	−1.69041
$746$	0	0
$747$	0	0
$748$	0	0
$749$	18.5072	0.676237
$750$	0	0
$751$	−10.1907	−0.371862	−0.185931	−	0.982563i	$-0.559530\pi$
$751$	−10.1907	−0.371862	−0.185931	+	0.982563i	$0.559530\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	50.1113	1.82374
$756$	0	0
$757$	−50.9880	−1.85319	−0.926595	−	0.376061i	$-0.877278\pi$
$757$	−50.9880	−1.85319	−0.926595	+	0.376061i	$0.877278\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	9.09138	0.329562	0.164781	−	0.986330i	$-0.447308\pi$
$761$	9.09138	0.329562	0.164781	+	0.986330i	$0.447308\pi$
$762$	0	0
$763$	2.67237	0.0967463
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−3.43321	−0.123966
$768$	0	0
$769$	20.9496	0.755462	0.377731	−	0.925915i	$-0.376704\pi$
$769$	20.9496	0.755462	0.377731	+	0.925915i	$0.376704\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0.323229	0.0116258	0.00581288	−	0.999983i	$-0.498150\pi$
$773$	0.323229	0.0116258	0.00581288	+	0.999983i	$0.498150\pi$
$774$	0	0
$775$	45.8268	1.64615
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−19.8735	−0.712041
$780$	0	0
$781$	41.3844	1.48085
$782$	0	0
$783$	0	0
$784$	0	0
$785$	34.3300	1.22529
$786$	0	0
$787$	54.7213	1.95060	0.975302	−	0.220874i	$-0.0708909\pi$
$787$	54.7213	1.95060	0.975302	+	0.220874i	$0.0708909\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−9.31211	−0.331100
$792$	0	0
$793$	36.7111	1.30365
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−30.0996	−1.06618	−0.533091	−	0.846058i	$-0.678970\pi$
$797$	−30.0996	−1.06618	−0.533091	+	0.846058i	$0.678970\pi$
$798$	0	0
$799$	−6.53212	−0.231090
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−10.4237	−0.367845
$804$	0	0
$805$	−31.6161	−1.11432
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−8.81348	−0.309866	−0.154933	−	0.987925i	$-0.549516\pi$
$809$	−8.81348	−0.309866	−0.154933	+	0.987925i	$0.549516\pi$
$810$	0	0
$811$	42.3997	1.48886	0.744428	−	0.667702i	$-0.232722\pi$
$811$	42.3997	1.48886	0.744428	+	0.667702i	$0.232722\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	64.1884	2.24842
$816$	0	0
$817$	−14.2463	−0.498413
$818$	0	0
$819$	0	0
$820$	0	0
$821$	19.3581	0.675602	0.337801	−	0.941218i	$-0.390317\pi$
$821$	19.3581	0.675602	0.337801	+	0.941218i	$0.390317\pi$
$822$	0	0
$823$	−24.5992	−0.857474	−0.428737	−	0.903429i	$-0.641041\pi$
$823$	−24.5992	−0.857474	−0.428737	+	0.903429i	$0.641041\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−39.3747	−1.36919	−0.684597	−	0.728922i	$-0.740022\pi$
$827$	−39.3747	−1.36919	−0.684597	+	0.728922i	$0.740022\pi$
$828$	0	0
$829$	7.25597	0.252010	0.126005	−	0.992030i	$-0.459784\pi$
$829$	7.25597	0.252010	0.126005	+	0.992030i	$0.459784\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−3.79546	−0.131505
$834$	0	0
$835$	55.0836	1.90625
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−11.4582	−0.395581	−0.197790	−	0.980244i	$-0.563377\pi$
$839$	−11.4582	−0.395581	−0.197790	+	0.980244i	$0.563377\pi$
$840$	0	0
$841$	−20.4611	−0.705555
$842$	0	0
$843$	0	0
$844$	0	0
$845$	13.2238	0.454913
$846$	0	0
$847$	6.36982	0.218869
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−41.8622	−1.43502
$852$	0	0
$853$	50.3957	1.72552	0.862758	−	0.505617i	$-0.168735\pi$
$853$	50.3957	1.72552	0.862758	+	0.505617i	$0.168735\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	48.0227	1.64042	0.820211	−	0.572060i	$-0.193856\pi$
$857$	48.0227	1.64042	0.820211	+	0.572060i	$0.193856\pi$
$858$	0	0
$859$	38.2844	1.30625	0.653123	−	0.757252i	$-0.273459\pi$
$859$	38.2844	1.30625	0.653123	+	0.757252i	$0.273459\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−13.2049	−0.449500	−0.224750	−	0.974416i	$-0.572157\pi$
$863$	−13.2049	−0.449500	−0.224750	+	0.974416i	$0.572157\pi$
$864$	0	0
$865$	−36.9344	−1.25581
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−26.4036	−0.895679
$870$	0	0
$871$	−6.85949	−0.232425
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−8.53213	−0.288438
$876$	0	0
$877$	23.9630	0.809173	0.404586	−	0.914500i	$-0.367416\pi$
$877$	23.9630	0.809173	0.404586	+	0.914500i	$0.367416\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−39.3964	−1.32730	−0.663650	−	0.748044i	$-0.730993\pi$
$881$	−39.3964	−1.32730	−0.663650	+	0.748044i	$0.730993\pi$
$882$	0	0
$883$	−18.3593	−0.617840	−0.308920	−	0.951088i	$-0.599967\pi$
$883$	−18.3593	−0.617840	−0.308920	+	0.951088i	$0.599967\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	30.6649	1.02963	0.514813	−	0.857303i	$-0.327861\pi$
$887$	30.6649	1.02963	0.514813	+	0.857303i	$0.327861\pi$
$888$	0	0
$889$	−16.7663	−0.562322
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−9.30296	−0.311312
$894$	0	0
$895$	63.7461	2.13080
$896$	0	0
$897$	0	0
$898$	0	0
$899$	18.0452	0.601842
$900$	0	0
$901$	43.3476	1.44412
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−78.9521	−2.62446
$906$	0	0
$907$	41.9799	1.39392	0.696960	−	0.717110i	$-0.254536\pi$
$907$	41.9799	1.39392	0.696960	+	0.717110i	$0.254536\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	37.1429	1.23060	0.615299	−	0.788294i	$-0.289035\pi$
$911$	37.1429	1.23060	0.615299	+	0.788294i	$0.289035\pi$
$912$	0	0
$913$	46.0381	1.52364
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−2.73131	−0.0901960
$918$	0	0
$919$	−16.5903	−0.547263	−0.273632	−	0.961835i	$-0.588225\pi$
$919$	−16.5903	−0.547263	−0.273632	+	0.961835i	$0.588225\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−30.1967	−0.993937
$924$	0	0
$925$	−34.6297	−1.13862
$926$	0	0
$927$	0	0
$928$	0	0
$929$	37.1599	1.21918	0.609589	−	0.792718i	$-0.291335\pi$
$929$	37.1599	1.21918	0.609589	+	0.792718i	$0.291335\pi$
$930$	0	0
$931$	−5.40544	−0.177156
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−55.7492	−1.82319
$936$	0	0
$937$	−54.5323	−1.78149	−0.890746	−	0.454502i	$-0.849817\pi$
$937$	−54.5323	−1.78149	−0.890746	+	0.454502i	$0.849817\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−41.8789	−1.36521	−0.682606	−	0.730787i	$-0.739153\pi$
$941$	−41.8789	−1.36521	−0.682606	+	0.730787i	$0.739153\pi$
$942$	0	0
$943$	32.9818	1.07403
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−13.4290	−0.436384	−0.218192	−	0.975906i	$-0.570016\pi$
$947$	−13.4290	−0.436384	−0.218192	+	0.975906i	$0.570016\pi$
$948$	0	0
$949$	7.60580	0.246895
$950$	0	0
$951$	0	0
$952$	0	0
$953$	22.1328	0.716950	0.358475	−	0.933539i	$-0.383297\pi$
$953$	22.1328	0.716950	0.358475	+	0.933539i	$0.383297\pi$
$954$	0	0
$955$	11.9468	0.386590
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−16.2574	−0.524979
$960$	0	0
$961$	7.13492	0.230159
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−56.3772	−1.81485
$966$	0	0
$967$	29.9032	0.961621	0.480811	−	0.876824i	$-0.340342\pi$
$967$	29.9032	0.961621	0.480811	+	0.876824i	$0.340342\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	17.0708	0.547827	0.273914	−	0.961754i	$-0.411682\pi$
$971$	17.0708	0.547827	0.273914	+	0.961754i	$0.411682\pi$
$972$	0	0
$973$	−1.30811	−0.0419362
$974$	0	0
$975$	0	0
$976$	0	0
$977$	33.2598	1.06407	0.532037	−	0.846721i	$-0.321427\pi$
$977$	33.2598	1.06407	0.532037	+	0.846721i	$0.321427\pi$
$978$	0	0
$979$	−56.1380	−1.79418
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−26.5588	−0.847095	−0.423548	−	0.905874i	$-0.639215\pi$
$983$	−26.5588	−0.847095	−0.423548	+	0.905874i	$0.639215\pi$
$984$	0	0
$985$	−23.7922	−0.758082
$986$	0	0
$987$	0	0
$988$	0	0
$989$	23.6429	0.751800
$990$	0	0
$991$	−48.1268	−1.52880	−0.764398	−	0.644744i	$-0.776964\pi$
$991$	−48.1268	−1.52880	−0.764398	+	0.644744i	$0.776964\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	40.2511	1.27605
$996$	0	0
$997$	19.3574	0.613054	0.306527	−	0.951862i	$-0.400833\pi$
$997$	19.3574	0.613054	0.306527	+	0.951862i	$0.400833\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9072.2.a.cm.1.1		5
3.2	odd	2		9072.2.a.cn.1.5		5
4.3	odd	2		4536.2.a.bc.1.1		5
9.2	odd	6		1008.2.r.n.337.2		10
9.4	even	3		3024.2.r.n.2017.5		10
9.5	odd	6		1008.2.r.n.673.2		10
9.7	even	3		3024.2.r.n.1009.5		10
12.11	even	2		4536.2.a.bd.1.5		5
36.7	odd	6		1512.2.r.f.1009.5		10
36.11	even	6		504.2.r.f.337.4	yes	10
36.23	even	6		504.2.r.f.169.4	✓	10
36.31	odd	6		1512.2.r.f.505.5		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.2.r.f.169.4	✓	10	36.23	even	6
504.2.r.f.337.4	yes	10	36.11	even	6
1008.2.r.n.337.2		10	9.2	odd	6
1008.2.r.n.673.2		10	9.5	odd	6
1512.2.r.f.505.5		10	36.31	odd	6
1512.2.r.f.1009.5		10	36.7	odd	6
3024.2.r.n.1009.5		10	9.7	even	3
3024.2.r.n.2017.5		10	9.4	even	3
4536.2.a.bc.1.1		5	4.3	odd	2
4536.2.a.bd.1.5		5	12.11	even	2
9072.2.a.cm.1.1		5	1.1	even	1	trivial
9072.2.a.cn.1.5		5	3.2	odd	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:	\( N \)	\(=\)	\( 9072 = 2^{4} \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9072.a (trivial)

\(f(q)\)	\(=\)	\(q-3.52433 q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q-3.52433 q^{5} +1.00000 q^{7} -4.16771 q^{11} +3.04103 q^{13} -3.79546 q^{17} -5.40544 q^{19} +8.97080 q^{23} +7.42092 q^{25} +2.92214 q^{29} +6.17535 q^{31} -3.52433 q^{35} -4.66649 q^{37} +3.67657 q^{41} +2.63554 q^{43} +1.72104 q^{47} +1.00000 q^{49} -11.4209 q^{53} +14.6884 q^{55} -1.12896 q^{59} +12.0719 q^{61} -10.7176 q^{65} -2.25565 q^{67} -9.92978 q^{71} +2.50106 q^{73} -4.16771 q^{77} +6.33527 q^{79} -11.0464 q^{83} +13.3765 q^{85} +13.4697 q^{89} +3.04103 q^{91} +19.0506 q^{95} +18.9374 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 3 q^{5} + 5 q^{7}+O(q^{10}) \) 5 * q - 3 * q^5 + 5 * q^7	\( 5 q - 3 q^{5} + 5 q^{7} - 4 q^{11} + 3 q^{13} - q^{19} - 8 q^{23} + 10 q^{25} - 9 q^{29} - 3 q^{31} - 3 q^{35} - 3 q^{37} - 12 q^{41} - 5 q^{43} - 3 q^{47} + 5 q^{49} - 30 q^{53} - 22 q^{55} - 7 q^{59} + 14 q^{61} - 11 q^{65} - 8 q^{67} - 9 q^{71} + 15 q^{73} - 4 q^{77} - 3 q^{79} - 20 q^{83} + 21 q^{85} + 12 q^{89} + 3 q^{91} + 12 q^{95} + 37 q^{97}+O(q^{100}) \) 5 * q - 3 * q^5 + 5 * q^7 - 4 * q^11 + 3 * q^13 - q^19 - 8 * q^23 + 10 * q^25 - 9 * q^29 - 3 * q^31 - 3 * q^35 - 3 * q^37 - 12 * q^41 - 5 * q^43 - 3 * q^47 + 5 * q^49 - 30 * q^53 - 22 * q^55 - 7 * q^59 + 14 * q^61 - 11 * q^65 - 8 * q^67 - 9 * q^71 + 15 * q^73 - 4 * q^77 - 3 * q^79 - 20 * q^83 + 21 * q^85 + 12 * q^89 + 3 * q^91 + 12 * q^95 + 37 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more