LMFDB - Embedded newform 540.4.i.a.361.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [540,4,Mod(181,540)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(540, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("540.181");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 540 = 2^{2} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	540.i (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$31.8610314031$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 180)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			361.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	540.361
Dual form			540.4.i.a.181.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.50000 + 4.33013i) q^{5} +(3.50000 + 6.06218i) q^{7} +O(q^{10})$	$q+(-2.50000 + 4.33013i) q^{5} +(3.50000 + 6.06218i) q^{7} +(-15.0000 - 25.9808i) q^{11} +(11.0000 - 19.0526i) q^{13} -48.0000 q^{17} +68.0000 q^{19} +(55.5000 - 96.1288i) q^{23} +(-12.5000 - 21.6506i) q^{25} +(43.5000 + 75.3442i) q^{29} +(-10.0000 + 17.3205i) q^{31} -35.0000 q^{35} +200.000 q^{37} +(34.5000 - 59.7558i) q^{41} +(116.000 + 200.918i) q^{43} +(121.500 + 210.444i) q^{47} +(147.000 - 254.611i) q^{49} +498.000 q^{53} +150.000 q^{55} +(33.0000 - 57.1577i) q^{59} +(-179.500 - 310.903i) q^{61} +(55.0000 + 95.2628i) q^{65} +(531.500 - 920.585i) q^{67} +618.000 q^{71} -532.000 q^{73} +(105.000 - 181.865i) q^{77} +(-205.000 - 355.070i) q^{79} +(346.500 + 600.156i) q^{83} +(120.000 - 207.846i) q^{85} +1599.00 q^{89} +154.000 q^{91} +(-170.000 + 294.449i) q^{95} +(-25.0000 - 43.3013i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 5 q^{5} + 7 q^{7}+O(q^{10}) $ 2 * q - 5 * q^5 + 7 * q^7	$ 2 q - 5 q^{5} + 7 q^{7} - 30 q^{11} + 22 q^{13} - 96 q^{17} + 136 q^{19} + 111 q^{23} - 25 q^{25} + 87 q^{29} - 20 q^{31} - 70 q^{35} + 400 q^{37} + 69 q^{41} + 232 q^{43} + 243 q^{47} + 294 q^{49} + 996 q^{53} + 300 q^{55} + 66 q^{59} - 359 q^{61} + 110 q^{65} + 1063 q^{67} + 1236 q^{71} - 1064 q^{73} + 210 q^{77} - 410 q^{79} + 693 q^{83} + 240 q^{85} + 3198 q^{89} + 308 q^{91} - 340 q^{95} - 50 q^{97}+O(q^{100}) $ 2 * q - 5 * q^5 + 7 * q^7 - 30 * q^11 + 22 * q^13 - 96 * q^17 + 136 * q^19 + 111 * q^23 - 25 * q^25 + 87 * q^29 - 20 * q^31 - 70 * q^35 + 400 * q^37 + 69 * q^41 + 232 * q^43 + 243 * q^47 + 294 * q^49 + 996 * q^53 + 300 * q^55 + 66 * q^59 - 359 * q^61 + 110 * q^65 + 1063 * q^67 + 1236 * q^71 - 1064 * q^73 + 210 * q^77 - 410 * q^79 + 693 * q^83 + 240 * q^85 + 3198 * q^89 + 308 * q^91 - 340 * q^95 - 50 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/540\mathbb{Z}\right)^\times$.

$n$	$217$	$271$	$461$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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
$5$	−2.50000	+	4.33013i	−0.223607	+	0.387298i
$5$	−2.50000	+	4.33013i	−0.223607	+	0.387298i
$6$		0			0
$7$	3.50000	+	6.06218i	0.188982	+	0.327327i	0.944911	−	0.327327i	$-0.106148\pi$
$7$	3.50000	+	6.06218i	0.188982	+	0.327327i	−0.755929	+	0.654654i	$0.772814\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−15.0000	−	25.9808i	−0.411152	−	0.712136i	0.583864	−	0.811851i	$-0.301540\pi$
$11$	−15.0000	−	25.9808i	−0.411152	−	0.712136i	−0.995016	+	0.0997155i	$0.968207\pi$
$12$		0			0
$13$	11.0000	−	19.0526i	0.234681	−	0.406479i	−0.724499	−	0.689276i	$-0.757929\pi$
$13$	11.0000	−	19.0526i	0.234681	−	0.406479i	0.959180	+	0.282797i	$0.0912622\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−48.0000			−0.684806			−0.342403	−	0.939553i	$-0.611241\pi$
$17$	−48.0000			−0.684806			−0.342403	+	0.939553i	$0.611241\pi$
$18$		0			0
$19$	68.0000			0.821067			0.410533	−	0.911846i	$-0.365343\pi$
$19$	68.0000			0.821067			0.410533	+	0.911846i	$0.365343\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	55.5000	−	96.1288i	0.503154	−	0.871489i	−0.496839	−	0.867843i	$-0.665506\pi$
$23$	55.5000	−	96.1288i	0.503154	−	0.871489i	0.999993	−	0.00364616i	$-0.00116061\pi$
$24$		0			0
$25$	−12.5000	−	21.6506i	−0.100000	−	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	43.5000	+	75.3442i	0.278543	+	0.482451i	0.971023	−	0.238987i	$-0.0768152\pi$
$29$	43.5000	+	75.3442i	0.278543	+	0.482451i	−0.692480	+	0.721437i	$0.743482\pi$
$30$		0			0
$31$	−10.0000	+	17.3205i	−0.0579372	+	0.100350i	−0.893539	−	0.448985i	$-0.851786\pi$
$31$	−10.0000	+	17.3205i	−0.0579372	+	0.100350i	0.835602	+	0.549335i	$0.185119\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−35.0000			−0.169031
$36$		0			0
$37$	200.000			0.888643			0.444322	−	0.895867i	$-0.353445\pi$
$37$	200.000			0.888643			0.444322	+	0.895867i	$0.353445\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	34.5000	−	59.7558i	0.131415	−	0.227617i	−0.792808	−	0.609472i	$-0.791381\pi$
$41$	34.5000	−	59.7558i	0.131415	−	0.227617i	0.924222	+	0.381855i	$0.124715\pi$
$42$		0			0
$43$	116.000	+	200.918i	0.411391	+	0.712551i	0.995042	−	0.0994539i	$-0.0317096\pi$
$43$	116.000	+	200.918i	0.411391	+	0.712551i	−0.583651	+	0.812005i	$0.698376\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	121.500	+	210.444i	0.377077	+	0.653116i	0.990635	−	0.136533i	$-0.0435960\pi$
$47$	121.500	+	210.444i	0.377077	+	0.653116i	−0.613559	+	0.789649i	$0.710263\pi$
$48$		0			0
$49$	147.000	−	254.611i	0.428571	−	0.742307i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	498.000			1.29067			0.645335	−	0.763899i	$-0.276718\pi$
$53$	498.000			1.29067			0.645335	+	0.763899i	$0.276718\pi$
$54$		0			0
$55$	150.000			0.367745
$56$		0			0
$57$		0			0
$58$		0			0
$59$	33.0000	−	57.1577i	0.0728175	−	0.126124i	−0.827318	−	0.561734i	$-0.810134\pi$
$59$	33.0000	−	57.1577i	0.0728175	−	0.126124i	0.900135	+	0.435611i	$0.143468\pi$
$60$		0			0
$61$	−179.500	−	310.903i	−0.376764	−	0.652575i	0.613825	−	0.789442i	$-0.289630\pi$
$61$	−179.500	−	310.903i	−0.376764	−	0.652575i	−0.990589	+	0.136867i	$0.956297\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	55.0000	+	95.2628i	0.104952	+	0.181783i
$66$		0			0
$67$	531.500	−	920.585i	0.969150	−	1.67862i	0.271125	−	0.962544i	$-0.412604\pi$
$67$	531.500	−	920.585i	0.969150	−	1.67862i	0.698025	−	0.716073i	$-0.254062\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	618.000			1.03300			0.516501	−	0.856287i	$-0.327234\pi$
$71$	618.000			1.03300			0.516501	+	0.856287i	$0.327234\pi$
$72$		0			0
$73$	−532.000			−0.852957			−0.426479	−	0.904498i	$-0.640246\pi$
$73$	−532.000			−0.852957			−0.426479	+	0.904498i	$0.640246\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	105.000	−	181.865i	0.155401	−	0.269162i
$78$		0			0
$79$	−205.000	−	355.070i	−0.291953	−	0.505678i	0.682318	−	0.731055i	$-0.260972\pi$
$79$	−205.000	−	355.070i	−0.291953	−	0.505678i	−0.974272	+	0.225377i	$0.927638\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	346.500	+	600.156i	0.458233	+	0.793682i	0.998868	−	0.0475749i	$-0.0151493\pi$
$83$	346.500	+	600.156i	0.458233	+	0.793682i	−0.540635	+	0.841257i	$0.681816\pi$
$84$		0			0
$85$	120.000	−	207.846i	0.153127	−	0.265224i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	1599.00			1.90442			0.952212	−	0.305439i	$-0.0988033\pi$
$89$	1599.00			1.90442			0.952212	+	0.305439i	$0.0988033\pi$
$90$		0			0
$91$	154.000			0.177402
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−170.000	+	294.449i	−0.183596	+	0.317998i
$96$		0			0
$97$	−25.0000	−	43.3013i	−0.0261687	−	0.0453255i	0.852644	−	0.522492i	$-0.174997\pi$
$97$	−25.0000	−	43.3013i	−0.0261687	−	0.0453255i	−0.878813	+	0.477166i	$0.841664\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	591.000	+	1023.64i	0.582245	+	1.00848i	0.995213	+	0.0977317i	$0.0311587\pi$
$101$	591.000	+	1023.64i	0.582245	+	1.00848i	−0.412968	+	0.910745i	$0.635508\pi$
$102$		0			0
$103$	788.000	−	1364.86i	0.753825	−	1.30566i	−0.192132	−	0.981369i	$-0.561540\pi$
$103$	788.000	−	1364.86i	0.753825	−	1.30566i	0.945956	−	0.324294i	$-0.105127\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1023.00			−0.924272			−0.462136	−	0.886809i	$-0.652917\pi$
$107$	−1023.00			−0.924272			−0.462136	+	0.886809i	$0.652917\pi$
$108$		0			0
$109$	−1051.00			−0.923555			−0.461778	−	0.886996i	$-0.652788\pi$
$109$	−1051.00			−0.923555			−0.461778	+	0.886996i	$0.652788\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	432.000	−	748.246i	0.359638	−	0.622912i	−0.628262	−	0.778002i	$-0.716233\pi$
$113$	432.000	−	748.246i	0.359638	−	0.622912i	0.987900	+	0.155090i	$0.0495667\pi$
$114$		0			0
$115$	277.500	+	480.644i	0.225017	+	0.389742i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−168.000	−	290.985i	−0.129416	−	0.224156i
$120$		0			0
$121$	215.500	−	373.257i	0.161908	−	0.280433i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	125.000			0.0894427
$126$		0			0
$127$	1991.00			1.39112			0.695562	−	0.718466i	$-0.255156\pi$
$127$	1991.00			1.39112			0.695562	+	0.718466i	$0.255156\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−627.000	+	1086.00i	−0.418177	+	0.724305i	−0.995756	−	0.0920304i	$-0.970664\pi$
$131$	−627.000	+	1086.00i	−0.418177	+	0.724305i	0.577579	+	0.816335i	$0.303998\pi$
$132$		0			0
$133$	238.000	+	412.228i	0.155167	+	0.268757i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−798.000	−	1382.18i	−0.497648	−	0.861951i	0.502349	−	0.864665i	$-0.332469\pi$
$137$	−798.000	−	1382.18i	−0.497648	−	0.861951i	−0.999996	+	0.00271402i	$0.999136\pi$
$138$		0			0
$139$	−640.000	+	1108.51i	−0.390533	+	0.676423i	−0.992520	−	0.122083i	$-0.961043\pi$
$139$	−640.000	+	1108.51i	−0.390533	+	0.676423i	0.601987	+	0.798506i	$0.294376\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−660.000			−0.385958
$144$		0			0
$145$	−435.000			−0.249136
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1360.50	−	2356.46i	0.748030	−	1.29563i	−0.200735	−	0.979646i	$-0.564333\pi$
$149$	1360.50	−	2356.46i	0.748030	−	1.29563i	0.948766	−	0.315981i	$-0.102334\pi$
$150$		0			0
$151$	1391.00	+	2409.28i	0.749655	+	1.29844i	0.947988	+	0.318307i	$0.103114\pi$
$151$	1391.00	+	2409.28i	0.749655	+	1.29844i	−0.198332	+	0.980135i	$0.563553\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−50.0000	−	86.6025i	−0.0259103	−	0.0448780i
$156$		0			0
$157$	1493.00	−	2585.95i	0.758945	−	1.31453i	−0.184444	−	0.982843i	$-0.559048\pi$
$157$	1493.00	−	2585.95i	0.758945	−	1.31453i	0.943389	−	0.331688i	$-0.107618\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	777.000			0.380349
$162$		0			0
$163$	−1132.00			−0.543958			−0.271979	−	0.962303i	$-0.587678\pi$
$163$	−1132.00			−0.543958			−0.271979	+	0.962303i	$0.587678\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−922.500	+	1597.82i	−0.427456	+	0.740376i	−0.996646	−	0.0818301i	$-0.973924\pi$
$167$	−922.500	+	1597.82i	−0.427456	+	0.740376i	0.569190	+	0.822206i	$0.307257\pi$
$168$		0			0
$169$	856.500	+	1483.50i	0.389850	+	0.675240i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−540.000	−	935.307i	−0.237315	−	0.411041i	0.722628	−	0.691237i	$-0.242934\pi$
$173$	−540.000	−	935.307i	−0.237315	−	0.411041i	−0.959943	+	0.280196i	$0.909601\pi$
$174$		0			0
$175$	87.5000	−	151.554i	0.0377964	−	0.0654654i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−474.000			−0.197924			−0.0989621	−	0.995091i	$-0.531552\pi$
$179$	−474.000			−0.197924			−0.0989621	+	0.995091i	$0.531552\pi$
$180$		0			0
$181$	−3895.00			−1.59952			−0.799760	−	0.600320i	$-0.795040\pi$
$181$	−3895.00			−1.59952			−0.799760	+	0.600320i	$0.795040\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−500.000	+	866.025i	−0.198707	+	0.344170i
$186$		0			0
$187$	720.000	+	1247.08i	0.281559	+	0.487675i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	78.0000	+	135.100i	0.0295491	+	0.0511806i	0.880422	−	0.474191i	$-0.157260\pi$
$191$	78.0000	+	135.100i	0.0295491	+	0.0511806i	−0.850873	+	0.525372i	$0.823926\pi$
$192$		0			0
$193$	−409.000	+	708.409i	−0.152541	+	0.264209i	−0.932161	−	0.362044i	$-0.882079\pi$
$193$	−409.000	+	708.409i	−0.152541	+	0.264209i	0.779620	+	0.626253i	$0.215412\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−2244.00			−0.811565			−0.405783	−	0.913970i	$-0.633001\pi$
$197$	−2244.00			−0.811565			−0.405783	+	0.913970i	$0.633001\pi$
$198$		0			0
$199$	−1912.00			−0.681096			−0.340548	−	0.940227i	$-0.610613\pi$
$199$	−1912.00			−0.681096			−0.340548	+	0.940227i	$0.610613\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−304.500	+	527.409i	−0.105279	+	0.182349i
$204$		0			0
$205$	172.500	+	298.779i	0.0587704	+	0.101793i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−1020.00	−	1766.69i	−0.337583	−	0.584711i
$210$		0			0
$211$	−2923.00	+	5062.78i	−0.953685	+	1.65183i	−0.216337	+	0.976319i	$0.569411\pi$
$211$	−2923.00	+	5062.78i	−0.953685	+	1.65183i	−0.737348	+	0.675513i	$0.763922\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−1160.00			−0.367960
$216$		0			0
$217$	−140.000			−0.0437964
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−528.000	+	914.523i	−0.160711	+	0.278360i
$222$		0			0
$223$	543.500	+	941.370i	0.163208	+	0.282685i	0.936018	−	0.351953i	$-0.114482\pi$
$223$	543.500	+	941.370i	0.163208	+	0.282685i	−0.772809	+	0.634638i	$0.781149\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−2178.00	−	3772.41i	−0.636824	−	1.10301i	−0.986126	−	0.166001i	$-0.946915\pi$
$227$	−2178.00	−	3772.41i	−0.636824	−	1.10301i	0.349302	−	0.937010i	$-0.386419\pi$
$228$		0			0
$229$	2784.50	−	4822.90i	0.803515	−	1.39173i	−0.113774	−	0.993507i	$-0.536294\pi$
$229$	2784.50	−	4822.90i	0.803515	−	1.39173i	0.917289	−	0.398222i	$-0.130373\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−2580.00			−0.725414			−0.362707	−	0.931903i	$-0.618147\pi$
$233$	−2580.00			−0.725414			−0.362707	+	0.931903i	$0.618147\pi$
$234$		0			0
$235$	−1215.00			−0.337267
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−1800.00	+	3117.69i	−0.487165	+	0.843794i	−0.999891	−	0.0147583i	$-0.995302\pi$
$239$	−1800.00	+	3117.69i	−0.487165	+	0.843794i	0.512727	+	0.858552i	$0.328635\pi$
$240$		0			0
$241$	−86.5000	−	149.822i	−0.0231201	−	0.0400453i	0.854234	−	0.519889i	$-0.174027\pi$
$241$	−86.5000	−	149.822i	−0.0231201	−	0.0400453i	−0.877354	+	0.479844i	$0.840693\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	735.000	+	1273.06i	0.191663	+	0.331970i
$246$		0			0
$247$	748.000	−	1295.57i	0.192689	−	0.333747i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−708.000			−0.178042			−0.0890210	−	0.996030i	$-0.528374\pi$
$251$	−708.000			−0.178042			−0.0890210	+	0.996030i	$0.528374\pi$
$252$		0			0
$253$	−3330.00			−0.827491
$254$		0			0
$255$		0			0
$256$		0			0
$257$	1161.00	−	2010.91i	0.281795	−	0.488082i	−0.690032	−	0.723779i	$-0.742404\pi$
$257$	1161.00	−	2010.91i	0.281795	−	0.488082i	0.971827	+	0.235696i	$0.0757370\pi$
$258$		0			0
$259$	700.000	+	1212.44i	0.167938	+	0.290877i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	1104.00	+	1912.18i	0.258842	+	0.448328i	0.965932	−	0.258796i	$-0.0833256\pi$
$263$	1104.00	+	1912.18i	0.258842	+	0.448328i	−0.707090	+	0.707124i	$0.749992\pi$
$264$		0			0
$265$	−1245.00	+	2156.40i	−0.288603	+	0.499875i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−2481.00			−0.562339			−0.281170	−	0.959658i	$-0.590722\pi$
$269$	−2481.00			−0.562339			−0.281170	+	0.959658i	$0.590722\pi$
$270$		0			0
$271$	−7516.00			−1.68474			−0.842370	−	0.538900i	$-0.818840\pi$
$271$	−7516.00			−1.68474			−0.842370	+	0.538900i	$0.818840\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−375.000	+	649.519i	−0.0822304	+	0.142427i
$276$		0			0
$277$	−172.000	−	297.913i	−0.0373086	−	0.0646204i	0.846768	−	0.531962i	$-0.178545\pi$
$277$	−172.000	−	297.913i	−0.0373086	−	0.0646204i	−0.884077	+	0.467342i	$0.845212\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	3796.50	+	6575.73i	0.805979	+	1.39600i	0.915628	+	0.402027i	$0.131694\pi$
$281$	3796.50	+	6575.73i	0.805979	+	1.39600i	−0.109649	+	0.993970i	$0.534973\pi$
$282$		0			0
$283$	−1328.50	+	2301.03i	−0.279050	+	0.483329i	−0.971149	−	0.238474i	$-0.923353\pi$
$283$	−1328.50	+	2301.03i	−0.279050	+	0.483329i	0.692099	+	0.721803i	$0.256686\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	483.000			0.0993400
$288$		0			0
$289$	−2609.00			−0.531040
$290$		0			0
$291$		0			0
$292$		0			0
$293$	3696.00	−	6401.66i	0.736937	−	1.27641i	−0.216931	−	0.976187i	$-0.569605\pi$
$293$	3696.00	−	6401.66i	0.736937	−	1.27641i	0.953868	−	0.300226i	$-0.0970620\pi$
$294$		0			0
$295$	165.000	+	285.788i	0.0325650	+	0.0564042i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1221.00	−	2114.83i	−0.236161	−	0.409044i
$300$		0			0
$301$	−812.000	+	1406.43i	−0.155491	+	0.269319i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	1795.00			0.336988
$306$		0			0
$307$	−3877.00			−0.720756			−0.360378	−	0.932806i	$-0.617352\pi$
$307$	−3877.00			−0.720756			−0.360378	+	0.932806i	$0.617352\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−2886.00	+	4998.70i	−0.526206	+	0.911415i	0.473328	+	0.880886i	$0.343053\pi$
$311$	−2886.00	+	4998.70i	−0.526206	+	0.911415i	−0.999534	+	0.0305291i	$0.990281\pi$
$312$		0			0
$313$	719.000	+	1245.34i	0.129841	+	0.224891i	0.923615	−	0.383322i	$-0.125220\pi$
$313$	719.000	+	1245.34i	0.129841	+	0.224891i	−0.793774	+	0.608213i	$0.791887\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−3207.00	−	5554.69i	−0.568212	−	0.984171i	−0.996743	−	0.0806444i	$-0.974302\pi$
$317$	−3207.00	−	5554.69i	−0.568212	−	0.984171i	0.428531	−	0.903527i	$-0.359031\pi$
$318$		0			0
$319$	1305.00	−	2260.33i	0.229047	−	0.396721i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−3264.00			−0.562272
$324$		0			0
$325$	−550.000			−0.0938723
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−850.500	+	1473.11i	−0.142522	+	0.246855i
$330$		0			0
$331$	5807.00	+	10058.0i	0.964295	+	1.67021i	0.711498	+	0.702688i	$0.248017\pi$
$331$	5807.00	+	10058.0i	0.964295	+	1.67021i	0.252797	+	0.967519i	$0.418650\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	2657.50	+	4602.93i	0.433417	+	0.750700i
$336$		0			0
$337$	4652.00	−	8057.50i	0.751960	−	1.30243i	−0.194911	−	0.980821i	$-0.562442\pi$
$337$	4652.00	−	8057.50i	0.751960	−	1.30243i	0.946871	−	0.321612i	$-0.104225\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	600.000			0.0952839
$342$		0			0
$343$	4459.00			0.701934
$344$		0			0
$345$		0			0
$346$		0			0
$347$	4914.00	−	8511.30i	0.760223	−	1.31674i	−0.182513	−	0.983204i	$-0.558423\pi$
$347$	4914.00	−	8511.30i	0.760223	−	1.31674i	0.942736	−	0.333541i	$-0.108244\pi$
$348$		0			0
$349$	786.500	+	1362.26i	0.120631	+	0.208940i	0.920017	−	0.391879i	$-0.128175\pi$
$349$	786.500	+	1362.26i	0.120631	+	0.208940i	−0.799385	+	0.600819i	$0.794841\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−1932.00	−	3346.32i	−0.291303	−	0.504552i	0.682815	−	0.730591i	$-0.260755\pi$
$353$	−1932.00	−	3346.32i	−0.291303	−	0.504552i	−0.974118	+	0.226040i	$0.927422\pi$
$354$		0			0
$355$	−1545.00	+	2676.02i	−0.230986	+	0.400080i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−7596.00			−1.11672			−0.558359	−	0.829600i	$-0.688569\pi$
$359$	−7596.00			−1.11672			−0.558359	+	0.829600i	$0.688569\pi$
$360$		0			0
$361$	−2235.00			−0.325849
$362$		0			0
$363$		0			0
$364$		0			0
$365$	1330.00	−	2303.63i	0.190727	−	0.330349i
$366$		0			0
$367$	1508.00	+	2611.93i	0.214488	+	0.371503i	0.953114	−	0.302612i	$-0.0978585\pi$
$367$	1508.00	+	2611.93i	0.214488	+	0.371503i	−0.738626	+	0.674115i	$0.764525\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	1743.00	+	3018.96i	0.243914	+	0.422471i
$372$		0			0
$373$	4799.00	−	8312.11i	0.666174	−	1.15385i	−0.312792	−	0.949822i	$-0.601264\pi$
$373$	4799.00	−	8312.11i	0.666174	−	1.15385i	0.978966	−	0.204025i	$-0.0654024\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	1914.00			0.261475
$378$		0			0
$379$	12926.0			1.75188			0.875942	−	0.482416i	$-0.160241\pi$
$379$	12926.0			1.75188			0.875942	+	0.482416i	$0.160241\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−4506.00	+	7804.62i	−0.601164	+	1.04125i	0.391481	+	0.920186i	$0.371963\pi$
$383$	−4506.00	+	7804.62i	−0.601164	+	1.04125i	−0.992645	+	0.121061i	$0.961370\pi$
$384$		0			0
$385$	525.000	+	909.327i	0.0694973	+	0.120373i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−4507.50	−	7807.22i	−0.587505	−	1.01759i	−0.994558	−	0.104184i	$-0.966777\pi$
$389$	−4507.50	−	7807.22i	−0.587505	−	1.01759i	0.407053	−	0.913404i	$-0.366556\pi$
$390$		0			0
$391$	−2664.00	+	4614.18i	−0.344563	+	0.596801i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	2050.00			0.261131
$396$		0			0
$397$	−3094.00			−0.391142			−0.195571	−	0.980690i	$-0.562656\pi$
$397$	−3094.00			−0.391142			−0.195571	+	0.980690i	$0.562656\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−6189.00	+	10719.7i	−0.770733	+	1.33495i	0.166429	+	0.986053i	$0.446776\pi$
$401$	−6189.00	+	10719.7i	−0.770733	+	1.33495i	−0.937162	+	0.348895i	$0.886557\pi$
$402$		0			0
$403$	220.000	+	381.051i	0.0271935	+	0.0471005i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−3000.00	−	5196.15i	−0.365367	−	0.632835i
$408$		0			0
$409$	4445.00	−	7698.97i	0.537387	−	0.930781i	−0.461657	−	0.887058i	$-0.652745\pi$
$409$	4445.00	−	7698.97i	0.537387	−	0.930781i	0.999044	−	0.0437224i	$-0.0139217\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	462.000			0.0550449
$414$		0			0
$415$	−3465.00			−0.409856
$416$		0			0
$417$		0			0
$418$		0			0
$419$	4059.00	−	7030.39i	0.473258	−	0.819707i	−0.526273	−	0.850315i	$-0.676411\pi$
$419$	4059.00	−	7030.39i	0.473258	−	0.819707i	0.999531	+	0.0306084i	$0.00974448\pi$
$420$		0			0
$421$	−1369.00	−	2371.18i	−0.158482	−	0.274499i	0.775839	−	0.630930i	$-0.217327\pi$
$421$	−1369.00	−	2371.18i	−0.158482	−	0.274499i	−0.934322	+	0.356431i	$0.883993\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	600.000	+	1039.23i	0.0684806	+	0.118612i
$426$		0			0
$427$	1256.50	−	2176.32i	0.142404	−	0.246650i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	16134.0			1.80313			0.901563	−	0.432648i	$-0.142421\pi$
$431$	16134.0			1.80313			0.901563	+	0.432648i	$0.142421\pi$
$432$		0			0
$433$	6608.00			0.733395			0.366698	−	0.930340i	$-0.380488\pi$
$433$	6608.00			0.733395			0.366698	+	0.930340i	$0.380488\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	3774.00	−	6536.76i	0.413123	−	0.715551i
$438$		0			0
$439$	−3214.00	−	5566.81i	−0.349421	−	0.605215i	0.636726	−	0.771090i	$-0.280288\pi$
$439$	−3214.00	−	5566.81i	−0.349421	−	0.605215i	−0.986147	+	0.165875i	$0.946955\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	5449.50	+	9438.81i	0.584455	+	1.01231i	0.994943	+	0.100440i	$0.0320250\pi$
$443$	5449.50	+	9438.81i	0.584455	+	1.01231i	−0.410488	+	0.911866i	$0.634642\pi$
$444$		0			0
$445$	−3997.50	+	6923.87i	−0.425842	+	0.737580i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	11610.0			1.22029			0.610145	−	0.792290i	$-0.291111\pi$
$449$	11610.0			1.22029			0.610145	+	0.792290i	$0.291111\pi$
$450$		0			0
$451$	−2070.00			−0.216125
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−385.000	+	666.840i	−0.0396683	+	0.0687075i
$456$		0			0
$457$	−7300.00	−	12644.0i	−0.747220	−	1.29422i	−0.949150	−	0.314823i	$-0.898055\pi$
$457$	−7300.00	−	12644.0i	−0.747220	−	1.29422i	0.201930	−	0.979400i	$-0.435279\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	2896.50	+	5016.89i	0.292632	+	0.506854i	0.974431	−	0.224686i	$-0.0721355\pi$
$461$	2896.50	+	5016.89i	0.292632	+	0.506854i	−0.681799	+	0.731539i	$0.738802\pi$
$462$		0			0
$463$	2846.00	−	4929.42i	0.285669	−	0.494794i	−0.687102	−	0.726561i	$-0.741117\pi$
$463$	2846.00	−	4929.42i	0.285669	−	0.494794i	0.972771	+	0.231767i	$0.0744508\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	6708.00			0.664688			0.332344	−	0.943158i	$-0.392161\pi$
$467$	6708.00			0.664688			0.332344	+	0.943158i	$0.392161\pi$
$468$		0			0
$469$	7441.00			0.732609
$470$		0			0
$471$		0			0
$472$		0			0
$473$	3480.00	−	6027.54i	0.338289	−	0.585933i
$474$		0			0
$475$	−850.000	−	1472.24i	−0.0821067	−	0.142213i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	1575.00	+	2727.98i	0.150237	+	0.260218i	0.931315	−	0.364216i	$-0.118663\pi$
$479$	1575.00	+	2727.98i	0.150237	+	0.260218i	−0.781077	+	0.624434i	$0.785330\pi$
$480$		0			0
$481$	2200.00	−	3810.51i	0.208548	−	0.361215i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	250.000			0.0234060
$486$		0			0
$487$	11768.0			1.09499			0.547494	−	0.836810i	$-0.315582\pi$
$487$	11768.0			1.09499			0.547494	+	0.836810i	$0.315582\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−3702.00	+	6412.05i	−0.340263	+	0.589352i	−0.984481	−	0.175489i	$-0.943849\pi$
$491$	−3702.00	+	6412.05i	−0.340263	+	0.589352i	0.644219	+	0.764841i	$0.277183\pi$
$492$		0			0
$493$	−2088.00	−	3616.52i	−0.190748	−	0.330385i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	2163.00	+	3746.43i	0.195219	+	0.338129i
$498$		0			0
$499$	2168.00	−	3755.09i	0.194495	−	0.336875i	−0.752240	−	0.658889i	$-0.771027\pi$
$499$	2168.00	−	3755.09i	0.194495	−	0.336875i	0.946735	+	0.322014i	$0.104360\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−3789.00			−0.335871			−0.167936	−	0.985798i	$-0.553710\pi$
$503$	−3789.00			−0.335871			−0.167936	+	0.985798i	$0.553710\pi$
$504$		0			0
$505$	−5910.00			−0.520775
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−550.500	+	953.494i	−0.0479381	+	0.0830312i	−0.888999	−	0.457910i	$-0.848598\pi$
$509$	−550.500	+	953.494i	−0.0479381	+	0.0830312i	0.841061	+	0.540941i	$0.181932\pi$
$510$		0			0
$511$	−1862.00	−	3225.08i	−0.161194	−	0.279196i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	3940.00	+	6824.28i	0.337121	+	0.583910i
$516$		0			0
$517$	3645.00	−	6313.33i	0.310071	−	0.537059i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	12813.0			1.07744			0.538721	−	0.842484i	$-0.318908\pi$
$521$	12813.0			1.07744			0.538721	+	0.842484i	$0.318908\pi$
$522$		0			0
$523$	9359.00			0.782487			0.391243	−	0.920287i	$-0.372045\pi$
$523$	9359.00			0.782487			0.391243	+	0.920287i	$0.372045\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	480.000	−	831.384i	0.0396758	−	0.0687204i
$528$		0			0
$529$	−77.0000	−	133.368i	−0.00632859	−	0.0109614i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−759.000	−	1314.63i	−0.0616809	−	0.106835i
$534$		0			0
$535$	2557.50	−	4429.72i	0.206674	−	0.357969i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−8820.00			−0.704832
$540$		0			0
$541$	−10807.0			−0.858834			−0.429417	−	0.903106i	$-0.641281\pi$
$541$	−10807.0			−0.858834			−0.429417	+	0.903106i	$0.641281\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	2627.50	−	4550.96i	0.206513	−	0.357691i
$546$		0			0
$547$	−1295.50	−	2243.87i	−0.101264	−	0.175395i	0.810941	−	0.585127i	$-0.198955\pi$
$547$	−1295.50	−	2243.87i	−0.101264	−	0.175395i	−0.912206	+	0.409732i	$0.865622\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	2958.00	+	5123.41i	0.228702	+	0.396124i
$552$		0			0
$553$	1435.00	−	2485.49i	0.110348	−	0.191128i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	19662.0			1.49570			0.747851	−	0.663867i	$-0.231086\pi$
$557$	19662.0			1.49570			0.747851	+	0.663867i	$0.231086\pi$
$558$		0			0
$559$	5104.00			0.386183
$560$		0			0
$561$		0			0
$562$		0			0
$563$	4531.50	−	7848.79i	0.339218	−	0.587543i	−0.645068	−	0.764126i	$-0.723171\pi$
$563$	4531.50	−	7848.79i	0.339218	−	0.587543i	0.984286	+	0.176582i	$0.0565041\pi$
$564$		0			0
$565$	2160.00	+	3741.23i	0.160835	+	0.278575i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	8277.00	+	14336.2i	0.609824	+	1.05625i	0.991269	+	0.131855i	$0.0420932\pi$
$569$	8277.00	+	14336.2i	0.609824	+	1.05625i	−0.381445	+	0.924392i	$0.624573\pi$
$570$		0			0
$571$	−13276.0	+	22994.7i	−0.973001	+	1.68529i	−0.286619	+	0.958045i	$0.592531\pi$
$571$	−13276.0	+	22994.7i	−0.973001	+	1.68529i	−0.686381	+	0.727242i	$0.740802\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−2775.00			−0.201262
$576$		0			0
$577$	17846.0			1.28759			0.643794	−	0.765199i	$-0.277359\pi$
$577$	17846.0			1.28759			0.643794	+	0.765199i	$0.277359\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−2425.50	+	4201.09i	−0.173196	+	0.299984i
$582$		0			0
$583$	−7470.00	−	12938.4i	−0.530662	−	0.919133i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−7234.50	−	12530.5i	−0.508688	−	0.881073i	−0.999949	−	0.0100611i	$-0.996797\pi$
$587$	−7234.50	−	12530.5i	−0.508688	−	0.881073i	0.491262	−	0.871012i	$-0.336536\pi$
$588$		0			0
$589$	−680.000	+	1177.79i	−0.0475703	+	0.0823942i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−5592.00			−0.387244			−0.193622	−	0.981076i	$-0.562024\pi$
$593$	−5592.00			−0.387244			−0.193622	+	0.981076i	$0.562024\pi$
$594$		0			0
$595$	1680.00			0.115753
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−1173.00	+	2031.70i	−0.0800125	+	0.138586i	−0.903255	−	0.429104i	$-0.858829\pi$
$599$	−1173.00	+	2031.70i	−0.0800125	+	0.138586i	0.823243	+	0.567690i	$0.192163\pi$
$600$		0			0
$601$	3131.00	+	5423.05i	0.212506	+	0.368071i	0.952498	−	0.304544i	$-0.0985041\pi$
$601$	3131.00	+	5423.05i	0.212506	+	0.368071i	−0.739992	+	0.672616i	$0.765171\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	1077.50	+	1866.28i	0.0724076	+	0.125414i
$606$		0			0
$607$	−2705.50	+	4686.06i	−0.180911	+	0.313347i	−0.942191	−	0.335076i	$-0.891238\pi$
$607$	−2705.50	+	4686.06i	−0.180911	+	0.313347i	0.761280	+	0.648423i	$0.224571\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	5346.00			0.353971
$612$		0			0
$613$	−23176.0			−1.52703			−0.763515	−	0.645790i	$-0.776528\pi$
$613$	−23176.0			−1.52703			−0.763515	+	0.645790i	$0.776528\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−2202.00	+	3813.98i	−0.143678	+	0.248857i	−0.928879	−	0.370384i	$-0.879226\pi$
$617$	−2202.00	+	3813.98i	−0.143678	+	0.248857i	0.785201	+	0.619241i	$0.212560\pi$
$618$		0			0
$619$	−6772.00	−	11729.4i	−0.439725	−	0.761626i	0.557943	−	0.829879i	$-0.311591\pi$
$619$	−6772.00	−	11729.4i	−0.439725	−	0.761626i	−0.997668	+	0.0682534i	$0.978257\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	5596.50	+	9693.42i	0.359902	+	0.623369i
$624$		0			0
$625$	−312.500	+	541.266i	−0.0200000	+	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−9600.00			−0.608549
$630$		0			0
$631$	14240.0			0.898392			0.449196	−	0.893433i	$-0.351710\pi$
$631$	14240.0			0.898392			0.449196	+	0.893433i	$0.351710\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−4977.50	+	8621.28i	−0.311065	+	0.538780i
$636$		0			0
$637$	−3234.00	−	5601.45i	−0.201155	−	0.348411i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−5599.50	−	9698.62i	−0.345034	−	0.597617i	0.640326	−	0.768103i	$-0.278799\pi$
$641$	−5599.50	−	9698.62i	−0.345034	−	0.597617i	−0.985360	+	0.170487i	$0.945466\pi$
$642$		0			0
$643$	−4433.50	+	7679.05i	−0.271913	+	0.470967i	−0.969352	−	0.245677i	$-0.920990\pi$
$643$	−4433.50	+	7679.05i	−0.271913	+	0.470967i	0.697439	+	0.716645i	$0.254323\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−24363.0			−1.48038			−0.740192	−	0.672396i	$-0.765265\pi$
$647$	−24363.0			−1.48038			−0.740192	+	0.672396i	$0.765265\pi$
$648$		0			0
$649$	−1980.00			−0.119756
$650$		0			0
$651$		0			0
$652$		0			0
$653$	8664.00	−	15006.5i	0.519217	−	0.899310i	−0.480534	−	0.876976i	$-0.659557\pi$
$653$	8664.00	−	15006.5i	0.519217	−	0.899310i	0.999751	−	0.0223336i	$-0.00710959\pi$
$654$		0			0
$655$	−3135.00	−	5429.98i	−0.187015	−	0.323919i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−10128.0	−	17542.2i	−0.598681	−	1.03695i	−0.993016	−	0.117979i	$-0.962358\pi$
$659$	−10128.0	−	17542.2i	−0.598681	−	1.03695i	0.394335	−	0.918967i	$-0.370975\pi$
$660$		0			0
$661$	−6205.00	+	10747.4i	−0.365123	+	0.632412i	−0.988796	−	0.149273i	$-0.952307\pi$
$661$	−6205.00	+	10747.4i	−0.365123	+	0.632412i	0.623673	+	0.781686i	$0.285640\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−2380.00			−0.138786
$666$		0			0
$667$	9657.00			0.560600
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−5385.00	+	9327.09i	−0.309815	+	0.536615i
$672$		0			0
$673$	11567.0	+	20034.6i	0.662519	+	1.14752i	0.979952	+	0.199235i	$0.0638458\pi$
$673$	11567.0	+	20034.6i	0.662519	+	1.14752i	−0.317433	+	0.948281i	$0.602821\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−10875.0	−	18836.1i	−0.617371	−	1.06932i	−0.989964	−	0.141323i	$-0.954864\pi$
$677$	−10875.0	−	18836.1i	−0.617371	−	1.06932i	0.372592	−	0.927995i	$-0.378469\pi$
$678$		0			0
$679$	175.000	−	303.109i	0.00989084	−	0.0171314i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−13284.0			−0.744214			−0.372107	−	0.928190i	$-0.621365\pi$
$683$	−13284.0			−0.744214			−0.372107	+	0.928190i	$0.621365\pi$
$684$		0			0
$685$	7980.00			0.445110
$686$		0			0
$687$		0			0
$688$		0			0
$689$	5478.00	−	9488.17i	0.302896	−	0.524631i
$690$		0			0
$691$	−4825.00	−	8357.15i	−0.265632	−	0.460088i	0.702097	−	0.712081i	$-0.252247\pi$
$691$	−4825.00	−	8357.15i	−0.265632	−	0.460088i	−0.967729	+	0.251993i	$0.918914\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−3200.00	−	5542.56i	−0.174652	−	0.302506i
$696$		0			0
$697$	−1656.00	+	2868.28i	−0.0899935	+	0.155873i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−519.000			−0.0279634			−0.0139817	−	0.999902i	$-0.504451\pi$
$701$	−519.000			−0.0279634			−0.0139817	+	0.999902i	$0.504451\pi$
$702$		0			0
$703$	13600.0			0.729635
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−4137.00	+	7165.49i	−0.220068	+	0.381169i
$708$		0			0
$709$	5172.50	+	8959.03i	0.273988	+	0.474561i	0.969879	−	0.243586i	$-0.0783239\pi$
$709$	5172.50	+	8959.03i	0.273988	+	0.474561i	−0.695891	+	0.718147i	$0.744991\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	1110.00	+	1922.58i	0.0583027	+	0.100983i
$714$		0			0
$715$	1650.00	−	2857.88i	0.0863028	−	0.149481i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−2502.00			−0.129776			−0.0648879	−	0.997893i	$-0.520669\pi$
$719$	−2502.00			−0.129776			−0.0648879	+	0.997893i	$0.520669\pi$
$720$		0			0
$721$	11032.0			0.569838
$722$		0			0
$723$		0			0
$724$		0			0
$725$	1087.50	−	1883.61i	0.0557086	−	0.0964901i
$726$		0			0
$727$	1326.50	+	2297.57i	0.0676715	+	0.117210i	0.897876	−	0.440249i	$-0.145110\pi$
$727$	1326.50	+	2297.57i	0.0676715	+	0.117210i	−0.830204	+	0.557459i	$0.811776\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−5568.00	−	9644.06i	−0.281724	−	0.487960i
$732$		0			0
$733$	−15571.0	+	26969.8i	−0.784622	+	1.35901i	0.144602	+	0.989490i	$0.453810\pi$
$733$	−15571.0	+	26969.8i	−0.784622	+	1.35901i	−0.929225	+	0.369516i	$0.879524\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−31890.0			−1.59387
$738$		0			0
$739$	−10042.0			−0.499866			−0.249933	−	0.968263i	$-0.580409\pi$
$739$	−10042.0			−0.499866			−0.249933	+	0.968263i	$0.580409\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	10219.5	−	17700.7i	0.504599	−	0.873991i	−0.495387	−	0.868672i	$-0.664974\pi$
$743$	10219.5	−	17700.7i	0.504599	−	0.873991i	0.999986	−	0.00531869i	$-0.00169300\pi$
$744$		0			0
$745$	6802.50	+	11782.3i	0.334529	+	0.579422i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−3580.50	−	6201.61i	−0.174671	−	0.302539i
$750$		0			0
$751$	−12007.0	+	20796.7i	−0.583411	+	1.01050i	0.411660	+	0.911337i	$0.364949\pi$
$751$	−12007.0	+	20796.7i	−0.583411	+	1.01050i	−0.995071	+	0.0991602i	$0.968384\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−13910.0			−0.670512
$756$		0			0
$757$	−10630.0			−0.510375			−0.255188	−	0.966892i	$-0.582137\pi$
$757$	−10630.0			−0.510375			−0.255188	+	0.966892i	$0.582137\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−12172.5	+	21083.4i	−0.579833	+	1.00430i	0.415665	+	0.909518i	$0.363549\pi$
$761$	−12172.5	+	21083.4i	−0.579833	+	1.00430i	−0.995498	+	0.0947822i	$0.969785\pi$
$762$		0			0
$763$	−3678.50	−	6371.35i	−0.174536	−	0.302304i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−726.000	−	1257.47i	−0.0341778	−	0.0591976i
$768$		0			0
$769$	−17390.5	+	30121.2i	−0.815497	+	1.41248i	0.0934729	+	0.995622i	$0.470203\pi$
$769$	−17390.5	+	30121.2i	−0.815497	+	1.41248i	−0.908970	+	0.416861i	$0.863130\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−29232.0			−1.36016			−0.680079	−	0.733139i	$-0.738054\pi$
$773$	−29232.0			−1.36016			−0.680079	+	0.733139i	$0.738054\pi$
$774$		0			0
$775$	500.000			0.0231749
$776$		0			0
$777$		0			0
$778$		0			0
$779$	2346.00	−	4063.39i	0.107900	−	0.186888i
$780$		0			0
$781$	−9270.00	−	16056.1i	−0.424720	−	0.735637i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	7465.00	+	12929.8i	0.339411	+	0.587876i
$786$		0			0
$787$	−2110.00	+	3654.63i	−0.0955697	+	0.165532i	−0.909846	−	0.414946i	$-0.863801\pi$
$787$	−2110.00	+	3654.63i	−0.0955697	+	0.165532i	0.814277	+	0.580477i	$0.197134\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	6048.00			0.271861
$792$		0			0
$793$	−7898.00			−0.353677
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−17649.0	+	30569.0i	−0.784391	+	1.35861i	0.144971	+	0.989436i	$0.453691\pi$
$797$	−17649.0	+	30569.0i	−0.784391	+	1.35861i	−0.929362	+	0.369169i	$0.879642\pi$
$798$		0			0
$799$	−5832.00	−	10101.3i	−0.258224	−	0.447258i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	7980.00	+	13821.8i	0.350695	+	0.607422i
$804$		0			0
$805$	−1942.50	+	3364.51i	−0.0850486	+	0.147309i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	39966.0			1.73687			0.868436	−	0.495801i	$-0.165125\pi$
$809$	39966.0			1.73687			0.868436	+	0.495801i	$0.165125\pi$
$810$		0			0
$811$	14438.0			0.625138			0.312569	−	0.949895i	$-0.398810\pi$
$811$	14438.0			0.625138			0.312569	+	0.949895i	$0.398810\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	2830.00	−	4901.70i	0.121633	−	0.210674i
$816$		0			0
$817$	7888.00	+	13662.4i	0.337780	+	0.585052i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−7828.50	−	13559.4i	−0.332785	−	0.576401i	0.650272	−	0.759702i	$-0.274655\pi$
$821$	−7828.50	−	13559.4i	−0.332785	−	0.576401i	−0.983057	+	0.183301i	$0.941322\pi$
$822$		0			0
$823$	−8336.50	+	14439.2i	−0.353089	+	0.611568i	−0.986789	−	0.162011i	$-0.948202\pi$
$823$	−8336.50	+	14439.2i	−0.353089	+	0.611568i	0.633700	+	0.773579i	$0.281535\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	17871.0			0.751434			0.375717	−	0.926735i	$-0.377397\pi$
$827$	17871.0			0.751434			0.375717	+	0.926735i	$0.377397\pi$
$828$		0			0
$829$	−21163.0			−0.886636			−0.443318	−	0.896364i	$-0.646199\pi$
$829$	−21163.0			−0.886636			−0.443318	+	0.896364i	$0.646199\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−7056.00	+	12221.4i	−0.293488	+	0.508337i
$834$		0			0
$835$	−4612.50	−	7989.08i	−0.191164	−	0.331106i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−450.000	−	779.423i	−0.0185170	−	0.0320723i	0.856618	−	0.515950i	$-0.172561\pi$
$839$	−450.000	−	779.423i	−0.0185170	−	0.0320723i	−0.875135	+	0.483878i	$0.839228\pi$
$840$		0			0
$841$	8410.00	−	14566.5i	0.344828	−	0.597259i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−8565.00			−0.348692
$846$		0			0
$847$	3017.00			0.122391
$848$		0			0
$849$		0			0
$850$		0			0
$851$	11100.0	−	19225.8i	0.447125	−	0.774443i
$852$		0			0
$853$	−10399.0	−	18011.6i	−0.417415	−	0.722984i	0.578264	−	0.815850i	$-0.303731\pi$
$853$	−10399.0	−	18011.6i	−0.417415	−	0.722984i	−0.995679	+	0.0928660i	$0.970397\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	27.0000	+	46.7654i	0.00107620	+	0.00186403i	0.866563	−	0.499068i	$-0.166324\pi$
$857$	27.0000	+	46.7654i	0.00107620	+	0.00186403i	−0.865487	+	0.500932i	$0.832991\pi$
$858$		0			0
$859$	−20845.0	+	36104.6i	−0.827965	+	1.43408i	0.0716664	+	0.997429i	$0.477168\pi$
$859$	−20845.0	+	36104.6i	−0.827965	+	1.43408i	−0.899632	+	0.436649i	$0.856165\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	26115.0			1.03009			0.515043	−	0.857164i	$-0.327776\pi$
$863$	26115.0			1.03009			0.515043	+	0.857164i	$0.327776\pi$
$864$		0			0
$865$	5400.00			0.212261
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−6150.00	+	10652.1i	−0.240074	+	0.415821i
$870$		0			0
$871$	−11693.0	−	20252.9i	−0.454882	−	0.787879i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	437.500	+	757.772i	0.0169031	+	0.0292770i
$876$		0			0
$877$	2159.00	−	3739.50i	0.0831291	−	0.143984i	−0.821463	−	0.570261i	$-0.806842\pi$
$877$	2159.00	−	3739.50i	0.0831291	−	0.143984i	0.904592	+	0.426277i	$0.140175\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−16485.0			−0.630413			−0.315206	−	0.949023i	$-0.602074\pi$
$881$	−16485.0			−0.630413			−0.315206	+	0.949023i	$0.602074\pi$
$882$		0			0
$883$	17267.0			0.658076			0.329038	−	0.944317i	$-0.393276\pi$
$883$	17267.0			0.658076			0.329038	+	0.944317i	$0.393276\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−3138.00	+	5435.18i	−0.118787	+	0.205744i	−0.919287	−	0.393588i	$-0.871234\pi$
$887$	−3138.00	+	5435.18i	−0.118787	+	0.205744i	0.800500	+	0.599332i	$0.204567\pi$
$888$		0			0
$889$	6968.50	+	12069.8i	0.262898	+	0.455352i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	8262.00	+	14310.2i	0.309605	+	0.536252i
$894$		0			0
$895$	1185.00	−	2052.48i	0.0442572	−	0.0766557i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−1740.00			−0.0645520
$900$		0			0
$901$	−23904.0			−0.883860
$902$		0			0
$903$		0			0
$904$		0			0
$905$	9737.50	−	16865.8i	0.357663	−	0.619491i
$906$		0			0
$907$	−19316.5	−	33457.2i	−0.707160	−	1.22484i	−0.965906	−	0.258891i	$-0.916643\pi$
$907$	−19316.5	−	33457.2i	−0.707160	−	1.22484i	0.258747	−	0.965945i	$-0.416690\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−26793.0	−	46406.8i	−0.974415	−	1.68774i	−0.681854	−	0.731489i	$-0.738826\pi$
$911$	−26793.0	−	46406.8i	−0.974415	−	1.68774i	−0.292561	−	0.956247i	$-0.594507\pi$
$912$		0			0
$913$	10395.0	−	18004.7i	0.376806	−	0.652648i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−8778.00			−0.316112
$918$		0			0
$919$	−16486.0			−0.591755			−0.295878	−	0.955226i	$-0.595612\pi$
$919$	−16486.0			−0.591755			−0.295878	+	0.955226i	$0.595612\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	6798.00	−	11774.5i	0.242426	−	0.419894i
$924$		0			0
$925$	−2500.00	−	4330.13i	−0.0888643	−	0.153918i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	9027.00	+	15635.2i	0.318801	+	0.552180i	0.980238	−	0.197821i	$-0.0633865\pi$
$929$	9027.00	+	15635.2i	0.318801	+	0.552180i	−0.661437	+	0.750001i	$0.730053\pi$
$930$		0			0
$931$	9996.00	−	17313.6i	0.351886	−	0.609484i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−7200.00			−0.251834
$936$		0			0
$937$	30728.0			1.07133			0.535667	−	0.844429i	$-0.320060\pi$
$937$	30728.0			1.07133			0.535667	+	0.844429i	$0.320060\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−8647.50	+	14977.9i	−0.299575	+	0.518880i	−0.976039	−	0.217596i	$-0.930178\pi$
$941$	−8647.50	+	14977.9i	−0.299575	+	0.518880i	0.676463	+	0.736476i	$0.263512\pi$
$942$		0			0
$943$	−3829.50	−	6632.89i	−0.132244	−	0.229053i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	16402.5	+	28410.0i	0.562840	+	0.974868i	0.997247	+	0.0741510i	$0.0236247\pi$
$947$	16402.5	+	28410.0i	0.562840	+	0.974868i	−0.434407	+	0.900717i	$0.643042\pi$
$948$		0			0
$949$	−5852.00	+	10136.0i	−0.200173	+	0.346709i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−55962.0			−1.90219			−0.951095	−	0.308899i	$-0.900040\pi$
$953$	−55962.0			−1.90219			−0.951095	+	0.308899i	$0.900040\pi$
$954$		0			0
$955$	−780.000			−0.0264295
$956$		0			0
$957$		0			0
$958$		0			0
$959$	5586.00	−	9675.24i	0.188093	−	0.325787i
$960$		0			0
$961$	14695.5	+	25453.4i	0.493287	+	0.854397i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−2045.00	−	3542.04i	−0.0682185	−	0.118158i
$966$		0			0
$967$	−15326.5	+	26546.3i	−0.509687	+	0.882803i	0.490250	+	0.871582i	$0.336905\pi$
$967$	−15326.5	+	26546.3i	−0.509687	+	0.882803i	−0.999937	+	0.0112216i	$0.996428\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	28008.0			0.925664			0.462832	−	0.886446i	$-0.346833\pi$
$971$	28008.0			0.925664			0.462832	+	0.886446i	$0.346833\pi$
$972$		0			0
$973$	−8960.00			−0.295215
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−19821.0	+	34331.0i	−0.649058	+	1.12420i	0.334290	+	0.942470i	$0.391504\pi$
$977$	−19821.0	+	34331.0i	−0.649058	+	1.12420i	−0.983348	+	0.181732i	$0.941830\pi$
$978$		0			0
$979$	−23985.0	−	41543.2i	−0.783007	−	1.35621i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	16903.5	+	29277.7i	0.548462	+	0.949964i	0.998380	+	0.0568940i	$0.0181197\pi$
$983$	16903.5	+	29277.7i	0.548462	+	0.949964i	−0.449918	+	0.893070i	$0.648547\pi$
$984$		0			0
$985$	5610.00	−	9716.81i	0.181472	−	0.314318i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	25752.0			0.827974
$990$		0			0
$991$	7082.00			0.227010			0.113505	−	0.993537i	$-0.463792\pi$
$991$	7082.00			0.227010			0.113505	+	0.993537i	$0.463792\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	4780.00	−	8279.20i	0.152298	−	0.263787i
$996$		0			0
$997$	21545.0	+	37317.0i	0.684390	+	1.18540i	0.973628	+	0.228141i	$0.0732648\pi$
$997$	21545.0	+	37317.0i	0.684390	+	1.18540i	−0.289238	+	0.957257i	$0.593402\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	540.4.i.a.361.1		2
3.2	odd	2		180.4.i.a.121.1	yes	2
9.2	odd	6		180.4.i.a.61.1	✓	2
9.4	even	3		1620.4.a.b.1.1		1
9.5	odd	6		1620.4.a.a.1.1		1
9.7	even	3	inner	540.4.i.a.181.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.4.i.a.61.1	✓	2	9.2	odd	6
180.4.i.a.121.1	yes	2	3.2	odd	2
540.4.i.a.181.1		2	9.7	even	3	inner
540.4.i.a.361.1		2	1.1	even	1	trivial
1620.4.a.a.1.1		1	9.5	odd	6
1620.4.a.b.1.1		1	9.4	even	3

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:	\( N \)	\(=\)	\( 540 = 2^{2} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	540.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.50000 + 4.33013i) q^{5} +(3.50000 + 6.06218i) q^{7} +O(q^{10})\)	\(q+(-2.50000 + 4.33013i) q^{5} +(3.50000 + 6.06218i) q^{7} +(-15.0000 - 25.9808i) q^{11} +(11.0000 - 19.0526i) q^{13} -48.0000 q^{17} +68.0000 q^{19} +(55.5000 - 96.1288i) q^{23} +(-12.5000 - 21.6506i) q^{25} +(43.5000 + 75.3442i) q^{29} +(-10.0000 + 17.3205i) q^{31} -35.0000 q^{35} +200.000 q^{37} +(34.5000 - 59.7558i) q^{41} +(116.000 + 200.918i) q^{43} +(121.500 + 210.444i) q^{47} +(147.000 - 254.611i) q^{49} +498.000 q^{53} +150.000 q^{55} +(33.0000 - 57.1577i) q^{59} +(-179.500 - 310.903i) q^{61} +(55.0000 + 95.2628i) q^{65} +(531.500 - 920.585i) q^{67} +618.000 q^{71} -532.000 q^{73} +(105.000 - 181.865i) q^{77} +(-205.000 - 355.070i) q^{79} +(346.500 + 600.156i) q^{83} +(120.000 - 207.846i) q^{85} +1599.00 q^{89} +154.000 q^{91} +(-170.000 + 294.449i) q^{95} +(-25.0000 - 43.3013i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 5 q^{5} + 7 q^{7}+O(q^{10}) \) 2 * q - 5 * q^5 + 7 * q^7	\( 2 q - 5 q^{5} + 7 q^{7} - 30 q^{11} + 22 q^{13} - 96 q^{17} + 136 q^{19} + 111 q^{23} - 25 q^{25} + 87 q^{29} - 20 q^{31} - 70 q^{35} + 400 q^{37} + 69 q^{41} + 232 q^{43} + 243 q^{47} + 294 q^{49} + 996 q^{53} + 300 q^{55} + 66 q^{59} - 359 q^{61} + 110 q^{65} + 1063 q^{67} + 1236 q^{71} - 1064 q^{73} + 210 q^{77} - 410 q^{79} + 693 q^{83} + 240 q^{85} + 3198 q^{89} + 308 q^{91} - 340 q^{95} - 50 q^{97}+O(q^{100}) \) 2 * q - 5 * q^5 + 7 * q^7 - 30 * q^11 + 22 * q^13 - 96 * q^17 + 136 * q^19 + 111 * q^23 - 25 * q^25 + 87 * q^29 - 20 * q^31 - 70 * q^35 + 400 * q^37 + 69 * q^41 + 232 * q^43 + 243 * q^47 + 294 * q^49 + 996 * q^53 + 300 * q^55 + 66 * q^59 - 359 * q^61 + 110 * q^65 + 1063 * q^67 + 1236 * q^71 - 1064 * q^73 + 210 * q^77 - 410 * q^79 + 693 * q^83 + 240 * q^85 + 3198 * q^89 + 308 * q^91 - 340 * q^95 - 50 * q^97

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