LMFDB - Embedded newform 480.3.j.a.319.10

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [480,3,Mod(319,480)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("480.319"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(480, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 480 = 2^{5} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	480.j (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,4,0,0,0,36,0,0,0,0,0,-24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)

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

Self dual:	no
Analytic conductor:	$13.0790526893$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 6 x^{11} + 49 x^{10} - 190 x^{9} + 792 x^{8} - 2094 x^{7} + 5517 x^{6} - 9954 x^{5} + \cdots + 5584 $ x^12 - 6x^11 + 49x^10 - 190x^9 + 792x^8 - 2094x^7 + 5517x^6 - 9954x^5 + 17189x^4 - 19884x^3 + 20996x^2 - 12416*x + 5584
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{22} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			319.10
Root			$0.500000 - 1.80359i$ of defining polynomial
Character	$\chi$	$=$	480.319
Dual form			480.3.j.a.319.9

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205 q^{3} +(0.621940 + 4.96117i) q^{5} -9.43574 q^{7} +3.00000 q^{9} +3.73048i q^{11} -12.5929i q^{13} +(1.07723 + 8.59300i) q^{15} +30.0484i q^{17} +10.7454i q^{19} -16.3432 q^{21} -38.0158 q^{23} +(-24.2264 + 6.17109i) q^{25} +5.19615 q^{27} -27.0083 q^{29} -16.9678i q^{31} +6.46138i q^{33} +(-5.86846 - 46.8123i) q^{35} +45.2400i q^{37} -21.8116i q^{39} +68.6923 q^{41} -81.0133 q^{43} +(1.86582 + 14.8835i) q^{45} +17.1188 q^{47} +40.0331 q^{49} +52.0454i q^{51} +50.3769i q^{53} +(-18.5075 + 2.32013i) q^{55} +18.6115i q^{57} -63.3848i q^{59} -12.1092 q^{61} -28.3072 q^{63} +(62.4756 - 7.83204i) q^{65} +10.4760 q^{67} -65.8452 q^{69} -1.92327i q^{71} +31.3735i q^{73} +(-41.9613 + 10.6886i) q^{75} -35.1998i q^{77} +69.1535i q^{79} +9.00000 q^{81} +67.3300 q^{83} +(-149.075 + 18.6883i) q^{85} -46.7798 q^{87} +63.1761 q^{89} +118.823i q^{91} -29.3890i q^{93} +(-53.3096 + 6.68297i) q^{95} +51.3786i q^{97} +11.1914i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 4 q^{5} + 36 q^{9} - 24 q^{15} - 80 q^{23} + 28 q^{25} - 40 q^{29} - 144 q^{35} + 136 q^{41} - 224 q^{43} + 12 q^{45} + 208 q^{47} + 212 q^{49} - 192 q^{55} + 40 q^{61} + 96 q^{65} + 352 q^{67} - 192 q^{75}+ \cdots - 176 q^{95}+O(q^{100}) $ 12 * q + 4 * q^5 + 36 * q^9 - 24 * q^15 - 80 * q^23 + 28 * q^25 - 40 * q^29 - 144 * q^35 + 136 * q^41 - 224 * q^43 + 12 * q^45 + 208 * q^47 + 212 * q^49 - 192 * q^55 + 40 * q^61 + 96 * q^65 + 352 * q^67 - 192 * q^75 + 108 * q^81 - 64 * q^85 + 48 * q^87 - 8 * q^89 - 176 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/480\mathbb{Z}\right)^\times$.

$n$	$31$	$97$	$161$	$421$
$\chi(n)$	$-1$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	1.73205			0.577350
$3$	1.73205			0.577350
$4$		0			0
$5$	0.621940	+	4.96117i	0.124388	+	0.992234i
$5$	0.621940	+	4.96117i	0.124388	+	0.992234i
$6$		0			0
$7$	−9.43574			−1.34796			−0.673981	−	0.738748i	$-0.735417\pi$
$7$	−9.43574			−1.34796			−0.673981	+	0.738748i	$0.735417\pi$
$8$		0			0
$9$	3.00000			0.333333
$10$		0			0
$11$			3.73048i			0.339134i	0.985519	+	0.169567i	$0.0542370\pi$
$11$			3.73048i			0.339134i	−0.985519	+	0.169567i	$0.945763\pi$
$12$		0			0
$13$		−	12.5929i		−	0.968686i	−0.874878	−	0.484343i	$-0.839059\pi$
$13$		−	12.5929i		−	0.968686i	0.874878	−	0.484343i	$-0.160941\pi$
$14$		0			0
$15$	1.07723	+	8.59300i	0.0718154	+	0.572866i
$16$		0			0
$17$			30.0484i			1.76755i	0.467907	+	0.883777i	$0.345008\pi$
$17$			30.0484i			1.76755i	−0.467907	+	0.883777i	$0.654992\pi$
$18$		0			0
$19$			10.7454i			0.565546i	0.959187	+	0.282773i	$0.0912543\pi$
$19$			10.7454i			0.565546i	−0.959187	+	0.282773i	$0.908746\pi$
$20$		0			0
$21$	−16.3432			−0.778246
$22$		0			0
$23$	−38.0158			−1.65286			−0.826430	−	0.563040i	$-0.809632\pi$
$23$	−38.0158			−1.65286			−0.826430	+	0.563040i	$0.809632\pi$
$24$		0			0
$25$	−24.2264	+	6.17109i	−0.969055	+	0.246844i
$26$		0			0
$27$	5.19615			0.192450
$28$		0			0
$29$	−27.0083			−0.931321			−0.465661	−	0.884963i	$-0.654183\pi$
$29$	−27.0083			−0.931321			−0.465661	+	0.884963i	$0.654183\pi$
$30$		0			0
$31$		−	16.9678i		−	0.547348i	−0.961823	−	0.273674i	$-0.911761\pi$
$31$		−	16.9678i		−	0.547348i	0.961823	−	0.273674i	$-0.0882389\pi$
$32$		0			0
$33$			6.46138i			0.195799i
$34$		0			0
$35$	−5.86846	−	46.8123i	−0.167670	−	1.33749i
$36$		0			0
$37$			45.2400i			1.22270i	0.791359	+	0.611352i	$0.209374\pi$
$37$			45.2400i			1.22270i	−0.791359	+	0.611352i	$0.790626\pi$
$38$		0			0
$39$		−	21.8116i		−	0.559271i
$40$		0			0
$41$	68.6923			1.67542			0.837711	−	0.546113i	$-0.183893\pi$
$41$	68.6923			1.67542			0.837711	+	0.546113i	$0.183893\pi$
$42$		0			0
$43$	−81.0133			−1.88403			−0.942016	−	0.335569i	$-0.891071\pi$
$43$	−81.0133			−1.88403			−0.942016	+	0.335569i	$0.891071\pi$
$44$		0			0
$45$	1.86582	+	14.8835i	0.0414626	+	0.330745i
$46$		0			0
$47$	17.1188			0.364230			0.182115	−	0.983277i	$-0.441706\pi$
$47$	17.1188			0.364230			0.182115	+	0.983277i	$0.441706\pi$
$48$		0			0
$49$	40.0331			0.817003
$50$		0			0
$51$			52.0454i			1.02050i
$52$		0			0
$53$			50.3769i			0.950507i	0.879849	+	0.475254i	$0.157644\pi$
$53$			50.3769i			0.950507i	−0.879849	+	0.475254i	$0.842356\pi$
$54$		0			0
$55$	−18.5075	+	2.32013i	−0.336501	+	0.0421842i
$56$		0			0
$57$			18.6115i			0.326518i
$58$		0			0
$59$		−	63.3848i		−	1.07432i	−0.843481	−	0.537160i	$-0.819497\pi$
$59$		−	63.3848i		−	1.07432i	0.843481	−	0.537160i	$-0.180503\pi$
$60$		0			0
$61$	−12.1092			−0.198512			−0.0992560	−	0.995062i	$-0.531646\pi$
$61$	−12.1092			−0.198512			−0.0992560	+	0.995062i	$0.531646\pi$
$62$		0			0
$63$	−28.3072			−0.449321
$64$		0			0
$65$	62.4756	−	7.83204i	0.961163	−	0.120493i
$66$		0			0
$67$	10.4760			0.156358			0.0781790	−	0.996939i	$-0.475089\pi$
$67$	10.4760			0.156358			0.0781790	+	0.996939i	$0.475089\pi$
$68$		0			0
$69$	−65.8452			−0.954279
$70$		0			0
$71$		−	1.92327i		−	0.0270883i	−0.999908	−	0.0135442i	$-0.995689\pi$
$71$		−	1.92327i		−	0.0270883i	0.999908	−	0.0135442i	$-0.00431137\pi$
$72$		0			0
$73$			31.3735i			0.429775i	0.976639	+	0.214887i	$0.0689384\pi$
$73$			31.3735i			0.429775i	−0.976639	+	0.214887i	$0.931062\pi$
$74$		0			0
$75$	−41.9613	+	10.6886i	−0.559484	+	0.142515i
$76$		0			0
$77$		−	35.1998i		−	0.457140i
$78$		0			0
$79$			69.1535i			0.875361i	0.899131	+	0.437681i	$0.144200\pi$
$79$			69.1535i			0.875361i	−0.899131	+	0.437681i	$0.855800\pi$
$80$		0			0
$81$	9.00000			0.111111
$82$		0			0
$83$	67.3300			0.811205			0.405603	−	0.914050i	$-0.367062\pi$
$83$	67.3300			0.811205			0.405603	+	0.914050i	$0.367062\pi$
$84$		0			0
$85$	−149.075	+	18.6883i	−1.75383	+	0.219862i
$86$		0			0
$87$	−46.7798			−0.537699
$88$		0			0
$89$	63.1761			0.709844			0.354922	−	0.934896i	$-0.384507\pi$
$89$	63.1761			0.709844			0.354922	+	0.934896i	$0.384507\pi$
$90$		0			0
$91$			118.823i			1.30575i
$92$		0			0
$93$		−	29.3890i		−	0.316011i
$94$		0			0
$95$	−53.3096	+	6.68297i	−0.561154	+	0.0703471i
$96$		0			0
$97$			51.3786i			0.529676i	0.964293	+	0.264838i	$0.0853185\pi$
$97$			51.3786i			0.529676i	−0.964293	+	0.264838i	$0.914681\pi$
$98$		0			0
$99$			11.1914i			0.113045i
$100$		0			0
$101$	68.0951			0.674209			0.337105	−	0.941467i	$-0.390552\pi$
$101$	68.0951			0.674209			0.337105	+	0.941467i	$0.390552\pi$
$102$		0			0
$103$	155.320			1.50796			0.753982	−	0.656895i	$-0.228131\pi$
$103$	155.320			1.50796			0.753982	+	0.656895i	$0.228131\pi$
$104$		0			0
$105$	−10.1645	−	81.0812i	−0.0968044	−	0.772202i
$106$		0			0
$107$	−21.7681			−0.203440			−0.101720	−	0.994813i	$-0.532435\pi$
$107$	−21.7681			−0.203440			−0.101720	+	0.994813i	$0.532435\pi$
$108$		0			0
$109$	−126.784			−1.16315			−0.581577	−	0.813492i	$-0.697564\pi$
$109$	−126.784			−1.16315			−0.581577	+	0.813492i	$0.697564\pi$
$110$		0			0
$111$			78.3580i			0.705928i
$112$		0			0
$113$		−	80.0613i		−	0.708507i	−0.935149	−	0.354253i	$-0.884735\pi$
$113$		−	80.0613i		−	0.708507i	0.935149	−	0.354253i	$-0.115265\pi$
$114$		0			0
$115$	−23.6435	−	188.603i	−0.205596	−	1.64002i
$116$		0			0
$117$		−	37.7788i		−	0.322895i
$118$		0			0
$119$		−	283.529i		−	2.38260i
$120$		0			0
$121$	107.084			0.884988
$122$		0			0
$123$	118.979			0.967306
$124$		0			0
$125$	−45.6832	−	116.353i	−0.365465	−	0.930825i
$126$		0			0
$127$	156.543			1.23262			0.616312	−	0.787502i	$-0.288626\pi$
$127$	156.543			1.23262			0.616312	+	0.787502i	$0.288626\pi$
$128$		0			0
$129$	−140.319			−1.08775
$130$		0			0
$131$		−	35.5220i		−	0.271160i	−0.990766	−	0.135580i	$-0.956710\pi$
$131$		−	35.5220i		−	0.271160i	0.990766	−	0.135580i	$-0.0432898\pi$
$132$		0			0
$133$		−	101.391i		−	0.762335i
$134$		0			0
$135$	3.23169	+	25.7790i	0.0239385	+	0.190955i
$136$		0			0
$137$			105.233i			0.768122i	0.923308	+	0.384061i	$0.125475\pi$
$137$			105.233i			0.768122i	−0.923308	+	0.384061i	$0.874525\pi$
$138$		0			0
$139$			181.835i			1.30817i	0.756423	+	0.654083i	$0.226945\pi$
$139$			181.835i			1.30817i	−0.756423	+	0.654083i	$0.773055\pi$
$140$		0			0
$141$	29.6506			0.210288
$142$		0			0
$143$	46.9776			0.328515
$144$		0			0
$145$	−16.7975	−	133.993i	−0.115845	−	0.924088i
$146$		0			0
$147$	69.3394			0.471697
$148$		0			0
$149$	132.241			0.887527			0.443763	−	0.896144i	$-0.353643\pi$
$149$	132.241			0.887527			0.443763	+	0.896144i	$0.353643\pi$
$150$		0			0
$151$			250.226i			1.65713i	0.559895	+	0.828564i	$0.310842\pi$
$151$			250.226i			1.65713i	−0.559895	+	0.828564i	$0.689158\pi$
$152$		0			0
$153$			90.1453i			0.589185i
$154$		0			0
$155$	84.1800	−	10.5529i	0.543097	−	0.0680834i
$156$		0			0
$157$		−	283.974i		−	1.80875i	−0.426735	−	0.904377i	$-0.640336\pi$
$157$		−	283.974i		−	1.80875i	0.426735	−	0.904377i	$-0.359664\pi$
$158$		0			0
$159$			87.2553i			0.548775i
$160$		0			0
$161$	358.707			2.22799
$162$		0			0
$163$	−142.259			−0.872753			−0.436376	−	0.899764i	$-0.643738\pi$
$163$	−142.259			−0.872753			−0.436376	+	0.899764i	$0.643738\pi$
$164$		0			0
$165$	−32.0560	+	4.01859i	−0.194279	+	0.0243551i
$166$		0			0
$167$	59.4298			0.355867			0.177934	−	0.984042i	$-0.443059\pi$
$167$	59.4298			0.355867			0.177934	+	0.984042i	$0.443059\pi$
$168$		0			0
$169$	10.4183			0.0616470
$170$		0			0
$171$			32.2361i			0.188515i
$172$		0			0
$173$			235.203i			1.35956i	0.733418	+	0.679778i	$0.237924\pi$
$173$			235.203i			1.35956i	−0.733418	+	0.679778i	$0.762076\pi$
$174$		0			0
$175$	228.594	−	58.2288i	1.30625	−	0.332736i
$176$		0			0
$177$		−	109.786i		−	0.620258i
$178$		0			0
$179$		−	52.4818i		−	0.293194i	−0.989196	−	0.146597i	$-0.953168\pi$
$179$		−	52.4818i		−	0.293194i	0.989196	−	0.146597i	$-0.0468321\pi$
$180$		0			0
$181$	−14.9662			−0.0826864			−0.0413432	−	0.999145i	$-0.513164\pi$
$181$	−14.9662			−0.0826864			−0.0413432	+	0.999145i	$0.513164\pi$
$182$		0			0
$183$	−20.9738			−0.114611
$184$		0			0
$185$	−224.443	+	28.1366i	−1.21321	+	0.152089i
$186$		0			0
$187$	−112.095			−0.599439
$188$		0			0
$189$	−49.0295			−0.259415
$190$		0			0
$191$			208.113i			1.08960i	0.838567	+	0.544798i	$0.183394\pi$
$191$			208.113i			1.08960i	−0.838567	+	0.544798i	$0.816606\pi$
$192$		0			0
$193$		−	303.975i		−	1.57500i	−0.616316	−	0.787499i	$-0.711375\pi$
$193$		−	303.975i		−	1.57500i	0.616316	−	0.787499i	$-0.288625\pi$
$194$		0			0
$195$	108.211	−	13.5655i	0.554928	−	0.0695666i
$196$		0			0
$197$		−	318.751i		−	1.61803i	−0.587790	−	0.809014i	$-0.700002\pi$
$197$		−	318.751i		−	1.61803i	0.587790	−	0.809014i	$-0.299998\pi$
$198$		0			0
$199$		−	180.444i		−	0.906756i	−0.891318	−	0.453378i	$-0.850219\pi$
$199$		−	180.444i		−	0.906756i	0.891318	−	0.453378i	$-0.149781\pi$
$200$		0			0
$201$	18.1449			0.0902733
$202$		0			0
$203$	254.843			1.25539
$204$		0			0
$205$	42.7225	+	340.794i	0.208402	+	1.66241i
$206$		0			0
$207$	−114.047			−0.550953
$208$		0			0
$209$	−40.0854			−0.191796
$210$		0			0
$211$		−	47.1478i		−	0.223449i	−0.993739	−	0.111725i	$-0.964363\pi$
$211$		−	47.1478i		−	0.223449i	0.993739	−	0.111725i	$-0.0356375\pi$
$212$		0			0
$213$		−	3.33120i		−	0.0156395i
$214$		0			0
$215$	−50.3854	−	401.921i	−0.234351	−	1.86940i
$216$		0			0
$217$			160.103i			0.737804i
$218$		0			0
$219$			54.3406i			0.248130i
$220$		0			0
$221$	378.398			1.71221
$222$		0			0
$223$	−281.448			−1.26210			−0.631050	−	0.775742i	$-0.717376\pi$
$223$	−281.448			−1.26210			−0.631050	+	0.775742i	$0.717376\pi$
$224$		0			0
$225$	−72.6791	+	18.5133i	−0.323018	+	0.0822812i
$226$		0			0
$227$	241.837			1.06536			0.532682	−	0.846316i	$-0.321184\pi$
$227$	241.837			1.06536			0.532682	+	0.846316i	$0.321184\pi$
$228$		0			0
$229$	103.354			0.451328			0.225664	−	0.974205i	$-0.427545\pi$
$229$	103.354			0.451328			0.225664	+	0.974205i	$0.427545\pi$
$230$		0			0
$231$		−	60.9679i		−	0.263930i
$232$		0			0
$233$		−	356.103i		−	1.52834i	−0.645015	−	0.764170i	$-0.723149\pi$
$233$		−	356.103i		−	1.52834i	0.645015	−	0.764170i	$-0.276851\pi$
$234$		0			0
$235$	10.6469	+	84.9292i	0.0453058	+	0.361401i
$236$		0			0
$237$			119.777i			0.505390i
$238$		0			0
$239$		−	61.3511i		−	0.256699i	−0.991729	−	0.128350i	$-0.959032\pi$
$239$		−	61.3511i		−	0.256699i	0.991729	−	0.128350i	$-0.0409680\pi$
$240$		0			0
$241$	−101.184			−0.419851			−0.209925	−	0.977717i	$-0.567322\pi$
$241$	−101.184			−0.419851			−0.209925	+	0.977717i	$0.567322\pi$
$242$		0			0
$243$	15.5885			0.0641500
$244$		0			0
$245$	24.8982	+	198.611i	0.101625	+	0.810658i
$246$		0			0
$247$	135.316			0.547837
$248$		0			0
$249$	116.619			0.468349
$250$		0			0
$251$			115.712i			0.461005i	0.973072	+	0.230502i	$0.0740370\pi$
$251$			115.712i			0.461005i	−0.973072	+	0.230502i	$0.925963\pi$
$252$		0			0
$253$		−	141.817i		−	0.560542i
$254$		0			0
$255$	−258.206	+	32.3691i	−1.01257	+	0.126938i
$256$		0			0
$257$			159.654i			0.621222i	0.950537	+	0.310611i	$0.100534\pi$
$257$			159.654i			0.621222i	−0.950537	+	0.310611i	$0.899466\pi$
$258$		0			0
$259$		−	426.873i		−	1.64816i
$260$		0			0
$261$	−81.0249			−0.310440
$262$		0			0
$263$	6.73896			0.0256234			0.0128117	−	0.999918i	$-0.495922\pi$
$263$	6.73896			0.0256234			0.0128117	+	0.999918i	$0.495922\pi$
$264$		0			0
$265$	−249.928	+	31.3314i	−0.943125	+	0.118232i
$266$		0			0
$267$	109.424			0.409829
$268$		0			0
$269$	−333.834			−1.24102			−0.620509	−	0.784199i	$-0.713074\pi$
$269$	−333.834			−1.24102			−0.620509	+	0.784199i	$0.713074\pi$
$270$		0			0
$271$			485.690i			1.79221i	0.443839	+	0.896107i	$0.353616\pi$
$271$			485.690i			1.79221i	−0.443839	+	0.896107i	$0.646384\pi$
$272$		0			0
$273$			205.808i			0.753877i
$274$		0			0
$275$	−23.0211	−	90.3760i	−0.0837132	−	0.328640i
$276$		0			0
$277$			132.443i			0.478134i	0.971003	+	0.239067i	$0.0768415\pi$
$277$			132.443i			0.478134i	−0.971003	+	0.239067i	$0.923158\pi$
$278$		0			0
$279$		−	50.9033i		−	0.182449i
$280$		0			0
$281$	−265.883			−0.946202			−0.473101	−	0.881008i	$-0.656865\pi$
$281$	−265.883			−0.946202			−0.473101	+	0.881008i	$0.656865\pi$
$282$		0			0
$283$	−352.065			−1.24405			−0.622024	−	0.782998i	$-0.713689\pi$
$283$	−352.065			−1.24405			−0.622024	+	0.782998i	$0.713689\pi$
$284$		0			0
$285$	−92.3350	+	11.5753i	−0.323982	+	0.0406149i
$286$		0			0
$287$	−648.163			−2.25841
$288$		0			0
$289$	−613.908			−2.12425
$290$		0			0
$291$			88.9904i			0.305809i
$292$		0			0
$293$			246.313i			0.840657i	0.907372	+	0.420329i	$0.138085\pi$
$293$			246.313i			0.840657i	−0.907372	+	0.420329i	$0.861915\pi$
$294$		0			0
$295$	314.463	−	39.4215i	1.06598	−	0.133632i
$296$		0			0
$297$			19.3841i			0.0652665i
$298$		0			0
$299$			478.730i			1.60110i
$300$		0			0
$301$	764.421			2.53960
$302$		0			0
$303$	117.944			0.389255
$304$		0			0
$305$	−7.53121	−	60.0760i	−0.0246925	−	0.196970i
$306$		0			0
$307$	−479.633			−1.56232			−0.781162	−	0.624329i	$-0.785373\pi$
$307$	−479.633			−1.56232			−0.781162	+	0.624329i	$0.785373\pi$
$308$		0			0
$309$	269.023			0.870623
$310$		0			0
$311$			238.106i			0.765613i	0.923828	+	0.382807i	$0.125042\pi$
$311$			238.106i			0.765613i	−0.923828	+	0.382807i	$0.874958\pi$
$312$		0			0
$313$			158.297i			0.505741i	0.967500	+	0.252870i	$0.0813746\pi$
$313$			158.297i			0.505741i	−0.967500	+	0.252870i	$0.918625\pi$
$314$		0			0
$315$	−17.6054	−	140.437i	−0.0558901	−	0.445831i
$316$		0			0
$317$		−	172.575i		−	0.544399i	−0.962241	−	0.272200i	$-0.912249\pi$
$317$		−	172.575i		−	0.544399i	0.962241	−	0.272200i	$-0.0877511\pi$
$318$		0			0
$319$		−	100.754i		−	0.315843i
$320$		0			0
$321$	−37.7035			−0.117456
$322$		0			0
$323$	−322.882			−0.999634
$324$		0			0
$325$	77.7121	+	305.081i	0.239114	+	0.938711i
$326$		0			0
$327$	−219.596			−0.671547
$328$		0			0
$329$	−161.528			−0.490968
$330$		0			0
$331$			394.897i			1.19304i	0.802598	+	0.596521i	$0.203451\pi$
$331$			394.897i			1.19304i	−0.802598	+	0.596521i	$0.796549\pi$
$332$		0			0
$333$			135.720i			0.407568i
$334$		0			0
$335$	6.51543	+	51.9731i	0.0194490	+	0.155144i
$336$		0			0
$337$			552.712i			1.64010i	0.572295	+	0.820048i	$0.306053\pi$
$337$			552.712i			1.64010i	−0.572295	+	0.820048i	$0.693947\pi$
$338$		0			0
$339$		−	138.670i		−	0.409057i
$340$		0			0
$341$	63.2979			0.185624
$342$		0			0
$343$	84.6090			0.246674
$344$		0			0
$345$	−40.9518	−	326.669i	−0.118701	−	0.946868i
$346$		0			0
$347$	−342.973			−0.988395			−0.494197	−	0.869350i	$-0.664538\pi$
$347$	−342.973			−0.988395			−0.494197	+	0.869350i	$0.664538\pi$
$348$		0			0
$349$	−21.4938			−0.0615869			−0.0307935	−	0.999526i	$-0.509803\pi$
$349$	−21.4938			−0.0615869			−0.0307935	+	0.999526i	$0.509803\pi$
$350$		0			0
$351$		−	65.4347i		−	0.186424i
$352$		0			0
$353$			623.505i			1.76630i	0.469086	+	0.883152i	$0.344583\pi$
$353$			623.505i			1.76630i	−0.469086	+	0.883152i	$0.655417\pi$
$354$		0			0
$355$	9.54167	−	1.19616i	0.0268780	−	0.00336946i
$356$		0			0
$357$		−	491.087i		−	1.37559i
$358$		0			0
$359$			283.468i			0.789605i	0.918766	+	0.394802i	$0.129187\pi$
$359$			283.468i			0.789605i	−0.918766	+	0.394802i	$0.870813\pi$
$360$		0			0
$361$	245.537			0.680158
$362$		0			0
$363$	185.474			0.510948
$364$		0			0
$365$	−155.649	+	19.5124i	−0.426437	+	0.0534588i
$366$		0			0
$367$	542.611			1.47850			0.739252	−	0.673429i	$-0.235179\pi$
$367$	542.611			1.47850			0.739252	+	0.673429i	$0.235179\pi$
$368$		0			0
$369$	206.077			0.558474
$370$		0			0
$371$		−	475.343i		−	1.28125i
$372$		0			0
$373$			248.299i			0.665682i	0.942983	+	0.332841i	$0.108007\pi$
$373$			248.299i			0.665682i	−0.942983	+	0.332841i	$0.891993\pi$
$374$		0			0
$375$	−79.1256	−	201.530i	−0.211002	−	0.537412i
$376$		0			0
$377$			340.114i			0.902158i
$378$		0			0
$379$		−	304.847i		−	0.804345i	−0.915564	−	0.402173i	$-0.868255\pi$
$379$		−	304.847i		−	0.804345i	0.915564	−	0.402173i	$-0.131745\pi$
$380$		0			0
$381$	271.141			0.711656
$382$		0			0
$383$	3.78168			0.00987383			0.00493692	−	0.999988i	$-0.498429\pi$
$383$	3.78168			0.00987383			0.00493692	+	0.999988i	$0.498429\pi$
$384$		0			0
$385$	174.632	−	21.8922i	0.453590	−	0.0568627i
$386$		0			0
$387$	−243.040			−0.628010
$388$		0			0
$389$	394.970			1.01535			0.507674	−	0.861549i	$-0.330505\pi$
$389$	394.970			1.01535			0.507674	+	0.861549i	$0.330505\pi$
$390$		0			0
$391$		−	1142.31i		−	2.92152i
$392$		0			0
$393$		−	61.5259i		−	0.156554i
$394$		0			0
$395$	−343.082	+	43.0093i	−0.868563	+	0.108884i
$396$		0			0
$397$		−	287.721i		−	0.724739i	−0.932035	−	0.362369i	$-0.881968\pi$
$397$		−	287.721i		−	0.724739i	0.932035	−	0.362369i	$-0.118032\pi$
$398$		0			0
$399$		−	175.614i		−	0.440134i
$400$		0			0
$401$	−640.735			−1.59784			−0.798922	−	0.601435i	$-0.794596\pi$
$401$	−640.735			−1.59784			−0.798922	+	0.601435i	$0.794596\pi$
$402$		0			0
$403$	−213.674			−0.530208
$404$		0			0
$405$	5.59746	+	44.6505i	0.0138209	+	0.110248i
$406$		0			0
$407$	−168.767			−0.414661
$408$		0			0
$409$	8.13388			0.0198872			0.00994361	−	0.999951i	$-0.496835\pi$
$409$	8.13388			0.0198872			0.00994361	+	0.999951i	$0.496835\pi$
$410$		0			0
$411$			182.268i			0.443476i
$412$		0			0
$413$			598.083i			1.44814i
$414$		0			0
$415$	41.8752	+	334.036i	0.100904	+	0.804905i
$416$		0			0
$417$			314.947i			0.755269i
$418$		0			0
$419$		−	335.898i		−	0.801665i	−0.916151	−	0.400832i	$-0.868721\pi$
$419$		−	335.898i		−	0.801665i	0.916151	−	0.400832i	$-0.131279\pi$
$420$		0			0
$421$	−101.764			−0.241720			−0.120860	−	0.992670i	$-0.538565\pi$
$421$	−101.764			−0.241720			−0.120860	+	0.992670i	$0.538565\pi$
$422$		0			0
$423$	51.3564			0.121410
$424$		0			0
$425$	−185.432	−	727.965i	−0.436310	−	1.71286i
$426$		0			0
$427$	114.260			0.267587
$428$		0			0
$429$	81.3676			0.189668
$430$		0			0
$431$			709.379i			1.64589i	0.568120	+	0.822946i	$0.307671\pi$
$431$			709.379i			1.64589i	−0.568120	+	0.822946i	$0.692329\pi$
$432$		0			0
$433$		−	775.129i		−	1.79014i	−0.445929	−	0.895068i	$-0.647127\pi$
$433$		−	775.129i		−	1.79014i	0.445929	−	0.895068i	$-0.352873\pi$
$434$		0			0
$435$	−29.0942	−	232.082i	−0.0668832	−	0.533523i
$436$		0			0
$437$		−	408.494i		−	0.934768i
$438$		0			0
$439$		−	49.7622i		−	0.113354i	−0.998393	−	0.0566768i	$-0.981950\pi$
$439$		−	49.7622i		−	0.113354i	0.998393	−	0.0566768i	$-0.0180505\pi$
$440$		0			0
$441$	120.099			0.272334
$442$		0			0
$443$	625.947			1.41297			0.706487	−	0.707726i	$-0.250279\pi$
$443$	625.947			1.41297			0.706487	+	0.707726i	$0.250279\pi$
$444$		0			0
$445$	39.2917	+	313.427i	0.0882960	+	0.704331i
$446$		0			0
$447$	229.049			0.512414
$448$		0			0
$449$	−444.155			−0.989209			−0.494605	−	0.869118i	$-0.664687\pi$
$449$	−444.155			−0.989209			−0.494605	+	0.869118i	$0.664687\pi$
$450$		0			0
$451$			256.255i			0.568194i
$452$		0			0
$453$			433.405i			0.956743i
$454$		0			0
$455$	−589.503	+	73.9010i	−1.29561	+	0.162420i
$456$		0			0
$457$			28.9532i			0.0633550i	0.999498	+	0.0316775i	$0.0100849\pi$
$457$			28.9532i			0.0633550i	−0.999498	+	0.0316775i	$0.989915\pi$
$458$		0			0
$459$			156.136i			0.340166i
$460$		0			0
$461$	−651.815			−1.41392			−0.706958	−	0.707256i	$-0.749933\pi$
$461$	−651.815			−1.41392			−0.706958	+	0.707256i	$0.749933\pi$
$462$		0			0
$463$	−329.121			−0.710845			−0.355422	−	0.934706i	$-0.615663\pi$
$463$	−329.121			−0.710845			−0.355422	+	0.934706i	$0.615663\pi$
$464$		0			0
$465$	145.804	−	18.2782i	0.313557	−	0.0393080i
$466$		0			0
$467$	760.999			1.62955			0.814775	−	0.579778i	$-0.196861\pi$
$467$	760.999			1.62955			0.814775	+	0.579778i	$0.196861\pi$
$468$		0			0
$469$	−98.8486			−0.210765
$470$		0			0
$471$		−	491.858i		−	1.04428i
$472$		0			0
$473$		−	302.219i		−	0.638940i
$474$		0			0
$475$	−66.3107	−	260.322i	−0.139602	−	0.548045i
$476$		0			0
$477$			151.131i			0.316836i
$478$		0			0
$479$			536.008i			1.11902i	0.828825	+	0.559508i	$0.189010\pi$
$479$			536.008i			1.11902i	−0.828825	+	0.559508i	$0.810990\pi$
$480$		0			0
$481$	569.704			1.18442
$482$		0			0
$483$	621.298			1.28633
$484$		0			0
$485$	−254.898	+	31.9544i	−0.525563	+	0.0658853i
$486$		0			0
$487$	−22.2364			−0.0456600			−0.0228300	−	0.999739i	$-0.507268\pi$
$487$	−22.2364			−0.0456600			−0.0228300	+	0.999739i	$0.507268\pi$
$488$		0			0
$489$	−246.399			−0.503884
$490$		0			0
$491$			762.724i			1.55341i	0.629865	+	0.776705i	$0.283110\pi$
$491$			762.724i			1.55341i	−0.629865	+	0.776705i	$0.716890\pi$
$492$		0			0
$493$		−	811.558i		−	1.64616i
$494$		0			0
$495$	−55.5226	+	6.96040i	−0.112167	+	0.0140614i
$496$		0			0
$497$			18.1475i			0.0365141i
$498$		0			0
$499$		−	162.461i		−	0.325572i	−0.986661	−	0.162786i	$-0.947952\pi$
$499$		−	162.461i		−	0.325572i	0.986661	−	0.162786i	$-0.0520481\pi$
$500$		0			0
$501$	102.935			0.205460
$502$		0			0
$503$	517.965			1.02975			0.514876	−	0.857265i	$-0.327838\pi$
$503$	517.965			1.02975			0.514876	+	0.857265i	$0.327838\pi$
$504$		0			0
$505$	42.3511	+	337.831i	0.0838635	+	0.668973i
$506$		0			0
$507$	18.0451			0.0355919
$508$		0			0
$509$	143.469			0.281865			0.140932	−	0.990019i	$-0.454990\pi$
$509$	143.469			0.281865			0.140932	+	0.990019i	$0.454990\pi$
$510$		0			0
$511$		−	296.033i		−	0.579320i
$512$		0			0
$513$			55.8346i			0.108839i
$514$		0			0
$515$	96.5998	+	770.570i	0.187572	+	1.49625i
$516$		0			0
$517$			63.8613i			0.123523i
$518$		0			0
$519$			407.384i			0.784940i
$520$		0			0
$521$	542.459			1.04119			0.520594	−	0.853804i	$-0.325710\pi$
$521$	542.459			1.04119			0.520594	+	0.853804i	$0.325710\pi$
$522$		0			0
$523$	−572.228			−1.09413			−0.547063	−	0.837091i	$-0.684254\pi$
$523$	−572.228			−1.09413			−0.547063	+	0.837091i	$0.684254\pi$
$524$		0			0
$525$	395.936	−	100.855i	0.754164	−	0.192105i
$526$		0			0
$527$	509.855			0.967467
$528$		0			0
$529$	916.199			1.73194
$530$		0			0
$531$		−	190.154i		−	0.358106i
$532$		0			0
$533$		−	865.037i		−	1.62296i
$534$		0			0
$535$	−13.5384	−	107.995i	−0.0253055	−	0.201860i
$536$		0			0
$537$		−	90.9011i		−	0.169276i
$538$		0			0
$539$			149.343i			0.277074i
$540$		0			0
$541$	−875.914			−1.61906			−0.809532	−	0.587075i	$-0.800279\pi$
$541$	−875.914			−1.61906			−0.809532	+	0.587075i	$0.800279\pi$
$542$		0			0
$543$	−25.9223			−0.0477390
$544$		0			0
$545$	−78.8518	−	628.995i	−0.144682	−	1.15412i
$546$		0			0
$547$	776.471			1.41951			0.709754	−	0.704449i	$-0.248806\pi$
$547$	776.471			1.41951			0.709754	+	0.704449i	$0.248806\pi$
$548$		0			0
$549$	−36.3277			−0.0661707
$550$		0			0
$551$		−	290.215i		−	0.526705i
$552$		0			0
$553$		−	652.514i		−	1.17995i
$554$		0			0
$555$	−388.747	+	48.7339i	−0.700446	+	0.0878089i
$556$		0			0
$557$		−	417.952i		−	0.750363i	−0.926951	−	0.375182i	$-0.877580\pi$
$557$		−	417.952i		−	0.750363i	0.926951	−	0.375182i	$-0.122420\pi$
$558$		0			0
$559$			1020.19i			1.82504i
$560$		0			0
$561$	−194.154			−0.346086
$562$		0			0
$563$	131.170			0.232984			0.116492	−	0.993192i	$-0.462835\pi$
$563$	131.170			0.232984			0.116492	+	0.993192i	$0.462835\pi$
$564$		0			0
$565$	397.197	−	49.7933i	0.703004	−	0.0881297i
$566$		0			0
$567$	−84.9216			−0.149774
$568$		0			0
$569$	241.675			0.424736			0.212368	−	0.977190i	$-0.431882\pi$
$569$	241.675			0.424736			0.212368	+	0.977190i	$0.431882\pi$
$570$		0			0
$571$		−	311.504i		−	0.545542i	−0.962079	−	0.272771i	$-0.912060\pi$
$571$		−	311.504i		−	0.545542i	0.962079	−	0.272771i	$-0.0879401\pi$
$572$		0			0
$573$			360.462i			0.629079i
$574$		0			0
$575$	920.985	−	234.599i	1.60171	−	0.407998i
$576$		0			0
$577$			293.075i			0.507929i	0.967214	+	0.253964i	$0.0817346\pi$
$577$			293.075i			0.507929i	−0.967214	+	0.253964i	$0.918265\pi$
$578$		0			0
$579$		−	526.500i		−	0.909326i
$580$		0			0
$581$	−635.308			−1.09347
$582$		0			0
$583$	−187.930			−0.322350
$584$		0			0
$585$	187.427	−	23.4961i	0.320388	−	0.0401643i
$586$		0			0
$587$	−528.326			−0.900044			−0.450022	−	0.893018i	$-0.648584\pi$
$587$	−528.326			−0.900044			−0.450022	+	0.893018i	$0.648584\pi$
$588$		0			0
$589$	182.325			0.309550
$590$		0			0
$591$		−	552.094i		−	0.934169i
$592$		0			0
$593$			216.255i			0.364680i	0.983236	+	0.182340i	$0.0583671\pi$
$593$			216.255i			0.364680i	−0.983236	+	0.182340i	$0.941633\pi$
$594$		0			0
$595$	1406.64	−	176.338i	2.36409	−	0.296366i
$596$		0			0
$597$		−	312.539i		−	0.523516i
$598$		0			0
$599$		−	300.236i		−	0.501229i	−0.968087	−	0.250615i	$-0.919367\pi$
$599$		−	300.236i		−	0.501229i	0.968087	−	0.250615i	$-0.0806327\pi$
$600$		0			0
$601$	−387.274			−0.644383			−0.322191	−	0.946675i	$-0.604419\pi$
$601$	−387.274			−0.644383			−0.322191	+	0.946675i	$0.604419\pi$
$602$		0			0
$603$	31.4279			0.0521193
$604$		0			0
$605$	66.5995	+	531.259i	0.110082	+	0.878115i
$606$		0			0
$607$	160.885			0.265049			0.132524	−	0.991180i	$-0.457692\pi$
$607$	160.885			0.265049			0.132524	+	0.991180i	$0.457692\pi$
$608$		0			0
$609$	441.402			0.724797
$610$		0			0
$611$		−	215.576i		−	0.352824i
$612$		0			0
$613$		−	150.755i		−	0.245930i	−0.992411	−	0.122965i	$-0.960760\pi$
$613$		−	150.755i		−	0.245930i	0.992411	−	0.122965i	$-0.0392404\pi$
$614$		0			0
$615$	73.9975	+	590.273i	0.120321	+	0.959793i
$616$		0			0
$617$			252.185i			0.408728i	0.978895	+	0.204364i	$0.0655127\pi$
$617$			252.185i			0.408728i	−0.978895	+	0.204364i	$0.934487\pi$
$618$		0			0
$619$			22.1097i			0.0357184i	0.999841	+	0.0178592i	$0.00568506\pi$
$619$			22.1097i			0.0357184i	−0.999841	+	0.0178592i	$0.994315\pi$
$620$		0			0
$621$	−197.536			−0.318093
$622$		0			0
$623$	−596.113			−0.956843
$624$		0			0
$625$	548.835	−	299.007i	0.878136	−	0.478410i
$626$		0			0
$627$	−69.4300			−0.110734
$628$		0			0
$629$	−1359.39			−2.16120
$630$		0			0
$631$			33.7222i			0.0534424i	0.999643	+	0.0267212i	$0.00850664\pi$
$631$			33.7222i			0.0534424i	−0.999643	+	0.0267212i	$0.991493\pi$
$632$		0			0
$633$		−	81.6624i		−	0.129009i
$634$		0			0
$635$	97.3605	+	776.638i	0.153324	+	1.22305i
$636$		0			0
$637$		−	504.134i		−	0.791419i
$638$		0			0
$639$		−	5.76982i		−	0.00902944i
$640$		0			0
$641$	−492.458			−0.768265			−0.384133	−	0.923278i	$-0.625499\pi$
$641$	−492.458			−0.768265			−0.384133	+	0.923278i	$0.625499\pi$
$642$		0			0
$643$	−981.203			−1.52598			−0.762988	−	0.646413i	$-0.776268\pi$
$643$	−981.203			−1.52598			−0.762988	+	0.646413i	$0.776268\pi$
$644$		0			0
$645$	−87.2701	−	696.147i	−0.135302	−	1.07930i
$646$		0			0
$647$	635.264			0.981861			0.490930	−	0.871199i	$-0.336657\pi$
$647$	635.264			0.981861			0.490930	+	0.871199i	$0.336657\pi$
$648$		0			0
$649$	236.456			0.364339
$650$		0			0
$651$			277.307i			0.425971i
$652$		0			0
$653$			12.2032i			0.0186879i	0.999956	+	0.00934396i	$0.00297432\pi$
$653$			12.2032i			0.0186879i	−0.999956	+	0.00934396i	$0.997026\pi$
$654$		0			0
$655$	176.231	−	22.0925i	0.269054	−	0.0337290i
$656$		0			0
$657$			94.1206i			0.143258i
$658$		0			0
$659$			420.257i			0.637720i	0.947802	+	0.318860i	$0.103300\pi$
$659$			420.257i			0.637720i	−0.947802	+	0.318860i	$0.896700\pi$
$660$		0			0
$661$	−344.639			−0.521390			−0.260695	−	0.965421i	$-0.583952\pi$
$661$	−344.639			−0.521390			−0.260695	+	0.965421i	$0.583952\pi$
$662$		0			0
$663$	655.404			0.988543
$664$		0			0
$665$	503.016	−	63.0588i	0.756414	−	0.0948252i
$666$		0			0
$667$	1026.74			1.53934
$668$		0			0
$669$	−487.483			−0.728674
$670$		0			0
$671$		−	45.1733i		−	0.0673223i
$672$		0			0
$673$		−	1307.73i		−	1.94314i	−0.236747	−	0.971571i	$-0.576081\pi$
$673$		−	1307.73i		−	1.94314i	0.236747	−	0.971571i	$-0.423919\pi$
$674$		0			0
$675$	−125.884	+	32.0659i	−0.186495	+	0.0475051i
$676$		0			0
$677$		−	247.644i		−	0.365796i	−0.983132	−	0.182898i	$-0.941452\pi$
$677$		−	247.644i		−	0.365796i	0.983132	−	0.182898i	$-0.0585479\pi$
$678$		0			0
$679$		−	484.795i		−	0.713984i
$680$		0			0
$681$	418.875			0.615088
$682$		0			0
$683$	242.635			0.355249			0.177624	−	0.984098i	$-0.443159\pi$
$683$	242.635			0.355249			0.177624	+	0.984098i	$0.443159\pi$
$684$		0			0
$685$	−522.077	+	65.4484i	−0.762157	+	0.0955451i
$686$		0			0
$687$	179.015			0.260574
$688$		0			0
$689$	634.392			0.920743
$690$		0			0
$691$		−	753.499i		−	1.09045i	−0.838291	−	0.545224i	$-0.816445\pi$
$691$		−	753.499i		−	1.09045i	0.838291	−	0.545224i	$-0.183555\pi$
$692$		0			0
$693$		−	105.599i		−	0.152380i
$694$		0			0
$695$	−902.114	+	113.090i	−1.29801	+	0.162720i
$696$		0			0
$697$			2064.10i			2.96140i
$698$		0			0
$699$		−	616.789i		−	0.882387i
$700$		0			0
$701$	367.412			0.524126			0.262063	−	0.965051i	$-0.415597\pi$
$701$	367.412			0.524126			0.262063	+	0.965051i	$0.415597\pi$
$702$		0			0
$703$	−486.121			−0.691495
$704$		0			0
$705$	18.4409	+	147.102i	0.0261573	+	0.208655i
$706$		0			0
$707$	−642.528			−0.908809
$708$		0			0
$709$	394.799			0.556839			0.278419	−	0.960460i	$-0.410190\pi$
$709$	394.799			0.556839			0.278419	+	0.960460i	$0.410190\pi$
$710$		0			0
$711$			207.461i			0.291787i
$712$		0			0
$713$			645.043i			0.904689i
$714$		0			0
$715$	29.2172	+	233.064i	0.0408633	+	0.325964i
$716$		0			0
$717$		−	106.263i		−	0.148205i
$718$		0			0
$719$		−	475.466i		−	0.661288i	−0.943756	−	0.330644i	$-0.892734\pi$
$719$		−	475.466i		−	0.661288i	0.943756	−	0.330644i	$-0.107266\pi$
$720$		0			0
$721$	−1465.56			−2.03268
$722$		0			0
$723$	−175.256			−0.242401
$724$		0			0
$725$	654.314	−	166.671i	0.902502	−	0.229891i
$726$		0			0
$727$	−452.918			−0.622996			−0.311498	−	0.950247i	$-0.600831\pi$
$727$	−452.918			−0.622996			−0.311498	+	0.950247i	$0.600831\pi$
$728$		0			0
$729$	27.0000			0.0370370
$730$		0			0
$731$		−	2434.32i		−	3.33013i
$732$		0			0
$733$		−	92.6649i		−	0.126419i	−0.998000	−	0.0632094i	$-0.979866\pi$
$733$		−	92.6649i		−	0.126419i	0.998000	−	0.0632094i	$-0.0201336\pi$
$734$		0			0
$735$	43.1249	+	344.005i	0.0586734	+	0.468033i
$736$		0			0
$737$			39.0804i			0.0530264i
$738$		0			0
$739$			1373.49i			1.85857i	0.369360	+	0.929286i	$0.379577\pi$
$739$			1373.49i			1.85857i	−0.369360	+	0.929286i	$0.620423\pi$
$740$		0			0
$741$	234.374			0.316294
$742$		0			0
$743$	423.832			0.570434			0.285217	−	0.958463i	$-0.407934\pi$
$743$	423.832			0.570434			0.285217	+	0.958463i	$0.407934\pi$
$744$		0			0
$745$	82.2462	+	656.072i	0.110398	+	0.880634i
$746$		0			0
$747$	201.990			0.270402
$748$		0			0
$749$	205.398			0.274230
$750$		0			0
$751$		−	781.600i		−	1.04075i	−0.853939	−	0.520373i	$-0.825793\pi$
$751$		−	781.600i		−	1.04075i	0.853939	−	0.520373i	$-0.174207\pi$
$752$		0			0
$753$			200.419i			0.266161i
$754$		0			0
$755$	−1241.41	+	155.626i	−1.64426	+	0.206127i
$756$		0			0
$757$			122.603i			0.161959i	0.996716	+	0.0809793i	$0.0258048\pi$
$757$			122.603i			0.161959i	−0.996716	+	0.0809793i	$0.974195\pi$
$758$		0			0
$759$		−	245.634i		−	0.323629i
$760$		0			0
$761$	481.169			0.632285			0.316142	−	0.948712i	$-0.397612\pi$
$761$	481.169			0.632285			0.316142	+	0.948712i	$0.397612\pi$
$762$		0			0
$763$	1196.30			1.56789
$764$		0			0
$765$	−447.226	+	56.0649i	−0.584609	+	0.0732875i
$766$		0			0
$767$	−798.200			−1.04068
$768$		0			0
$769$	1369.26			1.78057			0.890287	−	0.455400i	$-0.150504\pi$
$769$	1369.26			1.78057			0.890287	+	0.455400i	$0.150504\pi$
$770$		0			0
$771$			276.529i			0.358663i
$772$		0			0
$773$			248.772i			0.321826i	0.986969	+	0.160913i	$0.0514439\pi$
$773$			248.772i			0.321826i	−0.986969	+	0.160913i	$0.948556\pi$
$774$		0			0
$775$	104.710	+	411.068i	0.135109	+	0.530410i
$776$		0			0
$777$		−	739.366i		−	0.951564i
$778$		0			0
$779$			738.125i			0.947529i
$780$		0			0
$781$	7.17472			0.00918659
$782$		0			0
$783$	−140.339			−0.179233
$784$		0			0
$785$	1408.84	−	176.615i	1.79471	−	0.224987i
$786$		0			0
$787$	73.4073			0.0932749			0.0466374	−	0.998912i	$-0.485149\pi$
$787$	73.4073			0.0932749			0.0466374	+	0.998912i	$0.485149\pi$
$788$		0			0
$789$	11.6722			0.0147937
$790$		0			0
$791$			755.437i			0.955040i
$792$		0			0
$793$			152.491i			0.192296i
$794$		0			0
$795$	−432.888	+	54.2675i	−0.544514	+	0.0682610i
$796$		0			0
$797$			845.614i			1.06100i	0.847686	+	0.530498i	$0.177995\pi$
$797$			845.614i			1.06100i	−0.847686	+	0.530498i	$0.822005\pi$
$798$		0			0
$799$			514.393i			0.643796i
$800$		0			0
$801$	189.528			0.236615
$802$		0			0
$803$	−117.038			−0.145751
$804$		0			0
$805$	223.094	+	1779.60i	0.277135	+	2.21069i
$806$		0			0
$807$	−578.218			−0.716503
$808$		0			0
$809$	46.2307			0.0571455			0.0285728	−	0.999592i	$-0.490904\pi$
$809$	46.2307			0.0571455			0.0285728	+	0.999592i	$0.490904\pi$
$810$		0			0
$811$			614.528i			0.757742i	0.925450	+	0.378871i	$0.123688\pi$
$811$			614.528i			0.757742i	−0.925450	+	0.378871i	$0.876312\pi$
$812$		0			0
$813$			841.239i			1.03473i
$814$		0			0
$815$	−88.4763	−	705.769i	−0.108560	−	0.865975i
$816$		0			0
$817$		−	870.519i		−	1.06551i
$818$		0			0
$819$			356.470i			0.435251i
$820$		0			0
$821$	−1006.35			−1.22576			−0.612880	−	0.790176i	$-0.709989\pi$
$821$	−1006.35			−1.22576			−0.612880	+	0.790176i	$0.709989\pi$
$822$		0			0
$823$	−742.969			−0.902757			−0.451379	−	0.892332i	$-0.649068\pi$
$823$	−742.969			−0.902757			−0.451379	+	0.892332i	$0.649068\pi$
$824$		0			0
$825$	−39.8738	−	156.536i	−0.0483318	−	0.189740i
$826$		0			0
$827$	−1080.66			−1.30672			−0.653361	−	0.757047i	$-0.726642\pi$
$827$	−1080.66			−1.30672			−0.653361	+	0.757047i	$0.726642\pi$
$828$		0			0
$829$	33.5681			0.0404922			0.0202461	−	0.999795i	$-0.493555\pi$
$829$	33.5681			0.0404922			0.0202461	+	0.999795i	$0.493555\pi$
$830$		0			0
$831$			229.398i			0.276051i
$832$		0			0
$833$			1202.93i			1.44410i
$834$		0			0
$835$	36.9618	+	294.841i	0.0442656	+	0.353103i
$836$		0			0
$837$		−	88.1671i		−	0.105337i
$838$		0			0
$839$		−	228.003i		−	0.271756i	−0.990726	−	0.135878i	$-0.956614\pi$
$839$		−	228.003i		−	0.271756i	0.990726	−	0.135878i	$-0.0433855\pi$
$840$		0			0
$841$	−111.551			−0.132641
$842$		0			0
$843$	−460.522			−0.546290
$844$		0			0
$845$	6.47957	+	51.6871i	0.00766813	+	0.0611682i
$846$		0			0
$847$	−1010.41			−1.19293
$848$		0			0
$849$	−609.795			−0.718251
$850$		0			0
$851$		−	1719.83i		−	2.02096i
$852$		0			0
$853$			1499.35i			1.75774i	0.477065	+	0.878868i	$0.341701\pi$
$853$			1499.35i			1.75774i	−0.477065	+	0.878868i	$0.658299\pi$
$854$		0			0
$855$	−159.929	+	20.0489i	−0.187051	+	0.0234490i
$856$		0			0
$857$		−	1462.41i		−	1.70643i	−0.521558	−	0.853216i	$-0.674649\pi$
$857$		−	1462.41i		−	1.70643i	0.521558	−	0.853216i	$-0.325351\pi$
$858$		0			0
$859$		−	1082.73i		−	1.26045i	−0.776412	−	0.630225i	$-0.782963\pi$
$859$		−	1082.73i		−	1.26045i	0.776412	−	0.630225i	$-0.217037\pi$
$860$		0			0
$861$	−1122.65			−1.30389
$862$		0			0
$863$	1359.98			1.57587			0.787935	−	0.615758i	$-0.211150\pi$
$863$	1359.98			1.57587			0.787935	+	0.615758i	$0.211150\pi$
$864$		0			0
$865$	−1166.88	+	146.282i	−1.34900	+	0.169112i
$866$		0			0
$867$	−1063.32			−1.22644
$868$		0			0
$869$	−257.976			−0.296865
$870$		0			0
$871$		−	131.923i		−	0.151462i
$872$		0			0
$873$			154.136i			0.176559i
$874$		0			0
$875$	431.054	+	1097.88i	0.492634	+	1.25472i
$876$		0			0
$877$		−	814.748i		−	0.929018i	−0.885569	−	0.464509i	$-0.846231\pi$
$877$		−	814.748i		−	0.929018i	0.885569	−	0.464509i	$-0.153769\pi$
$878$		0			0
$879$			426.626i			0.485354i
$880$		0			0
$881$	−477.767			−0.542301			−0.271150	−	0.962537i	$-0.587404\pi$
$881$	−477.767			−0.542301			−0.271150	+	0.962537i	$0.587404\pi$
$882$		0			0
$883$	−1044.41			−1.18280			−0.591401	−	0.806378i	$-0.701425\pi$
$883$	−1044.41			−1.18280			−0.591401	+	0.806378i	$0.701425\pi$
$884$		0			0
$885$	544.666	−	68.2801i	0.615441	−	0.0771527i
$886$		0			0
$887$	1042.60			1.17543			0.587713	−	0.809069i	$-0.300028\pi$
$887$	1042.60			1.17543			0.587713	+	0.809069i	$0.300028\pi$
$888$		0			0
$889$	−1477.10			−1.66153
$890$		0			0
$891$			33.5743i			0.0376816i
$892$		0			0
$893$			183.948i			0.205989i
$894$		0			0
$895$	260.371	−	32.6405i	0.290917	−	0.0364698i
$896$		0			0
$897$			829.184i			0.924397i
$898$		0			0
$899$			458.271i			0.509756i
$900$		0			0
$901$	−1513.75			−1.68007
$902$		0			0
$903$	1324.02			1.46624
$904$		0			0
$905$	−9.30809	−	74.2500i	−0.0102852	−	0.0820442i
$906$		0			0
$907$	−615.703			−0.678835			−0.339417	−	0.940636i	$-0.610230\pi$
$907$	−615.703			−0.678835			−0.339417	+	0.940636i	$0.610230\pi$
$908$		0			0
$909$	204.285			0.224736
$910$		0			0
$911$		−	400.257i		−	0.439360i	−0.975572	−	0.219680i	$-0.929499\pi$
$911$		−	400.257i		−	0.439360i	0.975572	−	0.219680i	$-0.0705013\pi$
$912$		0			0
$913$			251.173i			0.275108i
$914$		0			0
$915$	−13.0444	−	104.055i	−0.0142562	−	0.113721i
$916$		0			0
$917$			335.176i			0.365514i
$918$		0			0
$919$			1614.36i			1.75665i	0.478066	+	0.878324i	$0.341338\pi$
$919$			1614.36i			1.75665i	−0.478066	+	0.878324i	$0.658662\pi$
$920$		0			0
$921$	−830.749			−0.902008
$922$		0			0
$923$	−24.2196			−0.0262401
$924$		0			0
$925$	−279.180	−	1096.00i	−0.301817	−	1.18487i
$926$		0			0
$927$	465.961			0.502654
$928$		0			0
$929$	153.105			0.164807			0.0824033	−	0.996599i	$-0.473740\pi$
$929$	153.105			0.164807			0.0824033	+	0.996599i	$0.473740\pi$
$930$		0			0
$931$			430.171i			0.462053i
$932$		0			0
$933$			412.411i			0.442027i
$934$		0			0
$935$	−69.7163	−	556.122i	−0.0745629	−	0.594783i
$936$		0			0
$937$			296.081i			0.315989i	0.987440	+	0.157994i	$0.0505028\pi$
$937$			296.081i			0.315989i	−0.987440	+	0.157994i	$0.949497\pi$
$938$		0			0
$939$			274.178i			0.291989i
$940$		0			0
$941$	129.275			0.137380			0.0686902	−	0.997638i	$-0.478118\pi$
$941$	129.275			0.137380			0.0686902	+	0.997638i	$0.478118\pi$
$942$		0			0
$943$	−2611.39			−2.76924
$944$		0			0
$945$	−30.4934	−	243.244i	−0.0322681	−	0.257401i
$946$		0			0
$947$	141.017			0.148910			0.0744548	−	0.997224i	$-0.476278\pi$
$947$	141.017			0.148910			0.0744548	+	0.997224i	$0.476278\pi$
$948$		0			0
$949$	395.085			0.416317
$950$		0			0
$951$		−	298.908i		−	0.314309i
$952$		0			0
$953$			806.942i			0.846738i	0.905957	+	0.423369i	$0.139153\pi$
$953$			806.942i			0.846738i	−0.905957	+	0.423369i	$0.860847\pi$
$954$		0			0
$955$	−1032.48	+	129.434i	−1.08113	+	0.135533i
$956$		0			0
$957$		−	174.511i		−	0.182352i
$958$		0			0
$959$		−	992.948i		−	1.03540i
$960$		0			0
$961$	673.095			0.700411
$962$		0			0
$963$	−65.3043			−0.0678134
$964$		0			0
$965$	1508.07	−	189.054i	1.56277	−	0.195911i
$966$		0			0
$967$	−889.210			−0.919555			−0.459778	−	0.888034i	$-0.652071\pi$
$967$	−889.210			−0.919555			−0.459778	+	0.888034i	$0.652071\pi$
$968$		0			0
$969$	−559.248			−0.577139
$970$		0			0
$971$		−	1331.48i		−	1.37125i	−0.727955	−	0.685625i	$-0.759529\pi$
$971$		−	1331.48i		−	1.37125i	0.727955	−	0.685625i	$-0.240471\pi$
$972$		0			0
$973$		−	1715.75i		−	1.76336i
$974$		0			0
$975$	134.601	+	528.416i	0.138053	+	0.541965i
$976$		0			0
$977$			1024.81i			1.04894i	0.851430	+	0.524468i	$0.175736\pi$
$977$			1024.81i			1.04894i	−0.851430	+	0.524468i	$0.824264\pi$
$978$		0			0
$979$			235.677i			0.240733i
$980$		0			0
$981$	−380.351			−0.387718
$982$		0			0
$983$	566.908			0.576712			0.288356	−	0.957523i	$-0.406891\pi$
$983$	566.908			0.576712			0.288356	+	0.957523i	$0.406891\pi$
$984$		0			0
$985$	1581.38	−	198.244i	1.60546	−	0.201263i
$986$		0			0
$987$	−279.775			−0.283460
$988$		0			0
$989$	3079.78			3.11404
$990$		0			0
$991$		−	1522.10i		−	1.53592i	−0.640497	−	0.767961i	$-0.721271\pi$
$991$		−	1522.10i		−	1.53592i	0.640497	−	0.767961i	$-0.278729\pi$
$992$		0			0
$993$			683.981i			0.688803i
$994$		0			0
$995$	895.215	−	112.226i	0.899714	−	0.112789i
$996$		0			0
$997$			1574.81i			1.57954i	0.613400	+	0.789772i	$0.289801\pi$
$997$			1574.81i			1.57954i	−0.613400	+	0.789772i	$0.710199\pi$
$998$		0			0
$999$			235.074i			0.235309i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	480.3.j.a.319.10	yes	12
3.2	odd	2		1440.3.j.d.1279.7		12
4.3	odd	2		480.3.j.b.319.4	yes	12
5.2	odd	4		2400.3.e.h.1951.1		12
5.3	odd	4		2400.3.e.i.1951.12		12
5.4	even	2		480.3.j.b.319.3	yes	12
8.3	odd	2		960.3.j.g.319.9		12
8.5	even	2		960.3.j.f.319.3		12
12.11	even	2		1440.3.j.c.1279.7		12
15.14	odd	2		1440.3.j.c.1279.8		12
20.3	even	4		2400.3.e.i.1951.1		12
20.7	even	4		2400.3.e.h.1951.12		12
20.19	odd	2	inner	480.3.j.a.319.9	✓	12
40.19	odd	2		960.3.j.f.319.4		12
40.29	even	2		960.3.j.g.319.10		12
60.59	even	2		1440.3.j.d.1279.8		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
480.3.j.a.319.9	✓	12	20.19	odd	2	inner
480.3.j.a.319.10	yes	12	1.1	even	1	trivial
480.3.j.b.319.3	yes	12	5.4	even	2
480.3.j.b.319.4	yes	12	4.3	odd	2
960.3.j.f.319.3		12	8.5	even	2
960.3.j.f.319.4		12	40.19	odd	2
960.3.j.g.319.9		12	8.3	odd	2
960.3.j.g.319.10		12	40.29	even	2
1440.3.j.c.1279.7		12	12.11	even	2
1440.3.j.c.1279.8		12	15.14	odd	2
1440.3.j.d.1279.7		12	3.2	odd	2
1440.3.j.d.1279.8		12	60.59	even	2
2400.3.e.h.1951.1		12	5.2	odd	4
2400.3.e.h.1951.12		12	20.7	even	4
2400.3.e.i.1951.1		12	20.3	even	4
2400.3.e.i.1951.12		12	5.3	odd	4

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 480 = 2^{5} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	480.j (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.73205 q^{3} +(0.621940 + 4.96117i) q^{5} -9.43574 q^{7} +3.00000 q^{9} +3.73048i q^{11} -12.5929i q^{13} +(1.07723 + 8.59300i) q^{15} +30.0484i q^{17} +10.7454i q^{19} -16.3432 q^{21} -38.0158 q^{23} +(-24.2264 + 6.17109i) q^{25} +5.19615 q^{27} -27.0083 q^{29} -16.9678i q^{31} +6.46138i q^{33} +(-5.86846 - 46.8123i) q^{35} +45.2400i q^{37} -21.8116i q^{39} +68.6923 q^{41} -81.0133 q^{43} +(1.86582 + 14.8835i) q^{45} +17.1188 q^{47} +40.0331 q^{49} +52.0454i q^{51} +50.3769i q^{53} +(-18.5075 + 2.32013i) q^{55} +18.6115i q^{57} -63.3848i q^{59} -12.1092 q^{61} -28.3072 q^{63} +(62.4756 - 7.83204i) q^{65} +10.4760 q^{67} -65.8452 q^{69} -1.92327i q^{71} +31.3735i q^{73} +(-41.9613 + 10.6886i) q^{75} -35.1998i q^{77} +69.1535i q^{79} +9.00000 q^{81} +67.3300 q^{83} +(-149.075 + 18.6883i) q^{85} -46.7798 q^{87} +63.1761 q^{89} +118.823i q^{91} -29.3890i q^{93} +(-53.3096 + 6.68297i) q^{95} +51.3786i q^{97} +11.1914i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 4 q^{5} + 36 q^{9} - 24 q^{15} - 80 q^{23} + 28 q^{25} - 40 q^{29} - 144 q^{35} + 136 q^{41} - 224 q^{43} + 12 q^{45} + 208 q^{47} + 212 q^{49} - 192 q^{55} + 40 q^{61} + 96 q^{65} + 352 q^{67} - 192 q^{75}+ \cdots - 176 q^{95}+O(q^{100}) \) 12 * q + 4 * q^5 + 36 * q^9 - 24 * q^15 - 80 * q^23 + 28 * q^25 - 40 * q^29 - 144 * q^35 + 136 * q^41 - 224 * q^43 + 12 * q^45 + 208 * q^47 + 212 * q^49 - 192 * q^55 + 40 * q^61 + 96 * q^65 + 352 * q^67 - 192 * q^75 + 108 * q^81 - 64 * q^85 + 48 * q^87 - 8 * q^89 - 176 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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