LMFDB - Embedded newform 1380.2.i.b.1241.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1380,2,Mod(1241,1380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1380, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0, 1]))

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

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

chi := DirichletCharacter("1380.1241");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1380.i (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$11.0193554789$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{14} - 4 x^{13} - 2 x^{12} + 12 x^{11} + 17 x^{10} + 16 x^{9} - 110 x^{8} + 48 x^{7} + \cdots + 6561 $ x^16 - x^14 - 4x^13 - 2x^12 + 12x^11 + 17x^10 + 16x^9 - 110x^8 + 48x^7 + 153x^6 + 324x^5 - 162x^4 - 972x^3 - 729x^2 + 6561
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1241.8
Root			$-0.0831550 - 1.73005i$ of defining polynomial
Character	$\chi$	$=$	1380.1241
Dual form			1380.2.i.b.1241.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.0831550 + 1.73005i) q^{3} +1.00000 q^{5} -4.02092i q^{7} +(-2.98617 + 0.287725i) q^{9} +O(q^{10})$	\(q+(0.0831550 + 1.73005i) q^{3} +1.00000 q^{5} -4.02092i q^{7} +(-2.98617 + 0.287725i) q^{9} -0.641443 q^{11} -2.89996 q^{13} +(0.0831550 + 1.73005i) q^{15} +3.10358 q^{17} -1.33629i q^{19} +(6.95640 - 0.334360i) q^{21} +(3.05339 - 3.69822i) q^{23} +1.00000 q^{25} +(-0.746095 - 5.14231i) q^{27} -3.14000i q^{29} +6.11022 q^{31} +(-0.0533392 - 1.10973i) q^{33} -4.02092i q^{35} -7.06427i q^{37} +(-0.241146 - 5.01709i) q^{39} -3.72186i q^{41} -1.20330i q^{43} +(-2.98617 + 0.287725i) q^{45} +3.57780i q^{47} -9.16778 q^{49} +(0.258078 + 5.36936i) q^{51} +5.12728 q^{53} -0.641443 q^{55} +(2.31186 - 0.111119i) q^{57} -11.3428i q^{59} +0.921789i q^{61} +(1.15692 + 12.0071i) q^{63} -2.89996 q^{65} +1.27284i q^{67} +(6.65202 + 4.97500i) q^{69} +1.44420i q^{71} -10.9062 q^{73} +(0.0831550 + 1.73005i) q^{75} +2.57919i q^{77} -3.08045i q^{79} +(8.83443 - 1.71839i) q^{81} +15.6650 q^{83} +3.10358 q^{85} +(5.43237 - 0.261107i) q^{87} -4.31384 q^{89} +11.6605i q^{91} +(0.508096 + 10.5710i) q^{93} -1.33629i q^{95} -13.1699i q^{97} +(1.91546 - 0.184559i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{5} + 2 q^{9}+O(q^{10}) $ 16 * q + 16 * q^5 + 2 * q^9	$ 16 q + 16 q^{5} + 2 q^{9} + 4 q^{21} - 10 q^{23} + 16 q^{25} - 12 q^{27} + 4 q^{31} - 2 q^{33} - 14 q^{39} + 2 q^{45} + 8 q^{49} + 2 q^{51} - 4 q^{53} + 2 q^{57} - 30 q^{63} + 8 q^{69} - 4 q^{73} + 10 q^{81} + 4 q^{83} - 10 q^{87} + 28 q^{89} - 10 q^{93} + 26 q^{99}+O(q^{100}) $ 16 * q + 16 * q^5 + 2 * q^9 + 4 * q^21 - 10 * q^23 + 16 * q^25 - 12 * q^27 + 4 * q^31 - 2 * q^33 - 14 * q^39 + 2 * q^45 + 8 * q^49 + 2 * q^51 - 4 * q^53 + 2 * q^57 - 30 * q^63 + 8 * q^69 - 4 * q^73 + 10 * q^81 + 4 * q^83 - 10 * q^87 + 28 * q^89 - 10 * q^93 + 26 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1380\mathbb{Z}\right)^\times$.

$n$	$277$	$461$	$691$	$1201$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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
$3$	0.0831550	+	1.73005i	0.0480096	+	0.998847i
$3$	0.0831550	+	1.73005i	0.0480096	+	0.998847i
$4$		0			0
$5$	1.00000			0.447214
$5$	1.00000			0.447214
$6$		0			0
$7$		−	4.02092i		−	1.51976i	−0.650061	−	0.759882i	$-0.725257\pi$
$7$		−	4.02092i		−	1.51976i	0.650061	−	0.759882i	$-0.274743\pi$
$8$		0			0
$9$	−2.98617	+	0.287725i	−0.995390	+	0.0959084i
$10$		0			0
$11$	−0.641443			−0.193402			−0.0967012	−	0.995313i	$-0.530829\pi$
$11$	−0.641443			−0.193402			−0.0967012	+	0.995313i	$0.530829\pi$
$12$		0			0
$13$	−2.89996			−0.804305			−0.402152	−	0.915573i	$-0.631738\pi$
$13$	−2.89996			−0.804305			−0.402152	+	0.915573i	$0.631738\pi$
$14$		0			0
$15$	0.0831550	+	1.73005i	0.0214705	+	0.446698i
$16$		0			0
$17$	3.10358			0.752729			0.376364	−	0.926472i	$-0.377174\pi$
$17$	3.10358			0.752729			0.376364	+	0.926472i	$0.377174\pi$
$18$		0			0
$19$		−	1.33629i		−	0.306567i	−0.988182	−	0.153283i	$-0.951015\pi$
$19$		−	1.33629i		−	0.306567i	0.988182	−	0.153283i	$-0.0489847\pi$
$20$		0			0
$21$	6.95640	−	0.334360i	1.51801	−	0.0729633i
$22$		0			0
$23$	3.05339	−	3.69822i	0.636676	−	0.771132i
$23$	3.05339	−	3.69822i	0.636676	−	0.771132i
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	−0.746095	−	5.14231i	−0.143586	−	0.989638i
$28$		0			0
$29$		−	3.14000i		−	0.583084i	−0.956558	−	0.291542i	$-0.905832\pi$
$29$		−	3.14000i		−	0.583084i	0.956558	−	0.291542i	$-0.0941683\pi$
$30$		0			0
$31$	6.11022			1.09743			0.548714	−	0.836010i	$-0.315118\pi$
$31$	6.11022			1.09743			0.548714	+	0.836010i	$0.315118\pi$
$32$		0			0
$33$	−0.0533392	−	1.10973i	−0.00928517	−	0.193179i
$34$		0			0
$35$		−	4.02092i		−	0.679659i
$36$		0			0
$37$		−	7.06427i		−	1.16136i	−0.814132	−	0.580679i	$-0.802787\pi$
$37$		−	7.06427i		−	1.16136i	0.814132	−	0.580679i	$-0.197213\pi$
$38$		0			0
$39$	−0.241146	−	5.01709i	−0.0386143	−	0.803377i
$40$		0			0
$41$		−	3.72186i		−	0.581256i	−0.956836	−	0.290628i	$-0.906136\pi$
$41$		−	3.72186i		−	0.581256i	0.956836	−	0.290628i	$-0.0938643\pi$
$42$		0			0
$43$		−	1.20330i		−	0.183501i	−0.995782	−	0.0917506i	$-0.970754\pi$
$43$		−	1.20330i		−	0.183501i	0.995782	−	0.0917506i	$-0.0292462\pi$
$44$		0			0
$45$	−2.98617	+	0.287725i	−0.445152	+	0.0428916i
$46$		0			0
$47$			3.57780i			0.521876i	0.965356	+	0.260938i	$0.0840318\pi$
$47$			3.57780i			0.521876i	−0.965356	+	0.260938i	$0.915968\pi$
$48$		0			0
$49$	−9.16778			−1.30968
$50$		0			0
$51$	0.258078	+	5.36936i	0.0361382	+	0.751861i
$52$		0			0
$53$	5.12728			0.704286			0.352143	−	0.935946i	$-0.385453\pi$
$53$	5.12728			0.704286			0.352143	+	0.935946i	$0.385453\pi$
$54$		0			0
$55$	−0.641443			−0.0864922
$56$		0			0
$57$	2.31186	−	0.111119i	0.306213	−	0.0147181i
$58$		0			0
$59$		−	11.3428i		−	1.47670i	−0.674415	−	0.738352i	$-0.735604\pi$
$59$		−	11.3428i		−	1.47670i	0.674415	−	0.738352i	$-0.264396\pi$
$60$		0			0
$61$			0.921789i			0.118023i	0.998257	+	0.0590115i	$0.0187949\pi$
$61$			0.921789i			0.118023i	−0.998257	+	0.0590115i	$0.981205\pi$
$62$		0			0
$63$	1.15692	+	12.0071i	0.145758	+	1.51276i
$64$		0			0
$65$	−2.89996			−0.359696
$66$		0			0
$67$			1.27284i			0.155502i	0.996973	+	0.0777509i	$0.0247739\pi$
$67$			1.27284i			0.155502i	−0.996973	+	0.0777509i	$0.975226\pi$
$68$		0			0
$69$	6.65202	+	4.97500i	0.800809	+	0.598920i
$70$		0			0
$71$			1.44420i			0.171395i	0.996321	+	0.0856976i	$0.0273119\pi$
$71$			1.44420i			0.171395i	−0.996321	+	0.0856976i	$0.972688\pi$
$72$		0			0
$73$	−10.9062			−1.27647			−0.638235	−	0.769841i	$-0.720335\pi$
$73$	−10.9062			−1.27647			−0.638235	+	0.769841i	$0.720335\pi$
$74$		0			0
$75$	0.0831550	+	1.73005i	0.00960192	+	0.199769i
$76$		0			0
$77$			2.57919i			0.293926i
$78$		0			0
$79$		−	3.08045i		−	0.346577i	−0.984871	−	0.173289i	$-0.944561\pi$
$79$		−	3.08045i		−	0.346577i	0.984871	−	0.173289i	$-0.0554393\pi$
$80$		0			0
$81$	8.83443	−	1.71839i	0.981603	−	0.190933i
$82$		0			0
$83$	15.6650			1.71946			0.859728	−	0.510752i	$-0.170633\pi$
$83$	15.6650			1.71946			0.859728	+	0.510752i	$0.170633\pi$
$84$		0			0
$85$	3.10358			0.336631
$86$		0			0
$87$	5.43237	−	0.261107i	0.582411	−	0.0279936i
$88$		0			0
$89$	−4.31384			−0.457266			−0.228633	−	0.973513i	$-0.573426\pi$
$89$	−4.31384			−0.457266			−0.228633	+	0.973513i	$0.573426\pi$
$90$		0			0
$91$			11.6605i			1.22235i
$92$		0			0
$93$	0.508096	+	10.5710i	0.0526871	+	1.09616i
$94$		0			0
$95$		−	1.33629i		−	0.137101i
$96$		0			0
$97$		−	13.1699i		−	1.33720i	−0.743620	−	0.668602i	$-0.766893\pi$
$97$		−	13.1699i		−	1.33720i	0.743620	−	0.668602i	$-0.233107\pi$
$98$		0			0
$99$	1.91546	−	0.184559i	0.192511	−	0.0185489i
$100$		0			0
$101$			7.15864i			0.712312i	0.934427	+	0.356156i	$0.115913\pi$
$101$			7.15864i			0.712312i	−0.934427	+	0.356156i	$0.884087\pi$
$102$		0			0
$103$			0.510025i			0.0502542i	0.999684	+	0.0251271i	$0.00799905\pi$
$103$			0.510025i			0.0502542i	−0.999684	+	0.0251271i	$0.992001\pi$
$104$		0			0
$105$	6.95640	−	0.334360i	0.678876	−	0.0326302i
$106$		0			0
$107$	14.0806			1.36122			0.680611	−	0.732645i	$-0.261714\pi$
$107$	14.0806			1.36122			0.680611	+	0.732645i	$0.261714\pi$
$108$		0			0
$109$			12.5760i			1.20456i	0.798283	+	0.602282i	$0.205742\pi$
$109$			12.5760i			1.20456i	−0.798283	+	0.602282i	$0.794258\pi$
$110$		0			0
$111$	12.2216	−	0.587430i	1.16002	−	0.0557563i
$112$		0			0
$113$	0.170942			0.0160809			0.00804045	−	0.999968i	$-0.497441\pi$
$113$	0.170942			0.0160809			0.00804045	+	0.999968i	$0.497441\pi$
$114$		0			0
$115$	3.05339	−	3.69822i	0.284730	−	0.344861i
$116$		0			0
$117$	8.65978	−	0.834393i	0.800597	−	0.0771396i
$118$		0			0
$119$		−	12.4792i		−	1.14397i
$120$		0			0
$121$	−10.5886			−0.962596
$122$		0			0
$123$	6.43901	−	0.309491i	0.580586	−	0.0279059i
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	15.8686			1.40811			0.704056	−	0.710145i	$-0.251371\pi$
$127$	15.8686			1.40811			0.704056	+	0.710145i	$0.251371\pi$
$128$		0			0
$129$	2.08177	−	0.100060i	0.183290	−	0.00880981i
$130$		0			0
$131$			8.60285i			0.751634i	0.926694	+	0.375817i	$0.122638\pi$
$131$			8.60285i			0.751634i	−0.926694	+	0.375817i	$0.877362\pi$
$132$		0			0
$133$	−5.37312			−0.465909
$134$		0			0
$135$	−0.746095	−	5.14231i	−0.0642137	−	0.442579i
$136$		0			0
$137$	−7.74794			−0.661951			−0.330975	−	0.943639i	$-0.607378\pi$
$137$	−7.74794			−0.661951			−0.330975	+	0.943639i	$0.607378\pi$
$138$		0			0
$139$	−10.2656			−0.870715			−0.435358	−	0.900258i	$-0.643378\pi$
$139$	−10.2656			−0.870715			−0.435358	+	0.900258i	$0.643378\pi$
$140$		0			0
$141$	−6.18979	+	0.297512i	−0.521274	+	0.0250551i
$142$		0			0
$143$	1.86016			0.155554
$144$		0			0
$145$		−	3.14000i		−	0.260763i
$146$		0			0
$147$	−0.762347	−	15.8608i	−0.0628774	−	1.30817i
$148$		0			0
$149$	−9.47232			−0.776003			−0.388001	−	0.921659i	$-0.626834\pi$
$149$	−9.47232			−0.776003			−0.388001	+	0.921659i	$0.626834\pi$
$150$		0			0
$151$	−13.8802			−1.12956			−0.564779	−	0.825242i	$-0.691038\pi$
$151$	−13.8802			−1.12956			−0.564779	+	0.825242i	$0.691038\pi$
$152$		0			0
$153$	−9.26782	+	0.892979i	−0.749259	+	0.0721931i
$154$		0			0
$155$	6.11022			0.490785
$156$		0			0
$157$			12.9188i			1.03103i	0.856880	+	0.515516i	$0.172400\pi$
$157$			12.9188i			1.03103i	−0.856880	+	0.515516i	$0.827600\pi$
$158$		0			0
$159$	0.426359	+	8.87047i	0.0338125	+	0.703474i
$160$		0			0
$161$	−14.8702	−	12.2774i	−1.17194	−	0.967597i
$162$		0			0
$163$	21.6758			1.69778			0.848892	−	0.528567i	$-0.177270\pi$
$163$	21.6758			1.69778			0.848892	+	0.528567i	$0.177270\pi$
$164$		0			0
$165$	−0.0533392	−	1.10973i	−0.00415245	−	0.0863924i
$166$		0			0
$167$		−	3.64677i		−	0.282196i	−0.989996	−	0.141098i	$-0.954937\pi$
$167$		−	3.64677i		−	0.282196i	0.989996	−	0.141098i	$-0.0450632\pi$
$168$		0			0
$169$	−4.59022			−0.353094
$170$		0			0
$171$	0.384485	+	3.99040i	0.0294023	+	0.305153i
$172$		0			0
$173$			9.85550i			0.749300i	0.927166	+	0.374650i	$0.122237\pi$
$173$			9.85550i			0.749300i	−0.927166	+	0.374650i	$0.877763\pi$
$174$		0			0
$175$		−	4.02092i		−	0.303953i
$176$		0			0
$177$	19.6236	−	0.943210i	1.47500	−	0.0708960i
$178$		0			0
$179$			16.9605i			1.26769i	0.773460	+	0.633845i	$0.218524\pi$
$179$			16.9605i			1.26769i	−0.773460	+	0.633845i	$0.781476\pi$
$180$		0			0
$181$		−	25.5650i		−	1.90023i	−0.311893	−	0.950117i	$-0.600963\pi$
$181$		−	25.5650i		−	1.90023i	0.311893	−	0.950117i	$-0.399037\pi$
$182$		0			0
$183$	−1.59474	+	0.0766514i	−0.117887	+	0.00566624i
$184$		0			0
$185$		−	7.06427i		−	0.519375i
$186$		0			0
$187$	−1.99077			−0.145580
$188$		0			0
$189$	−20.6768	+	2.99999i	−1.50402	+	0.218217i
$190$		0			0
$191$	12.4794			0.902981			0.451491	−	0.892276i	$-0.350892\pi$
$191$	12.4794			0.902981			0.451491	+	0.892276i	$0.350892\pi$
$192$		0			0
$193$	−21.0097			−1.51231			−0.756155	−	0.654392i	$-0.772924\pi$
$193$	−21.0097			−1.51231			−0.756155	+	0.654392i	$0.772924\pi$
$194$		0			0
$195$	−0.241146	−	5.01709i	−0.0172689	−	0.359281i
$196$		0			0
$197$			9.84641i			0.701528i	0.936464	+	0.350764i	$0.114078\pi$
$197$			9.84641i			0.701528i	−0.936464	+	0.350764i	$0.885922\pi$
$198$		0			0
$199$		−	8.69855i		−	0.616624i	−0.951285	−	0.308312i	$-0.900236\pi$
$199$		−	8.69855i		−	0.616624i	0.951285	−	0.308312i	$-0.0997640\pi$
$200$		0			0
$201$	−2.20208	+	0.105843i	−0.155323	+	0.00746558i
$202$		0			0
$203$	−12.6257			−0.886150
$204$		0			0
$205$		−	3.72186i		−	0.259946i
$206$		0			0
$207$	−8.05387	+	11.9220i	−0.559782	+	0.828640i
$208$		0			0
$209$			0.857156i			0.0592907i
$210$		0			0
$211$	−19.8422			−1.36600			−0.682998	−	0.730420i	$-0.739324\pi$
$211$	−19.8422			−1.36600			−0.682998	+	0.730420i	$0.739324\pi$
$212$		0			0
$213$	−2.49855	+	0.120093i	−0.171197	+	0.00822861i
$214$		0			0
$215$		−	1.20330i		−	0.0820642i
$216$		0			0
$217$		−	24.5687i		−	1.66783i
$218$		0			0
$219$	−0.906903	−	18.8683i	−0.0612828	−	1.27500i
$220$		0			0
$221$	−9.00027			−0.605423
$222$		0			0
$223$	−1.99786			−0.133786			−0.0668932	−	0.997760i	$-0.521309\pi$
$223$	−1.99786			−0.133786			−0.0668932	+	0.997760i	$0.521309\pi$
$224$		0			0
$225$	−2.98617	+	0.287725i	−0.199078	+	0.0191817i
$226$		0			0
$227$	−1.61196			−0.106990			−0.0534949	−	0.998568i	$-0.517036\pi$
$227$	−1.61196			−0.106990			−0.0534949	+	0.998568i	$0.517036\pi$
$228$		0			0
$229$			11.7830i			0.778641i	0.921102	+	0.389321i	$0.127290\pi$
$229$			11.7830i			0.778641i	−0.921102	+	0.389321i	$0.872710\pi$
$230$		0			0
$231$	−4.46214	+	0.214473i	−0.293587	+	0.0141113i
$232$		0			0
$233$		−	9.04517i		−	0.592569i	−0.955100	−	0.296285i	$-0.904252\pi$
$233$		−	9.04517i		−	0.592569i	0.955100	−	0.296285i	$-0.0957477\pi$
$234$		0			0
$235$			3.57780i			0.233390i
$236$		0			0
$237$	5.32934	−	0.256155i	0.346178	−	0.0166390i
$238$		0			0
$239$			13.0937i			0.846962i	0.905905	+	0.423481i	$0.139192\pi$
$239$			13.0937i			0.846962i	−0.905905	+	0.423481i	$0.860808\pi$
$240$		0			0
$241$		−	21.5418i		−	1.38763i	−0.720155	−	0.693813i	$-0.755929\pi$
$241$		−	21.5418i		−	1.38763i	0.720155	−	0.693813i	$-0.244071\pi$
$242$		0			0
$243$	3.70754	+	15.1411i	0.237839	+	0.971305i
$244$		0			0
$245$	−9.16778			−0.585708
$246$		0			0
$247$			3.87520i			0.246573i
$248$		0			0
$249$	1.30262	+	27.1013i	0.0825504	+	1.71747i
$250$		0			0
$251$	−2.74004			−0.172950			−0.0864750	−	0.996254i	$-0.527560\pi$
$251$	−2.74004			−0.172950			−0.0864750	+	0.996254i	$0.527560\pi$
$252$		0			0
$253$	−1.95858	+	2.37220i	−0.123135	+	0.149139i
$254$		0			0
$255$	0.258078	+	5.36936i	0.0161615	+	0.336242i
$256$		0			0
$257$			26.5539i			1.65639i	0.560442	+	0.828194i	$0.310631\pi$
$257$			26.5539i			1.65639i	−0.560442	+	0.828194i	$0.689369\pi$
$258$		0			0
$259$	−28.4048			−1.76499
$260$		0			0
$261$	0.903458	+	9.37658i	0.0559227	+	0.580396i
$262$		0			0
$263$	4.09330			0.252404			0.126202	−	0.992005i	$-0.459721\pi$
$263$	4.09330			0.252404			0.126202	+	0.992005i	$0.459721\pi$
$264$		0			0
$265$	5.12728			0.314966
$266$		0			0
$267$	−0.358718	−	7.46317i	−0.0219532	−	0.456739i
$268$		0			0
$269$			5.94400i			0.362412i	0.983445	+	0.181206i	$0.0580001\pi$
$269$			5.94400i			0.362412i	−0.983445	+	0.181206i	$0.942000\pi$
$270$		0			0
$271$	2.18049			0.132456			0.0662278	−	0.997805i	$-0.478904\pi$
$271$	2.18049			0.132456			0.0662278	+	0.997805i	$0.478904\pi$
$272$		0			0
$273$	−20.1733	+	0.969630i	−1.22094	+	0.0586847i
$274$		0			0
$275$	−0.641443			−0.0386805
$276$		0			0
$277$	25.4414			1.52862			0.764312	−	0.644846i	$-0.223079\pi$
$277$	25.4414			1.52862			0.764312	+	0.644846i	$0.223079\pi$
$278$		0			0
$279$	−18.2462	+	1.75807i	−1.09237	+	0.105253i
$280$		0			0
$281$	−26.1951			−1.56267			−0.781335	−	0.624112i	$-0.785461\pi$
$281$	−26.1951			−1.56267			−0.781335	+	0.624112i	$0.785461\pi$
$282$		0			0
$283$		−	10.9054i		−	0.648261i	−0.946012	−	0.324130i	$-0.894928\pi$
$283$		−	10.9054i		−	0.648261i	0.946012	−	0.324130i	$-0.105072\pi$
$284$		0			0
$285$	2.31186	−	0.111119i	0.136943	−	0.00658215i
$286$		0			0
$287$	−14.9653			−0.883373
$288$		0			0
$289$	−7.36779			−0.433399
$290$		0			0
$291$	22.7847	−	1.09515i	1.33566	−	0.0641986i
$292$		0			0
$293$	27.0813			1.58211			0.791054	−	0.611746i	$-0.209532\pi$
$293$	27.0813			1.58211			0.791054	+	0.611746i	$0.209532\pi$
$294$		0			0
$295$		−	11.3428i		−	0.660402i
$296$		0			0
$297$	0.478578	+	3.29850i	0.0277699	+	0.191398i
$298$		0			0
$299$	−8.85471	+	10.7247i	−0.512081	+	0.620225i
$300$		0			0
$301$	−4.83836			−0.278878
$302$		0			0
$303$	−12.3848	+	0.595277i	−0.711490	+	0.0341978i
$304$		0			0
$305$			0.921789i			0.0527815i
$306$		0			0
$307$	14.8147			0.845517			0.422759	−	0.906242i	$-0.361062\pi$
$307$	14.8147			0.845517			0.422759	+	0.906242i	$0.361062\pi$
$308$		0			0
$309$	−0.882370	+	0.0424111i	−0.0501963	+	0.00241269i
$310$		0			0
$311$			0.729791i			0.0413826i	0.999786	+	0.0206913i	$0.00658672\pi$
$311$			0.729791i			0.0413826i	−0.999786	+	0.0206913i	$0.993413\pi$
$312$		0			0
$313$		−	17.2658i		−	0.975921i	−0.872866	−	0.487960i	$-0.837741\pi$
$313$		−	17.2658i		−	0.975921i	0.872866	−	0.487960i	$-0.162259\pi$
$314$		0			0
$315$	1.15692	+	12.0071i	0.0651851	+	0.676526i
$316$		0			0
$317$			26.1142i			1.46672i	0.679841	+	0.733360i	$0.262049\pi$
$317$			26.1142i			1.46672i	−0.679841	+	0.733360i	$0.737951\pi$
$318$		0			0
$319$			2.01413i			0.112770i
$320$		0			0
$321$	1.17087	+	24.3602i	0.0653517	+	1.35965i
$322$		0			0
$323$		−	4.14729i		−	0.230762i
$324$		0			0
$325$	−2.89996			−0.160861
$326$		0			0
$327$	−21.7572	+	1.04576i	−1.20318	+	0.0578306i
$328$		0			0
$329$	14.3860			0.793129
$330$		0			0
$331$	3.80035			0.208886			0.104443	−	0.994531i	$-0.466694\pi$
$331$	3.80035			0.208886			0.104443	+	0.994531i	$0.466694\pi$
$332$		0			0
$333$	2.03257	+	21.0951i	0.111384	+	1.15601i
$334$		0			0
$335$			1.27284i			0.0695426i
$336$		0			0
$337$		−	31.5995i		−	1.72133i	−0.509169	−	0.860667i	$-0.670047\pi$
$337$		−	31.5995i		−	1.72133i	0.509169	−	0.860667i	$-0.329953\pi$
$338$		0			0
$339$	0.0142147	+	0.295740i	0.000772038	+	0.0160624i
$340$		0			0
$341$	−3.91936			−0.212245
$342$		0			0
$343$			8.71648i			0.470646i
$344$		0			0
$345$	6.65202	+	4.97500i	0.358133	+	0.267845i
$346$		0			0
$347$		−	35.4098i		−	1.90090i	−0.310879	−	0.950449i	$-0.600624\pi$
$347$		−	35.4098i		−	1.90090i	0.310879	−	0.950449i	$-0.399376\pi$
$348$		0			0
$349$	−6.70111			−0.358702			−0.179351	−	0.983785i	$-0.557400\pi$
$349$	−6.70111			−0.358702			−0.179351	+	0.983785i	$0.557400\pi$
$350$		0			0
$351$	2.16365	+	14.9125i	0.115487	+	0.795970i
$352$		0			0
$353$			16.1005i			0.856942i	0.903556	+	0.428471i	$0.140948\pi$
$353$			16.1005i			0.856942i	−0.903556	+	0.428471i	$0.859052\pi$
$354$		0			0
$355$			1.44420i			0.0766502i
$356$		0			0
$357$	21.5898	−	1.03771i	1.14265	−	0.0549215i
$358$		0			0
$359$	4.18835			0.221053			0.110526	−	0.993873i	$-0.464746\pi$
$359$	4.18835			0.221053			0.110526	+	0.993873i	$0.464746\pi$
$360$		0			0
$361$	17.2143			0.906017
$362$		0			0
$363$	−0.880491	−	18.3188i	−0.0462138	−	0.961486i
$364$		0			0
$365$	−10.9062			−0.570855
$366$		0			0
$367$		−	21.2504i		−	1.10926i	−0.832097	−	0.554630i	$-0.812860\pi$
$367$		−	21.2504i		−	1.10926i	0.832097	−	0.554630i	$-0.187140\pi$
$368$		0			0
$369$	1.07087	+	11.1141i	0.0557474	+	0.578577i
$370$		0			0
$371$		−	20.6164i		−	1.07035i
$372$		0			0
$373$			30.7001i			1.58959i	0.606877	+	0.794796i	$0.292422\pi$
$373$			30.7001i			1.58959i	−0.606877	+	0.794796i	$0.707578\pi$
$374$		0			0
$375$	0.0831550	+	1.73005i	0.00429411	+	0.0893396i
$376$		0			0
$377$			9.10589i			0.468977i
$378$		0			0
$379$			30.0529i			1.54371i	0.635797	+	0.771857i	$0.280672\pi$
$379$			30.0529i			1.54371i	−0.635797	+	0.771857i	$0.719328\pi$
$380$		0			0
$381$	1.31955	+	27.4535i	0.0676028	+	1.40649i
$382$		0			0
$383$	−26.4414			−1.35109			−0.675546	−	0.737318i	$-0.736092\pi$
$383$	−26.4414			−1.35109			−0.675546	+	0.737318i	$0.736092\pi$
$384$		0			0
$385$			2.57919i			0.131448i
$386$		0			0
$387$	0.346219	+	3.59325i	0.0175993	+	0.182655i
$388$		0			0
$389$	15.8089			0.801542			0.400771	−	0.916178i	$-0.368742\pi$
$389$	15.8089			0.801542			0.400771	+	0.916178i	$0.368742\pi$
$390$		0			0
$391$	9.47644	−	11.4777i	0.479244	−	0.580453i
$392$		0			0
$393$	−14.8834	+	0.715370i	−0.750767	+	0.0360856i
$394$		0			0
$395$		−	3.08045i		−	0.154994i
$396$		0			0
$397$	12.5795			0.631348			0.315674	−	0.948868i	$-0.397769\pi$
$397$	12.5795			0.631348			0.315674	+	0.948868i	$0.397769\pi$
$398$		0			0
$399$	−0.446802	−	9.29579i	−0.0223681	−	0.465372i
$400$		0			0
$401$	−34.7830			−1.73698			−0.868489	−	0.495708i	$-0.834909\pi$
$401$	−34.7830			−1.73698			−0.868489	+	0.495708i	$0.834909\pi$
$402$		0			0
$403$	−17.7194			−0.882667
$404$		0			0
$405$	8.83443	−	1.71839i	0.438986	−	0.0853877i
$406$		0			0
$407$			4.53133i			0.224610i
$408$		0			0
$409$	−5.98209			−0.295795			−0.147898	−	0.989003i	$-0.547251\pi$
$409$	−5.98209			−0.295795			−0.147898	+	0.989003i	$0.547251\pi$
$410$		0			0
$411$	−0.644280	−	13.4043i	−0.0317800	−	0.661188i
$412$		0			0
$413$	−45.6084			−2.24424
$414$		0			0
$415$	15.6650			0.768964
$416$		0			0
$417$	−0.853635	−	17.7600i	−0.0418027	−	0.869711i
$418$		0			0
$419$	8.04481			0.393015			0.196507	−	0.980502i	$-0.437040\pi$
$419$	8.04481			0.393015			0.196507	+	0.980502i	$0.437040\pi$
$420$		0			0
$421$			13.4364i			0.654849i	0.944877	+	0.327425i	$0.106181\pi$
$421$			13.4364i			0.654849i	−0.944877	+	0.327425i	$0.893819\pi$
$422$		0			0
$423$	−1.02942	−	10.6839i	−0.0500523	−	0.519470i
$424$		0			0
$425$	3.10358			0.150546
$426$		0			0
$427$	3.70644			0.179367
$428$		0			0
$429$	0.154682	+	3.21818i	0.00746811	+	0.155375i
$430$		0			0
$431$	2.73846			0.131907			0.0659534	−	0.997823i	$-0.478991\pi$
$431$	2.73846			0.131907			0.0659534	+	0.997823i	$0.478991\pi$
$432$		0			0
$433$			19.0187i			0.913979i	0.889472	+	0.456989i	$0.151072\pi$
$433$			19.0187i			0.913979i	−0.889472	+	0.456989i	$0.848928\pi$
$434$		0			0
$435$	5.43237	−	0.261107i	0.260462	−	0.0125191i
$436$		0			0
$437$	−4.94190	−	4.08022i	−0.236403	−	0.195183i
$438$		0			0
$439$	24.0581			1.14823			0.574114	−	0.818775i	$-0.305346\pi$
$439$	24.0581			1.14823			0.574114	+	0.818775i	$0.305346\pi$
$440$		0			0
$441$	27.3766	−	2.63780i	1.30365	−	0.125610i
$442$		0			0
$443$		−	16.5450i		−	0.786074i	−0.919523	−	0.393037i	$-0.871424\pi$
$443$		−	16.5450i		−	0.786074i	0.919523	−	0.393037i	$-0.128576\pi$
$444$		0			0
$445$	−4.31384			−0.204496
$446$		0			0
$447$	−0.787671	−	16.3876i	−0.0372556	−	0.775108i
$448$		0			0
$449$			18.9142i			0.892617i	0.894879	+	0.446309i	$0.147262\pi$
$449$			18.9142i			0.892617i	−0.894879	+	0.446309i	$0.852738\pi$
$450$		0			0
$451$			2.38736i			0.112416i
$452$		0			0
$453$	−1.15421	−	24.0135i	−0.0542296	−	1.12825i
$454$		0			0
$455$			11.6605i			0.546653i
$456$		0			0
$457$		−	3.91674i		−	0.183217i	−0.995795	−	0.0916087i	$-0.970799\pi$
$457$		−	3.91674i		−	0.183217i	0.995795	−	0.0916087i	$-0.0292009\pi$
$458$		0			0
$459$	−2.31557	−	15.9596i	−0.108081	−	0.744929i
$460$		0			0
$461$		−	21.7358i		−	1.01234i	−0.862435	−	0.506168i	$-0.831062\pi$
$461$		−	21.7358i		−	1.01234i	0.862435	−	0.506168i	$-0.168938\pi$
$462$		0			0
$463$	−7.15417			−0.332483			−0.166241	−	0.986085i	$-0.553163\pi$
$463$	−7.15417			−0.332483			−0.166241	+	0.986085i	$0.553163\pi$
$464$		0			0
$465$	0.508096	+	10.5710i	0.0235624	+	0.490219i
$466$		0			0
$467$	35.0147			1.62029			0.810143	−	0.586232i	$-0.199389\pi$
$467$	35.0147			1.62029			0.810143	+	0.586232i	$0.199389\pi$
$468$		0			0
$469$	5.11798			0.236326
$470$		0			0
$471$	−22.3502	+	1.07426i	−1.02984	+	0.0494994i
$472$		0			0
$473$			0.771847i			0.0354896i
$474$		0			0
$475$		−	1.33629i		−	0.0613133i
$476$		0			0
$477$	−15.3109	+	1.47525i	−0.701039	+	0.0675470i
$478$		0			0
$479$	−12.5386			−0.572903			−0.286451	−	0.958095i	$-0.592476\pi$
$479$	−12.5386			−0.572903			−0.286451	+	0.958095i	$0.592476\pi$
$480$		0			0
$481$			20.4861i			0.934086i
$482$		0			0
$483$	20.0041	−	26.7472i	0.910217	−	1.21704i
$484$		0			0
$485$		−	13.1699i		−	0.598016i
$486$		0			0
$487$	16.3582			0.741262			0.370631	−	0.928780i	$-0.379141\pi$
$487$	16.3582			0.741262			0.370631	+	0.928780i	$0.379141\pi$
$488$		0			0
$489$	1.80246	+	37.5004i	0.0815099	+	1.69583i
$490$		0			0
$491$		−	12.4930i		−	0.563803i	−0.959443	−	0.281902i	$-0.909035\pi$
$491$		−	12.4930i		−	0.563803i	0.959443	−	0.281902i	$-0.0909651\pi$
$492$		0			0
$493$		−	9.74525i		−	0.438904i
$494$		0			0
$495$	1.91546	−	0.184559i	0.0860935	−	0.00829533i
$496$		0			0
$497$	5.80701			0.260480
$498$		0			0
$499$	−25.5989			−1.14596			−0.572981	−	0.819568i	$-0.694213\pi$
$499$	−25.5989			−1.14596			−0.572981	+	0.819568i	$0.694213\pi$
$500$		0			0
$501$	6.30911	−	0.303247i	0.281870	−	0.0135481i
$502$		0			0
$503$	−38.4303			−1.71352			−0.856762	−	0.515712i	$-0.827528\pi$
$503$	−38.4303			−1.71352			−0.856762	+	0.515712i	$0.827528\pi$
$504$		0			0
$505$			7.15864i			0.318556i
$506$		0			0
$507$	−0.381700	−	7.94132i	−0.0169519	−	0.352687i
$508$		0			0
$509$			21.0597i			0.933455i	0.884401	+	0.466728i	$0.154567\pi$
$509$			21.0597i			0.933455i	−0.884401	+	0.466728i	$0.845433\pi$
$510$		0			0
$511$			43.8528i			1.93993i
$512$		0			0
$513$	−6.87163	+	0.997002i	−0.303390	+	0.0440187i
$514$		0			0
$515$			0.510025i			0.0224744i
$516$		0			0
$517$		−	2.29496i		−	0.100932i
$518$		0			0
$519$	−17.0505	+	0.819534i	−0.748436	+	0.0359736i
$520$		0			0
$521$	29.4599			1.29066			0.645331	−	0.763903i	$-0.276720\pi$
$521$	29.4599			1.29066			0.645331	+	0.763903i	$0.276720\pi$
$522$		0			0
$523$			24.3365i			1.06416i	0.846694	+	0.532080i	$0.178589\pi$
$523$			24.3365i			1.06416i	−0.846694	+	0.532080i	$0.821411\pi$
$524$		0			0
$525$	6.95640	−	0.334360i	0.303602	−	0.0145927i
$526$		0			0
$527$	18.9636			0.826066
$528$		0			0
$529$	−4.35364	−	22.5842i	−0.189289	−	0.981921i
$530$		0			0
$531$	3.26361	+	33.8715i	0.141628	+	1.46990i
$532$		0			0
$533$			10.7932i			0.467507i
$534$		0			0
$535$	14.0806			0.608757
$536$		0			0
$537$	−29.3427	+	1.41035i	−1.26623	+	0.0608613i
$538$		0			0
$539$	5.88061			0.253296
$540$		0			0
$541$	13.0056			0.559157			0.279578	−	0.960123i	$-0.409805\pi$
$541$	13.0056			0.559157			0.279578	+	0.960123i	$0.409805\pi$
$542$		0			0
$543$	44.2289	−	2.12586i	1.89804	−	0.0912295i
$544$		0			0
$545$			12.5760i			0.538698i
$546$		0			0
$547$	−16.2330			−0.694073			−0.347037	−	0.937852i	$-0.612812\pi$
$547$	−16.2330			−0.694073			−0.347037	+	0.937852i	$0.612812\pi$
$548$		0			0
$549$	−0.265222	−	2.75262i	−0.0113194	−	0.117479i
$550$		0			0
$551$	−4.19596			−0.178754
$552$		0			0
$553$	−12.3862			−0.526716
$554$		0			0
$555$	12.2216	−	0.587430i	0.518776	−	0.0249350i
$556$		0			0
$557$	2.76745			0.117261			0.0586303	−	0.998280i	$-0.481327\pi$
$557$	2.76745			0.117261			0.0586303	+	0.998280i	$0.481327\pi$
$558$		0			0
$559$			3.48952i			0.147591i
$560$		0			0
$561$	−0.165543	−	3.44414i	−0.00698921	−	0.145412i
$562$		0			0
$563$	32.3086			1.36165			0.680823	−	0.732448i	$-0.261622\pi$
$563$	32.3086			1.36165			0.680823	+	0.732448i	$0.261622\pi$
$564$		0			0
$565$	0.170942			0.00719160
$566$		0			0
$567$	−6.90952	−	35.5225i	−0.290173	−	1.49181i
$568$		0			0
$569$	43.6500			1.82990			0.914952	−	0.403563i	$-0.132228\pi$
$569$	43.6500			1.82990			0.914952	+	0.403563i	$0.132228\pi$
$570$		0			0
$571$			19.2922i			0.807352i	0.914902	+	0.403676i	$0.132268\pi$
$571$			19.2922i			0.807352i	−0.914902	+	0.403676i	$0.867732\pi$
$572$		0			0
$573$	1.03773	+	21.5901i	0.0433517	+	0.901940i
$574$		0			0
$575$	3.05339	−	3.69822i	0.127335	−	0.154226i
$576$		0			0
$577$	−32.1157			−1.33700			−0.668498	−	0.743714i	$-0.733062\pi$
$577$	−32.1157			−1.33700			−0.668498	+	0.743714i	$0.733062\pi$
$578$		0			0
$579$	−1.74706	−	36.3479i	−0.0726054	−	1.51057i
$580$		0			0
$581$		−	62.9877i		−	2.61317i
$582$		0			0
$583$	−3.28886			−0.136211
$584$		0			0
$585$	8.65978	−	0.834393i	0.358038	−	0.0344979i
$586$		0			0
$587$			7.67739i			0.316880i	0.987369	+	0.158440i	$0.0506464\pi$
$587$			7.67739i			0.316880i	−0.987369	+	0.158440i	$0.949354\pi$
$588$		0			0
$589$		−	8.16505i		−	0.336435i
$590$		0			0
$591$	−17.0348	+	0.818779i	−0.700719	+	0.0336800i
$592$		0			0
$593$		−	4.82648i		−	0.198200i	−0.995077	−	0.0990998i	$-0.968404\pi$
$593$		−	4.82648i		−	0.198200i	0.995077	−	0.0990998i	$-0.0315963\pi$
$594$		0			0
$595$		−	12.4792i		−	0.511599i
$596$		0			0
$597$	15.0490	−	0.723328i	0.615913	−	0.0296039i
$598$		0			0
$599$			32.9192i			1.34504i	0.740078	+	0.672521i	$0.234789\pi$
$599$			32.9192i			1.34504i	−0.740078	+	0.672521i	$0.765211\pi$
$600$		0			0
$601$	3.84513			0.156846			0.0784231	−	0.996920i	$-0.475011\pi$
$601$	3.84513			0.156846			0.0784231	+	0.996920i	$0.475011\pi$
$602$		0			0
$603$	−0.366228	−	3.80091i	−0.0149139	−	0.154785i
$604$		0			0
$605$	−10.5886			−0.430486
$606$		0			0
$607$	5.90559			0.239700			0.119850	−	0.992792i	$-0.461759\pi$
$607$	5.90559			0.239700			0.119850	+	0.992792i	$0.461759\pi$
$608$		0			0
$609$	−1.04989	−	21.8431i	−0.0425437	−	0.885128i
$610$		0			0
$611$		−	10.3755i		−	0.419747i
$612$		0			0
$613$			5.79086i			0.233890i	0.993138	+	0.116945i	$0.0373102\pi$
$613$			5.79086i			0.233890i	−0.993138	+	0.116945i	$0.962690\pi$
$614$		0			0
$615$	6.43901	−	0.309491i	0.259646	−	0.0124799i
$616$		0			0
$617$	22.1573			0.892018			0.446009	−	0.895028i	$-0.352845\pi$
$617$	22.1573			0.892018			0.446009	+	0.895028i	$0.352845\pi$
$618$		0			0
$619$			44.6942i			1.79641i	0.439574	+	0.898206i	$0.355129\pi$
$619$			44.6942i			1.79641i	−0.439574	+	0.898206i	$0.644871\pi$
$620$		0			0
$621$	−21.2955	−	12.9422i	−0.854559	−	0.519354i
$622$		0			0
$623$			17.3456i			0.694937i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−1.48293	+	0.0712768i	−0.0592223	+	0.00284652i
$628$		0			0
$629$		−	21.9245i		−	0.874188i
$630$		0			0
$631$			8.96253i			0.356793i	0.983959	+	0.178396i	$0.0570909\pi$
$631$			8.96253i			0.356793i	−0.983959	+	0.178396i	$0.942909\pi$
$632$		0			0
$633$	−1.64998	−	34.3282i	−0.0655809	−	1.36442i
$634$		0			0
$635$	15.8686			0.629727
$636$		0			0
$637$	26.5862			1.05338
$638$		0			0
$639$	−0.415533	−	4.31263i	−0.0164382	−	0.170605i
$640$		0			0
$641$	35.2004			1.39033			0.695166	−	0.718849i	$-0.255331\pi$
$641$	35.2004			1.39033			0.695166	+	0.718849i	$0.255331\pi$
$642$		0			0
$643$		−	38.1653i		−	1.50509i	−0.658538	−	0.752547i	$-0.728825\pi$
$643$		−	38.1653i		−	1.50509i	0.658538	−	0.752547i	$-0.271175\pi$
$644$		0			0
$645$	2.08177	−	0.100060i	0.0819696	−	0.00393987i
$646$		0			0
$647$		−	16.4962i		−	0.648533i	−0.945966	−	0.324266i	$-0.894883\pi$
$647$		−	16.4962i		−	0.648533i	0.945966	−	0.324266i	$-0.105117\pi$
$648$		0			0
$649$			7.27575i			0.285598i
$650$		0			0
$651$	42.5052	−	2.04301i	1.66591	−	0.0800719i
$652$		0			0
$653$			22.1280i			0.865934i	0.901410	+	0.432967i	$0.142533\pi$
$653$			22.1280i			0.865934i	−0.901410	+	0.432967i	$0.857467\pi$
$654$		0			0
$655$			8.60285i			0.336141i
$656$		0			0
$657$	32.5677	−	3.13798i	1.27059	−	0.122424i
$658$		0			0
$659$	42.8854			1.67058			0.835288	−	0.549813i	$-0.185301\pi$
$659$	42.8854			1.67058			0.835288	+	0.549813i	$0.185301\pi$
$660$		0			0
$661$		−	20.0140i		−	0.778455i	−0.921142	−	0.389228i	$-0.872742\pi$
$661$		−	20.0140i		−	0.778455i	0.921142	−	0.389228i	$-0.127258\pi$
$662$		0			0
$663$	−0.748418	−	15.5709i	−0.0290661	−	0.604725i
$664$		0			0
$665$	−5.37312			−0.208361
$666$		0			0
$667$	−11.6124	−	9.58764i	−0.449634	−	0.371235i
$668$		0			0
$669$	−0.166132	−	3.45640i	−0.00642303	−	0.133632i
$670$		0			0
$671$		−	0.591275i		−	0.0228259i
$672$		0			0
$673$	11.8502			0.456793			0.228396	−	0.973568i	$-0.426652\pi$
$673$	11.8502			0.456793			0.228396	+	0.973568i	$0.426652\pi$
$674$		0			0
$675$	−0.746095	−	5.14231i	−0.0287172	−	0.197928i
$676$		0			0
$677$	−28.1072			−1.08025			−0.540123	−	0.841586i	$-0.681622\pi$
$677$	−28.1072			−1.08025			−0.540123	+	0.841586i	$0.681622\pi$
$678$		0			0
$679$	−52.9553			−2.03224
$680$		0			0
$681$	−0.134043	−	2.78878i	−0.00513653	−	0.106866i
$682$		0			0
$683$		−	41.5454i		−	1.58969i	−0.606813	−	0.794844i	$-0.707552\pi$
$683$		−	41.5454i		−	1.58969i	0.606813	−	0.794844i	$-0.292448\pi$
$684$		0			0
$685$	−7.74794			−0.296033
$686$		0			0
$687$	−20.3852	+	0.979814i	−0.777744	+	0.0373823i
$688$		0			0
$689$	−14.8689			−0.566461
$690$		0			0
$691$	−13.0681			−0.497133			−0.248566	−	0.968615i	$-0.579959\pi$
$691$	−13.0681			−0.497133			−0.248566	+	0.968615i	$0.579959\pi$
$692$		0			0
$693$	−0.742098	−	7.70190i	−0.0281900	−	0.292571i
$694$		0			0
$695$	−10.2656			−0.389396
$696$		0			0
$697$		−	11.5511i		−	0.437529i
$698$		0			0
$699$	15.6486	−	0.752152i	0.591886	−	0.0284490i
$700$		0			0
$701$	−24.3220			−0.918629			−0.459314	−	0.888274i	$-0.651905\pi$
$701$	−24.3220			−0.918629			−0.459314	+	0.888274i	$0.651905\pi$
$702$		0			0
$703$	−9.43993			−0.356034
$704$		0			0
$705$	−6.18979	+	0.297512i	−0.233121	+	0.0112050i
$706$		0			0
$707$	28.7843			1.08255
$708$		0			0
$709$			19.2745i			0.723870i	0.932203	+	0.361935i	$0.117884\pi$
$709$			19.2745i			0.723870i	−0.932203	+	0.361935i	$0.882116\pi$
$710$		0			0
$711$	0.886322	+	9.19874i	0.0332397	+	0.344980i
$712$		0			0
$713$	18.6569	−	22.5969i	0.698706	−	0.846262i
$714$		0			0
$715$	1.86016			0.0695661
$716$		0			0
$717$	−22.6528	+	1.08881i	−0.845985	+	0.0406623i
$718$		0			0
$719$			11.2588i			0.419883i	0.977714	+	0.209942i	$0.0673274\pi$
$719$			11.2588i			0.419883i	−0.977714	+	0.209942i	$0.932673\pi$
$720$		0			0
$721$	2.05077			0.0763746
$722$		0			0
$723$	37.2684	−	1.79131i	1.38603	−	0.0666194i
$724$		0			0
$725$		−	3.14000i		−	0.116617i
$726$		0			0
$727$		−	14.9001i		−	0.552614i	−0.961069	−	0.276307i	$-0.910889\pi$
$727$		−	14.9001i		−	0.552614i	0.961069	−	0.276307i	$-0.0891107\pi$
$728$		0			0
$729$	−25.8867	+	7.67331i	−0.958766	+	0.284197i
$730$		0			0
$731$		−	3.73453i		−	0.138127i
$732$		0			0
$733$		−	22.6998i		−	0.838435i	−0.907886	−	0.419217i	$-0.862304\pi$
$733$		−	22.6998i		−	0.838435i	0.907886	−	0.419217i	$-0.137696\pi$
$734$		0			0
$735$	−0.762347	−	15.8608i	−0.0281196	−	0.585033i
$736$		0			0
$737$		−	0.816453i		−	0.0300744i
$738$		0			0
$739$	47.0904			1.73225			0.866124	−	0.499829i	$-0.166604\pi$
$739$	47.0904			1.73225			0.866124	+	0.499829i	$0.166604\pi$
$740$		0			0
$741$	−6.70430	+	0.322242i	−0.246289	+	0.0118379i
$742$		0			0
$743$	−46.1428			−1.69282			−0.846408	−	0.532536i	$-0.821239\pi$
$743$	−46.1428			−1.69282			−0.846408	+	0.532536i	$0.821239\pi$
$744$		0			0
$745$	−9.47232			−0.347039
$746$		0			0
$747$	−46.7783	+	4.50722i	−1.71153	+	0.164910i
$748$		0			0
$749$		−	56.6169i		−	2.06874i
$750$		0			0
$751$		−	29.8659i		−	1.08982i	−0.838494	−	0.544911i	$-0.816563\pi$
$751$		−	29.8659i		−	1.08982i	0.838494	−	0.544911i	$-0.183437\pi$
$752$		0			0
$753$	−0.227849	−	4.74042i	−0.00830326	−	0.172751i
$754$		0			0
$755$	−13.8802			−0.505153
$756$		0			0
$757$			39.1104i			1.42149i	0.703449	+	0.710746i	$0.251642\pi$
$757$			39.1104i			1.42149i	−0.703449	+	0.710746i	$0.748358\pi$
$758$		0			0
$759$	−4.26689	−	3.19118i	−0.154878	−	0.115832i
$760$		0			0
$761$			0.0699634i			0.00253617i	0.999999	+	0.00126809i	$0.000403644\pi$
$761$			0.0699634i			0.00253617i	−0.999999	+	0.00126809i	$0.999596\pi$
$762$		0			0
$763$	50.5672			1.83065
$764$		0			0
$765$	−9.26782	+	0.892979i	−0.335079	+	0.0322857i
$766$		0			0
$767$			32.8936i			1.18772i
$768$		0			0
$769$		−	22.1180i		−	0.797596i	−0.917039	−	0.398798i	$-0.869427\pi$
$769$		−	22.1180i		−	0.797596i	0.917039	−	0.398798i	$-0.130573\pi$
$770$		0			0
$771$	−45.9397	+	2.20809i	−1.65448	+	0.0795225i
$772$		0			0
$773$	32.8192			1.18043			0.590213	−	0.807248i	$-0.299044\pi$
$773$	32.8192			1.18043			0.590213	+	0.807248i	$0.299044\pi$
$774$		0			0
$775$	6.11022			0.219486
$776$		0			0
$777$	−2.36201	−	49.1419i	−0.0847365	−	1.76296i
$778$		0			0
$779$	−4.97349			−0.178194
$780$		0			0
$781$		−	0.926373i		−	0.0331482i
$782$		0			0
$783$	−16.1469	+	2.34274i	−0.577042	+	0.0837227i
$784$		0			0
$785$			12.9188i			0.461091i
$786$		0			0
$787$			34.8223i			1.24128i	0.784096	+	0.620640i	$0.213127\pi$
$787$			34.8223i			1.24128i	−0.784096	+	0.620640i	$0.786873\pi$
$788$		0			0
$789$	0.340379	+	7.08163i	0.0121178	+	0.252113i
$790$		0			0
$791$		−	0.687345i		−	0.0244392i
$792$		0			0
$793$		−	2.67315i		−	0.0949265i
$794$		0			0
$795$	0.426359	+	8.87047i	0.0151214	+	0.314603i
$796$		0			0
$797$	24.6425			0.872882			0.436441	−	0.899733i	$-0.356239\pi$
$797$	24.6425			0.872882			0.436441	+	0.899733i	$0.356239\pi$
$798$		0			0
$799$			11.1040i			0.392831i
$800$		0			0
$801$	12.8819	−	1.24120i	0.455158	−	0.0438557i
$802$		0			0
$803$	6.99569			0.246872
$804$		0			0
$805$	−14.8702	−	12.2774i	−0.524107	−	0.432722i
$806$		0			0
$807$	−10.2834	+	0.494274i	−0.361994	+	0.0173993i
$808$		0			0
$809$		−	17.7305i		−	0.623372i	−0.950185	−	0.311686i	$-0.899106\pi$
$809$		−	17.7305i		−	0.623372i	0.950185	−	0.311686i	$-0.100894\pi$
$810$		0			0
$811$	−11.1173			−0.390380			−0.195190	−	0.980765i	$-0.562532\pi$
$811$	−11.1173			−0.390380			−0.195190	+	0.980765i	$0.562532\pi$
$812$		0			0
$813$	0.181319	+	3.77237i	0.00635913	+	0.132303i
$814$		0			0
$815$	21.6758			0.759272
$816$		0			0
$817$	−1.60796			−0.0562553
$818$		0			0
$819$	−3.35502	−	34.8203i	−0.117234	−	1.21672i
$820$		0			0
$821$		−	35.5549i		−	1.24088i	−0.784256	−	0.620438i	$-0.786955\pi$
$821$		−	35.5549i		−	1.24088i	0.784256	−	0.620438i	$-0.213045\pi$
$822$		0			0
$823$	49.0792			1.71079			0.855397	−	0.517973i	$-0.173313\pi$
$823$	49.0792			1.71079			0.855397	+	0.517973i	$0.173313\pi$
$824$		0			0
$825$	−0.0533392	−	1.10973i	−0.00185703	−	0.0386359i
$826$		0			0
$827$	12.2238			0.425062			0.212531	−	0.977154i	$-0.431829\pi$
$827$	12.2238			0.425062			0.212531	+	0.977154i	$0.431829\pi$
$828$		0			0
$829$	−8.73797			−0.303482			−0.151741	−	0.988420i	$-0.548488\pi$
$829$	−8.73797			−0.303482			−0.151741	+	0.988420i	$0.548488\pi$
$830$		0			0
$831$	2.11558	+	44.0150i	0.0733886	+	1.52686i
$832$		0			0
$833$	−28.4530			−0.985837
$834$		0			0
$835$		−	3.64677i		−	0.126202i
$836$		0			0
$837$	−4.55881	−	31.4206i	−0.157575	−	1.08606i
$838$		0			0
$839$	−1.22017			−0.0421250			−0.0210625	−	0.999778i	$-0.506705\pi$
$839$	−1.22017			−0.0421250			−0.0210625	+	0.999778i	$0.506705\pi$
$840$		0			0
$841$	19.1404			0.660013
$842$		0			0
$843$	−2.17826	−	45.3190i	−0.0750231	−	1.56087i
$844$		0			0
$845$	−4.59022			−0.157908
$846$		0			0
$847$			42.5757i			1.46292i
$848$		0			0
$849$	18.8670	−	0.906842i	0.647513	−	0.0311227i
$850$		0			0
$851$	−26.1252	−	21.5700i	−0.895561	−	0.739409i
$852$		0			0
$853$	−21.6114			−0.739959			−0.369980	−	0.929040i	$-0.620635\pi$
$853$	−21.6114			−0.739959			−0.369980	+	0.929040i	$0.620635\pi$
$854$		0			0
$855$	0.384485	+	3.99040i	0.0131491	+	0.136469i
$856$		0			0
$857$			32.5300i			1.11120i	0.831448	+	0.555602i	$0.187512\pi$
$857$			32.5300i			1.11120i	−0.831448	+	0.555602i	$0.812488\pi$
$858$		0			0
$859$	52.7309			1.79916			0.899578	−	0.436760i	$-0.143874\pi$
$859$	52.7309			1.79916			0.899578	+	0.436760i	$0.143874\pi$
$860$		0			0
$861$	−1.24444	−	25.8907i	−0.0424104	−	0.882354i
$862$		0			0
$863$			44.1954i			1.50443i	0.658918	+	0.752214i	$0.271014\pi$
$863$			44.1954i			1.50443i	−0.658918	+	0.752214i	$0.728986\pi$
$864$		0			0
$865$			9.85550i			0.335097i
$866$		0			0
$867$	−0.612669	−	12.7467i	−0.0208073	−	0.432899i
$868$		0			0
$869$			1.97593i			0.0670289i
$870$		0			0
$871$		−	3.69118i		−	0.125071i
$872$		0			0
$873$	3.78933	+	39.3277i	0.128249	+	1.33104i
$874$		0			0
$875$		−	4.02092i		−	0.135932i
$876$		0			0
$877$	57.0543			1.92659			0.963293	−	0.268454i	$-0.0865126\pi$
$877$	57.0543			1.92659			0.963293	+	0.268454i	$0.0865126\pi$
$878$		0			0
$879$	2.25195	+	46.8522i	0.0759564	+	1.58028i
$880$		0			0
$881$	36.7818			1.23921			0.619605	−	0.784914i	$-0.287293\pi$
$881$	36.7818			1.23921			0.619605	+	0.784914i	$0.287293\pi$
$882$		0			0
$883$	46.4704			1.56385			0.781926	−	0.623371i	$-0.214237\pi$
$883$	46.4704			1.56385			0.781926	+	0.623371i	$0.214237\pi$
$884$		0			0
$885$	19.6236	−	0.943210i	0.659641	−	0.0317056i
$886$		0			0
$887$		−	21.1776i		−	0.711074i	−0.934662	−	0.355537i	$-0.884298\pi$
$887$		−	21.1776i		−	0.711074i	0.934662	−	0.355537i	$-0.115702\pi$
$888$		0			0
$889$		−	63.8064i		−	2.14000i
$890$		0			0
$891$	−5.66678	+	1.10225i	−0.189844	+	0.0369268i
$892$		0			0
$893$	4.78099			0.159990
$894$		0			0
$895$			16.9605i			0.566929i
$896$		0			0
$897$	−19.2906	−	14.4273i	−0.644095	−	0.481714i
$898$		0			0
$899$		−	19.1861i		−	0.639892i
$900$		0			0
$901$	15.9129			0.530136
$902$		0			0
$903$	−0.402334	−	8.37062i	−0.0133888	−	0.278557i
$904$		0			0
$905$		−	25.5650i		−	0.849811i
$906$		0			0
$907$			45.8979i			1.52402i	0.647567	+	0.762008i	$0.275786\pi$
$907$			45.8979i			1.52402i	−0.647567	+	0.762008i	$0.724214\pi$
$908$		0			0
$909$	−2.05972	−	21.3769i	−0.0683167	−	0.709028i
$910$		0			0
$911$	−38.9640			−1.29093			−0.645467	−	0.763788i	$-0.723337\pi$
$911$	−38.9640			−1.29093			−0.645467	+	0.763788i	$0.723337\pi$
$912$		0			0
$913$	−10.0482			−0.332547
$914$		0			0
$915$	−1.59474	+	0.0766514i	−0.0527206	+	0.00253402i
$916$		0			0
$917$	34.5913			1.14231
$918$		0			0
$919$			41.4674i			1.36788i	0.729537	+	0.683941i	$0.239735\pi$
$919$			41.4674i			1.36788i	−0.729537	+	0.683941i	$0.760265\pi$
$920$		0			0
$921$	1.23191	+	25.6302i	0.0405929	+	0.844542i
$922$		0			0
$923$		−	4.18813i		−	0.137854i
$924$		0			0
$925$		−	7.06427i		−	0.232272i
$926$		0			0
$927$	−0.146747	−	1.52302i	−0.00481981	−	0.0500226i
$928$		0			0
$929$			24.0813i			0.790082i	0.918663	+	0.395041i	$0.129270\pi$
$929$			24.0813i			0.790082i	−0.918663	+	0.395041i	$0.870730\pi$
$930$		0			0
$931$			12.2508i			0.401505i
$932$		0			0
$933$	−1.26258	+	0.0606858i	−0.0413349	+	0.00198676i
$934$		0			0
$935$	−1.99077			−0.0651052
$936$		0			0
$937$			3.45592i			0.112900i	0.998405	+	0.0564499i	$0.0179781\pi$
$937$			3.45592i			0.112900i	−0.998405	+	0.0564499i	$0.982022\pi$
$938$		0			0
$939$	29.8708	−	1.43574i	0.974795	−	0.0468535i
$940$		0			0
$941$	9.70843			0.316486			0.158243	−	0.987400i	$-0.449417\pi$
$941$	9.70843			0.316486			0.158243	+	0.987400i	$0.449417\pi$
$942$		0			0
$943$	−13.7642	−	11.3643i	−0.448225	−	0.370072i
$944$		0			0
$945$	−20.6768	+	2.99999i	−0.672616	+	0.0975896i
$946$		0			0
$947$			25.2341i			0.819998i	0.912086	+	0.409999i	$0.134471\pi$
$947$			25.2341i			0.819998i	−0.912086	+	0.409999i	$0.865529\pi$
$948$		0			0
$949$	31.6275			1.02667
$950$		0			0
$951$	−45.1790	+	2.17153i	−1.46503	+	0.0704166i
$952$		0			0
$953$	56.6762			1.83592			0.917960	−	0.396672i	$-0.129835\pi$
$953$	56.6762			1.83592			0.917960	+	0.396672i	$0.129835\pi$
$954$		0			0
$955$	12.4794			0.403825
$956$		0			0
$957$	−3.48456	+	0.167485i	−0.112640	+	0.00541403i
$958$		0			0
$959$			31.1538i			1.00601i
$960$		0			0
$961$	6.33481			0.204349
$962$		0			0
$963$	−42.0470	+	4.05134i	−1.35495	+	0.130553i
$964$		0			0
$965$	−21.0097			−0.676326
$966$		0			0
$967$	2.76298			0.0888513			0.0444257	−	0.999013i	$-0.485854\pi$
$967$	2.76298			0.0888513			0.0444257	+	0.999013i	$0.485854\pi$
$968$		0			0
$969$	7.17504	−	0.344868i	0.230495	−	0.0110788i
$970$		0			0
$971$	11.8285			0.379594			0.189797	−	0.981823i	$-0.439217\pi$
$971$	11.8285			0.379594			0.189797	+	0.981823i	$0.439217\pi$
$972$		0			0
$973$			41.2771i			1.32328i
$974$		0			0
$975$	−0.241146	−	5.01709i	−0.00772287	−	0.160675i
$976$		0			0
$977$	43.2315			1.38310			0.691549	−	0.722330i	$-0.256929\pi$
$977$	43.2315			1.38310			0.691549	+	0.722330i	$0.256929\pi$
$978$		0			0
$979$	2.76708			0.0884364
$980$		0			0
$981$	−3.61844	−	37.5541i	−0.115528	−	1.19901i
$982$		0			0
$983$	26.4684			0.844211			0.422105	−	0.906547i	$-0.361291\pi$
$983$	26.4684			0.844211			0.422105	+	0.906547i	$0.361291\pi$
$984$		0			0
$985$			9.84641i			0.313733i
$986$		0			0
$987$	1.19627	+	24.8886i	0.0380778	+	0.792214i
$988$		0			0
$989$	−4.45006	−	3.67413i	−0.141504	−	0.116831i
$990$		0			0
$991$	10.1890			0.323664			0.161832	−	0.986818i	$-0.448260\pi$
$991$	10.1890			0.323664			0.161832	+	0.986818i	$0.448260\pi$
$992$		0			0
$993$	0.316018	+	6.57480i	0.0100285	+	0.208645i
$994$		0			0
$995$		−	8.69855i		−	0.275763i
$996$		0			0
$997$	−60.4205			−1.91354			−0.956769	−	0.290849i	$-0.906062\pi$
$997$	−60.4205			−1.91354			−0.956769	+	0.290849i	$0.906062\pi$
$998$		0			0
$999$	−36.3267	+	5.27062i	−1.14932	+	0.166755i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1380.2.i.b.1241.8	yes	16
3.2	odd	2		1380.2.i.a.1241.7	✓	16
23.22	odd	2		1380.2.i.a.1241.8	yes	16
69.68	even	2	inner	1380.2.i.b.1241.7	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.2.i.a.1241.7	✓	16	3.2	odd	2
1380.2.i.a.1241.8	yes	16	23.22	odd	2
1380.2.i.b.1241.7	yes	16	69.68	even	2	inner
1380.2.i.b.1241.8	yes	16	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1380.i (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(0.0831550 + 1.73005i) q^{3} +1.00000 q^{5} -4.02092i q^{7} +(-2.98617 + 0.287725i) q^{9} +O(q^{10})\)	\(q+(0.0831550 + 1.73005i) q^{3} +1.00000 q^{5} -4.02092i q^{7} +(-2.98617 + 0.287725i) q^{9} -0.641443 q^{11} -2.89996 q^{13} +(0.0831550 + 1.73005i) q^{15} +3.10358 q^{17} -1.33629i q^{19} +(6.95640 - 0.334360i) q^{21} +(3.05339 - 3.69822i) q^{23} +1.00000 q^{25} +(-0.746095 - 5.14231i) q^{27} -3.14000i q^{29} +6.11022 q^{31} +(-0.0533392 - 1.10973i) q^{33} -4.02092i q^{35} -7.06427i q^{37} +(-0.241146 - 5.01709i) q^{39} -3.72186i q^{41} -1.20330i q^{43} +(-2.98617 + 0.287725i) q^{45} +3.57780i q^{47} -9.16778 q^{49} +(0.258078 + 5.36936i) q^{51} +5.12728 q^{53} -0.641443 q^{55} +(2.31186 - 0.111119i) q^{57} -11.3428i q^{59} +0.921789i q^{61} +(1.15692 + 12.0071i) q^{63} -2.89996 q^{65} +1.27284i q^{67} +(6.65202 + 4.97500i) q^{69} +1.44420i q^{71} -10.9062 q^{73} +(0.0831550 + 1.73005i) q^{75} +2.57919i q^{77} -3.08045i q^{79} +(8.83443 - 1.71839i) q^{81} +15.6650 q^{83} +3.10358 q^{85} +(5.43237 - 0.261107i) q^{87} -4.31384 q^{89} +11.6605i q^{91} +(0.508096 + 10.5710i) q^{93} -1.33629i q^{95} -13.1699i q^{97} +(1.91546 - 0.184559i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{5} + 2 q^{9}+O(q^{10}) \) 16 * q + 16 * q^5 + 2 * q^9	\( 16 q + 16 q^{5} + 2 q^{9} + 4 q^{21} - 10 q^{23} + 16 q^{25} - 12 q^{27} + 4 q^{31} - 2 q^{33} - 14 q^{39} + 2 q^{45} + 8 q^{49} + 2 q^{51} - 4 q^{53} + 2 q^{57} - 30 q^{63} + 8 q^{69} - 4 q^{73} + 10 q^{81} + 4 q^{83} - 10 q^{87} + 28 q^{89} - 10 q^{93} + 26 q^{99}+O(q^{100}) \) 16 * q + 16 * q^5 + 2 * q^9 + 4 * q^21 - 10 * q^23 + 16 * q^25 - 12 * q^27 + 4 * q^31 - 2 * q^33 - 14 * q^39 + 2 * q^45 + 8 * q^49 + 2 * q^51 - 4 * q^53 + 2 * q^57 - 30 * q^63 + 8 * q^69 - 4 * q^73 + 10 * q^81 + 4 * q^83 - 10 * q^87 + 28 * q^89 - 10 * q^93 + 26 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more