LMFDB - Embedded newform 930.2.h.b.371.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [930,2,Mod(371,930)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(930, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("930.371");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 930 = 2 \cdot 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	930.h (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:	$7.42608738798$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			371.3
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	930.371
Dual form			930.2.h.b.371.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.73205 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.73205i q^{6} +2.00000 q^{7} -1.00000i q^{8} +3.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} -1.73205 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.73205i q^{6} +2.00000 q^{7} -1.00000i q^{8} +3.00000 q^{9} -1.00000 q^{10} -3.46410 q^{11} +1.73205 q^{12} +2.00000i q^{14} -1.73205i q^{15} +1.00000 q^{16} +6.92820 q^{17} +3.00000i q^{18} +4.00000 q^{19} -1.00000i q^{20} -3.46410 q^{21} -3.46410i q^{22} -3.46410 q^{23} +1.73205i q^{24} -1.00000 q^{25} -5.19615 q^{27} -2.00000 q^{28} +3.46410 q^{29} +1.73205 q^{30} +(2.00000 + 5.19615i) q^{31} +1.00000i q^{32} +6.00000 q^{33} +6.92820i q^{34} +2.00000i q^{35} -3.00000 q^{36} +4.00000i q^{38} +1.00000 q^{40} +12.0000i q^{41} -3.46410i q^{42} +3.46410 q^{44} +3.00000i q^{45} -3.46410i q^{46} -1.73205 q^{48} -3.00000 q^{49} -1.00000i q^{50} -12.0000 q^{51} -5.19615i q^{54} -3.46410i q^{55} -2.00000i q^{56} -6.92820 q^{57} +3.46410i q^{58} +6.00000i q^{59} +1.73205i q^{60} +(-5.19615 + 2.00000i) q^{62} +6.00000 q^{63} -1.00000 q^{64} +6.00000i q^{66} -8.00000 q^{67} -6.92820 q^{68} +6.00000 q^{69} -2.00000 q^{70} +12.0000i q^{71} -3.00000i q^{72} +10.3923i q^{73} +1.73205 q^{75} -4.00000 q^{76} -6.92820 q^{77} +10.3923i q^{79} +1.00000i q^{80} +9.00000 q^{81} -12.0000 q^{82} -3.46410 q^{83} +3.46410 q^{84} +6.92820i q^{85} -6.00000 q^{87} +3.46410i q^{88} +13.8564 q^{89} -3.00000 q^{90} +3.46410 q^{92} +(-3.46410 - 9.00000i) q^{93} +4.00000i q^{95} -1.73205i q^{96} -10.0000 q^{97} -3.00000i q^{98} -10.3923 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} + 8 q^{7} + 12 q^{9}+O(q^{10}) $ 4 * q - 4 * q^4 + 8 * q^7 + 12 * q^9	$ 4 q - 4 q^{4} + 8 q^{7} + 12 q^{9} - 4 q^{10} + 4 q^{16} + 16 q^{19} - 4 q^{25} - 8 q^{28} + 8 q^{31} + 24 q^{33} - 12 q^{36} + 4 q^{40} - 12 q^{49} - 48 q^{51} + 24 q^{63} - 4 q^{64} - 32 q^{67} + 24 q^{69} - 8 q^{70} - 16 q^{76} + 36 q^{81} - 48 q^{82} - 24 q^{87} - 12 q^{90} - 40 q^{97}+O(q^{100}) $ 4 * q - 4 * q^4 + 8 * q^7 + 12 * q^9 - 4 * q^10 + 4 * q^16 + 16 * q^19 - 4 * q^25 - 8 * q^28 + 8 * q^31 + 24 * q^33 - 12 * q^36 + 4 * q^40 - 12 * q^49 - 48 * q^51 + 24 * q^63 - 4 * q^64 - 32 * q^67 + 24 * q^69 - 8 * q^70 - 16 * q^76 + 36 * q^81 - 48 * q^82 - 24 * q^87 - 12 * q^90 - 40 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/930\mathbb{Z}\right)^\times$.

$n$	$187$	$311$	$871$
$\chi(n)$	$1$	$-1$	$-1$

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 $

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$	−1.73205			−1.00000
$3$	−1.73205			−1.00000
$4$	−1.00000			−0.500000
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		−	1.73205i		−	0.707107i
$7$	2.00000			0.755929			0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000			0.755929			0.377964	+	0.925820i	$0.376624\pi$
$8$		−	1.00000i		−	0.353553i
$9$	3.00000			1.00000
$10$	−1.00000			−0.316228
$11$	−3.46410			−1.04447			−0.522233	−	0.852803i	$-0.674901\pi$
$11$	−3.46410			−1.04447			−0.522233	+	0.852803i	$0.674901\pi$
$12$	1.73205			0.500000
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$			2.00000i			0.534522i
$15$		−	1.73205i		−	0.447214i
$16$	1.00000			0.250000
$17$	6.92820			1.68034			0.840168	−	0.542326i	$-0.182456\pi$
$17$	6.92820			1.68034			0.840168	+	0.542326i	$0.182456\pi$
$18$			3.00000i			0.707107i
$19$	4.00000			0.917663			0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000			0.917663			0.458831	+	0.888523i	$0.348268\pi$
$20$		−	1.00000i		−	0.223607i
$21$	−3.46410			−0.755929
$22$		−	3.46410i		−	0.738549i
$23$	−3.46410			−0.722315			−0.361158	−	0.932505i	$-0.617618\pi$
$23$	−3.46410			−0.722315			−0.361158	+	0.932505i	$0.617618\pi$
$24$			1.73205i			0.353553i
$25$	−1.00000			−0.200000
$26$		0			0
$27$	−5.19615			−1.00000
$28$	−2.00000			−0.377964
$29$	3.46410			0.643268			0.321634	−	0.946864i	$-0.395768\pi$
$29$	3.46410			0.643268			0.321634	+	0.946864i	$0.395768\pi$
$30$	1.73205			0.316228
$31$	2.00000	+	5.19615i	0.359211	+	0.933257i
$31$	2.00000	+	5.19615i	0.359211	+	0.933257i
$32$			1.00000i			0.176777i
$33$	6.00000			1.04447
$34$			6.92820i			1.18818i
$35$			2.00000i			0.338062i
$36$	−3.00000			−0.500000
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$			4.00000i			0.648886i
$39$		0			0
$40$	1.00000			0.158114
$41$			12.0000i			1.87409i	0.349215	+	0.937043i	$0.386448\pi$
$41$			12.0000i			1.87409i	−0.349215	+	0.937043i	$0.613552\pi$
$42$		−	3.46410i		−	0.534522i
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$	3.46410			0.522233
$45$			3.00000i			0.447214i
$46$		−	3.46410i		−	0.510754i
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$	−1.73205			−0.250000
$49$	−3.00000			−0.428571
$50$		−	1.00000i		−	0.141421i
$51$	−12.0000			−1.68034
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		−	5.19615i		−	0.707107i
$55$		−	3.46410i		−	0.467099i
$56$		−	2.00000i		−	0.267261i
$57$	−6.92820			−0.917663
$58$			3.46410i			0.454859i
$59$			6.00000i			0.781133i	0.920575	+	0.390567i	$0.127721\pi$
$59$			6.00000i			0.781133i	−0.920575	+	0.390567i	$0.872279\pi$
$60$			1.73205i			0.223607i
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$	−5.19615	+	2.00000i	−0.659912	+	0.254000i
$63$	6.00000			0.755929
$64$	−1.00000			−0.125000
$65$		0			0
$66$			6.00000i			0.738549i
$67$	−8.00000			−0.977356			−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000			−0.977356			−0.488678	+	0.872464i	$0.662521\pi$
$68$	−6.92820			−0.840168
$69$	6.00000			0.722315
$70$	−2.00000			−0.239046
$71$			12.0000i			1.42414i	0.702109	+	0.712069i	$0.252242\pi$
$71$			12.0000i			1.42414i	−0.702109	+	0.712069i	$0.747758\pi$
$72$		−	3.00000i		−	0.353553i
$73$			10.3923i			1.21633i	0.793812	+	0.608164i	$0.208094\pi$
$73$			10.3923i			1.21633i	−0.793812	+	0.608164i	$0.791906\pi$
$74$		0			0
$75$	1.73205			0.200000
$76$	−4.00000			−0.458831
$77$	−6.92820			−0.789542
$78$		0			0
$79$			10.3923i			1.16923i	0.811312	+	0.584613i	$0.198754\pi$
$79$			10.3923i			1.16923i	−0.811312	+	0.584613i	$0.801246\pi$
$80$			1.00000i			0.111803i
$81$	9.00000			1.00000
$82$	−12.0000			−1.32518
$83$	−3.46410			−0.380235			−0.190117	−	0.981761i	$-0.560887\pi$
$83$	−3.46410			−0.380235			−0.190117	+	0.981761i	$0.560887\pi$
$84$	3.46410			0.377964
$85$			6.92820i			0.751469i
$86$		0			0
$87$	−6.00000			−0.643268
$88$			3.46410i			0.369274i
$89$	13.8564			1.46878			0.734388	−	0.678730i	$-0.237469\pi$
$89$	13.8564			1.46878			0.734388	+	0.678730i	$0.237469\pi$
$90$	−3.00000			−0.316228
$91$		0			0
$92$	3.46410			0.361158
$93$	−3.46410	−	9.00000i	−0.359211	−	0.933257i
$94$		0			0
$95$			4.00000i			0.410391i
$96$		−	1.73205i		−	0.176777i
$97$	−10.0000			−1.01535			−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000			−1.01535			−0.507673	+	0.861550i	$0.669494\pi$
$98$		−	3.00000i		−	0.303046i
$99$	−10.3923			−1.04447
$100$	1.00000			0.100000
$101$		−	6.00000i		−	0.597022i	−0.954406	−	0.298511i	$-0.903510\pi$
$101$		−	6.00000i		−	0.597022i	0.954406	−	0.298511i	$-0.0964900\pi$
$102$		−	12.0000i		−	1.18818i
$103$	10.0000			0.985329			0.492665	−	0.870219i	$-0.336023\pi$
$103$	10.0000			0.985329			0.492665	+	0.870219i	$0.336023\pi$
$104$		0			0
$105$		−	3.46410i		−	0.338062i
$106$		0			0
$107$		−	12.0000i		−	1.16008i	−0.814587	−	0.580042i	$-0.803036\pi$
$107$		−	12.0000i		−	1.16008i	0.814587	−	0.580042i	$-0.196964\pi$
$108$	5.19615			0.500000
$109$	14.0000			1.34096			0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000			1.34096			0.670478	+	0.741929i	$0.266089\pi$
$110$	3.46410			0.330289
$111$		0			0
$112$	2.00000			0.188982
$113$		−	6.00000i		−	0.564433i	−0.959351	−	0.282216i	$-0.908930\pi$
$113$		−	6.00000i		−	0.564433i	0.959351	−	0.282216i	$-0.0910696\pi$
$114$		−	6.92820i		−	0.648886i
$115$		−	3.46410i		−	0.323029i
$116$	−3.46410			−0.321634
$117$		0			0
$118$	−6.00000			−0.552345
$119$	13.8564			1.27021
$120$	−1.73205			−0.158114
$121$	1.00000			0.0909091
$122$		0			0
$123$		−	20.7846i		−	1.87409i
$124$	−2.00000	−	5.19615i	−0.179605	−	0.466628i
$125$		−	1.00000i		−	0.0894427i
$126$			6.00000i			0.534522i
$127$		−	10.3923i		−	0.922168i	−0.887357	−	0.461084i	$-0.847461\pi$
$127$		−	10.3923i		−	0.922168i	0.887357	−	0.461084i	$-0.152539\pi$
$128$		−	1.00000i		−	0.0883883i
$129$		0			0
$130$		0			0
$131$		−	18.0000i		−	1.57267i	−0.617802	−	0.786334i	$-0.711977\pi$
$131$		−	18.0000i		−	1.57267i	0.617802	−	0.786334i	$-0.288023\pi$
$132$	−6.00000			−0.522233
$133$	8.00000			0.693688
$134$		−	8.00000i		−	0.691095i
$135$		−	5.19615i		−	0.447214i
$136$		−	6.92820i		−	0.594089i
$137$	20.7846			1.77575			0.887875	−	0.460086i	$-0.152181\pi$
$137$	20.7846			1.77575			0.887875	+	0.460086i	$0.152181\pi$
$138$			6.00000i			0.510754i
$139$			10.3923i			0.881464i	0.897639	+	0.440732i	$0.145281\pi$
$139$			10.3923i			0.881464i	−0.897639	+	0.440732i	$0.854719\pi$
$140$		−	2.00000i		−	0.169031i
$141$		0			0
$142$	−12.0000			−1.00702
$143$		0			0
$144$	3.00000			0.250000
$145$			3.46410i			0.287678i
$146$	−10.3923			−0.860073
$147$	5.19615			0.428571
$148$		0			0
$149$			18.0000i			1.47462i	0.675556	+	0.737309i	$0.263904\pi$
$149$			18.0000i			1.47462i	−0.675556	+	0.737309i	$0.736096\pi$
$150$			1.73205i			0.141421i
$151$			10.3923i			0.845714i	0.906196	+	0.422857i	$0.138973\pi$
$151$			10.3923i			0.845714i	−0.906196	+	0.422857i	$0.861027\pi$
$152$		−	4.00000i		−	0.324443i
$153$	20.7846			1.68034
$154$		−	6.92820i		−	0.558291i
$155$	−5.19615	+	2.00000i	−0.417365	+	0.160644i
$156$		0			0
$157$	−8.00000			−0.638470			−0.319235	−	0.947676i	$-0.603426\pi$
$157$	−8.00000			−0.638470			−0.319235	+	0.947676i	$0.603426\pi$
$158$	−10.3923			−0.826767
$159$		0			0
$160$	−1.00000			−0.0790569
$161$	−6.92820			−0.546019
$162$			9.00000i			0.707107i
$163$	−4.00000			−0.313304			−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000			−0.313304			−0.156652	+	0.987654i	$0.550070\pi$
$164$		−	12.0000i		−	0.937043i
$165$			6.00000i			0.467099i
$166$		−	3.46410i		−	0.268866i
$167$	10.3923			0.804181			0.402090	−	0.915600i	$-0.368284\pi$
$167$	10.3923			0.804181			0.402090	+	0.915600i	$0.368284\pi$
$168$			3.46410i			0.267261i
$169$	13.0000			1.00000
$170$	−6.92820			−0.531369
$171$	12.0000			0.917663
$172$		0			0
$173$		−	18.0000i		−	1.36851i	−0.729241	−	0.684257i	$-0.760127\pi$
$173$		−	18.0000i		−	1.36851i	0.729241	−	0.684257i	$-0.239873\pi$
$174$		−	6.00000i		−	0.454859i
$175$	−2.00000			−0.151186
$176$	−3.46410			−0.261116
$177$		−	10.3923i		−	0.781133i
$178$			13.8564i			1.03858i
$179$	10.3923			0.776757			0.388379	−	0.921500i	$-0.373035\pi$
$179$	10.3923			0.776757			0.388379	+	0.921500i	$0.373035\pi$
$180$		−	3.00000i		−	0.223607i
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$			3.46410i			0.255377i
$185$		0			0
$186$	9.00000	−	3.46410i	0.659912	−	0.254000i
$187$	−24.0000			−1.75505
$188$		0			0
$189$	−10.3923			−0.755929
$190$	−4.00000			−0.290191
$191$		−	12.0000i		−	0.868290i	−0.900843	−	0.434145i	$-0.857051\pi$
$191$		−	12.0000i		−	0.868290i	0.900843	−	0.434145i	$-0.142949\pi$
$192$	1.73205			0.125000
$193$	10.0000			0.719816			0.359908	−	0.932988i	$-0.382808\pi$
$193$	10.0000			0.719816			0.359908	+	0.932988i	$0.382808\pi$
$194$		−	10.0000i		−	0.717958i
$195$		0			0
$196$	3.00000			0.214286
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		−	10.3923i		−	0.738549i
$199$		−	10.3923i		−	0.736691i	−0.929689	−	0.368345i	$-0.879924\pi$
$199$		−	10.3923i		−	0.736691i	0.929689	−	0.368345i	$-0.120076\pi$
$200$			1.00000i			0.0707107i
$201$	13.8564			0.977356
$202$	6.00000			0.422159
$203$	6.92820			0.486265
$204$	12.0000			0.840168
$205$	−12.0000			−0.838116
$206$			10.0000i			0.696733i
$207$	−10.3923			−0.722315
$208$		0			0
$209$	−13.8564			−0.958468
$210$	3.46410			0.239046
$211$	−4.00000			−0.275371			−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000			−0.275371			−0.137686	+	0.990476i	$0.543966\pi$
$212$		0			0
$213$		−	20.7846i		−	1.42414i
$214$	12.0000			0.820303
$215$		0			0
$216$			5.19615i			0.353553i
$217$	4.00000	+	10.3923i	0.271538	+	0.705476i
$218$			14.0000i			0.948200i
$219$		−	18.0000i		−	1.21633i
$220$			3.46410i			0.233550i
$221$		0			0
$222$		0			0
$223$			10.3923i			0.695920i	0.937509	+	0.347960i	$0.113126\pi$
$223$			10.3923i			0.695920i	−0.937509	+	0.347960i	$0.886874\pi$
$224$			2.00000i			0.133631i
$225$	−3.00000			−0.200000
$226$	6.00000			0.399114
$227$		0			0		1.00000			$0$
$227$		0			0		−1.00000			$\pi$
$228$	6.92820			0.458831
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$	3.46410			0.228416
$231$	12.0000			0.789542
$232$		−	3.46410i		−	0.227429i
$233$			6.00000i			0.393073i	0.980497	+	0.196537i	$0.0629694\pi$
$233$			6.00000i			0.393073i	−0.980497	+	0.196537i	$0.937031\pi$
$234$		0			0
$235$		0			0
$236$		−	6.00000i		−	0.390567i
$237$		−	18.0000i		−	1.16923i
$238$			13.8564i			0.898177i
$239$	−27.7128			−1.79259			−0.896296	−	0.443455i	$-0.853752\pi$
$239$	−27.7128			−1.79259			−0.896296	+	0.443455i	$0.853752\pi$
$240$		−	1.73205i		−	0.111803i
$241$			20.7846i			1.33885i	0.742878	+	0.669427i	$0.233460\pi$
$241$			20.7846i			1.33885i	−0.742878	+	0.669427i	$0.766540\pi$
$242$			1.00000i			0.0642824i
$243$	−15.5885			−1.00000
$244$		0			0
$245$		−	3.00000i		−	0.191663i
$246$	20.7846			1.32518
$247$		0			0
$248$	5.19615	−	2.00000i	0.329956	−	0.127000i
$249$	6.00000			0.380235
$250$	1.00000			0.0632456
$251$	−3.46410			−0.218652			−0.109326	−	0.994006i	$-0.534869\pi$
$251$	−3.46410			−0.218652			−0.109326	+	0.994006i	$0.534869\pi$
$252$	−6.00000			−0.377964
$253$	12.0000			0.754434
$254$	10.3923			0.652071
$255$		−	12.0000i		−	0.751469i
$256$	1.00000			0.0625000
$257$		−	30.0000i		−	1.87135i	−0.352865	−	0.935674i	$-0.614792\pi$
$257$		−	30.0000i		−	1.87135i	0.352865	−	0.935674i	$-0.385208\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	10.3923			0.643268
$262$	18.0000			1.11204
$263$	3.46410			0.213606			0.106803	−	0.994280i	$-0.465939\pi$
$263$	3.46410			0.213606			0.106803	+	0.994280i	$0.465939\pi$
$264$		−	6.00000i		−	0.369274i
$265$		0			0
$266$			8.00000i			0.490511i
$267$	−24.0000			−1.46878
$268$	8.00000			0.488678
$269$	−3.46410			−0.211210			−0.105605	−	0.994408i	$-0.533678\pi$
$269$	−3.46410			−0.211210			−0.105605	+	0.994408i	$0.533678\pi$
$270$	5.19615			0.316228
$271$		−	10.3923i		−	0.631288i	−0.948878	−	0.315644i	$-0.897780\pi$
$271$		−	10.3923i		−	0.631288i	0.948878	−	0.315644i	$-0.102220\pi$
$272$	6.92820			0.420084
$273$		0			0
$274$			20.7846i			1.25564i
$275$	3.46410			0.208893
$276$	−6.00000			−0.361158
$277$			20.7846i			1.24883i	0.781094	+	0.624413i	$0.214662\pi$
$277$			20.7846i			1.24883i	−0.781094	+	0.624413i	$0.785338\pi$
$278$	−10.3923			−0.623289
$279$	6.00000	+	15.5885i	0.359211	+	0.933257i
$280$	2.00000			0.119523
$281$			24.0000i			1.43172i	0.698244	+	0.715860i	$0.253965\pi$
$281$			24.0000i			1.43172i	−0.698244	+	0.715860i	$0.746035\pi$
$282$		0			0
$283$	−28.0000			−1.66443			−0.832214	−	0.554455i	$-0.812927\pi$
$283$	−28.0000			−1.66443			−0.832214	+	0.554455i	$0.812927\pi$
$284$		−	12.0000i		−	0.712069i
$285$		−	6.92820i		−	0.410391i
$286$		0			0
$287$			24.0000i			1.41668i
$288$			3.00000i			0.176777i
$289$	31.0000			1.82353
$290$	−3.46410			−0.203419
$291$	17.3205			1.01535
$292$		−	10.3923i		−	0.608164i
$293$		−	18.0000i		−	1.05157i	−0.850617	−	0.525786i	$-0.823771\pi$
$293$		−	18.0000i		−	1.05157i	0.850617	−	0.525786i	$-0.176229\pi$
$294$			5.19615i			0.303046i
$295$	−6.00000			−0.349334
$296$		0			0
$297$	18.0000			1.04447
$298$	−18.0000			−1.04271
$299$		0			0
$300$	−1.73205			−0.100000
$301$		0			0
$302$	−10.3923			−0.598010
$303$			10.3923i			0.597022i
$304$	4.00000			0.229416
$305$		0			0
$306$			20.7846i			1.18818i
$307$	4.00000			0.228292			0.114146	−	0.993464i	$-0.463587\pi$
$307$	4.00000			0.228292			0.114146	+	0.993464i	$0.463587\pi$
$308$	6.92820			0.394771
$309$	−17.3205			−0.985329
$310$	−2.00000	−	5.19615i	−0.113592	−	0.295122i
$311$		−	24.0000i		−	1.36092i	−0.732787	−	0.680458i	$-0.761781\pi$
$311$		−	24.0000i		−	1.36092i	0.732787	−	0.680458i	$-0.238219\pi$
$312$		0			0
$313$		−	10.3923i		−	0.587408i	−0.955896	−	0.293704i	$-0.905112\pi$
$313$		−	10.3923i		−	0.587408i	0.955896	−	0.293704i	$-0.0948880\pi$
$314$		−	8.00000i		−	0.451466i
$315$			6.00000i			0.338062i
$316$		−	10.3923i		−	0.584613i
$317$			30.0000i			1.68497i	0.538721	+	0.842484i	$0.318908\pi$
$317$			30.0000i			1.68497i	−0.538721	+	0.842484i	$0.681092\pi$
$318$		0			0
$319$	−12.0000			−0.671871
$320$		−	1.00000i		−	0.0559017i
$321$			20.7846i			1.16008i
$322$		−	6.92820i		−	0.386094i
$323$	27.7128			1.54198
$324$	−9.00000			−0.500000
$325$		0			0
$326$		−	4.00000i		−	0.221540i
$327$	−24.2487			−1.34096
$328$	12.0000			0.662589
$329$		0			0
$330$	−6.00000			−0.330289
$331$		−	31.1769i		−	1.71364i	−0.515617	−	0.856819i	$-0.672437\pi$
$331$		−	31.1769i		−	1.71364i	0.515617	−	0.856819i	$-0.327563\pi$
$332$	3.46410			0.190117
$333$		0			0
$334$			10.3923i			0.568642i
$335$		−	8.00000i		−	0.437087i
$336$	−3.46410			−0.188982
$337$			10.3923i			0.566105i	0.959104	+	0.283052i	$0.0913471\pi$
$337$			10.3923i			0.566105i	−0.959104	+	0.283052i	$0.908653\pi$
$338$			13.0000i			0.707107i
$339$			10.3923i			0.564433i
$340$		−	6.92820i		−	0.375735i
$341$	−6.92820	−	18.0000i	−0.375183	−	0.974755i
$342$			12.0000i			0.648886i
$343$	−20.0000			−1.07990
$344$		0			0
$345$			6.00000i			0.323029i
$346$	18.0000			0.967686
$347$	17.3205			0.929814			0.464907	−	0.885360i	$-0.346088\pi$
$347$	17.3205			0.929814			0.464907	+	0.885360i	$0.346088\pi$
$348$	6.00000			0.321634
$349$	−2.00000			−0.107058			−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000			−0.107058			−0.0535288	+	0.998566i	$0.517047\pi$
$350$		−	2.00000i		−	0.106904i
$351$		0			0
$352$		−	3.46410i		−	0.184637i
$353$	20.7846			1.10625			0.553127	−	0.833097i	$-0.313435\pi$
$353$	20.7846			1.10625			0.553127	+	0.833097i	$0.313435\pi$
$354$	10.3923			0.552345
$355$	−12.0000			−0.636894
$356$	−13.8564			−0.734388
$357$	−24.0000			−1.27021
$358$			10.3923i			0.549250i
$359$			12.0000i			0.633336i	0.948536	+	0.316668i	$0.102564\pi$
$359$			12.0000i			0.633336i	−0.948536	+	0.316668i	$0.897436\pi$
$360$	3.00000			0.158114
$361$	−3.00000			−0.157895
$362$		0			0
$363$	−1.73205			−0.0909091
$364$		0			0
$365$	−10.3923			−0.543958
$366$		0			0
$367$		−	31.1769i		−	1.62742i	−0.581270	−	0.813711i	$-0.697444\pi$
$367$		−	31.1769i		−	1.62742i	0.581270	−	0.813711i	$-0.302556\pi$
$368$	−3.46410			−0.180579
$369$			36.0000i			1.87409i
$370$		0			0
$371$		0			0
$372$	3.46410	+	9.00000i	0.179605	+	0.466628i
$373$	−4.00000			−0.207112			−0.103556	−	0.994624i	$-0.533022\pi$
$373$	−4.00000			−0.207112			−0.103556	+	0.994624i	$0.533022\pi$
$374$		−	24.0000i		−	1.24101i
$375$			1.73205i			0.0894427i
$376$		0			0
$377$		0			0
$378$		−	10.3923i		−	0.534522i
$379$	−4.00000			−0.205466			−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000			−0.205466			−0.102733	+	0.994709i	$0.532759\pi$
$380$		−	4.00000i		−	0.205196i
$381$			18.0000i			0.922168i
$382$	12.0000			0.613973
$383$	−3.46410			−0.177007			−0.0885037	−	0.996076i	$-0.528208\pi$
$383$	−3.46410			−0.177007			−0.0885037	+	0.996076i	$0.528208\pi$
$384$			1.73205i			0.0883883i
$385$		−	6.92820i		−	0.353094i
$386$			10.0000i			0.508987i
$387$		0			0
$388$	10.0000			0.507673
$389$	−3.46410			−0.175637			−0.0878185	−	0.996136i	$-0.527990\pi$
$389$	−3.46410			−0.175637			−0.0878185	+	0.996136i	$0.527990\pi$
$390$		0			0
$391$	−24.0000			−1.21373
$392$			3.00000i			0.151523i
$393$			31.1769i			1.57267i
$394$		0			0
$395$	−10.3923			−0.522894
$396$	10.3923			0.522233
$397$	−4.00000			−0.200754			−0.100377	−	0.994949i	$-0.532005\pi$
$397$	−4.00000			−0.200754			−0.100377	+	0.994949i	$0.532005\pi$
$398$	10.3923			0.520919
$399$	−13.8564			−0.693688
$400$	−1.00000			−0.0500000
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$			13.8564i			0.691095i
$403$		0			0
$404$			6.00000i			0.298511i
$405$			9.00000i			0.447214i
$406$			6.92820i			0.343841i
$407$		0			0
$408$			12.0000i			0.594089i
$409$		−	20.7846i		−	1.02773i	−0.857870	−	0.513866i	$-0.828213\pi$
$409$		−	20.7846i		−	1.02773i	0.857870	−	0.513866i	$-0.171787\pi$
$410$		−	12.0000i		−	0.592638i
$411$	−36.0000			−1.77575
$412$	−10.0000			−0.492665
$413$			12.0000i			0.590481i
$414$		−	10.3923i		−	0.510754i
$415$		−	3.46410i		−	0.170046i
$416$		0			0
$417$		−	18.0000i		−	0.881464i
$418$		−	13.8564i		−	0.677739i
$419$			30.0000i			1.46560i	0.680446	+	0.732798i	$0.261786\pi$
$419$			30.0000i			1.46560i	−0.680446	+	0.732798i	$0.738214\pi$
$420$			3.46410i			0.169031i
$421$	34.0000			1.65706			0.828529	−	0.559946i	$-0.189178\pi$
$421$	34.0000			1.65706			0.828529	+	0.559946i	$0.189178\pi$
$422$		−	4.00000i		−	0.194717i
$423$		0			0
$424$		0			0
$425$	−6.92820			−0.336067
$426$	20.7846			1.00702
$427$		0			0
$428$			12.0000i			0.580042i
$429$		0			0
$430$		0			0
$431$		−	36.0000i		−	1.73406i	−0.498257	−	0.867029i	$-0.666026\pi$
$431$		−	36.0000i		−	1.73406i	0.498257	−	0.867029i	$-0.333974\pi$
$432$	−5.19615			−0.250000
$433$		−	31.1769i		−	1.49827i	−0.662419	−	0.749133i	$-0.730470\pi$
$433$		−	31.1769i		−	1.49827i	0.662419	−	0.749133i	$-0.269530\pi$
$434$	−10.3923	+	4.00000i	−0.498847	+	0.192006i
$435$		−	6.00000i		−	0.287678i
$436$	−14.0000			−0.670478
$437$	−13.8564			−0.662842
$438$	18.0000			0.860073
$439$	16.0000			0.763638			0.381819	−	0.924237i	$-0.375298\pi$
$439$	16.0000			0.763638			0.381819	+	0.924237i	$0.375298\pi$
$440$	−3.46410			−0.165145
$441$	−9.00000			−0.428571
$442$		0			0
$443$			24.0000i			1.14027i	0.821549	+	0.570137i	$0.193110\pi$
$443$			24.0000i			1.14027i	−0.821549	+	0.570137i	$0.806890\pi$
$444$		0			0
$445$			13.8564i			0.656857i
$446$	−10.3923			−0.492090
$447$		−	31.1769i		−	1.47462i
$448$	−2.00000			−0.0944911
$449$	−27.7128			−1.30785			−0.653924	−	0.756560i	$-0.726879\pi$
$449$	−27.7128			−1.30785			−0.653924	+	0.756560i	$0.726879\pi$
$450$		−	3.00000i		−	0.141421i
$451$		−	41.5692i		−	1.95742i
$452$			6.00000i			0.282216i
$453$		−	18.0000i		−	0.845714i
$454$		0			0
$455$		0			0
$456$			6.92820i			0.324443i
$457$		−	31.1769i		−	1.45839i	−0.684304	−	0.729197i	$-0.739894\pi$
$457$		−	31.1769i		−	1.45839i	0.684304	−	0.729197i	$-0.260106\pi$
$458$		0			0
$459$	−36.0000			−1.68034
$460$			3.46410i			0.161515i
$461$	−38.1051			−1.77473			−0.887366	−	0.461065i	$-0.847467\pi$
$461$	−38.1051			−1.77473			−0.887366	+	0.461065i	$0.847467\pi$
$462$			12.0000i			0.558291i
$463$			31.1769i			1.44891i	0.689320	+	0.724457i	$0.257909\pi$
$463$			31.1769i			1.44891i	−0.689320	+	0.724457i	$0.742091\pi$
$464$	3.46410			0.160817
$465$	9.00000	−	3.46410i	0.417365	−	0.160644i
$466$	−6.00000			−0.277945
$467$			12.0000i			0.555294i	0.960683	+	0.277647i	$0.0895545\pi$
$467$			12.0000i			0.555294i	−0.960683	+	0.277647i	$0.910445\pi$
$468$		0			0
$469$	−16.0000			−0.738811
$470$		0			0
$471$	13.8564			0.638470
$472$	6.00000			0.276172
$473$		0			0
$474$	18.0000			0.826767
$475$	−4.00000			−0.183533
$476$	−13.8564			−0.635107
$477$		0			0
$478$		−	27.7128i		−	1.26755i
$479$			24.0000i			1.09659i	0.836286	+	0.548294i	$0.184723\pi$
$479$			24.0000i			1.09659i	−0.836286	+	0.548294i	$0.815277\pi$
$480$	1.73205			0.0790569
$481$		0			0
$482$	−20.7846			−0.946713
$483$	12.0000			0.546019
$484$	−1.00000			−0.0454545
$485$		−	10.0000i		−	0.454077i
$486$		−	15.5885i		−	0.707107i
$487$		−	10.3923i		−	0.470920i	−0.971884	−	0.235460i	$-0.924340\pi$
$487$		−	10.3923i		−	0.470920i	0.971884	−	0.235460i	$-0.0756597\pi$
$488$		0			0
$489$	6.92820			0.313304
$490$	3.00000			0.135526
$491$	24.2487			1.09433			0.547165	−	0.837025i	$-0.315707\pi$
$491$	24.2487			1.09433			0.547165	+	0.837025i	$0.315707\pi$
$492$			20.7846i			0.937043i
$493$	24.0000			1.08091
$494$		0			0
$495$		−	10.3923i		−	0.467099i
$496$	2.00000	+	5.19615i	0.0898027	+	0.233314i
$497$			24.0000i			1.07655i
$498$			6.00000i			0.268866i
$499$		−	31.1769i		−	1.39567i	−0.716258	−	0.697835i	$-0.754147\pi$
$499$		−	31.1769i		−	1.39567i	0.716258	−	0.697835i	$-0.245853\pi$
$500$			1.00000i			0.0447214i
$501$	−18.0000			−0.804181
$502$		−	3.46410i		−	0.154610i
$503$			24.0000i			1.07011i	0.844818	+	0.535054i	$0.179709\pi$
$503$			24.0000i			1.07011i	−0.844818	+	0.535054i	$0.820291\pi$
$504$		−	6.00000i		−	0.267261i
$505$	6.00000			0.266996
$506$			12.0000i			0.533465i
$507$	−22.5167			−1.00000
$508$			10.3923i			0.461084i
$509$	24.2487			1.07481			0.537403	−	0.843326i	$-0.319406\pi$
$509$	24.2487			1.07481			0.537403	+	0.843326i	$0.319406\pi$
$510$	12.0000			0.531369
$511$			20.7846i			0.919457i
$512$			1.00000i			0.0441942i
$513$	−20.7846			−0.917663
$514$	30.0000			1.32324
$515$			10.0000i			0.440653i
$516$		0			0
$517$		0			0
$518$		0			0
$519$			31.1769i			1.36851i
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$			10.3923i			0.454859i
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$			18.0000i			0.786334i
$525$	3.46410			0.151186
$526$			3.46410i			0.151042i
$527$	13.8564	+	36.0000i	0.603595	+	1.56818i
$528$	6.00000			0.261116
$529$	−11.0000			−0.478261
$530$		0			0
$531$			18.0000i			0.781133i
$532$	−8.00000			−0.346844
$533$		0			0
$534$		−	24.0000i		−	1.03858i
$535$	12.0000			0.518805
$536$			8.00000i			0.345547i
$537$	−18.0000			−0.776757
$538$		−	3.46410i		−	0.149348i
$539$	10.3923			0.447628
$540$			5.19615i			0.223607i
$541$	−14.0000			−0.601907			−0.300954	−	0.953639i	$-0.597305\pi$
$541$	−14.0000			−0.601907			−0.300954	+	0.953639i	$0.597305\pi$
$542$	10.3923			0.446388
$543$		0			0
$544$			6.92820i			0.297044i
$545$			14.0000i			0.599694i
$546$		0			0
$547$	−8.00000			−0.342055			−0.171028	−	0.985266i	$-0.554709\pi$
$547$	−8.00000			−0.342055			−0.171028	+	0.985266i	$0.554709\pi$
$548$	−20.7846			−0.887875
$549$		0			0
$550$			3.46410i			0.147710i
$551$	13.8564			0.590303
$552$		−	6.00000i		−	0.255377i
$553$			20.7846i			0.883852i
$554$	−20.7846			−0.883053
$555$		0			0
$556$		−	10.3923i		−	0.440732i
$557$	−27.7128			−1.17423			−0.587115	−	0.809504i	$-0.699736\pi$
$557$	−27.7128			−1.17423			−0.587115	+	0.809504i	$0.699736\pi$
$558$	−15.5885	+	6.00000i	−0.659912	+	0.254000i
$559$		0			0
$560$			2.00000i			0.0845154i
$561$	41.5692			1.75505
$562$	−24.0000			−1.01238
$563$		−	24.0000i		−	1.01148i	−0.862686	−	0.505740i	$-0.831220\pi$
$563$		−	24.0000i		−	1.01148i	0.862686	−	0.505740i	$-0.168780\pi$
$564$		0			0
$565$	6.00000			0.252422
$566$		−	28.0000i		−	1.17693i
$567$	18.0000			0.755929
$568$	12.0000			0.503509
$569$	−6.92820			−0.290445			−0.145223	−	0.989399i	$-0.546390\pi$
$569$	−6.92820			−0.290445			−0.145223	+	0.989399i	$0.546390\pi$
$570$	6.92820			0.290191
$571$			31.1769i			1.30471i	0.757912	+	0.652357i	$0.226220\pi$
$571$			31.1769i			1.30471i	−0.757912	+	0.652357i	$0.773780\pi$
$572$		0			0
$573$			20.7846i			0.868290i
$574$	−24.0000			−1.00174
$575$	3.46410			0.144463
$576$	−3.00000			−0.125000
$577$	−14.0000			−0.582828			−0.291414	−	0.956597i	$-0.594126\pi$
$577$	−14.0000			−0.582828			−0.291414	+	0.956597i	$0.594126\pi$
$578$			31.0000i			1.28943i
$579$	−17.3205			−0.719816
$580$		−	3.46410i		−	0.143839i
$581$	−6.92820			−0.287430
$582$			17.3205i			0.717958i
$583$		0			0
$584$	10.3923			0.430037
$585$		0			0
$586$	18.0000			0.743573
$587$	38.1051			1.57277			0.786383	−	0.617739i	$-0.211951\pi$
$587$	38.1051			1.57277			0.786383	+	0.617739i	$0.211951\pi$
$588$	−5.19615			−0.214286
$589$	8.00000	+	20.7846i	0.329634	+	0.856415i
$590$		−	6.00000i		−	0.247016i
$591$		0			0
$592$		0			0
$593$		−	30.0000i		−	1.23195i	−0.787765	−	0.615976i	$-0.788762\pi$
$593$		−	30.0000i		−	1.23195i	0.787765	−	0.615976i	$-0.211238\pi$
$594$			18.0000i			0.738549i
$595$			13.8564i			0.568057i
$596$		−	18.0000i		−	0.737309i
$597$			18.0000i			0.736691i
$598$		0			0
$599$		−	12.0000i		−	0.490307i	−0.969484	−	0.245153i	$-0.921162\pi$
$599$		−	12.0000i		−	0.490307i	0.969484	−	0.245153i	$-0.0788383\pi$
$600$		−	1.73205i		−	0.0707107i
$601$		−	41.5692i		−	1.69564i	−0.530281	−	0.847822i	$-0.677914\pi$
$601$		−	41.5692i		−	1.69564i	0.530281	−	0.847822i	$-0.322086\pi$
$602$		0			0
$603$	−24.0000			−0.977356
$604$		−	10.3923i		−	0.422857i
$605$			1.00000i			0.0406558i
$606$	−10.3923			−0.422159
$607$	22.0000			0.892952			0.446476	−	0.894795i	$-0.352679\pi$
$607$	22.0000			0.892952			0.446476	+	0.894795i	$0.352679\pi$
$608$			4.00000i			0.162221i
$609$	−12.0000			−0.486265
$610$		0			0
$611$		0			0
$612$	−20.7846			−0.840168
$613$		−	20.7846i		−	0.839482i	−0.907644	−	0.419741i	$-0.862121\pi$
$613$		−	20.7846i		−	0.839482i	0.907644	−	0.419741i	$-0.137879\pi$
$614$			4.00000i			0.161427i
$615$	20.7846			0.838116
$616$			6.92820i			0.279145i
$617$		−	18.0000i		−	0.724653i	−0.932051	−	0.362326i	$-0.881983\pi$
$617$		−	18.0000i		−	0.724653i	0.932051	−	0.362326i	$-0.118017\pi$
$618$		−	17.3205i		−	0.696733i
$619$		−	10.3923i		−	0.417702i	−0.977947	−	0.208851i	$-0.933028\pi$
$619$		−	10.3923i		−	0.417702i	0.977947	−	0.208851i	$-0.0669724\pi$
$620$	5.19615	−	2.00000i	0.208683	−	0.0803219i
$621$	18.0000			0.722315
$622$	24.0000			0.962312
$623$	27.7128			1.11029
$624$		0			0
$625$	1.00000			0.0400000
$626$	10.3923			0.415360
$627$	24.0000			0.958468
$628$	8.00000			0.319235
$629$		0			0
$630$	−6.00000			−0.239046
$631$		−	31.1769i		−	1.24113i	−0.784154	−	0.620567i	$-0.786903\pi$
$631$		−	31.1769i		−	1.24113i	0.784154	−	0.620567i	$-0.213097\pi$
$632$	10.3923			0.413384
$633$	6.92820			0.275371
$634$	−30.0000			−1.19145
$635$	10.3923			0.412406
$636$		0			0
$637$		0			0
$638$		−	12.0000i		−	0.475085i
$639$			36.0000i			1.42414i
$640$	1.00000			0.0395285
$641$	−41.5692			−1.64189			−0.820943	−	0.571011i	$-0.806552\pi$
$641$	−41.5692			−1.64189			−0.820943	+	0.571011i	$0.806552\pi$
$642$	−20.7846			−0.820303
$643$			41.5692i			1.63933i	0.572843	+	0.819665i	$0.305840\pi$
$643$			41.5692i			1.63933i	−0.572843	+	0.819665i	$0.694160\pi$
$644$	6.92820			0.273009
$645$		0			0
$646$			27.7128i			1.09035i
$647$	−17.3205			−0.680939			−0.340470	−	0.940255i	$-0.610586\pi$
$647$	−17.3205			−0.680939			−0.340470	+	0.940255i	$0.610586\pi$
$648$		−	9.00000i		−	0.353553i
$649$		−	20.7846i		−	0.815867i
$650$		0			0
$651$	−6.92820	−	18.0000i	−0.271538	−	0.705476i
$652$	4.00000			0.156652
$653$			18.0000i			0.704394i	0.935926	+	0.352197i	$0.114565\pi$
$653$			18.0000i			0.704394i	−0.935926	+	0.352197i	$0.885435\pi$
$654$		−	24.2487i		−	0.948200i
$655$	18.0000			0.703318
$656$			12.0000i			0.468521i
$657$			31.1769i			1.21633i
$658$		0			0
$659$			6.00000i			0.233727i	0.993148	+	0.116863i	$0.0372840\pi$
$659$			6.00000i			0.233727i	−0.993148	+	0.116863i	$0.962716\pi$
$660$		−	6.00000i		−	0.233550i
$661$	14.0000			0.544537			0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000			0.544537			0.272268	+	0.962221i	$0.412226\pi$
$662$	31.1769			1.21173
$663$		0			0
$664$			3.46410i			0.134433i
$665$			8.00000i			0.310227i
$666$		0			0
$667$	−12.0000			−0.464642
$668$	−10.3923			−0.402090
$669$		−	18.0000i		−	0.695920i
$670$	8.00000			0.309067
$671$		0			0
$672$		−	3.46410i		−	0.133631i
$673$		−	31.1769i		−	1.20178i	−0.799331	−	0.600891i	$-0.794813\pi$
$673$		−	31.1769i		−	1.20178i	0.799331	−	0.600891i	$-0.205187\pi$
$674$	−10.3923			−0.400297
$675$	5.19615			0.200000
$676$	−13.0000			−0.500000
$677$	48.4974			1.86391			0.931954	−	0.362577i	$-0.118103\pi$
$677$	48.4974			1.86391			0.931954	+	0.362577i	$0.118103\pi$
$678$	−10.3923			−0.399114
$679$	−20.0000			−0.767530
$680$	6.92820			0.265684
$681$		0			0
$682$	18.0000	−	6.92820i	0.689256	−	0.265295i
$683$		−	12.0000i		−	0.459167i	−0.973289	−	0.229584i	$-0.926264\pi$
$683$		−	12.0000i		−	0.459167i	0.973289	−	0.229584i	$-0.0737364\pi$
$684$	−12.0000			−0.458831
$685$			20.7846i			0.794139i
$686$		−	20.0000i		−	0.763604i
$687$		0			0
$688$		0			0
$689$		0			0
$690$	−6.00000			−0.228416
$691$	20.0000			0.760836			0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000			0.760836			0.380418	+	0.924815i	$0.375780\pi$
$692$			18.0000i			0.684257i
$693$	−20.7846			−0.789542
$694$			17.3205i			0.657477i
$695$	−10.3923			−0.394203
$696$			6.00000i			0.227429i
$697$			83.1384i			3.14909i
$698$		−	2.00000i		−	0.0757011i
$699$		−	10.3923i		−	0.393073i
$700$	2.00000			0.0755929
$701$		−	6.00000i		−	0.226617i	−0.993560	−	0.113308i	$-0.963855\pi$
$701$		−	6.00000i		−	0.226617i	0.993560	−	0.113308i	$-0.0361448\pi$
$702$		0			0
$703$		0			0
$704$	3.46410			0.130558
$705$		0			0
$706$			20.7846i			0.782239i
$707$		−	12.0000i		−	0.451306i
$708$			10.3923i			0.390567i
$709$		0			0		1.00000			$0$
$709$		0			0		−1.00000			$\pi$
$710$		−	12.0000i		−	0.450352i
$711$			31.1769i			1.16923i
$712$		−	13.8564i		−	0.519291i
$713$	−6.92820	−	18.0000i	−0.259463	−	0.674105i
$714$		−	24.0000i		−	0.898177i
$715$		0			0
$716$	−10.3923			−0.388379
$717$	48.0000			1.79259
$718$	−12.0000			−0.447836
$719$	48.4974			1.80865			0.904324	−	0.426846i	$-0.140375\pi$
$719$	48.4974			1.80865			0.904324	+	0.426846i	$0.140375\pi$
$720$			3.00000i			0.111803i
$721$	20.0000			0.744839
$722$		−	3.00000i		−	0.111648i
$723$		−	36.0000i		−	1.33885i
$724$		0			0
$725$	−3.46410			−0.128654
$726$		−	1.73205i		−	0.0642824i
$727$	2.00000			0.0741759			0.0370879	−	0.999312i	$-0.488192\pi$
$727$	2.00000			0.0741759			0.0370879	+	0.999312i	$0.488192\pi$
$728$		0			0
$729$	27.0000			1.00000
$730$		−	10.3923i		−	0.384636i
$731$		0			0
$732$		0			0
$733$	−44.0000			−1.62518			−0.812589	−	0.582838i	$-0.801942\pi$
$733$	−44.0000			−1.62518			−0.812589	+	0.582838i	$0.801942\pi$
$734$	31.1769			1.15076
$735$			5.19615i			0.191663i
$736$		−	3.46410i		−	0.127688i
$737$	27.7128			1.02081
$738$	−36.0000			−1.32518
$739$		−	10.3923i		−	0.382287i	−0.981562	−	0.191144i	$-0.938780\pi$
$739$		−	10.3923i		−	0.382287i	0.981562	−	0.191144i	$-0.0612196\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−38.1051			−1.39794			−0.698971	−	0.715150i	$-0.746358\pi$
$743$	−38.1051			−1.39794			−0.698971	+	0.715150i	$0.746358\pi$
$744$	−9.00000	+	3.46410i	−0.329956	+	0.127000i
$745$	−18.0000			−0.659469
$746$		−	4.00000i		−	0.146450i
$747$	−10.3923			−0.380235
$748$	24.0000			0.877527
$749$		−	24.0000i		−	0.876941i
$750$	−1.73205			−0.0632456
$751$	28.0000			1.02173			0.510867	−	0.859660i	$-0.329324\pi$
$751$	28.0000			1.02173			0.510867	+	0.859660i	$0.329324\pi$
$752$		0			0
$753$	6.00000			0.218652
$754$		0			0
$755$	−10.3923			−0.378215
$756$	10.3923			0.377964
$757$			20.7846i			0.755429i	0.925922	+	0.377715i	$0.123290\pi$
$757$			20.7846i			0.755429i	−0.925922	+	0.377715i	$0.876710\pi$
$758$		−	4.00000i		−	0.145287i
$759$	−20.7846			−0.754434
$760$	4.00000			0.145095
$761$	−41.5692			−1.50688			−0.753442	−	0.657515i	$-0.771608\pi$
$761$	−41.5692			−1.50688			−0.753442	+	0.657515i	$0.771608\pi$
$762$	−18.0000			−0.652071
$763$	28.0000			1.01367
$764$			12.0000i			0.434145i
$765$			20.7846i			0.751469i
$766$		−	3.46410i		−	0.125163i
$767$		0			0
$768$	−1.73205			−0.0625000
$769$	−14.0000			−0.504853			−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000			−0.504853			−0.252426	+	0.967616i	$0.581229\pi$
$770$	6.92820			0.249675
$771$			51.9615i			1.87135i
$772$	−10.0000			−0.359908
$773$	20.7846			0.747570			0.373785	−	0.927515i	$-0.378060\pi$
$773$	20.7846			0.747570			0.373785	+	0.927515i	$0.378060\pi$
$774$		0			0
$775$	−2.00000	−	5.19615i	−0.0718421	−	0.186651i
$776$			10.0000i			0.358979i
$777$		0			0
$778$		−	3.46410i		−	0.124194i
$779$			48.0000i			1.71978i
$780$		0			0
$781$		−	41.5692i		−	1.48746i
$782$		−	24.0000i		−	0.858238i
$783$	−18.0000			−0.643268
$784$	−3.00000			−0.107143
$785$		−	8.00000i		−	0.285532i
$786$	−31.1769			−1.11204
$787$		−	20.7846i		−	0.740891i	−0.928854	−	0.370446i	$-0.879205\pi$
$787$		−	20.7846i		−	0.740891i	0.928854	−	0.370446i	$-0.120795\pi$
$788$		0			0
$789$	−6.00000			−0.213606
$790$		−	10.3923i		−	0.369742i
$791$		−	12.0000i		−	0.426671i
$792$			10.3923i			0.369274i
$793$		0			0
$794$		−	4.00000i		−	0.141955i
$795$		0			0
$796$			10.3923i			0.368345i
$797$	−34.6410			−1.22705			−0.613524	−	0.789676i	$-0.710249\pi$
$797$	−34.6410			−1.22705			−0.613524	+	0.789676i	$0.710249\pi$
$798$		−	13.8564i		−	0.490511i
$799$		0			0
$800$		−	1.00000i		−	0.0353553i
$801$	41.5692			1.46878
$802$		0			0
$803$		−	36.0000i		−	1.27041i
$804$	−13.8564			−0.488678
$805$		−	6.92820i		−	0.244187i
$806$		0			0
$807$	6.00000			0.211210
$808$	−6.00000			−0.211079
$809$	20.7846			0.730748			0.365374	−	0.930861i	$-0.380941\pi$
$809$	20.7846			0.730748			0.365374	+	0.930861i	$0.380941\pi$
$810$	−9.00000			−0.316228
$811$	4.00000			0.140459			0.0702295	−	0.997531i	$-0.477627\pi$
$811$	4.00000			0.140459			0.0702295	+	0.997531i	$0.477627\pi$
$812$	−6.92820			−0.243132
$813$			18.0000i			0.631288i
$814$		0			0
$815$		−	4.00000i		−	0.140114i
$816$	−12.0000			−0.420084
$817$		0			0
$818$	20.7846			0.726717
$819$		0			0
$820$	12.0000			0.419058
$821$	31.1769			1.08808			0.544041	−	0.839059i	$-0.316894\pi$
$821$	31.1769			1.08808			0.544041	+	0.839059i	$0.316894\pi$
$822$		−	36.0000i		−	1.25564i
$823$			10.3923i			0.362253i	0.983460	+	0.181126i	$0.0579743\pi$
$823$			10.3923i			0.362253i	−0.983460	+	0.181126i	$0.942026\pi$
$824$		−	10.0000i		−	0.348367i
$825$	−6.00000			−0.208893
$826$	−12.0000			−0.417533
$827$	10.3923			0.361376			0.180688	−	0.983540i	$-0.442168\pi$
$827$	10.3923			0.361376			0.180688	+	0.983540i	$0.442168\pi$
$828$	10.3923			0.361158
$829$		−	41.5692i		−	1.44376i	−0.692019	−	0.721879i	$-0.743279\pi$
$829$		−	41.5692i		−	1.44376i	0.692019	−	0.721879i	$-0.256721\pi$
$830$	3.46410			0.120241
$831$		−	36.0000i		−	1.24883i
$832$		0			0
$833$	−20.7846			−0.720144
$834$	18.0000			0.623289
$835$			10.3923i			0.359641i
$836$	13.8564			0.479234
$837$	−10.3923	−	27.0000i	−0.359211	−	0.933257i
$838$	−30.0000			−1.03633
$839$		−	36.0000i		−	1.24286i	−0.783470	−	0.621429i	$-0.786552\pi$
$839$		−	36.0000i		−	1.24286i	0.783470	−	0.621429i	$-0.213448\pi$
$840$	−3.46410			−0.119523
$841$	−17.0000			−0.586207
$842$			34.0000i			1.17172i
$843$		−	41.5692i		−	1.43172i
$844$	4.00000			0.137686
$845$			13.0000i			0.447214i
$846$		0			0
$847$	2.00000			0.0687208
$848$		0			0
$849$	48.4974			1.66443
$850$		−	6.92820i		−	0.237635i
$851$		0			0
$852$			20.7846i			0.712069i
$853$	16.0000			0.547830			0.273915	−	0.961754i	$-0.411681\pi$
$853$	16.0000			0.547830			0.273915	+	0.961754i	$0.411681\pi$
$854$		0			0
$855$			12.0000i			0.410391i
$856$	−12.0000			−0.410152
$857$		−	18.0000i		−	0.614868i	−0.951569	−	0.307434i	$-0.900530\pi$
$857$		−	18.0000i		−	0.614868i	0.951569	−	0.307434i	$-0.0994704\pi$
$858$		0			0
$859$			10.3923i			0.354581i	0.984159	+	0.177290i	$0.0567332\pi$
$859$			10.3923i			0.354581i	−0.984159	+	0.177290i	$0.943267\pi$
$860$		0			0
$861$		−	41.5692i		−	1.41668i
$862$	36.0000			1.22616
$863$	3.46410			0.117919			0.0589597	−	0.998260i	$-0.481222\pi$
$863$	3.46410			0.117919			0.0589597	+	0.998260i	$0.481222\pi$
$864$		−	5.19615i		−	0.176777i
$865$	18.0000			0.612018
$866$	31.1769			1.05943
$867$	−53.6936			−1.82353
$868$	−4.00000	−	10.3923i	−0.135769	−	0.352738i
$869$		−	36.0000i		−	1.22122i
$870$	6.00000			0.203419
$871$		0			0
$872$		−	14.0000i		−	0.474100i
$873$	−30.0000			−1.01535
$874$		−	13.8564i		−	0.468700i
$875$		−	2.00000i		−	0.0676123i
$876$			18.0000i			0.608164i
$877$	40.0000			1.35070			0.675352	−	0.737496i	$-0.263992\pi$
$877$	40.0000			1.35070			0.675352	+	0.737496i	$0.263992\pi$
$878$			16.0000i			0.539974i
$879$			31.1769i			1.05157i
$880$		−	3.46410i		−	0.116775i
$881$	−34.6410			−1.16709			−0.583543	−	0.812082i	$-0.698334\pi$
$881$	−34.6410			−1.16709			−0.583543	+	0.812082i	$0.698334\pi$
$882$		−	9.00000i		−	0.303046i
$883$		−	41.5692i		−	1.39892i	−0.714674	−	0.699458i	$-0.753425\pi$
$883$		−	41.5692i		−	1.39892i	0.714674	−	0.699458i	$-0.246575\pi$
$884$		0			0
$885$	10.3923			0.349334
$886$	−24.0000			−0.806296
$887$			48.0000i			1.61168i	0.592132	+	0.805841i	$0.298286\pi$
$887$			48.0000i			1.61168i	−0.592132	+	0.805841i	$0.701714\pi$
$888$		0			0
$889$		−	20.7846i		−	0.697093i
$890$	−13.8564			−0.464468
$891$	−31.1769			−1.04447
$892$		−	10.3923i		−	0.347960i
$893$		0			0
$894$	31.1769			1.04271
$895$			10.3923i			0.347376i
$896$		−	2.00000i		−	0.0668153i
$897$		0			0
$898$		−	27.7128i		−	0.924789i
$899$	6.92820	+	18.0000i	0.231069	+	0.600334i
$900$	3.00000			0.100000
$901$		0			0
$902$	41.5692			1.38410
$903$		0			0
$904$	−6.00000			−0.199557
$905$		0			0
$906$	18.0000			0.598010
$907$	52.0000			1.72663			0.863316	−	0.504664i	$-0.168384\pi$
$907$	52.0000			1.72663			0.863316	+	0.504664i	$0.168384\pi$
$908$		0			0
$909$		−	18.0000i		−	0.597022i
$910$		0			0
$911$	27.7128			0.918166			0.459083	−	0.888393i	$-0.348178\pi$
$911$	27.7128			0.918166			0.459083	+	0.888393i	$0.348178\pi$
$912$	−6.92820			−0.229416
$913$	12.0000			0.397142
$914$	31.1769			1.03124
$915$		0			0
$916$		0			0
$917$		−	36.0000i		−	1.18882i
$918$		−	36.0000i		−	1.18818i
$919$	4.00000			0.131948			0.0659739	−	0.997821i	$-0.478985\pi$
$919$	4.00000			0.131948			0.0659739	+	0.997821i	$0.478985\pi$
$920$	−3.46410			−0.114208
$921$	−6.92820			−0.228292
$922$		−	38.1051i		−	1.25493i
$923$		0			0
$924$	−12.0000			−0.394771
$925$		0			0
$926$	−31.1769			−1.02454
$927$	30.0000			0.985329
$928$			3.46410i			0.113715i
$929$	6.92820			0.227307			0.113653	−	0.993520i	$-0.463745\pi$
$929$	6.92820			0.227307			0.113653	+	0.993520i	$0.463745\pi$
$930$	3.46410	+	9.00000i	0.113592	+	0.295122i
$931$	−12.0000			−0.393284
$932$		−	6.00000i		−	0.196537i
$933$			41.5692i			1.36092i
$934$	−12.0000			−0.392652
$935$		−	24.0000i		−	0.784884i
$936$		0			0
$937$	22.0000			0.718709			0.359354	−	0.933201i	$-0.382997\pi$
$937$	22.0000			0.718709			0.359354	+	0.933201i	$0.382997\pi$
$938$		−	16.0000i		−	0.522419i
$939$			18.0000i			0.587408i
$940$		0			0
$941$	−10.3923			−0.338779			−0.169390	−	0.985549i	$-0.554180\pi$
$941$	−10.3923			−0.338779			−0.169390	+	0.985549i	$0.554180\pi$
$942$			13.8564i			0.451466i
$943$		−	41.5692i		−	1.35368i
$944$			6.00000i			0.195283i
$945$		−	10.3923i		−	0.338062i
$946$		0			0
$947$	−31.1769			−1.01311			−0.506557	−	0.862207i	$-0.669082\pi$
$947$	−31.1769			−1.01311			−0.506557	+	0.862207i	$0.669082\pi$
$948$			18.0000i			0.584613i
$949$		0			0
$950$		−	4.00000i		−	0.129777i
$951$		−	51.9615i		−	1.68497i
$952$		−	13.8564i		−	0.449089i
$953$	−20.7846			−0.673280			−0.336640	−	0.941634i	$-0.609290\pi$
$953$	−20.7846			−0.673280			−0.336640	+	0.941634i	$0.609290\pi$
$954$		0			0
$955$	12.0000			0.388311
$956$	27.7128			0.896296
$957$	20.7846			0.671871
$958$	−24.0000			−0.775405
$959$	41.5692			1.34234
$960$			1.73205i			0.0559017i
$961$	−23.0000	+	20.7846i	−0.741935	+	0.670471i
$962$		0			0
$963$		−	36.0000i		−	1.16008i
$964$		−	20.7846i		−	0.669427i
$965$			10.0000i			0.321911i
$966$			12.0000i			0.386094i
$967$			10.3923i			0.334194i	0.985940	+	0.167097i	$0.0534393\pi$
$967$			10.3923i			0.334194i	−0.985940	+	0.167097i	$0.946561\pi$
$968$		−	1.00000i		−	0.0321412i
$969$	−48.0000			−1.54198
$970$	10.0000			0.321081
$971$		−	18.0000i		−	0.577647i	−0.957382	−	0.288824i	$-0.906736\pi$
$971$		−	18.0000i		−	0.577647i	0.957382	−	0.288824i	$-0.0932642\pi$
$972$	15.5885			0.500000
$973$			20.7846i			0.666324i
$974$	10.3923			0.332991
$975$		0			0
$976$		0			0
$977$		−	42.0000i		−	1.34370i	−0.740688	−	0.671850i	$-0.765500\pi$
$977$		−	42.0000i		−	1.34370i	0.740688	−	0.671850i	$-0.234500\pi$
$978$			6.92820i			0.221540i
$979$	−48.0000			−1.53409
$980$			3.00000i			0.0958315i
$981$	42.0000			1.34096
$982$			24.2487i			0.773807i
$983$	17.3205			0.552438			0.276219	−	0.961095i	$-0.410918\pi$
$983$	17.3205			0.552438			0.276219	+	0.961095i	$0.410918\pi$
$984$	−20.7846			−0.662589
$985$		0			0
$986$			24.0000i			0.764316i
$987$		0			0
$988$		0			0
$989$		0			0
$990$	10.3923			0.330289
$991$			31.1769i			0.990367i	0.868788	+	0.495184i	$0.164899\pi$
$991$			31.1769i			0.990367i	−0.868788	+	0.495184i	$0.835101\pi$
$992$	−5.19615	+	2.00000i	−0.164978	+	0.0635001i
$993$			54.0000i			1.71364i
$994$	−24.0000			−0.761234
$995$	10.3923			0.329458
$996$	−6.00000			−0.190117
$997$	20.0000			0.633406			0.316703	−	0.948525i	$-0.397424\pi$
$997$	20.0000			0.633406			0.316703	+	0.948525i	$0.397424\pi$
$998$	31.1769			0.986888
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	930.2.h.b.371.3	yes	4
3.2	odd	2	inner	930.2.h.b.371.2	yes	4
31.30	odd	2	inner	930.2.h.b.371.4	yes	4
93.92	even	2	inner	930.2.h.b.371.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.h.b.371.1	✓	4	93.92	even	2	inner
930.2.h.b.371.2	yes	4	3.2	odd	2	inner
930.2.h.b.371.3	yes	4	1.1	even	1	trivial
930.2.h.b.371.4	yes	4	31.30	odd	2	inner

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:	\( N \)	\(=\)	\( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	930.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.73205 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.73205i q^{6} +2.00000 q^{7} -1.00000i q^{8} +3.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -1.73205 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.73205i q^{6} +2.00000 q^{7} -1.00000i q^{8} +3.00000 q^{9} -1.00000 q^{10} -3.46410 q^{11} +1.73205 q^{12} +2.00000i q^{14} -1.73205i q^{15} +1.00000 q^{16} +6.92820 q^{17} +3.00000i q^{18} +4.00000 q^{19} -1.00000i q^{20} -3.46410 q^{21} -3.46410i q^{22} -3.46410 q^{23} +1.73205i q^{24} -1.00000 q^{25} -5.19615 q^{27} -2.00000 q^{28} +3.46410 q^{29} +1.73205 q^{30} +(2.00000 + 5.19615i) q^{31} +1.00000i q^{32} +6.00000 q^{33} +6.92820i q^{34} +2.00000i q^{35} -3.00000 q^{36} +4.00000i q^{38} +1.00000 q^{40} +12.0000i q^{41} -3.46410i q^{42} +3.46410 q^{44} +3.00000i q^{45} -3.46410i q^{46} -1.73205 q^{48} -3.00000 q^{49} -1.00000i q^{50} -12.0000 q^{51} -5.19615i q^{54} -3.46410i q^{55} -2.00000i q^{56} -6.92820 q^{57} +3.46410i q^{58} +6.00000i q^{59} +1.73205i q^{60} +(-5.19615 + 2.00000i) q^{62} +6.00000 q^{63} -1.00000 q^{64} +6.00000i q^{66} -8.00000 q^{67} -6.92820 q^{68} +6.00000 q^{69} -2.00000 q^{70} +12.0000i q^{71} -3.00000i q^{72} +10.3923i q^{73} +1.73205 q^{75} -4.00000 q^{76} -6.92820 q^{77} +10.3923i q^{79} +1.00000i q^{80} +9.00000 q^{81} -12.0000 q^{82} -3.46410 q^{83} +3.46410 q^{84} +6.92820i q^{85} -6.00000 q^{87} +3.46410i q^{88} +13.8564 q^{89} -3.00000 q^{90} +3.46410 q^{92} +(-3.46410 - 9.00000i) q^{93} +4.00000i q^{95} -1.73205i q^{96} -10.0000 q^{97} -3.00000i q^{98} -10.3923 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} + 8 q^{7} + 12 q^{9}+O(q^{10}) \) 4 * q - 4 * q^4 + 8 * q^7 + 12 * q^9	\( 4 q - 4 q^{4} + 8 q^{7} + 12 q^{9} - 4 q^{10} + 4 q^{16} + 16 q^{19} - 4 q^{25} - 8 q^{28} + 8 q^{31} + 24 q^{33} - 12 q^{36} + 4 q^{40} - 12 q^{49} - 48 q^{51} + 24 q^{63} - 4 q^{64} - 32 q^{67} + 24 q^{69} - 8 q^{70} - 16 q^{76} + 36 q^{81} - 48 q^{82} - 24 q^{87} - 12 q^{90} - 40 q^{97}+O(q^{100}) \) 4 * q - 4 * q^4 + 8 * q^7 + 12 * q^9 - 4 * q^10 + 4 * q^16 + 16 * q^19 - 4 * q^25 - 8 * q^28 + 8 * q^31 + 24 * q^33 - 12 * q^36 + 4 * q^40 - 12 * q^49 - 48 * q^51 + 24 * q^63 - 4 * q^64 - 32 * q^67 + 24 * q^69 - 8 * q^70 - 16 * q^76 + 36 * q^81 - 48 * q^82 - 24 * q^87 - 12 * q^90 - 40 * q^97

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