LMFDB - Embedded newform 2240.2.n.e.1119.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2240,2,Mod(1119,2240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2240, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2240.1119");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2240 = 2^{6} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2240.n (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:	$17.8864900528$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1119.6
Root			$-0.965926 + 0.258819i$ of defining polynomial
Character	$\chi$	$=$	2240.1119
Dual form			2240.2.n.e.1119.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.41421 - 1.73205i) q^{5} +(2.44949 + 1.00000i) q^{7} -3.00000 q^{9} +O(q^{10})$	$q+(1.41421 - 1.73205i) q^{5} +(2.44949 + 1.00000i) q^{7} -3.00000 q^{9} +3.46410 q^{11} +3.46410i q^{13} +4.00000 q^{17} +2.82843i q^{19} -4.89898 q^{23} +(-1.00000 - 4.89898i) q^{25} +6.92820i q^{29} +9.79796 q^{31} +(5.19615 - 2.82843i) q^{35} +5.65685 q^{37} +9.79796i q^{41} +2.82843i q^{43} +(-4.24264 + 5.19615i) q^{45} -10.0000i q^{47} +(5.00000 + 4.89898i) q^{49} -11.3137 q^{53} +(4.89898 - 6.00000i) q^{55} +8.48528i q^{59} -14.1421 q^{61} +(-7.34847 - 3.00000i) q^{63} +(6.00000 + 4.89898i) q^{65} -14.1421i q^{67} -4.00000i q^{71} +12.0000 q^{73} +(8.48528 + 3.46410i) q^{77} -12.0000i q^{79} +9.00000 q^{81} +13.8564 q^{83} +(5.65685 - 6.92820i) q^{85} +9.79796i q^{89} +(-3.46410 + 8.48528i) q^{91} +(4.89898 + 4.00000i) q^{95} +4.00000 q^{97} -10.3923 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 24 q^{9}+O(q^{10}) $ 8 * q - 24 * q^9	$ 8 q - 24 q^{9} + 32 q^{17} - 8 q^{25} + 40 q^{49} + 48 q^{65} + 96 q^{73} + 72 q^{81} + 32 q^{97}+O(q^{100}) $ 8 * q - 24 * q^9 + 32 * q^17 - 8 * q^25 + 40 * q^49 + 48 * q^65 + 96 * q^73 + 72 * q^81 + 32 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times$.

$n$	$897$	$1471$	$1541$	$1921$
$\chi(n)$	$-1$	$-1$	$-1$	$-1$

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			0			−	1.00000i	$-0.5\pi$
$3$		0			0				1.00000i	$0.5\pi$
$4$		0			0
$5$	1.41421	−	1.73205i	0.632456	−	0.774597i
$5$	1.41421	−	1.73205i	0.632456	−	0.774597i
$6$		0			0
$7$	2.44949	+	1.00000i	0.925820	+	0.377964i
$7$	2.44949	+	1.00000i	0.925820	+	0.377964i
$8$		0			0
$9$	−3.00000			−1.00000
$10$		0			0
$11$	3.46410			1.04447			0.522233	−	0.852803i	$-0.325099\pi$
$11$	3.46410			1.04447			0.522233	+	0.852803i	$0.325099\pi$
$12$		0			0
$13$			3.46410i			0.960769i	0.877058	+	0.480384i	$0.159503\pi$
$13$			3.46410i			0.960769i	−0.877058	+	0.480384i	$0.840497\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	4.00000			0.970143			0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000			0.970143			0.485071	+	0.874475i	$0.338794\pi$
$18$		0			0
$19$			2.82843i			0.648886i	0.945905	+	0.324443i	$0.105177\pi$
$19$			2.82843i			0.648886i	−0.945905	+	0.324443i	$0.894823\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.89898			−1.02151			−0.510754	−	0.859727i	$-0.670634\pi$
$23$	−4.89898			−1.02151			−0.510754	+	0.859727i	$0.670634\pi$
$24$		0			0
$25$	−1.00000	−	4.89898i	−0.200000	−	0.979796i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			6.92820i			1.28654i	0.765641	+	0.643268i	$0.222422\pi$
$29$			6.92820i			1.28654i	−0.765641	+	0.643268i	$0.777578\pi$
$30$		0			0
$31$	9.79796			1.75977			0.879883	−	0.475191i	$-0.157621\pi$
$31$	9.79796			1.75977			0.879883	+	0.475191i	$0.157621\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	5.19615	−	2.82843i	0.878310	−	0.478091i
$36$		0			0
$37$	5.65685			0.929981			0.464991	−	0.885316i	$-0.346058\pi$
$37$	5.65685			0.929981			0.464991	+	0.885316i	$0.346058\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			9.79796i			1.53018i	0.643921	+	0.765092i	$0.277307\pi$
$41$			9.79796i			1.53018i	−0.643921	+	0.765092i	$0.722693\pi$
$42$		0			0
$43$			2.82843i			0.431331i	0.976467	+	0.215666i	$0.0691921\pi$
$43$			2.82843i			0.431331i	−0.976467	+	0.215666i	$0.930808\pi$
$44$		0			0
$45$	−4.24264	+	5.19615i	−0.632456	+	0.774597i
$46$		0			0
$47$		−	10.0000i		−	1.45865i	−0.684167	−	0.729325i	$-0.739834\pi$
$47$		−	10.0000i		−	1.45865i	0.684167	−	0.729325i	$-0.260166\pi$
$48$		0			0
$49$	5.00000	+	4.89898i	0.714286	+	0.699854i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−11.3137			−1.55406			−0.777029	−	0.629465i	$-0.783274\pi$
$53$	−11.3137			−1.55406			−0.777029	+	0.629465i	$0.783274\pi$
$54$		0			0
$55$	4.89898	−	6.00000i	0.660578	−	0.809040i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			8.48528i			1.10469i	0.833616	+	0.552345i	$0.186267\pi$
$59$			8.48528i			1.10469i	−0.833616	+	0.552345i	$0.813733\pi$
$60$		0			0
$61$	−14.1421			−1.81071			−0.905357	−	0.424650i	$-0.860397\pi$
$61$	−14.1421			−1.81071			−0.905357	+	0.424650i	$0.860397\pi$
$62$		0			0
$63$	−7.34847	−	3.00000i	−0.925820	−	0.377964i
$64$		0			0
$65$	6.00000	+	4.89898i	0.744208	+	0.607644i
$66$		0			0
$67$		−	14.1421i		−	1.72774i	−0.503718	−	0.863868i	$-0.668035\pi$
$67$		−	14.1421i		−	1.72774i	0.503718	−	0.863868i	$-0.331965\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	4.00000i		−	0.474713i	−0.971423	−	0.237356i	$-0.923719\pi$
$71$		−	4.00000i		−	0.474713i	0.971423	−	0.237356i	$-0.0762809\pi$
$72$		0			0
$73$	12.0000			1.40449			0.702247	−	0.711934i	$-0.252180\pi$
$73$	12.0000			1.40449			0.702247	+	0.711934i	$0.252180\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	8.48528	+	3.46410i	0.966988	+	0.394771i
$78$		0			0
$79$		−	12.0000i		−	1.35011i	−0.737769	−	0.675053i	$-0.764121\pi$
$79$		−	12.0000i		−	1.35011i	0.737769	−	0.675053i	$-0.235879\pi$
$80$		0			0
$81$	9.00000			1.00000
$82$		0			0
$83$	13.8564			1.52094			0.760469	−	0.649374i	$-0.224969\pi$
$83$	13.8564			1.52094			0.760469	+	0.649374i	$0.224969\pi$
$84$		0			0
$85$	5.65685	−	6.92820i	0.613572	−	0.751469i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			9.79796i			1.03858i	0.854598	+	0.519291i	$0.173804\pi$
$89$			9.79796i			1.03858i	−0.854598	+	0.519291i	$0.826196\pi$
$90$		0			0
$91$	−3.46410	+	8.48528i	−0.363137	+	0.889499i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	4.89898	+	4.00000i	0.502625	+	0.410391i
$96$		0			0
$97$	4.00000			0.406138			0.203069	−	0.979164i	$-0.434908\pi$
$97$	4.00000			0.406138			0.203069	+	0.979164i	$0.434908\pi$
$98$		0			0
$99$	−10.3923			−1.04447
$100$		0			0
$101$	8.48528			0.844317			0.422159	−	0.906522i	$-0.361273\pi$
$101$	8.48528			0.844317			0.422159	+	0.906522i	$0.361273\pi$
$102$		0			0
$103$		−	6.00000i		−	0.591198i	−0.955312	−	0.295599i	$-0.904481\pi$
$103$		−	6.00000i		−	0.591198i	0.955312	−	0.295599i	$-0.0955191\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			2.82843i			0.273434i	0.990610	+	0.136717i	$0.0436552\pi$
$107$			2.82843i			0.273434i	−0.990610	+	0.136717i	$0.956345\pi$
$108$		0			0
$109$		−	6.92820i		−	0.663602i	−0.943349	−	0.331801i	$-0.892344\pi$
$109$		−	6.92820i		−	0.663602i	0.943349	−	0.331801i	$-0.107656\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	9.79796i		−	0.921714i	−0.887474	−	0.460857i	$-0.847542\pi$
$113$		−	9.79796i		−	0.921714i	0.887474	−	0.460857i	$-0.152458\pi$
$114$		0			0
$115$	−6.92820	+	8.48528i	−0.646058	+	0.791257i
$116$		0			0
$117$		−	10.3923i		−	0.960769i
$118$		0			0
$119$	9.79796	+	4.00000i	0.898177	+	0.366679i
$120$		0			0
$121$	1.00000			0.0909091
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−9.89949	−	5.19615i	−0.885438	−	0.464758i
$126$		0			0
$127$	−4.89898			−0.434714			−0.217357	−	0.976092i	$-0.569744\pi$
$127$	−4.89898			−0.434714			−0.217357	+	0.976092i	$0.569744\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			2.82843i			0.247121i	0.992337	+	0.123560i	$0.0394313\pi$
$131$			2.82843i			0.247121i	−0.992337	+	0.123560i	$0.960569\pi$
$132$		0			0
$133$	−2.82843	+	6.92820i	−0.245256	+	0.600751i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		1.00000			$0$
$137$		0			0		−1.00000			$\pi$
$138$		0			0
$139$		−	2.82843i		−	0.239904i	−0.992780	−	0.119952i	$-0.961726\pi$
$139$		−	2.82843i		−	0.239904i	0.992780	−	0.119952i	$-0.0382741\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			12.0000i			1.00349i
$144$		0			0
$145$	12.0000	+	9.79796i	0.996546	+	0.813676i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			6.92820i			0.567581i	0.958886	+	0.283790i	$0.0915919\pi$
$149$			6.92820i			0.567581i	−0.958886	+	0.283790i	$0.908408\pi$
$150$		0			0
$151$		−	12.0000i		−	0.976546i	−0.872691	−	0.488273i	$-0.837627\pi$
$151$		−	12.0000i		−	0.976546i	0.872691	−	0.488273i	$-0.162373\pi$
$152$		0			0
$153$	−12.0000			−0.970143
$154$		0			0
$155$	13.8564	−	16.9706i	1.11297	−	1.36311i
$156$		0			0
$157$			10.3923i			0.829396i	0.909959	+	0.414698i	$0.136113\pi$
$157$			10.3923i			0.829396i	−0.909959	+	0.414698i	$0.863887\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−12.0000	−	4.89898i	−0.945732	−	0.386094i
$162$		0			0
$163$			19.7990i			1.55078i	0.631485	+	0.775388i	$0.282446\pi$
$163$			19.7990i			1.55078i	−0.631485	+	0.775388i	$0.717554\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	10.0000i		−	0.773823i	−0.922117	−	0.386912i	$-0.873542\pi$
$167$		−	10.0000i		−	0.773823i	0.922117	−	0.386912i	$-0.126458\pi$
$168$		0			0
$169$	1.00000			0.0769231
$170$		0			0
$171$		−	8.48528i		−	0.648886i
$172$		0			0
$173$			17.3205i			1.31685i	0.752645	+	0.658427i	$0.228778\pi$
$173$			17.3205i			1.31685i	−0.752645	+	0.658427i	$0.771222\pi$
$174$		0			0
$175$	2.44949	−	13.0000i	0.185164	−	0.982708i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	10.3923			0.776757			0.388379	−	0.921500i	$-0.373035\pi$
$179$	10.3923			0.776757			0.388379	+	0.921500i	$0.373035\pi$
$180$		0			0
$181$	−2.82843			−0.210235			−0.105118	−	0.994460i	$-0.533522\pi$
$181$	−2.82843			−0.210235			−0.105118	+	0.994460i	$0.533522\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	8.00000	−	9.79796i	0.588172	−	0.720360i
$186$		0			0
$187$	13.8564			1.01328
$188$		0			0
$189$		0			0
$190$		0			0
$191$			20.0000i			1.44715i	0.690246	+	0.723575i	$0.257502\pi$
$191$			20.0000i			1.44715i	−0.690246	+	0.723575i	$0.742498\pi$
$192$		0			0
$193$		−	19.5959i		−	1.41055i	−0.708936	−	0.705273i	$-0.750825\pi$
$193$		−	19.5959i		−	1.41055i	0.708936	−	0.705273i	$-0.249175\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−11.3137			−0.806068			−0.403034	−	0.915185i	$-0.632044\pi$
$197$	−11.3137			−0.806068			−0.403034	+	0.915185i	$0.632044\pi$
$198$		0			0
$199$	19.5959			1.38912			0.694559	−	0.719436i	$-0.255600\pi$
$199$	19.5959			1.38912			0.694559	+	0.719436i	$0.255600\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−6.92820	+	16.9706i	−0.486265	+	1.19110i
$204$		0			0
$205$	16.9706	+	13.8564i	1.18528	+	0.967773i
$206$		0			0
$207$	14.6969			1.02151
$208$		0			0
$209$			9.79796i			0.677739i
$210$		0			0
$211$	10.3923			0.715436			0.357718	−	0.933830i	$-0.383555\pi$
$211$	10.3923			0.715436			0.357718	+	0.933830i	$0.383555\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	4.89898	+	4.00000i	0.334108	+	0.272798i
$216$		0			0
$217$	24.0000	+	9.79796i	1.62923	+	0.665129i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			13.8564i			0.932083i
$222$		0			0
$223$		−	18.0000i		−	1.20537i	−0.797980	−	0.602685i	$-0.794098\pi$
$223$		−	18.0000i		−	1.20537i	0.797980	−	0.602685i	$-0.205902\pi$
$224$		0			0
$225$	3.00000	+	14.6969i	0.200000	+	0.979796i
$226$		0			0
$227$	−13.8564			−0.919682			−0.459841	−	0.888001i	$-0.652094\pi$
$227$	−13.8564			−0.919682			−0.459841	+	0.888001i	$0.652094\pi$
$228$		0			0
$229$	2.82843			0.186908			0.0934539	−	0.995624i	$-0.470209\pi$
$229$	2.82843			0.186908			0.0934539	+	0.995624i	$0.470209\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	9.79796i		−	0.641886i	−0.947099	−	0.320943i	$-0.896000\pi$
$233$		−	9.79796i		−	0.641886i	0.947099	−	0.320943i	$-0.104000\pi$
$234$		0			0
$235$	−17.3205	−	14.1421i	−1.12987	−	0.922531i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	4.00000i		−	0.258738i	−0.991596	−	0.129369i	$-0.958705\pi$
$239$		−	4.00000i		−	0.258738i	0.991596	−	0.129369i	$-0.0412952\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	15.5563	−	1.73205i	0.993859	−	0.110657i
$246$		0			0
$247$	−9.79796			−0.623429
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	2.82843i		−	0.178529i	−0.996008	−	0.0892644i	$-0.971548\pi$
$251$		−	2.82843i		−	0.178529i	0.996008	−	0.0892644i	$-0.0284516\pi$
$252$		0			0
$253$	−16.9706			−1.06693
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−20.0000			−1.24757			−0.623783	−	0.781598i	$-0.714405\pi$
$257$	−20.0000			−1.24757			−0.623783	+	0.781598i	$0.714405\pi$
$258$		0			0
$259$	13.8564	+	5.65685i	0.860995	+	0.351500i
$260$		0			0
$261$		−	20.7846i		−	1.28654i
$262$		0			0
$263$	−14.6969			−0.906252			−0.453126	−	0.891446i	$-0.649691\pi$
$263$	−14.6969			−0.906252			−0.453126	+	0.891446i	$0.649691\pi$
$264$		0			0
$265$	−16.0000	+	19.5959i	−0.982872	+	1.20377i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−8.48528			−0.517357			−0.258678	−	0.965964i	$-0.583287\pi$
$269$	−8.48528			−0.517357			−0.258678	+	0.965964i	$0.583287\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−3.46410	−	16.9706i	−0.208893	−	1.02336i
$276$		0			0
$277$	5.65685			0.339887			0.169944	−	0.985454i	$-0.445641\pi$
$277$	5.65685			0.339887			0.169944	+	0.985454i	$0.445641\pi$
$278$		0			0
$279$	−29.3939			−1.75977
$280$		0			0
$281$	2.00000			0.119310			0.0596550	−	0.998219i	$-0.481000\pi$
$281$	2.00000			0.119310			0.0596550	+	0.998219i	$0.481000\pi$
$282$		0			0
$283$	27.7128			1.64736			0.823678	−	0.567058i	$-0.191918\pi$
$283$	27.7128			1.64736			0.823678	+	0.567058i	$0.191918\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−9.79796	+	24.0000i	−0.578355	+	1.41668i
$288$		0			0
$289$	−1.00000			−0.0588235
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	24.2487i		−	1.41662i	−0.705899	−	0.708312i	$-0.749457\pi$
$293$		−	24.2487i		−	1.41662i	0.705899	−	0.708312i	$-0.250543\pi$
$294$		0			0
$295$	14.6969	+	12.0000i	0.855689	+	0.698667i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	16.9706i		−	0.981433i
$300$		0			0
$301$	−2.82843	+	6.92820i	−0.163028	+	0.399335i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−20.0000	+	24.4949i	−1.14520	+	1.40257i
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−9.79796			−0.555591			−0.277796	−	0.960640i	$-0.589604\pi$
$311$	−9.79796			−0.555591			−0.277796	+	0.960640i	$0.589604\pi$
$312$		0			0
$313$	−12.0000			−0.678280			−0.339140	−	0.940736i	$-0.610136\pi$
$313$	−12.0000			−0.678280			−0.339140	+	0.940736i	$0.610136\pi$
$314$		0			0
$315$	−15.5885	+	8.48528i	−0.878310	+	0.478091i
$316$		0			0
$317$		0			0			−	1.00000i	$-0.5\pi$
$317$		0			0				1.00000i	$0.5\pi$
$318$		0			0
$319$			24.0000i			1.34374i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			11.3137i			0.629512i
$324$		0			0
$325$	16.9706	−	3.46410i	0.941357	−	0.192154i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	10.0000	−	24.4949i	0.551318	−	1.35045i
$330$		0			0
$331$	3.46410			0.190404			0.0952021	−	0.995458i	$-0.469650\pi$
$331$	3.46410			0.190404			0.0952021	+	0.995458i	$0.469650\pi$
$332$		0			0
$333$	−16.9706			−0.929981
$334$		0			0
$335$	−24.4949	−	20.0000i	−1.33830	−	1.09272i
$336$		0			0
$337$		−	9.79796i		−	0.533729i	−0.963734	−	0.266864i	$-0.914012\pi$
$337$		−	9.79796i		−	0.533729i	0.963734	−	0.266864i	$-0.0859876\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	33.9411			1.83801
$342$		0			0
$343$	7.34847	+	17.0000i	0.396780	+	0.917914i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			8.48528i			0.455514i	0.973718	+	0.227757i	$0.0731391\pi$
$347$			8.48528i			0.455514i	−0.973718	+	0.227757i	$0.926861\pi$
$348$		0			0
$349$	−19.7990			−1.05982			−0.529908	−	0.848055i	$-0.677773\pi$
$349$	−19.7990			−1.05982			−0.529908	+	0.848055i	$0.677773\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−4.00000			−0.212899			−0.106449	−	0.994318i	$-0.533948\pi$
$353$	−4.00000			−0.212899			−0.106449	+	0.994318i	$0.533948\pi$
$354$		0			0
$355$	−6.92820	−	5.65685i	−0.367711	−	0.300235i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			4.00000i			0.211112i	0.994413	+	0.105556i	$0.0336622\pi$
$359$			4.00000i			0.211112i	−0.994413	+	0.105556i	$0.966338\pi$
$360$		0			0
$361$	11.0000			0.578947
$362$		0			0
$363$		0			0
$364$		0			0
$365$	16.9706	−	20.7846i	0.888280	−	1.08792i
$366$		0			0
$367$			2.00000i			0.104399i	0.998637	+	0.0521996i	$0.0166232\pi$
$367$			2.00000i			0.104399i	−0.998637	+	0.0521996i	$0.983377\pi$
$368$		0			0
$369$		−	29.3939i		−	1.53018i
$370$		0			0
$371$	−27.7128	−	11.3137i	−1.43878	−	0.587378i
$372$		0			0
$373$	−28.2843			−1.46450			−0.732252	−	0.681034i	$-0.761531\pi$
$373$	−28.2843			−1.46450			−0.732252	+	0.681034i	$0.761531\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−24.0000			−1.23606
$378$		0			0
$379$	−3.46410			−0.177939			−0.0889695	−	0.996034i	$-0.528357\pi$
$379$	−3.46410			−0.177939			−0.0889695	+	0.996034i	$0.528357\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			22.0000i			1.12415i	0.827087	+	0.562074i	$0.189996\pi$
$383$			22.0000i			1.12415i	−0.827087	+	0.562074i	$0.810004\pi$
$384$		0			0
$385$	18.0000	−	9.79796i	0.917365	−	0.499350i
$386$		0			0
$387$		−	8.48528i		−	0.431331i
$388$		0			0
$389$		−	20.7846i		−	1.05382i	−0.849921	−	0.526911i	$-0.823350\pi$
$389$		−	20.7846i		−	1.05382i	0.849921	−	0.526911i	$-0.176650\pi$
$390$		0			0
$391$	−19.5959			−0.991008
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−20.7846	−	16.9706i	−1.04579	−	0.853882i
$396$		0			0
$397$		−	3.46410i		−	0.173858i	−0.996214	−	0.0869291i	$-0.972295\pi$
$397$		−	3.46410i		−	0.173858i	0.996214	−	0.0869291i	$-0.0277054\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−10.0000			−0.499376			−0.249688	−	0.968326i	$-0.580328\pi$
$401$	−10.0000			−0.499376			−0.249688	+	0.968326i	$0.580328\pi$
$402$		0			0
$403$			33.9411i			1.69073i
$404$		0			0
$405$	12.7279	−	15.5885i	0.632456	−	0.774597i
$406$		0			0
$407$	19.5959			0.971334
$408$		0			0
$409$		−	9.79796i		−	0.484478i	−0.970217	−	0.242239i	$-0.922118\pi$
$409$		−	9.79796i		−	0.484478i	0.970217	−	0.242239i	$-0.0778818\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−8.48528	+	20.7846i	−0.417533	+	1.02274i
$414$		0			0
$415$	19.5959	−	24.0000i	0.961926	−	1.17811i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	25.4558i		−	1.24360i	−0.783176	−	0.621800i	$-0.786402\pi$
$419$		−	25.4558i		−	1.24360i	0.783176	−	0.621800i	$-0.213598\pi$
$420$		0			0
$421$			6.92820i			0.337660i	0.985645	+	0.168830i	$0.0539989\pi$
$421$			6.92820i			0.337660i	−0.985645	+	0.168830i	$0.946001\pi$
$422$		0			0
$423$			30.0000i			1.45865i
$424$		0			0
$425$	−4.00000	−	19.5959i	−0.194029	−	0.950542i
$426$		0			0
$427$	−34.6410	−	14.1421i	−1.67640	−	0.684386i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	20.0000i		−	0.963366i	−0.876346	−	0.481683i	$-0.840026\pi$
$431$		−	20.0000i		−	0.963366i	0.876346	−	0.481683i	$-0.159974\pi$
$432$		0			0
$433$	−28.0000			−1.34559			−0.672797	−	0.739827i	$-0.734907\pi$
$433$	−28.0000			−1.34559			−0.672797	+	0.739827i	$0.734907\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	13.8564i		−	0.662842i
$438$		0			0
$439$	−9.79796			−0.467631			−0.233816	−	0.972281i	$-0.575121\pi$
$439$	−9.79796			−0.467631			−0.233816	+	0.972281i	$0.575121\pi$
$440$		0			0
$441$	−15.0000	−	14.6969i	−0.714286	−	0.699854i
$442$		0			0
$443$			8.48528i			0.403148i	0.979473	+	0.201574i	$0.0646056\pi$
$443$			8.48528i			0.403148i	−0.979473	+	0.201574i	$0.935394\pi$
$444$		0			0
$445$	16.9706	+	13.8564i	0.804482	+	0.656857i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−22.0000			−1.03824			−0.519122	−	0.854700i	$-0.673741\pi$
$449$	−22.0000			−1.03824			−0.519122	+	0.854700i	$0.673741\pi$
$450$		0			0
$451$			33.9411i			1.59823i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	9.79796	+	18.0000i	0.459335	+	0.843853i
$456$		0			0
$457$			29.3939i			1.37499i	0.726190	+	0.687494i	$0.241289\pi$
$457$			29.3939i			1.37499i	−0.726190	+	0.687494i	$0.758711\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−25.4558			−1.18560			−0.592798	−	0.805351i	$-0.701977\pi$
$461$	−25.4558			−1.18560			−0.592798	+	0.805351i	$0.701977\pi$
$462$		0			0
$463$	−34.2929			−1.59372			−0.796862	−	0.604161i	$-0.793508\pi$
$463$	−34.2929			−1.59372			−0.796862	+	0.604161i	$0.793508\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$		0			0
$469$	14.1421	−	34.6410i	0.653023	−	1.59957i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			9.79796i			0.450511i
$474$		0			0
$475$	13.8564	−	2.82843i	0.635776	−	0.129777i
$476$		0			0
$477$	33.9411			1.55406
$478$		0			0
$479$	−9.79796			−0.447680			−0.223840	−	0.974626i	$-0.571859\pi$
$479$	−9.79796			−0.447680			−0.223840	+	0.974626i	$0.571859\pi$
$480$		0			0
$481$			19.5959i			0.893497i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	5.65685	−	6.92820i	0.256865	−	0.314594i
$486$		0			0
$487$	−4.89898			−0.221994			−0.110997	−	0.993821i	$-0.535404\pi$
$487$	−4.89898			−0.221994			−0.110997	+	0.993821i	$0.535404\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−31.1769			−1.40699			−0.703497	−	0.710698i	$-0.748379\pi$
$491$	−31.1769			−1.40699			−0.703497	+	0.710698i	$0.748379\pi$
$492$		0			0
$493$			27.7128i			1.24812i
$494$		0			0
$495$	−14.6969	+	18.0000i	−0.660578	+	0.809040i
$496$		0			0
$497$	4.00000	−	9.79796i	0.179425	−	0.439499i
$498$		0			0
$499$	−17.3205			−0.775372			−0.387686	−	0.921791i	$-0.626726\pi$
$499$	−17.3205			−0.775372			−0.387686	+	0.921791i	$0.626726\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	14.0000i		−	0.624229i	−0.950044	−	0.312115i	$-0.898963\pi$
$503$		−	14.0000i		−	0.624229i	0.950044	−	0.312115i	$-0.101037\pi$
$504$		0			0
$505$	12.0000	−	14.6969i	0.533993	−	0.654005i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−14.1421			−0.626839			−0.313420	−	0.949615i	$-0.601475\pi$
$509$	−14.1421			−0.626839			−0.313420	+	0.949615i	$0.601475\pi$
$510$		0			0
$511$	29.3939	+	12.0000i	1.30031	+	0.530849i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−10.3923	−	8.48528i	−0.457940	−	0.373906i
$516$		0			0
$517$		−	34.6410i		−	1.52351i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	39.1918i		−	1.71703i	−0.512792	−	0.858513i	$-0.671389\pi$
$521$		−	39.1918i		−	1.71703i	0.512792	−	0.858513i	$-0.328611\pi$
$522$		0			0
$523$		0			0			−	1.00000i	$-0.5\pi$
$523$		0			0				1.00000i	$0.5\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	39.1918			1.70722
$528$		0			0
$529$	1.00000			0.0434783
$530$		0			0
$531$		−	25.4558i		−	1.10469i
$532$		0			0
$533$	−33.9411			−1.47015
$534$		0			0
$535$	4.89898	+	4.00000i	0.211801	+	0.172935i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	17.3205	+	16.9706i	0.746047	+	0.730974i
$540$		0			0
$541$		−	20.7846i		−	0.893600i	−0.894634	−	0.446800i	$-0.852564\pi$
$541$		−	20.7846i		−	0.893600i	0.894634	−	0.446800i	$-0.147436\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−12.0000	−	9.79796i	−0.514024	−	0.419698i
$546$		0			0
$547$		−	14.1421i		−	0.604674i	−0.953201	−	0.302337i	$-0.902233\pi$
$547$		−	14.1421i		−	0.604674i	0.953201	−	0.302337i	$-0.0977668\pi$
$548$		0			0
$549$	42.4264			1.81071
$550$		0			0
$551$	−19.5959			−0.834814
$552$		0			0
$553$	12.0000	−	29.3939i	0.510292	−	1.24995i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	16.9706			0.719066			0.359533	−	0.933132i	$-0.382936\pi$
$557$	16.9706			0.719066			0.359533	+	0.933132i	$0.382936\pi$
$558$		0			0
$559$	−9.79796			−0.414410
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−41.5692			−1.75193			−0.875967	−	0.482371i	$-0.839776\pi$
$563$	−41.5692			−1.75193			−0.875967	+	0.482371i	$0.839776\pi$
$564$		0			0
$565$	−16.9706	−	13.8564i	−0.713957	−	0.582943i
$566$		0			0
$567$	22.0454	+	9.00000i	0.925820	+	0.377964i
$568$		0			0
$569$	10.0000			0.419222			0.209611	−	0.977785i	$-0.432780\pi$
$569$	10.0000			0.419222			0.209611	+	0.977785i	$0.432780\pi$
$570$		0			0
$571$	24.2487			1.01478			0.507388	−	0.861717i	$-0.330611\pi$
$571$	24.2487			1.01478			0.507388	+	0.861717i	$0.330611\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	4.89898	+	24.0000i	0.204302	+	1.00087i
$576$		0			0
$577$	12.0000			0.499567			0.249783	−	0.968302i	$-0.419641\pi$
$577$	12.0000			0.499567			0.249783	+	0.968302i	$0.419641\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	33.9411	+	13.8564i	1.40812	+	0.574861i
$582$		0			0
$583$	−39.1918			−1.62316
$584$		0			0
$585$	−18.0000	−	14.6969i	−0.744208	−	0.607644i
$586$		0			0
$587$	13.8564			0.571915			0.285958	−	0.958242i	$-0.407688\pi$
$587$	13.8564			0.571915			0.285958	+	0.958242i	$0.407688\pi$
$588$		0			0
$589$			27.7128i			1.14189i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−4.00000			−0.164260			−0.0821302	−	0.996622i	$-0.526172\pi$
$593$	−4.00000			−0.164260			−0.0821302	+	0.996622i	$0.526172\pi$
$594$		0			0
$595$	20.7846	−	11.3137i	0.852086	−	0.463817i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	4.00000i		−	0.163436i	−0.996656	−	0.0817178i	$-0.973959\pi$
$599$		−	4.00000i		−	0.163436i	0.996656	−	0.0817178i	$-0.0260406\pi$
$600$		0			0
$601$			19.5959i			0.799334i	0.916660	+	0.399667i	$0.130874\pi$
$601$			19.5959i			0.799334i	−0.916660	+	0.399667i	$0.869126\pi$
$602$		0			0
$603$			42.4264i			1.72774i
$604$		0			0
$605$	1.41421	−	1.73205i	0.0574960	−	0.0704179i
$606$		0			0
$607$		−	30.0000i		−	1.21766i	−0.793300	−	0.608831i	$-0.791639\pi$
$607$		−	30.0000i		−	1.21766i	0.793300	−	0.608831i	$-0.208361\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	34.6410			1.40143
$612$		0			0
$613$	−11.3137			−0.456956			−0.228478	−	0.973549i	$-0.573375\pi$
$613$	−11.3137			−0.456956			−0.228478	+	0.973549i	$0.573375\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			39.1918i			1.57780i	0.614519	+	0.788902i	$0.289350\pi$
$617$			39.1918i			1.57780i	−0.614519	+	0.788902i	$0.710650\pi$
$618$		0			0
$619$		−	31.1127i		−	1.25052i	−0.780415	−	0.625262i	$-0.784992\pi$
$619$		−	31.1127i		−	1.25052i	0.780415	−	0.625262i	$-0.215008\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−9.79796	+	24.0000i	−0.392547	+	0.961540i
$624$		0			0
$625$	−23.0000	+	9.79796i	−0.920000	+	0.391918i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	22.6274			0.902214
$630$		0			0
$631$		−	4.00000i		−	0.159237i	−0.996825	−	0.0796187i	$-0.974630\pi$
$631$		−	4.00000i		−	0.159237i	0.996825	−	0.0796187i	$-0.0253703\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−6.92820	+	8.48528i	−0.274937	+	0.336728i
$636$		0			0
$637$	−16.9706	+	17.3205i	−0.672398	+	0.686264i
$638$		0			0
$639$			12.0000i			0.474713i
$640$		0			0
$641$	−2.00000			−0.0789953			−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000			−0.0789953			−0.0394976	+	0.999220i	$0.512576\pi$
$642$		0			0
$643$	41.5692			1.63933			0.819665	−	0.572843i	$-0.194160\pi$
$643$	41.5692			1.63933			0.819665	+	0.572843i	$0.194160\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			38.0000i			1.49393i	0.664861	+	0.746967i	$0.268491\pi$
$647$			38.0000i			1.49393i	−0.664861	+	0.746967i	$0.731509\pi$
$648$		0			0
$649$			29.3939i			1.15381i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−33.9411			−1.32822			−0.664109	−	0.747635i	$-0.731189\pi$
$653$	−33.9411			−1.32822			−0.664109	+	0.747635i	$0.731189\pi$
$654$		0			0
$655$	4.89898	+	4.00000i	0.191419	+	0.156293i
$656$		0			0
$657$	−36.0000			−1.40449
$658$		0			0
$659$	−10.3923			−0.404827			−0.202413	−	0.979300i	$-0.564878\pi$
$659$	−10.3923			−0.404827			−0.202413	+	0.979300i	$0.564878\pi$
$660$		0			0
$661$	−14.1421			−0.550065			−0.275033	−	0.961435i	$-0.588689\pi$
$661$	−14.1421			−0.550065			−0.275033	+	0.961435i	$0.588689\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	8.00000	+	14.6969i	0.310227	+	0.569923i
$666$		0			0
$667$		−	33.9411i		−	1.31421i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−48.9898			−1.89123
$672$		0			0
$673$			9.79796i			0.377684i	0.982008	+	0.188842i	$0.0604733\pi$
$673$			9.79796i			0.377684i	−0.982008	+	0.188842i	$0.939527\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	17.3205i		−	0.665681i	−0.942983	−	0.332841i	$-0.891993\pi$
$677$		−	17.3205i		−	0.665681i	0.942983	−	0.332841i	$-0.108007\pi$
$678$		0			0
$679$	9.79796	+	4.00000i	0.376011	+	0.153506i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			36.7696i			1.40695i	0.710721	+	0.703474i	$0.248369\pi$
$683$			36.7696i			1.40695i	−0.710721	+	0.703474i	$0.751631\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	39.1918i		−	1.49309i
$690$		0			0
$691$		−	31.1127i		−	1.18358i	−0.806091	−	0.591791i	$-0.798421\pi$
$691$		−	31.1127i		−	1.18358i	0.806091	−	0.591791i	$-0.201579\pi$
$692$		0			0
$693$	−25.4558	−	10.3923i	−0.966988	−	0.394771i
$694$		0			0
$695$	−4.89898	−	4.00000i	−0.185829	−	0.151729i
$696$		0			0
$697$			39.1918i			1.48450i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			20.7846i			0.785024i	0.919747	+	0.392512i	$0.128394\pi$
$701$			20.7846i			0.785024i	−0.919747	+	0.392512i	$0.871606\pi$
$702$		0			0
$703$			16.0000i			0.603451i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	20.7846	+	8.48528i	0.781686	+	0.319122i
$708$		0			0
$709$			48.4974i			1.82136i	0.413114	+	0.910679i	$0.364441\pi$
$709$			48.4974i			1.82136i	−0.413114	+	0.910679i	$0.635559\pi$
$710$		0			0
$711$			36.0000i			1.35011i
$712$		0			0
$713$	−48.0000			−1.79761
$714$		0			0
$715$	20.7846	+	16.9706i	0.777300	+	0.634663i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−9.79796			−0.365402			−0.182701	−	0.983169i	$-0.558484\pi$
$719$	−9.79796			−0.365402			−0.182701	+	0.983169i	$0.558484\pi$
$720$		0			0
$721$	6.00000	−	14.6969i	0.223452	−	0.547343i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	33.9411	−	6.92820i	1.26054	−	0.257307i
$726$		0			0
$727$		−	42.0000i		−	1.55769i	−0.627214	−	0.778847i	$-0.715805\pi$
$727$		−	42.0000i		−	1.55769i	0.627214	−	0.778847i	$-0.284195\pi$
$728$		0			0
$729$	−27.0000			−1.00000
$730$		0			0
$731$			11.3137i			0.418453i
$732$		0			0
$733$		−	17.3205i		−	0.639748i	−0.947460	−	0.319874i	$-0.896359\pi$
$733$		−	17.3205i		−	0.639748i	0.947460	−	0.319874i	$-0.103641\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	48.9898i		−	1.80456i
$738$		0			0
$739$	−10.3923			−0.382287			−0.191144	−	0.981562i	$-0.561220\pi$
$739$	−10.3923			−0.382287			−0.191144	+	0.981562i	$0.561220\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	4.89898			0.179726			0.0898631	−	0.995954i	$-0.471357\pi$
$743$	4.89898			0.179726			0.0898631	+	0.995954i	$0.471357\pi$
$744$		0			0
$745$	12.0000	+	9.79796i	0.439646	+	0.358969i
$746$		0			0
$747$	−41.5692			−1.52094
$748$		0			0
$749$	−2.82843	+	6.92820i	−0.103348	+	0.253151i
$750$		0			0
$751$			44.0000i			1.60558i	0.596260	+	0.802791i	$0.296653\pi$
$751$			44.0000i			1.60558i	−0.596260	+	0.802791i	$0.703347\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−20.7846	−	16.9706i	−0.756429	−	0.617622i
$756$		0			0
$757$	39.5980			1.43921			0.719607	−	0.694382i	$-0.244322\pi$
$757$	39.5980			1.43921			0.719607	+	0.694382i	$0.244322\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	19.5959i		−	0.710351i	−0.934800	−	0.355176i	$-0.884421\pi$
$761$		−	19.5959i		−	0.710351i	0.934800	−	0.355176i	$-0.115579\pi$
$762$		0			0
$763$	6.92820	−	16.9706i	0.250818	−	0.614376i
$764$		0			0
$765$	−16.9706	+	20.7846i	−0.613572	+	0.751469i
$766$		0			0
$767$	−29.3939			−1.06135
$768$		0			0
$769$			48.9898i			1.76662i	0.468792	+	0.883309i	$0.344689\pi$
$769$			48.9898i			1.76662i	−0.468792	+	0.883309i	$0.655311\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			10.3923i			0.373785i	0.982380	+	0.186893i	$0.0598416\pi$
$773$			10.3923i			0.373785i	−0.982380	+	0.186893i	$0.940158\pi$
$774$		0			0
$775$	−9.79796	−	48.0000i	−0.351953	−	1.72421i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−27.7128			−0.992915
$780$		0			0
$781$		−	13.8564i		−	0.495821i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	18.0000	+	14.6969i	0.642448	+	0.524556i
$786$		0			0
$787$	−27.7128			−0.987855			−0.493928	−	0.869503i	$-0.664439\pi$
$787$	−27.7128			−0.987855			−0.493928	+	0.869503i	$0.664439\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	9.79796	−	24.0000i	0.348375	−	0.853342i
$792$		0			0
$793$		−	48.9898i		−	1.73968i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	3.46410i		−	0.122705i	−0.998116	−	0.0613524i	$-0.980459\pi$
$797$		−	3.46410i		−	0.122705i	0.998116	−	0.0613524i	$-0.0195413\pi$
$798$		0			0
$799$		−	40.0000i		−	1.41510i
$800$		0			0
$801$		−	29.3939i		−	1.03858i
$802$		0			0
$803$	41.5692			1.46695
$804$		0			0
$805$	−25.4558	+	13.8564i	−0.897201	+	0.488374i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	2.00000			0.0703163			0.0351581	−	0.999382i	$-0.488807\pi$
$809$	2.00000			0.0703163			0.0351581	+	0.999382i	$0.488807\pi$
$810$		0			0
$811$		−	19.7990i		−	0.695237i	−0.937636	−	0.347618i	$-0.886991\pi$
$811$		−	19.7990i		−	0.695237i	0.937636	−	0.347618i	$-0.113009\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	34.2929	+	28.0000i	1.20123	+	0.980797i
$816$		0			0
$817$	−8.00000			−0.279885
$818$		0			0
$819$	10.3923	−	25.4558i	0.363137	−	0.889499i
$820$		0			0
$821$		−	48.4974i		−	1.69257i	−0.532729	−	0.846286i	$-0.678834\pi$
$821$		−	48.4974i		−	1.69257i	0.532729	−	0.846286i	$-0.321166\pi$
$822$		0			0
$823$	−4.89898			−0.170768			−0.0853838	−	0.996348i	$-0.527212\pi$
$823$	−4.89898			−0.170768			−0.0853838	+	0.996348i	$0.527212\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	25.4558i		−	0.885186i	−0.896723	−	0.442593i	$-0.854059\pi$
$827$		−	25.4558i		−	0.885186i	0.896723	−	0.442593i	$-0.145941\pi$
$828$		0			0
$829$	2.82843			0.0982353			0.0491177	−	0.998793i	$-0.484359\pi$
$829$	2.82843			0.0982353			0.0491177	+	0.998793i	$0.484359\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	20.0000	+	19.5959i	0.692959	+	0.678958i
$834$		0			0
$835$	−17.3205	−	14.1421i	−0.599401	−	0.489409i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	−19.0000			−0.655172
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1.41421	−	1.73205i	0.0486504	−	0.0595844i
$846$		0			0
$847$	2.44949	+	1.00000i	0.0841655	+	0.0343604i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−27.7128			−0.949983
$852$		0			0
$853$			38.1051i			1.30469i	0.757920	+	0.652347i	$0.226216\pi$
$853$			38.1051i			1.30469i	−0.757920	+	0.652347i	$0.773784\pi$
$854$		0			0
$855$	−14.6969	−	12.0000i	−0.502625	−	0.410391i
$856$		0			0
$857$	52.0000			1.77629			0.888143	−	0.459567i	$-0.151995\pi$
$857$	52.0000			1.77629			0.888143	+	0.459567i	$0.151995\pi$
$858$		0			0
$859$			53.7401i			1.83359i	0.399359	+	0.916795i	$0.369233\pi$
$859$			53.7401i			1.83359i	−0.399359	+	0.916795i	$0.630767\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	34.2929			1.16734			0.583671	−	0.811990i	$-0.301616\pi$
$863$	34.2929			1.16734			0.583671	+	0.811990i	$0.301616\pi$
$864$		0			0
$865$	30.0000	+	24.4949i	1.02003	+	0.832851i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	41.5692i		−	1.41014i
$870$		0			0
$871$	48.9898			1.65996
$872$		0			0
$873$	−12.0000			−0.406138
$874$		0			0
$875$	−19.0526	−	22.6274i	−0.644094	−	0.764946i
$876$		0			0
$877$	28.2843			0.955092			0.477546	−	0.878607i	$-0.341526\pi$
$877$	28.2843			0.955092			0.477546	+	0.878607i	$0.341526\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			29.3939i			0.990305i	0.868806	+	0.495152i	$0.164888\pi$
$881$			29.3939i			0.990305i	−0.868806	+	0.495152i	$0.835112\pi$
$882$		0			0
$883$			14.1421i			0.475921i	0.971275	+	0.237960i	$0.0764788\pi$
$883$			14.1421i			0.475921i	−0.971275	+	0.237960i	$0.923521\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			26.0000i			0.872995i	0.899706	+	0.436497i	$0.143781\pi$
$887$			26.0000i			0.872995i	−0.899706	+	0.436497i	$0.856219\pi$
$888$		0			0
$889$	−12.0000	−	4.89898i	−0.402467	−	0.164306i
$890$		0			0
$891$	31.1769			1.04447
$892$		0			0
$893$	28.2843			0.946497
$894$		0			0
$895$	14.6969	−	18.0000i	0.491264	−	0.601674i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			67.8823i			2.26400i
$900$		0			0
$901$	−45.2548			−1.50766
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−4.00000	+	4.89898i	−0.132964	+	0.162848i
$906$		0			0
$907$		−	2.82843i		−	0.0939164i	−0.998897	−	0.0469582i	$-0.985047\pi$
$907$		−	2.82843i		−	0.0939164i	0.998897	−	0.0469582i	$-0.0149528\pi$
$908$		0			0
$909$	−25.4558			−0.844317
$910$		0			0
$911$		−	20.0000i		−	0.662630i	−0.943520	−	0.331315i	$-0.892508\pi$
$911$		−	20.0000i		−	0.662630i	0.943520	−	0.331315i	$-0.107492\pi$
$912$		0			0
$913$	48.0000			1.58857
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−2.82843	+	6.92820i	−0.0934029	+	0.228789i
$918$		0			0
$919$		−	20.0000i		−	0.659739i	−0.944027	−	0.329870i	$-0.892995\pi$
$919$		−	20.0000i		−	0.659739i	0.944027	−	0.329870i	$-0.107005\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	13.8564			0.456089
$924$		0			0
$925$	−5.65685	−	27.7128i	−0.185996	−	0.911192i
$926$		0			0
$927$			18.0000i			0.591198i
$928$		0			0
$929$			19.5959i			0.642921i	0.946923	+	0.321461i	$0.104174\pi$
$929$			19.5959i			0.642921i	−0.946923	+	0.321461i	$0.895826\pi$
$930$		0			0
$931$	−13.8564	+	14.1421i	−0.454125	+	0.463490i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	19.5959	−	24.0000i	0.640855	−	0.784884i
$936$		0			0
$937$	28.0000			0.914720			0.457360	−	0.889282i	$-0.348795\pi$
$937$	28.0000			0.914720			0.457360	+	0.889282i	$0.348795\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	25.4558			0.829837			0.414918	−	0.909859i	$-0.363810\pi$
$941$	25.4558			0.829837			0.414918	+	0.909859i	$0.363810\pi$
$942$		0			0
$943$		−	48.0000i		−	1.56310i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			14.1421i			0.459558i	0.973243	+	0.229779i	$0.0738003\pi$
$947$			14.1421i			0.459558i	−0.973243	+	0.229779i	$0.926200\pi$
$948$		0			0
$949$			41.5692i			1.34939i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	58.7878i		−	1.90432i	−0.305597	−	0.952161i	$-0.598856\pi$
$953$		−	58.7878i		−	1.90432i	0.305597	−	0.952161i	$-0.401144\pi$
$954$		0			0
$955$	34.6410	+	28.2843i	1.12096	+	0.915258i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	65.0000			2.09677
$962$		0			0
$963$		−	8.48528i		−	0.273434i
$964$		0			0
$965$	−33.9411	−	27.7128i	−1.09260	−	0.892107i
$966$		0			0
$967$	4.89898			0.157541			0.0787703	−	0.996893i	$-0.474901\pi$
$967$	4.89898			0.157541			0.0787703	+	0.996893i	$0.474901\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			2.82843i			0.0907685i	0.998970	+	0.0453843i	$0.0144512\pi$
$971$			2.82843i			0.0907685i	−0.998970	+	0.0453843i	$0.985549\pi$
$972$		0			0
$973$	2.82843	−	6.92820i	0.0906752	−	0.222108i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	29.3939i		−	0.940393i	−0.882562	−	0.470197i	$-0.844183\pi$
$977$		−	29.3939i		−	0.940393i	0.882562	−	0.470197i	$-0.155817\pi$
$978$		0			0
$979$			33.9411i			1.08476i
$980$		0			0
$981$			20.7846i			0.663602i
$982$		0			0
$983$		−	26.0000i		−	0.829271i	−0.909988	−	0.414636i	$-0.863909\pi$
$983$		−	26.0000i		−	0.829271i	0.909988	−	0.414636i	$-0.136091\pi$
$984$		0			0
$985$	−16.0000	+	19.5959i	−0.509802	+	0.624378i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	13.8564i		−	0.440608i
$990$		0			0
$991$		−	44.0000i		−	1.39771i	−0.715265	−	0.698853i	$-0.753694\pi$
$991$		−	44.0000i		−	1.39771i	0.715265	−	0.698853i	$-0.246306\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	27.7128	−	33.9411i	0.878555	−	1.07601i
$996$		0			0
$997$			45.0333i			1.42622i	0.701052	+	0.713110i	$0.252714\pi$
$997$			45.0333i			1.42622i	−0.701052	+	0.713110i	$0.747286\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.2.n.e.1119.6	yes	8
4.3	odd	2	inner	2240.2.n.e.1119.5	yes	8
5.4	even	2		2240.2.n.c.1119.7	yes	8
7.6	odd	2		2240.2.n.c.1119.4	yes	8
8.3	odd	2	inner	2240.2.n.e.1119.3	yes	8
8.5	even	2	inner	2240.2.n.e.1119.4	yes	8
20.19	odd	2		2240.2.n.c.1119.8	yes	8
28.27	even	2		2240.2.n.c.1119.3	yes	8
35.34	odd	2	inner	2240.2.n.e.1119.1	yes	8
40.19	odd	2		2240.2.n.c.1119.2	yes	8
40.29	even	2		2240.2.n.c.1119.1	✓	8
56.13	odd	2		2240.2.n.c.1119.6	yes	8
56.27	even	2		2240.2.n.c.1119.5	yes	8
140.139	even	2	inner	2240.2.n.e.1119.2	yes	8
280.69	odd	2	inner	2240.2.n.e.1119.7	yes	8
280.139	even	2	inner	2240.2.n.e.1119.8	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2240.2.n.c.1119.1	✓	8	40.29	even	2
2240.2.n.c.1119.2	yes	8	40.19	odd	2
2240.2.n.c.1119.3	yes	8	28.27	even	2
2240.2.n.c.1119.4	yes	8	7.6	odd	2
2240.2.n.c.1119.5	yes	8	56.27	even	2
2240.2.n.c.1119.6	yes	8	56.13	odd	2
2240.2.n.c.1119.7	yes	8	5.4	even	2
2240.2.n.c.1119.8	yes	8	20.19	odd	2
2240.2.n.e.1119.1	yes	8	35.34	odd	2	inner
2240.2.n.e.1119.2	yes	8	140.139	even	2	inner
2240.2.n.e.1119.3	yes	8	8.3	odd	2	inner
2240.2.n.e.1119.4	yes	8	8.5	even	2	inner
2240.2.n.e.1119.5	yes	8	4.3	odd	2	inner
2240.2.n.e.1119.6	yes	8	1.1	even	1	trivial
2240.2.n.e.1119.7	yes	8	280.69	odd	2	inner
2240.2.n.e.1119.8	yes	8	280.139	even	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:	\( N \)	\(=\)	\( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2240.n (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.41421 - 1.73205i) q^{5} +(2.44949 + 1.00000i) q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q+(1.41421 - 1.73205i) q^{5} +(2.44949 + 1.00000i) q^{7} -3.00000 q^{9} +3.46410 q^{11} +3.46410i q^{13} +4.00000 q^{17} +2.82843i q^{19} -4.89898 q^{23} +(-1.00000 - 4.89898i) q^{25} +6.92820i q^{29} +9.79796 q^{31} +(5.19615 - 2.82843i) q^{35} +5.65685 q^{37} +9.79796i q^{41} +2.82843i q^{43} +(-4.24264 + 5.19615i) q^{45} -10.0000i q^{47} +(5.00000 + 4.89898i) q^{49} -11.3137 q^{53} +(4.89898 - 6.00000i) q^{55} +8.48528i q^{59} -14.1421 q^{61} +(-7.34847 - 3.00000i) q^{63} +(6.00000 + 4.89898i) q^{65} -14.1421i q^{67} -4.00000i q^{71} +12.0000 q^{73} +(8.48528 + 3.46410i) q^{77} -12.0000i q^{79} +9.00000 q^{81} +13.8564 q^{83} +(5.65685 - 6.92820i) q^{85} +9.79796i q^{89} +(-3.46410 + 8.48528i) q^{91} +(4.89898 + 4.00000i) q^{95} +4.00000 q^{97} -10.3923 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 24 q^{9}+O(q^{10}) \) 8 * q - 24 * q^9	\( 8 q - 24 q^{9} + 32 q^{17} - 8 q^{25} + 40 q^{49} + 48 q^{65} + 96 q^{73} + 72 q^{81} + 32 q^{97}+O(q^{100}) \) 8 * q - 24 * q^9 + 32 * q^17 - 8 * q^25 + 40 * q^49 + 48 * q^65 + 96 * q^73 + 72 * q^81 + 32 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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