LMFDB - Embedded newform 2240.2.k.e.1791.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2240,2,Mod(1791,2240)]

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, 0, 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("2240.1791");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2240 = 2^{6} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2240.k (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$17.8864900528$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.342102016.5
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + x^{6} + 4x^{4} + 4x^{2} + 16 $ x^8 + x^6 + 4x^4 + 4x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 140)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1791.6
Root			$0.599676 - 1.28078i$ of defining polynomial
Character	$\chi$	$=$	2240.1791
Dual form			2240.2.k.e.1791.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.936426 q^{3} +1.00000i q^{5} +(2.60399 + 0.468213i) q^{7} -2.12311 q^{9} +O(q^{10})$	$q+0.936426 q^{3} +1.00000i q^{5} +(2.60399 + 0.468213i) q^{7} -2.12311 q^{9} -2.39871i q^{11} +2.00000i q^{13} +0.936426i q^{15} +7.12311i q^{17} +2.39871 q^{19} +(2.43845 + 0.438447i) q^{21} +5.73384i q^{23} -1.00000 q^{25} -4.79741 q^{27} +2.00000 q^{29} -6.67026 q^{31} -2.24621i q^{33} +(-0.468213 + 2.60399i) q^{35} -2.00000 q^{37} +1.87285i q^{39} -7.12311i q^{41} +7.60669i q^{43} -2.12311i q^{45} +10.0054 q^{47} +(6.56155 + 2.43845i) q^{49} +6.67026i q^{51} -2.00000 q^{53} +2.39871 q^{55} +2.24621 q^{57} +10.9418 q^{59} +2.00000i q^{61} +(-5.52855 - 0.994066i) q^{63} -2.00000 q^{65} +14.2770i q^{67} +5.36932i q^{69} +6.14441i q^{71} +9.36932i q^{73} -0.936426 q^{75} +(1.12311 - 6.24621i) q^{77} -4.27156i q^{79} +1.87689 q^{81} +0.936426 q^{83} -7.12311 q^{85} +1.87285 q^{87} -12.0000i q^{89} +(-0.936426 + 5.20798i) q^{91} -6.24621 q^{93} +2.39871i q^{95} -7.12311i q^{97} +5.09271i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{9}+O(q^{10}) $ 8 * q + 16 * q^9	$ 8 q + 16 q^{9} + 36 q^{21} - 8 q^{25} + 16 q^{29} - 16 q^{37} + 36 q^{49} - 16 q^{53} - 48 q^{57} - 16 q^{65} - 24 q^{77} + 48 q^{81} - 24 q^{85} + 16 q^{93}+O(q^{100}) $ 8 * q + 16 * q^9 + 36 * q^21 - 8 * q^25 + 16 * q^29 - 16 * q^37 + 36 * q^49 - 16 * q^53 - 48 * q^57 - 16 * q^65 - 24 * q^77 + 48 * q^81 - 24 * q^85 + 16 * q^93

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
$3$	0.936426			0.540646			0.270323	−	0.962770i	$-0.412870\pi$
$3$	0.936426			0.540646			0.270323	+	0.962770i	$0.412870\pi$
$4$		0			0
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		0			0
$7$	2.60399	+	0.468213i	0.984217	+	0.176968i
$7$	2.60399	+	0.468213i	0.984217	+	0.176968i
$8$		0			0
$9$	−2.12311			−0.707702
$10$		0			0
$11$		−	2.39871i		−	0.723237i	−0.932326	−	0.361618i	$-0.882224\pi$
$11$		−	2.39871i		−	0.723237i	0.932326	−	0.361618i	$-0.117776\pi$
$12$		0			0
$13$			2.00000i			0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$			2.00000i			0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$		0			0
$15$			0.936426i			0.241784i
$16$		0			0
$17$			7.12311i			1.72761i	0.503829	+	0.863803i	$0.331924\pi$
$17$			7.12311i			1.72761i	−0.503829	+	0.863803i	$0.668076\pi$
$18$		0			0
$19$	2.39871			0.550301			0.275150	−	0.961401i	$-0.411272\pi$
$19$	2.39871			0.550301			0.275150	+	0.961401i	$0.411272\pi$
$20$		0			0
$21$	2.43845	+	0.438447i	0.532113	+	0.0956770i
$22$		0			0
$23$			5.73384i			1.19559i	0.801650	+	0.597794i	$0.203956\pi$
$23$			5.73384i			1.19559i	−0.801650	+	0.597794i	$0.796044\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	−4.79741			−0.923262
$28$		0			0
$29$	2.00000			0.371391			0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000			0.371391			0.185695	+	0.982607i	$0.440546\pi$
$30$		0			0
$31$	−6.67026			−1.19801			−0.599007	−	0.800743i	$-0.704438\pi$
$31$	−6.67026			−1.19801			−0.599007	+	0.800743i	$0.704438\pi$
$32$		0			0
$33$		−	2.24621i		−	0.391015i
$34$		0			0
$35$	−0.468213	+	2.60399i	−0.0791425	+	0.440155i
$36$		0			0
$37$	−2.00000			−0.328798			−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000			−0.328798			−0.164399	+	0.986394i	$0.552568\pi$
$38$		0			0
$39$			1.87285i			0.299896i
$40$		0			0
$41$		−	7.12311i		−	1.11244i	−0.831034	−	0.556221i	$-0.812251\pi$
$41$		−	7.12311i		−	1.11244i	0.831034	−	0.556221i	$-0.187749\pi$
$42$		0			0
$43$			7.60669i			1.16001i	0.814613	+	0.580005i	$0.196949\pi$
$43$			7.60669i			1.16001i	−0.814613	+	0.580005i	$0.803051\pi$
$44$		0			0
$45$		−	2.12311i		−	0.316494i
$46$		0			0
$47$	10.0054			1.45944			0.729719	−	0.683748i	$-0.239651\pi$
$47$	10.0054			1.45944			0.729719	+	0.683748i	$0.239651\pi$
$48$		0			0
$49$	6.56155	+	2.43845i	0.937365	+	0.348350i
$50$		0			0
$51$			6.67026i			0.934024i
$52$		0			0
$53$	−2.00000			−0.274721			−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000			−0.274721			−0.137361	+	0.990521i	$0.543862\pi$
$54$		0			0
$55$	2.39871			0.323441
$56$		0			0
$57$	2.24621			0.297518
$58$		0			0
$59$	10.9418			1.42450			0.712252	−	0.701924i	$-0.247675\pi$
$59$	10.9418			1.42450			0.712252	+	0.701924i	$0.247675\pi$
$60$		0			0
$61$			2.00000i			0.256074i	0.991769	+	0.128037i	$0.0408676\pi$
$61$			2.00000i			0.256074i	−0.991769	+	0.128037i	$0.959132\pi$
$62$		0			0
$63$	−5.52855	−	0.994066i	−0.696532	−	0.125241i
$64$		0			0
$65$	−2.00000			−0.248069
$66$		0			0
$67$			14.2770i			1.74421i	0.489321	+	0.872104i	$0.337245\pi$
$67$			14.2770i			1.74421i	−0.489321	+	0.872104i	$0.662755\pi$
$68$		0			0
$69$			5.36932i			0.646390i
$70$		0			0
$71$			6.14441i			0.729207i	0.931163	+	0.364604i	$0.118796\pi$
$71$			6.14441i			0.729207i	−0.931163	+	0.364604i	$0.881204\pi$
$72$		0			0
$73$			9.36932i			1.09660i	0.836283	+	0.548298i	$0.184724\pi$
$73$			9.36932i			1.09660i	−0.836283	+	0.548298i	$0.815276\pi$
$74$		0			0
$75$	−0.936426			−0.108129
$76$		0			0
$77$	1.12311	−	6.24621i	0.127990	−	0.711822i
$78$		0			0
$79$		−	4.27156i		−	0.480588i	−0.970700	−	0.240294i	$-0.922756\pi$
$79$		−	4.27156i		−	0.480588i	0.970700	−	0.240294i	$-0.0772438\pi$
$80$		0			0
$81$	1.87689			0.208544
$82$		0			0
$83$	0.936426			0.102786			0.0513931	−	0.998679i	$-0.483634\pi$
$83$	0.936426			0.102786			0.0513931	+	0.998679i	$0.483634\pi$
$84$		0			0
$85$	−7.12311			−0.772609
$86$		0			0
$87$	1.87285			0.200791
$88$		0			0
$89$		−	12.0000i		−	1.27200i	−0.771690	−	0.635999i	$-0.780588\pi$
$89$		−	12.0000i		−	1.27200i	0.771690	−	0.635999i	$-0.219412\pi$
$90$		0			0
$91$	−0.936426	+	5.20798i	−0.0981642	+	0.545945i
$92$		0			0
$93$	−6.24621			−0.647702
$94$		0			0
$95$			2.39871i			0.246102i
$96$		0			0
$97$		−	7.12311i		−	0.723242i	−0.932325	−	0.361621i	$-0.882223\pi$
$97$		−	7.12311i		−	0.723242i	0.932325	−	0.361621i	$-0.117777\pi$
$98$		0			0
$99$			5.09271i			0.511836i
$100$		0			0
$101$		−	6.87689i		−	0.684277i	−0.939650	−	0.342138i	$-0.888849\pi$
$101$		−	6.87689i		−	0.684277i	0.939650	−	0.342138i	$-0.111151\pi$
$102$		0			0
$103$	−1.46228			−0.144083			−0.0720413	−	0.997402i	$-0.522951\pi$
$103$	−1.46228			−0.144083			−0.0720413	+	0.997402i	$0.522951\pi$
$104$		0			0
$105$	−0.438447	+	2.43845i	−0.0427881	+	0.237968i
$106$		0			0
$107$		−	9.47954i		−	0.916422i	−0.888843	−	0.458211i	$-0.848490\pi$
$107$		−	9.47954i		−	0.916422i	0.888843	−	0.458211i	$-0.151510\pi$
$108$		0			0
$109$	−1.12311			−0.107574			−0.0537870	−	0.998552i	$-0.517129\pi$
$109$	−1.12311			−0.107574			−0.0537870	+	0.998552i	$0.517129\pi$
$110$		0			0
$111$	−1.87285			−0.177763
$112$		0			0
$113$	−14.4924			−1.36333			−0.681666	−	0.731663i	$-0.738744\pi$
$113$	−14.4924			−1.36333			−0.681666	+	0.731663i	$0.738744\pi$
$114$		0			0
$115$	−5.73384			−0.534683
$116$		0			0
$117$		−	4.24621i		−	0.392562i
$118$		0			0
$119$	−3.33513	+	18.5485i	−0.305731	+	1.70034i
$120$		0			0
$121$	5.24621			0.476928
$122$		0			0
$123$		−	6.67026i		−	0.601437i
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$			13.2252i			1.17355i	0.809750	+	0.586776i	$0.199603\pi$
$127$			13.2252i			1.17355i	−0.809750	+	0.586776i	$0.800397\pi$
$128$		0			0
$129$			7.12311i			0.627154i
$130$		0			0
$131$	−13.8664			−1.21151			−0.605756	−	0.795651i	$-0.707129\pi$
$131$	−13.8664			−1.21151			−0.605756	+	0.795651i	$0.707129\pi$
$132$		0			0
$133$	6.24621	+	1.12311i	0.541615	+	0.0973856i
$134$		0			0
$135$		−	4.79741i		−	0.412895i
$136$		0			0
$137$	−14.0000			−1.19610			−0.598050	−	0.801459i	$-0.704058\pi$
$137$	−14.0000			−1.19610			−0.598050	+	0.801459i	$0.704058\pi$
$138$		0			0
$139$	1.34700			0.114251			0.0571255	−	0.998367i	$-0.481806\pi$
$139$	1.34700			0.114251			0.0571255	+	0.998367i	$0.481806\pi$
$140$		0			0
$141$	9.36932			0.789039
$142$		0			0
$143$	4.79741			0.401180
$144$		0			0
$145$			2.00000i			0.166091i
$146$		0			0
$147$	6.14441	+	2.28343i	0.506782	+	0.188334i
$148$		0			0
$149$	19.3693			1.58680			0.793398	−	0.608703i	$-0.208310\pi$
$149$	19.3693			1.58680			0.793398	+	0.608703i	$0.208310\pi$
$150$		0			0
$151$			14.6875i			1.19525i	0.801774	+	0.597627i	$0.203890\pi$
$151$			14.6875i			1.19525i	−0.801774	+	0.597627i	$0.796110\pi$
$152$		0			0
$153$		−	15.1231i		−	1.22263i
$154$		0			0
$155$		−	6.67026i		−	0.535769i
$156$		0			0
$157$			16.2462i			1.29659i	0.761390	+	0.648294i	$0.224517\pi$
$157$			16.2462i			1.29659i	−0.761390	+	0.648294i	$0.775483\pi$
$158$		0			0
$159$	−1.87285			−0.148527
$160$		0			0
$161$	−2.68466	+	14.9309i	−0.211581	+	1.17672i
$162$		0			0
$163$			0.936426i			0.0733466i	0.999327	+	0.0366733i	$0.0116761\pi$
$163$			0.936426i			0.0733466i	−0.999327	+	0.0366733i	$0.988324\pi$
$164$		0			0
$165$	2.24621			0.174867
$166$		0			0
$167$	2.28343			0.176697			0.0883484	−	0.996090i	$-0.471841\pi$
$167$	2.28343			0.176697			0.0883484	+	0.996090i	$0.471841\pi$
$168$		0			0
$169$	9.00000			0.692308
$170$		0			0
$171$	−5.09271			−0.389449
$172$		0			0
$173$			0.246211i			0.0187191i	0.999956	+	0.00935955i	$0.00297928\pi$
$173$			0.246211i			0.0187191i	−0.999956	+	0.00935955i	$0.997021\pi$
$174$		0			0
$175$	−2.60399	−	0.468213i	−0.196843	−	0.0353936i
$176$		0			0
$177$	10.2462			0.770152
$178$		0			0
$179$			0.525853i			0.0393041i	0.999807	+	0.0196520i	$0.00625584\pi$
$179$			0.525853i			0.0393041i	−0.999807	+	0.0196520i	$0.993744\pi$
$180$		0			0
$181$		−	6.87689i		−	0.511156i	−0.966789	−	0.255578i	$-0.917734\pi$
$181$		−	6.87689i		−	0.511156i	0.966789	−	0.255578i	$-0.0822657\pi$
$182$		0			0
$183$			1.87285i			0.138445i
$184$		0			0
$185$		−	2.00000i		−	0.147043i
$186$		0			0
$187$	17.0862			1.24947
$188$		0			0
$189$	−12.4924	−	2.24621i	−0.908690	−	0.163388i
$190$		0			0
$191$		−	1.34700i		−	0.0974655i	−0.998812	−	0.0487327i	$-0.984482\pi$
$191$		−	1.34700i		−	0.0974655i	0.998812	−	0.0487327i	$-0.0155183\pi$
$192$		0			0
$193$	10.4924			0.755261			0.377631	−	0.925956i	$-0.376739\pi$
$193$	10.4924			0.755261			0.377631	+	0.925956i	$0.376739\pi$
$194$		0			0
$195$	−1.87285			−0.134118
$196$		0			0
$197$	−18.0000			−1.28245			−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000			−1.28245			−0.641223	+	0.767354i	$0.721573\pi$
$198$		0			0
$199$	16.0345			1.13666			0.568329	−	0.822802i	$-0.307590\pi$
$199$	16.0345			1.13666			0.568329	+	0.822802i	$0.307590\pi$
$200$		0			0
$201$			13.3693i			0.942999i
$202$		0			0
$203$	5.20798	+	0.936426i	0.365529	+	0.0657242i
$204$		0			0
$205$	7.12311			0.497499
$206$		0			0
$207$		−	12.1735i		−	0.846120i
$208$		0			0
$209$		−	5.75379i		−	0.397998i
$210$		0			0
$211$			18.4332i			1.26900i	0.772924	+	0.634498i	$0.218793\pi$
$211$			18.4332i			1.26900i	−0.772924	+	0.634498i	$0.781207\pi$
$212$		0			0
$213$			5.75379i			0.394243i
$214$		0			0
$215$	−7.60669			−0.518772
$216$		0			0
$217$	−17.3693	−	3.12311i	−1.17911	−	0.212010i
$218$		0			0
$219$			8.77368i			0.592870i
$220$		0			0
$221$	−14.2462			−0.958304
$222$		0			0
$223$	12.9300			0.865854			0.432927	−	0.901429i	$-0.357481\pi$
$223$	12.9300			0.865854			0.432927	+	0.901429i	$0.357481\pi$
$224$		0			0
$225$	2.12311			0.141540
$226$		0			0
$227$	14.2770			0.947595			0.473797	−	0.880634i	$-0.342883\pi$
$227$	14.2770			0.947595			0.473797	+	0.880634i	$0.342883\pi$
$228$		0			0
$229$		−	21.1231i		−	1.39585i	−0.716169	−	0.697927i	$-0.754106\pi$
$229$		−	21.1231i		−	1.39585i	0.716169	−	0.697927i	$-0.245894\pi$
$230$		0			0
$231$	1.05171	−	5.84912i	0.0691972	−	0.384844i
$232$		0			0
$233$	24.2462			1.58842			0.794211	−	0.607642i	$-0.207884\pi$
$233$	24.2462			1.58842			0.794211	+	0.607642i	$0.207884\pi$
$234$		0			0
$235$			10.0054i			0.652680i
$236$		0			0
$237$		−	4.00000i		−	0.259828i
$238$		0			0
$239$		−	12.8147i		−	0.828912i	−0.910069	−	0.414456i	$-0.863972\pi$
$239$		−	12.8147i		−	0.828912i	0.910069	−	0.414456i	$-0.136028\pi$
$240$		0			0
$241$		−	15.6155i		−	1.00588i	−0.864320	−	0.502942i	$-0.832251\pi$
$241$		−	15.6155i		−	1.00588i	0.864320	−	0.502942i	$-0.167749\pi$
$242$		0			0
$243$	16.1498			1.03601
$244$		0			0
$245$	−2.43845	+	6.56155i	−0.155787	+	0.419202i
$246$		0			0
$247$			4.79741i			0.305252i
$248$		0			0
$249$	0.876894			0.0555709
$250$		0			0
$251$	−16.5604			−1.04528			−0.522641	−	0.852553i	$-0.675053\pi$
$251$	−16.5604			−1.04528			−0.522641	+	0.852553i	$0.675053\pi$
$252$		0			0
$253$	13.7538			0.864693
$254$		0			0
$255$	−6.67026			−0.417708
$256$		0			0
$257$		−	13.3693i		−	0.833955i	−0.908917	−	0.416978i	$-0.863089\pi$
$257$		−	13.3693i		−	0.833955i	0.908917	−	0.416978i	$-0.136911\pi$
$258$		0			0
$259$	−5.20798	−	0.936426i	−0.323608	−	0.0581867i
$260$		0			0
$261$	−4.24621			−0.262834
$262$		0			0
$263$		−	1.98813i		−	0.122593i	−0.998120	−	0.0612967i	$-0.980476\pi$
$263$		−	1.98813i		−	0.122593i	0.998120	−	0.0612967i	$-0.0195236\pi$
$264$		0			0
$265$		−	2.00000i		−	0.122859i
$266$		0			0
$267$		−	11.2371i		−	0.687700i
$268$		0			0
$269$			16.2462i			0.990549i	0.868737	+	0.495274i	$0.164933\pi$
$269$			16.2462i			0.990549i	−0.868737	+	0.495274i	$0.835067\pi$
$270$		0			0
$271$	−23.7565			−1.44310			−0.721552	−	0.692360i	$-0.756571\pi$
$271$	−23.7565			−1.44310			−0.721552	+	0.692360i	$0.756571\pi$
$272$		0			0
$273$	−0.876894	+	4.87689i	−0.0530721	+	0.295163i
$274$		0			0
$275$			2.39871i			0.144647i
$276$		0			0
$277$	4.24621			0.255130			0.127565	−	0.991830i	$-0.459284\pi$
$277$	4.24621			0.255130			0.127565	+	0.991830i	$0.459284\pi$
$278$		0			0
$279$	14.1617			0.847837
$280$		0			0
$281$	−4.63068			−0.276243			−0.138122	−	0.990415i	$-0.544107\pi$
$281$	−4.63068			−0.276243			−0.138122	+	0.990415i	$0.544107\pi$
$282$		0			0
$283$	−24.6929			−1.46784			−0.733921	−	0.679235i	$-0.762312\pi$
$283$	−24.6929			−1.46784			−0.733921	+	0.679235i	$0.762312\pi$
$284$		0			0
$285$			2.24621i			0.133054i
$286$		0			0
$287$	3.33513	−	18.5485i	0.196867	−	1.09488i
$288$		0			0
$289$	−33.7386			−1.98463
$290$		0			0
$291$		−	6.67026i		−	0.391018i
$292$		0			0
$293$		−	32.2462i		−	1.88384i	−0.335832	−	0.941922i	$-0.609017\pi$
$293$		−	32.2462i		−	1.88384i	0.335832	−	0.941922i	$-0.390983\pi$
$294$		0			0
$295$			10.9418i			0.637058i
$296$		0			0
$297$			11.5076i			0.667737i
$298$		0			0
$299$	−11.4677			−0.663193
$300$		0			0
$301$	−3.56155	+	19.8078i	−0.205284	+	1.14170i
$302$		0			0
$303$		−	6.43971i		−	0.369951i
$304$		0			0
$305$	−2.00000			−0.114520
$306$		0			0
$307$	31.3632			1.78999			0.894996	−	0.446074i	$-0.147178\pi$
$307$	31.3632			1.78999			0.894996	+	0.446074i	$0.147178\pi$
$308$		0			0
$309$	−1.36932			−0.0778977
$310$		0			0
$311$	−9.36426			−0.530999			−0.265499	−	0.964111i	$-0.585537\pi$
$311$	−9.36426			−0.530999			−0.265499	+	0.964111i	$0.585537\pi$
$312$		0			0
$313$			7.61553i			0.430455i	0.976564	+	0.215228i	$0.0690493\pi$
$313$			7.61553i			0.430455i	−0.976564	+	0.215228i	$0.930951\pi$
$314$		0			0
$315$	0.994066	−	5.52855i	0.0560093	−	0.311499i
$316$		0			0
$317$	4.24621			0.238491			0.119245	−	0.992865i	$-0.461952\pi$
$317$	4.24621			0.238491			0.119245	+	0.992865i	$0.461952\pi$
$318$		0			0
$319$		−	4.79741i		−	0.268603i
$320$		0			0
$321$		−	8.87689i		−	0.495460i
$322$		0			0
$323$			17.0862i			0.950703i
$324$		0			0
$325$		−	2.00000i		−	0.110940i
$326$		0			0
$327$	−1.05171			−0.0581595
$328$		0			0
$329$	26.0540	+	4.68466i	1.43640	+	0.258274i
$330$		0			0
$331$		−	29.0798i		−	1.59837i	−0.601086	−	0.799184i	$-0.705265\pi$
$331$		−	29.0798i		−	1.59837i	0.601086	−	0.799184i	$-0.294735\pi$
$332$		0			0
$333$	4.24621			0.232691
$334$		0			0
$335$	−14.2770			−0.780033
$336$		0			0
$337$	4.24621			0.231306			0.115653	−	0.993290i	$-0.463104\pi$
$337$	4.24621			0.231306			0.115653	+	0.993290i	$0.463104\pi$
$338$		0			0
$339$	−13.5711			−0.737080
$340$		0			0
$341$			16.0000i			0.866449i
$342$		0			0
$343$	15.9445	+	9.42190i	0.860923	+	0.508735i
$344$		0			0
$345$	−5.36932			−0.289074
$346$		0			0
$347$		−	9.47954i		−	0.508889i	−0.967087	−	0.254444i	$-0.918107\pi$
$347$		−	9.47954i		−	0.508889i	0.967087	−	0.254444i	$-0.0818925\pi$
$348$		0			0
$349$		−	27.8617i		−	1.49140i	−0.666279	−	0.745702i	$-0.732114\pi$
$349$		−	27.8617i		−	1.49140i	0.666279	−	0.745702i	$-0.267886\pi$
$350$		0			0
$351$		−	9.59482i		−	0.512134i
$352$		0			0
$353$		−	5.36932i		−	0.285780i	−0.989739	−	0.142890i	$-0.954361\pi$
$353$		−	5.36932i		−	0.285780i	0.989739	−	0.142890i	$-0.0456395\pi$
$354$		0			0
$355$	−6.14441			−0.326111
$356$		0			0
$357$	−3.12311	+	17.3693i	−0.165292	+	0.919282i
$358$		0			0
$359$		−	16.5604i		−	0.874023i	−0.899456	−	0.437012i	$-0.856037\pi$
$359$		−	16.5604i		−	0.874023i	0.899456	−	0.437012i	$-0.143963\pi$
$360$		0			0
$361$	−13.2462			−0.697169
$362$		0			0
$363$	4.91269			0.257849
$364$		0			0
$365$	−9.36932			−0.490412
$366$		0			0
$367$	7.08084			0.369617			0.184808	−	0.982775i	$-0.440834\pi$
$367$	7.08084			0.369617			0.184808	+	0.982775i	$0.440834\pi$
$368$		0			0
$369$			15.1231i			0.787277i
$370$		0			0
$371$	−5.20798	−	0.936426i	−0.270385	−	0.0486168i
$372$		0			0
$373$	−22.4924			−1.16461			−0.582307	−	0.812969i	$-0.697850\pi$
$373$	−22.4924			−1.16461			−0.582307	+	0.812969i	$0.697850\pi$
$374$		0			0
$375$		−	0.936426i		−	0.0483569i
$376$		0			0
$377$			4.00000i			0.206010i
$378$		0			0
$379$		−	22.4095i		−	1.15110i	−0.817767	−	0.575549i	$-0.804788\pi$
$379$		−	22.4095i		−	1.15110i	0.817767	−	0.575549i	$-0.195212\pi$
$380$		0			0
$381$			12.3845i			0.634476i
$382$		0			0
$383$	17.4968			0.894045			0.447023	−	0.894523i	$-0.352484\pi$
$383$	17.4968			0.894045			0.447023	+	0.894523i	$0.352484\pi$
$384$		0			0
$385$	6.24621	+	1.12311i	0.318336	+	0.0572388i
$386$		0			0
$387$		−	16.1498i		−	0.820941i
$388$		0			0
$389$	−1.12311			−0.0569437			−0.0284719	−	0.999595i	$-0.509064\pi$
$389$	−1.12311			−0.0569437			−0.0284719	+	0.999595i	$0.509064\pi$
$390$		0			0
$391$	−40.8427			−2.06551
$392$		0			0
$393$	−12.9848			−0.654999
$394$		0			0
$395$	4.27156			0.214925
$396$		0			0
$397$		−	14.0000i		−	0.702640i	−0.936255	−	0.351320i	$-0.885733\pi$
$397$		−	14.0000i		−	0.702640i	0.936255	−	0.351320i	$-0.114267\pi$
$398$		0			0
$399$	5.84912	+	1.05171i	0.292822	+	0.0526511i
$400$		0			0
$401$	6.00000			0.299626			0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000			0.299626			0.149813	+	0.988714i	$0.452133\pi$
$402$		0			0
$403$		−	13.3405i		−	0.664539i
$404$		0			0
$405$			1.87689i			0.0932636i
$406$		0			0
$407$			4.79741i			0.237799i
$408$		0			0
$409$			8.87689i			0.438934i	0.975620	+	0.219467i	$0.0704319\pi$
$409$			8.87689i			0.438934i	−0.975620	+	0.219467i	$0.929568\pi$
$410$		0			0
$411$	−13.1100			−0.646667
$412$		0			0
$413$	28.4924	+	5.12311i	1.40202	+	0.252092i
$414$		0			0
$415$			0.936426i			0.0459674i
$416$		0			0
$417$	1.26137			0.0617694
$418$		0			0
$419$	11.9935			0.585922			0.292961	−	0.956124i	$-0.405359\pi$
$419$	11.9935			0.585922			0.292961	+	0.956124i	$0.405359\pi$
$420$		0			0
$421$	17.6155			0.858528			0.429264	−	0.903179i	$-0.358773\pi$
$421$	17.6155			0.858528			0.429264	+	0.903179i	$0.358773\pi$
$422$		0			0
$423$	−21.2425			−1.03285
$424$		0			0
$425$		−	7.12311i		−	0.345521i
$426$		0			0
$427$	−0.936426	+	5.20798i	−0.0453168	+	0.252032i
$428$		0			0
$429$	4.49242			0.216896
$430$		0			0
$431$		−	36.5712i		−	1.76157i	−0.473515	−	0.880786i	$-0.657015\pi$
$431$		−	36.5712i		−	1.76157i	0.473515	−	0.880786i	$-0.342985\pi$
$432$		0			0
$433$			11.6155i			0.558207i	0.960261	+	0.279103i	$0.0900372\pi$
$433$			11.6155i			0.558207i	−0.960261	+	0.279103i	$0.909963\pi$
$434$		0			0
$435$			1.87285i			0.0897964i
$436$		0			0
$437$			13.7538i			0.657933i
$438$		0			0
$439$	18.1379			0.865677			0.432838	−	0.901472i	$-0.357512\pi$
$439$	18.1379			0.865677			0.432838	+	0.901472i	$0.357512\pi$
$440$		0			0
$441$	−13.9309	−	5.17708i	−0.663375	−	0.246528i
$442$		0			0
$443$		−	30.3115i		−	1.44014i	−0.693900	−	0.720071i	$-0.744109\pi$
$443$		−	30.3115i		−	1.44014i	0.693900	−	0.720071i	$-0.255891\pi$
$444$		0			0
$445$	12.0000			0.568855
$446$		0			0
$447$	18.1379			0.857895
$448$		0			0
$449$	25.6155			1.20887			0.604436	−	0.796654i	$-0.293399\pi$
$449$	25.6155			1.20887			0.604436	+	0.796654i	$0.293399\pi$
$450$		0			0
$451$	−17.0862			−0.804559
$452$		0			0
$453$			13.7538i			0.646209i
$454$		0			0
$455$	−5.20798	−	0.936426i	−0.244154	−	0.0439003i
$456$		0			0
$457$	6.00000			0.280668			0.140334	−	0.990104i	$-0.455182\pi$
$457$	6.00000			0.280668			0.140334	+	0.990104i	$0.455182\pi$
$458$		0			0
$459$		−	34.1725i		−	1.59503i
$460$		0			0
$461$		−	37.6155i		−	1.75193i	−0.482375	−	0.875965i	$-0.660226\pi$
$461$		−	37.6155i		−	1.75193i	0.482375	−	0.875965i	$-0.339774\pi$
$462$		0			0
$463$		−	21.9989i		−	1.02238i	−0.859469	−	0.511188i	$-0.829205\pi$
$463$		−	21.9989i		−	1.02238i	0.859469	−	0.511188i	$-0.170795\pi$
$464$		0			0
$465$		−	6.24621i		−	0.289661i
$466$		0			0
$467$	−1.98813			−0.0919998			−0.0459999	−	0.998941i	$-0.514647\pi$
$467$	−1.98813			−0.0919998			−0.0459999	+	0.998941i	$0.514647\pi$
$468$		0			0
$469$	−6.68466	+	37.1771i	−0.308669	+	1.71668i
$470$		0			0
$471$			15.2134i			0.700996i
$472$		0			0
$473$	18.2462			0.838962
$474$		0			0
$475$	−2.39871			−0.110060
$476$		0			0
$477$	4.24621			0.194421
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		−	4.00000i		−	0.182384i
$482$		0			0
$483$	−2.51398	+	13.9817i	−0.114390	+	0.636188i
$484$		0			0
$485$	7.12311			0.323444
$486$		0			0
$487$			19.0744i			0.864342i	0.901792	+	0.432171i	$0.142252\pi$
$487$			19.0744i			0.864342i	−0.901792	+	0.432171i	$0.857748\pi$
$488$		0			0
$489$			0.876894i			0.0396545i
$490$		0			0
$491$		−	13.6358i		−	0.615376i	−0.951487	−	0.307688i	$-0.900445\pi$
$491$		−	13.6358i		−	0.615376i	0.951487	−	0.307688i	$-0.0995553\pi$
$492$		0			0
$493$			14.2462i			0.641617i
$494$		0			0
$495$	−5.09271			−0.228900
$496$		0			0
$497$	−2.87689	+	16.0000i	−0.129046	+	0.717698i
$498$		0			0
$499$			27.2069i			1.21795i	0.793190	+	0.608974i	$0.208419\pi$
$499$			27.2069i			1.21795i	−0.793190	+	0.608974i	$0.791581\pi$
$500$		0			0
$501$	2.13826			0.0955304
$502$		0			0
$503$	−21.4731			−0.957437			−0.478718	−	0.877968i	$-0.658899\pi$
$503$	−21.4731			−0.957437			−0.478718	+	0.877968i	$0.658899\pi$
$504$		0			0
$505$	6.87689			0.306018
$506$		0			0
$507$	8.42784			0.374293
$508$		0			0
$509$			17.6155i			0.780795i	0.920646	+	0.390397i	$0.127662\pi$
$509$			17.6155i			0.780795i	−0.920646	+	0.390397i	$0.872338\pi$
$510$		0			0
$511$	−4.38684	+	24.3976i	−0.194062	+	1.07929i
$512$		0			0
$513$	−11.5076			−0.508072
$514$		0			0
$515$		−	1.46228i		−	0.0644357i
$516$		0			0
$517$		−	24.0000i		−	1.05552i
$518$		0			0
$519$			0.230559i			0.0101204i
$520$		0			0
$521$			26.2462i			1.14987i	0.818200	+	0.574934i	$0.194972\pi$
$521$			26.2462i			1.14987i	−0.818200	+	0.574934i	$0.805028\pi$
$522$		0			0
$523$	−17.2015			−0.752170			−0.376085	−	0.926585i	$-0.622730\pi$
$523$	−17.2015			−0.752170			−0.376085	+	0.926585i	$0.622730\pi$
$524$		0			0
$525$	−2.43845	−	0.438447i	−0.106423	−	0.0191354i
$526$		0			0
$527$		−	47.5130i		−	2.06970i
$528$		0			0
$529$	−9.87689			−0.429430
$530$		0			0
$531$	−23.2306			−1.00812
$532$		0			0
$533$	14.2462			0.617072
$534$		0			0
$535$	9.47954			0.409836
$536$		0			0
$537$			0.492423i			0.0212496i
$538$		0			0
$539$	5.84912	−	15.7392i	0.251939	−	0.677937i
$540$		0			0
$541$	−30.0000			−1.28980			−0.644900	−	0.764267i	$-0.723101\pi$
$541$	−30.0000			−1.28980			−0.644900	+	0.764267i	$0.723101\pi$
$542$		0			0
$543$		−	6.43971i		−	0.276354i
$544$		0			0
$545$		−	1.12311i		−	0.0481086i
$546$		0			0
$547$			1.75757i			0.0751484i	0.999294	+	0.0375742i	$0.0119631\pi$
$547$			1.75757i			0.0751484i	−0.999294	+	0.0375742i	$0.988037\pi$
$548$		0			0
$549$		−	4.24621i		−	0.181224i
$550$		0			0
$551$	4.79741			0.204377
$552$		0			0
$553$	2.00000	−	11.1231i	0.0850487	−	0.473003i
$554$		0			0
$555$		−	1.87285i		−	0.0794982i
$556$		0			0
$557$	−6.49242			−0.275093			−0.137546	−	0.990495i	$-0.543922\pi$
$557$	−6.49242			−0.275093			−0.137546	+	0.990495i	$0.543922\pi$
$558$		0			0
$559$	−15.2134			−0.643457
$560$		0			0
$561$	16.0000			0.675521
$562$		0			0
$563$	26.7963			1.12933			0.564665	−	0.825320i	$-0.309005\pi$
$563$	26.7963			1.12933			0.564665	+	0.825320i	$0.309005\pi$
$564$		0			0
$565$		−	14.4924i		−	0.609701i
$566$		0			0
$567$	4.88742	+	0.878787i	0.205252	+	0.0369056i
$568$		0			0
$569$	−37.1231			−1.55628			−0.778141	−	0.628090i	$-0.783837\pi$
$569$	−37.1231			−1.55628			−0.778141	+	0.628090i	$0.783837\pi$
$570$		0			0
$571$			28.0281i			1.17294i	0.809972	+	0.586469i	$0.199482\pi$
$571$			28.0281i			1.17294i	−0.809972	+	0.586469i	$0.800518\pi$
$572$		0			0
$573$		−	1.26137i		−	0.0526943i
$574$		0			0
$575$		−	5.73384i		−	0.239118i
$576$		0			0
$577$		−	4.38447i		−	0.182528i	−0.995827	−	0.0912640i	$-0.970909\pi$
$577$		−	4.38447i		−	0.182528i	0.995827	−	0.0912640i	$-0.0290907\pi$
$578$		0			0
$579$	9.82538			0.408329
$580$		0			0
$581$	2.43845	+	0.438447i	0.101164	+	0.0181899i
$582$		0			0
$583$			4.79741i			0.198688i
$584$		0			0
$585$	4.24621			0.175559
$586$		0			0
$587$	8.65840			0.357370			0.178685	−	0.983906i	$-0.442816\pi$
$587$	8.65840			0.357370			0.178685	+	0.983906i	$0.442816\pi$
$588$		0			0
$589$	−16.0000			−0.659269
$590$		0			0
$591$	−16.8557			−0.693350
$592$		0			0
$593$		−	13.3693i		−	0.549012i	−0.961585	−	0.274506i	$-0.911486\pi$
$593$		−	13.3693i		−	0.549012i	0.961585	−	0.274506i	$-0.0885143\pi$
$594$		0			0
$595$	−18.5485	−	3.33513i	−0.760415	−	0.136727i
$596$		0			0
$597$	15.0152			0.614529
$598$		0			0
$599$		−	10.1207i		−	0.413520i	−0.978392	−	0.206760i	$-0.933708\pi$
$599$		−	10.1207i		−	0.413520i	0.978392	−	0.206760i	$-0.0662919\pi$
$600$		0			0
$601$		−	8.87689i		−	0.362096i	−0.983474	−	0.181048i	$-0.942051\pi$
$601$		−	8.87689i		−	0.362096i	0.983474	−	0.181048i	$-0.0579489\pi$
$602$		0			0
$603$		−	30.3115i		−	1.23438i
$604$		0			0
$605$			5.24621i			0.213289i
$606$		0			0
$607$	7.90198			0.320732			0.160366	−	0.987058i	$-0.448733\pi$
$607$	7.90198			0.320732			0.160366	+	0.987058i	$0.448733\pi$
$608$		0			0
$609$	4.87689	+	0.876894i	0.197622	+	0.0355336i
$610$		0			0
$611$			20.0108i			0.809550i
$612$		0			0
$613$	−11.7538			−0.474731			−0.237366	−	0.971420i	$-0.576284\pi$
$613$	−11.7538			−0.474731			−0.237366	+	0.971420i	$0.576284\pi$
$614$		0			0
$615$	6.67026			0.268971
$616$		0			0
$617$	−26.0000			−1.04672			−0.523360	−	0.852111i	$-0.675322\pi$
$617$	−26.0000			−1.04672			−0.523360	+	0.852111i	$0.675322\pi$
$618$		0			0
$619$	48.8600			1.96385			0.981925	−	0.189273i	$-0.0606131\pi$
$619$	48.8600			1.96385			0.981925	+	0.189273i	$0.0606131\pi$
$620$		0			0
$621$		−	27.5076i		−	1.10384i
$622$		0			0
$623$	5.61856	−	31.2479i	0.225103	−	1.25192i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		−	5.38800i		−	0.215176i
$628$		0			0
$629$		−	14.2462i		−	0.568034i
$630$		0			0
$631$		−	7.19612i		−	0.286473i	−0.989688	−	0.143236i	$-0.954249\pi$
$631$		−	7.19612i		−	0.286473i	0.989688	−	0.143236i	$-0.0457509\pi$
$632$		0			0
$633$			17.2614i			0.686078i
$634$		0			0
$635$	−13.2252			−0.524828
$636$		0			0
$637$	−4.87689	+	13.1231i	−0.193230	+	0.519956i
$638$		0			0
$639$		−	13.0452i		−	0.516061i
$640$		0			0
$641$	1.12311			0.0443600			0.0221800	−	0.999754i	$-0.492939\pi$
$641$	1.12311			0.0443600			0.0221800	+	0.999754i	$0.492939\pi$
$642$		0			0
$643$	−4.91269			−0.193738			−0.0968688	−	0.995297i	$-0.530883\pi$
$643$	−4.91269			−0.193738			−0.0968688	+	0.995297i	$0.530883\pi$
$644$		0			0
$645$	−7.12311			−0.280472
$646$		0			0
$647$	37.7382			1.48364			0.741820	−	0.670599i	$-0.233963\pi$
$647$	37.7382			1.48364			0.741820	+	0.670599i	$0.233963\pi$
$648$		0			0
$649$		−	26.2462i		−	1.03025i
$650$		0			0
$651$	−16.2651	−	2.92456i	−0.637479	−	0.114622i
$652$		0			0
$653$	−42.9848			−1.68213			−0.841063	−	0.540936i	$-0.818070\pi$
$653$	−42.9848			−1.68213			−0.841063	+	0.540936i	$0.818070\pi$
$654$		0			0
$655$		−	13.8664i		−	0.541804i
$656$		0			0
$657$		−	19.8920i		−	0.776063i
$658$		0			0
$659$			4.27156i			0.166396i	0.996533	+	0.0831981i	$0.0265134\pi$
$659$			4.27156i			0.166396i	−0.996533	+	0.0831981i	$0.973487\pi$
$660$		0			0
$661$			4.24621i			0.165158i	0.996585	+	0.0825792i	$0.0263158\pi$
$661$			4.24621i			0.165158i	−0.996585	+	0.0825792i	$0.973684\pi$
$662$		0			0
$663$	−13.3405			−0.518103
$664$		0			0
$665$	−1.12311	+	6.24621i	−0.0435522	+	0.242218i
$666$		0			0
$667$			11.4677i			0.444030i
$668$		0			0
$669$	12.1080			0.468120
$670$		0			0
$671$	4.79741			0.185202
$672$		0			0
$673$	−14.0000			−0.539660			−0.269830	−	0.962908i	$-0.586968\pi$
$673$	−14.0000			−0.539660			−0.269830	+	0.962908i	$0.586968\pi$
$674$		0			0
$675$	4.79741			0.184652
$676$		0			0
$677$			16.7386i			0.643318i	0.946856	+	0.321659i	$0.104240\pi$
$677$			16.7386i			0.643318i	−0.946856	+	0.321659i	$0.895760\pi$
$678$		0			0
$679$	3.33513	−	18.5485i	0.127991	−	0.711827i
$680$		0			0
$681$	13.3693			0.512313
$682$		0			0
$683$			7.60669i			0.291062i	0.989354	+	0.145531i	$0.0464890\pi$
$683$			7.60669i			0.291062i	−0.989354	+	0.145531i	$0.953511\pi$
$684$		0			0
$685$		−	14.0000i		−	0.534913i
$686$		0			0
$687$		−	19.7802i		−	0.754663i
$688$		0			0
$689$		−	4.00000i		−	0.152388i
$690$		0			0
$691$	22.4095			0.852497			0.426249	−	0.904606i	$-0.359835\pi$
$691$	22.4095			0.852497			0.426249	+	0.904606i	$0.359835\pi$
$692$		0			0
$693$	−2.38447	+	13.2614i	−0.0905786	+	0.503758i
$694$		0			0
$695$			1.34700i			0.0510946i
$696$		0			0
$697$	50.7386			1.92186
$698$		0			0
$699$	22.7048			0.858774
$700$		0			0
$701$	−2.87689			−0.108659			−0.0543294	−	0.998523i	$-0.517302\pi$
$701$	−2.87689			−0.108659			−0.0543294	+	0.998523i	$0.517302\pi$
$702$		0			0
$703$	−4.79741			−0.180938
$704$		0			0
$705$			9.36932i			0.352869i
$706$		0			0
$707$	3.21985	−	17.9074i	0.121095	−	0.673476i
$708$		0			0
$709$	4.73863			0.177963			0.0889816	−	0.996033i	$-0.471639\pi$
$709$	4.73863			0.177963			0.0889816	+	0.996033i	$0.471639\pi$
$710$		0			0
$711$			9.06897i			0.340113i
$712$		0			0
$713$		−	38.2462i		−	1.43233i
$714$		0			0
$715$			4.79741i			0.179413i
$716$		0			0
$717$		−	12.0000i		−	0.448148i
$718$		0			0
$719$	−37.9182			−1.41411			−0.707055	−	0.707159i	$-0.749977\pi$
$719$	−37.9182			−1.41411			−0.707055	+	0.707159i	$0.749977\pi$
$720$		0			0
$721$	−3.80776	−	0.684658i	−0.141809	−	0.0254980i
$722$		0			0
$723$		−	14.6228i		−	0.543828i
$724$		0			0
$725$	−2.00000			−0.0742781
$726$		0			0
$727$	22.2942			0.826847			0.413423	−	0.910539i	$-0.364333\pi$
$727$	22.2942			0.826847			0.413423	+	0.910539i	$0.364333\pi$
$728$		0			0
$729$	9.49242			0.351571
$730$		0			0
$731$	−54.1833			−2.00404
$732$		0			0
$733$		−	48.7386i		−	1.80020i	−0.435681	−	0.900101i	$-0.643492\pi$
$733$		−	48.7386i		−	1.80020i	0.435681	−	0.900101i	$-0.356508\pi$
$734$		0			0
$735$	−2.28343	+	6.14441i	−0.0842254	+	0.226640i
$736$		0			0
$737$	34.2462			1.26148
$738$		0			0
$739$			15.5087i			0.570496i	0.958454	+	0.285248i	$0.0920759\pi$
$739$			15.5087i			0.570496i	−0.958454	+	0.285248i	$0.907924\pi$
$740$		0			0
$741$			4.49242i			0.165033i
$742$		0			0
$743$		−	15.0981i		−	0.553896i	−0.960885	−	0.276948i	$-0.910677\pi$
$743$		−	15.0981i		−	0.553896i	0.960885	−	0.276948i	$-0.0893229\pi$
$744$		0			0
$745$			19.3693i			0.709637i
$746$		0			0
$747$	−1.98813			−0.0727420
$748$		0			0
$749$	4.43845	−	24.6847i	0.162177	−	0.901958i
$750$		0			0
$751$			5.09271i			0.185835i	0.995674	+	0.0929177i	$0.0296194\pi$
$751$			5.09271i			0.185835i	−0.995674	+	0.0929177i	$0.970381\pi$
$752$		0			0
$753$	−15.5076			−0.565128
$754$		0			0
$755$	−14.6875			−0.534534
$756$		0			0
$757$	12.2462			0.445096			0.222548	−	0.974922i	$-0.428563\pi$
$757$	12.2462			0.445096			0.222548	+	0.974922i	$0.428563\pi$
$758$		0			0
$759$	12.8794			0.467493
$760$		0			0
$761$		−	35.2311i		−	1.27712i	−0.769570	−	0.638562i	$-0.779529\pi$
$761$		−	35.2311i		−	1.27712i	0.769570	−	0.638562i	$-0.220471\pi$
$762$		0			0
$763$	−2.92456	−	0.525853i	−0.105876	−	0.0190372i
$764$		0			0
$765$	15.1231			0.546777
$766$		0			0
$767$			21.8836i			0.790173i
$768$		0			0
$769$		−	55.2311i		−	1.99168i	−0.0911037	−	0.995841i	$-0.529039\pi$
$769$		−	55.2311i		−	1.99168i	0.0911037	−	0.995841i	$-0.470961\pi$
$770$		0			0
$771$		−	12.5194i		−	0.450874i
$772$		0			0
$773$			16.7386i			0.602047i	0.953617	+	0.301023i	$0.0973282\pi$
$773$			16.7386i			0.602047i	−0.953617	+	0.301023i	$0.902672\pi$
$774$		0			0
$775$	6.67026			0.239603
$776$		0			0
$777$	−4.87689	−	0.876894i	−0.174958	−	0.0314584i
$778$		0			0
$779$		−	17.0862i		−	0.612178i
$780$		0			0
$781$	14.7386			0.527390
$782$		0			0
$783$	−9.59482			−0.342891
$784$		0			0
$785$	−16.2462			−0.579852
$786$		0			0
$787$	0.936426			0.0333800			0.0166900	−	0.999861i	$-0.494687\pi$
$787$	0.936426			0.0333800			0.0166900	+	0.999861i	$0.494687\pi$
$788$		0			0
$789$		−	1.86174i		−	0.0662797i
$790$		0			0
$791$	−37.7382	−	6.78554i	−1.34181	−	0.241266i
$792$		0			0
$793$	−4.00000			−0.142044
$794$		0			0
$795$		−	1.87285i		−	0.0664232i
$796$		0			0
$797$		−	20.2462i		−	0.717158i	−0.933499	−	0.358579i	$-0.883261\pi$
$797$		−	20.2462i		−	0.717158i	0.933499	−	0.358579i	$-0.116739\pi$
$798$		0			0
$799$			71.2695i			2.52133i
$800$		0			0
$801$			25.4773i			0.900195i
$802$		0			0
$803$	22.4742			0.793098
$804$		0			0
$805$	−14.9309	−	2.68466i	−0.526244	−	0.0946218i
$806$		0			0
$807$			15.2134i			0.535536i
$808$		0			0
$809$	−18.9848			−0.667472			−0.333736	−	0.942667i	$-0.608309\pi$
$809$	−18.9848			−0.667472			−0.333736	+	0.942667i	$0.608309\pi$
$810$		0			0
$811$	−9.06897			−0.318455			−0.159227	−	0.987242i	$-0.550900\pi$
$811$	−9.06897			−0.318455			−0.159227	+	0.987242i	$0.550900\pi$
$812$		0			0
$813$	−22.2462			−0.780209
$814$		0			0
$815$	−0.936426			−0.0328016
$816$		0			0
$817$			18.2462i			0.638354i
$818$		0			0
$819$	1.98813	−	11.0571i	0.0694710	−	0.386366i
$820$		0			0
$821$	25.6155			0.893988			0.446994	−	0.894537i	$-0.352495\pi$
$821$	25.6155			0.893988			0.446994	+	0.894537i	$0.352495\pi$
$822$		0			0
$823$			32.1843i			1.12188i	0.827858	+	0.560938i	$0.189559\pi$
$823$			32.1843i			1.12188i	−0.827858	+	0.560938i	$0.810441\pi$
$824$		0			0
$825$			2.24621i			0.0782030i
$826$		0			0
$827$			0.115279i			0.00400866i	0.999998	+	0.00200433i	$0.000637998\pi$
$827$			0.115279i			0.00400866i	−0.999998	+	0.00200433i	$0.999362\pi$
$828$		0			0
$829$			32.2462i			1.11996i	0.828507	+	0.559979i	$0.189191\pi$
$829$			32.2462i			1.11996i	−0.828507	+	0.559979i	$0.810809\pi$
$830$		0			0
$831$	3.97626			0.137935
$832$		0			0
$833$	−17.3693	+	46.7386i	−0.601811	+	1.61940i
$834$		0			0
$835$			2.28343i			0.0790212i
$836$		0			0
$837$	32.0000			1.10608
$838$		0			0
$839$	35.2242			1.21607			0.608037	−	0.793909i	$-0.291957\pi$
$839$	35.2242			1.21607			0.608037	+	0.793909i	$0.291957\pi$
$840$		0			0
$841$	−25.0000			−0.862069
$842$		0			0
$843$	−4.33629			−0.149350
$844$		0			0
$845$			9.00000i			0.309609i
$846$		0			0
$847$	13.6611	+	2.45635i	0.469401	+	0.0844010i
$848$		0			0
$849$	−23.1231			−0.793583
$850$		0			0
$851$		−	11.4677i		−	0.393107i
$852$		0			0
$853$			7.75379i			0.265485i	0.991151	+	0.132742i	$0.0423783\pi$
$853$			7.75379i			0.265485i	−0.991151	+	0.132742i	$0.957622\pi$
$854$		0			0
$855$		−	5.09271i		−	0.174167i
$856$		0			0
$857$			15.6155i			0.533416i	0.963777	+	0.266708i	$0.0859360\pi$
$857$			15.6155i			0.533416i	−0.963777	+	0.266708i	$0.914064\pi$
$858$		0			0
$859$	41.3686			1.41148			0.705739	−	0.708472i	$-0.250615\pi$
$859$	41.3686			1.41148			0.705739	+	0.708472i	$0.250615\pi$
$860$		0			0
$861$	3.12311	−	17.3693i	0.106435	−	0.591945i
$862$		0			0
$863$			41.7792i			1.42218i	0.703101	+	0.711090i	$0.251798\pi$
$863$			41.7792i			1.42218i	−0.703101	+	0.711090i	$0.748202\pi$
$864$		0			0
$865$	−0.246211			−0.00837143
$866$		0			0
$867$	−31.5937			−1.07298
$868$		0			0
$869$	−10.2462			−0.347579
$870$		0			0
$871$	−28.5539			−0.967512
$872$		0			0
$873$			15.1231i			0.511840i
$874$		0			0
$875$	0.468213	−	2.60399i	0.0158285	−	0.0880310i
$876$		0			0
$877$	48.7386			1.64579			0.822893	−	0.568196i	$-0.192358\pi$
$877$	48.7386			1.64579			0.822893	+	0.568196i	$0.192358\pi$
$878$		0			0
$879$		−	30.1962i		−	1.01849i
$880$		0			0
$881$		−	6.63068i		−	0.223393i	−0.993742	−	0.111697i	$-0.964371\pi$
$881$		−	6.63068i		−	0.223393i	0.993742	−	0.111697i	$-0.0356285\pi$
$882$		0			0
$883$			13.4558i			0.452824i	0.974032	+	0.226412i	$0.0726996\pi$
$883$			13.4558i			0.452824i	−0.974032	+	0.226412i	$0.927300\pi$
$884$		0			0
$885$			10.2462i			0.344423i
$886$		0			0
$887$	17.7274			0.595227			0.297613	−	0.954686i	$-0.403809\pi$
$887$	17.7274			0.595227			0.297613	+	0.954686i	$0.403809\pi$
$888$		0			0
$889$	−6.19224	+	34.4384i	−0.207681	+	1.15503i
$890$		0			0
$891$		−	4.50212i		−	0.150827i
$892$		0			0
$893$	24.0000			0.803129
$894$		0			0
$895$	−0.525853			−0.0175773
$896$		0			0
$897$	−10.7386			−0.358553
$898$		0			0
$899$	−13.3405			−0.444932
$900$		0			0
$901$		−	14.2462i		−	0.474610i
$902$		0			0
$903$	−3.33513	+	18.5485i	−0.110986	+	0.617256i
$904$		0			0
$905$	6.87689			0.228596
$906$		0			0
$907$		−	14.5075i		−	0.481714i	−0.970561	−	0.240857i	$-0.922572\pi$
$907$		−	14.5075i		−	0.481714i	0.970561	−	0.240857i	$-0.0774285\pi$
$908$		0			0
$909$			14.6004i			0.484264i
$910$		0			0
$911$			10.9418i			0.362519i	0.983435	+	0.181259i	$0.0580174\pi$
$911$			10.9418i			0.362519i	−0.983435	+	0.181259i	$0.941983\pi$
$912$		0			0
$913$		−	2.24621i		−	0.0743387i
$914$		0			0
$915$	−1.87285			−0.0619146
$916$		0			0
$917$	−36.1080	−	6.49242i	−1.19239	−	0.214399i
$918$		0			0
$919$			29.9009i			0.986340i	0.869933	+	0.493170i	$0.164162\pi$
$919$			29.9009i			0.986340i	−0.869933	+	0.493170i	$0.835838\pi$
$920$		0			0
$921$	29.3693			0.967752
$922$		0			0
$923$	−12.2888			−0.404492
$924$		0			0
$925$	2.00000			0.0657596
$926$		0			0
$927$	3.10457			0.101968
$928$		0			0
$929$			12.8769i			0.422477i	0.977435	+	0.211239i	$0.0677497\pi$
$929$			12.8769i			0.422477i	−0.977435	+	0.211239i	$0.932250\pi$
$930$		0			0
$931$	15.7392	+	5.84912i	0.515833	+	0.191697i
$932$		0			0
$933$	−8.76894			−0.287082
$934$		0			0
$935$			17.0862i			0.558780i
$936$		0			0
$937$		−	36.1080i		−	1.17960i	−0.807551	−	0.589798i	$-0.799207\pi$
$937$		−	36.1080i		−	1.17960i	0.807551	−	0.589798i	$-0.200793\pi$
$938$		0			0
$939$			7.13138i			0.232724i
$940$		0			0
$941$			51.3693i			1.67459i	0.546750	+	0.837296i	$0.315865\pi$
$941$			51.3693i			1.67459i	−0.546750	+	0.837296i	$0.684135\pi$
$942$		0			0
$943$	40.8427			1.33002
$944$		0			0
$945$	2.24621	−	12.4924i	0.0730693	−	0.406379i
$946$		0			0
$947$			18.8438i			0.612341i	0.951977	+	0.306171i	$0.0990478\pi$
$947$			18.8438i			0.612341i	−0.951977	+	0.306171i	$0.900952\pi$
$948$		0			0
$949$	−18.7386			−0.608282
$950$		0			0
$951$	3.97626			0.128939
$952$		0			0
$953$	11.7538			0.380743			0.190371	−	0.981712i	$-0.439031\pi$
$953$	11.7538			0.380743			0.190371	+	0.981712i	$0.439031\pi$
$954$		0			0
$955$	1.34700			0.0435879
$956$		0			0
$957$		−	4.49242i		−	0.145219i
$958$		0			0
$959$	−36.4559	−	6.55498i	−1.17722	−	0.211671i
$960$		0			0
$961$	13.4924			0.435239
$962$		0			0
$963$			20.1261i			0.648554i
$964$		0			0
$965$			10.4924i			0.337763i
$966$		0			0
$967$		−	55.1197i		−	1.77253i	−0.463179	−	0.886265i	$-0.653291\pi$
$967$		−	55.1197i		−	1.77253i	0.463179	−	0.886265i	$-0.346709\pi$
$968$		0			0
$969$			16.0000i			0.513994i
$970$		0			0
$971$	0.525853			0.0168754			0.00843771	−	0.999964i	$-0.497314\pi$
$971$	0.525853			0.0168754			0.00843771	+	0.999964i	$0.497314\pi$
$972$		0			0
$973$	3.50758	+	0.630683i	0.112448	+	0.0202188i
$974$		0			0
$975$		−	1.87285i		−	0.0599793i
$976$		0			0
$977$	−10.4924			−0.335682			−0.167841	−	0.985814i	$-0.553680\pi$
$977$	−10.4924			−0.335682			−0.167841	+	0.985814i	$0.553680\pi$
$978$		0			0
$979$	−28.7845			−0.919956
$980$		0			0
$981$	2.38447			0.0761303
$982$		0			0
$983$	−12.6994			−0.405048			−0.202524	−	0.979277i	$-0.564914\pi$
$983$	−12.6994			−0.405048			−0.202524	+	0.979277i	$0.564914\pi$
$984$		0			0
$985$		−	18.0000i		−	0.573528i
$986$		0			0
$987$	24.3976	+	4.38684i	0.776585	+	0.139635i
$988$		0			0
$989$	−43.6155			−1.38689
$990$		0			0
$991$			34.4678i			1.09490i	0.836837	+	0.547452i	$0.184402\pi$
$991$			34.4678i			1.09490i	−0.836837	+	0.547452i	$0.815598\pi$
$992$		0			0
$993$		−	27.2311i		−	0.864151i
$994$		0			0
$995$			16.0345i			0.508329i
$996$		0			0
$997$		−	38.4924i		−	1.21907i	−0.792760	−	0.609534i	$-0.791357\pi$
$997$		−	38.4924i		−	1.21907i	0.792760	−	0.609534i	$-0.208643\pi$
$998$		0			0
$999$	9.59482			0.303567

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.2.k.e.1791.6		8
4.3	odd	2	inner	2240.2.k.e.1791.4		8
7.6	odd	2	inner	2240.2.k.e.1791.3		8
8.3	odd	2		140.2.g.c.111.8	yes	8
8.5	even	2		140.2.g.c.111.5	✓	8
24.5	odd	2		1260.2.c.c.811.4		8
24.11	even	2		1260.2.c.c.811.2		8
28.27	even	2	inner	2240.2.k.e.1791.5		8
40.3	even	4		700.2.c.j.699.5		8
40.13	odd	4		700.2.c.j.699.3		8
40.19	odd	2		700.2.g.j.251.1		8
40.27	even	4		700.2.c.i.699.4		8
40.29	even	2		700.2.g.j.251.4		8
40.37	odd	4		700.2.c.i.699.6		8
56.3	even	6		980.2.o.e.411.2		16
56.5	odd	6		980.2.o.e.31.1		16
56.11	odd	6		980.2.o.e.411.1		16
56.13	odd	2		140.2.g.c.111.6	yes	8
56.19	even	6		980.2.o.e.31.6		16
56.27	even	2		140.2.g.c.111.7	yes	8
56.37	even	6		980.2.o.e.31.2		16
56.45	odd	6		980.2.o.e.411.5		16
56.51	odd	6		980.2.o.e.31.5		16
56.53	even	6		980.2.o.e.411.6		16
168.83	odd	2		1260.2.c.c.811.1		8
168.125	even	2		1260.2.c.c.811.3		8
280.13	even	4		700.2.c.i.699.3		8
280.27	odd	4		700.2.c.j.699.4		8
280.69	odd	2		700.2.g.j.251.3		8
280.83	odd	4		700.2.c.i.699.5		8
280.139	even	2		700.2.g.j.251.2		8
280.237	even	4		700.2.c.j.699.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.2.g.c.111.5	✓	8	8.5	even	2
140.2.g.c.111.6	yes	8	56.13	odd	2
140.2.g.c.111.7	yes	8	56.27	even	2
140.2.g.c.111.8	yes	8	8.3	odd	2
700.2.c.i.699.3		8	280.13	even	4
700.2.c.i.699.4		8	40.27	even	4
700.2.c.i.699.5		8	280.83	odd	4
700.2.c.i.699.6		8	40.37	odd	4
700.2.c.j.699.3		8	40.13	odd	4
700.2.c.j.699.4		8	280.27	odd	4
700.2.c.j.699.5		8	40.3	even	4
700.2.c.j.699.6		8	280.237	even	4
700.2.g.j.251.1		8	40.19	odd	2
700.2.g.j.251.2		8	280.139	even	2
700.2.g.j.251.3		8	280.69	odd	2
700.2.g.j.251.4		8	40.29	even	2
980.2.o.e.31.1		16	56.5	odd	6
980.2.o.e.31.2		16	56.37	even	6
980.2.o.e.31.5		16	56.51	odd	6
980.2.o.e.31.6		16	56.19	even	6
980.2.o.e.411.1		16	56.11	odd	6
980.2.o.e.411.2		16	56.3	even	6
980.2.o.e.411.5		16	56.45	odd	6
980.2.o.e.411.6		16	56.53	even	6
1260.2.c.c.811.1		8	168.83	odd	2
1260.2.c.c.811.2		8	24.11	even	2
1260.2.c.c.811.3		8	168.125	even	2
1260.2.c.c.811.4		8	24.5	odd	2
2240.2.k.e.1791.3		8	7.6	odd	2	inner
2240.2.k.e.1791.4		8	4.3	odd	2	inner
2240.2.k.e.1791.5		8	28.27	even	2	inner
2240.2.k.e.1791.6		8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2240.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.936426 q^{3} +1.00000i q^{5} +(2.60399 + 0.468213i) q^{7} -2.12311 q^{9} +O(q^{10})\)	\(q+0.936426 q^{3} +1.00000i q^{5} +(2.60399 + 0.468213i) q^{7} -2.12311 q^{9} -2.39871i q^{11} +2.00000i q^{13} +0.936426i q^{15} +7.12311i q^{17} +2.39871 q^{19} +(2.43845 + 0.438447i) q^{21} +5.73384i q^{23} -1.00000 q^{25} -4.79741 q^{27} +2.00000 q^{29} -6.67026 q^{31} -2.24621i q^{33} +(-0.468213 + 2.60399i) q^{35} -2.00000 q^{37} +1.87285i q^{39} -7.12311i q^{41} +7.60669i q^{43} -2.12311i q^{45} +10.0054 q^{47} +(6.56155 + 2.43845i) q^{49} +6.67026i q^{51} -2.00000 q^{53} +2.39871 q^{55} +2.24621 q^{57} +10.9418 q^{59} +2.00000i q^{61} +(-5.52855 - 0.994066i) q^{63} -2.00000 q^{65} +14.2770i q^{67} +5.36932i q^{69} +6.14441i q^{71} +9.36932i q^{73} -0.936426 q^{75} +(1.12311 - 6.24621i) q^{77} -4.27156i q^{79} +1.87689 q^{81} +0.936426 q^{83} -7.12311 q^{85} +1.87285 q^{87} -12.0000i q^{89} +(-0.936426 + 5.20798i) q^{91} -6.24621 q^{93} +2.39871i q^{95} -7.12311i q^{97} +5.09271i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{9}+O(q^{10}) \) 8 * q + 16 * q^9	\( 8 q + 16 q^{9} + 36 q^{21} - 8 q^{25} + 16 q^{29} - 16 q^{37} + 36 q^{49} - 16 q^{53} - 48 q^{57} - 16 q^{65} - 24 q^{77} + 48 q^{81} - 24 q^{85} + 16 q^{93}+O(q^{100}) \) 8 * q + 16 * q^9 + 36 * q^21 - 8 * q^25 + 16 * q^29 - 16 * q^37 + 36 * q^49 - 16 * q^53 - 48 * q^57 - 16 * q^65 - 24 * q^77 + 48 * q^81 - 24 * q^85 + 16 * q^93

Introduction

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Database

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