LMFDB - Embedded newform 2240.2.e.g.2239.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2240,2,Mod(2239,2240)]

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, 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("2240.2239");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2240 = 2^{6} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2240.e (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:	$48$
Twist minimal:	no (minimal twist has level 1120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2239.11
Character	$\chi$	$=$	2240.2239
Dual form			2240.2.e.g.2239.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.326439i q^{3} +(-0.278349 - 2.21868i) q^{5} +(-2.25431 - 1.38495i) q^{7} +2.89344 q^{9} +O(q^{10})$	\(q-0.326439i q^{3} +(-0.278349 - 2.21868i) q^{5} +(-2.25431 - 1.38495i) q^{7} +2.89344 q^{9} -4.91165i q^{11} -4.89435 q^{13} +(-0.724262 + 0.0908638i) q^{15} -6.14952 q^{17} +4.72104 q^{19} +(-0.452102 + 0.735894i) q^{21} +6.30675 q^{23} +(-4.84504 + 1.23513i) q^{25} -1.92385i q^{27} -4.66444 q^{29} -4.98821 q^{31} -1.60335 q^{33} +(-2.44528 + 5.38708i) q^{35} -2.88849i q^{37} +1.59771i q^{39} +7.38331i q^{41} +6.20558 q^{43} +(-0.805384 - 6.41960i) q^{45} +5.09495i q^{47} +(3.16381 + 6.24422i) q^{49} +2.00744i q^{51} +1.63690i q^{53} +(-10.8974 + 1.36715i) q^{55} -1.54113i q^{57} +1.30740 q^{59} -8.50226i q^{61} +(-6.52270 - 4.00727i) q^{63} +(1.36233 + 10.8590i) q^{65} -3.67526 q^{67} -2.05877i q^{69} +1.27752i q^{71} -8.00456 q^{73} +(0.403195 + 1.58161i) q^{75} +(-6.80240 + 11.0724i) q^{77} +14.0588i q^{79} +8.05229 q^{81} +0.484014i q^{83} +(1.71171 + 13.6438i) q^{85} +1.52265i q^{87} -11.3650i q^{89} +(11.0334 + 6.77844i) q^{91} +1.62835i q^{93} +(-1.31409 - 10.4745i) q^{95} -6.14952 q^{97} -14.2115i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 48 q^{9}+O(q^{10}) $ 48 * q - 48 * q^9	$ 48 q - 48 q^{9} - 8 q^{21} - 16 q^{25} - 16 q^{29} + 24 q^{49} - 16 q^{65} - 32 q^{81} - 32 q^{85}+O(q^{100}) $ 48 * q - 48 * q^9 - 8 * q^21 - 16 * q^25 - 16 * q^29 + 24 * q^49 - 16 * q^65 - 32 * q^81 - 32 * q^85

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
$3$		−	0.326439i		−	0.188470i	−0.995550	−	0.0942348i	$-0.969960\pi$
$3$		−	0.326439i		−	0.188470i	0.995550	−	0.0942348i	$-0.0300404\pi$
$4$		0			0
$5$	−0.278349	−	2.21868i	−0.124481	−	0.992222i
$5$	−0.278349	−	2.21868i	−0.124481	−	0.992222i
$6$		0			0
$7$	−2.25431	−	1.38495i	−0.852048	−	0.523463i
$7$	−2.25431	−	1.38495i	−0.852048	−	0.523463i
$8$		0			0
$9$	2.89344			0.964479
$10$		0			0
$11$		−	4.91165i		−	1.48092i	−0.672102	−	0.740459i	$-0.734608\pi$
$11$		−	4.91165i		−	1.48092i	0.672102	−	0.740459i	$-0.265392\pi$
$12$		0			0
$13$	−4.89435			−1.35745			−0.678724	−	0.734393i	$-0.737467\pi$
$13$	−4.89435			−1.35745			−0.678724	+	0.734393i	$0.737467\pi$
$14$		0			0
$15$	−0.724262	+	0.0908638i	−0.187004	+	0.0234609i
$16$		0			0
$17$	−6.14952			−1.49148			−0.745739	−	0.666238i	$-0.767903\pi$
$17$	−6.14952			−1.49148			−0.745739	+	0.666238i	$0.767903\pi$
$18$		0			0
$19$	4.72104			1.08308			0.541540	−	0.840675i	$-0.317841\pi$
$19$	4.72104			1.08308			0.541540	+	0.840675i	$0.317841\pi$
$20$		0			0
$21$	−0.452102	+	0.735894i	−0.0986568	+	0.160585i
$22$		0			0
$23$	6.30675			1.31505			0.657524	−	0.753434i	$-0.271604\pi$
$23$	6.30675			1.31505			0.657524	+	0.753434i	$0.271604\pi$
$24$		0			0
$25$	−4.84504	+	1.23513i	−0.969009	+	0.247026i
$26$		0			0
$27$		−	1.92385i		−	0.370245i
$28$		0			0
$29$	−4.66444			−0.866164			−0.433082	−	0.901354i	$-0.642574\pi$
$29$	−4.66444			−0.866164			−0.433082	+	0.901354i	$0.642574\pi$
$30$		0			0
$31$	−4.98821			−0.895909			−0.447955	−	0.894056i	$-0.647847\pi$
$31$	−4.98821			−0.895909			−0.447955	+	0.894056i	$0.647847\pi$
$32$		0			0
$33$	−1.60335			−0.279108
$34$		0			0
$35$	−2.44528	+	5.38708i	−0.413327	+	0.910583i
$36$		0			0
$37$		−	2.88849i		−	0.474865i	−0.971404	−	0.237432i	$-0.923694\pi$
$37$		−	2.88849i		−	0.474865i	0.971404	−	0.237432i	$-0.0763058\pi$
$38$		0			0
$39$			1.59771i			0.255838i
$40$		0			0
$41$			7.38331i			1.15308i	0.817069	+	0.576540i	$0.195597\pi$
$41$			7.38331i			1.15308i	−0.817069	+	0.576540i	$0.804403\pi$
$42$		0			0
$43$	6.20558			0.946343			0.473171	−	0.880970i	$-0.343109\pi$
$43$	6.20558			0.946343			0.473171	+	0.880970i	$0.343109\pi$
$44$		0			0
$45$	−0.805384	−	6.41960i	−0.120060	−	0.956977i
$46$		0			0
$47$			5.09495i			0.743174i	0.928398	+	0.371587i	$0.121186\pi$
$47$			5.09495i			0.743174i	−0.928398	+	0.371587i	$0.878814\pi$
$48$		0			0
$49$	3.16381	+	6.24422i	0.451973	+	0.892032i
$50$		0			0
$51$			2.00744i			0.281098i
$52$		0			0
$53$			1.63690i			0.224846i	0.993660	+	0.112423i	$0.0358612\pi$
$53$			1.63690i			0.224846i	−0.993660	+	0.112423i	$0.964139\pi$
$54$		0			0
$55$	−10.8974	+	1.36715i	−1.46940	+	0.184346i
$56$		0			0
$57$		−	1.54113i		−	0.204128i
$58$		0			0
$59$	1.30740			0.170209			0.0851047	−	0.996372i	$-0.472878\pi$
$59$	1.30740			0.170209			0.0851047	+	0.996372i	$0.472878\pi$
$60$		0			0
$61$		−	8.50226i		−	1.08860i	−0.838890	−	0.544301i	$-0.816795\pi$
$61$		−	8.50226i		−	1.08860i	0.838890	−	0.544301i	$-0.183205\pi$
$62$		0			0
$63$	−6.52270	−	4.00727i	−0.821783	−	0.504869i
$64$		0			0
$65$	1.36233	+	10.8590i	0.168977	+	1.34689i
$66$		0			0
$67$	−3.67526			−0.449005			−0.224502	−	0.974474i	$-0.572076\pi$
$67$	−3.67526			−0.449005			−0.224502	+	0.974474i	$0.572076\pi$
$68$		0			0
$69$		−	2.05877i		−	0.247846i
$70$		0			0
$71$			1.27752i			0.151614i	0.997123	+	0.0758070i	$0.0241533\pi$
$71$			1.27752i			0.151614i	−0.997123	+	0.0758070i	$0.975847\pi$
$72$		0			0
$73$	−8.00456			−0.936863			−0.468431	−	0.883500i	$-0.655181\pi$
$73$	−8.00456			−0.936863			−0.468431	+	0.883500i	$0.655181\pi$
$74$		0			0
$75$	0.403195	+	1.58161i	0.0465569	+	0.182629i
$76$		0			0
$77$	−6.80240	+	11.0724i	−0.775205	+	1.26181i
$78$		0			0
$79$			14.0588i			1.58174i	0.611986	+	0.790869i	$0.290371\pi$
$79$			14.0588i			1.58174i	−0.611986	+	0.790869i	$0.709629\pi$
$80$		0			0
$81$	8.05229			0.894699
$82$		0			0
$83$			0.484014i			0.0531274i	0.999647	+	0.0265637i	$0.00845649\pi$
$83$			0.484014i			0.0531274i	−0.999647	+	0.0265637i	$0.991544\pi$
$84$		0			0
$85$	1.71171	+	13.6438i	0.185661	+	1.47988i
$86$		0			0
$87$			1.52265i			0.163246i
$88$		0			0
$89$		−	11.3650i		−	1.20469i	−0.798236	−	0.602345i	$-0.794233\pi$
$89$		−	11.3650i		−	1.20469i	0.798236	−	0.602345i	$-0.205767\pi$
$90$		0			0
$91$	11.0334	+	6.77844i	1.15661	+	0.710574i
$92$		0			0
$93$			1.62835i			0.168852i
$94$		0			0
$95$	−1.31409	−	10.4745i	−0.134823	−	1.07466i
$96$		0			0
$97$	−6.14952			−0.624389			−0.312195	−	0.950018i	$-0.601064\pi$
$97$	−6.14952			−0.624389			−0.312195	+	0.950018i	$0.601064\pi$
$98$		0			0
$99$		−	14.2115i		−	1.42831i
$100$		0			0
$101$			4.52053i			0.449810i	0.974381	+	0.224905i	$0.0722072\pi$
$101$			4.52053i			0.449810i	−0.974381	+	0.224905i	$0.927793\pi$
$102$		0			0
$103$		−	16.9928i		−	1.67435i	−0.546935	−	0.837175i	$-0.684206\pi$
$103$		−	16.9928i		−	1.67435i	0.546935	−	0.837175i	$-0.315794\pi$
$104$		0			0
$105$	1.75855	+	0.798233i	0.171617	+	0.0778996i
$106$		0			0
$107$	3.67526			0.355301			0.177650	−	0.984094i	$-0.443150\pi$
$107$	3.67526			0.355301			0.177650	+	0.984094i	$0.443150\pi$
$108$		0			0
$109$	−8.99206			−0.861283			−0.430642	−	0.902523i	$-0.641713\pi$
$109$	−8.99206			−0.861283			−0.430642	+	0.902523i	$0.641713\pi$
$110$		0			0
$111$	−0.942915			−0.0894975
$112$		0			0
$113$			7.71199i			0.725483i	0.931890	+	0.362741i	$0.118159\pi$
$113$			7.71199i			0.725483i	−0.931890	+	0.362741i	$0.881841\pi$
$114$		0			0
$115$	−1.75547	−	13.9926i	−0.163699	−	1.30482i
$116$		0			0
$117$	−14.1615			−1.30923
$118$		0			0
$119$	13.8629	+	8.51679i	1.27081	+	0.780733i
$120$		0			0
$121$	−13.1243			−1.19312
$122$		0			0
$123$	2.41020			0.217320
$124$		0			0
$125$	4.08896	+	10.4058i	0.365728	+	0.930722i
$126$		0			0
$127$	3.06009			0.271539			0.135770	−	0.990740i	$-0.456649\pi$
$127$	3.06009			0.271539			0.135770	+	0.990740i	$0.456649\pi$
$128$		0			0
$129$		−	2.02574i		−	0.178357i
$130$		0			0
$131$	−8.14223			−0.711390			−0.355695	−	0.934602i	$-0.615756\pi$
$131$	−8.14223			−0.711390			−0.355695	+	0.934602i	$0.615756\pi$
$132$		0			0
$133$	−10.6427	−	6.53841i	−0.922837	−	0.566952i
$134$		0			0
$135$	−4.26839	+	0.535500i	−0.367365	+	0.0460885i
$136$		0			0
$137$			20.3791i			1.74111i	0.492073	+	0.870554i	$0.336239\pi$
$137$			20.3791i			1.74111i	−0.492073	+	0.870554i	$0.663761\pi$
$138$		0			0
$139$	−16.4420			−1.39459			−0.697296	−	0.716783i	$-0.745614\pi$
$139$	−16.4420			−1.39459			−0.697296	+	0.716783i	$0.745614\pi$
$140$		0			0
$141$	1.66319			0.140066
$142$		0			0
$143$			24.0393i			2.01027i
$144$		0			0
$145$	1.29834	+	10.3489i	0.107821	+	0.859427i
$146$		0			0
$147$	2.03836	−	1.03279i	0.168121	−	0.0851832i
$148$		0			0
$149$	−6.06121			−0.496554			−0.248277	−	0.968689i	$-0.579864\pi$
$149$	−6.06121			−0.496554			−0.248277	+	0.968689i	$0.579864\pi$
$150$		0			0
$151$		−	14.2405i		−	1.15888i	−0.815016	−	0.579439i	$-0.803272\pi$
$151$		−	14.2405i		−	1.15888i	0.815016	−	0.579439i	$-0.196728\pi$
$152$		0			0
$153$	−17.7933			−1.43850
$154$		0			0
$155$	1.38846	+	11.0672i	0.111524	+	0.888941i
$156$		0			0
$157$	13.2998			1.06144			0.530719	−	0.847548i	$-0.321922\pi$
$157$	13.2998			1.06144			0.530719	+	0.847548i	$0.321922\pi$
$158$		0			0
$159$	0.534349			0.0423767
$160$		0			0
$161$	−14.2174	−	8.73455i	−1.12048	−	0.688379i
$162$		0			0
$163$	−21.2293			−1.66281			−0.831403	−	0.555670i	$-0.812462\pi$
$163$	−21.2293			−1.66281			−0.831403	+	0.555670i	$0.812462\pi$
$164$		0			0
$165$	0.446291	+	3.55732i	0.0347437	+	0.276937i
$166$		0			0
$167$		−	10.0560i		−	0.778159i	−0.921204	−	0.389080i	$-0.872793\pi$
$167$		−	10.0560i		−	0.778159i	0.921204	−	0.389080i	$-0.127207\pi$
$168$		0			0
$169$	10.9547			0.842666
$170$		0			0
$171$	13.6600			1.04461
$172$		0			0
$173$	−18.4843			−1.40534			−0.702668	−	0.711518i	$-0.748008\pi$
$173$	−18.4843			−1.40534			−0.702668	+	0.711518i	$0.748008\pi$
$174$		0			0
$175$	12.6328	+	3.92579i	0.954951	+	0.296762i
$176$		0			0
$177$		−	0.426787i		−	0.0320793i
$178$		0			0
$179$		−	8.09763i		−	0.605245i	−0.953110	−	0.302623i	$-0.902138\pi$
$179$		−	8.09763i		−	0.605245i	0.953110	−	0.302623i	$-0.0978622\pi$
$180$		0			0
$181$		−	6.71553i		−	0.499161i	−0.968354	−	0.249581i	$-0.919707\pi$
$181$		−	6.71553i		−	0.499161i	0.968354	−	0.249581i	$-0.0802928\pi$
$182$		0			0
$183$	−2.77547			−0.205168
$184$		0			0
$185$	−6.40862	+	0.804007i	−0.471171	+	0.0591118i
$186$		0			0
$187$			30.2043i			2.20876i
$188$		0			0
$189$	−2.66444	+	4.33694i	−0.193809	+	0.315466i
$190$		0			0
$191$		−	8.50191i		−	0.615177i	−0.951520	−	0.307588i	$-0.900478\pi$
$191$		−	8.50191i		−	0.615177i	0.951520	−	0.307588i	$-0.0995220\pi$
$192$		0			0
$193$		−	25.7465i		−	1.85327i	−0.375962	−	0.926635i	$-0.622688\pi$
$193$		−	25.7465i		−	1.85327i	0.375962	−	0.926635i	$-0.377312\pi$
$194$		0			0
$195$	3.54479	−	0.444719i	0.253848	−	0.0318470i
$196$		0			0
$197$			4.94908i			0.352607i	0.984336	+	0.176304i	$0.0564141\pi$
$197$			4.94908i			0.352607i	−0.984336	+	0.176304i	$0.943586\pi$
$198$		0			0
$199$	15.1346			1.07286			0.536432	−	0.843944i	$-0.319772\pi$
$199$	15.1346			1.07286			0.536432	+	0.843944i	$0.319772\pi$
$200$		0			0
$201$			1.19975i			0.0846237i
$202$		0			0
$203$	10.5151	+	6.46002i	0.738014	+	0.453405i
$204$		0			0
$205$	16.3812	−	2.05513i	1.14411	−	0.143537i
$206$		0			0
$207$	18.2482			1.26834
$208$		0			0
$209$		−	23.1881i		−	1.60395i
$210$		0			0
$211$			10.2986i			0.708988i	0.935058	+	0.354494i	$0.115347\pi$
$211$			10.2986i			0.708988i	−0.935058	+	0.354494i	$0.884653\pi$
$212$		0			0
$213$	0.417033			0.0285746
$214$		0			0
$215$	−1.72732	−	13.7682i	−0.117802	−	0.938982i
$216$		0			0
$217$	11.2450	+	6.90844i	0.763358	+	0.468975i
$218$		0			0
$219$			2.61300i			0.176570i
$220$		0			0
$221$	30.0979			2.02460
$222$		0			0
$223$			15.4189i			1.03252i	0.856430	+	0.516262i	$0.172677\pi$
$223$			15.4189i			1.03252i	−0.856430	+	0.516262i	$0.827323\pi$
$224$		0			0
$225$	−14.0188	+	3.57377i	−0.934589	+	0.238252i
$226$		0			0
$227$		−	19.8139i		−	1.31510i	−0.753413	−	0.657548i	$-0.771594\pi$
$227$		−	19.8139i		−	1.31510i	0.753413	−	0.657548i	$-0.228406\pi$
$228$		0			0
$229$			16.2339i			1.07277i	0.843974	+	0.536385i	$0.180210\pi$
$229$			16.2339i			1.07277i	−0.843974	+	0.536385i	$0.819790\pi$
$230$		0			0
$231$	3.61445	+	2.22057i	0.237813	+	0.146103i
$232$		0			0
$233$		−	23.8114i		−	1.55994i	−0.625817	−	0.779970i	$-0.715234\pi$
$233$		−	23.8114i		−	1.55994i	0.625817	−	0.779970i	$-0.284766\pi$
$234$		0			0
$235$	11.3040	−	1.41817i	0.737394	−	0.0925113i
$236$		0			0
$237$	4.58934			0.298109
$238$		0			0
$239$		−	11.5200i		−	0.745166i	−0.927999	−	0.372583i	$-0.878472\pi$
$239$		−	11.5200i		−	0.745166i	0.927999	−	0.372583i	$-0.121528\pi$
$240$		0			0
$241$		−	9.99811i		−	0.644035i	−0.946734	−	0.322018i	$-0.895639\pi$
$241$		−	9.99811i		−	0.644035i	0.946734	−	0.322018i	$-0.104361\pi$
$242$		0			0
$243$		−	8.40012i		−	0.538868i
$244$		0			0
$245$	12.9733	−	8.75754i	0.828831	−	0.559499i
$246$		0			0
$247$	−23.1064			−1.47023
$248$		0			0
$249$	0.158001			0.0100129
$250$		0			0
$251$	−25.6120			−1.61662			−0.808309	−	0.588759i	$-0.799617\pi$
$251$	−25.6120			−1.61662			−0.808309	+	0.588759i	$0.799617\pi$
$252$		0			0
$253$		−	30.9765i		−	1.94748i
$254$		0			0
$255$	4.45386	−	0.558769i	0.278912	−	0.0349915i
$256$		0			0
$257$	16.8304			1.04985			0.524925	−	0.851148i	$-0.324093\pi$
$257$	16.8304			1.04985			0.524925	+	0.851148i	$0.324093\pi$
$258$		0			0
$259$	−4.00042	+	6.51155i	−0.248574	+	0.404608i
$260$		0			0
$261$	−13.4963			−0.835397
$262$		0			0
$263$	−10.8818			−0.671001			−0.335500	−	0.942040i	$-0.608905\pi$
$263$	−10.8818			−0.671001			−0.335500	+	0.942040i	$0.608905\pi$
$264$		0			0
$265$	3.63176	−	0.455630i	0.223097	−	0.0279891i
$266$		0			0
$267$	−3.70999			−0.227048
$268$		0			0
$269$			24.8631i			1.51593i	0.652297	+	0.757964i	$0.273806\pi$
$269$			24.8631i			1.51593i	−0.652297	+	0.757964i	$0.726194\pi$
$270$		0			0
$271$	5.36512			0.325908			0.162954	−	0.986634i	$-0.447898\pi$
$271$	5.36512			0.325908			0.162954	+	0.986634i	$0.447898\pi$
$272$		0			0
$273$	2.21275	−	3.60172i	0.133922	−	0.217986i
$274$		0			0
$275$	6.06652	+	23.7971i	0.365825	+	1.43502i
$276$		0			0
$277$			1.24303i			0.0746862i	0.999303	+	0.0373431i	$0.0118894\pi$
$277$			1.24303i			0.0746862i	−0.999303	+	0.0373431i	$0.988111\pi$
$278$		0			0
$279$	−14.4331			−0.864086
$280$		0			0
$281$	−3.53222			−0.210715			−0.105357	−	0.994434i	$-0.533599\pi$
$281$	−3.53222			−0.210715			−0.105357	+	0.994434i	$0.533599\pi$
$282$		0			0
$283$			2.70274i			0.160661i	0.996768	+	0.0803305i	$0.0255976\pi$
$283$			2.70274i			0.160661i	−0.996768	+	0.0803305i	$0.974402\pi$
$284$		0			0
$285$	−3.41927	+	0.428971i	−0.202540	+	0.0254101i
$286$		0			0
$287$	10.2255	−	16.6443i	0.603594	−	0.982479i
$288$		0			0
$289$	20.8166			1.22451
$290$		0			0
$291$			2.00744i			0.117678i
$292$		0			0
$293$	3.60343			0.210514			0.105257	−	0.994445i	$-0.466433\pi$
$293$	3.60343			0.210514			0.105257	+	0.994445i	$0.466433\pi$
$294$		0			0
$295$	−0.363914	−	2.90070i	−0.0211879	−	0.168885i
$296$		0			0
$297$	−9.44926			−0.548301
$298$		0			0
$299$	−30.8674			−1.78511
$300$		0			0
$301$	−13.9893	−	8.59444i	−0.806330	−	0.495375i
$302$		0			0
$303$	1.47568			0.0847755
$304$		0			0
$305$	−18.8637	+	2.36659i	−1.08014	+	0.135511i
$306$		0			0
$307$		−	31.2942i		−	1.78605i	−0.450004	−	0.893026i	$-0.648578\pi$
$307$		−	31.2942i		−	1.78605i	0.450004	−	0.893026i	$-0.351422\pi$
$308$		0			0
$309$	−5.54711			−0.315564
$310$		0			0
$311$	29.0305			1.64617			0.823085	−	0.567918i	$-0.192251\pi$
$311$	29.0305			1.64617			0.823085	+	0.567918i	$0.192251\pi$
$312$		0			0
$313$	13.2465			0.748735			0.374368	−	0.927280i	$-0.377860\pi$
$313$	13.2465			0.748735			0.374368	+	0.927280i	$0.377860\pi$
$314$		0			0
$315$	−7.07526	+	15.5872i	−0.398646	+	0.878238i
$316$		0			0
$317$			22.5513i			1.26661i	0.773903	+	0.633304i	$0.218302\pi$
$317$			22.5513i			1.26661i	−0.773903	+	0.633304i	$0.781698\pi$
$318$		0			0
$319$			22.9101i			1.28272i
$320$		0			0
$321$		−	1.19975i		−	0.0669634i
$322$		0			0
$323$	−29.0321			−1.61539
$324$		0			0
$325$	23.7133	−	6.04516i	1.31538	−	0.335325i
$326$		0			0
$327$			2.93536i			0.162326i
$328$		0			0
$329$	7.05626	−	11.4856i	0.389024	−	0.633220i
$330$		0			0
$331$		−	6.16680i		−	0.338958i	−0.985534	−	0.169479i	$-0.945792\pi$
$331$		−	6.16680i		−	0.338958i	0.985534	−	0.169479i	$-0.0542084\pi$
$332$		0			0
$333$		−	8.35766i		−	0.457997i
$334$		0			0
$335$	1.02300	+	8.15421i	0.0558927	+	0.445512i
$336$		0			0
$337$		−	5.74407i		−	0.312899i	−0.987686	−	0.156450i	$-0.949995\pi$
$337$		−	5.74407i		−	0.312899i	0.987686	−	0.156450i	$-0.0500049\pi$
$338$		0			0
$339$	2.51749			0.136731
$340$		0			0
$341$			24.5003i			1.32677i
$342$		0			0
$343$	1.51574	−	18.4581i	0.0818423	−	0.996645i
$344$		0			0
$345$	−4.56774	+	0.573055i	−0.245919	+	0.0308522i
$346$		0			0
$347$	−5.70870			−0.306459			−0.153229	−	0.988191i	$-0.548967\pi$
$347$	−5.70870			−0.306459			−0.153229	+	0.988191i	$0.548967\pi$
$348$		0			0
$349$			1.94494i			0.104110i	0.998644	+	0.0520552i	$0.0165772\pi$
$349$			1.94494i			0.104110i	−0.998644	+	0.0520552i	$0.983423\pi$
$350$		0			0
$351$			9.41598i			0.502588i
$352$		0			0
$353$	−36.5542			−1.94558			−0.972792	−	0.231678i	$-0.925578\pi$
$353$	−36.5542			−1.94558			−0.972792	+	0.231678i	$0.925578\pi$
$354$		0			0
$355$	2.83441	−	0.355597i	0.150435	−	0.0188731i
$356$		0			0
$357$	2.78021	−	4.52539i	0.147144	−	0.239509i
$358$		0			0
$359$		−	8.72750i		−	0.460620i	−0.973117	−	0.230310i	$-0.926026\pi$
$359$		−	8.72750i		−	0.460620i	0.973117	−	0.230310i	$-0.0739740\pi$
$360$		0			0
$361$	3.28820			0.173063
$362$		0			0
$363$			4.28427i			0.224866i
$364$		0			0
$365$	2.22806	+	17.7595i	0.116622	+	0.929576i
$366$		0			0
$367$			31.1442i			1.62571i	0.582465	+	0.812856i	$0.302088\pi$
$367$			31.1442i			1.62571i	−0.582465	+	0.812856i	$0.697912\pi$
$368$		0			0
$369$			21.3631i			1.11212i
$370$		0			0
$371$	2.26704	−	3.69009i	0.117699	−	0.191580i
$372$		0			0
$373$		−	26.2574i		−	1.35955i	−0.733419	−	0.679777i	$-0.762076\pi$
$373$		−	26.2574i		−	1.35955i	0.733419	−	0.679777i	$-0.237924\pi$
$374$		0			0
$375$	3.39685	−	1.33480i	0.175413	−	0.0689286i
$376$		0			0
$377$	22.8294			1.17577
$378$		0			0
$379$		−	35.1481i		−	1.80544i	−0.430233	−	0.902718i	$-0.641568\pi$
$379$		−	35.1481i		−	1.80544i	0.430233	−	0.902718i	$-0.358432\pi$
$380$		0			0
$381$		−	0.998933i		−	0.0511769i
$382$		0			0
$383$		−	12.2000i		−	0.623389i	−0.950182	−	0.311695i	$-0.899103\pi$
$383$		−	12.2000i		−	0.623389i	0.950182	−	0.311695i	$-0.100897\pi$
$384$		0			0
$385$	26.4594	+	12.0103i	1.34850	+	0.612104i
$386$		0			0
$387$	17.9555			0.912728
$388$		0			0
$389$	−17.8115			−0.903081			−0.451540	−	0.892251i	$-0.649125\pi$
$389$	−17.8115			−0.903081			−0.451540	+	0.892251i	$0.649125\pi$
$390$		0			0
$391$	−38.7835			−1.96136
$392$		0			0
$393$			2.65794i			0.134075i
$394$		0			0
$395$	31.1919	−	3.91324i	1.56943	−	0.196897i
$396$		0			0
$397$	19.9560			1.00157			0.500783	−	0.865573i	$-0.333046\pi$
$397$	19.9560			1.00157			0.500783	+	0.865573i	$0.333046\pi$
$398$		0			0
$399$	−2.13439	+	3.47418i	−0.106853	+	0.173927i
$400$		0			0
$401$	−1.25564			−0.0627039			−0.0313519	−	0.999508i	$-0.509981\pi$
$401$	−1.25564			−0.0627039			−0.0313519	+	0.999508i	$0.509981\pi$
$402$		0			0
$403$	24.4141			1.21615
$404$		0			0
$405$	−2.24134	−	17.8654i	−0.111373	−	0.887740i
$406$		0			0
$407$	−14.1872			−0.703235
$408$		0			0
$409$			0.503797i			0.0249112i	0.999922	+	0.0124556i	$0.00396484\pi$
$409$			0.503797i			0.0249112i	−0.999922	+	0.0124556i	$0.996035\pi$
$410$		0			0
$411$	6.65254			0.328146
$412$		0			0
$413$	−2.94729	−	1.81069i	−0.145027	−	0.0890983i
$414$		0			0
$415$	1.07387	−	0.134725i	0.0527142	−	0.00661337i
$416$		0			0
$417$			5.36731i			0.262838i
$418$		0			0
$419$	−19.8632			−0.970381			−0.485190	−	0.874409i	$-0.661250\pi$
$419$	−19.8632			−0.970381			−0.485190	+	0.874409i	$0.661250\pi$
$420$		0			0
$421$	24.6824			1.20295			0.601473	−	0.798893i	$-0.294581\pi$
$421$	24.6824			1.20295			0.601473	+	0.798893i	$0.294581\pi$
$422$		0			0
$423$			14.7419i			0.716776i
$424$		0			0
$425$	29.7947	−	7.59546i	1.44526	−	0.368434i
$426$		0			0
$427$	−11.7752	+	19.1667i	−0.569843	+	0.927542i
$428$		0			0
$429$	7.84737			0.378874
$430$		0			0
$431$		−	10.7931i		−	0.519884i	−0.965624	−	0.259942i	$-0.916297\pi$
$431$		−	10.7931i		−	0.519884i	0.965624	−	0.259942i	$-0.0837034\pi$
$432$		0			0
$433$	18.4181			0.885118			0.442559	−	0.896739i	$-0.354071\pi$
$433$	18.4181			0.885118			0.442559	+	0.896739i	$0.354071\pi$
$434$		0			0
$435$	3.37827	−	0.423828i	0.161976	−	0.0203210i
$436$		0			0
$437$	29.7744			1.42430
$438$		0			0
$439$	−7.73853			−0.369340			−0.184670	−	0.982801i	$-0.559122\pi$
$439$	−7.73853			−0.369340			−0.184670	+	0.982801i	$0.559122\pi$
$440$		0			0
$441$	9.15429	+	18.0673i	0.435919	+	0.860346i
$442$		0			0
$443$	27.8363			1.32254			0.661272	−	0.750147i	$-0.270017\pi$
$443$	27.8363			1.32254			0.661272	+	0.750147i	$0.270017\pi$
$444$		0			0
$445$	−25.2153	+	3.16344i	−1.19532	+	0.149961i
$446$		0			0
$447$			1.97862i			0.0935853i
$448$		0			0
$449$	−12.3517			−0.582913			−0.291456	−	0.956584i	$-0.594140\pi$
$449$	−12.3517			−0.582913			−0.291456	+	0.956584i	$0.594140\pi$
$450$		0			0
$451$	36.2642			1.70761
$452$		0			0
$453$	−4.64866			−0.218413
$454$		0			0
$455$	11.9680	−	26.3662i	0.561070	−	1.23607i
$456$		0			0
$457$		−	6.87553i		−	0.321624i	−0.986985	−	0.160812i	$-0.948589\pi$
$457$		−	6.87553i		−	0.321624i	0.986985	−	0.160812i	$-0.0514112\pi$
$458$		0			0
$459$			11.8307i			0.552211i
$460$		0			0
$461$			3.02229i			0.140762i	0.997520	+	0.0703811i	$0.0224215\pi$
$461$			3.02229i			0.140762i	−0.997520	+	0.0703811i	$0.977578\pi$
$462$		0			0
$463$	31.7368			1.47493			0.737467	−	0.675383i	$-0.236022\pi$
$463$	31.7368			1.47493			0.737467	+	0.675383i	$0.236022\pi$
$464$		0			0
$465$	3.61277	−	0.453248i	0.167538	−	0.0210189i
$466$		0			0
$467$			7.65250i			0.354116i	0.984200	+	0.177058i	$0.0566580\pi$
$467$			7.65250i			0.354116i	−0.984200	+	0.177058i	$0.943342\pi$
$468$		0			0
$469$	8.28517	+	5.09006i	0.382574	+	0.235037i
$470$		0			0
$471$		−	4.34157i		−	0.200049i
$472$		0			0
$473$		−	30.4796i		−	1.40146i
$474$		0			0
$475$	−22.8736	+	5.83110i	−1.04951	+	0.267549i
$476$		0			0
$477$			4.73628i			0.216859i
$478$		0			0
$479$	19.5885			0.895020			0.447510	−	0.894279i	$-0.352311\pi$
$479$	19.5885			0.895020			0.447510	+	0.894279i	$0.352311\pi$
$480$		0			0
$481$			14.1373i			0.644604i
$482$		0			0
$483$	−2.85130	+	4.64110i	−0.129738	+	0.211177i
$484$		0			0
$485$	1.71171	+	13.6438i	0.0777248	+	0.619533i
$486$		0			0
$487$	−14.7720			−0.669385			−0.334693	−	0.942327i	$-0.608632\pi$
$487$	−14.7720			−0.669385			−0.334693	+	0.942327i	$0.608632\pi$
$488$		0			0
$489$			6.93006i			0.313388i
$490$		0			0
$491$			27.1954i			1.22731i	0.789573	+	0.613657i	$0.210302\pi$
$491$			27.1954i			1.22731i	−0.789573	+	0.613657i	$0.789698\pi$
$492$		0			0
$493$	28.6841			1.29186
$494$		0			0
$495$	−31.5308	+	3.95576i	−1.41720	+	0.177798i
$496$		0			0
$497$	1.76931	−	2.87993i	0.0793643	−	0.129183i
$498$		0			0
$499$		−	35.2160i		−	1.57648i	−0.615365	−	0.788242i	$-0.710992\pi$
$499$		−	35.2160i		−	1.57648i	0.615365	−	0.788242i	$-0.289008\pi$
$500$		0			0
$501$	−3.28268			−0.146659
$502$		0			0
$503$		−	18.4439i		−	0.822374i	−0.911551	−	0.411187i	$-0.865114\pi$
$503$		−	18.4439i		−	0.822374i	0.911551	−	0.411187i	$-0.134886\pi$
$504$		0			0
$505$	10.0296	−	1.25828i	0.446311	−	0.0559929i
$506$		0			0
$507$		−	3.57602i		−	0.158817i
$508$		0			0
$509$		−	40.5671i		−	1.79811i	−0.437840	−	0.899053i	$-0.644256\pi$
$509$		−	40.5671i		−	1.79811i	0.437840	−	0.899053i	$-0.355744\pi$
$510$		0			0
$511$	18.0447	+	11.0859i	0.798252	+	0.490413i
$512$		0			0
$513$		−	9.08255i		−	0.401005i
$514$		0			0
$515$	−37.7015	+	4.72992i	−1.66133	+	0.208425i
$516$		0			0
$517$	25.0246			1.10058
$518$		0			0
$519$			6.03400i			0.264863i
$520$		0			0
$521$			22.6490i			0.992269i	0.868246	+	0.496134i	$0.165248\pi$
$521$			22.6490i			0.992269i	−0.868246	+	0.496134i	$0.834752\pi$
$522$		0			0
$523$			6.26038i			0.273748i	0.990588	+	0.136874i	$0.0437055\pi$
$523$			6.26038i			0.273748i	−0.990588	+	0.136874i	$0.956295\pi$
$524$		0			0
$525$	1.28153	−	4.12384i	0.0559306	−	0.179979i
$526$		0			0
$527$	30.6751			1.33623
$528$		0			0
$529$	16.7751			0.729351
$530$		0			0
$531$	3.78289			0.164163
$532$		0			0
$533$		−	36.1365i		−	1.56525i
$534$		0			0
$535$	−1.02300	−	8.15421i	−0.0442283	−	0.352537i
$536$		0			0
$537$	−2.64338			−0.114070
$538$		0			0
$539$	30.6694	−	15.5395i	1.32102	−	0.669335i
$540$		0			0
$541$	−1.94390			−0.0835746			−0.0417873	−	0.999127i	$-0.513305\pi$
$541$	−1.94390			−0.0835746			−0.0417873	+	0.999127i	$0.513305\pi$
$542$		0			0
$543$	−2.19221			−0.0940767
$544$		0			0
$545$	2.50293	+	19.9505i	0.107214	+	0.854584i
$546$		0			0
$547$	17.4686			0.746904			0.373452	−	0.927649i	$-0.378174\pi$
$547$	17.4686			0.746904			0.373452	+	0.927649i	$0.378174\pi$
$548$		0			0
$549$		−	24.6007i		−	1.04993i
$550$		0			0
$551$	−22.0210			−0.938125
$552$		0			0
$553$	19.4708	−	31.6929i	0.827981	−	1.34772i
$554$		0			0
$555$	0.262459	+	2.09202i	0.0111408	+	0.0888014i
$556$		0			0
$557$		−	16.6670i		−	0.706203i	−0.935585	−	0.353102i	$-0.885127\pi$
$557$		−	16.6670i		−	0.706203i	0.935585	−	0.353102i	$-0.114873\pi$
$558$		0			0
$559$	−30.3723			−1.28461
$560$		0			0
$561$	9.85985			0.416283
$562$		0			0
$563$		−	34.0091i		−	1.43331i	−0.697427	−	0.716656i	$-0.745672\pi$
$563$		−	34.0091i		−	1.43331i	0.697427	−	0.716656i	$-0.254328\pi$
$564$		0			0
$565$	17.1104	−	2.14662i	0.719840	−	0.0903090i
$566$		0			0
$567$	−18.1524	−	11.1520i	−0.762327	−	0.468342i
$568$		0			0
$569$	−14.3812			−0.602890			−0.301445	−	0.953484i	$-0.597469\pi$
$569$	−14.3812			−0.602890			−0.301445	+	0.953484i	$0.597469\pi$
$570$		0			0
$571$			5.54259i			0.231950i	0.993252	+	0.115975i	$0.0369993\pi$
$571$			5.54259i			0.231950i	−0.993252	+	0.115975i	$0.963001\pi$
$572$		0			0
$573$	−2.77535			−0.115942
$574$		0			0
$575$	−30.5565	+	7.78966i	−1.27429	+	0.324851i
$576$		0			0
$577$	32.9250			1.37069			0.685344	−	0.728220i	$-0.259652\pi$
$577$	32.9250			1.37069			0.685344	+	0.728220i	$0.259652\pi$
$578$		0			0
$579$	−8.40464			−0.349285
$580$		0			0
$581$	0.670336	−	1.09112i	0.0278102	−	0.0452671i
$582$		0			0
$583$	8.03990			0.332979
$584$		0			0
$585$	3.94183	+	31.4198i	0.162975	+	1.29905i
$586$		0			0
$587$			19.5426i			0.806608i	0.915066	+	0.403304i	$0.132138\pi$
$587$			19.5426i			0.806608i	−0.915066	+	0.403304i	$0.867862\pi$
$588$		0			0
$589$	−23.5495			−0.970342
$590$		0			0
$591$	1.61557			0.0664557
$592$		0			0
$593$	−5.59945			−0.229942			−0.114971	−	0.993369i	$-0.536677\pi$
$593$	−5.59945			−0.229942			−0.114971	+	0.993369i	$0.536677\pi$
$594$		0			0
$595$	15.0373	−	33.1280i	0.616469	−	1.35811i
$596$		0			0
$597$		−	4.94052i		−	0.202202i
$598$		0			0
$599$			14.7084i			0.600968i	0.953787	+	0.300484i	$0.0971483\pi$
$599$			14.7084i			0.600968i	−0.953787	+	0.300484i	$0.902852\pi$
$600$		0			0
$601$			1.78304i			0.0727317i	0.999339	+	0.0363659i	$0.0115782\pi$
$601$			1.78304i			0.0727317i	−0.999339	+	0.0363659i	$0.988422\pi$
$602$		0			0
$603$	−10.6341			−0.433056
$604$		0			0
$605$	3.65312	+	29.1185i	0.148521	+	1.18384i
$606$		0			0
$607$		−	18.0320i		−	0.731898i	−0.930635	−	0.365949i	$-0.880745\pi$
$607$		−	18.0320i		−	0.731898i	0.930635	−	0.365949i	$-0.119255\pi$
$608$		0			0
$609$	2.10880	−	3.43253i	0.0854530	−	0.139093i
$610$		0			0
$611$		−	24.9364i		−	1.00882i
$612$		0			0
$613$		−	8.15533i		−	0.329391i	−0.986344	−	0.164695i	$-0.947336\pi$
$613$		−	8.15533i		−	0.329391i	0.986344	−	0.164695i	$-0.0526641\pi$
$614$		0			0
$615$	−0.670875	−	5.34745i	−0.0270523	−	0.215630i
$616$		0			0
$617$			12.2733i			0.494104i	0.969002	+	0.247052i	$0.0794618\pi$
$617$			12.2733i			0.494104i	−0.969002	+	0.247052i	$0.920538\pi$
$618$		0			0
$619$	41.7552			1.67828			0.839142	−	0.543913i	$-0.183058\pi$
$619$	41.7552			1.67828			0.839142	+	0.543913i	$0.183058\pi$
$620$		0			0
$621$		−	12.1332i		−	0.486889i
$622$		0			0
$623$	−15.7400	+	25.6203i	−0.630611	+	1.02645i
$624$		0			0
$625$	21.9489	−	11.9685i	0.877956	−	0.478741i
$626$		0			0
$627$	−7.56949			−0.302296
$628$		0			0
$629$			17.7628i			0.708250i
$630$		0			0
$631$			8.39437i			0.334174i	0.985942	+	0.167087i	$0.0534362\pi$
$631$			8.39437i			0.334174i	−0.985942	+	0.167087i	$0.946564\pi$
$632$		0			0
$633$	3.36188			0.133623
$634$		0			0
$635$	−0.851772	−	6.78935i	−0.0338016	−	0.269427i
$636$		0			0
$637$	−15.4848	−	30.5614i	−0.613530	−	1.21089i
$638$		0			0
$639$			3.69643i			0.146229i
$640$		0			0
$641$	24.1465			0.953730			0.476865	−	0.878977i	$-0.341773\pi$
$641$	24.1465			0.953730			0.476865	+	0.878977i	$0.341773\pi$
$642$		0			0
$643$		−	19.4389i		−	0.766595i	−0.923625	−	0.383297i	$-0.874788\pi$
$643$		−	19.4389i		−	0.766595i	0.923625	−	0.383297i	$-0.125212\pi$
$644$		0			0
$645$	−4.49447	+	0.563863i	−0.176969	+	0.0222021i
$646$		0			0
$647$			23.0872i			0.907653i	0.891090	+	0.453826i	$0.149941\pi$
$647$			23.0872i			0.907653i	−0.891090	+	0.453826i	$0.850059\pi$
$648$		0			0
$649$		−	6.42150i		−	0.252066i
$650$		0			0
$651$	2.25518	−	3.67079i	0.0883876	−	0.143870i
$652$		0			0
$653$		−	46.5267i		−	1.82073i	−0.413805	−	0.910365i	$-0.635801\pi$
$653$		−	46.5267i		−	1.82073i	0.413805	−	0.910365i	$-0.364199\pi$
$654$		0			0
$655$	2.26638	+	18.0650i	0.0885547	+	0.705857i
$656$		0			0
$657$	−23.1607			−0.903584
$658$		0			0
$659$			33.4853i			1.30440i	0.758047	+	0.652200i	$0.226154\pi$
$659$			33.4853i			1.30440i	−0.758047	+	0.652200i	$0.773846\pi$
$660$		0			0
$661$		−	31.5566i		−	1.22741i	−0.789536	−	0.613705i	$-0.789679\pi$
$661$		−	31.5566i		−	1.22741i	0.789536	−	0.613705i	$-0.210321\pi$
$662$		0			0
$663$		−	9.82512i		−	0.381576i
$664$		0			0
$665$	−11.5442	+	25.4326i	−0.447667	+	0.986234i
$666$		0			0
$667$	−29.4174			−1.13905
$668$		0			0
$669$	5.03332			0.194599
$670$		0			0
$671$	−41.7601			−1.61213
$672$		0			0
$673$		−	30.6140i		−	1.18008i	−0.807373	−	0.590041i	$-0.799112\pi$
$673$		−	30.6140i		−	1.18008i	0.807373	−	0.590041i	$-0.200888\pi$
$674$		0			0
$675$	2.37620	+	9.32112i	0.0914600	+	0.358770i
$676$		0			0
$677$	−25.5961			−0.983738			−0.491869	−	0.870669i	$-0.663686\pi$
$677$	−25.5961			−0.983738			−0.491869	+	0.870669i	$0.663686\pi$
$678$		0			0
$679$	13.8629	+	8.51679i	0.532010	+	0.326845i
$680$		0			0
$681$	−6.46803			−0.247856
$682$		0			0
$683$	−4.23156			−0.161916			−0.0809581	−	0.996718i	$-0.525798\pi$
$683$	−4.23156			−0.161916			−0.0809581	+	0.996718i	$0.525798\pi$
$684$		0			0
$685$	45.2147	−	5.67251i	1.72757	−	0.216735i
$686$		0			0
$687$	5.29939			0.202184
$688$		0			0
$689$		−	8.01158i		−	0.305217i
$690$		0			0
$691$	−8.51158			−0.323796			−0.161898	−	0.986808i	$-0.551762\pi$
$691$	−8.51158			−0.323796			−0.161898	+	0.986808i	$0.551762\pi$
$692$		0			0
$693$	−19.6823	+	32.0372i	−0.747669	+	1.21699i
$694$		0			0
$695$	4.57661	+	36.4795i	0.173601	+	1.38375i
$696$		0			0
$697$		−	45.4038i		−	1.71979i
$698$		0			0
$699$	−7.77298			−0.294001
$700$		0			0
$701$	−24.0088			−0.906799			−0.453400	−	0.891307i	$-0.649789\pi$
$701$	−24.0088			−0.906799			−0.453400	+	0.891307i	$0.649789\pi$
$702$		0			0
$703$		−	13.6367i		−	0.514317i
$704$		0			0
$705$	−0.462946	−	3.69008i	−0.0174356	−	0.138976i
$706$		0			0
$707$	6.26073	−	10.1907i	0.235459	−	0.383260i
$708$		0			0
$709$	−15.8791			−0.596353			−0.298176	−	0.954511i	$-0.596378\pi$
$709$	−15.8791			−0.596353			−0.298176	+	0.954511i	$0.596378\pi$
$710$		0			0
$711$			40.6782i			1.52555i
$712$		0			0
$713$	−31.4594			−1.17816
$714$		0			0
$715$	53.3354	−	6.69131i	1.99463	−	0.250241i
$716$		0			0
$717$	−3.76057			−0.140441
$718$		0			0
$719$	15.1593			0.565345			0.282673	−	0.959216i	$-0.408779\pi$
$719$	15.1593			0.565345			0.282673	+	0.959216i	$0.408779\pi$
$720$		0			0
$721$	−23.5342	+	38.3070i	−0.876460	+	1.42663i
$722$		0			0
$723$	−3.26377			−0.121381
$724$		0			0
$725$	22.5994	−	5.76119i	0.839321	−	0.213965i
$726$		0			0
$727$		−	8.19122i		−	0.303795i	−0.988396	−	0.151898i	$-0.951462\pi$
$727$		−	8.19122i		−	0.303795i	0.988396	−	0.151898i	$-0.0485384\pi$
$728$		0			0
$729$	21.4148			0.793139
$730$		0			0
$731$	−38.1614			−1.41145
$732$		0			0
$733$	−10.0964			−0.372920			−0.186460	−	0.982463i	$-0.559702\pi$
$733$	−10.0964			−0.372920			−0.186460	+	0.982463i	$0.559702\pi$
$734$		0			0
$735$	−2.85880	−	4.23498i	−0.105449	−	0.156209i
$736$		0			0
$737$			18.0516i			0.664939i
$738$		0			0
$739$			25.6788i			0.944611i	0.881435	+	0.472305i	$0.156578\pi$
$739$			25.6788i			0.944611i	−0.881435	+	0.472305i	$0.843422\pi$
$740$		0			0
$741$			7.54283i			0.277093i
$742$		0			0
$743$	38.8321			1.42461			0.712306	−	0.701869i	$-0.247651\pi$
$743$	38.8321			1.42461			0.712306	+	0.701869i	$0.247651\pi$
$744$		0			0
$745$	1.68713	+	13.4479i	0.0618117	+	0.492692i
$746$		0			0
$747$			1.40046i			0.0512403i
$748$		0			0
$749$	−8.28517	−	5.09006i	−0.302734	−	0.185987i
$750$		0			0
$751$		−	43.1863i		−	1.57589i	−0.615745	−	0.787945i	$-0.711145\pi$
$751$		−	43.1863i		−	1.57589i	0.615745	−	0.787945i	$-0.288855\pi$
$752$		0			0
$753$			8.36076i			0.304683i
$754$		0			0
$755$	−31.5951	+	3.96383i	−1.14986	+	0.144258i
$756$		0			0
$757$			26.8639i			0.976385i	0.872736	+	0.488193i	$0.162344\pi$
$757$			26.8639i			0.976385i	−0.872736	+	0.488193i	$0.837656\pi$
$758$		0			0
$759$	−10.1119			−0.367040
$760$		0			0
$761$		−	45.1603i		−	1.63706i	−0.574464	−	0.818530i	$-0.694790\pi$
$761$		−	45.1603i		−	1.63706i	0.574464	−	0.818530i	$-0.305210\pi$
$762$		0			0
$763$	20.2709	+	12.4536i	0.733855	+	0.450850i
$764$		0			0
$765$	4.95273	+	39.4775i	0.179066	+	1.42731i
$766$		0			0
$767$	−6.39888			−0.231050
$768$		0			0
$769$			36.2174i			1.30603i	0.757343	+	0.653017i	$0.226497\pi$
$769$			36.2174i			1.30603i	−0.757343	+	0.653017i	$0.773503\pi$
$770$		0			0
$771$		−	5.49409i		−	0.197865i
$772$		0			0
$773$	−25.8957			−0.931404			−0.465702	−	0.884942i	$-0.654198\pi$
$773$	−25.8957			−0.931404			−0.465702	+	0.884942i	$0.654198\pi$
$774$		0			0
$775$	24.1681	−	6.16109i	0.868144	−	0.221313i
$776$		0			0
$777$	2.12562	+	1.30589i	0.0762562	+	0.0468486i
$778$		0			0
$779$			34.8569i			1.24888i
$780$		0			0
$781$	6.27474			0.224528
$782$		0			0
$783$			8.97366i			0.320693i
$784$		0			0
$785$	−3.70198	−	29.5079i	−0.132129	−	1.05318i
$786$		0			0
$787$			7.96322i			0.283858i	0.989877	+	0.141929i	$0.0453305\pi$
$787$			7.96322i			0.283858i	−0.989877	+	0.141929i	$0.954670\pi$
$788$		0			0
$789$			3.55224i			0.126463i
$790$		0			0
$791$	10.6807	−	17.3852i	0.379763	−	0.618146i
$792$		0			0
$793$			41.6130i			1.47772i
$794$		0			0
$795$	−0.148735	−	1.18555i	−0.00527510	−	0.0420470i
$796$		0			0
$797$	−20.1208			−0.712714			−0.356357	−	0.934350i	$-0.615981\pi$
$797$	−20.1208			−0.712714			−0.356357	+	0.934350i	$0.615981\pi$
$798$		0			0
$799$		−	31.3315i		−	1.10843i
$800$		0			0
$801$		−	32.8840i		−	1.16190i
$802$		0			0
$803$			39.3156i			1.38742i
$804$		0			0
$805$	−15.4217	+	33.9750i	−0.543545	+	1.19746i
$806$		0			0
$807$	8.11627			0.285706
$808$		0			0
$809$	−26.5369			−0.932987			−0.466494	−	0.884525i	$-0.654483\pi$
$809$	−26.5369			−0.932987			−0.466494	+	0.884525i	$0.654483\pi$
$810$		0			0
$811$	−8.16912			−0.286857			−0.143428	−	0.989661i	$-0.545813\pi$
$811$	−8.16912			−0.286857			−0.143428	+	0.989661i	$0.545813\pi$
$812$		0			0
$813$		−	1.75138i		−	0.0614237i
$814$		0			0
$815$	5.90914	+	47.1009i	0.206988	+	1.64987i
$816$		0			0
$817$	29.2968			1.02497
$818$		0			0
$819$	31.9244	+	19.6130i	1.11553	+	0.685334i
$820$		0			0
$821$	2.12119			0.0740300			0.0370150	−	0.999315i	$-0.488215\pi$
$821$	2.12119			0.0740300			0.0370150	+	0.999315i	$0.488215\pi$
$822$		0			0
$823$	−51.3306			−1.78927			−0.894637	−	0.446794i	$-0.852566\pi$
$823$	−51.3306			−1.78927			−0.894637	+	0.446794i	$0.852566\pi$
$824$		0			0
$825$	7.76831	−	1.98035i	0.270458	−	0.0689469i
$826$		0			0
$827$	25.0785			0.872066			0.436033	−	0.899931i	$-0.356383\pi$
$827$	25.0785			0.872066			0.436033	+	0.899931i	$0.356383\pi$
$828$		0			0
$829$		−	4.71308i		−	0.163692i	−0.996645	−	0.0818460i	$-0.973918\pi$
$829$		−	4.71308i		−	0.163692i	0.996645	−	0.0818460i	$-0.0260816\pi$
$830$		0			0
$831$	0.405772			0.0140761
$832$		0			0
$833$	−19.4559	−	38.3990i	−0.674108	−	1.33045i
$834$		0			0
$835$	−22.3111	+	2.79908i	−0.772107	+	0.0968662i
$836$		0			0
$837$			9.59656i			0.331706i
$838$		0			0
$839$	−13.4627			−0.464784			−0.232392	−	0.972622i	$-0.574655\pi$
$839$	−13.4627			−0.464784			−0.232392	+	0.972622i	$0.574655\pi$
$840$		0			0
$841$	−7.24303			−0.249760
$842$		0			0
$843$			1.15305i			0.0397133i
$844$		0			0
$845$	−3.04921	−	24.3048i	−0.104896	−	0.836111i
$846$		0			0
$847$	29.5862	+	18.1765i	1.01659	+	0.624552i
$848$		0			0
$849$	0.882278			0.0302797
$850$		0			0
$851$		−	18.2170i		−	0.624470i
$852$		0			0
$853$	9.20013			0.315006			0.157503	−	0.987518i	$-0.449656\pi$
$853$	9.20013			0.315006			0.157503	+	0.987518i	$0.449656\pi$
$854$		0			0
$855$	−3.80225	−	30.3072i	−0.130034	−	1.03648i
$856$		0			0
$857$	−26.8592			−0.917493			−0.458747	−	0.888567i	$-0.651701\pi$
$857$	−26.8592			−0.917493			−0.458747	+	0.888567i	$0.651701\pi$
$858$		0			0
$859$	35.7268			1.21898			0.609491	−	0.792793i	$-0.291374\pi$
$859$	35.7268			1.21898			0.609491	+	0.792793i	$0.291374\pi$
$860$		0			0
$861$	−5.43333	−	3.33801i	−0.185167	−	0.113759i
$862$		0			0
$863$	−33.6371			−1.14502			−0.572511	−	0.819897i	$-0.694031\pi$
$863$	−33.6371			−1.14502			−0.572511	+	0.819897i	$0.694031\pi$
$864$		0			0
$865$	5.14508	+	41.0107i	0.174938	+	1.39441i
$866$		0			0
$867$		−	6.79535i		−	0.230782i
$868$		0			0
$869$	69.0518			2.34242
$870$		0			0
$871$	17.9880			0.609500
$872$		0			0
$873$	−17.7933			−0.602210
$874$		0			0
$875$	5.19373	−	29.1209i	0.175580	−	0.984465i
$876$		0			0
$877$		−	24.2932i		−	0.820322i	−0.912013	−	0.410161i	$-0.865472\pi$
$877$		−	24.2932i		−	0.820322i	0.912013	−	0.410161i	$-0.134528\pi$
$878$		0			0
$879$		−	1.17630i		−	0.0396756i
$880$		0			0
$881$		−	29.1258i		−	0.981272i	−0.871365	−	0.490636i	$-0.836764\pi$
$881$		−	29.1258i		−	0.981272i	0.871365	−	0.490636i	$-0.163236\pi$
$882$		0			0
$883$	−35.6866			−1.20095			−0.600475	−	0.799643i	$-0.705022\pi$
$883$	−35.6866			−1.20095			−0.600475	+	0.799643i	$0.705022\pi$
$884$		0			0
$885$	−0.946902	+	0.118796i	−0.0318298	+	0.00399327i
$886$		0			0
$887$		−	16.8816i		−	0.566830i	−0.958997	−	0.283415i	$-0.908533\pi$
$887$		−	16.8816i		−	0.566830i	0.958997	−	0.283415i	$-0.0914674\pi$
$888$		0			0
$889$	−6.89839	−	4.23808i	−0.231365	−	0.142141i
$890$		0			0
$891$		−	39.5500i		−	1.32498i
$892$		0			0
$893$			24.0534i			0.804917i
$894$		0			0
$895$	−17.9660	+	2.25396i	−0.600538	+	0.0753417i
$896$		0			0
$897$			10.0763i			0.336439i
$898$		0			0
$899$	23.2672			0.776005
$900$		0			0
$901$		−	10.0662i		−	0.335353i
$902$		0			0
$903$	−2.80556	+	4.56665i	−0.0933631	+	0.151969i
$904$		0			0
$905$	−14.8996	+	1.86926i	−0.495279	+	0.0621362i
$906$		0			0
$907$	−6.57251			−0.218237			−0.109118	−	0.994029i	$-0.534803\pi$
$907$	−6.57251			−0.218237			−0.109118	+	0.994029i	$0.534803\pi$
$908$		0			0
$909$			13.0799i			0.433832i
$910$		0			0
$911$			56.4965i			1.87181i	0.352249	+	0.935907i	$0.385417\pi$
$911$			56.4965i			1.87181i	−0.352249	+	0.935907i	$0.614583\pi$
$912$		0			0
$913$	2.37731			0.0786773
$914$		0			0
$915$	0.772547	+	6.15786i	0.0255396	+	0.203573i
$916$		0			0
$917$	18.3551	+	11.2766i	0.606139	+	0.372386i
$918$		0			0
$919$		−	27.5287i		−	0.908087i	−0.890979	−	0.454044i	$-0.849981\pi$
$919$		−	27.5287i		−	0.908087i	0.890979	−	0.454044i	$-0.150019\pi$
$920$		0			0
$921$	−10.2156			−0.336617
$922$		0			0
$923$		−	6.25264i		−	0.205808i
$924$		0			0
$925$	3.56766	+	13.9949i	0.117304	+	0.460148i
$926$		0			0
$927$		−	49.1676i		−	1.61488i
$928$		0			0
$929$			6.50100i			0.213291i	0.994297	+	0.106645i	$0.0340110\pi$
$929$			6.50100i			0.213291i	−0.994297	+	0.106645i	$0.965989\pi$
$930$		0			0
$931$	14.9365	+	29.4792i	0.489523	+	0.966142i
$932$		0			0
$933$		−	9.47670i		−	0.310253i
$934$		0			0
$935$	67.0135	−	8.40732i	2.19158	−	0.274949i
$936$		0			0
$937$	15.9382			0.520679			0.260339	−	0.965517i	$-0.416166\pi$
$937$	15.9382			0.520679			0.260339	+	0.965517i	$0.416166\pi$
$938$		0			0
$939$		−	4.32417i		−	0.141114i
$940$		0			0
$941$			34.8946i			1.13753i	0.822499	+	0.568766i	$0.192579\pi$
$941$			34.8946i			1.13753i	−0.822499	+	0.568766i	$0.807421\pi$
$942$		0			0
$943$			46.5647i			1.51635i
$944$		0			0
$945$	10.3639	+	4.70434i	0.337138	+	0.153032i
$946$		0			0
$947$	−21.2357			−0.690067			−0.345033	−	0.938590i	$-0.612132\pi$
$947$	−21.2357			−0.690067			−0.345033	+	0.938590i	$0.612132\pi$
$948$		0			0
$949$	39.1771			1.27174
$950$		0			0
$951$	7.36162			0.238717
$952$		0			0
$953$			21.2557i			0.688540i	0.938871	+	0.344270i	$0.111874\pi$
$953$			21.2557i			0.688540i	−0.938871	+	0.344270i	$0.888126\pi$
$954$		0			0
$955$	−18.8630	+	2.36649i	−0.610392	+	0.0765780i
$956$		0			0
$957$	7.47874			0.241753
$958$		0			0
$959$	28.2242	−	45.9409i	0.911405	−	1.48351i
$960$		0			0
$961$	−6.11773			−0.197346
$962$		0			0
$963$	10.6341			0.342680
$964$		0			0
$965$	−57.1230	+	7.16649i	−1.83886	+	0.230697i
$966$		0			0
$967$	−36.9819			−1.18926			−0.594628	−	0.804001i	$-0.702701\pi$
$967$	−36.9819			−1.18926			−0.594628	+	0.804001i	$0.702701\pi$
$968$		0			0
$969$			9.47721i			0.304452i
$970$		0			0
$971$	−23.1431			−0.742697			−0.371349	−	0.928493i	$-0.621105\pi$
$971$	−23.1431			−0.742697			−0.371349	+	0.928493i	$0.621105\pi$
$972$		0			0
$973$	37.0653	+	22.7714i	1.18826	+	0.730017i
$974$		0			0
$975$	−1.97337	−	7.74095i	−0.0631986	−	0.247909i
$976$		0			0
$977$			39.5133i			1.26414i	0.774910	+	0.632071i	$0.217795\pi$
$977$			39.5133i			1.26414i	−0.774910	+	0.632071i	$0.782205\pi$
$978$		0			0
$979$	−55.8210			−1.78405
$980$		0			0
$981$	−26.0180			−0.830690
$982$		0			0
$983$		−	40.7330i		−	1.29918i	−0.760284	−	0.649591i	$-0.774940\pi$
$983$		−	40.7330i		−	1.29918i	0.760284	−	0.649591i	$-0.225060\pi$
$984$		0			0
$985$	10.9804	−	1.37757i	0.349865	−	0.0438930i
$986$		0			0
$987$	−3.74934	−	2.30344i	−0.119343	−	0.0733192i
$988$		0			0
$989$	39.1371			1.24449
$990$		0			0
$991$		−	5.37018i		−	0.170589i	−0.996356	−	0.0852947i	$-0.972817\pi$
$991$		−	5.37018i		−	0.170589i	0.996356	−	0.0852947i	$-0.0271832\pi$
$992$		0			0
$993$	−2.01308			−0.0638832
$994$		0			0
$995$	−4.21269	−	33.5788i	−0.133551	−	1.06452i
$996$		0			0
$997$	−1.49749			−0.0474260			−0.0237130	−	0.999719i	$-0.507549\pi$
$997$	−1.49749			−0.0474260			−0.0237130	+	0.999719i	$0.507549\pi$
$998$		0			0
$999$	−5.55701			−0.175816

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.2.e.g.2239.11		48
4.3	odd	2	inner	2240.2.e.g.2239.22		48
5.4	even	2	inner	2240.2.e.g.2239.24		48
7.6	odd	2	inner	2240.2.e.g.2239.14		48
8.3	odd	2		1120.2.e.a.1119.13	yes	48
8.5	even	2		1120.2.e.a.1119.24	yes	48
20.19	odd	2	inner	2240.2.e.g.2239.13		48
28.27	even	2	inner	2240.2.e.g.2239.23		48
35.34	odd	2	inner	2240.2.e.g.2239.21		48
40.19	odd	2		1120.2.e.a.1119.22	yes	48
40.29	even	2		1120.2.e.a.1119.11	✓	48
56.13	odd	2		1120.2.e.a.1119.21	yes	48
56.27	even	2		1120.2.e.a.1119.12	yes	48
140.139	even	2	inner	2240.2.e.g.2239.12		48
280.69	odd	2		1120.2.e.a.1119.14	yes	48
280.139	even	2		1120.2.e.a.1119.23	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1120.2.e.a.1119.11	✓	48	40.29	even	2
1120.2.e.a.1119.12	yes	48	56.27	even	2
1120.2.e.a.1119.13	yes	48	8.3	odd	2
1120.2.e.a.1119.14	yes	48	280.69	odd	2
1120.2.e.a.1119.21	yes	48	56.13	odd	2
1120.2.e.a.1119.22	yes	48	40.19	odd	2
1120.2.e.a.1119.23	yes	48	280.139	even	2
1120.2.e.a.1119.24	yes	48	8.5	even	2
2240.2.e.g.2239.11		48	1.1	even	1	trivial
2240.2.e.g.2239.12		48	140.139	even	2	inner
2240.2.e.g.2239.13		48	20.19	odd	2	inner
2240.2.e.g.2239.14		48	7.6	odd	2	inner
2240.2.e.g.2239.21		48	35.34	odd	2	inner
2240.2.e.g.2239.22		48	4.3	odd	2	inner
2240.2.e.g.2239.23		48	28.27	even	2	inner
2240.2.e.g.2239.24		48	5.4	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.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.326439i q^{3} +(-0.278349 - 2.21868i) q^{5} +(-2.25431 - 1.38495i) q^{7} +2.89344 q^{9} +O(q^{10})\)	\(q-0.326439i q^{3} +(-0.278349 - 2.21868i) q^{5} +(-2.25431 - 1.38495i) q^{7} +2.89344 q^{9} -4.91165i q^{11} -4.89435 q^{13} +(-0.724262 + 0.0908638i) q^{15} -6.14952 q^{17} +4.72104 q^{19} +(-0.452102 + 0.735894i) q^{21} +6.30675 q^{23} +(-4.84504 + 1.23513i) q^{25} -1.92385i q^{27} -4.66444 q^{29} -4.98821 q^{31} -1.60335 q^{33} +(-2.44528 + 5.38708i) q^{35} -2.88849i q^{37} +1.59771i q^{39} +7.38331i q^{41} +6.20558 q^{43} +(-0.805384 - 6.41960i) q^{45} +5.09495i q^{47} +(3.16381 + 6.24422i) q^{49} +2.00744i q^{51} +1.63690i q^{53} +(-10.8974 + 1.36715i) q^{55} -1.54113i q^{57} +1.30740 q^{59} -8.50226i q^{61} +(-6.52270 - 4.00727i) q^{63} +(1.36233 + 10.8590i) q^{65} -3.67526 q^{67} -2.05877i q^{69} +1.27752i q^{71} -8.00456 q^{73} +(0.403195 + 1.58161i) q^{75} +(-6.80240 + 11.0724i) q^{77} +14.0588i q^{79} +8.05229 q^{81} +0.484014i q^{83} +(1.71171 + 13.6438i) q^{85} +1.52265i q^{87} -11.3650i q^{89} +(11.0334 + 6.77844i) q^{91} +1.62835i q^{93} +(-1.31409 - 10.4745i) q^{95} -6.14952 q^{97} -14.2115i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 48 q^{9}+O(q^{10}) \) 48 * q - 48 * q^9	\( 48 q - 48 q^{9} - 8 q^{21} - 16 q^{25} - 16 q^{29} + 24 q^{49} - 16 q^{65} - 32 q^{81} - 32 q^{85}+O(q^{100}) \) 48 * q - 48 * q^9 - 8 * q^21 - 16 * q^25 - 16 * q^29 + 24 * q^49 - 16 * q^65 - 32 * q^81 - 32 * q^85

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