LMFDB - Embedded newform 2240.2.b.h.1121.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2240,2,Mod(1121,2240)]

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([0, 1, 0, 0]))

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

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

chi := DirichletCharacter("2240.1121");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2240 = 2^{6} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2240.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$17.8864900528$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 6 x^{11} + 35 x^{10} - 120 x^{9} + 362 x^{8} - 794 x^{7} + 1463 x^{6} - 2048 x^{5} + 2258 x^{4} + \cdots + 61 $ x^12 - 6x^11 + 35x^10 - 120x^9 + 362x^8 - 794x^7 + 1463x^6 - 2048x^5 + 2258x^4 - 1822x^3 + 1035x^2 - 364*x + 61
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1121.8
Root			$0.500000 + 1.16542i$ of defining polynomial
Character	$\chi$	$=$	2240.1121
Dual form			2240.2.b.h.1121.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.639640i q^{3} -1.00000i q^{5} +1.00000 q^{7} +2.59086 q^{9} +O(q^{10})$	$q+0.639640i q^{3} -1.00000i q^{5} +1.00000 q^{7} +2.59086 q^{9} +4.91036i q^{11} -5.17077i q^{13} +0.639640 q^{15} +6.70253 q^{17} -7.89632i q^{19} +0.639640i q^{21} -1.23465 q^{23} -1.00000 q^{25} +3.57614i q^{27} +2.97801i q^{29} -3.83343 q^{31} -3.14087 q^{33} -1.00000i q^{35} -5.97803i q^{37} +3.30743 q^{39} -7.86536 q^{41} +0.0848558i q^{43} -2.59086i q^{45} +9.45622 q^{47} +1.00000 q^{49} +4.28720i q^{51} +4.91519i q^{53} +4.91036 q^{55} +5.05080 q^{57} +0.943282i q^{59} -11.4824i q^{61} +2.59086 q^{63} -5.17077 q^{65} +10.4233i q^{67} -0.789730i q^{69} +10.0227 q^{71} +10.1567 q^{73} -0.639640i q^{75} +4.91036i q^{77} +4.36415 q^{79} +5.48514 q^{81} +2.56685i q^{83} -6.70253i q^{85} -1.90486 q^{87} +3.23465 q^{89} -5.17077i q^{91} -2.45202i q^{93} -7.89632 q^{95} +4.22568 q^{97} +12.7221i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{7} - 20 q^{9}+O(q^{10}) $ 12 * q + 12 * q^7 - 20 * q^9	$ 12 q + 12 q^{7} - 20 q^{9} + 16 q^{17} + 8 q^{23} - 12 q^{25} - 8 q^{31} + 72 q^{33} - 32 q^{39} - 32 q^{47} + 12 q^{49} + 8 q^{55} + 8 q^{57} - 20 q^{63} + 8 q^{65} + 48 q^{71} + 8 q^{73} - 16 q^{79} + 92 q^{81} - 32 q^{87} + 16 q^{89} + 32 q^{97}+O(q^{100}) $ 12 * q + 12 * q^7 - 20 * q^9 + 16 * q^17 + 8 * q^23 - 12 * q^25 - 8 * q^31 + 72 * q^33 - 32 * q^39 - 32 * q^47 + 12 * q^49 + 8 * q^55 + 8 * q^57 - 20 * q^63 + 8 * q^65 + 48 * q^71 + 8 * q^73 - 16 * q^79 + 92 * q^81 - 32 * q^87 + 16 * q^89 + 32 * q^97

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0.639640i	0.369296i	0.982805	+	0.184648i	$0.0591146\pi$
$3$	0.639640i	0.369296i	−0.982805	+	0.184648i	$0.940885\pi$
$4$	0	0
$5$	− 1.00000i	− 0.447214i
$5$	− 1.00000i	− 0.447214i
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	2.59086	0.863620
$10$	0	0
$11$	4.91036i	1.48053i	0.672315	+	0.740265i	$0.265300\pi$
$11$	4.91036i	1.48053i	−0.672315	+	0.740265i	$0.734700\pi$
$12$	0	0
$13$	− 5.17077i	− 1.43411i	−0.697014	−	0.717057i	$-0.745489\pi$
$13$	− 5.17077i	− 1.43411i	0.697014	−	0.717057i	$-0.254511\pi$
$14$	0	0
$15$	0.639640	0.165154
$16$	0	0
$17$	6.70253	1.62560	0.812801	−	0.582542i	$-0.197942\pi$
$17$	6.70253	1.62560	0.812801	+	0.582542i	$0.197942\pi$
$18$	0	0
$19$	− 7.89632i	− 1.81154i	−0.423770	−	0.905770i	$-0.639294\pi$
$19$	− 7.89632i	− 1.81154i	0.423770	−	0.905770i	$-0.360706\pi$
$20$	0	0
$21$	0.639640i	0.139581i
$22$	0	0
$23$	−1.23465	−0.257442	−0.128721	−	0.991681i	$-0.541087\pi$
$23$	−1.23465	−0.257442	−0.128721	+	0.991681i	$0.541087\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	3.57614i	0.688228i
$28$	0	0
$29$	2.97801i	0.553003i	0.961013	+	0.276502i	$0.0891751\pi$
$29$	2.97801i	0.553003i	−0.961013	+	0.276502i	$0.910825\pi$
$30$	0	0
$31$	−3.83343	−0.688505	−0.344252	−	0.938877i	$-0.611868\pi$
$31$	−3.83343	−0.688505	−0.344252	+	0.938877i	$0.611868\pi$
$32$	0	0
$33$	−3.14087	−0.546754
$34$	0	0
$35$	− 1.00000i	− 0.169031i
$36$	0	0
$37$	− 5.97803i	− 0.982782i	−0.870939	−	0.491391i	$-0.836489\pi$
$37$	− 5.97803i	− 0.982782i	0.870939	−	0.491391i	$-0.163511\pi$
$38$	0	0
$39$	3.30743	0.529613
$40$	0	0
$41$	−7.86536	−1.22836	−0.614182	−	0.789165i	$-0.710514\pi$
$41$	−7.86536	−1.22836	−0.614182	+	0.789165i	$0.710514\pi$
$42$	0	0
$43$	0.0848558i	0.0129404i	0.999979	+	0.00647019i	$0.00205954\pi$
$43$	0.0848558i	0.0129404i	−0.999979	+	0.00647019i	$0.997940\pi$
$44$	0	0
$45$	− 2.59086i	− 0.386223i
$46$	0	0
$47$	9.45622	1.37933	0.689666	−	0.724128i	$-0.257757\pi$
$47$	9.45622	1.37933	0.689666	+	0.724128i	$0.257757\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	4.28720i	0.600328i
$52$	0	0
$53$	4.91519i	0.675153i	0.941298	+	0.337576i	$0.109607\pi$
$53$	4.91519i	0.675153i	−0.941298	+	0.337576i	$0.890393\pi$
$54$	0	0
$55$	4.91036	0.662113
$56$	0	0
$57$	5.05080	0.668995
$58$	0	0
$59$	0.943282i	0.122805i	0.998113	+	0.0614025i	$0.0195573\pi$
$59$	0.943282i	0.122805i	−0.998113	+	0.0614025i	$0.980443\pi$
$60$	0	0
$61$	− 11.4824i	− 1.47016i	−0.677978	−	0.735082i	$-0.737143\pi$
$61$	− 11.4824i	− 1.47016i	0.677978	−	0.735082i	$-0.262857\pi$
$62$	0	0
$63$	2.59086	0.326418
$64$	0	0
$65$	−5.17077	−0.641356
$66$	0	0
$67$	10.4233i	1.27340i	0.771110	+	0.636702i	$0.219702\pi$
$67$	10.4233i	1.27340i	−0.771110	+	0.636702i	$0.780298\pi$
$68$	0	0
$69$	− 0.789730i	− 0.0950723i
$70$	0	0
$71$	10.0227	1.18947	0.594736	−	0.803921i	$-0.297257\pi$
$71$	10.0227	1.18947	0.594736	+	0.803921i	$0.297257\pi$
$72$	0	0
$73$	10.1567	1.18875	0.594377	−	0.804186i	$-0.297399\pi$
$73$	10.1567	1.18875	0.594377	+	0.804186i	$0.297399\pi$
$74$	0	0
$75$	− 0.639640i	− 0.0738593i
$76$	0	0
$77$	4.91036i	0.559588i
$78$	0	0
$79$	4.36415	0.491005	0.245503	−	0.969396i	$-0.421047\pi$
$79$	4.36415	0.491005	0.245503	+	0.969396i	$0.421047\pi$
$80$	0	0
$81$	5.48514	0.609460
$82$	0	0
$83$	2.56685i	0.281749i	0.990027	+	0.140874i	$0.0449914\pi$
$83$	2.56685i	0.281749i	−0.990027	+	0.140874i	$0.955009\pi$
$84$	0	0
$85$	− 6.70253i	− 0.726991i
$86$	0	0
$87$	−1.90486	−0.204222
$88$	0	0
$89$	3.23465	0.342872	0.171436	−	0.985195i	$-0.445159\pi$
$89$	3.23465	0.342872	0.171436	+	0.985195i	$0.445159\pi$
$90$	0	0
$91$	− 5.17077i	− 0.542044i
$92$	0	0
$93$	− 2.45202i	− 0.254262i
$94$	0	0
$95$	−7.89632	−0.810145
$96$	0	0
$97$	4.22568	0.429053	0.214526	−	0.976718i	$-0.431179\pi$
$97$	4.22568	0.429053	0.214526	+	0.976718i	$0.431179\pi$
$98$	0	0
$99$	12.7221i	1.27862i
$100$	0	0
$101$	− 10.8592i	− 1.08053i	−0.841495	−	0.540265i	$-0.818324\pi$
$101$	− 10.8592i	− 1.08053i	0.841495	−	0.540265i	$-0.181676\pi$
$102$	0	0
$103$	−9.45622	−0.931749	−0.465875	−	0.884851i	$-0.654260\pi$
$103$	−9.45622	−0.931749	−0.465875	+	0.884851i	$0.654260\pi$
$104$	0	0
$105$	0.639640	0.0624225
$106$	0	0
$107$	− 4.37929i	− 0.423362i	−0.977339	−	0.211681i	$-0.932106\pi$
$107$	− 4.37929i	− 0.423362i	0.977339	−	0.211681i	$-0.0678938\pi$
$108$	0	0
$109$	− 9.82450i	− 0.941017i	−0.882395	−	0.470508i	$-0.844071\pi$
$109$	− 9.82450i	− 0.941017i	0.882395	−	0.470508i	$-0.155929\pi$
$110$	0	0
$111$	3.82379	0.362938
$112$	0	0
$113$	14.7994	1.39221	0.696104	−	0.717941i	$-0.254915\pi$
$113$	14.7994	1.39221	0.696104	+	0.717941i	$0.254915\pi$
$114$	0	0
$115$	1.23465i	0.115132i
$116$	0	0
$117$	− 13.3968i	− 1.23853i
$118$	0	0
$119$	6.70253	0.614419
$120$	0	0
$121$	−13.1117	−1.19197
$122$	0	0
$123$	− 5.03100i	− 0.453630i
$124$	0	0
$125$	1.00000i	0.0894427i
$126$	0	0
$127$	−8.46100	−0.750792	−0.375396	−	0.926864i	$-0.622493\pi$
$127$	−8.46100	−0.750792	−0.375396	+	0.926864i	$0.622493\pi$
$128$	0	0
$129$	−0.0542771	−0.00477884
$130$	0	0
$131$	− 5.16595i	− 0.451351i	−0.974202	−	0.225676i	$-0.927541\pi$
$131$	− 5.16595i	− 0.451351i	0.974202	−	0.225676i	$-0.0724590\pi$
$132$	0	0
$133$	− 7.89632i	− 0.684698i
$134$	0	0
$135$	3.57614	0.307785
$136$	0	0
$137$	−11.4824	−0.981004	−0.490502	−	0.871440i	$-0.663187\pi$
$137$	−11.4824	−0.981004	−0.490502	+	0.871440i	$0.663187\pi$
$138$	0	0
$139$	− 6.22256i	− 0.527791i	−0.964551	−	0.263895i	$-0.914993\pi$
$139$	− 6.22256i	− 0.527791i	0.964551	−	0.263895i	$-0.0850074\pi$
$140$	0	0
$141$	6.04858i	0.509382i
$142$	0	0
$143$	25.3904	2.12325
$144$	0	0
$145$	2.97801	0.247311
$146$	0	0
$147$	0.639640i	0.0527566i
$148$	0	0
$149$	− 14.5951i	− 1.19567i	−0.801618	−	0.597837i	$-0.796027\pi$
$149$	− 14.5951i	− 1.19567i	0.801618	−	0.597837i	$-0.203973\pi$
$150$	0	0
$151$	20.1568	1.64034	0.820168	−	0.572123i	$-0.193880\pi$
$151$	20.1568	1.64034	0.820168	+	0.572123i	$0.193880\pi$
$152$	0	0
$153$	17.3653	1.40390
$154$	0	0
$155$	3.83343i	0.307909i
$156$	0	0
$157$	21.5610i	1.72076i	0.509656	+	0.860378i	$0.329772\pi$
$157$	21.5610i	1.72076i	−0.509656	+	0.860378i	$0.670228\pi$
$158$	0	0
$159$	−3.14395	−0.249331
$160$	0	0
$161$	−1.23465	−0.0973039
$162$	0	0
$163$	11.8124i	0.925221i	0.886562	+	0.462611i	$0.153087\pi$
$163$	11.8124i	0.925221i	−0.886562	+	0.462611i	$0.846913\pi$
$164$	0	0
$165$	3.14087i	0.244516i
$166$	0	0
$167$	0.823415	0.0637177	0.0318589	−	0.999492i	$-0.489857\pi$
$167$	0.823415	0.0637177	0.0318589	+	0.999492i	$0.489857\pi$
$168$	0	0
$169$	−13.7369	−1.05668
$170$	0	0
$171$	− 20.4583i	− 1.56448i
$172$	0	0
$173$	4.40608i	0.334988i	0.985873	+	0.167494i	$0.0535675\pi$
$173$	4.40608i	0.334988i	−0.985873	+	0.167494i	$0.946432\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	−0.603361	−0.0453514
$178$	0	0
$179$	23.0540i	1.72313i	0.507643	+	0.861567i	$0.330517\pi$
$179$	23.0540i	1.72313i	−0.507643	+	0.861567i	$0.669483\pi$
$180$	0	0
$181$	16.4329i	1.22145i	0.791843	+	0.610724i	$0.209122\pi$
$181$	16.4329i	1.22145i	−0.791843	+	0.610724i	$0.790878\pi$
$182$	0	0
$183$	7.34457	0.542926
$184$	0	0
$185$	−5.97803	−0.439513
$186$	0	0
$187$	32.9118i	2.40675i
$188$	0	0
$189$	3.57614i	0.260126i
$190$	0	0
$191$	24.8458	1.79778	0.898889	−	0.438177i	$-0.144376\pi$
$191$	24.8458	1.79778	0.898889	+	0.438177i	$0.144376\pi$
$192$	0	0
$193$	−8.76919	−0.631220	−0.315610	−	0.948889i	$-0.602209\pi$
$193$	−8.76919	−0.631220	−0.315610	+	0.948889i	$0.602209\pi$
$194$	0	0
$195$	− 3.30743i	− 0.236850i
$196$	0	0
$197$	− 11.2319i	− 0.800240i	−0.916463	−	0.400120i	$-0.868968\pi$
$197$	− 11.2319i	− 0.800240i	0.916463	−	0.400120i	$-0.131032\pi$
$198$	0	0
$199$	13.1377	0.931310	0.465655	−	0.884966i	$-0.345819\pi$
$199$	13.1377	0.931310	0.465655	+	0.884966i	$0.345819\pi$
$200$	0	0
$201$	−6.66713	−0.470263
$202$	0	0
$203$	2.97801i	0.209016i
$204$	0	0
$205$	7.86536i	0.549341i
$206$	0	0
$207$	−3.19880	−0.222332
$208$	0	0
$209$	38.7738	2.68204
$210$	0	0
$211$	− 24.7387i	− 1.70308i	−0.524289	−	0.851540i	$-0.675669\pi$
$211$	− 24.7387i	− 1.70308i	0.524289	−	0.851540i	$-0.324331\pi$
$212$	0	0
$213$	6.41089i	0.439267i
$214$	0	0
$215$	0.0848558	0.00578712
$216$	0	0
$217$	−3.83343	−0.260230
$218$	0	0
$219$	6.49665i	0.439003i
$220$	0	0
$221$	− 34.6572i	− 2.33130i
$222$	0	0
$223$	−0.631808	−0.0423090	−0.0211545	−	0.999776i	$-0.506734\pi$
$223$	−0.631808	−0.0423090	−0.0211545	+	0.999776i	$0.506734\pi$
$224$	0	0
$225$	−2.59086	−0.172724
$226$	0	0
$227$	3.88118i	0.257603i	0.991670	+	0.128801i	$0.0411130\pi$
$227$	3.88118i	0.257603i	−0.991670	+	0.128801i	$0.958887\pi$
$228$	0	0
$229$	22.7081i	1.50060i	0.661100	+	0.750298i	$0.270090\pi$
$229$	22.7081i	1.50060i	−0.661100	+	0.750298i	$0.729910\pi$
$230$	0	0
$231$	−3.14087	−0.206654
$232$	0	0
$233$	−23.7768	−1.55767	−0.778835	−	0.627229i	$-0.784189\pi$
$233$	−23.7768	−1.55767	−0.778835	+	0.627229i	$0.784189\pi$
$234$	0	0
$235$	− 9.45622i	− 0.616856i
$236$	0	0
$237$	2.79149i	0.181326i
$238$	0	0
$239$	−17.2849	−1.11807	−0.559033	−	0.829145i	$-0.688828\pi$
$239$	−17.2849	−1.11807	−0.559033	+	0.829145i	$0.688828\pi$
$240$	0	0
$241$	12.9749	0.835789	0.417894	−	0.908496i	$-0.362768\pi$
$241$	12.9749	0.835789	0.417894	+	0.908496i	$0.362768\pi$
$242$	0	0
$243$	14.2369i	0.913299i
$244$	0	0
$245$	− 1.00000i	− 0.0638877i
$246$	0	0
$247$	−40.8301	−2.59796
$248$	0	0
$249$	−1.64186	−0.104049
$250$	0	0
$251$	− 15.2023i	− 0.959560i	−0.877389	−	0.479780i	$-0.840716\pi$
$251$	− 15.2023i	− 0.959560i	0.877389	−	0.479780i	$-0.159284\pi$
$252$	0	0
$253$	− 6.06257i	− 0.381151i
$254$	0	0
$255$	4.28720	0.268475
$256$	0	0
$257$	24.1691	1.50762	0.753812	−	0.657090i	$-0.228213\pi$
$257$	24.1691	1.50762	0.753812	+	0.657090i	$0.228213\pi$
$258$	0	0
$259$	− 5.97803i	− 0.371457i
$260$	0	0
$261$	7.71562i	0.477585i
$262$	0	0
$263$	−27.6305	−1.70377	−0.851884	−	0.523731i	$-0.824540\pi$
$263$	−27.6305	−1.70377	−0.851884	+	0.523731i	$0.824540\pi$
$264$	0	0
$265$	4.91519	0.301938
$266$	0	0
$267$	2.06901i	0.126621i
$268$	0	0
$269$	7.92785i	0.483369i	0.970355	+	0.241685i	$0.0777000\pi$
$269$	7.92785i	0.483369i	−0.970355	+	0.241685i	$0.922300\pi$
$270$	0	0
$271$	−15.7874	−0.959016	−0.479508	−	0.877537i	$-0.659185\pi$
$271$	−15.7874	−0.959016	−0.479508	+	0.877537i	$0.659185\pi$
$272$	0	0
$273$	3.30743	0.200175
$274$	0	0
$275$	− 4.91036i	− 0.296106i
$276$	0	0
$277$	24.2206i	1.45527i	0.685962	+	0.727637i	$0.259381\pi$
$277$	24.2206i	1.45527i	−0.685962	+	0.727637i	$0.740619\pi$
$278$	0	0
$279$	−9.93189	−0.594607
$280$	0	0
$281$	−10.8241	−0.645712	−0.322856	−	0.946448i	$-0.604643\pi$
$281$	−10.8241	−0.645712	−0.322856	+	0.946448i	$0.604643\pi$
$282$	0	0
$283$	4.51399i	0.268329i	0.990959	+	0.134164i	$0.0428350\pi$
$283$	4.51399i	0.268329i	−0.990959	+	0.134164i	$0.957165\pi$
$284$	0	0
$285$	− 5.05080i	− 0.299184i
$286$	0	0
$287$	−7.86536	−0.464278
$288$	0	0
$289$	27.9238	1.64258
$290$	0	0
$291$	2.70291i	0.158448i
$292$	0	0
$293$	− 6.99150i	− 0.408448i	−0.978924	−	0.204224i	$-0.934533\pi$
$293$	− 6.99150i	− 0.408448i	0.978924	−	0.204224i	$-0.0654671\pi$
$294$	0	0
$295$	0.943282	0.0549200
$296$	0	0
$297$	−17.5601	−1.01894
$298$	0	0
$299$	6.38409i	0.369201i
$300$	0	0
$301$	0.0848558i	0.00489101i
$302$	0	0
$303$	6.94597	0.399036
$304$	0	0
$305$	−11.4824	−0.657478
$306$	0	0
$307$	11.4786i	0.655117i	0.944831	+	0.327559i	$0.106226\pi$
$307$	11.4786i	0.655117i	−0.944831	+	0.327559i	$0.893774\pi$
$308$	0	0
$309$	− 6.04858i	− 0.344092i
$310$	0	0
$311$	−9.83975	−0.557961	−0.278980	−	0.960297i	$-0.589996\pi$
$311$	−9.83975	−0.557961	−0.278980	+	0.960297i	$0.589996\pi$
$312$	0	0
$313$	−34.7390	−1.96357	−0.981783	−	0.190005i	$-0.939150\pi$
$313$	−34.7390	−1.96357	−0.981783	+	0.190005i	$0.939150\pi$
$314$	0	0
$315$	− 2.59086i	− 0.145978i
$316$	0	0
$317$	− 2.49816i	− 0.140311i	−0.997536	−	0.0701553i	$-0.977651\pi$
$317$	− 2.49816i	− 0.140311i	0.997536	−	0.0701553i	$-0.0223495\pi$
$318$	0	0
$319$	−14.6231	−0.818738
$320$	0	0
$321$	2.80117	0.156346
$322$	0	0
$323$	− 52.9253i	− 2.94484i
$324$	0	0
$325$	5.17077i	0.286823i
$326$	0	0
$327$	6.28415	0.347514
$328$	0	0
$329$	9.45622	0.521338
$330$	0	0
$331$	− 2.35731i	− 0.129569i	−0.997899	−	0.0647847i	$-0.979364\pi$
$331$	− 2.35731i	− 0.129569i	0.997899	−	0.0647847i	$-0.0206361\pi$
$332$	0	0
$333$	− 15.4882i	− 0.848750i
$334$	0	0
$335$	10.4233	0.569484
$336$	0	0
$337$	12.7564	0.694885	0.347443	−	0.937701i	$-0.387050\pi$
$337$	12.7564	0.694885	0.347443	+	0.937701i	$0.387050\pi$
$338$	0	0
$339$	9.46627i	0.514137i
$340$	0	0
$341$	− 18.8235i	− 1.01935i
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−0.789730	−0.0425176
$346$	0	0
$347$	24.3645i	1.30796i	0.756514	+	0.653978i	$0.226901\pi$
$347$	24.3645i	1.30796i	−0.756514	+	0.653978i	$0.773099\pi$
$348$	0	0
$349$	25.8375i	1.38305i	0.722353	+	0.691524i	$0.243060\pi$
$349$	25.8375i	1.38305i	−0.722353	+	0.691524i	$0.756940\pi$
$350$	0	0
$351$	18.4914	0.986998
$352$	0	0
$353$	1.38674	0.0738086	0.0369043	−	0.999319i	$-0.488250\pi$
$353$	1.38674	0.0738086	0.0369043	+	0.999319i	$0.488250\pi$
$354$	0	0
$355$	− 10.0227i	− 0.531948i
$356$	0	0
$357$	4.28720i	0.226903i
$358$	0	0
$359$	16.6542	0.878977	0.439488	−	0.898248i	$-0.355160\pi$
$359$	16.6542	0.878977	0.439488	+	0.898248i	$0.355160\pi$
$360$	0	0
$361$	−43.3518	−2.28167
$362$	0	0
$363$	− 8.38675i	− 0.440190i
$364$	0	0
$365$	− 10.1567i	− 0.531627i
$366$	0	0
$367$	−13.3003	−0.694270	−0.347135	−	0.937815i	$-0.612845\pi$
$367$	−13.3003	−0.694270	−0.347135	+	0.937815i	$0.612845\pi$
$368$	0	0
$369$	−20.3781	−1.06084
$370$	0	0
$371$	4.91519i	0.255184i
$372$	0	0
$373$	26.1307i	1.35299i	0.736445	+	0.676497i	$0.236503\pi$
$373$	26.1307i	1.35299i	−0.736445	+	0.676497i	$0.763497\pi$
$374$	0	0
$375$	−0.639640	−0.0330309
$376$	0	0
$377$	15.3986	0.793070
$378$	0	0
$379$	− 20.6212i	− 1.05924i	−0.848235	−	0.529620i	$-0.822335\pi$
$379$	− 20.6212i	− 1.05924i	0.848235	−	0.529620i	$-0.177665\pi$
$380$	0	0
$381$	− 5.41199i	− 0.277265i
$382$	0	0
$383$	−11.6050	−0.592985	−0.296493	−	0.955035i	$-0.595817\pi$
$383$	−11.6050	−0.592985	−0.296493	+	0.955035i	$0.595817\pi$
$384$	0	0
$385$	4.91036	0.250255
$386$	0	0
$387$	0.219850i	0.0111756i
$388$	0	0
$389$	− 10.2394i	− 0.519156i	−0.965722	−	0.259578i	$-0.916417\pi$
$389$	− 10.2394i	− 0.519156i	0.965722	−	0.259578i	$-0.0835834\pi$
$390$	0	0
$391$	−8.27526	−0.418498
$392$	0	0
$393$	3.30435	0.166682
$394$	0	0
$395$	− 4.36415i	− 0.219584i
$396$	0	0
$397$	− 25.6402i	− 1.28684i	−0.765512	−	0.643421i	$-0.777514\pi$
$397$	− 25.6402i	− 1.28684i	0.765512	−	0.643421i	$-0.222486\pi$
$398$	0	0
$399$	5.05080	0.252856
$400$	0	0
$401$	−18.7987	−0.938762	−0.469381	−	0.882996i	$-0.655523\pi$
$401$	−18.7987	−0.938762	−0.469381	+	0.882996i	$0.655523\pi$
$402$	0	0
$403$	19.8218i	0.987395i
$404$	0	0
$405$	− 5.48514i	− 0.272559i
$406$	0	0
$407$	29.3543	1.45504
$408$	0	0
$409$	6.95642	0.343973	0.171986	−	0.985099i	$-0.444982\pi$
$409$	6.95642	0.343973	0.171986	+	0.985099i	$0.444982\pi$
$410$	0	0
$411$	− 7.34457i	− 0.362281i
$412$	0	0
$413$	0.943282i	0.0464159i
$414$	0	0
$415$	2.56685	0.126002
$416$	0	0
$417$	3.98020	0.194911
$418$	0	0
$419$	− 26.7874i	− 1.30865i	−0.756213	−	0.654326i	$-0.772952\pi$
$419$	− 26.7874i	− 1.30865i	0.756213	−	0.654326i	$-0.227048\pi$
$420$	0	0
$421$	17.4445i	0.850193i	0.905148	+	0.425097i	$0.139760\pi$
$421$	17.4445i	0.850193i	−0.905148	+	0.425097i	$0.860240\pi$
$422$	0	0
$423$	24.4998	1.19122
$424$	0	0
$425$	−6.70253	−0.325120
$426$	0	0
$427$	− 11.4824i	− 0.555670i
$428$	0	0
$429$	16.2407i	0.784109i
$430$	0	0
$431$	−27.1667	−1.30857	−0.654287	−	0.756246i	$-0.727031\pi$
$431$	−27.1667	−1.30857	−0.654287	+	0.756246i	$0.727031\pi$
$432$	0	0
$433$	−9.15159	−0.439798	−0.219899	−	0.975523i	$-0.570573\pi$
$433$	−9.15159	−0.439798	−0.219899	+	0.975523i	$0.570573\pi$
$434$	0	0
$435$	1.90486i	0.0913309i
$436$	0	0
$437$	9.74917i	0.466366i
$438$	0	0
$439$	−30.7586	−1.46803	−0.734014	−	0.679134i	$-0.762355\pi$
$439$	−30.7586	−1.46803	−0.734014	+	0.679134i	$0.762355\pi$
$440$	0	0
$441$	2.59086	0.123374
$442$	0	0
$443$	− 4.36180i	− 0.207235i	−0.994617	−	0.103618i	$-0.966958\pi$
$443$	− 4.36180i	− 0.207235i	0.994617	−	0.103618i	$-0.0330419\pi$
$444$	0	0
$445$	− 3.23465i	− 0.153337i
$446$	0	0
$447$	9.33559	0.441558
$448$	0	0
$449$	−2.67230	−0.126114	−0.0630569	−	0.998010i	$-0.520085\pi$
$449$	−2.67230	−0.126114	−0.0630569	+	0.998010i	$0.520085\pi$
$450$	0	0
$451$	− 38.6218i	− 1.81863i
$452$	0	0
$453$	12.8931i	0.605770i
$454$	0	0
$455$	−5.17077	−0.242410
$456$	0	0
$457$	37.0593	1.73356	0.866781	−	0.498689i	$-0.166185\pi$
$457$	37.0593	1.73356	0.866781	+	0.498689i	$0.166185\pi$
$458$	0	0
$459$	23.9692i	1.11878i
$460$	0	0
$461$	2.85479i	0.132961i	0.997788	+	0.0664803i	$0.0211770\pi$
$461$	2.85479i	0.132961i	−0.997788	+	0.0664803i	$0.978823\pi$
$462$	0	0
$463$	−3.82007	−0.177534	−0.0887668	−	0.996052i	$-0.528293\pi$
$463$	−3.82007	−0.177534	−0.0887668	+	0.996052i	$0.528293\pi$
$464$	0	0
$465$	−2.45202	−0.113710
$466$	0	0
$467$	0.176544i	0.00816950i	0.999992	+	0.00408475i	$0.00130022\pi$
$467$	0.176544i	0.00816950i	−0.999992	+	0.00408475i	$0.998700\pi$
$468$	0	0
$469$	10.4233i	0.481301i
$470$	0	0
$471$	−13.7913	−0.635469
$472$	0	0
$473$	−0.416673	−0.0191586
$474$	0	0
$475$	7.89632i	0.362308i
$476$	0	0
$477$	12.7346i	0.583076i
$478$	0	0
$479$	−43.2393	−1.97566	−0.987828	−	0.155550i	$-0.950285\pi$
$479$	−43.2393	−1.97566	−0.987828	+	0.155550i	$0.950285\pi$
$480$	0	0
$481$	−30.9110	−1.40942
$482$	0	0
$483$	− 0.789730i	− 0.0359340i
$484$	0	0
$485$	− 4.22568i	− 0.191878i
$486$	0	0
$487$	−26.8370	−1.21610	−0.608049	−	0.793899i	$-0.708048\pi$
$487$	−26.8370	−1.21610	−0.608049	+	0.793899i	$0.708048\pi$
$488$	0	0
$489$	−7.55570	−0.341681
$490$	0	0
$491$	− 6.26171i	− 0.282587i	−0.989968	−	0.141294i	$-0.954874\pi$
$491$	− 6.26171i	− 0.282587i	0.989968	−	0.141294i	$-0.0451261\pi$
$492$	0	0
$493$	19.9602i	0.898963i
$494$	0	0
$495$	12.7221	0.571815
$496$	0	0
$497$	10.0227	0.449578
$498$	0	0
$499$	− 1.98216i	− 0.0887337i	−0.999015	−	0.0443669i	$-0.985873\pi$
$499$	− 1.98216i	− 0.0887337i	0.999015	−	0.0443669i	$-0.0141270\pi$
$500$	0	0
$501$	0.526689i	0.0235307i
$502$	0	0
$503$	−25.3099	−1.12851	−0.564257	−	0.825599i	$-0.690837\pi$
$503$	−25.3099	−1.12851	−0.564257	+	0.825599i	$0.690837\pi$
$504$	0	0
$505$	−10.8592	−0.483228
$506$	0	0
$507$	− 8.78667i	− 0.390230i
$508$	0	0
$509$	39.7089i	1.76006i	0.474914	+	0.880032i	$0.342479\pi$
$509$	39.7089i	1.76006i	−0.474914	+	0.880032i	$0.657521\pi$
$510$	0	0
$511$	10.1567	0.449307
$512$	0	0
$513$	28.2383	1.24675
$514$	0	0
$515$	9.45622i	0.416691i
$516$	0	0
$517$	46.4335i	2.04214i
$518$	0	0
$519$	−2.81831	−0.123710
$520$	0	0
$521$	−1.91886	−0.0840668	−0.0420334	−	0.999116i	$-0.513384\pi$
$521$	−1.91886	−0.0840668	−0.0420334	+	0.999116i	$0.513384\pi$
$522$	0	0
$523$	− 4.74890i	− 0.207655i	−0.994595	−	0.103827i	$-0.966891\pi$
$523$	− 4.74890i	− 0.207655i	0.994595	−	0.103827i	$-0.0331090\pi$
$524$	0	0
$525$	− 0.639640i	− 0.0279162i
$526$	0	0
$527$	−25.6937	−1.11923
$528$	0	0
$529$	−21.4756	−0.933724
$530$	0	0
$531$	2.44391i	0.106057i
$532$	0	0
$533$	40.6700i	1.76161i
$534$	0	0
$535$	−4.37929	−0.189333
$536$	0	0
$537$	−14.7462	−0.636347
$538$	0	0
$539$	4.91036i	0.211504i
$540$	0	0
$541$	− 30.2498i	− 1.30054i	−0.759702	−	0.650271i	$-0.774655\pi$
$541$	− 30.2498i	− 1.30054i	0.759702	−	0.650271i	$-0.225345\pi$
$542$	0	0
$543$	−10.5111	−0.451076
$544$	0	0
$545$	−9.82450	−0.420836
$546$	0	0
$547$	− 10.0944i	− 0.431606i	−0.976437	−	0.215803i	$-0.930763\pi$
$547$	− 10.0944i	− 0.431606i	0.976437	−	0.215803i	$-0.0692369\pi$
$548$	0	0
$549$	− 29.7492i	− 1.26966i
$550$	0	0
$551$	23.5153	1.00179
$552$	0	0
$553$	4.36415	0.185583
$554$	0	0
$555$	− 3.82379i	− 0.162311i
$556$	0	0
$557$	8.77989i	0.372016i	0.982548	+	0.186008i	$0.0595550\pi$
$557$	8.77989i	0.372016i	−0.982548	+	0.186008i	$0.940445\pi$
$558$	0	0
$559$	0.438770	0.0185580
$560$	0	0
$561$	−21.0517	−0.888805
$562$	0	0
$563$	− 3.15153i	− 0.132821i	−0.997792	−	0.0664106i	$-0.978845\pi$
$563$	− 3.15153i	− 0.132821i	0.997792	−	0.0664106i	$-0.0211547\pi$
$564$	0	0
$565$	− 14.7994i	− 0.622614i
$566$	0	0
$567$	5.48514	0.230354
$568$	0	0
$569$	−35.0267	−1.46840	−0.734198	−	0.678935i	$-0.762442\pi$
$569$	−35.0267	−1.46840	−0.734198	+	0.678935i	$0.762442\pi$
$570$	0	0
$571$	1.57603i	0.0659547i	0.999456	+	0.0329773i	$0.0104989\pi$
$571$	1.57603i	0.0659547i	−0.999456	+	0.0329773i	$0.989501\pi$
$572$	0	0
$573$	15.8924i	0.663913i
$574$	0	0
$575$	1.23465	0.0514884
$576$	0	0
$577$	40.8834	1.70200	0.850998	−	0.525168i	$-0.175998\pi$
$577$	40.8834	1.70200	0.850998	+	0.525168i	$0.175998\pi$
$578$	0	0
$579$	− 5.60912i	− 0.233107i
$580$	0	0
$581$	2.56685i	0.106491i
$582$	0	0
$583$	−24.1354	−0.999585
$584$	0	0
$585$	−13.3968	−0.553888
$586$	0	0
$587$	− 4.50371i	− 0.185888i	−0.995671	−	0.0929440i	$-0.970372\pi$
$587$	− 4.50371i	− 0.185888i	0.995671	−	0.0929440i	$-0.0296277\pi$
$588$	0	0
$589$	30.2700i	1.24725i
$590$	0	0
$591$	7.18437	0.295526
$592$	0	0
$593$	10.3694	0.425820	0.212910	−	0.977072i	$-0.431706\pi$
$593$	10.3694	0.425820	0.212910	+	0.977072i	$0.431706\pi$
$594$	0	0
$595$	− 6.70253i	− 0.274777i
$596$	0	0
$597$	8.40343i	0.343929i
$598$	0	0
$599$	−21.1034	−0.862263	−0.431132	−	0.902289i	$-0.641886\pi$
$599$	−21.1034	−0.862263	−0.431132	+	0.902289i	$0.641886\pi$
$600$	0	0
$601$	−22.9866	−0.937642	−0.468821	−	0.883293i	$-0.655321\pi$
$601$	−22.9866	−0.937642	−0.468821	+	0.883293i	$0.655321\pi$
$602$	0	0
$603$	27.0052i	1.09974i
$604$	0	0
$605$	13.1117i	0.533066i
$606$	0	0
$607$	−28.9965	−1.17693	−0.588466	−	0.808522i	$-0.700268\pi$
$607$	−28.9965	−1.17693	−0.588466	+	0.808522i	$0.700268\pi$
$608$	0	0
$609$	−1.90486	−0.0771887
$610$	0	0
$611$	− 48.8960i	− 1.97812i
$612$	0	0
$613$	− 25.6519i	− 1.03607i	−0.855360	−	0.518035i	$-0.826664\pi$
$613$	− 25.6519i	− 1.03607i	0.855360	−	0.518035i	$-0.173336\pi$
$614$	0	0
$615$	−5.03100	−0.202869
$616$	0	0
$617$	44.7732	1.80250	0.901251	−	0.433297i	$-0.142650\pi$
$617$	44.7732	1.80250	0.901251	+	0.433297i	$0.142650\pi$
$618$	0	0
$619$	− 20.7344i	− 0.833386i	−0.909047	−	0.416693i	$-0.863189\pi$
$619$	− 20.7344i	− 0.833386i	0.909047	−	0.416693i	$-0.136811\pi$
$620$	0	0
$621$	− 4.41527i	− 0.177179i
$622$	0	0
$623$	3.23465	0.129593
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	24.8013i	0.990467i
$628$	0	0
$629$	− 40.0679i	− 1.59761i
$630$	0	0
$631$	−16.0949	−0.640727	−0.320363	−	0.947295i	$-0.603805\pi$
$631$	−16.0949	−0.640727	−0.320363	+	0.947295i	$0.603805\pi$
$632$	0	0
$633$	15.8238	0.628941
$634$	0	0
$635$	8.46100i	0.335765i
$636$	0	0
$637$	− 5.17077i	− 0.204874i
$638$	0	0
$639$	25.9673	1.02725
$640$	0	0
$641$	−31.2615	−1.23475	−0.617377	−	0.786667i	$-0.711805\pi$
$641$	−31.2615	−1.23475	−0.617377	+	0.786667i	$0.711805\pi$
$642$	0	0
$643$	− 11.0724i	− 0.436654i	−0.975876	−	0.218327i	$-0.929940\pi$
$643$	− 11.0724i	− 0.436654i	0.975876	−	0.218327i	$-0.0700599\pi$
$644$	0	0
$645$	0.0542771i	0.00213716i
$646$	0	0
$647$	19.9259	0.783368	0.391684	−	0.920100i	$-0.371893\pi$
$647$	19.9259	0.783368	0.391684	+	0.920100i	$0.371893\pi$
$648$	0	0
$649$	−4.63186	−0.181816
$650$	0	0
$651$	− 2.45202i	− 0.0961021i
$652$	0	0
$653$	− 12.2954i	− 0.481155i	−0.970630	−	0.240578i	$-0.922663\pi$
$653$	− 12.2954i	− 0.481155i	0.970630	−	0.240578i	$-0.0773369\pi$
$654$	0	0
$655$	−5.16595	−0.201850
$656$	0	0
$657$	26.3147	1.02663
$658$	0	0
$659$	48.2900i	1.88111i	0.339637	+	0.940556i	$0.389696\pi$
$659$	48.2900i	1.88111i	−0.339637	+	0.940556i	$0.610304\pi$
$660$	0	0
$661$	− 8.92560i	− 0.347166i	−0.984819	−	0.173583i	$-0.944466\pi$
$661$	− 8.92560i	− 0.347166i	0.984819	−	0.173583i	$-0.0555344\pi$
$662$	0	0
$663$	22.1682	0.860940
$664$	0	0
$665$	−7.89632	−0.306206
$666$	0	0
$667$	− 3.67680i	− 0.142366i
$668$	0	0
$669$	− 0.404130i	− 0.0156246i
$670$	0	0
$671$	56.3825	2.17662
$672$	0	0
$673$	31.7085	1.22227	0.611136	−	0.791525i	$-0.290713\pi$
$673$	31.7085	1.22227	0.611136	+	0.791525i	$0.290713\pi$
$674$	0	0
$675$	− 3.57614i	− 0.137646i
$676$	0	0
$677$	15.3157i	0.588632i	0.955708	+	0.294316i	$0.0950918\pi$
$677$	15.3157i	0.588632i	−0.955708	+	0.294316i	$0.904908\pi$
$678$	0	0
$679$	4.22568	0.162167
$680$	0	0
$681$	−2.48256	−0.0951318
$682$	0	0
$683$	33.5876i	1.28519i	0.766204	+	0.642597i	$0.222143\pi$
$683$	33.5876i	1.28519i	−0.766204	+	0.642597i	$0.777857\pi$
$684$	0	0
$685$	11.4824i	0.438718i
$686$	0	0
$687$	−14.5250	−0.554164
$688$	0	0
$689$	25.4153	0.968247
$690$	0	0
$691$	19.9239i	0.757940i	0.925409	+	0.378970i	$0.123722\pi$
$691$	19.9239i	0.757940i	−0.925409	+	0.378970i	$0.876278\pi$
$692$	0	0
$693$	12.7221i	0.483272i
$694$	0	0
$695$	−6.22256	−0.236035
$696$	0	0
$697$	−52.7178	−1.99683
$698$	0	0
$699$	− 15.2086i	− 0.575241i
$700$	0	0
$701$	− 0.670878i	− 0.0253387i	−0.999920	−	0.0126694i	$-0.995967\pi$
$701$	− 0.670878i	− 0.0253387i	0.999920	−	0.0126694i	$-0.00403289\pi$
$702$	0	0
$703$	−47.2044	−1.78035
$704$	0	0
$705$	6.04858	0.227803
$706$	0	0
$707$	− 10.8592i	− 0.408402i
$708$	0	0
$709$	− 16.3811i	− 0.615206i	−0.951515	−	0.307603i	$-0.900473\pi$
$709$	− 16.3811i	− 0.615206i	0.951515	−	0.307603i	$-0.0995268\pi$
$710$	0	0
$711$	11.3069	0.424042
$712$	0	0
$713$	4.73294	0.177250
$714$	0	0
$715$	− 25.3904i	− 0.949547i
$716$	0	0
$717$	− 11.0561i	− 0.412898i
$718$	0	0
$719$	3.44866	0.128613	0.0643066	−	0.997930i	$-0.479516\pi$
$719$	3.44866	0.128613	0.0643066	+	0.997930i	$0.479516\pi$
$720$	0	0
$721$	−9.45622	−0.352168
$722$	0	0
$723$	8.29928i	0.308654i
$724$	0	0
$725$	− 2.97801i	− 0.110601i
$726$	0	0
$727$	47.3235	1.75513	0.877565	−	0.479457i	$-0.159166\pi$
$727$	47.3235	1.75513	0.877565	+	0.479457i	$0.159166\pi$
$728$	0	0
$729$	7.34892	0.272182
$730$	0	0
$731$	0.568748i	0.0210359i
$732$	0	0
$733$	− 15.6044i	− 0.576361i	−0.957576	−	0.288181i	$-0.906950\pi$
$733$	− 15.6044i	− 0.576361i	0.957576	−	0.288181i	$-0.0930504\pi$
$734$	0	0
$735$	0.639640	0.0235935
$736$	0	0
$737$	−51.1820	−1.88531
$738$	0	0
$739$	− 13.9114i	− 0.511738i	−0.966712	−	0.255869i	$-0.917638\pi$
$739$	− 13.9114i	− 0.511738i	0.966712	−	0.255869i	$-0.0823615\pi$
$740$	0	0
$741$	− 26.1165i	− 0.959415i
$742$	0	0
$743$	−20.4048	−0.748580	−0.374290	−	0.927312i	$-0.622114\pi$
$743$	−20.4048	−0.748580	−0.374290	+	0.927312i	$0.622114\pi$
$744$	0	0
$745$	−14.5951	−0.534722
$746$	0	0
$747$	6.65036i	0.243324i
$748$	0	0
$749$	− 4.37929i	− 0.160016i
$750$	0	0
$751$	−48.1239	−1.75607	−0.878034	−	0.478599i	$-0.841145\pi$
$751$	−48.1239	−1.75607	−0.878034	+	0.478599i	$0.841145\pi$
$752$	0	0
$753$	9.72399	0.354362
$754$	0	0
$755$	− 20.1568i	− 0.733581i
$756$	0	0
$757$	26.8795i	0.976954i	0.872577	+	0.488477i	$0.162447\pi$
$757$	26.8795i	0.976954i	−0.872577	+	0.488477i	$0.837553\pi$
$758$	0	0
$759$	3.87786	0.140758
$760$	0	0
$761$	−9.84742	−0.356969	−0.178484	−	0.983943i	$-0.557119\pi$
$761$	−9.84742	−0.356969	−0.178484	+	0.983943i	$0.557119\pi$
$762$	0	0
$763$	− 9.82450i	− 0.355671i
$764$	0	0
$765$	− 17.3653i	− 0.627844i
$766$	0	0
$767$	4.87750	0.176116
$768$	0	0
$769$	16.5173	0.595628	0.297814	−	0.954624i	$-0.403742\pi$
$769$	16.5173	0.595628	0.297814	+	0.954624i	$0.403742\pi$
$770$	0	0
$771$	15.4595i	0.556760i
$772$	0	0
$773$	9.77679i	0.351647i	0.984422	+	0.175823i	$0.0562588\pi$
$773$	9.77679i	0.351647i	−0.984422	+	0.175823i	$0.943741\pi$
$774$	0	0
$775$	3.83343	0.137701
$776$	0	0
$777$	3.82379	0.137178
$778$	0	0
$779$	62.1074i	2.22523i
$780$	0	0
$781$	49.2149i	1.76105i
$782$	0	0
$783$	−10.6498	−0.380592
$784$	0	0
$785$	21.5610	0.769545
$786$	0	0
$787$	− 12.9593i	− 0.461950i	−0.972960	−	0.230975i	$-0.925808\pi$
$787$	− 12.9593i	− 0.461950i	0.972960	−	0.230975i	$-0.0741916\pi$
$788$	0	0
$789$	− 17.6736i	− 0.629195i
$790$	0	0
$791$	14.7994	0.526205
$792$	0	0
$793$	−59.3727	−2.10839
$794$	0	0
$795$	3.14395i	0.111504i
$796$	0	0
$797$	50.3347i	1.78295i	0.453074	+	0.891473i	$0.350327\pi$
$797$	50.3347i	1.78295i	−0.453074	+	0.891473i	$0.649673\pi$
$798$	0	0
$799$	63.3806	2.24224
$800$	0	0
$801$	8.38052	0.296111
$802$	0	0
$803$	49.8732i	1.75999i
$804$	0	0
$805$	1.23465i	0.0435156i
$806$	0	0
$807$	−5.07097	−0.178507
$808$	0	0
$809$	9.88832	0.347655	0.173827	−	0.984776i	$-0.444387\pi$
$809$	9.88832	0.347655	0.173827	+	0.984776i	$0.444387\pi$
$810$	0	0
$811$	− 51.2753i	− 1.80052i	−0.435353	−	0.900260i	$-0.643377\pi$
$811$	− 51.2753i	− 1.80052i	0.435353	−	0.900260i	$-0.356623\pi$
$812$	0	0
$813$	− 10.0982i	− 0.354161i
$814$	0	0
$815$	11.8124	0.413772
$816$	0	0
$817$	0.670048	0.0234420
$818$	0	0
$819$	− 13.3968i	− 0.468121i
$820$	0	0
$821$	− 11.9562i	− 0.417275i	−0.977993	−	0.208637i	$-0.933097\pi$
$821$	− 11.9562i	− 0.417275i	0.977993	−	0.208637i	$-0.0669028\pi$
$822$	0	0
$823$	28.4173	0.990565	0.495283	−	0.868732i	$-0.335064\pi$
$823$	28.4173	0.990565	0.495283	+	0.868732i	$0.335064\pi$
$824$	0	0
$825$	3.14087	0.109351
$826$	0	0
$827$	39.5611i	1.37568i	0.725865	+	0.687838i	$0.241440\pi$
$827$	39.5611i	1.37568i	−0.725865	+	0.687838i	$0.758560\pi$
$828$	0	0
$829$	36.4368i	1.26550i	0.774355	+	0.632752i	$0.218075\pi$
$829$	36.4368i	1.26550i	−0.774355	+	0.632752i	$0.781925\pi$
$830$	0	0
$831$	−15.4925	−0.537427
$832$	0	0
$833$	6.70253	0.232229
$834$	0	0
$835$	− 0.823415i	− 0.0284954i
$836$	0	0
$837$	− 13.7089i	− 0.473848i
$838$	0	0
$839$	−9.44375	−0.326035	−0.163017	−	0.986623i	$-0.552123\pi$
$839$	−9.44375	−0.326035	−0.163017	+	0.986623i	$0.552123\pi$
$840$	0	0
$841$	20.1314	0.694187
$842$	0	0
$843$	− 6.92353i	− 0.238459i
$844$	0	0
$845$	13.7369i	0.472564i
$846$	0	0
$847$	−13.1117	−0.450523
$848$	0	0
$849$	−2.88733	−0.0990928
$850$	0	0
$851$	7.38076i	0.253009i
$852$	0	0
$853$	23.0644i	0.789709i	0.918744	+	0.394854i	$0.129205\pi$
$853$	23.0644i	0.789709i	−0.918744	+	0.394854i	$0.870795\pi$
$854$	0	0
$855$	−20.4583	−0.699658
$856$	0	0
$857$	41.3619	1.41289	0.706447	−	0.707766i	$-0.250297\pi$
$857$	41.3619	1.41289	0.706447	+	0.707766i	$0.250297\pi$
$858$	0	0
$859$	15.8674i	0.541390i	0.962665	+	0.270695i	$0.0872535\pi$
$859$	15.8674i	0.541390i	−0.962665	+	0.270695i	$0.912746\pi$
$860$	0	0
$861$	− 5.03100i	− 0.171456i
$862$	0	0
$863$	40.7413	1.38685	0.693425	−	0.720529i	$-0.256101\pi$
$863$	40.7413	1.38685	0.693425	+	0.720529i	$0.256101\pi$
$864$	0	0
$865$	4.40608	0.149811
$866$	0	0
$867$	17.8612i	0.606598i
$868$	0	0
$869$	21.4296i	0.726948i
$870$	0	0
$871$	53.8963	1.82621
$872$	0	0
$873$	10.9481	0.370539
$874$	0	0
$875$	1.00000i	0.0338062i
$876$	0	0
$877$	4.64454i	0.156835i	0.996921	+	0.0784174i	$0.0249867\pi$
$877$	4.64454i	0.156835i	−0.996921	+	0.0784174i	$0.975013\pi$
$878$	0	0
$879$	4.47204	0.150838
$880$	0	0
$881$	53.4306	1.80012	0.900061	−	0.435764i	$-0.143522\pi$
$881$	53.4306	1.80012	0.900061	+	0.435764i	$0.143522\pi$
$882$	0	0
$883$	26.1779i	0.880957i	0.897763	+	0.440478i	$0.145191\pi$
$883$	26.1779i	0.880957i	−0.897763	+	0.440478i	$0.854809\pi$
$884$	0	0
$885$	0.603361i	0.0202818i
$886$	0	0
$887$	−34.2696	−1.15066	−0.575330	−	0.817921i	$-0.695126\pi$
$887$	−34.2696	−1.15066	−0.575330	+	0.817921i	$0.695126\pi$
$888$	0	0
$889$	−8.46100	−0.283773
$890$	0	0
$891$	26.9340i	0.902324i
$892$	0	0
$893$	− 74.6693i	− 2.49871i
$894$	0	0
$895$	23.0540	0.770609
$896$	0	0
$897$	−4.08352	−0.136345
$898$	0	0
$899$	− 11.4160i	− 0.380745i
$900$	0	0
$901$	32.9442i	1.09753i
$902$	0	0
$903$	−0.0542771	−0.00180623
$904$	0	0
$905$	16.4329	0.546248
$906$	0	0
$907$	− 16.5550i	− 0.549700i	−0.961487	−	0.274850i	$-0.911372\pi$
$907$	− 16.5550i	− 0.549700i	0.961487	−	0.274850i	$-0.0886282\pi$
$908$	0	0
$909$	− 28.1347i	− 0.933168i
$910$	0	0
$911$	21.2951	0.705538	0.352769	−	0.935710i	$-0.385240\pi$
$911$	21.2951	0.705538	0.352769	+	0.935710i	$0.385240\pi$
$912$	0	0
$913$	−12.6042	−0.417138
$914$	0	0
$915$	− 7.34457i	− 0.242804i
$916$	0	0
$917$	− 5.16595i	− 0.170595i
$918$	0	0
$919$	−24.3263	−0.802450	−0.401225	−	0.915980i	$-0.631415\pi$
$919$	−24.3263	−0.802450	−0.401225	+	0.915980i	$0.631415\pi$
$920$	0	0
$921$	−7.34216	−0.241932
$922$	0	0
$923$	− 51.8249i	− 1.70584i
$924$	0	0
$925$	5.97803i	0.196556i
$926$	0	0
$927$	−24.4998	−0.804677
$928$	0	0
$929$	−14.9461	−0.490367	−0.245184	−	0.969477i	$-0.578848\pi$
$929$	−14.9461	−0.490367	−0.245184	+	0.969477i	$0.578848\pi$
$930$	0	0
$931$	− 7.89632i	− 0.258791i
$932$	0	0
$933$	− 6.29390i	− 0.206053i
$934$	0	0
$935$	32.9118	1.07633
$936$	0	0
$937$	−35.3491	−1.15481	−0.577403	−	0.816460i	$-0.695934\pi$
$937$	−35.3491	−1.15481	−0.577403	+	0.816460i	$0.695934\pi$
$938$	0	0
$939$	− 22.2205i	− 0.725138i
$940$	0	0
$941$	2.68116i	0.0874032i	0.999045	+	0.0437016i	$0.0139151\pi$
$941$	2.68116i	0.0874032i	−0.999045	+	0.0437016i	$0.986085\pi$
$942$	0	0
$943$	9.71095	0.316232
$944$	0	0
$945$	3.57614	0.116332
$946$	0	0
$947$	41.9276i	1.36246i	0.732068	+	0.681232i	$0.238555\pi$
$947$	41.9276i	1.36246i	−0.732068	+	0.681232i	$0.761445\pi$
$948$	0	0
$949$	− 52.5181i	− 1.70481i
$950$	0	0
$951$	1.59792	0.0518162
$952$	0	0
$953$	−56.9762	−1.84564	−0.922820	−	0.385231i	$-0.874122\pi$
$953$	−56.9762	−1.84564	−0.922820	+	0.385231i	$0.874122\pi$
$954$	0	0
$955$	− 24.8458i	− 0.803991i
$956$	0	0
$957$	− 9.35354i	− 0.302357i
$958$	0	0
$959$	−11.4824	−0.370785
$960$	0	0
$961$	−16.3048	−0.525961
$962$	0	0
$963$	− 11.3461i	− 0.365624i
$964$	0	0
$965$	8.76919i	0.282290i
$966$	0	0
$967$	−53.9431	−1.73469	−0.867347	−	0.497704i	$-0.834176\pi$
$967$	−53.9431	−1.73469	−0.867347	+	0.497704i	$0.834176\pi$
$968$	0	0
$969$	33.8531	1.08752
$970$	0	0
$971$	− 0.778454i	− 0.0249818i	−0.999922	−	0.0124909i	$-0.996024\pi$
$971$	− 0.778454i	− 0.0249818i	0.999922	−	0.0124909i	$-0.00397608\pi$
$972$	0	0
$973$	− 6.22256i	− 0.199486i
$974$	0	0
$975$	−3.30743	−0.105923
$976$	0	0
$977$	5.48442	0.175462	0.0877310	−	0.996144i	$-0.472038\pi$
$977$	5.48442	0.175462	0.0877310	+	0.996144i	$0.472038\pi$
$978$	0	0
$979$	15.8833i	0.507632i
$980$	0	0
$981$	− 25.4539i	− 0.812681i
$982$	0	0
$983$	36.0539	1.14994	0.574970	−	0.818174i	$-0.305014\pi$
$983$	36.0539	1.14994	0.574970	+	0.818174i	$0.305014\pi$
$984$	0	0
$985$	−11.2319	−0.357878
$986$	0	0
$987$	6.04858i	0.192528i
$988$	0	0
$989$	− 0.104767i	− 0.00333140i
$990$	0	0
$991$	−47.2390	−1.50060	−0.750298	−	0.661100i	$-0.770090\pi$
$991$	−47.2390	−1.50060	−0.750298	+	0.661100i	$0.770090\pi$
$992$	0	0
$993$	1.50783	0.0478495
$994$	0	0
$995$	− 13.1377i	− 0.416495i
$996$	0	0
$997$	− 6.99150i	− 0.221423i	−0.993853	−	0.110712i	$-0.964687\pi$
$997$	− 6.99150i	− 0.221423i	0.993853	−	0.110712i	$-0.0353130\pi$
$998$	0	0
$999$	21.3783	0.676378

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.2.b.h.1121.8	yes	12
4.3	odd	2		2240.2.b.g.1121.5	✓	12
8.3	odd	2		2240.2.b.g.1121.8	yes	12
8.5	even	2	inner	2240.2.b.h.1121.5	yes	12
16.3	odd	4		8960.2.a.cc.1.4		6
16.5	even	4		8960.2.a.ce.1.4		6
16.11	odd	4		8960.2.a.ch.1.3		6
16.13	even	4		8960.2.a.cb.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2240.2.b.g.1121.5	✓	12	4.3	odd	2
2240.2.b.g.1121.8	yes	12	8.3	odd	2
2240.2.b.h.1121.5	yes	12	8.5	even	2	inner
2240.2.b.h.1121.8	yes	12	1.1	even	1	trivial
8960.2.a.cb.1.3		6	16.13	even	4
8960.2.a.cc.1.4		6	16.3	odd	4
8960.2.a.ce.1.4		6	16.5	even	4
8960.2.a.ch.1.3		6	16.11	odd	4

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.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.639640i q^{3} -1.00000i q^{5} +1.00000 q^{7} +2.59086 q^{9} +O(q^{10})\)	\(q+0.639640i q^{3} -1.00000i q^{5} +1.00000 q^{7} +2.59086 q^{9} +4.91036i q^{11} -5.17077i q^{13} +0.639640 q^{15} +6.70253 q^{17} -7.89632i q^{19} +0.639640i q^{21} -1.23465 q^{23} -1.00000 q^{25} +3.57614i q^{27} +2.97801i q^{29} -3.83343 q^{31} -3.14087 q^{33} -1.00000i q^{35} -5.97803i q^{37} +3.30743 q^{39} -7.86536 q^{41} +0.0848558i q^{43} -2.59086i q^{45} +9.45622 q^{47} +1.00000 q^{49} +4.28720i q^{51} +4.91519i q^{53} +4.91036 q^{55} +5.05080 q^{57} +0.943282i q^{59} -11.4824i q^{61} +2.59086 q^{63} -5.17077 q^{65} +10.4233i q^{67} -0.789730i q^{69} +10.0227 q^{71} +10.1567 q^{73} -0.639640i q^{75} +4.91036i q^{77} +4.36415 q^{79} +5.48514 q^{81} +2.56685i q^{83} -6.70253i q^{85} -1.90486 q^{87} +3.23465 q^{89} -5.17077i q^{91} -2.45202i q^{93} -7.89632 q^{95} +4.22568 q^{97} +12.7221i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{7} - 20 q^{9}+O(q^{10}) \) 12 * q + 12 * q^7 - 20 * q^9	\( 12 q + 12 q^{7} - 20 q^{9} + 16 q^{17} + 8 q^{23} - 12 q^{25} - 8 q^{31} + 72 q^{33} - 32 q^{39} - 32 q^{47} + 12 q^{49} + 8 q^{55} + 8 q^{57} - 20 q^{63} + 8 q^{65} + 48 q^{71} + 8 q^{73} - 16 q^{79} + 92 q^{81} - 32 q^{87} + 16 q^{89} + 32 q^{97}+O(q^{100}) \) 12 * q + 12 * q^7 - 20 * q^9 + 16 * q^17 + 8 * q^23 - 12 * q^25 - 8 * q^31 + 72 * q^33 - 32 * q^39 - 32 * q^47 + 12 * q^49 + 8 * q^55 + 8 * q^57 - 20 * q^63 + 8 * q^65 + 48 * q^71 + 8 * q^73 - 16 * q^79 + 92 * q^81 - 32 * q^87 + 16 * q^89 + 32 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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