LMFDB - Embedded newform 2240.2.h.e.671.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2240,2,Mod(671,2240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

chi = DirichletCharacter(H, H._module([1, 1, 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("2240.671");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$17.8864900528$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			671.1
Character	$\chi$	$=$	2240.671
Dual form			2240.2.h.e.671.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.84405i q^{3} -1.00000 q^{5} +(-0.712435 - 2.54803i) q^{7} -5.08861 q^{9} +O(q^{10})$	$q-2.84405i q^{3} -1.00000 q^{5} +(-0.712435 - 2.54803i) q^{7} -5.08861 q^{9} -5.41989 q^{11} +0.641175 q^{13} +2.84405i q^{15} -5.16295i q^{17} -3.44743i q^{19} +(-7.24671 + 2.02620i) q^{21} -7.94993i q^{23} +1.00000 q^{25} +5.94010i q^{27} +10.0316i q^{29} +5.38283 q^{31} +15.4144i q^{33} +(0.712435 + 2.54803i) q^{35} -0.931011i q^{37} -1.82353i q^{39} +8.33276i q^{41} -6.76735 q^{43} +5.08861 q^{45} +8.86368 q^{47} +(-5.98487 + 3.63061i) q^{49} -14.6837 q^{51} -6.31384i q^{53} +5.41989 q^{55} -9.80466 q^{57} +11.5640i q^{59} +0.834217 q^{61} +(3.62530 + 12.9659i) q^{63} -0.641175 q^{65} -2.01317 q^{67} -22.6100 q^{69} -0.628104i q^{71} -10.3662i q^{73} -2.84405i q^{75} +(3.86132 + 13.8100i) q^{77} +2.98179i q^{79} +1.62810 q^{81} +11.2733i q^{83} +5.16295i q^{85} +28.5304 q^{87} +13.3729i q^{89} +(-0.456796 - 1.63373i) q^{91} -15.3090i q^{93} +3.44743i q^{95} -9.07453i q^{97} +27.5797 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 24 q^{5} - 36 q^{9}+O(q^{10}) $ 24 * q - 24 * q^5 - 36 * q^9	$ 24 q - 24 q^{5} - 36 q^{9} - 4 q^{13} - 8 q^{21} + 24 q^{25} + 36 q^{45} + 24 q^{57} - 56 q^{61} + 4 q^{65} - 96 q^{69} + 12 q^{77} + 24 q^{81}+O(q^{100}) $ 24 * q - 24 * q^5 - 36 * q^9 - 4 * q^13 - 8 * q^21 + 24 * q^25 + 36 * q^45 + 24 * q^57 - 56 * q^61 + 4 * q^65 - 96 * q^69 + 12 * q^77 + 24 * q^81

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
$3$		−	2.84405i		−	1.64201i	−0.570920	−	0.821006i	$-0.693413\pi$
$3$		−	2.84405i		−	1.64201i	0.570920	−	0.821006i	$-0.306587\pi$
$4$		0			0
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$	−0.712435	−	2.54803i	−0.269275	−	0.963063i
$7$	−0.712435	−	2.54803i	−0.269275	−	0.963063i
$8$		0			0
$9$	−5.08861			−1.69620
$10$		0			0
$11$	−5.41989			−1.63416			−0.817080	−	0.576525i	$-0.804408\pi$
$11$	−5.41989			−1.63416			−0.817080	+	0.576525i	$0.804408\pi$
$12$		0			0
$13$	0.641175			0.177830			0.0889150	−	0.996039i	$-0.471660\pi$
$13$	0.641175			0.177830			0.0889150	+	0.996039i	$0.471660\pi$
$14$		0			0
$15$			2.84405i			0.734330i
$16$		0			0
$17$		−	5.16295i		−	1.25220i	−0.779743	−	0.626099i	$-0.784650\pi$
$17$		−	5.16295i		−	1.25220i	0.779743	−	0.626099i	$-0.215350\pi$
$18$		0			0
$19$		−	3.44743i		−	0.790895i	−0.918489	−	0.395448i	$-0.870589\pi$
$19$		−	3.44743i		−	0.790895i	0.918489	−	0.395448i	$-0.129411\pi$
$20$		0			0
$21$	−7.24671	+	2.02620i	−1.58136	+	0.442153i
$22$		0			0
$23$		−	7.94993i		−	1.65768i	−0.559489	−	0.828838i	$-0.689003\pi$
$23$		−	7.94993i		−	1.65768i	0.559489	−	0.828838i	$-0.310997\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$			5.94010i			1.14317i
$28$		0			0
$29$			10.0316i			1.86282i	0.363968	+	0.931411i	$0.381422\pi$
$29$			10.0316i			1.86282i	−0.363968	+	0.931411i	$0.618578\pi$
$30$		0			0
$31$	5.38283			0.966785			0.483392	−	0.875404i	$-0.339404\pi$
$31$	5.38283			0.966785			0.483392	+	0.875404i	$0.339404\pi$
$32$		0			0
$33$			15.4144i			2.68331i
$34$		0			0
$35$	0.712435	+	2.54803i	0.120424	+	0.430695i
$36$		0			0
$37$		−	0.931011i		−	0.153057i	−0.997067	−	0.0765287i	$-0.975616\pi$
$37$		−	0.931011i		−	0.153057i	0.997067	−	0.0765287i	$-0.0243837\pi$
$38$		0			0
$39$		−	1.82353i		−	0.291999i
$40$		0			0
$41$			8.33276i			1.30136i	0.759353	+	0.650679i	$0.225516\pi$
$41$			8.33276i			1.30136i	−0.759353	+	0.650679i	$0.774484\pi$
$42$		0			0
$43$	−6.76735			−1.03201			−0.516006	−	0.856585i	$-0.672582\pi$
$43$	−6.76735			−1.03201			−0.516006	+	0.856585i	$0.672582\pi$
$44$		0			0
$45$	5.08861			0.758565
$46$		0			0
$47$	8.86368			1.29290			0.646451	−	0.762956i	$-0.276253\pi$
$47$	8.86368			1.29290			0.646451	+	0.762956i	$0.276253\pi$
$48$		0			0
$49$	−5.98487	+	3.63061i	−0.854982	+	0.518658i
$50$		0			0
$51$	−14.6837			−2.05612
$52$		0			0
$53$		−	6.31384i		−	0.867273i	−0.901088	−	0.433636i	$-0.857230\pi$
$53$		−	6.31384i		−	0.867273i	0.901088	−	0.433636i	$-0.142770\pi$
$54$		0			0
$55$	5.41989			0.730818
$56$		0			0
$57$	−9.80466			−1.29866
$58$		0			0
$59$			11.5640i			1.50550i	0.658304	+	0.752752i	$0.271274\pi$
$59$			11.5640i			1.50550i	−0.658304	+	0.752752i	$0.728726\pi$
$60$		0			0
$61$	0.834217			0.106811			0.0534053	−	0.998573i	$-0.482992\pi$
$61$	0.834217			0.106811			0.0534053	+	0.998573i	$0.482992\pi$
$62$		0			0
$63$	3.62530	+	12.9659i	0.456745	+	1.63355i
$64$		0			0
$65$	−0.641175			−0.0795280
$66$		0			0
$67$	−2.01317			−0.245948			−0.122974	−	0.992410i	$-0.539243\pi$
$67$	−2.01317			−0.245948			−0.122974	+	0.992410i	$0.539243\pi$
$68$		0			0
$69$	−22.6100			−2.72192
$70$		0			0
$71$		−	0.628104i		−	0.0745423i	−0.999305	−	0.0372711i	$-0.988133\pi$
$71$		−	0.628104i		−	0.0745423i	0.999305	−	0.0372711i	$-0.0118665\pi$
$72$		0			0
$73$		−	10.3662i		−	1.21328i	−0.794978	−	0.606638i	$-0.792518\pi$
$73$		−	10.3662i		−	1.21328i	0.794978	−	0.606638i	$-0.207482\pi$
$74$		0			0
$75$		−	2.84405i		−	0.328402i
$76$		0			0
$77$	3.86132	+	13.8100i	0.440039	+	1.57380i
$78$		0			0
$79$			2.98179i			0.335477i	0.985832	+	0.167739i	$0.0536465\pi$
$79$			2.98179i			0.335477i	−0.985832	+	0.167739i	$0.946354\pi$
$80$		0			0
$81$	1.62810			0.180900
$82$		0			0
$83$			11.2733i			1.23740i	0.785627	+	0.618701i	$0.212341\pi$
$83$			11.2733i			1.23740i	−0.785627	+	0.618701i	$0.787659\pi$
$84$		0			0
$85$			5.16295i			0.560000i
$86$		0			0
$87$	28.5304			3.05878
$88$		0			0
$89$			13.3729i			1.41752i	0.705449	+	0.708761i	$0.250746\pi$
$89$			13.3729i			1.41752i	−0.705449	+	0.708761i	$0.749254\pi$
$90$		0			0
$91$	−0.456796	−	1.63373i	−0.0478852	−	0.171262i
$92$		0			0
$93$		−	15.3090i		−	1.58747i
$94$		0			0
$95$			3.44743i			0.353699i
$96$		0			0
$97$		−	9.07453i		−	0.921379i	−0.887561	−	0.460690i	$-0.847602\pi$
$97$		−	9.07453i		−	0.921379i	0.887561	−	0.460690i	$-0.152398\pi$
$98$		0			0
$99$	27.5797			2.77187
$100$		0			0
$101$	4.22577			0.420480			0.210240	−	0.977650i	$-0.432575\pi$
$101$	4.22577			0.420480			0.210240	+	0.977650i	$0.432575\pi$
$102$		0			0
$103$	−1.34746			−0.132769			−0.0663846	−	0.997794i	$-0.521146\pi$
$103$	−1.34746			−0.132769			−0.0663846	+	0.997794i	$0.521146\pi$
$104$		0			0
$105$	7.24671	−	2.02620i	0.707206	−	0.197737i
$106$		0			0
$107$	−1.02545			−0.0991340			−0.0495670	−	0.998771i	$-0.515784\pi$
$107$	−1.02545			−0.0991340			−0.0495670	+	0.998771i	$0.515784\pi$
$108$		0			0
$109$		−	3.24422i		−	0.310740i	−0.987856	−	0.155370i	$-0.950343\pi$
$109$		−	3.24422i		−	0.310740i	0.987856	−	0.155370i	$-0.0496569\pi$
$110$		0			0
$111$	−2.64784			−0.251322
$112$		0			0
$113$	2.86877			0.269871			0.134935	−	0.990854i	$-0.456917\pi$
$113$	2.86877			0.269871			0.134935	+	0.990854i	$0.456917\pi$
$114$		0			0
$115$			7.94993i			0.741335i
$116$		0			0
$117$	−3.26269			−0.301636
$118$		0			0
$119$	−13.1553	+	3.67826i	−1.20595	+	0.337186i
$120$		0			0
$121$	18.3753			1.67048
$122$		0			0
$123$	23.6988			2.13685
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$			17.6790i			1.56876i	0.620281	+	0.784379i	$0.287018\pi$
$127$			17.6790i			1.56876i	−0.620281	+	0.784379i	$0.712982\pi$
$128$		0			0
$129$			19.2467i			1.69458i
$130$		0			0
$131$		−	18.5951i		−	1.62466i	−0.583197	−	0.812330i	$-0.698199\pi$
$131$		−	18.5951i		−	1.62466i	0.583197	−	0.812330i	$-0.301801\pi$
$132$		0			0
$133$	−8.78415	+	2.45607i	−0.761682	+	0.212968i
$134$		0			0
$135$		−	5.94010i		−	0.511242i
$136$		0			0
$137$	−9.51544			−0.812959			−0.406479	−	0.913660i	$-0.633244\pi$
$137$	−9.51544			−0.812959			−0.406479	+	0.913660i	$0.633244\pi$
$138$		0			0
$139$		−	21.7758i		−	1.84700i	−0.383603	−	0.923498i	$-0.625317\pi$
$139$		−	21.7758i		−	1.84700i	0.383603	−	0.923498i	$-0.374683\pi$
$140$		0			0
$141$		−	25.2087i		−	2.12296i
$142$		0			0
$143$	−3.47510			−0.290603
$144$		0			0
$145$		−	10.0316i		−	0.833080i
$146$		0			0
$147$	10.3256	+	17.0213i	0.851643	+	1.40389i
$148$		0			0
$149$		−	9.82601i		−	0.804978i	−0.915425	−	0.402489i	$-0.868145\pi$
$149$		−	9.82601i		−	0.804978i	0.915425	−	0.402489i	$-0.131855\pi$
$150$		0			0
$151$		−	11.3875i		−	0.926704i	−0.886174	−	0.463352i	$-0.846647\pi$
$151$		−	11.3875i		−	0.926704i	0.886174	−	0.463352i	$-0.153353\pi$
$152$		0			0
$153$			26.2722i			2.12398i
$154$		0			0
$155$	−5.38283			−0.432359
$156$		0			0
$157$	5.22166			0.416734			0.208367	−	0.978051i	$-0.433185\pi$
$157$	5.22166			0.416734			0.208367	+	0.978051i	$0.433185\pi$
$158$		0			0
$159$	−17.9569			−1.42407
$160$		0			0
$161$	−20.2566	+	5.66381i	−1.59645	+	0.446371i
$162$		0			0
$163$	−18.7934			−1.47202			−0.736008	−	0.676973i	$-0.763291\pi$
$163$	−18.7934			−1.47202			−0.736008	+	0.676973i	$0.763291\pi$
$164$		0			0
$165$		−	15.4144i		−	1.20001i
$166$		0			0
$167$	2.40930			0.186438			0.0932188	−	0.995646i	$-0.470284\pi$
$167$	2.40930			0.186438			0.0932188	+	0.995646i	$0.470284\pi$
$168$		0			0
$169$	−12.5889			−0.968376
$170$		0			0
$171$			17.5426i			1.34152i
$172$		0			0
$173$	−6.32927			−0.481206			−0.240603	−	0.970624i	$-0.577345\pi$
$173$	−6.32927			−0.481206			−0.240603	+	0.970624i	$0.577345\pi$
$174$		0			0
$175$	−0.712435	−	2.54803i	−0.0538550	−	0.192613i
$176$		0			0
$177$	32.8886			2.47206
$178$		0			0
$179$	−1.82168			−0.136158			−0.0680792	−	0.997680i	$-0.521687\pi$
$179$	−1.82168			−0.136158			−0.0680792	+	0.997680i	$0.521687\pi$
$180$		0			0
$181$	7.23721			0.537937			0.268969	−	0.963149i	$-0.413317\pi$
$181$	7.23721			0.537937			0.268969	+	0.963149i	$0.413317\pi$
$182$		0			0
$183$		−	2.37255i		−	0.175384i
$184$		0			0
$185$			0.931011i			0.0684493i
$186$		0			0
$187$			27.9826i			2.04629i
$188$		0			0
$189$	15.1355	−	4.23194i	1.10095	−	0.307828i
$190$		0			0
$191$		−	7.21032i		−	0.521720i	−0.965377	−	0.260860i	$-0.915994\pi$
$191$		−	7.21032i		−	0.521720i	0.965377	−	0.260860i	$-0.0840061\pi$
$192$		0			0
$193$	7.32334			0.527145			0.263573	−	0.964640i	$-0.415099\pi$
$193$	7.32334			0.527145			0.263573	+	0.964640i	$0.415099\pi$
$194$		0			0
$195$			1.82353i			0.130586i
$196$		0			0
$197$			12.5257i			0.892419i	0.894929	+	0.446210i	$0.147226\pi$
$197$			12.5257i			0.892419i	−0.894929	+	0.446210i	$0.852774\pi$
$198$		0			0
$199$	15.7926			1.11950			0.559752	−	0.828660i	$-0.310896\pi$
$199$	15.7926			1.11950			0.559752	+	0.828660i	$0.310896\pi$
$200$		0			0
$201$			5.72554i			0.403849i
$202$		0			0
$203$	25.5608	−	7.14687i	1.79402	−	0.501612i
$204$		0			0
$205$		−	8.33276i		−	0.581985i
$206$		0			0
$207$			40.4541i			2.81175i
$208$		0			0
$209$			18.6847i			1.29245i
$210$		0			0
$211$	18.7103			1.28807			0.644034	−	0.764997i	$-0.277259\pi$
$211$	18.7103			1.28807			0.644034	+	0.764997i	$0.277259\pi$
$212$		0			0
$213$	−1.78636			−0.122399
$214$		0			0
$215$	6.76735			0.461530
$216$		0			0
$217$	−3.83492	−	13.7156i	−0.260331	−	0.931075i
$218$		0			0
$219$	−29.4821			−1.99221
$220$		0			0
$221$		−	3.31035i		−	0.222678i
$222$		0			0
$223$	−21.7324			−1.45531			−0.727654	−	0.685945i	$-0.759389\pi$
$223$	−21.7324			−1.45531			−0.727654	+	0.685945i	$0.759389\pi$
$224$		0			0
$225$	−5.08861			−0.339241
$226$		0			0
$227$		−	1.17561i		−	0.0780282i	−0.999239	−	0.0390141i	$-0.987578\pi$
$227$		−	1.17561i		−	0.0780282i	0.999239	−	0.0390141i	$-0.0124217\pi$
$228$		0			0
$229$	−2.66909			−0.176378			−0.0881892	−	0.996104i	$-0.528108\pi$
$229$	−2.66909			−0.176378			−0.0881892	+	0.996104i	$0.528108\pi$
$230$		0			0
$231$	39.2764	−	10.9818i	2.58420	−	0.722549i
$232$		0			0
$233$	−20.3693			−1.33444			−0.667219	−	0.744861i	$-0.732516\pi$
$233$	−20.3693			−1.33444			−0.667219	+	0.744861i	$0.732516\pi$
$234$		0			0
$235$	−8.86368			−0.578203
$236$		0			0
$237$	8.48034			0.550858
$238$		0			0
$239$			15.4631i			1.00023i	0.865960	+	0.500113i	$0.166708\pi$
$239$			15.4631i			1.00023i	−0.865960	+	0.500113i	$0.833292\pi$
$240$		0			0
$241$		−	1.09761i		−	0.0707035i	−0.999375	−	0.0353517i	$-0.988745\pi$
$241$		−	1.09761i		−	0.0707035i	0.999375	−	0.0353517i	$-0.0112552\pi$
$242$		0			0
$243$			13.1899i			0.846132i
$244$		0			0
$245$	5.98487	−	3.63061i	0.382359	−	0.231951i
$246$		0			0
$247$		−	2.21041i		−	0.140645i
$248$		0			0
$249$	32.0617			2.03183
$250$		0			0
$251$		−	2.05500i		−	0.129710i	−0.997895	−	0.0648551i	$-0.979341\pi$
$251$		−	2.05500i		−	0.129710i	0.997895	−	0.0648551i	$-0.0206585\pi$
$252$		0			0
$253$			43.0878i			2.70891i
$254$		0			0
$255$	14.6837			0.919527
$256$		0			0
$257$		−	4.42576i		−	0.276071i	−0.990427	−	0.138036i	$-0.955921\pi$
$257$		−	4.42576i		−	0.276071i	0.990427	−	0.138036i	$-0.0440788\pi$
$258$		0			0
$259$	−2.37224	+	0.663285i	−0.147404	+	0.0412145i
$260$		0			0
$261$		−	51.0469i		−	3.15972i
$262$		0			0
$263$			6.67542i			0.411624i	0.978592	+	0.205812i	$0.0659835\pi$
$263$			6.67542i			0.411624i	−0.978592	+	0.205812i	$0.934016\pi$
$264$		0			0
$265$			6.31384i			0.387856i
$266$		0			0
$267$	38.0331			2.32759
$268$		0			0
$269$	−31.8245			−1.94037			−0.970186	−	0.242362i	$-0.922078\pi$
$269$	−31.8245			−1.94037			−0.970186	+	0.242362i	$0.922078\pi$
$270$		0			0
$271$	−2.41834			−0.146904			−0.0734519	−	0.997299i	$-0.523402\pi$
$271$	−2.41834			−0.146904			−0.0734519	+	0.997299i	$0.523402\pi$
$272$		0			0
$273$	−4.64641	+	1.29915i	−0.281213	+	0.0786281i
$274$		0			0
$275$	−5.41989			−0.326832
$276$		0			0
$277$			3.82317i			0.229712i	0.993382	+	0.114856i	$0.0366407\pi$
$277$			3.82317i			0.229712i	−0.993382	+	0.114856i	$0.963359\pi$
$278$		0			0
$279$	−27.3911			−1.63986
$280$		0			0
$281$	7.39364			0.441068			0.220534	−	0.975379i	$-0.429220\pi$
$281$	7.39364			0.441068			0.220534	+	0.975379i	$0.429220\pi$
$282$		0			0
$283$		−	13.6388i		−	0.810741i	−0.914152	−	0.405370i	$-0.867143\pi$
$283$		−	13.6388i		−	0.810741i	0.914152	−	0.405370i	$-0.132857\pi$
$284$		0			0
$285$	9.80466			0.580778
$286$		0			0
$287$	21.2321	−	5.93655i	1.25329	−	0.350424i
$288$		0			0
$289$	−9.65602			−0.568001
$290$		0			0
$291$	−25.8084			−1.51292
$292$		0			0
$293$	−13.8479			−0.809006			−0.404503	−	0.914537i	$-0.632555\pi$
$293$	−13.8479			−0.809006			−0.404503	+	0.914537i	$0.632555\pi$
$294$		0			0
$295$		−	11.5640i		−	0.673282i
$296$		0			0
$297$		−	32.1947i		−	1.86813i
$298$		0			0
$299$		−	5.09730i		−	0.294784i
$300$		0			0
$301$	4.82130	+	17.2434i	0.277895	+	0.993893i
$302$		0			0
$303$		−	12.0183i		−	0.690434i
$304$		0			0
$305$	−0.834217			−0.0477671
$306$		0			0
$307$		−	20.5060i		−	1.17034i	−0.810911	−	0.585170i	$-0.801028\pi$
$307$		−	20.5060i		−	1.17034i	0.810911	−	0.585170i	$-0.198972\pi$
$308$		0			0
$309$			3.83224i			0.218008i
$310$		0			0
$311$	−28.8437			−1.63558			−0.817789	−	0.575518i	$-0.804800\pi$
$311$	−28.8437			−1.63558			−0.817789	+	0.575518i	$0.804800\pi$
$312$		0			0
$313$		−	6.59043i		−	0.372513i	−0.982501	−	0.186257i	$-0.940364\pi$
$313$		−	6.59043i		−	0.372513i	0.982501	−	0.186257i	$-0.0596356\pi$
$314$		0			0
$315$	−3.62530	−	12.9659i	−0.204263	−	0.730546i
$316$		0			0
$317$			11.0984i			0.623348i	0.950189	+	0.311674i	$0.100890\pi$
$317$			11.0984i			0.623348i	−0.950189	+	0.311674i	$0.899110\pi$
$318$		0			0
$319$		−	54.3702i		−	3.04415i
$320$		0			0
$321$			2.91643i			0.162779i
$322$		0			0
$323$	−17.7989			−0.990358
$324$		0			0
$325$	0.641175			0.0355660
$326$		0			0
$327$	−9.22671			−0.510238
$328$		0			0
$329$	−6.31480	−	22.5849i	−0.348146	−	1.24515i
$330$		0			0
$331$	−22.3941			−1.23089			−0.615446	−	0.788179i	$-0.711024\pi$
$331$	−22.3941			−1.23089			−0.615446	+	0.788179i	$0.711024\pi$
$332$		0			0
$333$			4.73755i			0.259616i
$334$		0			0
$335$	2.01317			0.109991
$336$		0			0
$337$	9.40233			0.512178			0.256089	−	0.966653i	$-0.417566\pi$
$337$	9.40233			0.512178			0.256089	+	0.966653i	$0.417566\pi$
$338$		0			0
$339$		−	8.15891i		−	0.443131i
$340$		0			0
$341$	−29.1744			−1.57988
$342$		0			0
$343$	13.5147	+	12.6630i	0.729726	+	0.683740i
$344$		0			0
$345$	22.6100			1.21728
$346$		0			0
$347$	−20.6458			−1.10832			−0.554161	−	0.832409i	$-0.686961\pi$
$347$	−20.6458			−1.10832			−0.554161	+	0.832409i	$0.686961\pi$
$348$		0			0
$349$	19.5346			1.04566			0.522832	−	0.852436i	$-0.324875\pi$
$349$	19.5346			1.04566			0.522832	+	0.852436i	$0.324875\pi$
$350$		0			0
$351$			3.80864i			0.203290i
$352$		0			0
$353$		−	25.6220i		−	1.36372i	−0.731482	−	0.681860i	$-0.761171\pi$
$353$		−	25.6220i		−	1.36372i	0.731482	−	0.681860i	$-0.238829\pi$
$354$		0			0
$355$			0.628104i			0.0333363i
$356$		0			0
$357$	10.4612	+	37.4144i	0.553663	+	1.98018i
$358$		0			0
$359$		−	4.67391i		−	0.246679i	−0.992365	−	0.123340i	$-0.960640\pi$
$359$		−	4.67391i		−	0.246679i	0.992365	−	0.123340i	$-0.0393605\pi$
$360$		0			0
$361$	7.11521			0.374485
$362$		0			0
$363$		−	52.2601i		−	2.74294i
$364$		0			0
$365$			10.3662i			0.542594i
$366$		0			0
$367$	−28.6118			−1.49352			−0.746761	−	0.665092i	$-0.768392\pi$
$367$	−28.6118			−1.49352			−0.746761	+	0.665092i	$0.768392\pi$
$368$		0			0
$369$		−	42.4022i		−	2.20737i
$370$		0			0
$371$	−16.0878	+	4.49820i	−0.835239	+	0.233535i
$372$		0			0
$373$			27.6768i			1.43305i	0.697562	+	0.716524i	$0.254268\pi$
$373$			27.6768i			1.43305i	−0.697562	+	0.716524i	$0.745732\pi$
$374$		0			0
$375$			2.84405i			0.146866i
$376$		0			0
$377$			6.43202i			0.331266i
$378$		0			0
$379$	6.96855			0.357950			0.178975	−	0.983854i	$-0.442722\pi$
$379$	6.96855			0.357950			0.178975	+	0.983854i	$0.442722\pi$
$380$		0			0
$381$	50.2800			2.57592
$382$		0			0
$383$	22.7794			1.16397			0.581987	−	0.813198i	$-0.302275\pi$
$383$	22.7794			1.16397			0.581987	+	0.813198i	$0.302275\pi$
$384$		0			0
$385$	−3.86132	−	13.8100i	−0.196791	−	0.703824i
$386$		0			0
$387$	34.4364			1.75050
$388$		0			0
$389$			21.8852i			1.10962i	0.831976	+	0.554811i	$0.187210\pi$
$389$			21.8852i			1.10962i	−0.831976	+	0.554811i	$0.812790\pi$
$390$		0			0
$391$	−41.0451			−2.07574
$392$		0			0
$393$	−52.8853			−2.66771
$394$		0			0
$395$		−	2.98179i		−	0.150030i
$396$		0			0
$397$	15.6837			0.787144			0.393572	−	0.919294i	$-0.371239\pi$
$397$	15.6837			0.787144			0.393572	+	0.919294i	$0.371239\pi$
$398$		0			0
$399$	6.98519	+	24.9825i	0.349697	+	1.25069i
$400$		0			0
$401$	2.17338			0.108533			0.0542667	−	0.998526i	$-0.482718\pi$
$401$	2.17338			0.108533			0.0542667	+	0.998526i	$0.482718\pi$
$402$		0			0
$403$	3.45134			0.171923
$404$		0			0
$405$	−1.62810			−0.0809012
$406$		0			0
$407$			5.04598i			0.250120i
$408$		0			0
$409$			4.87458i			0.241032i	0.992711	+	0.120516i	$0.0384549\pi$
$409$			4.87458i			0.241032i	−0.992711	+	0.120516i	$0.961545\pi$
$410$		0			0
$411$			27.0624i			1.33489i
$412$		0			0
$413$	29.4654	−	8.23860i	1.44990	−	0.405395i
$414$		0			0
$415$		−	11.2733i		−	0.553383i
$416$		0			0
$417$	−61.9313			−3.03279
$418$		0			0
$419$			20.3998i			0.996593i	0.867007	+	0.498297i	$0.166041\pi$
$419$			20.3998i			0.996593i	−0.867007	+	0.498297i	$0.833959\pi$
$420$		0			0
$421$		−	14.4496i		−	0.704233i	−0.935956	−	0.352116i	$-0.885462\pi$
$421$		−	14.4496i		−	0.704233i	0.935956	−	0.352116i	$-0.114538\pi$
$422$		0			0
$423$	−45.1038			−2.19302
$424$		0			0
$425$		−	5.16295i		−	0.250440i
$426$		0			0
$427$	−0.594325	−	2.12561i	−0.0287614	−	0.102865i
$428$		0			0
$429$			9.88335i			0.477173i
$430$		0			0
$431$		−	25.3511i		−	1.22112i	−0.791970	−	0.610560i	$-0.790944\pi$
$431$		−	25.3511i		−	1.22112i	0.791970	−	0.610560i	$-0.209056\pi$
$432$		0			0
$433$		−	17.4460i		−	0.838402i	−0.907893	−	0.419201i	$-0.862310\pi$
$433$		−	17.4460i		−	0.838402i	0.907893	−	0.419201i	$-0.137690\pi$
$434$		0			0
$435$	−28.5304			−1.36793
$436$		0			0
$437$	−27.4068			−1.31105
$438$		0			0
$439$	−27.6601			−1.32015			−0.660073	−	0.751201i	$-0.729475\pi$
$439$	−27.6601			−1.32015			−0.660073	+	0.751201i	$0.729475\pi$
$440$		0			0
$441$	30.4547	−	18.4747i	1.45022	−	0.879749i
$442$		0			0
$443$	−26.2826			−1.24872			−0.624362	−	0.781135i	$-0.714641\pi$
$443$	−26.2826			−1.24872			−0.624362	+	0.781135i	$0.714641\pi$
$444$		0			0
$445$		−	13.3729i		−	0.633935i
$446$		0			0
$447$	−27.9456			−1.32178
$448$		0			0
$449$	7.17443			0.338582			0.169291	−	0.985566i	$-0.445852\pi$
$449$	7.17443			0.338582			0.169291	+	0.985566i	$0.445852\pi$
$450$		0			0
$451$		−	45.1627i		−	2.12663i
$452$		0			0
$453$	−32.3867			−1.52166
$454$		0			0
$455$	0.456796	+	1.63373i	0.0214149	+	0.0765905i
$456$		0			0
$457$	−14.0099			−0.655358			−0.327679	−	0.944789i	$-0.606266\pi$
$457$	−14.0099			−0.655358			−0.327679	+	0.944789i	$0.606266\pi$
$458$		0			0
$459$	30.6684			1.43148
$460$		0			0
$461$	−12.5644			−0.585182			−0.292591	−	0.956238i	$-0.594517\pi$
$461$	−12.5644			−0.585182			−0.292591	+	0.956238i	$0.594517\pi$
$462$		0			0
$463$		−	27.4324i		−	1.27489i	−0.770496	−	0.637445i	$-0.779991\pi$
$463$		−	27.4324i		−	1.27489i	0.770496	−	0.637445i	$-0.220009\pi$
$464$		0			0
$465$			15.3090i			0.709939i
$466$		0			0
$467$		−	13.4842i		−	0.623976i	−0.950086	−	0.311988i	$-0.899005\pi$
$467$		−	13.4842i		−	0.623976i	0.950086	−	0.311988i	$-0.100995\pi$
$468$		0			0
$469$	1.43425	+	5.12960i	0.0662276	+	0.236863i
$470$		0			0
$471$		−	14.8506i		−	0.684282i
$472$		0			0
$473$	36.6783			1.68647
$474$		0			0
$475$		−	3.44743i		−	0.158179i
$476$		0			0
$477$			32.1287i			1.47107i
$478$		0			0
$479$	38.0298			1.73763			0.868813	−	0.495141i	$-0.164884\pi$
$479$	38.0298			1.73763			0.868813	+	0.495141i	$0.164884\pi$
$480$		0			0
$481$		−	0.596941i		−	0.0272182i
$482$		0			0
$483$	16.1081	+	57.6108i	0.732946	+	2.62138i
$484$		0			0
$485$			9.07453i			0.412053i
$486$		0			0
$487$		−	7.81007i		−	0.353908i	−0.984219	−	0.176954i	$-0.943376\pi$
$487$		−	7.81007i		−	0.353908i	0.984219	−	0.176954i	$-0.0566244\pi$
$488$		0			0
$489$			53.4494i			2.41707i
$490$		0			0
$491$	−6.06004			−0.273486			−0.136743	−	0.990607i	$-0.543663\pi$
$491$	−6.06004			−0.273486			−0.136743	+	0.990607i	$0.543663\pi$
$492$		0			0
$493$	51.7927			2.33262
$494$		0			0
$495$	−27.5797			−1.23962
$496$		0			0
$497$	−1.60043	+	0.447484i	−0.0717889	+	0.0200724i
$498$		0			0
$499$	3.56253			0.159481			0.0797403	−	0.996816i	$-0.474591\pi$
$499$	3.56253			0.159481			0.0797403	+	0.996816i	$0.474591\pi$
$500$		0			0
$501$		−	6.85218i		−	0.306133i
$502$		0			0
$503$	7.55427			0.336828			0.168414	−	0.985716i	$-0.446135\pi$
$503$	7.55427			0.336828			0.168414	+	0.985716i	$0.446135\pi$
$504$		0			0
$505$	−4.22577			−0.188044
$506$		0			0
$507$			35.8034i			1.59009i
$508$		0			0
$509$	29.3030			1.29883			0.649417	−	0.760433i	$-0.275013\pi$
$509$	29.3030			1.29883			0.649417	+	0.760433i	$0.275013\pi$
$510$		0			0
$511$	−26.4135	+	7.38527i	−1.16846	+	0.326705i
$512$		0			0
$513$	20.4781			0.904130
$514$		0			0
$515$	1.34746			0.0593762
$516$		0			0
$517$	−48.0402			−2.11281
$518$		0			0
$519$			18.0007i			0.790145i
$520$		0			0
$521$		−	16.1046i		−	0.705553i	−0.935708	−	0.352777i	$-0.885238\pi$
$521$		−	16.1046i		−	0.705553i	0.935708	−	0.352777i	$-0.114762\pi$
$522$		0			0
$523$		−	23.8267i		−	1.04187i	−0.853597	−	0.520935i	$-0.825583\pi$
$523$		−	23.8267i		−	1.04187i	0.853597	−	0.520935i	$-0.174417\pi$
$524$		0			0
$525$	−7.24671	+	2.02620i	−0.316272	+	0.0884306i
$526$		0			0
$527$		−	27.7913i		−	1.21061i
$528$		0			0
$529$	−40.2014			−1.74789
$530$		0			0
$531$		−	58.8447i		−	2.55364i
$532$		0			0
$533$			5.34276i			0.231421i
$534$		0			0
$535$	1.02545			0.0443341
$536$		0			0
$537$			5.18093i			0.223574i
$538$		0			0
$539$	32.4374	−	19.6775i	1.39718	−	0.847570i
$540$		0			0
$541$		−	11.6194i		−	0.499557i	−0.968303	−	0.249779i	$-0.919642\pi$
$541$		−	11.6194i		−	0.499557i	0.968303	−	0.249779i	$-0.0803578\pi$
$542$		0			0
$543$		−	20.5830i		−	0.883299i
$544$		0			0
$545$			3.24422i			0.138967i
$546$		0			0
$547$	−10.6177			−0.453978			−0.226989	−	0.973897i	$-0.572888\pi$
$547$	−10.6177			−0.453978			−0.226989	+	0.973897i	$0.572888\pi$
$548$		0			0
$549$	−4.24500			−0.181172
$550$		0			0
$551$	34.5833			1.47330
$552$		0			0
$553$	7.59767	−	2.12433i	0.323086	−	0.0903357i
$554$		0			0
$555$	2.64784			0.112395
$556$		0			0
$557$		−	3.02433i		−	0.128145i	−0.997945	−	0.0640726i	$-0.979591\pi$
$557$		−	3.02433i		−	0.128145i	0.997945	−	0.0640726i	$-0.0204089\pi$
$558$		0			0
$559$	−4.33906			−0.183523
$560$		0			0
$561$	79.5839			3.36004
$562$		0			0
$563$			5.75591i			0.242583i	0.992617	+	0.121291i	$0.0387035\pi$
$563$			5.75591i			0.242583i	−0.992617	+	0.121291i	$0.961296\pi$
$564$		0			0
$565$	−2.86877			−0.120690
$566$		0			0
$567$	−1.15992	−	4.14845i	−0.0487120	−	0.174219i
$568$		0			0
$569$	−9.85322			−0.413069			−0.206534	−	0.978439i	$-0.566218\pi$
$569$	−9.85322			−0.413069			−0.206534	+	0.978439i	$0.566218\pi$
$570$		0			0
$571$	−45.2736			−1.89464			−0.947320	−	0.320288i	$-0.896220\pi$
$571$	−45.2736			−1.89464			−0.947320	+	0.320288i	$0.896220\pi$
$572$		0			0
$573$	−20.5065			−0.856671
$574$		0			0
$575$		−	7.94993i		−	0.331535i
$576$		0			0
$577$		−	8.14551i		−	0.339102i	−0.985521	−	0.169551i	$-0.945768\pi$
$577$		−	8.14551i		−	0.339102i	0.985521	−	0.169551i	$-0.0542317\pi$
$578$		0			0
$579$		−	20.8279i		−	0.865579i
$580$		0			0
$581$	28.7246	−	8.03147i	1.19170	−	0.333202i
$582$		0			0
$583$			34.2204i			1.41726i
$584$		0			0
$585$	3.26269			0.134896
$586$		0			0
$587$		−	14.0072i		−	0.578139i	−0.957308	−	0.289070i	$-0.906654\pi$
$587$		−	14.0072i		−	0.578139i	0.957308	−	0.289070i	$-0.0933459\pi$
$588$		0			0
$589$		−	18.5569i		−	0.764626i
$590$		0			0
$591$	35.6237			1.46536
$592$		0			0
$593$			36.9468i			1.51722i	0.651543	+	0.758612i	$0.274122\pi$
$593$			36.9468i			1.51722i	−0.651543	+	0.758612i	$0.725878\pi$
$594$		0			0
$595$	13.1553	−	3.67826i	0.539316	−	0.150794i
$596$		0			0
$597$		−	44.9148i		−	1.83824i
$598$		0			0
$599$			47.6418i			1.94659i	0.229553	+	0.973296i	$0.426273\pi$
$599$			47.6418i			1.94659i	−0.229553	+	0.973296i	$0.573727\pi$
$600$		0			0
$601$			7.18377i			0.293032i	0.989208	+	0.146516i	$0.0468060\pi$
$601$			7.18377i			0.293032i	−0.989208	+	0.146516i	$0.953194\pi$
$602$		0			0
$603$	10.2442			0.417177
$604$		0			0
$605$	−18.3753			−0.747060
$606$		0			0
$607$	−19.5887			−0.795082			−0.397541	−	0.917584i	$-0.630136\pi$
$607$	−19.5887			−0.795082			−0.397541	+	0.917584i	$0.630136\pi$
$608$		0			0
$609$	−20.3260	−	72.6961i	−0.823653	−	2.94580i
$610$		0			0
$611$	5.68317			0.229917
$612$		0			0
$613$		−	29.7747i		−	1.20259i	−0.799027	−	0.601295i	$-0.794652\pi$
$613$		−	29.7747i		−	1.20259i	0.799027	−	0.601295i	$-0.205348\pi$
$614$		0			0
$615$	−23.6988			−0.955627
$616$		0			0
$617$	−45.6621			−1.83829			−0.919144	−	0.393921i	$-0.871118\pi$
$617$	−45.6621			−1.83829			−0.919144	+	0.393921i	$0.871118\pi$
$618$		0			0
$619$			40.5087i			1.62818i	0.580737	+	0.814091i	$0.302764\pi$
$619$			40.5087i			1.62818i	−0.580737	+	0.814091i	$0.697236\pi$
$620$		0			0
$621$	47.2234			1.89501
$622$		0			0
$623$	34.0744	−	9.52731i	1.36516	−	0.381704i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	53.1402			2.12222
$628$		0			0
$629$	−4.80676			−0.191658
$630$		0			0
$631$			30.5287i			1.21533i	0.794194	+	0.607664i	$0.207893\pi$
$631$			30.5287i			1.21533i	−0.794194	+	0.607664i	$0.792107\pi$
$632$		0			0
$633$		−	53.2129i		−	2.11502i
$634$		0			0
$635$		−	17.6790i		−	0.701570i
$636$		0			0
$637$	−3.83735	+	2.32785i	−0.152041	+	0.0922329i
$638$		0			0
$639$			3.19618i			0.126439i
$640$		0			0
$641$	19.9086			0.786342			0.393171	−	0.919465i	$-0.371378\pi$
$641$	19.9086			0.786342			0.393171	+	0.919465i	$0.371378\pi$
$642$		0			0
$643$			6.16244i			0.243023i	0.992590	+	0.121512i	$0.0387741\pi$
$643$			6.16244i			0.243023i	−0.992590	+	0.121512i	$0.961226\pi$
$644$		0			0
$645$		−	19.2467i		−	0.757837i
$646$		0			0
$647$	−17.3528			−0.682210			−0.341105	−	0.940025i	$-0.610801\pi$
$647$	−17.3528			−0.682210			−0.341105	+	0.940025i	$0.610801\pi$
$648$		0			0
$649$		−	62.6757i		−	2.46023i
$650$		0			0
$651$	−39.0078	+	10.9067i	−1.52884	+	0.427467i
$652$		0			0
$653$		−	11.4104i		−	0.446523i	−0.974759	−	0.223262i	$-0.928330\pi$
$653$		−	11.4104i		−	0.446523i	0.974759	−	0.223262i	$-0.0716704\pi$
$654$		0			0
$655$			18.5951i			0.726570i
$656$		0			0
$657$			52.7497i			2.05796i
$658$		0			0
$659$	12.6731			0.493674			0.246837	−	0.969057i	$-0.420609\pi$
$659$	12.6731			0.493674			0.246837	+	0.969057i	$0.420609\pi$
$660$		0			0
$661$	−6.24632			−0.242954			−0.121477	−	0.992594i	$-0.538763\pi$
$661$	−6.24632			−0.242954			−0.121477	+	0.992594i	$0.538763\pi$
$662$		0			0
$663$	−9.41480			−0.365641
$664$		0			0
$665$	8.78415	−	2.45607i	0.340635	−	0.0952424i
$666$		0			0
$667$	79.7506			3.08795
$668$		0			0
$669$			61.8079i			2.38963i
$670$		0			0
$671$	−4.52137			−0.174545
$672$		0			0
$673$	44.5556			1.71749			0.858747	−	0.512400i	$-0.171243\pi$
$673$	44.5556			1.71749			0.858747	+	0.512400i	$0.171243\pi$
$674$		0			0
$675$			5.94010i			0.228635i
$676$		0			0
$677$	18.0160			0.692410			0.346205	−	0.938159i	$-0.387470\pi$
$677$	18.0160			0.692410			0.346205	+	0.938159i	$0.387470\pi$
$678$		0			0
$679$	−23.1221	+	6.46502i	−0.887346	+	0.248105i
$680$		0			0
$681$	−3.34350			−0.128123
$682$		0			0
$683$	−8.75563			−0.335025			−0.167512	−	0.985870i	$-0.553573\pi$
$683$	−8.75563			−0.335025			−0.167512	+	0.985870i	$0.553573\pi$
$684$		0			0
$685$	9.51544			0.363566
$686$		0			0
$687$			7.59102i			0.289616i
$688$		0			0
$689$		−	4.04828i		−	0.154227i
$690$		0			0
$691$		−	22.6145i		−	0.860298i	−0.902758	−	0.430149i	$-0.858461\pi$
$691$		−	22.6145i		−	0.860298i	0.902758	−	0.430149i	$-0.141539\pi$
$692$		0			0
$693$	−19.6488	−	70.2738i	−0.746395	−	2.66948i
$694$		0			0
$695$			21.7758i			0.826002i
$696$		0			0
$697$	43.0216			1.62956
$698$		0			0
$699$			57.9313i			2.19116i
$700$		0			0
$701$		−	45.4268i		−	1.71575i	−0.513862	−	0.857873i	$-0.671786\pi$
$701$		−	45.4268i		−	1.71575i	0.513862	−	0.857873i	$-0.328214\pi$
$702$		0			0
$703$	−3.20960			−0.121052
$704$		0			0
$705$			25.2087i			0.949416i
$706$		0			0
$707$	−3.01059	−	10.7674i	−0.113225	−	0.404949i
$708$		0			0
$709$			30.5313i			1.14663i	0.819336	+	0.573313i	$0.194342\pi$
$709$			30.5313i			1.14663i	−0.819336	+	0.573313i	$0.805658\pi$
$710$		0			0
$711$		−	15.1731i		−	0.569037i
$712$		0			0
$713$		−	42.7931i		−	1.60262i
$714$		0			0
$715$	3.47510			0.129961
$716$		0			0
$717$	43.9778			1.64238
$718$		0			0
$719$	−34.5967			−1.29024			−0.645119	−	0.764082i	$-0.723192\pi$
$719$	−34.5967			−1.29024			−0.645119	+	0.764082i	$0.723192\pi$
$720$		0			0
$721$	0.959977	+	3.43336i	0.0357514	+	0.127865i
$722$		0			0
$723$	−3.12166			−0.116096
$724$		0			0
$725$			10.0316i			0.372565i
$726$		0			0
$727$	−47.4130			−1.75845			−0.879225	−	0.476406i	$-0.841939\pi$
$727$	−47.4130			−1.75845			−0.879225	+	0.476406i	$0.841939\pi$
$728$		0			0
$729$	42.3970			1.57026
$730$		0			0
$731$			34.9395i			1.29228i
$732$		0			0
$733$	11.7854			0.435303			0.217651	−	0.976027i	$-0.430160\pi$
$733$	11.7854			0.435303			0.217651	+	0.976027i	$0.430160\pi$
$734$		0			0
$735$	−10.3256	−	17.0213i	−0.380866	−	0.627839i
$736$		0			0
$737$	10.9112			0.401918
$738$		0			0
$739$	−0.601286			−0.0221187			−0.0110593	−	0.999939i	$-0.503520\pi$
$739$	−0.601286			−0.0221187			−0.0110593	+	0.999939i	$0.503520\pi$
$740$		0			0
$741$	−6.28651			−0.230941
$742$		0			0
$743$		−	17.9303i		−	0.657799i	−0.944365	−	0.328900i	$-0.893322\pi$
$743$		−	17.9303i		−	0.657799i	0.944365	−	0.328900i	$-0.106678\pi$
$744$		0			0
$745$			9.82601i			0.359997i
$746$		0			0
$747$		−	57.3652i		−	2.09888i
$748$		0			0
$749$	0.730566	+	2.61287i	0.0266943	+	0.0954723i
$750$		0			0
$751$		−	33.8324i		−	1.23456i	−0.786742	−	0.617281i	$-0.788234\pi$
$751$		−	33.8324i		−	1.23456i	0.786742	−	0.617281i	$-0.211766\pi$
$752$		0			0
$753$	−5.84450			−0.212986
$754$		0			0
$755$			11.3875i			0.414435i
$756$		0			0
$757$		−	2.97221i		−	0.108027i	−0.998540	−	0.0540133i	$-0.982799\pi$
$757$		−	2.97221i		−	0.108027i	0.998540	−	0.0540133i	$-0.0172013\pi$
$758$		0			0
$759$	122.544			4.44805
$760$		0			0
$761$			51.9475i			1.88309i	0.336881	+	0.941547i	$0.390628\pi$
$761$			51.9475i			1.88309i	−0.336881	+	0.941547i	$0.609372\pi$
$762$		0			0
$763$	−8.26635	+	2.31129i	−0.299262	+	0.0836745i
$764$		0			0
$765$		−	26.2722i		−	0.949874i
$766$		0			0
$767$			7.41455i			0.267724i
$768$		0			0
$769$		−	26.4287i		−	0.953044i	−0.879163	−	0.476522i	$-0.841897\pi$
$769$		−	26.4287i		−	0.953044i	0.879163	−	0.476522i	$-0.158103\pi$
$770$		0			0
$771$	−12.5871			−0.453312
$772$		0			0
$773$	49.3274			1.77418			0.887091	−	0.461595i	$-0.152723\pi$
$773$	49.3274			1.77418			0.887091	+	0.461595i	$0.152723\pi$
$774$		0			0
$775$	5.38283			0.193357
$776$		0			0
$777$	1.88641	+	6.74677i	0.0676747	+	0.242039i
$778$		0			0
$779$	28.7266			1.02924
$780$		0			0
$781$			3.40426i			0.121814i
$782$		0			0
$783$	−59.5887			−2.12953
$784$		0			0
$785$	−5.22166			−0.186369
$786$		0			0
$787$		−	47.3351i		−	1.68731i	−0.536882	−	0.843657i	$-0.680398\pi$
$787$		−	47.3351i		−	1.68731i	0.536882	−	0.843657i	$-0.319602\pi$
$788$		0			0
$789$	18.9852			0.675891
$790$		0			0
$791$	−2.04381	−	7.30969i	−0.0726695	−	0.259903i
$792$		0			0
$793$	0.534879			0.0189941
$794$		0			0
$795$	17.9569			0.636864
$796$		0			0
$797$	−25.6507			−0.908594			−0.454297	−	0.890850i	$-0.650109\pi$
$797$	−25.6507			−0.908594			−0.454297	+	0.890850i	$0.650109\pi$
$798$		0			0
$799$		−	45.7627i		−	1.61897i
$800$		0			0
$801$		−	68.0493i		−	2.40440i
$802$		0			0
$803$			56.1839i			1.98269i
$804$		0			0
$805$	20.2566	−	5.66381i	0.713952	−	0.199623i
$806$		0			0
$807$			90.5103i			3.18611i
$808$		0			0
$809$	32.1867			1.13163			0.565813	−	0.824534i	$-0.308562\pi$
$809$	32.1867			1.13163			0.565813	+	0.824534i	$0.308562\pi$
$810$		0			0
$811$			19.9381i			0.700121i	0.936727	+	0.350061i	$0.113839\pi$
$811$			19.9381i			0.700121i	−0.936727	+	0.350061i	$0.886161\pi$
$812$		0			0
$813$			6.87788i			0.241218i
$814$		0			0
$815$	18.7934			0.658306
$816$		0			0
$817$			23.3300i			0.816213i
$818$		0			0
$819$	2.32445	+	8.31341i	0.0812230	+	0.290494i
$820$		0			0
$821$			10.1331i			0.353649i	0.984242	+	0.176824i	$0.0565825\pi$
$821$			10.1331i			0.353649i	−0.984242	+	0.176824i	$0.943418\pi$
$822$		0			0
$823$			39.0552i			1.36138i	0.732573	+	0.680689i	$0.238319\pi$
$823$			39.0552i			1.36138i	−0.732573	+	0.680689i	$0.761681\pi$
$824$		0			0
$825$			15.4144i			0.536662i
$826$		0			0
$827$	−23.6380			−0.821975			−0.410987	−	0.911641i	$-0.634816\pi$
$827$	−23.6380			−0.821975			−0.410987	+	0.911641i	$0.634816\pi$
$828$		0			0
$829$	18.6826			0.648873			0.324436	−	0.945908i	$-0.394825\pi$
$829$	18.6826			0.648873			0.324436	+	0.945908i	$0.394825\pi$
$830$		0			0
$831$	10.8733			0.377190
$832$		0			0
$833$	18.7446	+	30.8996i	0.649463	+	1.07061i
$834$		0			0
$835$	−2.40930			−0.0833774
$836$		0			0
$837$			31.9746i			1.10520i
$838$		0			0
$839$	8.07720			0.278856			0.139428	−	0.990232i	$-0.455474\pi$
$839$	8.07720			0.278856			0.139428	+	0.990232i	$0.455474\pi$
$840$		0			0
$841$	−71.6331			−2.47011
$842$		0			0
$843$		−	21.0279i		−	0.724238i
$844$		0			0
$845$	12.5889			0.433071
$846$		0			0
$847$	−13.0912	−	46.8206i	−0.449818	−	1.60878i
$848$		0			0
$849$	−38.7893			−1.33125
$850$		0			0
$851$	−7.40147			−0.253719
$852$		0			0
$853$	40.0440			1.37108			0.685541	−	0.728034i	$-0.259566\pi$
$853$	40.0440			1.37108			0.685541	+	0.728034i	$0.259566\pi$
$854$		0			0
$855$		−	17.5426i		−	0.599945i
$856$		0			0
$857$			35.5291i			1.21365i	0.794835	+	0.606826i	$0.207558\pi$
$857$			35.5291i			1.21365i	−0.794835	+	0.606826i	$0.792442\pi$
$858$		0			0
$859$		−	8.13535i		−	0.277575i	−0.990322	−	0.138787i	$-0.955680\pi$
$859$		−	8.13535i		−	0.277575i	0.990322	−	0.138787i	$-0.0443204\pi$
$860$		0			0
$861$	−16.8838	−	60.3851i	−0.575400	−	2.05792i
$862$		0			0
$863$			37.5054i			1.27670i	0.769747	+	0.638349i	$0.220382\pi$
$863$			37.5054i			1.27670i	−0.769747	+	0.638349i	$0.779618\pi$
$864$		0			0
$865$	6.32927			0.215202
$866$		0			0
$867$			27.4622i			0.932665i
$868$		0			0
$869$		−	16.1610i		−	0.548223i
$870$		0			0
$871$	−1.29079			−0.0437368
$872$		0			0
$873$			46.1767i			1.56285i
$874$		0			0
$875$	0.712435	+	2.54803i	0.0240847	+	0.0861390i
$876$		0			0
$877$			24.9278i			0.841752i	0.907118	+	0.420876i	$0.138277\pi$
$877$			24.9278i			0.841752i	−0.907118	+	0.420876i	$0.861723\pi$
$878$		0			0
$879$			39.3842i			1.32840i
$880$		0			0
$881$			34.7731i			1.17154i	0.810479	+	0.585768i	$0.199207\pi$
$881$			34.7731i			1.17154i	−0.810479	+	0.585768i	$0.800793\pi$
$882$		0			0
$883$	34.0838			1.14701			0.573505	−	0.819202i	$-0.305583\pi$
$883$	34.0838			1.14701			0.573505	+	0.819202i	$0.305583\pi$
$884$		0			0
$885$	−32.8886			−1.10554
$886$		0			0
$887$	−51.6722			−1.73498			−0.867491	−	0.497454i	$-0.834268\pi$
$887$	−51.6722			−1.73498			−0.867491	+	0.497454i	$0.834268\pi$
$888$		0			0
$889$	45.0466	−	12.5951i	1.51081	−	0.422428i
$890$		0			0
$891$	−8.82415			−0.295620
$892$		0			0
$893$		−	30.5570i		−	1.02255i
$894$		0			0
$895$	1.82168			0.0608919
$896$		0			0
$897$	−14.4970			−0.484039
$898$		0			0
$899$			53.9984i			1.80095i
$900$		0			0
$901$	−32.5980			−1.08600
$902$		0			0
$903$	49.0410	−	13.7120i	1.63198	−	0.456307i
$904$		0			0
$905$	−7.23721			−0.240573
$906$		0			0
$907$	−3.00402			−0.0997469			−0.0498735	−	0.998756i	$-0.515882\pi$
$907$	−3.00402			−0.0997469			−0.0498735	+	0.998756i	$0.515882\pi$
$908$		0			0
$909$	−21.5033			−0.713220
$910$		0			0
$911$		−	39.0502i		−	1.29379i	−0.762578	−	0.646896i	$-0.776067\pi$
$911$		−	39.0502i		−	1.29379i	0.762578	−	0.646896i	$-0.223933\pi$
$912$		0			0
$913$		−	61.0999i		−	2.02211i
$914$		0			0
$915$			2.37255i			0.0784342i
$916$		0			0
$917$	−47.3808	+	13.2478i	−1.56465	+	0.437481i
$918$		0			0
$919$			6.70141i			0.221059i	0.993873	+	0.110530i	$0.0352547\pi$
$919$			6.70141i			0.221059i	−0.993873	+	0.110530i	$0.964745\pi$
$920$		0			0
$921$	−58.3200			−1.92171
$922$		0			0
$923$		−	0.402725i		−	0.0132558i
$924$		0			0
$925$		−	0.931011i		−	0.0306115i
$926$		0			0
$927$	6.85669			0.225203
$928$		0			0
$929$			16.5761i			0.543843i	0.962319	+	0.271921i	$0.0876590\pi$
$929$			16.5761i			0.543843i	−0.962319	+	0.271921i	$0.912341\pi$
$930$		0			0
$931$	12.5163	+	20.6324i	0.410204	+	0.676201i
$932$		0			0
$933$			82.0330i			2.68564i
$934$		0			0
$935$		−	27.9826i		−	0.915130i
$936$		0			0
$937$			33.3090i			1.08816i	0.839035	+	0.544078i	$0.183120\pi$
$937$			33.3090i			1.08816i	−0.839035	+	0.544078i	$0.816880\pi$
$938$		0			0
$939$	−18.7435			−0.611671
$940$		0			0
$941$	−22.4583			−0.732121			−0.366061	−	0.930591i	$-0.619294\pi$
$941$	−22.4583			−0.732121			−0.366061	+	0.930591i	$0.619294\pi$
$942$		0			0
$943$	66.2449			2.15723
$944$		0			0
$945$	−15.1355	+	4.23194i	−0.492359	+	0.137665i
$946$		0			0
$947$	17.2969			0.562073			0.281036	−	0.959697i	$-0.409322\pi$
$947$	17.2969			0.562073			0.281036	+	0.959697i	$0.409322\pi$
$948$		0			0
$949$		−	6.64658i		−	0.215757i
$950$		0			0
$951$	31.5644			1.02354
$952$		0			0
$953$	−43.0097			−1.39322			−0.696611	−	0.717449i	$-0.745310\pi$
$953$	−43.0097			−1.39322			−0.696611	+	0.717449i	$0.745310\pi$
$954$		0			0
$955$			7.21032i			0.233320i
$956$		0			0
$957$	−154.632			−4.99853
$958$		0			0
$959$	6.77913	+	24.2456i	0.218910	+	0.782931i
$960$		0			0
$961$	−2.02513			−0.0653268
$962$		0			0
$963$	5.21811			0.168151
$964$		0			0
$965$	−7.32334			−0.235747
$966$		0			0
$967$		−	15.0026i		−	0.482450i	−0.970469	−	0.241225i	$-0.922451\pi$
$967$		−	15.0026i		−	0.482450i	0.970469	−	0.241225i	$-0.0775492\pi$
$968$		0			0
$969$			50.6210i			1.62618i
$970$		0			0
$971$		−	8.73468i		−	0.280309i	−0.990130	−	0.140155i	$-0.955240\pi$
$971$		−	8.73468i		−	0.280309i	0.990130	−	0.140155i	$-0.0447600\pi$
$972$		0			0
$973$	−55.4852	+	15.5138i	−1.77877	+	0.497350i
$974$		0			0
$975$		−	1.82353i		−	0.0583998i
$976$		0			0
$977$	−23.3974			−0.748549			−0.374274	−	0.927318i	$-0.622108\pi$
$977$	−23.3974			−0.748549			−0.374274	+	0.927318i	$0.622108\pi$
$978$		0			0
$979$		−	72.4796i		−	2.31646i
$980$		0			0
$981$			16.5086i			0.527077i
$982$		0			0
$983$	39.6577			1.26488			0.632442	−	0.774607i	$-0.282052\pi$
$983$	39.6577			1.26488			0.632442	+	0.774607i	$0.282052\pi$
$984$		0			0
$985$		−	12.5257i		−	0.399102i
$986$		0			0
$987$	−64.2325	+	17.9596i	−2.04454	+	0.571660i
$988$		0			0
$989$			53.8000i			1.71074i
$990$		0			0
$991$		−	20.4890i		−	0.650854i	−0.945567	−	0.325427i	$-0.894492\pi$
$991$		−	20.4890i		−	0.650854i	0.945567	−	0.325427i	$-0.105508\pi$
$992$		0			0
$993$			63.6900i			2.02114i
$994$		0			0
$995$	−15.7926			−0.500658
$996$		0			0
$997$	21.5967			0.683975			0.341988	−	0.939704i	$-0.388900\pi$
$997$	21.5967			0.683975			0.341988	+	0.939704i	$0.388900\pi$
$998$		0			0
$999$	5.53030			0.174971

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.2.h.e.671.1	✓	24
4.3	odd	2	inner	2240.2.h.e.671.24	yes	24
7.6	odd	2		2240.2.h.f.671.24	yes	24
8.3	odd	2		2240.2.h.f.671.23	yes	24
8.5	even	2		2240.2.h.f.671.2	yes	24
28.27	even	2		2240.2.h.f.671.1	yes	24
56.13	odd	2	inner	2240.2.h.e.671.23	yes	24
56.27	even	2	inner	2240.2.h.e.671.2	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2240.2.h.e.671.1	✓	24	1.1	even	1	trivial
2240.2.h.e.671.2	yes	24	56.27	even	2	inner
2240.2.h.e.671.23	yes	24	56.13	odd	2	inner
2240.2.h.e.671.24	yes	24	4.3	odd	2	inner
2240.2.h.f.671.1	yes	24	28.27	even	2
2240.2.h.f.671.2	yes	24	8.5	even	2
2240.2.h.f.671.23	yes	24	8.3	odd	2
2240.2.h.f.671.24	yes	24	7.6	odd	2

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

\(f(q)\)	\(=\)	\(q-2.84405i q^{3} -1.00000 q^{5} +(-0.712435 - 2.54803i) q^{7} -5.08861 q^{9} +O(q^{10})\)	\(q-2.84405i q^{3} -1.00000 q^{5} +(-0.712435 - 2.54803i) q^{7} -5.08861 q^{9} -5.41989 q^{11} +0.641175 q^{13} +2.84405i q^{15} -5.16295i q^{17} -3.44743i q^{19} +(-7.24671 + 2.02620i) q^{21} -7.94993i q^{23} +1.00000 q^{25} +5.94010i q^{27} +10.0316i q^{29} +5.38283 q^{31} +15.4144i q^{33} +(0.712435 + 2.54803i) q^{35} -0.931011i q^{37} -1.82353i q^{39} +8.33276i q^{41} -6.76735 q^{43} +5.08861 q^{45} +8.86368 q^{47} +(-5.98487 + 3.63061i) q^{49} -14.6837 q^{51} -6.31384i q^{53} +5.41989 q^{55} -9.80466 q^{57} +11.5640i q^{59} +0.834217 q^{61} +(3.62530 + 12.9659i) q^{63} -0.641175 q^{65} -2.01317 q^{67} -22.6100 q^{69} -0.628104i q^{71} -10.3662i q^{73} -2.84405i q^{75} +(3.86132 + 13.8100i) q^{77} +2.98179i q^{79} +1.62810 q^{81} +11.2733i q^{83} +5.16295i q^{85} +28.5304 q^{87} +13.3729i q^{89} +(-0.456796 - 1.63373i) q^{91} -15.3090i q^{93} +3.44743i q^{95} -9.07453i q^{97} +27.5797 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 24 q^{5} - 36 q^{9}+O(q^{10}) \) 24 * q - 24 * q^5 - 36 * q^9	\( 24 q - 24 q^{5} - 36 q^{9} - 4 q^{13} - 8 q^{21} + 24 q^{25} + 36 q^{45} + 24 q^{57} - 56 q^{61} + 4 q^{65} - 96 q^{69} + 12 q^{77} + 24 q^{81}+O(q^{100}) \) 24 * q - 24 * q^5 - 36 * q^9 - 4 * q^13 - 8 * q^21 + 24 * q^25 + 36 * q^45 + 24 * q^57 - 56 * q^61 + 4 * q^65 - 96 * q^69 + 12 * q^77 + 24 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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