LMFDB - Embedded newform 1248.2.g.b.625.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1248,2,Mod(625,1248)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1248, 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("1248.625");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1248 = 2^{5} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1248.g (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:	$9.96533017226$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} + 2 x^{14} - 4 x^{13} + 9 x^{12} - 10 x^{11} + 2 x^{10} - 8 x^{9} + 28 x^{8} + \cdots + 256 $ x^16 - 2x^15 + 2x^14 - 4x^13 + 9x^12 - 10x^11 + 2x^10 - 8x^9 + 28x^8 - 16x^7 + 8x^6 - 80x^5 + 144x^4 - 128x^3 + 128x^2 - 256*x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{20} $
Twist minimal:	no (minimal twist has level 312)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			625.7
Root			$1.27276 - 0.616518i$ of defining polynomial
Character	$\chi$	$=$	1248.625
Dual form			1248.2.g.b.625.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +3.29521i q^{5} -2.97802 q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +3.29521i q^{5} -2.97802 q^{7} -1.00000 q^{9} -4.64206i q^{11} -1.00000i q^{13} +3.29521 q^{15} +2.52410 q^{17} +2.18396i q^{19} +2.97802i q^{21} -7.18826 q^{23} -5.85840 q^{25} +1.00000i q^{27} -8.39420i q^{29} +5.60297 q^{31} -4.64206 q^{33} -9.81321i q^{35} -5.55709i q^{37} -1.00000 q^{39} -7.09899 q^{41} -6.08120i q^{43} -3.29521i q^{45} +4.27323 q^{47} +1.86863 q^{49} -2.52410i q^{51} -2.30359i q^{53} +15.2966 q^{55} +2.18396 q^{57} -13.8574i q^{59} -13.0161i q^{61} +2.97802 q^{63} +3.29521 q^{65} +2.39703i q^{67} +7.18826i q^{69} -10.0246 q^{71} +7.23131 q^{73} +5.85840i q^{75} +13.8242i q^{77} -10.1262 q^{79} +1.00000 q^{81} +11.4615i q^{83} +8.31745i q^{85} -8.39420 q^{87} -7.07469 q^{89} +2.97802i q^{91} -5.60297i q^{93} -7.19662 q^{95} +3.60104 q^{97} +4.64206i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 4 q^{7} - 16 q^{9}+O(q^{10}) $ 16 * q - 4 * q^7 - 16 * q^9	$ 16 q - 4 q^{7} - 16 q^{9} + 4 q^{15} + 16 q^{17} + 8 q^{23} - 32 q^{25} + 4 q^{31} - 16 q^{39} - 36 q^{41} - 24 q^{47} + 48 q^{49} - 24 q^{55} - 12 q^{57} + 4 q^{63} + 4 q^{65} - 32 q^{73} + 16 q^{81} - 8 q^{87} - 60 q^{89} + 24 q^{95} + 40 q^{97}+O(q^{100}) $ 16 * q - 4 * q^7 - 16 * q^9 + 4 * q^15 + 16 * q^17 + 8 * q^23 - 32 * q^25 + 4 * q^31 - 16 * q^39 - 36 * q^41 - 24 * q^47 + 48 * q^49 - 24 * q^55 - 12 * q^57 + 4 * q^63 + 4 * q^65 - 32 * q^73 + 16 * q^81 - 8 * q^87 - 60 * q^89 + 24 * q^95 + 40 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1248\mathbb{Z}\right)^\times$.

$n$	$703$	$769$	$833$	$1093$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	0	0
$5$	3.29521i	1.47366i	0.676077	+	0.736831i	$0.263679\pi$
$5$	3.29521i	1.47366i	−0.676077	+	0.736831i	$0.736321\pi$
$6$	0	0
$7$	−2.97802	−1.12559	−0.562794	−	0.826597i	$-0.690273\pi$
$7$	−2.97802	−1.12559	−0.562794	+	0.826597i	$0.690273\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	− 4.64206i	− 1.39963i	−0.714322	−	0.699817i	$-0.753265\pi$
$11$	− 4.64206i	− 1.39963i	0.714322	−	0.699817i	$-0.246735\pi$
$12$	0	0
$13$	− 1.00000i	− 0.277350i
$13$	− 1.00000i	− 0.277350i
$14$	0	0
$15$	3.29521	0.850819
$16$	0	0
$17$	2.52410	0.612185	0.306093	−	0.952002i	$-0.400978\pi$
$17$	2.52410	0.612185	0.306093	+	0.952002i	$0.400978\pi$
$18$	0	0
$19$	2.18396i	0.501036i	0.968112	+	0.250518i	$0.0806009\pi$
$19$	2.18396i	0.501036i	−0.968112	+	0.250518i	$0.919399\pi$
$20$	0	0
$21$	2.97802i	0.649858i
$22$	0	0
$23$	−7.18826	−1.49886	−0.749428	−	0.662086i	$-0.769672\pi$
$23$	−7.18826	−1.49886	−0.749428	+	0.662086i	$0.769672\pi$
$24$	0	0
$25$	−5.85840	−1.17168
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	− 8.39420i	− 1.55876i	−0.626549	−	0.779382i	$-0.715533\pi$
$29$	− 8.39420i	− 1.55876i	0.626549	−	0.779382i	$-0.284467\pi$
$30$	0	0
$31$	5.60297	1.00632	0.503162	−	0.864192i	$-0.332170\pi$
$31$	5.60297	1.00632	0.503162	+	0.864192i	$0.332170\pi$
$32$	0	0
$33$	−4.64206	−0.808079
$34$	0	0
$35$	− 9.81321i	− 1.65874i
$36$	0	0
$37$	− 5.55709i	− 0.913580i	−0.889574	−	0.456790i	$-0.848999\pi$
$37$	− 5.55709i	− 0.913580i	0.889574	−	0.456790i	$-0.151001\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−7.09899	−1.10868	−0.554338	−	0.832291i	$-0.687029\pi$
$41$	−7.09899	−1.10868	−0.554338	+	0.832291i	$0.687029\pi$
$42$	0	0
$43$	− 6.08120i	− 0.927374i	−0.885999	−	0.463687i	$-0.846526\pi$
$43$	− 6.08120i	− 0.927374i	0.885999	−	0.463687i	$-0.153474\pi$
$44$	0	0
$45$	− 3.29521i	− 0.491221i
$46$	0	0
$47$	4.27323	0.623315	0.311658	−	0.950194i	$-0.399116\pi$
$47$	4.27323	0.623315	0.311658	+	0.950194i	$0.399116\pi$
$48$	0	0
$49$	1.86863	0.266947
$50$	0	0
$51$	− 2.52410i	− 0.353445i
$52$	0	0
$53$	− 2.30359i	− 0.316422i	−0.987405	−	0.158211i	$-0.949427\pi$
$53$	− 2.30359i	− 0.316422i	0.987405	−	0.158211i	$-0.0505727\pi$
$54$	0	0
$55$	15.2966	2.06259
$56$	0	0
$57$	2.18396	0.289273
$58$	0	0
$59$	− 13.8574i	− 1.80408i	−0.431649	−	0.902042i	$-0.642068\pi$
$59$	− 13.8574i	− 1.80408i	0.431649	−	0.902042i	$-0.357932\pi$
$60$	0	0
$61$	− 13.0161i	− 1.66655i	−0.552862	−	0.833273i	$-0.686464\pi$
$61$	− 13.0161i	− 1.66655i	0.552862	−	0.833273i	$-0.313536\pi$
$62$	0	0
$63$	2.97802	0.375196
$64$	0	0
$65$	3.29521	0.408720
$66$	0	0
$67$	2.39703i	0.292844i	0.989222	+	0.146422i	$0.0467758\pi$
$67$	2.39703i	0.292844i	−0.989222	+	0.146422i	$0.953224\pi$
$68$	0	0
$69$	7.18826i	0.865365i
$70$	0	0
$71$	−10.0246	−1.18970	−0.594849	−	0.803838i	$-0.702788\pi$
$71$	−10.0246	−1.18970	−0.594849	+	0.803838i	$0.702788\pi$
$72$	0	0
$73$	7.23131	0.846361	0.423181	−	0.906045i	$-0.360914\pi$
$73$	7.23131	0.846361	0.423181	+	0.906045i	$0.360914\pi$
$74$	0	0
$75$	5.85840i	0.676470i
$76$	0	0
$77$	13.8242i	1.57541i
$78$	0	0
$79$	−10.1262	−1.13929	−0.569644	−	0.821891i	$-0.692919\pi$
$79$	−10.1262	−1.13929	−0.569644	+	0.821891i	$0.692919\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	11.4615i	1.25806i	0.777380	+	0.629031i	$0.216548\pi$
$83$	11.4615i	1.25806i	−0.777380	+	0.629031i	$0.783452\pi$
$84$	0	0
$85$	8.31745i	0.902154i
$86$	0	0
$87$	−8.39420	−0.899953
$88$	0	0
$89$	−7.07469	−0.749916	−0.374958	−	0.927042i	$-0.622343\pi$
$89$	−7.07469	−0.749916	−0.374958	+	0.927042i	$0.622343\pi$
$90$	0	0
$91$	2.97802i	0.312182i
$92$	0	0
$93$	− 5.60297i	− 0.581001i
$94$	0	0
$95$	−7.19662	−0.738357
$96$	0	0
$97$	3.60104	0.365630	0.182815	−	0.983147i	$-0.441479\pi$
$97$	3.60104	0.365630	0.182815	+	0.983147i	$0.441479\pi$
$98$	0	0
$99$	4.64206i	0.466545i
$100$	0	0
$101$	− 8.26799i	− 0.822696i	−0.911478	−	0.411348i	$-0.865058\pi$
$101$	− 8.26799i	− 0.822696i	0.911478	−	0.411348i	$-0.134942\pi$
$102$	0	0
$103$	9.79636	0.965264	0.482632	−	0.875823i	$-0.339681\pi$
$103$	9.79636	0.965264	0.482632	+	0.875823i	$0.339681\pi$
$104$	0	0
$105$	−9.81321	−0.957672
$106$	0	0
$107$	7.98315i	0.771760i	0.922549	+	0.385880i	$0.126102\pi$
$107$	7.98315i	0.771760i	−0.922549	+	0.385880i	$0.873898\pi$
$108$	0	0
$109$	− 5.98977i	− 0.573716i	−0.957973	−	0.286858i	$-0.907389\pi$
$109$	− 5.98977i	− 0.573716i	0.957973	−	0.286858i	$-0.0926108\pi$
$110$	0	0
$111$	−5.55709	−0.527456
$112$	0	0
$113$	−16.2287	−1.52667	−0.763334	−	0.646004i	$-0.776439\pi$
$113$	−16.2287	−1.52667	−0.763334	+	0.646004i	$0.776439\pi$
$114$	0	0
$115$	− 23.6868i	− 2.20881i
$116$	0	0
$117$	1.00000i	0.0924500i
$118$	0	0
$119$	−7.51684	−0.689068
$120$	0	0
$121$	−10.5487	−0.958976
$122$	0	0
$123$	7.09899i	0.640095i
$124$	0	0
$125$	− 2.82861i	− 0.252998i
$126$	0	0
$127$	−6.37434	−0.565631	−0.282815	−	0.959174i	$-0.591268\pi$
$127$	−6.37434	−0.565631	−0.282815	+	0.959174i	$0.591268\pi$
$128$	0	0
$129$	−6.08120	−0.535419
$130$	0	0
$131$	− 3.98315i	− 0.348010i	−0.984745	−	0.174005i	$-0.944329\pi$
$131$	− 3.98315i	− 0.348010i	0.984745	−	0.174005i	$-0.0556708\pi$
$132$	0	0
$133$	− 6.50390i	− 0.563960i
$134$	0	0
$135$	−3.29521	−0.283606
$136$	0	0
$137$	11.6295	0.993576	0.496788	−	0.867872i	$-0.334513\pi$
$137$	11.6295	0.993576	0.496788	+	0.867872i	$0.334513\pi$
$138$	0	0
$139$	17.3060i	1.46787i	0.679217	+	0.733937i	$0.262319\pi$
$139$	17.3060i	1.46787i	−0.679217	+	0.733937i	$0.737681\pi$
$140$	0	0
$141$	− 4.27323i	− 0.359871i
$142$	0	0
$143$	−4.64206	−0.388189
$144$	0	0
$145$	27.6607	2.29709
$146$	0	0
$147$	− 1.86863i	− 0.154122i
$148$	0	0
$149$	16.7970i	1.37606i	0.725682	+	0.688031i	$0.241525\pi$
$149$	16.7970i	1.37606i	−0.725682	+	0.688031i	$0.758475\pi$
$150$	0	0
$151$	1.20051	0.0976964	0.0488482	−	0.998806i	$-0.484445\pi$
$151$	1.20051	0.0976964	0.0488482	+	0.998806i	$0.484445\pi$
$152$	0	0
$153$	−2.52410	−0.204062
$154$	0	0
$155$	18.4630i	1.48298i
$156$	0	0
$157$	− 6.81445i	− 0.543852i	−0.962318	−	0.271926i	$-0.912339\pi$
$157$	− 6.81445i	− 0.543852i	0.962318	−	0.271926i	$-0.0876606\pi$
$158$	0	0
$159$	−2.30359	−0.182686
$160$	0	0
$161$	21.4068	1.68709
$162$	0	0
$163$	− 2.51916i	− 0.197316i	−0.995121	−	0.0986580i	$-0.968545\pi$
$163$	− 2.51916i	− 0.197316i	0.995121	−	0.0986580i	$-0.0314550\pi$
$164$	0	0
$165$	− 15.2966i	− 1.19084i
$166$	0	0
$167$	−5.53908	−0.428627	−0.214313	−	0.976765i	$-0.568751\pi$
$167$	−5.53908	−0.428627	−0.214313	+	0.976765i	$0.568751\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	− 2.18396i	− 0.167012i
$172$	0	0
$173$	16.9366i	1.28766i	0.765167	+	0.643832i	$0.222657\pi$
$173$	16.9366i	1.28766i	−0.765167	+	0.643832i	$0.777343\pi$
$174$	0	0
$175$	17.4465	1.31883
$176$	0	0
$177$	−13.8574	−1.04159
$178$	0	0
$179$	9.81321i	0.733474i	0.930325	+	0.366737i	$0.119525\pi$
$179$	9.81321i	0.733474i	−0.930325	+	0.366737i	$0.880475\pi$
$180$	0	0
$181$	− 3.95056i	− 0.293643i	−0.989163	−	0.146821i	$-0.953096\pi$
$181$	− 3.95056i	− 0.293643i	0.989163	−	0.146821i	$-0.0469042\pi$
$182$	0	0
$183$	−13.0161	−0.962181
$184$	0	0
$185$	18.3118	1.34631
$186$	0	0
$187$	− 11.7170i	− 0.856835i
$188$	0	0
$189$	− 2.97802i	− 0.216619i
$190$	0	0
$191$	−7.91757	−0.572895	−0.286448	−	0.958096i	$-0.592474\pi$
$191$	−7.91757	−0.572895	−0.286448	+	0.958096i	$0.592474\pi$
$192$	0	0
$193$	17.9334	1.29087	0.645436	−	0.763814i	$-0.276676\pi$
$193$	17.9334	1.29087	0.645436	+	0.763814i	$0.276676\pi$
$194$	0	0
$195$	− 3.29521i	− 0.235975i
$196$	0	0
$197$	1.95469i	0.139266i	0.997573	+	0.0696329i	$0.0221828\pi$
$197$	1.95469i	0.139266i	−0.997573	+	0.0696329i	$0.977817\pi$
$198$	0	0
$199$	−1.75434	−0.124362	−0.0621811	−	0.998065i	$-0.519806\pi$
$199$	−1.75434	−0.124362	−0.0621811	+	0.998065i	$0.519806\pi$
$200$	0	0
$201$	2.39703	0.169074
$202$	0	0
$203$	24.9981i	1.75453i
$204$	0	0
$205$	− 23.3927i	− 1.63381i
$206$	0	0
$207$	7.18826	0.499619
$208$	0	0
$209$	10.1381	0.701267
$210$	0	0
$211$	− 14.9125i	− 1.02662i	−0.858204	−	0.513309i	$-0.828419\pi$
$211$	− 14.9125i	− 1.02662i	0.858204	−	0.513309i	$-0.171581\pi$
$212$	0	0
$213$	10.0246i	0.686872i
$214$	0	0
$215$	20.0388	1.36664
$216$	0	0
$217$	−16.6858	−1.13271
$218$	0	0
$219$	− 7.23131i	− 0.488647i
$220$	0	0
$221$	− 2.52410i	− 0.169790i
$222$	0	0
$223$	9.51121	0.636918	0.318459	−	0.947937i	$-0.396835\pi$
$223$	9.51121	0.636918	0.318459	+	0.947937i	$0.396835\pi$
$224$	0	0
$225$	5.85840	0.390560
$226$	0	0
$227$	− 15.2139i	− 1.00978i	−0.863183	−	0.504891i	$-0.831533\pi$
$227$	− 15.2139i	− 1.00978i	0.863183	−	0.504891i	$-0.168467\pi$
$228$	0	0
$229$	28.1184i	1.85812i	0.369930	+	0.929059i	$0.379382\pi$
$229$	28.1184i	1.85812i	−0.369930	+	0.929059i	$0.620618\pi$
$230$	0	0
$231$	13.8242	0.909564
$232$	0	0
$233$	−15.4284	−1.01075	−0.505375	−	0.862900i	$-0.668646\pi$
$233$	−15.4284	−1.01075	−0.505375	+	0.862900i	$0.668646\pi$
$234$	0	0
$235$	14.0812i	0.918556i
$236$	0	0
$237$	10.1262i	0.657769i
$238$	0	0
$239$	4.14724	0.268263	0.134131	−	0.990964i	$-0.457176\pi$
$239$	4.14724	0.268263	0.134131	+	0.990964i	$0.457176\pi$
$240$	0	0
$241$	20.3323	1.30972	0.654860	−	0.755750i	$-0.272727\pi$
$241$	20.3323	1.30972	0.654860	+	0.755750i	$0.272727\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	6.15753i	0.393390i
$246$	0	0
$247$	2.18396	0.138962
$248$	0	0
$249$	11.4615	0.726343
$250$	0	0
$251$	− 16.7482i	− 1.05713i	−0.848891	−	0.528567i	$-0.822729\pi$
$251$	− 16.7482i	− 1.05713i	0.848891	−	0.528567i	$-0.177271\pi$
$252$	0	0
$253$	33.3684i	2.09785i
$254$	0	0
$255$	8.31745	0.520859
$256$	0	0
$257$	−27.6709	−1.72606	−0.863032	−	0.505149i	$-0.831437\pi$
$257$	−27.6709	−1.72606	−0.863032	+	0.505149i	$0.831437\pi$
$258$	0	0
$259$	16.5492i	1.02831i
$260$	0	0
$261$	8.39420i	0.519588i
$262$	0	0
$263$	29.4465	1.81575	0.907875	−	0.419241i	$-0.137704\pi$
$263$	29.4465	1.81575	0.907875	+	0.419241i	$0.137704\pi$
$264$	0	0
$265$	7.59080	0.466299
$266$	0	0
$267$	7.07469i	0.432964i
$268$	0	0
$269$	2.86814i	0.174874i	0.996170	+	0.0874368i	$0.0278676\pi$
$269$	2.86814i	0.174874i	−0.996170	+	0.0874368i	$0.972132\pi$
$270$	0	0
$271$	2.33228	0.141676	0.0708379	−	0.997488i	$-0.477433\pi$
$271$	2.33228	0.141676	0.0708379	+	0.997488i	$0.477433\pi$
$272$	0	0
$273$	2.97802	0.180238
$274$	0	0
$275$	27.1951i	1.63992i
$276$	0	0
$277$	− 3.26234i	− 0.196015i	−0.995186	−	0.0980076i	$-0.968753\pi$
$277$	− 3.26234i	− 0.196015i	0.995186	−	0.0980076i	$-0.0312469\pi$
$278$	0	0
$279$	−5.60297	−0.335441
$280$	0	0
$281$	−27.6350	−1.64856	−0.824282	−	0.566179i	$-0.808421\pi$
$281$	−27.6350	−1.64856	−0.824282	+	0.566179i	$0.808421\pi$
$282$	0	0
$283$	− 19.3872i	− 1.15245i	−0.817292	−	0.576224i	$-0.804526\pi$
$283$	− 19.3872i	− 1.15245i	0.817292	−	0.576224i	$-0.195474\pi$
$284$	0	0
$285$	7.19662i	0.426291i
$286$	0	0
$287$	21.1410	1.24791
$288$	0	0
$289$	−10.6289	−0.625229
$290$	0	0
$291$	− 3.60104i	− 0.211097i
$292$	0	0
$293$	− 14.4928i	− 0.846679i	−0.905971	−	0.423339i	$-0.860858\pi$
$293$	− 14.4928i	− 0.846679i	0.905971	−	0.423339i	$-0.139142\pi$
$294$	0	0
$295$	45.6631	2.65861
$296$	0	0
$297$	4.64206	0.269360
$298$	0	0
$299$	7.18826i	0.415708i
$300$	0	0
$301$	18.1100i	1.04384i
$302$	0	0
$303$	−8.26799	−0.474984
$304$	0	0
$305$	42.8909	2.45593
$306$	0	0
$307$	10.8749i	0.620663i	0.950628	+	0.310331i	$0.100440\pi$
$307$	10.8749i	0.620663i	−0.950628	+	0.310331i	$0.899560\pi$
$308$	0	0
$309$	− 9.79636i	− 0.557295i
$310$	0	0
$311$	−27.5186	−1.56044	−0.780219	−	0.625506i	$-0.784893\pi$
$311$	−27.5186	−1.56044	−0.780219	+	0.625506i	$0.784893\pi$
$312$	0	0
$313$	−31.7653	−1.79548	−0.897741	−	0.440523i	$-0.854793\pi$
$313$	−31.7653	−1.79548	−0.897741	+	0.440523i	$0.854793\pi$
$314$	0	0
$315$	9.81321i	0.552912i
$316$	0	0
$317$	− 12.7231i	− 0.714599i	−0.933990	−	0.357299i	$-0.883698\pi$
$317$	− 12.7231i	− 0.714599i	0.933990	−	0.357299i	$-0.116302\pi$
$318$	0	0
$319$	−38.9664	−2.18170
$320$	0	0
$321$	7.98315	0.445576
$322$	0	0
$323$	5.51255i	0.306727i
$324$	0	0
$325$	5.85840i	0.324966i
$326$	0	0
$327$	−5.98977	−0.331235
$328$	0	0
$329$	−12.7258	−0.701596
$330$	0	0
$331$	13.8265i	0.759975i	0.924992	+	0.379988i	$0.124072\pi$
$331$	13.8265i	0.759975i	−0.924992	+	0.379988i	$0.875928\pi$
$332$	0	0
$333$	5.55709i	0.304527i
$334$	0	0
$335$	−7.89873	−0.431554
$336$	0	0
$337$	27.3659	1.49072	0.745358	−	0.666665i	$-0.232279\pi$
$337$	27.3659	1.49072	0.745358	+	0.666665i	$0.232279\pi$
$338$	0	0
$339$	16.2287i	0.881422i
$340$	0	0
$341$	− 26.0094i	− 1.40849i
$342$	0	0
$343$	15.2813	0.825115
$344$	0	0
$345$	−23.6868	−1.27526
$346$	0	0
$347$	− 20.9603i	− 1.12521i	−0.826726	−	0.562604i	$-0.809800\pi$
$347$	− 20.9603i	− 1.12521i	0.826726	−	0.562604i	$-0.190200\pi$
$348$	0	0
$349$	9.49826i	0.508430i	0.967148	+	0.254215i	$0.0818171\pi$
$349$	9.49826i	0.508430i	−0.967148	+	0.254215i	$0.918183\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	−17.8275	−0.948862	−0.474431	−	0.880293i	$-0.657346\pi$
$353$	−17.8275	−0.948862	−0.474431	+	0.880293i	$0.657346\pi$
$354$	0	0
$355$	− 33.0330i	− 1.75321i
$356$	0	0
$357$	7.51684i	0.397834i
$358$	0	0
$359$	4.14652	0.218845	0.109423	−	0.993995i	$-0.465100\pi$
$359$	4.14652	0.218845	0.109423	+	0.993995i	$0.465100\pi$
$360$	0	0
$361$	14.2303	0.748963
$362$	0	0
$363$	10.5487i	0.553665i
$364$	0	0
$365$	23.8287i	1.24725i
$366$	0	0
$367$	6.05231	0.315928	0.157964	−	0.987445i	$-0.449507\pi$
$367$	6.05231	0.315928	0.157964	+	0.987445i	$0.449507\pi$
$368$	0	0
$369$	7.09899	0.369559
$370$	0	0
$371$	6.86014i	0.356161i
$372$	0	0
$373$	13.6264i	0.705550i	0.935708	+	0.352775i	$0.114762\pi$
$373$	13.6264i	0.705550i	−0.935708	+	0.352775i	$0.885238\pi$
$374$	0	0
$375$	−2.82861	−0.146069
$376$	0	0
$377$	−8.39420	−0.432323
$378$	0	0
$379$	− 4.49791i	− 0.231042i	−0.993305	−	0.115521i	$-0.963146\pi$
$379$	− 4.49791i	− 0.231042i	0.993305	−	0.115521i	$-0.0368538\pi$
$380$	0	0
$381$	6.37434i	0.326567i
$382$	0	0
$383$	29.8991	1.52777	0.763887	−	0.645351i	$-0.223289\pi$
$383$	29.8991	1.52777	0.763887	+	0.645351i	$0.223289\pi$
$384$	0	0
$385$	−45.5535	−2.32162
$386$	0	0
$387$	6.08120i	0.309125i
$388$	0	0
$389$	− 19.4869i	− 0.988025i	−0.869455	−	0.494012i	$-0.835530\pi$
$389$	− 19.4869i	− 0.988025i	0.869455	−	0.494012i	$-0.164470\pi$
$390$	0	0
$391$	−18.1439	−0.917578
$392$	0	0
$393$	−3.98315	−0.200923
$394$	0	0
$395$	− 33.3680i	− 1.67893i
$396$	0	0
$397$	27.1686i	1.36355i	0.731560	+	0.681777i	$0.238793\pi$
$397$	27.1686i	1.36355i	−0.731560	+	0.681777i	$0.761207\pi$
$398$	0	0
$399$	−6.50390	−0.325602
$400$	0	0
$401$	−20.2577	−1.01162	−0.505810	−	0.862645i	$-0.668806\pi$
$401$	−20.2577	−1.01162	−0.505810	+	0.862645i	$0.668806\pi$
$402$	0	0
$403$	− 5.60297i	− 0.279104i
$404$	0	0
$405$	3.29521i	0.163740i
$406$	0	0
$407$	−25.7964	−1.27868
$408$	0	0
$409$	18.4798	0.913769	0.456885	−	0.889526i	$-0.348965\pi$
$409$	18.4798	0.913769	0.456885	+	0.889526i	$0.348965\pi$
$410$	0	0
$411$	− 11.6295i	− 0.573642i
$412$	0	0
$413$	41.2678i	2.03065i
$414$	0	0
$415$	−37.7680	−1.85396
$416$	0	0
$417$	17.3060	0.847478
$418$	0	0
$419$	0.857313i	0.0418825i	0.999781	+	0.0209412i	$0.00666629\pi$
$419$	0.857313i	0.0418825i	−0.999781	+	0.0209412i	$0.993334\pi$
$420$	0	0
$421$	− 9.04512i	− 0.440832i	−0.975406	−	0.220416i	$-0.929258\pi$
$421$	− 9.04512i	− 0.440832i	0.975406	−	0.220416i	$-0.0707415\pi$
$422$	0	0
$423$	−4.27323	−0.207772
$424$	0	0
$425$	−14.7872	−0.717285
$426$	0	0
$427$	38.7624i	1.87584i
$428$	0	0
$429$	4.64206i	0.224121i
$430$	0	0
$431$	−27.6418	−1.33146	−0.665730	−	0.746193i	$-0.731880\pi$
$431$	−27.6418	−1.33146	−0.665730	+	0.746193i	$0.731880\pi$
$432$	0	0
$433$	1.05911	0.0508975	0.0254488	−	0.999676i	$-0.491899\pi$
$433$	1.05911	0.0508975	0.0254488	+	0.999676i	$0.491899\pi$
$434$	0	0
$435$	− 27.6607i	− 1.32623i
$436$	0	0
$437$	− 15.6989i	− 0.750981i
$438$	0	0
$439$	−1.16146	−0.0554334	−0.0277167	−	0.999616i	$-0.508824\pi$
$439$	−1.16146	−0.0554334	−0.0277167	+	0.999616i	$0.508824\pi$
$440$	0	0
$441$	−1.86863	−0.0889824
$442$	0	0
$443$	− 8.38160i	− 0.398222i	−0.979977	−	0.199111i	$-0.936195\pi$
$443$	− 8.38160i	− 0.398222i	0.979977	−	0.199111i	$-0.0638054\pi$
$444$	0	0
$445$	− 23.3126i	− 1.10512i
$446$	0	0
$447$	16.7970	0.794469
$448$	0	0
$449$	10.6747	0.503769	0.251884	−	0.967757i	$-0.418950\pi$
$449$	10.6747	0.503769	0.251884	+	0.967757i	$0.418950\pi$
$450$	0	0
$451$	32.9540i	1.55174i
$452$	0	0
$453$	− 1.20051i	− 0.0564051i
$454$	0	0
$455$	−9.81321	−0.460051
$456$	0	0
$457$	1.80521	0.0844440	0.0422220	−	0.999108i	$-0.486556\pi$
$457$	1.80521	0.0844440	0.0422220	+	0.999108i	$0.486556\pi$
$458$	0	0
$459$	2.52410i	0.117815i
$460$	0	0
$461$	− 25.6739i	− 1.19575i	−0.801589	−	0.597876i	$-0.796012\pi$
$461$	− 25.6739i	− 1.19575i	0.801589	−	0.597876i	$-0.203988\pi$
$462$	0	0
$463$	7.38853	0.343374	0.171687	−	0.985152i	$-0.445078\pi$
$463$	7.38853	0.343374	0.171687	+	0.985152i	$0.445078\pi$
$464$	0	0
$465$	18.4630	0.856200
$466$	0	0
$467$	3.35257i	0.155139i	0.996987	+	0.0775693i	$0.0247159\pi$
$467$	3.35257i	0.155139i	−0.996987	+	0.0775693i	$0.975284\pi$
$468$	0	0
$469$	− 7.13843i	− 0.329622i
$470$	0	0
$471$	−6.81445	−0.313993
$472$	0	0
$473$	−28.2293	−1.29798
$474$	0	0
$475$	− 12.7945i	− 0.587054i
$476$	0	0
$477$	2.30359i	0.105474i
$478$	0	0
$479$	13.5015	0.616900	0.308450	−	0.951241i	$-0.400190\pi$
$479$	13.5015	0.616900	0.308450	+	0.951241i	$0.400190\pi$
$480$	0	0
$481$	−5.55709	−0.253382
$482$	0	0
$483$	− 21.4068i	− 0.974044i
$484$	0	0
$485$	11.8662i	0.538816i
$486$	0	0
$487$	33.7023	1.52720	0.763599	−	0.645691i	$-0.223430\pi$
$487$	33.7023	1.52720	0.763599	+	0.645691i	$0.223430\pi$
$488$	0	0
$489$	−2.51916	−0.113920
$490$	0	0
$491$	0.546467i	0.0246617i	0.999924	+	0.0123309i	$0.00392513\pi$
$491$	0.546467i	0.0246617i	−0.999924	+	0.0123309i	$0.996075\pi$
$492$	0	0
$493$	− 21.1878i	− 0.954252i
$494$	0	0
$495$	−15.2966	−0.687529
$496$	0	0
$497$	29.8534	1.33911
$498$	0	0
$499$	4.10579i	0.183800i	0.995768	+	0.0919002i	$0.0292941\pi$
$499$	4.10579i	0.183800i	−0.995768	+	0.0919002i	$0.970706\pi$
$500$	0	0
$501$	5.53908i	0.247468i
$502$	0	0
$503$	16.8449	0.751079	0.375539	−	0.926806i	$-0.377457\pi$
$503$	16.8449	0.751079	0.375539	+	0.926806i	$0.377457\pi$
$504$	0	0
$505$	27.2448	1.21238
$506$	0	0
$507$	1.00000i	0.0444116i
$508$	0	0
$509$	− 6.65092i	− 0.294797i	−0.989077	−	0.147398i	$-0.952910\pi$
$509$	− 6.65092i	− 0.294797i	0.989077	−	0.147398i	$-0.0470899\pi$
$510$	0	0
$511$	−21.5350	−0.952654
$512$	0	0
$513$	−2.18396	−0.0964244
$514$	0	0
$515$	32.2810i	1.42247i
$516$	0	0
$517$	− 19.8366i	− 0.872413i
$518$	0	0
$519$	16.9366	0.743434
$520$	0	0
$521$	−27.3712	−1.19916	−0.599578	−	0.800316i	$-0.704665\pi$
$521$	−27.3712	−1.19916	−0.599578	+	0.800316i	$0.704665\pi$
$522$	0	0
$523$	6.59126i	0.288216i	0.989562	+	0.144108i	$0.0460312\pi$
$523$	6.59126i	0.288216i	−0.989562	+	0.144108i	$0.953969\pi$
$524$	0	0
$525$	− 17.4465i	− 0.761426i
$526$	0	0
$527$	14.1425	0.616056
$528$	0	0
$529$	28.6711	1.24657
$530$	0	0
$531$	13.8574i	0.601361i
$532$	0	0
$533$	7.09899i	0.307492i
$534$	0	0
$535$	−26.3062	−1.13731
$536$	0	0
$537$	9.81321	0.423471
$538$	0	0
$539$	− 8.67430i	− 0.373629i
$540$	0	0
$541$	− 32.3743i	− 1.39188i	−0.718099	−	0.695941i	$-0.754988\pi$
$541$	− 32.3743i	− 1.39188i	0.718099	−	0.695941i	$-0.245012\pi$
$542$	0	0
$543$	−3.95056	−0.169535
$544$	0	0
$545$	19.7375	0.845463
$546$	0	0
$547$	18.4957i	0.790817i	0.918505	+	0.395408i	$0.129397\pi$
$547$	18.4957i	0.790817i	−0.918505	+	0.395408i	$0.870603\pi$
$548$	0	0
$549$	13.0161i	0.555515i
$550$	0	0
$551$	18.3326	0.780997
$552$	0	0
$553$	30.1561	1.28237
$554$	0	0
$555$	− 18.3118i	− 0.777292i
$556$	0	0
$557$	23.4875i	0.995198i	0.867407	+	0.497599i	$0.165785\pi$
$557$	23.4875i	0.995198i	−0.867407	+	0.497599i	$0.834215\pi$
$558$	0	0
$559$	−6.08120	−0.257207
$560$	0	0
$561$	−11.7170	−0.494694
$562$	0	0
$563$	16.3616i	0.689560i	0.938683	+	0.344780i	$0.112047\pi$
$563$	16.3616i	0.689560i	−0.938683	+	0.344780i	$0.887953\pi$
$564$	0	0
$565$	− 53.4770i	− 2.24979i
$566$	0	0
$567$	−2.97802	−0.125065
$568$	0	0
$569$	22.0818	0.925717	0.462859	−	0.886432i	$-0.346824\pi$
$569$	22.0818	0.925717	0.462859	+	0.886432i	$0.346824\pi$
$570$	0	0
$571$	− 36.3287i	− 1.52031i	−0.649743	−	0.760154i	$-0.725124\pi$
$571$	− 36.3287i	− 1.52031i	0.649743	−	0.760154i	$-0.274876\pi$
$572$	0	0
$573$	7.91757i	0.330761i
$574$	0	0
$575$	42.1117	1.75618
$576$	0	0
$577$	32.7126	1.36184	0.680921	−	0.732357i	$-0.261580\pi$
$577$	32.7126	1.36184	0.680921	+	0.732357i	$0.261580\pi$
$578$	0	0
$579$	− 17.9334i	− 0.745286i
$580$	0	0
$581$	− 34.1326i	− 1.41606i
$582$	0	0
$583$	−10.6934	−0.442875
$584$	0	0
$585$	−3.29521	−0.136240
$586$	0	0
$587$	− 21.3134i	− 0.879699i	−0.898071	−	0.439850i	$-0.855032\pi$
$587$	− 21.3134i	− 0.879699i	0.898071	−	0.439850i	$-0.144968\pi$
$588$	0	0
$589$	12.2367i	0.504204i
$590$	0	0
$591$	1.95469	0.0804052
$592$	0	0
$593$	−7.67265	−0.315078	−0.157539	−	0.987513i	$-0.550356\pi$
$593$	−7.67265	−0.315078	−0.157539	+	0.987513i	$0.550356\pi$
$594$	0	0
$595$	− 24.7696i	− 1.01545i
$596$	0	0
$597$	1.75434i	0.0718005i
$598$	0	0
$599$	−23.1225	−0.944761	−0.472381	−	0.881395i	$-0.656605\pi$
$599$	−23.1225	−0.944761	−0.472381	+	0.881395i	$0.656605\pi$
$600$	0	0
$601$	−24.4418	−0.997001	−0.498501	−	0.866889i	$-0.666116\pi$
$601$	−24.4418	−0.997001	−0.498501	+	0.866889i	$0.666116\pi$
$602$	0	0
$603$	− 2.39703i	− 0.0976148i
$604$	0	0
$605$	− 34.7603i	− 1.41321i
$606$	0	0
$607$	−29.5925	−1.20112	−0.600561	−	0.799579i	$-0.705056\pi$
$607$	−29.5925	−1.20112	−0.600561	+	0.799579i	$0.705056\pi$
$608$	0	0
$609$	24.9981	1.01298
$610$	0	0
$611$	− 4.27323i	− 0.172877i
$612$	0	0
$613$	15.3430i	0.619696i	0.950786	+	0.309848i	$0.100278\pi$
$613$	15.3430i	0.619696i	−0.950786	+	0.309848i	$0.899722\pi$
$614$	0	0
$615$	−23.3927	−0.943283
$616$	0	0
$617$	−0.365995	−0.0147344	−0.00736720	−	0.999973i	$-0.502345\pi$
$617$	−0.365995	−0.0147344	−0.00736720	+	0.999973i	$0.502345\pi$
$618$	0	0
$619$	− 11.5651i	− 0.464839i	−0.972616	−	0.232419i	$-0.925336\pi$
$619$	− 11.5651i	− 0.464839i	0.972616	−	0.232419i	$-0.0746642\pi$
$620$	0	0
$621$	− 7.18826i	− 0.288455i
$622$	0	0
$623$	21.0686	0.844096
$624$	0	0
$625$	−19.9711	−0.798846
$626$	0	0
$627$	− 10.1381i	− 0.404877i
$628$	0	0
$629$	− 14.0267i	− 0.559280i
$630$	0	0
$631$	−22.2627	−0.886264	−0.443132	−	0.896456i	$-0.646133\pi$
$631$	−22.2627	−0.886264	−0.443132	+	0.896456i	$0.646133\pi$
$632$	0	0
$633$	−14.9125	−0.592718
$634$	0	0
$635$	− 21.0048i	− 0.833549i
$636$	0	0
$637$	− 1.86863i	− 0.0740379i
$638$	0	0
$639$	10.0246	0.396566
$640$	0	0
$641$	40.0104	1.58032	0.790159	−	0.612902i	$-0.209998\pi$
$641$	40.0104	1.58032	0.790159	+	0.612902i	$0.209998\pi$
$642$	0	0
$643$	− 37.1465i	− 1.46492i	−0.680813	−	0.732458i	$-0.738373\pi$
$643$	− 37.1465i	− 1.46492i	0.680813	−	0.732458i	$-0.261627\pi$
$644$	0	0
$645$	− 20.0388i	− 0.789027i
$646$	0	0
$647$	41.1525	1.61787	0.808936	−	0.587897i	$-0.200044\pi$
$647$	41.1525	1.61787	0.808936	+	0.587897i	$0.200044\pi$
$648$	0	0
$649$	−64.3270	−2.52506
$650$	0	0
$651$	16.6858i	0.653968i
$652$	0	0
$653$	− 5.05283i	− 0.197732i	−0.995101	−	0.0988662i	$-0.968478\pi$
$653$	− 5.05283i	− 0.197732i	0.995101	−	0.0988662i	$-0.0315216\pi$
$654$	0	0
$655$	13.1253	0.512849
$656$	0	0
$657$	−7.23131	−0.282120
$658$	0	0
$659$	31.3892i	1.22275i	0.791341	+	0.611375i	$0.209383\pi$
$659$	31.3892i	1.22275i	−0.791341	+	0.611375i	$0.790617\pi$
$660$	0	0
$661$	− 25.0378i	− 0.973859i	−0.873441	−	0.486929i	$-0.838117\pi$
$661$	− 25.0378i	− 0.973859i	0.873441	−	0.486929i	$-0.161883\pi$
$662$	0	0
$663$	−2.52410	−0.0980281
$664$	0	0
$665$	21.4317	0.831086
$666$	0	0
$667$	60.3397i	2.33636i
$668$	0	0
$669$	− 9.51121i	− 0.367725i
$670$	0	0
$671$	−60.4217	−2.33256
$672$	0	0
$673$	−28.8507	−1.11211	−0.556056	−	0.831145i	$-0.687686\pi$
$673$	−28.8507	−1.11211	−0.556056	+	0.831145i	$0.687686\pi$
$674$	0	0
$675$	− 5.85840i	− 0.225490i
$676$	0	0
$677$	25.8847i	0.994830i	0.867513	+	0.497415i	$0.165717\pi$
$677$	25.8847i	0.994830i	−0.867513	+	0.497415i	$0.834283\pi$
$678$	0	0
$679$	−10.7240	−0.411549
$680$	0	0
$681$	−15.2139	−0.582998
$682$	0	0
$683$	− 29.0440i	− 1.11134i	−0.831404	−	0.555668i	$-0.812463\pi$
$683$	− 29.0440i	− 1.11134i	0.831404	−	0.555668i	$-0.187537\pi$
$684$	0	0
$685$	38.3217i	1.46420i
$686$	0	0
$687$	28.1184	1.07279
$688$	0	0
$689$	−2.30359	−0.0877597
$690$	0	0
$691$	31.6523i	1.20411i	0.798454	+	0.602056i	$0.205652\pi$
$691$	31.6523i	1.20411i	−0.798454	+	0.602056i	$0.794348\pi$
$692$	0	0
$693$	− 13.8242i	− 0.525137i
$694$	0	0
$695$	−57.0268	−2.16315
$696$	0	0
$697$	−17.9186	−0.678715
$698$	0	0
$699$	15.4284i	0.583557i
$700$	0	0
$701$	− 38.8106i	− 1.46586i	−0.680306	−	0.732928i	$-0.738153\pi$
$701$	− 38.8106i	− 1.46586i	0.680306	−	0.732928i	$-0.261847\pi$
$702$	0	0
$703$	12.1365	0.457736
$704$	0	0
$705$	14.0812	0.530329
$706$	0	0
$707$	24.6223i	0.926016i
$708$	0	0
$709$	1.18538i	0.0445178i	0.999752	+	0.0222589i	$0.00708581\pi$
$709$	1.18538i	0.0445178i	−0.999752	+	0.0222589i	$0.992914\pi$
$710$	0	0
$711$	10.1262	0.379763
$712$	0	0
$713$	−40.2757	−1.50834
$714$	0	0
$715$	− 15.2966i	− 0.572059i
$716$	0	0
$717$	− 4.14724i	− 0.154882i
$718$	0	0
$719$	3.27616	0.122180	0.0610902	−	0.998132i	$-0.480542\pi$
$719$	3.27616	0.122180	0.0610902	+	0.998132i	$0.480542\pi$
$720$	0	0
$721$	−29.1738	−1.08649
$722$	0	0
$723$	− 20.3323i	− 0.756168i
$724$	0	0
$725$	49.1766i	1.82637i
$726$	0	0
$727$	−21.4797	−0.796636	−0.398318	−	0.917247i	$-0.630406\pi$
$727$	−21.4797	−0.796636	−0.398318	+	0.917247i	$0.630406\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	− 15.3496i	− 0.567724i
$732$	0	0
$733$	4.61402i	0.170423i	0.996363	+	0.0852114i	$0.0271566\pi$
$733$	4.61402i	0.170423i	−0.996363	+	0.0852114i	$0.972843\pi$
$734$	0	0
$735$	6.15753	0.227124
$736$	0	0
$737$	11.1272	0.409875
$738$	0	0
$739$	− 44.9923i	− 1.65507i	−0.561416	−	0.827534i	$-0.689743\pi$
$739$	− 44.9923i	− 1.65507i	0.561416	−	0.827534i	$-0.310257\pi$
$740$	0	0
$741$	− 2.18396i	− 0.0802299i
$742$	0	0
$743$	−17.1359	−0.628654	−0.314327	−	0.949315i	$-0.601779\pi$
$743$	−17.1359	−0.628654	−0.314327	+	0.949315i	$0.601779\pi$
$744$	0	0
$745$	−55.3495	−2.02785
$746$	0	0
$747$	− 11.4615i	− 0.419354i
$748$	0	0
$749$	− 23.7740i	− 0.868684i
$750$	0	0
$751$	10.5517	0.385037	0.192518	−	0.981293i	$-0.438335\pi$
$751$	10.5517	0.385037	0.192518	+	0.981293i	$0.438335\pi$
$752$	0	0
$753$	−16.7482	−0.610337
$754$	0	0
$755$	3.95594i	0.143972i
$756$	0	0
$757$	− 45.4487i	− 1.65186i	−0.563773	−	0.825930i	$-0.690651\pi$
$757$	− 45.4487i	− 1.65186i	0.563773	−	0.825930i	$-0.309349\pi$
$758$	0	0
$759$	33.3684	1.21119
$760$	0	0
$761$	33.3202	1.20786	0.603928	−	0.797039i	$-0.293601\pi$
$761$	33.3202	1.20786	0.603928	+	0.797039i	$0.293601\pi$
$762$	0	0
$763$	17.8377i	0.645767i
$764$	0	0
$765$	− 8.31745i	− 0.300718i
$766$	0	0
$767$	−13.8574	−0.500363
$768$	0	0
$769$	−14.6076	−0.526762	−0.263381	−	0.964692i	$-0.584838\pi$
$769$	−14.6076	−0.526762	−0.263381	+	0.964692i	$0.584838\pi$
$770$	0	0
$771$	27.6709i	0.996543i
$772$	0	0
$773$	− 18.9736i	− 0.682433i	−0.939985	−	0.341217i	$-0.889161\pi$
$773$	− 18.9736i	− 0.682433i	0.939985	−	0.341217i	$-0.110839\pi$
$774$	0	0
$775$	−32.8245	−1.17909
$776$	0	0
$777$	16.5492	0.593698
$778$	0	0
$779$	− 15.5039i	− 0.555487i
$780$	0	0
$781$	46.5347i	1.66514i
$782$	0	0
$783$	8.39420	0.299984
$784$	0	0
$785$	22.4550	0.801455
$786$	0	0
$787$	27.1995i	0.969558i	0.874637	+	0.484779i	$0.161100\pi$
$787$	27.1995i	0.969558i	−0.874637	+	0.484779i	$0.838900\pi$
$788$	0	0
$789$	− 29.4465i	− 1.04832i
$790$	0	0
$791$	48.3295	1.71840
$792$	0	0
$793$	−13.0161	−0.462217
$794$	0	0
$795$	− 7.59080i	− 0.269218i
$796$	0	0
$797$	− 16.9799i	− 0.601458i	−0.953710	−	0.300729i	$-0.902770\pi$
$797$	− 16.9799i	− 0.601458i	0.953710	−	0.300729i	$-0.0972300\pi$
$798$	0	0
$799$	10.7861	0.381584
$800$	0	0
$801$	7.07469	0.249972
$802$	0	0
$803$	− 33.5682i	− 1.18460i
$804$	0	0
$805$	70.5400i	2.48621i
$806$	0	0
$807$	2.86814	0.100963
$808$	0	0
$809$	26.1302	0.918688	0.459344	−	0.888259i	$-0.348085\pi$
$809$	26.1302	0.918688	0.459344	+	0.888259i	$0.348085\pi$
$810$	0	0
$811$	− 30.5983i	− 1.07445i	−0.843438	−	0.537226i	$-0.819472\pi$
$811$	− 30.5983i	− 1.07445i	0.843438	−	0.537226i	$-0.180528\pi$
$812$	0	0
$813$	− 2.33228i	− 0.0817965i
$814$	0	0
$815$	8.30116	0.290777
$816$	0	0
$817$	13.2811	0.464647
$818$	0	0
$819$	− 2.97802i	− 0.104061i
$820$	0	0
$821$	13.2629i	0.462878i	0.972849	+	0.231439i	$0.0743434\pi$
$821$	13.2629i	0.462878i	−0.972849	+	0.231439i	$0.925657\pi$
$822$	0	0
$823$	−26.7159	−0.931257	−0.465629	−	0.884980i	$-0.654172\pi$
$823$	−26.7159	−0.931257	−0.465629	+	0.884980i	$0.654172\pi$
$824$	0	0
$825$	27.1951	0.946810
$826$	0	0
$827$	− 8.39923i	− 0.292070i	−0.989279	−	0.146035i	$-0.953349\pi$
$827$	− 8.39923i	− 0.292070i	0.989279	−	0.146035i	$-0.0466512\pi$
$828$	0	0
$829$	20.1364i	0.699365i	0.936868	+	0.349683i	$0.113711\pi$
$829$	20.1364i	0.699365i	−0.936868	+	0.349683i	$0.886289\pi$
$830$	0	0
$831$	−3.26234	−0.113169
$832$	0	0
$833$	4.71662	0.163421
$834$	0	0
$835$	− 18.2524i	− 0.631651i
$836$	0	0
$837$	5.60297i	0.193667i
$838$	0	0
$839$	−32.2636	−1.11386	−0.556931	−	0.830559i	$-0.688021\pi$
$839$	−32.2636	−1.11386	−0.556931	+	0.830559i	$0.688021\pi$
$840$	0	0
$841$	−41.4626	−1.42975
$842$	0	0
$843$	27.6350i	0.951799i
$844$	0	0
$845$	− 3.29521i	− 0.113359i
$846$	0	0
$847$	31.4144	1.07941
$848$	0	0
$849$	−19.3872	−0.665366
$850$	0	0
$851$	39.9458i	1.36933i
$852$	0	0
$853$	32.5703i	1.11519i	0.830114	+	0.557593i	$0.188275\pi$
$853$	32.5703i	1.11519i	−0.830114	+	0.557593i	$0.811725\pi$
$854$	0	0
$855$	7.19662	0.246119
$856$	0	0
$857$	−32.7173	−1.11760	−0.558801	−	0.829302i	$-0.688738\pi$
$857$	−32.7173	−1.11760	−0.558801	+	0.829302i	$0.688738\pi$
$858$	0	0
$859$	12.3546i	0.421533i	0.977536	+	0.210767i	$0.0675960\pi$
$859$	12.3546i	0.421533i	−0.977536	+	0.210767i	$0.932404\pi$
$860$	0	0
$861$	− 21.1410i	− 0.720483i
$862$	0	0
$863$	−34.3760	−1.17017	−0.585086	−	0.810971i	$-0.698939\pi$
$863$	−34.3760	−1.17017	−0.585086	+	0.810971i	$0.698939\pi$
$864$	0	0
$865$	−55.8096	−1.89758
$866$	0	0
$867$	10.6289i	0.360976i
$868$	0	0
$869$	47.0065i	1.59459i
$870$	0	0
$871$	2.39703	0.0812204
$872$	0	0
$873$	−3.60104	−0.121877
$874$	0	0
$875$	8.42367i	0.284772i
$876$	0	0
$877$	− 30.2395i	− 1.02112i	−0.859843	−	0.510558i	$-0.829439\pi$
$877$	− 30.2395i	− 1.02112i	0.859843	−	0.510558i	$-0.170561\pi$
$878$	0	0
$879$	−14.4928	−0.488830
$880$	0	0
$881$	−32.4097	−1.09191	−0.545956	−	0.837814i	$-0.683833\pi$
$881$	−32.4097	−1.09191	−0.545956	+	0.837814i	$0.683833\pi$
$882$	0	0
$883$	− 42.6147i	− 1.43410i	−0.697023	−	0.717049i	$-0.745492\pi$
$883$	− 42.6147i	− 1.43410i	0.697023	−	0.717049i	$-0.254508\pi$
$884$	0	0
$885$	− 45.6631i	− 1.53495i
$886$	0	0
$887$	33.9294	1.13924	0.569619	−	0.821909i	$-0.307091\pi$
$887$	33.9294	1.13924	0.569619	+	0.821909i	$0.307091\pi$
$888$	0	0
$889$	18.9829	0.636667
$890$	0	0
$891$	− 4.64206i	− 0.155515i
$892$	0	0
$893$	9.33259i	0.312303i
$894$	0	0
$895$	−32.3366	−1.08089
$896$	0	0
$897$	7.18826	0.240009
$898$	0	0
$899$	− 47.0325i	− 1.56862i
$900$	0	0
$901$	− 5.81450i	− 0.193709i
$902$	0	0
$903$	18.1100	0.602661
$904$	0	0
$905$	13.0179	0.432730
$906$	0	0
$907$	52.6048i	1.74671i	0.487081	+	0.873357i	$0.338062\pi$
$907$	52.6048i	1.74671i	−0.487081	+	0.873357i	$0.661938\pi$
$908$	0	0
$909$	8.26799i	0.274232i
$910$	0	0
$911$	20.6215	0.683222	0.341611	−	0.939841i	$-0.389028\pi$
$911$	20.6215	0.683222	0.341611	+	0.939841i	$0.389028\pi$
$912$	0	0
$913$	53.2050	1.76083
$914$	0	0
$915$	− 42.8909i	− 1.41793i
$916$	0	0
$917$	11.8619i	0.391715i
$918$	0	0
$919$	11.0660	0.365034	0.182517	−	0.983203i	$-0.441576\pi$
$919$	11.0660	0.365034	0.182517	+	0.983203i	$0.441576\pi$
$920$	0	0
$921$	10.8749	0.358340
$922$	0	0
$923$	10.0246i	0.329963i
$924$	0	0
$925$	32.5557i	1.07042i
$926$	0	0
$927$	−9.79636	−0.321755
$928$	0	0
$929$	26.4090	0.866450	0.433225	−	0.901286i	$-0.357376\pi$
$929$	26.4090	0.866450	0.433225	+	0.901286i	$0.357376\pi$
$930$	0	0
$931$	4.08102i	0.133750i
$932$	0	0
$933$	27.5186i	0.900920i
$934$	0	0
$935$	38.6101	1.26269
$936$	0	0
$937$	56.6713	1.85137	0.925685	−	0.378295i	$-0.123489\pi$
$937$	56.6713	1.85137	0.925685	+	0.378295i	$0.123489\pi$
$938$	0	0
$939$	31.7653i	1.03662i
$940$	0	0
$941$	− 19.8715i	− 0.647793i	−0.946092	−	0.323897i	$-0.895007\pi$
$941$	− 19.8715i	− 0.647793i	0.946092	−	0.323897i	$-0.104993\pi$
$942$	0	0
$943$	51.0294	1.66175
$944$	0	0
$945$	9.81321	0.319224
$946$	0	0
$947$	30.3284i	0.985540i	0.870160	+	0.492770i	$0.164016\pi$
$947$	30.3284i	0.985540i	−0.870160	+	0.492770i	$0.835984\pi$
$948$	0	0
$949$	− 7.23131i	− 0.234738i
$950$	0	0
$951$	−12.7231	−0.412574
$952$	0	0
$953$	44.0687	1.42753	0.713763	−	0.700387i	$-0.246989\pi$
$953$	44.0687	1.42753	0.713763	+	0.700387i	$0.246989\pi$
$954$	0	0
$955$	− 26.0900i	− 0.844254i
$956$	0	0
$957$	38.9664i	1.25961i
$958$	0	0
$959$	−34.6330	−1.11836
$960$	0	0
$961$	0.393317	0.0126877
$962$	0	0
$963$	− 7.98315i	− 0.257253i
$964$	0	0
$965$	59.0942i	1.90231i
$966$	0	0
$967$	9.31361	0.299506	0.149753	−	0.988723i	$-0.452152\pi$
$967$	9.31361	0.299506	0.149753	+	0.988723i	$0.452152\pi$
$968$	0	0
$969$	5.51255	0.177089
$970$	0	0
$971$	− 38.4042i	− 1.23245i	−0.787570	−	0.616225i	$-0.788661\pi$
$971$	− 38.4042i	− 1.23245i	0.787570	−	0.616225i	$-0.211339\pi$
$972$	0	0
$973$	− 51.5376i	− 1.65222i
$974$	0	0
$975$	5.85840	0.187619
$976$	0	0
$977$	−3.84116	−0.122890	−0.0614448	−	0.998110i	$-0.519571\pi$
$977$	−3.84116	−0.122890	−0.0614448	+	0.998110i	$0.519571\pi$
$978$	0	0
$979$	32.8412i	1.04961i
$980$	0	0
$981$	5.98977i	0.191239i
$982$	0	0
$983$	−32.0851	−1.02336	−0.511678	−	0.859177i	$-0.670976\pi$
$983$	−32.0851	−1.02336	−0.511678	+	0.859177i	$0.670976\pi$
$984$	0	0
$985$	−6.44111	−0.205231
$986$	0	0
$987$	12.7258i	0.405067i
$988$	0	0
$989$	43.7132i	1.39000i
$990$	0	0
$991$	13.5890	0.431669	0.215835	−	0.976430i	$-0.430753\pi$
$991$	13.5890	0.431669	0.215835	+	0.976430i	$0.430753\pi$
$992$	0	0
$993$	13.8265	0.438772
$994$	0	0
$995$	− 5.78093i	− 0.183268i
$996$	0	0
$997$	20.7619i	0.657535i	0.944411	+	0.328767i	$0.106633\pi$
$997$	20.7619i	0.657535i	−0.944411	+	0.328767i	$0.893367\pi$
$998$	0	0
$999$	5.55709	0.175819

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1248.2.g.b.625.7		16
3.2	odd	2		3744.2.g.e.1873.3		16
4.3	odd	2		312.2.g.b.157.9	✓	16
8.3	odd	2		312.2.g.b.157.10	yes	16
8.5	even	2	inner	1248.2.g.b.625.10		16
12.11	even	2		936.2.g.e.469.8		16
16.3	odd	4		9984.2.a.bt.1.7		8
16.5	even	4		9984.2.a.bs.1.2		8
16.11	odd	4		9984.2.a.bu.1.2		8
16.13	even	4		9984.2.a.bv.1.7		8
24.5	odd	2		3744.2.g.e.1873.14		16
24.11	even	2		936.2.g.e.469.7		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.2.g.b.157.9	✓	16	4.3	odd	2
312.2.g.b.157.10	yes	16	8.3	odd	2
936.2.g.e.469.7		16	24.11	even	2
936.2.g.e.469.8		16	12.11	even	2
1248.2.g.b.625.7		16	1.1	even	1	trivial
1248.2.g.b.625.10		16	8.5	even	2	inner
3744.2.g.e.1873.3		16	3.2	odd	2
3744.2.g.e.1873.14		16	24.5	odd	2
9984.2.a.bs.1.2		8	16.5	even	4
9984.2.a.bt.1.7		8	16.3	odd	4
9984.2.a.bu.1.2		8	16.11	odd	4
9984.2.a.bv.1.7		8	16.13	even	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:	\( N \)	\(=\)	\( 1248 = 2^{5} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1248.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +3.29521i q^{5} -2.97802 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +3.29521i q^{5} -2.97802 q^{7} -1.00000 q^{9} -4.64206i q^{11} -1.00000i q^{13} +3.29521 q^{15} +2.52410 q^{17} +2.18396i q^{19} +2.97802i q^{21} -7.18826 q^{23} -5.85840 q^{25} +1.00000i q^{27} -8.39420i q^{29} +5.60297 q^{31} -4.64206 q^{33} -9.81321i q^{35} -5.55709i q^{37} -1.00000 q^{39} -7.09899 q^{41} -6.08120i q^{43} -3.29521i q^{45} +4.27323 q^{47} +1.86863 q^{49} -2.52410i q^{51} -2.30359i q^{53} +15.2966 q^{55} +2.18396 q^{57} -13.8574i q^{59} -13.0161i q^{61} +2.97802 q^{63} +3.29521 q^{65} +2.39703i q^{67} +7.18826i q^{69} -10.0246 q^{71} +7.23131 q^{73} +5.85840i q^{75} +13.8242i q^{77} -10.1262 q^{79} +1.00000 q^{81} +11.4615i q^{83} +8.31745i q^{85} -8.39420 q^{87} -7.07469 q^{89} +2.97802i q^{91} -5.60297i q^{93} -7.19662 q^{95} +3.60104 q^{97} +4.64206i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{7} - 16 q^{9}+O(q^{10}) \) 16 * q - 4 * q^7 - 16 * q^9	\( 16 q - 4 q^{7} - 16 q^{9} + 4 q^{15} + 16 q^{17} + 8 q^{23} - 32 q^{25} + 4 q^{31} - 16 q^{39} - 36 q^{41} - 24 q^{47} + 48 q^{49} - 24 q^{55} - 12 q^{57} + 4 q^{63} + 4 q^{65} - 32 q^{73} + 16 q^{81} - 8 q^{87} - 60 q^{89} + 24 q^{95} + 40 q^{97}+O(q^{100}) \) 16 * q - 4 * q^7 - 16 * q^9 + 4 * q^15 + 16 * q^17 + 8 * q^23 - 32 * q^25 + 4 * q^31 - 16 * q^39 - 36 * q^41 - 24 * q^47 + 48 * q^49 - 24 * q^55 - 12 * q^57 + 4 * q^63 + 4 * q^65 - 32 * q^73 + 16 * q^81 - 8 * q^87 - 60 * q^89 + 24 * q^95 + 40 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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