LMFDB - Embedded newform 128.12.b.f.65.10

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [128,12,Mod(65,128)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("128.65"); S:= CuspForms(chi, 12); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(128, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 12, names="a")

Level:	$ N $	$=$	$ 128 = 2^{7} $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	128.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,0,0,0,0,-316732] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$98.3479271116$
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} + 3286 x^{10} + 3725205 x^{8} + 1773266980 x^{6} + 401838244180 x^{4} + 42969249696816 x^{2} + 17\!\cdots\!96 $ x^12 + 3286x^10 + 3725205x^8 + 1773266980x^6 + 401838244180x^4 + 42969249696816*x^2 + 1736910980997696
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{120}\cdot 3^{2}\cdot 5^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			65.10
Root			$16.5733i$ of defining polynomial
Character	$\chi$	$=$	128.65
Dual form			128.12.b.f.65.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+501.129i q^{3} +10865.5i q^{5} +32440.9 q^{7} -73983.7 q^{9} +786304. i q^{11} +1.13752e6i q^{13} -5.44504e6 q^{15} -1.54131e6 q^{17} -1.43638e7i q^{19} +1.62571e7i q^{21} +1.37586e7 q^{23} -6.92317e7 q^{25} +5.16982e7i q^{27} +1.65630e8i q^{29} -1.74967e8 q^{31} -3.94040e8 q^{33} +3.52487e8i q^{35} +8.41310e7i q^{37} -5.70046e8 q^{39} +1.19255e9 q^{41} +1.69021e9i q^{43} -8.03872e8i q^{45} +1.59759e8 q^{47} -9.24918e8 q^{49} -7.72397e8i q^{51} -2.72276e9i q^{53} -8.54361e9 q^{55} +7.19811e9 q^{57} +2.99614e9i q^{59} -5.22269e9i q^{61} -2.40009e9 q^{63} -1.23598e10 q^{65} +1.47878e10i q^{67} +6.89484e9i q^{69} +1.81623e10 q^{71} -7.96108e9 q^{73} -3.46940e10i q^{75} +2.55084e10i q^{77} +5.33558e10 q^{79} -3.90135e10 q^{81} -4.38742e10i q^{83} -1.67472e10i q^{85} -8.30023e10 q^{87} -5.45016e10 q^{89} +3.69022e10i q^{91} -8.76809e10i q^{93} +1.56070e11 q^{95} +1.19088e11 q^{97} -5.81737e10i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 316732 q^{9} + 11301736 q^{17} - 100402276 q^{25} - 370222336 q^{33} + 1458786872 q^{41} - 2425030420 q^{49} - 8422115584 q^{57} - 18057782080 q^{65} - 41779508088 q^{73} - 213089686484 q^{81} - 331639752632 q^{89}+ \cdots - 396775590616 q^{97}+O(q^{100}) $ 12 * q - 316732 * q^9 + 11301736 * q^17 - 100402276 * q^25 - 370222336 * q^33 + 1458786872 * q^41 - 2425030420 * q^49 - 8422115584 * q^57 - 18057782080 * q^65 - 41779508088 * q^73 - 213089686484 * q^81 - 331639752632 * q^89 - 396775590616 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/128\mathbb{Z}\right)^\times$.

$n$	$5$	$127$
$\chi(n)$	$-1$	$1$

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
$3$	501.129i	1.19065i	0.803486	+	0.595323i	$0.202976\pi$
$3$	501.129i	1.19065i	−0.803486	+	0.595323i	$0.797024\pi$
$4$	0	0
$5$	10865.5i	1.55495i	0.628915	+	0.777474i	$0.283499\pi$
$5$	10865.5i	1.55495i	−0.628915	+	0.777474i	$0.716501\pi$
$6$	0	0
$7$	32440.9	0.729547	0.364773	−	0.931096i	$-0.381146\pi$
$7$	32440.9	0.729547	0.364773	+	0.931096i	$0.381146\pi$
$8$	0	0
$9$	−73983.7	−0.417640
$10$	0	0
$11$	786304.i	1.47208i	0.676939	+	0.736039i	$0.263306\pi$
$11$	786304.i	1.47208i	−0.676939	+	0.736039i	$0.736694\pi$
$12$	0	0
$13$	1.13752e6i	0.849712i	0.905261	+	0.424856i	$0.139675\pi$
$13$	1.13752e6i	0.849712i	−0.905261	+	0.424856i	$0.860325\pi$
$14$	0	0
$15$	−5.44504e6	−1.85139
$16$	0	0
$17$	−1.54131e6	−0.263282	−0.131641	−	0.991297i	$-0.542025\pi$
$17$	−1.54131e6	−0.263282	−0.131641	+	0.991297i	$0.542025\pi$
$18$	0	0
$19$	− 1.43638e7i	− 1.33083i	−0.746472	−	0.665417i	$-0.768254\pi$
$19$	− 1.43638e7i	− 1.33083i	0.746472	−	0.665417i	$-0.231746\pi$
$20$	0	0
$21$	1.62571e7i	0.868632i
$22$	0	0
$23$	1.37586e7	0.445729	0.222865	−	0.974849i	$-0.428459\pi$
$23$	1.37586e7	0.445729	0.222865	+	0.974849i	$0.428459\pi$
$24$	0	0
$25$	−6.92317e7	−1.41787
$26$	0	0
$27$	5.16982e7i	0.693385i
$28$	0	0
$29$	1.65630e8i	1.49952i	0.661712	+	0.749758i	$0.269830\pi$
$29$	1.65630e8i	1.49952i	−0.661712	+	0.749758i	$0.730170\pi$
$30$	0	0
$31$	−1.74967e8	−1.09765	−0.548827	−	0.835936i	$-0.684926\pi$
$31$	−1.74967e8	−1.09765	−0.548827	+	0.835936i	$0.684926\pi$
$32$	0	0
$33$	−3.94040e8	−1.75272
$34$	0	0
$35$	3.52487e8i	1.13441i
$36$	0	0
$37$	8.41310e7i	0.199456i	0.995015	+	0.0997279i	$0.0317972\pi$
$37$	8.41310e7i	0.199456i	−0.995015	+	0.0997279i	$0.968203\pi$
$38$	0	0
$39$	−5.70046e8	−1.01171
$40$	0	0
$41$	1.19255e9	1.60755	0.803776	−	0.594933i	$-0.202821\pi$
$41$	1.19255e9	1.60755	0.803776	+	0.594933i	$0.202821\pi$
$42$	0	0
$43$	1.69021e9i	1.75334i	0.481095	+	0.876668i	$0.340239\pi$
$43$	1.69021e9i	1.75334i	−0.481095	+	0.876668i	$0.659761\pi$
$44$	0	0
$45$	− 8.03872e8i	− 0.649409i
$46$	0	0
$47$	1.59759e8	0.101608	0.0508040	−	0.998709i	$-0.483822\pi$
$47$	1.59759e8	0.101608	0.0508040	+	0.998709i	$0.483822\pi$
$48$	0	0
$49$	−9.24918e8	−0.467762
$50$	0	0
$51$	− 7.72397e8i	− 0.313476i
$52$	0	0
$53$	− 2.72276e9i	− 0.894319i	−0.894454	−	0.447159i	$-0.852436\pi$
$53$	− 2.72276e9i	− 0.894319i	0.894454	−	0.447159i	$-0.147564\pi$
$54$	0	0
$55$	−8.54361e9	−2.28901
$56$	0	0
$57$	7.19811e9	1.58455
$58$	0	0
$59$	2.99614e9i	0.545602i	0.962071	+	0.272801i	$0.0879500\pi$
$59$	2.99614e9i	0.545602i	−0.962071	+	0.272801i	$0.912050\pi$
$60$	0	0
$61$	− 5.22269e9i	− 0.791735i	−0.918308	−	0.395868i	$-0.870444\pi$
$61$	− 5.22269e9i	− 0.791735i	0.918308	−	0.395868i	$-0.129556\pi$
$62$	0	0
$63$	−2.40009e9	−0.304688
$64$	0	0
$65$	−1.23598e10	−1.32126
$66$	0	0
$67$	1.47878e10i	1.33811i	0.743213	+	0.669055i	$0.233301\pi$
$67$	1.47878e10i	1.33811i	−0.743213	+	0.669055i	$0.766699\pi$
$68$	0	0
$69$	6.89484e9i	0.530706i
$70$	0	0
$71$	1.81623e10	1.19468	0.597338	−	0.801989i	$-0.296225\pi$
$71$	1.81623e10	1.19468	0.597338	+	0.801989i	$0.296225\pi$
$72$	0	0
$73$	−7.96108e9	−0.449465	−0.224733	−	0.974420i	$-0.572151\pi$
$73$	−7.96108e9	−0.449465	−0.224733	+	0.974420i	$0.572151\pi$
$74$	0	0
$75$	− 3.46940e10i	− 1.68818i
$76$	0	0
$77$	2.55084e10i	1.07395i
$78$	0	0
$79$	5.33558e10	1.95089	0.975445	−	0.220245i	$-0.0706856\pi$
$79$	5.33558e10	1.95089	0.975445	+	0.220245i	$0.0706856\pi$
$80$	0	0
$81$	−3.90135e10	−1.24322
$82$	0	0
$83$	− 4.38742e10i	− 1.22259i	−0.791404	−	0.611293i	$-0.790650\pi$
$83$	− 4.38742e10i	− 1.22259i	0.791404	−	0.611293i	$-0.209350\pi$
$84$	0	0
$85$	− 1.67472e10i	− 0.409391i
$86$	0	0
$87$	−8.30023e10	−1.78539
$88$	0	0
$89$	−5.45016e10	−1.03458	−0.517290	−	0.855810i	$-0.673059\pi$
$89$	−5.45016e10	−1.03458	−0.517290	+	0.855810i	$0.673059\pi$
$90$	0	0
$91$	3.69022e10i	0.619905i
$92$	0	0
$93$	− 8.76809e10i	− 1.30692i
$94$	0	0
$95$	1.56070e11	2.06938
$96$	0	0
$97$	1.19088e11	1.40807	0.704036	−	0.710164i	$-0.251379\pi$
$97$	1.19088e11	1.40807	0.704036	+	0.710164i	$0.251379\pi$
$98$	0	0
$99$	− 5.81737e10i	− 0.614799i
$100$	0	0
$101$	− 1.75332e11i	− 1.65995i	−0.557802	−	0.829974i	$-0.688355\pi$
$101$	− 1.75332e11i	− 1.65995i	0.557802	−	0.829974i	$-0.311645\pi$
$102$	0	0
$103$	−8.21061e10	−0.697863	−0.348932	−	0.937148i	$-0.613455\pi$
$103$	−8.21061e10	−0.697863	−0.348932	+	0.937148i	$0.613455\pi$
$104$	0	0
$105$	−1.76642e11	−1.35068
$106$	0	0
$107$	− 1.39285e11i	− 0.960050i	−0.877255	−	0.480025i	$-0.840628\pi$
$107$	− 1.39285e11i	− 0.960050i	0.877255	−	0.480025i	$-0.159372\pi$
$108$	0	0
$109$	− 1.49353e11i	− 0.929755i	−0.885375	−	0.464877i	$-0.846098\pi$
$109$	− 1.49353e11i	− 0.929755i	0.885375	−	0.464877i	$-0.153902\pi$
$110$	0	0
$111$	−4.21605e10	−0.237482
$112$	0	0
$113$	4.23408e10	0.216186	0.108093	−	0.994141i	$-0.465526\pi$
$113$	4.23408e10	0.216186	0.108093	+	0.994141i	$0.465526\pi$
$114$	0	0
$115$	1.49495e11i	0.693086i
$116$	0	0
$117$	− 8.41581e10i	− 0.354874i
$118$	0	0
$119$	−5.00015e10	−0.192077
$120$	0	0
$121$	−3.32963e11	−1.16701
$122$	0	0
$123$	5.97621e11i	1.91403i
$124$	0	0
$125$	− 2.21696e11i	− 0.649759i
$126$	0	0
$127$	−5.91241e10	−0.158798	−0.0793988	−	0.996843i	$-0.525300\pi$
$127$	−5.91241e10	−0.158798	−0.0793988	+	0.996843i	$0.525300\pi$
$128$	0	0
$129$	−8.47016e11	−2.08761
$130$	0	0
$131$	2.96036e11i	0.670427i	0.942142	+	0.335214i	$0.108809\pi$
$131$	2.96036e11i	0.670427i	−0.942142	+	0.335214i	$0.891191\pi$
$132$	0	0
$133$	− 4.65973e11i	− 0.970905i
$134$	0	0
$135$	−5.61728e11	−1.07818
$136$	0	0
$137$	−4.48098e11	−0.793249	−0.396624	−	0.917981i	$-0.629818\pi$
$137$	−4.48098e11	−0.793249	−0.396624	+	0.917981i	$0.629818\pi$
$138$	0	0
$139$	1.68440e10i	0.0275337i	0.999905	+	0.0137668i	$0.00438226\pi$
$139$	1.68440e10i	0.0275337i	−0.999905	+	0.0137668i	$0.995618\pi$
$140$	0	0
$141$	8.00601e10i	0.120979i
$142$	0	0
$143$	−8.94439e11	−1.25084
$144$	0	0
$145$	−1.79966e12	−2.33167
$146$	0	0
$147$	− 4.63503e11i	− 0.556939i
$148$	0	0
$149$	− 6.87972e11i	− 0.767443i	−0.923449	−	0.383721i	$-0.874642\pi$
$149$	− 6.87972e11i	− 0.767443i	0.923449	−	0.383721i	$-0.125358\pi$
$150$	0	0
$151$	7.82198e11	0.810855	0.405428	−	0.914127i	$-0.367123\pi$
$151$	7.82198e11	0.810855	0.405428	+	0.914127i	$0.367123\pi$
$152$	0	0
$153$	1.14032e11	0.109957
$154$	0	0
$155$	− 1.90111e12i	− 1.70680i
$156$	0	0
$157$	− 7.46684e11i	− 0.624724i	−0.949963	−	0.312362i	$-0.898880\pi$
$157$	− 7.46684e11i	− 0.624724i	0.949963	−	0.312362i	$-0.101120\pi$
$158$	0	0
$159$	1.36446e12	1.06482
$160$	0	0
$161$	4.46341e11	0.325180
$162$	0	0
$163$	1.04376e12i	0.710511i	0.934769	+	0.355255i	$0.115606\pi$
$163$	1.04376e12i	0.710511i	−0.934769	+	0.355255i	$0.884394\pi$
$164$	0	0
$165$	− 4.28146e12i	− 2.72540i
$166$	0	0
$167$	2.33077e12	1.38854	0.694271	−	0.719713i	$-0.255727\pi$
$167$	2.33077e12	1.38854	0.694271	+	0.719713i	$0.255727\pi$
$168$	0	0
$169$	4.98201e11	0.277989
$170$	0	0
$171$	1.06268e12i	0.555809i
$172$	0	0
$173$	− 2.09455e12i	− 1.02763i	−0.857900	−	0.513816i	$-0.828231\pi$
$173$	− 2.09455e12i	− 1.02763i	0.857900	−	0.513816i	$-0.171769\pi$
$174$	0	0
$175$	−2.24594e12	−1.03440
$176$	0	0
$177$	−1.50145e12	−0.649619
$178$	0	0
$179$	− 1.85912e12i	− 0.756165i	−0.925772	−	0.378083i	$-0.876584\pi$
$179$	− 1.85912e12i	− 0.756165i	0.925772	−	0.378083i	$-0.123416\pi$
$180$	0	0
$181$	1.92913e12i	0.738125i	0.929405	+	0.369063i	$0.120321\pi$
$181$	1.92913e12i	0.738125i	−0.929405	+	0.369063i	$0.879679\pi$
$182$	0	0
$183$	2.61724e12	0.942677
$184$	0	0
$185$	−9.14129e11	−0.310144
$186$	0	0
$187$	− 1.21194e12i	− 0.387572i
$188$	0	0
$189$	1.67713e12i	0.505857i
$190$	0	0
$191$	−2.35682e12	−0.670878	−0.335439	−	0.942062i	$-0.608885\pi$
$191$	−2.35682e12	−0.670878	−0.335439	+	0.942062i	$0.608885\pi$
$192$	0	0
$193$	−2.78673e12	−0.749083	−0.374541	−	0.927210i	$-0.622200\pi$
$193$	−2.78673e12	−0.749083	−0.374541	+	0.927210i	$0.622200\pi$
$194$	0	0
$195$	− 6.19386e12i	− 1.57315i
$196$	0	0
$197$	− 7.51506e12i	− 1.80455i	−0.431163	−	0.902274i	$-0.641897\pi$
$197$	− 7.51506e12i	− 1.80455i	0.431163	−	0.902274i	$-0.358103\pi$
$198$	0	0
$199$	7.06255e12	1.60424	0.802120	−	0.597163i	$-0.203705\pi$
$199$	7.06255e12	1.60424	0.802120	+	0.597163i	$0.203705\pi$
$200$	0	0
$201$	−7.41060e12	−1.59322
$202$	0	0
$203$	5.37319e12i	1.09397i
$204$	0	0
$205$	1.29577e13i	2.49966i
$206$	0	0
$207$	−1.01791e12	−0.186154
$208$	0	0
$209$	1.12943e13	1.95909
$210$	0	0
$211$	6.99105e12i	1.15077i	0.817883	+	0.575385i	$0.195148\pi$
$211$	6.99105e12i	1.15077i	−0.817883	+	0.575385i	$0.804852\pi$
$212$	0	0
$213$	9.10167e12i	1.42244i
$214$	0	0
$215$	−1.83651e13	−2.72635
$216$	0	0
$217$	−5.67607e12	−0.800791
$218$	0	0
$219$	− 3.98953e12i	− 0.535154i
$220$	0	0
$221$	− 1.75328e12i	− 0.223714i
$222$	0	0
$223$	−9.85048e11	−0.119614	−0.0598068	−	0.998210i	$-0.519048\pi$
$223$	−9.85048e11	−0.119614	−0.0598068	+	0.998210i	$0.519048\pi$
$224$	0	0
$225$	5.12202e12	0.592157
$226$	0	0
$227$	1.66599e13i	1.83455i	0.398251	+	0.917276i	$0.369617\pi$
$227$	1.66599e13i	1.83455i	−0.398251	+	0.917276i	$0.630383\pi$
$228$	0	0
$229$	− 1.50916e13i	− 1.58358i	−0.610793	−	0.791790i	$-0.709149\pi$
$229$	− 1.50916e13i	− 1.58358i	0.610793	−	0.791790i	$-0.290851\pi$
$230$	0	0
$231$	−1.27830e13	−1.27869
$232$	0	0
$233$	1.73091e13	1.65127	0.825634	−	0.564206i	$-0.190818\pi$
$233$	1.73091e13	1.65127	0.825634	+	0.564206i	$0.190818\pi$
$234$	0	0
$235$	1.73587e12i	0.157995i
$236$	0	0
$237$	2.67382e13i	2.32282i
$238$	0	0
$239$	1.47419e13	1.22283	0.611414	−	0.791311i	$-0.290601\pi$
$239$	1.47419e13	1.22283	0.611414	+	0.791311i	$0.290601\pi$
$240$	0	0
$241$	3.20248e12	0.253743	0.126871	−	0.991919i	$-0.459506\pi$
$241$	3.20248e12	0.253743	0.126871	+	0.991919i	$0.459506\pi$
$242$	0	0
$243$	− 1.03926e13i	− 0.786847i
$244$	0	0
$245$	− 1.00497e13i	− 0.727345i
$246$	0	0
$247$	1.63391e13	1.13083
$248$	0	0
$249$	2.19867e13	1.45567
$250$	0	0
$251$	1.18861e13i	0.753070i	0.926402	+	0.376535i	$0.122885\pi$
$251$	1.18861e13i	0.753070i	−0.926402	+	0.376535i	$0.877115\pi$
$252$	0	0
$253$	1.08184e13i	0.656148i
$254$	0	0
$255$	8.39251e12	0.487440
$256$	0	0
$257$	1.56011e13	0.868006	0.434003	−	0.900911i	$-0.357101\pi$
$257$	1.56011e13	0.868006	0.434003	+	0.900911i	$0.357101\pi$
$258$	0	0
$259$	2.72928e12i	0.145512i
$260$	0	0
$261$	− 1.22539e13i	− 0.626258i
$262$	0	0
$263$	−1.67572e13	−0.821191	−0.410595	−	0.911818i	$-0.634679\pi$
$263$	−1.67572e13	−0.821191	−0.410595	+	0.911818i	$0.634679\pi$
$264$	0	0
$265$	2.95843e13	1.39062
$266$	0	0
$267$	− 2.73124e13i	− 1.23182i
$268$	0	0
$269$	2.99367e13i	1.29589i	0.761689	+	0.647943i	$0.224370\pi$
$269$	2.99367e13i	1.29589i	−0.761689	+	0.647943i	$0.775630\pi$
$270$	0	0
$271$	−1.18798e13	−0.493719	−0.246859	−	0.969051i	$-0.579399\pi$
$271$	−1.18798e13	−0.493719	−0.246859	+	0.969051i	$0.579399\pi$
$272$	0	0
$273$	−1.84928e13	−0.738088
$274$	0	0
$275$	− 5.44372e13i	− 2.08721i
$276$	0	0
$277$	4.50126e13i	1.65842i	0.558934	+	0.829212i	$0.311211\pi$
$277$	4.50126e13i	1.65842i	−0.558934	+	0.829212i	$0.688789\pi$
$278$	0	0
$279$	1.29447e13	0.458425
$280$	0	0
$281$	2.10660e13	0.717295	0.358647	−	0.933473i	$-0.383238\pi$
$281$	2.10660e13	0.717295	0.358647	+	0.933473i	$0.383238\pi$
$282$	0	0
$283$	− 8.94389e12i	− 0.292888i	−0.989219	−	0.146444i	$-0.953217\pi$
$283$	− 8.94389e12i	− 0.292888i	0.989219	−	0.146444i	$-0.0467827\pi$
$284$	0	0
$285$	7.82113e13i	2.46390i
$286$	0	0
$287$	3.86873e13	1.17278
$288$	0	0
$289$	−3.18962e13	−0.930682
$290$	0	0
$291$	5.96787e13i	1.67652i
$292$	0	0
$293$	− 5.20452e13i	− 1.40802i	−0.710190	−	0.704010i	$-0.751391\pi$
$293$	− 5.20452e13i	− 1.40802i	0.710190	−	0.704010i	$-0.248609\pi$
$294$	0	0
$295$	−3.25546e13	−0.848382
$296$	0	0
$297$	−4.06505e13	−1.02072
$298$	0	0
$299$	1.56507e13i	0.378742i
$300$	0	0
$301$	5.48320e13i	1.27914i
$302$	0	0
$303$	8.78642e13	1.97641
$304$	0	0
$305$	5.67473e13	1.23111
$306$	0	0
$307$	− 5.63253e13i	− 1.17881i	−0.807839	−	0.589403i	$-0.799363\pi$
$307$	− 5.63253e13i	− 1.17881i	0.807839	−	0.589403i	$-0.200637\pi$
$308$	0	0
$309$	− 4.11458e13i	− 0.830909i
$310$	0	0
$311$	3.30973e13	0.645075	0.322537	−	0.946557i	$-0.395464\pi$
$311$	3.30973e13	0.645075	0.322537	+	0.946557i	$0.395464\pi$
$312$	0	0
$313$	−1.70502e12	−0.0320801	−0.0160401	−	0.999871i	$-0.505106\pi$
$313$	−1.70502e12	−0.0320801	−0.0160401	+	0.999871i	$0.505106\pi$
$314$	0	0
$315$	− 2.60783e13i	− 0.473774i
$316$	0	0
$317$	− 8.88080e12i	− 0.155821i	−0.996960	−	0.0779105i	$-0.975175\pi$
$317$	− 8.88080e12i	− 0.155821i	0.996960	−	0.0779105i	$-0.0248249\pi$
$318$	0	0
$319$	−1.30236e14	−2.20740
$320$	0	0
$321$	6.97999e13	1.14308
$322$	0	0
$323$	2.21391e13i	0.350385i
$324$	0	0
$325$	− 7.87527e13i	− 1.20478i
$326$	0	0
$327$	7.48452e13	1.10701
$328$	0	0
$329$	5.18273e12	0.0741278
$330$	0	0
$331$	1.27911e14i	1.76951i	0.466056	+	0.884755i	$0.345675\pi$
$331$	1.27911e14i	1.76951i	−0.466056	+	0.884755i	$0.654325\pi$
$332$	0	0
$333$	− 6.22432e12i	− 0.0833007i
$334$	0	0
$335$	−1.60677e14	−2.08069
$336$	0	0
$337$	−2.04853e13	−0.256731	−0.128366	−	0.991727i	$-0.540973\pi$
$337$	−2.04853e13	−0.256731	−0.128366	+	0.991727i	$0.540973\pi$
$338$	0	0
$339$	2.12182e13i	0.257401i
$340$	0	0
$341$	− 1.37577e14i	− 1.61583i
$342$	0	0
$343$	−9.41513e13	−1.07080
$344$	0	0
$345$	−7.49161e13	−0.825221
$346$	0	0
$347$	8.05992e12i	0.0860040i	0.999075	+	0.0430020i	$0.0136922\pi$
$347$	8.05992e12i	0.0860040i	−0.999075	+	0.0430020i	$0.986308\pi$
$348$	0	0
$349$	5.76676e13i	0.596200i	0.954535	+	0.298100i	$0.0963529\pi$
$349$	5.76676e13i	0.596200i	−0.954535	+	0.298100i	$0.903647\pi$
$350$	0	0
$351$	−5.88079e13	−0.589178
$352$	0	0
$353$	1.59086e13	0.154479	0.0772397	−	0.997013i	$-0.475389\pi$
$353$	1.59086e13	0.154479	0.0772397	+	0.997013i	$0.475389\pi$
$354$	0	0
$355$	1.97343e14i	1.85766i
$356$	0	0
$357$	− 2.50572e13i	− 0.228696i
$358$	0	0
$359$	1.05958e14	0.937810	0.468905	−	0.883249i	$-0.344649\pi$
$359$	1.05958e14	0.937810	0.468905	+	0.883249i	$0.344649\pi$
$360$	0	0
$361$	−8.98277e13	−0.771118
$362$	0	0
$363$	− 1.66857e14i	− 1.38950i
$364$	0	0
$365$	− 8.65014e13i	− 0.698895i
$366$	0	0
$367$	2.29994e13	0.180324	0.0901620	−	0.995927i	$-0.471262\pi$
$367$	2.29994e13	0.180324	0.0901620	+	0.995927i	$0.471262\pi$
$368$	0	0
$369$	−8.82291e13	−0.671378
$370$	0	0
$371$	− 8.83287e13i	− 0.652447i
$372$	0	0
$373$	− 1.64439e14i	− 1.17925i	−0.807677	−	0.589625i	$-0.799276\pi$
$373$	− 1.64439e14i	− 1.17925i	0.807677	−	0.589625i	$-0.200724\pi$
$374$	0	0
$375$	1.11098e14	0.773634
$376$	0	0
$377$	−1.88409e14	−1.27416
$378$	0	0
$379$	− 2.18392e14i	− 1.43456i	−0.696783	−	0.717282i	$-0.745386\pi$
$379$	− 2.18392e14i	− 1.43456i	0.696783	−	0.717282i	$-0.254614\pi$
$380$	0	0
$381$	− 2.96288e13i	− 0.189072i
$382$	0	0
$383$	−4.30682e13	−0.267032	−0.133516	−	0.991047i	$-0.542627\pi$
$383$	−4.30682e13	−0.267032	−0.133516	+	0.991047i	$0.542627\pi$
$384$	0	0
$385$	−2.77162e14	−1.66994
$386$	0	0
$387$	− 1.25048e14i	− 0.732264i
$388$	0	0
$389$	2.52330e14i	1.43630i	0.695886	+	0.718152i	$0.255012\pi$
$389$	2.52330e14i	1.43630i	−0.695886	+	0.718152i	$0.744988\pi$
$390$	0	0
$391$	−2.12063e13	−0.117353
$392$	0	0
$393$	−1.48352e14	−0.798242
$394$	0	0
$395$	5.79739e14i	3.03353i
$396$	0	0
$397$	− 6.09074e13i	− 0.309972i	−0.987917	−	0.154986i	$-0.950467\pi$
$397$	− 6.09074e13i	− 0.309972i	0.987917	−	0.154986i	$-0.0495332\pi$
$398$	0	0
$399$	2.33513e14	1.15601
$400$	0	0
$401$	−3.25665e14	−1.56847	−0.784237	−	0.620462i	$-0.786945\pi$
$401$	−3.25665e14	−1.56847	−0.784237	+	0.620462i	$0.786945\pi$
$402$	0	0
$403$	− 1.99029e14i	− 0.932691i
$404$	0	0
$405$	− 4.23902e14i	− 1.93314i
$406$	0	0
$407$	−6.61526e13	−0.293615
$408$	0	0
$409$	4.00594e14	1.73072	0.865359	−	0.501153i	$-0.167091\pi$
$409$	4.00594e14	1.73072	0.865359	+	0.501153i	$0.167091\pi$
$410$	0	0
$411$	− 2.24555e14i	− 0.944479i
$412$	0	0
$413$	9.71972e13i	0.398042i
$414$	0	0
$415$	4.76717e14	1.90106
$416$	0	0
$417$	−8.44104e12	−0.0327829
$418$	0	0
$419$	2.22124e14i	0.840269i	0.907462	+	0.420135i	$0.138017\pi$
$419$	2.22124e14i	0.840269i	−0.907462	+	0.420135i	$0.861983\pi$
$420$	0	0
$421$	7.38645e13i	0.272198i	0.990695	+	0.136099i	$0.0434565\pi$
$421$	7.38645e13i	0.272198i	−0.990695	+	0.136099i	$0.956544\pi$
$422$	0	0
$423$	−1.18196e13	−0.0424356
$424$	0	0
$425$	1.06708e14	0.373299
$426$	0	0
$427$	− 1.69428e14i	− 0.577608i
$428$	0	0
$429$	− 4.48230e14i	− 1.48931i
$430$	0	0
$431$	−3.32792e14	−1.07782	−0.538912	−	0.842362i	$-0.681164\pi$
$431$	−3.32792e14	−1.07782	−0.538912	+	0.842362i	$0.681164\pi$
$432$	0	0
$433$	−3.88388e12	−0.0122626	−0.00613130	−	0.999981i	$-0.501952\pi$
$433$	−3.88388e12	−0.0122626	−0.00613130	+	0.999981i	$0.501952\pi$
$434$	0	0
$435$	− 9.01864e14i	− 2.77620i
$436$	0	0
$437$	− 1.97625e14i	− 0.593192i
$438$	0	0
$439$	−1.96680e14	−0.575712	−0.287856	−	0.957674i	$-0.592942\pi$
$439$	−1.96680e14	−0.575712	−0.287856	+	0.957674i	$0.592942\pi$
$440$	0	0
$441$	6.84288e13	0.195356
$442$	0	0
$443$	− 2.45009e14i	− 0.682278i	−0.940013	−	0.341139i	$-0.889187\pi$
$443$	− 2.45009e14i	− 0.682278i	0.940013	−	0.341139i	$-0.110813\pi$
$444$	0	0
$445$	− 5.92189e14i	− 1.60872i
$446$	0	0
$447$	3.44763e14	0.913753
$448$	0	0
$449$	−3.42410e13	−0.0885505	−0.0442753	−	0.999019i	$-0.514098\pi$
$449$	−3.42410e13	−0.0885505	−0.0442753	+	0.999019i	$0.514098\pi$
$450$	0	0
$451$	9.37706e14i	2.36644i
$452$	0	0
$453$	3.91982e14i	0.965442i
$454$	0	0
$455$	−4.00962e14	−0.963920
$456$	0	0
$457$	3.54108e14	0.830993	0.415496	−	0.909595i	$-0.363608\pi$
$457$	3.54108e14	0.830993	0.415496	+	0.909595i	$0.363608\pi$
$458$	0	0
$459$	− 7.96831e13i	− 0.182556i
$460$	0	0
$461$	2.64036e14i	0.590619i	0.955402	+	0.295309i	$0.0954227\pi$
$461$	2.64036e14i	0.590619i	−0.955402	+	0.295309i	$0.904577\pi$
$462$	0	0
$463$	−3.47749e14	−0.759575	−0.379788	−	0.925074i	$-0.624003\pi$
$463$	−3.47749e14	−0.759575	−0.379788	+	0.925074i	$0.624003\pi$
$464$	0	0
$465$	9.52700e14	2.03219
$466$	0	0
$467$	5.68016e14i	1.18336i	0.806172	+	0.591681i	$0.201535\pi$
$467$	5.68016e14i	1.18336i	−0.806172	+	0.591681i	$0.798465\pi$
$468$	0	0
$469$	4.79729e14i	0.976214i
$470$	0	0
$471$	3.74185e14	0.743826
$472$	0	0
$473$	−1.32902e15	−2.58105
$474$	0	0
$475$	9.94429e14i	1.88694i
$476$	0	0
$477$	2.01440e14i	0.373503i
$478$	0	0
$479$	−1.05384e14	−0.190954	−0.0954770	−	0.995432i	$-0.530438\pi$
$479$	−1.05384e14	−0.190954	−0.0954770	+	0.995432i	$0.530438\pi$
$480$	0	0
$481$	−9.57010e13	−0.169480
$482$	0	0
$483$	2.23675e14i	0.387175i
$484$	0	0
$485$	1.29396e15i	2.18948i
$486$	0	0
$487$	−8.76991e14	−1.45073	−0.725364	−	0.688366i	$-0.758328\pi$
$487$	−8.76991e14	−1.45073	−0.725364	+	0.688366i	$0.758328\pi$
$488$	0	0
$489$	−5.23061e14	−0.845968
$490$	0	0
$491$	− 2.30296e14i	− 0.364198i	−0.983280	−	0.182099i	$-0.941711\pi$
$491$	− 2.30296e14i	− 0.364198i	0.983280	−	0.182099i	$-0.0582892\pi$
$492$	0	0
$493$	− 2.55288e14i	− 0.394796i
$494$	0	0
$495$	6.32088e14	0.955980
$496$	0	0
$497$	5.89201e14	0.871572
$498$	0	0
$499$	− 1.58906e14i	− 0.229925i	−0.993370	−	0.114962i	$-0.963325\pi$
$499$	− 1.58906e14i	− 0.229925i	0.993370	−	0.114962i	$-0.0366748\pi$
$500$	0	0
$501$	1.16802e15i	1.65326i
$502$	0	0
$503$	−1.08015e15	−1.49575	−0.747874	−	0.663841i	$-0.768925\pi$
$503$	−1.08015e15	−1.49575	−0.747874	+	0.663841i	$0.768925\pi$
$504$	0	0
$505$	1.90508e15	2.58113
$506$	0	0
$507$	2.49663e14i	0.330987i
$508$	0	0
$509$	5.50264e14i	0.713877i	0.934128	+	0.356939i	$0.116179\pi$
$509$	5.50264e14i	0.713877i	−0.934128	+	0.356939i	$0.883821\pi$
$510$	0	0
$511$	−2.58264e14	−0.327906
$512$	0	0
$513$	7.42581e14	0.922780
$514$	0	0
$515$	− 8.92126e14i	− 1.08514i
$516$	0	0
$517$	1.25619e14i	0.149575i
$518$	0	0
$519$	1.04964e15	1.22355
$520$	0	0
$521$	−5.41098e14	−0.617545	−0.308773	−	0.951136i	$-0.599918\pi$
$521$	−5.41098e14	−0.617545	−0.308773	+	0.951136i	$0.599918\pi$
$522$	0	0
$523$	− 3.17878e14i	− 0.355223i	−0.984101	−	0.177612i	$-0.943163\pi$
$523$	− 3.17878e14i	− 0.355223i	0.984101	−	0.177612i	$-0.0568371\pi$
$524$	0	0
$525$	− 1.12550e15i	− 1.23160i
$526$	0	0
$527$	2.69678e14	0.288993
$528$	0	0
$529$	−7.63511e14	−0.801325
$530$	0	0
$531$	− 2.21665e14i	− 0.227865i
$532$	0	0
$533$	1.35655e15i	1.36596i
$534$	0	0
$535$	1.51341e15	1.49283
$536$	0	0
$537$	9.31662e14	0.900326
$538$	0	0
$539$	− 7.27267e14i	− 0.688582i
$540$	0	0
$541$	− 1.93152e14i	− 0.179190i	−0.995978	−	0.0895952i	$-0.971443\pi$
$541$	− 1.93152e14i	− 0.179190i	0.995978	−	0.0895952i	$-0.0285573\pi$
$542$	0	0
$543$	−9.66745e14	−0.878847
$544$	0	0
$545$	1.62280e15	1.44572
$546$	0	0
$547$	1.98918e14i	0.173678i	0.996222	+	0.0868389i	$0.0276766\pi$
$547$	1.98918e14i	0.173678i	−0.996222	+	0.0868389i	$0.972323\pi$
$548$	0	0
$549$	3.86394e14i	0.330660i
$550$	0	0
$551$	2.37908e15	1.99561
$552$	0	0
$553$	1.73091e15	1.42326
$554$	0	0
$555$	− 4.58097e14i	− 0.369272i
$556$	0	0
$557$	− 6.03702e14i	− 0.477110i	−0.971129	−	0.238555i	$-0.923326\pi$
$557$	− 6.03702e14i	− 0.477110i	0.971129	−	0.238555i	$-0.0766738\pi$
$558$	0	0
$559$	−1.92266e15	−1.48983
$560$	0	0
$561$	6.07339e14	0.461462
$562$	0	0
$563$	4.54016e13i	0.0338279i	0.999857	+	0.0169139i	$0.00538413\pi$
$563$	4.54016e13i	0.0338279i	−0.999857	+	0.0169139i	$0.994616\pi$
$564$	0	0
$565$	4.60055e14i	0.336158i
$566$	0	0
$567$	−1.26563e15	−0.906985
$568$	0	0
$569$	7.08796e14	0.498200	0.249100	−	0.968478i	$-0.419865\pi$
$569$	7.08796e14	0.498200	0.249100	+	0.968478i	$0.419865\pi$
$570$	0	0
$571$	− 1.29941e14i	− 0.0895872i	−0.998996	−	0.0447936i	$-0.985737\pi$
$571$	− 1.29941e14i	− 0.0895872i	0.998996	−	0.0447936i	$-0.0142630\pi$
$572$	0	0
$573$	− 1.18107e15i	− 0.798779i
$574$	0	0
$575$	−9.52532e14	−0.631984
$576$	0	0
$577$	−1.90841e15	−1.24224	−0.621119	−	0.783717i	$-0.713321\pi$
$577$	−1.90841e15	−1.24224	−0.621119	+	0.783717i	$0.713321\pi$
$578$	0	0
$579$	− 1.39651e15i	− 0.891893i
$580$	0	0
$581$	− 1.42332e15i	− 0.891934i
$582$	0	0
$583$	2.14092e15	1.31651
$584$	0	0
$585$	9.14423e14	0.551810
$586$	0	0
$587$	1.63145e15i	0.966191i	0.875568	+	0.483096i	$0.160488\pi$
$587$	1.63145e15i	0.966191i	−0.875568	+	0.483096i	$0.839512\pi$
$588$	0	0
$589$	2.51318e15i	1.46080i
$590$	0	0
$591$	3.76602e15	2.14858
$592$	0	0
$593$	−7.10762e14	−0.398037	−0.199019	−	0.979996i	$-0.563775\pi$
$593$	−7.10762e14	−0.398037	−0.199019	+	0.979996i	$0.563775\pi$
$594$	0	0
$595$	− 5.43293e14i	− 0.298670i
$596$	0	0
$597$	3.53925e15i	1.91008i
$598$	0	0
$599$	2.39588e15	1.26946	0.634729	−	0.772735i	$-0.281112\pi$
$599$	2.39588e15	1.26946	0.634729	+	0.772735i	$0.281112\pi$
$600$	0	0
$601$	4.29375e13	0.0223371	0.0111686	−	0.999938i	$-0.496445\pi$
$601$	4.29375e13	0.0223371	0.0111686	+	0.999938i	$0.496445\pi$
$602$	0	0
$603$	− 1.09405e15i	− 0.558848i
$604$	0	0
$605$	− 3.61782e15i	− 1.81465i
$606$	0	0
$607$	1.01523e14	0.0500065	0.0250032	−	0.999687i	$-0.492040\pi$
$607$	1.01523e14	0.0500065	0.0250032	+	0.999687i	$0.492040\pi$
$608$	0	0
$609$	−2.69267e15	−1.30253
$610$	0	0
$611$	1.81730e14i	0.0863376i
$612$	0	0
$613$	2.25359e15i	1.05158i	0.850614	+	0.525791i	$0.176231\pi$
$613$	2.25359e15i	1.05158i	−0.850614	+	0.525791i	$0.823769\pi$
$614$	0	0
$615$	−6.49347e15	−2.97621
$616$	0	0
$617$	−3.34496e15	−1.50599	−0.752997	−	0.658024i	$-0.771392\pi$
$617$	−3.34496e15	−1.50599	−0.752997	+	0.658024i	$0.771392\pi$
$618$	0	0
$619$	1.68101e15i	0.743484i	0.928336	+	0.371742i	$0.121239\pi$
$619$	1.68101e15i	0.743484i	−0.928336	+	0.371742i	$0.878761\pi$
$620$	0	0
$621$	7.11295e14i	0.309062i
$622$	0	0
$623$	−1.76808e15	−0.754775
$624$	0	0
$625$	−9.71611e14	−0.407523
$626$	0	0
$627$	5.65990e15i	2.33259i
$628$	0	0
$629$	− 1.29672e14i	− 0.0525132i
$630$	0	0
$631$	−8.85236e14	−0.352288	−0.176144	−	0.984364i	$-0.556362\pi$
$631$	−8.85236e14	−0.352288	−0.176144	+	0.984364i	$0.556362\pi$
$632$	0	0
$633$	−3.50342e15	−1.37016
$634$	0	0
$635$	− 6.42415e14i	− 0.246922i
$636$	0	0
$637$	− 1.05212e15i	− 0.397463i
$638$	0	0
$639$	−1.34371e15	−0.498945
$640$	0	0
$641$	1.28306e15	0.468305	0.234153	−	0.972200i	$-0.424768\pi$
$641$	1.28306e15	0.468305	0.234153	+	0.972200i	$0.424768\pi$
$642$	0	0
$643$	− 1.55382e15i	− 0.557493i	−0.960365	−	0.278746i	$-0.910081\pi$
$643$	− 1.55382e15i	− 0.557493i	0.960365	−	0.278746i	$-0.0899189\pi$
$644$	0	0
$645$	− 9.20328e15i	− 3.24612i
$646$	0	0
$647$	−4.07796e15	−1.41407	−0.707033	−	0.707180i	$-0.749967\pi$
$647$	−4.07796e15	−1.41407	−0.707033	+	0.707180i	$0.749967\pi$
$648$	0	0
$649$	−2.35587e15	−0.803168
$650$	0	0
$651$	− 2.84444e15i	− 0.953459i
$652$	0	0
$653$	− 1.32534e14i	− 0.0436823i	−0.999761	−	0.0218411i	$-0.993047\pi$
$653$	− 1.32534e14i	− 0.0436823i	0.999761	−	0.0218411i	$-0.00695280\pi$
$654$	0	0
$655$	−3.21658e15	−1.04248
$656$	0	0
$657$	5.88990e14	0.187715
$658$	0	0
$659$	3.94510e15i	1.23648i	0.785988	+	0.618242i	$0.212155\pi$
$659$	3.94510e15i	1.23648i	−0.785988	+	0.618242i	$0.787845\pi$
$660$	0	0
$661$	− 7.20164e14i	− 0.221985i	−0.993821	−	0.110992i	$-0.964597\pi$
$661$	− 7.20164e14i	− 0.221985i	0.993821	−	0.110992i	$-0.0354029\pi$
$662$	0	0
$663$	8.78620e14	0.266365
$664$	0	0
$665$	5.06305e15	1.50971
$666$	0	0
$667$	2.27884e15i	0.668379i
$668$	0	0
$669$	− 4.93637e14i	− 0.142418i
$670$	0	0
$671$	4.10662e15	1.16550
$672$	0	0
$673$	1.24892e15	0.348699	0.174350	−	0.984684i	$-0.444218\pi$
$673$	1.24892e15	0.348699	0.174350	+	0.984684i	$0.444218\pi$
$674$	0	0
$675$	− 3.57915e15i	− 0.983127i
$676$	0	0
$677$	− 3.69331e15i	− 0.998108i	−0.866571	−	0.499054i	$-0.833681\pi$
$677$	− 3.69331e15i	− 0.998108i	0.866571	−	0.499054i	$-0.166319\pi$
$678$	0	0
$679$	3.86333e15	1.02725
$680$	0	0
$681$	−8.34877e15	−2.18430
$682$	0	0
$683$	7.43075e14i	0.191302i	0.995415	+	0.0956508i	$0.0304932\pi$
$683$	7.43075e14i	0.191302i	−0.995415	+	0.0956508i	$0.969507\pi$
$684$	0	0
$685$	− 4.86882e15i	− 1.23346i
$686$	0	0
$687$	7.56284e15	1.88548
$688$	0	0
$689$	3.09720e15	0.759914
$690$	0	0
$691$	− 2.45739e15i	− 0.593396i	−0.954971	−	0.296698i	$-0.904115\pi$
$691$	− 2.45739e15i	− 0.593396i	0.954971	−	0.296698i	$-0.0958855\pi$
$692$	0	0
$693$	− 1.88720e15i	− 0.448524i
$694$	0	0
$695$	−1.83019e14	−0.0428135
$696$	0	0
$697$	−1.83809e15	−0.423240
$698$	0	0
$699$	8.67411e15i	1.96608i
$700$	0	0
$701$	− 5.13293e14i	− 0.114529i	−0.998359	−	0.0572646i	$-0.981762\pi$
$701$	− 5.13293e14i	− 0.114529i	0.998359	−	0.0572646i	$-0.0182379\pi$
$702$	0	0
$703$	1.20844e15	0.265443
$704$	0	0
$705$	−8.69896e14	−0.188117
$706$	0	0
$707$	− 5.68793e15i	− 1.21101i
$708$	0	0
$709$	− 3.87859e15i	− 0.813054i	−0.913639	−	0.406527i	$-0.866740\pi$
$709$	− 3.87859e15i	− 0.813054i	0.913639	−	0.406527i	$-0.133260\pi$
$710$	0	0
$711$	−3.94746e15	−0.814769
$712$	0	0
$713$	−2.40730e15	−0.489257
$714$	0	0
$715$	− 9.71856e15i	− 1.94500i
$716$	0	0
$717$	7.38760e15i	1.45596i
$718$	0	0
$719$	−1.13007e15	−0.219329	−0.109664	−	0.993969i	$-0.534978\pi$
$719$	−1.13007e15	−0.219329	−0.109664	+	0.993969i	$0.534978\pi$
$720$	0	0
$721$	−2.66359e15	−0.509124
$722$	0	0
$723$	1.60486e15i	0.302118i
$724$	0	0
$725$	− 1.14669e16i	− 2.12611i
$726$	0	0
$727$	−4.83082e15	−0.882229	−0.441114	−	0.897451i	$-0.645417\pi$
$727$	−4.83082e15	−0.882229	−0.441114	+	0.897451i	$0.645417\pi$
$728$	0	0
$729$	−1.70307e15	−0.306360
$730$	0	0
$731$	− 2.60515e15i	− 0.461623i
$732$	0	0
$733$	1.10087e16i	1.92160i	0.277243	+	0.960800i	$0.410579\pi$
$733$	1.10087e16i	1.92160i	−0.277243	+	0.960800i	$0.589421\pi$
$734$	0	0
$735$	5.03621e15	0.866011
$736$	0	0
$737$	−1.16277e16	−1.96980
$738$	0	0
$739$	− 7.28702e15i	− 1.21620i	−0.793860	−	0.608101i	$-0.791932\pi$
$739$	− 7.28702e15i	− 1.21620i	0.793860	−	0.608101i	$-0.208068\pi$
$740$	0	0
$741$	8.18802e15i	1.34641i
$742$	0	0
$743$	8.82914e15	1.43047	0.715237	−	0.698882i	$-0.246319\pi$
$743$	8.82914e15	1.43047	0.715237	+	0.698882i	$0.246319\pi$
$744$	0	0
$745$	7.47518e15	1.19333
$746$	0	0
$747$	3.24597e15i	0.510601i
$748$	0	0
$749$	− 4.51853e15i	− 0.700402i
$750$	0	0
$751$	8.58508e14	0.131137	0.0655684	−	0.997848i	$-0.479114\pi$
$751$	8.58508e14	0.131137	0.0655684	+	0.997848i	$0.479114\pi$
$752$	0	0
$753$	−5.95649e15	−0.896640
$754$	0	0
$755$	8.49900e15i	1.26084i
$756$	0	0
$757$	2.15377e15i	0.314900i	0.987527	+	0.157450i	$0.0503273\pi$
$757$	2.15377e15i	0.314900i	−0.987527	+	0.157450i	$0.949673\pi$
$758$	0	0
$759$	−5.42144e15	−0.781241
$760$	0	0
$761$	9.15972e15	1.30097	0.650484	−	0.759520i	$-0.274566\pi$
$761$	9.15972e15	1.30097	0.650484	+	0.759520i	$0.274566\pi$
$762$	0	0
$763$	− 4.84514e15i	− 0.678300i
$764$	0	0
$765$	1.23902e15i	0.170978i
$766$	0	0
$767$	−3.40818e15	−0.463604
$768$	0	0
$769$	−3.21505e15	−0.431115	−0.215557	−	0.976491i	$-0.569157\pi$
$769$	−3.21505e15	−0.431115	−0.215557	+	0.976491i	$0.569157\pi$
$770$	0	0
$771$	7.81817e15i	1.03349i
$772$	0	0
$773$	− 2.64257e15i	− 0.344381i	−0.985064	−	0.172190i	$-0.944916\pi$
$773$	− 2.64257e15i	− 0.344381i	0.985064	−	0.172190i	$-0.0550844\pi$
$774$	0	0
$775$	1.21132e16	1.55633
$776$	0	0
$777$	−1.36772e15	−0.173254
$778$	0	0
$779$	− 1.71295e16i	− 2.13938i
$780$	0	0
$781$	1.42811e16i	1.75866i
$782$	0	0
$783$	−8.56279e15	−1.03974
$784$	0	0
$785$	8.11312e15	0.971414
$786$	0	0
$787$	− 3.48027e14i	− 0.0410915i	−0.999789	−	0.0205457i	$-0.993460\pi$
$787$	− 3.48027e14i	− 0.0410915i	0.999789	−	0.0205457i	$-0.00654037\pi$
$788$	0	0
$789$	− 8.39751e15i	− 0.977748i
$790$	0	0
$791$	1.37357e15	0.157718
$792$	0	0
$793$	5.94093e15	0.672747
$794$	0	0
$795$	1.48255e16i	1.65574i
$796$	0	0
$797$	1.68376e16i	1.85464i	0.374270	+	0.927320i	$0.377893\pi$
$797$	1.68376e16i	1.85464i	−0.374270	+	0.927320i	$0.622107\pi$
$798$	0	0
$799$	−2.46239e14	−0.0267516
$800$	0	0
$801$	4.03223e15	0.432082
$802$	0	0
$803$	− 6.25983e15i	− 0.661648i
$804$	0	0
$805$	4.84973e15i	0.505639i
$806$	0	0
$807$	−1.50022e16	−1.54294
$808$	0	0
$809$	−1.54229e15	−0.156477	−0.0782384	−	0.996935i	$-0.524930\pi$
$809$	−1.54229e15	−0.156477	−0.0782384	+	0.996935i	$0.524930\pi$
$810$	0	0
$811$	1.59301e16i	1.59442i	0.603699	+	0.797212i	$0.293693\pi$
$811$	1.59301e16i	1.59442i	−0.603699	+	0.797212i	$0.706307\pi$
$812$	0	0
$813$	− 5.95334e15i	− 0.587845i
$814$	0	0
$815$	−1.13411e16	−1.10481
$816$	0	0
$817$	2.42779e16	2.33340
$818$	0	0
$819$	− 2.73016e15i	− 0.258897i
$820$	0	0
$821$	− 1.46372e16i	− 1.36953i	−0.728765	−	0.684763i	$-0.759906\pi$
$821$	− 1.46372e16i	− 1.36953i	0.728765	−	0.684763i	$-0.240094\pi$
$822$	0	0
$823$	1.56959e16	1.44906	0.724531	−	0.689242i	$-0.242056\pi$
$823$	1.56959e16	1.44906	0.724531	+	0.689242i	$0.242056\pi$
$824$	0	0
$825$	2.72801e16	2.48513
$826$	0	0
$827$	6.76206e15i	0.607853i	0.952695	+	0.303927i	$0.0982977\pi$
$827$	6.76206e15i	0.607853i	−0.952695	+	0.303927i	$0.901702\pi$
$828$	0	0
$829$	1.35976e15i	0.120618i	0.998180	+	0.0603092i	$0.0192087\pi$
$829$	1.35976e15i	0.120618i	−0.998180	+	0.0603092i	$0.980791\pi$
$830$	0	0
$831$	−2.25572e16	−1.97460
$832$	0	0
$833$	1.42559e15	0.123153
$834$	0	0
$835$	2.53251e16i	2.15911i
$836$	0	0
$837$	− 9.04545e15i	− 0.761098i
$838$	0	0
$839$	2.14715e16	1.78309	0.891543	−	0.452936i	$-0.149623\pi$
$839$	2.14715e16	1.78309	0.891543	+	0.452936i	$0.149623\pi$
$840$	0	0
$841$	−1.52329e16	−1.24855
$842$	0	0
$843$	1.05568e16i	0.854045i
$844$	0	0
$845$	5.41322e15i	0.432259i
$846$	0	0
$847$	−1.08016e16	−0.851391
$848$	0	0
$849$	4.48204e15	0.348726
$850$	0	0
$851$	1.15753e15i	0.0889033i
$852$	0	0
$853$	9.40361e15i	0.712976i	0.934300	+	0.356488i	$0.116026\pi$
$853$	9.40361e15i	0.712976i	−0.934300	+	0.356488i	$0.883974\pi$
$854$	0	0
$855$	−1.15466e16	−0.864255
$856$	0	0
$857$	4.08283e13	0.00301694	0.00150847	−	0.999999i	$-0.499520\pi$
$857$	4.08283e13	0.00301694	0.00150847	+	0.999999i	$0.499520\pi$
$858$	0	0
$859$	− 1.92371e16i	− 1.40339i	−0.712479	−	0.701693i	$-0.752428\pi$
$859$	− 1.92371e16i	− 1.40339i	0.712479	−	0.701693i	$-0.247572\pi$
$860$	0	0
$861$	1.93873e16i	1.39637i
$862$	0	0
$863$	1.14013e16	0.810768	0.405384	−	0.914146i	$-0.367138\pi$
$863$	1.14013e16	0.810768	0.405384	+	0.914146i	$0.367138\pi$
$864$	0	0
$865$	2.27584e16	1.59791
$866$	0	0
$867$	− 1.59841e16i	− 1.10811i
$868$	0	0
$869$	4.19539e16i	2.87186i
$870$	0	0
$871$	−1.68215e16	−1.13701
$872$	0	0
$873$	−8.81060e15	−0.588067
$874$	0	0
$875$	− 7.19200e15i	− 0.474030i
$876$	0	0
$877$	1.08353e16i	0.705253i	0.935764	+	0.352627i	$0.114711\pi$
$877$	1.08353e16i	0.705253i	−0.935764	+	0.352627i	$0.885289\pi$
$878$	0	0
$879$	2.60814e16	1.67646
$880$	0	0
$881$	−1.77432e16	−1.12633	−0.563163	−	0.826346i	$-0.690416\pi$
$881$	−1.77432e16	−1.12633	−0.563163	+	0.826346i	$0.690416\pi$
$882$	0	0
$883$	− 1.35940e16i	− 0.852245i	−0.904665	−	0.426123i	$-0.859879\pi$
$883$	− 1.35940e16i	− 0.852245i	0.904665	−	0.426123i	$-0.140121\pi$
$884$	0	0
$885$	− 1.63141e16i	− 1.01012i
$886$	0	0
$887$	7.74640e15	0.473718	0.236859	−	0.971544i	$-0.423882\pi$
$887$	7.74640e15	0.473718	0.236859	+	0.971544i	$0.423882\pi$
$888$	0	0
$889$	−1.91804e15	−0.115850
$890$	0	0
$891$	− 3.06764e16i	− 1.83011i
$892$	0	0
$893$	− 2.29475e15i	− 0.135223i
$894$	0	0
$895$	2.02004e16	1.17580
$896$	0	0
$897$	−7.84304e15	−0.450948
$898$	0	0
$899$	− 2.89798e16i	− 1.64595i
$900$	0	0
$901$	4.19663e15i	0.235458i
$902$	0	0
$903$	−2.74779e16	−1.52301
$904$	0	0
$905$	−2.09611e16	−1.14775
$906$	0	0
$907$	2.39540e16i	1.29580i	0.761725	+	0.647901i	$0.224353\pi$
$907$	2.39540e16i	1.29580i	−0.761725	+	0.647901i	$0.775647\pi$
$908$	0	0
$909$	1.29717e16i	0.693261i
$910$	0	0
$911$	−2.28362e15	−0.120579	−0.0602897	−	0.998181i	$-0.519202\pi$
$911$	−2.28362e15	−0.120579	−0.0602897	+	0.998181i	$0.519202\pi$
$912$	0	0
$913$	3.44985e16	1.79974
$914$	0	0
$915$	2.84377e16i	1.46581i
$916$	0	0
$917$	9.60365e15i	0.489108i
$918$	0	0
$919$	2.07508e16	1.04424	0.522120	−	0.852872i	$-0.325141\pi$
$919$	2.07508e16	1.04424	0.522120	+	0.852872i	$0.325141\pi$
$920$	0	0
$921$	2.82263e16	1.40354
$922$	0	0
$923$	2.06601e16i	1.01513i
$924$	0	0
$925$	− 5.82454e15i	− 0.282802i
$926$	0	0
$927$	6.07451e15	0.291456
$928$	0	0
$929$	−1.65947e16	−0.786834	−0.393417	−	0.919360i	$-0.628707\pi$
$929$	−1.65947e16	−0.786834	−0.393417	+	0.919360i	$0.628707\pi$
$930$	0	0
$931$	1.32853e16i	0.622513i
$932$	0	0
$933$	1.65860e16i	0.768056i
$934$	0	0
$935$	1.31684e16	0.602655
$936$	0	0
$937$	−1.53690e16	−0.695148	−0.347574	−	0.937653i	$-0.612994\pi$
$937$	−1.53690e16	−0.695148	−0.347574	+	0.937653i	$0.612994\pi$
$938$	0	0
$939$	− 8.54437e14i	− 0.0381961i
$940$	0	0
$941$	− 3.47960e16i	− 1.53740i	−0.639610	−	0.768700i	$-0.720904\pi$
$941$	− 3.47960e16i	− 1.53740i	0.639610	−	0.768700i	$-0.279096\pi$
$942$	0	0
$943$	1.64078e16	0.716533
$944$	0	0
$945$	−1.82229e16	−0.786581
$946$	0	0
$947$	− 2.25273e16i	− 0.961137i	−0.876957	−	0.480568i	$-0.840430\pi$
$947$	− 2.25273e16i	− 0.961137i	0.876957	−	0.480568i	$-0.159570\pi$
$948$	0	0
$949$	− 9.05591e15i	− 0.381916i
$950$	0	0
$951$	4.45043e15	0.185528
$952$	0	0
$953$	2.65174e15	0.109275	0.0546375	−	0.998506i	$-0.482600\pi$
$953$	2.65174e15	0.109275	0.0546375	+	0.998506i	$0.482600\pi$
$954$	0	0
$955$	− 2.56081e16i	− 1.04318i
$956$	0	0
$957$	− 6.52650e16i	− 2.62824i
$958$	0	0
$959$	−1.45367e16	−0.578712
$960$	0	0
$961$	5.20483e15	0.204846
$962$	0	0
$963$	1.03048e16i	0.400955i
$964$	0	0
$965$	− 3.02793e16i	− 1.16478i
$966$	0	0
$967$	−8.03109e15	−0.305442	−0.152721	−	0.988269i	$-0.548804\pi$
$967$	−8.03109e15	−0.305442	−0.152721	+	0.988269i	$0.548804\pi$
$968$	0	0
$969$	−1.10945e16	−0.417185
$970$	0	0
$971$	− 1.33261e16i	− 0.495447i	−0.968831	−	0.247724i	$-0.920318\pi$
$971$	− 1.33261e16i	− 0.495447i	0.968831	−	0.247724i	$-0.0796825\pi$
$972$	0	0
$973$	5.46435e14i	0.0200871i
$974$	0	0
$975$	3.94653e16	1.43446
$976$	0	0
$977$	−1.81968e16	−0.653997	−0.326998	−	0.945025i	$-0.606037\pi$
$977$	−1.81968e16	−0.653997	−0.326998	+	0.945025i	$0.606037\pi$
$978$	0	0
$979$	− 4.28548e16i	− 1.52298i
$980$	0	0
$981$	1.10497e16i	0.388303i
$982$	0	0
$983$	−3.08304e16	−1.07136	−0.535679	−	0.844422i	$-0.679944\pi$
$983$	−3.08304e16	−1.07136	−0.535679	+	0.844422i	$0.679944\pi$
$984$	0	0
$985$	8.16552e16	2.80598
$986$	0	0
$987$	2.59722e15i	0.0882601i
$988$	0	0
$989$	2.32550e16i	0.781514i
$990$	0	0
$991$	−3.24999e16	−1.08013	−0.540066	−	0.841623i	$-0.681601\pi$
$991$	−3.24999e16	−1.08013	−0.540066	+	0.841623i	$0.681601\pi$
$992$	0	0
$993$	−6.40999e16	−2.10686
$994$	0	0
$995$	7.67384e16i	2.49451i
$996$	0	0
$997$	− 3.35074e15i	− 0.107725i	−0.998548	−	0.0538626i	$-0.982847\pi$
$997$	− 3.35074e15i	− 0.107725i	0.998548	−	0.0538626i	$-0.0171533\pi$
$998$	0	0
$999$	−4.34942e15	−0.138300

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	128.12.b.f.65.10	yes	12
4.3	odd	2	inner	128.12.b.f.65.4	yes	12
8.3	odd	2	inner	128.12.b.f.65.9	yes	12
8.5	even	2	inner	128.12.b.f.65.3	✓	12
16.3	odd	4		256.12.a.k.1.5		6
16.5	even	4		256.12.a.j.1.5		6
16.11	odd	4		256.12.a.j.1.2		6
16.13	even	4		256.12.a.k.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.12.b.f.65.3	✓	12	8.5	even	2	inner
128.12.b.f.65.4	yes	12	4.3	odd	2	inner
128.12.b.f.65.9	yes	12	8.3	odd	2	inner
128.12.b.f.65.10	yes	12	1.1	even	1	trivial
256.12.a.j.1.2		6	16.11	odd	4
256.12.a.j.1.5		6	16.5	even	4
256.12.a.k.1.2		6	16.13	even	4
256.12.a.k.1.5		6	16.3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 128 = 2^{7} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	128.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+501.129i q^{3} +10865.5i q^{5} +32440.9 q^{7} -73983.7 q^{9} +786304. i q^{11} +1.13752e6i q^{13} -5.44504e6 q^{15} -1.54131e6 q^{17} -1.43638e7i q^{19} +1.62571e7i q^{21} +1.37586e7 q^{23} -6.92317e7 q^{25} +5.16982e7i q^{27} +1.65630e8i q^{29} -1.74967e8 q^{31} -3.94040e8 q^{33} +3.52487e8i q^{35} +8.41310e7i q^{37} -5.70046e8 q^{39} +1.19255e9 q^{41} +1.69021e9i q^{43} -8.03872e8i q^{45} +1.59759e8 q^{47} -9.24918e8 q^{49} -7.72397e8i q^{51} -2.72276e9i q^{53} -8.54361e9 q^{55} +7.19811e9 q^{57} +2.99614e9i q^{59} -5.22269e9i q^{61} -2.40009e9 q^{63} -1.23598e10 q^{65} +1.47878e10i q^{67} +6.89484e9i q^{69} +1.81623e10 q^{71} -7.96108e9 q^{73} -3.46940e10i q^{75} +2.55084e10i q^{77} +5.33558e10 q^{79} -3.90135e10 q^{81} -4.38742e10i q^{83} -1.67472e10i q^{85} -8.30023e10 q^{87} -5.45016e10 q^{89} +3.69022e10i q^{91} -8.76809e10i q^{93} +1.56070e11 q^{95} +1.19088e11 q^{97} -5.81737e10i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 316732 q^{9} + 11301736 q^{17} - 100402276 q^{25} - 370222336 q^{33} + 1458786872 q^{41} - 2425030420 q^{49} - 8422115584 q^{57} - 18057782080 q^{65} - 41779508088 q^{73} - 213089686484 q^{81} - 331639752632 q^{89}+ \cdots - 396775590616 q^{97}+O(q^{100}) \) 12 * q - 316732 * q^9 + 11301736 * q^17 - 100402276 * q^25 - 370222336 * q^33 + 1458786872 * q^41 - 2425030420 * q^49 - 8422115584 * q^57 - 18057782080 * q^65 - 41779508088 * q^73 - 213089686484 * q^81 - 331639752632 * q^89 - 396775590616 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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