LMFDB - Embedded newform 576.7.g.g.127.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [576,7,Mod(127,576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(576, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0]))

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("576.127");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 576 = 2^{6} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	576.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:	$132.511152165$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 48)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.2
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	576.127
Dual form			576.7.g.g.127.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+6.00000 q^{5} +200.918i q^{7} +O(q^{10})$	$q+6.00000 q^{5} +200.918i q^{7} +644.323i q^{11} +2654.00 q^{13} +7206.00 q^{17} -3540.31i q^{19} -17459.1i q^{23} -15589.0 q^{25} +11550.0 q^{29} -8542.47i q^{31} +1205.51i q^{35} -22346.0 q^{37} -103626. q^{41} +127389. i q^{43} -160831. i q^{47} +77281.0 q^{49} +168462. q^{53} +3865.94i q^{55} +111593. i q^{59} +260470. q^{61} +15924.0 q^{65} +317083. i q^{67} -708423. i q^{71} -395918. q^{73} -129456. q^{77} +556743. i q^{79} +89145.2i q^{83} +43236.0 q^{85} +251886. q^{89} +533236. i q^{91} -21241.9i q^{95} +517474. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 12 q^{5}+O(q^{10}) $ 2 * q + 12 * q^5	$ 2 q + 12 q^{5} + 5308 q^{13} + 14412 q^{17} - 31178 q^{25} + 23100 q^{29} - 44692 q^{37} - 207252 q^{41} + 154562 q^{49} + 336924 q^{53} + 520940 q^{61} + 31848 q^{65} - 791836 q^{73} - 258912 q^{77} + 86472 q^{85} + 503772 q^{89} + 1034948 q^{97}+O(q^{100}) $ 2 * q + 12 * q^5 + 5308 * q^13 + 14412 * q^17 - 31178 * q^25 + 23100 * q^29 - 44692 * q^37 - 207252 * q^41 + 154562 * q^49 + 336924 * q^53 + 520940 * q^61 + 31848 * q^65 - 791836 * q^73 - 258912 * q^77 + 86472 * q^85 + 503772 * q^89 + 1034948 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/576\mathbb{Z}\right)^\times$.

$n$	$65$	$127$	$325$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	6.00000	0.0480000	0.0240000	−	0.999712i	$-0.492360\pi$
$5$	6.00000	0.0480000	0.0240000	+	0.999712i	$0.492360\pi$
$6$	0	0
$7$	200.918i	0.585766i	0.956148	+	0.292883i	$0.0946147\pi$
$7$	200.918i	0.585766i	−0.956148	+	0.292883i	$0.905385\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	644.323i	0.484089i	0.970265	+	0.242045i	$0.0778181\pi$
$11$	644.323i	0.484089i	−0.970265	+	0.242045i	$0.922182\pi$
$12$	0	0
$13$	2654.00	1.20801	0.604005	−	0.796980i	$-0.293571\pi$
$13$	2654.00	1.20801	0.604005	+	0.796980i	$0.293571\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	7206.00	1.46672	0.733360	−	0.679840i	$-0.237951\pi$
$17$	7206.00	1.46672	0.733360	+	0.679840i	$0.237951\pi$
$18$	0	0
$19$	− 3540.31i	− 0.516156i	−0.966124	−	0.258078i	$-0.916911\pi$
$19$	− 3540.31i	− 0.516156i	0.966124	−	0.258078i	$-0.0830891\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 17459.1i	− 1.43495i	−0.696583	−	0.717476i	$-0.745297\pi$
$23$	− 17459.1i	− 1.43495i	0.696583	−	0.717476i	$-0.254703\pi$
$24$	0	0
$25$	−15589.0	−0.997696
$26$	0	0
$27$	0	0
$28$	0	0
$29$	11550.0	0.473574	0.236787	−	0.971562i	$-0.423906\pi$
$29$	11550.0	0.473574	0.236787	+	0.971562i	$0.423906\pi$
$30$	0	0
$31$	− 8542.47i	− 0.286747i	−0.989669	−	0.143373i	$-0.954205\pi$
$31$	− 8542.47i	− 0.286747i	0.989669	−	0.143373i	$-0.0457950\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1205.51i	0.0281168i
$36$	0	0
$37$	−22346.0	−0.441158	−0.220579	−	0.975369i	$-0.570795\pi$
$37$	−22346.0	−0.441158	−0.220579	+	0.975369i	$0.570795\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−103626.	−1.50355	−0.751774	−	0.659421i	$-0.770801\pi$
$41$	−103626.	−1.50355	−0.751774	+	0.659421i	$0.770801\pi$
$42$	0	0
$43$	127389.i	1.60223i	0.598507	+	0.801117i	$0.295761\pi$
$43$	127389.i	1.60223i	−0.598507	+	0.801117i	$0.704239\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 160831.i	− 1.54909i	−0.632518	−	0.774546i	$-0.717979\pi$
$47$	− 160831.i	− 1.54909i	0.632518	−	0.774546i	$-0.282021\pi$
$48$	0	0
$49$	77281.0	0.656878
$50$	0	0
$51$	0	0
$52$	0	0
$53$	168462.	1.13155	0.565776	−	0.824559i	$-0.308577\pi$
$53$	168462.	1.13155	0.565776	+	0.824559i	$0.308577\pi$
$54$	0	0
$55$	3865.94i	0.0232363i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	111593.i	0.543349i	0.962389	+	0.271675i	$0.0875775\pi$
$59$	111593.i	0.543349i	−0.962389	+	0.271675i	$0.912423\pi$
$60$	0	0
$61$	260470.	1.14754	0.573770	−	0.819016i	$-0.305480\pi$
$61$	260470.	1.14754	0.573770	+	0.819016i	$0.305480\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	15924.0	0.0579845
$66$	0	0
$67$	317083.i	1.05426i	0.849784	+	0.527131i	$0.176732\pi$
$67$	317083.i	1.05426i	−0.849784	+	0.527131i	$0.823268\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 708423.i	− 1.97933i	−0.143412	−	0.989663i	$-0.545807\pi$
$71$	− 708423.i	− 1.97933i	0.143412	−	0.989663i	$-0.454193\pi$
$72$	0	0
$73$	−395918.	−1.01774	−0.508870	−	0.860844i	$-0.669937\pi$
$73$	−395918.	−1.01774	−0.508870	+	0.860844i	$0.669937\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−129456.	−0.283563
$78$	0	0
$79$	556743.i	1.12921i	0.825362	+	0.564604i	$0.190971\pi$
$79$	556743.i	1.12921i	−0.825362	+	0.564604i	$0.809029\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	89145.2i	0.155906i	0.996957	+	0.0779531i	$0.0248385\pi$
$83$	89145.2i	0.155906i	−0.996957	+	0.0779531i	$0.975162\pi$
$84$	0	0
$85$	43236.0	0.0704026
$86$	0	0
$87$	0	0
$88$	0	0
$89$	251886.	0.357301	0.178650	−	0.983913i	$-0.442827\pi$
$89$	251886.	0.357301	0.178650	+	0.983913i	$0.442827\pi$
$90$	0	0
$91$	533236.i	0.707612i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 21241.9i	− 0.0247755i
$96$	0	0
$97$	517474.	0.566987	0.283494	−	0.958974i	$-0.408507\pi$
$97$	517474.	0.566987	0.283494	+	0.958974i	$0.408507\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1.02382e6	0.993712	0.496856	−	0.867833i	$-0.334488\pi$
$101$	1.02382e6	0.993712	0.496856	+	0.867833i	$0.334488\pi$
$102$	0	0
$103$	− 90960.4i	− 0.0832416i	−0.999133	−	0.0416208i	$-0.986748\pi$
$103$	− 90960.4i	− 0.0832416i	0.999133	−	0.0416208i	$-0.0132521\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 2.20282e6i	− 1.79815i	−0.437791	−	0.899077i	$-0.644239\pi$
$107$	− 2.20282e6i	− 1.79815i	0.437791	−	0.899077i	$-0.355761\pi$
$108$	0	0
$109$	−952594.	−0.735577	−0.367789	−	0.929909i	$-0.619885\pi$
$109$	−952594.	−0.735577	−0.367789	+	0.929909i	$0.619885\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	121038.	0.0838854	0.0419427	−	0.999120i	$-0.486645\pi$
$113$	121038.	0.0838854	0.0419427	+	0.999120i	$0.486645\pi$
$114$	0	0
$115$	− 104754.i	− 0.0688777i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.44781e6i	0.859156i
$120$	0	0
$121$	1.35641e6	0.765658
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−187284.	−0.0958894
$126$	0	0
$127$	1.44941e6i	0.707590i	0.935323	+	0.353795i	$0.115109\pi$
$127$	1.44941e6i	0.707590i	−0.935323	+	0.353795i	$0.884891\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 632787.i	− 0.281478i	−0.990047	−	0.140739i	$-0.955052\pi$
$131$	− 632787.i	− 0.281478i	0.990047	−	0.140739i	$-0.0449478\pi$
$132$	0	0
$133$	711312.	0.302347
$134$	0	0
$135$	0	0
$136$	0	0
$137$	4.47821e6	1.74158	0.870789	−	0.491656i	$-0.163608\pi$
$137$	4.47821e6	1.74158	0.870789	+	0.491656i	$0.163608\pi$
$138$	0	0
$139$	− 2.24905e6i	− 0.837443i	−0.908115	−	0.418722i	$-0.862478\pi$
$139$	− 2.24905e6i	− 0.837443i	0.908115	−	0.418722i	$-0.137522\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1.71003e6i	0.584785i
$144$	0	0
$145$	69300.0	0.0227316
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1.84051e6	−0.556389	−0.278194	−	0.960525i	$-0.589736\pi$
$149$	−1.84051e6	−0.556389	−0.278194	+	0.960525i	$0.589736\pi$
$150$	0	0
$151$	− 1.14464e6i	− 0.332460i	−0.986087	−	0.166230i	$-0.946841\pi$
$151$	− 1.14464e6i	− 0.332460i	0.986087	−	0.166230i	$-0.0531594\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 51254.8i	− 0.0137638i
$156$	0	0
$157$	5.28775e6	1.36638	0.683191	−	0.730240i	$-0.260592\pi$
$157$	5.28775e6	1.36638	0.683191	+	0.730240i	$0.260592\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	3.50784e6	0.840547
$162$	0	0
$163$	− 6.45869e6i	− 1.49136i	−0.666306	−	0.745678i	$-0.732126\pi$
$163$	− 6.45869e6i	− 1.49136i	0.666306	−	0.745678i	$-0.267874\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3.97714e6i	0.853927i	0.904269	+	0.426964i	$0.140417\pi$
$167$	3.97714e6i	0.853927i	−0.904269	+	0.426964i	$0.859583\pi$
$168$	0	0
$169$	2.21691e6	0.459290
$170$	0	0
$171$	0	0
$172$	0	0
$173$	5.73548e6	1.10772	0.553862	−	0.832609i	$-0.313154\pi$
$173$	5.73548e6	1.10772	0.553862	+	0.832609i	$0.313154\pi$
$174$	0	0
$175$	− 3.13211e6i	− 0.584417i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	8.16883e6i	1.42430i	0.702029	+	0.712149i	$0.252278\pi$
$179$	8.16883e6i	1.42430i	−0.702029	+	0.712149i	$0.747722\pi$
$180$	0	0
$181$	−3.24656e6	−0.547505	−0.273752	−	0.961800i	$-0.588265\pi$
$181$	−3.24656e6	−0.547505	−0.273752	+	0.961800i	$0.588265\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−134076.	−0.0211756
$186$	0	0
$187$	4.64299e6i	0.710024i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1.11705e6i	0.160314i	0.996782	+	0.0801571i	$0.0255422\pi$
$191$	1.11705e6i	0.160314i	−0.996782	+	0.0801571i	$0.974458\pi$
$192$	0	0
$193$	3.12375e6	0.434514	0.217257	−	0.976114i	$-0.430289\pi$
$193$	3.12375e6	0.434514	0.217257	+	0.976114i	$0.430289\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1.08530e7	1.41955	0.709777	−	0.704426i	$-0.248796\pi$
$197$	1.08530e7	1.41955	0.709777	+	0.704426i	$0.248796\pi$
$198$	0	0
$199$	− 8.99733e6i	− 1.14171i	−0.821052	−	0.570853i	$-0.806612\pi$
$199$	− 8.99733e6i	− 1.14171i	0.821052	−	0.570853i	$-0.193388\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2.32060e6i	0.277404i
$204$	0	0
$205$	−621756.	−0.0721703
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2.28110e6	0.249865
$210$	0	0
$211$	3.84129e6i	0.408912i	0.978876	+	0.204456i	$0.0655426\pi$
$211$	3.84129e6i	0.408912i	−0.978876	+	0.204456i	$0.934457\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	764333.i	0.0769073i
$216$	0	0
$217$	1.71634e6	0.167967
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1.91247e7	1.77181
$222$	0	0
$223$	3.51732e6i	0.317174i	0.987345	+	0.158587i	$0.0506938\pi$
$223$	3.51732e6i	0.317174i	−0.987345	+	0.158587i	$0.949306\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.62115e7i	1.38595i	0.720964	+	0.692973i	$0.243699\pi$
$227$	1.62115e7i	1.38595i	−0.720964	+	0.692973i	$0.756301\pi$
$228$	0	0
$229$	1.46861e7	1.22293	0.611463	−	0.791273i	$-0.290582\pi$
$229$	1.46861e7	1.22293	0.611463	+	0.791273i	$0.290582\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6.10117e6	0.482331	0.241165	−	0.970484i	$-0.422470\pi$
$233$	6.10117e6	0.482331	0.241165	+	0.970484i	$0.422470\pi$
$234$	0	0
$235$	− 964988.i	− 0.0743564i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 2.63536e7i	− 1.93039i	−0.261528	−	0.965196i	$-0.584226\pi$
$239$	− 2.63536e7i	− 1.93039i	0.261528	−	0.965196i	$-0.415774\pi$
$240$	0	0
$241$	−2.15214e6	−0.153752	−0.0768758	−	0.997041i	$-0.524495\pi$
$241$	−2.15214e6	−0.153752	−0.0768758	+	0.997041i	$0.524495\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	463686.	0.0315301
$246$	0	0
$247$	− 9.39599e6i	− 0.623522i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 7.99802e6i	− 0.505780i	−0.967495	−	0.252890i	$-0.918619\pi$
$251$	− 7.99802e6i	− 0.505780i	0.967495	−	0.252890i	$-0.0813810\pi$
$252$	0	0
$253$	1.12493e7	0.694645
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2.86154e7	1.68578	0.842890	−	0.538086i	$-0.180853\pi$
$257$	2.86154e7	1.68578	0.842890	+	0.538086i	$0.180853\pi$
$258$	0	0
$259$	− 4.48971e6i	− 0.258416i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 1.49487e6i	− 0.0821744i	−0.999156	−	0.0410872i	$-0.986918\pi$
$263$	− 1.49487e6i	− 0.0821744i	0.999156	−	0.0410872i	$-0.0130821\pi$
$264$	0	0
$265$	1.01077e6	0.0543145
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−532290.	−0.0273459	−0.0136729	−	0.999907i	$-0.504352\pi$
$269$	−532290.	−0.0273459	−0.0136729	+	0.999907i	$0.504352\pi$
$270$	0	0
$271$	2.88563e7i	1.44988i	0.688811	+	0.724941i	$0.258133\pi$
$271$	2.88563e7i	1.44988i	−0.688811	+	0.724941i	$0.741867\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 1.00443e7i	− 0.482974i
$276$	0	0
$277$	1.82479e7	0.858567	0.429284	−	0.903170i	$-0.358766\pi$
$277$	1.82479e7	0.858567	0.429284	+	0.903170i	$0.358766\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−9.52019e6	−0.429069	−0.214534	−	0.976716i	$-0.568823\pi$
$281$	−9.52019e6	−0.429069	−0.214534	+	0.976716i	$0.568823\pi$
$282$	0	0
$283$	3.81369e7i	1.68262i	0.540552	+	0.841310i	$0.318215\pi$
$283$	3.81369e7i	1.68262i	−0.540552	+	0.841310i	$0.681785\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 2.08203e7i	− 0.880728i
$288$	0	0
$289$	2.77889e7	1.15127
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1.68345e7	0.669266	0.334633	−	0.942349i	$-0.391388\pi$
$293$	1.68345e7	0.669266	0.334633	+	0.942349i	$0.391388\pi$
$294$	0	0
$295$	669555.i	0.0260808i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 4.63364e7i	− 1.73344i
$300$	0	0
$301$	−2.55947e7	−0.938535
$302$	0	0
$303$	0	0
$304$	0	0
$305$	1.56282e6	0.0550820
$306$	0	0
$307$	− 4.39154e7i	− 1.51775i	−0.651234	−	0.758877i	$-0.725748\pi$
$307$	− 4.39154e7i	− 1.51775i	0.651234	−	0.758877i	$-0.274252\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	5.08371e7i	1.69005i	0.534726	+	0.845025i	$0.320415\pi$
$311$	5.08371e7i	1.69005i	−0.534726	+	0.845025i	$0.679585\pi$
$312$	0	0
$313$	1.30721e7	0.426298	0.213149	−	0.977020i	$-0.431628\pi$
$313$	1.30721e7	0.426298	0.213149	+	0.977020i	$0.431628\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	5.07723e6	0.159386	0.0796928	−	0.996819i	$-0.474606\pi$
$317$	5.07723e6	0.159386	0.0796928	+	0.996819i	$0.474606\pi$
$318$	0	0
$319$	7.44193e6i	0.229252i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 2.55115e7i	− 0.757056i
$324$	0	0
$325$	−4.13732e7	−1.20523
$326$	0	0
$327$	0	0
$328$	0	0
$329$	3.23139e7	0.907406
$330$	0	0
$331$	5.59418e7i	1.54260i	0.636474	+	0.771298i	$0.280392\pi$
$331$	5.59418e7i	1.54260i	−0.636474	+	0.771298i	$0.719608\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	1.90250e6i	0.0506046i
$336$	0	0
$337$	−2.55737e7	−0.668197	−0.334099	−	0.942538i	$-0.608432\pi$
$337$	−2.55737e7	−0.668197	−0.334099	+	0.942538i	$0.608432\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	5.50411e6	0.138811
$342$	0	0
$343$	3.91649e7i	0.970543i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 4.96031e6i	− 0.118719i	−0.998237	−	0.0593595i	$-0.981094\pi$
$347$	− 4.96031e6i	− 0.118719i	0.998237	−	0.0593595i	$-0.0189058\pi$
$348$	0	0
$349$	3.06563e7	0.721179	0.360590	−	0.932725i	$-0.382576\pi$
$349$	3.06563e7	0.721179	0.360590	+	0.932725i	$0.382576\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	5.33353e7	1.21252	0.606262	−	0.795265i	$-0.292668\pi$
$353$	5.33353e7	1.21252	0.606262	+	0.795265i	$0.292668\pi$
$354$	0	0
$355$	− 4.25054e6i	− 0.0950077i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	2.83885e7i	0.613564i	0.951780	+	0.306782i	$0.0992522\pi$
$359$	2.83885e7i	0.613564i	−0.951780	+	0.306782i	$0.900748\pi$
$360$	0	0
$361$	3.45121e7	0.733583
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−2.37551e6	−0.0488515
$366$	0	0
$367$	5.75671e7i	1.16460i	0.812974	+	0.582299i	$0.197847\pi$
$367$	5.75671e7i	1.16460i	−0.812974	+	0.582299i	$0.802153\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	3.38470e7i	0.662825i
$372$	0	0
$373$	4.68328e7	0.902451	0.451226	−	0.892410i	$-0.350987\pi$
$373$	4.68328e7	0.902451	0.451226	+	0.892410i	$0.350987\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3.06537e7	0.572083
$378$	0	0
$379$	− 7.92476e7i	− 1.45569i	−0.685742	−	0.727845i	$-0.740522\pi$
$379$	− 7.92476e7i	− 1.45569i	0.685742	−	0.727845i	$-0.259478\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 2.00575e7i	− 0.357010i	−0.983939	−	0.178505i	$-0.942874\pi$
$383$	− 2.00575e7i	− 0.357010i	0.983939	−	0.178505i	$-0.0571260\pi$
$384$	0	0
$385$	−776736.	−0.0136110
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−4.79345e7	−0.814328	−0.407164	−	0.913355i	$-0.633482\pi$
$389$	−4.79345e7	−0.814328	−0.407164	+	0.913355i	$0.633482\pi$
$390$	0	0
$391$	− 1.25810e8i	− 2.10468i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	3.34046e6i	0.0542020i
$396$	0	0
$397$	3.43785e7	0.549434	0.274717	−	0.961525i	$-0.411416\pi$
$397$	3.43785e7	0.549434	0.274717	+	0.961525i	$0.411416\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−9.09234e7	−1.41008	−0.705038	−	0.709170i	$-0.749070\pi$
$401$	−9.09234e7	−1.41008	−0.705038	+	0.709170i	$0.749070\pi$
$402$	0	0
$403$	− 2.26717e7i	− 0.346393i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 1.43980e7i	− 0.213560i
$408$	0	0
$409$	−1.15454e8	−1.68748	−0.843739	−	0.536753i	$-0.819651\pi$
$409$	−1.15454e8	−1.68748	−0.843739	+	0.536753i	$0.819651\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−2.24209e7	−0.318276
$414$	0	0
$415$	534871.i	0.00748350i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	7.40305e7i	1.00640i	0.864171	+	0.503198i	$0.167843\pi$
$419$	7.40305e7i	1.00640i	−0.864171	+	0.503198i	$0.832157\pi$
$420$	0	0
$421$	−1.14811e8	−1.53863	−0.769317	−	0.638867i	$-0.779404\pi$
$421$	−1.14811e8	−1.53863	−0.769317	+	0.638867i	$0.779404\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−1.12334e8	−1.46334
$426$	0	0
$427$	5.23331e7i	0.672191i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 3.42559e6i	− 0.0427862i	−0.999771	−	0.0213931i	$-0.993190\pi$
$431$	− 3.42559e6i	− 0.0427862i	0.999771	−	0.0213931i	$-0.00681016\pi$
$432$	0	0
$433$	1.08786e8	1.34002	0.670008	−	0.742354i	$-0.266291\pi$
$433$	1.08786e8	1.34002	0.670008	+	0.742354i	$0.266291\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−6.18106e7	−0.740659
$438$	0	0
$439$	− 1.13400e8i	− 1.34036i	−0.742199	−	0.670180i	$-0.766217\pi$
$439$	− 1.13400e8i	− 1.34036i	0.742199	−	0.670180i	$-0.233783\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	2.44463e7i	0.281191i	0.990067	+	0.140595i	$0.0449017\pi$
$443$	2.44463e7i	0.281191i	−0.990067	+	0.140595i	$0.955098\pi$
$444$	0	0
$445$	1.51132e6	0.0171504
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−9.03699e7	−0.998354	−0.499177	−	0.866500i	$-0.666364\pi$
$449$	−9.03699e7	−0.998354	−0.499177	+	0.866500i	$0.666364\pi$
$450$	0	0
$451$	− 6.67686e7i	− 0.727851i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	3.19942e6i	0.0339654i
$456$	0	0
$457$	−5.82048e7	−0.609832	−0.304916	−	0.952379i	$-0.598628\pi$
$457$	−5.82048e7	−0.609832	−0.304916	+	0.952379i	$0.598628\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	1.73271e8	1.76858	0.884288	−	0.466942i	$-0.154644\pi$
$461$	1.73271e8	1.76858	0.884288	+	0.466942i	$0.154644\pi$
$462$	0	0
$463$	− 1.25400e8i	− 1.26344i	−0.775197	−	0.631719i	$-0.782350\pi$
$463$	− 1.25400e8i	− 1.26344i	0.775197	−	0.631719i	$-0.217650\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1.45905e8i	1.43258i	0.697800	+	0.716292i	$0.254162\pi$
$467$	1.45905e8i	1.43258i	−0.697800	+	0.716292i	$0.745838\pi$
$468$	0	0
$469$	−6.37077e7	−0.617551
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−8.20796e7	−0.775625
$474$	0	0
$475$	5.51899e7i	0.514966i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 7.62417e6i	− 0.0693723i	−0.999398	−	0.0346861i	$-0.988957\pi$
$479$	− 7.62417e6i	− 0.0693723i	0.999398	−	0.0346861i	$-0.0110432\pi$
$480$	0	0
$481$	−5.93063e7	−0.532924
$482$	0	0
$483$	0	0
$484$	0	0
$485$	3.10484e6	0.0272154
$486$	0	0
$487$	1.71349e8i	1.48352i	0.670665	+	0.741760i	$0.266009\pi$
$487$	1.71349e8i	1.48352i	−0.670665	+	0.741760i	$0.733991\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 3.30473e6i	− 0.0279185i	−0.999903	−	0.0139592i	$-0.995556\pi$
$491$	− 3.30473e6i	− 0.0279185i	0.999903	−	0.0139592i	$-0.00444351\pi$
$492$	0	0
$493$	8.32293e7	0.694601
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.42335e8	1.15942
$498$	0	0
$499$	1.26207e8i	1.01574i	0.861435	+	0.507868i	$0.169566\pi$
$499$	1.26207e8i	1.01574i	−0.861435	+	0.507868i	$0.830434\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	212336.i	0.00166847i	1.00000		0.000834236i	$0.000265546\pi$
$503$	212336.i	0.00166847i	−1.00000		0.000834236i	$0.999734\pi$
$504$	0	0
$505$	6.14293e6	0.0476982
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−1.49792e8	−1.13589	−0.567944	−	0.823067i	$-0.692261\pi$
$509$	−1.49792e8	−1.13589	−0.567944	+	0.823067i	$0.692261\pi$
$510$	0	0
$511$	− 7.95470e7i	− 0.596158i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 545762.i	− 0.00399560i
$516$	0	0
$517$	1.03627e8	0.749899
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−1.11422e8	−0.787876	−0.393938	−	0.919137i	$-0.628887\pi$
$521$	−1.11422e8	−0.787876	−0.393938	+	0.919137i	$0.628887\pi$
$522$	0	0
$523$	− 2.69486e8i	− 1.88379i	−0.335911	−	0.941894i	$-0.609044\pi$
$523$	− 2.69486e8i	− 1.88379i	0.335911	−	0.941894i	$-0.390956\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 6.15571e7i	− 0.420578i
$528$	0	0
$529$	−1.56783e8	−1.05909
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−2.75023e8	−1.81630
$534$	0	0
$535$	− 1.32169e7i	− 0.0863114i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	4.97939e7i	0.317987i
$540$	0	0
$541$	−3.03425e8	−1.91628	−0.958142	−	0.286292i	$-0.907577\pi$
$541$	−3.03425e8	−1.91628	−0.958142	+	0.286292i	$0.907577\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−5.71556e6	−0.0353077
$546$	0	0
$547$	1.92937e7i	0.117884i	0.998261	+	0.0589418i	$0.0187726\pi$
$547$	1.92937e7i	0.117884i	−0.998261	+	0.0589418i	$0.981227\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 4.08906e7i	− 0.244438i
$552$	0	0
$553$	−1.11860e8	−0.661452
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−1.71693e8	−0.993544	−0.496772	−	0.867881i	$-0.665482\pi$
$557$	−1.71693e8	−0.993544	−0.496772	+	0.867881i	$0.665482\pi$
$558$	0	0
$559$	3.38090e8i	1.93552i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 2.21392e8i	− 1.24062i	−0.784358	−	0.620308i	$-0.787008\pi$
$563$	− 2.21392e8i	− 1.24062i	0.784358	−	0.620308i	$-0.212992\pi$
$564$	0	0
$565$	726228.	0.00402650
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−2.56991e8	−1.39502	−0.697511	−	0.716574i	$-0.745709\pi$
$569$	−2.56991e8	−1.39502	−0.697511	+	0.716574i	$0.745709\pi$
$570$	0	0
$571$	2.71249e8i	1.45700i	0.685045	+	0.728501i	$0.259782\pi$
$571$	2.71249e8i	1.45700i	−0.685045	+	0.728501i	$0.740218\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	2.72169e8i	1.43165i
$576$	0	0
$577$	−3.41573e6	−0.0177810	−0.00889049	−	0.999960i	$-0.502830\pi$
$577$	−3.41573e6	−0.0177810	−0.00889049	+	0.999960i	$0.502830\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−1.79109e7	−0.0913247
$582$	0	0
$583$	1.08544e8i	0.547772i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 2.32848e8i	− 1.15122i	−0.817725	−	0.575609i	$-0.804765\pi$
$587$	− 2.32848e8i	− 1.15122i	0.817725	−	0.575609i	$-0.195235\pi$
$588$	0	0
$589$	−3.02430e7	−0.148006
$590$	0	0
$591$	0	0
$592$	0	0
$593$	8.86693e7	0.425216	0.212608	−	0.977138i	$-0.431804\pi$
$593$	8.86693e7	0.425216	0.212608	+	0.977138i	$0.431804\pi$
$594$	0	0
$595$	8.68689e6i	0.0412395i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 2.37591e7i	− 0.110547i	−0.998471	−	0.0552737i	$-0.982397\pi$
$599$	− 2.37591e7i	− 0.110547i	0.998471	−	0.0552737i	$-0.0176031\pi$
$600$	0	0
$601$	2.56421e8	1.18122	0.590609	−	0.806958i	$-0.298888\pi$
$601$	2.56421e8	1.18122	0.590609	+	0.806958i	$0.298888\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	8.13845e6	0.0367516
$606$	0	0
$607$	− 2.08193e8i	− 0.930895i	−0.885076	−	0.465447i	$-0.845893\pi$
$607$	− 2.08193e8i	− 0.930895i	0.885076	−	0.465447i	$-0.154107\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 4.26846e8i	− 1.87132i
$612$	0	0
$613$	−2.59662e8	−1.12727	−0.563634	−	0.826024i	$-0.690597\pi$
$613$	−2.59662e8	−1.12727	−0.563634	+	0.826024i	$0.690597\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1.40534e8	−0.598309	−0.299155	−	0.954205i	$-0.596705\pi$
$617$	−1.40534e8	−0.598309	−0.299155	+	0.954205i	$0.596705\pi$
$618$	0	0
$619$	− 1.58033e8i	− 0.666307i	−0.942873	−	0.333154i	$-0.891887\pi$
$619$	− 1.58033e8i	− 0.666307i	0.942873	−	0.333154i	$-0.108113\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	5.06084e7i	0.209295i
$624$	0	0
$625$	2.42454e8	0.993093
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−1.61025e8	−0.647056
$630$	0	0
$631$	2.49902e8i	0.994677i	0.867556	+	0.497339i	$0.165689\pi$
$631$	2.49902e8i	0.994677i	−0.867556	+	0.497339i	$0.834311\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	8.69649e6i	0.0339643i
$636$	0	0
$637$	2.05104e8	0.793515
$638$	0	0
$639$	0	0
$640$	0	0
$641$	2.94095e7	0.111664	0.0558321	−	0.998440i	$-0.482219\pi$
$641$	2.94095e7	0.111664	0.0558321	+	0.998440i	$0.482219\pi$
$642$	0	0
$643$	2.55925e8i	0.962675i	0.876535	+	0.481337i	$0.159849\pi$
$643$	2.55925e8i	0.962675i	−0.876535	+	0.481337i	$0.840151\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 4.08342e8i	− 1.50769i	−0.657054	−	0.753844i	$-0.728197\pi$
$647$	− 4.08342e8i	− 1.50769i	0.657054	−	0.753844i	$-0.271803\pi$
$648$	0	0
$649$	−7.19016e7	−0.263030
$650$	0	0
$651$	0	0
$652$	0	0
$653$	2.06095e8	0.740164	0.370082	−	0.928999i	$-0.379330\pi$
$653$	2.06095e8	0.740164	0.370082	+	0.928999i	$0.379330\pi$
$654$	0	0
$655$	− 3.79672e6i	− 0.0135109i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 7.85056e7i	− 0.274312i	−0.990549	−	0.137156i	$-0.956204\pi$
$659$	− 7.85056e7i	− 0.274312i	0.990549	−	0.137156i	$-0.0437961\pi$
$660$	0	0
$661$	5.50505e7	0.190615	0.0953074	−	0.995448i	$-0.469617\pi$
$661$	5.50505e7	0.190615	0.0953074	+	0.995448i	$0.469617\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	4.26787e6	0.0145126
$666$	0	0
$667$	− 2.01652e8i	− 0.679557i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1.67827e8i	0.555512i
$672$	0	0
$673$	−4.82818e7	−0.158394	−0.0791969	−	0.996859i	$-0.525236\pi$
$673$	−4.82818e7	−0.158394	−0.0791969	+	0.996859i	$0.525236\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−4.02892e8	−1.29844	−0.649221	−	0.760600i	$-0.724905\pi$
$677$	−4.02892e8	−1.29844	−0.649221	+	0.760600i	$0.724905\pi$
$678$	0	0
$679$	1.03970e8i	0.332122i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	1.26946e8i	0.398434i	0.979955	+	0.199217i	$0.0638398\pi$
$683$	1.26946e8i	0.398434i	−0.979955	+	0.199217i	$0.936160\pi$
$684$	0	0
$685$	2.68693e7	0.0835958
$686$	0	0
$687$	0	0
$688$	0	0
$689$	4.47098e8	1.36693
$690$	0	0
$691$	1.39216e8i	0.421944i	0.977492	+	0.210972i	$0.0676629\pi$
$691$	1.39216e8i	0.421944i	−0.977492	+	0.210972i	$0.932337\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 1.34943e7i	− 0.0401973i
$696$	0	0
$697$	−7.46729e8	−2.20528
$698$	0	0
$699$	0	0
$700$	0	0
$701$	4.45981e8	1.29468	0.647340	−	0.762201i	$-0.275881\pi$
$701$	4.45981e8	1.29468	0.647340	+	0.762201i	$0.275881\pi$
$702$	0	0
$703$	7.91118e7i	0.227706i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2.05704e8i	0.582083i
$708$	0	0
$709$	−1.59057e8	−0.446288	−0.223144	−	0.974786i	$-0.571632\pi$
$709$	−1.59057e8	−0.446288	−0.223144	+	0.974786i	$0.571632\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−1.49144e8	−0.411468
$714$	0	0
$715$	1.02602e7i	0.0280697i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 3.34049e8i	− 0.898718i	−0.893351	−	0.449359i	$-0.851652\pi$
$719$	− 3.34049e8i	− 0.898718i	0.893351	−	0.449359i	$-0.148348\pi$
$720$	0	0
$721$	1.82756e7	0.0487602
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−1.80053e8	−0.472483
$726$	0	0
$727$	− 2.60960e8i	− 0.679157i	−0.940578	−	0.339579i	$-0.889716\pi$
$727$	− 2.60960e8i	− 0.679157i	0.940578	−	0.339579i	$-0.110284\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	9.17964e8i	2.35003i
$732$	0	0
$733$	3.56906e8	0.906238	0.453119	−	0.891450i	$-0.350311\pi$
$733$	3.56906e8	0.906238	0.453119	+	0.891450i	$0.350311\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−2.04304e8	−0.510357
$738$	0	0
$739$	− 8.61653e7i	− 0.213501i	−0.994286	−	0.106750i	$-0.965955\pi$
$739$	− 8.61653e7i	− 0.213501i	0.994286	−	0.106750i	$-0.0340446\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 2.61364e7i	− 0.0637204i	−0.999492	−	0.0318602i	$-0.989857\pi$
$743$	− 2.61364e7i	− 0.0637204i	0.999492	−	0.0318602i	$-0.0101431\pi$
$744$	0	0
$745$	−1.10430e7	−0.0267067
$746$	0	0
$747$	0	0
$748$	0	0
$749$	4.42585e8	1.05330
$750$	0	0
$751$	− 5.80052e8i	− 1.36945i	−0.728800	−	0.684727i	$-0.759922\pi$
$751$	− 5.80052e8i	− 1.36945i	0.728800	−	0.684727i	$-0.240078\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 6.86786e6i	− 0.0159581i
$756$	0	0
$757$	−3.35602e8	−0.773636	−0.386818	−	0.922156i	$-0.626426\pi$
$757$	−3.35602e8	−0.773636	−0.386818	+	0.922156i	$0.626426\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−2.34765e8	−0.532696	−0.266348	−	0.963877i	$-0.585817\pi$
$761$	−2.34765e8	−0.532696	−0.266348	+	0.963877i	$0.585817\pi$
$762$	0	0
$763$	− 1.91393e8i	− 0.430877i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.96167e8i	0.656372i
$768$	0	0
$769$	3.78738e8	0.832837	0.416419	−	0.909173i	$-0.363285\pi$
$769$	3.78738e8	0.832837	0.416419	+	0.909173i	$0.363285\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−1.34904e8	−0.292070	−0.146035	−	0.989279i	$-0.546651\pi$
$773$	−1.34904e8	−0.292070	−0.146035	+	0.989279i	$0.546651\pi$
$774$	0	0
$775$	1.33169e8i	0.286086i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	3.66868e8i	0.776065i
$780$	0	0
$781$	4.56453e8	0.958171
$782$	0	0
$783$	0	0
$784$	0	0
$785$	3.17265e7	0.0655863
$786$	0	0
$787$	8.58679e7i	0.176160i	0.996113	+	0.0880799i	$0.0280731\pi$
$787$	8.58679e7i	0.176160i	−0.996113	+	0.0880799i	$0.971927\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	2.43187e7i	0.0491373i
$792$	0	0
$793$	6.91287e8	1.38624
$794$	0	0
$795$	0	0
$796$	0	0
$797$	8.82735e8	1.74363	0.871817	−	0.489832i	$-0.162942\pi$
$797$	8.82735e8	1.74363	0.871817	+	0.489832i	$0.162942\pi$
$798$	0	0
$799$	− 1.15895e9i	− 2.27208i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 2.55099e8i	− 0.492677i
$804$	0	0
$805$	2.10470e7	0.0403463
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−3.19181e7	−0.0602825	−0.0301412	−	0.999546i	$-0.509596\pi$
$809$	−3.19181e7	−0.0602825	−0.0301412	+	0.999546i	$0.509596\pi$
$810$	0	0
$811$	8.59397e7i	0.161113i	0.996750	+	0.0805567i	$0.0256698\pi$
$811$	8.59397e7i	0.161113i	−0.996750	+	0.0805567i	$0.974330\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 3.87521e7i	− 0.0715851i
$816$	0	0
$817$	4.50996e8	0.827003
$818$	0	0
$819$	0	0
$820$	0	0
$821$	7.71027e8	1.39328	0.696642	−	0.717419i	$-0.254676\pi$
$821$	7.71027e8	1.39328	0.696642	+	0.717419i	$0.254676\pi$
$822$	0	0
$823$	2.04669e8i	0.367158i	0.983005	+	0.183579i	$0.0587683\pi$
$823$	2.04669e8i	0.367158i	−0.983005	+	0.183579i	$0.941232\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	2.55414e8i	0.451574i	0.974177	+	0.225787i	$0.0724953\pi$
$827$	2.55414e8i	0.451574i	−0.974177	+	0.225787i	$0.927505\pi$
$828$	0	0
$829$	6.29211e8	1.10442	0.552208	−	0.833706i	$-0.313785\pi$
$829$	6.29211e8	1.10442	0.552208	+	0.833706i	$0.313785\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	5.56887e8	0.963456
$834$	0	0
$835$	2.38628e7i	0.0409885i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	2.24118e8i	0.379481i	0.981834	+	0.189741i	$0.0607647\pi$
$839$	2.24118e8i	0.379481i	−0.981834	+	0.189741i	$0.939235\pi$
$840$	0	0
$841$	−4.61421e8	−0.775728
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1.33014e7	0.0220459
$846$	0	0
$847$	2.72527e8i	0.448496i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	3.90140e8i	0.633042i
$852$	0	0
$853$	−2.09382e8	−0.337359	−0.168679	−	0.985671i	$-0.553950\pi$
$853$	−2.09382e8	−0.337359	−0.168679	+	0.985671i	$0.553950\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	3.32297e8	0.527940	0.263970	−	0.964531i	$-0.414968\pi$
$857$	3.32297e8	0.527940	0.263970	+	0.964531i	$0.414968\pi$
$858$	0	0
$859$	− 6.58920e8i	− 1.03957i	−0.854297	−	0.519784i	$-0.826012\pi$
$859$	− 6.58920e8i	− 1.03957i	0.854297	−	0.519784i	$-0.173988\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1.00694e9i	1.56665i	0.621610	+	0.783327i	$0.286479\pi$
$863$	1.00694e9i	1.56665i	−0.621610	+	0.783327i	$0.713521\pi$
$864$	0	0
$865$	3.44129e7	0.0531707
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−3.58723e8	−0.546637
$870$	0	0
$871$	8.41538e8i	1.27356i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 3.76287e7i	− 0.0561688i
$876$	0	0
$877$	−7.31416e8	−1.08434	−0.542170	−	0.840269i	$-0.682397\pi$
$877$	−7.31416e8	−1.08434	−0.542170	+	0.840269i	$0.682397\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1.16606e9	1.70528	0.852638	−	0.522502i	$-0.175001\pi$
$881$	1.16606e9	1.70528	0.852638	+	0.522502i	$0.175001\pi$
$882$	0	0
$883$	7.42439e7i	0.107840i	0.998545	+	0.0539198i	$0.0171715\pi$
$883$	7.42439e7i	0.107840i	−0.998545	+	0.0539198i	$0.982828\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 2.35897e8i	− 0.338027i	−0.985614	−	0.169014i	$-0.945942\pi$
$887$	− 2.35897e8i	− 0.338027i	0.985614	−	0.169014i	$-0.0540582\pi$
$888$	0	0
$889$	−2.91213e8	−0.414482
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−5.69393e8	−0.799572
$894$	0	0
$895$	4.90130e7i	0.0683663i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 9.86656e7i	− 0.135796i
$900$	0	0
$901$	1.21394e9	1.65967
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1.94794e7	−0.0262802
$906$	0	0
$907$	− 2.54036e8i	− 0.340466i	−0.985404	−	0.170233i	$-0.945548\pi$
$907$	− 2.54036e8i	− 0.340466i	0.985404	−	0.170233i	$-0.0544520\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 1.03209e9i	− 1.36509i	−0.730844	−	0.682544i	$-0.760874\pi$
$911$	− 1.03209e9i	− 1.36509i	0.730844	−	0.682544i	$-0.239126\pi$
$912$	0	0
$913$	−5.74383e7	−0.0754726
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.27138e8	0.164880
$918$	0	0
$919$	− 1.13004e9i	− 1.45595i	−0.685602	−	0.727976i	$-0.740461\pi$
$919$	− 1.13004e9i	− 1.45595i	0.685602	−	0.727976i	$-0.259539\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 1.88015e9i	− 2.39105i
$924$	0	0
$925$	3.48352e8	0.440142
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1.40049e9	1.74676	0.873382	−	0.487036i	$-0.161922\pi$
$929$	1.40049e9	1.74676	0.873382	+	0.487036i	$0.161922\pi$
$930$	0	0
$931$	− 2.73599e8i	− 0.339051i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	2.78579e7i	0.0340812i
$936$	0	0
$937$	3.55654e8	0.432324	0.216162	−	0.976357i	$-0.430646\pi$
$937$	3.55654e8	0.432324	0.216162	+	0.976357i	$0.430646\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1.04392e8	0.125285	0.0626427	−	0.998036i	$-0.480047\pi$
$941$	1.04392e8	0.125285	0.0626427	+	0.998036i	$0.480047\pi$
$942$	0	0
$943$	1.80921e9i	2.15752i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 9.83576e8i	− 1.15813i	−0.815281	−	0.579066i	$-0.803417\pi$
$947$	− 9.83576e8i	− 1.15813i	0.815281	−	0.579066i	$-0.196583\pi$
$948$	0	0
$949$	−1.05077e9	−1.22944
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−3.13478e8	−0.362184	−0.181092	−	0.983466i	$-0.557963\pi$
$953$	−3.13478e8	−0.362184	−0.181092	+	0.983466i	$0.557963\pi$
$954$	0	0
$955$	6.70229e6i	0.00769508i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	8.99753e8i	1.02016i
$960$	0	0
$961$	8.14530e8	0.917776
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1.87425e7	0.0208567
$966$	0	0
$967$	3.95187e8i	0.437042i	0.975832	+	0.218521i	$0.0701232\pi$
$967$	3.95187e8i	0.437042i	−0.975832	+	0.218521i	$0.929877\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	2.04334e8i	0.223194i	0.993754	+	0.111597i	$0.0355966\pi$
$971$	2.04334e8i	0.223194i	−0.993754	+	0.111597i	$0.964403\pi$
$972$	0	0
$973$	4.51875e8	0.490546
$974$	0	0
$975$	0	0
$976$	0	0
$977$	3.87988e7	0.0416040	0.0208020	−	0.999784i	$-0.493378\pi$
$977$	3.87988e7	0.0416040	0.0208020	+	0.999784i	$0.493378\pi$
$978$	0	0
$979$	1.62296e8i	0.172966i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	7.59099e8i	0.799168i	0.916697	+	0.399584i	$0.130845\pi$
$983$	7.59099e8i	0.799168i	−0.916697	+	0.399584i	$0.869155\pi$
$984$	0	0
$985$	6.51181e7	0.0681386
$986$	0	0
$987$	0	0
$988$	0	0
$989$	2.22409e9	2.29913
$990$	0	0
$991$	− 5.08413e8i	− 0.522391i	−0.965286	−	0.261195i	$-0.915883\pi$
$991$	− 5.08413e8i	− 0.522391i	0.965286	−	0.261195i	$-0.0841167\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 5.39840e7i	− 0.0548019i
$996$	0	0
$997$	−1.50556e8	−0.151919	−0.0759594	−	0.997111i	$-0.524202\pi$
$997$	−1.50556e8	−0.151919	−0.0759594	+	0.997111i	$0.524202\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.7.g.g.127.2		2
3.2	odd	2		192.7.g.b.127.1		2
4.3	odd	2	inner	576.7.g.g.127.1		2
8.3	odd	2		144.7.g.c.127.1		2
8.5	even	2		144.7.g.c.127.2		2
12.11	even	2		192.7.g.b.127.2		2
24.5	odd	2		48.7.g.b.31.2	yes	2
24.11	even	2		48.7.g.b.31.1	✓	2
48.5	odd	4		768.7.b.b.127.1		4
48.11	even	4		768.7.b.b.127.3		4
48.29	odd	4		768.7.b.b.127.4		4
48.35	even	4		768.7.b.b.127.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.7.g.b.31.1	✓	2	24.11	even	2
48.7.g.b.31.2	yes	2	24.5	odd	2
144.7.g.c.127.1		2	8.3	odd	2
144.7.g.c.127.2		2	8.5	even	2
192.7.g.b.127.1		2	3.2	odd	2
192.7.g.b.127.2		2	12.11	even	2
576.7.g.g.127.1		2	4.3	odd	2	inner
576.7.g.g.127.2		2	1.1	even	1	trivial
768.7.b.b.127.1		4	48.5	odd	4
768.7.b.b.127.2		4	48.35	even	4
768.7.b.b.127.3		4	48.11	even	4
768.7.b.b.127.4		4	48.29	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 576 = 2^{6} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	576.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+6.00000 q^{5} +200.918i q^{7} +O(q^{10})\)	\(q+6.00000 q^{5} +200.918i q^{7} +644.323i q^{11} +2654.00 q^{13} +7206.00 q^{17} -3540.31i q^{19} -17459.1i q^{23} -15589.0 q^{25} +11550.0 q^{29} -8542.47i q^{31} +1205.51i q^{35} -22346.0 q^{37} -103626. q^{41} +127389. i q^{43} -160831. i q^{47} +77281.0 q^{49} +168462. q^{53} +3865.94i q^{55} +111593. i q^{59} +260470. q^{61} +15924.0 q^{65} +317083. i q^{67} -708423. i q^{71} -395918. q^{73} -129456. q^{77} +556743. i q^{79} +89145.2i q^{83} +43236.0 q^{85} +251886. q^{89} +533236. i q^{91} -21241.9i q^{95} +517474. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 12 q^{5}+O(q^{10}) \) 2 * q + 12 * q^5	\( 2 q + 12 q^{5} + 5308 q^{13} + 14412 q^{17} - 31178 q^{25} + 23100 q^{29} - 44692 q^{37} - 207252 q^{41} + 154562 q^{49} + 336924 q^{53} + 520940 q^{61} + 31848 q^{65} - 791836 q^{73} - 258912 q^{77} + 86472 q^{85} + 503772 q^{89} + 1034948 q^{97}+O(q^{100}) \) 2 * q + 12 * q^5 + 5308 * q^13 + 14412 * q^17 - 31178 * q^25 + 23100 * q^29 - 44692 * q^37 - 207252 * q^41 + 154562 * q^49 + 336924 * q^53 + 520940 * q^61 + 31848 * q^65 - 791836 * q^73 - 258912 * q^77 + 86472 * q^85 + 503772 * q^89 + 1034948 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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