LMFDB - Embedded newform 4896.2.l.d.3025.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4896,2,Mod(3025,4896)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4896, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4896.3025");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4896 = 2^{5} \cdot 3^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4896.l (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:	$39.0947568296$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - x^{16} - 2x^{14} + 2x^{12} - 4x^{11} + 4x^{10} + 8x^{8} - 16x^{7} + 16x^{6} - 64x^{4} - 128x^{2} + 512 $ x^18 - x^16 - 2x^14 + 2x^12 - 4x^11 + 4x^10 + 8x^8 - 16x^7 + 16x^6 - 64x^4 - 128*x^2 + 512
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{17} $
Twist minimal:	no (minimal twist has level 408)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3025.5
Root			$1.40931 - 0.117654i$ of defining polynomial
Character	$\chi$	$=$	4896.3025
Dual form			4896.2.l.d.3025.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.46472 q^{5} -3.26018i q^{7} +0.312167 q^{11} +6.39254i q^{13} +(3.01195 - 2.81570i) q^{17} +2.24353i q^{19} -7.34630i q^{23} -2.85460 q^{25} -5.39236 q^{29} -0.606910i q^{31} +4.77524i q^{35} +11.2816 q^{37} +5.36742i q^{41} -7.51199i q^{43} -4.24122 q^{47} -3.62875 q^{49} -8.91746i q^{53} -0.457236 q^{55} -5.15579i q^{59} +6.41607 q^{61} -9.36327i q^{65} +11.8427i q^{67} +6.62295i q^{71} -4.81332i q^{73} -1.01772i q^{77} +11.5068i q^{79} -13.3845i q^{83} +(-4.41165 + 4.12421i) q^{85} -9.54739 q^{89} +20.8408 q^{91} -3.28614i q^{95} -13.0210i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 4 q^{5} + 2 q^{17} + 22 q^{25} - 12 q^{29} - 16 q^{37} - 18 q^{49} + 16 q^{55} - 16 q^{61} + 16 q^{85} - 20 q^{89}+O(q^{100}) $ 18 * q + 4 * q^5 + 2 * q^17 + 22 * q^25 - 12 * q^29 - 16 * q^37 - 18 * q^49 + 16 * q^55 - 16 * q^61 + 16 * q^85 - 20 * q^89

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4896\mathbb{Z}\right)^\times$.

$n$	$613$	$2143$	$3809$	$4321$
$\chi(n)$	$-1$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−1.46472			−0.655041			−0.327521	−	0.944844i	$-0.606213\pi$
$5$	−1.46472			−0.655041			−0.327521	+	0.944844i	$0.606213\pi$
$6$		0			0
$7$		−	3.26018i		−	1.23223i	−0.787656	−	0.616115i	$-0.788705\pi$
$7$		−	3.26018i		−	1.23223i	0.787656	−	0.616115i	$-0.211295\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	0.312167			0.0941219			0.0470609	−	0.998892i	$-0.485015\pi$
$11$	0.312167			0.0941219			0.0470609	+	0.998892i	$0.485015\pi$
$12$		0			0
$13$			6.39254i			1.77297i	0.462755	+	0.886486i	$0.346861\pi$
$13$			6.39254i			1.77297i	−0.462755	+	0.886486i	$0.653139\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.01195	−	2.81570i	0.730505	−	0.682908i
$17$	3.01195	−	2.81570i	0.730505	−	0.682908i
$18$		0			0
$19$			2.24353i			0.514702i	0.966318	+	0.257351i	$0.0828496\pi$
$19$			2.24353i			0.514702i	−0.966318	+	0.257351i	$0.917150\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	7.34630i		−	1.53181i	−0.642954	−	0.765905i	$-0.722291\pi$
$23$		−	7.34630i		−	1.53181i	0.642954	−	0.765905i	$-0.277709\pi$
$24$		0			0
$25$	−2.85460			−0.570921
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−5.39236			−1.00134			−0.500668	−	0.865639i	$-0.666912\pi$
$29$	−5.39236			−1.00134			−0.500668	+	0.865639i	$0.666912\pi$
$30$		0			0
$31$		−	0.606910i		−	0.109004i	−0.998514	−	0.0545021i	$-0.982643\pi$
$31$		−	0.606910i		−	0.109004i	0.998514	−	0.0545021i	$-0.0173572\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			4.77524i			0.807162i
$36$		0			0
$37$	11.2816			1.85469			0.927344	−	0.374211i	$-0.122086\pi$
$37$	11.2816			1.85469			0.927344	+	0.374211i	$0.122086\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			5.36742i			0.838249i	0.907929	+	0.419125i	$0.137663\pi$
$41$			5.36742i			0.838249i	−0.907929	+	0.419125i	$0.862337\pi$
$42$		0			0
$43$		−	7.51199i		−	1.14557i	−0.819707	−	0.572784i	$-0.805863\pi$
$43$		−	7.51199i		−	1.14557i	0.819707	−	0.572784i	$-0.194137\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−4.24122			−0.618646			−0.309323	−	0.950957i	$-0.600102\pi$
$47$	−4.24122			−0.618646			−0.309323	+	0.950957i	$0.600102\pi$
$48$		0			0
$49$	−3.62875			−0.518393
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	8.91746i		−	1.22491i	−0.790507	−	0.612454i	$-0.790183\pi$
$53$		−	8.91746i		−	1.22491i	0.790507	−	0.612454i	$-0.209817\pi$
$54$		0			0
$55$	−0.457236			−0.0616537
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	5.15579i		−	0.671226i	−0.942000	−	0.335613i	$-0.891057\pi$
$59$		−	5.15579i		−	0.671226i	0.942000	−	0.335613i	$-0.108943\pi$
$60$		0			0
$61$	6.41607			0.821493			0.410747	−	0.911750i	$-0.365268\pi$
$61$	6.41607			0.821493			0.410747	+	0.911750i	$0.365268\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	9.36327i		−	1.16137i
$66$		0			0
$67$			11.8427i			1.44682i	0.690421	+	0.723408i	$0.257425\pi$
$67$			11.8427i			1.44682i	−0.690421	+	0.723408i	$0.742575\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			6.62295i			0.785999i	0.919539	+	0.393000i	$0.128563\pi$
$71$			6.62295i			0.785999i	−0.919539	+	0.393000i	$0.871437\pi$
$72$		0			0
$73$		−	4.81332i		−	0.563356i	−0.959509	−	0.281678i	$-0.909109\pi$
$73$		−	4.81332i		−	0.563356i	0.959509	−	0.281678i	$-0.0908910\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	1.01772i		−	0.115980i
$78$		0			0
$79$			11.5068i			1.29461i	0.762230	+	0.647306i	$0.224104\pi$
$79$			11.5068i			1.29461i	−0.762230	+	0.647306i	$0.775896\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	13.3845i		−	1.46914i	−0.678535	−	0.734568i	$-0.737385\pi$
$83$		−	13.3845i		−	1.46914i	0.678535	−	0.734568i	$-0.262615\pi$
$84$		0			0
$85$	−4.41165	+	4.12421i	−0.478511	+	0.447333i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−9.54739			−1.01202			−0.506011	−	0.862527i	$-0.668880\pi$
$89$	−9.54739			−1.01202			−0.506011	+	0.862527i	$0.668880\pi$
$90$		0			0
$91$	20.8408			2.18471
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	3.28614i		−	0.337151i
$96$		0			0
$97$		−	13.0210i		−	1.32208i	−0.750351	−	0.661040i	$-0.770115\pi$
$97$		−	13.0210i		−	1.32208i	0.750351	−	0.661040i	$-0.229885\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			1.94347i			0.193383i	0.995314	+	0.0966913i	$0.0308259\pi$
$101$			1.94347i			0.193383i	−0.995314	+	0.0966913i	$0.969174\pi$
$102$		0			0
$103$	−6.97083			−0.686856			−0.343428	−	0.939179i	$-0.611588\pi$
$103$	−6.97083			−0.686856			−0.343428	+	0.939179i	$0.611588\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−7.39946			−0.715333			−0.357667	−	0.933849i	$-0.616428\pi$
$107$	−7.39946			−0.715333			−0.357667	+	0.933849i	$0.616428\pi$
$108$		0			0
$109$	−10.6455			−1.01966			−0.509828	−	0.860277i	$-0.670291\pi$
$109$	−10.6455			−1.01966			−0.509828	+	0.860277i	$0.670291\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	11.0887i		−	1.04313i	−0.853211	−	0.521567i	$-0.825348\pi$
$113$		−	11.0887i		−	1.04313i	0.853211	−	0.521567i	$-0.174652\pi$
$114$		0			0
$115$			10.7603i			1.00340i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−9.17968	−	9.81948i	−0.841500	−	0.900150i
$120$		0			0
$121$	−10.9026			−0.991141
$122$		0			0
$123$		0			0
$124$		0			0
$125$	11.5048			1.02902
$126$		0			0
$127$	16.9758			1.50636			0.753179	−	0.657816i	$-0.228520\pi$
$127$	16.9758			1.50636			0.753179	+	0.657816i	$0.228520\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−0.168506			−0.0147225			−0.00736123	−	0.999973i	$-0.502343\pi$
$131$	−0.168506			−0.0147225			−0.00736123	+	0.999973i	$0.502343\pi$
$132$		0			0
$133$	7.31432			0.634232
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−6.02753			−0.514966			−0.257483	−	0.966283i	$-0.582893\pi$
$137$	−6.02753			−0.514966			−0.257483	+	0.966283i	$0.582893\pi$
$138$		0			0
$139$	−3.56380			−0.302278			−0.151139	−	0.988513i	$-0.548294\pi$
$139$	−3.56380			−0.302278			−0.151139	+	0.988513i	$0.548294\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			1.99554i			0.166875i
$144$		0			0
$145$	7.89829			0.655917
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	5.50380i		−	0.450889i	−0.974256	−	0.225444i	$-0.927617\pi$
$149$		−	5.50380i		−	0.450889i	0.974256	−	0.225444i	$-0.0723834\pi$
$150$		0			0
$151$	−0.619725			−0.0504325			−0.0252162	−	0.999682i	$-0.508027\pi$
$151$	−0.619725			−0.0504325			−0.0252162	+	0.999682i	$0.508027\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			0.888952i			0.0714023i
$156$		0			0
$157$		−	18.7631i		−	1.49746i	−0.662875	−	0.748730i	$-0.730664\pi$
$157$		−	18.7631i		−	1.49746i	0.662875	−	0.748730i	$-0.269336\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−23.9502			−1.88754
$162$		0			0
$163$	−15.9222			−1.24712			−0.623561	−	0.781775i	$-0.714315\pi$
$163$	−15.9222			−1.24712			−0.623561	+	0.781775i	$0.714315\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	7.08898i		−	0.548561i	−0.961650	−	0.274281i	$-0.911560\pi$
$167$		−	7.08898i		−	0.548561i	0.961650	−	0.274281i	$-0.0884397\pi$
$168$		0			0
$169$	−27.8646			−2.14343
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−10.1123			−0.768827			−0.384414	−	0.923161i	$-0.625596\pi$
$173$	−10.1123			−0.768827			−0.384414	+	0.923161i	$0.625596\pi$
$174$		0			0
$175$			9.30651i			0.703506i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	2.38313i		−	0.178124i	−0.996026	−	0.0890618i	$-0.971613\pi$
$179$		−	2.38313i		−	0.178124i	0.996026	−	0.0890618i	$-0.0283869\pi$
$180$		0			0
$181$	−10.5481			−0.784038			−0.392019	−	0.919957i	$-0.628223\pi$
$181$	−10.5481			−0.784038			−0.392019	+	0.919957i	$0.628223\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−16.5244			−1.21490
$186$		0			0
$187$	0.940230	−	0.878969i	0.0687565	−	0.0642765i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−16.9179			−1.22414			−0.612069	−	0.790804i	$-0.709662\pi$
$191$	−16.9179			−1.22414			−0.612069	+	0.790804i	$0.709662\pi$
$192$		0			0
$193$			3.08185i			0.221836i	0.993830	+	0.110918i	$0.0353791\pi$
$193$			3.08185i			0.221836i	−0.993830	+	0.110918i	$0.964621\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−22.7820			−1.62315			−0.811577	−	0.584246i	$-0.801390\pi$
$197$	−22.7820			−1.62315			−0.811577	+	0.584246i	$0.801390\pi$
$198$		0			0
$199$		−	14.9021i		−	1.05638i	−0.849126	−	0.528190i	$-0.822871\pi$
$199$		−	14.9021i		−	1.05638i	0.849126	−	0.528190i	$-0.177129\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			17.5801i			1.23388i
$204$		0			0
$205$		−	7.86175i		−	0.549088i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			0.700357i			0.0484447i
$210$		0			0
$211$	−11.1308			−0.766278			−0.383139	−	0.923691i	$-0.625157\pi$
$211$	−11.1308			−0.766278			−0.383139	+	0.923691i	$0.625157\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			11.0029i			0.750394i
$216$		0			0
$217$	−1.97863			−0.134318
$218$		0			0
$219$		0			0
$220$		0			0
$221$	17.9995	+	19.2540i	1.21078	+	1.29516i
$222$		0			0
$223$	0.320676			0.0214741			0.0107370	−	0.999942i	$-0.496582\pi$
$223$	0.320676			0.0214741			0.0107370	+	0.999942i	$0.496582\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−26.1137			−1.73323			−0.866613	−	0.498980i	$-0.833708\pi$
$227$	−26.1137			−1.73323			−0.866613	+	0.498980i	$0.833708\pi$
$228$		0			0
$229$			14.6905i			0.970772i	0.874300	+	0.485386i	$0.161321\pi$
$229$			14.6905i			0.970772i	−0.874300	+	0.485386i	$0.838679\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	27.5228i		−	1.80308i	−0.432694	−	0.901541i	$-0.642437\pi$
$233$		−	27.5228i		−	1.80308i	0.432694	−	0.901541i	$-0.357563\pi$
$234$		0			0
$235$	6.21219			0.405239
$236$		0			0
$237$		0			0
$238$		0			0
$239$	5.80155			0.375271			0.187636	−	0.982239i	$-0.439918\pi$
$239$	5.80155			0.375271			0.187636	+	0.982239i	$0.439918\pi$
$240$		0			0
$241$		−	17.1627i		−	1.10554i	−0.833333	−	0.552772i	$-0.813570\pi$
$241$		−	17.1627i		−	1.10554i	0.833333	−	0.552772i	$-0.186430\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	5.31509			0.339569
$246$		0			0
$247$	−14.3419			−0.912552
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	6.91981i		−	0.436774i	−0.975862	−	0.218387i	$-0.929920\pi$
$251$		−	6.91981i		−	0.436774i	0.975862	−	0.218387i	$-0.0700796\pi$
$252$		0			0
$253$		−	2.29327i		−	0.144177i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	0.288958			0.0180247			0.00901234	−	0.999959i	$-0.497131\pi$
$257$	0.288958			0.0180247			0.00901234	+	0.999959i	$0.497131\pi$
$258$		0			0
$259$		−	36.7801i		−	2.28540i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−1.38062			−0.0851326			−0.0425663	−	0.999094i	$-0.513553\pi$
$263$	−1.38062			−0.0851326			−0.0425663	+	0.999094i	$0.513553\pi$
$264$		0			0
$265$			13.0616i			0.802365i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	17.5243			1.06847			0.534237	−	0.845335i	$-0.320599\pi$
$269$	17.5243			1.06847			0.534237	+	0.845335i	$0.320599\pi$
$270$		0			0
$271$	−0.927991			−0.0563715			−0.0281857	−	0.999603i	$-0.508973\pi$
$271$	−0.927991			−0.0563715			−0.0281857	+	0.999603i	$0.508973\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−0.891113			−0.0537361
$276$		0			0
$277$	9.97435			0.599301			0.299650	−	0.954049i	$-0.403130\pi$
$277$	9.97435			0.599301			0.299650	+	0.954049i	$0.403130\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	19.4670			1.16131			0.580653	−	0.814151i	$-0.302797\pi$
$281$	19.4670			1.16131			0.580653	+	0.814151i	$0.302797\pi$
$282$		0			0
$283$	−2.02405			−0.120317			−0.0601587	−	0.998189i	$-0.519161\pi$
$283$	−2.02405			−0.120317			−0.0601587	+	0.998189i	$0.519161\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	17.4987			1.03292
$288$		0			0
$289$	1.14366	−	16.9615i	0.0672742	−	0.997735i
$290$		0			0
$291$		0			0
$292$		0			0
$293$			1.29056i			0.0753952i	0.999289	+	0.0376976i	$0.0120024\pi$
$293$			1.29056i			0.0753952i	−0.999289	+	0.0376976i	$0.987998\pi$
$294$		0			0
$295$			7.55177i			0.439681i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	46.9616			2.71586
$300$		0			0
$301$	−24.4904			−1.41160
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−9.39773			−0.538112
$306$		0			0
$307$		−	7.42682i		−	0.423871i	−0.977284	−	0.211936i	$-0.932023\pi$
$307$		−	7.42682i		−	0.423871i	0.977284	−	0.211936i	$-0.0679767\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	3.73279i		−	0.211667i	−0.994384	−	0.105834i	$-0.966249\pi$
$311$		−	3.73279i		−	0.211667i	0.994384	−	0.105834i	$-0.0337511\pi$
$312$		0			0
$313$			1.09676i			0.0619925i	0.999520	+	0.0309963i	$0.00986800\pi$
$313$			1.09676i			0.0619925i	−0.999520	+	0.0309963i	$0.990132\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	19.9349			1.11965			0.559827	−	0.828610i	$-0.310868\pi$
$317$	19.9349			1.11965			0.559827	+	0.828610i	$0.310868\pi$
$318$		0			0
$319$	−1.68332			−0.0942477
$320$		0			0
$321$		0			0
$322$		0			0
$323$	6.31712	+	6.75741i	0.351494	+	0.375992i
$324$		0			0
$325$		−	18.2482i		−	1.01223i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			13.8271i			0.762314i
$330$		0			0
$331$			26.9222i			1.47978i	0.672728	+	0.739890i	$0.265122\pi$
$331$			26.9222i			1.47978i	−0.672728	+	0.739890i	$0.734878\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	17.3462i		−	0.947724i
$336$		0			0
$337$		−	12.2109i		−	0.665170i	−0.943073	−	0.332585i	$-0.892079\pi$
$337$		−	12.2109i		−	0.665170i	0.943073	−	0.332585i	$-0.107921\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	0.189457i		−	0.0102597i
$342$		0			0
$343$		−	10.9909i		−	0.593451i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	14.2507			0.765018			0.382509	−	0.923952i	$-0.375060\pi$
$347$	14.2507			0.765018			0.382509	+	0.923952i	$0.375060\pi$
$348$		0			0
$349$		−	18.5830i		−	0.994725i	−0.867543	−	0.497363i	$-0.834302\pi$
$349$		−	18.5830i		−	0.994725i	0.867543	−	0.497363i	$-0.165698\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	3.89576			0.207350			0.103675	−	0.994611i	$-0.466940\pi$
$353$	3.89576			0.207350			0.103675	+	0.994611i	$0.466940\pi$
$354$		0			0
$355$		−	9.70074i		−	0.514862i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	8.66787			0.457472			0.228736	−	0.973488i	$-0.426541\pi$
$359$	8.66787			0.457472			0.228736	+	0.973488i	$0.426541\pi$
$360$		0			0
$361$	13.9666			0.735082
$362$		0			0
$363$		0			0
$364$		0			0
$365$			7.05015i			0.369022i
$366$		0			0
$367$		−	32.3753i		−	1.68998i	−0.534785	−	0.844988i	$-0.679608\pi$
$367$		−	32.3753i		−	1.68998i	0.534785	−	0.844988i	$-0.320392\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−29.0725			−1.50937
$372$		0			0
$373$			11.7221i			0.606947i	0.952840	+	0.303473i	$0.0981464\pi$
$373$			11.7221i			0.606947i	−0.952840	+	0.303473i	$0.901854\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	34.4709i		−	1.77534i
$378$		0			0
$379$	−27.7442			−1.42512			−0.712562	−	0.701609i	$-0.752465\pi$
$379$	−27.7442			−1.42512			−0.712562	+	0.701609i	$0.752465\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−25.2288			−1.28913			−0.644566	−	0.764549i	$-0.722962\pi$
$383$	−25.2288			−1.28913			−0.644566	+	0.764549i	$0.722962\pi$
$384$		0			0
$385$			1.49067i			0.0759716i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	26.3923i		−	1.33814i	−0.743198	−	0.669072i	$-0.766692\pi$
$389$		−	26.3923i		−	1.33814i	0.743198	−	0.669072i	$-0.233308\pi$
$390$		0			0
$391$	−20.6850	−	22.1267i	−1.04608	−	1.11899i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	16.8542i		−	0.848025i
$396$		0			0
$397$	−21.2037			−1.06418			−0.532091	−	0.846687i	$-0.678594\pi$
$397$	−21.2037			−1.06418			−0.532091	+	0.846687i	$0.678594\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			7.21971i			0.360535i	0.983618	+	0.180268i	$0.0576964\pi$
$401$			7.21971i			0.360535i	−0.983618	+	0.180268i	$0.942304\pi$
$402$		0			0
$403$	3.87970			0.193262
$404$		0			0
$405$		0			0
$406$		0			0
$407$	3.52175			0.174567
$408$		0			0
$409$	24.5533			1.21408			0.607042	−	0.794670i	$-0.292356\pi$
$409$	24.5533			1.21408			0.607042	+	0.794670i	$0.292356\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−16.8088			−0.827106
$414$		0			0
$415$			19.6045i			0.962345i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	39.0863			1.90949			0.954745	−	0.297426i	$-0.0961283\pi$
$419$	39.0863			1.90949			0.954745	+	0.297426i	$0.0961283\pi$
$420$		0			0
$421$			9.66214i			0.470904i	0.971886	+	0.235452i	$0.0756570\pi$
$421$			9.66214i			0.470904i	−0.971886	+	0.235452i	$0.924343\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−8.59792	+	8.03771i	−0.417060	+	0.389886i
$426$		0			0
$427$		−	20.9175i		−	1.01227i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	6.07054i		−	0.292408i	−0.989254	−	0.146204i	$-0.953294\pi$
$431$		−	6.07054i		−	0.292408i	0.989254	−	0.146204i	$-0.0467055\pi$
$432$		0			0
$433$	20.5097			0.985634			0.492817	−	0.870133i	$-0.335967\pi$
$433$	20.5097			0.985634			0.492817	+	0.870133i	$0.335967\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	16.4817			0.788426
$438$		0			0
$439$		−	7.36907i		−	0.351706i	−0.984416	−	0.175853i	$-0.943732\pi$
$439$		−	7.36907i		−	0.351706i	0.984416	−	0.175853i	$-0.0562684\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			18.9079i			0.898342i	0.893446	+	0.449171i	$0.148281\pi$
$443$			18.9079i			0.898342i	−0.893446	+	0.449171i	$0.851719\pi$
$444$		0			0
$445$	13.9842			0.662916
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	19.0187i		−	0.897550i	−0.893645	−	0.448775i	$-0.851861\pi$
$449$		−	19.0187i		−	0.897550i	0.893645	−	0.448775i	$-0.148139\pi$
$450$		0			0
$451$			1.67553i			0.0788976i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−30.5259			−1.43108
$456$		0			0
$457$	−20.5126			−0.959539			−0.479769	−	0.877395i	$-0.659280\pi$
$457$	−20.5126			−0.959539			−0.479769	+	0.877395i	$0.659280\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			8.07937i			0.376294i	0.982141	+	0.188147i	$0.0602481\pi$
$461$			8.07937i			0.376294i	−0.982141	+	0.188147i	$0.939752\pi$
$462$		0			0
$463$	−12.6813			−0.589350			−0.294675	−	0.955597i	$-0.595211\pi$
$463$	−12.6813			−0.589350			−0.294675	+	0.955597i	$0.595211\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			32.1472i			1.48760i	0.668405	+	0.743798i	$0.266977\pi$
$467$			32.1472i			1.48760i	−0.668405	+	0.743798i	$0.733023\pi$
$468$		0			0
$469$	38.6093			1.78281
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	2.34499i		−	0.107823i
$474$		0			0
$475$		−	6.40440i		−	0.293854i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	25.4835i		−	1.16437i	−0.813056	−	0.582185i	$-0.802198\pi$
$479$		−	25.4835i		−	1.16437i	0.813056	−	0.582185i	$-0.197802\pi$
$480$		0			0
$481$			72.1183i			3.28831i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			19.0721i			0.866017i
$486$		0			0
$487$			3.09673i			0.140326i	0.997536	+	0.0701630i	$0.0223520\pi$
$487$			3.09673i			0.140326i	−0.997536	+	0.0701630i	$0.977648\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			37.4349i			1.68941i	0.535231	+	0.844706i	$0.320225\pi$
$491$			37.4349i			1.68941i	−0.535231	+	0.844706i	$0.679775\pi$
$492$		0			0
$493$	−16.2415	+	15.1833i	−0.731481	+	0.683820i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	21.5920			0.968532
$498$		0			0
$499$	19.7963			0.886204			0.443102	−	0.896471i	$-0.353878\pi$
$499$	19.7963			0.886204			0.443102	+	0.896471i	$0.353878\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	10.8162i		−	0.482268i	−0.970492	−	0.241134i	$-0.922481\pi$
$503$		−	10.8162i		−	0.482268i	0.970492	−	0.241134i	$-0.0775194\pi$
$504$		0			0
$505$		−	2.84663i		−	0.126674i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	2.63717i		−	0.116890i	−0.998291	−	0.0584452i	$-0.981386\pi$
$509$		−	2.63717i		−	0.116890i	0.998291	−	0.0584452i	$-0.0186143\pi$
$510$		0			0
$511$	−15.6923			−0.694185
$512$		0			0
$513$		0			0
$514$		0			0
$515$	10.2103			0.449919
$516$		0			0
$517$	−1.32397			−0.0582281
$518$		0			0
$519$		0			0
$520$		0			0
$521$			38.5194i			1.68757i	0.536684	+	0.843783i	$0.319677\pi$
$521$			38.5194i			1.68757i	−0.536684	+	0.843783i	$0.680323\pi$
$522$		0			0
$523$		−	0.305975i		−	0.0133793i	−0.999978	−	0.00668967i	$-0.997871\pi$
$523$		−	0.305975i		−	0.0133793i	0.999978	−	0.00668967i	$-0.00212940\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1.70888	−	1.82798i	−0.0744399	−	0.0796281i
$528$		0			0
$529$	−30.9682			−1.34644
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−34.3114			−1.48619
$534$		0			0
$535$	10.8381			0.468573
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−1.13278			−0.0487921
$540$		0			0
$541$	1.50556			0.0647292			0.0323646	−	0.999476i	$-0.489696\pi$
$541$	1.50556			0.0647292			0.0323646	+	0.999476i	$0.489696\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	15.5927			0.667916
$546$		0			0
$547$	3.68763			0.157672			0.0788358	−	0.996888i	$-0.474880\pi$
$547$	3.68763			0.157672			0.0788358	+	0.996888i	$0.474880\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	12.0979i		−	0.515390i
$552$		0			0
$553$	37.5141			1.59526
$554$		0			0
$555$		0			0
$556$		0			0
$557$			29.3032i			1.24162i	0.783962	+	0.620809i	$0.213196\pi$
$557$			29.3032i			1.24162i	−0.783962	+	0.620809i	$0.786804\pi$
$558$		0			0
$559$	48.0207			2.03106
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	27.2285i		−	1.14754i	−0.819015	−	0.573772i	$-0.805480\pi$
$563$		−	27.2285i		−	1.14754i	0.819015	−	0.573772i	$-0.194520\pi$
$564$		0			0
$565$			16.2417i			0.683296i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−8.50233			−0.356436			−0.178218	−	0.983991i	$-0.557033\pi$
$569$	−8.50233			−0.356436			−0.178218	+	0.983991i	$0.557033\pi$
$570$		0			0
$571$	−3.89571			−0.163030			−0.0815151	−	0.996672i	$-0.525976\pi$
$571$	−3.89571			−0.163030			−0.0815151	+	0.996672i	$0.525976\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			20.9708i			0.874542i
$576$		0			0
$577$	3.69131			0.153671			0.0768355	−	0.997044i	$-0.475518\pi$
$577$	3.69131			0.153671			0.0768355	+	0.997044i	$0.475518\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−43.6357			−1.81032
$582$		0			0
$583$		−	2.78374i		−	0.115291i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	18.6727i		−	0.770704i	−0.922770	−	0.385352i	$-0.874080\pi$
$587$		−	18.6727i		−	0.770704i	0.922770	−	0.385352i	$-0.125920\pi$
$588$		0			0
$589$	1.36162			0.0561047
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−18.0541			−0.741393			−0.370697	−	0.928754i	$-0.620881\pi$
$593$	−18.0541			−0.741393			−0.370697	+	0.928754i	$0.620881\pi$
$594$		0			0
$595$	13.4456	+	14.3828i	0.551217	+	0.589636i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−43.6498			−1.78348			−0.891741	−	0.452546i	$-0.850516\pi$
$599$	−43.6498			−1.78348			−0.891741	+	0.452546i	$0.850516\pi$
$600$		0			0
$601$			10.1680i			0.414760i	0.978260	+	0.207380i	$0.0664937\pi$
$601$			10.1680i			0.414760i	−0.978260	+	0.207380i	$0.933506\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	15.9692			0.649239
$606$		0			0
$607$			10.9378i			0.443952i	0.975052	+	0.221976i	$0.0712506\pi$
$607$			10.9378i			0.443952i	−0.975052	+	0.221976i	$0.928749\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	27.1122i		−	1.09684i
$612$		0			0
$613$			35.0085i			1.41398i	0.707224	+	0.706990i	$0.249947\pi$
$613$			35.0085i			1.41398i	−0.707224	+	0.706990i	$0.750053\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			24.5518i			0.988419i	0.869343	+	0.494209i	$0.164542\pi$
$617$			24.5518i			0.988419i	−0.869343	+	0.494209i	$0.835458\pi$
$618$		0			0
$619$	−37.2040			−1.49536			−0.747678	−	0.664061i	$-0.768832\pi$
$619$	−37.2040			−1.49536			−0.747678	+	0.664061i	$0.768832\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			31.1262i			1.24704i
$624$		0			0
$625$	−2.57822			−0.103129
$626$		0			0
$627$		0			0
$628$		0			0
$629$	33.9797	−	31.7657i	1.35486	−	1.26658i
$630$		0			0
$631$	−12.3656			−0.492266			−0.246133	−	0.969236i	$-0.579160\pi$
$631$	−12.3656			−0.492266			−0.246133	+	0.969236i	$0.579160\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−24.8647			−0.986726
$636$		0			0
$637$		−	23.1969i		−	0.919096i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			37.6527i			1.48719i	0.668629	+	0.743596i	$0.266882\pi$
$641$			37.6527i			1.48719i	−0.668629	+	0.743596i	$0.733118\pi$
$642$		0			0
$643$	29.1814			1.15080			0.575400	−	0.817872i	$-0.304846\pi$
$643$	29.1814			1.15080			0.575400	+	0.817872i	$0.304846\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	32.6928			1.28529			0.642643	−	0.766166i	$-0.277838\pi$
$647$	32.6928			1.28529			0.642643	+	0.766166i	$0.277838\pi$
$648$		0			0
$649$		−	1.60947i		−	0.0631771i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	5.75472			0.225200			0.112600	−	0.993640i	$-0.464082\pi$
$653$	5.75472			0.225200			0.112600	+	0.993640i	$0.464082\pi$
$654$		0			0
$655$	0.246814			0.00964382
$656$		0			0
$657$		0			0
$658$		0			0
$659$			16.3764i			0.637933i	0.947766	+	0.318966i	$0.103336\pi$
$659$			16.3764i			0.637933i	−0.947766	+	0.318966i	$0.896664\pi$
$660$		0			0
$661$		−	26.9352i		−	1.04766i	−0.851823	−	0.523829i	$-0.824503\pi$
$661$		−	26.9352i		−	1.04766i	0.851823	−	0.523829i	$-0.175497\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−10.7134			−0.415448
$666$		0			0
$667$			39.6139i			1.53386i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	2.00288			0.0773205
$672$		0			0
$673$		−	48.5604i		−	1.87187i	−0.352179	−	0.935933i	$-0.614559\pi$
$673$		−	48.5604i		−	1.87187i	0.352179	−	0.935933i	$-0.385441\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−18.4217			−0.708002			−0.354001	−	0.935245i	$-0.615179\pi$
$677$	−18.4217			−0.708002			−0.354001	+	0.935245i	$0.615179\pi$
$678$		0			0
$679$	−42.4507			−1.62911
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−35.1819			−1.34620			−0.673099	−	0.739552i	$-0.735037\pi$
$683$	−35.1819			−1.34620			−0.673099	+	0.739552i	$0.735037\pi$
$684$		0			0
$685$	8.82862			0.337324
$686$		0			0
$687$		0			0
$688$		0			0
$689$	57.0052			2.17173
$690$		0			0
$691$	−16.9759			−0.645792			−0.322896	−	0.946434i	$-0.604656\pi$
$691$	−16.9759			−0.645792			−0.322896	+	0.946434i	$0.604656\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	5.21996			0.198004
$696$		0			0
$697$	15.1130	+	16.1664i	0.572447	+	0.612345i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	25.1119i		−	0.948462i	−0.880400	−	0.474231i	$-0.842726\pi$
$701$		−	25.1119i		−	0.948462i	0.880400	−	0.474231i	$-0.157274\pi$
$702$		0			0
$703$			25.3107i			0.954611i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	6.33606			0.238292
$708$		0			0
$709$	−32.3694			−1.21566			−0.607830	−	0.794067i	$-0.707960\pi$
$709$	−32.3694			−1.21566			−0.607830	+	0.794067i	$0.707960\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−4.45855			−0.166974
$714$		0			0
$715$		−	2.92290i		−	0.109310i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			14.6515i			0.546410i	0.961956	+	0.273205i	$0.0880838\pi$
$719$			14.6515i			0.546410i	−0.961956	+	0.273205i	$0.911916\pi$
$720$		0			0
$721$			22.7261i			0.846366i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	15.3931			0.571684
$726$		0			0
$727$	−5.77315			−0.214114			−0.107057	−	0.994253i	$-0.534143\pi$
$727$	−5.77315			−0.214114			−0.107057	+	0.994253i	$0.534143\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−21.1515	−	22.6257i	−0.782317	−	0.836842i
$732$		0			0
$733$		−	9.97415i		−	0.368403i	−0.982888	−	0.184202i	$-0.941030\pi$
$733$		−	9.97415i		−	0.368403i	0.982888	−	0.184202i	$-0.0589700\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			3.69690i			0.136177i
$738$		0			0
$739$		−	31.2897i		−	1.15101i	−0.817798	−	0.575506i	$-0.804805\pi$
$739$		−	31.2897i		−	1.15101i	0.817798	−	0.575506i	$-0.195195\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	0.432434i		−	0.0158645i	−0.999969	−	0.00793223i	$-0.997475\pi$
$743$		−	0.432434i		−	0.0158645i	0.999969	−	0.00793223i	$-0.00252493\pi$
$744$		0			0
$745$			8.06151i			0.295351i
$746$		0			0
$747$		0			0
$748$		0			0
$749$			24.1236i			0.881455i
$750$		0			0
$751$		−	44.0064i		−	1.60582i	−0.596103	−	0.802908i	$-0.703285\pi$
$751$		−	44.0064i		−	1.60582i	0.596103	−	0.802908i	$-0.296715\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	0.907721			0.0330354
$756$		0			0
$757$			48.4044i			1.75929i	0.475634	+	0.879643i	$0.342219\pi$
$757$			48.4044i			1.75929i	−0.475634	+	0.879643i	$0.657781\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	37.9787			1.37673			0.688363	−	0.725366i	$-0.258329\pi$
$761$	37.9787			1.37673			0.688363	+	0.725366i	$0.258329\pi$
$762$		0			0
$763$			34.7062i			1.25645i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	32.9586			1.19007
$768$		0			0
$769$	−3.04684			−0.109872			−0.0549358	−	0.998490i	$-0.517495\pi$
$769$	−3.04684			−0.109872			−0.0549358	+	0.998490i	$0.517495\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	21.3814i		−	0.769036i	−0.923118	−	0.384518i	$-0.874368\pi$
$773$		−	21.3814i		−	0.769036i	0.923118	−	0.384518i	$-0.125632\pi$
$774$		0			0
$775$			1.73249i			0.0622328i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−12.0420			−0.431449
$780$		0			0
$781$			2.06746i			0.0739797i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			27.4827i			0.980899i
$786$		0			0
$787$	38.9140			1.38713			0.693567	−	0.720392i	$-0.256038\pi$
$787$	38.9140			1.38713			0.693567	+	0.720392i	$0.256038\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−36.1510			−1.28538
$792$		0			0
$793$			41.0150i			1.45649i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			18.6530i			0.660722i	0.943855	+	0.330361i	$0.107171\pi$
$797$			18.6530i			0.660722i	−0.943855	+	0.330361i	$0.892829\pi$
$798$		0			0
$799$	−12.7743	+	11.9420i	−0.451924	+	0.422478i
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	1.50256i		−	0.0530241i
$804$		0			0
$805$	35.0803			1.23642
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	32.7083i		−	1.14996i	−0.818166	−	0.574982i	$-0.805009\pi$
$809$		−	32.7083i		−	1.14996i	0.818166	−	0.574982i	$-0.194991\pi$
$810$		0			0
$811$	31.1576			1.09409			0.547045	−	0.837103i	$-0.315753\pi$
$811$	31.1576			1.09409			0.547045	+	0.837103i	$0.315753\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	23.3215			0.816916
$816$		0			0
$817$	16.8534			0.589626
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−28.5075			−0.994917			−0.497459	−	0.867488i	$-0.665733\pi$
$821$	−28.5075			−0.994917			−0.497459	+	0.867488i	$0.665733\pi$
$822$		0			0
$823$		−	23.6353i		−	0.823876i	−0.911212	−	0.411938i	$-0.864852\pi$
$823$		−	23.6353i		−	0.823876i	0.911212	−	0.411938i	$-0.135148\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	33.1756			1.15363			0.576814	−	0.816875i	$-0.304296\pi$
$827$	33.1756			1.15363			0.576814	+	0.816875i	$0.304296\pi$
$828$		0			0
$829$		−	24.0031i		−	0.833662i	−0.908984	−	0.416831i	$-0.863141\pi$
$829$		−	24.0031i		−	0.833662i	0.908984	−	0.416831i	$-0.136859\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−10.9296	+	10.2175i	−0.378688	+	0.354014i
$834$		0			0
$835$			10.3833i			0.359330i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			27.2563i			0.940993i	0.882402	+	0.470496i	$0.155925\pi$
$839$			27.2563i			0.940993i	−0.882402	+	0.470496i	$0.844075\pi$
$840$		0			0
$841$	0.0775701			0.00267483
$842$		0			0
$843$		0			0
$844$		0			0
$845$	40.8138			1.40404
$846$		0			0
$847$			35.5442i			1.22131i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	82.8782i		−	2.84103i
$852$		0			0
$853$	35.1320			1.20290			0.601449	−	0.798911i	$-0.294590\pi$
$853$	35.1320			1.20290			0.601449	+	0.798911i	$0.294590\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			27.2840i			0.932005i	0.884783	+	0.466002i	$0.154306\pi$
$857$			27.2840i			0.932005i	−0.884783	+	0.466002i	$0.845694\pi$
$858$		0			0
$859$		−	3.90819i		−	0.133346i	−0.997775	−	0.0666728i	$-0.978762\pi$
$859$		−	3.90819i		−	0.133346i	0.997775	−	0.0666728i	$-0.0212384\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	30.8940			1.05164			0.525821	−	0.850595i	$-0.323758\pi$
$863$	30.8940			1.05164			0.525821	+	0.850595i	$0.323758\pi$
$864$		0			0
$865$	14.8117			0.503614
$866$		0			0
$867$		0			0
$868$		0			0
$869$			3.59203i			0.121851i
$870$		0			0
$871$	−75.7049			−2.56516
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	37.5076i		−	1.26799i
$876$		0			0
$877$	−17.7726			−0.600137			−0.300068	−	0.953918i	$-0.597010\pi$
$877$	−17.7726			−0.600137			−0.300068	+	0.953918i	$0.597010\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			34.8223i			1.17319i	0.809879	+	0.586597i	$0.199533\pi$
$881$			34.8223i			1.17319i	−0.809879	+	0.586597i	$0.800467\pi$
$882$		0			0
$883$			7.89791i			0.265786i	0.991130	+	0.132893i	$0.0424266\pi$
$883$			7.89791i			0.265786i	−0.991130	+	0.132893i	$0.957573\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	21.4557i		−	0.720412i	−0.932873	−	0.360206i	$-0.882706\pi$
$887$		−	21.4557i		−	0.720412i	0.932873	−	0.360206i	$-0.117294\pi$
$888$		0			0
$889$		−	55.3440i		−	1.85618i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	9.51532i		−	0.318418i
$894$		0			0
$895$			3.49061i			0.116678i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			3.27268i			0.109150i
$900$		0			0
$901$	−25.1089	−	26.8589i	−0.836499	−	0.894800i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	15.4501			0.513577
$906$		0			0
$907$	46.9549			1.55911			0.779557	−	0.626332i	$-0.215444\pi$
$907$	46.9549			1.55911			0.779557	+	0.626332i	$0.215444\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			20.5297i			0.680181i	0.940393	+	0.340090i	$0.110458\pi$
$911$			20.5297i			0.680181i	−0.940393	+	0.340090i	$0.889542\pi$
$912$		0			0
$913$		−	4.17819i		−	0.138278i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			0.549360i			0.0181415i
$918$		0			0
$919$	53.2755			1.75740			0.878698	−	0.477379i	$-0.158413\pi$
$919$	53.2755			1.75740			0.878698	+	0.477379i	$0.158413\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−42.3375			−1.39355
$924$		0			0
$925$	−32.2046			−1.05888
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	40.7355i		−	1.33649i	−0.743942	−	0.668244i	$-0.767046\pi$
$929$		−	40.7355i		−	1.33649i	0.743942	−	0.668244i	$-0.232954\pi$
$930$		0			0
$931$		−	8.14122i		−	0.266818i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−1.37717	+	1.28744i	−0.0450383	+	0.0421038i
$936$		0			0
$937$	−21.3186			−0.696449			−0.348225	−	0.937411i	$-0.613215\pi$
$937$	−21.3186			−0.696449			−0.348225	+	0.937411i	$0.613215\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−27.9682			−0.911738			−0.455869	−	0.890047i	$-0.650671\pi$
$941$	−27.9682			−0.911738			−0.455869	+	0.890047i	$0.650671\pi$
$942$		0			0
$943$	39.4307			1.28404
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−50.2968			−1.63443			−0.817214	−	0.576335i	$-0.804482\pi$
$947$	−50.2968			−1.63443			−0.817214	+	0.576335i	$0.804482\pi$
$948$		0			0
$949$	30.7693			0.998815
$950$		0			0
$951$		0			0
$952$		0			0
$953$	15.4664			0.501006			0.250503	−	0.968116i	$-0.419404\pi$
$953$	15.4664			0.501006			0.250503	+	0.968116i	$0.419404\pi$
$954$		0			0
$955$	24.7800			0.801861
$956$		0			0
$957$		0			0
$958$		0			0
$959$			19.6508i			0.634557i
$960$		0			0
$961$	30.6317			0.988118
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	4.51403i		−	0.145312i
$966$		0			0
$967$	50.1822			1.61375			0.806876	−	0.590721i	$-0.201157\pi$
$967$	50.1822			1.61375			0.806876	+	0.590721i	$0.201157\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			30.8673i			0.990579i	0.868728	+	0.495290i	$0.164938\pi$
$971$			30.8673i			0.990579i	−0.868728	+	0.495290i	$0.835062\pi$
$972$		0			0
$973$			11.6186i			0.372476i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	31.1825			0.997617			0.498809	−	0.866712i	$-0.333771\pi$
$977$	31.1825			0.997617			0.498809	+	0.866712i	$0.333771\pi$
$978$		0			0
$979$	−2.98038			−0.0952533
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	2.61279i		−	0.0833350i	−0.999132	−	0.0416675i	$-0.986733\pi$
$983$		−	2.61279i		−	0.0833350i	0.999132	−	0.0416675i	$-0.0132670\pi$
$984$		0			0
$985$	33.3693			1.06323
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−55.1853			−1.75479
$990$		0			0
$991$			35.6087i			1.13115i	0.824698	+	0.565573i	$0.191345\pi$
$991$			35.6087i			1.13115i	−0.824698	+	0.565573i	$0.808655\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			21.8273i			0.691973i
$996$		0			0
$997$	−4.13227			−0.130870			−0.0654351	−	0.997857i	$-0.520844\pi$
$997$	−4.13227			−0.130870			−0.0654351	+	0.997857i	$0.520844\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4896.2.l.d.3025.5		18
3.2	odd	2		1632.2.l.a.1393.13		18
4.3	odd	2		1224.2.l.d.1189.17		18
8.3	odd	2		1224.2.l.c.1189.18		18
8.5	even	2		4896.2.l.c.3025.13		18
12.11	even	2		408.2.l.b.373.2	yes	18
17.16	even	2		4896.2.l.c.3025.14		18
24.5	odd	2		1632.2.l.b.1393.5		18
24.11	even	2		408.2.l.a.373.1	✓	18
51.50	odd	2		1632.2.l.b.1393.6		18
68.67	odd	2		1224.2.l.c.1189.17		18
136.67	odd	2		1224.2.l.d.1189.18		18
136.101	even	2	inner	4896.2.l.d.3025.6		18
204.203	even	2		408.2.l.a.373.2	yes	18
408.101	odd	2		1632.2.l.a.1393.14		18
408.203	even	2		408.2.l.b.373.1	yes	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
408.2.l.a.373.1	✓	18	24.11	even	2
408.2.l.a.373.2	yes	18	204.203	even	2
408.2.l.b.373.1	yes	18	408.203	even	2
408.2.l.b.373.2	yes	18	12.11	even	2
1224.2.l.c.1189.17		18	68.67	odd	2
1224.2.l.c.1189.18		18	8.3	odd	2
1224.2.l.d.1189.17		18	4.3	odd	2
1224.2.l.d.1189.18		18	136.67	odd	2
1632.2.l.a.1393.13		18	3.2	odd	2
1632.2.l.a.1393.14		18	408.101	odd	2
1632.2.l.b.1393.5		18	24.5	odd	2
1632.2.l.b.1393.6		18	51.50	odd	2
4896.2.l.c.3025.13		18	8.5	even	2
4896.2.l.c.3025.14		18	17.16	even	2
4896.2.l.d.3025.5		18	1.1	even	1	trivial
4896.2.l.d.3025.6		18	136.101	even	2	inner

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:	\( N \)	\(=\)	\( 4896 = 2^{5} \cdot 3^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4896.l (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.46472 q^{5} -3.26018i q^{7} +0.312167 q^{11} +6.39254i q^{13} +(3.01195 - 2.81570i) q^{17} +2.24353i q^{19} -7.34630i q^{23} -2.85460 q^{25} -5.39236 q^{29} -0.606910i q^{31} +4.77524i q^{35} +11.2816 q^{37} +5.36742i q^{41} -7.51199i q^{43} -4.24122 q^{47} -3.62875 q^{49} -8.91746i q^{53} -0.457236 q^{55} -5.15579i q^{59} +6.41607 q^{61} -9.36327i q^{65} +11.8427i q^{67} +6.62295i q^{71} -4.81332i q^{73} -1.01772i q^{77} +11.5068i q^{79} -13.3845i q^{83} +(-4.41165 + 4.12421i) q^{85} -9.54739 q^{89} +20.8408 q^{91} -3.28614i q^{95} -13.0210i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 4 q^{5} + 2 q^{17} + 22 q^{25} - 12 q^{29} - 16 q^{37} - 18 q^{49} + 16 q^{55} - 16 q^{61} + 16 q^{85} - 20 q^{89}+O(q^{100}) \) 18 * q + 4 * q^5 + 2 * q^17 + 22 * q^25 - 12 * q^29 - 16 * q^37 - 18 * q^49 + 16 * q^55 - 16 * q^61 + 16 * q^85 - 20 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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