LMFDB - Embedded newform 4560.2.d.j.2431.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4560,2,Mod(2431,4560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4560, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0, 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("4560.2431");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4560.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$36.4117833217$
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} + 35x^{10} + 202x^{8} + 362x^{6} + 245x^{4} + 63x^{2} + 4 $ x^12 + 35x^10 + 202x^8 + 362x^6 + 245x^4 + 63*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{11}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2431.3
Root			$5.32019i$ of defining polynomial
Character	$\chi$	$=$	4560.2431
Dual form			4560.2.d.j.2431.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} -3.21222i q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.00000 q^{5} -3.21222i q^{7} +1.00000 q^{9} +2.53893i q^{11} -1.54784i q^{13} -1.00000 q^{15} -2.67107 q^{17} +(-1.05331 + 4.22972i) q^{19} +3.21222i q^{21} +5.98643i q^{23} +1.00000 q^{25} -1.00000 q^{27} -0.925176i q^{29} -3.27279 q^{31} -2.53893i q^{33} -3.21222i q^{35} -10.3117i q^{37} +1.54784i q^{39} -9.19865i q^{41} -11.9254i q^{43} +1.00000 q^{45} +4.08676i q^{47} -3.31834 q^{49} +2.67107 q^{51} -2.52233i q^{53} +2.53893i q^{55} +(1.05331 - 4.22972i) q^{57} -9.59113 q^{59} -3.68166 q^{61} -3.21222i q^{63} -1.54784i q^{65} +14.6323 q^{67} -5.98643i q^{69} -7.94385 q^{71} +7.83724 q^{73} -1.00000 q^{75} +8.15558 q^{77} -16.2951 q^{79} +1.00000 q^{81} +11.0509i q^{83} -2.67107 q^{85} +0.925176i q^{87} +2.71016i q^{89} -4.97199 q^{91} +3.27279 q^{93} +(-1.05331 + 4.22972i) q^{95} -8.46130i q^{97} +2.53893i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{3} + 12 q^{5} + 12 q^{9}+O(q^{10}) $ 12 * q - 12 * q^3 + 12 * q^5 + 12 * q^9	$ 12 q - 12 q^{3} + 12 q^{5} + 12 q^{9} - 12 q^{15} + 4 q^{17} + 12 q^{25} - 12 q^{27} + 12 q^{31} + 12 q^{45} - 28 q^{49} - 4 q^{51} - 52 q^{59} - 56 q^{61} + 32 q^{67} - 8 q^{71} + 32 q^{73} - 12 q^{75} + 24 q^{77} + 28 q^{79} + 12 q^{81} + 4 q^{85} - 32 q^{91} - 12 q^{93}+O(q^{100}) $ 12 * q - 12 * q^3 + 12 * q^5 + 12 * q^9 - 12 * q^15 + 4 * q^17 + 12 * q^25 - 12 * q^27 + 12 * q^31 + 12 * q^45 - 28 * q^49 - 4 * q^51 - 52 * q^59 - 56 * q^61 + 32 * q^67 - 8 * q^71 + 32 * q^73 - 12 * q^75 + 24 * q^77 + 28 * q^79 + 12 * q^81 + 4 * q^85 - 32 * q^91 - 12 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4560\mathbb{Z}\right)^\times$.

$n$	$1141$	$1711$	$1921$	$2737$	$3041$
$\chi(n)$	$1$	$-1$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−1.00000			−0.577350
$3$	−1.00000			−0.577350
$4$		0			0
$5$	1.00000			0.447214
$5$	1.00000			0.447214
$6$		0			0
$7$		−	3.21222i		−	1.21410i	−0.794662	−	0.607052i	$-0.792352\pi$
$7$		−	3.21222i		−	1.21410i	0.794662	−	0.607052i	$-0.207648\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$			2.53893i			0.765515i	0.923849	+	0.382757i	$0.125025\pi$
$11$			2.53893i			0.765515i	−0.923849	+	0.382757i	$0.874975\pi$
$12$		0			0
$13$		−	1.54784i		−	0.429293i	−0.976692	−	0.214646i	$-0.931140\pi$
$13$		−	1.54784i		−	0.429293i	0.976692	−	0.214646i	$-0.0688599\pi$
$14$		0			0
$15$	−1.00000			−0.258199
$16$		0			0
$17$	−2.67107			−0.647829			−0.323914	−	0.946086i	$-0.604999\pi$
$17$	−2.67107			−0.647829			−0.323914	+	0.946086i	$0.604999\pi$
$18$		0			0
$19$	−1.05331	+	4.22972i	−0.241645	+	0.970365i
$19$	−1.05331	+	4.22972i	−0.241645	+	0.970365i
$20$		0			0
$21$			3.21222i			0.700963i
$22$		0			0
$23$			5.98643i			1.24826i	0.781322	+	0.624129i	$0.214546\pi$
$23$			5.98643i			1.24826i	−0.781322	+	0.624129i	$0.785454\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	−1.00000			−0.192450
$28$		0			0
$29$		−	0.925176i		−	0.171801i	−0.996304	−	0.0859005i	$-0.972623\pi$
$29$		−	0.925176i		−	0.171801i	0.996304	−	0.0859005i	$-0.0273767\pi$
$30$		0			0
$31$	−3.27279			−0.587810			−0.293905	−	0.955835i	$-0.594955\pi$
$31$	−3.27279			−0.587810			−0.293905	+	0.955835i	$0.594955\pi$
$32$		0			0
$33$		−	2.53893i		−	0.441970i
$34$		0			0
$35$		−	3.21222i		−	0.542964i
$36$		0			0
$37$		−	10.3117i		−	1.69522i	−0.530616	−	0.847612i	$-0.678039\pi$
$37$		−	10.3117i		−	1.69522i	0.530616	−	0.847612i	$-0.321961\pi$
$38$		0			0
$39$			1.54784i			0.247852i
$40$		0			0
$41$		−	9.19865i		−	1.43659i	−0.695740	−	0.718294i	$-0.744923\pi$
$41$		−	9.19865i		−	1.43659i	0.695740	−	0.718294i	$-0.255077\pi$
$42$		0			0
$43$		−	11.9254i		−	1.81861i	−0.416134	−	0.909303i	$-0.636615\pi$
$43$		−	11.9254i		−	1.81861i	0.416134	−	0.909303i	$-0.363385\pi$
$44$		0			0
$45$	1.00000			0.149071
$46$		0			0
$47$			4.08676i			0.596115i	0.954548	+	0.298058i	$0.0963388\pi$
$47$			4.08676i			0.596115i	−0.954548	+	0.298058i	$0.903661\pi$
$48$		0			0
$49$	−3.31834			−0.474048
$50$		0			0
$51$	2.67107			0.374024
$52$		0			0
$53$		−	2.52233i		−	0.346469i	−0.984881	−	0.173234i	$-0.944578\pi$
$53$		−	2.52233i		−	0.346469i	0.984881	−	0.173234i	$-0.0554218\pi$
$54$		0			0
$55$			2.53893i			0.342349i
$56$		0			0
$57$	1.05331	−	4.22972i	0.139514	−	0.560240i
$58$		0			0
$59$	−9.59113			−1.24866			−0.624329	−	0.781161i	$-0.714628\pi$
$59$	−9.59113			−1.24866			−0.624329	+	0.781161i	$0.714628\pi$
$60$		0			0
$61$	−3.68166			−0.471388			−0.235694	−	0.971827i	$-0.575736\pi$
$61$	−3.68166			−0.471388			−0.235694	+	0.971827i	$0.575736\pi$
$62$		0			0
$63$		−	3.21222i		−	0.404701i
$64$		0			0
$65$		−	1.54784i		−	0.191985i
$66$		0			0
$67$	14.6323			1.78762			0.893812	−	0.448441i	$-0.148021\pi$
$67$	14.6323			1.78762			0.893812	+	0.448441i	$0.148021\pi$
$68$		0			0
$69$		−	5.98643i		−	0.720682i
$70$		0			0
$71$	−7.94385			−0.942762			−0.471381	−	0.881930i	$-0.656244\pi$
$71$	−7.94385			−0.942762			−0.471381	+	0.881930i	$0.656244\pi$
$72$		0			0
$73$	7.83724			0.917280			0.458640	−	0.888622i	$-0.348337\pi$
$73$	7.83724			0.917280			0.458640	+	0.888622i	$0.348337\pi$
$74$		0			0
$75$	−1.00000			−0.115470
$76$		0			0
$77$	8.15558			0.929415
$78$		0			0
$79$	−16.2951			−1.83334			−0.916672	−	0.399640i	$-0.869135\pi$
$79$	−16.2951			−1.83334			−0.916672	+	0.399640i	$0.869135\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			11.0509i			1.21299i	0.795088	+	0.606494i	$0.207425\pi$
$83$			11.0509i			1.21299i	−0.795088	+	0.606494i	$0.792575\pi$
$84$		0			0
$85$	−2.67107			−0.289718
$86$		0			0
$87$			0.925176i			0.0991893i
$88$		0			0
$89$			2.71016i			0.287276i	0.989630	+	0.143638i	$0.0458801\pi$
$89$			2.71016i			0.287276i	−0.989630	+	0.143638i	$0.954120\pi$
$90$		0			0
$91$	−4.97199			−0.521206
$92$		0			0
$93$	3.27279			0.339372
$94$		0			0
$95$	−1.05331	+	4.22972i	−0.108067	+	0.433960i
$96$		0			0
$97$		−	8.46130i		−	0.859115i	−0.903040	−	0.429557i	$-0.858670\pi$
$97$		−	8.46130i		−	0.859115i	0.903040	−	0.429557i	$-0.141330\pi$
$98$		0			0
$99$			2.53893i			0.255172i
$100$		0			0
$101$	−13.3407			−1.32745			−0.663723	−	0.747978i	$-0.731025\pi$
$101$	−13.3407			−1.32745			−0.663723	+	0.747978i	$0.731025\pi$
$102$		0			0
$103$	−14.6323			−1.44177			−0.720883	−	0.693056i	$-0.756264\pi$
$103$	−14.6323			−1.44177			−0.720883	+	0.693056i	$0.756264\pi$
$104$		0			0
$105$			3.21222i			0.313480i
$106$		0			0
$107$	14.5762			1.40913			0.704567	−	0.709638i	$-0.251141\pi$
$107$	14.5762			1.40913			0.704567	+	0.709638i	$0.251141\pi$
$108$		0			0
$109$			10.1423i			0.971454i	0.874111	+	0.485727i	$0.161445\pi$
$109$			10.1423i			0.971454i	−0.874111	+	0.485727i	$0.838555\pi$
$110$		0			0
$111$			10.3117i			0.978739i
$112$		0			0
$113$			17.2216i			1.62007i	0.586383	+	0.810034i	$0.300551\pi$
$113$			17.2216i			1.62007i	−0.586383	+	0.810034i	$0.699449\pi$
$114$		0			0
$115$			5.98643i			0.558238i
$116$		0			0
$117$		−	1.54784i		−	0.143098i
$118$		0			0
$119$			8.58004i			0.786531i
$120$		0			0
$121$	4.55386			0.413987
$122$		0			0
$123$			9.19865i			0.829414i
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	−20.4696			−1.81638			−0.908191	−	0.418557i	$-0.862536\pi$
$127$	−20.4696			−1.81638			−0.908191	+	0.418557i	$0.862536\pi$
$128$		0			0
$129$			11.9254i			1.04997i
$130$		0			0
$131$		−	19.6268i		−	1.71481i	−0.514646	−	0.857403i	$-0.672077\pi$
$131$		−	19.6268i		−	1.71481i	0.514646	−	0.857403i	$-0.327923\pi$
$132$		0			0
$133$	13.5868	+	3.38345i	1.17812	+	0.293382i
$134$		0			0
$135$	−1.00000			−0.0860663
$136$		0			0
$137$	−18.0819			−1.54484			−0.772420	−	0.635112i	$-0.780954\pi$
$137$	−18.0819			−1.54484			−0.772420	+	0.635112i	$0.780954\pi$
$138$		0			0
$139$		−	0.978154i		−	0.0829659i	−0.999139	−	0.0414830i	$-0.986792\pi$
$139$		−	0.978154i		−	0.0829659i	0.999139	−	0.0414830i	$-0.0132082\pi$
$140$		0			0
$141$		−	4.08676i		−	0.344167i
$142$		0			0
$143$	3.92984			0.328630
$144$		0			0
$145$		−	0.925176i		−	0.0768317i
$146$		0			0
$147$	3.31834			0.273692
$148$		0			0
$149$	12.4894			1.02317			0.511587	−	0.859232i	$-0.329058\pi$
$149$	12.4894			1.02317			0.511587	+	0.859232i	$0.329058\pi$
$150$		0			0
$151$	7.14498			0.581451			0.290725	−	0.956807i	$-0.406103\pi$
$151$	7.14498			0.581451			0.290725	+	0.956807i	$0.406103\pi$
$152$		0			0
$153$	−2.67107			−0.215943
$154$		0			0
$155$	−3.27279			−0.262877
$156$		0			0
$157$	−24.2739			−1.93727			−0.968635	−	0.248487i	$-0.920067\pi$
$157$	−24.2739			−1.93727			−0.968635	+	0.248487i	$0.920067\pi$
$158$		0			0
$159$			2.52233i			0.200034i
$160$		0			0
$161$	19.2297			1.51551
$162$		0			0
$163$		−	15.2689i		−	1.19595i	−0.801514	−	0.597976i	$-0.795972\pi$
$163$		−	15.2689i		−	1.19595i	0.801514	−	0.597976i	$-0.204028\pi$
$164$		0			0
$165$		−	2.53893i		−	0.197655i
$166$		0			0
$167$	−14.4754			−1.12014			−0.560071	−	0.828445i	$-0.689226\pi$
$167$	−14.4754			−1.12014			−0.560071	+	0.828445i	$0.689226\pi$
$168$		0			0
$169$	10.6042			0.815708
$170$		0			0
$171$	−1.05331	+	4.22972i	−0.0805483	+	0.323455i
$172$		0			0
$173$		−	17.7746i		−	1.35138i	−0.737186	−	0.675690i	$-0.763846\pi$
$173$		−	17.7746i		−	1.35138i	0.737186	−	0.675690i	$-0.236154\pi$
$174$		0			0
$175$		−	3.21222i		−	0.242821i
$176$		0			0
$177$	9.59113			0.720913
$178$		0			0
$179$	−25.6372			−1.91622			−0.958109	−	0.286404i	$-0.907540\pi$
$179$	−25.6372			−1.91622			−0.958109	+	0.286404i	$0.907540\pi$
$180$		0			0
$181$			2.28875i			0.170121i	0.996376	+	0.0850607i	$0.0271084\pi$
$181$			2.28875i			0.170121i	−0.996376	+	0.0850607i	$0.972892\pi$
$182$		0			0
$183$	3.68166			0.272156
$184$		0			0
$185$		−	10.3117i		−	0.758128i
$186$		0			0
$187$		−	6.78164i		−	0.495922i
$188$		0			0
$189$			3.21222i			0.233654i
$190$		0			0
$191$			8.12456i			0.587872i	0.955825	+	0.293936i	$0.0949653\pi$
$191$			8.12456i			0.587872i	−0.955825	+	0.293936i	$0.905035\pi$
$192$		0			0
$193$		−	7.21598i		−	0.519417i	−0.965687	−	0.259709i	$-0.916373\pi$
$193$		−	7.21598i		−	0.519417i	0.965687	−	0.259709i	$-0.0836266\pi$
$194$		0			0
$195$			1.54784i			0.110843i
$196$		0			0
$197$	0.319681			0.0227763			0.0113882	−	0.999935i	$-0.496375\pi$
$197$	0.319681			0.0227763			0.0113882	+	0.999935i	$0.496375\pi$
$198$		0			0
$199$		−	18.2126i		−	1.29106i	−0.763735	−	0.645530i	$-0.776637\pi$
$199$		−	18.2126i		−	1.29106i	0.763735	−	0.645530i	$-0.223363\pi$
$200$		0			0
$201$	−14.6323			−1.03209
$202$		0			0
$203$	−2.97187			−0.208584
$204$		0			0
$205$		−	9.19865i		−	0.642461i
$206$		0			0
$207$			5.98643i			0.416086i
$208$		0			0
$209$	−10.7389	−	2.67427i	−0.742829	−	0.184983i
$210$		0			0
$211$	−14.7936			−1.01844			−0.509218	−	0.860638i	$-0.670065\pi$
$211$	−14.7936			−1.01844			−0.509218	+	0.860638i	$0.670065\pi$
$212$		0			0
$213$	7.94385			0.544304
$214$		0			0
$215$		−	11.9254i		−	0.813306i
$216$		0			0
$217$			10.5129i			0.713662i
$218$		0			0
$219$	−7.83724			−0.529592
$220$		0			0
$221$			4.13437i			0.278108i
$222$		0			0
$223$	21.7598			1.45714			0.728571	−	0.684970i	$-0.240185\pi$
$223$	21.7598			1.45714			0.728571	+	0.684970i	$0.240185\pi$
$224$		0			0
$225$	1.00000			0.0666667
$226$		0			0
$227$	1.41229			0.0937370			0.0468685	−	0.998901i	$-0.485076\pi$
$227$	1.41229			0.0937370			0.0468685	+	0.998901i	$0.485076\pi$
$228$		0			0
$229$	−16.9705			−1.12144			−0.560722	−	0.828004i	$-0.689476\pi$
$229$	−16.9705			−1.12144			−0.560722	+	0.828004i	$0.689476\pi$
$230$		0			0
$231$	−8.15558			−0.536598
$232$		0			0
$233$	−0.281071			−0.0184135			−0.00920677	−	0.999958i	$-0.502931\pi$
$233$	−0.281071			−0.0184135			−0.00920677	+	0.999958i	$0.502931\pi$
$234$		0			0
$235$			4.08676i			0.266591i
$236$		0			0
$237$	16.2951			1.05848
$238$		0			0
$239$			6.10518i			0.394911i	0.980312	+	0.197455i	$0.0632678\pi$
$239$			6.10518i			0.394911i	−0.980312	+	0.197455i	$0.936732\pi$
$240$		0			0
$241$			2.30218i			0.148296i	0.997247	+	0.0741481i	$0.0236237\pi$
$241$			2.30218i			0.148296i	−0.997247	+	0.0741481i	$0.976376\pi$
$242$		0			0
$243$	−1.00000			−0.0641500
$244$		0			0
$245$	−3.31834			−0.212001
$246$		0			0
$247$	6.54692	+	1.63034i	0.416570	+	0.103736i
$248$		0			0
$249$		−	11.0509i		−	0.700319i
$250$		0			0
$251$		−	13.8121i		−	0.871814i	−0.899992	−	0.435907i	$-0.856428\pi$
$251$		−	13.8121i		−	0.871814i	0.899992	−	0.435907i	$-0.143572\pi$
$252$		0			0
$253$	−15.1991			−0.955559
$254$		0			0
$255$	2.67107			0.167269
$256$		0			0
$257$		−	2.47301i		−	0.154262i	−0.997021	−	0.0771311i	$-0.975424\pi$
$257$		−	2.47301i		−	0.154262i	0.997021	−	0.0771311i	$-0.0245760\pi$
$258$		0			0
$259$	−33.1233			−2.05818
$260$		0			0
$261$		−	0.925176i		−	0.0572670i
$262$		0			0
$263$			24.2851i			1.49748i	0.662862	+	0.748742i	$0.269342\pi$
$263$			24.2851i			1.49748i	−0.662862	+	0.748742i	$0.730658\pi$
$264$		0			0
$265$		−	2.52233i		−	0.154945i
$266$		0			0
$267$		−	2.71016i		−	0.165859i
$268$		0			0
$269$		−	17.7800i		−	1.08407i	−0.840357	−	0.542033i	$-0.817655\pi$
$269$		−	17.7800i		−	1.08407i	0.840357	−	0.542033i	$-0.182345\pi$
$270$		0			0
$271$			12.5298i			0.761129i	0.924754	+	0.380564i	$0.124270\pi$
$271$			12.5298i			0.761129i	−0.924754	+	0.380564i	$0.875730\pi$
$272$		0			0
$273$	4.97199			0.300918
$274$		0			0
$275$			2.53893i			0.153103i
$276$		0			0
$277$	0.337218			0.0202615			0.0101307	−	0.999949i	$-0.496775\pi$
$277$	0.337218			0.0202615			0.0101307	+	0.999949i	$0.496775\pi$
$278$		0			0
$279$	−3.27279			−0.195937
$280$		0			0
$281$			24.9732i			1.48977i	0.667190	+	0.744887i	$0.267497\pi$
$281$			24.9732i			1.48977i	−0.667190	+	0.744887i	$0.732503\pi$
$282$		0			0
$283$			30.2093i			1.79576i	0.440243	+	0.897879i	$0.354892\pi$
$283$			30.2093i			1.79576i	−0.440243	+	0.897879i	$0.645108\pi$
$284$		0			0
$285$	1.05331	−	4.22972i	0.0623925	−	0.250547i
$286$		0			0
$287$	−29.5481			−1.74417
$288$		0			0
$289$	−9.86541			−0.580318
$290$		0			0
$291$			8.46130i			0.496010i
$292$		0			0
$293$			3.55233i			0.207529i	0.994602	+	0.103765i	$0.0330889\pi$
$293$			3.55233i			0.207529i	−0.994602	+	0.103765i	$0.966911\pi$
$294$		0			0
$295$	−9.59113			−0.558417
$296$		0			0
$297$		−	2.53893i		−	0.147323i
$298$		0			0
$299$	9.26601			0.535867
$300$		0			0
$301$	−38.3070			−2.20798
$302$		0			0
$303$	13.3407			0.766401
$304$		0			0
$305$	−3.68166			−0.210811
$306$		0			0
$307$	−17.5146			−0.999611			−0.499806	−	0.866138i	$-0.666595\pi$
$307$	−17.5146			−0.999611			−0.499806	+	0.866138i	$0.666595\pi$
$308$		0			0
$309$	14.6323			0.832405
$310$		0			0
$311$		−	23.1622i		−	1.31341i	−0.754148	−	0.656705i	$-0.771950\pi$
$311$		−	23.1622i		−	1.31341i	0.754148	−	0.656705i	$-0.228050\pi$
$312$		0			0
$313$	−28.3068			−1.60000			−0.799998	−	0.600003i	$-0.795166\pi$
$313$	−28.3068			−1.60000			−0.799998	+	0.600003i	$0.795166\pi$
$314$		0			0
$315$		−	3.21222i		−	0.180988i
$316$		0			0
$317$		−	30.4729i		−	1.71153i	−0.517365	−	0.855765i	$-0.673087\pi$
$317$		−	30.4729i		−	1.71153i	0.517365	−	0.855765i	$-0.326913\pi$
$318$		0			0
$319$	2.34895			0.131516
$320$		0			0
$321$	−14.5762			−0.813564
$322$		0			0
$323$	2.81345	−	11.2979i	0.156544	−	0.628630i
$324$		0			0
$325$		−	1.54784i		−	0.0858585i
$326$		0			0
$327$		−	10.1423i		−	0.560869i
$328$		0			0
$329$	13.1276			0.723746
$330$		0			0
$331$	15.3547			0.843970			0.421985	−	0.906603i	$-0.361334\pi$
$331$	15.3547			0.843970			0.421985	+	0.906603i	$0.361334\pi$
$332$		0			0
$333$		−	10.3117i		−	0.565075i
$334$		0			0
$335$	14.6323			0.799450
$336$		0			0
$337$			19.1023i			1.04057i	0.853993	+	0.520285i	$0.174174\pi$
$337$			19.1023i			1.04057i	−0.853993	+	0.520285i	$0.825826\pi$
$338$		0			0
$339$		−	17.2216i		−	0.935346i
$340$		0			0
$341$		−	8.30936i		−	0.449977i
$342$		0			0
$343$		−	11.8263i		−	0.638560i
$344$		0			0
$345$		−	5.98643i		−	0.322299i
$346$		0			0
$347$		−	4.94300i		−	0.265354i	−0.991159	−	0.132677i	$-0.957643\pi$
$347$		−	4.94300i		−	0.265354i	0.991159	−	0.132677i	$-0.0423573\pi$
$348$		0			0
$349$	−24.1725			−1.29392			−0.646961	−	0.762523i	$-0.723960\pi$
$349$	−24.1725			−1.29392			−0.646961	+	0.762523i	$0.723960\pi$
$350$		0			0
$351$			1.54784i			0.0826174i
$352$		0			0
$353$	29.8335			1.58788			0.793939	−	0.607998i	$-0.208027\pi$
$353$	29.8335			1.58788			0.793939	+	0.607998i	$0.208027\pi$
$354$		0			0
$355$	−7.94385			−0.421616
$356$		0			0
$357$		−	8.58004i		−	0.454104i
$358$		0			0
$359$			21.0003i			1.10835i	0.832399	+	0.554176i	$0.186967\pi$
$359$			21.0003i			1.10835i	−0.832399	+	0.554176i	$0.813033\pi$
$360$		0			0
$361$	−16.7811	−	8.91038i	−0.883215	−	0.468967i
$362$		0			0
$363$	−4.55386			−0.239016
$364$		0			0
$365$	7.83724			0.410220
$366$		0			0
$367$			3.70124i			0.193203i	0.995323	+	0.0966017i	$0.0307973\pi$
$367$			3.70124i			0.193203i	−0.995323	+	0.0966017i	$0.969203\pi$
$368$		0			0
$369$		−	9.19865i		−	0.478863i
$370$		0			0
$371$	−8.10227			−0.420649
$372$		0			0
$373$		−	17.7289i		−	0.917967i	−0.888445	−	0.458983i	$-0.848214\pi$
$373$		−	17.7289i		−	0.917967i	0.888445	−	0.458983i	$-0.151786\pi$
$374$		0			0
$375$	−1.00000			−0.0516398
$376$		0			0
$377$	−1.43202			−0.0737528
$378$		0			0
$379$	6.78792			0.348672			0.174336	−	0.984686i	$-0.444222\pi$
$379$	6.78792			0.348672			0.174336	+	0.984686i	$0.444222\pi$
$380$		0			0
$381$	20.4696			1.04869
$382$		0			0
$383$	20.8106			1.06337			0.531686	−	0.846941i	$-0.321559\pi$
$383$	20.8106			1.06337			0.531686	+	0.846941i	$0.321559\pi$
$384$		0			0
$385$	8.15558			0.415647
$386$		0			0
$387$		−	11.9254i		−	0.606202i
$388$		0			0
$389$	−7.55535			−0.383072			−0.191536	−	0.981486i	$-0.561347\pi$
$389$	−7.55535			−0.383072			−0.191536	+	0.981486i	$0.561347\pi$
$390$		0			0
$391$		−	15.9901i		−	0.808657i
$392$		0			0
$393$			19.6268i			0.990043i
$394$		0			0
$395$	−16.2951			−0.819896
$396$		0			0
$397$	8.03143			0.403086			0.201543	−	0.979480i	$-0.435404\pi$
$397$	8.03143			0.403086			0.201543	+	0.979480i	$0.435404\pi$
$398$		0			0
$399$	−13.5868	−	3.38345i	−0.680190	−	0.169384i
$400$		0			0
$401$		−	28.1366i		−	1.40508i	−0.711646	−	0.702538i	$-0.752050\pi$
$401$		−	28.1366i		−	1.40508i	0.711646	−	0.702538i	$-0.247950\pi$
$402$		0			0
$403$			5.06574i			0.252342i
$404$		0			0
$405$	1.00000			0.0496904
$406$		0			0
$407$	26.1805			1.29772
$408$		0			0
$409$		−	6.71758i		−	0.332163i	−0.986112	−	0.166081i	$-0.946889\pi$
$409$		−	6.71758i		−	0.332163i	0.986112	−	0.166081i	$-0.0531115\pi$
$410$		0			0
$411$	18.0819			0.891914
$412$		0			0
$413$			30.8088i			1.51600i
$414$		0			0
$415$			11.0509i			0.542465i
$416$		0			0
$417$			0.978154i			0.0479004i
$418$		0			0
$419$			22.6853i			1.10825i	0.832433	+	0.554125i	$0.186947\pi$
$419$			22.6853i			1.10825i	−0.832433	+	0.554125i	$0.813053\pi$
$420$		0			0
$421$			9.23308i			0.449993i	0.974360	+	0.224996i	$0.0722370\pi$
$421$			9.23308i			0.449993i	−0.974360	+	0.224996i	$0.927763\pi$
$422$		0			0
$423$			4.08676i			0.198705i
$424$		0			0
$425$	−2.67107			−0.129566
$426$		0			0
$427$			11.8263i			0.572315i
$428$		0			0
$429$	−3.92984			−0.189734
$430$		0			0
$431$	−18.8019			−0.905657			−0.452829	−	0.891598i	$-0.649585\pi$
$431$	−18.8019			−0.905657			−0.452829	+	0.891598i	$0.649585\pi$
$432$		0			0
$433$			30.5862i			1.46988i	0.678134	+	0.734938i	$0.262789\pi$
$433$			30.5862i			1.46988i	−0.678134	+	0.734938i	$0.737211\pi$
$434$		0			0
$435$			0.925176i			0.0443588i
$436$		0			0
$437$	−25.3209	−	6.30554i	−1.21126	−	0.301635i
$438$		0			0
$439$	−15.7175			−0.750154			−0.375077	−	0.926994i	$-0.622384\pi$
$439$	−15.7175			−0.750154			−0.375077	+	0.926994i	$0.622384\pi$
$440$		0			0
$441$	−3.31834			−0.158016
$442$		0			0
$443$		−	18.4828i		−	0.878146i	−0.898451	−	0.439073i	$-0.855307\pi$
$443$		−	18.4828i		−	0.878146i	0.898451	−	0.439073i	$-0.144693\pi$
$444$		0			0
$445$			2.71016i			0.128474i
$446$		0			0
$447$	−12.4894			−0.590730
$448$		0			0
$449$		−	38.2554i		−	1.80538i	−0.430287	−	0.902692i	$-0.641587\pi$
$449$		−	38.2554i		−	1.80538i	0.430287	−	0.902692i	$-0.358413\pi$
$450$		0			0
$451$	23.3547			1.09973
$452$		0			0
$453$	−7.14498			−0.335701
$454$		0			0
$455$	−4.97199			−0.233090
$456$		0			0
$457$	−20.4936			−0.958652			−0.479326	−	0.877637i	$-0.659119\pi$
$457$	−20.4936			−0.958652			−0.479326	+	0.877637i	$0.659119\pi$
$458$		0			0
$459$	2.67107			0.124675
$460$		0			0
$461$	32.4120			1.50958			0.754788	−	0.655969i	$-0.227740\pi$
$461$	32.4120			1.50958			0.754788	+	0.655969i	$0.227740\pi$
$462$		0			0
$463$		−	16.8208i		−	0.781729i	−0.920448	−	0.390864i	$-0.872176\pi$
$463$		−	16.8208i		−	0.781729i	0.920448	−	0.390864i	$-0.127824\pi$
$464$		0			0
$465$	3.27279			0.151772
$466$		0			0
$467$			6.42141i			0.297148i	0.988901	+	0.148574i	$0.0474682\pi$
$467$			6.42141i			0.297148i	−0.988901	+	0.148574i	$0.952532\pi$
$468$		0			0
$469$		−	47.0022i		−	2.17036i
$470$		0			0
$471$	24.2739			1.11848
$472$		0			0
$473$	30.2777			1.39217
$474$		0			0
$475$	−1.05331	+	4.22972i	−0.0483290	+	0.194073i
$476$		0			0
$477$		−	2.52233i		−	0.115490i
$478$		0			0
$479$			20.3372i			0.929232i	0.885512	+	0.464616i	$0.153808\pi$
$479$			20.3372i			0.929232i	−0.885512	+	0.464616i	$0.846192\pi$
$480$		0			0
$481$	−15.9607			−0.727747
$482$		0			0
$483$	−19.2297			−0.874982
$484$		0			0
$485$		−	8.46130i		−	0.384208i
$486$		0			0
$487$	34.8567			1.57951			0.789755	−	0.613423i	$-0.210208\pi$
$487$	34.8567			1.57951			0.789755	+	0.613423i	$0.210208\pi$
$488$		0			0
$489$			15.2689i			0.690484i
$490$		0			0
$491$		−	9.84278i		−	0.444198i	−0.975024	−	0.222099i	$-0.928709\pi$
$491$		−	9.84278i		−	0.444198i	0.975024	−	0.222099i	$-0.0712909\pi$
$492$		0			0
$493$			2.47121i			0.111298i
$494$		0			0
$495$			2.53893i			0.114116i
$496$		0			0
$497$			25.5174i			1.14461i
$498$		0			0
$499$		−	30.0869i		−	1.34687i	−0.739245	−	0.673437i	$-0.764817\pi$
$499$		−	30.0869i		−	1.34687i	0.739245	−	0.673437i	$-0.235183\pi$
$500$		0			0
$501$	14.4754			0.646714
$502$		0			0
$503$			3.61248i			0.161072i	0.996752	+	0.0805362i	$0.0256632\pi$
$503$			3.61248i			0.161072i	−0.996752	+	0.0805362i	$0.974337\pi$
$504$		0			0
$505$	−13.3407			−0.593652
$506$		0			0
$507$	−10.6042			−0.470949
$508$		0			0
$509$		−	23.6635i		−	1.04886i	−0.851452	−	0.524432i	$-0.824278\pi$
$509$		−	23.6635i		−	1.04886i	0.851452	−	0.524432i	$-0.175722\pi$
$510$		0			0
$511$		−	25.1749i		−	1.11367i
$512$		0			0
$513$	1.05331	−	4.22972i	0.0465046	−	0.186747i
$514$		0			0
$515$	−14.6323			−0.644778
$516$		0			0
$517$	−10.3760			−0.456335
$518$		0			0
$519$			17.7746i			0.780220i
$520$		0			0
$521$			14.9835i			0.656438i	0.944602	+	0.328219i	$0.106448\pi$
$521$			14.9835i			0.656438i	−0.944602	+	0.328219i	$0.893552\pi$
$522$		0			0
$523$	−4.41765			−0.193170			−0.0965852	−	0.995325i	$-0.530792\pi$
$523$	−4.41765			−0.193170			−0.0965852	+	0.995325i	$0.530792\pi$
$524$		0			0
$525$			3.21222i			0.140193i
$526$		0			0
$527$	8.74183			0.380800
$528$		0			0
$529$	−12.8374			−0.558146
$530$		0			0
$531$	−9.59113			−0.416219
$532$		0			0
$533$	−14.2380			−0.616716
$534$		0			0
$535$	14.5762			0.630184
$536$		0			0
$537$	25.6372			1.10633
$538$		0			0
$539$		−	8.42501i		−	0.362891i
$540$		0			0
$541$	−18.1358			−0.779717			−0.389859	−	0.920875i	$-0.627476\pi$
$541$	−18.1358			−0.779717			−0.389859	+	0.920875i	$0.627476\pi$
$542$		0			0
$543$		−	2.28875i		−	0.0982196i
$544$		0			0
$545$			10.1423i			0.434447i
$546$		0			0
$547$	−27.4794			−1.17493			−0.587466	−	0.809249i	$-0.699874\pi$
$547$	−27.4794			−1.17493			−0.587466	+	0.809249i	$0.699874\pi$
$548$		0			0
$549$	−3.68166			−0.157129
$550$		0			0
$551$	3.91324	+	0.974494i	0.166710	+	0.0415148i
$552$		0			0
$553$			52.3435i			2.22587i
$554$		0			0
$555$			10.3117i			0.437705i
$556$		0			0
$557$	14.2719			0.604718			0.302359	−	0.953194i	$-0.402226\pi$
$557$	14.2719			0.604718			0.302359	+	0.953194i	$0.402226\pi$
$558$		0			0
$559$	−18.4586			−0.780714
$560$		0			0
$561$			6.78164i			0.286321i
$562$		0			0
$563$	−10.1877			−0.429358			−0.214679	−	0.976685i	$-0.568871\pi$
$563$	−10.1877			−0.429358			−0.214679	+	0.976685i	$0.568871\pi$
$564$		0			0
$565$			17.2216i			0.724516i
$566$		0			0
$567$		−	3.21222i		−	0.134900i
$568$		0			0
$569$			14.9875i			0.628308i	0.949372	+	0.314154i	$0.101721\pi$
$569$			14.9875i			0.628308i	−0.949372	+	0.314154i	$0.898279\pi$
$570$		0			0
$571$		−	11.3338i		−	0.474303i	−0.971473	−	0.237151i	$-0.923786\pi$
$571$		−	11.3338i		−	0.474303i	0.971473	−	0.237151i	$-0.0762138\pi$
$572$		0			0
$573$		−	8.12456i		−	0.339408i
$574$		0			0
$575$			5.98643i			0.249651i
$576$		0			0
$577$	−7.05725			−0.293797			−0.146899	−	0.989152i	$-0.546929\pi$
$577$	−7.05725			−0.293797			−0.146899	+	0.989152i	$0.546929\pi$
$578$		0			0
$579$			7.21598i			0.299886i
$580$		0			0
$581$	35.4977			1.47269
$582$		0			0
$583$	6.40401			0.265227
$584$		0			0
$585$		−	1.54784i		−	0.0639951i
$586$		0			0
$587$		−	6.81923i		−	0.281460i	−0.990048	−	0.140730i	$-0.955055\pi$
$587$		−	6.81923i		−	0.281460i	0.990048	−	0.140730i	$-0.0449449\pi$
$588$		0			0
$589$	3.44725	−	13.8430i	0.142041	−	0.570390i
$590$		0			0
$591$	−0.319681			−0.0131499
$592$		0			0
$593$	34.6470			1.42278			0.711391	−	0.702796i	$-0.248065\pi$
$593$	34.6470			1.42278			0.711391	+	0.702796i	$0.248065\pi$
$594$		0			0
$595$			8.58004i			0.351747i
$596$		0			0
$597$			18.2126i			0.745394i
$598$		0			0
$599$	9.55670			0.390476			0.195238	−	0.980756i	$-0.437452\pi$
$599$	9.55670			0.390476			0.195238	+	0.980756i	$0.437452\pi$
$600$		0			0
$601$		−	13.3033i		−	0.542654i	−0.962487	−	0.271327i	$-0.912538\pi$
$601$		−	13.3033i		−	0.542654i	0.962487	−	0.271327i	$-0.0874624\pi$
$602$		0			0
$603$	14.6323			0.595875
$604$		0			0
$605$	4.55386			0.185141
$606$		0			0
$607$	−32.2367			−1.30845			−0.654224	−	0.756301i	$-0.727004\pi$
$607$	−32.2367			−1.30845			−0.654224	+	0.756301i	$0.727004\pi$
$608$		0			0
$609$	2.97187			0.120426
$610$		0			0
$611$	6.32564			0.255908
$612$		0			0
$613$	21.8868			0.883998			0.441999	−	0.897015i	$-0.354269\pi$
$613$	21.8868			0.883998			0.441999	+	0.897015i	$0.354269\pi$
$614$		0			0
$615$			9.19865i			0.370925i
$616$		0			0
$617$	−18.9822			−0.764196			−0.382098	−	0.924122i	$-0.624798\pi$
$617$	−18.9822			−0.764196			−0.382098	+	0.924122i	$0.624798\pi$
$618$		0			0
$619$			6.97762i			0.280454i	0.990119	+	0.140227i	$0.0447833\pi$
$619$			6.97762i			0.280454i	−0.990119	+	0.140227i	$0.955217\pi$
$620$		0			0
$621$		−	5.98643i		−	0.240227i
$622$		0			0
$623$	8.70561			0.348783
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	10.7389	+	2.67427i	0.428872	+	0.106800i
$628$		0			0
$629$			27.5431i			1.09822i
$630$		0			0
$631$			29.7974i			1.18622i	0.805123	+	0.593109i	$0.202100\pi$
$631$			29.7974i			1.18622i	−0.805123	+	0.593109i	$0.797900\pi$
$632$		0			0
$633$	14.7936			0.587994
$634$		0			0
$635$	−20.4696			−0.812310
$636$		0			0
$637$			5.13624i			0.203505i
$638$		0			0
$639$	−7.94385			−0.314254
$640$		0			0
$641$			36.0432i			1.42362i	0.702372	+	0.711810i	$0.252124\pi$
$641$			36.0432i			1.42362i	−0.702372	+	0.711810i	$0.747876\pi$
$642$		0			0
$643$		−	1.79655i		−	0.0708491i	−0.999372	−	0.0354245i	$-0.988722\pi$
$643$		−	1.79655i		−	0.0708491i	0.999372	−	0.0354245i	$-0.0112783\pi$
$644$		0			0
$645$			11.9254i			0.469562i
$646$		0			0
$647$		−	1.72893i		−	0.0679713i	−0.999422	−	0.0339857i	$-0.989180\pi$
$647$		−	1.72893i		−	0.0679713i	0.999422	−	0.0339857i	$-0.0108201\pi$
$648$		0			0
$649$		−	24.3512i		−	0.955866i
$650$		0			0
$651$		−	10.5129i		−	0.412033i
$652$		0			0
$653$	15.4671			0.605274			0.302637	−	0.953106i	$-0.402133\pi$
$653$	15.4671			0.605274			0.302637	+	0.953106i	$0.402133\pi$
$654$		0			0
$655$		−	19.6268i		−	0.766884i
$656$		0			0
$657$	7.83724			0.305760
$658$		0			0
$659$	−34.7171			−1.35239			−0.676193	−	0.736724i	$-0.736372\pi$
$659$	−34.7171			−1.35239			−0.676193	+	0.736724i	$0.736372\pi$
$660$		0			0
$661$			40.1807i			1.56285i	0.624000	+	0.781424i	$0.285506\pi$
$661$			40.1807i			1.56285i	−0.624000	+	0.781424i	$0.714494\pi$
$662$		0			0
$663$		−	4.13437i		−	0.160566i
$664$		0			0
$665$	13.5868	+	3.38345i	0.526873	+	0.131204i
$666$		0			0
$667$	5.53850			0.214452
$668$		0			0
$669$	−21.7598			−0.841282
$670$		0			0
$671$		−	9.34746i		−	0.360855i
$672$		0			0
$673$		−	8.94691i		−	0.344878i	−0.985020	−	0.172439i	$-0.944835\pi$
$673$		−	8.94691i		−	0.344878i	0.985020	−	0.172439i	$-0.0551648\pi$
$674$		0			0
$675$	−1.00000			−0.0384900
$676$		0			0
$677$			40.4341i			1.55401i	0.629495	+	0.777005i	$0.283262\pi$
$677$			40.4341i			1.55401i	−0.629495	+	0.777005i	$0.716738\pi$
$678$		0			0
$679$	−27.1795			−1.04305
$680$		0			0
$681$	−1.41229			−0.0541191
$682$		0			0
$683$	42.3252			1.61953			0.809765	−	0.586755i	$-0.199595\pi$
$683$	42.3252			1.61953			0.809765	+	0.586755i	$0.199595\pi$
$684$		0			0
$685$	−18.0819			−0.690874
$686$		0			0
$687$	16.9705			0.647466
$688$		0			0
$689$	−3.90415			−0.148736
$690$		0			0
$691$			11.8439i			0.450562i	0.974294	+	0.225281i	$0.0723300\pi$
$691$			11.8439i			0.450562i	−0.974294	+	0.225281i	$0.927670\pi$
$692$		0			0
$693$	8.15558			0.309805
$694$		0			0
$695$		−	0.978154i		−	0.0371035i
$696$		0			0
$697$			24.5702i			0.930662i
$698$		0			0
$699$	0.281071			0.0106311
$700$		0			0
$701$	−7.57075			−0.285943			−0.142972	−	0.989727i	$-0.545666\pi$
$701$	−7.57075			−0.285943			−0.142972	+	0.989727i	$0.545666\pi$
$702$		0			0
$703$	43.6154	+	10.8613i	1.64499	+	0.409643i
$704$		0			0
$705$		−	4.08676i		−	0.153916i
$706$		0			0
$707$			42.8531i			1.61166i
$708$		0			0
$709$	−15.7826			−0.592728			−0.296364	−	0.955075i	$-0.595774\pi$
$709$	−15.7826			−0.592728			−0.296364	+	0.955075i	$0.595774\pi$
$710$		0			0
$711$	−16.2951			−0.611115
$712$		0			0
$713$		−	19.5923i		−	0.733738i
$714$		0			0
$715$	3.92984			0.146968
$716$		0			0
$717$		−	6.10518i		−	0.228002i
$718$		0			0
$719$			5.52963i			0.206220i	0.994670	+	0.103110i	$0.0328794\pi$
$719$			5.52963i			0.206220i	−0.994670	+	0.103110i	$0.967121\pi$
$720$		0			0
$721$			47.0022i			1.75045i
$722$		0			0
$723$		−	2.30218i		−	0.0856188i
$724$		0			0
$725$		−	0.925176i		−	0.0343602i
$726$		0			0
$727$		−	49.6316i		−	1.84073i	−0.391056	−	0.920367i	$-0.627890\pi$
$727$		−	49.6316i		−	1.84073i	0.391056	−	0.920367i	$-0.372110\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$			31.8535i			1.17815i
$732$		0			0
$733$	−42.1979			−1.55862			−0.779308	−	0.626641i	$-0.784429\pi$
$733$	−42.1979			−1.55862			−0.779308	+	0.626641i	$0.784429\pi$
$734$		0			0
$735$	3.31834			0.122399
$736$		0			0
$737$			37.1504i			1.36845i
$738$		0			0
$739$		−	11.8402i		−	0.435547i	−0.975999	−	0.217774i	$-0.930121\pi$
$739$		−	11.8402i		−	0.435547i	0.975999	−	0.217774i	$-0.0698794\pi$
$740$		0			0
$741$	−6.54692	−	1.63034i	−0.240507	−	0.0598922i
$742$		0			0
$743$	22.2273			0.815440			0.407720	−	0.913107i	$-0.366324\pi$
$743$	22.2273			0.815440			0.407720	+	0.913107i	$0.366324\pi$
$744$		0			0
$745$	12.4894			0.457577
$746$		0			0
$747$			11.0509i			0.404330i
$748$		0			0
$749$		−	46.8219i		−	1.71083i
$750$		0			0
$751$	14.2573			0.520257			0.260129	−	0.965574i	$-0.416235\pi$
$751$	14.2573			0.520257			0.260129	+	0.965574i	$0.416235\pi$
$752$		0			0
$753$			13.8121i			0.503342i
$754$		0			0
$755$	7.14498			0.260033
$756$		0			0
$757$	25.3860			0.922669			0.461335	−	0.887226i	$-0.347371\pi$
$757$	25.3860			0.922669			0.461335	+	0.887226i	$0.347371\pi$
$758$		0			0
$759$	15.1991			0.551692
$760$		0			0
$761$	20.7027			0.750470			0.375235	−	0.926930i	$-0.377562\pi$
$761$	20.7027			0.750470			0.375235	+	0.926930i	$0.377562\pi$
$762$		0			0
$763$	32.5792			1.17945
$764$		0			0
$765$	−2.67107			−0.0965726
$766$		0			0
$767$			14.8455i			0.536040i
$768$		0			0
$769$	−7.80505			−0.281457			−0.140729	−	0.990048i	$-0.544944\pi$
$769$	−7.80505			−0.281457			−0.140729	+	0.990048i	$0.544944\pi$
$770$		0			0
$771$			2.47301i			0.0890634i
$772$		0			0
$773$		−	8.05834i		−	0.289838i	−0.989443	−	0.144919i	$-0.953708\pi$
$773$		−	8.05834i		−	0.289838i	0.989443	−	0.144919i	$-0.0462922\pi$
$774$		0			0
$775$	−3.27279			−0.117562
$776$		0			0
$777$	33.1233			1.18829
$778$		0			0
$779$	38.9077	+	9.68899i	1.39401	+	0.347144i
$780$		0			0
$781$		−	20.1688i		−	0.721698i
$782$		0			0
$783$			0.925176i			0.0330631i
$784$		0			0
$785$	−24.2739			−0.866374
$786$		0			0
$787$	42.0475			1.49883			0.749417	−	0.662099i	$-0.230334\pi$
$787$	42.0475			1.49883			0.749417	+	0.662099i	$0.230334\pi$
$788$		0			0
$789$		−	24.2851i		−	0.864572i
$790$		0			0
$791$	55.3194			1.96693
$792$		0			0
$793$			5.69861i			0.202364i
$794$		0			0
$795$			2.52233i			0.0894578i
$796$		0			0
$797$		−	34.5098i		−	1.22240i	−0.791477	−	0.611199i	$-0.790687\pi$
$797$		−	34.5098i		−	1.22240i	0.791477	−	0.611199i	$-0.209313\pi$
$798$		0			0
$799$		−	10.9160i		−	0.386181i
$800$		0			0
$801$			2.71016i			0.0957586i
$802$		0			0
$803$			19.8982i			0.702191i
$804$		0			0
$805$	19.2297			0.677758
$806$		0			0
$807$			17.7800i			0.625886i
$808$		0			0
$809$	42.5441			1.49577			0.747886	−	0.663828i	$-0.231069\pi$
$809$	42.5441			1.49577			0.747886	+	0.663828i	$0.231069\pi$
$810$		0			0
$811$	−0.649193			−0.0227963			−0.0113981	−	0.999935i	$-0.503628\pi$
$811$	−0.649193			−0.0227963			−0.0113981	+	0.999935i	$0.503628\pi$
$812$		0			0
$813$		−	12.5298i		−	0.439438i
$814$		0			0
$815$		−	15.2689i		−	0.534846i
$816$		0			0
$817$	50.4411	+	12.5611i	1.76471	+	0.439457i
$818$		0			0
$819$	−4.97199			−0.173735
$820$		0			0
$821$	−31.1750			−1.08802			−0.544008	−	0.839080i	$-0.683094\pi$
$821$	−31.1750			−1.08802			−0.544008	+	0.839080i	$0.683094\pi$
$822$		0			0
$823$		−	12.1123i		−	0.422207i	−0.977464	−	0.211103i	$-0.932294\pi$
$823$		−	12.1123i		−	0.422207i	0.977464	−	0.211103i	$-0.0677057\pi$
$824$		0			0
$825$		−	2.53893i		−	0.0883940i
$826$		0			0
$827$	0.573827			0.0199539			0.00997696	−	0.999950i	$-0.496824\pi$
$827$	0.573827			0.0199539			0.00997696	+	0.999950i	$0.496824\pi$
$828$		0			0
$829$		−	3.66481i		−	0.127284i	−0.997973	−	0.0636421i	$-0.979728\pi$
$829$		−	3.66481i		−	0.127284i	0.997973	−	0.0636421i	$-0.0202716\pi$
$830$		0			0
$831$	−0.337218			−0.0116980
$832$		0			0
$833$	8.86350			0.307102
$834$		0			0
$835$	−14.4754			−0.500942
$836$		0			0
$837$	3.27279			0.113124
$838$		0			0
$839$	−54.6910			−1.88814			−0.944071	−	0.329742i	$-0.893038\pi$
$839$	−54.6910			−1.88814			−0.944071	+	0.329742i	$0.893038\pi$
$840$		0			0
$841$	28.1440			0.970484
$842$		0			0
$843$		−	24.9732i		−	0.860122i
$844$		0			0
$845$	10.6042			0.364796
$846$		0			0
$847$		−	14.6280i		−	0.502623i
$848$		0			0
$849$		−	30.2093i		−	1.03678i
$850$		0			0
$851$	61.7300			2.11608
$852$		0			0
$853$	0.337218			0.0115461			0.00577306	−	0.999983i	$-0.498162\pi$
$853$	0.337218			0.0115461			0.00577306	+	0.999983i	$0.498162\pi$
$854$		0			0
$855$	−1.05331	+	4.22972i	−0.0360223	+	0.144653i
$856$		0			0
$857$			48.8699i			1.66936i	0.550732	+	0.834682i	$0.314349\pi$
$857$			48.8699i			1.66936i	−0.550732	+	0.834682i	$0.685651\pi$
$858$		0			0
$859$		−	17.5842i		−	0.599964i	−0.953945	−	0.299982i	$-0.903019\pi$
$859$		−	17.5842i		−	0.599964i	0.953945	−	0.299982i	$-0.0969807\pi$
$860$		0			0
$861$	29.5481			1.00700
$862$		0			0
$863$	4.17839			0.142234			0.0711170	−	0.997468i	$-0.477344\pi$
$863$	4.17839			0.142234			0.0711170	+	0.997468i	$0.477344\pi$
$864$		0			0
$865$		−	17.7746i		−	0.604356i
$866$		0			0
$867$	9.86541			0.335047
$868$		0			0
$869$		−	41.3721i		−	1.40345i
$870$		0			0
$871$		−	22.6485i		−	0.767414i
$872$		0			0
$873$		−	8.46130i		−	0.286372i
$874$		0			0
$875$		−	3.21222i		−	0.108593i
$876$		0			0
$877$		−	28.9762i		−	0.978457i	−0.872156	−	0.489229i	$-0.837278\pi$
$877$		−	28.9762i		−	0.978457i	0.872156	−	0.489229i	$-0.162722\pi$
$878$		0			0
$879$		−	3.55233i		−	0.119817i
$880$		0			0
$881$	−15.9701			−0.538047			−0.269024	−	0.963134i	$-0.586701\pi$
$881$	−15.9701			−0.538047			−0.269024	+	0.963134i	$0.586701\pi$
$882$		0			0
$883$		−	34.4453i		−	1.15918i	−0.814909	−	0.579589i	$-0.803213\pi$
$883$		−	34.4453i		−	1.15918i	0.814909	−	0.579589i	$-0.196787\pi$
$884$		0			0
$885$	9.59113			0.322402
$886$		0			0
$887$	−4.13999			−0.139007			−0.0695036	−	0.997582i	$-0.522142\pi$
$887$	−4.13999			−0.139007			−0.0695036	+	0.997582i	$0.522142\pi$
$888$		0			0
$889$			65.7527i			2.20528i
$890$		0			0
$891$			2.53893i			0.0850572i
$892$		0			0
$893$	−17.2859	−	4.30461i	−0.578449	−	0.144048i
$894$		0			0
$895$	−25.6372			−0.856959
$896$		0			0
$897$	−9.26601			−0.309383
$898$		0			0
$899$			3.02790i			0.100986i
$900$		0			0
$901$			6.73731i			0.224452i
$902$		0			0
$903$	38.3070			1.27478
$904$		0			0
$905$			2.28875i			0.0760806i
$906$		0			0
$907$	22.7881			0.756665			0.378333	−	0.925670i	$-0.376498\pi$
$907$	22.7881			0.756665			0.378333	+	0.925670i	$0.376498\pi$
$908$		0			0
$909$	−13.3407			−0.442482
$910$		0			0
$911$	21.2496			0.704030			0.352015	−	0.935994i	$-0.385497\pi$
$911$	21.2496			0.704030			0.352015	+	0.935994i	$0.385497\pi$
$912$		0			0
$913$	−28.0573			−0.928561
$914$		0			0
$915$	3.68166			0.121712
$916$		0			0
$917$	−63.0457			−2.08195
$918$		0			0
$919$		−	37.4400i		−	1.23503i	−0.786558	−	0.617516i	$-0.788139\pi$
$919$		−	37.4400i		−	1.23503i	0.786558	−	0.617516i	$-0.211861\pi$
$920$		0			0
$921$	17.5146			0.577126
$922$		0			0
$923$			12.2958i			0.404721i
$924$		0			0
$925$		−	10.3117i		−	0.339045i
$926$		0			0
$927$	−14.6323			−0.480589
$928$		0			0
$929$	−22.2199			−0.729012			−0.364506	−	0.931201i	$-0.618762\pi$
$929$	−22.2199			−0.729012			−0.364506	+	0.931201i	$0.618762\pi$
$930$		0			0
$931$	3.49523	−	14.0357i	0.114551	−	0.460000i
$932$		0			0
$933$			23.1622i			0.758297i
$934$		0			0
$935$		−	6.78164i		−	0.221783i
$936$		0			0
$937$	6.44618			0.210587			0.105294	−	0.994441i	$-0.466422\pi$
$937$	6.44618			0.210587			0.105294	+	0.994441i	$0.466422\pi$
$938$		0			0
$939$	28.3068			0.923758
$940$		0			0
$941$			7.11955i			0.232091i	0.993244	+	0.116045i	$0.0370218\pi$
$941$			7.11955i			0.232091i	−0.993244	+	0.116045i	$0.962978\pi$
$942$		0			0
$943$	55.0671			1.79323
$944$		0			0
$945$			3.21222i			0.104493i
$946$		0			0
$947$		−	30.9148i		−	1.00460i	−0.864694	−	0.502299i	$-0.832488\pi$
$947$		−	30.9148i		−	1.00460i	0.864694	−	0.502299i	$-0.167512\pi$
$948$		0			0
$949$		−	12.1308i		−	0.393781i
$950$		0			0
$951$			30.4729i			0.988152i
$952$		0			0
$953$		−	9.97245i		−	0.323039i	−0.986870	−	0.161520i	$-0.948360\pi$
$953$		−	9.97245i		−	0.323039i	0.986870	−	0.161520i	$-0.0516395\pi$
$954$		0			0
$955$			8.12456i			0.262905i
$956$		0			0
$957$	−2.34895			−0.0759309
$958$		0			0
$959$			58.0830i			1.87560i
$960$		0			0
$961$	−20.2889			−0.654479
$962$		0			0
$963$	14.5762			0.469711
$964$		0			0
$965$		−	7.21598i		−	0.232291i
$966$		0			0
$967$		−	35.2362i		−	1.13312i	−0.824020	−	0.566561i	$-0.808274\pi$
$967$		−	35.2362i		−	1.13312i	0.824020	−	0.566561i	$-0.191726\pi$
$968$		0			0
$969$	−2.81345	+	11.2979i	−0.0903810	+	0.362940i
$970$		0			0
$971$	13.0751			0.419600			0.209800	−	0.977744i	$-0.432719\pi$
$971$	13.0751			0.419600			0.209800	+	0.977744i	$0.432719\pi$
$972$		0			0
$973$	−3.14204			−0.100729
$974$		0			0
$975$			1.54784i			0.0495704i
$976$		0			0
$977$			9.04169i			0.289269i	0.989485	+	0.144635i	$0.0462007\pi$
$977$			9.04169i			0.289269i	−0.989485	+	0.144635i	$0.953799\pi$
$978$		0			0
$979$	−6.88088			−0.219914
$980$		0			0
$981$			10.1423i			0.323818i
$982$		0			0
$983$	56.8593			1.81353			0.906764	−	0.421638i	$-0.138545\pi$
$983$	56.8593			1.81353			0.906764	+	0.421638i	$0.138545\pi$
$984$		0			0
$985$	0.319681			0.0101859
$986$		0			0
$987$	−13.1276			−0.417855
$988$		0			0
$989$	71.3906			2.27009
$990$		0			0
$991$	−26.0365			−0.827077			−0.413539	−	0.910487i	$-0.635707\pi$
$991$	−26.0365			−0.827077			−0.413539	+	0.910487i	$0.635707\pi$
$992$		0			0
$993$	−15.3547			−0.487266
$994$		0			0
$995$		−	18.2126i		−	0.577379i
$996$		0			0
$997$	8.19864			0.259654			0.129827	−	0.991537i	$-0.458558\pi$
$997$	8.19864			0.259654			0.129827	+	0.991537i	$0.458558\pi$
$998$		0			0
$999$			10.3117i			0.326246i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4560.2.d.j.2431.3	✓	12
4.3	odd	2		4560.2.d.l.2431.10	yes	12
19.18	odd	2		4560.2.d.l.2431.3	yes	12
76.75	even	2	inner	4560.2.d.j.2431.10	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4560.2.d.j.2431.3	✓	12	1.1	even	1	trivial
4560.2.d.j.2431.10	yes	12	76.75	even	2	inner
4560.2.d.l.2431.3	yes	12	19.18	odd	2
4560.2.d.l.2431.10	yes	12	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4560.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} -3.21222i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.00000 q^{5} -3.21222i q^{7} +1.00000 q^{9} +2.53893i q^{11} -1.54784i q^{13} -1.00000 q^{15} -2.67107 q^{17} +(-1.05331 + 4.22972i) q^{19} +3.21222i q^{21} +5.98643i q^{23} +1.00000 q^{25} -1.00000 q^{27} -0.925176i q^{29} -3.27279 q^{31} -2.53893i q^{33} -3.21222i q^{35} -10.3117i q^{37} +1.54784i q^{39} -9.19865i q^{41} -11.9254i q^{43} +1.00000 q^{45} +4.08676i q^{47} -3.31834 q^{49} +2.67107 q^{51} -2.52233i q^{53} +2.53893i q^{55} +(1.05331 - 4.22972i) q^{57} -9.59113 q^{59} -3.68166 q^{61} -3.21222i q^{63} -1.54784i q^{65} +14.6323 q^{67} -5.98643i q^{69} -7.94385 q^{71} +7.83724 q^{73} -1.00000 q^{75} +8.15558 q^{77} -16.2951 q^{79} +1.00000 q^{81} +11.0509i q^{83} -2.67107 q^{85} +0.925176i q^{87} +2.71016i q^{89} -4.97199 q^{91} +3.27279 q^{93} +(-1.05331 + 4.22972i) q^{95} -8.46130i q^{97} +2.53893i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{3} + 12 q^{5} + 12 q^{9}+O(q^{10}) \) 12 * q - 12 * q^3 + 12 * q^5 + 12 * q^9	\( 12 q - 12 q^{3} + 12 q^{5} + 12 q^{9} - 12 q^{15} + 4 q^{17} + 12 q^{25} - 12 q^{27} + 12 q^{31} + 12 q^{45} - 28 q^{49} - 4 q^{51} - 52 q^{59} - 56 q^{61} + 32 q^{67} - 8 q^{71} + 32 q^{73} - 12 q^{75} + 24 q^{77} + 28 q^{79} + 12 q^{81} + 4 q^{85} - 32 q^{91} - 12 q^{93}+O(q^{100}) \) 12 * q - 12 * q^3 + 12 * q^5 + 12 * q^9 - 12 * q^15 + 4 * q^17 + 12 * q^25 - 12 * q^27 + 12 * q^31 + 12 * q^45 - 28 * q^49 - 4 * q^51 - 52 * q^59 - 56 * q^61 + 32 * q^67 - 8 * q^71 + 32 * q^73 - 12 * q^75 + 24 * q^77 + 28 * q^79 + 12 * q^81 + 4 * q^85 - 32 * q^91 - 12 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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