LMFDB - Embedded newform 560.2.g.b.449.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [560,2,Mod(449,560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(560, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("560.449");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 560 = 2^{4} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	560.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:	$4.47162251319$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 35)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	560.449
Dual form			560.2.g.b.449.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +(-2.00000 - 1.00000i) q^{5} +1.00000i q^{7} +2.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +(-2.00000 - 1.00000i) q^{5} +1.00000i q^{7} +2.00000 q^{9} +3.00000 q^{11} -1.00000i q^{13} +(1.00000 - 2.00000i) q^{15} +7.00000i q^{17} -1.00000 q^{21} +6.00000i q^{23} +(3.00000 + 4.00000i) q^{25} +5.00000i q^{27} +5.00000 q^{29} -2.00000 q^{31} +3.00000i q^{33} +(1.00000 - 2.00000i) q^{35} +2.00000i q^{37} +1.00000 q^{39} +2.00000 q^{41} -4.00000i q^{43} +(-4.00000 - 2.00000i) q^{45} +3.00000i q^{47} -1.00000 q^{49} -7.00000 q^{51} -6.00000i q^{53} +(-6.00000 - 3.00000i) q^{55} +10.0000 q^{59} -8.00000 q^{61} +2.00000i q^{63} +(-1.00000 + 2.00000i) q^{65} -2.00000i q^{67} -6.00000 q^{69} +8.00000 q^{71} -6.00000i q^{73} +(-4.00000 + 3.00000i) q^{75} +3.00000i q^{77} -5.00000 q^{79} +1.00000 q^{81} -4.00000i q^{83} +(7.00000 - 14.0000i) q^{85} +5.00000i q^{87} +1.00000 q^{91} -2.00000i q^{93} +7.00000i q^{97} +6.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{5} + 4 q^{9}+O(q^{10}) $ 2 * q - 4 * q^5 + 4 * q^9	$ 2 q - 4 q^{5} + 4 q^{9} + 6 q^{11} + 2 q^{15} - 2 q^{21} + 6 q^{25} + 10 q^{29} - 4 q^{31} + 2 q^{35} + 2 q^{39} + 4 q^{41} - 8 q^{45} - 2 q^{49} - 14 q^{51} - 12 q^{55} + 20 q^{59} - 16 q^{61} - 2 q^{65} - 12 q^{69} + 16 q^{71} - 8 q^{75} - 10 q^{79} + 2 q^{81} + 14 q^{85} + 2 q^{91} + 12 q^{99}+O(q^{100}) $ 2 * q - 4 * q^5 + 4 * q^9 + 6 * q^11 + 2 * q^15 - 2 * q^21 + 6 * q^25 + 10 * q^29 - 4 * q^31 + 2 * q^35 + 2 * q^39 + 4 * q^41 - 8 * q^45 - 2 * q^49 - 14 * q^51 - 12 * q^55 + 20 * q^59 - 16 * q^61 - 2 * q^65 - 12 * q^69 + 16 * q^71 - 8 * q^75 - 10 * q^79 + 2 * q^81 + 14 * q^85 + 2 * q^91 + 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/560\mathbb{Z}\right)^\times$.

$n$	$241$	$337$	$351$	$421$
$\chi(n)$	$1$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$			1.00000i			0.577350i	0.957427	+	0.288675i	$0.0932147\pi$
$3$			1.00000i			0.577350i	−0.957427	+	0.288675i	$0.906785\pi$
$4$		0			0
$5$	−2.00000	−	1.00000i	−0.894427	−	0.447214i
$5$	−2.00000	−	1.00000i	−0.894427	−	0.447214i
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$	2.00000			0.666667
$10$		0			0
$11$	3.00000			0.904534			0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000			0.904534			0.452267	+	0.891883i	$0.350615\pi$
$12$		0			0
$13$		−	1.00000i		−	0.277350i	−0.990338	−	0.138675i	$-0.955716\pi$
$13$		−	1.00000i		−	0.277350i	0.990338	−	0.138675i	$-0.0442844\pi$
$14$		0			0
$15$	1.00000	−	2.00000i	0.258199	−	0.516398i
$16$		0			0
$17$			7.00000i			1.69775i	0.528594	+	0.848875i	$0.322719\pi$
$17$			7.00000i			1.69775i	−0.528594	+	0.848875i	$0.677281\pi$
$18$		0			0
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		0			0
$21$	−1.00000			−0.218218
$22$		0			0
$23$			6.00000i			1.25109i	0.780189	+	0.625543i	$0.215123\pi$
$23$			6.00000i			1.25109i	−0.780189	+	0.625543i	$0.784877\pi$
$24$		0			0
$25$	3.00000	+	4.00000i	0.600000	+	0.800000i
$26$		0			0
$27$			5.00000i			0.962250i
$28$		0			0
$29$	5.00000			0.928477			0.464238	−	0.885710i	$-0.346328\pi$
$29$	5.00000			0.928477			0.464238	+	0.885710i	$0.346328\pi$
$30$		0			0
$31$	−2.00000			−0.359211			−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000			−0.359211			−0.179605	+	0.983739i	$0.557482\pi$
$32$		0			0
$33$			3.00000i			0.522233i
$34$		0			0
$35$	1.00000	−	2.00000i	0.169031	−	0.338062i
$36$		0			0
$37$			2.00000i			0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$			2.00000i			0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$		0			0
$39$	1.00000			0.160128
$40$		0			0
$41$	2.00000			0.312348			0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000			0.312348			0.156174	+	0.987730i	$0.450084\pi$
$42$		0			0
$43$		−	4.00000i		−	0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$		−	4.00000i		−	0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$		0			0
$45$	−4.00000	−	2.00000i	−0.596285	−	0.298142i
$46$		0			0
$47$			3.00000i			0.437595i	0.975770	+	0.218797i	$0.0702134\pi$
$47$			3.00000i			0.437595i	−0.975770	+	0.218797i	$0.929787\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	−7.00000			−0.980196
$52$		0			0
$53$		−	6.00000i		−	0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$		−	6.00000i		−	0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$		0			0
$55$	−6.00000	−	3.00000i	−0.809040	−	0.404520i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	10.0000			1.30189			0.650945	−	0.759125i	$-0.274373\pi$
$59$	10.0000			1.30189			0.650945	+	0.759125i	$0.274373\pi$
$60$		0			0
$61$	−8.00000			−1.02430			−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000			−1.02430			−0.512148	+	0.858898i	$0.671150\pi$
$62$		0			0
$63$			2.00000i			0.251976i
$64$		0			0
$65$	−1.00000	+	2.00000i	−0.124035	+	0.248069i
$66$		0			0
$67$		−	2.00000i		−	0.244339i	−0.992509	−	0.122169i	$-0.961015\pi$
$67$		−	2.00000i		−	0.244339i	0.992509	−	0.122169i	$-0.0389851\pi$
$68$		0			0
$69$	−6.00000			−0.722315
$70$		0			0
$71$	8.00000			0.949425			0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000			0.949425			0.474713	+	0.880141i	$0.342552\pi$
$72$		0			0
$73$		−	6.00000i		−	0.702247i	−0.936329	−	0.351123i	$-0.885800\pi$
$73$		−	6.00000i		−	0.702247i	0.936329	−	0.351123i	$-0.114200\pi$
$74$		0			0
$75$	−4.00000	+	3.00000i	−0.461880	+	0.346410i
$76$		0			0
$77$			3.00000i			0.341882i
$78$		0			0
$79$	−5.00000			−0.562544			−0.281272	−	0.959628i	$-0.590756\pi$
$79$	−5.00000			−0.562544			−0.281272	+	0.959628i	$0.590756\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	4.00000i		−	0.439057i	−0.975606	−	0.219529i	$-0.929548\pi$
$83$		−	4.00000i		−	0.439057i	0.975606	−	0.219529i	$-0.0704519\pi$
$84$		0			0
$85$	7.00000	−	14.0000i	0.759257	−	1.51851i
$86$		0			0
$87$			5.00000i			0.536056i
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$	1.00000			0.104828
$92$		0			0
$93$		−	2.00000i		−	0.207390i
$94$		0			0
$95$		0			0
$96$		0			0
$97$			7.00000i			0.710742i	0.934725	+	0.355371i	$0.115646\pi$
$97$			7.00000i			0.710742i	−0.934725	+	0.355371i	$0.884354\pi$
$98$		0			0
$99$	6.00000			0.603023
$100$		0			0
$101$	12.0000			1.19404			0.597022	−	0.802225i	$-0.296350\pi$
$101$	12.0000			1.19404			0.597022	+	0.802225i	$0.296350\pi$
$102$		0			0
$103$		−	19.0000i		−	1.87213i	−0.351833	−	0.936063i	$-0.614441\pi$
$103$		−	19.0000i		−	1.87213i	0.351833	−	0.936063i	$-0.385559\pi$
$104$		0			0
$105$	2.00000	+	1.00000i	0.195180	+	0.0975900i
$106$		0			0
$107$			8.00000i			0.773389i	0.922208	+	0.386695i	$0.126383\pi$
$107$			8.00000i			0.773389i	−0.922208	+	0.386695i	$0.873617\pi$
$108$		0			0
$109$	−5.00000			−0.478913			−0.239457	−	0.970907i	$-0.576969\pi$
$109$	−5.00000			−0.478913			−0.239457	+	0.970907i	$0.576969\pi$
$110$		0			0
$111$	−2.00000			−0.189832
$112$		0			0
$113$		−	6.00000i		−	0.564433i	−0.959351	−	0.282216i	$-0.908930\pi$
$113$		−	6.00000i		−	0.564433i	0.959351	−	0.282216i	$-0.0910696\pi$
$114$		0			0
$115$	6.00000	−	12.0000i	0.559503	−	1.11901i
$116$		0			0
$117$		−	2.00000i		−	0.184900i
$118$		0			0
$119$	−7.00000			−0.641689
$120$		0			0
$121$	−2.00000			−0.181818
$122$		0			0
$123$			2.00000i			0.180334i
$124$		0			0
$125$	−2.00000	−	11.0000i	−0.178885	−	0.983870i
$126$		0			0
$127$		−	2.00000i		−	0.177471i	−0.996055	−	0.0887357i	$-0.971717\pi$
$127$		−	2.00000i		−	0.177471i	0.996055	−	0.0887357i	$-0.0282826\pi$
$128$		0			0
$129$	4.00000			0.352180
$130$		0			0
$131$	−22.0000			−1.92215			−0.961074	−	0.276289i	$-0.910895\pi$
$131$	−22.0000			−1.92215			−0.961074	+	0.276289i	$0.910895\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	5.00000	−	10.0000i	0.430331	−	0.860663i
$136$		0			0
$137$			12.0000i			1.02523i	0.858619	+	0.512615i	$0.171323\pi$
$137$			12.0000i			1.02523i	−0.858619	+	0.512615i	$0.828677\pi$
$138$		0			0
$139$	−10.0000			−0.848189			−0.424094	−	0.905618i	$-0.639408\pi$
$139$	−10.0000			−0.848189			−0.424094	+	0.905618i	$0.639408\pi$
$140$		0			0
$141$	−3.00000			−0.252646
$142$		0			0
$143$		−	3.00000i		−	0.250873i
$144$		0			0
$145$	−10.0000	−	5.00000i	−0.830455	−	0.415227i
$146$		0			0
$147$		−	1.00000i		−	0.0824786i
$148$		0			0
$149$	−10.0000			−0.819232			−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000			−0.819232			−0.409616	+	0.912258i	$0.634337\pi$
$150$		0			0
$151$	13.0000			1.05792			0.528962	−	0.848645i	$-0.322581\pi$
$151$	13.0000			1.05792			0.528962	+	0.848645i	$0.322581\pi$
$152$		0			0
$153$			14.0000i			1.13183i
$154$		0			0
$155$	4.00000	+	2.00000i	0.321288	+	0.160644i
$156$		0			0
$157$		−	18.0000i		−	1.43656i	−0.695756	−	0.718278i	$-0.744931\pi$
$157$		−	18.0000i		−	1.43656i	0.695756	−	0.718278i	$-0.255069\pi$
$158$		0			0
$159$	6.00000			0.475831
$160$		0			0
$161$	−6.00000			−0.472866
$162$		0			0
$163$		−	14.0000i		−	1.09656i	−0.836293	−	0.548282i	$-0.815282\pi$
$163$		−	14.0000i		−	1.09656i	0.836293	−	0.548282i	$-0.184718\pi$
$164$		0			0
$165$	3.00000	−	6.00000i	0.233550	−	0.467099i
$166$		0			0
$167$			3.00000i			0.232147i	0.993241	+	0.116073i	$0.0370308\pi$
$167$			3.00000i			0.232147i	−0.993241	+	0.116073i	$0.962969\pi$
$168$		0			0
$169$	12.0000			0.923077
$170$		0			0
$171$		0			0
$172$		0			0
$173$			9.00000i			0.684257i	0.939653	+	0.342129i	$0.111148\pi$
$173$			9.00000i			0.684257i	−0.939653	+	0.342129i	$0.888852\pi$
$174$		0			0
$175$	−4.00000	+	3.00000i	−0.302372	+	0.226779i
$176$		0			0
$177$			10.0000i			0.751646i
$178$		0			0
$179$	−20.0000			−1.49487			−0.747435	−	0.664335i	$-0.768715\pi$
$179$	−20.0000			−1.49487			−0.747435	+	0.664335i	$0.768715\pi$
$180$		0			0
$181$	−18.0000			−1.33793			−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−18.0000			−1.33793			−0.668965	+	0.743294i	$0.733262\pi$
$182$		0			0
$183$		−	8.00000i		−	0.591377i
$184$		0			0
$185$	2.00000	−	4.00000i	0.147043	−	0.294086i
$186$		0			0
$187$			21.0000i			1.53567i
$188$		0			0
$189$	−5.00000			−0.363696
$190$		0			0
$191$	3.00000			0.217072			0.108536	−	0.994092i	$-0.465384\pi$
$191$	3.00000			0.217072			0.108536	+	0.994092i	$0.465384\pi$
$192$		0			0
$193$		−	16.0000i		−	1.15171i	−0.817554	−	0.575853i	$-0.804670\pi$
$193$		−	16.0000i		−	1.15171i	0.817554	−	0.575853i	$-0.195330\pi$
$194$		0			0
$195$	−2.00000	−	1.00000i	−0.143223	−	0.0716115i
$196$		0			0
$197$			2.00000i			0.142494i	0.997459	+	0.0712470i	$0.0226979\pi$
$197$			2.00000i			0.142494i	−0.997459	+	0.0712470i	$0.977302\pi$
$198$		0			0
$199$	−10.0000			−0.708881			−0.354441	−	0.935079i	$-0.615329\pi$
$199$	−10.0000			−0.708881			−0.354441	+	0.935079i	$0.615329\pi$
$200$		0			0
$201$	2.00000			0.141069
$202$		0			0
$203$			5.00000i			0.350931i
$204$		0			0
$205$	−4.00000	−	2.00000i	−0.279372	−	0.139686i
$206$		0			0
$207$			12.0000i			0.834058i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	13.0000			0.894957			0.447478	−	0.894295i	$-0.352322\pi$
$211$	13.0000			0.894957			0.447478	+	0.894295i	$0.352322\pi$
$212$		0			0
$213$			8.00000i			0.548151i
$214$		0			0
$215$	−4.00000	+	8.00000i	−0.272798	+	0.545595i
$216$		0			0
$217$		−	2.00000i		−	0.135769i
$218$		0			0
$219$	6.00000			0.405442
$220$		0			0
$221$	7.00000			0.470871
$222$		0			0
$223$			21.0000i			1.40626i	0.711059	+	0.703132i	$0.248216\pi$
$223$			21.0000i			1.40626i	−0.711059	+	0.703132i	$0.751784\pi$
$224$		0			0
$225$	6.00000	+	8.00000i	0.400000	+	0.533333i
$226$		0			0
$227$		−	17.0000i		−	1.12833i	−0.825662	−	0.564165i	$-0.809198\pi$
$227$		−	17.0000i		−	1.12833i	0.825662	−	0.564165i	$-0.190802\pi$
$228$		0			0
$229$	10.0000			0.660819			0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000			0.660819			0.330409	+	0.943838i	$0.392813\pi$
$230$		0			0
$231$	−3.00000			−0.197386
$232$		0			0
$233$		−	16.0000i		−	1.04819i	−0.851658	−	0.524097i	$-0.824403\pi$
$233$		−	16.0000i		−	1.04819i	0.851658	−	0.524097i	$-0.175597\pi$
$234$		0			0
$235$	3.00000	−	6.00000i	0.195698	−	0.391397i
$236$		0			0
$237$		−	5.00000i		−	0.324785i
$238$		0			0
$239$	15.0000			0.970269			0.485135	−	0.874439i	$-0.338771\pi$
$239$	15.0000			0.970269			0.485135	+	0.874439i	$0.338771\pi$
$240$		0			0
$241$	22.0000			1.41714			0.708572	−	0.705638i	$-0.249340\pi$
$241$	22.0000			1.41714			0.708572	+	0.705638i	$0.249340\pi$
$242$		0			0
$243$			16.0000i			1.02640i
$244$		0			0
$245$	2.00000	+	1.00000i	0.127775	+	0.0638877i
$246$		0			0
$247$		0			0
$248$		0			0
$249$	4.00000			0.253490
$250$		0			0
$251$	18.0000			1.13615			0.568075	−	0.822977i	$-0.307688\pi$
$251$	18.0000			1.13615			0.568075	+	0.822977i	$0.307688\pi$
$252$		0			0
$253$			18.0000i			1.13165i
$254$		0			0
$255$	14.0000	+	7.00000i	0.876714	+	0.438357i
$256$		0			0
$257$			22.0000i			1.37232i	0.727450	+	0.686161i	$0.240706\pi$
$257$			22.0000i			1.37232i	−0.727450	+	0.686161i	$0.759294\pi$
$258$		0			0
$259$	−2.00000			−0.124274
$260$		0			0
$261$	10.0000			0.618984
$262$		0			0
$263$		−	24.0000i		−	1.47990i	−0.672660	−	0.739952i	$-0.734848\pi$
$263$		−	24.0000i		−	1.47990i	0.672660	−	0.739952i	$-0.265152\pi$
$264$		0			0
$265$	−6.00000	+	12.0000i	−0.368577	+	0.737154i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−10.0000			−0.609711			−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000			−0.609711			−0.304855	+	0.952399i	$0.598608\pi$
$270$		0			0
$271$	8.00000			0.485965			0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000			0.485965			0.242983	+	0.970031i	$0.421874\pi$
$272$		0			0
$273$			1.00000i			0.0605228i
$274$		0			0
$275$	9.00000	+	12.0000i	0.542720	+	0.723627i
$276$		0			0
$277$			2.00000i			0.120168i	0.998193	+	0.0600842i	$0.0191369\pi$
$277$			2.00000i			0.120168i	−0.998193	+	0.0600842i	$0.980863\pi$
$278$		0			0
$279$	−4.00000			−0.239474
$280$		0			0
$281$	7.00000			0.417585			0.208792	−	0.977960i	$-0.433047\pi$
$281$	7.00000			0.417585			0.208792	+	0.977960i	$0.433047\pi$
$282$		0			0
$283$			11.0000i			0.653882i	0.945045	+	0.326941i	$0.106018\pi$
$283$			11.0000i			0.653882i	−0.945045	+	0.326941i	$0.893982\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			2.00000i			0.118056i
$288$		0			0
$289$	−32.0000			−1.88235
$290$		0			0
$291$	−7.00000			−0.410347
$292$		0			0
$293$			9.00000i			0.525786i	0.964825	+	0.262893i	$0.0846766\pi$
$293$			9.00000i			0.525786i	−0.964825	+	0.262893i	$0.915323\pi$
$294$		0			0
$295$	−20.0000	−	10.0000i	−1.16445	−	0.582223i
$296$		0			0
$297$			15.0000i			0.870388i
$298$		0			0
$299$	6.00000			0.346989
$300$		0			0
$301$	4.00000			0.230556
$302$		0			0
$303$			12.0000i			0.689382i
$304$		0			0
$305$	16.0000	+	8.00000i	0.916157	+	0.458079i
$306$		0			0
$307$		−	7.00000i		−	0.399511i	−0.979846	−	0.199756i	$-0.935985\pi$
$307$		−	7.00000i		−	0.399511i	0.979846	−	0.199756i	$-0.0640148\pi$
$308$		0			0
$309$	19.0000			1.08087
$310$		0			0
$311$	−12.0000			−0.680458			−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000			−0.680458			−0.340229	+	0.940343i	$0.610505\pi$
$312$		0			0
$313$		−	21.0000i		−	1.18699i	−0.804838	−	0.593495i	$-0.797748\pi$
$313$		−	21.0000i		−	1.18699i	0.804838	−	0.593495i	$-0.202252\pi$
$314$		0			0
$315$	2.00000	−	4.00000i	0.112687	−	0.225374i
$316$		0			0
$317$			12.0000i			0.673987i	0.941507	+	0.336994i	$0.109410\pi$
$317$			12.0000i			0.673987i	−0.941507	+	0.336994i	$0.890590\pi$
$318$		0			0
$319$	15.0000			0.839839
$320$		0			0
$321$	−8.00000			−0.446516
$322$		0			0
$323$		0			0
$324$		0			0
$325$	4.00000	−	3.00000i	0.221880	−	0.166410i
$326$		0			0
$327$		−	5.00000i		−	0.276501i
$328$		0			0
$329$	−3.00000			−0.165395
$330$		0			0
$331$	−12.0000			−0.659580			−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000			−0.659580			−0.329790	+	0.944054i	$0.606978\pi$
$332$		0			0
$333$			4.00000i			0.219199i
$334$		0			0
$335$	−2.00000	+	4.00000i	−0.109272	+	0.218543i
$336$		0			0
$337$		−	18.0000i		−	0.980522i	−0.871576	−	0.490261i	$-0.836901\pi$
$337$		−	18.0000i		−	0.980522i	0.871576	−	0.490261i	$-0.163099\pi$
$338$		0			0
$339$	6.00000			0.325875
$340$		0			0
$341$	−6.00000			−0.324918
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$	12.0000	+	6.00000i	0.646058	+	0.323029i
$346$		0			0
$347$			18.0000i			0.966291i	0.875540	+	0.483145i	$0.160506\pi$
$347$			18.0000i			0.966291i	−0.875540	+	0.483145i	$0.839494\pi$
$348$		0			0
$349$	20.0000			1.07058			0.535288	−	0.844670i	$-0.320203\pi$
$349$	20.0000			1.07058			0.535288	+	0.844670i	$0.320203\pi$
$350$		0			0
$351$	5.00000			0.266880
$352$		0			0
$353$		−	11.0000i		−	0.585471i	−0.956193	−	0.292735i	$-0.905434\pi$
$353$		−	11.0000i		−	0.585471i	0.956193	−	0.292735i	$-0.0945655\pi$
$354$		0			0
$355$	−16.0000	−	8.00000i	−0.849192	−	0.424596i
$356$		0			0
$357$		−	7.00000i		−	0.370479i
$358$		0			0
$359$	−20.0000			−1.05556			−0.527780	−	0.849381i	$-0.676975\pi$
$359$	−20.0000			−1.05556			−0.527780	+	0.849381i	$0.676975\pi$
$360$		0			0
$361$	−19.0000			−1.00000
$362$		0			0
$363$		−	2.00000i		−	0.104973i
$364$		0			0
$365$	−6.00000	+	12.0000i	−0.314054	+	0.628109i
$366$		0			0
$367$			3.00000i			0.156599i	0.996930	+	0.0782994i	$0.0249490\pi$
$367$			3.00000i			0.156599i	−0.996930	+	0.0782994i	$0.975051\pi$
$368$		0			0
$369$	4.00000			0.208232
$370$		0			0
$371$	6.00000			0.311504
$372$		0			0
$373$			24.0000i			1.24267i	0.783544	+	0.621336i	$0.213410\pi$
$373$			24.0000i			1.24267i	−0.783544	+	0.621336i	$0.786590\pi$
$374$		0			0
$375$	11.0000	−	2.00000i	0.568038	−	0.103280i
$376$		0			0
$377$		−	5.00000i		−	0.257513i
$378$		0			0
$379$	20.0000			1.02733			0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000			1.02733			0.513665	+	0.857991i	$0.328287\pi$
$380$		0			0
$381$	2.00000			0.102463
$382$		0			0
$383$			16.0000i			0.817562i	0.912633	+	0.408781i	$0.134046\pi$
$383$			16.0000i			0.817562i	−0.912633	+	0.408781i	$0.865954\pi$
$384$		0			0
$385$	3.00000	−	6.00000i	0.152894	−	0.305788i
$386$		0			0
$387$		−	8.00000i		−	0.406663i
$388$		0			0
$389$	−5.00000			−0.253510			−0.126755	−	0.991934i	$-0.540456\pi$
$389$	−5.00000			−0.253510			−0.126755	+	0.991934i	$0.540456\pi$
$390$		0			0
$391$	−42.0000			−2.12403
$392$		0			0
$393$		−	22.0000i		−	1.10975i
$394$		0			0
$395$	10.0000	+	5.00000i	0.503155	+	0.251577i
$396$		0			0
$397$			7.00000i			0.351320i	0.984451	+	0.175660i	$0.0562059\pi$
$397$			7.00000i			0.351320i	−0.984451	+	0.175660i	$0.943794\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−3.00000			−0.149813			−0.0749064	−	0.997191i	$-0.523866\pi$
$401$	−3.00000			−0.149813			−0.0749064	+	0.997191i	$0.523866\pi$
$402$		0			0
$403$			2.00000i			0.0996271i
$404$		0			0
$405$	−2.00000	−	1.00000i	−0.0993808	−	0.0496904i
$406$		0			0
$407$			6.00000i			0.297409i
$408$		0			0
$409$	−20.0000			−0.988936			−0.494468	−	0.869196i	$-0.664637\pi$
$409$	−20.0000			−0.988936			−0.494468	+	0.869196i	$0.664637\pi$
$410$		0			0
$411$	−12.0000			−0.591916
$412$		0			0
$413$			10.0000i			0.492068i
$414$		0			0
$415$	−4.00000	+	8.00000i	−0.196352	+	0.392705i
$416$		0			0
$417$		−	10.0000i		−	0.489702i
$418$		0			0
$419$	30.0000			1.46560			0.732798	−	0.680446i	$-0.238214\pi$
$419$	30.0000			1.46560			0.732798	+	0.680446i	$0.238214\pi$
$420$		0			0
$421$	−3.00000			−0.146211			−0.0731055	−	0.997324i	$-0.523291\pi$
$421$	−3.00000			−0.146211			−0.0731055	+	0.997324i	$0.523291\pi$
$422$		0			0
$423$			6.00000i			0.291730i
$424$		0			0
$425$	−28.0000	+	21.0000i	−1.35820	+	1.01865i
$426$		0			0
$427$		−	8.00000i		−	0.387147i
$428$		0			0
$429$	3.00000			0.144841
$430$		0			0
$431$	23.0000			1.10787			0.553936	−	0.832560i	$-0.313125\pi$
$431$	23.0000			1.10787			0.553936	+	0.832560i	$0.313125\pi$
$432$		0			0
$433$		−	26.0000i		−	1.24948i	−0.780833	−	0.624740i	$-0.785205\pi$
$433$		−	26.0000i		−	1.24948i	0.780833	−	0.624740i	$-0.214795\pi$
$434$		0			0
$435$	5.00000	−	10.0000i	0.239732	−	0.479463i
$436$		0			0
$437$		0			0
$438$		0			0
$439$	30.0000			1.43182			0.715911	−	0.698192i	$-0.246012\pi$
$439$	30.0000			1.43182			0.715911	+	0.698192i	$0.246012\pi$
$440$		0			0
$441$	−2.00000			−0.0952381
$442$		0			0
$443$		−	4.00000i		−	0.190046i	−0.995475	−	0.0950229i	$-0.969708\pi$
$443$		−	4.00000i		−	0.190046i	0.995475	−	0.0950229i	$-0.0302924\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		−	10.0000i		−	0.472984i
$448$		0			0
$449$	5.00000			0.235965			0.117982	−	0.993016i	$-0.462357\pi$
$449$	5.00000			0.235965			0.117982	+	0.993016i	$0.462357\pi$
$450$		0			0
$451$	6.00000			0.282529
$452$		0			0
$453$			13.0000i			0.610793i
$454$		0			0
$455$	−2.00000	−	1.00000i	−0.0937614	−	0.0468807i
$456$		0			0
$457$		−	38.0000i		−	1.77757i	−0.458329	−	0.888783i	$-0.651552\pi$
$457$		−	38.0000i		−	1.77757i	0.458329	−	0.888783i	$-0.348448\pi$
$458$		0			0
$459$	−35.0000			−1.63366
$460$		0			0
$461$	12.0000			0.558896			0.279448	−	0.960161i	$-0.409849\pi$
$461$	12.0000			0.558896			0.279448	+	0.960161i	$0.409849\pi$
$462$		0			0
$463$			36.0000i			1.67306i	0.547920	+	0.836531i	$0.315420\pi$
$463$			36.0000i			1.67306i	−0.547920	+	0.836531i	$0.684580\pi$
$464$		0			0
$465$	−2.00000	+	4.00000i	−0.0927478	+	0.185496i
$466$		0			0
$467$		−	27.0000i		−	1.24941i	−0.780860	−	0.624705i	$-0.785219\pi$
$467$		−	27.0000i		−	1.24941i	0.780860	−	0.624705i	$-0.214781\pi$
$468$		0			0
$469$	2.00000			0.0923514
$470$		0			0
$471$	18.0000			0.829396
$472$		0			0
$473$		−	12.0000i		−	0.551761i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		−	12.0000i		−	0.549442i
$478$		0			0
$479$	−30.0000			−1.37073			−0.685367	−	0.728197i	$-0.740358\pi$
$479$	−30.0000			−1.37073			−0.685367	+	0.728197i	$0.740358\pi$
$480$		0			0
$481$	2.00000			0.0911922
$482$		0			0
$483$		−	6.00000i		−	0.273009i
$484$		0			0
$485$	7.00000	−	14.0000i	0.317854	−	0.635707i
$486$		0			0
$487$		−	42.0000i		−	1.90320i	−0.307337	−	0.951601i	$-0.599438\pi$
$487$		−	42.0000i		−	1.90320i	0.307337	−	0.951601i	$-0.400562\pi$
$488$		0			0
$489$	14.0000			0.633102
$490$		0			0
$491$	−7.00000			−0.315906			−0.157953	−	0.987447i	$-0.550489\pi$
$491$	−7.00000			−0.315906			−0.157953	+	0.987447i	$0.550489\pi$
$492$		0			0
$493$			35.0000i			1.57632i
$494$		0			0
$495$	−12.0000	−	6.00000i	−0.539360	−	0.269680i
$496$		0			0
$497$			8.00000i			0.358849i
$498$		0			0
$499$	−35.0000			−1.56682			−0.783408	−	0.621508i	$-0.786520\pi$
$499$	−35.0000			−1.56682			−0.783408	+	0.621508i	$0.786520\pi$
$500$		0			0
$501$	−3.00000			−0.134030
$502$		0			0
$503$		−	9.00000i		−	0.401290i	−0.979664	−	0.200645i	$-0.935696\pi$
$503$		−	9.00000i		−	0.401290i	0.979664	−	0.200645i	$-0.0643038\pi$
$504$		0			0
$505$	−24.0000	−	12.0000i	−1.06799	−	0.533993i
$506$		0			0
$507$			12.0000i			0.532939i
$508$		0			0
$509$	30.0000			1.32973			0.664863	−	0.746965i	$-0.268490\pi$
$509$	30.0000			1.32973			0.664863	+	0.746965i	$0.268490\pi$
$510$		0			0
$511$	6.00000			0.265424
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−19.0000	+	38.0000i	−0.837240	+	1.67448i
$516$		0			0
$517$			9.00000i			0.395820i
$518$		0			0
$519$	−9.00000			−0.395056
$520$		0			0
$521$	−28.0000			−1.22670			−0.613351	−	0.789810i	$-0.710179\pi$
$521$	−28.0000			−1.22670			−0.613351	+	0.789810i	$0.710179\pi$
$522$		0			0
$523$		−	4.00000i		−	0.174908i	−0.996169	−	0.0874539i	$-0.972127\pi$
$523$		−	4.00000i		−	0.174908i	0.996169	−	0.0874539i	$-0.0278730\pi$
$524$		0			0
$525$	−3.00000	−	4.00000i	−0.130931	−	0.174574i
$526$		0			0
$527$		−	14.0000i		−	0.609850i
$528$		0			0
$529$	−13.0000			−0.565217
$530$		0			0
$531$	20.0000			0.867926
$532$		0			0
$533$		−	2.00000i		−	0.0866296i
$534$		0			0
$535$	8.00000	−	16.0000i	0.345870	−	0.691740i
$536$		0			0
$537$		−	20.0000i		−	0.863064i
$538$		0			0
$539$	−3.00000			−0.129219
$540$		0			0
$541$	27.0000			1.16082			0.580410	−	0.814324i	$-0.302892\pi$
$541$	27.0000			1.16082			0.580410	+	0.814324i	$0.302892\pi$
$542$		0			0
$543$		−	18.0000i		−	0.772454i
$544$		0			0
$545$	10.0000	+	5.00000i	0.428353	+	0.214176i
$546$		0			0
$547$		−	32.0000i		−	1.36822i	−0.729378	−	0.684111i	$-0.760191\pi$
$547$		−	32.0000i		−	1.36822i	0.729378	−	0.684111i	$-0.239809\pi$
$548$		0			0
$549$	−16.0000			−0.682863
$550$		0			0
$551$		0			0
$552$		0			0
$553$		−	5.00000i		−	0.212622i
$554$		0			0
$555$	4.00000	+	2.00000i	0.169791	+	0.0848953i
$556$		0			0
$557$		−	28.0000i		−	1.18640i	−0.805056	−	0.593199i	$-0.797865\pi$
$557$		−	28.0000i		−	1.18640i	0.805056	−	0.593199i	$-0.202135\pi$
$558$		0			0
$559$	−4.00000			−0.169182
$560$		0			0
$561$	−21.0000			−0.886621
$562$		0			0
$563$		−	24.0000i		−	1.01148i	−0.862686	−	0.505740i	$-0.831220\pi$
$563$		−	24.0000i		−	1.01148i	0.862686	−	0.505740i	$-0.168780\pi$
$564$		0			0
$565$	−6.00000	+	12.0000i	−0.252422	+	0.504844i
$566$		0			0
$567$			1.00000i			0.0419961i
$568$		0			0
$569$	30.0000			1.25767			0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000			1.25767			0.628833	+	0.777541i	$0.283533\pi$
$570$		0			0
$571$	−12.0000			−0.502184			−0.251092	−	0.967963i	$-0.580790\pi$
$571$	−12.0000			−0.502184			−0.251092	+	0.967963i	$0.580790\pi$
$572$		0			0
$573$			3.00000i			0.125327i
$574$		0			0
$575$	−24.0000	+	18.0000i	−1.00087	+	0.750652i
$576$		0			0
$577$		−	43.0000i		−	1.79011i	−0.445952	−	0.895057i	$-0.647135\pi$
$577$		−	43.0000i		−	1.79011i	0.445952	−	0.895057i	$-0.352865\pi$
$578$		0			0
$579$	16.0000			0.664937
$580$		0			0
$581$	4.00000			0.165948
$582$		0			0
$583$		−	18.0000i		−	0.745484i
$584$		0			0
$585$	−2.00000	+	4.00000i	−0.0826898	+	0.165380i
$586$		0			0
$587$		−	12.0000i		−	0.495293i	−0.968850	−	0.247647i	$-0.920343\pi$
$587$		−	12.0000i		−	0.495293i	0.968850	−	0.247647i	$-0.0796572\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$	−2.00000			−0.0822690
$592$		0			0
$593$		−	41.0000i		−	1.68367i	−0.539736	−	0.841834i	$-0.681476\pi$
$593$		−	41.0000i		−	1.68367i	0.539736	−	0.841834i	$-0.318524\pi$
$594$		0			0
$595$	14.0000	+	7.00000i	0.573944	+	0.286972i
$596$		0			0
$597$		−	10.0000i		−	0.409273i
$598$		0			0
$599$	25.0000			1.02147			0.510736	−	0.859738i	$-0.329373\pi$
$599$	25.0000			1.02147			0.510736	+	0.859738i	$0.329373\pi$
$600$		0			0
$601$	−8.00000			−0.326327			−0.163163	−	0.986599i	$-0.552170\pi$
$601$	−8.00000			−0.326327			−0.163163	+	0.986599i	$0.552170\pi$
$602$		0			0
$603$		−	4.00000i		−	0.162893i
$604$		0			0
$605$	4.00000	+	2.00000i	0.162623	+	0.0813116i
$606$		0			0
$607$		−	27.0000i		−	1.09590i	−0.836512	−	0.547948i	$-0.815409\pi$
$607$		−	27.0000i		−	1.09590i	0.836512	−	0.547948i	$-0.184591\pi$
$608$		0			0
$609$	−5.00000			−0.202610
$610$		0			0
$611$	3.00000			0.121367
$612$		0			0
$613$			44.0000i			1.77714i	0.458738	+	0.888572i	$0.348302\pi$
$613$			44.0000i			1.77714i	−0.458738	+	0.888572i	$0.651698\pi$
$614$		0			0
$615$	2.00000	−	4.00000i	0.0806478	−	0.161296i
$616$		0			0
$617$			22.0000i			0.885687i	0.896599	+	0.442843i	$0.146030\pi$
$617$			22.0000i			0.885687i	−0.896599	+	0.442843i	$0.853970\pi$
$618$		0			0
$619$	10.0000			0.401934			0.200967	−	0.979598i	$-0.435592\pi$
$619$	10.0000			0.401934			0.200967	+	0.979598i	$0.435592\pi$
$620$		0			0
$621$	−30.0000			−1.20386
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−7.00000	+	24.0000i	−0.280000	+	0.960000i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−14.0000			−0.558217
$630$		0			0
$631$	−37.0000			−1.47295			−0.736473	−	0.676467i	$-0.763510\pi$
$631$	−37.0000			−1.47295			−0.736473	+	0.676467i	$0.763510\pi$
$632$		0			0
$633$			13.0000i			0.516704i
$634$		0			0
$635$	−2.00000	+	4.00000i	−0.0793676	+	0.158735i
$636$		0			0
$637$			1.00000i			0.0396214i
$638$		0			0
$639$	16.0000			0.632950
$640$		0			0
$641$	22.0000			0.868948			0.434474	−	0.900684i	$-0.356934\pi$
$641$	22.0000			0.868948			0.434474	+	0.900684i	$0.356934\pi$
$642$		0			0
$643$			1.00000i			0.0394362i	0.999806	+	0.0197181i	$0.00627687\pi$
$643$			1.00000i			0.0394362i	−0.999806	+	0.0197181i	$0.993723\pi$
$644$		0			0
$645$	−8.00000	−	4.00000i	−0.315000	−	0.157500i
$646$		0			0
$647$			8.00000i			0.314512i	0.987558	+	0.157256i	$0.0502649\pi$
$647$			8.00000i			0.314512i	−0.987558	+	0.157256i	$0.949735\pi$
$648$		0			0
$649$	30.0000			1.17760
$650$		0			0
$651$	2.00000			0.0783862
$652$		0			0
$653$			4.00000i			0.156532i	0.996933	+	0.0782660i	$0.0249384\pi$
$653$			4.00000i			0.156532i	−0.996933	+	0.0782660i	$0.975062\pi$
$654$		0			0
$655$	44.0000	+	22.0000i	1.71922	+	0.859611i
$656$		0			0
$657$		−	12.0000i		−	0.468165i
$658$		0			0
$659$	−15.0000			−0.584317			−0.292159	−	0.956370i	$-0.594373\pi$
$659$	−15.0000			−0.584317			−0.292159	+	0.956370i	$0.594373\pi$
$660$		0			0
$661$	12.0000			0.466746			0.233373	−	0.972387i	$-0.425024\pi$
$661$	12.0000			0.466746			0.233373	+	0.972387i	$0.425024\pi$
$662$		0			0
$663$			7.00000i			0.271857i
$664$		0			0
$665$		0			0
$666$		0			0
$667$			30.0000i			1.16160i
$668$		0			0
$669$	−21.0000			−0.811907
$670$		0			0
$671$	−24.0000			−0.926510
$672$		0			0
$673$			24.0000i			0.925132i	0.886585	+	0.462566i	$0.153071\pi$
$673$			24.0000i			0.925132i	−0.886585	+	0.462566i	$0.846929\pi$
$674$		0			0
$675$	−20.0000	+	15.0000i	−0.769800	+	0.577350i
$676$		0			0
$677$		−	43.0000i		−	1.65262i	−0.563212	−	0.826312i	$-0.690435\pi$
$677$		−	43.0000i		−	1.65262i	0.563212	−	0.826312i	$-0.309565\pi$
$678$		0			0
$679$	−7.00000			−0.268635
$680$		0			0
$681$	17.0000			0.651441
$682$		0			0
$683$			16.0000i			0.612223i	0.951996	+	0.306111i	$0.0990280\pi$
$683$			16.0000i			0.612223i	−0.951996	+	0.306111i	$0.900972\pi$
$684$		0			0
$685$	12.0000	−	24.0000i	0.458496	−	0.916993i
$686$		0			0
$687$			10.0000i			0.381524i
$688$		0			0
$689$	−6.00000			−0.228582
$690$		0			0
$691$	8.00000			0.304334			0.152167	−	0.988355i	$-0.451375\pi$
$691$	8.00000			0.304334			0.152167	+	0.988355i	$0.451375\pi$
$692$		0			0
$693$			6.00000i			0.227921i
$694$		0			0
$695$	20.0000	+	10.0000i	0.758643	+	0.379322i
$696$		0			0
$697$			14.0000i			0.530288i
$698$		0			0
$699$	16.0000			0.605176
$700$		0			0
$701$	−13.0000			−0.491003			−0.245502	−	0.969396i	$-0.578953\pi$
$701$	−13.0000			−0.491003			−0.245502	+	0.969396i	$0.578953\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	6.00000	+	3.00000i	0.225973	+	0.112987i
$706$		0			0
$707$			12.0000i			0.451306i
$708$		0			0
$709$	−35.0000			−1.31445			−0.657226	−	0.753693i	$-0.728270\pi$
$709$	−35.0000			−1.31445			−0.657226	+	0.753693i	$0.728270\pi$
$710$		0			0
$711$	−10.0000			−0.375029
$712$		0			0
$713$		−	12.0000i		−	0.449404i
$714$		0			0
$715$	−3.00000	+	6.00000i	−0.112194	+	0.224387i
$716$		0			0
$717$			15.0000i			0.560185i
$718$		0			0
$719$	30.0000			1.11881			0.559406	−	0.828894i	$-0.311029\pi$
$719$	30.0000			1.11881			0.559406	+	0.828894i	$0.311029\pi$
$720$		0			0
$721$	19.0000			0.707597
$722$		0			0
$723$			22.0000i			0.818189i
$724$		0			0
$725$	15.0000	+	20.0000i	0.557086	+	0.742781i
$726$		0			0
$727$			28.0000i			1.03846i	0.854634	+	0.519231i	$0.173782\pi$
$727$			28.0000i			1.03846i	−0.854634	+	0.519231i	$0.826218\pi$
$728$		0			0
$729$	−13.0000			−0.481481
$730$		0			0
$731$	28.0000			1.03562
$732$		0			0
$733$		−	1.00000i		−	0.0369358i	−0.999829	−	0.0184679i	$-0.994121\pi$
$733$		−	1.00000i		−	0.0369358i	0.999829	−	0.0184679i	$-0.00587886\pi$
$734$		0			0
$735$	−1.00000	+	2.00000i	−0.0368856	+	0.0737711i
$736$		0			0
$737$		−	6.00000i		−	0.221013i
$738$		0			0
$739$	−35.0000			−1.28750			−0.643748	−	0.765238i	$-0.722621\pi$
$739$	−35.0000			−1.28750			−0.643748	+	0.765238i	$0.722621\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	14.0000i		−	0.513610i	−0.966463	−	0.256805i	$-0.917330\pi$
$743$		−	14.0000i		−	0.513610i	0.966463	−	0.256805i	$-0.0826698\pi$
$744$		0			0
$745$	20.0000	+	10.0000i	0.732743	+	0.366372i
$746$		0			0
$747$		−	8.00000i		−	0.292705i
$748$		0			0
$749$	−8.00000			−0.292314
$750$		0			0
$751$	33.0000			1.20419			0.602094	−	0.798426i	$-0.294333\pi$
$751$	33.0000			1.20419			0.602094	+	0.798426i	$0.294333\pi$
$752$		0			0
$753$			18.0000i			0.655956i
$754$		0			0
$755$	−26.0000	−	13.0000i	−0.946237	−	0.473118i
$756$		0			0
$757$			32.0000i			1.16306i	0.813525	+	0.581530i	$0.197546\pi$
$757$			32.0000i			1.16306i	−0.813525	+	0.581530i	$0.802454\pi$
$758$		0			0
$759$	−18.0000			−0.653359
$760$		0			0
$761$	2.00000			0.0724999			0.0362500	−	0.999343i	$-0.488459\pi$
$761$	2.00000			0.0724999			0.0362500	+	0.999343i	$0.488459\pi$
$762$		0			0
$763$		−	5.00000i		−	0.181012i
$764$		0			0
$765$	14.0000	−	28.0000i	0.506171	−	1.01234i
$766$		0			0
$767$		−	10.0000i		−	0.361079i
$768$		0			0
$769$		0			0			−	1.00000i	$-0.5\pi$
$769$		0			0				1.00000i	$0.5\pi$
$770$		0			0
$771$	−22.0000			−0.792311
$772$		0			0
$773$		−	21.0000i		−	0.755318i	−0.925945	−	0.377659i	$-0.876729\pi$
$773$		−	21.0000i		−	0.755318i	0.925945	−	0.377659i	$-0.123271\pi$
$774$		0			0
$775$	−6.00000	−	8.00000i	−0.215526	−	0.287368i
$776$		0			0
$777$		−	2.00000i		−	0.0717496i
$778$		0			0
$779$		0			0
$780$		0			0
$781$	24.0000			0.858788
$782$		0			0
$783$			25.0000i			0.893427i
$784$		0			0
$785$	−18.0000	+	36.0000i	−0.642448	+	1.28490i
$786$		0			0
$787$		−	17.0000i		−	0.605985i	−0.952993	−	0.302992i	$-0.902014\pi$
$787$		−	17.0000i		−	0.605985i	0.952993	−	0.302992i	$-0.0979856\pi$
$788$		0			0
$789$	24.0000			0.854423
$790$		0			0
$791$	6.00000			0.213335
$792$		0			0
$793$			8.00000i			0.284088i
$794$		0			0
$795$	−12.0000	−	6.00000i	−0.425596	−	0.212798i
$796$		0			0
$797$		−	13.0000i		−	0.460484i	−0.973133	−	0.230242i	$-0.926048\pi$
$797$		−	13.0000i		−	0.460484i	0.973133	−	0.230242i	$-0.0739517\pi$
$798$		0			0
$799$	−21.0000			−0.742927
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	18.0000i		−	0.635206i
$804$		0			0
$805$	12.0000	+	6.00000i	0.422944	+	0.211472i
$806$		0			0
$807$		−	10.0000i		−	0.352017i
$808$		0			0
$809$	−45.0000			−1.58212			−0.791058	−	0.611741i	$-0.790469\pi$
$809$	−45.0000			−1.58212			−0.791058	+	0.611741i	$0.790469\pi$
$810$		0			0
$811$	38.0000			1.33436			0.667180	−	0.744896i	$-0.267501\pi$
$811$	38.0000			1.33436			0.667180	+	0.744896i	$0.267501\pi$
$812$		0			0
$813$			8.00000i			0.280572i
$814$		0			0
$815$	−14.0000	+	28.0000i	−0.490399	+	0.980797i
$816$		0			0
$817$		0			0
$818$		0			0
$819$	2.00000			0.0698857
$820$		0			0
$821$	−23.0000			−0.802706			−0.401353	−	0.915924i	$-0.631460\pi$
$821$	−23.0000			−0.802706			−0.401353	+	0.915924i	$0.631460\pi$
$822$		0			0
$823$			26.0000i			0.906303i	0.891434	+	0.453152i	$0.149700\pi$
$823$			26.0000i			0.906303i	−0.891434	+	0.453152i	$0.850300\pi$
$824$		0			0
$825$	−12.0000	+	9.00000i	−0.417786	+	0.313340i
$826$		0			0
$827$			18.0000i			0.625921i	0.949766	+	0.312961i	$0.101321\pi$
$827$			18.0000i			0.625921i	−0.949766	+	0.312961i	$0.898679\pi$
$828$		0			0
$829$	30.0000			1.04194			0.520972	−	0.853574i	$-0.325570\pi$
$829$	30.0000			1.04194			0.520972	+	0.853574i	$0.325570\pi$
$830$		0			0
$831$	−2.00000			−0.0693792
$832$		0			0
$833$		−	7.00000i		−	0.242536i
$834$		0			0
$835$	3.00000	−	6.00000i	0.103819	−	0.207639i
$836$		0			0
$837$		−	10.0000i		−	0.345651i
$838$		0			0
$839$	30.0000			1.03572			0.517858	−	0.855467i	$-0.326730\pi$
$839$	30.0000			1.03572			0.517858	+	0.855467i	$0.326730\pi$
$840$		0			0
$841$	−4.00000			−0.137931
$842$		0			0
$843$			7.00000i			0.241093i
$844$		0			0
$845$	−24.0000	−	12.0000i	−0.825625	−	0.412813i
$846$		0			0
$847$		−	2.00000i		−	0.0687208i
$848$		0			0
$849$	−11.0000			−0.377519
$850$		0			0
$851$	−12.0000			−0.411355
$852$		0			0
$853$			54.0000i			1.84892i	0.381273	+	0.924462i	$0.375486\pi$
$853$			54.0000i			1.84892i	−0.381273	+	0.924462i	$0.624514\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			42.0000i			1.43469i	0.696717	+	0.717346i	$0.254643\pi$
$857$			42.0000i			1.43469i	−0.696717	+	0.717346i	$0.745357\pi$
$858$		0			0
$859$	40.0000			1.36478			0.682391	−	0.730987i	$-0.260940\pi$
$859$	40.0000			1.36478			0.682391	+	0.730987i	$0.260940\pi$
$860$		0			0
$861$	−2.00000			−0.0681598
$862$		0			0
$863$		−	54.0000i		−	1.83818i	−0.394046	−	0.919091i	$-0.628925\pi$
$863$		−	54.0000i		−	1.83818i	0.394046	−	0.919091i	$-0.371075\pi$
$864$		0			0
$865$	9.00000	−	18.0000i	0.306009	−	0.612018i
$866$		0			0
$867$		−	32.0000i		−	1.08678i
$868$		0			0
$869$	−15.0000			−0.508840
$870$		0			0
$871$	−2.00000			−0.0677674
$872$		0			0
$873$			14.0000i			0.473828i
$874$		0			0
$875$	11.0000	−	2.00000i	0.371868	−	0.0676123i
$876$		0			0
$877$			32.0000i			1.08056i	0.841484	+	0.540282i	$0.181682\pi$
$877$			32.0000i			1.08056i	−0.841484	+	0.540282i	$0.818318\pi$
$878$		0			0
$879$	−9.00000			−0.303562
$880$		0			0
$881$	32.0000			1.07811			0.539054	−	0.842271i	$-0.318782\pi$
$881$	32.0000			1.07811			0.539054	+	0.842271i	$0.318782\pi$
$882$		0			0
$883$			36.0000i			1.21150i	0.795656	+	0.605748i	$0.207126\pi$
$883$			36.0000i			1.21150i	−0.795656	+	0.605748i	$0.792874\pi$
$884$		0			0
$885$	10.0000	−	20.0000i	0.336146	−	0.672293i
$886$		0			0
$887$			28.0000i			0.940148i	0.882627	+	0.470074i	$0.155773\pi$
$887$			28.0000i			0.940148i	−0.882627	+	0.470074i	$0.844227\pi$
$888$		0			0
$889$	2.00000			0.0670778
$890$		0			0
$891$	3.00000			0.100504
$892$		0			0
$893$		0			0
$894$		0			0
$895$	40.0000	+	20.0000i	1.33705	+	0.668526i
$896$		0			0
$897$			6.00000i			0.200334i
$898$		0			0
$899$	−10.0000			−0.333519
$900$		0			0
$901$	42.0000			1.39922
$902$		0			0
$903$			4.00000i			0.133112i
$904$		0			0
$905$	36.0000	+	18.0000i	1.19668	+	0.598340i
$906$		0			0
$907$			38.0000i			1.26177i	0.775877	+	0.630885i	$0.217308\pi$
$907$			38.0000i			1.26177i	−0.775877	+	0.630885i	$0.782692\pi$
$908$		0			0
$909$	24.0000			0.796030
$910$		0			0
$911$	−12.0000			−0.397578			−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000			−0.397578			−0.198789	+	0.980042i	$0.563701\pi$
$912$		0			0
$913$		−	12.0000i		−	0.397142i
$914$		0			0
$915$	−8.00000	+	16.0000i	−0.264472	+	0.528944i
$916$		0			0
$917$		−	22.0000i		−	0.726504i
$918$		0			0
$919$	−5.00000			−0.164935			−0.0824674	−	0.996594i	$-0.526280\pi$
$919$	−5.00000			−0.164935			−0.0824674	+	0.996594i	$0.526280\pi$
$920$		0			0
$921$	7.00000			0.230658
$922$		0			0
$923$		−	8.00000i		−	0.263323i
$924$		0			0
$925$	−8.00000	+	6.00000i	−0.263038	+	0.197279i
$926$		0			0
$927$		−	38.0000i		−	1.24808i
$928$		0			0
$929$	50.0000			1.64045			0.820223	−	0.572043i	$-0.193849\pi$
$929$	50.0000			1.64045			0.820223	+	0.572043i	$0.193849\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		−	12.0000i		−	0.392862i
$934$		0			0
$935$	21.0000	−	42.0000i	0.686773	−	1.37355i
$936$		0			0
$937$		−	13.0000i		−	0.424691i	−0.977195	−	0.212346i	$-0.931890\pi$
$937$		−	13.0000i		−	0.424691i	0.977195	−	0.212346i	$-0.0681103\pi$
$938$		0			0
$939$	21.0000			0.685309
$940$		0			0
$941$	−48.0000			−1.56476			−0.782378	−	0.622804i	$-0.785993\pi$
$941$	−48.0000			−1.56476			−0.782378	+	0.622804i	$0.785993\pi$
$942$		0			0
$943$			12.0000i			0.390774i
$944$		0			0
$945$	10.0000	+	5.00000i	0.325300	+	0.162650i
$946$		0			0
$947$		−	42.0000i		−	1.36482i	−0.730971	−	0.682408i	$-0.760933\pi$
$947$		−	42.0000i		−	1.36482i	0.730971	−	0.682408i	$-0.239067\pi$
$948$		0			0
$949$	−6.00000			−0.194768
$950$		0			0
$951$	−12.0000			−0.389127
$952$		0			0
$953$		−	6.00000i		−	0.194359i	−0.995267	−	0.0971795i	$-0.969018\pi$
$953$		−	6.00000i		−	0.194359i	0.995267	−	0.0971795i	$-0.0309821\pi$
$954$		0			0
$955$	−6.00000	−	3.00000i	−0.194155	−	0.0970777i
$956$		0			0
$957$			15.0000i			0.484881i
$958$		0			0
$959$	−12.0000			−0.387500
$960$		0			0
$961$	−27.0000			−0.870968
$962$		0			0
$963$			16.0000i			0.515593i
$964$		0			0
$965$	−16.0000	+	32.0000i	−0.515058	+	1.03012i
$966$		0			0
$967$			8.00000i			0.257263i	0.991692	+	0.128631i	$0.0410584\pi$
$967$			8.00000i			0.257263i	−0.991692	+	0.128631i	$0.958942\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−22.0000			−0.706014			−0.353007	−	0.935621i	$-0.614841\pi$
$971$	−22.0000			−0.706014			−0.353007	+	0.935621i	$0.614841\pi$
$972$		0			0
$973$		−	10.0000i		−	0.320585i
$974$		0			0
$975$	3.00000	+	4.00000i	0.0960769	+	0.128103i
$976$		0			0
$977$			12.0000i			0.383914i	0.981403	+	0.191957i	$0.0614834\pi$
$977$			12.0000i			0.383914i	−0.981403	+	0.191957i	$0.938517\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−10.0000			−0.319275
$982$		0			0
$983$			51.0000i			1.62665i	0.581811	+	0.813324i	$0.302344\pi$
$983$			51.0000i			1.62665i	−0.581811	+	0.813324i	$0.697656\pi$
$984$		0			0
$985$	2.00000	−	4.00000i	0.0637253	−	0.127451i
$986$		0			0
$987$		−	3.00000i		−	0.0954911i
$988$		0			0
$989$	24.0000			0.763156
$990$		0			0
$991$	8.00000			0.254128			0.127064	−	0.991894i	$-0.459445\pi$
$991$	8.00000			0.254128			0.127064	+	0.991894i	$0.459445\pi$
$992$		0			0
$993$		−	12.0000i		−	0.380808i
$994$		0			0
$995$	20.0000	+	10.0000i	0.634043	+	0.317021i
$996$		0			0
$997$		−	13.0000i		−	0.411714i	−0.978582	−	0.205857i	$-0.934002\pi$
$997$		−	13.0000i		−	0.411714i	0.978582	−	0.205857i	$-0.0659982\pi$
$998$		0			0
$999$	−10.0000			−0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	560.2.g.b.449.2		2
3.2	odd	2		5040.2.t.p.1009.2		2
4.3	odd	2		35.2.b.a.29.2	yes	2
5.2	odd	4		2800.2.a.w.1.1		1
5.3	odd	4		2800.2.a.l.1.1		1
5.4	even	2	inner	560.2.g.b.449.1		2
8.3	odd	2		2240.2.g.h.449.2		2
8.5	even	2		2240.2.g.g.449.1		2
12.11	even	2		315.2.d.a.64.1		2
15.14	odd	2		5040.2.t.p.1009.1		2
20.3	even	4		175.2.a.c.1.1		1
20.7	even	4		175.2.a.a.1.1		1
20.19	odd	2		35.2.b.a.29.1	✓	2
28.3	even	6		245.2.j.d.79.1		4
28.11	odd	6		245.2.j.e.79.1		4
28.19	even	6		245.2.j.d.214.2		4
28.23	odd	6		245.2.j.e.214.2		4
28.27	even	2		245.2.b.a.99.2		2
40.19	odd	2		2240.2.g.h.449.1		2
40.29	even	2		2240.2.g.g.449.2		2
60.23	odd	4		1575.2.a.a.1.1		1
60.47	odd	4		1575.2.a.k.1.1		1
60.59	even	2		315.2.d.a.64.2		2
84.83	odd	2		2205.2.d.b.1324.1		2
140.19	even	6		245.2.j.d.214.1		4
140.27	odd	4		1225.2.a.a.1.1		1
140.39	odd	6		245.2.j.e.79.2		4
140.59	even	6		245.2.j.d.79.2		4
140.79	odd	6		245.2.j.e.214.1		4
140.83	odd	4		1225.2.a.i.1.1		1
140.139	even	2		245.2.b.a.99.1		2
420.419	odd	2		2205.2.d.b.1324.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.2.b.a.29.1	✓	2	20.19	odd	2
35.2.b.a.29.2	yes	2	4.3	odd	2
175.2.a.a.1.1		1	20.7	even	4
175.2.a.c.1.1		1	20.3	even	4
245.2.b.a.99.1		2	140.139	even	2
245.2.b.a.99.2		2	28.27	even	2
245.2.j.d.79.1		4	28.3	even	6
245.2.j.d.79.2		4	140.59	even	6
245.2.j.d.214.1		4	140.19	even	6
245.2.j.d.214.2		4	28.19	even	6
245.2.j.e.79.1		4	28.11	odd	6
245.2.j.e.79.2		4	140.39	odd	6
245.2.j.e.214.1		4	140.79	odd	6
245.2.j.e.214.2		4	28.23	odd	6
315.2.d.a.64.1		2	12.11	even	2
315.2.d.a.64.2		2	60.59	even	2
560.2.g.b.449.1		2	5.4	even	2	inner
560.2.g.b.449.2		2	1.1	even	1	trivial
1225.2.a.a.1.1		1	140.27	odd	4
1225.2.a.i.1.1		1	140.83	odd	4
1575.2.a.a.1.1		1	60.23	odd	4
1575.2.a.k.1.1		1	60.47	odd	4
2205.2.d.b.1324.1		2	84.83	odd	2
2205.2.d.b.1324.2		2	420.419	odd	2
2240.2.g.g.449.1		2	8.5	even	2
2240.2.g.g.449.2		2	40.29	even	2
2240.2.g.h.449.1		2	40.19	odd	2
2240.2.g.h.449.2		2	8.3	odd	2
2800.2.a.l.1.1		1	5.3	odd	4
2800.2.a.w.1.1		1	5.2	odd	4
5040.2.t.p.1009.1		2	15.14	odd	2
5040.2.t.p.1009.2		2	3.2	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:	\( N \)	\(=\)	\( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	560.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +(-2.00000 - 1.00000i) q^{5} +1.00000i q^{7} +2.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +(-2.00000 - 1.00000i) q^{5} +1.00000i q^{7} +2.00000 q^{9} +3.00000 q^{11} -1.00000i q^{13} +(1.00000 - 2.00000i) q^{15} +7.00000i q^{17} -1.00000 q^{21} +6.00000i q^{23} +(3.00000 + 4.00000i) q^{25} +5.00000i q^{27} +5.00000 q^{29} -2.00000 q^{31} +3.00000i q^{33} +(1.00000 - 2.00000i) q^{35} +2.00000i q^{37} +1.00000 q^{39} +2.00000 q^{41} -4.00000i q^{43} +(-4.00000 - 2.00000i) q^{45} +3.00000i q^{47} -1.00000 q^{49} -7.00000 q^{51} -6.00000i q^{53} +(-6.00000 - 3.00000i) q^{55} +10.0000 q^{59} -8.00000 q^{61} +2.00000i q^{63} +(-1.00000 + 2.00000i) q^{65} -2.00000i q^{67} -6.00000 q^{69} +8.00000 q^{71} -6.00000i q^{73} +(-4.00000 + 3.00000i) q^{75} +3.00000i q^{77} -5.00000 q^{79} +1.00000 q^{81} -4.00000i q^{83} +(7.00000 - 14.0000i) q^{85} +5.00000i q^{87} +1.00000 q^{91} -2.00000i q^{93} +7.00000i q^{97} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{5} + 4 q^{9}+O(q^{10}) \) 2 * q - 4 * q^5 + 4 * q^9	\( 2 q - 4 q^{5} + 4 q^{9} + 6 q^{11} + 2 q^{15} - 2 q^{21} + 6 q^{25} + 10 q^{29} - 4 q^{31} + 2 q^{35} + 2 q^{39} + 4 q^{41} - 8 q^{45} - 2 q^{49} - 14 q^{51} - 12 q^{55} + 20 q^{59} - 16 q^{61} - 2 q^{65} - 12 q^{69} + 16 q^{71} - 8 q^{75} - 10 q^{79} + 2 q^{81} + 14 q^{85} + 2 q^{91} + 12 q^{99}+O(q^{100}) \) 2 * q - 4 * q^5 + 4 * q^9 + 6 * q^11 + 2 * q^15 - 2 * q^21 + 6 * q^25 + 10 * q^29 - 4 * q^31 + 2 * q^35 + 2 * q^39 + 4 * q^41 - 8 * q^45 - 2 * q^49 - 14 * q^51 - 12 * q^55 + 20 * q^59 - 16 * q^61 - 2 * q^65 - 12 * q^69 + 16 * q^71 - 8 * q^75 - 10 * q^79 + 2 * q^81 + 14 * q^85 + 2 * q^91 + 12 * q^99

Introduction

L-functions

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