LMFDB - Embedded newform 3120.2.l.p.1249.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3120,2,Mod(1249,3120)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3120, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3120.1249"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3120.l (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [10,0,0,0,-2,0,0,0,-10,0,-10,0,0,0,0,0,0,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$24.9133254306$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	10.0.13266844647424.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 16x^{8} + 90x^{6} + 208x^{4} + 169x^{2} + 16 $ x^10 + 16x^8 + 90x^6 + 208x^4 + 169x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 195)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.7
Root			$2.47948i$ of defining polynomial
Character	$\chi$	$=$	3120.1249
Dual form			3120.2.l.p.1249.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +(-1.72481 - 1.42303i) q^{5} -0.949959i q^{7} -1.00000 q^{9} -3.89611 q^{11} -1.00000i q^{13} +(1.42303 - 1.72481i) q^{15} -5.90893i q^{17} +6.35541 q^{19} +0.949959 q^{21} +3.30537i q^{23} +(0.949959 + 4.90893i) q^{25} -1.00000i q^{27} +4.29569 q^{29} -7.56253 q^{31} -3.89611i q^{33} +(-1.35182 + 1.63850i) q^{35} -1.30537i q^{37} +1.00000 q^{39} -6.75499 q^{41} +8.29569i q^{43} +(1.72481 + 1.42303i) q^{45} -0.0128228i q^{47} +6.09758 q^{49} +5.90893 q^{51} -2.51249i q^{53} +(6.72005 + 5.54428i) q^{55} +6.35541i q^{57} -6.30851 q^{59} +1.61324 q^{61} +0.949959i q^{63} +(-1.42303 + 1.72481i) q^{65} -4.66328i q^{67} -3.30537 q^{69} -7.92175 q^{71} +14.5914i q^{73} +(-4.90893 + 0.949959i) q^{75} +3.70114i q^{77} -16.6004 q^{79} +1.00000 q^{81} +12.8044i q^{83} +(-8.40860 + 10.1918i) q^{85} +4.29569i q^{87} -17.6542 q^{89} -0.949959 q^{91} -7.56253i q^{93} +(-10.9619 - 9.04395i) q^{95} +18.1226i q^{97} +3.89611 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 2 q^{5} - 10 q^{9} - 10 q^{11} + 16 q^{19} + 10 q^{21} + 10 q^{25} - 16 q^{29} - 24 q^{31} - 12 q^{35} + 10 q^{39} + 10 q^{41} + 2 q^{45} - 44 q^{49} + 10 q^{51} - 2 q^{55} + 16 q^{59} + 26 q^{61}+ \cdots + 10 q^{99}+O(q^{100}) $ 10 * q - 2 * q^5 - 10 * q^9 - 10 * q^11 + 16 * q^19 + 10 * q^21 + 10 * q^25 - 16 * q^29 - 24 * q^31 - 12 * q^35 + 10 * q^39 + 10 * q^41 + 2 * q^45 - 44 * q^49 + 10 * q^51 - 2 * q^55 + 16 * q^59 + 26 * q^61 + 14 * q^69 - 10 * q^71 + 2 * q^79 + 10 * q^81 - 4 * q^85 - 38 * q^89 - 10 * q^91 + 48 * q^95 + 10 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3120\mathbb{Z}\right)^\times$.

$n$	$1951$	$2081$	$2341$	$2497$	$2641$
$\chi(n)$	$1$	$1$	$1$	$-1$	$1$

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
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$		0			0
$5$	−1.72481	−	1.42303i	−0.771360	−	0.636399i
$5$	−1.72481	−	1.42303i	−0.771360	−	0.636399i
$6$		0			0
$7$		−	0.949959i		−	0.359051i	−0.983753	−	0.179525i	$-0.942544\pi$
$7$		−	0.949959i		−	0.359051i	0.983753	−	0.179525i	$-0.0574562\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−3.89611			−1.17472			−0.587360	−	0.809326i	$-0.699833\pi$
$11$	−3.89611			−1.17472			−0.587360	+	0.809326i	$0.699833\pi$
$12$		0			0
$13$		−	1.00000i		−	0.277350i
$13$		−	1.00000i		−	0.277350i
$14$		0			0
$15$	1.42303	−	1.72481i	0.367425	−	0.445345i
$16$		0			0
$17$		−	5.90893i		−	1.43313i	−0.697523	−	0.716563i	$-0.745714\pi$
$17$		−	5.90893i		−	1.43313i	0.697523	−	0.716563i	$-0.254286\pi$
$18$		0			0
$19$	6.35541			1.45803			0.729015	−	0.684497i	$-0.239978\pi$
$19$	6.35541			1.45803			0.729015	+	0.684497i	$0.239978\pi$
$20$		0			0
$21$	0.949959			0.207298
$22$		0			0
$23$			3.30537i			0.689217i	0.938747	+	0.344608i	$0.111988\pi$
$23$			3.30537i			0.689217i	−0.938747	+	0.344608i	$0.888012\pi$
$24$		0			0
$25$	0.949959	+	4.90893i	0.189992	+	0.981786i
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$	4.29569			0.797690			0.398845	−	0.917018i	$-0.369411\pi$
$29$	4.29569			0.797690			0.398845	+	0.917018i	$0.369411\pi$
$30$		0			0
$31$	−7.56253			−1.35827			−0.679135	−	0.734013i	$-0.737645\pi$
$31$	−7.56253			−1.35827			−0.679135	+	0.734013i	$0.737645\pi$
$32$		0			0
$33$		−	3.89611i		−	0.678225i
$34$		0			0
$35$	−1.35182	+	1.63850i	−0.228500	+	0.276957i
$36$		0			0
$37$		−	1.30537i		−	0.214601i	−0.994227	−	0.107301i	$-0.965779\pi$
$37$		−	1.30537i		−	0.214601i	0.994227	−	0.107301i	$-0.0342207\pi$
$38$		0			0
$39$	1.00000			0.160128
$40$		0			0
$41$	−6.75499			−1.05495			−0.527476	−	0.849570i	$-0.676862\pi$
$41$	−6.75499			−1.05495			−0.527476	+	0.849570i	$0.676862\pi$
$42$		0			0
$43$			8.29569i			1.26508i	0.774527	+	0.632540i	$0.217988\pi$
$43$			8.29569i			1.26508i	−0.774527	+	0.632540i	$0.782012\pi$
$44$		0			0
$45$	1.72481	+	1.42303i	0.257120	+	0.212133i
$46$		0			0
$47$		−	0.0128228i		−	0.00187040i	−1.00000		0.000935200i	$-0.999702\pi$
$47$		−	0.0128228i		−	0.00187040i	1.00000		0.000935200i	$-0.000297683\pi$
$48$		0			0
$49$	6.09758			0.871083
$50$		0			0
$51$	5.90893			0.827415
$52$		0			0
$53$		−	2.51249i		−	0.345117i	−0.984999	−	0.172559i	$-0.944797\pi$
$53$		−	2.51249i		−	0.345117i	0.984999	−	0.172559i	$-0.0552034\pi$
$54$		0			0
$55$	6.72005	+	5.54428i	0.906132	+	0.747591i
$56$		0			0
$57$			6.35541i			0.841794i
$58$		0			0
$59$	−6.30851			−0.821298			−0.410649	−	0.911793i	$-0.634698\pi$
$59$	−6.30851			−0.821298			−0.410649	+	0.911793i	$0.634698\pi$
$60$		0			0
$61$	1.61324			0.206554			0.103277	−	0.994653i	$-0.467067\pi$
$61$	1.61324			0.206554			0.103277	+	0.994653i	$0.467067\pi$
$62$		0			0
$63$			0.949959i			0.119684i
$64$		0			0
$65$	−1.42303	+	1.72481i	−0.176505	+	0.213937i
$66$		0			0
$67$		−	4.66328i		−	0.569710i	−0.958571	−	0.284855i	$-0.908055\pi$
$67$		−	4.66328i		−	0.569710i	0.958571	−	0.284855i	$-0.0919455\pi$
$68$		0			0
$69$	−3.30537			−0.397919
$70$		0			0
$71$	−7.92175			−0.940139			−0.470069	−	0.882629i	$-0.655771\pi$
$71$	−7.92175			−0.940139			−0.470069	+	0.882629i	$0.655771\pi$
$72$		0			0
$73$			14.5914i			1.70779i	0.520444	+	0.853896i	$0.325767\pi$
$73$			14.5914i			1.70779i	−0.520444	+	0.853896i	$0.674233\pi$
$74$		0			0
$75$	−4.90893	+	0.949959i	−0.566834	+	0.109692i
$76$		0			0
$77$			3.70114i			0.421784i
$78$		0			0
$79$	−16.6004			−1.86769			−0.933845	−	0.357678i	$-0.883569\pi$
$79$	−16.6004			−1.86769			−0.933845	+	0.357678i	$0.883569\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			12.8044i			1.40546i	0.711456	+	0.702731i	$0.248036\pi$
$83$			12.8044i			1.40546i	−0.711456	+	0.702731i	$0.751964\pi$
$84$		0			0
$85$	−8.40860	+	10.1918i	−0.912040	+	1.10546i
$86$		0			0
$87$			4.29569i			0.460546i
$88$		0			0
$89$	−17.6542			−1.87135			−0.935673	−	0.352868i	$-0.885206\pi$
$89$	−17.6542			−1.87135			−0.935673	+	0.352868i	$0.885206\pi$
$90$		0			0
$91$	−0.949959			−0.0995827
$92$		0			0
$93$		−	7.56253i		−	0.784198i
$94$		0			0
$95$	−10.9619	−	9.04395i	−1.12467	−	0.927890i
$96$		0			0
$97$			18.1226i			1.84007i	0.391840	+	0.920033i	$0.371839\pi$
$97$			18.1226i			1.84007i	−0.391840	+	0.920033i	$0.628161\pi$
$98$		0			0
$99$	3.89611			0.391573
$100$		0			0
$101$	11.6921			1.16341			0.581705	−	0.813400i	$-0.302386\pi$
$101$	11.6921			1.16341			0.581705	+	0.813400i	$0.302386\pi$
$102$		0			0
$103$			7.52217i			0.741181i	0.928796	+	0.370591i	$0.120845\pi$
$103$			7.52217i			0.741181i	−0.928796	+	0.370591i	$0.879155\pi$
$104$		0			0
$105$	−1.63850	−	1.35182i	−0.159901	−	0.131924i
$106$		0			0
$107$		−	6.00901i		−	0.580913i	−0.956888	−	0.290457i	$-0.906193\pi$
$107$		−	6.00901i		−	0.580913i	0.956888	−	0.290457i	$-0.0938072\pi$
$108$		0			0
$109$	−1.08139			−0.103579			−0.0517894	−	0.998658i	$-0.516492\pi$
$109$	−1.08139			−0.103579			−0.0517894	+	0.998658i	$0.516492\pi$
$110$		0			0
$111$	1.30537			0.123900
$112$		0			0
$113$			7.29636i			0.686383i	0.939265	+	0.343192i	$0.111508\pi$
$113$			7.29636i			0.686383i	−0.939265	+	0.343192i	$0.888492\pi$
$114$		0			0
$115$	4.70364	−	5.70114i	0.438617	−	0.531634i
$116$		0			0
$117$			1.00000i			0.0924500i
$118$		0			0
$119$	−5.61324			−0.514565
$120$		0			0
$121$	4.17964			0.379967
$122$		0			0
$123$		−	6.75499i		−	0.609077i
$124$		0			0
$125$	5.34706	−	9.81880i	0.478256	−	0.878221i
$126$		0			0
$127$		−	10.6915i		−	0.948714i	−0.880333	−	0.474357i	$-0.842681\pi$
$127$		−	10.6915i		−	0.948714i	0.880333	−	0.474357i	$-0.157319\pi$
$128$		0			0
$129$	−8.29569			−0.730395
$130$		0			0
$131$	0.225811			0.0197292			0.00986458	−	0.999951i	$-0.496860\pi$
$131$	0.225811			0.0197292			0.00986458	+	0.999951i	$0.496860\pi$
$132$		0			0
$133$		−	6.03738i		−	0.523507i
$134$		0			0
$135$	−1.42303	+	1.72481i	−0.122475	+	0.148448i
$136$		0			0
$137$			6.63828i			0.567146i	0.958951	+	0.283573i	$0.0915199\pi$
$137$			6.63828i			0.567146i	−0.958951	+	0.283573i	$0.908480\pi$
$138$		0			0
$139$	3.09758			0.262733			0.131367	−	0.991334i	$-0.458064\pi$
$139$	3.09758			0.262733			0.131367	+	0.991334i	$0.458064\pi$
$140$		0			0
$141$	0.0128228			0.00107988
$142$		0			0
$143$			3.89611i			0.325809i
$144$		0			0
$145$	−7.40926	−	6.11291i	−0.615306	−	0.507649i
$146$		0			0
$147$			6.09758i			0.502920i
$148$		0			0
$149$	−10.9808			−0.899582			−0.449791	−	0.893134i	$-0.648502\pi$
$149$	−10.9808			−0.899582			−0.449791	+	0.893134i	$0.648502\pi$
$150$		0			0
$151$	2.78901			0.226966			0.113483	−	0.993540i	$-0.463799\pi$
$151$	2.78901			0.226966			0.113483	+	0.993540i	$0.463799\pi$
$152$		0			0
$153$			5.90893i			0.477709i
$154$		0			0
$155$	13.0440	+	10.7617i	1.04772	+	0.864403i
$156$		0			0
$157$		−	11.3843i		−	0.908563i	−0.890858	−	0.454281i	$-0.849896\pi$
$157$		−	11.3843i		−	0.908563i	0.890858	−	0.454281i	$-0.150104\pi$
$158$		0			0
$159$	2.51249			0.199253
$160$		0			0
$161$	3.13996			0.247464
$162$		0			0
$163$			2.13144i			0.166947i	0.996510	+	0.0834735i	$0.0266014\pi$
$163$			2.13144i			0.166947i	−0.996510	+	0.0834735i	$0.973399\pi$
$164$		0			0
$165$	−5.54428	+	6.72005i	−0.431622	+	0.523155i
$166$		0			0
$167$			3.00587i			0.232601i	0.993214	+	0.116300i	$0.0371035\pi$
$167$			3.00587i			0.232601i	−0.993214	+	0.116300i	$0.962896\pi$
$168$		0			0
$169$	−1.00000			−0.0769231
$170$		0			0
$171$	−6.35541			−0.486010
$172$		0			0
$173$		−	0.673441i		−	0.0512008i	−0.999672	−	0.0256004i	$-0.991850\pi$
$173$		−	0.673441i		−	0.0512008i	0.999672	−	0.0256004i	$-0.00814974\pi$
$174$		0			0
$175$	4.66328	−	0.902422i	0.352511	−	0.0682167i
$176$		0			0
$177$		−	6.30851i		−	0.474177i
$178$		0			0
$179$	7.81786			0.584334			0.292167	−	0.956367i	$-0.405624\pi$
$179$	7.81786			0.584334			0.292167	+	0.956367i	$0.405624\pi$
$180$		0			0
$181$	−9.53118			−0.708447			−0.354223	−	0.935161i	$-0.615255\pi$
$181$	−9.53118			−0.708447			−0.354223	+	0.935161i	$0.615255\pi$
$182$		0			0
$183$			1.61324i			0.119254i
$184$		0			0
$185$	−1.85758	+	2.25151i	−0.136572	+	0.165535i
$186$		0			0
$187$			23.0218i			1.68352i
$188$		0			0
$189$	−0.949959			−0.0690993
$190$		0			0
$191$	16.1073			1.16548			0.582740	−	0.812659i	$-0.301981\pi$
$191$	16.1073			1.16548			0.582740	+	0.812659i	$0.301981\pi$
$192$		0			0
$193$			13.8967i			1.00031i	0.865936	+	0.500155i	$0.166724\pi$
$193$			13.8967i			1.00031i	−0.865936	+	0.500155i	$0.833276\pi$
$194$		0			0
$195$	−1.72481	−	1.42303i	−0.123516	−	0.101905i
$196$		0			0
$197$			11.7647i			0.838198i	0.907941	+	0.419099i	$0.137654\pi$
$197$			11.7647i			0.838198i	−0.907941	+	0.419099i	$0.862346\pi$
$198$		0			0
$199$	−8.41513			−0.596533			−0.298266	−	0.954483i	$-0.596408\pi$
$199$	−8.41513			−0.596533			−0.298266	+	0.954483i	$0.596408\pi$
$200$		0			0
$201$	4.66328			0.328922
$202$		0			0
$203$		−	4.08073i		−	0.286411i
$204$		0			0
$205$	11.6511	+	9.61257i	0.813748	+	0.671371i
$206$		0			0
$207$		−	3.30537i		−	0.229739i
$208$		0			0
$209$	−24.7613			−1.71278
$210$		0			0
$211$	−11.5402			−0.794459			−0.397230	−	0.917719i	$-0.630028\pi$
$211$	−11.5402			−0.794459			−0.397230	+	0.917719i	$0.630028\pi$
$212$		0			0
$213$		−	7.92175i		−	0.542789i
$214$		0			0
$215$	11.8050	−	14.3085i	0.805097	−	0.975832i
$216$		0			0
$217$			7.18409i			0.487688i
$218$		0			0
$219$	−14.5914			−0.985994
$220$		0			0
$221$	−5.90893			−0.397478
$222$		0			0
$223$			7.82173i			0.523782i	0.965098	+	0.261891i	$0.0843460\pi$
$223$			7.82173i			0.523782i	−0.965098	+	0.261891i	$0.915654\pi$
$224$		0			0
$225$	−0.949959	−	4.90893i	−0.0633306	−	0.327262i
$226$		0			0
$227$			21.1316i			1.40255i	0.712889	+	0.701277i	$0.247386\pi$
$227$			21.1316i			1.40255i	−0.712889	+	0.701277i	$0.752614\pi$
$228$		0			0
$229$	23.5280			1.55477			0.777387	−	0.629022i	$-0.216545\pi$
$229$	23.5280			1.55477			0.777387	+	0.629022i	$0.216545\pi$
$230$		0			0
$231$	−3.70114			−0.243517
$232$		0			0
$233$			6.40612i			0.419679i	0.977736	+	0.209839i	$0.0672941\pi$
$233$			6.40612i			0.419679i	−0.977736	+	0.209839i	$0.932706\pi$
$234$		0			0
$235$	−0.0182473	+	0.0221169i	−0.00119032	+	0.00144275i
$236$		0			0
$237$		−	16.6004i		−	1.07831i
$238$		0			0
$239$	−12.6696			−0.819530			−0.409765	−	0.912191i	$-0.634389\pi$
$239$	−12.6696			−0.819530			−0.409765	+	0.912191i	$0.634389\pi$
$240$		0			0
$241$	−0.206457			−0.0132990			−0.00664952	−	0.999978i	$-0.502117\pi$
$241$	−0.206457			−0.0132990			−0.00664952	+	0.999978i	$0.502117\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$	−10.5172	−	8.67705i	−0.671918	−	0.554356i
$246$		0			0
$247$		−	6.35541i		−	0.404385i
$248$		0			0
$249$	−12.8044			−0.811444
$250$		0			0
$251$	−15.5028			−0.978529			−0.489264	−	0.872135i	$-0.662735\pi$
$251$	−15.5028			−0.978529			−0.489264	+	0.872135i	$0.662735\pi$
$252$		0			0
$253$		−	12.8781i		−	0.809637i
$254$		0			0
$255$	−10.1918	−	8.40860i	−0.638235	−	0.526567i
$256$		0			0
$257$		−	6.11944i		−	0.381720i	−0.981617	−	0.190860i	$-0.938872\pi$
$257$		−	6.11944i		−	0.381720i	0.981617	−	0.190860i	$-0.0611276\pi$
$258$		0			0
$259$	−1.24004			−0.0770526
$260$		0			0
$261$	−4.29569			−0.265897
$262$		0			0
$263$		−	17.1066i		−	1.05484i	−0.849605	−	0.527419i	$-0.823160\pi$
$263$		−	17.1066i		−	1.05484i	0.849605	−	0.527419i	$-0.176840\pi$
$264$		0			0
$265$	−3.57535	+	4.33357i	−0.219632	+	0.266209i
$266$		0			0
$267$		−	17.6542i		−	1.08042i
$268$		0			0
$269$	−27.8981			−1.70098			−0.850490	−	0.525992i	$-0.823694\pi$
$269$	−27.8981			−1.70098			−0.850490	+	0.525992i	$0.823694\pi$
$270$		0			0
$271$	28.6569			1.74079			0.870393	−	0.492358i	$-0.163865\pi$
$271$	28.6569			1.74079			0.870393	+	0.492358i	$0.163865\pi$
$272$		0			0
$273$		−	0.949959i		−	0.0574941i
$274$		0			0
$275$	−3.70114	−	19.1257i	−0.223187	−	1.15332i
$276$		0			0
$277$		−	0.433599i		−	0.0260525i	−0.999915	−	0.0130262i	$-0.995854\pi$
$277$		−	0.433599i		−	0.0260525i	0.999915	−	0.0130262i	$-0.00414649\pi$
$278$		0			0
$279$	7.56253			0.452757
$280$		0			0
$281$	−18.5747			−1.10807			−0.554036	−	0.832492i	$-0.686913\pi$
$281$	−18.5747			−1.10807			−0.554036	+	0.832492i	$0.686913\pi$
$282$		0			0
$283$			21.9360i			1.30396i	0.758237	+	0.651979i	$0.226061\pi$
$283$			21.9360i			1.30396i	−0.758237	+	0.651979i	$0.773939\pi$
$284$		0			0
$285$	9.04395	−	10.9619i	0.535717	−	0.649326i
$286$		0			0
$287$			6.41696i			0.378781i
$288$		0			0
$289$	−17.9154			−1.05385
$290$		0			0
$291$	−18.1226			−1.06236
$292$		0			0
$293$			9.26031i			0.540993i	0.962721	+	0.270497i	$0.0871879\pi$
$293$			9.26031i			0.540993i	−0.962721	+	0.270497i	$0.912812\pi$
$294$		0			0
$295$	10.8810	+	8.97722i	0.633517	+	0.522674i
$296$		0			0
$297$			3.89611i			0.226075i
$298$		0			0
$299$	3.30537			0.191154
$300$		0			0
$301$	7.88056			0.454228
$302$		0			0
$303$			11.6921i			0.671695i
$304$		0			0
$305$	−2.78253	−	2.29569i	−0.159327	−	0.131451i
$306$		0			0
$307$		−	9.15012i		−	0.522225i	−0.965308	−	0.261113i	$-0.915911\pi$
$307$		−	9.15012i		−	0.522225i	0.965308	−	0.261113i	$-0.0840894\pi$
$308$		0			0
$309$	−7.52217			−0.427921
$310$		0			0
$311$	−8.11488			−0.460153			−0.230076	−	0.973173i	$-0.573898\pi$
$311$	−8.11488			−0.460153			−0.230076	+	0.973173i	$0.573898\pi$
$312$		0			0
$313$			4.21952i			0.238501i	0.992864	+	0.119251i	$0.0380492\pi$
$313$			4.21952i			0.238501i	−0.992864	+	0.119251i	$0.961951\pi$
$314$		0			0
$315$	1.35182	−	1.63850i	0.0761665	−	0.0923191i
$316$		0			0
$317$		−	15.4374i		−	0.867053i	−0.901141	−	0.433527i	$-0.857269\pi$
$317$		−	15.4374i		−	0.867053i	0.901141	−	0.433527i	$-0.142731\pi$
$318$		0			0
$319$	−16.7365			−0.937062
$320$		0			0
$321$	6.00901			0.335390
$322$		0			0
$323$		−	37.5537i		−	2.08954i
$324$		0			0
$325$	4.90893	−	0.949959i	0.272298	−	0.0526942i
$326$		0			0
$327$		−	1.08139i		−	0.0598013i
$328$		0			0
$329$	−0.0121811			−0.000671568
$330$		0			0
$331$	−14.6619			−0.805894			−0.402947	−	0.915223i	$-0.632014\pi$
$331$	−14.6619			−0.805894			−0.402947	+	0.915223i	$0.632014\pi$
$332$		0			0
$333$			1.30537i			0.0715337i
$334$		0			0
$335$	−6.63600	+	8.04328i	−0.362563	+	0.439452i
$336$		0			0
$337$		−	26.6094i		−	1.44951i	−0.689009	−	0.724753i	$-0.741954\pi$
$337$		−	26.6094i		−	1.44951i	0.689009	−	0.724753i	$-0.258046\pi$
$338$		0			0
$339$	−7.29636			−0.396284
$340$		0			0
$341$	29.4644			1.59559
$342$		0			0
$343$		−	12.4422i		−	0.671813i
$344$		0			0
$345$	5.70114	+	4.70364i	0.306939	+	0.253236i
$346$		0			0
$347$			3.06153i			0.164352i	0.996618	+	0.0821759i	$0.0261869\pi$
$347$			3.06153i			0.164352i	−0.996618	+	0.0821759i	$0.973813\pi$
$348$		0			0
$349$	22.4273			1.20050			0.600252	−	0.799811i	$-0.295067\pi$
$349$	22.4273			1.20050			0.600252	+	0.799811i	$0.295067\pi$
$350$		0			0
$351$	−1.00000			−0.0533761
$352$		0			0
$353$			23.7147i			1.26220i	0.775700	+	0.631102i	$0.217397\pi$
$353$			23.7147i			1.26220i	−0.775700	+	0.631102i	$0.782603\pi$
$354$		0			0
$355$	13.6635	+	11.2729i	0.725185	+	0.598304i
$356$		0			0
$357$		−	5.61324i		−	0.297084i
$358$		0			0
$359$	−18.6484			−0.984227			−0.492113	−	0.870531i	$-0.663775\pi$
$359$	−18.6484			−0.984227			−0.492113	+	0.870531i	$0.663775\pi$
$360$		0			0
$361$	21.3912			1.12585
$362$		0			0
$363$			4.17964i			0.219374i
$364$		0			0
$365$	20.7640	−	25.1674i	1.08684	−	1.31732i
$366$		0			0
$367$			10.7870i			0.563076i	0.959550	+	0.281538i	$0.0908446\pi$
$367$			10.7870i			0.563076i	−0.959550	+	0.281538i	$0.909155\pi$
$368$		0			0
$369$	6.75499			0.351651
$370$		0			0
$371$	−2.38676			−0.123914
$372$		0			0
$373$			19.0430i			0.986009i	0.870027	+	0.493005i	$0.164101\pi$
$373$			19.0430i			0.986009i	−0.870027	+	0.493005i	$0.835899\pi$
$374$		0			0
$375$	9.81880	+	5.34706i	0.507041	+	0.276121i
$376$		0			0
$377$		−	4.29569i		−	0.221239i
$378$		0			0
$379$	−15.9537			−0.819489			−0.409744	−	0.912200i	$-0.634382\pi$
$379$	−15.9537			−0.819489			−0.409744	+	0.912200i	$0.634382\pi$
$380$		0			0
$381$	10.6915			0.547740
$382$		0			0
$383$		−	24.8044i		−	1.26744i	−0.773561	−	0.633722i	$-0.781526\pi$
$383$		−	24.8044i		−	1.26744i	0.773561	−	0.633722i	$-0.218474\pi$
$384$		0			0
$385$	5.26684	−	6.38377i	0.268423	−	0.325347i
$386$		0			0
$387$		−	8.29569i		−	0.421694i
$388$		0			0
$389$	36.2379			1.83734			0.918668	−	0.395030i	$-0.129266\pi$
$389$	36.2379			1.83734			0.918668	+	0.395030i	$0.129266\pi$
$390$		0			0
$391$	19.5312			0.987734
$392$		0			0
$393$			0.225811i			0.0113906i
$394$		0			0
$395$	28.6326	+	23.6229i	1.44066	+	1.18860i
$396$		0			0
$397$			18.4111i			0.924025i	0.886873	+	0.462013i	$0.152873\pi$
$397$			18.4111i			0.924025i	−0.886873	+	0.462013i	$0.847127\pi$
$398$		0			0
$399$	6.03738			0.302247
$400$		0			0
$401$	24.0667			1.20183			0.600916	−	0.799312i	$-0.294803\pi$
$401$	24.0667			1.20183			0.600916	+	0.799312i	$0.294803\pi$
$402$		0			0
$403$			7.56253i			0.376717i
$404$		0			0
$405$	−1.72481	−	1.42303i	−0.0857066	−	0.0707110i
$406$		0			0
$407$			5.08585i			0.252096i
$408$		0			0
$409$	−35.7102			−1.76575			−0.882877	−	0.469605i	$-0.844396\pi$
$409$	−35.7102			−1.76575			−0.882877	+	0.469605i	$0.844396\pi$
$410$		0			0
$411$	−6.63828			−0.327442
$412$		0			0
$413$			5.99283i			0.294888i
$414$		0			0
$415$	18.2210	−	22.0851i	0.894435	−	1.08412i
$416$		0			0
$417$			3.09758i			0.151689i
$418$		0			0
$419$	21.8241			1.06618			0.533090	−	0.846059i	$-0.321031\pi$
$419$	21.8241			1.06618			0.533090	+	0.846059i	$0.321031\pi$
$420$		0			0
$421$	−0.107706			−0.00524928			−0.00262464	−	0.999997i	$-0.500835\pi$
$421$	−0.107706			−0.00524928			−0.00262464	+	0.999997i	$0.500835\pi$
$422$		0			0
$423$			0.0128228i			0.000623466i
$424$		0			0
$425$	29.0065	−	5.61324i	1.40702	−	0.272282i
$426$		0			0
$427$		−	1.53251i		−	0.0741634i
$428$		0			0
$429$	−3.89611			−0.188106
$430$		0			0
$431$	−32.3215			−1.55687			−0.778437	−	0.627723i	$-0.783987\pi$
$431$	−32.3215			−1.55687			−0.778437	+	0.627723i	$0.783987\pi$
$432$		0			0
$433$			14.5914i			0.701217i	0.936522	+	0.350608i	$0.114025\pi$
$433$			14.5914i			0.701217i	−0.936522	+	0.350608i	$0.885975\pi$
$434$		0			0
$435$	6.11291	−	7.40926i	0.293091	−	0.355247i
$436$		0			0
$437$			21.0070i			1.00490i
$438$		0			0
$439$	−27.4117			−1.30829			−0.654146	−	0.756369i	$-0.726972\pi$
$439$	−27.4117			−1.30829			−0.654146	+	0.756369i	$0.726972\pi$
$440$		0			0
$441$	−6.09758			−0.290361
$442$		0			0
$443$		−	7.19811i		−	0.341993i	−0.985272	−	0.170996i	$-0.945301\pi$
$443$		−	7.19811i		−	0.341993i	0.985272	−	0.170996i	$-0.0546986\pi$
$444$		0			0
$445$	30.4503	+	25.1226i	1.44348	+	1.19092i
$446$		0			0
$447$		−	10.9808i		−	0.519374i
$448$		0			0
$449$	−18.9884			−0.896119			−0.448060	−	0.894004i	$-0.647885\pi$
$449$	−18.9884			−0.896119			−0.448060	+	0.894004i	$0.647885\pi$
$450$		0			0
$451$	26.3182			1.23927
$452$		0			0
$453$			2.78901i			0.131039i
$454$		0			0
$455$	1.63850	+	1.35182i	0.0768141	+	0.0633744i
$456$		0			0
$457$		−	26.3998i		−	1.23493i	−0.786599	−	0.617465i	$-0.788160\pi$
$457$		−	26.3998i		−	1.23493i	0.786599	−	0.617465i	$-0.211840\pi$
$458$		0			0
$459$	−5.90893			−0.275805
$460$		0			0
$461$	−22.0585			−1.02737			−0.513684	−	0.857980i	$-0.671720\pi$
$461$	−22.0585			−1.02737			−0.513684	+	0.857980i	$0.671720\pi$
$462$		0			0
$463$			1.31425i			0.0610781i	0.999534	+	0.0305391i	$0.00972240\pi$
$463$			1.31425i			0.0610781i	−0.999534	+	0.0305391i	$0.990278\pi$
$464$		0			0
$465$	−10.7617	+	13.0440i	−0.499063	+	0.604899i
$466$		0			0
$467$		−	14.8898i		−	0.689017i	−0.938783	−	0.344509i	$-0.888046\pi$
$467$		−	14.8898i		−	0.689017i	0.938783	−	0.344509i	$-0.111954\pi$
$468$		0			0
$469$	−4.42992			−0.204555
$470$		0			0
$471$	11.3843			0.524559
$472$		0			0
$473$		−	32.3209i		−	1.48612i
$474$		0			0
$475$	6.03738	+	31.1982i	0.277014	+	1.43147i
$476$		0			0
$477$			2.51249i			0.115039i
$478$		0			0
$479$	3.03653			0.138743			0.0693713	−	0.997591i	$-0.477901\pi$
$479$	3.03653			0.138743			0.0693713	+	0.997591i	$0.477901\pi$
$480$		0			0
$481$	−1.30537			−0.0595196
$482$		0			0
$483$			3.13996i			0.142873i
$484$		0			0
$485$	25.7890	−	31.2580i	1.17102	−	1.41935i
$486$		0			0
$487$		−	16.2584i		−	0.736741i	−0.929679	−	0.368370i	$-0.879916\pi$
$487$		−	16.2584i		−	0.736741i	0.929679	−	0.368370i	$-0.120084\pi$
$488$		0			0
$489$	−2.13144			−0.0963869
$490$		0			0
$491$	12.5098			0.564558			0.282279	−	0.959332i	$-0.408910\pi$
$491$	12.5098			0.564558			0.282279	+	0.959332i	$0.408910\pi$
$492$		0			0
$493$		−	25.3829i		−	1.14319i
$494$		0			0
$495$	−6.72005	−	5.54428i	−0.302044	−	0.249197i
$496$		0			0
$497$			7.52534i			0.337557i
$498$		0			0
$499$	−0.977442			−0.0437563			−0.0218782	−	0.999761i	$-0.506965\pi$
$499$	−0.977442			−0.0437563			−0.0218782	+	0.999761i	$0.506965\pi$
$500$		0			0
$501$	−3.00587			−0.134292
$502$		0			0
$503$			22.9051i			1.02129i	0.859792	+	0.510644i	$0.170593\pi$
$503$			22.9051i			1.02129i	−0.859792	+	0.510644i	$0.829407\pi$
$504$		0			0
$505$	−20.1667	−	16.6383i	−0.897408	−	0.740394i
$506$		0			0
$507$		−	1.00000i		−	0.0444116i
$508$		0			0
$509$	13.2463			0.587132			0.293566	−	0.955939i	$-0.405158\pi$
$509$	13.2463			0.587132			0.293566	+	0.955939i	$0.405158\pi$
$510$		0			0
$511$	13.8612			0.613184
$512$		0			0
$513$		−	6.35541i		−	0.280598i
$514$		0			0
$515$	10.7043	−	12.9743i	0.471687	−	0.571717i
$516$		0			0
$517$			0.0499590i			0.00219720i
$518$		0			0
$519$	0.673441			0.0295608
$520$		0			0
$521$	−8.70493			−0.381370			−0.190685	−	0.981651i	$-0.561071\pi$
$521$	−8.70493			−0.381370			−0.190685	+	0.981651i	$0.561071\pi$
$522$		0			0
$523$			29.4014i			1.28563i	0.766020	+	0.642817i	$0.222234\pi$
$523$			29.4014i			1.28563i	−0.766020	+	0.642817i	$0.777766\pi$
$524$		0			0
$525$	0.902422	+	4.66328i	0.0393849	+	0.203522i
$526$		0			0
$527$			44.6865i			1.94657i
$528$		0			0
$529$	12.0745			0.524980
$530$		0			0
$531$	6.30851			0.273766
$532$		0			0
$533$			6.75499i			0.292591i
$534$		0			0
$535$	−8.55102	+	10.3644i	−0.369693	+	0.448093i
$536$		0			0
$537$			7.81786i			0.337365i
$538$		0			0
$539$	−23.7568			−1.02328
$540$		0			0
$541$	−1.40100			−0.0602335			−0.0301168	−	0.999546i	$-0.509588\pi$
$541$	−1.40100			−0.0602335			−0.0301168	+	0.999546i	$0.509588\pi$
$542$		0			0
$543$		−	9.53118i		−	0.409022i
$544$		0			0
$545$	1.86520	+	1.53886i	0.0798965	+	0.0659175i
$546$		0			0
$547$			2.42904i			0.103858i	0.998651	+	0.0519292i	$0.0165370\pi$
$547$			2.42904i			0.103858i	−0.998651	+	0.0519292i	$0.983463\pi$
$548$		0			0
$549$	−1.61324			−0.0688513
$550$		0			0
$551$	27.3009			1.16306
$552$		0			0
$553$			15.7697i			0.670595i
$554$		0			0
$555$	−2.25151	−	1.85758i	−0.0955714	−	0.0788499i
$556$		0			0
$557$		−	14.2429i		−	0.603491i	−0.953388	−	0.301746i	$-0.902431\pi$
$557$		−	14.2429i		−	0.603491i	0.953388	−	0.301746i	$-0.0975693\pi$
$558$		0			0
$559$	8.29569			0.350870
$560$		0			0
$561$	−23.0218			−0.971982
$562$		0			0
$563$		−	38.1924i		−	1.60962i	−0.593533	−	0.804810i	$-0.702267\pi$
$563$		−	38.1924i		−	1.60962i	0.593533	−	0.804810i	$-0.297733\pi$
$564$		0			0
$565$	10.3829	−	12.5848i	0.436814	−	0.529449i
$566$		0			0
$567$		−	0.949959i		−	0.0398945i
$568$		0			0
$569$	22.1387			0.928104			0.464052	−	0.885808i	$-0.346395\pi$
$569$	22.1387			0.928104			0.464052	+	0.885808i	$0.346395\pi$
$570$		0			0
$571$	44.8956			1.87882			0.939412	−	0.342791i	$-0.111372\pi$
$571$	44.8956			1.87882			0.939412	+	0.342791i	$0.111372\pi$
$572$		0			0
$573$			16.1073i			0.672890i
$574$		0			0
$575$	−16.2258	+	3.13996i	−0.676663	+	0.130945i
$576$		0			0
$577$		−	4.07248i		−	0.169540i	−0.996401	−	0.0847698i	$-0.972984\pi$
$577$		−	4.07248i		−	0.169540i	0.996401	−	0.0847698i	$-0.0270155\pi$
$578$		0			0
$579$	−13.8967			−0.577529
$580$		0			0
$581$	12.1636			0.504632
$582$		0			0
$583$			9.78893i			0.405416i
$584$		0			0
$585$	1.42303	−	1.72481i	0.0588351	−	0.0713122i
$586$		0			0
$587$		−	24.5786i		−	1.01447i	−0.861809	−	0.507233i	$-0.830668\pi$
$587$		−	24.5786i		−	1.01447i	0.861809	−	0.507233i	$-0.169332\pi$
$588$		0			0
$589$	−48.0630			−1.98040
$590$		0			0
$591$	−11.7647			−0.483934
$592$		0			0
$593$		−	0.191663i		−	0.00787065i	−0.999992	−	0.00393532i	$-0.998747\pi$
$593$		−	0.191663i		−	0.00787065i	0.999992	−	0.00393532i	$-0.00125266\pi$
$594$		0			0
$595$	9.68179	+	7.98782i	0.396915	+	0.327469i
$596$		0			0
$597$		−	8.41513i		−	0.344408i
$598$		0			0
$599$	22.5977			0.923316			0.461658	−	0.887058i	$-0.347255\pi$
$599$	22.5977			0.923316			0.461658	+	0.887058i	$0.347255\pi$
$600$		0			0
$601$	17.0155			0.694077			0.347039	−	0.937851i	$-0.387187\pi$
$601$	17.0155			0.694077			0.347039	+	0.937851i	$0.387187\pi$
$602$		0			0
$603$			4.66328i			0.189903i
$604$		0			0
$605$	−7.20910	−	5.94776i	−0.293091	−	0.241811i
$606$		0			0
$607$			34.4651i			1.39889i	0.714684	+	0.699447i	$0.246570\pi$
$607$			34.4651i			1.39889i	−0.714684	+	0.699447i	$0.753430\pi$
$608$		0			0
$609$	4.08073			0.165359
$610$		0			0
$611$	−0.0128228			−0.000518755
$612$		0			0
$613$			7.02682i			0.283810i	0.989880	+	0.141905i	$0.0453228\pi$
$613$			7.02682i			0.283810i	−0.989880	+	0.141905i	$0.954677\pi$
$614$		0			0
$615$	−9.61257	+	11.6511i	−0.387616	+	0.469818i
$616$		0			0
$617$			12.1122i			0.487620i	0.969823	+	0.243810i	$0.0783973\pi$
$617$			12.1122i			0.487620i	−0.969823	+	0.243810i	$0.921603\pi$
$618$		0			0
$619$	30.8710			1.24081			0.620406	−	0.784281i	$-0.286968\pi$
$619$	30.8710			1.24081			0.620406	+	0.784281i	$0.286968\pi$
$620$		0			0
$621$	3.30537			0.132640
$622$		0			0
$623$			16.7708i			0.671908i
$624$		0			0
$625$	−23.1952	+	9.32656i	−0.927806	+	0.373062i
$626$		0			0
$627$		−	24.7613i		−	0.988873i
$628$		0			0
$629$	−7.71332			−0.307550
$630$		0			0
$631$	−9.48047			−0.377412			−0.188706	−	0.982034i	$-0.560429\pi$
$631$	−9.48047			−0.377412			−0.188706	+	0.982034i	$0.560429\pi$
$632$		0			0
$633$		−	11.5402i		−	0.458681i
$634$		0			0
$635$	−15.2143	+	18.4408i	−0.603761	+	0.731800i
$636$		0			0
$637$		−	6.09758i		−	0.241595i
$638$		0			0
$639$	7.92175			0.313380
$640$		0			0
$641$	17.3964			0.687118			0.343559	−	0.939131i	$-0.388367\pi$
$641$	17.3964			0.687118			0.343559	+	0.939131i	$0.388367\pi$
$642$		0			0
$643$		−	33.9506i		−	1.33888i	−0.742866	−	0.669440i	$-0.766534\pi$
$643$		−	33.9506i		−	1.33888i	0.742866	−	0.669440i	$-0.233466\pi$
$644$		0			0
$645$	14.3085	+	11.8050i	0.563397	+	0.464823i
$646$		0			0
$647$		−	9.18548i		−	0.361118i	−0.983564	−	0.180559i	$-0.942209\pi$
$647$		−	9.18548i		−	0.361118i	0.983564	−	0.180559i	$-0.0577908\pi$
$648$		0			0
$649$	24.5786			0.964796
$650$		0			0
$651$	−7.18409			−0.281567
$652$		0			0
$653$			3.54781i			0.138837i	0.997588	+	0.0694183i	$0.0221143\pi$
$653$			3.54781i			0.138837i	−0.997588	+	0.0694183i	$0.977886\pi$
$654$		0			0
$655$	−0.389481	−	0.321336i	−0.0152183	−	0.0125556i
$656$		0			0
$657$		−	14.5914i		−	0.569264i
$658$		0			0
$659$	−15.5339			−0.605115			−0.302557	−	0.953131i	$-0.597840\pi$
$659$	−15.5339			−0.605115			−0.302557	+	0.953131i	$0.597840\pi$
$660$		0			0
$661$	1.60878			0.0625745			0.0312872	−	0.999510i	$-0.490039\pi$
$661$	1.60878			0.0625745			0.0312872	+	0.999510i	$0.490039\pi$
$662$		0			0
$663$		−	5.90893i		−	0.229484i
$664$		0			0
$665$	−8.59138	+	10.4133i	−0.333159	+	0.403812i
$666$		0			0
$667$			14.1988i			0.549781i
$668$		0			0
$669$	−7.82173			−0.302405
$670$		0			0
$671$	−6.28535			−0.242643
$672$		0			0
$673$			8.18214i			0.315398i	0.987487	+	0.157699i	$0.0504076\pi$
$673$			8.18214i			0.315398i	−0.987487	+	0.157699i	$0.949592\pi$
$674$		0			0
$675$	4.90893	−	0.949959i	0.188945	−	0.0365639i
$676$		0			0
$677$		−	42.7063i		−	1.64134i	−0.571405	−	0.820668i	$-0.693601\pi$
$677$		−	42.7063i		−	1.64134i	0.571405	−	0.820668i	$-0.306399\pi$
$678$		0			0
$679$	17.2157			0.660677
$680$		0			0
$681$	−21.1316			−0.809764
$682$		0			0
$683$			9.14962i			0.350100i	0.984560	+	0.175050i	$0.0560088\pi$
$683$			9.14962i			0.350100i	−0.984560	+	0.175050i	$0.943991\pi$
$684$		0			0
$685$	9.44648	−	11.4498i	0.360931	−	0.437474i
$686$		0			0
$687$			23.5280i			0.897649i
$688$		0			0
$689$	−2.51249			−0.0957182
$690$		0			0
$691$	16.9918			0.646398			0.323199	−	0.946331i	$-0.395242\pi$
$691$	16.9918			0.646398			0.323199	+	0.946331i	$0.395242\pi$
$692$		0			0
$693$		−	3.70114i		−	0.140595i
$694$		0			0
$695$	−5.34274	−	4.40795i	−0.202662	−	0.167203i
$696$		0			0
$697$			39.9148i			1.51188i
$698$		0			0
$699$	−6.40612			−0.242302
$700$		0			0
$701$	−40.8320			−1.54220			−0.771101	−	0.636712i	$-0.780294\pi$
$701$	−40.8320			−1.54220			−0.771101	+	0.636712i	$0.780294\pi$
$702$		0			0
$703$		−	8.29614i		−	0.312895i
$704$		0			0
$705$	−0.0221169	−	0.0182473i	−0.000832972	−	0.000687232i
$706$		0			0
$707$		−	11.1070i		−	0.417723i
$708$		0			0
$709$	38.0193			1.42784			0.713922	−	0.700225i	$-0.246917\pi$
$709$	38.0193			1.42784			0.713922	+	0.700225i	$0.246917\pi$
$710$		0			0
$711$	16.6004			0.622563
$712$		0			0
$713$		−	24.9969i		−	0.936143i
$714$		0			0
$715$	5.54428	−	6.72005i	0.207344	−	0.251316i
$716$		0			0
$717$		−	12.6696i		−	0.473156i
$718$		0			0
$719$	27.9175			1.04115			0.520573	−	0.853817i	$-0.325718\pi$
$719$	27.9175			1.04115			0.520573	+	0.853817i	$0.325718\pi$
$720$		0			0
$721$	7.14575			0.266122
$722$		0			0
$723$		−	0.206457i		−	0.00767820i
$724$		0			0
$725$	4.08073	+	21.0872i	0.151554	+	0.783160i
$726$		0			0
$727$			4.31960i			0.160205i	0.996787	+	0.0801026i	$0.0255248\pi$
$727$			4.31960i			0.160205i	−0.996787	+	0.0801026i	$0.974475\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	49.0186			1.81302
$732$		0			0
$733$			13.1669i			0.486330i	0.969985	+	0.243165i	$0.0781857\pi$
$733$			13.1669i			0.486330i	−0.969985	+	0.243165i	$0.921814\pi$
$734$		0			0
$735$	8.67705	−	10.5172i	0.320058	−	0.387932i
$736$		0			0
$737$			18.1686i			0.669250i
$738$		0			0
$739$	2.14218			0.0788014			0.0394007	−	0.999223i	$-0.487455\pi$
$739$	2.14218			0.0788014			0.0394007	+	0.999223i	$0.487455\pi$
$740$		0			0
$741$	6.35541			0.233472
$742$		0			0
$743$		−	24.4771i		−	0.897979i	−0.893537	−	0.448990i	$-0.851784\pi$
$743$		−	24.4771i		−	0.897979i	0.893537	−	0.448990i	$-0.148216\pi$
$744$		0			0
$745$	18.9398	+	15.6260i	0.693902	+	0.572494i
$746$		0			0
$747$		−	12.8044i		−	0.468487i
$748$		0			0
$749$	−5.70831			−0.208577
$750$		0			0
$751$	12.0058			0.438097			0.219049	−	0.975714i	$-0.429705\pi$
$751$	12.0058			0.438097			0.219049	+	0.975714i	$0.429705\pi$
$752$		0			0
$753$		−	15.5028i		−	0.564954i
$754$		0			0
$755$	−4.81052	−	3.96885i	−0.175073	−	0.144441i
$756$		0			0
$757$			16.8442i			0.612212i	0.951997	+	0.306106i	$0.0990262\pi$
$757$			16.8442i			0.612212i	−0.951997	+	0.306106i	$0.900974\pi$
$758$		0			0
$759$	12.8781			0.467444
$760$		0			0
$761$	3.97932			0.144250			0.0721251	−	0.997396i	$-0.477022\pi$
$761$	3.97932			0.144250			0.0721251	+	0.997396i	$0.477022\pi$
$762$		0			0
$763$			1.02728i			0.0371900i
$764$		0			0
$765$	8.40860	−	10.1918i	0.304013	−	0.368485i
$766$		0			0
$767$			6.30851i			0.227787i
$768$		0			0
$769$	−15.5176			−0.559579			−0.279790	−	0.960061i	$-0.590265\pi$
$769$	−15.5176			−0.559579			−0.279790	+	0.960061i	$0.590265\pi$
$770$		0			0
$771$	6.11944			0.220386
$772$		0			0
$773$		−	44.4624i		−	1.59920i	−0.600531	−	0.799601i	$-0.705044\pi$
$773$		−	44.4624i		−	1.59920i	0.600531	−	0.799601i	$-0.294956\pi$
$774$		0			0
$775$	−7.18409	−	37.1239i	−0.258060	−	1.33353i
$776$		0			0
$777$		−	1.24004i		−	0.0444864i
$778$		0			0
$779$	−42.9307			−1.53815
$780$		0			0
$781$	30.8640			1.10440
$782$		0			0
$783$		−	4.29569i		−	0.153515i
$784$		0			0
$785$	−16.2002	+	19.6357i	−0.578209	+	0.700829i
$786$		0			0
$787$		−	32.0619i		−	1.14288i	−0.820643	−	0.571441i	$-0.806384\pi$
$787$		−	32.0619i		−	1.14288i	0.820643	−	0.571441i	$-0.193616\pi$
$788$		0			0
$789$	17.1066			0.609011
$790$		0			0
$791$	6.93124			0.246446
$792$		0			0
$793$		−	1.61324i		−	0.0572878i
$794$		0			0
$795$	−4.33357	−	3.57535i	−0.153696	−	0.126805i
$796$		0			0
$797$			24.7378i			0.876260i	0.898912	+	0.438130i	$0.144359\pi$
$797$			24.7378i			0.876260i	−0.898912	+	0.438130i	$0.855641\pi$
$798$		0			0
$799$	−0.0757691			−0.00268052
$800$		0			0
$801$	17.6542			0.623782
$802$		0			0
$803$		−	56.8496i		−	2.00618i
$804$		0			0
$805$	−5.41585	−	4.46827i	−0.190884	−	0.157486i
$806$		0			0
$807$		−	27.8981i		−	0.982061i
$808$		0			0
$809$	−11.3400			−0.398694			−0.199347	−	0.979929i	$-0.563882\pi$
$809$	−11.3400			−0.398694			−0.199347	+	0.979929i	$0.563882\pi$
$810$		0			0
$811$	−8.96070			−0.314653			−0.157326	−	0.987547i	$-0.550287\pi$
$811$	−8.96070			−0.314653			−0.157326	+	0.987547i	$0.550287\pi$
$812$		0			0
$813$			28.6569i			1.00504i
$814$		0			0
$815$	3.03310	−	3.67633i	0.106245	−	0.128776i
$816$		0			0
$817$			52.7225i			1.84453i
$818$		0			0
$819$	0.949959			0.0331942
$820$		0			0
$821$	5.28500			0.184448			0.0922239	−	0.995738i	$-0.470602\pi$
$821$	5.28500			0.184448			0.0922239	+	0.995738i	$0.470602\pi$
$822$		0			0
$823$		−	11.2463i		−	0.392023i	−0.980602	−	0.196012i	$-0.937201\pi$
$823$		−	11.2463i		−	0.392023i	0.980602	−	0.196012i	$-0.0627990\pi$
$824$		0			0
$825$	19.1257	−	3.70114i	0.665872	−	0.128857i
$826$		0			0
$827$			42.9625i			1.49395i	0.664851	+	0.746976i	$0.268495\pi$
$827$			42.9625i			1.49395i	−0.664851	+	0.746976i	$0.731505\pi$
$828$		0			0
$829$	−21.1444			−0.734376			−0.367188	−	0.930147i	$-0.619679\pi$
$829$	−21.1444			−0.734376			−0.367188	+	0.930147i	$0.619679\pi$
$830$		0			0
$831$	0.433599			0.0150414
$832$		0			0
$833$		−	36.0302i		−	1.24837i
$834$		0			0
$835$	4.27744	−	5.18455i	0.148027	−	0.179419i
$836$		0			0
$837$			7.56253i			0.261399i
$838$		0			0
$839$	−26.3671			−0.910293			−0.455146	−	0.890417i	$-0.650413\pi$
$839$	−26.3671			−0.910293			−0.455146	+	0.890417i	$0.650413\pi$
$840$		0			0
$841$	−10.5470			−0.363691
$842$		0			0
$843$		−	18.5747i		−	0.639746i
$844$		0			0
$845$	1.72481	+	1.42303i	0.0593354	+	0.0489538i
$846$		0			0
$847$		−	3.97048i		−	0.136427i
$848$		0			0
$849$	−21.9360			−0.752840
$850$		0			0
$851$	4.31472			0.147907
$852$		0			0
$853$		−	52.4061i		−	1.79435i	−0.441676	−	0.897175i	$-0.645616\pi$
$853$		−	52.4061i		−	1.79435i	0.441676	−	0.897175i	$-0.354384\pi$
$854$		0			0
$855$	10.9619	+	9.04395i	0.374889	+	0.309297i
$856$		0			0
$857$			24.6382i			0.841625i	0.907148	+	0.420813i	$0.138255\pi$
$857$			24.6382i			0.841625i	−0.907148	+	0.420813i	$0.861745\pi$
$858$		0			0
$859$	−47.0110			−1.60399			−0.801997	−	0.597329i	$-0.796229\pi$
$859$	−47.0110			−1.60399			−0.801997	+	0.597329i	$0.796229\pi$
$860$		0			0
$861$	−6.41696			−0.218690
$862$		0			0
$863$		−	14.9378i		−	0.508490i	−0.967140	−	0.254245i	$-0.918173\pi$
$863$		−	14.9378i		−	0.508490i	0.967140	−	0.254245i	$-0.0818269\pi$
$864$		0			0
$865$	−0.958328	+	1.16156i	−0.0325841	+	0.0394942i
$866$		0			0
$867$		−	17.9154i		−	0.608440i
$868$		0			0
$869$	64.6769			2.19401
$870$		0			0
$871$	−4.66328			−0.158009
$872$		0			0
$873$		−	18.1226i		−	0.613356i
$874$		0			0
$875$	−9.32746	−	5.07949i	−0.315326	−	0.171718i
$876$		0			0
$877$		−	29.3418i		−	0.990801i	−0.868665	−	0.495400i	$-0.835021\pi$
$877$		−	29.3418i		−	0.990801i	0.868665	−	0.495400i	$-0.164979\pi$
$878$		0			0
$879$	−9.26031			−0.312342
$880$		0			0
$881$	12.1590			0.409647			0.204824	−	0.978799i	$-0.434338\pi$
$881$	12.1590			0.409647			0.204824	+	0.978799i	$0.434338\pi$
$882$		0			0
$883$			31.1431i			1.04805i	0.851703	+	0.524024i	$0.175570\pi$
$883$			31.1431i			1.04805i	−0.851703	+	0.524024i	$0.824430\pi$
$884$		0			0
$885$	−8.97722	+	10.8810i	−0.301766	+	0.365761i
$886$		0			0
$887$			2.56117i			0.0859955i	0.999075	+	0.0429978i	$0.0136908\pi$
$887$			2.56117i			0.0859955i	−0.999075	+	0.0429978i	$0.986309\pi$
$888$		0			0
$889$	−10.1564			−0.340636
$890$		0			0
$891$	−3.89611			−0.130524
$892$		0			0
$893$		−	0.0814942i		−	0.00272710i
$894$		0			0
$895$	−13.4843	−	11.1251i	−0.450732	−	0.371870i
$896$		0			0
$897$			3.30537i			0.110363i
$898$		0			0
$899$	−32.4863			−1.08348
$900$		0			0
$901$	−14.8461			−0.494596
$902$		0			0
$903$			7.88056i			0.262249i
$904$		0			0
$905$	16.4395	+	13.5632i	0.546467	+	0.450855i
$906$		0			0
$907$		−	34.5273i		−	1.14646i	−0.819394	−	0.573231i	$-0.805690\pi$
$907$		−	34.5273i		−	1.14646i	0.819394	−	0.573231i	$-0.194310\pi$
$908$		0			0
$909$	−11.6921			−0.387803
$910$		0			0
$911$	16.4985			0.546619			0.273309	−	0.961926i	$-0.411882\pi$
$911$	16.4985			0.546619			0.273309	+	0.961926i	$0.411882\pi$
$912$		0			0
$913$		−	49.8872i		−	1.65102i
$914$		0			0
$915$	2.29569	−	2.78253i	0.0758932	−	0.0919877i
$916$		0			0
$917$		−	0.214511i		−	0.00708377i
$918$		0			0
$919$	29.8399			0.984327			0.492163	−	0.870503i	$-0.336206\pi$
$919$	29.8399			0.984327			0.492163	+	0.870503i	$0.336206\pi$
$920$		0			0
$921$	9.15012			0.301507
$922$		0			0
$923$			7.92175i			0.260748i
$924$		0			0
$925$	6.40795	−	1.24004i	0.210692	−	0.0407724i
$926$		0			0
$927$		−	7.52217i		−	0.247060i
$928$		0			0
$929$	−29.9391			−0.982270			−0.491135	−	0.871084i	$-0.663418\pi$
$929$	−29.9391			−0.982270			−0.491135	+	0.871084i	$0.663418\pi$
$930$		0			0
$931$	38.7526			1.27007
$932$		0			0
$933$		−	8.11488i		−	0.265669i
$934$		0			0
$935$	32.7608	−	39.7083i	1.07139	−	1.29860i
$936$		0			0
$937$		−	32.7542i		−	1.07003i	−0.844842	−	0.535016i	$-0.820306\pi$
$937$		−	32.7542i		−	1.07003i	0.844842	−	0.535016i	$-0.179694\pi$
$938$		0			0
$939$	−4.21952			−0.137699
$940$		0			0
$941$	−26.6679			−0.869349			−0.434675	−	0.900588i	$-0.643137\pi$
$941$	−26.6679			−0.869349			−0.434675	+	0.900588i	$0.643137\pi$
$942$		0			0
$943$		−	22.3277i		−	0.727091i
$944$		0			0
$945$	1.63850	+	1.35182i	0.0533004	+	0.0439748i
$946$		0			0
$947$			16.9685i			0.551402i	0.961243	+	0.275701i	$0.0889099\pi$
$947$			16.9685i			0.551402i	−0.961243	+	0.275701i	$0.911090\pi$
$948$		0			0
$949$	14.5914			0.473656
$950$		0			0
$951$	15.4374			0.500593
$952$		0			0
$953$			45.6056i			1.47731i	0.674084	+	0.738655i	$0.264539\pi$
$953$			45.6056i			1.47731i	−0.674084	+	0.738655i	$0.735461\pi$
$954$		0			0
$955$	−27.7820	−	22.9211i	−0.899004	−	0.741711i
$956$		0			0
$957$		−	16.7365i		−	0.541013i
$958$		0			0
$959$	6.30609			0.203634
$960$		0			0
$961$	26.1919			0.844899
$962$		0			0
$963$			6.00901i			0.193638i
$964$		0			0
$965$	19.7755	−	23.9693i	0.636596	−	0.771599i
$966$		0			0
$967$		−	7.77794i		−	0.250122i	−0.992149	−	0.125061i	$-0.960087\pi$
$967$		−	7.77794i		−	0.250122i	0.992149	−	0.125061i	$-0.0399126\pi$
$968$		0			0
$969$	37.5537			1.20640
$970$		0			0
$971$	−14.1572			−0.454327			−0.227163	−	0.973857i	$-0.572945\pi$
$971$	−14.1572			−0.454327			−0.227163	+	0.973857i	$0.572945\pi$
$972$		0			0
$973$		−	2.94257i		−	0.0943345i
$974$		0			0
$975$	0.949959	+	4.90893i	0.0304230	+	0.157212i
$976$		0			0
$977$		−	13.0985i		−	0.419058i	−0.977802	−	0.209529i	$-0.932807\pi$
$977$		−	13.0985i		−	0.419058i	0.977802	−	0.209529i	$-0.0671930\pi$
$978$		0			0
$979$	68.7828			2.19831
$980$		0			0
$981$	1.08139			0.0345263
$982$		0			0
$983$			49.1176i			1.56661i	0.621638	+	0.783305i	$0.286467\pi$
$983$			49.1176i			1.56661i	−0.621638	+	0.783305i	$0.713533\pi$
$984$		0			0
$985$	16.7415	−	20.2919i	0.533429	−	0.646552i
$986$		0			0
$987$		−	0.0121811i		−	0.000387730i
$988$		0			0
$989$	−27.4203			−0.871915
$990$		0			0
$991$	−17.0358			−0.541161			−0.270581	−	0.962697i	$-0.587216\pi$
$991$	−17.0358			−0.541161			−0.270581	+	0.962697i	$0.587216\pi$
$992$		0			0
$993$		−	14.6619i		−	0.465283i
$994$		0			0
$995$	14.5145	+	11.9750i	0.460141	+	0.379633i
$996$		0			0
$997$		−	8.01802i		−	0.253933i	−0.991907	−	0.126967i	$-0.959476\pi$
$997$		−	8.01802i		−	0.253933i	0.991907	−	0.126967i	$-0.0405241\pi$
$998$		0			0
$999$	−1.30537			−0.0413000

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3120.2.l.p.1249.7		10
4.3	odd	2		195.2.c.b.79.1	✓	10
5.4	even	2	inner	3120.2.l.p.1249.2		10
12.11	even	2		585.2.c.c.469.10		10
20.3	even	4		975.2.a.s.1.1		5
20.7	even	4		975.2.a.r.1.5		5
20.19	odd	2		195.2.c.b.79.10	yes	10
60.23	odd	4		2925.2.a.bm.1.5		5
60.47	odd	4		2925.2.a.bl.1.1		5
60.59	even	2		585.2.c.c.469.1		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
195.2.c.b.79.1	✓	10	4.3	odd	2
195.2.c.b.79.10	yes	10	20.19	odd	2
585.2.c.c.469.1		10	60.59	even	2
585.2.c.c.469.10		10	12.11	even	2
975.2.a.r.1.5		5	20.7	even	4
975.2.a.s.1.1		5	20.3	even	4
2925.2.a.bl.1.1		5	60.47	odd	4
2925.2.a.bm.1.5		5	60.23	odd	4
3120.2.l.p.1249.2		10	5.4	even	2	inner
3120.2.l.p.1249.7		10	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3120.l (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +(-1.72481 - 1.42303i) q^{5} -0.949959i q^{7} -1.00000 q^{9} -3.89611 q^{11} -1.00000i q^{13} +(1.42303 - 1.72481i) q^{15} -5.90893i q^{17} +6.35541 q^{19} +0.949959 q^{21} +3.30537i q^{23} +(0.949959 + 4.90893i) q^{25} -1.00000i q^{27} +4.29569 q^{29} -7.56253 q^{31} -3.89611i q^{33} +(-1.35182 + 1.63850i) q^{35} -1.30537i q^{37} +1.00000 q^{39} -6.75499 q^{41} +8.29569i q^{43} +(1.72481 + 1.42303i) q^{45} -0.0128228i q^{47} +6.09758 q^{49} +5.90893 q^{51} -2.51249i q^{53} +(6.72005 + 5.54428i) q^{55} +6.35541i q^{57} -6.30851 q^{59} +1.61324 q^{61} +0.949959i q^{63} +(-1.42303 + 1.72481i) q^{65} -4.66328i q^{67} -3.30537 q^{69} -7.92175 q^{71} +14.5914i q^{73} +(-4.90893 + 0.949959i) q^{75} +3.70114i q^{77} -16.6004 q^{79} +1.00000 q^{81} +12.8044i q^{83} +(-8.40860 + 10.1918i) q^{85} +4.29569i q^{87} -17.6542 q^{89} -0.949959 q^{91} -7.56253i q^{93} +(-10.9619 - 9.04395i) q^{95} +18.1226i q^{97} +3.89611 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 2 q^{5} - 10 q^{9} - 10 q^{11} + 16 q^{19} + 10 q^{21} + 10 q^{25} - 16 q^{29} - 24 q^{31} - 12 q^{35} + 10 q^{39} + 10 q^{41} + 2 q^{45} - 44 q^{49} + 10 q^{51} - 2 q^{55} + 16 q^{59} + 26 q^{61}+ \cdots + 10 q^{99}+O(q^{100}) \) 10 * q - 2 * q^5 - 10 * q^9 - 10 * q^11 + 16 * q^19 + 10 * q^21 + 10 * q^25 - 16 * q^29 - 24 * q^31 - 12 * q^35 + 10 * q^39 + 10 * q^41 + 2 * q^45 - 44 * q^49 + 10 * q^51 - 2 * q^55 + 16 * q^59 + 26 * q^61 + 14 * q^69 - 10 * q^71 + 2 * q^79 + 10 * q^81 - 4 * q^85 - 38 * q^89 - 10 * q^91 + 48 * q^95 + 10 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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