LMFDB - Embedded newform 731.2.d.b.560.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [731,2,Mod(560,731)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(731, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("731.560");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 731 = 17 \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	731.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$5.83706438776$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 20x^{6} + 129x^{4} + 323x^{2} + 256 $ x^8 + 20x^6 + 129x^4 + 323*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			560.7
Root			$2.15629i$ of defining polynomial
Character	$\chi$	$=$	731.560
Dual form			731.2.d.b.560.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.15629i q^{3} -2.00000 q^{4} +2.88045i q^{5} +2.59528i q^{7} -1.64959 q^{9} +O(q^{10})$	\(q+2.15629i q^{3} -2.00000 q^{4} +2.88045i q^{5} +2.59528i q^{7} -1.64959 q^{9} +4.55588i q^{11} -4.31258i q^{12} +4.56150 q^{13} -6.21109 q^{15} +4.00000 q^{16} +(-4.07885 + 0.602510i) q^{17} +2.00000 q^{19} -5.76090i q^{20} -5.59619 q^{21} -3.67790i q^{23} -3.29700 q^{25} +2.91187i q^{27} -5.19057i q^{28} -5.76090i q^{29} +3.10756i q^{31} -9.82381 q^{33} -7.47559 q^{35} +3.29919 q^{36} -1.27831i q^{37} +9.83593i q^{39} -11.1948i q^{41} +1.00000 q^{43} -9.11177i q^{44} -4.75158i q^{45} -2.24578 q^{47} +8.62517i q^{48} +0.264498 q^{49} +(-1.29919 - 8.79518i) q^{51} -9.12300 q^{52} +11.9119 q^{53} -13.1230 q^{55} +4.31258i q^{57} +0.297003 q^{59} +12.4222 q^{60} +8.38761i q^{61} -4.28117i q^{63} -8.00000 q^{64} +13.1392i q^{65} +2.01871 q^{67} +(8.15769 - 1.20502i) q^{68} +7.93062 q^{69} -3.58842i q^{71} -3.19704i q^{73} -7.10930i q^{75} -4.00000 q^{76} -11.8238 q^{77} -1.02456i q^{79} +11.5218i q^{80} -11.2276 q^{81} +5.61490 q^{83} +11.1924 q^{84} +(-1.73550 - 11.7489i) q^{85} +12.4222 q^{87} -11.2298 q^{89} +11.8384i q^{91} +7.35579i q^{92} -6.70081 q^{93} +5.76090i q^{95} -2.08301i q^{97} -7.51536i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 16 q^{4} - 16 q^{9}+O(q^{10}) $ 8 * q - 16 * q^4 - 16 * q^9	$ 8 q - 16 q^{4} - 16 q^{9} - 10 q^{13} - 6 q^{15} + 32 q^{16} - 3 q^{17} + 16 q^{19} - 32 q^{21} - 8 q^{25} + 20 q^{33} + 12 q^{35} + 32 q^{36} + 8 q^{43} - 8 q^{47} - 26 q^{49} - 16 q^{51} + 20 q^{52} + 46 q^{53} - 12 q^{55} - 16 q^{59} + 12 q^{60} - 64 q^{64} - 2 q^{67} + 6 q^{68} - 4 q^{69} - 32 q^{76} + 4 q^{77} - 4 q^{81} + 14 q^{83} + 64 q^{84} - 42 q^{85} + 12 q^{87} - 28 q^{89} - 48 q^{93}+O(q^{100}) $ 8 * q - 16 * q^4 - 16 * q^9 - 10 * q^13 - 6 * q^15 + 32 * q^16 - 3 * q^17 + 16 * q^19 - 32 * q^21 - 8 * q^25 + 20 * q^33 + 12 * q^35 + 32 * q^36 + 8 * q^43 - 8 * q^47 - 26 * q^49 - 16 * q^51 + 20 * q^52 + 46 * q^53 - 12 * q^55 - 16 * q^59 + 12 * q^60 - 64 * q^64 - 2 * q^67 + 6 * q^68 - 4 * q^69 - 32 * q^76 + 4 * q^77 - 4 * q^81 + 14 * q^83 + 64 * q^84 - 42 * q^85 + 12 * q^87 - 28 * q^89 - 48 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/731\mathbb{Z}\right)^\times$.

$n$	$173$	$562$
$\chi(n)$	$-1$	$1$

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			−	1.00000i	$-0.5\pi$
$2$		0			0				1.00000i	$0.5\pi$
$3$			2.15629i			1.24494i	0.782645	+	0.622468i	$0.213870\pi$
$3$			2.15629i			1.24494i	−0.782645	+	0.622468i	$0.786130\pi$
$4$	−2.00000			−1.00000
$5$			2.88045i			1.28818i	0.764951	+	0.644089i	$0.222763\pi$
$5$			2.88045i			1.28818i	−0.764951	+	0.644089i	$0.777237\pi$
$6$		0			0
$7$			2.59528i			0.980925i	0.871462	+	0.490463i	$0.163172\pi$
$7$			2.59528i			0.980925i	−0.871462	+	0.490463i	$0.836828\pi$
$8$		0			0
$9$	−1.64959			−0.549865
$10$		0			0
$11$			4.55588i			1.37365i	0.726823	+	0.686825i	$0.240996\pi$
$11$			4.55588i			1.37365i	−0.726823	+	0.686825i	$0.759004\pi$
$12$		−	4.31258i		−	1.24494i
$13$	4.56150			1.26513			0.632566	−	0.774506i	$-0.282002\pi$
$13$	4.56150			1.26513			0.632566	+	0.774506i	$0.282002\pi$
$14$		0			0
$15$	−6.21109			−1.60370
$16$	4.00000			1.00000
$17$	−4.07885	+	0.602510i	−0.989265	+	0.146130i
$17$	−4.07885	+	0.602510i	−0.989265	+	0.146130i
$18$		0			0
$19$	2.00000			0.458831			0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000			0.458831			0.229416	+	0.973329i	$0.426318\pi$
$20$		−	5.76090i		−	1.28818i
$21$	−5.59619			−1.22119
$22$		0			0
$23$		−	3.67790i		−	0.766895i	−0.923563	−	0.383447i	$-0.874737\pi$
$23$		−	3.67790i		−	0.766895i	0.923563	−	0.383447i	$-0.125263\pi$
$24$		0			0
$25$	−3.29700			−0.659401
$26$		0			0
$27$			2.91187i			0.560389i
$28$		−	5.19057i		−	0.980925i
$29$		−	5.76090i		−	1.06977i	−0.844924	−	0.534886i	$-0.820354\pi$
$29$		−	5.76090i		−	1.06977i	0.844924	−	0.534886i	$-0.179646\pi$
$30$		0			0
$31$			3.10756i			0.558135i	0.960272	+	0.279067i	$0.0900253\pi$
$31$			3.10756i			0.558135i	−0.960272	+	0.279067i	$0.909975\pi$
$32$		0			0
$33$	−9.82381			−1.71011
$34$		0			0
$35$	−7.47559			−1.26361
$36$	3.29919			0.549865
$37$		−	1.27831i		−	0.210152i	−0.994464	−	0.105076i	$-0.966491\pi$
$37$		−	1.27831i		−	0.210152i	0.994464	−	0.105076i	$-0.0335086\pi$
$38$		0			0
$39$			9.83593i			1.57501i
$40$		0			0
$41$		−	11.1948i		−	1.74833i	−0.485629	−	0.874165i	$-0.661410\pi$
$41$		−	11.1948i		−	1.74833i	0.485629	−	0.874165i	$-0.338590\pi$
$42$		0			0
$43$	1.00000			0.152499
$43$	1.00000			0.152499
$44$		−	9.11177i		−	1.37365i
$45$		−	4.75158i		−	0.708323i
$46$		0			0
$47$	−2.24578			−0.327581			−0.163791	−	0.986495i	$-0.552372\pi$
$47$	−2.24578			−0.327581			−0.163791	+	0.986495i	$0.552372\pi$
$48$			8.62517i			1.24494i
$49$	0.264498			0.0377854
$50$		0			0
$51$	−1.29919	−	8.79518i	−0.181923	−	1.23157i
$52$	−9.12300			−1.26513
$53$	11.9119			1.63623			0.818113	−	0.575057i	$-0.195020\pi$
$53$	11.9119			1.63623			0.818113	+	0.575057i	$0.195020\pi$
$54$		0			0
$55$	−13.1230			−1.76951
$56$		0			0
$57$			4.31258i			0.571216i
$58$		0			0
$59$	0.297003			0.0386664			0.0193332	−	0.999813i	$-0.493846\pi$
$59$	0.297003			0.0386664			0.0193332	+	0.999813i	$0.493846\pi$
$60$	12.4222			1.60370
$61$			8.38761i			1.07392i	0.843607	+	0.536961i	$0.180428\pi$
$61$			8.38761i			1.07392i	−0.843607	+	0.536961i	$0.819572\pi$
$62$		0			0
$63$		−	4.28117i		−	0.539376i
$64$	−8.00000			−1.00000
$65$			13.1392i			1.62971i
$66$		0			0
$67$	2.01871			0.246625			0.123313	−	0.992368i	$-0.460648\pi$
$67$	2.01871			0.246625			0.123313	+	0.992368i	$0.460648\pi$
$68$	8.15769	−	1.20502i	0.989265	−	0.146130i
$69$	7.93062			0.954734
$70$		0			0
$71$		−	3.58842i		−	0.425868i	−0.977067	−	0.212934i	$-0.931698\pi$
$71$		−	3.58842i		−	0.425868i	0.977067	−	0.212934i	$-0.0683019\pi$
$72$		0			0
$73$		−	3.19704i		−	0.374185i	−0.982342	−	0.187092i	$-0.940094\pi$
$73$		−	3.19704i		−	0.374185i	0.982342	−	0.187092i	$-0.0599064\pi$
$74$		0			0
$75$		−	7.10930i		−	0.820911i
$76$	−4.00000			−0.458831
$77$	−11.8238			−1.34745
$78$		0			0
$79$		−	1.02456i		−	0.115272i	−0.998338	−	0.0576358i	$-0.981644\pi$
$79$		−	1.02456i		−	0.115272i	0.998338	−	0.0576358i	$-0.0183562\pi$
$80$			11.5218i			1.28818i
$81$	−11.2276			−1.24751
$82$		0			0
$83$	5.61490			0.616316			0.308158	−	0.951335i	$-0.400287\pi$
$83$	5.61490			0.616316			0.308158	+	0.951335i	$0.400287\pi$
$84$	11.1924			1.22119
$85$	−1.73550	−	11.7489i	−0.188242	−	1.27435i
$86$		0			0
$87$	12.4222			1.33180
$88$		0			0
$89$	−11.2298			−1.19036			−0.595179	−	0.803593i	$-0.702919\pi$
$89$	−11.2298			−1.19036			−0.595179	+	0.803593i	$0.702919\pi$
$90$		0			0
$91$			11.8384i			1.24100i
$92$			7.35579i			0.766895i
$93$	−6.70081			−0.694842
$94$		0			0
$95$			5.76090i			0.591056i
$96$		0			0
$97$		−	2.08301i		−	0.211497i	−0.994393	−	0.105749i	$-0.966276\pi$
$97$		−	2.08301i		−	0.211497i	0.994393	−	0.105749i	$-0.0337239\pi$
$98$		0			0
$99$		−	7.51536i		−	0.755322i
$100$	6.59401			0.659401
$101$	14.8238			1.47502			0.737512	−	0.675334i	$-0.236000\pi$
$101$	14.8238			1.47502			0.737512	+	0.675334i	$0.236000\pi$
$102$		0			0
$103$	17.9303			1.76672			0.883362	−	0.468692i	$-0.155274\pi$
$103$	17.9303			1.76672			0.883362	+	0.468692i	$0.155274\pi$
$104$		0			0
$105$		−	16.1196i		−	1.57311i
$106$		0			0
$107$			9.98975i			0.965746i	0.875690	+	0.482873i	$0.160407\pi$
$107$			9.98975i			0.965746i	−0.875690	+	0.482873i	$0.839593\pi$
$108$		−	5.82374i		−	0.560389i
$109$			20.3903i			1.95303i	0.215440	+	0.976517i	$0.430881\pi$
$109$			20.3903i			1.95303i	−0.215440	+	0.976517i	$0.569119\pi$
$110$		0			0
$111$	2.75640			0.261626
$112$			10.3811i			0.980925i
$113$		−	6.93929i		−	0.652793i	−0.945233	−	0.326397i	$-0.894166\pi$
$113$		−	6.93929i		−	0.652793i	0.945233	−	0.326397i	$-0.105834\pi$
$114$		0			0
$115$	10.5940			0.987896
$116$			11.5218i			1.06977i
$117$	−7.52462			−0.695652
$118$		0			0
$119$	−1.56369	−	10.5858i	−0.143343	−	0.970396i
$120$		0			0
$121$	−9.75607			−0.886915
$122$		0			0
$123$	24.1392			2.17656
$124$		−	6.21513i		−	0.558135i
$125$			4.90540i			0.438752i
$126$		0			0
$127$	−10.3157			−0.915372			−0.457686	−	0.889114i	$-0.651322\pi$
$127$	−10.3157			−0.915372			−0.457686	+	0.889114i	$0.651322\pi$
$128$		0			0
$129$			2.15629i			0.189851i
$130$		0			0
$131$		−	10.2362i		−	0.894345i	−0.894448	−	0.447172i	$-0.852431\pi$
$131$		−	10.2362i		−	0.894345i	0.894448	−	0.447172i	$-0.147569\pi$
$132$	19.6476			1.71011
$133$			5.19057i			0.450079i
$134$		0			0
$135$	−8.38750			−0.721881
$136$		0			0
$137$	−6.59401			−0.563364			−0.281682	−	0.959508i	$-0.590892\pi$
$137$	−6.59401			−0.563364			−0.281682	+	0.959508i	$0.590892\pi$
$138$		0			0
$139$			7.27358i			0.616937i	0.951235	+	0.308468i	$0.0998164\pi$
$139$			7.27358i			0.616937i	−0.951235	+	0.308468i	$0.900184\pi$
$140$	14.9512			1.26361
$141$		−	4.84257i		−	0.407818i
$142$		0			0
$143$			20.7817i			1.73785i
$144$	−6.59838			−0.549865
$145$	16.5940			1.37806
$146$		0			0
$147$			0.570334i			0.0470404i
$148$			2.55661i			0.210152i
$149$	−10.3154			−0.845069			−0.422535	−	0.906347i	$-0.638859\pi$
$149$	−10.3154			−0.845069			−0.422535	+	0.906347i	$0.638859\pi$
$150$		0			0
$151$	2.59401			0.211097			0.105549	−	0.994414i	$-0.466340\pi$
$151$	2.59401			0.211097			0.105549	+	0.994414i	$0.466340\pi$
$152$		0			0
$153$	6.72844	−	0.993898i	0.543962	−	0.0803519i
$154$		0			0
$155$	−8.95118			−0.718976
$156$		−	19.6719i		−	1.57501i
$157$	−8.52900			−0.680688			−0.340344	−	0.940301i	$-0.610544\pi$
$157$	−8.52900			−0.680688			−0.340344	+	0.940301i	$0.610544\pi$
$158$		0			0
$159$			25.6855i			2.03700i
$160$		0			0
$161$	9.54519			0.752266
$162$		0			0
$163$		−	5.83099i		−	0.456719i	−0.973577	−	0.228359i	$-0.926664\pi$
$163$		−	5.83099i		−	0.456719i	0.973577	−	0.228359i	$-0.0733361\pi$
$164$			22.3895i			1.74833i
$165$		−	28.2970i		−	2.20292i
$166$		0			0
$167$		−	3.67790i		−	0.284604i	−0.989823	−	0.142302i	$-0.954550\pi$
$167$		−	3.67790i		−	0.284604i	0.989823	−	0.142302i	$-0.0454505\pi$
$168$		0			0
$169$	7.80729			0.600560
$170$		0			0
$171$	−3.29919			−0.252295
$172$	−2.00000			−0.152499
$173$		−	21.3326i		−	1.62189i	−0.585123	−	0.810944i	$-0.698954\pi$
$173$		−	21.3326i		−	1.62189i	0.585123	−	0.810944i	$-0.301046\pi$
$174$		0			0
$175$		−	8.55666i		−	0.646823i
$176$			18.2235i			1.37365i
$177$			0.640424i			0.0481372i
$178$		0			0
$179$	11.8238			0.883753			0.441877	−	0.897076i	$-0.354313\pi$
$179$	11.8238			0.883753			0.441877	+	0.897076i	$0.354313\pi$
$180$			9.50315i			0.708323i
$181$		−	9.53553i		−	0.708770i	−0.935100	−	0.354385i	$-0.884690\pi$
$181$		−	9.53553i		−	0.708770i	0.935100	−	0.354385i	$-0.115310\pi$
$182$		0			0
$183$	−18.0861			−1.33696
$184$		0			0
$185$	3.68210			0.270713
$186$		0			0
$187$	−2.74497	−	18.5827i	−0.200732	−	1.35890i
$188$	4.49157			0.327581
$189$	−7.55713			−0.549700
$190$		0			0
$191$	−22.9094			−1.65767			−0.828833	−	0.559497i	$-0.810994\pi$
$191$	−22.9094			−1.65767			−0.828833	+	0.559497i	$0.810994\pi$
$192$		−	17.2503i		−	1.24494i
$193$			3.61506i			0.260218i	0.991500	+	0.130109i	$0.0415327\pi$
$193$			3.61506i			0.260218i	−0.991500	+	0.130109i	$0.958467\pi$
$194$		0			0
$195$	−28.3319			−2.02889
$196$	−0.528996			−0.0377854
$197$		−	18.1607i		−	1.29390i	−0.762534	−	0.646948i	$-0.776045\pi$
$197$		−	18.1607i		−	1.29390i	0.762534	−	0.646948i	$-0.223955\pi$
$198$		0			0
$199$			22.2195i			1.57510i	0.616250	+	0.787550i	$0.288651\pi$
$199$			22.2195i			1.57510i	−0.616250	+	0.787550i	$0.711349\pi$
$200$		0			0
$201$			4.35293i			0.307032i
$202$		0			0
$203$	14.9512			1.04937
$204$	2.59838	+	17.5904i	0.181923	+	1.23157i
$205$	32.2460			2.25216
$206$		0			0
$207$			6.06704i			0.421688i
$208$	18.2460			1.26513
$209$			9.11177i			0.630274i
$210$		0			0
$211$			21.5791i			1.48557i	0.669532	+	0.742783i	$0.266495\pi$
$211$			21.5791i			1.48557i	−0.669532	+	0.742783i	$0.733505\pi$
$212$	−23.8238			−1.63623
$213$	7.73769			0.530178
$214$		0			0
$215$			2.88045i			0.196445i
$216$		0			0
$217$	−8.06501			−0.547489
$218$		0			0
$219$	6.89374			0.465836
$220$	26.2460			1.76951
$221$	−18.6057	+	2.74835i	−1.25155	+	0.184874i
$222$		0			0
$223$	17.6146			1.17956			0.589779	−	0.807564i	$-0.299215\pi$
$223$	17.6146			1.17956			0.589779	+	0.807564i	$0.299215\pi$
$224$		0			0
$225$	5.43872			0.362581
$226$		0			0
$227$		−	5.22295i		−	0.346659i	−0.984864	−	0.173330i	$-0.944547\pi$
$227$		−	5.22295i		−	0.346659i	0.984864	−	0.173330i	$-0.0554526\pi$
$228$		−	8.62517i		−	0.571216i
$229$	24.2273			1.60098			0.800492	−	0.599343i	$-0.204572\pi$
$229$	24.2273			1.60098			0.800492	+	0.599343i	$0.204572\pi$
$230$		0			0
$231$		−	25.4956i		−	1.67749i
$232$		0			0
$233$			4.27391i			0.279993i	0.990152	+	0.139997i	$0.0447092\pi$
$233$			4.27391i			0.279993i	−0.990152	+	0.139997i	$0.955291\pi$
$234$		0			0
$235$		−	6.46888i		−	0.421983i
$236$	−0.594005			−0.0386664
$237$	2.20924			0.143506
$238$		0			0
$239$	−0.842309			−0.0544844			−0.0272422	−	0.999629i	$-0.508673\pi$
$239$	−0.842309			−0.0544844			−0.0272422	+	0.999629i	$0.508673\pi$
$240$	−24.8444			−1.60370
$241$			5.40819i			0.348372i	0.984713	+	0.174186i	$0.0557294\pi$
$241$			5.40819i			0.348372i	−0.984713	+	0.174186i	$0.944271\pi$
$242$		0			0
$243$		−	15.4744i		−	0.992685i
$244$		−	16.7752i		−	1.07392i
$245$			0.761873i			0.0486743i
$246$		0			0
$247$	9.12300			0.580483
$248$		0			0
$249$			12.1074i			0.767273i
$250$		0			0
$251$	5.17367			0.326559			0.163280	−	0.986580i	$-0.447793\pi$
$251$	5.17367			0.326559			0.163280	+	0.986580i	$0.447793\pi$
$252$			8.56233i			0.539376i
$253$	16.7561			1.05344
$254$		0			0
$255$	25.3341	−	3.74225i	1.58648	−	0.234349i
$256$	16.0000			1.00000
$257$	5.40599			0.337217			0.168608	−	0.985683i	$-0.446073\pi$
$257$	5.40599			0.337217			0.168608	+	0.985683i	$0.446073\pi$
$258$		0			0
$259$	3.31757			0.206144
$260$		−	26.2784i		−	1.62971i
$261$			9.50315i			0.588230i
$262$		0			0
$263$	−27.7214			−1.70937			−0.854687	−	0.519144i	$-0.826251\pi$
$263$	−27.7214			−1.70937			−0.854687	+	0.519144i	$0.826251\pi$
$264$		0			0
$265$			34.3117i			2.10775i
$266$		0			0
$267$		−	24.2147i		−	1.48192i
$268$	−4.03743			−0.246625
$269$		−	6.31185i		−	0.384841i	−0.981313	−	0.192420i	$-0.938366\pi$
$269$		−	6.31185i		−	0.384841i	0.981313	−	0.192420i	$-0.0616337\pi$
$270$		0			0
$271$	−11.0905			−0.673700			−0.336850	−	0.941558i	$-0.609361\pi$
$271$	−11.0905			−0.673700			−0.336850	+	0.941558i	$0.609361\pi$
$272$	−16.3154	+	2.41004i	−0.989265	+	0.146130i
$273$	−25.5270			−1.54497
$274$		0			0
$275$		−	15.0208i		−	0.905786i
$276$	−15.8612			−0.954734
$277$		−	17.0290i		−	1.02317i	−0.859232	−	0.511586i	$-0.829058\pi$
$277$		−	17.0290i		−	1.02317i	0.859232	−	0.511586i	$-0.170942\pi$
$278$		0			0
$279$		−	5.12622i		−	0.306899i
$280$		0			0
$281$	−2.63088			−0.156945			−0.0784726	−	0.996916i	$-0.525004\pi$
$281$	−2.63088			−0.156945			−0.0784726	+	0.996916i	$0.525004\pi$
$282$		0			0
$283$		−	24.5563i		−	1.45972i	−0.683597	−	0.729860i	$-0.739585\pi$
$283$		−	24.5563i		−	1.45972i	0.683597	−	0.729860i	$-0.260415\pi$
$284$			7.17685i			0.425868i
$285$	−12.4222			−0.735827
$286$		0			0
$287$	29.0536			1.71498
$288$		0			0
$289$	16.2740	−	4.91509i	0.957292	−	0.289123i
$290$		0			0
$291$	4.49157			0.263300
$292$			6.39407i			0.374185i
$293$	−27.4731			−1.60499			−0.802497	−	0.596656i	$-0.796496\pi$
$293$	−27.4731			−1.60499			−0.802497	+	0.596656i	$0.796496\pi$
$294$		0			0
$295$			0.855502i			0.0498092i
$296$		0			0
$297$	−13.2661			−0.769779
$298$		0			0
$299$		−	16.7767i		−	0.970223i
$300$			14.2186i			0.820911i
$301$			2.59528i			0.149590i
$302$		0			0
$303$			31.9645i			1.83631i
$304$	8.00000			0.458831
$305$	−24.1601			−1.38340
$306$		0			0
$307$	3.49157			0.199274			0.0996372	−	0.995024i	$-0.468232\pi$
$307$	3.49157			0.199274			0.0996372	+	0.995024i	$0.468232\pi$
$308$	23.6476			1.34745
$309$			38.6629i			2.19946i
$310$		0			0
$311$		−	23.3174i		−	1.32221i	−0.750295	−	0.661103i	$-0.770088\pi$
$311$		−	23.3174i		−	1.32221i	0.750295	−	0.661103i	$-0.229912\pi$
$312$		0			0
$313$		−	1.91873i		−	0.108453i	−0.998529	−	0.0542265i	$-0.982731\pi$
$313$		−	1.91873i		−	0.108453i	0.998529	−	0.0542265i	$-0.0172693\pi$
$314$		0			0
$315$	12.3317			0.694812
$316$			2.04911i			0.115272i
$317$			16.1405i			0.906542i	0.891373	+	0.453271i	$0.149743\pi$
$317$			16.1405i			0.906542i	−0.891373	+	0.453271i	$0.850257\pi$
$318$		0			0
$319$	26.2460			1.46949
$320$		−	23.0436i		−	1.28818i
$321$	−21.5408			−1.20229
$322$		0			0
$323$	−8.15769	+	1.20502i	−0.453906	+	0.0670492i
$324$	22.4552			1.24751
$325$	−15.0393			−0.834229
$326$		0			0
$327$	−43.9674			−2.43140
$328$		0			0
$329$		−	5.82845i		−	0.321333i
$330$		0			0
$331$	−19.6146			−1.07811			−0.539057	−	0.842269i	$-0.681219\pi$
$331$	−19.6146			−1.07811			−0.539057	+	0.842269i	$0.681219\pi$
$332$	−11.2298			−0.616316
$333$			2.10869i			0.115555i
$334$		0			0
$335$			5.81481i			0.317697i
$336$	−22.3848			−1.22119
$337$		−	23.0242i		−	1.25421i	−0.778935	−	0.627105i	$-0.784240\pi$
$337$		−	23.0242i		−	1.25421i	0.778935	−	0.627105i	$-0.215760\pi$
$338$		0			0
$339$	14.9631			0.812685
$340$	3.47100	+	23.4978i	0.188242	+	1.27435i
$341$	−14.1577			−0.766682
$342$		0			0
$343$			18.8534i			1.01799i
$344$		0			0
$345$			22.8438i			1.22987i
$346$		0			0
$347$			33.8717i			1.81833i	0.416437	+	0.909165i	$0.363279\pi$
$347$			33.8717i			1.81833i	−0.416437	+	0.909165i	$0.636721\pi$
$348$	−24.8444			−1.33180
$349$	−25.3690			−1.35797			−0.678986	−	0.734151i	$-0.737580\pi$
$349$	−25.3690			−1.35797			−0.678986	+	0.734151i	$0.737580\pi$
$350$		0			0
$351$			13.2825i			0.708967i
$352$		0			0
$353$	−0.208909			−0.0111191			−0.00555955	−	0.999985i	$-0.501770\pi$
$353$	−0.208909			−0.0111191			−0.00555955	+	0.999985i	$0.501770\pi$
$354$		0			0
$355$	10.3363			0.548593
$356$	22.4596			1.19036
$357$	22.8260	−	3.37176i	1.20808	−	0.178453i
$358$		0			0
$359$	33.1601			1.75012			0.875061	−	0.484012i	$-0.160821\pi$
$359$	33.1601			1.75012			0.875061	+	0.484012i	$0.160821\pi$
$360$		0			0
$361$	−15.0000			−0.789474
$362$		0			0
$363$		−	21.0369i		−	1.10415i
$364$		−	23.6768i		−	1.24100i
$365$	9.20891			0.482016
$366$		0			0
$367$		−	7.92613i		−	0.413741i	−0.978368	−	0.206870i	$-0.933672\pi$
$367$		−	7.92613i		−	0.413741i	0.978368	−	0.206870i	$-0.0663278\pi$
$368$		−	14.7116i		−	0.766895i
$369$			18.4668i			0.961345i
$370$		0			0
$371$			30.9148i			1.60502i
$372$	13.4016			0.694842
$373$	−32.2834			−1.67157			−0.835786	−	0.549055i	$-0.814988\pi$
$373$	−32.2834			−1.67157			−0.835786	+	0.549055i	$0.814988\pi$
$374$		0			0
$375$	−10.5775			−0.546219
$376$		0			0
$377$		−	26.2784i		−	1.35340i
$378$		0			0
$379$			13.4887i			0.692868i	0.938074	+	0.346434i	$0.112608\pi$
$379$			13.4887i			0.692868i	−0.938074	+	0.346434i	$0.887392\pi$
$380$		−	11.5218i		−	0.591056i
$381$		−	22.2437i		−	1.13958i
$382$		0			0
$383$	−17.0856			−0.873032			−0.436516	−	0.899696i	$-0.643788\pi$
$383$	−17.0856			−0.873032			−0.436516	+	0.899696i	$0.643788\pi$
$384$		0			0
$385$		−	34.0579i		−	1.73575i
$386$		0			0
$387$	−1.64959			−0.0838536
$388$			4.16601i			0.211497i
$389$	−26.7750			−1.35755			−0.678773	−	0.734348i	$-0.737488\pi$
$389$	−26.7750			−1.35755			−0.678773	+	0.734348i	$0.737488\pi$
$390$		0			0
$391$	2.21597	+	15.0016i	0.112066	+	0.758662i
$392$		0			0
$393$	22.0723			1.11340
$394$		0			0
$395$	2.95118			0.148490
$396$			15.0307i			0.755322i
$397$			15.4944i			0.777640i	0.921314	+	0.388820i	$0.127117\pi$
$397$			15.4944i			0.777640i	−0.921314	+	0.388820i	$0.872883\pi$
$398$		0			0
$399$	−11.1924			−0.560320
$400$	−13.1880			−0.659401
$401$			16.4177i			0.819862i	0.912117	+	0.409931i	$0.134447\pi$
$401$			16.4177i			0.819862i	−0.912117	+	0.409931i	$0.865553\pi$
$402$		0			0
$403$			14.1751i			0.706114i
$404$	−29.6476			−1.47502
$405$		−	32.3406i		−	1.60702i
$406$		0			0
$407$	5.82381			0.288676
$408$		0			0
$409$	−15.1604			−0.749635			−0.374817	−	0.927099i	$-0.622295\pi$
$409$	−15.1604			−0.749635			−0.374817	+	0.927099i	$0.622295\pi$
$410$		0			0
$411$		−	14.2186i		−	0.701352i
$412$	−35.8606			−1.76672
$413$			0.770806i			0.0379289i
$414$		0			0
$415$			16.1735i			0.793924i
$416$		0			0
$417$	−15.6840			−0.768046
$418$		0			0
$419$			29.0735i			1.42033i	0.704034	+	0.710167i	$0.251380\pi$
$419$			29.0735i			1.42033i	−0.704034	+	0.710167i	$0.748620\pi$
$420$			32.2391i			1.57311i
$421$	23.3690			1.13894			0.569468	−	0.822014i	$-0.307149\pi$
$421$	23.3690			1.13894			0.569468	+	0.822014i	$0.307149\pi$
$422$		0			0
$423$	3.70463			0.180125
$424$		0			0
$425$	13.4480	−	1.98648i	0.652322	−	0.0963584i
$426$		0			0
$427$	−21.7682			−1.05344
$428$		−	19.9795i		−	0.965746i
$429$	−44.8113			−2.16351
$430$		0			0
$431$			21.8386i			1.05193i	0.850507	+	0.525964i	$0.176295\pi$
$431$			21.8386i			1.05193i	−0.850507	+	0.525964i	$0.823705\pi$
$432$			11.6475i			0.560389i
$433$	16.1436			0.775810			0.387905	−	0.921699i	$-0.373199\pi$
$433$	16.1436			0.775810			0.387905	+	0.921699i	$0.373199\pi$
$434$		0			0
$435$			35.7815i			1.71559i
$436$		−	40.7805i		−	1.95303i
$437$		−	7.35579i		−	0.351875i
$438$		0			0
$439$		−	34.6089i		−	1.65179i	−0.563823	−	0.825896i	$-0.690670\pi$
$439$		−	34.6089i		−	1.65179i	0.563823	−	0.825896i	$-0.309330\pi$
$440$		0			0
$441$	−0.436314			−0.0207769
$442$		0			0
$443$	−34.2825			−1.62881			−0.814406	−	0.580295i	$-0.802937\pi$
$443$	−34.2825			−1.62881			−0.814406	+	0.580295i	$0.802937\pi$
$444$	−5.51280			−0.261626
$445$		−	32.3469i		−	1.53339i
$446$		0			0
$447$		−	22.2430i		−	1.05206i
$448$		−	20.7623i		−	0.980925i
$449$			12.2926i			0.580124i	0.957008	+	0.290062i	$0.0936760\pi$
$449$			12.2926i			0.580124i	−0.957008	+	0.290062i	$0.906324\pi$
$450$		0			0
$451$	51.0021			2.40159
$452$			13.8786i			0.652793i
$453$			5.59343i			0.262802i
$454$		0			0
$455$	−34.0999			−1.59863
$456$		0			0
$457$	15.5084			0.725454			0.362727	−	0.931895i	$-0.381846\pi$
$457$	15.5084			0.725454			0.362727	+	0.931895i	$0.381846\pi$
$458$		0			0
$459$	−1.75443	−	11.8771i	−0.0818898	−	0.554374i
$460$	−21.1880			−0.987896
$461$	−35.1416			−1.63671			−0.818354	−	0.574715i	$-0.805113\pi$
$461$	−35.1416			−1.63671			−0.818354	+	0.574715i	$0.805113\pi$
$462$		0			0
$463$	39.8606			1.85248			0.926239	−	0.376937i	$-0.123023\pi$
$463$	39.8606			1.85248			0.926239	+	0.376937i	$0.123023\pi$
$464$		−	23.0436i		−	1.06977i
$465$		−	19.3014i		−	0.895079i
$466$		0			0
$467$	25.4428			1.17735			0.588675	−	0.808370i	$-0.299650\pi$
$467$	25.4428			1.17735			0.588675	+	0.808370i	$0.299650\pi$
$468$	15.0492			0.695652
$469$			5.23913i			0.241921i
$470$		0			0
$471$		−	18.3910i		−	0.847413i
$472$		0			0
$473$			4.55588i			0.209480i
$474$		0			0
$475$	−6.59401			−0.302554
$476$	3.12737	+	21.1715i	0.143343	+	0.970396i
$477$	−19.6498			−0.899703
$478$		0			0
$479$			25.7274i			1.17552i	0.809037	+	0.587758i	$0.199989\pi$
$479$			25.7274i			1.17552i	−0.809037	+	0.587758i	$0.800011\pi$
$480$		0			0
$481$		−	5.83099i		−	0.265870i
$482$		0			0
$483$			20.5822i			0.936523i
$484$	19.5121			0.886915
$485$	6.00000			0.272446
$486$		0			0
$487$			14.5472i			0.659194i	0.944122	+	0.329597i	$0.106913\pi$
$487$			14.5472i			0.659194i	−0.944122	+	0.329597i	$0.893087\pi$
$488$		0			0
$489$	12.5733			0.568585
$490$		0			0
$491$	34.3154			1.54863			0.774316	−	0.632799i	$-0.218094\pi$
$491$	34.3154			1.54863			0.774316	+	0.632799i	$0.218094\pi$
$492$	−48.2784			−2.17656
$493$	3.47100	+	23.4978i	0.156326	+	1.05829i
$494$		0			0
$495$	21.6476			0.972988
$496$			12.4303i			0.558135i
$497$	9.31298			0.417744
$498$		0			0
$499$		−	34.0094i		−	1.52247i	−0.648477	−	0.761234i	$-0.724594\pi$
$499$		−	34.0094i		−	1.52247i	0.648477	−	0.761234i	$-0.275406\pi$
$500$		−	9.81080i		−	0.438752i
$501$	7.93062			0.354314
$502$		0			0
$503$		−	19.1601i		−	0.854308i	−0.904179	−	0.427154i	$-0.859516\pi$
$503$		−	19.1601i		−	0.854308i	0.904179	−	0.427154i	$-0.140484\pi$
$504$		0			0
$505$			42.6993i			1.90009i
$506$		0			0
$507$			16.8348i			0.747659i
$508$	20.6314			0.915372
$509$	16.7333			0.741691			0.370846	−	0.928695i	$-0.379068\pi$
$509$	16.7333			0.741691			0.370846	+	0.928695i	$0.379068\pi$
$510$		0			0
$511$	8.29722			0.367047
$512$		0			0
$513$			5.82374i			0.257124i
$514$		0			0
$515$			51.6473i			2.27585i
$516$		−	4.31258i		−	0.189851i
$517$		−	10.2315i		−	0.449982i
$518$		0			0
$519$	45.9993			2.01915
$520$		0			0
$521$		−	16.3597i		−	0.716730i	−0.933582	−	0.358365i	$-0.883334\pi$
$521$		−	16.3597i		−	0.716730i	0.933582	−	0.358365i	$-0.116666\pi$
$522$		0			0
$523$	−5.50843			−0.240867			−0.120433	−	0.992721i	$-0.538428\pi$
$523$	−5.50843			−0.240867			−0.120433	+	0.992721i	$0.538428\pi$
$524$			20.4725i			0.894345i
$525$	18.4507			0.805253
$526$		0			0
$527$	−1.87234	−	12.6753i	−0.0815604	−	0.552143i
$528$	−39.2952			−1.71011
$529$	9.47307			0.411873
$530$		0			0
$531$	−0.489934			−0.0212613
$532$		−	10.3811i		−	0.450079i
$533$		−	51.0650i		−	2.21187i
$534$		0			0
$535$	−28.7750			−1.24405
$536$		0			0
$537$			25.4956i			1.10022i
$538$		0			0
$539$			1.20502i			0.0519039i
$540$	16.7750			0.721881
$541$		−	20.3903i		−	0.876646i	−0.898817	−	0.438323i	$-0.855573\pi$
$541$		−	20.3903i		−	0.876646i	0.898817	−	0.438323i	$-0.144427\pi$
$542$		0			0
$543$	20.5614			0.882373
$544$		0			0
$545$	−58.7332			−2.51585
$546$		0			0
$547$		−	28.2148i		−	1.20638i	−0.797598	−	0.603189i	$-0.793897\pi$
$547$		−	28.2148i		−	1.20638i	0.797598	−	0.603189i	$-0.206103\pi$
$548$	13.1880			0.563364
$549$		−	13.8361i		−	0.590512i
$550$		0			0
$551$		−	11.5218i		−	0.490845i
$552$		0			0
$553$	2.65901			0.113073
$554$		0			0
$555$			7.93968i			0.337021i
$556$		−	14.5472i		−	0.616937i
$557$	−27.6333			−1.17086			−0.585430	−	0.810723i	$-0.699074\pi$
$557$	−27.6333			−1.17086			−0.585430	+	0.810723i	$0.699074\pi$
$558$		0			0
$559$	4.56150			0.192931
$560$	−29.9024			−1.26361
$561$	40.0698	−	5.91895i	1.69175	−	0.249898i
$562$		0			0
$563$	29.7750			1.25487			0.627433	−	0.778670i	$-0.284106\pi$
$563$	29.7750			1.25487			0.627433	+	0.778670i	$0.284106\pi$
$564$			9.68514i			0.407818i
$565$	19.9883			0.840913
$566$		0			0
$567$		−	29.1389i		−	1.22372i
$568$		0			0
$569$	−22.9655			−0.962765			−0.481382	−	0.876511i	$-0.659865\pi$
$569$	−22.9655			−0.962765			−0.481382	+	0.876511i	$0.659865\pi$
$570$		0			0
$571$		−	22.9517i		−	0.960497i	−0.877132	−	0.480249i	$-0.840546\pi$
$571$		−	22.9517i		−	0.960497i	0.877132	−	0.480249i	$-0.159454\pi$
$572$		−	41.5633i		−	1.73785i
$573$		−	49.3993i		−	2.06369i
$574$		0			0
$575$			12.1260i			0.505691i
$576$	13.1968			0.549865
$577$	28.7376			1.19636			0.598180	−	0.801361i	$-0.295891\pi$
$577$	28.7376			1.19636			0.598180	+	0.801361i	$0.295891\pi$
$578$		0			0
$579$	−7.79513			−0.323955
$580$	−33.1880			−1.37806
$581$			14.5723i			0.604560i
$582$		0			0
$583$			54.2692i			2.24760i
$584$		0			0
$585$		−	21.6743i		−	0.896123i
$586$		0			0
$587$	24.4178			1.00783			0.503916	−	0.863753i	$-0.331892\pi$
$587$	24.4178			1.00783			0.503916	+	0.863753i	$0.331892\pi$
$588$		−	1.14067i		−	0.0470404i
$589$			6.21513i			0.256090i
$590$		0			0
$591$	39.1598			1.61082
$592$		−	5.11322i		−	0.210152i
$593$	40.3154			1.65555			0.827777	−	0.561057i	$-0.189605\pi$
$593$	40.3154			1.65555			0.827777	+	0.561057i	$0.189605\pi$
$594$		0			0
$595$	30.4918	−	4.50412i	1.25004	−	0.184651i
$596$	20.6308			0.845069
$597$	−47.9118			−1.96090
$598$		0			0
$599$	27.0487			1.10518			0.552590	−	0.833453i	$-0.313640\pi$
$599$	27.0487			1.10518			0.552590	+	0.833453i	$0.313640\pi$
$600$		0			0
$601$		−	4.47534i		−	0.182553i	−0.995826	−	0.0912765i	$-0.970905\pi$
$601$		−	4.47534i		−	0.182553i	0.995826	−	0.0912765i	$-0.0290947\pi$
$602$		0			0
$603$	−3.33006			−0.135610
$604$	−5.18801			−0.211097
$605$		−	28.1019i		−	1.14250i
$606$		0			0
$607$		−	27.7942i		−	1.12813i	−0.825729	−	0.564067i	$-0.809236\pi$
$607$		−	27.7942i		−	1.12813i	0.825729	−	0.564067i	$-0.190764\pi$
$608$		0			0
$609$			32.2391i			1.30639i
$610$		0			0
$611$	−10.2441			−0.414434
$612$	−13.4569	+	1.98780i	−0.543962	+	0.0803519i
$613$	29.5105			1.19192			0.595959	−	0.803015i	$-0.296772\pi$
$613$	29.5105			1.19192			0.595959	+	0.803015i	$0.296772\pi$
$614$		0			0
$615$			69.5318i			2.80379i
$616$		0			0
$617$		−	12.1889i		−	0.490705i	−0.969434	−	0.245353i	$-0.921096\pi$
$617$		−	12.1889i		−	0.490705i	0.969434	−	0.245353i	$-0.0789038\pi$
$618$		0			0
$619$			6.64528i			0.267096i	0.991042	+	0.133548i	$0.0426371\pi$
$619$			6.64528i			0.267096i	−0.991042	+	0.133548i	$0.957363\pi$
$620$	17.9024			0.718976
$621$	10.7096			0.429760
$622$		0			0
$623$		−	29.1445i		−	1.16765i
$624$			39.3437i			1.57501i
$625$	−30.6148			−1.22459
$626$		0			0
$627$	−19.6476			−0.784651
$628$	17.0580			0.680688
$629$	0.770193	+	5.21401i	0.0307096	+	0.207896i
$630$		0			0
$631$	22.7750			0.906658			0.453329	−	0.891343i	$-0.350236\pi$
$631$	22.7750			0.906658			0.453329	+	0.891343i	$0.350236\pi$
$632$		0			0
$633$	−46.5308			−1.84944
$634$		0			0
$635$		−	29.7139i		−	1.17916i
$636$		−	51.3711i		−	2.03700i
$637$	1.20651			0.0478035
$638$		0			0
$639$			5.91944i			0.234170i
$640$		0			0
$641$			12.7984i			0.505508i	0.967531	+	0.252754i	$0.0813363\pi$
$641$			12.7984i			0.505508i	−0.967531	+	0.252754i	$0.918664\pi$
$642$		0			0
$643$			7.41863i			0.292562i	0.989243	+	0.146281i	$0.0467304\pi$
$643$			7.41863i			0.292562i	−0.989243	+	0.146281i	$0.953270\pi$
$644$	−19.0904			−0.752266
$645$	−6.21109			−0.244562
$646$		0			0
$647$	46.5570			1.83035			0.915173	−	0.403062i	$-0.132054\pi$
$647$	46.5570			1.83035			0.915173	+	0.403062i	$0.132054\pi$
$648$		0			0
$649$			1.35311i			0.0531142i
$650$		0			0
$651$		−	17.3905i		−	0.681588i
$652$			11.6620i			0.456719i
$653$		−	23.8506i		−	0.933344i	−0.884430	−	0.466672i	$-0.845453\pi$
$653$		−	23.8506i		−	0.933344i	0.884430	−	0.466672i	$-0.154547\pi$
$654$		0			0
$655$	29.4850			1.15207
$656$		−	44.7791i		−	1.74833i
$657$			5.27381i			0.205751i
$658$		0			0
$659$	13.8053			0.537779			0.268889	−	0.963171i	$-0.413343\pi$
$659$	13.8053			0.537779			0.268889	+	0.963171i	$0.413343\pi$
$660$			56.5940i			2.20292i
$661$	35.0533			1.36341			0.681707	−	0.731625i	$-0.261238\pi$
$661$	35.0533			1.36341			0.681707	+	0.731625i	$0.261238\pi$
$662$		0			0
$663$	−5.92625	−	40.1192i	−0.230156	−	1.55810i
$664$		0			0
$665$	−14.9512			−0.579782
$666$		0			0
$667$	−21.1880			−0.820403
$668$			7.35579i			0.284604i
$669$			37.9822i			1.46847i
$670$		0			0
$671$	−38.2129			−1.47519
$672$		0			0
$673$			32.1392i			1.23887i	0.785046	+	0.619437i	$0.212639\pi$
$673$			32.1392i			1.23887i	−0.785046	+	0.619437i	$0.787361\pi$
$674$		0			0
$675$		−	9.60044i		−	0.369521i
$676$	−15.6146			−0.600560
$677$			1.48699i			0.0571498i	0.999592	+	0.0285749i	$0.00909691\pi$
$677$			1.48699i			0.0571498i	−0.999592	+	0.0285749i	$0.990903\pi$
$678$		0			0
$679$	5.40599			0.207463
$680$		0			0
$681$	11.2622			0.431568
$682$		0			0
$683$			36.8723i			1.41088i	0.708769	+	0.705441i	$0.249251\pi$
$683$			36.8723i			1.41088i	−0.708769	+	0.705441i	$0.750749\pi$
$684$	6.59838			0.252295
$685$		−	18.9937i		−	0.725713i
$686$		0			0
$687$			52.2411i			1.99312i
$688$	4.00000			0.152499
$689$	54.3362			2.07004
$690$		0			0
$691$		−	39.4274i		−	1.49989i	−0.661500	−	0.749945i	$-0.730080\pi$
$691$		−	39.4274i		−	1.49989i	0.661500	−	0.749945i	$-0.269920\pi$
$692$			42.6652i			1.62189i
$693$	19.5045			0.740914
$694$		0			0
$695$	−20.9512			−0.794724
$696$		0			0
$697$	6.74497	+	45.6617i	0.255484	+	1.72956i
$698$		0			0
$699$	−9.21580			−0.348574
$700$			17.1133i			0.646823i
$701$	30.2642			1.14306			0.571531	−	0.820581i	$-0.306350\pi$
$701$	30.2642			1.14306			0.571531	+	0.820581i	$0.306350\pi$
$702$		0			0
$703$		−	2.55661i		−	0.0964245i
$704$		−	36.4471i		−	1.37365i
$705$	13.9488			0.525342
$706$		0			0
$707$			38.4720i			1.44689i
$708$		−	1.28085i		−	0.0481372i
$709$			22.4394i			0.842729i	0.906891	+	0.421364i	$0.138449\pi$
$709$			22.4394i			0.842729i	−0.906891	+	0.421364i	$0.861551\pi$
$710$		0			0
$711$			1.69010i			0.0633838i
$712$		0			0
$713$	11.4293			0.428030
$714$		0			0
$715$	−59.8606			−2.23866
$716$	−23.6476			−0.883753
$717$		−	1.81626i		−	0.0678296i
$718$		0			0
$719$			9.98975i			0.372555i	0.982497	+	0.186277i	$0.0596423\pi$
$719$			9.98975i			0.372555i	−0.982497	+	0.186277i	$0.940358\pi$
$720$		−	19.0063i		−	0.708323i
$721$			46.5342i			1.73302i
$722$		0			0
$723$	−11.6616			−0.433701
$724$			19.0711i			0.708770i
$725$			18.9937i			0.705409i
$726$		0			0
$727$	−29.5496			−1.09593			−0.547966	−	0.836500i	$-0.684598\pi$
$727$	−29.5496			−1.09593			−0.547966	+	0.836500i	$0.684598\pi$
$728$		0			0
$729$	−0.315499			−0.0116851
$730$		0			0
$731$	−4.07885	+	0.602510i	−0.150862	+	0.0222847i
$732$	36.1722			1.33696
$733$	−19.6894			−0.727245			−0.363623	−	0.931546i	$-0.618460\pi$
$733$	−19.6894			−0.727245			−0.363623	+	0.931546i	$0.618460\pi$
$734$		0			0
$735$	−1.64282			−0.0605964
$736$		0			0
$737$			9.19702i			0.338777i
$738$		0			0
$739$	18.8026			0.691664			0.345832	−	0.938296i	$-0.387597\pi$
$739$	18.8026			0.691664			0.345832	+	0.938296i	$0.387597\pi$
$740$	−7.36420			−0.270713
$741$			19.6719i			0.722664i
$742$		0			0
$743$		−	11.6918i		−	0.428931i	−0.976732	−	0.214466i	$-0.931199\pi$
$743$		−	11.6918i		−	0.428931i	0.976732	−	0.214466i	$-0.0688010\pi$
$744$		0			0
$745$		−	29.7130i		−	1.08860i
$746$		0			0
$747$	−9.26231			−0.338890
$748$	5.48993	+	37.1655i	0.200732	+	1.35890i
$749$	−25.9262			−0.947324
$750$		0			0
$751$			39.4562i			1.43978i	0.694089	+	0.719889i	$0.255807\pi$
$751$			39.4562i			1.43978i	−0.694089	+	0.719889i	$0.744193\pi$
$752$	−8.98314			−0.327581
$753$			11.1559i			0.406545i
$754$		0			0
$755$			7.47191i			0.271931i
$756$	15.1143			0.549700
$757$	−34.3521			−1.24855			−0.624275	−	0.781205i	$-0.714605\pi$
$757$	−34.3521			−1.24855			−0.624275	+	0.781205i	$0.714605\pi$
$758$		0			0
$759$			36.1310i			1.31147i
$760$		0			0
$761$	−11.0536			−0.400693			−0.200347	−	0.979725i	$-0.564207\pi$
$761$	−11.0536			−0.400693			−0.200347	+	0.979725i	$0.564207\pi$
$762$		0			0
$763$	−52.9186			−1.91578
$764$	45.8188			1.65767
$765$	2.86287	+	19.3809i	0.103507	+	0.700720i
$766$		0			0
$767$	1.35478			0.0489182
$768$			34.5007i			1.24494i
$769$	33.1232			1.19445			0.597227	−	0.802072i	$-0.296269\pi$
$769$	33.1232			1.19445			0.597227	+	0.802072i	$0.296269\pi$
$770$		0			0
$771$			11.6569i			0.419813i
$772$		−	7.23013i		−	0.260218i
$773$	−9.89756			−0.355991			−0.177995	−	0.984031i	$-0.556961\pi$
$773$	−9.89756			−0.355991			−0.177995	+	0.984031i	$0.556961\pi$
$774$		0			0
$775$		−	10.2456i		−	0.368034i
$776$		0			0
$777$			7.15364i			0.256636i
$778$		0			0
$779$		−	22.3895i		−	0.802189i
$780$	56.6638			2.02889
$781$	16.3484			0.584993
$782$		0			0
$783$	16.7750			0.599489
$784$	1.05799			0.0377854
$785$		−	24.5674i		−	0.876847i
$786$		0			0
$787$			32.4485i			1.15666i	0.815801	+	0.578332i	$0.196296\pi$
$787$			32.4485i			1.15666i	−0.815801	+	0.578332i	$0.803704\pi$
$788$			36.3214i			1.29390i
$789$		−	59.7754i		−	2.12806i
$790$		0			0
$791$	18.0094			0.640341
$792$		0			0
$793$			38.2601i			1.35865i
$794$		0			0
$795$	−73.9860			−2.62401
$796$		−	44.4391i		−	1.57510i
$797$	−14.3666			−0.508891			−0.254446	−	0.967087i	$-0.581893\pi$
$797$	−14.3666			−0.508891			−0.254446	+	0.967087i	$0.581893\pi$
$798$		0			0
$799$	9.16021	−	1.35311i	0.324065	−	0.0478695i
$800$		0			0
$801$	18.5246			0.654535
$802$		0			0
$803$	14.5653			0.513999
$804$		−	8.70587i		−	0.307032i
$805$			27.4945i			0.969052i
$806$		0			0
$807$	13.6102			0.479102
$808$		0			0
$809$		−	49.3848i		−	1.73628i	−0.496322	−	0.868139i	$-0.665316\pi$
$809$		−	49.3848i		−	1.73628i	0.496322	−	0.868139i	$-0.334684\pi$
$810$		0			0
$811$		−	15.7120i		−	0.551722i	−0.961197	−	0.275861i	$-0.911037\pi$
$811$		−	15.7120i		−	0.551722i	0.961197	−	0.275861i	$-0.0889630\pi$
$812$	−29.9024			−1.04937
$813$		−	23.9143i		−	0.838713i
$814$		0			0
$815$	16.7959			0.588335
$816$	−5.19675	−	35.1807i	−0.181923	−	1.23157i
$817$	2.00000			0.0699711
$818$		0			0
$819$		−	19.5285i		−	0.682382i
$820$	−64.4920			−2.25216
$821$			33.2443i			1.16023i	0.814533	+	0.580117i	$0.196993\pi$
$821$			33.2443i			1.16023i	−0.814533	+	0.580117i	$0.803007\pi$
$822$		0			0
$823$		−	37.3444i		−	1.30175i	−0.759187	−	0.650873i	$-0.774403\pi$
$823$		−	37.3444i		−	1.30175i	0.759187	−	0.650873i	$-0.225597\pi$
$824$		0			0
$825$	32.3891			1.12764
$826$		0			0
$827$		−	20.6157i		−	0.716878i	−0.933553	−	0.358439i	$-0.883309\pi$
$827$		−	20.6157i		−	0.716878i	0.933553	−	0.358439i	$-0.116691\pi$
$828$		−	12.1341i		−	0.421688i
$829$	−38.5570			−1.33914			−0.669570	−	0.742749i	$-0.733522\pi$
$829$	−38.5570			−1.33914			−0.669570	+	0.742749i	$0.733522\pi$
$830$		0			0
$831$	36.7194			1.27378
$832$	−36.4920			−1.26513
$833$	−1.07885	+	0.159363i	−0.0373798	+	0.00552159i
$834$		0			0
$835$	10.5940			0.366621
$836$		−	18.2235i		−	0.630274i
$837$	−9.04882			−0.312773
$838$		0			0
$839$		−	16.8034i		−	0.580117i	−0.957009	−	0.290058i	$-0.906325\pi$
$839$		−	16.8034i		−	0.580117i	0.957009	−	0.290058i	$-0.0936747\pi$
$840$		0			0
$841$	−4.18801			−0.144414
$842$		0			0
$843$		−	5.67295i		−	0.195387i
$844$		−	43.1582i		−	1.48557i
$845$			22.4885i			0.773628i
$846$		0			0
$847$		−	25.3198i		−	0.869998i
$848$	47.6476			1.63623
$849$	52.9505			1.81726
$850$		0			0
$851$	−4.70148			−0.161165
$852$	−15.4754			−0.530178
$853$			5.48523i			0.187811i	0.995581	+	0.0939053i	$0.0299351\pi$
$853$			5.48523i			0.187811i	−0.995581	+	0.0939053i	$0.970065\pi$
$854$		0			0
$855$		−	9.50315i		−	0.325001i
$856$		0			0
$857$		−	51.6338i		−	1.76378i	−0.471459	−	0.881888i	$-0.656272\pi$
$857$		−	51.6338i		−	1.76378i	0.471459	−	0.881888i	$-0.343728\pi$
$858$		0			0
$859$	56.9836			1.94425			0.972127	−	0.234454i	$-0.0753302\pi$
$859$	56.9836			1.94425			0.972127	+	0.234454i	$0.0753302\pi$
$860$		−	5.76090i		−	0.196445i
$861$			62.6481i			2.13504i
$862$		0			0
$863$	34.1392			1.16211			0.581056	−	0.813864i	$-0.302640\pi$
$863$	34.1392			1.16211			0.581056	+	0.813864i	$0.302640\pi$
$864$		0			0
$865$	61.4476			2.08928
$866$		0			0
$867$	10.5984	+	35.0914i	0.359940	+	1.19177i
$868$	16.1300			0.547489
$869$	4.66776			0.158343
$870$		0			0
$871$	9.20836			0.312013
$872$		0			0
$873$			3.43612i			0.116295i
$874$		0			0
$875$	−12.7309			−0.430383
$876$	−13.7875			−0.465836
$877$			2.60006i			0.0877979i	0.999036	+	0.0438989i	$0.0139780\pi$
$877$			2.60006i			0.0877979i	−0.999036	+	0.0438989i	$0.986022\pi$
$878$		0			0
$879$		−	59.2400i		−	1.99811i
$880$	−52.4920			−1.76951
$881$		−	44.6678i		−	1.50490i	−0.658650	−	0.752449i	$-0.728872\pi$
$881$		−	44.6678i		−	1.50490i	0.658650	−	0.752449i	$-0.271128\pi$
$882$		0			0
$883$	−8.98806			−0.302472			−0.151236	−	0.988498i	$-0.548325\pi$
$883$	−8.98806			−0.302472			−0.151236	+	0.988498i	$0.548325\pi$
$884$	37.2113	−	5.49670i	1.25155	−	0.184874i
$885$	−1.84471			−0.0620093
$886$		0			0
$887$			22.0946i			0.741863i	0.928660	+	0.370932i	$0.120962\pi$
$887$			22.0946i			0.741863i	−0.928660	+	0.370932i	$0.879038\pi$
$888$		0			0
$889$		−	26.7722i		−	0.897911i
$890$		0			0
$891$		−	51.1517i		−	1.71365i
$892$	−35.2291			−1.17956
$893$	−4.49157			−0.150305
$894$		0			0
$895$			34.0579i			1.13843i
$896$		0			0
$897$	36.1755			1.20787
$898$		0			0
$899$	17.9024			0.597077
$900$	−10.8774			−0.362581
$901$	−48.5868	+	7.17705i	−1.61866	+	0.239102i
$902$		0			0
$903$	−5.59619			−0.186230
$904$		0			0
$905$	27.4666			0.913022
$906$		0			0
$907$			19.1868i			0.637086i	0.947908	+	0.318543i	$0.103194\pi$
$907$			19.1868i			0.637086i	−0.947908	+	0.318543i	$0.896806\pi$
$908$			10.4459i			0.346659i
$909$	−24.4533			−0.811064
$910$		0			0
$911$			7.04646i			0.233460i	0.993164	+	0.116730i	$0.0372412\pi$
$911$			7.04646i			0.233460i	−0.993164	+	0.116730i	$0.962759\pi$
$912$			17.2503i			0.571216i
$913$			25.5808i			0.846602i
$914$		0			0
$915$		−	52.0962i		−	1.72225i
$916$	−48.4546			−1.60098
$917$	26.5660			0.877286
$918$		0			0
$919$	28.4593			0.938785			0.469393	−	0.882990i	$-0.344473\pi$
$919$	28.4593			0.938785			0.469393	+	0.882990i	$0.344473\pi$
$920$		0			0
$921$			7.52884i			0.248084i
$922$		0			0
$923$		−	16.3686i		−	0.538779i
$924$			50.9912i			1.67749i
$925$			4.21458i			0.138574i
$926$		0			0
$927$	−29.5777			−0.971459
$928$		0			0
$929$			14.8727i			0.487956i	0.969781	+	0.243978i	$0.0784525\pi$
$929$			14.8727i			0.487956i	−0.969781	+	0.243978i	$0.921547\pi$
$930$		0			0
$931$	0.528996			0.0173371
$932$		−	8.54782i		−	0.279993i
$933$	50.2791			1.64606
$934$		0			0
$935$	53.5267	−	7.90674i	1.75051	−	0.258578i
$936$		0			0
$937$	−5.51214			−0.180074			−0.0900368	−	0.995938i	$-0.528698\pi$
$937$	−5.51214			−0.180074			−0.0900368	+	0.995938i	$0.528698\pi$
$938$		0			0
$939$	4.13734			0.135017
$940$			12.9378i			0.421983i
$941$			5.45173i			0.177721i	0.996044	+	0.0888607i	$0.0283226\pi$
$941$			5.45173i			0.177721i	−0.996044	+	0.0888607i	$0.971677\pi$
$942$		0			0
$943$	−41.1732			−1.34078
$944$	1.18801			0.0386664
$945$		−	21.7679i		−	0.708111i
$946$		0			0
$947$		−	2.42303i		−	0.0787380i	−0.999225	−	0.0393690i	$-0.987465\pi$
$947$		−	2.42303i		−	0.0787380i	0.999225	−	0.0393690i	$-0.0125348\pi$
$948$	−4.41848			−0.143506
$949$		−	14.5833i		−	0.473393i
$950$		0			0
$951$	−34.8037			−1.12859
$952$		0			0
$953$	6.41782			0.207894			0.103947	−	0.994583i	$-0.466853\pi$
$953$	6.41782			0.207894			0.103947	+	0.994583i	$0.466853\pi$
$954$		0			0
$955$		−	65.9894i		−	2.13537i
$956$	1.68462			0.0544844
$957$			56.5940i			1.82943i
$958$		0			0
$959$		−	17.1133i		−	0.552618i
$960$	49.6888			1.60370
$961$	21.3431			0.688486
$962$		0			0
$963$		−	16.4790i		−	0.531029i
$964$		−	10.8164i		−	0.348372i
$965$	−10.4130			−0.335207
$966$		0			0
$967$	−39.5294			−1.27118			−0.635590	−	0.772026i	$-0.719243\pi$
$967$	−39.5294			−1.27118			−0.635590	+	0.772026i	$0.719243\pi$
$968$		0			0
$969$	−2.59838	−	17.5904i	−0.0834719	−	0.565084i
$970$		0			0
$971$	13.2761			0.426050			0.213025	−	0.977047i	$-0.431668\pi$
$971$	13.2761			0.426050			0.213025	+	0.977047i	$0.431668\pi$
$972$			30.9488i			0.992685i
$973$	−18.8770			−0.605169
$974$		0			0
$975$		−	32.4291i		−	1.03856i
$976$			33.5504i			1.07392i
$977$	−46.4849			−1.48718			−0.743592	−	0.668634i	$-0.766879\pi$
$977$	−46.4849			−1.48718			−0.743592	+	0.668634i	$0.766879\pi$
$978$		0			0
$979$		−	51.1617i		−	1.63513i
$980$		−	1.52375i		−	0.0486743i
$981$		−	33.6357i		−	1.07390i
$982$		0			0
$983$			2.51722i			0.0802869i	0.999194	+	0.0401434i	$0.0127815\pi$
$983$			2.51722i			0.0802869i	−0.999194	+	0.0401434i	$0.987219\pi$
$984$		0			0
$985$	52.3110			1.66677
$986$		0			0
$987$	12.5678			0.400039
$988$	−18.2460			−0.580483
$989$		−	3.67790i		−	0.116950i
$990$		0			0
$991$			41.0343i			1.30350i	0.758435	+	0.651749i	$0.225964\pi$
$991$			41.0343i			1.30350i	−0.758435	+	0.651749i	$0.774036\pi$
$992$		0			0
$993$		−	42.2947i		−	1.34218i
$994$		0			0
$995$	−64.0023			−2.02901
$996$		−	24.2147i		−	0.767273i
$997$		−	15.0255i		−	0.475863i	−0.971282	−	0.237932i	$-0.923531\pi$
$997$		−	15.0255i		−	0.475863i	0.971282	−	0.237932i	$-0.0764695\pi$
$998$		0			0
$999$	3.72226			0.117767

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.d.b.560.7	yes	8
17.16	even	2	inner	731.2.d.b.560.2	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.d.b.560.2	✓	8	17.16	even	2	inner
731.2.d.b.560.7	yes	8	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.15629i q^{3} -2.00000 q^{4} +2.88045i q^{5} +2.59528i q^{7} -1.64959 q^{9} +O(q^{10})\)	\(q+2.15629i q^{3} -2.00000 q^{4} +2.88045i q^{5} +2.59528i q^{7} -1.64959 q^{9} +4.55588i q^{11} -4.31258i q^{12} +4.56150 q^{13} -6.21109 q^{15} +4.00000 q^{16} +(-4.07885 + 0.602510i) q^{17} +2.00000 q^{19} -5.76090i q^{20} -5.59619 q^{21} -3.67790i q^{23} -3.29700 q^{25} +2.91187i q^{27} -5.19057i q^{28} -5.76090i q^{29} +3.10756i q^{31} -9.82381 q^{33} -7.47559 q^{35} +3.29919 q^{36} -1.27831i q^{37} +9.83593i q^{39} -11.1948i q^{41} +1.00000 q^{43} -9.11177i q^{44} -4.75158i q^{45} -2.24578 q^{47} +8.62517i q^{48} +0.264498 q^{49} +(-1.29919 - 8.79518i) q^{51} -9.12300 q^{52} +11.9119 q^{53} -13.1230 q^{55} +4.31258i q^{57} +0.297003 q^{59} +12.4222 q^{60} +8.38761i q^{61} -4.28117i q^{63} -8.00000 q^{64} +13.1392i q^{65} +2.01871 q^{67} +(8.15769 - 1.20502i) q^{68} +7.93062 q^{69} -3.58842i q^{71} -3.19704i q^{73} -7.10930i q^{75} -4.00000 q^{76} -11.8238 q^{77} -1.02456i q^{79} +11.5218i q^{80} -11.2276 q^{81} +5.61490 q^{83} +11.1924 q^{84} +(-1.73550 - 11.7489i) q^{85} +12.4222 q^{87} -11.2298 q^{89} +11.8384i q^{91} +7.35579i q^{92} -6.70081 q^{93} +5.76090i q^{95} -2.08301i q^{97} -7.51536i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{4} - 16 q^{9}+O(q^{10}) \) 8 * q - 16 * q^4 - 16 * q^9	\( 8 q - 16 q^{4} - 16 q^{9} - 10 q^{13} - 6 q^{15} + 32 q^{16} - 3 q^{17} + 16 q^{19} - 32 q^{21} - 8 q^{25} + 20 q^{33} + 12 q^{35} + 32 q^{36} + 8 q^{43} - 8 q^{47} - 26 q^{49} - 16 q^{51} + 20 q^{52} + 46 q^{53} - 12 q^{55} - 16 q^{59} + 12 q^{60} - 64 q^{64} - 2 q^{67} + 6 q^{68} - 4 q^{69} - 32 q^{76} + 4 q^{77} - 4 q^{81} + 14 q^{83} + 64 q^{84} - 42 q^{85} + 12 q^{87} - 28 q^{89} - 48 q^{93}+O(q^{100}) \) 8 * q - 16 * q^4 - 16 * q^9 - 10 * q^13 - 6 * q^15 + 32 * q^16 - 3 * q^17 + 16 * q^19 - 32 * q^21 - 8 * q^25 + 20 * q^33 + 12 * q^35 + 32 * q^36 + 8 * q^43 - 8 * q^47 - 26 * q^49 - 16 * q^51 + 20 * q^52 + 46 * q^53 - 12 * q^55 - 16 * q^59 + 12 * q^60 - 64 * q^64 - 2 * q^67 + 6 * q^68 - 4 * q^69 - 32 * q^76 + 4 * q^77 - 4 * q^81 + 14 * q^83 + 64 * q^84 - 42 * q^85 + 12 * q^87 - 28 * q^89 - 48 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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