LMFDB - Embedded newform 688.5.b.d.257.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [688,5,Mod(257,688)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(688, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("688.257");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 688 = 2^{4} \cdot 43 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	688.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$71.1185346017$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 142x^{10} + 7173x^{8} + 157368x^{6} + 1510016x^{4} + 5098688x^{2} + 90352 $ x^12 + 142x^10 + 7173x^8 + 157368x^6 + 1510016x^4 + 5098688*x^2 + 90352
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{11} $
Twist minimal:	no (minimal twist has level 43)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			257.3
Root			$2.75662i$ of defining polynomial
Character	$\chi$	$=$	688.257
Dual form			688.5.b.d.257.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-12.3317i q^{3} +21.9831i q^{5} +63.3272i q^{7} -71.0696 q^{9} +O(q^{10})$	$q-12.3317i q^{3} +21.9831i q^{5} +63.3272i q^{7} -71.0696 q^{9} +1.17035 q^{11} -173.901 q^{13} +271.087 q^{15} +469.315 q^{17} +27.8220i q^{19} +780.929 q^{21} -79.3541 q^{23} +141.745 q^{25} -122.458i q^{27} -696.895i q^{29} -1190.63 q^{31} -14.4324i q^{33} -1392.12 q^{35} +1535.36i q^{37} +2144.49i q^{39} -2455.37 q^{41} +(1435.68 + 1165.17i) q^{43} -1562.33i q^{45} -1694.64 q^{47} -1609.33 q^{49} -5787.42i q^{51} -1990.15 q^{53} +25.7279i q^{55} +343.091 q^{57} -3563.55 q^{59} -5447.36i q^{61} -4500.64i q^{63} -3822.88i q^{65} -4930.23 q^{67} +978.567i q^{69} +9660.68i q^{71} -5609.18i q^{73} -1747.95i q^{75} +74.1151i q^{77} +11423.8 q^{79} -7266.75 q^{81} -5524.25 q^{83} +10317.0i q^{85} -8593.86 q^{87} -2559.89i q^{89} -11012.7i q^{91} +14682.5i q^{93} -611.612 q^{95} +3659.79 q^{97} -83.1766 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 462 q^{9}+O(q^{10}) $ 12 * q - 462 * q^9	$ 12 q - 462 q^{9} + 180 q^{11} - 216 q^{13} + 92 q^{15} + 678 q^{17} - 2392 q^{21} - 1566 q^{23} - 174 q^{25} - 5710 q^{31} - 936 q^{35} + 4878 q^{41} + 1108 q^{43} + 5526 q^{47} - 8544 q^{49} + 1212 q^{53} - 7692 q^{57} - 14016 q^{59} + 1088 q^{67} - 24302 q^{79} - 23660 q^{81} + 7032 q^{83} - 17850 q^{87} - 606 q^{95} - 5842 q^{97} + 25924 q^{99}+O(q^{100}) $ 12 * q - 462 * q^9 + 180 * q^11 - 216 * q^13 + 92 * q^15 + 678 * q^17 - 2392 * q^21 - 1566 * q^23 - 174 * q^25 - 5710 * q^31 - 936 * q^35 + 4878 * q^41 + 1108 * q^43 + 5526 * q^47 - 8544 * q^49 + 1212 * q^53 - 7692 * q^57 - 14016 * q^59 + 1088 * q^67 - 24302 * q^79 - 23660 * q^81 + 7032 * q^83 - 17850 * q^87 - 606 * q^95 - 5842 * q^97 + 25924 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/688\mathbb{Z}\right)^\times$.

$n$	$431$	$433$	$517$
$\chi(n)$	$1$	$-1$	$1$

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
$3$		−	12.3317i		−	1.37018i	−0.728457	−	0.685092i	$-0.759762\pi$
$3$		−	12.3317i		−	1.37018i	0.728457	−	0.685092i	$-0.240238\pi$
$4$		0			0
$5$			21.9831i			0.879322i	0.898164	+	0.439661i	$0.144901\pi$
$5$			21.9831i			0.879322i	−0.898164	+	0.439661i	$0.855099\pi$
$6$		0			0
$7$			63.3272i			1.29239i	0.763172	+	0.646196i	$0.223641\pi$
$7$			63.3272i			1.29239i	−0.763172	+	0.646196i	$0.776359\pi$
$8$		0			0
$9$	−71.0696			−0.877403
$10$		0			0
$11$	1.17035			0.00967234			0.00483617	−	0.999988i	$-0.498461\pi$
$11$	1.17035			0.00967234			0.00483617	+	0.999988i	$0.498461\pi$
$12$		0			0
$13$	−173.901			−1.02900			−0.514501	−	0.857490i	$-0.672023\pi$
$13$	−173.901			−1.02900			−0.514501	+	0.857490i	$0.672023\pi$
$14$		0			0
$15$	271.087			1.20483
$16$		0			0
$17$	469.315			1.62393			0.811963	−	0.583709i	$-0.198399\pi$
$17$	469.315			1.62393			0.811963	+	0.583709i	$0.198399\pi$
$18$		0			0
$19$			27.8220i			0.0770692i	0.999257	+	0.0385346i	$0.0122690\pi$
$19$			27.8220i			0.0770692i	−0.999257	+	0.0385346i	$0.987731\pi$
$20$		0			0
$21$	780.929			1.77081
$22$		0			0
$23$	−79.3541			−0.150008			−0.0750039	−	0.997183i	$-0.523897\pi$
$23$	−79.3541			−0.150008			−0.0750039	+	0.997183i	$0.523897\pi$
$24$		0			0
$25$	141.745			0.226792
$26$		0			0
$27$		−	122.458i		−	0.167980i
$28$		0			0
$29$		−	696.895i		−	0.828650i	−0.910129	−	0.414325i	$-0.864018\pi$
$29$		−	696.895i		−	0.828650i	0.910129	−	0.414325i	$-0.135982\pi$
$30$		0			0
$31$	−1190.63			−1.23895			−0.619477	−	0.785015i	$-0.712655\pi$
$31$	−1190.63			−1.23895			−0.619477	+	0.785015i	$0.712655\pi$
$32$		0			0
$33$		−	14.4324i		−	0.0132529i
$34$		0			0
$35$	−1392.12			−1.13643
$36$		0			0
$37$			1535.36i			1.12152i	0.827980	+	0.560758i	$0.189490\pi$
$37$			1535.36i			1.12152i	−0.827980	+	0.560758i	$0.810510\pi$
$38$		0			0
$39$			2144.49i			1.40992i
$40$		0			0
$41$	−2455.37			−1.46066			−0.730329	−	0.683095i	$-0.760633\pi$
$41$	−2455.37			−1.46066			−0.730329	+	0.683095i	$0.760633\pi$
$42$		0			0
$43$	1435.68	+	1165.17i	0.776463	+	0.630163i
$43$	1435.68	+	1165.17i	0.776463	+	0.630163i
$44$		0			0
$45$		−	1562.33i		−	0.771520i
$46$		0			0
$47$	−1694.64			−0.767154			−0.383577	−	0.923509i	$-0.625308\pi$
$47$	−1694.64			−0.767154			−0.383577	+	0.923509i	$0.625308\pi$
$48$		0			0
$49$	−1609.33			−0.670275
$50$		0			0
$51$		−	5787.42i		−	2.22508i
$52$		0			0
$53$	−1990.15			−0.708489			−0.354245	−	0.935153i	$-0.615262\pi$
$53$	−1990.15			−0.708489			−0.354245	+	0.935153i	$0.615262\pi$
$54$		0			0
$55$			25.7279i			0.00850510i
$56$		0			0
$57$	343.091			0.105599
$58$		0			0
$59$	−3563.55			−1.02371			−0.511857	−	0.859071i	$-0.671042\pi$
$59$	−3563.55			−1.02371			−0.511857	+	0.859071i	$0.671042\pi$
$60$		0			0
$61$		−	5447.36i		−	1.46395i	−0.681331	−	0.731975i	$-0.738599\pi$
$61$		−	5447.36i		−	1.46395i	0.681331	−	0.731975i	$-0.261401\pi$
$62$		0			0
$63$		−	4500.64i		−	1.13395i
$64$		0			0
$65$		−	3822.88i		−	0.904824i
$66$		0			0
$67$	−4930.23			−1.09829			−0.549145	−	0.835727i	$-0.685047\pi$
$67$	−4930.23			−1.09829			−0.549145	+	0.835727i	$0.685047\pi$
$68$		0			0
$69$			978.567i			0.205538i
$70$		0			0
$71$			9660.68i			1.91642i	0.286065	+	0.958210i	$0.407653\pi$
$71$			9660.68i			1.91642i	−0.286065	+	0.958210i	$0.592347\pi$
$72$		0			0
$73$		−	5609.18i		−	1.05258i	−0.850306	−	0.526288i	$-0.823583\pi$
$73$		−	5609.18i		−	1.05258i	0.850306	−	0.526288i	$-0.176417\pi$
$74$		0			0
$75$		−	1747.95i		−	0.310747i
$76$		0			0
$77$			74.1151i			0.0125004i
$78$		0			0
$79$	11423.8			1.83044			0.915218	−	0.402959i	$-0.132018\pi$
$79$	11423.8			1.83044			0.915218	+	0.402959i	$0.132018\pi$
$80$		0			0
$81$	−7266.75			−1.10757
$82$		0			0
$83$	−5524.25			−0.801894			−0.400947	−	0.916101i	$-0.631319\pi$
$83$	−5524.25			−0.801894			−0.400947	+	0.916101i	$0.631319\pi$
$84$		0			0
$85$			10317.0i			1.42795i
$86$		0			0
$87$	−8593.86			−1.13540
$88$		0			0
$89$		−	2559.89i		−	0.323177i	−0.986858	−	0.161589i	$-0.948338\pi$
$89$		−	2559.89i		−	0.323177i	0.986858	−	0.161589i	$-0.0516618\pi$
$90$		0			0
$91$		−	11012.7i		−	1.32987i
$92$		0			0
$93$			14682.5i			1.69759i
$94$		0			0
$95$	−611.612			−0.0677686
$96$		0			0
$97$	3659.79			0.388967			0.194484	−	0.980906i	$-0.437697\pi$
$97$	3659.79			0.388967			0.194484	+	0.980906i	$0.437697\pi$
$98$		0			0
$99$	−83.1766			−0.00848654
$100$		0			0
$101$	−10505.1			−1.02981			−0.514907	−	0.857246i	$-0.672173\pi$
$101$	−10505.1			−1.02981			−0.514907	+	0.857246i	$0.672173\pi$
$102$		0			0
$103$	−2014.90			−0.189924			−0.0949619	−	0.995481i	$-0.530273\pi$
$103$	−2014.90			−0.189924			−0.0949619	+	0.995481i	$0.530273\pi$
$104$		0			0
$105$			17167.2i			1.55712i
$106$		0			0
$107$	−8728.48			−0.762379			−0.381190	−	0.924497i	$-0.624486\pi$
$107$	−8728.48			−0.762379			−0.381190	+	0.924497i	$0.624486\pi$
$108$		0			0
$109$	−9286.44			−0.781621			−0.390811	−	0.920471i	$-0.627805\pi$
$109$	−9286.44			−0.781621			−0.390811	+	0.920471i	$0.627805\pi$
$110$		0			0
$111$	18933.5			1.53668
$112$		0			0
$113$			16690.4i			1.30711i	0.756881	+	0.653553i	$0.226722\pi$
$113$			16690.4i			1.30711i	−0.756881	+	0.653553i	$0.773278\pi$
$114$		0			0
$115$		−	1744.45i		−	0.131905i
$116$		0			0
$117$	12359.1			0.902849
$118$		0			0
$119$			29720.4i			2.09875i
$120$		0			0
$121$	−14639.6			−0.999906
$122$		0			0
$123$			30278.7i			2.00137i
$124$		0			0
$125$			16855.4i			1.07875i
$126$		0			0
$127$	17016.4			1.05502			0.527510	−	0.849549i	$-0.323126\pi$
$127$	17016.4			1.05502			0.527510	+	0.849549i	$0.323126\pi$
$128$		0			0
$129$	14368.5	−	17704.3i	0.863440	−	1.06390i
$130$		0			0
$131$		−	13378.5i		−	0.779590i	−0.920902	−	0.389795i	$-0.872546\pi$
$131$		−	13378.5i		−	0.779590i	0.920902	−	0.389795i	$-0.127454\pi$
$132$		0			0
$133$	−1761.89			−0.0996035
$134$		0			0
$135$	2691.99			0.147709
$136$		0			0
$137$		−	9129.47i		−	0.486412i	−0.969975	−	0.243206i	$-0.921801\pi$
$137$		−	9129.47i		−	0.486412i	0.969975	−	0.243206i	$-0.0781991\pi$
$138$		0			0
$139$	−28220.0			−1.46059			−0.730293	−	0.683134i	$-0.760616\pi$
$139$	−28220.0			−1.46059			−0.730293	+	0.683134i	$0.760616\pi$
$140$		0			0
$141$			20897.8i			1.05114i
$142$		0			0
$143$	−203.526			−0.00995285
$144$		0			0
$145$	15319.9			0.728651
$146$		0			0
$147$			19845.7i			0.918400i
$148$		0			0
$149$		−	11221.8i		−	0.505464i	−0.967536	−	0.252732i	$-0.918671\pi$
$149$		−	11221.8i		−	0.505464i	0.967536	−	0.252732i	$-0.0813291\pi$
$150$		0			0
$151$			37652.1i			1.65133i	0.564158	+	0.825667i	$0.309201\pi$
$151$			37652.1i			1.65133i	−0.564158	+	0.825667i	$0.690799\pi$
$152$		0			0
$153$	−33354.0			−1.42484
$154$		0			0
$155$		−	26173.8i		−	1.08944i
$156$		0			0
$157$			9904.16i			0.401808i	0.979611	+	0.200904i	$0.0643879\pi$
$157$			9904.16i			0.401808i	−0.979611	+	0.200904i	$0.935612\pi$
$158$		0			0
$159$			24541.8i			0.970761i
$160$		0			0
$161$		−	5025.27i		−	0.193869i
$162$		0			0
$163$			20673.8i			0.778117i	0.921213	+	0.389059i	$0.127200\pi$
$163$			20673.8i			0.778117i	−0.921213	+	0.389059i	$0.872800\pi$
$164$		0			0
$165$	317.268			0.0116536
$166$		0			0
$167$	313.151			0.0112285			0.00561424	−	0.999984i	$-0.498213\pi$
$167$	313.151			0.0112285			0.00561424	+	0.999984i	$0.498213\pi$
$168$		0			0
$169$	1680.65			0.0588441
$170$		0			0
$171$		−	1977.30i		−	0.0676207i
$172$		0			0
$173$	−5952.11			−0.198874			−0.0994372	−	0.995044i	$-0.531704\pi$
$173$	−5952.11			−0.198874			−0.0994372	+	0.995044i	$0.531704\pi$
$174$		0			0
$175$			8976.32i			0.293104i
$176$		0			0
$177$			43944.4i			1.40268i
$178$		0			0
$179$		−	39430.1i		−	1.23062i	−0.788287	−	0.615308i	$-0.789032\pi$
$179$		−	39430.1i		−	1.23062i	0.788287	−	0.615308i	$-0.210968\pi$
$180$		0			0
$181$	29726.1			0.907361			0.453681	−	0.891164i	$-0.350111\pi$
$181$	29726.1			0.907361			0.453681	+	0.891164i	$0.350111\pi$
$182$		0			0
$183$	−67174.9			−2.00588
$184$		0			0
$185$	−33751.8			−0.986174
$186$		0			0
$187$	549.264			0.0157072
$188$		0			0
$189$	7754.90			0.217096
$190$		0			0
$191$		−	35347.9i		−	0.968940i	−0.874808	−	0.484470i	$-0.839012\pi$
$191$		−	35347.9i		−	0.968940i	0.874808	−	0.484470i	$-0.160988\pi$
$192$		0			0
$193$	−74147.3			−1.99058			−0.995292	−	0.0969181i	$-0.969102\pi$
$193$	−74147.3			−1.99058			−0.995292	+	0.0969181i	$0.969102\pi$
$194$		0			0
$195$	−47142.4			−1.23978
$196$		0			0
$197$	33924.2			0.874133			0.437066	−	0.899429i	$-0.356017\pi$
$197$	33924.2			0.874133			0.437066	+	0.899429i	$0.356017\pi$
$198$		0			0
$199$			20327.0i			0.513296i	0.966505	+	0.256648i	$0.0826182\pi$
$199$			20327.0i			0.513296i	−0.966505	+	0.256648i	$0.917382\pi$
$200$		0			0
$201$			60797.9i			1.50486i
$202$		0			0
$203$	44132.4			1.07094
$204$		0			0
$205$		−	53976.5i		−	1.28439i
$206$		0			0
$207$	5639.67			0.131617
$208$		0			0
$209$			32.5615i			0.000745439i
$210$		0			0
$211$			6957.00i			0.156263i	0.996943	+	0.0781316i	$0.0248954\pi$
$211$			6957.00i			0.156263i	−0.996943	+	0.0781316i	$0.975105\pi$
$212$		0			0
$213$	119132.			2.62585
$214$		0			0
$215$	−25614.1	+	31560.6i	−0.554117	+	0.682761i
$216$		0			0
$217$		−	75399.5i		−	1.60121i
$218$		0			0
$219$	−69170.4			−1.44222
$220$		0			0
$221$	−81614.4			−1.67102
$222$		0			0
$223$			2228.35i			0.0448099i	0.999749	+	0.0224050i	$0.00713232\pi$
$223$			2228.35i			0.0448099i	−0.999749	+	0.0224050i	$0.992868\pi$
$224$		0			0
$225$	−10073.8			−0.198988
$226$		0			0
$227$		−	61573.4i		−	1.19493i	−0.801896	−	0.597464i	$-0.796175\pi$
$227$		−	61573.4i		−	1.19493i	0.801896	−	0.597464i	$-0.203825\pi$
$228$		0			0
$229$	−40330.0			−0.769054			−0.384527	−	0.923114i	$-0.625635\pi$
$229$	−40330.0			−0.769054			−0.384527	+	0.923114i	$0.625635\pi$
$230$		0			0
$231$	913.962			0.0171279
$232$		0			0
$233$			74382.7i			1.37013i	0.728484	+	0.685063i	$0.240225\pi$
$233$			74382.7i			1.37013i	−0.728484	+	0.685063i	$0.759775\pi$
$234$		0			0
$235$		−	37253.4i		−	0.674576i
$236$		0			0
$237$		−	140874.i		−	2.50803i
$238$		0			0
$239$	−11874.5			−0.207883			−0.103942	−	0.994583i	$-0.533146\pi$
$239$	−11874.5			−0.207883			−0.103942	+	0.994583i	$0.533146\pi$
$240$		0			0
$241$			23536.3i			0.405233i	0.979258	+	0.202616i	$0.0649444\pi$
$241$			23536.3i			0.405233i	−0.979258	+	0.202616i	$0.935056\pi$
$242$		0			0
$243$			79691.9i			1.34959i
$244$		0			0
$245$		−	35378.0i		−	0.589388i
$246$		0			0
$247$		−	4838.28i		−	0.0793043i
$248$		0			0
$249$			68123.1i			1.09874i
$250$		0			0
$251$	54757.4			0.869151			0.434575	−	0.900635i	$-0.356898\pi$
$251$	54757.4			0.869151			0.434575	+	0.900635i	$0.356898\pi$
$252$		0			0
$253$	−92.8723			−0.00145093
$254$		0			0
$255$	127225.			1.95656
$256$		0			0
$257$		−	57759.9i		−	0.874501i	−0.899340	−	0.437250i	$-0.855952\pi$
$257$		−	57759.9i		−	0.874501i	0.899340	−	0.437250i	$-0.144048\pi$
$258$		0			0
$259$	−97229.7			−1.44944
$260$		0			0
$261$			49528.1i			0.727060i
$262$		0			0
$263$		−	7567.24i		−	0.109402i	−0.998503	−	0.0547011i	$-0.982579\pi$
$263$		−	7567.24i		−	0.109402i	0.998503	−	0.0547011i	$-0.0174206\pi$
$264$		0			0
$265$		−	43749.5i		−	0.622991i
$266$		0			0
$267$	−31567.6			−0.442812
$268$		0			0
$269$	−39923.2			−0.551723			−0.275861	−	0.961197i	$-0.588963\pi$
$269$	−39923.2			−0.551723			−0.275861	+	0.961197i	$0.588963\pi$
$270$		0			0
$271$	−119787.			−1.63107			−0.815533	−	0.578710i	$-0.803556\pi$
$271$	−119787.			−1.63107			−0.815533	+	0.578710i	$0.803556\pi$
$272$		0			0
$273$	−135804.			−1.82217
$274$		0			0
$275$	165.892			0.00219361
$276$		0			0
$277$			37526.7i			0.489081i	0.969639	+	0.244540i	$0.0786371\pi$
$277$			37526.7i			0.489081i	−0.969639	+	0.244540i	$0.921363\pi$
$278$		0			0
$279$	84618.0			1.08706
$280$		0			0
$281$	−29935.1			−0.379113			−0.189556	−	0.981870i	$-0.560705\pi$
$281$	−29935.1			−0.379113			−0.189556	+	0.981870i	$0.560705\pi$
$282$		0			0
$283$	54918.3			0.685716			0.342858	−	0.939387i	$-0.388605\pi$
$283$	54918.3			0.685716			0.342858	+	0.939387i	$0.388605\pi$
$284$		0			0
$285$			7542.19i			0.0928555i
$286$		0			0
$287$		−	155491.i		−	1.88774i
$288$		0			0
$289$	136735.			1.63713
$290$		0			0
$291$		−	45131.3i		−	0.532956i
$292$		0			0
$293$	−155740.			−1.81412			−0.907060	−	0.421002i	$-0.861678\pi$
$293$	−155740.			−1.81412			−0.907060	+	0.421002i	$0.861678\pi$
$294$		0			0
$295$		−	78337.6i		−	0.900174i
$296$		0			0
$297$		−	143.319i		−	0.00162476i
$298$		0			0
$299$	13799.8			0.154358
$300$		0			0
$301$	−73787.1	+	90917.5i	−0.814418	+	1.00349i
$302$		0			0
$303$			129546.i			1.41103i
$304$		0			0
$305$	119750.			1.28728
$306$		0			0
$307$	−70887.6			−0.752131			−0.376065	−	0.926593i	$-0.622723\pi$
$307$	−70887.6			−0.752131			−0.376065	+	0.926593i	$0.622723\pi$
$308$		0			0
$309$			24847.1i			0.260230i
$310$		0			0
$311$	27909.5			0.288557			0.144278	−	0.989537i	$-0.453914\pi$
$311$	27909.5			0.288557			0.144278	+	0.989537i	$0.453914\pi$
$312$		0			0
$313$		−	50613.7i		−	0.516630i	−0.966061	−	0.258315i	$-0.916833\pi$
$313$		−	50613.7i		−	0.516630i	0.966061	−	0.258315i	$-0.0831672\pi$
$314$		0			0
$315$	98937.8			0.997106
$316$		0			0
$317$	111161.			1.10620			0.553099	−	0.833116i	$-0.313445\pi$
$317$	111161.			1.10620			0.553099	+	0.833116i	$0.313445\pi$
$318$		0			0
$319$		−	815.613i		−	0.00801498i
$320$		0			0
$321$			107637.i			1.04460i
$322$		0			0
$323$			13057.3i			0.125155i
$324$		0			0
$325$	−24649.7			−0.233369
$326$		0			0
$327$			114517.i			1.07096i
$328$		0			0
$329$		−	107317.i		−	0.991463i
$330$		0			0
$331$		−	199722.i		−	1.82293i	−0.411376	−	0.911466i	$-0.634952\pi$
$331$		−	199722.i		−	1.82293i	0.411376	−	0.911466i	$-0.365048\pi$
$332$		0			0
$333$		−	109117.i		−	0.984021i
$334$		0			0
$335$		−	108381.i		−	0.965752i
$336$		0			0
$337$	−19032.7			−0.167588			−0.0837938	−	0.996483i	$-0.526704\pi$
$337$	−19032.7			−0.167588			−0.0837938	+	0.996483i	$0.526704\pi$
$338$		0			0
$339$	205821.			1.79097
$340$		0			0
$341$	−1393.46			−0.0119836
$342$		0			0
$343$			50134.2i			0.426134i
$344$		0			0
$345$	−21511.9			−0.180734
$346$		0			0
$347$		−	13299.5i		−	0.110453i	−0.998474	−	0.0552263i	$-0.982412\pi$
$347$		−	13299.5i		−	0.110453i	0.998474	−	0.0552263i	$-0.0175880\pi$
$348$		0			0
$349$		−	207713.i		−	1.70535i	−0.522442	−	0.852675i	$-0.674979\pi$
$349$		−	207713.i		−	1.70535i	0.522442	−	0.852675i	$-0.325021\pi$
$350$		0			0
$351$			21295.5i			0.172852i
$352$		0			0
$353$	69576.5			0.558358			0.279179	−	0.960239i	$-0.409938\pi$
$353$	69576.5			0.558358			0.279179	+	0.960239i	$0.409938\pi$
$354$		0			0
$355$	−212371.			−1.68515
$356$		0			0
$357$	366501.			2.87567
$358$		0			0
$359$	165095.			1.28099			0.640495	−	0.767963i	$-0.278729\pi$
$359$	165095.			1.28099			0.640495	+	0.767963i	$0.278729\pi$
$360$		0			0
$361$	129547.			0.994060
$362$		0			0
$363$			180531.i			1.37006i
$364$		0			0
$365$	123307.			0.925554
$366$		0			0
$367$	58660.7			0.435527			0.217764	−	0.976002i	$-0.430124\pi$
$367$	58660.7			0.435527			0.217764	+	0.976002i	$0.430124\pi$
$368$		0			0
$369$	174502.			1.28159
$370$		0			0
$371$		−	126030.i		−	0.915645i
$372$		0			0
$373$			179031.i			1.28680i	0.765530	+	0.643401i	$0.222477\pi$
$373$			179031.i			1.28680i	−0.765530	+	0.643401i	$0.777523\pi$
$374$		0			0
$375$	207855.			1.47808
$376$		0			0
$377$			121191.i			0.852682i
$378$		0			0
$379$	47513.9			0.330783			0.165391	−	0.986228i	$-0.447111\pi$
$379$	47513.9			0.330783			0.165391	+	0.986228i	$0.447111\pi$
$380$		0			0
$381$		−	209840.i		−	1.44557i
$382$		0			0
$383$			59722.7i			0.407138i	0.979061	+	0.203569i	$0.0652541\pi$
$383$			59722.7i			0.407138i	−0.979061	+	0.203569i	$0.934746\pi$
$384$		0			0
$385$	−1629.28			−0.0109919
$386$		0			0
$387$	−102033.	−	82808.4i	−0.681271	−	0.552907i
$388$		0			0
$389$			254851.i			1.68417i	0.539343	+	0.842086i	$0.318673\pi$
$389$			254851.i			1.68417i	−0.539343	+	0.842086i	$0.681327\pi$
$390$		0			0
$391$	−37242.0			−0.243601
$392$		0			0
$393$	−164980.			−1.06818
$394$		0			0
$395$			251129.i			1.60954i
$396$		0			0
$397$	−79062.9			−0.501639			−0.250820	−	0.968034i	$-0.580700\pi$
$397$	−79062.9			−0.501639			−0.250820	+	0.968034i	$0.580700\pi$
$398$		0			0
$399$			21727.0i			0.136475i
$400$		0			0
$401$	−196389.			−1.22132			−0.610658	−	0.791894i	$-0.709095\pi$
$401$	−196389.			−1.22132			−0.610658	+	0.791894i	$0.709095\pi$
$402$		0			0
$403$	207053.			1.27489
$404$		0			0
$405$		−	159745.i		−	0.973908i
$406$		0			0
$407$			1796.91i			0.0108477i
$408$		0			0
$409$			115044.i			0.687729i	0.939019	+	0.343864i	$0.111736\pi$
$409$			115044.i			0.687729i	−0.939019	+	0.343864i	$0.888264\pi$
$410$		0			0
$411$	−112581.			−0.666474
$412$		0			0
$413$		−	225669.i		−	1.32304i
$414$		0			0
$415$		−	121440.i		−	0.705123i
$416$		0			0
$417$			347999.i			2.00127i
$418$		0			0
$419$			281568.i			1.60382i	0.597446	+	0.801909i	$0.296182\pi$
$419$			281568.i			1.60382i	−0.597446	+	0.801909i	$0.703818\pi$
$420$		0			0
$421$		−	182603.i		−	1.03025i	−0.857115	−	0.515125i	$-0.827745\pi$
$421$		−	182603.i		−	1.03025i	0.857115	−	0.515125i	$-0.172255\pi$
$422$		0			0
$423$	120438.			0.673103
$424$		0			0
$425$	66523.0			0.368294
$426$		0			0
$427$	344966.			1.89200
$428$		0			0
$429$			2509.81i			0.0136372i
$430$		0			0
$431$	−295412.			−1.59028			−0.795139	−	0.606427i	$-0.792602\pi$
$431$	−295412.			−1.59028			−0.795139	+	0.606427i	$0.792602\pi$
$432$		0			0
$433$			315628.i			1.68345i	0.539908	+	0.841724i	$0.318459\pi$
$433$			315628.i			1.68345i	−0.539908	+	0.841724i	$0.681541\pi$
$434$		0			0
$435$		−	188919.i		−	0.998385i
$436$		0			0
$437$		−	2207.79i		−	0.0115610i
$438$		0			0
$439$	−289252.			−1.50088			−0.750442	−	0.660937i	$-0.770159\pi$
$439$	−289252.			−1.50088			−0.750442	+	0.660937i	$0.770159\pi$
$440$		0			0
$441$	114375.			0.588101
$442$		0			0
$443$	13612.9			0.0693655			0.0346827	−	0.999398i	$-0.488958\pi$
$443$	13612.9			0.0693655			0.0346827	+	0.999398i	$0.488958\pi$
$444$		0			0
$445$	56274.1			0.284177
$446$		0			0
$447$	−138383.			−0.692578
$448$		0			0
$449$		−	164810.i		−	0.817506i	−0.912645	−	0.408753i	$-0.865964\pi$
$449$		−	164810.i		−	0.817506i	0.912645	−	0.408753i	$-0.134036\pi$
$450$		0			0
$451$	−2873.65			−0.0141280
$452$		0			0
$453$	464312.			2.26263
$454$		0			0
$455$	242092.			1.16939
$456$		0			0
$457$		−	157048.i		−	0.751971i	−0.926626	−	0.375985i	$-0.877304\pi$
$457$		−	157048.i		−	0.751971i	0.926626	−	0.375985i	$-0.122696\pi$
$458$		0			0
$459$		−	57471.2i		−	0.272788i
$460$		0			0
$461$	332439.			1.56426			0.782131	−	0.623114i	$-0.214133\pi$
$461$	332439.			1.56426			0.782131	+	0.623114i	$0.214133\pi$
$462$		0			0
$463$			197697.i			0.922226i	0.887341	+	0.461113i	$0.152550\pi$
$463$			197697.i			0.922226i	−0.887341	+	0.461113i	$0.847450\pi$
$464$		0			0
$465$	−322766.			−1.49273
$466$		0			0
$467$		−	121639.i		−	0.557749i	−0.960328	−	0.278874i	$-0.910039\pi$
$467$		−	121639.i		−	0.557749i	0.960328	−	0.278874i	$-0.0899613\pi$
$468$		0			0
$469$		−	312217.i		−	1.41942i
$470$		0			0
$471$	122135.			0.550550
$472$		0			0
$473$	1680.25	+	1363.66i	0.00751021	+	0.00609515i
$474$		0			0
$475$			3943.63i			0.0174787i
$476$		0			0
$477$	141439.			0.621631
$478$		0			0
$479$	141700.			0.617586			0.308793	−	0.951129i	$-0.400075\pi$
$479$	141700.			0.617586			0.308793	+	0.951129i	$0.400075\pi$
$480$		0			0
$481$		−	267000.i		−	1.15404i
$482$		0			0
$483$	−61969.9			−0.265636
$484$		0			0
$485$			80453.4i			0.342027i
$486$		0			0
$487$	−194637.			−0.820668			−0.410334	−	0.911935i	$-0.634588\pi$
$487$	−194637.			−0.820668			−0.410334	+	0.911935i	$0.634588\pi$
$488$		0			0
$489$	254942.			1.06616
$490$		0			0
$491$		−	143694.i		−	0.596040i	−0.954560	−	0.298020i	$-0.903674\pi$
$491$		−	143694.i		−	0.596040i	0.954560	−	0.298020i	$-0.0963263\pi$
$492$		0			0
$493$		−	327063.i		−	1.34567i
$494$		0			0
$495$		−	1828.48i		−	0.00746240i
$496$		0			0
$497$	−611783.			−2.47676
$498$		0			0
$499$		−	328853.i		−	1.32069i	−0.750962	−	0.660345i	$-0.770410\pi$
$499$		−	328853.i		−	1.32069i	0.750962	−	0.660345i	$-0.229590\pi$
$500$		0			0
$501$		−	3861.67i		−	0.0153851i
$502$		0			0
$503$		−	278129.i		−	1.09929i	−0.835399	−	0.549643i	$-0.814764\pi$
$503$		−	278129.i		−	1.09929i	0.835399	−	0.549643i	$-0.185236\pi$
$504$		0			0
$505$		−	230935.i		−	0.905538i
$506$		0			0
$507$		−	20725.2i		−	0.0806273i
$508$		0			0
$509$	53485.5			0.206443			0.103222	−	0.994658i	$-0.467085\pi$
$509$	53485.5			0.206443			0.103222	+	0.994658i	$0.467085\pi$
$510$		0			0
$511$	355213.			1.36034
$512$		0			0
$513$	3407.01			0.0129461
$514$		0			0
$515$		−	44293.7i		−	0.167004i
$516$		0			0
$517$	−1983.33			−0.00742017
$518$		0			0
$519$			73399.4i			0.272494i
$520$		0			0
$521$		−	43791.8i		−	0.161331i	−0.996741	−	0.0806655i	$-0.974295\pi$
$521$		−	43791.8i		−	0.161331i	0.996741	−	0.0806655i	$-0.0257045\pi$
$522$		0			0
$523$		−	104920.i		−	0.383578i	−0.981436	−	0.191789i	$-0.938571\pi$
$523$		−	104920.i		−	0.383578i	0.981436	−	0.191789i	$-0.0614290\pi$
$524$		0			0
$525$	110693.			0.401607
$526$		0			0
$527$	−558782.			−2.01197
$528$		0			0
$529$	−273544.			−0.977498
$530$		0			0
$531$	253260.			0.898209
$532$		0			0
$533$	426991.			1.50302
$534$		0			0
$535$		−	191879.i		−	0.670377i
$536$		0			0
$537$	−486239.			−1.68617
$538$		0			0
$539$	−1883.48			−0.00648312
$540$		0			0
$541$	−60893.0			−0.208052			−0.104026	−	0.994575i	$-0.533173\pi$
$541$	−60893.0			−0.208052			−0.104026	+	0.994575i	$0.533173\pi$
$542$		0			0
$543$		−	366571.i		−	1.24325i
$544$		0			0
$545$		−	204144.i		−	0.687297i
$546$		0			0
$547$	−77111.4			−0.257717			−0.128859	−	0.991663i	$-0.541131\pi$
$547$	−77111.4			−0.257717			−0.128859	+	0.991663i	$0.541131\pi$
$548$		0			0
$549$			387142.i			1.28447i
$550$		0			0
$551$	19389.0			0.0638634
$552$		0			0
$553$			723434.i			2.36564i
$554$		0			0
$555$			416215.i			1.35124i
$556$		0			0
$557$	−496754.			−1.60115			−0.800573	−	0.599235i	$-0.795471\pi$
$557$	−496754.			−1.60115			−0.800573	+	0.599235i	$0.795471\pi$
$558$		0			0
$559$	−249666.	−	202625.i	−0.798981	−	0.648439i
$560$		0			0
$561$		−	6773.33i		−	0.0215217i
$562$		0			0
$563$	510115.			1.60935			0.804676	−	0.593714i	$-0.202339\pi$
$563$	510115.			1.60935			0.804676	+	0.593714i	$0.202339\pi$
$564$		0			0
$565$	−366907.			−1.14937
$566$		0			0
$567$		−	460182.i		−	1.43141i
$568$		0			0
$569$	−18540.2			−0.0572650			−0.0286325	−	0.999590i	$-0.509115\pi$
$569$	−18540.2			−0.0572650			−0.0286325	+	0.999590i	$0.509115\pi$
$570$		0			0
$571$			145098.i			0.445029i	0.974929	+	0.222515i	$0.0714266\pi$
$571$			145098.i			0.445029i	−0.974929	+	0.222515i	$0.928573\pi$
$572$		0			0
$573$	−435898.			−1.32763
$574$		0			0
$575$	−11248.1			−0.0340206
$576$		0			0
$577$		−	492406.i		−	1.47901i	−0.673150	−	0.739506i	$-0.735059\pi$
$577$		−	492406.i		−	1.47901i	0.673150	−	0.739506i	$-0.264941\pi$
$578$		0			0
$579$			914359.i			2.72747i
$580$		0			0
$581$		−	349835.i		−	1.03636i
$582$		0			0
$583$	−2329.17			−0.00685275
$584$		0			0
$585$			271691.i			0.793895i
$586$		0			0
$587$			113204.i			0.328539i	0.986416	+	0.164269i	$0.0525266\pi$
$587$			113204.i			0.328539i	−0.986416	+	0.164269i	$0.947473\pi$
$588$		0			0
$589$		−	33125.8i		−	0.0954852i
$590$		0			0
$591$		−	418342.i		−	1.19772i
$592$		0			0
$593$			378754.i			1.07708i	0.842600	+	0.538540i	$0.181024\pi$
$593$			378754.i			1.07708i	−0.842600	+	0.538540i	$0.818976\pi$
$594$		0			0
$595$	−653344.			−1.84548
$596$		0			0
$597$	250666.			0.703310
$598$		0			0
$599$	70600.5			0.196768			0.0983840	−	0.995149i	$-0.468633\pi$
$599$	70600.5			0.196768			0.0983840	+	0.995149i	$0.468633\pi$
$600$		0			0
$601$			576111.i			1.59499i	0.603327	+	0.797494i	$0.293841\pi$
$601$			576111.i			1.59499i	−0.603327	+	0.797494i	$0.706159\pi$
$602$		0			0
$603$	350390.			0.963644
$604$		0			0
$605$		−	321824.i		−	0.879240i
$606$		0			0
$607$		−	179183.i		−	0.486317i	−0.969987	−	0.243159i	$-0.921816\pi$
$607$		−	179183.i		−	0.486317i	0.969987	−	0.243159i	$-0.0781835\pi$
$608$		0			0
$609$		−	544225.i		−	1.46738i
$610$		0			0
$611$	294701.			0.789403
$612$		0			0
$613$	527970.			1.40504			0.702519	−	0.711665i	$-0.252059\pi$
$613$	527970.			1.40504			0.702519	+	0.711665i	$0.252059\pi$
$614$		0			0
$615$	−665619.			−1.75985
$616$		0			0
$617$	21720.1			0.0570547			0.0285274	−	0.999593i	$-0.490918\pi$
$617$	21720.1			0.0570547			0.0285274	+	0.999593i	$0.490918\pi$
$618$		0			0
$619$	64840.3			0.169225			0.0846124	−	0.996414i	$-0.473035\pi$
$619$	64840.3			0.169225			0.0846124	+	0.996414i	$0.473035\pi$
$620$		0			0
$621$			9717.52i			0.0251984i
$622$		0			0
$623$	162110.			0.417671
$624$		0			0
$625$	−281943.			−0.721773
$626$		0			0
$627$	401.537			0.00102139
$628$		0			0
$629$			720564.i			1.82126i
$630$		0			0
$631$			639737.i			1.60673i	0.595488	+	0.803364i	$0.296959\pi$
$631$			639737.i			1.60673i	−0.595488	+	0.803364i	$0.703041\pi$
$632$		0			0
$633$	85791.3			0.214109
$634$		0			0
$635$			374073.i			0.927702i
$636$		0			0
$637$	279865.			0.689714
$638$		0			0
$639$		−	686581.i		−	1.68147i
$640$		0			0
$641$		−	92614.4i		−	0.225404i	−0.993629	−	0.112702i	$-0.964049\pi$
$641$		−	92614.4i		−	0.225404i	0.993629	−	0.112702i	$-0.0359506\pi$
$642$		0			0
$643$	−517514.			−1.25170			−0.625850	−	0.779943i	$-0.715248\pi$
$643$	−517514.			−1.25170			−0.625850	+	0.779943i	$0.715248\pi$
$644$		0			0
$645$	389195.	+	315864.i	0.935508	+	0.759242i
$646$		0			0
$647$		−	320760.i		−	0.766253i	−0.923696	−	0.383126i	$-0.874847\pi$
$647$		−	320760.i		−	0.766253i	0.923696	−	0.383126i	$-0.125153\pi$
$648$		0			0
$649$	−4170.61			−0.00990170
$650$		0			0
$651$	−929801.			−2.19396
$652$		0			0
$653$		−	271078.i		−	0.635724i	−0.948137	−	0.317862i	$-0.897035\pi$
$653$		−	271078.i		−	0.635724i	0.948137	−	0.317862i	$-0.102965\pi$
$654$		0			0
$655$	294101.			0.685511
$656$		0			0
$657$			398642.i			0.923534i
$658$		0			0
$659$	297067.			0.684044			0.342022	−	0.939692i	$-0.388888\pi$
$659$	297067.			0.684044			0.342022	+	0.939692i	$0.388888\pi$
$660$		0			0
$661$	91628.4			0.209714			0.104857	−	0.994487i	$-0.466562\pi$
$661$	91628.4			0.209714			0.104857	+	0.994487i	$0.466562\pi$
$662$		0			0
$663$			1.00644e6i			2.28961i
$664$		0			0
$665$		−	38731.7i		−	0.0875836i
$666$		0			0
$667$			55301.5i			0.124304i
$668$		0			0
$669$	27479.3			0.0613978
$670$		0			0
$671$		−	6375.33i		−	0.0141598i
$672$		0			0
$673$			601565.i			1.32817i	0.747658	+	0.664084i	$0.231178\pi$
$673$			601565.i			1.32817i	−0.747658	+	0.664084i	$0.768822\pi$
$674$		0			0
$675$		−	17357.8i		−	0.0380966i
$676$		0			0
$677$		−	211198.i		−	0.460800i	−0.973096	−	0.230400i	$-0.925997\pi$
$677$		−	211198.i		−	0.460800i	0.973096	−	0.230400i	$-0.0740034\pi$
$678$		0			0
$679$			231764.i			0.502698i
$680$		0			0
$681$	−759302.			−1.63727
$682$		0			0
$683$	260659.			0.558768			0.279384	−	0.960179i	$-0.409870\pi$
$683$	260659.			0.558768			0.279384	+	0.960179i	$0.409870\pi$
$684$		0			0
$685$	200694.			0.427713
$686$		0			0
$687$			497335.i			1.05375i
$688$		0			0
$689$	346089.			0.729037
$690$		0			0
$691$			670752.i			1.40477i	0.711797	+	0.702386i	$0.247882\pi$
$691$			670752.i			1.40477i	−0.711797	+	0.702386i	$0.752118\pi$
$692$		0			0
$693$		−	5267.34i		−	0.0109679i
$694$		0			0
$695$		−	620361.i		−	1.28433i
$696$		0			0
$697$	−1.15234e6			−2.37200
$698$		0			0
$699$	917262.			1.87732
$700$		0			0
$701$	104357.			0.212366			0.106183	−	0.994347i	$-0.466137\pi$
$701$	104357.			0.212366			0.106183	+	0.994347i	$0.466137\pi$
$702$		0			0
$703$	−42716.6			−0.0864343
$704$		0			0
$705$	−459396.			−0.924293
$706$		0			0
$707$		−	665260.i		−	1.33092i
$708$		0			0
$709$	135186.			0.268930			0.134465	−	0.990918i	$-0.457068\pi$
$709$	135186.			0.268930			0.134465	+	0.990918i	$0.457068\pi$
$710$		0			0
$711$	−811882.			−1.60603
$712$		0			0
$713$	94481.8			0.185853
$714$		0			0
$715$		−	4474.12i		−	0.00875176i
$716$		0			0
$717$			146432.i			0.284838i
$718$		0			0
$719$	−99461.8			−0.192397			−0.0961985	−	0.995362i	$-0.530668\pi$
$719$	−99461.8			−0.192397			−0.0961985	+	0.995362i	$0.530668\pi$
$720$		0			0
$721$		−	127598.i		−	0.245456i
$722$		0			0
$723$	290242.			0.555243
$724$		0			0
$725$		−	98781.4i		−	0.187931i
$726$		0			0
$727$			440731.i			0.833882i	0.908933	+	0.416941i	$0.136898\pi$
$727$			440731.i			0.833882i	−0.908933	+	0.416941i	$0.863102\pi$
$728$		0			0
$729$	394127.			0.741619
$730$		0			0
$731$	673785.	+	546832.i	1.26092	+	1.02334i
$732$		0			0
$733$			353122.i			0.657229i	0.944464	+	0.328615i	$0.106582\pi$
$733$			353122.i			0.657229i	−0.944464	+	0.328615i	$0.893418\pi$
$734$		0			0
$735$	−436269.			−0.807569
$736$		0			0
$737$	−5770.11			−0.0106230
$738$		0			0
$739$			511160.i			0.935983i	0.883733	+	0.467992i	$0.155022\pi$
$739$			511160.i			0.935983i	−0.883733	+	0.467992i	$0.844978\pi$
$740$		0			0
$741$	−59663.9			−0.108661
$742$		0			0
$743$		−	162933.i		−	0.295141i	−0.989052	−	0.147571i	$-0.952855\pi$
$743$		−	162933.i		−	0.295141i	0.989052	−	0.147571i	$-0.0471454\pi$
$744$		0			0
$745$	246690.			0.444466
$746$		0			0
$747$	392606.			0.703584
$748$		0			0
$749$		−	552750.i		−	0.985292i
$750$		0			0
$751$			476066.i			0.844087i	0.906576	+	0.422044i	$0.138687\pi$
$751$			476066.i			0.844087i	−0.906576	+	0.422044i	$0.861313\pi$
$752$		0			0
$753$		−	675249.i		−	1.19090i
$754$		0			0
$755$	−827708.			−1.45205
$756$		0			0
$757$		−	513071.i		−	0.895335i	−0.894200	−	0.447667i	$-0.852255\pi$
$757$		−	513071.i		−	0.895335i	0.894200	−	0.447667i	$-0.147745\pi$
$758$		0			0
$759$			1145.27i			0.00198803i
$760$		0			0
$761$		−	132174.i		−	0.228233i	−0.993467	−	0.114116i	$-0.963596\pi$
$761$		−	132174.i		−	0.228233i	0.993467	−	0.114116i	$-0.0364036\pi$
$762$		0			0
$763$		−	588084.i		−	1.01016i
$764$		0			0
$765$		−	733223.i		−	1.25289i
$766$		0			0
$767$	619705.			1.05340
$768$		0			0
$769$	−549403.			−0.929048			−0.464524	−	0.885561i	$-0.653775\pi$
$769$	−549403.			−0.929048			−0.464524	+	0.885561i	$0.653775\pi$
$770$		0			0
$771$	−712275.			−1.19823
$772$		0			0
$773$			357526.i			0.598340i	0.954200	+	0.299170i	$0.0967098\pi$
$773$			357526.i			0.598340i	−0.954200	+	0.299170i	$0.903290\pi$
$774$		0			0
$775$	−168767.			−0.280985
$776$		0			0
$777$			1.19900e6i			1.98599i
$778$		0			0
$779$		−	68313.2i		−	0.112572i
$780$		0			0
$781$			11306.4i			0.0185363i
$782$		0			0
$783$	−85340.1			−0.139197
$784$		0			0
$785$	−217724.			−0.353319
$786$		0			0
$787$	−11905.5			−0.0192221			−0.00961103	−	0.999954i	$-0.503059\pi$
$787$	−11905.5			−0.0192221			−0.00961103	+	0.999954i	$0.503059\pi$
$788$		0			0
$789$	−93316.5			−0.149901
$790$		0			0
$791$	−1.05696e6			−1.68929
$792$		0			0
$793$			947303.i			1.50641i
$794$		0			0
$795$	−539504.			−0.853611
$796$		0			0
$797$	−54534.3			−0.0858526			−0.0429263	−	0.999078i	$-0.513668\pi$
$797$	−54534.3			−0.0858526			−0.0429263	+	0.999078i	$0.513668\pi$
$798$		0			0
$799$	−795321.			−1.24580
$800$		0			0
$801$			181930.i			0.283557i
$802$		0			0
$803$		−	6564.72i		−	0.0101809i
$804$		0			0
$805$	110471.			0.170473
$806$		0			0
$807$			492319.i			0.755962i
$808$		0			0
$809$	1.14580e6			1.75070			0.875350	−	0.483489i	$-0.160631\pi$
$809$	1.14580e6			1.75070			0.875350	+	0.483489i	$0.160631\pi$
$810$		0			0
$811$			56880.8i			0.0864816i	0.999065	+	0.0432408i	$0.0137683\pi$
$811$			56880.8i			0.0864816i	−0.999065	+	0.0432408i	$0.986232\pi$
$812$		0			0
$813$			1.47717e6i			2.23486i
$814$		0			0
$815$	−454473.			−0.684216
$816$		0			0
$817$	−32417.4	+	39943.4i	−0.0485662	+	0.0598413i
$818$		0			0
$819$			782667.i			1.16683i
$820$		0			0
$821$	200304.			0.297169			0.148584	−	0.988900i	$-0.452528\pi$
$821$	200304.			0.297169			0.148584	+	0.988900i	$0.452528\pi$
$822$		0			0
$823$	1.07152e6			1.58198			0.790989	−	0.611830i	$-0.209566\pi$
$823$	1.07152e6			1.58198			0.790989	+	0.611830i	$0.209566\pi$
$824$		0			0
$825$		−	2045.72i		−	0.00300565i
$826$		0			0
$827$	−767100.			−1.12161			−0.560803	−	0.827949i	$-0.689508\pi$
$827$	−767100.			−1.12161			−0.560803	+	0.827949i	$0.689508\pi$
$828$		0			0
$829$		−	358430.i		−	0.521549i	−0.965400	−	0.260774i	$-0.916022\pi$
$829$		−	358430.i		−	0.521549i	0.965400	−	0.260774i	$-0.0839778\pi$
$830$		0			0
$831$	462766.			0.670130
$832$		0			0
$833$	−755282.			−1.08848
$834$		0			0
$835$			6884.02i			0.00987345i
$836$		0			0
$837$			145802.i			0.208120i
$838$		0			0
$839$		−	93647.6i		−	0.133037i	−0.997785	−	0.0665185i	$-0.978811\pi$
$839$		−	93647.6i		−	0.133037i	0.997785	−	0.0665185i	$-0.0211891\pi$
$840$		0			0
$841$	221619.			0.313339
$842$		0			0
$843$			369150.i			0.519454i
$844$		0			0
$845$			36945.8i			0.0517430i
$846$		0			0
$847$		−	927086.i		−	1.29227i
$848$		0			0
$849$		−	677233.i		−	0.939556i
$850$		0			0
$851$		−	121837.i		−	0.168236i
$852$		0			0
$853$	368391.			0.506304			0.253152	−	0.967427i	$-0.418533\pi$
$853$	368391.			0.506304			0.253152	+	0.967427i	$0.418533\pi$
$854$		0			0
$855$	43467.1			0.0594604
$856$		0			0
$857$	851457.			1.15931			0.579657	−	0.814860i	$-0.303186\pi$
$857$	851457.			1.15931			0.579657	+	0.814860i	$0.303186\pi$
$858$		0			0
$859$		−	236916.i		−	0.321076i	−0.987030	−	0.160538i	$-0.948677\pi$
$859$		−	236916.i		−	0.321076i	0.987030	−	0.160538i	$-0.0513229\pi$
$860$		0			0
$861$	−1.91747e6			−2.58655
$862$		0			0
$863$			22247.8i			0.0298721i	0.999888	+	0.0149361i	$0.00475447\pi$
$863$			22247.8i			0.0298721i	−0.999888	+	0.0149361i	$0.995246\pi$
$864$		0			0
$865$		−	130846.i		−	0.174875i
$866$		0			0
$867$		−	1.68617e6i		−	2.24317i
$868$		0			0
$869$	13369.8			0.0177046
$870$		0			0
$871$	857373.			1.13014
$872$		0			0
$873$	−260100.			−0.341281
$874$		0			0
$875$	−1.06740e6			−1.39416
$876$		0			0
$877$	2045.84			0.00265995			0.00132998	−	0.999999i	$-0.499577\pi$
$877$	2045.84			0.00265995			0.00132998	+	0.999999i	$0.499577\pi$
$878$		0			0
$879$			1.92054e6i			2.48568i
$880$		0			0
$881$	−196655.			−0.253369			−0.126684	−	0.991943i	$-0.540434\pi$
$881$	−196655.			−0.253369			−0.126684	+	0.991943i	$0.540434\pi$
$882$		0			0
$883$	−59837.7			−0.0767456			−0.0383728	−	0.999263i	$-0.512217\pi$
$883$	−59837.7			−0.0767456			−0.0383728	+	0.999263i	$0.512217\pi$
$884$		0			0
$885$	−966033.			−1.23340
$886$		0			0
$887$			577234.i			0.733676i	0.930285	+	0.366838i	$0.119560\pi$
$887$			577234.i			0.733676i	−0.930285	+	0.366838i	$0.880440\pi$
$888$		0			0
$889$			1.07760e6i			1.36350i
$890$		0			0
$891$	−8504.66			−0.0107128
$892$		0			0
$893$		−	47148.3i		−	0.0591239i
$894$		0			0
$895$	866795.			1.08211
$896$		0			0
$897$		−	170174.i		−	0.211499i
$898$		0			0
$899$			829747.i			1.02666i
$900$		0			0
$901$	−934005.			−1.15053
$902$		0			0
$903$	1.12116e6	+	909916.i	1.37497	+	1.11590i
$904$		0			0
$905$			653470.i			0.797863i
$906$		0			0
$907$	543357.			0.660496			0.330248	−	0.943894i	$-0.392868\pi$
$907$	543357.			0.660496			0.330248	+	0.943894i	$0.392868\pi$
$908$		0			0
$909$	746596.			0.903561
$910$		0			0
$911$			119998.i			0.144590i	0.997383	+	0.0722948i	$0.0230322\pi$
$911$			119998.i			0.144590i	−0.997383	+	0.0722948i	$0.976968\pi$
$912$		0			0
$913$	−6465.32			−0.00775619
$914$		0			0
$915$		−	1.47671e6i		−	1.76382i
$916$		0			0
$917$	847225.			1.00753
$918$		0			0
$919$	−198081.			−0.234537			−0.117268	−	0.993100i	$-0.537414\pi$
$919$	−198081.			−0.234537			−0.117268	+	0.993100i	$0.537414\pi$
$920$		0			0
$921$			874161.i			1.03056i
$922$		0			0
$923$		−	1.68000e6i		−	1.97200i
$924$		0			0
$925$			217629.i			0.254351i
$926$		0			0
$927$	143198.			0.166640
$928$		0			0
$929$		−	1.28224e6i		−	1.48573i	−0.669443	−	0.742864i	$-0.733467\pi$
$929$		−	1.28224e6i		−	1.48573i	0.669443	−	0.742864i	$-0.266533\pi$
$930$		0			0
$931$		−	44774.7i		−	0.0516575i
$932$		0			0
$933$		−	344170.i		−	0.395376i
$934$		0			0
$935$			12074.5i			0.0138117i
$936$		0			0
$937$			1.00382e6i			1.14334i	0.820483	+	0.571671i	$0.193705\pi$
$937$			1.00382e6i			1.14334i	−0.820483	+	0.571671i	$0.806295\pi$
$938$		0			0
$939$	−624151.			−0.707878
$940$		0			0
$941$	893241.			1.00876			0.504382	−	0.863481i	$-0.331720\pi$
$941$	893241.			1.00876			0.504382	+	0.863481i	$0.331720\pi$
$942$		0			0
$943$	194844.			0.219110
$944$		0			0
$945$			170476.i			0.190898i
$946$		0			0
$947$	66944.6			0.0746475			0.0373238	−	0.999303i	$-0.488117\pi$
$947$	66944.6			0.0746475			0.0373238	+	0.999303i	$0.488117\pi$
$948$		0			0
$949$			975443.i			1.08310i
$950$		0			0
$951$		−	1.37079e6i		−	1.51569i
$952$		0			0
$953$		−	863195.i		−	0.950436i	−0.879868	−	0.475218i	$-0.842369\pi$
$953$		−	863195.i		−	0.950436i	0.879868	−	0.475218i	$-0.157631\pi$
$954$		0			0
$955$	777055.			0.852011
$956$		0			0
$957$	−10057.9			−0.0109820
$958$		0			0
$959$	578143.			0.628635
$960$		0			0
$961$	494090.			0.535007
$962$		0			0
$963$	620330.			0.668914
$964$		0			0
$965$		−	1.62998e6i		−	1.75037i
$966$		0			0
$967$	1.55611e6			1.66413			0.832065	−	0.554677i	$-0.187158\pi$
$967$	1.55611e6			1.66413			0.832065	+	0.554677i	$0.187158\pi$
$968$		0			0
$969$	161018.			0.171485
$970$		0			0
$971$	734703.			0.779243			0.389622	−	0.920975i	$-0.372606\pi$
$971$	734703.			0.779243			0.389622	+	0.920975i	$0.372606\pi$
$972$		0			0
$973$		−	1.78709e6i		−	1.88765i
$974$		0			0
$975$			303971.i			0.319759i
$976$		0			0
$977$	−857114.			−0.897944			−0.448972	−	0.893546i	$-0.648210\pi$
$977$	−857114.			−0.897944			−0.448972	+	0.893546i	$0.648210\pi$
$978$		0			0
$979$		−	2995.97i		−	0.00312588i
$980$		0			0
$981$	659984.			0.685797
$982$		0			0
$983$			116039.i			0.120087i	0.998196	+	0.0600437i	$0.0191240\pi$
$983$			116039.i			0.120087i	−0.998196	+	0.0600437i	$0.980876\pi$
$984$		0			0
$985$			745758.i			0.768645i
$986$		0			0
$987$	−1.32340e6			−1.35849
$988$		0			0
$989$	−113927.	−	92461.2i	−0.116475	−	0.0945294i
$990$		0			0
$991$			1.12664e6i			1.14720i	0.819136	+	0.573600i	$0.194454\pi$
$991$			1.12664e6i			1.14720i	−0.819136	+	0.573600i	$0.805546\pi$
$992$		0			0
$993$	−2.46290e6			−2.49775
$994$		0			0
$995$	−446851.			−0.451353
$996$		0			0
$997$		−	1.20881e6i		−	1.21610i	−0.793900	−	0.608048i	$-0.791953\pi$
$997$		−	1.20881e6i		−	1.21610i	0.793900	−	0.608048i	$-0.208047\pi$
$998$		0			0
$999$	188016.			0.188393

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	688.5.b.d.257.3		12
4.3	odd	2		43.5.b.b.42.8	yes	12
12.11	even	2		387.5.b.c.343.5		12
43.42	odd	2	inner	688.5.b.d.257.10		12
172.171	even	2		43.5.b.b.42.5	✓	12
516.515	odd	2		387.5.b.c.343.8		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.5.b.b.42.5	✓	12	172.171	even	2
43.5.b.b.42.8	yes	12	4.3	odd	2
387.5.b.c.343.5		12	12.11	even	2
387.5.b.c.343.8		12	516.515	odd	2
688.5.b.d.257.3		12	1.1	even	1	trivial
688.5.b.d.257.10		12	43.42	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 688 = 2^{4} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	688.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-12.3317i q^{3} +21.9831i q^{5} +63.3272i q^{7} -71.0696 q^{9} +O(q^{10})\)	\(q-12.3317i q^{3} +21.9831i q^{5} +63.3272i q^{7} -71.0696 q^{9} +1.17035 q^{11} -173.901 q^{13} +271.087 q^{15} +469.315 q^{17} +27.8220i q^{19} +780.929 q^{21} -79.3541 q^{23} +141.745 q^{25} -122.458i q^{27} -696.895i q^{29} -1190.63 q^{31} -14.4324i q^{33} -1392.12 q^{35} +1535.36i q^{37} +2144.49i q^{39} -2455.37 q^{41} +(1435.68 + 1165.17i) q^{43} -1562.33i q^{45} -1694.64 q^{47} -1609.33 q^{49} -5787.42i q^{51} -1990.15 q^{53} +25.7279i q^{55} +343.091 q^{57} -3563.55 q^{59} -5447.36i q^{61} -4500.64i q^{63} -3822.88i q^{65} -4930.23 q^{67} +978.567i q^{69} +9660.68i q^{71} -5609.18i q^{73} -1747.95i q^{75} +74.1151i q^{77} +11423.8 q^{79} -7266.75 q^{81} -5524.25 q^{83} +10317.0i q^{85} -8593.86 q^{87} -2559.89i q^{89} -11012.7i q^{91} +14682.5i q^{93} -611.612 q^{95} +3659.79 q^{97} -83.1766 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 462 q^{9}+O(q^{10}) \) 12 * q - 462 * q^9	\( 12 q - 462 q^{9} + 180 q^{11} - 216 q^{13} + 92 q^{15} + 678 q^{17} - 2392 q^{21} - 1566 q^{23} - 174 q^{25} - 5710 q^{31} - 936 q^{35} + 4878 q^{41} + 1108 q^{43} + 5526 q^{47} - 8544 q^{49} + 1212 q^{53} - 7692 q^{57} - 14016 q^{59} + 1088 q^{67} - 24302 q^{79} - 23660 q^{81} + 7032 q^{83} - 17850 q^{87} - 606 q^{95} - 5842 q^{97} + 25924 q^{99}+O(q^{100}) \) 12 * q - 462 * q^9 + 180 * q^11 - 216 * q^13 + 92 * q^15 + 678 * q^17 - 2392 * q^21 - 1566 * q^23 - 174 * q^25 - 5710 * q^31 - 936 * q^35 + 4878 * q^41 + 1108 * q^43 + 5526 * q^47 - 8544 * q^49 + 1212 * q^53 - 7692 * q^57 - 14016 * q^59 + 1088 * q^67 - 24302 * q^79 - 23660 * q^81 + 7032 * q^83 - 17850 * q^87 - 606 * q^95 - 5842 * q^97 + 25924 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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