LMFDB - Embedded newform 688.5.b.d.257.2

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			257.2
Root			$-7.49282i$ of defining polynomial
Character	$\chi$	$=$	688.257
Dual form			688.5.b.d.257.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-14.5182i q^{3} +9.40180i q^{5} -53.5326i q^{7} -129.777 q^{9} +O(q^{10})$	$q-14.5182i q^{3} +9.40180i q^{5} -53.5326i q^{7} -129.777 q^{9} -151.146 q^{11} -319.508 q^{13} +136.497 q^{15} +32.2585 q^{17} -304.131i q^{19} -777.194 q^{21} +199.312 q^{23} +536.606 q^{25} +708.150i q^{27} +268.672i q^{29} -665.439 q^{31} +2194.36i q^{33} +503.303 q^{35} -1176.50i q^{37} +4638.66i q^{39} +212.488 q^{41} +(-1352.49 + 1260.78i) q^{43} -1220.14i q^{45} +3100.78 q^{47} -464.736 q^{49} -468.334i q^{51} +2662.72 q^{53} -1421.04i q^{55} -4415.43 q^{57} -2748.48 q^{59} +5880.73i q^{61} +6947.29i q^{63} -3003.95i q^{65} +1093.36 q^{67} -2893.64i q^{69} -5841.22i q^{71} -663.511i q^{73} -7790.53i q^{75} +8091.21i q^{77} -6328.79 q^{79} -230.889 q^{81} -8356.34 q^{83} +303.288i q^{85} +3900.62 q^{87} +4257.21i q^{89} +17104.1i q^{91} +9660.95i q^{93} +2859.38 q^{95} -3585.40 q^{97} +19615.2 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$		−	14.5182i		−	1.61313i	−0.591146	−	0.806564i	$-0.701324\pi$
$3$		−	14.5182i		−	1.61313i	0.591146	−	0.806564i	$-0.298676\pi$
$4$		0			0
$5$			9.40180i			0.376072i	0.982162	+	0.188036i	$0.0602122\pi$
$5$			9.40180i			0.376072i	−0.982162	+	0.188036i	$0.939788\pi$
$6$		0			0
$7$		−	53.5326i		−	1.09250i	−0.837622	−	0.546251i	$-0.816055\pi$
$7$		−	53.5326i		−	1.09250i	0.837622	−	0.546251i	$-0.183945\pi$
$8$		0			0
$9$	−129.777			−1.60218
$10$		0			0
$11$	−151.146			−1.24914			−0.624569	−	0.780970i	$-0.714725\pi$
$11$	−151.146			−1.24914			−0.624569	+	0.780970i	$0.714725\pi$
$12$		0			0
$13$	−319.508			−1.89058			−0.945289	−	0.326233i	$-0.894221\pi$
$13$	−319.508			−1.89058			−0.945289	+	0.326233i	$0.894221\pi$
$14$		0			0
$15$	136.497			0.606653
$16$		0			0
$17$	32.2585			0.111621			0.0558106	−	0.998441i	$-0.482226\pi$
$17$	32.2585			0.111621			0.0558106	+	0.998441i	$0.482226\pi$
$18$		0			0
$19$		−	304.131i		−	0.842469i	−0.906952	−	0.421235i	$-0.861597\pi$
$19$		−	304.131i		−	0.842469i	0.906952	−	0.421235i	$-0.138403\pi$
$20$		0			0
$21$	−777.194			−1.76235
$22$		0			0
$23$	199.312			0.376771			0.188386	−	0.982095i	$-0.439675\pi$
$23$	199.312			0.376771			0.188386	+	0.982095i	$0.439675\pi$
$24$		0			0
$25$	536.606			0.858570
$26$		0			0
$27$			708.150i			0.971400i
$28$		0			0
$29$			268.672i			0.319467i	0.987160	+	0.159734i	$0.0510635\pi$
$29$			268.672i			0.319467i	−0.987160	+	0.159734i	$0.948937\pi$
$30$		0			0
$31$	−665.439			−0.692444			−0.346222	−	0.938153i	$-0.612536\pi$
$31$	−665.439			−0.692444			−0.346222	+	0.938153i	$0.612536\pi$
$32$		0			0
$33$			2194.36i			2.01502i
$34$		0			0
$35$	503.303			0.410859
$36$		0			0
$37$		−	1176.50i		−	0.859387i	−0.902975	−	0.429693i	$-0.858622\pi$
$37$		−	1176.50i		−	0.859387i	0.902975	−	0.429693i	$-0.141378\pi$
$38$		0			0
$39$			4638.66i			3.04975i
$40$		0			0
$41$	212.488			0.126406			0.0632028	−	0.998001i	$-0.479868\pi$
$41$	212.488			0.126406			0.0632028	+	0.998001i	$0.479868\pi$
$42$		0			0
$43$	−1352.49	+	1260.78i	−0.731470	+	0.681873i
$43$	−1352.49	+	1260.78i	−0.731470	+	0.681873i
$44$		0			0
$45$		−	1220.14i		−	0.602537i
$46$		0			0
$47$	3100.78			1.40371			0.701853	−	0.712322i	$-0.252357\pi$
$47$	3100.78			1.40371			0.701853	+	0.712322i	$0.252357\pi$
$48$		0			0
$49$	−464.736			−0.193559
$50$		0			0
$51$		−	468.334i		−	0.180059i
$52$		0			0
$53$	2662.72			0.947924			0.473962	−	0.880545i	$-0.342823\pi$
$53$	2662.72			0.947924			0.473962	+	0.880545i	$0.342823\pi$
$54$		0			0
$55$		−	1421.04i		−	0.469766i
$56$		0			0
$57$	−4415.43			−1.35901
$58$		0			0
$59$	−2748.48			−0.789565			−0.394783	−	0.918775i	$-0.629180\pi$
$59$	−2748.48			−0.789565			−0.394783	+	0.918775i	$0.629180\pi$
$60$		0			0
$61$			5880.73i			1.58042i	0.612839	+	0.790208i	$0.290028\pi$
$61$			5880.73i			1.58042i	−0.612839	+	0.790208i	$0.709972\pi$
$62$		0			0
$63$			6947.29i			1.75039i
$64$		0			0
$65$		−	3003.95i		−	0.710994i
$66$		0			0
$67$	1093.36			0.243565			0.121782	−	0.992557i	$-0.461139\pi$
$67$	1093.36			0.243565			0.121782	+	0.992557i	$0.461139\pi$
$68$		0			0
$69$		−	2893.64i		−	0.607780i
$70$		0			0
$71$		−	5841.22i		−	1.15874i	−0.815064	−	0.579371i	$-0.803298\pi$
$71$		−	5841.22i		−	1.15874i	0.815064	−	0.579371i	$-0.196702\pi$
$72$		0			0
$73$		−	663.511i		−	0.124509i	−0.998060	−	0.0622547i	$-0.980171\pi$
$73$		−	663.511i		−	0.124509i	0.998060	−	0.0622547i	$-0.0198291\pi$
$74$		0			0
$75$		−	7790.53i		−	1.38498i
$76$		0			0
$77$			8091.21i			1.36468i
$78$		0			0
$79$	−6328.79			−1.01407			−0.507034	−	0.861926i	$-0.669258\pi$
$79$	−6328.79			−1.01407			−0.507034	+	0.861926i	$0.669258\pi$
$80$		0			0
$81$	−230.889			−0.0351911
$82$		0			0
$83$	−8356.34			−1.21300			−0.606499	−	0.795084i	$-0.707427\pi$
$83$	−8356.34			−1.21300			−0.606499	+	0.795084i	$0.707427\pi$
$84$		0			0
$85$			303.288i			0.0419776i
$86$		0			0
$87$	3900.62			0.515341
$88$		0			0
$89$			4257.21i			0.537459i	0.963216	+	0.268729i	$0.0866037\pi$
$89$			4257.21i			0.537459i	−0.963216	+	0.268729i	$0.913396\pi$
$90$		0			0
$91$			17104.1i			2.06546i
$92$		0			0
$93$			9660.95i			1.11700i
$94$		0			0
$95$	2859.38			0.316829
$96$		0			0
$97$	−3585.40			−0.381061			−0.190530	−	0.981681i	$-0.561021\pi$
$97$	−3585.40			−0.381061			−0.190530	+	0.981681i	$0.561021\pi$
$98$		0			0
$99$	19615.2			2.00135
$100$		0			0
$101$	3138.84			0.307699			0.153850	−	0.988094i	$-0.450833\pi$
$101$	3138.84			0.307699			0.153850	+	0.988094i	$0.450833\pi$
$102$		0			0
$103$	7225.57			0.681079			0.340539	−	0.940230i	$-0.389390\pi$
$103$	7225.57			0.681079			0.340539	+	0.940230i	$0.389390\pi$
$104$		0			0
$105$		−	7307.03i		−	0.662769i
$106$		0			0
$107$	11372.2			0.993291			0.496645	−	0.867954i	$-0.334565\pi$
$107$	11372.2			0.993291			0.496645	+	0.867954i	$0.334565\pi$
$108$		0			0
$109$	6830.22			0.574886			0.287443	−	0.957798i	$-0.407195\pi$
$109$	6830.22			0.574886			0.287443	+	0.957798i	$0.407195\pi$
$110$		0			0
$111$	−17080.6			−1.38630
$112$		0			0
$113$			14950.0i			1.17081i	0.810742	+	0.585403i	$0.199064\pi$
$113$			14950.0i			1.17081i	−0.810742	+	0.585403i	$0.800936\pi$
$114$		0			0
$115$			1873.89i			0.141693i
$116$		0			0
$117$	41464.7			3.02905
$118$		0			0
$119$		−	1726.88i		−	0.121946i
$120$		0			0
$121$	8204.00			0.560344
$122$		0			0
$123$		−	3084.93i		−	0.203909i
$124$		0			0
$125$			10921.2i			0.698956i
$126$		0			0
$127$	11093.6			0.687806			0.343903	−	0.939005i	$-0.388251\pi$
$127$	11093.6			0.687806			0.343903	+	0.939005i	$0.388251\pi$
$128$		0			0
$129$	18304.3	+	19635.6i	1.09995	+	1.17996i
$130$		0			0
$131$		−	7521.37i		−	0.438283i	−0.975693	−	0.219141i	$-0.929674\pi$
$131$		−	7521.37i		−	0.438283i	0.975693	−	0.219141i	$-0.0703256\pi$
$132$		0			0
$133$	−16280.9			−0.920399
$134$		0			0
$135$	−6657.89			−0.365316
$136$		0			0
$137$		−	15580.8i		−	0.830137i	−0.909790	−	0.415069i	$-0.863758\pi$
$137$		−	15580.8i		−	0.830137i	0.909790	−	0.415069i	$-0.136242\pi$
$138$		0			0
$139$	−9394.31			−0.486223			−0.243111	−	0.969998i	$-0.578168\pi$
$139$	−9394.31			−0.486223			−0.243111	+	0.969998i	$0.578168\pi$
$140$		0			0
$141$		−	45017.7i		−	2.26436i
$142$		0			0
$143$	48292.2			2.36159
$144$		0			0
$145$	−2526.00			−0.120143
$146$		0			0
$147$			6747.11i			0.312236i
$148$		0			0
$149$		−	2012.81i		−	0.0906630i	−0.998972	−	0.0453315i	$-0.985566\pi$
$149$		−	2012.81i		−	0.0906630i	0.998972	−	0.0453315i	$-0.0144344\pi$
$150$		0			0
$151$		−	7447.35i		−	0.326624i	−0.986574	−	0.163312i	$-0.947782\pi$
$151$		−	7447.35i		−	0.326624i	0.986574	−	0.163312i	$-0.0522177\pi$
$152$		0			0
$153$	−4186.41			−0.178838
$154$		0			0
$155$		−	6256.33i		−	0.260409i
$156$		0			0
$157$		−	16371.4i		−	0.664179i	−0.943248	−	0.332090i	$-0.892246\pi$
$157$		−	16371.4i		−	0.664179i	0.943248	−	0.332090i	$-0.107754\pi$
$158$		0			0
$159$		−	38657.8i		−	1.52912i
$160$		0			0
$161$		−	10669.7i		−	0.411623i
$162$		0			0
$163$			45112.4i			1.69793i	0.528445	+	0.848967i	$0.322775\pi$
$163$			45112.4i			1.69793i	−0.528445	+	0.848967i	$0.677225\pi$
$164$		0			0
$165$	−20630.9			−0.757793
$166$		0			0
$167$	24211.5			0.868139			0.434069	−	0.900879i	$-0.357077\pi$
$167$	24211.5			0.868139			0.434069	+	0.900879i	$0.357077\pi$
$168$		0			0
$169$	73524.2			2.57429
$170$		0			0
$171$			39469.2i			1.34979i
$172$		0			0
$173$	36697.5			1.22615			0.613076	−	0.790024i	$-0.289932\pi$
$173$	36697.5			1.22615			0.613076	+	0.790024i	$0.289932\pi$
$174$		0			0
$175$		−	28725.9i		−	0.937989i
$176$		0			0
$177$			39902.8i			1.27367i
$178$		0			0
$179$			50619.1i			1.57982i	0.613220	+	0.789912i	$0.289874\pi$
$179$			50619.1i			1.57982i	−0.613220	+	0.789912i	$0.710126\pi$
$180$		0			0
$181$	−31719.0			−0.968194			−0.484097	−	0.875014i	$-0.660852\pi$
$181$	−31719.0			−0.968194			−0.484097	+	0.875014i	$0.660852\pi$
$182$		0			0
$183$	85377.3			2.54941
$184$		0			0
$185$	11061.2			0.323191
$186$		0			0
$187$	−4875.73			−0.139430
$188$		0			0
$189$	37909.1			1.06126
$190$		0			0
$191$		−	49921.7i		−	1.36843i	−0.729280	−	0.684215i	$-0.760145\pi$
$191$		−	49921.7i		−	1.36843i	0.729280	−	0.684215i	$-0.239855\pi$
$192$		0			0
$193$	541.363			0.0145336			0.00726681	−	0.999974i	$-0.497687\pi$
$193$	541.363			0.0145336			0.00726681	+	0.999974i	$0.497687\pi$
$194$		0			0
$195$	−43611.8			−1.14692
$196$		0			0
$197$	−31142.6			−0.802458			−0.401229	−	0.915978i	$-0.631417\pi$
$197$	−31142.6			−0.802458			−0.401229	+	0.915978i	$0.631417\pi$
$198$		0			0
$199$			59618.5i			1.50548i	0.658319	+	0.752739i	$0.271268\pi$
$199$			59618.5i			1.50548i	−0.658319	+	0.752739i	$0.728732\pi$
$200$		0			0
$201$		−	15873.6i		−	0.392902i
$202$		0			0
$203$	14382.7			0.349018
$204$		0			0
$205$			1997.77i			0.0475377i
$206$		0			0
$207$	−25866.1			−0.603656
$208$		0			0
$209$			45968.1i			1.05236i
$210$		0			0
$211$			45278.4i			1.01701i	0.861059	+	0.508506i	$0.169802\pi$
$211$			45278.4i			1.01701i	−0.861059	+	0.508506i	$0.830198\pi$
$212$		0			0
$213$	−84803.8			−1.86920
$214$		0			0
$215$	−11853.6	−	12715.8i	−0.256434	−	0.275086i
$216$		0			0
$217$			35622.7i			0.756496i
$218$		0			0
$219$	−9632.95			−0.200850
$220$		0			0
$221$	−10306.8			−0.211029
$222$		0			0
$223$		−	26029.7i		−	0.523431i	−0.965145	−	0.261716i	$-0.915712\pi$
$223$		−	26029.7i		−	0.523431i	0.965145	−	0.261716i	$-0.0842883\pi$
$224$		0			0
$225$	−69639.1			−1.37559
$226$		0			0
$227$		−	57788.9i		−	1.12148i	−0.827991	−	0.560741i	$-0.810516\pi$
$227$		−	57788.9i		−	1.12148i	0.827991	−	0.560741i	$-0.189484\pi$
$228$		0			0
$229$	−15622.4			−0.297905			−0.148953	−	0.988844i	$-0.547590\pi$
$229$	−15622.4			−0.297905			−0.148953	+	0.988844i	$0.547590\pi$
$230$		0			0
$231$	117470.			2.20141
$232$		0			0
$233$		−	60252.0i		−	1.10984i	−0.831904	−	0.554919i	$-0.812749\pi$
$233$		−	60252.0i		−	1.10984i	0.831904	−	0.554919i	$-0.187251\pi$
$234$		0			0
$235$			29153.0i			0.527894i
$236$		0			0
$237$			91882.4i			1.63582i
$238$		0			0
$239$	−91262.4			−1.59770			−0.798851	−	0.601528i	$-0.794559\pi$
$239$	−91262.4			−1.59770			−0.798851	+	0.601528i	$0.794559\pi$
$240$		0			0
$241$			11694.8i			0.201354i	0.994919	+	0.100677i	$0.0321008\pi$
$241$			11694.8i			0.201354i	−0.994919	+	0.100677i	$0.967899\pi$
$242$		0			0
$243$			60712.3i			1.02817i
$244$		0			0
$245$		−	4369.36i		−	0.0727923i
$246$		0			0
$247$			97172.4i			1.59275i
$248$		0			0
$249$			121319.i			1.95672i
$250$		0			0
$251$	−95330.3			−1.51316			−0.756578	−	0.653904i	$-0.773130\pi$
$251$	−95330.3			−1.51316			−0.756578	+	0.653904i	$0.773130\pi$
$252$		0			0
$253$	−30125.1			−0.470639
$254$		0			0
$255$	4403.18			0.0677153
$256$		0			0
$257$			98138.1i			1.48584i	0.669382	+	0.742919i	$0.266559\pi$
$257$			98138.1i			1.48584i	−0.669382	+	0.742919i	$0.733441\pi$
$258$		0			0
$259$	−62981.1			−0.938881
$260$		0			0
$261$		−	34867.4i		−	0.511845i
$262$		0			0
$263$			7996.23i			0.115604i	0.998328	+	0.0578022i	$0.0184093\pi$
$263$			7996.23i			0.115604i	−0.998328	+	0.0578022i	$0.981591\pi$
$264$		0			0
$265$			25034.4i			0.356488i
$266$		0			0
$267$	61806.8			0.866990
$268$		0			0
$269$	−42975.4			−0.593903			−0.296951	−	0.954893i	$-0.595970\pi$
$269$	−42975.4			−0.593903			−0.296951	+	0.954893i	$0.595970\pi$
$270$		0			0
$271$	−22222.2			−0.302585			−0.151293	−	0.988489i	$-0.548344\pi$
$271$	−22222.2			−0.302585			−0.151293	+	0.988489i	$0.548344\pi$
$272$		0			0
$273$	248320.			3.33185
$274$		0			0
$275$	−81105.7			−1.07247
$276$		0			0
$277$		−	67337.1i		−	0.877596i	−0.898586	−	0.438798i	$-0.855404\pi$
$277$		−	67337.1i		−	0.877596i	0.898586	−	0.438798i	$-0.144596\pi$
$278$		0			0
$279$	86358.6			1.10942
$280$		0			0
$281$	−14673.1			−0.185828			−0.0929138	−	0.995674i	$-0.529618\pi$
$281$	−14673.1			−0.185828			−0.0929138	+	0.995674i	$0.529618\pi$
$282$		0			0
$283$	−25223.6			−0.314944			−0.157472	−	0.987523i	$-0.550334\pi$
$283$	−25223.6			−0.314944			−0.157472	+	0.987523i	$0.550334\pi$
$284$		0			0
$285$		−	41513.0i		−	0.511086i
$286$		0			0
$287$		−	11375.0i		−	0.138098i
$288$		0			0
$289$	−82480.4			−0.987541
$290$		0			0
$291$			52053.4i			0.614700i
$292$		0			0
$293$	112480.			1.31020			0.655102	−	0.755540i	$-0.272626\pi$
$293$	112480.			1.31020			0.655102	+	0.755540i	$0.272626\pi$
$294$		0			0
$295$		−	25840.6i		−	0.296933i
$296$		0			0
$297$		−	107034.i		−	1.21341i
$298$		0			0
$299$	−63681.7			−0.712315
$300$		0			0
$301$	67493.0	+	72402.2i	0.744948	+	0.799132i
$302$		0			0
$303$		−	45570.2i		−	0.496359i
$304$		0			0
$305$	−55289.5			−0.594351
$306$		0			0
$307$	−136783.			−1.45130			−0.725648	−	0.688067i	$-0.758460\pi$
$307$	−136783.			−1.45130			−0.725648	+	0.688067i	$0.758460\pi$
$308$		0			0
$309$		−	104902.i		−	1.09867i
$310$		0			0
$311$	46529.6			0.481070			0.240535	−	0.970640i	$-0.422677\pi$
$311$	46529.6			0.481070			0.240535	+	0.970640i	$0.422677\pi$
$312$		0			0
$313$			68823.9i			0.702507i	0.936280	+	0.351253i	$0.114244\pi$
$313$			68823.9i			0.702507i	−0.936280	+	0.351253i	$0.885756\pi$
$314$		0			0
$315$	−65317.1			−0.658272
$316$		0			0
$317$	−48726.3			−0.484892			−0.242446	−	0.970165i	$-0.577950\pi$
$317$	−48726.3			−0.484892			−0.242446	+	0.970165i	$0.577950\pi$
$318$		0			0
$319$		−	40608.6i		−	0.399058i
$320$		0			0
$321$		−	165103.i		−	1.60231i
$322$		0			0
$323$		−	9810.83i		−	0.0940374i
$324$		0			0
$325$	−171450.			−1.62319
$326$		0			0
$327$		−	99162.3i		−	0.927366i
$328$		0			0
$329$		−	165993.i		−	1.53355i
$330$		0			0
$331$			61590.4i			0.562156i	0.959685	+	0.281078i	$0.0906919\pi$
$331$			61590.4i			0.562156i	−0.959685	+	0.281078i	$0.909308\pi$
$332$		0			0
$333$			152683.i			1.37690i
$334$		0			0
$335$			10279.6i			0.0915980i
$336$		0			0
$337$	53084.4			0.467420			0.233710	−	0.972306i	$-0.424913\pi$
$337$	53084.4			0.467420			0.233710	+	0.972306i	$0.424913\pi$
$338$		0			0
$339$	217047.			1.88866
$340$		0			0
$341$	100578.			0.864958
$342$		0			0
$343$		−	103653.i		−	0.881038i
$344$		0			0
$345$	27205.4			0.228569
$346$		0			0
$347$		−	80407.5i		−	0.667787i	−0.942611	−	0.333893i	$-0.891637\pi$
$347$		−	80407.5i		−	0.667787i	0.942611	−	0.333893i	$-0.108363\pi$
$348$		0			0
$349$		−	173808.i		−	1.42698i	−0.700663	−	0.713492i	$-0.747112\pi$
$349$		−	173808.i		−	1.42698i	0.700663	−	0.713492i	$-0.252888\pi$
$350$		0			0
$351$		−	226260.i		−	1.83651i
$352$		0			0
$353$	−80068.2			−0.642556			−0.321278	−	0.946985i	$-0.604112\pi$
$353$	−80068.2			−0.642556			−0.321278	+	0.946985i	$0.604112\pi$
$354$		0			0
$355$	54918.0			0.435771
$356$		0			0
$357$	−25071.1			−0.196715
$358$		0			0
$359$	−83187.2			−0.645457			−0.322729	−	0.946492i	$-0.604600\pi$
$359$	−83187.2			−0.645457			−0.322729	+	0.946492i	$0.604600\pi$
$360$		0			0
$361$	37825.1			0.290245
$362$		0			0
$363$		−	119107.i		−	0.903907i
$364$		0			0
$365$	6238.20			0.0468245
$366$		0			0
$367$	−248797.			−1.84719			−0.923597	−	0.383365i	$-0.874765\pi$
$367$	−248797.			−1.84719			−0.923597	+	0.383365i	$0.874765\pi$
$368$		0			0
$369$	−27576.0			−0.202525
$370$		0			0
$371$		−	142542.i		−	1.03561i
$372$		0			0
$373$			209386.i			1.50498i	0.658603	+	0.752490i	$0.271148\pi$
$373$			209386.i			1.50498i	−0.658603	+	0.752490i	$0.728852\pi$
$374$		0			0
$375$	158556.			1.12751
$376$		0			0
$377$		−	85842.7i		−	0.603978i
$378$		0			0
$379$	92058.9			0.640895			0.320448	−	0.947266i	$-0.396167\pi$
$379$	92058.9			0.640895			0.320448	+	0.947266i	$0.396167\pi$
$380$		0			0
$381$		−	161059.i		−	1.10952i
$382$		0			0
$383$			51836.8i			0.353379i	0.984267	+	0.176690i	$0.0565388\pi$
$383$			51836.8i			0.353379i	−0.984267	+	0.176690i	$0.943461\pi$
$384$		0			0
$385$	−76072.0			−0.513220
$386$		0			0
$387$	175522.	−	163621.i	1.17195	−	1.09249i
$388$		0			0
$389$			170164.i			1.12452i	0.826959	+	0.562262i	$0.190069\pi$
$389$			170164.i			1.12452i	−0.826959	+	0.562262i	$0.809931\pi$
$390$		0			0
$391$	6429.50			0.0420556
$392$		0			0
$393$	−109196.			−0.707007
$394$		0			0
$395$		−	59502.1i		−	0.381362i
$396$		0			0
$397$	−201719.			−1.27987			−0.639936	−	0.768428i	$-0.721039\pi$
$397$	−201719.			−1.27987			−0.639936	+	0.768428i	$0.721039\pi$
$398$		0			0
$399$			236369.i			1.48472i
$400$		0			0
$401$	38112.2			0.237015			0.118507	−	0.992953i	$-0.462189\pi$
$401$	38112.2			0.237015			0.118507	+	0.992953i	$0.462189\pi$
$402$		0			0
$403$	212613.			1.30912
$404$		0			0
$405$		−	2170.77i		−	0.0132344i
$406$		0			0
$407$			177823.i			1.07349i
$408$		0			0
$409$		−	301746.i		−	1.80383i	−0.431918	−	0.901913i	$-0.642163\pi$
$409$		−	301746.i		−	1.80383i	0.431918	−	0.901913i	$-0.357837\pi$
$410$		0			0
$411$	−226205.			−1.33912
$412$		0			0
$413$			147133.i			0.862601i
$414$		0			0
$415$		−	78564.7i		−	0.456175i
$416$		0			0
$417$			136388.i			0.784340i
$418$		0			0
$419$		−	15464.5i		−	0.0880859i	−0.999030	−	0.0440430i	$-0.985976\pi$
$419$		−	15464.5i		−	0.0880859i	0.999030	−	0.0440430i	$-0.0140238\pi$
$420$		0			0
$421$		−	218060.i		−	1.23030i	−0.788409	−	0.615151i	$-0.789095\pi$
$421$		−	218060.i		−	1.23030i	0.788409	−	0.615151i	$-0.210905\pi$
$422$		0			0
$423$	−402410.			−2.24899
$424$		0			0
$425$	17310.1			0.0958345
$426$		0			0
$427$	314811.			1.72661
$428$		0			0
$429$		−	701114.i		−	3.80955i
$430$		0			0
$431$	173738.			0.935277			0.467639	−	0.883920i	$-0.345105\pi$
$431$	173738.			0.935277			0.467639	+	0.883920i	$0.345105\pi$
$432$		0			0
$433$			252649.i			1.34754i	0.738942	+	0.673769i	$0.235326\pi$
$433$			252649.i			1.34754i	−0.738942	+	0.673769i	$0.764674\pi$
$434$		0			0
$435$			36672.9i			0.193806i
$436$		0			0
$437$		−	60617.0i		−	0.317418i
$438$		0			0
$439$	197576.			1.02519			0.512597	−	0.858630i	$-0.328684\pi$
$439$	197576.			1.02519			0.512597	+	0.858630i	$0.328684\pi$
$440$		0			0
$441$	60312.0			0.310118
$442$		0			0
$443$	−116118.			−0.591685			−0.295842	−	0.955237i	$-0.595600\pi$
$443$	−116118.			−0.591685			−0.295842	+	0.955237i	$0.595600\pi$
$444$		0			0
$445$	−40025.4			−0.202123
$446$		0			0
$447$	−29222.3			−0.146251
$448$		0			0
$449$			221303.i			1.09773i	0.835911	+	0.548864i	$0.184940\pi$
$449$			221303.i			1.09773i	−0.835911	+	0.548864i	$0.815060\pi$
$450$		0			0
$451$	−32116.6			−0.157898
$452$		0			0
$453$	−108122.			−0.526886
$454$		0			0
$455$	−160809.			−0.776762
$456$		0			0
$457$		−	79170.3i		−	0.379079i	−0.981873	−	0.189540i	$-0.939300\pi$
$457$		−	79170.3i		−	0.379079i	0.981873	−	0.189540i	$-0.0606996\pi$
$458$		0			0
$459$			22843.9i			0.108429i
$460$		0			0
$461$	272584.			1.28262			0.641311	−	0.767281i	$-0.278391\pi$
$461$	272584.			1.28262			0.641311	+	0.767281i	$0.278391\pi$
$462$		0			0
$463$			324723.i			1.51478i	0.652960	+	0.757392i	$0.273527\pi$
$463$			324723.i			1.51478i	−0.652960	+	0.757392i	$0.726473\pi$
$464$		0			0
$465$	−90830.3			−0.420073
$466$		0			0
$467$			48808.8i			0.223802i	0.993719	+	0.111901i	$0.0356940\pi$
$467$			48808.8i			0.223802i	−0.993719	+	0.111901i	$0.964306\pi$
$468$		0			0
$469$		−	58530.5i		−	0.266095i
$470$		0			0
$471$	−237682.			−1.07141
$472$		0			0
$473$	204423.	−	190562.i	0.913707	−	0.851754i
$474$		0			0
$475$		−	163199.i		−	0.723319i
$476$		0			0
$477$	−345559.			−1.51875
$478$		0			0
$479$	−383151.			−1.66993			−0.834966	−	0.550301i	$-0.814513\pi$
$479$	−383151.			−1.66993			−0.834966	+	0.550301i	$0.814513\pi$
$480$		0			0
$481$			375901.i			1.62474i
$482$		0			0
$483$	−154904.			−0.664001
$484$		0			0
$485$		−	33709.2i		−	0.143306i
$486$		0			0
$487$	−296539.			−1.25033			−0.625164	−	0.780494i	$-0.714968\pi$
$487$	−296539.			−1.25033			−0.625164	+	0.780494i	$0.714968\pi$
$488$		0			0
$489$	654949.			2.73899
$490$		0			0
$491$			234793.i			0.973916i	0.873425	+	0.486958i	$0.161894\pi$
$491$			234793.i			0.973916i	−0.873425	+	0.486958i	$0.838106\pi$
$492$		0			0
$493$			8666.95i			0.0356593i
$494$		0			0
$495$			184418.i			0.752651i
$496$		0			0
$497$	−312696.			−1.26593
$498$		0			0
$499$			223696.i			0.898372i	0.893438	+	0.449186i	$0.148286\pi$
$499$			223696.i			0.898372i	−0.893438	+	0.449186i	$0.851714\pi$
$500$		0			0
$501$		−	351507.i		−	1.40042i
$502$		0			0
$503$			171252.i			0.676861i	0.940991	+	0.338430i	$0.109896\pi$
$503$			171252.i			0.676861i	−0.940991	+	0.338430i	$0.890104\pi$
$504$		0			0
$505$			29510.8i			0.115717i
$506$		0			0
$507$		−	1.06744e6i		−	4.15266i
$508$		0			0
$509$	−489934.			−1.89105			−0.945524	−	0.325554i	$-0.894449\pi$
$509$	−489934.			−1.89105			−0.945524	+	0.325554i	$0.894449\pi$
$510$		0			0
$511$	−35519.4			−0.136027
$512$		0			0
$513$	215371.			0.818374
$514$		0			0
$515$			67933.4i			0.256135i
$516$		0			0
$517$	−468670.			−1.75342
$518$		0			0
$519$		−	532780.i		−	1.97794i
$520$		0			0
$521$			286304.i			1.05476i	0.849630	+	0.527378i	$0.176825\pi$
$521$			286304.i			1.05476i	−0.849630	+	0.527378i	$0.823175\pi$
$522$		0			0
$523$		−	132186.i		−	0.483263i	−0.970368	−	0.241631i	$-0.922318\pi$
$523$		−	132186.i		−	0.483263i	0.970368	−	0.241631i	$-0.0776825\pi$
$524$		0			0
$525$	−417047.			−1.51310
$526$		0			0
$527$	−21466.1			−0.0772914
$528$		0			0
$529$	−240116.			−0.858044
$530$		0			0
$531$	356689.			1.26503
$532$		0			0
$533$	−67891.6			−0.238980
$534$		0			0
$535$			106919.i			0.373549i
$536$		0			0
$537$	734897.			2.54846
$538$		0			0
$539$	70242.8			0.241782
$540$		0			0
$541$	114352.			0.390704			0.195352	−	0.980733i	$-0.437415\pi$
$541$	114352.			0.390704			0.195352	+	0.980733i	$0.437415\pi$
$542$		0			0
$543$			460501.i			1.56182i
$544$		0			0
$545$			64216.4i			0.216199i
$546$		0			0
$547$	−48485.5			−0.162046			−0.0810228	−	0.996712i	$-0.525819\pi$
$547$	−48485.5			−0.162046			−0.0810228	+	0.996712i	$0.525819\pi$
$548$		0			0
$549$		−	763183.i		−	2.53212i
$550$		0			0
$551$	81711.5			0.269141
$552$		0			0
$553$			338797.i			1.10787i
$554$		0			0
$555$		−	160589.i		−	0.521349i
$556$		0			0
$557$	−96856.1			−0.312188			−0.156094	−	0.987742i	$-0.549890\pi$
$557$	−96856.1			−0.312188			−0.156094	+	0.987742i	$0.549890\pi$
$558$		0			0
$559$	432131.	−	402830.i	1.38290	−	1.28914i
$560$		0			0
$561$			70786.6i			0.224919i
$562$		0			0
$563$	26855.0			0.0847244			0.0423622	−	0.999102i	$-0.486512\pi$
$563$	26855.0			0.0847244			0.0423622	+	0.999102i	$0.486512\pi$
$564$		0			0
$565$	−140557.			−0.440308
$566$		0			0
$567$			12360.1i			0.0384464i
$568$		0			0
$569$	228140.			0.704654			0.352327	−	0.935877i	$-0.385390\pi$
$569$	228140.			0.704654			0.352327	+	0.935877i	$0.385390\pi$
$570$		0			0
$571$		−	319649.i		−	0.980396i	−0.871611	−	0.490198i	$-0.836924\pi$
$571$		−	319649.i		−	0.980396i	0.871611	−	0.490198i	$-0.163076\pi$
$572$		0			0
$573$	−724771.			−2.20745
$574$		0			0
$575$	106952.			0.323484
$576$		0			0
$577$			527850.i			1.58547i	0.609565	+	0.792736i	$0.291344\pi$
$577$			527850.i			1.58547i	−0.609565	+	0.792736i	$0.708656\pi$
$578$		0			0
$579$		−	7859.59i		−	0.0234446i
$580$		0			0
$581$			447336.i			1.32520i
$582$		0			0
$583$	−402458.			−1.18409
$584$		0			0
$585$			389843.i			1.13914i
$586$		0			0
$587$			571037.i			1.65725i	0.559804	+	0.828625i	$0.310876\pi$
$587$			571037.i			1.65725i	−0.559804	+	0.828625i	$0.689124\pi$
$588$		0			0
$589$			202381.i			0.583363i
$590$		0			0
$591$			452133.i			1.29447i
$592$		0			0
$593$			564829.i			1.60623i	0.595824	+	0.803115i	$0.296825\pi$
$593$			564829.i			1.60623i	−0.595824	+	0.803115i	$0.703175\pi$
$594$		0			0
$595$	16235.8			0.0458606
$596$		0			0
$597$	865550.			2.42853
$598$		0			0
$599$	−323806.			−0.902467			−0.451233	−	0.892406i	$-0.649016\pi$
$599$	−323806.			−0.902467			−0.451233	+	0.892406i	$0.649016\pi$
$600$		0			0
$601$			239522.i			0.663126i	0.943433	+	0.331563i	$0.107576\pi$
$601$			239522.i			0.663126i	−0.943433	+	0.331563i	$0.892424\pi$
$602$		0			0
$603$	−141893.			−0.390236
$604$		0			0
$605$			77132.4i			0.210730i
$606$		0			0
$607$			344432.i			0.934816i	0.884042	+	0.467408i	$0.154812\pi$
$607$			344432.i			0.934816i	−0.884042	+	0.467408i	$0.845188\pi$
$608$		0			0
$609$		−	208810.i		−	0.563011i
$610$		0			0
$611$	−990725.			−2.65382
$612$		0			0
$613$	446348.			1.18783			0.593913	−	0.804529i	$-0.297582\pi$
$613$	446348.			1.18783			0.593913	+	0.804529i	$0.297582\pi$
$614$		0			0
$615$	29003.9			0.0766844
$616$		0			0
$617$	−663045.			−1.74170			−0.870848	−	0.491552i	$-0.836430\pi$
$617$	−663045.			−1.74170			−0.870848	+	0.491552i	$0.836430\pi$
$618$		0			0
$619$	298590.			0.779281			0.389641	−	0.920967i	$-0.372599\pi$
$619$	298590.			0.779281			0.389641	+	0.920967i	$0.372599\pi$
$620$		0			0
$621$			141143.i			0.365995i
$622$		0			0
$623$	227899.			0.587174
$624$		0			0
$625$	232700.			0.595712
$626$		0			0
$627$	667373.			1.69759
$628$		0			0
$629$		−	37952.1i		−	0.0959257i
$630$		0			0
$631$		−	204687.i		−	0.514082i	−0.966400	−	0.257041i	$-0.917252\pi$
$631$		−	204687.i		−	0.514082i	0.966400	−	0.257041i	$-0.0827475\pi$
$632$		0			0
$633$	657358.			1.64057
$634$		0			0
$635$			104300.i			0.258665i
$636$		0			0
$637$	148487.			0.365939
$638$		0			0
$639$			758056.i			1.85652i
$640$		0			0
$641$		−	293809.i		−	0.715070i	−0.933900	−	0.357535i	$-0.883617\pi$
$641$		−	293809.i		−	0.715070i	0.933900	−	0.357535i	$-0.116383\pi$
$642$		0			0
$643$	376164.			0.909820			0.454910	−	0.890537i	$-0.349671\pi$
$643$	376164.			0.909820			0.454910	+	0.890537i	$0.349671\pi$
$644$		0			0
$645$	−184610.	+	172093.i	−0.443748	+	0.413660i
$646$		0			0
$647$		−	72724.1i		−	0.173728i	−0.996220	−	0.0868640i	$-0.972315\pi$
$647$		−	72724.1i		−	0.173728i	0.996220	−	0.0868640i	$-0.0276846\pi$
$648$		0			0
$649$	415420.			0.986275
$650$		0			0
$651$	517175.			1.22033
$652$		0			0
$653$		−	338825.i		−	0.794602i	−0.917688	−	0.397301i	$-0.869947\pi$
$653$		−	338825.i		−	0.794602i	0.917688	−	0.397301i	$-0.130053\pi$
$654$		0			0
$655$	70714.5			0.164826
$656$		0			0
$657$			86108.3i			0.199487i
$658$		0			0
$659$	−548017.			−1.26189			−0.630947	−	0.775826i	$-0.717334\pi$
$659$	−548017.			−1.26189			−0.630947	+	0.775826i	$0.717334\pi$
$660$		0			0
$661$	487460.			1.11567			0.557836	−	0.829951i	$-0.311632\pi$
$661$	487460.			1.11567			0.557836	+	0.829951i	$0.311632\pi$
$662$		0			0
$663$			149636.i			0.340416i
$664$		0			0
$665$		−	153070.i		−	0.346136i
$666$		0			0
$667$			53549.5i			0.120366i
$668$		0			0
$669$	−377904.			−0.844362
$670$		0			0
$671$		−	888846.i		−	1.97416i
$672$		0			0
$673$		−	738888.i		−	1.63136i	−0.578507	−	0.815678i	$-0.696364\pi$
$673$		−	738888.i		−	1.63136i	0.578507	−	0.815678i	$-0.303636\pi$
$674$		0			0
$675$			379998.i			0.834014i
$676$		0			0
$677$		−	227328.i		−	0.495994i	−0.968761	−	0.247997i	$-0.920228\pi$
$677$		−	227328.i		−	0.495994i	0.968761	−	0.247997i	$-0.0797723\pi$
$678$		0			0
$679$			191936.i			0.416309i
$680$		0			0
$681$	−838988.			−1.80910
$682$		0			0
$683$	−161434.			−0.346061			−0.173031	−	0.984916i	$-0.555356\pi$
$683$	−161434.			−0.346061			−0.173031	+	0.984916i	$0.555356\pi$
$684$		0			0
$685$	146488.			0.312191
$686$		0			0
$687$			226809.i			0.480559i
$688$		0			0
$689$	−850760.			−1.79213
$690$		0			0
$691$			327344.i			0.685565i	0.939415	+	0.342783i	$0.111369\pi$
$691$			327344.i			0.685565i	−0.939415	+	0.342783i	$0.888631\pi$
$692$		0			0
$693$		−	1.05005e6i		−	2.18648i
$694$		0			0
$695$		−	88323.4i		−	0.182855i
$696$		0			0
$697$	6854.54			0.0141095
$698$		0			0
$699$	−874749.			−1.79031
$700$		0			0
$701$	436072.			0.887406			0.443703	−	0.896174i	$-0.353665\pi$
$701$	436072.			0.887406			0.443703	+	0.896174i	$0.353665\pi$
$702$		0			0
$703$	−357811.			−0.724007
$704$		0			0
$705$	423247.			0.851562
$706$		0			0
$707$		−	168030.i		−	0.336162i
$708$		0			0
$709$	−453704.			−0.902569			−0.451285	−	0.892380i	$-0.649034\pi$
$709$	−453704.			−0.902569			−0.451285	+	0.892380i	$0.649034\pi$
$710$		0			0
$711$	821331.			1.62472
$712$		0			0
$713$	−132630.			−0.260893
$714$		0			0
$715$			454034.i			0.888129i
$716$		0			0
$717$			1.32496e6i			2.57730i
$718$		0			0
$719$	677749.			1.31103			0.655513	−	0.755184i	$-0.272453\pi$
$719$	677749.			1.31103			0.655513	+	0.755184i	$0.272453\pi$
$720$		0			0
$721$		−	386803.i		−	0.744080i
$722$		0			0
$723$	169787.			0.324810
$724$		0			0
$725$			144171.i			0.274285i
$726$		0			0
$727$			421634.i			0.797751i	0.917005	+	0.398875i	$0.130599\pi$
$727$			421634.i			0.797751i	−0.917005	+	0.398875i	$0.869401\pi$
$728$		0			0
$729$	862728.			1.62338
$730$		0			0
$731$	−43629.2	+	40671.0i	−0.0816475	+	0.0761115i
$732$		0			0
$733$			545085.i			1.01451i	0.861796	+	0.507255i	$0.169340\pi$
$733$			545085.i			1.01451i	−0.861796	+	0.507255i	$0.830660\pi$
$734$		0			0
$735$	−63435.0			−0.117423
$736$		0			0
$737$	−165257.			−0.304246
$738$		0			0
$739$		−	308914.i		−	0.565651i	−0.959171	−	0.282826i	$-0.908728\pi$
$739$		−	308914.i		−	0.565651i	0.959171	−	0.282826i	$-0.0912718\pi$
$740$		0			0
$741$	1.41076e6			2.56932
$742$		0			0
$743$		−	1.03791e6i		−	1.88011i	−0.341026	−	0.940054i	$-0.610775\pi$
$743$		−	1.03791e6i		−	1.88011i	0.341026	−	0.940054i	$-0.389225\pi$
$744$		0			0
$745$	18924.0			0.0340958
$746$		0			0
$747$	1.08446e6			1.94345
$748$		0			0
$749$		−	608782.i		−	1.08517i
$750$		0			0
$751$		−	162809.i		−	0.288669i	−0.989529	−	0.144334i	$-0.953896\pi$
$751$		−	162809.i		−	0.288669i	0.989529	−	0.144334i	$-0.0461041\pi$
$752$		0			0
$753$			1.38402e6i			2.44091i
$754$		0			0
$755$	70018.5			0.122834
$756$		0			0
$757$		−	24344.0i		−	0.0424816i	−0.999774	−	0.0212408i	$-0.993238\pi$
$757$		−	24344.0i		−	0.0424816i	0.999774	−	0.0212408i	$-0.00676166\pi$
$758$		0			0
$759$			437361.i			0.759201i
$760$		0			0
$761$		−	935074.i		−	1.61464i	−0.590111	−	0.807322i	$-0.700916\pi$
$761$		−	935074.i		−	1.61464i	0.590111	−	0.807322i	$-0.299084\pi$
$762$		0			0
$763$		−	365639.i		−	0.628064i
$764$		0			0
$765$		−	39359.8i		−	0.0672558i
$766$		0			0
$767$	878159.			1.49273
$768$		0			0
$769$	314204.			0.531323			0.265662	−	0.964066i	$-0.414410\pi$
$769$	314204.			0.531323			0.265662	+	0.964066i	$0.414410\pi$
$770$		0			0
$771$	1.42478e6			2.39685
$772$		0			0
$773$		−	786232.i		−	1.31581i	−0.753103	−	0.657903i	$-0.771444\pi$
$773$		−	786232.i		−	1.31581i	0.753103	−	0.657903i	$-0.228556\pi$
$774$		0			0
$775$	−357079.			−0.594512
$776$		0			0
$777$			914369.i			1.51454i
$778$		0			0
$779$		−	64624.3i		−	0.106493i
$780$		0			0
$781$			882875.i			1.44743i
$782$		0			0
$783$	−190260.			−0.310330
$784$		0			0
$785$	153920.			0.249779
$786$		0			0
$787$	−229414.			−0.370400			−0.185200	−	0.982701i	$-0.559293\pi$
$787$	−229414.			−0.370400			−0.185200	+	0.982701i	$0.559293\pi$
$788$		0			0
$789$	116091.			0.186485
$790$		0			0
$791$	800313.			1.27911
$792$		0			0
$793$		−	1.87894e6i		−	2.98790i
$794$		0			0
$795$	363453.			0.575061
$796$		0			0
$797$	−1.10312e6			−1.73663			−0.868316	−	0.496011i	$-0.834798\pi$
$797$	−1.10312e6			−1.73663			−0.868316	+	0.496011i	$0.834798\pi$
$798$		0			0
$799$	100027.			0.156683
$800$		0			0
$801$		−	552487.i		−	0.861107i
$802$		0			0
$803$			100287.i			0.155529i
$804$		0			0
$805$	100314.			0.154800
$806$		0			0
$807$			623924.i			0.958042i
$808$		0			0
$809$	602078.			0.919932			0.459966	−	0.887937i	$-0.347862\pi$
$809$	602078.			0.919932			0.459966	+	0.887937i	$0.347862\pi$
$810$		0			0
$811$			930391.i			1.41457i	0.706930	+	0.707284i	$0.250080\pi$
$811$			930391.i			1.41457i	−0.706930	+	0.707284i	$0.749920\pi$
$812$		0			0
$813$			322625.i			0.488109i
$814$		0			0
$815$	−424138.			−0.638546
$816$		0			0
$817$	383444.	+	411334.i	0.574458	+	0.616241i
$818$		0			0
$819$		−	2.21971e6i		−	3.30925i
$820$		0			0
$821$	−857913.			−1.27279			−0.636395	−	0.771363i	$-0.719575\pi$
$821$	−857913.			−1.27279			−0.636395	+	0.771363i	$0.719575\pi$
$822$		0			0
$823$	1.18057e6			1.74298			0.871488	−	0.490417i	$-0.163155\pi$
$823$	1.18057e6			1.74298			0.871488	+	0.490417i	$0.163155\pi$
$824$		0			0
$825$			1.17750e6i			1.73003i
$826$		0			0
$827$	−48059.2			−0.0702693			−0.0351347	−	0.999383i	$-0.511186\pi$
$827$	−48059.2			−0.0702693			−0.0351347	+	0.999383i	$0.511186\pi$
$828$		0			0
$829$		−	640828.i		−	0.932465i	−0.884662	−	0.466233i	$-0.845611\pi$
$829$		−	640828.i		−	0.932465i	0.884662	−	0.466233i	$-0.154389\pi$
$830$		0			0
$831$	−977610.			−1.41568
$832$		0			0
$833$	−14991.7			−0.0216053
$834$		0			0
$835$			227632.i			0.326483i
$836$		0			0
$837$		−	471231.i		−	0.672640i
$838$		0			0
$839$			380215.i			0.540138i	0.962841	+	0.270069i	$0.0870465\pi$
$839$			380215.i			0.540138i	−0.962841	+	0.270069i	$0.912953\pi$
$840$		0			0
$841$	635096.			0.897941
$842$		0			0
$843$			213027.i			0.299764i
$844$		0			0
$845$			691260.i			0.968118i
$846$		0			0
$847$		−	439181.i		−	0.612177i
$848$		0			0
$849$			366200.i			0.508045i
$850$		0			0
$851$		−	234490.i		−	0.323792i
$852$		0			0
$853$	374276.			0.514392			0.257196	−	0.966359i	$-0.417201\pi$
$853$	374276.			0.514392			0.257196	+	0.966359i	$0.417201\pi$
$854$		0			0
$855$	−371082.			−0.507619
$856$		0			0
$857$	581564.			0.791837			0.395918	−	0.918286i	$-0.370426\pi$
$857$	581564.			0.791837			0.395918	+	0.918286i	$0.370426\pi$
$858$		0			0
$859$			1.14416e6i			1.55061i	0.631589	+	0.775303i	$0.282403\pi$
$859$			1.14416e6i			1.55061i	−0.631589	+	0.775303i	$0.717597\pi$
$860$		0			0
$861$	−165144.			−0.222770
$862$		0			0
$863$		−	48167.9i		−	0.0646750i	−0.999477	−	0.0323375i	$-0.989705\pi$
$863$		−	48167.9i		−	0.0646750i	0.999477	−	0.0323375i	$-0.0102951\pi$
$864$		0			0
$865$			345022.i			0.461121i
$866$		0			0
$867$			1.19746e6i			1.59303i
$868$		0			0
$869$	956570.			1.26671
$870$		0			0
$871$	−349338.			−0.460479
$872$		0			0
$873$	465302.			0.610529
$874$		0			0
$875$	584640.			0.763611
$876$		0			0
$877$	−481656.			−0.626235			−0.313118	−	0.949714i	$-0.601373\pi$
$877$	−481656.			−0.626235			−0.313118	+	0.949714i	$0.601373\pi$
$878$		0			0
$879$		−	1.63300e6i		−	2.11353i
$880$		0			0
$881$	−771331.			−0.993777			−0.496889	−	0.867814i	$-0.665524\pi$
$881$	−771331.			−0.993777			−0.496889	+	0.867814i	$0.665524\pi$
$882$		0			0
$883$	−1.36525e6			−1.75101			−0.875507	−	0.483205i	$-0.839473\pi$
$883$	−1.36525e6			−1.75101			−0.875507	+	0.483205i	$0.839473\pi$
$884$		0			0
$885$	−375158.			−0.478992
$886$		0			0
$887$			395935.i			0.503242i	0.967826	+	0.251621i	$0.0809637\pi$
$887$			395935.i			0.503242i	−0.967826	+	0.251621i	$0.919036\pi$
$888$		0			0
$889$		−	593870.i		−	0.751429i
$890$		0			0
$891$	34897.9			0.0439586
$892$		0			0
$893$		−	943046.i		−	1.18258i
$894$		0			0
$895$	−475911.			−0.594128
$896$		0			0
$897$			924541.i			1.14906i
$898$		0			0
$899$		−	178785.i		−	0.221213i
$900$		0			0
$901$	85895.3			0.105808
$902$		0			0
$903$	1.05115e6	−	979874.i	1.28910	−	1.20170i
$904$		0			0
$905$		−	298216.i		−	0.364111i
$906$		0			0
$907$	−1.02709e6			−1.24852			−0.624260	−	0.781216i	$-0.714600\pi$
$907$	−1.02709e6			−1.24852			−0.624260	+	0.781216i	$0.714600\pi$
$908$		0			0
$909$	−407349.			−0.492991
$910$		0			0
$911$		−	1.39342e6i		−	1.67898i	−0.543374	−	0.839491i	$-0.682853\pi$
$911$		−	1.39342e6i		−	1.67898i	0.543374	−	0.839491i	$-0.317147\pi$
$912$		0			0
$913$	1.26302e6			1.51520
$914$		0			0
$915$			802701.i			0.958764i
$916$		0			0
$917$	−402638.			−0.478825
$918$		0			0
$919$	1.19770e6			1.41813			0.709064	−	0.705144i	$-0.249118\pi$
$919$	1.19770e6			1.41813			0.709064	+	0.705144i	$0.249118\pi$
$920$		0			0
$921$			1.98584e6i			2.34113i
$922$		0			0
$923$			1.86632e6i			2.19069i
$924$		0			0
$925$		−	631317.i		−	0.737843i
$926$		0			0
$927$	−937711.			−1.09121
$928$		0			0
$929$			235931.i			0.273371i	0.990614	+	0.136686i	$0.0436450\pi$
$929$			235931.i			0.273371i	−0.990614	+	0.136686i	$0.956355\pi$
$930$		0			0
$931$			141341.i			0.163068i
$932$		0			0
$933$		−	675523.i		−	0.776028i
$934$		0			0
$935$		−	45840.7i		−	0.0524358i
$936$		0			0
$937$			1.54482e6i			1.75954i	0.475401	+	0.879769i	$0.342303\pi$
$937$			1.54482e6i			1.75954i	−0.475401	+	0.879769i	$0.657697\pi$
$938$		0			0
$939$	999196.			1.13323
$940$		0			0
$941$	−573364.			−0.647517			−0.323759	−	0.946140i	$-0.604947\pi$
$941$	−573364.			−0.647517			−0.323759	+	0.946140i	$0.604947\pi$
$942$		0			0
$943$	42351.4			0.0476260
$944$		0			0
$945$			356414.i			0.399109i
$946$		0			0
$947$	−1.66137e6			−1.85253			−0.926265	−	0.376872i	$-0.877000\pi$
$947$	−1.66137e6			−1.85253			−0.926265	+	0.376872i	$0.877000\pi$
$948$		0			0
$949$			211997.i			0.235395i
$950$		0			0
$951$			707417.i			0.782193i
$952$		0			0
$953$			1.10535e6i			1.21707i	0.793527	+	0.608535i	$0.208242\pi$
$953$			1.10535e6i			1.21707i	−0.793527	+	0.608535i	$0.791758\pi$
$954$		0			0
$955$	469354.			0.514629
$956$		0			0
$957$	−589562.			−0.643732
$958$		0			0
$959$	−834083.			−0.906926
$960$		0			0
$961$	−480712.			−0.520521
$962$		0			0
$963$	−1.47585e6			−1.59143
$964$		0			0
$965$			5089.79i			0.00546569i
$966$		0			0
$967$	−352269.			−0.376723			−0.188361	−	0.982100i	$-0.560318\pi$
$967$	−352269.			−0.376723			−0.188361	+	0.982100i	$0.560318\pi$
$968$		0			0
$969$	−142435.			−0.151694
$970$		0			0
$971$	697949.			0.740262			0.370131	−	0.928980i	$-0.379313\pi$
$971$	697949.			0.740262			0.370131	+	0.928980i	$0.379313\pi$
$972$		0			0
$973$			502901.i			0.531199i
$974$		0			0
$975$			2.48914e6i			2.61842i
$976$		0			0
$977$	84775.3			0.0888137			0.0444069	−	0.999014i	$-0.485860\pi$
$977$	84775.3			0.0888137			0.0444069	+	0.999014i	$0.485860\pi$
$978$		0			0
$979$		−	643459.i		−	0.671360i
$980$		0			0
$981$	−886405.			−0.921073
$982$		0			0
$983$			528881.i			0.547333i	0.961825	+	0.273666i	$0.0882364\pi$
$983$			528881.i			0.547333i	−0.961825	+	0.273666i	$0.911764\pi$
$984$		0			0
$985$		−	292796.i		−	0.301782i
$986$		0			0
$987$	−2.40991e6			−2.47381
$988$		0			0
$989$	−269567.	+	251289.i	−0.275597	+	0.256910i
$990$		0			0
$991$			766986.i			0.780980i	0.920607	+	0.390490i	$0.127694\pi$
$991$			766986.i			0.780980i	−0.920607	+	0.390490i	$0.872306\pi$
$992$		0			0
$993$	894178.			0.906830
$994$		0			0
$995$	−560521.			−0.566169
$996$		0			0
$997$		−	932501.i		−	0.938121i	−0.883166	−	0.469060i	$-0.844593\pi$
$997$		−	932501.i		−	0.938121i	0.883166	−	0.469060i	$-0.155407\pi$
$998$		0			0
$999$	833139.			0.834808

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	688.5.b.d.257.2		12
4.3	odd	2		43.5.b.b.42.1	✓	12
12.11	even	2		387.5.b.c.343.12		12
43.42	odd	2	inner	688.5.b.d.257.11		12
172.171	even	2		43.5.b.b.42.12	yes	12
516.515	odd	2		387.5.b.c.343.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.5.b.b.42.1	✓	12	4.3	odd	2
43.5.b.b.42.12	yes	12	172.171	even	2
387.5.b.c.343.1		12	516.515	odd	2
387.5.b.c.343.12		12	12.11	even	2
688.5.b.d.257.2		12	1.1	even	1	trivial
688.5.b.d.257.11		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)

\(f(q)\)	\(=\)	\(q-14.5182i q^{3} +9.40180i q^{5} -53.5326i q^{7} -129.777 q^{9} +O(q^{10})\)	\(q-14.5182i q^{3} +9.40180i q^{5} -53.5326i q^{7} -129.777 q^{9} -151.146 q^{11} -319.508 q^{13} +136.497 q^{15} +32.2585 q^{17} -304.131i q^{19} -777.194 q^{21} +199.312 q^{23} +536.606 q^{25} +708.150i q^{27} +268.672i q^{29} -665.439 q^{31} +2194.36i q^{33} +503.303 q^{35} -1176.50i q^{37} +4638.66i q^{39} +212.488 q^{41} +(-1352.49 + 1260.78i) q^{43} -1220.14i q^{45} +3100.78 q^{47} -464.736 q^{49} -468.334i q^{51} +2662.72 q^{53} -1421.04i q^{55} -4415.43 q^{57} -2748.48 q^{59} +5880.73i q^{61} +6947.29i q^{63} -3003.95i q^{65} +1093.36 q^{67} -2893.64i q^{69} -5841.22i q^{71} -663.511i q^{73} -7790.53i q^{75} +8091.21i q^{77} -6328.79 q^{79} -230.889 q^{81} -8356.34 q^{83} +303.288i q^{85} +3900.62 q^{87} +4257.21i q^{89} +17104.1i q^{91} +9660.95i q^{93} +2859.38 q^{95} -3585.40 q^{97} +19615.2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 462 q^{9}+O(q^{10}) \) 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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