LMFDB - Embedded newform 342.5.d.a.37.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [342,5,Mod(37,342)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(342, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("342.37");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 342 = 2 \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	342.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:	$35.3525273747$
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} + 450x^{6} + 68229x^{4} + 4001228x^{2} + 77475204 $ x^8 + 450x^6 + 68229x^4 + 4001228*x^2 + 77475204
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 38)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			37.3
Root			$12.2418i$ of defining polynomial
Character	$\chi$	$=$	342.37
Dual form			342.5.d.a.37.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.82843i q^{2} -8.00000 q^{4} +26.2296 q^{5} -86.0910 q^{7} +22.6274i q^{8} +O(q^{10})$	\(q-2.82843i q^{2} -8.00000 q^{4} +26.2296 q^{5} -86.0910 q^{7} +22.6274i q^{8} -74.1886i q^{10} -114.743 q^{11} -231.848i q^{13} +243.502i q^{14} +64.0000 q^{16} +244.466 q^{17} +(253.655 + 256.866i) q^{19} -209.837 q^{20} +324.543i q^{22} +269.052 q^{23} +62.9931 q^{25} -655.766 q^{26} +688.728 q^{28} +1131.05i q^{29} +1037.59i q^{31} -181.019i q^{32} -691.455i q^{34} -2258.14 q^{35} +302.815i q^{37} +(726.527 - 717.444i) q^{38} +593.509i q^{40} +1769.82i q^{41} -2316.56 q^{43} +917.946 q^{44} -760.993i q^{46} +835.757 q^{47} +5010.66 q^{49} -178.171i q^{50} +1854.79i q^{52} +656.208i q^{53} -3009.67 q^{55} -1948.02i q^{56} +3199.09 q^{58} +4923.33i q^{59} -1576.71 q^{61} +2934.75 q^{62} -512.000 q^{64} -6081.30i q^{65} +7000.44i q^{67} -1955.73 q^{68} +6386.97i q^{70} +4101.26i q^{71} +7018.02 q^{73} +856.490 q^{74} +(-2029.24 - 2054.93i) q^{76} +9878.36 q^{77} +1264.00i q^{79} +1678.70 q^{80} +5005.79 q^{82} -1020.09 q^{83} +6412.25 q^{85} +6552.21i q^{86} -2596.34i q^{88} +3746.30i q^{89} +19960.1i q^{91} -2152.41 q^{92} -2363.88i q^{94} +(6653.27 + 6737.50i) q^{95} -13041.0i q^{97} -14172.3i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 64 q^{4} - 18 q^{5} - 162 q^{7}+O(q^{10}) $ 8 * q - 64 * q^4 - 18 * q^5 - 162 * q^7	$ 8 q - 64 q^{4} - 18 q^{5} - 162 q^{7} + 6 q^{11} + 512 q^{16} - 510 q^{17} - 12 q^{19} + 144 q^{20} + 396 q^{23} + 3458 q^{25} + 192 q^{26} + 1296 q^{28} - 1002 q^{35} + 3216 q^{38} - 8654 q^{43} - 48 q^{44} - 3210 q^{47} + 9222 q^{49} + 17146 q^{55} - 960 q^{58} + 1314 q^{61} + 15168 q^{62} - 4096 q^{64} + 4080 q^{68} + 23398 q^{73} - 13152 q^{74} + 96 q^{76} + 44622 q^{77} - 1152 q^{80} + 16512 q^{82} + 10440 q^{83} + 21274 q^{85} - 3168 q^{92} + 34686 q^{95}+O(q^{100}) $ 8 * q - 64 * q^4 - 18 * q^5 - 162 * q^7 + 6 * q^11 + 512 * q^16 - 510 * q^17 - 12 * q^19 + 144 * q^20 + 396 * q^23 + 3458 * q^25 + 192 * q^26 + 1296 * q^28 - 1002 * q^35 + 3216 * q^38 - 8654 * q^43 - 48 * q^44 - 3210 * q^47 + 9222 * q^49 + 17146 * q^55 - 960 * q^58 + 1314 * q^61 + 15168 * q^62 - 4096 * q^64 + 4080 * q^68 + 23398 * q^73 - 13152 * q^74 + 96 * q^76 + 44622 * q^77 - 1152 * q^80 + 16512 * q^82 + 10440 * q^83 + 21274 * q^85 - 3168 * q^92 + 34686 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/342\mathbb{Z}\right)^\times$.

$n$	$191$	$325$
$\chi(n)$	$1$	$-1$

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 $

$2$		−	2.82843i		−	0.707107i
$2$		−	2.82843i		−	0.707107i
$3$		0			0
$3$		0			0
$4$	−8.00000			−0.500000
$5$	26.2296			1.04918			0.524592	−	0.851353i	$-0.324218\pi$
$5$	26.2296			1.04918			0.524592	+	0.851353i	$0.324218\pi$
$6$		0			0
$7$	−86.0910			−1.75696			−0.878480	−	0.477779i	$-0.841442\pi$
$7$	−86.0910			−1.75696			−0.878480	+	0.477779i	$0.841442\pi$
$8$			22.6274i			0.353553i
$9$		0			0
$10$		−	74.1886i		−	0.741886i
$11$	−114.743			−0.948291			−0.474145	−	0.880447i	$-0.657243\pi$
$11$	−114.743			−0.948291			−0.474145	+	0.880447i	$0.657243\pi$
$12$		0			0
$13$		−	231.848i		−	1.37188i	−0.727656	−	0.685942i	$-0.759390\pi$
$13$		−	231.848i		−	1.37188i	0.727656	−	0.685942i	$-0.240610\pi$
$14$			243.502i			1.24236i
$15$		0			0
$16$	64.0000			0.250000
$17$	244.466			0.845903			0.422952	−	0.906152i	$-0.360994\pi$
$17$	244.466			0.845903			0.422952	+	0.906152i	$0.360994\pi$
$18$		0			0
$19$	253.655	+	256.866i	0.702645	+	0.711540i
$19$	253.655	+	256.866i	0.702645	+	0.711540i
$20$	−209.837			−0.524592
$21$		0			0
$22$			324.543i			0.670543i
$23$	269.052			0.508604			0.254302	−	0.967125i	$-0.418154\pi$
$23$	269.052			0.508604			0.254302	+	0.967125i	$0.418154\pi$
$24$		0			0
$25$	62.9931			0.100789
$26$	−655.766			−0.970069
$27$		0			0
$28$	688.728			0.878480
$29$			1131.05i			1.34489i	0.740148	+	0.672443i	$0.234755\pi$
$29$			1131.05i			1.34489i	−0.740148	+	0.672443i	$0.765245\pi$
$30$		0			0
$31$			1037.59i			1.07970i	0.841762	+	0.539850i	$0.181519\pi$
$31$			1037.59i			1.07970i	−0.841762	+	0.539850i	$0.818481\pi$
$32$		−	181.019i		−	0.176777i
$33$		0			0
$34$		−	691.455i		−	0.598144i
$35$	−2258.14			−1.84338
$36$		0			0
$37$			302.815i			0.221194i	0.993865	+	0.110597i	$0.0352763\pi$
$37$			302.815i			0.221194i	−0.993865	+	0.110597i	$0.964724\pi$
$38$	726.527	−	717.444i	0.503135	−	0.496845i
$39$		0			0
$40$			593.509i			0.370943i
$41$			1769.82i			1.05283i	0.850226	+	0.526417i	$0.176465\pi$
$41$			1769.82i			1.05283i	−0.850226	+	0.526417i	$0.823535\pi$
$42$		0			0
$43$	−2316.56			−1.25287			−0.626435	−	0.779474i	$-0.715487\pi$
$43$	−2316.56			−1.25287			−0.626435	+	0.779474i	$0.715487\pi$
$44$	917.946			0.474145
$45$		0			0
$46$		−	760.993i		−	0.359638i
$47$	835.757			0.378342			0.189171	−	0.981944i	$-0.439420\pi$
$47$	835.757			0.378342			0.189171	+	0.981944i	$0.439420\pi$
$48$		0			0
$49$	5010.66			2.08691
$50$		−	178.171i		−	0.0712685i
$51$		0			0
$52$			1854.79i			0.685942i
$53$			656.208i			0.233609i	0.993155	+	0.116805i	$0.0372651\pi$
$53$			656.208i			0.233609i	−0.993155	+	0.116805i	$0.962735\pi$
$54$		0			0
$55$	−3009.67			−0.994932
$56$		−	1948.02i		−	0.621179i
$57$		0			0
$58$	3199.09			0.950979
$59$			4923.33i			1.41434i	0.707042	+	0.707171i	$0.250029\pi$
$59$			4923.33i			1.41434i	−0.707042	+	0.707171i	$0.749971\pi$
$60$		0			0
$61$	−1576.71			−0.423733			−0.211866	−	0.977299i	$-0.567954\pi$
$61$	−1576.71			−0.423733			−0.211866	+	0.977299i	$0.567954\pi$
$62$	2934.75			0.763463
$63$		0			0
$64$	−512.000			−0.125000
$65$		−	6081.30i		−	1.43936i
$66$		0			0
$67$			7000.44i			1.55946i	0.626113	+	0.779732i	$0.284645\pi$
$67$			7000.44i			1.55946i	−0.626113	+	0.779732i	$0.715355\pi$
$68$	−1955.73			−0.422952
$69$		0			0
$70$			6386.97i			1.30346i
$71$			4101.26i			0.813581i	0.913521	+	0.406790i	$0.133352\pi$
$71$			4101.26i			0.813581i	−0.913521	+	0.406790i	$0.866648\pi$
$72$		0			0
$73$	7018.02			1.31695			0.658474	−	0.752603i	$-0.271202\pi$
$73$	7018.02			1.31695			0.658474	+	0.752603i	$0.271202\pi$
$74$	856.490			0.156408
$75$		0			0
$76$	−2029.24	−	2054.93i	−0.351323	−	0.355770i
$77$	9878.36			1.66611
$78$		0			0
$79$			1264.00i			0.202532i	0.994859	+	0.101266i	$0.0322893\pi$
$79$			1264.00i			0.202532i	−0.994859	+	0.101266i	$0.967711\pi$
$80$	1678.70			0.262296
$81$		0			0
$82$	5005.79			0.744467
$83$	−1020.09			−0.148075			−0.0740377	−	0.997255i	$-0.523589\pi$
$83$	−1020.09			−0.148075			−0.0740377	+	0.997255i	$0.523589\pi$
$84$		0			0
$85$	6412.25			0.887509
$86$			6552.21i			0.885913i
$87$		0			0
$88$		−	2596.34i		−	0.335271i
$89$			3746.30i			0.472958i	0.971637	+	0.236479i	$0.0759934\pi$
$89$			3746.30i			0.472958i	−0.971637	+	0.236479i	$0.924007\pi$
$90$		0			0
$91$			19960.1i			2.41034i
$92$	−2152.41			−0.254302
$93$		0			0
$94$		−	2363.88i		−	0.267528i
$95$	6653.27	+	6737.50i	0.737205	+	0.746537i
$96$		0			0
$97$		−	13041.0i		−	1.38601i	−0.720933	−	0.693005i	$-0.756287\pi$
$97$		−	13041.0i		−	1.38601i	0.720933	−	0.693005i	$-0.243713\pi$
$98$		−	14172.3i		−	1.47567i
$99$		0			0
$100$	−503.945			−0.0503945
$101$	4755.84			0.466213			0.233106	−	0.972451i	$-0.425111\pi$
$101$	4755.84			0.466213			0.233106	+	0.972451i	$0.425111\pi$
$102$		0			0
$103$		−	8727.79i		−	0.822678i	−0.911483	−	0.411339i	$-0.865061\pi$
$103$		−	8727.79i		−	0.822678i	0.911483	−	0.411339i	$-0.134939\pi$
$104$	5246.13			0.485034
$105$		0			0
$106$	1856.04			0.165187
$107$		−	16884.9i		−	1.47479i	−0.675462	−	0.737395i	$-0.736056\pi$
$107$		−	16884.9i		−	1.47479i	0.675462	−	0.737395i	$-0.263944\pi$
$108$		0			0
$109$			205.403i			0.0172883i	0.999963	+	0.00864416i	$0.00275156\pi$
$109$			205.403i			0.0172883i	−0.999963	+	0.00864416i	$0.997248\pi$
$110$			8512.63i			0.703524i
$111$		0			0
$112$	−5509.83			−0.439240
$113$		−	4276.02i		−	0.334875i	−0.985883	−	0.167438i	$-0.946451\pi$
$113$		−	4276.02i		−	0.334875i	0.985883	−	0.167438i	$-0.0535492\pi$
$114$		0			0
$115$	7057.13			0.533620
$116$		−	9048.40i		−	0.672443i
$117$		0			0
$118$	13925.3			1.00009
$119$	−21046.3			−1.48622
$120$		0			0
$121$	−1475.00			−0.100744
$122$			4459.61i			0.299624i
$123$		0			0
$124$		−	8300.73i		−	0.539850i
$125$	−14741.2			−0.943439
$126$		0			0
$127$			4501.00i			0.279063i	0.990218	+	0.139531i	$0.0445596\pi$
$127$			4501.00i			0.279063i	−0.990218	+	0.139531i	$0.955440\pi$
$128$			1448.15i			0.0883883i
$129$		0			0
$130$	−17200.5			−1.01778
$131$	−27606.3			−1.60866			−0.804332	−	0.594180i	$-0.797477\pi$
$131$	−27606.3			−1.60866			−0.804332	+	0.594180i	$0.797477\pi$
$132$		0			0
$133$	−21837.4	−	22113.9i	−1.23452	−	1.25015i
$134$	19800.2			1.10271
$135$		0			0
$136$			5531.64i			0.299072i
$137$	−8250.29			−0.439570			−0.219785	−	0.975548i	$-0.570536\pi$
$137$	−8250.29			−0.439570			−0.219785	+	0.975548i	$0.570536\pi$
$138$		0			0
$139$	27435.0			1.41996			0.709978	−	0.704224i	$-0.248705\pi$
$139$	27435.0			1.41996			0.709978	+	0.704224i	$0.248705\pi$
$140$	18065.1			0.921688
$141$		0			0
$142$	11600.1			0.575289
$143$			26603.0i			1.30095i
$144$		0			0
$145$			29667.0i			1.41104i
$146$		−	19850.0i		−	0.931223i
$147$		0			0
$148$		−	2422.52i		−	0.110597i
$149$	40247.3			1.81286			0.906430	−	0.422355i	$-0.138796\pi$
$149$	40247.3			1.81286			0.906430	+	0.422355i	$0.138796\pi$
$150$		0			0
$151$			16932.1i			0.742605i	0.928512	+	0.371303i	$0.121089\pi$
$151$			16932.1i			0.742605i	−0.928512	+	0.371303i	$0.878911\pi$
$152$	−5812.22	+	5739.56i	−0.251568	+	0.248423i
$153$		0			0
$154$		−	27940.2i		−	1.17812i
$155$			27215.6i			1.13280i
$156$		0			0
$157$	5684.97			0.230637			0.115319	−	0.993329i	$-0.463211\pi$
$157$	5684.97			0.230637			0.115319	+	0.993329i	$0.463211\pi$
$158$	3575.14			0.143212
$159$		0			0
$160$		−	4748.07i		−	0.185471i
$161$	−23162.9			−0.893597
$162$		0			0
$163$	−30550.6			−1.14986			−0.574929	−	0.818203i	$-0.694970\pi$
$163$	−30550.6			−1.14986			−0.574929	+	0.818203i	$0.694970\pi$
$164$		−	14158.5i		−	0.526417i
$165$		0			0
$166$			2885.26i			0.104705i
$167$			28241.5i			1.01264i	0.862345	+	0.506320i	$0.168995\pi$
$167$			28241.5i			1.01264i	−0.862345	+	0.506320i	$0.831005\pi$
$168$		0			0
$169$	−25192.7			−0.882066
$170$		−	18136.6i		−	0.627564i
$171$		0			0
$172$	18532.4			0.626435
$173$			35673.0i			1.19192i	0.803013	+	0.595961i	$0.203229\pi$
$173$			35673.0i			1.19192i	−0.803013	+	0.595961i	$0.796771\pi$
$174$		0			0
$175$	−5423.14			−0.177082
$176$	−7343.56			−0.237073
$177$		0			0
$178$	10596.1			0.334432
$179$		−	34371.5i		−	1.07273i	−0.843985	−	0.536367i	$-0.819796\pi$
$179$		−	34371.5i		−	1.07273i	0.843985	−	0.536367i	$-0.180204\pi$
$180$		0			0
$181$			29195.1i			0.891153i	0.895244	+	0.445577i	$0.147001\pi$
$181$			29195.1i			0.891153i	−0.895244	+	0.445577i	$0.852999\pi$
$182$	56455.6			1.70437
$183$		0			0
$184$			6087.95i			0.179819i
$185$			7942.72i			0.232074i
$186$		0			0
$187$	−28050.8			−0.802163
$188$	−6686.05			−0.189171
$189$		0			0
$190$	19056.5	−	18818.3i	0.527882	−	0.521282i
$191$	−13498.2			−0.370008			−0.185004	−	0.982738i	$-0.559230\pi$
$191$	−13498.2			−0.370008			−0.185004	+	0.982738i	$0.559230\pi$
$192$		0			0
$193$		−	20819.7i		−	0.558933i	−0.960155	−	0.279467i	$-0.909842\pi$
$193$		−	20819.7i		−	0.558933i	0.960155	−	0.279467i	$-0.0901577\pi$
$194$	−36885.4			−0.980056
$195$		0			0
$196$	−40085.3			−1.04345
$197$	−34233.5			−0.882101			−0.441051	−	0.897482i	$-0.645394\pi$
$197$	−34233.5			−0.882101			−0.441051	+	0.897482i	$0.645394\pi$
$198$		0			0
$199$	−55504.1			−1.40158			−0.700792	−	0.713366i	$-0.747170\pi$
$199$	−55504.1			−1.40158			−0.700792	+	0.713366i	$0.747170\pi$
$200$			1425.37i			0.0356343i
$201$		0			0
$202$		−	13451.5i		−	0.329662i
$203$		−	97373.2i		−	2.36291i
$204$		0			0
$205$			46421.6i			1.10462i
$206$	−24685.9			−0.581721
$207$		0			0
$208$		−	14838.3i		−	0.342971i
$209$	−29105.2	−	29473.6i	−0.666312	−	0.674747i
$210$		0			0
$211$		−	77552.5i		−	1.74193i	−0.491345	−	0.870965i	$-0.663495\pi$
$211$		−	77552.5i		−	1.74193i	0.491345	−	0.870965i	$-0.336505\pi$
$212$		−	5249.66i		−	0.116805i
$213$		0			0
$214$	−47757.6			−1.04283
$215$	−60762.4			−1.31449
$216$		0			0
$217$		−	89327.3i		−	1.89699i
$218$	580.966			0.0122247
$219$		0			0
$220$	24077.4			0.497466
$221$		−	56679.1i		−	1.16048i
$222$		0			0
$223$			30754.3i			0.618438i	0.950991	+	0.309219i	$0.100068\pi$
$223$			30754.3i			0.618438i	−0.950991	+	0.309219i	$0.899932\pi$
$224$			15584.1i			0.310590i
$225$		0			0
$226$	−12094.4			−0.236792
$227$			81299.3i			1.57774i	0.614561	+	0.788869i	$0.289333\pi$
$227$			81299.3i			1.57774i	−0.614561	+	0.788869i	$0.710667\pi$
$228$		0			0
$229$	−93187.0			−1.77699			−0.888494	−	0.458889i	$-0.848248\pi$
$229$	−93187.0			−1.77699			−0.888494	+	0.458889i	$0.848248\pi$
$230$		−	19960.6i		−	0.377326i
$231$		0			0
$232$	−25592.7			−0.475489
$233$	3379.09			0.0622426			0.0311213	−	0.999516i	$-0.490092\pi$
$233$	3379.09			0.0622426			0.0311213	+	0.999516i	$0.490092\pi$
$234$		0			0
$235$	21921.6			0.396950
$236$		−	39386.6i		−	0.707171i
$237$		0			0
$238$			59528.0i			1.05091i
$239$	−35917.8			−0.628802			−0.314401	−	0.949290i	$-0.601804\pi$
$239$	−35917.8			−0.628802			−0.314401	+	0.949290i	$0.601804\pi$
$240$		0			0
$241$		−	57352.1i		−	0.987451i	−0.869618	−	0.493726i	$-0.835635\pi$
$241$		−	57352.1i		−	0.987451i	0.869618	−	0.493726i	$-0.164365\pi$
$242$			4171.92i			0.0712370i
$243$		0			0
$244$	12613.7			0.211866
$245$	131428.			2.18955
$246$		0			0
$247$	59554.0	−	58809.5i	0.976151	−	0.963948i
$248$	−23478.0			−0.381731
$249$		0			0
$250$			41694.5i			0.667112i
$251$	52486.8			0.833110			0.416555	−	0.909110i	$-0.363237\pi$
$251$	52486.8			0.833110			0.416555	+	0.909110i	$0.363237\pi$
$252$		0			0
$253$	−30871.9			−0.482305
$254$	12730.8			0.197327
$255$		0			0
$256$	4096.00			0.0625000
$257$			85222.5i			1.29029i	0.764059	+	0.645146i	$0.223204\pi$
$257$			85222.5i			1.29029i	−0.764059	+	0.645146i	$0.776796\pi$
$258$		0			0
$259$		−	26069.6i		−	0.388629i
$260$			48650.4i			0.719680i
$261$		0			0
$262$			78082.4i			1.13750i
$263$	27539.5			0.398149			0.199074	−	0.979984i	$-0.436206\pi$
$263$	27539.5			0.398149			0.199074	+	0.979984i	$0.436206\pi$
$264$		0			0
$265$			17212.1i			0.245099i
$266$	−62547.5	+	61765.5i	−0.883988	+	0.872937i
$267$		0			0
$268$		−	56003.5i		−	0.779732i
$269$			58880.0i			0.813697i	0.913496	+	0.406849i	$0.133372\pi$
$269$			58880.0i			0.813697i	−0.913496	+	0.406849i	$0.866628\pi$
$270$		0			0
$271$	99955.1			1.36103			0.680513	−	0.732736i	$-0.261757\pi$
$271$	99955.1			1.36103			0.680513	+	0.732736i	$0.261757\pi$
$272$	15645.8			0.211476
$273$		0			0
$274$			23335.4i			0.310823i
$275$	−7228.03			−0.0955772
$276$		0			0
$277$	66298.2			0.864056			0.432028	−	0.901860i	$-0.357798\pi$
$277$	66298.2			0.864056			0.432028	+	0.901860i	$0.357798\pi$
$278$		−	77597.8i		−	1.00406i
$279$		0			0
$280$		−	51095.8i		−	0.651732i
$281$			55021.9i			0.696824i	0.937341	+	0.348412i	$0.113279\pi$
$281$			55021.9i			0.696824i	−0.937341	+	0.348412i	$0.886721\pi$
$282$		0			0
$283$	10575.2			0.132044			0.0660219	−	0.997818i	$-0.478969\pi$
$283$	10575.2			0.132044			0.0660219	+	0.997818i	$0.478969\pi$
$284$		−	32810.1i		−	0.406790i
$285$		0			0
$286$	75244.7			0.919907
$287$		−	152365.i		−	1.84979i
$288$		0			0
$289$	−23757.3			−0.284447
$290$	83911.0			0.997752
$291$		0			0
$292$	−56144.2			−0.658474
$293$		−	9449.70i		−	0.110074i	−0.998484	−	0.0550368i	$-0.982472\pi$
$293$		−	9449.70i		−	0.110074i	0.998484	−	0.0550368i	$-0.0175276\pi$
$294$		0			0
$295$			129137.i			1.48391i
$296$	−6851.92			−0.0782040
$297$		0			0
$298$		−	113837.i		−	1.28189i
$299$		−	62379.2i		−	0.697746i
$300$		0			0
$301$	199435.			2.20124
$302$	47891.3			0.525101
$303$		0			0
$304$	16233.9	+	16439.4i	0.175661	+	0.177885i
$305$	−41356.5			−0.444574
$306$		0			0
$307$			112622.i			1.19494i	0.801890	+	0.597472i	$0.203828\pi$
$307$			112622.i			1.19494i	−0.801890	+	0.597472i	$0.796172\pi$
$308$	−79026.9			−0.833054
$309$		0			0
$310$	76977.4			0.801013
$311$	75878.9			0.784513			0.392257	−	0.919856i	$-0.371695\pi$
$311$	75878.9			0.784513			0.392257	+	0.919856i	$0.371695\pi$
$312$		0			0
$313$	−110960.			−1.13260			−0.566301	−	0.824199i	$-0.691626\pi$
$313$	−110960.			−1.13260			−0.566301	+	0.824199i	$0.691626\pi$
$314$		−	16079.5i		−	0.163085i
$315$		0			0
$316$		−	10112.0i		−	0.101266i
$317$		−	63970.1i		−	0.636589i	−0.947992	−	0.318294i	$-0.896890\pi$
$317$		−	63970.1i		−	0.636589i	0.947992	−	0.318294i	$-0.103110\pi$
$318$		0			0
$319$		−	129780.i		−	1.27534i
$320$	−13429.6			−0.131148
$321$		0			0
$322$			65514.7i			0.631869i
$323$	62010.0	+	62795.0i	0.594370	+	0.601894i
$324$		0			0
$325$		−	14604.8i		−	0.138271i
$326$			86410.0i			0.813072i
$327$		0			0
$328$	−40046.4			−0.372233
$329$	−71951.1			−0.664731
$330$		0			0
$331$			79462.6i			0.725282i	0.931929	+	0.362641i	$0.118125\pi$
$331$			79462.6i			0.725282i	−0.931929	+	0.362641i	$0.881875\pi$
$332$	8160.74			0.0740377
$333$		0			0
$334$	79879.1			0.716045
$335$			183619.i			1.63617i
$336$		0			0
$337$			2779.80i			0.0244767i	0.999925	+	0.0122384i	$0.00389569\pi$
$337$			2779.80i			0.0244767i	−0.999925	+	0.0122384i	$0.996104\pi$
$338$			71255.7i			0.623715i
$339$		0			0
$340$	−51298.0			−0.443755
$341$		−	119057.i		−	1.02387i
$342$		0			0
$343$	−224669.			−1.90965
$344$		−	52417.7i		−	0.442956i
$345$		0			0
$346$	100899.			0.842816
$347$	15728.1			0.130622			0.0653110	−	0.997865i	$-0.479196\pi$
$347$	15728.1			0.130622			0.0653110	+	0.997865i	$0.479196\pi$
$348$		0			0
$349$	−41782.5			−0.343039			−0.171519	−	0.985181i	$-0.554868\pi$
$349$	−41782.5			−0.343039			−0.171519	+	0.985181i	$0.554868\pi$
$350$			15339.0i			0.125216i
$351$		0			0
$352$			20770.7i			0.167636i
$353$	−39292.1			−0.315323			−0.157661	−	0.987493i	$-0.550395\pi$
$353$	−39292.1			−0.315323			−0.157661	+	0.987493i	$0.550395\pi$
$354$		0			0
$355$			107575.i			0.853597i
$356$		−	29970.4i		−	0.236479i
$357$		0			0
$358$	−97217.2			−0.758537
$359$	−160964.			−1.24894			−0.624468	−	0.781050i	$-0.714684\pi$
$359$	−160964.			−1.24894			−0.624468	+	0.781050i	$0.714684\pi$
$360$		0			0
$361$	−1639.37	+	130311.i	−0.0125794	+	0.999921i
$362$	82576.1			0.630141
$363$		0			0
$364$		−	159681.i		−	1.20517i
$365$	184080.			1.38172
$366$		0			0
$367$	19835.4			0.147268			0.0736340	−	0.997285i	$-0.476540\pi$
$367$	19835.4			0.147268			0.0736340	+	0.997285i	$0.476540\pi$
$368$	17219.3			0.127151
$369$		0			0
$370$	22465.4			0.164101
$371$		−	56493.6i		−	0.410442i
$372$		0			0
$373$		−	14463.5i		−	0.103958i	−0.998648	−	0.0519788i	$-0.983447\pi$
$373$		−	14463.5i		−	0.103958i	0.998648	−	0.0519788i	$-0.0165528\pi$
$374$			79339.7i			0.567215i
$375$		0			0
$376$			18911.0i			0.133764i
$377$	262232.			1.84503
$378$		0			0
$379$		−	25932.7i		−	0.180539i	−0.995917	−	0.0902693i	$-0.971227\pi$
$379$		−	25932.7i		−	0.180539i	0.995917	−	0.0902693i	$-0.0287728\pi$
$380$	−53226.2	−	53900.0i	−0.368602	−	0.373269i
$381$		0			0
$382$			38178.8i			0.261635i
$383$			9634.98i			0.0656831i	0.999461	+	0.0328415i	$0.0104557\pi$
$383$			9634.98i			0.0656831i	−0.999461	+	0.0328415i	$0.989544\pi$
$384$		0			0
$385$	259106.			1.74806
$386$	−58887.0			−0.395226
$387$		0			0
$388$			104328.i			0.693005i
$389$	−101890.			−0.673335			−0.336668	−	0.941624i	$-0.609300\pi$
$389$	−101890.			−0.673335			−0.336668	+	0.941624i	$0.609300\pi$
$390$		0			0
$391$	65774.0			0.430230
$392$			113378.i			0.737833i
$393$		0			0
$394$			96826.9i			0.623740i
$395$			33154.3i			0.212494i
$396$		0			0
$397$	100161.			0.635503			0.317752	−	0.948174i	$-0.397072\pi$
$397$	100161.			0.635503			0.317752	+	0.948174i	$0.397072\pi$
$398$			156989.i			0.991069i
$399$		0			0
$400$	4031.56			0.0251972
$401$		−	16696.5i		−	0.103833i	−0.998651	−	0.0519165i	$-0.983467\pi$
$401$		−	16696.5i		−	0.103833i	0.998651	−	0.0519165i	$-0.0165330\pi$
$402$		0			0
$403$	240564.			1.48122
$404$	−38046.7			−0.233106
$405$		0			0
$406$	−275413.			−1.67083
$407$		−	34745.9i		−	0.209756i
$408$		0			0
$409$			1173.94i			0.00701780i	0.999994	+	0.00350890i	$0.00111692\pi$
$409$			1173.94i			0.00701780i	−0.999994	+	0.00350890i	$0.998883\pi$
$410$	131300.			0.781083
$411$		0			0
$412$			69822.3i			0.411339i
$413$		−	423854.i		−	2.48494i
$414$		0			0
$415$	−26756.6			−0.155359
$416$	−41969.0			−0.242517
$417$		0			0
$418$	−83364.0	+	82321.9i	−0.477118	+	0.471154i
$419$	209961.			1.19595			0.597973	−	0.801517i	$-0.295973\pi$
$419$	209961.			1.19595			0.597973	+	0.801517i	$0.295973\pi$
$420$		0			0
$421$		−	249117.i		−	1.40552i	−0.711425	−	0.702762i	$-0.751950\pi$
$421$		−	249117.i		−	1.40552i	0.711425	−	0.702762i	$-0.248050\pi$
$422$	−219351.			−1.23173
$423$		0			0
$424$	−14848.3			−0.0825933
$425$	15399.7			0.0852577
$426$		0			0
$427$	135741.			0.744481
$428$			135079.i			0.737395i
$429$		0			0
$430$			171862.i			0.929486i
$431$			126947.i			0.683387i	0.939811	+	0.341694i	$0.111001\pi$
$431$			126947.i			0.683387i	−0.939811	+	0.341694i	$0.888999\pi$
$432$		0			0
$433$			327343.i			1.74593i	0.487783	+	0.872965i	$0.337806\pi$
$433$			327343.i			1.74593i	−0.487783	+	0.872965i	$0.662194\pi$
$434$	−252656.			−1.34137
$435$		0			0
$436$		−	1643.22i		−	0.00864416i
$437$	68246.3	+	69110.3i	0.357368	+	0.361893i
$438$		0			0
$439$			323562.i			1.67891i	0.543425	+	0.839457i	$0.317127\pi$
$439$			323562.i			1.67891i	−0.543425	+	0.839457i	$0.682873\pi$
$440$		−	68101.1i		−	0.351762i
$441$		0			0
$442$	−160313.			−0.820584
$443$	−188338.			−0.959689			−0.479845	−	0.877354i	$-0.659307\pi$
$443$	−188338.			−0.959689			−0.479845	+	0.877354i	$0.659307\pi$
$444$		0			0
$445$			98264.0i			0.496220i
$446$	86986.3			0.437302
$447$		0			0
$448$	44078.6			0.219620
$449$			31205.6i			0.154789i	0.997001	+	0.0773945i	$0.0246601\pi$
$449$			31205.6i			0.154789i	−0.997001	+	0.0773945i	$0.975340\pi$
$450$		0			0
$451$		−	203074.i		−	0.998394i
$452$			34208.2i			0.167438i
$453$		0			0
$454$	229949.			1.11563
$455$			523545.i			2.52890i
$456$		0			0
$457$	89688.2			0.429440			0.214720	−	0.976676i	$-0.431116\pi$
$457$	89688.2			0.429440			0.214720	+	0.976676i	$0.431116\pi$
$458$			263573.i			1.25652i
$459$		0			0
$460$	−56457.0			−0.266810
$461$	71209.7			0.335071			0.167536	−	0.985866i	$-0.446419\pi$
$461$	71209.7			0.335071			0.167536	+	0.985866i	$0.446419\pi$
$462$		0			0
$463$	−96472.6			−0.450030			−0.225015	−	0.974355i	$-0.572243\pi$
$463$	−96472.6			−0.450030			−0.225015	+	0.974355i	$0.572243\pi$
$464$			72387.2i			0.336222i
$465$		0			0
$466$		−	9557.51i		−	0.0440122i
$467$	−337202.			−1.54617			−0.773084	−	0.634304i	$-0.781287\pi$
$467$	−337202.			−1.54617			−0.773084	+	0.634304i	$0.781287\pi$
$468$		0			0
$469$		−	602675.i		−	2.73992i
$470$		−	62003.6i		−	0.280686i
$471$		0			0
$472$	−111402.			−0.500046
$473$	265809.			1.18808
$474$		0			0
$475$	15978.5	+	16180.8i	0.0708189	+	0.0717154i
$476$	168371.			0.743109
$477$		0			0
$478$			101591.i			0.444630i
$479$	−126684.			−0.552143			−0.276071	−	0.961137i	$-0.589033\pi$
$479$	−126684.			−0.552143			−0.276071	+	0.961137i	$0.589033\pi$
$480$		0			0
$481$	70207.1			0.303453
$482$	−162216.			−0.698233
$483$		0			0
$484$	11800.0			0.0503722
$485$		−	342059.i		−	1.45418i
$486$		0			0
$487$		−	233861.i		−	0.986053i	−0.870014	−	0.493027i	$-0.835891\pi$
$487$		−	233861.i		−	0.986053i	0.870014	−	0.493027i	$-0.164109\pi$
$488$		−	35676.9i		−	0.149812i
$489$		0			0
$490$		−	371734.i		−	1.54825i
$491$	423929.			1.75845			0.879226	−	0.476406i	$-0.158061\pi$
$491$	423929.			1.75845			0.879226	+	0.476406i	$0.158061\pi$
$492$		0			0
$493$			276503.i			1.13764i
$494$	−166338.	−	168444.i	−0.681614	−	0.690243i
$495$		0			0
$496$			66405.8i			0.269925i
$497$		−	353082.i		−	1.42943i
$498$		0			0
$499$	−175944.			−0.706600			−0.353300	−	0.935510i	$-0.614941\pi$
$499$	−175944.			−0.706600			−0.353300	+	0.935510i	$0.614941\pi$
$500$	117930.			0.471719
$501$		0			0
$502$		−	148455.i		−	0.589098i
$503$	−108798.			−0.430017			−0.215009	−	0.976612i	$-0.568978\pi$
$503$	−108798.			−0.430017			−0.215009	+	0.976612i	$0.568978\pi$
$504$		0			0
$505$	124744.			0.489143
$506$			87318.8i			0.341041i
$507$		0			0
$508$		−	36008.0i		−	0.139531i
$509$			72826.1i			0.281094i	0.990074	+	0.140547i	$0.0448861\pi$
$509$			72826.1i			0.281094i	−0.990074	+	0.140547i	$0.955114\pi$
$510$		0			0
$511$	−604188.			−2.31383
$512$		−	11585.2i		−	0.0441942i
$513$		0			0
$514$	241046.			0.912375
$515$		−	228927.i		−	0.863141i
$516$		0			0
$517$	−95897.4			−0.358778
$518$	−73736.1			−0.274802
$519$		0			0
$520$	137604.			0.508891
$521$			93748.1i			0.345372i	0.984977	+	0.172686i	$0.0552446\pi$
$521$			93748.1i			0.345372i	−0.984977	+	0.172686i	$0.944755\pi$
$522$		0			0
$523$		−	87245.8i		−	0.318964i	−0.987201	−	0.159482i	$-0.949018\pi$
$523$		−	87245.8i		−	0.318964i	0.987201	−	0.159482i	$-0.0509823\pi$
$524$	220850.			0.804332
$525$		0			0
$526$		−	77893.6i		−	0.281534i
$527$			253656.i			0.913321i
$528$		0			0
$529$	−207452.			−0.741322
$530$	48683.1			0.173311
$531$		0			0
$532$	174699.	+	176911.i	0.617260	+	0.625074i
$533$	410329.			1.44437
$534$		0			0
$535$		−	442884.i		−	1.54733i
$536$	−158402.			−0.551354
$537$		0			0
$538$	166538.			0.575371
$539$	−574940.			−1.97900
$540$		0			0
$541$	374060.			1.27805			0.639024	−	0.769187i	$-0.279338\pi$
$541$	374060.			1.27805			0.639024	+	0.769187i	$0.279338\pi$
$542$		−	282716.i		−	0.962391i
$543$		0			0
$544$		−	44253.1i		−	0.149536i
$545$			5387.63i			0.0181386i
$546$		0			0
$547$		−	160411.i		−	0.536118i	−0.963402	−	0.268059i	$-0.913618\pi$
$547$		−	160411.i		−	0.536118i	0.963402	−	0.268059i	$-0.0863823\pi$
$548$	66002.3			0.219785
$549$		0			0
$550$			20443.9i			0.0675833i
$551$	−290528.	+	286896.i	−0.956941	+	0.944978i
$552$		0			0
$553$		−	108819.i		−	0.355841i
$554$		−	187520.i		−	0.610980i
$555$		0			0
$556$	−219480.			−0.709978
$557$	−460605.			−1.48463			−0.742314	−	0.670052i	$-0.766272\pi$
$557$	−460605.			−1.48463			−0.742314	+	0.670052i	$0.766272\pi$
$558$		0			0
$559$			537090.i			1.71879i
$560$	−144521.			−0.460844
$561$		0			0
$562$	155626.			0.492729
$563$		−	73084.0i		−	0.230571i	−0.993332	−	0.115286i	$-0.963222\pi$
$563$		−	73084.0i		−	0.230571i	0.993332	−	0.115286i	$-0.0367783\pi$
$564$		0			0
$565$		−	112158.i		−	0.351346i
$566$		−	29911.3i		−	0.0933690i
$567$		0			0
$568$	−92801.0			−0.287644
$569$		−	609592.i		−	1.88285i	−0.337227	−	0.941424i	$-0.609489\pi$
$569$		−	609592.i		−	1.88285i	0.337227	−	0.941424i	$-0.390511\pi$
$570$		0			0
$571$	−11273.0			−0.0345755			−0.0172877	−	0.999851i	$-0.505503\pi$
$571$	−11273.0			−0.0345755			−0.0172877	+	0.999851i	$0.505503\pi$
$572$		−	212824.i		−	0.650473i
$573$		0			0
$574$	−430954.			−1.30800
$575$	16948.4			0.0512617
$576$		0			0
$577$	−186962.			−0.561567			−0.280783	−	0.959771i	$-0.590594\pi$
$577$	−186962.			−0.561567			−0.280783	+	0.959771i	$0.590594\pi$
$578$			67195.9i			0.201135i
$579$		0			0
$580$		−	237336.i		−	0.705518i
$581$	87820.8			0.260163
$582$		0			0
$583$		−	75295.4i		−	0.221529i
$584$			158800.i			0.465612i
$585$		0			0
$586$	−26727.8			−0.0778337
$587$	258778.			0.751020			0.375510	−	0.926818i	$-0.377468\pi$
$587$	258778.			0.751020			0.375510	+	0.926818i	$0.377468\pi$
$588$		0			0
$589$	−266522.	+	263190.i	−0.768250	+	0.758645i
$590$	365255.			1.04928
$591$		0			0
$592$			19380.2i			0.0552986i
$593$	603214.			1.71539			0.857693	−	0.514162i	$-0.171897\pi$
$593$	603214.			1.71539			0.857693	+	0.514162i	$0.171897\pi$
$594$		0			0
$595$	−552037.			−1.55932
$596$	−321979.			−0.906430
$597$		0			0
$598$	−176435.			−0.493381
$599$			32971.2i			0.0918927i	0.998944	+	0.0459464i	$0.0146303\pi$
$599$			32971.2i			0.0918927i	−0.998944	+	0.0459464i	$0.985370\pi$
$600$		0			0
$601$		−	61000.3i		−	0.168882i	−0.996428	−	0.0844410i	$-0.973090\pi$
$601$		−	61000.3i		−	0.168882i	0.996428	−	0.0844410i	$-0.0269104\pi$
$602$		−	564086.i		−	1.55651i
$603$		0			0
$604$		−	135457.i		−	0.371303i
$605$	−38688.6			−0.105699
$606$		0			0
$607$		−	252238.i		−	0.684594i	−0.939592	−	0.342297i	$-0.888795\pi$
$607$		−	252238.i		−	0.684594i	0.939592	−	0.342297i	$-0.111205\pi$
$608$	46497.7	−	45916.4i	0.125784	−	0.124211i
$609$		0			0
$610$			116974.i			0.314361i
$611$		−	193769.i		−	0.519041i
$612$		0			0
$613$	384604.			1.02351			0.511757	−	0.859131i	$-0.328995\pi$
$613$	384604.			1.02351			0.511757	+	0.859131i	$0.328995\pi$
$614$	318544.			0.844953
$615$		0			0
$616$			223522.i			0.589058i
$617$	−89917.4			−0.236197			−0.118098	−	0.993002i	$-0.537680\pi$
$617$	−89917.4			−0.236197			−0.118098	+	0.993002i	$0.537680\pi$
$618$		0			0
$619$	−133253.			−0.347772			−0.173886	−	0.984766i	$-0.555632\pi$
$619$	−133253.			−0.347772			−0.173886	+	0.984766i	$0.555632\pi$
$620$		−	217725.i		−	0.566402i
$621$		0			0
$622$		−	214618.i		−	0.554735i
$623$		−	322523.i		−	0.830968i
$624$		0			0
$625$	−426028.			−1.09063
$626$			313842.i			0.800870i
$627$		0			0
$628$	−45479.8			−0.115319
$629$			74028.0i			0.187109i
$630$		0			0
$631$	250184.			0.628348			0.314174	−	0.949365i	$-0.398273\pi$
$631$	250184.			0.628348			0.314174	+	0.949365i	$0.398273\pi$
$632$	−28601.1			−0.0716059
$633$		0			0
$634$	−180935.			−0.450136
$635$			118060.i			0.292788i
$636$		0			0
$637$		−	1.16171e6i		−	2.86299i
$638$	−367074.			−0.901804
$639$		0			0
$640$			37984.6i			0.0927357i
$641$			601164.i			1.46311i	0.681782	+	0.731555i	$0.261205\pi$
$641$			601164.i			1.46311i	−0.681782	+	0.731555i	$0.738795\pi$
$642$		0			0
$643$	379057.			0.916816			0.458408	−	0.888742i	$-0.348420\pi$
$643$	379057.			0.916816			0.458408	+	0.888742i	$0.348420\pi$
$644$	185304.			0.446799
$645$		0			0
$646$	177611.	−	175391.i	0.425604	−	0.420283i
$647$	−245678.			−0.586892			−0.293446	−	0.955976i	$-0.594802\pi$
$647$	−245678.			−0.586892			−0.293446	+	0.955976i	$0.594802\pi$
$648$		0			0
$649$		−	564918.i		−	1.34121i
$650$	−41308.7			−0.0977722
$651$		0			0
$652$	244405.			0.574929
$653$	15905.7			0.0373016			0.0186508	−	0.999826i	$-0.494063\pi$
$653$	15905.7			0.0373016			0.0186508	+	0.999826i	$0.494063\pi$
$654$		0			0
$655$	−724102.			−1.68779
$656$			113268.i			0.263209i
$657$		0			0
$658$			203509.i			0.470036i
$659$			481053.i			1.10770i	0.832617	+	0.553850i	$0.186842\pi$
$659$			481053.i			1.10770i	−0.832617	+	0.553850i	$0.813158\pi$
$660$		0			0
$661$			468775.i			1.07290i	0.843931	+	0.536452i	$0.180236\pi$
$661$			468775.i			1.07290i	−0.843931	+	0.536452i	$0.819764\pi$
$662$	224754.			0.512852
$663$		0			0
$664$		−	23082.0i		−	0.0523526i
$665$	−572787.	−	580038.i	−1.29524	−	1.31164i
$666$		0			0
$667$			304311.i			0.684015i
$668$		−	225932.i		−	0.506320i
$669$		0			0
$670$	519352.			1.15694
$671$	180917.			0.401822
$672$		0			0
$673$		−	229158.i		−	0.505947i	−0.967473	−	0.252973i	$-0.918592\pi$
$673$		−	229158.i		−	0.505947i	0.967473	−	0.252973i	$-0.0814085\pi$
$674$	7862.46			0.0173077
$675$		0			0
$676$	201541.			0.441033
$677$		−	38248.1i		−	0.0834512i	−0.999129	−	0.0417256i	$-0.986714\pi$
$677$		−	38248.1i		−	0.0834512i	0.999129	−	0.0417256i	$-0.0132855\pi$
$678$		0			0
$679$			1.12271e6i			2.43516i
$680$			145093.i			0.313782i
$681$		0			0
$682$	−336743.			−0.723985
$683$			31694.5i			0.0679426i	0.999423	+	0.0339713i	$0.0108155\pi$
$683$			31694.5i			0.0679426i	−0.999423	+	0.0339713i	$0.989185\pi$
$684$		0			0
$685$	−216402.			−0.461190
$686$			635459.i			1.35033i
$687$		0			0
$688$	−148260.			−0.313217
$689$	152141.			0.320484
$690$		0			0
$691$	140291.			0.293815			0.146907	−	0.989150i	$-0.453068\pi$
$691$	140291.			0.293815			0.146907	+	0.989150i	$0.453068\pi$
$692$		−	285384.i		−	0.595961i
$693$		0			0
$694$		−	44485.7i		−	0.0923637i
$695$	719609.			1.48980
$696$		0			0
$697$			432660.i			0.890597i
$698$			118179.i			0.242565i
$699$		0			0
$700$	43385.1			0.0885410
$701$	−550797.			−1.12087			−0.560436	−	0.828198i	$-0.689366\pi$
$701$	−550797.			−1.12087			−0.560436	+	0.828198i	$0.689366\pi$
$702$		0			0
$703$	−77782.9	+	76810.5i	−0.157389	+	0.155421i
$704$	58748.5			0.118536
$705$		0			0
$706$			111135.i			0.222967i
$707$	−409435.			−0.819117
$708$		0			0
$709$	725719.			1.44370			0.721848	−	0.692051i	$-0.243293\pi$
$709$	725719.			1.44370			0.721848	+	0.692051i	$0.243293\pi$
$710$	304267.			0.603584
$711$		0			0
$712$	−84769.1			−0.167216
$713$			279166.i			0.549140i
$714$		0			0
$715$			697787.i			1.36493i
$716$			274972.i			0.536367i
$717$		0			0
$718$			455275.i			0.883131i
$719$	−184282.			−0.356472			−0.178236	−	0.983988i	$-0.557039\pi$
$719$	−184282.			−0.356472			−0.178236	+	0.983988i	$0.557039\pi$
$720$		0			0
$721$			751384.i			1.44541i
$722$	368574.	+	4636.83i	0.707051	+	0.00889501i
$723$		0			0
$724$		−	233561.i		−	0.445577i
$725$			71248.3i			0.135550i
$726$		0			0
$727$	−30302.5			−0.0573336			−0.0286668	−	0.999589i	$-0.509126\pi$
$727$	−30302.5			−0.0573336			−0.0286668	+	0.999589i	$0.509126\pi$
$728$	−451645.			−0.852186
$729$		0			0
$730$		−	520657.i		−	0.977025i
$731$	−566319.			−1.05981
$732$		0			0
$733$	210592.			0.391954			0.195977	−	0.980609i	$-0.437212\pi$
$733$	210592.			0.391954			0.195977	+	0.980609i	$0.437212\pi$
$734$		−	56102.9i		−	0.104134i
$735$		0			0
$736$		−	48703.6i		−	0.0899094i
$737$		−	803252.i		−	1.47883i
$738$		0			0
$739$	−257029.			−0.470645			−0.235323	−	0.971917i	$-0.575615\pi$
$739$	−257029.			−0.470645			−0.235323	+	0.971917i	$0.575615\pi$
$740$		−	63541.8i		−	0.116037i
$741$		0			0
$742$	−159788.			−0.290226
$743$			459249.i			0.831899i	0.909388	+	0.415949i	$0.136551\pi$
$743$			459249.i			0.831899i	−0.909388	+	0.415949i	$0.863449\pi$
$744$		0			0
$745$	1.05567e6			1.90203
$746$	−40909.0			−0.0735091
$747$		0			0
$748$	224407.			0.401081
$749$			1.45364e6i			2.59115i
$750$		0			0
$751$		−	705805.i		−	1.25142i	−0.780054	−	0.625712i	$-0.784808\pi$
$751$		−	705805.i		−	1.25142i	0.780054	−	0.625712i	$-0.215192\pi$
$752$	53488.4			0.0945854
$753$		0			0
$754$		−	741704.i		−	1.30463i
$755$			444124.i			0.779130i
$756$		0			0
$757$	128249.			0.223801			0.111901	−	0.993719i	$-0.464306\pi$
$757$	128249.			0.223801			0.111901	+	0.993719i	$0.464306\pi$
$758$	−73348.8			−0.127660
$759$		0			0
$760$	−152452.	+	150546.i	−0.263941	+	0.260641i
$761$	−779962.			−1.34680			−0.673402	−	0.739277i	$-0.735167\pi$
$761$	−779962.			−1.34680			−0.673402	+	0.739277i	$0.735167\pi$
$762$		0			0
$763$		−	17683.3i		−	0.0303749i
$764$	107986.			0.185004
$765$		0			0
$766$	27251.8			0.0464449
$767$	1.14147e6			1.94031
$768$		0			0
$769$	−50722.8			−0.0857730			−0.0428865	−	0.999080i	$-0.513655\pi$
$769$	−50722.8			−0.0857730			−0.0428865	+	0.999080i	$0.513655\pi$
$770$		−	732861.i		−	1.23606i
$771$		0			0
$772$			166558.i			0.279467i
$773$		−	942086.i		−	1.57664i	−0.615267	−	0.788318i	$-0.710952\pi$
$773$		−	942086.i		−	1.57664i	0.615267	−	0.788318i	$-0.289048\pi$
$774$		0			0
$775$			65361.0i			0.108822i
$776$	295083.			0.490028
$777$		0			0
$778$			288188.i			0.476120i
$779$	−454606.	+	448922.i	−0.749135	+	0.739769i
$780$		0			0
$781$		−	470592.i		−	0.771511i
$782$		−	186037.i		−	0.304219i
$783$		0			0
$784$	320683.			0.521727
$785$	149115.			0.241981
$786$		0			0
$787$		−	764737.i		−	1.23470i	−0.786687	−	0.617352i	$-0.788206\pi$
$787$		−	764737.i		−	1.23470i	0.786687	−	0.617352i	$-0.211794\pi$
$788$	273868.			0.441051
$789$		0			0
$790$	93774.6			0.150256
$791$			368127.i			0.588362i
$792$		0			0
$793$			365557.i			0.581312i
$794$		−	283298.i		−	0.449368i
$795$		0			0
$796$	444033.			0.700792
$797$			555222.i			0.874077i	0.899443	+	0.437039i	$0.143973\pi$
$797$			555222.i			0.874077i	−0.899443	+	0.437039i	$0.856027\pi$
$798$		0			0
$799$	204314.			0.320040
$800$		−	11403.0i		−	0.0178171i
$801$		0			0
$802$	−47224.7			−0.0734211
$803$	−805270.			−1.24885
$804$		0			0
$805$	−607555.			−0.937549
$806$		−	680417.i		−	1.04738i
$807$		0			0
$808$			107612.i			0.164831i
$809$	94777.3			0.144813			0.0724065	−	0.997375i	$-0.476932\pi$
$809$	94777.3			0.144813			0.0724065	+	0.997375i	$0.476932\pi$
$810$		0			0
$811$			68642.4i			0.104364i	0.998638	+	0.0521820i	$0.0166176\pi$
$811$			68642.4i			0.104364i	−0.998638	+	0.0521820i	$0.983382\pi$
$812$			778986.i			1.18146i
$813$		0			0
$814$	−98276.4			−0.148320
$815$	−801330.			−1.20641
$816$		0			0
$817$	−587606.	−	595045.i	−0.880323	−	0.891467i
$818$	3320.42			0.00496233
$819$		0			0
$820$		−	371373.i		−	0.552309i
$821$	−601916.			−0.892996			−0.446498	−	0.894785i	$-0.647329\pi$
$821$	−601916.			−0.892996			−0.446498	+	0.894785i	$0.647329\pi$
$822$		0			0
$823$	−307730.			−0.454329			−0.227165	−	0.973856i	$-0.572946\pi$
$823$	−307730.			−0.454329			−0.227165	+	0.973856i	$0.572946\pi$
$824$	197487.			0.290860
$825$		0			0
$826$	−1.19884e6			−1.75712
$827$		−	194825.i		−	0.284862i	−0.989805	−	0.142431i	$-0.954508\pi$
$827$		−	194825.i		−	0.284862i	0.989805	−	0.142431i	$-0.0454919\pi$
$828$		0			0
$829$			1.14738e6i			1.66955i	0.550592	+	0.834774i	$0.314402\pi$
$829$			1.14738e6i			1.66955i	−0.550592	+	0.834774i	$0.685598\pi$
$830$			75679.2i			0.109855i
$831$		0			0
$832$			118706.i			0.171486i
$833$	1.22494e6			1.76532
$834$		0			0
$835$			740765.i			1.06245i
$836$	232841.	+	235789.i	0.333156	+	0.337374i
$837$		0			0
$838$		−	593860.i		−	0.845661i
$839$			283733.i			0.403075i	0.979481	+	0.201538i	$0.0645938\pi$
$839$			283733.i			0.403075i	−0.979481	+	0.201538i	$0.935406\pi$
$840$		0			0
$841$	−571993.			−0.808721
$842$	−704608.			−0.993856
$843$		0			0
$844$			620420.i			0.870965i
$845$	−660795.			−0.925450
$846$		0			0
$847$	126984.			0.177004
$848$			41997.3i			0.0584023i
$849$		0			0
$850$		−	43556.8i		−	0.0602863i
$851$			81472.9i			0.112500i
$852$		0			0
$853$	975341.			1.34047			0.670237	−	0.742147i	$-0.266193\pi$
$853$	975341.			1.34047			0.670237	+	0.742147i	$0.266193\pi$
$854$		−	383932.i		−	0.526428i
$855$		0			0
$856$	382061.			0.521417
$857$			5569.67i			0.00758347i	0.999993	+	0.00379173i	$0.00120695\pi$
$857$			5569.67i			0.00758347i	−0.999993	+	0.00379173i	$0.998793\pi$
$858$		0			0
$859$	312767.			0.423871			0.211936	−	0.977284i	$-0.432023\pi$
$859$	312767.			0.423871			0.211936	+	0.977284i	$0.432023\pi$
$860$	486099.			0.657246
$861$		0			0
$862$	359060.			0.483228
$863$		−	359441.i		−	0.482621i	−0.970448	−	0.241311i	$-0.922423\pi$
$863$		−	359441.i		−	0.482621i	0.970448	−	0.241311i	$-0.0775773\pi$
$864$		0			0
$865$			935690.i			1.25055i
$866$	925865.			1.23456
$867$		0			0
$868$			714618.i			0.948494i
$869$		−	145036.i		−	0.192059i
$870$		0			0
$871$	1.62304e6			2.13940
$872$	−4647.73			−0.00611234
$873$		0			0
$874$	195473.	−	193030.i	0.255897	−	0.252698i
$875$	1.26909e6			1.65758
$876$		0			0
$877$			415375.i			0.540059i	0.962852	+	0.270030i	$0.0870335\pi$
$877$			415375.i			0.540059i	−0.962852	+	0.270030i	$0.912967\pi$
$878$	915172.			1.18717
$879$		0			0
$880$	−192619.			−0.248733
$881$	403508.			0.519876			0.259938	−	0.965625i	$-0.416298\pi$
$881$	403508.			0.519876			0.259938	+	0.965625i	$0.416298\pi$
$882$		0			0
$883$	265307.			0.340273			0.170137	−	0.985420i	$-0.445579\pi$
$883$	265307.			0.340273			0.170137	+	0.985420i	$0.445579\pi$
$884$			453433.i			0.580241i
$885$		0			0
$886$			532700.i			0.678603i
$887$		−	264831.i		−	0.336605i	−0.985735	−	0.168303i	$-0.946171\pi$
$887$		−	264831.i		−	0.336605i	0.985735	−	0.168303i	$-0.0538286\pi$
$888$		0			0
$889$		−	387496.i		−	0.490302i
$890$	277933.			0.350881
$891$		0			0
$892$		−	246034.i		−	0.309219i
$893$	211994.	+	214678.i	0.265840	+	0.269205i
$894$		0			0
$895$		−	901551.i		−	1.12550i
$896$		−	124673.i		−	0.155295i
$897$		0			0
$898$	88262.9			0.109452
$899$	−1.17357e6			−1.45207
$900$		0			0
$901$			160421.i			0.197611i
$902$	−574381.			−0.705971
$903$		0			0
$904$	96755.3			0.118396
$905$			765776.i			0.934985i
$906$		0			0
$907$			1.43037e6i			1.73874i	0.494165	+	0.869368i	$0.335474\pi$
$907$			1.43037e6i			1.73874i	−0.494165	+	0.869368i	$0.664526\pi$
$908$		−	650394.i		−	0.788869i
$909$		0			0
$910$	1.48081e6			1.78820
$911$		−	1.14296e6i		−	1.37719i	−0.725148	−	0.688593i	$-0.758229\pi$
$911$		−	1.14296e6i		−	1.37719i	0.725148	−	0.688593i	$-0.241771\pi$
$912$		0			0
$913$	117049.			0.140419
$914$		−	253676.i		−	0.303660i
$915$		0			0
$916$	745496.			0.888494
$917$	2.37665e6			2.82636
$918$		0			0
$919$	−627508.			−0.742999			−0.371500	−	0.928433i	$-0.621156\pi$
$919$	−627508.			−0.742999			−0.371500	+	0.928433i	$0.621156\pi$
$920$			159685.i			0.188663i
$921$		0			0
$922$		−	201411.i		−	0.236931i
$923$	950871.			1.11614
$924$		0			0
$925$			19075.2i			0.0222939i
$926$			272866.i			0.318220i
$927$		0			0
$928$	204742.			0.237745
$929$	1.63447e6			1.89385			0.946924	−	0.321458i	$-0.104173\pi$
$929$	1.63447e6			1.89385			0.946924	+	0.321458i	$0.104173\pi$
$930$		0			0
$931$	1.27098e6	+	1.28707e6i	1.46636	+	1.48492i
$932$	−27032.7			−0.0311213
$933$		0			0
$934$			953752.i			1.09331i
$935$	−735762.			−0.841617
$936$		0			0
$937$	−55506.2			−0.0632211			−0.0316106	−	0.999500i	$-0.510064\pi$
$937$	−55506.2			−0.0632211			−0.0316106	+	0.999500i	$0.510064\pi$
$938$	−1.70462e6			−1.93741
$939$		0			0
$940$	−175373.			−0.198475
$941$			1.60327e6i			1.81062i	0.424747	+	0.905312i	$0.360363\pi$
$941$			1.60327e6i			1.81062i	−0.424747	+	0.905312i	$0.639637\pi$
$942$		0			0
$943$			476172.i			0.535476i
$944$			315093.i			0.353586i
$945$		0			0
$946$		−	751822.i		−	0.840103i
$947$	−165986.			−0.185086			−0.0925428	−	0.995709i	$-0.529499\pi$
$947$	−165986.			−0.185086			−0.0925428	+	0.995709i	$0.529499\pi$
$948$		0			0
$949$		−	1.62712e6i		−	1.80670i
$950$	45766.2	−	45194.0i	0.0507104	−	0.0500765i
$951$		0			0
$952$		−	476224.i		−	0.525457i
$953$		−	821794.i		−	0.904851i	−0.891802	−	0.452426i	$-0.850559\pi$
$953$		−	821794.i		−	0.904851i	0.891802	−	0.452426i	$-0.149441\pi$
$954$		0			0
$955$	−354054.			−0.388206
$956$	287342.			0.314401
$957$		0			0
$958$			358317.i			0.390424i
$959$	710276.			0.772307
$960$		0			0
$961$	−153074.			−0.165750
$962$		−	198576.i		−	0.214574i
$963$		0			0
$964$			458817.i			0.493726i
$965$		−	546093.i		−	0.586424i
$966$		0			0
$967$	−859901.			−0.919593			−0.459796	−	0.888024i	$-0.652078\pi$
$967$	−859901.			−0.919593			−0.459796	+	0.888024i	$0.652078\pi$
$968$		−	33375.4i		−	0.0356185i
$969$		0			0
$970$	−967490.			−1.02826
$971$		−	1.15671e6i		−	1.22684i	−0.789759	−	0.613418i	$-0.789794\pi$
$971$		−	1.15671e6i		−	1.22684i	0.789759	−	0.613418i	$-0.210206\pi$
$972$		0			0
$973$	−2.36191e6			−2.49481
$974$	−661459.			−0.697245
$975$		0			0
$976$	−100909.			−0.105933
$977$		−	627667.i		−	0.657567i	−0.944405	−	0.328784i	$-0.893361\pi$
$977$		−	627667.i		−	0.657567i	0.944405	−	0.328784i	$-0.106639\pi$
$978$		0			0
$979$		−	429862.i		−	0.448502i
$980$	−1.05142e6			−1.09478
$981$		0			0
$982$		−	1.19905e6i		−	1.24341i
$983$			1.45983e6i			1.51075i	0.655290	+	0.755377i	$0.272546\pi$
$983$			1.45983e6i			1.51075i	−0.655290	+	0.755377i	$0.727454\pi$
$984$		0			0
$985$	−897931.			−0.925487
$986$	782070.			0.804436
$987$		0			0
$988$	−476432.	+	470476.i	−0.488075	+	0.481974i
$989$	−623273.			−0.637215
$990$		0			0
$991$		−	977409.i		−	0.995243i	−0.867394	−	0.497621i	$-0.834207\pi$
$991$		−	977409.i		−	0.995243i	0.867394	−	0.497621i	$-0.165793\pi$
$992$	187824.			0.190866
$993$		0			0
$994$	−998666.			−1.01076
$995$	−1.45585e6			−1.47052
$996$		0			0
$997$	1.39372e6			1.40212			0.701059	−	0.713104i	$-0.252711\pi$
$997$	1.39372e6			1.40212			0.701059	+	0.713104i	$0.252711\pi$
$998$			497645.i			0.499642i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	342.5.d.a.37.3		8
3.2	odd	2		38.5.b.a.37.5	yes	8
12.11	even	2		304.5.e.e.113.7		8
19.18	odd	2	inner	342.5.d.a.37.7		8
57.56	even	2		38.5.b.a.37.4	✓	8
228.227	odd	2		304.5.e.e.113.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.5.b.a.37.4	✓	8	57.56	even	2
38.5.b.a.37.5	yes	8	3.2	odd	2
304.5.e.e.113.2		8	228.227	odd	2
304.5.e.e.113.7		8	12.11	even	2
342.5.d.a.37.3		8	1.1	even	1	trivial
342.5.d.a.37.7		8	19.18	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:	\( N \)	\(=\)	\( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	342.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.82843i q^{2} -8.00000 q^{4} +26.2296 q^{5} -86.0910 q^{7} +22.6274i q^{8} +O(q^{10})\)	\(q-2.82843i q^{2} -8.00000 q^{4} +26.2296 q^{5} -86.0910 q^{7} +22.6274i q^{8} -74.1886i q^{10} -114.743 q^{11} -231.848i q^{13} +243.502i q^{14} +64.0000 q^{16} +244.466 q^{17} +(253.655 + 256.866i) q^{19} -209.837 q^{20} +324.543i q^{22} +269.052 q^{23} +62.9931 q^{25} -655.766 q^{26} +688.728 q^{28} +1131.05i q^{29} +1037.59i q^{31} -181.019i q^{32} -691.455i q^{34} -2258.14 q^{35} +302.815i q^{37} +(726.527 - 717.444i) q^{38} +593.509i q^{40} +1769.82i q^{41} -2316.56 q^{43} +917.946 q^{44} -760.993i q^{46} +835.757 q^{47} +5010.66 q^{49} -178.171i q^{50} +1854.79i q^{52} +656.208i q^{53} -3009.67 q^{55} -1948.02i q^{56} +3199.09 q^{58} +4923.33i q^{59} -1576.71 q^{61} +2934.75 q^{62} -512.000 q^{64} -6081.30i q^{65} +7000.44i q^{67} -1955.73 q^{68} +6386.97i q^{70} +4101.26i q^{71} +7018.02 q^{73} +856.490 q^{74} +(-2029.24 - 2054.93i) q^{76} +9878.36 q^{77} +1264.00i q^{79} +1678.70 q^{80} +5005.79 q^{82} -1020.09 q^{83} +6412.25 q^{85} +6552.21i q^{86} -2596.34i q^{88} +3746.30i q^{89} +19960.1i q^{91} -2152.41 q^{92} -2363.88i q^{94} +(6653.27 + 6737.50i) q^{95} -13041.0i q^{97} -14172.3i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 64 q^{4} - 18 q^{5} - 162 q^{7}+O(q^{10}) \) 8 * q - 64 * q^4 - 18 * q^5 - 162 * q^7	\( 8 q - 64 q^{4} - 18 q^{5} - 162 q^{7} + 6 q^{11} + 512 q^{16} - 510 q^{17} - 12 q^{19} + 144 q^{20} + 396 q^{23} + 3458 q^{25} + 192 q^{26} + 1296 q^{28} - 1002 q^{35} + 3216 q^{38} - 8654 q^{43} - 48 q^{44} - 3210 q^{47} + 9222 q^{49} + 17146 q^{55} - 960 q^{58} + 1314 q^{61} + 15168 q^{62} - 4096 q^{64} + 4080 q^{68} + 23398 q^{73} - 13152 q^{74} + 96 q^{76} + 44622 q^{77} - 1152 q^{80} + 16512 q^{82} + 10440 q^{83} + 21274 q^{85} - 3168 q^{92} + 34686 q^{95}+O(q^{100}) \) 8 * q - 64 * q^4 - 18 * q^5 - 162 * q^7 + 6 * q^11 + 512 * q^16 - 510 * q^17 - 12 * q^19 + 144 * q^20 + 396 * q^23 + 3458 * q^25 + 192 * q^26 + 1296 * q^28 - 1002 * q^35 + 3216 * q^38 - 8654 * q^43 - 48 * q^44 - 3210 * q^47 + 9222 * q^49 + 17146 * q^55 - 960 * q^58 + 1314 * q^61 + 15168 * q^62 - 4096 * q^64 + 4080 * q^68 + 23398 * q^73 - 13152 * q^74 + 96 * q^76 + 44622 * q^77 - 1152 * q^80 + 16512 * q^82 + 10440 * q^83 + 21274 * q^85 - 3168 * q^92 + 34686 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more