LMFDB - Embedded newform 304.5.e.e.113.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [304,5,Mod(113,304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(304, 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("304.113");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 304 = 2^{4} \cdot 19 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	304.e (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:	$31.4244687775$
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:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{7}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 38)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			113.7
Root			$12.2418i$ of defining polynomial
Character	$\chi$	$=$	304.113
Dual form			304.5.e.e.113.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+10.8276i q^{3} -26.2296 q^{5} +86.0910 q^{7} -36.2364 q^{9} +O(q^{10})$	$q+10.8276i q^{3} -26.2296 q^{5} +86.0910 q^{7} -36.2364 q^{9} -114.743 q^{11} -231.848i q^{13} -284.003i q^{15} -244.466 q^{17} +(-253.655 - 256.866i) q^{19} +932.157i q^{21} +269.052 q^{23} +62.9931 q^{25} +484.681i q^{27} -1131.05i q^{29} -1037.59i q^{31} -1242.39i q^{33} -2258.14 q^{35} +302.815i q^{37} +2510.36 q^{39} -1769.82i q^{41} +2316.56 q^{43} +950.467 q^{45} +835.757 q^{47} +5010.66 q^{49} -2646.98i q^{51} -656.208i q^{53} +3009.67 q^{55} +(2781.24 - 2746.47i) q^{57} +4923.33i q^{59} -1576.71 q^{61} -3119.63 q^{63} +6081.30i q^{65} -7000.44i q^{67} +2913.18i q^{69} +4101.26i q^{71} +7018.02 q^{73} +682.062i q^{75} -9878.36 q^{77} -1264.00i q^{79} -8183.07 q^{81} -1020.09 q^{83} +6412.25 q^{85} +12246.5 q^{87} -3746.30i q^{89} -19960.1i q^{91} +11234.6 q^{93} +(6653.27 + 6737.50i) q^{95} -13041.0i q^{97} +4157.88 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 18 q^{5} + 162 q^{7} - 268 q^{9}+O(q^{10}) $ 8 * q + 18 * q^5 + 162 * q^7 - 268 * q^9	$ 8 q + 18 q^{5} + 162 q^{7} - 268 q^{9} + 6 q^{11} + 510 q^{17} + 12 q^{19} + 396 q^{23} + 3458 q^{25} - 1002 q^{35} + 6588 q^{39} + 8654 q^{43} - 10334 q^{45} - 3210 q^{47} + 9222 q^{49} - 17146 q^{55} - 14076 q^{57} + 1314 q^{61} - 29938 q^{63} + 23398 q^{73} - 44622 q^{77} - 20368 q^{81} + 10440 q^{83} + 21274 q^{85} + 14316 q^{87} + 19416 q^{93} + 34686 q^{95} + 56798 q^{99}+O(q^{100}) $ 8 * q + 18 * q^5 + 162 * q^7 - 268 * q^9 + 6 * q^11 + 510 * q^17 + 12 * q^19 + 396 * q^23 + 3458 * q^25 - 1002 * q^35 + 6588 * q^39 + 8654 * q^43 - 10334 * q^45 - 3210 * q^47 + 9222 * q^49 - 17146 * q^55 - 14076 * q^57 + 1314 * q^61 - 29938 * q^63 + 23398 * q^73 - 44622 * q^77 - 20368 * q^81 + 10440 * q^83 + 21274 * q^85 + 14316 * q^87 + 19416 * q^93 + 34686 * q^95 + 56798 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/304\mathbb{Z}\right)^\times$.

$n$	$97$	$191$	$229$
$\chi(n)$	$-1$	$1$	$1$

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
$3$			10.8276i			1.20306i	0.798849	+	0.601532i	$0.205443\pi$
$3$			10.8276i			1.20306i	−0.798849	+	0.601532i	$0.794557\pi$
$4$		0			0
$5$	−26.2296			−1.04918			−0.524592	−	0.851353i	$-0.675782\pi$
$5$	−26.2296			−1.04918			−0.524592	+	0.851353i	$0.675782\pi$
$6$		0			0
$7$	86.0910			1.75696			0.878480	−	0.477779i	$-0.158558\pi$
$7$	86.0910			1.75696			0.878480	+	0.477779i	$0.158558\pi$
$8$		0			0
$9$	−36.2364			−0.447363
$10$		0			0
$11$	−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	$-0.240610\pi$
$14$		0			0
$15$		−	284.003i		−	1.26224i
$16$		0			0
$17$	−244.466			−0.845903			−0.422952	−	0.906152i	$-0.639006\pi$
$17$	−244.466			−0.845903			−0.422952	+	0.906152i	$0.639006\pi$
$18$		0			0
$19$	−253.655	−	256.866i	−0.702645	−	0.711540i
$19$	−253.655	−	256.866i	−0.702645	−	0.711540i
$20$		0			0
$21$			932.157i			2.11373i
$22$		0			0
$23$	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
$26$		0			0
$27$			484.681i			0.664858i
$28$		0			0
$29$		−	1131.05i		−	1.34489i	−0.740148	−	0.672443i	$-0.765245\pi$
$29$		−	1131.05i		−	1.34489i	0.740148	−	0.672443i	$-0.234755\pi$
$30$		0			0
$31$		−	1037.59i		−	1.07970i	−0.841762	−	0.539850i	$-0.818481\pi$
$31$		−	1037.59i		−	1.07970i	0.841762	−	0.539850i	$-0.181519\pi$
$32$		0			0
$33$		−	1242.39i		−	1.14085i
$34$		0			0
$35$	−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	$0.964724\pi$
$38$		0			0
$39$	2510.36			1.65046
$40$		0			0
$41$		−	1769.82i		−	1.05283i	−0.850226	−	0.526417i	$-0.823535\pi$
$41$		−	1769.82i		−	1.05283i	0.850226	−	0.526417i	$-0.176465\pi$
$42$		0			0
$43$	2316.56			1.25287			0.626435	−	0.779474i	$-0.284513\pi$
$43$	2316.56			1.25287			0.626435	+	0.779474i	$0.284513\pi$
$44$		0			0
$45$	950.467			0.469366
$46$		0			0
$47$	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
$50$		0			0
$51$		−	2646.98i		−	1.01768i
$52$		0			0
$53$		−	656.208i		−	0.233609i	−0.993155	−	0.116805i	$-0.962735\pi$
$53$		−	656.208i		−	0.233609i	0.993155	−	0.116805i	$-0.0372651\pi$
$54$		0			0
$55$	3009.67			0.994932
$56$		0			0
$57$	2781.24	−	2746.47i	0.856029	−	0.845327i
$58$		0			0
$59$			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	$0.567954\pi$
$62$		0			0
$63$	−3119.63			−0.785999
$64$		0			0
$65$			6081.30i			1.43936i
$66$		0			0
$67$		−	7000.44i		−	1.55946i	−0.626113	−	0.779732i	$-0.715355\pi$
$67$		−	7000.44i		−	1.55946i	0.626113	−	0.779732i	$-0.284645\pi$
$68$		0			0
$69$			2913.18i			0.611884i
$70$		0			0
$71$			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	$0.271202\pi$
$74$		0			0
$75$			682.062i			0.121256i
$76$		0			0
$77$	−9878.36			−1.66611
$78$		0			0
$79$		−	1264.00i		−	0.202532i	−0.994859	−	0.101266i	$-0.967711\pi$
$79$		−	1264.00i		−	0.202532i	0.994859	−	0.101266i	$-0.0322893\pi$
$80$		0			0
$81$	−8183.07			−1.24723
$82$		0			0
$83$	−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
$86$		0			0
$87$	12246.5			1.61799
$88$		0			0
$89$		−	3746.30i		−	0.472958i	−0.971637	−	0.236479i	$-0.924007\pi$
$89$		−	3746.30i		−	0.472958i	0.971637	−	0.236479i	$-0.0759934\pi$
$90$		0			0
$91$		−	19960.1i		−	2.41034i
$92$		0			0
$93$	11234.6			1.29895
$94$		0			0
$95$	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	$-0.243713\pi$
$98$		0			0
$99$	4157.88			0.424230
$100$		0			0
$101$	−4755.84			−0.466213			−0.233106	−	0.972451i	$-0.574889\pi$
$101$	−4755.84			−0.466213			−0.233106	+	0.972451i	$0.574889\pi$
$102$		0			0
$103$			8727.79i			0.822678i	0.911483	+	0.411339i	$0.134939\pi$
$103$			8727.79i			0.822678i	−0.911483	+	0.411339i	$0.865061\pi$
$104$		0			0
$105$		−	24450.1i		−	2.21770i
$106$		0			0
$107$		−	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	$0.997248\pi$
$110$		0			0
$111$	−3278.75			−0.266111
$112$		0			0
$113$			4276.02i			0.334875i	0.985883	+	0.167438i	$0.0535492\pi$
$113$			4276.02i			0.334875i	−0.985883	+	0.167438i	$0.946451\pi$
$114$		0			0
$115$	−7057.13			−0.533620
$116$		0			0
$117$			8401.35i			0.613730i
$118$		0			0
$119$	−21046.3			−1.48622
$120$		0			0
$121$	−1475.00			−0.100744
$122$		0			0
$123$	19162.8			1.26663
$124$		0			0
$125$	14741.2			0.943439
$126$		0			0
$127$		−	4501.00i		−	0.279063i	−0.990218	−	0.139531i	$-0.955440\pi$
$127$		−	4501.00i		−	0.279063i	0.990218	−	0.139531i	$-0.0445596\pi$
$128$		0			0
$129$			25082.7i			1.50728i
$130$		0			0
$131$	−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
$134$		0			0
$135$		−	12713.0i		−	0.697559i
$136$		0			0
$137$	8250.29			0.439570			0.219785	−	0.975548i	$-0.429464\pi$
$137$	8250.29			0.439570			0.219785	+	0.975548i	$0.429464\pi$
$138$		0			0
$139$	−27435.0			−1.41996			−0.709978	−	0.704224i	$-0.751295\pi$
$139$	−27435.0			−1.41996			−0.709978	+	0.704224i	$0.751295\pi$
$140$		0			0
$141$			9049.22i			0.455169i
$142$		0			0
$143$			26603.0i			1.30095i
$144$		0			0
$145$			29667.0i			1.41104i
$146$		0			0
$147$			54253.3i			2.51068i
$148$		0			0
$149$	−40247.3			−1.81286			−0.906430	−	0.422355i	$-0.861204\pi$
$149$	−40247.3			−1.81286			−0.906430	+	0.422355i	$0.861204\pi$
$150$		0			0
$151$		−	16932.1i		−	0.742605i	−0.928512	−	0.371303i	$-0.878911\pi$
$151$		−	16932.1i		−	0.742605i	0.928512	−	0.371303i	$-0.121089\pi$
$152$		0			0
$153$	8858.57			0.378426
$154$		0			0
$155$			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	$0.463211\pi$
$158$		0			0
$159$	7105.14			0.281047
$160$		0			0
$161$	23162.9			0.893597
$162$		0			0
$163$	30550.6			1.14986			0.574929	−	0.818203i	$-0.305030\pi$
$163$	30550.6			1.14986			0.574929	+	0.818203i	$0.305030\pi$
$164$		0			0
$165$			32587.4i			1.19697i
$166$		0			0
$167$			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
$170$		0			0
$171$	9191.54	+	9307.90i	0.314337	+	0.318317i
$172$		0			0
$173$		−	35673.0i		−	1.19192i	−0.803013	−	0.595961i	$-0.796771\pi$
$173$		−	35673.0i		−	1.19192i	0.803013	−	0.595961i	$-0.203229\pi$
$174$		0			0
$175$	5423.14			0.177082
$176$		0			0
$177$	−53307.7			−1.70154
$178$		0			0
$179$		−	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	$0.852999\pi$
$182$		0			0
$183$		−	17071.9i		−	0.509777i
$184$		0			0
$185$		−	7942.72i		−	0.232074i
$186$		0			0
$187$	28050.8			0.802163
$188$		0			0
$189$			41726.7i			1.16813i
$190$		0			0
$191$	−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	$-0.0901577\pi$
$194$		0			0
$195$	−65845.7			−1.73164
$196$		0			0
$197$	34233.5			0.882101			0.441051	−	0.897482i	$-0.354606\pi$
$197$	34233.5			0.882101			0.441051	+	0.897482i	$0.354606\pi$
$198$		0			0
$199$	55504.1			1.40158			0.700792	−	0.713366i	$-0.252830\pi$
$199$	55504.1			1.40158			0.700792	+	0.713366i	$0.252830\pi$
$200$		0			0
$201$	75797.8			1.87614
$202$		0			0
$203$		−	97373.2i		−	2.36291i
$204$		0			0
$205$			46421.6i			1.10462i
$206$		0			0
$207$	−9749.46			−0.227531
$208$		0			0
$209$	29105.2	+	29473.6i	0.666312	+	0.674747i
$210$		0			0
$211$			77552.5i			1.74193i	0.491345	+	0.870965i	$0.336505\pi$
$211$			77552.5i			1.74193i	−0.491345	+	0.870965i	$0.663495\pi$
$212$		0			0
$213$	−44406.7			−0.978790
$214$		0			0
$215$	−60762.4			−1.31449
$216$		0			0
$217$		−	89327.3i		−	1.89699i
$218$		0			0
$219$			75988.1i			1.58437i
$220$		0			0
$221$			56679.1i			1.16048i
$222$		0			0
$223$		−	30754.3i		−	0.618438i	−0.950991	−	0.309219i	$-0.899932\pi$
$223$		−	30754.3i		−	0.618438i	0.950991	−	0.309219i	$-0.100068\pi$
$224$		0			0
$225$	−2282.64			−0.0450892
$226$		0			0
$227$			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	$0.848248\pi$
$230$		0			0
$231$		−	106959.i		−	2.00444i
$232$		0			0
$233$	−3379.09			−0.0622426			−0.0311213	−	0.999516i	$-0.509908\pi$
$233$	−3379.09			−0.0622426			−0.0311213	+	0.999516i	$0.509908\pi$
$234$		0			0
$235$	−21921.6			−0.396950
$236$		0			0
$237$	13686.1			0.243659
$238$		0			0
$239$	−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	$-0.164365\pi$
$242$		0			0
$243$		−	49343.6i		−	0.835639i
$244$		0			0
$245$	−131428.			−2.18955
$246$		0			0
$247$	−59554.0	+	58809.5i	−0.976151	+	0.963948i
$248$		0			0
$249$		−	11045.1i		−	0.178144i
$250$		0			0
$251$	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
$254$		0			0
$255$			69429.2i			1.06773i
$256$		0			0
$257$		−	85222.5i		−	1.29029i	−0.764059	−	0.645146i	$-0.776796\pi$
$257$		−	85222.5i		−	1.29029i	0.764059	−	0.645146i	$-0.223204\pi$
$258$		0			0
$259$			26069.6i			0.388629i
$260$		0			0
$261$			40985.2i			0.601653i
$262$		0			0
$263$	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
$266$		0			0
$267$	40563.3			0.568999
$268$		0			0
$269$		−	58880.0i		−	0.813697i	−0.913496	−	0.406849i	$-0.866628\pi$
$269$		−	58880.0i		−	0.813697i	0.913496	−	0.406849i	$-0.133372\pi$
$270$		0			0
$271$	−99955.1			−1.36103			−0.680513	−	0.732736i	$-0.738243\pi$
$271$	−99955.1			−1.36103			−0.680513	+	0.732736i	$0.738243\pi$
$272$		0			0
$273$	216119.			2.89980
$274$		0			0
$275$	−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	$0.357798\pi$
$278$		0			0
$279$			37598.6i			0.483017i
$280$		0			0
$281$		−	55021.9i		−	0.696824i	−0.937341	−	0.348412i	$-0.886721\pi$
$281$		−	55021.9i		−	0.696824i	0.937341	−	0.348412i	$-0.113279\pi$
$282$		0			0
$283$	−10575.2			−0.132044			−0.0660219	−	0.997818i	$-0.521031\pi$
$283$	−10575.2			−0.132044			−0.0660219	+	0.997818i	$0.521031\pi$
$284$		0			0
$285$	−72950.8	+	72038.8i	−0.898132	+	0.886904i
$286$		0			0
$287$		−	152365.i		−	1.84979i
$288$		0			0
$289$	−23757.3			−0.284447
$290$		0			0
$291$	141202.			1.66746
$292$		0			0
$293$			9449.70i			0.110074i	0.998484	+	0.0550368i	$0.0175276\pi$
$293$			9449.70i			0.110074i	−0.998484	+	0.0550368i	$0.982472\pi$
$294$		0			0
$295$		−	129137.i		−	1.48391i
$296$		0			0
$297$		−	55613.9i		−	0.630479i
$298$		0			0
$299$		−	62379.2i		−	0.697746i
$300$		0			0
$301$	199435.			2.20124
$302$		0			0
$303$		−	51494.2i		−	0.560884i
$304$		0			0
$305$	41356.5			0.444574
$306$		0			0
$307$		−	112622.i		−	1.19494i	−0.801890	−	0.597472i	$-0.796172\pi$
$307$		−	112622.i		−	1.19494i	0.801890	−	0.597472i	$-0.203828\pi$
$308$		0			0
$309$	−94500.8			−0.989734
$310$		0			0
$311$	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	$0.691626\pi$
$314$		0			0
$315$	81826.7			0.824658
$316$		0			0
$317$			63970.1i			0.636589i	0.947992	+	0.318294i	$0.103110\pi$
$317$			63970.1i			0.636589i	−0.947992	+	0.318294i	$0.896890\pi$
$318$		0			0
$319$			129780.i			1.27534i
$320$		0			0
$321$	182822.			1.77427
$322$		0			0
$323$	62010.0	+	62795.0i	0.594370	+	0.601894i
$324$		0			0
$325$		−	14604.8i		−	0.138271i
$326$		0			0
$327$	−2224.01			−0.0207990
$328$		0			0
$329$	71951.1			0.664731
$330$		0			0
$331$		−	79462.6i		−	0.725282i	−0.931929	−	0.362641i	$-0.881875\pi$
$331$		−	79462.6i		−	0.725282i	0.931929	−	0.362641i	$-0.118125\pi$
$332$		0			0
$333$		−	10972.9i		−	0.0989541i
$334$		0			0
$335$			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	$0.996104\pi$
$338$		0			0
$339$	−46298.9			−0.402876
$340$		0			0
$341$			119057.i			1.02387i
$342$		0			0
$343$	224669.			1.90965
$344$		0			0
$345$		−	76411.6i		−	0.641979i
$346$		0			0
$347$	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	$0.554868\pi$
$350$		0			0
$351$	112373.			0.912108
$352$		0			0
$353$	39292.1			0.315323			0.157661	−	0.987493i	$-0.449605\pi$
$353$	39292.1			0.315323			0.157661	+	0.987493i	$0.449605\pi$
$354$		0			0
$355$		−	107575.i		−	0.853597i
$356$		0			0
$357$		−	227881.i		−	1.78802i
$358$		0			0
$359$	−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
$362$		0			0
$363$		−	15970.7i		−	0.121202i
$364$		0			0
$365$	−184080.			−1.38172
$366$		0			0
$367$	−19835.4			−0.147268			−0.0736340	−	0.997285i	$-0.523460\pi$
$367$	−19835.4			−0.147268			−0.0736340	+	0.997285i	$0.523460\pi$
$368$		0			0
$369$			64131.7i			0.470999i
$370$		0			0
$371$		−	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	$-0.0165528\pi$
$374$		0			0
$375$			159612.i			1.13502i
$376$		0			0
$377$	−262232.			−1.84503
$378$		0			0
$379$			25932.7i			0.180539i	0.995917	+	0.0902693i	$0.0287728\pi$
$379$			25932.7i			0.180539i	−0.995917	+	0.0902693i	$0.971227\pi$
$380$		0			0
$381$	48734.9			0.335730
$382$		0			0
$383$			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
$386$		0			0
$387$	−83943.6			−0.560487
$388$		0			0
$389$	101890.			0.673335			0.336668	−	0.941624i	$-0.390700\pi$
$389$	101890.			0.673335			0.336668	+	0.941624i	$0.390700\pi$
$390$		0			0
$391$	−65774.0			−0.430230
$392$		0			0
$393$		−	298909.i		−	1.93533i
$394$		0			0
$395$			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	$0.397072\pi$
$398$		0			0
$399$	239440.	−	236446.i	1.50401	−	1.48521i
$400$		0			0
$401$			16696.5i			0.103833i	0.998651	+	0.0519165i	$0.0165330\pi$
$401$			16696.5i			0.103833i	−0.998651	+	0.0519165i	$0.983467\pi$
$402$		0			0
$403$	−240564.			−1.48122
$404$		0			0
$405$	214639.			1.30857
$406$		0			0
$407$		−	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	$0.998883\pi$
$410$		0			0
$411$			89330.7i			0.528831i
$412$		0			0
$413$			423854.i			2.48494i
$414$		0			0
$415$	26756.6			0.155359
$416$		0			0
$417$		−	297054.i		−	1.70830i
$418$		0			0
$419$	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	$-0.248050\pi$
$422$		0			0
$423$	−30284.8			−0.169256
$424$		0			0
$425$	−15399.7			−0.0852577
$426$		0			0
$427$	−135741.			−0.744481
$428$		0			0
$429$	−288046.			−1.56512
$430$		0			0
$431$			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	$0.662194\pi$
$434$		0			0
$435$	−321222.			−1.69757
$436$		0			0
$437$	−68246.3	−	69110.3i	−0.357368	−	0.361893i
$438$		0			0
$439$		−	323562.i		−	1.67891i	−0.543425	−	0.839457i	$-0.682873\pi$
$439$		−	323562.i		−	1.67891i	0.543425	−	0.839457i	$-0.317127\pi$
$440$		0			0
$441$	−181568.			−0.933605
$442$		0			0
$443$	−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
$446$		0			0
$447$		−	435781.i		−	2.18099i
$448$		0			0
$449$		−	31205.6i		−	0.154789i	−0.997001	−	0.0773945i	$-0.975340\pi$
$449$		−	31205.6i		−	0.154789i	0.997001	−	0.0773945i	$-0.0246601\pi$
$450$		0			0
$451$			203074.i			0.998394i
$452$		0			0
$453$	183334.			0.893402
$454$		0			0
$455$			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	$0.431116\pi$
$458$		0			0
$459$		−	118488.i		−	0.562405i
$460$		0			0
$461$	−71209.7			−0.335071			−0.167536	−	0.985866i	$-0.553581\pi$
$461$	−71209.7			−0.335071			−0.167536	+	0.985866i	$0.553581\pi$
$462$		0			0
$463$	96472.6			0.450030			0.225015	−	0.974355i	$-0.427757\pi$
$463$	96472.6			0.450030			0.225015	+	0.974355i	$0.427757\pi$
$464$		0			0
$465$	−294679.			−1.36284
$466$		0			0
$467$	−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
$470$		0			0
$471$			61554.5i			0.277471i
$472$		0			0
$473$	−265809.			−1.18808
$474$		0			0
$475$	−15978.5	−	16180.8i	−0.0708189	−	0.0717154i
$476$		0			0
$477$			23778.6i			0.104508i
$478$		0			0
$479$	−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
$482$		0			0
$483$			250798.i			1.07505i
$484$		0			0
$485$			342059.i			1.45418i
$486$		0			0
$487$			233861.i			0.986053i	0.870014	+	0.493027i	$0.164109\pi$
$487$			233861.i			0.986053i	−0.870014	+	0.493027i	$0.835891\pi$
$488$		0			0
$489$			330789.i			1.38335i
$490$		0			0
$491$	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
$494$		0			0
$495$	−109060.			−0.445096
$496$		0			0
$497$			353082.i			1.42943i
$498$		0			0
$499$	175944.			0.706600			0.353300	−	0.935510i	$-0.385059\pi$
$499$	175944.			0.706600			0.353300	+	0.935510i	$0.385059\pi$
$500$		0			0
$501$	−305787.			−1.21827
$502$		0			0
$503$	−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
$506$		0			0
$507$		−	272776.i		−	1.06118i
$508$		0			0
$509$		−	72826.1i		−	0.281094i	−0.990074	−	0.140547i	$-0.955114\pi$
$509$		−	72826.1i		−	0.281094i	0.990074	−	0.140547i	$-0.0448861\pi$
$510$		0			0
$511$	604188.			2.31383
$512$		0			0
$513$	124498.	−	122942.i	0.473073	−	0.467159i
$514$		0			0
$515$		−	228927.i		−	0.863141i
$516$		0			0
$517$	−95897.4			−0.358778
$518$		0			0
$519$	386253.			1.43396
$520$		0			0
$521$		−	93748.1i		−	0.345372i	−0.984977	−	0.172686i	$-0.944755\pi$
$521$		−	93748.1i		−	0.345372i	0.984977	−	0.172686i	$-0.0552446\pi$
$522$		0			0
$523$			87245.8i			0.318964i	0.987201	+	0.159482i	$0.0509823\pi$
$523$			87245.8i			0.318964i	−0.987201	+	0.159482i	$0.949018\pi$
$524$		0			0
$525$			58719.4i			0.213041i
$526$		0			0
$527$			253656.i			0.913321i
$528$		0			0
$529$	−207452.			−0.741322
$530$		0			0
$531$		−	178404.i		−	0.632724i
$532$		0			0
$533$	−410329.			−1.44437
$534$		0			0
$535$			442884.i			1.54733i
$536$		0			0
$537$	372160.			1.29057
$538$		0			0
$539$	−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	$0.279338\pi$
$542$		0			0
$543$	−316112.			−1.07211
$544$		0			0
$545$		−	5387.63i		−	0.0181386i
$546$		0			0
$547$			160411.i			0.536118i	0.963402	+	0.268059i	$0.0863823\pi$
$547$			160411.i			0.536118i	−0.963402	+	0.268059i	$0.913618\pi$
$548$		0			0
$549$	57134.3			0.189562
$550$		0			0
$551$	−290528.	+	286896.i	−0.956941	+	0.944978i
$552$		0			0
$553$		−	108819.i		−	0.355841i
$554$		0			0
$555$	86000.4			0.279199
$556$		0			0
$557$	460605.			1.48463			0.742314	−	0.670052i	$-0.233728\pi$
$557$	460605.			1.48463			0.742314	+	0.670052i	$0.233728\pi$
$558$		0			0
$559$		−	537090.i		−	1.71879i
$560$		0			0
$561$			303722.i			0.965053i
$562$		0			0
$563$		−	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
$566$		0			0
$567$	−704489.			−2.19133
$568$		0			0
$569$			609592.i			1.88285i	0.337227	+	0.941424i	$0.390511\pi$
$569$			609592.i			1.88285i	−0.337227	+	0.941424i	$0.609489\pi$
$570$		0			0
$571$	11273.0			0.0345755			0.0172877	−	0.999851i	$-0.494497\pi$
$571$	11273.0			0.0345755			0.0172877	+	0.999851i	$0.494497\pi$
$572$		0			0
$573$		−	146153.i		−	0.445143i
$574$		0			0
$575$	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	$0.590594\pi$
$578$		0			0
$579$	225427.			0.672433
$580$		0			0
$581$	−87820.8			−0.260163
$582$		0			0
$583$			75295.4i			0.221529i
$584$		0			0
$585$		−	220364.i		−	0.643916i
$586$		0			0
$587$	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
$590$		0			0
$591$			370665.i			1.06122i
$592$		0			0
$593$	−603214.			−1.71539			−0.857693	−	0.514162i	$-0.828103\pi$
$593$	−603214.			−1.71539			−0.857693	+	0.514162i	$0.828103\pi$
$594$		0			0
$595$	552037.			1.55932
$596$		0			0
$597$			600975.i			1.68619i
$598$		0			0
$599$			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	$-0.0269104\pi$
$602$		0			0
$603$			253671.i			0.697647i
$604$		0			0
$605$	38688.6			0.105699
$606$		0			0
$607$			252238.i			0.684594i	0.939592	+	0.342297i	$0.111205\pi$
$607$			252238.i			0.684594i	−0.939592	+	0.342297i	$0.888795\pi$
$608$		0			0
$609$	1.05432e6			2.84273
$610$		0			0
$611$		−	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	$0.328995\pi$
$614$		0			0
$615$	−502633.			−1.32893
$616$		0			0
$617$	89917.4			0.236197			0.118098	−	0.993002i	$-0.462320\pi$
$617$	89917.4			0.236197			0.118098	+	0.993002i	$0.462320\pi$
$618$		0			0
$619$	133253.			0.347772			0.173886	−	0.984766i	$-0.444368\pi$
$619$	133253.			0.347772			0.173886	+	0.984766i	$0.444368\pi$
$620$		0			0
$621$			130404.i			0.338150i
$622$		0			0
$623$		−	322523.i		−	0.830968i
$624$		0			0
$625$	−426028.			−1.09063
$626$		0			0
$627$	−319128.	+	315139.i	−0.811764	+	0.801616i
$628$		0			0
$629$		−	74028.0i		−	0.187109i
$630$		0			0
$631$	−250184.			−0.628348			−0.314174	−	0.949365i	$-0.601727\pi$
$631$	−250184.			−0.628348			−0.314174	+	0.949365i	$0.601727\pi$
$632$		0			0
$633$	−839705.			−2.09565
$634$		0			0
$635$			118060.i			0.292788i
$636$		0			0
$637$		−	1.16171e6i		−	2.86299i
$638$		0			0
$639$		−	148615.i		−	0.363966i
$640$		0			0
$641$		−	601164.i		−	1.46311i	−0.681782	−	0.731555i	$-0.738795\pi$
$641$		−	601164.i		−	1.46311i	0.681782	−	0.731555i	$-0.261205\pi$
$642$		0			0
$643$	−379057.			−0.916816			−0.458408	−	0.888742i	$-0.651580\pi$
$643$	−379057.			−0.916816			−0.458408	+	0.888742i	$0.651580\pi$
$644$		0			0
$645$		−	657909.i		−	1.58142i
$646$		0			0
$647$	−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
$650$		0			0
$651$	967198.			2.28220
$652$		0			0
$653$	−15905.7			−0.0373016			−0.0186508	−	0.999826i	$-0.505937\pi$
$653$	−15905.7			−0.0373016			−0.0186508	+	0.999826i	$0.505937\pi$
$654$		0			0
$655$	724102.			1.68779
$656$		0			0
$657$	−254308.			−0.589154
$658$		0			0
$659$			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	$0.819764\pi$
$662$		0			0
$663$	−613697.			−1.39613
$664$		0			0
$665$	572787.	+	580038.i	1.29524	+	1.31164i
$666$		0			0
$667$		−	304311.i		−	0.684015i
$668$		0			0
$669$	332994.			0.744020
$670$		0			0
$671$	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	$-0.0814085\pi$
$674$		0			0
$675$			30531.6i			0.0670103i
$676$		0			0
$677$			38248.1i			0.0834512i	0.999129	+	0.0417256i	$0.0132855\pi$
$677$			38248.1i			0.0834512i	−0.999129	+	0.0417256i	$0.986714\pi$
$678$		0			0
$679$		−	1.12271e6i		−	2.43516i
$680$		0			0
$681$	−880274.			−1.89812
$682$		0			0
$683$			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
$686$		0			0
$687$		−	1.00899e6i		−	2.13783i
$688$		0			0
$689$	−152141.			−0.320484
$690$		0			0
$691$	−140291.			−0.293815			−0.146907	−	0.989150i	$-0.546932\pi$
$691$	−140291.			−0.293815			−0.146907	+	0.989150i	$0.546932\pi$
$692$		0			0
$693$	357956.			0.745355
$694$		0			0
$695$	719609.			1.48980
$696$		0			0
$697$			432660.i			0.890597i
$698$		0			0
$699$		−	36587.3i		−	0.0748818i
$700$		0			0
$701$	550797.			1.12087			0.560436	−	0.828198i	$-0.310634\pi$
$701$	550797.			1.12087			0.560436	+	0.828198i	$0.310634\pi$
$702$		0			0
$703$	77782.9	−	76810.5i	0.157389	−	0.155421i
$704$		0			0
$705$		−	237358.i		−	0.477557i
$706$		0			0
$707$	−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	$0.243293\pi$
$710$		0			0
$711$			45802.9i			0.0906053i
$712$		0			0
$713$		−	279166.i		−	0.549140i
$714$		0			0
$715$		−	697787.i		−	1.36493i
$716$		0			0
$717$		−	388903.i		−	0.756489i
$718$		0			0
$719$	−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
$722$		0			0
$723$	620985.			1.18797
$724$		0			0
$725$		−	71248.3i		−	0.135550i
$726$		0			0
$727$	30302.5			0.0573336			0.0286668	−	0.999589i	$-0.490874\pi$
$727$	30302.5			0.0573336			0.0286668	+	0.999589i	$0.490874\pi$
$728$		0			0
$729$	−128557.			−0.241902
$730$		0			0
$731$	−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	$0.437212\pi$
$734$		0			0
$735$		−	1.42304e6i		−	2.63417i
$736$		0			0
$737$			803252.i			1.47883i
$738$		0			0
$739$	257029.			0.470645			0.235323	−	0.971917i	$-0.424385\pi$
$739$	257029.			0.470645			0.235323	+	0.971917i	$0.424385\pi$
$740$		0			0
$741$	−636764.	−	644825.i	−1.15969	−	1.17437i
$742$		0			0
$743$			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
$746$		0			0
$747$	36964.5			0.0662435
$748$		0			0
$749$		−	1.45364e6i		−	2.59115i
$750$		0			0
$751$			705805.i			1.25142i	0.780054	+	0.625712i	$0.215192\pi$
$751$			705805.i			1.25142i	−0.780054	+	0.625712i	$0.784808\pi$
$752$		0			0
$753$			568305.i			1.00228i
$754$		0			0
$755$			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	$0.464306\pi$
$758$		0			0
$759$		−	334267.i		−	0.580244i
$760$		0			0
$761$	779962.			1.34680			0.673402	−	0.739277i	$-0.264833\pi$
$761$	779962.			1.34680			0.673402	+	0.739277i	$0.264833\pi$
$762$		0			0
$763$			17683.3i			0.0303749i
$764$		0			0
$765$	−232357.			−0.397039
$766$		0			0
$767$	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	$0.513655\pi$
$770$		0			0
$771$	922754.			1.55230
$772$		0			0
$773$			942086.i			1.57664i	0.615267	+	0.788318i	$0.289048\pi$
$773$			942086.i			1.57664i	−0.615267	+	0.788318i	$0.710952\pi$
$774$		0			0
$775$		−	65361.0i		−	0.108822i
$776$		0			0
$777$	−282271.			−0.467546
$778$		0			0
$779$	−454606.	+	448922.i	−0.749135	+	0.739769i
$780$		0			0
$781$		−	470592.i		−	0.771511i
$782$		0			0
$783$	548199.			0.894159
$784$		0			0
$785$	−149115.			−0.241981
$786$		0			0
$787$			764737.i			1.23470i	0.786687	+	0.617352i	$0.211794\pi$
$787$			764737.i			1.23470i	−0.786687	+	0.617352i	$0.788206\pi$
$788$		0			0
$789$			298186.i			0.478998i
$790$		0			0
$791$			368127.i			0.588362i
$792$		0			0
$793$			365557.i			0.581312i
$794$		0			0
$795$	−186365.			−0.294870
$796$		0			0
$797$		−	555222.i		−	0.874077i	−0.899443	−	0.437039i	$-0.856027\pi$
$797$		−	555222.i		−	0.874077i	0.899443	−	0.437039i	$-0.143973\pi$
$798$		0			0
$799$	−204314.			−0.320040
$800$		0			0
$801$			135752.i			0.211584i
$802$		0			0
$803$	−805270.			−1.24885
$804$		0			0
$805$	−607555.			−0.937549
$806$		0			0
$807$	637527.			0.978930
$808$		0			0
$809$	−94777.3			−0.144813			−0.0724065	−	0.997375i	$-0.523068\pi$
$809$	−94777.3			−0.144813			−0.0724065	+	0.997375i	$0.523068\pi$
$810$		0			0
$811$		−	68642.4i		−	0.104364i	−0.998638	−	0.0521820i	$-0.983382\pi$
$811$		−	68642.4i		−	0.104364i	0.998638	−	0.0521820i	$-0.0166176\pi$
$812$		0			0
$813$		−	1.08227e6i		−	1.63740i
$814$		0			0
$815$	−801330.			−1.20641
$816$		0			0
$817$	−587606.	−	595045.i	−0.880323	−	0.891467i
$818$		0			0
$819$			723281.i			1.07830i
$820$		0			0
$821$	601916.			0.892996			0.446498	−	0.894785i	$-0.352671\pi$
$821$	601916.			0.892996			0.446498	+	0.894785i	$0.352671\pi$
$822$		0			0
$823$	307730.			0.454329			0.227165	−	0.973856i	$-0.427054\pi$
$823$	307730.			0.454329			0.227165	+	0.973856i	$0.427054\pi$
$824$		0			0
$825$		−	78262.0i		−	0.114986i
$826$		0			0
$827$		−	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	$0.685598\pi$
$830$		0			0
$831$			717848.i			1.03951i
$832$		0			0
$833$	−1.22494e6			−1.76532
$834$		0			0
$835$		−	740765.i		−	1.06245i
$836$		0			0
$837$	502901.			0.717846
$838$		0			0
$839$			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
$842$		0			0
$843$	595754.			0.838324
$844$		0			0
$845$	660795.			0.925450
$846$		0			0
$847$	−126984.			−0.177004
$848$		0			0
$849$		−	114504.i		−	0.158857i
$850$		0			0
$851$			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	$0.266193\pi$
$854$		0			0
$855$	−241091.	−	244143.i	−0.329798	−	0.333973i
$856$		0			0
$857$		−	5569.67i		−	0.00758347i	−0.999993	−	0.00379173i	$-0.998793\pi$
$857$		−	5569.67i		−	0.00758347i	0.999993	−	0.00379173i	$-0.00120695\pi$
$858$		0			0
$859$	−312767.			−0.423871			−0.211936	−	0.977284i	$-0.567977\pi$
$859$	−312767.			−0.423871			−0.211936	+	0.977284i	$0.567977\pi$
$860$		0			0
$861$	1.64975e6			2.22541
$862$		0			0
$863$		−	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
$866$		0			0
$867$		−	257234.i		−	0.342208i
$868$		0			0
$869$			145036.i			0.192059i
$870$		0			0
$871$	−1.62304e6			−2.13940
$872$		0			0
$873$			472557.i			0.620049i
$874$		0			0
$875$	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	$0.912967\pi$
$878$		0			0
$879$	−102317.			−0.132425
$880$		0			0
$881$	−403508.			−0.519876			−0.259938	−	0.965625i	$-0.583702\pi$
$881$	−403508.			−0.519876			−0.259938	+	0.965625i	$0.583702\pi$
$882$		0			0
$883$	−265307.			−0.340273			−0.170137	−	0.985420i	$-0.554421\pi$
$883$	−265307.			−0.340273			−0.170137	+	0.985420i	$0.554421\pi$
$884$		0			0
$885$	1.39824e6			1.78523
$886$		0			0
$887$		−	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
$890$		0			0
$891$	938952.			1.18274
$892$		0			0
$893$	−211994.	−	214678.i	−0.265840	−	0.269205i
$894$		0			0
$895$			901551.i			1.12550i
$896$		0			0
$897$	675416.			0.839433
$898$		0			0
$899$	−1.17357e6			−1.45207
$900$		0			0
$901$			160421.i			0.197611i
$902$		0			0
$903$			2.15939e6i			2.64823i
$904$		0			0
$905$		−	765776.i		−	0.934985i
$906$		0			0
$907$		−	1.43037e6i		−	1.73874i	−0.494165	−	0.869368i	$-0.664526\pi$
$907$		−	1.43037e6i		−	1.73874i	0.494165	−	0.869368i	$-0.335474\pi$
$908$		0			0
$909$	172334.			0.208566
$910$		0			0
$911$		−	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
$914$		0			0
$915$			447790.i			0.534851i
$916$		0			0
$917$	−2.37665e6			−2.82636
$918$		0			0
$919$	627508.			0.742999			0.371500	−	0.928433i	$-0.378844\pi$
$919$	627508.			0.742999			0.371500	+	0.928433i	$0.378844\pi$
$920$		0			0
$921$	1.21943e6			1.43759
$922$		0			0
$923$	950871.			1.11614
$924$		0			0
$925$			19075.2i			0.0222939i
$926$		0			0
$927$		−	316263.i		−	0.368035i
$928$		0			0
$929$	−1.63447e6			−1.89385			−0.946924	−	0.321458i	$-0.895827\pi$
$929$	−1.63447e6			−1.89385			−0.946924	+	0.321458i	$0.895827\pi$
$930$		0			0
$931$	−1.27098e6	−	1.28707e6i	−1.46636	−	1.48492i
$932$		0			0
$933$			821585.i			0.943820i
$934$		0			0
$935$	−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	$0.510064\pi$
$938$		0			0
$939$		−	1.20143e6i		−	1.36259i
$940$		0			0
$941$		−	1.60327e6i		−	1.81062i	−0.424747	−	0.905312i	$-0.639637\pi$
$941$		−	1.60327e6i		−	1.81062i	0.424747	−	0.905312i	$-0.360363\pi$
$942$		0			0
$943$		−	476172.i		−	0.535476i
$944$		0			0
$945$		−	1.09448e6i		−	1.22558i
$946$		0			0
$947$	−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
$950$		0			0
$951$	−692642.			−0.765857
$952$		0			0
$953$			821794.i			0.904851i	0.891802	+	0.452426i	$0.149441\pi$
$953$			821794.i			0.904851i	−0.891802	+	0.452426i	$0.850559\pi$
$954$		0			0
$955$	354054.			0.388206
$956$		0			0
$957$	−1.40521e6			−1.53432
$958$		0			0
$959$	710276.			0.772307
$960$		0			0
$961$	−153074.			−0.165750
$962$		0			0
$963$			611847.i			0.659766i
$964$		0			0
$965$			546093.i			0.586424i
$966$		0			0
$967$	859901.			0.919593			0.459796	−	0.888024i	$-0.347922\pi$
$967$	859901.			0.919593			0.459796	+	0.888024i	$0.347922\pi$
$968$		0			0
$969$	−679918.	+	671418.i	−0.724118	+	0.715065i
$970$		0			0
$971$		−	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
$974$		0			0
$975$	158135.			0.166349
$976$		0			0
$977$			627667.i			0.657567i	0.944405	+	0.328784i	$0.106639\pi$
$977$			627667.i			0.657567i	−0.944405	+	0.328784i	$0.893361\pi$
$978$		0			0
$979$			429862.i			0.448502i
$980$		0			0
$981$		−	7443.05i		−	0.00773415i
$982$		0			0
$983$			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
$986$		0			0
$987$			779056.i			0.799714i
$988$		0			0
$989$	623273.			0.637215
$990$		0			0
$991$			977409.i			0.995243i	0.867394	+	0.497621i	$0.165793\pi$
$991$			977409.i			0.995243i	−0.867394	+	0.497621i	$0.834207\pi$
$992$		0			0
$993$	860388.			0.872561
$994$		0			0
$995$	−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	$0.252711\pi$
$998$		0			0
$999$	−146769.			−0.147063

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.5.b.a.37.4	✓	8	76.75	even	2
38.5.b.a.37.5	yes	8	4.3	odd	2
304.5.e.e.113.2		8	19.18	odd	2	inner
304.5.e.e.113.7		8	1.1	even	1	trivial
342.5.d.a.37.3		8	12.11	even	2
342.5.d.a.37.7		8	228.227	odd	2

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:	\( N \)	\(=\)	\( 304 = 2^{4} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	304.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+10.8276i q^{3} -26.2296 q^{5} +86.0910 q^{7} -36.2364 q^{9} +O(q^{10})\)	\(q+10.8276i q^{3} -26.2296 q^{5} +86.0910 q^{7} -36.2364 q^{9} -114.743 q^{11} -231.848i q^{13} -284.003i q^{15} -244.466 q^{17} +(-253.655 - 256.866i) q^{19} +932.157i q^{21} +269.052 q^{23} +62.9931 q^{25} +484.681i q^{27} -1131.05i q^{29} -1037.59i q^{31} -1242.39i q^{33} -2258.14 q^{35} +302.815i q^{37} +2510.36 q^{39} -1769.82i q^{41} +2316.56 q^{43} +950.467 q^{45} +835.757 q^{47} +5010.66 q^{49} -2646.98i q^{51} -656.208i q^{53} +3009.67 q^{55} +(2781.24 - 2746.47i) q^{57} +4923.33i q^{59} -1576.71 q^{61} -3119.63 q^{63} +6081.30i q^{65} -7000.44i q^{67} +2913.18i q^{69} +4101.26i q^{71} +7018.02 q^{73} +682.062i q^{75} -9878.36 q^{77} -1264.00i q^{79} -8183.07 q^{81} -1020.09 q^{83} +6412.25 q^{85} +12246.5 q^{87} -3746.30i q^{89} -19960.1i q^{91} +11234.6 q^{93} +(6653.27 + 6737.50i) q^{95} -13041.0i q^{97} +4157.88 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 18 q^{5} + 162 q^{7} - 268 q^{9}+O(q^{10}) \) 8 * q + 18 * q^5 + 162 * q^7 - 268 * q^9	\( 8 q + 18 q^{5} + 162 q^{7} - 268 q^{9} + 6 q^{11} + 510 q^{17} + 12 q^{19} + 396 q^{23} + 3458 q^{25} - 1002 q^{35} + 6588 q^{39} + 8654 q^{43} - 10334 q^{45} - 3210 q^{47} + 9222 q^{49} - 17146 q^{55} - 14076 q^{57} + 1314 q^{61} - 29938 q^{63} + 23398 q^{73} - 44622 q^{77} - 20368 q^{81} + 10440 q^{83} + 21274 q^{85} + 14316 q^{87} + 19416 q^{93} + 34686 q^{95} + 56798 q^{99}+O(q^{100}) \) 8 * q + 18 * q^5 + 162 * q^7 - 268 * q^9 + 6 * q^11 + 510 * q^17 + 12 * q^19 + 396 * q^23 + 3458 * q^25 - 1002 * q^35 + 6588 * q^39 + 8654 * q^43 - 10334 * q^45 - 3210 * q^47 + 9222 * q^49 - 17146 * q^55 - 14076 * q^57 + 1314 * q^61 - 29938 * q^63 + 23398 * q^73 - 44622 * q^77 - 20368 * q^81 + 10440 * q^83 + 21274 * q^85 + 14316 * q^87 + 19416 * q^93 + 34686 * q^95 + 56798 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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