LMFDB - Embedded newform 304.5.e.f.113.8

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:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 996 x^{18} + 408854 x^{16} + 89661524 x^{14} + 11414409521 x^{12} + 861580608848 x^{10} + \cdots + 34\!\cdots\!64 $ x^20 + 996x^18 + 408854x^16 + 89661524x^14 + 11414409521x^12 + 861580608848x^10 + 37911701888064x^8 + 915101881952256x^6 + 10365057927217152x^4 + 34336742150504448*x^2 + 34273490123096064
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{50} $
Twist minimal:	no (minimal twist has level 152)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			113.8
Root			$-5.59979i$ of defining polynomial
Character	$\chi$	$=$	304.113
Dual form			304.5.e.f.113.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.59979i q^{3} -33.0135 q^{5} +43.4389 q^{7} +49.6423 q^{9} +O(q^{10})$	$q-5.59979i q^{3} -33.0135 q^{5} +43.4389 q^{7} +49.6423 q^{9} -13.5135 q^{11} -132.348i q^{13} +184.869i q^{15} +368.451 q^{17} +(-230.370 + 277.940i) q^{19} -243.249i q^{21} -14.2094 q^{23} +464.892 q^{25} -731.570i q^{27} -307.715i q^{29} -273.075i q^{31} +75.6728i q^{33} -1434.07 q^{35} -1926.93i q^{37} -741.121 q^{39} -1399.62i q^{41} -2116.02 q^{43} -1638.87 q^{45} -1452.33 q^{47} -514.065 q^{49} -2063.25i q^{51} -4057.36i q^{53} +446.128 q^{55} +(1556.41 + 1290.02i) q^{57} +5227.64i q^{59} -2749.67 q^{61} +2156.41 q^{63} +4369.27i q^{65} -3422.30i q^{67} +79.5697i q^{69} +102.568i q^{71} -3647.48 q^{73} -2603.30i q^{75} -587.011 q^{77} -7136.82i q^{79} -75.6139 q^{81} -2017.01 q^{83} -12163.9 q^{85} -1723.14 q^{87} +7511.38i q^{89} -5749.04i q^{91} -1529.17 q^{93} +(7605.31 - 9175.79i) q^{95} -17507.9i q^{97} -670.841 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 32 q^{7} - 372 q^{9}+O(q^{10}) $ 20 * q - 32 * q^7 - 372 * q^9	$ 20 q - 32 q^{7} - 372 q^{9} + 24 q^{11} + 216 q^{17} + 596 q^{19} - 576 q^{23} + 1412 q^{25} + 144 q^{35} + 520 q^{39} + 1256 q^{43} + 7232 q^{45} + 3768 q^{47} - 2740 q^{49} + 10128 q^{55} - 728 q^{57} + 352 q^{61} - 6104 q^{63} + 1352 q^{73} + 9288 q^{77} - 4220 q^{81} + 16104 q^{83} + 10232 q^{85} - 2936 q^{87} + 36432 q^{93} - 14232 q^{95} - 760 q^{99}+O(q^{100}) $ 20 * q - 32 * q^7 - 372 * q^9 + 24 * q^11 + 216 * q^17 + 596 * q^19 - 576 * q^23 + 1412 * q^25 + 144 * q^35 + 520 * q^39 + 1256 * q^43 + 7232 * q^45 + 3768 * q^47 - 2740 * q^49 + 10128 * q^55 - 728 * q^57 + 352 * q^61 - 6104 * q^63 + 1352 * q^73 + 9288 * q^77 - 4220 * q^81 + 16104 * q^83 + 10232 * q^85 - 2936 * q^87 + 36432 * q^93 - 14232 * q^95 - 760 * 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
$3$		−	5.59979i		−	0.622199i	−0.950377	−	0.311100i	$-0.899303\pi$
$3$		−	5.59979i		−	0.622199i	0.950377	−	0.311100i	$-0.100697\pi$
$4$		0			0
$5$	−33.0135			−1.32054			−0.660270	−	0.751028i	$-0.729558\pi$
$5$	−33.0135			−1.32054			−0.660270	+	0.751028i	$0.729558\pi$
$6$		0			0
$7$	43.4389			0.886507			0.443254	−	0.896396i	$-0.353824\pi$
$7$	43.4389			0.886507			0.443254	+	0.896396i	$0.353824\pi$
$8$		0			0
$9$	49.6423			0.612868
$10$		0			0
$11$	−13.5135			−0.111682			−0.0558409	−	0.998440i	$-0.517784\pi$
$11$	−13.5135			−0.111682			−0.0558409	+	0.998440i	$0.517784\pi$
$12$		0			0
$13$		−	132.348i		−	0.783123i	−0.920152	−	0.391562i	$-0.871935\pi$
$13$		−	132.348i		−	0.783123i	0.920152	−	0.391562i	$-0.128065\pi$
$14$		0			0
$15$			184.869i			0.821640i
$16$		0			0
$17$	368.451			1.27492			0.637459	−	0.770484i	$-0.279986\pi$
$17$	368.451			1.27492			0.637459	+	0.770484i	$0.279986\pi$
$18$		0			0
$19$	−230.370	+	277.940i	−0.638143	+	0.769918i
$19$	−230.370	+	277.940i	−0.638143	+	0.769918i
$20$		0			0
$21$		−	243.249i		−	0.551584i
$22$		0			0
$23$	−14.2094			−0.0268609			−0.0134304	−	0.999910i	$-0.504275\pi$
$23$	−14.2094			−0.0268609			−0.0134304	+	0.999910i	$0.504275\pi$
$24$		0			0
$25$	464.892			0.743828
$26$		0			0
$27$		−	731.570i		−	1.00353i
$28$		0			0
$29$		−	307.715i		−	0.365892i	−0.983123	−	0.182946i	$-0.941437\pi$
$29$		−	307.715i		−	0.365892i	0.983123	−	0.182946i	$-0.0585633\pi$
$30$		0			0
$31$		−	273.075i		−	0.284157i	−0.989855	−	0.142079i	$-0.954621\pi$
$31$		−	273.075i		−	0.284157i	0.989855	−	0.142079i	$-0.0453786\pi$
$32$		0			0
$33$			75.6728i			0.0694883i
$34$		0			0
$35$	−1434.07			−1.17067
$36$		0			0
$37$		−	1926.93i		−	1.40754i	−0.710426	−	0.703772i	$-0.751498\pi$
$37$		−	1926.93i		−	1.40754i	0.710426	−	0.703772i	$-0.248502\pi$
$38$		0			0
$39$	−741.121			−0.487259
$40$		0			0
$41$		−	1399.62i		−	0.832611i	−0.909225	−	0.416306i	$-0.863325\pi$
$41$		−	1399.62i		−	0.832611i	0.909225	−	0.416306i	$-0.136675\pi$
$42$		0			0
$43$	−2116.02			−1.14441			−0.572207	−	0.820109i	$-0.693913\pi$
$43$	−2116.02			−1.14441			−0.572207	+	0.820109i	$0.693913\pi$
$44$		0			0
$45$	−1638.87			−0.809317
$46$		0			0
$47$	−1452.33			−0.657459			−0.328730	−	0.944424i	$-0.606620\pi$
$47$	−1452.33			−0.657459			−0.328730	+	0.944424i	$0.606620\pi$
$48$		0			0
$49$	−514.065			−0.214105
$50$		0			0
$51$		−	2063.25i		−	0.793253i
$52$		0			0
$53$		−	4057.36i		−	1.44442i	−0.691676	−	0.722208i	$-0.743127\pi$
$53$		−	4057.36i		−	1.44442i	0.691676	−	0.722208i	$-0.256873\pi$
$54$		0			0
$55$	446.128			0.147480
$56$		0			0
$57$	1556.41	+	1290.02i	0.479042	+	0.397052i
$58$		0			0
$59$			5227.64i			1.50176i	0.660437	+	0.750882i	$0.270371\pi$
$59$			5227.64i			1.50176i	−0.660437	+	0.750882i	$0.729629\pi$
$60$		0			0
$61$	−2749.67			−0.738961			−0.369480	−	0.929239i	$-0.620464\pi$
$61$	−2749.67			−0.738961			−0.369480	+	0.929239i	$0.620464\pi$
$62$		0			0
$63$	2156.41			0.543312
$64$		0			0
$65$			4369.27i			1.03415i
$66$		0			0
$67$		−	3422.30i		−	0.762374i	−0.924498	−	0.381187i	$-0.875515\pi$
$67$		−	3422.30i		−	0.762374i	0.924498	−	0.381187i	$-0.124485\pi$
$68$		0			0
$69$			79.5697i			0.0167128i
$70$		0			0
$71$			102.568i			0.0203468i	0.999948	+	0.0101734i	$0.00323835\pi$
$71$			102.568i			0.0203468i	−0.999948	+	0.0101734i	$0.996762\pi$
$72$		0			0
$73$	−3647.48			−0.684459			−0.342229	−	0.939616i	$-0.611182\pi$
$73$	−3647.48			−0.684459			−0.342229	+	0.939616i	$0.611182\pi$
$74$		0			0
$75$		−	2603.30i		−	0.462809i
$76$		0			0
$77$	−587.011			−0.0990067
$78$		0			0
$79$		−	7136.82i		−	1.14354i	−0.820415	−	0.571769i	$-0.806257\pi$
$79$		−	7136.82i		−	1.14354i	0.820415	−	0.571769i	$-0.193743\pi$
$80$		0			0
$81$	−75.6139			−0.0115247
$82$		0			0
$83$	−2017.01			−0.292786			−0.146393	−	0.989226i	$-0.546766\pi$
$83$	−2017.01			−0.292786			−0.146393	+	0.989226i	$0.546766\pi$
$84$		0			0
$85$	−12163.9			−1.68358
$86$		0			0
$87$	−1723.14			−0.227658
$88$		0			0
$89$			7511.38i			0.948287i	0.880448	+	0.474143i	$0.157242\pi$
$89$			7511.38i			0.948287i	−0.880448	+	0.474143i	$0.842758\pi$
$90$		0			0
$91$		−	5749.04i		−	0.694245i
$92$		0			0
$93$	−1529.17			−0.176803
$94$		0			0
$95$	7605.31	−	9175.79i	0.842693	−	1.01671i
$96$		0			0
$97$		−	17507.9i		−	1.86076i	−0.366596	−	0.930380i	$-0.619477\pi$
$97$		−	17507.9i		−	1.86076i	0.366596	−	0.930380i	$-0.380523\pi$
$98$		0			0
$99$	−670.841			−0.0684462
$100$		0			0
$101$	3897.81			0.382101			0.191050	−	0.981580i	$-0.438811\pi$
$101$	3897.81			0.382101			0.191050	+	0.981580i	$0.438811\pi$
$102$		0			0
$103$		−	16132.9i		−	1.52068i	−0.649525	−	0.760340i	$-0.725032\pi$
$103$		−	16132.9i		−	1.52068i	0.649525	−	0.760340i	$-0.274968\pi$
$104$		0			0
$105$			8030.49i			0.728390i
$106$		0			0
$107$			13844.6i			1.20924i	0.796515	+	0.604618i	$0.206674\pi$
$107$			13844.6i			1.20924i	−0.796515	+	0.604618i	$0.793326\pi$
$108$		0			0
$109$		−	4124.31i		−	0.347135i	−0.984822	−	0.173568i	$-0.944470\pi$
$109$		−	4124.31i		−	0.347135i	0.984822	−	0.173568i	$-0.0555295\pi$
$110$		0			0
$111$	−10790.4			−0.875773
$112$		0			0
$113$		−	10090.1i		−	0.790202i	−0.918638	−	0.395101i	$-0.870710\pi$
$113$		−	10090.1i		−	0.790202i	0.918638	−	0.395101i	$-0.129290\pi$
$114$		0			0
$115$	469.102			0.0354709
$116$		0			0
$117$		−	6570.05i		−	0.479951i
$118$		0			0
$119$	16005.1			1.13022
$120$		0			0
$121$	−14458.4			−0.987527
$122$		0			0
$123$	−7837.58			−0.518050
$124$		0			0
$125$	5285.72			0.338286
$126$		0			0
$127$		−	5561.06i		−	0.344786i	−0.985028	−	0.172393i	$-0.944850\pi$
$127$		−	5561.06i		−	0.344786i	0.985028	−	0.172393i	$-0.0551499\pi$
$128$		0			0
$129$			11849.3i			0.712054i
$130$		0			0
$131$	9099.49			0.530242			0.265121	−	0.964215i	$-0.414588\pi$
$131$	9099.49			0.530242			0.265121	+	0.964215i	$0.414588\pi$
$132$		0			0
$133$	−10007.0	+	12073.4i	−0.565718	+	0.682538i
$134$		0			0
$135$			24151.7i			1.32520i
$136$		0			0
$137$	25680.9			1.36826			0.684132	−	0.729359i	$-0.260181\pi$
$137$	25680.9			1.36826			0.684132	+	0.729359i	$0.260181\pi$
$138$		0			0
$139$	23519.1			1.21728			0.608641	−	0.793445i	$-0.291715\pi$
$139$	23519.1			1.21728			0.608641	+	0.793445i	$0.291715\pi$
$140$		0			0
$141$			8132.73i			0.409071i
$142$		0			0
$143$			1788.48i			0.0874606i
$144$		0			0
$145$			10158.8i			0.483175i
$146$		0			0
$147$			2878.66i			0.133216i
$148$		0			0
$149$	11034.3			0.497020			0.248510	−	0.968629i	$-0.420059\pi$
$149$	11034.3			0.497020			0.248510	+	0.968629i	$0.420059\pi$
$150$		0			0
$151$			37716.8i			1.65417i	0.562074	+	0.827087i	$0.310004\pi$
$151$			37716.8i			1.65417i	−0.562074	+	0.827087i	$0.689996\pi$
$152$		0			0
$153$	18290.8			0.781356
$154$		0			0
$155$			9015.18i			0.375242i
$156$		0			0
$157$	34740.4			1.40940			0.704702	−	0.709503i	$-0.251081\pi$
$157$	34740.4			1.40940			0.704702	+	0.709503i	$0.251081\pi$
$158$		0			0
$159$	−22720.4			−0.898714
$160$		0			0
$161$	−617.240			−0.0238124
$162$		0			0
$163$	−38280.9			−1.44081			−0.720405	−	0.693553i	$-0.756044\pi$
$163$	−38280.9			−1.44081			−0.720405	+	0.693553i	$0.756044\pi$
$164$		0			0
$165$		−	2498.23i		−	0.0917622i
$166$		0			0
$167$			10534.4i			0.377725i	0.982004	+	0.188862i	$0.0604800\pi$
$167$			10534.4i			0.377725i	−0.982004	+	0.188862i	$0.939520\pi$
$168$		0			0
$169$	11045.0			0.386718
$170$		0			0
$171$	−11436.1	+	13797.6i	−0.391097	+	0.471858i
$172$		0			0
$173$		−	14002.2i		−	0.467847i	−0.972255	−	0.233923i	$-0.924844\pi$
$173$		−	14002.2i		−	0.467847i	0.972255	−	0.233923i	$-0.0751565\pi$
$174$		0			0
$175$	20194.4			0.659409
$176$		0			0
$177$	29273.7			0.934396
$178$		0			0
$179$			7303.14i			0.227931i	0.993485	+	0.113966i	$0.0363554\pi$
$179$			7303.14i			0.227931i	−0.993485	+	0.113966i	$0.963645\pi$
$180$		0			0
$181$		−	24214.3i		−	0.739120i	−0.929207	−	0.369560i	$-0.879508\pi$
$181$		−	24214.3i		−	0.739120i	0.929207	−	0.369560i	$-0.120492\pi$
$182$		0			0
$183$			15397.6i			0.459781i
$184$		0			0
$185$			63614.7i			1.85872i
$186$		0			0
$187$	−4979.07			−0.142385
$188$		0			0
$189$		−	31778.6i		−	0.889633i
$190$		0			0
$191$	−38117.1			−1.04485			−0.522424	−	0.852686i	$-0.674972\pi$
$191$	−38117.1			−1.04485			−0.522424	+	0.852686i	$0.674972\pi$
$192$		0			0
$193$			18613.1i			0.499695i	0.968285	+	0.249847i	$0.0803805\pi$
$193$			18613.1i			0.499695i	−0.968285	+	0.249847i	$0.919620\pi$
$194$		0			0
$195$	24467.0			0.643445
$196$		0			0
$197$	−57877.0			−1.49133			−0.745665	−	0.666321i	$-0.767868\pi$
$197$	−57877.0			−1.49133			−0.745665	+	0.666321i	$0.767868\pi$
$198$		0			0
$199$	22962.6			0.579848			0.289924	−	0.957050i	$-0.406370\pi$
$199$	22962.6			0.579848			0.289924	+	0.957050i	$0.406370\pi$
$200$		0			0
$201$	−19164.2			−0.474348
$202$		0			0
$203$		−	13366.8i		−	0.324366i
$204$		0			0
$205$			46206.4i			1.09950i
$206$		0			0
$207$	−705.387			−0.0164622
$208$		0			0
$209$	3113.10	−	3755.95i	0.0712689	−	0.0859858i
$210$		0			0
$211$		−	21390.9i		−	0.480468i	−0.970715	−	0.240234i	$-0.922776\pi$
$211$		−	21390.9i		−	0.480468i	0.970715	−	0.240234i	$-0.0772241\pi$
$212$		0			0
$213$	574.361			0.0126598
$214$		0			0
$215$	69857.3			1.51125
$216$		0			0
$217$		−	11862.1i		−	0.251908i
$218$		0			0
$219$			20425.1i			0.425870i
$220$		0			0
$221$		−	48763.7i		−	0.998418i
$222$		0			0
$223$		−	33873.4i		−	0.681161i	−0.940215	−	0.340580i	$-0.889376\pi$
$223$		−	33873.4i		−	0.681161i	0.940215	−	0.340580i	$-0.110624\pi$
$224$		0			0
$225$	23078.3			0.455868
$226$		0			0
$227$			34256.6i			0.664802i	0.943138	+	0.332401i	$0.107859\pi$
$227$			34256.6i			0.664802i	−0.943138	+	0.332401i	$0.892141\pi$
$228$		0			0
$229$	−37770.7			−0.720251			−0.360126	−	0.932904i	$-0.617266\pi$
$229$	−37770.7			−0.720251			−0.360126	+	0.932904i	$0.617266\pi$
$230$		0			0
$231$			3287.14i			0.0616019i
$232$		0			0
$233$	97710.6			1.79982			0.899912	−	0.436072i	$-0.143631\pi$
$233$	97710.6			1.79982			0.899912	+	0.436072i	$0.143631\pi$
$234$		0			0
$235$	47946.4			0.868202
$236$		0			0
$237$	−39964.7			−0.711508
$238$		0			0
$239$	−52760.4			−0.923660			−0.461830	−	0.886969i	$-0.652807\pi$
$239$	−52760.4			−0.923660			−0.461830	+	0.886969i	$0.652807\pi$
$240$		0			0
$241$		−	12018.5i		−	0.206926i	−0.994633	−	0.103463i	$-0.967008\pi$
$241$		−	12018.5i		−	0.206926i	0.994633	−	0.103463i	$-0.0329924\pi$
$242$		0			0
$243$		−	58833.7i		−	0.996355i
$244$		0			0
$245$	16971.1			0.282734
$246$		0			0
$247$	36784.8	+	30488.9i	0.602941	+	0.499744i
$248$		0			0
$249$			11294.8i			0.182171i
$250$		0			0
$251$	72886.6			1.15691			0.578456	−	0.815714i	$-0.303655\pi$
$251$	72886.6			1.15691			0.578456	+	0.815714i	$0.303655\pi$
$252$		0			0
$253$	192.019			0.00299987
$254$		0			0
$255$			68115.2i			1.04752i
$256$		0			0
$257$			106172.i			1.60748i	0.594981	+	0.803740i	$0.297160\pi$
$257$			106172.i			1.60748i	−0.594981	+	0.803740i	$0.702840\pi$
$258$		0			0
$259$		−	83703.6i		−	1.24780i
$260$		0			0
$261$		−	15275.7i		−	0.224243i
$262$		0			0
$263$	−93242.0			−1.34803			−0.674016	−	0.738717i	$-0.735432\pi$
$263$	−93242.0			−1.34803			−0.674016	+	0.738717i	$0.735432\pi$
$264$		0			0
$265$			133948.i			1.90741i
$266$		0			0
$267$	42062.2			0.590023
$268$		0			0
$269$		−	11505.1i		−	0.158995i	−0.996835	−	0.0794976i	$-0.974668\pi$
$269$		−	11505.1i		−	0.158995i	0.996835	−	0.0794976i	$-0.0253316\pi$
$270$		0			0
$271$	48554.0			0.661129			0.330565	−	0.943783i	$-0.392761\pi$
$271$	48554.0			0.661129			0.330565	+	0.943783i	$0.392761\pi$
$272$		0			0
$273$	−32193.4			−0.431959
$274$		0			0
$275$	−6282.32			−0.0830720
$276$		0			0
$277$	−97597.0			−1.27197			−0.635985	−	0.771701i	$-0.719406\pi$
$277$	−97597.0			−1.27197			−0.635985	+	0.771701i	$0.719406\pi$
$278$		0			0
$279$		−	13556.1i		−	0.174151i
$280$		0			0
$281$			12611.1i			0.159713i	0.996806	+	0.0798563i	$0.0254462\pi$
$281$			12611.1i			0.159713i	−0.996806	+	0.0798563i	$0.974554\pi$
$282$		0			0
$283$	109327.			1.36506			0.682532	−	0.730855i	$-0.260879\pi$
$283$	109327.			1.36506			0.682532	+	0.730855i	$0.260879\pi$
$284$		0			0
$285$	−51382.5	−	42588.2i	−0.632595	−	0.524323i
$286$		0			0
$287$		−	60797.9i		−	0.738116i
$288$		0			0
$289$	52235.4			0.625416
$290$		0			0
$291$	−98040.6			−1.15776
$292$		0			0
$293$		−	131519.i		−	1.53198i	−0.642851	−	0.765991i	$-0.722249\pi$
$293$		−	131519.i		−	1.53198i	0.642851	−	0.765991i	$-0.277751\pi$
$294$		0			0
$295$		−	172583.i		−	1.98314i
$296$		0			0
$297$			9886.07i			0.112076i
$298$		0			0
$299$			1880.58i			0.0210354i
$300$		0			0
$301$	−91917.6			−1.01453
$302$		0			0
$303$		−	21826.9i		−	0.237743i
$304$		0			0
$305$	90776.3			0.975827
$306$		0			0
$307$		−	135453.i		−	1.43719i	−0.695430	−	0.718594i	$-0.744786\pi$
$307$		−	135453.i		−	1.43719i	0.695430	−	0.718594i	$-0.255214\pi$
$308$		0			0
$309$	−90340.9			−0.946166
$310$		0			0
$311$	−92130.5			−0.952539			−0.476269	−	0.879299i	$-0.658011\pi$
$311$	−92130.5			−0.952539			−0.476269	+	0.879299i	$0.658011\pi$
$312$		0			0
$313$	−108339.			−1.10585			−0.552927	−	0.833230i	$-0.686489\pi$
$313$	−108339.			−1.10585			−0.552927	+	0.833230i	$0.686489\pi$
$314$		0			0
$315$	−71190.5			−0.717466
$316$		0			0
$317$		−	176919.i		−	1.76058i	−0.474433	−	0.880291i	$-0.657347\pi$
$317$		−	176919.i		−	1.76058i	0.474433	−	0.880291i	$-0.342653\pi$
$318$		0			0
$319$			4158.30i			0.0408634i
$320$		0			0
$321$	77526.6			0.752386
$322$		0			0
$323$	−84880.0	+	102408.i	−0.813580	+	0.981582i
$324$		0			0
$325$		−	61527.5i		−	0.582509i
$326$		0			0
$327$	−23095.3			−0.215987
$328$		0			0
$329$	−63087.5			−0.582842
$330$		0			0
$331$			151026.i			1.37846i	0.724541	+	0.689232i	$0.242052\pi$
$331$			151026.i			1.37846i	−0.724541	+	0.689232i	$0.757948\pi$
$332$		0			0
$333$		−	95657.1i		−	0.862639i
$334$		0			0
$335$			112982.i			1.00675i
$336$		0			0
$337$			84147.4i			0.740936i	0.928845	+	0.370468i	$0.120803\pi$
$337$			84147.4i			0.740936i	−0.928845	+	0.370468i	$0.879197\pi$
$338$		0			0
$339$	−56502.4			−0.491663
$340$		0			0
$341$			3690.20i			0.0317352i
$342$		0			0
$343$	−126627.			−1.07631
$344$		0			0
$345$		−	2626.88i		−	0.0220699i
$346$		0			0
$347$	−60745.3			−0.504491			−0.252245	−	0.967663i	$-0.581169\pi$
$347$	−60745.3			−0.504491			−0.252245	+	0.967663i	$0.581169\pi$
$348$		0			0
$349$	−62942.7			−0.516767			−0.258383	−	0.966042i	$-0.583190\pi$
$349$	−62942.7			−0.516767			−0.258383	+	0.966042i	$0.583190\pi$
$350$		0			0
$351$	−96821.7			−0.785884
$352$		0			0
$353$	106908.			0.857946			0.428973	−	0.903317i	$-0.358876\pi$
$353$	106908.			0.857946			0.428973	+	0.903317i	$0.358876\pi$
$354$		0			0
$355$		−	3386.14i		−	0.0268688i
$356$		0			0
$357$		−	89625.3i		−	0.703225i
$358$		0			0
$359$	87383.3			0.678015			0.339008	−	0.940784i	$-0.389909\pi$
$359$	87383.3			0.678015			0.339008	+	0.940784i	$0.389909\pi$
$360$		0			0
$361$	−24180.8	−	128058.i	−0.185548	−	0.982635i
$362$		0			0
$363$			80964.0i			0.614439i
$364$		0			0
$365$	120416.			0.903856
$366$		0			0
$367$	85274.3			0.633120			0.316560	−	0.948573i	$-0.397472\pi$
$367$	85274.3			0.633120			0.316560	+	0.948573i	$0.397472\pi$
$368$		0			0
$369$		−	69480.3i		−	0.510281i
$370$		0			0
$371$		−	176247.i		−	1.28049i
$372$		0			0
$373$			194985.i			1.40147i	0.713422	+	0.700734i	$0.247144\pi$
$373$			194985.i			1.40147i	−0.713422	+	0.700734i	$0.752856\pi$
$374$		0			0
$375$		−	29598.9i		−	0.210481i
$376$		0			0
$377$	−40725.4			−0.286538
$378$		0			0
$379$		−	48409.0i		−	0.337013i	−0.985701	−	0.168507i	$-0.946106\pi$
$379$		−	48409.0i		−	0.337013i	0.985701	−	0.168507i	$-0.0538945\pi$
$380$		0			0
$381$	−31140.8			−0.214526
$382$		0			0
$383$		−	97079.7i		−	0.661806i	−0.943665	−	0.330903i	$-0.892647\pi$
$383$		−	97079.7i		−	0.661806i	0.943665	−	0.330903i	$-0.107353\pi$
$384$		0			0
$385$	19379.3			0.130742
$386$		0			0
$387$	−105044.			−0.701375
$388$		0			0
$389$	42408.2			0.280254			0.140127	−	0.990134i	$-0.455249\pi$
$389$	42408.2			0.280254			0.140127	+	0.990134i	$0.455249\pi$
$390$		0			0
$391$	−5235.47			−0.0342454
$392$		0			0
$393$		−	50955.2i		−	0.329916i
$394$		0			0
$395$			235611.i			1.51009i
$396$		0			0
$397$	161847.			1.02689			0.513443	−	0.858123i	$-0.328370\pi$
$397$	161847.			1.02689			0.513443	+	0.858123i	$0.328370\pi$
$398$		0			0
$399$	67608.6	+	56037.1i	0.424675	+	0.351990i
$400$		0			0
$401$		−	163782.i		−	1.01854i	−0.860608	−	0.509268i	$-0.829916\pi$
$401$		−	163782.i		−	1.01854i	0.860608	−	0.509268i	$-0.170084\pi$
$402$		0			0
$403$	−36140.9			−0.222530
$404$		0			0
$405$	2496.28			0.0152189
$406$		0			0
$407$			26039.5i			0.157197i
$408$		0			0
$409$			96082.2i			0.574376i	0.957874	+	0.287188i	$0.0927205\pi$
$409$			96082.2i			0.574376i	−0.957874	+	0.287188i	$0.907280\pi$
$410$		0			0
$411$		−	143808.i		−	0.851332i
$412$		0			0
$413$			227083.i			1.33132i
$414$		0			0
$415$	66588.4			0.386636
$416$		0			0
$417$		−	131702.i		−	0.757393i
$418$		0			0
$419$	154730.			0.881347			0.440673	−	0.897668i	$-0.354740\pi$
$419$	154730.			0.881347			0.440673	+	0.897668i	$0.354740\pi$
$420$		0			0
$421$			173315.i			0.977852i	0.872325	+	0.488926i	$0.162611\pi$
$421$			173315.i			0.977852i	−0.872325	+	0.488926i	$0.837389\pi$
$422$		0			0
$423$	−72096.9			−0.402936
$424$		0			0
$425$	171290.			0.948319
$426$		0			0
$427$	−119443.			−0.655094
$428$		0			0
$429$	10015.1			0.0544179
$430$		0			0
$431$			145250.i			0.781916i	0.920408	+	0.390958i	$0.127856\pi$
$431$			145250.i			0.781916i	−0.920408	+	0.390958i	$0.872144\pi$
$432$		0			0
$433$		−	104948.i		−	0.559754i	−0.960036	−	0.279877i	$-0.909706\pi$
$433$		−	104948.i		−	0.559754i	0.960036	−	0.279877i	$-0.0902938\pi$
$434$		0			0
$435$	56886.9			0.300631
$436$		0			0
$437$	3273.41	−	3949.37i	0.0171411	−	0.0206807i
$438$		0			0
$439$			7268.99i			0.0377177i	0.999822	+	0.0188588i	$0.00600331\pi$
$439$			7268.99i			0.0377177i	−0.999822	+	0.0188588i	$0.993997\pi$
$440$		0			0
$441$	−25519.4			−0.131218
$442$		0			0
$443$	−244091.			−1.24378			−0.621892	−	0.783103i	$-0.713636\pi$
$443$	−244091.			−1.24378			−0.621892	+	0.783103i	$0.713636\pi$
$444$		0			0
$445$		−	247977.i		−	1.25225i
$446$		0			0
$447$		−	61790.1i		−	0.309246i
$448$		0			0
$449$			258942.i			1.28443i	0.766526	+	0.642213i	$0.221983\pi$
$449$			258942.i			1.28443i	−0.766526	+	0.642213i	$0.778017\pi$
$450$		0			0
$451$			18913.8i			0.0929875i
$452$		0			0
$453$	211206.			1.02923
$454$		0			0
$455$			189796.i			0.916778i
$456$		0			0
$457$	−178433.			−0.854362			−0.427181	−	0.904166i	$-0.640493\pi$
$457$	−178433.			−0.854362			−0.427181	+	0.904166i	$0.640493\pi$
$458$		0			0
$459$		−	269548.i		−	1.27941i
$460$		0			0
$461$	−194872.			−0.916956			−0.458478	−	0.888706i	$-0.651605\pi$
$461$	−194872.			−0.916956			−0.458478	+	0.888706i	$0.651605\pi$
$462$		0			0
$463$	306514.			1.42984			0.714921	−	0.699205i	$-0.246463\pi$
$463$	306514.			1.42984			0.714921	+	0.699205i	$0.246463\pi$
$464$		0			0
$465$	50483.1			0.233475
$466$		0			0
$467$	283160.			1.29837			0.649186	−	0.760630i	$-0.275110\pi$
$467$	283160.			1.29837			0.649186	+	0.760630i	$0.275110\pi$
$468$		0			0
$469$		−	148661.i		−	0.675850i
$470$		0			0
$471$		−	194539.i		−	0.876930i
$472$		0			0
$473$	28594.9			0.127810
$474$		0			0
$475$	−107097.	+	129212.i	−0.474668	+	0.572686i
$476$		0			0
$477$		−	201417.i		−	0.885236i
$478$		0			0
$479$	348624.			1.51945			0.759726	−	0.650244i	$-0.225333\pi$
$479$	348624.			1.51945			0.759726	+	0.650244i	$0.225333\pi$
$480$		0			0
$481$	−255025.			−1.10228
$482$		0			0
$483$			3456.42i			0.0148160i
$484$		0			0
$485$			577997.i			2.45721i
$486$		0			0
$487$			3191.16i			0.0134552i	0.999977	+	0.00672760i	$0.00214148\pi$
$487$			3191.16i			0.0134552i	−0.999977	+	0.00672760i	$0.997859\pi$
$488$		0			0
$489$			214365.i			0.896472i
$490$		0			0
$491$	364011.			1.50991			0.754955	−	0.655777i	$-0.227659\pi$
$491$	364011.			1.50991			0.754955	+	0.655777i	$0.227659\pi$
$492$		0			0
$493$		−	113378.i		−	0.466482i
$494$		0			0
$495$	22146.8			0.0903860
$496$		0			0
$497$			4455.45i			0.0180376i
$498$		0			0
$499$	7596.44			0.0305077			0.0152538	−	0.999884i	$-0.495144\pi$
$499$	7596.44			0.0305077			0.0152538	+	0.999884i	$0.495144\pi$
$500$		0			0
$501$	58990.3			0.235020
$502$		0			0
$503$	132493.			0.523670			0.261835	−	0.965113i	$-0.415672\pi$
$503$	132493.			0.523670			0.261835	+	0.965113i	$0.415672\pi$
$504$		0			0
$505$	−128680.			−0.504580
$506$		0			0
$507$		−	61850.0i		−	0.240616i
$508$		0			0
$509$			412802.i			1.59333i	0.604420	+	0.796666i	$0.293405\pi$
$509$			412802.i			1.59333i	−0.604420	+	0.796666i	$0.706595\pi$
$510$		0			0
$511$	−158442.			−0.606778
$512$		0			0
$513$	203333.	+	168531.i	0.772632	+	0.640392i
$514$		0			0
$515$			532604.i			2.00812i
$516$		0			0
$517$	19626.0			0.0734262
$518$		0			0
$519$	−78409.3			−0.291094
$520$		0			0
$521$			183709.i			0.676792i	0.941004	+	0.338396i	$0.109884\pi$
$521$			183709.i			0.676792i	−0.941004	+	0.338396i	$0.890116\pi$
$522$		0			0
$523$			15258.3i			0.0557832i	0.999611	+	0.0278916i	$0.00887933\pi$
$523$			15258.3i			0.0557832i	−0.999611	+	0.0278916i	$0.991121\pi$
$524$		0			0
$525$		−	113084.i		−	0.410284i
$526$		0			0
$527$		−	100615.i		−	0.362277i
$528$		0			0
$529$	−279639.			−0.999278
$530$		0			0
$531$			259512.i			0.920383i
$532$		0			0
$533$	−185237.			−0.652037
$534$		0			0
$535$		−	457057.i		−	1.59685i
$536$		0			0
$537$	40896.1			0.141819
$538$		0			0
$539$	6946.82			0.0239116
$540$		0			0
$541$	145622.			0.497546			0.248773	−	0.968562i	$-0.419973\pi$
$541$	145622.			0.497546			0.248773	+	0.968562i	$0.419973\pi$
$542$		0			0
$543$	−135595.			−0.459880
$544$		0			0
$545$			136158.i			0.458406i
$546$		0			0
$547$		−	508520.i		−	1.69955i	−0.527147	−	0.849774i	$-0.676738\pi$
$547$		−	508520.i		−	1.69955i	0.527147	−	0.849774i	$-0.323262\pi$
$548$		0			0
$549$	−136500.			−0.452885
$550$		0			0
$551$	85526.4	+	70888.1i	0.281707	+	0.233491i
$552$		0			0
$553$		−	310015.i		−	1.01375i
$554$		0			0
$555$	356229.			1.15649
$556$		0			0
$557$	38107.9			0.122830			0.0614151	−	0.998112i	$-0.480439\pi$
$557$	38107.9			0.122830			0.0614151	+	0.998112i	$0.480439\pi$
$558$		0			0
$559$			280051.i			0.896218i
$560$		0			0
$561$			27881.7i			0.0885919i
$562$		0			0
$563$			450461.i			1.42115i	0.703621	+	0.710575i	$0.251565\pi$
$563$			450461.i			1.42115i	−0.703621	+	0.710575i	$0.748435\pi$
$564$		0			0
$565$			333109.i			1.04349i
$566$		0			0
$567$	−3284.58			−0.0102168
$568$		0			0
$569$		−	40952.5i		−	0.126490i	−0.997998	−	0.0632449i	$-0.979855\pi$
$569$		−	40952.5i		−	0.126490i	0.997998	−	0.0632449i	$-0.0201449\pi$
$570$		0			0
$571$	−400930.			−1.22969			−0.614846	−	0.788647i	$-0.710782\pi$
$571$	−400930.			−1.22969			−0.614846	+	0.788647i	$0.710782\pi$
$572$		0			0
$573$			213448.i			0.650103i
$574$		0			0
$575$	−6605.84			−0.0199799
$576$		0			0
$577$	494717.			1.48595			0.742977	−	0.669316i	$-0.233413\pi$
$577$	494717.			1.48595			0.742977	+	0.669316i	$0.233413\pi$
$578$		0			0
$579$	104230.			0.310910
$580$		0			0
$581$	−87616.4			−0.259557
$582$		0			0
$583$			54829.2i			0.161315i
$584$		0			0
$585$			216901.i			0.633795i
$586$		0			0
$587$	−26197.7			−0.0760304			−0.0380152	−	0.999277i	$-0.512104\pi$
$587$	−26197.7			−0.0760304			−0.0380152	+	0.999277i	$0.512104\pi$
$588$		0			0
$589$	75898.7	+	62908.2i	0.218778	+	0.181333i
$590$		0			0
$591$			324100.i			0.927905i
$592$		0			0
$593$	430078.			1.22303			0.611516	−	0.791232i	$-0.290560\pi$
$593$	430078.			1.22303			0.611516	+	0.791232i	$0.290560\pi$
$594$		0			0
$595$	−528385.			−1.49251
$596$		0			0
$597$		−	128586.i		−	0.360781i
$598$		0			0
$599$		−	457770.i		−	1.27583i	−0.770106	−	0.637916i	$-0.779796\pi$
$599$		−	457770.i		−	1.27583i	0.770106	−	0.637916i	$-0.220204\pi$
$600$		0			0
$601$		−	250433.i		−	0.693333i	−0.937988	−	0.346666i	$-0.887314\pi$
$601$		−	250433.i		−	0.693333i	0.937988	−	0.346666i	$-0.112686\pi$
$602$		0			0
$603$		−	169891.i		−	0.467235i
$604$		0			0
$605$	477322.			1.30407
$606$		0			0
$607$			523844.i			1.42176i	0.703316	+	0.710878i	$0.251702\pi$
$607$			523844.i			1.42176i	−0.703316	+	0.710878i	$0.748298\pi$
$608$		0			0
$609$	−74851.2			−0.201820
$610$		0			0
$611$			192212.i			0.514872i
$612$		0			0
$613$	43702.8			0.116302			0.0581512	−	0.998308i	$-0.481479\pi$
$613$	43702.8			0.116302			0.0581512	+	0.998308i	$0.481479\pi$
$614$		0			0
$615$	258746.			0.684106
$616$		0			0
$617$	74270.8			0.195096			0.0975479	−	0.995231i	$-0.468900\pi$
$617$	74270.8			0.195096			0.0975479	+	0.995231i	$0.468900\pi$
$618$		0			0
$619$	713118.			1.86114			0.930572	−	0.366110i	$-0.119311\pi$
$619$	713118.			1.86114			0.930572	+	0.366110i	$0.119311\pi$
$620$		0			0
$621$			10395.2i			0.0269556i
$622$		0			0
$623$			326286.i			0.840663i
$624$		0			0
$625$	−465058.			−1.19055
$626$		0			0
$627$	−21032.5	−	17432.7i	−0.0535003	−	0.0443435i
$628$		0			0
$629$		−	709979.i		−	1.79450i
$630$		0			0
$631$	−639765.			−1.60680			−0.803399	−	0.595441i	$-0.796977\pi$
$631$	−639765.			−1.60680			−0.803399	+	0.595441i	$0.796977\pi$
$632$		0			0
$633$	−119785.			−0.298947
$634$		0			0
$635$			183590.i			0.455304i
$636$		0			0
$637$			68035.4i			0.167670i
$638$		0			0
$639$			5091.72i			0.0124699i
$640$		0			0
$641$			213077.i			0.518586i	0.965799	+	0.259293i	$0.0834895\pi$
$641$			213077.i			0.518586i	−0.965799	+	0.259293i	$0.916511\pi$
$642$		0			0
$643$	517082.			1.25065			0.625327	−	0.780363i	$-0.284966\pi$
$643$	517082.			1.25065			0.625327	+	0.780363i	$0.284966\pi$
$644$		0			0
$645$		−	391187.i		−	0.940296i
$646$		0			0
$647$	−276646.			−0.660869			−0.330435	−	0.943829i	$-0.607195\pi$
$647$	−276646.			−0.660869			−0.330435	+	0.943829i	$0.607195\pi$
$648$		0			0
$649$		−	70643.7i		−	0.167720i
$650$		0			0
$651$	−66425.2			−0.156737
$652$		0			0
$653$	399123.			0.936010			0.468005	−	0.883726i	$-0.344973\pi$
$653$	399123.			0.936010			0.468005	+	0.883726i	$0.344973\pi$
$654$		0			0
$655$	−300406.			−0.700206
$656$		0			0
$657$	−181069.			−0.419483
$658$		0			0
$659$		−	719067.i		−	1.65576i	−0.560902	−	0.827882i	$-0.689545\pi$
$659$		−	719067.i		−	1.65576i	0.560902	−	0.827882i	$-0.310455\pi$
$660$		0			0
$661$			535751.i			1.22620i	0.790006	+	0.613099i	$0.210077\pi$
$661$			535751.i			1.22620i	−0.790006	+	0.613099i	$0.789923\pi$
$662$		0			0
$663$	−273067.			−0.621215
$664$		0			0
$665$	330366.	−	398586.i	0.747054	−	0.901319i
$666$		0			0
$667$			4372.44i			0.00982817i
$668$		0			0
$669$	−189684.			−0.423818
$670$		0			0
$671$	37157.7			0.0825284
$672$		0			0
$673$		−	869817.i		−	1.92043i	−0.279268	−	0.960213i	$-0.590092\pi$
$673$		−	869817.i		−	1.92043i	0.279268	−	0.960213i	$-0.409908\pi$
$674$		0			0
$675$		−	340101.i		−	0.746450i
$676$		0			0
$677$		−	207393.i		−	0.452499i	−0.974069	−	0.226250i	$-0.927354\pi$
$677$		−	207393.i		−	0.452499i	0.974069	−	0.226250i	$-0.0726465\pi$
$678$		0			0
$679$		−	760523.i		−	1.64958i
$680$		0			0
$681$	191830.			0.413640
$682$		0			0
$683$			597649.i			1.28116i	0.767890	+	0.640582i	$0.221307\pi$
$683$			597649.i			1.28116i	−0.767890	+	0.640582i	$0.778693\pi$
$684$		0			0
$685$	−847818.			−1.80685
$686$		0			0
$687$			211508.i			0.448140i
$688$		0			0
$689$	−536983.			−1.13116
$690$		0			0
$691$	−751655.			−1.57421			−0.787104	−	0.616820i	$-0.788421\pi$
$691$	−751655.			−1.57421			−0.787104	+	0.616820i	$0.788421\pi$
$692$		0			0
$693$	−29140.6			−0.0606781
$694$		0			0
$695$	−776449.			−1.60747
$696$		0			0
$697$		−	515692.i		−	1.06151i
$698$		0			0
$699$		−	547159.i		−	1.11985i
$700$		0			0
$701$	375431.			0.764001			0.382001	−	0.924162i	$-0.375235\pi$
$701$	375431.			0.764001			0.382001	+	0.924162i	$0.375235\pi$
$702$		0			0
$703$	535571.	+	443905.i	1.08369	+	0.898214i
$704$		0			0
$705$		−	268490.i		−	0.540194i
$706$		0			0
$707$	169317.			0.338735
$708$		0			0
$709$	343580.			0.683494			0.341747	−	0.939792i	$-0.388981\pi$
$709$	343580.			0.683494			0.341747	+	0.939792i	$0.388981\pi$
$710$		0			0
$711$		−	354288.i		−	0.700838i
$712$		0			0
$713$			3880.24i			0.00763271i
$714$		0			0
$715$		−	59044.1i		−	0.115495i
$716$		0			0
$717$			295447.i			0.574700i
$718$		0			0
$719$	570002.			1.10260			0.551300	−	0.834307i	$-0.314132\pi$
$719$	570002.			1.10260			0.551300	+	0.834307i	$0.314132\pi$
$720$		0			0
$721$		−	700795.i		−	1.34809i
$722$		0			0
$723$	−67301.0			−0.128749
$724$		0			0
$725$		−	143054.i		−	0.272160i
$726$		0			0
$727$	−323658.			−0.612375			−0.306188	−	0.951971i	$-0.599054\pi$
$727$	−323658.			−0.612375			−0.306188	+	0.951971i	$0.599054\pi$
$728$		0			0
$729$	−335582.			−0.631456
$730$		0			0
$731$	−779651.			−1.45903
$732$		0			0
$733$	958654.			1.78424			0.892121	−	0.451796i	$-0.149217\pi$
$733$	958654.			1.78424			0.892121	+	0.451796i	$0.149217\pi$
$734$		0			0
$735$		−	95034.6i		−	0.175917i
$736$		0			0
$737$			46247.2i			0.0851433i
$738$		0			0
$739$	−766051.			−1.40271			−0.701356	−	0.712811i	$-0.747422\pi$
$739$	−766051.			−1.40271			−0.701356	+	0.712811i	$0.747422\pi$
$740$		0			0
$741$	170732.	−	205987.i	0.310941	−	0.375149i
$742$		0			0
$743$			700181.i			1.26833i	0.773197	+	0.634165i	$0.218656\pi$
$743$			700181.i			1.26833i	−0.773197	+	0.634165i	$0.781344\pi$
$744$		0			0
$745$	−364283.			−0.656336
$746$		0			0
$747$	−100129.			−0.179439
$748$		0			0
$749$			601392.i			1.07200i
$750$		0			0
$751$		−	550564.i		−	0.976176i	−0.872794	−	0.488088i	$-0.837695\pi$
$751$		−	550564.i		−	0.976176i	0.872794	−	0.488088i	$-0.162305\pi$
$752$		0			0
$753$		−	408150.i		−	0.719830i
$754$		0			0
$755$		−	1.24516e6i		−	2.18440i
$756$		0			0
$757$	784390.			1.36880			0.684400	−	0.729106i	$-0.260064\pi$
$757$	784390.			1.36880			0.684400	+	0.729106i	$0.260064\pi$
$758$		0			0
$759$		−	1075.26i		−	0.00186652i
$760$		0			0
$761$	−123152.			−0.212653			−0.106327	−	0.994331i	$-0.533909\pi$
$761$	−123152.			−0.212653			−0.106327	+	0.994331i	$0.533909\pi$
$762$		0			0
$763$		−	179156.i		−	0.307738i
$764$		0			0
$765$	−603843.			−1.03181
$766$		0			0
$767$	691867.			1.17607
$768$		0			0
$769$	−1.04880e6			−1.77353			−0.886766	−	0.462218i	$-0.847054\pi$
$769$	−1.04880e6			−1.77353			−0.886766	+	0.462218i	$0.847054\pi$
$770$		0			0
$771$	594544.			1.00017
$772$		0			0
$773$			1.06230e6i			1.77783i	0.458075	+	0.888913i	$0.348539\pi$
$773$			1.06230e6i			1.77783i	−0.458075	+	0.888913i	$0.651461\pi$
$774$		0			0
$775$		−	126951.i		−	0.211364i
$776$		0			0
$777$	−468723.			−0.776379
$778$		0			0
$779$	389011.	+	322430.i	0.641042	+	0.531325i
$780$		0			0
$781$		−	1386.06i		−	0.00227237i
$782$		0			0
$783$	−225115.			−0.367182
$784$		0			0
$785$	−1.14690e6			−1.86118
$786$		0			0
$787$			277829.i			0.448569i	0.974524	+	0.224284i	$0.0720044\pi$
$787$			277829.i			0.448569i	−0.974524	+	0.224284i	$0.927996\pi$
$788$		0			0
$789$			522136.i			0.838744i
$790$		0			0
$791$		−	438302.i		−	0.700520i
$792$		0			0
$793$			363913.i			0.578697i
$794$		0			0
$795$	750080.			1.18679
$796$		0			0
$797$			1.10778e6i			1.74397i	0.489536	+	0.871983i	$0.337166\pi$
$797$			1.10778e6i			1.74397i	−0.489536	+	0.871983i	$0.662834\pi$
$798$		0			0
$799$	−535112.			−0.838207
$800$		0			0
$801$			372882.i			0.581175i
$802$		0			0
$803$	49290.2			0.0764416
$804$		0			0
$805$	20377.3			0.0314452
$806$		0			0
$807$	−64425.9			−0.0989267
$808$		0			0
$809$	−601103.			−0.918443			−0.459221	−	0.888322i	$-0.651872\pi$
$809$	−601103.			−0.918443			−0.459221	+	0.888322i	$0.651872\pi$
$810$		0			0
$811$			1.20244e6i			1.82819i	0.405496	+	0.914097i	$0.367099\pi$
$811$			1.20244e6i			1.82819i	−0.405496	+	0.914097i	$0.632901\pi$
$812$		0			0
$813$		−	271892.i		−	0.411354i
$814$		0			0
$815$	1.26379e6			1.90265
$816$		0			0
$817$	487467.	−	588128.i	0.730300	−	0.881105i
$818$		0			0
$819$		−	285396.i		−	0.425480i
$820$		0			0
$821$	143119.			0.212329			0.106165	−	0.994349i	$-0.466143\pi$
$821$	143119.			0.212329			0.106165	+	0.994349i	$0.466143\pi$
$822$		0			0
$823$	43823.2			0.0647000			0.0323500	−	0.999477i	$-0.489701\pi$
$823$	43823.2			0.0647000			0.0323500	+	0.999477i	$0.489701\pi$
$824$		0			0
$825$			35179.7i			0.0516874i
$826$		0			0
$827$		−	784710.i		−	1.14736i	−0.819081	−	0.573678i	$-0.805516\pi$
$827$		−	784710.i		−	1.14736i	0.819081	−	0.573678i	$-0.194484\pi$
$828$		0			0
$829$			170879.i			0.248645i	0.992242	+	0.124323i	$0.0396758\pi$
$829$			170879.i			0.248645i	−0.992242	+	0.124323i	$0.960324\pi$
$830$		0			0
$831$			546523.i			0.791419i
$832$		0			0
$833$	−189408.			−0.272966
$834$		0			0
$835$		−	347776.i		−	0.498801i
$836$		0			0
$837$	−199774.			−0.285159
$838$		0			0
$839$		−	264804.i		−	0.376184i	−0.982151	−	0.188092i	$-0.939770\pi$
$839$		−	264804.i		−	0.376184i	0.982151	−	0.188092i	$-0.0602304\pi$
$840$		0			0
$841$	612593.			0.866123
$842$		0			0
$843$	70619.4			0.0993731
$844$		0			0
$845$	−364636.			−0.510677
$846$		0			0
$847$	−628056.			−0.875450
$848$		0			0
$849$		−	612207.i		−	0.849342i
$850$		0			0
$851$			27380.5i			0.0378078i
$852$		0			0
$853$	445113.			0.611747			0.305874	−	0.952072i	$-0.401052\pi$
$853$	445113.			0.611747			0.305874	+	0.952072i	$0.401052\pi$
$854$		0			0
$855$	377545.	−	455507.i	0.516460	−	0.623108i
$856$		0			0
$857$		−	622452.i		−	0.847509i	−0.905777	−	0.423755i	$-0.860712\pi$
$857$		−	622452.i		−	0.847509i	0.905777	−	0.423755i	$-0.139288\pi$
$858$		0			0
$859$	618227.			0.837841			0.418921	−	0.908023i	$-0.362409\pi$
$859$	618227.			0.837841			0.418921	+	0.908023i	$0.362409\pi$
$860$		0			0
$861$	−340456.			−0.459255
$862$		0			0
$863$			1.36857e6i			1.83758i	0.394750	+	0.918789i	$0.370831\pi$
$863$			1.36857e6i			1.83758i	−0.394750	+	0.918789i	$0.629169\pi$
$864$		0			0
$865$			462261.i			0.617811i
$866$		0			0
$867$		−	292507.i		−	0.389133i
$868$		0			0
$869$			96443.4i			0.127712i
$870$		0			0
$871$	−452934.			−0.597033
$872$		0			0
$873$		−	869132.i		−	1.14040i
$874$		0			0
$875$	229605.			0.299893
$876$		0			0
$877$			830305.i			1.07954i	0.841813	+	0.539769i	$0.181489\pi$
$877$			830305.i			1.07954i	−0.841813	+	0.539769i	$0.818511\pi$
$878$		0			0
$879$	−736480.			−0.953198
$880$		0			0
$881$	−229860.			−0.296150			−0.148075	−	0.988976i	$-0.547308\pi$
$881$	−229860.			−0.296150			−0.148075	+	0.988976i	$0.547308\pi$
$882$		0			0
$883$	−1.48774e6			−1.90813			−0.954063	−	0.299606i	$-0.903145\pi$
$883$	−1.48774e6			−1.90813			−0.954063	+	0.299606i	$0.903145\pi$
$884$		0			0
$885$	−966428.			−1.23391
$886$		0			0
$887$		−	604766.i		−	0.768670i	−0.923194	−	0.384335i	$-0.874431\pi$
$887$		−	604766.i		−	0.768670i	0.923194	−	0.384335i	$-0.125569\pi$
$888$		0			0
$889$		−	241566.i		−	0.305655i
$890$		0			0
$891$	1021.81			0.00128710
$892$		0			0
$893$	334572.	−	403661.i	0.419553	−	0.506190i
$894$		0			0
$895$		−	241102.i		−	0.300992i
$896$		0			0
$897$	10530.9			0.0130882
$898$		0			0
$899$	−84029.3			−0.103971
$900$		0			0
$901$		−	1.49494e6i		−	1.84151i
$902$		0			0
$903$			514720.i			0.631241i
$904$		0			0
$905$			799399.i			0.976038i
$906$		0			0
$907$			960117.i			1.16710i	0.812076	+	0.583552i	$0.198338\pi$
$907$			960117.i			1.16710i	−0.812076	+	0.583552i	$0.801662\pi$
$908$		0			0
$909$	193496.			0.234177
$910$		0			0
$911$		−	1.30057e6i		−	1.56710i	−0.621331	−	0.783548i	$-0.713408\pi$
$911$		−	1.30057e6i		−	1.56710i	0.621331	−	0.783548i	$-0.286592\pi$
$912$		0			0
$913$	27256.8			0.0326989
$914$		0			0
$915$		−	508329.i		−	0.607159i
$916$		0			0
$917$	395271.			0.470064
$918$		0			0
$919$	1.11252e6			1.31728			0.658638	−	0.752460i	$-0.271133\pi$
$919$	1.11252e6			1.31728			0.658638	+	0.752460i	$0.271133\pi$
$920$		0			0
$921$	−758511.			−0.894217
$922$		0			0
$923$	13574.7			0.0159341
$924$		0			0
$925$		−	895814.i		−	1.04697i
$926$		0			0
$927$		−	800874.i		−	0.931977i
$928$		0			0
$929$	713119.			0.826287			0.413143	−	0.910666i	$-0.364431\pi$
$929$	713119.			0.826287			0.413143	+	0.910666i	$0.364431\pi$
$930$		0			0
$931$	118425.	−	142879.i	0.136629	−	0.164843i
$932$		0			0
$933$			515912.i			0.592669i
$934$		0			0
$935$	164376.			0.188025
$936$		0			0
$937$	105273.			0.119905			0.0599523	−	0.998201i	$-0.480905\pi$
$937$	105273.			0.119905			0.0599523	+	0.998201i	$0.480905\pi$
$938$		0			0
$939$			606678.i			0.688061i
$940$		0			0
$941$			1.31342e6i			1.48328i	0.670796	+	0.741642i	$0.265953\pi$
$941$			1.31342e6i			1.48328i	−0.670796	+	0.741642i	$0.734047\pi$
$942$		0			0
$943$			19887.7i			0.0223647i
$944$		0			0
$945$			1.04912e6i			1.17480i
$946$		0			0
$947$	−14348.9			−0.0160000			−0.00799998	−	0.999968i	$-0.502546\pi$
$947$	−14348.9			−0.0160000			−0.00799998	+	0.999968i	$0.502546\pi$
$948$		0			0
$949$			482736.i			0.536016i
$950$		0			0
$951$	−990711.			−1.09543
$952$		0			0
$953$			536179.i			0.590369i	0.955440	+	0.295185i	$0.0953812\pi$
$953$			536179.i			0.590369i	−0.955440	+	0.295185i	$0.904619\pi$
$954$		0			0
$955$	1.25838e6			1.37976
$956$		0			0
$957$	23285.6			0.0254252
$958$		0			0
$959$	1.11555e6			1.21298
$960$		0			0
$961$	848951.			0.919255
$962$		0			0
$963$			687275.i			0.741102i
$964$		0			0
$965$		−	614485.i		−	0.659867i
$966$		0			0
$967$	671096.			0.717682			0.358841	−	0.933399i	$-0.383172\pi$
$967$	671096.			0.717682			0.358841	+	0.933399i	$0.383172\pi$
$968$		0			0
$969$	573461.	+	475310.i	0.610740	+	0.506209i
$970$		0			0
$971$			1.45602e6i			1.54429i	0.635447	+	0.772144i	$0.280816\pi$
$971$			1.45602e6i			1.54429i	−0.635447	+	0.772144i	$0.719184\pi$
$972$		0			0
$973$	1.02164e6			1.07913
$974$		0			0
$975$	−344541.			−0.362437
$976$		0			0
$977$		−	193421.i		−	0.202635i	−0.994854	−	0.101317i	$-0.967694\pi$
$977$		−	193421.i		−	0.202635i	0.994854	−	0.101317i	$-0.0323058\pi$
$978$		0			0
$979$		−	101505.i		−	0.105906i
$980$		0			0
$981$		−	204740.i		−	0.212748i
$982$		0			0
$983$		−	21764.8i		−	0.0225241i	−0.999937	−	0.0112621i	$-0.996415\pi$
$983$		−	21764.8i		−	0.0225241i	0.999937	−	0.0112621i	$-0.00358490\pi$
$984$		0			0
$985$	1.91072e6			1.96936
$986$		0			0
$987$			353277.i			0.362644i
$988$		0			0
$989$	30067.4			0.0307400
$990$		0			0
$991$		−	374486.i		−	0.381319i	−0.981656	−	0.190659i	$-0.938937\pi$
$991$		−	374486.i		−	0.381319i	0.981656	−	0.190659i	$-0.0610626\pi$
$992$		0			0
$993$	845713.			0.857679
$994$		0			0
$995$	−758075.			−0.765713
$996$		0			0
$997$	−572532.			−0.575983			−0.287991	−	0.957633i	$-0.592987\pi$
$997$	−572532.			−0.575983			−0.287991	+	0.957633i	$0.592987\pi$
$998$		0			0
$999$	−1.40968e6			−1.41251

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.5.e.f.113.8		20
4.3	odd	2		152.5.e.a.113.13	yes	20
12.11	even	2		1368.5.o.a.721.17		20
19.18	odd	2	inner	304.5.e.f.113.13		20
76.75	even	2		152.5.e.a.113.8	✓	20
228.227	odd	2		1368.5.o.a.721.18		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.5.e.a.113.8	✓	20	76.75	even	2
152.5.e.a.113.13	yes	20	4.3	odd	2
304.5.e.f.113.8		20	1.1	even	1	trivial
304.5.e.f.113.13		20	19.18	odd	2	inner
1368.5.o.a.721.17		20	12.11	even	2
1368.5.o.a.721.18		20	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)

\(f(q)\)	\(=\)	\(q-5.59979i q^{3} -33.0135 q^{5} +43.4389 q^{7} +49.6423 q^{9} +O(q^{10})\)	\(q-5.59979i q^{3} -33.0135 q^{5} +43.4389 q^{7} +49.6423 q^{9} -13.5135 q^{11} -132.348i q^{13} +184.869i q^{15} +368.451 q^{17} +(-230.370 + 277.940i) q^{19} -243.249i q^{21} -14.2094 q^{23} +464.892 q^{25} -731.570i q^{27} -307.715i q^{29} -273.075i q^{31} +75.6728i q^{33} -1434.07 q^{35} -1926.93i q^{37} -741.121 q^{39} -1399.62i q^{41} -2116.02 q^{43} -1638.87 q^{45} -1452.33 q^{47} -514.065 q^{49} -2063.25i q^{51} -4057.36i q^{53} +446.128 q^{55} +(1556.41 + 1290.02i) q^{57} +5227.64i q^{59} -2749.67 q^{61} +2156.41 q^{63} +4369.27i q^{65} -3422.30i q^{67} +79.5697i q^{69} +102.568i q^{71} -3647.48 q^{73} -2603.30i q^{75} -587.011 q^{77} -7136.82i q^{79} -75.6139 q^{81} -2017.01 q^{83} -12163.9 q^{85} -1723.14 q^{87} +7511.38i q^{89} -5749.04i q^{91} -1529.17 q^{93} +(7605.31 - 9175.79i) q^{95} -17507.9i q^{97} -670.841 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 32 q^{7} - 372 q^{9}+O(q^{10}) \) 20 * q - 32 * q^7 - 372 * q^9	\( 20 q - 32 q^{7} - 372 q^{9} + 24 q^{11} + 216 q^{17} + 596 q^{19} - 576 q^{23} + 1412 q^{25} + 144 q^{35} + 520 q^{39} + 1256 q^{43} + 7232 q^{45} + 3768 q^{47} - 2740 q^{49} + 10128 q^{55} - 728 q^{57} + 352 q^{61} - 6104 q^{63} + 1352 q^{73} + 9288 q^{77} - 4220 q^{81} + 16104 q^{83} + 10232 q^{85} - 2936 q^{87} + 36432 q^{93} - 14232 q^{95} - 760 q^{99}+O(q^{100}) \) 20 * q - 32 * q^7 - 372 * q^9 + 24 * q^11 + 216 * q^17 + 596 * q^19 - 576 * q^23 + 1412 * q^25 + 144 * q^35 + 520 * q^39 + 1256 * q^43 + 7232 * q^45 + 3768 * q^47 - 2740 * q^49 + 10128 * q^55 - 728 * q^57 + 352 * q^61 - 6104 * q^63 + 1352 * q^73 + 9288 * q^77 - 4220 * q^81 + 16104 * q^83 + 10232 * q^85 - 2936 * q^87 + 36432 * q^93 - 14232 * q^95 - 760 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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