LMFDB - Embedded newform 304.5.e.f.113.18

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [304,5,Mod(113,304)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [20,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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			113.18
Root			$13.8615i$ of defining polynomial
Character	$\chi$	$=$	304.113
Dual form			304.5.e.f.113.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+13.8615i q^{3} +16.6305 q^{5} +34.1005 q^{7} -111.141 q^{9} +198.897 q^{11} -164.629i q^{13} +230.524i q^{15} +463.763 q^{17} +(176.546 - 314.885i) q^{19} +472.684i q^{21} +807.248 q^{23} -348.425 q^{25} -417.796i q^{27} -424.371i q^{29} -1283.92i q^{31} +2757.01i q^{33} +567.110 q^{35} -1126.53i q^{37} +2282.01 q^{39} +408.190i q^{41} -1546.41 q^{43} -1848.33 q^{45} -198.561 q^{47} -1238.16 q^{49} +6428.45i q^{51} +1512.97i q^{53} +3307.77 q^{55} +(4364.77 + 2447.19i) q^{57} +2476.47i q^{59} -4116.57 q^{61} -3789.96 q^{63} -2737.87i q^{65} +8501.12i q^{67} +11189.7i q^{69} +4053.08i q^{71} -7.79120 q^{73} -4829.69i q^{75} +6782.50 q^{77} -4348.12i q^{79} -3211.13 q^{81} -8742.26 q^{83} +7712.63 q^{85} +5882.42 q^{87} -8132.18i q^{89} -5613.94i q^{91} +17797.1 q^{93} +(2936.06 - 5236.70i) q^{95} -70.4206i q^{97} -22105.6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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}+ \cdots - 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

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
$3$			13.8615i			1.54017i	0.637944	+	0.770083i	$0.279785\pi$
$3$			13.8615i			1.54017i	−0.637944	+	0.770083i	$0.720215\pi$
$4$		0			0
$5$	16.6305			0.665221			0.332611	−	0.943064i	$-0.392070\pi$
$5$	16.6305			0.665221			0.332611	+	0.943064i	$0.392070\pi$
$6$		0			0
$7$	34.1005			0.695929			0.347964	−	0.937508i	$-0.386873\pi$
$7$	34.1005			0.695929			0.347964	+	0.937508i	$0.386873\pi$
$8$		0			0
$9$	−111.141			−1.37211
$10$		0			0
$11$	198.897			1.64378			0.821890	−	0.569646i	$-0.192920\pi$
$11$	198.897			1.64378			0.821890	+	0.569646i	$0.192920\pi$
$12$		0			0
$13$		−	164.629i		−	0.974138i	−0.873364	−	0.487069i	$-0.838066\pi$
$13$		−	164.629i		−	0.974138i	0.873364	−	0.487069i	$-0.161934\pi$
$14$		0			0
$15$			230.524i			1.02455i
$16$		0			0
$17$	463.763			1.60472			0.802358	−	0.596842i	$-0.203578\pi$
$17$	463.763			1.60472			0.802358	+	0.596842i	$0.203578\pi$
$18$		0			0
$19$	176.546	−	314.885i	0.489048	−	0.872257i
$19$	176.546	−	314.885i	0.489048	−	0.872257i
$20$		0			0
$21$			472.684i			1.07184i
$22$		0			0
$23$	807.248			1.52599			0.762994	−	0.646405i	$-0.223728\pi$
$23$	807.248			1.52599			0.762994	+	0.646405i	$0.223728\pi$
$24$		0			0
$25$	−348.425			−0.557480
$26$		0			0
$27$		−	417.796i		−	0.573109i
$28$		0			0
$29$		−	424.371i		−	0.504603i	−0.967649	−	0.252302i	$-0.918812\pi$
$29$		−	424.371i		−	0.504603i	0.967649	−	0.252302i	$-0.0811875\pi$
$30$		0			0
$31$		−	1283.92i		−	1.33603i	−0.744148	−	0.668015i	$-0.767144\pi$
$31$		−	1283.92i		−	1.33603i	0.744148	−	0.668015i	$-0.232856\pi$
$32$		0			0
$33$			2757.01i			2.53169i
$34$		0			0
$35$	567.110			0.462947
$36$		0			0
$37$		−	1126.53i		−	0.822883i	−0.911436	−	0.411441i	$-0.865025\pi$
$37$		−	1126.53i		−	0.822883i	0.911436	−	0.411441i	$-0.134975\pi$
$38$		0			0
$39$	2282.01			1.50033
$40$		0			0
$41$			408.190i			0.242826i	0.992602	+	0.121413i	$0.0387425\pi$
$41$			408.190i			0.242826i	−0.992602	+	0.121413i	$0.961257\pi$
$42$		0			0
$43$	−1546.41			−0.836350			−0.418175	−	0.908366i	$-0.637330\pi$
$43$	−1546.41			−0.836350			−0.418175	+	0.908366i	$0.637330\pi$
$44$		0			0
$45$	−1848.33			−0.912756
$46$		0			0
$47$	−198.561			−0.0898874			−0.0449437	−	0.998990i	$-0.514311\pi$
$47$	−198.561			−0.0898874			−0.0449437	+	0.998990i	$0.514311\pi$
$48$		0			0
$49$	−1238.16			−0.515683
$50$		0			0
$51$			6428.45i			2.47153i
$52$		0			0
$53$			1512.97i			0.538617i	0.963054	+	0.269308i	$0.0867951\pi$
$53$			1512.97i			0.538617i	−0.963054	+	0.269308i	$0.913205\pi$
$54$		0			0
$55$	3307.77			1.09348
$56$		0			0
$57$	4364.77	+	2447.19i	1.34342	+	0.753214i
$58$		0			0
$59$			2476.47i			0.711426i	0.934595	+	0.355713i	$0.115762\pi$
$59$			2476.47i			0.711426i	−0.934595	+	0.355713i	$0.884238\pi$
$60$		0			0
$61$	−4116.57			−1.10631			−0.553153	−	0.833080i	$-0.686576\pi$
$61$	−4116.57			−1.10631			−0.553153	+	0.833080i	$0.686576\pi$
$62$		0			0
$63$	−3789.96			−0.954890
$64$		0			0
$65$		−	2737.87i		−	0.648017i
$66$		0			0
$67$			8501.12i			1.89377i	0.321576	+	0.946884i	$0.395788\pi$
$67$			8501.12i			1.89377i	−0.321576	+	0.946884i	$0.604212\pi$
$68$		0			0
$69$			11189.7i			2.35027i
$70$		0			0
$71$			4053.08i			0.804024i	0.915634	+	0.402012i	$0.131689\pi$
$71$			4053.08i			0.804024i	−0.915634	+	0.402012i	$0.868311\pi$
$72$		0			0
$73$	−7.79120			−0.00146204			−0.000731019	−	1.00000i	$-0.500233\pi$
$73$	−7.79120			−0.00146204			−0.000731019		1.00000i	$0.500233\pi$
$74$		0			0
$75$		−	4829.69i		−	0.858612i
$76$		0			0
$77$	6782.50			1.14395
$78$		0			0
$79$		−	4348.12i		−	0.696703i	−0.937364	−	0.348351i	$-0.886742\pi$
$79$		−	4348.12i		−	0.696703i	0.937364	−	0.348351i	$-0.113258\pi$
$80$		0			0
$81$	−3211.13			−0.489426
$82$		0			0
$83$	−8742.26			−1.26902			−0.634509	−	0.772916i	$-0.718797\pi$
$83$	−8742.26			−1.26902			−0.634509	+	0.772916i	$0.718797\pi$
$84$		0			0
$85$	7712.63			1.06749
$86$		0			0
$87$	5882.42			0.777172
$88$		0			0
$89$		−	8132.18i		−	1.02666i	−0.858191	−	0.513331i	$-0.828411\pi$
$89$		−	8132.18i		−	1.02666i	0.858191	−	0.513331i	$-0.171589\pi$
$90$		0			0
$91$		−	5613.94i		−	0.677930i
$92$		0			0
$93$	17797.1			2.05771
$94$		0			0
$95$	2936.06	−	5236.70i	0.325325	−	0.580244i
$96$		0			0
$97$		−	70.4206i		−	0.00748438i	−0.999993	−	0.00374219i	$-0.998809\pi$
$97$		−	70.4206i		−	0.00748438i	0.999993	−	0.00374219i	$-0.00119118\pi$
$98$		0			0
$99$	−22105.6			−2.25545
$100$		0			0
$101$	249.501			0.0244585			0.0122292	−	0.999925i	$-0.496107\pi$
$101$	249.501			0.0244585			0.0122292	+	0.999925i	$0.496107\pi$
$102$		0			0
$103$			14423.2i			1.35952i	0.733433	+	0.679762i	$0.237917\pi$
$103$			14423.2i			1.35952i	−0.733433	+	0.679762i	$0.762083\pi$
$104$		0			0
$105$			7860.98i			0.713014i
$106$		0			0
$107$			11154.7i			0.974297i	0.873319	+	0.487148i	$0.161963\pi$
$107$			11154.7i			0.974297i	−0.873319	+	0.487148i	$0.838037\pi$
$108$		0			0
$109$		−	15364.9i		−	1.29323i	−0.762815	−	0.646617i	$-0.776183\pi$
$109$		−	15364.9i		−	1.29323i	0.762815	−	0.646617i	$-0.223817\pi$
$110$		0			0
$111$	15615.3			1.26738
$112$		0			0
$113$		−	5434.97i		−	0.425638i	−0.977092	−	0.212819i	$-0.931736\pi$
$113$		−	5434.97i		−	0.425638i	0.977092	−	0.212819i	$-0.0682645\pi$
$114$		0			0
$115$	13425.0			1.01512
$116$		0			0
$117$			18297.0i			1.33662i
$118$		0			0
$119$	15814.6			1.11677
$120$		0			0
$121$	24919.2			1.70202
$122$		0			0
$123$	−5658.12			−0.373992
$124$		0			0
$125$	−16188.6			−1.03607
$126$		0			0
$127$			6910.69i			0.428464i	0.976783	+	0.214232i	$0.0687248\pi$
$127$			6910.69i			0.428464i	−0.976783	+	0.214232i	$0.931275\pi$
$128$		0			0
$129$		−	21435.6i		−	1.28812i
$130$		0			0
$131$	25573.0			1.49018			0.745091	−	0.666962i	$-0.232406\pi$
$131$	25573.0			1.49018			0.745091	+	0.666962i	$0.232406\pi$
$132$		0			0
$133$	6020.32	−	10737.7i	0.340342	−	0.607029i
$134$		0			0
$135$		−	6948.18i		−	0.381244i
$136$		0			0
$137$	−26984.5			−1.43772			−0.718859	−	0.695156i	$-0.755335\pi$
$137$	−26984.5			−1.43772			−0.718859	+	0.695156i	$0.755335\pi$
$138$		0			0
$139$	−31644.5			−1.63783			−0.818914	−	0.573916i	$-0.805424\pi$
$139$	−31644.5			−1.63783			−0.818914	+	0.573916i	$0.805424\pi$
$140$		0			0
$141$		−	2752.35i		−	0.138441i
$142$		0			0
$143$		−	32744.4i		−	1.60127i
$144$		0			0
$145$		−	7057.52i		−	0.335673i
$146$		0			0
$147$		−	17162.7i		−	0.794238i
$148$		0			0
$149$	−25545.4			−1.15064			−0.575320	−	0.817928i	$-0.695122\pi$
$149$	−25545.4			−1.15064			−0.575320	+	0.817928i	$0.695122\pi$
$150$		0			0
$151$		−	10722.8i		−	0.470280i	−0.971962	−	0.235140i	$-0.924445\pi$
$151$		−	10722.8i		−	0.470280i	0.971962	−	0.235140i	$-0.0755548\pi$
$152$		0			0
$153$	−51543.0			−2.20185
$154$		0			0
$155$		−	21352.3i		−	0.888755i
$156$		0			0
$157$	−1380.26			−0.0559964			−0.0279982	−	0.999608i	$-0.508913\pi$
$157$	−1380.26			−0.0559964			−0.0279982	+	0.999608i	$0.508913\pi$
$158$		0			0
$159$	−20972.1			−0.829559
$160$		0			0
$161$	27527.6			1.06198
$162$		0			0
$163$	−18266.6			−0.687516			−0.343758	−	0.939058i	$-0.611700\pi$
$163$	−18266.6			−0.687516			−0.343758	+	0.939058i	$0.611700\pi$
$164$		0			0
$165$			45850.6i			1.68414i
$166$		0			0
$167$			34186.3i			1.22580i	0.790160	+	0.612900i	$0.209997\pi$
$167$			34186.3i			1.22580i	−0.790160	+	0.612900i	$0.790003\pi$
$168$		0			0
$169$	1458.19			0.0510553
$170$		0			0
$171$	−19621.5	+	34996.5i	−0.671027	+	1.19683i
$172$		0			0
$173$			47486.7i			1.58665i	0.608801	+	0.793323i	$0.291651\pi$
$173$			47486.7i			1.58665i	−0.608801	+	0.793323i	$0.708349\pi$
$174$		0			0
$175$	−11881.5			−0.387967
$176$		0			0
$177$	−34327.6			−1.09571
$178$		0			0
$179$		−	38613.4i		−	1.20513i	−0.798072	−	0.602563i	$-0.794146\pi$
$179$		−	38613.4i		−	1.20513i	0.798072	−	0.602563i	$-0.205854\pi$
$180$		0			0
$181$			12404.6i			0.378639i	0.981915	+	0.189320i	$0.0606283\pi$
$181$			12404.6i			0.378639i	−0.981915	+	0.189320i	$0.939372\pi$
$182$		0			0
$183$		−	57061.7i		−	1.70389i
$184$		0			0
$185$		−	18734.7i		−	0.547399i
$186$		0			0
$187$	92241.3			2.63780
$188$		0			0
$189$		−	14247.1i		−	0.398843i
$190$		0			0
$191$	9053.96			0.248183			0.124091	−	0.992271i	$-0.460398\pi$
$191$	9053.96			0.248183			0.124091	+	0.992271i	$0.460398\pi$
$192$		0			0
$193$		−	29253.9i		−	0.785362i	−0.919675	−	0.392681i	$-0.871548\pi$
$193$		−	29253.9i		−	0.785362i	0.919675	−	0.392681i	$-0.128452\pi$
$194$		0			0
$195$	37951.0			0.998054
$196$		0			0
$197$	37953.7			0.977962			0.488981	−	0.872294i	$-0.337369\pi$
$197$	37953.7			0.977962			0.488981	+	0.872294i	$0.337369\pi$
$198$		0			0
$199$	−41610.7			−1.05075			−0.525375	−	0.850871i	$-0.676075\pi$
$199$	−41610.7			−1.05075			−0.525375	+	0.850871i	$0.676075\pi$
$200$		0			0
$201$	−117838.			−2.91672
$202$		0			0
$203$		−	14471.3i		−	0.351168i
$204$		0			0
$205$			6788.42i			0.161533i
$206$		0			0
$207$	−89718.2			−2.09382
$208$		0			0
$209$	35114.6	−	62629.8i	0.803887	−	1.43380i
$210$		0			0
$211$			57280.1i			1.28659i	0.765620	+	0.643293i	$0.222432\pi$
$211$			57280.1i			1.28659i	−0.765620	+	0.643293i	$0.777568\pi$
$212$		0			0
$213$	−56181.8			−1.23833
$214$		0			0
$215$	−25717.6			−0.556358
$216$		0			0
$217$		−	43782.4i		−	0.929781i
$218$		0			0
$219$		−	107.998i		−	0.00225178i
$220$		0			0
$221$		−	76349.0i		−	1.56322i
$222$		0			0
$223$		−	71207.6i		−	1.43191i	−0.698145	−	0.715957i	$-0.745991\pi$
$223$		−	71207.6i		−	1.43191i	0.698145	−	0.715957i	$-0.254009\pi$
$224$		0			0
$225$	38724.3			0.764924
$226$		0			0
$227$		−	20746.3i		−	0.402615i	−0.979528	−	0.201307i	$-0.935481\pi$
$227$		−	20746.3i		−	0.402615i	0.979528	−	0.201307i	$-0.0645190\pi$
$228$		0			0
$229$	33103.0			0.631242			0.315621	−	0.948885i	$-0.397787\pi$
$229$	33103.0			0.631242			0.315621	+	0.948885i	$0.397787\pi$
$230$		0			0
$231$			94015.6i			1.76188i
$232$		0			0
$233$	6773.66			0.124770			0.0623852	−	0.998052i	$-0.480129\pi$
$233$	6773.66			0.124770			0.0623852	+	0.998052i	$0.480129\pi$
$234$		0			0
$235$	−3302.18			−0.0597950
$236$		0			0
$237$	60271.4			1.07304
$238$		0			0
$239$	99987.5			1.75045			0.875225	−	0.483716i	$-0.160713\pi$
$239$	99987.5			1.75045			0.875225	+	0.483716i	$0.160713\pi$
$240$		0			0
$241$			9387.50i			0.161628i	0.996729	+	0.0808139i	$0.0257519\pi$
$241$			9387.50i			0.161628i	−0.996729	+	0.0808139i	$0.974248\pi$
$242$		0			0
$243$		−	78352.5i		−	1.32691i
$244$		0			0
$245$	−20591.2			−0.343044
$246$		0			0
$247$	−51839.3	−	29064.7i	−0.849699	−	0.476400i
$248$		0			0
$249$		−	121181.i		−	1.95450i
$250$		0			0
$251$	−20871.7			−0.331292			−0.165646	−	0.986185i	$-0.552971\pi$
$251$	−20871.7			−0.331292			−0.165646	+	0.986185i	$0.552971\pi$
$252$		0			0
$253$	160560.			2.50839
$254$		0			0
$255$			106909.i			1.64411i
$256$		0			0
$257$			51138.9i			0.774258i	0.922026	+	0.387129i	$0.126533\pi$
$257$			51138.9i			0.774258i	−0.922026	+	0.387129i	$0.873467\pi$
$258$		0			0
$259$		−	38415.1i		−	0.572668i
$260$		0			0
$261$			47165.0i			0.692371i
$262$		0			0
$263$	52658.8			0.761306			0.380653	−	0.924718i	$-0.375699\pi$
$263$	52658.8			0.761306			0.380653	+	0.924718i	$0.375699\pi$
$264$		0			0
$265$			25161.6i			0.358299i
$266$		0			0
$267$	112724.			1.58123
$268$		0			0
$269$			142239.i			1.96569i	0.184440	+	0.982844i	$0.440953\pi$
$269$			142239.i			1.96569i	−0.184440	+	0.982844i	$0.559047\pi$
$270$		0			0
$271$	−87901.2			−1.19690			−0.598448	−	0.801162i	$-0.704216\pi$
$271$	−87901.2			−1.19690			−0.598448	+	0.801162i	$0.704216\pi$
$272$		0			0
$273$	77817.6			1.04412
$274$		0			0
$275$	−69300.9			−0.916376
$276$		0			0
$277$	43465.3			0.566478			0.283239	−	0.959049i	$-0.408591\pi$
$277$	43465.3			0.566478			0.283239	+	0.959049i	$0.408591\pi$
$278$		0			0
$279$			142696.i			1.83318i
$280$		0			0
$281$			44636.1i			0.565293i	0.959224	+	0.282647i	$0.0912124\pi$
$281$			44636.1i			0.565293i	−0.959224	+	0.282647i	$0.908788\pi$
$282$		0			0
$283$	−28005.9			−0.349685			−0.174842	−	0.984596i	$-0.555942\pi$
$283$	−28005.9			−0.349685			−0.174842	+	0.984596i	$0.555942\pi$
$284$		0			0
$285$	72588.5	+	40698.1i	0.893672	+	0.501054i
$286$		0			0
$287$			13919.5i			0.168989i
$288$		0			0
$289$	131555.			1.57512
$290$		0			0
$291$	976.134			0.0115272
$292$		0			0
$293$			46994.9i			0.547413i	0.961813	+	0.273707i	$0.0882497\pi$
$293$			46994.9i			0.547413i	−0.961813	+	0.273707i	$0.911750\pi$
$294$		0			0
$295$			41185.1i			0.473256i
$296$		0			0
$297$		−	83098.7i		−	0.942066i
$298$		0			0
$299$		−	132897.i		−	1.48652i
$300$		0			0
$301$	−52733.4			−0.582040
$302$		0			0
$303$			3458.46i			0.0376701i
$304$		0			0
$305$	−68460.7			−0.735939
$306$		0			0
$307$			181833.i			1.92928i	0.263573	+	0.964640i	$0.415099\pi$
$307$			181833.i			1.92928i	−0.263573	+	0.964640i	$0.584901\pi$
$308$		0			0
$309$	−199927.			−2.09389
$310$		0			0
$311$	−85515.3			−0.884144			−0.442072	−	0.896980i	$-0.645756\pi$
$311$	−85515.3			−0.884144			−0.442072	+	0.896980i	$0.645756\pi$
$312$		0			0
$313$	−84722.7			−0.864790			−0.432395	−	0.901684i	$-0.642331\pi$
$313$	−84722.7			−0.864790			−0.432395	+	0.901684i	$0.642331\pi$
$314$		0			0
$315$	−63029.0			−0.635213
$316$		0			0
$317$		−	52210.6i		−	0.519565i	−0.965667	−	0.259783i	$-0.916349\pi$
$317$		−	52210.6i		−	0.519565i	0.965667	−	0.259783i	$-0.0836509\pi$
$318$		0			0
$319$		−	84406.4i		−	0.829457i
$320$		0			0
$321$	−154621.			−1.50058
$322$		0			0
$323$	81875.7	−	146032.i	0.784783	−	1.39973i
$324$		0			0
$325$			57361.0i			0.543063i
$326$		0			0
$327$	212981.			1.99179
$328$		0			0
$329$	−6771.04			−0.0625552
$330$		0			0
$331$			118412.i			1.08079i	0.841412	+	0.540394i	$0.181725\pi$
$331$			118412.i			1.08079i	−0.841412	+	0.540394i	$0.818275\pi$
$332$		0			0
$333$			125203.i			1.12908i
$334$		0			0
$335$			141378.i			1.25977i
$336$		0			0
$337$		−	59595.5i		−	0.524752i	−0.964966	−	0.262376i	$-0.915494\pi$
$337$		−	59595.5i		−	0.524752i	0.964966	−	0.262376i	$-0.0845060\pi$
$338$		0			0
$339$	75336.8			0.655553
$340$		0			0
$341$		−	255369.i		−	2.19614i
$342$		0			0
$343$	−124097.			−1.05481
$344$		0			0
$345$			186090.i			1.56345i
$346$		0			0
$347$	16642.0			0.138212			0.0691062	−	0.997609i	$-0.477985\pi$
$347$	16642.0			0.138212			0.0691062	+	0.997609i	$0.477985\pi$
$348$		0			0
$349$	153463.			1.25995			0.629973	−	0.776617i	$-0.283066\pi$
$349$	153463.			1.25995			0.629973	+	0.776617i	$0.283066\pi$
$350$		0			0
$351$	−68781.5			−0.558287
$352$		0			0
$353$	−1766.45			−0.0141760			−0.00708799	−	0.999975i	$-0.502256\pi$
$353$	−1766.45			−0.0141760			−0.00708799	+	0.999975i	$0.502256\pi$
$354$		0			0
$355$			67405.0i			0.534854i
$356$		0			0
$357$			219213.i			1.72001i
$358$		0			0
$359$	−43498.4			−0.337508			−0.168754	−	0.985658i	$-0.553974\pi$
$359$	−43498.4			−0.337508			−0.168754	+	0.985658i	$0.553974\pi$
$360$		0			0
$361$	−67983.8	−	111183.i	−0.521665	−	0.853151i
$362$		0			0
$363$			345417.i			2.62139i
$364$		0			0
$365$	−129.572			−0.000972579
$366$		0			0
$367$	123157.			0.914377			0.457189	−	0.889370i	$-0.348856\pi$
$367$	123157.			0.914377			0.457189	+	0.889370i	$0.348856\pi$
$368$		0			0
$369$		−	45366.6i		−	0.333183i
$370$		0			0
$371$			51593.2i			0.374839i
$372$		0			0
$373$			251344.i			1.80655i	0.429059	+	0.903277i	$0.358845\pi$
$373$			251344.i			1.80655i	−0.429059	+	0.903277i	$0.641155\pi$
$374$		0			0
$375$		−	224398.i		−	1.59572i
$376$		0			0
$377$	−69864.0			−0.491553
$378$		0			0
$379$			97859.0i			0.681275i	0.940195	+	0.340637i	$0.110643\pi$
$379$			97859.0i			0.681275i	−0.940195	+	0.340637i	$0.889357\pi$
$380$		0			0
$381$	−95792.4			−0.659905
$382$		0			0
$383$		−	255398.i		−	1.74108i	−0.492095	−	0.870541i	$-0.663769\pi$
$383$		−	255398.i		−	1.74108i	0.492095	−	0.870541i	$-0.336231\pi$
$384$		0			0
$385$	112797.			0.760983
$386$		0			0
$387$	171869.			1.14756
$388$		0			0
$389$	−250026.			−1.65229			−0.826145	−	0.563458i	$-0.809471\pi$
$389$	−250026.			−1.65229			−0.826145	+	0.563458i	$0.809471\pi$
$390$		0			0
$391$	374372.			2.44878
$392$		0			0
$393$			354480.i			2.29513i
$394$		0			0
$395$		−	72311.6i		−	0.463462i
$396$		0			0
$397$	219854.			1.39493			0.697466	−	0.716618i	$-0.254311\pi$
$397$	219854.			1.39493			0.697466	+	0.716618i	$0.254311\pi$
$398$		0			0
$399$	148841.	+	83450.5i	0.934924	+	0.524183i
$400$		0			0
$401$			88448.2i			0.550047i	0.961437	+	0.275024i	$0.0886857\pi$
$401$			88448.2i			0.550047i	−0.961437	+	0.275024i	$0.911314\pi$
$402$		0			0
$403$	−211372.			−1.30148
$404$		0			0
$405$	−53402.8			−0.325577
$406$		0			0
$407$		−	224063.i		−	1.35264i
$408$		0			0
$409$			104881.i			0.626977i	0.949592	+	0.313488i	$0.101498\pi$
$409$			104881.i			0.626977i	−0.949592	+	0.313488i	$0.898502\pi$
$410$		0			0
$411$		−	374045.i		−	2.21432i
$412$		0			0
$413$			84449.0i			0.495102i
$414$		0			0
$415$	−145388.			−0.844177
$416$		0			0
$417$		−	438640.i		−	2.52253i
$418$		0			0
$419$	−112863.			−0.642868			−0.321434	−	0.946932i	$-0.604165\pi$
$419$	−112863.			−0.642868			−0.321434	+	0.946932i	$0.604165\pi$
$420$		0			0
$421$			216141.i			1.21947i	0.792604	+	0.609737i	$0.208725\pi$
$421$			216141.i			1.21947i	−0.792604	+	0.609737i	$0.791275\pi$
$422$		0			0
$423$	22068.3			0.123335
$424$		0			0
$425$	−161587.			−0.894598
$426$		0			0
$427$	−140377.			−0.769910
$428$		0			0
$429$	453885.			2.46622
$430$		0			0
$431$			274109.i			1.47560i	0.675019	+	0.737800i	$0.264135\pi$
$431$			274109.i			1.47560i	−0.675019	+	0.737800i	$0.735865\pi$
$432$		0			0
$433$		−	304373.i		−	1.62342i	−0.584062	−	0.811709i	$-0.698538\pi$
$433$		−	304373.i		−	1.62342i	0.584062	−	0.811709i	$-0.301462\pi$
$434$		0			0
$435$	97827.8			0.516992
$436$		0			0
$437$	142517.	−	254190.i	0.746281	−	1.33105i
$438$		0			0
$439$			237360.i			1.23163i	0.787892	+	0.615814i	$0.211173\pi$
$439$			237360.i			1.23163i	−0.787892	+	0.615814i	$0.788827\pi$
$440$		0			0
$441$	137610.			0.707574
$442$		0			0
$443$	158727.			0.808804			0.404402	−	0.914581i	$-0.367480\pi$
$443$	158727.			0.808804			0.404402	+	0.914581i	$0.367480\pi$
$444$		0			0
$445$		−	135243.i		−	0.682957i
$446$		0			0
$447$		−	354097.i		−	1.77218i
$448$		0			0
$449$		−	348261.i		−	1.72747i	−0.503942	−	0.863737i	$-0.668118\pi$
$449$		−	348261.i		−	1.72747i	0.503942	−	0.863737i	$-0.331882\pi$
$450$		0			0
$451$			81188.0i			0.399152i
$452$		0			0
$453$	148635.			0.724308
$454$		0			0
$455$		−	93362.9i		−	0.450974i
$456$		0			0
$457$	−75791.2			−0.362900			−0.181450	−	0.983400i	$-0.558079\pi$
$457$	−75791.2			−0.362900			−0.181450	+	0.983400i	$0.558079\pi$
$458$		0			0
$459$		−	193759.i		−	0.919678i
$460$		0			0
$461$	−80904.8			−0.380691			−0.190346	−	0.981717i	$-0.560961\pi$
$461$	−80904.8			−0.380691			−0.190346	+	0.981717i	$0.560961\pi$
$462$		0			0
$463$	265052.			1.23643			0.618215	−	0.786009i	$-0.287856\pi$
$463$	265052.			1.23643			0.618215	+	0.786009i	$0.287856\pi$
$464$		0			0
$465$	295975.			1.36883
$466$		0			0
$467$	−414041.			−1.89850			−0.949248	−	0.314529i	$-0.898153\pi$
$467$	−414041.			−1.89850			−0.949248	+	0.314529i	$0.898153\pi$
$468$		0			0
$469$			289893.i			1.31793i
$470$		0			0
$471$		−	19132.4i		−	0.0862438i
$472$		0			0
$473$	−307577.			−1.37478
$474$		0			0
$475$	−61513.2	+	109714.i	−0.272635	+	0.486266i
$476$		0			0
$477$		−	168153.i		−	0.739041i
$478$		0			0
$479$	209176.			0.911678			0.455839	−	0.890062i	$-0.349339\pi$
$479$	209176.			0.911678			0.455839	+	0.890062i	$0.349339\pi$
$480$		0			0
$481$	−185459.			−0.801601
$482$		0			0
$483$			381573.i			1.63562i
$484$		0			0
$485$		−	1171.13i		−	0.00497877i
$486$		0			0
$487$			88387.7i			0.372678i	0.982485	+	0.186339i	$0.0596623\pi$
$487$			88387.7i			0.372678i	−0.982485	+	0.186339i	$0.940338\pi$
$488$		0			0
$489$		−	253202.i		−	1.05889i
$490$		0			0
$491$	−362034.			−1.50171			−0.750856	−	0.660466i	$-0.770359\pi$
$491$	−362034.			−1.50171			−0.750856	+	0.660466i	$0.770359\pi$
$492$		0			0
$493$		−	196808.i		−	0.809745i
$494$		0			0
$495$	−367628.			−1.50037
$496$		0			0
$497$			138212.i			0.559543i
$498$		0			0
$499$	−163151.			−0.655222			−0.327611	−	0.944813i	$-0.606244\pi$
$499$	−163151.			−0.655222			−0.327611	+	0.944813i	$0.606244\pi$
$500$		0			0
$501$	−473873.			−1.88793
$502$		0			0
$503$	437918.			1.73084			0.865420	−	0.501048i	$-0.167052\pi$
$503$	437918.			1.73084			0.865420	+	0.501048i	$0.167052\pi$
$504$		0			0
$505$	4149.34			0.0162703
$506$		0			0
$507$			20212.7i			0.0786336i
$508$		0			0
$509$		−	321145.i		−	1.23955i	−0.784778	−	0.619777i	$-0.787223\pi$
$509$		−	321145.i		−	1.23955i	0.784778	−	0.619777i	$-0.212777\pi$
$510$		0			0
$511$	−265.684			−0.00101747
$512$		0			0
$513$	−131558.	−	73760.4i	−0.499898	−	0.280278i
$514$		0			0
$515$			239865.i			0.904384i
$516$		0			0
$517$	−39493.3			−0.147755
$518$		0			0
$519$	−658236.			−2.44370
$520$		0			0
$521$		−	57254.3i		−	0.210927i	−0.994423	−	0.105464i	$-0.966367\pi$
$521$		−	57254.3i		−	0.210927i	0.994423	−	0.105464i	$-0.0336327\pi$
$522$		0			0
$523$		−	346981.i		−	1.26854i	−0.773113	−	0.634268i	$-0.781302\pi$
$523$		−	346981.i		−	1.26854i	0.773113	−	0.634268i	$-0.218698\pi$
$524$		0			0
$525$		−	164695.i		−	0.597532i
$526$		0			0
$527$		−	595437.i		−	2.14395i
$528$		0			0
$529$	371808.			1.32864
$530$		0			0
$531$		−	275237.i		−	0.976154i
$532$		0			0
$533$	67200.1			0.236546
$534$		0			0
$535$			185509.i			0.648123i
$536$		0			0
$537$	535239.			1.85609
$538$		0			0
$539$	−246266.			−0.847671
$540$		0			0
$541$	−419291.			−1.43259			−0.716294	−	0.697798i	$-0.754163\pi$
$541$	−419291.			−1.43259			−0.716294	+	0.697798i	$0.754163\pi$
$542$		0			0
$543$	−171946.			−0.583167
$544$		0			0
$545$		−	255527.i		−	0.860287i
$546$		0			0
$547$		−	100244.i		−	0.335029i	−0.985870	−	0.167515i	$-0.946426\pi$
$547$		−	100244.i		−	0.335029i	0.985870	−	0.167515i	$-0.0535741\pi$
$548$		0			0
$549$	457518.			1.51797
$550$		0			0
$551$	−133628.	−	74921.2i	−0.440144	−	0.246775i
$552$		0			0
$553$		−	148273.i		−	0.484855i
$554$		0			0
$555$	259691.			0.843085
$556$		0			0
$557$	−472339.			−1.52245			−0.761225	−	0.648487i	$-0.775402\pi$
$557$	−472339.			−1.52245			−0.761225	+	0.648487i	$0.775402\pi$
$558$		0			0
$559$			254585.i			0.814720i
$560$		0			0
$561$			1.27860e6i			4.06265i
$562$		0			0
$563$			116539.i			0.367667i	0.982957	+	0.183833i	$0.0588507\pi$
$563$			116539.i			0.367667i	−0.982957	+	0.183833i	$0.941149\pi$
$564$		0			0
$565$		−	90386.5i		−	0.283144i
$566$		0			0
$567$	−109501.			−0.340606
$568$		0			0
$569$			356049.i			1.09973i	0.835254	+	0.549864i	$0.185320\pi$
$569$			356049.i			1.09973i	−0.835254	+	0.549864i	$0.814680\pi$
$570$		0			0
$571$	−543882.			−1.66814			−0.834069	−	0.551660i	$-0.813995\pi$
$571$	−543882.			−1.66814			−0.834069	+	0.551660i	$0.813995\pi$
$572$		0			0
$573$			125501.i			0.382243i
$574$		0			0
$575$	−281266.			−0.850709
$576$		0			0
$577$	−168627.			−0.506497			−0.253248	−	0.967401i	$-0.581499\pi$
$577$	−168627.			−0.506497			−0.253248	+	0.967401i	$0.581499\pi$
$578$		0			0
$579$	405503.			1.20959
$580$		0			0
$581$	−298115.			−0.883145
$582$		0			0
$583$			300927.i			0.885368i
$584$		0			0
$585$			304290.i			0.889150i
$586$		0			0
$587$	226344.			0.656890			0.328445	−	0.944523i	$-0.393475\pi$
$587$	226344.			0.656890			0.328445	+	0.944523i	$0.393475\pi$
$588$		0			0
$589$	−404288.	−	226672.i	−1.16536	−	0.653382i
$590$		0			0
$591$			526095.i			1.50622i
$592$		0			0
$593$	138124.			0.392790			0.196395	−	0.980525i	$-0.437077\pi$
$593$	138124.			0.392790			0.196395	+	0.980525i	$0.437077\pi$
$594$		0			0
$595$	263005.			0.742898
$596$		0			0
$597$		−	576787.i		−	1.61833i
$598$		0			0
$599$			85172.8i			0.237382i	0.992931	+	0.118691i	$0.0378698\pi$
$599$			85172.8i			0.237382i	−0.992931	+	0.118691i	$0.962130\pi$
$600$		0			0
$601$			323788.i			0.896420i	0.893928	+	0.448210i	$0.147938\pi$
$601$			323788.i			0.896420i	−0.893928	+	0.448210i	$0.852062\pi$
$602$		0			0
$603$		−	944822.i		−	2.59846i
$604$		0			0
$605$	414420.			1.13222
$606$		0			0
$607$		−	312727.i		−	0.848766i	−0.905483	−	0.424383i	$-0.860491\pi$
$607$		−	312727.i		−	0.848766i	0.905483	−	0.424383i	$-0.139509\pi$
$608$		0			0
$609$	200593.			0.540856
$610$		0			0
$611$			32689.0i			0.0875627i
$612$		0			0
$613$	−87167.0			−0.231970			−0.115985	−	0.993251i	$-0.537002\pi$
$613$	−87167.0			−0.231970			−0.115985	+	0.993251i	$0.537002\pi$
$614$		0			0
$615$	−94097.6			−0.248787
$616$		0			0
$617$	74773.6			0.196417			0.0982083	−	0.995166i	$-0.468689\pi$
$617$	74773.6			0.196417			0.0982083	+	0.995166i	$0.468689\pi$
$618$		0			0
$619$	−517474.			−1.35054			−0.675269	−	0.737571i	$-0.735973\pi$
$619$	−517474.			−1.35054			−0.675269	+	0.737571i	$0.735973\pi$
$620$		0			0
$621$		−	337265.i		−	0.874558i
$622$		0			0
$623$		−	277311.i		−	0.714483i
$624$		0			0
$625$	−51459.1			−0.131735
$626$		0			0
$627$	868142.	+	486741.i	2.20829	+	1.23812i
$628$		0			0
$629$		−	522442.i		−	1.32049i
$630$		0			0
$631$	234598.			0.589205			0.294602	−	0.955620i	$-0.404813\pi$
$631$	234598.			0.589205			0.294602	+	0.955620i	$0.404813\pi$
$632$		0			0
$633$	−793987.			−1.98156
$634$		0			0
$635$			114928.i			0.285023i
$636$		0			0
$637$			203837.i			0.502347i
$638$		0			0
$639$		−	450463.i		−	1.10321i
$640$		0			0
$641$			201204.i			0.489690i	0.969562	+	0.244845i	$0.0787371\pi$
$641$			201204.i			0.489690i	−0.969562	+	0.244845i	$0.921263\pi$
$642$		0			0
$643$	397450.			0.961304			0.480652	−	0.876911i	$-0.340400\pi$
$643$	397450.			0.961304			0.480652	+	0.876911i	$0.340400\pi$
$644$		0			0
$645$		−	356485.i		−	0.856883i
$646$		0			0
$647$	319131.			0.762360			0.381180	−	0.924501i	$-0.375518\pi$
$647$	319131.			0.762360			0.381180	+	0.924501i	$0.375518\pi$
$648$		0			0
$649$			492565.i			1.16943i
$650$		0			0
$651$	606890.			1.43202
$652$		0			0
$653$	−615311.			−1.44301			−0.721504	−	0.692411i	$-0.756549\pi$
$653$	−615311.			−1.44301			−0.721504	+	0.692411i	$0.756549\pi$
$654$		0			0
$655$	425293.			0.991302
$656$		0			0
$657$	865.920			0.00200608
$658$		0			0
$659$			389964.i			0.897953i	0.893544	+	0.448977i	$0.148211\pi$
$659$			389964.i			0.897953i	−0.893544	+	0.448977i	$0.851789\pi$
$660$		0			0
$661$			202339.i			0.463101i	0.972823	+	0.231551i	$0.0743799\pi$
$661$			202339.i			0.463101i	−0.972823	+	0.231551i	$0.925620\pi$
$662$		0			0
$663$	1.05831e6			2.40761
$664$		0			0
$665$	100121.	−	178574.i	0.226403	−	0.403808i
$666$		0			0
$667$		−	342573.i		−	0.770019i
$668$		0			0
$669$	987043.			2.20538
$670$		0			0
$671$	−818775.			−1.81853
$672$		0			0
$673$		−	564672.i		−	1.24671i	−0.781938	−	0.623356i	$-0.785769\pi$
$673$		−	564672.i		−	1.24671i	0.781938	−	0.623356i	$-0.214231\pi$
$674$		0			0
$675$			145571.i			0.319497i
$676$		0			0
$677$			157666.i			0.344002i	0.985097	+	0.172001i	$0.0550232\pi$
$677$			157666.i			0.344002i	−0.985097	+	0.172001i	$0.944977\pi$
$678$		0			0
$679$		−	2401.38i		−	0.00520860i
$680$		0			0
$681$	287575.			0.620093
$682$		0			0
$683$		−	132840.i		−	0.284765i	−0.989812	−	0.142382i	$-0.954524\pi$
$683$		−	132840.i		−	0.284765i	0.989812	−	0.142382i	$-0.0454762\pi$
$684$		0			0
$685$	−448767.			−0.956400
$686$		0			0
$687$			458856.i			0.972217i
$688$		0			0
$689$	249080.			0.524687
$690$		0			0
$691$	822098.			1.72174			0.860870	−	0.508826i	$-0.169920\pi$
$691$	822098.			1.72174			0.860870	+	0.508826i	$0.169920\pi$
$692$		0			0
$693$	−753813.			−1.56963
$694$		0			0
$695$	−526265.			−1.08952
$696$		0			0
$697$			189304.i			0.389667i
$698$		0			0
$699$			93893.0i			0.192167i
$700$		0			0
$701$	524576.			1.06751			0.533755	−	0.845639i	$-0.320780\pi$
$701$	524576.			1.06751			0.533755	+	0.845639i	$0.320780\pi$
$702$		0			0
$703$	−354726.	−	198884.i	−0.717765	−	0.402429i
$704$		0			0
$705$		−	45773.1i		−	0.0920942i
$706$		0			0
$707$	8508.11			0.0170214
$708$		0			0
$709$	350588.			0.697435			0.348718	−	0.937228i	$-0.386617\pi$
$709$	350588.			0.697435			0.348718	+	0.937228i	$0.386617\pi$
$710$		0			0
$711$			483254.i			0.955952i
$712$		0			0
$713$		−	1.03645e6i		−	2.03877i
$714$		0			0
$715$		−	544556.i		−	1.06520i
$716$		0			0
$717$			1.38598e6i			2.69598i
$718$		0			0
$719$	−105280.			−0.203652			−0.101826	−	0.994802i	$-0.532468\pi$
$719$	−105280.			−0.203652			−0.101826	+	0.994802i	$0.532468\pi$
$720$		0			0
$721$			491838.i			0.946131i
$722$		0			0
$723$	−130125.			−0.248933
$724$		0			0
$725$			147862.i			0.281306i
$726$		0			0
$727$	−142455.			−0.269531			−0.134766	−	0.990878i	$-0.543028\pi$
$727$	−142455.			−0.269531			−0.134766	+	0.990878i	$0.543028\pi$
$728$		0			0
$729$	825981.			1.55423
$730$		0			0
$731$	−717169.			−1.34211
$732$		0			0
$733$	−46226.2			−0.0860361			−0.0430180	−	0.999074i	$-0.513697\pi$
$733$	−46226.2			−0.0860361			−0.0430180	+	0.999074i	$0.513697\pi$
$734$		0			0
$735$		−	285425.i		−	0.528344i
$736$		0			0
$737$			1.69085e6i			3.11294i
$738$		0			0
$739$	383816.			0.702803			0.351402	−	0.936225i	$-0.385705\pi$
$739$	383816.			0.702803			0.351402	+	0.936225i	$0.385705\pi$
$740$		0			0
$741$	402880.	−	718569.i	0.733735	−	1.30868i
$742$		0			0
$743$			289968.i			0.525258i	0.964897	+	0.262629i	$0.0845895\pi$
$743$			289968.i			0.525258i	−0.964897	+	0.262629i	$0.915410\pi$
$744$		0			0
$745$	−424833.			−0.765430
$746$		0			0
$747$	971622.			1.74123
$748$		0			0
$749$			380382.i			0.678041i
$750$		0			0
$751$		−	701251.i		−	1.24335i	−0.783275	−	0.621675i	$-0.786452\pi$
$751$		−	701251.i		−	1.24335i	0.783275	−	0.621675i	$-0.213548\pi$
$752$		0			0
$753$		−	289313.i		−	0.510244i
$754$		0			0
$755$		−	178327.i		−	0.312840i
$756$		0			0
$757$	193035.			0.336856			0.168428	−	0.985714i	$-0.446131\pi$
$757$	193035.			0.336856			0.168428	+	0.985714i	$0.446131\pi$
$758$		0			0
$759$			2.22559e6i			3.86334i
$760$		0			0
$761$	882197.			1.52334			0.761669	−	0.647966i	$-0.224380\pi$
$761$	882197.			1.52334			0.761669	+	0.647966i	$0.224380\pi$
$762$		0			0
$763$		−	523951.i		−	0.899998i
$764$		0			0
$765$	−857188.			−1.46472
$766$		0			0
$767$	407700.			0.693027
$768$		0			0
$769$	−50360.6			−0.0851605			−0.0425803	−	0.999093i	$-0.513558\pi$
$769$	−50360.6			−0.0851605			−0.0425803	+	0.999093i	$0.513558\pi$
$770$		0			0
$771$	−708862.			−1.19248
$772$		0			0
$773$		−	186023.i		−	0.311320i	−0.987811	−	0.155660i	$-0.950250\pi$
$773$		−	186023.i		−	0.311320i	0.987811	−	0.155660i	$-0.0497505\pi$
$774$		0			0
$775$			447352.i			0.744810i
$776$		0			0
$777$	532491.			0.882003
$778$		0			0
$779$	128533.	+	72064.4i	0.211806	+	0.118753i
$780$		0			0
$781$			806148.i			1.32164i
$782$		0			0
$783$	−177301.			−0.289193
$784$		0			0
$785$	−22954.4			−0.0372500
$786$		0			0
$787$			496717.i			0.801973i	0.916084	+	0.400987i	$0.131333\pi$
$787$			496717.i			0.801973i	−0.916084	+	0.400987i	$0.868667\pi$
$788$		0			0
$789$			729929.i			1.17254i
$790$		0			0
$791$		−	185335.i		−	0.296214i
$792$		0			0
$793$			677707.i			1.07769i
$794$		0			0
$795$	−348777.			−0.551840
$796$		0			0
$797$		−	1.16084e6i		−	1.82749i	−0.406289	−	0.913745i	$-0.633177\pi$
$797$		−	1.16084e6i		−	1.82749i	0.406289	−	0.913745i	$-0.366823\pi$
$798$		0			0
$799$	−92085.4			−0.144244
$800$		0			0
$801$			903817.i			1.40869i
$802$		0			0
$803$	−1549.65			−0.00240327
$804$		0			0
$805$	457798.			0.706451
$806$		0			0
$807$	−1.97165e6			−3.02748
$808$		0			0
$809$	−327930.			−0.501054			−0.250527	−	0.968110i	$-0.580604\pi$
$809$	−327930.			−0.501054			−0.250527	+	0.968110i	$0.580604\pi$
$810$		0			0
$811$			921655.i			1.40129i	0.713512	+	0.700643i	$0.247103\pi$
$811$			921655.i			1.40129i	−0.713512	+	0.700643i	$0.752897\pi$
$812$		0			0
$813$		−	1.21844e6i		−	1.84342i
$814$		0			0
$815$	−303783.			−0.457350
$816$		0			0
$817$	−273013.	+	486941.i	−0.409015	+	0.729512i
$818$		0			0
$819$			623938.i			0.930194i
$820$		0			0
$821$	−285899.			−0.424157			−0.212078	−	0.977253i	$-0.568023\pi$
$821$	−285899.			−0.424157			−0.212078	+	0.977253i	$0.568023\pi$
$822$		0			0
$823$	127934.			0.188880			0.0944398	−	0.995531i	$-0.469894\pi$
$823$	127934.			0.188880			0.0944398	+	0.995531i	$0.469894\pi$
$824$		0			0
$825$		−	960614.i		−	1.41137i
$826$		0			0
$827$		−	244346.i		−	0.357268i	−0.983916	−	0.178634i	$-0.942832\pi$
$827$		−	244346.i		−	0.357268i	0.983916	−	0.178634i	$-0.0571679\pi$
$828$		0			0
$829$		−	575618.i		−	0.837579i	−0.908083	−	0.418789i	$-0.862455\pi$
$829$		−	575618.i		−	0.837579i	0.908083	−	0.418789i	$-0.137545\pi$
$830$		0			0
$831$			602494.i			0.872470i
$832$		0			0
$833$	−574211.			−0.827526
$834$		0			0
$835$			568537.i			0.815428i
$836$		0			0
$837$	−536419.			−0.765690
$838$		0			0
$839$		−	1.19220e6i		−	1.69366i	−0.531866	−	0.846829i	$-0.678509\pi$
$839$		−	1.19220e6i		−	1.69366i	0.531866	−	0.846829i	$-0.321491\pi$
$840$		0			0
$841$	527190.			0.745376
$842$		0			0
$843$	−618723.			−0.870645
$844$		0			0
$845$	24250.5			0.0339631
$846$		0			0
$847$	849758.			1.18448
$848$		0			0
$849$		−	388204.i		−	0.538572i
$850$		0			0
$851$		−	909386.i		−	1.25571i
$852$		0			0
$853$	661907.			0.909702			0.454851	−	0.890568i	$-0.349693\pi$
$853$	661907.			0.909702			0.454851	+	0.890568i	$0.349693\pi$
$854$		0			0
$855$	−326316.	+	582011.i	−0.446381	+	0.796158i
$856$		0			0
$857$		−	982841.i		−	1.33820i	−0.743171	−	0.669101i	$-0.766679\pi$
$857$		−	982841.i		−	1.33820i	0.743171	−	0.669101i	$-0.233321\pi$
$858$		0			0
$859$	521534.			0.706800			0.353400	−	0.935472i	$-0.385026\pi$
$859$	521534.			0.706800			0.353400	+	0.935472i	$0.385026\pi$
$860$		0			0
$861$	−192945.			−0.260272
$862$		0			0
$863$			941751.i			1.26449i	0.774770	+	0.632243i	$0.217866\pi$
$863$			941751.i			1.26449i	−0.774770	+	0.632243i	$0.782134\pi$
$864$		0			0
$865$			789730.i			1.05547i
$866$		0			0
$867$			1.82355e6i			2.42594i
$868$		0			0
$869$		−	864831.i		−	1.14523i
$870$		0			0
$871$	1.39953e6			1.84479
$872$		0			0
$873$			7826.60i			0.0102694i
$874$		0			0
$875$	−552039.			−0.721030
$876$		0			0
$877$		−	343716.i		−	0.446890i	−0.974717	−	0.223445i	$-0.928270\pi$
$877$		−	343716.i		−	0.446890i	0.974717	−	0.223445i	$-0.0717303\pi$
$878$		0			0
$879$	−651419.			−0.843107
$880$		0			0
$881$	322791.			0.415882			0.207941	−	0.978141i	$-0.433324\pi$
$881$	322791.			0.415882			0.207941	+	0.978141i	$0.433324\pi$
$882$		0			0
$883$	−555429.			−0.712372			−0.356186	−	0.934415i	$-0.615923\pi$
$883$	−555429.			−0.712372			−0.356186	+	0.934415i	$0.615923\pi$
$884$		0			0
$885$	−570887.			−0.728892
$886$		0			0
$887$		−	1.26195e6i		−	1.60396i	−0.597349	−	0.801981i	$-0.703779\pi$
$887$		−	1.26195e6i		−	1.60396i	0.597349	−	0.801981i	$-0.296221\pi$
$888$		0			0
$889$			235658.i			0.298180i
$890$		0			0
$891$	−638685.			−0.804510
$892$		0			0
$893$	−35055.2	+	62523.9i	−0.0439592	+	0.0784049i
$894$		0			0
$895$		−	642162.i		−	0.801675i
$896$		0			0
$897$	1.84215e6			2.28949
$898$		0			0
$899$	−544861.			−0.674165
$900$		0			0
$901$			701662.i			0.864327i
$902$		0			0
$903$		−	730963.i		−	0.896438i
$904$		0			0
$905$			206295.i			0.251879i
$906$		0			0
$907$			713110.i			0.866846i	0.901191	+	0.433423i	$0.142694\pi$
$907$			713110.i			0.866846i	−0.901191	+	0.433423i	$0.857306\pi$
$908$		0			0
$909$	−27729.7			−0.0335597
$910$		0			0
$911$		−	613356.i		−	0.739054i	−0.929220	−	0.369527i	$-0.879520\pi$
$911$		−	613356.i		−	0.739054i	0.929220	−	0.369527i	$-0.120480\pi$
$912$		0			0
$913$	−1.73881e6			−2.08599
$914$		0			0
$915$		−	948967.i		−	1.13347i
$916$		0			0
$917$	872053.			1.03706
$918$		0			0
$919$	341024.			0.403788			0.201894	−	0.979407i	$-0.435290\pi$
$919$	341024.			0.403788			0.201894	+	0.979407i	$0.435290\pi$
$920$		0			0
$921$	−2.52047e6			−2.97141
$922$		0			0
$923$	667257.			0.783230
$924$		0			0
$925$			392510.i			0.458741i
$926$		0			0
$927$		−	1.60300e6i		−	1.86541i
$928$		0			0
$929$	396594.			0.459531			0.229765	−	0.973246i	$-0.426204\pi$
$929$	396594.			0.459531			0.229765	+	0.973246i	$0.426204\pi$
$930$		0			0
$931$	−218592.	+	389876.i	−0.252194	+	0.449809i
$932$		0			0
$933$		−	1.18537e6i		−	1.36173i
$934$		0			0
$935$	1.53402e6			1.75472
$936$		0			0
$937$	1.60617e6			1.82941			0.914707	−	0.404119i	$-0.132422\pi$
$937$	1.60617e6			1.82941			0.914707	+	0.404119i	$0.132422\pi$
$938$		0			0
$939$		−	1.17438e6i		−	1.33192i
$940$		0			0
$941$		−	767970.i		−	0.867292i	−0.901083	−	0.433646i	$-0.857227\pi$
$941$		−	767970.i		−	0.867292i	0.901083	−	0.433646i	$-0.142773\pi$
$942$		0			0
$943$			329511.i			0.370549i
$944$		0			0
$945$		−	236936.i		−	0.265319i
$946$		0			0
$947$	66938.4			0.0746406			0.0373203	−	0.999303i	$-0.488118\pi$
$947$	66938.4			0.0746406			0.0373203	+	0.999303i	$0.488118\pi$
$948$		0			0
$949$			1282.66i			0.00142423i
$950$		0			0
$951$	723716.			0.800216
$952$		0			0
$953$		−	237880.i		−	0.261922i	−0.991388	−	0.130961i	$-0.958194\pi$
$953$		−	237880.i		−	0.261922i	0.991388	−	0.130961i	$-0.0418063\pi$
$954$		0			0
$955$	150572.			0.165097
$956$		0			0
$957$	1.17000e6			1.27750
$958$		0			0
$959$	−920185.			−1.00055
$960$		0			0
$961$	−724940.			−0.784974
$962$		0			0
$963$		−	1.23974e6i		−	1.33684i
$964$		0			0
$965$		−	486509.i		−	0.522440i
$966$		0			0
$967$	1.36716e6			1.46206			0.731031	−	0.682345i	$-0.239040\pi$
$967$	1.36716e6			1.46206			0.731031	+	0.682345i	$0.239040\pi$
$968$		0			0
$969$	2.02422e6	+	1.13492e6i	2.15581	+	1.20870i
$970$		0			0
$971$		−	72775.5i		−	0.0771875i	−0.999255	−	0.0385937i	$-0.987712\pi$
$971$		−	72775.5i		−	0.0771875i	0.999255	−	0.0385937i	$-0.0122878\pi$
$972$		0			0
$973$	−1.07909e6			−1.13981
$974$		0			0
$975$	−795109.			−0.836406
$976$		0			0
$977$			496154.i			0.519789i	0.965637	+	0.259895i	$0.0836879\pi$
$977$			496154.i			0.519789i	−0.965637	+	0.259895i	$0.916312\pi$
$978$		0			0
$979$		−	1.61747e6i		−	1.68761i
$980$		0			0
$981$			1.70767e6i			1.77446i
$982$		0			0
$983$		−	354053.i		−	0.366405i	−0.983075	−	0.183203i	$-0.941354\pi$
$983$		−	354053.i		−	0.366405i	0.983075	−	0.183203i	$-0.0586465\pi$
$984$		0			0
$985$	631191.			0.650561
$986$		0			0
$987$		−	93856.6i		−	0.0963453i
$988$		0			0
$989$	−1.24834e6			−1.27626
$990$		0			0
$991$		−	431313.i		−	0.439183i	−0.975592	−	0.219591i	$-0.929528\pi$
$991$		−	431313.i		−	0.439183i	0.975592	−	0.219591i	$-0.0704724\pi$
$992$		0			0
$993$	−1.64137e6			−1.66459
$994$		0			0
$995$	−692009.			−0.698981
$996$		0			0
$997$	522673.			0.525823			0.262912	−	0.964820i	$-0.415317\pi$
$997$	522673.			0.525823			0.262912	+	0.964820i	$0.415317\pi$
$998$		0			0
$999$	−470659.			−0.471602

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.5.e.f.113.18		20
4.3	odd	2		152.5.e.a.113.3	✓	20
12.11	even	2		1368.5.o.a.721.5		20
19.18	odd	2	inner	304.5.e.f.113.3		20
76.75	even	2		152.5.e.a.113.18	yes	20
228.227	odd	2		1368.5.o.a.721.6		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.5.e.a.113.3	✓	20	4.3	odd	2
152.5.e.a.113.18	yes	20	76.75	even	2
304.5.e.f.113.3		20	19.18	odd	2	inner
304.5.e.f.113.18		20	1.1	even	1	trivial
1368.5.o.a.721.5		20	12.11	even	2
1368.5.o.a.721.6		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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+13.8615i q^{3} +16.6305 q^{5} +34.1005 q^{7} -111.141 q^{9} +198.897 q^{11} -164.629i q^{13} +230.524i q^{15} +463.763 q^{17} +(176.546 - 314.885i) q^{19} +472.684i q^{21} +807.248 q^{23} -348.425 q^{25} -417.796i q^{27} -424.371i q^{29} -1283.92i q^{31} +2757.01i q^{33} +567.110 q^{35} -1126.53i q^{37} +2282.01 q^{39} +408.190i q^{41} -1546.41 q^{43} -1848.33 q^{45} -198.561 q^{47} -1238.16 q^{49} +6428.45i q^{51} +1512.97i q^{53} +3307.77 q^{55} +(4364.77 + 2447.19i) q^{57} +2476.47i q^{59} -4116.57 q^{61} -3789.96 q^{63} -2737.87i q^{65} +8501.12i q^{67} +11189.7i q^{69} +4053.08i q^{71} -7.79120 q^{73} -4829.69i q^{75} +6782.50 q^{77} -4348.12i q^{79} -3211.13 q^{81} -8742.26 q^{83} +7712.63 q^{85} +5882.42 q^{87} -8132.18i q^{89} -5613.94i q^{91} +17797.1 q^{93} +(2936.06 - 5236.70i) q^{95} -70.4206i q^{97} -22105.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 32 q^{7} - 372 q^{9} + 24 q^{11} + 216 q^{17} + 596 q^{19} - 576 q^{23} + 1412 q^{25} + 144 q^{35} + 520 q^{39} + 1256 q^{43} + 7232 q^{45} + 3768 q^{47} - 2740 q^{49} + 10128 q^{55} - 728 q^{57}+ \cdots - 760 q^{99}+O(q^{100}) \) 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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