LMFDB - Embedded newform 304.5.e.f.113.11

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			113.11
Root			$1.48358i$ of defining polynomial
Character	$\chi$	$=$	304.113
Dual form			304.5.e.f.113.10

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.48358i q^{3} -22.0416 q^{5} -25.0930 q^{7} +78.7990 q^{9} +90.5788 q^{11} -89.9607i q^{13} -32.7006i q^{15} -493.355 q^{17} +(74.3224 - 353.266i) q^{19} -37.2276i q^{21} +781.178 q^{23} -139.167 q^{25} +237.075i q^{27} +1270.23i q^{29} +1620.99i q^{31} +134.381i q^{33} +553.090 q^{35} +2410.90i q^{37} +133.464 q^{39} -1675.60i q^{41} +633.074 q^{43} -1736.86 q^{45} +3088.77 q^{47} -1771.34 q^{49} -731.934i q^{51} +2109.89i q^{53} -1996.50 q^{55} +(524.101 + 110.264i) q^{57} +5338.68i q^{59} +2965.04 q^{61} -1977.30 q^{63} +1982.88i q^{65} +1715.97i q^{67} +1158.94i q^{69} -3689.95i q^{71} +2412.51 q^{73} -206.466i q^{75} -2272.89 q^{77} +10187.8i q^{79} +6031.00 q^{81} -8009.63 q^{83} +10874.3 q^{85} -1884.50 q^{87} +11847.4i q^{89} +2257.38i q^{91} -2404.88 q^{93} +(-1638.19 + 7786.56i) q^{95} -4008.24i q^{97} +7137.51 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$			1.48358i			0.164843i	0.996598	+	0.0824214i	$0.0262653\pi$
$3$			1.48358i			0.164843i	−0.996598	+	0.0824214i	$0.973735\pi$
$4$		0			0
$5$	−22.0416			−0.881665			−0.440832	−	0.897589i	$-0.645317\pi$
$5$	−22.0416			−0.881665			−0.440832	+	0.897589i	$0.645317\pi$
$6$		0			0
$7$	−25.0930			−0.512102			−0.256051	−	0.966663i	$-0.582421\pi$
$7$	−25.0930			−0.512102			−0.256051	+	0.966663i	$0.582421\pi$
$8$		0			0
$9$	78.7990			0.972827
$10$		0			0
$11$	90.5788			0.748585			0.374292	−	0.927311i	$-0.377886\pi$
$11$	90.5788			0.748585			0.374292	+	0.927311i	$0.377886\pi$
$12$		0			0
$13$		−	89.9607i		−	0.532312i	−0.963930	−	0.266156i	$-0.914246\pi$
$13$		−	89.9607i		−	0.532312i	0.963930	−	0.266156i	$-0.0857536\pi$
$14$		0			0
$15$		−	32.7006i		−	0.145336i
$16$		0			0
$17$	−493.355			−1.70711			−0.853556	−	0.521002i	$-0.825559\pi$
$17$	−493.355			−1.70711			−0.853556	+	0.521002i	$0.825559\pi$
$18$		0			0
$19$	74.3224	−	353.266i	0.205879	−	0.978577i
$19$	74.3224	−	353.266i	0.205879	−	0.978577i
$20$		0			0
$21$		−	37.2276i		−	0.0844162i
$22$		0			0
$23$	781.178			1.47671			0.738354	−	0.674414i	$-0.235603\pi$
$23$	781.178			1.47671			0.738354	+	0.674414i	$0.235603\pi$
$24$		0			0
$25$	−139.167			−0.222667
$26$		0			0
$27$			237.075i			0.325206i
$28$		0			0
$29$			1270.23i			1.51038i	0.655504	+	0.755192i	$0.272456\pi$
$29$			1270.23i			1.51038i	−0.655504	+	0.755192i	$0.727544\pi$
$30$		0			0
$31$			1620.99i			1.68678i	0.537303	+	0.843389i	$0.319443\pi$
$31$			1620.99i			1.68678i	−0.537303	+	0.843389i	$0.680557\pi$
$32$		0			0
$33$			134.381i			0.123399i
$34$		0			0
$35$	553.090			0.451502
$36$		0			0
$37$			2410.90i			1.76107i	0.473983	+	0.880534i	$0.342816\pi$
$37$			2410.90i			1.76107i	−0.473983	+	0.880534i	$0.657184\pi$
$38$		0			0
$39$	133.464			0.0877477
$40$		0			0
$41$		−	1675.60i		−	0.996788i	−0.866951	−	0.498394i	$-0.833923\pi$
$41$		−	1675.60i		−	0.996788i	0.866951	−	0.498394i	$-0.166077\pi$
$42$		0			0
$43$	633.074			0.342387			0.171194	−	0.985237i	$-0.445238\pi$
$43$	633.074			0.342387			0.171194	+	0.985237i	$0.445238\pi$
$44$		0			0
$45$	−1736.86			−0.857707
$46$		0			0
$47$	3088.77			1.39827			0.699134	−	0.714991i	$-0.253569\pi$
$47$	3088.77			1.39827			0.699134	+	0.714991i	$0.253569\pi$
$48$		0			0
$49$	−1771.34			−0.737752
$50$		0			0
$51$		−	731.934i		−	0.281405i
$52$		0			0
$53$			2109.89i			0.751117i	0.926799	+	0.375558i	$0.122549\pi$
$53$			2109.89i			0.751117i	−0.926799	+	0.375558i	$0.877451\pi$
$54$		0			0
$55$	−1996.50			−0.660001
$56$		0			0
$57$	524.101	+	110.264i	0.161311	+	0.0339377i
$58$		0			0
$59$			5338.68i			1.53366i	0.641848	+	0.766832i	$0.278168\pi$
$59$			5338.68i			1.53366i	−0.641848	+	0.766832i	$0.721832\pi$
$60$		0			0
$61$	2965.04			0.796840			0.398420	−	0.917203i	$-0.369559\pi$
$61$	2965.04			0.796840			0.398420	+	0.917203i	$0.369559\pi$
$62$		0			0
$63$	−1977.30			−0.498186
$64$		0			0
$65$			1982.88i			0.469321i
$66$		0			0
$67$			1715.97i			0.382261i	0.981565	+	0.191131i	$0.0612154\pi$
$67$			1715.97i			0.382261i	−0.981565	+	0.191131i	$0.938785\pi$
$68$		0			0
$69$			1158.94i			0.243425i
$70$		0			0
$71$		−	3689.95i		−	0.731988i	−0.930617	−	0.365994i	$-0.880729\pi$
$71$		−	3689.95i		−	0.731988i	0.930617	−	0.365994i	$-0.119271\pi$
$72$		0			0
$73$	2412.51			0.452714			0.226357	−	0.974044i	$-0.427318\pi$
$73$	2412.51			0.452714			0.226357	+	0.974044i	$0.427318\pi$
$74$		0			0
$75$		−	206.466i		−	0.0367051i
$76$		0			0
$77$	−2272.89			−0.383351
$78$		0			0
$79$			10187.8i			1.63240i	0.577767	+	0.816202i	$0.303924\pi$
$79$			10187.8i			1.63240i	−0.577767	+	0.816202i	$0.696076\pi$
$80$		0			0
$81$	6031.00			0.919219
$82$		0			0
$83$	−8009.63			−1.16267			−0.581335	−	0.813664i	$-0.697469\pi$
$83$	−8009.63			−1.16267			−0.581335	+	0.813664i	$0.697469\pi$
$84$		0			0
$85$	10874.3			1.50510
$86$		0			0
$87$	−1884.50			−0.248976
$88$		0			0
$89$			11847.4i			1.49570i	0.663870	+	0.747848i	$0.268913\pi$
$89$			11847.4i			1.49570i	−0.663870	+	0.747848i	$0.731087\pi$
$90$		0			0
$91$			2257.38i			0.272598i
$92$		0			0
$93$	−2404.88			−0.278053
$94$		0			0
$95$	−1638.19	+	7786.56i	−0.181517	+	0.862777i
$96$		0			0
$97$		−	4008.24i		−	0.426001i	−0.977052	−	0.213000i	$-0.931676\pi$
$97$		−	4008.24i		−	0.426001i	0.977052	−	0.213000i	$-0.0683235\pi$
$98$		0			0
$99$	7137.51			0.728243
$100$		0			0
$101$	719.191			0.0705021			0.0352510	−	0.999378i	$-0.488777\pi$
$101$	719.191			0.0705021			0.0352510	+	0.999378i	$0.488777\pi$
$102$		0			0
$103$		−	10439.2i		−	0.983995i	−0.870597	−	0.491998i	$-0.836267\pi$
$103$		−	10439.2i		−	0.983995i	0.870597	−	0.491998i	$-0.163733\pi$
$104$		0			0
$105$			820.556i			0.0744268i
$106$		0			0
$107$			5911.14i			0.516302i	0.966105	+	0.258151i	$0.0831132\pi$
$107$			5911.14i			0.516302i	−0.966105	+	0.258151i	$0.916887\pi$
$108$		0			0
$109$			14685.2i			1.23602i	0.786169	+	0.618012i	$0.212062\pi$
$109$			14685.2i			1.23602i	−0.786169	+	0.618012i	$0.787938\pi$
$110$		0			0
$111$	−3576.78			−0.290299
$112$		0			0
$113$		−	19906.4i		−	1.55896i	−0.626428	−	0.779480i	$-0.715484\pi$
$113$		−	19906.4i		−	1.55896i	0.626428	−	0.779480i	$-0.284516\pi$
$114$		0			0
$115$	−17218.4			−1.30196
$116$		0			0
$117$		−	7088.81i		−	0.517847i
$118$		0			0
$119$	12379.8			0.874215
$120$		0			0
$121$	−6436.49			−0.439621
$122$		0			0
$123$	2485.89			0.164313
$124$		0			0
$125$	16843.5			1.07798
$126$		0			0
$127$		−	2693.51i		−	0.166998i	−0.996508	−	0.0834990i	$-0.973390\pi$
$127$		−	2693.51i		−	0.166998i	0.996508	−	0.0834990i	$-0.0266095\pi$
$128$		0			0
$129$			939.219i			0.0564401i
$130$		0			0
$131$	−514.799			−0.0299982			−0.0149991	−	0.999888i	$-0.504775\pi$
$131$	−514.799			−0.0299982			−0.0149991	+	0.999888i	$0.504775\pi$
$132$		0			0
$133$	−1864.97	+	8864.51i	−0.105431	+	0.501131i
$134$		0			0
$135$		−	5225.52i		−	0.286723i
$136$		0			0
$137$	12605.6			0.671619			0.335810	−	0.941930i	$-0.390990\pi$
$137$	12605.6			0.671619			0.335810	+	0.941930i	$0.390990\pi$
$138$		0			0
$139$	7368.93			0.381395			0.190698	−	0.981649i	$-0.438925\pi$
$139$	7368.93			0.381395			0.190698	+	0.981649i	$0.438925\pi$
$140$		0			0
$141$			4582.46i			0.230494i
$142$		0			0
$143$		−	8148.53i		−	0.398481i
$144$		0			0
$145$		−	27998.0i		−	1.33165i
$146$		0			0
$147$		−	2627.94i		−	0.121613i
$148$		0			0
$149$	−41469.6			−1.86791			−0.933957	−	0.357385i	$-0.883669\pi$
$149$	−41469.6			−1.86791			−0.933957	+	0.357385i	$0.883669\pi$
$150$		0			0
$151$			21320.9i			0.935087i	0.883970	+	0.467543i	$0.154861\pi$
$151$			21320.9i			0.935087i	−0.883970	+	0.467543i	$0.845139\pi$
$152$		0			0
$153$	−38875.9			−1.66072
$154$		0			0
$155$		−	35729.3i		−	1.48717i
$156$		0			0
$157$	−2105.40			−0.0854152			−0.0427076	−	0.999088i	$-0.513598\pi$
$157$	−2105.40			−0.0854152			−0.0427076	+	0.999088i	$0.513598\pi$
$158$		0			0
$159$	−3130.20			−0.123816
$160$		0			0
$161$	−19602.1			−0.756224
$162$		0			0
$163$	47528.0			1.78885			0.894425	−	0.447218i	$-0.147585\pi$
$163$	47528.0			1.78885			0.894425	+	0.447218i	$0.147585\pi$
$164$		0			0
$165$		−	2961.98i		−	0.108796i
$166$		0			0
$167$		−	44354.6i		−	1.59040i	−0.606349	−	0.795199i	$-0.707366\pi$
$167$		−	44354.6i		−	1.59040i	0.606349	−	0.795199i	$-0.292634\pi$
$168$		0			0
$169$	20468.1			0.716644
$170$		0			0
$171$	5856.53	−	27837.0i	0.200285	−	0.951986i
$172$		0			0
$173$		−	23078.2i		−	0.771100i	−0.922687	−	0.385550i	$-0.874012\pi$
$173$		−	23078.2i		−	0.771100i	0.922687	−	0.385550i	$-0.125988\pi$
$174$		0			0
$175$	3492.11			0.114028
$176$		0			0
$177$	−7920.39			−0.252813
$178$		0			0
$179$			12699.6i			0.396355i	0.980166	+	0.198177i	$0.0635022\pi$
$179$			12699.6i			0.396355i	−0.980166	+	0.198177i	$0.936498\pi$
$180$		0			0
$181$			12065.9i			0.368302i	0.982898	+	0.184151i	$0.0589535\pi$
$181$			12065.9i			0.368302i	−0.982898	+	0.184151i	$0.941046\pi$
$182$		0			0
$183$			4398.89i			0.131353i
$184$		0			0
$185$		−	53140.2i		−	1.55267i
$186$		0			0
$187$	−44687.5			−1.27792
$188$		0			0
$189$		−	5948.93i		−	0.166539i
$190$		0			0
$191$	18113.1			0.496507			0.248254	−	0.968695i	$-0.420143\pi$
$191$	18113.1			0.496507			0.248254	+	0.968695i	$0.420143\pi$
$192$		0			0
$193$			22968.4i			0.616618i	0.951286	+	0.308309i	$0.0997631\pi$
$193$			22968.4i			0.616618i	−0.951286	+	0.308309i	$0.900237\pi$
$194$		0			0
$195$	−2941.77			−0.0773641
$196$		0			0
$197$	−17113.6			−0.440970			−0.220485	−	0.975390i	$-0.570764\pi$
$197$	−17113.6			−0.440970			−0.220485	+	0.975390i	$0.570764\pi$
$198$		0			0
$199$	−35012.4			−0.884129			−0.442064	−	0.896983i	$-0.645754\pi$
$199$	−35012.4			−0.884129			−0.442064	+	0.896983i	$0.645754\pi$
$200$		0			0
$201$	−2545.79			−0.0630130
$202$		0			0
$203$		−	31873.9i		−	0.773470i
$204$		0			0
$205$			36932.9i			0.878833i
$206$		0			0
$207$	61556.1			1.43658
$208$		0			0
$209$	6732.04	−	31998.4i	0.154118	−	0.732548i
$210$		0			0
$211$			43106.1i			0.968220i	0.875007	+	0.484110i	$0.160857\pi$
$211$			43106.1i			0.968220i	−0.875007	+	0.484110i	$0.839143\pi$
$212$		0			0
$213$	5474.35			0.120663
$214$		0			0
$215$	−13954.0			−0.301871
$216$		0			0
$217$		−	40675.6i		−	0.863802i
$218$		0			0
$219$			3579.17i			0.0746266i
$220$		0			0
$221$			44382.6i			0.908716i
$222$		0			0
$223$		−	20353.6i		−	0.409291i	−0.978836	−	0.204645i	$-0.934396\pi$
$223$		−	20353.6i		−	0.409291i	0.978836	−	0.204645i	$-0.0656041\pi$
$224$		0			0
$225$	−10966.2			−0.216617
$226$		0			0
$227$		−	1145.39i		−	0.0222280i	−0.999938	−	0.0111140i	$-0.996462\pi$
$227$		−	1145.39i		−	0.0222280i	0.999938	−	0.0111140i	$-0.00353778\pi$
$228$		0			0
$229$	75560.6			1.44087			0.720435	−	0.693523i	$-0.243942\pi$
$229$	75560.6			1.44087			0.720435	+	0.693523i	$0.243942\pi$
$230$		0			0
$231$		−	3372.03i		−	0.0631927i
$232$		0			0
$233$	−15230.4			−0.280542			−0.140271	−	0.990113i	$-0.544797\pi$
$233$	−15230.4			−0.280542			−0.140271	+	0.990113i	$0.544797\pi$
$234$		0			0
$235$	−68081.6			−1.23280
$236$		0			0
$237$	−15114.5			−0.269090
$238$		0			0
$239$	−63021.4			−1.10330			−0.551648	−	0.834077i	$-0.686001\pi$
$239$	−63021.4			−1.10330			−0.551648	+	0.834077i	$0.686001\pi$
$240$		0			0
$241$		−	69567.0i		−	1.19776i	−0.800839	−	0.598879i	$-0.795613\pi$
$241$		−	69567.0i		−	1.19776i	0.800839	−	0.598879i	$-0.204387\pi$
$242$		0			0
$243$			28150.6i			0.476733i
$244$		0			0
$245$	39043.3			0.650450
$246$		0			0
$247$	−31780.1	−	6686.10i	−0.520908	−	0.109592i
$248$		0			0
$249$		−	11883.0i		−	0.191658i
$250$		0			0
$251$	108698.			1.72534			0.862671	−	0.505766i	$-0.168790\pi$
$251$	108698.			1.72534			0.862671	+	0.505766i	$0.168790\pi$
$252$		0			0
$253$	70758.2			1.10544
$254$		0			0
$255$			16133.0i			0.248105i
$256$		0			0
$257$		−	29231.5i		−	0.442573i	−0.975209	−	0.221287i	$-0.928974\pi$
$257$		−	29231.5i		−	0.442573i	0.975209	−	0.221287i	$-0.0710257\pi$
$258$		0			0
$259$		−	60496.7i		−	0.901845i
$260$		0			0
$261$			100093.i			1.46934i
$262$		0			0
$263$	6152.13			0.0889435			0.0444718	−	0.999011i	$-0.485840\pi$
$263$	6152.13			0.0889435			0.0444718	+	0.999011i	$0.485840\pi$
$264$		0			0
$265$		−	46505.3i		−	0.662233i
$266$		0			0
$267$	−17576.6			−0.246555
$268$		0			0
$269$			89153.5i			1.23207i	0.787721	+	0.616033i	$0.211261\pi$
$269$			89153.5i			1.23207i	−0.787721	+	0.616033i	$0.788739\pi$
$270$		0			0
$271$	−65642.1			−0.893808			−0.446904	−	0.894582i	$-0.647473\pi$
$271$	−65642.1			−0.893808			−0.446904	+	0.894582i	$0.647473\pi$
$272$		0			0
$273$	−3349.02			−0.0449358
$274$		0			0
$275$	−12605.6			−0.166685
$276$		0			0
$277$	−24522.7			−0.319601			−0.159800	−	0.987149i	$-0.551085\pi$
$277$	−24522.7			−0.319601			−0.159800	+	0.987149i	$0.551085\pi$
$278$		0			0
$279$			127733.i			1.64094i
$280$		0			0
$281$		−	83782.6i		−	1.06106i	−0.847665	−	0.530532i	$-0.821992\pi$
$281$		−	83782.6i		−	1.06106i	0.847665	−	0.530532i	$-0.178008\pi$
$282$		0			0
$283$	−74948.5			−0.935816			−0.467908	−	0.883777i	$-0.654992\pi$
$283$	−74948.5			−0.935816			−0.467908	+	0.883777i	$0.654992\pi$
$284$		0			0
$285$	−11552.0	−	2430.39i	−0.142223	−	0.0299217i
$286$		0			0
$287$			42045.8i			0.510457i
$288$		0			0
$289$	159878.			1.91423
$290$		0			0
$291$	5946.56			0.0702231
$292$		0			0
$293$			24472.6i			0.285066i	0.989790	+	0.142533i	$0.0455247\pi$
$293$			24472.6i			0.285066i	−0.989790	+	0.142533i	$0.954475\pi$
$294$		0			0
$295$		−	117673.i		−	1.35218i
$296$		0			0
$297$			21474.0i			0.243444i
$298$		0			0
$299$		−	70275.4i		−	0.786069i
$300$		0			0
$301$	−15885.7			−0.175337
$302$		0			0
$303$			1066.98i			0.0116218i
$304$		0			0
$305$	−65354.3			−0.702546
$306$		0			0
$307$			67113.8i			0.712091i	0.934469	+	0.356045i	$0.115875\pi$
$307$			67113.8i			0.712091i	−0.934469	+	0.356045i	$0.884125\pi$
$308$		0			0
$309$	15487.4			0.162204
$310$		0			0
$311$	75928.7			0.785028			0.392514	−	0.919746i	$-0.371605\pi$
$311$	75928.7			0.785028			0.392514	+	0.919746i	$0.371605\pi$
$312$		0			0
$313$	29188.4			0.297935			0.148968	−	0.988842i	$-0.452405\pi$
$313$	29188.4			0.297935			0.148968	+	0.988842i	$0.452405\pi$
$314$		0			0
$315$	43582.9			0.439233
$316$		0			0
$317$			66441.5i			0.661182i	0.943774	+	0.330591i	$0.107248\pi$
$317$			66441.5i			0.661182i	−0.943774	+	0.330591i	$0.892752\pi$
$318$		0			0
$319$			115056.i			1.13065i
$320$		0			0
$321$	−8769.68			−0.0851086
$322$		0			0
$323$	−36667.4	+	174286.i	−0.351459	+	1.67054i
$324$		0			0
$325$			12519.6i			0.118528i
$326$		0			0
$327$	−21786.7			−0.203750
$328$		0			0
$329$	−77506.5			−0.716055
$330$		0			0
$331$			136481.i			1.24571i	0.782338	+	0.622854i	$0.214027\pi$
$331$			136481.i			1.24571i	−0.782338	+	0.622854i	$0.785973\pi$
$332$		0			0
$333$			189977.i			1.71321i
$334$		0			0
$335$		−	37822.8i		−	0.337026i
$336$		0			0
$337$			140060.i			1.23326i	0.787255	+	0.616628i	$0.211502\pi$
$337$			140060.i			1.23326i	−0.787255	+	0.616628i	$0.788498\pi$
$338$		0			0
$339$	29532.8			0.256983
$340$		0			0
$341$			146828.i			1.26270i
$342$		0			0
$343$	104696.			0.889906
$344$		0			0
$345$		−	25545.0i		−	0.214619i
$346$		0			0
$347$	128168.			1.06444			0.532218	−	0.846607i	$-0.321359\pi$
$347$	128168.			1.06444			0.532218	+	0.846607i	$0.321359\pi$
$348$		0			0
$349$	49366.2			0.405302			0.202651	−	0.979251i	$-0.435044\pi$
$349$	49366.2			0.405302			0.202651	+	0.979251i	$0.435044\pi$
$350$		0			0
$351$	21327.5			0.173111
$352$		0			0
$353$	−193639.			−1.55397			−0.776985	−	0.629519i	$-0.783252\pi$
$353$	−193639.			−1.55397			−0.776985	+	0.629519i	$0.783252\pi$
$354$		0			0
$355$			81332.5i			0.645368i
$356$		0			0
$357$			18366.4i			0.144108i
$358$		0			0
$359$	151082.			1.17226			0.586129	−	0.810218i	$-0.300651\pi$
$359$	151082.			1.17226			0.586129	+	0.810218i	$0.300651\pi$
$360$		0			0
$361$	−119273.	−	52511.3i	−0.915227	−	0.402938i
$362$		0			0
$363$		−	9549.07i		−	0.0724683i
$364$		0			0
$365$	−53175.7			−0.399142
$366$		0			0
$367$	41282.3			0.306501			0.153251	−	0.988187i	$-0.451026\pi$
$367$	41282.3			0.306501			0.153251	+	0.988187i	$0.451026\pi$
$368$		0			0
$369$		−	132036.i		−	0.969702i
$370$		0			0
$371$		−	52943.3i		−	0.384648i
$372$		0			0
$373$		−	14176.2i		−	0.101892i	−0.998701	−	0.0509461i	$-0.983776\pi$
$373$		−	14176.2i		−	0.101892i	0.998701	−	0.0509461i	$-0.0162237\pi$
$374$		0			0
$375$			24988.7i			0.177698i
$376$		0			0
$377$	114271.			0.803995
$378$		0			0
$379$		−	22732.7i		−	0.158261i	−0.996864	−	0.0791304i	$-0.974786\pi$
$379$		−	22732.7i		−	0.158261i	0.996864	−	0.0791304i	$-0.0252144\pi$
$380$		0			0
$381$	3996.05			0.0275284
$382$		0			0
$383$		−	183863.i		−	1.25342i	−0.779253	−	0.626709i	$-0.784401\pi$
$383$		−	183863.i		−	1.25342i	0.779253	−	0.626709i	$-0.215599\pi$
$384$		0			0
$385$	50098.2			0.337988
$386$		0			0
$387$	49885.6			0.333084
$388$		0			0
$389$	−92679.5			−0.612470			−0.306235	−	0.951956i	$-0.599069\pi$
$389$	−92679.5			−0.612470			−0.306235	+	0.951956i	$0.599069\pi$
$390$		0			0
$391$	−385398.			−2.52090
$392$		0			0
$393$		−	763.747i		−	0.00494498i
$394$		0			0
$395$		−	224556.i		−	1.43923i
$396$		0			0
$397$	60594.9			0.384464			0.192232	−	0.981350i	$-0.438427\pi$
$397$	60594.9			0.384464			0.192232	+	0.981350i	$0.438427\pi$
$398$		0			0
$399$	−13151.2	−	2766.84i	−0.0826078	−	0.0173796i
$400$		0			0
$401$		−	217565.i		−	1.35301i	−0.736440	−	0.676503i	$-0.763494\pi$
$401$		−	217565.i		−	1.35301i	0.736440	−	0.676503i	$-0.236506\pi$
$402$		0			0
$403$	145826.			0.897892
$404$		0			0
$405$	−132933.			−0.810443
$406$		0			0
$407$			218376.i			1.31831i
$408$		0			0
$409$			222271.i			1.32873i	0.747408	+	0.664365i	$0.231298\pi$
$409$			222271.i			1.32873i	−0.747408	+	0.664365i	$0.768702\pi$
$410$		0			0
$411$			18701.5i			0.110712i
$412$		0			0
$413$		−	133963.i		−	0.785391i
$414$		0			0
$415$	176545.			1.02509
$416$		0			0
$417$			10932.4i			0.0628702i
$418$		0			0
$419$	−169872.			−0.967593			−0.483796	−	0.875181i	$-0.660742\pi$
$419$	−169872.			−0.967593			−0.483796	+	0.875181i	$0.660742\pi$
$420$		0			0
$421$			172925.i			0.975650i	0.872941	+	0.487825i	$0.162210\pi$
$421$			172925.i			0.975650i	−0.872941	+	0.487825i	$0.837790\pi$
$422$		0			0
$423$	243392.			1.36027
$424$		0			0
$425$	68658.8			0.380118
$426$		0			0
$427$	−74401.7			−0.408063
$428$		0			0
$429$	12089.0			0.0656866
$430$		0			0
$431$		−	140649.i		−	0.757152i	−0.925570	−	0.378576i	$-0.876414\pi$
$431$		−	140649.i		−	0.757152i	0.925570	−	0.378576i	$-0.123586\pi$
$432$		0			0
$433$			245244.i			1.30805i	0.756475	+	0.654023i	$0.226920\pi$
$433$			245244.i			1.30805i	−0.756475	+	0.654023i	$0.773080\pi$
$434$		0			0
$435$	41537.4			0.219513
$436$		0			0
$437$	58059.1	−	275964.i	0.304024	−	1.44507i
$438$		0			0
$439$			202225.i			1.04931i	0.851314	+	0.524657i	$0.175806\pi$
$439$			202225.i			1.04931i	−0.851314	+	0.524657i	$0.824194\pi$
$440$		0			0
$441$	−139580.			−0.717705
$442$		0			0
$443$	−168391.			−0.858048			−0.429024	−	0.903293i	$-0.641142\pi$
$443$	−168391.			−0.858048			−0.429024	+	0.903293i	$0.641142\pi$
$444$		0			0
$445$		−	261136.i		−	1.31870i
$446$		0			0
$447$		−	61523.6i		−	0.307912i
$448$		0			0
$449$		−	36906.9i		−	0.183069i	−0.995802	−	0.0915346i	$-0.970823\pi$
$449$		−	36906.9i		−	0.183069i	0.995802	−	0.0915346i	$-0.0291772\pi$
$450$		0			0
$451$		−	151774.i		−	0.746180i
$452$		0			0
$453$	−31631.4			−0.154142
$454$		0			0
$455$		−	49756.4i		−	0.240340i
$456$		0			0
$457$	−27462.6			−0.131495			−0.0657475	−	0.997836i	$-0.520943\pi$
$457$	−27462.6			−0.131495			−0.0657475	+	0.997836i	$0.520943\pi$
$458$		0			0
$459$		−	116962.i		−	0.555163i
$460$		0			0
$461$	144326.			0.679116			0.339558	−	0.940585i	$-0.389723\pi$
$461$	144326.			0.679116			0.339558	+	0.940585i	$0.389723\pi$
$462$		0			0
$463$	73496.6			0.342851			0.171425	−	0.985197i	$-0.445163\pi$
$463$	73496.6			0.342851			0.171425	+	0.985197i	$0.445163\pi$
$464$		0			0
$465$	53007.5			0.245150
$466$		0			0
$467$	205290.			0.941311			0.470656	−	0.882317i	$-0.344017\pi$
$467$	205290.			0.941311			0.470656	+	0.882317i	$0.344017\pi$
$468$		0			0
$469$		−	43058.8i		−	0.195757i
$470$		0			0
$471$		−	3123.54i		−	0.0140801i
$472$		0			0
$473$	57343.1			0.256306
$474$		0			0
$475$	−10343.2	+	49163.0i	−0.0458426	+	0.217897i
$476$		0			0
$477$			166257.i			0.730707i
$478$		0			0
$479$	−360535.			−1.57136			−0.785681	−	0.618632i	$-0.787687\pi$
$479$	−360535.			−1.57136			−0.785681	+	0.618632i	$0.787687\pi$
$480$		0			0
$481$	216886.			0.937437
$482$		0			0
$483$		−	29081.4i		−	0.124658i
$484$		0			0
$485$			88348.1i			0.375590i
$486$		0			0
$487$			217780.i			0.918249i	0.888372	+	0.459125i	$0.151837\pi$
$487$			217780.i			0.918249i	−0.888372	+	0.459125i	$0.848163\pi$
$488$		0			0
$489$			70511.7i			0.294879i
$490$		0			0
$491$	−68766.9			−0.285244			−0.142622	−	0.989777i	$-0.545553\pi$
$491$	−68766.9			−0.285244			−0.142622	+	0.989777i	$0.545553\pi$
$492$		0			0
$493$		−	626676.i		−	2.57839i
$494$		0			0
$495$	−157322.			−0.642067
$496$		0			0
$497$			92591.8i			0.374852i
$498$		0			0
$499$	−439361.			−1.76450			−0.882248	−	0.470784i	$-0.843971\pi$
$499$	−439361.			−1.76450			−0.882248	+	0.470784i	$0.843971\pi$
$500$		0			0
$501$	65803.8			0.262166
$502$		0			0
$503$	−165528.			−0.654239			−0.327120	−	0.944983i	$-0.606078\pi$
$503$	−165528.			−0.654239			−0.327120	+	0.944983i	$0.606078\pi$
$504$		0			0
$505$	−15852.1			−0.0621592
$506$		0			0
$507$			30366.1i			0.118134i
$508$		0			0
$509$		−	65330.2i		−	0.252161i	−0.992020	−	0.126081i	$-0.959760\pi$
$509$		−	65330.2i		−	0.252161i	0.992020	−	0.126081i	$-0.0402398\pi$
$510$		0			0
$511$	−60537.2			−0.231836
$512$		0			0
$513$	83750.7	+	17620.0i	0.318239	+	0.0669532i
$514$		0			0
$515$			230097.i			0.867554i
$516$		0			0
$517$	279777.			1.04672
$518$		0			0
$519$	34238.5			0.127110
$520$		0			0
$521$		−	237321.i		−	0.874301i	−0.899388	−	0.437151i	$-0.855988\pi$
$521$		−	237321.i		−	0.874301i	0.899388	−	0.437151i	$-0.144012\pi$
$522$		0			0
$523$		−	321195.i		−	1.17426i	−0.809491	−	0.587132i	$-0.800257\pi$
$523$		−	321195.i		−	1.17426i	0.809491	−	0.587132i	$-0.199743\pi$
$524$		0			0
$525$			5180.85i			0.0187967i
$526$		0			0
$527$		−	799726.i		−	2.87952i
$528$		0			0
$529$	330399.			1.18067
$530$		0			0
$531$			420683.i			1.49199i
$532$		0			0
$533$	−150738.			−0.530602
$534$		0			0
$535$		−	130291.i		−	0.455205i
$536$		0			0
$537$	−18840.9			−0.0653362
$538$		0			0
$539$	−160446.			−0.552270
$540$		0			0
$541$	−478895.			−1.63624			−0.818118	−	0.575051i	$-0.804982\pi$
$541$	−478895.			−1.63624			−0.818118	+	0.575051i	$0.804982\pi$
$542$		0			0
$543$	−17900.8			−0.0607119
$544$		0			0
$545$		−	323686.i		−	1.08976i
$546$		0			0
$547$		−	276283.i		−	0.923376i	−0.887042	−	0.461688i	$-0.847244\pi$
$547$		−	276283.i		−	0.923376i	0.887042	−	0.461688i	$-0.152756\pi$
$548$		0			0
$549$	233642.			0.775187
$550$		0			0
$551$	448730.	+	94406.8i	1.47803	+	0.310957i
$552$		0			0
$553$		−	255643.i		−	0.835956i
$554$		0			0
$555$	78837.9			0.255947
$556$		0			0
$557$	−322734.			−1.04024			−0.520121	−	0.854092i	$-0.674113\pi$
$557$	−322734.			−1.04024			−0.520121	+	0.854092i	$0.674113\pi$
$558$		0			0
$559$		−	56951.8i		−	0.182257i
$560$		0			0
$561$		−	66297.7i		−	0.210655i
$562$		0			0
$563$		−	224027.i		−	0.706780i	−0.935476	−	0.353390i	$-0.885029\pi$
$563$		−	224027.i		−	0.706780i	0.935476	−	0.353390i	$-0.114971\pi$
$564$		0			0
$565$			438768.i			1.37448i
$566$		0			0
$567$	−151336.			−0.470734
$568$		0			0
$569$		−	305755.i		−	0.944385i	−0.881495	−	0.472193i	$-0.843463\pi$
$569$		−	305755.i		−	0.944385i	0.881495	−	0.472193i	$-0.156537\pi$
$570$		0			0
$571$	118145.			0.362362			0.181181	−	0.983450i	$-0.442008\pi$
$571$	118145.			0.362362			0.181181	+	0.983450i	$0.442008\pi$
$572$		0			0
$573$			26872.3i			0.0818456i
$574$		0			0
$575$	−108714.			−0.328814
$576$		0			0
$577$	−443650.			−1.33257			−0.666283	−	0.745699i	$-0.732116\pi$
$577$	−443650.			−1.33257			−0.666283	+	0.745699i	$0.732116\pi$
$578$		0			0
$579$	−34075.6			−0.101645
$580$		0			0
$581$	200986.			0.595405
$582$		0			0
$583$			191111.i			0.562275i
$584$		0			0
$585$			156249.i			0.456568i
$586$		0			0
$587$	228853.			0.664172			0.332086	−	0.943249i	$-0.392248\pi$
$587$	228853.			0.664172			0.332086	+	0.943249i	$0.392248\pi$
$588$		0			0
$589$	572643.	+	120476.i	1.65064	+	0.347273i
$590$		0			0
$591$		−	25389.5i		−	0.0726907i
$592$		0			0
$593$	−390241.			−1.10975			−0.554873	−	0.831935i	$-0.687233\pi$
$593$	−390241.			−1.10975			−0.554873	+	0.831935i	$0.687233\pi$
$594$		0			0
$595$	−272870.			−0.770764
$596$		0			0
$597$		−	51943.8i		−	0.145742i
$598$		0			0
$599$		−	241501.i		−	0.673078i	−0.941670	−	0.336539i	$-0.890744\pi$
$599$		−	241501.i		−	0.673078i	0.941670	−	0.336539i	$-0.109256\pi$
$600$		0			0
$601$			153315.i			0.424459i	0.977220	+	0.212230i	$0.0680725\pi$
$601$			153315.i			0.424459i	−0.977220	+	0.212230i	$0.931927\pi$
$602$		0			0
$603$			135217.i			0.371874i
$604$		0			0
$605$	141871.			0.387598
$606$		0			0
$607$			570939.i			1.54958i	0.632222	+	0.774788i	$0.282143\pi$
$607$			570939.i			1.54958i	−0.632222	+	0.774788i	$0.717857\pi$
$608$		0			0
$609$	47287.7			0.127501
$610$		0			0
$611$		−	277868.i		−	0.744314i
$612$		0			0
$613$	−734286.			−1.95409			−0.977045	−	0.213034i	$-0.931666\pi$
$613$	−734286.			−1.95409			−0.977045	+	0.213034i	$0.931666\pi$
$614$		0			0
$615$	−54793.1			−0.144869
$616$		0			0
$617$	126400.			0.332030			0.166015	−	0.986123i	$-0.446910\pi$
$617$	126400.			0.332030			0.166015	+	0.986123i	$0.446910\pi$
$618$		0			0
$619$	223622.			0.583625			0.291812	−	0.956476i	$-0.405742\pi$
$619$	223622.			0.583625			0.291812	+	0.956476i	$0.405742\pi$
$620$		0			0
$621$			185198.i			0.480234i
$622$		0			0
$623$		−	297287.i		−	0.765949i
$624$		0			0
$625$	−284278.			−0.727752
$626$		0			0
$627$	47472.4	+	9987.54i	0.120755	+	0.0254053i
$628$		0			0
$629$		−	1.18943e6i		−	3.00634i
$630$		0			0
$631$	446767.			1.12208			0.561038	−	0.827790i	$-0.310402\pi$
$631$	446767.			1.12208			0.561038	+	0.827790i	$0.310402\pi$
$632$		0			0
$633$	−63951.6			−0.159604
$634$		0			0
$635$			59369.3i			0.147236i
$636$		0			0
$637$			159351.i			0.392714i
$638$		0			0
$639$		−	290764.i		−	0.712097i
$640$		0			0
$641$			715029.i			1.74023i	0.492846	+	0.870116i	$0.335957\pi$
$641$			715029.i			1.74023i	−0.492846	+	0.870116i	$0.664043\pi$
$642$		0			0
$643$	772328.			1.86801			0.934007	−	0.357255i	$-0.116287\pi$
$643$	772328.			1.86801			0.934007	+	0.357255i	$0.116287\pi$
$644$		0			0
$645$		−	20701.9i		−	0.0497612i
$646$		0			0
$647$	−453919.			−1.08435			−0.542175	−	0.840266i	$-0.682399\pi$
$647$	−453919.			−1.08435			−0.542175	+	0.840266i	$0.682399\pi$
$648$		0			0
$649$			483571.i			1.14808i
$650$		0			0
$651$	60345.6			0.142391
$652$		0			0
$653$	41499.5			0.0973233			0.0486616	−	0.998815i	$-0.484504\pi$
$653$	41499.5			0.0973233			0.0486616	+	0.998815i	$0.484504\pi$
$654$		0			0
$655$	11347.0			0.0264483
$656$		0			0
$657$	190104.			0.440413
$658$		0			0
$659$			234021.i			0.538871i	0.963018	+	0.269435i	$0.0868371\pi$
$659$			234021.i			0.538871i	−0.963018	+	0.269435i	$0.913163\pi$
$660$		0			0
$661$			278550.i			0.637529i	0.947834	+	0.318765i	$0.103268\pi$
$661$			278550.i			0.637529i	−0.947834	+	0.318765i	$0.896732\pi$
$662$		0			0
$663$	−65845.3			−0.149795
$664$		0			0
$665$	41107.0	−	195388.i	0.0929549	−	0.441830i
$666$		0			0
$667$			992278.i			2.23039i
$668$		0			0
$669$	30196.3			0.0674686
$670$		0			0
$671$	268570.			0.596502
$672$		0			0
$673$			230099.i			0.508024i	0.967201	+	0.254012i	$0.0817504\pi$
$673$			230099.i			0.508024i	−0.967201	+	0.254012i	$0.918250\pi$
$674$		0			0
$675$		−	32993.1i		−	0.0724128i
$676$		0			0
$677$		−	385800.i		−	0.841753i	−0.907118	−	0.420876i	$-0.861723\pi$
$677$		−	385800.i		−	0.841753i	0.907118	−	0.420876i	$-0.138277\pi$
$678$		0			0
$679$			100579.i			0.218156i
$680$		0			0
$681$	1699.28			0.00366413
$682$		0			0
$683$		−	482596.i		−	1.03453i	−0.855826	−	0.517264i	$-0.826951\pi$
$683$		−	482596.i		−	1.03453i	0.855826	−	0.517264i	$-0.173049\pi$
$684$		0			0
$685$	−277848.			−0.592143
$686$		0			0
$687$			112101.i			0.237517i
$688$		0			0
$689$	189807.			0.399828
$690$		0			0
$691$	264485.			0.553917			0.276959	−	0.960882i	$-0.410673\pi$
$691$	264485.			0.553917			0.276959	+	0.960882i	$0.410673\pi$
$692$		0			0
$693$	−179101.			−0.372935
$694$		0			0
$695$	−162423.			−0.336263
$696$		0			0
$697$			826666.i			1.70163i
$698$		0			0
$699$		−	22595.5i		−	0.0462453i
$700$		0			0
$701$	−221716.			−0.451192			−0.225596	−	0.974221i	$-0.572433\pi$
$701$	−221716.			−0.451192			−0.225596	+	0.974221i	$0.572433\pi$
$702$		0			0
$703$	851691.	+	179184.i	1.72334	+	0.362567i
$704$		0			0
$705$		−	101005.i		−	0.203219i
$706$		0			0
$707$	−18046.7			−0.0361042
$708$		0			0
$709$	−386790.			−0.769454			−0.384727	−	0.923030i	$-0.625704\pi$
$709$	−386790.			−0.769454			−0.384727	+	0.923030i	$0.625704\pi$
$710$		0			0
$711$			802791.i			1.58805i
$712$		0			0
$713$			1.26629e6i			2.49088i
$714$		0			0
$715$			179607.i			0.351326i
$716$		0			0
$717$		−	93497.6i		−	0.181870i
$718$		0			0
$719$	178094.			0.344501			0.172250	−	0.985053i	$-0.444896\pi$
$719$	178094.			0.344501			0.172250	+	0.985053i	$0.444896\pi$
$720$		0			0
$721$			261951.i			0.503906i
$722$		0			0
$723$	103209.			0.197442
$724$		0			0
$725$		−	176774.i		−	0.336313i
$726$		0			0
$727$	−84944.4			−0.160719			−0.0803593	−	0.996766i	$-0.525607\pi$
$727$	−84944.4			−0.160719			−0.0803593	+	0.996766i	$0.525607\pi$
$728$		0			0
$729$	446747.			0.840633
$730$		0			0
$731$	−312330.			−0.584493
$732$		0			0
$733$	−442203.			−0.823026			−0.411513	−	0.911404i	$-0.635000\pi$
$733$	−442203.			−0.823026			−0.411513	+	0.911404i	$0.635000\pi$
$734$		0			0
$735$			57924.0i			0.107222i
$736$		0			0
$737$			155431.i			0.286155i
$738$		0			0
$739$	−135010.			−0.247217			−0.123609	−	0.992331i	$-0.539447\pi$
$739$	−135010.			−0.247217			−0.123609	+	0.992331i	$0.539447\pi$
$740$		0			0
$741$	9919.39	−	47148.5i	0.0180654	−	0.0858680i
$742$		0			0
$743$			768452.i			1.39200i	0.718042	+	0.695999i	$0.245038\pi$
$743$			768452.i			1.39200i	−0.718042	+	0.695999i	$0.754962\pi$
$744$		0			0
$745$	914057.			1.64687
$746$		0			0
$747$	−631151.			−1.13108
$748$		0			0
$749$		−	148328.i		−	0.264399i
$750$		0			0
$751$		−	496024.i		−	0.879474i	−0.898127	−	0.439737i	$-0.855072\pi$
$751$		−	496024.i		−	0.879474i	0.898127	−	0.439737i	$-0.144928\pi$
$752$		0			0
$753$			161263.i			0.284410i
$754$		0			0
$755$		−	469947.i		−	0.824433i
$756$		0			0
$757$	446372.			0.778942			0.389471	−	0.921039i	$-0.372658\pi$
$757$	446372.			0.778942			0.389471	+	0.921039i	$0.372658\pi$
$758$		0			0
$759$			104976.i			0.182224i
$760$		0			0
$761$	−980410.			−1.69293			−0.846464	−	0.532446i	$-0.821273\pi$
$761$	−980410.			−1.69293			−0.846464	+	0.532446i	$0.821273\pi$
$762$		0			0
$763$		−	368495.i		−	0.632970i
$764$		0			0
$765$	856888.			1.46420
$766$		0			0
$767$	480271.			0.816387
$768$		0			0
$769$	5545.25			0.00937710			0.00468855	−	0.999989i	$-0.498508\pi$
$769$	5545.25			0.00937710			0.00468855	+	0.999989i	$0.498508\pi$
$770$		0			0
$771$	43367.4			0.0729550
$772$		0			0
$773$		−	1.07628e6i		−	1.80122i	−0.434627	−	0.900611i	$-0.643120\pi$
$773$		−	1.07628e6i		−	1.80122i	0.434627	−	0.900611i	$-0.356880\pi$
$774$		0			0
$775$		−	225589.i		−	0.375590i
$776$		0			0
$777$	89752.0			0.148663
$778$		0			0
$779$	−591933.	−	124535.i	−0.975434	−	0.205218i
$780$		0			0
$781$		−	334231.i		−	0.547955i
$782$		0			0
$783$	−301141.			−0.491186
$784$		0			0
$785$	46406.4			0.0753075
$786$		0			0
$787$			717648.i			1.15868i	0.815087	+	0.579338i	$0.196689\pi$
$787$			717648.i			1.15868i	−0.815087	+	0.579338i	$0.803311\pi$
$788$		0			0
$789$			9127.21i			0.0146617i
$790$		0			0
$791$			499510.i			0.798346i
$792$		0			0
$793$		−	266737.i		−	0.424167i
$794$		0			0
$795$	68994.6			0.109164
$796$		0			0
$797$		−	42888.1i		−	0.0675181i	−0.999430	−	0.0337590i	$-0.989252\pi$
$797$		−	42888.1i		−	0.0675181i	0.999430	−	0.0337590i	$-0.0107479\pi$
$798$		0			0
$799$	−1.52386e6			−2.38700
$800$		0			0
$801$			933564.i			1.45505i
$802$		0			0
$803$	218523.			0.338895
$804$		0			0
$805$	432062.			0.666736
$806$		0			0
$807$	−132267.			−0.203097
$808$		0			0
$809$	754628.			1.15302			0.576509	−	0.817091i	$-0.304415\pi$
$809$	754628.			1.15302			0.576509	+	0.817091i	$0.304415\pi$
$810$		0			0
$811$		−	566538.i		−	0.861366i	−0.902503	−	0.430683i	$-0.858273\pi$
$811$		−	566538.i		−	0.861366i	0.902503	−	0.430683i	$-0.141727\pi$
$812$		0			0
$813$		−	97385.6i		−	0.147338i
$814$		0			0
$815$	−1.04759e6			−1.57717
$816$		0			0
$817$	47051.6	−	223644.i	0.0704905	−	0.335053i
$818$		0			0
$819$			177879.i			0.265190i
$820$		0			0
$821$	574408.			0.852186			0.426093	−	0.904679i	$-0.359890\pi$
$821$	574408.			0.852186			0.426093	+	0.904679i	$0.359890\pi$
$822$		0			0
$823$	716685.			1.05810			0.529052	−	0.848589i	$-0.322547\pi$
$823$	716685.			1.05810			0.529052	+	0.848589i	$0.322547\pi$
$824$		0			0
$825$		−	18701.4i		−	0.0274769i
$826$		0			0
$827$		−	1.16092e6i		−	1.69743i	−0.528851	−	0.848715i	$-0.677377\pi$
$827$		−	1.16092e6i		−	1.69743i	0.528851	−	0.848715i	$-0.322623\pi$
$828$		0			0
$829$			261267.i			0.380167i	0.981768	+	0.190084i	$0.0608759\pi$
$829$			261267.i			0.380167i	−0.981768	+	0.190084i	$0.939124\pi$
$830$		0			0
$831$		−	36381.4i		−	0.0526839i
$832$		0			0
$833$	873901.			1.25942
$834$		0			0
$835$			977647.i			1.40220i
$836$		0			0
$837$	−384298.			−0.548551
$838$		0			0
$839$			205092.i			0.291356i	0.989332	+	0.145678i	$0.0465364\pi$
$839$			205092.i			0.291356i	−0.989332	+	0.145678i	$0.953464\pi$
$840$		0			0
$841$	−906209.			−1.28126
$842$		0			0
$843$	124299.			0.174909
$844$		0			0
$845$	−451149.			−0.631840
$846$		0			0
$847$	161511.			0.225130
$848$		0			0
$849$		−	111193.i		−	0.154262i
$850$		0			0
$851$			1.88334e6i			2.60058i
$852$		0			0
$853$	−393930.			−0.541403			−0.270701	−	0.962663i	$-0.587256\pi$
$853$	−393930.			−0.541403			−0.270701	+	0.962663i	$0.587256\pi$
$854$		0			0
$855$	−129087.	+	613573.i	−0.176584	+	0.839333i
$856$		0			0
$857$			667640.i			0.909035i	0.890738	+	0.454518i	$0.150188\pi$
$857$			667640.i			0.909035i	−0.890738	+	0.454518i	$0.849812\pi$
$858$		0			0
$859$	1.31768e6			1.78576			0.892879	−	0.450297i	$-0.148682\pi$
$859$	1.31768e6			1.78576			0.892879	+	0.450297i	$0.148682\pi$
$860$		0			0
$861$	−62378.5			−0.0841451
$862$		0			0
$863$		−	756574.i		−	1.01585i	−0.861401	−	0.507925i	$-0.830413\pi$
$863$		−	756574.i		−	1.01585i	0.861401	−	0.507925i	$-0.169587\pi$
$864$		0			0
$865$			508682.i			0.679851i
$866$		0			0
$867$			237193.i			0.315547i
$868$		0			0
$869$			922801.i			1.22199i
$870$		0			0
$871$	154370.			0.203482
$872$		0			0
$873$		−	315845.i		−	0.414425i
$874$		0			0
$875$	−422653.			−0.552037
$876$		0			0
$877$			193421.i			0.251481i	0.992063	+	0.125741i	$0.0401307\pi$
$877$			193421.i			0.251481i	−0.992063	+	0.125741i	$0.959869\pi$
$878$		0			0
$879$	−36307.2			−0.0469911
$880$		0			0
$881$	973793.			1.25463			0.627314	−	0.778766i	$-0.284154\pi$
$881$	973793.			1.25463			0.627314	+	0.778766i	$0.284154\pi$
$882$		0			0
$883$	−327032.			−0.419439			−0.209720	−	0.977762i	$-0.567255\pi$
$883$	−327032.			−0.419439			−0.209720	+	0.977762i	$0.567255\pi$
$884$		0			0
$885$	174578.			0.222897
$886$		0			0
$887$		−	171406.i		−	0.217861i	−0.994049	−	0.108931i	$-0.965257\pi$
$887$		−	171406.i		−	0.217861i	0.994049	−	0.108931i	$-0.0347426\pi$
$888$		0			0
$889$			67588.2i			0.0855199i
$890$		0			0
$891$	546280.			0.688113
$892$		0			0
$893$	229565.	−	1.09116e6i	0.287874	−	1.36831i
$894$		0			0
$895$		−	279920.i		−	0.349452i
$896$		0			0
$897$	104259.			0.129578
$898$		0			0
$899$	−2.05904e6			−2.54768
$900$		0			0
$901$		−	1.04092e6i		−	1.28224i
$902$		0			0
$903$		−	23567.8i		−	0.0289030i
$904$		0			0
$905$		−	265953.i		−	0.324719i
$906$		0			0
$907$		−	1.09015e6i		−	1.32517i	−0.748986	−	0.662586i	$-0.769459\pi$
$907$		−	1.09015e6i		−	1.32517i	0.748986	−	0.662586i	$-0.230541\pi$
$908$		0			0
$909$	56671.5			0.0685863
$910$		0			0
$911$		−	269302.i		−	0.324491i	−0.986750	−	0.162246i	$-0.948126\pi$
$911$		−	269302.i		−	0.324491i	0.986750	−	0.162246i	$-0.0518737\pi$
$912$		0			0
$913$	−725503.			−0.870357
$914$		0			0
$915$		−	96958.6i		−	0.115810i
$916$		0			0
$917$	12917.8			0.0153621
$918$		0			0
$919$	32738.8			0.0387643			0.0193821	−	0.999812i	$-0.493830\pi$
$919$	32738.8			0.0387643			0.0193821	+	0.999812i	$0.493830\pi$
$920$		0			0
$921$	−99569.0			−0.117383
$922$		0			0
$923$	−331950.			−0.389646
$924$		0			0
$925$		−	335518.i		−	0.392132i
$926$		0			0
$927$		−	822599.i		−	0.957257i
$928$		0			0
$929$	606224.			0.702427			0.351214	−	0.936295i	$-0.385769\pi$
$929$	606224.			0.702427			0.351214	+	0.936295i	$0.385769\pi$
$930$		0			0
$931$	−131651.	+	625756.i	−0.151888	+	0.721947i
$932$		0			0
$933$			112647.i			0.129406i
$934$		0			0
$935$	984985.			1.12670
$936$		0			0
$937$	−599917.			−0.683301			−0.341650	−	0.939827i	$-0.610986\pi$
$937$	−599917.			−0.683301			−0.341650	+	0.939827i	$0.610986\pi$
$938$		0			0
$939$			43303.5i			0.0491125i
$940$		0			0
$941$			9550.71i			0.0107859i	0.999985	+	0.00539295i	$0.00171664\pi$
$941$			9550.71i			0.0107859i	−0.999985	+	0.00539295i	$0.998283\pi$
$942$		0			0
$943$		−	1.30894e6i		−	1.47196i
$944$		0			0
$945$			131124.i			0.146831i
$946$		0			0
$947$	1.60087e6			1.78508			0.892538	−	0.450973i	$-0.148923\pi$
$947$	1.60087e6			1.78508			0.892538	+	0.450973i	$0.148923\pi$
$948$		0			0
$949$		−	217031.i		−	0.240985i
$950$		0			0
$951$	−98571.6			−0.108991
$952$		0			0
$953$		−	451633.i		−	0.497279i	−0.968596	−	0.248639i	$-0.920017\pi$
$953$		−	451633.i		−	0.497279i	0.968596	−	0.248639i	$-0.0799834\pi$
$954$		0			0
$955$	−399242.			−0.437753
$956$		0			0
$957$	−170695.			−0.186379
$958$		0			0
$959$	−316313.			−0.343937
$960$		0			0
$961$	−1.70410e6			−1.84522
$962$		0			0
$963$			465792.i			0.502272i
$964$		0			0
$965$		−	506261.i		−	0.543650i
$966$		0			0
$967$	1.30999e6			1.40093			0.700463	−	0.713689i	$-0.252977\pi$
$967$	1.30999e6			1.40093			0.700463	+	0.713689i	$0.252977\pi$
$968$		0			0
$969$	−258568.	−	54399.1i	−0.275377	−	0.0579355i
$970$		0			0
$971$		−	1.68075e6i		−	1.78265i	−0.453368	−	0.891323i	$-0.649778\pi$
$971$		−	1.68075e6i		−	1.78265i	0.453368	−	0.891323i	$-0.350222\pi$
$972$		0			0
$973$	−184908.			−0.195313
$974$		0			0
$975$	−18573.8			−0.0195385
$976$		0			0
$977$		−	977079.i		−	1.02362i	−0.859097	−	0.511812i	$-0.828974\pi$
$977$		−	977079.i		−	1.02362i	0.859097	−	0.511812i	$-0.171026\pi$
$978$		0			0
$979$			1.07312e6i			1.11966i
$980$		0			0
$981$			1.15718e6i			1.20244i
$982$		0			0
$983$			800046.i			0.827957i	0.910287	+	0.413978i	$0.135861\pi$
$983$			800046.i			0.827957i	−0.910287	+	0.413978i	$0.864139\pi$
$984$		0			0
$985$	377212.			0.388788
$986$		0			0
$987$		−	114987.i		−	0.118036i
$988$		0			0
$989$	494544.			0.505606
$990$		0			0
$991$		−	1.43409e6i		−	1.46026i	−0.683308	−	0.730131i	$-0.739459\pi$
$991$		−	1.43409e6i		−	1.46026i	0.683308	−	0.730131i	$-0.260541\pi$
$992$		0			0
$993$	−202481.			−0.205346
$994$		0			0
$995$	771730.			0.779505
$996$		0			0
$997$	−29754.6			−0.0299340			−0.0149670	−	0.999888i	$-0.504764\pi$
$997$	−29754.6			−0.0299340			−0.0149670	+	0.999888i	$0.504764\pi$
$998$		0			0
$999$	−571565.			−0.572710

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.5.e.f.113.11		20
4.3	odd	2		152.5.e.a.113.10	✓	20
12.11	even	2		1368.5.o.a.721.15		20
19.18	odd	2	inner	304.5.e.f.113.10		20
76.75	even	2		152.5.e.a.113.11	yes	20
228.227	odd	2		1368.5.o.a.721.16		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.5.e.a.113.10	✓	20	4.3	odd	2
152.5.e.a.113.11	yes	20	76.75	even	2
304.5.e.f.113.10		20	19.18	odd	2	inner
304.5.e.f.113.11		20	1.1	even	1	trivial
1368.5.o.a.721.15		20	12.11	even	2
1368.5.o.a.721.16		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)

\(f(q)\)	\(=\)	\(q+1.48358i q^{3} -22.0416 q^{5} -25.0930 q^{7} +78.7990 q^{9} +90.5788 q^{11} -89.9607i q^{13} -32.7006i q^{15} -493.355 q^{17} +(74.3224 - 353.266i) q^{19} -37.2276i q^{21} +781.178 q^{23} -139.167 q^{25} +237.075i q^{27} +1270.23i q^{29} +1620.99i q^{31} +134.381i q^{33} +553.090 q^{35} +2410.90i q^{37} +133.464 q^{39} -1675.60i q^{41} +633.074 q^{43} -1736.86 q^{45} +3088.77 q^{47} -1771.34 q^{49} -731.934i q^{51} +2109.89i q^{53} -1996.50 q^{55} +(524.101 + 110.264i) q^{57} +5338.68i q^{59} +2965.04 q^{61} -1977.30 q^{63} +1982.88i q^{65} +1715.97i q^{67} +1158.94i q^{69} -3689.95i q^{71} +2412.51 q^{73} -206.466i q^{75} -2272.89 q^{77} +10187.8i q^{79} +6031.00 q^{81} -8009.63 q^{83} +10874.3 q^{85} -1884.50 q^{87} +11847.4i q^{89} +2257.38i q^{91} -2404.88 q^{93} +(-1638.19 + 7786.56i) q^{95} -4008.24i q^{97} +7137.51 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 32 q^{7} - 372 q^{9} + 24 q^{11} + 216 q^{17} + 596 q^{19} - 576 q^{23} + 1412 q^{25} + 144 q^{35} + 520 q^{39} + 1256 q^{43} + 7232 q^{45} + 3768 q^{47} - 2740 q^{49} + 10128 q^{55} - 728 q^{57}+ \cdots - 760 q^{99}+O(q^{100}) \) 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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