LMFDB - Embedded newform 304.5.e.f.113.10

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.48358i q^{3} -22.0416 q^{5} -25.0930 q^{7} +78.7990 q^{9} +O(q^{10})$	$q-1.48358i q^{3} -22.0416 q^{5} -25.0930 q^{7} +78.7990 q^{9} +90.5788 q^{11} +89.9607i q^{13} +32.7006i q^{15} -493.355 q^{17} +(74.3224 + 353.266i) q^{19} +37.2276i q^{21} +781.178 q^{23} -139.167 q^{25} -237.075i q^{27} -1270.23i q^{29} -1620.99i q^{31} -134.381i q^{33} +553.090 q^{35} -2410.90i q^{37} +133.464 q^{39} +1675.60i q^{41} +633.074 q^{43} -1736.86 q^{45} +3088.77 q^{47} -1771.34 q^{49} +731.934i q^{51} -2109.89i q^{53} -1996.50 q^{55} +(524.101 - 110.264i) q^{57} -5338.68i q^{59} +2965.04 q^{61} -1977.30 q^{63} -1982.88i q^{65} -1715.97i q^{67} -1158.94i q^{69} +3689.95i q^{71} +2412.51 q^{73} +206.466i q^{75} -2272.89 q^{77} -10187.8i q^{79} +6031.00 q^{81} -8009.63 q^{83} +10874.3 q^{85} -1884.50 q^{87} -11847.4i q^{89} -2257.38i q^{91} -2404.88 q^{93} +(-1638.19 - 7786.56i) q^{95} +4008.24i q^{97} +7137.51 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 32 q^{7} - 372 q^{9}+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
$3$		−	1.48358i		−	0.164843i	−0.996598	−	0.0824214i	$-0.973735\pi$
$3$		−	1.48358i		−	0.164843i	0.996598	−	0.0824214i	$-0.0262653\pi$
$4$		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	$0.0857536\pi$
$13$			89.9607i			0.532312i	−0.963930	+	0.266156i	$0.914246\pi$
$14$		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	$-0.727544\pi$
$29$		−	1270.23i		−	1.51038i	0.655504	−	0.755192i	$-0.272456\pi$
$30$		0			0
$31$		−	1620.99i		−	1.68678i	−0.537303	−	0.843389i	$-0.680557\pi$
$31$		−	1620.99i		−	1.68678i	0.537303	−	0.843389i	$-0.319443\pi$
$32$		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	$-0.657184\pi$
$37$		−	2410.90i		−	1.76107i	0.473983	−	0.880534i	$-0.342816\pi$
$38$		0			0
$39$	133.464			0.0877477
$40$		0			0
$41$			1675.60i			0.996788i	0.866951	+	0.498394i	$0.166077\pi$
$41$			1675.60i			0.996788i	−0.866951	+	0.498394i	$0.833923\pi$
$42$		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	$-0.877451\pi$
$53$		−	2109.89i		−	0.751117i	0.926799	−	0.375558i	$-0.122549\pi$
$54$		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	$-0.721832\pi$
$59$		−	5338.68i		−	1.53366i	0.641848	−	0.766832i	$-0.278168\pi$
$60$		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	$-0.938785\pi$
$67$		−	1715.97i		−	0.382261i	0.981565	−	0.191131i	$-0.0612154\pi$
$68$		0			0
$69$		−	1158.94i		−	0.243425i
$70$		0			0
$71$			3689.95i			0.731988i	0.930617	+	0.365994i	$0.119271\pi$
$71$			3689.95i			0.731988i	−0.930617	+	0.365994i	$0.880729\pi$
$72$		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	$-0.696076\pi$
$79$		−	10187.8i		−	1.63240i	0.577767	−	0.816202i	$-0.303924\pi$
$80$		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	$-0.731087\pi$
$89$		−	11847.4i		−	1.49570i	0.663870	−	0.747848i	$-0.268913\pi$
$90$		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	$0.0683235\pi$
$97$			4008.24i			0.426001i	−0.977052	+	0.213000i	$0.931676\pi$
$98$		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	$0.163733\pi$
$103$			10439.2i			0.983995i	−0.870597	+	0.491998i	$0.836267\pi$
$104$		0			0
$105$		−	820.556i		−	0.0744268i
$106$		0			0
$107$		−	5911.14i		−	0.516302i	−0.966105	−	0.258151i	$-0.916887\pi$
$107$		−	5911.14i		−	0.516302i	0.966105	−	0.258151i	$-0.0831132\pi$
$108$		0			0
$109$		−	14685.2i		−	1.23602i	−0.786169	−	0.618012i	$-0.787938\pi$
$109$		−	14685.2i		−	1.23602i	0.786169	−	0.618012i	$-0.212062\pi$
$110$		0			0
$111$	−3576.78			−0.290299
$112$		0			0
$113$			19906.4i			1.55896i	0.626428	+	0.779480i	$0.284516\pi$
$113$			19906.4i			1.55896i	−0.626428	+	0.779480i	$0.715484\pi$
$114$		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	$0.0266095\pi$
$127$			2693.51i			0.166998i	−0.996508	+	0.0834990i	$0.973390\pi$
$128$		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	$-0.845139\pi$
$151$		−	21320.9i		−	0.935087i	0.883970	−	0.467543i	$-0.154861\pi$
$152$		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	$0.292634\pi$
$167$			44354.6i			1.59040i	−0.606349	+	0.795199i	$0.707366\pi$
$168$		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	$0.125988\pi$
$173$			23078.2i			0.771100i	−0.922687	+	0.385550i	$0.874012\pi$
$174$		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	$-0.936498\pi$
$179$		−	12699.6i		−	0.396355i	0.980166	−	0.198177i	$-0.0635022\pi$
$180$		0			0
$181$		−	12065.9i		−	0.368302i	−0.982898	−	0.184151i	$-0.941046\pi$
$181$		−	12065.9i		−	0.368302i	0.982898	−	0.184151i	$-0.0589535\pi$
$182$		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	$-0.900237\pi$
$193$		−	22968.4i		−	0.616618i	0.951286	−	0.308309i	$-0.0997631\pi$
$194$		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	$-0.839143\pi$
$211$		−	43106.1i		−	0.968220i	0.875007	−	0.484110i	$-0.160857\pi$
$212$		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	$0.0656041\pi$
$223$			20353.6i			0.409291i	−0.978836	+	0.204645i	$0.934396\pi$
$224$		0			0
$225$	−10966.2			−0.216617
$226$		0			0
$227$			1145.39i			0.0222280i	0.999938	+	0.0111140i	$0.00353778\pi$
$227$			1145.39i			0.0222280i	−0.999938	+	0.0111140i	$0.996462\pi$
$228$		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	$0.204387\pi$
$241$			69567.0i			1.19776i	−0.800839	+	0.598879i	$0.795613\pi$
$242$		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	$0.0710257\pi$
$257$			29231.5i			0.442573i	−0.975209	+	0.221287i	$0.928974\pi$
$258$		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	$-0.788739\pi$
$269$		−	89153.5i		−	1.23207i	0.787721	−	0.616033i	$-0.211261\pi$
$270$		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	$0.178008\pi$
$281$			83782.6i			1.06106i	−0.847665	+	0.530532i	$0.821992\pi$
$282$		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	$-0.954475\pi$
$293$		−	24472.6i		−	0.285066i	0.989790	−	0.142533i	$-0.0455247\pi$
$294$		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	$-0.884125\pi$
$307$		−	67113.8i		−	0.712091i	0.934469	−	0.356045i	$-0.115875\pi$
$308$		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	$-0.892752\pi$
$317$		−	66441.5i		−	0.661182i	0.943774	−	0.330591i	$-0.107248\pi$
$318$		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	$-0.785973\pi$
$331$		−	136481.i		−	1.24571i	0.782338	−	0.622854i	$-0.214027\pi$
$332$		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	$-0.788498\pi$
$337$		−	140060.i		−	1.23326i	0.787255	−	0.616628i	$-0.211502\pi$
$338$		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	$0.0162237\pi$
$373$			14176.2i			0.101892i	−0.998701	+	0.0509461i	$0.983776\pi$
$374$		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	$0.0252144\pi$
$379$			22732.7i			0.158261i	−0.996864	+	0.0791304i	$0.974786\pi$
$380$		0			0
$381$	3996.05			0.0275284
$382$		0			0
$383$			183863.i			1.25342i	0.779253	+	0.626709i	$0.215599\pi$
$383$			183863.i			1.25342i	−0.779253	+	0.626709i	$0.784401\pi$
$384$		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	$0.236506\pi$
$401$			217565.i			1.35301i	−0.736440	+	0.676503i	$0.763494\pi$
$402$		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	$-0.768702\pi$
$409$		−	222271.i		−	1.32873i	0.747408	−	0.664365i	$-0.231298\pi$
$410$		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	$-0.837790\pi$
$421$		−	172925.i		−	0.975650i	0.872941	−	0.487825i	$-0.162210\pi$
$422$		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	$0.123586\pi$
$431$			140649.i			0.757152i	−0.925570	+	0.378576i	$0.876414\pi$
$432$		0			0
$433$		−	245244.i		−	1.30805i	−0.756475	−	0.654023i	$-0.773080\pi$
$433$		−	245244.i		−	1.30805i	0.756475	−	0.654023i	$-0.226920\pi$
$434$		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	$-0.824194\pi$
$439$		−	202225.i		−	1.04931i	0.851314	−	0.524657i	$-0.175806\pi$
$440$		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	$0.0291772\pi$
$449$			36906.9i			0.183069i	−0.995802	+	0.0915346i	$0.970823\pi$
$450$		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	$-0.848163\pi$
$487$		−	217780.i		−	0.918249i	0.888372	−	0.459125i	$-0.151837\pi$
$488$		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	$0.0402398\pi$
$509$			65330.2i			0.252161i	−0.992020	+	0.126081i	$0.959760\pi$
$510$		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	$0.144012\pi$
$521$			237321.i			0.874301i	−0.899388	+	0.437151i	$0.855988\pi$
$522$		0			0
$523$			321195.i			1.17426i	0.809491	+	0.587132i	$0.199743\pi$
$523$			321195.i			1.17426i	−0.809491	+	0.587132i	$0.800257\pi$
$524$		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	$0.152756\pi$
$547$			276283.i			0.923376i	−0.887042	+	0.461688i	$0.847244\pi$
$548$		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	$0.114971\pi$
$563$			224027.i			0.706780i	−0.935476	+	0.353390i	$0.885029\pi$
$564$		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	$0.156537\pi$
$569$			305755.i			0.944385i	−0.881495	+	0.472193i	$0.843463\pi$
$570$		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	$0.109256\pi$
$599$			241501.i			0.673078i	−0.941670	+	0.336539i	$0.890744\pi$
$600$		0			0
$601$		−	153315.i		−	0.424459i	−0.977220	−	0.212230i	$-0.931927\pi$
$601$		−	153315.i		−	0.424459i	0.977220	−	0.212230i	$-0.0680725\pi$
$602$		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	$-0.717857\pi$
$607$		−	570939.i		−	1.54958i	0.632222	−	0.774788i	$-0.282143\pi$
$608$		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	$-0.664043\pi$
$641$		−	715029.i		−	1.74023i	0.492846	−	0.870116i	$-0.335957\pi$
$642$		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	$-0.913163\pi$
$659$		−	234021.i		−	0.538871i	0.963018	−	0.269435i	$-0.0868371\pi$
$660$		0			0
$661$		−	278550.i		−	0.637529i	−0.947834	−	0.318765i	$-0.896732\pi$
$661$		−	278550.i		−	0.637529i	0.947834	−	0.318765i	$-0.103268\pi$
$662$		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	$-0.918250\pi$
$673$		−	230099.i		−	0.508024i	0.967201	−	0.254012i	$-0.0817504\pi$
$674$		0			0
$675$			32993.1i			0.0724128i
$676$		0			0
$677$			385800.i			0.841753i	0.907118	+	0.420876i	$0.138277\pi$
$677$			385800.i			0.841753i	−0.907118	+	0.420876i	$0.861723\pi$
$678$		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	$0.173049\pi$
$683$			482596.i			1.03453i	−0.855826	+	0.517264i	$0.826951\pi$
$684$		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	$-0.754962\pi$
$743$		−	768452.i		−	1.39200i	0.718042	−	0.695999i	$-0.245038\pi$
$744$		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	$0.144928\pi$
$751$			496024.i			0.879474i	−0.898127	+	0.439737i	$0.855072\pi$
$752$		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	$0.356880\pi$
$773$			1.07628e6i			1.80122i	−0.434627	+	0.900611i	$0.643120\pi$
$774$		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	$-0.803311\pi$
$787$		−	717648.i		−	1.15868i	0.815087	−	0.579338i	$-0.196689\pi$
$788$		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	$0.0107479\pi$
$797$			42888.1i			0.0675181i	−0.999430	+	0.0337590i	$0.989252\pi$
$798$		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	$0.141727\pi$
$811$			566538.i			0.861366i	−0.902503	+	0.430683i	$0.858273\pi$
$812$		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	$0.322623\pi$
$827$			1.16092e6i			1.69743i	−0.528851	+	0.848715i	$0.677377\pi$
$828$		0			0
$829$		−	261267.i		−	0.380167i	−0.981768	−	0.190084i	$-0.939124\pi$
$829$		−	261267.i		−	0.380167i	0.981768	−	0.190084i	$-0.0608759\pi$
$830$		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	$-0.953464\pi$
$839$		−	205092.i		−	0.291356i	0.989332	−	0.145678i	$-0.0465364\pi$
$840$		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	$-0.849812\pi$
$857$		−	667640.i		−	0.909035i	0.890738	−	0.454518i	$-0.150188\pi$
$858$		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	$0.169587\pi$
$863$			756574.i			1.01585i	−0.861401	+	0.507925i	$0.830413\pi$
$864$		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	$-0.959869\pi$
$877$		−	193421.i		−	0.251481i	0.992063	−	0.125741i	$-0.0401307\pi$
$878$		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	$0.0347426\pi$
$887$			171406.i			0.217861i	−0.994049	+	0.108931i	$0.965257\pi$
$888$		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	$0.230541\pi$
$907$			1.09015e6i			1.32517i	−0.748986	+	0.662586i	$0.769459\pi$
$908$		0			0
$909$	56671.5			0.0685863
$910$		0			0
$911$			269302.i			0.324491i	0.986750	+	0.162246i	$0.0518737\pi$
$911$			269302.i			0.324491i	−0.986750	+	0.162246i	$0.948126\pi$
$912$		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	$-0.998283\pi$
$941$		−	9550.71i		−	0.0107859i	0.999985	−	0.00539295i	$-0.00171664\pi$
$942$		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	$0.0799834\pi$
$953$			451633.i			0.497279i	−0.968596	+	0.248639i	$0.920017\pi$
$954$		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	$0.350222\pi$
$971$			1.68075e6i			1.78265i	−0.453368	+	0.891323i	$0.649778\pi$
$972$		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	$0.171026\pi$
$977$			977079.i			1.02362i	−0.859097	+	0.511812i	$0.828974\pi$
$978$		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	$-0.864139\pi$
$983$		−	800046.i		−	0.827957i	0.910287	−	0.413978i	$-0.135861\pi$
$984$		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	$0.260541\pi$
$991$			1.43409e6i			1.46026i	−0.683308	+	0.730131i	$0.739459\pi$
$992$		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.113.10		20
4.3	odd	2		152.5.e.a.113.11	yes	20
12.11	even	2		1368.5.o.a.721.16		20
19.18	odd	2	inner	304.5.e.f.113.11		20
76.75	even	2		152.5.e.a.113.10	✓	20
228.227	odd	2		1368.5.o.a.721.15		20

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

\(f(q)\)	\(=\)	\(q-1.48358i q^{3} -22.0416 q^{5} -25.0930 q^{7} +78.7990 q^{9} +O(q^{10})\)	\(q-1.48358i q^{3} -22.0416 q^{5} -25.0930 q^{7} +78.7990 q^{9} +90.5788 q^{11} +89.9607i q^{13} +32.7006i q^{15} -493.355 q^{17} +(74.3224 + 353.266i) q^{19} +37.2276i q^{21} +781.178 q^{23} -139.167 q^{25} -237.075i q^{27} -1270.23i q^{29} -1620.99i q^{31} -134.381i q^{33} +553.090 q^{35} -2410.90i q^{37} +133.464 q^{39} +1675.60i q^{41} +633.074 q^{43} -1736.86 q^{45} +3088.77 q^{47} -1771.34 q^{49} +731.934i q^{51} -2109.89i q^{53} -1996.50 q^{55} +(524.101 - 110.264i) q^{57} -5338.68i q^{59} +2965.04 q^{61} -1977.30 q^{63} -1982.88i q^{65} -1715.97i q^{67} -1158.94i q^{69} +3689.95i q^{71} +2412.51 q^{73} +206.466i q^{75} -2272.89 q^{77} -10187.8i q^{79} +6031.00 q^{81} -8009.63 q^{83} +10874.3 q^{85} -1884.50 q^{87} -11847.4i q^{89} -2257.38i q^{91} -2404.88 q^{93} +(-1638.19 - 7786.56i) q^{95} +4008.24i q^{97} +7137.51 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 32 q^{7} - 372 q^{9}+O(q^{10}) \) 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

Related objects

Downloads

Learn more