LMFDB - Embedded newform 8004.2.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8004,2,Mod(1,8004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8004, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8004.a (trivial)

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:	yes
Analytic conductor:	$63.9122617778$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 8x^{6} + 19x^{5} + 19x^{4} - 35x^{3} - 10x^{2} + 18x - 4 $ x^8 - 3x^7 - 8x^6 + 19x^5 + 19x^4 - 35x^3 - 10x^2 + 18*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.386133$ of defining polynomial
Character	$\chi$	$=$	8004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -2.77471 q^{5} +0.556399 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -2.77471 q^{5} +0.556399 q^{7} +1.00000 q^{9} +3.82791 q^{11} -0.0549651 q^{13} -2.77471 q^{15} -2.00765 q^{17} -2.76410 q^{19} +0.556399 q^{21} +1.00000 q^{23} +2.69903 q^{25} +1.00000 q^{27} +1.00000 q^{29} -3.02923 q^{31} +3.82791 q^{33} -1.54385 q^{35} +1.22025 q^{37} -0.0549651 q^{39} -6.65903 q^{41} -4.40578 q^{43} -2.77471 q^{45} -4.00567 q^{47} -6.69042 q^{49} -2.00765 q^{51} -3.40057 q^{53} -10.6213 q^{55} -2.76410 q^{57} +10.2440 q^{59} +3.87008 q^{61} +0.556399 q^{63} +0.152512 q^{65} -0.00393019 q^{67} +1.00000 q^{69} -3.59047 q^{71} +7.82504 q^{73} +2.69903 q^{75} +2.12984 q^{77} +3.03633 q^{79} +1.00000 q^{81} +12.5062 q^{83} +5.57065 q^{85} +1.00000 q^{87} -15.2154 q^{89} -0.0305825 q^{91} -3.02923 q^{93} +7.66958 q^{95} -9.71227 q^{97} +3.82791 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{3} - 5 q^{5} - 4 q^{7} + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^3 - 5 * q^5 - 4 * q^7 + 8 * q^9	$ 8 q + 8 q^{3} - 5 q^{5} - 4 q^{7} + 8 q^{9} - 5 q^{11} - 4 q^{13} - 5 q^{15} - 3 q^{17} - 5 q^{19} - 4 q^{21} + 8 q^{23} - 5 q^{25} + 8 q^{27} + 8 q^{29} - 2 q^{31} - 5 q^{33} - 15 q^{35} - 10 q^{37} - 4 q^{39} - 11 q^{41} - 7 q^{43} - 5 q^{45} - 14 q^{47} - 18 q^{49} - 3 q^{51} - 15 q^{53} - 17 q^{55} - 5 q^{57} + 4 q^{59} + q^{61} - 4 q^{63} - 5 q^{67} + 8 q^{69} - q^{71} - 21 q^{73} - 5 q^{75} - 8 q^{79} + 8 q^{81} + 3 q^{83} + 8 q^{87} - 20 q^{89} - 7 q^{91} - 2 q^{93} - 3 q^{95} - 7 q^{97} - 5 q^{99}+O(q^{100}) $ 8 * q + 8 * q^3 - 5 * q^5 - 4 * q^7 + 8 * q^9 - 5 * q^11 - 4 * q^13 - 5 * q^15 - 3 * q^17 - 5 * q^19 - 4 * q^21 + 8 * q^23 - 5 * q^25 + 8 * q^27 + 8 * q^29 - 2 * q^31 - 5 * q^33 - 15 * q^35 - 10 * q^37 - 4 * q^39 - 11 * q^41 - 7 * q^43 - 5 * q^45 - 14 * q^47 - 18 * q^49 - 3 * q^51 - 15 * q^53 - 17 * q^55 - 5 * q^57 + 4 * q^59 + q^61 - 4 * q^63 - 5 * q^67 + 8 * q^69 - q^71 - 21 * q^73 - 5 * q^75 - 8 * q^79 + 8 * q^81 + 3 * q^83 + 8 * q^87 - 20 * q^89 - 7 * q^91 - 2 * q^93 - 3 * q^95 - 7 * q^97 - 5 * q^99

Display 100 coefficients 10 coefficients

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−2.77471	−1.24089	−0.620445	−	0.784250i	$-0.713048\pi$
$5$	−2.77471	−1.24089	−0.620445	+	0.784250i	$0.713048\pi$
$6$	0	0
$7$	0.556399	0.210299	0.105149	−	0.994456i	$-0.466468\pi$
$7$	0.556399	0.210299	0.105149	+	0.994456i	$0.466468\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.82791	1.15416	0.577079	−	0.816689i	$-0.304193\pi$
$11$	3.82791	1.15416	0.577079	+	0.816689i	$0.304193\pi$
$12$	0	0
$13$	−0.0549651	−0.0152446	−0.00762228	−	0.999971i	$-0.502426\pi$
$13$	−0.0549651	−0.0152446	−0.00762228	+	0.999971i	$0.502426\pi$
$14$	0	0
$15$	−2.77471	−0.716428
$16$	0	0
$17$	−2.00765	−0.486926	−0.243463	−	0.969910i	$-0.578284\pi$
$17$	−2.00765	−0.486926	−0.243463	+	0.969910i	$0.578284\pi$
$18$	0	0
$19$	−2.76410	−0.634128	−0.317064	−	0.948404i	$-0.602697\pi$
$19$	−2.76410	−0.634128	−0.317064	+	0.948404i	$0.602697\pi$
$20$	0	0
$21$	0.556399	0.121416
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	2.69903	0.539806
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−3.02923	−0.544066	−0.272033	−	0.962288i	$-0.587696\pi$
$31$	−3.02923	−0.544066	−0.272033	+	0.962288i	$0.587696\pi$
$32$	0	0
$33$	3.82791	0.666353
$34$	0	0
$35$	−1.54385	−0.260958
$36$	0	0
$37$	1.22025	0.200608	0.100304	−	0.994957i	$-0.468018\pi$
$37$	1.22025	0.200608	0.100304	+	0.994957i	$0.468018\pi$
$38$	0	0
$39$	−0.0549651	−0.00880145
$40$	0	0
$41$	−6.65903	−1.03997	−0.519983	−	0.854177i	$-0.674062\pi$
$41$	−6.65903	−1.03997	−0.519983	+	0.854177i	$0.674062\pi$
$42$	0	0
$43$	−4.40578	−0.671875	−0.335938	−	0.941884i	$-0.609053\pi$
$43$	−4.40578	−0.671875	−0.335938	+	0.941884i	$0.609053\pi$
$44$	0	0
$45$	−2.77471	−0.413630
$46$	0	0
$47$	−4.00567	−0.584287	−0.292143	−	0.956375i	$-0.594368\pi$
$47$	−4.00567	−0.584287	−0.292143	+	0.956375i	$0.594368\pi$
$48$	0	0
$49$	−6.69042	−0.955774
$50$	0	0
$51$	−2.00765	−0.281127
$52$	0	0
$53$	−3.40057	−0.467105	−0.233552	−	0.972344i	$-0.575035\pi$
$53$	−3.40057	−0.467105	−0.233552	+	0.972344i	$0.575035\pi$
$54$	0	0
$55$	−10.6213	−1.43218
$56$	0	0
$57$	−2.76410	−0.366114
$58$	0	0
$59$	10.2440	1.33366	0.666830	−	0.745210i	$-0.267651\pi$
$59$	10.2440	1.33366	0.666830	+	0.745210i	$0.267651\pi$
$60$	0	0
$61$	3.87008	0.495513	0.247757	−	0.968822i	$-0.420307\pi$
$61$	3.87008	0.495513	0.247757	+	0.968822i	$0.420307\pi$
$62$	0	0
$63$	0.556399	0.0700997
$64$	0	0
$65$	0.152512	0.0189168
$66$	0	0
$67$	−0.00393019	−0.000480149 0	−0.000240075	−	1.00000i	$-0.500076\pi$
$67$	−0.00393019	−0.000480149 0	−0.000240075		1.00000i	$0.500076\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−3.59047	−0.426111	−0.213055	−	0.977040i	$-0.568341\pi$
$71$	−3.59047	−0.426111	−0.213055	+	0.977040i	$0.568341\pi$
$72$	0	0
$73$	7.82504	0.915852	0.457926	−	0.888990i	$-0.348592\pi$
$73$	7.82504	0.915852	0.457926	+	0.888990i	$0.348592\pi$
$74$	0	0
$75$	2.69903	0.311657
$76$	0	0
$77$	2.12984	0.242718
$78$	0	0
$79$	3.03633	0.341614	0.170807	−	0.985305i	$-0.445363\pi$
$79$	3.03633	0.341614	0.170807	+	0.985305i	$0.445363\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	12.5062	1.37273	0.686367	−	0.727255i	$-0.259204\pi$
$83$	12.5062	1.37273	0.686367	+	0.727255i	$0.259204\pi$
$84$	0	0
$85$	5.57065	0.604222
$86$	0	0
$87$	1.00000	0.107211
$88$	0	0
$89$	−15.2154	−1.61283	−0.806417	−	0.591347i	$-0.798596\pi$
$89$	−15.2154	−1.61283	−0.806417	+	0.591347i	$0.798596\pi$
$90$	0	0
$91$	−0.0305825	−0.00320592
$92$	0	0
$93$	−3.02923	−0.314117
$94$	0	0
$95$	7.66958	0.786883
$96$	0	0
$97$	−9.71227	−0.986132	−0.493066	−	0.869992i	$-0.664124\pi$
$97$	−9.71227	−0.986132	−0.493066	+	0.869992i	$0.664124\pi$
$98$	0	0
$99$	3.82791	0.384719
$100$	0	0
$101$	15.1476	1.50725	0.753623	−	0.657307i	$-0.228305\pi$
$101$	15.1476	1.50725	0.753623	+	0.657307i	$0.228305\pi$
$102$	0	0
$103$	1.83698	0.181003	0.0905013	−	0.995896i	$-0.471153\pi$
$103$	1.83698	0.181003	0.0905013	+	0.995896i	$0.471153\pi$
$104$	0	0
$105$	−1.54385	−0.150664
$106$	0	0
$107$	−17.5327	−1.69495	−0.847476	−	0.530834i	$-0.821879\pi$
$107$	−17.5327	−1.69495	−0.847476	+	0.530834i	$0.821879\pi$
$108$	0	0
$109$	1.30868	0.125349	0.0626743	−	0.998034i	$-0.480037\pi$
$109$	1.30868	0.125349	0.0626743	+	0.998034i	$0.480037\pi$
$110$	0	0
$111$	1.22025	0.115821
$112$	0	0
$113$	5.19652	0.488847	0.244424	−	0.969669i	$-0.421401\pi$
$113$	5.19652	0.488847	0.244424	+	0.969669i	$0.421401\pi$
$114$	0	0
$115$	−2.77471	−0.258743
$116$	0	0
$117$	−0.0549651	−0.00508152
$118$	0	0
$119$	−1.11705	−0.102400
$120$	0	0
$121$	3.65286	0.332079
$122$	0	0
$123$	−6.65903	−0.600424
$124$	0	0
$125$	6.38453	0.571050
$126$	0	0
$127$	−12.3688	−1.09756	−0.548778	−	0.835968i	$-0.684907\pi$
$127$	−12.3688	−1.09756	−0.548778	+	0.835968i	$0.684907\pi$
$128$	0	0
$129$	−4.40578	−0.387907
$130$	0	0
$131$	−5.84132	−0.510358	−0.255179	−	0.966894i	$-0.582134\pi$
$131$	−5.84132	−0.510358	−0.255179	+	0.966894i	$0.582134\pi$
$132$	0	0
$133$	−1.53794	−0.133356
$134$	0	0
$135$	−2.77471	−0.238809
$136$	0	0
$137$	−2.29571	−0.196136	−0.0980679	−	0.995180i	$-0.531266\pi$
$137$	−2.29571	−0.196136	−0.0980679	+	0.995180i	$0.531266\pi$
$138$	0	0
$139$	−7.99948	−0.678507	−0.339254	−	0.940695i	$-0.610175\pi$
$139$	−7.99948	−0.678507	−0.339254	+	0.940695i	$0.610175\pi$
$140$	0	0
$141$	−4.00567	−0.337338
$142$	0	0
$143$	−0.210401	−0.0175946
$144$	0	0
$145$	−2.77471	−0.230427
$146$	0	0
$147$	−6.69042	−0.551817
$148$	0	0
$149$	−7.73558	−0.633723	−0.316862	−	0.948472i	$-0.602629\pi$
$149$	−7.73558	−0.633723	−0.316862	+	0.948472i	$0.602629\pi$
$150$	0	0
$151$	−14.9583	−1.21729	−0.608644	−	0.793444i	$-0.708286\pi$
$151$	−14.9583	−1.21729	−0.608644	+	0.793444i	$0.708286\pi$
$152$	0	0
$153$	−2.00765	−0.162309
$154$	0	0
$155$	8.40525	0.675126
$156$	0	0
$157$	10.3517	0.826154	0.413077	−	0.910696i	$-0.364454\pi$
$157$	10.3517	0.826154	0.413077	+	0.910696i	$0.364454\pi$
$158$	0	0
$159$	−3.40057	−0.269683
$160$	0	0
$161$	0.556399	0.0438504
$162$	0	0
$163$	−12.8884	−1.00949	−0.504747	−	0.863267i	$-0.668414\pi$
$163$	−12.8884	−1.00949	−0.504747	+	0.863267i	$0.668414\pi$
$164$	0	0
$165$	−10.6213	−0.826870
$166$	0	0
$167$	9.20220	0.712088	0.356044	−	0.934469i	$-0.384125\pi$
$167$	9.20220	0.712088	0.356044	+	0.934469i	$0.384125\pi$
$168$	0	0
$169$	−12.9970	−0.999768
$170$	0	0
$171$	−2.76410	−0.211376
$172$	0	0
$173$	−7.71770	−0.586766	−0.293383	−	0.955995i	$-0.594781\pi$
$173$	−7.71770	−0.586766	−0.293383	+	0.955995i	$0.594781\pi$
$174$	0	0
$175$	1.50174	0.113521
$176$	0	0
$177$	10.2440	0.769989
$178$	0	0
$179$	3.03175	0.226604	0.113302	−	0.993561i	$-0.463857\pi$
$179$	3.03175	0.226604	0.113302	+	0.993561i	$0.463857\pi$
$180$	0	0
$181$	8.48454	0.630651	0.315326	−	0.948984i	$-0.397886\pi$
$181$	8.48454	0.630651	0.315326	+	0.948984i	$0.397886\pi$
$182$	0	0
$183$	3.87008	0.286085
$184$	0	0
$185$	−3.38585	−0.248933
$186$	0	0
$187$	−7.68509	−0.561990
$188$	0	0
$189$	0.556399	0.0404721
$190$	0	0
$191$	12.3424	0.893067	0.446533	−	0.894767i	$-0.352658\pi$
$191$	12.3424	0.893067	0.446533	+	0.894767i	$0.352658\pi$
$192$	0	0
$193$	−14.4947	−1.04335	−0.521677	−	0.853143i	$-0.674693\pi$
$193$	−14.4947	−1.04335	−0.521677	+	0.853143i	$0.674693\pi$
$194$	0	0
$195$	0.152512	0.0109216
$196$	0	0
$197$	6.96683	0.496366	0.248183	−	0.968713i	$-0.420167\pi$
$197$	6.96683	0.496366	0.248183	+	0.968713i	$0.420167\pi$
$198$	0	0
$199$	−12.4386	−0.881749	−0.440875	−	0.897569i	$-0.645332\pi$
$199$	−12.4386	−0.881749	−0.440875	+	0.897569i	$0.645332\pi$
$200$	0	0
$201$	−0.00393019	−0.000277214 0
$202$	0	0
$203$	0.556399	0.0390515
$204$	0	0
$205$	18.4769	1.29048
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	−10.5807	−0.731883
$210$	0	0
$211$	−8.83834	−0.608456	−0.304228	−	0.952599i	$-0.598399\pi$
$211$	−8.83834	−0.608456	−0.304228	+	0.952599i	$0.598399\pi$
$212$	0	0
$213$	−3.59047	−0.246015
$214$	0	0
$215$	12.2248	0.833723
$216$	0	0
$217$	−1.68546	−0.114417
$218$	0	0
$219$	7.82504	0.528768
$220$	0	0
$221$	0.110351	0.00742298
$222$	0	0
$223$	3.52468	0.236030	0.118015	−	0.993012i	$-0.462347\pi$
$223$	3.52468	0.236030	0.118015	+	0.993012i	$0.462347\pi$
$224$	0	0
$225$	2.69903	0.179935
$226$	0	0
$227$	−19.7084	−1.30809	−0.654045	−	0.756456i	$-0.726929\pi$
$227$	−19.7084	−1.30809	−0.654045	+	0.756456i	$0.726929\pi$
$228$	0	0
$229$	−8.86659	−0.585921	−0.292961	−	0.956125i	$-0.594640\pi$
$229$	−8.86659	−0.585921	−0.292961	+	0.956125i	$0.594640\pi$
$230$	0	0
$231$	2.12984	0.140133
$232$	0	0
$233$	−8.26987	−0.541777	−0.270889	−	0.962611i	$-0.587317\pi$
$233$	−8.26987	−0.541777	−0.270889	+	0.962611i	$0.587317\pi$
$234$	0	0
$235$	11.1146	0.725035
$236$	0	0
$237$	3.03633	0.197231
$238$	0	0
$239$	−25.2837	−1.63547	−0.817734	−	0.575596i	$-0.804770\pi$
$239$	−25.2837	−1.63547	−0.817734	+	0.575596i	$0.804770\pi$
$240$	0	0
$241$	23.1164	1.48906	0.744530	−	0.667589i	$-0.232674\pi$
$241$	23.1164	1.48906	0.744530	+	0.667589i	$0.232674\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	18.5640	1.18601
$246$	0	0
$247$	0.151929	0.00966701
$248$	0	0
$249$	12.5062	0.792548
$250$	0	0
$251$	−27.7022	−1.74855	−0.874274	−	0.485432i	$-0.838662\pi$
$251$	−27.7022	−1.74855	−0.874274	+	0.485432i	$0.838662\pi$
$252$	0	0
$253$	3.82791	0.240658
$254$	0	0
$255$	5.57065	0.348848
$256$	0	0
$257$	21.3254	1.33024	0.665121	−	0.746736i	$-0.268380\pi$
$257$	21.3254	1.33024	0.665121	+	0.746736i	$0.268380\pi$
$258$	0	0
$259$	0.678947	0.0421877
$260$	0	0
$261$	1.00000	0.0618984
$262$	0	0
$263$	14.6435	0.902956	0.451478	−	0.892282i	$-0.350897\pi$
$263$	14.6435	0.902956	0.451478	+	0.892282i	$0.350897\pi$
$264$	0	0
$265$	9.43561	0.579625
$266$	0	0
$267$	−15.2154	−0.931170
$268$	0	0
$269$	7.19583	0.438737	0.219369	−	0.975642i	$-0.429600\pi$
$269$	7.19583	0.438737	0.219369	+	0.975642i	$0.429600\pi$
$270$	0	0
$271$	−13.1008	−0.795814	−0.397907	−	0.917426i	$-0.630263\pi$
$271$	−13.1008	−0.795814	−0.397907	+	0.917426i	$0.630263\pi$
$272$	0	0
$273$	−0.0305825	−0.00185094
$274$	0	0
$275$	10.3316	0.623021
$276$	0	0
$277$	9.17284	0.551143	0.275571	−	0.961281i	$-0.411133\pi$
$277$	9.17284	0.551143	0.275571	+	0.961281i	$0.411133\pi$
$278$	0	0
$279$	−3.02923	−0.181355
$280$	0	0
$281$	23.5095	1.40246	0.701230	−	0.712936i	$-0.252635\pi$
$281$	23.5095	1.40246	0.701230	+	0.712936i	$0.252635\pi$
$282$	0	0
$283$	−33.2249	−1.97501	−0.987507	−	0.157574i	$-0.949633\pi$
$283$	−33.2249	−1.97501	−0.987507	+	0.157574i	$0.949633\pi$
$284$	0	0
$285$	7.66958	0.454307
$286$	0	0
$287$	−3.70507	−0.218704
$288$	0	0
$289$	−12.9693	−0.762903
$290$	0	0
$291$	−9.71227	−0.569343
$292$	0	0
$293$	10.7203	0.626286	0.313143	−	0.949706i	$-0.398618\pi$
$293$	10.7203	0.626286	0.313143	+	0.949706i	$0.398618\pi$
$294$	0	0
$295$	−28.4243	−1.65492
$296$	0	0
$297$	3.82791	0.222118
$298$	0	0
$299$	−0.0549651	−0.00317871
$300$	0	0
$301$	−2.45137	−0.141295
$302$	0	0
$303$	15.1476	0.870209
$304$	0	0
$305$	−10.7384	−0.614877
$306$	0	0
$307$	−19.2238	−1.09716	−0.548580	−	0.836098i	$-0.684831\pi$
$307$	−19.2238	−1.09716	−0.548580	+	0.836098i	$0.684831\pi$
$308$	0	0
$309$	1.83698	0.104502
$310$	0	0
$311$	11.6337	0.659687	0.329844	−	0.944036i	$-0.393004\pi$
$311$	11.6337	0.659687	0.329844	+	0.944036i	$0.393004\pi$
$312$	0	0
$313$	−18.2765	−1.03305	−0.516523	−	0.856273i	$-0.672774\pi$
$313$	−18.2765	−1.03305	−0.516523	+	0.856273i	$0.672774\pi$
$314$	0	0
$315$	−1.54385	−0.0869859
$316$	0	0
$317$	−32.0811	−1.80185	−0.900926	−	0.433972i	$-0.857112\pi$
$317$	−32.0811	−1.80185	−0.900926	+	0.433972i	$0.857112\pi$
$318$	0	0
$319$	3.82791	0.214322
$320$	0	0
$321$	−17.5327	−0.978581
$322$	0	0
$323$	5.54934	0.308774
$324$	0	0
$325$	−0.148352	−0.00822911
$326$	0	0
$327$	1.30868	0.0723701
$328$	0	0
$329$	−2.22875	−0.122875
$330$	0	0
$331$	−19.6188	−1.07835	−0.539174	−	0.842194i	$-0.681264\pi$
$331$	−19.6188	−1.07835	−0.539174	+	0.842194i	$0.681264\pi$
$332$	0	0
$333$	1.22025	0.0668695
$334$	0	0
$335$	0.0109051	0.000595812 0
$336$	0	0
$337$	−20.1982	−1.10027	−0.550133	−	0.835077i	$-0.685423\pi$
$337$	−20.1982	−1.10027	−0.550133	+	0.835077i	$0.685423\pi$
$338$	0	0
$339$	5.19652	0.282236
$340$	0	0
$341$	−11.5956	−0.627938
$342$	0	0
$343$	−7.61733	−0.411297
$344$	0	0
$345$	−2.77471	−0.149386
$346$	0	0
$347$	15.0133	0.805954	0.402977	−	0.915210i	$-0.367975\pi$
$347$	15.0133	0.805954	0.402977	+	0.915210i	$0.367975\pi$
$348$	0	0
$349$	−30.4729	−1.63118	−0.815589	−	0.578631i	$-0.803587\pi$
$349$	−30.4729	−1.63118	−0.815589	+	0.578631i	$0.803587\pi$
$350$	0	0
$351$	−0.0549651	−0.00293382
$352$	0	0
$353$	−24.9894	−1.33005	−0.665026	−	0.746820i	$-0.731580\pi$
$353$	−24.9894	−1.33005	−0.665026	+	0.746820i	$0.731580\pi$
$354$	0	0
$355$	9.96253	0.528756
$356$	0	0
$357$	−1.11705	−0.0591207
$358$	0	0
$359$	13.3685	0.705561	0.352780	−	0.935706i	$-0.385236\pi$
$359$	13.3685	0.705561	0.352780	+	0.935706i	$0.385236\pi$
$360$	0	0
$361$	−11.3598	−0.597882
$362$	0	0
$363$	3.65286	0.191726
$364$	0	0
$365$	−21.7123	−1.13647
$366$	0	0
$367$	14.5984	0.762031	0.381016	−	0.924569i	$-0.375574\pi$
$367$	14.5984	0.762031	0.381016	+	0.924569i	$0.375574\pi$
$368$	0	0
$369$	−6.65903	−0.346655
$370$	0	0
$371$	−1.89207	−0.0982316
$372$	0	0
$373$	−18.8792	−0.977526	−0.488763	−	0.872417i	$-0.662552\pi$
$373$	−18.8792	−0.977526	−0.488763	+	0.872417i	$0.662552\pi$
$374$	0	0
$375$	6.38453	0.329696
$376$	0	0
$377$	−0.0549651	−0.00283084
$378$	0	0
$379$	1.55072	0.0796552	0.0398276	−	0.999207i	$-0.487319\pi$
$379$	1.55072	0.0796552	0.0398276	+	0.999207i	$0.487319\pi$
$380$	0	0
$381$	−12.3688	−0.633674
$382$	0	0
$383$	−11.2587	−0.575293	−0.287647	−	0.957737i	$-0.592873\pi$
$383$	−11.2587	−0.575293	−0.287647	+	0.957737i	$0.592873\pi$
$384$	0	0
$385$	−5.90970	−0.301186
$386$	0	0
$387$	−4.40578	−0.223958
$388$	0	0
$389$	−19.8046	−1.00413	−0.502066	−	0.864829i	$-0.667426\pi$
$389$	−19.8046	−1.00413	−0.502066	+	0.864829i	$0.667426\pi$
$390$	0	0
$391$	−2.00765	−0.101531
$392$	0	0
$393$	−5.84132	−0.294656
$394$	0	0
$395$	−8.42495	−0.423905
$396$	0	0
$397$	−8.76623	−0.439965	−0.219982	−	0.975504i	$-0.570600\pi$
$397$	−8.76623	−0.439965	−0.219982	+	0.975504i	$0.570600\pi$
$398$	0	0
$399$	−1.53794	−0.0769934
$400$	0	0
$401$	21.1717	1.05726	0.528631	−	0.848852i	$-0.322706\pi$
$401$	21.1717	1.05726	0.528631	+	0.848852i	$0.322706\pi$
$402$	0	0
$403$	0.166502	0.00829405
$404$	0	0
$405$	−2.77471	−0.137877
$406$	0	0
$407$	4.67101	0.231534
$408$	0	0
$409$	34.6111	1.71141	0.855704	−	0.517465i	$-0.173124\pi$
$409$	34.6111	1.71141	0.855704	+	0.517465i	$0.173124\pi$
$410$	0	0
$411$	−2.29571	−0.113239
$412$	0	0
$413$	5.69977	0.280467
$414$	0	0
$415$	−34.7011	−1.70341
$416$	0	0
$417$	−7.99948	−0.391736
$418$	0	0
$419$	−1.43999	−0.0703482	−0.0351741	−	0.999381i	$-0.511199\pi$
$419$	−1.43999	−0.0703482	−0.0351741	+	0.999381i	$0.511199\pi$
$420$	0	0
$421$	−21.8112	−1.06301	−0.531507	−	0.847054i	$-0.678374\pi$
$421$	−21.8112	−1.06301	−0.531507	+	0.847054i	$0.678374\pi$
$422$	0	0
$423$	−4.00567	−0.194762
$424$	0	0
$425$	−5.41871	−0.262846
$426$	0	0
$427$	2.15331	0.104206
$428$	0	0
$429$	−0.210401	−0.0101583
$430$	0	0
$431$	−18.2824	−0.880633	−0.440316	−	0.897843i	$-0.645134\pi$
$431$	−18.2824	−0.880633	−0.440316	+	0.897843i	$0.645134\pi$
$432$	0	0
$433$	2.58010	0.123992	0.0619959	−	0.998076i	$-0.480253\pi$
$433$	2.58010	0.123992	0.0619959	+	0.998076i	$0.480253\pi$
$434$	0	0
$435$	−2.77471	−0.133037
$436$	0	0
$437$	−2.76410	−0.132225
$438$	0	0
$439$	−30.9993	−1.47951	−0.739757	−	0.672874i	$-0.765059\pi$
$439$	−30.9993	−1.47951	−0.739757	+	0.672874i	$0.765059\pi$
$440$	0	0
$441$	−6.69042	−0.318591
$442$	0	0
$443$	−14.6452	−0.695813	−0.347906	−	0.937529i	$-0.613107\pi$
$443$	−14.6452	−0.695813	−0.347906	+	0.937529i	$0.613107\pi$
$444$	0	0
$445$	42.2185	2.00135
$446$	0	0
$447$	−7.73558	−0.365880
$448$	0	0
$449$	32.3776	1.52800	0.763998	−	0.645219i	$-0.223234\pi$
$449$	32.3776	1.52800	0.763998	+	0.645219i	$0.223234\pi$
$450$	0	0
$451$	−25.4901	−1.20028
$452$	0	0
$453$	−14.9583	−0.702801
$454$	0	0
$455$	0.0848576	0.00397819
$456$	0	0
$457$	−14.7830	−0.691519	−0.345759	−	0.938323i	$-0.612379\pi$
$457$	−14.7830	−0.691519	−0.345759	+	0.938323i	$0.612379\pi$
$458$	0	0
$459$	−2.00765	−0.0937090
$460$	0	0
$461$	17.8061	0.829312	0.414656	−	0.909978i	$-0.363902\pi$
$461$	17.8061	0.829312	0.414656	+	0.909978i	$0.363902\pi$
$462$	0	0
$463$	9.26773	0.430708	0.215354	−	0.976536i	$-0.430909\pi$
$463$	9.26773	0.430708	0.215354	+	0.976536i	$0.430909\pi$
$464$	0	0
$465$	8.40525	0.389784
$466$	0	0
$467$	0.449880	0.0208180	0.0104090	−	0.999946i	$-0.496687\pi$
$467$	0.449880	0.0208180	0.0104090	+	0.999946i	$0.496687\pi$
$468$	0	0
$469$	−0.00218675	−0.000100975 0
$470$	0	0
$471$	10.3517	0.476980
$472$	0	0
$473$	−16.8649	−0.775450
$474$	0	0
$475$	−7.46039	−0.342306
$476$	0	0
$477$	−3.40057	−0.155702
$478$	0	0
$479$	32.3675	1.47891	0.739455	−	0.673206i	$-0.235083\pi$
$479$	32.3675	1.47891	0.739455	+	0.673206i	$0.235083\pi$
$480$	0	0
$481$	−0.0670713	−0.00305819
$482$	0	0
$483$	0.556399	0.0253170
$484$	0	0
$485$	26.9488	1.22368
$486$	0	0
$487$	−13.8049	−0.625559	−0.312779	−	0.949826i	$-0.601260\pi$
$487$	−13.8049	−0.625559	−0.312779	+	0.949826i	$0.601260\pi$
$488$	0	0
$489$	−12.8884	−0.582832
$490$	0	0
$491$	−18.2392	−0.823124	−0.411562	−	0.911382i	$-0.635017\pi$
$491$	−18.2392	−0.823124	−0.411562	+	0.911382i	$0.635017\pi$
$492$	0	0
$493$	−2.00765	−0.0904200
$494$	0	0
$495$	−10.6213	−0.477394
$496$	0	0
$497$	−1.99774	−0.0896107
$498$	0	0
$499$	−30.0883	−1.34694	−0.673468	−	0.739216i	$-0.735196\pi$
$499$	−30.0883	−1.34694	−0.673468	+	0.739216i	$0.735196\pi$
$500$	0	0
$501$	9.20220	0.411124
$502$	0	0
$503$	33.6546	1.50058	0.750292	−	0.661106i	$-0.229913\pi$
$503$	33.6546	1.50058	0.750292	+	0.661106i	$0.229913\pi$
$504$	0	0
$505$	−42.0303	−1.87032
$506$	0	0
$507$	−12.9970	−0.577216
$508$	0	0
$509$	43.6065	1.93283	0.966413	−	0.256994i	$-0.0827320\pi$
$509$	43.6065	1.93283	0.966413	+	0.256994i	$0.0827320\pi$
$510$	0	0
$511$	4.35385	0.192603
$512$	0	0
$513$	−2.76410	−0.122038
$514$	0	0
$515$	−5.09708	−0.224604
$516$	0	0
$517$	−15.3333	−0.674359
$518$	0	0
$519$	−7.71770	−0.338770
$520$	0	0
$521$	−13.9841	−0.612656	−0.306328	−	0.951926i	$-0.599100\pi$
$521$	−13.9841	−0.612656	−0.306328	+	0.951926i	$0.599100\pi$
$522$	0	0
$523$	9.57656	0.418754	0.209377	−	0.977835i	$-0.432856\pi$
$523$	9.57656	0.418754	0.209377	+	0.977835i	$0.432856\pi$
$524$	0	0
$525$	1.50174	0.0655412
$526$	0	0
$527$	6.08163	0.264920
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	10.2440	0.444553
$532$	0	0
$533$	0.366014	0.0158538
$534$	0	0
$535$	48.6482	2.10325
$536$	0	0
$537$	3.03175	0.130830
$538$	0	0
$539$	−25.6103	−1.10311
$540$	0	0
$541$	−5.99859	−0.257899	−0.128950	−	0.991651i	$-0.541161\pi$
$541$	−5.99859	−0.257899	−0.128950	+	0.991651i	$0.541161\pi$
$542$	0	0
$543$	8.48454	0.364107
$544$	0	0
$545$	−3.63121	−0.155544
$546$	0	0
$547$	18.8422	0.805633	0.402816	−	0.915281i	$-0.368031\pi$
$547$	18.8422	0.805633	0.402816	+	0.915281i	$0.368031\pi$
$548$	0	0
$549$	3.87008	0.165171
$550$	0	0
$551$	−2.76410	−0.117755
$552$	0	0
$553$	1.68941	0.0718411
$554$	0	0
$555$	−3.38585	−0.143721
$556$	0	0
$557$	−42.0209	−1.78048	−0.890241	−	0.455490i	$-0.849464\pi$
$557$	−42.0209	−1.78048	−0.890241	+	0.455490i	$0.849464\pi$
$558$	0	0
$559$	0.242164	0.0102424
$560$	0	0
$561$	−7.68509	−0.324465
$562$	0	0
$563$	13.5776	0.572226	0.286113	−	0.958196i	$-0.407637\pi$
$563$	13.5776	0.572226	0.286113	+	0.958196i	$0.407637\pi$
$564$	0	0
$565$	−14.4188	−0.606605
$566$	0	0
$567$	0.556399	0.0233666
$568$	0	0
$569$	−20.6650	−0.866321	−0.433161	−	0.901317i	$-0.642602\pi$
$569$	−20.6650	−0.866321	−0.433161	+	0.901317i	$0.642602\pi$
$570$	0	0
$571$	1.29913	0.0543671	0.0271835	−	0.999630i	$-0.491346\pi$
$571$	1.29913	0.0543671	0.0271835	+	0.999630i	$0.491346\pi$
$572$	0	0
$573$	12.3424	0.515612
$574$	0	0
$575$	2.69903	0.112557
$576$	0	0
$577$	27.5948	1.14879	0.574393	−	0.818580i	$-0.305238\pi$
$577$	27.5948	1.14879	0.574393	+	0.818580i	$0.305238\pi$
$578$	0	0
$579$	−14.4947	−0.602381
$580$	0	0
$581$	6.95844	0.288685
$582$	0	0
$583$	−13.0171	−0.539112
$584$	0	0
$585$	0.152512	0.00630561
$586$	0	0
$587$	−4.30710	−0.177773	−0.0888865	−	0.996042i	$-0.528331\pi$
$587$	−4.30710	−0.177773	−0.0888865	+	0.996042i	$0.528331\pi$
$588$	0	0
$589$	8.37310	0.345007
$590$	0	0
$591$	6.96683	0.286577
$592$	0	0
$593$	22.7594	0.934616	0.467308	−	0.884094i	$-0.345224\pi$
$593$	22.7594	0.934616	0.467308	+	0.884094i	$0.345224\pi$
$594$	0	0
$595$	3.09950	0.127067
$596$	0	0
$597$	−12.4386	−0.509078
$598$	0	0
$599$	33.7658	1.37963	0.689817	−	0.723984i	$-0.257691\pi$
$599$	33.7658	1.37963	0.689817	+	0.723984i	$0.257691\pi$
$600$	0	0
$601$	−26.4457	−1.07874	−0.539372	−	0.842068i	$-0.681338\pi$
$601$	−26.4457	−1.07874	−0.539372	+	0.842068i	$0.681338\pi$
$602$	0	0
$603$	−0.00393019	−0.000160050 0
$604$	0	0
$605$	−10.1356	−0.412073
$606$	0	0
$607$	5.95175	0.241574	0.120787	−	0.992678i	$-0.461458\pi$
$607$	5.95175	0.241574	0.120787	+	0.992678i	$0.461458\pi$
$608$	0	0
$609$	0.556399	0.0225464
$610$	0	0
$611$	0.220172	0.00890720
$612$	0	0
$613$	25.2789	1.02101	0.510503	−	0.859876i	$-0.329459\pi$
$613$	25.2789	1.02101	0.510503	+	0.859876i	$0.329459\pi$
$614$	0	0
$615$	18.4769	0.745060
$616$	0	0
$617$	41.6316	1.67603	0.838013	−	0.545650i	$-0.183717\pi$
$617$	41.6316	1.67603	0.838013	+	0.545650i	$0.183717\pi$
$618$	0	0
$619$	34.9531	1.40488	0.702442	−	0.711741i	$-0.252093\pi$
$619$	34.9531	1.40488	0.702442	+	0.711741i	$0.252093\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	−8.46586	−0.339177
$624$	0	0
$625$	−31.2104	−1.24842
$626$	0	0
$627$	−10.5807	−0.422553
$628$	0	0
$629$	−2.44984	−0.0976815
$630$	0	0
$631$	−3.25055	−0.129402	−0.0647012	−	0.997905i	$-0.520609\pi$
$631$	−3.25055	−0.129402	−0.0647012	+	0.997905i	$0.520609\pi$
$632$	0	0
$633$	−8.83834	−0.351292
$634$	0	0
$635$	34.3199	1.36195
$636$	0	0
$637$	0.367739	0.0145704
$638$	0	0
$639$	−3.59047	−0.142037
$640$	0	0
$641$	−8.69172	−0.343302	−0.171651	−	0.985158i	$-0.554910\pi$
$641$	−8.69172	−0.343302	−0.171651	+	0.985158i	$0.554910\pi$
$642$	0	0
$643$	8.15011	0.321409	0.160704	−	0.987003i	$-0.448623\pi$
$643$	8.15011	0.321409	0.160704	+	0.987003i	$0.448623\pi$
$644$	0	0
$645$	12.2248	0.481350
$646$	0	0
$647$	−8.14432	−0.320186	−0.160093	−	0.987102i	$-0.551179\pi$
$647$	−8.14432	−0.320186	−0.160093	+	0.987102i	$0.551179\pi$
$648$	0	0
$649$	39.2132	1.53925
$650$	0	0
$651$	−1.68546	−0.0660584
$652$	0	0
$653$	−26.1352	−1.02275	−0.511376	−	0.859357i	$-0.670864\pi$
$653$	−26.1352	−1.02275	−0.511376	+	0.859357i	$0.670864\pi$
$654$	0	0
$655$	16.2080	0.633298
$656$	0	0
$657$	7.82504	0.305284
$658$	0	0
$659$	−25.3965	−0.989307	−0.494653	−	0.869090i	$-0.664705\pi$
$659$	−25.3965	−0.989307	−0.494653	+	0.869090i	$0.664705\pi$
$660$	0	0
$661$	−8.48368	−0.329977	−0.164988	−	0.986296i	$-0.552759\pi$
$661$	−8.48368	−0.329977	−0.164988	+	0.986296i	$0.552759\pi$
$662$	0	0
$663$	0.110351	0.00428566
$664$	0	0
$665$	4.26735	0.165481
$666$	0	0
$667$	1.00000	0.0387202
$668$	0	0
$669$	3.52468	0.136272
$670$	0	0
$671$	14.8143	0.571900
$672$	0	0
$673$	33.8562	1.30506	0.652530	−	0.757763i	$-0.273708\pi$
$673$	33.8562	1.30506	0.652530	+	0.757763i	$0.273708\pi$
$674$	0	0
$675$	2.69903	0.103886
$676$	0	0
$677$	−37.0178	−1.42271	−0.711356	−	0.702832i	$-0.751918\pi$
$677$	−37.0178	−1.42271	−0.711356	+	0.702832i	$0.751918\pi$
$678$	0	0
$679$	−5.40390	−0.207383
$680$	0	0
$681$	−19.7084	−0.755226
$682$	0	0
$683$	−0.824117	−0.0315340	−0.0157670	−	0.999876i	$-0.505019\pi$
$683$	−0.824117	−0.0315340	−0.0157670	+	0.999876i	$0.505019\pi$
$684$	0	0
$685$	6.36994	0.243383
$686$	0	0
$687$	−8.86659	−0.338282
$688$	0	0
$689$	0.186913	0.00712081
$690$	0	0
$691$	6.23305	0.237116	0.118558	−	0.992947i	$-0.462173\pi$
$691$	6.23305	0.237116	0.118558	+	0.992947i	$0.462173\pi$
$692$	0	0
$693$	2.12984	0.0809060
$694$	0	0
$695$	22.1963	0.841953
$696$	0	0
$697$	13.3690	0.506387
$698$	0	0
$699$	−8.26987	−0.312795
$700$	0	0
$701$	7.41067	0.279897	0.139949	−	0.990159i	$-0.455306\pi$
$701$	7.41067	0.279897	0.139949	+	0.990159i	$0.455306\pi$
$702$	0	0
$703$	−3.37290	−0.127211
$704$	0	0
$705$	11.1146	0.418599
$706$	0	0
$707$	8.42812	0.316972
$708$	0	0
$709$	39.5048	1.48364	0.741818	−	0.670602i	$-0.233964\pi$
$709$	39.5048	1.48364	0.741818	+	0.670602i	$0.233964\pi$
$710$	0	0
$711$	3.03633	0.113871
$712$	0	0
$713$	−3.02923	−0.113446
$714$	0	0
$715$	0.583803	0.0218330
$716$	0	0
$717$	−25.2837	−0.944238
$718$	0	0
$719$	19.1065	0.712552	0.356276	−	0.934381i	$-0.384046\pi$
$719$	19.1065	0.712552	0.356276	+	0.934381i	$0.384046\pi$
$720$	0	0
$721$	1.02209	0.0380647
$722$	0	0
$723$	23.1164	0.859709
$724$	0	0
$725$	2.69903	0.100239
$726$	0	0
$727$	9.79292	0.363199	0.181600	−	0.983373i	$-0.441872\pi$
$727$	9.79292	0.363199	0.181600	+	0.983373i	$0.441872\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	8.84526	0.327154
$732$	0	0
$733$	−31.5855	−1.16664	−0.583319	−	0.812243i	$-0.698246\pi$
$733$	−31.5855	−1.16664	−0.583319	+	0.812243i	$0.698246\pi$
$734$	0	0
$735$	18.5640	0.684743
$736$	0	0
$737$	−0.0150444	−0.000554168 0
$738$	0	0
$739$	−12.3989	−0.456100	−0.228050	−	0.973649i	$-0.573235\pi$
$739$	−12.3989	−0.456100	−0.228050	+	0.973649i	$0.573235\pi$
$740$	0	0
$741$	0.151929	0.00558125
$742$	0	0
$743$	15.0421	0.551840	0.275920	−	0.961181i	$-0.411018\pi$
$743$	15.0421	0.551840	0.275920	+	0.961181i	$0.411018\pi$
$744$	0	0
$745$	21.4640	0.786380
$746$	0	0
$747$	12.5062	0.457578
$748$	0	0
$749$	−9.75518	−0.356447
$750$	0	0
$751$	−41.9261	−1.52991	−0.764953	−	0.644086i	$-0.777238\pi$
$751$	−41.9261	−1.52991	−0.764953	+	0.644086i	$0.777238\pi$
$752$	0	0
$753$	−27.7022	−1.00952
$754$	0	0
$755$	41.5049	1.51052
$756$	0	0
$757$	22.7886	0.828266	0.414133	−	0.910216i	$-0.364085\pi$
$757$	22.7886	0.828266	0.414133	+	0.910216i	$0.364085\pi$
$758$	0	0
$759$	3.82791	0.138944
$760$	0	0
$761$	30.5620	1.10787	0.553935	−	0.832560i	$-0.313125\pi$
$761$	30.5620	1.10787	0.553935	+	0.832560i	$0.313125\pi$
$762$	0	0
$763$	0.728147	0.0263607
$764$	0	0
$765$	5.57065	0.201407
$766$	0	0
$767$	−0.563064	−0.0203311
$768$	0	0
$769$	−35.6392	−1.28518	−0.642591	−	0.766209i	$-0.722141\pi$
$769$	−35.6392	−1.28518	−0.642591	+	0.766209i	$0.722141\pi$
$770$	0	0
$771$	21.3254	0.768016
$772$	0	0
$773$	34.7781	1.25088	0.625441	−	0.780271i	$-0.284919\pi$
$773$	34.7781	1.25088	0.625441	+	0.780271i	$0.284919\pi$
$774$	0	0
$775$	−8.17599	−0.293690
$776$	0	0
$777$	0.678947	0.0243571
$778$	0	0
$779$	18.4062	0.659471
$780$	0	0
$781$	−13.7440	−0.491799
$782$	0	0
$783$	1.00000	0.0357371
$784$	0	0
$785$	−28.7229	−1.02517
$786$	0	0
$787$	7.64623	0.272559	0.136279	−	0.990670i	$-0.456486\pi$
$787$	7.64623	0.272559	0.136279	+	0.990670i	$0.456486\pi$
$788$	0	0
$789$	14.6435	0.521322
$790$	0	0
$791$	2.89134	0.102804
$792$	0	0
$793$	−0.212719	−0.00755389
$794$	0	0
$795$	9.43561	0.334647
$796$	0	0
$797$	−4.58525	−0.162418	−0.0812089	−	0.996697i	$-0.525878\pi$
$797$	−4.58525	−0.162418	−0.0812089	+	0.996697i	$0.525878\pi$
$798$	0	0
$799$	8.04198	0.284505
$800$	0	0
$801$	−15.2154	−0.537611
$802$	0	0
$803$	29.9535	1.05704
$804$	0	0
$805$	−1.54385	−0.0544135
$806$	0	0
$807$	7.19583	0.253305
$808$	0	0
$809$	−50.7530	−1.78438	−0.892190	−	0.451660i	$-0.850832\pi$
$809$	−50.7530	−1.78438	−0.892190	+	0.451660i	$0.850832\pi$
$810$	0	0
$811$	33.6815	1.18272	0.591360	−	0.806408i	$-0.298591\pi$
$811$	33.6815	1.18272	0.591360	+	0.806408i	$0.298591\pi$
$812$	0	0
$813$	−13.1008	−0.459463
$814$	0	0
$815$	35.7615	1.25267
$816$	0	0
$817$	12.1780	0.426055
$818$	0	0
$819$	−0.0305825	−0.00106864
$820$	0	0
$821$	−23.8622	−0.832797	−0.416399	−	0.909182i	$-0.636708\pi$
$821$	−23.8622	−0.832797	−0.416399	+	0.909182i	$0.636708\pi$
$822$	0	0
$823$	32.1028	1.11903	0.559516	−	0.828819i	$-0.310987\pi$
$823$	32.1028	1.11903	0.559516	+	0.828819i	$0.310987\pi$
$824$	0	0
$825$	10.3316	0.359701
$826$	0	0
$827$	−16.9028	−0.587768	−0.293884	−	0.955841i	$-0.594948\pi$
$827$	−16.9028	−0.587768	−0.293884	+	0.955841i	$0.594948\pi$
$828$	0	0
$829$	5.70185	0.198034	0.0990168	−	0.995086i	$-0.468430\pi$
$829$	5.70185	0.198034	0.0990168	+	0.995086i	$0.468430\pi$
$830$	0	0
$831$	9.17284	0.318202
$832$	0	0
$833$	13.4320	0.465392
$834$	0	0
$835$	−25.5335	−0.883622
$836$	0	0
$837$	−3.02923	−0.104706
$838$	0	0
$839$	37.4186	1.29183	0.645917	−	0.763408i	$-0.276475\pi$
$839$	37.4186	1.29183	0.645917	+	0.763408i	$0.276475\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	23.5095	0.809710
$844$	0	0
$845$	36.0629	1.24060
$846$	0	0
$847$	2.03245	0.0698358
$848$	0	0
$849$	−33.2249	−1.14028
$850$	0	0
$851$	1.22025	0.0418297
$852$	0	0
$853$	53.9024	1.84558	0.922792	−	0.385298i	$-0.125901\pi$
$853$	53.9024	1.84558	0.922792	+	0.385298i	$0.125901\pi$
$854$	0	0
$855$	7.66958	0.262294
$856$	0	0
$857$	21.1594	0.722790	0.361395	−	0.932413i	$-0.382301\pi$
$857$	21.1594	0.722790	0.361395	+	0.932413i	$0.382301\pi$
$858$	0	0
$859$	−7.49079	−0.255582	−0.127791	−	0.991801i	$-0.540789\pi$
$859$	−7.49079	−0.255582	−0.127791	+	0.991801i	$0.540789\pi$
$860$	0	0
$861$	−3.70507	−0.126269
$862$	0	0
$863$	−34.0074	−1.15763	−0.578813	−	0.815460i	$-0.696484\pi$
$863$	−34.0074	−1.15763	−0.578813	+	0.815460i	$0.696484\pi$
$864$	0	0
$865$	21.4144	0.728112
$866$	0	0
$867$	−12.9693	−0.440462
$868$	0	0
$869$	11.6228	0.394276
$870$	0	0
$871$	0.000216023 0	7.31967e−6 0
$872$	0	0
$873$	−9.71227	−0.328711
$874$	0	0
$875$	3.55235	0.120091
$876$	0	0
$877$	−4.02078	−0.135772	−0.0678860	−	0.997693i	$-0.521625\pi$
$877$	−4.02078	−0.135772	−0.0678860	+	0.997693i	$0.521625\pi$
$878$	0	0
$879$	10.7203	0.361587
$880$	0	0
$881$	−46.1329	−1.55426	−0.777128	−	0.629343i	$-0.783324\pi$
$881$	−46.1329	−1.55426	−0.777128	+	0.629343i	$0.783324\pi$
$882$	0	0
$883$	−7.94209	−0.267272	−0.133636	−	0.991030i	$-0.542665\pi$
$883$	−7.94209	−0.267272	−0.133636	+	0.991030i	$0.542665\pi$
$884$	0	0
$885$	−28.4243	−0.955471
$886$	0	0
$887$	−10.1411	−0.340505	−0.170252	−	0.985401i	$-0.554458\pi$
$887$	−10.1411	−0.340505	−0.170252	+	0.985401i	$0.554458\pi$
$888$	0	0
$889$	−6.88200	−0.230815
$890$	0	0
$891$	3.82791	0.128240
$892$	0	0
$893$	11.0721	0.370513
$894$	0	0
$895$	−8.41224	−0.281190
$896$	0	0
$897$	−0.0549651	−0.00183523
$898$	0	0
$899$	−3.02923	−0.101031
$900$	0	0
$901$	6.82716	0.227446
$902$	0	0
$903$	−2.45137	−0.0815765
$904$	0	0
$905$	−23.5422	−0.782568
$906$	0	0
$907$	−11.1257	−0.369423	−0.184712	−	0.982793i	$-0.559135\pi$
$907$	−11.1257	−0.369423	−0.184712	+	0.982793i	$0.559135\pi$
$908$	0	0
$909$	15.1476	0.502415
$910$	0	0
$911$	28.3903	0.940613	0.470306	−	0.882503i	$-0.344143\pi$
$911$	28.3903	0.940613	0.470306	+	0.882503i	$0.344143\pi$
$912$	0	0
$913$	47.8726	1.58435
$914$	0	0
$915$	−10.7384	−0.355000
$916$	0	0
$917$	−3.25010	−0.107328
$918$	0	0
$919$	32.3739	1.06792	0.533958	−	0.845511i	$-0.320704\pi$
$919$	32.3739	1.06792	0.533958	+	0.845511i	$0.320704\pi$
$920$	0	0
$921$	−19.2238	−0.633446
$922$	0	0
$923$	0.197351	0.00649587
$924$	0	0
$925$	3.29350	0.108290
$926$	0	0
$927$	1.83698	0.0603342
$928$	0	0
$929$	−52.7456	−1.73053	−0.865264	−	0.501317i	$-0.832849\pi$
$929$	−52.7456	−1.73053	−0.865264	+	0.501317i	$0.832849\pi$
$930$	0	0
$931$	18.4930	0.606083
$932$	0	0
$933$	11.6337	0.380871
$934$	0	0
$935$	21.3239	0.697367
$936$	0	0
$937$	10.9498	0.357715	0.178857	−	0.983875i	$-0.442760\pi$
$937$	10.9498	0.357715	0.178857	+	0.983875i	$0.442760\pi$
$938$	0	0
$939$	−18.2765	−0.596430
$940$	0	0
$941$	32.9537	1.07426	0.537130	−	0.843499i	$-0.319508\pi$
$941$	32.9537	1.07426	0.537130	+	0.843499i	$0.319508\pi$
$942$	0	0
$943$	−6.65903	−0.216848
$944$	0	0
$945$	−1.54385	−0.0502213
$946$	0	0
$947$	2.12566	0.0690746	0.0345373	−	0.999403i	$-0.489004\pi$
$947$	2.12566	0.0690746	0.0345373	+	0.999403i	$0.489004\pi$
$948$	0	0
$949$	−0.430104	−0.0139618
$950$	0	0
$951$	−32.0811	−1.04030
$952$	0	0
$953$	37.8431	1.22586	0.612929	−	0.790138i	$-0.289991\pi$
$953$	37.8431	1.22586	0.612929	+	0.790138i	$0.289991\pi$
$954$	0	0
$955$	−34.2467	−1.10820
$956$	0	0
$957$	3.82791	0.123739
$958$	0	0
$959$	−1.27733	−0.0412472
$960$	0	0
$961$	−21.8238	−0.703992
$962$	0	0
$963$	−17.5327	−0.564984
$964$	0	0
$965$	40.2187	1.29469
$966$	0	0
$967$	26.9996	0.868249	0.434125	−	0.900853i	$-0.357058\pi$
$967$	26.9996	0.868249	0.434125	+	0.900853i	$0.357058\pi$
$968$	0	0
$969$	5.54934	0.178271
$970$	0	0
$971$	11.3380	0.363855	0.181928	−	0.983312i	$-0.441766\pi$
$971$	11.3380	0.363855	0.181928	+	0.983312i	$0.441766\pi$
$972$	0	0
$973$	−4.45090	−0.142689
$974$	0	0
$975$	−0.148352	−0.00475108
$976$	0	0
$977$	−17.5348	−0.560988	−0.280494	−	0.959856i	$-0.590498\pi$
$977$	−17.5348	−0.560988	−0.280494	+	0.959856i	$0.590498\pi$
$978$	0	0
$979$	−58.2433	−1.86146
$980$	0	0
$981$	1.30868	0.0417829
$982$	0	0
$983$	33.8666	1.08018	0.540088	−	0.841609i	$-0.318391\pi$
$983$	33.8666	1.08018	0.540088	+	0.841609i	$0.318391\pi$
$984$	0	0
$985$	−19.3309	−0.615935
$986$	0	0
$987$	−2.22875	−0.0709419
$988$	0	0
$989$	−4.40578	−0.140096
$990$	0	0
$991$	−3.41219	−0.108392	−0.0541959	−	0.998530i	$-0.517260\pi$
$991$	−3.41219	−0.108392	−0.0541959	+	0.998530i	$0.517260\pi$
$992$	0	0
$993$	−19.6188	−0.622585
$994$	0	0
$995$	34.5136	1.09415
$996$	0	0
$997$	−48.0159	−1.52068	−0.760340	−	0.649525i	$-0.774968\pi$
$997$	−48.0159	−1.52068	−0.760340	+	0.649525i	$0.774968\pi$
$998$	0	0
$999$	1.22025	0.0386071

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.d.1.2	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.d.1.2	✓	8	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8004.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -2.77471 q^{5} +0.556399 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -2.77471 q^{5} +0.556399 q^{7} +1.00000 q^{9} +3.82791 q^{11} -0.0549651 q^{13} -2.77471 q^{15} -2.00765 q^{17} -2.76410 q^{19} +0.556399 q^{21} +1.00000 q^{23} +2.69903 q^{25} +1.00000 q^{27} +1.00000 q^{29} -3.02923 q^{31} +3.82791 q^{33} -1.54385 q^{35} +1.22025 q^{37} -0.0549651 q^{39} -6.65903 q^{41} -4.40578 q^{43} -2.77471 q^{45} -4.00567 q^{47} -6.69042 q^{49} -2.00765 q^{51} -3.40057 q^{53} -10.6213 q^{55} -2.76410 q^{57} +10.2440 q^{59} +3.87008 q^{61} +0.556399 q^{63} +0.152512 q^{65} -0.00393019 q^{67} +1.00000 q^{69} -3.59047 q^{71} +7.82504 q^{73} +2.69903 q^{75} +2.12984 q^{77} +3.03633 q^{79} +1.00000 q^{81} +12.5062 q^{83} +5.57065 q^{85} +1.00000 q^{87} -15.2154 q^{89} -0.0305825 q^{91} -3.02923 q^{93} +7.66958 q^{95} -9.71227 q^{97} +3.82791 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{3} - 5 q^{5} - 4 q^{7} + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^3 - 5 * q^5 - 4 * q^7 + 8 * q^9	\( 8 q + 8 q^{3} - 5 q^{5} - 4 q^{7} + 8 q^{9} - 5 q^{11} - 4 q^{13} - 5 q^{15} - 3 q^{17} - 5 q^{19} - 4 q^{21} + 8 q^{23} - 5 q^{25} + 8 q^{27} + 8 q^{29} - 2 q^{31} - 5 q^{33} - 15 q^{35} - 10 q^{37} - 4 q^{39} - 11 q^{41} - 7 q^{43} - 5 q^{45} - 14 q^{47} - 18 q^{49} - 3 q^{51} - 15 q^{53} - 17 q^{55} - 5 q^{57} + 4 q^{59} + q^{61} - 4 q^{63} - 5 q^{67} + 8 q^{69} - q^{71} - 21 q^{73} - 5 q^{75} - 8 q^{79} + 8 q^{81} + 3 q^{83} + 8 q^{87} - 20 q^{89} - 7 q^{91} - 2 q^{93} - 3 q^{95} - 7 q^{97} - 5 q^{99}+O(q^{100}) \) 8 * q + 8 * q^3 - 5 * q^5 - 4 * q^7 + 8 * q^9 - 5 * q^11 - 4 * q^13 - 5 * q^15 - 3 * q^17 - 5 * q^19 - 4 * q^21 + 8 * q^23 - 5 * q^25 + 8 * q^27 + 8 * q^29 - 2 * q^31 - 5 * q^33 - 15 * q^35 - 10 * q^37 - 4 * q^39 - 11 * q^41 - 7 * q^43 - 5 * q^45 - 14 * q^47 - 18 * q^49 - 3 * q^51 - 15 * q^53 - 17 * q^55 - 5 * q^57 + 4 * q^59 + q^61 - 4 * q^63 - 5 * q^67 + 8 * q^69 - q^71 - 21 * q^73 - 5 * q^75 - 8 * q^79 + 8 * q^81 + 3 * q^83 + 8 * q^87 - 20 * q^89 - 7 * q^91 - 2 * q^93 - 3 * q^95 - 7 * q^97 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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