LMFDB - Embedded newform 8004.2.a.j.1.11

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:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 3 x^{15} - 49 x^{14} + 130 x^{13} + 932 x^{12} - 2028 x^{11} - 8965 x^{10} + 14400 x^{9} + \cdots + 3888 $ x^16 - 3x^15 - 49x^14 + 130x^13 + 932x^12 - 2028x^11 - 8965x^10 + 14400x^9 + 46229x^8 - 47547x^7 - 122604x^6 + 65278x^5 + 151028x^4 - 17988x^3 - 67608x^2 - 8424*x + 3888
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$1.63007$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +1.63007 q^{5} -0.577524 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.63007 q^{5} -0.577524 q^{7} +1.00000 q^{9} +5.51763 q^{11} +0.530107 q^{13} +1.63007 q^{15} +6.30808 q^{17} +0.869087 q^{19} -0.577524 q^{21} +1.00000 q^{23} -2.34288 q^{25} +1.00000 q^{27} -1.00000 q^{29} -4.68209 q^{31} +5.51763 q^{33} -0.941403 q^{35} -2.51955 q^{37} +0.530107 q^{39} +7.33380 q^{41} +7.82962 q^{43} +1.63007 q^{45} +4.94492 q^{47} -6.66647 q^{49} +6.30808 q^{51} +2.53916 q^{53} +8.99412 q^{55} +0.869087 q^{57} +14.0049 q^{59} -5.51000 q^{61} -0.577524 q^{63} +0.864111 q^{65} -11.2304 q^{67} +1.00000 q^{69} -9.56805 q^{71} -3.25416 q^{73} -2.34288 q^{75} -3.18656 q^{77} +0.347989 q^{79} +1.00000 q^{81} +10.1590 q^{83} +10.2826 q^{85} -1.00000 q^{87} +14.8738 q^{89} -0.306149 q^{91} -4.68209 q^{93} +1.41667 q^{95} +2.03129 q^{97} +5.51763 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{3} + 3 q^{5} + 4 q^{7} + 16 q^{9}+O(q^{10}) $ 16 * q + 16 * q^3 + 3 * q^5 + 4 * q^7 + 16 * q^9	$ 16 q + 16 q^{3} + 3 q^{5} + 4 q^{7} + 16 q^{9} + 5 q^{11} + 6 q^{13} + 3 q^{15} + 3 q^{17} + 11 q^{19} + 4 q^{21} + 16 q^{23} + 27 q^{25} + 16 q^{27} - 16 q^{29} + 14 q^{31} + 5 q^{33} + 11 q^{35} + 4 q^{37} + 6 q^{39} + 11 q^{41} + 23 q^{43} + 3 q^{45} - 2 q^{47} + 34 q^{49} + 3 q^{51} + 19 q^{53} + 31 q^{55} + 11 q^{57} + 32 q^{59} + 19 q^{61} + 4 q^{63} + 6 q^{65} + 33 q^{67} + 16 q^{69} - 5 q^{71} + 23 q^{73} + 27 q^{75} + 42 q^{77} + 24 q^{79} + 16 q^{81} + 7 q^{83} - 16 q^{87} - 2 q^{89} + 25 q^{91} + 14 q^{93} + 7 q^{95} + 33 q^{97} + 5 q^{99}+O(q^{100}) $ 16 * q + 16 * q^3 + 3 * q^5 + 4 * q^7 + 16 * q^9 + 5 * q^11 + 6 * q^13 + 3 * q^15 + 3 * q^17 + 11 * q^19 + 4 * q^21 + 16 * q^23 + 27 * q^25 + 16 * q^27 - 16 * q^29 + 14 * q^31 + 5 * q^33 + 11 * q^35 + 4 * q^37 + 6 * q^39 + 11 * q^41 + 23 * q^43 + 3 * q^45 - 2 * q^47 + 34 * q^49 + 3 * q^51 + 19 * q^53 + 31 * q^55 + 11 * q^57 + 32 * q^59 + 19 * q^61 + 4 * q^63 + 6 * q^65 + 33 * q^67 + 16 * q^69 - 5 * q^71 + 23 * q^73 + 27 * q^75 + 42 * q^77 + 24 * q^79 + 16 * q^81 + 7 * q^83 - 16 * q^87 - 2 * q^89 + 25 * q^91 + 14 * q^93 + 7 * q^95 + 33 * 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
$5$	1.63007	0.728989	0.364494	−	0.931206i	$-0.381242\pi$
$5$	1.63007	0.728989	0.364494	+	0.931206i	$0.381242\pi$
$6$	0	0
$7$	−0.577524	−0.218283	−0.109142	−	0.994026i	$-0.534810\pi$
$7$	−0.577524	−0.218283	−0.109142	+	0.994026i	$0.534810\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.51763	1.66363	0.831814	−	0.555054i	$-0.187302\pi$
$11$	5.51763	1.66363	0.831814	+	0.555054i	$0.187302\pi$
$12$	0	0
$13$	0.530107	0.147025	0.0735126	−	0.997294i	$-0.476579\pi$
$13$	0.530107	0.147025	0.0735126	+	0.997294i	$0.476579\pi$
$14$	0	0
$15$	1.63007	0.420882
$16$	0	0
$17$	6.30808	1.52993	0.764967	−	0.644069i	$-0.222755\pi$
$17$	6.30808	1.52993	0.764967	+	0.644069i	$0.222755\pi$
$18$	0	0
$19$	0.869087	0.199382	0.0996911	−	0.995018i	$-0.468215\pi$
$19$	0.869087	0.199382	0.0996911	+	0.995018i	$0.468215\pi$
$20$	0	0
$21$	−0.577524	−0.126026
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−2.34288	−0.468575
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−4.68209	−0.840929	−0.420464	−	0.907309i	$-0.638133\pi$
$31$	−4.68209	−0.840929	−0.420464	+	0.907309i	$0.638133\pi$
$32$	0	0
$33$	5.51763	0.960497
$34$	0	0
$35$	−0.941403	−0.159126
$36$	0	0
$37$	−2.51955	−0.414211	−0.207106	−	0.978319i	$-0.566404\pi$
$37$	−2.51955	−0.414211	−0.207106	+	0.978319i	$0.566404\pi$
$38$	0	0
$39$	0.530107	0.0848850
$40$	0	0
$41$	7.33380	1.14535	0.572673	−	0.819784i	$-0.305906\pi$
$41$	7.33380	1.14535	0.572673	+	0.819784i	$0.305906\pi$
$42$	0	0
$43$	7.82962	1.19401	0.597003	−	0.802239i	$-0.296358\pi$
$43$	7.82962	1.19401	0.597003	+	0.802239i	$0.296358\pi$
$44$	0	0
$45$	1.63007	0.242996
$46$	0	0
$47$	4.94492	0.721290	0.360645	−	0.932703i	$-0.382557\pi$
$47$	4.94492	0.721290	0.360645	+	0.932703i	$0.382557\pi$
$48$	0	0
$49$	−6.66647	−0.952352
$50$	0	0
$51$	6.30808	0.883308
$52$	0	0
$53$	2.53916	0.348780	0.174390	−	0.984677i	$-0.444205\pi$
$53$	2.53916	0.348780	0.174390	+	0.984677i	$0.444205\pi$
$54$	0	0
$55$	8.99412	1.21277
$56$	0	0
$57$	0.869087	0.115113
$58$	0	0
$59$	14.0049	1.82328	0.911642	−	0.410984i	$-0.134815\pi$
$59$	14.0049	1.82328	0.911642	+	0.410984i	$0.134815\pi$
$60$	0	0
$61$	−5.51000	−0.705484	−0.352742	−	0.935721i	$-0.614751\pi$
$61$	−5.51000	−0.705484	−0.352742	+	0.935721i	$0.614751\pi$
$62$	0	0
$63$	−0.577524	−0.0727612
$64$	0	0
$65$	0.864111	0.107180
$66$	0	0
$67$	−11.2304	−1.37201	−0.686006	−	0.727596i	$-0.740638\pi$
$67$	−11.2304	−1.37201	−0.686006	+	0.727596i	$0.740638\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−9.56805	−1.13552	−0.567759	−	0.823195i	$-0.692190\pi$
$71$	−9.56805	−1.13552	−0.567759	+	0.823195i	$0.692190\pi$
$72$	0	0
$73$	−3.25416	−0.380871	−0.190435	−	0.981700i	$-0.560990\pi$
$73$	−3.25416	−0.380871	−0.190435	+	0.981700i	$0.560990\pi$
$74$	0	0
$75$	−2.34288	−0.270532
$76$	0	0
$77$	−3.18656	−0.363143
$78$	0	0
$79$	0.347989	0.0391518	0.0195759	−	0.999808i	$-0.493768\pi$
$79$	0.347989	0.0391518	0.0195759	+	0.999808i	$0.493768\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	10.1590	1.11509	0.557546	−	0.830146i	$-0.311743\pi$
$83$	10.1590	1.11509	0.557546	+	0.830146i	$0.311743\pi$
$84$	0	0
$85$	10.2826	1.11531
$86$	0	0
$87$	−1.00000	−0.107211
$88$	0	0
$89$	14.8738	1.57662	0.788309	−	0.615280i	$-0.210957\pi$
$89$	14.8738	1.57662	0.788309	+	0.615280i	$0.210957\pi$
$90$	0	0
$91$	−0.306149	−0.0320932
$92$	0	0
$93$	−4.68209	−0.485510
$94$	0	0
$95$	1.41667	0.145347
$96$	0	0
$97$	2.03129	0.206246	0.103123	−	0.994669i	$-0.467116\pi$
$97$	2.03129	0.206246	0.103123	+	0.994669i	$0.467116\pi$
$98$	0	0
$99$	5.51763	0.554543
$100$	0	0
$101$	−7.88165	−0.784253	−0.392127	−	0.919911i	$-0.628260\pi$
$101$	−7.88165	−0.784253	−0.392127	+	0.919911i	$0.628260\pi$
$102$	0	0
$103$	3.55532	0.350316	0.175158	−	0.984540i	$-0.443956\pi$
$103$	3.55532	0.350316	0.175158	+	0.984540i	$0.443956\pi$
$104$	0	0
$105$	−0.941403	−0.0918716
$106$	0	0
$107$	−3.84295	−0.371512	−0.185756	−	0.982596i	$-0.559473\pi$
$107$	−3.84295	−0.371512	−0.185756	+	0.982596i	$0.559473\pi$
$108$	0	0
$109$	16.9800	1.62639	0.813195	−	0.581992i	$-0.197726\pi$
$109$	16.9800	1.62639	0.813195	+	0.581992i	$0.197726\pi$
$110$	0	0
$111$	−2.51955	−0.239145
$112$	0	0
$113$	−4.39778	−0.413708	−0.206854	−	0.978372i	$-0.566323\pi$
$113$	−4.39778	−0.413708	−0.206854	+	0.978372i	$0.566323\pi$
$114$	0	0
$115$	1.63007	0.152005
$116$	0	0
$117$	0.530107	0.0490084
$118$	0	0
$119$	−3.64307	−0.333959
$120$	0	0
$121$	19.4443	1.76766
$122$	0	0
$123$	7.33380	0.661266
$124$	0	0
$125$	−11.9694	−1.07058
$126$	0	0
$127$	−12.8595	−1.14110	−0.570548	−	0.821264i	$-0.693269\pi$
$127$	−12.8595	−1.14110	−0.570548	+	0.821264i	$0.693269\pi$
$128$	0	0
$129$	7.82962	0.689360
$130$	0	0
$131$	−12.0820	−1.05561	−0.527807	−	0.849365i	$-0.676985\pi$
$131$	−12.0820	−1.05561	−0.527807	+	0.849365i	$0.676985\pi$
$132$	0	0
$133$	−0.501919	−0.0435219
$134$	0	0
$135$	1.63007	0.140294
$136$	0	0
$137$	1.50378	0.128477	0.0642383	−	0.997935i	$-0.479538\pi$
$137$	1.50378	0.128477	0.0642383	+	0.997935i	$0.479538\pi$
$138$	0	0
$139$	−18.4802	−1.56747	−0.783735	−	0.621095i	$-0.786688\pi$
$139$	−18.4802	−1.56747	−0.783735	+	0.621095i	$0.786688\pi$
$140$	0	0
$141$	4.94492	0.416437
$142$	0	0
$143$	2.92494	0.244595
$144$	0	0
$145$	−1.63007	−0.135370
$146$	0	0
$147$	−6.66647	−0.549841
$148$	0	0
$149$	5.69566	0.466607	0.233304	−	0.972404i	$-0.425046\pi$
$149$	5.69566	0.466607	0.233304	+	0.972404i	$0.425046\pi$
$150$	0	0
$151$	−11.4329	−0.930395	−0.465198	−	0.885207i	$-0.654017\pi$
$151$	−11.4329	−0.930395	−0.465198	+	0.885207i	$0.654017\pi$
$152$	0	0
$153$	6.30808	0.509978
$154$	0	0
$155$	−7.63213	−0.613028
$156$	0	0
$157$	−2.38936	−0.190691	−0.0953457	−	0.995444i	$-0.530396\pi$
$157$	−2.38936	−0.190691	−0.0953457	+	0.995444i	$0.530396\pi$
$158$	0	0
$159$	2.53916	0.201369
$160$	0	0
$161$	−0.577524	−0.0455153
$162$	0	0
$163$	−23.0628	−1.80642	−0.903208	−	0.429203i	$-0.858794\pi$
$163$	−23.0628	−1.80642	−0.903208	+	0.429203i	$0.858794\pi$
$164$	0	0
$165$	8.99412	0.700191
$166$	0	0
$167$	−2.42660	−0.187776	−0.0938878	−	0.995583i	$-0.529930\pi$
$167$	−2.42660	−0.187776	−0.0938878	+	0.995583i	$0.529930\pi$
$168$	0	0
$169$	−12.7190	−0.978384
$170$	0	0
$171$	0.869087	0.0664608
$172$	0	0
$173$	−15.0262	−1.14242	−0.571209	−	0.820805i	$-0.693525\pi$
$173$	−15.0262	−1.14242	−0.571209	+	0.820805i	$0.693525\pi$
$174$	0	0
$175$	1.35307	0.102282
$176$	0	0
$177$	14.0049	1.05267
$178$	0	0
$179$	15.9353	1.19106	0.595531	−	0.803332i	$-0.296942\pi$
$179$	15.9353	1.19106	0.595531	+	0.803332i	$0.296942\pi$
$180$	0	0
$181$	−12.1265	−0.901359	−0.450680	−	0.892686i	$-0.648818\pi$
$181$	−12.1265	−0.901359	−0.450680	+	0.892686i	$0.648818\pi$
$182$	0	0
$183$	−5.51000	−0.407311
$184$	0	0
$185$	−4.10704	−0.301955
$186$	0	0
$187$	34.8057	2.54524
$188$	0	0
$189$	−0.577524	−0.0420087
$190$	0	0
$191$	2.12503	0.153762	0.0768808	−	0.997040i	$-0.475504\pi$
$191$	2.12503	0.153762	0.0768808	+	0.997040i	$0.475504\pi$
$192$	0	0
$193$	−13.0166	−0.936959	−0.468479	−	0.883474i	$-0.655198\pi$
$193$	−13.0166	−0.936959	−0.468479	+	0.883474i	$0.655198\pi$
$194$	0	0
$195$	0.864111	0.0618802
$196$	0	0
$197$	11.2634	0.802483	0.401242	−	0.915972i	$-0.368579\pi$
$197$	11.2634	0.802483	0.401242	+	0.915972i	$0.368579\pi$
$198$	0	0
$199$	−9.31732	−0.660487	−0.330244	−	0.943896i	$-0.607131\pi$
$199$	−9.31732	−0.660487	−0.330244	+	0.943896i	$0.607131\pi$
$200$	0	0
$201$	−11.2304	−0.792132
$202$	0	0
$203$	0.577524	0.0405342
$204$	0	0
$205$	11.9546	0.834945
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	4.79530	0.331698
$210$	0	0
$211$	7.76189	0.534350	0.267175	−	0.963648i	$-0.413910\pi$
$211$	7.76189	0.534350	0.267175	+	0.963648i	$0.413910\pi$
$212$	0	0
$213$	−9.56805	−0.655592
$214$	0	0
$215$	12.7628	0.870417
$216$	0	0
$217$	2.70402	0.183561
$218$	0	0
$219$	−3.25416	−0.219896
$220$	0	0
$221$	3.34396	0.224939
$222$	0	0
$223$	9.00853	0.603256	0.301628	−	0.953426i	$-0.402470\pi$
$223$	9.00853	0.603256	0.301628	+	0.953426i	$0.402470\pi$
$224$	0	0
$225$	−2.34288	−0.156192
$226$	0	0
$227$	18.9075	1.25493	0.627467	−	0.778643i	$-0.284092\pi$
$227$	18.9075	1.25493	0.627467	+	0.778643i	$0.284092\pi$
$228$	0	0
$229$	0.0153065	0.00101148	0.000505742	−	1.00000i	$-0.499839\pi$
$229$	0.0153065	0.00101148	0.000505742		1.00000i	$0.499839\pi$
$230$	0	0
$231$	−3.18656	−0.209661
$232$	0	0
$233$	10.9974	0.720461	0.360230	−	0.932863i	$-0.382698\pi$
$233$	10.9974	0.720461	0.360230	+	0.932863i	$0.382698\pi$
$234$	0	0
$235$	8.06056	0.525813
$236$	0	0
$237$	0.347989	0.0226043
$238$	0	0
$239$	17.4397	1.12808	0.564039	−	0.825748i	$-0.309247\pi$
$239$	17.4397	1.12808	0.564039	+	0.825748i	$0.309247\pi$
$240$	0	0
$241$	30.6019	1.97124	0.985622	−	0.168967i	$-0.0540431\pi$
$241$	30.6019	1.97124	0.985622	+	0.168967i	$0.0540431\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−10.8668	−0.694254
$246$	0	0
$247$	0.460709	0.0293142
$248$	0	0
$249$	10.1590	0.643799
$250$	0	0
$251$	13.7596	0.868497	0.434248	−	0.900793i	$-0.357014\pi$
$251$	13.7596	0.868497	0.434248	+	0.900793i	$0.357014\pi$
$252$	0	0
$253$	5.51763	0.346891
$254$	0	0
$255$	10.2826	0.643922
$256$	0	0
$257$	18.0156	1.12378	0.561890	−	0.827212i	$-0.310074\pi$
$257$	18.0156	1.12378	0.561890	+	0.827212i	$0.310074\pi$
$258$	0	0
$259$	1.45510	0.0904154
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	0	0
$263$	−8.65502	−0.533691	−0.266846	−	0.963739i	$-0.585981\pi$
$263$	−8.65502	−0.533691	−0.266846	+	0.963739i	$0.585981\pi$
$264$	0	0
$265$	4.13901	0.254257
$266$	0	0
$267$	14.8738	0.910260
$268$	0	0
$269$	−12.1540	−0.741043	−0.370521	−	0.928824i	$-0.620821\pi$
$269$	−12.1540	−0.741043	−0.370521	+	0.928824i	$0.620821\pi$
$270$	0	0
$271$	9.41238	0.571761	0.285881	−	0.958265i	$-0.407714\pi$
$271$	9.41238	0.571761	0.285881	+	0.958265i	$0.407714\pi$
$272$	0	0
$273$	−0.306149	−0.0185290
$274$	0	0
$275$	−12.9271	−0.779535
$276$	0	0
$277$	−22.4479	−1.34877	−0.674383	−	0.738381i	$-0.735590\pi$
$277$	−22.4479	−1.34877	−0.674383	+	0.738381i	$0.735590\pi$
$278$	0	0
$279$	−4.68209	−0.280310
$280$	0	0
$281$	−4.81073	−0.286984	−0.143492	−	0.989651i	$-0.545833\pi$
$281$	−4.81073	−0.286984	−0.143492	+	0.989651i	$0.545833\pi$
$282$	0	0
$283$	15.2976	0.909349	0.454675	−	0.890658i	$-0.349756\pi$
$283$	15.2976	0.909349	0.454675	+	0.890658i	$0.349756\pi$
$284$	0	0
$285$	1.41667	0.0839164
$286$	0	0
$287$	−4.23544	−0.250010
$288$	0	0
$289$	22.7919	1.34070
$290$	0	0
$291$	2.03129	0.119076
$292$	0	0
$293$	28.3226	1.65463	0.827313	−	0.561741i	$-0.189868\pi$
$293$	28.3226	1.65463	0.827313	+	0.561741i	$0.189868\pi$
$294$	0	0
$295$	22.8290	1.32915
$296$	0	0
$297$	5.51763	0.320166
$298$	0	0
$299$	0.530107	0.0306569
$300$	0	0
$301$	−4.52179	−0.260632
$302$	0	0
$303$	−7.88165	−0.452789
$304$	0	0
$305$	−8.98169	−0.514290
$306$	0	0
$307$	9.67118	0.551963	0.275982	−	0.961163i	$-0.410997\pi$
$307$	9.67118	0.551963	0.275982	+	0.961163i	$0.410997\pi$
$308$	0	0
$309$	3.55532	0.202255
$310$	0	0
$311$	−25.4302	−1.44202	−0.721008	−	0.692927i	$-0.756321\pi$
$311$	−25.4302	−1.44202	−0.721008	+	0.692927i	$0.756321\pi$
$312$	0	0
$313$	−0.926444	−0.0523657	−0.0261829	−	0.999657i	$-0.508335\pi$
$313$	−0.926444	−0.0523657	−0.0261829	+	0.999657i	$0.508335\pi$
$314$	0	0
$315$	−0.941403	−0.0530421
$316$	0	0
$317$	−5.52724	−0.310441	−0.155220	−	0.987880i	$-0.549609\pi$
$317$	−5.52724	−0.310441	−0.155220	+	0.987880i	$0.549609\pi$
$318$	0	0
$319$	−5.51763	−0.308928
$320$	0	0
$321$	−3.84295	−0.214492
$322$	0	0
$323$	5.48227	0.305042
$324$	0	0
$325$	−1.24197	−0.0688924
$326$	0	0
$327$	16.9800	0.938996
$328$	0	0
$329$	−2.85581	−0.157446
$330$	0	0
$331$	7.45576	0.409805	0.204903	−	0.978782i	$-0.434312\pi$
$331$	7.45576	0.409805	0.204903	+	0.978782i	$0.434312\pi$
$332$	0	0
$333$	−2.51955	−0.138070
$334$	0	0
$335$	−18.3063	−1.00018
$336$	0	0
$337$	8.64930	0.471157	0.235579	−	0.971855i	$-0.424301\pi$
$337$	8.64930	0.471157	0.235579	+	0.971855i	$0.424301\pi$
$338$	0	0
$339$	−4.39778	−0.238854
$340$	0	0
$341$	−25.8341	−1.39899
$342$	0	0
$343$	7.89271	0.426166
$344$	0	0
$345$	1.63007	0.0877600
$346$	0	0
$347$	−13.9560	−0.749199	−0.374599	−	0.927187i	$-0.622220\pi$
$347$	−13.9560	−0.749199	−0.374599	+	0.927187i	$0.622220\pi$
$348$	0	0
$349$	27.1835	1.45510	0.727551	−	0.686054i	$-0.240659\pi$
$349$	27.1835	1.45510	0.727551	+	0.686054i	$0.240659\pi$
$350$	0	0
$351$	0.530107	0.0282950
$352$	0	0
$353$	24.2745	1.29200	0.646002	−	0.763336i	$-0.276440\pi$
$353$	24.2745	1.29200	0.646002	+	0.763336i	$0.276440\pi$
$354$	0	0
$355$	−15.5966	−0.827781
$356$	0	0
$357$	−3.64307	−0.192812
$358$	0	0
$359$	−25.7342	−1.35820	−0.679099	−	0.734046i	$-0.737629\pi$
$359$	−25.7342	−1.35820	−0.679099	+	0.734046i	$0.737629\pi$
$360$	0	0
$361$	−18.2447	−0.960247
$362$	0	0
$363$	19.4443	1.02056
$364$	0	0
$365$	−5.30450	−0.277650
$366$	0	0
$367$	24.5849	1.28332	0.641661	−	0.766988i	$-0.278246\pi$
$367$	24.5849	1.28332	0.641661	+	0.766988i	$0.278246\pi$
$368$	0	0
$369$	7.33380	0.381782
$370$	0	0
$371$	−1.46643	−0.0761330
$372$	0	0
$373$	11.4513	0.592927	0.296463	−	0.955044i	$-0.404193\pi$
$373$	11.4513	0.592927	0.296463	+	0.955044i	$0.404193\pi$
$374$	0	0
$375$	−11.9694	−0.618097
$376$	0	0
$377$	−0.530107	−0.0273019
$378$	0	0
$379$	−27.8423	−1.43016	−0.715081	−	0.699042i	$-0.753610\pi$
$379$	−27.8423	−1.43016	−0.715081	+	0.699042i	$0.753610\pi$
$380$	0	0
$381$	−12.8595	−0.658812
$382$	0	0
$383$	−20.9540	−1.07070	−0.535349	−	0.844631i	$-0.679820\pi$
$383$	−20.9540	−1.07070	−0.535349	+	0.844631i	$0.679820\pi$
$384$	0	0
$385$	−5.19432	−0.264727
$386$	0	0
$387$	7.82962	0.398002
$388$	0	0
$389$	15.5047	0.786122	0.393061	−	0.919512i	$-0.371416\pi$
$389$	15.5047	0.786122	0.393061	+	0.919512i	$0.371416\pi$
$390$	0	0
$391$	6.30808	0.319013
$392$	0	0
$393$	−12.0820	−0.609459
$394$	0	0
$395$	0.567246	0.0285412
$396$	0	0
$397$	21.8236	1.09530	0.547649	−	0.836709i	$-0.315523\pi$
$397$	21.8236	1.09530	0.547649	+	0.836709i	$0.315523\pi$
$398$	0	0
$399$	−0.501919	−0.0251274
$400$	0	0
$401$	26.5683	1.32676	0.663379	−	0.748283i	$-0.269122\pi$
$401$	26.5683	1.32676	0.663379	+	0.748283i	$0.269122\pi$
$402$	0	0
$403$	−2.48201	−0.123638
$404$	0	0
$405$	1.63007	0.0809988
$406$	0	0
$407$	−13.9019	−0.689093
$408$	0	0
$409$	−8.65898	−0.428159	−0.214079	−	0.976816i	$-0.568675\pi$
$409$	−8.65898	−0.428159	−0.214079	+	0.976816i	$0.568675\pi$
$410$	0	0
$411$	1.50378	0.0741760
$412$	0	0
$413$	−8.08817	−0.397993
$414$	0	0
$415$	16.5598	0.812890
$416$	0	0
$417$	−18.4802	−0.904979
$418$	0	0
$419$	−8.94615	−0.437048	−0.218524	−	0.975832i	$-0.570124\pi$
$419$	−8.94615	−0.437048	−0.218524	+	0.975832i	$0.570124\pi$
$420$	0	0
$421$	2.15954	0.105249	0.0526247	−	0.998614i	$-0.483241\pi$
$421$	2.15954	0.105249	0.0526247	+	0.998614i	$0.483241\pi$
$422$	0	0
$423$	4.94492	0.240430
$424$	0	0
$425$	−14.7791	−0.716889
$426$	0	0
$427$	3.18216	0.153995
$428$	0	0
$429$	2.92494	0.141217
$430$	0	0
$431$	40.7851	1.96455	0.982275	−	0.187447i	$-0.0600212\pi$
$431$	40.7851	1.96455	0.982275	+	0.187447i	$0.0600212\pi$
$432$	0	0
$433$	−15.8505	−0.761727	−0.380863	−	0.924631i	$-0.624373\pi$
$433$	−15.8505	−0.761727	−0.380863	+	0.924631i	$0.624373\pi$
$434$	0	0
$435$	−1.63007	−0.0781558
$436$	0	0
$437$	0.869087	0.0415741
$438$	0	0
$439$	−10.6378	−0.507713	−0.253856	−	0.967242i	$-0.581699\pi$
$439$	−10.6378	−0.507713	−0.253856	+	0.967242i	$0.581699\pi$
$440$	0	0
$441$	−6.66647	−0.317451
$442$	0	0
$443$	−36.4732	−1.73289	−0.866446	−	0.499271i	$-0.833601\pi$
$443$	−36.4732	−1.73289	−0.866446	+	0.499271i	$0.833601\pi$
$444$	0	0
$445$	24.2453	1.14934
$446$	0	0
$447$	5.69566	0.269396
$448$	0	0
$449$	16.9657	0.800661	0.400330	−	0.916371i	$-0.368895\pi$
$449$	16.9657	0.800661	0.400330	+	0.916371i	$0.368895\pi$
$450$	0	0
$451$	40.4652	1.90543
$452$	0	0
$453$	−11.4329	−0.537164
$454$	0	0
$455$	−0.499044	−0.0233956
$456$	0	0
$457$	9.34001	0.436907	0.218454	−	0.975847i	$-0.429899\pi$
$457$	9.34001	0.436907	0.218454	+	0.975847i	$0.429899\pi$
$458$	0	0
$459$	6.30808	0.294436
$460$	0	0
$461$	21.6088	1.00642	0.503210	−	0.864164i	$-0.332152\pi$
$461$	21.6088	1.00642	0.503210	+	0.864164i	$0.332152\pi$
$462$	0	0
$463$	−17.4907	−0.812863	−0.406431	−	0.913681i	$-0.633227\pi$
$463$	−17.4907	−0.812863	−0.406431	+	0.913681i	$0.633227\pi$
$464$	0	0
$465$	−7.63213	−0.353932
$466$	0	0
$467$	16.7098	0.773240	0.386620	−	0.922239i	$-0.373643\pi$
$467$	16.7098	0.773240	0.386620	+	0.922239i	$0.373643\pi$
$468$	0	0
$469$	6.48583	0.299488
$470$	0	0
$471$	−2.38936	−0.110096
$472$	0	0
$473$	43.2010	1.98638
$474$	0	0
$475$	−2.03616	−0.0934256
$476$	0	0
$477$	2.53916	0.116260
$478$	0	0
$479$	−7.66553	−0.350247	−0.175123	−	0.984546i	$-0.556032\pi$
$479$	−7.66553	−0.350247	−0.175123	+	0.984546i	$0.556032\pi$
$480$	0	0
$481$	−1.33563	−0.0608995
$482$	0	0
$483$	−0.577524	−0.0262782
$484$	0	0
$485$	3.31115	0.150351
$486$	0	0
$487$	22.7388	1.03039	0.515196	−	0.857073i	$-0.327719\pi$
$487$	22.7388	1.03039	0.515196	+	0.857073i	$0.327719\pi$
$488$	0	0
$489$	−23.0628	−1.04293
$490$	0	0
$491$	−6.66675	−0.300866	−0.150433	−	0.988620i	$-0.548067\pi$
$491$	−6.66675	−0.300866	−0.150433	+	0.988620i	$0.548067\pi$
$492$	0	0
$493$	−6.30808	−0.284102
$494$	0	0
$495$	8.99412	0.404256
$496$	0	0
$497$	5.52578	0.247865
$498$	0	0
$499$	−36.1611	−1.61879	−0.809397	−	0.587262i	$-0.800206\pi$
$499$	−36.1611	−1.61879	−0.809397	+	0.587262i	$0.800206\pi$
$500$	0	0
$501$	−2.42660	−0.108412
$502$	0	0
$503$	13.1400	0.585886	0.292943	−	0.956130i	$-0.405365\pi$
$503$	13.1400	0.585886	0.292943	+	0.956130i	$0.405365\pi$
$504$	0	0
$505$	−12.8476	−0.571712
$506$	0	0
$507$	−12.7190	−0.564870
$508$	0	0
$509$	−33.1123	−1.46767	−0.733837	−	0.679325i	$-0.762273\pi$
$509$	−33.1123	−1.46767	−0.733837	+	0.679325i	$0.762273\pi$
$510$	0	0
$511$	1.87935	0.0831377
$512$	0	0
$513$	0.869087	0.0383711
$514$	0	0
$515$	5.79541	0.255376
$516$	0	0
$517$	27.2842	1.19996
$518$	0	0
$519$	−15.0262	−0.659575
$520$	0	0
$521$	−9.99806	−0.438023	−0.219012	−	0.975722i	$-0.570283\pi$
$521$	−9.99806	−0.438023	−0.219012	+	0.975722i	$0.570283\pi$
$522$	0	0
$523$	2.52154	0.110259	0.0551297	−	0.998479i	$-0.482443\pi$
$523$	2.52154	0.110259	0.0551297	+	0.998479i	$0.482443\pi$
$524$	0	0
$525$	1.35307	0.0590527
$526$	0	0
$527$	−29.5350	−1.28657
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	14.0049	0.607762
$532$	0	0
$533$	3.88770	0.168395
$534$	0	0
$535$	−6.26427	−0.270828
$536$	0	0
$537$	15.9353	0.687660
$538$	0	0
$539$	−36.7831	−1.58436
$540$	0	0
$541$	−13.5362	−0.581968	−0.290984	−	0.956728i	$-0.593983\pi$
$541$	−13.5362	−0.581968	−0.290984	+	0.956728i	$0.593983\pi$
$542$	0	0
$543$	−12.1265	−0.520400
$544$	0	0
$545$	27.6786	1.18562
$546$	0	0
$547$	−8.42273	−0.360130	−0.180065	−	0.983655i	$-0.557631\pi$
$547$	−8.42273	−0.360130	−0.180065	+	0.983655i	$0.557631\pi$
$548$	0	0
$549$	−5.51000	−0.235161
$550$	0	0
$551$	−0.869087	−0.0370244
$552$	0	0
$553$	−0.200972	−0.00854619
$554$	0	0
$555$	−4.10704	−0.174334
$556$	0	0
$557$	−29.9001	−1.26691	−0.633454	−	0.773781i	$-0.718363\pi$
$557$	−29.9001	−1.26691	−0.633454	+	0.773781i	$0.718363\pi$
$558$	0	0
$559$	4.15054	0.175549
$560$	0	0
$561$	34.8057	1.46950
$562$	0	0
$563$	−27.4066	−1.15505	−0.577526	−	0.816373i	$-0.695982\pi$
$563$	−27.4066	−1.15505	−0.577526	+	0.816373i	$0.695982\pi$
$564$	0	0
$565$	−7.16868	−0.301588
$566$	0	0
$567$	−0.577524	−0.0242537
$568$	0	0
$569$	−19.5145	−0.818089	−0.409044	−	0.912514i	$-0.634138\pi$
$569$	−19.5145	−0.818089	−0.409044	+	0.912514i	$0.634138\pi$
$570$	0	0
$571$	32.2698	1.35045	0.675224	−	0.737613i	$-0.264047\pi$
$571$	32.2698	1.35045	0.675224	+	0.737613i	$0.264047\pi$
$572$	0	0
$573$	2.12503	0.0887743
$574$	0	0
$575$	−2.34288	−0.0977047
$576$	0	0
$577$	−11.5878	−0.482407	−0.241203	−	0.970475i	$-0.577542\pi$
$577$	−11.5878	−0.482407	−0.241203	+	0.970475i	$0.577542\pi$
$578$	0	0
$579$	−13.0166	−0.540953
$580$	0	0
$581$	−5.86705	−0.243406
$582$	0	0
$583$	14.0102	0.580241
$584$	0	0
$585$	0.864111	0.0357266
$586$	0	0
$587$	21.7791	0.898919	0.449460	−	0.893301i	$-0.351617\pi$
$587$	21.7791	0.898919	0.449460	+	0.893301i	$0.351617\pi$
$588$	0	0
$589$	−4.06915	−0.167666
$590$	0	0
$591$	11.2634	0.463314
$592$	0	0
$593$	−13.9266	−0.571898	−0.285949	−	0.958245i	$-0.592309\pi$
$593$	−13.9266	−0.571898	−0.285949	+	0.958245i	$0.592309\pi$
$594$	0	0
$595$	−5.93845	−0.243453
$596$	0	0
$597$	−9.31732	−0.381333
$598$	0	0
$599$	−40.5647	−1.65743	−0.828714	−	0.559672i	$-0.810927\pi$
$599$	−40.5647	−1.65743	−0.828714	+	0.559672i	$0.810927\pi$
$600$	0	0
$601$	17.5871	0.717391	0.358695	−	0.933455i	$-0.383222\pi$
$601$	17.5871	0.717391	0.358695	+	0.933455i	$0.383222\pi$
$602$	0	0
$603$	−11.2304	−0.457337
$604$	0	0
$605$	31.6955	1.28861
$606$	0	0
$607$	−2.45923	−0.0998170	−0.0499085	−	0.998754i	$-0.515893\pi$
$607$	−2.45923	−0.0998170	−0.0499085	+	0.998754i	$0.515893\pi$
$608$	0	0
$609$	0.577524	0.0234024
$610$	0	0
$611$	2.62134	0.106048
$612$	0	0
$613$	10.4024	0.420148	0.210074	−	0.977686i	$-0.432630\pi$
$613$	10.4024	0.420148	0.210074	+	0.977686i	$0.432630\pi$
$614$	0	0
$615$	11.9546	0.482056
$616$	0	0
$617$	37.0896	1.49317	0.746586	−	0.665288i	$-0.231691\pi$
$617$	37.0896	1.49317	0.746586	+	0.665288i	$0.231691\pi$
$618$	0	0
$619$	27.3901	1.10090	0.550451	−	0.834868i	$-0.314456\pi$
$619$	27.3901	1.10090	0.550451	+	0.834868i	$0.314456\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	−8.58996	−0.344149
$624$	0	0
$625$	−7.79655	−0.311862
$626$	0	0
$627$	4.79530	0.191506
$628$	0	0
$629$	−15.8935	−0.633716
$630$	0	0
$631$	−21.7665	−0.866509	−0.433254	−	0.901272i	$-0.642635\pi$
$631$	−21.7665	−0.866509	−0.433254	+	0.901272i	$0.642635\pi$
$632$	0	0
$633$	7.76189	0.308507
$634$	0	0
$635$	−20.9619	−0.831847
$636$	0	0
$637$	−3.53394	−0.140020
$638$	0	0
$639$	−9.56805	−0.378506
$640$	0	0
$641$	−14.5732	−0.575605	−0.287802	−	0.957690i	$-0.592925\pi$
$641$	−14.5732	−0.575605	−0.287802	+	0.957690i	$0.592925\pi$
$642$	0	0
$643$	−22.3695	−0.882166	−0.441083	−	0.897466i	$-0.645405\pi$
$643$	−22.3695	−0.882166	−0.441083	+	0.897466i	$0.645405\pi$
$644$	0	0
$645$	12.7628	0.502535
$646$	0	0
$647$	−37.6735	−1.48110	−0.740550	−	0.672002i	$-0.765435\pi$
$647$	−37.6735	−1.48110	−0.740550	+	0.672002i	$0.765435\pi$
$648$	0	0
$649$	77.2740	3.03327
$650$	0	0
$651$	2.70402	0.105979
$652$	0	0
$653$	5.53979	0.216789	0.108394	−	0.994108i	$-0.465429\pi$
$653$	5.53979	0.216789	0.108394	+	0.994108i	$0.465429\pi$
$654$	0	0
$655$	−19.6946	−0.769530
$656$	0	0
$657$	−3.25416	−0.126957
$658$	0	0
$659$	1.64161	0.0639479	0.0319739	−	0.999489i	$-0.489821\pi$
$659$	1.64161	0.0639479	0.0319739	+	0.999489i	$0.489821\pi$
$660$	0	0
$661$	21.5960	0.839986	0.419993	−	0.907527i	$-0.362033\pi$
$661$	21.5960	0.839986	0.419993	+	0.907527i	$0.362033\pi$
$662$	0	0
$663$	3.34396	0.129869
$664$	0	0
$665$	−0.818162	−0.0317270
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	0	0
$669$	9.00853	0.348290
$670$	0	0
$671$	−30.4022	−1.17366
$672$	0	0
$673$	−19.1413	−0.737845	−0.368922	−	0.929460i	$-0.620273\pi$
$673$	−19.1413	−0.737845	−0.368922	+	0.929460i	$0.620273\pi$
$674$	0	0
$675$	−2.34288	−0.0901773
$676$	0	0
$677$	41.5452	1.59671	0.798356	−	0.602185i	$-0.205703\pi$
$677$	41.5452	1.59671	0.798356	+	0.602185i	$0.205703\pi$
$678$	0	0
$679$	−1.17312	−0.0450202
$680$	0	0
$681$	18.9075	0.724536
$682$	0	0
$683$	−17.7308	−0.678450	−0.339225	−	0.940705i	$-0.610165\pi$
$683$	−17.7308	−0.678450	−0.339225	+	0.940705i	$0.610165\pi$
$684$	0	0
$685$	2.45127	0.0936580
$686$	0	0
$687$	0.0153065	0.000583981 0
$688$	0	0
$689$	1.34603	0.0512795
$690$	0	0
$691$	31.0918	1.18279	0.591393	−	0.806383i	$-0.298578\pi$
$691$	31.0918	1.18279	0.591393	+	0.806383i	$0.298578\pi$
$692$	0	0
$693$	−3.18656	−0.121048
$694$	0	0
$695$	−30.1240	−1.14267
$696$	0	0
$697$	46.2622	1.75231
$698$	0	0
$699$	10.9974	0.415958
$700$	0	0
$701$	14.1464	0.534301	0.267151	−	0.963655i	$-0.413918\pi$
$701$	14.1464	0.534301	0.267151	+	0.963655i	$0.413918\pi$
$702$	0	0
$703$	−2.18971	−0.0825863
$704$	0	0
$705$	8.06056	0.303578
$706$	0	0
$707$	4.55184	0.171190
$708$	0	0
$709$	−32.5729	−1.22330	−0.611650	−	0.791129i	$-0.709494\pi$
$709$	−32.5729	−1.22330	−0.611650	+	0.791129i	$0.709494\pi$
$710$	0	0
$711$	0.347989	0.0130506
$712$	0	0
$713$	−4.68209	−0.175346
$714$	0	0
$715$	4.76785	0.178307
$716$	0	0
$717$	17.4397	0.651296
$718$	0	0
$719$	31.3035	1.16742	0.583712	−	0.811961i	$-0.301600\pi$
$719$	31.3035	1.16742	0.583712	+	0.811961i	$0.301600\pi$
$720$	0	0
$721$	−2.05328	−0.0764682
$722$	0	0
$723$	30.6019	1.13810
$724$	0	0
$725$	2.34288	0.0870122
$726$	0	0
$727$	−34.3196	−1.27284	−0.636421	−	0.771342i	$-0.719586\pi$
$727$	−34.3196	−1.27284	−0.636421	+	0.771342i	$0.719586\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	49.3899	1.82675
$732$	0	0
$733$	−26.2172	−0.968355	−0.484177	−	0.874970i	$-0.660881\pi$
$733$	−26.2172	−0.968355	−0.484177	+	0.874970i	$0.660881\pi$
$734$	0	0
$735$	−10.8668	−0.400828
$736$	0	0
$737$	−61.9653	−2.28252
$738$	0	0
$739$	−37.3977	−1.37570	−0.687848	−	0.725855i	$-0.741444\pi$
$739$	−37.3977	−1.37570	−0.687848	+	0.725855i	$0.741444\pi$
$740$	0	0
$741$	0.460709	0.0169246
$742$	0	0
$743$	36.4872	1.33858	0.669292	−	0.742999i	$-0.266597\pi$
$743$	36.4872	1.33858	0.669292	+	0.742999i	$0.266597\pi$
$744$	0	0
$745$	9.28432	0.340151
$746$	0	0
$747$	10.1590	0.371698
$748$	0	0
$749$	2.21939	0.0810949
$750$	0	0
$751$	38.5428	1.40645	0.703223	−	0.710970i	$-0.251744\pi$
$751$	38.5428	1.40645	0.703223	+	0.710970i	$0.251744\pi$
$752$	0	0
$753$	13.7596	0.501427
$754$	0	0
$755$	−18.6364	−0.678248
$756$	0	0
$757$	30.5679	1.11101	0.555505	−	0.831513i	$-0.312525\pi$
$757$	30.5679	1.11101	0.555505	+	0.831513i	$0.312525\pi$
$758$	0	0
$759$	5.51763	0.200277
$760$	0	0
$761$	−13.3966	−0.485627	−0.242813	−	0.970073i	$-0.578070\pi$
$761$	−13.3966	−0.485627	−0.242813	+	0.970073i	$0.578070\pi$
$762$	0	0
$763$	−9.80635	−0.355014
$764$	0	0
$765$	10.2826	0.371768
$766$	0	0
$767$	7.42410	0.268069
$768$	0	0
$769$	−8.23363	−0.296912	−0.148456	−	0.988919i	$-0.547430\pi$
$769$	−8.23363	−0.296912	−0.148456	+	0.988919i	$0.547430\pi$
$770$	0	0
$771$	18.0156	0.648815
$772$	0	0
$773$	26.6397	0.958164	0.479082	−	0.877770i	$-0.340970\pi$
$773$	26.6397	0.958164	0.479082	+	0.877770i	$0.340970\pi$
$774$	0	0
$775$	10.9696	0.394038
$776$	0	0
$777$	1.45510	0.0522014
$778$	0	0
$779$	6.37371	0.228362
$780$	0	0
$781$	−52.7930	−1.88908
$782$	0	0
$783$	−1.00000	−0.0357371
$784$	0	0
$785$	−3.89481	−0.139012
$786$	0	0
$787$	18.7171	0.667193	0.333597	−	0.942716i	$-0.391738\pi$
$787$	18.7171	0.667193	0.333597	+	0.942716i	$0.391738\pi$
$788$	0	0
$789$	−8.65502	−0.308127
$790$	0	0
$791$	2.53982	0.0903056
$792$	0	0
$793$	−2.92089	−0.103724
$794$	0	0
$795$	4.13901	0.146795
$796$	0	0
$797$	−37.8850	−1.34196	−0.670978	−	0.741477i	$-0.734125\pi$
$797$	−37.8850	−1.34196	−0.670978	+	0.741477i	$0.734125\pi$
$798$	0	0
$799$	31.1929	1.10353
$800$	0	0
$801$	14.8738	0.525539
$802$	0	0
$803$	−17.9553	−0.633627
$804$	0	0
$805$	−0.941403	−0.0331801
$806$	0	0
$807$	−12.1540	−0.427841
$808$	0	0
$809$	−6.20298	−0.218085	−0.109043	−	0.994037i	$-0.534778\pi$
$809$	−6.20298	−0.218085	−0.109043	+	0.994037i	$0.534778\pi$
$810$	0	0
$811$	−50.6931	−1.78008	−0.890038	−	0.455885i	$-0.849323\pi$
$811$	−50.6931	−1.78008	−0.890038	+	0.455885i	$0.849323\pi$
$812$	0	0
$813$	9.41238	0.330106
$814$	0	0
$815$	−37.5939	−1.31686
$816$	0	0
$817$	6.80462	0.238064
$818$	0	0
$819$	−0.306149	−0.0106977
$820$	0	0
$821$	−23.1105	−0.806561	−0.403280	−	0.915076i	$-0.632130\pi$
$821$	−23.1105	−0.806561	−0.403280	+	0.915076i	$0.632130\pi$
$822$	0	0
$823$	−9.83204	−0.342723	−0.171362	−	0.985208i	$-0.554817\pi$
$823$	−9.83204	−0.342723	−0.171362	+	0.985208i	$0.554817\pi$
$824$	0	0
$825$	−12.9271	−0.450065
$826$	0	0
$827$	−13.3080	−0.462766	−0.231383	−	0.972863i	$-0.574325\pi$
$827$	−13.3080	−0.462766	−0.231383	+	0.972863i	$0.574325\pi$
$828$	0	0
$829$	−17.0463	−0.592043	−0.296021	−	0.955181i	$-0.595660\pi$
$829$	−17.0463	−0.592043	−0.296021	+	0.955181i	$0.595660\pi$
$830$	0	0
$831$	−22.4479	−0.778711
$832$	0	0
$833$	−42.0526	−1.45704
$834$	0	0
$835$	−3.95552	−0.136886
$836$	0	0
$837$	−4.68209	−0.161837
$838$	0	0
$839$	3.46221	0.119529	0.0597644	−	0.998213i	$-0.480965\pi$
$839$	3.46221	0.119529	0.0597644	+	0.998213i	$0.480965\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−4.81073	−0.165690
$844$	0	0
$845$	−20.7328	−0.713231
$846$	0	0
$847$	−11.2295	−0.385851
$848$	0	0
$849$	15.2976	0.525013
$850$	0	0
$851$	−2.51955	−0.0863690
$852$	0	0
$853$	−23.8575	−0.816865	−0.408432	−	0.912789i	$-0.633924\pi$
$853$	−23.8575	−0.816865	−0.408432	+	0.912789i	$0.633924\pi$
$854$	0	0
$855$	1.41667	0.0484492
$856$	0	0
$857$	13.1060	0.447691	0.223846	−	0.974625i	$-0.428139\pi$
$857$	13.1060	0.447691	0.223846	+	0.974625i	$0.428139\pi$
$858$	0	0
$859$	20.8685	0.712024	0.356012	−	0.934481i	$-0.384136\pi$
$859$	20.8685	0.712024	0.356012	+	0.934481i	$0.384136\pi$
$860$	0	0
$861$	−4.23544	−0.144343
$862$	0	0
$863$	54.2789	1.84767	0.923837	−	0.382785i	$-0.125035\pi$
$863$	54.2789	1.84767	0.923837	+	0.382785i	$0.125035\pi$
$864$	0	0
$865$	−24.4937	−0.832810
$866$	0	0
$867$	22.7919	0.774053
$868$	0	0
$869$	1.92008	0.0651341
$870$	0	0
$871$	−5.95332	−0.201720
$872$	0	0
$873$	2.03129	0.0687488
$874$	0	0
$875$	6.91261	0.233689
$876$	0	0
$877$	−42.0972	−1.42152	−0.710761	−	0.703433i	$-0.751649\pi$
$877$	−42.0972	−1.42152	−0.710761	+	0.703433i	$0.751649\pi$
$878$	0	0
$879$	28.3226	0.955299
$880$	0	0
$881$	10.7828	0.363283	0.181642	−	0.983365i	$-0.441859\pi$
$881$	10.7828	0.363283	0.181642	+	0.983365i	$0.441859\pi$
$882$	0	0
$883$	−48.5709	−1.63454	−0.817270	−	0.576255i	$-0.804514\pi$
$883$	−48.5709	−1.63454	−0.817270	+	0.576255i	$0.804514\pi$
$884$	0	0
$885$	22.8290	0.767388
$886$	0	0
$887$	−30.2426	−1.01545	−0.507723	−	0.861520i	$-0.669513\pi$
$887$	−30.2426	−1.01545	−0.507723	+	0.861520i	$0.669513\pi$
$888$	0	0
$889$	7.42667	0.249083
$890$	0	0
$891$	5.51763	0.184848
$892$	0	0
$893$	4.29757	0.143813
$894$	0	0
$895$	25.9757	0.868271
$896$	0	0
$897$	0.530107	0.0176998
$898$	0	0
$899$	4.68209	0.156157
$900$	0	0
$901$	16.0172	0.533611
$902$	0	0
$903$	−4.52179	−0.150476
$904$	0	0
$905$	−19.7671	−0.657081
$906$	0	0
$907$	−3.52279	−0.116972	−0.0584862	−	0.998288i	$-0.518627\pi$
$907$	−3.52279	−0.116972	−0.0584862	+	0.998288i	$0.518627\pi$
$908$	0	0
$909$	−7.88165	−0.261418
$910$	0	0
$911$	34.7657	1.15184	0.575919	−	0.817507i	$-0.304644\pi$
$911$	34.7657	1.15184	0.575919	+	0.817507i	$0.304644\pi$
$912$	0	0
$913$	56.0535	1.85510
$914$	0	0
$915$	−8.98169	−0.296925
$916$	0	0
$917$	6.97767	0.230423
$918$	0	0
$919$	36.2219	1.19485	0.597425	−	0.801925i	$-0.296191\pi$
$919$	36.2219	1.19485	0.597425	+	0.801925i	$0.296191\pi$
$920$	0	0
$921$	9.67118	0.318676
$922$	0	0
$923$	−5.07209	−0.166950
$924$	0	0
$925$	5.90299	0.194089
$926$	0	0
$927$	3.55532	0.116772
$928$	0	0
$929$	−1.45222	−0.0476458	−0.0238229	−	0.999716i	$-0.507584\pi$
$929$	−1.45222	−0.0476458	−0.0238229	+	0.999716i	$0.507584\pi$
$930$	0	0
$931$	−5.79374	−0.189882
$932$	0	0
$933$	−25.4302	−0.832548
$934$	0	0
$935$	56.7356	1.85545
$936$	0	0
$937$	21.0101	0.686372	0.343186	−	0.939268i	$-0.388494\pi$
$937$	21.0101	0.686372	0.343186	+	0.939268i	$0.388494\pi$
$938$	0	0
$939$	−0.926444	−0.0302334
$940$	0	0
$941$	−34.7983	−1.13439	−0.567196	−	0.823583i	$-0.691972\pi$
$941$	−34.7983	−1.13439	−0.567196	+	0.823583i	$0.691972\pi$
$942$	0	0
$943$	7.33380	0.238821
$944$	0	0
$945$	−0.941403	−0.0306239
$946$	0	0
$947$	−49.7719	−1.61737	−0.808685	−	0.588242i	$-0.799820\pi$
$947$	−49.7719	−1.61737	−0.808685	+	0.588242i	$0.799820\pi$
$948$	0	0
$949$	−1.72505	−0.0559976
$950$	0	0
$951$	−5.52724	−0.179233
$952$	0	0
$953$	−7.49439	−0.242767	−0.121384	−	0.992606i	$-0.538733\pi$
$953$	−7.49439	−0.242767	−0.121384	+	0.992606i	$0.538733\pi$
$954$	0	0
$955$	3.46394	0.112090
$956$	0	0
$957$	−5.51763	−0.178360
$958$	0	0
$959$	−0.868469	−0.0280443
$960$	0	0
$961$	−9.07800	−0.292839
$962$	0	0
$963$	−3.84295	−0.123837
$964$	0	0
$965$	−21.2180	−0.683033
$966$	0	0
$967$	28.7907	0.925848	0.462924	−	0.886398i	$-0.346800\pi$
$967$	28.7907	0.925848	0.462924	+	0.886398i	$0.346800\pi$
$968$	0	0
$969$	5.48227	0.176116
$970$	0	0
$971$	−28.6646	−0.919892	−0.459946	−	0.887947i	$-0.652131\pi$
$971$	−28.6646	−0.919892	−0.459946	+	0.887947i	$0.652131\pi$
$972$	0	0
$973$	10.6728	0.342153
$974$	0	0
$975$	−1.24197	−0.0397750
$976$	0	0
$977$	13.1205	0.419761	0.209881	−	0.977727i	$-0.432692\pi$
$977$	13.1205	0.419761	0.209881	+	0.977727i	$0.432692\pi$
$978$	0	0
$979$	82.0680	2.62291
$980$	0	0
$981$	16.9800	0.542130
$982$	0	0
$983$	13.4989	0.430547	0.215273	−	0.976554i	$-0.430936\pi$
$983$	13.4989	0.430547	0.215273	+	0.976554i	$0.430936\pi$
$984$	0	0
$985$	18.3601	0.585001
$986$	0	0
$987$	−2.85581	−0.0909014
$988$	0	0
$989$	7.82962	0.248967
$990$	0	0
$991$	−26.8488	−0.852881	−0.426440	−	0.904516i	$-0.640233\pi$
$991$	−26.8488	−0.852881	−0.426440	+	0.904516i	$0.640233\pi$
$992$	0	0
$993$	7.45576	0.236601
$994$	0	0
$995$	−15.1879	−0.481488
$996$	0	0
$997$	26.4714	0.838359	0.419179	−	0.907903i	$-0.362318\pi$
$997$	26.4714	0.838359	0.419179	+	0.907903i	$0.362318\pi$
$998$	0	0
$999$	−2.51955	−0.0797150

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.j.1.11	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.j.1.11	✓	16	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.63007 q^{5} -0.577524 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.63007 q^{5} -0.577524 q^{7} +1.00000 q^{9} +5.51763 q^{11} +0.530107 q^{13} +1.63007 q^{15} +6.30808 q^{17} +0.869087 q^{19} -0.577524 q^{21} +1.00000 q^{23} -2.34288 q^{25} +1.00000 q^{27} -1.00000 q^{29} -4.68209 q^{31} +5.51763 q^{33} -0.941403 q^{35} -2.51955 q^{37} +0.530107 q^{39} +7.33380 q^{41} +7.82962 q^{43} +1.63007 q^{45} +4.94492 q^{47} -6.66647 q^{49} +6.30808 q^{51} +2.53916 q^{53} +8.99412 q^{55} +0.869087 q^{57} +14.0049 q^{59} -5.51000 q^{61} -0.577524 q^{63} +0.864111 q^{65} -11.2304 q^{67} +1.00000 q^{69} -9.56805 q^{71} -3.25416 q^{73} -2.34288 q^{75} -3.18656 q^{77} +0.347989 q^{79} +1.00000 q^{81} +10.1590 q^{83} +10.2826 q^{85} -1.00000 q^{87} +14.8738 q^{89} -0.306149 q^{91} -4.68209 q^{93} +1.41667 q^{95} +2.03129 q^{97} +5.51763 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{3} + 3 q^{5} + 4 q^{7} + 16 q^{9}+O(q^{10}) \) 16 * q + 16 * q^3 + 3 * q^5 + 4 * q^7 + 16 * q^9	\( 16 q + 16 q^{3} + 3 q^{5} + 4 q^{7} + 16 q^{9} + 5 q^{11} + 6 q^{13} + 3 q^{15} + 3 q^{17} + 11 q^{19} + 4 q^{21} + 16 q^{23} + 27 q^{25} + 16 q^{27} - 16 q^{29} + 14 q^{31} + 5 q^{33} + 11 q^{35} + 4 q^{37} + 6 q^{39} + 11 q^{41} + 23 q^{43} + 3 q^{45} - 2 q^{47} + 34 q^{49} + 3 q^{51} + 19 q^{53} + 31 q^{55} + 11 q^{57} + 32 q^{59} + 19 q^{61} + 4 q^{63} + 6 q^{65} + 33 q^{67} + 16 q^{69} - 5 q^{71} + 23 q^{73} + 27 q^{75} + 42 q^{77} + 24 q^{79} + 16 q^{81} + 7 q^{83} - 16 q^{87} - 2 q^{89} + 25 q^{91} + 14 q^{93} + 7 q^{95} + 33 q^{97} + 5 q^{99}+O(q^{100}) \) 16 * q + 16 * q^3 + 3 * q^5 + 4 * q^7 + 16 * q^9 + 5 * q^11 + 6 * q^13 + 3 * q^15 + 3 * q^17 + 11 * q^19 + 4 * q^21 + 16 * q^23 + 27 * q^25 + 16 * q^27 - 16 * q^29 + 14 * q^31 + 5 * q^33 + 11 * q^35 + 4 * q^37 + 6 * q^39 + 11 * q^41 + 23 * q^43 + 3 * q^45 - 2 * q^47 + 34 * q^49 + 3 * q^51 + 19 * q^53 + 31 * q^55 + 11 * q^57 + 32 * q^59 + 19 * q^61 + 4 * q^63 + 6 * q^65 + 33 * q^67 + 16 * q^69 - 5 * q^71 + 23 * q^73 + 27 * q^75 + 42 * q^77 + 24 * q^79 + 16 * q^81 + 7 * q^83 - 16 * q^87 - 2 * q^89 + 25 * q^91 + 14 * q^93 + 7 * q^95 + 33 * q^97 + 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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