LMFDB - Embedded newform 8004.2.a.f.1.9

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:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 17x^{7} + 4x^{6} + 75x^{5} + x^{4} - 118x^{3} - 26x^{2} + 60x + 24 $ x^9 - x^8 - 17x^7 + 4x^6 + 75x^5 + x^4 - 118x^3 - 26x^2 + 60x + 24
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$-2.77196$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +2.77196 q^{5} +1.06236 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +2.77196 q^{5} +1.06236 q^{7} +1.00000 q^{9} -4.79059 q^{11} +1.09288 q^{13} +2.77196 q^{15} -4.90857 q^{17} -5.29888 q^{19} +1.06236 q^{21} -1.00000 q^{23} +2.68377 q^{25} +1.00000 q^{27} -1.00000 q^{29} +0.542549 q^{31} -4.79059 q^{33} +2.94482 q^{35} -9.49623 q^{37} +1.09288 q^{39} -0.882388 q^{41} -0.495447 q^{43} +2.77196 q^{45} -5.71872 q^{47} -5.87139 q^{49} -4.90857 q^{51} -0.364263 q^{53} -13.2793 q^{55} -5.29888 q^{57} -6.03067 q^{59} +3.37600 q^{61} +1.06236 q^{63} +3.02941 q^{65} -4.59575 q^{67} -1.00000 q^{69} +13.3019 q^{71} -14.8315 q^{73} +2.68377 q^{75} -5.08933 q^{77} +16.4055 q^{79} +1.00000 q^{81} +13.0805 q^{83} -13.6064 q^{85} -1.00000 q^{87} -7.11644 q^{89} +1.16103 q^{91} +0.542549 q^{93} -14.6883 q^{95} +2.03992 q^{97} -4.79059 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 9 q^{3} - q^{5} - 5 q^{7} + 9 q^{9}+O(q^{10}) $ 9 * q + 9 * q^3 - q^5 - 5 * q^7 + 9 * q^9	$ 9 q + 9 q^{3} - q^{5} - 5 q^{7} + 9 q^{9} - 8 q^{11} - q^{13} - q^{15} - 2 q^{17} - 11 q^{19} - 5 q^{21} - 9 q^{23} - 10 q^{25} + 9 q^{27} - 9 q^{29} - 8 q^{33} + q^{35} - 2 q^{37} - q^{39} - 3 q^{41} - 19 q^{43} - q^{45} - 3 q^{47} - 6 q^{49} - 2 q^{51} - 9 q^{53} - 7 q^{55} - 11 q^{57} - 2 q^{59} - 25 q^{61} - 5 q^{63} - 12 q^{65} - 20 q^{67} - 9 q^{69} + 9 q^{71} - 11 q^{73} - 10 q^{75} - 19 q^{77} + 4 q^{79} + 9 q^{81} - 9 q^{83} - 50 q^{85} - 9 q^{87} - 29 q^{89} - 38 q^{91} + 23 q^{95} - 43 q^{97} - 8 q^{99}+O(q^{100}) $ 9 * q + 9 * q^3 - q^5 - 5 * q^7 + 9 * q^9 - 8 * q^11 - q^13 - q^15 - 2 * q^17 - 11 * q^19 - 5 * q^21 - 9 * q^23 - 10 * q^25 + 9 * q^27 - 9 * q^29 - 8 * q^33 + q^35 - 2 * q^37 - q^39 - 3 * q^41 - 19 * q^43 - q^45 - 3 * q^47 - 6 * q^49 - 2 * q^51 - 9 * q^53 - 7 * q^55 - 11 * q^57 - 2 * q^59 - 25 * q^61 - 5 * q^63 - 12 * q^65 - 20 * q^67 - 9 * q^69 + 9 * q^71 - 11 * q^73 - 10 * q^75 - 19 * q^77 + 4 * q^79 + 9 * q^81 - 9 * q^83 - 50 * q^85 - 9 * q^87 - 29 * q^89 - 38 * q^91 + 23 * q^95 - 43 * q^97 - 8 * 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
$5$	2.77196	1.23966	0.619829	−	0.784737i	$-0.287202\pi$
$5$	2.77196	1.23966	0.619829	+	0.784737i	$0.287202\pi$
$6$	0	0
$7$	1.06236	0.401534	0.200767	−	0.979639i	$-0.435657\pi$
$7$	1.06236	0.401534	0.200767	+	0.979639i	$0.435657\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.79059	−1.44442	−0.722209	−	0.691675i	$-0.756873\pi$
$11$	−4.79059	−1.44442	−0.722209	+	0.691675i	$0.756873\pi$
$12$	0	0
$13$	1.09288	0.303109	0.151555	−	0.988449i	$-0.451572\pi$
$13$	1.09288	0.303109	0.151555	+	0.988449i	$0.451572\pi$
$14$	0	0
$15$	2.77196	0.715717
$16$	0	0
$17$	−4.90857	−1.19050	−0.595251	−	0.803540i	$-0.702947\pi$
$17$	−4.90857	−1.19050	−0.595251	+	0.803540i	$0.702947\pi$
$18$	0	0
$19$	−5.29888	−1.21565	−0.607823	−	0.794073i	$-0.707957\pi$
$19$	−5.29888	−1.21565	−0.607823	+	0.794073i	$0.707957\pi$
$20$	0	0
$21$	1.06236	0.231826
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	2.68377	0.536753
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	0.542549	0.0974448	0.0487224	−	0.998812i	$-0.484485\pi$
$31$	0.542549	0.0974448	0.0487224	+	0.998812i	$0.484485\pi$
$32$	0	0
$33$	−4.79059	−0.833935
$34$	0	0
$35$	2.94482	0.497765
$36$	0	0
$37$	−9.49623	−1.56117	−0.780586	−	0.625049i	$-0.785079\pi$
$37$	−9.49623	−1.56117	−0.780586	+	0.625049i	$0.785079\pi$
$38$	0	0
$39$	1.09288	0.175000
$40$	0	0
$41$	−0.882388	−0.137806	−0.0689029	−	0.997623i	$-0.521950\pi$
$41$	−0.882388	−0.137806	−0.0689029	+	0.997623i	$0.521950\pi$
$42$	0	0
$43$	−0.495447	−0.0755549	−0.0377775	−	0.999286i	$-0.512028\pi$
$43$	−0.495447	−0.0755549	−0.0377775	+	0.999286i	$0.512028\pi$
$44$	0	0
$45$	2.77196	0.413219
$46$	0	0
$47$	−5.71872	−0.834161	−0.417081	−	0.908869i	$-0.636947\pi$
$47$	−5.71872	−0.834161	−0.417081	+	0.908869i	$0.636947\pi$
$48$	0	0
$49$	−5.87139	−0.838770
$50$	0	0
$51$	−4.90857	−0.687337
$52$	0	0
$53$	−0.364263	−0.0500354	−0.0250177	−	0.999687i	$-0.507964\pi$
$53$	−0.364263	−0.0500354	−0.0250177	+	0.999687i	$0.507964\pi$
$54$	0	0
$55$	−13.2793	−1.79058
$56$	0	0
$57$	−5.29888	−0.701853
$58$	0	0
$59$	−6.03067	−0.785126	−0.392563	−	0.919725i	$-0.628412\pi$
$59$	−6.03067	−0.785126	−0.392563	+	0.919725i	$0.628412\pi$
$60$	0	0
$61$	3.37600	0.432253	0.216126	−	0.976365i	$-0.430658\pi$
$61$	3.37600	0.432253	0.216126	+	0.976365i	$0.430658\pi$
$62$	0	0
$63$	1.06236	0.133845
$64$	0	0
$65$	3.02941	0.375752
$66$	0	0
$67$	−4.59575	−0.561460	−0.280730	−	0.959787i	$-0.590577\pi$
$67$	−4.59575	−0.561460	−0.280730	+	0.959787i	$0.590577\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	13.3019	1.57865	0.789323	−	0.613978i	$-0.210432\pi$
$71$	13.3019	1.57865	0.789323	+	0.613978i	$0.210432\pi$
$72$	0	0
$73$	−14.8315	−1.73590	−0.867949	−	0.496654i	$-0.834562\pi$
$73$	−14.8315	−1.73590	−0.867949	+	0.496654i	$0.834562\pi$
$74$	0	0
$75$	2.68377	0.309895
$76$	0	0
$77$	−5.08933	−0.579983
$78$	0	0
$79$	16.4055	1.84576	0.922882	−	0.385082i	$-0.125827\pi$
$79$	16.4055	1.84576	0.922882	+	0.385082i	$0.125827\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	13.0805	1.43577	0.717886	−	0.696161i	$-0.245110\pi$
$83$	13.0805	1.43577	0.717886	+	0.696161i	$0.245110\pi$
$84$	0	0
$85$	−13.6064	−1.47582
$86$	0	0
$87$	−1.00000	−0.107211
$88$	0	0
$89$	−7.11644	−0.754342	−0.377171	−	0.926144i	$-0.623103\pi$
$89$	−7.11644	−0.754342	−0.377171	+	0.926144i	$0.623103\pi$
$90$	0	0
$91$	1.16103	0.121709
$92$	0	0
$93$	0.542549	0.0562598
$94$	0	0
$95$	−14.6883	−1.50699
$96$	0	0
$97$	2.03992	0.207122	0.103561	−	0.994623i	$-0.466976\pi$
$97$	2.03992	0.207122	0.103561	+	0.994623i	$0.466976\pi$
$98$	0	0
$99$	−4.79059	−0.481473
$100$	0	0
$101$	−9.54324	−0.949588	−0.474794	−	0.880097i	$-0.657477\pi$
$101$	−9.54324	−0.949588	−0.474794	+	0.880097i	$0.657477\pi$
$102$	0	0
$103$	−6.47638	−0.638137	−0.319068	−	0.947732i	$-0.603370\pi$
$103$	−6.47638	−0.638137	−0.319068	+	0.947732i	$0.603370\pi$
$104$	0	0
$105$	2.94482	0.287385
$106$	0	0
$107$	4.96297	0.479788	0.239894	−	0.970799i	$-0.422887\pi$
$107$	4.96297	0.479788	0.239894	+	0.970799i	$0.422887\pi$
$108$	0	0
$109$	−19.0139	−1.82120	−0.910601	−	0.413286i	$-0.864381\pi$
$109$	−19.0139	−1.82120	−0.910601	+	0.413286i	$0.864381\pi$
$110$	0	0
$111$	−9.49623	−0.901343
$112$	0	0
$113$	19.6197	1.84567	0.922833	−	0.385199i	$-0.125867\pi$
$113$	19.6197	1.84567	0.922833	+	0.385199i	$0.125867\pi$
$114$	0	0
$115$	−2.77196	−0.258487
$116$	0	0
$117$	1.09288	0.101036
$118$	0	0
$119$	−5.21466	−0.478027
$120$	0	0
$121$	11.9498	1.08634
$122$	0	0
$123$	−0.882388	−0.0795623
$124$	0	0
$125$	−6.42051	−0.574268
$126$	0	0
$127$	−14.6045	−1.29594	−0.647968	−	0.761668i	$-0.724381\pi$
$127$	−14.6045	−1.29594	−0.647968	+	0.761668i	$0.724381\pi$
$128$	0	0
$129$	−0.495447	−0.0436216
$130$	0	0
$131$	4.09151	0.357477	0.178738	−	0.983897i	$-0.442798\pi$
$131$	4.09151	0.357477	0.178738	+	0.983897i	$0.442798\pi$
$132$	0	0
$133$	−5.62931	−0.488123
$134$	0	0
$135$	2.77196	0.238572
$136$	0	0
$137$	−13.0627	−1.11602	−0.558010	−	0.829835i	$-0.688435\pi$
$137$	−13.0627	−1.11602	−0.558010	+	0.829835i	$0.688435\pi$
$138$	0	0
$139$	3.27110	0.277451	0.138726	−	0.990331i	$-0.455699\pi$
$139$	3.27110	0.277451	0.138726	+	0.990331i	$0.455699\pi$
$140$	0	0
$141$	−5.71872	−0.481603
$142$	0	0
$143$	−5.23552	−0.437817
$144$	0	0
$145$	−2.77196	−0.230199
$146$	0	0
$147$	−5.87139	−0.484264
$148$	0	0
$149$	−9.36059	−0.766850	−0.383425	−	0.923572i	$-0.625255\pi$
$149$	−9.36059	−0.766850	−0.383425	+	0.923572i	$0.625255\pi$
$150$	0	0
$151$	−10.4467	−0.850143	−0.425072	−	0.905160i	$-0.639751\pi$
$151$	−10.4467	−0.850143	−0.425072	+	0.905160i	$0.639751\pi$
$152$	0	0
$153$	−4.90857	−0.396834
$154$	0	0
$155$	1.50393	0.120798
$156$	0	0
$157$	9.48672	0.757123	0.378561	−	0.925576i	$-0.376419\pi$
$157$	9.48672	0.757123	0.378561	+	0.925576i	$0.376419\pi$
$158$	0	0
$159$	−0.364263	−0.0288879
$160$	0	0
$161$	−1.06236	−0.0837256
$162$	0	0
$163$	6.81600	0.533870	0.266935	−	0.963714i	$-0.413989\pi$
$163$	6.81600	0.533870	0.266935	+	0.963714i	$0.413989\pi$
$164$	0	0
$165$	−13.2793	−1.03379
$166$	0	0
$167$	10.6820	0.826596	0.413298	−	0.910596i	$-0.364377\pi$
$167$	10.6820	0.826596	0.413298	+	0.910596i	$0.364377\pi$
$168$	0	0
$169$	−11.8056	−0.908125
$170$	0	0
$171$	−5.29888	−0.405215
$172$	0	0
$173$	−0.0197633	−0.00150257	−0.000751287	−	1.00000i	$-0.500239\pi$
$173$	−0.0197633	−0.00150257	−0.000751287		1.00000i	$0.500239\pi$
$174$	0	0
$175$	2.85112	0.215525
$176$	0	0
$177$	−6.03067	−0.453293
$178$	0	0
$179$	−1.79432	−0.134114	−0.0670568	−	0.997749i	$-0.521361\pi$
$179$	−1.79432	−0.134114	−0.0670568	+	0.997749i	$0.521361\pi$
$180$	0	0
$181$	−20.4591	−1.52071	−0.760356	−	0.649507i	$-0.774975\pi$
$181$	−20.4591	−1.52071	−0.760356	+	0.649507i	$0.774975\pi$
$182$	0	0
$183$	3.37600	0.249561
$184$	0	0
$185$	−26.3232	−1.93532
$186$	0	0
$187$	23.5149	1.71958
$188$	0	0
$189$	1.06236	0.0772753
$190$	0	0
$191$	−3.09084	−0.223645	−0.111823	−	0.993728i	$-0.535669\pi$
$191$	−3.09084	−0.223645	−0.111823	+	0.993728i	$0.535669\pi$
$192$	0	0
$193$	−2.36627	−0.170328	−0.0851639	−	0.996367i	$-0.527141\pi$
$193$	−2.36627	−0.170328	−0.0851639	+	0.996367i	$0.527141\pi$
$194$	0	0
$195$	3.02941	0.216941
$196$	0	0
$197$	3.27592	0.233400	0.116700	−	0.993167i	$-0.462768\pi$
$197$	3.27592	0.233400	0.116700	+	0.993167i	$0.462768\pi$
$198$	0	0
$199$	−5.38451	−0.381698	−0.190849	−	0.981619i	$-0.561124\pi$
$199$	−5.38451	−0.381698	−0.190849	+	0.981619i	$0.561124\pi$
$200$	0	0
$201$	−4.59575	−0.324159
$202$	0	0
$203$	−1.06236	−0.0745630
$204$	0	0
$205$	−2.44595	−0.170832
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	25.3848	1.75590
$210$	0	0
$211$	13.7011	0.943225	0.471613	−	0.881806i	$-0.343672\pi$
$211$	13.7011	0.943225	0.471613	+	0.881806i	$0.343672\pi$
$212$	0	0
$213$	13.3019	0.911432
$214$	0	0
$215$	−1.37336	−0.0936623
$216$	0	0
$217$	0.576382	0.0391274
$218$	0	0
$219$	−14.8315	−1.00222
$220$	0	0
$221$	−5.36446	−0.360852
$222$	0	0
$223$	20.0766	1.34443	0.672215	−	0.740356i	$-0.265343\pi$
$223$	20.0766	1.34443	0.672215	+	0.740356i	$0.265343\pi$
$224$	0	0
$225$	2.68377	0.178918
$226$	0	0
$227$	2.37670	0.157747	0.0788736	−	0.996885i	$-0.474868\pi$
$227$	2.37670	0.157747	0.0788736	+	0.996885i	$0.474868\pi$
$228$	0	0
$229$	10.8234	0.715232	0.357616	−	0.933869i	$-0.383590\pi$
$229$	10.8234	0.715232	0.357616	+	0.933869i	$0.383590\pi$
$230$	0	0
$231$	−5.08933	−0.334853
$232$	0	0
$233$	15.3739	1.00718	0.503590	−	0.863943i	$-0.332012\pi$
$233$	15.3739	1.00718	0.503590	+	0.863943i	$0.332012\pi$
$234$	0	0
$235$	−15.8521	−1.03408
$236$	0	0
$237$	16.4055	1.06565
$238$	0	0
$239$	19.0941	1.23509	0.617547	−	0.786534i	$-0.288127\pi$
$239$	19.0941	1.23509	0.617547	+	0.786534i	$0.288127\pi$
$240$	0	0
$241$	10.3969	0.669722	0.334861	−	0.942268i	$-0.391311\pi$
$241$	10.3969	0.669722	0.334861	+	0.942268i	$0.391311\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−16.2753	−1.03979
$246$	0	0
$247$	−5.79102	−0.368474
$248$	0	0
$249$	13.0805	0.828943
$250$	0	0
$251$	11.3982	0.719450	0.359725	−	0.933058i	$-0.382870\pi$
$251$	11.3982	0.719450	0.359725	+	0.933058i	$0.382870\pi$
$252$	0	0
$253$	4.79059	0.301182
$254$	0	0
$255$	−13.6064	−0.852063
$256$	0	0
$257$	13.9938	0.872911	0.436456	−	0.899726i	$-0.356234\pi$
$257$	13.9938	0.872911	0.436456	+	0.899726i	$0.356234\pi$
$258$	0	0
$259$	−10.0884	−0.626863
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	0	0
$263$	0.164010	0.0101133	0.00505665	−	0.999987i	$-0.498390\pi$
$263$	0.164010	0.0101133	0.00505665	+	0.999987i	$0.498390\pi$
$264$	0	0
$265$	−1.00972	−0.0620268
$266$	0	0
$267$	−7.11644	−0.435519
$268$	0	0
$269$	−26.8224	−1.63539	−0.817697	−	0.575649i	$-0.804749\pi$
$269$	−26.8224	−1.63539	−0.817697	+	0.575649i	$0.804749\pi$
$270$	0	0
$271$	8.32336	0.505608	0.252804	−	0.967518i	$-0.418647\pi$
$271$	8.32336	0.505608	0.252804	+	0.967518i	$0.418647\pi$
$272$	0	0
$273$	1.16103	0.0702686
$274$	0	0
$275$	−12.8568	−0.775296
$276$	0	0
$277$	20.1954	1.21342	0.606711	−	0.794922i	$-0.292488\pi$
$277$	20.1954	1.21342	0.606711	+	0.794922i	$0.292488\pi$
$278$	0	0
$279$	0.542549	0.0324816
$280$	0	0
$281$	−7.62809	−0.455054	−0.227527	−	0.973772i	$-0.573064\pi$
$281$	−7.62809	−0.455054	−0.227527	+	0.973772i	$0.573064\pi$
$282$	0	0
$283$	20.8851	1.24149	0.620744	−	0.784013i	$-0.286831\pi$
$283$	20.8851	1.24149	0.620744	+	0.784013i	$0.286831\pi$
$284$	0	0
$285$	−14.6883	−0.870059
$286$	0	0
$287$	−0.937413	−0.0553337
$288$	0	0
$289$	7.09402	0.417295
$290$	0	0
$291$	2.03992	0.119582
$292$	0	0
$293$	5.22302	0.305132	0.152566	−	0.988293i	$-0.451246\pi$
$293$	5.22302	0.305132	0.152566	+	0.988293i	$0.451246\pi$
$294$	0	0
$295$	−16.7168	−0.973288
$296$	0	0
$297$	−4.79059	−0.277978
$298$	0	0
$299$	−1.09288	−0.0632027
$300$	0	0
$301$	−0.526342	−0.0303379
$302$	0	0
$303$	−9.54324	−0.548245
$304$	0	0
$305$	9.35814	0.535846
$306$	0	0
$307$	−0.551792	−0.0314924	−0.0157462	−	0.999876i	$-0.505012\pi$
$307$	−0.551792	−0.0314924	−0.0157462	+	0.999876i	$0.505012\pi$
$308$	0	0
$309$	−6.47638	−0.368428
$310$	0	0
$311$	10.5599	0.598796	0.299398	−	0.954128i	$-0.403214\pi$
$311$	10.5599	0.598796	0.299398	+	0.954128i	$0.403214\pi$
$312$	0	0
$313$	0.909310	0.0513972	0.0256986	−	0.999670i	$-0.491819\pi$
$313$	0.909310	0.0513972	0.0256986	+	0.999670i	$0.491819\pi$
$314$	0	0
$315$	2.94482	0.165922
$316$	0	0
$317$	−14.8865	−0.836110	−0.418055	−	0.908422i	$-0.637288\pi$
$317$	−14.8865	−0.836110	−0.418055	+	0.908422i	$0.637288\pi$
$318$	0	0
$319$	4.79059	0.268222
$320$	0	0
$321$	4.96297	0.277006
$322$	0	0
$323$	26.0099	1.44723
$324$	0	0
$325$	2.93302	0.162695
$326$	0	0
$327$	−19.0139	−1.05147
$328$	0	0
$329$	−6.07534	−0.334944
$330$	0	0
$331$	25.0196	1.37520	0.687600	−	0.726090i	$-0.258664\pi$
$331$	25.0196	1.37520	0.687600	+	0.726090i	$0.258664\pi$
$332$	0	0
$333$	−9.49623	−0.520390
$334$	0	0
$335$	−12.7392	−0.696018
$336$	0	0
$337$	−31.7815	−1.73125	−0.865623	−	0.500696i	$-0.833078\pi$
$337$	−31.7815	−1.73125	−0.865623	+	0.500696i	$0.833078\pi$
$338$	0	0
$339$	19.6197	1.06560
$340$	0	0
$341$	−2.59913	−0.140751
$342$	0	0
$343$	−13.6740	−0.738329
$344$	0	0
$345$	−2.77196	−0.149237
$346$	0	0
$347$	−19.2450	−1.03312	−0.516562	−	0.856250i	$-0.672788\pi$
$347$	−19.2450	−1.03312	−0.516562	+	0.856250i	$0.672788\pi$
$348$	0	0
$349$	−19.7314	−1.05620	−0.528098	−	0.849183i	$-0.677095\pi$
$349$	−19.7314	−1.05620	−0.528098	+	0.849183i	$0.677095\pi$
$350$	0	0
$351$	1.09288	0.0583334
$352$	0	0
$353$	23.4511	1.24818	0.624088	−	0.781354i	$-0.285471\pi$
$353$	23.4511	1.24818	0.624088	+	0.781354i	$0.285471\pi$
$354$	0	0
$355$	36.8724	1.95698
$356$	0	0
$357$	−5.21466	−0.275989
$358$	0	0
$359$	−12.9640	−0.684216	−0.342108	−	0.939661i	$-0.611141\pi$
$359$	−12.9640	−0.684216	−0.342108	+	0.939661i	$0.611141\pi$
$360$	0	0
$361$	9.07810	0.477795
$362$	0	0
$363$	11.9498	0.627200
$364$	0	0
$365$	−41.1124	−2.15192
$366$	0	0
$367$	−5.20008	−0.271442	−0.135721	−	0.990747i	$-0.543335\pi$
$367$	−5.20008	−0.271442	−0.135721	+	0.990747i	$0.543335\pi$
$368$	0	0
$369$	−0.882388	−0.0459353
$370$	0	0
$371$	−0.386978	−0.0200909
$372$	0	0
$373$	−21.1196	−1.09353	−0.546765	−	0.837286i	$-0.684141\pi$
$373$	−21.1196	−1.09353	−0.546765	+	0.837286i	$0.684141\pi$
$374$	0	0
$375$	−6.42051	−0.331554
$376$	0	0
$377$	−1.09288	−0.0562860
$378$	0	0
$379$	6.34650	0.325998	0.162999	−	0.986626i	$-0.447883\pi$
$379$	6.34650	0.325998	0.162999	+	0.986626i	$0.447883\pi$
$380$	0	0
$381$	−14.6045	−0.748209
$382$	0	0
$383$	0.888978	0.0454247	0.0227123	−	0.999742i	$-0.492770\pi$
$383$	0.888978	0.0454247	0.0227123	+	0.999742i	$0.492770\pi$
$384$	0	0
$385$	−14.1074	−0.718981
$386$	0	0
$387$	−0.495447	−0.0251850
$388$	0	0
$389$	9.14473	0.463656	0.231828	−	0.972757i	$-0.425529\pi$
$389$	9.14473	0.463656	0.231828	+	0.972757i	$0.425529\pi$
$390$	0	0
$391$	4.90857	0.248237
$392$	0	0
$393$	4.09151	0.206389
$394$	0	0
$395$	45.4754	2.28812
$396$	0	0
$397$	17.9801	0.902397	0.451198	−	0.892424i	$-0.350997\pi$
$397$	17.9801	0.902397	0.451198	+	0.892424i	$0.350997\pi$
$398$	0	0
$399$	−5.62931	−0.281818
$400$	0	0
$401$	22.1179	1.10451	0.552257	−	0.833674i	$-0.313767\pi$
$401$	22.1179	1.10451	0.552257	+	0.833674i	$0.313767\pi$
$402$	0	0
$403$	0.592940	0.0295364
$404$	0	0
$405$	2.77196	0.137740
$406$	0	0
$407$	45.4926	2.25498
$408$	0	0
$409$	17.7145	0.875927	0.437963	−	0.898993i	$-0.355700\pi$
$409$	17.7145	0.875927	0.437963	+	0.898993i	$0.355700\pi$
$410$	0	0
$411$	−13.0627	−0.644334
$412$	0	0
$413$	−6.40673	−0.315255
$414$	0	0
$415$	36.2586	1.77987
$416$	0	0
$417$	3.27110	0.160187
$418$	0	0
$419$	−16.7068	−0.816180	−0.408090	−	0.912942i	$-0.633805\pi$
$419$	−16.7068	−0.816180	−0.408090	+	0.912942i	$0.633805\pi$
$420$	0	0
$421$	−2.13644	−0.104124	−0.0520619	−	0.998644i	$-0.516579\pi$
$421$	−2.13644	−0.104124	−0.0520619	+	0.998644i	$0.516579\pi$
$422$	0	0
$423$	−5.71872	−0.278054
$424$	0	0
$425$	−13.1734	−0.639006
$426$	0	0
$427$	3.58653	0.173564
$428$	0	0
$429$	−5.23552	−0.252774
$430$	0	0
$431$	5.27931	0.254296	0.127148	−	0.991884i	$-0.459418\pi$
$431$	5.27931	0.254296	0.127148	+	0.991884i	$0.459418\pi$
$432$	0	0
$433$	−25.4361	−1.22238	−0.611189	−	0.791484i	$-0.709309\pi$
$433$	−25.4361	−1.22238	−0.611189	+	0.791484i	$0.709309\pi$
$434$	0	0
$435$	−2.77196	−0.132905
$436$	0	0
$437$	5.29888	0.253480
$438$	0	0
$439$	6.82721	0.325845	0.162922	−	0.986639i	$-0.447908\pi$
$439$	6.82721	0.325845	0.162922	+	0.986639i	$0.447908\pi$
$440$	0	0
$441$	−5.87139	−0.279590
$442$	0	0
$443$	−9.13075	−0.433815	−0.216907	−	0.976192i	$-0.569597\pi$
$443$	−9.13075	−0.433815	−0.216907	+	0.976192i	$0.569597\pi$
$444$	0	0
$445$	−19.7265	−0.935126
$446$	0	0
$447$	−9.36059	−0.442741
$448$	0	0
$449$	−15.0537	−0.710430	−0.355215	−	0.934785i	$-0.615592\pi$
$449$	−15.0537	−0.710430	−0.355215	+	0.934785i	$0.615592\pi$
$450$	0	0
$451$	4.22716	0.199049
$452$	0	0
$453$	−10.4467	−0.490831
$454$	0	0
$455$	3.21832	0.150877
$456$	0	0
$457$	36.7379	1.71852	0.859262	−	0.511535i	$-0.170923\pi$
$457$	36.7379	1.71852	0.859262	+	0.511535i	$0.170923\pi$
$458$	0	0
$459$	−4.90857	−0.229112
$460$	0	0
$461$	32.4619	1.51190	0.755951	−	0.654629i	$-0.227175\pi$
$461$	32.4619	1.51190	0.755951	+	0.654629i	$0.227175\pi$
$462$	0	0
$463$	−13.9183	−0.646839	−0.323420	−	0.946256i	$-0.604833\pi$
$463$	−13.9183	−0.646839	−0.323420	+	0.946256i	$0.604833\pi$
$464$	0	0
$465$	1.50393	0.0697429
$466$	0	0
$467$	−6.14803	−0.284497	−0.142248	−	0.989831i	$-0.545433\pi$
$467$	−6.14803	−0.284497	−0.142248	+	0.989831i	$0.545433\pi$
$468$	0	0
$469$	−4.88233	−0.225445
$470$	0	0
$471$	9.48672	0.437125
$472$	0	0
$473$	2.37348	0.109133
$474$	0	0
$475$	−14.2209	−0.652502
$476$	0	0
$477$	−0.364263	−0.0166785
$478$	0	0
$479$	−5.81648	−0.265762	−0.132881	−	0.991132i	$-0.542423\pi$
$479$	−5.81648	−0.265762	−0.132881	+	0.991132i	$0.542423\pi$
$480$	0	0
$481$	−10.3782	−0.473206
$482$	0	0
$483$	−1.06236	−0.0483390
$484$	0	0
$485$	5.65457	0.256761
$486$	0	0
$487$	13.2145	0.598808	0.299404	−	0.954126i	$-0.403212\pi$
$487$	13.2145	0.598808	0.299404	+	0.954126i	$0.403212\pi$
$488$	0	0
$489$	6.81600	0.308230
$490$	0	0
$491$	−27.7863	−1.25398	−0.626989	−	0.779028i	$-0.715713\pi$
$491$	−27.7863	−1.25398	−0.626989	+	0.779028i	$0.715713\pi$
$492$	0	0
$493$	4.90857	0.221071
$494$	0	0
$495$	−13.2793	−0.596862
$496$	0	0
$497$	14.1314	0.633880
$498$	0	0
$499$	−26.8648	−1.20263	−0.601316	−	0.799011i	$-0.705357\pi$
$499$	−26.8648	−1.20263	−0.601316	+	0.799011i	$0.705357\pi$
$500$	0	0
$501$	10.6820	0.477235
$502$	0	0
$503$	−8.81883	−0.393212	−0.196606	−	0.980483i	$-0.562992\pi$
$503$	−8.81883	−0.393212	−0.196606	+	0.980483i	$0.562992\pi$
$504$	0	0
$505$	−26.4535	−1.17716
$506$	0	0
$507$	−11.8056	−0.524306
$508$	0	0
$509$	2.54789	0.112933	0.0564665	−	0.998404i	$-0.482017\pi$
$509$	2.54789	0.112933	0.0564665	+	0.998404i	$0.482017\pi$
$510$	0	0
$511$	−15.7564	−0.697022
$512$	0	0
$513$	−5.29888	−0.233951
$514$	0	0
$515$	−17.9523	−0.791072
$516$	0	0
$517$	27.3961	1.20488
$518$	0	0
$519$	−0.0197633	−0.000867512 0
$520$	0	0
$521$	−6.92709	−0.303481	−0.151741	−	0.988420i	$-0.548488\pi$
$521$	−6.92709	−0.303481	−0.151741	+	0.988420i	$0.548488\pi$
$522$	0	0
$523$	−24.6659	−1.07856	−0.539282	−	0.842125i	$-0.681304\pi$
$523$	−24.6659	−1.07856	−0.539282	+	0.842125i	$0.681304\pi$
$524$	0	0
$525$	2.85112	0.124433
$526$	0	0
$527$	−2.66314	−0.116008
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−6.03067	−0.261709
$532$	0	0
$533$	−0.964341	−0.0417703
$534$	0	0
$535$	13.7572	0.594773
$536$	0	0
$537$	−1.79432	−0.0774306
$538$	0	0
$539$	28.1274	1.21153
$540$	0	0
$541$	22.0668	0.948725	0.474362	−	0.880330i	$-0.342679\pi$
$541$	22.0668	0.948725	0.474362	+	0.880330i	$0.342679\pi$
$542$	0	0
$543$	−20.4591	−0.877983
$544$	0	0
$545$	−52.7058	−2.25767
$546$	0	0
$547$	−31.5701	−1.34984	−0.674920	−	0.737891i	$-0.735822\pi$
$547$	−31.5701	−1.34984	−0.674920	+	0.737891i	$0.735822\pi$
$548$	0	0
$549$	3.37600	0.144084
$550$	0	0
$551$	5.29888	0.225740
$552$	0	0
$553$	17.4285	0.741137
$554$	0	0
$555$	−26.3232	−1.11736
$556$	0	0
$557$	−15.0650	−0.638324	−0.319162	−	0.947700i	$-0.603401\pi$
$557$	−15.0650	−0.638324	−0.319162	+	0.947700i	$0.603401\pi$
$558$	0	0
$559$	−0.541462	−0.0229014
$560$	0	0
$561$	23.5149	0.992801
$562$	0	0
$563$	−22.2276	−0.936780	−0.468390	−	0.883522i	$-0.655166\pi$
$563$	−22.2276	−0.936780	−0.468390	+	0.883522i	$0.655166\pi$
$564$	0	0
$565$	54.3851	2.28800
$566$	0	0
$567$	1.06236	0.0446149
$568$	0	0
$569$	−1.81969	−0.0762853	−0.0381427	−	0.999272i	$-0.512144\pi$
$569$	−1.81969	−0.0762853	−0.0381427	+	0.999272i	$0.512144\pi$
$570$	0	0
$571$	−2.88896	−0.120899	−0.0604497	−	0.998171i	$-0.519253\pi$
$571$	−2.88896	−0.120899	−0.0604497	+	0.998171i	$0.519253\pi$
$572$	0	0
$573$	−3.09084	−0.129122
$574$	0	0
$575$	−2.68377	−0.111921
$576$	0	0
$577$	−29.6657	−1.23500	−0.617499	−	0.786572i	$-0.711854\pi$
$577$	−29.6657	−1.23500	−0.617499	+	0.786572i	$0.711854\pi$
$578$	0	0
$579$	−2.36627	−0.0983388
$580$	0	0
$581$	13.8962	0.576511
$582$	0	0
$583$	1.74504	0.0722720
$584$	0	0
$585$	3.02941	0.125251
$586$	0	0
$587$	14.1027	0.582080	0.291040	−	0.956711i	$-0.405999\pi$
$587$	14.1027	0.582080	0.291040	+	0.956711i	$0.405999\pi$
$588$	0	0
$589$	−2.87490	−0.118458
$590$	0	0
$591$	3.27592	0.134753
$592$	0	0
$593$	14.5196	0.596248	0.298124	−	0.954527i	$-0.403639\pi$
$593$	14.5196	0.596248	0.298124	+	0.954527i	$0.403639\pi$
$594$	0	0
$595$	−14.4548	−0.592590
$596$	0	0
$597$	−5.38451	−0.220373
$598$	0	0
$599$	22.0635	0.901492	0.450746	−	0.892652i	$-0.351158\pi$
$599$	22.0635	0.901492	0.450746	+	0.892652i	$0.351158\pi$
$600$	0	0
$601$	22.8957	0.933936	0.466968	−	0.884274i	$-0.345346\pi$
$601$	22.8957	0.933936	0.466968	+	0.884274i	$0.345346\pi$
$602$	0	0
$603$	−4.59575	−0.187153
$604$	0	0
$605$	33.1243	1.34669
$606$	0	0
$607$	−26.2674	−1.06616	−0.533080	−	0.846065i	$-0.678966\pi$
$607$	−26.2674	−1.06616	−0.533080	+	0.846065i	$0.678966\pi$
$608$	0	0
$609$	−1.06236	−0.0430490
$610$	0	0
$611$	−6.24986	−0.252842
$612$	0	0
$613$	−16.6603	−0.672903	−0.336452	−	0.941701i	$-0.609227\pi$
$613$	−16.6603	−0.672903	−0.336452	+	0.941701i	$0.609227\pi$
$614$	0	0
$615$	−2.44595	−0.0986300
$616$	0	0
$617$	37.7862	1.52121	0.760607	−	0.649212i	$-0.224901\pi$
$617$	37.7862	1.52121	0.760607	+	0.649212i	$0.224901\pi$
$618$	0	0
$619$	19.5344	0.785154	0.392577	−	0.919719i	$-0.371584\pi$
$619$	19.5344	0.785154	0.392577	+	0.919719i	$0.371584\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	−7.56022	−0.302894
$624$	0	0
$625$	−31.2162	−1.24865
$626$	0	0
$627$	25.3848	1.01377
$628$	0	0
$629$	46.6129	1.85858
$630$	0	0
$631$	6.63009	0.263940	0.131970	−	0.991254i	$-0.457870\pi$
$631$	6.63009	0.263940	0.131970	+	0.991254i	$0.457870\pi$
$632$	0	0
$633$	13.7011	0.544571
$634$	0	0
$635$	−40.4830	−1.60652
$636$	0	0
$637$	−6.41671	−0.254239
$638$	0	0
$639$	13.3019	0.526215
$640$	0	0
$641$	29.0861	1.14883	0.574415	−	0.818564i	$-0.305230\pi$
$641$	29.0861	1.14883	0.574415	+	0.818564i	$0.305230\pi$
$642$	0	0
$643$	−16.4605	−0.649138	−0.324569	−	0.945862i	$-0.605219\pi$
$643$	−16.4605	−0.649138	−0.324569	+	0.945862i	$0.605219\pi$
$644$	0	0
$645$	−1.37336	−0.0540759
$646$	0	0
$647$	−31.4328	−1.23575	−0.617876	−	0.786276i	$-0.712007\pi$
$647$	−31.4328	−1.23575	−0.617876	+	0.786276i	$0.712007\pi$
$648$	0	0
$649$	28.8905	1.13405
$650$	0	0
$651$	0.576382	0.0225902
$652$	0	0
$653$	−23.0715	−0.902859	−0.451430	−	0.892307i	$-0.649086\pi$
$653$	−23.0715	−0.902859	−0.451430	+	0.892307i	$0.649086\pi$
$654$	0	0
$655$	11.3415	0.443149
$656$	0	0
$657$	−14.8315	−0.578632
$658$	0	0
$659$	−16.0960	−0.627012	−0.313506	−	0.949586i	$-0.601504\pi$
$659$	−16.0960	−0.627012	−0.313506	+	0.949586i	$0.601504\pi$
$660$	0	0
$661$	−42.7943	−1.66451	−0.832253	−	0.554396i	$-0.812949\pi$
$661$	−42.7943	−1.66451	−0.832253	+	0.554396i	$0.812949\pi$
$662$	0	0
$663$	−5.36446	−0.208338
$664$	0	0
$665$	−15.6042	−0.605106
$666$	0	0
$667$	1.00000	0.0387202
$668$	0	0
$669$	20.0766	0.776207
$670$	0	0
$671$	−16.1730	−0.624353
$672$	0	0
$673$	−34.6304	−1.33490	−0.667451	−	0.744654i	$-0.732615\pi$
$673$	−34.6304	−1.33490	−0.667451	+	0.744654i	$0.732615\pi$
$674$	0	0
$675$	2.68377	0.103298
$676$	0	0
$677$	7.83632	0.301174	0.150587	−	0.988597i	$-0.451884\pi$
$677$	7.83632	0.301174	0.150587	+	0.988597i	$0.451884\pi$
$678$	0	0
$679$	2.16713	0.0831667
$680$	0	0
$681$	2.37670	0.0910754
$682$	0	0
$683$	33.5977	1.28558	0.642790	−	0.766042i	$-0.277777\pi$
$683$	33.5977	1.28558	0.642790	+	0.766042i	$0.277777\pi$
$684$	0	0
$685$	−36.2092	−1.38348
$686$	0	0
$687$	10.8234	0.412940
$688$	0	0
$689$	−0.398095	−0.0151662
$690$	0	0
$691$	18.3044	0.696334	0.348167	−	0.937433i	$-0.386804\pi$
$691$	18.3044	0.696334	0.348167	+	0.937433i	$0.386804\pi$
$692$	0	0
$693$	−5.08933	−0.193328
$694$	0	0
$695$	9.06737	0.343945
$696$	0	0
$697$	4.33126	0.164058
$698$	0	0
$699$	15.3739	0.581496
$700$	0	0
$701$	−26.7837	−1.01161	−0.505804	−	0.862649i	$-0.668804\pi$
$701$	−26.7837	−1.01161	−0.505804	+	0.862649i	$0.668804\pi$
$702$	0	0
$703$	50.3194	1.89783
$704$	0	0
$705$	−15.8521	−0.597024
$706$	0	0
$707$	−10.1383	−0.381292
$708$	0	0
$709$	−14.8997	−0.559570	−0.279785	−	0.960063i	$-0.590263\pi$
$709$	−14.8997	−0.559570	−0.279785	+	0.960063i	$0.590263\pi$
$710$	0	0
$711$	16.4055	0.615255
$712$	0	0
$713$	−0.542549	−0.0203186
$714$	0	0
$715$	−14.5127	−0.542743
$716$	0	0
$717$	19.0941	0.713082
$718$	0	0
$719$	46.4036	1.73056	0.865282	−	0.501286i	$-0.167140\pi$
$719$	46.4036	1.73056	0.865282	+	0.501286i	$0.167140\pi$
$720$	0	0
$721$	−6.88024	−0.256234
$722$	0	0
$723$	10.3969	0.386664
$724$	0	0
$725$	−2.68377	−0.0996726
$726$	0	0
$727$	−41.3047	−1.53191	−0.765953	−	0.642897i	$-0.777732\pi$
$727$	−41.3047	−1.53191	−0.765953	+	0.642897i	$0.777732\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	2.43193	0.0899483
$732$	0	0
$733$	−21.4216	−0.791226	−0.395613	−	0.918417i	$-0.629468\pi$
$733$	−21.4216	−0.791226	−0.395613	+	0.918417i	$0.629468\pi$
$734$	0	0
$735$	−16.2753	−0.600322
$736$	0	0
$737$	22.0163	0.810982
$738$	0	0
$739$	35.6468	1.31129	0.655645	−	0.755069i	$-0.272397\pi$
$739$	35.6468	1.31129	0.655645	+	0.755069i	$0.272397\pi$
$740$	0	0
$741$	−5.79102	−0.212738
$742$	0	0
$743$	−40.9631	−1.50279	−0.751396	−	0.659852i	$-0.770619\pi$
$743$	−40.9631	−1.50279	−0.751396	+	0.659852i	$0.770619\pi$
$744$	0	0
$745$	−25.9472	−0.950632
$746$	0	0
$747$	13.0805	0.478591
$748$	0	0
$749$	5.27245	0.192651
$750$	0	0
$751$	−8.30308	−0.302984	−0.151492	−	0.988458i	$-0.548408\pi$
$751$	−8.30308	−0.302984	−0.151492	+	0.988458i	$0.548408\pi$
$752$	0	0
$753$	11.3982	0.415375
$754$	0	0
$755$	−28.9579	−1.05389
$756$	0	0
$757$	9.72098	0.353315	0.176658	−	0.984272i	$-0.443471\pi$
$757$	9.72098	0.353315	0.176658	+	0.984272i	$0.443471\pi$
$758$	0	0
$759$	4.79059	0.173887
$760$	0	0
$761$	48.0755	1.74274	0.871368	−	0.490629i	$-0.163233\pi$
$761$	48.0755	1.74274	0.871368	+	0.490629i	$0.163233\pi$
$762$	0	0
$763$	−20.1996	−0.731275
$764$	0	0
$765$	−13.6064	−0.491939
$766$	0	0
$767$	−6.59077	−0.237979
$768$	0	0
$769$	−18.1500	−0.654504	−0.327252	−	0.944937i	$-0.606123\pi$
$769$	−18.1500	−0.654504	−0.327252	+	0.944937i	$0.606123\pi$
$770$	0	0
$771$	13.9938	0.503976
$772$	0	0
$773$	−30.7130	−1.10467	−0.552336	−	0.833622i	$-0.686263\pi$
$773$	−30.7130	−1.10467	−0.552336	+	0.833622i	$0.686263\pi$
$774$	0	0
$775$	1.45608	0.0523038
$776$	0	0
$777$	−10.0884	−0.361920
$778$	0	0
$779$	4.67567	0.167523
$780$	0	0
$781$	−63.7240	−2.28022
$782$	0	0
$783$	−1.00000	−0.0357371
$784$	0	0
$785$	26.2968	0.938574
$786$	0	0
$787$	−31.3256	−1.11664	−0.558319	−	0.829626i	$-0.688554\pi$
$787$	−31.3256	−1.11664	−0.558319	+	0.829626i	$0.688554\pi$
$788$	0	0
$789$	0.164010	0.00583892
$790$	0	0
$791$	20.8432	0.741098
$792$	0	0
$793$	3.68955	0.131020
$794$	0	0
$795$	−1.00972	−0.0358112
$796$	0	0
$797$	−17.3747	−0.615445	−0.307722	−	0.951476i	$-0.599567\pi$
$797$	−17.3747	−0.615445	−0.307722	+	0.951476i	$0.599567\pi$
$798$	0	0
$799$	28.0707	0.993071
$800$	0	0
$801$	−7.11644	−0.251447
$802$	0	0
$803$	71.0517	2.50736
$804$	0	0
$805$	−2.94482	−0.103791
$806$	0	0
$807$	−26.8224	−0.944195
$808$	0	0
$809$	−16.8754	−0.593309	−0.296655	−	0.954985i	$-0.595871\pi$
$809$	−16.8754	−0.593309	−0.296655	+	0.954985i	$0.595871\pi$
$810$	0	0
$811$	33.8459	1.18849	0.594246	−	0.804283i	$-0.297451\pi$
$811$	33.8459	1.18849	0.594246	+	0.804283i	$0.297451\pi$
$812$	0	0
$813$	8.32336	0.291913
$814$	0	0
$815$	18.8937	0.661817
$816$	0	0
$817$	2.62531	0.0918480
$818$	0	0
$819$	1.16103	0.0405696
$820$	0	0
$821$	4.57345	0.159615	0.0798073	−	0.996810i	$-0.474570\pi$
$821$	4.57345	0.159615	0.0798073	+	0.996810i	$0.474570\pi$
$822$	0	0
$823$	−2.24515	−0.0782610	−0.0391305	−	0.999234i	$-0.512459\pi$
$823$	−2.24515	−0.0782610	−0.0391305	+	0.999234i	$0.512459\pi$
$824$	0	0
$825$	−12.8568	−0.447617
$826$	0	0
$827$	−0.810001	−0.0281665	−0.0140833	−	0.999901i	$-0.504483\pi$
$827$	−0.810001	−0.0281665	−0.0140833	+	0.999901i	$0.504483\pi$
$828$	0	0
$829$	−47.2548	−1.64123	−0.820614	−	0.571483i	$-0.806368\pi$
$829$	−47.2548	−1.64123	−0.820614	+	0.571483i	$0.806368\pi$
$830$	0	0
$831$	20.1954	0.700570
$832$	0	0
$833$	28.8201	0.998558
$834$	0	0
$835$	29.6100	1.02470
$836$	0	0
$837$	0.542549	0.0187533
$838$	0	0
$839$	−8.52942	−0.294468	−0.147234	−	0.989102i	$-0.547037\pi$
$839$	−8.52942	−0.294468	−0.147234	+	0.989102i	$0.547037\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−7.62809	−0.262725
$844$	0	0
$845$	−32.7247	−1.12576
$846$	0	0
$847$	12.6949	0.436203
$848$	0	0
$849$	20.8851	0.716774
$850$	0	0
$851$	9.49623	0.325527
$852$	0	0
$853$	−41.1412	−1.40865	−0.704323	−	0.709879i	$-0.748749\pi$
$853$	−41.1412	−1.40865	−0.704323	+	0.709879i	$0.748749\pi$
$854$	0	0
$855$	−14.6883	−0.502329
$856$	0	0
$857$	29.6895	1.01417	0.507086	−	0.861895i	$-0.330723\pi$
$857$	29.6895	1.01417	0.507086	+	0.861895i	$0.330723\pi$
$858$	0	0
$859$	−54.4822	−1.85891	−0.929454	−	0.368937i	$-0.879722\pi$
$859$	−54.4822	−1.85891	−0.929454	+	0.368937i	$0.879722\pi$
$860$	0	0
$861$	−0.937413	−0.0319470
$862$	0	0
$863$	2.30451	0.0784465	0.0392232	−	0.999230i	$-0.487512\pi$
$863$	2.30451	0.0784465	0.0392232	+	0.999230i	$0.487512\pi$
$864$	0	0
$865$	−0.0547830	−0.00186268
$866$	0	0
$867$	7.09402	0.240925
$868$	0	0
$869$	−78.5921	−2.66605
$870$	0	0
$871$	−5.02258	−0.170184
$872$	0	0
$873$	2.03992	0.0690408
$874$	0	0
$875$	−6.82089	−0.230588
$876$	0	0
$877$	46.1343	1.55785	0.778923	−	0.627120i	$-0.215766\pi$
$877$	46.1343	1.55785	0.778923	+	0.627120i	$0.215766\pi$
$878$	0	0
$879$	5.22302	0.176168
$880$	0	0
$881$	−11.7060	−0.394385	−0.197192	−	0.980365i	$-0.563182\pi$
$881$	−11.7060	−0.394385	−0.197192	+	0.980365i	$0.563182\pi$
$882$	0	0
$883$	−31.9688	−1.07584	−0.537918	−	0.842997i	$-0.680789\pi$
$883$	−31.9688	−1.07584	−0.537918	+	0.842997i	$0.680789\pi$
$884$	0	0
$885$	−16.7168	−0.561928
$886$	0	0
$887$	−22.3236	−0.749553	−0.374776	−	0.927115i	$-0.622281\pi$
$887$	−22.3236	−0.749553	−0.374776	+	0.927115i	$0.622281\pi$
$888$	0	0
$889$	−15.5152	−0.520362
$890$	0	0
$891$	−4.79059	−0.160491
$892$	0	0
$893$	30.3028	1.01405
$894$	0	0
$895$	−4.97378	−0.166255
$896$	0	0
$897$	−1.09288	−0.0364901
$898$	0	0
$899$	−0.542549	−0.0180950
$900$	0	0
$901$	1.78801	0.0595672
$902$	0	0
$903$	−0.526342	−0.0175156
$904$	0	0
$905$	−56.7118	−1.88516
$906$	0	0
$907$	−21.6864	−0.720085	−0.360042	−	0.932936i	$-0.617238\pi$
$907$	−21.6864	−0.720085	−0.360042	+	0.932936i	$0.617238\pi$
$908$	0	0
$909$	−9.54324	−0.316529
$910$	0	0
$911$	28.3014	0.937668	0.468834	−	0.883286i	$-0.344674\pi$
$911$	28.3014	0.937668	0.468834	+	0.883286i	$0.344674\pi$
$912$	0	0
$913$	−62.6634	−2.07385
$914$	0	0
$915$	9.35814	0.309371
$916$	0	0
$917$	4.34665	0.143539
$918$	0	0
$919$	−31.9843	−1.05507	−0.527533	−	0.849535i	$-0.676883\pi$
$919$	−31.9843	−1.05507	−0.527533	+	0.849535i	$0.676883\pi$
$920$	0	0
$921$	−0.551792	−0.0181822
$922$	0	0
$923$	14.5373	0.478503
$924$	0	0
$925$	−25.4857	−0.837964
$926$	0	0
$927$	−6.47638	−0.212712
$928$	0	0
$929$	−35.0582	−1.15022	−0.575111	−	0.818076i	$-0.695041\pi$
$929$	−35.0582	−1.15022	−0.575111	+	0.818076i	$0.695041\pi$
$930$	0	0
$931$	31.1118	1.01965
$932$	0	0
$933$	10.5599	0.345715
$934$	0	0
$935$	65.1825	2.13169
$936$	0	0
$937$	−14.0341	−0.458474	−0.229237	−	0.973371i	$-0.573623\pi$
$937$	−14.0341	−0.458474	−0.229237	+	0.973371i	$0.573623\pi$
$938$	0	0
$939$	0.909310	0.0296742
$940$	0	0
$941$	21.3710	0.696674	0.348337	−	0.937369i	$-0.386747\pi$
$941$	21.3710	0.696674	0.348337	+	0.937369i	$0.386747\pi$
$942$	0	0
$943$	0.882388	0.0287345
$944$	0	0
$945$	2.94482	0.0957949
$946$	0	0
$947$	11.7716	0.382525	0.191263	−	0.981539i	$-0.438742\pi$
$947$	11.7716	0.382525	0.191263	+	0.981539i	$0.438742\pi$
$948$	0	0
$949$	−16.2090	−0.526167
$950$	0	0
$951$	−14.8865	−0.482728
$952$	0	0
$953$	22.0014	0.712697	0.356348	−	0.934353i	$-0.384022\pi$
$953$	22.0014	0.712697	0.356348	+	0.934353i	$0.384022\pi$
$954$	0	0
$955$	−8.56769	−0.277244
$956$	0	0
$957$	4.79059	0.154858
$958$	0	0
$959$	−13.8772	−0.448120
$960$	0	0
$961$	−30.7056	−0.990505
$962$	0	0
$963$	4.96297	0.159929
$964$	0	0
$965$	−6.55921	−0.211148
$966$	0	0
$967$	−5.74246	−0.184665	−0.0923326	−	0.995728i	$-0.529432\pi$
$967$	−5.74246	−0.184665	−0.0923326	+	0.995728i	$0.529432\pi$
$968$	0	0
$969$	26.0099	0.835558
$970$	0	0
$971$	57.7953	1.85474	0.927369	−	0.374148i	$-0.122065\pi$
$971$	57.7953	1.85474	0.927369	+	0.374148i	$0.122065\pi$
$972$	0	0
$973$	3.47509	0.111406
$974$	0	0
$975$	2.93302	0.0939320
$976$	0	0
$977$	−17.5829	−0.562528	−0.281264	−	0.959630i	$-0.590754\pi$
$977$	−17.5829	−0.562528	−0.281264	+	0.959630i	$0.590754\pi$
$978$	0	0
$979$	34.0920	1.08958
$980$	0	0
$981$	−19.0139	−0.607068
$982$	0	0
$983$	−42.2450	−1.34741	−0.673703	−	0.739002i	$-0.735297\pi$
$983$	−42.2450	−1.34741	−0.673703	+	0.739002i	$0.735297\pi$
$984$	0	0
$985$	9.08073	0.289336
$986$	0	0
$987$	−6.07534	−0.193380
$988$	0	0
$989$	0.495447	0.0157543
$990$	0	0
$991$	22.0170	0.699392	0.349696	−	0.936863i	$-0.386285\pi$
$991$	22.0170	0.699392	0.349696	+	0.936863i	$0.386285\pi$
$992$	0	0
$993$	25.0196	0.793972
$994$	0	0
$995$	−14.9256	−0.473175
$996$	0	0
$997$	−30.4317	−0.963782	−0.481891	−	0.876231i	$-0.660050\pi$
$997$	−30.4317	−0.963782	−0.481891	+	0.876231i	$0.660050\pi$
$998$	0	0
$999$	−9.49623	−0.300448

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.f.1.9	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.f.1.9	✓	9	1.1	even	1	trivial

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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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +2.77196 q^{5} +1.06236 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +2.77196 q^{5} +1.06236 q^{7} +1.00000 q^{9} -4.79059 q^{11} +1.09288 q^{13} +2.77196 q^{15} -4.90857 q^{17} -5.29888 q^{19} +1.06236 q^{21} -1.00000 q^{23} +2.68377 q^{25} +1.00000 q^{27} -1.00000 q^{29} +0.542549 q^{31} -4.79059 q^{33} +2.94482 q^{35} -9.49623 q^{37} +1.09288 q^{39} -0.882388 q^{41} -0.495447 q^{43} +2.77196 q^{45} -5.71872 q^{47} -5.87139 q^{49} -4.90857 q^{51} -0.364263 q^{53} -13.2793 q^{55} -5.29888 q^{57} -6.03067 q^{59} +3.37600 q^{61} +1.06236 q^{63} +3.02941 q^{65} -4.59575 q^{67} -1.00000 q^{69} +13.3019 q^{71} -14.8315 q^{73} +2.68377 q^{75} -5.08933 q^{77} +16.4055 q^{79} +1.00000 q^{81} +13.0805 q^{83} -13.6064 q^{85} -1.00000 q^{87} -7.11644 q^{89} +1.16103 q^{91} +0.542549 q^{93} -14.6883 q^{95} +2.03992 q^{97} -4.79059 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{3} - q^{5} - 5 q^{7} + 9 q^{9}+O(q^{10}) \) 9 * q + 9 * q^3 - q^5 - 5 * q^7 + 9 * q^9	\( 9 q + 9 q^{3} - q^{5} - 5 q^{7} + 9 q^{9} - 8 q^{11} - q^{13} - q^{15} - 2 q^{17} - 11 q^{19} - 5 q^{21} - 9 q^{23} - 10 q^{25} + 9 q^{27} - 9 q^{29} - 8 q^{33} + q^{35} - 2 q^{37} - q^{39} - 3 q^{41} - 19 q^{43} - q^{45} - 3 q^{47} - 6 q^{49} - 2 q^{51} - 9 q^{53} - 7 q^{55} - 11 q^{57} - 2 q^{59} - 25 q^{61} - 5 q^{63} - 12 q^{65} - 20 q^{67} - 9 q^{69} + 9 q^{71} - 11 q^{73} - 10 q^{75} - 19 q^{77} + 4 q^{79} + 9 q^{81} - 9 q^{83} - 50 q^{85} - 9 q^{87} - 29 q^{89} - 38 q^{91} + 23 q^{95} - 43 q^{97} - 8 q^{99}+O(q^{100}) \) 9 * q + 9 * q^3 - q^5 - 5 * q^7 + 9 * q^9 - 8 * q^11 - q^13 - q^15 - 2 * q^17 - 11 * q^19 - 5 * q^21 - 9 * q^23 - 10 * q^25 + 9 * q^27 - 9 * q^29 - 8 * q^33 + q^35 - 2 * q^37 - q^39 - 3 * q^41 - 19 * q^43 - q^45 - 3 * q^47 - 6 * q^49 - 2 * q^51 - 9 * q^53 - 7 * q^55 - 11 * q^57 - 2 * q^59 - 25 * q^61 - 5 * q^63 - 12 * q^65 - 20 * q^67 - 9 * q^69 + 9 * q^71 - 11 * q^73 - 10 * q^75 - 19 * q^77 + 4 * q^79 + 9 * q^81 - 9 * q^83 - 50 * q^85 - 9 * q^87 - 29 * q^89 - 38 * q^91 + 23 * q^95 - 43 * q^97 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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