LMFDB - Embedded newform 8004.2.a.i.1.14

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} - 5 x^{15} - 43 x^{14} + 234 x^{13} + 634 x^{12} - 4048 x^{11} - 3483 x^{10} + 32512 x^{9} + \cdots - 208 $ x^16 - 5x^15 - 43x^14 + 234x^13 + 634x^12 - 4048x^11 - 3483x^10 + 32512x^9 - 137x^8 - 121665x^7 + 60168x^6 + 172218x^5 - 138024x^4 - 17844x^3 + 14352x^2 + 72*x - 208
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			1.14
Root			$3.56414$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +3.56414 q^{5} -0.955621 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +3.56414 q^{5} -0.955621 q^{7} +1.00000 q^{9} -5.80108 q^{11} +2.78668 q^{13} -3.56414 q^{15} -4.80496 q^{17} -2.83599 q^{19} +0.955621 q^{21} +1.00000 q^{23} +7.70306 q^{25} -1.00000 q^{27} +1.00000 q^{29} +1.35740 q^{31} +5.80108 q^{33} -3.40596 q^{35} -2.67980 q^{37} -2.78668 q^{39} +10.9216 q^{41} -5.88946 q^{43} +3.56414 q^{45} -0.527133 q^{47} -6.08679 q^{49} +4.80496 q^{51} -7.00804 q^{53} -20.6758 q^{55} +2.83599 q^{57} +2.89886 q^{59} +8.42802 q^{61} -0.955621 q^{63} +9.93209 q^{65} +11.0194 q^{67} -1.00000 q^{69} +12.6124 q^{71} +5.58199 q^{73} -7.70306 q^{75} +5.54363 q^{77} -9.15554 q^{79} +1.00000 q^{81} +5.98706 q^{83} -17.1255 q^{85} -1.00000 q^{87} +8.05698 q^{89} -2.66300 q^{91} -1.35740 q^{93} -10.1079 q^{95} +8.87633 q^{97} -5.80108 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{3} + 5 q^{5} - 4 q^{7} + 16 q^{9}+O(q^{10}) $ 16 * q - 16 * q^3 + 5 * q^5 - 4 * q^7 + 16 * q^9	$ 16 q - 16 q^{3} + 5 q^{5} - 4 q^{7} + 16 q^{9} + 5 q^{11} + 8 q^{13} - 5 q^{15} + 7 q^{17} + q^{19} + 4 q^{21} + 16 q^{23} + 31 q^{25} - 16 q^{27} + 16 q^{29} - 2 q^{31} - 5 q^{33} + 5 q^{35} + 14 q^{37} - 8 q^{39} - q^{41} - 13 q^{43} + 5 q^{45} - 4 q^{47} + 30 q^{49} - 7 q^{51} + 19 q^{53} - 37 q^{55} - q^{57} + 12 q^{59} + 21 q^{61} - 4 q^{63} + 26 q^{65} - 11 q^{67} - 16 q^{69} + 7 q^{71} - 13 q^{73} - 31 q^{75} + 4 q^{77} - 18 q^{79} + 16 q^{81} + 25 q^{83} + 48 q^{85} - 16 q^{87} + 12 q^{89} - 11 q^{91} + 2 q^{93} - q^{95} + 5 q^{97} + 5 q^{99}+O(q^{100}) $ 16 * q - 16 * q^3 + 5 * q^5 - 4 * q^7 + 16 * q^9 + 5 * q^11 + 8 * q^13 - 5 * q^15 + 7 * q^17 + q^19 + 4 * q^21 + 16 * q^23 + 31 * q^25 - 16 * q^27 + 16 * q^29 - 2 * q^31 - 5 * q^33 + 5 * q^35 + 14 * q^37 - 8 * q^39 - q^41 - 13 * q^43 + 5 * q^45 - 4 * q^47 + 30 * q^49 - 7 * q^51 + 19 * q^53 - 37 * q^55 - q^57 + 12 * q^59 + 21 * q^61 - 4 * q^63 + 26 * q^65 - 11 * q^67 - 16 * q^69 + 7 * q^71 - 13 * q^73 - 31 * q^75 + 4 * q^77 - 18 * q^79 + 16 * q^81 + 25 * q^83 + 48 * q^85 - 16 * q^87 + 12 * q^89 - 11 * q^91 + 2 * q^93 - q^95 + 5 * 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
$5$	3.56414	1.59393	0.796965	−	0.604025i	$-0.206437\pi$
$5$	3.56414	1.59393	0.796965	+	0.604025i	$0.206437\pi$
$6$	0	0
$7$	−0.955621	−0.361191	−0.180595	−	0.983557i	$-0.557802\pi$
$7$	−0.955621	−0.361191	−0.180595	+	0.983557i	$0.557802\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.80108	−1.74909	−0.874546	−	0.484943i	$-0.838840\pi$
$11$	−5.80108	−1.74909	−0.874546	+	0.484943i	$0.838840\pi$
$12$	0	0
$13$	2.78668	0.772885	0.386442	−	0.922314i	$-0.373704\pi$
$13$	2.78668	0.772885	0.386442	+	0.922314i	$0.373704\pi$
$14$	0	0
$15$	−3.56414	−0.920256
$16$	0	0
$17$	−4.80496	−1.16537	−0.582687	−	0.812697i	$-0.697999\pi$
$17$	−4.80496	−1.16537	−0.582687	+	0.812697i	$0.697999\pi$
$18$	0	0
$19$	−2.83599	−0.650621	−0.325311	−	0.945607i	$-0.605469\pi$
$19$	−2.83599	−0.650621	−0.325311	+	0.945607i	$0.605469\pi$
$20$	0	0
$21$	0.955621	0.208533
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	7.70306	1.54061
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	1.35740	0.243796	0.121898	−	0.992543i	$-0.461102\pi$
$31$	1.35740	0.243796	0.121898	+	0.992543i	$0.461102\pi$
$32$	0	0
$33$	5.80108	1.00984
$34$	0	0
$35$	−3.40596	−0.575713
$36$	0	0
$37$	−2.67980	−0.440556	−0.220278	−	0.975437i	$-0.570697\pi$
$37$	−2.67980	−0.440556	−0.220278	+	0.975437i	$0.570697\pi$
$38$	0	0
$39$	−2.78668	−0.446225
$40$	0	0
$41$	10.9216	1.70566	0.852832	−	0.522186i	$-0.174883\pi$
$41$	10.9216	1.70566	0.852832	+	0.522186i	$0.174883\pi$
$42$	0	0
$43$	−5.88946	−0.898134	−0.449067	−	0.893498i	$-0.648243\pi$
$43$	−5.88946	−0.898134	−0.449067	+	0.893498i	$0.648243\pi$
$44$	0	0
$45$	3.56414	0.531310
$46$	0	0
$47$	−0.527133	−0.0768903	−0.0384452	−	0.999261i	$-0.512240\pi$
$47$	−0.527133	−0.0768903	−0.0384452	+	0.999261i	$0.512240\pi$
$48$	0	0
$49$	−6.08679	−0.869541
$50$	0	0
$51$	4.80496	0.672829
$52$	0	0
$53$	−7.00804	−0.962628	−0.481314	−	0.876548i	$-0.659840\pi$
$53$	−7.00804	−0.962628	−0.481314	+	0.876548i	$0.659840\pi$
$54$	0	0
$55$	−20.6758	−2.78793
$56$	0	0
$57$	2.83599	0.375636
$58$	0	0
$59$	2.89886	0.377400	0.188700	−	0.982035i	$-0.439573\pi$
$59$	2.89886	0.377400	0.188700	+	0.982035i	$0.439573\pi$
$60$	0	0
$61$	8.42802	1.07910	0.539549	−	0.841954i	$-0.318595\pi$
$61$	8.42802	1.07910	0.539549	+	0.841954i	$0.318595\pi$
$62$	0	0
$63$	−0.955621	−0.120397
$64$	0	0
$65$	9.93209	1.23192
$66$	0	0
$67$	11.0194	1.34624	0.673120	−	0.739534i	$-0.264954\pi$
$67$	11.0194	1.34624	0.673120	+	0.739534i	$0.264954\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	12.6124	1.49681	0.748407	−	0.663240i	$-0.230819\pi$
$71$	12.6124	1.49681	0.748407	+	0.663240i	$0.230819\pi$
$72$	0	0
$73$	5.58199	0.653322	0.326661	−	0.945142i	$-0.394076\pi$
$73$	5.58199	0.653322	0.326661	+	0.945142i	$0.394076\pi$
$74$	0	0
$75$	−7.70306	−0.889473
$76$	0	0
$77$	5.54363	0.631755
$78$	0	0
$79$	−9.15554	−1.03008	−0.515039	−	0.857167i	$-0.672223\pi$
$79$	−9.15554	−1.03008	−0.515039	+	0.857167i	$0.672223\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	5.98706	0.657166	0.328583	−	0.944475i	$-0.393429\pi$
$83$	5.98706	0.657166	0.328583	+	0.944475i	$0.393429\pi$
$84$	0	0
$85$	−17.1255	−1.85752
$86$	0	0
$87$	−1.00000	−0.107211
$88$	0	0
$89$	8.05698	0.854038	0.427019	−	0.904243i	$-0.359564\pi$
$89$	8.05698	0.854038	0.427019	+	0.904243i	$0.359564\pi$
$90$	0	0
$91$	−2.66300	−0.279159
$92$	0	0
$93$	−1.35740	−0.140756
$94$	0	0
$95$	−10.1079	−1.03704
$96$	0	0
$97$	8.87633	0.901255	0.450627	−	0.892712i	$-0.351200\pi$
$97$	8.87633	0.901255	0.450627	+	0.892712i	$0.351200\pi$
$98$	0	0
$99$	−5.80108	−0.583030
$100$	0	0
$101$	15.8122	1.57337	0.786687	−	0.617352i	$-0.211795\pi$
$101$	15.8122	1.57337	0.786687	+	0.617352i	$0.211795\pi$
$102$	0	0
$103$	6.02264	0.593429	0.296714	−	0.954966i	$-0.404109\pi$
$103$	6.02264	0.593429	0.296714	+	0.954966i	$0.404109\pi$
$104$	0	0
$105$	3.40596	0.332388
$106$	0	0
$107$	10.9137	1.05506	0.527531	−	0.849536i	$-0.323118\pi$
$107$	10.9137	1.05506	0.527531	+	0.849536i	$0.323118\pi$
$108$	0	0
$109$	7.69560	0.737104	0.368552	−	0.929607i	$-0.379854\pi$
$109$	7.69560	0.737104	0.368552	+	0.929607i	$0.379854\pi$
$110$	0	0
$111$	2.67980	0.254355
$112$	0	0
$113$	−9.48154	−0.891948	−0.445974	−	0.895046i	$-0.647143\pi$
$113$	−9.48154	−0.891948	−0.445974	+	0.895046i	$0.647143\pi$
$114$	0	0
$115$	3.56414	0.332357
$116$	0	0
$117$	2.78668	0.257628
$118$	0	0
$119$	4.59172	0.420922
$120$	0	0
$121$	22.6525	2.05932
$122$	0	0
$123$	−10.9216	−0.984765
$124$	0	0
$125$	9.63409	0.861699
$126$	0	0
$127$	−3.02443	−0.268375	−0.134188	−	0.990956i	$-0.542842\pi$
$127$	−3.02443	−0.268375	−0.134188	+	0.990956i	$0.542842\pi$
$128$	0	0
$129$	5.88946	0.518538
$130$	0	0
$131$	8.68706	0.758992	0.379496	−	0.925193i	$-0.376097\pi$
$131$	8.68706	0.758992	0.379496	+	0.925193i	$0.376097\pi$
$132$	0	0
$133$	2.71013	0.234998
$134$	0	0
$135$	−3.56414	−0.306752
$136$	0	0
$137$	3.01854	0.257891	0.128946	−	0.991652i	$-0.458841\pi$
$137$	3.01854	0.257891	0.128946	+	0.991652i	$0.458841\pi$
$138$	0	0
$139$	−5.16341	−0.437955	−0.218978	−	0.975730i	$-0.570272\pi$
$139$	−5.16341	−0.437955	−0.218978	+	0.975730i	$0.570272\pi$
$140$	0	0
$141$	0.527133	0.0443926
$142$	0	0
$143$	−16.1657	−1.35185
$144$	0	0
$145$	3.56414	0.295985
$146$	0	0
$147$	6.08679	0.502030
$148$	0	0
$149$	3.58064	0.293338	0.146669	−	0.989186i	$-0.453145\pi$
$149$	3.58064	0.293338	0.146669	+	0.989186i	$0.453145\pi$
$150$	0	0
$151$	2.96648	0.241408	0.120704	−	0.992689i	$-0.461485\pi$
$151$	2.96648	0.241408	0.120704	+	0.992689i	$0.461485\pi$
$152$	0	0
$153$	−4.80496	−0.388458
$154$	0	0
$155$	4.83796	0.388594
$156$	0	0
$157$	18.2771	1.45867	0.729334	−	0.684158i	$-0.239830\pi$
$157$	18.2771	1.45867	0.729334	+	0.684158i	$0.239830\pi$
$158$	0	0
$159$	7.00804	0.555773
$160$	0	0
$161$	−0.955621	−0.0753134
$162$	0	0
$163$	7.06758	0.553575	0.276788	−	0.960931i	$-0.410730\pi$
$163$	7.06758	0.553575	0.276788	+	0.960931i	$0.410730\pi$
$164$	0	0
$165$	20.6758	1.60961
$166$	0	0
$167$	−0.993252	−0.0768602	−0.0384301	−	0.999261i	$-0.512236\pi$
$167$	−0.993252	−0.0768602	−0.0384301	+	0.999261i	$0.512236\pi$
$168$	0	0
$169$	−5.23444	−0.402649
$170$	0	0
$171$	−2.83599	−0.216874
$172$	0	0
$173$	10.1738	0.773502	0.386751	−	0.922184i	$-0.373597\pi$
$173$	10.1738	0.773502	0.386751	+	0.922184i	$0.373597\pi$
$174$	0	0
$175$	−7.36121	−0.556455
$176$	0	0
$177$	−2.89886	−0.217892
$178$	0	0
$179$	13.2458	0.990036	0.495018	−	0.868883i	$-0.335161\pi$
$179$	13.2458	0.990036	0.495018	+	0.868883i	$0.335161\pi$
$180$	0	0
$181$	−6.67164	−0.495899	−0.247950	−	0.968773i	$-0.579757\pi$
$181$	−6.67164	−0.495899	−0.247950	+	0.968773i	$0.579757\pi$
$182$	0	0
$183$	−8.42802	−0.623017
$184$	0	0
$185$	−9.55117	−0.702216
$186$	0	0
$187$	27.8740	2.03835
$188$	0	0
$189$	0.955621	0.0695112
$190$	0	0
$191$	−8.22295	−0.594991	−0.297496	−	0.954723i	$-0.596151\pi$
$191$	−8.22295	−0.594991	−0.297496	+	0.954723i	$0.596151\pi$
$192$	0	0
$193$	−9.08660	−0.654068	−0.327034	−	0.945013i	$-0.606049\pi$
$193$	−9.08660	−0.654068	−0.327034	+	0.945013i	$0.606049\pi$
$194$	0	0
$195$	−9.93209	−0.711252
$196$	0	0
$197$	9.97613	0.710770	0.355385	−	0.934720i	$-0.384350\pi$
$197$	9.97613	0.710770	0.355385	+	0.934720i	$0.384350\pi$
$198$	0	0
$199$	12.4190	0.880361	0.440180	−	0.897909i	$-0.354914\pi$
$199$	12.4190	0.880361	0.440180	+	0.897909i	$0.354914\pi$
$200$	0	0
$201$	−11.0194	−0.777252
$202$	0	0
$203$	−0.955621	−0.0670714
$204$	0	0
$205$	38.9260	2.71871
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	16.4518	1.13800
$210$	0	0
$211$	−20.1635	−1.38811	−0.694057	−	0.719920i	$-0.744178\pi$
$211$	−20.1635	−1.38811	−0.694057	+	0.719920i	$0.744178\pi$
$212$	0	0
$213$	−12.6124	−0.864186
$214$	0	0
$215$	−20.9908	−1.43156
$216$	0	0
$217$	−1.29716	−0.0880569
$218$	0	0
$219$	−5.58199	−0.377196
$220$	0	0
$221$	−13.3899	−0.900700
$222$	0	0
$223$	21.7862	1.45891	0.729455	−	0.684028i	$-0.239774\pi$
$223$	21.7862	1.45891	0.729455	+	0.684028i	$0.239774\pi$
$224$	0	0
$225$	7.70306	0.513538
$226$	0	0
$227$	0.0750304	0.00497994	0.00248997	−	0.999997i	$-0.499207\pi$
$227$	0.0750304	0.00497994	0.00248997	+	0.999997i	$0.499207\pi$
$228$	0	0
$229$	15.9524	1.05417	0.527084	−	0.849813i	$-0.323285\pi$
$229$	15.9524	1.05417	0.527084	+	0.849813i	$0.323285\pi$
$230$	0	0
$231$	−5.54363	−0.364744
$232$	0	0
$233$	−13.0981	−0.858082	−0.429041	−	0.903285i	$-0.641148\pi$
$233$	−13.0981	−0.858082	−0.429041	+	0.903285i	$0.641148\pi$
$234$	0	0
$235$	−1.87877	−0.122558
$236$	0	0
$237$	9.15554	0.594716
$238$	0	0
$239$	−27.9484	−1.80783	−0.903916	−	0.427710i	$-0.859321\pi$
$239$	−27.9484	−1.80783	−0.903916	+	0.427710i	$0.859321\pi$
$240$	0	0
$241$	−23.4964	−1.51353	−0.756767	−	0.653685i	$-0.773222\pi$
$241$	−23.4964	−1.51353	−0.756767	+	0.653685i	$0.773222\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−21.6941	−1.38599
$246$	0	0
$247$	−7.90299	−0.502855
$248$	0	0
$249$	−5.98706	−0.379415
$250$	0	0
$251$	6.46207	0.407882	0.203941	−	0.978983i	$-0.434625\pi$
$251$	6.46207	0.407882	0.203941	+	0.978983i	$0.434625\pi$
$252$	0	0
$253$	−5.80108	−0.364711
$254$	0	0
$255$	17.1255	1.07244
$256$	0	0
$257$	−9.92325	−0.618995	−0.309498	−	0.950900i	$-0.600161\pi$
$257$	−9.92325	−0.618995	−0.309498	+	0.950900i	$0.600161\pi$
$258$	0	0
$259$	2.56087	0.159125
$260$	0	0
$261$	1.00000	0.0618984
$262$	0	0
$263$	5.08891	0.313795	0.156898	−	0.987615i	$-0.449851\pi$
$263$	5.08891	0.313795	0.156898	+	0.987615i	$0.449851\pi$
$264$	0	0
$265$	−24.9776	−1.53436
$266$	0	0
$267$	−8.05698	−0.493079
$268$	0	0
$269$	−4.38799	−0.267540	−0.133770	−	0.991012i	$-0.542708\pi$
$269$	−4.38799	−0.267540	−0.133770	+	0.991012i	$0.542708\pi$
$270$	0	0
$271$	10.9204	0.663367	0.331684	−	0.943391i	$-0.392383\pi$
$271$	10.9204	0.663367	0.331684	+	0.943391i	$0.392383\pi$
$272$	0	0
$273$	2.66300	0.161172
$274$	0	0
$275$	−44.6861	−2.69467
$276$	0	0
$277$	1.80890	0.108686	0.0543432	−	0.998522i	$-0.482693\pi$
$277$	1.80890	0.108686	0.0543432	+	0.998522i	$0.482693\pi$
$278$	0	0
$279$	1.35740	0.0812654
$280$	0	0
$281$	23.3353	1.39207	0.696035	−	0.718008i	$-0.254946\pi$
$281$	23.3353	1.39207	0.696035	+	0.718008i	$0.254946\pi$
$282$	0	0
$283$	−25.4070	−1.51029	−0.755144	−	0.655559i	$-0.772433\pi$
$283$	−25.4070	−1.51029	−0.755144	+	0.655559i	$0.772433\pi$
$284$	0	0
$285$	10.1079	0.598738
$286$	0	0
$287$	−10.4369	−0.616070
$288$	0	0
$289$	6.08765	0.358097
$290$	0	0
$291$	−8.87633	−0.520340
$292$	0	0
$293$	−23.0688	−1.34769	−0.673846	−	0.738872i	$-0.735359\pi$
$293$	−23.0688	−1.34769	−0.673846	+	0.738872i	$0.735359\pi$
$294$	0	0
$295$	10.3319	0.601549
$296$	0	0
$297$	5.80108	0.336613
$298$	0	0
$299$	2.78668	0.161158
$300$	0	0
$301$	5.62809	0.324397
$302$	0	0
$303$	−15.8122	−0.908388
$304$	0	0
$305$	30.0386	1.72001
$306$	0	0
$307$	−22.2938	−1.27237	−0.636187	−	0.771535i	$-0.719489\pi$
$307$	−22.2938	−1.27237	−0.636187	+	0.771535i	$0.719489\pi$
$308$	0	0
$309$	−6.02264	−0.342616
$310$	0	0
$311$	11.0508	0.626635	0.313317	−	0.949648i	$-0.398560\pi$
$311$	11.0508	0.626635	0.313317	+	0.949648i	$0.398560\pi$
$312$	0	0
$313$	−2.95462	−0.167005	−0.0835024	−	0.996508i	$-0.526611\pi$
$313$	−2.95462	−0.167005	−0.0835024	+	0.996508i	$0.526611\pi$
$314$	0	0
$315$	−3.40596	−0.191904
$316$	0	0
$317$	3.09880	0.174046	0.0870230	−	0.996206i	$-0.472265\pi$
$317$	3.09880	0.174046	0.0870230	+	0.996206i	$0.472265\pi$
$318$	0	0
$319$	−5.80108	−0.324798
$320$	0	0
$321$	−10.9137	−0.609141
$322$	0	0
$323$	13.6268	0.758217
$324$	0	0
$325$	21.4659	1.19072
$326$	0	0
$327$	−7.69560	−0.425567
$328$	0	0
$329$	0.503739	0.0277721
$330$	0	0
$331$	−16.6986	−0.917838	−0.458919	−	0.888478i	$-0.651763\pi$
$331$	−16.6986	−0.917838	−0.458919	+	0.888478i	$0.651763\pi$
$332$	0	0
$333$	−2.67980	−0.146852
$334$	0	0
$335$	39.2748	2.14581
$336$	0	0
$337$	13.8811	0.756151	0.378075	−	0.925775i	$-0.376586\pi$
$337$	13.8811	0.756151	0.378075	+	0.925775i	$0.376586\pi$
$338$	0	0
$339$	9.48154	0.514966
$340$	0	0
$341$	−7.87438	−0.426422
$342$	0	0
$343$	12.5060	0.675261
$344$	0	0
$345$	−3.56414	−0.191887
$346$	0	0
$347$	−15.9087	−0.854026	−0.427013	−	0.904245i	$-0.640434\pi$
$347$	−15.9087	−0.854026	−0.427013	+	0.904245i	$0.640434\pi$
$348$	0	0
$349$	19.4711	1.04227	0.521133	−	0.853476i	$-0.325510\pi$
$349$	19.4711	1.04227	0.521133	+	0.853476i	$0.325510\pi$
$350$	0	0
$351$	−2.78668	−0.148742
$352$	0	0
$353$	11.5300	0.613677	0.306839	−	0.951762i	$-0.400729\pi$
$353$	11.5300	0.613677	0.306839	+	0.951762i	$0.400729\pi$
$354$	0	0
$355$	44.9522	2.38582
$356$	0	0
$357$	−4.59172	−0.243020
$358$	0	0
$359$	2.90372	0.153253	0.0766263	−	0.997060i	$-0.475585\pi$
$359$	2.90372	0.153253	0.0766263	+	0.997060i	$0.475585\pi$
$360$	0	0
$361$	−10.9571	−0.576692
$362$	0	0
$363$	−22.6525	−1.18895
$364$	0	0
$365$	19.8950	1.04135
$366$	0	0
$367$	24.7354	1.29118	0.645588	−	0.763686i	$-0.276613\pi$
$367$	24.7354	1.29118	0.645588	+	0.763686i	$0.276613\pi$
$368$	0	0
$369$	10.9216	0.568554
$370$	0	0
$371$	6.69702	0.347692
$372$	0	0
$373$	18.0826	0.936283	0.468142	−	0.883654i	$-0.344924\pi$
$373$	18.0826	0.936283	0.468142	+	0.883654i	$0.344924\pi$
$374$	0	0
$375$	−9.63409	−0.497502
$376$	0	0
$377$	2.78668	0.143521
$378$	0	0
$379$	−16.0998	−0.826990	−0.413495	−	0.910506i	$-0.635692\pi$
$379$	−16.0998	−0.826990	−0.413495	+	0.910506i	$0.635692\pi$
$380$	0	0
$381$	3.02443	0.154946
$382$	0	0
$383$	−13.8440	−0.707395	−0.353697	−	0.935360i	$-0.615076\pi$
$383$	−13.8440	−0.707395	−0.353697	+	0.935360i	$0.615076\pi$
$384$	0	0
$385$	19.7583	1.00697
$386$	0	0
$387$	−5.88946	−0.299378
$388$	0	0
$389$	1.10286	0.0559171	0.0279585	−	0.999609i	$-0.491099\pi$
$389$	1.10286	0.0559171	0.0279585	+	0.999609i	$0.491099\pi$
$390$	0	0
$391$	−4.80496	−0.242997
$392$	0	0
$393$	−8.68706	−0.438204
$394$	0	0
$395$	−32.6316	−1.64187
$396$	0	0
$397$	15.7315	0.789544	0.394772	−	0.918779i	$-0.370824\pi$
$397$	15.7315	0.789544	0.394772	+	0.918779i	$0.370824\pi$
$398$	0	0
$399$	−2.71013	−0.135676
$400$	0	0
$401$	19.8224	0.989882	0.494941	−	0.868926i	$-0.335190\pi$
$401$	19.8224	0.989882	0.494941	+	0.868926i	$0.335190\pi$
$402$	0	0
$403$	3.78263	0.188426
$404$	0	0
$405$	3.56414	0.177103
$406$	0	0
$407$	15.5457	0.770573
$408$	0	0
$409$	17.2715	0.854022	0.427011	−	0.904246i	$-0.359567\pi$
$409$	17.2715	0.854022	0.427011	+	0.904246i	$0.359567\pi$
$410$	0	0
$411$	−3.01854	−0.148894
$412$	0	0
$413$	−2.77021	−0.136313
$414$	0	0
$415$	21.3387	1.04748
$416$	0	0
$417$	5.16341	0.252853
$418$	0	0
$419$	−8.01063	−0.391345	−0.195672	−	0.980669i	$-0.562689\pi$
$419$	−8.01063	−0.391345	−0.195672	+	0.980669i	$0.562689\pi$
$420$	0	0
$421$	−30.3538	−1.47935	−0.739676	−	0.672963i	$-0.765021\pi$
$421$	−30.3538	−1.47935	−0.739676	+	0.672963i	$0.765021\pi$
$422$	0	0
$423$	−0.527133	−0.0256301
$424$	0	0
$425$	−37.0129	−1.79539
$426$	0	0
$427$	−8.05399	−0.389760
$428$	0	0
$429$	16.1657	0.780489
$430$	0	0
$431$	9.19842	0.443072	0.221536	−	0.975152i	$-0.428893\pi$
$431$	9.19842	0.443072	0.221536	+	0.975152i	$0.428893\pi$
$432$	0	0
$433$	−0.834370	−0.0400973	−0.0200486	−	0.999799i	$-0.506382\pi$
$433$	−0.834370	−0.0400973	−0.0200486	+	0.999799i	$0.506382\pi$
$434$	0	0
$435$	−3.56414	−0.170887
$436$	0	0
$437$	−2.83599	−0.135664
$438$	0	0
$439$	22.2395	1.06143	0.530716	−	0.847550i	$-0.321923\pi$
$439$	22.2395	1.06143	0.530716	+	0.847550i	$0.321923\pi$
$440$	0	0
$441$	−6.08679	−0.289847
$442$	0	0
$443$	13.1992	0.627115	0.313558	−	0.949569i	$-0.398479\pi$
$443$	13.1992	0.627115	0.313558	+	0.949569i	$0.398479\pi$
$444$	0	0
$445$	28.7162	1.36128
$446$	0	0
$447$	−3.58064	−0.169359
$448$	0	0
$449$	−16.8450	−0.794964	−0.397482	−	0.917610i	$-0.630116\pi$
$449$	−16.8450	−0.794964	−0.397482	+	0.917610i	$0.630116\pi$
$450$	0	0
$451$	−63.3569	−2.98336
$452$	0	0
$453$	−2.96648	−0.139377
$454$	0	0
$455$	−9.49131	−0.444960
$456$	0	0
$457$	−41.8557	−1.95792	−0.978962	−	0.204041i	$-0.934593\pi$
$457$	−41.8557	−1.95792	−0.978962	+	0.204041i	$0.934593\pi$
$458$	0	0
$459$	4.80496	0.224276
$460$	0	0
$461$	11.4646	0.533961	0.266981	−	0.963702i	$-0.413974\pi$
$461$	11.4646	0.533961	0.266981	+	0.963702i	$0.413974\pi$
$462$	0	0
$463$	9.04554	0.420382	0.210191	−	0.977660i	$-0.432591\pi$
$463$	9.04554	0.420382	0.210191	+	0.977660i	$0.432591\pi$
$464$	0	0
$465$	−4.83796	−0.224355
$466$	0	0
$467$	9.38461	0.434268	0.217134	−	0.976142i	$-0.430329\pi$
$467$	9.38461	0.434268	0.217134	+	0.976142i	$0.430329\pi$
$468$	0	0
$469$	−10.5304	−0.486249
$470$	0	0
$471$	−18.2771	−0.842162
$472$	0	0
$473$	34.1652	1.57092
$474$	0	0
$475$	−21.8458	−1.00236
$476$	0	0
$477$	−7.00804	−0.320876
$478$	0	0
$479$	−1.45125	−0.0663091	−0.0331546	−	0.999450i	$-0.510555\pi$
$479$	−1.45125	−0.0663091	−0.0331546	+	0.999450i	$0.510555\pi$
$480$	0	0
$481$	−7.46773	−0.340499
$482$	0	0
$483$	0.955621	0.0434822
$484$	0	0
$485$	31.6364	1.43654
$486$	0	0
$487$	21.0211	0.952555	0.476277	−	0.879295i	$-0.341986\pi$
$487$	21.0211	0.952555	0.476277	+	0.879295i	$0.341986\pi$
$488$	0	0
$489$	−7.06758	−0.319607
$490$	0	0
$491$	−23.5710	−1.06374	−0.531871	−	0.846825i	$-0.678511\pi$
$491$	−23.5710	−1.06374	−0.531871	+	0.846825i	$0.678511\pi$
$492$	0	0
$493$	−4.80496	−0.216405
$494$	0	0
$495$	−20.6758	−0.929310
$496$	0	0
$497$	−12.0527	−0.540635
$498$	0	0
$499$	25.1831	1.12735	0.563676	−	0.825996i	$-0.309387\pi$
$499$	25.1831	1.12735	0.563676	+	0.825996i	$0.309387\pi$
$500$	0	0
$501$	0.993252	0.0443752
$502$	0	0
$503$	39.2545	1.75027	0.875136	−	0.483877i	$-0.160772\pi$
$503$	39.2545	1.75027	0.875136	+	0.483877i	$0.160772\pi$
$504$	0	0
$505$	56.3569	2.50785
$506$	0	0
$507$	5.23444	0.232469
$508$	0	0
$509$	29.5193	1.30842	0.654209	−	0.756314i	$-0.273002\pi$
$509$	29.5193	1.30842	0.654209	+	0.756314i	$0.273002\pi$
$510$	0	0
$511$	−5.33426	−0.235974
$512$	0	0
$513$	2.83599	0.125212
$514$	0	0
$515$	21.4655	0.945884
$516$	0	0
$517$	3.05794	0.134488
$518$	0	0
$519$	−10.1738	−0.446582
$520$	0	0
$521$	−0.834372	−0.0365545	−0.0182773	−	0.999833i	$-0.505818\pi$
$521$	−0.834372	−0.0365545	−0.0182773	+	0.999833i	$0.505818\pi$
$522$	0	0
$523$	−35.1951	−1.53898	−0.769488	−	0.638662i	$-0.779488\pi$
$523$	−35.1951	−1.53898	−0.769488	+	0.638662i	$0.779488\pi$
$524$	0	0
$525$	7.36121	0.321269
$526$	0	0
$527$	−6.52225	−0.284114
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	2.89886	0.125800
$532$	0	0
$533$	30.4349	1.31828
$534$	0	0
$535$	38.8977	1.68170
$536$	0	0
$537$	−13.2458	−0.571597
$538$	0	0
$539$	35.3099	1.52091
$540$	0	0
$541$	26.0125	1.11836	0.559181	−	0.829045i	$-0.311116\pi$
$541$	26.0125	1.11836	0.559181	+	0.829045i	$0.311116\pi$
$542$	0	0
$543$	6.67164	0.286308
$544$	0	0
$545$	27.4281	1.17489
$546$	0	0
$547$	25.3017	1.08182	0.540910	−	0.841080i	$-0.318080\pi$
$547$	25.3017	1.08182	0.540910	+	0.841080i	$0.318080\pi$
$548$	0	0
$549$	8.42802	0.359699
$550$	0	0
$551$	−2.83599	−0.120817
$552$	0	0
$553$	8.74922	0.372055
$554$	0	0
$555$	9.55117	0.405425
$556$	0	0
$557$	−23.6524	−1.00219	−0.501093	−	0.865393i	$-0.667069\pi$
$557$	−23.6524	−1.00219	−0.501093	+	0.865393i	$0.667069\pi$
$558$	0	0
$559$	−16.4120	−0.694154
$560$	0	0
$561$	−27.8740	−1.17684
$562$	0	0
$563$	−3.37852	−0.142388	−0.0711939	−	0.997462i	$-0.522681\pi$
$563$	−3.37852	−0.142388	−0.0711939	+	0.997462i	$0.522681\pi$
$564$	0	0
$565$	−33.7935	−1.42170
$566$	0	0
$567$	−0.955621	−0.0401323
$568$	0	0
$569$	−2.95456	−0.123862	−0.0619308	−	0.998080i	$-0.519726\pi$
$569$	−2.95456	−0.123862	−0.0619308	+	0.998080i	$0.519726\pi$
$570$	0	0
$571$	−3.06701	−0.128350	−0.0641752	−	0.997939i	$-0.520442\pi$
$571$	−3.06701	−0.128350	−0.0641752	+	0.997939i	$0.520442\pi$
$572$	0	0
$573$	8.22295	0.343518
$574$	0	0
$575$	7.70306	0.321240
$576$	0	0
$577$	−23.4116	−0.974637	−0.487319	−	0.873224i	$-0.662025\pi$
$577$	−23.4116	−0.974637	−0.487319	+	0.873224i	$0.662025\pi$
$578$	0	0
$579$	9.08660	0.377626
$580$	0	0
$581$	−5.72136	−0.237362
$582$	0	0
$583$	40.6542	1.68372
$584$	0	0
$585$	9.93209	0.410641
$586$	0	0
$587$	31.7728	1.31140	0.655702	−	0.755019i	$-0.272373\pi$
$587$	31.7728	1.31140	0.655702	+	0.755019i	$0.272373\pi$
$588$	0	0
$589$	−3.84957	−0.158619
$590$	0	0
$591$	−9.97613	−0.410363
$592$	0	0
$593$	29.9630	1.23043	0.615216	−	0.788358i	$-0.289069\pi$
$593$	29.9630	1.23043	0.615216	+	0.788358i	$0.289069\pi$
$594$	0	0
$595$	16.3655	0.670921
$596$	0	0
$597$	−12.4190	−0.508277
$598$	0	0
$599$	15.7292	0.642677	0.321339	−	0.946964i	$-0.395867\pi$
$599$	15.7292	0.642677	0.321339	+	0.946964i	$0.395867\pi$
$600$	0	0
$601$	6.85702	0.279703	0.139852	−	0.990172i	$-0.455337\pi$
$601$	6.85702	0.279703	0.139852	+	0.990172i	$0.455337\pi$
$602$	0	0
$603$	11.0194	0.448746
$604$	0	0
$605$	80.7366	3.28241
$606$	0	0
$607$	−7.61547	−0.309102	−0.154551	−	0.987985i	$-0.549393\pi$
$607$	−7.61547	−0.309102	−0.154551	+	0.987985i	$0.549393\pi$
$608$	0	0
$609$	0.955621	0.0387237
$610$	0	0
$611$	−1.46895	−0.0594274
$612$	0	0
$613$	−29.1497	−1.17734	−0.588672	−	0.808372i	$-0.700349\pi$
$613$	−29.1497	−1.17734	−0.588672	+	0.808372i	$0.700349\pi$
$614$	0	0
$615$	−38.9260	−1.56965
$616$	0	0
$617$	−25.8141	−1.03924	−0.519618	−	0.854399i	$-0.673926\pi$
$617$	−25.8141	−1.03924	−0.519618	+	0.854399i	$0.673926\pi$
$618$	0	0
$619$	7.90003	0.317529	0.158764	−	0.987316i	$-0.449249\pi$
$619$	7.90003	0.317529	0.158764	+	0.987316i	$0.449249\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	−7.69941	−0.308470
$624$	0	0
$625$	−4.17812	−0.167125
$626$	0	0
$627$	−16.4518	−0.657022
$628$	0	0
$629$	12.8763	0.513413
$630$	0	0
$631$	−0.734933	−0.0292572	−0.0146286	−	0.999893i	$-0.504657\pi$
$631$	−0.734933	−0.0292572	−0.0146286	+	0.999893i	$0.504657\pi$
$632$	0	0
$633$	20.1635	0.801428
$634$	0	0
$635$	−10.7795	−0.427771
$636$	0	0
$637$	−16.9619	−0.672055
$638$	0	0
$639$	12.6124	0.498938
$640$	0	0
$641$	−1.91285	−0.0755529	−0.0377765	−	0.999286i	$-0.512027\pi$
$641$	−1.91285	−0.0755529	−0.0377765	+	0.999286i	$0.512027\pi$
$642$	0	0
$643$	−35.0815	−1.38348	−0.691739	−	0.722148i	$-0.743155\pi$
$643$	−35.0815	−1.38348	−0.691739	+	0.722148i	$0.743155\pi$
$644$	0	0
$645$	20.9908	0.826513
$646$	0	0
$647$	48.3884	1.90234	0.951172	−	0.308661i	$-0.0998807\pi$
$647$	48.3884	1.90234	0.951172	+	0.308661i	$0.0998807\pi$
$648$	0	0
$649$	−16.8165	−0.660106
$650$	0	0
$651$	1.29716	0.0508397
$652$	0	0
$653$	−28.9349	−1.13231	−0.566154	−	0.824299i	$-0.691569\pi$
$653$	−28.9349	−1.13231	−0.566154	+	0.824299i	$0.691569\pi$
$654$	0	0
$655$	30.9618	1.20978
$656$	0	0
$657$	5.58199	0.217774
$658$	0	0
$659$	13.9584	0.543740	0.271870	−	0.962334i	$-0.412358\pi$
$659$	13.9584	0.543740	0.271870	+	0.962334i	$0.412358\pi$
$660$	0	0
$661$	−1.51563	−0.0589514	−0.0294757	−	0.999565i	$-0.509384\pi$
$661$	−1.51563	−0.0589514	−0.0294757	+	0.999565i	$0.509384\pi$
$662$	0	0
$663$	13.3899	0.520019
$664$	0	0
$665$	9.65928	0.374571
$666$	0	0
$667$	1.00000	0.0387202
$668$	0	0
$669$	−21.7862	−0.842303
$670$	0	0
$671$	−48.8916	−1.88744
$672$	0	0
$673$	17.6349	0.679777	0.339888	−	0.940466i	$-0.389611\pi$
$673$	17.6349	0.679777	0.339888	+	0.940466i	$0.389611\pi$
$674$	0	0
$675$	−7.70306	−0.296491
$676$	0	0
$677$	4.50314	0.173070	0.0865348	−	0.996249i	$-0.472421\pi$
$677$	4.50314	0.173070	0.0865348	+	0.996249i	$0.472421\pi$
$678$	0	0
$679$	−8.48240	−0.325525
$680$	0	0
$681$	−0.0750304	−0.00287517
$682$	0	0
$683$	26.3201	1.00711	0.503556	−	0.863963i	$-0.332025\pi$
$683$	26.3201	1.00711	0.503556	+	0.863963i	$0.332025\pi$
$684$	0	0
$685$	10.7585	0.411061
$686$	0	0
$687$	−15.9524	−0.608624
$688$	0	0
$689$	−19.5291	−0.744000
$690$	0	0
$691$	−7.65638	−0.291262	−0.145631	−	0.989339i	$-0.546521\pi$
$691$	−7.65638	−0.291262	−0.145631	+	0.989339i	$0.546521\pi$
$692$	0	0
$693$	5.54363	0.210585
$694$	0	0
$695$	−18.4031	−0.698070
$696$	0	0
$697$	−52.4777	−1.98774
$698$	0	0
$699$	13.0981	0.495414
$700$	0	0
$701$	36.4761	1.37768	0.688842	−	0.724912i	$-0.258120\pi$
$701$	36.4761	1.37768	0.688842	+	0.724912i	$0.258120\pi$
$702$	0	0
$703$	7.59989	0.286635
$704$	0	0
$705$	1.87877	0.0707588
$706$	0	0
$707$	−15.1105	−0.568288
$708$	0	0
$709$	−7.48088	−0.280950	−0.140475	−	0.990084i	$-0.544863\pi$
$709$	−7.48088	−0.280950	−0.140475	+	0.990084i	$0.544863\pi$
$710$	0	0
$711$	−9.15554	−0.343360
$712$	0	0
$713$	1.35740	0.0508350
$714$	0	0
$715$	−57.6169	−2.15475
$716$	0	0
$717$	27.9484	1.04375
$718$	0	0
$719$	−0.709976	−0.0264776	−0.0132388	−	0.999912i	$-0.504214\pi$
$719$	−0.709976	−0.0264776	−0.0132388	+	0.999912i	$0.504214\pi$
$720$	0	0
$721$	−5.75536	−0.214341
$722$	0	0
$723$	23.4964	0.873839
$724$	0	0
$725$	7.70306	0.286085
$726$	0	0
$727$	−1.81386	−0.0672724	−0.0336362	−	0.999434i	$-0.510709\pi$
$727$	−1.81386	−0.0672724	−0.0336362	+	0.999434i	$0.510709\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	28.2986	1.04666
$732$	0	0
$733$	38.4328	1.41955	0.709774	−	0.704430i	$-0.248797\pi$
$733$	38.4328	1.41955	0.709774	+	0.704430i	$0.248797\pi$
$734$	0	0
$735$	21.6941	0.800201
$736$	0	0
$737$	−63.9247	−2.35469
$738$	0	0
$739$	9.80458	0.360667	0.180334	−	0.983605i	$-0.442282\pi$
$739$	9.80458	0.360667	0.180334	+	0.983605i	$0.442282\pi$
$740$	0	0
$741$	7.90299	0.290324
$742$	0	0
$743$	−34.2477	−1.25643	−0.628214	−	0.778041i	$-0.716214\pi$
$743$	−34.2477	−1.25643	−0.628214	+	0.778041i	$0.716214\pi$
$744$	0	0
$745$	12.7619	0.467560
$746$	0	0
$747$	5.98706	0.219055
$748$	0	0
$749$	−10.4293	−0.381079
$750$	0	0
$751$	−27.2508	−0.994398	−0.497199	−	0.867637i	$-0.665638\pi$
$751$	−27.2508	−0.994398	−0.497199	+	0.867637i	$0.665638\pi$
$752$	0	0
$753$	−6.46207	−0.235491
$754$	0	0
$755$	10.5729	0.384788
$756$	0	0
$757$	−2.11360	−0.0768202	−0.0384101	−	0.999262i	$-0.512229\pi$
$757$	−2.11360	−0.0768202	−0.0384101	+	0.999262i	$0.512229\pi$
$758$	0	0
$759$	5.80108	0.210566
$760$	0	0
$761$	−34.5154	−1.25118	−0.625590	−	0.780152i	$-0.715142\pi$
$761$	−34.5154	−1.25118	−0.625590	+	0.780152i	$0.715142\pi$
$762$	0	0
$763$	−7.35407	−0.266235
$764$	0	0
$765$	−17.1255	−0.619175
$766$	0	0
$767$	8.07819	0.291687
$768$	0	0
$769$	−48.9236	−1.76423	−0.882115	−	0.471035i	$-0.843881\pi$
$769$	−48.9236	−1.76423	−0.882115	+	0.471035i	$0.843881\pi$
$770$	0	0
$771$	9.92325	0.357377
$772$	0	0
$773$	25.5483	0.918909	0.459454	−	0.888201i	$-0.348045\pi$
$773$	25.5483	0.918909	0.459454	+	0.888201i	$0.348045\pi$
$774$	0	0
$775$	10.4561	0.375596
$776$	0	0
$777$	−2.56087	−0.0918707
$778$	0	0
$779$	−30.9735	−1.10974
$780$	0	0
$781$	−73.1654	−2.61806
$782$	0	0
$783$	−1.00000	−0.0357371
$784$	0	0
$785$	65.1419	2.32501
$786$	0	0
$787$	12.2817	0.437794	0.218897	−	0.975748i	$-0.429754\pi$
$787$	12.2817	0.437794	0.218897	+	0.975748i	$0.429754\pi$
$788$	0	0
$789$	−5.08891	−0.181170
$790$	0	0
$791$	9.06075	0.322163
$792$	0	0
$793$	23.4862	0.834018
$794$	0	0
$795$	24.9776	0.885864
$796$	0	0
$797$	5.91892	0.209659	0.104829	−	0.994490i	$-0.466570\pi$
$797$	5.91892	0.209659	0.104829	+	0.994490i	$0.466570\pi$
$798$	0	0
$799$	2.53286	0.0896060
$800$	0	0
$801$	8.05698	0.284679
$802$	0	0
$803$	−32.3815	−1.14272
$804$	0	0
$805$	−3.40596	−0.120044
$806$	0	0
$807$	4.38799	0.154465
$808$	0	0
$809$	−17.4302	−0.612813	−0.306406	−	0.951901i	$-0.599127\pi$
$809$	−17.4302	−0.612813	−0.306406	+	0.951901i	$0.599127\pi$
$810$	0	0
$811$	44.5700	1.56506	0.782532	−	0.622611i	$-0.213928\pi$
$811$	44.5700	1.56506	0.782532	+	0.622611i	$0.213928\pi$
$812$	0	0
$813$	−10.9204	−0.382995
$814$	0	0
$815$	25.1898	0.882360
$816$	0	0
$817$	16.7025	0.584345
$818$	0	0
$819$	−2.66300	−0.0930529
$820$	0	0
$821$	18.6106	0.649513	0.324757	−	0.945798i	$-0.394718\pi$
$821$	18.6106	0.649513	0.324757	+	0.945798i	$0.394718\pi$
$822$	0	0
$823$	−39.8324	−1.38847	−0.694235	−	0.719749i	$-0.744257\pi$
$823$	−39.8324	−1.38847	−0.694235	+	0.719749i	$0.744257\pi$
$824$	0	0
$825$	44.6861	1.55577
$826$	0	0
$827$	−28.0395	−0.975030	−0.487515	−	0.873115i	$-0.662097\pi$
$827$	−28.0395	−0.975030	−0.487515	+	0.873115i	$0.662097\pi$
$828$	0	0
$829$	35.5688	1.23535	0.617677	−	0.786432i	$-0.288074\pi$
$829$	35.5688	1.23535	0.617677	+	0.786432i	$0.288074\pi$
$830$	0	0
$831$	−1.80890	−0.0627502
$832$	0	0
$833$	29.2468	1.01334
$834$	0	0
$835$	−3.54009	−0.122510
$836$	0	0
$837$	−1.35740	−0.0469186
$838$	0	0
$839$	51.4430	1.77601	0.888005	−	0.459834i	$-0.152091\pi$
$839$	51.4430	1.77601	0.888005	+	0.459834i	$0.152091\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−23.3353	−0.803712
$844$	0	0
$845$	−18.6562	−0.641794
$846$	0	0
$847$	−21.6472	−0.743807
$848$	0	0
$849$	25.4070	0.871965
$850$	0	0
$851$	−2.67980	−0.0918623
$852$	0	0
$853$	−37.6496	−1.28910	−0.644550	−	0.764562i	$-0.722955\pi$
$853$	−37.6496	−1.28910	−0.644550	+	0.764562i	$0.722955\pi$
$854$	0	0
$855$	−10.1079	−0.345682
$856$	0	0
$857$	33.4469	1.14252	0.571262	−	0.820768i	$-0.306454\pi$
$857$	33.4469	1.14252	0.571262	+	0.820768i	$0.306454\pi$
$858$	0	0
$859$	5.19282	0.177177	0.0885884	−	0.996068i	$-0.471764\pi$
$859$	5.19282	0.177177	0.0885884	+	0.996068i	$0.471764\pi$
$860$	0	0
$861$	10.4369	0.355688
$862$	0	0
$863$	28.6534	0.975374	0.487687	−	0.873019i	$-0.337841\pi$
$863$	28.6534	0.975374	0.487687	+	0.873019i	$0.337841\pi$
$864$	0	0
$865$	36.2609	1.23291
$866$	0	0
$867$	−6.08765	−0.206747
$868$	0	0
$869$	53.1120	1.80170
$870$	0	0
$871$	30.7076	1.04049
$872$	0	0
$873$	8.87633	0.300418
$874$	0	0
$875$	−9.20653	−0.311238
$876$	0	0
$877$	1.06764	0.0360517	0.0180259	−	0.999838i	$-0.494262\pi$
$877$	1.06764	0.0360517	0.0180259	+	0.999838i	$0.494262\pi$
$878$	0	0
$879$	23.0688	0.778091
$880$	0	0
$881$	28.4201	0.957499	0.478749	−	0.877952i	$-0.341090\pi$
$881$	28.4201	0.957499	0.478749	+	0.877952i	$0.341090\pi$
$882$	0	0
$883$	−9.84307	−0.331246	−0.165623	−	0.986189i	$-0.552963\pi$
$883$	−9.84307	−0.331246	−0.165623	+	0.986189i	$0.552963\pi$
$884$	0	0
$885$	−10.3319	−0.347304
$886$	0	0
$887$	42.3392	1.42161	0.710806	−	0.703388i	$-0.248330\pi$
$887$	42.3392	1.42161	0.710806	+	0.703388i	$0.248330\pi$
$888$	0	0
$889$	2.89021	0.0969346
$890$	0	0
$891$	−5.80108	−0.194343
$892$	0	0
$893$	1.49495	0.0500265
$894$	0	0
$895$	47.2097	1.57805
$896$	0	0
$897$	−2.78668	−0.0930444
$898$	0	0
$899$	1.35740	0.0452718
$900$	0	0
$901$	33.6733	1.12182
$902$	0	0
$903$	−5.62809	−0.187291
$904$	0	0
$905$	−23.7786	−0.790429
$906$	0	0
$907$	21.7926	0.723610	0.361805	−	0.932254i	$-0.382161\pi$
$907$	21.7926	0.723610	0.361805	+	0.932254i	$0.382161\pi$
$908$	0	0
$909$	15.8122	0.524458
$910$	0	0
$911$	−32.6889	−1.08303	−0.541517	−	0.840690i	$-0.682150\pi$
$911$	−32.6889	−1.08303	−0.541517	+	0.840690i	$0.682150\pi$
$912$	0	0
$913$	−34.7314	−1.14944
$914$	0	0
$915$	−30.0386	−0.993046
$916$	0	0
$917$	−8.30153	−0.274141
$918$	0	0
$919$	6.57313	0.216828	0.108414	−	0.994106i	$-0.465423\pi$
$919$	6.57313	0.216828	0.108414	+	0.994106i	$0.465423\pi$
$920$	0	0
$921$	22.2938	0.734606
$922$	0	0
$923$	35.1466	1.15687
$924$	0	0
$925$	−20.6427	−0.678727
$926$	0	0
$927$	6.02264	0.197810
$928$	0	0
$929$	−20.7768	−0.681664	−0.340832	−	0.940124i	$-0.610709\pi$
$929$	−20.7768	−0.681664	−0.340832	+	0.940124i	$0.610709\pi$
$930$	0	0
$931$	17.2621	0.565742
$932$	0	0
$933$	−11.0508	−0.361788
$934$	0	0
$935$	99.3466	3.24898
$936$	0	0
$937$	−24.7579	−0.808805	−0.404403	−	0.914581i	$-0.632521\pi$
$937$	−24.7579	−0.808805	−0.404403	+	0.914581i	$0.632521\pi$
$938$	0	0
$939$	2.95462	0.0964203
$940$	0	0
$941$	6.82478	0.222481	0.111241	−	0.993793i	$-0.464518\pi$
$941$	6.82478	0.222481	0.111241	+	0.993793i	$0.464518\pi$
$942$	0	0
$943$	10.9216	0.355655
$944$	0	0
$945$	3.40596	0.110796
$946$	0	0
$947$	38.4065	1.24804	0.624022	−	0.781407i	$-0.285497\pi$
$947$	38.4065	1.24804	0.624022	+	0.781407i	$0.285497\pi$
$948$	0	0
$949$	15.5552	0.504943
$950$	0	0
$951$	−3.09880	−0.100485
$952$	0	0
$953$	−37.7845	−1.22396	−0.611981	−	0.790873i	$-0.709627\pi$
$953$	−37.7845	−1.22396	−0.611981	+	0.790873i	$0.709627\pi$
$954$	0	0
$955$	−29.3077	−0.948375
$956$	0	0
$957$	5.80108	0.187522
$958$	0	0
$959$	−2.88458	−0.0931480
$960$	0	0
$961$	−29.1575	−0.940563
$962$	0	0
$963$	10.9137	0.351687
$964$	0	0
$965$	−32.3859	−1.04254
$966$	0	0
$967$	40.8040	1.31217	0.656085	−	0.754687i	$-0.272211\pi$
$967$	40.8040	1.31217	0.656085	+	0.754687i	$0.272211\pi$
$968$	0	0
$969$	−13.6268	−0.437757
$970$	0	0
$971$	−27.4923	−0.882271	−0.441136	−	0.897440i	$-0.645424\pi$
$971$	−27.4923	−0.882271	−0.441136	+	0.897440i	$0.645424\pi$
$972$	0	0
$973$	4.93426	0.158185
$974$	0	0
$975$	−21.4659	−0.687460
$976$	0	0
$977$	6.64545	0.212607	0.106303	−	0.994334i	$-0.466099\pi$
$977$	6.64545	0.212607	0.106303	+	0.994334i	$0.466099\pi$
$978$	0	0
$979$	−46.7392	−1.49379
$980$	0	0
$981$	7.69560	0.245701
$982$	0	0
$983$	−16.1250	−0.514307	−0.257153	−	0.966371i	$-0.582785\pi$
$983$	−16.1250	−0.514307	−0.257153	+	0.966371i	$0.582785\pi$
$984$	0	0
$985$	35.5563	1.13292
$986$	0	0
$987$	−0.503739	−0.0160342
$988$	0	0
$989$	−5.88946	−0.187274
$990$	0	0
$991$	−31.9866	−1.01609	−0.508044	−	0.861331i	$-0.669631\pi$
$991$	−31.9866	−1.01609	−0.508044	+	0.861331i	$0.669631\pi$
$992$	0	0
$993$	16.6986	0.529914
$994$	0	0
$995$	44.2631	1.40323
$996$	0	0
$997$	−57.6602	−1.82612	−0.913058	−	0.407829i	$-0.866286\pi$
$997$	−57.6602	−1.82612	−0.913058	+	0.407829i	$0.866286\pi$
$998$	0	0
$999$	2.67980	0.0847851

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.i.1.14	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.i.1.14	✓	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +3.56414 q^{5} -0.955621 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +3.56414 q^{5} -0.955621 q^{7} +1.00000 q^{9} -5.80108 q^{11} +2.78668 q^{13} -3.56414 q^{15} -4.80496 q^{17} -2.83599 q^{19} +0.955621 q^{21} +1.00000 q^{23} +7.70306 q^{25} -1.00000 q^{27} +1.00000 q^{29} +1.35740 q^{31} +5.80108 q^{33} -3.40596 q^{35} -2.67980 q^{37} -2.78668 q^{39} +10.9216 q^{41} -5.88946 q^{43} +3.56414 q^{45} -0.527133 q^{47} -6.08679 q^{49} +4.80496 q^{51} -7.00804 q^{53} -20.6758 q^{55} +2.83599 q^{57} +2.89886 q^{59} +8.42802 q^{61} -0.955621 q^{63} +9.93209 q^{65} +11.0194 q^{67} -1.00000 q^{69} +12.6124 q^{71} +5.58199 q^{73} -7.70306 q^{75} +5.54363 q^{77} -9.15554 q^{79} +1.00000 q^{81} +5.98706 q^{83} -17.1255 q^{85} -1.00000 q^{87} +8.05698 q^{89} -2.66300 q^{91} -1.35740 q^{93} -10.1079 q^{95} +8.87633 q^{97} -5.80108 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{3} + 5 q^{5} - 4 q^{7} + 16 q^{9}+O(q^{10}) \) 16 * q - 16 * q^3 + 5 * q^5 - 4 * q^7 + 16 * q^9	\( 16 q - 16 q^{3} + 5 q^{5} - 4 q^{7} + 16 q^{9} + 5 q^{11} + 8 q^{13} - 5 q^{15} + 7 q^{17} + q^{19} + 4 q^{21} + 16 q^{23} + 31 q^{25} - 16 q^{27} + 16 q^{29} - 2 q^{31} - 5 q^{33} + 5 q^{35} + 14 q^{37} - 8 q^{39} - q^{41} - 13 q^{43} + 5 q^{45} - 4 q^{47} + 30 q^{49} - 7 q^{51} + 19 q^{53} - 37 q^{55} - q^{57} + 12 q^{59} + 21 q^{61} - 4 q^{63} + 26 q^{65} - 11 q^{67} - 16 q^{69} + 7 q^{71} - 13 q^{73} - 31 q^{75} + 4 q^{77} - 18 q^{79} + 16 q^{81} + 25 q^{83} + 48 q^{85} - 16 q^{87} + 12 q^{89} - 11 q^{91} + 2 q^{93} - q^{95} + 5 q^{97} + 5 q^{99}+O(q^{100}) \) 16 * q - 16 * q^3 + 5 * q^5 - 4 * q^7 + 16 * q^9 + 5 * q^11 + 8 * q^13 - 5 * q^15 + 7 * q^17 + q^19 + 4 * q^21 + 16 * q^23 + 31 * q^25 - 16 * q^27 + 16 * q^29 - 2 * q^31 - 5 * q^33 + 5 * q^35 + 14 * q^37 - 8 * q^39 - q^41 - 13 * q^43 + 5 * q^45 - 4 * q^47 + 30 * q^49 - 7 * q^51 + 19 * q^53 - 37 * q^55 - q^57 + 12 * q^59 + 21 * q^61 - 4 * q^63 + 26 * q^65 - 11 * q^67 - 16 * q^69 + 7 * q^71 - 13 * q^73 - 31 * q^75 + 4 * q^77 - 18 * q^79 + 16 * q^81 + 25 * q^83 + 48 * q^85 - 16 * q^87 + 12 * q^89 - 11 * q^91 + 2 * q^93 - q^95 + 5 * q^97 + 5 * q^99

Introduction

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