LMFDB - Embedded newform 8004.2.a.g.1.6

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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 31 x^{10} + 80 x^{9} + 347 x^{8} - 697 x^{7} - 1714 x^{6} + 2146 x^{5} + 3304 x^{4} + \cdots + 16 $ x^12 - 3x^11 - 31x^10 + 80x^9 + 347x^8 - 697x^7 - 1714x^6 + 2146x^5 + 3304x^4 - 1156x^3 - 616x^2 + 136*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$0.305886$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -0.305886 q^{5} -0.140026 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -0.305886 q^{5} -0.140026 q^{7} +1.00000 q^{9} +1.85849 q^{11} -1.94243 q^{13} +0.305886 q^{15} +5.66444 q^{17} -1.04650 q^{19} +0.140026 q^{21} +1.00000 q^{23} -4.90643 q^{25} -1.00000 q^{27} -1.00000 q^{29} -6.72475 q^{31} -1.85849 q^{33} +0.0428320 q^{35} -1.42575 q^{37} +1.94243 q^{39} +0.522747 q^{41} +4.27739 q^{43} -0.305886 q^{45} +7.07300 q^{47} -6.98039 q^{49} -5.66444 q^{51} -11.0449 q^{53} -0.568487 q^{55} +1.04650 q^{57} -10.0473 q^{59} +7.67091 q^{61} -0.140026 q^{63} +0.594162 q^{65} -12.0677 q^{67} -1.00000 q^{69} +8.29946 q^{71} -0.0686045 q^{73} +4.90643 q^{75} -0.260237 q^{77} +8.58428 q^{79} +1.00000 q^{81} +7.65463 q^{83} -1.73267 q^{85} +1.00000 q^{87} +8.93734 q^{89} +0.271991 q^{91} +6.72475 q^{93} +0.320111 q^{95} +4.43366 q^{97} +1.85849 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{3} - 3 q^{5} + 4 q^{7} + 12 q^{9}+O(q^{10}) $ 12 * q - 12 * q^3 - 3 * q^5 + 4 * q^7 + 12 * q^9	$ 12 q - 12 q^{3} - 3 q^{5} + 4 q^{7} + 12 q^{9} - 5 q^{11} - 6 q^{13} + 3 q^{15} - 7 q^{17} - 3 q^{19} - 4 q^{21} + 12 q^{23} + 11 q^{25} - 12 q^{27} - 12 q^{29} + 2 q^{31} + 5 q^{33} - 9 q^{35} - 20 q^{37} + 6 q^{39} - 3 q^{41} + 5 q^{43} - 3 q^{45} - 2 q^{49} + 7 q^{51} - 3 q^{53} + 19 q^{55} + 3 q^{57} - 20 q^{59} - 17 q^{61} + 4 q^{63} - 4 q^{65} - 9 q^{67} - 12 q^{69} + 7 q^{71} - 9 q^{73} - 11 q^{75} - 34 q^{77} + 14 q^{79} + 12 q^{81} + 5 q^{83} - 12 q^{85} + 12 q^{87} - 22 q^{89} - 3 q^{91} - 2 q^{93} - 27 q^{95} + 17 q^{97} - 5 q^{99}+O(q^{100}) $ 12 * q - 12 * q^3 - 3 * q^5 + 4 * q^7 + 12 * q^9 - 5 * q^11 - 6 * q^13 + 3 * q^15 - 7 * q^17 - 3 * q^19 - 4 * q^21 + 12 * q^23 + 11 * q^25 - 12 * q^27 - 12 * q^29 + 2 * q^31 + 5 * q^33 - 9 * q^35 - 20 * q^37 + 6 * q^39 - 3 * q^41 + 5 * q^43 - 3 * q^45 - 2 * q^49 + 7 * q^51 - 3 * q^53 + 19 * q^55 + 3 * q^57 - 20 * q^59 - 17 * q^61 + 4 * q^63 - 4 * q^65 - 9 * q^67 - 12 * q^69 + 7 * q^71 - 9 * q^73 - 11 * q^75 - 34 * q^77 + 14 * q^79 + 12 * q^81 + 5 * q^83 - 12 * q^85 + 12 * q^87 - 22 * q^89 - 3 * q^91 - 2 * q^93 - 27 * q^95 + 17 * 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
$5$	−0.305886	−0.136796	−0.0683982	−	0.997658i	$-0.521789\pi$
$5$	−0.305886	−0.136796	−0.0683982	+	0.997658i	$0.521789\pi$
$6$	0	0
$7$	−0.140026	−0.0529249	−0.0264624	−	0.999650i	$-0.508424\pi$
$7$	−0.140026	−0.0529249	−0.0264624	+	0.999650i	$0.508424\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.85849	0.560356	0.280178	−	0.959948i	$-0.409606\pi$
$11$	1.85849	0.560356	0.280178	+	0.959948i	$0.409606\pi$
$12$	0	0
$13$	−1.94243	−0.538733	−0.269367	−	0.963038i	$-0.586814\pi$
$13$	−1.94243	−0.538733	−0.269367	+	0.963038i	$0.586814\pi$
$14$	0	0
$15$	0.305886	0.0789795
$16$	0	0
$17$	5.66444	1.37383	0.686914	−	0.726738i	$-0.258965\pi$
$17$	5.66444	1.37383	0.686914	+	0.726738i	$0.258965\pi$
$18$	0	0
$19$	−1.04650	−0.240084	−0.120042	−	0.992769i	$-0.538303\pi$
$19$	−1.04650	−0.240084	−0.120042	+	0.992769i	$0.538303\pi$
$20$	0	0
$21$	0.140026	0.0305562
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−4.90643	−0.981287
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−6.72475	−1.20780	−0.603901	−	0.797060i	$-0.706388\pi$
$31$	−6.72475	−1.20780	−0.603901	+	0.797060i	$0.706388\pi$
$32$	0	0
$33$	−1.85849	−0.323522
$34$	0	0
$35$	0.0428320	0.00723993
$36$	0	0
$37$	−1.42575	−0.234392	−0.117196	−	0.993109i	$-0.537391\pi$
$37$	−1.42575	−0.234392	−0.117196	+	0.993109i	$0.537391\pi$
$38$	0	0
$39$	1.94243	0.311038
$40$	0	0
$41$	0.522747	0.0816394	0.0408197	−	0.999167i	$-0.487003\pi$
$41$	0.522747	0.0816394	0.0408197	+	0.999167i	$0.487003\pi$
$42$	0	0
$43$	4.27739	0.652296	0.326148	−	0.945319i	$-0.394249\pi$
$43$	4.27739	0.652296	0.326148	+	0.945319i	$0.394249\pi$
$44$	0	0
$45$	−0.305886	−0.0455988
$46$	0	0
$47$	7.07300	1.03170	0.515852	−	0.856678i	$-0.327476\pi$
$47$	7.07300	1.03170	0.515852	+	0.856678i	$0.327476\pi$
$48$	0	0
$49$	−6.98039	−0.997199
$50$	0	0
$51$	−5.66444	−0.793180
$52$	0	0
$53$	−11.0449	−1.51713	−0.758567	−	0.651595i	$-0.774100\pi$
$53$	−11.0449	−1.51713	−0.758567	+	0.651595i	$0.774100\pi$
$54$	0	0
$55$	−0.568487	−0.0766548
$56$	0	0
$57$	1.04650	0.138613
$58$	0	0
$59$	−10.0473	−1.30804	−0.654021	−	0.756476i	$-0.726919\pi$
$59$	−10.0473	−1.30804	−0.654021	+	0.756476i	$0.726919\pi$
$60$	0	0
$61$	7.67091	0.982159	0.491080	−	0.871115i	$-0.336602\pi$
$61$	7.67091	0.982159	0.491080	+	0.871115i	$0.336602\pi$
$62$	0	0
$63$	−0.140026	−0.0176416
$64$	0	0
$65$	0.594162	0.0736968
$66$	0	0
$67$	−12.0677	−1.47430	−0.737149	−	0.675730i	$-0.763829\pi$
$67$	−12.0677	−1.47430	−0.737149	+	0.675730i	$0.763829\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	8.29946	0.984965	0.492482	−	0.870322i	$-0.336090\pi$
$71$	8.29946	0.984965	0.492482	+	0.870322i	$0.336090\pi$
$72$	0	0
$73$	−0.0686045	−0.00802955	−0.00401478	−	0.999992i	$-0.501278\pi$
$73$	−0.0686045	−0.00802955	−0.00401478	+	0.999992i	$0.501278\pi$
$74$	0	0
$75$	4.90643	0.566546
$76$	0	0
$77$	−0.260237	−0.0296568
$78$	0	0
$79$	8.58428	0.965807	0.482903	−	0.875674i	$-0.339582\pi$
$79$	8.58428	0.965807	0.482903	+	0.875674i	$0.339582\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	7.65463	0.840205	0.420102	−	0.907477i	$-0.361994\pi$
$83$	7.65463	0.840205	0.420102	+	0.907477i	$0.361994\pi$
$84$	0	0
$85$	−1.73267	−0.187935
$86$	0	0
$87$	1.00000	0.107211
$88$	0	0
$89$	8.93734	0.947357	0.473678	−	0.880698i	$-0.342926\pi$
$89$	8.93734	0.947357	0.473678	+	0.880698i	$0.342926\pi$
$90$	0	0
$91$	0.271991	0.0285124
$92$	0	0
$93$	6.72475	0.697324
$94$	0	0
$95$	0.320111	0.0328427
$96$	0	0
$97$	4.43366	0.450170	0.225085	−	0.974339i	$-0.427734\pi$
$97$	4.43366	0.450170	0.225085	+	0.974339i	$0.427734\pi$
$98$	0	0
$99$	1.85849	0.186785
$100$	0	0
$101$	5.01727	0.499237	0.249618	−	0.968344i	$-0.419695\pi$
$101$	5.01727	0.499237	0.249618	+	0.968344i	$0.419695\pi$
$102$	0	0
$103$	−13.5050	−1.33068	−0.665341	−	0.746539i	$-0.731714\pi$
$103$	−13.5050	−1.33068	−0.665341	+	0.746539i	$0.731714\pi$
$104$	0	0
$105$	−0.0428320	−0.00417998
$106$	0	0
$107$	−10.1993	−0.986007	−0.493003	−	0.870027i	$-0.664101\pi$
$107$	−10.1993	−0.986007	−0.493003	+	0.870027i	$0.664101\pi$
$108$	0	0
$109$	0.150542	0.0144193	0.00720965	−	0.999974i	$-0.497705\pi$
$109$	0.150542	0.0144193	0.00720965	+	0.999974i	$0.497705\pi$
$110$	0	0
$111$	1.42575	0.135327
$112$	0	0
$113$	11.3408	1.06685	0.533427	−	0.845846i	$-0.320904\pi$
$113$	11.3408	1.06685	0.533427	+	0.845846i	$0.320904\pi$
$114$	0	0
$115$	−0.305886	−0.0285240
$116$	0	0
$117$	−1.94243	−0.179578
$118$	0	0
$119$	−0.793169	−0.0727097
$120$	0	0
$121$	−7.54601	−0.686001
$122$	0	0
$123$	−0.522747	−0.0471345
$124$	0	0
$125$	3.03024	0.271033
$126$	0	0
$127$	4.72184	0.418995	0.209498	−	0.977809i	$-0.432817\pi$
$127$	4.72184	0.418995	0.209498	+	0.977809i	$0.432817\pi$
$128$	0	0
$129$	−4.27739	−0.376603
$130$	0	0
$131$	−11.9635	−1.04526	−0.522629	−	0.852560i	$-0.675049\pi$
$131$	−11.9635	−1.04526	−0.522629	+	0.852560i	$0.675049\pi$
$132$	0	0
$133$	0.146538	0.0127064
$134$	0	0
$135$	0.305886	0.0263265
$136$	0	0
$137$	−13.1712	−1.12529	−0.562645	−	0.826698i	$-0.690216\pi$
$137$	−13.1712	−1.12529	−0.562645	+	0.826698i	$0.690216\pi$
$138$	0	0
$139$	11.8454	1.00472	0.502358	−	0.864660i	$-0.332466\pi$
$139$	11.8454	1.00472	0.502358	+	0.864660i	$0.332466\pi$
$140$	0	0
$141$	−7.07300	−0.595654
$142$	0	0
$143$	−3.60999	−0.301882
$144$	0	0
$145$	0.305886	0.0254025
$146$	0	0
$147$	6.98039	0.575733
$148$	0	0
$149$	−15.1656	−1.24241	−0.621207	−	0.783646i	$-0.713358\pi$
$149$	−15.1656	−1.24241	−0.621207	+	0.783646i	$0.713358\pi$
$150$	0	0
$151$	0.571642	0.0465196	0.0232598	−	0.999729i	$-0.492596\pi$
$151$	0.571642	0.0465196	0.0232598	+	0.999729i	$0.492596\pi$
$152$	0	0
$153$	5.66444	0.457943
$154$	0	0
$155$	2.05701	0.165223
$156$	0	0
$157$	−17.2615	−1.37762	−0.688810	−	0.724941i	$-0.741867\pi$
$157$	−17.2615	−1.37762	−0.688810	+	0.724941i	$0.741867\pi$
$158$	0	0
$159$	11.0449	0.875918
$160$	0	0
$161$	−0.140026	−0.0110356
$162$	0	0
$163$	24.3324	1.90586	0.952929	−	0.303195i	$-0.0980532\pi$
$163$	24.3324	1.90586	0.952929	+	0.303195i	$0.0980532\pi$
$164$	0	0
$165$	0.568487	0.0442566
$166$	0	0
$167$	1.90162	0.147152	0.0735758	−	0.997290i	$-0.476559\pi$
$167$	1.90162	0.147152	0.0735758	+	0.997290i	$0.476559\pi$
$168$	0	0
$169$	−9.22697	−0.709767
$170$	0	0
$171$	−1.04650	−0.0800281
$172$	0	0
$173$	−6.53138	−0.496572	−0.248286	−	0.968687i	$-0.579867\pi$
$173$	−6.53138	−0.496572	−0.248286	+	0.968687i	$0.579867\pi$
$174$	0	0
$175$	0.687028	0.0519345
$176$	0	0
$177$	10.0473	0.755199
$178$	0	0
$179$	15.8879	1.18752	0.593760	−	0.804642i	$-0.297643\pi$
$179$	15.8879	1.18752	0.593760	+	0.804642i	$0.297643\pi$
$180$	0	0
$181$	−7.85906	−0.584160	−0.292080	−	0.956394i	$-0.594347\pi$
$181$	−7.85906	−0.584160	−0.292080	+	0.956394i	$0.594347\pi$
$182$	0	0
$183$	−7.67091	−0.567050
$184$	0	0
$185$	0.436118	0.0320641
$186$	0	0
$187$	10.5273	0.769834
$188$	0	0
$189$	0.140026	0.0101854
$190$	0	0
$191$	−3.03406	−0.219537	−0.109769	−	0.993957i	$-0.535011\pi$
$191$	−3.03406	−0.219537	−0.109769	+	0.993957i	$0.535011\pi$
$192$	0	0
$193$	−3.82892	−0.275612	−0.137806	−	0.990459i	$-0.544005\pi$
$193$	−3.82892	−0.275612	−0.137806	+	0.990459i	$0.544005\pi$
$194$	0	0
$195$	−0.594162	−0.0425489
$196$	0	0
$197$	−6.61415	−0.471238	−0.235619	−	0.971845i	$-0.575712\pi$
$197$	−6.61415	−0.471238	−0.235619	+	0.971845i	$0.575712\pi$
$198$	0	0
$199$	2.10692	0.149356	0.0746779	−	0.997208i	$-0.476207\pi$
$199$	2.10692	0.149356	0.0746779	+	0.997208i	$0.476207\pi$
$200$	0	0
$201$	12.0677	0.851187
$202$	0	0
$203$	0.140026	0.00982790
$204$	0	0
$205$	−0.159901	−0.0111680
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	−1.94492	−0.134533
$210$	0	0
$211$	3.66115	0.252044	0.126022	−	0.992027i	$-0.459779\pi$
$211$	3.66115	0.252044	0.126022	+	0.992027i	$0.459779\pi$
$212$	0	0
$213$	−8.29946	−0.568670
$214$	0	0
$215$	−1.30840	−0.0892318
$216$	0	0
$217$	0.941641	0.0639227
$218$	0	0
$219$	0.0686045	0.00463586
$220$	0	0
$221$	−11.0028	−0.740127
$222$	0	0
$223$	8.80133	0.589381	0.294690	−	0.955593i	$-0.404784\pi$
$223$	8.80133	0.589381	0.294690	+	0.955593i	$0.404784\pi$
$224$	0	0
$225$	−4.90643	−0.327096
$226$	0	0
$227$	−21.2093	−1.40771	−0.703854	−	0.710344i	$-0.748539\pi$
$227$	−21.2093	−1.40771	−0.703854	+	0.710344i	$0.748539\pi$
$228$	0	0
$229$	15.9425	1.05351	0.526754	−	0.850018i	$-0.323409\pi$
$229$	15.9425	1.05351	0.526754	+	0.850018i	$0.323409\pi$
$230$	0	0
$231$	0.260237	0.0171223
$232$	0	0
$233$	6.42481	0.420903	0.210452	−	0.977604i	$-0.432507\pi$
$233$	6.42481	0.420903	0.210452	+	0.977604i	$0.432507\pi$
$234$	0	0
$235$	−2.16353	−0.141133
$236$	0	0
$237$	−8.58428	−0.557609
$238$	0	0
$239$	−7.49563	−0.484852	−0.242426	−	0.970170i	$-0.577943\pi$
$239$	−7.49563	−0.484852	−0.242426	+	0.970170i	$0.577943\pi$
$240$	0	0
$241$	0.687449	0.0442825	0.0221412	−	0.999755i	$-0.492952\pi$
$241$	0.687449	0.0442825	0.0221412	+	0.999755i	$0.492952\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	2.13521	0.136413
$246$	0	0
$247$	2.03276	0.129341
$248$	0	0
$249$	−7.65463	−0.485092
$250$	0	0
$251$	−17.9404	−1.13239	−0.566193	−	0.824273i	$-0.691584\pi$
$251$	−17.9404	−1.13239	−0.566193	+	0.824273i	$0.691584\pi$
$252$	0	0
$253$	1.85849	0.116842
$254$	0	0
$255$	1.73267	0.108504
$256$	0	0
$257$	−21.1889	−1.32172	−0.660862	−	0.750507i	$-0.729809\pi$
$257$	−21.1889	−1.32172	−0.660862	+	0.750507i	$0.729809\pi$
$258$	0	0
$259$	0.199643	0.0124052
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	0	0
$263$	0.533746	0.0329122	0.0164561	−	0.999865i	$-0.494762\pi$
$263$	0.533746	0.0329122	0.0164561	+	0.999865i	$0.494762\pi$
$264$	0	0
$265$	3.37848	0.207539
$266$	0	0
$267$	−8.93734	−0.546957
$268$	0	0
$269$	−30.3949	−1.85321	−0.926605	−	0.376035i	$-0.877287\pi$
$269$	−30.3949	−1.85321	−0.926605	+	0.376035i	$0.877287\pi$
$270$	0	0
$271$	−19.7790	−1.20149	−0.600743	−	0.799442i	$-0.705128\pi$
$271$	−19.7790	−1.20149	−0.600743	+	0.799442i	$0.705128\pi$
$272$	0	0
$273$	−0.271991	−0.0164616
$274$	0	0
$275$	−9.11856	−0.549870
$276$	0	0
$277$	32.4269	1.94835	0.974173	−	0.225802i	$-0.0725002\pi$
$277$	32.4269	1.94835	0.974173	+	0.225802i	$0.0725002\pi$
$278$	0	0
$279$	−6.72475	−0.402600
$280$	0	0
$281$	−25.0927	−1.49691	−0.748454	−	0.663187i	$-0.769203\pi$
$281$	−25.0927	−1.49691	−0.748454	+	0.663187i	$0.769203\pi$
$282$	0	0
$283$	−26.0505	−1.54854	−0.774270	−	0.632856i	$-0.781883\pi$
$283$	−26.0505	−1.54854	−0.774270	+	0.632856i	$0.781883\pi$
$284$	0	0
$285$	−0.320111	−0.0189617
$286$	0	0
$287$	−0.0731982	−0.00432075
$288$	0	0
$289$	15.0859	0.887406
$290$	0	0
$291$	−4.43366	−0.259906
$292$	0	0
$293$	−26.2399	−1.53295	−0.766474	−	0.642275i	$-0.777991\pi$
$293$	−26.2399	−1.53295	−0.766474	+	0.642275i	$0.777991\pi$
$294$	0	0
$295$	3.07332	0.178936
$296$	0	0
$297$	−1.85849	−0.107841
$298$	0	0
$299$	−1.94243	−0.112334
$300$	0	0
$301$	−0.598946	−0.0345227
$302$	0	0
$303$	−5.01727	−0.288234
$304$	0	0
$305$	−2.34642	−0.134356
$306$	0	0
$307$	0.970427	0.0553852	0.0276926	−	0.999616i	$-0.491184\pi$
$307$	0.970427	0.0553852	0.0276926	+	0.999616i	$0.491184\pi$
$308$	0	0
$309$	13.5050	0.768270
$310$	0	0
$311$	10.0703	0.571036	0.285518	−	0.958373i	$-0.407834\pi$
$311$	10.0703	0.571036	0.285518	+	0.958373i	$0.407834\pi$
$312$	0	0
$313$	−29.9323	−1.69188	−0.845938	−	0.533281i	$-0.820959\pi$
$313$	−29.9323	−1.69188	−0.845938	+	0.533281i	$0.820959\pi$
$314$	0	0
$315$	0.0428320	0.00241331
$316$	0	0
$317$	20.1859	1.13375	0.566876	−	0.823803i	$-0.308152\pi$
$317$	20.1859	1.13375	0.566876	+	0.823803i	$0.308152\pi$
$318$	0	0
$319$	−1.85849	−0.104056
$320$	0	0
$321$	10.1993	0.569271
$322$	0	0
$323$	−5.92786	−0.329835
$324$	0	0
$325$	9.53040	0.528652
$326$	0	0
$327$	−0.150542	−0.00832498
$328$	0	0
$329$	−0.990404	−0.0546028
$330$	0	0
$331$	−28.2007	−1.55005	−0.775026	−	0.631930i	$-0.782263\pi$
$331$	−28.2007	−1.55005	−0.775026	+	0.631930i	$0.782263\pi$
$332$	0	0
$333$	−1.42575	−0.0781308
$334$	0	0
$335$	3.69133	0.201679
$336$	0	0
$337$	1.55048	0.0844602	0.0422301	−	0.999108i	$-0.486554\pi$
$337$	1.55048	0.0844602	0.0422301	+	0.999108i	$0.486554\pi$
$338$	0	0
$339$	−11.3408	−0.615949
$340$	0	0
$341$	−12.4979	−0.676799
$342$	0	0
$343$	1.95762	0.105701
$344$	0	0
$345$	0.305886	0.0164684
$346$	0	0
$347$	−19.2286	−1.03225	−0.516123	−	0.856515i	$-0.672625\pi$
$347$	−19.2286	−1.03225	−0.516123	+	0.856515i	$0.672625\pi$
$348$	0	0
$349$	6.23325	0.333658	0.166829	−	0.985986i	$-0.446647\pi$
$349$	6.23325	0.333658	0.166829	+	0.985986i	$0.446647\pi$
$350$	0	0
$351$	1.94243	0.103679
$352$	0	0
$353$	−28.2992	−1.50621	−0.753106	−	0.657899i	$-0.771445\pi$
$353$	−28.2992	−1.50621	−0.753106	+	0.657899i	$0.771445\pi$
$354$	0	0
$355$	−2.53869	−0.134740
$356$	0	0
$357$	0.793169	0.0419790
$358$	0	0
$359$	32.8506	1.73379	0.866895	−	0.498491i	$-0.166112\pi$
$359$	32.8506	1.73379	0.866895	+	0.498491i	$0.166112\pi$
$360$	0	0
$361$	−17.9048	−0.942359
$362$	0	0
$363$	7.54601	0.396063
$364$	0	0
$365$	0.0209852	0.00109841
$366$	0	0
$367$	7.67381	0.400570	0.200285	−	0.979738i	$-0.435813\pi$
$367$	7.67381	0.400570	0.200285	+	0.979738i	$0.435813\pi$
$368$	0	0
$369$	0.522747	0.0272131
$370$	0	0
$371$	1.54657	0.0802941
$372$	0	0
$373$	−7.94179	−0.411210	−0.205605	−	0.978635i	$-0.565916\pi$
$373$	−7.94179	−0.411210	−0.205605	+	0.978635i	$0.565916\pi$
$374$	0	0
$375$	−3.03024	−0.156481
$376$	0	0
$377$	1.94243	0.100040
$378$	0	0
$379$	−13.9632	−0.717243	−0.358621	−	0.933483i	$-0.616753\pi$
$379$	−13.9632	−0.717243	−0.358621	+	0.933483i	$0.616753\pi$
$380$	0	0
$381$	−4.72184	−0.241907
$382$	0	0
$383$	1.11226	0.0568337	0.0284169	−	0.999596i	$-0.490953\pi$
$383$	1.11226	0.0568337	0.0284169	+	0.999596i	$0.490953\pi$
$384$	0	0
$385$	0.0796030	0.00405694
$386$	0	0
$387$	4.27739	0.217432
$388$	0	0
$389$	32.0187	1.62341	0.811707	−	0.584065i	$-0.198539\pi$
$389$	32.0187	1.62341	0.811707	+	0.584065i	$0.198539\pi$
$390$	0	0
$391$	5.66444	0.286463
$392$	0	0
$393$	11.9635	0.603480
$394$	0	0
$395$	−2.62581	−0.132119
$396$	0	0
$397$	25.6356	1.28661	0.643306	−	0.765609i	$-0.277562\pi$
$397$	25.6356	1.28661	0.643306	+	0.765609i	$0.277562\pi$
$398$	0	0
$399$	−0.146538	−0.00733606
$400$	0	0
$401$	10.0646	0.502603	0.251301	−	0.967909i	$-0.419141\pi$
$401$	10.0646	0.502603	0.251301	+	0.967909i	$0.419141\pi$
$402$	0	0
$403$	13.0624	0.650683
$404$	0	0
$405$	−0.305886	−0.0151996
$406$	0	0
$407$	−2.64975	−0.131343
$408$	0	0
$409$	−16.3638	−0.809140	−0.404570	−	0.914507i	$-0.632579\pi$
$409$	−16.3638	−0.809140	−0.404570	+	0.914507i	$0.632579\pi$
$410$	0	0
$411$	13.1712	0.649687
$412$	0	0
$413$	1.40688	0.0692280
$414$	0	0
$415$	−2.34145	−0.114937
$416$	0	0
$417$	−11.8454	−0.580073
$418$	0	0
$419$	32.4761	1.58656	0.793282	−	0.608855i	$-0.208371\pi$
$419$	32.4761	1.58656	0.793282	+	0.608855i	$0.208371\pi$
$420$	0	0
$421$	−0.972317	−0.0473878	−0.0236939	−	0.999719i	$-0.507543\pi$
$421$	−0.972317	−0.0473878	−0.0236939	+	0.999719i	$0.507543\pi$
$422$	0	0
$423$	7.07300	0.343901
$424$	0	0
$425$	−27.7922	−1.34812
$426$	0	0
$427$	−1.07413	−0.0519806
$428$	0	0
$429$	3.60999	0.174292
$430$	0	0
$431$	−23.2012	−1.11756	−0.558781	−	0.829315i	$-0.688731\pi$
$431$	−23.2012	−1.11756	−0.558781	+	0.829315i	$0.688731\pi$
$432$	0	0
$433$	−7.39839	−0.355544	−0.177772	−	0.984072i	$-0.556889\pi$
$433$	−7.39839	−0.355544	−0.177772	+	0.984072i	$0.556889\pi$
$434$	0	0
$435$	−0.305886	−0.0146661
$436$	0	0
$437$	−1.04650	−0.0500611
$438$	0	0
$439$	30.8260	1.47124	0.735622	−	0.677392i	$-0.236890\pi$
$439$	30.8260	1.47124	0.735622	+	0.677392i	$0.236890\pi$
$440$	0	0
$441$	−6.98039	−0.332400
$442$	0	0
$443$	11.1200	0.528328	0.264164	−	0.964478i	$-0.414904\pi$
$443$	11.1200	0.528328	0.264164	+	0.964478i	$0.414904\pi$
$444$	0	0
$445$	−2.73381	−0.129595
$446$	0	0
$447$	15.1656	0.717308
$448$	0	0
$449$	−5.32321	−0.251218	−0.125609	−	0.992080i	$-0.540088\pi$
$449$	−5.32321	−0.251218	−0.125609	+	0.992080i	$0.540088\pi$
$450$	0	0
$451$	0.971521	0.0457471
$452$	0	0
$453$	−0.571642	−0.0268581
$454$	0	0
$455$	−0.0831982	−0.00390039
$456$	0	0
$457$	−25.1085	−1.17453	−0.587264	−	0.809396i	$-0.699795\pi$
$457$	−25.1085	−1.17453	−0.587264	+	0.809396i	$0.699795\pi$
$458$	0	0
$459$	−5.66444	−0.264393
$460$	0	0
$461$	−25.0268	−1.16562	−0.582808	−	0.812610i	$-0.698046\pi$
$461$	−25.0268	−1.16562	−0.582808	+	0.812610i	$0.698046\pi$
$462$	0	0
$463$	12.1410	0.564242	0.282121	−	0.959379i	$-0.408962\pi$
$463$	12.1410	0.564242	0.282121	+	0.959379i	$0.408962\pi$
$464$	0	0
$465$	−2.05701	−0.0953915
$466$	0	0
$467$	−31.1080	−1.43951	−0.719754	−	0.694229i	$-0.755746\pi$
$467$	−31.1080	−1.43951	−0.719754	+	0.694229i	$0.755746\pi$
$468$	0	0
$469$	1.68979	0.0780271
$470$	0	0
$471$	17.2615	0.795370
$472$	0	0
$473$	7.94950	0.365518
$474$	0	0
$475$	5.13460	0.235592
$476$	0	0
$477$	−11.0449	−0.505711
$478$	0	0
$479$	0.963078	0.0440042	0.0220021	−	0.999758i	$-0.492996\pi$
$479$	0.963078	0.0440042	0.0220021	+	0.999758i	$0.492996\pi$
$480$	0	0
$481$	2.76943	0.126275
$482$	0	0
$483$	0.140026	0.00637140
$484$	0	0
$485$	−1.35619	−0.0615816
$486$	0	0
$487$	−17.3396	−0.785734	−0.392867	−	0.919595i	$-0.628517\pi$
$487$	−17.3396	−0.785734	−0.392867	+	0.919595i	$0.628517\pi$
$488$	0	0
$489$	−24.3324	−1.10035
$490$	0	0
$491$	−29.5883	−1.33530	−0.667650	−	0.744475i	$-0.732700\pi$
$491$	−29.5883	−1.33530	−0.667650	+	0.744475i	$0.732700\pi$
$492$	0	0
$493$	−5.66444	−0.255114
$494$	0	0
$495$	−0.568487	−0.0255516
$496$	0	0
$497$	−1.16214	−0.0521291
$498$	0	0
$499$	15.6338	0.699864	0.349932	−	0.936775i	$-0.386205\pi$
$499$	15.6338	0.699864	0.349932	+	0.936775i	$0.386205\pi$
$500$	0	0
$501$	−1.90162	−0.0849580
$502$	0	0
$503$	−1.19850	−0.0534386	−0.0267193	−	0.999643i	$-0.508506\pi$
$503$	−1.19850	−0.0534386	−0.0267193	+	0.999643i	$0.508506\pi$
$504$	0	0
$505$	−1.53471	−0.0682938
$506$	0	0
$507$	9.22697	0.409784
$508$	0	0
$509$	−10.8016	−0.478770	−0.239385	−	0.970925i	$-0.576946\pi$
$509$	−10.8016	−0.478770	−0.239385	+	0.970925i	$0.576946\pi$
$510$	0	0
$511$	0.00960642	0.000424963 0
$512$	0	0
$513$	1.04650	0.0462043
$514$	0	0
$515$	4.13098	0.182033
$516$	0	0
$517$	13.1451	0.578121
$518$	0	0
$519$	6.53138	0.286696
$520$	0	0
$521$	23.6412	1.03574	0.517869	−	0.855460i	$-0.326725\pi$
$521$	23.6412	1.03574	0.517869	+	0.855460i	$0.326725\pi$
$522$	0	0
$523$	−14.1143	−0.617175	−0.308587	−	0.951196i	$-0.599856\pi$
$523$	−14.1143	−0.617175	−0.308587	+	0.951196i	$0.599856\pi$
$524$	0	0
$525$	−0.687028	−0.0299844
$526$	0	0
$527$	−38.0920	−1.65931
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−10.0473	−0.436014
$532$	0	0
$533$	−1.01540	−0.0439818
$534$	0	0
$535$	3.11984	0.134882
$536$	0	0
$537$	−15.8879	−0.685615
$538$	0	0
$539$	−12.9730	−0.558787
$540$	0	0
$541$	−13.4317	−0.577473	−0.288736	−	0.957409i	$-0.593235\pi$
$541$	−13.4317	−0.577473	−0.288736	+	0.957409i	$0.593235\pi$
$542$	0	0
$543$	7.85906	0.337265
$544$	0	0
$545$	−0.0460487	−0.00197251
$546$	0	0
$547$	−9.51664	−0.406902	−0.203451	−	0.979085i	$-0.565216\pi$
$547$	−9.51664	−0.406902	−0.203451	+	0.979085i	$0.565216\pi$
$548$	0	0
$549$	7.67091	0.327386
$550$	0	0
$551$	1.04650	0.0445826
$552$	0	0
$553$	−1.20202	−0.0511152
$554$	0	0
$555$	−0.436118	−0.0185122
$556$	0	0
$557$	−19.2946	−0.817538	−0.408769	−	0.912638i	$-0.634042\pi$
$557$	−19.2946	−0.817538	−0.408769	+	0.912638i	$0.634042\pi$
$558$	0	0
$559$	−8.30854	−0.351414
$560$	0	0
$561$	−10.5273	−0.444464
$562$	0	0
$563$	−26.4275	−1.11379	−0.556893	−	0.830584i	$-0.688007\pi$
$563$	−26.4275	−1.11379	−0.556893	+	0.830584i	$0.688007\pi$
$564$	0	0
$565$	−3.46900	−0.145942
$566$	0	0
$567$	−0.140026	−0.00588054
$568$	0	0
$569$	−20.2718	−0.849839	−0.424919	−	0.905231i	$-0.639698\pi$
$569$	−20.2718	−0.849839	−0.424919	+	0.905231i	$0.639698\pi$
$570$	0	0
$571$	20.4605	0.856244	0.428122	−	0.903721i	$-0.359175\pi$
$571$	20.4605	0.856244	0.428122	+	0.903721i	$0.359175\pi$
$572$	0	0
$573$	3.03406	0.126750
$574$	0	0
$575$	−4.90643	−0.204612
$576$	0	0
$577$	1.16293	0.0484132	0.0242066	−	0.999707i	$-0.492294\pi$
$577$	1.16293	0.0484132	0.0242066	+	0.999707i	$0.492294\pi$
$578$	0	0
$579$	3.82892	0.159125
$580$	0	0
$581$	−1.07185	−0.0444677
$582$	0	0
$583$	−20.5269	−0.850135
$584$	0	0
$585$	0.594162	0.0245656
$586$	0	0
$587$	39.1994	1.61793	0.808965	−	0.587856i	$-0.200028\pi$
$587$	39.1994	1.61793	0.808965	+	0.587856i	$0.200028\pi$
$588$	0	0
$589$	7.03748	0.289974
$590$	0	0
$591$	6.61415	0.272070
$592$	0	0
$593$	−16.1639	−0.663772	−0.331886	−	0.943320i	$-0.607685\pi$
$593$	−16.1639	−0.663772	−0.331886	+	0.943320i	$0.607685\pi$
$594$	0	0
$595$	0.242620	0.00994643
$596$	0	0
$597$	−2.10692	−0.0862306
$598$	0	0
$599$	−28.5186	−1.16524	−0.582619	−	0.812745i	$-0.697972\pi$
$599$	−28.5186	−1.16524	−0.582619	+	0.812745i	$0.697972\pi$
$600$	0	0
$601$	−7.67695	−0.313149	−0.156575	−	0.987666i	$-0.550045\pi$
$601$	−7.67695	−0.313149	−0.156575	+	0.987666i	$0.550045\pi$
$602$	0	0
$603$	−12.0677	−0.491433
$604$	0	0
$605$	2.30822	0.0938425
$606$	0	0
$607$	23.1764	0.940703	0.470351	−	0.882479i	$-0.344127\pi$
$607$	23.1764	0.940703	0.470351	+	0.882479i	$0.344127\pi$
$608$	0	0
$609$	−0.140026	−0.00567414
$610$	0	0
$611$	−13.7388	−0.555813
$612$	0	0
$613$	−41.7207	−1.68508	−0.842542	−	0.538631i	$-0.818942\pi$
$613$	−41.7207	−1.68508	−0.842542	+	0.538631i	$0.818942\pi$
$614$	0	0
$615$	0.159901	0.00644783
$616$	0	0
$617$	12.8239	0.516270	0.258135	−	0.966109i	$-0.416892\pi$
$617$	12.8239	0.516270	0.258135	+	0.966109i	$0.416892\pi$
$618$	0	0
$619$	−21.1322	−0.849373	−0.424687	−	0.905340i	$-0.639616\pi$
$619$	−21.1322	−0.849373	−0.424687	+	0.905340i	$0.639616\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	−1.25146	−0.0501387
$624$	0	0
$625$	23.6053	0.944210
$626$	0	0
$627$	1.94492	0.0776726
$628$	0	0
$629$	−8.07610	−0.322015
$630$	0	0
$631$	−3.19211	−0.127076	−0.0635380	−	0.997979i	$-0.520238\pi$
$631$	−3.19211	−0.127076	−0.0635380	+	0.997979i	$0.520238\pi$
$632$	0	0
$633$	−3.66115	−0.145518
$634$	0	0
$635$	−1.44435	−0.0573171
$636$	0	0
$637$	13.5589	0.537224
$638$	0	0
$639$	8.29946	0.328322
$640$	0	0
$641$	19.7436	0.779825	0.389912	−	0.920852i	$-0.372505\pi$
$641$	19.7436	0.779825	0.389912	+	0.920852i	$0.372505\pi$
$642$	0	0
$643$	8.39805	0.331187	0.165593	−	0.986194i	$-0.447046\pi$
$643$	8.39805	0.331187	0.165593	+	0.986194i	$0.447046\pi$
$644$	0	0
$645$	1.30840	0.0515180
$646$	0	0
$647$	0.653272	0.0256828	0.0128414	−	0.999918i	$-0.495912\pi$
$647$	0.653272	0.0256828	0.0128414	+	0.999918i	$0.495912\pi$
$648$	0	0
$649$	−18.6728	−0.732970
$650$	0	0
$651$	−0.941641	−0.0369058
$652$	0	0
$653$	11.5695	0.452749	0.226374	−	0.974040i	$-0.427313\pi$
$653$	11.5695	0.452749	0.226374	+	0.974040i	$0.427313\pi$
$654$	0	0
$655$	3.65948	0.142988
$656$	0	0
$657$	−0.0686045	−0.00267652
$658$	0	0
$659$	−24.8891	−0.969543	−0.484771	−	0.874641i	$-0.661097\pi$
$659$	−24.8891	−0.969543	−0.484771	+	0.874641i	$0.661097\pi$
$660$	0	0
$661$	−25.3623	−0.986480	−0.493240	−	0.869893i	$-0.664188\pi$
$661$	−25.3623	−0.986480	−0.493240	+	0.869893i	$0.664188\pi$
$662$	0	0
$663$	11.0028	0.427313
$664$	0	0
$665$	−0.0448239	−0.00173820
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	0	0
$669$	−8.80133	−0.340279
$670$	0	0
$671$	14.2563	0.550359
$672$	0	0
$673$	−17.9669	−0.692572	−0.346286	−	0.938129i	$-0.612557\pi$
$673$	−17.9669	−0.692572	−0.346286	+	0.938129i	$0.612557\pi$
$674$	0	0
$675$	4.90643	0.188849
$676$	0	0
$677$	−30.9240	−1.18850	−0.594252	−	0.804279i	$-0.702552\pi$
$677$	−30.9240	−1.18850	−0.594252	+	0.804279i	$0.702552\pi$
$678$	0	0
$679$	−0.620827	−0.0238252
$680$	0	0
$681$	21.2093	0.812741
$682$	0	0
$683$	−8.21255	−0.314244	−0.157122	−	0.987579i	$-0.550222\pi$
$683$	−8.21255	−0.314244	−0.157122	+	0.987579i	$0.550222\pi$
$684$	0	0
$685$	4.02888	0.153936
$686$	0	0
$687$	−15.9425	−0.608243
$688$	0	0
$689$	21.4539	0.817330
$690$	0	0
$691$	15.9772	0.607801	0.303901	−	0.952704i	$-0.401711\pi$
$691$	15.9772	0.607801	0.303901	+	0.952704i	$0.401711\pi$
$692$	0	0
$693$	−0.260237	−0.00988559
$694$	0	0
$695$	−3.62335	−0.137442
$696$	0	0
$697$	2.96107	0.112158
$698$	0	0
$699$	−6.42481	−0.243009
$700$	0	0
$701$	−50.8260	−1.91967	−0.959835	−	0.280565i	$-0.909478\pi$
$701$	−50.8260	−1.91967	−0.959835	+	0.280565i	$0.909478\pi$
$702$	0	0
$703$	1.49206	0.0562740
$704$	0	0
$705$	2.16353	0.0814834
$706$	0	0
$707$	−0.702548	−0.0264220
$708$	0	0
$709$	−27.8310	−1.04522	−0.522608	−	0.852573i	$-0.675041\pi$
$709$	−27.8310	−1.04522	−0.522608	+	0.852573i	$0.675041\pi$
$710$	0	0
$711$	8.58428	0.321936
$712$	0	0
$713$	−6.72475	−0.251844
$714$	0	0
$715$	1.10425	0.0412965
$716$	0	0
$717$	7.49563	0.279929
$718$	0	0
$719$	0.498348	0.0185852	0.00929262	−	0.999957i	$-0.497042\pi$
$719$	0.498348	0.0185852	0.00929262	+	0.999957i	$0.497042\pi$
$720$	0	0
$721$	1.89105	0.0704262
$722$	0	0
$723$	−0.687449	−0.0255665
$724$	0	0
$725$	4.90643	0.182220
$726$	0	0
$727$	48.4281	1.79610	0.898049	−	0.439895i	$-0.144985\pi$
$727$	48.4281	1.79610	0.898049	+	0.439895i	$0.144985\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	24.2290	0.896144
$732$	0	0
$733$	34.3149	1.26745	0.633725	−	0.773558i	$-0.281525\pi$
$733$	34.3149	1.26745	0.633725	+	0.773558i	$0.281525\pi$
$734$	0	0
$735$	−2.13521	−0.0787583
$736$	0	0
$737$	−22.4276	−0.826133
$738$	0	0
$739$	−17.0461	−0.627050	−0.313525	−	0.949580i	$-0.601510\pi$
$739$	−17.0461	−0.627050	−0.313525	+	0.949580i	$0.601510\pi$
$740$	0	0
$741$	−2.03276	−0.0746753
$742$	0	0
$743$	26.9392	0.988304	0.494152	−	0.869375i	$-0.335479\pi$
$743$	26.9392	0.988304	0.494152	+	0.869375i	$0.335479\pi$
$744$	0	0
$745$	4.63895	0.169958
$746$	0	0
$747$	7.65463	0.280068
$748$	0	0
$749$	1.42817	0.0521843
$750$	0	0
$751$	−36.4238	−1.32912	−0.664562	−	0.747233i	$-0.731382\pi$
$751$	−36.4238	−1.32912	−0.664562	+	0.747233i	$0.731382\pi$
$752$	0	0
$753$	17.9404	0.653783
$754$	0	0
$755$	−0.174857	−0.00636371
$756$	0	0
$757$	−3.92322	−0.142592	−0.0712959	−	0.997455i	$-0.522713\pi$
$757$	−3.92322	−0.142592	−0.0712959	+	0.997455i	$0.522713\pi$
$758$	0	0
$759$	−1.85849	−0.0674590
$760$	0	0
$761$	8.71135	0.315786	0.157893	−	0.987456i	$-0.449530\pi$
$761$	8.71135	0.315786	0.157893	+	0.987456i	$0.449530\pi$
$762$	0	0
$763$	−0.0210798	−0.000763139 0
$764$	0	0
$765$	−1.73267	−0.0626450
$766$	0	0
$767$	19.5161	0.704686
$768$	0	0
$769$	−40.4343	−1.45810	−0.729049	−	0.684461i	$-0.760037\pi$
$769$	−40.4343	−1.45810	−0.729049	+	0.684461i	$0.760037\pi$
$770$	0	0
$771$	21.1889	0.763098
$772$	0	0
$773$	6.95931	0.250309	0.125155	−	0.992137i	$-0.460057\pi$
$773$	6.95931	0.250309	0.125155	+	0.992137i	$0.460057\pi$
$774$	0	0
$775$	32.9946	1.18520
$776$	0	0
$777$	−0.199643	−0.00716214
$778$	0	0
$779$	−0.547057	−0.0196003
$780$	0	0
$781$	15.4245	0.551931
$782$	0	0
$783$	1.00000	0.0357371
$784$	0	0
$785$	5.28007	0.188454
$786$	0	0
$787$	16.8140	0.599355	0.299677	−	0.954041i	$-0.403121\pi$
$787$	16.8140	0.599355	0.299677	+	0.954041i	$0.403121\pi$
$788$	0	0
$789$	−0.533746	−0.0190019
$790$	0	0
$791$	−1.58801	−0.0564631
$792$	0	0
$793$	−14.9002	−0.529122
$794$	0	0
$795$	−3.37848	−0.119822
$796$	0	0
$797$	−5.59612	−0.198225	−0.0991123	−	0.995076i	$-0.531600\pi$
$797$	−5.59612	−0.198225	−0.0991123	+	0.995076i	$0.531600\pi$
$798$	0	0
$799$	40.0646	1.41738
$800$	0	0
$801$	8.93734	0.315786
$802$	0	0
$803$	−0.127501	−0.00449941
$804$	0	0
$805$	0.0428320	0.00150963
$806$	0	0
$807$	30.3949	1.06995
$808$	0	0
$809$	−4.38279	−0.154091	−0.0770454	−	0.997028i	$-0.524549\pi$
$809$	−4.38279	−0.154091	−0.0770454	+	0.997028i	$0.524549\pi$
$810$	0	0
$811$	40.9682	1.43859	0.719294	−	0.694706i	$-0.244466\pi$
$811$	40.9682	1.43859	0.719294	+	0.694706i	$0.244466\pi$
$812$	0	0
$813$	19.7790	0.693678
$814$	0	0
$815$	−7.44293	−0.260715
$816$	0	0
$817$	−4.47631	−0.156606
$818$	0	0
$819$	0.271991	0.00950412
$820$	0	0
$821$	−27.8672	−0.972573	−0.486286	−	0.873800i	$-0.661649\pi$
$821$	−27.8672	−0.972573	−0.486286	+	0.873800i	$0.661649\pi$
$822$	0	0
$823$	−14.8442	−0.517436	−0.258718	−	0.965953i	$-0.583300\pi$
$823$	−14.8442	−0.517436	−0.258718	+	0.965953i	$0.583300\pi$
$824$	0	0
$825$	9.11856	0.317468
$826$	0	0
$827$	−18.9515	−0.659009	−0.329504	−	0.944154i	$-0.606882\pi$
$827$	−18.9515	−0.659009	−0.329504	+	0.944154i	$0.606882\pi$
$828$	0	0
$829$	−30.5368	−1.06059	−0.530294	−	0.847814i	$-0.677918\pi$
$829$	−30.5368	−1.06059	−0.530294	+	0.847814i	$0.677918\pi$
$830$	0	0
$831$	−32.4269	−1.12488
$832$	0	0
$833$	−39.5400	−1.36998
$834$	0	0
$835$	−0.581679	−0.0201298
$836$	0	0
$837$	6.72475	0.232441
$838$	0	0
$839$	28.8921	0.997466	0.498733	−	0.866756i	$-0.333799\pi$
$839$	28.8921	0.997466	0.498733	+	0.866756i	$0.333799\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	25.0927	0.864240
$844$	0	0
$845$	2.82240	0.0970936
$846$	0	0
$847$	1.05664	0.0363065
$848$	0	0
$849$	26.0505	0.894049
$850$	0	0
$851$	−1.42575	−0.0488742
$852$	0	0
$853$	−7.03925	−0.241019	−0.120510	−	0.992712i	$-0.538453\pi$
$853$	−7.03925	−0.241019	−0.120510	+	0.992712i	$0.538453\pi$
$854$	0	0
$855$	0.320111	0.0109476
$856$	0	0
$857$	−0.221845	−0.00757809	−0.00378904	−	0.999993i	$-0.501206\pi$
$857$	−0.221845	−0.00757809	−0.00378904	+	0.999993i	$0.501206\pi$
$858$	0	0
$859$	36.2549	1.23700	0.618500	−	0.785785i	$-0.287741\pi$
$859$	36.2549	1.23700	0.618500	+	0.785785i	$0.287741\pi$
$860$	0	0
$861$	0.0731982	0.00249459
$862$	0	0
$863$	0.0417367	0.00142074	0.000710368	−	1.00000i	$-0.499774\pi$
$863$	0.0417367	0.00142074	0.000710368		1.00000i	$0.499774\pi$
$864$	0	0
$865$	1.99786	0.0679292
$866$	0	0
$867$	−15.0859	−0.512344
$868$	0	0
$869$	15.9538	0.541196
$870$	0	0
$871$	23.4406	0.794254
$872$	0	0
$873$	4.43366	0.150057
$874$	0	0
$875$	−0.424313	−0.0143444
$876$	0	0
$877$	−44.8890	−1.51579	−0.757897	−	0.652374i	$-0.773773\pi$
$877$	−44.8890	−1.51579	−0.757897	+	0.652374i	$0.773773\pi$
$878$	0	0
$879$	26.2399	0.885048
$880$	0	0
$881$	−4.54343	−0.153072	−0.0765360	−	0.997067i	$-0.524386\pi$
$881$	−4.54343	−0.153072	−0.0765360	+	0.997067i	$0.524386\pi$
$882$	0	0
$883$	−23.5745	−0.793345	−0.396672	−	0.917960i	$-0.629835\pi$
$883$	−23.5745	−0.793345	−0.396672	+	0.917960i	$0.629835\pi$
$884$	0	0
$885$	−3.07332	−0.103309
$886$	0	0
$887$	28.4093	0.953891	0.476945	−	0.878933i	$-0.341744\pi$
$887$	28.4093	0.953891	0.476945	+	0.878933i	$0.341744\pi$
$888$	0	0
$889$	−0.661180	−0.0221753
$890$	0	0
$891$	1.85849	0.0622618
$892$	0	0
$893$	−7.40192	−0.247696
$894$	0	0
$895$	−4.85990	−0.162449
$896$	0	0
$897$	1.94243	0.0648558
$898$	0	0
$899$	6.72475	0.224283
$900$	0	0
$901$	−62.5632	−2.08428
$902$	0	0
$903$	0.598946	0.0199317
$904$	0	0
$905$	2.40398	0.0799110
$906$	0	0
$907$	27.4950	0.912956	0.456478	−	0.889735i	$-0.349111\pi$
$907$	27.4950	0.912956	0.456478	+	0.889735i	$0.349111\pi$
$908$	0	0
$909$	5.01727	0.166412
$910$	0	0
$911$	−10.1181	−0.335227	−0.167613	−	0.985853i	$-0.553606\pi$
$911$	−10.1181	−0.335227	−0.167613	+	0.985853i	$0.553606\pi$
$912$	0	0
$913$	14.2261	0.470814
$914$	0	0
$915$	2.34642	0.0775704
$916$	0	0
$917$	1.67521	0.0553202
$918$	0	0
$919$	22.5438	0.743651	0.371826	−	0.928303i	$-0.378732\pi$
$919$	22.5438	0.743651	0.371826	+	0.928303i	$0.378732\pi$
$920$	0	0
$921$	−0.970427	−0.0319766
$922$	0	0
$923$	−16.1211	−0.530633
$924$	0	0
$925$	6.99536	0.230006
$926$	0	0
$927$	−13.5050	−0.443561
$928$	0	0
$929$	−38.5125	−1.26355	−0.631777	−	0.775150i	$-0.717674\pi$
$929$	−38.5125	−1.26355	−0.631777	+	0.775150i	$0.717674\pi$
$930$	0	0
$931$	7.30501	0.239412
$932$	0	0
$933$	−10.0703	−0.329688
$934$	0	0
$935$	−3.22016	−0.105311
$936$	0	0
$937$	−9.71092	−0.317242	−0.158621	−	0.987340i	$-0.550705\pi$
$937$	−9.71092	−0.317242	−0.158621	+	0.987340i	$0.550705\pi$
$938$	0	0
$939$	29.9323	0.976805
$940$	0	0
$941$	−29.5549	−0.963462	−0.481731	−	0.876319i	$-0.659992\pi$
$941$	−29.5549	−0.963462	−0.481731	+	0.876319i	$0.659992\pi$
$942$	0	0
$943$	0.522747	0.0170230
$944$	0	0
$945$	−0.0428320	−0.00139333
$946$	0	0
$947$	−48.0594	−1.56172	−0.780861	−	0.624705i	$-0.785219\pi$
$947$	−48.0594	−1.56172	−0.780861	+	0.624705i	$0.785219\pi$
$948$	0	0
$949$	0.133259	0.00432578
$950$	0	0
$951$	−20.1859	−0.654572
$952$	0	0
$953$	50.2867	1.62895	0.814473	−	0.580201i	$-0.197026\pi$
$953$	50.2867	1.62895	0.814473	+	0.580201i	$0.197026\pi$
$954$	0	0
$955$	0.928078	0.0300319
$956$	0	0
$957$	1.85849	0.0600765
$958$	0	0
$959$	1.84431	0.0595558
$960$	0	0
$961$	14.2223	0.458784
$962$	0	0
$963$	−10.1993	−0.328669
$964$	0	0
$965$	1.17122	0.0377027
$966$	0	0
$967$	41.3154	1.32861	0.664306	−	0.747461i	$-0.268727\pi$
$967$	41.3154	1.32861	0.664306	+	0.747461i	$0.268727\pi$
$968$	0	0
$969$	5.92786	0.190430
$970$	0	0
$971$	−45.3278	−1.45464	−0.727319	−	0.686299i	$-0.759234\pi$
$971$	−45.3278	−1.45464	−0.727319	+	0.686299i	$0.759234\pi$
$972$	0	0
$973$	−1.65867	−0.0531745
$974$	0	0
$975$	−9.53040	−0.305217
$976$	0	0
$977$	43.0997	1.37888	0.689441	−	0.724342i	$-0.257856\pi$
$977$	43.0997	1.37888	0.689441	+	0.724342i	$0.257856\pi$
$978$	0	0
$979$	16.6100	0.530857
$980$	0	0
$981$	0.150542	0.00480643
$982$	0	0
$983$	17.4186	0.555567	0.277783	−	0.960644i	$-0.410400\pi$
$983$	17.4186	0.555567	0.277783	+	0.960644i	$0.410400\pi$
$984$	0	0
$985$	2.02318	0.0644638
$986$	0	0
$987$	0.990404	0.0315249
$988$	0	0
$989$	4.27739	0.136013
$990$	0	0
$991$	−31.7050	−1.00714	−0.503571	−	0.863954i	$-0.667981\pi$
$991$	−31.7050	−1.00714	−0.503571	+	0.863954i	$0.667981\pi$
$992$	0	0
$993$	28.2007	0.894922
$994$	0	0
$995$	−0.644479	−0.0204313
$996$	0	0
$997$	33.2432	1.05282	0.526412	−	0.850230i	$-0.323537\pi$
$997$	33.2432	1.05282	0.526412	+	0.850230i	$0.323537\pi$
$998$	0	0
$999$	1.42575	0.0451088

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.g.1.6	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.g.1.6	✓	12	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -0.305886 q^{5} -0.140026 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -0.305886 q^{5} -0.140026 q^{7} +1.00000 q^{9} +1.85849 q^{11} -1.94243 q^{13} +0.305886 q^{15} +5.66444 q^{17} -1.04650 q^{19} +0.140026 q^{21} +1.00000 q^{23} -4.90643 q^{25} -1.00000 q^{27} -1.00000 q^{29} -6.72475 q^{31} -1.85849 q^{33} +0.0428320 q^{35} -1.42575 q^{37} +1.94243 q^{39} +0.522747 q^{41} +4.27739 q^{43} -0.305886 q^{45} +7.07300 q^{47} -6.98039 q^{49} -5.66444 q^{51} -11.0449 q^{53} -0.568487 q^{55} +1.04650 q^{57} -10.0473 q^{59} +7.67091 q^{61} -0.140026 q^{63} +0.594162 q^{65} -12.0677 q^{67} -1.00000 q^{69} +8.29946 q^{71} -0.0686045 q^{73} +4.90643 q^{75} -0.260237 q^{77} +8.58428 q^{79} +1.00000 q^{81} +7.65463 q^{83} -1.73267 q^{85} +1.00000 q^{87} +8.93734 q^{89} +0.271991 q^{91} +6.72475 q^{93} +0.320111 q^{95} +4.43366 q^{97} +1.85849 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{3} - 3 q^{5} + 4 q^{7} + 12 q^{9}+O(q^{10}) \) 12 * q - 12 * q^3 - 3 * q^5 + 4 * q^7 + 12 * q^9	\( 12 q - 12 q^{3} - 3 q^{5} + 4 q^{7} + 12 q^{9} - 5 q^{11} - 6 q^{13} + 3 q^{15} - 7 q^{17} - 3 q^{19} - 4 q^{21} + 12 q^{23} + 11 q^{25} - 12 q^{27} - 12 q^{29} + 2 q^{31} + 5 q^{33} - 9 q^{35} - 20 q^{37} + 6 q^{39} - 3 q^{41} + 5 q^{43} - 3 q^{45} - 2 q^{49} + 7 q^{51} - 3 q^{53} + 19 q^{55} + 3 q^{57} - 20 q^{59} - 17 q^{61} + 4 q^{63} - 4 q^{65} - 9 q^{67} - 12 q^{69} + 7 q^{71} - 9 q^{73} - 11 q^{75} - 34 q^{77} + 14 q^{79} + 12 q^{81} + 5 q^{83} - 12 q^{85} + 12 q^{87} - 22 q^{89} - 3 q^{91} - 2 q^{93} - 27 q^{95} + 17 q^{97} - 5 q^{99}+O(q^{100}) \) 12 * q - 12 * q^3 - 3 * q^5 + 4 * q^7 + 12 * q^9 - 5 * q^11 - 6 * q^13 + 3 * q^15 - 7 * q^17 - 3 * q^19 - 4 * q^21 + 12 * q^23 + 11 * q^25 - 12 * q^27 - 12 * q^29 + 2 * q^31 + 5 * q^33 - 9 * q^35 - 20 * q^37 + 6 * q^39 - 3 * q^41 + 5 * q^43 - 3 * q^45 - 2 * q^49 + 7 * q^51 - 3 * q^53 + 19 * q^55 + 3 * q^57 - 20 * q^59 - 17 * q^61 + 4 * q^63 - 4 * q^65 - 9 * q^67 - 12 * q^69 + 7 * q^71 - 9 * q^73 - 11 * q^75 - 34 * q^77 + 14 * q^79 + 12 * q^81 + 5 * q^83 - 12 * q^85 + 12 * q^87 - 22 * q^89 - 3 * q^91 - 2 * q^93 - 27 * q^95 + 17 * q^97 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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