LMFDB - Embedded newform 8004.2.a.i.1.7

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			1.7
Root			$-0.123882$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -0.123882 q^{5} +3.12124 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -0.123882 q^{5} +3.12124 q^{7} +1.00000 q^{9} -3.34537 q^{11} -4.64227 q^{13} +0.123882 q^{15} -4.27779 q^{17} -2.36905 q^{19} -3.12124 q^{21} +1.00000 q^{23} -4.98465 q^{25} -1.00000 q^{27} +1.00000 q^{29} -4.48181 q^{31} +3.34537 q^{33} -0.386665 q^{35} -5.19624 q^{37} +4.64227 q^{39} -7.96483 q^{41} +3.81430 q^{43} -0.123882 q^{45} +2.73638 q^{47} +2.74217 q^{49} +4.27779 q^{51} +11.8801 q^{53} +0.414430 q^{55} +2.36905 q^{57} +11.8045 q^{59} +1.29173 q^{61} +3.12124 q^{63} +0.575092 q^{65} +13.6476 q^{67} -1.00000 q^{69} -14.5132 q^{71} -1.55686 q^{73} +4.98465 q^{75} -10.4417 q^{77} +9.29227 q^{79} +1.00000 q^{81} +16.0061 q^{83} +0.529940 q^{85} -1.00000 q^{87} +9.47785 q^{89} -14.4897 q^{91} +4.48181 q^{93} +0.293481 q^{95} +11.4680 q^{97} -3.34537 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−0.123882	−0.0554016	−0.0277008	−	0.999616i	$-0.508819\pi$
$5$	−0.123882	−0.0554016	−0.0277008	+	0.999616i	$0.508819\pi$
$6$	0	0
$7$	3.12124	1.17972	0.589860	−	0.807506i	$-0.299183\pi$
$7$	3.12124	1.17972	0.589860	+	0.807506i	$0.299183\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.34537	−1.00867	−0.504333	−	0.863509i	$-0.668262\pi$
$11$	−3.34537	−1.00867	−0.504333	+	0.863509i	$0.668262\pi$
$12$	0	0
$13$	−4.64227	−1.28753	−0.643767	−	0.765221i	$-0.722630\pi$
$13$	−4.64227	−1.28753	−0.643767	+	0.765221i	$0.722630\pi$
$14$	0	0
$15$	0.123882	0.0319861
$16$	0	0
$17$	−4.27779	−1.03752	−0.518759	−	0.854921i	$-0.673606\pi$
$17$	−4.27779	−1.03752	−0.518759	+	0.854921i	$0.673606\pi$
$18$	0	0
$19$	−2.36905	−0.543497	−0.271748	−	0.962368i	$-0.587602\pi$
$19$	−2.36905	−0.543497	−0.271748	+	0.962368i	$0.587602\pi$
$20$	0	0
$21$	−3.12124	−0.681111
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−4.98465	−0.996931
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−4.48181	−0.804956	−0.402478	−	0.915430i	$-0.631851\pi$
$31$	−4.48181	−0.804956	−0.402478	+	0.915430i	$0.631851\pi$
$32$	0	0
$33$	3.34537	0.582354
$34$	0	0
$35$	−0.386665	−0.0653583
$36$	0	0
$37$	−5.19624	−0.854257	−0.427128	−	0.904191i	$-0.640475\pi$
$37$	−5.19624	−0.854257	−0.427128	+	0.904191i	$0.640475\pi$
$38$	0	0
$39$	4.64227	0.743358
$40$	0	0
$41$	−7.96483	−1.24390	−0.621949	−	0.783058i	$-0.713659\pi$
$41$	−7.96483	−1.24390	−0.621949	+	0.783058i	$0.713659\pi$
$42$	0	0
$43$	3.81430	0.581676	0.290838	−	0.956772i	$-0.406066\pi$
$43$	3.81430	0.581676	0.290838	+	0.956772i	$0.406066\pi$
$44$	0	0
$45$	−0.123882	−0.0184672
$46$	0	0
$47$	2.73638	0.399143	0.199571	−	0.979883i	$-0.436045\pi$
$47$	2.73638	0.399143	0.199571	+	0.979883i	$0.436045\pi$
$48$	0	0
$49$	2.74217	0.391738
$50$	0	0
$51$	4.27779	0.599011
$52$	0	0
$53$	11.8801	1.63186	0.815928	−	0.578153i	$-0.196226\pi$
$53$	11.8801	1.63186	0.815928	+	0.578153i	$0.196226\pi$
$54$	0	0
$55$	0.414430	0.0558817
$56$	0	0
$57$	2.36905	0.313788
$58$	0	0
$59$	11.8045	1.53682	0.768410	−	0.639957i	$-0.221048\pi$
$59$	11.8045	1.53682	0.768410	+	0.639957i	$0.221048\pi$
$60$	0	0
$61$	1.29173	0.165389	0.0826945	−	0.996575i	$-0.473647\pi$
$61$	1.29173	0.165389	0.0826945	+	0.996575i	$0.473647\pi$
$62$	0	0
$63$	3.12124	0.393240
$64$	0	0
$65$	0.575092	0.0713314
$66$	0	0
$67$	13.6476	1.66732	0.833662	−	0.552274i	$-0.186240\pi$
$67$	13.6476	1.66732	0.833662	+	0.552274i	$0.186240\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−14.5132	−1.72240	−0.861201	−	0.508264i	$-0.830287\pi$
$71$	−14.5132	−1.72240	−0.861201	+	0.508264i	$0.830287\pi$
$72$	0	0
$73$	−1.55686	−0.182217	−0.0911083	−	0.995841i	$-0.529041\pi$
$73$	−1.55686	−0.182217	−0.0911083	+	0.995841i	$0.529041\pi$
$74$	0	0
$75$	4.98465	0.575578
$76$	0	0
$77$	−10.4417	−1.18994
$78$	0	0
$79$	9.29227	1.04546	0.522731	−	0.852498i	$-0.324913\pi$
$79$	9.29227	1.04546	0.522731	+	0.852498i	$0.324913\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	16.0061	1.75690	0.878450	−	0.477834i	$-0.158578\pi$
$83$	16.0061	1.75690	0.878450	+	0.477834i	$0.158578\pi$
$84$	0	0
$85$	0.529940	0.0574801
$86$	0	0
$87$	−1.00000	−0.107211
$88$	0	0
$89$	9.47785	1.00465	0.502325	−	0.864679i	$-0.332478\pi$
$89$	9.47785	1.00465	0.502325	+	0.864679i	$0.332478\pi$
$90$	0	0
$91$	−14.4897	−1.51893
$92$	0	0
$93$	4.48181	0.464742
$94$	0	0
$95$	0.293481	0.0301106
$96$	0	0
$97$	11.4680	1.16439	0.582197	−	0.813048i	$-0.302193\pi$
$97$	11.4680	1.16439	0.582197	+	0.813048i	$0.302193\pi$
$98$	0	0
$99$	−3.34537	−0.336222
$100$	0	0
$101$	5.00291	0.497808	0.248904	−	0.968528i	$-0.419930\pi$
$101$	5.00291	0.497808	0.248904	+	0.968528i	$0.419930\pi$
$102$	0	0
$103$	6.43917	0.634471	0.317235	−	0.948347i	$-0.397245\pi$
$103$	6.43917	0.634471	0.317235	+	0.948347i	$0.397245\pi$
$104$	0	0
$105$	0.386665	0.0377346
$106$	0	0
$107$	4.23096	0.409022	0.204511	−	0.978864i	$-0.434440\pi$
$107$	4.23096	0.409022	0.204511	+	0.978864i	$0.434440\pi$
$108$	0	0
$109$	3.01374	0.288664	0.144332	−	0.989529i	$-0.453897\pi$
$109$	3.01374	0.288664	0.144332	+	0.989529i	$0.453897\pi$
$110$	0	0
$111$	5.19624	0.493205
$112$	0	0
$113$	14.5321	1.36707	0.683534	−	0.729919i	$-0.260442\pi$
$113$	14.5321	1.36707	0.683534	+	0.729919i	$0.260442\pi$
$114$	0	0
$115$	−0.123882	−0.0115520
$116$	0	0
$117$	−4.64227	−0.429178
$118$	0	0
$119$	−13.3520	−1.22398
$120$	0	0
$121$	0.191499	0.0174090
$122$	0	0
$123$	7.96483	0.718165
$124$	0	0
$125$	1.23692	0.110633
$126$	0	0
$127$	−14.1554	−1.25609	−0.628043	−	0.778179i	$-0.716144\pi$
$127$	−14.1554	−1.25609	−0.628043	+	0.778179i	$0.716144\pi$
$128$	0	0
$129$	−3.81430	−0.335831
$130$	0	0
$131$	−13.8029	−1.20596	−0.602981	−	0.797756i	$-0.706021\pi$
$131$	−13.8029	−1.20596	−0.602981	+	0.797756i	$0.706021\pi$
$132$	0	0
$133$	−7.39437	−0.641173
$134$	0	0
$135$	0.123882	0.0106620
$136$	0	0
$137$	15.5918	1.33209	0.666047	−	0.745910i	$-0.267985\pi$
$137$	15.5918	1.33209	0.666047	+	0.745910i	$0.267985\pi$
$138$	0	0
$139$	1.46483	0.124245	0.0621226	−	0.998069i	$-0.480213\pi$
$139$	1.46483	0.124245	0.0621226	+	0.998069i	$0.480213\pi$
$140$	0	0
$141$	−2.73638	−0.230445
$142$	0	0
$143$	15.5301	1.29869
$144$	0	0
$145$	−0.123882	−0.0102878
$146$	0	0
$147$	−2.74217	−0.226170
$148$	0	0
$149$	20.0755	1.64465	0.822323	−	0.569021i	$-0.192678\pi$
$149$	20.0755	1.64465	0.822323	+	0.569021i	$0.192678\pi$
$150$	0	0
$151$	−15.8842	−1.29263	−0.646317	−	0.763069i	$-0.723692\pi$
$151$	−15.8842	−1.29263	−0.646317	+	0.763069i	$0.723692\pi$
$152$	0	0
$153$	−4.27779	−0.345839
$154$	0	0
$155$	0.555214	0.0445958
$156$	0	0
$157$	−12.6535	−1.00986	−0.504931	−	0.863159i	$-0.668482\pi$
$157$	−12.6535	−1.00986	−0.504931	+	0.863159i	$0.668482\pi$
$158$	0	0
$159$	−11.8801	−0.942153
$160$	0	0
$161$	3.12124	0.245989
$162$	0	0
$163$	−15.0684	−1.18025	−0.590126	−	0.807311i	$-0.700922\pi$
$163$	−15.0684	−1.18025	−0.590126	+	0.807311i	$0.700922\pi$
$164$	0	0
$165$	−0.414430	−0.0322633
$166$	0	0
$167$	21.1441	1.63618	0.818090	−	0.575091i	$-0.195033\pi$
$167$	21.1441	1.63618	0.818090	+	0.575091i	$0.195033\pi$
$168$	0	0
$169$	8.55069	0.657745
$170$	0	0
$171$	−2.36905	−0.181166
$172$	0	0
$173$	1.19786	0.0910714	0.0455357	−	0.998963i	$-0.485501\pi$
$173$	1.19786	0.0910714	0.0455357	+	0.998963i	$0.485501\pi$
$174$	0	0
$175$	−15.5583	−1.17610
$176$	0	0
$177$	−11.8045	−0.887284
$178$	0	0
$179$	−3.76867	−0.281684	−0.140842	−	0.990032i	$-0.544981\pi$
$179$	−3.76867	−0.281684	−0.140842	+	0.990032i	$0.544981\pi$
$180$	0	0
$181$	26.2502	1.95116	0.975581	−	0.219642i	$-0.0704889\pi$
$181$	26.2502	1.95116	0.975581	+	0.219642i	$0.0704889\pi$
$182$	0	0
$183$	−1.29173	−0.0954874
$184$	0	0
$185$	0.643719	0.0473272
$186$	0	0
$187$	14.3108	1.04651
$188$	0	0
$189$	−3.12124	−0.227037
$190$	0	0
$191$	9.44833	0.683657	0.341828	−	0.939762i	$-0.388954\pi$
$191$	9.44833	0.683657	0.341828	+	0.939762i	$0.388954\pi$
$192$	0	0
$193$	−19.5334	−1.40604	−0.703022	−	0.711168i	$-0.748166\pi$
$193$	−19.5334	−1.40604	−0.703022	+	0.711168i	$0.748166\pi$
$194$	0	0
$195$	−0.575092	−0.0411832
$196$	0	0
$197$	−16.1253	−1.14888	−0.574438	−	0.818548i	$-0.694779\pi$
$197$	−16.1253	−1.14888	−0.574438	+	0.818548i	$0.694779\pi$
$198$	0	0
$199$	−0.786826	−0.0557766	−0.0278883	−	0.999611i	$-0.508878\pi$
$199$	−0.786826	−0.0557766	−0.0278883	+	0.999611i	$0.508878\pi$
$200$	0	0
$201$	−13.6476	−0.962630
$202$	0	0
$203$	3.12124	0.219068
$204$	0	0
$205$	0.986697	0.0689139
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	7.92534	0.548207
$210$	0	0
$211$	17.5450	1.20785	0.603924	−	0.797042i	$-0.293603\pi$
$211$	17.5450	1.20785	0.603924	+	0.797042i	$0.293603\pi$
$212$	0	0
$213$	14.5132	0.994430
$214$	0	0
$215$	−0.472522	−0.0322257
$216$	0	0
$217$	−13.9888	−0.949622
$218$	0	0
$219$	1.55686	0.105203
$220$	0	0
$221$	19.8587	1.33584
$222$	0	0
$223$	14.1765	0.949326	0.474663	−	0.880168i	$-0.342570\pi$
$223$	14.1765	0.949326	0.474663	+	0.880168i	$0.342570\pi$
$224$	0	0
$225$	−4.98465	−0.332310
$226$	0	0
$227$	−23.0451	−1.52955	−0.764777	−	0.644295i	$-0.777151\pi$
$227$	−23.0451	−1.52955	−0.764777	+	0.644295i	$0.777151\pi$
$228$	0	0
$229$	9.81224	0.648411	0.324206	−	0.945987i	$-0.394903\pi$
$229$	9.81224	0.648411	0.324206	+	0.945987i	$0.394903\pi$
$230$	0	0
$231$	10.4417	0.687015
$232$	0	0
$233$	27.3650	1.79274	0.896371	−	0.443304i	$-0.146194\pi$
$233$	27.3650	1.79274	0.896371	+	0.443304i	$0.146194\pi$
$234$	0	0
$235$	−0.338988	−0.0221131
$236$	0	0
$237$	−9.29227	−0.603598
$238$	0	0
$239$	12.3849	0.801112	0.400556	−	0.916272i	$-0.368817\pi$
$239$	12.3849	0.801112	0.400556	+	0.916272i	$0.368817\pi$
$240$	0	0
$241$	−28.5715	−1.84045	−0.920226	−	0.391388i	$-0.871995\pi$
$241$	−28.5715	−1.84045	−0.920226	+	0.391388i	$0.871995\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−0.339704	−0.0217029
$246$	0	0
$247$	10.9978	0.699771
$248$	0	0
$249$	−16.0061	−1.01435
$250$	0	0
$251$	−13.2296	−0.835047	−0.417523	−	0.908666i	$-0.637102\pi$
$251$	−13.2296	−0.835047	−0.417523	+	0.908666i	$0.637102\pi$
$252$	0	0
$253$	−3.34537	−0.210322
$254$	0	0
$255$	−0.529940	−0.0331862
$256$	0	0
$257$	−2.30623	−0.143858	−0.0719292	−	0.997410i	$-0.522916\pi$
$257$	−2.30623	−0.143858	−0.0719292	+	0.997410i	$0.522916\pi$
$258$	0	0
$259$	−16.2187	−1.00778
$260$	0	0
$261$	1.00000	0.0618984
$262$	0	0
$263$	15.0755	0.929595	0.464797	−	0.885417i	$-0.346127\pi$
$263$	15.0755	0.929595	0.464797	+	0.885417i	$0.346127\pi$
$264$	0	0
$265$	−1.47173	−0.0904074
$266$	0	0
$267$	−9.47785	−0.580035
$268$	0	0
$269$	10.7910	0.657938	0.328969	−	0.944341i	$-0.393299\pi$
$269$	10.7910	0.657938	0.328969	+	0.944341i	$0.393299\pi$
$270$	0	0
$271$	−5.46497	−0.331973	−0.165987	−	0.986128i	$-0.553081\pi$
$271$	−5.46497	−0.331973	−0.165987	+	0.986128i	$0.553081\pi$
$272$	0	0
$273$	14.4897	0.876954
$274$	0	0
$275$	16.6755	1.00557
$276$	0	0
$277$	19.5186	1.17276	0.586379	−	0.810037i	$-0.300553\pi$
$277$	19.5186	1.17276	0.586379	+	0.810037i	$0.300553\pi$
$278$	0	0
$279$	−4.48181	−0.268319
$280$	0	0
$281$	−22.9300	−1.36789	−0.683945	−	0.729533i	$-0.739737\pi$
$281$	−22.9300	−1.36789	−0.683945	+	0.729533i	$0.739737\pi$
$282$	0	0
$283$	25.4692	1.51399	0.756994	−	0.653422i	$-0.226667\pi$
$283$	25.4692	1.51399	0.756994	+	0.653422i	$0.226667\pi$
$284$	0	0
$285$	−0.293481	−0.0173843
$286$	0	0
$287$	−24.8602	−1.46745
$288$	0	0
$289$	1.29952	0.0764423
$290$	0	0
$291$	−11.4680	−0.672263
$292$	0	0
$293$	−14.1564	−0.827026	−0.413513	−	0.910498i	$-0.635698\pi$
$293$	−14.1564	−0.827026	−0.413513	+	0.910498i	$0.635698\pi$
$294$	0	0
$295$	−1.46237	−0.0851423
$296$	0	0
$297$	3.34537	0.194118
$298$	0	0
$299$	−4.64227	−0.268470
$300$	0	0
$301$	11.9054	0.686214
$302$	0	0
$303$	−5.00291	−0.287410
$304$	0	0
$305$	−0.160022	−0.00916281
$306$	0	0
$307$	−24.3935	−1.39221	−0.696106	−	0.717939i	$-0.745085\pi$
$307$	−24.3935	−1.39221	−0.696106	+	0.717939i	$0.745085\pi$
$308$	0	0
$309$	−6.43917	−0.366312
$310$	0	0
$311$	−6.36759	−0.361073	−0.180536	−	0.983568i	$-0.557783\pi$
$311$	−6.36759	−0.361073	−0.180536	+	0.983568i	$0.557783\pi$
$312$	0	0
$313$	−28.0338	−1.58457	−0.792283	−	0.610154i	$-0.791108\pi$
$313$	−28.0338	−1.58457	−0.792283	+	0.610154i	$0.791108\pi$
$314$	0	0
$315$	−0.386665	−0.0217861
$316$	0	0
$317$	8.70755	0.489065	0.244532	−	0.969641i	$-0.421366\pi$
$317$	8.70755	0.489065	0.244532	+	0.969641i	$0.421366\pi$
$318$	0	0
$319$	−3.34537	−0.187305
$320$	0	0
$321$	−4.23096	−0.236149
$322$	0	0
$323$	10.1343	0.563887
$324$	0	0
$325$	23.1401	1.28358
$326$	0	0
$327$	−3.01374	−0.166660
$328$	0	0
$329$	8.54092	0.470876
$330$	0	0
$331$	−24.5067	−1.34701	−0.673506	−	0.739181i	$-0.735213\pi$
$331$	−24.5067	−1.34701	−0.673506	+	0.739181i	$0.735213\pi$
$332$	0	0
$333$	−5.19624	−0.284752
$334$	0	0
$335$	−1.69069	−0.0923724
$336$	0	0
$337$	0.131842	0.00718190	0.00359095	−	0.999994i	$-0.498857\pi$
$337$	0.131842	0.00718190	0.00359095	+	0.999994i	$0.498857\pi$
$338$	0	0
$339$	−14.5321	−0.789277
$340$	0	0
$341$	14.9933	0.811933
$342$	0	0
$343$	−13.2897	−0.717578
$344$	0	0
$345$	0.123882	0.00666957
$346$	0	0
$347$	−2.80296	−0.150471	−0.0752354	−	0.997166i	$-0.523971\pi$
$347$	−2.80296	−0.150471	−0.0752354	+	0.997166i	$0.523971\pi$
$348$	0	0
$349$	8.54612	0.457464	0.228732	−	0.973489i	$-0.426542\pi$
$349$	8.54612	0.457464	0.228732	+	0.973489i	$0.426542\pi$
$350$	0	0
$351$	4.64227	0.247786
$352$	0	0
$353$	−2.12642	−0.113178	−0.0565889	−	0.998398i	$-0.518022\pi$
$353$	−2.12642	−0.113178	−0.0565889	+	0.998398i	$0.518022\pi$
$354$	0	0
$355$	1.79792	0.0954238
$356$	0	0
$357$	13.3520	0.706665
$358$	0	0
$359$	8.06665	0.425741	0.212871	−	0.977080i	$-0.431719\pi$
$359$	8.06665	0.425741	0.212871	+	0.977080i	$0.431719\pi$
$360$	0	0
$361$	−13.3876	−0.704612
$362$	0	0
$363$	−0.191499	−0.0100511
$364$	0	0
$365$	0.192866	0.0100951
$366$	0	0
$367$	16.9668	0.885658	0.442829	−	0.896606i	$-0.353975\pi$
$367$	16.9668	0.885658	0.442829	+	0.896606i	$0.353975\pi$
$368$	0	0
$369$	−7.96483	−0.414633
$370$	0	0
$371$	37.0807	1.92513
$372$	0	0
$373$	−5.91353	−0.306191	−0.153095	−	0.988211i	$-0.548924\pi$
$373$	−5.91353	−0.306191	−0.153095	+	0.988211i	$0.548924\pi$
$374$	0	0
$375$	−1.23692	−0.0638741
$376$	0	0
$377$	−4.64227	−0.239089
$378$	0	0
$379$	15.6879	0.805831	0.402916	−	0.915237i	$-0.367997\pi$
$379$	15.6879	0.805831	0.402916	+	0.915237i	$0.367997\pi$
$380$	0	0
$381$	14.1554	0.725201
$382$	0	0
$383$	−17.3356	−0.885808	−0.442904	−	0.896569i	$-0.646052\pi$
$383$	−17.3356	−0.885808	−0.442904	+	0.896569i	$0.646052\pi$
$384$	0	0
$385$	1.29354	0.0659248
$386$	0	0
$387$	3.81430	0.193892
$388$	0	0
$389$	−0.709337	−0.0359648	−0.0179824	−	0.999838i	$-0.505724\pi$
$389$	−0.709337	−0.0359648	−0.0179824	+	0.999838i	$0.505724\pi$
$390$	0	0
$391$	−4.27779	−0.216337
$392$	0	0
$393$	13.8029	0.696262
$394$	0	0
$395$	−1.15114	−0.0579203
$396$	0	0
$397$	29.5867	1.48491	0.742456	−	0.669895i	$-0.233661\pi$
$397$	29.5867	1.48491	0.742456	+	0.669895i	$0.233661\pi$
$398$	0	0
$399$	7.39437	0.370182
$400$	0	0
$401$	32.6563	1.63078	0.815389	−	0.578914i	$-0.196523\pi$
$401$	32.6563	1.63078	0.815389	+	0.578914i	$0.196523\pi$
$402$	0	0
$403$	20.8058	1.03641
$404$	0	0
$405$	−0.123882	−0.00615573
$406$	0	0
$407$	17.3833	0.861661
$408$	0	0
$409$	6.65832	0.329233	0.164616	−	0.986358i	$-0.447361\pi$
$409$	6.65832	0.329233	0.164616	+	0.986358i	$0.447361\pi$
$410$	0	0
$411$	−15.5918	−0.769085
$412$	0	0
$413$	36.8449	1.81302
$414$	0	0
$415$	−1.98287	−0.0973350
$416$	0	0
$417$	−1.46483	−0.0717330
$418$	0	0
$419$	−37.8148	−1.84738	−0.923688	−	0.383146i	$-0.874841\pi$
$419$	−37.8148	−1.84738	−0.923688	+	0.383146i	$0.874841\pi$
$420$	0	0
$421$	29.1024	1.41837	0.709183	−	0.705025i	$-0.249064\pi$
$421$	29.1024	1.41837	0.709183	+	0.705025i	$0.249064\pi$
$422$	0	0
$423$	2.73638	0.133048
$424$	0	0
$425$	21.3233	1.03433
$426$	0	0
$427$	4.03180	0.195113
$428$	0	0
$429$	−15.5301	−0.749801
$430$	0	0
$431$	−12.8846	−0.620628	−0.310314	−	0.950634i	$-0.600434\pi$
$431$	−12.8846	−0.620628	−0.310314	+	0.950634i	$0.600434\pi$
$432$	0	0
$433$	−16.0657	−0.772069	−0.386034	−	0.922484i	$-0.626155\pi$
$433$	−16.0657	−0.772069	−0.386034	+	0.922484i	$0.626155\pi$
$434$	0	0
$435$	0.123882	0.00593967
$436$	0	0
$437$	−2.36905	−0.113327
$438$	0	0
$439$	−0.642410	−0.0306605	−0.0153303	−	0.999882i	$-0.504880\pi$
$439$	−0.642410	−0.0306605	−0.0153303	+	0.999882i	$0.504880\pi$
$440$	0	0
$441$	2.74217	0.130579
$442$	0	0
$443$	20.6330	0.980304	0.490152	−	0.871637i	$-0.336941\pi$
$443$	20.6330	0.980304	0.490152	+	0.871637i	$0.336941\pi$
$444$	0	0
$445$	−1.17413	−0.0556592
$446$	0	0
$447$	−20.0755	−0.949537
$448$	0	0
$449$	16.1685	0.763037	0.381519	−	0.924361i	$-0.375401\pi$
$449$	16.1685	0.763037	0.381519	+	0.924361i	$0.375401\pi$
$450$	0	0
$451$	26.6453	1.25468
$452$	0	0
$453$	15.8842	0.746303
$454$	0	0
$455$	1.79500	0.0841511
$456$	0	0
$457$	−21.8185	−1.02063	−0.510313	−	0.859989i	$-0.670470\pi$
$457$	−21.8185	−1.02063	−0.510313	+	0.859989i	$0.670470\pi$
$458$	0	0
$459$	4.27779	0.199670
$460$	0	0
$461$	−13.2053	−0.615034	−0.307517	−	0.951543i	$-0.599498\pi$
$461$	−13.2053	−0.615034	−0.307517	+	0.951543i	$0.599498\pi$
$462$	0	0
$463$	−33.1684	−1.54147	−0.770734	−	0.637157i	$-0.780110\pi$
$463$	−33.1684	−1.54147	−0.770734	+	0.637157i	$0.780110\pi$
$464$	0	0
$465$	−0.555214	−0.0257474
$466$	0	0
$467$	37.1989	1.72136	0.860679	−	0.509148i	$-0.170039\pi$
$467$	37.1989	1.72136	0.860679	+	0.509148i	$0.170039\pi$
$468$	0	0
$469$	42.5976	1.96698
$470$	0	0
$471$	12.6535	0.583045
$472$	0	0
$473$	−12.7603	−0.586717
$474$	0	0
$475$	11.8089	0.541828
$476$	0	0
$477$	11.8801	0.543952
$478$	0	0
$479$	15.4721	0.706938	0.353469	−	0.935446i	$-0.385002\pi$
$479$	15.4721	0.706938	0.353469	+	0.935446i	$0.385002\pi$
$480$	0	0
$481$	24.1224	1.09989
$482$	0	0
$483$	−3.12124	−0.142022
$484$	0	0
$485$	−1.42067	−0.0645093
$486$	0	0
$487$	18.2873	0.828676	0.414338	−	0.910123i	$-0.364013\pi$
$487$	18.2873	0.828676	0.414338	+	0.910123i	$0.364013\pi$
$488$	0	0
$489$	15.0684	0.681418
$490$	0	0
$491$	−15.4741	−0.698338	−0.349169	−	0.937060i	$-0.613536\pi$
$491$	−15.4741	−0.698338	−0.349169	+	0.937060i	$0.613536\pi$
$492$	0	0
$493$	−4.27779	−0.192662
$494$	0	0
$495$	0.414430	0.0186272
$496$	0	0
$497$	−45.2993	−2.03195
$498$	0	0
$499$	32.6480	1.46153	0.730763	−	0.682632i	$-0.239165\pi$
$499$	32.6480	1.46153	0.730763	+	0.682632i	$0.239165\pi$
$500$	0	0
$501$	−21.1441	−0.944649
$502$	0	0
$503$	−25.1863	−1.12300	−0.561501	−	0.827476i	$-0.689776\pi$
$503$	−25.1863	−1.12300	−0.561501	+	0.827476i	$0.689776\pi$
$504$	0	0
$505$	−0.619769	−0.0275794
$506$	0	0
$507$	−8.55069	−0.379749
$508$	0	0
$509$	12.4673	0.552605	0.276303	−	0.961071i	$-0.410891\pi$
$509$	12.4673	0.552605	0.276303	+	0.961071i	$0.410891\pi$
$510$	0	0
$511$	−4.85934	−0.214964
$512$	0	0
$513$	2.36905	0.104596
$514$	0	0
$515$	−0.797696	−0.0351507
$516$	0	0
$517$	−9.15422	−0.402602
$518$	0	0
$519$	−1.19786	−0.0525801
$520$	0	0
$521$	35.7753	1.56734	0.783672	−	0.621175i	$-0.213344\pi$
$521$	35.7753	1.56734	0.783672	+	0.621175i	$0.213344\pi$
$522$	0	0
$523$	−12.4454	−0.544199	−0.272099	−	0.962269i	$-0.587718\pi$
$523$	−12.4454	−0.544199	−0.272099	+	0.962269i	$0.587718\pi$
$524$	0	0
$525$	15.5583	0.679021
$526$	0	0
$527$	19.1722	0.835156
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	11.8045	0.512274
$532$	0	0
$533$	36.9749	1.60156
$534$	0	0
$535$	−0.524138	−0.0226605
$536$	0	0
$537$	3.76867	0.162630
$538$	0	0
$539$	−9.17356	−0.395133
$540$	0	0
$541$	27.6453	1.18856	0.594282	−	0.804257i	$-0.297436\pi$
$541$	27.6453	1.18856	0.594282	+	0.804257i	$0.297436\pi$
$542$	0	0
$543$	−26.2502	−1.12650
$544$	0	0
$545$	−0.373347	−0.0159924
$546$	0	0
$547$	17.4057	0.744215	0.372108	−	0.928190i	$-0.378635\pi$
$547$	17.4057	0.744215	0.372108	+	0.928190i	$0.378635\pi$
$548$	0	0
$549$	1.29173	0.0551297
$550$	0	0
$551$	−2.36905	−0.100925
$552$	0	0
$553$	29.0035	1.23335
$554$	0	0
$555$	−0.643719	−0.0273244
$556$	0	0
$557$	−12.9472	−0.548592	−0.274296	−	0.961645i	$-0.588445\pi$
$557$	−12.9472	−0.548592	−0.274296	+	0.961645i	$0.588445\pi$
$558$	0	0
$559$	−17.7070	−0.748927
$560$	0	0
$561$	−14.3108	−0.604203
$562$	0	0
$563$	−18.5120	−0.780189	−0.390094	−	0.920775i	$-0.627558\pi$
$563$	−18.5120	−0.780189	−0.390094	+	0.920775i	$0.627558\pi$
$564$	0	0
$565$	−1.80027	−0.0757377
$566$	0	0
$567$	3.12124	0.131080
$568$	0	0
$569$	19.1094	0.801108	0.400554	−	0.916273i	$-0.368818\pi$
$569$	19.1094	0.801108	0.400554	+	0.916273i	$0.368818\pi$
$570$	0	0
$571$	15.9646	0.668099	0.334050	−	0.942555i	$-0.391585\pi$
$571$	15.9646	0.668099	0.334050	+	0.942555i	$0.391585\pi$
$572$	0	0
$573$	−9.44833	−0.394709
$574$	0	0
$575$	−4.98465	−0.207874
$576$	0	0
$577$	−37.3694	−1.55571	−0.777855	−	0.628444i	$-0.783692\pi$
$577$	−37.3694	−1.55571	−0.777855	+	0.628444i	$0.783692\pi$
$578$	0	0
$579$	19.5334	0.811780
$580$	0	0
$581$	49.9590	2.07265
$582$	0	0
$583$	−39.7433	−1.64600
$584$	0	0
$585$	0.575092	0.0237771
$586$	0	0
$587$	19.4465	0.802645	0.401322	−	0.915937i	$-0.368551\pi$
$587$	19.4465	0.802645	0.401322	+	0.915937i	$0.368551\pi$
$588$	0	0
$589$	10.6176	0.437491
$590$	0	0
$591$	16.1253	0.663304
$592$	0	0
$593$	8.79417	0.361133	0.180567	−	0.983563i	$-0.442207\pi$
$593$	8.79417	0.361133	0.180567	+	0.983563i	$0.442207\pi$
$594$	0	0
$595$	1.65407	0.0678104
$596$	0	0
$597$	0.786826	0.0322026
$598$	0	0
$599$	−37.0331	−1.51313	−0.756565	−	0.653919i	$-0.773124\pi$
$599$	−37.0331	−1.51313	−0.756565	+	0.653919i	$0.773124\pi$
$600$	0	0
$601$	−28.5398	−1.16416	−0.582080	−	0.813131i	$-0.697761\pi$
$601$	−28.5398	−1.16416	−0.582080	+	0.813131i	$0.697761\pi$
$602$	0	0
$603$	13.6476	0.555775
$604$	0	0
$605$	−0.0237232	−0.000964487 0
$606$	0	0
$607$	−5.79933	−0.235388	−0.117694	−	0.993050i	$-0.537550\pi$
$607$	−5.79933	−0.235388	−0.117694	+	0.993050i	$0.537550\pi$
$608$	0	0
$609$	−3.12124	−0.126479
$610$	0	0
$611$	−12.7030	−0.513910
$612$	0	0
$613$	20.0488	0.809763	0.404881	−	0.914369i	$-0.367313\pi$
$613$	20.0488	0.809763	0.404881	+	0.914369i	$0.367313\pi$
$614$	0	0
$615$	−0.986697	−0.0397875
$616$	0	0
$617$	6.23955	0.251195	0.125597	−	0.992081i	$-0.459915\pi$
$617$	6.23955	0.251195	0.125597	+	0.992081i	$0.459915\pi$
$618$	0	0
$619$	−4.24672	−0.170690	−0.0853450	−	0.996351i	$-0.527199\pi$
$619$	−4.24672	−0.170690	−0.0853450	+	0.996351i	$0.527199\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	29.5827	1.18521
$624$	0	0
$625$	24.7700	0.990801
$626$	0	0
$627$	−7.92534	−0.316507
$628$	0	0
$629$	22.2285	0.886306
$630$	0	0
$631$	22.1747	0.882762	0.441381	−	0.897320i	$-0.354489\pi$
$631$	22.1747	0.882762	0.441381	+	0.897320i	$0.354489\pi$
$632$	0	0
$633$	−17.5450	−0.697352
$634$	0	0
$635$	1.75359	0.0695891
$636$	0	0
$637$	−12.7299	−0.504376
$638$	0	0
$639$	−14.5132	−0.574134
$640$	0	0
$641$	37.6945	1.48884	0.744421	−	0.667710i	$-0.232725\pi$
$641$	37.6945	1.48884	0.744421	+	0.667710i	$0.232725\pi$
$642$	0	0
$643$	28.7934	1.13550	0.567750	−	0.823201i	$-0.307814\pi$
$643$	28.7934	1.13550	0.567750	+	0.823201i	$0.307814\pi$
$644$	0	0
$645$	0.472522	0.0186055
$646$	0	0
$647$	25.4215	0.999424	0.499712	−	0.866192i	$-0.333439\pi$
$647$	25.4215	0.999424	0.499712	+	0.866192i	$0.333439\pi$
$648$	0	0
$649$	−39.4906	−1.55014
$650$	0	0
$651$	13.9888	0.548265
$652$	0	0
$653$	18.6157	0.728487	0.364244	−	0.931304i	$-0.381328\pi$
$653$	18.6157	0.728487	0.364244	+	0.931304i	$0.381328\pi$
$654$	0	0
$655$	1.70992	0.0668122
$656$	0	0
$657$	−1.55686	−0.0607389
$658$	0	0
$659$	22.0488	0.858899	0.429449	−	0.903091i	$-0.358708\pi$
$659$	22.0488	0.858899	0.429449	+	0.903091i	$0.358708\pi$
$660$	0	0
$661$	21.3375	0.829931	0.414966	−	0.909837i	$-0.363794\pi$
$661$	21.3375	0.829931	0.414966	+	0.909837i	$0.363794\pi$
$662$	0	0
$663$	−19.8587	−0.771247
$664$	0	0
$665$	0.916027	0.0355220
$666$	0	0
$667$	1.00000	0.0387202
$668$	0	0
$669$	−14.1765	−0.548093
$670$	0	0
$671$	−4.32131	−0.166822
$672$	0	0
$673$	7.24529	0.279285	0.139643	−	0.990202i	$-0.455405\pi$
$673$	7.24529	0.279285	0.139643	+	0.990202i	$0.455405\pi$
$674$	0	0
$675$	4.98465	0.191859
$676$	0	0
$677$	2.27849	0.0875696	0.0437848	−	0.999041i	$-0.486058\pi$
$677$	2.27849	0.0875696	0.0437848	+	0.999041i	$0.486058\pi$
$678$	0	0
$679$	35.7943	1.37366
$680$	0	0
$681$	23.0451	0.883088
$682$	0	0
$683$	−7.55361	−0.289031	−0.144515	−	0.989503i	$-0.546162\pi$
$683$	−7.55361	−0.289031	−0.144515	+	0.989503i	$0.546162\pi$
$684$	0	0
$685$	−1.93153	−0.0738001
$686$	0	0
$687$	−9.81224	−0.374360
$688$	0	0
$689$	−55.1506	−2.10107
$690$	0	0
$691$	−14.0336	−0.533862	−0.266931	−	0.963716i	$-0.586010\pi$
$691$	−14.0336	−0.533862	−0.266931	+	0.963716i	$0.586010\pi$
$692$	0	0
$693$	−10.4417	−0.396648
$694$	0	0
$695$	−0.181466	−0.00688338
$696$	0	0
$697$	34.0719	1.29057
$698$	0	0
$699$	−27.3650	−1.03504
$700$	0	0
$701$	23.2228	0.877112	0.438556	−	0.898704i	$-0.355490\pi$
$701$	23.2228	0.877112	0.438556	+	0.898704i	$0.355490\pi$
$702$	0	0
$703$	12.3101	0.464286
$704$	0	0
$705$	0.338988	0.0127670
$706$	0	0
$707$	15.6153	0.587274
$708$	0	0
$709$	−13.7848	−0.517699	−0.258850	−	0.965918i	$-0.583343\pi$
$709$	−13.7848	−0.517699	−0.258850	+	0.965918i	$0.583343\pi$
$710$	0	0
$711$	9.29227	0.348488
$712$	0	0
$713$	−4.48181	−0.167845
$714$	0	0
$715$	−1.92390	−0.0719497
$716$	0	0
$717$	−12.3849	−0.462522
$718$	0	0
$719$	−7.06763	−0.263578	−0.131789	−	0.991278i	$-0.542072\pi$
$719$	−7.06763	−0.263578	−0.131789	+	0.991278i	$0.542072\pi$
$720$	0	0
$721$	20.0982	0.748498
$722$	0	0
$723$	28.5715	1.06259
$724$	0	0
$725$	−4.98465	−0.185125
$726$	0	0
$727$	−48.8742	−1.81264	−0.906322	−	0.422589i	$-0.861122\pi$
$727$	−48.8742	−1.81264	−0.906322	+	0.422589i	$0.861122\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−16.3168	−0.603499
$732$	0	0
$733$	0.544142	0.0200984	0.0100492	−	0.999950i	$-0.496801\pi$
$733$	0.544142	0.0200984	0.0100492	+	0.999950i	$0.496801\pi$
$734$	0	0
$735$	0.339704	0.0125302
$736$	0	0
$737$	−45.6564	−1.68178
$738$	0	0
$739$	0.446724	0.0164330	0.00821650	−	0.999966i	$-0.497385\pi$
$739$	0.446724	0.0164330	0.00821650	+	0.999966i	$0.497385\pi$
$740$	0	0
$741$	−10.9978	−0.404013
$742$	0	0
$743$	−44.9734	−1.64992	−0.824958	−	0.565194i	$-0.808801\pi$
$743$	−44.9734	−1.64992	−0.824958	+	0.565194i	$0.808801\pi$
$744$	0	0
$745$	−2.48698	−0.0911160
$746$	0	0
$747$	16.0061	0.585633
$748$	0	0
$749$	13.2059	0.482531
$750$	0	0
$751$	52.7829	1.92608	0.963038	−	0.269365i	$-0.0868138\pi$
$751$	52.7829	1.92608	0.963038	+	0.269365i	$0.0868138\pi$
$752$	0	0
$753$	13.2296	0.482114
$754$	0	0
$755$	1.96776	0.0716140
$756$	0	0
$757$	52.7256	1.91634	0.958172	−	0.286193i	$-0.0923900\pi$
$757$	52.7256	1.91634	0.958172	+	0.286193i	$0.0923900\pi$
$758$	0	0
$759$	3.34537	0.121429
$760$	0	0
$761$	17.9115	0.649293	0.324646	−	0.945835i	$-0.394755\pi$
$761$	17.9115	0.649293	0.324646	+	0.945835i	$0.394755\pi$
$762$	0	0
$763$	9.40661	0.340542
$764$	0	0
$765$	0.529940	0.0191600
$766$	0	0
$767$	−54.7999	−1.97871
$768$	0	0
$769$	−6.93378	−0.250039	−0.125019	−	0.992154i	$-0.539899\pi$
$769$	−6.93378	−0.250039	−0.125019	+	0.992154i	$0.539899\pi$
$770$	0	0
$771$	2.30623	0.0830567
$772$	0	0
$773$	44.3154	1.59391	0.796956	−	0.604037i	$-0.206442\pi$
$773$	44.3154	1.59391	0.796956	+	0.604037i	$0.206442\pi$
$774$	0	0
$775$	22.3402	0.802485
$776$	0	0
$777$	16.2187	0.581844
$778$	0	0
$779$	18.8691	0.676054
$780$	0	0
$781$	48.5521	1.73733
$782$	0	0
$783$	−1.00000	−0.0357371
$784$	0	0
$785$	1.56754	0.0559480
$786$	0	0
$787$	50.0015	1.78236	0.891181	−	0.453647i	$-0.149877\pi$
$787$	50.0015	1.78236	0.891181	+	0.453647i	$0.149877\pi$
$788$	0	0
$789$	−15.0755	−0.536702
$790$	0	0
$791$	45.3583	1.61276
$792$	0	0
$793$	−5.99656	−0.212944
$794$	0	0
$795$	1.47173	0.0521968
$796$	0	0
$797$	−28.0461	−0.993446	−0.496723	−	0.867909i	$-0.665463\pi$
$797$	−28.0461	−0.993446	−0.496723	+	0.867909i	$0.665463\pi$
$798$	0	0
$799$	−11.7057	−0.414117
$800$	0	0
$801$	9.47785	0.334884
$802$	0	0
$803$	5.20827	0.183796
$804$	0	0
$805$	−0.386665	−0.0136282
$806$	0	0
$807$	−10.7910	−0.379860
$808$	0	0
$809$	45.4593	1.59826	0.799132	−	0.601155i	$-0.205293\pi$
$809$	45.4593	1.59826	0.799132	+	0.601155i	$0.205293\pi$
$810$	0	0
$811$	−41.6591	−1.46285	−0.731425	−	0.681922i	$-0.761144\pi$
$811$	−41.6591	−1.46285	−0.731425	+	0.681922i	$0.761144\pi$
$812$	0	0
$813$	5.46497	0.191665
$814$	0	0
$815$	1.86670	0.0653878
$816$	0	0
$817$	−9.03626	−0.316139
$818$	0	0
$819$	−14.4897	−0.506310
$820$	0	0
$821$	−28.1607	−0.982815	−0.491407	−	0.870930i	$-0.663517\pi$
$821$	−28.1607	−0.982815	−0.491407	+	0.870930i	$0.663517\pi$
$822$	0	0
$823$	−35.5358	−1.23870	−0.619350	−	0.785115i	$-0.712604\pi$
$823$	−35.5358	−1.23870	−0.619350	+	0.785115i	$0.712604\pi$
$824$	0	0
$825$	−16.6755	−0.580567
$826$	0	0
$827$	−1.01185	−0.0351856	−0.0175928	−	0.999845i	$-0.505600\pi$
$827$	−1.01185	−0.0351856	−0.0175928	+	0.999845i	$0.505600\pi$
$828$	0	0
$829$	−49.5249	−1.72007	−0.860036	−	0.510234i	$-0.829559\pi$
$829$	−49.5249	−1.72007	−0.860036	+	0.510234i	$0.829559\pi$
$830$	0	0
$831$	−19.5186	−0.677093
$832$	0	0
$833$	−11.7304	−0.406435
$834$	0	0
$835$	−2.61937	−0.0906469
$836$	0	0
$837$	4.48181	0.154914
$838$	0	0
$839$	−22.5539	−0.778649	−0.389324	−	0.921101i	$-0.627291\pi$
$839$	−22.5539	−0.778649	−0.389324	+	0.921101i	$0.627291\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	22.9300	0.789752
$844$	0	0
$845$	−1.05927	−0.0364401
$846$	0	0
$847$	0.597716	0.0205378
$848$	0	0
$849$	−25.4692	−0.874101
$850$	0	0
$851$	−5.19624	−0.178125
$852$	0	0
$853$	−4.68276	−0.160335	−0.0801673	−	0.996781i	$-0.525545\pi$
$853$	−4.68276	−0.160335	−0.0801673	+	0.996781i	$0.525545\pi$
$854$	0	0
$855$	0.293481	0.0100369
$856$	0	0
$857$	31.1764	1.06497	0.532483	−	0.846441i	$-0.321259\pi$
$857$	31.1764	1.06497	0.532483	+	0.846441i	$0.321259\pi$
$858$	0	0
$859$	−54.1840	−1.84873	−0.924367	−	0.381504i	$-0.875406\pi$
$859$	−54.1840	−1.84873	−0.924367	+	0.381504i	$0.875406\pi$
$860$	0	0
$861$	24.8602	0.847233
$862$	0	0
$863$	−28.3776	−0.965986	−0.482993	−	0.875624i	$-0.660450\pi$
$863$	−28.3776	−0.965986	−0.482993	+	0.875624i	$0.660450\pi$
$864$	0	0
$865$	−0.148393	−0.00504550
$866$	0	0
$867$	−1.29952	−0.0441340
$868$	0	0
$869$	−31.0861	−1.05452
$870$	0	0
$871$	−63.3561	−2.14674
$872$	0	0
$873$	11.4680	0.388131
$874$	0	0
$875$	3.86072	0.130516
$876$	0	0
$877$	34.9027	1.17858	0.589290	−	0.807921i	$-0.299407\pi$
$877$	34.9027	1.17858	0.589290	+	0.807921i	$0.299407\pi$
$878$	0	0
$879$	14.1564	0.477483
$880$	0	0
$881$	1.50481	0.0506983	0.0253491	−	0.999679i	$-0.491930\pi$
$881$	1.50481	0.0506983	0.0253491	+	0.999679i	$0.491930\pi$
$882$	0	0
$883$	−52.3603	−1.76206	−0.881031	−	0.473058i	$-0.843150\pi$
$883$	−52.3603	−1.76206	−0.881031	+	0.473058i	$0.843150\pi$
$884$	0	0
$885$	1.46237	0.0491569
$886$	0	0
$887$	45.6573	1.53302	0.766510	−	0.642232i	$-0.221991\pi$
$887$	45.6573	1.53302	0.766510	+	0.642232i	$0.221991\pi$
$888$	0	0
$889$	−44.1824	−1.48183
$890$	0	0
$891$	−3.34537	−0.112074
$892$	0	0
$893$	−6.48262	−0.216933
$894$	0	0
$895$	0.466869	0.0156057
$896$	0	0
$897$	4.64227	0.155001
$898$	0	0
$899$	−4.48181	−0.149477
$900$	0	0
$901$	−50.8206	−1.69308
$902$	0	0
$903$	−11.9054	−0.396186
$904$	0	0
$905$	−3.25192	−0.108097
$906$	0	0
$907$	−42.4726	−1.41028	−0.705140	−	0.709068i	$-0.749116\pi$
$907$	−42.4726	−1.41028	−0.705140	+	0.709068i	$0.749116\pi$
$908$	0	0
$909$	5.00291	0.165936
$910$	0	0
$911$	−13.9406	−0.461874	−0.230937	−	0.972969i	$-0.574179\pi$
$911$	−13.9406	−0.461874	−0.230937	+	0.972969i	$0.574179\pi$
$912$	0	0
$913$	−53.5464	−1.77213
$914$	0	0
$915$	0.160022	0.00529015
$916$	0	0
$917$	−43.0821	−1.42270
$918$	0	0
$919$	−16.5900	−0.547253	−0.273627	−	0.961836i	$-0.588223\pi$
$919$	−16.5900	−0.547253	−0.273627	+	0.961836i	$0.588223\pi$
$920$	0	0
$921$	24.3935	0.803793
$922$	0	0
$923$	67.3743	2.21765
$924$	0	0
$925$	25.9015	0.851635
$926$	0	0
$927$	6.43917	0.211490
$928$	0	0
$929$	−34.0242	−1.11630	−0.558148	−	0.829741i	$-0.688488\pi$
$929$	−34.0242	−1.11630	−0.558148	+	0.829741i	$0.688488\pi$
$930$	0	0
$931$	−6.49632	−0.212908
$932$	0	0
$933$	6.36759	0.208466
$934$	0	0
$935$	−1.77285	−0.0579783
$936$	0	0
$937$	−32.1050	−1.04882	−0.524412	−	0.851465i	$-0.675715\pi$
$937$	−32.1050	−1.04882	−0.524412	+	0.851465i	$0.675715\pi$
$938$	0	0
$939$	28.0338	0.914849
$940$	0	0
$941$	8.01141	0.261165	0.130582	−	0.991437i	$-0.458315\pi$
$941$	8.01141	0.261165	0.130582	+	0.991437i	$0.458315\pi$
$942$	0	0
$943$	−7.96483	−0.259371
$944$	0	0
$945$	0.386665	0.0125782
$946$	0	0
$947$	−18.6384	−0.605668	−0.302834	−	0.953043i	$-0.597933\pi$
$947$	−18.6384	−0.605668	−0.302834	+	0.953043i	$0.597933\pi$
$948$	0	0
$949$	7.22736	0.234610
$950$	0	0
$951$	−8.70755	−0.282362
$952$	0	0
$953$	−24.8735	−0.805733	−0.402867	−	0.915259i	$-0.631986\pi$
$953$	−24.8735	−0.805733	−0.402867	+	0.915259i	$0.631986\pi$
$954$	0	0
$955$	−1.17047	−0.0378757
$956$	0	0
$957$	3.34537	0.108140
$958$	0	0
$959$	48.6657	1.57150
$960$	0	0
$961$	−10.9134	−0.352046
$962$	0	0
$963$	4.23096	0.136341
$964$	0	0
$965$	2.41983	0.0778970
$966$	0	0
$967$	20.7833	0.668347	0.334173	−	0.942512i	$-0.391543\pi$
$967$	20.7833	0.668347	0.334173	+	0.942512i	$0.391543\pi$
$968$	0	0
$969$	−10.1343	−0.325560
$970$	0	0
$971$	−25.5827	−0.820988	−0.410494	−	0.911863i	$-0.634644\pi$
$971$	−25.5827	−0.820988	−0.410494	+	0.911863i	$0.634644\pi$
$972$	0	0
$973$	4.57209	0.146575
$974$	0	0
$975$	−23.1401	−0.741077
$976$	0	0
$977$	50.3427	1.61060	0.805302	−	0.592865i	$-0.202003\pi$
$977$	50.3427	1.61060	0.805302	+	0.592865i	$0.202003\pi$
$978$	0	0
$979$	−31.7069	−1.01336
$980$	0	0
$981$	3.01374	0.0962213
$982$	0	0
$983$	−4.80568	−0.153277	−0.0766387	−	0.997059i	$-0.524419\pi$
$983$	−4.80568	−0.153277	−0.0766387	+	0.997059i	$0.524419\pi$
$984$	0	0
$985$	1.99762	0.0636496
$986$	0	0
$987$	−8.54092	−0.271861
$988$	0	0
$989$	3.81430	0.121288
$990$	0	0
$991$	55.5752	1.76540	0.882701	−	0.469934i	$-0.155722\pi$
$991$	55.5752	1.76540	0.882701	+	0.469934i	$0.155722\pi$
$992$	0	0
$993$	24.5067	0.777698
$994$	0	0
$995$	0.0974733	0.00309011
$996$	0	0
$997$	39.5059	1.25117	0.625583	−	0.780158i	$-0.284861\pi$
$997$	39.5059	1.25117	0.625583	+	0.780158i	$0.284861\pi$
$998$	0	0
$999$	5.19624	0.164402

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.i.1.7	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.i.1.7	✓	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -0.123882 q^{5} +3.12124 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -0.123882 q^{5} +3.12124 q^{7} +1.00000 q^{9} -3.34537 q^{11} -4.64227 q^{13} +0.123882 q^{15} -4.27779 q^{17} -2.36905 q^{19} -3.12124 q^{21} +1.00000 q^{23} -4.98465 q^{25} -1.00000 q^{27} +1.00000 q^{29} -4.48181 q^{31} +3.34537 q^{33} -0.386665 q^{35} -5.19624 q^{37} +4.64227 q^{39} -7.96483 q^{41} +3.81430 q^{43} -0.123882 q^{45} +2.73638 q^{47} +2.74217 q^{49} +4.27779 q^{51} +11.8801 q^{53} +0.414430 q^{55} +2.36905 q^{57} +11.8045 q^{59} +1.29173 q^{61} +3.12124 q^{63} +0.575092 q^{65} +13.6476 q^{67} -1.00000 q^{69} -14.5132 q^{71} -1.55686 q^{73} +4.98465 q^{75} -10.4417 q^{77} +9.29227 q^{79} +1.00000 q^{81} +16.0061 q^{83} +0.529940 q^{85} -1.00000 q^{87} +9.47785 q^{89} -14.4897 q^{91} +4.48181 q^{93} +0.293481 q^{95} +11.4680 q^{97} -3.34537 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{3} + 5 q^{5} - 4 q^{7} + 16 q^{9}+O(q^{10}) \) 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

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more