LMFDB - Embedded newform 8040.2.a.t.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8040,2,Mod(1,8040)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8040, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8040.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8040 = 2^{3} \cdot 3 \cdot 5 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8040.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:	$64.1997232251$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 15x^{5} + 3x^{4} + 43x^{3} - 6x^{2} - 29x + 6 $ x^7 - x^6 - 15x^5 + 3x^4 + 43x^3 - 6x^2 - 29*x + 6
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.70642$ of defining polynomial
Character	$\chi$	$=$	8040.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} -2.80016 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.00000 q^{5} -2.80016 q^{7} +1.00000 q^{9} +0.770919 q^{11} -3.59031 q^{13} -1.00000 q^{15} -0.618268 q^{17} -6.93864 q^{19} -2.80016 q^{21} +1.25243 q^{23} +1.00000 q^{25} +1.00000 q^{27} +3.30276 q^{29} +3.43206 q^{31} +0.770919 q^{33} +2.80016 q^{35} -5.13260 q^{37} -3.59031 q^{39} -1.43206 q^{41} -4.84274 q^{43} -1.00000 q^{45} +10.8741 q^{47} +0.840913 q^{49} -0.618268 q^{51} +12.6800 q^{53} -0.770919 q^{55} -6.93864 q^{57} +0.397624 q^{59} -12.9478 q^{61} -2.80016 q^{63} +3.59031 q^{65} +1.00000 q^{67} +1.25243 q^{69} -1.10374 q^{71} -2.18458 q^{73} +1.00000 q^{75} -2.15870 q^{77} +9.90420 q^{79} +1.00000 q^{81} -15.1850 q^{83} +0.618268 q^{85} +3.30276 q^{87} +10.2586 q^{89} +10.0535 q^{91} +3.43206 q^{93} +6.93864 q^{95} +18.2174 q^{97} +0.770919 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 7 q^{3} - 7 q^{5} + 10 q^{7} + 7 q^{9}+O(q^{10}) $ 7 * q + 7 * q^3 - 7 * q^5 + 10 * q^7 + 7 * q^9	$ 7 q + 7 q^{3} - 7 q^{5} + 10 q^{7} + 7 q^{9} - q^{13} - 7 q^{15} - 2 q^{17} + 9 q^{19} + 10 q^{21} + 2 q^{23} + 7 q^{25} + 7 q^{27} - q^{29} + 9 q^{31} - 10 q^{35} + 23 q^{37} - q^{39} + 5 q^{41} - 3 q^{43} - 7 q^{45} + 11 q^{47} + 13 q^{49} - 2 q^{51} + 13 q^{53} + 9 q^{57} + q^{59} + 4 q^{61} + 10 q^{63} + q^{65} + 7 q^{67} + 2 q^{69} + q^{71} + 14 q^{73} + 7 q^{75} + 18 q^{77} + 25 q^{79} + 7 q^{81} - 29 q^{83} + 2 q^{85} - q^{87} + 7 q^{89} + 27 q^{91} + 9 q^{93} - 9 q^{95} + 38 q^{97}+O(q^{100}) $ 7 * q + 7 * q^3 - 7 * q^5 + 10 * q^7 + 7 * q^9 - q^13 - 7 * q^15 - 2 * q^17 + 9 * q^19 + 10 * q^21 + 2 * q^23 + 7 * q^25 + 7 * q^27 - q^29 + 9 * q^31 - 10 * q^35 + 23 * q^37 - q^39 + 5 * q^41 - 3 * q^43 - 7 * q^45 + 11 * q^47 + 13 * q^49 - 2 * q^51 + 13 * q^53 + 9 * q^57 + q^59 + 4 * q^61 + 10 * q^63 + q^65 + 7 * q^67 + 2 * q^69 + q^71 + 14 * q^73 + 7 * q^75 + 18 * q^77 + 25 * q^79 + 7 * q^81 - 29 * q^83 + 2 * q^85 - q^87 + 7 * q^89 + 27 * q^91 + 9 * q^93 - 9 * q^95 + 38 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−2.80016	−1.05836	−0.529181	−	0.848509i	$-0.677501\pi$
$7$	−2.80016	−1.05836	−0.529181	+	0.848509i	$0.677501\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0.770919	0.232441	0.116220	−	0.993223i	$-0.462922\pi$
$11$	0.770919	0.232441	0.116220	+	0.993223i	$0.462922\pi$
$12$	0	0
$13$	−3.59031	−0.995774	−0.497887	−	0.867242i	$-0.665891\pi$
$13$	−3.59031	−0.995774	−0.497887	+	0.867242i	$0.665891\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−0.618268	−0.149952	−0.0749760	−	0.997185i	$-0.523888\pi$
$17$	−0.618268	−0.149952	−0.0749760	+	0.997185i	$0.523888\pi$
$18$	0	0
$19$	−6.93864	−1.59183	−0.795917	−	0.605406i	$-0.793011\pi$
$19$	−6.93864	−1.59183	−0.795917	+	0.605406i	$0.793011\pi$
$20$	0	0
$21$	−2.80016	−0.611046
$22$	0	0
$23$	1.25243	0.261150	0.130575	−	0.991438i	$-0.458318\pi$
$23$	1.25243	0.261150	0.130575	+	0.991438i	$0.458318\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	3.30276	0.613307	0.306654	−	0.951821i	$-0.400791\pi$
$29$	3.30276	0.613307	0.306654	+	0.951821i	$0.400791\pi$
$30$	0	0
$31$	3.43206	0.616417	0.308208	−	0.951319i	$-0.400271\pi$
$31$	3.43206	0.616417	0.308208	+	0.951319i	$0.400271\pi$
$32$	0	0
$33$	0.770919	0.134200
$34$	0	0
$35$	2.80016	0.473314
$36$	0	0
$37$	−5.13260	−0.843794	−0.421897	−	0.906644i	$-0.638636\pi$
$37$	−5.13260	−0.843794	−0.421897	+	0.906644i	$0.638636\pi$
$38$	0	0
$39$	−3.59031	−0.574910
$40$	0	0
$41$	−1.43206	−0.223651	−0.111825	−	0.993728i	$-0.535670\pi$
$41$	−1.43206	−0.223651	−0.111825	+	0.993728i	$0.535670\pi$
$42$	0	0
$43$	−4.84274	−0.738512	−0.369256	−	0.929328i	$-0.620387\pi$
$43$	−4.84274	−0.738512	−0.369256	+	0.929328i	$0.620387\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	10.8741	1.58616	0.793078	−	0.609120i	$-0.208477\pi$
$47$	10.8741	1.58616	0.793078	+	0.609120i	$0.208477\pi$
$48$	0	0
$49$	0.840913	0.120130
$50$	0	0
$51$	−0.618268	−0.0865748
$52$	0	0
$53$	12.6800	1.74174	0.870868	−	0.491516i	$-0.163557\pi$
$53$	12.6800	1.74174	0.870868	+	0.491516i	$0.163557\pi$
$54$	0	0
$55$	−0.770919	−0.103951
$56$	0	0
$57$	−6.93864	−0.919046
$58$	0	0
$59$	0.397624	0.0517662	0.0258831	−	0.999665i	$-0.491760\pi$
$59$	0.397624	0.0517662	0.0258831	+	0.999665i	$0.491760\pi$
$60$	0	0
$61$	−12.9478	−1.65780	−0.828899	−	0.559398i	$-0.811032\pi$
$61$	−12.9478	−1.65780	−0.828899	+	0.559398i	$0.811032\pi$
$62$	0	0
$63$	−2.80016	−0.352787
$64$	0	0
$65$	3.59031	0.445324
$66$	0	0
$67$	1.00000	0.122169
$67$	1.00000	0.122169
$68$	0	0
$69$	1.25243	0.150775
$70$	0	0
$71$	−1.10374	−0.130990	−0.0654951	−	0.997853i	$-0.520863\pi$
$71$	−1.10374	−0.130990	−0.0654951	+	0.997853i	$0.520863\pi$
$72$	0	0
$73$	−2.18458	−0.255686	−0.127843	−	0.991794i	$-0.540805\pi$
$73$	−2.18458	−0.255686	−0.127843	+	0.991794i	$0.540805\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−2.15870	−0.246006
$78$	0	0
$79$	9.90420	1.11431	0.557155	−	0.830409i	$-0.311893\pi$
$79$	9.90420	1.11431	0.557155	+	0.830409i	$0.311893\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−15.1850	−1.66677	−0.833387	−	0.552689i	$-0.813602\pi$
$83$	−15.1850	−1.66677	−0.833387	+	0.552689i	$0.813602\pi$
$84$	0	0
$85$	0.618268	0.0670606
$86$	0	0
$87$	3.30276	0.354093
$88$	0	0
$89$	10.2586	1.08741	0.543703	−	0.839278i	$-0.317022\pi$
$89$	10.2586	1.08741	0.543703	+	0.839278i	$0.317022\pi$
$90$	0	0
$91$	10.0535	1.05389
$92$	0	0
$93$	3.43206	0.355888
$94$	0	0
$95$	6.93864	0.711890
$96$	0	0
$97$	18.2174	1.84969	0.924847	−	0.380340i	$-0.124193\pi$
$97$	18.2174	1.84969	0.924847	+	0.380340i	$0.124193\pi$
$98$	0	0
$99$	0.770919	0.0774802
$100$	0	0
$101$	−5.98598	−0.595627	−0.297814	−	0.954624i	$-0.596257\pi$
$101$	−5.98598	−0.595627	−0.297814	+	0.954624i	$0.596257\pi$
$102$	0	0
$103$	17.8331	1.75715	0.878573	−	0.477607i	$-0.158496\pi$
$103$	17.8331	1.75715	0.878573	+	0.477607i	$0.158496\pi$
$104$	0	0
$105$	2.80016	0.273268
$106$	0	0
$107$	−4.03839	−0.390406	−0.195203	−	0.980763i	$-0.562537\pi$
$107$	−4.03839	−0.390406	−0.195203	+	0.980763i	$0.562537\pi$
$108$	0	0
$109$	4.95193	0.474309	0.237154	−	0.971472i	$-0.423785\pi$
$109$	4.95193	0.474309	0.237154	+	0.971472i	$0.423785\pi$
$110$	0	0
$111$	−5.13260	−0.487165
$112$	0	0
$113$	−2.11233	−0.198711	−0.0993557	−	0.995052i	$-0.531678\pi$
$113$	−2.11233	−0.198711	−0.0993557	+	0.995052i	$0.531678\pi$
$114$	0	0
$115$	−1.25243	−0.116790
$116$	0	0
$117$	−3.59031	−0.331925
$118$	0	0
$119$	1.73125	0.158704
$120$	0	0
$121$	−10.4057	−0.945971
$122$	0	0
$123$	−1.43206	−0.129125
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	6.97601	0.619021	0.309510	−	0.950896i	$-0.399835\pi$
$127$	6.97601	0.619021	0.309510	+	0.950896i	$0.399835\pi$
$128$	0	0
$129$	−4.84274	−0.426380
$130$	0	0
$131$	19.5046	1.70413	0.852064	−	0.523437i	$-0.175350\pi$
$131$	19.5046	1.70413	0.852064	+	0.523437i	$0.175350\pi$
$132$	0	0
$133$	19.4293	1.68474
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	2.24286	0.191621	0.0958103	−	0.995400i	$-0.469456\pi$
$137$	2.24286	0.191621	0.0958103	+	0.995400i	$0.469456\pi$
$138$	0	0
$139$	16.5268	1.40179	0.700894	−	0.713265i	$-0.252784\pi$
$139$	16.5268	1.40179	0.700894	+	0.713265i	$0.252784\pi$
$140$	0	0
$141$	10.8741	0.915768
$142$	0	0
$143$	−2.76784	−0.231458
$144$	0	0
$145$	−3.30276	−0.274279
$146$	0	0
$147$	0.840913	0.0693574
$148$	0	0
$149$	4.86809	0.398810	0.199405	−	0.979917i	$-0.436099\pi$
$149$	4.86809	0.398810	0.199405	+	0.979917i	$0.436099\pi$
$150$	0	0
$151$	−5.59330	−0.455176	−0.227588	−	0.973757i	$-0.573084\pi$
$151$	−5.59330	−0.455176	−0.227588	+	0.973757i	$0.573084\pi$
$152$	0	0
$153$	−0.618268	−0.0499840
$154$	0	0
$155$	−3.43206	−0.275670
$156$	0	0
$157$	−1.77705	−0.141824	−0.0709120	−	0.997483i	$-0.522591\pi$
$157$	−1.77705	−0.141824	−0.0709120	+	0.997483i	$0.522591\pi$
$158$	0	0
$159$	12.6800	1.00559
$160$	0	0
$161$	−3.50701	−0.276391
$162$	0	0
$163$	18.1332	1.42030	0.710149	−	0.704051i	$-0.248628\pi$
$163$	18.1332	1.42030	0.710149	+	0.704051i	$0.248628\pi$
$164$	0	0
$165$	−0.770919	−0.0600159
$166$	0	0
$167$	−2.83587	−0.219446	−0.109723	−	0.993962i	$-0.534996\pi$
$167$	−2.83587	−0.219446	−0.109723	+	0.993962i	$0.534996\pi$
$168$	0	0
$169$	−0.109639	−0.00843381
$170$	0	0
$171$	−6.93864	−0.530611
$172$	0	0
$173$	17.5907	1.33739	0.668696	−	0.743535i	$-0.266853\pi$
$173$	17.5907	1.33739	0.668696	+	0.743535i	$0.266853\pi$
$174$	0	0
$175$	−2.80016	−0.211672
$176$	0	0
$177$	0.397624	0.0298872
$178$	0	0
$179$	−21.0294	−1.57181	−0.785904	−	0.618348i	$-0.787802\pi$
$179$	−21.0294	−1.57181	−0.785904	+	0.618348i	$0.787802\pi$
$180$	0	0
$181$	−14.9936	−1.11447	−0.557234	−	0.830356i	$-0.688137\pi$
$181$	−14.9936	−1.11447	−0.557234	+	0.830356i	$0.688137\pi$
$182$	0	0
$183$	−12.9478	−0.957130
$184$	0	0
$185$	5.13260	0.377356
$186$	0	0
$187$	−0.476634	−0.0348549
$188$	0	0
$189$	−2.80016	−0.203682
$190$	0	0
$191$	2.98651	0.216097	0.108048	−	0.994146i	$-0.465540\pi$
$191$	2.98651	0.216097	0.108048	+	0.994146i	$0.465540\pi$
$192$	0	0
$193$	11.9299	0.858735	0.429367	−	0.903130i	$-0.358737\pi$
$193$	11.9299	0.858735	0.429367	+	0.903130i	$0.358737\pi$
$194$	0	0
$195$	3.59031	0.257108
$196$	0	0
$197$	23.3272	1.66199	0.830996	−	0.556279i	$-0.187771\pi$
$197$	23.3272	1.66199	0.830996	+	0.556279i	$0.187771\pi$
$198$	0	0
$199$	10.3504	0.733717	0.366859	−	0.930277i	$-0.380433\pi$
$199$	10.3504	0.733717	0.366859	+	0.930277i	$0.380433\pi$
$200$	0	0
$201$	1.00000	0.0705346
$202$	0	0
$203$	−9.24827	−0.649101
$204$	0	0
$205$	1.43206	0.100020
$206$	0	0
$207$	1.25243	0.0870499
$208$	0	0
$209$	−5.34913	−0.370007
$210$	0	0
$211$	−27.0627	−1.86307	−0.931536	−	0.363650i	$-0.881531\pi$
$211$	−27.0627	−1.86307	−0.931536	+	0.363650i	$0.881531\pi$
$212$	0	0
$213$	−1.10374	−0.0756272
$214$	0	0
$215$	4.84274	0.330272
$216$	0	0
$217$	−9.61034	−0.652392
$218$	0	0
$219$	−2.18458	−0.147621
$220$	0	0
$221$	2.21978	0.149318
$222$	0	0
$223$	18.1399	1.21474	0.607369	−	0.794420i	$-0.292225\pi$
$223$	18.1399	1.21474	0.607369	+	0.794420i	$0.292225\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	10.8027	0.716997	0.358499	−	0.933530i	$-0.383289\pi$
$227$	10.8027	0.716997	0.358499	+	0.933530i	$0.383289\pi$
$228$	0	0
$229$	−21.9118	−1.44797	−0.723987	−	0.689813i	$-0.757693\pi$
$229$	−21.9118	−1.44797	−0.723987	+	0.689813i	$0.757693\pi$
$230$	0	0
$231$	−2.15870	−0.142032
$232$	0	0
$233$	−2.19444	−0.143762	−0.0718811	−	0.997413i	$-0.522900\pi$
$233$	−2.19444	−0.143762	−0.0718811	+	0.997413i	$0.522900\pi$
$234$	0	0
$235$	−10.8741	−0.709351
$236$	0	0
$237$	9.90420	0.643347
$238$	0	0
$239$	21.6147	1.39814	0.699070	−	0.715053i	$-0.253598\pi$
$239$	21.6147	1.39814	0.699070	+	0.715053i	$0.253598\pi$
$240$	0	0
$241$	19.7227	1.27045	0.635226	−	0.772326i	$-0.280907\pi$
$241$	19.7227	1.27045	0.635226	+	0.772326i	$0.280907\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−0.840913	−0.0537240
$246$	0	0
$247$	24.9119	1.58511
$248$	0	0
$249$	−15.1850	−0.962313
$250$	0	0
$251$	12.8631	0.811911	0.405956	−	0.913893i	$-0.366939\pi$
$251$	12.8631	0.811911	0.405956	+	0.913893i	$0.366939\pi$
$252$	0	0
$253$	0.965521	0.0607018
$254$	0	0
$255$	0.618268	0.0387174
$256$	0	0
$257$	−14.3381	−0.894384	−0.447192	−	0.894438i	$-0.647576\pi$
$257$	−14.3381	−0.894384	−0.447192	+	0.894438i	$0.647576\pi$
$258$	0	0
$259$	14.3721	0.893040
$260$	0	0
$261$	3.30276	0.204436
$262$	0	0
$263$	−16.7140	−1.03063	−0.515314	−	0.857001i	$-0.672325\pi$
$263$	−16.7140	−1.03063	−0.515314	+	0.857001i	$0.672325\pi$
$264$	0	0
$265$	−12.6800	−0.778928
$266$	0	0
$267$	10.2586	0.627814
$268$	0	0
$269$	10.2574	0.625405	0.312702	−	0.949851i	$-0.398766\pi$
$269$	10.2574	0.625405	0.312702	+	0.949851i	$0.398766\pi$
$270$	0	0
$271$	0.523534	0.0318024	0.0159012	−	0.999874i	$-0.494938\pi$
$271$	0.523534	0.0318024	0.0159012	+	0.999874i	$0.494938\pi$
$272$	0	0
$273$	10.0535	0.608464
$274$	0	0
$275$	0.770919	0.0464881
$276$	0	0
$277$	4.58867	0.275706	0.137853	−	0.990453i	$-0.455980\pi$
$277$	4.58867	0.275706	0.137853	+	0.990453i	$0.455980\pi$
$278$	0	0
$279$	3.43206	0.205472
$280$	0	0
$281$	−5.89896	−0.351902	−0.175951	−	0.984399i	$-0.556300\pi$
$281$	−5.89896	−0.351902	−0.175951	+	0.984399i	$0.556300\pi$
$282$	0	0
$283$	4.28874	0.254939	0.127469	−	0.991842i	$-0.459315\pi$
$283$	4.28874	0.254939	0.127469	+	0.991842i	$0.459315\pi$
$284$	0	0
$285$	6.93864	0.411010
$286$	0	0
$287$	4.01001	0.236703
$288$	0	0
$289$	−16.6177	−0.977514
$290$	0	0
$291$	18.2174	1.06792
$292$	0	0
$293$	11.5034	0.672038	0.336019	−	0.941855i	$-0.390919\pi$
$293$	11.5034	0.672038	0.336019	+	0.941855i	$0.390919\pi$
$294$	0	0
$295$	−0.397624	−0.0231505
$296$	0	0
$297$	0.770919	0.0447332
$298$	0	0
$299$	−4.49662	−0.260046
$300$	0	0
$301$	13.5605	0.781613
$302$	0	0
$303$	−5.98598	−0.343885
$304$	0	0
$305$	12.9478	0.741390
$306$	0	0
$307$	−18.3432	−1.04690	−0.523451	−	0.852056i	$-0.675356\pi$
$307$	−18.3432	−1.04690	−0.523451	+	0.852056i	$0.675356\pi$
$308$	0	0
$309$	17.8331	1.01449
$310$	0	0
$311$	−21.8167	−1.23711	−0.618555	−	0.785742i	$-0.712281\pi$
$311$	−21.8167	−1.23711	−0.618555	+	0.785742i	$0.712281\pi$
$312$	0	0
$313$	15.8966	0.898526	0.449263	−	0.893400i	$-0.351687\pi$
$313$	15.8966	0.898526	0.449263	+	0.893400i	$0.351687\pi$
$314$	0	0
$315$	2.80016	0.157771
$316$	0	0
$317$	−14.6686	−0.823870	−0.411935	−	0.911213i	$-0.635147\pi$
$317$	−14.6686	−0.823870	−0.411935	+	0.911213i	$0.635147\pi$
$318$	0	0
$319$	2.54616	0.142558
$320$	0	0
$321$	−4.03839	−0.225401
$322$	0	0
$323$	4.28994	0.238699
$324$	0	0
$325$	−3.59031	−0.199155
$326$	0	0
$327$	4.95193	0.273842
$328$	0	0
$329$	−30.4494	−1.67873
$330$	0	0
$331$	−5.88278	−0.323347	−0.161674	−	0.986844i	$-0.551689\pi$
$331$	−5.88278	−0.323347	−0.161674	+	0.986844i	$0.551689\pi$
$332$	0	0
$333$	−5.13260	−0.281265
$334$	0	0
$335$	−1.00000	−0.0546358
$336$	0	0
$337$	−17.3684	−0.946117	−0.473059	−	0.881031i	$-0.656850\pi$
$337$	−17.3684	−0.946117	−0.473059	+	0.881031i	$0.656850\pi$
$338$	0	0
$339$	−2.11233	−0.114726
$340$	0	0
$341$	2.64584	0.143280
$342$	0	0
$343$	17.2464	0.931221
$344$	0	0
$345$	−1.25243	−0.0674285
$346$	0	0
$347$	−12.1492	−0.652203	−0.326102	−	0.945335i	$-0.605735\pi$
$347$	−12.1492	−0.652203	−0.326102	+	0.945335i	$0.605735\pi$
$348$	0	0
$349$	−3.61774	−0.193653	−0.0968267	−	0.995301i	$-0.530869\pi$
$349$	−3.61774	−0.193653	−0.0968267	+	0.995301i	$0.530869\pi$
$350$	0	0
$351$	−3.59031	−0.191637
$352$	0	0
$353$	−7.48416	−0.398341	−0.199171	−	0.979965i	$-0.563825\pi$
$353$	−7.48416	−0.398341	−0.199171	+	0.979965i	$0.563825\pi$
$354$	0	0
$355$	1.10374	0.0585806
$356$	0	0
$357$	1.73125	0.0916275
$358$	0	0
$359$	13.5067	0.712856	0.356428	−	0.934323i	$-0.383995\pi$
$359$	13.5067	0.712856	0.356428	+	0.934323i	$0.383995\pi$
$360$	0	0
$361$	29.1447	1.53393
$362$	0	0
$363$	−10.4057	−0.546157
$364$	0	0
$365$	2.18458	0.114346
$366$	0	0
$367$	17.2686	0.901414	0.450707	−	0.892672i	$-0.351172\pi$
$367$	17.2686	0.901414	0.450707	+	0.892672i	$0.351172\pi$
$368$	0	0
$369$	−1.43206	−0.0745502
$370$	0	0
$371$	−35.5062	−1.84339
$372$	0	0
$373$	30.7796	1.59371	0.796853	−	0.604173i	$-0.206497\pi$
$373$	30.7796	1.59371	0.796853	+	0.604173i	$0.206497\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−11.8580	−0.610716
$378$	0	0
$379$	22.8477	1.17361	0.586804	−	0.809729i	$-0.300386\pi$
$379$	22.8477	1.17361	0.586804	+	0.809729i	$0.300386\pi$
$380$	0	0
$381$	6.97601	0.357392
$382$	0	0
$383$	32.0531	1.63783	0.818917	−	0.573912i	$-0.194575\pi$
$383$	32.0531	1.63783	0.818917	+	0.573912i	$0.194575\pi$
$384$	0	0
$385$	2.15870	0.110017
$386$	0	0
$387$	−4.84274	−0.246171
$388$	0	0
$389$	−7.36726	−0.373535	−0.186768	−	0.982404i	$-0.559801\pi$
$389$	−7.36726	−0.373535	−0.186768	+	0.982404i	$0.559801\pi$
$390$	0	0
$391$	−0.774337	−0.0391599
$392$	0	0
$393$	19.5046	0.983879
$394$	0	0
$395$	−9.90420	−0.498334
$396$	0	0
$397$	5.39602	0.270819	0.135409	−	0.990790i	$-0.456765\pi$
$397$	5.39602	0.270819	0.135409	+	0.990790i	$0.456765\pi$
$398$	0	0
$399$	19.4293	0.972683
$400$	0	0
$401$	−9.41085	−0.469955	−0.234978	−	0.972001i	$-0.575502\pi$
$401$	−9.41085	−0.469955	−0.234978	+	0.972001i	$0.575502\pi$
$402$	0	0
$403$	−12.3222	−0.613812
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−3.95682	−0.196132
$408$	0	0
$409$	21.0386	1.04029	0.520145	−	0.854078i	$-0.325878\pi$
$409$	21.0386	1.04029	0.520145	+	0.854078i	$0.325878\pi$
$410$	0	0
$411$	2.24286	0.110632
$412$	0	0
$413$	−1.11341	−0.0547874
$414$	0	0
$415$	15.1850	0.745404
$416$	0	0
$417$	16.5268	0.809323
$418$	0	0
$419$	28.8120	1.40756	0.703779	−	0.710419i	$-0.251495\pi$
$419$	28.8120	1.40756	0.703779	+	0.710419i	$0.251495\pi$
$420$	0	0
$421$	−7.54037	−0.367495	−0.183748	−	0.982973i	$-0.558823\pi$
$421$	−7.54037	−0.367495	−0.183748	+	0.982973i	$0.558823\pi$
$422$	0	0
$423$	10.8741	0.528719
$424$	0	0
$425$	−0.618268	−0.0299904
$426$	0	0
$427$	36.2560	1.75455
$428$	0	0
$429$	−2.76784	−0.133633
$430$	0	0
$431$	20.2233	0.974124	0.487062	−	0.873367i	$-0.338069\pi$
$431$	20.2233	0.974124	0.487062	+	0.873367i	$0.338069\pi$
$432$	0	0
$433$	22.3192	1.07259	0.536297	−	0.844029i	$-0.319823\pi$
$433$	22.3192	1.07259	0.536297	+	0.844029i	$0.319823\pi$
$434$	0	0
$435$	−3.30276	−0.158355
$436$	0	0
$437$	−8.69016	−0.415707
$438$	0	0
$439$	37.2371	1.77723	0.888616	−	0.458653i	$-0.151668\pi$
$439$	37.2371	1.77723	0.888616	+	0.458653i	$0.151668\pi$
$440$	0	0
$441$	0.840913	0.0400435
$442$	0	0
$443$	2.59870	0.123468	0.0617339	−	0.998093i	$-0.480337\pi$
$443$	2.59870	0.123468	0.0617339	+	0.998093i	$0.480337\pi$
$444$	0	0
$445$	−10.2586	−0.486302
$446$	0	0
$447$	4.86809	0.230253
$448$	0	0
$449$	−10.8092	−0.510117	−0.255059	−	0.966926i	$-0.582095\pi$
$449$	−10.8092	−0.510117	−0.255059	+	0.966926i	$0.582095\pi$
$450$	0	0
$451$	−1.10400	−0.0519855
$452$	0	0
$453$	−5.59330	−0.262796
$454$	0	0
$455$	−10.0535	−0.471314
$456$	0	0
$457$	1.08113	0.0505732	0.0252866	−	0.999680i	$-0.491950\pi$
$457$	1.08113	0.0505732	0.0252866	+	0.999680i	$0.491950\pi$
$458$	0	0
$459$	−0.618268	−0.0288583
$460$	0	0
$461$	20.8420	0.970710	0.485355	−	0.874317i	$-0.338690\pi$
$461$	20.8420	0.970710	0.485355	+	0.874317i	$0.338690\pi$
$462$	0	0
$463$	39.2230	1.82285	0.911423	−	0.411471i	$-0.134985\pi$
$463$	39.2230	1.82285	0.911423	+	0.411471i	$0.134985\pi$
$464$	0	0
$465$	−3.43206	−0.159158
$466$	0	0
$467$	14.6977	0.680127	0.340064	−	0.940402i	$-0.389551\pi$
$467$	14.6977	0.680127	0.340064	+	0.940402i	$0.389551\pi$
$468$	0	0
$469$	−2.80016	−0.129300
$470$	0	0
$471$	−1.77705	−0.0818821
$472$	0	0
$473$	−3.73336	−0.171660
$474$	0	0
$475$	−6.93864	−0.318367
$476$	0	0
$477$	12.6800	0.580579
$478$	0	0
$479$	−36.0434	−1.64687	−0.823433	−	0.567413i	$-0.807944\pi$
$479$	−36.0434	−1.64687	−0.823433	+	0.567413i	$0.807944\pi$
$480$	0	0
$481$	18.4276	0.840228
$482$	0	0
$483$	−3.50701	−0.159574
$484$	0	0
$485$	−18.2174	−0.827208
$486$	0	0
$487$	−14.5604	−0.659794	−0.329897	−	0.944017i	$-0.607014\pi$
$487$	−14.5604	−0.659794	−0.329897	+	0.944017i	$0.607014\pi$
$488$	0	0
$489$	18.1332	0.820010
$490$	0	0
$491$	−29.3926	−1.32647	−0.663234	−	0.748412i	$-0.730816\pi$
$491$	−29.3926	−1.32647	−0.663234	+	0.748412i	$0.730816\pi$
$492$	0	0
$493$	−2.04199	−0.0919666
$494$	0	0
$495$	−0.770919	−0.0346502
$496$	0	0
$497$	3.09066	0.138635
$498$	0	0
$499$	0.0999720	0.00447536	0.00223768	−	0.999997i	$-0.499288\pi$
$499$	0.0999720	0.00447536	0.00223768	+	0.999997i	$0.499288\pi$
$500$	0	0
$501$	−2.83587	−0.126697
$502$	0	0
$503$	4.73233	0.211004	0.105502	−	0.994419i	$-0.466355\pi$
$503$	4.73233	0.211004	0.105502	+	0.994419i	$0.466355\pi$
$504$	0	0
$505$	5.98598	0.266373
$506$	0	0
$507$	−0.109639	−0.00486926
$508$	0	0
$509$	29.6276	1.31322	0.656610	−	0.754231i	$-0.271990\pi$
$509$	29.6276	1.31322	0.656610	+	0.754231i	$0.271990\pi$
$510$	0	0
$511$	6.11719	0.270609
$512$	0	0
$513$	−6.93864	−0.306349
$514$	0	0
$515$	−17.8331	−0.785820
$516$	0	0
$517$	8.38308	0.368687
$518$	0	0
$519$	17.5907	0.772144
$520$	0	0
$521$	−27.5380	−1.20646	−0.603232	−	0.797566i	$-0.706121\pi$
$521$	−27.5380	−1.20646	−0.603232	+	0.797566i	$0.706121\pi$
$522$	0	0
$523$	−16.7245	−0.731310	−0.365655	−	0.930750i	$-0.619155\pi$
$523$	−16.7245	−0.731310	−0.365655	+	0.930750i	$0.619155\pi$
$524$	0	0
$525$	−2.80016	−0.122209
$526$	0	0
$527$	−2.12193	−0.0924329
$528$	0	0
$529$	−21.4314	−0.931801
$530$	0	0
$531$	0.397624	0.0172554
$532$	0	0
$533$	5.14156	0.222706
$534$	0	0
$535$	4.03839	0.174595
$536$	0	0
$537$	−21.0294	−0.907484
$538$	0	0
$539$	0.648276	0.0279232
$540$	0	0
$541$	−12.0047	−0.516122	−0.258061	−	0.966129i	$-0.583084\pi$
$541$	−12.0047	−0.516122	−0.258061	+	0.966129i	$0.583084\pi$
$542$	0	0
$543$	−14.9936	−0.643438
$544$	0	0
$545$	−4.95193	−0.212117
$546$	0	0
$547$	−17.1256	−0.732239	−0.366120	−	0.930568i	$-0.619314\pi$
$547$	−17.1256	−0.732239	−0.366120	+	0.930568i	$0.619314\pi$
$548$	0	0
$549$	−12.9478	−0.552599
$550$	0	0
$551$	−22.9167	−0.976283
$552$	0	0
$553$	−27.7334	−1.17934
$554$	0	0
$555$	5.13260	0.217867
$556$	0	0
$557$	−25.4222	−1.07717	−0.538586	−	0.842571i	$-0.681041\pi$
$557$	−25.4222	−1.07717	−0.538586	+	0.842571i	$0.681041\pi$
$558$	0	0
$559$	17.3870	0.735391
$560$	0	0
$561$	−0.476634	−0.0201235
$562$	0	0
$563$	15.1871	0.640060	0.320030	−	0.947407i	$-0.396307\pi$
$563$	15.1871	0.640060	0.320030	+	0.947407i	$0.396307\pi$
$564$	0	0
$565$	2.11233	0.0888664
$566$	0	0
$567$	−2.80016	−0.117596
$568$	0	0
$569$	−4.93336	−0.206817	−0.103409	−	0.994639i	$-0.532975\pi$
$569$	−4.93336	−0.206817	−0.103409	+	0.994639i	$0.532975\pi$
$570$	0	0
$571$	2.91412	0.121952	0.0609761	−	0.998139i	$-0.480579\pi$
$571$	2.91412	0.121952	0.0609761	+	0.998139i	$0.480579\pi$
$572$	0	0
$573$	2.98651	0.124763
$574$	0	0
$575$	1.25243	0.0522299
$576$	0	0
$577$	−18.7838	−0.781981	−0.390991	−	0.920395i	$-0.627868\pi$
$577$	−18.7838	−0.781981	−0.390991	+	0.920395i	$0.627868\pi$
$578$	0	0
$579$	11.9299	0.495791
$580$	0	0
$581$	42.5206	1.76405
$582$	0	0
$583$	9.77528	0.404851
$584$	0	0
$585$	3.59031	0.148441
$586$	0	0
$587$	36.0169	1.48658	0.743288	−	0.668972i	$-0.233265\pi$
$587$	36.0169	1.48658	0.743288	+	0.668972i	$0.233265\pi$
$588$	0	0
$589$	−23.8139	−0.981233
$590$	0	0
$591$	23.3272	0.959551
$592$	0	0
$593$	−6.75562	−0.277420	−0.138710	−	0.990333i	$-0.544296\pi$
$593$	−6.75562	−0.277420	−0.138710	+	0.990333i	$0.544296\pi$
$594$	0	0
$595$	−1.73125	−0.0709744
$596$	0	0
$597$	10.3504	0.423612
$598$	0	0
$599$	−0.367910	−0.0150324	−0.00751619	−	0.999972i	$-0.502393\pi$
$599$	−0.367910	−0.0150324	−0.00751619	+	0.999972i	$0.502393\pi$
$600$	0	0
$601$	−47.7932	−1.94953	−0.974764	−	0.223240i	$-0.928337\pi$
$601$	−47.7932	−1.94953	−0.974764	+	0.223240i	$0.928337\pi$
$602$	0	0
$603$	1.00000	0.0407231
$604$	0	0
$605$	10.4057	0.423051
$606$	0	0
$607$	−41.7573	−1.69488	−0.847438	−	0.530895i	$-0.821856\pi$
$607$	−41.7573	−1.69488	−0.847438	+	0.530895i	$0.821856\pi$
$608$	0	0
$609$	−9.24827	−0.374759
$610$	0	0
$611$	−39.0416	−1.57945
$612$	0	0
$613$	−0.687236	−0.0277572	−0.0138786	−	0.999904i	$-0.504418\pi$
$613$	−0.687236	−0.0277572	−0.0138786	+	0.999904i	$0.504418\pi$
$614$	0	0
$615$	1.43206	0.0577464
$616$	0	0
$617$	39.4001	1.58619	0.793094	−	0.609099i	$-0.208469\pi$
$617$	39.4001	1.58619	0.793094	+	0.609099i	$0.208469\pi$
$618$	0	0
$619$	45.9164	1.84554	0.922768	−	0.385356i	$-0.125921\pi$
$619$	45.9164	1.84554	0.922768	+	0.385356i	$0.125921\pi$
$620$	0	0
$621$	1.25243	0.0502583
$622$	0	0
$623$	−28.7256	−1.15087
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−5.34913	−0.213624
$628$	0	0
$629$	3.17332	0.126529
$630$	0	0
$631$	−16.1120	−0.641409	−0.320705	−	0.947179i	$-0.603920\pi$
$631$	−16.1120	−0.641409	−0.320705	+	0.947179i	$0.603920\pi$
$632$	0	0
$633$	−27.0627	−1.07564
$634$	0	0
$635$	−6.97601	−0.276835
$636$	0	0
$637$	−3.01914	−0.119623
$638$	0	0
$639$	−1.10374	−0.0436634
$640$	0	0
$641$	−6.04522	−0.238772	−0.119386	−	0.992848i	$-0.538093\pi$
$641$	−6.04522	−0.238772	−0.119386	+	0.992848i	$0.538093\pi$
$642$	0	0
$643$	36.2149	1.42817	0.714087	−	0.700056i	$-0.246842\pi$
$643$	36.2149	1.42817	0.714087	+	0.700056i	$0.246842\pi$
$644$	0	0
$645$	4.84274	0.190683
$646$	0	0
$647$	−30.5047	−1.19926	−0.599632	−	0.800276i	$-0.704686\pi$
$647$	−30.5047	−1.19926	−0.599632	+	0.800276i	$0.704686\pi$
$648$	0	0
$649$	0.306535	0.0120326
$650$	0	0
$651$	−9.61034	−0.376659
$652$	0	0
$653$	−8.39308	−0.328446	−0.164223	−	0.986423i	$-0.552512\pi$
$653$	−8.39308	−0.328446	−0.164223	+	0.986423i	$0.552512\pi$
$654$	0	0
$655$	−19.5046	−0.762110
$656$	0	0
$657$	−2.18458	−0.0852288
$658$	0	0
$659$	−19.0519	−0.742157	−0.371078	−	0.928602i	$-0.621012\pi$
$659$	−19.0519	−0.742157	−0.371078	+	0.928602i	$0.621012\pi$
$660$	0	0
$661$	11.3170	0.440182	0.220091	−	0.975479i	$-0.429365\pi$
$661$	11.3170	0.440182	0.220091	+	0.975479i	$0.429365\pi$
$662$	0	0
$663$	2.21978	0.0862090
$664$	0	0
$665$	−19.4293	−0.753437
$666$	0	0
$667$	4.13647	0.160165
$668$	0	0
$669$	18.1399	0.701330
$670$	0	0
$671$	−9.98171	−0.385340
$672$	0	0
$673$	−35.8117	−1.38044	−0.690219	−	0.723600i	$-0.742486\pi$
$673$	−35.8117	−1.38044	−0.690219	+	0.723600i	$0.742486\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	5.01871	0.192885	0.0964423	−	0.995339i	$-0.469254\pi$
$677$	5.01871	0.192885	0.0964423	+	0.995339i	$0.469254\pi$
$678$	0	0
$679$	−51.0116	−1.95765
$680$	0	0
$681$	10.8027	0.413959
$682$	0	0
$683$	−37.0204	−1.41655	−0.708274	−	0.705938i	$-0.750526\pi$
$683$	−37.0204	−1.41655	−0.708274	+	0.705938i	$0.750526\pi$
$684$	0	0
$685$	−2.24286	−0.0856954
$686$	0	0
$687$	−21.9118	−0.835989
$688$	0	0
$689$	−45.5253	−1.73438
$690$	0	0
$691$	−11.0593	−0.420716	−0.210358	−	0.977624i	$-0.567463\pi$
$691$	−11.0593	−0.420716	−0.210358	+	0.977624i	$0.567463\pi$
$692$	0	0
$693$	−2.15870	−0.0820021
$694$	0	0
$695$	−16.5268	−0.626899
$696$	0	0
$697$	0.885399	0.0335369
$698$	0	0
$699$	−2.19444	−0.0830012
$700$	0	0
$701$	−6.85470	−0.258898	−0.129449	−	0.991586i	$-0.541321\pi$
$701$	−6.85470	−0.258898	−0.129449	+	0.991586i	$0.541321\pi$
$702$	0	0
$703$	35.6133	1.34318
$704$	0	0
$705$	−10.8741	−0.409544
$706$	0	0
$707$	16.7617	0.630389
$708$	0	0
$709$	52.0169	1.95354	0.976768	−	0.214298i	$-0.0687463\pi$
$709$	52.0169	1.95354	0.976768	+	0.214298i	$0.0687463\pi$
$710$	0	0
$711$	9.90420	0.371437
$712$	0	0
$713$	4.29842	0.160977
$714$	0	0
$715$	2.76784	0.103511
$716$	0	0
$717$	21.6147	0.807216
$718$	0	0
$719$	−7.65778	−0.285587	−0.142794	−	0.989752i	$-0.545609\pi$
$719$	−7.65778	−0.285587	−0.142794	+	0.989752i	$0.545609\pi$
$720$	0	0
$721$	−49.9356	−1.85970
$722$	0	0
$723$	19.7227	0.733496
$724$	0	0
$725$	3.30276	0.122661
$726$	0	0
$727$	28.0287	1.03953	0.519764	−	0.854310i	$-0.326020\pi$
$727$	28.0287	1.03953	0.519764	+	0.854310i	$0.326020\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	2.99411	0.110741
$732$	0	0
$733$	−17.5927	−0.649802	−0.324901	−	0.945748i	$-0.605331\pi$
$733$	−17.5927	−0.649802	−0.324901	+	0.945748i	$0.605331\pi$
$734$	0	0
$735$	−0.840913	−0.0310176
$736$	0	0
$737$	0.770919	0.0283972
$738$	0	0
$739$	−11.5493	−0.424846	−0.212423	−	0.977178i	$-0.568136\pi$
$739$	−11.5493	−0.424846	−0.212423	+	0.977178i	$0.568136\pi$
$740$	0	0
$741$	24.9119	0.915162
$742$	0	0
$743$	51.2577	1.88046	0.940232	−	0.340535i	$-0.110608\pi$
$743$	51.2577	1.88046	0.940232	+	0.340535i	$0.110608\pi$
$744$	0	0
$745$	−4.86809	−0.178353
$746$	0	0
$747$	−15.1850	−0.555592
$748$	0	0
$749$	11.3082	0.413191
$750$	0	0
$751$	1.34401	0.0490438	0.0245219	−	0.999699i	$-0.492194\pi$
$751$	1.34401	0.0490438	0.0245219	+	0.999699i	$0.492194\pi$
$752$	0	0
$753$	12.8631	0.468757
$754$	0	0
$755$	5.59330	0.203561
$756$	0	0
$757$	−13.0627	−0.474770	−0.237385	−	0.971416i	$-0.576290\pi$
$757$	−13.0627	−0.474770	−0.237385	+	0.971416i	$0.576290\pi$
$758$	0	0
$759$	0.965521	0.0350462
$760$	0	0
$761$	−41.9765	−1.52165	−0.760823	−	0.648959i	$-0.775205\pi$
$761$	−41.9765	−1.52165	−0.760823	+	0.648959i	$0.775205\pi$
$762$	0	0
$763$	−13.8662	−0.501990
$764$	0	0
$765$	0.618268	0.0223535
$766$	0	0
$767$	−1.42759	−0.0515474
$768$	0	0
$769$	−23.7149	−0.855182	−0.427591	−	0.903972i	$-0.640638\pi$
$769$	−23.7149	−0.855182	−0.427591	+	0.903972i	$0.640638\pi$
$770$	0	0
$771$	−14.3381	−0.516373
$772$	0	0
$773$	−14.6775	−0.527912	−0.263956	−	0.964535i	$-0.585027\pi$
$773$	−14.6775	−0.527912	−0.263956	+	0.964535i	$0.585027\pi$
$774$	0	0
$775$	3.43206	0.123283
$776$	0	0
$777$	14.3721	0.515597
$778$	0	0
$779$	9.93658	0.356015
$780$	0	0
$781$	−0.850896	−0.0304475
$782$	0	0
$783$	3.30276	0.118031
$784$	0	0
$785$	1.77705	0.0634256
$786$	0	0
$787$	10.7688	0.383865	0.191932	−	0.981408i	$-0.438525\pi$
$787$	10.7688	0.383865	0.191932	+	0.981408i	$0.438525\pi$
$788$	0	0
$789$	−16.7140	−0.595033
$790$	0	0
$791$	5.91487	0.210309
$792$	0	0
$793$	46.4867	1.65079
$794$	0	0
$795$	−12.6800	−0.449715
$796$	0	0
$797$	−16.0887	−0.569892	−0.284946	−	0.958544i	$-0.591976\pi$
$797$	−16.0887	−0.569892	−0.284946	+	0.958544i	$0.591976\pi$
$798$	0	0
$799$	−6.72313	−0.237847
$800$	0	0
$801$	10.2586	0.362468
$802$	0	0
$803$	−1.68414	−0.0594319
$804$	0	0
$805$	3.50701	0.123606
$806$	0	0
$807$	10.2574	0.361078
$808$	0	0
$809$	4.31327	0.151647	0.0758233	−	0.997121i	$-0.475842\pi$
$809$	4.31327	0.151647	0.0758233	+	0.997121i	$0.475842\pi$
$810$	0	0
$811$	52.3909	1.83969	0.919847	−	0.392278i	$-0.128313\pi$
$811$	52.3909	1.83969	0.919847	+	0.392278i	$0.128313\pi$
$812$	0	0
$813$	0.523534	0.0183611
$814$	0	0
$815$	−18.1332	−0.635177
$816$	0	0
$817$	33.6021	1.17559
$818$	0	0
$819$	10.0535	0.351297
$820$	0	0
$821$	−26.3661	−0.920184	−0.460092	−	0.887871i	$-0.652184\pi$
$821$	−26.3661	−0.920184	−0.460092	+	0.887871i	$0.652184\pi$
$822$	0	0
$823$	−44.2733	−1.54327	−0.771635	−	0.636065i	$-0.780561\pi$
$823$	−44.2733	−1.54327	−0.771635	+	0.636065i	$0.780561\pi$
$824$	0	0
$825$	0.770919	0.0268399
$826$	0	0
$827$	17.8720	0.621469	0.310734	−	0.950497i	$-0.399425\pi$
$827$	17.8720	0.621469	0.310734	+	0.950497i	$0.399425\pi$
$828$	0	0
$829$	37.7800	1.31215	0.656077	−	0.754694i	$-0.272215\pi$
$829$	37.7800	1.31215	0.656077	+	0.754694i	$0.272215\pi$
$830$	0	0
$831$	4.58867	0.159179
$832$	0	0
$833$	−0.519910	−0.0180138
$834$	0	0
$835$	2.83587	0.0981393
$836$	0	0
$837$	3.43206	0.118629
$838$	0	0
$839$	31.4349	1.08525	0.542627	−	0.839974i	$-0.317430\pi$
$839$	31.4349	1.08525	0.542627	+	0.839974i	$0.317430\pi$
$840$	0	0
$841$	−18.0918	−0.623854
$842$	0	0
$843$	−5.89896	−0.203171
$844$	0	0
$845$	0.109639	0.00377171
$846$	0	0
$847$	29.1376	1.00118
$848$	0	0
$849$	4.28874	0.147189
$850$	0	0
$851$	−6.42822	−0.220356
$852$	0	0
$853$	13.6990	0.469044	0.234522	−	0.972111i	$-0.424647\pi$
$853$	13.6990	0.469044	0.234522	+	0.972111i	$0.424647\pi$
$854$	0	0
$855$	6.93864	0.237297
$856$	0	0
$857$	48.3392	1.65124	0.825618	−	0.564230i	$-0.190827\pi$
$857$	48.3392	1.65124	0.825618	+	0.564230i	$0.190827\pi$
$858$	0	0
$859$	22.8014	0.777973	0.388986	−	0.921244i	$-0.372825\pi$
$859$	22.8014	0.777973	0.388986	+	0.921244i	$0.372825\pi$
$860$	0	0
$861$	4.01001	0.136661
$862$	0	0
$863$	55.5191	1.88989	0.944946	−	0.327227i	$-0.106114\pi$
$863$	55.5191	1.88989	0.944946	+	0.327227i	$0.106114\pi$
$864$	0	0
$865$	−17.5907	−0.598100
$866$	0	0
$867$	−16.6177	−0.564368
$868$	0	0
$869$	7.63533	0.259011
$870$	0	0
$871$	−3.59031	−0.121653
$872$	0	0
$873$	18.2174	0.616565
$874$	0	0
$875$	2.80016	0.0946628
$876$	0	0
$877$	−10.8801	−0.367393	−0.183697	−	0.982983i	$-0.558806\pi$
$877$	−10.8801	−0.367393	−0.183697	+	0.982983i	$0.558806\pi$
$878$	0	0
$879$	11.5034	0.388001
$880$	0	0
$881$	10.9986	0.370553	0.185277	−	0.982686i	$-0.440682\pi$
$881$	10.9986	0.370553	0.185277	+	0.982686i	$0.440682\pi$
$882$	0	0
$883$	24.0520	0.809414	0.404707	−	0.914446i	$-0.367374\pi$
$883$	24.0520	0.809414	0.404707	+	0.914446i	$0.367374\pi$
$884$	0	0
$885$	−0.397624	−0.0133660
$886$	0	0
$887$	−18.1313	−0.608790	−0.304395	−	0.952546i	$-0.598454\pi$
$887$	−18.1313	−0.608790	−0.304395	+	0.952546i	$0.598454\pi$
$888$	0	0
$889$	−19.5340	−0.655148
$890$	0	0
$891$	0.770919	0.0258267
$892$	0	0
$893$	−75.4518	−2.52490
$894$	0	0
$895$	21.0294	0.702934
$896$	0	0
$897$	−4.49662	−0.150138
$898$	0	0
$899$	11.3353	0.378053
$900$	0	0
$901$	−7.83966	−0.261177
$902$	0	0
$903$	13.5605	0.451264
$904$	0	0
$905$	14.9936	0.498405
$906$	0	0
$907$	−25.6799	−0.852686	−0.426343	−	0.904562i	$-0.640198\pi$
$907$	−25.6799	−0.852686	−0.426343	+	0.904562i	$0.640198\pi$
$908$	0	0
$909$	−5.98598	−0.198542
$910$	0	0
$911$	26.1355	0.865909	0.432954	−	0.901416i	$-0.357471\pi$
$911$	26.1355	0.865909	0.432954	+	0.901416i	$0.357471\pi$
$912$	0	0
$913$	−11.7064	−0.387426
$914$	0	0
$915$	12.9478	0.428042
$916$	0	0
$917$	−54.6162	−1.80359
$918$	0	0
$919$	−6.71219	−0.221415	−0.110707	−	0.993853i	$-0.535312\pi$
$919$	−6.71219	−0.221415	−0.110707	+	0.993853i	$0.535312\pi$
$920$	0	0
$921$	−18.3432	−0.604429
$922$	0	0
$923$	3.96278	0.130437
$924$	0	0
$925$	−5.13260	−0.168759
$926$	0	0
$927$	17.8331	0.585716
$928$	0	0
$929$	29.1764	0.957248	0.478624	−	0.878020i	$-0.341136\pi$
$929$	29.1764	0.957248	0.478624	+	0.878020i	$0.341136\pi$
$930$	0	0
$931$	−5.83480	−0.191228
$932$	0	0
$933$	−21.8167	−0.714245
$934$	0	0
$935$	0.476634	0.0155876
$936$	0	0
$937$	41.9706	1.37112	0.685559	−	0.728017i	$-0.259558\pi$
$937$	41.9706	1.37112	0.685559	+	0.728017i	$0.259558\pi$
$938$	0	0
$939$	15.8966	0.518764
$940$	0	0
$941$	−15.9652	−0.520451	−0.260225	−	0.965548i	$-0.583797\pi$
$941$	−15.9652	−0.520451	−0.260225	+	0.965548i	$0.583797\pi$
$942$	0	0
$943$	−1.79356	−0.0584063
$944$	0	0
$945$	2.80016	0.0910893
$946$	0	0
$947$	−20.1280	−0.654073	−0.327037	−	0.945012i	$-0.606050\pi$
$947$	−20.1280	−0.654073	−0.327037	+	0.945012i	$0.606050\pi$
$948$	0	0
$949$	7.84335	0.254606
$950$	0	0
$951$	−14.6686	−0.475662
$952$	0	0
$953$	−3.96459	−0.128426	−0.0642128	−	0.997936i	$-0.520454\pi$
$953$	−3.96459	−0.128426	−0.0642128	+	0.997936i	$0.520454\pi$
$954$	0	0
$955$	−2.98651	−0.0966414
$956$	0	0
$957$	2.54616	0.0823057
$958$	0	0
$959$	−6.28038	−0.202804
$960$	0	0
$961$	−19.2209	−0.620030
$962$	0	0
$963$	−4.03839	−0.130135
$964$	0	0
$965$	−11.9299	−0.384038
$966$	0	0
$967$	34.1500	1.09819	0.549096	−	0.835759i	$-0.314972\pi$
$967$	34.1500	1.09819	0.549096	+	0.835759i	$0.314972\pi$
$968$	0	0
$969$	4.28994	0.137813
$970$	0	0
$971$	−49.4362	−1.58648	−0.793242	−	0.608907i	$-0.791608\pi$
$971$	−49.4362	−1.58648	−0.793242	+	0.608907i	$0.791608\pi$
$972$	0	0
$973$	−46.2779	−1.48360
$974$	0	0
$975$	−3.59031	−0.114982
$976$	0	0
$977$	−52.4778	−1.67891	−0.839457	−	0.543426i	$-0.817127\pi$
$977$	−52.4778	−1.67891	−0.839457	+	0.543426i	$0.817127\pi$
$978$	0	0
$979$	7.90851	0.252757
$980$	0	0
$981$	4.95193	0.158103
$982$	0	0
$983$	−15.7583	−0.502611	−0.251306	−	0.967908i	$-0.580860\pi$
$983$	−15.7583	−0.502611	−0.251306	+	0.967908i	$0.580860\pi$
$984$	0	0
$985$	−23.3272	−0.743265
$986$	0	0
$987$	−30.4494	−0.969214
$988$	0	0
$989$	−6.06519	−0.192862
$990$	0	0
$991$	30.7377	0.976414	0.488207	−	0.872728i	$-0.337651\pi$
$991$	30.7377	0.976414	0.488207	+	0.872728i	$0.337651\pi$
$992$	0	0
$993$	−5.88278	−0.186685
$994$	0	0
$995$	−10.3504	−0.328128
$996$	0	0
$997$	−25.6134	−0.811184	−0.405592	−	0.914054i	$-0.632935\pi$
$997$	−25.6134	−0.811184	−0.405592	+	0.914054i	$0.632935\pi$
$998$	0	0
$999$	−5.13260	−0.162388

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8040.2.a.t.1.1	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8040.2.a.t.1.1	✓	7	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:	\( N \)	\(=\)	\( 8040 = 2^{3} \cdot 3 \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8040.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} -2.80016 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.00000 q^{5} -2.80016 q^{7} +1.00000 q^{9} +0.770919 q^{11} -3.59031 q^{13} -1.00000 q^{15} -0.618268 q^{17} -6.93864 q^{19} -2.80016 q^{21} +1.25243 q^{23} +1.00000 q^{25} +1.00000 q^{27} +3.30276 q^{29} +3.43206 q^{31} +0.770919 q^{33} +2.80016 q^{35} -5.13260 q^{37} -3.59031 q^{39} -1.43206 q^{41} -4.84274 q^{43} -1.00000 q^{45} +10.8741 q^{47} +0.840913 q^{49} -0.618268 q^{51} +12.6800 q^{53} -0.770919 q^{55} -6.93864 q^{57} +0.397624 q^{59} -12.9478 q^{61} -2.80016 q^{63} +3.59031 q^{65} +1.00000 q^{67} +1.25243 q^{69} -1.10374 q^{71} -2.18458 q^{73} +1.00000 q^{75} -2.15870 q^{77} +9.90420 q^{79} +1.00000 q^{81} -15.1850 q^{83} +0.618268 q^{85} +3.30276 q^{87} +10.2586 q^{89} +10.0535 q^{91} +3.43206 q^{93} +6.93864 q^{95} +18.2174 q^{97} +0.770919 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 7 q^{3} - 7 q^{5} + 10 q^{7} + 7 q^{9}+O(q^{10}) \) 7 * q + 7 * q^3 - 7 * q^5 + 10 * q^7 + 7 * q^9	\( 7 q + 7 q^{3} - 7 q^{5} + 10 q^{7} + 7 q^{9} - q^{13} - 7 q^{15} - 2 q^{17} + 9 q^{19} + 10 q^{21} + 2 q^{23} + 7 q^{25} + 7 q^{27} - q^{29} + 9 q^{31} - 10 q^{35} + 23 q^{37} - q^{39} + 5 q^{41} - 3 q^{43} - 7 q^{45} + 11 q^{47} + 13 q^{49} - 2 q^{51} + 13 q^{53} + 9 q^{57} + q^{59} + 4 q^{61} + 10 q^{63} + q^{65} + 7 q^{67} + 2 q^{69} + q^{71} + 14 q^{73} + 7 q^{75} + 18 q^{77} + 25 q^{79} + 7 q^{81} - 29 q^{83} + 2 q^{85} - q^{87} + 7 q^{89} + 27 q^{91} + 9 q^{93} - 9 q^{95} + 38 q^{97}+O(q^{100}) \) 7 * q + 7 * q^3 - 7 * q^5 + 10 * q^7 + 7 * q^9 - q^13 - 7 * q^15 - 2 * q^17 + 9 * q^19 + 10 * q^21 + 2 * q^23 + 7 * q^25 + 7 * q^27 - q^29 + 9 * q^31 - 10 * q^35 + 23 * q^37 - q^39 + 5 * q^41 - 3 * q^43 - 7 * q^45 + 11 * q^47 + 13 * q^49 - 2 * q^51 + 13 * q^53 + 9 * q^57 + q^59 + 4 * q^61 + 10 * q^63 + q^65 + 7 * q^67 + 2 * q^69 + q^71 + 14 * q^73 + 7 * q^75 + 18 * q^77 + 25 * q^79 + 7 * q^81 - 29 * q^83 + 2 * q^85 - q^87 + 7 * q^89 + 27 * q^91 + 9 * q^93 - 9 * q^95 + 38 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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