LMFDB - Embedded newform 8004.2.a.k.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8004,2,Mod(1,8004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8004, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8004.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$63.9122617778$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 5 x^{17} - 55 x^{16} + 288 x^{15} + 1222 x^{14} - 6888 x^{13} - 13745 x^{12} + 88434 x^{11} + \cdots - 115488 $ x^18 - 5x^17 - 55x^16 + 288x^15 + 1222x^14 - 6888x^13 - 13745x^12 + 88434x^11 + 76571x^10 - 655793x^9 - 114554x^8 + 2789438x^7 - 855636x^6 - 6184176x^5 + 4260960x^4 + 5001480x^3 - 5659296x^2 + 1509552*x - 115488
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Root			$2.45476$ of defining polynomial
Character	$\chi$	$=$	8004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +2.45476 q^{5} +0.345256 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +2.45476 q^{5} +0.345256 q^{7} +1.00000 q^{9} +3.27763 q^{11} +0.782475 q^{13} +2.45476 q^{15} +4.36410 q^{17} +2.85807 q^{19} +0.345256 q^{21} -1.00000 q^{23} +1.02586 q^{25} +1.00000 q^{27} +1.00000 q^{29} -3.97475 q^{31} +3.27763 q^{33} +0.847521 q^{35} +11.6972 q^{37} +0.782475 q^{39} +9.72779 q^{41} -8.57414 q^{43} +2.45476 q^{45} +2.54146 q^{47} -6.88080 q^{49} +4.36410 q^{51} -5.89348 q^{53} +8.04581 q^{55} +2.85807 q^{57} -4.40581 q^{59} +13.6104 q^{61} +0.345256 q^{63} +1.92079 q^{65} -7.73051 q^{67} -1.00000 q^{69} +1.43293 q^{71} -9.99601 q^{73} +1.02586 q^{75} +1.13162 q^{77} +6.95290 q^{79} +1.00000 q^{81} -13.5079 q^{83} +10.7128 q^{85} +1.00000 q^{87} +6.69015 q^{89} +0.270154 q^{91} -3.97475 q^{93} +7.01589 q^{95} -6.85116 q^{97} +3.27763 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 18 q^{3} + 5 q^{5} + 6 q^{7} + 18 q^{9}+O(q^{10}) $ 18 * q + 18 * q^3 + 5 * q^5 + 6 * q^7 + 18 * q^9	$ 18 q + 18 q^{3} + 5 q^{5} + 6 q^{7} + 18 q^{9} + 5 q^{11} + 6 q^{13} + 5 q^{15} + 7 q^{17} + 15 q^{19} + 6 q^{21} - 18 q^{23} + 45 q^{25} + 18 q^{27} + 18 q^{29} + 10 q^{31} + 5 q^{33} - 7 q^{35} + 22 q^{37} + 6 q^{39} + 17 q^{41} + 25 q^{43} + 5 q^{45} - 6 q^{47} + 64 q^{49} + 7 q^{51} + 21 q^{53} + 3 q^{55} + 15 q^{57} + 6 q^{59} + 7 q^{61} + 6 q^{63} + 44 q^{65} + 35 q^{67} - 18 q^{69} + q^{71} + 27 q^{73} + 45 q^{75} + 4 q^{77} + 18 q^{81} + 13 q^{83} + 24 q^{85} + 18 q^{87} + 30 q^{89} + 53 q^{91} + 10 q^{93} + 17 q^{95} + 27 q^{97} + 5 q^{99}+O(q^{100}) $ 18 * q + 18 * q^3 + 5 * q^5 + 6 * q^7 + 18 * q^9 + 5 * q^11 + 6 * q^13 + 5 * q^15 + 7 * q^17 + 15 * q^19 + 6 * q^21 - 18 * q^23 + 45 * q^25 + 18 * q^27 + 18 * q^29 + 10 * q^31 + 5 * q^33 - 7 * q^35 + 22 * q^37 + 6 * q^39 + 17 * q^41 + 25 * q^43 + 5 * q^45 - 6 * q^47 + 64 * q^49 + 7 * q^51 + 21 * q^53 + 3 * q^55 + 15 * q^57 + 6 * q^59 + 7 * q^61 + 6 * q^63 + 44 * q^65 + 35 * q^67 - 18 * q^69 + q^71 + 27 * q^73 + 45 * q^75 + 4 * q^77 + 18 * q^81 + 13 * q^83 + 24 * q^85 + 18 * q^87 + 30 * q^89 + 53 * q^91 + 10 * q^93 + 17 * q^95 + 27 * 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
$5$	2.45476	1.09780	0.548902	−	0.835887i	$-0.315046\pi$
$5$	2.45476	1.09780	0.548902	+	0.835887i	$0.315046\pi$
$6$	0	0
$7$	0.345256	0.130494	0.0652472	−	0.997869i	$-0.479216\pi$
$7$	0.345256	0.130494	0.0652472	+	0.997869i	$0.479216\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.27763	0.988243	0.494121	−	0.869393i	$-0.335490\pi$
$11$	3.27763	0.988243	0.494121	+	0.869393i	$0.335490\pi$
$12$	0	0
$13$	0.782475	0.217020	0.108510	−	0.994095i	$-0.465392\pi$
$13$	0.782475	0.217020	0.108510	+	0.994095i	$0.465392\pi$
$14$	0	0
$15$	2.45476	0.633817
$16$	0	0
$17$	4.36410	1.05845	0.529225	−	0.848481i	$-0.322483\pi$
$17$	4.36410	1.05845	0.529225	+	0.848481i	$0.322483\pi$
$18$	0	0
$19$	2.85807	0.655687	0.327843	−	0.944732i	$-0.393678\pi$
$19$	2.85807	0.655687	0.327843	+	0.944732i	$0.393678\pi$
$20$	0	0
$21$	0.345256	0.0753409
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	1.02586	0.205173
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−3.97475	−0.713886	−0.356943	−	0.934126i	$-0.616181\pi$
$31$	−3.97475	−0.713886	−0.356943	+	0.934126i	$0.616181\pi$
$32$	0	0
$33$	3.27763	0.570562
$34$	0	0
$35$	0.847521	0.143257
$36$	0	0
$37$	11.6972	1.92301	0.961505	−	0.274788i	$-0.0886077\pi$
$37$	11.6972	1.92301	0.961505	+	0.274788i	$0.0886077\pi$
$38$	0	0
$39$	0.782475	0.125296
$40$	0	0
$41$	9.72779	1.51923	0.759613	−	0.650375i	$-0.225388\pi$
$41$	9.72779	1.51923	0.759613	+	0.650375i	$0.225388\pi$
$42$	0	0
$43$	−8.57414	−1.30754	−0.653772	−	0.756692i	$-0.726814\pi$
$43$	−8.57414	−1.30754	−0.653772	+	0.756692i	$0.726814\pi$
$44$	0	0
$45$	2.45476	0.365935
$46$	0	0
$47$	2.54146	0.370710	0.185355	−	0.982672i	$-0.440657\pi$
$47$	2.54146	0.370710	0.185355	+	0.982672i	$0.440657\pi$
$48$	0	0
$49$	−6.88080	−0.982971
$50$	0	0
$51$	4.36410	0.611096
$52$	0	0
$53$	−5.89348	−0.809532	−0.404766	−	0.914420i	$-0.632647\pi$
$53$	−5.89348	−0.809532	−0.404766	+	0.914420i	$0.632647\pi$
$54$	0	0
$55$	8.04581	1.08490
$56$	0	0
$57$	2.85807	0.378561
$58$	0	0
$59$	−4.40581	−0.573588	−0.286794	−	0.957992i	$-0.592590\pi$
$59$	−4.40581	−0.573588	−0.286794	+	0.957992i	$0.592590\pi$
$60$	0	0
$61$	13.6104	1.74263	0.871314	−	0.490726i	$-0.163268\pi$
$61$	13.6104	1.74263	0.871314	+	0.490726i	$0.163268\pi$
$62$	0	0
$63$	0.345256	0.0434981
$64$	0	0
$65$	1.92079	0.238245
$66$	0	0
$67$	−7.73051	−0.944432	−0.472216	−	0.881483i	$-0.656546\pi$
$67$	−7.73051	−0.944432	−0.472216	+	0.881483i	$0.656546\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	1.43293	0.170058	0.0850290	−	0.996378i	$-0.472902\pi$
$71$	1.43293	0.170058	0.0850290	+	0.996378i	$0.472902\pi$
$72$	0	0
$73$	−9.99601	−1.16994	−0.584972	−	0.811053i	$-0.698895\pi$
$73$	−9.99601	−1.16994	−0.584972	+	0.811053i	$0.698895\pi$
$74$	0	0
$75$	1.02586	0.118456
$76$	0	0
$77$	1.13162	0.128960
$78$	0	0
$79$	6.95290	0.782262	0.391131	−	0.920335i	$-0.372084\pi$
$79$	6.95290	0.782262	0.391131	+	0.920335i	$0.372084\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−13.5079	−1.48269	−0.741344	−	0.671125i	$-0.765811\pi$
$83$	−13.5079	−1.48269	−0.741344	+	0.671125i	$0.765811\pi$
$84$	0	0
$85$	10.7128	1.16197
$86$	0	0
$87$	1.00000	0.107211
$88$	0	0
$89$	6.69015	0.709155	0.354577	−	0.935027i	$-0.384625\pi$
$89$	6.69015	0.709155	0.354577	+	0.935027i	$0.384625\pi$
$90$	0	0
$91$	0.270154	0.0283198
$92$	0	0
$93$	−3.97475	−0.412162
$94$	0	0
$95$	7.01589	0.719815
$96$	0	0
$97$	−6.85116	−0.695629	−0.347815	−	0.937563i	$-0.613076\pi$
$97$	−6.85116	−0.695629	−0.347815	+	0.937563i	$0.613076\pi$
$98$	0	0
$99$	3.27763	0.329414
$100$	0	0
$101$	16.1725	1.60922	0.804611	−	0.593803i	$-0.202374\pi$
$101$	16.1725	1.60922	0.804611	+	0.593803i	$0.202374\pi$
$102$	0	0
$103$	−8.98714	−0.885530	−0.442765	−	0.896638i	$-0.646002\pi$
$103$	−8.98714	−0.885530	−0.442765	+	0.896638i	$0.646002\pi$
$104$	0	0
$105$	0.847521	0.0827095
$106$	0	0
$107$	14.1162	1.36467	0.682335	−	0.731040i	$-0.260965\pi$
$107$	14.1162	1.36467	0.682335	+	0.731040i	$0.260965\pi$
$108$	0	0
$109$	−9.19043	−0.880284	−0.440142	−	0.897928i	$-0.645072\pi$
$109$	−9.19043	−0.880284	−0.440142	+	0.897928i	$0.645072\pi$
$110$	0	0
$111$	11.6972	1.11025
$112$	0	0
$113$	5.12559	0.482175	0.241088	−	0.970503i	$-0.422496\pi$
$113$	5.12559	0.482175	0.241088	+	0.970503i	$0.422496\pi$
$114$	0	0
$115$	−2.45476	−0.228908
$116$	0	0
$117$	0.782475	0.0723399
$118$	0	0
$119$	1.50673	0.138122
$120$	0	0
$121$	−0.257140	−0.0233764
$122$	0	0
$123$	9.72779	0.877126
$124$	0	0
$125$	−9.75557	−0.872564
$126$	0	0
$127$	−7.05704	−0.626211	−0.313106	−	0.949718i	$-0.601369\pi$
$127$	−7.05704	−0.626211	−0.313106	+	0.949718i	$0.601369\pi$
$128$	0	0
$129$	−8.57414	−0.754911
$130$	0	0
$131$	14.1542	1.23666	0.618328	−	0.785920i	$-0.287810\pi$
$131$	14.1542	1.23666	0.618328	+	0.785920i	$0.287810\pi$
$132$	0	0
$133$	0.986765	0.0855634
$134$	0	0
$135$	2.45476	0.211272
$136$	0	0
$137$	−6.95261	−0.594001	−0.297001	−	0.954877i	$-0.595986\pi$
$137$	−6.95261	−0.594001	−0.297001	+	0.954877i	$0.595986\pi$
$138$	0	0
$139$	17.1385	1.45367	0.726834	−	0.686813i	$-0.240991\pi$
$139$	17.1385	1.45367	0.726834	+	0.686813i	$0.240991\pi$
$140$	0	0
$141$	2.54146	0.214029
$142$	0	0
$143$	2.56466	0.214468
$144$	0	0
$145$	2.45476	0.203857
$146$	0	0
$147$	−6.88080	−0.567519
$148$	0	0
$149$	−12.0438	−0.986664	−0.493332	−	0.869841i	$-0.664221\pi$
$149$	−12.0438	−0.986664	−0.493332	+	0.869841i	$0.664221\pi$
$150$	0	0
$151$	14.2201	1.15721	0.578606	−	0.815607i	$-0.303597\pi$
$151$	14.2201	1.15721	0.578606	+	0.815607i	$0.303597\pi$
$152$	0	0
$153$	4.36410	0.352817
$154$	0	0
$155$	−9.75707	−0.783707
$156$	0	0
$157$	−12.7816	−1.02008	−0.510040	−	0.860150i	$-0.670370\pi$
$157$	−12.7816	−1.02008	−0.510040	+	0.860150i	$0.670370\pi$
$158$	0	0
$159$	−5.89348	−0.467383
$160$	0	0
$161$	−0.345256	−0.0272099
$162$	0	0
$163$	−11.2222	−0.878991	−0.439495	−	0.898245i	$-0.644843\pi$
$163$	−11.2222	−0.878991	−0.439495	+	0.898245i	$0.644843\pi$
$164$	0	0
$165$	8.04581	0.626365
$166$	0	0
$167$	−23.5082	−1.81912	−0.909558	−	0.415577i	$-0.863580\pi$
$167$	−23.5082	−1.81912	−0.909558	+	0.415577i	$0.863580\pi$
$168$	0	0
$169$	−12.3877	−0.952902
$170$	0	0
$171$	2.85807	0.218562
$172$	0	0
$173$	−11.6604	−0.886527	−0.443263	−	0.896391i	$-0.646179\pi$
$173$	−11.6604	−0.886527	−0.443263	+	0.896391i	$0.646179\pi$
$174$	0	0
$175$	0.354185	0.0267739
$176$	0	0
$177$	−4.40581	−0.331161
$178$	0	0
$179$	−3.92046	−0.293029	−0.146515	−	0.989209i	$-0.546806\pi$
$179$	−3.92046	−0.293029	−0.146515	+	0.989209i	$0.546806\pi$
$180$	0	0
$181$	−8.30805	−0.617532	−0.308766	−	0.951138i	$-0.599916\pi$
$181$	−8.30805	−0.617532	−0.308766	+	0.951138i	$0.599916\pi$
$182$	0	0
$183$	13.6104	1.00611
$184$	0	0
$185$	28.7139	2.11109
$186$	0	0
$187$	14.3039	1.04601
$188$	0	0
$189$	0.345256	0.0251136
$190$	0	0
$191$	21.1483	1.53024	0.765119	−	0.643889i	$-0.222680\pi$
$191$	21.1483	1.53024	0.765119	+	0.643889i	$0.222680\pi$
$192$	0	0
$193$	19.7546	1.42197	0.710984	−	0.703208i	$-0.248250\pi$
$193$	19.7546	1.42197	0.710984	+	0.703208i	$0.248250\pi$
$194$	0	0
$195$	1.92079	0.137551
$196$	0	0
$197$	1.55348	0.110681	0.0553403	−	0.998468i	$-0.482376\pi$
$197$	1.55348	0.110681	0.0553403	+	0.998468i	$0.482376\pi$
$198$	0	0
$199$	−1.87175	−0.132685	−0.0663424	−	0.997797i	$-0.521133\pi$
$199$	−1.87175	−0.132685	−0.0663424	+	0.997797i	$0.521133\pi$
$200$	0	0
$201$	−7.73051	−0.545268
$202$	0	0
$203$	0.345256	0.0242322
$204$	0	0
$205$	23.8794	1.66781
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	9.36770	0.647978
$210$	0	0
$211$	−2.74428	−0.188924	−0.0944620	−	0.995528i	$-0.530113\pi$
$211$	−2.74428	−0.188924	−0.0944620	+	0.995528i	$0.530113\pi$
$212$	0	0
$213$	1.43293	0.0981830
$214$	0	0
$215$	−21.0475	−1.43543
$216$	0	0
$217$	−1.37230	−0.0931581
$218$	0	0
$219$	−9.99601	−0.675468
$220$	0	0
$221$	3.41480	0.229704
$222$	0	0
$223$	−0.667247	−0.0446821	−0.0223411	−	0.999750i	$-0.507112\pi$
$223$	−0.667247	−0.0446821	−0.0223411	+	0.999750i	$0.507112\pi$
$224$	0	0
$225$	1.02586	0.0683909
$226$	0	0
$227$	3.54062	0.234999	0.117500	−	0.993073i	$-0.462512\pi$
$227$	3.54062	0.234999	0.117500	+	0.993073i	$0.462512\pi$
$228$	0	0
$229$	1.60693	0.106189	0.0530944	−	0.998589i	$-0.483092\pi$
$229$	1.60693	0.106189	0.0530944	+	0.998589i	$0.483092\pi$
$230$	0	0
$231$	1.13162	0.0744551
$232$	0	0
$233$	0.536681	0.0351591	0.0175796	−	0.999845i	$-0.494404\pi$
$233$	0.536681	0.0351591	0.0175796	+	0.999845i	$0.494404\pi$
$234$	0	0
$235$	6.23868	0.406967
$236$	0	0
$237$	6.95290	0.451639
$238$	0	0
$239$	−22.8313	−1.47683	−0.738417	−	0.674344i	$-0.764426\pi$
$239$	−22.8313	−1.47683	−0.738417	+	0.674344i	$0.764426\pi$
$240$	0	0
$241$	18.1389	1.16843	0.584213	−	0.811600i	$-0.301403\pi$
$241$	18.1389	1.16843	0.584213	+	0.811600i	$0.301403\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−16.8907	−1.07911
$246$	0	0
$247$	2.23637	0.142297
$248$	0	0
$249$	−13.5079	−0.856031
$250$	0	0
$251$	0.676745	0.0427158	0.0213579	−	0.999772i	$-0.493201\pi$
$251$	0.676745	0.0427158	0.0213579	+	0.999772i	$0.493201\pi$
$252$	0	0
$253$	−3.27763	−0.206063
$254$	0	0
$255$	10.7128	0.670864
$256$	0	0
$257$	−6.12135	−0.381839	−0.190920	−	0.981606i	$-0.561147\pi$
$257$	−6.12135	−0.381839	−0.190920	+	0.981606i	$0.561147\pi$
$258$	0	0
$259$	4.03853	0.250942
$260$	0	0
$261$	1.00000	0.0618984
$262$	0	0
$263$	30.0720	1.85432	0.927160	−	0.374665i	$-0.122242\pi$
$263$	30.0720	1.85432	0.927160	+	0.374665i	$0.122242\pi$
$264$	0	0
$265$	−14.4671	−0.888707
$266$	0	0
$267$	6.69015	0.409431
$268$	0	0
$269$	19.3489	1.17972	0.589861	−	0.807505i	$-0.299182\pi$
$269$	19.3489	1.17972	0.589861	+	0.807505i	$0.299182\pi$
$270$	0	0
$271$	−4.57531	−0.277930	−0.138965	−	0.990297i	$-0.544378\pi$
$271$	−4.57531	−0.277930	−0.138965	+	0.990297i	$0.544378\pi$
$272$	0	0
$273$	0.270154	0.0163505
$274$	0	0
$275$	3.36240	0.202760
$276$	0	0
$277$	30.6791	1.84333	0.921663	−	0.387991i	$-0.126831\pi$
$277$	30.6791	1.84333	0.921663	+	0.387991i	$0.126831\pi$
$278$	0	0
$279$	−3.97475	−0.237962
$280$	0	0
$281$	−10.8430	−0.646840	−0.323420	−	0.946255i	$-0.604833\pi$
$281$	−10.8430	−0.646840	−0.323420	+	0.946255i	$0.604833\pi$
$282$	0	0
$283$	−16.7742	−0.997122	−0.498561	−	0.866855i	$-0.666138\pi$
$283$	−16.7742	−0.997122	−0.498561	+	0.866855i	$0.666138\pi$
$284$	0	0
$285$	7.01589	0.415585
$286$	0	0
$287$	3.35857	0.198250
$288$	0	0
$289$	2.04538	0.120317
$290$	0	0
$291$	−6.85116	−0.401622
$292$	0	0
$293$	5.94969	0.347585	0.173792	−	0.984782i	$-0.444398\pi$
$293$	5.94969	0.347585	0.173792	+	0.984782i	$0.444398\pi$
$294$	0	0
$295$	−10.8152	−0.629687
$296$	0	0
$297$	3.27763	0.190187
$298$	0	0
$299$	−0.782475	−0.0452517
$300$	0	0
$301$	−2.96027	−0.170627
$302$	0	0
$303$	16.1725	0.929084
$304$	0	0
$305$	33.4102	1.91306
$306$	0	0
$307$	−12.9321	−0.738075	−0.369037	−	0.929415i	$-0.620313\pi$
$307$	−12.9321	−0.738075	−0.369037	+	0.929415i	$0.620313\pi$
$308$	0	0
$309$	−8.98714	−0.511261
$310$	0	0
$311$	25.9767	1.47300	0.736502	−	0.676435i	$-0.236476\pi$
$311$	25.9767	1.47300	0.736502	+	0.676435i	$0.236476\pi$
$312$	0	0
$313$	22.5617	1.27526	0.637631	−	0.770342i	$-0.279915\pi$
$313$	22.5617	1.27526	0.637631	+	0.770342i	$0.279915\pi$
$314$	0	0
$315$	0.847521	0.0477524
$316$	0	0
$317$	−25.2686	−1.41922	−0.709612	−	0.704592i	$-0.751130\pi$
$317$	−25.2686	−1.41922	−0.709612	+	0.704592i	$0.751130\pi$
$318$	0	0
$319$	3.27763	0.183512
$320$	0	0
$321$	14.1162	0.787892
$322$	0	0
$323$	12.4729	0.694012
$324$	0	0
$325$	0.802712	0.0445265
$326$	0	0
$327$	−9.19043	−0.508232
$328$	0	0
$329$	0.877453	0.0483755
$330$	0	0
$331$	23.8716	1.31210	0.656051	−	0.754716i	$-0.272226\pi$
$331$	23.8716	1.31210	0.656051	+	0.754716i	$0.272226\pi$
$332$	0	0
$333$	11.6972	0.641003
$334$	0	0
$335$	−18.9766	−1.03680
$336$	0	0
$337$	−29.6420	−1.61470	−0.807352	−	0.590070i	$-0.799100\pi$
$337$	−29.6420	−1.61470	−0.807352	+	0.590070i	$0.799100\pi$
$338$	0	0
$339$	5.12559	0.278384
$340$	0	0
$341$	−13.0278	−0.705493
$342$	0	0
$343$	−4.79242	−0.258767
$344$	0	0
$345$	−2.45476	−0.132160
$346$	0	0
$347$	21.5943	1.15924	0.579620	−	0.814887i	$-0.303201\pi$
$347$	21.5943	1.15924	0.579620	+	0.814887i	$0.303201\pi$
$348$	0	0
$349$	−6.05457	−0.324094	−0.162047	−	0.986783i	$-0.551810\pi$
$349$	−6.05457	−0.324094	−0.162047	+	0.986783i	$0.551810\pi$
$350$	0	0
$351$	0.782475	0.0417654
$352$	0	0
$353$	21.9607	1.16885	0.584425	−	0.811448i	$-0.301320\pi$
$353$	21.9607	1.16885	0.584425	+	0.811448i	$0.301320\pi$
$354$	0	0
$355$	3.51751	0.186690
$356$	0	0
$357$	1.50673	0.0797446
$358$	0	0
$359$	−31.4824	−1.66158	−0.830789	−	0.556587i	$-0.812110\pi$
$359$	−31.4824	−1.66158	−0.830789	+	0.556587i	$0.812110\pi$
$360$	0	0
$361$	−10.8314	−0.570075
$362$	0	0
$363$	−0.257140	−0.0134963
$364$	0	0
$365$	−24.5378	−1.28437
$366$	0	0
$367$	18.6901	0.975613	0.487806	−	0.872952i	$-0.337797\pi$
$367$	18.6901	0.975613	0.487806	+	0.872952i	$0.337797\pi$
$368$	0	0
$369$	9.72779	0.506409
$370$	0	0
$371$	−2.03476	−0.105639
$372$	0	0
$373$	−16.8943	−0.874752	−0.437376	−	0.899279i	$-0.644092\pi$
$373$	−16.8943	−0.874752	−0.437376	+	0.899279i	$0.644092\pi$
$374$	0	0
$375$	−9.75557	−0.503775
$376$	0	0
$377$	0.782475	0.0402995
$378$	0	0
$379$	−15.5709	−0.799824	−0.399912	−	0.916554i	$-0.630959\pi$
$379$	−15.5709	−0.799824	−0.399912	+	0.916554i	$0.630959\pi$
$380$	0	0
$381$	−7.05704	−0.361543
$382$	0	0
$383$	−6.97928	−0.356625	−0.178312	−	0.983974i	$-0.557064\pi$
$383$	−6.97928	−0.356625	−0.178312	+	0.983974i	$0.557064\pi$
$384$	0	0
$385$	2.77786	0.141573
$386$	0	0
$387$	−8.57414	−0.435848
$388$	0	0
$389$	−1.19792	−0.0607368	−0.0303684	−	0.999539i	$-0.509668\pi$
$389$	−1.19792	−0.0607368	−0.0303684	+	0.999539i	$0.509668\pi$
$390$	0	0
$391$	−4.36410	−0.220702
$392$	0	0
$393$	14.1542	0.713984
$394$	0	0
$395$	17.0677	0.858770
$396$	0	0
$397$	30.0509	1.50821	0.754106	−	0.656753i	$-0.228070\pi$
$397$	30.0509	1.50821	0.754106	+	0.656753i	$0.228070\pi$
$398$	0	0
$399$	0.986765	0.0494000
$400$	0	0
$401$	16.6425	0.831088	0.415544	−	0.909573i	$-0.363591\pi$
$401$	16.6425	0.831088	0.415544	+	0.909573i	$0.363591\pi$
$402$	0	0
$403$	−3.11014	−0.154927
$404$	0	0
$405$	2.45476	0.121978
$406$	0	0
$407$	38.3391	1.90040
$408$	0	0
$409$	13.0065	0.643131	0.321566	−	0.946887i	$-0.395791\pi$
$409$	13.0065	0.643131	0.321566	+	0.946887i	$0.395791\pi$
$410$	0	0
$411$	−6.95261	−0.342947
$412$	0	0
$413$	−1.52113	−0.0748500
$414$	0	0
$415$	−33.1588	−1.62770
$416$	0	0
$417$	17.1385	0.839276
$418$	0	0
$419$	−2.52602	−0.123404	−0.0617020	−	0.998095i	$-0.519653\pi$
$419$	−2.52602	−0.123404	−0.0617020	+	0.998095i	$0.519653\pi$
$420$	0	0
$421$	−37.4393	−1.82468	−0.912339	−	0.409435i	$-0.865726\pi$
$421$	−37.4393	−1.82468	−0.912339	+	0.409435i	$0.865726\pi$
$422$	0	0
$423$	2.54146	0.123570
$424$	0	0
$425$	4.47697	0.217165
$426$	0	0
$427$	4.69905	0.227403
$428$	0	0
$429$	2.56466	0.123823
$430$	0	0
$431$	−41.2313	−1.98604	−0.993020	−	0.117948i	$-0.962368\pi$
$431$	−41.2313	−1.98604	−0.993020	+	0.117948i	$0.962368\pi$
$432$	0	0
$433$	29.9820	1.44084	0.720422	−	0.693535i	$-0.243948\pi$
$433$	29.9820	1.44084	0.720422	+	0.693535i	$0.243948\pi$
$434$	0	0
$435$	2.45476	0.117697
$436$	0	0
$437$	−2.85807	−0.136720
$438$	0	0
$439$	20.9254	0.998714	0.499357	−	0.866396i	$-0.333570\pi$
$439$	20.9254	0.998714	0.499357	+	0.866396i	$0.333570\pi$
$440$	0	0
$441$	−6.88080	−0.327657
$442$	0	0
$443$	−15.4654	−0.734782	−0.367391	−	0.930067i	$-0.619749\pi$
$443$	−15.4654	−0.734782	−0.367391	+	0.930067i	$0.619749\pi$
$444$	0	0
$445$	16.4227	0.778512
$446$	0	0
$447$	−12.0438	−0.569651
$448$	0	0
$449$	34.3077	1.61908	0.809540	−	0.587065i	$-0.199717\pi$
$449$	34.3077	1.61908	0.809540	+	0.587065i	$0.199717\pi$
$450$	0	0
$451$	31.8841	1.50136
$452$	0	0
$453$	14.2201	0.668117
$454$	0	0
$455$	0.663164	0.0310896
$456$	0	0
$457$	18.8464	0.881597	0.440798	−	0.897606i	$-0.354695\pi$
$457$	18.8464	0.881597	0.440798	+	0.897606i	$0.354695\pi$
$458$	0	0
$459$	4.36410	0.203699
$460$	0	0
$461$	−32.6352	−1.51997	−0.759987	−	0.649938i	$-0.774795\pi$
$461$	−32.6352	−1.51997	−0.759987	+	0.649938i	$0.774795\pi$
$462$	0	0
$463$	−15.8707	−0.737573	−0.368786	−	0.929514i	$-0.620227\pi$
$463$	−15.8707	−0.737573	−0.368786	+	0.929514i	$0.620227\pi$
$464$	0	0
$465$	−9.75707	−0.452473
$466$	0	0
$467$	−8.90876	−0.412248	−0.206124	−	0.978526i	$-0.566085\pi$
$467$	−8.90876	−0.412248	−0.206124	+	0.978526i	$0.566085\pi$
$468$	0	0
$469$	−2.66900	−0.123243
$470$	0	0
$471$	−12.7816	−0.588944
$472$	0	0
$473$	−28.1028	−1.29217
$474$	0	0
$475$	2.93199	0.134529
$476$	0	0
$477$	−5.89348	−0.269844
$478$	0	0
$479$	−34.6862	−1.58485	−0.792426	−	0.609968i	$-0.791182\pi$
$479$	−34.6862	−1.58485	−0.792426	+	0.609968i	$0.791182\pi$
$480$	0	0
$481$	9.15278	0.417331
$482$	0	0
$483$	−0.345256	−0.0157097
$484$	0	0
$485$	−16.8180	−0.763664
$486$	0	0
$487$	−31.9260	−1.44671	−0.723353	−	0.690478i	$-0.757400\pi$
$487$	−31.9260	−1.44671	−0.723353	+	0.690478i	$0.757400\pi$
$488$	0	0
$489$	−11.2222	−0.507486
$490$	0	0
$491$	−2.73711	−0.123524	−0.0617621	−	0.998091i	$-0.519672\pi$
$491$	−2.73711	−0.123524	−0.0617621	+	0.998091i	$0.519672\pi$
$492$	0	0
$493$	4.36410	0.196549
$494$	0	0
$495$	8.04581	0.361632
$496$	0	0
$497$	0.494728	0.0221916
$498$	0	0
$499$	24.5906	1.10083	0.550414	−	0.834892i	$-0.314470\pi$
$499$	24.5906	1.10083	0.550414	+	0.834892i	$0.314470\pi$
$500$	0	0
$501$	−23.5082	−1.05027
$502$	0	0
$503$	−32.4549	−1.44709	−0.723546	−	0.690277i	$-0.757489\pi$
$503$	−32.4549	−1.44709	−0.723546	+	0.690277i	$0.757489\pi$
$504$	0	0
$505$	39.6996	1.76661
$506$	0	0
$507$	−12.3877	−0.550159
$508$	0	0
$509$	−33.2507	−1.47381	−0.736906	−	0.675995i	$-0.763714\pi$
$509$	−33.2507	−1.47381	−0.736906	+	0.675995i	$0.763714\pi$
$510$	0	0
$511$	−3.45118	−0.152671
$512$	0	0
$513$	2.85807	0.126187
$514$	0	0
$515$	−22.0613	−0.972137
$516$	0	0
$517$	8.32996	0.366351
$518$	0	0
$519$	−11.6604	−0.511836
$520$	0	0
$521$	−1.14153	−0.0500115	−0.0250058	−	0.999687i	$-0.507960\pi$
$521$	−1.14153	−0.0500115	−0.0250058	+	0.999687i	$0.507960\pi$
$522$	0	0
$523$	−34.2750	−1.49874	−0.749371	−	0.662150i	$-0.769644\pi$
$523$	−34.2750	−1.49874	−0.749371	+	0.662150i	$0.769644\pi$
$524$	0	0
$525$	0.354185	0.0154579
$526$	0	0
$527$	−17.3462	−0.755613
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−4.40581	−0.191196
$532$	0	0
$533$	7.61176	0.329702
$534$	0	0
$535$	34.6520	1.49814
$536$	0	0
$537$	−3.92046	−0.169181
$538$	0	0
$539$	−22.5527	−0.971414
$540$	0	0
$541$	33.9120	1.45799	0.728995	−	0.684519i	$-0.239988\pi$
$541$	33.9120	1.45799	0.728995	+	0.684519i	$0.239988\pi$
$542$	0	0
$543$	−8.30805	−0.356532
$544$	0	0
$545$	−22.5603	−0.966379
$546$	0	0
$547$	24.8521	1.06260	0.531298	−	0.847185i	$-0.321704\pi$
$547$	24.8521	1.06260	0.531298	+	0.847185i	$0.321704\pi$
$548$	0	0
$549$	13.6104	0.580876
$550$	0	0
$551$	2.85807	0.121758
$552$	0	0
$553$	2.40053	0.102081
$554$	0	0
$555$	28.7139	1.21884
$556$	0	0
$557$	28.9608	1.22711	0.613555	−	0.789652i	$-0.289739\pi$
$557$	28.9608	1.22711	0.613555	+	0.789652i	$0.289739\pi$
$558$	0	0
$559$	−6.70905	−0.283763
$560$	0	0
$561$	14.3039	0.603912
$562$	0	0
$563$	12.3523	0.520588	0.260294	−	0.965529i	$-0.416180\pi$
$563$	12.3523	0.520588	0.260294	+	0.965529i	$0.416180\pi$
$564$	0	0
$565$	12.5821	0.529333
$566$	0	0
$567$	0.345256	0.0144994
$568$	0	0
$569$	27.5593	1.15535	0.577674	−	0.816268i	$-0.303961\pi$
$569$	27.5593	1.15535	0.577674	+	0.816268i	$0.303961\pi$
$570$	0	0
$571$	−27.2551	−1.14059	−0.570296	−	0.821439i	$-0.693172\pi$
$571$	−27.2551	−1.14059	−0.570296	+	0.821439i	$0.693172\pi$
$572$	0	0
$573$	21.1483	0.883483
$574$	0	0
$575$	−1.02586	−0.0427814
$576$	0	0
$577$	−6.91006	−0.287670	−0.143835	−	0.989602i	$-0.545943\pi$
$577$	−6.91006	−0.287670	−0.143835	+	0.989602i	$0.545943\pi$
$578$	0	0
$579$	19.7546	0.820974
$580$	0	0
$581$	−4.66369	−0.193482
$582$	0	0
$583$	−19.3166	−0.800014
$584$	0	0
$585$	1.92079	0.0794150
$586$	0	0
$587$	−15.3087	−0.631856	−0.315928	−	0.948783i	$-0.602316\pi$
$587$	−15.3087	−0.631856	−0.315928	+	0.948783i	$0.602316\pi$
$588$	0	0
$589$	−11.3601	−0.468086
$590$	0	0
$591$	1.55348	0.0639015
$592$	0	0
$593$	39.5138	1.62264	0.811318	−	0.584605i	$-0.198750\pi$
$593$	39.5138	1.62264	0.811318	+	0.584605i	$0.198750\pi$
$594$	0	0
$595$	3.69867	0.151631
$596$	0	0
$597$	−1.87175	−0.0766056
$598$	0	0
$599$	−6.63515	−0.271105	−0.135553	−	0.990770i	$-0.543281\pi$
$599$	−6.63515	−0.271105	−0.135553	+	0.990770i	$0.543281\pi$
$600$	0	0
$601$	23.9941	0.978739	0.489369	−	0.872077i	$-0.337227\pi$
$601$	23.9941	0.978739	0.489369	+	0.872077i	$0.337227\pi$
$602$	0	0
$603$	−7.73051	−0.314811
$604$	0	0
$605$	−0.631218	−0.0256626
$606$	0	0
$607$	34.5857	1.40379	0.701895	−	0.712280i	$-0.252338\pi$
$607$	34.5857	1.40379	0.701895	+	0.712280i	$0.252338\pi$
$608$	0	0
$609$	0.345256	0.0139905
$610$	0	0
$611$	1.98863	0.0804513
$612$	0	0
$613$	25.7066	1.03828	0.519139	−	0.854690i	$-0.326253\pi$
$613$	25.7066	1.03828	0.519139	+	0.854690i	$0.326253\pi$
$614$	0	0
$615$	23.8794	0.962912
$616$	0	0
$617$	−12.2335	−0.492500	−0.246250	−	0.969206i	$-0.579198\pi$
$617$	−12.2335	−0.492500	−0.246250	+	0.969206i	$0.579198\pi$
$618$	0	0
$619$	−14.6930	−0.590563	−0.295282	−	0.955410i	$-0.595413\pi$
$619$	−14.6930	−0.590563	−0.295282	+	0.955410i	$0.595413\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	2.30981	0.0925407
$624$	0	0
$625$	−29.0769	−1.16308
$626$	0	0
$627$	9.36770	0.374110
$628$	0	0
$629$	51.0478	2.03541
$630$	0	0
$631$	−16.6847	−0.664209	−0.332104	−	0.943243i	$-0.607759\pi$
$631$	−16.6847	−0.664209	−0.332104	+	0.943243i	$0.607759\pi$
$632$	0	0
$633$	−2.74428	−0.109075
$634$	0	0
$635$	−17.3234	−0.687457
$636$	0	0
$637$	−5.38406	−0.213324
$638$	0	0
$639$	1.43293	0.0566860
$640$	0	0
$641$	−2.20477	−0.0870833	−0.0435416	−	0.999052i	$-0.513864\pi$
$641$	−2.20477	−0.0870833	−0.0435416	+	0.999052i	$0.513864\pi$
$642$	0	0
$643$	−6.00566	−0.236840	−0.118420	−	0.992964i	$-0.537783\pi$
$643$	−6.00566	−0.236840	−0.118420	+	0.992964i	$0.537783\pi$
$644$	0	0
$645$	−21.0475	−0.828744
$646$	0	0
$647$	9.27138	0.364496	0.182248	−	0.983253i	$-0.441663\pi$
$647$	9.27138	0.364496	0.182248	+	0.983253i	$0.441663\pi$
$648$	0	0
$649$	−14.4406	−0.566844
$650$	0	0
$651$	−1.37230	−0.0537849
$652$	0	0
$653$	−16.3652	−0.640419	−0.320209	−	0.947347i	$-0.603753\pi$
$653$	−16.3652	−0.640419	−0.320209	+	0.947347i	$0.603753\pi$
$654$	0	0
$655$	34.7452	1.35761
$656$	0	0
$657$	−9.99601	−0.389981
$658$	0	0
$659$	18.2350	0.710334	0.355167	−	0.934803i	$-0.384424\pi$
$659$	18.2350	0.710334	0.355167	+	0.934803i	$0.384424\pi$
$660$	0	0
$661$	−6.58335	−0.256063	−0.128031	−	0.991770i	$-0.540866\pi$
$661$	−6.58335	−0.256063	−0.128031	+	0.991770i	$0.540866\pi$
$662$	0	0
$663$	3.41480	0.132620
$664$	0	0
$665$	2.42227	0.0939318
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	0	0
$669$	−0.667247	−0.0257972
$670$	0	0
$671$	44.6097	1.72214
$672$	0	0
$673$	1.83461	0.0707191	0.0353595	−	0.999375i	$-0.488742\pi$
$673$	1.83461	0.0707191	0.0353595	+	0.999375i	$0.488742\pi$
$674$	0	0
$675$	1.02586	0.0394855
$676$	0	0
$677$	−2.92935	−0.112584	−0.0562920	−	0.998414i	$-0.517928\pi$
$677$	−2.92935	−0.112584	−0.0562920	+	0.998414i	$0.517928\pi$
$678$	0	0
$679$	−2.36540	−0.0907757
$680$	0	0
$681$	3.54062	0.135677
$682$	0	0
$683$	−4.43743	−0.169793	−0.0848967	−	0.996390i	$-0.527056\pi$
$683$	−4.43743	−0.169793	−0.0848967	+	0.996390i	$0.527056\pi$
$684$	0	0
$685$	−17.0670	−0.652097
$686$	0	0
$687$	1.60693	0.0613081
$688$	0	0
$689$	−4.61150	−0.175684
$690$	0	0
$691$	−37.3600	−1.42124	−0.710621	−	0.703575i	$-0.751586\pi$
$691$	−37.3600	−1.42124	−0.710621	+	0.703575i	$0.751586\pi$
$692$	0	0
$693$	1.13162	0.0429867
$694$	0	0
$695$	42.0710	1.59584
$696$	0	0
$697$	42.4531	1.60803
$698$	0	0
$699$	0.536681	0.0202991
$700$	0	0
$701$	−47.7179	−1.80228	−0.901140	−	0.433527i	$-0.857269\pi$
$701$	−47.7179	−1.80228	−0.901140	+	0.433527i	$0.857269\pi$
$702$	0	0
$703$	33.4315	1.26089
$704$	0	0
$705$	6.23868	0.234962
$706$	0	0
$707$	5.58364	0.209994
$708$	0	0
$709$	38.8734	1.45992	0.729960	−	0.683490i	$-0.239539\pi$
$709$	38.8734	1.45992	0.729960	+	0.683490i	$0.239539\pi$
$710$	0	0
$711$	6.95290	0.260754
$712$	0	0
$713$	3.97475	0.148856
$714$	0	0
$715$	6.29565	0.235444
$716$	0	0
$717$	−22.8313	−0.852650
$718$	0	0
$719$	−13.9068	−0.518635	−0.259317	−	0.965792i	$-0.583498\pi$
$719$	−13.9068	−0.518635	−0.259317	+	0.965792i	$0.583498\pi$
$720$	0	0
$721$	−3.10286	−0.115557
$722$	0	0
$723$	18.1389	0.674592
$724$	0	0
$725$	1.02586	0.0380996
$726$	0	0
$727$	6.41680	0.237986	0.118993	−	0.992895i	$-0.462033\pi$
$727$	6.41680	0.237986	0.118993	+	0.992895i	$0.462033\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−37.4184	−1.38397
$732$	0	0
$733$	50.6778	1.87183	0.935913	−	0.352230i	$-0.114577\pi$
$733$	50.6778	1.87183	0.935913	+	0.352230i	$0.114577\pi$
$734$	0	0
$735$	−16.8907	−0.623024
$736$	0	0
$737$	−25.3377	−0.933328
$738$	0	0
$739$	−20.1084	−0.739701	−0.369850	−	0.929091i	$-0.620591\pi$
$739$	−20.1084	−0.739701	−0.369850	+	0.929091i	$0.620591\pi$
$740$	0	0
$741$	2.23637	0.0821551
$742$	0	0
$743$	−5.08409	−0.186517	−0.0932585	−	0.995642i	$-0.529728\pi$
$743$	−5.08409	−0.186517	−0.0932585	+	0.995642i	$0.529728\pi$
$744$	0	0
$745$	−29.5646	−1.08316
$746$	0	0
$747$	−13.5079	−0.494230
$748$	0	0
$749$	4.87371	0.178082
$750$	0	0
$751$	15.2826	0.557668	0.278834	−	0.960339i	$-0.410052\pi$
$751$	15.2826	0.557668	0.278834	+	0.960339i	$0.410052\pi$
$752$	0	0
$753$	0.676745	0.0246620
$754$	0	0
$755$	34.9069	1.27039
$756$	0	0
$757$	−27.2779	−0.991431	−0.495716	−	0.868485i	$-0.665094\pi$
$757$	−27.2779	−0.991431	−0.495716	+	0.868485i	$0.665094\pi$
$758$	0	0
$759$	−3.27763	−0.118970
$760$	0	0
$761$	49.3878	1.79030	0.895152	−	0.445760i	$-0.147067\pi$
$761$	49.3878	1.79030	0.895152	+	0.445760i	$0.147067\pi$
$762$	0	0
$763$	−3.17305	−0.114872
$764$	0	0
$765$	10.7128	0.387323
$766$	0	0
$767$	−3.44744	−0.124480
$768$	0	0
$769$	53.2550	1.92042	0.960212	−	0.279274i	$-0.0900937\pi$
$769$	53.2550	1.92042	0.960212	+	0.279274i	$0.0900937\pi$
$770$	0	0
$771$	−6.12135	−0.220455
$772$	0	0
$773$	−22.9581	−0.825746	−0.412873	−	0.910789i	$-0.635475\pi$
$773$	−22.9581	−0.825746	−0.412873	+	0.910789i	$0.635475\pi$
$774$	0	0
$775$	−4.07755	−0.146470
$776$	0	0
$777$	4.03853	0.144881
$778$	0	0
$779$	27.8027	0.996136
$780$	0	0
$781$	4.69663	0.168059
$782$	0	0
$783$	1.00000	0.0357371
$784$	0	0
$785$	−31.3757	−1.11985
$786$	0	0
$787$	26.4835	0.944034	0.472017	−	0.881589i	$-0.343526\pi$
$787$	26.4835	0.944034	0.472017	+	0.881589i	$0.343526\pi$
$788$	0	0
$789$	30.0720	1.07059
$790$	0	0
$791$	1.76964	0.0629211
$792$	0	0
$793$	10.6498	0.378185
$794$	0	0
$795$	−14.4671	−0.513095
$796$	0	0
$797$	13.0120	0.460907	0.230454	−	0.973083i	$-0.425979\pi$
$797$	13.0120	0.460907	0.230454	+	0.973083i	$0.425979\pi$
$798$	0	0
$799$	11.0912	0.392378
$800$	0	0
$801$	6.69015	0.236385
$802$	0	0
$803$	−32.7632	−1.15619
$804$	0	0
$805$	−0.847521	−0.0298712
$806$	0	0
$807$	19.3489	0.681113
$808$	0	0
$809$	−14.0247	−0.493083	−0.246542	−	0.969132i	$-0.579294\pi$
$809$	−14.0247	−0.493083	−0.246542	+	0.969132i	$0.579294\pi$
$810$	0	0
$811$	4.64964	0.163271	0.0816355	−	0.996662i	$-0.473986\pi$
$811$	4.64964	0.163271	0.0816355	+	0.996662i	$0.473986\pi$
$812$	0	0
$813$	−4.57531	−0.160463
$814$	0	0
$815$	−27.5478	−0.964959
$816$	0	0
$817$	−24.5055	−0.857339
$818$	0	0
$819$	0.270154	0.00943994
$820$	0	0
$821$	−43.7912	−1.52832	−0.764161	−	0.645025i	$-0.776847\pi$
$821$	−43.7912	−1.52832	−0.764161	+	0.645025i	$0.776847\pi$
$822$	0	0
$823$	−34.7343	−1.21076	−0.605380	−	0.795936i	$-0.706979\pi$
$823$	−34.7343	−1.21076	−0.605380	+	0.795936i	$0.706979\pi$
$824$	0	0
$825$	3.36240	0.117064
$826$	0	0
$827$	9.17001	0.318873	0.159436	−	0.987208i	$-0.449032\pi$
$827$	9.17001	0.318873	0.159436	+	0.987208i	$0.449032\pi$
$828$	0	0
$829$	−24.9106	−0.865182	−0.432591	−	0.901590i	$-0.642401\pi$
$829$	−24.9106	−0.865182	−0.432591	+	0.901590i	$0.642401\pi$
$830$	0	0
$831$	30.6791	1.06424
$832$	0	0
$833$	−30.0285	−1.04043
$834$	0	0
$835$	−57.7070	−1.99703
$836$	0	0
$837$	−3.97475	−0.137387
$838$	0	0
$839$	1.09625	0.0378466	0.0189233	−	0.999821i	$-0.493976\pi$
$839$	1.09625	0.0378466	0.0189233	+	0.999821i	$0.493976\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−10.8430	−0.373454
$844$	0	0
$845$	−30.4090	−1.04610
$846$	0	0
$847$	−0.0887790	−0.00305048
$848$	0	0
$849$	−16.7742	−0.575689
$850$	0	0
$851$	−11.6972	−0.400975
$852$	0	0
$853$	−9.63447	−0.329878	−0.164939	−	0.986304i	$-0.552743\pi$
$853$	−9.63447	−0.329878	−0.164939	+	0.986304i	$0.552743\pi$
$854$	0	0
$855$	7.01589	0.239938
$856$	0	0
$857$	1.79247	0.0612296	0.0306148	−	0.999531i	$-0.490253\pi$
$857$	1.79247	0.0612296	0.0306148	+	0.999531i	$0.490253\pi$
$858$	0	0
$859$	38.1729	1.30244	0.651221	−	0.758888i	$-0.274257\pi$
$859$	38.1729	1.30244	0.651221	+	0.758888i	$0.274257\pi$
$860$	0	0
$861$	3.35857	0.114460
$862$	0	0
$863$	−1.37282	−0.0467314	−0.0233657	−	0.999727i	$-0.507438\pi$
$863$	−1.37282	−0.0467314	−0.0233657	+	0.999727i	$0.507438\pi$
$864$	0	0
$865$	−28.6236	−0.973232
$866$	0	0
$867$	2.04538	0.0694649
$868$	0	0
$869$	22.7890	0.773065
$870$	0	0
$871$	−6.04893	−0.204960
$872$	0	0
$873$	−6.85116	−0.231876
$874$	0	0
$875$	−3.36816	−0.113865
$876$	0	0
$877$	36.5435	1.23399	0.616994	−	0.786968i	$-0.288350\pi$
$877$	36.5435	1.23399	0.616994	+	0.786968i	$0.288350\pi$
$878$	0	0
$879$	5.94969	0.200678
$880$	0	0
$881$	−8.91852	−0.300473	−0.150236	−	0.988650i	$-0.548003\pi$
$881$	−8.91852	−0.300473	−0.150236	+	0.988650i	$0.548003\pi$
$882$	0	0
$883$	12.9741	0.436614	0.218307	−	0.975880i	$-0.429947\pi$
$883$	12.9741	0.436614	0.218307	+	0.975880i	$0.429947\pi$
$884$	0	0
$885$	−10.8152	−0.363550
$886$	0	0
$887$	−43.0336	−1.44493	−0.722464	−	0.691409i	$-0.756990\pi$
$887$	−43.0336	−1.44493	−0.722464	+	0.691409i	$0.756990\pi$
$888$	0	0
$889$	−2.43648	−0.0817170
$890$	0	0
$891$	3.27763	0.109805
$892$	0	0
$893$	7.26367	0.243070
$894$	0	0
$895$	−9.62381	−0.321689
$896$	0	0
$897$	−0.782475	−0.0261261
$898$	0	0
$899$	−3.97475	−0.132565
$900$	0	0
$901$	−25.7197	−0.856849
$902$	0	0
$903$	−2.96027	−0.0985116
$904$	0	0
$905$	−20.3943	−0.677929
$906$	0	0
$907$	−3.74838	−0.124463	−0.0622315	−	0.998062i	$-0.519822\pi$
$907$	−3.74838	−0.124463	−0.0622315	+	0.998062i	$0.519822\pi$
$908$	0	0
$909$	16.1725	0.536407
$910$	0	0
$911$	−0.775631	−0.0256978	−0.0128489	−	0.999917i	$-0.504090\pi$
$911$	−0.775631	−0.0256978	−0.0128489	+	0.999917i	$0.504090\pi$
$912$	0	0
$913$	−44.2740	−1.46526
$914$	0	0
$915$	33.4102	1.10451
$916$	0	0
$917$	4.88681	0.161377
$918$	0	0
$919$	31.8477	1.05056	0.525280	−	0.850929i	$-0.323961\pi$
$919$	31.8477	1.05056	0.525280	+	0.850929i	$0.323961\pi$
$920$	0	0
$921$	−12.9321	−0.426128
$922$	0	0
$923$	1.12124	0.0369059
$924$	0	0
$925$	11.9997	0.394549
$926$	0	0
$927$	−8.98714	−0.295177
$928$	0	0
$929$	−34.1424	−1.12018	−0.560089	−	0.828433i	$-0.689233\pi$
$929$	−34.1424	−1.12018	−0.560089	+	0.828433i	$0.689233\pi$
$930$	0	0
$931$	−19.6658	−0.644521
$932$	0	0
$933$	25.9767	0.850440
$934$	0	0
$935$	35.1127	1.14831
$936$	0	0
$937$	37.7418	1.23297	0.616485	−	0.787367i	$-0.288556\pi$
$937$	37.7418	1.23297	0.616485	+	0.787367i	$0.288556\pi$
$938$	0	0
$939$	22.5617	0.736273
$940$	0	0
$941$	29.1854	0.951415	0.475708	−	0.879603i	$-0.342192\pi$
$941$	29.1854	0.951415	0.475708	+	0.879603i	$0.342192\pi$
$942$	0	0
$943$	−9.72779	−0.316781
$944$	0	0
$945$	0.847521	0.0275698
$946$	0	0
$947$	21.6616	0.703906	0.351953	−	0.936018i	$-0.385518\pi$
$947$	21.6616	0.703906	0.351953	+	0.936018i	$0.385518\pi$
$948$	0	0
$949$	−7.82163	−0.253901
$950$	0	0
$951$	−25.2686	−0.819390
$952$	0	0
$953$	−37.0017	−1.19860	−0.599302	−	0.800523i	$-0.704555\pi$
$953$	−37.0017	−1.19860	−0.599302	+	0.800523i	$0.704555\pi$
$954$	0	0
$955$	51.9141	1.67990
$956$	0	0
$957$	3.27763	0.105951
$958$	0	0
$959$	−2.40043	−0.0775138
$960$	0	0
$961$	−15.2014	−0.490366
$962$	0	0
$963$	14.1162	0.454890
$964$	0	0
$965$	48.4929	1.56104
$966$	0	0
$967$	25.3479	0.815134	0.407567	−	0.913175i	$-0.366377\pi$
$967$	25.3479	0.815134	0.407567	+	0.913175i	$0.366377\pi$
$968$	0	0
$969$	12.4729	0.400688
$970$	0	0
$971$	−7.41861	−0.238075	−0.119037	−	0.992890i	$-0.537981\pi$
$971$	−7.41861	−0.238075	−0.119037	+	0.992890i	$0.537981\pi$
$972$	0	0
$973$	5.91716	0.189695
$974$	0	0
$975$	0.802712	0.0257074
$976$	0	0
$977$	41.4351	1.32563	0.662813	−	0.748785i	$-0.269362\pi$
$977$	41.4351	1.32563	0.662813	+	0.748785i	$0.269362\pi$
$978$	0	0
$979$	21.9278	0.700817
$980$	0	0
$981$	−9.19043	−0.293428
$982$	0	0
$983$	−0.851469	−0.0271576	−0.0135788	−	0.999908i	$-0.504322\pi$
$983$	−0.851469	−0.0271576	−0.0135788	+	0.999908i	$0.504322\pi$
$984$	0	0
$985$	3.81342	0.121506
$986$	0	0
$987$	0.877453	0.0279296
$988$	0	0
$989$	8.57414	0.272642
$990$	0	0
$991$	54.7072	1.73783	0.868915	−	0.494961i	$-0.164817\pi$
$991$	54.7072	1.73783	0.868915	+	0.494961i	$0.164817\pi$
$992$	0	0
$993$	23.8716	0.757543
$994$	0	0
$995$	−4.59470	−0.145662
$996$	0	0
$997$	−26.5210	−0.839929	−0.419964	−	0.907541i	$-0.637957\pi$
$997$	−26.5210	−0.839929	−0.419964	+	0.907541i	$0.637957\pi$
$998$	0	0
$999$	11.6972	0.370083

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.k.1.14	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.k.1.14	✓	18	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +2.45476 q^{5} +0.345256 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +2.45476 q^{5} +0.345256 q^{7} +1.00000 q^{9} +3.27763 q^{11} +0.782475 q^{13} +2.45476 q^{15} +4.36410 q^{17} +2.85807 q^{19} +0.345256 q^{21} -1.00000 q^{23} +1.02586 q^{25} +1.00000 q^{27} +1.00000 q^{29} -3.97475 q^{31} +3.27763 q^{33} +0.847521 q^{35} +11.6972 q^{37} +0.782475 q^{39} +9.72779 q^{41} -8.57414 q^{43} +2.45476 q^{45} +2.54146 q^{47} -6.88080 q^{49} +4.36410 q^{51} -5.89348 q^{53} +8.04581 q^{55} +2.85807 q^{57} -4.40581 q^{59} +13.6104 q^{61} +0.345256 q^{63} +1.92079 q^{65} -7.73051 q^{67} -1.00000 q^{69} +1.43293 q^{71} -9.99601 q^{73} +1.02586 q^{75} +1.13162 q^{77} +6.95290 q^{79} +1.00000 q^{81} -13.5079 q^{83} +10.7128 q^{85} +1.00000 q^{87} +6.69015 q^{89} +0.270154 q^{91} -3.97475 q^{93} +7.01589 q^{95} -6.85116 q^{97} +3.27763 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 18 q^{3} + 5 q^{5} + 6 q^{7} + 18 q^{9}+O(q^{10}) \) 18 * q + 18 * q^3 + 5 * q^5 + 6 * q^7 + 18 * q^9	\( 18 q + 18 q^{3} + 5 q^{5} + 6 q^{7} + 18 q^{9} + 5 q^{11} + 6 q^{13} + 5 q^{15} + 7 q^{17} + 15 q^{19} + 6 q^{21} - 18 q^{23} + 45 q^{25} + 18 q^{27} + 18 q^{29} + 10 q^{31} + 5 q^{33} - 7 q^{35} + 22 q^{37} + 6 q^{39} + 17 q^{41} + 25 q^{43} + 5 q^{45} - 6 q^{47} + 64 q^{49} + 7 q^{51} + 21 q^{53} + 3 q^{55} + 15 q^{57} + 6 q^{59} + 7 q^{61} + 6 q^{63} + 44 q^{65} + 35 q^{67} - 18 q^{69} + q^{71} + 27 q^{73} + 45 q^{75} + 4 q^{77} + 18 q^{81} + 13 q^{83} + 24 q^{85} + 18 q^{87} + 30 q^{89} + 53 q^{91} + 10 q^{93} + 17 q^{95} + 27 q^{97} + 5 q^{99}+O(q^{100}) \) 18 * q + 18 * q^3 + 5 * q^5 + 6 * q^7 + 18 * q^9 + 5 * q^11 + 6 * q^13 + 5 * q^15 + 7 * q^17 + 15 * q^19 + 6 * q^21 - 18 * q^23 + 45 * q^25 + 18 * q^27 + 18 * q^29 + 10 * q^31 + 5 * q^33 - 7 * q^35 + 22 * q^37 + 6 * q^39 + 17 * q^41 + 25 * q^43 + 5 * q^45 - 6 * q^47 + 64 * q^49 + 7 * q^51 + 21 * q^53 + 3 * q^55 + 15 * q^57 + 6 * q^59 + 7 * q^61 + 6 * q^63 + 44 * q^65 + 35 * q^67 - 18 * q^69 + q^71 + 27 * q^73 + 45 * q^75 + 4 * q^77 + 18 * q^81 + 13 * q^83 + 24 * q^85 + 18 * q^87 + 30 * q^89 + 53 * q^91 + 10 * q^93 + 17 * q^95 + 27 * q^97 + 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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