LMFDB - Embedded newform 4005.2.a.n.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4005,2,Mod(1,4005)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4005, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4005.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4005 = 3^{2} \cdot 5 \cdot 89 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4005.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:	$31.9800860095$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.10407557.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 8x^{4} + 2x^{3} + 18x^{2} + 7x - 2 $ x^6 - x^5 - 8x^4 + 2x^3 + 18x^2 + 7x - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1335)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.09854$ of defining polynomial
Character	$\chi$	$=$	4005.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.305330 q^{2} -1.90677 q^{4} +1.00000 q^{5} +1.72660 q^{7} -1.19286 q^{8} +O(q^{10})$	$q+0.305330 q^{2} -1.90677 q^{4} +1.00000 q^{5} +1.72660 q^{7} -1.19286 q^{8} +0.305330 q^{10} -5.65487 q^{13} +0.527184 q^{14} +3.44933 q^{16} -4.85429 q^{17} +2.00000 q^{19} -1.90677 q^{20} +2.20917 q^{23} +1.00000 q^{25} -1.72660 q^{26} -3.29224 q^{28} -4.24628 q^{29} -1.45321 q^{31} +3.43890 q^{32} -1.48216 q^{34} +1.72660 q^{35} +6.85429 q^{37} +0.610660 q^{38} -1.19286 q^{40} +12.1911 q^{41} +4.03712 q^{43} +0.674526 q^{46} +6.69760 q^{47} -4.01884 q^{49} +0.305330 q^{50} +10.7826 q^{52} +10.7230 q^{53} -2.05959 q^{56} -1.29652 q^{58} -5.44570 q^{59} +2.63603 q^{61} -0.443707 q^{62} -5.84867 q^{64} -5.65487 q^{65} -15.1259 q^{67} +9.25603 q^{68} +0.527184 q^{70} +10.4496 q^{71} +8.92588 q^{73} +2.09282 q^{74} -3.81355 q^{76} -7.79231 q^{79} +3.44933 q^{80} +3.72232 q^{82} -1.24404 q^{83} -4.85429 q^{85} +1.23265 q^{86} +1.00000 q^{89} -9.76372 q^{91} -4.21239 q^{92} +2.04498 q^{94} +2.00000 q^{95} -12.7402 q^{97} -1.22707 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{2} + 8 q^{4} + 6 q^{5} + q^{7} + 9 q^{8}+O(q^{10}) $ 6 * q + 4 * q^2 + 8 * q^4 + 6 * q^5 + q^7 + 9 * q^8	$ 6 q + 4 q^{2} + 8 q^{4} + 6 q^{5} + q^{7} + 9 q^{8} + 4 q^{10} - 3 q^{13} + 5 q^{14} + 12 q^{16} + 13 q^{17} + 12 q^{19} + 8 q^{20} + 19 q^{23} + 6 q^{25} - q^{26} - 6 q^{28} + 10 q^{31} + 17 q^{32} - 2 q^{34} + q^{35} - q^{37} + 8 q^{38} + 9 q^{40} + 4 q^{41} - 7 q^{43} - 6 q^{46} + 15 q^{47} - q^{49} + 4 q^{50} + q^{52} + 27 q^{53} + 14 q^{56} - 6 q^{58} + 4 q^{59} + 8 q^{61} - 2 q^{62} - q^{64} - 3 q^{65} - 11 q^{67} + 47 q^{68} + 5 q^{70} + 16 q^{71} + q^{73} + 10 q^{74} + 16 q^{76} + 12 q^{80} + q^{82} + 17 q^{83} + 13 q^{85} + 20 q^{86} + 6 q^{89} - 18 q^{91} + 36 q^{92} + 17 q^{94} + 12 q^{95} - 29 q^{97} + 23 q^{98}+O(q^{100}) $ 6 * q + 4 * q^2 + 8 * q^4 + 6 * q^5 + q^7 + 9 * q^8 + 4 * q^10 - 3 * q^13 + 5 * q^14 + 12 * q^16 + 13 * q^17 + 12 * q^19 + 8 * q^20 + 19 * q^23 + 6 * q^25 - q^26 - 6 * q^28 + 10 * q^31 + 17 * q^32 - 2 * q^34 + q^35 - q^37 + 8 * q^38 + 9 * q^40 + 4 * q^41 - 7 * q^43 - 6 * q^46 + 15 * q^47 - q^49 + 4 * q^50 + q^52 + 27 * q^53 + 14 * q^56 - 6 * q^58 + 4 * q^59 + 8 * q^61 - 2 * q^62 - q^64 - 3 * q^65 - 11 * q^67 + 47 * q^68 + 5 * q^70 + 16 * q^71 + q^73 + 10 * q^74 + 16 * q^76 + 12 * q^80 + q^82 + 17 * q^83 + 13 * q^85 + 20 * q^86 + 6 * q^89 - 18 * q^91 + 36 * q^92 + 17 * q^94 + 12 * q^95 - 29 * q^97 + 23 * q^98

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 $

$2$	0.305330	0.215901	0.107950	−	0.994156i	$-0.465571\pi$
$2$	0.305330	0.215901	0.107950	+	0.994156i	$0.465571\pi$
$3$	0	0
$3$	0	0
$4$	−1.90677	−0.953387
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.72660	0.652594	0.326297	−	0.945267i	$-0.394199\pi$
$7$	1.72660	0.652594	0.326297	+	0.945267i	$0.394199\pi$
$8$	−1.19286	−0.421738
$9$	0	0
$10$	0.305330	0.0965539
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	−5.65487	−1.56838	−0.784190	−	0.620521i	$-0.786921\pi$
$13$	−5.65487	−1.56838	−0.784190	+	0.620521i	$0.786921\pi$
$14$	0.527184	0.140896
$15$	0	0
$16$	3.44933	0.862333
$17$	−4.85429	−1.17734	−0.588669	−	0.808374i	$-0.700348\pi$
$17$	−4.85429	−1.17734	−0.588669	+	0.808374i	$0.700348\pi$
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	−1.90677	−0.426368
$21$	0	0
$22$	0	0
$23$	2.20917	0.460644	0.230322	−	0.973115i	$-0.426022\pi$
$23$	2.20917	0.460644	0.230322	+	0.973115i	$0.426022\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−1.72660	−0.338615
$27$	0	0
$28$	−3.29224	−0.622175
$29$	−4.24628	−0.788515	−0.394258	−	0.919000i	$-0.628998\pi$
$29$	−4.24628	−0.788515	−0.394258	+	0.919000i	$0.628998\pi$
$30$	0	0
$31$	−1.45321	−0.261003	−0.130502	−	0.991448i	$-0.541659\pi$
$31$	−1.45321	−0.261003	−0.130502	+	0.991448i	$0.541659\pi$
$32$	3.43890	0.607917
$33$	0	0
$34$	−1.48216	−0.254189
$35$	1.72660	0.291849
$36$	0	0
$37$	6.85429	1.12684	0.563419	−	0.826171i	$-0.309486\pi$
$37$	6.85429	1.12684	0.563419	+	0.826171i	$0.309486\pi$
$38$	0.610660	0.0990622
$39$	0	0
$40$	−1.19286	−0.188607
$41$	12.1911	1.90394	0.951969	−	0.306195i	$-0.0990559\pi$
$41$	12.1911	1.90394	0.951969	+	0.306195i	$0.0990559\pi$
$42$	0	0
$43$	4.03712	0.615654	0.307827	−	0.951442i	$-0.400398\pi$
$43$	4.03712	0.615654	0.307827	+	0.951442i	$0.400398\pi$
$44$	0	0
$45$	0	0
$46$	0.674526	0.0994534
$47$	6.69760	0.976946	0.488473	−	0.872579i	$-0.337554\pi$
$47$	6.69760	0.976946	0.488473	+	0.872579i	$0.337554\pi$
$48$	0	0
$49$	−4.01884	−0.574120
$50$	0.305330	0.0431802
$51$	0	0
$52$	10.7826	1.49527
$53$	10.7230	1.47291	0.736457	−	0.676485i	$-0.236497\pi$
$53$	10.7230	1.47291	0.736457	+	0.676485i	$0.236497\pi$
$54$	0	0
$55$	0	0
$56$	−2.05959	−0.275224
$57$	0	0
$58$	−1.29652	−0.170241
$59$	−5.44570	−0.708970	−0.354485	−	0.935062i	$-0.615344\pi$
$59$	−5.44570	−0.708970	−0.354485	+	0.935062i	$0.615344\pi$
$60$	0	0
$61$	2.63603	0.337509	0.168754	−	0.985658i	$-0.446025\pi$
$61$	2.63603	0.337509	0.168754	+	0.985658i	$0.446025\pi$
$62$	−0.443707	−0.0563509
$63$	0	0
$64$	−5.84867	−0.731083
$65$	−5.65487	−0.701401
$66$	0	0
$67$	−15.1259	−1.84793	−0.923964	−	0.382479i	$-0.875070\pi$
$67$	−15.1259	−1.84793	−0.923964	+	0.382479i	$0.875070\pi$
$68$	9.25603	1.12246
$69$	0	0
$70$	0.527184	0.0630105
$71$	10.4496	1.24014	0.620068	−	0.784548i	$-0.287105\pi$
$71$	10.4496	1.24014	0.620068	+	0.784548i	$0.287105\pi$
$72$	0	0
$73$	8.92588	1.04469	0.522347	−	0.852733i	$-0.325056\pi$
$73$	8.92588	1.04469	0.522347	+	0.852733i	$0.325056\pi$
$74$	2.09282	0.243286
$75$	0	0
$76$	−3.81355	−0.437444
$77$	0	0
$78$	0	0
$79$	−7.79231	−0.876704	−0.438352	−	0.898803i	$-0.644438\pi$
$79$	−7.79231	−0.876704	−0.438352	+	0.898803i	$0.644438\pi$
$80$	3.44933	0.385647
$81$	0	0
$82$	3.72232	0.411062
$83$	−1.24404	−0.136551	−0.0682754	−	0.997667i	$-0.521750\pi$
$83$	−1.24404	−0.136551	−0.0682754	+	0.997667i	$0.521750\pi$
$84$	0	0
$85$	−4.85429	−0.526522
$86$	1.23265	0.132920
$87$	0	0
$88$	0	0
$89$	1.00000	0.106000
$89$	1.00000	0.106000
$90$	0	0
$91$	−9.76372	−1.02352
$92$	−4.21239	−0.439171
$93$	0	0
$94$	2.04498	0.210924
$95$	2.00000	0.205196
$96$	0	0
$97$	−12.7402	−1.29357	−0.646787	−	0.762670i	$-0.723888\pi$
$97$	−12.7402	−1.29357	−0.646787	+	0.762670i	$0.723888\pi$
$98$	−1.22707	−0.123953
$99$	0	0
$100$	−1.90677	−0.190677
$101$	3.37999	0.336322	0.168161	−	0.985760i	$-0.446217\pi$
$101$	3.37999	0.336322	0.168161	+	0.985760i	$0.446217\pi$
$102$	0	0
$103$	−7.83602	−0.772106	−0.386053	−	0.922477i	$-0.626162\pi$
$103$	−7.83602	−0.772106	−0.386053	+	0.922477i	$0.626162\pi$
$104$	6.74545	0.661445
$105$	0	0
$106$	3.27405	0.318004
$107$	19.7203	1.90644	0.953218	−	0.302283i	$-0.0977489\pi$
$107$	19.7203	1.90644	0.953218	+	0.302283i	$0.0977489\pi$
$108$	0	0
$109$	4.45708	0.426911	0.213455	−	0.976953i	$-0.431528\pi$
$109$	4.45708	0.426911	0.213455	+	0.976953i	$0.431528\pi$
$110$	0	0
$111$	0	0
$112$	5.95563	0.562754
$113$	19.4609	1.83073	0.915365	−	0.402626i	$-0.131902\pi$
$113$	19.4609	1.83073	0.915365	+	0.402626i	$0.131902\pi$
$114$	0	0
$115$	2.20917	0.206006
$116$	8.09670	0.751760
$117$	0	0
$118$	−1.66274	−0.153067
$119$	−8.38143	−0.768325
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0.804859	0.0728685
$123$	0	0
$124$	2.77093	0.248837
$125$	1.00000	0.0894427
$126$	0	0
$127$	9.59447	0.851372	0.425686	−	0.904871i	$-0.360033\pi$
$127$	9.59447	0.851372	0.425686	+	0.904871i	$0.360033\pi$
$128$	−8.66357	−0.765758
$129$	0	0
$130$	−1.72660	−0.151433
$131$	−11.5339	−1.00772	−0.503860	−	0.863785i	$-0.668087\pi$
$131$	−11.5339	−1.00772	−0.503860	+	0.863785i	$0.668087\pi$
$132$	0	0
$133$	3.45321	0.299431
$134$	−4.61841	−0.398970
$135$	0	0
$136$	5.79047	0.496529
$137$	13.4962	1.15306	0.576529	−	0.817077i	$-0.304407\pi$
$137$	13.4962	1.15306	0.576529	+	0.817077i	$0.304407\pi$
$138$	0	0
$139$	18.0809	1.53360	0.766802	−	0.641884i	$-0.221847\pi$
$139$	18.0809	1.53360	0.766802	+	0.641884i	$0.221847\pi$
$140$	−3.29224	−0.278245
$141$	0	0
$142$	3.19057	0.267747
$143$	0	0
$144$	0	0
$145$	−4.24628	−0.352635
$146$	2.72534	0.225551
$147$	0	0
$148$	−13.0696	−1.07431
$149$	−1.74784	−0.143188	−0.0715942	−	0.997434i	$-0.522809\pi$
$149$	−1.74784	−0.143188	−0.0715942	+	0.997434i	$0.522809\pi$
$150$	0	0
$151$	10.5922	0.861983	0.430992	−	0.902356i	$-0.358164\pi$
$151$	10.5922	0.861983	0.430992	+	0.902356i	$0.358164\pi$
$152$	−2.38571	−0.193507
$153$	0	0
$154$	0	0
$155$	−1.45321	−0.116724
$156$	0	0
$157$	23.9677	1.91283	0.956415	−	0.292012i	$-0.0943248\pi$
$157$	23.9677	1.91283	0.956415	+	0.292012i	$0.0943248\pi$
$158$	−2.37923	−0.189281
$159$	0	0
$160$	3.43890	0.271869
$161$	3.81436	0.300613
$162$	0	0
$163$	−4.15867	−0.325733	−0.162866	−	0.986648i	$-0.552074\pi$
$163$	−4.15867	−0.325733	−0.162866	+	0.986648i	$0.552074\pi$
$164$	−23.2458	−1.81519
$165$	0	0
$166$	−0.379842	−0.0294814
$167$	−1.63400	−0.126443	−0.0632213	−	0.998000i	$-0.520137\pi$
$167$	−1.63400	−0.126443	−0.0632213	+	0.998000i	$0.520137\pi$
$168$	0	0
$169$	18.9776	1.45981
$170$	−1.48216	−0.113677
$171$	0	0
$172$	−7.69787	−0.586957
$173$	15.5114	1.17931	0.589655	−	0.807655i	$-0.299264\pi$
$173$	15.5114	1.17931	0.589655	+	0.807655i	$0.299264\pi$
$174$	0	0
$175$	1.72660	0.130519
$176$	0	0
$177$	0	0
$178$	0.305330	0.0228855
$179$	−2.85567	−0.213443	−0.106721	−	0.994289i	$-0.534035\pi$
$179$	−2.85567	−0.213443	−0.106721	+	0.994289i	$0.534035\pi$
$180$	0	0
$181$	23.7540	1.76562	0.882811	−	0.469728i	$-0.155648\pi$
$181$	23.7540	1.76562	0.882811	+	0.469728i	$0.155648\pi$
$182$	−2.98116	−0.220978
$183$	0	0
$184$	−2.63522	−0.194271
$185$	6.85429	0.503938
$186$	0	0
$187$	0	0
$188$	−12.7708	−0.931407
$189$	0	0
$190$	0.610660	0.0443019
$191$	−0.965377	−0.0698522	−0.0349261	−	0.999390i	$-0.511120\pi$
$191$	−0.965377	−0.0698522	−0.0349261	+	0.999390i	$0.511120\pi$
$192$	0	0
$193$	−18.2139	−1.31106	−0.655531	−	0.755168i	$-0.727555\pi$
$193$	−18.2139	−1.31106	−0.655531	+	0.755168i	$0.727555\pi$
$194$	−3.88998	−0.279284
$195$	0	0
$196$	7.66302	0.547359
$197$	−8.20841	−0.584825	−0.292412	−	0.956292i	$-0.594458\pi$
$197$	−8.20841	−0.584825	−0.292412	+	0.956292i	$0.594458\pi$
$198$	0	0
$199$	0.570891	0.0404694	0.0202347	−	0.999795i	$-0.493559\pi$
$199$	0.570891	0.0404694	0.0202347	+	0.999795i	$0.493559\pi$
$200$	−1.19286	−0.0843476
$201$	0	0
$202$	1.03201	0.0726123
$203$	−7.33165	−0.514581
$204$	0	0
$205$	12.1911	0.851467
$206$	−2.39257	−0.166698
$207$	0	0
$208$	−19.5055	−1.35247
$209$	0	0
$210$	0	0
$211$	−0.499827	−0.0344095	−0.0172048	−	0.999852i	$-0.505477\pi$
$211$	−0.499827	−0.0344095	−0.0172048	+	0.999852i	$0.505477\pi$
$212$	−20.4463	−1.40426
$213$	0	0
$214$	6.02121	0.411601
$215$	4.03712	0.275329
$216$	0	0
$217$	−2.50911	−0.170329
$218$	1.36088	0.0921704
$219$	0	0
$220$	0	0
$221$	27.4504	1.84651
$222$	0	0
$223$	−17.0405	−1.14112	−0.570558	−	0.821258i	$-0.693273\pi$
$223$	−17.0405	−1.14112	−0.570558	+	0.821258i	$0.693273\pi$
$224$	5.93761	0.396723
$225$	0	0
$226$	5.94200	0.395256
$227$	15.8711	1.05341	0.526703	−	0.850050i	$-0.323428\pi$
$227$	15.8711	1.05341	0.526703	+	0.850050i	$0.323428\pi$
$228$	0	0
$229$	−10.9956	−0.726607	−0.363304	−	0.931671i	$-0.618351\pi$
$229$	−10.9956	−0.726607	−0.363304	+	0.931671i	$0.618351\pi$
$230$	0.674526	0.0444769
$231$	0	0
$232$	5.06520	0.332547
$233$	−1.74034	−0.114013	−0.0570066	−	0.998374i	$-0.518156\pi$
$233$	−1.74034	−0.114013	−0.0570066	+	0.998374i	$0.518156\pi$
$234$	0	0
$235$	6.69760	0.436904
$236$	10.3837	0.675923
$237$	0	0
$238$	−2.55910	−0.165882
$239$	−0.557608	−0.0360686	−0.0180343	−	0.999837i	$-0.505741\pi$
$239$	−0.557608	−0.0360686	−0.0180343	+	0.999837i	$0.505741\pi$
$240$	0	0
$241$	4.99106	0.321503	0.160751	−	0.986995i	$-0.448608\pi$
$241$	4.99106	0.321503	0.160751	+	0.986995i	$0.448608\pi$
$242$	−3.35863	−0.215901
$243$	0	0
$244$	−5.02631	−0.321777
$245$	−4.01884	−0.256754
$246$	0	0
$247$	−11.3097	−0.719622
$248$	1.73346	0.110075
$249$	0	0
$250$	0.305330	0.0193108
$251$	17.6661	1.11508	0.557538	−	0.830152i	$-0.311746\pi$
$251$	17.6661	1.11508	0.557538	+	0.830152i	$0.311746\pi$
$252$	0	0
$253$	0	0
$254$	2.92948	0.183812
$255$	0	0
$256$	9.05209	0.565755
$257$	14.7177	0.918063	0.459031	−	0.888420i	$-0.348197\pi$
$257$	14.7177	0.918063	0.459031	+	0.888420i	$0.348197\pi$
$258$	0	0
$259$	11.8346	0.735369
$260$	10.7826	0.668706
$261$	0	0
$262$	−3.52164	−0.217568
$263$	19.5574	1.20596	0.602979	−	0.797757i	$-0.293980\pi$
$263$	19.5574	1.20596	0.602979	+	0.797757i	$0.293980\pi$
$264$	0	0
$265$	10.7230	0.658707
$266$	1.05437	0.0646474
$267$	0	0
$268$	28.8418	1.76179
$269$	4.58906	0.279800	0.139900	−	0.990166i	$-0.455322\pi$
$269$	4.58906	0.279800	0.139900	+	0.990166i	$0.455322\pi$
$270$	0	0
$271$	24.4231	1.48360	0.741798	−	0.670623i	$-0.233973\pi$
$271$	24.4231	1.48360	0.741798	+	0.670623i	$0.233973\pi$
$272$	−16.7441	−1.01526
$273$	0	0
$274$	4.12080	0.248946
$275$	0	0
$276$	0	0
$277$	13.4224	0.806475	0.403238	−	0.915095i	$-0.367885\pi$
$277$	13.4224	0.806475	0.403238	+	0.915095i	$0.367885\pi$
$278$	5.52065	0.331107
$279$	0	0
$280$	−2.05959	−0.123084
$281$	6.41972	0.382968	0.191484	−	0.981496i	$-0.438670\pi$
$281$	6.41972	0.382968	0.191484	+	0.981496i	$0.438670\pi$
$282$	0	0
$283$	−2.36621	−0.140657	−0.0703283	−	0.997524i	$-0.522405\pi$
$283$	−2.36621	−0.140657	−0.0703283	+	0.997524i	$0.522405\pi$
$284$	−19.9250	−1.18233
$285$	0	0
$286$	0	0
$287$	21.0493	1.24250
$288$	0	0
$289$	6.56415	0.386126
$290$	−1.29652	−0.0761342
$291$	0	0
$292$	−17.0196	−0.995998
$293$	−7.50472	−0.438430	−0.219215	−	0.975677i	$-0.570350\pi$
$293$	−7.50472	−0.438430	−0.219215	+	0.975677i	$0.570350\pi$
$294$	0	0
$295$	−5.44570	−0.317061
$296$	−8.17618	−0.475231
$297$	0	0
$298$	−0.533668	−0.0309145
$299$	−12.4926	−0.722464
$300$	0	0
$301$	6.97049	0.401773
$302$	3.23413	0.186103
$303$	0	0
$304$	6.89866	0.395666
$305$	2.63603	0.150939
$306$	0	0
$307$	28.1750	1.60803	0.804016	−	0.594607i	$-0.202692\pi$
$307$	28.1750	1.60803	0.804016	+	0.594607i	$0.202692\pi$
$308$	0	0
$309$	0	0
$310$	−0.443707	−0.0252009
$311$	−9.72359	−0.551374	−0.275687	−	0.961247i	$-0.588905\pi$
$311$	−9.72359	−0.551374	−0.275687	+	0.961247i	$0.588905\pi$
$312$	0	0
$313$	−9.99781	−0.565110	−0.282555	−	0.959251i	$-0.591182\pi$
$313$	−9.99781	−0.565110	−0.282555	+	0.959251i	$0.591182\pi$
$314$	7.31805	0.412982
$315$	0	0
$316$	14.8582	0.835838
$317$	12.4125	0.697153	0.348577	−	0.937280i	$-0.386665\pi$
$317$	12.4125	0.697153	0.348577	+	0.937280i	$0.386665\pi$
$318$	0	0
$319$	0	0
$320$	−5.84867	−0.326950
$321$	0	0
$322$	1.16464	0.0649027
$323$	−9.70858	−0.540200
$324$	0	0
$325$	−5.65487	−0.313676
$326$	−1.26977	−0.0703260
$327$	0	0
$328$	−14.5423	−0.802963
$329$	11.5641	0.637550
$330$	0	0
$331$	−14.9059	−0.819302	−0.409651	−	0.912242i	$-0.634350\pi$
$331$	−14.9059	−0.819302	−0.409651	+	0.912242i	$0.634350\pi$
$332$	2.37210	0.130186
$333$	0	0
$334$	−0.498909	−0.0272991
$335$	−15.1259	−0.826419
$336$	0	0
$337$	−27.5719	−1.50194	−0.750969	−	0.660338i	$-0.770413\pi$
$337$	−27.5719	−1.50194	−0.750969	+	0.660338i	$0.770413\pi$
$338$	5.79443	0.315175
$339$	0	0
$340$	9.25603	0.501979
$341$	0	0
$342$	0	0
$343$	−19.0252	−1.02726
$344$	−4.81570	−0.259645
$345$	0	0
$346$	4.73610	0.254614
$347$	−13.5913	−0.729619	−0.364810	−	0.931082i	$-0.618866\pi$
$347$	−13.5913	−0.729619	−0.364810	+	0.931082i	$0.618866\pi$
$348$	0	0
$349$	20.6072	1.10308	0.551540	−	0.834148i	$-0.314040\pi$
$349$	20.6072	1.10308	0.551540	+	0.834148i	$0.314040\pi$
$350$	0.527184	0.0281792
$351$	0	0
$352$	0	0
$353$	28.5720	1.52073	0.760366	−	0.649495i	$-0.225019\pi$
$353$	28.5720	1.52073	0.760366	+	0.649495i	$0.225019\pi$
$354$	0	0
$355$	10.4496	0.554606
$356$	−1.90677	−0.101059
$357$	0	0
$358$	−0.871923	−0.0460825
$359$	−31.9654	−1.68707	−0.843536	−	0.537073i	$-0.819530\pi$
$359$	−31.9654	−1.68707	−0.843536	+	0.537073i	$0.819530\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	7.25282	0.381200
$363$	0	0
$364$	18.6172	0.975806
$365$	8.92588	0.467202
$366$	0	0
$367$	14.7461	0.769738	0.384869	−	0.922971i	$-0.374247\pi$
$367$	14.7461	0.769738	0.384869	+	0.922971i	$0.374247\pi$
$368$	7.62016	0.397228
$369$	0	0
$370$	2.09282	0.108801
$371$	18.5143	0.961215
$372$	0	0
$373$	−30.9903	−1.60462	−0.802309	−	0.596909i	$-0.796395\pi$
$373$	−30.9903	−1.60462	−0.802309	+	0.596909i	$0.796395\pi$
$374$	0	0
$375$	0	0
$376$	−7.98927	−0.412015
$377$	24.0122	1.23669
$378$	0	0
$379$	−10.8837	−0.559056	−0.279528	−	0.960138i	$-0.590178\pi$
$379$	−10.8837	−0.559056	−0.279528	+	0.960138i	$0.590178\pi$
$380$	−3.81355	−0.195631
$381$	0	0
$382$	−0.294759	−0.0150812
$383$	−29.7390	−1.51959	−0.759796	−	0.650161i	$-0.774701\pi$
$383$	−29.7390	−1.51959	−0.759796	+	0.650161i	$0.774701\pi$
$384$	0	0
$385$	0	0
$386$	−5.56124	−0.283060
$387$	0	0
$388$	24.2927	1.23328
$389$	23.2727	1.17997	0.589987	−	0.807413i	$-0.299133\pi$
$389$	23.2727	1.17997	0.589987	+	0.807413i	$0.299133\pi$
$390$	0	0
$391$	−10.7240	−0.542333
$392$	4.79390	0.242128
$393$	0	0
$394$	−2.50627	−0.126264
$395$	−7.79231	−0.392074
$396$	0	0
$397$	2.04070	0.102420	0.0512100	−	0.998688i	$-0.483692\pi$
$397$	2.04070	0.102420	0.0512100	+	0.998688i	$0.483692\pi$
$398$	0.174310	0.00873739
$399$	0	0
$400$	3.44933	0.172467
$401$	6.85372	0.342258	0.171129	−	0.985249i	$-0.445258\pi$
$401$	6.85372	0.342258	0.171129	+	0.985249i	$0.445258\pi$
$402$	0	0
$403$	8.21769	0.409352
$404$	−6.44488	−0.320645
$405$	0	0
$406$	−2.23857	−0.111098
$407$	0	0
$408$	0	0
$409$	13.6943	0.677139	0.338569	−	0.940941i	$-0.390057\pi$
$409$	13.6943	0.677139	0.338569	+	0.940941i	$0.390057\pi$
$410$	3.72232	0.183833
$411$	0	0
$412$	14.9415	0.736116
$413$	−9.40257	−0.462670
$414$	0	0
$415$	−1.24404	−0.0610673
$416$	−19.4465	−0.953444
$417$	0	0
$418$	0	0
$419$	30.4661	1.48837	0.744184	−	0.667975i	$-0.232839\pi$
$419$	30.4661	1.48837	0.744184	+	0.667975i	$0.232839\pi$
$420$	0	0
$421$	0.498495	0.0242952	0.0121476	−	0.999926i	$-0.496133\pi$
$421$	0.498495	0.0242952	0.0121476	+	0.999926i	$0.496133\pi$
$422$	−0.152612	−0.00742905
$423$	0	0
$424$	−12.7910	−0.621184
$425$	−4.85429	−0.235468
$426$	0	0
$427$	4.55138	0.220257
$428$	−37.6022	−1.81757
$429$	0	0
$430$	1.23265	0.0594438
$431$	−16.5536	−0.797358	−0.398679	−	0.917091i	$-0.630531\pi$
$431$	−16.5536	−0.797358	−0.398679	+	0.917091i	$0.630531\pi$
$432$	0	0
$433$	−16.1616	−0.776679	−0.388339	−	0.921516i	$-0.626951\pi$
$433$	−16.1616	−0.776679	−0.388339	+	0.921516i	$0.626951\pi$
$434$	−0.766106	−0.0367743
$435$	0	0
$436$	−8.49864	−0.407011
$437$	4.41834	0.211358
$438$	0	0
$439$	−21.4610	−1.02428	−0.512138	−	0.858903i	$-0.671146\pi$
$439$	−21.4610	−1.02428	−0.512138	+	0.858903i	$0.671146\pi$
$440$	0	0
$441$	0	0
$442$	8.38143	0.398664
$443$	−27.1412	−1.28952	−0.644759	−	0.764386i	$-0.723042\pi$
$443$	−27.1412	−1.28952	−0.644759	+	0.764386i	$0.723042\pi$
$444$	0	0
$445$	1.00000	0.0474045
$446$	−5.20297	−0.246368
$447$	0	0
$448$	−10.0983	−0.477101
$449$	−22.7284	−1.07262	−0.536311	−	0.844021i	$-0.680183\pi$
$449$	−22.7284	−1.07262	−0.536311	+	0.844021i	$0.680183\pi$
$450$	0	0
$451$	0	0
$452$	−37.1076	−1.74539
$453$	0	0
$454$	4.84594	0.227431
$455$	−9.76372	−0.457730
$456$	0	0
$457$	−29.3514	−1.37300	−0.686500	−	0.727129i	$-0.740854\pi$
$457$	−29.3514	−1.37300	−0.686500	+	0.727129i	$0.740854\pi$
$458$	−3.35728	−0.156875
$459$	0	0
$460$	−4.21239	−0.196403
$461$	−26.2538	−1.22276	−0.611382	−	0.791336i	$-0.709386\pi$
$461$	−26.2538	−1.22276	−0.611382	+	0.791336i	$0.709386\pi$
$462$	0	0
$463$	40.2087	1.86866	0.934328	−	0.356414i	$-0.116001\pi$
$463$	40.2087	1.86866	0.934328	+	0.356414i	$0.116001\pi$
$464$	−14.6468	−0.679963
$465$	0	0
$466$	−0.531377	−0.0246156
$467$	−6.25706	−0.289542	−0.144771	−	0.989465i	$-0.546245\pi$
$467$	−6.25706	−0.289542	−0.144771	+	0.989465i	$0.546245\pi$
$468$	0	0
$469$	−26.1165	−1.20595
$470$	2.04498	0.0943279
$471$	0	0
$472$	6.49594	0.299000
$473$	0	0
$474$	0	0
$475$	2.00000	0.0917663
$476$	15.9815	0.732511
$477$	0	0
$478$	−0.170254	−0.00778725
$479$	14.0734	0.643031	0.321516	−	0.946904i	$-0.395808\pi$
$479$	14.0734	0.643031	0.321516	+	0.946904i	$0.395808\pi$
$480$	0	0
$481$	−38.7601	−1.76731
$482$	1.52392	0.0694128
$483$	0	0
$484$	20.9745	0.953387
$485$	−12.7402	−0.578504
$486$	0	0
$487$	21.9314	0.993805	0.496903	−	0.867806i	$-0.334471\pi$
$487$	21.9314	0.993805	0.496903	+	0.867806i	$0.334471\pi$
$488$	−3.14440	−0.142340
$489$	0	0
$490$	−1.22707	−0.0554335
$491$	1.76081	0.0794641	0.0397320	−	0.999210i	$-0.487350\pi$
$491$	1.76081	0.0794641	0.0397320	+	0.999210i	$0.487350\pi$
$492$	0	0
$493$	20.6127	0.928349
$494$	−3.45321	−0.155367
$495$	0	0
$496$	−5.01259	−0.225072
$497$	18.0423	0.809306
$498$	0	0
$499$	12.6360	0.565667	0.282834	−	0.959169i	$-0.408726\pi$
$499$	12.6360	0.565667	0.282834	+	0.959169i	$0.408726\pi$
$500$	−1.90677	−0.0852735
$501$	0	0
$502$	5.39400	0.240746
$503$	30.5530	1.36229	0.681146	−	0.732148i	$-0.261482\pi$
$503$	30.5530	1.36229	0.681146	+	0.732148i	$0.261482\pi$
$504$	0	0
$505$	3.37999	0.150408
$506$	0	0
$507$	0	0
$508$	−18.2945	−0.811687
$509$	−12.8582	−0.569930	−0.284965	−	0.958538i	$-0.591982\pi$
$509$	−12.8582	−0.569930	−0.284965	+	0.958538i	$0.591982\pi$
$510$	0	0
$511$	15.4114	0.681762
$512$	20.0910	0.887905
$513$	0	0
$514$	4.49375	0.198211
$515$	−7.83602	−0.345296
$516$	0	0
$517$	0	0
$518$	3.61347	0.158767
$519$	0	0
$520$	6.74545	0.295807
$521$	−7.08985	−0.310612	−0.155306	−	0.987866i	$-0.549636\pi$
$521$	−7.08985	−0.310612	−0.155306	+	0.987866i	$0.549636\pi$
$522$	0	0
$523$	−1.19442	−0.0522281	−0.0261141	−	0.999659i	$-0.508313\pi$
$523$	−1.19442	−0.0522281	−0.0261141	+	0.999659i	$0.508313\pi$
$524$	21.9925	0.960747
$525$	0	0
$526$	5.97145	0.260367
$527$	7.05428	0.307289
$528$	0	0
$529$	−18.1196	−0.787807
$530$	3.27405	0.142215
$531$	0	0
$532$	−6.58448	−0.285473
$533$	−68.9394	−2.98610
$534$	0	0
$535$	19.7203	0.852584
$536$	18.0431	0.779342
$537$	0	0
$538$	1.40118	0.0604091
$539$	0	0
$540$	0	0
$541$	−35.6998	−1.53485	−0.767427	−	0.641136i	$-0.778463\pi$
$541$	−35.6998	−1.53485	−0.767427	+	0.641136i	$0.778463\pi$
$542$	7.45710	0.320310
$543$	0	0
$544$	−16.6934	−0.715724
$545$	4.45708	0.190920
$546$	0	0
$547$	30.0709	1.28574	0.642869	−	0.765976i	$-0.277744\pi$
$547$	30.0709	1.28574	0.642869	+	0.765976i	$0.277744\pi$
$548$	−25.7342	−1.09931
$549$	0	0
$550$	0	0
$551$	−8.49257	−0.361796
$552$	0	0
$553$	−13.4542	−0.572132
$554$	4.09827	0.174119
$555$	0	0
$556$	−34.4762	−1.46212
$557$	−27.8485	−1.17998	−0.589990	−	0.807411i	$-0.700868\pi$
$557$	−27.8485	−1.17998	−0.589990	+	0.807411i	$0.700868\pi$
$558$	0	0
$559$	−22.8294	−0.965580
$560$	5.95563	0.251671
$561$	0	0
$562$	1.96013	0.0826832
$563$	24.2214	1.02081	0.510406	−	0.859934i	$-0.329495\pi$
$563$	24.2214	1.02081	0.510406	+	0.859934i	$0.329495\pi$
$564$	0	0
$565$	19.4609	0.818727
$566$	−0.722475	−0.0303679
$567$	0	0
$568$	−12.4648	−0.523013
$569$	36.8881	1.54643	0.773214	−	0.634145i	$-0.218648\pi$
$569$	36.8881	1.54643	0.773214	+	0.634145i	$0.218648\pi$
$570$	0	0
$571$	14.5318	0.608137	0.304068	−	0.952650i	$-0.401655\pi$
$571$	14.5318	0.608137	0.304068	+	0.952650i	$0.401655\pi$
$572$	0	0
$573$	0	0
$574$	6.42698	0.268257
$575$	2.20917	0.0921287
$576$	0	0
$577$	−21.2370	−0.884108	−0.442054	−	0.896989i	$-0.645750\pi$
$577$	−21.2370	−0.884108	−0.442054	+	0.896989i	$0.645750\pi$
$578$	2.00423	0.0833650
$579$	0	0
$580$	8.09670	0.336197
$581$	−2.14796	−0.0891123
$582$	0	0
$583$	0	0
$584$	−10.6473	−0.440588
$585$	0	0
$586$	−2.29142	−0.0946576
$587$	−4.10553	−0.169453	−0.0847266	−	0.996404i	$-0.527002\pi$
$587$	−4.10553	−0.169453	−0.0847266	+	0.996404i	$0.527002\pi$
$588$	0	0
$589$	−2.90641	−0.119757
$590$	−1.66274	−0.0684538
$591$	0	0
$592$	23.6427	0.971710
$593$	36.5810	1.50220	0.751100	−	0.660188i	$-0.229524\pi$
$593$	36.5810	1.50220	0.751100	+	0.660188i	$0.229524\pi$
$594$	0	0
$595$	−8.38143	−0.343605
$596$	3.33273	0.136514
$597$	0	0
$598$	−3.81436	−0.155981
$599$	6.65701	0.271998	0.135999	−	0.990709i	$-0.456576\pi$
$599$	6.65701	0.271998	0.135999	+	0.990709i	$0.456576\pi$
$600$	0	0
$601$	−20.4469	−0.834048	−0.417024	−	0.908896i	$-0.636927\pi$
$601$	−20.4469	−0.834048	−0.417024	+	0.908896i	$0.636927\pi$
$602$	2.12830	0.0867431
$603$	0	0
$604$	−20.1970	−0.821803
$605$	−11.0000	−0.447214
$606$	0	0
$607$	24.6196	0.999279	0.499639	−	0.866234i	$-0.333466\pi$
$607$	24.6196	0.999279	0.499639	+	0.866234i	$0.333466\pi$
$608$	6.87779	0.278931
$609$	0	0
$610$	0.804859	0.0325878
$611$	−37.8741	−1.53222
$612$	0	0
$613$	−12.7442	−0.514734	−0.257367	−	0.966314i	$-0.582855\pi$
$613$	−12.7442	−0.514734	−0.257367	+	0.966314i	$0.582855\pi$
$614$	8.60268	0.347176
$615$	0	0
$616$	0	0
$617$	−6.16995	−0.248393	−0.124196	−	0.992258i	$-0.539635\pi$
$617$	−6.16995	−0.248393	−0.124196	+	0.992258i	$0.539635\pi$
$618$	0	0
$619$	−44.8662	−1.80332	−0.901662	−	0.432441i	$-0.857652\pi$
$619$	−44.8662	−1.80332	−0.901662	+	0.432441i	$0.857652\pi$
$620$	2.77093	0.111283
$621$	0	0
$622$	−2.96890	−0.119042
$623$	1.72660	0.0691749
$624$	0	0
$625$	1.00000	0.0400000
$626$	−3.05263	−0.122008
$627$	0	0
$628$	−45.7009	−1.82367
$629$	−33.2727	−1.32667
$630$	0	0
$631$	−37.8357	−1.50622	−0.753109	−	0.657896i	$-0.771447\pi$
$631$	−37.8357	−1.50622	−0.753109	+	0.657896i	$0.771447\pi$
$632$	9.29510	0.369739
$633$	0	0
$634$	3.78990	0.150516
$635$	9.59447	0.380745
$636$	0	0
$637$	22.7260	0.900439
$638$	0	0
$639$	0	0
$640$	−8.66357	−0.342458
$641$	−4.88660	−0.193009	−0.0965046	−	0.995333i	$-0.530766\pi$
$641$	−4.88660	−0.193009	−0.0965046	+	0.995333i	$0.530766\pi$
$642$	0	0
$643$	8.95931	0.353321	0.176660	−	0.984272i	$-0.443471\pi$
$643$	8.95931	0.353321	0.176660	+	0.984272i	$0.443471\pi$
$644$	−7.27312	−0.286601
$645$	0	0
$646$	−2.96432	−0.116630
$647$	9.99555	0.392966	0.196483	−	0.980507i	$-0.437048\pi$
$647$	9.99555	0.392966	0.196483	+	0.980507i	$0.437048\pi$
$648$	0	0
$649$	0	0
$650$	−1.72660	−0.0677229
$651$	0	0
$652$	7.92965	0.310549
$653$	31.6341	1.23794	0.618969	−	0.785416i	$-0.287551\pi$
$653$	31.6341	1.23794	0.618969	+	0.785416i	$0.287551\pi$
$654$	0	0
$655$	−11.5339	−0.450666
$656$	42.0513	1.64183
$657$	0	0
$658$	3.53087	0.137648
$659$	−47.3581	−1.84481	−0.922406	−	0.386223i	$-0.873780\pi$
$659$	−47.3581	−1.84481	−0.922406	+	0.386223i	$0.873780\pi$
$660$	0	0
$661$	6.59356	0.256460	0.128230	−	0.991744i	$-0.459070\pi$
$661$	6.59356	0.256460	0.128230	+	0.991744i	$0.459070\pi$
$662$	−4.55122	−0.176888
$663$	0	0
$664$	1.48396	0.0575886
$665$	3.45321	0.133910
$666$	0	0
$667$	−9.38076	−0.363224
$668$	3.11567	0.120549
$669$	0	0
$670$	−4.61841	−0.178425
$671$	0	0
$672$	0	0
$673$	−15.9243	−0.613836	−0.306918	−	0.951736i	$-0.599298\pi$
$673$	−15.9243	−0.613836	−0.306918	+	0.951736i	$0.599298\pi$
$674$	−8.41853	−0.324270
$675$	0	0
$676$	−36.1860	−1.39177
$677$	18.7969	0.722422	0.361211	−	0.932484i	$-0.382363\pi$
$677$	18.7969	0.722422	0.361211	+	0.932484i	$0.382363\pi$
$678$	0	0
$679$	−21.9973	−0.844180
$680$	5.79047	0.222054
$681$	0	0
$682$	0	0
$683$	30.7031	1.17482	0.587410	−	0.809290i	$-0.300148\pi$
$683$	30.7031	1.17482	0.587410	+	0.809290i	$0.300148\pi$
$684$	0	0
$685$	13.4962	0.515663
$686$	−5.80896	−0.221787
$687$	0	0
$688$	13.9254	0.530899
$689$	−60.6371	−2.31009
$690$	0	0
$691$	32.2966	1.22862	0.614310	−	0.789065i	$-0.289434\pi$
$691$	32.2966	1.22862	0.614310	+	0.789065i	$0.289434\pi$
$692$	−29.5768	−1.12434
$693$	0	0
$694$	−4.14983	−0.157526
$695$	18.0809	0.685848
$696$	0	0
$697$	−59.1794	−2.24158
$698$	6.29201	0.238156
$699$	0	0
$700$	−3.29224	−0.124435
$701$	5.96805	0.225410	0.112705	−	0.993628i	$-0.464048\pi$
$701$	5.96805	0.225410	0.112705	+	0.993628i	$0.464048\pi$
$702$	0	0
$703$	13.7086	0.517029
$704$	0	0
$705$	0	0
$706$	8.72388	0.328328
$707$	5.83591	0.219482
$708$	0	0
$709$	−17.8853	−0.671698	−0.335849	−	0.941916i	$-0.609023\pi$
$709$	−17.8853	−0.671698	−0.335849	+	0.941916i	$0.609023\pi$
$710$	3.19057	0.119740
$711$	0	0
$712$	−1.19286	−0.0447041
$713$	−3.21038	−0.120230
$714$	0	0
$715$	0	0
$716$	5.44512	0.203494
$717$	0	0
$718$	−9.76001	−0.364240
$719$	−29.1519	−1.08718	−0.543592	−	0.839350i	$-0.682936\pi$
$719$	−29.1519	−1.08718	−0.543592	+	0.839350i	$0.682936\pi$
$720$	0	0
$721$	−13.5297	−0.503872
$722$	−4.57995	−0.170448
$723$	0	0
$724$	−45.2935	−1.68332
$725$	−4.24628	−0.157703
$726$	0	0
$727$	18.7417	0.695090	0.347545	−	0.937663i	$-0.387015\pi$
$727$	18.7417	0.695090	0.347545	+	0.937663i	$0.387015\pi$
$728$	11.6467	0.431656
$729$	0	0
$730$	2.72534	0.100869
$731$	−19.5973	−0.724834
$732$	0	0
$733$	5.01143	0.185101	0.0925506	−	0.995708i	$-0.470498\pi$
$733$	5.01143	0.185101	0.0925506	+	0.995708i	$0.470498\pi$
$734$	4.50241	0.166187
$735$	0	0
$736$	7.59710	0.280033
$737$	0	0
$738$	0	0
$739$	24.1619	0.888811	0.444405	−	0.895826i	$-0.353415\pi$
$739$	24.1619	0.888811	0.444405	+	0.895826i	$0.353415\pi$
$740$	−13.0696	−0.480447
$741$	0	0
$742$	5.65298	0.207527
$743$	−30.6288	−1.12366	−0.561831	−	0.827252i	$-0.689903\pi$
$743$	−30.6288	−1.12366	−0.561831	+	0.827252i	$0.689903\pi$
$744$	0	0
$745$	−1.74784	−0.0640358
$746$	−9.46227	−0.346438
$747$	0	0
$748$	0	0
$749$	34.0492	1.24413
$750$	0	0
$751$	−26.6476	−0.972385	−0.486193	−	0.873852i	$-0.661615\pi$
$751$	−26.6476	−0.972385	−0.486193	+	0.873852i	$0.661615\pi$
$752$	23.1023	0.842453
$753$	0	0
$754$	7.33165	0.267003
$755$	10.5922	0.385491
$756$	0	0
$757$	−0.683774	−0.0248522	−0.0124261	−	0.999923i	$-0.503955\pi$
$757$	−0.683774	−0.0248522	−0.0124261	+	0.999923i	$0.503955\pi$
$758$	−3.32311	−0.120701
$759$	0	0
$760$	−2.38571	−0.0865388
$761$	−38.5529	−1.39754	−0.698771	−	0.715345i	$-0.746270\pi$
$761$	−38.5529	−1.39754	−0.698771	+	0.715345i	$0.746270\pi$
$762$	0	0
$763$	7.69560	0.278600
$764$	1.84075	0.0665962
$765$	0	0
$766$	−9.08021	−0.328081
$767$	30.7948	1.11193
$768$	0	0
$769$	46.1123	1.66285	0.831426	−	0.555635i	$-0.187525\pi$
$769$	46.1123	1.66285	0.831426	+	0.555635i	$0.187525\pi$
$770$	0	0
$771$	0	0
$772$	34.7297	1.24995
$773$	−20.1269	−0.723914	−0.361957	−	0.932195i	$-0.617891\pi$
$773$	−20.1269	−0.723914	−0.361957	+	0.932195i	$0.617891\pi$
$774$	0	0
$775$	−1.45321	−0.0522007
$776$	15.1973	0.545550
$777$	0	0
$778$	7.10586	0.254758
$779$	24.3823	0.873587
$780$	0	0
$781$	0	0
$782$	−3.27434	−0.117090
$783$	0	0
$784$	−13.8623	−0.495083
$785$	23.9677	0.855443
$786$	0	0
$787$	51.2380	1.82644	0.913219	−	0.407470i	$-0.133589\pi$
$787$	51.2380	1.82644	0.913219	+	0.407470i	$0.133589\pi$
$788$	15.6516	0.557564
$789$	0	0
$790$	−2.37923	−0.0846491
$791$	33.6013	1.19472
$792$	0	0
$793$	−14.9064	−0.529342
$794$	0.623088	0.0221126
$795$	0	0
$796$	−1.08856	−0.0385830
$797$	−13.0874	−0.463581	−0.231790	−	0.972766i	$-0.574458\pi$
$797$	−13.0874	−0.463581	−0.231790	+	0.972766i	$0.574458\pi$
$798$	0	0
$799$	−32.5121	−1.15020
$800$	3.43890	0.121583
$801$	0	0
$802$	2.09265	0.0738939
$803$	0	0
$804$	0	0
$805$	3.81436	0.134438
$806$	2.50911	0.0883796
$807$	0	0
$808$	−4.03185	−0.141840
$809$	−15.4249	−0.542311	−0.271156	−	0.962536i	$-0.587406\pi$
$809$	−15.4249	−0.542311	−0.271156	+	0.962536i	$0.587406\pi$
$810$	0	0
$811$	−0.782988	−0.0274944	−0.0137472	−	0.999906i	$-0.504376\pi$
$811$	−0.782988	−0.0274944	−0.0137472	+	0.999906i	$0.504376\pi$
$812$	13.9798	0.490594
$813$	0	0
$814$	0	0
$815$	−4.15867	−0.145672
$816$	0	0
$817$	8.07423	0.282482
$818$	4.18128	0.146195
$819$	0	0
$820$	−23.2458	−0.811777
$821$	−23.9952	−0.837439	−0.418719	−	0.908116i	$-0.637521\pi$
$821$	−23.9952	−0.837439	−0.418719	+	0.908116i	$0.637521\pi$
$822$	0	0
$823$	−1.65905	−0.0578309	−0.0289154	−	0.999582i	$-0.509205\pi$
$823$	−1.65905	−0.0578309	−0.0289154	+	0.999582i	$0.509205\pi$
$824$	9.34724	0.325626
$825$	0	0
$826$	−2.87089	−0.0998909
$827$	35.1712	1.22302	0.611511	−	0.791236i	$-0.290562\pi$
$827$	35.1712	1.22302	0.611511	+	0.791236i	$0.290562\pi$
$828$	0	0
$829$	−10.3798	−0.360506	−0.180253	−	0.983620i	$-0.557692\pi$
$829$	−10.3798	−0.360506	−0.180253	+	0.983620i	$0.557692\pi$
$830$	−0.379842	−0.0131845
$831$	0	0
$832$	33.0735	1.14662
$833$	19.5086	0.675934
$834$	0	0
$835$	−1.63400	−0.0565469
$836$	0	0
$837$	0	0
$838$	9.30223	0.321340
$839$	−17.7981	−0.614459	−0.307229	−	0.951635i	$-0.599402\pi$
$839$	−17.7981	−0.614459	−0.307229	+	0.951635i	$0.599402\pi$
$840$	0	0
$841$	−10.9691	−0.378244
$842$	0.152206	0.00524535
$843$	0	0
$844$	0.953057	0.0328056
$845$	18.9776	0.652849
$846$	0	0
$847$	−18.9926	−0.652594
$848$	36.9871	1.27014
$849$	0	0
$850$	−1.48216	−0.0508377
$851$	15.1423	0.519071
$852$	0	0
$853$	−37.3797	−1.27986	−0.639929	−	0.768434i	$-0.721036\pi$
$853$	−37.3797	−1.27986	−0.639929	+	0.768434i	$0.721036\pi$
$854$	1.38967	0.0475536
$855$	0	0
$856$	−23.5235	−0.804017
$857$	−23.8521	−0.814773	−0.407387	−	0.913256i	$-0.633560\pi$
$857$	−23.8521	−0.814773	−0.407387	+	0.913256i	$0.633560\pi$
$858$	0	0
$859$	−52.0985	−1.77758	−0.888790	−	0.458315i	$-0.848453\pi$
$859$	−52.0985	−1.77758	−0.888790	+	0.458315i	$0.848453\pi$
$860$	−7.69787	−0.262495
$861$	0	0
$862$	−5.05430	−0.172150
$863$	−23.3422	−0.794579	−0.397289	−	0.917693i	$-0.630049\pi$
$863$	−23.3422	−0.794579	−0.397289	+	0.917693i	$0.630049\pi$
$864$	0	0
$865$	15.5114	0.527404
$866$	−4.93463	−0.167686
$867$	0	0
$868$	4.78430	0.162390
$869$	0	0
$870$	0	0
$871$	85.5353	2.89825
$872$	−5.31665	−0.180045
$873$	0	0
$874$	1.34905	0.0456323
$875$	1.72660	0.0583698
$876$	0	0
$877$	−42.9373	−1.44989	−0.724944	−	0.688808i	$-0.758134\pi$
$877$	−42.9373	−1.44989	−0.724944	+	0.688808i	$0.758134\pi$
$878$	−6.55268	−0.221142
$879$	0	0
$880$	0	0
$881$	−22.0454	−0.742729	−0.371365	−	0.928487i	$-0.621110\pi$
$881$	−22.0454	−0.742729	−0.371365	+	0.928487i	$0.621110\pi$
$882$	0	0
$883$	26.7482	0.900148	0.450074	−	0.892991i	$-0.351398\pi$
$883$	26.7482	0.900148	0.450074	+	0.892991i	$0.351398\pi$
$884$	−52.3417	−1.76044
$885$	0	0
$886$	−8.28703	−0.278408
$887$	−37.6920	−1.26557	−0.632787	−	0.774326i	$-0.718089\pi$
$887$	−37.6920	−1.26557	−0.632787	+	0.774326i	$0.718089\pi$
$888$	0	0
$889$	16.5658	0.555601
$890$	0.305330	0.0102347
$891$	0	0
$892$	32.4923	1.08792
$893$	13.3952	0.448254
$894$	0	0
$895$	−2.85567	−0.0954546
$896$	−14.9585	−0.499730
$897$	0	0
$898$	−6.93968	−0.231580
$899$	6.17072	0.205805
$900$	0	0
$901$	−52.0524	−1.73412
$902$	0	0
$903$	0	0
$904$	−23.2141	−0.772088
$905$	23.7540	0.789610
$906$	0	0
$907$	15.2863	0.507572	0.253786	−	0.967260i	$-0.418324\pi$
$907$	15.2863	0.507572	0.253786	+	0.967260i	$0.418324\pi$
$908$	−30.2627	−1.00430
$909$	0	0
$910$	−2.98116	−0.0988244
$911$	5.72155	0.189563	0.0947817	−	0.995498i	$-0.469785\pi$
$911$	5.72155	0.189563	0.0947817	+	0.995498i	$0.469785\pi$
$912$	0	0
$913$	0	0
$914$	−8.96187	−0.296432
$915$	0	0
$916$	20.9660	0.692738
$917$	−19.9144	−0.657633
$918$	0	0
$919$	−20.0972	−0.662946	−0.331473	−	0.943465i	$-0.607546\pi$
$919$	−20.0972	−0.662946	−0.331473	+	0.943465i	$0.607546\pi$
$920$	−2.63522	−0.0868806
$921$	0	0
$922$	−8.01609	−0.263996
$923$	−59.0910	−1.94500
$924$	0	0
$925$	6.85429	0.225368
$926$	12.2769	0.403445
$927$	0	0
$928$	−14.6025	−0.479352
$929$	59.1700	1.94131	0.970653	−	0.240484i	$-0.0773063\pi$
$929$	59.1700	1.94131	0.970653	+	0.240484i	$0.0773063\pi$
$930$	0	0
$931$	−8.03769	−0.263425
$932$	3.31843	0.108699
$933$	0	0
$934$	−1.91047	−0.0625125
$935$	0	0
$936$	0	0
$937$	50.0704	1.63573	0.817864	−	0.575411i	$-0.195158\pi$
$937$	50.0704	1.63573	0.817864	+	0.575411i	$0.195158\pi$
$938$	−7.97415	−0.260365
$939$	0	0
$940$	−12.7708	−0.416538
$941$	25.3522	0.826459	0.413229	−	0.910627i	$-0.364401\pi$
$941$	25.3522	0.826459	0.413229	+	0.910627i	$0.364401\pi$
$942$	0	0
$943$	26.9323	0.877037
$944$	−18.7840	−0.611368
$945$	0	0
$946$	0	0
$947$	−12.9700	−0.421468	−0.210734	−	0.977543i	$-0.567585\pi$
$947$	−12.9700	−0.421468	−0.210734	+	0.977543i	$0.567585\pi$
$948$	0	0
$949$	−50.4747	−1.63848
$950$	0.610660	0.0198124
$951$	0	0
$952$	9.99784	0.324032
$953$	−26.1638	−0.847530	−0.423765	−	0.905772i	$-0.639292\pi$
$953$	−26.1638	−0.847530	−0.423765	+	0.905772i	$0.639292\pi$
$954$	0	0
$955$	−0.965377	−0.0312389
$956$	1.06323	0.0343874
$957$	0	0
$958$	4.29704	0.138831
$959$	23.3026	0.752479
$960$	0	0
$961$	−28.8882	−0.931877
$962$	−11.8346	−0.381564
$963$	0	0
$964$	−9.51683	−0.306516
$965$	−18.2139	−0.586325
$966$	0	0
$967$	25.1856	0.809913	0.404956	−	0.914336i	$-0.367287\pi$
$967$	25.1856	0.809913	0.404956	+	0.914336i	$0.367287\pi$
$968$	13.1214	0.421738
$969$	0	0
$970$	−3.88998	−0.124900
$971$	15.0629	0.483393	0.241696	−	0.970352i	$-0.422296\pi$
$971$	15.0629	0.483393	0.241696	+	0.970352i	$0.422296\pi$
$972$	0	0
$973$	31.2186	1.00082
$974$	6.69631	0.214563
$975$	0	0
$976$	9.09254	0.291045
$977$	13.9654	0.446793	0.223397	−	0.974728i	$-0.428285\pi$
$977$	13.9654	0.446793	0.223397	+	0.974728i	$0.428285\pi$
$978$	0	0
$979$	0	0
$980$	7.66302	0.244786
$981$	0	0
$982$	0.537627	0.0171564
$983$	44.4497	1.41773	0.708863	−	0.705347i	$-0.249209\pi$
$983$	44.4497	1.41773	0.708863	+	0.705347i	$0.249209\pi$
$984$	0	0
$985$	−8.20841	−0.261542
$986$	6.29368	0.200432
$987$	0	0
$988$	21.5651	0.686078
$989$	8.91867	0.283597
$990$	0	0
$991$	19.3926	0.616025	0.308013	−	0.951382i	$-0.400336\pi$
$991$	19.3926	0.616025	0.308013	+	0.951382i	$0.400336\pi$
$992$	−4.99742	−0.158668
$993$	0	0
$994$	5.50885	0.174730
$995$	0.570891	0.0180985
$996$	0	0
$997$	−51.4756	−1.63025	−0.815125	−	0.579285i	$-0.803332\pi$
$997$	−51.4756	−1.63025	−0.815125	+	0.579285i	$0.803332\pi$
$998$	3.85817	0.122128
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4005.2.a.n.1.3		6
3.2	odd	2		1335.2.a.g.1.4	✓	6
15.14	odd	2		6675.2.a.u.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1335.2.a.g.1.4	✓	6	3.2	odd	2
4005.2.a.n.1.3		6	1.1	even	1	trivial
6675.2.a.u.1.3		6	15.14	odd	2

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:	\( N \)	\(=\)	\( 4005 = 3^{2} \cdot 5 \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4005.a (trivial)

\(f(q)\)	\(=\)	\(q+0.305330 q^{2} -1.90677 q^{4} +1.00000 q^{5} +1.72660 q^{7} -1.19286 q^{8} +O(q^{10})\)	\(q+0.305330 q^{2} -1.90677 q^{4} +1.00000 q^{5} +1.72660 q^{7} -1.19286 q^{8} +0.305330 q^{10} -5.65487 q^{13} +0.527184 q^{14} +3.44933 q^{16} -4.85429 q^{17} +2.00000 q^{19} -1.90677 q^{20} +2.20917 q^{23} +1.00000 q^{25} -1.72660 q^{26} -3.29224 q^{28} -4.24628 q^{29} -1.45321 q^{31} +3.43890 q^{32} -1.48216 q^{34} +1.72660 q^{35} +6.85429 q^{37} +0.610660 q^{38} -1.19286 q^{40} +12.1911 q^{41} +4.03712 q^{43} +0.674526 q^{46} +6.69760 q^{47} -4.01884 q^{49} +0.305330 q^{50} +10.7826 q^{52} +10.7230 q^{53} -2.05959 q^{56} -1.29652 q^{58} -5.44570 q^{59} +2.63603 q^{61} -0.443707 q^{62} -5.84867 q^{64} -5.65487 q^{65} -15.1259 q^{67} +9.25603 q^{68} +0.527184 q^{70} +10.4496 q^{71} +8.92588 q^{73} +2.09282 q^{74} -3.81355 q^{76} -7.79231 q^{79} +3.44933 q^{80} +3.72232 q^{82} -1.24404 q^{83} -4.85429 q^{85} +1.23265 q^{86} +1.00000 q^{89} -9.76372 q^{91} -4.21239 q^{92} +2.04498 q^{94} +2.00000 q^{95} -12.7402 q^{97} -1.22707 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{2} + 8 q^{4} + 6 q^{5} + q^{7} + 9 q^{8}+O(q^{10}) \) 6 * q + 4 * q^2 + 8 * q^4 + 6 * q^5 + q^7 + 9 * q^8	\( 6 q + 4 q^{2} + 8 q^{4} + 6 q^{5} + q^{7} + 9 q^{8} + 4 q^{10} - 3 q^{13} + 5 q^{14} + 12 q^{16} + 13 q^{17} + 12 q^{19} + 8 q^{20} + 19 q^{23} + 6 q^{25} - q^{26} - 6 q^{28} + 10 q^{31} + 17 q^{32} - 2 q^{34} + q^{35} - q^{37} + 8 q^{38} + 9 q^{40} + 4 q^{41} - 7 q^{43} - 6 q^{46} + 15 q^{47} - q^{49} + 4 q^{50} + q^{52} + 27 q^{53} + 14 q^{56} - 6 q^{58} + 4 q^{59} + 8 q^{61} - 2 q^{62} - q^{64} - 3 q^{65} - 11 q^{67} + 47 q^{68} + 5 q^{70} + 16 q^{71} + q^{73} + 10 q^{74} + 16 q^{76} + 12 q^{80} + q^{82} + 17 q^{83} + 13 q^{85} + 20 q^{86} + 6 q^{89} - 18 q^{91} + 36 q^{92} + 17 q^{94} + 12 q^{95} - 29 q^{97} + 23 q^{98}+O(q^{100}) \) 6 * q + 4 * q^2 + 8 * q^4 + 6 * q^5 + q^7 + 9 * q^8 + 4 * q^10 - 3 * q^13 + 5 * q^14 + 12 * q^16 + 13 * q^17 + 12 * q^19 + 8 * q^20 + 19 * q^23 + 6 * q^25 - q^26 - 6 * q^28 + 10 * q^31 + 17 * q^32 - 2 * q^34 + q^35 - q^37 + 8 * q^38 + 9 * q^40 + 4 * q^41 - 7 * q^43 - 6 * q^46 + 15 * q^47 - q^49 + 4 * q^50 + q^52 + 27 * q^53 + 14 * q^56 - 6 * q^58 + 4 * q^59 + 8 * q^61 - 2 * q^62 - q^64 - 3 * q^65 - 11 * q^67 + 47 * q^68 + 5 * q^70 + 16 * q^71 + q^73 + 10 * q^74 + 16 * q^76 + 12 * q^80 + q^82 + 17 * q^83 + 13 * q^85 + 20 * q^86 + 6 * q^89 - 18 * q^91 + 36 * q^92 + 17 * q^94 + 12 * q^95 - 29 * q^97 + 23 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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