LMFDB - Embedded newform 1003.2.a.i.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1003,2,Mod(1,1003)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1003, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1003.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1003 = 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1003.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:	$8.00899532273$
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} - 16 x^{16} + 112 x^{15} + 52 x^{14} - 984 x^{13} + 431 x^{12} + 4304 x^{11} + \cdots + 28 $ x^18 - 5x^17 - 16x^16 + 112x^15 + 52x^14 - 984x^13 + 431x^12 + 4304x^11 - 3825x^10 - 9826x^9 + 11533x^8 + 11182x^7 - 15697x^6 - 5604x^5 + 9201x^4 + 1189x^3 - 1952x^2 - 100*x + 28
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-0.690432$ of defining polynomial
Character	$\chi$	$=$	1003.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.690432 q^{2} +1.58357 q^{3} -1.52330 q^{4} +3.69903 q^{5} -1.09335 q^{6} +2.15550 q^{7} +2.43260 q^{8} -0.492301 q^{9} +O(q^{10})$	\(q-0.690432 q^{2} +1.58357 q^{3} -1.52330 q^{4} +3.69903 q^{5} -1.09335 q^{6} +2.15550 q^{7} +2.43260 q^{8} -0.492301 q^{9} -2.55393 q^{10} -2.39016 q^{11} -2.41226 q^{12} +2.73136 q^{13} -1.48823 q^{14} +5.85768 q^{15} +1.36706 q^{16} +1.00000 q^{17} +0.339901 q^{18} -1.26135 q^{19} -5.63475 q^{20} +3.41339 q^{21} +1.65025 q^{22} +9.19362 q^{23} +3.85220 q^{24} +8.68283 q^{25} -1.88582 q^{26} -5.53031 q^{27} -3.28348 q^{28} +1.12396 q^{29} -4.04433 q^{30} -6.63399 q^{31} -5.80907 q^{32} -3.78499 q^{33} -0.690432 q^{34} +7.97325 q^{35} +0.749925 q^{36} +3.23871 q^{37} +0.870874 q^{38} +4.32530 q^{39} +8.99827 q^{40} +0.0108344 q^{41} -2.35671 q^{42} -0.666989 q^{43} +3.64094 q^{44} -1.82104 q^{45} -6.34757 q^{46} -8.17111 q^{47} +2.16484 q^{48} -2.35383 q^{49} -5.99491 q^{50} +1.58357 q^{51} -4.16069 q^{52} -10.0373 q^{53} +3.81830 q^{54} -8.84129 q^{55} +5.24347 q^{56} -1.99743 q^{57} -0.776020 q^{58} -1.00000 q^{59} -8.92302 q^{60} +11.5971 q^{61} +4.58032 q^{62} -1.06115 q^{63} +1.27664 q^{64} +10.1034 q^{65} +2.61328 q^{66} +6.43357 q^{67} -1.52330 q^{68} +14.5587 q^{69} -5.50499 q^{70} +4.64132 q^{71} -1.19757 q^{72} +14.9991 q^{73} -2.23611 q^{74} +13.7499 q^{75} +1.92141 q^{76} -5.15199 q^{77} -2.98633 q^{78} +7.75146 q^{79} +5.05680 q^{80} -7.28073 q^{81} -0.00748039 q^{82} +4.55202 q^{83} -5.19962 q^{84} +3.69903 q^{85} +0.460510 q^{86} +1.77988 q^{87} -5.81431 q^{88} +14.4648 q^{89} +1.25730 q^{90} +5.88744 q^{91} -14.0047 q^{92} -10.5054 q^{93} +5.64160 q^{94} -4.66576 q^{95} -9.19907 q^{96} -5.58407 q^{97} +1.62516 q^{98} +1.17668 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 5 q^{2} + q^{3} + 21 q^{4} + 21 q^{5} + 5 q^{6} - q^{7} + 9 q^{8} + 17 q^{9}+O(q^{10}) $ 18 * q + 5 * q^2 + q^3 + 21 * q^4 + 21 * q^5 + 5 * q^6 - q^7 + 9 * q^8 + 17 * q^9	\( 18 q + 5 q^{2} + q^{3} + 21 q^{4} + 21 q^{5} + 5 q^{6} - q^{7} + 9 q^{8} + 17 q^{9} + 10 q^{10} + 6 q^{11} + 20 q^{12} + 12 q^{13} + 9 q^{14} - q^{15} + 19 q^{16} + 18 q^{17} + 10 q^{18} - 14 q^{19} + 17 q^{20} - 14 q^{21} - 4 q^{22} + 7 q^{23} - 3 q^{24} + 35 q^{25} + 3 q^{26} + 13 q^{27} + q^{28} + 23 q^{29} + 17 q^{30} - q^{31} + 13 q^{32} + 7 q^{33} + 5 q^{34} + 5 q^{35} - 16 q^{36} + 36 q^{37} - 19 q^{38} + 8 q^{39} - 4 q^{40} + 37 q^{41} + 9 q^{42} - 15 q^{43} - 9 q^{44} + 17 q^{45} + q^{46} + 23 q^{47} + 17 q^{48} + 15 q^{49} + 32 q^{50} + q^{51} - 11 q^{52} + 35 q^{53} - 17 q^{54} - 6 q^{55} + 37 q^{56} - 16 q^{57} + 2 q^{58} - 18 q^{59} - 6 q^{60} + 39 q^{61} - q^{62} + 15 q^{63} + 5 q^{64} + 15 q^{65} - 36 q^{66} + 14 q^{67} + 21 q^{68} + 16 q^{69} - 45 q^{70} + 3 q^{71} - 71 q^{72} + 56 q^{73} - 27 q^{74} + 16 q^{75} - q^{76} + 39 q^{77} - 71 q^{78} - 22 q^{79} + 45 q^{80} - 18 q^{81} + 5 q^{82} + 12 q^{83} + 2 q^{84} + 21 q^{85} + 4 q^{86} + 8 q^{87} - 18 q^{88} + 34 q^{89} + 48 q^{90} + 3 q^{91} + 67 q^{92} - 19 q^{93} - 38 q^{94} - 72 q^{95} + 6 q^{96} + 31 q^{97} + 19 q^{98} + 44 q^{99}+O(q^{100}) \) 18 * q + 5 * q^2 + q^3 + 21 * q^4 + 21 * q^5 + 5 * q^6 - q^7 + 9 * q^8 + 17 * q^9 + 10 * q^10 + 6 * q^11 + 20 * q^12 + 12 * q^13 + 9 * q^14 - q^15 + 19 * q^16 + 18 * q^17 + 10 * q^18 - 14 * q^19 + 17 * q^20 - 14 * q^21 - 4 * q^22 + 7 * q^23 - 3 * q^24 + 35 * q^25 + 3 * q^26 + 13 * q^27 + q^28 + 23 * q^29 + 17 * q^30 - q^31 + 13 * q^32 + 7 * q^33 + 5 * q^34 + 5 * q^35 - 16 * q^36 + 36 * q^37 - 19 * q^38 + 8 * q^39 - 4 * q^40 + 37 * q^41 + 9 * q^42 - 15 * q^43 - 9 * q^44 + 17 * q^45 + q^46 + 23 * q^47 + 17 * q^48 + 15 * q^49 + 32 * q^50 + q^51 - 11 * q^52 + 35 * q^53 - 17 * q^54 - 6 * q^55 + 37 * q^56 - 16 * q^57 + 2 * q^58 - 18 * q^59 - 6 * q^60 + 39 * q^61 - q^62 + 15 * q^63 + 5 * q^64 + 15 * q^65 - 36 * q^66 + 14 * q^67 + 21 * q^68 + 16 * q^69 - 45 * q^70 + 3 * q^71 - 71 * q^72 + 56 * q^73 - 27 * q^74 + 16 * q^75 - q^76 + 39 * q^77 - 71 * q^78 - 22 * q^79 + 45 * q^80 - 18 * q^81 + 5 * q^82 + 12 * q^83 + 2 * q^84 + 21 * q^85 + 4 * q^86 + 8 * q^87 - 18 * q^88 + 34 * q^89 + 48 * q^90 + 3 * q^91 + 67 * q^92 - 19 * q^93 - 38 * q^94 - 72 * q^95 + 6 * q^96 + 31 * q^97 + 19 * q^98 + 44 * 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 $

$2$	−0.690432	−0.488209	−0.244105	−	0.969749i	$-0.578494\pi$
$2$	−0.690432	−0.488209	−0.244105	+	0.969749i	$0.578494\pi$
$3$	1.58357	0.914275	0.457138	−	0.889396i	$-0.348875\pi$
$3$	1.58357	0.914275	0.457138	+	0.889396i	$0.348875\pi$
$4$	−1.52330	−0.761652
$5$	3.69903	1.65426	0.827128	−	0.562013i	$-0.189973\pi$
$5$	3.69903	1.65426	0.827128	+	0.562013i	$0.189973\pi$
$6$	−1.09335	−0.446358
$7$	2.15550	0.814702	0.407351	−	0.913272i	$-0.366453\pi$
$7$	2.15550	0.814702	0.407351	+	0.913272i	$0.366453\pi$
$8$	2.43260	0.860055
$9$	−0.492301	−0.164100
$10$	−2.55393	−0.807624
$11$	−2.39016	−0.720661	−0.360331	−	0.932825i	$-0.617336\pi$
$11$	−2.39016	−0.720661	−0.360331	+	0.932825i	$0.617336\pi$
$12$	−2.41226	−0.696359
$13$	2.73136	0.757543	0.378771	−	0.925490i	$-0.376347\pi$
$13$	2.73136	0.757543	0.378771	+	0.925490i	$0.376347\pi$
$14$	−1.48823	−0.397745
$15$	5.85768	1.51245
$16$	1.36706	0.341765
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0.339901	0.0801154
$19$	−1.26135	−0.289373	−0.144686	−	0.989478i	$-0.546217\pi$
$19$	−1.26135	−0.289373	−0.144686	+	0.989478i	$0.546217\pi$
$20$	−5.63475	−1.25997
$21$	3.41339	0.744862
$22$	1.65025	0.351833
$23$	9.19362	1.91700	0.958501	−	0.285090i	$-0.0920236\pi$
$23$	9.19362	1.91700	0.958501	+	0.285090i	$0.0920236\pi$
$24$	3.85220	0.786327
$25$	8.68283	1.73657
$26$	−1.88582	−0.369839
$27$	−5.53031	−1.06431
$28$	−3.28348	−0.620519
$29$	1.12396	0.208715	0.104357	−	0.994540i	$-0.466721\pi$
$29$	1.12396	0.208715	0.104357	+	0.994540i	$0.466721\pi$
$30$	−4.04433	−0.738390
$31$	−6.63399	−1.19150	−0.595750	−	0.803170i	$-0.703145\pi$
$31$	−6.63399	−1.19150	−0.595750	+	0.803170i	$0.703145\pi$
$32$	−5.80907	−1.02691
$33$	−3.78499	−0.658883
$34$	−0.690432	−0.118408
$35$	7.97325	1.34773
$36$	0.749925	0.124987
$37$	3.23871	0.532441	0.266220	−	0.963912i	$-0.414225\pi$
$37$	3.23871	0.532441	0.266220	+	0.963912i	$0.414225\pi$
$38$	0.870874	0.141274
$39$	4.32530	0.692603
$40$	8.99827	1.42275
$41$	0.0108344	0.00169204	0.000846021	−	1.00000i	$-0.499731\pi$
$41$	0.0108344	0.00169204	0.000846021		1.00000i	$0.499731\pi$
$42$	−2.35671	−0.363648
$43$	−0.666989	−0.101715	−0.0508574	−	0.998706i	$-0.516195\pi$
$43$	−0.666989	−0.101715	−0.0508574	+	0.998706i	$0.516195\pi$
$44$	3.64094	0.548893
$45$	−1.82104	−0.271464
$46$	−6.34757	−0.935898
$47$	−8.17111	−1.19188	−0.595939	−	0.803029i	$-0.703220\pi$
$47$	−8.17111	−1.19188	−0.595939	+	0.803029i	$0.703220\pi$
$48$	2.16484	0.312467
$49$	−2.35383	−0.336261
$50$	−5.99491	−0.847808
$51$	1.58357	0.221744
$52$	−4.16069	−0.576984
$53$	−10.0373	−1.37873	−0.689363	−	0.724416i	$-0.742110\pi$
$53$	−10.0373	−1.37873	−0.689363	+	0.724416i	$0.742110\pi$
$54$	3.81830	0.519605
$55$	−8.84129	−1.19216
$56$	5.24347	0.700688
$57$	−1.99743	−0.264566
$58$	−0.776020	−0.101896
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	−8.92302	−1.15196
$61$	11.5971	1.48485	0.742427	−	0.669927i	$-0.233675\pi$
$61$	11.5971	1.48485	0.742427	+	0.669927i	$0.233675\pi$
$62$	4.58032	0.581701
$63$	−1.06115	−0.133693
$64$	1.27664	0.159581
$65$	10.1034	1.25317
$66$	2.61328	0.321673
$67$	6.43357	0.785986	0.392993	−	0.919542i	$-0.371440\pi$
$67$	6.43357	0.785986	0.392993	+	0.919542i	$0.371440\pi$
$68$	−1.52330	−0.184728
$69$	14.5587	1.75267
$70$	−5.50499	−0.657972
$71$	4.64132	0.550824	0.275412	−	0.961326i	$-0.411186\pi$
$71$	4.64132	0.550824	0.275412	+	0.961326i	$0.411186\pi$
$72$	−1.19757	−0.141135
$73$	14.9991	1.75552	0.877758	−	0.479104i	$-0.159038\pi$
$73$	14.9991	1.75552	0.877758	+	0.479104i	$0.159038\pi$
$74$	−2.23611	−0.259942
$75$	13.7499	1.58770
$76$	1.92141	0.220401
$77$	−5.15199	−0.587124
$78$	−2.98633	−0.338135
$79$	7.75146	0.872107	0.436054	−	0.899921i	$-0.356376\pi$
$79$	7.75146	0.872107	0.436054	+	0.899921i	$0.356376\pi$
$80$	5.05680	0.565367
$81$	−7.28073	−0.808971
$82$	−0.00748039	−0.000826071 0
$83$	4.55202	0.499649	0.249825	−	0.968291i	$-0.419627\pi$
$83$	4.55202	0.499649	0.249825	+	0.968291i	$0.419627\pi$
$84$	−5.19962	−0.567325
$85$	3.69903	0.401216
$86$	0.460510	0.0496581
$87$	1.77988	0.190823
$88$	−5.81431	−0.619808
$89$	14.4648	1.53327	0.766634	−	0.642084i	$-0.221930\pi$
$89$	14.4648	1.53327	0.766634	+	0.642084i	$0.221930\pi$
$90$	1.25730	0.132531
$91$	5.88744	0.617172
$92$	−14.0047	−1.46009
$93$	−10.5054	−1.08936
$94$	5.64160	0.581886
$95$	−4.66576	−0.478697
$96$	−9.19907	−0.938876
$97$	−5.58407	−0.566977	−0.283488	−	0.958976i	$-0.591492\pi$
$97$	−5.58407	−0.566977	−0.283488	+	0.958976i	$0.591492\pi$
$98$	1.62516	0.164166
$99$	1.17668	0.118261
$100$	−13.2266	−1.32266
$101$	−3.31002	−0.329360	−0.164680	−	0.986347i	$-0.552659\pi$
$101$	−3.31002	−0.329360	−0.164680	+	0.986347i	$0.552659\pi$
$102$	−1.09335	−0.108258
$103$	−0.921581	−0.0908060	−0.0454030	−	0.998969i	$-0.514457\pi$
$103$	−0.921581	−0.0908060	−0.0454030	+	0.998969i	$0.514457\pi$
$104$	6.64431	0.651528
$105$	12.6262	1.23219
$106$	6.93006	0.673107
$107$	−17.6170	−1.70310	−0.851551	−	0.524272i	$-0.824338\pi$
$107$	−17.6170	−1.70310	−0.851551	+	0.524272i	$0.824338\pi$
$108$	8.42434	0.810632
$109$	−17.6610	−1.69161	−0.845807	−	0.533488i	$-0.820881\pi$
$109$	−17.6610	−1.69161	−0.845807	+	0.533488i	$0.820881\pi$
$110$	6.10431	0.582023
$111$	5.12873	0.486797
$112$	2.94670	0.278437
$113$	−11.0773	−1.04206	−0.521032	−	0.853537i	$-0.674453\pi$
$113$	−11.0773	−1.04206	−0.521032	+	0.853537i	$0.674453\pi$
$114$	1.37909	0.129164
$115$	34.0075	3.17121
$116$	−1.71214	−0.158968
$117$	−1.34465	−0.124313
$118$	0.690432	0.0635594
$119$	2.15550	0.197594
$120$	14.2494	1.30079
$121$	−5.28712	−0.480647
$122$	−8.00699	−0.724919
$123$	0.0171570	0.00154699
$124$	10.1056	0.907508
$125$	13.6229	1.21847
$126$	0.732655	0.0652701
$127$	−9.32241	−0.827230	−0.413615	−	0.910452i	$-0.635734\pi$
$127$	−9.32241	−0.827230	−0.413615	+	0.910452i	$0.635734\pi$
$128$	10.7367	0.948999
$129$	−1.05622	−0.0929954
$130$	−6.97570	−0.611810
$131$	1.54023	0.134571	0.0672853	−	0.997734i	$-0.478566\pi$
$131$	1.54023	0.134571	0.0672853	+	0.997734i	$0.478566\pi$
$132$	5.76569	0.501839
$133$	−2.71883	−0.235752
$134$	−4.44194	−0.383725
$135$	−20.4568	−1.76064
$136$	2.43260	0.208594
$137$	15.4564	1.32053	0.660265	−	0.751032i	$-0.270444\pi$
$137$	15.4564	1.32053	0.660265	+	0.751032i	$0.270444\pi$
$138$	−10.0518	−0.855668
$139$	2.54992	0.216282	0.108141	−	0.994136i	$-0.465510\pi$
$139$	2.54992	0.216282	0.108141	+	0.994136i	$0.465510\pi$
$140$	−12.1457	−1.02650
$141$	−12.9395	−1.08971
$142$	−3.20452	−0.268917
$143$	−6.52840	−0.545932
$144$	−0.673006	−0.0560838
$145$	4.15758	0.345268
$146$	−10.3559	−0.857059
$147$	−3.72745	−0.307435
$148$	−4.93354	−0.405534
$149$	22.3300	1.82935	0.914674	−	0.404193i	$-0.132448\pi$
$149$	22.3300	1.82935	0.914674	+	0.404193i	$0.132448\pi$
$150$	−9.49336	−0.775130
$151$	−16.6616	−1.35590	−0.677950	−	0.735108i	$-0.737131\pi$
$151$	−16.6616	−1.35590	−0.677950	+	0.735108i	$0.737131\pi$
$152$	−3.06835	−0.248876
$153$	−0.492301	−0.0398002
$154$	3.55710	0.286639
$155$	−24.5393	−1.97105
$156$	−6.58875	−0.527522
$157$	11.2885	0.900923	0.450462	−	0.892796i	$-0.351259\pi$
$157$	11.2885	0.900923	0.450462	+	0.892796i	$0.351259\pi$
$158$	−5.35186	−0.425771
$159$	−15.8948	−1.26054
$160$	−21.4879	−1.69877
$161$	19.8168	1.56178
$162$	5.02685	0.394947
$163$	−3.28257	−0.257111	−0.128555	−	0.991702i	$-0.541034\pi$
$163$	−3.28257	−0.257111	−0.128555	+	0.991702i	$0.541034\pi$
$164$	−0.0165040	−0.00128875
$165$	−14.0008	−1.08996
$166$	−3.14286	−0.243933
$167$	−21.4924	−1.66313	−0.831566	−	0.555425i	$-0.812555\pi$
$167$	−21.4924	−1.66313	−0.831566	+	0.555425i	$0.812555\pi$
$168$	8.30341	0.640622
$169$	−5.53967	−0.426129
$170$	−2.55393	−0.195877
$171$	0.620962	0.0474862
$172$	1.01603	0.0774713
$173$	−12.8694	−0.978439	−0.489220	−	0.872161i	$-0.662718\pi$
$173$	−12.8694	−0.978439	−0.489220	+	0.872161i	$0.662718\pi$
$174$	−1.22888	−0.0931614
$175$	18.7158	1.41478
$176$	−3.26750	−0.246297
$177$	−1.58357	−0.119029
$178$	−9.98698	−0.748556
$179$	10.5152	0.785946	0.392973	−	0.919550i	$-0.371447\pi$
$179$	10.5152	0.785946	0.392973	+	0.919550i	$0.371447\pi$
$180$	2.77399	0.206761
$181$	−14.3238	−1.06468	−0.532341	−	0.846530i	$-0.678688\pi$
$181$	−14.3238	−1.06468	−0.532341	+	0.846530i	$0.678688\pi$
$182$	−4.06488	−0.301309
$183$	18.3648	1.35756
$184$	22.3644	1.64873
$185$	11.9801	0.880794
$186$	7.25326	0.531835
$187$	−2.39016	−0.174786
$188$	12.4471	0.907797
$189$	−11.9206	−0.867094
$190$	3.22139	0.233704
$191$	−2.83226	−0.204935	−0.102467	−	0.994736i	$-0.532674\pi$
$191$	−2.83226	−0.204935	−0.102467	+	0.994736i	$0.532674\pi$
$192$	2.02166	0.145901
$193$	−18.7090	−1.34670	−0.673350	−	0.739324i	$-0.735145\pi$
$193$	−18.7090	−1.34670	−0.673350	+	0.739324i	$0.735145\pi$
$194$	3.85542	0.276803
$195$	15.9994	1.14574
$196$	3.58559	0.256114
$197$	14.7354	1.04985	0.524927	−	0.851147i	$-0.324093\pi$
$197$	14.7354	1.04985	0.524927	+	0.851147i	$0.324093\pi$
$198$	−0.812418	−0.0577360
$199$	11.9578	0.847669	0.423834	−	0.905740i	$-0.360684\pi$
$199$	11.9578	0.847669	0.423834	+	0.905740i	$0.360684\pi$
$200$	21.1219	1.49354
$201$	10.1880	0.718607
$202$	2.28535	0.160796
$203$	2.42270	0.170040
$204$	−2.41226	−0.168892
$205$	0.0400766	0.00279907
$206$	0.636289	0.0443323
$207$	−4.52603	−0.314581
$208$	3.73393	0.258902
$209$	3.01482	0.208540
$210$	−8.71755	−0.601568
$211$	2.25649	0.155343	0.0776717	−	0.996979i	$-0.475251\pi$
$211$	2.25649	0.155343	0.0776717	+	0.996979i	$0.475251\pi$
$212$	15.2898	1.05011
$213$	7.34986	0.503604
$214$	12.1634	0.831470
$215$	−2.46721	−0.168262
$216$	−13.4530	−0.915363
$217$	−14.2995	−0.970716
$218$	12.1937	0.825862
$219$	23.7522	1.60503
$220$	13.4680	0.908010
$221$	2.73136	0.183731
$222$	−3.54104	−0.237659
$223$	−18.2313	−1.22086	−0.610428	−	0.792072i	$-0.709003\pi$
$223$	−18.2313	−1.22086	−0.610428	+	0.792072i	$0.709003\pi$
$224$	−12.5214	−0.836623
$225$	−4.27457	−0.284971
$226$	7.64812	0.508745
$227$	−3.86914	−0.256804	−0.128402	−	0.991722i	$-0.540985\pi$
$227$	−3.86914	−0.256804	−0.128402	+	0.991722i	$0.540985\pi$
$228$	3.04269	0.201507
$229$	−29.3404	−1.93887	−0.969434	−	0.245352i	$-0.921097\pi$
$229$	−29.3404	−1.93887	−0.969434	+	0.245352i	$0.921097\pi$
$230$	−23.4798	−1.54822
$231$	−8.15855	−0.536793
$232$	2.73416	0.179506
$233$	2.45543	0.160861	0.0804303	−	0.996760i	$-0.474371\pi$
$233$	2.45543	0.160861	0.0804303	+	0.996760i	$0.474371\pi$
$234$	0.928391	0.0606908
$235$	−30.2252	−1.97167
$236$	1.52330	0.0991586
$237$	12.2750	0.797346
$238$	−1.48823	−0.0964673
$239$	−23.2666	−1.50499	−0.752496	−	0.658597i	$-0.771150\pi$
$239$	−23.2666	−1.50499	−0.752496	+	0.658597i	$0.771150\pi$
$240$	8.00780	0.516901
$241$	−10.4832	−0.675283	−0.337642	−	0.941275i	$-0.609629\pi$
$241$	−10.4832	−0.675283	−0.337642	+	0.941275i	$0.609629\pi$
$242$	3.65040	0.234657
$243$	5.06136	0.324687
$244$	−17.6659	−1.13094
$245$	−8.70688	−0.556262
$246$	−0.0118457	−0.000755256 0
$247$	−3.44519	−0.219212
$248$	−16.1378	−1.02475
$249$	7.20845	0.456817
$250$	−9.40569	−0.594868
$251$	19.9522	1.25937	0.629685	−	0.776850i	$-0.283184\pi$
$251$	19.9522	1.25937	0.629685	+	0.776850i	$0.283184\pi$
$252$	1.61646	0.101827
$253$	−21.9742	−1.38151
$254$	6.43649	0.403861
$255$	5.85768	0.366822
$256$	−9.96625	−0.622891
$257$	−31.6278	−1.97289	−0.986443	−	0.164105i	$-0.947527\pi$
$257$	−31.6278	−1.97289	−0.986443	+	0.164105i	$0.947527\pi$
$258$	0.729251	0.0454012
$259$	6.98103	0.433780
$260$	−15.3905	−0.954480
$261$	−0.553329	−0.0342502
$262$	−1.06342	−0.0656986
$263$	1.85624	0.114460	0.0572302	−	0.998361i	$-0.481773\pi$
$263$	1.85624	0.114460	0.0572302	+	0.998361i	$0.481773\pi$
$264$	−9.20738	−0.566675
$265$	−37.1282	−2.28077
$266$	1.87717	0.115096
$267$	22.9061	1.40183
$268$	−9.80028	−0.598647
$269$	16.4682	1.00409	0.502043	−	0.864843i	$-0.332582\pi$
$269$	16.4682	1.00409	0.502043	+	0.864843i	$0.332582\pi$
$270$	14.1240	0.859561
$271$	−22.6667	−1.37691	−0.688453	−	0.725281i	$-0.741710\pi$
$271$	−22.6667	−1.37691	−0.688453	+	0.725281i	$0.741710\pi$
$272$	1.36706	0.0828902
$273$	9.32318	0.564265
$274$	−10.6716	−0.644695
$275$	−20.7534	−1.25148
$276$	−22.1774	−1.33492
$277$	9.48359	0.569814	0.284907	−	0.958555i	$-0.408037\pi$
$277$	9.48359	0.569814	0.284907	+	0.958555i	$0.408037\pi$
$278$	−1.76055	−0.105591
$279$	3.26592	0.195526
$280$	19.3958	1.15912
$281$	22.5580	1.34570	0.672849	−	0.739780i	$-0.265071\pi$
$281$	22.5580	1.34570	0.672849	+	0.739780i	$0.265071\pi$
$282$	8.93387	0.532004
$283$	−17.0400	−1.01292	−0.506461	−	0.862263i	$-0.669047\pi$
$283$	−17.0400	−1.01292	−0.506461	+	0.862263i	$0.669047\pi$
$284$	−7.07014	−0.419536
$285$	−7.38856	−0.437660
$286$	4.50741	0.266529
$287$	0.0233534	0.00137851
$288$	2.85981	0.168516
$289$	1.00000	0.0588235
$290$	−2.87052	−0.168563
$291$	−8.84278	−0.518373
$292$	−22.8482	−1.33709
$293$	−7.70128	−0.449914	−0.224957	−	0.974369i	$-0.572224\pi$
$293$	−7.70128	−0.449914	−0.224957	+	0.974369i	$0.572224\pi$
$294$	2.57355	0.150093
$295$	−3.69903	−0.215366
$296$	7.87849	0.457928
$297$	13.2183	0.767006
$298$	−15.4174	−0.893104
$299$	25.1111	1.45221
$300$	−20.9452	−1.20927
$301$	−1.43769	−0.0828672
$302$	11.5037	0.661963
$303$	−5.24166	−0.301125
$304$	−1.72434	−0.0988974
$305$	42.8979	2.45633
$306$	0.339901	0.0194308
$307$	−2.79844	−0.159716	−0.0798578	−	0.996806i	$-0.525447\pi$
$307$	−2.79844	−0.159716	−0.0798578	+	0.996806i	$0.525447\pi$
$308$	7.84805	0.447184
$309$	−1.45939	−0.0830217
$310$	16.9427	0.962283
$311$	−1.14492	−0.0649222	−0.0324611	−	0.999473i	$-0.510335\pi$
$311$	−1.14492	−0.0649222	−0.0324611	+	0.999473i	$0.510335\pi$
$312$	10.5217	0.595676
$313$	26.8263	1.51631	0.758157	−	0.652072i	$-0.226100\pi$
$313$	26.8263	1.51631	0.758157	+	0.652072i	$0.226100\pi$
$314$	−7.79397	−0.439839
$315$	−3.92525	−0.221163
$316$	−11.8078	−0.664242
$317$	2.26634	0.127290	0.0636450	−	0.997973i	$-0.479727\pi$
$317$	2.26634	0.127290	0.0636450	+	0.997973i	$0.479727\pi$
$318$	10.9742	0.615405
$319$	−2.68646	−0.150413
$320$	4.72235	0.263987
$321$	−27.8978	−1.55710
$322$	−13.6822	−0.762477
$323$	−1.26135	−0.0701831
$324$	11.0908	0.616154
$325$	23.7159	1.31552
$326$	2.26639	0.125524
$327$	−27.9674	−1.54660
$328$	0.0263557	0.00145525
$329$	−17.6128	−0.971026
$330$	9.66661	0.532129
$331$	11.1404	0.612332	0.306166	−	0.951978i	$-0.400954\pi$
$331$	11.1404	0.612332	0.306166	+	0.951978i	$0.400954\pi$
$332$	−6.93411	−0.380559
$333$	−1.59442	−0.0873738
$334$	14.8391	0.811957
$335$	23.7980	1.30022
$336$	4.66630	0.254568
$337$	−5.07188	−0.276283	−0.138142	−	0.990412i	$-0.544113\pi$
$337$	−5.07188	−0.276283	−0.138142	+	0.990412i	$0.544113\pi$
$338$	3.82477	0.208040
$339$	−17.5417	−0.952733
$340$	−5.63475	−0.305587
$341$	15.8563	0.858667
$342$	−0.428732	−0.0231832
$343$	−20.1622	−1.08865
$344$	−1.62252	−0.0874803
$345$	53.8533	2.89936
$346$	8.88542	0.477683
$347$	−35.7145	−1.91726	−0.958628	−	0.284662i	$-0.908119\pi$
$347$	−35.7145	−1.91726	−0.958628	+	0.284662i	$0.908119\pi$
$348$	−2.71129	−0.145341
$349$	−12.4438	−0.666100	−0.333050	−	0.942909i	$-0.608078\pi$
$349$	−12.4438	−0.666100	−0.333050	+	0.942909i	$0.608078\pi$
$350$	−12.9220	−0.690710
$351$	−15.1053	−0.806259
$352$	13.8846	0.740052
$353$	12.5630	0.668659	0.334330	−	0.942456i	$-0.391490\pi$
$353$	12.5630	0.668659	0.334330	+	0.942456i	$0.391490\pi$
$354$	1.09335	0.0581108
$355$	17.1684	0.911204
$356$	−22.0343	−1.16782
$357$	3.41339	0.180656
$358$	−7.26006	−0.383706
$359$	19.8461	1.04744	0.523718	−	0.851892i	$-0.324544\pi$
$359$	19.8461	1.04744	0.523718	+	0.851892i	$0.324544\pi$
$360$	−4.42986	−0.233474
$361$	−17.4090	−0.916264
$362$	9.88964	0.519788
$363$	−8.37254	−0.439444
$364$	−8.96836	−0.470070
$365$	55.4823	2.90408
$366$	−12.6796	−0.662776
$367$	−29.8863	−1.56005	−0.780025	−	0.625748i	$-0.784794\pi$
$367$	−29.8863	−1.56005	−0.780025	+	0.625748i	$0.784794\pi$
$368$	12.5682	0.655164
$369$	−0.00533377	−0.000277665 0
$370$	−8.27144	−0.430012
$371$	−21.6353	−1.12325
$372$	16.0029	0.829712
$373$	10.7256	0.555350	0.277675	−	0.960675i	$-0.410436\pi$
$373$	10.7256	0.555350	0.277675	+	0.960675i	$0.410436\pi$
$374$	1.65025	0.0853321
$375$	21.5728	1.11402
$376$	−19.8771	−1.02508
$377$	3.06995	0.158110
$378$	8.23034	0.423323
$379$	−27.9659	−1.43651	−0.718256	−	0.695778i	$-0.755060\pi$
$379$	−27.9659	−1.43651	−0.718256	+	0.695778i	$0.755060\pi$
$380$	7.10736	0.364600
$381$	−14.7627	−0.756316
$382$	1.95548	0.100051
$383$	29.4715	1.50592	0.752961	−	0.658065i	$-0.228625\pi$
$383$	29.4715	1.50592	0.752961	+	0.658065i	$0.228625\pi$
$384$	17.0023	0.867646
$385$	−19.0574	−0.971254
$386$	12.9173	0.657472
$387$	0.328360	0.0166915
$388$	8.50624	0.431839
$389$	−27.8786	−1.41350	−0.706750	−	0.707464i	$-0.749839\pi$
$389$	−27.8786	−1.41350	−0.706750	+	0.707464i	$0.749839\pi$
$390$	−11.0465	−0.559362
$391$	9.19362	0.464941
$392$	−5.72593	−0.289203
$393$	2.43906	0.123035
$394$	−10.1738	−0.512548
$395$	28.6729	1.44269
$396$	−1.79244	−0.0900736
$397$	−25.1263	−1.26106	−0.630528	−	0.776167i	$-0.717161\pi$
$397$	−25.1263	−1.26106	−0.630528	+	0.776167i	$0.717161\pi$
$398$	−8.25607	−0.413840
$399$	−4.30546	−0.215543
$400$	11.8700	0.593498
$401$	0.0803360	0.00401179	0.00200589	−	0.999998i	$-0.499362\pi$
$401$	0.0803360	0.00401179	0.00200589	+	0.999998i	$0.499362\pi$
$402$	−7.03413	−0.350831
$403$	−18.1198	−0.902612
$404$	5.04217	0.250857
$405$	−26.9317	−1.33825
$406$	−1.67271	−0.0830152
$407$	−7.74105	−0.383709
$408$	3.85220	0.190712
$409$	1.09816	0.0543007	0.0271504	−	0.999631i	$-0.491357\pi$
$409$	1.09816	0.0543007	0.0271504	+	0.999631i	$0.491357\pi$
$410$	−0.0276702	−0.00136653
$411$	24.4763	1.20733
$412$	1.40385	0.0691626
$413$	−2.15550	−0.106065
$414$	3.12492	0.153581
$415$	16.8381	0.826548
$416$	−15.8667	−0.777927
$417$	4.03799	0.197741
$418$	−2.08153	−0.101811
$419$	34.8529	1.70267	0.851337	−	0.524620i	$-0.175792\pi$
$419$	34.8529	1.70267	0.851337	+	0.524620i	$0.175792\pi$
$420$	−19.2336	−0.938502
$421$	19.8874	0.969251	0.484626	−	0.874722i	$-0.338956\pi$
$421$	19.8874	0.969251	0.484626	+	0.874722i	$0.338956\pi$
$422$	−1.55796	−0.0758401
$423$	4.02265	0.195588
$424$	−24.4167	−1.18578
$425$	8.68283	0.421179
$426$	−5.07458	−0.245864
$427$	24.9975	1.20971
$428$	26.8361	1.29717
$429$	−10.3382	−0.499132
$430$	1.70344	0.0821473
$431$	−21.8770	−1.05378	−0.526889	−	0.849934i	$-0.676642\pi$
$431$	−21.8770	−1.05378	−0.526889	+	0.849934i	$0.676642\pi$
$432$	−7.56027	−0.363744
$433$	21.1657	1.01716	0.508580	−	0.861015i	$-0.330171\pi$
$433$	21.1657	1.01716	0.508580	+	0.861015i	$0.330171\pi$
$434$	9.87287	0.473913
$435$	6.58382	0.315670
$436$	26.9030	1.28842
$437$	−11.5963	−0.554728
$438$	−16.3993	−0.783588
$439$	−2.75114	−0.131305	−0.0656524	−	0.997843i	$-0.520913\pi$
$439$	−2.75114	−0.131305	−0.0656524	+	0.997843i	$0.520913\pi$
$440$	−21.5073	−1.02532
$441$	1.15879	0.0551806
$442$	−1.88582	−0.0896992
$443$	−12.6831	−0.602594	−0.301297	−	0.953530i	$-0.597420\pi$
$443$	−12.6831	−0.602594	−0.301297	+	0.953530i	$0.597420\pi$
$444$	−7.81261	−0.370770
$445$	53.5058	2.53642
$446$	12.5875	0.596033
$447$	35.3612	1.67253
$448$	2.75181	0.130011
$449$	8.92634	0.421260	0.210630	−	0.977566i	$-0.432448\pi$
$449$	8.92634	0.421260	0.210630	+	0.977566i	$0.432448\pi$
$450$	2.95130	0.139126
$451$	−0.0258959	−0.00121939
$452$	16.8741	0.793690
$453$	−26.3848	−1.23967
$454$	2.67138	0.125374
$455$	21.7778	1.02096
$456$	−4.85895	−0.227541
$457$	25.1790	1.17783	0.588913	−	0.808197i	$-0.299556\pi$
$457$	25.1790	1.17783	0.588913	+	0.808197i	$0.299556\pi$
$458$	20.2576	0.946573
$459$	−5.53031	−0.258133
$460$	−51.8037	−2.41536
$461$	9.73306	0.453314	0.226657	−	0.973975i	$-0.427220\pi$
$461$	9.73306	0.453314	0.226657	+	0.973975i	$0.427220\pi$
$462$	5.63292	0.262067
$463$	−16.7158	−0.776850	−0.388425	−	0.921480i	$-0.626981\pi$
$463$	−16.7158	−0.776850	−0.388425	+	0.921480i	$0.626981\pi$
$464$	1.53653	0.0713314
$465$	−38.8598	−1.80208
$466$	−1.69531	−0.0785336
$467$	−2.55818	−0.118378	−0.0591892	−	0.998247i	$-0.518852\pi$
$467$	−2.55818	−0.118378	−0.0591892	+	0.998247i	$0.518852\pi$
$468$	2.04831	0.0946834
$469$	13.8675	0.640344
$470$	20.8684	0.962589
$471$	17.8762	0.823692
$472$	−2.43260	−0.111970
$473$	1.59421	0.0733019
$474$	−8.47505	−0.389272
$475$	−10.9521	−0.502515
$476$	−3.28348	−0.150498
$477$	4.94137	0.226250
$478$	16.0640	0.734751
$479$	11.0934	0.506871	0.253435	−	0.967352i	$-0.418439\pi$
$479$	11.0934	0.506871	0.253435	+	0.967352i	$0.418439\pi$
$480$	−34.0277	−1.55314
$481$	8.84608	0.403347
$482$	7.23795	0.329680
$483$	31.3814	1.42790
$484$	8.05389	0.366086
$485$	−20.6557	−0.937925
$486$	−3.49453	−0.158515
$487$	3.71163	0.168190	0.0840949	−	0.996458i	$-0.473200\pi$
$487$	3.71163	0.168190	0.0840949	+	0.996458i	$0.473200\pi$
$488$	28.2111	1.27705
$489$	−5.19819	−0.235070
$490$	6.01151	0.271572
$491$	−23.5331	−1.06203	−0.531017	−	0.847361i	$-0.678190\pi$
$491$	−23.5331	−1.06203	−0.531017	+	0.847361i	$0.678190\pi$
$492$	−0.0261353	−0.00117827
$493$	1.12396	0.0506208
$494$	2.37867	0.107021
$495$	4.35258	0.195634
$496$	−9.06906	−0.407213
$497$	10.0044	0.448757
$498$	−4.97694	−0.223022
$499$	6.48449	0.290285	0.145143	−	0.989411i	$-0.453636\pi$
$499$	6.48449	0.290285	0.145143	+	0.989411i	$0.453636\pi$
$500$	−20.7518	−0.928050
$501$	−34.0348	−1.52056
$502$	−13.7756	−0.614836
$503$	9.89986	0.441413	0.220706	−	0.975340i	$-0.429164\pi$
$503$	9.89986	0.441413	0.220706	+	0.975340i	$0.429164\pi$
$504$	−2.58137	−0.114983
$505$	−12.2439	−0.544845
$506$	15.1717	0.674465
$507$	−8.77247	−0.389599
$508$	14.2009	0.630061
$509$	37.2158	1.64956	0.824780	−	0.565453i	$-0.191299\pi$
$509$	37.2158	1.64956	0.824780	+	0.565453i	$0.191299\pi$
$510$	−4.04433	−0.179086
$511$	32.3306	1.43022
$512$	−14.5924	−0.644898
$513$	6.97563	0.307982
$514$	21.8368	0.963181
$515$	−3.40896	−0.150217
$516$	1.60895	0.0708301
$517$	19.5303	0.858941
$518$	−4.81993	−0.211776
$519$	−20.3796	−0.894563
$520$	24.5775	1.07780
$521$	−4.07235	−0.178413	−0.0892063	−	0.996013i	$-0.528433\pi$
$521$	−4.07235	−0.178413	−0.0892063	+	0.996013i	$0.528433\pi$
$522$	0.382036	0.0167213
$523$	−12.6589	−0.553537	−0.276768	−	0.960937i	$-0.589264\pi$
$523$	−12.6589	−0.553537	−0.276768	+	0.960937i	$0.589264\pi$
$524$	−2.34624	−0.102496
$525$	29.6378	1.29350
$526$	−1.28161	−0.0558806
$527$	−6.63399	−0.288981
$528$	−5.17432	−0.225183
$529$	61.5226	2.67489
$530$	25.6345	1.11349
$531$	0.492301	0.0213641
$532$	4.14160	0.179561
$533$	0.0295925	0.00128179
$534$	−15.8151	−0.684386
$535$	−65.1659	−2.81737
$536$	15.6503	0.675991
$537$	16.6516	0.718571
$538$	−11.3702	−0.490204
$539$	5.62603	0.242330
$540$	31.1619	1.34099
$541$	0.0148051	0.000636520 0	0.000318260	−	1.00000i	$-0.499899\pi$
$541$	0.0148051	0.000636520 0	0.000318260		1.00000i	$0.499899\pi$
$542$	15.6498	0.672218
$543$	−22.6828	−0.973413
$544$	−5.80907	−0.249062
$545$	−65.3285	−2.79837
$546$	−6.43703	−0.275479
$547$	−22.9225	−0.980096	−0.490048	−	0.871695i	$-0.663021\pi$
$547$	−22.9225	−0.980096	−0.490048	+	0.871695i	$0.663021\pi$
$548$	−23.5448	−1.00578
$549$	−5.70926	−0.243665
$550$	14.3288	0.610982
$551$	−1.41771	−0.0603963
$552$	35.4156	1.50739
$553$	16.7083	0.710507
$554$	−6.54778	−0.278188
$555$	18.9713	0.805288
$556$	−3.88431	−0.164731
$557$	7.76249	0.328907	0.164454	−	0.986385i	$-0.447414\pi$
$557$	7.76249	0.328907	0.164454	+	0.986385i	$0.447414\pi$
$558$	−2.25490	−0.0954574
$559$	−1.82179	−0.0770534
$560$	10.8999	0.460606
$561$	−3.78499	−0.159803
$562$	−15.5748	−0.656982
$563$	34.3675	1.44842	0.724209	−	0.689580i	$-0.242205\pi$
$563$	34.3675	1.44842	0.724209	+	0.689580i	$0.242205\pi$
$564$	19.7108	0.829976
$565$	−40.9752	−1.72384
$566$	11.7650	0.494518
$567$	−15.6936	−0.659070
$568$	11.2905	0.473738
$569$	26.9128	1.12824	0.564122	−	0.825691i	$-0.309215\pi$
$569$	26.9128	1.12824	0.564122	+	0.825691i	$0.309215\pi$
$570$	5.10130	0.213670
$571$	1.77589	0.0743186	0.0371593	−	0.999309i	$-0.488169\pi$
$571$	1.77589	0.0743186	0.0371593	+	0.999309i	$0.488169\pi$
$572$	9.94473	0.415810
$573$	−4.48508	−0.187367
$574$	−0.0161240	−0.000673001 0
$575$	79.8266	3.32900
$576$	−0.628494	−0.0261873
$577$	12.4139	0.516798	0.258399	−	0.966038i	$-0.416805\pi$
$577$	12.4139	0.516798	0.258399	+	0.966038i	$0.416805\pi$
$578$	−0.690432	−0.0287182
$579$	−29.6270	−1.23126
$580$	−6.33325	−0.262974
$581$	9.81187	0.407065
$582$	6.10534	0.253074
$583$	23.9907	0.993595
$584$	36.4869	1.50984
$585$	−4.97391	−0.205646
$586$	5.31721	0.219652
$587$	13.4203	0.553916	0.276958	−	0.960882i	$-0.410674\pi$
$587$	13.4203	0.553916	0.276958	+	0.960882i	$0.410674\pi$
$588$	5.67804	0.234159
$589$	8.36775	0.344787
$590$	2.55393	0.105144
$591$	23.3346	0.959856
$592$	4.42751	0.181970
$593$	3.88292	0.159452	0.0797262	−	0.996817i	$-0.474595\pi$
$593$	3.88292	0.159452	0.0797262	+	0.996817i	$0.474595\pi$
$594$	−9.12637	−0.374459
$595$	7.97325	0.326872
$596$	−34.0154	−1.39333
$597$	18.9361	0.775002
$598$	−17.3375	−0.708983
$599$	17.6852	0.722599	0.361299	−	0.932450i	$-0.382333\pi$
$599$	17.6852	0.722599	0.361299	+	0.932450i	$0.382333\pi$
$600$	33.4480	1.36551
$601$	−26.0113	−1.06102	−0.530512	−	0.847677i	$-0.678000\pi$
$601$	−26.0113	−1.06102	−0.530512	+	0.847677i	$0.678000\pi$
$602$	0.992629	0.0404566
$603$	−3.16726	−0.128981
$604$	25.3806	1.03272
$605$	−19.5572	−0.795114
$606$	3.61901	0.147012
$607$	20.8890	0.847859	0.423930	−	0.905695i	$-0.360650\pi$
$607$	20.8890	0.847859	0.423930	+	0.905695i	$0.360650\pi$
$608$	7.32724	0.297159
$609$	3.83652	0.155464
$610$	−29.6181	−1.19920
$611$	−22.3182	−0.902900
$612$	0.749925	0.0303139
$613$	21.3954	0.864151	0.432075	−	0.901838i	$-0.357781\pi$
$613$	21.3954	0.864151	0.432075	+	0.901838i	$0.357781\pi$
$614$	1.93213	0.0779746
$615$	0.0634642	0.00255912
$616$	−12.5327	−0.504959
$617$	16.1803	0.651394	0.325697	−	0.945474i	$-0.394401\pi$
$617$	16.1803	0.651394	0.325697	+	0.945474i	$0.394401\pi$
$618$	1.00761	0.0405320
$619$	19.8764	0.798898	0.399449	−	0.916755i	$-0.369201\pi$
$619$	19.8764	0.798898	0.399449	+	0.916755i	$0.369201\pi$
$620$	37.3808	1.50125
$621$	−50.8435	−2.04028
$622$	0.790486	0.0316956
$623$	31.1789	1.24916
$624$	5.91295	0.236708
$625$	6.97740	0.279096
$626$	−18.5218	−0.740279
$627$	4.77418	0.190663
$628$	−17.1959	−0.686190
$629$	3.23871	0.129136
$630$	2.71012	0.107974
$631$	26.7626	1.06540	0.532701	−	0.846304i	$-0.321177\pi$
$631$	26.7626	1.06540	0.532701	+	0.846304i	$0.321177\pi$
$632$	18.8562	0.750060
$633$	3.57332	0.142027
$634$	−1.56475	−0.0621442
$635$	−34.4839	−1.36845
$636$	24.2125	0.960089
$637$	−6.42915	−0.254732
$638$	1.85482	0.0734328
$639$	−2.28493	−0.0903904
$640$	39.7154	1.56989
$641$	−29.1736	−1.15229	−0.576145	−	0.817348i	$-0.695444\pi$
$641$	−29.1736	−1.15229	−0.576145	+	0.817348i	$0.695444\pi$
$642$	19.2615	0.760193
$643$	−19.4773	−0.768111	−0.384055	−	0.923310i	$-0.625473\pi$
$643$	−19.4773	−0.768111	−0.384055	+	0.923310i	$0.625473\pi$
$644$	−30.1870	−1.18954
$645$	−3.90701	−0.153838
$646$	0.870874	0.0342641
$647$	25.8006	1.01433	0.507164	−	0.861850i	$-0.330694\pi$
$647$	25.8006	1.01433	0.507164	+	0.861850i	$0.330694\pi$
$648$	−17.7111	−0.695759
$649$	2.39016	0.0938221
$650$	−16.3742	−0.642251
$651$	−22.6444	−0.887502
$652$	5.00035	0.195829
$653$	1.24620	0.0487674	0.0243837	−	0.999703i	$-0.492238\pi$
$653$	1.24620	0.0487674	0.0243837	+	0.999703i	$0.492238\pi$
$654$	19.3096	0.755065
$655$	5.69736	0.222614
$656$	0.0148112	0.000578281 0
$657$	−7.38410	−0.288081
$658$	12.1605	0.474064
$659$	0.662644	0.0258130	0.0129065	−	0.999917i	$-0.495892\pi$
$659$	0.662644	0.0258130	0.0129065	+	0.999917i	$0.495892\pi$
$660$	21.3275	0.830171
$661$	−40.0895	−1.55930	−0.779651	−	0.626214i	$-0.784604\pi$
$661$	−40.0895	−1.55930	−0.779651	+	0.626214i	$0.784604\pi$
$662$	−7.69169	−0.298946
$663$	4.32530	0.167981
$664$	11.0733	0.429725
$665$	−10.0570	−0.389995
$666$	1.10084	0.0426567
$667$	10.3333	0.400107
$668$	32.7395	1.26673
$669$	−28.8705	−1.11620
$670$	−16.4309	−0.634781
$671$	−27.7189	−1.07008
$672$	−19.8286	−0.764904
$673$	4.35579	0.167904	0.0839518	−	0.996470i	$-0.473246\pi$
$673$	4.35579	0.167904	0.0839518	+	0.996470i	$0.473246\pi$
$674$	3.50179	0.134884
$675$	−48.0187	−1.84824
$676$	8.43860	0.324562
$677$	−10.8805	−0.418171	−0.209086	−	0.977897i	$-0.567049\pi$
$677$	−10.8805	−0.418171	−0.209086	+	0.977897i	$0.567049\pi$
$678$	12.1113	0.465133
$679$	−12.0365	−0.461917
$680$	8.99827	0.345068
$681$	−6.12705	−0.234789
$682$	−10.9477	−0.419209
$683$	−2.58250	−0.0988165	−0.0494083	−	0.998779i	$-0.515734\pi$
$683$	−2.58250	−0.0988165	−0.0494083	+	0.998779i	$0.515734\pi$
$684$	−0.945914	−0.0361679
$685$	57.1738	2.18450
$686$	13.9206	0.531491
$687$	−46.4626	−1.77266
$688$	−0.911814	−0.0347626
$689$	−27.4154	−1.04444
$690$	−37.1820	−1.41550
$691$	−18.8154	−0.715770	−0.357885	−	0.933766i	$-0.616502\pi$
$691$	−18.8154	−0.715770	−0.357885	+	0.933766i	$0.616502\pi$
$692$	19.6039	0.745230
$693$	2.53633	0.0963473
$694$	24.6584	0.936022
$695$	9.43225	0.357786
$696$	4.32973	0.164118
$697$	0.0108344	0.000410381 0
$698$	8.59159	0.325196
$699$	3.88835	0.147071
$700$	−28.5099	−1.07757
$701$	−17.2719	−0.652352	−0.326176	−	0.945309i	$-0.605760\pi$
$701$	−17.2719	−0.652352	−0.326176	+	0.945309i	$0.605760\pi$
$702$	10.4292	0.393623
$703$	−4.08513	−0.154074
$704$	−3.05139	−0.115004
$705$	−47.8637	−1.80265
$706$	−8.67387	−0.326446
$707$	−7.13475	−0.268330
$708$	2.41226	0.0906583
$709$	7.61289	0.285908	0.142954	−	0.989729i	$-0.454340\pi$
$709$	7.61289	0.285908	0.142954	+	0.989729i	$0.454340\pi$
$710$	−11.8536	−0.444858
$711$	−3.81606	−0.143113
$712$	35.1872	1.31869
$713$	−60.9903	−2.28411
$714$	−2.35671	−0.0881977
$715$	−24.1487	−0.903112
$716$	−16.0179	−0.598617
$717$	−36.8443	−1.37598
$718$	−13.7024	−0.511368
$719$	−2.58087	−0.0962502	−0.0481251	−	0.998841i	$-0.515325\pi$
$719$	−2.58087	−0.0962502	−0.0481251	+	0.998841i	$0.515325\pi$
$720$	−2.48947	−0.0927771
$721$	−1.98647	−0.0739798
$722$	12.0197	0.447328
$723$	−16.6009	−0.617395
$724$	21.8196	0.810917
$725$	9.75919	0.362447
$726$	5.78067	0.214541
$727$	15.5751	0.577650	0.288825	−	0.957382i	$-0.406735\pi$
$727$	15.5751	0.577650	0.288825	+	0.957382i	$0.406735\pi$
$728$	14.3218	0.530801
$729$	29.8572	1.10582
$730$	−38.3067	−1.41780
$731$	−0.666989	−0.0246695
$732$	−27.9752	−1.03399
$733$	−34.6966	−1.28155	−0.640774	−	0.767729i	$-0.721386\pi$
$733$	−34.6966	−1.28155	−0.640774	+	0.767729i	$0.721386\pi$
$734$	20.6344	0.761631
$735$	−13.7880	−0.508577
$736$	−53.4063	−1.96858
$737$	−15.3773	−0.566429
$738$	0.00368261	0.000135559 0
$739$	−1.36978	−0.0503880	−0.0251940	−	0.999683i	$-0.508020\pi$
$739$	−1.36978	−0.0503880	−0.0251940	+	0.999683i	$0.508020\pi$
$740$	−18.2493	−0.670858
$741$	−5.45570	−0.200420
$742$	14.9377	0.548382
$743$	49.8594	1.82916	0.914581	−	0.404402i	$-0.132520\pi$
$743$	49.8594	1.82916	0.914581	+	0.404402i	$0.132520\pi$
$744$	−25.5554	−0.936908
$745$	82.5995	3.02621
$746$	−7.40529	−0.271127
$747$	−2.24097	−0.0819927
$748$	3.64094	0.133126
$749$	−37.9735	−1.38752
$750$	−14.8946	−0.543873
$751$	34.7462	1.26791	0.633954	−	0.773371i	$-0.281431\pi$
$751$	34.7462	1.26791	0.633954	+	0.773371i	$0.281431\pi$
$752$	−11.1704	−0.407343
$753$	31.5957	1.15141
$754$	−2.11959	−0.0771910
$755$	−61.6317	−2.24301
$756$	18.1586	0.660424
$757$	24.2984	0.883142	0.441571	−	0.897226i	$-0.354421\pi$
$757$	24.2984	0.883142	0.441571	+	0.897226i	$0.354421\pi$
$758$	19.3086	0.701319
$759$	−34.7978	−1.26308
$760$	−11.3499	−0.411705
$761$	−11.4389	−0.414660	−0.207330	−	0.978271i	$-0.566477\pi$
$761$	−11.4389	−0.414660	−0.207330	+	0.978271i	$0.566477\pi$
$762$	10.1926	0.369241
$763$	−38.0682	−1.37816
$764$	4.31439	0.156089
$765$	−1.82104	−0.0658398
$766$	−20.3480	−0.735205
$767$	−2.73136	−0.0986237
$768$	−15.7823	−0.569494
$769$	8.50739	0.306784	0.153392	−	0.988165i	$-0.450980\pi$
$769$	8.50739	0.306784	0.153392	+	0.988165i	$0.450980\pi$
$770$	13.1578	0.474175
$771$	−50.0848	−1.80376
$772$	28.4994	1.02572
$773$	37.2372	1.33933	0.669665	−	0.742664i	$-0.266438\pi$
$773$	37.2372	1.33933	0.669665	+	0.742664i	$0.266438\pi$
$774$	−0.226710	−0.00814892
$775$	−57.6018	−2.06912
$776$	−13.5838	−0.487631
$777$	11.0550	0.396595
$778$	19.2483	0.690083
$779$	−0.0136659	−0.000489631 0
$780$	−24.3720	−0.872657
$781$	−11.0935	−0.396957
$782$	−6.34757	−0.226989
$783$	−6.21587	−0.222137
$784$	−3.21783	−0.114922
$785$	41.7567	1.49036
$786$	−1.68401	−0.0600666
$787$	5.83484	0.207990	0.103995	−	0.994578i	$-0.466837\pi$
$787$	5.83484	0.207990	0.103995	+	0.994578i	$0.466837\pi$
$788$	−22.4465	−0.799623
$789$	2.93948	0.104648
$790$	−19.7967	−0.704334
$791$	−23.8771	−0.848971
$792$	2.86240	0.101711
$793$	31.6758	1.12484
$794$	17.3480	0.615659
$795$	−58.7952	−2.08525
$796$	−18.2154	−0.645628
$797$	−4.00038	−0.141701	−0.0708504	−	0.997487i	$-0.522571\pi$
$797$	−4.00038	−0.141701	−0.0708504	+	0.997487i	$0.522571\pi$
$798$	2.97263	0.105230
$799$	−8.17111	−0.289073
$800$	−50.4391	−1.78329
$801$	−7.12105	−0.251610
$802$	−0.0554665	−0.00195859
$803$	−35.8504	−1.26513
$804$	−15.5194	−0.547329
$805$	73.3030	2.58359
$806$	12.5105	0.440663
$807$	26.0786	0.918011
$808$	−8.05197	−0.283267
$809$	48.9881	1.72233	0.861165	−	0.508325i	$-0.169735\pi$
$809$	48.9881	1.72233	0.861165	+	0.508325i	$0.169735\pi$
$810$	18.5945	0.653344
$811$	13.5603	0.476168	0.238084	−	0.971245i	$-0.423481\pi$
$811$	13.5603	0.476168	0.238084	+	0.971245i	$0.423481\pi$
$812$	−3.69051	−0.129511
$813$	−35.8944	−1.25887
$814$	5.34467	0.187330
$815$	−12.1423	−0.425328
$816$	2.16484	0.0757845
$817$	0.841303	0.0294335
$818$	−0.758208	−0.0265101
$819$	−2.89840	−0.101278
$820$	−0.0610489	−0.00213192
$821$	−4.88839	−0.170606	−0.0853030	−	0.996355i	$-0.527186\pi$
$821$	−4.88839	−0.170606	−0.0853030	+	0.996355i	$0.527186\pi$
$822$	−16.8992	−0.589429
$823$	12.1969	0.425157	0.212578	−	0.977144i	$-0.431814\pi$
$823$	12.1969	0.425157	0.212578	+	0.977144i	$0.431814\pi$
$824$	−2.24184	−0.0780982
$825$	−32.8645	−1.14419
$826$	1.48823	0.0517820
$827$	−44.4417	−1.54539	−0.772694	−	0.634779i	$-0.781091\pi$
$827$	−44.4417	−1.54539	−0.772694	+	0.634779i	$0.781091\pi$
$828$	6.89452	0.239601
$829$	14.3685	0.499038	0.249519	−	0.968370i	$-0.419727\pi$
$829$	14.3685	0.499038	0.249519	+	0.968370i	$0.419727\pi$
$830$	−11.6255	−0.403528
$831$	15.0179	0.520967
$832$	3.48698	0.120889
$833$	−2.35383	−0.0815553
$834$	−2.78796	−0.0965390
$835$	−79.5011	−2.75125
$836$	−4.59249	−0.158835
$837$	36.6880	1.26812
$838$	−24.0635	−0.831261
$839$	−28.5404	−0.985322	−0.492661	−	0.870221i	$-0.663976\pi$
$839$	−28.5404	−0.985322	−0.492661	+	0.870221i	$0.663976\pi$
$840$	30.7146	1.05975
$841$	−27.7367	−0.956438
$842$	−13.7309	−0.473197
$843$	35.7222	1.23034
$844$	−3.43732	−0.118318
$845$	−20.4914	−0.704926
$846$	−2.77737	−0.0954878
$847$	−11.3964	−0.391584
$848$	−13.7216	−0.471201
$849$	−26.9840	−0.926090
$850$	−5.99491	−0.205624
$851$	29.7755	1.02069
$852$	−11.1961	−0.383571
$853$	2.16452	0.0741116	0.0370558	−	0.999313i	$-0.488202\pi$
$853$	2.16452	0.0741116	0.0370558	+	0.999313i	$0.488202\pi$
$854$	−17.2591	−0.590593
$855$	2.29696	0.0785543
$856$	−42.8552	−1.46476
$857$	16.5500	0.565337	0.282669	−	0.959218i	$-0.408780\pi$
$857$	16.5500	0.565337	0.282669	+	0.959218i	$0.408780\pi$
$858$	7.13781	0.243681
$859$	−8.42586	−0.287487	−0.143743	−	0.989615i	$-0.545914\pi$
$859$	−8.42586	−0.287487	−0.143743	+	0.989615i	$0.545914\pi$
$860$	3.75831	0.128157
$861$	0.0369818	0.00126034
$862$	15.1046	0.514464
$863$	8.04949	0.274008	0.137004	−	0.990570i	$-0.456253\pi$
$863$	8.04949	0.274008	0.137004	+	0.990570i	$0.456253\pi$
$864$	32.1259	1.09295
$865$	−47.6042	−1.61859
$866$	−14.6135	−0.496587
$867$	1.58357	0.0537809
$868$	21.7825	0.739348
$869$	−18.5272	−0.628494
$870$	−4.54568	−0.154113
$871$	17.5724	0.595418
$872$	−42.9621	−1.45488
$873$	2.74905	0.0930412
$874$	8.00648	0.270823
$875$	29.3641	0.992689
$876$	−36.1818	−1.22247
$877$	−3.03442	−0.102465	−0.0512326	−	0.998687i	$-0.516315\pi$
$877$	−3.03442	−0.102465	−0.0512326	+	0.998687i	$0.516315\pi$
$878$	1.89948	0.0641043
$879$	−12.1955	−0.411345
$880$	−12.0866	−0.407438
$881$	26.0050	0.876131	0.438065	−	0.898943i	$-0.355664\pi$
$881$	26.0050	0.876131	0.438065	+	0.898943i	$0.355664\pi$
$882$	−0.800068	−0.0269397
$883$	15.9029	0.535175	0.267587	−	0.963534i	$-0.413774\pi$
$883$	15.9029	0.535175	0.267587	+	0.963534i	$0.413774\pi$
$884$	−4.16069	−0.139939
$885$	−5.85768	−0.196904
$886$	8.75685	0.294192
$887$	−1.21100	−0.0406615	−0.0203308	−	0.999793i	$-0.506472\pi$
$887$	−1.21100	−0.0406615	−0.0203308	+	0.999793i	$0.506472\pi$
$888$	12.4762	0.418672
$889$	−20.0944	−0.673946
$890$	−36.9421	−1.23830
$891$	17.4021	0.582994
$892$	27.7718	0.929867
$893$	10.3066	0.344897
$894$	−24.4145	−0.816543
$895$	38.8962	1.30016
$896$	23.1429	0.773151
$897$	39.7652	1.32772
$898$	−6.16303	−0.205663
$899$	−7.45636	−0.248684
$900$	6.51147	0.217049
$901$	−10.0373	−0.334390
$902$	0.0178793	0.000595317 0
$903$	−2.27669	−0.0757635
$904$	−26.9466	−0.896232
$905$	−52.9843	−1.76126
$906$	18.2169	0.605216
$907$	−43.6688	−1.45000	−0.724999	−	0.688750i	$-0.758160\pi$
$907$	−43.6688	−1.45000	−0.724999	+	0.688750i	$0.758160\pi$
$908$	5.89387	0.195595
$909$	1.62953	0.0540481
$910$	−15.0361	−0.498442
$911$	20.5675	0.681433	0.340717	−	0.940166i	$-0.389330\pi$
$911$	20.5675	0.681433	0.340717	+	0.940166i	$0.389330\pi$
$912$	−2.73061	−0.0904195
$913$	−10.8801	−0.360078
$914$	−17.3844	−0.575025
$915$	67.9320	2.24576
$916$	44.6943	1.47674
$917$	3.31996	0.109635
$918$	3.81830	0.126023
$919$	20.6135	0.679977	0.339988	−	0.940430i	$-0.389577\pi$
$919$	20.6135	0.679977	0.339988	+	0.940430i	$0.389577\pi$
$920$	82.7266	2.72742
$921$	−4.43153	−0.146024
$922$	−6.72001	−0.221312
$923$	12.6771	0.417272
$924$	12.4279	0.408849
$925$	28.1212	0.924619
$926$	11.5411	0.379266
$927$	0.453696	0.0149013
$928$	−6.52918	−0.214331
$929$	−9.36751	−0.307338	−0.153669	−	0.988122i	$-0.549109\pi$
$929$	−9.36751	−0.307338	−0.153669	+	0.988122i	$0.549109\pi$
$930$	26.8300	0.879792
$931$	2.96899	0.0973047
$932$	−3.74037	−0.122520
$933$	−1.81306	−0.0593568
$934$	1.76625	0.0577934
$935$	−8.84129	−0.289141
$936$	−3.27100	−0.106916
$937$	22.4282	0.732696	0.366348	−	0.930478i	$-0.380608\pi$
$937$	22.4282	0.732696	0.366348	+	0.930478i	$0.380608\pi$
$938$	−9.57460	−0.312622
$939$	42.4814	1.38633
$940$	46.0421	1.50173
$941$	21.5116	0.701257	0.350628	−	0.936515i	$-0.385968\pi$
$941$	21.5116	0.701257	0.350628	+	0.936515i	$0.385968\pi$
$942$	−12.3423	−0.402134
$943$	0.0996069	0.00324365
$944$	−1.36706	−0.0444940
$945$	−44.0946	−1.43440
$946$	−1.10069	−0.0357867
$947$	−29.9577	−0.973495	−0.486748	−	0.873543i	$-0.661817\pi$
$947$	−29.9577	−0.973495	−0.486748	+	0.873543i	$0.661817\pi$
$948$	−18.6985	−0.607300
$949$	40.9680	1.32988
$950$	7.56165	0.245332
$951$	3.58890	0.116378
$952$	5.24347	0.169942
$953$	35.8541	1.16143	0.580715	−	0.814107i	$-0.302773\pi$
$953$	35.8541	1.16143	0.580715	+	0.814107i	$0.302773\pi$
$954$	−3.41168	−0.110457
$955$	−10.4766	−0.339015
$956$	35.4421	1.14628
$957$	−4.25419	−0.137519
$958$	−7.65924	−0.247459
$959$	33.3163	1.07584
$960$	7.47818	0.241357
$961$	13.0098	0.419670
$962$	−6.10762	−0.196918
$963$	8.67289	0.279480
$964$	15.9691	0.514331
$965$	−69.2050	−2.22779
$966$	−21.6667	−0.697114
$967$	−6.01711	−0.193497	−0.0967486	−	0.995309i	$-0.530844\pi$
$967$	−6.01711	−0.193497	−0.0967486	+	0.995309i	$0.530844\pi$
$968$	−12.8615	−0.413383
$969$	−1.99743	−0.0641667
$970$	14.2613	0.457904
$971$	−39.0700	−1.25382	−0.626908	−	0.779093i	$-0.715680\pi$
$971$	−39.0700	−1.25382	−0.626908	+	0.779093i	$0.715680\pi$
$972$	−7.70999	−0.247298
$973$	5.49636	0.176205
$974$	−2.56263	−0.0821118
$975$	37.5559	1.20275
$976$	15.8539	0.507471
$977$	21.8991	0.700613	0.350307	−	0.936635i	$-0.386077\pi$
$977$	21.8991	0.700613	0.350307	+	0.936635i	$0.386077\pi$
$978$	3.58900	0.114763
$979$	−34.5733	−1.10497
$980$	13.2632	0.423678
$981$	8.69453	0.277595
$982$	16.2480	0.518495
$983$	−44.6264	−1.42336	−0.711681	−	0.702503i	$-0.752066\pi$
$983$	−44.6264	−1.42336	−0.711681	+	0.702503i	$0.752066\pi$
$984$	0.0417361	0.00133050
$985$	54.5067	1.73673
$986$	−0.776020	−0.0247135
$987$	−27.8911	−0.887785
$988$	5.24807	0.166963
$989$	−6.13204	−0.194987
$990$	−3.00516	−0.0955103
$991$	−21.6138	−0.686584	−0.343292	−	0.939229i	$-0.611542\pi$
$991$	−21.6138	−0.686584	−0.343292	+	0.939229i	$0.611542\pi$
$992$	38.5373	1.22356
$993$	17.6416	0.559840
$994$	−6.90733	−0.219087
$995$	44.2324	1.40226
$996$	−10.9807	−0.347935
$997$	−2.28952	−0.0725097	−0.0362549	−	0.999343i	$-0.511543\pi$
$997$	−2.28952	−0.0725097	−0.0362549	+	0.999343i	$0.511543\pi$
$998$	−4.47710	−0.141720
$999$	−17.9111	−0.566681

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1003.2.a.i.1.7	✓	18
3.2	odd	2		9027.2.a.q.1.12		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1003.2.a.i.1.7	✓	18	1.1	even	1	trivial
9027.2.a.q.1.12		18	3.2	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:	\( N \)	\(=\)	\( 1003 = 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1003.a (trivial)

\(f(q)\)	\(=\)	\(q-0.690432 q^{2} +1.58357 q^{3} -1.52330 q^{4} +3.69903 q^{5} -1.09335 q^{6} +2.15550 q^{7} +2.43260 q^{8} -0.492301 q^{9} +O(q^{10})\)	\(q-0.690432 q^{2} +1.58357 q^{3} -1.52330 q^{4} +3.69903 q^{5} -1.09335 q^{6} +2.15550 q^{7} +2.43260 q^{8} -0.492301 q^{9} -2.55393 q^{10} -2.39016 q^{11} -2.41226 q^{12} +2.73136 q^{13} -1.48823 q^{14} +5.85768 q^{15} +1.36706 q^{16} +1.00000 q^{17} +0.339901 q^{18} -1.26135 q^{19} -5.63475 q^{20} +3.41339 q^{21} +1.65025 q^{22} +9.19362 q^{23} +3.85220 q^{24} +8.68283 q^{25} -1.88582 q^{26} -5.53031 q^{27} -3.28348 q^{28} +1.12396 q^{29} -4.04433 q^{30} -6.63399 q^{31} -5.80907 q^{32} -3.78499 q^{33} -0.690432 q^{34} +7.97325 q^{35} +0.749925 q^{36} +3.23871 q^{37} +0.870874 q^{38} +4.32530 q^{39} +8.99827 q^{40} +0.0108344 q^{41} -2.35671 q^{42} -0.666989 q^{43} +3.64094 q^{44} -1.82104 q^{45} -6.34757 q^{46} -8.17111 q^{47} +2.16484 q^{48} -2.35383 q^{49} -5.99491 q^{50} +1.58357 q^{51} -4.16069 q^{52} -10.0373 q^{53} +3.81830 q^{54} -8.84129 q^{55} +5.24347 q^{56} -1.99743 q^{57} -0.776020 q^{58} -1.00000 q^{59} -8.92302 q^{60} +11.5971 q^{61} +4.58032 q^{62} -1.06115 q^{63} +1.27664 q^{64} +10.1034 q^{65} +2.61328 q^{66} +6.43357 q^{67} -1.52330 q^{68} +14.5587 q^{69} -5.50499 q^{70} +4.64132 q^{71} -1.19757 q^{72} +14.9991 q^{73} -2.23611 q^{74} +13.7499 q^{75} +1.92141 q^{76} -5.15199 q^{77} -2.98633 q^{78} +7.75146 q^{79} +5.05680 q^{80} -7.28073 q^{81} -0.00748039 q^{82} +4.55202 q^{83} -5.19962 q^{84} +3.69903 q^{85} +0.460510 q^{86} +1.77988 q^{87} -5.81431 q^{88} +14.4648 q^{89} +1.25730 q^{90} +5.88744 q^{91} -14.0047 q^{92} -10.5054 q^{93} +5.64160 q^{94} -4.66576 q^{95} -9.19907 q^{96} -5.58407 q^{97} +1.62516 q^{98} +1.17668 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 5 q^{2} + q^{3} + 21 q^{4} + 21 q^{5} + 5 q^{6} - q^{7} + 9 q^{8} + 17 q^{9}+O(q^{10}) \) 18 * q + 5 * q^2 + q^3 + 21 * q^4 + 21 * q^5 + 5 * q^6 - q^7 + 9 * q^8 + 17 * q^9	\( 18 q + 5 q^{2} + q^{3} + 21 q^{4} + 21 q^{5} + 5 q^{6} - q^{7} + 9 q^{8} + 17 q^{9} + 10 q^{10} + 6 q^{11} + 20 q^{12} + 12 q^{13} + 9 q^{14} - q^{15} + 19 q^{16} + 18 q^{17} + 10 q^{18} - 14 q^{19} + 17 q^{20} - 14 q^{21} - 4 q^{22} + 7 q^{23} - 3 q^{24} + 35 q^{25} + 3 q^{26} + 13 q^{27} + q^{28} + 23 q^{29} + 17 q^{30} - q^{31} + 13 q^{32} + 7 q^{33} + 5 q^{34} + 5 q^{35} - 16 q^{36} + 36 q^{37} - 19 q^{38} + 8 q^{39} - 4 q^{40} + 37 q^{41} + 9 q^{42} - 15 q^{43} - 9 q^{44} + 17 q^{45} + q^{46} + 23 q^{47} + 17 q^{48} + 15 q^{49} + 32 q^{50} + q^{51} - 11 q^{52} + 35 q^{53} - 17 q^{54} - 6 q^{55} + 37 q^{56} - 16 q^{57} + 2 q^{58} - 18 q^{59} - 6 q^{60} + 39 q^{61} - q^{62} + 15 q^{63} + 5 q^{64} + 15 q^{65} - 36 q^{66} + 14 q^{67} + 21 q^{68} + 16 q^{69} - 45 q^{70} + 3 q^{71} - 71 q^{72} + 56 q^{73} - 27 q^{74} + 16 q^{75} - q^{76} + 39 q^{77} - 71 q^{78} - 22 q^{79} + 45 q^{80} - 18 q^{81} + 5 q^{82} + 12 q^{83} + 2 q^{84} + 21 q^{85} + 4 q^{86} + 8 q^{87} - 18 q^{88} + 34 q^{89} + 48 q^{90} + 3 q^{91} + 67 q^{92} - 19 q^{93} - 38 q^{94} - 72 q^{95} + 6 q^{96} + 31 q^{97} + 19 q^{98} + 44 q^{99}+O(q^{100}) \) 18 * q + 5 * q^2 + q^3 + 21 * q^4 + 21 * q^5 + 5 * q^6 - q^7 + 9 * q^8 + 17 * q^9 + 10 * q^10 + 6 * q^11 + 20 * q^12 + 12 * q^13 + 9 * q^14 - q^15 + 19 * q^16 + 18 * q^17 + 10 * q^18 - 14 * q^19 + 17 * q^20 - 14 * q^21 - 4 * q^22 + 7 * q^23 - 3 * q^24 + 35 * q^25 + 3 * q^26 + 13 * q^27 + q^28 + 23 * q^29 + 17 * q^30 - q^31 + 13 * q^32 + 7 * q^33 + 5 * q^34 + 5 * q^35 - 16 * q^36 + 36 * q^37 - 19 * q^38 + 8 * q^39 - 4 * q^40 + 37 * q^41 + 9 * q^42 - 15 * q^43 - 9 * q^44 + 17 * q^45 + q^46 + 23 * q^47 + 17 * q^48 + 15 * q^49 + 32 * q^50 + q^51 - 11 * q^52 + 35 * q^53 - 17 * q^54 - 6 * q^55 + 37 * q^56 - 16 * q^57 + 2 * q^58 - 18 * q^59 - 6 * q^60 + 39 * q^61 - q^62 + 15 * q^63 + 5 * q^64 + 15 * q^65 - 36 * q^66 + 14 * q^67 + 21 * q^68 + 16 * q^69 - 45 * q^70 + 3 * q^71 - 71 * q^72 + 56 * q^73 - 27 * q^74 + 16 * q^75 - q^76 + 39 * q^77 - 71 * q^78 - 22 * q^79 + 45 * q^80 - 18 * q^81 + 5 * q^82 + 12 * q^83 + 2 * q^84 + 21 * q^85 + 4 * q^86 + 8 * q^87 - 18 * q^88 + 34 * q^89 + 48 * q^90 + 3 * q^91 + 67 * q^92 - 19 * q^93 - 38 * q^94 - 72 * q^95 + 6 * q^96 + 31 * q^97 + 19 * q^98 + 44 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more