LMFDB - Embedded newform 6034.2.a.k.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6034,2,Mod(1,6034)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6034, 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("6034.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6034 = 2 \cdot 7 \cdot 431 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6034.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:	$48.1817325796$
Analytic rank:	$1$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 3 x^{19} - 32 x^{18} + 106 x^{17} + 382 x^{16} - 1495 x^{15} - 1963 x^{14} + 10784 x^{13} + \cdots - 44 $ x^20 - 3x^19 - 32x^18 + 106x^17 + 382x^16 - 1495x^15 - 1963x^14 + 10784x^13 + 2170x^12 - 42069x^11 + 19553x^10 + 84697x^9 - 82713x^8 - 68589x^7 + 118845x^6 - 10051x^5 - 53572x^4 + 25598x^3 - 557x^2 - 1040*x - 44
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.57684$ of defining polynomial
Character	$\chi$	$=$	6034.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -2.57684 q^{3} +1.00000 q^{4} +4.04233 q^{5} +2.57684 q^{6} +1.00000 q^{7} -1.00000 q^{8} +3.64010 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.57684 q^{3} +1.00000 q^{4} +4.04233 q^{5} +2.57684 q^{6} +1.00000 q^{7} -1.00000 q^{8} +3.64010 q^{9} -4.04233 q^{10} -6.22537 q^{11} -2.57684 q^{12} -3.30637 q^{13} -1.00000 q^{14} -10.4164 q^{15} +1.00000 q^{16} +3.51139 q^{17} -3.64010 q^{18} -3.74588 q^{19} +4.04233 q^{20} -2.57684 q^{21} +6.22537 q^{22} -8.78740 q^{23} +2.57684 q^{24} +11.3404 q^{25} +3.30637 q^{26} -1.64943 q^{27} +1.00000 q^{28} -0.0905660 q^{29} +10.4164 q^{30} +9.21443 q^{31} -1.00000 q^{32} +16.0418 q^{33} -3.51139 q^{34} +4.04233 q^{35} +3.64010 q^{36} +9.62250 q^{37} +3.74588 q^{38} +8.51999 q^{39} -4.04233 q^{40} +1.10715 q^{41} +2.57684 q^{42} +1.77302 q^{43} -6.22537 q^{44} +14.7145 q^{45} +8.78740 q^{46} -6.26009 q^{47} -2.57684 q^{48} +1.00000 q^{49} -11.3404 q^{50} -9.04829 q^{51} -3.30637 q^{52} -0.187269 q^{53} +1.64943 q^{54} -25.1650 q^{55} -1.00000 q^{56} +9.65253 q^{57} +0.0905660 q^{58} -4.44265 q^{59} -10.4164 q^{60} +0.849612 q^{61} -9.21443 q^{62} +3.64010 q^{63} +1.00000 q^{64} -13.3655 q^{65} -16.0418 q^{66} +13.5699 q^{67} +3.51139 q^{68} +22.6437 q^{69} -4.04233 q^{70} +10.2441 q^{71} -3.64010 q^{72} +0.441416 q^{73} -9.62250 q^{74} -29.2225 q^{75} -3.74588 q^{76} -6.22537 q^{77} -8.51999 q^{78} +4.90674 q^{79} +4.04233 q^{80} -6.66998 q^{81} -1.10715 q^{82} +7.93719 q^{83} -2.57684 q^{84} +14.1942 q^{85} -1.77302 q^{86} +0.233374 q^{87} +6.22537 q^{88} -12.0685 q^{89} -14.7145 q^{90} -3.30637 q^{91} -8.78740 q^{92} -23.7441 q^{93} +6.26009 q^{94} -15.1421 q^{95} +2.57684 q^{96} -4.98514 q^{97} -1.00000 q^{98} -22.6610 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 20 q^{2} + 3 q^{3} + 20 q^{4} - 3 q^{5} - 3 q^{6} + 20 q^{7} - 20 q^{8} + 13 q^{9}+O(q^{10}) $ 20 * q - 20 * q^2 + 3 * q^3 + 20 * q^4 - 3 * q^5 - 3 * q^6 + 20 * q^7 - 20 * q^8 + 13 * q^9	\( 20 q - 20 q^{2} + 3 q^{3} + 20 q^{4} - 3 q^{5} - 3 q^{6} + 20 q^{7} - 20 q^{8} + 13 q^{9} + 3 q^{10} - 8 q^{11} + 3 q^{12} - 4 q^{13} - 20 q^{14} - 25 q^{15} + 20 q^{16} + 9 q^{17} - 13 q^{18} - 14 q^{19} - 3 q^{20} + 3 q^{21} + 8 q^{22} - 23 q^{23} - 3 q^{24} + 31 q^{25} + 4 q^{26} - 21 q^{27} + 20 q^{28} - 48 q^{29} + 25 q^{30} - q^{31} - 20 q^{32} - 29 q^{33} - 9 q^{34} - 3 q^{35} + 13 q^{36} - q^{37} + 14 q^{38} - q^{39} + 3 q^{40} - 27 q^{41} - 3 q^{42} - 3 q^{43} - 8 q^{44} - 12 q^{45} + 23 q^{46} - 26 q^{47} + 3 q^{48} + 20 q^{49} - 31 q^{50} - 17 q^{51} - 4 q^{52} - 43 q^{53} + 21 q^{54} - 16 q^{55} - 20 q^{56} - 25 q^{57} + 48 q^{58} - 19 q^{59} - 25 q^{60} + 9 q^{61} + q^{62} + 13 q^{63} + 20 q^{64} - 87 q^{65} + 29 q^{66} + 32 q^{67} + 9 q^{68} - 23 q^{69} + 3 q^{70} - 63 q^{71} - 13 q^{72} + 2 q^{73} + q^{74} - 8 q^{75} - 14 q^{76} - 8 q^{77} + q^{78} - 51 q^{79} - 3 q^{80} + 4 q^{81} + 27 q^{82} - 24 q^{83} + 3 q^{84} + 31 q^{85} + 3 q^{86} - 33 q^{87} + 8 q^{88} - 35 q^{89} + 12 q^{90} - 4 q^{91} - 23 q^{92} + 17 q^{93} + 26 q^{94} - 30 q^{95} - 3 q^{96} + 5 q^{97} - 20 q^{98} - 31 q^{99}+O(q^{100}) \) 20 * q - 20 * q^2 + 3 * q^3 + 20 * q^4 - 3 * q^5 - 3 * q^6 + 20 * q^7 - 20 * q^8 + 13 * q^9 + 3 * q^10 - 8 * q^11 + 3 * q^12 - 4 * q^13 - 20 * q^14 - 25 * q^15 + 20 * q^16 + 9 * q^17 - 13 * q^18 - 14 * q^19 - 3 * q^20 + 3 * q^21 + 8 * q^22 - 23 * q^23 - 3 * q^24 + 31 * q^25 + 4 * q^26 - 21 * q^27 + 20 * q^28 - 48 * q^29 + 25 * q^30 - q^31 - 20 * q^32 - 29 * q^33 - 9 * q^34 - 3 * q^35 + 13 * q^36 - q^37 + 14 * q^38 - q^39 + 3 * q^40 - 27 * q^41 - 3 * q^42 - 3 * q^43 - 8 * q^44 - 12 * q^45 + 23 * q^46 - 26 * q^47 + 3 * q^48 + 20 * q^49 - 31 * q^50 - 17 * q^51 - 4 * q^52 - 43 * q^53 + 21 * q^54 - 16 * q^55 - 20 * q^56 - 25 * q^57 + 48 * q^58 - 19 * q^59 - 25 * q^60 + 9 * q^61 + q^62 + 13 * q^63 + 20 * q^64 - 87 * q^65 + 29 * q^66 + 32 * q^67 + 9 * q^68 - 23 * q^69 + 3 * q^70 - 63 * q^71 - 13 * q^72 + 2 * q^73 + q^74 - 8 * q^75 - 14 * q^76 - 8 * q^77 + q^78 - 51 * q^79 - 3 * q^80 + 4 * q^81 + 27 * q^82 - 24 * q^83 + 3 * q^84 + 31 * q^85 + 3 * q^86 - 33 * q^87 + 8 * q^88 - 35 * q^89 + 12 * q^90 - 4 * q^91 - 23 * q^92 + 17 * q^93 + 26 * q^94 - 30 * q^95 - 3 * q^96 + 5 * q^97 - 20 * q^98 - 31 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−2.57684	−1.48774	−0.743869	−	0.668325i	$-0.767012\pi$
$3$	−2.57684	−1.48774	−0.743869	+	0.668325i	$0.767012\pi$
$4$	1.00000	0.500000
$5$	4.04233	1.80779	0.903893	−	0.427759i	$-0.140697\pi$
$5$	4.04233	1.80779	0.903893	+	0.427759i	$0.140697\pi$
$6$	2.57684	1.05199
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.00000	−0.353553
$9$	3.64010	1.21337
$10$	−4.04233	−1.27830
$11$	−6.22537	−1.87702	−0.938511	−	0.345251i	$-0.887794\pi$
$11$	−6.22537	−1.87702	−0.938511	+	0.345251i	$0.887794\pi$
$12$	−2.57684	−0.743869
$13$	−3.30637	−0.917023	−0.458512	−	0.888688i	$-0.651617\pi$
$13$	−3.30637	−0.917023	−0.458512	+	0.888688i	$0.651617\pi$
$14$	−1.00000	−0.267261
$15$	−10.4164	−2.68951
$16$	1.00000	0.250000
$17$	3.51139	0.851638	0.425819	−	0.904808i	$-0.359986\pi$
$17$	3.51139	0.851638	0.425819	+	0.904808i	$0.359986\pi$
$18$	−3.64010	−0.857980
$19$	−3.74588	−0.859364	−0.429682	−	0.902980i	$-0.641374\pi$
$19$	−3.74588	−0.859364	−0.429682	+	0.902980i	$0.641374\pi$
$20$	4.04233	0.903893
$21$	−2.57684	−0.562312
$22$	6.22537	1.32725
$23$	−8.78740	−1.83230	−0.916150	−	0.400835i	$-0.868720\pi$
$23$	−8.78740	−1.83230	−0.916150	+	0.400835i	$0.868720\pi$
$24$	2.57684	0.525995
$25$	11.3404	2.26809
$26$	3.30637	0.648433
$27$	−1.64943	−0.317433
$28$	1.00000	0.188982
$29$	−0.0905660	−0.0168177	−0.00840884	−	0.999965i	$-0.502677\pi$
$29$	−0.0905660	−0.0168177	−0.00840884	+	0.999965i	$0.502677\pi$
$30$	10.4164	1.90177
$31$	9.21443	1.65496	0.827480	−	0.561495i	$-0.189774\pi$
$31$	9.21443	1.65496	0.827480	+	0.561495i	$0.189774\pi$
$32$	−1.00000	−0.176777
$33$	16.0418	2.79252
$34$	−3.51139	−0.602199
$35$	4.04233	0.683279
$36$	3.64010	0.606683
$37$	9.62250	1.58193	0.790965	−	0.611862i	$-0.209579\pi$
$37$	9.62250	1.58193	0.790965	+	0.611862i	$0.209579\pi$
$38$	3.74588	0.607662
$39$	8.51999	1.36429
$40$	−4.04233	−0.639149
$41$	1.10715	0.172908	0.0864541	−	0.996256i	$-0.472446\pi$
$41$	1.10715	0.172908	0.0864541	+	0.996256i	$0.472446\pi$
$42$	2.57684	0.397615
$43$	1.77302	0.270383	0.135192	−	0.990819i	$-0.456835\pi$
$43$	1.77302	0.270383	0.135192	+	0.990819i	$0.456835\pi$
$44$	−6.22537	−0.938511
$45$	14.7145	2.19351
$46$	8.78740	1.29563
$47$	−6.26009	−0.913129	−0.456564	−	0.889690i	$-0.650920\pi$
$47$	−6.26009	−0.913129	−0.456564	+	0.889690i	$0.650920\pi$
$48$	−2.57684	−0.371935
$49$	1.00000	0.142857
$50$	−11.3404	−1.60378
$51$	−9.04829	−1.26701
$52$	−3.30637	−0.458512
$53$	−0.187269	−0.0257233	−0.0128617	−	0.999917i	$-0.504094\pi$
$53$	−0.187269	−0.0257233	−0.0128617	+	0.999917i	$0.504094\pi$
$54$	1.64943	0.224459
$55$	−25.1650	−3.39325
$56$	−1.00000	−0.133631
$57$	9.65253	1.27851
$58$	0.0905660	0.0118919
$59$	−4.44265	−0.578383	−0.289192	−	0.957271i	$-0.593387\pi$
$59$	−4.44265	−0.578383	−0.289192	+	0.957271i	$0.593387\pi$
$60$	−10.4164	−1.34476
$61$	0.849612	0.108782	0.0543908	−	0.998520i	$-0.482678\pi$
$61$	0.849612	0.108782	0.0543908	+	0.998520i	$0.482678\pi$
$62$	−9.21443	−1.17023
$63$	3.64010	0.458609
$64$	1.00000	0.125000
$65$	−13.3655	−1.65778
$66$	−16.0418	−1.97461
$67$	13.5699	1.65782	0.828911	−	0.559380i	$-0.188961\pi$
$67$	13.5699	1.65782	0.828911	+	0.559380i	$0.188961\pi$
$68$	3.51139	0.425819
$69$	22.6437	2.72598
$70$	−4.04233	−0.483151
$71$	10.2441	1.21576	0.607878	−	0.794030i	$-0.292021\pi$
$71$	10.2441	1.21576	0.607878	+	0.794030i	$0.292021\pi$
$72$	−3.64010	−0.428990
$73$	0.441416	0.0516639	0.0258319	−	0.999666i	$-0.491777\pi$
$73$	0.441416	0.0516639	0.0258319	+	0.999666i	$0.491777\pi$
$74$	−9.62250	−1.11859
$75$	−29.2225	−3.37432
$76$	−3.74588	−0.429682
$77$	−6.22537	−0.709447
$78$	−8.51999	−0.964699
$79$	4.90674	0.552051	0.276025	−	0.961150i	$-0.410983\pi$
$79$	4.90674	0.552051	0.276025	+	0.961150i	$0.410983\pi$
$80$	4.04233	0.451946
$81$	−6.66998	−0.741108
$82$	−1.10715	−0.122265
$83$	7.93719	0.871220	0.435610	−	0.900136i	$-0.356533\pi$
$83$	7.93719	0.871220	0.435610	+	0.900136i	$0.356533\pi$
$84$	−2.57684	−0.281156
$85$	14.1942	1.53958
$86$	−1.77302	−0.191190
$87$	0.233374	0.0250203
$88$	6.22537	0.663627
$89$	−12.0685	−1.27926	−0.639631	−	0.768682i	$-0.720913\pi$
$89$	−12.0685	−1.27926	−0.639631	+	0.768682i	$0.720913\pi$
$90$	−14.7145	−1.55104
$91$	−3.30637	−0.346602
$92$	−8.78740	−0.916150
$93$	−23.7441	−2.46215
$94$	6.26009	0.645679
$95$	−15.1421	−1.55355
$96$	2.57684	0.262998
$97$	−4.98514	−0.506164	−0.253082	−	0.967445i	$-0.581444\pi$
$97$	−4.98514	−0.506164	−0.253082	+	0.967445i	$0.581444\pi$
$98$	−1.00000	−0.101015
$99$	−22.6610	−2.27751
$100$	11.3404	1.13404
$101$	−11.1297	−1.10745	−0.553724	−	0.832700i	$-0.686794\pi$
$101$	−11.1297	−1.10745	−0.553724	+	0.832700i	$0.686794\pi$
$102$	9.04829	0.895915
$103$	−16.1188	−1.58823	−0.794116	−	0.607766i	$-0.792066\pi$
$103$	−16.1188	−1.58823	−0.794116	+	0.607766i	$0.792066\pi$
$104$	3.30637	0.324217
$105$	−10.4164	−1.01654
$106$	0.187269	0.0181891
$107$	−3.85335	−0.372518	−0.186259	−	0.982501i	$-0.559636\pi$
$107$	−3.85335	−0.372518	−0.186259	+	0.982501i	$0.559636\pi$
$108$	−1.64943	−0.158717
$109$	−9.35454	−0.896003	−0.448001	−	0.894033i	$-0.647864\pi$
$109$	−9.35454	−0.896003	−0.448001	+	0.894033i	$0.647864\pi$
$110$	25.1650	2.39939
$111$	−24.7956	−2.35350
$112$	1.00000	0.0944911
$113$	−6.12263	−0.575968	−0.287984	−	0.957635i	$-0.592985\pi$
$113$	−6.12263	−0.575968	−0.287984	+	0.957635i	$0.592985\pi$
$114$	−9.65253	−0.904042
$115$	−35.5216	−3.31241
$116$	−0.0905660	−0.00840884
$117$	−12.0355	−1.11269
$118$	4.44265	0.408979
$119$	3.51139	0.321889
$120$	10.4164	0.950886
$121$	27.7553	2.52321
$122$	−0.849612	−0.0769202
$123$	−2.85295	−0.257242
$124$	9.21443	0.827480
$125$	25.6302	2.29243
$126$	−3.64010	−0.324286
$127$	−18.5230	−1.64365	−0.821824	−	0.569742i	$-0.807043\pi$
$127$	−18.5230	−1.64365	−0.821824	+	0.569742i	$0.807043\pi$
$128$	−1.00000	−0.0883883
$129$	−4.56879	−0.402260
$130$	13.3655	1.17223
$131$	−15.1539	−1.32400	−0.662002	−	0.749502i	$-0.730293\pi$
$131$	−15.1539	−1.32400	−0.662002	+	0.749502i	$0.730293\pi$
$132$	16.0418	1.39626
$133$	−3.74588	−0.324809
$134$	−13.5699	−1.17226
$135$	−6.66755	−0.573852
$136$	−3.51139	−0.301099
$137$	−9.91303	−0.846927	−0.423464	−	0.905913i	$-0.639186\pi$
$137$	−9.91303	−0.846927	−0.423464	+	0.905913i	$0.639186\pi$
$138$	−22.6437	−1.92756
$139$	8.88111	0.753286	0.376643	−	0.926359i	$-0.377078\pi$
$139$	8.88111	0.753286	0.376643	+	0.926359i	$0.377078\pi$
$140$	4.04233	0.341639
$141$	16.1313	1.35850
$142$	−10.2441	−0.859669
$143$	20.5834	1.72127
$144$	3.64010	0.303342
$145$	−0.366098	−0.0304028
$146$	−0.441416	−0.0365319
$147$	−2.57684	−0.212534
$148$	9.62250	0.790965
$149$	18.0838	1.48148	0.740742	−	0.671789i	$-0.234474\pi$
$149$	18.0838	1.48148	0.740742	+	0.671789i	$0.234474\pi$
$150$	29.2225	2.38601
$151$	−16.6528	−1.35518	−0.677591	−	0.735439i	$-0.736976\pi$
$151$	−16.6528	−1.35518	−0.677591	+	0.735439i	$0.736976\pi$
$152$	3.74588	0.303831
$153$	12.7818	1.03335
$154$	6.22537	0.501655
$155$	37.2478	2.99181
$156$	8.51999	0.682146
$157$	9.78656	0.781053	0.390526	−	0.920592i	$-0.372293\pi$
$157$	9.78656	0.781053	0.390526	+	0.920592i	$0.372293\pi$
$158$	−4.90674	−0.390359
$159$	0.482561	0.0382696
$160$	−4.04233	−0.319574
$161$	−8.78740	−0.692544
$162$	6.66998	0.524043
$163$	2.83266	0.221871	0.110936	−	0.993828i	$-0.464615\pi$
$163$	2.83266	0.221871	0.110936	+	0.993828i	$0.464615\pi$
$164$	1.10715	0.0864541
$165$	64.8462	5.04827
$166$	−7.93719	−0.616045
$167$	16.8375	1.30293	0.651463	−	0.758681i	$-0.274156\pi$
$167$	16.8375	1.30293	0.651463	+	0.758681i	$0.274156\pi$
$168$	2.57684	0.198807
$169$	−2.06789	−0.159068
$170$	−14.1942	−1.08865
$171$	−13.6354	−1.04272
$172$	1.77302	0.135192
$173$	−10.6207	−0.807476	−0.403738	−	0.914875i	$-0.632289\pi$
$173$	−10.6207	−0.807476	−0.403738	+	0.914875i	$0.632289\pi$
$174$	−0.233374	−0.0176920
$175$	11.3404	0.857257
$176$	−6.22537	−0.469255
$177$	11.4480	0.860483
$178$	12.0685	0.904575
$179$	2.70955	0.202521	0.101261	−	0.994860i	$-0.467712\pi$
$179$	2.70955	0.202521	0.101261	+	0.994860i	$0.467712\pi$
$180$	14.7145	1.09675
$181$	−11.8433	−0.880309	−0.440154	−	0.897922i	$-0.645076\pi$
$181$	−11.8433	−0.880309	−0.440154	+	0.897922i	$0.645076\pi$
$182$	3.30637	0.245085
$183$	−2.18931	−0.161839
$184$	8.78740	0.647816
$185$	38.8973	2.85979
$186$	23.7441	1.74100
$187$	−21.8597	−1.59854
$188$	−6.26009	−0.456564
$189$	−1.64943	−0.119979
$190$	15.1421	1.09852
$191$	5.60110	0.405281	0.202641	−	0.979253i	$-0.435048\pi$
$191$	5.60110	0.405281	0.202641	+	0.979253i	$0.435048\pi$
$192$	−2.57684	−0.185967
$193$	−1.62687	−0.117105	−0.0585524	−	0.998284i	$-0.518648\pi$
$193$	−1.62687	−0.117105	−0.0585524	+	0.998284i	$0.518648\pi$
$194$	4.98514	0.357912
$195$	34.4406	2.46635
$196$	1.00000	0.0714286
$197$	14.3187	1.02017	0.510084	−	0.860124i	$-0.329614\pi$
$197$	14.3187	1.02017	0.510084	+	0.860124i	$0.329614\pi$
$198$	22.6610	1.61045
$199$	−9.88135	−0.700470	−0.350235	−	0.936662i	$-0.613898\pi$
$199$	−9.88135	−0.700470	−0.350235	+	0.936662i	$0.613898\pi$
$200$	−11.3404	−0.801891
$201$	−34.9674	−2.46641
$202$	11.1297	0.783085
$203$	−0.0905660	−0.00635649
$204$	−9.04829	−0.633507
$205$	4.47548	0.312581
$206$	16.1188	1.12305
$207$	−31.9870	−2.22325
$208$	−3.30637	−0.229256
$209$	23.3195	1.61304
$210$	10.4164	0.718803
$211$	−9.00051	−0.619621	−0.309810	−	0.950798i	$-0.600266\pi$
$211$	−9.00051	−0.619621	−0.309810	+	0.950798i	$0.600266\pi$
$212$	−0.187269	−0.0128617
$213$	−26.3975	−1.80873
$214$	3.85335	0.263410
$215$	7.16714	0.488795
$216$	1.64943	0.112230
$217$	9.21443	0.625516
$218$	9.35454	0.633570
$219$	−1.13746	−0.0768624
$220$	−25.1650	−1.69663
$221$	−11.6100	−0.780972
$222$	24.7956	1.66417
$223$	−13.7831	−0.922986	−0.461493	−	0.887144i	$-0.652686\pi$
$223$	−13.7831	−0.922986	−0.461493	+	0.887144i	$0.652686\pi$
$224$	−1.00000	−0.0668153
$225$	41.2804	2.75202
$226$	6.12263	0.407271
$227$	7.48357	0.496702	0.248351	−	0.968670i	$-0.420111\pi$
$227$	7.48357	0.496702	0.248351	+	0.968670i	$0.420111\pi$
$228$	9.65253	0.639254
$229$	−15.0672	−0.995671	−0.497836	−	0.867271i	$-0.665872\pi$
$229$	−15.0672	−0.995671	−0.497836	+	0.867271i	$0.665872\pi$
$230$	35.5216	2.34223
$231$	16.0418	1.05547
$232$	0.0905660	0.00594595
$233$	−9.20522	−0.603054	−0.301527	−	0.953458i	$-0.597496\pi$
$233$	−9.20522	−0.603054	−0.301527	+	0.953458i	$0.597496\pi$
$234$	12.0355	0.786787
$235$	−25.3054	−1.65074
$236$	−4.44265	−0.289192
$237$	−12.6439	−0.821307
$238$	−3.51139	−0.227610
$239$	−19.3351	−1.25069	−0.625343	−	0.780350i	$-0.715041\pi$
$239$	−19.3351	−1.25069	−0.625343	+	0.780350i	$0.715041\pi$
$240$	−10.4164	−0.672378
$241$	−4.20718	−0.271008	−0.135504	−	0.990777i	$-0.543265\pi$
$241$	−4.20718	−0.271008	−0.135504	+	0.990777i	$0.543265\pi$
$242$	−27.7553	−1.78418
$243$	22.1358	1.42001
$244$	0.849612	0.0543908
$245$	4.04233	0.258255
$246$	2.85295	0.181898
$247$	12.3853	0.788056
$248$	−9.21443	−0.585117
$249$	−20.4529	−1.29615
$250$	−25.6302	−1.62100
$251$	2.96491	0.187143	0.0935717	−	0.995613i	$-0.470172\pi$
$251$	2.96491	0.187143	0.0935717	+	0.995613i	$0.470172\pi$
$252$	3.64010	0.229305
$253$	54.7049	3.43927
$254$	18.5230	1.16223
$255$	−36.5762	−2.29049
$256$	1.00000	0.0625000
$257$	−19.9336	−1.24342	−0.621711	−	0.783247i	$-0.713562\pi$
$257$	−19.9336	−1.24342	−0.621711	+	0.783247i	$0.713562\pi$
$258$	4.56879	0.284441
$259$	9.62250	0.597913
$260$	−13.3655	−0.828891
$261$	−0.329669	−0.0204060
$262$	15.1539	0.936212
$263$	−27.7946	−1.71389	−0.856944	−	0.515410i	$-0.827640\pi$
$263$	−27.7946	−1.71389	−0.856944	+	0.515410i	$0.827640\pi$
$264$	−16.0418	−0.987304
$265$	−0.757002	−0.0465022
$266$	3.74588	0.229675
$267$	31.0987	1.90321
$268$	13.5699	0.828911
$269$	−23.8861	−1.45636	−0.728181	−	0.685385i	$-0.759634\pi$
$269$	−23.8861	−1.45636	−0.728181	+	0.685385i	$0.759634\pi$
$270$	6.66755	0.405774
$271$	−9.97898	−0.606180	−0.303090	−	0.952962i	$-0.598018\pi$
$271$	−9.97898	−0.606180	−0.303090	+	0.952962i	$0.598018\pi$
$272$	3.51139	0.212909
$273$	8.51999	0.515654
$274$	9.91303	0.598868
$275$	−70.5985	−4.25725
$276$	22.6437	1.36299
$277$	−20.5650	−1.23563	−0.617816	−	0.786322i	$-0.711983\pi$
$277$	−20.5650	−1.23563	−0.617816	+	0.786322i	$0.711983\pi$
$278$	−8.88111	−0.532653
$279$	33.5414	2.00807
$280$	−4.04233	−0.241576
$281$	14.6841	0.875980	0.437990	−	0.898980i	$-0.355691\pi$
$281$	14.6841	0.875980	0.437990	+	0.898980i	$0.355691\pi$
$282$	−16.1313	−0.960602
$283$	27.4765	1.63331	0.816655	−	0.577125i	$-0.195826\pi$
$283$	27.4765	1.63331	0.816655	+	0.577125i	$0.195826\pi$
$284$	10.2441	0.607878
$285$	39.0187	2.31127
$286$	−20.5834	−1.21712
$287$	1.10715	0.0653532
$288$	−3.64010	−0.214495
$289$	−4.67012	−0.274713
$290$	0.366098	0.0214980
$291$	12.8459	0.753041
$292$	0.441416	0.0258319
$293$	1.64138	0.0958903	0.0479451	−	0.998850i	$-0.484733\pi$
$293$	1.64138	0.0958903	0.0479451	+	0.998850i	$0.484733\pi$
$294$	2.57684	0.150284
$295$	−17.9587	−1.04559
$296$	−9.62250	−0.559296
$297$	10.2683	0.595829
$298$	−18.0838	−1.04757
$299$	29.0545	1.68026
$300$	−29.2225	−1.68716
$301$	1.77302	0.102195
$302$	16.6528	0.958258
$303$	28.6795	1.64759
$304$	−3.74588	−0.214841
$305$	3.43441	0.196654
$306$	−12.7818	−0.730688
$307$	29.2411	1.66888	0.834439	−	0.551101i	$-0.185792\pi$
$307$	29.2411	1.66888	0.834439	+	0.551101i	$0.185792\pi$
$308$	−6.22537	−0.354724
$309$	41.5355	2.36287
$310$	−37.2478	−2.11553
$311$	−30.4958	−1.72926	−0.864628	−	0.502413i	$-0.832446\pi$
$311$	−30.4958	−1.72926	−0.864628	+	0.502413i	$0.832446\pi$
$312$	−8.51999	−0.482350
$313$	−11.6242	−0.657040	−0.328520	−	0.944497i	$-0.606550\pi$
$313$	−11.6242	−0.657040	−0.328520	+	0.944497i	$0.606550\pi$
$314$	−9.78656	−0.552288
$315$	14.7145	0.829068
$316$	4.90674	0.276025
$317$	−20.8486	−1.17097	−0.585486	−	0.810683i	$-0.699096\pi$
$317$	−20.8486	−1.17097	−0.585486	+	0.810683i	$0.699096\pi$
$318$	−0.482561	−0.0270607
$319$	0.563807	0.0315671
$320$	4.04233	0.225973
$321$	9.92947	0.554209
$322$	8.78740	0.489703
$323$	−13.1533	−0.731867
$324$	−6.66998	−0.370554
$325$	−37.4958	−2.07989
$326$	−2.83266	−0.156887
$327$	24.1052	1.33302
$328$	−1.10715	−0.0611323
$329$	−6.26009	−0.345130
$330$	−64.8462	−3.56967
$331$	0.118124	0.00649270	0.00324635	−	0.999995i	$-0.498967\pi$
$331$	0.118124	0.00649270	0.00324635	+	0.999995i	$0.498967\pi$
$332$	7.93719	0.435610
$333$	35.0269	1.91946
$334$	−16.8375	−0.921308
$335$	54.8539	2.99699
$336$	−2.57684	−0.140578
$337$	−9.21858	−0.502168	−0.251084	−	0.967965i	$-0.580787\pi$
$337$	−9.21858	−0.502168	−0.251084	+	0.967965i	$0.580787\pi$
$338$	2.06789	0.112478
$339$	15.7770	0.856890
$340$	14.1942	0.769789
$341$	−57.3633	−3.10639
$342$	13.6354	0.737316
$343$	1.00000	0.0539949
$344$	−1.77302	−0.0955949
$345$	91.5335	4.92800
$346$	10.6207	0.570972
$347$	25.1396	1.34957	0.674783	−	0.738016i	$-0.264237\pi$
$347$	25.1396	1.34957	0.674783	+	0.738016i	$0.264237\pi$
$348$	0.233374	0.0125102
$349$	28.6642	1.53436	0.767179	−	0.641433i	$-0.221660\pi$
$349$	28.6642	1.53436	0.767179	+	0.641433i	$0.221660\pi$
$350$	−11.3404	−0.606172
$351$	5.45364	0.291094
$352$	6.22537	0.331814
$353$	−4.67765	−0.248966	−0.124483	−	0.992222i	$-0.539727\pi$
$353$	−4.67765	−0.248966	−0.124483	+	0.992222i	$0.539727\pi$
$354$	−11.4480	−0.608454
$355$	41.4102	2.19783
$356$	−12.0685	−0.639631
$357$	−9.04829	−0.478887
$358$	−2.70955	−0.143204
$359$	−5.11298	−0.269853	−0.134926	−	0.990856i	$-0.543080\pi$
$359$	−5.11298	−0.269853	−0.134926	+	0.990856i	$0.543080\pi$
$360$	−14.7145	−0.775522
$361$	−4.96839	−0.261494
$362$	11.8433	0.622472
$363$	−71.5209	−3.75387
$364$	−3.30637	−0.173301
$365$	1.78435	0.0933972
$366$	2.18931	0.114437
$367$	−18.7670	−0.979631	−0.489815	−	0.871826i	$-0.662936\pi$
$367$	−18.7670	−0.979631	−0.489815	+	0.871826i	$0.662936\pi$
$368$	−8.78740	−0.458075
$369$	4.03015	0.209801
$370$	−38.8973	−2.02218
$371$	−0.187269	−0.00972250
$372$	−23.7441	−1.23107
$373$	−10.3973	−0.538351	−0.269175	−	0.963091i	$-0.586751\pi$
$373$	−10.3973	−0.538351	−0.269175	+	0.963091i	$0.586751\pi$
$374$	21.8597	1.13034
$375$	−66.0449	−3.41054
$376$	6.26009	0.322840
$377$	0.299445	0.0154222
$378$	1.64943	0.0848377
$379$	−9.24147	−0.474702	−0.237351	−	0.971424i	$-0.576279\pi$
$379$	−9.24147	−0.474702	−0.237351	+	0.971424i	$0.576279\pi$
$380$	−15.1421	−0.776773
$381$	47.7307	2.44532
$382$	−5.60110	−0.286577
$383$	−23.8224	−1.21727	−0.608633	−	0.793452i	$-0.708282\pi$
$383$	−23.8224	−1.21727	−0.608633	+	0.793452i	$0.708282\pi$
$384$	2.57684	0.131499
$385$	−25.1650	−1.28253
$386$	1.62687	0.0828057
$387$	6.45398	0.328074
$388$	−4.98514	−0.253082
$389$	−31.0495	−1.57427	−0.787136	−	0.616779i	$-0.788437\pi$
$389$	−31.0495	−1.57427	−0.787136	+	0.616779i	$0.788437\pi$
$390$	−34.4406	−1.74397
$391$	−30.8560	−1.56046
$392$	−1.00000	−0.0505076
$393$	39.0492	1.96977
$394$	−14.3187	−0.721368
$395$	19.8347	0.997990
$396$	−22.6610	−1.13876
$397$	24.8942	1.24940	0.624701	−	0.780864i	$-0.285221\pi$
$397$	24.8942	1.24940	0.624701	+	0.780864i	$0.285221\pi$
$398$	9.88135	0.495307
$399$	9.65253	0.483231
$400$	11.3404	0.567022
$401$	−38.4663	−1.92091	−0.960457	−	0.278427i	$-0.910187\pi$
$401$	−38.4663	−1.92091	−0.960457	+	0.278427i	$0.910187\pi$
$402$	34.9674	1.74401
$403$	−30.4663	−1.51764
$404$	−11.1297	−0.553724
$405$	−26.9623	−1.33977
$406$	0.0905660	0.00449472
$407$	−59.9037	−2.96931
$408$	9.04829	0.447957
$409$	3.92667	0.194161	0.0970806	−	0.995277i	$-0.469050\pi$
$409$	3.92667	0.194161	0.0970806	+	0.995277i	$0.469050\pi$
$410$	−4.47548	−0.221028
$411$	25.5443	1.26001
$412$	−16.1188	−0.794116
$413$	−4.44265	−0.218608
$414$	31.9870	1.57208
$415$	32.0848	1.57498
$416$	3.30637	0.162108
$417$	−22.8852	−1.12069
$418$	−23.3195	−1.14059
$419$	−10.0452	−0.490739	−0.245369	−	0.969430i	$-0.578909\pi$
$419$	−10.0452	−0.490739	−0.245369	+	0.969430i	$0.578909\pi$
$420$	−10.4164	−0.508270
$421$	−0.901592	−0.0439409	−0.0219705	−	0.999759i	$-0.506994\pi$
$421$	−0.901592	−0.0439409	−0.0219705	+	0.999759i	$0.506994\pi$
$422$	9.00051	0.438138
$423$	−22.7874	−1.10796
$424$	0.187269	0.00909456
$425$	39.8208	1.93159
$426$	26.3975	1.27896
$427$	0.849612	0.0411156
$428$	−3.85335	−0.186259
$429$	−53.0402	−2.56080
$430$	−7.16714	−0.345630
$431$	1.00000	0.0481683
$431$	1.00000	0.0481683
$432$	−1.64943	−0.0793584
$433$	−16.7323	−0.804103	−0.402052	−	0.915617i	$-0.631703\pi$
$433$	−16.7323	−0.804103	−0.402052	+	0.915617i	$0.631703\pi$
$434$	−9.21443	−0.442307
$435$	0.943375	0.0452314
$436$	−9.35454	−0.448001
$437$	32.9166	1.57461
$438$	1.13746	0.0543499
$439$	−6.49334	−0.309910	−0.154955	−	0.987922i	$-0.549523\pi$
$439$	−6.49334	−0.309910	−0.154955	+	0.987922i	$0.549523\pi$
$440$	25.1650	1.19970
$441$	3.64010	0.173338
$442$	11.6100	0.552230
$443$	12.2502	0.582026	0.291013	−	0.956719i	$-0.406008\pi$
$443$	12.2502	0.582026	0.291013	+	0.956719i	$0.406008\pi$
$444$	−24.7956	−1.17675
$445$	−48.7850	−2.31263
$446$	13.7831	0.652649
$447$	−46.5991	−2.20406
$448$	1.00000	0.0472456
$449$	−2.37950	−0.112295	−0.0561477	−	0.998422i	$-0.517882\pi$
$449$	−2.37950	−0.112295	−0.0561477	+	0.998422i	$0.517882\pi$
$450$	−41.2804	−1.94597
$451$	−6.89244	−0.324552
$452$	−6.12263	−0.287984
$453$	42.9115	2.01616
$454$	−7.48357	−0.351221
$455$	−13.3655	−0.626583
$456$	−9.65253	−0.452021
$457$	−4.92576	−0.230418	−0.115209	−	0.993341i	$-0.536754\pi$
$457$	−4.92576	−0.230418	−0.115209	+	0.993341i	$0.536754\pi$
$458$	15.0672	0.704046
$459$	−5.79181	−0.270338
$460$	−35.5216	−1.65620
$461$	29.4557	1.37189	0.685945	−	0.727654i	$-0.259389\pi$
$461$	29.4557	1.37189	0.685945	+	0.727654i	$0.259389\pi$
$462$	−16.0418	−0.746332
$463$	7.97821	0.370779	0.185389	−	0.982665i	$-0.440645\pi$
$463$	7.97821	0.370779	0.185389	+	0.982665i	$0.440645\pi$
$464$	−0.0905660	−0.00420442
$465$	−95.9815	−4.45104
$466$	9.20522	0.426424
$467$	4.02212	0.186122	0.0930608	−	0.995660i	$-0.470335\pi$
$467$	4.02212	0.186122	0.0930608	+	0.995660i	$0.470335\pi$
$468$	−12.0355	−0.556343
$469$	13.5699	0.626598
$470$	25.3054	1.16725
$471$	−25.2184	−1.16200
$472$	4.44265	0.204489
$473$	−11.0377	−0.507515
$474$	12.6439	0.580752
$475$	−42.4800	−1.94911
$476$	3.51139	0.160944
$477$	−0.681676	−0.0312118
$478$	19.3351	0.884368
$479$	−11.1633	−0.510063	−0.255031	−	0.966933i	$-0.582086\pi$
$479$	−11.1633	−0.510063	−0.255031	+	0.966933i	$0.582086\pi$
$480$	10.4164	0.475443
$481$	−31.8156	−1.45067
$482$	4.20718	0.191632
$483$	22.6437	1.03033
$484$	27.7553	1.26160
$485$	−20.1516	−0.915037
$486$	−22.1358	−1.00410
$487$	27.7716	1.25845	0.629226	−	0.777223i	$-0.283372\pi$
$487$	27.7716	1.25845	0.629226	+	0.777223i	$0.283372\pi$
$488$	−0.849612	−0.0384601
$489$	−7.29931	−0.330086
$490$	−4.04233	−0.182614
$491$	21.3443	0.963257	0.481628	−	0.876376i	$-0.340045\pi$
$491$	21.3443	0.963257	0.481628	+	0.876376i	$0.340045\pi$
$492$	−2.85295	−0.128621
$493$	−0.318013	−0.0143226
$494$	−12.3853	−0.557240
$495$	−91.6032	−4.11726
$496$	9.21443	0.413740
$497$	10.2441	0.459513
$498$	20.4529	0.916515
$499$	10.9610	0.490682	0.245341	−	0.969437i	$-0.421100\pi$
$499$	10.9610	0.490682	0.245341	+	0.969437i	$0.421100\pi$
$500$	25.6302	1.14622
$501$	−43.3876	−1.93841
$502$	−2.96491	−0.132330
$503$	−26.5676	−1.18459	−0.592295	−	0.805721i	$-0.701778\pi$
$503$	−26.5676	−1.18459	−0.592295	+	0.805721i	$0.701778\pi$
$504$	−3.64010	−0.162143
$505$	−44.9900	−2.00203
$506$	−54.7049	−2.43193
$507$	5.32861	0.236652
$508$	−18.5230	−0.821824
$509$	−14.4471	−0.640357	−0.320179	−	0.947357i	$-0.603743\pi$
$509$	−14.4471	−0.640357	−0.320179	+	0.947357i	$0.603743\pi$
$510$	36.5762	1.61962
$511$	0.441416	0.0195271
$512$	−1.00000	−0.0441942
$513$	6.17858	0.272791
$514$	19.9336	0.879232
$515$	−65.1575	−2.87118
$516$	−4.56879	−0.201130
$517$	38.9714	1.71396
$518$	−9.62250	−0.422788
$519$	27.3678	1.20131
$520$	13.3655	0.586114
$521$	4.63095	0.202886	0.101443	−	0.994841i	$-0.467654\pi$
$521$	4.63095	0.202886	0.101443	+	0.994841i	$0.467654\pi$
$522$	0.329669	0.0144292
$523$	11.2882	0.493600	0.246800	−	0.969066i	$-0.420621\pi$
$523$	11.2882	0.493600	0.246800	+	0.969066i	$0.420621\pi$
$524$	−15.1539	−0.662002
$525$	−29.2225	−1.27537
$526$	27.7946	1.21190
$527$	32.3555	1.40943
$528$	16.0418	0.698129
$529$	54.2185	2.35733
$530$	0.757002	0.0328820
$531$	−16.1717	−0.701791
$532$	−3.74588	−0.162404
$533$	−3.66066	−0.158561
$534$	−31.0987	−1.34577
$535$	−15.5765	−0.673432
$536$	−13.5699	−0.586129
$537$	−6.98207	−0.301298
$538$	23.8861	1.02980
$539$	−6.22537	−0.268146
$540$	−6.66755	−0.286926
$541$	−16.6134	−0.714268	−0.357134	−	0.934053i	$-0.616246\pi$
$541$	−16.6134	−0.714268	−0.357134	+	0.934053i	$0.616246\pi$
$542$	9.97898	0.428634
$543$	30.5184	1.30967
$544$	−3.51139	−0.150550
$545$	−37.8142	−1.61978
$546$	−8.51999	−0.364622
$547$	−25.2250	−1.07854	−0.539272	−	0.842132i	$-0.681301\pi$
$547$	−25.2250	−1.07854	−0.539272	+	0.842132i	$0.681301\pi$
$548$	−9.91303	−0.423464
$549$	3.09267	0.131992
$550$	70.5985	3.01033
$551$	0.339249	0.0144525
$552$	−22.6437	−0.963781
$553$	4.90674	0.208656
$554$	20.5650	0.873724
$555$	−100.232	−4.25462
$556$	8.88111	0.376643
$557$	7.79259	0.330183	0.165091	−	0.986278i	$-0.447208\pi$
$557$	7.79259	0.330183	0.165091	+	0.986278i	$0.447208\pi$
$558$	−33.5414	−1.41992
$559$	−5.86227	−0.247948
$560$	4.04233	0.170820
$561$	56.3290	2.37821
$562$	−14.6841	−0.619411
$563$	35.3370	1.48928	0.744639	−	0.667467i	$-0.232622\pi$
$563$	35.3370	1.48928	0.744639	+	0.667467i	$0.232622\pi$
$564$	16.1313	0.679248
$565$	−24.7497	−1.04123
$566$	−27.4765	−1.15493
$567$	−6.66998	−0.280113
$568$	−10.2441	−0.429835
$569$	−3.68815	−0.154615	−0.0773077	−	0.997007i	$-0.524632\pi$
$569$	−3.68815	−0.154615	−0.0773077	+	0.997007i	$0.524632\pi$
$570$	−39.0187	−1.63431
$571$	−28.3286	−1.18551	−0.592757	−	0.805381i	$-0.701961\pi$
$571$	−28.3286	−1.18551	−0.592757	+	0.805381i	$0.701961\pi$
$572$	20.5834	0.860636
$573$	−14.4331	−0.602953
$574$	−1.10715	−0.0462117
$575$	−99.6531	−4.15582
$576$	3.64010	0.151671
$577$	33.7570	1.40532	0.702661	−	0.711525i	$-0.251995\pi$
$577$	33.7570	1.40532	0.702661	+	0.711525i	$0.251995\pi$
$578$	4.67012	0.194251
$579$	4.19219	0.174221
$580$	−0.366098	−0.0152014
$581$	7.93719	0.329290
$582$	−12.8459	−0.532480
$583$	1.16582	0.0482832
$584$	−0.441416	−0.0182659
$585$	−48.6516	−2.01150
$586$	−1.64138	−0.0678047
$587$	22.1321	0.913491	0.456745	−	0.889598i	$-0.349015\pi$
$587$	22.1321	0.913491	0.456745	+	0.889598i	$0.349015\pi$
$588$	−2.57684	−0.106267
$589$	−34.5161	−1.42221
$590$	17.9587	0.739346
$591$	−36.8971	−1.51774
$592$	9.62250	0.395482
$593$	25.1662	1.03345	0.516725	−	0.856151i	$-0.327151\pi$
$593$	25.1662	1.03345	0.516725	+	0.856151i	$0.327151\pi$
$594$	−10.2683	−0.421315
$595$	14.1942	0.581906
$596$	18.0838	0.740742
$597$	25.4626	1.04212
$598$	−29.0545	−1.18812
$599$	−12.0314	−0.491592	−0.245796	−	0.969322i	$-0.579049\pi$
$599$	−12.0314	−0.491592	−0.245796	+	0.969322i	$0.579049\pi$
$600$	29.2225	1.19300
$601$	−26.4258	−1.07793	−0.538966	−	0.842328i	$-0.681185\pi$
$601$	−26.4258	−1.07793	−0.538966	+	0.842328i	$0.681185\pi$
$602$	−1.77302	−0.0722630
$603$	49.3957	2.01155
$604$	−16.6528	−0.677591
$605$	112.196	4.56142
$606$	−28.6795	−1.16503
$607$	−11.4242	−0.463694	−0.231847	−	0.972752i	$-0.574477\pi$
$607$	−11.4242	−0.463694	−0.231847	+	0.972752i	$0.574477\pi$
$608$	3.74588	0.151915
$609$	0.233374	0.00945679
$610$	−3.43441	−0.139055
$611$	20.6982	0.837360
$612$	12.7818	0.516674
$613$	13.4846	0.544639	0.272319	−	0.962207i	$-0.412209\pi$
$613$	13.4846	0.544639	0.272319	+	0.962207i	$0.412209\pi$
$614$	−29.2411	−1.18007
$615$	−11.5326	−0.465039
$616$	6.22537	0.250827
$617$	47.1014	1.89623	0.948116	−	0.317923i	$-0.102985\pi$
$617$	47.1014	1.89623	0.948116	+	0.317923i	$0.102985\pi$
$618$	−41.5355	−1.67080
$619$	47.8830	1.92458	0.962290	−	0.272026i	$-0.0876936\pi$
$619$	47.8830	1.92458	0.962290	+	0.272026i	$0.0876936\pi$
$620$	37.2478	1.49591
$621$	14.4942	0.581634
$622$	30.4958	1.22277
$623$	−12.0685	−0.483516
$624$	8.51999	0.341073
$625$	46.9035	1.87614
$626$	11.6242	0.464598
$627$	−60.0906	−2.39979
$628$	9.78656	0.390526
$629$	33.7884	1.34723
$630$	−14.7145	−0.586239
$631$	8.69080	0.345975	0.172988	−	0.984924i	$-0.444658\pi$
$631$	8.69080	0.345975	0.172988	+	0.984924i	$0.444658\pi$
$632$	−4.90674	−0.195179
$633$	23.1929	0.921834
$634$	20.8486	0.828002
$635$	−74.8760	−2.97136
$636$	0.482561	0.0191348
$637$	−3.30637	−0.131003
$638$	−0.563807	−0.0223213
$639$	37.2897	1.47516
$640$	−4.04233	−0.159787
$641$	−47.3938	−1.87194	−0.935971	−	0.352077i	$-0.885476\pi$
$641$	−47.3938	−1.87194	−0.935971	+	0.352077i	$0.885476\pi$
$642$	−9.92947	−0.391885
$643$	−4.33163	−0.170823	−0.0854114	−	0.996346i	$-0.527220\pi$
$643$	−4.33163	−0.170823	−0.0854114	+	0.996346i	$0.527220\pi$
$644$	−8.78740	−0.346272
$645$	−18.4686	−0.727199
$646$	13.1533	0.517508
$647$	28.2914	1.11225	0.556125	−	0.831099i	$-0.312288\pi$
$647$	28.2914	1.11225	0.556125	+	0.831099i	$0.312288\pi$
$648$	6.66998	0.262021
$649$	27.6571	1.08564
$650$	37.4958	1.47071
$651$	−23.7441	−0.930604
$652$	2.83266	0.110936
$653$	−29.4679	−1.15317	−0.576584	−	0.817038i	$-0.695615\pi$
$653$	−29.4679	−1.15317	−0.576584	+	0.817038i	$0.695615\pi$
$654$	−24.1052	−0.942586
$655$	−61.2572	−2.39352
$656$	1.10715	0.0432270
$657$	1.60680	0.0626872
$658$	6.26009	0.244044
$659$	4.13095	0.160919	0.0804595	−	0.996758i	$-0.474361\pi$
$659$	4.13095	0.160919	0.0804595	+	0.996758i	$0.474361\pi$
$660$	64.8462	2.52414
$661$	−47.4990	−1.84750	−0.923748	−	0.383000i	$-0.874891\pi$
$661$	−47.4990	−1.84750	−0.923748	+	0.383000i	$0.874891\pi$
$662$	−0.118124	−0.00459103
$663$	29.9171	1.16188
$664$	−7.93719	−0.308023
$665$	−15.1421	−0.587185
$666$	−35.0269	−1.35726
$667$	0.795840	0.0308151
$668$	16.8375	0.651463
$669$	35.5169	1.37316
$670$	−54.8539	−2.11919
$671$	−5.28915	−0.204185
$672$	2.57684	0.0994037
$673$	−41.6797	−1.60664	−0.803318	−	0.595551i	$-0.796934\pi$
$673$	−41.6797	−1.60664	−0.803318	+	0.595551i	$0.796934\pi$
$674$	9.21858	0.355087
$675$	−18.7053	−0.719968
$676$	−2.06789	−0.0795341
$677$	30.2268	1.16171	0.580855	−	0.814007i	$-0.302718\pi$
$677$	30.2268	1.16171	0.580855	+	0.814007i	$0.302718\pi$
$678$	−15.7770	−0.605913
$679$	−4.98514	−0.191312
$680$	−14.1942	−0.544323
$681$	−19.2840	−0.738963
$682$	57.3633	2.19655
$683$	16.7449	0.640726	0.320363	−	0.947295i	$-0.396195\pi$
$683$	16.7449	0.640726	0.320363	+	0.947295i	$0.396195\pi$
$684$	−13.6354	−0.521361
$685$	−40.0717	−1.53106
$686$	−1.00000	−0.0381802
$687$	38.8259	1.48130
$688$	1.77302	0.0675958
$689$	0.619180	0.0235889
$690$	−91.5335	−3.48462
$691$	−34.6104	−1.31664	−0.658322	−	0.752737i	$-0.728733\pi$
$691$	−34.6104	−1.31664	−0.658322	+	0.752737i	$0.728733\pi$
$692$	−10.6207	−0.403738
$693$	−22.6610	−0.860820
$694$	−25.1396	−0.954288
$695$	35.9004	1.36178
$696$	−0.233374	−0.00884602
$697$	3.88765	0.147255
$698$	−28.6642	−1.08496
$699$	23.7204	0.897187
$700$	11.3404	0.428629
$701$	−40.6829	−1.53657	−0.768286	−	0.640107i	$-0.778890\pi$
$701$	−40.6829	−1.53657	−0.768286	+	0.640107i	$0.778890\pi$
$702$	−5.45364	−0.205834
$703$	−36.0447	−1.35945
$704$	−6.22537	−0.234628
$705$	65.2079	2.45587
$706$	4.67765	0.176046
$707$	−11.1297	−0.418576
$708$	11.4480	0.430242
$709$	−40.1701	−1.50862	−0.754310	−	0.656518i	$-0.772028\pi$
$709$	−40.1701	−1.50862	−0.754310	+	0.656518i	$0.772028\pi$
$710$	−41.4102	−1.55410
$711$	17.8610	0.669840
$712$	12.0685	0.452288
$713$	−80.9709	−3.03238
$714$	9.04829	0.338624
$715$	83.2050	3.11169
$716$	2.70955	0.101261
$717$	49.8235	1.86069
$718$	5.11298	0.190815
$719$	−19.8743	−0.741187	−0.370593	−	0.928795i	$-0.620846\pi$
$719$	−19.8743	−0.741187	−0.370593	+	0.928795i	$0.620846\pi$
$720$	14.7145	0.548377
$721$	−16.1188	−0.600295
$722$	4.96839	0.184904
$723$	10.8412	0.403189
$724$	−11.8433	−0.440154
$725$	−1.02706	−0.0381440
$726$	71.5209	2.65439
$727$	28.9454	1.07353	0.536763	−	0.843733i	$-0.319647\pi$
$727$	28.9454	1.07353	0.536763	+	0.843733i	$0.319647\pi$
$728$	3.30637	0.122542
$729$	−37.0303	−1.37149
$730$	−1.78435	−0.0660418
$731$	6.22578	0.230269
$732$	−2.18931	−0.0809193
$733$	24.7637	0.914670	0.457335	−	0.889295i	$-0.348804\pi$
$733$	24.7637	0.914670	0.457335	+	0.889295i	$0.348804\pi$
$734$	18.7670	0.692704
$735$	−10.4164	−0.384216
$736$	8.78740	0.323908
$737$	−84.4775	−3.11177
$738$	−4.03015	−0.148352
$739$	46.0981	1.69575	0.847873	−	0.530199i	$-0.177883\pi$
$739$	46.0981	1.69575	0.847873	+	0.530199i	$0.177883\pi$
$740$	38.8973	1.42989
$741$	−31.9149	−1.17242
$742$	0.187269	0.00687484
$743$	−4.20064	−0.154107	−0.0770533	−	0.997027i	$-0.524551\pi$
$743$	−4.20064	−0.154107	−0.0770533	+	0.997027i	$0.524551\pi$
$744$	23.7441	0.870501
$745$	73.1008	2.67821
$746$	10.3973	0.380671
$747$	28.8922	1.05711
$748$	−21.8597	−0.799271
$749$	−3.85335	−0.140798
$750$	66.0449	2.41162
$751$	14.9298	0.544795	0.272397	−	0.962185i	$-0.412184\pi$
$751$	14.9298	0.544795	0.272397	+	0.962185i	$0.412184\pi$
$752$	−6.26009	−0.228282
$753$	−7.64009	−0.278420
$754$	−0.299445	−0.0109051
$755$	−67.3160	−2.44988
$756$	−1.64943	−0.0599893
$757$	−25.0897	−0.911900	−0.455950	−	0.890006i	$-0.650700\pi$
$757$	−25.0897	−0.911900	−0.455950	+	0.890006i	$0.650700\pi$
$758$	9.24147	0.335665
$759$	−140.966	−5.11673
$760$	15.1421	0.549261
$761$	30.0113	1.08791	0.543955	−	0.839114i	$-0.316926\pi$
$761$	30.0113	1.08791	0.543955	+	0.839114i	$0.316926\pi$
$762$	−47.7307	−1.72910
$763$	−9.35454	−0.338657
$764$	5.60110	0.202641
$765$	51.6684	1.86807
$766$	23.8224	0.860737
$767$	14.6891	0.530391
$768$	−2.57684	−0.0929837
$769$	−40.0548	−1.44441	−0.722207	−	0.691677i	$-0.756872\pi$
$769$	−40.0548	−1.44441	−0.722207	+	0.691677i	$0.756872\pi$
$770$	25.1650	0.906885
$771$	51.3656	1.84989
$772$	−1.62687	−0.0585524
$773$	9.31017	0.334864	0.167432	−	0.985884i	$-0.446453\pi$
$773$	9.31017	0.334864	0.167432	+	0.985884i	$0.446453\pi$
$774$	−6.45398	−0.231983
$775$	104.496	3.75360
$776$	4.98514	0.178956
$777$	−24.7956	−0.889538
$778$	31.0495	1.11318
$779$	−4.14726	−0.148591
$780$	34.4406	1.23317
$781$	−63.7736	−2.28200
$782$	30.8560	1.10341
$783$	0.149383	0.00533850
$784$	1.00000	0.0357143
$785$	39.5605	1.41198
$786$	−39.0492	−1.39284
$787$	35.0944	1.25098	0.625490	−	0.780232i	$-0.284899\pi$
$787$	35.0944	1.25098	0.625490	+	0.780232i	$0.284899\pi$
$788$	14.3187	0.510084
$789$	71.6222	2.54982
$790$	−19.8347	−0.705685
$791$	−6.12263	−0.217696
$792$	22.6610	0.805223
$793$	−2.80913	−0.0997553
$794$	−24.8942	−0.883461
$795$	1.95067	0.0691832
$796$	−9.88135	−0.350235
$797$	−12.1614	−0.430779	−0.215389	−	0.976528i	$-0.569102\pi$
$797$	−12.1614	−0.430779	−0.215389	+	0.976528i	$0.569102\pi$
$798$	−9.65253	−0.341696
$799$	−21.9816	−0.777655
$800$	−11.3404	−0.400945
$801$	−43.9307	−1.55221
$802$	38.4663	1.35829
$803$	−2.74798	−0.0969742
$804$	−34.9674	−1.23320
$805$	−35.5216	−1.25197
$806$	30.4663	1.07313
$807$	61.5506	2.16668
$808$	11.1297	0.391542
$809$	41.3954	1.45538	0.727692	−	0.685904i	$-0.240593\pi$
$809$	41.3954	1.45538	0.727692	+	0.685904i	$0.240593\pi$
$810$	26.9623	0.947357
$811$	29.1038	1.02197	0.510986	−	0.859589i	$-0.329280\pi$
$811$	29.1038	1.02197	0.510986	+	0.859589i	$0.329280\pi$
$812$	−0.0905660	−0.00317824
$813$	25.7142	0.901837
$814$	59.9037	2.09962
$815$	11.4506	0.401095
$816$	−9.04829	−0.316754
$817$	−6.64153	−0.232358
$818$	−3.92667	−0.137293
$819$	−12.0355	−0.420556
$820$	4.47548	0.156290
$821$	−4.54897	−0.158760	−0.0793801	−	0.996844i	$-0.525294\pi$
$821$	−4.54897	−0.158760	−0.0793801	+	0.996844i	$0.525294\pi$
$822$	−25.5443	−0.890959
$823$	−39.6969	−1.38375	−0.691873	−	0.722019i	$-0.743214\pi$
$823$	−39.6969	−1.38375	−0.691873	+	0.722019i	$0.743214\pi$
$824$	16.1188	0.561525
$825$	181.921	6.33368
$826$	4.44265	0.154579
$827$	−13.7574	−0.478391	−0.239196	−	0.970971i	$-0.576884\pi$
$827$	−13.7574	−0.478391	−0.239196	+	0.970971i	$0.576884\pi$
$828$	−31.9870	−1.11163
$829$	49.3082	1.71254	0.856272	−	0.516526i	$-0.172775\pi$
$829$	49.3082	1.71254	0.856272	+	0.516526i	$0.172775\pi$
$830$	−32.0848	−1.11368
$831$	52.9928	1.83830
$832$	−3.30637	−0.114628
$833$	3.51139	0.121663
$834$	22.8852	0.792449
$835$	68.0628	2.35541
$836$	23.3195	0.806522
$837$	−15.1986	−0.525340
$838$	10.0452	0.347005
$839$	−5.57568	−0.192494	−0.0962470	−	0.995357i	$-0.530684\pi$
$839$	−5.57568	−0.192494	−0.0962470	+	0.995357i	$0.530684\pi$
$840$	10.4164	0.359401
$841$	−28.9918	−0.999717
$842$	0.901592	0.0310709
$843$	−37.8385	−1.30323
$844$	−9.00051	−0.309810
$845$	−8.35909	−0.287561
$846$	22.7874	0.783446
$847$	27.7553	0.953683
$848$	−0.187269	−0.00643083
$849$	−70.8026	−2.42994
$850$	−39.8208	−1.36584
$851$	−84.5568	−2.89857
$852$	−26.3975	−0.904364
$853$	−3.85924	−0.132138	−0.0660690	−	0.997815i	$-0.521046\pi$
$853$	−3.85924	−0.132138	−0.0660690	+	0.997815i	$0.521046\pi$
$854$	−0.849612	−0.0290731
$855$	−55.1187	−1.88502
$856$	3.85335	0.131705
$857$	−10.3770	−0.354470	−0.177235	−	0.984169i	$-0.556715\pi$
$857$	−10.3770	−0.354470	−0.177235	+	0.984169i	$0.556715\pi$
$858$	53.0402	1.81076
$859$	−13.2946	−0.453607	−0.226804	−	0.973940i	$-0.572828\pi$
$859$	−13.2946	−0.453607	−0.226804	+	0.973940i	$0.572828\pi$
$860$	7.16714	0.244398
$861$	−2.85295	−0.0972284
$862$	−1.00000	−0.0340601
$863$	51.8006	1.76331	0.881657	−	0.471891i	$-0.156428\pi$
$863$	51.8006	1.76331	0.881657	+	0.471891i	$0.156428\pi$
$864$	1.64943	0.0561148
$865$	−42.9323	−1.45974
$866$	16.7323	0.568587
$867$	12.0341	0.408701
$868$	9.21443	0.312758
$869$	−30.5463	−1.03621
$870$	−0.943375	−0.0319834
$871$	−44.8671	−1.52026
$872$	9.35454	0.316785
$873$	−18.1464	−0.614163
$874$	−32.9166	−1.11342
$875$	25.6302	0.866459
$876$	−1.13746	−0.0384312
$877$	20.2396	0.683441	0.341721	−	0.939802i	$-0.388990\pi$
$877$	20.2396	0.683441	0.341721	+	0.939802i	$0.388990\pi$
$878$	6.49334	0.219140
$879$	−4.22957	−0.142660
$880$	−25.1650	−0.848313
$881$	54.0798	1.82200	0.910998	−	0.412410i	$-0.135313\pi$
$881$	54.0798	1.82200	0.910998	+	0.412410i	$0.135313\pi$
$882$	−3.64010	−0.122569
$883$	22.2669	0.749340	0.374670	−	0.927158i	$-0.377756\pi$
$883$	22.2669	0.749340	0.374670	+	0.927158i	$0.377756\pi$
$884$	−11.6100	−0.390486
$885$	46.2766	1.55557
$886$	−12.2502	−0.411555
$887$	−5.04970	−0.169552	−0.0847761	−	0.996400i	$-0.527017\pi$
$887$	−5.04970	−0.169552	−0.0847761	+	0.996400i	$0.527017\pi$
$888$	24.7956	0.832087
$889$	−18.5230	−0.621240
$890$	48.7850	1.63528
$891$	41.5231	1.39108
$892$	−13.7831	−0.461493
$893$	23.4496	0.784709
$894$	46.5991	1.55851
$895$	10.9529	0.366115
$896$	−1.00000	−0.0334077
$897$	−74.8686	−2.49979
$898$	2.37950	0.0794048
$899$	−0.834514	−0.0278326
$900$	41.2804	1.37601
$901$	−0.657573	−0.0219069
$902$	6.89244	0.229493
$903$	−4.56879	−0.152040
$904$	6.12263	0.203636
$905$	−47.8747	−1.59141
$906$	−42.9115	−1.42564
$907$	−6.84423	−0.227259	−0.113629	−	0.993523i	$-0.536248\pi$
$907$	−6.84423	−0.227259	−0.113629	+	0.993523i	$0.536248\pi$
$908$	7.48357	0.248351
$909$	−40.5133	−1.34374
$910$	13.3655	0.443061
$911$	9.37310	0.310545	0.155272	−	0.987872i	$-0.450375\pi$
$911$	9.37310	0.310545	0.155272	+	0.987872i	$0.450375\pi$
$912$	9.65253	0.319627
$913$	−49.4120	−1.63530
$914$	4.92576	0.162930
$915$	−8.84993	−0.292570
$916$	−15.0672	−0.497836
$917$	−15.1539	−0.500427
$918$	5.79181	0.191158
$919$	−12.0290	−0.396799	−0.198400	−	0.980121i	$-0.563574\pi$
$919$	−12.0290	−0.396799	−0.198400	+	0.980121i	$0.563574\pi$
$920$	35.5216	1.17111
$921$	−75.3496	−2.48285
$922$	−29.4557	−0.970072
$923$	−33.8710	−1.11488
$924$	16.0418	0.527736
$925$	109.123	3.58796
$926$	−7.97821	−0.262180
$927$	−58.6740	−1.92711
$928$	0.0905660	0.00297297
$929$	−19.5932	−0.642832	−0.321416	−	0.946938i	$-0.604159\pi$
$929$	−19.5932	−0.642832	−0.321416	+	0.946938i	$0.604159\pi$
$930$	95.9815	3.14736
$931$	−3.74588	−0.122766
$932$	−9.20522	−0.301527
$933$	78.5827	2.57268
$934$	−4.02212	−0.131608
$935$	−88.3643	−2.88982
$936$	12.0355	0.393394
$937$	16.4504	0.537412	0.268706	−	0.963222i	$-0.413404\pi$
$937$	16.4504	0.537412	0.268706	+	0.963222i	$0.413404\pi$
$938$	−13.5699	−0.443072
$939$	29.9538	0.977504
$940$	−25.3054	−0.825370
$941$	−11.9997	−0.391178	−0.195589	−	0.980686i	$-0.562662\pi$
$941$	−11.9997	−0.391178	−0.195589	+	0.980686i	$0.562662\pi$
$942$	25.2184	0.821660
$943$	−9.72900	−0.316820
$944$	−4.44265	−0.144596
$945$	−6.66755	−0.216896
$946$	11.0377	0.358867
$947$	−22.1752	−0.720596	−0.360298	−	0.932837i	$-0.617325\pi$
$947$	−22.1752	−0.720596	−0.360298	+	0.932837i	$0.617325\pi$
$948$	−12.6439	−0.410654
$949$	−1.45949	−0.0473770
$950$	42.4800	1.37823
$951$	53.7234	1.74210
$952$	−3.51139	−0.113805
$953$	12.2147	0.395672	0.197836	−	0.980235i	$-0.436609\pi$
$953$	12.2147	0.395672	0.197836	+	0.980235i	$0.436609\pi$
$954$	0.681676	0.0220701
$955$	22.6415	0.732662
$956$	−19.3351	−0.625343
$957$	−1.45284	−0.0469637
$958$	11.1633	0.360669
$959$	−9.91303	−0.320108
$960$	−10.4164	−0.336189
$961$	53.9057	1.73889
$962$	31.8156	1.02578
$963$	−14.0266	−0.452000
$964$	−4.20718	−0.135504
$965$	−6.57636	−0.211701
$966$	−22.6437	−0.728550
$967$	52.0019	1.67227	0.836135	−	0.548524i	$-0.184810\pi$
$967$	52.0019	1.67227	0.836135	+	0.548524i	$0.184810\pi$
$968$	−27.7553	−0.892089
$969$	33.8938	1.08883
$970$	20.1516	0.647029
$971$	17.7326	0.569065	0.284532	−	0.958666i	$-0.408162\pi$
$971$	17.7326	0.569065	0.284532	+	0.958666i	$0.408162\pi$
$972$	22.1358	0.710005
$973$	8.88111	0.284715
$974$	−27.7716	−0.889860
$975$	96.6206	3.09433
$976$	0.849612	0.0271954
$977$	17.7315	0.567282	0.283641	−	0.958931i	$-0.408458\pi$
$977$	17.7315	0.567282	0.283641	+	0.958931i	$0.408458\pi$
$978$	7.29931	0.233406
$979$	75.1312	2.40120
$980$	4.04233	0.129128
$981$	−34.0515	−1.08718
$982$	−21.3443	−0.681125
$983$	−54.6088	−1.74175	−0.870876	−	0.491503i	$-0.836448\pi$
$983$	−54.6088	−1.74175	−0.870876	+	0.491503i	$0.836448\pi$
$984$	2.85295	0.0909489
$985$	57.8811	1.84425
$986$	0.318013	0.0101276
$987$	16.1313	0.513463
$988$	12.3853	0.394028
$989$	−15.5803	−0.495423
$990$	91.6032	2.91134
$991$	−44.9326	−1.42733	−0.713665	−	0.700487i	$-0.752966\pi$
$991$	−44.9326	−1.42733	−0.713665	+	0.700487i	$0.752966\pi$
$992$	−9.21443	−0.292558
$993$	−0.304387	−0.00965944
$994$	−10.2441	−0.324924
$995$	−39.9437	−1.26630
$996$	−20.4529	−0.648074
$997$	29.7072	0.940837	0.470418	−	0.882443i	$-0.344103\pi$
$997$	29.7072	0.940837	0.470418	+	0.882443i	$0.344103\pi$
$998$	−10.9610	−0.346964
$999$	−15.8717	−0.502157

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6034.2.a.k.1.3	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6034.2.a.k.1.3	✓	20	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6034 = 2 \cdot 7 \cdot 431 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6034.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.57684 q^{3} +1.00000 q^{4} +4.04233 q^{5} +2.57684 q^{6} +1.00000 q^{7} -1.00000 q^{8} +3.64010 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.57684 q^{3} +1.00000 q^{4} +4.04233 q^{5} +2.57684 q^{6} +1.00000 q^{7} -1.00000 q^{8} +3.64010 q^{9} -4.04233 q^{10} -6.22537 q^{11} -2.57684 q^{12} -3.30637 q^{13} -1.00000 q^{14} -10.4164 q^{15} +1.00000 q^{16} +3.51139 q^{17} -3.64010 q^{18} -3.74588 q^{19} +4.04233 q^{20} -2.57684 q^{21} +6.22537 q^{22} -8.78740 q^{23} +2.57684 q^{24} +11.3404 q^{25} +3.30637 q^{26} -1.64943 q^{27} +1.00000 q^{28} -0.0905660 q^{29} +10.4164 q^{30} +9.21443 q^{31} -1.00000 q^{32} +16.0418 q^{33} -3.51139 q^{34} +4.04233 q^{35} +3.64010 q^{36} +9.62250 q^{37} +3.74588 q^{38} +8.51999 q^{39} -4.04233 q^{40} +1.10715 q^{41} +2.57684 q^{42} +1.77302 q^{43} -6.22537 q^{44} +14.7145 q^{45} +8.78740 q^{46} -6.26009 q^{47} -2.57684 q^{48} +1.00000 q^{49} -11.3404 q^{50} -9.04829 q^{51} -3.30637 q^{52} -0.187269 q^{53} +1.64943 q^{54} -25.1650 q^{55} -1.00000 q^{56} +9.65253 q^{57} +0.0905660 q^{58} -4.44265 q^{59} -10.4164 q^{60} +0.849612 q^{61} -9.21443 q^{62} +3.64010 q^{63} +1.00000 q^{64} -13.3655 q^{65} -16.0418 q^{66} +13.5699 q^{67} +3.51139 q^{68} +22.6437 q^{69} -4.04233 q^{70} +10.2441 q^{71} -3.64010 q^{72} +0.441416 q^{73} -9.62250 q^{74} -29.2225 q^{75} -3.74588 q^{76} -6.22537 q^{77} -8.51999 q^{78} +4.90674 q^{79} +4.04233 q^{80} -6.66998 q^{81} -1.10715 q^{82} +7.93719 q^{83} -2.57684 q^{84} +14.1942 q^{85} -1.77302 q^{86} +0.233374 q^{87} +6.22537 q^{88} -12.0685 q^{89} -14.7145 q^{90} -3.30637 q^{91} -8.78740 q^{92} -23.7441 q^{93} +6.26009 q^{94} -15.1421 q^{95} +2.57684 q^{96} -4.98514 q^{97} -1.00000 q^{98} -22.6610 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 20 q^{2} + 3 q^{3} + 20 q^{4} - 3 q^{5} - 3 q^{6} + 20 q^{7} - 20 q^{8} + 13 q^{9}+O(q^{10}) \) 20 * q - 20 * q^2 + 3 * q^3 + 20 * q^4 - 3 * q^5 - 3 * q^6 + 20 * q^7 - 20 * q^8 + 13 * q^9	\( 20 q - 20 q^{2} + 3 q^{3} + 20 q^{4} - 3 q^{5} - 3 q^{6} + 20 q^{7} - 20 q^{8} + 13 q^{9} + 3 q^{10} - 8 q^{11} + 3 q^{12} - 4 q^{13} - 20 q^{14} - 25 q^{15} + 20 q^{16} + 9 q^{17} - 13 q^{18} - 14 q^{19} - 3 q^{20} + 3 q^{21} + 8 q^{22} - 23 q^{23} - 3 q^{24} + 31 q^{25} + 4 q^{26} - 21 q^{27} + 20 q^{28} - 48 q^{29} + 25 q^{30} - q^{31} - 20 q^{32} - 29 q^{33} - 9 q^{34} - 3 q^{35} + 13 q^{36} - q^{37} + 14 q^{38} - q^{39} + 3 q^{40} - 27 q^{41} - 3 q^{42} - 3 q^{43} - 8 q^{44} - 12 q^{45} + 23 q^{46} - 26 q^{47} + 3 q^{48} + 20 q^{49} - 31 q^{50} - 17 q^{51} - 4 q^{52} - 43 q^{53} + 21 q^{54} - 16 q^{55} - 20 q^{56} - 25 q^{57} + 48 q^{58} - 19 q^{59} - 25 q^{60} + 9 q^{61} + q^{62} + 13 q^{63} + 20 q^{64} - 87 q^{65} + 29 q^{66} + 32 q^{67} + 9 q^{68} - 23 q^{69} + 3 q^{70} - 63 q^{71} - 13 q^{72} + 2 q^{73} + q^{74} - 8 q^{75} - 14 q^{76} - 8 q^{77} + q^{78} - 51 q^{79} - 3 q^{80} + 4 q^{81} + 27 q^{82} - 24 q^{83} + 3 q^{84} + 31 q^{85} + 3 q^{86} - 33 q^{87} + 8 q^{88} - 35 q^{89} + 12 q^{90} - 4 q^{91} - 23 q^{92} + 17 q^{93} + 26 q^{94} - 30 q^{95} - 3 q^{96} + 5 q^{97} - 20 q^{98} - 31 q^{99}+O(q^{100}) \) 20 * q - 20 * q^2 + 3 * q^3 + 20 * q^4 - 3 * q^5 - 3 * q^6 + 20 * q^7 - 20 * q^8 + 13 * q^9 + 3 * q^10 - 8 * q^11 + 3 * q^12 - 4 * q^13 - 20 * q^14 - 25 * q^15 + 20 * q^16 + 9 * q^17 - 13 * q^18 - 14 * q^19 - 3 * q^20 + 3 * q^21 + 8 * q^22 - 23 * q^23 - 3 * q^24 + 31 * q^25 + 4 * q^26 - 21 * q^27 + 20 * q^28 - 48 * q^29 + 25 * q^30 - q^31 - 20 * q^32 - 29 * q^33 - 9 * q^34 - 3 * q^35 + 13 * q^36 - q^37 + 14 * q^38 - q^39 + 3 * q^40 - 27 * q^41 - 3 * q^42 - 3 * q^43 - 8 * q^44 - 12 * q^45 + 23 * q^46 - 26 * q^47 + 3 * q^48 + 20 * q^49 - 31 * q^50 - 17 * q^51 - 4 * q^52 - 43 * q^53 + 21 * q^54 - 16 * q^55 - 20 * q^56 - 25 * q^57 + 48 * q^58 - 19 * q^59 - 25 * q^60 + 9 * q^61 + q^62 + 13 * q^63 + 20 * q^64 - 87 * q^65 + 29 * q^66 + 32 * q^67 + 9 * q^68 - 23 * q^69 + 3 * q^70 - 63 * q^71 - 13 * q^72 + 2 * q^73 + q^74 - 8 * q^75 - 14 * q^76 - 8 * q^77 + q^78 - 51 * q^79 - 3 * q^80 + 4 * q^81 + 27 * q^82 - 24 * q^83 + 3 * q^84 + 31 * q^85 + 3 * q^86 - 33 * q^87 + 8 * q^88 - 35 * q^89 + 12 * q^90 - 4 * q^91 - 23 * q^92 + 17 * q^93 + 26 * q^94 - 30 * q^95 - 3 * q^96 + 5 * q^97 - 20 * q^98 - 31 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more