LMFDB - Embedded newform 4033.2.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4033,2,Mod(1,4033)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4033, 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("4033.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4033 = 37 \cdot 109 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4033.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:	$32.2036671352$
Analytic rank:	$1$
Dimension:	$79$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Character	$\chi$	$=$	4033.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.77527 q^{2} +1.86757 q^{3} +5.70214 q^{4} +1.50597 q^{5} -5.18302 q^{6} -3.04702 q^{7} -10.2744 q^{8} +0.487817 q^{9} +O(q^{10})$	\(q-2.77527 q^{2} +1.86757 q^{3} +5.70214 q^{4} +1.50597 q^{5} -5.18302 q^{6} -3.04702 q^{7} -10.2744 q^{8} +0.487817 q^{9} -4.17947 q^{10} -4.97460 q^{11} +10.6491 q^{12} +4.84844 q^{13} +8.45631 q^{14} +2.81250 q^{15} +17.1101 q^{16} -1.65425 q^{17} -1.35383 q^{18} +7.46249 q^{19} +8.58723 q^{20} -5.69052 q^{21} +13.8059 q^{22} -5.06186 q^{23} -19.1882 q^{24} -2.73206 q^{25} -13.4557 q^{26} -4.69168 q^{27} -17.3745 q^{28} -3.53076 q^{29} -7.80545 q^{30} +7.56226 q^{31} -26.9363 q^{32} -9.29040 q^{33} +4.59099 q^{34} -4.58871 q^{35} +2.78160 q^{36} -1.00000 q^{37} -20.7105 q^{38} +9.05480 q^{39} -15.4730 q^{40} +9.30932 q^{41} +15.7928 q^{42} +3.86049 q^{43} -28.3658 q^{44} +0.734637 q^{45} +14.0480 q^{46} -3.71430 q^{47} +31.9543 q^{48} +2.28433 q^{49} +7.58222 q^{50} -3.08943 q^{51} +27.6465 q^{52} +2.18318 q^{53} +13.0207 q^{54} -7.49157 q^{55} +31.3064 q^{56} +13.9367 q^{57} +9.79883 q^{58} -7.82573 q^{59} +16.0372 q^{60} +6.93522 q^{61} -20.9873 q^{62} -1.48639 q^{63} +40.5353 q^{64} +7.30159 q^{65} +25.7834 q^{66} -13.1252 q^{67} -9.43276 q^{68} -9.45338 q^{69} +12.7349 q^{70} +9.15497 q^{71} -5.01205 q^{72} -11.8430 q^{73} +2.77527 q^{74} -5.10232 q^{75} +42.5522 q^{76} +15.1577 q^{77} -25.1295 q^{78} -6.00754 q^{79} +25.7672 q^{80} -10.2255 q^{81} -25.8359 q^{82} +17.2265 q^{83} -32.4481 q^{84} -2.49124 q^{85} -10.7139 q^{86} -6.59395 q^{87} +51.1112 q^{88} -15.1661 q^{89} -2.03882 q^{90} -14.7733 q^{91} -28.8634 q^{92} +14.1230 q^{93} +10.3082 q^{94} +11.2383 q^{95} -50.3054 q^{96} -7.08860 q^{97} -6.33964 q^{98} -2.42669 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 79 q - 11 q^{2} - 11 q^{3} + 79 q^{4} - 16 q^{5} - 14 q^{6} - 15 q^{7} - 42 q^{8} + 76 q^{9}+O(q^{10}) $ 79 * q - 11 * q^2 - 11 * q^3 + 79 * q^4 - 16 * q^5 - 14 * q^6 - 15 * q^7 - 42 * q^8 + 76 * q^9	\( 79 q - 11 q^{2} - 11 q^{3} + 79 q^{4} - 16 q^{5} - 14 q^{6} - 15 q^{7} - 42 q^{8} + 76 q^{9} - 13 q^{10} - 5 q^{11} - 40 q^{12} - 18 q^{13} - 42 q^{14} - 49 q^{15} + 83 q^{16} - 62 q^{17} - 33 q^{18} - 25 q^{19} - 39 q^{20} - 15 q^{21} - 31 q^{22} - 94 q^{23} - 39 q^{24} + 71 q^{25} - 35 q^{26} - 47 q^{27} - 13 q^{28} - 17 q^{29} + 15 q^{30} - 37 q^{31} - 105 q^{32} - 60 q^{33} + 9 q^{34} - 60 q^{35} + 43 q^{36} - 79 q^{37} - 80 q^{38} - 41 q^{39} - 64 q^{40} - 37 q^{41} - 30 q^{42} - 20 q^{43} - 14 q^{44} - 8 q^{45} + 61 q^{46} - 148 q^{47} - 39 q^{48} + 82 q^{49} - 90 q^{50} - 45 q^{51} - 27 q^{52} - 70 q^{53} - 41 q^{54} - 105 q^{55} - 68 q^{56} - 31 q^{57} - 14 q^{58} - 96 q^{59} - 74 q^{60} - 21 q^{61} + 4 q^{62} - 60 q^{63} + 132 q^{64} - 15 q^{65} + 71 q^{66} - 44 q^{67} - 166 q^{68} - 72 q^{69} - 9 q^{70} - 55 q^{71} - 126 q^{72} - 27 q^{73} + 11 q^{74} - 39 q^{75} - 4 q^{76} - 104 q^{77} - 47 q^{78} - 49 q^{79} - 82 q^{80} + 55 q^{81} + 20 q^{82} - 52 q^{83} - 29 q^{84} + 3 q^{85} - 32 q^{86} - 113 q^{87} + 14 q^{88} - 68 q^{89} - 39 q^{90} - 30 q^{91} - 179 q^{92} - 53 q^{93} - 33 q^{94} - 86 q^{95} + 33 q^{96} - 57 q^{97} - 116 q^{98} + q^{99}+O(q^{100}) \) 79 * q - 11 * q^2 - 11 * q^3 + 79 * q^4 - 16 * q^5 - 14 * q^6 - 15 * q^7 - 42 * q^8 + 76 * q^9 - 13 * q^10 - 5 * q^11 - 40 * q^12 - 18 * q^13 - 42 * q^14 - 49 * q^15 + 83 * q^16 - 62 * q^17 - 33 * q^18 - 25 * q^19 - 39 * q^20 - 15 * q^21 - 31 * q^22 - 94 * q^23 - 39 * q^24 + 71 * q^25 - 35 * q^26 - 47 * q^27 - 13 * q^28 - 17 * q^29 + 15 * q^30 - 37 * q^31 - 105 * q^32 - 60 * q^33 + 9 * q^34 - 60 * q^35 + 43 * q^36 - 79 * q^37 - 80 * q^38 - 41 * q^39 - 64 * q^40 - 37 * q^41 - 30 * q^42 - 20 * q^43 - 14 * q^44 - 8 * q^45 + 61 * q^46 - 148 * q^47 - 39 * q^48 + 82 * q^49 - 90 * q^50 - 45 * q^51 - 27 * q^52 - 70 * q^53 - 41 * q^54 - 105 * q^55 - 68 * q^56 - 31 * q^57 - 14 * q^58 - 96 * q^59 - 74 * q^60 - 21 * q^61 + 4 * q^62 - 60 * q^63 + 132 * q^64 - 15 * q^65 + 71 * q^66 - 44 * q^67 - 166 * q^68 - 72 * q^69 - 9 * q^70 - 55 * q^71 - 126 * q^72 - 27 * q^73 + 11 * q^74 - 39 * q^75 - 4 * q^76 - 104 * q^77 - 47 * q^78 - 49 * q^79 - 82 * q^80 + 55 * q^81 + 20 * q^82 - 52 * q^83 - 29 * q^84 + 3 * q^85 - 32 * q^86 - 113 * q^87 + 14 * q^88 - 68 * q^89 - 39 * q^90 - 30 * q^91 - 179 * q^92 - 53 * q^93 - 33 * q^94 - 86 * q^95 + 33 * q^96 - 57 * q^97 - 116 * q^98 + 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 $

$2$	−2.77527	−1.96241	−0.981207	−	0.192958i	$-0.938192\pi$
$2$	−2.77527	−1.96241	−0.981207	+	0.192958i	$0.938192\pi$
$3$	1.86757	1.07824	0.539121	−	0.842228i	$-0.318757\pi$
$3$	1.86757	1.07824	0.539121	+	0.842228i	$0.318757\pi$
$4$	5.70214	2.85107
$5$	1.50597	0.673489	0.336744	−	0.941596i	$-0.390674\pi$
$5$	1.50597	0.673489	0.336744	+	0.941596i	$0.390674\pi$
$6$	−5.18302	−2.11596
$7$	−3.04702	−1.15167	−0.575833	−	0.817568i	$-0.695322\pi$
$7$	−3.04702	−1.15167	−0.575833	+	0.817568i	$0.695322\pi$
$8$	−10.2744	−3.63256
$9$	0.487817	0.162606
$10$	−4.17947	−1.32166
$11$	−4.97460	−1.49990	−0.749948	−	0.661496i	$-0.769922\pi$
$11$	−4.97460	−1.49990	−0.749948	+	0.661496i	$0.769922\pi$
$12$	10.6491	3.07414
$13$	4.84844	1.34472	0.672358	−	0.740226i	$-0.265282\pi$
$13$	4.84844	1.34472	0.672358	+	0.740226i	$0.265282\pi$
$14$	8.45631	2.26004
$15$	2.81250	0.726184
$16$	17.1101	4.27752
$17$	−1.65425	−0.401214	−0.200607	−	0.979672i	$-0.564292\pi$
$17$	−1.65425	−0.401214	−0.200607	+	0.979672i	$0.564292\pi$
$18$	−1.35383	−0.319100
$19$	7.46249	1.71201	0.856007	−	0.516965i	$-0.172938\pi$
$19$	7.46249	1.71201	0.856007	+	0.516965i	$0.172938\pi$
$20$	8.58723	1.92016
$21$	−5.69052	−1.24177
$22$	13.8059	2.94342
$23$	−5.06186	−1.05547	−0.527736	−	0.849409i	$-0.676959\pi$
$23$	−5.06186	−1.05547	−0.527736	+	0.849409i	$0.676959\pi$
$24$	−19.1882	−3.91678
$25$	−2.73206	−0.546413
$26$	−13.4557	−2.63889
$27$	−4.69168	−0.902914
$28$	−17.3745	−3.28348
$29$	−3.53076	−0.655646	−0.327823	−	0.944739i	$-0.606315\pi$
$29$	−3.53076	−0.655646	−0.327823	+	0.944739i	$0.606315\pi$
$30$	−7.80545	−1.42507
$31$	7.56226	1.35822	0.679111	−	0.734036i	$-0.262366\pi$
$31$	7.56226	1.35822	0.679111	+	0.734036i	$0.262366\pi$
$32$	−26.9363	−4.76171
$33$	−9.29040	−1.61725
$34$	4.59099	0.787349
$35$	−4.58871	−0.775634
$36$	2.78160	0.463600
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	−20.7105	−3.35968
$39$	9.05480	1.44993
$40$	−15.4730	−2.44649
$41$	9.30932	1.45387	0.726936	−	0.686705i	$-0.240944\pi$
$41$	9.30932	1.45387	0.726936	+	0.686705i	$0.240944\pi$
$42$	15.7928	2.43687
$43$	3.86049	0.588719	0.294359	−	0.955695i	$-0.404894\pi$
$43$	3.86049	0.588719	0.294359	+	0.955695i	$0.404894\pi$
$44$	−28.3658	−4.27631
$45$	0.734637	0.109513
$46$	14.0480	2.07127
$47$	−3.71430	−0.541787	−0.270893	−	0.962609i	$-0.587319\pi$
$47$	−3.71430	−0.541787	−0.270893	+	0.962609i	$0.587319\pi$
$48$	31.9543	4.61220
$49$	2.28433	0.326333
$50$	7.58222	1.07229
$51$	−3.08943	−0.432606
$52$	27.6465	3.83388
$53$	2.18318	0.299883	0.149941	−	0.988695i	$-0.452092\pi$
$53$	2.18318	0.299883	0.149941	+	0.988695i	$0.452092\pi$
$54$	13.0207	1.77189
$55$	−7.49157	−1.01016
$56$	31.3064	4.18350
$57$	13.9367	1.84596
$58$	9.79883	1.28665
$59$	−7.82573	−1.01882	−0.509412	−	0.860523i	$-0.670137\pi$
$59$	−7.82573	−1.01882	−0.509412	+	0.860523i	$0.670137\pi$
$60$	16.0372	2.07040
$61$	6.93522	0.887964	0.443982	−	0.896036i	$-0.353565\pi$
$61$	6.93522	0.887964	0.443982	+	0.896036i	$0.353565\pi$
$62$	−20.9873	−2.66539
$63$	−1.48639	−0.187267
$64$	40.5353	5.06692
$65$	7.30159	0.905651
$66$	25.7834	3.17372
$67$	−13.1252	−1.60349	−0.801747	−	0.597664i	$-0.796096\pi$
$67$	−13.1252	−1.60349	−0.801747	+	0.597664i	$0.796096\pi$
$68$	−9.43276	−1.14389
$69$	−9.45338	−1.13805
$70$	12.7349	1.52211
$71$	9.15497	1.08650	0.543248	−	0.839573i	$-0.317195\pi$
$71$	9.15497	1.08650	0.543248	+	0.839573i	$0.317195\pi$
$72$	−5.01205	−0.590676
$73$	−11.8430	−1.38612	−0.693059	−	0.720881i	$-0.743738\pi$
$73$	−11.8430	−1.38612	−0.693059	+	0.720881i	$0.743738\pi$
$74$	2.77527	0.322619
$75$	−5.10232	−0.589165
$76$	42.5522	4.88107
$77$	15.1577	1.72738
$78$	−25.1295	−2.84536
$79$	−6.00754	−0.675902	−0.337951	−	0.941164i	$-0.609734\pi$
$79$	−6.00754	−0.675902	−0.337951	+	0.941164i	$0.609734\pi$
$80$	25.7672	2.88086
$81$	−10.2255	−1.13617
$82$	−25.8359	−2.85310
$83$	17.2265	1.89086	0.945430	−	0.325827i	$-0.105643\pi$
$83$	17.2265	1.89086	0.945430	+	0.325827i	$0.105643\pi$
$84$	−32.4481	−3.54038
$85$	−2.49124	−0.270213
$86$	−10.7139	−1.15531
$87$	−6.59395	−0.706945
$88$	51.1112	5.44847
$89$	−15.1661	−1.60760	−0.803800	−	0.594900i	$-0.797192\pi$
$89$	−15.1661	−1.60760	−0.803800	+	0.594900i	$0.797192\pi$
$90$	−2.03882	−0.214910
$91$	−14.7733	−1.54866
$92$	−28.8634	−3.00922
$93$	14.1230	1.46449
$94$	10.3082	1.06321
$95$	11.2383	1.15302
$96$	−50.3054	−5.13427
$97$	−7.08860	−0.719738	−0.359869	−	0.933003i	$-0.617179\pi$
$97$	−7.08860	−0.719738	−0.359869	+	0.933003i	$0.617179\pi$
$98$	−6.33964	−0.640400
$99$	−2.42669	−0.243892
$100$	−15.5786	−1.55786
$101$	−6.28711	−0.625591	−0.312796	−	0.949820i	$-0.601265\pi$
$101$	−6.28711	−0.625591	−0.312796	+	0.949820i	$0.601265\pi$
$102$	8.57400	0.848953
$103$	−9.26369	−0.912778	−0.456389	−	0.889780i	$-0.650857\pi$
$103$	−9.26369	−0.912778	−0.456389	+	0.889780i	$0.650857\pi$
$104$	−49.8150	−4.88476
$105$	−8.56974	−0.836321
$106$	−6.05891	−0.588494
$107$	−5.38708	−0.520789	−0.260394	−	0.965502i	$-0.583853\pi$
$107$	−5.38708	−0.520789	−0.260394	+	0.965502i	$0.583853\pi$
$108$	−26.7526	−2.57427
$109$	−1.00000	−0.0957826
$109$	−1.00000	−0.0957826
$110$	20.7912	1.98236
$111$	−1.86757	−0.177262
$112$	−52.1348	−4.92627
$113$	−1.86056	−0.175027	−0.0875135	−	0.996163i	$-0.527892\pi$
$113$	−1.86056	−0.175027	−0.0875135	+	0.996163i	$0.527892\pi$
$114$	−38.6782	−3.62255
$115$	−7.62300	−0.710848
$116$	−20.1329	−1.86929
$117$	2.36515	0.218659
$118$	21.7185	1.99935
$119$	5.04053	0.462065
$120$	−28.8968	−2.63791
$121$	13.7466	1.24969
$122$	−19.2471	−1.74255
$123$	17.3858	1.56763
$124$	43.1210	3.87238
$125$	−11.6442	−1.04149
$126$	4.12514	0.367496
$127$	−5.25893	−0.466654	−0.233327	−	0.972398i	$-0.574961\pi$
$127$	−5.25893	−0.466654	−0.233327	+	0.972398i	$0.574961\pi$
$128$	−58.6241	−5.18168
$129$	7.20973	0.634782
$130$	−20.2639	−1.77726
$131$	−5.59196	−0.488572	−0.244286	−	0.969703i	$-0.578553\pi$
$131$	−5.59196	−0.488572	−0.244286	+	0.969703i	$0.578553\pi$
$132$	−52.9752	−4.61090
$133$	−22.7384	−1.97167
$134$	36.4259	3.14672
$135$	−7.06551	−0.608102
$136$	16.9965	1.45744
$137$	8.44838	0.721794	0.360897	−	0.932606i	$-0.382471\pi$
$137$	8.44838	0.721794	0.360897	+	0.932606i	$0.382471\pi$
$138$	26.2357	2.23333
$139$	−17.2359	−1.46193	−0.730964	−	0.682416i	$-0.760929\pi$
$139$	−17.2359	−1.46193	−0.730964	+	0.682416i	$0.760929\pi$
$140$	−26.1655	−2.21138
$141$	−6.93672	−0.584177
$142$	−25.4075	−2.13215
$143$	−24.1190	−2.01694
$144$	8.34660	0.695550
$145$	−5.31721	−0.441570
$146$	32.8675	2.72014
$147$	4.26615	0.351866
$148$	−5.70214	−0.468713
$149$	−1.99641	−0.163552	−0.0817761	−	0.996651i	$-0.526059\pi$
$149$	−1.99641	−0.163552	−0.0817761	+	0.996651i	$0.526059\pi$
$150$	14.1603	1.15619
$151$	−5.23292	−0.425849	−0.212924	−	0.977069i	$-0.568299\pi$
$151$	−5.23292	−0.425849	−0.212924	+	0.977069i	$0.568299\pi$
$152$	−76.6729	−6.21900
$153$	−0.806972	−0.0652398
$154$	−42.0667	−3.38983
$155$	11.3885	0.914747
$156$	51.6317	4.13385
$157$	1.21537	0.0969972	0.0484986	−	0.998823i	$-0.484556\pi$
$157$	1.21537	0.0969972	0.0484986	+	0.998823i	$0.484556\pi$
$158$	16.6726	1.32640
$159$	4.07724	0.323346
$160$	−40.5651	−3.20696
$161$	15.4236	1.21555
$162$	28.3785	2.22963
$163$	−2.79396	−0.218840	−0.109420	−	0.993996i	$-0.534899\pi$
$163$	−2.79396	−0.218840	−0.109420	+	0.993996i	$0.534899\pi$
$164$	53.0830	4.14509
$165$	−13.9910	−1.08920
$166$	−47.8084	−3.71065
$167$	−1.39187	−0.107706	−0.0538529	−	0.998549i	$-0.517150\pi$
$167$	−1.39187	−0.107706	−0.0538529	+	0.998549i	$0.517150\pi$
$168$	58.4669	4.51082
$169$	10.5074	0.808261
$170$	6.91388	0.530271
$171$	3.64033	0.278383
$172$	22.0130	1.67848
$173$	13.8539	1.05329	0.526646	−	0.850085i	$-0.323449\pi$
$173$	13.8539	1.05329	0.526646	+	0.850085i	$0.323449\pi$
$174$	18.3000	1.38732
$175$	8.32466	0.629285
$176$	−85.1158	−6.41584
$177$	−14.6151	−1.09854
$178$	42.0900	3.15478
$179$	21.1116	1.57796	0.788978	−	0.614421i	$-0.210610\pi$
$179$	21.1116	1.57796	0.788978	+	0.614421i	$0.210610\pi$
$180$	4.18900	0.312230
$181$	−23.0872	−1.71606	−0.858029	−	0.513601i	$-0.828311\pi$
$181$	−23.0872	−1.71606	−0.858029	+	0.513601i	$0.828311\pi$
$182$	40.9999	3.03912
$183$	12.9520	0.957440
$184$	52.0078	3.83407
$185$	−1.50597	−0.110721
$186$	−39.1953	−2.87394
$187$	8.22922	0.601780
$188$	−21.1795	−1.54467
$189$	14.2956	1.03985
$190$	−31.1892	−2.26271
$191$	−13.3647	−0.967037	−0.483518	−	0.875334i	$-0.660641\pi$
$191$	−13.3647	−0.967037	−0.483518	+	0.875334i	$0.660641\pi$
$192$	75.7026	5.46336
$193$	−5.31023	−0.382238	−0.191119	−	0.981567i	$-0.561212\pi$
$193$	−5.31023	−0.382238	−0.191119	+	0.981567i	$0.561212\pi$
$194$	19.6728	1.41242
$195$	13.6362	0.976511
$196$	13.0256	0.930397
$197$	−15.4164	−1.09838	−0.549188	−	0.835699i	$-0.685063\pi$
$197$	−15.4164	−1.09838	−0.549188	+	0.835699i	$0.685063\pi$
$198$	6.73474	0.478617
$199$	−12.3582	−0.876049	−0.438025	−	0.898963i	$-0.644322\pi$
$199$	−12.3582	−0.876049	−0.438025	+	0.898963i	$0.644322\pi$
$200$	28.0704	1.98488
$201$	−24.5122	−1.72895
$202$	17.4485	1.22767
$203$	10.7583	0.755085
$204$	−17.6163	−1.23339
$205$	14.0195	0.979166
$206$	25.7093	1.79125
$207$	−2.46927	−0.171626
$208$	82.9573	5.75205
$209$	−37.1229	−2.56784
$210$	23.7834	1.64121
$211$	13.6874	0.942280	0.471140	−	0.882058i	$-0.343843\pi$
$211$	13.6874	0.942280	0.471140	+	0.882058i	$0.343843\pi$
$212$	12.4488	0.854986
$213$	17.0975	1.17150
$214$	14.9506	1.02200
$215$	5.81377	0.396496
$216$	48.2043	3.27989
$217$	−23.0423	−1.56422
$218$	2.77527	0.187965
$219$	−22.1176	−1.49457
$220$	−42.7180	−2.88005
$221$	−8.02053	−0.539519
$222$	5.18302	0.347861
$223$	−7.15168	−0.478912	−0.239456	−	0.970907i	$-0.576969\pi$
$223$	−7.15168	−0.478912	−0.239456	+	0.970907i	$0.576969\pi$
$224$	82.0754	5.48389
$225$	−1.33275	−0.0888499
$226$	5.16357	0.343475
$227$	−23.2683	−1.54437	−0.772186	−	0.635397i	$-0.780837\pi$
$227$	−23.2683	−1.54437	−0.772186	+	0.635397i	$0.780837\pi$
$228$	79.4691	5.26297
$229$	−21.3020	−1.40767	−0.703837	−	0.710362i	$-0.748531\pi$
$229$	−21.3020	−1.40767	−0.703837	+	0.710362i	$0.748531\pi$
$230$	21.1559	1.39498
$231$	28.3080	1.86253
$232$	36.2766	2.38168
$233$	−27.9747	−1.83268	−0.916342	−	0.400397i	$-0.868872\pi$
$233$	−27.9747	−1.83268	−0.916342	+	0.400397i	$0.868872\pi$
$234$	−6.56395	−0.429099
$235$	−5.59362	−0.364887
$236$	−44.6234	−2.90473
$237$	−11.2195	−0.728785
$238$	−13.9888	−0.906762
$239$	0.737685	0.0477169	0.0238585	−	0.999715i	$-0.492405\pi$
$239$	0.737685	0.0477169	0.0238585	+	0.999715i	$0.492405\pi$
$240$	48.1221	3.10627
$241$	10.3748	0.668297	0.334148	−	0.942520i	$-0.391551\pi$
$241$	10.3748	0.668297	0.334148	+	0.942520i	$0.391551\pi$
$242$	−38.1505	−2.45241
$243$	−5.02178	−0.322147
$244$	39.5456	2.53165
$245$	3.44013	0.219782
$246$	−48.2504	−3.07633
$247$	36.1815	2.30217
$248$	−77.6979	−4.93382
$249$	32.1718	2.03880
$250$	32.3159	2.04384
$251$	−23.8826	−1.50746	−0.753729	−	0.657185i	$-0.771747\pi$
$251$	−23.8826	−1.50746	−0.753729	+	0.657185i	$0.771747\pi$
$252$	−8.47560	−0.533912
$253$	25.1807	1.58310
$254$	14.5950	0.915769
$255$	−4.65257	−0.291355
$256$	81.6270	5.10169
$257$	−22.1270	−1.38024	−0.690121	−	0.723694i	$-0.742443\pi$
$257$	−22.1270	−1.38024	−0.690121	+	0.723694i	$0.742443\pi$
$258$	−20.0090	−1.24570
$259$	3.04702	0.189333
$260$	41.6347	2.58207
$261$	−1.72237	−0.106612
$262$	15.5192	0.958780
$263$	−23.4109	−1.44358	−0.721790	−	0.692112i	$-0.756681\pi$
$263$	−23.4109	−1.44358	−0.721790	+	0.692112i	$0.756681\pi$
$264$	95.4537	5.87477
$265$	3.28779	0.201968
$266$	63.1052	3.86923
$267$	−28.3237	−1.73338
$268$	−74.8415	−4.57167
$269$	25.2480	1.53940	0.769700	−	0.638406i	$-0.220406\pi$
$269$	25.2480	1.53940	0.769700	+	0.638406i	$0.220406\pi$
$270$	19.6087	1.19335
$271$	−10.0829	−0.612490	−0.306245	−	0.951953i	$-0.599073\pi$
$271$	−10.0829	−0.612490	−0.306245	+	0.951953i	$0.599073\pi$
$272$	−28.3044	−1.71620
$273$	−27.5902	−1.66983
$274$	−23.4466	−1.41646
$275$	13.5909	0.819563
$276$	−53.9045	−3.24467
$277$	5.96381	0.358330	0.179165	−	0.983819i	$-0.442660\pi$
$277$	5.96381	0.358330	0.179165	+	0.983819i	$0.442660\pi$
$278$	47.8343	2.86891
$279$	3.68900	0.220855
$280$	47.1464	2.81754
$281$	14.4865	0.864190	0.432095	−	0.901828i	$-0.357774\pi$
$281$	14.4865	0.864190	0.432095	+	0.901828i	$0.357774\pi$
$282$	19.2513	1.14640
$283$	−15.6869	−0.932491	−0.466245	−	0.884655i	$-0.654394\pi$
$283$	−15.6869	−0.932491	−0.466245	+	0.884655i	$0.654394\pi$
$284$	52.2029	3.09767
$285$	20.9882	1.24324
$286$	66.9369	3.95806
$287$	−28.3657	−1.67437
$288$	−13.1400	−0.774281
$289$	−14.2635	−0.839027
$290$	14.7567	0.866544
$291$	−13.2385	−0.776052
$292$	−67.5304	−3.95192
$293$	11.0611	0.646198	0.323099	−	0.946365i	$-0.395275\pi$
$293$	11.0611	0.646198	0.323099	+	0.946365i	$0.395275\pi$
$294$	−11.8397	−0.690506
$295$	−11.7853	−0.686166
$296$	10.2744	0.597190
$297$	23.3392	1.35428
$298$	5.54058	0.320957
$299$	−24.5422	−1.41931
$300$	−29.0941	−1.67975
$301$	−11.7630	−0.678007
$302$	14.5228	0.835692
$303$	−11.7416	−0.674539
$304$	127.684	7.32318
$305$	10.4442	0.598034
$306$	2.23957	0.128028
$307$	21.5534	1.23012	0.615058	−	0.788482i	$-0.289133\pi$
$307$	21.5534	1.23012	0.615058	+	0.788482i	$0.289133\pi$
$308$	86.4312	4.92488
$309$	−17.3006	−0.984196
$310$	−31.6062	−1.79511
$311$	−11.2228	−0.636385	−0.318193	−	0.948026i	$-0.603076\pi$
$311$	−11.2228	−0.636385	−0.318193	+	0.948026i	$0.603076\pi$
$312$	−93.0330	−5.26696
$313$	18.4336	1.04193	0.520964	−	0.853579i	$-0.325573\pi$
$313$	18.4336	1.04193	0.520964	+	0.853579i	$0.325573\pi$
$314$	−3.37299	−0.190349
$315$	−2.23845	−0.126123
$316$	−34.2558	−1.92704
$317$	28.9466	1.62581	0.812903	−	0.582399i	$-0.197886\pi$
$317$	28.9466	1.62581	0.812903	+	0.582399i	$0.197886\pi$
$318$	−11.3154	−0.634539
$319$	17.5641	0.983402
$320$	61.0449	3.41251
$321$	−10.0607	−0.561536
$322$	−42.8047	−2.38541
$323$	−12.3448	−0.686884
$324$	−58.3071	−3.23928
$325$	−13.2463	−0.734770
$326$	7.75400	0.429455
$327$	−1.86757	−0.103277
$328$	−95.6481	−5.28128
$329$	11.3176	0.623957
$330$	38.8289	2.13746
$331$	−16.4158	−0.902294	−0.451147	−	0.892450i	$-0.648985\pi$
$331$	−16.4158	−0.902294	−0.451147	+	0.892450i	$0.648985\pi$
$332$	98.2281	5.39097
$333$	−0.487817	−0.0267322
$334$	3.86281	0.211364
$335$	−19.7661	−1.07994
$336$	−97.3654	−5.31172
$337$	6.21803	0.338718	0.169359	−	0.985554i	$-0.445830\pi$
$337$	6.21803	0.338718	0.169359	+	0.985554i	$0.445830\pi$
$338$	−29.1609	−1.58614
$339$	−3.47473	−0.188721
$340$	−14.2054	−0.770397
$341$	−37.6192	−2.03719
$342$	−10.1029	−0.546303
$343$	14.3687	0.775839
$344$	−39.6643	−2.13856
$345$	−14.2365	−0.766466
$346$	−38.4483	−2.06700
$347$	−17.5959	−0.944596	−0.472298	−	0.881439i	$-0.656575\pi$
$347$	−17.5959	−0.944596	−0.472298	+	0.881439i	$0.656575\pi$
$348$	−37.5996	−2.01555
$349$	22.2994	1.19366	0.596830	−	0.802367i	$-0.296426\pi$
$349$	22.2994	1.19366	0.596830	+	0.802367i	$0.296426\pi$
$350$	−23.1032	−1.23492
$351$	−22.7473	−1.21416
$352$	133.997	7.14207
$353$	−13.1549	−0.700166	−0.350083	−	0.936719i	$-0.613847\pi$
$353$	−13.1549	−0.700166	−0.350083	+	0.936719i	$0.613847\pi$
$354$	40.5609	2.15579
$355$	13.7871	0.731742
$356$	−86.4790	−4.58338
$357$	9.41355	0.498218
$358$	−58.5905	−3.09660
$359$	13.0453	0.688506	0.344253	−	0.938877i	$-0.388132\pi$
$359$	13.0453	0.688506	0.344253	+	0.938877i	$0.388132\pi$
$360$	−7.54798	−0.397813
$361$	36.6888	1.93099
$362$	64.0733	3.36762
$363$	25.6727	1.34747
$364$	−84.2394	−4.41534
$365$	−17.8352	−0.933535
$366$	−35.9454	−1.87889
$367$	−26.4008	−1.37811	−0.689056	−	0.724708i	$-0.741975\pi$
$367$	−26.4008	−1.37811	−0.689056	+	0.724708i	$0.741975\pi$
$368$	−86.6089	−4.51480
$369$	4.54125	0.236408
$370$	4.17947	0.217280
$371$	−6.65219	−0.345364
$372$	80.5315	4.17537
$373$	28.5611	1.47884	0.739419	−	0.673245i	$-0.235100\pi$
$373$	28.5611	1.47884	0.739419	+	0.673245i	$0.235100\pi$
$374$	−22.8383	−1.18094
$375$	−21.7464	−1.12298
$376$	38.1624	1.96807
$377$	−17.1187	−0.881658
$378$	−39.6743	−2.04062
$379$	−12.3531	−0.634535	−0.317267	−	0.948336i	$-0.602765\pi$
$379$	−12.3531	−0.634535	−0.317267	+	0.948336i	$0.602765\pi$
$380$	64.0821	3.28734
$381$	−9.82142	−0.503166
$382$	37.0907	1.89773
$383$	35.7315	1.82579	0.912896	−	0.408192i	$-0.133841\pi$
$383$	35.7315	1.82579	0.912896	+	0.408192i	$0.133841\pi$
$384$	−109.485	−5.58711
$385$	22.8270	1.16337
$386$	14.7373	0.750110
$387$	1.88321	0.0957291
$388$	−40.4202	−2.05202
$389$	8.03533	0.407407	0.203704	−	0.979033i	$-0.434702\pi$
$389$	8.03533	0.407407	0.203704	+	0.979033i	$0.434702\pi$
$390$	−37.8443	−1.91632
$391$	8.37359	0.423470
$392$	−23.4702	−1.18542
$393$	−10.4434	−0.526798
$394$	42.7848	2.15547
$395$	−9.04716	−0.455212
$396$	−13.8373	−0.695353
$397$	15.9147	0.798738	0.399369	−	0.916790i	$-0.369229\pi$
$397$	15.9147	0.798738	0.399369	+	0.916790i	$0.369229\pi$
$398$	34.2974	1.71917
$399$	−42.4655	−2.12593
$400$	−46.7459	−2.33729
$401$	14.8140	0.739776	0.369888	−	0.929076i	$-0.379396\pi$
$401$	14.8140	0.739776	0.369888	+	0.929076i	$0.379396\pi$
$402$	68.0279	3.39292
$403$	36.6652	1.82642
$404$	−35.8500	−1.78360
$405$	−15.3992	−0.765194
$406$	−29.8572	−1.48179
$407$	4.97460	0.246582
$408$	31.7421	1.57147
$409$	9.07411	0.448686	0.224343	−	0.974510i	$-0.427976\pi$
$409$	9.07411	0.448686	0.224343	+	0.974510i	$0.427976\pi$
$410$	−38.9080	−1.92153
$411$	15.7779	0.778269
$412$	−52.8228	−2.60239
$413$	23.8452	1.17334
$414$	6.85288	0.336801
$415$	25.9426	1.27347
$416$	−130.599	−6.40314
$417$	−32.1892	−1.57631
$418$	103.026	5.03917
$419$	−4.83572	−0.236240	−0.118120	−	0.992999i	$-0.537687\pi$
$419$	−4.83572	−0.236240	−0.118120	+	0.992999i	$0.537687\pi$
$420$	−48.8658	−2.38441
$421$	21.7162	1.05838	0.529191	−	0.848503i	$-0.322496\pi$
$421$	21.7162	1.05838	0.529191	+	0.848503i	$0.322496\pi$
$422$	−37.9863	−1.84914
$423$	−1.81190	−0.0880977
$424$	−22.4309	−1.08934
$425$	4.51952	0.219229
$426$	−47.4504	−2.29898
$427$	−21.1318	−1.02264
$428$	−30.7179	−1.48480
$429$	−45.0440	−2.17474
$430$	−16.1348	−0.778088
$431$	12.6370	0.608704	0.304352	−	0.952560i	$-0.401560\pi$
$431$	12.6370	0.608704	0.304352	+	0.952560i	$0.401560\pi$
$432$	−80.2750	−3.86223
$433$	25.1637	1.20929	0.604645	−	0.796495i	$-0.293315\pi$
$433$	25.1637	1.20929	0.604645	+	0.796495i	$0.293315\pi$
$434$	63.9488	3.06964
$435$	−9.93027	−0.476120
$436$	−5.70214	−0.273083
$437$	−37.7741	−1.80698
$438$	61.3824	2.93297
$439$	24.4025	1.16467	0.582334	−	0.812949i	$-0.302140\pi$
$439$	24.4025	1.16467	0.582334	+	0.812949i	$0.302140\pi$
$440$	76.9717	3.66948
$441$	1.11434	0.0530636
$442$	22.2592	1.05876
$443$	30.1894	1.43434	0.717170	−	0.696899i	$-0.245437\pi$
$443$	30.1894	1.43434	0.717170	+	0.696899i	$0.245437\pi$
$444$	−10.6491	−0.505386
$445$	−22.8396	−1.08270
$446$	19.8478	0.939823
$447$	−3.72843	−0.176349
$448$	−123.512	−5.83539
$449$	14.3744	0.678371	0.339185	−	0.940720i	$-0.389849\pi$
$449$	14.3744	0.678371	0.339185	+	0.940720i	$0.389849\pi$
$450$	3.69874	0.174360
$451$	−46.3101	−2.18066
$452$	−10.6092	−0.499014
$453$	−9.77284	−0.459168
$454$	64.5759	3.03070
$455$	−22.2481	−1.04301
$456$	−143.192	−6.70558
$457$	−17.0020	−0.795321	−0.397660	−	0.917533i	$-0.630178\pi$
$457$	−17.0020	−0.795321	−0.397660	+	0.917533i	$0.630178\pi$
$458$	59.1188	2.76244
$459$	7.76120	0.362262
$460$	−43.4674	−2.02668
$461$	30.0299	1.39863	0.699316	−	0.714812i	$-0.253488\pi$
$461$	30.0299	1.39863	0.699316	+	0.714812i	$0.253488\pi$
$462$	−78.5625	−3.65506
$463$	−11.6971	−0.543611	−0.271806	−	0.962352i	$-0.587621\pi$
$463$	−11.6971	−0.543611	−0.271806	+	0.962352i	$0.587621\pi$
$464$	−60.4117	−2.80454
$465$	21.2688	0.986319
$466$	77.6374	3.59648
$467$	−5.69237	−0.263412	−0.131706	−	0.991289i	$-0.542045\pi$
$467$	−5.69237	−0.263412	−0.131706	+	0.991289i	$0.542045\pi$
$468$	13.4864	0.623411
$469$	39.9926	1.84669
$470$	15.5238	0.716060
$471$	2.26979	0.104586
$472$	80.4050	3.70094
$473$	−19.2044	−0.883018
$474$	31.1372	1.43018
$475$	−20.3880	−0.935466
$476$	28.7418	1.31738
$477$	1.06499	0.0487627
$478$	−2.04728	−0.0936403
$479$	17.6713	0.807423	0.403712	−	0.914886i	$-0.367720\pi$
$479$	17.6713	0.807423	0.403712	+	0.914886i	$0.367720\pi$
$480$	−75.7582	−3.45787
$481$	−4.84844	−0.221070
$482$	−28.7928	−1.31148
$483$	28.8046	1.31066
$484$	78.3850	3.56295
$485$	−10.6752	−0.484736
$486$	13.9368	0.632187
$487$	−34.4430	−1.56076	−0.780380	−	0.625305i	$-0.784974\pi$
$487$	−34.4430	−1.56076	−0.780380	+	0.625305i	$0.784974\pi$
$488$	−71.2555	−3.22558
$489$	−5.21792	−0.235962
$490$	−9.54728	−0.431302
$491$	−18.2239	−0.822432	−0.411216	−	0.911538i	$-0.634896\pi$
$491$	−18.2239	−0.822432	−0.411216	+	0.911538i	$0.634896\pi$
$492$	99.1363	4.46941
$493$	5.84076	0.263055
$494$	−100.413	−4.51781
$495$	−3.65452	−0.164258
$496$	129.391	5.80982
$497$	−27.8954	−1.25128
$498$	−89.2854	−4.00098
$499$	−22.4715	−1.00596	−0.502981	−	0.864297i	$-0.667763\pi$
$499$	−22.4715	−1.00596	−0.502981	+	0.864297i	$0.667763\pi$
$500$	−66.3970	−2.96936
$501$	−2.59941	−0.116133
$502$	66.2808	2.95826
$503$	−37.0854	−1.65355	−0.826777	−	0.562530i	$-0.809828\pi$
$503$	−37.0854	−1.65355	−0.826777	+	0.562530i	$0.809828\pi$
$504$	15.2718	0.680261
$505$	−9.46818	−0.421329
$506$	−69.8834	−3.10669
$507$	19.6233	0.871501
$508$	−29.9871	−1.33046
$509$	18.3451	0.813131	0.406566	−	0.913622i	$-0.366726\pi$
$509$	18.3451	0.813131	0.406566	+	0.913622i	$0.366726\pi$
$510$	12.9122	0.571760
$511$	36.0859	1.59634
$512$	−109.289	−4.82994
$513$	−35.0116	−1.54580
$514$	61.4083	2.70860
$515$	−13.9508	−0.614746
$516$	41.1109	1.80981
$517$	18.4772	0.812624
$518$	−8.45631	−0.371549
$519$	25.8731	1.13570
$520$	−75.0197	−3.28983
$521$	−11.0598	−0.484537	−0.242269	−	0.970209i	$-0.577892\pi$
$521$	−11.0598	−0.484537	−0.242269	+	0.970209i	$0.577892\pi$
$522$	4.78004	0.209217
$523$	17.0460	0.745371	0.372686	−	0.927958i	$-0.378437\pi$
$523$	17.0460	0.745371	0.372686	+	0.927958i	$0.378437\pi$
$524$	−31.8861	−1.39295
$525$	15.5469	0.678521
$526$	64.9718	2.83290
$527$	−12.5099	−0.544938
$528$	−158.960	−6.91783
$529$	2.62246	0.114020
$530$	−9.12452	−0.396344
$531$	−3.81753	−0.165667
$532$	−129.657	−5.62136
$533$	45.1357	1.95504
$534$	78.6060	3.40161
$535$	−8.11276	−0.350745
$536$	134.854	5.82479
$537$	39.4274	1.70142
$538$	−70.0702	−3.02094
$539$	−11.3636	−0.489466
$540$	−40.2885	−1.73374
$541$	−29.0154	−1.24747	−0.623736	−	0.781635i	$-0.714386\pi$
$541$	−29.0154	−1.24747	−0.623736	+	0.781635i	$0.714386\pi$
$542$	27.9827	1.20196
$543$	−43.1170	−1.85033
$544$	44.5593	1.91047
$545$	−1.50597	−0.0645085
$546$	76.5702	3.27690
$547$	0.327963	0.0140227	0.00701134	−	0.999975i	$-0.497768\pi$
$547$	0.327963	0.0140227	0.00701134	+	0.999975i	$0.497768\pi$
$548$	48.1738	2.05788
$549$	3.38312	0.144388
$550$	−37.7185	−1.60832
$551$	−26.3483	−1.12248
$552$	97.1282	4.13405
$553$	18.3051	0.778412
$554$	−16.5512	−0.703193
$555$	−2.81250	−0.119384
$556$	−98.2814	−4.16806
$557$	20.3230	0.861113	0.430557	−	0.902564i	$-0.358317\pi$
$557$	20.3230	0.861113	0.430557	+	0.902564i	$0.358317\pi$
$558$	−10.2380	−0.433408
$559$	18.7174	0.791660
$560$	−78.5132	−3.31779
$561$	15.3686	0.648865
$562$	−40.2039	−1.69590
$563$	−18.1328	−0.764208	−0.382104	−	0.924119i	$-0.624800\pi$
$563$	−18.1328	−0.764208	−0.382104	+	0.924119i	$0.624800\pi$
$564$	−39.5541	−1.66553
$565$	−2.80195	−0.117879
$566$	43.5355	1.82993
$567$	31.1573	1.30848
$568$	−94.0622	−3.94676
$569$	−1.26921	−0.0532080	−0.0266040	−	0.999646i	$-0.508469\pi$
$569$	−1.26921	−0.0532080	−0.0266040	+	0.999646i	$0.508469\pi$
$570$	−58.2481	−2.43974
$571$	−4.86424	−0.203562	−0.101781	−	0.994807i	$-0.532454\pi$
$571$	−4.86424	−0.203562	−0.101781	+	0.994807i	$0.532454\pi$
$572$	−137.530	−5.75042
$573$	−24.9595	−1.04270
$574$	78.7225	3.28581
$575$	13.8293	0.576723
$576$	19.7739	0.823910
$577$	−24.5522	−1.02212	−0.511062	−	0.859544i	$-0.670748\pi$
$577$	−24.5522	−1.02212	−0.511062	+	0.859544i	$0.670748\pi$
$578$	39.5850	1.64652
$579$	−9.91722	−0.412146
$580$	−30.3195	−1.25895
$581$	−52.4896	−2.17764
$582$	36.7403	1.52294
$583$	−10.8604	−0.449793
$584$	121.680	5.03516
$585$	3.56184	0.147264
$586$	−30.6976	−1.26811
$587$	−16.4595	−0.679358	−0.339679	−	0.940541i	$-0.610318\pi$
$587$	−16.4595	−0.679358	−0.339679	+	0.940541i	$0.610318\pi$
$588$	24.3262	1.00319
$589$	56.4333	2.32529
$590$	32.7074	1.34654
$591$	−28.7913	−1.18432
$592$	−17.1101	−0.703220
$593$	0.822481	0.0337753	0.0168876	−	0.999857i	$-0.494624\pi$
$593$	0.822481	0.0337753	0.0168876	+	0.999857i	$0.494624\pi$
$594$	−64.7726	−2.65765
$595$	7.59087	0.311195
$596$	−11.3838	−0.466298
$597$	−23.0798	−0.944593
$598$	68.1112	2.78527
$599$	−23.6924	−0.968043	−0.484022	−	0.875056i	$-0.660824\pi$
$599$	−23.6924	−0.968043	−0.484022	+	0.875056i	$0.660824\pi$
$600$	52.4235	2.14018
$601$	−12.4631	−0.508381	−0.254190	−	0.967154i	$-0.581809\pi$
$601$	−12.4631	−0.508381	−0.254190	+	0.967154i	$0.581809\pi$
$602$	32.6455	1.33053
$603$	−6.40268	−0.260737
$604$	−29.8388	−1.21412
$605$	20.7019	0.841652
$606$	32.5862	1.32372
$607$	−11.4328	−0.464043	−0.232022	−	0.972711i	$-0.574534\pi$
$607$	−11.4328	−0.464043	−0.232022	+	0.972711i	$0.574534\pi$
$608$	−201.012	−8.15211
$609$	20.0919	0.814165
$610$	−28.9855	−1.17359
$611$	−18.0086	−0.728549
$612$	−4.60146	−0.186003
$613$	−16.0318	−0.647517	−0.323758	−	0.946140i	$-0.604946\pi$
$613$	−16.0318	−0.647517	−0.323758	+	0.946140i	$0.604946\pi$
$614$	−59.8165	−2.41400
$615$	26.1824	1.05578
$616$	−155.737	−6.27481
$617$	29.9747	1.20674	0.603368	−	0.797463i	$-0.293825\pi$
$617$	29.9747	1.20674	0.603368	+	0.797463i	$0.293825\pi$
$618$	48.0138	1.93140
$619$	11.7212	0.471115	0.235557	−	0.971860i	$-0.424308\pi$
$619$	11.7212	0.471115	0.235557	+	0.971860i	$0.424308\pi$
$620$	64.9388	2.60801
$621$	23.7486	0.953000
$622$	31.1463	1.24885
$623$	46.2113	1.85142
$624$	154.929	6.20210
$625$	−3.87550	−0.155020
$626$	−51.1582	−2.04469
$627$	−69.3296	−2.76876
$628$	6.93021	0.276546
$629$	1.65425	0.0659593
$630$	6.21232	0.247505
$631$	−8.35781	−0.332719	−0.166360	−	0.986065i	$-0.553201\pi$
$631$	−8.35781	−0.332719	−0.166360	+	0.986065i	$0.553201\pi$
$632$	61.7241	2.45525
$633$	25.5622	1.01601
$634$	−80.3348	−3.19050
$635$	−7.91977	−0.314287
$636$	23.2490	0.921881
$637$	11.0754	0.438825
$638$	−48.7452	−1.92984
$639$	4.46595	0.176670
$640$	−88.2859	−3.48981
$641$	25.9920	1.02662	0.513311	−	0.858203i	$-0.328419\pi$
$641$	25.9920	1.02662	0.513311	+	0.858203i	$0.328419\pi$
$642$	27.9213	1.10197
$643$	−19.9678	−0.787452	−0.393726	−	0.919228i	$-0.628814\pi$
$643$	−19.9678	−0.787452	−0.393726	+	0.919228i	$0.628814\pi$
$644$	87.9475	3.46562
$645$	10.8576	0.427518
$646$	34.2603	1.34795
$647$	−38.6465	−1.51935	−0.759675	−	0.650303i	$-0.774642\pi$
$647$	−38.6465	−1.51935	−0.759675	+	0.650303i	$0.774642\pi$
$648$	105.061	4.12719
$649$	38.9298	1.52813
$650$	36.7620	1.44192
$651$	−43.0332	−1.68660
$652$	−15.9316	−0.623928
$653$	−15.2668	−0.597438	−0.298719	−	0.954341i	$-0.596559\pi$
$653$	−15.2668	−0.597438	−0.298719	+	0.954341i	$0.596559\pi$
$654$	5.18302	0.202672
$655$	−8.42130	−0.329047
$656$	159.283	6.21897
$657$	−5.77722	−0.225391
$658$	−31.4093	−1.22446
$659$	−39.6208	−1.54341	−0.771703	−	0.635984i	$-0.780595\pi$
$659$	−39.6208	−1.54341	−0.771703	+	0.635984i	$0.780595\pi$
$660$	−79.7788	−3.10539
$661$	33.5220	1.30386	0.651928	−	0.758281i	$-0.273960\pi$
$661$	33.5220	1.30386	0.651928	+	0.758281i	$0.273960\pi$
$662$	45.5583	1.77067
$663$	−14.9789	−0.581733
$664$	−176.993	−6.86866
$665$	−34.2432	−1.32789
$666$	1.35383	0.0524597
$667$	17.8722	0.692016
$668$	−7.93661	−0.307077
$669$	−13.3563	−0.516383
$670$	54.8562	2.11928
$671$	−34.4999	−1.33185
$672$	153.282	5.91296
$673$	21.6676	0.835224	0.417612	−	0.908625i	$-0.362867\pi$
$673$	21.6676	0.835224	0.417612	+	0.908625i	$0.362867\pi$
$674$	−17.2567	−0.664705
$675$	12.8180	0.493364
$676$	59.9146	2.30441
$677$	−18.4380	−0.708631	−0.354315	−	0.935126i	$-0.615286\pi$
$677$	−18.4380	−0.708631	−0.354315	+	0.935126i	$0.615286\pi$
$678$	9.64333	0.370350
$679$	21.5991	0.828898
$680$	25.5961	0.981567
$681$	−43.4552	−1.66521
$682$	104.403	3.99781
$683$	16.6759	0.638084	0.319042	−	0.947741i	$-0.396639\pi$
$683$	16.6759	0.638084	0.319042	+	0.947741i	$0.396639\pi$
$684$	20.7577	0.793690
$685$	12.7230	0.486120
$686$	−39.8772	−1.52252
$687$	−39.7829	−1.51781
$688$	66.0533	2.51826
$689$	10.5850	0.403257
$690$	39.5101	1.50412
$691$	−22.5250	−0.856891	−0.428446	−	0.903568i	$-0.640939\pi$
$691$	−22.5250	−0.856891	−0.428446	+	0.903568i	$0.640939\pi$
$692$	78.9968	3.00301
$693$	7.39419	0.280882
$694$	48.8334	1.85369
$695$	−25.9567	−0.984592
$696$	67.7491	2.56802
$697$	−15.3999	−0.583314
$698$	−61.8870	−2.34246
$699$	−52.2447	−1.97608
$700$	47.4683	1.79413
$701$	−32.8914	−1.24229	−0.621146	−	0.783695i	$-0.713332\pi$
$701$	−32.8914	−1.24229	−0.621146	+	0.783695i	$0.713332\pi$
$702$	63.1300	2.38269
$703$	−7.46249	−0.281453
$704$	−201.647	−7.59985
$705$	−10.4465	−0.393437
$706$	36.5085	1.37402
$707$	19.1570	0.720472
$708$	−83.3373	−3.13201
$709$	23.8091	0.894171	0.447086	−	0.894491i	$-0.352462\pi$
$709$	23.8091	0.894171	0.447086	+	0.894491i	$0.352462\pi$
$710$	−38.2629	−1.43598
$711$	−2.93059	−0.109906
$712$	155.823	5.83971
$713$	−38.2791	−1.43356
$714$	−26.1252	−0.977709
$715$	−36.3225	−1.35838
$716$	120.381	4.49886
$717$	1.37768	0.0514504
$718$	−36.2043	−1.35113
$719$	21.9492	0.818565	0.409283	−	0.912408i	$-0.365779\pi$
$719$	21.9492	0.818565	0.409283	+	0.912408i	$0.365779\pi$
$720$	12.5697	0.468445
$721$	28.2266	1.05121
$722$	−101.821	−3.78940
$723$	19.3756	0.720586
$724$	−131.646	−4.89260
$725$	9.64628	0.358254
$726$	−71.2488	−2.64429
$727$	2.07020	0.0767794	0.0383897	−	0.999263i	$-0.487777\pi$
$727$	2.07020	0.0767794	0.0383897	+	0.999263i	$0.487777\pi$
$728$	151.787	5.62561
$729$	21.2979	0.788812
$730$	49.4974	1.83198
$731$	−6.38621	−0.236203
$732$	73.8541	2.72973
$733$	42.4892	1.56937	0.784687	−	0.619892i	$-0.212824\pi$
$733$	42.4892	1.56937	0.784687	+	0.619892i	$0.212824\pi$
$734$	73.2695	2.70443
$735$	6.42467	0.236978
$736$	136.348	5.02585
$737$	65.2924	2.40508
$738$	−12.6032	−0.463930
$739$	−1.78901	−0.0658099	−0.0329050	−	0.999458i	$-0.510476\pi$
$739$	−1.78901	−0.0658099	−0.0329050	+	0.999458i	$0.510476\pi$
$740$	−8.58723	−0.315673
$741$	67.5714	2.48230
$742$	18.4616	0.677748
$743$	16.4932	0.605075	0.302538	−	0.953137i	$-0.402166\pi$
$743$	16.4932	0.605075	0.302538	+	0.953137i	$0.402166\pi$
$744$	−145.106	−5.31986
$745$	−3.00652	−0.110151
$746$	−79.2649	−2.90209
$747$	8.40341	0.307465
$748$	46.9241	1.71572
$749$	16.4145	0.599774
$750$	60.3522	2.20375
$751$	−30.6686	−1.11911	−0.559556	−	0.828792i	$-0.689028\pi$
$751$	−30.6686	−1.11911	−0.559556	+	0.828792i	$0.689028\pi$
$752$	−63.5521	−2.31751
$753$	−44.6025	−1.62540
$754$	47.5091	1.73018
$755$	−7.88060	−0.286804
$756$	81.5156	2.96470
$757$	−39.1436	−1.42270	−0.711349	−	0.702839i	$-0.751915\pi$
$757$	−39.1436	−1.42270	−0.711349	+	0.702839i	$0.751915\pi$
$758$	34.2832	1.24522
$759$	47.0268	1.70696
$760$	−115.467	−4.18842
$761$	42.9294	1.55619	0.778096	−	0.628146i	$-0.216186\pi$
$761$	42.9294	1.55619	0.778096	+	0.628146i	$0.216186\pi$
$762$	27.2571	0.987421
$763$	3.04702	0.110310
$764$	−76.2074	−2.75709
$765$	−1.21527	−0.0439383
$766$	−99.1645	−3.58296
$767$	−37.9426	−1.37003
$768$	152.444	5.50086
$769$	49.8051	1.79602	0.898009	−	0.439978i	$-0.145014\pi$
$769$	49.8051	1.79602	0.898009	+	0.439978i	$0.145014\pi$
$770$	−63.3511	−2.28301
$771$	−41.3236	−1.48823
$772$	−30.2796	−1.08979
$773$	−39.4379	−1.41848	−0.709242	−	0.704965i	$-0.750963\pi$
$773$	−39.4379	−1.41848	−0.709242	+	0.704965i	$0.750963\pi$
$774$	−5.22643	−0.187860
$775$	−20.6606	−0.742150
$776$	72.8314	2.61449
$777$	5.69052	0.204146
$778$	−22.3002	−0.799502
$779$	69.4708	2.48905
$780$	77.7557	2.78410
$781$	−45.5423	−1.62963
$782$	−23.2390	−0.831024
$783$	16.5652	0.591992
$784$	39.0851	1.39590
$785$	1.83031	0.0653265
$786$	28.9832	1.03380
$787$	29.4680	1.05042	0.525211	−	0.850972i	$-0.323987\pi$
$787$	29.4680	1.05042	0.525211	+	0.850972i	$0.323987\pi$
$788$	−87.9067	−3.13155
$789$	−43.7216	−1.55653
$790$	25.1083	0.893314
$791$	5.66917	0.201573
$792$	24.9329	0.885953
$793$	33.6250	1.19406
$794$	−44.1677	−1.56745
$795$	6.14018	0.217770
$796$	−70.4681	−2.49768
$797$	−36.8191	−1.30420	−0.652099	−	0.758133i	$-0.726112\pi$
$797$	−36.8191	−1.30420	−0.652099	+	0.758133i	$0.726112\pi$
$798$	117.853	4.17196
$799$	6.14439	0.217373
$800$	73.5917	2.60186
$801$	−7.39827	−0.261405
$802$	−41.1129	−1.45175
$803$	58.9141	2.07903
$804$	−139.772	−4.92937
$805$	23.2274	0.818659
$806$	−101.756	−3.58420
$807$	47.1525	1.65985
$808$	64.5966	2.27250
$809$	30.5530	1.07418	0.537092	−	0.843523i	$-0.319523\pi$
$809$	30.5530	1.07418	0.537092	+	0.843523i	$0.319523\pi$
$810$	42.7371	1.50163
$811$	1.46555	0.0514625	0.0257312	−	0.999669i	$-0.491809\pi$
$811$	1.46555	0.0514625	0.0257312	+	0.999669i	$0.491809\pi$
$812$	61.3453	2.15280
$813$	−18.8305	−0.660413
$814$	−13.8059	−0.483895
$815$	−4.20761	−0.147386
$816$	−52.8604	−1.85048
$817$	28.8089	1.00789
$818$	−25.1831	−0.880507
$819$	−7.20667	−0.251822
$820$	79.9413	2.79167
$821$	10.2438	0.357512	0.178756	−	0.983893i	$-0.442793\pi$
$821$	10.2438	0.357512	0.178756	+	0.983893i	$0.442793\pi$
$822$	−43.7881	−1.52729
$823$	33.9663	1.18399	0.591996	−	0.805941i	$-0.298340\pi$
$823$	33.9663	1.18399	0.591996	+	0.805941i	$0.298340\pi$
$824$	95.1792	3.31572
$825$	25.3820	0.883687
$826$	−66.1768	−2.30259
$827$	−12.8306	−0.446163	−0.223081	−	0.974800i	$-0.571612\pi$
$827$	−12.8306	−0.446163	−0.223081	+	0.974800i	$0.571612\pi$
$828$	−14.0801	−0.489317
$829$	−9.41832	−0.327112	−0.163556	−	0.986534i	$-0.552296\pi$
$829$	−9.41832	−0.327112	−0.163556	+	0.986534i	$0.552296\pi$
$830$	−71.9978	−2.49908
$831$	11.1378	0.386367
$832$	196.533	6.81357
$833$	−3.77885	−0.130929
$834$	89.3339	3.09338
$835$	−2.09610	−0.0725387
$836$	−211.680	−7.32110
$837$	−35.4797	−1.22636
$838$	13.4204	0.463601
$839$	37.7212	1.30228	0.651141	−	0.758957i	$-0.274291\pi$
$839$	37.7212	1.30228	0.651141	+	0.758957i	$0.274291\pi$
$840$	88.0492	3.03799
$841$	−16.5337	−0.570128
$842$	−60.2683	−2.07698
$843$	27.0545	0.931806
$844$	78.0475	2.68650
$845$	15.8238	0.544354
$846$	5.02852	0.172884
$847$	−41.8862	−1.43923
$848$	37.3544	1.28275
$849$	−29.2964	−1.00545
$850$	−12.5429	−0.430218
$851$	5.06186	0.173518
$852$	97.4926	3.34004
$853$	−14.1384	−0.484090	−0.242045	−	0.970265i	$-0.577818\pi$
$853$	−14.1384	−0.484090	−0.242045	+	0.970265i	$0.577818\pi$
$854$	58.6464	2.00684
$855$	5.48222	0.187488
$856$	55.3492	1.89180
$857$	20.7096	0.707425	0.353712	−	0.935354i	$-0.384919\pi$
$857$	20.7096	0.707425	0.353712	+	0.935354i	$0.384919\pi$
$858$	125.009	4.26775
$859$	−16.7201	−0.570481	−0.285241	−	0.958456i	$-0.592074\pi$
$859$	−16.7201	−0.570481	−0.285241	+	0.958456i	$0.592074\pi$
$860$	33.1509	1.13044
$861$	−52.9749	−1.80538
$862$	−35.0712	−1.19453
$863$	37.3327	1.27082	0.635409	−	0.772175i	$-0.280831\pi$
$863$	37.3327	1.27082	0.635409	+	0.772175i	$0.280831\pi$
$864$	126.376	4.29941
$865$	20.8635	0.709380
$866$	−69.8361	−2.37313
$867$	−26.6380	−0.904674
$868$	−131.391	−4.45969
$869$	29.8851	1.01378
$870$	27.5592	0.934344
$871$	−63.6366	−2.15624
$872$	10.2744	0.347936
$873$	−3.45794	−0.117034
$874$	104.833	3.54605
$875$	35.4802	1.19945
$876$	−126.118	−4.26112
$877$	−41.0087	−1.38476	−0.692382	−	0.721531i	$-0.743439\pi$
$877$	−41.0087	−1.38476	−0.692382	+	0.721531i	$0.743439\pi$
$878$	−67.7236	−2.28556
$879$	20.6574	0.696757
$880$	−128.182	−4.32100
$881$	39.2597	1.32269	0.661346	−	0.750081i	$-0.269985\pi$
$881$	39.2597	1.32269	0.661346	+	0.750081i	$0.269985\pi$
$882$	−3.09259	−0.104133
$883$	12.3124	0.414345	0.207173	−	0.978304i	$-0.433574\pi$
$883$	12.3124	0.414345	0.207173	+	0.978304i	$0.433574\pi$
$884$	−45.7342	−1.53821
$885$	−22.0098	−0.739853
$886$	−83.7837	−2.81477
$887$	−18.4172	−0.618391	−0.309195	−	0.950999i	$-0.600060\pi$
$887$	−18.4172	−0.618391	−0.309195	+	0.950999i	$0.600060\pi$
$888$	19.1882	0.643915
$889$	16.0241	0.537430
$890$	63.3861	2.12471
$891$	50.8677	1.70413
$892$	−40.7798	−1.36541
$893$	−27.7180	−0.927546
$894$	10.3474	0.346069
$895$	31.7934	1.06274
$896$	178.629	5.96757
$897$	−45.8342	−1.53036
$898$	−39.8929	−1.33124
$899$	−26.7005	−0.890513
$900$	−7.59952	−0.253317
$901$	−3.61152	−0.120317
$902$	128.523	4.27935
$903$	−21.9682	−0.731056
$904$	19.1162	0.635797
$905$	−34.7685	−1.15575
$906$	27.1223	0.901078
$907$	7.27336	0.241508	0.120754	−	0.992682i	$-0.461469\pi$
$907$	7.27336	0.241508	0.120754	+	0.992682i	$0.461469\pi$
$908$	−132.679	−4.40311
$909$	−3.06696	−0.101725
$910$	61.7445	2.04681
$911$	−0.691119	−0.0228978	−0.0114489	−	0.999934i	$-0.503644\pi$
$911$	−0.691119	−0.0228978	−0.0114489	+	0.999934i	$0.503644\pi$
$912$	238.459	7.89616
$913$	−85.6951	−2.83609
$914$	47.1852	1.56075
$915$	19.5053	0.644825
$916$	−121.467	−4.01337
$917$	17.0388	0.562671
$918$	−21.5395	−0.710908
$919$	−24.7904	−0.817761	−0.408881	−	0.912588i	$-0.634081\pi$
$919$	−24.7904	−0.817761	−0.408881	+	0.912588i	$0.634081\pi$
$920$	78.3220	2.58220
$921$	40.2524	1.32636
$922$	−83.3412	−2.74470
$923$	44.3873	1.46103
$924$	161.416	5.31021
$925$	2.73206	0.0898297
$926$	32.4627	1.06679
$927$	−4.51899	−0.148423
$928$	95.1057	3.12200
$929$	10.9340	0.358733	0.179367	−	0.983782i	$-0.442595\pi$
$929$	10.9340	0.358733	0.179367	+	0.983782i	$0.442595\pi$
$930$	−59.0268	−1.93557
$931$	17.0468	0.558686
$932$	−159.516	−5.22510
$933$	−20.9593	−0.686177
$934$	15.7979	0.516922
$935$	12.3929	0.405292
$936$	−24.3006	−0.794291
$937$	−8.64073	−0.282280	−0.141140	−	0.989990i	$-0.545077\pi$
$937$	−8.64073	−0.282280	−0.141140	+	0.989990i	$0.545077\pi$
$938$	−110.990	−3.62397
$939$	34.4260	1.12345
$940$	−31.8956	−1.04032
$941$	−13.9654	−0.455260	−0.227630	−	0.973748i	$-0.573098\pi$
$941$	−13.9654	−0.455260	−0.227630	+	0.973748i	$0.573098\pi$
$942$	−6.29929	−0.205242
$943$	−47.1225	−1.53452
$944$	−133.899	−4.35804
$945$	21.5287	0.700330
$946$	53.2974	1.73285
$947$	−20.2021	−0.656479	−0.328239	−	0.944595i	$-0.606455\pi$
$947$	−20.2021	−0.656479	−0.328239	+	0.944595i	$0.606455\pi$
$948$	−63.9752	−2.07782
$949$	−57.4201	−1.86394
$950$	56.5823	1.83577
$951$	54.0599	1.75301
$952$	−51.7886	−1.67848
$953$	−20.7886	−0.673408	−0.336704	−	0.941611i	$-0.609312\pi$
$953$	−20.7886	−0.673408	−0.336704	+	0.941611i	$0.609312\pi$
$954$	−2.95564	−0.0956925
$955$	−20.1268	−0.651288
$956$	4.20638	0.136044
$957$	32.8022	1.06035
$958$	−49.0427	−1.58450
$959$	−25.7424	−0.831265
$960$	114.006	3.67951
$961$	26.1877	0.844766
$962$	13.4557	0.433831
$963$	−2.62791	−0.0846833
$964$	59.1583	1.90536
$965$	−7.99702	−0.257433
$966$	−79.9407	−2.57205
$967$	−23.6220	−0.759633	−0.379817	−	0.925062i	$-0.624013\pi$
$967$	−23.6220	−0.759633	−0.379817	+	0.925062i	$0.624013\pi$
$968$	−141.239	−4.53958
$969$	−23.0548	−0.740628
$970$	29.6266	0.951252
$971$	59.8678	1.92125	0.960624	−	0.277851i	$-0.0896221\pi$
$971$	59.8678	1.92125	0.960624	+	0.277851i	$0.0896221\pi$
$972$	−28.6349	−0.918464
$973$	52.5181	1.68365
$974$	95.5886	3.06286
$975$	−24.7383	−0.792260
$976$	118.662	3.79829
$977$	−17.5645	−0.561938	−0.280969	−	0.959717i	$-0.590656\pi$
$977$	−17.5645	−0.561938	−0.280969	+	0.959717i	$0.590656\pi$
$978$	14.4811	0.463056
$979$	75.4450	2.41123
$980$	19.6161	0.626612
$981$	−0.487817	−0.0155748
$982$	50.5762	1.61395
$983$	20.2279	0.645170	0.322585	−	0.946540i	$-0.395448\pi$
$983$	20.2279	0.645170	0.322585	+	0.946540i	$0.395448\pi$
$984$	−178.629	−5.69450
$985$	−23.2167	−0.739744
$986$	−16.2097	−0.516222
$987$	21.1363	0.672777
$988$	206.312	6.56365
$989$	−19.5413	−0.621376
$990$	10.1423	0.322343
$991$	8.58612	0.272747	0.136374	−	0.990657i	$-0.456455\pi$
$991$	8.58612	0.272747	0.136374	+	0.990657i	$0.456455\pi$
$992$	−203.699	−6.46745
$993$	−30.6576	−0.972891
$994$	77.4173	2.45553
$995$	−18.6110	−0.590009
$996$	183.448	5.81277
$997$	−15.8439	−0.501782	−0.250891	−	0.968015i	$-0.580724\pi$
$997$	−15.8439	−0.501782	−0.250891	+	0.968015i	$0.580724\pi$
$998$	62.3645	1.97411
$999$	4.69168	0.148438

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4033.2.a.d.1.2	✓	79

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4033.2.a.d.1.2	✓	79	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:	\( N \)	\(=\)	\( 4033 = 37 \cdot 109 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4033.a (trivial)

\(f(q)\)	\(=\)	\(q-2.77527 q^{2} +1.86757 q^{3} +5.70214 q^{4} +1.50597 q^{5} -5.18302 q^{6} -3.04702 q^{7} -10.2744 q^{8} +0.487817 q^{9} +O(q^{10})\)	\(q-2.77527 q^{2} +1.86757 q^{3} +5.70214 q^{4} +1.50597 q^{5} -5.18302 q^{6} -3.04702 q^{7} -10.2744 q^{8} +0.487817 q^{9} -4.17947 q^{10} -4.97460 q^{11} +10.6491 q^{12} +4.84844 q^{13} +8.45631 q^{14} +2.81250 q^{15} +17.1101 q^{16} -1.65425 q^{17} -1.35383 q^{18} +7.46249 q^{19} +8.58723 q^{20} -5.69052 q^{21} +13.8059 q^{22} -5.06186 q^{23} -19.1882 q^{24} -2.73206 q^{25} -13.4557 q^{26} -4.69168 q^{27} -17.3745 q^{28} -3.53076 q^{29} -7.80545 q^{30} +7.56226 q^{31} -26.9363 q^{32} -9.29040 q^{33} +4.59099 q^{34} -4.58871 q^{35} +2.78160 q^{36} -1.00000 q^{37} -20.7105 q^{38} +9.05480 q^{39} -15.4730 q^{40} +9.30932 q^{41} +15.7928 q^{42} +3.86049 q^{43} -28.3658 q^{44} +0.734637 q^{45} +14.0480 q^{46} -3.71430 q^{47} +31.9543 q^{48} +2.28433 q^{49} +7.58222 q^{50} -3.08943 q^{51} +27.6465 q^{52} +2.18318 q^{53} +13.0207 q^{54} -7.49157 q^{55} +31.3064 q^{56} +13.9367 q^{57} +9.79883 q^{58} -7.82573 q^{59} +16.0372 q^{60} +6.93522 q^{61} -20.9873 q^{62} -1.48639 q^{63} +40.5353 q^{64} +7.30159 q^{65} +25.7834 q^{66} -13.1252 q^{67} -9.43276 q^{68} -9.45338 q^{69} +12.7349 q^{70} +9.15497 q^{71} -5.01205 q^{72} -11.8430 q^{73} +2.77527 q^{74} -5.10232 q^{75} +42.5522 q^{76} +15.1577 q^{77} -25.1295 q^{78} -6.00754 q^{79} +25.7672 q^{80} -10.2255 q^{81} -25.8359 q^{82} +17.2265 q^{83} -32.4481 q^{84} -2.49124 q^{85} -10.7139 q^{86} -6.59395 q^{87} +51.1112 q^{88} -15.1661 q^{89} -2.03882 q^{90} -14.7733 q^{91} -28.8634 q^{92} +14.1230 q^{93} +10.3082 q^{94} +11.2383 q^{95} -50.3054 q^{96} -7.08860 q^{97} -6.33964 q^{98} -2.42669 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 79 q - 11 q^{2} - 11 q^{3} + 79 q^{4} - 16 q^{5} - 14 q^{6} - 15 q^{7} - 42 q^{8} + 76 q^{9}+O(q^{10}) \) 79 * q - 11 * q^2 - 11 * q^3 + 79 * q^4 - 16 * q^5 - 14 * q^6 - 15 * q^7 - 42 * q^8 + 76 * q^9	\( 79 q - 11 q^{2} - 11 q^{3} + 79 q^{4} - 16 q^{5} - 14 q^{6} - 15 q^{7} - 42 q^{8} + 76 q^{9} - 13 q^{10} - 5 q^{11} - 40 q^{12} - 18 q^{13} - 42 q^{14} - 49 q^{15} + 83 q^{16} - 62 q^{17} - 33 q^{18} - 25 q^{19} - 39 q^{20} - 15 q^{21} - 31 q^{22} - 94 q^{23} - 39 q^{24} + 71 q^{25} - 35 q^{26} - 47 q^{27} - 13 q^{28} - 17 q^{29} + 15 q^{30} - 37 q^{31} - 105 q^{32} - 60 q^{33} + 9 q^{34} - 60 q^{35} + 43 q^{36} - 79 q^{37} - 80 q^{38} - 41 q^{39} - 64 q^{40} - 37 q^{41} - 30 q^{42} - 20 q^{43} - 14 q^{44} - 8 q^{45} + 61 q^{46} - 148 q^{47} - 39 q^{48} + 82 q^{49} - 90 q^{50} - 45 q^{51} - 27 q^{52} - 70 q^{53} - 41 q^{54} - 105 q^{55} - 68 q^{56} - 31 q^{57} - 14 q^{58} - 96 q^{59} - 74 q^{60} - 21 q^{61} + 4 q^{62} - 60 q^{63} + 132 q^{64} - 15 q^{65} + 71 q^{66} - 44 q^{67} - 166 q^{68} - 72 q^{69} - 9 q^{70} - 55 q^{71} - 126 q^{72} - 27 q^{73} + 11 q^{74} - 39 q^{75} - 4 q^{76} - 104 q^{77} - 47 q^{78} - 49 q^{79} - 82 q^{80} + 55 q^{81} + 20 q^{82} - 52 q^{83} - 29 q^{84} + 3 q^{85} - 32 q^{86} - 113 q^{87} + 14 q^{88} - 68 q^{89} - 39 q^{90} - 30 q^{91} - 179 q^{92} - 53 q^{93} - 33 q^{94} - 86 q^{95} + 33 q^{96} - 57 q^{97} - 116 q^{98} + q^{99}+O(q^{100}) \) 79 * q - 11 * q^2 - 11 * q^3 + 79 * q^4 - 16 * q^5 - 14 * q^6 - 15 * q^7 - 42 * q^8 + 76 * q^9 - 13 * q^10 - 5 * q^11 - 40 * q^12 - 18 * q^13 - 42 * q^14 - 49 * q^15 + 83 * q^16 - 62 * q^17 - 33 * q^18 - 25 * q^19 - 39 * q^20 - 15 * q^21 - 31 * q^22 - 94 * q^23 - 39 * q^24 + 71 * q^25 - 35 * q^26 - 47 * q^27 - 13 * q^28 - 17 * q^29 + 15 * q^30 - 37 * q^31 - 105 * q^32 - 60 * q^33 + 9 * q^34 - 60 * q^35 + 43 * q^36 - 79 * q^37 - 80 * q^38 - 41 * q^39 - 64 * q^40 - 37 * q^41 - 30 * q^42 - 20 * q^43 - 14 * q^44 - 8 * q^45 + 61 * q^46 - 148 * q^47 - 39 * q^48 + 82 * q^49 - 90 * q^50 - 45 * q^51 - 27 * q^52 - 70 * q^53 - 41 * q^54 - 105 * q^55 - 68 * q^56 - 31 * q^57 - 14 * q^58 - 96 * q^59 - 74 * q^60 - 21 * q^61 + 4 * q^62 - 60 * q^63 + 132 * q^64 - 15 * q^65 + 71 * q^66 - 44 * q^67 - 166 * q^68 - 72 * q^69 - 9 * q^70 - 55 * q^71 - 126 * q^72 - 27 * q^73 + 11 * q^74 - 39 * q^75 - 4 * q^76 - 104 * q^77 - 47 * q^78 - 49 * q^79 - 82 * q^80 + 55 * q^81 + 20 * q^82 - 52 * q^83 - 29 * q^84 + 3 * q^85 - 32 * q^86 - 113 * q^87 + 14 * q^88 - 68 * q^89 - 39 * q^90 - 30 * q^91 - 179 * q^92 - 53 * q^93 - 33 * q^94 - 86 * q^95 + 33 * q^96 - 57 * q^97 - 116 * q^98 + q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more