LMFDB - Embedded newform 1474.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1474,2,Mod(1,1474)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1474, 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("1474.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1474 = 2 \cdot 11 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1474.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:	$11.7699492579$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	1474.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -0.618034 q^{3} +1.00000 q^{4} -3.23607 q^{5} -0.618034 q^{6} -1.23607 q^{7} +1.00000 q^{8} -2.61803 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -0.618034 q^{3} +1.00000 q^{4} -3.23607 q^{5} -0.618034 q^{6} -1.23607 q^{7} +1.00000 q^{8} -2.61803 q^{9} -3.23607 q^{10} +1.00000 q^{11} -0.618034 q^{12} +5.85410 q^{13} -1.23607 q^{14} +2.00000 q^{15} +1.00000 q^{16} +5.23607 q^{17} -2.61803 q^{18} -4.85410 q^{19} -3.23607 q^{20} +0.763932 q^{21} +1.00000 q^{22} +2.00000 q^{23} -0.618034 q^{24} +5.47214 q^{25} +5.85410 q^{26} +3.47214 q^{27} -1.23607 q^{28} -4.76393 q^{29} +2.00000 q^{30} -5.61803 q^{31} +1.00000 q^{32} -0.618034 q^{33} +5.23607 q^{34} +4.00000 q^{35} -2.61803 q^{36} +7.85410 q^{37} -4.85410 q^{38} -3.61803 q^{39} -3.23607 q^{40} +10.0902 q^{41} +0.763932 q^{42} +9.23607 q^{43} +1.00000 q^{44} +8.47214 q^{45} +2.00000 q^{46} +13.2361 q^{47} -0.618034 q^{48} -5.47214 q^{49} +5.47214 q^{50} -3.23607 q^{51} +5.85410 q^{52} +0.763932 q^{53} +3.47214 q^{54} -3.23607 q^{55} -1.23607 q^{56} +3.00000 q^{57} -4.76393 q^{58} +3.23607 q^{59} +2.00000 q^{60} +13.7984 q^{61} -5.61803 q^{62} +3.23607 q^{63} +1.00000 q^{64} -18.9443 q^{65} -0.618034 q^{66} +1.00000 q^{67} +5.23607 q^{68} -1.23607 q^{69} +4.00000 q^{70} +11.7082 q^{71} -2.61803 q^{72} +4.47214 q^{73} +7.85410 q^{74} -3.38197 q^{75} -4.85410 q^{76} -1.23607 q^{77} -3.61803 q^{78} -8.00000 q^{79} -3.23607 q^{80} +5.70820 q^{81} +10.0902 q^{82} -4.00000 q^{83} +0.763932 q^{84} -16.9443 q^{85} +9.23607 q^{86} +2.94427 q^{87} +1.00000 q^{88} -13.6180 q^{89} +8.47214 q^{90} -7.23607 q^{91} +2.00000 q^{92} +3.47214 q^{93} +13.2361 q^{94} +15.7082 q^{95} -0.618034 q^{96} -4.94427 q^{97} -5.47214 q^{98} -2.61803 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + q^{3} + 2 q^{4} - 2 q^{5} + q^{6} + 2 q^{7} + 2 q^{8} - 3 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 + q^3 + 2 * q^4 - 2 * q^5 + q^6 + 2 * q^7 + 2 * q^8 - 3 * q^9	\( 2 q + 2 q^{2} + q^{3} + 2 q^{4} - 2 q^{5} + q^{6} + 2 q^{7} + 2 q^{8} - 3 q^{9} - 2 q^{10} + 2 q^{11} + q^{12} + 5 q^{13} + 2 q^{14} + 4 q^{15} + 2 q^{16} + 6 q^{17} - 3 q^{18} - 3 q^{19} - 2 q^{20} + 6 q^{21} + 2 q^{22} + 4 q^{23} + q^{24} + 2 q^{25} + 5 q^{26} - 2 q^{27} + 2 q^{28} - 14 q^{29} + 4 q^{30} - 9 q^{31} + 2 q^{32} + q^{33} + 6 q^{34} + 8 q^{35} - 3 q^{36} + 9 q^{37} - 3 q^{38} - 5 q^{39} - 2 q^{40} + 9 q^{41} + 6 q^{42} + 14 q^{43} + 2 q^{44} + 8 q^{45} + 4 q^{46} + 22 q^{47} + q^{48} - 2 q^{49} + 2 q^{50} - 2 q^{51} + 5 q^{52} + 6 q^{53} - 2 q^{54} - 2 q^{55} + 2 q^{56} + 6 q^{57} - 14 q^{58} + 2 q^{59} + 4 q^{60} + 3 q^{61} - 9 q^{62} + 2 q^{63} + 2 q^{64} - 20 q^{65} + q^{66} + 2 q^{67} + 6 q^{68} + 2 q^{69} + 8 q^{70} + 10 q^{71} - 3 q^{72} + 9 q^{74} - 9 q^{75} - 3 q^{76} + 2 q^{77} - 5 q^{78} - 16 q^{79} - 2 q^{80} - 2 q^{81} + 9 q^{82} - 8 q^{83} + 6 q^{84} - 16 q^{85} + 14 q^{86} - 12 q^{87} + 2 q^{88} - 25 q^{89} + 8 q^{90} - 10 q^{91} + 4 q^{92} - 2 q^{93} + 22 q^{94} + 18 q^{95} + q^{96} + 8 q^{97} - 2 q^{98} - 3 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 + q^3 + 2 * q^4 - 2 * q^5 + q^6 + 2 * q^7 + 2 * q^8 - 3 * q^9 - 2 * q^10 + 2 * q^11 + q^12 + 5 * q^13 + 2 * q^14 + 4 * q^15 + 2 * q^16 + 6 * q^17 - 3 * q^18 - 3 * q^19 - 2 * q^20 + 6 * q^21 + 2 * q^22 + 4 * q^23 + q^24 + 2 * q^25 + 5 * q^26 - 2 * q^27 + 2 * q^28 - 14 * q^29 + 4 * q^30 - 9 * q^31 + 2 * q^32 + q^33 + 6 * q^34 + 8 * q^35 - 3 * q^36 + 9 * q^37 - 3 * q^38 - 5 * q^39 - 2 * q^40 + 9 * q^41 + 6 * q^42 + 14 * q^43 + 2 * q^44 + 8 * q^45 + 4 * q^46 + 22 * q^47 + q^48 - 2 * q^49 + 2 * q^50 - 2 * q^51 + 5 * q^52 + 6 * q^53 - 2 * q^54 - 2 * q^55 + 2 * q^56 + 6 * q^57 - 14 * q^58 + 2 * q^59 + 4 * q^60 + 3 * q^61 - 9 * q^62 + 2 * q^63 + 2 * q^64 - 20 * q^65 + q^66 + 2 * q^67 + 6 * q^68 + 2 * q^69 + 8 * q^70 + 10 * q^71 - 3 * q^72 + 9 * q^74 - 9 * q^75 - 3 * q^76 + 2 * q^77 - 5 * q^78 - 16 * q^79 - 2 * q^80 - 2 * q^81 + 9 * q^82 - 8 * q^83 + 6 * q^84 - 16 * q^85 + 14 * q^86 - 12 * q^87 + 2 * q^88 - 25 * q^89 + 8 * q^90 - 10 * q^91 + 4 * q^92 - 2 * q^93 + 22 * q^94 + 18 * q^95 + q^96 + 8 * q^97 - 2 * q^98 - 3 * 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
$3$	−0.618034	−0.356822	−0.178411	−	0.983956i	$-0.557096\pi$
$3$	−0.618034	−0.356822	−0.178411	+	0.983956i	$0.557096\pi$
$4$	1.00000	0.500000
$5$	−3.23607	−1.44721	−0.723607	−	0.690212i	$-0.757517\pi$
$5$	−3.23607	−1.44721	−0.723607	+	0.690212i	$0.757517\pi$
$6$	−0.618034	−0.252311
$7$	−1.23607	−0.467190	−0.233595	−	0.972334i	$-0.575049\pi$
$7$	−1.23607	−0.467190	−0.233595	+	0.972334i	$0.575049\pi$
$8$	1.00000	0.353553
$9$	−2.61803	−0.872678
$10$	−3.23607	−1.02333
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−0.618034	−0.178411
$13$	5.85410	1.62364	0.811818	−	0.583911i	$-0.198478\pi$
$13$	5.85410	1.62364	0.811818	+	0.583911i	$0.198478\pi$
$14$	−1.23607	−0.330353
$15$	2.00000	0.516398
$16$	1.00000	0.250000
$17$	5.23607	1.26993	0.634967	−	0.772540i	$-0.281014\pi$
$17$	5.23607	1.26993	0.634967	+	0.772540i	$0.281014\pi$
$18$	−2.61803	−0.617077
$19$	−4.85410	−1.11361	−0.556804	−	0.830644i	$-0.687972\pi$
$19$	−4.85410	−1.11361	−0.556804	+	0.830644i	$0.687972\pi$
$20$	−3.23607	−0.723607
$21$	0.763932	0.166704
$22$	1.00000	0.213201
$23$	2.00000	0.417029	0.208514	−	0.978019i	$-0.433137\pi$
$23$	2.00000	0.417029	0.208514	+	0.978019i	$0.433137\pi$
$24$	−0.618034	−0.126156
$25$	5.47214	1.09443
$26$	5.85410	1.14808
$27$	3.47214	0.668213
$28$	−1.23607	−0.233595
$29$	−4.76393	−0.884640	−0.442320	−	0.896857i	$-0.645844\pi$
$29$	−4.76393	−0.884640	−0.442320	+	0.896857i	$0.645844\pi$
$30$	2.00000	0.365148
$31$	−5.61803	−1.00903	−0.504514	−	0.863403i	$-0.668328\pi$
$31$	−5.61803	−1.00903	−0.504514	+	0.863403i	$0.668328\pi$
$32$	1.00000	0.176777
$33$	−0.618034	−0.107586
$34$	5.23607	0.897978
$35$	4.00000	0.676123
$36$	−2.61803	−0.436339
$37$	7.85410	1.29121	0.645603	−	0.763673i	$-0.276606\pi$
$37$	7.85410	1.29121	0.645603	+	0.763673i	$0.276606\pi$
$38$	−4.85410	−0.787439
$39$	−3.61803	−0.579349
$40$	−3.23607	−0.511667
$41$	10.0902	1.57582	0.787910	−	0.615791i	$-0.211163\pi$
$41$	10.0902	1.57582	0.787910	+	0.615791i	$0.211163\pi$
$42$	0.763932	0.117877
$43$	9.23607	1.40849	0.704244	−	0.709958i	$-0.251286\pi$
$43$	9.23607	1.40849	0.704244	+	0.709958i	$0.251286\pi$
$44$	1.00000	0.150756
$45$	8.47214	1.26295
$46$	2.00000	0.294884
$47$	13.2361	1.93068	0.965339	−	0.260998i	$-0.0840514\pi$
$47$	13.2361	1.93068	0.965339	+	0.260998i	$0.0840514\pi$
$48$	−0.618034	−0.0892055
$49$	−5.47214	−0.781734
$50$	5.47214	0.773877
$51$	−3.23607	−0.453140
$52$	5.85410	0.811818
$53$	0.763932	0.104934	0.0524671	−	0.998623i	$-0.483292\pi$
$53$	0.763932	0.104934	0.0524671	+	0.998623i	$0.483292\pi$
$54$	3.47214	0.472498
$55$	−3.23607	−0.436351
$56$	−1.23607	−0.165177
$57$	3.00000	0.397360
$58$	−4.76393	−0.625535
$59$	3.23607	0.421300	0.210650	−	0.977562i	$-0.432442\pi$
$59$	3.23607	0.421300	0.210650	+	0.977562i	$0.432442\pi$
$60$	2.00000	0.258199
$61$	13.7984	1.76670	0.883350	−	0.468713i	$-0.155282\pi$
$61$	13.7984	1.76670	0.883350	+	0.468713i	$0.155282\pi$
$62$	−5.61803	−0.713491
$63$	3.23607	0.407706
$64$	1.00000	0.125000
$65$	−18.9443	−2.34975
$66$	−0.618034	−0.0760747
$67$	1.00000	0.122169
$67$	1.00000	0.122169
$68$	5.23607	0.634967
$69$	−1.23607	−0.148805
$70$	4.00000	0.478091
$71$	11.7082	1.38951	0.694754	−	0.719247i	$-0.255513\pi$
$71$	11.7082	1.38951	0.694754	+	0.719247i	$0.255513\pi$
$72$	−2.61803	−0.308538
$73$	4.47214	0.523424	0.261712	−	0.965146i	$-0.415713\pi$
$73$	4.47214	0.523424	0.261712	+	0.965146i	$0.415713\pi$
$74$	7.85410	0.913021
$75$	−3.38197	−0.390516
$76$	−4.85410	−0.556804
$77$	−1.23607	−0.140863
$78$	−3.61803	−0.409662
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	−3.23607	−0.361803
$81$	5.70820	0.634245
$82$	10.0902	1.11427
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	0.763932	0.0833518
$85$	−16.9443	−1.83786
$86$	9.23607	0.995951
$87$	2.94427	0.315659
$88$	1.00000	0.106600
$89$	−13.6180	−1.44351	−0.721754	−	0.692149i	$-0.756664\pi$
$89$	−13.6180	−1.44351	−0.721754	+	0.692149i	$0.756664\pi$
$90$	8.47214	0.893042
$91$	−7.23607	−0.758546
$92$	2.00000	0.208514
$93$	3.47214	0.360044
$94$	13.2361	1.36520
$95$	15.7082	1.61163
$96$	−0.618034	−0.0630778
$97$	−4.94427	−0.502015	−0.251007	−	0.967985i	$-0.580762\pi$
$97$	−4.94427	−0.502015	−0.251007	+	0.967985i	$0.580762\pi$
$98$	−5.47214	−0.552769
$99$	−2.61803	−0.263122
$100$	5.47214	0.547214
$101$	−14.7984	−1.47249	−0.736247	−	0.676713i	$-0.763404\pi$
$101$	−14.7984	−1.47249	−0.736247	+	0.676713i	$0.763404\pi$
$102$	−3.23607	−0.320418
$103$	−8.18034	−0.806033	−0.403016	−	0.915193i	$-0.632038\pi$
$103$	−8.18034	−0.806033	−0.403016	+	0.915193i	$0.632038\pi$
$104$	5.85410	0.574042
$105$	−2.47214	−0.241256
$106$	0.763932	0.0741996
$107$	7.90983	0.764672	0.382336	−	0.924023i	$-0.375120\pi$
$107$	7.90983	0.764672	0.382336	+	0.924023i	$0.375120\pi$
$108$	3.47214	0.334106
$109$	−12.0902	−1.15803	−0.579014	−	0.815318i	$-0.696562\pi$
$109$	−12.0902	−1.15803	−0.579014	+	0.815318i	$0.696562\pi$
$110$	−3.23607	−0.308547
$111$	−4.85410	−0.460731
$112$	−1.23607	−0.116797
$113$	15.4164	1.45025	0.725127	−	0.688615i	$-0.241781\pi$
$113$	15.4164	1.45025	0.725127	+	0.688615i	$0.241781\pi$
$114$	3.00000	0.280976
$115$	−6.47214	−0.603530
$116$	−4.76393	−0.442320
$117$	−15.3262	−1.41691
$118$	3.23607	0.297904
$119$	−6.47214	−0.593300
$120$	2.00000	0.182574
$121$	1.00000	0.0909091
$122$	13.7984	1.24925
$123$	−6.23607	−0.562287
$124$	−5.61803	−0.504514
$125$	−1.52786	−0.136656
$126$	3.23607	0.288292
$127$	−17.8541	−1.58430	−0.792148	−	0.610329i	$-0.791037\pi$
$127$	−17.8541	−1.58430	−0.792148	+	0.610329i	$0.791037\pi$
$128$	1.00000	0.0883883
$129$	−5.70820	−0.502579
$130$	−18.9443	−1.66152
$131$	−10.3262	−0.902208	−0.451104	−	0.892471i	$-0.648970\pi$
$131$	−10.3262	−0.902208	−0.451104	+	0.892471i	$0.648970\pi$
$132$	−0.618034	−0.0537930
$133$	6.00000	0.520266
$134$	1.00000	0.0863868
$135$	−11.2361	−0.967047
$136$	5.23607	0.448989
$137$	8.18034	0.698894	0.349447	−	0.936956i	$-0.386370\pi$
$137$	8.18034	0.698894	0.349447	+	0.936956i	$0.386370\pi$
$138$	−1.23607	−0.105221
$139$	4.47214	0.379322	0.189661	−	0.981850i	$-0.439261\pi$
$139$	4.47214	0.379322	0.189661	+	0.981850i	$0.439261\pi$
$140$	4.00000	0.338062
$141$	−8.18034	−0.688909
$142$	11.7082	0.982531
$143$	5.85410	0.489545
$144$	−2.61803	−0.218169
$145$	15.4164	1.28026
$146$	4.47214	0.370117
$147$	3.38197	0.278940
$148$	7.85410	0.645603
$149$	−6.18034	−0.506313	−0.253157	−	0.967425i	$-0.581469\pi$
$149$	−6.18034	−0.506313	−0.253157	+	0.967425i	$0.581469\pi$
$150$	−3.38197	−0.276136
$151$	−3.09017	−0.251474	−0.125737	−	0.992064i	$-0.540130\pi$
$151$	−3.09017	−0.251474	−0.125737	+	0.992064i	$0.540130\pi$
$152$	−4.85410	−0.393720
$153$	−13.7082	−1.10824
$154$	−1.23607	−0.0996052
$155$	18.1803	1.46028
$156$	−3.61803	−0.289675
$157$	18.2705	1.45815	0.729073	−	0.684436i	$-0.239952\pi$
$157$	18.2705	1.45815	0.729073	+	0.684436i	$0.239952\pi$
$158$	−8.00000	−0.636446
$159$	−0.472136	−0.0374428
$160$	−3.23607	−0.255834
$161$	−2.47214	−0.194832
$162$	5.70820	0.448479
$163$	−2.76393	−0.216488	−0.108244	−	0.994124i	$-0.534523\pi$
$163$	−2.76393	−0.216488	−0.108244	+	0.994124i	$0.534523\pi$
$164$	10.0902	0.787910
$165$	2.00000	0.155700
$166$	−4.00000	−0.310460
$167$	1.90983	0.147787	0.0738935	−	0.997266i	$-0.476457\pi$
$167$	1.90983	0.147787	0.0738935	+	0.997266i	$0.476457\pi$
$168$	0.763932	0.0589386
$169$	21.2705	1.63619
$170$	−16.9443	−1.29957
$171$	12.7082	0.971821
$172$	9.23607	0.704244
$173$	16.4721	1.25235	0.626177	−	0.779681i	$-0.284619\pi$
$173$	16.4721	1.25235	0.626177	+	0.779681i	$0.284619\pi$
$174$	2.94427	0.223205
$175$	−6.76393	−0.511305
$176$	1.00000	0.0753778
$177$	−2.00000	−0.150329
$178$	−13.6180	−1.02071
$179$	12.5623	0.938951	0.469475	−	0.882946i	$-0.344443\pi$
$179$	12.5623	0.938951	0.469475	+	0.882946i	$0.344443\pi$
$180$	8.47214	0.631476
$181$	−17.0902	−1.27030	−0.635151	−	0.772388i	$-0.719062\pi$
$181$	−17.0902	−1.27030	−0.635151	+	0.772388i	$0.719062\pi$
$182$	−7.23607	−0.536373
$183$	−8.52786	−0.630398
$184$	2.00000	0.147442
$185$	−25.4164	−1.86865
$186$	3.47214	0.254589
$187$	5.23607	0.382899
$188$	13.2361	0.965339
$189$	−4.29180	−0.312182
$190$	15.7082	1.13959
$191$	−20.2705	−1.46672	−0.733361	−	0.679839i	$-0.762050\pi$
$191$	−20.2705	−1.46672	−0.733361	+	0.679839i	$0.762050\pi$
$192$	−0.618034	−0.0446028
$193$	−21.1246	−1.52058	−0.760291	−	0.649582i	$-0.774944\pi$
$193$	−21.1246	−1.52058	−0.760291	+	0.649582i	$0.774944\pi$
$194$	−4.94427	−0.354978
$195$	11.7082	0.838442
$196$	−5.47214	−0.390867
$197$	6.38197	0.454696	0.227348	−	0.973814i	$-0.426994\pi$
$197$	6.38197	0.454696	0.227348	+	0.973814i	$0.426994\pi$
$198$	−2.61803	−0.186056
$199$	16.9443	1.20115	0.600574	−	0.799569i	$-0.294939\pi$
$199$	16.9443	1.20115	0.600574	+	0.799569i	$0.294939\pi$
$200$	5.47214	0.386938
$201$	−0.618034	−0.0435928
$202$	−14.7984	−1.04121
$203$	5.88854	0.413295
$204$	−3.23607	−0.226570
$205$	−32.6525	−2.28055
$206$	−8.18034	−0.569951
$207$	−5.23607	−0.363932
$208$	5.85410	0.405909
$209$	−4.85410	−0.335765
$210$	−2.47214	−0.170594
$211$	−18.3262	−1.26163	−0.630815	−	0.775933i	$-0.717279\pi$
$211$	−18.3262	−1.26163	−0.630815	+	0.775933i	$0.717279\pi$
$212$	0.763932	0.0524671
$213$	−7.23607	−0.495807
$214$	7.90983	0.540705
$215$	−29.8885	−2.03838
$216$	3.47214	0.236249
$217$	6.94427	0.471408
$218$	−12.0902	−0.818850
$219$	−2.76393	−0.186769
$220$	−3.23607	−0.218176
$221$	30.6525	2.06191
$222$	−4.85410	−0.325786
$223$	0.472136	0.0316166	0.0158083	−	0.999875i	$-0.494968\pi$
$223$	0.472136	0.0316166	0.0158083	+	0.999875i	$0.494968\pi$
$224$	−1.23607	−0.0825883
$225$	−14.3262	−0.955083
$226$	15.4164	1.02548
$227$	20.5066	1.36107	0.680535	−	0.732716i	$-0.261748\pi$
$227$	20.5066	1.36107	0.680535	+	0.732716i	$0.261748\pi$
$228$	3.00000	0.198680
$229$	2.94427	0.194563	0.0972815	−	0.995257i	$-0.468985\pi$
$229$	2.94427	0.194563	0.0972815	+	0.995257i	$0.468985\pi$
$230$	−6.47214	−0.426760
$231$	0.763932	0.0502630
$232$	−4.76393	−0.312767
$233$	−11.2705	−0.738356	−0.369178	−	0.929359i	$-0.620361\pi$
$233$	−11.2705	−0.738356	−0.369178	+	0.929359i	$0.620361\pi$
$234$	−15.3262	−1.00191
$235$	−42.8328	−2.79410
$236$	3.23607	0.210650
$237$	4.94427	0.321165
$238$	−6.47214	−0.419526
$239$	3.70820	0.239864	0.119932	−	0.992782i	$-0.461732\pi$
$239$	3.70820	0.239864	0.119932	+	0.992782i	$0.461732\pi$
$240$	2.00000	0.129099
$241$	18.4721	1.18989	0.594947	−	0.803765i	$-0.297173\pi$
$241$	18.4721	1.18989	0.594947	+	0.803765i	$0.297173\pi$
$242$	1.00000	0.0642824
$243$	−13.9443	−0.894525
$244$	13.7984	0.883350
$245$	17.7082	1.13134
$246$	−6.23607	−0.397597
$247$	−28.4164	−1.80809
$248$	−5.61803	−0.356746
$249$	2.47214	0.156665
$250$	−1.52786	−0.0966306
$251$	4.03444	0.254652	0.127326	−	0.991861i	$-0.459361\pi$
$251$	4.03444	0.254652	0.127326	+	0.991861i	$0.459361\pi$
$252$	3.23607	0.203853
$253$	2.00000	0.125739
$254$	−17.8541	−1.12027
$255$	10.4721	0.655791
$256$	1.00000	0.0625000
$257$	2.14590	0.133857	0.0669287	−	0.997758i	$-0.478680\pi$
$257$	2.14590	0.133857	0.0669287	+	0.997758i	$0.478680\pi$
$258$	−5.70820	−0.355377
$259$	−9.70820	−0.603238
$260$	−18.9443	−1.17487
$261$	12.4721	0.772006
$262$	−10.3262	−0.637957
$263$	−13.8541	−0.854281	−0.427140	−	0.904185i	$-0.640479\pi$
$263$	−13.8541	−0.854281	−0.427140	+	0.904185i	$0.640479\pi$
$264$	−0.618034	−0.0380374
$265$	−2.47214	−0.151862
$266$	6.00000	0.367884
$267$	8.41641	0.515076
$268$	1.00000	0.0610847
$269$	22.0344	1.34346	0.671732	−	0.740794i	$-0.265551\pi$
$269$	22.0344	1.34346	0.671732	+	0.740794i	$0.265551\pi$
$270$	−11.2361	−0.683805
$271$	−2.58359	−0.156942	−0.0784710	−	0.996916i	$-0.525004\pi$
$271$	−2.58359	−0.156942	−0.0784710	+	0.996916i	$0.525004\pi$
$272$	5.23607	0.317483
$273$	4.47214	0.270666
$274$	8.18034	0.494192
$275$	5.47214	0.329982
$276$	−1.23607	−0.0744025
$277$	−13.7082	−0.823646	−0.411823	−	0.911264i	$-0.635108\pi$
$277$	−13.7082	−0.823646	−0.411823	+	0.911264i	$0.635108\pi$
$278$	4.47214	0.268221
$279$	14.7082	0.880557
$280$	4.00000	0.239046
$281$	24.7984	1.47935	0.739673	−	0.672966i	$-0.234980\pi$
$281$	24.7984	1.47935	0.739673	+	0.672966i	$0.234980\pi$
$282$	−8.18034	−0.487132
$283$	30.3262	1.80271	0.901354	−	0.433083i	$-0.142574\pi$
$283$	30.3262	1.80271	0.901354	+	0.433083i	$0.142574\pi$
$284$	11.7082	0.694754
$285$	−9.70820	−0.575064
$286$	5.85410	0.346160
$287$	−12.4721	−0.736207
$288$	−2.61803	−0.154269
$289$	10.4164	0.612730
$290$	15.4164	0.905283
$291$	3.05573	0.179130
$292$	4.47214	0.261712
$293$	22.9443	1.34042	0.670209	−	0.742172i	$-0.266204\pi$
$293$	22.9443	1.34042	0.670209	+	0.742172i	$0.266204\pi$
$294$	3.38197	0.197240
$295$	−10.4721	−0.609711
$296$	7.85410	0.456510
$297$	3.47214	0.201474
$298$	−6.18034	−0.358017
$299$	11.7082	0.677103
$300$	−3.38197	−0.195258
$301$	−11.4164	−0.658031
$302$	−3.09017	−0.177819
$303$	9.14590	0.525418
$304$	−4.85410	−0.278402
$305$	−44.6525	−2.55679
$306$	−13.7082	−0.783646
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	−1.23607	−0.0704315
$309$	5.05573	0.287610
$310$	18.1803	1.03257
$311$	2.90983	0.165001	0.0825007	−	0.996591i	$-0.473709\pi$
$311$	2.90983	0.165001	0.0825007	+	0.996591i	$0.473709\pi$
$312$	−3.61803	−0.204831
$313$	−10.2918	−0.581727	−0.290863	−	0.956765i	$-0.593943\pi$
$313$	−10.2918	−0.581727	−0.290863	+	0.956765i	$0.593943\pi$
$314$	18.2705	1.03106
$315$	−10.4721	−0.590038
$316$	−8.00000	−0.450035
$317$	27.7984	1.56131	0.780656	−	0.624961i	$-0.214885\pi$
$317$	27.7984	1.56131	0.780656	+	0.624961i	$0.214885\pi$
$318$	−0.472136	−0.0264761
$319$	−4.76393	−0.266729
$320$	−3.23607	−0.180902
$321$	−4.88854	−0.272852
$322$	−2.47214	−0.137767
$323$	−25.4164	−1.41421
$324$	5.70820	0.317122
$325$	32.0344	1.77695
$326$	−2.76393	−0.153080
$327$	7.47214	0.413210
$328$	10.0902	0.557136
$329$	−16.3607	−0.901993
$330$	2.00000	0.110096
$331$	−19.7426	−1.08515	−0.542577	−	0.840006i	$-0.682551\pi$
$331$	−19.7426	−1.08515	−0.542577	+	0.840006i	$0.682551\pi$
$332$	−4.00000	−0.219529
$333$	−20.5623	−1.12681
$334$	1.90983	0.104501
$335$	−3.23607	−0.176805
$336$	0.763932	0.0416759
$337$	22.2705	1.21315	0.606576	−	0.795026i	$-0.292543\pi$
$337$	22.2705	1.21315	0.606576	+	0.795026i	$0.292543\pi$
$338$	21.2705	1.15696
$339$	−9.52786	−0.517483
$340$	−16.9443	−0.918932
$341$	−5.61803	−0.304234
$342$	12.7082	0.687181
$343$	15.4164	0.832408
$344$	9.23607	0.497975
$345$	4.00000	0.215353
$346$	16.4721	0.885548
$347$	−2.65248	−0.142392	−0.0711962	−	0.997462i	$-0.522682\pi$
$347$	−2.65248	−0.142392	−0.0711962	+	0.997462i	$0.522682\pi$
$348$	2.94427	0.157830
$349$	10.7639	0.576180	0.288090	−	0.957603i	$-0.406980\pi$
$349$	10.7639	0.576180	0.288090	+	0.957603i	$0.406980\pi$
$350$	−6.76393	−0.361547
$351$	20.3262	1.08493
$352$	1.00000	0.0533002
$353$	35.4164	1.88503	0.942513	−	0.334171i	$-0.108456\pi$
$353$	35.4164	1.88503	0.942513	+	0.334171i	$0.108456\pi$
$354$	−2.00000	−0.106299
$355$	−37.8885	−2.01092
$356$	−13.6180	−0.721754
$357$	4.00000	0.211702
$358$	12.5623	0.663938
$359$	−31.7984	−1.67825	−0.839127	−	0.543936i	$-0.816934\pi$
$359$	−31.7984	−1.67825	−0.839127	+	0.543936i	$0.816934\pi$
$360$	8.47214	0.446521
$361$	4.56231	0.240121
$362$	−17.0902	−0.898239
$363$	−0.618034	−0.0324384
$364$	−7.23607	−0.379273
$365$	−14.4721	−0.757506
$366$	−8.52786	−0.445759
$367$	−20.9443	−1.09328	−0.546641	−	0.837367i	$-0.684094\pi$
$367$	−20.9443	−1.09328	−0.546641	+	0.837367i	$0.684094\pi$
$368$	2.00000	0.104257
$369$	−26.4164	−1.37518
$370$	−25.4164	−1.32134
$371$	−0.944272	−0.0490242
$372$	3.47214	0.180022
$373$	−7.88854	−0.408453	−0.204227	−	0.978924i	$-0.565468\pi$
$373$	−7.88854	−0.408453	−0.204227	+	0.978924i	$0.565468\pi$
$374$	5.23607	0.270751
$375$	0.944272	0.0487620
$376$	13.2361	0.682598
$377$	−27.8885	−1.43633
$378$	−4.29180	−0.220746
$379$	−20.6180	−1.05908	−0.529539	−	0.848286i	$-0.677635\pi$
$379$	−20.6180	−1.05908	−0.529539	+	0.848286i	$0.677635\pi$
$380$	15.7082	0.805814
$381$	11.0344	0.565312
$382$	−20.2705	−1.03713
$383$	−18.3820	−0.939275	−0.469637	−	0.882859i	$-0.655615\pi$
$383$	−18.3820	−0.939275	−0.469637	+	0.882859i	$0.655615\pi$
$384$	−0.618034	−0.0315389
$385$	4.00000	0.203859
$386$	−21.1246	−1.07521
$387$	−24.1803	−1.22916
$388$	−4.94427	−0.251007
$389$	0.798374	0.0404792	0.0202396	−	0.999795i	$-0.493557\pi$
$389$	0.798374	0.0404792	0.0202396	+	0.999795i	$0.493557\pi$
$390$	11.7082	0.592868
$391$	10.4721	0.529599
$392$	−5.47214	−0.276385
$393$	6.38197	0.321928
$394$	6.38197	0.321519
$395$	25.8885	1.30259
$396$	−2.61803	−0.131561
$397$	8.38197	0.420679	0.210339	−	0.977628i	$-0.432543\pi$
$397$	8.38197	0.420679	0.210339	+	0.977628i	$0.432543\pi$
$398$	16.9443	0.849340
$399$	−3.70820	−0.185642
$400$	5.47214	0.273607
$401$	4.65248	0.232334	0.116167	−	0.993230i	$-0.462939\pi$
$401$	4.65248	0.232334	0.116167	+	0.993230i	$0.462939\pi$
$402$	−0.618034	−0.0308247
$403$	−32.8885	−1.63830
$404$	−14.7984	−0.736247
$405$	−18.4721	−0.917888
$406$	5.88854	0.292244
$407$	7.85410	0.389313
$408$	−3.23607	−0.160209
$409$	6.94427	0.343372	0.171686	−	0.985152i	$-0.445079\pi$
$409$	6.94427	0.343372	0.171686	+	0.985152i	$0.445079\pi$
$410$	−32.6525	−1.61259
$411$	−5.05573	−0.249381
$412$	−8.18034	−0.403016
$413$	−4.00000	−0.196827
$414$	−5.23607	−0.257339
$415$	12.9443	0.635409
$416$	5.85410	0.287021
$417$	−2.76393	−0.135350
$418$	−4.85410	−0.237422
$419$	20.4721	1.00013	0.500065	−	0.865988i	$-0.333310\pi$
$419$	20.4721	1.00013	0.500065	+	0.865988i	$0.333310\pi$
$420$	−2.47214	−0.120628
$421$	12.5623	0.612249	0.306125	−	0.951991i	$-0.400968\pi$
$421$	12.5623	0.612249	0.306125	+	0.951991i	$0.400968\pi$
$422$	−18.3262	−0.892107
$423$	−34.6525	−1.68486
$424$	0.763932	0.0370998
$425$	28.6525	1.38985
$426$	−7.23607	−0.350589
$427$	−17.0557	−0.825385
$428$	7.90983	0.382336
$429$	−3.61803	−0.174680
$430$	−29.8885	−1.44135
$431$	17.2705	0.831891	0.415946	−	0.909390i	$-0.363451\pi$
$431$	17.2705	0.831891	0.415946	+	0.909390i	$0.363451\pi$
$432$	3.47214	0.167053
$433$	34.4721	1.65663	0.828313	−	0.560266i	$-0.189301\pi$
$433$	34.4721	1.65663	0.828313	+	0.560266i	$0.189301\pi$
$434$	6.94427	0.333336
$435$	−9.52786	−0.456826
$436$	−12.0902	−0.579014
$437$	−9.70820	−0.464406
$438$	−2.76393	−0.132066
$439$	29.8885	1.42650	0.713251	−	0.700909i	$-0.247222\pi$
$439$	29.8885	1.42650	0.713251	+	0.700909i	$0.247222\pi$
$440$	−3.23607	−0.154273
$441$	14.3262	0.682202
$442$	30.6525	1.45799
$443$	−5.90983	−0.280784	−0.140392	−	0.990096i	$-0.544836\pi$
$443$	−5.90983	−0.280784	−0.140392	+	0.990096i	$0.544836\pi$
$444$	−4.85410	−0.230365
$445$	44.0689	2.08907
$446$	0.472136	0.0223563
$447$	3.81966	0.180664
$448$	−1.23607	−0.0583987
$449$	18.1459	0.856358	0.428179	−	0.903694i	$-0.359155\pi$
$449$	18.1459	0.856358	0.428179	+	0.903694i	$0.359155\pi$
$450$	−14.3262	−0.675345
$451$	10.0902	0.475128
$452$	15.4164	0.725127
$453$	1.90983	0.0897316
$454$	20.5066	0.962421
$455$	23.4164	1.09778
$456$	3.00000	0.140488
$457$	−17.5967	−0.823141	−0.411571	−	0.911378i	$-0.635020\pi$
$457$	−17.5967	−0.823141	−0.411571	+	0.911378i	$0.635020\pi$
$458$	2.94427	0.137577
$459$	18.1803	0.848586
$460$	−6.47214	−0.301765
$461$	14.9443	0.696024	0.348012	−	0.937490i	$-0.386857\pi$
$461$	14.9443	0.696024	0.348012	+	0.937490i	$0.386857\pi$
$462$	0.763932	0.0355413
$463$	−12.9443	−0.601571	−0.300786	−	0.953692i	$-0.597249\pi$
$463$	−12.9443	−0.601571	−0.300786	+	0.953692i	$0.597249\pi$
$464$	−4.76393	−0.221160
$465$	−11.2361	−0.521060
$466$	−11.2705	−0.522096
$467$	37.2361	1.72308	0.861540	−	0.507690i	$-0.169500\pi$
$467$	37.2361	1.72308	0.861540	+	0.507690i	$0.169500\pi$
$468$	−15.3262	−0.708456
$469$	−1.23607	−0.0570763
$470$	−42.8328	−1.97573
$471$	−11.2918	−0.520298
$472$	3.23607	0.148952
$473$	9.23607	0.424675
$474$	4.94427	0.227098
$475$	−26.5623	−1.21876
$476$	−6.47214	−0.296650
$477$	−2.00000	−0.0915737
$478$	3.70820	0.169609
$479$	−33.8885	−1.54841	−0.774204	−	0.632937i	$-0.781849\pi$
$479$	−33.8885	−1.54841	−0.774204	+	0.632937i	$0.781849\pi$
$480$	2.00000	0.0912871
$481$	45.9787	2.09645
$482$	18.4721	0.841383
$483$	1.52786	0.0695202
$484$	1.00000	0.0454545
$485$	16.0000	0.726523
$486$	−13.9443	−0.632525
$487$	4.67376	0.211788	0.105894	−	0.994377i	$-0.466230\pi$
$487$	4.67376	0.211788	0.105894	+	0.994377i	$0.466230\pi$
$488$	13.7984	0.624623
$489$	1.70820	0.0772477
$490$	17.7082	0.799975
$491$	−20.7984	−0.938617	−0.469309	−	0.883034i	$-0.655497\pi$
$491$	−20.7984	−0.938617	−0.469309	+	0.883034i	$0.655497\pi$
$492$	−6.23607	−0.281144
$493$	−24.9443	−1.12343
$494$	−28.4164	−1.27851
$495$	8.47214	0.380794
$496$	−5.61803	−0.252257
$497$	−14.4721	−0.649164
$498$	2.47214	0.110779
$499$	−30.8541	−1.38122	−0.690610	−	0.723228i	$-0.742658\pi$
$499$	−30.8541	−1.38122	−0.690610	+	0.723228i	$0.742658\pi$
$500$	−1.52786	−0.0683282
$501$	−1.18034	−0.0527337
$502$	4.03444	0.180066
$503$	−19.1246	−0.852724	−0.426362	−	0.904553i	$-0.640205\pi$
$503$	−19.1246	−0.852724	−0.426362	+	0.904553i	$0.640205\pi$
$504$	3.23607	0.144146
$505$	47.8885	2.13101
$506$	2.00000	0.0889108
$507$	−13.1459	−0.583830
$508$	−17.8541	−0.792148
$509$	−17.3262	−0.767972	−0.383986	−	0.923339i	$-0.625449\pi$
$509$	−17.3262	−0.767972	−0.383986	+	0.923339i	$0.625449\pi$
$510$	10.4721	0.463714
$511$	−5.52786	−0.244538
$512$	1.00000	0.0441942
$513$	−16.8541	−0.744127
$514$	2.14590	0.0946515
$515$	26.4721	1.16650
$516$	−5.70820	−0.251290
$517$	13.2361	0.582122
$518$	−9.70820	−0.426554
$519$	−10.1803	−0.446867
$520$	−18.9443	−0.830761
$521$	1.05573	0.0462523	0.0231261	−	0.999733i	$-0.492638\pi$
$521$	1.05573	0.0462523	0.0231261	+	0.999733i	$0.492638\pi$
$522$	12.4721	0.545891
$523$	22.6180	0.989018	0.494509	−	0.869173i	$-0.335348\pi$
$523$	22.6180	0.989018	0.494509	+	0.869173i	$0.335348\pi$
$524$	−10.3262	−0.451104
$525$	4.18034	0.182445
$526$	−13.8541	−0.604068
$527$	−29.4164	−1.28140
$528$	−0.618034	−0.0268965
$529$	−19.0000	−0.826087
$530$	−2.47214	−0.107383
$531$	−8.47214	−0.367659
$532$	6.00000	0.260133
$533$	59.0689	2.55856
$534$	8.41641	0.364214
$535$	−25.5967	−1.10664
$536$	1.00000	0.0431934
$537$	−7.76393	−0.335038
$538$	22.0344	0.949972
$539$	−5.47214	−0.235702
$540$	−11.2361	−0.483523
$541$	−21.8541	−0.939581	−0.469791	−	0.882778i	$-0.655671\pi$
$541$	−21.8541	−0.939581	−0.469791	+	0.882778i	$0.655671\pi$
$542$	−2.58359	−0.110975
$543$	10.5623	0.453272
$544$	5.23607	0.224495
$545$	39.1246	1.67591
$546$	4.47214	0.191390
$547$	−15.5967	−0.666869	−0.333434	−	0.942773i	$-0.608208\pi$
$547$	−15.5967	−0.666869	−0.333434	+	0.942773i	$0.608208\pi$
$548$	8.18034	0.349447
$549$	−36.1246	−1.54176
$550$	5.47214	0.233333
$551$	23.1246	0.985142
$552$	−1.23607	−0.0526105
$553$	9.88854	0.420504
$554$	−13.7082	−0.582406
$555$	15.7082	0.666776
$556$	4.47214	0.189661
$557$	−34.6525	−1.46827	−0.734136	−	0.679002i	$-0.762413\pi$
$557$	−34.6525	−1.46827	−0.734136	+	0.679002i	$0.762413\pi$
$558$	14.7082	0.622648
$559$	54.0689	2.28687
$560$	4.00000	0.169031
$561$	−3.23607	−0.136627
$562$	24.7984	1.04606
$563$	12.6525	0.533238	0.266619	−	0.963802i	$-0.414093\pi$
$563$	12.6525	0.533238	0.266619	+	0.963802i	$0.414093\pi$
$564$	−8.18034	−0.344454
$565$	−49.8885	−2.09883
$566$	30.3262	1.27471
$567$	−7.05573	−0.296313
$568$	11.7082	0.491265
$569$	20.0000	0.838444	0.419222	−	0.907884i	$-0.362303\pi$
$569$	20.0000	0.838444	0.419222	+	0.907884i	$0.362303\pi$
$570$	−9.70820	−0.406632
$571$	36.9443	1.54607	0.773035	−	0.634364i	$-0.218738\pi$
$571$	36.9443	1.54607	0.773035	+	0.634364i	$0.218738\pi$
$572$	5.85410	0.244772
$573$	12.5279	0.523359
$574$	−12.4721	−0.520577
$575$	10.9443	0.456408
$576$	−2.61803	−0.109085
$577$	−31.7771	−1.32290	−0.661449	−	0.749990i	$-0.730058\pi$
$577$	−31.7771	−1.32290	−0.661449	+	0.749990i	$0.730058\pi$
$578$	10.4164	0.433265
$579$	13.0557	0.542578
$580$	15.4164	0.640131
$581$	4.94427	0.205123
$582$	3.05573	0.126664
$583$	0.763932	0.0316388
$584$	4.47214	0.185058
$585$	49.5967	2.05057
$586$	22.9443	0.947819
$587$	9.20163	0.379792	0.189896	−	0.981804i	$-0.439185\pi$
$587$	9.20163	0.379792	0.189896	+	0.981804i	$0.439185\pi$
$588$	3.38197	0.139470
$589$	27.2705	1.12366
$590$	−10.4721	−0.431131
$591$	−3.94427	−0.162246
$592$	7.85410	0.322802
$593$	−23.9230	−0.982399	−0.491200	−	0.871047i	$-0.663441\pi$
$593$	−23.9230	−0.982399	−0.491200	+	0.871047i	$0.663441\pi$
$594$	3.47214	0.142463
$595$	20.9443	0.858631
$596$	−6.18034	−0.253157
$597$	−10.4721	−0.428596
$598$	11.7082	0.478784
$599$	−35.1459	−1.43602	−0.718011	−	0.696032i	$-0.754947\pi$
$599$	−35.1459	−1.43602	−0.718011	+	0.696032i	$0.754947\pi$
$600$	−3.38197	−0.138068
$601$	2.29180	0.0934843	0.0467422	−	0.998907i	$-0.485116\pi$
$601$	2.29180	0.0934843	0.0467422	+	0.998907i	$0.485116\pi$
$602$	−11.4164	−0.465298
$603$	−2.61803	−0.106615
$604$	−3.09017	−0.125737
$605$	−3.23607	−0.131565
$606$	9.14590	0.371527
$607$	−40.7426	−1.65369	−0.826846	−	0.562428i	$-0.809867\pi$
$607$	−40.7426	−1.65369	−0.826846	+	0.562428i	$0.809867\pi$
$608$	−4.85410	−0.196860
$609$	−3.63932	−0.147473
$610$	−44.6525	−1.80793
$611$	77.4853	3.13472
$612$	−13.7082	−0.554121
$613$	31.4164	1.26890	0.634448	−	0.772965i	$-0.281227\pi$
$613$	31.4164	1.26890	0.634448	+	0.772965i	$0.281227\pi$
$614$	0	0
$615$	20.1803	0.813750
$616$	−1.23607	−0.0498026
$617$	37.8541	1.52395	0.761974	−	0.647607i	$-0.224230\pi$
$617$	37.8541	1.52395	0.761974	+	0.647607i	$0.224230\pi$
$618$	5.05573	0.203371
$619$	33.8885	1.36210	0.681048	−	0.732239i	$-0.261525\pi$
$619$	33.8885	1.36210	0.681048	+	0.732239i	$0.261525\pi$
$620$	18.1803	0.730140
$621$	6.94427	0.278664
$622$	2.90983	0.116674
$623$	16.8328	0.674393
$624$	−3.61803	−0.144837
$625$	−22.4164	−0.896656
$626$	−10.2918	−0.411343
$627$	3.00000	0.119808
$628$	18.2705	0.729073
$629$	41.1246	1.63975
$630$	−10.4721	−0.417220
$631$	−26.9098	−1.07126	−0.535632	−	0.844452i	$-0.679926\pi$
$631$	−26.9098	−1.07126	−0.535632	+	0.844452i	$0.679926\pi$
$632$	−8.00000	−0.318223
$633$	11.3262	0.450178
$634$	27.7984	1.10401
$635$	57.7771	2.29281
$636$	−0.472136	−0.0187214
$637$	−32.0344	−1.26925
$638$	−4.76393	−0.188606
$639$	−30.6525	−1.21259
$640$	−3.23607	−0.127917
$641$	−24.5410	−0.969312	−0.484656	−	0.874705i	$-0.661055\pi$
$641$	−24.5410	−0.969312	−0.484656	+	0.874705i	$0.661055\pi$
$642$	−4.88854	−0.192935
$643$	−30.9443	−1.22032	−0.610161	−	0.792277i	$-0.708895\pi$
$643$	−30.9443	−1.22032	−0.610161	+	0.792277i	$0.708895\pi$
$644$	−2.47214	−0.0974158
$645$	18.4721	0.727340
$646$	−25.4164	−0.999995
$647$	−4.94427	−0.194379	−0.0971897	−	0.995266i	$-0.530985\pi$
$647$	−4.94427	−0.194379	−0.0971897	+	0.995266i	$0.530985\pi$
$648$	5.70820	0.224239
$649$	3.23607	0.127027
$650$	32.0344	1.25649
$651$	−4.29180	−0.168209
$652$	−2.76393	−0.108244
$653$	−8.18034	−0.320121	−0.160061	−	0.987107i	$-0.551169\pi$
$653$	−8.18034	−0.320121	−0.160061	+	0.987107i	$0.551169\pi$
$654$	7.47214	0.292184
$655$	33.4164	1.30569
$656$	10.0902	0.393955
$657$	−11.7082	−0.456781
$658$	−16.3607	−0.637806
$659$	−40.6312	−1.58277	−0.791383	−	0.611320i	$-0.790639\pi$
$659$	−40.6312	−1.58277	−0.791383	+	0.611320i	$0.790639\pi$
$660$	2.00000	0.0778499
$661$	29.7082	1.15552	0.577758	−	0.816208i	$-0.303928\pi$
$661$	29.7082	1.15552	0.577758	+	0.816208i	$0.303928\pi$
$662$	−19.7426	−0.767320
$663$	−18.9443	−0.735735
$664$	−4.00000	−0.155230
$665$	−19.4164	−0.752936
$666$	−20.5623	−0.796773
$667$	−9.52786	−0.368920
$668$	1.90983	0.0738935
$669$	−0.291796	−0.0112815
$670$	−3.23607	−0.125020
$671$	13.7984	0.532680
$672$	0.763932	0.0294693
$673$	−11.2705	−0.434446	−0.217223	−	0.976122i	$-0.569700\pi$
$673$	−11.2705	−0.434446	−0.217223	+	0.976122i	$0.569700\pi$
$674$	22.2705	0.857828
$675$	19.0000	0.731310
$676$	21.2705	0.818097
$677$	−45.9787	−1.76711	−0.883553	−	0.468332i	$-0.844855\pi$
$677$	−45.9787	−1.76711	−0.883553	+	0.468332i	$0.844855\pi$
$678$	−9.52786	−0.365915
$679$	6.11146	0.234536
$680$	−16.9443	−0.649783
$681$	−12.6738	−0.485660
$682$	−5.61803	−0.215126
$683$	−4.38197	−0.167671	−0.0838356	−	0.996480i	$-0.526717\pi$
$683$	−4.38197	−0.167671	−0.0838356	+	0.996480i	$0.526717\pi$
$684$	12.7082	0.485910
$685$	−26.4721	−1.01145
$686$	15.4164	0.588601
$687$	−1.81966	−0.0694244
$688$	9.23607	0.352122
$689$	4.47214	0.170375
$690$	4.00000	0.152277
$691$	−4.11146	−0.156407	−0.0782036	−	0.996937i	$-0.524918\pi$
$691$	−4.11146	−0.156407	−0.0782036	+	0.996937i	$0.524918\pi$
$692$	16.4721	0.626177
$693$	3.23607	0.122928
$694$	−2.65248	−0.100687
$695$	−14.4721	−0.548959
$696$	2.94427	0.111602
$697$	52.8328	2.00119
$698$	10.7639	0.407421
$699$	6.96556	0.263462
$700$	−6.76393	−0.255653
$701$	20.6738	0.780837	0.390419	−	0.920637i	$-0.372330\pi$
$701$	20.6738	0.780837	0.390419	+	0.920637i	$0.372330\pi$
$702$	20.3262	0.767164
$703$	−38.1246	−1.43790
$704$	1.00000	0.0376889
$705$	26.4721	0.996998
$706$	35.4164	1.33291
$707$	18.2918	0.687934
$708$	−2.00000	−0.0751646
$709$	−7.85410	−0.294967	−0.147483	−	0.989065i	$-0.547117\pi$
$709$	−7.85410	−0.294967	−0.147483	+	0.989065i	$0.547117\pi$
$710$	−37.8885	−1.42193
$711$	20.9443	0.785472
$712$	−13.6180	−0.510357
$713$	−11.2361	−0.420794
$714$	4.00000	0.149696
$715$	−18.9443	−0.708476
$716$	12.5623	0.469475
$717$	−2.29180	−0.0855887
$718$	−31.7984	−1.18670
$719$	−28.8328	−1.07528	−0.537641	−	0.843174i	$-0.680685\pi$
$719$	−28.8328	−1.07528	−0.537641	+	0.843174i	$0.680685\pi$
$720$	8.47214	0.315738
$721$	10.1115	0.376570
$722$	4.56231	0.169791
$723$	−11.4164	−0.424581
$724$	−17.0902	−0.635151
$725$	−26.0689	−0.968174
$726$	−0.618034	−0.0229374
$727$	−16.2148	−0.601373	−0.300686	−	0.953723i	$-0.597216\pi$
$727$	−16.2148	−0.601373	−0.300686	+	0.953723i	$0.597216\pi$
$728$	−7.23607	−0.268187
$729$	−8.50658	−0.315058
$730$	−14.4721	−0.535638
$731$	48.3607	1.78868
$732$	−8.52786	−0.315199
$733$	−14.0000	−0.517102	−0.258551	−	0.965998i	$-0.583245\pi$
$733$	−14.0000	−0.517102	−0.258551	+	0.965998i	$0.583245\pi$
$734$	−20.9443	−0.773067
$735$	−10.9443	−0.403686
$736$	2.00000	0.0737210
$737$	1.00000	0.0368355
$738$	−26.4164	−0.972401
$739$	25.4164	0.934958	0.467479	−	0.884004i	$-0.345162\pi$
$739$	25.4164	0.934958	0.467479	+	0.884004i	$0.345162\pi$
$740$	−25.4164	−0.934326
$741$	17.5623	0.645167
$742$	−0.944272	−0.0346653
$743$	−15.6738	−0.575015	−0.287507	−	0.957778i	$-0.592827\pi$
$743$	−15.6738	−0.575015	−0.287507	+	0.957778i	$0.592827\pi$
$744$	3.47214	0.127295
$745$	20.0000	0.732743
$746$	−7.88854	−0.288820
$747$	10.4721	0.383155
$748$	5.23607	0.191450
$749$	−9.77709	−0.357247
$750$	0.944272	0.0344799
$751$	−27.2361	−0.993858	−0.496929	−	0.867791i	$-0.665539\pi$
$751$	−27.2361	−0.993858	−0.496929	+	0.867791i	$0.665539\pi$
$752$	13.2361	0.482670
$753$	−2.49342	−0.0908654
$754$	−27.8885	−1.01564
$755$	10.0000	0.363937
$756$	−4.29180	−0.156091
$757$	27.1246	0.985861	0.492930	−	0.870069i	$-0.335926\pi$
$757$	27.1246	0.985861	0.492930	+	0.870069i	$0.335926\pi$
$758$	−20.6180	−0.748881
$759$	−1.23607	−0.0448664
$760$	15.7082	0.569796
$761$	21.8885	0.793459	0.396730	−	0.917936i	$-0.370145\pi$
$761$	21.8885	0.793459	0.396730	+	0.917936i	$0.370145\pi$
$762$	11.0344	0.399736
$763$	14.9443	0.541019
$764$	−20.2705	−0.733361
$765$	44.3607	1.60386
$766$	−18.3820	−0.664167
$767$	18.9443	0.684038
$768$	−0.618034	−0.0223014
$769$	−26.6180	−0.959871	−0.479935	−	0.877304i	$-0.659340\pi$
$769$	−26.6180	−0.959871	−0.479935	+	0.877304i	$0.659340\pi$
$770$	4.00000	0.144150
$771$	−1.32624	−0.0477633
$772$	−21.1246	−0.760291
$773$	−47.8885	−1.72243	−0.861216	−	0.508240i	$-0.830296\pi$
$773$	−47.8885	−1.72243	−0.861216	+	0.508240i	$0.830296\pi$
$774$	−24.1803	−0.869144
$775$	−30.7426	−1.10431
$776$	−4.94427	−0.177489
$777$	6.00000	0.215249
$778$	0.798374	0.0286231
$779$	−48.9787	−1.75484
$780$	11.7082	0.419221
$781$	11.7082	0.418952
$782$	10.4721	0.374483
$783$	−16.5410	−0.591128
$784$	−5.47214	−0.195433
$785$	−59.1246	−2.11025
$786$	6.38197	0.227637
$787$	50.0689	1.78476	0.892381	−	0.451282i	$-0.149033\pi$
$787$	50.0689	1.78476	0.892381	+	0.451282i	$0.149033\pi$
$788$	6.38197	0.227348
$789$	8.56231	0.304826
$790$	25.8885	0.921073
$791$	−19.0557	−0.677544
$792$	−2.61803	−0.0930278
$793$	80.7771	2.86848
$794$	8.38197	0.297465
$795$	1.52786	0.0541878
$796$	16.9443	0.600574
$797$	−34.6312	−1.22670	−0.613350	−	0.789811i	$-0.710178\pi$
$797$	−34.6312	−1.22670	−0.613350	+	0.789811i	$0.710178\pi$
$798$	−3.70820	−0.131269
$799$	69.3050	2.45183
$800$	5.47214	0.193469
$801$	35.6525	1.25972
$802$	4.65248	0.164285
$803$	4.47214	0.157818
$804$	−0.618034	−0.0217964
$805$	8.00000	0.281963
$806$	−32.8885	−1.15845
$807$	−13.6180	−0.479378
$808$	−14.7984	−0.520605
$809$	22.6180	0.795208	0.397604	−	0.917557i	$-0.369842\pi$
$809$	22.6180	0.795208	0.397604	+	0.917557i	$0.369842\pi$
$810$	−18.4721	−0.649045
$811$	10.7639	0.377973	0.188986	−	0.981980i	$-0.439480\pi$
$811$	10.7639	0.377973	0.188986	+	0.981980i	$0.439480\pi$
$812$	5.88854	0.206647
$813$	1.59675	0.0560004
$814$	7.85410	0.275286
$815$	8.94427	0.313304
$816$	−3.23607	−0.113285
$817$	−44.8328	−1.56850
$818$	6.94427	0.242801
$819$	18.9443	0.661966
$820$	−32.6525	−1.14027
$821$	20.7639	0.724666	0.362333	−	0.932049i	$-0.381980\pi$
$821$	20.7639	0.724666	0.362333	+	0.932049i	$0.381980\pi$
$822$	−5.05573	−0.176339
$823$	49.9574	1.74141	0.870703	−	0.491809i	$-0.163664\pi$
$823$	49.9574	1.74141	0.870703	+	0.491809i	$0.163664\pi$
$824$	−8.18034	−0.284976
$825$	−3.38197	−0.117745
$826$	−4.00000	−0.139178
$827$	32.2148	1.12022	0.560109	−	0.828419i	$-0.310759\pi$
$827$	32.2148	1.12022	0.560109	+	0.828419i	$0.310759\pi$
$828$	−5.23607	−0.181966
$829$	−6.67376	−0.231789	−0.115895	−	0.993262i	$-0.536974\pi$
$829$	−6.67376	−0.231789	−0.115895	+	0.993262i	$0.536974\pi$
$830$	12.9443	0.449302
$831$	8.47214	0.293895
$832$	5.85410	0.202954
$833$	−28.6525	−0.992749
$834$	−2.76393	−0.0957071
$835$	−6.18034	−0.213879
$836$	−4.85410	−0.167883
$837$	−19.5066	−0.674246
$838$	20.4721	0.707198
$839$	−13.1246	−0.453112	−0.226556	−	0.973998i	$-0.572747\pi$
$839$	−13.1246	−0.453112	−0.226556	+	0.973998i	$0.572747\pi$
$840$	−2.47214	−0.0852968
$841$	−6.30495	−0.217412
$842$	12.5623	0.432926
$843$	−15.3262	−0.527864
$844$	−18.3262	−0.630815
$845$	−68.8328	−2.36792
$846$	−34.6525	−1.19138
$847$	−1.23607	−0.0424718
$848$	0.763932	0.0262335
$849$	−18.7426	−0.643246
$850$	28.6525	0.982772
$851$	15.7082	0.538470
$852$	−7.23607	−0.247904
$853$	15.3050	0.524032	0.262016	−	0.965064i	$-0.415613\pi$
$853$	15.3050	0.524032	0.262016	+	0.965064i	$0.415613\pi$
$854$	−17.0557	−0.583635
$855$	−41.1246	−1.40643
$856$	7.90983	0.270352
$857$	3.78522	0.129301	0.0646503	−	0.997908i	$-0.479407\pi$
$857$	3.78522	0.129301	0.0646503	+	0.997908i	$0.479407\pi$
$858$	−3.61803	−0.123518
$859$	−0.180340	−0.00615312	−0.00307656	−	0.999995i	$-0.500979\pi$
$859$	−0.180340	−0.00615312	−0.00307656	+	0.999995i	$0.500979\pi$
$860$	−29.8885	−1.01919
$861$	7.70820	0.262695
$862$	17.2705	0.588236
$863$	−41.0132	−1.39610	−0.698052	−	0.716047i	$-0.745950\pi$
$863$	−41.0132	−1.39610	−0.698052	+	0.716047i	$0.745950\pi$
$864$	3.47214	0.118124
$865$	−53.3050	−1.81242
$866$	34.4721	1.17141
$867$	−6.43769	−0.218636
$868$	6.94427	0.235704
$869$	−8.00000	−0.271381
$870$	−9.52786	−0.323025
$871$	5.85410	0.198359
$872$	−12.0902	−0.409425
$873$	12.9443	0.438097
$874$	−9.70820	−0.328385
$875$	1.88854	0.0638444
$876$	−2.76393	−0.0933846
$877$	10.7639	0.363472	0.181736	−	0.983347i	$-0.441828\pi$
$877$	10.7639	0.363472	0.181736	+	0.983347i	$0.441828\pi$
$878$	29.8885	1.00869
$879$	−14.1803	−0.478291
$880$	−3.23607	−0.109088
$881$	13.4164	0.452010	0.226005	−	0.974126i	$-0.427433\pi$
$881$	13.4164	0.452010	0.226005	+	0.974126i	$0.427433\pi$
$882$	14.3262	0.482390
$883$	−48.5623	−1.63425	−0.817126	−	0.576459i	$-0.804434\pi$
$883$	−48.5623	−1.63425	−0.817126	+	0.576459i	$0.804434\pi$
$884$	30.6525	1.03095
$885$	6.47214	0.217558
$886$	−5.90983	−0.198545
$887$	−36.0000	−1.20876	−0.604381	−	0.796696i	$-0.706579\pi$
$887$	−36.0000	−1.20876	−0.604381	+	0.796696i	$0.706579\pi$
$888$	−4.85410	−0.162893
$889$	22.0689	0.740167
$890$	44.0689	1.47719
$891$	5.70820	0.191232
$892$	0.472136	0.0158083
$893$	−64.2492	−2.15002
$894$	3.81966	0.127749
$895$	−40.6525	−1.35886
$896$	−1.23607	−0.0412941
$897$	−7.23607	−0.241605
$898$	18.1459	0.605536
$899$	26.7639	0.892627
$900$	−14.3262	−0.477541
$901$	4.00000	0.133259
$902$	10.0902	0.335966
$903$	7.05573	0.234800
$904$	15.4164	0.512742
$905$	55.3050	1.83840
$906$	1.90983	0.0634499
$907$	26.6525	0.884981	0.442490	−	0.896773i	$-0.354095\pi$
$907$	26.6525	0.884981	0.442490	+	0.896773i	$0.354095\pi$
$908$	20.5066	0.680535
$909$	38.7426	1.28501
$910$	23.4164	0.776246
$911$	32.0000	1.06021	0.530104	−	0.847933i	$-0.322153\pi$
$911$	32.0000	1.06021	0.530104	+	0.847933i	$0.322153\pi$
$912$	3.00000	0.0993399
$913$	−4.00000	−0.132381
$914$	−17.5967	−0.582049
$915$	27.5967	0.912320
$916$	2.94427	0.0972815
$917$	12.7639	0.421502
$918$	18.1803	0.600041
$919$	−45.3050	−1.49447	−0.747236	−	0.664559i	$-0.768620\pi$
$919$	−45.3050	−1.49447	−0.747236	+	0.664559i	$0.768620\pi$
$920$	−6.47214	−0.213380
$921$	0	0
$922$	14.9443	0.492163
$923$	68.5410	2.25606
$924$	0.763932	0.0251315
$925$	42.9787	1.41313
$926$	−12.9443	−0.425375
$927$	21.4164	0.703407
$928$	−4.76393	−0.156384
$929$	37.2361	1.22168	0.610838	−	0.791756i	$-0.290833\pi$
$929$	37.2361	1.22168	0.610838	+	0.791756i	$0.290833\pi$
$930$	−11.2361	−0.368445
$931$	26.5623	0.870544
$932$	−11.2705	−0.369178
$933$	−1.79837	−0.0588761
$934$	37.2361	1.21840
$935$	−16.9443	−0.554137
$936$	−15.3262	−0.500954
$937$	6.27051	0.204849	0.102424	−	0.994741i	$-0.467340\pi$
$937$	6.27051	0.204849	0.102424	+	0.994741i	$0.467340\pi$
$938$	−1.23607	−0.0403591
$939$	6.36068	0.207573
$940$	−42.8328	−1.39705
$941$	37.5623	1.22450	0.612248	−	0.790666i	$-0.290265\pi$
$941$	37.5623	1.22450	0.612248	+	0.790666i	$0.290265\pi$
$942$	−11.2918	−0.367907
$943$	20.1803	0.657162
$944$	3.23607	0.105325
$945$	13.8885	0.451794
$946$	9.23607	0.300290
$947$	−3.70820	−0.120500	−0.0602502	−	0.998183i	$-0.519190\pi$
$947$	−3.70820	−0.120500	−0.0602502	+	0.998183i	$0.519190\pi$
$948$	4.94427	0.160582
$949$	26.1803	0.849850
$950$	−26.5623	−0.861795
$951$	−17.1803	−0.557111
$952$	−6.47214	−0.209763
$953$	−20.5410	−0.665389	−0.332694	−	0.943035i	$-0.607958\pi$
$953$	−20.5410	−0.665389	−0.332694	+	0.943035i	$0.607958\pi$
$954$	−2.00000	−0.0647524
$955$	65.5967	2.12266
$956$	3.70820	0.119932
$957$	2.94427	0.0951748
$958$	−33.8885	−1.09489
$959$	−10.1115	−0.326516
$960$	2.00000	0.0645497
$961$	0.562306	0.0181389
$962$	45.9787	1.48241
$963$	−20.7082	−0.667313
$964$	18.4721	0.594947
$965$	68.3607	2.20061
$966$	1.52786	0.0491582
$967$	11.4164	0.367127	0.183563	−	0.983008i	$-0.441237\pi$
$967$	11.4164	0.367127	0.183563	+	0.983008i	$0.441237\pi$
$968$	1.00000	0.0321412
$969$	15.7082	0.504620
$970$	16.0000	0.513729
$971$	−31.5967	−1.01399	−0.506994	−	0.861950i	$-0.669243\pi$
$971$	−31.5967	−1.01399	−0.506994	+	0.861950i	$0.669243\pi$
$972$	−13.9443	−0.447263
$973$	−5.52786	−0.177215
$974$	4.67376	0.149757
$975$	−19.7984	−0.634055
$976$	13.7984	0.441675
$977$	−32.3951	−1.03641	−0.518206	−	0.855256i	$-0.673400\pi$
$977$	−32.3951	−1.03641	−0.518206	+	0.855256i	$0.673400\pi$
$978$	1.70820	0.0546223
$979$	−13.6180	−0.435234
$980$	17.7082	0.565668
$981$	31.6525	1.01059
$982$	−20.7984	−0.663703
$983$	37.7426	1.20380	0.601902	−	0.798570i	$-0.294410\pi$
$983$	37.7426	1.20380	0.601902	+	0.798570i	$0.294410\pi$
$984$	−6.23607	−0.198799
$985$	−20.6525	−0.658043
$986$	−24.9443	−0.794387
$987$	10.1115	0.321851
$988$	−28.4164	−0.904046
$989$	18.4721	0.587380
$990$	8.47214	0.269262
$991$	−34.6869	−1.10187	−0.550933	−	0.834549i	$-0.685728\pi$
$991$	−34.6869	−1.10187	−0.550933	+	0.834549i	$0.685728\pi$
$992$	−5.61803	−0.178373
$993$	12.2016	0.387207
$994$	−14.4721	−0.459028
$995$	−54.8328	−1.73832
$996$	2.47214	0.0783326
$997$	40.0689	1.26899	0.634497	−	0.772925i	$-0.281207\pi$
$997$	40.0689	1.26899	0.634497	+	0.772925i	$0.281207\pi$
$998$	−30.8541	−0.976670
$999$	27.2705	0.862801

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1474.2.a.b.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1474.2.a.b.1.1	✓	2	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:	\( N \)	\(=\)	\( 1474 = 2 \cdot 11 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1474.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -0.618034 q^{3} +1.00000 q^{4} -3.23607 q^{5} -0.618034 q^{6} -1.23607 q^{7} +1.00000 q^{8} -2.61803 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -0.618034 q^{3} +1.00000 q^{4} -3.23607 q^{5} -0.618034 q^{6} -1.23607 q^{7} +1.00000 q^{8} -2.61803 q^{9} -3.23607 q^{10} +1.00000 q^{11} -0.618034 q^{12} +5.85410 q^{13} -1.23607 q^{14} +2.00000 q^{15} +1.00000 q^{16} +5.23607 q^{17} -2.61803 q^{18} -4.85410 q^{19} -3.23607 q^{20} +0.763932 q^{21} +1.00000 q^{22} +2.00000 q^{23} -0.618034 q^{24} +5.47214 q^{25} +5.85410 q^{26} +3.47214 q^{27} -1.23607 q^{28} -4.76393 q^{29} +2.00000 q^{30} -5.61803 q^{31} +1.00000 q^{32} -0.618034 q^{33} +5.23607 q^{34} +4.00000 q^{35} -2.61803 q^{36} +7.85410 q^{37} -4.85410 q^{38} -3.61803 q^{39} -3.23607 q^{40} +10.0902 q^{41} +0.763932 q^{42} +9.23607 q^{43} +1.00000 q^{44} +8.47214 q^{45} +2.00000 q^{46} +13.2361 q^{47} -0.618034 q^{48} -5.47214 q^{49} +5.47214 q^{50} -3.23607 q^{51} +5.85410 q^{52} +0.763932 q^{53} +3.47214 q^{54} -3.23607 q^{55} -1.23607 q^{56} +3.00000 q^{57} -4.76393 q^{58} +3.23607 q^{59} +2.00000 q^{60} +13.7984 q^{61} -5.61803 q^{62} +3.23607 q^{63} +1.00000 q^{64} -18.9443 q^{65} -0.618034 q^{66} +1.00000 q^{67} +5.23607 q^{68} -1.23607 q^{69} +4.00000 q^{70} +11.7082 q^{71} -2.61803 q^{72} +4.47214 q^{73} +7.85410 q^{74} -3.38197 q^{75} -4.85410 q^{76} -1.23607 q^{77} -3.61803 q^{78} -8.00000 q^{79} -3.23607 q^{80} +5.70820 q^{81} +10.0902 q^{82} -4.00000 q^{83} +0.763932 q^{84} -16.9443 q^{85} +9.23607 q^{86} +2.94427 q^{87} +1.00000 q^{88} -13.6180 q^{89} +8.47214 q^{90} -7.23607 q^{91} +2.00000 q^{92} +3.47214 q^{93} +13.2361 q^{94} +15.7082 q^{95} -0.618034 q^{96} -4.94427 q^{97} -5.47214 q^{98} -2.61803 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + q^{3} + 2 q^{4} - 2 q^{5} + q^{6} + 2 q^{7} + 2 q^{8} - 3 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 + q^3 + 2 * q^4 - 2 * q^5 + q^6 + 2 * q^7 + 2 * q^8 - 3 * q^9	\( 2 q + 2 q^{2} + q^{3} + 2 q^{4} - 2 q^{5} + q^{6} + 2 q^{7} + 2 q^{8} - 3 q^{9} - 2 q^{10} + 2 q^{11} + q^{12} + 5 q^{13} + 2 q^{14} + 4 q^{15} + 2 q^{16} + 6 q^{17} - 3 q^{18} - 3 q^{19} - 2 q^{20} + 6 q^{21} + 2 q^{22} + 4 q^{23} + q^{24} + 2 q^{25} + 5 q^{26} - 2 q^{27} + 2 q^{28} - 14 q^{29} + 4 q^{30} - 9 q^{31} + 2 q^{32} + q^{33} + 6 q^{34} + 8 q^{35} - 3 q^{36} + 9 q^{37} - 3 q^{38} - 5 q^{39} - 2 q^{40} + 9 q^{41} + 6 q^{42} + 14 q^{43} + 2 q^{44} + 8 q^{45} + 4 q^{46} + 22 q^{47} + q^{48} - 2 q^{49} + 2 q^{50} - 2 q^{51} + 5 q^{52} + 6 q^{53} - 2 q^{54} - 2 q^{55} + 2 q^{56} + 6 q^{57} - 14 q^{58} + 2 q^{59} + 4 q^{60} + 3 q^{61} - 9 q^{62} + 2 q^{63} + 2 q^{64} - 20 q^{65} + q^{66} + 2 q^{67} + 6 q^{68} + 2 q^{69} + 8 q^{70} + 10 q^{71} - 3 q^{72} + 9 q^{74} - 9 q^{75} - 3 q^{76} + 2 q^{77} - 5 q^{78} - 16 q^{79} - 2 q^{80} - 2 q^{81} + 9 q^{82} - 8 q^{83} + 6 q^{84} - 16 q^{85} + 14 q^{86} - 12 q^{87} + 2 q^{88} - 25 q^{89} + 8 q^{90} - 10 q^{91} + 4 q^{92} - 2 q^{93} + 22 q^{94} + 18 q^{95} + q^{96} + 8 q^{97} - 2 q^{98} - 3 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 + q^3 + 2 * q^4 - 2 * q^5 + q^6 + 2 * q^7 + 2 * q^8 - 3 * q^9 - 2 * q^10 + 2 * q^11 + q^12 + 5 * q^13 + 2 * q^14 + 4 * q^15 + 2 * q^16 + 6 * q^17 - 3 * q^18 - 3 * q^19 - 2 * q^20 + 6 * q^21 + 2 * q^22 + 4 * q^23 + q^24 + 2 * q^25 + 5 * q^26 - 2 * q^27 + 2 * q^28 - 14 * q^29 + 4 * q^30 - 9 * q^31 + 2 * q^32 + q^33 + 6 * q^34 + 8 * q^35 - 3 * q^36 + 9 * q^37 - 3 * q^38 - 5 * q^39 - 2 * q^40 + 9 * q^41 + 6 * q^42 + 14 * q^43 + 2 * q^44 + 8 * q^45 + 4 * q^46 + 22 * q^47 + q^48 - 2 * q^49 + 2 * q^50 - 2 * q^51 + 5 * q^52 + 6 * q^53 - 2 * q^54 - 2 * q^55 + 2 * q^56 + 6 * q^57 - 14 * q^58 + 2 * q^59 + 4 * q^60 + 3 * q^61 - 9 * q^62 + 2 * q^63 + 2 * q^64 - 20 * q^65 + q^66 + 2 * q^67 + 6 * q^68 + 2 * q^69 + 8 * q^70 + 10 * q^71 - 3 * q^72 + 9 * q^74 - 9 * q^75 - 3 * q^76 + 2 * q^77 - 5 * q^78 - 16 * q^79 - 2 * q^80 - 2 * q^81 + 9 * q^82 - 8 * q^83 + 6 * q^84 - 16 * q^85 + 14 * q^86 - 12 * q^87 + 2 * q^88 - 25 * q^89 + 8 * q^90 - 10 * q^91 + 4 * q^92 - 2 * q^93 + 22 * q^94 + 18 * q^95 + q^96 + 8 * q^97 - 2 * q^98 - 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more