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Properties

Label 637.2.k.a.569.1
Level $637$
Weight $2$
Character 637.569
Analytic conductor $5.086$
Analytic rank $1$
Dimension $2$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [637,2,Mod(459,637)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("637.459"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(637, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.k (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-2,-2,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.08647060876\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 13)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 569.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 637.569
Dual form 637.2.k.a.459.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.73205i q^{2} +(-1.00000 + 1.73205i) q^{3} -1.00000 q^{4} +(-1.50000 - 0.866025i) q^{5} +(3.00000 + 1.73205i) q^{6} -1.73205i q^{8} +(-0.500000 - 0.866025i) q^{9} +(-1.50000 + 2.59808i) q^{10} +(1.00000 - 1.73205i) q^{12} +(-2.50000 + 2.59808i) q^{13} +(3.00000 - 1.73205i) q^{15} -5.00000 q^{16} -3.00000 q^{17} +(-1.50000 + 0.866025i) q^{18} +(3.00000 - 1.73205i) q^{19} +(1.50000 + 0.866025i) q^{20} -6.00000 q^{23} +(3.00000 + 1.73205i) q^{24} +(-1.00000 - 1.73205i) q^{25} +(4.50000 + 4.33013i) q^{26} -4.00000 q^{27} +(-1.50000 - 2.59808i) q^{29} +(-3.00000 - 5.19615i) q^{30} +(-3.00000 + 1.73205i) q^{31} +5.19615i q^{32} +5.19615i q^{34} +(0.500000 + 0.866025i) q^{36} +8.66025i q^{37} +(-3.00000 - 5.19615i) q^{38} +(-2.00000 - 6.92820i) q^{39} +(-1.50000 + 2.59808i) q^{40} +(-4.50000 + 2.59808i) q^{41} +(-4.00000 + 6.92820i) q^{43} +1.73205i q^{45} +10.3923i q^{46} +(-3.00000 - 1.73205i) q^{47} +(5.00000 - 8.66025i) q^{48} +(-3.00000 + 1.73205i) q^{50} +(3.00000 - 5.19615i) q^{51} +(2.50000 - 2.59808i) q^{52} +(1.50000 + 2.59808i) q^{53} +6.92820i q^{54} +6.92820i q^{57} +(-4.50000 + 2.59808i) q^{58} -6.92820i q^{59} +(-3.00000 + 1.73205i) q^{60} +(-0.500000 - 0.866025i) q^{61} +(3.00000 + 5.19615i) q^{62} -1.00000 q^{64} +(6.00000 - 1.73205i) q^{65} +(-3.00000 - 1.73205i) q^{67} +3.00000 q^{68} +(6.00000 - 10.3923i) q^{69} +(3.00000 + 1.73205i) q^{71} +(-1.50000 + 0.866025i) q^{72} +(-1.50000 + 0.866025i) q^{73} +15.0000 q^{74} +4.00000 q^{75} +(-3.00000 + 1.73205i) q^{76} +(-12.0000 + 3.46410i) q^{78} +(-2.00000 + 3.46410i) q^{79} +(7.50000 + 4.33013i) q^{80} +(5.50000 - 9.52628i) q^{81} +(4.50000 + 7.79423i) q^{82} -13.8564i q^{83} +(4.50000 + 2.59808i) q^{85} +(12.0000 + 6.92820i) q^{86} +6.00000 q^{87} -6.92820i q^{89} +3.00000 q^{90} +6.00000 q^{92} -6.92820i q^{93} +(-3.00000 + 5.19615i) q^{94} -6.00000 q^{95} +(-9.00000 - 5.19615i) q^{96} +(6.00000 + 3.46410i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 2 q^{4} - 3 q^{5} + 6 q^{6} - q^{9} - 3 q^{10} + 2 q^{12} - 5 q^{13} + 6 q^{15} - 10 q^{16} - 6 q^{17} - 3 q^{18} + 6 q^{19} + 3 q^{20} - 12 q^{23} + 6 q^{24} - 2 q^{25} + 9 q^{26} - 8 q^{27}+ \cdots + 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/637\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(248\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{3}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.73205i − 1.22474i −0.790569 − 0.612372i \(-0.790215\pi\)
0.790569 − 0.612372i \(-0.209785\pi\)
\(3\) −1.00000 + 1.73205i −0.577350 + 1.00000i 0.418432 + 0.908248i \(0.362580\pi\)
−0.995782 + 0.0917517i \(0.970753\pi\)
\(4\) −1.00000 −0.500000
\(5\) −1.50000 − 0.866025i −0.670820 − 0.387298i 0.125567 − 0.992085i \(-0.459925\pi\)
−0.796387 + 0.604787i \(0.793258\pi\)
\(6\) 3.00000 + 1.73205i 1.22474 + 0.707107i
\(7\) 0 0
\(8\) − 1.73205i − 0.612372i
\(9\) −0.500000 − 0.866025i −0.166667 − 0.288675i
\(10\) −1.50000 + 2.59808i −0.474342 + 0.821584i
\(11\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(12\) 1.00000 − 1.73205i 0.288675 − 0.500000i
\(13\) −2.50000 + 2.59808i −0.693375 + 0.720577i
\(14\) 0 0
\(15\) 3.00000 − 1.73205i 0.774597 − 0.447214i
\(16\) −5.00000 −1.25000
\(17\) −3.00000 −0.727607 −0.363803 − 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) −1.50000 + 0.866025i −0.353553 + 0.204124i
\(19\) 3.00000 − 1.73205i 0.688247 − 0.397360i −0.114708 − 0.993399i \(-0.536593\pi\)
0.802955 + 0.596040i \(0.203260\pi\)
\(20\) 1.50000 + 0.866025i 0.335410 + 0.193649i
\(21\) 0 0
\(22\) 0 0
\(23\) −6.00000 −1.25109 −0.625543 − 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 3.00000 + 1.73205i 0.612372 + 0.353553i
\(25\) −1.00000 − 1.73205i −0.200000 − 0.346410i
\(26\) 4.50000 + 4.33013i 0.882523 + 0.849208i
\(27\) −4.00000 −0.769800
\(28\) 0 0
\(29\) −1.50000 − 2.59808i −0.278543 − 0.482451i 0.692480 − 0.721437i \(-0.256518\pi\)
−0.971023 + 0.238987i \(0.923185\pi\)
\(30\) −3.00000 − 5.19615i −0.547723 − 0.948683i
\(31\) −3.00000 + 1.73205i −0.538816 + 0.311086i −0.744599 − 0.667512i \(-0.767359\pi\)
0.205783 + 0.978598i \(0.434026\pi\)
\(32\) 5.19615i 0.918559i
\(33\) 0 0
\(34\) 5.19615i 0.891133i
\(35\) 0 0
\(36\) 0.500000 + 0.866025i 0.0833333 + 0.144338i
\(37\) 8.66025i 1.42374i 0.702313 + 0.711868i \(0.252151\pi\)
−0.702313 + 0.711868i \(0.747849\pi\)
\(38\) −3.00000 − 5.19615i −0.486664 − 0.842927i
\(39\) −2.00000 − 6.92820i −0.320256 − 1.10940i
\(40\) −1.50000 + 2.59808i −0.237171 + 0.410792i
\(41\) −4.50000 + 2.59808i −0.702782 + 0.405751i −0.808383 − 0.588657i \(-0.799657\pi\)
0.105601 + 0.994409i \(0.466323\pi\)
\(42\) 0 0
\(43\) −4.00000 + 6.92820i −0.609994 + 1.05654i 0.381246 + 0.924473i \(0.375495\pi\)
−0.991241 + 0.132068i \(0.957838\pi\)
\(44\) 0 0
\(45\) 1.73205i 0.258199i
\(46\) 10.3923i 1.53226i
\(47\) −3.00000 − 1.73205i −0.437595 − 0.252646i 0.264982 − 0.964253i \(-0.414634\pi\)
−0.702577 + 0.711608i \(0.747967\pi\)
\(48\) 5.00000 − 8.66025i 0.721688 − 1.25000i
\(49\) 0 0
\(50\) −3.00000 + 1.73205i −0.424264 + 0.244949i
\(51\) 3.00000 − 5.19615i 0.420084 − 0.727607i
\(52\) 2.50000 − 2.59808i 0.346688 − 0.360288i
\(53\) 1.50000 + 2.59808i 0.206041 + 0.356873i 0.950464 − 0.310835i \(-0.100609\pi\)
−0.744423 + 0.667708i \(0.767275\pi\)
\(54\) 6.92820i 0.942809i
\(55\) 0 0
\(56\) 0 0
\(57\) 6.92820i 0.917663i
\(58\) −4.50000 + 2.59808i −0.590879 + 0.341144i
\(59\) − 6.92820i − 0.901975i −0.892530 − 0.450988i \(-0.851072\pi\)
0.892530 − 0.450988i \(-0.148928\pi\)
\(60\) −3.00000 + 1.73205i −0.387298 + 0.223607i
\(61\) −0.500000 − 0.866025i −0.0640184 − 0.110883i 0.832240 − 0.554416i \(-0.187058\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) 3.00000 + 5.19615i 0.381000 + 0.659912i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 6.00000 − 1.73205i 0.744208 − 0.214834i
\(66\) 0 0
\(67\) −3.00000 − 1.73205i −0.366508 − 0.211604i 0.305424 − 0.952217i \(-0.401202\pi\)
−0.671932 + 0.740613i \(0.734535\pi\)
\(68\) 3.00000 0.363803
\(69\) 6.00000 − 10.3923i 0.722315 − 1.25109i
\(70\) 0 0
\(71\) 3.00000 + 1.73205i 0.356034 + 0.205557i 0.667340 − 0.744753i \(-0.267433\pi\)
−0.311305 + 0.950310i \(0.600766\pi\)
\(72\) −1.50000 + 0.866025i −0.176777 + 0.102062i
\(73\) −1.50000 + 0.866025i −0.175562 + 0.101361i −0.585206 − 0.810885i \(-0.698986\pi\)
0.409644 + 0.912245i \(0.365653\pi\)
\(74\) 15.0000 1.74371
\(75\) 4.00000 0.461880
\(76\) −3.00000 + 1.73205i −0.344124 + 0.198680i
\(77\) 0 0
\(78\) −12.0000 + 3.46410i −1.35873 + 0.392232i
\(79\) −2.00000 + 3.46410i −0.225018 + 0.389742i −0.956325 − 0.292306i \(-0.905577\pi\)
0.731307 + 0.682048i \(0.238911\pi\)
\(80\) 7.50000 + 4.33013i 0.838525 + 0.484123i
\(81\) 5.50000 − 9.52628i 0.611111 − 1.05848i
\(82\) 4.50000 + 7.79423i 0.496942 + 0.860729i
\(83\) − 13.8564i − 1.52094i −0.649374 − 0.760469i \(-0.724969\pi\)
0.649374 − 0.760469i \(-0.275031\pi\)
\(84\) 0 0
\(85\) 4.50000 + 2.59808i 0.488094 + 0.281801i
\(86\) 12.0000 + 6.92820i 1.29399 + 0.747087i
\(87\) 6.00000 0.643268
\(88\) 0 0
\(89\) − 6.92820i − 0.734388i −0.930144 − 0.367194i \(-0.880318\pi\)
0.930144 − 0.367194i \(-0.119682\pi\)
\(90\) 3.00000 0.316228
\(91\) 0 0
\(92\) 6.00000 0.625543
\(93\) − 6.92820i − 0.718421i
\(94\) −3.00000 + 5.19615i −0.309426 + 0.535942i
\(95\) −6.00000 −0.615587
\(96\) −9.00000 − 5.19615i −0.918559 − 0.530330i
\(97\) 6.00000 + 3.46410i 0.609208 + 0.351726i 0.772655 − 0.634826i \(-0.218928\pi\)
−0.163448 + 0.986552i \(0.552261\pi\)
\(98\) 0 0
\(99\) 0 0
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 637.2.k.a.569.1 2
7.2 even 3 13.2.e.a.10.1 yes 2
7.3 odd 6 637.2.u.b.361.1 2
7.4 even 3 637.2.u.c.361.1 2
7.5 odd 6 637.2.q.a.491.1 2
7.6 odd 2 637.2.k.c.569.1 2
13.4 even 6 637.2.u.c.30.1 2
21.2 odd 6 117.2.q.c.10.1 2
28.23 odd 6 208.2.w.b.49.1 2
35.2 odd 12 325.2.m.a.49.1 4
35.9 even 6 325.2.n.a.101.1 2
35.23 odd 12 325.2.m.a.49.2 4
56.37 even 6 832.2.w.d.257.1 2
56.51 odd 6 832.2.w.a.257.1 2
84.23 even 6 1872.2.by.d.1297.1 2
91.2 odd 12 169.2.a.a.1.1 2
91.4 even 6 inner 637.2.k.a.459.1 2
91.9 even 3 169.2.e.a.147.1 2
91.16 even 3 169.2.b.a.168.1 2
91.17 odd 6 637.2.k.c.459.1 2
91.23 even 6 169.2.b.a.168.2 2
91.30 even 6 13.2.e.a.4.1 ✓ 2
91.37 odd 12 169.2.a.a.1.2 2
91.44 odd 12 169.2.c.a.146.2 4
91.51 even 6 169.2.e.a.23.1 2
91.54 even 12 8281.2.a.q.1.1 2
91.58 odd 12 169.2.c.a.22.2 4
91.69 odd 6 637.2.u.b.30.1 2
91.72 odd 12 169.2.c.a.22.1 4
91.82 odd 6 637.2.q.a.589.1 2
91.86 odd 12 169.2.c.a.146.1 4
91.89 even 12 8281.2.a.q.1.2 2
273.2 even 12 1521.2.a.k.1.2 2
273.23 odd 6 1521.2.b.a.1351.1 2
273.107 odd 6 1521.2.b.a.1351.2 2
273.128 even 12 1521.2.a.k.1.1 2
273.212 odd 6 117.2.q.c.82.1 2
364.23 odd 6 2704.2.f.b.337.1 2
364.107 odd 6 2704.2.f.b.337.2 2
364.219 even 12 2704.2.a.o.1.1 2
364.275 even 12 2704.2.a.o.1.2 2
364.303 odd 6 208.2.w.b.17.1 2
455.184 odd 12 4225.2.a.v.1.2 2
455.212 odd 12 325.2.m.a.199.2 4
455.219 odd 12 4225.2.a.v.1.1 2
455.303 odd 12 325.2.m.a.199.1 4
455.394 even 6 325.2.n.a.251.1 2
728.485 even 6 832.2.w.d.641.1 2
728.667 odd 6 832.2.w.a.641.1 2
1092.1031 even 6 1872.2.by.d.433.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
13.2.e.a.4.1 ✓ 2 91.30 even 6
13.2.e.a.10.1 yes 2 7.2 even 3
117.2.q.c.10.1 2 21.2 odd 6
117.2.q.c.82.1 2 273.212 odd 6
169.2.a.a.1.1 2 91.2 odd 12
169.2.a.a.1.2 2 91.37 odd 12
169.2.b.a.168.1 2 91.16 even 3
169.2.b.a.168.2 2 91.23 even 6
169.2.c.a.22.1 4 91.72 odd 12
169.2.c.a.22.2 4 91.58 odd 12
169.2.c.a.146.1 4 91.86 odd 12
169.2.c.a.146.2 4 91.44 odd 12
169.2.e.a.23.1 2 91.51 even 6
169.2.e.a.147.1 2 91.9 even 3
208.2.w.b.17.1 2 364.303 odd 6
208.2.w.b.49.1 2 28.23 odd 6
325.2.m.a.49.1 4 35.2 odd 12
325.2.m.a.49.2 4 35.23 odd 12
325.2.m.a.199.1 4 455.303 odd 12
325.2.m.a.199.2 4 455.212 odd 12
325.2.n.a.101.1 2 35.9 even 6
325.2.n.a.251.1 2 455.394 even 6
637.2.k.a.459.1 2 91.4 even 6 inner
637.2.k.a.569.1 2 1.1 even 1 trivial
637.2.k.c.459.1 2 91.17 odd 6
637.2.k.c.569.1 2 7.6 odd 2
637.2.q.a.491.1 2 7.5 odd 6
637.2.q.a.589.1 2 91.82 odd 6
637.2.u.b.30.1 2 91.69 odd 6
637.2.u.b.361.1 2 7.3 odd 6
637.2.u.c.30.1 2 13.4 even 6
637.2.u.c.361.1 2 7.4 even 3
832.2.w.a.257.1 2 56.51 odd 6
832.2.w.a.641.1 2 728.667 odd 6
832.2.w.d.257.1 2 56.37 even 6
832.2.w.d.641.1 2 728.485 even 6
1521.2.a.k.1.1 2 273.128 even 12
1521.2.a.k.1.2 2 273.2 even 12
1521.2.b.a.1351.1 2 273.23 odd 6
1521.2.b.a.1351.2 2 273.107 odd 6
1872.2.by.d.433.1 2 1092.1031 even 6
1872.2.by.d.1297.1 2 84.23 even 6
2704.2.a.o.1.1 2 364.219 even 12
2704.2.a.o.1.2 2 364.275 even 12
2704.2.f.b.337.1 2 364.23 odd 6
2704.2.f.b.337.2 2 364.107 odd 6
4225.2.a.v.1.1 2 455.219 odd 12
4225.2.a.v.1.2 2 455.184 odd 12
8281.2.a.q.1.1 2 91.54 even 12
8281.2.a.q.1.2 2 91.89 even 12