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Properties

Label 637.2.u.b.30.1
Level $637$
Weight $2$
Character 637.30
Analytic conductor $5.086$
Analytic rank $0$
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(30,637)] 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("637.30"); 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([2, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.u (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,3,-4,1,-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: \(0\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 13)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 30.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 637.30
Dual form 637.2.u.b.361.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.50000 - 0.866025i) q^{2} -2.00000 q^{3} +(0.500000 - 0.866025i) q^{4} +(-1.50000 - 0.866025i) q^{5} +(-3.00000 + 1.73205i) q^{6} +1.73205i q^{8} +1.00000 q^{9} -3.00000 q^{10} +(-1.00000 + 1.73205i) q^{12} +(2.50000 + 2.59808i) q^{13} +(3.00000 + 1.73205i) q^{15} +(2.50000 + 4.33013i) q^{16} +(-1.50000 + 2.59808i) q^{17} +(1.50000 - 0.866025i) q^{18} +3.46410i q^{19} +(-1.50000 + 0.866025i) q^{20} +(3.00000 + 5.19615i) q^{23} -3.46410i q^{24} +(-1.00000 - 1.73205i) q^{25} +(6.00000 + 1.73205i) q^{26} +4.00000 q^{27} +(-1.50000 + 2.59808i) q^{29} +6.00000 q^{30} +(-3.00000 + 1.73205i) q^{31} +(4.50000 + 2.59808i) q^{32} +5.19615i q^{34} +(0.500000 - 0.866025i) q^{36} +(-7.50000 + 4.33013i) q^{37} +(3.00000 + 5.19615i) q^{38} +(-5.00000 - 5.19615i) q^{39} +(1.50000 - 2.59808i) q^{40} +(4.50000 + 2.59808i) q^{41} +(-4.00000 - 6.92820i) q^{43} +(-1.50000 - 0.866025i) q^{45} +(9.00000 + 5.19615i) q^{46} +(-3.00000 - 1.73205i) q^{47} +(-5.00000 - 8.66025i) q^{48} +(-3.00000 - 1.73205i) q^{50} +(3.00000 - 5.19615i) q^{51} +(3.50000 - 0.866025i) q^{52} +(1.50000 + 2.59808i) q^{53} +(6.00000 - 3.46410i) q^{54} -6.92820i q^{57} +5.19615i q^{58} +(6.00000 + 3.46410i) q^{59} +(3.00000 - 1.73205i) q^{60} -1.00000 q^{61} +(-3.00000 + 5.19615i) q^{62} -1.00000 q^{64} +(-1.50000 - 6.06218i) q^{65} -3.46410i q^{67} +(1.50000 + 2.59808i) q^{68} +(-6.00000 - 10.3923i) q^{69} +(3.00000 - 1.73205i) q^{71} +1.73205i q^{72} +(-1.50000 + 0.866025i) q^{73} +(-7.50000 + 12.9904i) q^{74} +(2.00000 + 3.46410i) q^{75} +(3.00000 + 1.73205i) q^{76} +(-12.0000 - 3.46410i) q^{78} +(-2.00000 + 3.46410i) q^{79} -8.66025i q^{80} -11.0000 q^{81} +9.00000 q^{82} -13.8564i q^{83} +(4.50000 - 2.59808i) q^{85} +(-12.0000 - 6.92820i) q^{86} +(3.00000 - 5.19615i) q^{87} +(-6.00000 + 3.46410i) q^{89} -3.00000 q^{90} +6.00000 q^{92} +(6.00000 - 3.46410i) q^{93} -6.00000 q^{94} +(3.00000 - 5.19615i) q^{95} +(-9.00000 - 5.19615i) q^{96} +(-6.00000 + 3.46410i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{2} - 4 q^{3} + q^{4} - 3 q^{5} - 6 q^{6} + 2 q^{9} - 6 q^{10} - 2 q^{12} + 5 q^{13} + 6 q^{15} + 5 q^{16} - 3 q^{17} + 3 q^{18} - 3 q^{20} + 6 q^{23} - 2 q^{25} + 12 q^{26} + 8 q^{27} - 3 q^{29}+ \cdots - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.50000 − 0.866025i 1.06066 − 0.612372i 0.135045 − 0.990839i \(-0.456882\pi\)
0.925615 + 0.378467i \(0.123549\pi\)
\(3\) −2.00000 −1.15470 −0.577350 − 0.816497i \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) 0.500000 − 0.866025i 0.250000 − 0.433013i
\(5\) −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
\(9\) 1.00000 0.333333
\(10\) −3.00000 −0.948683
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) −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
\(16\) 2.50000 + 4.33013i 0.625000 + 1.08253i
\(17\) −1.50000 + 2.59808i −0.363803 + 0.630126i −0.988583 − 0.150675i \(-0.951855\pi\)
0.624780 + 0.780801i \(0.285189\pi\)
\(18\) 1.50000 − 0.866025i 0.353553 − 0.204124i
\(19\) 3.46410i 0.794719i 0.917663 + 0.397360i \(0.130073\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) −1.50000 + 0.866025i −0.335410 + 0.193649i
\(21\) 0 0
\(22\) 0 0
\(23\) 3.00000 + 5.19615i 0.625543 + 1.08347i 0.988436 + 0.151642i \(0.0484560\pi\)
−0.362892 + 0.931831i \(0.618211\pi\)
\(24\) − 3.46410i − 0.707107i
\(25\) −1.00000 − 1.73205i −0.200000 − 0.346410i
\(26\) 6.00000 + 1.73205i 1.17670 + 0.339683i
\(27\) 4.00000 0.769800
\(28\) 0 0
\(29\) −1.50000 + 2.59808i −0.278543 + 0.482451i −0.971023 − 0.238987i \(-0.923185\pi\)
0.692480 + 0.721437i \(0.256518\pi\)
\(30\) 6.00000 1.09545
\(31\) −3.00000 + 1.73205i −0.538816 + 0.311086i −0.744599 − 0.667512i \(-0.767359\pi\)
0.205783 + 0.978598i \(0.434026\pi\)
\(32\) 4.50000 + 2.59808i 0.795495 + 0.459279i
\(33\) 0 0
\(34\) 5.19615i 0.891133i
\(35\) 0 0
\(36\) 0.500000 − 0.866025i 0.0833333 − 0.144338i
\(37\) −7.50000 + 4.33013i −1.23299 + 0.711868i −0.967653 − 0.252286i \(-0.918817\pi\)
−0.265340 + 0.964155i \(0.585484\pi\)
\(38\) 3.00000 + 5.19615i 0.486664 + 0.842927i
\(39\) −5.00000 − 5.19615i −0.800641 − 0.832050i
\(40\) 1.50000 − 2.59808i 0.237171 − 0.410792i
\(41\) 4.50000 + 2.59808i 0.702782 + 0.405751i 0.808383 − 0.588657i \(-0.200343\pi\)
−0.105601 + 0.994409i \(0.533677\pi\)
\(42\) 0 0
\(43\) −4.00000 − 6.92820i −0.609994 − 1.05654i −0.991241 − 0.132068i \(-0.957838\pi\)
0.381246 − 0.924473i \(-0.375495\pi\)
\(44\) 0 0
\(45\) −1.50000 − 0.866025i −0.223607 − 0.129099i
\(46\) 9.00000 + 5.19615i 1.32698 + 0.766131i
\(47\) −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
\(52\) 3.50000 − 0.866025i 0.485363 − 0.120096i
\(53\) 1.50000 + 2.59808i 0.206041 + 0.356873i 0.950464 − 0.310835i \(-0.100609\pi\)
−0.744423 + 0.667708i \(0.767275\pi\)
\(54\) 6.00000 − 3.46410i 0.816497 − 0.471405i
\(55\) 0 0
\(56\) 0 0
\(57\) − 6.92820i − 0.917663i
\(58\) 5.19615i 0.682288i
\(59\) 6.00000 + 3.46410i 0.781133 + 0.450988i 0.836832 − 0.547460i \(-0.184405\pi\)
−0.0556984 + 0.998448i \(0.517739\pi\)
\(60\) 3.00000 − 1.73205i 0.387298 − 0.223607i
\(61\) −1.00000 −0.128037 −0.0640184 − 0.997949i \(-0.520392\pi\)
−0.0640184 + 0.997949i \(0.520392\pi\)
\(62\) −3.00000 + 5.19615i −0.381000 + 0.659912i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) −1.50000 − 6.06218i −0.186052 − 0.751921i
\(66\) 0 0
\(67\) − 3.46410i − 0.423207i −0.977356 − 0.211604i \(-0.932131\pi\)
0.977356 − 0.211604i \(-0.0678686\pi\)
\(68\) 1.50000 + 2.59808i 0.181902 + 0.315063i
\(69\) −6.00000 − 10.3923i −0.722315 − 1.25109i
\(70\) 0 0
\(71\) 3.00000 − 1.73205i 0.356034 − 0.205557i −0.311305 − 0.950310i \(-0.600766\pi\)
0.667340 + 0.744753i \(0.267433\pi\)
\(72\) 1.73205i 0.204124i
\(73\) −1.50000 + 0.866025i −0.175562 + 0.101361i −0.585206 − 0.810885i \(-0.698986\pi\)
0.409644 + 0.912245i \(0.365653\pi\)
\(74\) −7.50000 + 12.9904i −0.871857 + 1.51010i
\(75\) 2.00000 + 3.46410i 0.230940 + 0.400000i
\(76\) 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 \(0.238911\pi\)
\(80\) − 8.66025i − 0.968246i
\(81\) −11.0000 −1.22222
\(82\) 9.00000 0.993884
\(83\) − 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
\(87\) 3.00000 − 5.19615i 0.321634 − 0.557086i
\(88\) 0 0
\(89\) −6.00000 + 3.46410i −0.635999 + 0.367194i −0.783072 − 0.621932i \(-0.786348\pi\)
0.147073 + 0.989126i \(0.453015\pi\)
\(90\) −3.00000 −0.316228
\(91\) 0 0
\(92\) 6.00000 0.625543
\(93\) 6.00000 − 3.46410i 0.622171 − 0.359211i
\(94\) −6.00000 −0.618853
\(95\) 3.00000 − 5.19615i 0.307794 − 0.533114i
\(96\) −9.00000 − 5.19615i −0.918559 − 0.530330i
\(97\) −6.00000 + 3.46410i −0.609208 + 0.351726i −0.772655 − 0.634826i \(-0.781072\pi\)
0.163448 + 0.986552i \(0.447739\pi\)
\(98\) 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.u.b.30.1 2
7.2 even 3 637.2.q.a.589.1 2
7.3 odd 6 637.2.k.a.459.1 2
7.4 even 3 637.2.k.c.459.1 2
7.5 odd 6 13.2.e.a.4.1 ✓ 2
7.6 odd 2 637.2.u.c.30.1 2
13.10 even 6 637.2.k.c.569.1 2
21.5 even 6 117.2.q.c.82.1 2
28.19 even 6 208.2.w.b.17.1 2
35.12 even 12 325.2.m.a.199.2 4
35.19 odd 6 325.2.n.a.251.1 2
35.33 even 12 325.2.m.a.199.1 4
56.5 odd 6 832.2.w.d.641.1 2
56.19 even 6 832.2.w.a.641.1 2
84.47 odd 6 1872.2.by.d.433.1 2
91.5 even 12 169.2.c.a.22.1 4
91.10 odd 6 637.2.u.c.361.1 2
91.12 odd 6 169.2.e.a.147.1 2
91.19 even 12 169.2.a.a.1.2 2
91.23 even 6 637.2.q.a.491.1 2
91.33 even 12 169.2.a.a.1.1 2
91.47 even 12 169.2.c.a.22.2 4
91.54 even 12 169.2.c.a.146.1 4
91.58 odd 12 8281.2.a.q.1.2 2
91.61 odd 6 169.2.b.a.168.2 2
91.62 odd 6 637.2.k.a.569.1 2
91.68 odd 6 169.2.e.a.23.1 2
91.72 odd 12 8281.2.a.q.1.1 2
91.75 odd 6 13.2.e.a.10.1 yes 2
91.82 odd 6 169.2.b.a.168.1 2
91.88 even 6 inner 637.2.u.b.361.1 2
91.89 even 12 169.2.c.a.146.2 4
273.110 odd 12 1521.2.a.k.1.1 2
273.152 even 6 1521.2.b.a.1351.1 2
273.173 even 6 1521.2.b.a.1351.2 2
273.215 odd 12 1521.2.a.k.1.2 2
273.257 even 6 117.2.q.c.10.1 2
364.19 odd 12 2704.2.a.o.1.1 2
364.75 even 6 208.2.w.b.49.1 2
364.215 odd 12 2704.2.a.o.1.2 2
364.243 even 6 2704.2.f.b.337.1 2
364.355 even 6 2704.2.f.b.337.2 2
455.19 even 12 4225.2.a.v.1.1 2
455.124 even 12 4225.2.a.v.1.2 2
455.257 even 12 325.2.m.a.49.1 4
455.348 even 12 325.2.m.a.49.2 4
455.439 odd 6 325.2.n.a.101.1 2
728.75 even 6 832.2.w.a.257.1 2
728.621 odd 6 832.2.w.d.257.1 2
1092.803 odd 6 1872.2.by.d.1297.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
13.2.e.a.4.1 ✓ 2 7.5 odd 6
13.2.e.a.10.1 yes 2 91.75 odd 6
117.2.q.c.10.1 2 273.257 even 6
117.2.q.c.82.1 2 21.5 even 6
169.2.a.a.1.1 2 91.33 even 12
169.2.a.a.1.2 2 91.19 even 12
169.2.b.a.168.1 2 91.82 odd 6
169.2.b.a.168.2 2 91.61 odd 6
169.2.c.a.22.1 4 91.5 even 12
169.2.c.a.22.2 4 91.47 even 12
169.2.c.a.146.1 4 91.54 even 12
169.2.c.a.146.2 4 91.89 even 12
169.2.e.a.23.1 2 91.68 odd 6
169.2.e.a.147.1 2 91.12 odd 6
208.2.w.b.17.1 2 28.19 even 6
208.2.w.b.49.1 2 364.75 even 6
325.2.m.a.49.1 4 455.257 even 12
325.2.m.a.49.2 4 455.348 even 12
325.2.m.a.199.1 4 35.33 even 12
325.2.m.a.199.2 4 35.12 even 12
325.2.n.a.101.1 2 455.439 odd 6
325.2.n.a.251.1 2 35.19 odd 6
637.2.k.a.459.1 2 7.3 odd 6
637.2.k.a.569.1 2 91.62 odd 6
637.2.k.c.459.1 2 7.4 even 3
637.2.k.c.569.1 2 13.10 even 6
637.2.q.a.491.1 2 91.23 even 6
637.2.q.a.589.1 2 7.2 even 3
637.2.u.b.30.1 2 1.1 even 1 trivial
637.2.u.b.361.1 2 91.88 even 6 inner
637.2.u.c.30.1 2 7.6 odd 2
637.2.u.c.361.1 2 91.10 odd 6
832.2.w.a.257.1 2 728.75 even 6
832.2.w.a.641.1 2 56.19 even 6
832.2.w.d.257.1 2 728.621 odd 6
832.2.w.d.641.1 2 56.5 odd 6
1521.2.a.k.1.1 2 273.110 odd 12
1521.2.a.k.1.2 2 273.215 odd 12
1521.2.b.a.1351.1 2 273.152 even 6
1521.2.b.a.1351.2 2 273.173 even 6
1872.2.by.d.433.1 2 84.47 odd 6
1872.2.by.d.1297.1 2 1092.803 odd 6
2704.2.a.o.1.1 2 364.19 odd 12
2704.2.a.o.1.2 2 364.215 odd 12
2704.2.f.b.337.1 2 364.243 even 6
2704.2.f.b.337.2 2 364.355 even 6
4225.2.a.v.1.1 2 455.19 even 12
4225.2.a.v.1.2 2 455.124 even 12
8281.2.a.q.1.1 2 91.72 odd 12
8281.2.a.q.1.2 2 91.58 odd 12