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Properties

Label 50.12.d.b
Level $50$
Weight $12$
Character orbit 50.d
Analytic conductor $38.417$
Analytic rank $0$
Dimension $56$
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] = [50,12,Mod(11,50)] 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("50.11"); S:= CuspForms(chi, 12); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(50, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([8])) N = Newforms(chi, 12, names="a")
 
Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 50.d (of order \(5\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [56] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(38.4171590280\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(14\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 56 q - 448 q^{2} - 263 q^{3} - 14336 q^{4} + 1770 q^{5} - 8416 q^{6} - 111844 q^{7} - 458752 q^{8} - 1174523 q^{9} + 304960 q^{10} + 207277 q^{11} + 1026048 q^{12} + 893677 q^{13} - 1270048 q^{14} + 4696640 q^{15}+ \cdots - 505737997606 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −25.8885 + 18.8091i −246.102 + 757.423i 316.433 − 973.882i −6031.65 − 3528.08i −7875.25 − 24237.5i 73958.0 10125.9 + 31164.2i −369808. − 268681.i 222511. − 22113.3i
11.2 −25.8885 + 18.8091i −200.084 + 615.795i 316.433 − 973.882i −232.661 + 6983.84i −6402.68 − 19705.4i −50541.7 10125.9 + 31164.2i −195855. − 142297.i −125337. − 185178.i
11.3 −25.8885 + 18.8091i −199.816 + 614.971i 316.433 − 973.882i −615.922 − 6960.51i −6394.12 − 19679.1i −82801.1 10125.9 + 31164.2i −194948. − 141638.i 146867. + 168613.i
11.4 −25.8885 + 18.8091i −170.771 + 525.580i 316.433 − 973.882i 6904.85 + 1072.90i −5464.68 − 16818.6i 1017.46 10125.9 + 31164.2i −103757. − 75383.7i −198937. + 102098.i
11.5 −25.8885 + 18.8091i −123.506 + 380.112i 316.433 − 973.882i −4417.52 + 5414.21i −3952.18 − 12163.6i 6670.69 10125.9 + 31164.2i 14083.8 + 10232.5i 12526.6 − 223256.i
11.6 −25.8885 + 18.8091i −84.0143 + 258.570i 316.433 − 973.882i 3827.94 − 5845.94i −2688.46 − 8274.23i 54397.6 10125.9 + 31164.2i 83515.1 + 60677.3i 10857.1 + 223343.i
11.7 −25.8885 + 18.8091i −18.4264 + 56.7107i 316.433 − 973.882i −6941.67 − 800.879i −589.645 − 1814.74i −8223.94 10125.9 + 31164.2i 140438. + 102034.i 194773. − 109833.i
11.8 −25.8885 + 18.8091i 22.6894 − 69.8307i 316.433 − 973.882i 4847.01 + 5033.35i 726.060 + 2234.58i −16025.8 10125.9 + 31164.2i 138953. + 100956.i −220155. − 39137.9i
11.9 −25.8885 + 18.8091i 42.9958 − 132.328i 316.433 − 973.882i 5249.33 − 4612.23i 1375.87 + 4234.48i −79523.0 10125.9 + 31164.2i 127653. + 92745.3i −49145.4 + 218139.i
11.10 −25.8885 + 18.8091i 53.1095 − 163.454i 316.433 − 973.882i 1.40075 + 6987.71i 1699.50 + 5230.53i 86927.7 10125.9 + 31164.2i 119418. + 86762.5i −131469. − 180875.i
11.11 −25.8885 + 18.8091i 111.492 − 343.137i 316.433 − 973.882i 528.793 − 6967.68i 3567.74 + 10980.4i 17695.9 10125.9 + 31164.2i 38002.6 + 27610.5i 117366. + 190329.i
11.12 −25.8885 + 18.8091i 179.007 − 550.928i 316.433 − 973.882i −4657.15 − 5209.52i 5728.23 + 17629.7i −160.425 10125.9 + 31164.2i −128163. − 93115.7i 218553. + 47270.1i
11.13 −25.8885 + 18.8091i 203.811 − 627.266i 316.433 − 973.882i 6676.95 + 2060.70i 6521.96 + 20072.5i −9341.29 10125.9 + 31164.2i −208609. − 151563.i −211616. + 72239.1i
11.14 −25.8885 + 18.8091i 228.024 − 701.784i 316.433 − 973.882i −4899.58 + 4982.19i 7296.75 + 22457.1i 8243.01 10125.9 + 31164.2i −297192. − 215922.i 33132.4 − 221139.i
21.1 9.88854 − 30.4338i −619.017 − 449.742i −828.433 − 601.892i −1807.38 + 6749.93i −19808.6 + 14391.8i −82655.9 −26509.9 + 19260.5i 126173. + 388320.i 187554. + 121752.i
21.2 9.88854 − 30.4338i −505.161 − 367.021i −828.433 − 601.892i 6683.35 − 2039.85i −16165.1 + 11744.7i 20699.7 −26509.9 + 19260.5i 65741.6 + 202332.i 4008.11 − 223571.i
21.3 9.88854 − 30.4338i −486.674 − 353.590i −828.433 − 601.892i −1629.59 − 6795.04i −15573.6 + 11314.9i 3121.65 −26509.9 + 19260.5i 57084.8 + 175689.i −222913. − 17598.3i
21.4 9.88854 − 30.4338i −342.324 − 248.713i −828.433 − 601.892i −4234.41 + 5558.59i −10954.4 + 7958.81i 40593.8 −26509.9 + 19260.5i 586.044 + 1803.66i 127297. + 183836.i
21.5 9.88854 − 30.4338i −255.372 − 185.539i −828.433 − 601.892i 5959.51 + 3648.62i −8171.90 + 5937.24i 53413.9 −26509.9 + 19260.5i −23951.2 − 73714.1i 169972. − 145291.i
21.6 9.88854 − 30.4338i −85.6566 − 62.2332i −828.433 − 601.892i 4809.61 + 5069.10i −2741.01 + 1991.46i −45440.5 −26509.9 + 19260.5i −51277.3 − 157815.i 201832. − 96248.8i
See all 56 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 11.14
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 50.12.d.b ✓ 56
25.d even 5 1 inner 50.12.d.b ✓ 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.12.d.b ✓ 56 1.a even 1 1 trivial
50.12.d.b ✓ 56 25.d even 5 1 inner