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Properties

Label 50.12.e.a
Level $50$
Weight $12$
Character orbit 50.e
Analytic conductor $38.417$
Analytic rank $0$
Dimension $112$
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(9,50)] 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("50.9"); 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([7])) N = Newforms(chi, 12, names="a")
 
Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 50.e (of order \(10\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(38.4171590280\)
Analytic rank: \(0\)
Dimension: \(112\)
Relative dimension: \(28\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 112 q + 28672 q^{4} - 10320 q^{5} - 15552 q^{6} + 1703666 q^{9} + 385920 q^{10} - 1791796 q^{11} - 2590720 q^{12} + 2151296 q^{14} - 14457100 q^{15} - 29360128 q^{16} - 4194730 q^{17} - 27928990 q^{19}+ \cdots - 844498378268 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1 −18.8091 + 25.8885i −690.467 + 224.346i −316.433 − 973.882i −270.720 + 6982.47i 7179.08 − 22094.9i 75739.2i 31164.2 + 10125.9i 283099. − 205683.i −175674. − 138343.i
9.2 −18.8091 + 25.8885i −661.498 + 214.934i −316.433 − 973.882i 6714.92 − 1933.38i 6877.88 − 21167.9i − 27199.0i 31164.2 + 10125.9i 248068. − 180232.i −76249.5 + 210205.i
9.3 −18.8091 + 25.8885i −602.128 + 195.643i −316.433 − 973.882i −6677.09 − 2060.24i 6260.58 − 19268.1i − 64353.7i 31164.2 + 10125.9i 180967. − 131480.i 178927. − 134109.i
9.4 −18.8091 + 25.8885i −329.617 + 107.099i −316.433 − 973.882i −6987.46 − 59.3651i 3427.17 − 10547.7i 57449.7i 31164.2 + 10125.9i −46137.9 + 33521.2i 132965. − 179779.i
9.5 −18.8091 + 25.8885i −234.676 + 76.2508i −316.433 − 973.882i 3423.13 − 6091.82i 2440.03 − 7509.63i 23640.6i 31164.2 + 10125.9i −94056.3 + 68335.9i 93322.4 + 203202.i
9.6 −18.8091 + 25.8885i −198.368 + 64.4536i −316.433 − 973.882i −1693.20 + 6779.47i 2062.52 − 6347.77i − 25705.9i 31164.2 + 10125.9i −108119. + 78553.4i −143663. − 171350.i
9.7 −18.8091 + 25.8885i −131.148 + 42.6127i −316.433 − 973.882i −2996.30 − 6312.71i 1363.61 − 4196.75i 6390.34i 31164.2 + 10125.9i −127931. + 92947.2i 219785. + 41166.7i
9.8 −18.8091 + 25.8885i 214.434 − 69.6740i −316.433 − 973.882i 6483.89 + 2605.25i −2229.57 + 6861.90i − 79228.9i 31164.2 + 10125.9i −102187. + 74243.4i −189402. + 118856.i
9.9 −18.8091 + 25.8885i 219.267 − 71.2442i −316.433 − 973.882i −4919.82 + 4962.21i −2279.81 + 7016.55i − 10514.2i 31164.2 + 10125.9i −100313. + 72881.4i −35926.9 − 220702.i
9.10 −18.8091 + 25.8885i 257.067 − 83.5261i −316.433 − 973.882i 5992.74 + 3593.76i −2672.83 + 8226.14i 70567.4i 31164.2 + 10125.9i −84208.2 + 61180.8i −205756. + 87547.8i
9.11 −18.8091 + 25.8885i 415.587 − 135.032i −316.433 − 973.882i 4139.55 − 5629.59i −4321.04 + 13298.8i − 19159.3i 31164.2 + 10125.9i 11163.8 − 8110.99i 67880.5 + 213055.i
9.12 −18.8091 + 25.8885i 452.195 − 146.927i −316.433 − 973.882i −6531.51 − 2483.45i −4701.67 + 14470.3i − 53247.3i 31164.2 + 10125.9i 39578.2 − 28755.2i 187145. − 122380.i
9.13 −18.8091 + 25.8885i 597.074 − 194.001i −316.433 − 973.882i −4836.36 − 5043.59i −6208.03 + 19106.4i 60016.1i 31164.2 + 10125.9i 175546. − 127541.i 221539. − 30340.8i
9.14 −18.8091 + 25.8885i 744.033 − 241.751i −316.433 − 973.882i −537.083 + 6967.04i −7736.03 + 23809.1i − 10475.0i 31164.2 + 10125.9i 351827. − 255617.i −170264. − 144948.i
9.15 18.8091 − 25.8885i −754.471 + 245.142i −316.433 − 973.882i −3806.08 + 5860.19i −7844.56 + 24143.1i − 22725.5i −31164.2 − 10125.9i 365816. − 265781.i 80122.8 + 208759.i
9.16 18.8091 − 25.8885i −640.316 + 208.051i −316.433 − 973.882i 6935.35 − 853.832i −6657.64 + 20490.1i 24644.1i −31164.2 − 10125.9i 223404. − 162312.i 108343. − 195606.i
9.17 18.8091 − 25.8885i −465.751 + 151.332i −316.433 − 973.882i −3359.83 − 6126.96i −4842.62 + 14904.0i 69237.1i −31164.2 − 10125.9i 50708.2 − 36841.7i −221814. − 28261.7i
9.18 18.8091 − 25.8885i −281.592 + 91.4948i −316.433 − 973.882i 3840.59 + 5837.64i −2927.83 + 9010.94i − 32590.5i −31164.2 − 10125.9i −72392.2 + 52596.0i 223366. + 10373.7i
9.19 18.8091 − 25.8885i −271.306 + 88.1526i −316.433 − 973.882i −6951.66 − 708.910i −2820.88 + 8681.79i − 46355.1i −31164.2 − 10125.9i −77478.9 + 56291.7i −149107. + 166634.i
9.20 18.8091 − 25.8885i −226.802 + 73.6926i −316.433 − 973.882i −980.872 + 6918.53i −2358.16 + 7257.68i 58045.3i −31164.2 − 10125.9i −97306.2 + 70697.1i 160661. + 155525.i
See next 80 embeddings (of 112 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.28
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.e even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 50.12.e.a ✓ 112
25.e even 10 1 inner 50.12.e.a ✓ 112
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.12.e.a ✓ 112 1.a even 1 1 trivial
50.12.e.a ✓ 112 25.e even 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{12}^{\mathrm{new}}(50, [\chi])\).