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Properties

Label 50.12.d.a
Level $50$
Weight $12$
Character orbit 50.d
Analytic conductor $38.417$
Analytic rank $0$
Dimension $52$
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 = [52] 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: \(52\)
Relative dimension: \(13\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 52 q + 416 q^{2} - 749 q^{3} - 13312 q^{4} + 4895 q^{5} + 23968 q^{6} + 157068 q^{7} + 425984 q^{8} - 369506 q^{9} - 404960 q^{10} + 1584519 q^{11} + 528384 q^{12} + 1297741 q^{13} + 3421344 q^{14} + 9921640 q^{15}+ \cdots + 111931179778 q^{99}+O(q^{100}) \) 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 −200.859 + 618.179i 316.433 − 973.882i −6568.03 − 2385.17i 6427.48 + 19781.7i −63638.5 −10125.9 − 31164.2i −198486. − 144209.i −214900. + 61790.3i
11.2 25.8885 − 18.8091i −178.479 + 549.303i 316.433 − 973.882i 811.858 − 6940.39i 5711.34 + 17577.7i 26053.3 −10125.9 − 31164.2i −126564. − 91954.0i −109525. − 194947.i
11.3 25.8885 − 18.8091i −172.109 + 529.697i 316.433 − 973.882i 6756.44 + 1782.89i 5507.49 + 16950.3i 41872.1 −10125.9 − 31164.2i −107643. − 78206.9i 208449. − 80926.4i
11.4 25.8885 − 18.8091i −157.385 + 484.381i 316.433 − 973.882i −5353.18 + 4491.28i 5036.32 + 15500.2i 57436.7 −10125.9 − 31164.2i −66540.2 − 48344.3i −54108.9 + 216961.i
11.5 25.8885 − 18.8091i −73.2948 + 225.578i 316.433 − 973.882i 6685.37 − 2033.22i 2345.43 + 7218.51i −64150.3 −10125.9 − 31164.2i 97801.5 + 71056.9i 134831. − 178383.i
11.6 25.8885 − 18.8091i −56.7778 + 174.744i 316.433 − 973.882i −856.896 + 6934.97i 1816.89 + 5591.81i −36588.0 −10125.9 − 31164.2i 116003. + 84281.2i 108257. + 195654.i
11.7 25.8885 − 18.8091i −12.0459 + 37.0735i 316.433 − 973.882i −6951.46 − 710.866i 385.470 + 1186.35i 51221.0 −10125.9 − 31164.2i 142086. + 103231.i −193334. + 112348.i
11.8 25.8885 − 18.8091i 63.8890 − 196.630i 316.433 − 973.882i −2221.22 − 6625.28i −2044.45 − 6292.16i 61934.8 −10125.9 − 31164.2i 108733. + 78999.4i −182120. − 129740.i
11.9 25.8885 − 18.8091i 71.7267 − 220.752i 316.433 − 973.882i 2303.38 − 6597.16i −2295.26 − 7064.07i −28129.5 −10125.9 − 31164.2i 99728.1 + 72456.7i −64455.6 − 214116.i
11.10 25.8885 − 18.8091i 86.3549 − 265.773i 316.433 − 973.882i 5592.14 + 4190.00i −2763.36 − 8504.74i 18434.5 −10125.9 − 31164.2i 80136.8 + 58222.8i 223583. + 3289.76i
11.11 25.8885 − 18.8091i 168.897 − 519.813i 316.433 − 973.882i −6875.71 + 1246.08i −5404.72 − 16634.0i −43998.3 −10125.9 − 31164.2i −98363.9 − 71465.6i −154564. + 161585.i
11.12 25.8885 − 18.8091i 196.602 − 605.080i 316.433 − 973.882i −1173.57 + 6888.46i −6291.27 − 19362.5i 52259.7 −10125.9 − 31164.2i −184154. − 133796.i 99183.9 + 200406.i
11.13 25.8885 − 18.8091i 212.071 − 652.688i 316.433 − 973.882i 4269.88 − 5531.39i −6786.28 − 20886.0i −3186.55 −10125.9 − 31164.2i −237713. − 172708.i 6500.48 − 223512.i
21.1 −9.88854 + 30.4338i −652.917 − 474.372i −828.433 − 601.892i −6791.79 − 1643.07i 20893.3 − 15179.9i −27952.8 26509.9 − 19260.5i 146530. + 450974.i 117166. − 190453.i
21.2 −9.88854 + 30.4338i −539.879 − 392.245i −828.433 − 601.892i 6192.58 − 3237.29i 17276.1 − 12551.9i −13924.8 26509.9 − 19260.5i 82871.9 + 255054.i 37287.4 + 220476.i
21.3 −9.88854 + 30.4338i −412.484 − 299.687i −828.433 − 601.892i 2233.66 + 6621.09i 13199.5 − 9589.99i −29775.1 26509.9 − 19260.5i 25589.2 + 78755.6i −223593. + 2505.96i
21.4 −9.88854 + 30.4338i −247.049 − 179.491i −828.433 − 601.892i −6720.69 − 1913.23i 7905.56 − 5743.73i 68265.0 26509.9 − 19260.5i −25925.5 − 79790.5i 124685. − 185617.i
21.5 −9.88854 + 30.4338i −228.184 − 165.785i −828.433 − 601.892i 6928.13 − 910.535i 7301.89 − 5305.14i 80711.3 26509.9 − 19260.5i −30158.3 − 92817.6i −40798.1 + 219853.i
21.6 −9.88854 + 30.4338i −199.891 − 145.229i −828.433 − 601.892i −2149.44 − 6648.91i 6396.52 − 4647.34i −27228.5 26509.9 − 19260.5i −35876.5 − 110417.i 223607. + 332.368i
21.7 −9.88854 + 30.4338i −54.9054 − 39.8911i −828.433 − 601.892i −5186.73 + 4682.52i 1756.97 − 1276.52i −62679.9 26509.9 − 19260.5i −53318.1 − 164096.i −91217.8 − 204155.i
See all 52 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.13
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.a ✓ 52
25.d even 5 1 inner 50.12.d.a ✓ 52
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.12.d.a ✓ 52 1.a even 1 1 trivial
50.12.d.a ✓ 52 25.d even 5 1 inner