↑

Properties

Label 22.13.d.a
Level $22$
Weight $13$
Character orbit 22.d
Analytic conductor $20.108$
Analytic rank $0$
Dimension $48$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [22,13,Mod(7,22)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(22, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([7]))
 
N = Newforms(chi, 13, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("22.7");
 
S:= CuspForms(chi, 13);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 22 = 2 \cdot 11 \)
Weight: \( k \) \(=\) \( 13 \)
Character orbit: \([\chi]\) \(=\) 22.d (of order \(10\), degree \(4\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(20.1078639801\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(12\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 1240 q^{3} + 24576 q^{4} + 1440 q^{5} - 147200 q^{6} + 439200 q^{7} + 563348 q^{9} - 7317468 q^{11} + 737280 q^{12} + 13341600 q^{13} - 7524864 q^{14} - 29808888 q^{15} - 50331648 q^{16} + 29458800 q^{17}+ \cdots - 5181480396620 q^{99}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −26.6001 + 36.6119i −319.732 + 984.033i −632.867 − 1947.76i −20129.5 + 14624.9i −27522.4 − 37881.4i −68815.6 + 22359.6i 88145.7 + 28640.3i −436148. − 316880.i − 1.12601e6i
7.2 −26.6001 + 36.6119i −191.068 + 588.048i −632.867 − 1947.76i −2891.36 + 2100.69i −16447.1 − 22637.5i 136173. − 44245.2i 88145.7 + 28640.3i 120651. + 87658.4i − 161737.i
7.3 −26.6001 + 36.6119i −123.108 + 378.886i −632.867 − 1947.76i 9050.01 − 6575.22i −10597.1 − 14585.6i −73754.4 + 23964.3i 88145.7 + 28640.3i 301546. + 219086.i 506240.i
7.4 −26.6001 + 36.6119i 103.669 − 319.061i −632.867 − 1947.76i −5380.08 + 3908.86i 8923.84 + 12282.6i −84352.8 + 27407.9i 88145.7 + 28640.3i 338892. + 246219.i − 300951.i
7.5 −26.6001 + 36.6119i 212.654 − 654.482i −632.867 − 1947.76i 23105.0 − 16786.8i 18305.2 + 25195.0i 58390.5 − 18972.2i 88145.7 + 28640.3i 46820.2 + 34016.8i 1.29245e6i
7.6 −26.6001 + 36.6119i 264.981 − 815.527i −632.867 − 1947.76i −19531.7 + 14190.6i 22809.5 + 31394.6i 187247. − 60840.2i 88145.7 + 28640.3i −164924. − 119824.i − 1.09256e6i
7.7 26.6001 − 36.6119i −334.229 + 1028.65i −632.867 − 1947.76i 14882.4 − 10812.7i 28770.4 + 39599.1i −20015.5 + 6503.44i −88145.7 − 28640.3i −516471. − 375238.i − 832491.i
7.8 26.6001 − 36.6119i −215.213 + 662.356i −632.867 − 1947.76i −6730.35 + 4889.89i 18525.5 + 25498.1i 127422. − 41402.0i −88145.7 − 28640.3i 37545.4 + 27278.3i 376483.i
7.9 26.6001 − 36.6119i 48.3765 − 148.888i −632.867 − 1947.76i −17010.2 + 12358.6i −4164.24 − 5731.59i 19106.9 − 6208.20i −88145.7 − 28640.3i 410118. + 297968.i 951517.i
7.10 26.6001 − 36.6119i 90.5465 − 278.674i −632.867 − 1947.76i 20021.7 − 14546.6i −7794.23 − 10727.8i 95102.6 − 30900.7i −88145.7 − 28640.3i 360484. + 261907.i − 1.11997e6i
7.11 26.6001 − 36.6119i 163.298 − 502.579i −632.867 − 1947.76i 5397.24 − 3921.32i −14056.6 − 19347.3i −214881. + 69819.2i −88145.7 − 28640.3i 204025. + 148233.i − 301911.i
7.12 26.6001 − 36.6119i 430.939 − 1326.29i −632.867 − 1947.76i −6380.03 + 4635.37i −37095.2 − 51057.1i 104479. − 33947.2i −88145.7 − 28640.3i −1.14340e6 − 830731.i 356887.i
13.1 −43.0399 − 13.9845i −974.141 + 707.755i 1656.87 + 1203.78i 1693.39 + 5211.73i 51824.5 − 16838.8i 23282.7 − 32045.9i −54477.1 − 74981.2i 283809. − 873474.i − 247993.i
13.2 −43.0399 − 13.9845i −450.921 + 327.613i 1656.87 + 1203.78i −2750.10 − 8463.94i 23989.1 − 7794.53i −63696.9 + 87671.2i −54477.1 − 74981.2i −68225.1 + 209975.i 402746.i
13.3 −43.0399 − 13.9845i 178.144 − 129.429i 1656.87 + 1203.78i 636.648 + 1959.40i −9477.29 + 3079.36i 74339.7 − 102320.i −54477.1 − 74981.2i −149241. + 459317.i − 93235.6i
13.4 −43.0399 − 13.9845i 296.805 − 215.641i 1656.87 + 1203.78i 8358.15 + 25723.7i −15790.1 + 5130.51i −25261.8 + 34770.0i −54477.1 − 74981.2i −122632. + 377424.i − 1.22403e6i
13.5 −43.0399 − 13.9845i 547.628 − 397.875i 1656.87 + 1203.78i −7656.05 − 23562.9i −29134.0 + 9466.20i −86058.9 + 118450.i −54477.1 − 74981.2i −22632.3 + 69655.1i 1.12121e6i
13.6 −43.0399 − 13.9845i 1147.47 − 833.682i 1656.87 + 1203.78i −187.556 − 577.239i −61045.4 + 19834.9i 79191.6 − 108998.i −54477.1 − 74981.2i 457426. − 1.40781e6i 27467.2i
13.7 43.0399 + 13.9845i −907.619 + 659.424i 1656.87 + 1203.78i −5263.46 − 16199.3i −48285.5 + 15688.9i 2402.42 − 3306.65i 54477.1 + 74981.2i 224708. − 691580.i − 770822.i
13.8 43.0399 + 13.9845i −884.677 + 642.756i 1656.87 + 1203.78i 8997.22 + 27690.6i −47065.0 + 15292.4i −64750.0 + 89120.7i 54477.1 + 74981.2i 205295. − 631832.i 1.31762e6i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 22.13.d.a ✓ 48
11.d odd 10 1 inner 22.13.d.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.13.d.a ✓ 48 1.a even 1 1 trivial
22.13.d.a ✓ 48 11.d odd 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{13}^{\mathrm{new}}(22, [\chi])\).