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Properties

Label 37.10.b.a
Level $37$
Weight $10$
Character orbit 37.b
Analytic conductor $19.056$
Analytic rank $0$
Dimension $28$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [37,10,Mod(36,37)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(37, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("37.36");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 37 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 37.b (of order \(2\), degree \(1\), 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: \(19.0563259381\)
Analytic rank: \(0\)
Dimension: \(28\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 150 q^{3} - 7340 q^{4} + 5940 q^{7} + 203834 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q - 150 q^{3} - 7340 q^{4} + 5940 q^{7} + 203834 q^{9} - 37794 q^{10} + 64218 q^{11} + 127066 q^{12} + 1640020 q^{16} - 40760 q^{21} - 19052702 q^{25} - 6123570 q^{26} - 1640160 q^{27} + 7625000 q^{28} - 19333840 q^{30} + 15698940 q^{33} + 708476 q^{34} - 51365130 q^{36} + 16215912 q^{37} - 28583988 q^{38} + 105594566 q^{40} - 7076754 q^{41} - 103376142 q^{44} + 77028630 q^{46} + 227172960 q^{47} - 65810002 q^{48} + 204785368 q^{49} + 50216184 q^{53} - 535678486 q^{58} - 438751290 q^{62} + 78365380 q^{63} + 177466492 q^{64} - 778712844 q^{65} + 631746366 q^{67} - 160866964 q^{70} + 1006626660 q^{71} - 83180574 q^{73} - 1466328336 q^{74} - 381338576 q^{75} + 408926592 q^{77} + 2562172434 q^{78} + 2121646388 q^{81} + 2834717148 q^{83} - 2459748244 q^{84} - 1400055912 q^{85} - 1148298888 q^{86} - 1425076412 q^{90} - 3422824284 q^{95} - 2703697668 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
36.1 − 42.2789i −80.1183 −1275.50 815.869i 3387.31i 10597.9 32280.0i −13264.1 34494.0
36.2 − 41.4868i −141.326 −1209.16 − 2156.92i 5863.16i −9603.13 28922.7i 290.033 −89483.6
36.3 − 39.1761i 143.787 −1022.76 1267.14i − 5633.02i −8206.15 20009.8i 991.782 49641.5
36.4 − 36.6273i 218.781 −829.562 − 2715.49i − 8013.37i 4758.22 11631.5i 28182.1 −99461.4
36.5 − 33.0590i 135.672 −580.901 361.548i − 4485.17i 3757.48 2277.79i −1276.21 11952.4
36.6 − 32.8810i −268.496 −569.158 1775.64i 8828.42i −3759.38 1879.42i 52407.3 58384.8
36.7 − 27.1629i −60.9392 −225.823 − 732.887i 1655.29i −919.752 − 7773.41i −15969.4 −19907.3
36.8 − 22.8718i −66.4996 −11.1170 2205.98i 1520.96i −4665.50 − 11456.1i −15260.8 50454.7
36.9 − 21.1013i −233.250 66.7348 − 1577.22i 4921.87i 9882.66 − 12212.1i 34722.4 −33281.3
36.10 − 16.0659i 264.988 253.887 1518.28i − 4257.26i 218.903 − 12304.7i 50535.4 24392.6
36.11 − 12.3956i 115.163 358.349 − 220.872i − 1427.51i 6117.18 − 10788.5i −6420.50 −2737.84
36.12 − 10.6764i 115.620 398.015 − 1685.85i − 1234.40i −9624.92 − 9715.67i −6315.04 −17998.7
36.13 − 6.09679i −42.8023 474.829 2485.27i 260.956i 8705.80 − 6016.49i −17851.0 15152.1
36.14 − 3.13565i −175.579 502.168 − 159.129i 550.554i −4289.25 − 3180.07i 11145.0 −498.971
36.15 3.13565i −175.579 502.168 159.129i − 550.554i −4289.25 3180.07i 11145.0 −498.971
36.16 6.09679i −42.8023 474.829 − 2485.27i − 260.956i 8705.80 6016.49i −17851.0 15152.1
36.17 10.6764i 115.620 398.015 1685.85i 1234.40i −9624.92 9715.67i −6315.04 −17998.7
36.18 12.3956i 115.163 358.349 220.872i 1427.51i 6117.18 10788.5i −6420.50 −2737.84
36.19 16.0659i 264.988 253.887 − 1518.28i 4257.26i 218.903 12304.7i 50535.4 24392.6
36.20 21.1013i −233.250 66.7348 1577.22i − 4921.87i 9882.66 12212.1i 34722.4 −33281.3
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 36.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 37.10.b.a ✓ 28
37.b even 2 1 inner 37.10.b.a ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.10.b.a ✓ 28 1.a even 1 1 trivial
37.10.b.a ✓ 28 37.b even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(37, [\chi])\).