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Properties

Label 30.8.e.a
Level $30$
Weight $8$
Character orbit 30.e
Analytic conductor $9.372$
Analytic rank $0$
Dimension $28$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [30,8,Mod(17,30)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(30, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("30.17");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 30 = 2 \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 30.e (of order \(4\), degree \(2\), 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: \(9.37155076452\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(14\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 52 q^{3} - 928 q^{6} - 1348 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q - 52 q^{3} - 928 q^{6} - 1348 q^{7} - 416 q^{10} + 3328 q^{12} - 21840 q^{13} - 37672 q^{15} - 114688 q^{16} + 50624 q^{18} - 142744 q^{21} - 47008 q^{22} + 419744 q^{25} + 212972 q^{27} - 86272 q^{28} - 301152 q^{30} - 44936 q^{31} + 1398788 q^{33} + 50432 q^{36} + 608088 q^{37} + 339968 q^{40} - 624032 q^{42} - 1288968 q^{43} - 656648 q^{45} - 1486208 q^{46} + 212992 q^{48} + 4263064 q^{51} + 1397760 q^{52} - 3985428 q^{55} - 9921952 q^{57} - 416480 q^{58} - 365824 q^{60} - 14602248 q^{61} + 4050428 q^{63} + 10106816 q^{66} + 22755248 q^{67} + 551584 q^{70} - 3239936 q^{72} - 522380 q^{73} - 17815892 q^{75} - 4901888 q^{76} + 8835840 q^{78} + 33339556 q^{81} + 22502528 q^{82} + 27842408 q^{85} - 38505100 q^{87} - 3008512 q^{88} - 26143552 q^{90} - 77623680 q^{91} + 70200824 q^{93} + 3801088 q^{96} + 82948404 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 −5.65685 + 5.65685i −46.7594 − 0.750092i − 64.0000i 268.955 − 76.0794i 268.754 − 260.268i −661.697 − 661.697i 362.039 + 362.039i 2185.87 + 70.1477i −1091.07 + 1951.81i
17.2 −5.65685 + 5.65685i −22.8135 + 40.8233i − 64.0000i −279.492 − 3.02623i −101.879 − 359.984i −103.844 − 103.844i 362.039 + 362.039i −1146.09 − 1862.65i 1598.17 − 1563.93i
17.3 −5.65685 + 5.65685i −16.5441 − 43.7412i − 64.0000i −157.986 − 230.576i 341.025 + 153.850i 330.070 + 330.070i 362.039 + 362.039i −1639.59 + 1447.32i 2198.04 + 410.630i
17.4 −5.65685 + 5.65685i −2.77769 + 46.6828i − 64.0000i 263.295 + 93.8130i −248.365 − 279.791i 1279.41 + 1279.41i 362.039 + 362.039i −2171.57 − 259.341i −2020.11 + 958.733i
17.5 −5.65685 + 5.65685i 16.7888 − 43.6478i − 64.0000i 212.251 + 181.864i 151.937 + 341.882i −469.949 − 469.949i 362.039 + 362.039i −1623.27 − 1465.59i −2229.45 + 171.893i
17.6 −5.65685 + 5.65685i 32.8544 + 33.2804i − 64.0000i 9.36750 − 279.351i −374.115 − 2.40995i −919.616 − 919.616i 362.039 + 362.039i −28.1750 + 2186.82i 1527.26 + 1633.24i
17.7 −5.65685 + 5.65685i 46.7575 + 0.858698i − 64.0000i −189.818 + 205.169i −269.358 + 259.643i 208.621 + 208.621i 362.039 + 362.039i 2185.53 + 80.3011i −86.8377 − 2234.38i
17.8 5.65685 − 5.65685i −46.6828 + 2.77769i − 64.0000i −263.295 − 93.8130i −248.365 + 279.791i 1279.41 + 1279.41i −362.039 − 362.039i 2171.57 − 259.341i −2020.11 + 958.733i
17.9 5.65685 − 5.65685i −40.8233 + 22.8135i − 64.0000i 279.492 + 3.02623i −101.879 + 359.984i −103.844 − 103.844i −362.039 − 362.039i 1146.09 − 1862.65i 1598.17 − 1563.93i
17.10 5.65685 − 5.65685i −33.2804 − 32.8544i − 64.0000i −9.36750 + 279.351i −374.115 + 2.40995i −919.616 − 919.616i −362.039 − 362.039i 28.1750 + 2186.82i 1527.26 + 1633.24i
17.11 5.65685 − 5.65685i −0.858698 − 46.7575i − 64.0000i 189.818 − 205.169i −269.358 − 259.643i 208.621 + 208.621i −362.039 − 362.039i −2185.53 + 80.3011i −86.8377 − 2234.38i
17.12 5.65685 − 5.65685i 0.750092 + 46.7594i − 64.0000i −268.955 + 76.0794i 268.754 + 260.268i −661.697 − 661.697i −362.039 − 362.039i −2185.87 + 70.1477i −1091.07 + 1951.81i
17.13 5.65685 − 5.65685i 43.6478 − 16.7888i − 64.0000i −212.251 − 181.864i 151.937 − 341.882i −469.949 − 469.949i −362.039 − 362.039i 1623.27 − 1465.59i −2229.45 + 171.893i
17.14 5.65685 − 5.65685i 43.7412 + 16.5441i − 64.0000i 157.986 + 230.576i 341.025 − 153.850i 330.070 + 330.070i −362.039 − 362.039i 1639.59 + 1447.32i 2198.04 + 410.630i
23.1 −5.65685 − 5.65685i −46.7594 + 0.750092i 64.0000i 268.955 + 76.0794i 268.754 + 260.268i −661.697 + 661.697i 362.039 − 362.039i 2185.87 − 70.1477i −1091.07 − 1951.81i
23.2 −5.65685 − 5.65685i −22.8135 − 40.8233i 64.0000i −279.492 + 3.02623i −101.879 + 359.984i −103.844 + 103.844i 362.039 − 362.039i −1146.09 + 1862.65i 1598.17 + 1563.93i
23.3 −5.65685 − 5.65685i −16.5441 + 43.7412i 64.0000i −157.986 + 230.576i 341.025 − 153.850i 330.070 − 330.070i 362.039 − 362.039i −1639.59 − 1447.32i 2198.04 − 410.630i
23.4 −5.65685 − 5.65685i −2.77769 − 46.6828i 64.0000i 263.295 − 93.8130i −248.365 + 279.791i 1279.41 − 1279.41i 362.039 − 362.039i −2171.57 + 259.341i −2020.11 − 958.733i
23.5 −5.65685 − 5.65685i 16.7888 + 43.6478i 64.0000i 212.251 − 181.864i 151.937 − 341.882i −469.949 + 469.949i 362.039 − 362.039i −1623.27 + 1465.59i −2229.45 − 171.893i
23.6 −5.65685 − 5.65685i 32.8544 − 33.2804i 64.0000i 9.36750 + 279.351i −374.115 + 2.40995i −919.616 + 919.616i 362.039 − 362.039i −28.1750 − 2186.82i 1527.26 − 1633.24i
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 30.8.e.a ✓ 28
3.b odd 2 1 inner 30.8.e.a ✓ 28
5.b even 2 1 150.8.e.b 28
5.c odd 4 1 inner 30.8.e.a ✓ 28
5.c odd 4 1 150.8.e.b 28
15.d odd 2 1 150.8.e.b 28
15.e even 4 1 inner 30.8.e.a ✓ 28
15.e even 4 1 150.8.e.b 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.8.e.a ✓ 28 1.a even 1 1 trivial
30.8.e.a ✓ 28 3.b odd 2 1 inner
30.8.e.a ✓ 28 5.c odd 4 1 inner
30.8.e.a ✓ 28 15.e even 4 1 inner
150.8.e.b 28 5.b even 2 1
150.8.e.b 28 5.c odd 4 1
150.8.e.b 28 15.d odd 2 1
150.8.e.b 28 15.e even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(30, [\chi])\).