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Properties

Label 304.7.d.b
Level $304$
Weight $7$
Character orbit 304.d
Analytic conductor $69.936$
Analytic rank $0$
Dimension $36$
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] = [304,7,Mod(191,304)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(304, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("304.191");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 304.d (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: \(69.9364414204\)
Analytic rank: \(0\)
Dimension: \(36\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 488 q^{5} - 7724 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q + 488 q^{5} - 7724 q^{9} + 216 q^{13} - 8152 q^{17} + 68928 q^{21} + 81852 q^{25} + 67992 q^{29} - 25248 q^{33} + 142200 q^{37} + 26632 q^{41} - 321688 q^{45} - 247548 q^{49} - 692072 q^{53} - 997368 q^{61} - 717808 q^{65} + 1525120 q^{69} - 313176 q^{73} - 513184 q^{77} + 1377380 q^{81} + 3408336 q^{85} + 1587624 q^{89} + 3108896 q^{93} - 2000568 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
191.1 0 − 53.3181i 0 122.517 0 447.644i 0 −2113.82 0
191.2 0 − 45.7950i 0 −127.753 0 151.323i 0 −1368.18 0
191.3 0 − 44.3039i 0 −59.5728 0 − 533.499i 0 −1233.84 0
191.4 0 − 42.0622i 0 37.8953 0 72.9335i 0 −1040.23 0
191.5 0 − 39.7908i 0 226.575 0 − 37.2143i 0 −854.308 0
191.6 0 − 38.0727i 0 −114.082 0 91.7459i 0 −720.531 0
191.7 0 − 32.7255i 0 122.284 0 − 368.060i 0 −341.959 0
191.8 0 − 31.4259i 0 −152.655 0 626.350i 0 −258.585 0
191.9 0 − 31.2747i 0 32.6328 0 − 5.44618i 0 −249.105 0
191.10 0 − 27.6501i 0 172.166 0 − 311.569i 0 −35.5275 0
191.11 0 − 24.3880i 0 −163.286 0 − 22.2626i 0 134.227 0
191.12 0 − 17.7377i 0 163.680 0 607.739i 0 414.373 0
191.13 0 − 13.7179i 0 −24.3481 0 553.112i 0 540.820 0
191.14 0 − 13.2607i 0 198.425 0 413.760i 0 553.153 0
191.15 0 − 9.85074i 0 −53.8332 0 − 45.5418i 0 631.963 0
191.16 0 − 6.94735i 0 −226.881 0 318.897i 0 680.734 0
191.17 0 − 5.61636i 0 26.6433 0 46.7757i 0 697.456 0
191.18 0 − 5.25753i 0 63.5922 0 − 374.983i 0 701.358 0
191.19 0 5.25753i 0 63.5922 0 374.983i 0 701.358 0
191.20 0 5.61636i 0 26.6433 0 − 46.7757i 0 697.456 0
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 191.36
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 304.7.d.b ✓ 36
4.b odd 2 1 inner 304.7.d.b ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
304.7.d.b ✓ 36 1.a even 1 1 trivial
304.7.d.b ✓ 36 4.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{36} + 16984 T_{3}^{34} + 130094748 T_{3}^{32} + 595196302152 T_{3}^{30} + \cdots + 95\!\cdots\!56 \) acting on \(S_{7}^{\mathrm{new}}(304, [\chi])\). Copy content Toggle raw display