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Properties

Label 304.7.e.f
Level $304$
Weight $7$
Character orbit 304.e
Analytic conductor $69.936$
Analytic rank $0$
Dimension $30$
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(113,304)]
 
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([0, 0, 1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("304.113");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 304.e (of order \(2\), degree \(1\), not 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: \(30\)
Twist minimal: no (minimal twist has level 152)
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.

\(\operatorname{Tr}(f)(q) = \) \( 30 q + 720 q^{7} - 8670 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q + 720 q^{7} - 8670 q^{9} + 2524 q^{11} + 9700 q^{17} - 4014 q^{19} + 39376 q^{23} + 110742 q^{25} + 19976 q^{35} + 266500 q^{39} + 106788 q^{43} - 91360 q^{45} - 222756 q^{47} + 593586 q^{49} - 540936 q^{55} - 545972 q^{57} - 242640 q^{61} + 377716 q^{63} + 545964 q^{73} - 272356 q^{77} + 2189926 q^{81} - 1542652 q^{83} - 826908 q^{85} + 2729572 q^{87} - 2139912 q^{93} - 2142716 q^{95} + 293012 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
113.1 0 − 50.6470i 0 207.318 0 −216.532 0 −1836.12 0
113.2 0 − 48.7582i 0 −200.006 0 536.711 0 −1648.37 0
113.3 0 − 47.0707i 0 −22.4613 0 196.082 0 −1486.65 0
113.4 0 − 43.2780i 0 −100.391 0 −640.715 0 −1143.99 0
113.5 0 − 36.8625i 0 9.31317 0 −333.649 0 −629.840 0
113.6 0 − 33.4715i 0 71.5877 0 −51.8109 0 −391.339 0
113.7 0 − 33.1604i 0 225.972 0 256.290 0 −370.612 0
113.8 0 − 28.4003i 0 32.4520 0 263.602 0 −77.5754 0
113.9 0 − 27.9511i 0 −141.368 0 −302.893 0 −52.2651 0
113.10 0 − 20.4841i 0 −148.208 0 486.917 0 309.402 0
113.11 0 − 17.1584i 0 166.964 0 600.271 0 434.588 0
113.12 0 − 11.4888i 0 −77.3334 0 256.225 0 597.007 0
113.13 0 − 9.57785i 0 39.7533 0 24.5800 0 637.265 0
113.14 0 − 9.20554i 0 −204.194 0 −264.373 0 644.258 0
113.15 0 − 7.05423i 0 140.601 0 −450.706 0 679.238 0
113.16 0 7.05423i 0 140.601 0 −450.706 0 679.238 0
113.17 0 9.20554i 0 −204.194 0 −264.373 0 644.258 0
113.18 0 9.57785i 0 39.7533 0 24.5800 0 637.265 0
113.19 0 11.4888i 0 −77.3334 0 256.225 0 597.007 0
113.20 0 17.1584i 0 166.964 0 600.271 0 434.588 0
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 113.30
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.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.e.f 30
4.b odd 2 1 152.7.e.a ✓ 30
19.b odd 2 1 inner 304.7.e.f 30
76.d even 2 1 152.7.e.a ✓ 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.7.e.a ✓ 30 4.b odd 2 1
152.7.e.a ✓ 30 76.d even 2 1
304.7.e.f 30 1.a even 1 1 trivial
304.7.e.f 30 19.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{7}^{\mathrm{new}}(304, [\chi])\):

\( T_{3}^{30} + 15270 T_{3}^{28} + 103327031 T_{3}^{26} + 408990954956 T_{3}^{24} + \cdots + 16\!\cdots\!68 \) Copy content Toggle raw display
\( T_{5}^{15} - 144873 T_{5}^{13} - 1422438 T_{5}^{12} + 7991401275 T_{5}^{11} + 137484061124 T_{5}^{10} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display