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Properties

Label 7381.2.a.v
Level $7381$
Weight $2$
Character orbit 7381.a
Self dual yes
Analytic conductor $58.938$
Analytic rank $0$
Dimension $64$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7381,2,Mod(1,7381)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7381, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("7381.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7381 = 11^{2} \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7381.a (trivial)

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: yes
Analytic conductor: \(58.9375817319\)
Analytic rank: \(0\)
Dimension: \(64\)
Twist minimal: no (minimal twist has level 671)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 64 q + 5 q^{2} + 19 q^{3} + 73 q^{4} + 21 q^{5} - q^{6} + 6 q^{7} + 3 q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q + 5 q^{2} + 19 q^{3} + 73 q^{4} + 21 q^{5} - q^{6} + 6 q^{7} + 3 q^{8} + 81 q^{9} - 4 q^{10} + 41 q^{12} - q^{13} + 41 q^{14} + 28 q^{15} + 107 q^{16} + 13 q^{17} + 38 q^{18} - q^{19} + 65 q^{20} + q^{21} + 52 q^{23} + 3 q^{24} + 87 q^{25} + 37 q^{26} + 88 q^{27} - q^{28} + 19 q^{29} - 19 q^{30} + 45 q^{31} - 24 q^{32} - 23 q^{34} + 10 q^{35} + 146 q^{36} + 26 q^{37} - 4 q^{38} - 6 q^{39} + 84 q^{40} - 12 q^{41} + 28 q^{42} + 5 q^{43} + 71 q^{45} - 5 q^{46} + 63 q^{47} + 85 q^{48} + 78 q^{49} + 14 q^{50} + 22 q^{51} - 24 q^{52} + 86 q^{53} - 114 q^{54} + 119 q^{56} - 29 q^{57} + q^{58} + 119 q^{59} - 22 q^{60} + 64 q^{61} + 13 q^{62} - 28 q^{63} + 135 q^{64} - 30 q^{65} + 2 q^{67} + 59 q^{68} + 63 q^{69} - 2 q^{70} + 126 q^{71} + 48 q^{72} + 8 q^{73} - 27 q^{74} + 107 q^{75} - 82 q^{76} - 13 q^{78} - 5 q^{79} + 119 q^{80} + 96 q^{81} - 27 q^{82} + 14 q^{83} + 182 q^{84} - 52 q^{85} + 60 q^{86} - 8 q^{87} + 59 q^{89} - 45 q^{90} + 83 q^{91} + 111 q^{92} - 7 q^{93} + 21 q^{94} - 26 q^{95} - 86 q^{96} - 39 q^{97} - 57 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −2.80175 2.27857 5.84979 0.434211 −6.38397 −2.03379 −10.7861 2.19187 −1.21655
1.2 −2.69570 0.353590 5.26682 0.958298 −0.953175 −0.547384 −8.80638 −2.87497 −2.58329
1.3 −2.68320 3.30101 5.19955 −2.29730 −8.85726 −1.02709 −8.58501 7.89668 6.16412
1.4 −2.65712 −0.921980 5.06029 −3.49961 2.44981 −3.95363 −8.13155 −2.14995 9.29889
1.5 −2.64263 −2.02261 4.98348 3.07958 5.34500 −3.18832 −7.88423 1.09094 −8.13819
1.6 −2.50914 1.70568 4.29577 2.67647 −4.27979 0.628541 −5.76040 −0.0906485 −6.71563
1.7 −2.36361 −0.120226 3.58664 −1.13694 0.284166 2.88472 −3.75020 −2.98555 2.68729
1.8 −2.30350 2.86548 3.30610 4.11838 −6.60062 4.52556 −3.00859 5.21098 −9.48668
1.9 −2.29697 −2.58320 3.27607 1.33468 5.93353 −4.76262 −2.93111 3.67290 −3.06571
1.10 −2.29405 −1.62402 3.26266 −3.07361 3.72559 1.92994 −2.89660 −0.362545 7.05102
1.11 −2.10438 −2.04639 2.42841 2.55130 4.30637 3.75742 −0.901543 1.18769 −5.36890
1.12 −1.98800 2.79573 1.95214 2.00486 −5.55790 −4.55995 0.0951428 4.81608 −3.98567
1.13 −1.86589 3.07680 1.48156 3.54844 −5.74097 −1.55335 0.967361 6.46667 −6.62101
1.14 −1.82924 −1.21646 1.34613 −1.80381 2.22520 1.81436 1.19608 −1.52023 3.29961
1.15 −1.79564 −0.368963 1.22433 4.17396 0.662525 −1.01733 1.39283 −2.86387 −7.49493
1.16 −1.49906 1.05312 0.247194 −0.565778 −1.57870 4.58937 2.62757 −1.89094 0.848138
1.17 −1.29757 3.18255 −0.316304 −1.60573 −4.12959 −2.41010 3.00557 7.12864 2.08355
1.18 −1.20104 0.570170 −0.557498 0.389712 −0.684798 4.86716 3.07166 −2.67491 −0.468060
1.19 −1.14254 −1.36210 −0.694601 −1.77754 1.55625 −3.06707 3.07869 −1.14469 2.03092
1.20 −1.07990 −0.374304 −0.833814 −2.56091 0.404211 −1.05897 3.06024 −2.85990 2.76553
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.64
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(11\) \(1\)
\(61\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7381.2.a.v 64
11.b odd 2 1 7381.2.a.u 64
11.c even 5 2 671.2.j.c ✓ 128
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
671.2.j.c ✓ 128 11.c even 5 2
7381.2.a.u 64 11.b odd 2 1
7381.2.a.v 64 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7381))\):

\( T_{2}^{64} - 5 T_{2}^{63} - 88 T_{2}^{62} + 474 T_{2}^{61} + 3594 T_{2}^{60} - 21207 T_{2}^{59} + \cdots - 2556 \) Copy content Toggle raw display
\( T_{7}^{64} - 6 T_{7}^{63} - 245 T_{7}^{62} + 1451 T_{7}^{61} + 28508 T_{7}^{60} - 165578 T_{7}^{59} + \cdots - 15\!\cdots\!61 \) Copy content Toggle raw display