Properties

Label 7381.2.a.j
Level $7381$
Weight $2$
Character orbit 7381.a
Self dual yes
Analytic conductor $58.938$
Analytic rank $0$
Dimension $21$
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: \(21\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 21 q + 3 q^{3} + 32 q^{4} + 7 q^{5} - 5 q^{6} - 5 q^{7} + 6 q^{8} + 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 21 q + 3 q^{3} + 32 q^{4} + 7 q^{5} - 5 q^{6} - 5 q^{7} + 6 q^{8} + 40 q^{9} - q^{10} + 6 q^{12} - 20 q^{13} + 17 q^{14} + 18 q^{15} + 50 q^{16} - q^{17} + 5 q^{18} - 15 q^{19} - 2 q^{20} - 16 q^{21} + 11 q^{23} + 16 q^{24} + 48 q^{25} - 5 q^{26} + 12 q^{27} + 16 q^{28} + 9 q^{29} - 16 q^{30} + 22 q^{31} - 3 q^{32} + 33 q^{34} + 39 q^{35} + 57 q^{36} + 21 q^{37} + 11 q^{38} + 28 q^{39} + 16 q^{40} - 7 q^{41} - 55 q^{42} - 16 q^{43} + 44 q^{45} + 3 q^{46} + 5 q^{47} - 71 q^{48} + 80 q^{49} + 33 q^{50} + 19 q^{51} - 60 q^{52} + 9 q^{53} - 13 q^{54} + 44 q^{56} - 39 q^{57} - 27 q^{58} + 13 q^{59} + 70 q^{60} - 21 q^{61} + 23 q^{62} - 24 q^{63} + 66 q^{64} - 25 q^{65} + 38 q^{67} + 74 q^{68} - 17 q^{69} - 33 q^{70} + 12 q^{71} + 75 q^{72} - 20 q^{73} + 12 q^{74} - 10 q^{75} - 59 q^{76} - 14 q^{78} - q^{79} - 38 q^{80} + 89 q^{81} + 7 q^{82} + 19 q^{83} + 14 q^{84} - 38 q^{85} - 3 q^{86} - 4 q^{87} + 37 q^{89} + 174 q^{90} + 24 q^{91} + 31 q^{92} - 15 q^{93} + 64 q^{94} + 43 q^{95} + 38 q^{96} + 68 q^{97} + 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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 −2.68161 −2.98584 5.19104 −3.56342 8.00685 −3.18744 −8.55713 5.91522 9.55570
1.2 −2.67686 1.01551 5.16560 1.61394 −2.71838 4.48909 −8.47390 −1.96874 −4.32029
1.3 −2.54333 0.588033 4.46852 −0.268217 −1.49556 −2.27426 −6.27827 −2.65422 0.682163
1.4 −2.16302 −0.957308 2.67865 3.42546 2.07068 −1.52570 −1.46794 −2.08356 −7.40934
1.5 −2.08860 3.36235 2.36225 −0.0546248 −7.02260 2.53368 −0.756597 8.30537 0.114089
1.6 −1.61525 2.06891 0.609041 2.50050 −3.34180 −4.06392 2.24675 1.28037 −4.03894
1.7 −1.29392 2.84985 −0.325780 −3.15298 −3.68746 −5.03078 3.00937 5.12162 4.07969
1.8 −0.942064 2.26946 −1.11252 4.16220 −2.13797 0.914635 2.93219 2.15044 −3.92106
1.9 −0.731918 −2.97561 −1.46430 2.76447 2.17790 1.34613 2.53558 5.85424 −2.02337
1.10 −0.543169 −1.77928 −1.70497 −0.382346 0.966447 4.84691 2.01242 0.165824 0.207678
1.11 −0.472572 −1.64615 −1.77668 −3.20957 0.777925 −4.06194 1.78475 −0.290187 1.51675
1.12 0.404310 1.66838 −1.83653 0.0975766 0.674545 −1.38056 −1.55115 −0.216497 0.0394512
1.13 0.634118 −1.50740 −1.59789 −1.64555 −0.955868 3.41198 −2.28149 −0.727754 −1.04347
1.14 1.29453 3.40741 −0.324186 2.94198 4.41100 0.962580 −3.00873 8.61042 3.80849
1.15 1.45442 −2.91272 0.115325 3.82594 −4.23631 −3.26042 −2.74110 5.48395 5.56450
1.16 1.47389 −0.831223 0.172365 0.593969 −1.22513 −3.31761 −2.69374 −2.30907 0.875448
1.17 2.19302 1.13601 2.80936 −3.87665 2.49129 −3.34814 1.77494 −1.70949 −8.50160
1.18 2.40042 −0.672797 3.76203 −3.90458 −1.61500 3.05635 4.22962 −2.54734 −9.37264
1.19 2.55546 1.15142 4.53040 4.10969 2.94242 4.64385 6.46634 −1.67423 10.5022
1.20 2.60374 2.97894 4.77947 0.851432 7.75638 −3.04314 7.23702 5.87407 2.21691
See all 21 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.21
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.j 21
11.b odd 2 1 671.2.a.d 21
33.d even 2 1 6039.2.a.l 21
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
671.2.a.d 21 11.b odd 2 1
6039.2.a.l 21 33.d even 2 1
7381.2.a.j 21 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7381))\):

\( T_{2}^{21} - 37 T_{2}^{19} - 2 T_{2}^{18} + 582 T_{2}^{17} + 65 T_{2}^{16} - 5074 T_{2}^{15} - 876 T_{2}^{14} + 26808 T_{2}^{13} + 6340 T_{2}^{12} - 88228 T_{2}^{11} - 26710 T_{2}^{10} + 179236 T_{2}^{9} + 66498 T_{2}^{8} + \cdots + 2082 \) Copy content Toggle raw display
\( T_{7}^{21} + 5 T_{7}^{20} - 101 T_{7}^{19} - 540 T_{7}^{18} + 4187 T_{7}^{17} + 24602 T_{7}^{16} - 90995 T_{7}^{15} - 616649 T_{7}^{14} + 1079283 T_{7}^{13} + 9307761 T_{7}^{12} - 6077091 T_{7}^{11} + \cdots + 1454145536 \) Copy content Toggle raw display