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Properties

Label 7381.2.a.u
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 algebraic \(q\)-expansion of this newform has not been computed, 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 −2.74633 −2.76163 5.54233 3.30523 7.58435 −0.211115 −9.72840 4.62660 −9.07725
1.2 −2.74495 1.78715 5.53475 −0.775802 −4.90563 −4.69516 −9.70270 0.193900 2.12954
1.3 −2.69219 0.924537 5.24786 1.65974 −2.48903 −4.06555 −8.74385 −2.14523 −4.46833
1.4 −2.69115 −2.10951 5.24226 −0.278397 5.67700 0.628167 −8.72540 1.45003 0.749206
1.5 −2.63465 3.15802 4.94136 2.77920 −8.32025 1.24885 −7.74944 6.97306 −7.32222
1.6 −2.49445 −0.575321 4.22230 3.63247 1.43511 3.17846 −5.54342 −2.66901 −9.06102
1.7 −2.48481 2.56608 4.17426 3.73698 −6.37622 −0.422093 −5.40262 3.58478 −9.28568
1.8 −2.34518 1.95527 3.49986 −2.73288 −4.58546 0.384983 −3.51743 0.823092 6.40908
1.9 −2.32237 3.21773 3.39339 −3.99960 −7.47274 −2.64706 −3.23597 7.35376 9.28854
1.10 −2.15768 3.04589 2.65560 1.13882 −6.57207 2.02952 −1.41458 6.27746 −2.45722
1.11 −2.12501 −0.601302 2.51567 3.56711 1.27777 −2.77751 −1.09581 −2.63844 −7.58016
1.12 −2.10740 −3.11278 2.44112 −0.200658 6.55986 2.77586 −0.929619 6.68940 0.422867
1.13 −2.03956 −0.0696537 2.15982 −2.00769 0.142063 0.0145029 −0.325958 −2.99515 4.09481
1.14 −1.91934 −1.60550 1.68387 −0.201028 3.08150 −4.89182 0.606756 −0.422380 0.385841
1.15 −1.90630 2.59472 1.63399 1.21405 −4.94632 −3.79073 0.697722 3.73257 −2.31435
1.16 −1.89213 −2.41769 1.58017 −0.628775 4.57460 4.16714 0.794374 2.84525 1.18973
1.17 −1.58725 −1.31423 0.519363 −2.37841 2.08602 −1.37520 2.35014 −1.27279 3.77513
1.18 −1.48179 1.65514 0.195716 −1.97691 −2.45257 −0.807083 2.67358 −0.260526 2.92937
1.19 −1.44753 0.712677 0.0953450 −3.29893 −1.03162 3.22042 2.75705 −2.49209 4.77531
1.20 −1.30962 1.51164 −0.284895 2.14581 −1.97968 −4.42411 2.99234 −0.714935 −2.81019
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.u 64
11.b odd 2 1 7381.2.a.v 64
11.d odd 10 2 671.2.j.c ✓ 128
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
671.2.j.c ✓ 128 11.d odd 10 2
7381.2.a.u 64 1.a even 1 1 trivial
7381.2.a.v 64 11.b odd 2 1

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