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Properties

Label 7007.2.a.bi
Level $7007$
Weight $2$
Character orbit 7007.a
Self dual yes
Analytic conductor $55.951$
Analytic rank $0$
Dimension $25$
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] = [7007,2,Mod(1,7007)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7007, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("7007.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7007 = 7^{2} \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7007.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: \(55.9511766963\)
Analytic rank: \(0\)
Dimension: \(25\)
Twist minimal: no (minimal twist has level 1001)
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.

\(\operatorname{Tr}(f)(q) = \) \( 25 q + 6 q^{2} + 2 q^{3} + 30 q^{4} + q^{5} - 2 q^{6} + 21 q^{8} + 37 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 25 q + 6 q^{2} + 2 q^{3} + 30 q^{4} + q^{5} - 2 q^{6} + 21 q^{8} + 37 q^{9} + 3 q^{10} + 25 q^{11} - 9 q^{12} + 25 q^{13} + 32 q^{16} + q^{17} + 44 q^{18} - 5 q^{19} - 4 q^{20} + 6 q^{22} + 15 q^{23} + 4 q^{24} + 50 q^{25} + 6 q^{26} + 17 q^{27} + 24 q^{29} + q^{30} + 12 q^{31} + 48 q^{32} + 2 q^{33} + 8 q^{34} + 30 q^{36} + 33 q^{37} - 16 q^{38} + 2 q^{39} + 21 q^{40} - 12 q^{41} + 38 q^{43} + 30 q^{44} + 22 q^{45} + 39 q^{46} - 4 q^{47} - 82 q^{48} + 16 q^{50} + 51 q^{51} + 30 q^{52} + 2 q^{53} - 10 q^{54} + q^{55} + 38 q^{57} + 17 q^{58} + 4 q^{59} - 33 q^{60} + 22 q^{61} - 42 q^{62} + 41 q^{64} + q^{65} - 2 q^{66} + 24 q^{67} - 14 q^{68} + 30 q^{69} + 9 q^{71} + 102 q^{72} - 11 q^{73} + 39 q^{74} + 16 q^{75} - 58 q^{76} - 2 q^{78} + 19 q^{79} + 33 q^{80} + 73 q^{81} + 32 q^{82} - 16 q^{83} + 14 q^{85} + 27 q^{86} + 11 q^{87} + 21 q^{88} - 13 q^{89} - 40 q^{90} + 17 q^{93} + 56 q^{94} + 15 q^{95} - 55 q^{96} - 34 q^{97} + 37 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −2.63981 −1.55395 4.96858 0.218386 4.10214 0 −7.83647 −0.585225 −0.576498
1.2 −2.27102 −0.785231 3.15753 −4.14311 1.78328 0 −2.62878 −2.38341 9.40907
1.3 −2.23494 −0.814970 2.99497 2.65780 1.82141 0 −2.22370 −2.33582 −5.94003
1.4 −2.15940 1.84247 2.66302 1.44384 −3.97863 0 −1.43174 0.394684 −3.11784
1.5 −1.73112 0.0633923 0.996768 −3.28047 −0.109739 0 1.73671 −2.99598 5.67887
1.6 −1.66285 3.41862 0.765065 −1.23738 −5.68465 0 2.05351 8.68696 2.05757
1.7 −1.35435 0.853008 −0.165725 −1.11633 −1.15527 0 2.93316 −2.27238 1.51190
1.8 −1.21726 −3.08327 −0.518274 3.44467 3.75314 0 3.06540 6.50654 −4.19307
1.9 −0.869091 2.46495 −1.24468 4.31820 −2.14227 0 2.81992 3.07598 −3.75291
1.10 −0.337064 1.85392 −1.88639 −3.04745 −0.624888 0 1.30996 0.437003 1.02718
1.11 −0.110087 −1.11088 −1.98788 1.81203 0.122293 0 0.439013 −1.76595 −0.199480
1.12 0.0484145 1.85996 −1.99766 2.21431 0.0900488 0 −0.193545 0.459434 0.107205
1.13 0.108636 −2.05068 −1.98820 0.246533 −0.222777 0 −0.433262 1.20528 0.0267823
1.14 0.288494 −3.12424 −1.91677 −2.87666 −0.901324 0 −1.12997 6.76085 −0.829900
1.15 1.02090 1.38596 −0.957757 −3.23836 1.41493 0 −3.01958 −1.07912 −3.30606
1.16 1.40578 3.29751 −0.0237751 2.16852 4.63558 0 −2.84499 7.87357 3.04847
1.17 1.47788 0.505829 0.184135 1.49526 0.747555 0 −2.68363 −2.74414 2.20981
1.18 1.64181 −0.819427 0.695542 3.58866 −1.34534 0 −2.14167 −2.32854 5.89190
1.19 1.67728 −2.71463 0.813275 −0.785433 −4.55320 0 −1.99047 4.36921 −1.31739
1.20 2.13945 −2.30334 2.57723 −2.85369 −4.92786 0 1.23494 2.30536 −6.10531
See all 25 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.25
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(-1\)
\(11\) \(-1\)
\(13\) \(-1\)

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 7007.2.a.bi 25
7.b odd 2 1 7007.2.a.bh 25
7.d odd 6 2 1001.2.i.d ✓ 50
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1001.2.i.d ✓ 50 7.d odd 6 2
7007.2.a.bh 25 7.b odd 2 1
7007.2.a.bi 25 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7007))\):

\( T_{2}^{25} - 6 T_{2}^{24} - 22 T_{2}^{23} + 189 T_{2}^{22} + 121 T_{2}^{21} - 2544 T_{2}^{20} + 953 T_{2}^{19} + \cdots + 9 \) Copy content Toggle raw display
\( T_{3}^{25} - 2 T_{3}^{24} - 54 T_{3}^{23} + 101 T_{3}^{22} + 1262 T_{3}^{21} - 2188 T_{3}^{20} + \cdots - 23805 \) Copy content Toggle raw display
\( T_{5}^{25} - T_{5}^{24} - 87 T_{5}^{23} + 97 T_{5}^{22} + 3283 T_{5}^{21} - 4078 T_{5}^{20} + \cdots + 10091520 \) Copy content Toggle raw display