↑

Properties

Label 6790.2.a.bd
Level $6790$
Weight $2$
Character orbit 6790.a
Self dual yes
Analytic conductor $54.218$
Analytic rank $0$
Dimension $15$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6790,2,Mod(1,6790)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6790, 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("6790.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6790 = 2 \cdot 5 \cdot 7 \cdot 97 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6790.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: \(54.2184229724\)
Analytic rank: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - 2 x^{14} - 33 x^{13} + 62 x^{12} + 417 x^{11} - 720 x^{10} - 2524 x^{9} + 3856 x^{8} + \cdots + 32 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + \beta_1 q^{3} + q^{4} - q^{5} + \beta_1 q^{6} + q^{7} + q^{8} + (\beta_{2} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + \beta_1 q^{3} + q^{4} - q^{5} + \beta_1 q^{6} + q^{7} + q^{8} + (\beta_{2} + 2) q^{9} - q^{10} + ( - \beta_{10} + 1) q^{11} + \beta_1 q^{12} - \beta_{7} q^{13} + q^{14} - \beta_1 q^{15} + q^{16} + (\beta_{3} + 1) q^{17} + (\beta_{2} + 2) q^{18} + \beta_{13} q^{19} - q^{20} + \beta_1 q^{21} + ( - \beta_{10} + 1) q^{22} + (\beta_{14} + \beta_{9} + 2) q^{23} + \beta_1 q^{24} + q^{25} - \beta_{7} q^{26} + (\beta_{9} - \beta_{8} + \beta_{4} + \cdots + 1) q^{27}+ \cdots + (\beta_{14} - \beta_{12} - \beta_{9} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q + 15 q^{2} + 2 q^{3} + 15 q^{4} - 15 q^{5} + 2 q^{6} + 15 q^{7} + 15 q^{8} + 25 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q + 15 q^{2} + 2 q^{3} + 15 q^{4} - 15 q^{5} + 2 q^{6} + 15 q^{7} + 15 q^{8} + 25 q^{9} - 15 q^{10} + 13 q^{11} + 2 q^{12} + 3 q^{13} + 15 q^{14} - 2 q^{15} + 15 q^{16} + 11 q^{17} + 25 q^{18} + 4 q^{19} - 15 q^{20} + 2 q^{21} + 13 q^{22} + 21 q^{23} + 2 q^{24} + 15 q^{25} + 3 q^{26} + 8 q^{27} + 15 q^{28} + 18 q^{29} - 2 q^{30} + 4 q^{31} + 15 q^{32} + 5 q^{33} + 11 q^{34} - 15 q^{35} + 25 q^{36} + 16 q^{37} + 4 q^{38} + 12 q^{39} - 15 q^{40} + 3 q^{41} + 2 q^{42} + 17 q^{43} + 13 q^{44} - 25 q^{45} + 21 q^{46} + 4 q^{47} + 2 q^{48} + 15 q^{49} + 15 q^{50} + 17 q^{51} + 3 q^{52} + 30 q^{53} + 8 q^{54} - 13 q^{55} + 15 q^{56} + 10 q^{57} + 18 q^{58} + 10 q^{59} - 2 q^{60} - 5 q^{61} + 4 q^{62} + 25 q^{63} + 15 q^{64} - 3 q^{65} + 5 q^{66} + 25 q^{67} + 11 q^{68} - 26 q^{69} - 15 q^{70} + 44 q^{71} + 25 q^{72} + 11 q^{73} + 16 q^{74} + 2 q^{75} + 4 q^{76} + 13 q^{77} + 12 q^{78} + 3 q^{79} - 15 q^{80} + 63 q^{81} + 3 q^{82} + 4 q^{83} + 2 q^{84} - 11 q^{85} + 17 q^{86} - 6 q^{87} + 13 q^{88} + q^{89} - 25 q^{90} + 3 q^{91} + 21 q^{92} + 16 q^{93} + 4 q^{94} - 4 q^{95} + 2 q^{96} + 15 q^{97} + 15 q^{98} - 5 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 2 x^{14} - 33 x^{13} + 62 x^{12} + 417 x^{11} - 720 x^{10} - 2524 x^{9} + 3856 x^{8} + \cdots + 32 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 9356921 \nu^{14} + 36262578 \nu^{13} + 266610795 \nu^{12} - 1195884924 \nu^{11} + \cdots + 21654015976 ) / 5031676936 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5189134 \nu^{14} - 14299208 \nu^{13} - 213943466 \nu^{12} + 449885735 \nu^{11} + \cdots + 4736940876 ) / 2515838468 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 11612265 \nu^{14} - 36969284 \nu^{13} - 352278663 \nu^{12} + 1126054514 \nu^{11} + \cdots + 9181046256 ) / 5031676936 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1698387 \nu^{14} - 1455080 \nu^{13} + 58746171 \nu^{12} + 45589932 \nu^{11} - 791048199 \nu^{10} + \cdots + 29944872 ) / 387052072 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 27928803 \nu^{14} - 46618770 \nu^{13} - 963353743 \nu^{12} + 1526512676 \nu^{11} + \cdots + 3006200368 ) / 5031676936 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 59442299 \nu^{14} + 81761768 \nu^{13} + 1943048327 \nu^{12} - 2427482096 \nu^{11} + \cdots + 18597796336 ) / 5031676936 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 69820567 \nu^{14} + 110360184 \nu^{13} + 2370935259 \nu^{12} - 3327253566 \nu^{11} + \cdots + 4092237648 ) / 5031676936 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 108904417 \nu^{14} - 216910342 \nu^{13} - 3579431691 \nu^{12} + 6622650164 \nu^{11} + \cdots - 5644803832 ) / 5031676936 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 5016372 \nu^{14} - 6993810 \nu^{13} - 170011315 \nu^{12} + 212368052 \nu^{11} + 2227894640 \nu^{10} + \cdots - 41751828 ) / 228712588 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 9035887 \nu^{14} + 15508482 \nu^{13} + 299346663 \nu^{12} - 477704996 \nu^{11} + \cdots - 538245528 ) / 387052072 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 59919980 \nu^{14} + 98438153 \nu^{13} + 2019356086 \nu^{12} - 3016319189 \nu^{11} + \cdots - 6020141440 ) / 2515838468 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 94191509 \nu^{14} - 142882465 \nu^{13} - 3161248904 \nu^{12} + 4337267662 \nu^{11} + \cdots - 16349879296 ) / 2515838468 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} - \beta_{8} + \beta_{4} + 8\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{13} + 2\beta_{10} + 2\beta_{9} + \beta_{7} - \beta_{6} + \beta_{4} + 2\beta_{3} + 10\beta_{2} + 42 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 3 \beta_{14} + 2 \beta_{13} + 4 \beta_{11} + 2 \beta_{10} + 12 \beta_{9} - 14 \beta_{8} + \beta_{7} + \cdots + 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 3 \beta_{14} + 21 \beta_{13} - 2 \beta_{12} + 4 \beta_{11} + 30 \beta_{10} + 36 \beta_{9} + 16 \beta_{7} + \cdots + 386 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 54 \beta_{14} + 35 \beta_{13} + 2 \beta_{12} + 76 \beta_{11} + 38 \beta_{10} + 136 \beta_{9} + \cdots + 136 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 53 \beta_{14} + 308 \beta_{13} - 48 \beta_{12} + 100 \beta_{11} + 360 \beta_{10} + 492 \beta_{9} + \cdots + 3712 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 721 \beta_{14} + 471 \beta_{13} + 50 \beta_{12} + 1072 \beta_{11} + 554 \beta_{10} + 1566 \beta_{9} + \cdots + 1610 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 680 \beta_{14} + 3977 \beta_{13} - 754 \beta_{12} + 1712 \beta_{11} + 4070 \beta_{10} + 6138 \beta_{9} + \cdots + 36862 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 8651 \beta_{14} + 5832 \beta_{13} + 804 \beta_{12} + 13556 \beta_{11} + 7316 \beta_{10} + \cdots + 19728 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 7675 \beta_{14} + 48347 \beta_{13} - 10014 \beta_{12} + 24972 \beta_{11} + 45230 \beta_{10} + \cdots + 375210 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 98824 \beta_{14} + 69945 \beta_{13} + 10706 \beta_{12} + 162804 \beta_{11} + 91842 \beta_{10} + \cdots + 245550 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 80641 \beta_{14} + 568852 \beta_{13} - 122464 \beta_{12} + 334420 \beta_{11} + 500980 \beta_{10} + \cdots + 3893956 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−3.22551
−2.97641
−2.54371
−2.00862
−1.13078
−0.925257
−0.0749568
0.151712
0.569572
0.766858
1.85766
2.47014
2.81151
2.88486
3.37294
1.00000 −3.22551 1.00000 −1.00000 −3.22551 1.00000 1.00000 7.40394 −1.00000
1.2 1.00000 −2.97641 1.00000 −1.00000 −2.97641 1.00000 1.00000 5.85899 −1.00000
1.3 1.00000 −2.54371 1.00000 −1.00000 −2.54371 1.00000 1.00000 3.47045 −1.00000
1.4 1.00000 −2.00862 1.00000 −1.00000 −2.00862 1.00000 1.00000 1.03456 −1.00000
1.5 1.00000 −1.13078 1.00000 −1.00000 −1.13078 1.00000 1.00000 −1.72134 −1.00000
1.6 1.00000 −0.925257 1.00000 −1.00000 −0.925257 1.00000 1.00000 −2.14390 −1.00000
1.7 1.00000 −0.0749568 1.00000 −1.00000 −0.0749568 1.00000 1.00000 −2.99438 −1.00000
1.8 1.00000 0.151712 1.00000 −1.00000 0.151712 1.00000 1.00000 −2.97698 −1.00000
1.9 1.00000 0.569572 1.00000 −1.00000 0.569572 1.00000 1.00000 −2.67559 −1.00000
1.10 1.00000 0.766858 1.00000 −1.00000 0.766858 1.00000 1.00000 −2.41193 −1.00000
1.11 1.00000 1.85766 1.00000 −1.00000 1.85766 1.00000 1.00000 0.450890 −1.00000
1.12 1.00000 2.47014 1.00000 −1.00000 2.47014 1.00000 1.00000 3.10159 −1.00000
1.13 1.00000 2.81151 1.00000 −1.00000 2.81151 1.00000 1.00000 4.90458 −1.00000
1.14 1.00000 2.88486 1.00000 −1.00000 2.88486 1.00000 1.00000 5.32240 −1.00000
1.15 1.00000 3.37294 1.00000 −1.00000 3.37294 1.00000 1.00000 8.37671 −1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(7\) \(-1\)
\(97\) \(-1\)

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 6790.2.a.bd ✓ 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6790.2.a.bd ✓ 15 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6790))\):

\( T_{3}^{15} - 2 T_{3}^{14} - 33 T_{3}^{13} + 62 T_{3}^{12} + 417 T_{3}^{11} - 720 T_{3}^{10} - 2524 T_{3}^{9} + \cdots + 32 \) Copy content Toggle raw display
\( T_{11}^{15} - 13 T_{11}^{14} - 14 T_{11}^{13} + 785 T_{11}^{12} - 1787 T_{11}^{11} - 15808 T_{11}^{10} + \cdots - 22464 \) Copy content Toggle raw display
\( T_{23}^{15} - 21 T_{23}^{14} + 71 T_{23}^{13} + 1614 T_{23}^{12} - 17419 T_{23}^{11} + 41155 T_{23}^{10} + \cdots - 34752 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{15} \) Copy content Toggle raw display
$3$ \( T^{15} - 2 T^{14} + \cdots + 32 \) Copy content Toggle raw display
$5$ \( (T + 1)^{15} \) Copy content Toggle raw display
$7$ \( (T - 1)^{15} \) Copy content Toggle raw display
$11$ \( T^{15} - 13 T^{14} + \cdots - 22464 \) Copy content Toggle raw display
$13$ \( T^{15} - 3 T^{14} + \cdots - 704 \) Copy content Toggle raw display
$17$ \( T^{15} - 11 T^{14} + \cdots + 78936192 \) Copy content Toggle raw display
$19$ \( T^{15} - 4 T^{14} + \cdots + 2571008 \) Copy content Toggle raw display
$23$ \( T^{15} - 21 T^{14} + \cdots - 34752 \) Copy content Toggle raw display
$29$ \( T^{15} + \cdots - 2400813024 \) Copy content Toggle raw display
$31$ \( T^{15} - 4 T^{14} + \cdots - 2523136 \) Copy content Toggle raw display
$37$ \( T^{15} + \cdots + 11550907904 \) Copy content Toggle raw display
$41$ \( T^{15} + \cdots + 15202231968 \) Copy content Toggle raw display
$43$ \( T^{15} + \cdots - 3891041888 \) Copy content Toggle raw display
$47$ \( T^{15} + \cdots - 88625032848 \) Copy content Toggle raw display
$53$ \( T^{15} + \cdots + 3558400512 \) Copy content Toggle raw display
$59$ \( T^{15} + \cdots + 6533615616 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots + 14140238704 \) Copy content Toggle raw display
$67$ \( T^{15} + \cdots + 44308224896 \) Copy content Toggle raw display
$71$ \( T^{15} + \cdots - 40287467470848 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots - 182853077632 \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots + 6477272128 \) Copy content Toggle raw display
$83$ \( T^{15} - 4 T^{14} + \cdots - 1022208 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots + 512785421184 \) Copy content Toggle raw display
$97$ \( (T - 1)^{15} \) Copy content Toggle raw display
show more
show less