↑

Properties

Label 6790.2.a.ba
Level $6790$
Weight $2$
Character orbit 6790.a
Self dual yes
Analytic conductor $54.218$
Analytic rank $1$
Dimension $14$
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: \(1\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 30 x^{12} - 2 x^{11} + 340 x^{10} + 30 x^{9} - 1821 x^{8} - 106 x^{7} + 4638 x^{6} - 172 x^{5} + \cdots - 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3 \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 30 x^{12} - 2 x^{11} + 340 x^{10} + 30 x^{9} - 1821 x^{8} - 106 x^{7} + 4638 x^{6} - 172 x^{5} + \cdots - 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 42409 \nu^{13} - 37238 \nu^{12} - 1256418 \nu^{11} + 894090 \nu^{10} + 14009944 \nu^{9} + \cdots - 14225016 ) / 3656304 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 33092 \nu^{13} - 4897 \nu^{12} - 960315 \nu^{11} + 6810 \nu^{10} + 10418018 \nu^{9} + \cdots + 2799144 ) / 914076 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 90825 \nu^{13} + 85778 \nu^{12} - 2826088 \nu^{11} - 2465826 \nu^{10} + 33209832 \nu^{9} + \cdots - 6896264 ) / 1828152 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 108097 \nu^{13} - 25246 \nu^{12} + 3168234 \nu^{11} + 1017222 \nu^{10} - 34728832 \nu^{9} + \cdots - 6067464 ) / 1828152 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 100587 \nu^{13} - 37018 \nu^{12} + 2979698 \nu^{11} + 1365786 \nu^{10} - 33113568 \nu^{9} + \cdots - 6648248 ) / 1218768 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 323321 \nu^{13} + 21946 \nu^{12} - 9631006 \nu^{11} - 1389438 \nu^{10} + 108025280 \nu^{9} + \cdots - 14145176 ) / 3656304 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 360223 \nu^{13} + 99286 \nu^{12} - 10766118 \nu^{11} - 3729138 \nu^{10} + 121054600 \nu^{9} + \cdots + 8104392 ) / 3656304 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 102222 \nu^{13} + 20627 \nu^{12} - 3058408 \nu^{11} - 863088 \nu^{10} + 34477080 \nu^{9} + \cdots + 2072464 ) / 914076 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 461465 \nu^{13} + 213986 \nu^{12} - 13848474 \nu^{11} - 7125150 \nu^{10} + 156300296 \nu^{9} + \cdots + 15604296 ) / 3656304 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 204261 \nu^{13} + 93824 \nu^{12} - 6142948 \nu^{11} - 3147126 \nu^{10} + 69503736 \nu^{9} + \cdots + 12653428 ) / 914076 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 219179 \nu^{13} + 80434 \nu^{12} - 6547489 \nu^{11} - 2827422 \nu^{10} + 73502618 \nu^{9} + \cdots + 7664632 ) / 914076 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} - \beta_{8} + \beta_{7} - \beta_{6} + \beta_{2} + 8\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 2 \beta_{13} + 2 \beta_{12} - \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} - \beta_{6} + \beta_{4} + \cdots + 27 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 2 \beta_{13} - \beta_{11} + 14 \beta_{10} + 5 \beta_{9} - 16 \beta_{8} + 14 \beta_{7} - 16 \beta_{6} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 34 \beta_{13} + 29 \beta_{12} - 9 \beta_{11} - 12 \beta_{10} + 23 \beta_{9} + 8 \beta_{8} + \cdots + 210 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 42 \beta_{13} + 3 \beta_{12} - 18 \beta_{11} + 162 \beta_{10} + 94 \beta_{9} - 200 \beta_{8} + \cdots + 19 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 440 \beta_{13} + 339 \beta_{12} - 62 \beta_{11} - 105 \beta_{10} + 352 \beta_{9} + 19 \beta_{8} + \cdots + 1801 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 656 \beta_{13} + 84 \beta_{12} - 236 \beta_{11} + 1789 \beta_{10} + 1328 \beta_{9} - 2311 \beta_{8} + \cdots + 308 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 5212 \beta_{13} + 3736 \beta_{12} - 387 \beta_{11} - 741 \beta_{10} + 4655 \beta_{9} - 565 \beta_{8} + \cdots + 16591 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 9062 \beta_{13} + 1574 \beta_{12} - 2753 \beta_{11} + 19448 \beta_{10} + 16913 \beta_{9} + \cdots + 4807 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 59686 \beta_{13} + 40463 \beta_{12} - 2343 \beta_{11} - 3386 \beta_{10} + 57617 \beta_{9} + \cdots + 161002 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 117198 \beta_{13} + 24835 \beta_{12} - 30530 \beta_{11} + 210218 \beta_{10} + 205222 \beta_{9} + \cdots + 71915 \) 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
3.38590
2.66092
2.53214
1.84888
1.25856
0.386022
0.341448
−0.0817552
−0.398220
−1.76292
−2.11470
−2.25605
−2.65584
−3.14438
−1.00000 −3.38590 1.00000 −1.00000 3.38590 1.00000 −1.00000 8.46433 1.00000
1.2 −1.00000 −2.66092 1.00000 −1.00000 2.66092 1.00000 −1.00000 4.08048 1.00000
1.3 −1.00000 −2.53214 1.00000 −1.00000 2.53214 1.00000 −1.00000 3.41175 1.00000
1.4 −1.00000 −1.84888 1.00000 −1.00000 1.84888 1.00000 −1.00000 0.418360 1.00000
1.5 −1.00000 −1.25856 1.00000 −1.00000 1.25856 1.00000 −1.00000 −1.41603 1.00000
1.6 −1.00000 −0.386022 1.00000 −1.00000 0.386022 1.00000 −1.00000 −2.85099 1.00000
1.7 −1.00000 −0.341448 1.00000 −1.00000 0.341448 1.00000 −1.00000 −2.88341 1.00000
1.8 −1.00000 0.0817552 1.00000 −1.00000 −0.0817552 1.00000 −1.00000 −2.99332 1.00000
1.9 −1.00000 0.398220 1.00000 −1.00000 −0.398220 1.00000 −1.00000 −2.84142 1.00000
1.10 −1.00000 1.76292 1.00000 −1.00000 −1.76292 1.00000 −1.00000 0.107898 1.00000
1.11 −1.00000 2.11470 1.00000 −1.00000 −2.11470 1.00000 −1.00000 1.47196 1.00000
1.12 −1.00000 2.25605 1.00000 −1.00000 −2.25605 1.00000 −1.00000 2.08977 1.00000
1.13 −1.00000 2.65584 1.00000 −1.00000 −2.65584 1.00000 −1.00000 4.05350 1.00000
1.14 −1.00000 3.14438 1.00000 −1.00000 −3.14438 1.00000 −1.00000 6.88712 1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
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.ba ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6790.2.a.ba ✓ 14 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6790))\):

\( T_{3}^{14} - 30 T_{3}^{12} + 2 T_{3}^{11} + 340 T_{3}^{10} - 30 T_{3}^{9} - 1821 T_{3}^{8} + 106 T_{3}^{7} + \cdots - 16 \) Copy content Toggle raw display
\( T_{11}^{14} + 6 T_{11}^{13} - 68 T_{11}^{12} - 439 T_{11}^{11} + 1494 T_{11}^{10} + 11944 T_{11}^{9} + \cdots + 1294848 \) Copy content Toggle raw display
\( T_{23}^{14} + 20 T_{23}^{13} + 26 T_{23}^{12} - 1868 T_{23}^{11} - 11387 T_{23}^{10} + 35328 T_{23}^{9} + \cdots + 9819648 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{14} \) Copy content Toggle raw display
$3$ \( T^{14} - 30 T^{12} + \cdots - 16 \) Copy content Toggle raw display
$5$ \( (T + 1)^{14} \) Copy content Toggle raw display
$7$ \( (T - 1)^{14} \) Copy content Toggle raw display
$11$ \( T^{14} + 6 T^{13} + \cdots + 1294848 \) Copy content Toggle raw display
$13$ \( T^{14} + 4 T^{13} + \cdots - 88935424 \) Copy content Toggle raw display
$17$ \( T^{14} + 10 T^{13} + \cdots + 198144 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots - 233344768 \) Copy content Toggle raw display
$23$ \( T^{14} + 20 T^{13} + \cdots + 9819648 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 3078563328 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 2514255872 \) Copy content Toggle raw display
$37$ \( T^{14} + 22 T^{13} + \cdots + 54073856 \) Copy content Toggle raw display
$41$ \( T^{14} + 14 T^{13} + \cdots + 20172816 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 300617984 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 6119399232 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 104656896 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 34083569664 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots - 48746480128 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots - 201105903136 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots - 407337172992 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots - 145223166208 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots - 2906389952 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots - 1536653435904 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 6619723776 \) Copy content Toggle raw display
$97$ \( (T - 1)^{14} \) Copy content Toggle raw display
show more
show less