Properties

Label 6900.2.f.r
Level $6900$
Weight $2$
Character orbit 6900.f
Analytic conductor $55.097$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 6900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6900.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(55.0967773947\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.158155776.1
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 24x^{3} + 81x^{2} + 54x + 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1380)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + ( - \beta_{5} - \beta_{2}) q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} + ( - \beta_{5} - \beta_{2}) q^{7} - q^{9} + (\beta_1 + 1) q^{11} + (\beta_{4} + \beta_{2}) q^{13} + (\beta_{5} - \beta_{4}) q^{17} + ( - \beta_1 - 3) q^{19} + ( - \beta_{3} + 1) q^{21} + \beta_{2} q^{23} - \beta_{2} q^{27} + (\beta_{3} - \beta_1) q^{29} + (\beta_{3} + 3) q^{31} + ( - \beta_{4} + \beta_{2}) q^{33} + (\beta_{5} - 2 \beta_{4} - \beta_{2}) q^{37} + (\beta_1 - 1) q^{39} + (3 \beta_{3} - \beta_1 + 2) q^{41} + (2 \beta_{5} + 6 \beta_{2}) q^{43} + ( - 2 \beta_{5} + \beta_{4} + \beta_{2}) q^{47} + (\beta_{3} - 2 \beta_1 - 4) q^{49} + (\beta_{3} - \beta_1) q^{51} + ( - 3 \beta_{5} + \beta_{4} + 2 \beta_{2}) q^{53} + (\beta_{4} - 3 \beta_{2}) q^{57} + ( - \beta_{3} + \beta_1) q^{59} + (2 \beta_{3} - \beta_1 + 3) q^{61} + (\beta_{5} + \beta_{2}) q^{63} + (\beta_{5} - 3 \beta_{2}) q^{67} - q^{69} + ( - 3 \beta_{3} + \beta_1 - 2) q^{71} + ( - 3 \beta_{4} + 5 \beta_{2}) q^{73} + ( - 4 \beta_{5} + \beta_{4} + 3 \beta_{2}) q^{77} + ( - 2 \beta_{3} - 4) q^{79} + q^{81} + (\beta_{5} + \beta_{4} + 4 \beta_{2}) q^{83} + ( - \beta_{5} + \beta_{4}) q^{87} + (2 \beta_{3} + 2 \beta_1 - 2) q^{89} + (2 \beta_{3} - \beta_1 + 5) q^{91} + ( - \beta_{5} + 3 \beta_{2}) q^{93} + ( - 4 \beta_{5} + 2 \beta_{4} + 6 \beta_{2}) q^{97} + ( - \beta_1 - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{9} + 8 q^{11} - 20 q^{19} + 4 q^{21} + 20 q^{31} - 4 q^{39} + 16 q^{41} - 26 q^{49} + 20 q^{61} - 6 q^{69} - 16 q^{71} - 28 q^{79} + 6 q^{81} - 4 q^{89} + 32 q^{91} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 2x^{4} + 24x^{3} + 81x^{2} + 54x + 18 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 11\nu^{5} - 34\nu^{4} + 121\nu^{3} + 132\nu^{2} + 66\nu + 1203 ) / 681 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -29\nu^{5} + 69\nu^{4} - 92\nu^{3} - 575\nu^{2} - 2217\nu - 819 ) / 681 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 34\nu^{5} - 167\nu^{4} + 374\nu^{3} + 408\nu^{2} + 204\nu - 2163 ) / 681 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 40\nu^{5} - 103\nu^{4} + 213\nu^{3} + 707\nu^{2} + 3645\nu + 1341 ) / 681 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -123\nu^{5} + 277\nu^{4} - 218\nu^{3} - 3292\nu^{2} - 8229\nu - 3051 ) / 681 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{4} + \beta_{2} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} + 2\beta_{4} + 7\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{5} + 13\beta_{4} - 2\beta_{3} + 29\beta_{2} + 13\beta _1 - 29 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -11\beta_{3} + 34\beta _1 - 95 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 46\beta_{5} - 197\beta_{4} - 46\beta_{3} - 493\beta_{2} + 197\beta _1 - 493 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6900\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(1201\) \(3451\) \(4601\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
6349.1
−1.42234 + 1.42234i
2.79911 2.79911i
−0.376763 + 0.376763i
−0.376763 0.376763i
2.79911 + 2.79911i
−1.42234 1.42234i
0 1.00000i 0 0 0 3.73549i 0 −1.00000 0
6349.2 0 1.00000i 0 0 0 1.52644i 0 −1.00000 0
6349.3 0 1.00000i 0 0 0 4.20905i 0 −1.00000 0
6349.4 0 1.00000i 0 0 0 4.20905i 0 −1.00000 0
6349.5 0 1.00000i 0 0 0 1.52644i 0 −1.00000 0
6349.6 0 1.00000i 0 0 0 3.73549i 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 6349.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6900.2.f.r 6
5.b even 2 1 inner 6900.2.f.r 6
5.c odd 4 1 1380.2.a.j 3
5.c odd 4 1 6900.2.a.x 3
15.e even 4 1 4140.2.a.s 3
20.e even 4 1 5520.2.a.bv 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1380.2.a.j 3 5.c odd 4 1
4140.2.a.s 3 15.e even 4 1
5520.2.a.bv 3 20.e even 4 1
6900.2.a.x 3 5.c odd 4 1
6900.2.f.r 6 1.a even 1 1 trivial
6900.2.f.r 6 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(6900, [\chi])\):

\( T_{7}^{6} + 34T_{7}^{4} + 321T_{7}^{2} + 576 \) Copy content Toggle raw display
\( T_{11}^{3} - 4T_{11}^{2} - 14T_{11} + 48 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 34 T^{4} + 321 T^{2} + \cdots + 576 \) Copy content Toggle raw display
$11$ \( (T^{3} - 4 T^{2} - 14 T + 48)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 40 T^{4} + 276 T^{2} + \cdots + 144 \) Copy content Toggle raw display
$17$ \( T^{6} + 42 T^{4} + 441 T^{2} + \cdots + 324 \) Copy content Toggle raw display
$19$ \( (T^{3} + 10 T^{2} + 14 T - 52)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$29$ \( (T^{3} - 21 T + 18)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 10 T^{2} + 17 T + 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 130 T^{4} + 3537 T^{2} + \cdots + 11664 \) Copy content Toggle raw display
$41$ \( (T^{3} - 8 T^{2} - 101 T + 582)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 216 T^{4} + 10128 T^{2} + \cdots + 92416 \) Copy content Toggle raw display
$47$ \( T^{6} + 116 T^{4} + 4228 T^{2} + \cdots + 46656 \) Copy content Toggle raw display
$53$ \( T^{6} + 266 T^{4} + 19513 T^{2} + \cdots + 338724 \) Copy content Toggle raw display
$59$ \( (T^{3} - 21 T - 18)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 10 T^{2} - 22 T + 292)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 66 T^{4} + 369 T^{2} + \cdots + 16 \) Copy content Toggle raw display
$71$ \( (T^{3} + 8 T^{2} - 101 T - 582)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 456 T^{4} + 58068 T^{2} + \cdots + 2226064 \) Copy content Toggle raw display
$79$ \( (T^{3} + 14 T^{2} - 96)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 134 T^{4} + 4849 T^{2} + \cdots + 51984 \) Copy content Toggle raw display
$89$ \( (T^{3} + 2 T^{2} - 200 T - 912)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 576 T^{4} + 101184 T^{2} + \cdots + 5456896 \) Copy content Toggle raw display
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