Properties

Label 2205.4.a.bx
Level $2205$
Weight $4$
Character orbit 2205.a
Self dual yes
Analytic conductor $130.099$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2205.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(130.099211563\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \( x^{6} - 34x^{4} + 241x^{2} - 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{9} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 3) q^{4} - 5 q^{5} + (\beta_{3} + 4 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 3) q^{4} - 5 q^{5} + (\beta_{3} + 4 \beta_1) q^{8} - 5 \beta_1 q^{10} + \beta_{5} q^{11} + ( - \beta_{5} - \beta_{3} - 3 \beta_1) q^{13} + (\beta_{4} + \beta_{2} + 16) q^{16} + ( - \beta_{4} - 7) q^{17} + ( - 2 \beta_{5} + 3 \beta_{3} - \beta_1) q^{19} + ( - 5 \beta_{2} - 15) q^{20} + (\beta_{4} + 2 \beta_{2} - 5) q^{22} + (\beta_{5} - 4 \beta_{3} + 6 \beta_1) q^{23} + 25 q^{25} + ( - 2 \beta_{4} - 10 \beta_{2} - 24) q^{26} + (2 \beta_{5} - \beta_{3} + 17 \beta_1) q^{29} + (2 \beta_{5} - 8 \beta_{3} - 34 \beta_1) q^{31} + (2 \beta_{5} - 3 \beta_1) q^{32} + ( - 2 \beta_{5} - 7 \beta_{3} - 11 \beta_1) q^{34} + ( - 3 \beta_{4} - 8 \beta_{2} + 67) q^{37} + (\beta_{4} + 10 \beta_{2} - 13) q^{38} + ( - 5 \beta_{3} - 20 \beta_1) q^{40} + (2 \beta_{4} - 20 \beta_{2} - 96) q^{41} + ( - \beta_{4} - 4 \beta_{2} - 53) q^{43} + ( - 6 \beta_{5} + 9 \beta_{3} + 17 \beta_1) q^{44} + ( - 3 \beta_{4} - 12 \beta_{2} + 77) q^{46} + (5 \beta_{4} - 20 \beta_{2} - 15) q^{47} + 25 \beta_1 q^{50} + (4 \beta_{5} - 16 \beta_{3} - 98 \beta_1) q^{52} + ( - 7 \beta_{5} + 14 \beta_{3} - 10 \beta_1) q^{53} - 5 \beta_{5} q^{55} + (\beta_{4} + 16 \beta_{2} + 181) q^{58} + (6 \beta_{4} + 40 \beta_{2} - 38) q^{59} + (15 \beta_{3} - 75 \beta_1) q^{61} + ( - 6 \beta_{4} - 70 \beta_{2} - 352) q^{62} + ( - 6 \beta_{4} - 7 \beta_{2} - 171) q^{64} + (5 \beta_{5} + 5 \beta_{3} + 15 \beta_1) q^{65} + (11 \beta_{4} + 8 \beta_{2} + 179) q^{67} + ( - \beta_{4} - 50 \beta_{2} - 27) q^{68} + ( - 5 \beta_{5} - \beta_{3} - 43 \beta_1) q^{71} + ( - 19 \beta_{5} + 21 \beta_{3} - 137 \beta_1) q^{73} + ( - 6 \beta_{5} - 29 \beta_{3} - 17 \beta_1) q^{74} + (18 \beta_{5} - 7 \beta_{3} + 89 \beta_1) q^{76} + (2 \beta_{4} - 4 \beta_{2} + 234) q^{79} + ( - 5 \beta_{4} - 5 \beta_{2} - 80) q^{80} + (4 \beta_{5} - 6 \beta_{3} - 268 \beta_1) q^{82} + (2 \beta_{4} - 60 \beta_{2} - 466) q^{83} + (5 \beta_{4} + 35) q^{85} + ( - 2 \beta_{5} - 11 \beta_{3} - 93 \beta_1) q^{86} + ( - 5 \beta_{4} + 34 \beta_{2} + 221) q^{88} + (2 \beta_{4} - 80 \beta_{2} - 36) q^{89} + ( - 14 \beta_{5} - \beta_{3} - 91 \beta_1) q^{92} + (10 \beta_{5} + 15 \beta_{3} - 175 \beta_1) q^{94} + (10 \beta_{5} - 15 \beta_{3} + 5 \beta_1) q^{95} + (11 \beta_{5} + \beta_{3} + 203 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 20 q^{4} - 30 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 20 q^{4} - 30 q^{5} + 100 q^{16} - 44 q^{17} - 100 q^{20} - 24 q^{22} + 150 q^{25} - 168 q^{26} + 380 q^{37} - 56 q^{38} - 612 q^{41} - 328 q^{43} + 432 q^{46} - 120 q^{47} + 1120 q^{58} - 136 q^{59} - 2264 q^{62} - 1052 q^{64} + 1112 q^{67} - 264 q^{68} + 1400 q^{79} - 500 q^{80} - 2912 q^{83} + 220 q^{85} + 1384 q^{88} - 372 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 34x^{4} + 241x^{2} - 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 11 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 20\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 25\nu^{2} + 59 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} - 32\nu^{3} + 195\nu ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 20\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 25\beta_{2} + 216 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{5} + 32\beta_{3} + 445\beta_1 \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−4.91092
−3.09945
−0.525584
0.525584
3.09945
4.91092
−4.91092 0 16.1171 −5.00000 0 0 −39.8626 0 24.5546
1.2 −3.09945 0 1.60662 −5.00000 0 0 19.8160 0 15.4973
1.3 −0.525584 0 −7.72376 −5.00000 0 0 8.26415 0 2.62792
1.4 0.525584 0 −7.72376 −5.00000 0 0 −8.26415 0 −2.62792
1.5 3.09945 0 1.60662 −5.00000 0 0 −19.8160 0 −15.4973
1.6 4.91092 0 16.1171 −5.00000 0 0 39.8626 0 −24.5546
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(1\)
\(7\) \(-1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2205.4.a.bx 6
3.b odd 2 1 2205.4.a.by yes 6
7.b odd 2 1 2205.4.a.by yes 6
21.c even 2 1 inner 2205.4.a.bx 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2205.4.a.bx 6 1.a even 1 1 trivial
2205.4.a.bx 6 21.c even 2 1 inner
2205.4.a.by yes 6 3.b odd 2 1
2205.4.a.by yes 6 7.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2205))\):

\( T_{2}^{6} - 34T_{2}^{4} + 241T_{2}^{2} - 64 \) Copy content Toggle raw display
\( T_{11}^{6} - 3512T_{11}^{4} + 2814992T_{11}^{2} - 335622400 \) Copy content Toggle raw display
\( T_{13}^{6} - 6744T_{13}^{4} + 14726800T_{13}^{2} - 10404000000 \) Copy content Toggle raw display
\( T_{17}^{3} + 22T_{17}^{2} - 5860T_{17} - 216600 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - 34 T^{4} + 241 T^{2} + \cdots - 64 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( (T + 5)^{6} \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( T^{6} - 3512 T^{4} + \cdots - 335622400 \) Copy content Toggle raw display
$13$ \( T^{6} - 6744 T^{4} + \cdots - 10404000000 \) Copy content Toggle raw display
$17$ \( (T^{3} + 22 T^{2} - 5860 T - 216600)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} - 19176 T^{4} + \cdots - 23592960000 \) Copy content Toggle raw display
$23$ \( T^{6} - 23824 T^{4} + \cdots - 183759683584 \) Copy content Toggle raw display
$29$ \( T^{6} - 21856 T^{4} + \cdots - 53084160000 \) Copy content Toggle raw display
$31$ \( T^{6} - 127496 T^{4} + \cdots - 19948728960000 \) Copy content Toggle raw display
$37$ \( (T^{3} - 190 T^{2} - 52804 T - 193800)^{2} \) Copy content Toggle raw display
$41$ \( (T^{3} + 306 T^{2} - 48260 T - 15501000)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} + 164 T^{2} + 400 T - 369600)^{2} \) Copy content Toggle raw display
$47$ \( (T^{3} + 60 T^{2} - 201200 T - 17640000)^{2} \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots - 379858852641024 \) Copy content Toggle raw display
$59$ \( (T^{3} + 68 T^{2} - 460240 T - 68808000)^{2} \) Copy content Toggle raw display
$61$ \( T^{6} - 554400 T^{4} + \cdots - 82301184000000 \) Copy content Toggle raw display
$67$ \( (T^{3} - 556 T^{2} - 639936 T + 377458560)^{2} \) Copy content Toggle raw display
$71$ \( T^{6} - 153944 T^{4} + \cdots - 16881580038400 \) Copy content Toggle raw display
$73$ \( T^{6} - 1972824 T^{4} + \cdots - 87\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{3} - 700 T^{2} + 137408 T - 6055424)^{2} \) Copy content Toggle raw display
$83$ \( (T^{3} + 1456 T^{2} + 170000 T - 185865600)^{2} \) Copy content Toggle raw display
$89$ \( (T^{3} + 186 T^{2} - 926900 T - 95685000)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} - 1787864 T^{4} + \cdots - 34\!\cdots\!00 \) Copy content Toggle raw display
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