Properties

Label 9025.2.a.cd
Level $9025$
Weight $2$
Character orbit 9025.a
Self dual yes
Analytic conductor $72.065$
Analytic rank $0$
Dimension $9$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9025 = 5^{2} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9025.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(72.0649878242\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \( x^{9} - 12x^{7} - 4x^{6} + 48x^{5} + 27x^{4} - 72x^{3} - 51x^{2} + 27x + 19 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 95)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 12x^{7} - 4x^{6} + 48x^{5} + 27x^{4} - 72x^{3} - 51x^{2} + 27x + 19 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{6} - \nu^{5} - 8\nu^{4} + 4\nu^{3} + 17\nu^{2} - 2\nu - 7 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{7} - \nu^{6} - 8\nu^{5} + 4\nu^{4} + 17\nu^{3} - 2\nu^{2} - 7\nu \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{8} - \nu^{7} - 10\nu^{6} + 6\nu^{5} + 33\nu^{4} - 10\nu^{3} - 41\nu^{2} + 4\nu + 14 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 2\nu^{8} - 3\nu^{7} - 19\nu^{6} + 21\nu^{5} + 60\nu^{4} - 42\nu^{3} - 72\nu^{2} + 21\nu + 22 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -3\nu^{8} + 5\nu^{7} + 29\nu^{6} - 36\nu^{5} - 96\nu^{4} + 72\nu^{3} + 123\nu^{2} - 32\nu - 38 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( -4\nu^{8} + 6\nu^{7} + 39\nu^{6} - 42\nu^{5} - 129\nu^{4} + 81\nu^{3} + 165\nu^{2} - 33\nu - 53 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{8} + \beta_{7} - \beta_{5} + \beta_{2} + 4\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{8} + 2\beta_{7} + \beta_{6} - \beta_{4} - \beta_{3} + 6\beta_{2} + 7\beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -7\beta_{8} + 9\beta_{7} + 3\beta_{6} - 7\beta_{5} - \beta_{4} - 2\beta_{3} + 9\beta_{2} + 20\beta _1 + 16 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -11\beta_{8} + 21\beta_{7} + 11\beta_{6} - 3\beta_{5} - 9\beta_{4} - 9\beta_{3} + 36\beta_{2} + 45\beta _1 + 60 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -46\beta_{8} + 68\beta_{7} + 31\beta_{6} - 42\beta_{5} - 12\beta_{4} - 21\beta_{3} + 69\beta_{2} + 118\beta _1 + 112 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 91 \beta_{8} + 168 \beta_{7} + 90 \beta_{6} - 39 \beta_{5} - 63 \beta_{4} - 66 \beta_{3} + 228 \beta_{2} + 294 \beta _1 + 349 \) 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
−2.02162
−1.57047
−1.46231
−1.19408
−0.593847
0.719457
1.68361
1.81702
2.62224
−2.02162 −1.90055 2.08694 0 3.84218 −3.09218 −0.175759 0.612075 0
1.2 −1.57047 2.28502 0.466387 0 −3.58856 4.01337 2.40850 2.22131 0
1.3 −1.46231 1.51036 0.138346 0 −2.20860 −4.07172 2.72231 −0.718827 0
1.4 −1.19408 2.27318 −0.574177 0 −2.71435 2.19649 3.07377 2.16734 0
1.5 −0.593847 −1.93003 −1.64735 0 1.14614 −1.06052 2.16596 0.725033 0
1.6 0.719457 −1.23428 −1.48238 0 −0.888013 −1.29194 −2.50542 −1.47655 0
1.7 1.68361 −3.25202 0.834534 0 −5.47512 0.548389 −1.96219 7.57562 0
1.8 1.81702 0.177104 1.30157 0 0.321803 −1.07346 −1.26906 −2.96863 0
1.9 2.62224 −0.928776 4.87613 0 −2.43547 3.83157 7.54188 −2.13737 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(19\) \(-1\)

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 9025.2.a.cd 9
5.b even 2 1 1805.2.a.u 9
19.b odd 2 1 9025.2.a.ce 9
19.f odd 18 2 475.2.l.b 18
95.d odd 2 1 1805.2.a.t 9
95.o odd 18 2 95.2.k.b 18
95.r even 36 4 475.2.u.c 36
285.bf even 18 2 855.2.bs.b 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.k.b 18 95.o odd 18 2
475.2.l.b 18 19.f odd 18 2
475.2.u.c 36 95.r even 36 4
855.2.bs.b 18 285.bf even 18 2
1805.2.a.t 9 95.d odd 2 1
1805.2.a.u 9 5.b even 2 1
9025.2.a.cd 9 1.a even 1 1 trivial
9025.2.a.ce 9 19.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9025))\):

\( T_{2}^{9} - 12T_{2}^{7} - 4T_{2}^{6} + 48T_{2}^{5} + 27T_{2}^{4} - 72T_{2}^{3} - 51T_{2}^{2} + 27T_{2} + 19 \) Copy content Toggle raw display
\( T_{3}^{9} + 3T_{3}^{8} - 12T_{3}^{7} - 37T_{3}^{6} + 39T_{3}^{5} + 147T_{3}^{4} - 6T_{3}^{3} - 186T_{3}^{2} - 75T_{3} + 19 \) Copy content Toggle raw display
\( T_{7}^{9} - 33T_{7}^{7} - 10T_{7}^{6} + 327T_{7}^{5} + 228T_{7}^{4} - 911T_{7}^{3} - 1029T_{7}^{2} + 147T_{7} + 343 \) Copy content Toggle raw display
\( T_{11}^{9} - 36T_{11}^{7} - 40T_{11}^{6} + 282T_{11}^{5} + 321T_{11}^{4} - 517T_{11}^{3} - 108T_{11}^{2} + 135T_{11} - 19 \) Copy content Toggle raw display
\( T_{29}^{9} - 15 T_{29}^{8} - 54 T_{29}^{7} + 1519 T_{29}^{6} - 2274 T_{29}^{5} - 33729 T_{29}^{4} + 67033 T_{29}^{3} + 248334 T_{29}^{2} - 329271 T_{29} - 702001 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} - 12 T^{7} - 4 T^{6} + 48 T^{5} + \cdots + 19 \) Copy content Toggle raw display
$3$ \( T^{9} + 3 T^{8} - 12 T^{7} - 37 T^{6} + \cdots + 19 \) Copy content Toggle raw display
$5$ \( T^{9} \) Copy content Toggle raw display
$7$ \( T^{9} - 33 T^{7} - 10 T^{6} + \cdots + 343 \) Copy content Toggle raw display
$11$ \( T^{9} - 36 T^{7} - 40 T^{6} + 282 T^{5} + \cdots - 19 \) Copy content Toggle raw display
$13$ \( T^{9} - 3 T^{8} - 57 T^{7} + \cdots - 9937 \) Copy content Toggle raw display
$17$ \( T^{9} - 9 T^{8} - 12 T^{7} + 256 T^{6} + \cdots - 1 \) Copy content Toggle raw display
$19$ \( T^{9} \) Copy content Toggle raw display
$23$ \( T^{9} - 114 T^{7} - 170 T^{6} + \cdots + 63197 \) Copy content Toggle raw display
$29$ \( T^{9} - 15 T^{8} - 54 T^{7} + \cdots - 702001 \) Copy content Toggle raw display
$31$ \( T^{9} - 30 T^{8} + 285 T^{7} + \cdots - 996623 \) Copy content Toggle raw display
$37$ \( T^{9} + 30 T^{8} + 267 T^{7} + \cdots + 27721 \) Copy content Toggle raw display
$41$ \( T^{9} - 18 T^{8} - 18 T^{7} + \cdots - 363977 \) Copy content Toggle raw display
$43$ \( T^{9} - 6 T^{8} - 99 T^{7} + \cdots + 10099 \) Copy content Toggle raw display
$47$ \( T^{9} + 21 T^{8} - 60 T^{7} + \cdots - 5721697 \) Copy content Toggle raw display
$53$ \( T^{9} - 9 T^{8} - 147 T^{7} + \cdots + 4387499 \) Copy content Toggle raw display
$59$ \( T^{9} - 27 T^{8} + 138 T^{7} + \cdots - 577711 \) Copy content Toggle raw display
$61$ \( T^{9} - 12 T^{8} - 219 T^{7} + \cdots + 1862369 \) Copy content Toggle raw display
$67$ \( T^{9} + 36 T^{8} + 183 T^{7} + \cdots - 60058259 \) Copy content Toggle raw display
$71$ \( T^{9} + 6 T^{8} - 177 T^{7} + \cdots + 92683 \) Copy content Toggle raw display
$73$ \( T^{9} - 9 T^{8} - 267 T^{7} + \cdots + 1023553 \) Copy content Toggle raw display
$79$ \( T^{9} - 45 T^{8} + 627 T^{7} + \cdots - 17803297 \) Copy content Toggle raw display
$83$ \( T^{9} - 171 T^{7} + 507 T^{6} + \cdots + 9829 \) Copy content Toggle raw display
$89$ \( T^{9} + 9 T^{8} - 474 T^{7} + \cdots - 11971 \) Copy content Toggle raw display
$97$ \( T^{9} + 45 T^{8} + \cdots + 191335897 \) Copy content Toggle raw display
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