Properties

Label 9025.2.a.ce
Level $9025$
Weight $2$
Character orbit 9025.a
Self dual yes
Analytic conductor $72.065$
Analytic rank $1$
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: \(1\)
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
2.62224
1.81702
1.68361
0.719457
−0.593847
−1.19408
−1.46231
−1.57047
−2.02162
−2.62224 0.928776 4.87613 0 −2.43547 3.83157 −7.54188 −2.13737 0
1.2 −1.81702 −0.177104 1.30157 0 0.321803 −1.07346 1.26906 −2.96863 0
1.3 −1.68361 3.25202 0.834534 0 −5.47512 0.548389 1.96219 7.57562 0
1.4 −0.719457 1.23428 −1.48238 0 −0.888013 −1.29194 2.50542 −1.47655 0
1.5 0.593847 1.93003 −1.64735 0 1.14614 −1.06052 −2.16596 0.725033 0
1.6 1.19408 −2.27318 −0.574177 0 −2.71435 2.19649 −3.07377 2.16734 0
1.7 1.46231 −1.51036 0.138346 0 −2.20860 −4.07172 −2.72231 −0.718827 0
1.8 1.57047 −2.28502 0.466387 0 −3.58856 4.01337 −2.40850 2.22131 0
1.9 2.02162 1.90055 2.08694 0 3.84218 −3.09218 0.175759 0.612075 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.ce 9
5.b even 2 1 1805.2.a.t 9
19.b odd 2 1 9025.2.a.cd 9
19.e even 9 2 475.2.l.b 18
95.d odd 2 1 1805.2.a.u 9
95.p even 18 2 95.2.k.b 18
95.q odd 36 4 475.2.u.c 36
285.bd odd 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.p even 18 2
475.2.l.b 18 19.e even 9 2
475.2.u.c 36 95.q odd 36 4
855.2.bs.b 18 285.bd odd 18 2
1805.2.a.t 9 5.b even 2 1
1805.2.a.u 9 95.d odd 2 1
9025.2.a.cd 9 19.b odd 2 1
9025.2.a.ce 9 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9025))\):

\( T_{2}^{9} - 12T_{2}^{7} + 4T_{2}^{6} + 48T_{2}^{5} - 27T_{2}^{4} - 72T_{2}^{3} + 51T_{2}^{2} + 27T_{2} - 19 \) 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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