Properties

Label 9025.2.a.cf
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: \( x^{9} - 3x^{8} - 6x^{7} + 16x^{6} + 12x^{5} - 27x^{4} - 8x^{3} + 15x^{2} - 1 \) 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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} + ( - \beta_{7} + 1) q^{3} + (\beta_{6} + \beta_{4}) q^{4} + ( - \beta_{7} + \beta_{3} - \beta_1 + 2) q^{6} + (\beta_{7} + \beta_{6} + \beta_{3} - \beta_{2}) q^{7} + ( - \beta_{8} + \beta_{7} + \beta_{4} - \beta_{3} - \beta_1) q^{8} + ( - 2 \beta_{7} + \beta_{5} + \beta_{3} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{2} + ( - \beta_{7} + 1) q^{3} + (\beta_{6} + \beta_{4}) q^{4} + ( - \beta_{7} + \beta_{3} - \beta_1 + 2) q^{6} + (\beta_{7} + \beta_{6} + \beta_{3} - \beta_{2}) q^{7} + ( - \beta_{8} + \beta_{7} + \beta_{4} - \beta_{3} - \beta_1) q^{8} + ( - 2 \beta_{7} + \beta_{5} + \beta_{3} + 1) q^{9} + (\beta_{8} + \beta_{7} + \beta_{6} + \beta_{4} - \beta_{2} - \beta_1) q^{11} + (\beta_{6} + \beta_{4} + 2 \beta_{3} - \beta_{2} + 2) q^{12} + (\beta_{8} - \beta_{7} + \beta_{6} + 2 \beta_{5} + \beta_{4} + 2 \beta_1) q^{13} + ( - \beta_{8} + \beta_{7} + \beta_{5} - 2 \beta_{2} - \beta_1) q^{14} + ( - \beta_{8} + 2 \beta_{7} - \beta_{3} - \beta_1 + 1) q^{16} + (\beta_{8} + \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} + 1) q^{17} + ( - 3 \beta_{7} + \beta_{6} + \beta_{5} + 3 \beta_{3} - \beta_{2} + \beta_1 + 3) q^{18} + ( - \beta_{8} + 2 \beta_{7} + 2 \beta_{6} + \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - 2) q^{21} + (2 \beta_{7} - \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - 2 \beta_1 + 2) q^{22} + (\beta_{8} + 2 \beta_{5} + 2 \beta_{4} + \beta_{3} - 2 \beta_{2} + \beta_1 + 1) q^{23} + ( - \beta_{8} + \beta_{7} + \beta_{4} - 3 \beta_{2} - \beta_1) q^{24} + ( - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - 3 \beta_1 + 1) q^{26} + ( - \beta_{7} + \beta_{6} + 3 \beta_{5} - \beta_{4} + 3 \beta_{3} - \beta_{2} - \beta_1 + 4) q^{27} + ( - \beta_{8} - \beta_{7} + \beta_{6} + 2 \beta_{5} + \beta_{4} - \beta_{2} + 2 \beta_1) q^{28} + (2 \beta_{8} - \beta_{7} + 2 \beta_{5} + 2 \beta_{3} + \beta_1) q^{29} + (\beta_{8} + \beta_{6} + 2 \beta_{5} - \beta_{4} - \beta_{2} - 2 \beta_1 - 1) q^{31} + (\beta_{8} + \beta_{7} + 2 \beta_{6} + \beta_{5} - \beta_{4} + \beta_1) q^{32} + (2 \beta_{7} - 3 \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 - 1) q^{33} + ( - \beta_{6} - \beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_{2} - 3 \beta_1 + 3) q^{34} + ( - \beta_{8} - \beta_{7} - \beta_{4} + 4 \beta_{3} - 4 \beta_{2} - 2 \beta_1 + 4) q^{36} + (\beta_{8} + \beta_{7} - 2 \beta_{6} - 3 \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - 3 \beta_1 + 4) q^{37} + (\beta_{8} - 3 \beta_{7} + \beta_{5} - \beta_{4} + 2 \beta_{2} + 2 \beta_1 + 2) q^{39} + ( - \beta_{7} - \beta_{5} + 2 \beta_{4} + 2 \beta_{2} + 2 \beta_1 - 2) q^{41} + ( - 3 \beta_{8} + 2 \beta_{7} + \beta_{6} + 2 \beta_{5} + \beta_{4} + \beta_{3} - 5 \beta_{2} + \cdots - 1) q^{42}+ \cdots + ( - 2 \beta_{8} + 2 \beta_{7} - 2 \beta_{6} - 7 \beta_{5} + \beta_{4} + \beta_{2} - \beta_1 - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 6 q^{2} + 9 q^{3} + 6 q^{4} + 12 q^{6} + 6 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 6 q^{2} + 9 q^{3} + 6 q^{4} + 12 q^{6} + 6 q^{8} + 6 q^{9} + 18 q^{12} + 9 q^{13} + 12 q^{16} + 9 q^{17} + 24 q^{18} - 12 q^{21} + 24 q^{22} + 12 q^{23} + 3 q^{24} - 3 q^{26} + 24 q^{27} + 15 q^{28} - 9 q^{29} - 18 q^{31} + 3 q^{32} - 9 q^{33} + 24 q^{34} + 18 q^{36} + 18 q^{37} + 18 q^{39} - 6 q^{41} + 12 q^{43} + 48 q^{44} + 9 q^{46} - 15 q^{47} - 21 q^{48} - 9 q^{49} + 6 q^{51} + 33 q^{52} + 15 q^{53} + 63 q^{54} - 6 q^{58} - 21 q^{59} - 12 q^{61} + 36 q^{62} - 21 q^{63} - 36 q^{64} + 3 q^{66} + 60 q^{67} + 51 q^{68} + 15 q^{69} + 18 q^{71} - 27 q^{73} + 27 q^{74} + 30 q^{77} - 6 q^{78} - 15 q^{79} + 33 q^{81} - 24 q^{82} + 48 q^{84} + 39 q^{86} - 15 q^{87} + 27 q^{88} + 39 q^{89} - 21 q^{91} + 6 q^{92} - 15 q^{93} - 15 q^{94} - 33 q^{96} + 15 q^{97} - 15 q^{98} - 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 3x^{8} - 6x^{7} + 16x^{6} + 12x^{5} - 27x^{4} - 8x^{3} + 15x^{2} - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{6} - 3\nu^{5} - 4\nu^{4} + 10\nu^{3} + 4\nu^{2} - 7\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{7} - 3\nu^{6} - 4\nu^{5} + 10\nu^{4} + 4\nu^{3} - 7\nu^{2} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{8} - 3\nu^{7} - 5\nu^{6} + 13\nu^{5} + 7\nu^{4} - 14\nu^{3} - \nu^{2} + \nu - 1 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{7} - 3\nu^{6} - 5\nu^{5} + 13\nu^{4} + 7\nu^{3} - 14\nu^{2} - 2\nu + 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\nu^{8} + 3\nu^{7} + 5\nu^{6} - 13\nu^{5} - 7\nu^{4} + 14\nu^{3} + 2\nu^{2} - 3\nu \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \nu^{8} - 3\nu^{7} - 5\nu^{6} + 13\nu^{5} + 8\nu^{4} - 17\nu^{3} - 4\nu^{2} + 8\nu \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( 2\nu^{8} - 7\nu^{7} - 7\nu^{6} + 30\nu^{5} + 5\nu^{4} - 34\nu^{3} - \nu^{2} + 7\nu + 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{4} + 2\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} - \beta_{7} + 3\beta_{6} + 2\beta_{4} + \beta_{3} + 8\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{8} - 2\beta_{7} + 12\beta_{6} + 8\beta_{4} + 3\beta_{3} + 23\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 12\beta_{8} - 9\beta_{7} + 38\beta_{6} - \beta_{5} + 23\beta_{4} + 13\beta_{3} + 77\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 38\beta_{8} - 25\beta_{7} + 128\beta_{6} - 3\beta_{5} + 77\beta_{4} + 41\beta_{3} + \beta_{2} + 242\beta _1 + 7 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 128\beta_{8} - 87\beta_{7} + 411\beta_{6} - 13\beta_{5} + 242\beta_{4} + 142\beta_{3} + 3\beta_{2} + 786\beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 411 \beta_{8} - 269 \beta_{7} + 1338 \beta_{6} - 41 \beta_{5} + 786 \beta_{4} + 455 \beta_{3} + 14 \beta_{2} + 2519 \beta _1 + 46 \) 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
3.22274
1.63278
1.30799
0.789016
0.309891
−0.256961
−1.13237
−1.28997
−1.58312
−2.22274 1.03700 2.94057 0 −2.30498 2.02508 −2.09064 −1.92463 0
1.2 −0.632780 1.91964 −1.59959 0 −1.21471 −4.08895 2.27775 0.685005 0
1.3 −0.307988 −1.64392 −1.90514 0 0.506308 −0.0891959 1.20274 −0.297533 0
1.4 0.210984 0.0798955 −1.95549 0 0.0168566 1.68723 −0.834543 −2.99362 0
1.5 0.690109 −0.694850 −1.52375 0 −0.479522 2.33464 −2.43177 −2.51718 0
1.6 1.25696 3.01225 −0.420048 0 3.78628 −3.72392 −3.04191 6.07366 0
1.7 2.13237 2.23040 2.54700 0 4.75604 −1.48562 1.16642 1.97468 0
1.8 2.28997 3.30730 3.24395 0 7.57362 2.93910 2.84861 7.93825 0
1.9 2.58312 −0.247720 4.67249 0 −0.639888 0.401640 6.90335 −2.93864 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.cf 9
5.b even 2 1 1805.2.a.s 9
19.b odd 2 1 9025.2.a.cc 9
19.f odd 18 2 475.2.l.c 18
95.d odd 2 1 1805.2.a.v 9
95.o odd 18 2 95.2.k.a 18
95.r even 36 4 475.2.u.b 36
285.bf even 18 2 855.2.bs.c 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.k.a 18 95.o odd 18 2
475.2.l.c 18 19.f odd 18 2
475.2.u.b 36 95.r even 36 4
855.2.bs.c 18 285.bf even 18 2
1805.2.a.s 9 5.b even 2 1
1805.2.a.v 9 95.d odd 2 1
9025.2.a.cc 9 19.b odd 2 1
9025.2.a.cf 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9025))\):

\( T_{2}^{9} - 6T_{2}^{8} + 6T_{2}^{7} + 26T_{2}^{6} - 60T_{2}^{5} + 21T_{2}^{4} + 30T_{2}^{3} - 15T_{2}^{2} - 3T_{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{9} - 9T_{3}^{8} + 24T_{3}^{7} + T_{3}^{6} - 87T_{3}^{5} + 81T_{3}^{4} + 44T_{3}^{3} - 48T_{3}^{2} - 9T_{3} + 1 \) Copy content Toggle raw display
\( T_{7}^{9} - 27T_{7}^{7} + 24T_{7}^{6} + 213T_{7}^{5} - 342T_{7}^{4} - 277T_{7}^{3} + 651T_{7}^{2} - 153T_{7} - 19 \) Copy content Toggle raw display
\( T_{11}^{9} - 60T_{11}^{7} - 52T_{11}^{6} + 1218T_{11}^{5} + 2037T_{11}^{4} - 7981T_{11}^{3} - 20100T_{11}^{2} - 7353T_{11} + 773 \) Copy content Toggle raw display
\( T_{29}^{9} + 9 T_{29}^{8} - 66 T_{29}^{7} - 673 T_{29}^{6} + 1278 T_{29}^{5} + 16863 T_{29}^{4} - 1959 T_{29}^{3} - 149670 T_{29}^{2} - 111699 T_{29} + 210403 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} - 6 T^{8} + 6 T^{7} + 26 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{9} - 9 T^{8} + 24 T^{7} + T^{6} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{9} \) Copy content Toggle raw display
$7$ \( T^{9} - 27 T^{7} + 24 T^{6} + 213 T^{5} + \cdots - 19 \) Copy content Toggle raw display
$11$ \( T^{9} - 60 T^{7} - 52 T^{6} + \cdots + 773 \) Copy content Toggle raw display
$13$ \( T^{9} - 9 T^{8} - 15 T^{7} + 358 T^{6} + \cdots + 53 \) Copy content Toggle raw display
$17$ \( T^{9} - 9 T^{8} - 18 T^{7} + \cdots - 1691 \) Copy content Toggle raw display
$19$ \( T^{9} \) Copy content Toggle raw display
$23$ \( T^{9} - 12 T^{8} - 36 T^{7} + \cdots + 45667 \) Copy content Toggle raw display
$29$ \( T^{9} + 9 T^{8} - 66 T^{7} + \cdots + 210403 \) Copy content Toggle raw display
$31$ \( T^{9} + 18 T^{8} + 21 T^{7} + \cdots - 216991 \) Copy content Toggle raw display
$37$ \( T^{9} - 18 T^{8} - 15 T^{7} + \cdots - 11125 \) Copy content Toggle raw display
$41$ \( T^{9} + 6 T^{8} - 78 T^{7} - 377 T^{6} + \cdots - 361 \) Copy content Toggle raw display
$43$ \( T^{9} - 12 T^{8} - 153 T^{7} + \cdots - 1591019 \) Copy content Toggle raw display
$47$ \( T^{9} + 15 T^{8} - 108 T^{7} + \cdots - 1425943 \) Copy content Toggle raw display
$53$ \( T^{9} - 15 T^{8} - 159 T^{7} + \cdots - 211859 \) Copy content Toggle raw display
$59$ \( T^{9} + 21 T^{8} + 6 T^{7} + \cdots - 733771 \) Copy content Toggle raw display
$61$ \( T^{9} + 12 T^{8} - 123 T^{7} + \cdots + 61561 \) Copy content Toggle raw display
$67$ \( T^{9} - 60 T^{8} + 1467 T^{7} + \cdots - 6493589 \) Copy content Toggle raw display
$71$ \( T^{9} - 18 T^{8} - 225 T^{7} + \cdots + 53239843 \) Copy content Toggle raw display
$73$ \( T^{9} + 27 T^{8} - 33 T^{7} + \cdots + 24197203 \) Copy content Toggle raw display
$79$ \( T^{9} + 15 T^{8} - 141 T^{7} + \cdots - 3833 \) Copy content Toggle raw display
$83$ \( T^{9} - 507 T^{7} + \cdots - 189817057 \) Copy content Toggle raw display
$89$ \( T^{9} - 39 T^{8} + 438 T^{7} + \cdots - 957419 \) Copy content Toggle raw display
$97$ \( T^{9} - 15 T^{8} - 243 T^{7} + \cdots - 1914625 \) Copy content Toggle raw display
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