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Properties

Label 325.2.n.e
Level $325$
Weight $2$
Character orbit 325.n
Analytic conductor $2.595$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 325 = 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 325.n (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.59513806569\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
Defining polynomial: \( x^{10} + 16x^{8} + 84x^{6} + 163x^{4} + 118x^{2} + 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} + (\beta_{9} - \beta_{8} - \beta_{2} - 1) q^{3} + (\beta_{9} - \beta_{7} + \beta_{6} + \beta_{4} - 2 \beta_{2}) q^{4} + (\beta_{7} - \beta_{5} + \beta_{3} + \beta_{2} + \beta_1 + 2) q^{6} + (\beta_{7} - \beta_{5}) q^{7} + ( - \beta_{5} - \beta_{4} + 2 \beta_{2} + \beta_1 + 1) q^{8} + ( - \beta_{7} - \beta_{5} + 2 \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} + (\beta_{9} - \beta_{8} - \beta_{2} - 1) q^{3} + (\beta_{9} - \beta_{7} + \beta_{6} + \beta_{4} - 2 \beta_{2}) q^{4} + (\beta_{7} - \beta_{5} + \beta_{3} + \beta_{2} + \beta_1 + 2) q^{6} + (\beta_{7} - \beta_{5}) q^{7} + ( - \beta_{5} - \beta_{4} + 2 \beta_{2} + \beta_1 + 1) q^{8} + ( - \beta_{7} - \beta_{5} + 2 \beta_{2}) q^{9} + ( - \beta_{6} - \beta_{4} + \beta_{3} + \beta_{2} - 1) q^{11} + ( - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} - 2 \beta_{3} - \beta_1 - 4) q^{12} + ( - \beta_{9} + \beta_{8} - \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + 1) q^{13} + ( - \beta_{8} - \beta_{7} - \beta_{6} - 2) q^{14} + (\beta_{9} - \beta_{8} - 2 \beta_{7} + \beta_{6} + 2 \beta_{5} + \beta_{4} - 4 \beta_{2} - 4) q^{16} + ( - \beta_{9} + \beta_{6} + \beta_{5} + \beta_{4} - 2 \beta_{2}) q^{17} + (2 \beta_{9} - \beta_{8} - 3 \beta_{7} + 3 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} - 4 \beta_{2} + \cdots - 2) q^{18}+ \cdots + ( - 4 \beta_{9} + 2 \beta_{8} + \beta_{7} - \beta_{6} - 5 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 3 q^{3} + 6 q^{4} + 9 q^{6} - 6 q^{7} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 3 q^{3} + 6 q^{4} + 9 q^{6} - 6 q^{7} - 8 q^{9} - 9 q^{11} - 28 q^{12} + 8 q^{13} - 8 q^{14} - 12 q^{16} + 8 q^{17} - 12 q^{22} + 13 q^{23} + 42 q^{24} - 17 q^{26} + 30 q^{27} + 33 q^{28} + 7 q^{29} + 3 q^{32} - 6 q^{33} - 3 q^{36} + 3 q^{37} - 62 q^{38} + 8 q^{39} + 12 q^{41} + 32 q^{42} + 4 q^{43} + 39 q^{46} - 26 q^{48} - q^{49} + 16 q^{51} + 61 q^{52} + 24 q^{53} - 9 q^{54} - 21 q^{56} + 18 q^{58} - 48 q^{59} + 13 q^{61} + 17 q^{62} - 34 q^{64} - 42 q^{66} + 6 q^{67} - 13 q^{68} + 20 q^{69} - 27 q^{71} - 141 q^{72} - 26 q^{74} - 12 q^{76} - 48 q^{77} + 56 q^{78} + 4 q^{79} - 17 q^{81} - q^{82} - 90 q^{84} - 49 q^{87} - 6 q^{88} + 24 q^{89} + 13 q^{91} + 34 q^{92} - 63 q^{93} + 5 q^{94} - 15 q^{97} - 45 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 16x^{8} + 84x^{6} + 163x^{4} + 118x^{2} + 27 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{9} + 13\nu^{7} + 45\nu^{5} + 13\nu^{3} - 41\nu - 15 ) / 30 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{8} + 13\nu^{6} + 50\nu^{4} + 53\nu^{2} - 5\nu + 9 ) / 10 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{9} + 6\nu^{8} + 13\nu^{7} + 93\nu^{6} + 45\nu^{5} + 450\nu^{4} + 28\nu^{3} + 678\nu^{2} + 34\nu + 219 ) / 30 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{9} - 6\nu^{8} + 13\nu^{7} - 93\nu^{6} + 45\nu^{5} - 450\nu^{4} + 28\nu^{3} - 678\nu^{2} + 34\nu - 219 ) / 30 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2\nu^{9} + 31\nu^{7} + 150\nu^{5} + 5\nu^{4} + 226\nu^{3} + 35\nu^{2} + 68\nu + 20 ) / 10 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -2\nu^{9} - 31\nu^{7} - 150\nu^{5} + 5\nu^{4} - 226\nu^{3} + 35\nu^{2} - 68\nu + 20 ) / 10 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -2\nu^{8} - 31\nu^{6} - 150\nu^{4} - 231\nu^{2} - 93 ) / 5 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -4\nu^{9} - 2\nu^{8} - 62\nu^{7} - 31\nu^{6} - 305\nu^{5} - 150\nu^{4} - 497\nu^{3} - 231\nu^{2} - 211\nu - 93 ) / 10 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{8} + \beta_{5} - \beta_{4} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{4} - 2\beta_{2} - 5\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7\beta_{8} + \beta_{7} + \beta_{6} - 7\beta_{5} + 7\beta_{4} + 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -2\beta_{9} + \beta_{8} + 2\beta_{7} - 2\beta_{6} - 9\beta_{5} - 9\beta_{4} + 18\beta_{2} + 30\beta _1 + 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -46\beta_{8} - 10\beta_{7} - 10\beta_{6} + 45\beta_{5} - 45\beta_{4} - 4\beta_{3} - 2\beta _1 - 155 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 24\beta_{9} - 12\beta_{8} - 25\beta_{7} + 25\beta_{6} + 68\beta_{5} + 68\beta_{4} - 148\beta_{2} - 190\beta _1 - 74 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 301\beta_{8} + 80\beta_{7} + 80\beta_{6} - 288\beta_{5} + 288\beta_{4} + 62\beta_{3} + 31\beta _1 + 1018 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 222 \beta_{9} + 111 \beta_{8} + 235 \beta_{7} - 235 \beta_{6} - 492 \beta_{5} - 492 \beta_{4} + 1170 \beta_{2} + 1226 \beta _1 + 585 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/325\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(301\)
\(\chi(n)\) \(1\) \(-\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1
2.63252i
1.36551i
− 0.666048i
− 0.887996i
− 2.44399i
− 2.63252i
− 1.36551i
0.666048i
0.887996i
2.44399i
−2.27983 + 1.31626i −1.34953 − 2.33745i 2.46508 − 4.26964i 0 6.15337 + 3.55265i 2.37354 + 1.37037i 7.71370i −2.14244 + 3.71081i 0
101.2 −1.18257 + 0.682755i 0.199959 + 0.346339i −0.0676905 + 0.117243i 0 −0.472929 − 0.273046i −3.15550 − 1.82183i − 2.91589i 1.42003 − 2.45957i 0
101.3 0.576815 − 0.333024i 1.24146 + 2.15028i −0.778190 + 1.34786i 0 1.43219 + 0.826874i 0.509002 + 0.293872i 2.36872i −1.58246 + 2.74090i 0
101.4 0.769027 − 0.443998i −1.53097 − 2.65171i −0.605732 + 1.04916i 0 −2.35471 − 1.35949i −3.08568 − 1.78152i 2.85177i −3.18771 + 5.52128i 0
101.5 2.11655 − 1.22199i −0.0609297 − 0.105533i 1.98653 − 3.44078i 0 −0.257922 − 0.148911i 0.358632 + 0.207056i − 4.82215i 1.49258 − 2.58522i 0
251.1 −2.27983 − 1.31626i −1.34953 + 2.33745i 2.46508 + 4.26964i 0 6.15337 − 3.55265i 2.37354 − 1.37037i − 7.71370i −2.14244 − 3.71081i 0
251.2 −1.18257 − 0.682755i 0.199959 − 0.346339i −0.0676905 − 0.117243i 0 −0.472929 + 0.273046i −3.15550 + 1.82183i 2.91589i 1.42003 + 2.45957i 0
251.3 0.576815 + 0.333024i 1.24146 − 2.15028i −0.778190 − 1.34786i 0 1.43219 − 0.826874i 0.509002 − 0.293872i − 2.36872i −1.58246 − 2.74090i 0
251.4 0.769027 + 0.443998i −1.53097 + 2.65171i −0.605732 − 1.04916i 0 −2.35471 + 1.35949i −3.08568 + 1.78152i − 2.85177i −3.18771 − 5.52128i 0
251.5 2.11655 + 1.22199i −0.0609297 + 0.105533i 1.98653 + 3.44078i 0 −0.257922 + 0.148911i 0.358632 − 0.207056i 4.82215i 1.49258 + 2.58522i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 251.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 325.2.n.e ✓ 10
5.b even 2 1 325.2.n.f yes 10
5.c odd 4 2 325.2.m.d 20
13.e even 6 1 inner 325.2.n.e ✓ 10
13.f odd 12 2 4225.2.a.bv 10
65.l even 6 1 325.2.n.f yes 10
65.r odd 12 2 325.2.m.d 20
65.s odd 12 2 4225.2.a.bu 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
325.2.m.d 20 5.c odd 4 2
325.2.m.d 20 65.r odd 12 2
325.2.n.e ✓ 10 1.a even 1 1 trivial
325.2.n.e ✓ 10 13.e even 6 1 inner
325.2.n.f yes 10 5.b even 2 1
325.2.n.f yes 10 65.l even 6 1
4225.2.a.bu 10 65.s odd 12 2
4225.2.a.bv 10 13.f odd 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} - 8T_{2}^{8} + 54T_{2}^{6} - 15T_{2}^{5} - 77T_{2}^{4} + 30T_{2}^{3} + 91T_{2}^{2} - 90T_{2} + 27 \) acting on \(S_{2}^{\mathrm{new}}(325, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} - 8 T^{8} + 54 T^{6} - 15 T^{5} + \cdots + 27 \) Copy content Toggle raw display
$3$ \( T^{10} + 3 T^{9} + 16 T^{8} + 17 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( T^{10} + 6 T^{9} + T^{8} - 66 T^{7} + \cdots + 75 \) Copy content Toggle raw display
$11$ \( T^{10} + 9 T^{9} + 18 T^{8} + \cdots + 6075 \) Copy content Toggle raw display
$13$ \( T^{10} - 8 T^{9} + 2 T^{8} + \cdots + 371293 \) Copy content Toggle raw display
$17$ \( T^{10} - 8 T^{9} + 64 T^{8} + \cdots + 1521 \) Copy content Toggle raw display
$19$ \( T^{10} - 68 T^{8} + 3708 T^{6} + \cdots + 531723 \) Copy content Toggle raw display
$23$ \( T^{10} - 13 T^{9} + 127 T^{8} - 556 T^{7} + \cdots + 9 \) Copy content Toggle raw display
$29$ \( T^{10} - 7 T^{9} + 112 T^{8} + \cdots + 1766241 \) Copy content Toggle raw display
$31$ \( T^{10} + 154 T^{8} + \cdots + 10546875 \) Copy content Toggle raw display
$37$ \( T^{10} - 3 T^{9} - 140 T^{8} + \cdots + 112914675 \) Copy content Toggle raw display
$41$ \( T^{10} - 12 T^{9} - 14 T^{8} + \cdots + 45387 \) Copy content Toggle raw display
$43$ \( T^{10} - 4 T^{9} + 108 T^{8} + \cdots + 525625 \) Copy content Toggle raw display
$47$ \( T^{10} + 160 T^{8} + 6108 T^{6} + \cdots + 771147 \) Copy content Toggle raw display
$53$ \( (T^{5} - 12 T^{4} - 51 T^{3} + 819 T^{2} + \cdots - 8469)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} + 48 T^{9} + 951 T^{8} + \cdots + 3102867 \) Copy content Toggle raw display
$61$ \( T^{10} - 13 T^{9} + \cdots + 2238709225 \) Copy content Toggle raw display
$67$ \( T^{10} - 6 T^{9} - 33 T^{8} + \cdots + 16875 \) Copy content Toggle raw display
$71$ \( T^{10} + 27 T^{9} + \cdots + 116251875 \) Copy content Toggle raw display
$73$ \( T^{10} + 490 T^{8} + \cdots + 408916875 \) Copy content Toggle raw display
$79$ \( (T^{5} - 2 T^{4} - 227 T^{3} - 94 T^{2} + \cdots - 9475)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + 393 T^{8} + 36615 T^{6} + \cdots + 243 \) Copy content Toggle raw display
$89$ \( T^{10} - 24 T^{9} + 165 T^{8} + \cdots + 45139923 \) Copy content Toggle raw display
$97$ \( T^{10} + 15 T^{9} - 143 T^{8} + \cdots + 5738067 \) Copy content Toggle raw display
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