Properties

Label 325.2.n.d
Level $325$
Weight $2$
Character orbit 325.n
Analytic conductor $2.595$
Analytic rank $0$
Dimension $8$
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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.22581504.2
Defining polynomial: \( x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 65)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{7} + \beta_{5} + \beta_{3}) q^{2} + (\beta_{7} + \beta_{4} + 2 \beta_{2} + \beta_1 - 1) q^{3} + (2 \beta_{7} + \beta_{6} + \beta_{3} + \beta_{2} + \beta_1 - 1) q^{4} + (2 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 - 5) q^{6} + ( - \beta_{5} + \beta_{4} + \beta_{3} - 3 \beta_{2}) q^{7} + (\beta_{7} + 4 \beta_{6} - \beta_{5} + \beta_{3} + \beta_{2} + \beta_1 - 3) q^{8} + ( - 2 \beta_{6} - 2 \beta_{3} - 2 \beta_1 + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{7} + \beta_{5} + \beta_{3}) q^{2} + (\beta_{7} + \beta_{4} + 2 \beta_{2} + \beta_1 - 1) q^{3} + (2 \beta_{7} + \beta_{6} + \beta_{3} + \beta_{2} + \beta_1 - 1) q^{4} + (2 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 - 5) q^{6} + ( - \beta_{5} + \beta_{4} + \beta_{3} - 3 \beta_{2}) q^{7} + (\beta_{7} + 4 \beta_{6} - \beta_{5} + \beta_{3} + \beta_{2} + \beta_1 - 3) q^{8} + ( - 2 \beta_{6} - 2 \beta_{3} - 2 \beta_1 + 2) q^{9} + (2 \beta_{7} - \beta_{6} + 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - 2 \beta_1 + 1) q^{11} + ( - 2 \beta_{7} - 3 \beta_{5} + 3 \beta_{4} + 2 \beta_{2} - 4) q^{12} + (\beta_{7} - 2 \beta_{6} + \beta_{5} - \beta_{3} + \beta_{2} - 3 \beta_1 + 3) q^{13} + ( - \beta_{7} + 3 \beta_{5} - 3 \beta_{4} + \beta_{2} + 2) q^{14} + (\beta_{7} - \beta_{5} + 3 \beta_{4} + \beta_{3} + 2 \beta_{2} + 3 \beta_1 - 3) q^{16} + ( - 2 \beta_{7} + \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2}) q^{17} + ( - 6 \beta_{7} - 4 \beta_{6} - 2 \beta_{4} - 2 \beta_{3} - 6 \beta_{2} - 2 \beta_1 + 4) q^{18} + ( - \beta_{6} - 4 \beta_{2} + 2) q^{19} + ( - 2 \beta_{7} + 6 \beta_{6} - 2 \beta_{5} + 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 5) q^{21} + ( - 2 \beta_{7} + 4 \beta_{6} - \beta_{5} - \beta_{4} - \beta_{2} + \beta_1 - 1) q^{22} + (\beta_{7} - 2 \beta_{6} - 2 \beta_{5} + 3 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 3 \beta_1 - 1) q^{23} + ( - 8 \beta_{7} - \beta_{6} - 4 \beta_{5} - 4 \beta_{3} - 1) q^{24} + ( - 5 \beta_{7} - 2 \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - 6 \beta_{2} - \beta_1 + 3) q^{26} + (\beta_{7} + \beta_{5} - 2 \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 + 1) q^{27} + (2 \beta_{7} + 2 \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_1 + 1) q^{28} + (2 \beta_{7} + 3 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 4 \beta_{2} + \cdots - 5) q^{29}+ \cdots + ( - 8 \beta_{7} + 4 \beta_{6} - 2 \beta_{5} - 2 \beta_{4} - 8 \beta_{2} - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{3} + 2 q^{4} - 18 q^{6} + 6 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{3} + 2 q^{4} - 18 q^{6} + 6 q^{7} - 4 q^{9} - 20 q^{12} + 4 q^{14} - 2 q^{16} + 2 q^{17} + 12 q^{19} + 12 q^{22} + 10 q^{23} - 12 q^{24} + 10 q^{26} + 4 q^{27} + 18 q^{28} - 8 q^{29} - 6 q^{32} - 42 q^{33} + 20 q^{36} - 6 q^{37} + 16 q^{38} + 12 q^{41} - 4 q^{42} + 2 q^{43} - 42 q^{46} - 28 q^{48} + 12 q^{49} - 8 q^{51} + 6 q^{52} + 24 q^{53} + 18 q^{54} + 12 q^{56} - 36 q^{58} - 12 q^{59} - 28 q^{61} - 4 q^{62} + 24 q^{63} - 8 q^{64} + 12 q^{66} - 6 q^{67} + 14 q^{68} - 16 q^{69} + 48 q^{72} + 10 q^{74} + 54 q^{76} + 36 q^{77} + 56 q^{78} - 16 q^{79} + 8 q^{81} - 4 q^{82} - 30 q^{84} - 22 q^{87} + 18 q^{88} + 24 q^{89} + 28 q^{91} - 44 q^{92} + 32 q^{94} + 30 q^{97} - 72 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} - 2\nu^{6} + \nu^{5} + 4\nu^{4} - 3\nu^{3} - 2\nu^{2} + 8\nu - 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} + 2\nu^{6} - \nu^{5} - 4\nu^{4} + 3\nu^{3} + 10\nu^{2} - 16\nu + 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} - 3\nu^{6} + 3\nu^{5} + 3\nu^{4} - 7\nu^{3} - 3\nu^{2} + 18\nu - 16 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{7} - 5\nu^{6} + 2\nu^{5} + 7\nu^{4} - 8\nu^{3} - 9\nu^{2} + 28\nu - 20 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -3\nu^{7} + 7\nu^{6} - 3\nu^{5} - 11\nu^{4} + 15\nu^{3} + 11\nu^{2} - 40\nu + 32 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 7\nu^{7} - 20\nu^{6} + 11\nu^{5} + 30\nu^{4} - 45\nu^{3} - 28\nu^{2} + 116\nu - 88 ) / 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} + \beta_{5} + 2\beta_{2} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + 4\beta_{2} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{6} - \beta_{5} + 2\beta_{4} - 2\beta_{3} + 4\beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -\beta_{7} - 3\beta_{6} - 5\beta_{5} + \beta_{4} - 4\beta_{3} + 3\beta_{2} + 4\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -6\beta_{7} - 8\beta_{6} - 2\beta_{5} + 4\beta_{4} + 2\beta_{2} + \beta _1 + 6 \) Copy content Toggle raw display

Character values

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

\(n\) \(27\) \(301\)
\(\chi(n)\) \(1\) \(\beta_{6}\)

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
0.665665 + 1.24775i
1.40994 + 0.109843i
−1.27597 0.609843i
1.20036 0.747754i
0.665665 1.24775i
1.40994 0.109843i
−1.27597 + 0.609843i
1.20036 + 0.747754i
−1.29515 + 0.747754i 0.0473938 + 0.0820885i 0.118272 0.204852i 0 −0.122764 0.0708778i 4.18016 + 2.41342i 2.63726i 1.49551 2.59030i 0
101.2 −1.05628 + 0.609843i 1.16612 + 2.01978i −0.256182 + 0.443720i 0 −2.46350 1.42231i −3.11786 1.80010i 3.06430i −1.21969 + 2.11256i 0
101.3 0.190254 0.109843i −0.800098 1.38581i −0.975869 + 1.69025i 0 −0.304444 0.175771i 0.287734 + 0.166123i 0.868145i 0.219687 0.380509i 0
101.4 2.16117 1.24775i −1.41342 2.44811i 2.11378 3.66117i 0 −6.10929 3.52720i 1.64996 + 0.952606i 5.55889i −2.49551 + 4.32235i 0
251.1 −1.29515 0.747754i 0.0473938 0.0820885i 0.118272 + 0.204852i 0 −0.122764 + 0.0708778i 4.18016 2.41342i 2.63726i 1.49551 + 2.59030i 0
251.2 −1.05628 0.609843i 1.16612 2.01978i −0.256182 0.443720i 0 −2.46350 + 1.42231i −3.11786 + 1.80010i 3.06430i −1.21969 2.11256i 0
251.3 0.190254 + 0.109843i −0.800098 + 1.38581i −0.975869 1.69025i 0 −0.304444 + 0.175771i 0.287734 0.166123i 0.868145i 0.219687 + 0.380509i 0
251.4 2.16117 + 1.24775i −1.41342 + 2.44811i 2.11378 + 3.66117i 0 −6.10929 + 3.52720i 1.64996 0.952606i 5.55889i −2.49551 4.32235i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 251.4
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.d 8
5.b even 2 1 65.2.m.a 8
5.c odd 4 1 325.2.m.b 8
5.c odd 4 1 325.2.m.c 8
13.e even 6 1 inner 325.2.n.d 8
13.f odd 12 1 4225.2.a.bi 4
13.f odd 12 1 4225.2.a.bl 4
15.d odd 2 1 585.2.bu.c 8
20.d odd 2 1 1040.2.da.b 8
65.d even 2 1 845.2.m.g 8
65.g odd 4 1 845.2.e.m 8
65.g odd 4 1 845.2.e.n 8
65.l even 6 1 65.2.m.a 8
65.l even 6 1 845.2.c.g 8
65.n even 6 1 845.2.c.g 8
65.n even 6 1 845.2.m.g 8
65.r odd 12 1 325.2.m.b 8
65.r odd 12 1 325.2.m.c 8
65.s odd 12 1 845.2.a.l 4
65.s odd 12 1 845.2.a.m 4
65.s odd 12 1 845.2.e.m 8
65.s odd 12 1 845.2.e.n 8
195.y odd 6 1 585.2.bu.c 8
195.bh even 12 1 7605.2.a.cf 4
195.bh even 12 1 7605.2.a.cj 4
260.w odd 6 1 1040.2.da.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.m.a 8 5.b even 2 1
65.2.m.a 8 65.l even 6 1
325.2.m.b 8 5.c odd 4 1
325.2.m.b 8 65.r odd 12 1
325.2.m.c 8 5.c odd 4 1
325.2.m.c 8 65.r odd 12 1
325.2.n.d 8 1.a even 1 1 trivial
325.2.n.d 8 13.e even 6 1 inner
585.2.bu.c 8 15.d odd 2 1
585.2.bu.c 8 195.y odd 6 1
845.2.a.l 4 65.s odd 12 1
845.2.a.m 4 65.s odd 12 1
845.2.c.g 8 65.l even 6 1
845.2.c.g 8 65.n even 6 1
845.2.e.m 8 65.g odd 4 1
845.2.e.m 8 65.s odd 12 1
845.2.e.n 8 65.g odd 4 1
845.2.e.n 8 65.s odd 12 1
845.2.m.g 8 65.d even 2 1
845.2.m.g 8 65.n even 6 1
1040.2.da.b 8 20.d odd 2 1
1040.2.da.b 8 260.w odd 6 1
4225.2.a.bi 4 13.f odd 12 1
4225.2.a.bl 4 13.f odd 12 1
7605.2.a.cf 4 195.bh even 12 1
7605.2.a.cj 4 195.bh even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 5T_{2}^{6} + 24T_{2}^{4} + 30T_{2}^{3} + 7T_{2}^{2} - 6T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(325, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 5 T^{6} + 24 T^{4} + 30 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{8} + 2 T^{7} + 10 T^{6} + 8 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 6 T^{7} - 2 T^{6} + 84 T^{5} + \cdots + 121 \) Copy content Toggle raw display
$11$ \( T^{8} - 30 T^{6} + 867 T^{4} + \cdots + 1089 \) Copy content Toggle raw display
$13$ \( T^{8} + 16 T^{6} - 96 T^{5} + \cdots + 28561 \) Copy content Toggle raw display
$17$ \( T^{8} - 2 T^{7} + 22 T^{6} + 16 T^{5} + \cdots + 169 \) Copy content Toggle raw display
$19$ \( (T^{4} - 6 T^{3} - T^{2} + 78 T + 169)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} - 10 T^{7} + 94 T^{6} + \cdots + 89401 \) Copy content Toggle raw display
$29$ \( T^{8} + 8 T^{7} + 82 T^{6} - 64 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$31$ \( (T^{4} + 32 T^{2} + 64)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 6 T^{7} - 38 T^{6} - 300 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( (T^{4} - 6 T^{3} + 11 T^{2} + 6 T + 1)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} - 2 T^{7} + 22 T^{6} + 16 T^{5} + \cdots + 169 \) Copy content Toggle raw display
$47$ \( T^{8} + 208 T^{6} + 14304 T^{4} + \cdots + 1763584 \) Copy content Toggle raw display
$53$ \( (T^{4} - 12 T^{3} + 36 T^{2} - 48)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 12 T^{7} + 30 T^{6} - 216 T^{5} + \cdots + 9 \) Copy content Toggle raw display
$61$ \( T^{8} + 28 T^{7} + 526 T^{6} + \cdots + 1590121 \) Copy content Toggle raw display
$67$ \( T^{8} + 6 T^{7} - 102 T^{6} + \cdots + 7667361 \) Copy content Toggle raw display
$71$ \( T^{8} - 218 T^{6} + \cdots + 109767529 \) Copy content Toggle raw display
$73$ \( T^{8} + 232 T^{6} + 16944 T^{4} + \cdots + 2930944 \) Copy content Toggle raw display
$79$ \( (T^{4} + 8 T^{3} - 132 T^{2} - 640 T + 4432)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 192 T^{6} + 7104 T^{4} + \cdots + 36864 \) Copy content Toggle raw display
$89$ \( T^{8} - 24 T^{7} - 18 T^{6} + \cdots + 78375609 \) Copy content Toggle raw display
$97$ \( T^{8} - 30 T^{7} + 358 T^{6} + \cdots + 196249 \) Copy content Toggle raw display
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