Properties

Label 325.2.n.b
Level $325$
Weight $2$
Character orbit 325.n
Analytic conductor $2.595$
Analytic rank $0$
Dimension $4$
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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-7})\)
Defining polynomial: \( x^{4} - x^{3} - x^{2} - 2x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} - 1) q^{2} + \beta_{2} q^{3} + ( - 2 \beta_{3} + \beta_{2} + \beta_1) q^{4} + (\beta_1 - 1) q^{6} + (\beta_{2} + 1) q^{7} + (2 \beta_{2} - 1) q^{8} + ( - 2 \beta_{2} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} - 1) q^{2} + \beta_{2} q^{3} + ( - 2 \beta_{3} + \beta_{2} + \beta_1) q^{4} + (\beta_1 - 1) q^{6} + (\beta_{2} + 1) q^{7} + (2 \beta_{2} - 1) q^{8} + ( - 2 \beta_{2} + 2) q^{9} + ( - 2 \beta_{3} + \beta_{2}) q^{11} + ( - \beta_{3} - \beta_1 + 1) q^{12} + (4 \beta_{2} - 1) q^{13} + (\beta_{3} + \beta_1 - 2) q^{14} + (\beta_{3} - 2 \beta_{2} - 2 \beta_1 + 1) q^{16} + ( - 4 \beta_{3} + 3 \beta_{2} + 2 \beta_1 - 1) q^{17} + (2 \beta_{3} - 2 \beta_1) q^{18} + (\beta_{2} + 1) q^{19} + (2 \beta_{2} - 1) q^{21} + (2 \beta_{3} + 2 \beta_{2} - \beta_1 - 3) q^{22} + ( - 2 \beta_{3} + 3 \beta_{2} + 4 \beta_1 - 2) q^{23} + (\beta_{2} - 2) q^{24} + ( - \beta_{3} + 4 \beta_1 - 3) q^{26} + 5 q^{27} + ( - 3 \beta_{3} + \beta_{2} + 1) q^{28} + (2 \beta_{3} - 3 \beta_{2} - 4 \beta_1 + 2) q^{29} + (6 \beta_{3} - 4 \beta_{2} - 6 \beta_1 + 2) q^{31} + ( - 3 \beta_{2} + \beta_1 - 4) q^{32} + ( - \beta_{2} - 2 \beta_1 + 1) q^{33} + (3 \beta_{3} + 4 \beta_{2} - 3 \beta_1 - 2) q^{34} + ( - 2 \beta_{3} + 2 \beta_{2} + 4 \beta_1 - 2) q^{36} + (6 \beta_{3} - 3 \beta_{2}) q^{37} + (\beta_{3} + \beta_1 - 2) q^{38} + (3 \beta_{2} - 4) q^{39} + (2 \beta_{3} - \beta_{2}) q^{41} + ( - \beta_{3} + 2 \beta_1 - 1) q^{42} + (4 \beta_{3} - 9 \beta_{2} - 2 \beta_1 + 7) q^{43} + ( - \beta_{3} - 6 \beta_{2} + \beta_1 + 3) q^{44} + (2 \beta_{2} - 3 \beta_1 + 5) q^{46} + (4 \beta_{3} - 4 \beta_1) q^{47} + (2 \beta_{3} - 2 \beta_{2} - \beta_1 + 1) q^{48} - 4 \beta_{2} q^{49} + ( - 2 \beta_{3} - 2 \beta_1 + 1) q^{51} + ( - 2 \beta_{3} - \beta_{2} - 5 \beta_1 + 4) q^{52} + ( - 2 \beta_{3} - 2 \beta_1 - 2) q^{53} + (5 \beta_{3} - 5) q^{54} + (3 \beta_{2} - 3) q^{56} + (2 \beta_{2} - 1) q^{57} + ( - 2 \beta_{2} + 3 \beta_1 - 5) q^{58} + (7 \beta_{2} + 4 \beta_1 + 3) q^{59} + ( - 4 \beta_{3} - 3 \beta_{2} + 2 \beta_1 + 5) q^{61} + ( - 4 \beta_{3} - 6 \beta_{2} + 8 \beta_1 - 4) q^{62} + ( - 2 \beta_{2} + 4) q^{63} + ( - 2 \beta_{3} - 2 \beta_1 + 9) q^{64} + (\beta_{3} + \beta_1 - 4) q^{66} + ( - 6 \beta_{3} - \beta_{2} + 8) q^{67} + ( - \beta_{3} - 9 \beta_{2} + 2 \beta_1 - 1) q^{68} + ( - 4 \beta_{3} + 3 \beta_{2} + 2 \beta_1 - 1) q^{69} + (3 \beta_{2} + 4 \beta_1 - 1) q^{71} + (2 \beta_{2} + 2) q^{72} + ( - 6 \beta_{3} - 6 \beta_{2} + 3 \beta_1 + 9) q^{74} + ( - 3 \beta_{3} + \beta_{2} + 1) q^{76} + ( - 2 \beta_{3} - 2 \beta_1 + 1) q^{77} + ( - 4 \beta_{3} + 3 \beta_1 + 1) q^{78} - 6 q^{79} - \beta_{2} q^{81} + ( - 2 \beta_{3} - 2 \beta_{2} + \beta_1 + 3) q^{82} + ( - 2 \beta_{3} - 8 \beta_{2} + 2 \beta_1 + 4) q^{83} + ( - \beta_{2} - 3 \beta_1 + 2) q^{84} + (3 \beta_{3} - 4 \beta_{2} - 3 \beta_1 + 2) q^{86} + (4 \beta_{3} - 3 \beta_{2} - 2 \beta_1 + 1) q^{87} + (2 \beta_{3} - 3 \beta_{2} - 4 \beta_1 + 2) q^{88} + ( - 2 \beta_{3} + 5 \beta_{2} - 8) q^{89} + (7 \beta_{2} - 5) q^{91} + (\beta_{3} + \beta_1 - 11) q^{92} + (6 \beta_{3} - 2 \beta_{2} - 2) q^{93} + ( - 4 \beta_{3} - 4 \beta_{2} + 8 \beta_1 - 4) q^{94} + ( - \beta_{3} - 6 \beta_{2} + \beta_1 + 3) q^{96} + (5 \beta_{2} + 6 \beta_1 - 1) q^{97} + ( - 4 \beta_1 + 4) q^{98} + ( - 4 \beta_{3} + 4 \beta_{2} + 4 \beta_1 - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 3 q^{2} + 2 q^{3} + q^{4} - 3 q^{6} + 6 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 3 q^{2} + 2 q^{3} + q^{4} - 3 q^{6} + 6 q^{7} + 4 q^{9} + 2 q^{12} + 4 q^{13} - 6 q^{14} - q^{16} + 6 q^{19} - 7 q^{22} - 6 q^{24} - 9 q^{26} + 20 q^{27} + 3 q^{28} - 21 q^{32} - 2 q^{36} - 6 q^{38} - 10 q^{39} - 3 q^{42} + 12 q^{43} + 21 q^{46} + q^{48} - 8 q^{49} + 7 q^{52} - 12 q^{53} - 15 q^{54} - 6 q^{56} - 21 q^{58} + 30 q^{59} + 12 q^{61} - 24 q^{62} + 12 q^{63} + 32 q^{64} - 14 q^{66} + 24 q^{67} - 21 q^{68} + 6 q^{71} + 12 q^{72} + 21 q^{74} + 3 q^{76} + 3 q^{78} - 24 q^{79} - 2 q^{81} + 7 q^{82} + 3 q^{84} - 24 q^{89} - 6 q^{91} - 42 q^{92} - 6 q^{93} - 20 q^{94} + 12 q^{97} + 12 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - x^{2} - 2x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + \nu^{2} - \nu - 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + \nu^{2} + \nu + 2 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{3} + \beta_{2} + \beta _1 + 2 \) Copy content Toggle raw display

Character values

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

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

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
−0.895644 1.09445i
1.39564 + 0.228425i
−0.895644 + 1.09445i
1.39564 0.228425i
−1.89564 + 1.09445i 0.500000 + 0.866025i 1.39564 2.41733i 0 −1.89564 1.09445i 1.50000 + 0.866025i 1.73205i 1.00000 1.73205i 0
101.2 0.395644 0.228425i 0.500000 + 0.866025i −0.895644 + 1.55130i 0 0.395644 + 0.228425i 1.50000 + 0.866025i 1.73205i 1.00000 1.73205i 0
251.1 −1.89564 1.09445i 0.500000 0.866025i 1.39564 + 2.41733i 0 −1.89564 + 1.09445i 1.50000 0.866025i 1.73205i 1.00000 + 1.73205i 0
251.2 0.395644 + 0.228425i 0.500000 0.866025i −0.895644 1.55130i 0 0.395644 0.228425i 1.50000 0.866025i 1.73205i 1.00000 + 1.73205i 0
\(n\): e.g. 2-40 or 990-1000
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.b 4
5.b even 2 1 325.2.n.c 4
5.c odd 4 2 65.2.l.a 8
13.e even 6 1 inner 325.2.n.b 4
13.f odd 12 2 4225.2.a.bj 4
15.e even 4 2 585.2.bf.a 8
20.e even 4 2 1040.2.df.b 8
65.f even 4 2 845.2.n.d 8
65.h odd 4 2 845.2.l.c 8
65.k even 4 2 845.2.n.c 8
65.l even 6 1 325.2.n.c 4
65.o even 12 2 845.2.b.f 8
65.o even 12 2 845.2.n.d 8
65.q odd 12 2 845.2.d.c 8
65.q odd 12 2 845.2.l.c 8
65.r odd 12 2 65.2.l.a 8
65.r odd 12 2 845.2.d.c 8
65.s odd 12 2 4225.2.a.bk 4
65.t even 12 2 845.2.b.f 8
65.t even 12 2 845.2.n.c 8
195.bf even 12 2 585.2.bf.a 8
260.bg even 12 2 1040.2.df.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.l.a 8 5.c odd 4 2
65.2.l.a 8 65.r odd 12 2
325.2.n.b 4 1.a even 1 1 trivial
325.2.n.b 4 13.e even 6 1 inner
325.2.n.c 4 5.b even 2 1
325.2.n.c 4 65.l even 6 1
585.2.bf.a 8 15.e even 4 2
585.2.bf.a 8 195.bf even 12 2
845.2.b.f 8 65.o even 12 2
845.2.b.f 8 65.t even 12 2
845.2.d.c 8 65.q odd 12 2
845.2.d.c 8 65.r odd 12 2
845.2.l.c 8 65.h odd 4 2
845.2.l.c 8 65.q odd 12 2
845.2.n.c 8 65.k even 4 2
845.2.n.c 8 65.t even 12 2
845.2.n.d 8 65.f even 4 2
845.2.n.d 8 65.o even 12 2
1040.2.df.b 8 20.e even 4 2
1040.2.df.b 8 260.bg even 12 2
4225.2.a.bj 4 13.f odd 12 2
4225.2.a.bk 4 65.s odd 12 2

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 3 T^{3} + 2 T^{2} - 3 T + 1 \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} - 3 T + 3)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} - 7T^{2} + 49 \) Copy content Toggle raw display
$13$ \( (T^{2} - 2 T + 13)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 21T^{2} + 441 \) Copy content Toggle raw display
$19$ \( (T^{2} - 3 T + 3)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 21T^{2} + 441 \) Copy content Toggle raw display
$29$ \( T^{4} + 21T^{2} + 441 \) Copy content Toggle raw display
$31$ \( T^{4} + 132T^{2} + 3600 \) Copy content Toggle raw display
$37$ \( T^{4} - 63T^{2} + 3969 \) Copy content Toggle raw display
$41$ \( T^{4} - 7T^{2} + 49 \) Copy content Toggle raw display
$43$ \( T^{4} - 12 T^{3} + 129 T^{2} + \cdots + 225 \) Copy content Toggle raw display
$47$ \( T^{4} + 80T^{2} + 256 \) Copy content Toggle raw display
$53$ \( (T^{2} + 6 T - 12)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 30 T^{3} + 347 T^{2} + \cdots + 2209 \) Copy content Toggle raw display
$61$ \( T^{4} - 12 T^{3} + 129 T^{2} + \cdots + 225 \) Copy content Toggle raw display
$67$ \( T^{4} - 24 T^{3} + 177 T^{2} + \cdots + 225 \) Copy content Toggle raw display
$71$ \( T^{4} - 6 T^{3} - 13 T^{2} + 150 T + 625 \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( (T + 6)^{4} \) Copy content Toggle raw display
$83$ \( T^{4} + 164T^{2} + 4624 \) Copy content Toggle raw display
$89$ \( T^{4} + 24 T^{3} + 233 T^{2} + \cdots + 1681 \) Copy content Toggle raw display
$97$ \( T^{4} - 12 T^{3} - 3 T^{2} + \cdots + 2601 \) Copy content Toggle raw display
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