Properties

Label 325.2.n.c
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: \(x^{4} - x^{3} - x^{2} - 2 x + 4\)
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.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + \nu^{2} - \nu - 2 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + \nu^{2} + \nu + 2 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + \beta_{2}\)
\(\nu^{3}\)\(=\)\(-\beta_{3} + \beta_{2} + \beta_{1} + 2\)

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
1.39564 + 0.228425i
−0.895644 1.09445i
1.39564 0.228425i
−0.895644 + 1.09445i
−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
101.2 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
251.1 −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
251.2 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
\(n\): 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.c 4
5.b even 2 1 325.2.n.b 4
5.c odd 4 2 65.2.l.a 8
13.e even 6 1 inner 325.2.n.c 4
13.f odd 12 2 4225.2.a.bk 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.c 8
65.h odd 4 2 845.2.l.c 8
65.k even 4 2 845.2.n.d 8
65.l even 6 1 325.2.n.b 4
65.o even 12 2 845.2.b.f 8
65.o even 12 2 845.2.n.c 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.bj 4
65.t even 12 2 845.2.b.f 8
65.t even 12 2 845.2.n.d 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 5.b even 2 1
325.2.n.b 4 65.l even 6 1
325.2.n.c 4 1.a even 1 1 trivial
325.2.n.c 4 13.e even 6 1 inner
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.f even 4 2
845.2.n.c 8 65.o even 12 2
845.2.n.d 8 65.k even 4 2
845.2.n.d 8 65.t 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 65.s odd 12 2
4225.2.a.bk 4 13.f odd 12 2

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 3 T + 2 T^{2} - 3 T^{3} + T^{4} \)
$3$ \( ( 1 + T + T^{2} )^{2} \)
$5$ \( T^{4} \)
$7$ \( ( 3 + 3 T + T^{2} )^{2} \)
$11$ \( 49 - 7 T^{2} + T^{4} \)
$13$ \( ( 13 + 2 T + T^{2} )^{2} \)
$17$ \( 441 + 21 T^{2} + T^{4} \)
$19$ \( ( 3 - 3 T + T^{2} )^{2} \)
$23$ \( 441 + 21 T^{2} + T^{4} \)
$29$ \( 441 + 21 T^{2} + T^{4} \)
$31$ \( 3600 + 132 T^{2} + T^{4} \)
$37$ \( 3969 - 63 T^{2} + T^{4} \)
$41$ \( 49 - 7 T^{2} + T^{4} \)
$43$ \( 225 + 180 T + 129 T^{2} + 12 T^{3} + T^{4} \)
$47$ \( 256 + 80 T^{2} + T^{4} \)
$53$ \( ( -12 - 6 T + T^{2} )^{2} \)
$59$ \( 2209 - 1410 T + 347 T^{2} - 30 T^{3} + T^{4} \)
$61$ \( 225 - 180 T + 129 T^{2} - 12 T^{3} + T^{4} \)
$67$ \( 225 - 360 T + 177 T^{2} + 24 T^{3} + T^{4} \)
$71$ \( 625 + 150 T - 13 T^{2} - 6 T^{3} + T^{4} \)
$73$ \( T^{4} \)
$79$ \( ( 6 + T )^{4} \)
$83$ \( 4624 + 164 T^{2} + T^{4} \)
$89$ \( 1681 + 984 T + 233 T^{2} + 24 T^{3} + T^{4} \)
$97$ \( 2601 - 612 T - 3 T^{2} + 12 T^{3} + T^{4} \)
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