Properties

Label 325.2.b.f
Level $325$
Weight $2$
Character orbit 325.b
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.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.59513806569\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 65)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{2} + \beta_1) q^{2} - \beta_{2} q^{3} + ( - 2 \beta_{3} - 1) q^{4} + (\beta_{3} + 2) q^{6} + ( - 2 \beta_{2} - 2 \beta_1) q^{7} + ( - \beta_{2} - 3 \beta_1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + \beta_1) q^{2} - \beta_{2} q^{3} + ( - 2 \beta_{3} - 1) q^{4} + (\beta_{3} + 2) q^{6} + ( - 2 \beta_{2} - 2 \beta_1) q^{7} + ( - \beta_{2} - 3 \beta_1) q^{8} + q^{9} + (\beta_{3} + 2) q^{11} + (\beta_{2} + 4 \beta_1) q^{12} - \beta_1 q^{13} + (4 \beta_{3} + 6) q^{14} + 3 q^{16} + ( - 2 \beta_{2} + 2 \beta_1) q^{17} + (\beta_{2} + \beta_1) q^{18} + (\beta_{3} - 2) q^{19} + ( - 2 \beta_{3} - 4) q^{21} + (3 \beta_{2} + 4 \beta_1) q^{22} + \beta_{2} q^{23} + ( - 3 \beta_{3} - 2) q^{24} + (\beta_{3} + 1) q^{26} - 4 \beta_{2} q^{27} + (6 \beta_{2} + 10 \beta_1) q^{28} + 4 \beta_{3} q^{29} + ( - 3 \beta_{3} + 6) q^{31} + (\beta_{2} - 3 \beta_1) q^{32} + ( - 2 \beta_{2} - 2 \beta_1) q^{33} + 2 q^{34} + ( - 2 \beta_{3} - 1) q^{36} + 6 \beta_{2} q^{37} - \beta_{2} q^{38} - \beta_{3} q^{39} + (2 \beta_{3} - 6) q^{41} + ( - 6 \beta_{2} - 8 \beta_1) q^{42} + ( - 5 \beta_{2} - 4 \beta_1) q^{43} + ( - 5 \beta_{3} - 6) q^{44} + ( - \beta_{3} - 2) q^{46} + (2 \beta_{2} + 2 \beta_1) q^{47} - 3 \beta_{2} q^{48} + ( - 8 \beta_{3} - 5) q^{49} + (2 \beta_{3} - 4) q^{51} + (2 \beta_{2} + \beta_1) q^{52} + (6 \beta_{2} - 6 \beta_1) q^{53} + (4 \beta_{3} + 8) q^{54} + ( - 8 \beta_{3} - 10) q^{56} + (2 \beta_{2} - 2 \beta_1) q^{57} + (4 \beta_{2} + 8 \beta_1) q^{58} + (3 \beta_{3} - 6) q^{59} - 8 q^{61} + 3 \beta_{2} q^{62} + ( - 2 \beta_{2} - 2 \beta_1) q^{63} + (2 \beta_{3} + 7) q^{64} + (4 \beta_{3} + 6) q^{66} + 2 \beta_1 q^{67} + ( - 2 \beta_{2} + 6 \beta_1) q^{68} + 2 q^{69} + (7 \beta_{3} + 2) q^{71} + ( - \beta_{2} - 3 \beta_1) q^{72} + 6 \beta_{2} q^{73} + ( - 6 \beta_{3} - 12) q^{74} + (3 \beta_{3} - 2) q^{76} + ( - 6 \beta_{2} - 8 \beta_1) q^{77} + ( - \beta_{2} - 2 \beta_1) q^{78} + 6 \beta_{3} q^{79} - 5 q^{81} + ( - 4 \beta_{2} - 2 \beta_1) q^{82} + (2 \beta_{2} - 6 \beta_1) q^{83} + (10 \beta_{3} + 12) q^{84} + (9 \beta_{3} + 14) q^{86} - 8 \beta_1 q^{87} + ( - 5 \beta_{2} - 8 \beta_1) q^{88} - 6 q^{89} + ( - 2 \beta_{3} - 2) q^{91} + ( - \beta_{2} - 4 \beta_1) q^{92} + ( - 6 \beta_{2} + 6 \beta_1) q^{93} + ( - 4 \beta_{3} - 6) q^{94} + ( - 3 \beta_{3} + 2) q^{96} + (4 \beta_{2} + 2 \beta_1) q^{97} + ( - 13 \beta_{2} - 21 \beta_1) q^{98} + (\beta_{3} + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 8 q^{6} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} + 8 q^{6} + 4 q^{9} + 8 q^{11} + 24 q^{14} + 12 q^{16} - 8 q^{19} - 16 q^{21} - 8 q^{24} + 4 q^{26} + 24 q^{31} + 8 q^{34} - 4 q^{36} - 24 q^{41} - 24 q^{44} - 8 q^{46} - 20 q^{49} - 16 q^{51} + 32 q^{54} - 40 q^{56} - 24 q^{59} - 32 q^{61} + 28 q^{64} + 24 q^{66} + 8 q^{69} + 8 q^{71} - 48 q^{74} - 8 q^{76} - 20 q^{81} + 48 q^{84} + 56 q^{86} - 24 q^{89} - 8 q^{91} - 24 q^{94} + 8 q^{96} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{8}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{8}^{3} + \zeta_{8} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{8}^{3} + \zeta_{8} \) Copy content Toggle raw display
\(\zeta_{8}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{8}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{8}^{3}\)\(=\) \( ( -\beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(27\) \(301\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
274.1
0.707107 0.707107i
−0.707107 0.707107i
−0.707107 + 0.707107i
0.707107 + 0.707107i
2.41421i 1.41421i −3.82843 0 3.41421 4.82843i 4.41421i 1.00000 0
274.2 0.414214i 1.41421i 1.82843 0 0.585786 0.828427i 1.58579i 1.00000 0
274.3 0.414214i 1.41421i 1.82843 0 0.585786 0.828427i 1.58579i 1.00000 0
274.4 2.41421i 1.41421i −3.82843 0 3.41421 4.82843i 4.41421i 1.00000 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
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 325.2.b.f 4
3.b odd 2 1 2925.2.c.r 4
5.b even 2 1 inner 325.2.b.f 4
5.c odd 4 1 65.2.a.b 2
5.c odd 4 1 325.2.a.i 2
15.d odd 2 1 2925.2.c.r 4
15.e even 4 1 585.2.a.m 2
15.e even 4 1 2925.2.a.u 2
20.e even 4 1 1040.2.a.j 2
20.e even 4 1 5200.2.a.bu 2
35.f even 4 1 3185.2.a.j 2
40.i odd 4 1 4160.2.a.bf 2
40.k even 4 1 4160.2.a.z 2
55.e even 4 1 7865.2.a.j 2
60.l odd 4 1 9360.2.a.cd 2
65.f even 4 1 845.2.c.b 4
65.h odd 4 1 845.2.a.g 2
65.h odd 4 1 4225.2.a.r 2
65.k even 4 1 845.2.c.b 4
65.o even 12 2 845.2.m.f 8
65.q odd 12 2 845.2.e.h 4
65.r odd 12 2 845.2.e.c 4
65.t even 12 2 845.2.m.f 8
195.s even 4 1 7605.2.a.x 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.a.b 2 5.c odd 4 1
325.2.a.i 2 5.c odd 4 1
325.2.b.f 4 1.a even 1 1 trivial
325.2.b.f 4 5.b even 2 1 inner
585.2.a.m 2 15.e even 4 1
845.2.a.g 2 65.h odd 4 1
845.2.c.b 4 65.f even 4 1
845.2.c.b 4 65.k even 4 1
845.2.e.c 4 65.r odd 12 2
845.2.e.h 4 65.q odd 12 2
845.2.m.f 8 65.o even 12 2
845.2.m.f 8 65.t even 12 2
1040.2.a.j 2 20.e even 4 1
2925.2.a.u 2 15.e even 4 1
2925.2.c.r 4 3.b odd 2 1
2925.2.c.r 4 15.d odd 2 1
3185.2.a.j 2 35.f even 4 1
4160.2.a.z 2 40.k even 4 1
4160.2.a.bf 2 40.i odd 4 1
4225.2.a.r 2 65.h odd 4 1
5200.2.a.bu 2 20.e even 4 1
7605.2.a.x 2 195.s even 4 1
7865.2.a.j 2 55.e even 4 1
9360.2.a.cd 2 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(325, [\chi])\):

\( T_{2}^{4} + 6T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{2} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 6T^{2} + 1 \) Copy content Toggle raw display
$3$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 24T^{2} + 16 \) Copy content Toggle raw display
$11$ \( (T^{2} - 4 T + 2)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 24T^{2} + 16 \) Copy content Toggle raw display
$19$ \( (T^{2} + 4 T + 2)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 12 T + 18)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 72)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 12 T + 28)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 132T^{2} + 1156 \) Copy content Toggle raw display
$47$ \( T^{4} + 24T^{2} + 16 \) Copy content Toggle raw display
$53$ \( T^{4} + 216T^{2} + 1296 \) Copy content Toggle raw display
$59$ \( (T^{2} + 12 T + 18)^{2} \) Copy content Toggle raw display
$61$ \( (T + 8)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 4 T - 94)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 72)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 72)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 88T^{2} + 784 \) Copy content Toggle raw display
$89$ \( (T + 6)^{4} \) Copy content Toggle raw display
$97$ \( T^{4} + 72T^{2} + 784 \) Copy content Toggle raw display
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