Properties

Label 425.2.b.b
Level $425$
Weight $2$
Character orbit 425.b
Analytic conductor $3.394$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 425.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.39364208590\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 17)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + i q^{2} + q^{4} - 4 i q^{7} + 3 i q^{8} + 3 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + i q^{2} + q^{4} - 4 i q^{7} + 3 i q^{8} + 3 q^{9} - 2 i q^{13} + 4 q^{14} - q^{16} - i q^{17} + 3 i q^{18} + 4 q^{19} + 4 i q^{23} + 2 q^{26} - 4 i q^{28} - 6 q^{29} + 4 q^{31} + 5 i q^{32} + q^{34} + 3 q^{36} + 2 i q^{37} + 4 i q^{38} - 6 q^{41} + 4 i q^{43} - 4 q^{46} - 9 q^{49} - 2 i q^{52} + 6 i q^{53} + 12 q^{56} - 6 i q^{58} + 12 q^{59} - 10 q^{61} + 4 i q^{62} - 12 i q^{63} - 7 q^{64} - 4 i q^{67} - i q^{68} - 4 q^{71} + 9 i q^{72} - 6 i q^{73} - 2 q^{74} + 4 q^{76} - 12 q^{79} + 9 q^{81} - 6 i q^{82} - 4 i q^{83} - 4 q^{86} - 10 q^{89} - 8 q^{91} + 4 i q^{92} - 2 i q^{97} - 9 i q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{4} + 6 q^{9} + 8 q^{14} - 2 q^{16} + 8 q^{19} + 4 q^{26} - 12 q^{29} + 8 q^{31} + 2 q^{34} + 6 q^{36} - 12 q^{41} - 8 q^{46} - 18 q^{49} + 24 q^{56} + 24 q^{59} - 20 q^{61} - 14 q^{64} - 8 q^{71} - 4 q^{74} + 8 q^{76} - 24 q^{79} + 18 q^{81} - 8 q^{86} - 20 q^{89} - 16 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/425\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(326\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
324.1
1.00000i
1.00000i
1.00000i 0 1.00000 0 0 4.00000i 3.00000i 3.00000 0
324.2 1.00000i 0 1.00000 0 0 4.00000i 3.00000i 3.00000 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 425.2.b.b 2
5.b even 2 1 inner 425.2.b.b 2
5.c odd 4 1 17.2.a.a 1
5.c odd 4 1 425.2.a.d 1
15.e even 4 1 153.2.a.c 1
15.e even 4 1 3825.2.a.d 1
20.e even 4 1 272.2.a.b 1
20.e even 4 1 6800.2.a.n 1
35.f even 4 1 833.2.a.a 1
35.k even 12 2 833.2.e.a 2
35.l odd 12 2 833.2.e.b 2
40.i odd 4 1 1088.2.a.i 1
40.k even 4 1 1088.2.a.h 1
55.e even 4 1 2057.2.a.e 1
60.l odd 4 1 2448.2.a.o 1
65.h odd 4 1 2873.2.a.c 1
85.f odd 4 1 289.2.b.a 2
85.g odd 4 1 289.2.a.a 1
85.g odd 4 1 7225.2.a.g 1
85.i odd 4 1 289.2.b.a 2
85.k odd 8 2 289.2.c.a 4
85.n odd 8 2 289.2.c.a 4
85.o even 16 4 289.2.d.d 8
85.r even 16 4 289.2.d.d 8
95.g even 4 1 6137.2.a.b 1
105.k odd 4 1 7497.2.a.l 1
115.e even 4 1 8993.2.a.a 1
120.q odd 4 1 9792.2.a.i 1
120.w even 4 1 9792.2.a.n 1
255.o even 4 1 2601.2.a.g 1
340.r even 4 1 4624.2.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.2.a.a 1 5.c odd 4 1
153.2.a.c 1 15.e even 4 1
272.2.a.b 1 20.e even 4 1
289.2.a.a 1 85.g odd 4 1
289.2.b.a 2 85.f odd 4 1
289.2.b.a 2 85.i odd 4 1
289.2.c.a 4 85.k odd 8 2
289.2.c.a 4 85.n odd 8 2
289.2.d.d 8 85.o even 16 4
289.2.d.d 8 85.r even 16 4
425.2.a.d 1 5.c odd 4 1
425.2.b.b 2 1.a even 1 1 trivial
425.2.b.b 2 5.b even 2 1 inner
833.2.a.a 1 35.f even 4 1
833.2.e.a 2 35.k even 12 2
833.2.e.b 2 35.l odd 12 2
1088.2.a.h 1 40.k even 4 1
1088.2.a.i 1 40.i odd 4 1
2057.2.a.e 1 55.e even 4 1
2448.2.a.o 1 60.l odd 4 1
2601.2.a.g 1 255.o even 4 1
2873.2.a.c 1 65.h odd 4 1
3825.2.a.d 1 15.e even 4 1
4624.2.a.d 1 340.r even 4 1
6137.2.a.b 1 95.g even 4 1
6800.2.a.n 1 20.e even 4 1
7225.2.a.g 1 85.g odd 4 1
7497.2.a.l 1 105.k odd 4 1
8993.2.a.a 1 115.e even 4 1
9792.2.a.i 1 120.q odd 4 1
9792.2.a.n 1 120.w even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(425, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 16 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 4 \) Copy content Toggle raw display
$17$ \( T^{2} + 1 \) Copy content Toggle raw display
$19$ \( (T - 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 16 \) Copy content Toggle raw display
$29$ \( (T + 6)^{2} \) Copy content Toggle raw display
$31$ \( (T - 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 4 \) Copy content Toggle raw display
$41$ \( (T + 6)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 16 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 36 \) Copy content Toggle raw display
$59$ \( (T - 12)^{2} \) Copy content Toggle raw display
$61$ \( (T + 10)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 16 \) Copy content Toggle raw display
$71$ \( (T + 4)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 36 \) Copy content Toggle raw display
$79$ \( (T + 12)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 16 \) Copy content Toggle raw display
$89$ \( (T + 10)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 4 \) Copy content Toggle raw display
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