Properties

Label 1225.2.b.f
Level $1225$
Weight $2$
Character orbit 1225.b
Analytic conductor $9.782$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1225 = 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1225.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.78167424761\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{17})\)
Defining polynomial: \( x^{4} + 9x^{2} + 16 \) 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 35)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 9x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 5\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{2} - 5\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1225\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(1177\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1
2.56155i
1.56155i
1.56155i
2.56155i
2.56155i 1.56155i −4.56155 0 4.00000 0 6.56155i 0.561553 0
99.2 1.56155i 2.56155i −0.438447 0 4.00000 0 2.43845i −3.56155 0
99.3 1.56155i 2.56155i −0.438447 0 4.00000 0 2.43845i −3.56155 0
99.4 2.56155i 1.56155i −4.56155 0 4.00000 0 6.56155i 0.561553 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 1225.2.b.f 4
5.b even 2 1 inner 1225.2.b.f 4
5.c odd 4 1 245.2.a.d 2
5.c odd 4 1 1225.2.a.s 2
7.b odd 2 1 175.2.b.b 4
15.e even 4 1 2205.2.a.x 2
20.e even 4 1 3920.2.a.bs 2
21.c even 2 1 1575.2.d.e 4
28.d even 2 1 2800.2.g.t 4
35.c odd 2 1 175.2.b.b 4
35.f even 4 1 35.2.a.b 2
35.f even 4 1 175.2.a.f 2
35.k even 12 2 245.2.e.i 4
35.l odd 12 2 245.2.e.h 4
105.g even 2 1 1575.2.d.e 4
105.k odd 4 1 315.2.a.e 2
105.k odd 4 1 1575.2.a.p 2
140.c even 2 1 2800.2.g.t 4
140.j odd 4 1 560.2.a.i 2
140.j odd 4 1 2800.2.a.bi 2
280.s even 4 1 2240.2.a.bh 2
280.y odd 4 1 2240.2.a.bd 2
385.l odd 4 1 4235.2.a.m 2
420.w even 4 1 5040.2.a.bt 2
455.s even 4 1 5915.2.a.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.a.b 2 35.f even 4 1
175.2.a.f 2 35.f even 4 1
175.2.b.b 4 7.b odd 2 1
175.2.b.b 4 35.c odd 2 1
245.2.a.d 2 5.c odd 4 1
245.2.e.h 4 35.l odd 12 2
245.2.e.i 4 35.k even 12 2
315.2.a.e 2 105.k odd 4 1
560.2.a.i 2 140.j odd 4 1
1225.2.a.s 2 5.c odd 4 1
1225.2.b.f 4 1.a even 1 1 trivial
1225.2.b.f 4 5.b even 2 1 inner
1575.2.a.p 2 105.k odd 4 1
1575.2.d.e 4 21.c even 2 1
1575.2.d.e 4 105.g even 2 1
2205.2.a.x 2 15.e even 4 1
2240.2.a.bd 2 280.y odd 4 1
2240.2.a.bh 2 280.s even 4 1
2800.2.a.bi 2 140.j odd 4 1
2800.2.g.t 4 28.d even 2 1
2800.2.g.t 4 140.c even 2 1
3920.2.a.bs 2 20.e even 4 1
4235.2.a.m 2 385.l odd 4 1
5040.2.a.bt 2 420.w even 4 1
5915.2.a.l 2 455.s even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1225, [\chi])\):

\( T_{2}^{4} + 9T_{2}^{2} + 16 \) Copy content Toggle raw display
\( T_{19}^{2} + 6T_{19} - 8 \) Copy content Toggle raw display
\( T_{31} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 9T^{2} + 16 \) Copy content Toggle raw display
$3$ \( T^{4} + 9T^{2} + 16 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - T - 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 21T^{2} + 4 \) Copy content Toggle raw display
$17$ \( T^{4} + 21T^{2} + 4 \) Copy content Toggle raw display
$19$ \( (T^{2} + 6 T - 8)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 36T^{2} + 256 \) Copy content Toggle raw display
$29$ \( (T^{2} + T - 38)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 2 T - 16)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 84T^{2} + 64 \) Copy content Toggle raw display
$47$ \( T^{4} + 89T^{2} + 1024 \) Copy content Toggle raw display
$53$ \( T^{4} + 36T^{2} + 256 \) Copy content Toggle raw display
$59$ \( (T + 4)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 6 T - 144)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 144T^{2} + 4096 \) Copy content Toggle raw display
$71$ \( (T - 8)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 168T^{2} + 2704 \) Copy content Toggle raw display
$79$ \( (T^{2} - 9 T + 16)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 6 T - 8)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 253T^{2} + 7396 \) Copy content Toggle raw display
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