Properties

Label 275.2.b.c
Level $275$
Weight $2$
Character orbit 275.b
Analytic conductor $2.196$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 275 = 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 275.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.19588605559\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{13})\)
Defining polynomial: \( x^{4} + 7x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + \beta_1) q^{3} + (\beta_{3} - 2) q^{4} - 3 q^{6} + ( - 2 \beta_{2} + \beta_1) q^{7} + 3 \beta_{2} q^{8} - \beta_{3} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + \beta_1) q^{3} + (\beta_{3} - 2) q^{4} - 3 q^{6} + ( - 2 \beta_{2} + \beta_1) q^{7} + 3 \beta_{2} q^{8} - \beta_{3} q^{9} - q^{11} + (2 \beta_{2} - \beta_1) q^{12} + 5 \beta_{2} q^{13} + (3 \beta_{3} - 6) q^{14} + ( - \beta_{3} - 1) q^{16} + ( - 3 \beta_{2} - 3 \beta_1) q^{17} - 3 \beta_{2} q^{18} + q^{19} + (2 \beta_{3} - 3) q^{21} - \beta_1 q^{22} + ( - 6 \beta_{2} - \beta_1) q^{23} - 3 \beta_{3} q^{24} + ( - 5 \beta_{3} + 5) q^{26} + ( - \beta_{2} + 2 \beta_1) q^{27} + (5 \beta_{2} - 4 \beta_1) q^{28} + ( - 3 \beta_{3} + 6) q^{29} + (4 \beta_{3} + 1) q^{31} + (3 \beta_{2} - \beta_1) q^{32} + ( - \beta_{2} - \beta_1) q^{33} + 9 q^{34} + (\beta_{3} - 3) q^{36} + ( - 5 \beta_{2} + 2 \beta_1) q^{37} + \beta_1 q^{38} - 5 \beta_{3} q^{39} + ( - 2 \beta_{3} - 1) q^{41} + (6 \beta_{2} - 3 \beta_1) q^{42} + (2 \beta_{2} + 4 \beta_1) q^{43} + ( - \beta_{3} + 2) q^{44} + (5 \beta_{3} - 2) q^{46} + 3 \beta_{2} q^{47} + ( - 5 \beta_{2} - 2 \beta_1) q^{48} + (5 \beta_{3} - 5) q^{49} + (3 \beta_{3} + 9) q^{51} + ( - 5 \beta_{2} + 5 \beta_1) q^{52} - \beta_1 q^{53} + (3 \beta_{3} - 9) q^{54} + ( - 3 \beta_{3} + 9) q^{56} + (\beta_{2} + \beta_1) q^{57} + ( - 9 \beta_{2} + 6 \beta_1) q^{58} + (4 \beta_{3} + 5) q^{59} + ( - 3 \beta_{3} - 1) q^{61} + (12 \beta_{2} + \beta_1) q^{62} + ( - \beta_{2} + 2 \beta_1) q^{63} + ( - 6 \beta_{3} + 5) q^{64} + 3 q^{66} + 4 \beta_{2} q^{67} + ( - 6 \beta_{2} + 3 \beta_1) q^{68} + (6 \beta_{3} + 3) q^{69} + ( - 2 \beta_{3} + 2) q^{71} + ( - 3 \beta_{2} - 3 \beta_1) q^{72} + ( - 4 \beta_{2} - 3 \beta_1) q^{73} + (7 \beta_{3} - 13) q^{74} + (\beta_{3} - 2) q^{76} + (2 \beta_{2} - \beta_1) q^{77} - 15 \beta_{2} q^{78} + (3 \beta_{3} + 4) q^{79} + ( - 2 \beta_{3} - 6) q^{81} + ( - 6 \beta_{2} - \beta_1) q^{82} + (3 \beta_{2} - 5 \beta_1) q^{83} + ( - 5 \beta_{3} + 12) q^{84} + (2 \beta_{3} - 14) q^{86} + ( - 6 \beta_{2} + 3 \beta_1) q^{87} - 3 \beta_{2} q^{88} + (\beta_{3} - 4) q^{89} + ( - 5 \beta_{3} + 15) q^{91} + (3 \beta_{2} - 4 \beta_1) q^{92} + (17 \beta_{2} + 5 \beta_1) q^{93} + ( - 3 \beta_{3} + 3) q^{94} + ( - 3 \beta_{3} + 3) q^{96} + ( - 14 \beta_{2} - \beta_1) q^{97} + (15 \beta_{2} - 5 \beta_1) q^{98} + \beta_{3} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{4} - 12 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 6 q^{4} - 12 q^{6} - 2 q^{9} - 4 q^{11} - 18 q^{14} - 6 q^{16} + 4 q^{19} - 8 q^{21} - 6 q^{24} + 10 q^{26} + 18 q^{29} + 12 q^{31} + 36 q^{34} - 10 q^{36} - 10 q^{39} - 8 q^{41} + 6 q^{44} + 2 q^{46} - 10 q^{49} + 42 q^{51} - 30 q^{54} + 30 q^{56} + 28 q^{59} - 10 q^{61} + 8 q^{64} + 12 q^{66} + 24 q^{69} + 4 q^{71} - 38 q^{74} - 6 q^{76} + 22 q^{79} - 28 q^{81} + 38 q^{84} - 52 q^{86} - 14 q^{89} + 50 q^{91} + 6 q^{94} + 6 q^{96} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 7x^{2} + 9 \) : Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/275\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
2.30278i
1.30278i
1.30278i
2.30278i
2.30278i 1.30278i −3.30278 0 −3.00000 4.30278i 3.00000i 1.30278 0
199.2 1.30278i 2.30278i 0.302776 0 −3.00000 0.697224i 3.00000i −2.30278 0
199.3 1.30278i 2.30278i 0.302776 0 −3.00000 0.697224i 3.00000i −2.30278 0
199.4 2.30278i 1.30278i −3.30278 0 −3.00000 4.30278i 3.00000i 1.30278 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 275.2.b.c 4
3.b odd 2 1 2475.2.c.k 4
4.b odd 2 1 4400.2.b.y 4
5.b even 2 1 inner 275.2.b.c 4
5.c odd 4 1 275.2.a.e 2
5.c odd 4 1 275.2.a.f yes 2
15.d odd 2 1 2475.2.c.k 4
15.e even 4 1 2475.2.a.o 2
15.e even 4 1 2475.2.a.t 2
20.d odd 2 1 4400.2.b.y 4
20.e even 4 1 4400.2.a.bh 2
20.e even 4 1 4400.2.a.bs 2
55.e even 4 1 3025.2.a.h 2
55.e even 4 1 3025.2.a.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
275.2.a.e 2 5.c odd 4 1
275.2.a.f yes 2 5.c odd 4 1
275.2.b.c 4 1.a even 1 1 trivial
275.2.b.c 4 5.b even 2 1 inner
2475.2.a.o 2 15.e even 4 1
2475.2.a.t 2 15.e even 4 1
2475.2.c.k 4 3.b odd 2 1
2475.2.c.k 4 15.d odd 2 1
3025.2.a.h 2 55.e even 4 1
3025.2.a.n 2 55.e even 4 1
4400.2.a.bh 2 20.e even 4 1
4400.2.a.bs 2 20.e even 4 1
4400.2.b.y 4 4.b odd 2 1
4400.2.b.y 4 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 7T_{2}^{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(275, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 7T^{2} + 9 \) Copy content Toggle raw display
$3$ \( T^{4} + 7T^{2} + 9 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 19T^{2} + 9 \) Copy content Toggle raw display
$11$ \( (T + 1)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 25)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 63T^{2} + 729 \) Copy content Toggle raw display
$19$ \( (T - 1)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + 67T^{2} + 729 \) Copy content Toggle raw display
$29$ \( (T^{2} - 9 T - 9)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 6 T - 43)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 98T^{2} + 529 \) Copy content Toggle raw display
$41$ \( (T^{2} + 4 T - 9)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 52)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 7T^{2} + 9 \) Copy content Toggle raw display
$59$ \( (T^{2} - 14 T - 3)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 5 T - 23)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 2 T - 12)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 71T^{2} + 529 \) Copy content Toggle raw display
$79$ \( (T^{2} - 11 T + 1)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 223T^{2} + 2601 \) Copy content Toggle raw display
$89$ \( (T^{2} + 7 T + 9)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 371 T^{2} + 32041 \) Copy content Toggle raw display
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