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: \(x^{4} + 7 x^{2} + 9\)
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.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 4 \nu \)\()/3\)
\(\beta_{3}\)\(=\)\( \nu^{2} + 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} - 4\)
\(\nu^{3}\)\(=\)\(3 \beta_{2} - 4 \beta_{1}\)

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 \( T_{2}^{4} + 7 T_{2}^{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(275, [\chi])\).

Hecke characteristic polynomials

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