Properties

Label 275.2.b.e
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: \(\Q(i, \sqrt{5})\)
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{3} - \beta_1) q^{3} + (\beta_{2} + 1) q^{4} + ( - 2 \beta_{2} + 1) q^{6} + ( - 2 \beta_{3} - 3 \beta_1) q^{7} + (\beta_{3} + 2 \beta_1) q^{8} + (3 \beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{3} - \beta_1) q^{3} + (\beta_{2} + 1) q^{4} + ( - 2 \beta_{2} + 1) q^{6} + ( - 2 \beta_{3} - 3 \beta_1) q^{7} + (\beta_{3} + 2 \beta_1) q^{8} + (3 \beta_{2} + 1) q^{9} + q^{11} + \beta_1 q^{12} + (3 \beta_{3} - 2 \beta_1) q^{13} + ( - \beta_{2} + 3) q^{14} + 3 \beta_{2} q^{16} + ( - \beta_{3} - \beta_1) q^{17} + (3 \beta_{3} - 2 \beta_1) q^{18} + ( - 6 \beta_{2} - 3) q^{19} + (4 \beta_{2} - 1) q^{21} + \beta_1 q^{22} + (4 \beta_{3} + 5 \beta_1) q^{23} + ( - 3 \beta_{2} + 1) q^{24} + ( - 5 \beta_{2} + 2) q^{26} + (\beta_{3} + 2 \beta_1) q^{27} + ( - 5 \beta_{3} - 2 \beta_1) q^{28} + (\beta_{2} + 3) q^{29} - 3 q^{31} + (5 \beta_{3} + \beta_1) q^{32} + (\beta_{3} - \beta_1) q^{33} + q^{34} + (\beta_{2} + 4) q^{36} + ( - 7 \beta_{3} + 2 \beta_1) q^{37} + ( - 6 \beta_{3} + 3 \beta_1) q^{38} + (7 \beta_{2} - 5) q^{39} - 3 q^{41} + (4 \beta_{3} - 5 \beta_1) q^{42} - 6 \beta_{3} q^{43} + (\beta_{2} + 1) q^{44} + (\beta_{2} - 5) q^{46} + (\beta_{3} + 8 \beta_1) q^{47} + ( - 3 \beta_{3} + 6 \beta_1) q^{48} + ( - 3 \beta_{2} - 6) q^{49} + \beta_{2} q^{51} + (\beta_{3} + 3 \beta_1) q^{52} + ( - 2 \beta_{3} - 7 \beta_1) q^{53} + (\beta_{2} - 2) q^{54} + (\beta_{2} + 8) q^{56} + (3 \beta_{3} - 9 \beta_1) q^{57} + (\beta_{3} + 2 \beta_1) q^{58} + ( - 4 \beta_{2} - 7) q^{59} + ( - 5 \beta_{2} - 8) q^{61} - 3 \beta_1 q^{62} - 11 \beta_{3} q^{63} + (2 \beta_{2} - 1) q^{64} + ( - 2 \beta_{2} + 1) q^{66} - 8 \beta_{3} q^{67} + ( - 2 \beta_{3} - \beta_1) q^{68} + ( - 6 \beta_{2} + 1) q^{69} + ( - 10 \beta_{2} - 8) q^{71} + (7 \beta_{3} - \beta_1) q^{72} + (12 \beta_{3} + \beta_1) q^{73} + (9 \beta_{2} - 2) q^{74} + ( - 3 \beta_{2} - 9) q^{76} + ( - 2 \beta_{3} - 3 \beta_1) q^{77} + (7 \beta_{3} - 12 \beta_1) q^{78} + (3 \beta_{2} - 1) q^{79} + (6 \beta_{2} + 4) q^{81} - 3 \beta_1 q^{82} + ( - 15 \beta_{3} - 3 \beta_1) q^{83} + ( - \beta_{2} + 3) q^{84} + 6 \beta_{2} q^{86} + (2 \beta_{3} - \beta_1) q^{87} + (\beta_{3} + 2 \beta_1) q^{88} + (5 \beta_{2} + 15) q^{89} + 11 \beta_{2} q^{91} + (9 \beta_{3} + 4 \beta_1) q^{92} + ( - 3 \beta_{3} + 3 \beta_1) q^{93} + (7 \beta_{2} - 8) q^{94} + (3 \beta_{2} - 4) q^{96} + \beta_1 q^{97} + ( - 3 \beta_{3} - 3 \beta_1) q^{98} + (3 \beta_{2} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} + 8 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{4} + 8 q^{6} - 2 q^{9} + 4 q^{11} + 14 q^{14} - 6 q^{16} - 12 q^{21} + 10 q^{24} + 18 q^{26} + 10 q^{29} - 12 q^{31} + 4 q^{34} + 14 q^{36} - 34 q^{39} - 12 q^{41} + 2 q^{44} - 22 q^{46} - 18 q^{49} - 2 q^{51} - 10 q^{54} + 30 q^{56} - 20 q^{59} - 22 q^{61} - 8 q^{64} + 8 q^{66} + 16 q^{69} - 12 q^{71} - 26 q^{74} - 30 q^{76} - 10 q^{79} + 4 q^{81} + 14 q^{84} - 12 q^{86} + 50 q^{89} - 22 q^{91} - 46 q^{94} - 22 q^{96} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 3x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 2\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 2\beta_1 \) 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
1.61803i
0.618034i
0.618034i
1.61803i
1.61803i 2.61803i −0.618034 0 4.23607 2.85410i 2.23607i −3.85410 0
199.2 0.618034i 0.381966i 1.61803 0 −0.236068 3.85410i 2.23607i 2.85410 0
199.3 0.618034i 0.381966i 1.61803 0 −0.236068 3.85410i 2.23607i 2.85410 0
199.4 1.61803i 2.61803i −0.618034 0 4.23607 2.85410i 2.23607i −3.85410 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.e 4
3.b odd 2 1 2475.2.c.p 4
4.b odd 2 1 4400.2.b.x 4
5.b even 2 1 inner 275.2.b.e 4
5.c odd 4 1 275.2.a.d 2
5.c odd 4 1 275.2.a.g yes 2
15.d odd 2 1 2475.2.c.p 4
15.e even 4 1 2475.2.a.n 2
15.e even 4 1 2475.2.a.s 2
20.d odd 2 1 4400.2.b.x 4
20.e even 4 1 4400.2.a.bg 2
20.e even 4 1 4400.2.a.bv 2
55.e even 4 1 3025.2.a.i 2
55.e even 4 1 3025.2.a.m 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
275.2.a.d 2 5.c odd 4 1
275.2.a.g yes 2 5.c odd 4 1
275.2.b.e 4 1.a even 1 1 trivial
275.2.b.e 4 5.b even 2 1 inner
2475.2.a.n 2 15.e even 4 1
2475.2.a.s 2 15.e even 4 1
2475.2.c.p 4 3.b odd 2 1
2475.2.c.p 4 15.d odd 2 1
3025.2.a.i 2 55.e even 4 1
3025.2.a.m 2 55.e even 4 1
4400.2.a.bg 2 20.e even 4 1
4400.2.a.bv 2 20.e even 4 1
4400.2.b.x 4 4.b odd 2 1
4400.2.b.x 4 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 3T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(275, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 3T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} + 7T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 23T^{2} + 121 \) Copy content Toggle raw display
$11$ \( (T - 1)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 42T^{2} + 121 \) Copy content Toggle raw display
$17$ \( T^{4} + 3T^{2} + 1 \) Copy content Toggle raw display
$19$ \( (T^{2} - 45)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 67T^{2} + 841 \) Copy content Toggle raw display
$29$ \( (T^{2} - 5 T + 5)^{2} \) Copy content Toggle raw display
$31$ \( (T + 3)^{4} \) Copy content Toggle raw display
$37$ \( T^{4} + 138T^{2} + 3481 \) Copy content Toggle raw display
$41$ \( (T + 3)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 178T^{2} + 5041 \) Copy content Toggle raw display
$53$ \( T^{4} + 127T^{2} + 3481 \) Copy content Toggle raw display
$59$ \( (T^{2} + 10 T + 5)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 11 T - 1)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 6 T - 116)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 267 T^{2} + 17161 \) Copy content Toggle raw display
$79$ \( (T^{2} + 5 T - 5)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 387 T^{2} + 29241 \) Copy content Toggle raw display
$89$ \( (T^{2} - 25 T + 125)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 3T^{2} + 1 \) Copy content Toggle raw display
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