Properties

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

Related objects

Downloads

Learn more

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: \(2\)
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 55)
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.

\(f(q)\) \(=\) \( q + i q^{2} + q^{4} + 3 i q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + i q^{2} + q^{4} + 3 i q^{8} + 3 q^{9} - q^{11} - 2 i q^{13} - q^{16} + 6 i q^{17} + 3 i q^{18} + 4 q^{19} - i q^{22} - 4 i q^{23} + 2 q^{26} - 6 q^{29} - 8 q^{31} + 5 i q^{32} - 6 q^{34} + 3 q^{36} - 2 i q^{37} + 4 i q^{38} + 2 q^{41} - 4 i q^{43} - q^{44} + 4 q^{46} - 12 i q^{47} + 7 q^{49} - 2 i q^{52} + 2 i q^{53} - 6 i q^{58} - 4 q^{59} - 10 q^{61} - 8 i q^{62} - 7 q^{64} - 16 i q^{67} + 6 i q^{68} + 8 q^{71} + 9 i q^{72} - 14 i q^{73} + 2 q^{74} + 4 q^{76} - 8 q^{79} + 9 q^{81} + 2 i q^{82} + 4 i q^{83} + 4 q^{86} - 3 i q^{88} - 10 q^{89} - 4 i q^{92} + 12 q^{94} + 10 i q^{97} + 7 i q^{98} - 3 q^{99} +O(q^{100}) \) 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 \( 2 q + 2 q^{4} + 6 q^{9} - 2 q^{11} - 2 q^{16} + 8 q^{19} + 4 q^{26} - 12 q^{29} - 16 q^{31} - 12 q^{34} + 6 q^{36} + 4 q^{41} - 2 q^{44} + 8 q^{46} + 14 q^{49} - 8 q^{59} - 20 q^{61} - 14 q^{64} + 16 q^{71} + 4 q^{74} + 8 q^{76} - 16 q^{79} + 18 q^{81} + 8 q^{86} - 20 q^{89} + 24 q^{94} - 6 q^{99}+O(q^{100}) \) 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
1.00000i
1.00000i
1.00000i 0 1.00000 0 0 0 3.00000i 3.00000 0
199.2 1.00000i 0 1.00000 0 0 0 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 275.2.b.b 2
3.b odd 2 1 2475.2.c.f 2
4.b odd 2 1 4400.2.b.n 2
5.b even 2 1 inner 275.2.b.b 2
5.c odd 4 1 55.2.a.a 1
5.c odd 4 1 275.2.a.a 1
15.d odd 2 1 2475.2.c.f 2
15.e even 4 1 495.2.a.a 1
15.e even 4 1 2475.2.a.i 1
20.d odd 2 1 4400.2.b.n 2
20.e even 4 1 880.2.a.h 1
20.e even 4 1 4400.2.a.p 1
35.f even 4 1 2695.2.a.c 1
40.i odd 4 1 3520.2.a.p 1
40.k even 4 1 3520.2.a.n 1
55.e even 4 1 605.2.a.b 1
55.e even 4 1 3025.2.a.f 1
55.k odd 20 4 605.2.g.a 4
55.l even 20 4 605.2.g.c 4
60.l odd 4 1 7920.2.a.i 1
65.h odd 4 1 9295.2.a.b 1
165.l odd 4 1 5445.2.a.i 1
220.i odd 4 1 9680.2.a.r 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.a.a 1 5.c odd 4 1
275.2.a.a 1 5.c odd 4 1
275.2.b.b 2 1.a even 1 1 trivial
275.2.b.b 2 5.b even 2 1 inner
495.2.a.a 1 15.e even 4 1
605.2.a.b 1 55.e even 4 1
605.2.g.a 4 55.k odd 20 4
605.2.g.c 4 55.l even 20 4
880.2.a.h 1 20.e even 4 1
2475.2.a.i 1 15.e even 4 1
2475.2.c.f 2 3.b odd 2 1
2475.2.c.f 2 15.d odd 2 1
2695.2.a.c 1 35.f even 4 1
3025.2.a.f 1 55.e even 4 1
3520.2.a.n 1 40.k even 4 1
3520.2.a.p 1 40.i odd 4 1
4400.2.a.p 1 20.e even 4 1
4400.2.b.n 2 4.b odd 2 1
4400.2.b.n 2 20.d odd 2 1
5445.2.a.i 1 165.l odd 4 1
7920.2.a.i 1 60.l odd 4 1
9295.2.a.b 1 65.h odd 4 1
9680.2.a.r 1 220.i odd 4 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( (T + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 4 \) Copy content Toggle raw display
$17$ \( T^{2} + 36 \) 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
$31$ \( (T + 8)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 4 \) Copy content Toggle raw display
$41$ \( (T - 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 16 \) Copy content Toggle raw display
$47$ \( T^{2} + 144 \) Copy content Toggle raw display
$53$ \( T^{2} + 4 \) Copy content Toggle raw display
$59$ \( (T + 4)^{2} \) Copy content Toggle raw display
$61$ \( (T + 10)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 256 \) Copy content Toggle raw display
$71$ \( (T - 8)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 196 \) Copy content Toggle raw display
$79$ \( (T + 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 16 \) Copy content Toggle raw display
$89$ \( (T + 10)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 100 \) Copy content Toggle raw display
show more
show less