Properties

Label 775.2.b.b
Level $775$
Weight $2$
Character orbit 775.b
Analytic conductor $6.188$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 775 = 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 775.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.18840615665\)
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 155)
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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/775\mathbb{Z}\right)^\times\).

\(n\) \(251\) \(652\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
249.1
1.00000i
1.00000i
1.00000i 2.00000i 1.00000 0 −2.00000 4.00000i 3.00000i −1.00000 0
249.2 1.00000i 2.00000i 1.00000 0 −2.00000 4.00000i 3.00000i −1.00000 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 775.2.b.b 2
5.b even 2 1 inner 775.2.b.b 2
5.c odd 4 1 155.2.a.b 1
5.c odd 4 1 775.2.a.b 1
15.e even 4 1 1395.2.a.d 1
15.e even 4 1 6975.2.a.d 1
20.e even 4 1 2480.2.a.b 1
35.f even 4 1 7595.2.a.c 1
40.i odd 4 1 9920.2.a.g 1
40.k even 4 1 9920.2.a.bd 1
155.f even 4 1 4805.2.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
155.2.a.b 1 5.c odd 4 1
775.2.a.b 1 5.c odd 4 1
775.2.b.b 2 1.a even 1 1 trivial
775.2.b.b 2 5.b even 2 1 inner
1395.2.a.d 1 15.e even 4 1
2480.2.a.b 1 20.e even 4 1
4805.2.a.d 1 155.f even 4 1
6975.2.a.d 1 15.e even 4 1
7595.2.a.c 1 35.f even 4 1
9920.2.a.g 1 40.i odd 4 1
9920.2.a.bd 1 40.k even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(775, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{2} + 4 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 16 \) Copy content Toggle raw display
$11$ \( (T - 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 64 \) Copy content Toggle raw display
$19$ \( (T + 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 4 \) Copy content Toggle raw display
$29$ \( (T - 6)^{2} \) Copy content Toggle raw display
$31$ \( (T - 1)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 16 \) Copy content Toggle raw display
$41$ \( (T + 6)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 36 \) Copy content Toggle raw display
$47$ \( T^{2} + 64 \) Copy content Toggle raw display
$53$ \( T^{2} + 144 \) Copy content Toggle raw display
$59$ \( (T - 4)^{2} \) Copy content Toggle raw display
$61$ \( (T - 10)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 64 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 16 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 4 \) Copy content Toggle raw display
$89$ \( (T + 14)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 324 \) Copy content Toggle raw display
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