Properties

Label 775.2.e.e
Level $775$
Weight $2$
Character orbit 775.e
Analytic conductor $6.188$
Analytic rank $0$
Dimension $4$
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.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.18840615665\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 31)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{3} + 1) q^{2} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{3} + (2 \beta_{3} + 1) q^{4} + ( - 2 \beta_{3} + 3 \beta_{2} - 2 \beta_1) q^{6} + (\beta_{3} - \beta_{2} + \beta_1) q^{7} + (\beta_{3} + 3) q^{8} + 2 \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + 1) q^{2} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{3} + (2 \beta_{3} + 1) q^{4} + ( - 2 \beta_{3} + 3 \beta_{2} - 2 \beta_1) q^{6} + (\beta_{3} - \beta_{2} + \beta_1) q^{7} + (\beta_{3} + 3) q^{8} + 2 \beta_1 q^{9} + (\beta_{2} - 3 \beta_1 + 1) q^{11} + ( - 3 \beta_{3} + 5 \beta_{2} - 3 \beta_1) q^{12} + ( - \beta_{2} - 2 \beta_1 - 1) q^{13} + (2 \beta_{3} - 3 \beta_{2} + 2 \beta_1) q^{14} + 3 q^{16} + (2 \beta_{3} + 3 \beta_{2} + 2 \beta_1) q^{17} + ( - 4 \beta_{2} + 2 \beta_1 - 4) q^{18} + (\beta_{3} + 3 \beta_{2} + \beta_1) q^{19} + (3 \beta_{2} - 2 \beta_1 + 3) q^{21} + (7 \beta_{2} - 4 \beta_1 + 7) q^{22} + 4 q^{23} + ( - 4 \beta_{3} + 5 \beta_{2} - 4 \beta_1) q^{24} + (3 \beta_{2} - \beta_1 + 3) q^{26} + ( - \beta_{3} + 1) q^{27} + (3 \beta_{3} - 5 \beta_{2} + 3 \beta_1) q^{28} + (2 \beta_{3} - 4) q^{29} + (\beta_{3} + 2 \beta_1 - 5) q^{31} + (\beta_{3} - 3) q^{32} + ( - 4 \beta_{3} - 7) q^{33} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{34} + ( - 8 \beta_{2} + 2 \beta_1 - 8) q^{36} - \beta_{2} q^{37} + ( - 2 \beta_{3} + \beta_{2} - 2 \beta_1) q^{38} + ( - \beta_{3} - 3) q^{39} + ( - \beta_{2} + 6 \beta_1 - 1) q^{41} + (7 \beta_{2} - 5 \beta_1 + 7) q^{42} + (7 \beta_{3} + \beta_{2} + 7 \beta_1) q^{43} + (13 \beta_{2} - 5 \beta_1 + 13) q^{44} + (4 \beta_{3} + 4) q^{46} + (4 \beta_{3} - 4) q^{47} + ( - 3 \beta_{3} + 3 \beta_{2} - 3 \beta_1) q^{48} + (4 \beta_{2} + 2 \beta_1 + 4) q^{49} + (\beta_{2} + \beta_1 + 1) q^{51} + (7 \beta_{2} + 7) q^{52} + (3 \beta_{2} + 2 \beta_1 + 3) q^{53} - q^{54} + (4 \beta_{3} - 5 \beta_{2} + 4 \beta_1) q^{56} + ( - \beta_{2} + 2 \beta_1 - 1) q^{57} - 2 \beta_{3} q^{58} + (5 \beta_{3} - 3 \beta_{2} + 5 \beta_1) q^{59} + 2 \beta_{3} q^{61} + ( - 4 \beta_{3} - 4 \beta_{2} + 2 \beta_1 - 7) q^{62} + ( - 2 \beta_{3} - 4) q^{63} + ( - 2 \beta_{3} - 7) q^{64} + ( - 11 \beta_{3} - 15) q^{66} + ( - \beta_{2} + 3 \beta_1 - 1) q^{67} + ( - 4 \beta_{3} - 5 \beta_{2} - 4 \beta_1) q^{68} + ( - 4 \beta_{3} + 4 \beta_{2} - 4 \beta_1) q^{69} + (7 \beta_{2} - 5 \beta_1 + 7) q^{71} + ( - 4 \beta_{2} + 6 \beta_1 - 4) q^{72} + (\beta_{2} - 2 \beta_1 + 1) q^{73} + (\beta_{3} - \beta_{2} + \beta_1) q^{74} + ( - 5 \beta_{3} - \beta_{2} - 5 \beta_1) q^{76} + (4 \beta_{3} + 7) q^{77} + ( - 4 \beta_{3} - 5) q^{78} + (3 \beta_{3} - 11 \beta_{2} + 3 \beta_1) q^{79} + (6 \beta_{3} - \beta_{2} + 6 \beta_1) q^{81} + ( - 13 \beta_{2} + 7 \beta_1 - 13) q^{82} + ( - 3 \beta_{2} - 5 \beta_1 - 3) q^{83} + (11 \beta_{2} - 8 \beta_1 + 11) q^{84} + (6 \beta_{3} - 13 \beta_{2} + 6 \beta_1) q^{86} + (2 \beta_{3} + 2 \beta_1) q^{87} + (9 \beta_{2} - 10 \beta_1 + 9) q^{88} + ( - 6 \beta_{3} - 4) q^{89} + (\beta_{3} + 3) q^{91} + (8 \beta_{3} + 4) q^{92} + (6 \beta_{3} - 3 \beta_{2} + 4 \beta_1 + 4) q^{93} + 4 q^{94} + (2 \beta_{3} - \beta_{2} + 2 \beta_1) q^{96} + ( - 2 \beta_{3} - 8) q^{97} - 2 \beta_1 q^{98} + (2 \beta_{3} - 12 \beta_{2} + 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} - 2 q^{3} + 4 q^{4} - 6 q^{6} + 2 q^{7} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} - 2 q^{3} + 4 q^{4} - 6 q^{6} + 2 q^{7} + 12 q^{8} + 2 q^{11} - 10 q^{12} - 2 q^{13} + 6 q^{14} + 12 q^{16} - 6 q^{17} - 8 q^{18} - 6 q^{19} + 6 q^{21} + 14 q^{22} + 16 q^{23} - 10 q^{24} + 6 q^{26} + 4 q^{27} + 10 q^{28} - 16 q^{29} - 20 q^{31} - 12 q^{32} - 28 q^{33} + 2 q^{34} - 16 q^{36} + 2 q^{37} - 2 q^{38} - 12 q^{39} - 2 q^{41} + 14 q^{42} - 2 q^{43} + 26 q^{44} + 16 q^{46} - 16 q^{47} - 6 q^{48} + 8 q^{49} + 2 q^{51} + 14 q^{52} + 6 q^{53} - 4 q^{54} + 10 q^{56} - 2 q^{57} + 6 q^{59} - 20 q^{62} - 16 q^{63} - 28 q^{64} - 60 q^{66} - 2 q^{67} + 10 q^{68} - 8 q^{69} + 14 q^{71} - 8 q^{72} + 2 q^{73} + 2 q^{74} + 2 q^{76} + 28 q^{77} - 20 q^{78} + 22 q^{79} + 2 q^{81} - 26 q^{82} - 6 q^{83} + 22 q^{84} + 26 q^{86} + 18 q^{88} - 16 q^{89} + 12 q^{91} + 16 q^{92} + 22 q^{93} + 16 q^{94} + 2 q^{96} - 32 q^{97} + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

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

Character values

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

\(n\) \(251\) \(652\)
\(\chi(n)\) \(-1 - \beta_{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
501.1
0.707107 + 1.22474i
−0.707107 1.22474i
0.707107 1.22474i
−0.707107 + 1.22474i
−0.414214 0.207107 0.358719i −1.82843 0 −0.0857864 + 0.148586i −0.207107 + 0.358719i 1.58579 1.41421 + 2.44949i 0
501.2 2.41421 −1.20711 + 2.09077i 3.82843 0 −2.91421 + 5.04757i 1.20711 2.09077i 4.41421 −1.41421 2.44949i 0
676.1 −0.414214 0.207107 + 0.358719i −1.82843 0 −0.0857864 0.148586i −0.207107 0.358719i 1.58579 1.41421 2.44949i 0
676.2 2.41421 −1.20711 2.09077i 3.82843 0 −2.91421 5.04757i 1.20711 + 2.09077i 4.41421 −1.41421 + 2.44949i 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
31.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 775.2.e.e 4
5.b even 2 1 31.2.c.a 4
5.c odd 4 2 775.2.o.d 8
15.d odd 2 1 279.2.h.c 4
20.d odd 2 1 496.2.i.h 4
31.c even 3 1 inner 775.2.e.e 4
155.c odd 2 1 961.2.c.a 4
155.i odd 6 1 961.2.a.c 2
155.i odd 6 1 961.2.c.a 4
155.j even 6 1 31.2.c.a 4
155.j even 6 1 961.2.a.a 2
155.m odd 10 4 961.2.g.r 16
155.n even 10 4 961.2.g.o 16
155.o odd 12 2 775.2.o.d 8
155.u even 30 4 961.2.d.l 8
155.u even 30 4 961.2.g.o 16
155.v odd 30 4 961.2.d.i 8
155.v odd 30 4 961.2.g.r 16
465.t even 6 1 8649.2.a.k 2
465.u odd 6 1 279.2.h.c 4
465.u odd 6 1 8649.2.a.l 2
620.o odd 6 1 496.2.i.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.2.c.a 4 5.b even 2 1
31.2.c.a 4 155.j even 6 1
279.2.h.c 4 15.d odd 2 1
279.2.h.c 4 465.u odd 6 1
496.2.i.h 4 20.d odd 2 1
496.2.i.h 4 620.o odd 6 1
775.2.e.e 4 1.a even 1 1 trivial
775.2.e.e 4 31.c even 3 1 inner
775.2.o.d 8 5.c odd 4 2
775.2.o.d 8 155.o odd 12 2
961.2.a.a 2 155.j even 6 1
961.2.a.c 2 155.i odd 6 1
961.2.c.a 4 155.c odd 2 1
961.2.c.a 4 155.i odd 6 1
961.2.d.i 8 155.v odd 30 4
961.2.d.l 8 155.u even 30 4
961.2.g.o 16 155.n even 10 4
961.2.g.o 16 155.u even 30 4
961.2.g.r 16 155.m odd 10 4
961.2.g.r 16 155.v odd 30 4
8649.2.a.k 2 465.t even 6 1
8649.2.a.l 2 465.u odd 6 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2 T - 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} + 2 T^{3} + 5 T^{2} - 2 T + 1 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 2 T^{3} + 5 T^{2} + 2 T + 1 \) Copy content Toggle raw display
$11$ \( T^{4} - 2 T^{3} + 21 T^{2} + 34 T + 289 \) Copy content Toggle raw display
$13$ \( T^{4} + 2 T^{3} + 11 T^{2} - 14 T + 49 \) Copy content Toggle raw display
$17$ \( T^{4} + 6 T^{3} + 35 T^{2} + 6 T + 1 \) Copy content Toggle raw display
$19$ \( T^{4} + 6 T^{3} + 29 T^{2} + 42 T + 49 \) Copy content Toggle raw display
$23$ \( (T - 4)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + 8 T + 8)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 10 T + 31)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 2 T^{3} + 75 T^{2} + \cdots + 5041 \) Copy content Toggle raw display
$43$ \( T^{4} + 2 T^{3} + 101 T^{2} + \cdots + 9409 \) Copy content Toggle raw display
$47$ \( (T^{2} + 8 T - 16)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} - 6 T^{3} + 35 T^{2} - 6 T + 1 \) Copy content Toggle raw display
$59$ \( T^{4} - 6 T^{3} + 77 T^{2} + \cdots + 1681 \) Copy content Toggle raw display
$61$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 2 T^{3} + 21 T^{2} - 34 T + 289 \) Copy content Toggle raw display
$71$ \( T^{4} - 14 T^{3} + 197 T^{2} + 14 T + 1 \) Copy content Toggle raw display
$73$ \( T^{4} - 2 T^{3} + 11 T^{2} + 14 T + 49 \) Copy content Toggle raw display
$79$ \( T^{4} - 22 T^{3} + 381 T^{2} + \cdots + 10609 \) Copy content Toggle raw display
$83$ \( T^{4} + 6 T^{3} + 77 T^{2} + \cdots + 1681 \) Copy content Toggle raw display
$89$ \( (T^{2} + 8 T - 56)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 16 T + 56)^{2} \) Copy content Toggle raw display
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