Properties

Label 152.2.c.a
Level $152$
Weight $2$
Character orbit 152.c
Analytic conductor $1.214$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 152 = 2^{3} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 152.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.21372611072\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + i ) q^{2} -2 i q^{4} + 2 q^{7} + ( 2 + 2 i ) q^{8} + 3 q^{9} +O(q^{10})\) \( q + ( -1 + i ) q^{2} -2 i q^{4} + 2 q^{7} + ( 2 + 2 i ) q^{8} + 3 q^{9} + 4 i q^{11} -2 i q^{13} + ( -2 + 2 i ) q^{14} -4 q^{16} + 2 q^{17} + ( -3 + 3 i ) q^{18} + i q^{19} + ( -4 - 4 i ) q^{22} -2 q^{23} + 5 q^{25} + ( 2 + 2 i ) q^{26} -4 i q^{28} -2 i q^{29} -8 q^{31} + ( 4 - 4 i ) q^{32} + ( -2 + 2 i ) q^{34} -6 i q^{36} -10 i q^{37} + ( -1 - i ) q^{38} -6 q^{41} -4 i q^{43} + 8 q^{44} + ( 2 - 2 i ) q^{46} -2 q^{47} -3 q^{49} + ( -5 + 5 i ) q^{50} -4 q^{52} + 2 i q^{53} + ( 4 + 4 i ) q^{56} + ( 2 + 2 i ) q^{58} + 12 i q^{59} + 8 i q^{61} + ( 8 - 8 i ) q^{62} + 6 q^{63} + 8 i q^{64} -4 i q^{67} -4 i q^{68} -12 q^{71} + ( 6 + 6 i ) q^{72} -10 q^{73} + ( 10 + 10 i ) q^{74} + 2 q^{76} + 8 i q^{77} + 4 q^{79} + 9 q^{81} + ( 6 - 6 i ) q^{82} -4 i q^{83} + ( 4 + 4 i ) q^{86} + ( -8 + 8 i ) q^{88} + 6 q^{89} -4 i q^{91} + 4 i q^{92} + ( 2 - 2 i ) q^{94} -10 q^{97} + ( 3 - 3 i ) q^{98} + 12 i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 4q^{7} + 4q^{8} + 6q^{9} + O(q^{10}) \) \( 2q - 2q^{2} + 4q^{7} + 4q^{8} + 6q^{9} - 4q^{14} - 8q^{16} + 4q^{17} - 6q^{18} - 8q^{22} - 4q^{23} + 10q^{25} + 4q^{26} - 16q^{31} + 8q^{32} - 4q^{34} - 2q^{38} - 12q^{41} + 16q^{44} + 4q^{46} - 4q^{47} - 6q^{49} - 10q^{50} - 8q^{52} + 8q^{56} + 4q^{58} + 16q^{62} + 12q^{63} - 24q^{71} + 12q^{72} - 20q^{73} + 20q^{74} + 4q^{76} + 8q^{79} + 18q^{81} + 12q^{82} + 8q^{86} - 16q^{88} + 12q^{89} + 4q^{94} - 20q^{97} + 6q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/152\mathbb{Z}\right)^\times\).

\(n\) \(39\) \(77\) \(97\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
77.1
1.00000i
1.00000i
−1.00000 1.00000i 0 2.00000i 0 0 2.00000 2.00000 2.00000i 3.00000 0
77.2 −1.00000 + 1.00000i 0 2.00000i 0 0 2.00000 2.00000 + 2.00000i 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
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 152.2.c.a 2
3.b odd 2 1 1368.2.g.a 2
4.b odd 2 1 608.2.c.a 2
8.b even 2 1 inner 152.2.c.a 2
8.d odd 2 1 608.2.c.a 2
12.b even 2 1 5472.2.g.a 2
16.e even 4 1 4864.2.a.g 1
16.e even 4 1 4864.2.a.h 1
16.f odd 4 1 4864.2.a.i 1
16.f odd 4 1 4864.2.a.j 1
24.f even 2 1 5472.2.g.a 2
24.h odd 2 1 1368.2.g.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.2.c.a 2 1.a even 1 1 trivial
152.2.c.a 2 8.b even 2 1 inner
608.2.c.a 2 4.b odd 2 1
608.2.c.a 2 8.d odd 2 1
1368.2.g.a 2 3.b odd 2 1
1368.2.g.a 2 24.h odd 2 1
4864.2.a.g 1 16.e even 4 1
4864.2.a.h 1 16.e even 4 1
4864.2.a.i 1 16.f odd 4 1
4864.2.a.j 1 16.f odd 4 1
5472.2.g.a 2 12.b even 2 1
5472.2.g.a 2 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{2}^{\mathrm{new}}(152, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2 + 2 T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( ( -2 + T )^{2} \)
$11$ \( 16 + T^{2} \)
$13$ \( 4 + T^{2} \)
$17$ \( ( -2 + T )^{2} \)
$19$ \( 1 + T^{2} \)
$23$ \( ( 2 + T )^{2} \)
$29$ \( 4 + T^{2} \)
$31$ \( ( 8 + T )^{2} \)
$37$ \( 100 + T^{2} \)
$41$ \( ( 6 + T )^{2} \)
$43$ \( 16 + T^{2} \)
$47$ \( ( 2 + T )^{2} \)
$53$ \( 4 + T^{2} \)
$59$ \( 144 + T^{2} \)
$61$ \( 64 + T^{2} \)
$67$ \( 16 + T^{2} \)
$71$ \( ( 12 + T )^{2} \)
$73$ \( ( 10 + T )^{2} \)
$79$ \( ( -4 + T )^{2} \)
$83$ \( 16 + T^{2} \)
$89$ \( ( -6 + T )^{2} \)
$97$ \( ( 10 + T )^{2} \)
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