Properties

Label 156.2.k.d
Level $156$
Weight $2$
Character orbit 156.k
Analytic conductor $1.246$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 156 = 2^{2} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 156.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.24566627153\)
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: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - i + 1) q^{2} - i q^{3} - 2 i q^{4} + (2 i + 2) q^{5} + ( - i - 1) q^{6} + ( - 2 i - 2) q^{8} - q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - i + 1) q^{2} - i q^{3} - 2 i q^{4} + (2 i + 2) q^{5} + ( - i - 1) q^{6} + ( - 2 i - 2) q^{8} - q^{9} + 4 q^{10} + ( - i - 1) q^{11} - 2 q^{12} + (2 i - 3) q^{13} + ( - 2 i + 2) q^{15} - 4 q^{16} + 2 i q^{17} + (i - 1) q^{18} + (2 i - 2) q^{19} + ( - 4 i + 4) q^{20} - 2 q^{22} + 6 q^{23} + (2 i - 2) q^{24} + 3 i q^{25} + (5 i - 1) q^{26} + i q^{27} + 10 q^{29} - 4 i q^{30} + (6 i - 6) q^{31} + (4 i - 4) q^{32} + (i - 1) q^{33} + (2 i + 2) q^{34} + 2 i q^{36} + ( - 3 i + 3) q^{37} + 4 i q^{38} + (3 i + 2) q^{39} - 8 i q^{40} + ( - 4 i - 4) q^{41} - 4 q^{43} + (2 i - 2) q^{44} + ( - 2 i - 2) q^{45} + ( - 6 i + 6) q^{46} + ( - 5 i - 5) q^{47} + 4 i q^{48} - 7 i q^{49} + (3 i + 3) q^{50} + 2 q^{51} + (6 i + 4) q^{52} - 6 q^{53} + (i + 1) q^{54} - 4 i q^{55} + (2 i + 2) q^{57} + ( - 10 i + 10) q^{58} + (7 i + 7) q^{59} + ( - 4 i - 4) q^{60} - 8 q^{61} + 12 i q^{62} + 8 i q^{64} + ( - 2 i - 10) q^{65} + 2 i q^{66} + (8 i - 8) q^{67} + 4 q^{68} - 6 i q^{69} + ( - 9 i + 9) q^{71} + (2 i + 2) q^{72} + ( - 5 i + 5) q^{73} - 6 i q^{74} + 3 q^{75} + (4 i + 4) q^{76} + (i + 5) q^{78} - 16 i q^{79} + ( - 8 i - 8) q^{80} + q^{81} - 8 q^{82} + (5 i - 5) q^{83} + (4 i - 4) q^{85} + (4 i - 4) q^{86} - 10 i q^{87} + 4 i q^{88} + ( - 2 i + 2) q^{89} - 4 q^{90} - 12 i q^{92} + (6 i + 6) q^{93} - 10 q^{94} - 8 q^{95} + (4 i + 4) q^{96} + (5 i + 5) q^{97} + ( - 7 i - 7) q^{98} + (i + 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 4 q^{5} - 2 q^{6} - 4 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 4 q^{5} - 2 q^{6} - 4 q^{8} - 2 q^{9} + 8 q^{10} - 2 q^{11} - 4 q^{12} - 6 q^{13} + 4 q^{15} - 8 q^{16} - 2 q^{18} - 4 q^{19} + 8 q^{20} - 4 q^{22} + 12 q^{23} - 4 q^{24} - 2 q^{26} + 20 q^{29} - 12 q^{31} - 8 q^{32} - 2 q^{33} + 4 q^{34} + 6 q^{37} + 4 q^{39} - 8 q^{41} - 8 q^{43} - 4 q^{44} - 4 q^{45} + 12 q^{46} - 10 q^{47} + 6 q^{50} + 4 q^{51} + 8 q^{52} - 12 q^{53} + 2 q^{54} + 4 q^{57} + 20 q^{58} + 14 q^{59} - 8 q^{60} - 16 q^{61} - 20 q^{65} - 16 q^{67} + 8 q^{68} + 18 q^{71} + 4 q^{72} + 10 q^{73} + 6 q^{75} + 8 q^{76} + 10 q^{78} - 16 q^{80} + 2 q^{81} - 16 q^{82} - 10 q^{83} - 8 q^{85} - 8 q^{86} + 4 q^{89} - 8 q^{90} + 12 q^{93} - 20 q^{94} - 16 q^{95} + 8 q^{96} + 10 q^{97} - 14 q^{98} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/156\mathbb{Z}\right)^\times\).

\(n\) \(53\) \(79\) \(145\)
\(\chi(n)\) \(1\) \(-1\) \(i\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
1.00000i
1.00000i
1.00000 + 1.00000i 1.00000i 2.00000i 2.00000 2.00000i −1.00000 + 1.00000i 0 −2.00000 + 2.00000i −1.00000 4.00000
151.1 1.00000 1.00000i 1.00000i 2.00000i 2.00000 + 2.00000i −1.00000 1.00000i 0 −2.00000 2.00000i −1.00000 4.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
52.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 156.2.k.d yes 2
3.b odd 2 1 468.2.n.b 2
4.b odd 2 1 156.2.k.c 2
12.b even 2 1 468.2.n.d 2
13.d odd 4 1 156.2.k.c 2
39.f even 4 1 468.2.n.d 2
52.f even 4 1 inner 156.2.k.d yes 2
156.l odd 4 1 468.2.n.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.2.k.c 2 4.b odd 2 1
156.2.k.c 2 13.d odd 4 1
156.2.k.d yes 2 1.a even 1 1 trivial
156.2.k.d yes 2 52.f even 4 1 inner
468.2.n.b 2 3.b odd 2 1
468.2.n.b 2 156.l odd 4 1
468.2.n.d 2 12.b even 2 1
468.2.n.d 2 39.f even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(156, [\chi])\):

\( T_{5}^{2} - 4T_{5} + 8 \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display
\( T_{11}^{2} + 2T_{11} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$3$ \( T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{2} - 4T + 8 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$13$ \( T^{2} + 6T + 13 \) Copy content Toggle raw display
$17$ \( T^{2} + 4 \) Copy content Toggle raw display
$19$ \( T^{2} + 4T + 8 \) Copy content Toggle raw display
$23$ \( (T - 6)^{2} \) Copy content Toggle raw display
$29$ \( (T - 10)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 12T + 72 \) Copy content Toggle raw display
$37$ \( T^{2} - 6T + 18 \) Copy content Toggle raw display
$41$ \( T^{2} + 8T + 32 \) Copy content Toggle raw display
$43$ \( (T + 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 10T + 50 \) Copy content Toggle raw display
$53$ \( (T + 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 14T + 98 \) Copy content Toggle raw display
$61$ \( (T + 8)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 16T + 128 \) Copy content Toggle raw display
$71$ \( T^{2} - 18T + 162 \) Copy content Toggle raw display
$73$ \( T^{2} - 10T + 50 \) Copy content Toggle raw display
$79$ \( T^{2} + 256 \) Copy content Toggle raw display
$83$ \( T^{2} + 10T + 50 \) Copy content Toggle raw display
$89$ \( T^{2} - 4T + 8 \) Copy content Toggle raw display
$97$ \( T^{2} - 10T + 50 \) Copy content Toggle raw display
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