Properties

Label 784.2.m.a
Level $784$
Weight $2$
Character orbit 784.m
Analytic conductor $6.260$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 784.m (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.26027151847\)
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 112)
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.

\(f(q)\) \(=\) \( q + (i - 1) q^{2} - 2 i q^{4} + ( - 2 i - 2) q^{5} + (2 i + 2) q^{8} + 3 i q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (i - 1) q^{2} - 2 i q^{4} + ( - 2 i - 2) q^{5} + (2 i + 2) q^{8} + 3 i q^{9} + 4 q^{10} + (i + 1) q^{11} - 4 q^{16} + 2 q^{17} + ( - 3 i - 3) q^{18} + (2 i - 2) q^{19} + (4 i - 4) q^{20} - 2 q^{22} - 6 i q^{23} + 3 i q^{25} + ( - 7 i + 7) q^{29} + 8 q^{31} + ( - 4 i + 4) q^{32} + (2 i - 2) q^{34} + 6 q^{36} + ( - 5 i - 5) q^{37} - 4 i q^{38} - 8 i q^{40} - 10 i q^{41} + ( - i - 1) q^{43} + ( - 2 i + 2) q^{44} + ( - 6 i + 6) q^{45} + (6 i + 6) q^{46} + 12 q^{47} + ( - 3 i - 3) q^{50} + ( - i - 1) q^{53} - 4 i q^{55} + 14 i q^{58} + ( - 8 i - 8) q^{59} + (6 i - 6) q^{61} + (8 i - 8) q^{62} + 8 i q^{64} + ( - 3 i + 3) q^{67} - 4 i q^{68} + (6 i - 6) q^{72} + 6 i q^{73} + 10 q^{74} + (4 i + 4) q^{76} + 10 q^{79} + (8 i + 8) q^{80} - 9 q^{81} + (10 i + 10) q^{82} + ( - 10 i + 10) q^{83} + ( - 4 i - 4) q^{85} + 2 q^{86} + 4 i q^{88} + 14 i q^{89} + 12 i q^{90} - 12 q^{92} + (12 i - 12) q^{94} + 8 q^{95} + 2 q^{97} + (3 i - 3) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 4 q^{5} + 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 4 q^{5} + 4 q^{8} + 8 q^{10} + 2 q^{11} - 8 q^{16} + 4 q^{17} - 6 q^{18} - 4 q^{19} - 8 q^{20} - 4 q^{22} + 14 q^{29} + 16 q^{31} + 8 q^{32} - 4 q^{34} + 12 q^{36} - 10 q^{37} - 2 q^{43} + 4 q^{44} + 12 q^{45} + 12 q^{46} + 24 q^{47} - 6 q^{50} - 2 q^{53} - 16 q^{59} - 12 q^{61} - 16 q^{62} + 6 q^{67} - 12 q^{72} + 20 q^{74} + 8 q^{76} + 20 q^{79} + 16 q^{80} - 18 q^{81} + 20 q^{82} + 20 q^{83} - 8 q^{85} + 4 q^{86} - 24 q^{92} - 24 q^{94} + 16 q^{95} + 4 q^{97} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/784\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(687\) \(689\)
\(\chi(n)\) \(i\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
197.1
1.00000i
1.00000i
−1.00000 + 1.00000i 0 2.00000i −2.00000 2.00000i 0 0 2.00000 + 2.00000i 3.00000i 4.00000
589.1 −1.00000 1.00000i 0 2.00000i −2.00000 + 2.00000i 0 0 2.00000 2.00000i 3.00000i 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
16.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.2.m.a 2
7.b odd 2 1 112.2.m.b 2
7.c even 3 2 784.2.x.e 4
7.d odd 6 2 784.2.x.d 4
16.e even 4 1 inner 784.2.m.a 2
28.d even 2 1 448.2.m.a 2
56.e even 2 1 896.2.m.c 2
56.h odd 2 1 896.2.m.b 2
112.j even 4 1 448.2.m.a 2
112.j even 4 1 896.2.m.c 2
112.l odd 4 1 112.2.m.b 2
112.l odd 4 1 896.2.m.b 2
112.w even 12 2 784.2.x.e 4
112.x odd 12 2 784.2.x.d 4
224.v odd 8 2 7168.2.a.b 2
224.x even 8 2 7168.2.a.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.m.b 2 7.b odd 2 1
112.2.m.b 2 112.l odd 4 1
448.2.m.a 2 28.d even 2 1
448.2.m.a 2 112.j even 4 1
784.2.m.a 2 1.a even 1 1 trivial
784.2.m.a 2 16.e even 4 1 inner
784.2.x.d 4 7.d odd 6 2
784.2.x.d 4 112.x odd 12 2
784.2.x.e 4 7.c even 3 2
784.2.x.e 4 112.w even 12 2
896.2.m.b 2 56.h odd 2 1
896.2.m.b 2 112.l odd 4 1
896.2.m.c 2 56.e even 2 1
896.2.m.c 2 112.j even 4 1
7168.2.a.b 2 224.v odd 8 2
7168.2.a.k 2 224.x even 8 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(784, [\chi])\):

\( T_{3} \) Copy content Toggle raw display
\( T_{5}^{2} + 4T_{5} + 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$3$ \( T^{2} \) 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
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T - 2)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 4T + 8 \) Copy content Toggle raw display
$23$ \( T^{2} + 36 \) Copy content Toggle raw display
$29$ \( T^{2} - 14T + 98 \) Copy content Toggle raw display
$31$ \( (T - 8)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 10T + 50 \) Copy content Toggle raw display
$41$ \( T^{2} + 100 \) Copy content Toggle raw display
$43$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$47$ \( (T - 12)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$59$ \( T^{2} + 16T + 128 \) Copy content Toggle raw display
$61$ \( T^{2} + 12T + 72 \) Copy content Toggle raw display
$67$ \( T^{2} - 6T + 18 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 36 \) Copy content Toggle raw display
$79$ \( (T - 10)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 20T + 200 \) Copy content Toggle raw display
$89$ \( T^{2} + 196 \) Copy content Toggle raw display
$97$ \( (T - 2)^{2} \) Copy content Toggle raw display
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