Properties

Label 1274.2.d.c
Level $1274$
Weight $2$
Character orbit 1274.d
Analytic conductor $10.173$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1274 = 2 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1274.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.1729412175\)
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: no (minimal twist has level 26)
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.

\(f(q)\) \(=\) \( q + i q^{2} + q^{3} - q^{4} + 3 i q^{5} + i q^{6} -i q^{8} -2 q^{9} +O(q^{10})\) \( q + i q^{2} + q^{3} - q^{4} + 3 i q^{5} + i q^{6} -i q^{8} -2 q^{9} -3 q^{10} - q^{12} + ( -2 - 3 i ) q^{13} + 3 i q^{15} + q^{16} -3 q^{17} -2 i q^{18} + 6 i q^{19} -3 i q^{20} -6 q^{23} -i q^{24} -4 q^{25} + ( 3 - 2 i ) q^{26} -5 q^{27} -3 q^{30} + i q^{32} -3 i q^{34} + 2 q^{36} + 3 i q^{37} -6 q^{38} + ( -2 - 3 i ) q^{39} + 3 q^{40} - q^{43} -6 i q^{45} -6 i q^{46} -3 i q^{47} + q^{48} -4 i q^{50} -3 q^{51} + ( 2 + 3 i ) q^{52} -6 q^{53} -5 i q^{54} + 6 i q^{57} + 6 i q^{59} -3 i q^{60} + 8 q^{61} - q^{64} + ( 9 - 6 i ) q^{65} -12 i q^{67} + 3 q^{68} -6 q^{69} + 15 i q^{71} + 2 i q^{72} -6 i q^{73} -3 q^{74} -4 q^{75} -6 i q^{76} + ( 3 - 2 i ) q^{78} + 10 q^{79} + 3 i q^{80} + q^{81} -6 i q^{83} -9 i q^{85} -i q^{86} + 6 i q^{89} + 6 q^{90} + 6 q^{92} + 3 q^{94} -18 q^{95} + i q^{96} + 12 i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} - 2q^{4} - 4q^{9} + O(q^{10}) \) \( 2q + 2q^{3} - 2q^{4} - 4q^{9} - 6q^{10} - 2q^{12} - 4q^{13} + 2q^{16} - 6q^{17} - 12q^{23} - 8q^{25} + 6q^{26} - 10q^{27} - 6q^{30} + 4q^{36} - 12q^{38} - 4q^{39} + 6q^{40} - 2q^{43} + 2q^{48} - 6q^{51} + 4q^{52} - 12q^{53} + 16q^{61} - 2q^{64} + 18q^{65} + 6q^{68} - 12q^{69} - 6q^{74} - 8q^{75} + 6q^{78} + 20q^{79} + 2q^{81} + 12q^{90} + 12q^{92} + 6q^{94} - 36q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1274\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(885\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
883.1
1.00000i
1.00000i
1.00000i 1.00000 −1.00000 3.00000i 1.00000i 0 1.00000i −2.00000 −3.00000
883.2 1.00000i 1.00000 −1.00000 3.00000i 1.00000i 0 1.00000i −2.00000 −3.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1274.2.d.c 2
7.b odd 2 1 26.2.b.a 2
7.c even 3 2 1274.2.n.c 4
7.d odd 6 2 1274.2.n.d 4
13.b even 2 1 inner 1274.2.d.c 2
21.c even 2 1 234.2.b.b 2
28.d even 2 1 208.2.f.a 2
35.c odd 2 1 650.2.d.b 2
35.f even 4 1 650.2.c.a 2
35.f even 4 1 650.2.c.d 2
56.e even 2 1 832.2.f.b 2
56.h odd 2 1 832.2.f.d 2
84.h odd 2 1 1872.2.c.f 2
91.b odd 2 1 26.2.b.a 2
91.i even 4 1 338.2.a.b 1
91.i even 4 1 338.2.a.d 1
91.n odd 6 2 338.2.e.c 4
91.r even 6 2 1274.2.n.c 4
91.s odd 6 2 1274.2.n.d 4
91.t odd 6 2 338.2.e.c 4
91.bc even 12 2 338.2.c.b 2
91.bc even 12 2 338.2.c.f 2
273.g even 2 1 234.2.b.b 2
273.o odd 4 1 3042.2.a.g 1
273.o odd 4 1 3042.2.a.j 1
364.h even 2 1 208.2.f.a 2
364.p odd 4 1 2704.2.a.j 1
364.p odd 4 1 2704.2.a.k 1
455.h odd 2 1 650.2.d.b 2
455.s even 4 1 650.2.c.a 2
455.s even 4 1 650.2.c.d 2
455.u even 4 1 8450.2.a.h 1
455.u even 4 1 8450.2.a.u 1
728.b even 2 1 832.2.f.b 2
728.l odd 2 1 832.2.f.d 2
1092.d odd 2 1 1872.2.c.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.b.a 2 7.b odd 2 1
26.2.b.a 2 91.b odd 2 1
208.2.f.a 2 28.d even 2 1
208.2.f.a 2 364.h even 2 1
234.2.b.b 2 21.c even 2 1
234.2.b.b 2 273.g even 2 1
338.2.a.b 1 91.i even 4 1
338.2.a.d 1 91.i even 4 1
338.2.c.b 2 91.bc even 12 2
338.2.c.f 2 91.bc even 12 2
338.2.e.c 4 91.n odd 6 2
338.2.e.c 4 91.t odd 6 2
650.2.c.a 2 35.f even 4 1
650.2.c.a 2 455.s even 4 1
650.2.c.d 2 35.f even 4 1
650.2.c.d 2 455.s even 4 1
650.2.d.b 2 35.c odd 2 1
650.2.d.b 2 455.h odd 2 1
832.2.f.b 2 56.e even 2 1
832.2.f.b 2 728.b even 2 1
832.2.f.d 2 56.h odd 2 1
832.2.f.d 2 728.l odd 2 1
1274.2.d.c 2 1.a even 1 1 trivial
1274.2.d.c 2 13.b even 2 1 inner
1274.2.n.c 4 7.c even 3 2
1274.2.n.c 4 91.r even 6 2
1274.2.n.d 4 7.d odd 6 2
1274.2.n.d 4 91.s odd 6 2
1872.2.c.f 2 84.h odd 2 1
1872.2.c.f 2 1092.d odd 2 1
2704.2.a.j 1 364.p odd 4 1
2704.2.a.k 1 364.p odd 4 1
3042.2.a.g 1 273.o odd 4 1
3042.2.a.j 1 273.o odd 4 1
8450.2.a.h 1 455.u even 4 1
8450.2.a.u 1 455.u even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1274, [\chi])\):

\( T_{3} - 1 \)
\( T_{5}^{2} + 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( 9 + T^{2} \)
$7$ \( T^{2} \)
$11$ \( T^{2} \)
$13$ \( 13 + 4 T + T^{2} \)
$17$ \( ( 3 + T )^{2} \)
$19$ \( 36 + T^{2} \)
$23$ \( ( 6 + T )^{2} \)
$29$ \( T^{2} \)
$31$ \( T^{2} \)
$37$ \( 9 + T^{2} \)
$41$ \( T^{2} \)
$43$ \( ( 1 + T )^{2} \)
$47$ \( 9 + T^{2} \)
$53$ \( ( 6 + T )^{2} \)
$59$ \( 36 + T^{2} \)
$61$ \( ( -8 + T )^{2} \)
$67$ \( 144 + T^{2} \)
$71$ \( 225 + T^{2} \)
$73$ \( 36 + T^{2} \)
$79$ \( ( -10 + T )^{2} \)
$83$ \( 36 + T^{2} \)
$89$ \( 36 + T^{2} \)
$97$ \( 144 + T^{2} \)
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