Properties

Label 1274.2.n.c
Level $1274$
Weight $2$
Character orbit 1274.n
Analytic conductor $10.173$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1274 = 2 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1274.n (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.1729412175\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 26)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{12} q^{2} -\zeta_{12}^{2} q^{3} + \zeta_{12}^{2} q^{4} + 3 \zeta_{12} q^{5} -\zeta_{12}^{3} q^{6} + \zeta_{12}^{3} q^{8} + ( 2 - 2 \zeta_{12}^{2} ) q^{9} +O(q^{10})\) \( q + \zeta_{12} q^{2} -\zeta_{12}^{2} q^{3} + \zeta_{12}^{2} q^{4} + 3 \zeta_{12} q^{5} -\zeta_{12}^{3} q^{6} + \zeta_{12}^{3} q^{8} + ( 2 - 2 \zeta_{12}^{2} ) q^{9} + 3 \zeta_{12}^{2} q^{10} + ( 1 - \zeta_{12}^{2} ) q^{12} + ( -2 + 3 \zeta_{12}^{3} ) q^{13} -3 \zeta_{12}^{3} q^{15} + ( -1 + \zeta_{12}^{2} ) q^{16} + 3 \zeta_{12}^{2} q^{17} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{18} + 6 \zeta_{12} q^{19} + 3 \zeta_{12}^{3} q^{20} + ( 6 - 6 \zeta_{12}^{2} ) q^{23} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{24} + 4 \zeta_{12}^{2} q^{25} + ( -3 - 2 \zeta_{12} + 3 \zeta_{12}^{2} ) q^{26} -5 q^{27} + ( 3 - 3 \zeta_{12}^{2} ) q^{30} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{32} + 3 \zeta_{12}^{3} q^{34} + 2 q^{36} + 3 \zeta_{12} q^{37} + 6 \zeta_{12}^{2} q^{38} + ( 3 \zeta_{12} + 2 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{39} + ( -3 + 3 \zeta_{12}^{2} ) q^{40} - q^{43} + ( 6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{45} + ( 6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{46} -3 \zeta_{12} q^{47} + q^{48} + 4 \zeta_{12}^{3} q^{50} + ( 3 - 3 \zeta_{12}^{2} ) q^{51} + ( -3 \zeta_{12} - 2 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{52} + 6 \zeta_{12}^{2} q^{53} -5 \zeta_{12} q^{54} -6 \zeta_{12}^{3} q^{57} + ( -6 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{59} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{60} + ( -8 + 8 \zeta_{12}^{2} ) q^{61} - q^{64} + ( -9 - 6 \zeta_{12} + 9 \zeta_{12}^{2} ) q^{65} + ( 12 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{67} + ( -3 + 3 \zeta_{12}^{2} ) q^{68} -6 q^{69} -15 \zeta_{12}^{3} q^{71} + 2 \zeta_{12} q^{72} + ( 6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{73} + 3 \zeta_{12}^{2} q^{74} + ( 4 - 4 \zeta_{12}^{2} ) q^{75} + 6 \zeta_{12}^{3} q^{76} + ( 3 + 2 \zeta_{12}^{3} ) q^{78} + ( -10 + 10 \zeta_{12}^{2} ) q^{79} + ( -3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{80} -\zeta_{12}^{2} q^{81} + 6 \zeta_{12}^{3} q^{83} + 9 \zeta_{12}^{3} q^{85} -\zeta_{12} q^{86} + 6 \zeta_{12} q^{89} + 6 q^{90} + 6 q^{92} -3 \zeta_{12}^{2} q^{94} + 18 \zeta_{12}^{2} q^{95} + \zeta_{12} q^{96} -12 \zeta_{12}^{3} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{3} + 2q^{4} + 4q^{9} + O(q^{10}) \) \( 4q - 2q^{3} + 2q^{4} + 4q^{9} + 6q^{10} + 2q^{12} - 8q^{13} - 2q^{16} + 6q^{17} + 12q^{23} + 8q^{25} - 6q^{26} - 20q^{27} + 6q^{30} + 8q^{36} + 12q^{38} + 4q^{39} - 6q^{40} - 4q^{43} + 4q^{48} + 6q^{51} - 4q^{52} + 12q^{53} - 16q^{61} - 4q^{64} - 18q^{65} - 6q^{68} - 24q^{69} + 6q^{74} + 8q^{75} + 12q^{78} - 20q^{79} - 2q^{81} + 24q^{90} + 24q^{92} - 6q^{94} + 36q^{95} + O(q^{100}) \)

Character values

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

\(n\) \(197\) \(885\)
\(\chi(n)\) \(-1\) \(-1 + \zeta_{12}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
753.1
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i −0.500000 + 0.866025i 0.500000 0.866025i −2.59808 + 1.50000i 1.00000i 0 1.00000i 1.00000 + 1.73205i 1.50000 2.59808i
753.2 0.866025 0.500000i −0.500000 + 0.866025i 0.500000 0.866025i 2.59808 1.50000i 1.00000i 0 1.00000i 1.00000 + 1.73205i 1.50000 2.59808i
961.1 −0.866025 0.500000i −0.500000 0.866025i 0.500000 + 0.866025i −2.59808 1.50000i 1.00000i 0 1.00000i 1.00000 1.73205i 1.50000 + 2.59808i
961.2 0.866025 + 0.500000i −0.500000 0.866025i 0.500000 + 0.866025i 2.59808 + 1.50000i 1.00000i 0 1.00000i 1.00000 1.73205i 1.50000 + 2.59808i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
13.b even 2 1 inner
91.r even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1274.2.n.c 4
7.b odd 2 1 1274.2.n.d 4
7.c even 3 1 1274.2.d.c 2
7.c even 3 1 inner 1274.2.n.c 4
7.d odd 6 1 26.2.b.a 2
7.d odd 6 1 1274.2.n.d 4
13.b even 2 1 inner 1274.2.n.c 4
21.g even 6 1 234.2.b.b 2
28.f even 6 1 208.2.f.a 2
35.i odd 6 1 650.2.d.b 2
35.k even 12 1 650.2.c.a 2
35.k even 12 1 650.2.c.d 2
56.j odd 6 1 832.2.f.d 2
56.m even 6 1 832.2.f.b 2
84.j odd 6 1 1872.2.c.f 2
91.b odd 2 1 1274.2.n.d 4
91.l odd 6 1 338.2.e.c 4
91.m odd 6 1 338.2.e.c 4
91.p odd 6 1 338.2.e.c 4
91.r even 6 1 1274.2.d.c 2
91.r even 6 1 inner 1274.2.n.c 4
91.s odd 6 1 26.2.b.a 2
91.s odd 6 1 1274.2.n.d 4
91.v odd 6 1 338.2.e.c 4
91.w even 12 1 338.2.c.b 2
91.w even 12 1 338.2.c.f 2
91.ba even 12 1 338.2.c.b 2
91.ba even 12 1 338.2.c.f 2
91.bb even 12 1 338.2.a.b 1
91.bb even 12 1 338.2.a.d 1
273.ba even 6 1 234.2.b.b 2
273.cb odd 12 1 3042.2.a.g 1
273.cb odd 12 1 3042.2.a.j 1
364.x even 6 1 208.2.f.a 2
364.bw odd 12 1 2704.2.a.j 1
364.bw odd 12 1 2704.2.a.k 1
455.bf odd 6 1 650.2.d.b 2
455.co even 12 1 8450.2.a.h 1
455.co even 12 1 8450.2.a.u 1
455.df even 12 1 650.2.c.a 2
455.df even 12 1 650.2.c.d 2
728.bv odd 6 1 832.2.f.d 2
728.cy even 6 1 832.2.f.b 2
1092.ct odd 6 1 1872.2.c.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.b.a 2 7.d odd 6 1
26.2.b.a 2 91.s odd 6 1
208.2.f.a 2 28.f even 6 1
208.2.f.a 2 364.x even 6 1
234.2.b.b 2 21.g even 6 1
234.2.b.b 2 273.ba even 6 1
338.2.a.b 1 91.bb even 12 1
338.2.a.d 1 91.bb even 12 1
338.2.c.b 2 91.w even 12 1
338.2.c.b 2 91.ba even 12 1
338.2.c.f 2 91.w even 12 1
338.2.c.f 2 91.ba even 12 1
338.2.e.c 4 91.l odd 6 1
338.2.e.c 4 91.m odd 6 1
338.2.e.c 4 91.p odd 6 1
338.2.e.c 4 91.v odd 6 1
650.2.c.a 2 35.k even 12 1
650.2.c.a 2 455.df even 12 1
650.2.c.d 2 35.k even 12 1
650.2.c.d 2 455.df even 12 1
650.2.d.b 2 35.i odd 6 1
650.2.d.b 2 455.bf odd 6 1
832.2.f.b 2 56.m even 6 1
832.2.f.b 2 728.cy even 6 1
832.2.f.d 2 56.j odd 6 1
832.2.f.d 2 728.bv odd 6 1
1274.2.d.c 2 7.c even 3 1
1274.2.d.c 2 91.r even 6 1
1274.2.n.c 4 1.a even 1 1 trivial
1274.2.n.c 4 7.c even 3 1 inner
1274.2.n.c 4 13.b even 2 1 inner
1274.2.n.c 4 91.r even 6 1 inner
1274.2.n.d 4 7.b odd 2 1
1274.2.n.d 4 7.d odd 6 1
1274.2.n.d 4 91.b odd 2 1
1274.2.n.d 4 91.s odd 6 1
1872.2.c.f 2 84.j odd 6 1
1872.2.c.f 2 1092.ct odd 6 1
2704.2.a.j 1 364.bw odd 12 1
2704.2.a.k 1 364.bw odd 12 1
3042.2.a.g 1 273.cb odd 12 1
3042.2.a.j 1 273.cb odd 12 1
8450.2.a.h 1 455.co even 12 1
8450.2.a.u 1 455.co even 12 1

Hecke kernels

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

\( T_{3}^{2} + T_{3} + 1 \)
\( T_{5}^{4} - 9 T_{5}^{2} + 81 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} + T^{4} \)
$3$ \( ( 1 + T + T^{2} )^{2} \)
$5$ \( 81 - 9 T^{2} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( T^{4} \)
$13$ \( ( 13 + 4 T + T^{2} )^{2} \)
$17$ \( ( 9 - 3 T + T^{2} )^{2} \)
$19$ \( 1296 - 36 T^{2} + T^{4} \)
$23$ \( ( 36 - 6 T + T^{2} )^{2} \)
$29$ \( T^{4} \)
$31$ \( T^{4} \)
$37$ \( 81 - 9 T^{2} + T^{4} \)
$41$ \( T^{4} \)
$43$ \( ( 1 + T )^{4} \)
$47$ \( 81 - 9 T^{2} + T^{4} \)
$53$ \( ( 36 - 6 T + T^{2} )^{2} \)
$59$ \( 1296 - 36 T^{2} + T^{4} \)
$61$ \( ( 64 + 8 T + T^{2} )^{2} \)
$67$ \( 20736 - 144 T^{2} + T^{4} \)
$71$ \( ( 225 + T^{2} )^{2} \)
$73$ \( 1296 - 36 T^{2} + T^{4} \)
$79$ \( ( 100 + 10 T + T^{2} )^{2} \)
$83$ \( ( 36 + T^{2} )^{2} \)
$89$ \( 1296 - 36 T^{2} + T^{4} \)
$97$ \( ( 144 + T^{2} )^{2} \)
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