Properties

Label 1274.2.f.p
Level $1274$
Weight $2$
Character orbit 1274.f
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.f (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.1729412175\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 26)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(197\) \(885\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
0.500000 0.866025i
0.500000 + 0.866025i
0.500000 0.866025i −0.500000 0.866025i −0.500000 0.866025i 1.50000 2.59808i −1.00000 0 −1.00000 1.00000 1.73205i −1.50000 2.59808i
1145.1 0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 1.50000 + 2.59808i −1.00000 0 −1.00000 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1274.2.f.p 2
7.b odd 2 1 1274.2.f.r 2
7.c even 3 1 26.2.a.a 1
7.c even 3 1 inner 1274.2.f.p 2
7.d odd 6 1 1274.2.a.d 1
7.d odd 6 1 1274.2.f.r 2
21.h odd 6 1 234.2.a.e 1
28.g odd 6 1 208.2.a.a 1
35.j even 6 1 650.2.a.j 1
35.l odd 12 2 650.2.b.d 2
56.k odd 6 1 832.2.a.i 1
56.p even 6 1 832.2.a.d 1
63.g even 3 1 2106.2.e.ba 2
63.h even 3 1 2106.2.e.ba 2
63.j odd 6 1 2106.2.e.b 2
63.n odd 6 1 2106.2.e.b 2
77.h odd 6 1 3146.2.a.n 1
84.n even 6 1 1872.2.a.q 1
91.g even 3 1 338.2.c.d 2
91.h even 3 1 338.2.c.d 2
91.k even 6 1 338.2.c.a 2
91.r even 6 1 338.2.a.f 1
91.u even 6 1 338.2.c.a 2
91.x odd 12 2 338.2.e.a 4
91.z odd 12 2 338.2.b.c 2
91.bd odd 12 2 338.2.e.a 4
105.o odd 6 1 5850.2.a.p 1
105.x even 12 2 5850.2.e.a 2
112.u odd 12 2 3328.2.b.j 2
112.w even 12 2 3328.2.b.m 2
119.j even 6 1 7514.2.a.c 1
133.r odd 6 1 9386.2.a.j 1
140.p odd 6 1 5200.2.a.x 1
168.s odd 6 1 7488.2.a.g 1
168.v even 6 1 7488.2.a.h 1
273.w odd 6 1 3042.2.a.a 1
273.cd even 12 2 3042.2.b.a 2
364.bl odd 6 1 2704.2.a.f 1
364.ce even 12 2 2704.2.f.d 2
455.bh even 6 1 8450.2.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.a.a 1 7.c even 3 1
208.2.a.a 1 28.g odd 6 1
234.2.a.e 1 21.h odd 6 1
338.2.a.f 1 91.r even 6 1
338.2.b.c 2 91.z odd 12 2
338.2.c.a 2 91.k even 6 1
338.2.c.a 2 91.u even 6 1
338.2.c.d 2 91.g even 3 1
338.2.c.d 2 91.h even 3 1
338.2.e.a 4 91.x odd 12 2
338.2.e.a 4 91.bd odd 12 2
650.2.a.j 1 35.j even 6 1
650.2.b.d 2 35.l odd 12 2
832.2.a.d 1 56.p even 6 1
832.2.a.i 1 56.k odd 6 1
1274.2.a.d 1 7.d odd 6 1
1274.2.f.p 2 1.a even 1 1 trivial
1274.2.f.p 2 7.c even 3 1 inner
1274.2.f.r 2 7.b odd 2 1
1274.2.f.r 2 7.d odd 6 1
1872.2.a.q 1 84.n even 6 1
2106.2.e.b 2 63.j odd 6 1
2106.2.e.b 2 63.n odd 6 1
2106.2.e.ba 2 63.g even 3 1
2106.2.e.ba 2 63.h even 3 1
2704.2.a.f 1 364.bl odd 6 1
2704.2.f.d 2 364.ce even 12 2
3042.2.a.a 1 273.w odd 6 1
3042.2.b.a 2 273.cd even 12 2
3146.2.a.n 1 77.h odd 6 1
3328.2.b.j 2 112.u odd 12 2
3328.2.b.m 2 112.w even 12 2
5200.2.a.x 1 140.p odd 6 1
5850.2.a.p 1 105.o odd 6 1
5850.2.e.a 2 105.x even 12 2
7488.2.a.g 1 168.s odd 6 1
7488.2.a.h 1 168.v even 6 1
7514.2.a.c 1 119.j even 6 1
8450.2.a.c 1 455.bh even 6 1
9386.2.a.j 1 133.r odd 6 1

Hecke kernels

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

\( T_{3}^{2} + T_{3} + 1 \) Copy content Toggle raw display
\( T_{5}^{2} - 3T_{5} + 9 \) Copy content Toggle raw display
\( T_{11}^{2} + 6T_{11} + 36 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$13$ \( (T - 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$19$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( (T - 6)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$37$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( (T + 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$61$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$67$ \( T^{2} + 14T + 196 \) Copy content Toggle raw display
$71$ \( (T + 3)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$79$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$83$ \( (T - 12)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$97$ \( (T + 10)^{2} \) Copy content Toggle raw display
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