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Properties

Label 1274.2.n.m
Level $1274$
Weight $2$
Character orbit 1274.n
Analytic conductor $10.173$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1274,2,Mod(753,1274)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1274, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1274.753");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
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

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.1729412175\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} + 2 x^{10} - 20 x^{9} - 5 x^{8} + 106 x^{7} - 2 x^{6} + 236 x^{5} + 701 x^{4} + \cdots + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 182)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{8} - \beta_{3}) q^{2} + ( - \beta_{11} - \beta_{2}) q^{3} + (\beta_{9} + 1) q^{4} + ( - \beta_{7} - \beta_{4} + \beta_1) q^{5} + \beta_{10} q^{6} + \beta_{8} q^{8} + ( - \beta_{11} + 5 \beta_{9} + \cdots + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{8} - \beta_{3}) q^{2} + ( - \beta_{11} - \beta_{2}) q^{3} + (\beta_{9} + 1) q^{4} + ( - \beta_{7} - \beta_{4} + \beta_1) q^{5} + \beta_{10} q^{6} + \beta_{8} q^{8} + ( - \beta_{11} + 5 \beta_{9} + \cdots + \beta_1) q^{9}+ \cdots + (10 \beta_{10} + 12 \beta_{8}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{3} + 6 q^{4} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 2 q^{3} + 6 q^{4} - 24 q^{9} + 4 q^{10} + 2 q^{12} - 20 q^{13} - 6 q^{16} + 4 q^{17} + 28 q^{22} + 6 q^{23} + 18 q^{25} + 4 q^{26} - 68 q^{27} + 32 q^{29} - 4 q^{30} - 48 q^{36} + 8 q^{38} + 6 q^{39} - 4 q^{40} + 32 q^{43} + 4 q^{48} + 12 q^{51} - 10 q^{52} + 36 q^{53} + 120 q^{55} - 10 q^{61} + 28 q^{62} - 12 q^{64} + 20 q^{65} + 42 q^{66} - 4 q^{68} + 68 q^{69} - 18 q^{74} + 78 q^{75} - 80 q^{78} + 14 q^{79} - 46 q^{81} - 30 q^{82} - 88 q^{87} + 14 q^{88} + 48 q^{90} + 12 q^{92} - 34 q^{94} - 40 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} + 2 x^{10} - 20 x^{9} - 5 x^{8} + 106 x^{7} - 2 x^{6} + 236 x^{5} + 701 x^{4} + \cdots + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 1016873 \nu^{11} - 2071538 \nu^{10} + 5317271 \nu^{9} - 35882779 \nu^{8} + 16697975 \nu^{7} + \cdots - 4551539153 ) / 1377938599 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 14100012 \nu^{11} - 70688489 \nu^{10} + 96085304 \nu^{9} - 336026168 \nu^{8} + 748569404 \nu^{7} + \cdots + 839688 ) / 2755877198 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 42488465 \nu^{11} - 67885280 \nu^{10} + 54025928 \nu^{9} - 819069464 \nu^{8} - 551995161 \nu^{7} + \cdots + 56400048 ) / 2755877198 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 251723 \nu^{11} + 404408 \nu^{10} - 346478 \nu^{9} + 4933516 \nu^{8} + 3121603 \nu^{7} + \cdots + 5658396 ) / 15929926 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 48715918 \nu^{11} - 243004937 \nu^{10} + 330308912 \nu^{9} - 1158579232 \nu^{8} + \cdots + 2821400 ) / 2755877198 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 49963961 \nu^{11} - 100902054 \nu^{10} + 101993694 \nu^{9} - 1003998996 \nu^{8} + \cdots - 978600172 ) / 2755877198 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1287989 \nu^{11} + 2059484 \nu^{10} - 1633032 \nu^{9} + 24888162 \nu^{8} + 16624685 \nu^{7} + \cdots + 13677170 ) / 15395962 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1414599 \nu^{11} + 2577475 \nu^{10} - 2424790 \nu^{9} + 27945502 \nu^{8} + 12006511 \nu^{7} + \cdots + 14099646 ) / 15929926 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 760553063 \nu^{11} + 1216635308 \nu^{10} - 973441494 \nu^{9} + 14715213802 \nu^{8} + \cdots + 8092950434 ) / 2755877198 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 809537657 \nu^{11} - 1475545857 \nu^{10} + 1383785796 \nu^{9} - 15987231162 \nu^{8} + \cdots - 8069962538 ) / 2755877198 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} + \beta_{6} - \beta_{5} - 3\beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} - 2\beta_{8} + 7\beta_{4} + \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -7\beta_{11} - 19\beta_{9} + 11\beta_{7} + 11\beta_{4} + 11\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -11\beta_{11} - 26\beta_{9} + 11\beta_{6} - 55\beta_{5} - 26\beta_{3} - 11\beta_{2} - 26 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 55\beta_{10} - 143\beta_{8} - 103\beta_{5} + 103\beta_{4} - 103\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 103 \beta_{11} + 103 \beta_{10} - 254 \beta_{9} - 254 \beta_{8} + 459 \beta_{7} + \cdots + 254 \beta_{3} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -459\beta_{11} - 1171\beta_{9} + 919\beta_{7} - 919\beta_{5} - 459\beta_{2} - 1171 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 919\beta_{10} - 2298\beta_{8} - 3927\beta_{5} - 919\beta_{2} - 3927\beta _1 - 2298 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 3927\beta_{10} - 9943\beta_{8} + 8063\beta_{7} - 3927\beta_{6} + 8063\beta_{4} + 9943\beta_{3} - 8063\beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -8063\beta_{11} - 20262\beta_{9} + 33923\beta_{7} - 8063\beta_{6} + 20262\beta_{3} - 8063\beta_{2} - 20262 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(197\) \(885\)
\(\chi(n)\) \(-1\) \(\beta_{9}\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
753.1
0.465879 − 1.73868i
−0.762919 + 2.84725i
−0.0689846 + 0.257454i
−1.73868 − 0.465879i
2.84725 + 0.762919i
0.257454 + 0.0689846i
0.465879 + 1.73868i
−0.762919 − 2.84725i
−0.0689846 − 0.257454i
−1.73868 + 0.465879i
2.84725 − 0.762919i
0.257454 − 0.0689846i
−0.866025 + 0.500000i −1.39283 + 2.41246i 0.500000 − 0.866025i 2.20456 − 1.27280i − 2.78567i 0 1.00000i −2.37997 − 4.12223i −1.27280 + 2.20456i
753.2 −0.866025 + 0.500000i −0.760115 + 1.31656i 0.500000 − 0.866025i −3.61017 + 2.08433i − 1.52023i 0 1.00000i 0.344450 + 0.596604i 2.08433 − 3.61017i
753.3 −0.866025 + 0.500000i 1.65295 − 2.86299i 0.500000 − 0.866025i −0.326439 + 0.188470i 3.30590i 0 1.00000i −3.96448 − 6.86668i 0.188470 − 0.326439i
753.4 0.866025 − 0.500000i −1.39283 + 2.41246i 0.500000 − 0.866025i −2.20456 + 1.27280i 2.78567i 0 − 1.00000i −2.37997 − 4.12223i −1.27280 + 2.20456i
753.5 0.866025 − 0.500000i −0.760115 + 1.31656i 0.500000 − 0.866025i 3.61017 − 2.08433i 1.52023i 0 − 1.00000i 0.344450 + 0.596604i 2.08433 − 3.61017i
753.6 0.866025 − 0.500000i 1.65295 − 2.86299i 0.500000 − 0.866025i 0.326439 − 0.188470i − 3.30590i 0 − 1.00000i −3.96448 − 6.86668i 0.188470 − 0.326439i
961.1 −0.866025 − 0.500000i −1.39283 − 2.41246i 0.500000 + 0.866025i 2.20456 + 1.27280i 2.78567i 0 − 1.00000i −2.37997 + 4.12223i −1.27280 − 2.20456i
961.2 −0.866025 − 0.500000i −0.760115 − 1.31656i 0.500000 + 0.866025i −3.61017 − 2.08433i 1.52023i 0 − 1.00000i 0.344450 − 0.596604i 2.08433 + 3.61017i
961.3 −0.866025 − 0.500000i 1.65295 + 2.86299i 0.500000 + 0.866025i −0.326439 − 0.188470i − 3.30590i 0 − 1.00000i −3.96448 + 6.86668i 0.188470 + 0.326439i
961.4 0.866025 + 0.500000i −1.39283 − 2.41246i 0.500000 + 0.866025i −2.20456 − 1.27280i − 2.78567i 0 1.00000i −2.37997 + 4.12223i −1.27280 − 2.20456i
961.5 0.866025 + 0.500000i −0.760115 − 1.31656i 0.500000 + 0.866025i 3.61017 + 2.08433i − 1.52023i 0 1.00000i 0.344450 − 0.596604i 2.08433 + 3.61017i
961.6 0.866025 + 0.500000i 1.65295 + 2.86299i 0.500000 + 0.866025i 0.326439 + 0.188470i 3.30590i 0 1.00000i −3.96448 + 6.86668i 0.188470 + 0.326439i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 753.6
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.m 12
7.b odd 2 1 1274.2.n.n 12
7.c even 3 1 182.2.d.b ✓ 6
7.c even 3 1 inner 1274.2.n.m 12
7.d odd 6 1 1274.2.d.l 6
7.d odd 6 1 1274.2.n.n 12
13.b even 2 1 inner 1274.2.n.m 12
21.h odd 6 1 1638.2.c.i 6
28.g odd 6 1 1456.2.k.b 6
91.b odd 2 1 1274.2.n.n 12
91.r even 6 1 182.2.d.b ✓ 6
91.r even 6 1 inner 1274.2.n.m 12
91.s odd 6 1 1274.2.d.l 6
91.s odd 6 1 1274.2.n.n 12
91.z odd 12 1 2366.2.a.x 3
91.z odd 12 1 2366.2.a.bc 3
273.w odd 6 1 1638.2.c.i 6
364.bl odd 6 1 1456.2.k.b 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
182.2.d.b ✓ 6 7.c even 3 1
182.2.d.b ✓ 6 91.r even 6 1
1274.2.d.l 6 7.d odd 6 1
1274.2.d.l 6 91.s odd 6 1
1274.2.n.m 12 1.a even 1 1 trivial
1274.2.n.m 12 7.c even 3 1 inner
1274.2.n.m 12 13.b even 2 1 inner
1274.2.n.m 12 91.r even 6 1 inner
1274.2.n.n 12 7.b odd 2 1
1274.2.n.n 12 7.d odd 6 1
1274.2.n.n 12 91.b odd 2 1
1274.2.n.n 12 91.s odd 6 1
1456.2.k.b 6 28.g odd 6 1
1456.2.k.b 6 364.bl odd 6 1
1638.2.c.i 6 21.h odd 6 1
1638.2.c.i 6 273.w odd 6 1
2366.2.a.x 3 91.z odd 12 1
2366.2.a.bc 3 91.z odd 12 1

Hecke kernels

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

\( T_{3}^{6} + T_{3}^{5} + 11T_{3}^{4} + 18T_{3}^{3} + 114T_{3}^{2} + 140T_{3} + 196 \) Copy content Toggle raw display
\( T_{5}^{12} - 24T_{5}^{10} + 460T_{5}^{8} - 2752T_{5}^{6} + 13072T_{5}^{4} - 1856T_{5}^{2} + 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{3} \) Copy content Toggle raw display
$3$ \( (T^{6} + T^{5} + 11 T^{4} + \cdots + 196)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} - 24 T^{10} + \cdots + 256 \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( T^{12} - 65 T^{10} + \cdots + 59969536 \) Copy content Toggle raw display
$13$ \( (T^{6} + 10 T^{5} + \cdots + 2197)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} - 2 T^{5} + \cdots + 3136)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} - 28 T^{10} + \cdots + 65536 \) Copy content Toggle raw display
$23$ \( (T^{6} - 3 T^{5} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} - 8 T^{2} + \cdots + 176)^{4} \) Copy content Toggle raw display
$31$ \( T^{12} - 65 T^{10} + \cdots + 59969536 \) Copy content Toggle raw display
$37$ \( T^{12} - 65 T^{10} + \cdots + 3748096 \) Copy content Toggle raw display
$41$ \( (T^{6} + 113 T^{4} + \cdots + 1936)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} - 8 T^{2} + \cdots + 128)^{4} \) Copy content Toggle raw display
$47$ \( T^{12} - 145 T^{10} + \cdots + 59969536 \) Copy content Toggle raw display
$53$ \( (T^{2} - 6 T + 36)^{6} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 610673479936 \) Copy content Toggle raw display
$61$ \( (T^{6} + 5 T^{5} + 27 T^{4} + \cdots + 4)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} - 113 T^{10} + \cdots + 65536 \) Copy content Toggle raw display
$71$ \( (T^{6} + 336 T^{4} + \cdots + 802816)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} - 89 T^{10} + \cdots + 256 \) Copy content Toggle raw display
$79$ \( (T^{6} - 7 T^{5} + \cdots + 21904)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + 352 T^{4} + \cdots + 1567504)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 2897022976 \) Copy content Toggle raw display
$97$ \( (T^{6} + 185 T^{4} + \cdots + 38416)^{2} \) Copy content Toggle raw display
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