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Properties

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

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1274,2,Mod(459,1274)]
 
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, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1274.459");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1274 = 2 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1274.o (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: \( x^{12} - 6 x^{11} + 39 x^{10} - 140 x^{9} + 460 x^{8} - 1066 x^{7} + 2127 x^{6} - 3172 x^{5} + \cdots + 169 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{6} + \beta_{5}) q^{2} - \beta_{10} q^{3} - q^{4} + (\beta_{8} + \beta_{7} - \beta_{6} + \cdots - \beta_1) q^{5}+ \cdots + ( - \beta_{11} + \beta_{10} + \cdots + 2 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{6} + \beta_{5}) q^{2} - \beta_{10} q^{3} - q^{4} + (\beta_{8} + \beta_{7} - \beta_{6} + \cdots - \beta_1) q^{5}+ \cdots + (2 \beta_{11} + 4 \beta_{10} + \cdots + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{3} - 12 q^{4} + 6 q^{6} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 2 q^{3} - 12 q^{4} + 6 q^{6} - 6 q^{9} - 2 q^{10} + 18 q^{11} + 2 q^{12} - 8 q^{13} - 6 q^{15} + 12 q^{16} - 8 q^{17} - 12 q^{19} - 2 q^{22} + 12 q^{23} - 6 q^{24} + 12 q^{25} - 2 q^{26} + 40 q^{27} - 10 q^{29} + 14 q^{30} + 18 q^{31} - 12 q^{33} + 6 q^{36} - 4 q^{38} + 24 q^{39} + 2 q^{40} - 24 q^{41} + 26 q^{43} - 18 q^{44} + 48 q^{47} - 2 q^{48} - 12 q^{50} + 18 q^{51} + 8 q^{52} - 18 q^{53} - 6 q^{55} - 24 q^{58} + 6 q^{60} - 28 q^{61} - 2 q^{62} - 12 q^{64} - 4 q^{65} + 42 q^{67} + 8 q^{68} + 32 q^{69} + 48 q^{71} - 48 q^{73} + 96 q^{75} + 12 q^{76} - 8 q^{78} - 22 q^{79} - 34 q^{81} + 6 q^{82} + 54 q^{85} + 6 q^{86} - 4 q^{87} + 2 q^{88} + 12 q^{90} - 12 q^{92} + 8 q^{94} - 64 q^{95} + 6 q^{96} + 60 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} + 39 x^{10} - 140 x^{9} + 460 x^{8} - 1066 x^{7} + 2127 x^{6} - 3172 x^{5} + \cdots + 169 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 24 \nu^{10} - 120 \nu^{9} + 751 \nu^{8} - 2284 \nu^{7} + 6728 \nu^{6} - 12694 \nu^{5} + 20323 \nu^{4} + \cdots + 2041 ) / 286 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 31 \nu^{10} + 155 \nu^{9} - 976 \nu^{8} + 2974 \nu^{7} - 8881 \nu^{6} + 16885 \nu^{5} - 28044 \nu^{4} + \cdots - 5252 ) / 286 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 162 \nu^{11} + 186 \nu^{10} - 51 \nu^{9} + 15887 \nu^{8} - 51264 \nu^{7} + 192268 \nu^{6} + \cdots + 66227 ) / 7898 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2323 \nu^{11} - 22649 \nu^{10} + 132943 \nu^{9} - 590285 \nu^{8} + 1852053 \nu^{7} - 4762085 \nu^{6} + \cdots - 1517893 ) / 102674 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 2323 \nu^{11} + 2904 \nu^{10} - 34218 \nu^{9} - 29708 \nu^{8} + 35569 \nu^{7} - 841546 \nu^{6} + \cdots - 689598 ) / 102674 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 324 \nu^{11} + 1782 \nu^{10} - 10668 \nu^{9} + 34641 \nu^{8} - 98512 \nu^{7} + 195608 \nu^{6} + \cdots - 3573 ) / 7898 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 5977 \nu^{11} + 33053 \nu^{10} - 223218 \nu^{9} + 741326 \nu^{8} - 2485739 \nu^{7} + \cdots - 527956 ) / 102674 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 5977 \nu^{11} - 32694 \nu^{10} + 221423 \nu^{9} - 766456 \nu^{8} + 2597029 \nu^{7} - 6035626 \nu^{6} + \cdots - 3597672 ) / 102674 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 6388 \nu^{11} + 30826 \nu^{10} - 187767 \nu^{9} + 543572 \nu^{8} - 1501380 \nu^{7} + \cdots - 435708 ) / 102674 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 9240 \nu^{11} + 55128 \nu^{10} - 344643 \nu^{9} + 1207618 \nu^{8} - 3759676 \nu^{7} + \cdots + 1667016 ) / 102674 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{5} - \beta_{3} + \beta_{2} + \beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{11} + \beta_{10} + \beta_{9} - \beta_{8} - 2 \beta_{7} + 6 \beta_{6} - 4 \beta_{5} - \beta_{3} + \cdots - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 2 \beta_{11} + 2 \beta_{10} + 3 \beta_{9} - \beta_{8} - 6 \beta_{7} - \beta_{6} - 21 \beta_{5} + \cdots + 29 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8 \beta_{11} - 15 \beta_{10} - 8 \beta_{9} + 13 \beta_{8} + 25 \beta_{7} - 73 \beta_{6} + 26 \beta_{5} + \cdots + 34 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 29 \beta_{11} - 50 \beta_{10} - 48 \beta_{9} + 25 \beta_{8} + 124 \beta_{7} - 81 \beta_{6} + 266 \beta_{5} + \cdots - 247 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 60 \beta_{11} + 135 \beta_{10} + 17 \beta_{9} - 115 \beta_{8} - 145 \beta_{7} + 668 \beta_{6} + \cdots - 526 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 380 \beta_{11} + 778 \beta_{10} + 515 \beta_{9} - 363 \beta_{8} - 1632 \beta_{7} + 1517 \beta_{6} + \cdots + 2008 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 264 \beta_{11} - 673 \beta_{10} + 454 \beta_{9} + 839 \beta_{8} - 287 \beta_{7} - 5265 \beta_{6} + \cdots + 7116 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 4383 \beta_{11} - 9560 \beta_{10} - 4630 \beta_{9} + 4411 \beta_{8} + 17550 \beta_{7} - 20512 \beta_{6} + \cdots - 13713 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1759 \beta_{11} - 3186 \beta_{10} - 9684 \beta_{9} - 4506 \beta_{8} + 22413 \beta_{7} + 33458 \beta_{6} + \cdots - 85518 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(197\) \(885\)
\(\chi(n)\) \(1 - \beta_{7}\) \(-1 + \beta_{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
459.1
0.500000 − 3.15681i
0.500000 − 0.399480i
0.500000 + 1.69027i
0.500000 + 2.47866i
0.500000 − 0.613147i
0.500000 − 1.73154i
0.500000 − 2.47866i
0.500000 + 0.613147i
0.500000 + 1.73154i
0.500000 + 3.15681i
0.500000 + 0.399480i
0.500000 − 1.69027i
− 1.00000i −1.14539 − 1.98388i −1.00000 0.781015 − 0.450919i −1.98388 + 1.14539i 0 1.00000i −1.12385 + 1.94657i −0.450919 − 0.781015i
459.2 − 1.00000i 0.233273 + 0.404040i −1.00000 −2.93529 + 1.69469i 0.404040 − 0.233273i 0 1.00000i 1.39117 − 2.40957i 1.69469 + 2.93529i
459.3 − 1.00000i 1.27815 + 2.21381i −1.00000 3.02030 − 1.74377i 2.21381 − 1.27815i 0 1.00000i −1.76732 + 3.06108i −1.74377 − 3.02030i
459.4 1.00000i −1.67234 − 2.89658i −1.00000 1.35408 − 0.781779i 2.89658 − 1.67234i 0 − 1.00000i −4.09347 + 7.09010i 0.781779 + 1.35408i
459.5 1.00000i −0.126439 − 0.218999i −1.00000 0.993985 − 0.573878i 0.218999 − 0.126439i 0 − 1.00000i 1.46803 − 2.54270i 0.573878 + 0.993985i
459.6 1.00000i 0.432757 + 0.749558i −1.00000 −3.21409 + 1.85566i −0.749558 + 0.432757i 0 − 1.00000i 1.12544 − 1.94932i −1.85566 − 3.21409i
569.1 − 1.00000i −1.67234 + 2.89658i −1.00000 1.35408 + 0.781779i 2.89658 + 1.67234i 0 1.00000i −4.09347 − 7.09010i 0.781779 − 1.35408i
569.2 − 1.00000i −0.126439 + 0.218999i −1.00000 0.993985 + 0.573878i 0.218999 + 0.126439i 0 1.00000i 1.46803 + 2.54270i 0.573878 − 0.993985i
569.3 − 1.00000i 0.432757 − 0.749558i −1.00000 −3.21409 − 1.85566i −0.749558 − 0.432757i 0 1.00000i 1.12544 + 1.94932i −1.85566 + 3.21409i
569.4 1.00000i −1.14539 + 1.98388i −1.00000 0.781015 + 0.450919i −1.98388 − 1.14539i 0 − 1.00000i −1.12385 − 1.94657i −0.450919 + 0.781015i
569.5 1.00000i 0.233273 − 0.404040i −1.00000 −2.93529 − 1.69469i 0.404040 + 0.233273i 0 − 1.00000i 1.39117 + 2.40957i 1.69469 − 2.93529i
569.6 1.00000i 1.27815 − 2.21381i −1.00000 3.02030 + 1.74377i 2.21381 + 1.27815i 0 − 1.00000i −1.76732 − 3.06108i −1.74377 + 3.02030i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 459.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.k 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.o.d 12
7.b odd 2 1 1274.2.o.e 12
7.c even 3 1 182.2.m.b ✓ 12
7.c even 3 1 1274.2.v.e 12
7.d odd 6 1 1274.2.m.c 12
7.d odd 6 1 1274.2.v.d 12
13.e even 6 1 1274.2.v.e 12
21.h odd 6 1 1638.2.bj.g 12
28.g odd 6 1 1456.2.cc.d 12
91.h even 3 1 2366.2.d.r 12
91.k even 6 1 inner 1274.2.o.d 12
91.k even 6 1 2366.2.d.r 12
91.l odd 6 1 1274.2.o.e 12
91.p odd 6 1 1274.2.m.c 12
91.t odd 6 1 1274.2.v.d 12
91.u even 6 1 182.2.m.b ✓ 12
91.x odd 12 1 2366.2.a.bf 6
91.x odd 12 1 2366.2.a.bh 6
273.x odd 6 1 1638.2.bj.g 12
364.s odd 6 1 1456.2.cc.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
182.2.m.b ✓ 12 7.c even 3 1
182.2.m.b ✓ 12 91.u even 6 1
1274.2.m.c 12 7.d odd 6 1
1274.2.m.c 12 91.p odd 6 1
1274.2.o.d 12 1.a even 1 1 trivial
1274.2.o.d 12 91.k even 6 1 inner
1274.2.o.e 12 7.b odd 2 1
1274.2.o.e 12 91.l odd 6 1
1274.2.v.d 12 7.d odd 6 1
1274.2.v.d 12 91.t odd 6 1
1274.2.v.e 12 7.c even 3 1
1274.2.v.e 12 13.e even 6 1
1456.2.cc.d 12 28.g odd 6 1
1456.2.cc.d 12 364.s odd 6 1
1638.2.bj.g 12 21.h odd 6 1
1638.2.bj.g 12 273.x odd 6 1
2366.2.a.bf 6 91.x odd 12 1
2366.2.a.bh 6 91.x odd 12 1
2366.2.d.r 12 91.h even 3 1
2366.2.d.r 12 91.k even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 2 T_{3}^{11} + 14 T_{3}^{10} + 4 T_{3}^{9} + 103 T_{3}^{8} + 34 T_{3}^{7} + 354 T_{3}^{6} + \cdots + 4 \) acting on \(S_{2}^{\mathrm{new}}(1274, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$3$ \( T^{12} + 2 T^{11} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( T^{12} - 21 T^{10} + \cdots + 5041 \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( T^{12} - 18 T^{11} + \cdots + 495616 \) Copy content Toggle raw display
$13$ \( T^{12} + 8 T^{11} + \cdots + 4826809 \) Copy content Toggle raw display
$17$ \( (T^{6} + 4 T^{5} + \cdots - 176)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 369869824 \) Copy content Toggle raw display
$23$ \( (T^{6} - 6 T^{5} + \cdots + 3142)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + 10 T^{11} + \cdots + 135424 \) Copy content Toggle raw display
$31$ \( T^{12} - 18 T^{11} + \cdots + 1024 \) Copy content Toggle raw display
$37$ \( T^{12} + 310 T^{10} + \cdots + 1024 \) Copy content Toggle raw display
$41$ \( T^{12} + 24 T^{11} + \cdots + 1024 \) Copy content Toggle raw display
$43$ \( T^{12} - 26 T^{11} + \cdots + 8667136 \) Copy content Toggle raw display
$47$ \( T^{12} - 48 T^{11} + \cdots + 31719424 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 2018525184 \) Copy content Toggle raw display
$59$ \( T^{12} + 168 T^{10} + \cdots + 4 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 80364879169 \) Copy content Toggle raw display
$67$ \( T^{12} - 42 T^{11} + \cdots + 1024 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 750321664 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 2708994304 \) Copy content Toggle raw display
$79$ \( T^{12} + 22 T^{11} + \cdots + 64513024 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 879478336 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 10303062016 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 6400000000 \) Copy content Toggle raw display
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