Properties

Label 1352.2.i.n
Level $1352$
Weight $2$
Character orbit 1352.i
Analytic conductor $10.796$
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] = [1352,2,Mod(529,1352)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1352, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1352.529");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1352 = 2^{3} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1352.i (of order \(3\), 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: \(10.7957743533\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} + 11 x^{10} - 12 x^{9} + 89 x^{8} - 84 x^{7} + 297 x^{6} - 105 x^{5} + 552 x^{4} - 110 x^{3} + 610 x^{2} + 234 x + 169 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{10} + \beta_{7} + \beta_{2} - \beta_1) q^{3} + (\beta_{5} + \beta_{4} + \beta_1) q^{5} + ( - \beta_{9} - \beta_{5} + \beta_{3} - \beta_{2}) q^{7} + ( - \beta_{11} - 2 \beta_{9} - \beta_{7} - 2 \beta_{5} - 2 \beta_{3} - 2 \beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{10} + \beta_{7} + \beta_{2} - \beta_1) q^{3} + (\beta_{5} + \beta_{4} + \beta_1) q^{5} + ( - \beta_{9} - \beta_{5} + \beta_{3} - \beta_{2}) q^{7} + ( - \beta_{11} - 2 \beta_{9} - \beta_{7} - 2 \beta_{5} - 2 \beta_{3} - 2 \beta_{2} - 1) q^{9} + ( - \beta_{11} - \beta_{10} + \beta_{9} - \beta_{7} - \beta_{6} - \beta_{4} + 2 \beta_{2} + 1) q^{11} + ( - \beta_{11} + \beta_{10} + \beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} - \beta_{4} - 2 \beta_{2} - \beta_1 + 1) q^{15} + (\beta_{11} - \beta_{10} - \beta_{9} - 2 \beta_{7} + \beta_{6} + \beta_{3} - 3) q^{17} + (\beta_{10} + 3 \beta_{9} + 2 \beta_{7} - \beta_{6} + \beta_{5} + 2 \beta_{3} + 2 \beta_{2} + 3) q^{19} + (6 \beta_{9} + \beta_{8} - 4 \beta_{6} - 2 \beta_{4} + \beta_{3} + 7) q^{21} + (\beta_{11} - 3 \beta_{10} - \beta_{9} + \beta_{8} + 4 \beta_{7} + \beta_{6} + \beta_{4} + \beta_{2} - 1) q^{23} + ( - 6 \beta_{9} - \beta_{8} + 2 \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_1) q^{25} + (3 \beta_{9} + \beta_{8} + 3 \beta_{6} + 2 \beta_{5} - \beta_{4} + \beta_{3} + 2 \beta_1 + 3) q^{27} + ( - 2 \beta_{11} - 4 \beta_{10} + 2 \beta_{9} + \beta_{8} + 4 \beta_{7} - 2 \beta_{6} + \cdots + 2) q^{29}+ \cdots + ( - 12 \beta_{9} - 6 \beta_{8} + 6 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} - 6 \beta_{3} + \cdots - 7) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 3 q^{3} + 4 q^{5} + 3 q^{7} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 3 q^{3} + 4 q^{5} + 3 q^{7} - 9 q^{9} + 13 q^{11} - 14 q^{15} - 11 q^{17} + 15 q^{19} + 44 q^{21} - 15 q^{23} + 32 q^{25} + 42 q^{27} - 9 q^{29} + 6 q^{31} - 2 q^{33} - 14 q^{35} + 14 q^{37} - 20 q^{41} - 10 q^{43} - 9 q^{45} + 20 q^{47} - 27 q^{49} + 10 q^{51} + 2 q^{53} - 14 q^{55} - 18 q^{57} + 50 q^{59} + 2 q^{63} + 6 q^{67} - 32 q^{69} + 9 q^{71} + 12 q^{73} - 40 q^{75} - 4 q^{77} + 26 q^{79} - 42 q^{81} - 72 q^{83} - 31 q^{85} - 22 q^{87} - 18 q^{89} - 12 q^{93} + 21 q^{95} - 14 q^{97} - 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{11} + 11 x^{10} - 12 x^{9} + 89 x^{8} - 84 x^{7} + 297 x^{6} - 105 x^{5} + 552 x^{4} - 110 x^{3} + 610 x^{2} + 234 x + 169 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 52311476 \nu^{11} + 16043125687 \nu^{10} - 29345587172 \nu^{9} + 151233419242 \nu^{8} - 320688592710 \nu^{7} + \cdots + 1653563018471 ) / 2918043821108 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 29557613 \nu^{11} + 38707150 \nu^{10} - 4362711104 \nu^{9} - 1784030870 \nu^{8} - 34956937715 \nu^{7} + 7370292117 \nu^{6} + \cdots - 164748264096 ) / 224464909316 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 17578882 \nu^{11} + 1230983680 \nu^{10} + 238980100 \nu^{9} + 9869808053 \nu^{8} - 2562199107 \nu^{7} + 69180219708 \nu^{6} + \cdots - 340583949997 ) / 112232454658 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 49158605 \nu^{11} - 63880065 \nu^{10} - 444476667 \nu^{9} - 464422355 \nu^{8} - 3496753260 \nu^{7} - 4165748199 \nu^{6} + \cdots - 30417801258 ) / 56116227329 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 2936598 \nu^{11} - 14626024 \nu^{10} - 33065230 \nu^{9} - 114268773 \nu^{8} - 197576661 \nu^{7} - 819775828 \nu^{6} - 614956135 \nu^{5} + \cdots + 288763303 ) / 1580738798 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2339830866 \nu^{11} - 2978892731 \nu^{10} + 24907698681 \nu^{9} - 33856167063 \nu^{8} + 202207456459 \nu^{7} - 242003585124 \nu^{6} + \cdots - 343288132863 ) / 729510955277 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 11016608 \nu^{11} - 17449087 \nu^{10} - 66381158 \nu^{9} - 99270656 \nu^{8} - 318753650 \nu^{7} - 979605793 \nu^{6} + 942885975 \nu^{5} + \cdots - 1210992991 ) / 3161477596 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 13038835 \nu^{11} - 11069801 \nu^{10} - 127503402 \nu^{9} - 76168720 \nu^{8} - 933624537 \nu^{7} - 718884330 \nu^{6} - 2378501260 \nu^{5} + \cdots - 6822402521 ) / 3161477596 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 12672943392 \nu^{11} + 13057192361 \nu^{10} - 139905570262 \nu^{9} + 208790565056 \nu^{8} - 1104699560578 \nu^{7} + \cdots + 2137142385001 ) / 2918043821108 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 25454760627 \nu^{11} - 41836584332 \nu^{10} + 251491951678 \nu^{9} - 460768982326 \nu^{8} + 2052895262983 \nu^{7} + \cdots + 1536050650212 ) / 2918043821108 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} - \beta_{9} - 4\beta_{7} - \beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} - \beta_{8} + 2\beta_{6} - 5\beta_{5} + \beta_{4} - \beta_{3} - 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -8\beta_{11} - 3\beta_{10} + 8\beta_{9} + \beta_{8} + 24\beta_{7} - 8\beta_{6} - 8\beta_{4} + 8\beta_{2} + 3\beta _1 + 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 11\beta_{11} + 10\beta_{10} - 17\beta_{9} - 35\beta_{7} - 10\beta_{6} + 31\beta_{5} + 10\beta_{3} - 27\beta_{2} - 25 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -25\beta_{9} - 11\beta_{8} + 89\beta_{6} - 36\beta_{5} + 58\beta_{4} - 11\beta_{3} - 36\beta _1 + 74 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 100 \beta_{11} - 80 \beta_{10} + 100 \beta_{9} + 78 \beta_{8} + 313 \beta_{7} - 100 \beta_{6} - 100 \beta_{4} + 210 \beta_{2} + 210 \beta _1 + 100 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 420 \beta_{11} + 256 \beta_{10} - 254 \beta_{9} - 1172 \beta_{7} - 256 \beta_{6} + 335 \beta_{5} + 102 \beta_{3} - 510 \beta_{2} - 916 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 126\beta_{9} - 574\beta_{8} + 1469\beta_{6} - 1490\beta_{5} + 845\beta_{4} - 574\beta_{3} - 1490\beta _1 + 1119 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 3085 \beta_{11} - 1993 \beta_{10} + 3085 \beta_{9} + 895 \beta_{8} + 8671 \beta_{7} - 3085 \beta_{6} - 3085 \beta_{4} + 4025 \beta_{2} + 2859 \beta _1 + 3085 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 6884 \beta_{11} + 4875 \beta_{10} - 7239 \beta_{9} - 20736 \beta_{7} - 4875 \beta_{6} + 10861 \beta_{5} + 4183 \beta_{3} - 12114 \beta_{2} - 15861 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1352\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(1015\) \(1185\)
\(\chi(n)\) \(1\) \(1\) \(-1 - \beta_{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
529.1
0.764791 1.32466i
1.15932 2.00801i
0.918180 1.59033i
−0.258354 + 0.447482i
−1.38828 + 2.40457i
−0.695659 + 1.20492i
0.764791 + 1.32466i
1.15932 + 2.00801i
0.918180 + 1.59033i
−0.258354 0.447482i
−1.38828 2.40457i
−0.695659 1.20492i
0 −1.66576 + 2.88518i 0 −0.932734 0 −1.72864 2.99410i 0 −4.04951 7.01396i 0
529.2 0 −1.38184 + 2.39342i 0 4.24972 0 1.18730 + 2.05646i 0 −2.31898 4.01660i 0
529.3 0 −0.294690 + 0.510418i 0 3.45555 0 −2.35016 4.07060i 0 1.32632 + 2.29725i 0
529.4 0 0.0358330 0.0620645i 0 −3.69476 0 0.837161 + 1.45000i 0 1.49743 + 2.59363i 0
529.5 0 0.487312 0.844049i 0 0.130796 0 1.38263 + 2.39479i 0 1.02505 + 1.77545i 0
529.6 0 1.31915 2.28483i 0 −1.20857 0 2.17171 + 3.76152i 0 −1.98031 3.42999i 0
1329.1 0 −1.66576 2.88518i 0 −0.932734 0 −1.72864 + 2.99410i 0 −4.04951 + 7.01396i 0
1329.2 0 −1.38184 2.39342i 0 4.24972 0 1.18730 2.05646i 0 −2.31898 + 4.01660i 0
1329.3 0 −0.294690 0.510418i 0 3.45555 0 −2.35016 + 4.07060i 0 1.32632 2.29725i 0
1329.4 0 0.0358330 + 0.0620645i 0 −3.69476 0 0.837161 1.45000i 0 1.49743 2.59363i 0
1329.5 0 0.487312 + 0.844049i 0 0.130796 0 1.38263 2.39479i 0 1.02505 1.77545i 0
1329.6 0 1.31915 + 2.28483i 0 −1.20857 0 2.17171 3.76152i 0 −1.98031 + 3.42999i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 529.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1352.2.i.n 12
13.b even 2 1 1352.2.i.m 12
13.c even 3 1 1352.2.a.n yes 6
13.c even 3 1 inner 1352.2.i.n 12
13.d odd 4 2 1352.2.o.h 24
13.e even 6 1 1352.2.a.m 6
13.e even 6 1 1352.2.i.m 12
13.f odd 12 2 1352.2.f.g 12
13.f odd 12 2 1352.2.o.h 24
52.i odd 6 1 2704.2.a.bf 6
52.j odd 6 1 2704.2.a.bg 6
52.l even 12 2 2704.2.f.r 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1352.2.a.m 6 13.e even 6 1
1352.2.a.n yes 6 13.c even 3 1
1352.2.f.g 12 13.f odd 12 2
1352.2.i.m 12 13.b even 2 1
1352.2.i.m 12 13.e even 6 1
1352.2.i.n 12 1.a even 1 1 trivial
1352.2.i.n 12 13.c even 3 1 inner
1352.2.o.h 24 13.d odd 4 2
1352.2.o.h 24 13.f odd 12 2
2704.2.a.bf 6 52.i odd 6 1
2704.2.a.bg 6 52.j odd 6 1
2704.2.f.r 12 52.l even 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1352, [\chi])\):

\( T_{3}^{12} + 3 T_{3}^{11} + 18 T_{3}^{10} + 19 T_{3}^{9} + 135 T_{3}^{8} + 130 T_{3}^{7} + 623 T_{3}^{6} - 114 T_{3}^{5} + 515 T_{3}^{4} + 149 T_{3}^{3} + 184 T_{3}^{2} - 13 T_{3} + 1 \) Copy content Toggle raw display
\( T_{5}^{6} - 2T_{5}^{5} - 21T_{5}^{4} + 23T_{5}^{3} + 98T_{5}^{2} + 48T_{5} - 8 \) Copy content Toggle raw display
\( T_{7}^{12} - 3 T_{7}^{11} + 39 T_{7}^{10} - 124 T_{7}^{9} + 1073 T_{7}^{8} - 3182 T_{7}^{7} + 14861 T_{7}^{6} - 38092 T_{7}^{5} + 137204 T_{7}^{4} - 293344 T_{7}^{3} + 624752 T_{7}^{2} - 667360 T_{7} + 602176 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 3 T^{11} + 18 T^{10} + 19 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( (T^{6} - 2 T^{5} - 21 T^{4} + 23 T^{3} + \cdots - 8)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} - 3 T^{11} + 39 T^{10} + \cdots + 602176 \) Copy content Toggle raw display
$11$ \( T^{12} - 13 T^{11} + 128 T^{10} + \cdots + 16129 \) Copy content Toggle raw display
$13$ \( T^{12} \) Copy content Toggle raw display
$17$ \( T^{12} + 11 T^{11} + 92 T^{10} + 313 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{12} - 15 T^{11} + 184 T^{10} + \cdots + 49970761 \) Copy content Toggle raw display
$23$ \( T^{12} + 15 T^{11} + 182 T^{10} + \cdots + 64 \) Copy content Toggle raw display
$29$ \( T^{12} + 9 T^{11} + 175 T^{10} + \cdots + 584865856 \) Copy content Toggle raw display
$31$ \( (T^{6} - 3 T^{5} - 54 T^{4} + 43 T^{3} + \cdots - 664)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} - 14 T^{11} + 155 T^{10} + \cdots + 18181696 \) Copy content Toggle raw display
$41$ \( T^{12} + 20 T^{11} + 324 T^{10} + \cdots + 28561 \) Copy content Toggle raw display
$43$ \( T^{12} + 10 T^{11} + 151 T^{10} + \cdots + 4826809 \) Copy content Toggle raw display
$47$ \( (T^{6} - 10 T^{5} - 55 T^{4} + 451 T^{3} + \cdots + 2696)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} - T^{5} - 106 T^{4} + 201 T^{3} + \cdots - 8632)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} - 50 T^{11} + \cdots + 71247887929 \) Copy content Toggle raw display
$61$ \( T^{12} + 135 T^{10} + \cdots + 28217344 \) Copy content Toggle raw display
$67$ \( T^{12} - 6 T^{11} + \cdots + 4561246369 \) Copy content Toggle raw display
$71$ \( T^{12} - 9 T^{11} + \cdots + 12592430656 \) Copy content Toggle raw display
$73$ \( (T^{6} - 6 T^{5} - 321 T^{4} + \cdots - 282449)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} - 13 T^{5} - 174 T^{4} + \cdots - 246616)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + 36 T^{5} + 357 T^{4} + \cdots + 70783)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + 18 T^{11} + 231 T^{10} + \cdots + 1164241 \) Copy content Toggle raw display
$97$ \( T^{12} + 14 T^{11} + \cdots + 15507471841 \) Copy content Toggle raw display
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