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Properties

Label 1710.2.p.b
Level $1710$
Weight $2$
Character orbit 1710.p
Analytic conductor $13.654$
Analytic rank $0$
Dimension $16$
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] = [1710,2,Mod(37,1710)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1710, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1710.37");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1710 = 2 \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1710.p (of order \(4\), degree \(2\), 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: \(13.6544187456\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 24x^{14} + 212x^{12} + 880x^{10} + 1858x^{8} + 1960x^{6} + 892x^{4} + 96x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 190)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{9} q^{2} - \beta_{2} q^{4} + ( - \beta_{4} - \beta_{2}) q^{5} + (\beta_{15} - \beta_{4}) q^{7} - \beta_{8} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{9} q^{2} - \beta_{2} q^{4} + ( - \beta_{4} - \beta_{2}) q^{5} + (\beta_{15} - \beta_{4}) q^{7} - \beta_{8} q^{8} + ( - \beta_{11} - \beta_{8}) q^{10} + (\beta_{6} - \beta_{3} + \beta_{2}) q^{11} + (\beta_{13} - \beta_{11} + \cdots - \beta_1) q^{13}+ \cdots + ( - \beta_{13} + \beta_{12} + \cdots + \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{7} - 16 q^{16} + 32 q^{17} - 8 q^{20} + 8 q^{23} + 32 q^{25} + 32 q^{26} + 8 q^{28} + 24 q^{35} - 8 q^{38} + 24 q^{43} - 32 q^{47} - 56 q^{55} - 64 q^{61} + 8 q^{62} + 32 q^{68} + 16 q^{73} - 16 q^{76} - 72 q^{77} + 40 q^{82} + 16 q^{83} + 32 q^{85} - 8 q^{92} - 24 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 24x^{14} + 212x^{12} + 880x^{10} + 1858x^{8} + 1960x^{6} + 892x^{4} + 96x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 35 \nu^{15} + 154 \nu^{14} + 615 \nu^{13} + 3634 \nu^{12} + 2265 \nu^{11} + 31136 \nu^{10} + \cdots + 5958 ) / 2320 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{15} - 26\nu^{13} - 259\nu^{11} - 1278\nu^{9} - 3369\nu^{7} - 4598\nu^{5} - 2743\nu^{3} - 402\nu ) / 40 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 61 \nu^{15} + 130 \nu^{14} + 1511 \nu^{13} + 3030 \nu^{12} + 13849 \nu^{11} + 25730 \nu^{10} + \cdots + 7180 ) / 2320 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 51 \nu^{15} - 210 \nu^{14} - 1211 \nu^{13} - 4850 \nu^{12} - 10509 \nu^{11} - 40270 \nu^{10} + \cdots - 5040 ) / 2320 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 109 \nu^{15} + 210 \nu^{14} - 2719 \nu^{13} + 4850 \nu^{12} - 25531 \nu^{11} + 40270 \nu^{10} + \cdots + 5040 ) / 2320 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 119 \nu^{15} - 130 \nu^{14} + 3019 \nu^{13} - 3030 \nu^{12} + 28871 \nu^{11} - 25730 \nu^{10} + \cdots - 7180 ) / 2320 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 165 \nu^{15} + 154 \nu^{14} - 3935 \nu^{13} + 3634 \nu^{12} - 34375 \nu^{11} + 31136 \nu^{10} + \cdots + 5958 ) / 2320 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 85 \nu^{15} + 18 \nu^{14} - 2115 \nu^{13} + 453 \nu^{12} - 19690 \nu^{11} + 4272 \nu^{10} + \cdots + 1041 ) / 1160 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 85 \nu^{15} - 18 \nu^{14} - 2115 \nu^{13} - 453 \nu^{12} - 19690 \nu^{11} - 4272 \nu^{10} + \cdots - 1041 ) / 1160 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 165 \nu^{15} - 154 \nu^{14} - 3935 \nu^{13} - 3634 \nu^{12} - 34375 \nu^{11} - 31136 \nu^{10} + \cdots - 5958 ) / 2320 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 525 \nu^{15} + 76 \nu^{14} - 12415 \nu^{13} + 1816 \nu^{12} - 107055 \nu^{11} + 15814 \nu^{10} + \cdots - 1888 ) / 2320 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 525 \nu^{15} - 76 \nu^{14} - 12415 \nu^{13} - 1816 \nu^{12} - 107055 \nu^{11} - 15814 \nu^{10} + \cdots + 1888 ) / 2320 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 525 \nu^{15} - 376 \nu^{14} - 12415 \nu^{13} - 8496 \nu^{12} - 107055 \nu^{11} - 67874 \nu^{10} + \cdots + 4548 ) / 2320 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 953 \nu^{15} + 150 \nu^{14} - 22413 \nu^{13} + 3340 \nu^{12} - 191717 \nu^{11} + 26030 \nu^{10} + \cdots + 3310 ) / 2320 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 1123 \nu^{15} - 230 \nu^{14} - 26643 \nu^{13} - 5160 \nu^{12} - 231097 \nu^{11} - 40570 \nu^{10} + \cdots + 1150 ) / 2320 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{9} + \beta_{8} + \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{15} + \beta_{14} + \beta_{13} - \beta_{12} - \beta_{6} + \beta_{4} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - \beta_{15} - \beta_{14} + 3 \beta_{12} + 3 \beta_{11} + 3 \beta_{10} - 8 \beta_{9} - 8 \beta_{8} + \cdots + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 8 \beta_{15} - 8 \beta_{14} - 8 \beta_{13} + 10 \beta_{12} - 2 \beta_{11} - 2 \beta_{9} + 2 \beta_{8} + \cdots + 19 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 10 \beta_{15} + 10 \beta_{14} - 30 \beta_{12} - 30 \beta_{11} - 40 \beta_{10} + 69 \beta_{9} + 69 \beta_{8} + \cdots - 10 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 66 \beta_{15} + 66 \beta_{14} + 67 \beta_{13} - 93 \beta_{12} + 26 \beta_{11} - 6 \beta_{10} + \cdots - 150 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 85 \beta_{15} - 85 \beta_{14} + 259 \beta_{12} + 259 \beta_{11} + 427 \beta_{10} - 610 \beta_{9} + \cdots + 85 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 564 \beta_{15} - 564 \beta_{14} - 576 \beta_{13} + 856 \beta_{12} - 280 \beta_{11} + 108 \beta_{10} + \cdots + 1283 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 712 \beta_{15} + 712 \beta_{14} - 2208 \beta_{12} - 2208 \beta_{11} - 4284 \beta_{10} + 5479 \beta_{9} + \cdots - 712 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 4945 \beta_{15} + 4945 \beta_{14} + 5055 \beta_{13} - 7903 \beta_{12} + 2848 \beta_{11} + \cdots - 11363 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 6055 \beta_{15} - 6055 \beta_{14} + 19085 \beta_{12} + 19085 \beta_{11} + 41965 \beta_{10} + \cdots + 6055 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 44200 \beta_{15} - 44200 \beta_{14} - 45144 \beta_{13} + 73366 \beta_{12} - 28222 \beta_{11} + \cdots + 102625 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 52558 \beta_{15} + 52558 \beta_{14} - 167986 \beta_{12} - 167986 \beta_{11} - 406276 \beta_{10} + \cdots - 52558 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 400944 \beta_{15} + 400944 \beta_{14} + 408985 \beta_{13} - 684547 \beta_{12} + 275562 \beta_{11} + \cdots - 939032 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 465071 \beta_{15} - 465071 \beta_{14} + 1503585 \beta_{12} + 1503585 \beta_{11} + 3906321 \beta_{10} + \cdots + 465071 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1710\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(1027\) \(1351\)
\(\chi(n)\) \(1\) \(\beta_{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
− 2.60402i
− 0.366085i
− 1.52212i
1.66380i
− 1.18980i
1.04813i
− 0.107911i
3.07801i
2.60402i
0.366085i
1.52212i
− 1.66380i
1.18980i
− 1.04813i
0.107911i
− 3.07801i
−0.707107 + 0.707107i 0 − 1.00000i −2.23553 + 0.0491316i 0 0.710512 + 0.710512i 0.707107 + 0.707107i 0 1.54602 − 1.61550i
37.2 −0.707107 + 0.707107i 0 − 1.00000i −1.46207 − 1.69185i 0 −2.49494 − 2.49494i 0.707107 + 0.707107i 0 2.23016 + 0.162483i
37.3 −0.707107 + 0.707107i 0 − 1.00000i 1.72906 − 1.41787i 0 2.12620 + 2.12620i 0.707107 + 0.707107i 0 −0.220040 + 2.22522i
37.4 −0.707107 + 0.707107i 0 − 1.00000i 1.96854 + 1.06059i 0 1.65823 + 1.65823i 0.707107 + 0.707107i 0 −2.14192 + 0.642014i
37.5 0.707107 − 0.707107i 0 − 1.00000i −2.23553 + 0.0491316i 0 0.710512 + 0.710512i −0.707107 − 0.707107i 0 −1.54602 + 1.61550i
37.6 0.707107 − 0.707107i 0 − 1.00000i −1.46207 − 1.69185i 0 −2.49494 − 2.49494i −0.707107 − 0.707107i 0 −2.23016 − 0.162483i
37.7 0.707107 − 0.707107i 0 − 1.00000i 1.72906 − 1.41787i 0 2.12620 + 2.12620i −0.707107 − 0.707107i 0 0.220040 − 2.22522i
37.8 0.707107 − 0.707107i 0 − 1.00000i 1.96854 + 1.06059i 0 1.65823 + 1.65823i −0.707107 − 0.707107i 0 2.14192 − 0.642014i
1063.1 −0.707107 − 0.707107i 0 1.00000i −2.23553 − 0.0491316i 0 0.710512 − 0.710512i 0.707107 − 0.707107i 0 1.54602 + 1.61550i
1063.2 −0.707107 − 0.707107i 0 1.00000i −1.46207 + 1.69185i 0 −2.49494 + 2.49494i 0.707107 − 0.707107i 0 2.23016 − 0.162483i
1063.3 −0.707107 − 0.707107i 0 1.00000i 1.72906 + 1.41787i 0 2.12620 − 2.12620i 0.707107 − 0.707107i 0 −0.220040 − 2.22522i
1063.4 −0.707107 − 0.707107i 0 1.00000i 1.96854 − 1.06059i 0 1.65823 − 1.65823i 0.707107 − 0.707107i 0 −2.14192 − 0.642014i
1063.5 0.707107 + 0.707107i 0 1.00000i −2.23553 − 0.0491316i 0 0.710512 − 0.710512i −0.707107 + 0.707107i 0 −1.54602 − 1.61550i
1063.6 0.707107 + 0.707107i 0 1.00000i −1.46207 + 1.69185i 0 −2.49494 + 2.49494i −0.707107 + 0.707107i 0 −2.23016 + 0.162483i
1063.7 0.707107 + 0.707107i 0 1.00000i 1.72906 + 1.41787i 0 2.12620 − 2.12620i −0.707107 + 0.707107i 0 0.220040 + 2.22522i
1063.8 0.707107 + 0.707107i 0 1.00000i 1.96854 − 1.06059i 0 1.65823 − 1.65823i −0.707107 + 0.707107i 0 2.14192 + 0.642014i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
19.b odd 2 1 inner
95.g even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1710.2.p.b 16
3.b odd 2 1 190.2.f.b ✓ 16
5.c odd 4 1 inner 1710.2.p.b 16
15.d odd 2 1 950.2.f.c 16
15.e even 4 1 190.2.f.b ✓ 16
15.e even 4 1 950.2.f.c 16
19.b odd 2 1 inner 1710.2.p.b 16
57.d even 2 1 190.2.f.b ✓ 16
95.g even 4 1 inner 1710.2.p.b 16
285.b even 2 1 950.2.f.c 16
285.j odd 4 1 190.2.f.b ✓ 16
285.j odd 4 1 950.2.f.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.f.b ✓ 16 3.b odd 2 1
190.2.f.b ✓ 16 15.e even 4 1
190.2.f.b ✓ 16 57.d even 2 1
190.2.f.b ✓ 16 285.j odd 4 1
950.2.f.c 16 15.d odd 2 1
950.2.f.c 16 15.e even 4 1
950.2.f.c 16 285.b even 2 1
950.2.f.c 16 285.j odd 4 1
1710.2.p.b 16 1.a even 1 1 trivial
1710.2.p.b 16 5.c odd 4 1 inner
1710.2.p.b 16 19.b odd 2 1 inner
1710.2.p.b 16 95.g even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} - 4T_{7}^{7} + 8T_{7}^{6} - 12T_{7}^{5} + 150T_{7}^{4} - 620T_{7}^{3} + 1352T_{7}^{2} - 1300T_{7} + 625 \) acting on \(S_{2}^{\mathrm{new}}(1710, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} - 8 T^{6} + \cdots + 625)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} - 4 T^{7} + \cdots + 625)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 20 T^{2} + \cdots - 10)^{4} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 2300257521 \) Copy content Toggle raw display
$17$ \( (T^{8} - 16 T^{7} + \cdots + 25)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 16983563041 \) Copy content Toggle raw display
$23$ \( (T^{8} - 4 T^{7} + \cdots + 140625)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} - 148 T^{6} + \cdots + 50625)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 96 T^{6} + \cdots + 152100)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 136048896 \) Copy content Toggle raw display
$41$ \( (T^{8} + 96 T^{6} + \cdots + 22500)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} - 12 T^{7} + \cdots + 2500)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 16 T^{7} + \cdots + 57600)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 3486784401 \) Copy content Toggle raw display
$59$ \( (T^{8} - 244 T^{6} + \cdots + 3515625)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 16 T^{3} + \cdots - 1910)^{4} \) Copy content Toggle raw display
$67$ \( T^{16} + 19724 T^{12} + \cdots + 43046721 \) Copy content Toggle raw display
$71$ \( (T^{8} + 368 T^{6} + \cdots + 14516100)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} - 8 T^{7} + \cdots + 15625)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} - 784 T^{6} + \cdots + 729000000)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 8 T^{7} + \cdots + 1600)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} - 528 T^{6} + \cdots + 70056900)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 9475854336 \) Copy content Toggle raw display
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