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Properties

Label 1710.2.t.d
Level $1710$
Weight $2$
Character orbit 1710.t
Analytic conductor $13.654$
Analytic rank $0$
Dimension $20$
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(919,1710)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1710, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1710.919");
 
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.t (of order \(6\), 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: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 20 x^{18} + 270 x^{16} - 1928 x^{14} + 9835 x^{12} - 29980 x^{10} + 66046 x^{8} - 89920 x^{6} + \cdots + 10000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 190)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{13} + \beta_{2}) q^{2} + ( - \beta_{3} + 1) q^{4} + \beta_{14} q^{5} + ( - \beta_{13} + \beta_{9}) q^{7} + \beta_{13} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{13} + \beta_{2}) q^{2} + ( - \beta_{3} + 1) q^{4} + \beta_{14} q^{5} + ( - \beta_{13} + \beta_{9}) q^{7} + \beta_{13} q^{8} + \beta_{16} q^{10} + (\beta_{19} + \beta_{15} - \beta_{11} + \cdots + 1) q^{11}+ \cdots + ( - \beta_{15} + \beta_{14} + \cdots + 3 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 10 q^{4} + 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 10 q^{4} + 2 q^{5} + 12 q^{11} + 10 q^{14} - 10 q^{16} - 22 q^{19} + 4 q^{20} - 6 q^{25} - 8 q^{26} + 4 q^{29} - 16 q^{31} - 8 q^{34} + 2 q^{35} - 2 q^{41} + 6 q^{44} - 52 q^{46} + 40 q^{49} - 40 q^{50} + 18 q^{55} + 20 q^{56} + 44 q^{59} - 4 q^{61} - 20 q^{64} - 48 q^{65} - 8 q^{70} + 44 q^{71} - 10 q^{74} + 4 q^{76} - 4 q^{79} + 2 q^{80} + 12 q^{85} - 2 q^{89} + 20 q^{91} - 40 q^{94} + 4 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 20 x^{18} + 270 x^{16} - 1928 x^{14} + 9835 x^{12} - 29980 x^{10} + 66046 x^{8} - 89920 x^{6} + \cdots + 10000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 67521462807 \nu^{19} + 3129986432010 \nu^{17} - 52636278341640 \nu^{15} + \cdots + 25\!\cdots\!00 \nu ) / 99\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 54067767641 \nu^{18} + 1052563531840 \nu^{16} - 14058915653070 \nu^{14} + \cdots + 15\!\cdots\!00 ) / 995735151006500 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1689612 \nu^{18} + 26942635 \nu^{16} - 323563365 \nu^{14} + 1511616336 \nu^{12} + \cdots - 30857811000 ) / 22157483500 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 102026243919 \nu^{18} + 2229773264110 \nu^{16} - 30750494419705 \nu^{14} + \cdots + 13\!\cdots\!00 ) / 995735151006500 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 244351455749 \nu^{19} - 1553004285275 \nu^{18} - 7051030576145 \nu^{17} + \cdots + 14\!\cdots\!00 ) / 99\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 149747246509 \nu^{19} - 1149846030380 \nu^{18} - 2721738955500 \nu^{17} + \cdots - 20\!\cdots\!00 ) / 49\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 446324990017 \nu^{19} + 1317043616260 \nu^{18} + 9062355730875 \nu^{17} + \cdots - 20\!\cdots\!00 ) / 99\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 54067767641 \nu^{19} - 1052563531840 \nu^{17} + 14058915653070 \nu^{15} + \cdots - 507053781808000 \nu ) / 995735151006500 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 244351455749 \nu^{19} - 1553004285275 \nu^{18} + 7051030576145 \nu^{17} + \cdots + 14\!\cdots\!00 ) / 99\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 203038749068 \nu^{19} - 1607322513115 \nu^{18} - 3807391628420 \nu^{17} + \cdots + 60\!\cdots\!00 ) / 49\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 446324990017 \nu^{19} + 1317043616260 \nu^{18} - 9062355730875 \nu^{17} + \cdots - 20\!\cdots\!00 ) / 99\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 30857811 \nu^{19} - 634052340 \nu^{17} + 8601035320 \nu^{15} - 62729493258 \nu^{13} + \cdots - 338972268000 \nu ) / 221574835000 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 1748616823509 \nu^{19} - 824090674470 \nu^{18} - 33524641658625 \nu^{17} + \cdots + 29\!\cdots\!00 ) / 99\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 1748616823509 \nu^{19} - 824090674470 \nu^{18} + 33524641658625 \nu^{17} + \cdots + 29\!\cdots\!00 ) / 99\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 1964304297788 \nu^{19} + 571224133225 \nu^{18} + 37105178622365 \nu^{17} + \cdots - 35\!\cdots\!00 ) / 99\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 1964304297788 \nu^{19} - 571224133225 \nu^{18} + 37105178622365 \nu^{17} + \cdots + 35\!\cdots\!00 ) / 99\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 2048111316527 \nu^{19} + 1475601386290 \nu^{18} - 38968119569625 \nu^{17} + \cdots + 33\!\cdots\!00 ) / 99\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 2399045777394 \nu^{19} + 3943558637035 \nu^{18} - 48190455491610 \nu^{17} + \cdots + 31\!\cdots\!00 ) / 99\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{17} - \beta_{16} + \beta_{5} - 4\beta_{3} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{19} + \beta_{17} + \beta_{16} - \beta_{15} - 4\beta_{13} + \beta_{12} + \beta_{11} + 7\beta_{9} - \beta_{8} + \beta_{6} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{18} - 2\beta_{15} - \beta_{14} - 9\beta_{12} - 9\beta_{8} + \beta_{7} - 13\beta_{4} - 28\beta_{3} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 10 \beta_{19} - 10 \beta_{18} + 10 \beta_{17} + 10 \beta_{16} + 10 \beta_{11} - 6 \beta_{10} + \cdots - 61 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 16 \beta_{19} - 81 \beta_{17} + 81 \beta_{16} - 26 \beta_{15} - 10 \beta_{14} - 81 \beta_{12} + \cdots - 244 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 91 \beta_{18} + 189 \beta_{15} - 98 \beta_{14} + 608 \beta_{13} - 97 \beta_{12} + 97 \beta_{8} + \cdots - 573 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 195 \beta_{19} + 195 \beta_{18} - 761 \beta_{17} + 761 \beta_{16} + 195 \beta_{11} + 85 \beta_{10} + \cdots - 2304 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 846 \beta_{19} - 956 \beta_{17} - 956 \beta_{16} + 2046 \beta_{15} - 1200 \beta_{14} + \cdots - 2046 \beta_{6} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2156 \beta_{18} + 2892 \beta_{15} + 736 \beta_{14} + 7369 \beta_{12} + 7369 \beta_{8} + \cdots + 22488 \beta_{3} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 8105 \beta_{19} + 8105 \beta_{18} - 9525 \beta_{17} - 9525 \beta_{16} - 8105 \beta_{11} + \cdots + 55031 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 22801 \beta_{19} + 72661 \beta_{17} - 72661 \beta_{16} + 29486 \beta_{15} + 6685 \beta_{14} + \cdots + 222964 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 79346 \beta_{18} - 219724 \beta_{15} + 140378 \beta_{14} - 683118 \beta_{13} + 95462 \beta_{12} + \cdots + 549093 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 235840 \beta_{19} - 235840 \beta_{18} + 723901 \beta_{17} - 723901 \beta_{16} - 235840 \beta_{11} + \cdots + 2228604 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 787131 \beta_{19} + 959741 \beta_{17} + 959741 \beta_{16} - 2238801 \beta_{15} + \cdots + 2238801 \beta_{6} \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 2411411 \beta_{18} - 3025932 \beta_{15} - 614521 \beta_{14} - 7253629 \beta_{12} + \cdots - 22372248 \beta_{3} \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 7868150 \beta_{19} - 7868150 \beta_{18} + 9665040 \beta_{17} + 9665040 \beta_{16} + \cdots - 55380151 \beta_1 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 24505836 \beta_{19} - 72913341 \beta_{17} + 72913341 \beta_{16} - 30577096 \beta_{15} + \cdots - 225114384 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 78984601 \beta_{18} + 229790209 \beta_{15} - 150805608 \beta_{14} + 710971628 \beta_{13} + \cdots - 557790863 \beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(191\) \(1027\) \(1351\)
\(\chi(n)\) \(1\) \(-1\) \(-1 + \beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
919.1
1.15118 − 0.664633i
−1.28416 + 0.741409i
−0.590953 + 0.341187i
2.75095 − 1.58826i
−2.02701 + 1.17030i
2.02701 − 1.17030i
1.28416 − 0.741409i
−2.75095 + 1.58826i
−1.15118 + 0.664633i
0.590953 − 0.341187i
1.15118 + 0.664633i
−1.28416 − 0.741409i
−0.590953 − 0.341187i
2.75095 + 1.58826i
−2.02701 − 1.17030i
2.02701 + 1.17030i
1.28416 + 0.741409i
−2.75095 − 1.58826i
−1.15118 − 0.664633i
0.590953 + 0.341187i
−0.866025 + 0.500000i 0 0.500000 − 0.866025i −2.21845 − 0.280148i 0 − 2.32927i 1.00000i 0 2.06131 − 0.866610i
919.2 −0.866025 + 0.500000i 0 0.500000 − 0.866025i −1.25308 − 1.85197i 0 0.482818i 1.00000i 0 2.01118 + 0.977310i
919.3 −0.866025 + 0.500000i 0 0.500000 − 0.866025i −0.373767 + 2.20461i 0 − 0.317626i 1.00000i 0 −0.778613 − 2.09613i
919.4 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 2.10986 + 0.740612i 0 − 4.17652i 1.00000i 0 −2.19750 + 0.413539i
919.5 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 2.23544 + 0.0529205i 0 1.34059i 1.00000i 0 −1.96241 + 1.07189i
919.6 0.866025 − 0.500000i 0 0.500000 − 0.866025i −1.07189 + 1.96241i 0 − 1.34059i − 1.00000i 0 0.0529205 + 2.23544i
919.7 0.866025 − 0.500000i 0 0.500000 − 0.866025i −0.977310 − 2.01118i 0 − 0.482818i − 1.00000i 0 −1.85197 − 1.25308i
919.8 0.866025 − 0.500000i 0 0.500000 − 0.866025i −0.413539 + 2.19750i 0 4.17652i − 1.00000i 0 0.740612 + 2.10986i
919.9 0.866025 − 0.500000i 0 0.500000 − 0.866025i 0.866610 − 2.06131i 0 2.32927i − 1.00000i 0 −0.280148 − 2.21845i
919.10 0.866025 − 0.500000i 0 0.500000 − 0.866025i 2.09613 + 0.778613i 0 0.317626i − 1.00000i 0 2.20461 − 0.373767i
1189.1 −0.866025 − 0.500000i 0 0.500000 + 0.866025i −2.21845 + 0.280148i 0 2.32927i − 1.00000i 0 2.06131 + 0.866610i
1189.2 −0.866025 − 0.500000i 0 0.500000 + 0.866025i −1.25308 + 1.85197i 0 − 0.482818i − 1.00000i 0 2.01118 − 0.977310i
1189.3 −0.866025 − 0.500000i 0 0.500000 + 0.866025i −0.373767 − 2.20461i 0 0.317626i − 1.00000i 0 −0.778613 + 2.09613i
1189.4 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 2.10986 − 0.740612i 0 4.17652i − 1.00000i 0 −2.19750 − 0.413539i
1189.5 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 2.23544 − 0.0529205i 0 − 1.34059i − 1.00000i 0 −1.96241 − 1.07189i
1189.6 0.866025 + 0.500000i 0 0.500000 + 0.866025i −1.07189 − 1.96241i 0 1.34059i 1.00000i 0 0.0529205 − 2.23544i
1189.7 0.866025 + 0.500000i 0 0.500000 + 0.866025i −0.977310 + 2.01118i 0 0.482818i 1.00000i 0 −1.85197 + 1.25308i
1189.8 0.866025 + 0.500000i 0 0.500000 + 0.866025i −0.413539 − 2.19750i 0 − 4.17652i 1.00000i 0 0.740612 − 2.10986i
1189.9 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0.866610 + 2.06131i 0 − 2.32927i 1.00000i 0 −0.280148 + 2.21845i
1189.10 0.866025 + 0.500000i 0 0.500000 + 0.866025i 2.09613 − 0.778613i 0 − 0.317626i 1.00000i 0 2.20461 + 0.373767i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 919.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
19.c even 3 1 inner
95.i even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1710.2.t.d 20
3.b odd 2 1 190.2.i.a ✓ 20
5.b even 2 1 inner 1710.2.t.d 20
15.d odd 2 1 190.2.i.a ✓ 20
15.e even 4 1 950.2.e.n 10
15.e even 4 1 950.2.e.o 10
19.c even 3 1 inner 1710.2.t.d 20
57.h odd 6 1 190.2.i.a ✓ 20
95.i even 6 1 inner 1710.2.t.d 20
285.n odd 6 1 190.2.i.a ✓ 20
285.v even 12 1 950.2.e.n 10
285.v even 12 1 950.2.e.o 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.i.a ✓ 20 3.b odd 2 1
190.2.i.a ✓ 20 15.d odd 2 1
190.2.i.a ✓ 20 57.h odd 6 1
190.2.i.a ✓ 20 285.n odd 6 1
950.2.e.n 10 15.e even 4 1
950.2.e.n 10 285.v even 12 1
950.2.e.o 10 15.e even 4 1
950.2.e.o 10 285.v even 12 1
1710.2.t.d 20 1.a even 1 1 trivial
1710.2.t.d 20 5.b even 2 1 inner
1710.2.t.d 20 19.c even 3 1 inner
1710.2.t.d 20 95.i even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{10} + 25T_{7}^{8} + 144T_{7}^{6} + 216T_{7}^{4} + 60T_{7}^{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(1710, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{5} \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( T^{20} - 2 T^{19} + \cdots + 9765625 \) Copy content Toggle raw display
$7$ \( (T^{10} + 25 T^{8} + \cdots + 4)^{2} \) Copy content Toggle raw display
$11$ \( (T^{5} - 3 T^{4} + \cdots - 1555)^{4} \) Copy content Toggle raw display
$13$ \( T^{20} - 60 T^{18} + \cdots + 65536 \) Copy content Toggle raw display
$17$ \( T^{20} - 128 T^{18} + \cdots + 65536 \) Copy content Toggle raw display
$19$ \( (T^{10} + 11 T^{9} + \cdots + 2476099)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 842290759696 \) Copy content Toggle raw display
$29$ \( (T^{10} - 2 T^{9} + \cdots + 4000000)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} + 4 T^{4} + \cdots + 1252)^{4} \) Copy content Toggle raw display
$37$ \( (T^{10} + 233 T^{8} + \cdots + 60186564)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} + T^{9} + \cdots + 6561)^{2} \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 1761205026816 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 16796160000 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 39\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{10} - 22 T^{9} + \cdots + 15376)^{2} \) Copy content Toggle raw display
$61$ \( (T^{10} + 2 T^{9} + \cdots + 62853184)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 430467210000 \) Copy content Toggle raw display
$71$ \( (T^{10} - 22 T^{9} + \cdots + 952576)^{2} \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{10} + 2 T^{9} + \cdots + 19360000)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} + 272 T^{8} + \cdots + 627264)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + T^{9} + \cdots + 397922704)^{2} \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 70\!\cdots\!00 \) Copy content Toggle raw display
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