↑

Properties

Label 35.8.e.b
Level $35$
Weight $8$
Character orbit 35.e
Analytic conductor $10.933$
Analytic rank $0$
Dimension $20$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [35,8,Mod(11,35)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("35.11"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(35, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 35.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.9334758919\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 5 x^{19} + 990 x^{18} - 1065 x^{17} + 626958 x^{16} - 59773 x^{15} + 229358265 x^{14} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{20}\cdot 7^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q + (2 \beta_{2} + \beta_1) q^{2} + (\beta_{7} + 2 \beta_{2} + 2) q^{3} + (\beta_{5} - 2 \beta_{3} - 71 \beta_{2} - 71) q^{4} - 125 \beta_{2} q^{5} + (\beta_{12} - \beta_{9} - 4 \beta_{8} + \cdots - 12) q^{6}+ \cdots + (2547 \beta_{19} - 5225 \beta_{18} + \cdots + 4192464) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 15 q^{2} + 20 q^{3} - 695 q^{4} + 1250 q^{5} - 256 q^{6} + 1500 q^{7} + 4290 q^{8} - 5082 q^{9} + 1875 q^{10} - 8744 q^{11} - 11470 q^{12} - 3020 q^{13} - 30531 q^{14} + 5000 q^{15} - 30947 q^{16}+ \cdots + 83778400 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 5 x^{19} + 990 x^{18} - 1065 x^{17} + 626958 x^{16} - 59773 x^{15} + 229358265 x^{14} + \cdots + 12\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 93\!\cdots\!33 \nu^{19} + \cdots - 43\!\cdots\!00 ) / 26\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 10\!\cdots\!25 \nu^{19} + \cdots - 15\!\cdots\!00 ) / 36\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11\!\cdots\!66 \nu^{19} + \cdots - 30\!\cdots\!00 ) / 18\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 65\!\cdots\!87 \nu^{19} + \cdots - 64\!\cdots\!00 ) / 53\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 55\!\cdots\!67 \nu^{19} + \cdots + 11\!\cdots\!00 ) / 41\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 74\!\cdots\!93 \nu^{19} + \cdots - 31\!\cdots\!00 ) / 33\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 12\!\cdots\!07 \nu^{19} + \cdots + 42\!\cdots\!00 ) / 33\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 32\!\cdots\!43 \nu^{19} + \cdots + 12\!\cdots\!00 ) / 33\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 70\!\cdots\!67 \nu^{19} + \cdots - 34\!\cdots\!00 ) / 41\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 25\!\cdots\!42 \nu^{19} + \cdots + 98\!\cdots\!00 ) / 69\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 13\!\cdots\!57 \nu^{19} + \cdots + 13\!\cdots\!00 ) / 33\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 35\!\cdots\!17 \nu^{19} + \cdots + 85\!\cdots\!00 ) / 69\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 85\!\cdots\!87 \nu^{19} + \cdots + 19\!\cdots\!00 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 11\!\cdots\!17 \nu^{19} + \cdots - 23\!\cdots\!00 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 27\!\cdots\!27 \nu^{19} + \cdots - 47\!\cdots\!00 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 28\!\cdots\!09 \nu^{19} + \cdots + 72\!\cdots\!00 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 99\!\cdots\!91 \nu^{19} + \cdots - 61\!\cdots\!00 ) / 55\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 89\!\cdots\!11 \nu^{19} + \cdots + 35\!\cdots\!00 ) / 33\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + 2\beta_{3} - 195\beta_{2} - 195 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{13} - \beta_{11} + 12\beta_{8} + 12\beta_{7} - 2\beta_{4} + 317\beta_{3} - 317\beta _1 - 419 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 2 \beta_{19} + 2 \beta_{18} - 6 \beta_{16} + 6 \beta_{15} + 8 \beta_{12} - 2 \beta_{10} + \cdots - 1354 \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 76 \beta_{19} - 76 \beta_{18} - 8 \beta_{17} + 64 \beta_{16} + 72 \beta_{15} - 4 \beta_{14} + \cdots + 278451 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 1010 \beta_{19} - 1806 \beta_{18} - 3510 \beta_{17} + 232 \beta_{16} - 1836 \beta_{15} - 1110 \beta_{14} + \cdots + 23380475 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 45972 \beta_{19} - 58772 \beta_{18} + 51088 \beta_{17} + 4144 \beta_{16} - 1576 \beta_{15} + \cdots - 48520 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 64888 \beta_{19} - 64888 \beta_{18} + 1767550 \beta_{17} + 1503726 \beta_{16} - 992898 \beta_{15} + \cdots - 9439950811 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 850068 \beta_{19} + 60822984 \beta_{18} - 25783520 \beta_{17} - 27793720 \beta_{16} + \cdots - 72387020007 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 173225694 \beta_{19} + 794176482 \beta_{18} - 198457752 \beta_{17} - 768436350 \beta_{16} + \cdots - 25577532 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 10367815920 \beta_{19} - 10367815920 \beta_{18} - 1066391328 \beta_{17} + 12831529632 \beta_{16} + \cdots + 34834513999919 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 52595823098 \beta_{19} - 433959715370 \beta_{18} - 200019268350 \beta_{17} + 125040189696 \beta_{16} + \cdots + 16\!\cdots\!03 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 4521001566120 \beta_{19} - 11153655541268 \beta_{18} + 6710755057320 \beta_{17} + \cdots - 4457608064572 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 436963957484 \beta_{19} + 436963957484 \beta_{18} + 136546290225022 \beta_{17} + \cdots - 72\!\cdots\!75 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 79395997661900 \beta_{19} + \cdots - 78\!\cdots\!71 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 42\!\cdots\!62 \beta_{19} + \cdots + 18\!\cdots\!76 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 80\!\cdots\!56 \beta_{19} + \cdots + 37\!\cdots\!95 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 19\!\cdots\!94 \beta_{19} + \cdots + 13\!\cdots\!59 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 37\!\cdots\!00 \beta_{19} + \cdots - 34\!\cdots\!28 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/35\mathbb{Z}\right)^\times\).

\(n\) \(22\) \(31\)
\(\chi(n)\) \(1\) \(\beta_{2}\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
−10.0234 − 17.3611i
−7.41760 − 12.8477i
−6.33448 − 10.9716i
−3.86494 − 6.69427i
−2.91627 − 5.05112i
2.75257 + 4.76759i
3.51170 + 6.08245i
6.36936 + 11.0321i
9.63570 + 16.6895i
10.7874 + 18.6843i
−10.0234 + 17.3611i
−7.41760 + 12.8477i
−6.33448 + 10.9716i
−3.86494 + 6.69427i
−2.91627 + 5.05112i
2.75257 − 4.76759i
3.51170 − 6.08245i
6.36936 − 11.0321i
9.63570 − 16.6895i
10.7874 − 18.6843i
−11.0234 − 19.0932i 14.0283 − 24.2977i −179.033 + 310.094i 62.5000 + 108.253i −618.559 857.215 − 297.868i 5072.22 699.915 + 1212.29i 1377.93 − 2386.65i
11.2 −8.41760 − 14.5797i −43.2794 + 74.9621i −77.7119 + 134.601i 62.5000 + 108.253i 1457.23 347.899 + 838.158i 461.684 −2652.71 − 4594.62i 1052.20 − 1822.46i
11.3 −7.33448 − 12.7037i 34.1017 − 59.0659i −43.5890 + 75.4984i 62.5000 + 108.253i −1000.47 −864.013 + 277.533i −598.814 −1232.35 − 2134.50i 916.809 − 1587.96i
11.4 −4.86494 − 8.42632i −12.1221 + 20.9961i 16.6648 − 28.8642i 62.5000 + 108.253i 235.893 −726.993 − 543.161i −1569.72 799.609 + 1384.96i 608.117 − 1053.29i
11.5 −3.91627 − 6.78317i −2.71140 + 4.69629i 33.3257 − 57.7218i 62.5000 + 108.253i 42.4743 901.388 − 105.085i −1524.61 1078.80 + 1868.53i 489.533 − 847.897i
11.6 1.75257 + 3.03553i 40.7059 − 70.5046i 57.8570 − 100.211i 62.5000 + 108.253i 285.359 889.208 − 181.249i 854.250 −2220.43 − 3845.90i −219.071 + 379.442i
11.7 2.51170 + 4.35039i −19.1729 + 33.2085i 51.3827 − 88.9975i 62.5000 + 108.253i −192.626 −469.294 + 776.728i 1159.23 358.299 + 620.592i −313.963 + 543.799i
11.8 5.36936 + 9.30001i 11.1313 − 19.2799i 6.33984 − 10.9809i 62.5000 + 108.253i 239.072 −238.930 − 875.475i 1510.72 845.689 + 1464.78i −671.171 + 1162.50i
11.9 8.63570 + 14.9575i 17.3448 − 30.0420i −85.1505 + 147.485i 62.5000 + 108.253i 599.136 −636.507 + 646.840i −730.596 491.819 + 851.855i −1079.46 + 1869.68i
11.10 9.78739 + 16.9523i −30.0261 + 52.0067i −127.586 + 220.985i 62.5000 + 108.253i −1175.51 690.027 + 589.412i −2489.37 −709.633 − 1229.12i −1223.42 + 2119.03i
16.1 −11.0234 + 19.0932i 14.0283 + 24.2977i −179.033 − 310.094i 62.5000 − 108.253i −618.559 857.215 + 297.868i 5072.22 699.915 − 1212.29i 1377.93 + 2386.65i
16.2 −8.41760 + 14.5797i −43.2794 − 74.9621i −77.7119 − 134.601i 62.5000 − 108.253i 1457.23 347.899 − 838.158i 461.684 −2652.71 + 4594.62i 1052.20 + 1822.46i
16.3 −7.33448 + 12.7037i 34.1017 + 59.0659i −43.5890 − 75.4984i 62.5000 − 108.253i −1000.47 −864.013 − 277.533i −598.814 −1232.35 + 2134.50i 916.809 + 1587.96i
16.4 −4.86494 + 8.42632i −12.1221 − 20.9961i 16.6648 + 28.8642i 62.5000 − 108.253i 235.893 −726.993 + 543.161i −1569.72 799.609 − 1384.96i 608.117 + 1053.29i
16.5 −3.91627 + 6.78317i −2.71140 − 4.69629i 33.3257 + 57.7218i 62.5000 − 108.253i 42.4743 901.388 + 105.085i −1524.61 1078.80 − 1868.53i 489.533 + 847.897i
16.6 1.75257 − 3.03553i 40.7059 + 70.5046i 57.8570 + 100.211i 62.5000 − 108.253i 285.359 889.208 + 181.249i 854.250 −2220.43 + 3845.90i −219.071 − 379.442i
16.7 2.51170 − 4.35039i −19.1729 − 33.2085i 51.3827 + 88.9975i 62.5000 − 108.253i −192.626 −469.294 − 776.728i 1159.23 358.299 − 620.592i −313.963 − 543.799i
16.8 5.36936 − 9.30001i 11.1313 + 19.2799i 6.33984 + 10.9809i 62.5000 − 108.253i 239.072 −238.930 + 875.475i 1510.72 845.689 − 1464.78i −671.171 − 1162.50i
16.9 8.63570 − 14.9575i 17.3448 + 30.0420i −85.1505 − 147.485i 62.5000 − 108.253i 599.136 −636.507 − 646.840i −730.596 491.819 − 851.855i −1079.46 − 1869.68i
16.10 9.78739 − 16.9523i −30.0261 − 52.0067i −127.586 − 220.985i 62.5000 − 108.253i −1175.51 690.027 − 589.412i −2489.37 −709.633 + 1229.12i −1223.42 − 2119.03i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.10
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 35.8.e.b ✓ 20
7.c even 3 1 inner 35.8.e.b ✓ 20
7.c even 3 1 245.8.a.k 10
7.d odd 6 1 245.8.a.l 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.8.e.b ✓ 20 1.a even 1 1 trivial
35.8.e.b ✓ 20 7.c even 3 1 inner
245.8.a.k 10 7.c even 3 1
245.8.a.l 10 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} + 15 T_{2}^{19} + 1100 T_{2}^{18} + 11385 T_{2}^{17} + 697308 T_{2}^{16} + \cdots + 70\!\cdots\!16 \) acting on \(S_{8}^{\mathrm{new}}(35, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + \cdots + 70\!\cdots\!16 \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots + 99\!\cdots\!16 \) Copy content Toggle raw display
$5$ \( (T^{2} - 125 T + 15625)^{10} \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 14\!\cdots\!49 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 52\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{10} + \cdots + 18\!\cdots\!04)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 17\!\cdots\!36 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 59\!\cdots\!16 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 22\!\cdots\!41 \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots - 33\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 22\!\cdots\!96 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots - 66\!\cdots\!75)^{2} \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots + 21\!\cdots\!36)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 72\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 89\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 19\!\cdots\!76 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 62\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots + 72\!\cdots\!84)^{2} \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 19\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 88\!\cdots\!76 \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots + 12\!\cdots\!96)^{2} \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 12\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots + 18\!\cdots\!16)^{2} \) Copy content Toggle raw display
show more
show less