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Properties

Label 66.4.e.d
Level $66$
Weight $4$
Character orbit 66.e
Analytic conductor $3.894$
Analytic rank $0$
Dimension $8$
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] = [66,4,Mod(25,66)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(66, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 8]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("66.25");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 66 = 2 \cdot 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 66.e (of order \(5\), degree \(4\), 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: \(3.89412606038\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 10x^{6} + 32x^{5} + 929x^{4} + 1208x^{3} + 29240x^{2} + 34568x + 355216 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 5 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \beta_{2} q^{2} + ( - 3 \beta_{4} + 3 \beta_{3} + \cdots + 3) q^{3}+ \cdots - 9 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 \beta_{2} q^{2} + ( - 3 \beta_{4} + 3 \beta_{3} + \cdots + 3) q^{3}+ \cdots + ( - 27 \beta_{7} + 9 \beta_{6} + \cdots + 27) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} + 6 q^{3} - 8 q^{4} - 27 q^{5} - 12 q^{6} + 27 q^{7} + 16 q^{8} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{2} + 6 q^{3} - 8 q^{4} - 27 q^{5} - 12 q^{6} + 27 q^{7} + 16 q^{8} - 18 q^{9} + 24 q^{10} + 93 q^{11} - 96 q^{12} + 39 q^{13} - 54 q^{14} - 99 q^{15} - 32 q^{16} + 102 q^{17} + 36 q^{18} + 390 q^{19} - 108 q^{20} - 6 q^{21} - 186 q^{22} - 116 q^{23} - 48 q^{24} + 123 q^{25} + 42 q^{26} + 54 q^{27} - 112 q^{28} - 195 q^{29} + 198 q^{30} - 289 q^{31} - 256 q^{32} - 174 q^{33} - 44 q^{34} + 652 q^{35} - 72 q^{36} - 943 q^{37} + 630 q^{38} - 117 q^{39} - 264 q^{40} + 1511 q^{41} - 168 q^{42} - 1216 q^{43} - 488 q^{44} - 108 q^{45} + 1002 q^{46} - 283 q^{47} + 96 q^{48} + 309 q^{49} + 794 q^{50} + 339 q^{51} - 84 q^{52} + 19 q^{53} + 432 q^{54} - 2 q^{55} - 16 q^{56} + 945 q^{57} + 390 q^{58} + 670 q^{59} + 324 q^{60} - 1399 q^{61} - 522 q^{62} + 243 q^{63} - 128 q^{64} - 5926 q^{65} - 732 q^{66} + 3222 q^{67} + 408 q^{68} - 1677 q^{69} - 594 q^{70} - 799 q^{71} + 144 q^{72} - 401 q^{73} + 1886 q^{74} + 1191 q^{75} - 600 q^{76} + 2237 q^{77} - 216 q^{78} + 1655 q^{79} + 528 q^{80} - 162 q^{81} + 1998 q^{82} + 939 q^{83} - 324 q^{84} + 3327 q^{85} - 1638 q^{86} - 1470 q^{87} - 464 q^{88} + 560 q^{89} + 486 q^{90} - 5159 q^{91} + 2236 q^{92} + 867 q^{93} - 824 q^{94} - 5015 q^{95} - 192 q^{96} + 1377 q^{97} - 5668 q^{98} + 837 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} + 10x^{6} + 32x^{5} + 929x^{4} + 1208x^{3} + 29240x^{2} + 34568x + 355216 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 69294855 \nu^{7} + 10026058851 \nu^{6} - 53726589054 \nu^{5} - 272179338156 \nu^{4} + \cdots + 92382637283312 ) / 139765762224968 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 210082753 \nu^{7} - 1392119529 \nu^{6} - 2170826352 \nu^{5} + 61715703984 \nu^{4} + \cdots + 35022926541336 ) / 139765762224968 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 692576415 \nu^{7} - 5754404969 \nu^{6} - 1266466834 \nu^{5} + 235961161964 \nu^{4} + \cdots - 17757882093664 ) / 139765762224968 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 720576123 \nu^{7} + 8488214729 \nu^{6} - 23942680910 \nu^{5} + 80238269584 \nu^{4} + \cdots + 176195013432128 ) / 139765762224968 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3235477209 \nu^{7} + 25314385035 \nu^{6} - 293638813000 \nu^{5} + 393139997404 \nu^{4} + \cdots + 409849038862448 ) / 139765762224968 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 110789307 \nu^{7} + 307271851 \nu^{6} - 1713997952 \nu^{5} - 2687320592 \nu^{4} + \cdots - 4931916858456 ) / 2368911224152 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 12197390415 \nu^{7} - 3864103135 \nu^{6} + 293250352870 \nu^{5} + \cdots - 849558874011328 ) / 139765762224968 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} - \beta_{6} + 2\beta_{5} - \beta_{4} - \beta_{3} - 3\beta_{2} - 3\beta _1 + 2 ) / 5 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{7} - 2\beta_{6} + 4\beta_{5} - 2\beta_{4} + 43\beta_{3} - 156\beta_{2} - \beta _1 + 44 ) / 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 10\beta_{7} - 10\beta_{6} + 11\beta_{5} + 7\beta_{4} + 52\beta_{3} - 51\beta_{2} + 11\beta _1 + 11 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 107\beta_{7} - 192\beta_{6} - 96\beta_{5} + 2473\beta_{4} + 3093\beta_{3} - 96\beta_{2} + 299\beta _1 - 2366 ) / 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 426 \beta_{7} - 2239 \beta_{6} - 4052 \beta_{5} + 13606 \beta_{4} + 426 \beta_{3} + 11793 \beta_{2} + \cdots - 18382 ) / 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 2182 \beta_{7} - 987 \beta_{6} - 4364 \beta_{5} + 20526 \beta_{4} - 19331 \beta_{3} + 42831 \beta_{2} + \cdots - 43818 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 137416 \beta_{7} + 29851 \beta_{6} - 167267 \beta_{5} + 29851 \beta_{4} - 630029 \beta_{3} + \cdots - 698737 ) / 5 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/66\mathbb{Z}\right)^\times\).

\(n\) \(13\) \(23\)
\(\chi(n)\) \(\beta_{3}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1
1.37526 + 4.23260i
−1.18427 − 3.64482i
5.48149 − 3.98253i
−4.17247 + 3.03148i
1.37526 − 4.23260i
−1.18427 + 3.64482i
5.48149 + 3.98253i
−4.17247 − 3.03148i
1.61803 − 1.17557i −0.927051 − 2.85317i 1.23607 − 3.80423i −16.3356 − 11.8685i −4.85410 − 3.52671i −4.42136 + 13.6076i −2.47214 − 7.60845i −7.28115 + 5.29007i −40.3839
25.2 1.61803 − 1.17557i −0.927051 − 2.85317i 1.23607 − 3.80423i 7.90857 + 5.74591i −4.85410 − 3.52671i 8.37628 − 25.7795i −2.47214 − 7.60845i −7.28115 + 5.29007i 19.5511
31.1 −0.618034 + 1.90211i 2.42705 − 1.76336i −3.23607 − 2.35114i −5.08446 − 15.6484i 1.85410 + 5.70634i −19.3623 − 14.0676i 6.47214 − 4.70228i 2.78115 − 8.55951i 32.9073
31.2 −0.618034 + 1.90211i 2.42705 − 1.76336i −3.23607 − 2.35114i 0.0115129 + 0.0354332i 1.85410 + 5.70634i 28.9074 + 21.0025i 6.47214 − 4.70228i 2.78115 − 8.55951i −0.0745134
37.1 1.61803 + 1.17557i −0.927051 + 2.85317i 1.23607 + 3.80423i −16.3356 + 11.8685i −4.85410 + 3.52671i −4.42136 − 13.6076i −2.47214 + 7.60845i −7.28115 − 5.29007i −40.3839
37.2 1.61803 + 1.17557i −0.927051 + 2.85317i 1.23607 + 3.80423i 7.90857 − 5.74591i −4.85410 + 3.52671i 8.37628 + 25.7795i −2.47214 + 7.60845i −7.28115 − 5.29007i 19.5511
49.1 −0.618034 − 1.90211i 2.42705 + 1.76336i −3.23607 + 2.35114i −5.08446 + 15.6484i 1.85410 − 5.70634i −19.3623 + 14.0676i 6.47214 + 4.70228i 2.78115 + 8.55951i 32.9073
49.2 −0.618034 − 1.90211i 2.42705 + 1.76336i −3.23607 + 2.35114i 0.0115129 − 0.0354332i 1.85410 − 5.70634i 28.9074 − 21.0025i 6.47214 + 4.70228i 2.78115 + 8.55951i −0.0745134
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 66.4.e.d ✓ 8
3.b odd 2 1 198.4.f.e 8
11.c even 5 1 inner 66.4.e.d ✓ 8
11.c even 5 1 726.4.a.u 4
11.d odd 10 1 726.4.a.x 4
33.f even 10 1 2178.4.a.bw 4
33.h odd 10 1 198.4.f.e 8
33.h odd 10 1 2178.4.a.cc 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
66.4.e.d ✓ 8 1.a even 1 1 trivial
66.4.e.d ✓ 8 11.c even 5 1 inner
198.4.f.e 8 3.b odd 2 1
198.4.f.e 8 33.h odd 10 1
726.4.a.u 4 11.c even 5 1
726.4.a.x 4 11.d odd 10 1
2178.4.a.bw 4 33.f even 10 1
2178.4.a.cc 4 33.h odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 27T_{5}^{7} + 428T_{5}^{6} + 1089T_{5}^{5} + 1455T_{5}^{4} - 504471T_{5}^{3} + 10559428T_{5}^{2} - 243573T_{5} + 14641 \) acting on \(S_{4}^{\mathrm{new}}(66, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 2 T^{3} + 4 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$3$ \( (T^{4} - 3 T^{3} + 9 T^{2} + \cdots + 81)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} + 27 T^{7} + \cdots + 14641 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 109999018921 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 3138428376721 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 198092000634256 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 93643864504576 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T^{4} + 58 T^{3} + \cdots + 177995876)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 215917133186161 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 25\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 66\!\cdots\!76 \) Copy content Toggle raw display
$43$ \( (T^{4} + 608 T^{3} + \cdots - 3026465604)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 30\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 20\!\cdots\!01 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 12\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 18\!\cdots\!76 \) Copy content Toggle raw display
$67$ \( (T^{4} - 1611 T^{3} + \cdots - 117551155664)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 17\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 45\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 23\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 62\!\cdots\!01 \) Copy content Toggle raw display
$89$ \( (T^{4} - 280 T^{3} + \cdots + 820553900)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 13\!\cdots\!01 \) Copy content Toggle raw display
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