Properties

Label 62.4.c.b
Level $62$
Weight $4$
Character orbit 62.c
Analytic conductor $3.658$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [62,4,Mod(5,62)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(62, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("62.5");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 62 = 2 \cdot 31 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 62.c (of order \(3\), 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: \(3.65811842036\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 12x^{6} + 99x^{5} + 64x^{4} - 3051x^{3} + 22683x^{2} - 85202x + 132796 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 3 \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} + (\beta_{5} - 3 \beta_1) q^{3} + 4 q^{4} + ( - \beta_{4} - \beta_{2} - 1) q^{5} + (2 \beta_{5} - 6 \beta_1) q^{6} + ( - 2 \beta_{6} - \beta_{5} - \beta_{4} + 4 \beta_1) q^{7} + 8 q^{8} + ( - 3 \beta_{7} + 3 \beta_{6} + 2 \beta_{4} - 5 \beta_{3} + 2 \beta_{2} - 7 \beta_1 - 5) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + (\beta_{5} - 3 \beta_1) q^{3} + 4 q^{4} + ( - \beta_{4} - \beta_{2} - 1) q^{5} + (2 \beta_{5} - 6 \beta_1) q^{6} + ( - 2 \beta_{6} - \beta_{5} - \beta_{4} + 4 \beta_1) q^{7} + 8 q^{8} + ( - 3 \beta_{7} + 3 \beta_{6} + 2 \beta_{4} - 5 \beta_{3} + 2 \beta_{2} - 7 \beta_1 - 5) q^{9} + ( - 2 \beta_{4} - 2 \beta_{2} - 2) q^{10} + (\beta_{7} - \beta_{6} + 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + 5 \beta_1 + 7) q^{11} + (4 \beta_{5} - 12 \beta_1) q^{12} + (3 \beta_{7} - 3 \beta_{6} - \beta_{4} + 5 \beta_{3} - \beta_{2} + 7 \beta_1 + 6) q^{13} + ( - 4 \beta_{6} - 2 \beta_{5} - 2 \beta_{4} + 8 \beta_1) q^{14} + (5 \beta_{5} + 5 \beta_{3} - 3 \beta_{2} - 5 \beta_1 - 12) q^{15} + 16 q^{16} + (7 \beta_{6} - 7 \beta_{5} - 2 \beta_{4} - 7 \beta_1) q^{17} + ( - 6 \beta_{7} + 6 \beta_{6} + 4 \beta_{4} - 10 \beta_{3} + 4 \beta_{2} + \cdots - 10) q^{18}+ \cdots + ( - 47 \beta_{6} - 71 \beta_{5} + 16 \beta_{4} + 294 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 16 q^{2} + 10 q^{3} + 32 q^{4} - 2 q^{5} + 20 q^{6} - 16 q^{7} + 64 q^{8} - 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16 q^{2} + 10 q^{3} + 32 q^{4} - 2 q^{5} + 20 q^{6} - 16 q^{7} + 64 q^{8} - 40 q^{9} - 4 q^{10} + 22 q^{11} + 40 q^{12} + 42 q^{13} - 32 q^{14} - 64 q^{15} + 128 q^{16} + 60 q^{17} - 80 q^{18} - 30 q^{19} - 8 q^{20} + 138 q^{21} + 44 q^{22} - 112 q^{23} + 80 q^{24} - 104 q^{25} + 84 q^{26} - 788 q^{27} - 64 q^{28} - 296 q^{29} - 128 q^{30} - 788 q^{31} + 256 q^{32} - 272 q^{33} + 120 q^{34} - 412 q^{35} - 160 q^{36} - 186 q^{37} - 60 q^{38} + 1392 q^{39} - 16 q^{40} + 380 q^{41} + 276 q^{42} + 366 q^{43} + 88 q^{44} + 508 q^{45} - 224 q^{46} + 792 q^{47} + 160 q^{48} - 192 q^{49} - 208 q^{50} + 650 q^{51} + 168 q^{52} - 250 q^{53} - 1576 q^{54} + 1344 q^{55} - 128 q^{56} + 548 q^{57} - 592 q^{58} + 982 q^{59} - 256 q^{60} - 1496 q^{61} - 1576 q^{62} + 5720 q^{63} + 512 q^{64} + 96 q^{65} - 544 q^{66} - 418 q^{67} + 240 q^{68} - 520 q^{69} - 824 q^{70} + 860 q^{71} - 320 q^{72} - 1100 q^{73} - 372 q^{74} + 440 q^{75} - 120 q^{76} - 116 q^{77} + 2784 q^{78} + 608 q^{79} - 32 q^{80} - 2692 q^{81} + 760 q^{82} + 306 q^{83} + 552 q^{84} - 4628 q^{85} + 732 q^{86} + 636 q^{87} + 176 q^{88} - 2544 q^{89} + 1016 q^{90} - 5308 q^{91} - 448 q^{92} - 6210 q^{93} + 1584 q^{94} + 3568 q^{95} + 320 q^{96} - 4440 q^{97} - 384 q^{98} - 1160 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 12x^{6} + 99x^{5} + 64x^{4} - 3051x^{3} + 22683x^{2} - 85202x + 132796 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 3921 \nu^{7} - 39803 \nu^{6} - 164260 \nu^{5} - 709901 \nu^{4} - 3445900 \nu^{3} - 5718133 \nu^{2} + 22130765 \nu - 110957968 ) / 200366454 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 547888 \nu^{7} - 1708382 \nu^{6} + 12257544 \nu^{5} + 19319478 \nu^{4} - 99877682 \nu^{3} - 1040650746 \nu^{2} + 4442680268 \nu - 74248452940 ) / 6645487391 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 5598961 \nu^{7} + 452075 \nu^{6} + 37153324 \nu^{5} - 132079747 \nu^{4} + 73454076 \nu^{3} + 18536966553 \nu^{2} + \cdots + 146313298790 ) / 39872924346 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 6984509 \nu^{7} + 55473731 \nu^{6} + 235039744 \nu^{5} + 1148703293 \nu^{4} + 7869721644 \nu^{3} - 5994340011 \nu^{2} + \cdots + 27725524352 ) / 39872924346 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 3528706 \nu^{7} - 14224769 \nu^{6} - 13625023 \nu^{5} - 288918905 \nu^{4} - 1909086371 \nu^{3} + 9490569277 \nu^{2} + \cdots - 59563714368 ) / 19936462173 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 6200848 \nu^{7} - 670253 \nu^{6} - 85401445 \nu^{5} - 1106457053 \nu^{4} - 3926259719 \nu^{3} + 9904854175 \nu^{2} + \cdots + 238594505328 ) / 19936462173 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2591955 \nu^{7} - 628395 \nu^{6} + 15565387 \nu^{5} + 210397579 \nu^{4} - 21247099 \nu^{3} - 10707929729 \nu^{2} + 39425542654 \nu - 146961254306 ) / 6645487391 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 2\beta_{5} + \beta_{4} - 2\beta_{3} - \beta_{2} + \beta _1 + 2 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -6\beta_{7} - 3\beta_{6} + \beta_{5} - 7\beta_{4} - 10\beta_{3} + 4\beta_{2} - 55\beta _1 - 38 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -24\beta_{7} - 6\beta_{6} - 38\beta_{5} + 44\beta_{4} + 14\beta_{3} + 37\beta_{2} + 410\beta _1 - 14 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 43\beta_{7} - 39\beta_{6} - 28\beta_{5} - 105\beta_{4} + 119\beta_{3} - 105\beta_{2} - 506\beta _1 - 483 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 270\beta_{7} + 366\beta_{6} + 880\beta_{5} + 1157\beta_{4} + 668\beta_{3} + 1384\beta_{2} + 2999\beta _1 + 18088 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 1623 \beta_{7} + 1986 \beta_{6} + 4487 \beta_{5} + 1048 \beta_{4} - 5567 \beta_{3} - 9067 \beta_{2} - 28673 \beta _1 - 71953 ) / 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 21300 \beta_{7} - 4662 \beta_{6} - 23980 \beta_{5} - 24893 \beta_{4} - 45116 \beta_{3} + 47099 \beta_{2} - 140807 \beta _1 + 144224 ) / 6 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/62\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1 - \beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
1.61817 + 3.95158i
2.64685 1.02474i
−5.12581 + 2.87225i
1.36079 4.93306i
1.61817 3.95158i
2.64685 + 1.02474i
−5.12581 2.87225i
1.36079 + 4.93306i
2.00000 −2.73125 + 4.73066i 4.00000 2.73634 + 4.73947i −5.46250 + 9.46132i −7.90710 + 13.6955i 8.00000 −1.41944 2.45855i 5.47267 + 9.47895i
5.2 2.00000 1.06402 1.84294i 4.00000 4.79370 + 8.30294i 2.12805 3.68589i 10.0711 17.4436i 8.00000 11.2357 + 19.4608i 9.58741 + 16.6059i
5.3 2.00000 1.57546 2.72878i 4.00000 −10.7516 18.6224i 3.15092 5.45756i 4.28276 7.41796i 8.00000 8.53584 + 14.7845i −21.5032 37.2447i
5.4 2.00000 5.09176 8.81919i 4.00000 2.22159 + 3.84790i 10.1835 17.6384i −14.4468 + 25.0225i 8.00000 −38.3521 66.4278i 4.44317 + 7.69580i
25.1 2.00000 −2.73125 4.73066i 4.00000 2.73634 4.73947i −5.46250 9.46132i −7.90710 13.6955i 8.00000 −1.41944 + 2.45855i 5.47267 9.47895i
25.2 2.00000 1.06402 + 1.84294i 4.00000 4.79370 8.30294i 2.12805 + 3.68589i 10.0711 + 17.4436i 8.00000 11.2357 19.4608i 9.58741 16.6059i
25.3 2.00000 1.57546 + 2.72878i 4.00000 −10.7516 + 18.6224i 3.15092 + 5.45756i 4.28276 + 7.41796i 8.00000 8.53584 14.7845i −21.5032 + 37.2447i
25.4 2.00000 5.09176 + 8.81919i 4.00000 2.22159 3.84790i 10.1835 + 17.6384i −14.4468 25.0225i 8.00000 −38.3521 + 66.4278i 4.44317 7.69580i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.4
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 62.4.c.b 8
3.b odd 2 1 558.4.e.d 8
4.b odd 2 1 496.4.i.c 8
31.c even 3 1 inner 62.4.c.b 8
31.c even 3 1 1922.4.a.h 4
31.e odd 6 1 1922.4.a.j 4
93.h odd 6 1 558.4.e.d 8
124.i odd 6 1 496.4.i.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
62.4.c.b 8 1.a even 1 1 trivial
62.4.c.b 8 31.c even 3 1 inner
496.4.i.c 8 4.b odd 2 1
496.4.i.c 8 124.i odd 6 1
558.4.e.d 8 3.b odd 2 1
558.4.e.d 8 93.h odd 6 1
1922.4.a.h 4 31.c even 3 1
1922.4.a.j 4 31.e odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 10T_{3}^{7} + 124T_{3}^{6} - 284T_{3}^{5} + 3569T_{3}^{4} - 13748T_{3}^{3} + 59692T_{3}^{2} - 97726T_{3} + 139129 \) acting on \(S_{4}^{\mathrm{new}}(62, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 2)^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 10 T^{7} + 124 T^{6} + \cdots + 139129 \) Copy content Toggle raw display
$5$ \( T^{8} + 2 T^{7} + 304 T^{6} + \cdots + 25130169 \) Copy content Toggle raw display
$7$ \( T^{8} + 16 T^{7} + \cdots + 6214641889 \) Copy content Toggle raw display
$11$ \( T^{8} - 22 T^{7} + \cdots + 118985673249 \) Copy content Toggle raw display
$13$ \( T^{8} - 42 T^{7} + \cdots + 731107521 \) Copy content Toggle raw display
$17$ \( T^{8} - 60 T^{7} + \cdots + 35162065517121 \) Copy content Toggle raw display
$19$ \( T^{8} + 30 T^{7} + \cdots + 6026450543689 \) Copy content Toggle raw display
$23$ \( (T^{4} + 56 T^{3} - 10176 T^{2} + \cdots + 8626176)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 148 T^{3} - 5096 T^{2} + \cdots - 36022608)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 788 T^{7} + \cdots + 78\!\cdots\!61 \) Copy content Toggle raw display
$37$ \( T^{8} + 186 T^{7} + \cdots + 68\!\cdots\!49 \) Copy content Toggle raw display
$41$ \( T^{8} - 380 T^{7} + \cdots + 15\!\cdots\!21 \) Copy content Toggle raw display
$43$ \( T^{8} - 366 T^{7} + \cdots + 20\!\cdots\!69 \) Copy content Toggle raw display
$47$ \( (T^{4} - 396 T^{3} - 135904 T^{2} + \cdots - 172657152)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 250 T^{7} + \cdots + 27\!\cdots\!49 \) Copy content Toggle raw display
$59$ \( T^{8} - 982 T^{7} + \cdots + 16\!\cdots\!21 \) Copy content Toggle raw display
$61$ \( (T^{4} + 748 T^{3} + \cdots - 26488681104)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 418 T^{7} + \cdots + 51\!\cdots\!69 \) Copy content Toggle raw display
$71$ \( T^{8} - 860 T^{7} + \cdots + 19\!\cdots\!09 \) Copy content Toggle raw display
$73$ \( T^{8} + 1100 T^{7} + \cdots + 10\!\cdots\!89 \) Copy content Toggle raw display
$79$ \( T^{8} - 608 T^{7} + \cdots + 14\!\cdots\!29 \) Copy content Toggle raw display
$83$ \( T^{8} - 306 T^{7} + \cdots + 20\!\cdots\!01 \) Copy content Toggle raw display
$89$ \( (T^{4} + 1272 T^{3} + \cdots - 1251333571248)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 2220 T^{3} + \cdots - 203751443472)^{2} \) Copy content Toggle raw display
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