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Properties

Label 62.8.c.a
Level $62$
Weight $8$
Character orbit 62.c
Analytic conductor $19.368$
Analytic rank $0$
Dimension $18$
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] = [62,8,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, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("62.5");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 62 = 2 \cdot 31 \)
Weight: \( k \) \(=\) \( 8 \)
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: \(19.3678715800\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(9\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - x^{17} + 11381 x^{16} + 73148 x^{15} + 92644250 x^{14} + 576491498 x^{13} + \cdots + 22\!\cdots\!25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{26}\cdot 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 8 q^{2} + (\beta_{3} - 6 \beta_{2} - \beta_1 - 6) q^{3} + 64 q^{4} + ( - \beta_{7} - \beta_{5} + \cdots - \beta_1) q^{5}+ \cdots + ( - \beta_{10} + 2 \beta_{7} + \cdots + 6 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 8 q^{2} + (\beta_{3} - 6 \beta_{2} - \beta_1 - 6) q^{3} + 64 q^{4} + ( - \beta_{7} - \beta_{5} + \cdots - \beta_1) q^{5}+ \cdots + ( - 749 \beta_{17} - 683 \beta_{16} + \cdots - 738382) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 144 q^{2} - 53 q^{3} + 1152 q^{4} - 97 q^{5} + 424 q^{6} + 701 q^{7} - 9216 q^{8} - 3390 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 144 q^{2} - 53 q^{3} + 1152 q^{4} - 97 q^{5} + 424 q^{6} + 701 q^{7} - 9216 q^{8} - 3390 q^{9} + 776 q^{10} + 1425 q^{11} - 3392 q^{12} - 4857 q^{13} - 5608 q^{14} - 48334 q^{15} + 73728 q^{16} - 17459 q^{17} + 27120 q^{18} + 98923 q^{19} - 6208 q^{20} + 51157 q^{21} - 11400 q^{22} - 145776 q^{23} + 27136 q^{24} - 174284 q^{25} + 38856 q^{26} + 105838 q^{27} + 44864 q^{28} - 545828 q^{29} + 386672 q^{30} - 283940 q^{31} - 589824 q^{32} - 80650 q^{33} + 139672 q^{34} + 245918 q^{35} - 216960 q^{36} - 418149 q^{37} - 791384 q^{38} - 959342 q^{39} + 49664 q^{40} - 208739 q^{41} - 409256 q^{42} - 520809 q^{43} + 91200 q^{44} + 2114018 q^{45} + 1166208 q^{46} - 1404168 q^{47} - 217088 q^{48} - 1881938 q^{49} + 1394272 q^{50} + 3508721 q^{51} - 310848 q^{52} + 2380443 q^{53} - 846704 q^{54} - 357691 q^{55} - 358912 q^{56} + 1540711 q^{57} + 4366624 q^{58} - 44489 q^{59} - 3093376 q^{60} + 2307612 q^{61} + 2271520 q^{62} - 4076796 q^{63} + 4718592 q^{64} + 3008231 q^{65} + 645200 q^{66} - 6216511 q^{67} - 1117376 q^{68} + 2724736 q^{69} - 1967344 q^{70} - 4789145 q^{71} + 1735680 q^{72} + 5403165 q^{73} + 3345192 q^{74} + 2976100 q^{75} + 6331072 q^{76} - 19326478 q^{77} + 7674736 q^{78} - 11215787 q^{79} - 397312 q^{80} - 5409705 q^{81} + 1669912 q^{82} - 9453995 q^{83} + 3274048 q^{84} + 20789302 q^{85} + 4166472 q^{86} + 10844658 q^{87} - 729600 q^{88} + 652924 q^{89} - 16912144 q^{90} - 202018 q^{91} - 9329664 q^{92} - 17989059 q^{93} + 11233344 q^{94} - 18910382 q^{95} + 1736704 q^{96} + 62124196 q^{97} + 15055504 q^{98} - 6717146 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - x^{17} + 11381 x^{16} + 73148 x^{15} + 92644250 x^{14} + 576491498 x^{13} + \cdots + 22\!\cdots\!25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 96\!\cdots\!34 \nu^{17} + \cdots + 34\!\cdots\!50 ) / 62\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 24\!\cdots\!77 \nu^{17} + \cdots + 53\!\cdots\!50 ) / 15\!\cdots\!65 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 58\!\cdots\!89 \nu^{17} + \cdots + 91\!\cdots\!00 ) / 60\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 32\!\cdots\!63 \nu^{17} + \cdots + 63\!\cdots\!60 ) / 12\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 55\!\cdots\!76 \nu^{17} + \cdots - 13\!\cdots\!25 ) / 10\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 59\!\cdots\!12 \nu^{17} + \cdots - 29\!\cdots\!25 ) / 52\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 42\!\cdots\!73 \nu^{17} + \cdots - 22\!\cdots\!90 ) / 30\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 12\!\cdots\!21 \nu^{17} + \cdots - 22\!\cdots\!40 ) / 76\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 87\!\cdots\!16 \nu^{17} + \cdots + 83\!\cdots\!75 ) / 52\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 23\!\cdots\!74 \nu^{17} + \cdots - 17\!\cdots\!75 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 66\!\cdots\!69 \nu^{17} + \cdots - 14\!\cdots\!20 ) / 30\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 29\!\cdots\!35 \nu^{17} + \cdots + 92\!\cdots\!00 ) / 12\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 67\!\cdots\!12 \nu^{17} + \cdots - 15\!\cdots\!25 ) / 26\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 55\!\cdots\!31 \nu^{17} + \cdots - 33\!\cdots\!75 ) / 11\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 42\!\cdots\!48 \nu^{17} + \cdots - 30\!\cdots\!75 ) / 39\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 55\!\cdots\!74 \nu^{17} + \cdots - 36\!\cdots\!25 ) / 48\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} - 2\beta_{7} - 6\beta_{3} - 2528\beta_{2} + 6\beta _1 - 2528 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 7 \beta_{13} - 13 \beta_{12} + 13 \beta_{9} - 15 \beta_{8} - 21 \beta_{6} - 75 \beta_{5} + \cdots - 13563 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 96 \beta_{17} - 801 \beta_{16} + 351 \beta_{15} - 1299 \beta_{14} - 801 \beta_{13} + \cdots - 67392 \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 118212 \beta_{17} - 74344 \beta_{16} - 114373 \beta_{15} + 197112 \beta_{14} + 125956 \beta_{12} + \cdots + 159376533 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 7594923 \beta_{13} + 4851915 \beta_{12} + 2837733 \beta_{9} + 57648 \beta_{8} - 11304129 \beta_{6} + \cdots + 77202496166 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 829644603 \beta_{17} + 599207647 \beta_{16} + 832400812 \beta_{15} - 1425539253 \beta_{14} + \cdots + 205892903873 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 5940655536 \beta_{17} + 59222933970 \beta_{16} - 17029419798 \beta_{15} + 77886446646 \beta_{14} + \cdots - 521462239675853 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 4551853278856 \beta_{13} - 7100244088480 \beta_{12} + 5763026182114 \beta_{9} - 5749312575912 \beta_{8} + \cdots - 13\!\cdots\!90 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 77214028518192 \beta_{17} - 441282792204774 \beta_{16} + 88052790548610 \beta_{15} + \cdots - 46\!\cdots\!30 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 39\!\cdots\!79 \beta_{17} + \cdots + 11\!\cdots\!25 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 32\!\cdots\!43 \beta_{13} + \cdots + 25\!\cdots\!07 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 27\!\cdots\!76 \beta_{17} + \cdots + 71\!\cdots\!45 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 70\!\cdots\!48 \beta_{17} + \cdots - 18\!\cdots\!94 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 18\!\cdots\!63 \beta_{13} + \cdots - 74\!\cdots\!32 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 59\!\cdots\!08 \beta_{17} + \cdots - 23\!\cdots\!20 \beta_1 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 13\!\cdots\!76 \beta_{17} + \cdots + 59\!\cdots\!88 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(3\)
\(\chi(n)\) \(\beta_{2}\)

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.

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
−39.9608 − 69.2142i
−23.8709 − 41.3456i
−18.5330 − 32.1001i
−15.3585 − 26.6017i
1.93866 + 3.35787i
7.87905 + 13.6469i
22.3132 + 38.6476i
23.1367 + 40.0740i
42.9556 + 74.4012i
−39.9608 + 69.2142i
−23.8709 + 41.3456i
−18.5330 + 32.1001i
−15.3585 + 26.6017i
1.93866 − 3.35787i
7.87905 − 13.6469i
22.3132 − 38.6476i
23.1367 − 40.0740i
42.9556 − 74.4012i
−8.00000 −42.9608 + 74.4104i 64.0000 164.834 + 285.501i 343.687 − 595.283i 499.698 − 865.503i −512.000 −2597.77 − 4499.46i −1318.67 − 2284.01i
5.2 −8.00000 −26.8709 + 46.5418i 64.0000 28.6565 + 49.6345i 214.967 − 372.334i −651.326 + 1128.13i −512.000 −350.590 − 607.239i −229.252 − 397.076i
5.3 −8.00000 −21.5330 + 37.2963i 64.0000 −14.2500 − 24.6818i 172.264 − 298.370i −14.5862 + 25.2641i −512.000 166.159 + 287.795i 114.000 + 197.454i
5.4 −8.00000 −18.3585 + 31.7978i 64.0000 −125.732 − 217.774i 146.868 − 254.382i 258.387 − 447.540i −512.000 419.433 + 726.479i 1005.86 + 1742.20i
5.5 −8.00000 −1.06134 + 1.83829i 64.0000 −239.603 − 415.004i 8.49068 − 14.7063i 624.172 − 1081.10i −512.000 1091.25 + 1890.10i 1916.82 + 3320.03i
5.6 −8.00000 4.87905 − 8.45076i 64.0000 251.975 + 436.433i −39.0324 + 67.6061i −315.897 + 547.150i −512.000 1045.89 + 1811.53i −2015.80 − 3491.46i
5.7 −8.00000 19.3132 − 33.4514i 64.0000 94.3503 + 163.419i −154.506 + 267.611i 728.718 − 1262.18i −512.000 347.501 + 601.890i −754.802 − 1307.36i
5.8 −8.00000 20.1367 − 34.8779i 64.0000 −213.366 − 369.561i −161.094 + 279.023i −749.182 + 1297.62i −512.000 282.524 + 489.345i 1706.93 + 2956.49i
5.9 −8.00000 39.9556 − 69.2051i 64.0000 4.63499 + 8.02804i −319.645 + 553.641i −29.4841 + 51.0679i −512.000 −2099.40 − 3636.26i −37.0799 − 64.2243i
25.1 −8.00000 −42.9608 − 74.4104i 64.0000 164.834 − 285.501i 343.687 + 595.283i 499.698 + 865.503i −512.000 −2597.77 + 4499.46i −1318.67 + 2284.01i
25.2 −8.00000 −26.8709 − 46.5418i 64.0000 28.6565 − 49.6345i 214.967 + 372.334i −651.326 − 1128.13i −512.000 −350.590 + 607.239i −229.252 + 397.076i
25.3 −8.00000 −21.5330 − 37.2963i 64.0000 −14.2500 + 24.6818i 172.264 + 298.370i −14.5862 − 25.2641i −512.000 166.159 − 287.795i 114.000 − 197.454i
25.4 −8.00000 −18.3585 − 31.7978i 64.0000 −125.732 + 217.774i 146.868 + 254.382i 258.387 + 447.540i −512.000 419.433 − 726.479i 1005.86 − 1742.20i
25.5 −8.00000 −1.06134 − 1.83829i 64.0000 −239.603 + 415.004i 8.49068 + 14.7063i 624.172 + 1081.10i −512.000 1091.25 − 1890.10i 1916.82 − 3320.03i
25.6 −8.00000 4.87905 + 8.45076i 64.0000 251.975 − 436.433i −39.0324 − 67.6061i −315.897 − 547.150i −512.000 1045.89 − 1811.53i −2015.80 + 3491.46i
25.7 −8.00000 19.3132 + 33.4514i 64.0000 94.3503 − 163.419i −154.506 − 267.611i 728.718 + 1262.18i −512.000 347.501 − 601.890i −754.802 + 1307.36i
25.8 −8.00000 20.1367 + 34.8779i 64.0000 −213.366 + 369.561i −161.094 − 279.023i −749.182 − 1297.62i −512.000 282.524 − 489.345i 1706.93 − 2956.49i
25.9 −8.00000 39.9556 + 69.2051i 64.0000 4.63499 − 8.02804i −319.645 − 553.641i −29.4841 − 51.0679i −512.000 −2099.40 + 3636.26i −37.0799 + 64.2243i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.9
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 62.8.c.a ✓ 18
31.c even 3 1 inner 62.8.c.a ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
62.8.c.a ✓ 18 1.a even 1 1 trivial
62.8.c.a ✓ 18 31.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{18} + 53 T_{3}^{17} + 12941 T_{3}^{16} + 446420 T_{3}^{15} + 101452586 T_{3}^{14} + \cdots + 35\!\cdots\!25 \) acting on \(S_{8}^{\mathrm{new}}(62, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 8)^{18} \) Copy content Toggle raw display
$3$ \( T^{18} + \cdots + 35\!\cdots\!25 \) Copy content Toggle raw display
$5$ \( T^{18} + \cdots + 59\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{18} + \cdots + 39\!\cdots\!81 \) Copy content Toggle raw display
$11$ \( T^{18} + \cdots + 13\!\cdots\!81 \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots + 29\!\cdots\!25 \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots + 15\!\cdots\!29 \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 12\!\cdots\!61 \) Copy content Toggle raw display
$23$ \( (T^{9} + \cdots - 27\!\cdots\!60)^{2} \) Copy content Toggle raw display
$29$ \( (T^{9} + \cdots - 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 90\!\cdots\!91 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 34\!\cdots\!61 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 92\!\cdots\!01 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 17\!\cdots\!49 \) Copy content Toggle raw display
$47$ \( (T^{9} + \cdots + 12\!\cdots\!84)^{2} \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 18\!\cdots\!25 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 35\!\cdots\!21 \) Copy content Toggle raw display
$61$ \( (T^{9} + \cdots - 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 10\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 14\!\cdots\!61 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 28\!\cdots\!09 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 57\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 12\!\cdots\!25 \) Copy content Toggle raw display
$89$ \( (T^{9} + \cdots - 90\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{9} + \cdots - 21\!\cdots\!04)^{2} \) Copy content Toggle raw display
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