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Properties

Label 351.2.bd.e
Level $351$
Weight $2$
Character orbit 351.bd
Analytic conductor $2.803$
Analytic rank $0$
Dimension $20$
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] = [351,2,Mod(80,351)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(351, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("351.80");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 351 = 3^{3} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 351.bd (of order \(12\), 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: \(2.80274911095\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(5\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 88 x^{16} - 6 x^{15} + 48 x^{13} + 1980 x^{12} - 204 x^{11} + 18 x^{10} + 2076 x^{9} + \cdots + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} + (\beta_{15} - \beta_{13} + \cdots - \beta_{3}) q^{4}+ \cdots + ( - \beta_{17} + 2 \beta_{16} + \cdots - 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{2} + (\beta_{15} - \beta_{13} + \cdots - \beta_{3}) q^{4}+ \cdots + (2 \beta_{19} + \beta_{15} + \cdots + 2 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 6 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 6 q^{5} - 12 q^{10} - 8 q^{13} + 24 q^{16} - 12 q^{17} - 12 q^{19} + 36 q^{20} + 8 q^{22} - 42 q^{26} + 2 q^{28} - 6 q^{29} - 22 q^{31} - 36 q^{32} - 6 q^{34} - 36 q^{35} + 8 q^{37} + 72 q^{38} - 36 q^{40} + 30 q^{41} - 30 q^{43} + 36 q^{44} - 48 q^{46} + 6 q^{47} + 30 q^{49} + 54 q^{50} + 4 q^{52} - 28 q^{55} - 60 q^{56} + 44 q^{58} + 30 q^{59} - 16 q^{61} - 30 q^{62} - 78 q^{65} + 18 q^{67} + 6 q^{68} + 38 q^{70} - 60 q^{71} - 72 q^{74} - 8 q^{76} - 12 q^{77} - 16 q^{79} + 126 q^{80} + 78 q^{82} + 12 q^{83} + 12 q^{85} + 18 q^{86} + 84 q^{89} + 30 q^{91} - 22 q^{94} - 66 q^{95} + 26 q^{97} + 42 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 88 x^{16} - 6 x^{15} + 48 x^{13} + 1980 x^{12} - 204 x^{11} + 18 x^{10} + 2076 x^{9} + \cdots + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 92\!\cdots\!68 \nu^{19} + \cdots + 25\!\cdots\!15 ) / 19\!\cdots\!78 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 55\!\cdots\!50 \nu^{19} + \cdots - 30\!\cdots\!93 ) / 59\!\cdots\!34 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 33189979741536 \nu^{19} + 6504226812868 \nu^{18} - 835272479316 \nu^{17} + \cdots + 12\!\cdots\!50 ) / 89\!\cdots\!62 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 28\!\cdots\!39 \nu^{19} + \cdots - 44\!\cdots\!50 ) / 59\!\cdots\!34 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 64\!\cdots\!55 \nu^{19} + \cdots - 18\!\cdots\!20 ) / 37\!\cdots\!66 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 13\!\cdots\!50 \nu^{19} - 298709817673824 \nu^{18} + 58538041315812 \nu^{17} + \cdots - 23\!\cdots\!00 ) / 80\!\cdots\!58 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 12\!\cdots\!70 \nu^{19} + \cdots + 21\!\cdots\!67 ) / 59\!\cdots\!34 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 58\!\cdots\!85 \nu^{19} + \cdots + 18\!\cdots\!45 ) / 17\!\cdots\!02 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 26\!\cdots\!93 \nu^{19} + \cdots - 76\!\cdots\!32 ) / 59\!\cdots\!34 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 28\!\cdots\!82 \nu^{19} + \cdots + 10\!\cdots\!59 ) / 59\!\cdots\!34 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 14\!\cdots\!00 \nu^{19} + \cdots + 74\!\cdots\!81 ) / 19\!\cdots\!78 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 46\!\cdots\!22 \nu^{19} + \cdots + 10\!\cdots\!77 ) / 59\!\cdots\!34 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 23\!\cdots\!47 \nu^{19} + \cdots - 34\!\cdots\!26 ) / 19\!\cdots\!78 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 25\!\cdots\!11 \nu^{19} + \cdots - 22\!\cdots\!10 ) / 19\!\cdots\!78 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 24\!\cdots\!97 \nu^{19} + \cdots - 42\!\cdots\!15 ) / 17\!\cdots\!02 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 27\!\cdots\!67 \nu^{19} + \cdots - 72\!\cdots\!81 ) / 19\!\cdots\!78 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 26\!\cdots\!04 \nu^{19} + \cdots + 22\!\cdots\!83 ) / 17\!\cdots\!02 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 11\!\cdots\!92 \nu^{19} + \cdots - 46\!\cdots\!00 ) / 59\!\cdots\!34 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{15} - \beta_{14} - \beta_{10} + 3\beta_{6} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{17} - 2\beta_{16} - \beta_{13} - \beta_{9} + 2\beta_{7} - \beta_{6} + 5\beta_{5} - 2\beta_{3} - 5\beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 8\beta_{19} + \beta_{18} - 7\beta_{17} + 7\beta_{12} - \beta_{11} + \beta_{9} - 8\beta_{8} + \beta_{4} + \beta_{2} - 17 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{19} + 8 \beta_{18} - \beta_{17} + \beta_{15} - \beta_{14} - 2 \beta_{13} - 9 \beta_{12} + \cdots - 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 2 \beta_{19} - 11 \beta_{18} - \beta_{17} + 56 \beta_{15} + 56 \beta_{14} + 9 \beta_{13} + \cdots + 13 \beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 14 \beta_{19} - 71 \beta_{17} + 112 \beta_{16} - 14 \beta_{15} + 11 \beta_{14} + 82 \beta_{13} + \cdots - 95 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 387 \beta_{19} - 96 \beta_{18} + 320 \beta_{17} - 3 \beta_{16} + 29 \beta_{15} - 29 \beta_{14} + \cdots + 774 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 95 \beta_{19} - 387 \beta_{18} + 140 \beta_{17} - 95 \beta_{15} + 143 \beta_{14} + 283 \beta_{13} + \cdots + 477 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 302 \beta_{19} + 826 \beta_{18} + 118 \beta_{17} + 48 \beta_{16} - 2698 \beta_{15} - 2698 \beta_{14} + \cdots - 105 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1290 \beta_{19} - 66 \beta_{18} + 4100 \beta_{17} - 5396 \beta_{16} + 1290 \beta_{15} - 762 \beta_{14} + \cdots + 5938 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 19030 \beta_{19} + 6086 \beta_{18} - 15570 \beta_{17} + 528 \beta_{16} - 2755 \beta_{15} + 2755 \beta_{14} + \cdots - 39016 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 5902 \beta_{19} + 19030 \beta_{18} - 9973 \beta_{17} + 5902 \beta_{15} - 10930 \beta_{14} - 20903 \beta_{13} + \cdots - 27019 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 23464 \beta_{19} - 51651 \beta_{18} - 6491 \beta_{17} - 5028 \beta_{16} + 135626 \beta_{15} + \cdots + 12711 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 89371 \beta_{19} + 10482 \beta_{18} - 229715 \beta_{17} + 271252 \beta_{16} - 89371 \beta_{15} + \cdots - 348218 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 974724 \beta_{19} - 352303 \beta_{18} + 794409 \beta_{17} - 44682 \beta_{16} + 191926 \beta_{15} + \cdots + 2036492 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 332576 \beta_{19} - 974724 \beta_{18} + 592007 \beta_{17} - 332576 \beta_{15} + 715043 \beta_{14} + \cdots + 1459149 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 1530183 \beta_{19} + 3019640 \beta_{18} + 262470 \beta_{17} + 382467 \beta_{16} - 7050951 \beta_{15} + \cdots - 1051866 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 5641401 \beta_{19} - 1005243 \beta_{18} + 12754512 \beta_{17} - 14101902 \beta_{16} + 5641401 \beta_{15} + \cdots + 20029857 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/351\mathbb{Z}\right)^\times\).

\(n\) \(28\) \(326\)
\(\chi(n)\) \(\beta_{3} + \beta_{6}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
80.1
1.90127 + 1.90127i
0.943572 + 0.943572i
−0.561693 − 0.561693i
−0.606342 − 0.606342i
−1.67681 − 1.67681i
−1.93433 − 1.93433i
−0.799987 − 0.799987i
0.107781 + 0.107781i
1.03270 + 1.03270i
1.59384 + 1.59384i
1.90127 − 1.90127i
0.943572 − 0.943572i
−0.561693 + 0.561693i
−0.606342 + 0.606342i
−1.67681 + 1.67681i
−1.93433 + 1.93433i
−0.799987 + 0.799987i
0.107781 − 0.107781i
1.03270 − 1.03270i
1.59384 − 1.59384i
−0.695914 + 2.59719i 0 −4.52903 − 2.61483i 1.12850 + 1.12850i 0 −2.90139 + 0.777424i 6.14048 − 6.14048i 0 −3.71626 + 2.14558i
80.2 −0.345371 + 1.28894i 0 0.189956 + 0.109671i 0.198323 + 0.198323i 0 1.42634 − 0.382186i −2.09411 + 2.09411i 0 −0.324123 + 0.187132i
80.3 0.205594 − 0.767287i 0 1.18559 + 0.684501i −2.75236 − 2.75236i 0 1.54508 − 0.414002i 1.89235 − 1.89235i 0 −2.67772 + 1.54598i
80.4 0.221937 − 0.828279i 0 1.09526 + 0.632349i 2.06553 + 2.06553i 0 −3.61723 + 0.969233i 1.97952 − 1.97952i 0 2.16926 − 1.25242i
80.5 0.613754 − 2.29056i 0 −3.13793 − 1.81169i 1.72603 + 1.72603i 0 4.41323 − 1.18252i −2.72209 + 2.72209i 0 5.01295 − 2.89423i
188.1 −2.64234 + 0.708013i 0 4.74863 − 2.74162i 2.27842 + 2.27842i 0 −0.703212 + 2.62442i −6.73775 + 6.73775i 0 −7.63353 − 4.40722i
188.2 −1.09280 + 0.292816i 0 −0.623573 + 0.360020i −1.12325 − 1.12325i 0 −1.05984 + 3.95536i 2.17600 − 2.17600i 0 1.55640 + 0.898588i
188.3 0.147232 − 0.0394506i 0 −1.71193 + 0.988383i 0.671392 + 0.671392i 0 0.383498 − 1.43124i −0.428620 + 0.428620i 0 0.125337 + 0.0723632i
188.4 1.41069 − 0.377994i 0 0.115125 − 0.0664674i −2.86027 − 2.86027i 0 0.575104 − 2.14632i −1.92812 + 1.92812i 0 −5.11614 − 2.95380i
188.5 2.17722 − 0.583384i 0 2.66790 − 1.54031i 1.66769 + 1.66769i 0 −0.0615804 + 0.229821i 1.72233 − 1.72233i 0 4.60382 + 2.65802i
215.1 −0.695914 − 2.59719i 0 −4.52903 + 2.61483i 1.12850 − 1.12850i 0 −2.90139 − 0.777424i 6.14048 + 6.14048i 0 −3.71626 − 2.14558i
215.2 −0.345371 − 1.28894i 0 0.189956 − 0.109671i 0.198323 − 0.198323i 0 1.42634 + 0.382186i −2.09411 − 2.09411i 0 −0.324123 − 0.187132i
215.3 0.205594 + 0.767287i 0 1.18559 − 0.684501i −2.75236 + 2.75236i 0 1.54508 + 0.414002i 1.89235 + 1.89235i 0 −2.67772 − 1.54598i
215.4 0.221937 + 0.828279i 0 1.09526 − 0.632349i 2.06553 − 2.06553i 0 −3.61723 − 0.969233i 1.97952 + 1.97952i 0 2.16926 + 1.25242i
215.5 0.613754 + 2.29056i 0 −3.13793 + 1.81169i 1.72603 − 1.72603i 0 4.41323 + 1.18252i −2.72209 − 2.72209i 0 5.01295 + 2.89423i
323.1 −2.64234 − 0.708013i 0 4.74863 + 2.74162i 2.27842 − 2.27842i 0 −0.703212 − 2.62442i −6.73775 − 6.73775i 0 −7.63353 + 4.40722i
323.2 −1.09280 − 0.292816i 0 −0.623573 − 0.360020i −1.12325 + 1.12325i 0 −1.05984 − 3.95536i 2.17600 + 2.17600i 0 1.55640 − 0.898588i
323.3 0.147232 + 0.0394506i 0 −1.71193 − 0.988383i 0.671392 − 0.671392i 0 0.383498 + 1.43124i −0.428620 − 0.428620i 0 0.125337 − 0.0723632i
323.4 1.41069 + 0.377994i 0 0.115125 + 0.0664674i −2.86027 + 2.86027i 0 0.575104 + 2.14632i −1.92812 − 1.92812i 0 −5.11614 + 2.95380i
323.5 2.17722 + 0.583384i 0 2.66790 + 1.54031i 1.66769 − 1.66769i 0 −0.0615804 − 0.229821i 1.72233 + 1.72233i 0 4.60382 − 2.65802i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 80.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
39.k even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 351.2.bd.e yes 20
3.b odd 2 1 351.2.bd.d ✓ 20
13.f odd 12 1 351.2.bd.d ✓ 20
39.k even 12 1 inner 351.2.bd.e yes 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
351.2.bd.d ✓ 20 3.b odd 2 1
351.2.bd.d ✓ 20 13.f odd 12 1
351.2.bd.e yes 20 1.a even 1 1 trivial
351.2.bd.e yes 20 39.k even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} - 44 T_{2}^{16} + 36 T_{2}^{15} - 228 T_{2}^{13} + 1914 T_{2}^{12} - 1908 T_{2}^{11} + \cdots + 81 \) acting on \(S_{2}^{\mathrm{new}}(351, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} - 44 T^{16} + \cdots + 81 \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( T^{20} - 6 T^{19} + \cdots + 331776 \) Copy content Toggle raw display
$7$ \( T^{20} - 15 T^{18} + \cdots + 1119364 \) Copy content Toggle raw display
$11$ \( T^{20} + 36 T^{17} + \cdots + 9801 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 137858491849 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 118396551744 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 1076233636 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 799136361 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 24732594756 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 7118672231056 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 2497657838404 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 6249851136 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 795228763536 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 586837198809 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 35\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 170406576596169 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 55777213584 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 89\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 149381243954244 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 16\!\cdots\!41 \) Copy content Toggle raw display
$79$ \( (T^{10} + 8 T^{9} + \cdots - 89058)^{2} \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 82\!\cdots\!09 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 12\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 30824604064081 \) Copy content Toggle raw display
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