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Properties

Label 36.8.e.a
Level $36$
Weight $8$
Character orbit 36.e
Analytic conductor $11.246$
Analytic rank $0$
Dimension $14$
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] = [36,8,Mod(13,36)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(36, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("36.13");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 36 = 2^{2} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 36.e (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: \(11.2458609174\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 7 x^{13} + 2516 x^{12} - 15005 x^{11} + 2504148 x^{10} - 12383361 x^{9} + 1253722962 x^{8} + \cdots + 24\!\cdots\!76 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{20}\cdot 3^{21} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{4} - 4 \beta_{2} - 2) q^{3} + ( - \beta_{8} - 46 \beta_{2} + \beta_1) q^{5} + (\beta_{7} + \beta_{5} + 2 \beta_{4} + \cdots - 12) q^{7}+ \cdots + (\beta_{10} + \beta_{8} + 4 \beta_{5} + \cdots + 35) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} - 4 \beta_{2} - 2) q^{3} + ( - \beta_{8} - 46 \beta_{2} + \beta_1) q^{5} + (\beta_{7} + \beta_{5} + 2 \beta_{4} + \cdots - 12) q^{7}+ \cdots + ( - 396 \beta_{13} + 609 \beta_{12} + \cdots + 3928767) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 321 q^{5} - 83 q^{7} + 264 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 321 q^{5} - 83 q^{7} + 264 q^{9} - 111 q^{11} - 1847 q^{13} - 7875 q^{15} - 96 q^{17} + 20248 q^{19} + 9003 q^{21} + 19119 q^{23} - 73378 q^{25} + 158760 q^{27} + 6045 q^{29} - 153089 q^{31} - 332487 q^{33} + 27426 q^{35} + 139348 q^{37} + 76641 q^{39} + 446631 q^{41} - 384347 q^{43} + 996309 q^{45} + 298413 q^{47} - 351876 q^{49} - 967464 q^{51} - 908076 q^{53} - 2526966 q^{55} + 12150 q^{57} + 2619543 q^{59} - 146231 q^{61} + 4217721 q^{63} + 2535735 q^{65} + 1637419 q^{67} - 3024117 q^{69} - 8706984 q^{71} - 4264520 q^{73} - 4432044 q^{75} + 9785451 q^{77} + 2402185 q^{79} + 16357320 q^{81} + 12936357 q^{83} - 1015002 q^{85} - 24976413 q^{87} - 39369660 q^{89} - 984406 q^{91} - 4054125 q^{93} + 22685196 q^{95} - 2853257 q^{97} + 69035013 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 7 x^{13} + 2516 x^{12} - 15005 x^{11} + 2504148 x^{10} - 12383361 x^{9} + 1253722962 x^{8} + \cdots + 24\!\cdots\!76 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 52\!\cdots\!39 \nu^{12} + \cdots + 10\!\cdots\!90 ) / 91\!\cdots\!14 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 136216215213000 \nu^{13} - 885405398884500 \nu^{12} + \cdots - 48\!\cdots\!86 ) / 15\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 23\!\cdots\!00 \nu^{13} + \cdots - 11\!\cdots\!30 ) / 66\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 17\!\cdots\!34 \nu^{13} + \cdots - 42\!\cdots\!44 ) / 59\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 17\!\cdots\!34 \nu^{13} + \cdots + 66\!\cdots\!34 ) / 59\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 22\!\cdots\!42 \nu^{13} + \cdots + 39\!\cdots\!14 ) / 77\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 27\!\cdots\!36 \nu^{13} + \cdots - 13\!\cdots\!06 ) / 25\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 13\!\cdots\!20 \nu^{13} + \cdots + 14\!\cdots\!58 ) / 23\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 17\!\cdots\!34 \nu^{13} + \cdots - 32\!\cdots\!28 ) / 22\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 24\!\cdots\!92 \nu^{13} + \cdots - 66\!\cdots\!06 ) / 23\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 38\!\cdots\!18 \nu^{13} + \cdots - 32\!\cdots\!08 ) / 23\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 29\!\cdots\!30 \nu^{13} + \cdots - 16\!\cdots\!16 ) / 11\!\cdots\!66 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 98\!\cdots\!98 \nu^{13} + \cdots - 18\!\cdots\!82 ) / 23\!\cdots\!32 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 9\beta_{5} - 9\beta_{4} + \beta_{3} + 4\beta_{2} + 27 ) / 54 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 3 \beta_{13} - 6 \beta_{12} + 3 \beta_{11} + 7 \beta_{9} + 3 \beta_{7} + 6 \beta_{6} - 93 \beta_{5} + \cdots - 38422 ) / 108 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 78 \beta_{13} + 105 \beta_{12} - 75 \beta_{11} + 153 \beta_{10} + 209 \beta_{9} - 2124 \beta_{8} + \cdots - 297530 ) / 216 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 5283 \beta_{13} + 9405 \beta_{12} - 3906 \beta_{11} + 1989 \beta_{10} - 15123 \beta_{9} + \cdots + 39303246 ) / 216 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 83319 \beta_{13} - 79413 \beta_{12} + 97554 \beta_{11} - 172458 \beta_{10} - 114962 \beta_{9} + \cdots + 291534383 ) / 216 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 4055421 \beta_{13} - 6688797 \beta_{12} + 2437302 \beta_{11} - 2679642 \beta_{10} + \cdots - 22527807157 ) / 216 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 74023956 \beta_{13} + 52896510 \beta_{12} - 94396563 \beta_{11} + 153348903 \beta_{10} + \cdots - 243755971095 ) / 216 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 3017758665 \beta_{13} + 4734547764 \beta_{12} - 1585777971 \beta_{11} + 2724013584 \beta_{10} + \cdots + 13703777050228 ) / 216 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 62237475939 \beta_{13} - 35762976993 \beta_{12} + 81319263603 \beta_{11} - 124764300252 \beta_{10} + \cdots + 192469650796927 ) / 216 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 2214751149387 \beta_{13} - 3389006558781 \beta_{12} + 1089865505691 \beta_{11} + \cdots - 86\!\cdots\!03 ) / 216 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 50784961568142 \beta_{13} + 24589427634417 \beta_{12} - 65949166664403 \beta_{11} + \cdots - 14\!\cdots\!67 ) / 216 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 16\!\cdots\!00 \beta_{13} + \cdots + 56\!\cdots\!96 ) / 216 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 20\!\cdots\!85 \beta_{13} + \cdots + 56\!\cdots\!92 ) / 108 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/36\mathbb{Z}\right)^\times\).

\(n\) \(19\) \(29\)
\(\chi(n)\) \(1\) \(\beta_{2}\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
0.500000 + 11.0768i
0.500000 + 27.1699i
0.500000 − 13.9166i
0.500000 − 23.1573i
0.500000 + 22.6756i
0.500000 − 19.3696i
0.500000 − 3.61272i
0.500000 − 11.0768i
0.500000 − 27.1699i
0.500000 + 13.9166i
0.500000 + 23.1573i
0.500000 − 22.6756i
0.500000 + 19.3696i
0.500000 + 3.61272i
0 −46.0898 − 7.92053i 0 −131.630 + 227.990i 0 −85.0938 − 147.387i 0 2061.53 + 730.110i 0
13.2 0 −35.6058 + 30.3188i 0 154.861 − 268.227i 0 42.3923 + 73.4256i 0 348.546 − 2159.05i 0
13.3 0 −19.4519 − 42.5279i 0 255.357 − 442.292i 0 376.715 + 652.489i 0 −1430.25 + 1654.50i 0
13.4 0 5.85702 − 46.3971i 0 −111.524 + 193.165i 0 −518.114 − 897.400i 0 −2118.39 − 543.498i 0
13.5 0 7.42261 + 46.1726i 0 −100.077 + 173.339i 0 144.274 + 249.889i 0 −2076.81 + 685.442i 0
13.6 0 43.5405 − 17.0652i 0 −57.4598 + 99.5233i 0 737.052 + 1276.61i 0 1604.56 − 1486.06i 0
13.7 0 44.3273 + 14.9028i 0 150.973 − 261.492i 0 −738.725 − 1279.51i 0 1742.81 + 1321.20i 0
25.1 0 −46.0898 + 7.92053i 0 −131.630 − 227.990i 0 −85.0938 + 147.387i 0 2061.53 − 730.110i 0
25.2 0 −35.6058 − 30.3188i 0 154.861 + 268.227i 0 42.3923 − 73.4256i 0 348.546 + 2159.05i 0
25.3 0 −19.4519 + 42.5279i 0 255.357 + 442.292i 0 376.715 − 652.489i 0 −1430.25 − 1654.50i 0
25.4 0 5.85702 + 46.3971i 0 −111.524 − 193.165i 0 −518.114 + 897.400i 0 −2118.39 + 543.498i 0
25.5 0 7.42261 − 46.1726i 0 −100.077 − 173.339i 0 144.274 − 249.889i 0 −2076.81 − 685.442i 0
25.6 0 43.5405 + 17.0652i 0 −57.4598 − 99.5233i 0 737.052 − 1276.61i 0 1604.56 + 1486.06i 0
25.7 0 44.3273 − 14.9028i 0 150.973 + 261.492i 0 −738.725 + 1279.51i 0 1742.81 − 1321.20i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.7
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 36.8.e.a ✓ 14
3.b odd 2 1 108.8.e.a 14
4.b odd 2 1 144.8.i.d 14
9.c even 3 1 inner 36.8.e.a ✓ 14
9.c even 3 1 324.8.a.c 7
9.d odd 6 1 108.8.e.a 14
9.d odd 6 1 324.8.a.d 7
12.b even 2 1 432.8.i.d 14
36.f odd 6 1 144.8.i.d 14
36.h even 6 1 432.8.i.d 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.8.e.a ✓ 14 1.a even 1 1 trivial
36.8.e.a ✓ 14 9.c even 3 1 inner
108.8.e.a 14 3.b odd 2 1
108.8.e.a 14 9.d odd 6 1
144.8.i.d 14 4.b odd 2 1
144.8.i.d 14 36.f odd 6 1
324.8.a.c 7 9.c even 3 1
324.8.a.d 7 9.d odd 6 1
432.8.i.d 14 12.b even 2 1
432.8.i.d 14 36.h even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(36, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( T^{14} + \cdots + 23\!\cdots\!83 \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots + 41\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots + 50\!\cdots\!24 \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 49\!\cdots\!89 \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 35\!\cdots\!84 \) Copy content Toggle raw display
$17$ \( (T^{7} + \cdots - 99\!\cdots\!88)^{2} \) Copy content Toggle raw display
$19$ \( (T^{7} + \cdots + 39\!\cdots\!84)^{2} \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 21\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 49\!\cdots\!04 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 41\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{7} + \cdots + 54\!\cdots\!96)^{2} \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 92\!\cdots\!09 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 18\!\cdots\!69 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 97\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( (T^{7} + \cdots - 25\!\cdots\!32)^{2} \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 40\!\cdots\!89 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 12\!\cdots\!64 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 97\!\cdots\!01 \) Copy content Toggle raw display
$71$ \( (T^{7} + \cdots + 28\!\cdots\!88)^{2} \) Copy content Toggle raw display
$73$ \( (T^{7} + \cdots - 64\!\cdots\!40)^{2} \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 21\!\cdots\!36 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 12\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( (T^{7} + \cdots + 43\!\cdots\!04)^{2} \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 30\!\cdots\!69 \) Copy content Toggle raw display
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