↑

Properties

Label 33.6.d.b
Level $33$
Weight $6$
Character orbit 33.d
Analytic conductor $5.293$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [33,6,Mod(32,33)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(33, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("33.32");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 33 = 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 33.d (of order \(2\), degree \(1\), 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: \(5.29266605383\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 6 x^{15} - 195 x^{14} - 642 x^{13} + 89670 x^{12} + 53946 x^{11} + 91115757 x^{10} - 2121785838 x^{9} + 37710373995 x^{8} - 835758339660 x^{7} + \cdots + 92\!\cdots\!40 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{11}\cdot 3^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + ( - \beta_{2} - 3) q^{3} + (\beta_{9} + 20) q^{4} + \beta_{8} q^{5} + (\beta_{11} - 3 \beta_{3}) q^{6} - \beta_{13} q^{7} + ( - \beta_{14} + 14 \beta_{3}) q^{8} + ( - 2 \beta_{10} + \beta_{9} - 2 \beta_{8} + \beta_{6} + 2 \beta_{2} - 15) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} + ( - \beta_{2} - 3) q^{3} + (\beta_{9} + 20) q^{4} + \beta_{8} q^{5} + (\beta_{11} - 3 \beta_{3}) q^{6} - \beta_{13} q^{7} + ( - \beta_{14} + 14 \beta_{3}) q^{8} + ( - 2 \beta_{10} + \beta_{9} - 2 \beta_{8} + \beta_{6} + 2 \beta_{2} - 15) q^{9} + (\beta_{13} + \beta_{11} + \beta_{5}) q^{10} + ( - \beta_{14} + 2 \beta_{10} - 2 \beta_{8} + \beta_{4} + 9 \beta_{3} - \beta_{2} - \beta_1 + 1) q^{11} + (5 \beta_{10} - 7 \beta_{9} + 5 \beta_{8} + 2 \beta_{6} - 18 \beta_{2} + 3 \beta_1 - 27) q^{12} + ( - \beta_{13} - \beta_{11} + \beta_{7}) q^{13} + ( - 13 \beta_{10} - 7 \beta_{8} + 2 \beta_{2} + 5 \beta_1 - 5) q^{14} + (3 \beta_{15} - 9 \beta_{9} - 9 \beta_{8} - 7 \beta_{2} + 7 \beta_1 - 103) q^{15} + ( - 2 \beta_{15} + 17 \beta_{9} - 2 \beta_{6} + 26 \beta_{2} - 10 \beta_1 + 98) q^{16} + (\beta_{15} + \beta_{14} - \beta_{13} - \beta_{12} + 3 \beta_{11} - \beta_{6} + 2 \beta_{4} - 42 \beta_{3}) q^{17} + ( - \beta_{15} + \beta_{14} - 6 \beta_{13} + \beta_{12} - 2 \beta_{11} - \beta_{7} + \beta_{6} - 3 \beta_{5} + \cdots + 9 \beta_{3}) q^{18}+ \cdots + ( - 8 \beta_{15} - 181 \beta_{14} - 72 \beta_{13} - 70 \beta_{12} + \cdots - 15783) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 54 q^{3} + 316 q^{4} - 222 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 54 q^{3} + 316 q^{4} - 222 q^{9} - 552 q^{12} - 1674 q^{15} + 1684 q^{16} + 7932 q^{22} - 1356 q^{25} - 3240 q^{27} - 11980 q^{31} - 5106 q^{33} - 34032 q^{34} + 14016 q^{36} + 9356 q^{37} + 45912 q^{42} + 77430 q^{45} - 78012 q^{48} - 1136 q^{49} + 117308 q^{55} + 31848 q^{58} - 220548 q^{60} + 5860 q^{64} - 164796 q^{66} - 364132 q^{67} + 113790 q^{69} + 231144 q^{70} + 320364 q^{75} + 296088 q^{78} - 251334 q^{81} + 4824 q^{82} + 586836 q^{88} - 209184 q^{91} - 521046 q^{93} + 119852 q^{97} - 243894 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 6 x^{15} - 195 x^{14} - 642 x^{13} + 89670 x^{12} + 53946 x^{11} + 91115757 x^{10} - 2121785838 x^{9} + 37710373995 x^{8} - 835758339660 x^{7} + \cdots + 92\!\cdots\!40 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 15\!\cdots\!79 \nu^{15} + \cdots - 29\!\cdots\!92 ) / 36\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 11\!\cdots\!21 \nu^{15} + \cdots + 16\!\cdots\!20 ) / 11\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 11\!\cdots\!21 \nu^{15} + \cdots - 16\!\cdots\!20 ) / 11\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 32\!\cdots\!69 \nu^{15} + \cdots + 33\!\cdots\!64 ) / 19\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 67\!\cdots\!86 \nu^{15} + \cdots - 14\!\cdots\!80 ) / 33\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 11\!\cdots\!67 \nu^{15} + \cdots + 30\!\cdots\!36 ) / 49\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 15\!\cdots\!23 \nu^{15} + \cdots + 23\!\cdots\!20 ) / 49\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 17\!\cdots\!31 \nu^{15} + \cdots + 16\!\cdots\!28 ) / 29\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 10\!\cdots\!42 \nu^{15} + \cdots + 15\!\cdots\!72 ) / 16\!\cdots\!99 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 10\!\cdots\!55 \nu^{15} + \cdots + 87\!\cdots\!24 ) / 14\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 89\!\cdots\!17 \nu^{15} + \cdots - 19\!\cdots\!40 ) / 55\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 17\!\cdots\!65 \nu^{15} + \cdots - 30\!\cdots\!60 ) / 99\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 49\!\cdots\!76 \nu^{15} + \cdots - 18\!\cdots\!60 ) / 23\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 54\!\cdots\!65 \nu^{15} + \cdots - 32\!\cdots\!20 ) / 18\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 73\!\cdots\!99 \nu^{15} + \cdots + 73\!\cdots\!52 ) / 24\!\cdots\!72 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -2\beta_{11} - 2\beta_{10} + 2\beta_{9} - 2\beta_{8} + \beta_{6} - 4\beta_{2} + 28 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 6 \beta_{15} + 2 \beta_{14} - 18 \beta_{13} + 3 \beta_{12} + 12 \beta_{11} - 51 \beta_{10} - 24 \beta_{9} + 3 \beta_{8} - 3 \beta_{7} - 12 \beta_{6} - 9 \beta_{5} - 6 \beta_{4} + 78 \beta_{3} + 99 \beta_{2} + 30 \beta _1 + 294 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 372 \beta_{15} + 112 \beta_{14} - 420 \beta_{13} - 224 \beta_{12} - 356 \beta_{11} - 330 \beta_{10} - 802 \beta_{9} - 1140 \beta_{8} + 80 \beta_{7} + 317 \beta_{6} + 132 \beta_{5} + 16 \beta_{4} - 1512 \beta_{3} + \cdots - 14861 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 10546 \beta_{15} + 2264 \beta_{14} + 385 \beta_{13} + 3670 \beta_{12} + 2685 \beta_{11} - 8331 \beta_{10} - 6777 \beta_{9} + 11703 \beta_{8} - 2205 \beta_{7} - 3157 \beta_{6} - 3825 \beta_{5} + 1718 \beta_{4} + \cdots - 46695 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 216177 \beta_{15} + 28482 \beta_{14} - 103968 \beta_{13} - 175044 \beta_{12} + 159864 \beta_{11} - 16848 \beta_{10} - 723825 \beta_{9} - 181116 \beta_{8} + 19110 \beta_{7} + \cdots - 40028783 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 3944952 \beta_{15} + 377721 \beta_{14} + 2513083 \beta_{13} + 3029053 \beta_{12} - 3622161 \beta_{11} + 2633169 \beta_{10} + 3119583 \beta_{9} + 7272363 \beta_{8} - 128142 \beta_{7} + \cdots + 633552018 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 41284029 \beta_{15} + 5620056 \beta_{14} - 29124024 \beta_{13} - 45453096 \beta_{12} + 285952776 \beta_{11} + 73430730 \beta_{10} - 266860902 \beta_{9} + \cdots - 23981003268 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 596148018 \beta_{15} - 293960496 \beta_{14} + 2183715858 \beta_{13} + 310073349 \beta_{12} - 5993803422 \beta_{11} + 2392866063 \beta_{10} + 5813350506 \beta_{9} + \cdots + 459497229300 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 44698815648 \beta_{15} - 9499750164 \beta_{14} + 1636782426 \beta_{13} + 25378702308 \beta_{12} + 160693447338 \beta_{11} + 18933574878 \beta_{10} + \cdots - 5057485863405 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 2073828990276 \beta_{15} - 235257555618 \beta_{14} + 48686983167 \beta_{13} - 963707954640 \beta_{12} - 3008734006437 \beta_{11} + \cdots + 62609026750263 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 59813051150205 \beta_{15} - 5027460218316 \beta_{14} + 14118283742760 \beta_{13} + 37146417204600 \beta_{12} + 27891054717336 \beta_{11} + \cdots + 41\!\cdots\!95 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 15\!\cdots\!04 \beta_{15} - 53168082834627 \beta_{14} - 664617308228793 \beta_{13} - 983911535888265 \beta_{12} + 48723963283467 \beta_{11} + \cdots - 14\!\cdots\!66 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 31\!\cdots\!79 \beta_{15} - 688692625062714 \beta_{14} + \cdots + 54\!\cdots\!26 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 45\!\cdots\!68 \beta_{15} + \cdots - 14\!\cdots\!70 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/33\mathbb{Z}\right)^\times\).

\(n\) \(13\) \(23\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
32.1
−2.15316 + 11.0840i
−2.15316 − 11.0840i
−23.0756 + 10.4175i
−23.0756 − 10.4175i
6.80653 + 6.27606i
6.80653 − 6.27606i
−6.72395 + 15.5879i
−6.72395 − 15.5879i
0.456001 + 15.5879i
0.456001 − 15.5879i
15.7319 + 6.27606i
15.7319 − 6.27606i
−6.11708 + 10.4175i
−6.11708 − 10.4175i
18.0754 + 11.0840i
18.0754 − 11.0840i
−10.1143 −10.9611 − 11.0840i 70.2982 − 88.5196i 110.863 + 112.106i − 126.220i −387.358 −2.70883 + 242.985i 895.310i
32.2 −10.1143 −10.9611 + 11.0840i 70.2982 88.5196i 110.863 − 112.106i 126.220i −387.358 −2.70883 − 242.985i − 895.310i
32.3 −8.47928 11.5964 − 10.4175i 39.8981 35.7023i −98.3287 + 88.3330i − 11.5666i −66.9703 25.9508 − 241.610i − 302.729i
32.4 −8.47928 11.5964 + 10.4175i 39.8981 − 35.7023i −98.3287 − 88.3330i 11.5666i −66.9703 25.9508 + 241.610i 302.729i
32.5 −4.46270 −14.2692 − 6.27606i −12.0843 59.8956i 63.6794 + 28.0082i − 169.425i 196.735 164.222 + 179.109i − 267.296i
32.6 −4.46270 −14.2692 + 6.27606i −12.0843 − 59.8956i 63.6794 − 28.0082i 169.425i 196.735 164.222 − 179.109i 267.296i
32.7 −3.58998 0.133977 − 15.5879i −19.1121 − 11.8803i −0.480974 + 55.9601i 150.804i 183.491 −242.964 − 4.17684i 42.6500i
32.8 −3.58998 0.133977 + 15.5879i −19.1121 11.8803i −0.480974 − 55.9601i − 150.804i 183.491 −242.964 + 4.17684i − 42.6500i
32.9 3.58998 0.133977 − 15.5879i −19.1121 − 11.8803i 0.480974 − 55.9601i − 150.804i −183.491 −242.964 − 4.17684i − 42.6500i
32.10 3.58998 0.133977 + 15.5879i −19.1121 11.8803i 0.480974 + 55.9601i 150.804i −183.491 −242.964 + 4.17684i 42.6500i
32.11 4.46270 −14.2692 − 6.27606i −12.0843 59.8956i −63.6794 − 28.0082i 169.425i −196.735 164.222 + 179.109i 267.296i
32.12 4.46270 −14.2692 + 6.27606i −12.0843 − 59.8956i −63.6794 + 28.0082i − 169.425i −196.735 164.222 − 179.109i − 267.296i
32.13 8.47928 11.5964 − 10.4175i 39.8981 35.7023i 98.3287 − 88.3330i 11.5666i 66.9703 25.9508 − 241.610i 302.729i
32.14 8.47928 11.5964 + 10.4175i 39.8981 − 35.7023i 98.3287 + 88.3330i − 11.5666i 66.9703 25.9508 + 241.610i − 302.729i
32.15 10.1143 −10.9611 − 11.0840i 70.2982 − 88.5196i −110.863 − 112.106i 126.220i 387.358 −2.70883 + 242.985i − 895.310i
32.16 10.1143 −10.9611 + 11.0840i 70.2982 88.5196i −110.863 + 112.106i − 126.220i 387.358 −2.70883 − 242.985i 895.310i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 32.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.b odd 2 1 inner
33.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 33.6.d.b ✓ 16
3.b odd 2 1 inner 33.6.d.b ✓ 16
11.b odd 2 1 inner 33.6.d.b ✓ 16
33.d even 2 1 inner 33.6.d.b ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.6.d.b ✓ 16 1.a even 1 1 trivial
33.6.d.b ✓ 16 3.b odd 2 1 inner
33.6.d.b ✓ 16 11.b odd 2 1 inner
33.6.d.b ✓ 16 33.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 207T_{2}^{6} + 13326T_{2}^{4} - 285984T_{2}^{2} + 1887840 \) acting on \(S_{6}^{\mathrm{new}}(33, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - 207 T^{6} + 13326 T^{4} + \cdots + 1887840)^{2} \) Copy content Toggle raw display
$3$ \( (T^{8} + 27 T^{7} + 420 T^{6} + \cdots + 3486784401)^{2} \) Copy content Toggle raw display
$5$ \( (T^{8} + 12839 T^{6} + \cdots + 5057252969536)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} + 67512 T^{6} + \cdots + 13\!\cdots\!40)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} - 523232 T^{14} + \cdots + 45\!\cdots\!01 \) Copy content Toggle raw display
$13$ \( (T^{8} + 2453916 T^{6} + \cdots + 10\!\cdots\!60)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 2918916 T^{6} + \cdots + 28\!\cdots\!60)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 6264828 T^{6} + \cdots + 20\!\cdots\!60)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 18393395 T^{6} + \cdots + 18\!\cdots\!56)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} - 40561356 T^{6} + \cdots + 36\!\cdots\!60)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 2995 T^{3} + \cdots - 28686372143360)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} - 2339 T^{3} + \cdots + 17\!\cdots\!84)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} - 738524844 T^{6} + \cdots + 22\!\cdots\!40)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 1049669388 T^{6} + \cdots + 39\!\cdots\!60)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 628768868 T^{6} + \cdots + 49\!\cdots\!96)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 1078834028 T^{6} + \cdots + 24\!\cdots\!24)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 2794298423 T^{6} + \cdots + 21\!\cdots\!64)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 2308041276 T^{6} + \cdots + 67\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 91033 T^{3} + \cdots - 90\!\cdots\!68)^{4} \) Copy content Toggle raw display
$71$ \( (T^{8} + 8140749275 T^{6} + \cdots + 38\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 11461184064 T^{6} + \cdots + 12\!\cdots\!40)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 12257928120 T^{6} + \cdots + 67\!\cdots\!40)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 5152546224 T^{6} + \cdots + 11\!\cdots\!40)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 11404518611 T^{6} + \cdots + 47\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 29963 T^{3} + \cdots + 73\!\cdots\!92)^{4} \) Copy content Toggle raw display
show more
show less