Properties

Label 33.3.h.a
Level $33$
Weight $3$
Character orbit 33.h
Analytic conductor $0.899$
Analytic rank $0$
Dimension $8$
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] = [33,3,Mod(5,33)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(33, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("33.5");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 33 = 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 33.h (of order \(10\), 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: \(0.899184872389\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{20})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-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 primitive root of unity \(\zeta_{20}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{20} q^{2} + ( - 2 \zeta_{20}^{6} + 2 \zeta_{20}^{5} + 2 \zeta_{20}^{4} - \zeta_{20}^{3} - 2 \zeta_{20}^{2} + 2 \zeta_{20} + 2) q^{3} - 3 \zeta_{20}^{2} q^{4} + (5 \zeta_{20}^{7} - 6 \zeta_{20}^{5} + 6 \zeta_{20}^{3} - 5 \zeta_{20}) q^{5} + ( - 2 \zeta_{20}^{7} + 2 \zeta_{20}^{6} + 2 \zeta_{20}^{5} - \zeta_{20}^{4} - 2 \zeta_{20}^{3} + 2 \zeta_{20}^{2} + 2 \zeta_{20}) q^{6} + ( - 3 \zeta_{20}^{4} + 7 \zeta_{20}^{2} - 3) q^{7} - 7 \zeta_{20}^{3} q^{8} + ( - 8 \zeta_{20}^{7} + \zeta_{20}^{6} + 8 \zeta_{20}^{5} + 4 \zeta_{20}) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{20} q^{2} + ( - 2 \zeta_{20}^{6} + 2 \zeta_{20}^{5} + 2 \zeta_{20}^{4} - \zeta_{20}^{3} - 2 \zeta_{20}^{2} + 2 \zeta_{20} + 2) q^{3} - 3 \zeta_{20}^{2} q^{4} + (5 \zeta_{20}^{7} - 6 \zeta_{20}^{5} + 6 \zeta_{20}^{3} - 5 \zeta_{20}) q^{5} + ( - 2 \zeta_{20}^{7} + 2 \zeta_{20}^{6} + 2 \zeta_{20}^{5} - \zeta_{20}^{4} - 2 \zeta_{20}^{3} + 2 \zeta_{20}^{2} + 2 \zeta_{20}) q^{6} + ( - 3 \zeta_{20}^{4} + 7 \zeta_{20}^{2} - 3) q^{7} - 7 \zeta_{20}^{3} q^{8} + ( - 8 \zeta_{20}^{7} + \zeta_{20}^{6} + 8 \zeta_{20}^{5} + 4 \zeta_{20}) q^{9} + ( - \zeta_{20}^{6} + \zeta_{20}^{4} - 5) q^{10} + (2 \zeta_{20}^{7} - 9 \zeta_{20}^{5} + 6 \zeta_{20}^{3} - 12 \zeta_{20}) q^{11} + ( - 6 \zeta_{20}^{7} + 3 \zeta_{20}^{5} - 6 \zeta_{20}^{3} - 6) q^{12} + (10 \zeta_{20}^{6} - 2 \zeta_{20}^{2} + 2) q^{13} + ( - 3 \zeta_{20}^{5} + 7 \zeta_{20}^{3} - 3 \zeta_{20}) q^{14} + (10 \zeta_{20}^{7} - 11 \zeta_{20}^{4} - 2 \zeta_{20}^{3} + 8 \zeta_{20}^{2} + 2 \zeta_{20} - 11) q^{15} + 5 \zeta_{20}^{4} q^{16} + ( - 6 \zeta_{20}^{7} + 6 \zeta_{20}^{5} - 6 \zeta_{20}^{3} + 6 \zeta_{20}) q^{17} + (\zeta_{20}^{7} + 8 \zeta_{20}^{4} - 4 \zeta_{20}^{2} + 8) q^{18} + ( - 20 \zeta_{20}^{6} + 2 \zeta_{20}^{4} - 2 \zeta_{20}^{2} + 20) q^{19} + (3 \zeta_{20}^{7} - 3 \zeta_{20}^{5} + 15 \zeta_{20}) q^{20} + (11 \zeta_{20}^{7} + 6 \zeta_{20}^{6} - 13 \zeta_{20}^{5} - 6 \zeta_{20}^{4} + 11 \zeta_{20}^{3} + 8) q^{21} + ( - 7 \zeta_{20}^{6} + 4 \zeta_{20}^{4} - 10 \zeta_{20}^{2} - 2) q^{22} + (4 \zeta_{20}^{7} + 20 \zeta_{20}^{5} + 4 \zeta_{20}^{3}) q^{23} + ( - 7 \zeta_{20}^{6} - 14 \zeta_{20}^{2} - 14 \zeta_{20} + 14) q^{24} + ( - \zeta_{20}^{6} + 12 \zeta_{20}^{4} - 12 \zeta_{20}^{2} + 1) q^{25} + (10 \zeta_{20}^{7} - 2 \zeta_{20}^{3} + 2 \zeta_{20}) q^{26} + ( - 7 \zeta_{20}^{7} - 7 \zeta_{20}^{5} + 22 \zeta_{20}^{4} + 7 \zeta_{20}^{3} + 7 \zeta_{20}) q^{27} + (9 \zeta_{20}^{6} - 21 \zeta_{20}^{4} + 9 \zeta_{20}^{2}) q^{28} + ( - 27 \zeta_{20}^{7} - 33 \zeta_{20}^{3} + 33 \zeta_{20}) q^{29} + (10 \zeta_{20}^{6} - 11 \zeta_{20}^{5} - 12 \zeta_{20}^{4} + 8 \zeta_{20}^{3} + 12 \zeta_{20}^{2} + \cdots - 10) q^{30} + \cdots + ( - 6 \zeta_{20}^{7} + 20 \zeta_{20}^{6} - 6 \zeta_{20}^{5} - 112 \zeta_{20}^{4} + 4 \zeta_{20}^{3} + \cdots - 32) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{3} - 6 q^{4} + 10 q^{6} - 4 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{3} - 6 q^{4} + 10 q^{6} - 4 q^{7} + 2 q^{9} - 44 q^{10} - 48 q^{12} + 32 q^{13} - 50 q^{15} - 10 q^{16} + 40 q^{18} + 112 q^{19} + 88 q^{21} - 58 q^{22} + 70 q^{24} - 42 q^{25} - 44 q^{27} + 78 q^{28} - 12 q^{30} - 18 q^{31} - 90 q^{33} + 48 q^{34} + 6 q^{36} - 120 q^{37} - 64 q^{39} + 42 q^{40} - 70 q^{42} - 264 q^{43} + 80 q^{45} + 24 q^{46} + 20 q^{48} - 150 q^{49} + 60 q^{51} + 36 q^{52} + 316 q^{55} + 136 q^{57} + 186 q^{58} + 180 q^{60} + 336 q^{61} + 4 q^{63} + 26 q^{64} - 124 q^{66} - 24 q^{67} - 240 q^{69} + 42 q^{70} - 280 q^{72} - 182 q^{73} - 136 q^{75} - 264 q^{76} + 40 q^{78} - 460 q^{79} + 158 q^{81} - 72 q^{82} + 24 q^{84} - 36 q^{85} + 660 q^{87} - 266 q^{88} - 16 q^{90} + 84 q^{91} + 36 q^{93} + 52 q^{94} - 330 q^{96} + 516 q^{97} + 40 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(13\) \(23\)
\(\chi(n)\) \(-\zeta_{20}^{2}\) \(-1\)

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
−0.951057 + 0.309017i
0.951057 0.309017i
−0.587785 0.809017i
0.587785 + 0.809017i
−0.951057 0.309017i
0.951057 + 0.309017i
−0.587785 + 0.809017i
0.587785 0.809017i
−0.951057 + 0.309017i 0.303706 + 2.98459i −2.42705 + 1.76336i 4.16750 + 1.35410i −1.21113 2.74466i 1.73607 1.26133i 4.11450 5.66312i −8.81553 + 1.81288i −4.38197
5.2 0.951057 0.309017i 2.93236 0.633446i −2.42705 + 1.76336i −4.16750 1.35410i 2.59310 1.50859i 1.73607 1.26133i −4.11450 + 5.66312i 8.19749 3.71499i −4.38197
14.1 −0.587785 0.809017i −2.74466 1.21113i 0.927051 2.85317i 3.88998 5.35410i 0.633446 + 2.93236i −2.73607 + 8.42075i −6.65740 + 2.16312i 6.06633 + 6.64828i −6.61803
14.2 0.587785 + 0.809017i 1.50859 2.59310i 0.927051 2.85317i −3.88998 + 5.35410i 2.98459 0.303706i −2.73607 + 8.42075i 6.65740 2.16312i −4.44829 7.82385i −6.61803
20.1 −0.951057 0.309017i 0.303706 2.98459i −2.42705 1.76336i 4.16750 1.35410i −1.21113 + 2.74466i 1.73607 + 1.26133i 4.11450 + 5.66312i −8.81553 1.81288i −4.38197
20.2 0.951057 + 0.309017i 2.93236 + 0.633446i −2.42705 1.76336i −4.16750 + 1.35410i 2.59310 + 1.50859i 1.73607 + 1.26133i −4.11450 5.66312i 8.19749 + 3.71499i −4.38197
26.1 −0.587785 + 0.809017i −2.74466 + 1.21113i 0.927051 + 2.85317i 3.88998 + 5.35410i 0.633446 2.93236i −2.73607 8.42075i −6.65740 2.16312i 6.06633 6.64828i −6.61803
26.2 0.587785 0.809017i 1.50859 + 2.59310i 0.927051 + 2.85317i −3.88998 5.35410i 2.98459 + 0.303706i −2.73607 8.42075i 6.65740 + 2.16312i −4.44829 + 7.82385i −6.61803
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.c even 5 1 inner
33.h odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 33.3.h.a 8
3.b odd 2 1 inner 33.3.h.a 8
11.b odd 2 1 363.3.h.g 8
11.c even 5 1 inner 33.3.h.a 8
11.c even 5 1 363.3.b.f 4
11.c even 5 2 363.3.h.h 8
11.d odd 10 1 363.3.b.g 4
11.d odd 10 1 363.3.h.g 8
11.d odd 10 2 363.3.h.i 8
33.d even 2 1 363.3.h.g 8
33.f even 10 1 363.3.b.g 4
33.f even 10 1 363.3.h.g 8
33.f even 10 2 363.3.h.i 8
33.h odd 10 1 inner 33.3.h.a 8
33.h odd 10 1 363.3.b.f 4
33.h odd 10 2 363.3.h.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.3.h.a 8 1.a even 1 1 trivial
33.3.h.a 8 3.b odd 2 1 inner
33.3.h.a 8 11.c even 5 1 inner
33.3.h.a 8 33.h odd 10 1 inner
363.3.b.f 4 11.c even 5 1
363.3.b.f 4 33.h odd 10 1
363.3.b.g 4 11.d odd 10 1
363.3.b.g 4 33.f even 10 1
363.3.h.g 8 11.b odd 2 1
363.3.h.g 8 11.d odd 10 1
363.3.h.g 8 33.d even 2 1
363.3.h.g 8 33.f even 10 1
363.3.h.h 8 11.c even 5 2
363.3.h.h 8 33.h odd 10 2
363.3.h.i 8 11.d odd 10 2
363.3.h.i 8 33.f even 10 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - T_{2}^{6} + T_{2}^{4} - T_{2}^{2} + 1 \) acting on \(S_{3}^{\mathrm{new}}(33, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - T^{6} + T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{8} - 4 T^{7} + 7 T^{6} + 8 T^{5} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( T^{8} - 4 T^{6} + 1446 T^{4} + \cdots + 707281 \) Copy content Toggle raw display
$7$ \( (T^{4} + 2 T^{3} + 64 T^{2} - 247 T + 361)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} - 316 T^{6} + \cdots + 214358881 \) Copy content Toggle raw display
$13$ \( (T^{4} - 16 T^{3} + 136 T^{2} + \cdots + 13456)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} - 36 T^{6} + 1296 T^{4} + \cdots + 1679616 \) Copy content Toggle raw display
$19$ \( (T^{4} - 56 T^{3} + 1336 T^{2} + \cdots + 80656)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 1008 T^{2} + 215296)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 261 T^{6} + \cdots + 2449228130001 \) Copy content Toggle raw display
$31$ \( (T^{4} + 9 T^{3} + 796 T^{2} - 8786 T + 36481)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 60 T^{3} + 1840 T^{2} + \cdots + 336400)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 1124 T^{6} + \cdots + 3544535296 \) Copy content Toggle raw display
$43$ \( (T^{2} + 66 T + 1044)^{4} \) Copy content Toggle raw display
$47$ \( T^{8} - 496 T^{6} + \cdots + 33362176 \) Copy content Toggle raw display
$53$ \( T^{8} + 101 T^{6} + \cdots + 3969126001 \) Copy content Toggle raw display
$59$ \( T^{8} - 100 T^{6} + \cdots + 276281640625 \) Copy content Toggle raw display
$61$ \( (T^{4} - 168 T^{3} + 12024 T^{2} + \cdots + 6533136)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 6 T - 5436)^{4} \) Copy content Toggle raw display
$71$ \( T^{8} - 9904 T^{6} + \cdots + 11\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( (T^{4} + 91 T^{3} + 3136 T^{2} + \cdots + 866761)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 230 T^{3} + 25360 T^{2} + \cdots + 55428025)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} - 12004 T^{6} + \cdots + 22\!\cdots\!01 \) Copy content Toggle raw display
$89$ \( (T^{4} + 9632 T^{2} + 891136)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 258 T^{3} + 36864 T^{2} + \cdots + 147403881)^{2} \) Copy content Toggle raw display
show more
show less