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Properties

Label 33.4.d.b
Level $33$
Weight $4$
Character orbit 33.d
Analytic conductor $1.947$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [33,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("33.32");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 33 = 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
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: \(1.94706303019\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 35x^{6} + 10x^{5} + 2614x^{4} + 16258x^{3} + 120841x^{2} + 205270x + 821047 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + ( - \beta_{3} + 1) q^{3} + (\beta_{3} + \beta_1 + 5) q^{4} + (\beta_{5} + \beta_{3} - \beta_1) q^{5} + (\beta_{6} - \beta_{2}) q^{6} + \beta_{7} q^{7} + ( - \beta_{6} - \beta_{4} + \beta_{2}) q^{8} + (3 \beta_{5} - 3 \beta_1 - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + ( - \beta_{3} + 1) q^{3} + (\beta_{3} + \beta_1 + 5) q^{4} + (\beta_{5} + \beta_{3} - \beta_1) q^{5} + (\beta_{6} - \beta_{2}) q^{6} + \beta_{7} q^{7} + ( - \beta_{6} - \beta_{4} + \beta_{2}) q^{8} + (3 \beta_{5} - 3 \beta_1 - 3) q^{9} + ( - \beta_{7} - \beta_{6} + \beta_{4}) q^{10} + (\beta_{6} - \beta_{5} + \beta_{4} + 3 \beta_{3} - 3 \beta_{2} - 3 \beta_1) q^{11} + ( - 3 \beta_{5} - 7 \beta_{3} + 3 \beta_1 - 17) q^{12} - \beta_{7} q^{13} + ( - 9 \beta_{5} - 4 \beta_{3} + 4 \beta_1) q^{14} + (2 \beta_{3} + 9 \beta_1 + 16) q^{15} + (3 \beta_{3} + 3 \beta_1 - 35) q^{16} + ( - 4 \beta_{6} - 4 \beta_{4} + 8 \beta_{2}) q^{17} + ( - 3 \beta_{7} + 3 \beta_{4} - 9 \beta_{2}) q^{18} + (4 \beta_{6} - 4 \beta_{4}) q^{19} + (7 \beta_{5} + 14 \beta_{3} - 14 \beta_1) q^{20} + (3 \beta_{7} + 2 \beta_{6} + 6 \beta_{4} - 2 \beta_{2}) q^{21} + (\beta_{7} - 3 \beta_{6} + 3 \beta_{4} - 13 \beta_{3} - 13 \beta_1 - 31) q^{22} + (\beta_{5} - 3 \beta_{3} + 3 \beta_1) q^{23} + (3 \beta_{7} - \beta_{6} - 3 \beta_{4} - 17 \beta_{2}) q^{24} + ( - 9 \beta_{3} - 9 \beta_1 + 39) q^{25} + (9 \beta_{5} + 4 \beta_{3} - 4 \beta_1) q^{26} + (9 \beta_{5} + 12 \beta_{3} + 18 \beta_1 + 33) q^{27} + (\beta_{7} + 4 \beta_{6} - 4 \beta_{4}) q^{28} + (2 \beta_{6} + 2 \beta_{4} + 18 \beta_{2}) q^{29} + ( - 2 \beta_{6} - 9 \beta_{4} + 38 \beta_{2}) q^{30} + ( - 3 \beta_{3} - 3 \beta_1 + 104) q^{31} + (5 \beta_{6} + 5 \beta_{4} - 31 \beta_{2}) q^{32} + ( - 3 \beta_{7} - \beta_{6} - 12 \beta_{5} + 3 \beta_{4} - 2 \beta_{3} + 19 \beta_{2} + \cdots + 104) q^{33}+ \cdots + ( - 6 \beta_{7} + 15 \beta_{6} - 30 \beta_{5} - 21 \beta_{4} - 129 \beta_{3} + \cdots + 186) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 6 q^{3} + 44 q^{4} - 30 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 6 q^{3} + 44 q^{4} - 30 q^{9} - 144 q^{12} + 150 q^{15} - 268 q^{16} - 300 q^{22} + 276 q^{25} + 324 q^{27} + 820 q^{31} + 834 q^{33} + 768 q^{34} - 696 q^{36} - 884 q^{37} - 120 q^{42} - 1722 q^{45} - 732 q^{48} - 2032 q^{49} - 476 q^{55} + 1992 q^{58} + 2772 q^{60} - 1084 q^{64} + 2076 q^{66} + 172 q^{67} - 834 q^{69} + 5016 q^{70} + 1800 q^{75} + 120 q^{78} - 4014 q^{81} - 6408 q^{82} - 3852 q^{88} + 4776 q^{91} + 1146 q^{93} - 3836 q^{97} + 1074 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 35x^{6} + 10x^{5} + 2614x^{4} + 16258x^{3} + 120841x^{2} + 205270x + 821047 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 654449 \nu^{7} - 4621500 \nu^{6} - 37910704 \nu^{5} + 258375310 \nu^{4} + 2191936298 \nu^{3} - 3509536790 \nu^{2} - 27353124741 \nu - 251237072504 ) / 43097961858 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1117339 \nu^{7} + 7661354 \nu^{6} + 34074596 \nu^{5} - 310366016 \nu^{4} - 2938713578 \nu^{3} + 1344980044 \nu^{2} + \cdots + 204737740374 ) / 43097961858 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1117339 \nu^{7} - 7661354 \nu^{6} - 34074596 \nu^{5} + 310366016 \nu^{4} + 2938713578 \nu^{3} - 1344980044 \nu^{2} + 66274294521 \nu - 204737740374 ) / 43097961858 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2849712 \nu^{7} + 18044843 \nu^{6} - 255666928 \nu^{5} - 533197318 \nu^{4} + 13402791352 \nu^{3} + 104272839604 \nu^{2} + \cdots + 1611381064281 ) / 43097961858 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 9352895 \nu^{7} - 760108 \nu^{6} + 480817032 \nu^{5} + 759114088 \nu^{4} - 33192323398 \nu^{3} - 193283264130 \nu^{2} + \cdots - 1902320865922 ) / 129293885574 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 5889566 \nu^{7} + 8072123 \nu^{6} + 295356518 \nu^{5} - 34001274 \nu^{4} - 20257454786 \nu^{3} - 114008586182 \nu^{2} + \cdots - 403613661909 ) / 43097961858 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 345396 \nu^{7} - 399831 \nu^{6} - 16270822 \nu^{5} + 12503088 \nu^{4} + 886340512 \nu^{3} + 6517009006 \nu^{2} + 37980909510 \nu + 64671640283 ) / 2268313782 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -2\beta_{6} + 3\beta_{5} + 3\beta_{3} + 4\beta_{2} - 2\beta _1 + 9 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -9\beta_{7} - 7\beta_{6} + 9\beta_{5} + 8\beta_{4} + 39\beta_{3} - \beta_{2} - 33\beta _1 + 22 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -12\beta_{7} - 28\beta_{6} + 225\beta_{5} + 116\beta_{4} + 214\beta_{3} - 264\beta_{2} - 381\beta _1 - 897 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -285\beta_{7} - 42\beta_{6} + 732\beta_{5} + 653\beta_{4} - 660\beta_{3} - 5079\beta_{2} - 4421\beta _1 - 10692 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 288 \beta_{7} + 5656 \beta_{6} + 1206 \beta_{5} + 6910 \beta_{4} - 18636 \beta_{3} - 48980 \beta_{2} - 22362 \beta _1 - 165641 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 16338 \beta_{7} + 61282 \beta_{6} - 51966 \beta_{5} + 14032 \beta_{4} - 327059 \beta_{3} - 469273 \beta_{2} - 97938 \beta _1 - 1076460 \) 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
−0.913072 + 4.51265i
−0.913072 − 4.51265i
−5.69289 + 3.22272i
−5.69289 − 3.22272i
−0.459178 + 3.22272i
−0.459178 − 3.22272i
8.06514 + 4.51265i
8.06514 − 4.51265i
−4.48911 −2.57603 − 4.51265i 12.1521 12.2625i 11.5641 + 20.2578i 14.5320i −18.6391 −13.7281 + 23.2495i − 55.0476i
32.2 −4.48911 −2.57603 + 4.51265i 12.1521 − 12.2625i 11.5641 − 20.2578i − 14.5320i −18.6391 −13.7281 − 23.2495i 55.0476i
32.3 −2.61686 4.07603 − 3.22272i −1.15207 − 5.53456i −10.6664 + 8.43340i − 31.3500i 23.9496 6.22810 − 26.2719i 14.4832i
32.4 −2.61686 4.07603 + 3.22272i −1.15207 5.53456i −10.6664 − 8.43340i 31.3500i 23.9496 6.22810 + 26.2719i − 14.4832i
32.5 2.61686 4.07603 − 3.22272i −1.15207 − 5.53456i 10.6664 − 8.43340i 31.3500i −23.9496 6.22810 − 26.2719i − 14.4832i
32.6 2.61686 4.07603 + 3.22272i −1.15207 5.53456i 10.6664 + 8.43340i − 31.3500i −23.9496 6.22810 + 26.2719i 14.4832i
32.7 4.48911 −2.57603 − 4.51265i 12.1521 12.2625i −11.5641 − 20.2578i − 14.5320i 18.6391 −13.7281 + 23.2495i 55.0476i
32.8 4.48911 −2.57603 + 4.51265i 12.1521 − 12.2625i −11.5641 + 20.2578i 14.5320i 18.6391 −13.7281 − 23.2495i − 55.0476i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 32.8
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 33.4.d.b ✓ 8
3.b odd 2 1 inner 33.4.d.b ✓ 8
4.b odd 2 1 528.4.b.e 8
11.b odd 2 1 inner 33.4.d.b ✓ 8
12.b even 2 1 528.4.b.e 8
33.d even 2 1 inner 33.4.d.b ✓ 8
44.c even 2 1 528.4.b.e 8
132.d odd 2 1 528.4.b.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.4.d.b ✓ 8 1.a even 1 1 trivial
33.4.d.b ✓ 8 3.b odd 2 1 inner
33.4.d.b ✓ 8 11.b odd 2 1 inner
33.4.d.b ✓ 8 33.d even 2 1 inner
528.4.b.e 8 4.b odd 2 1
528.4.b.e 8 12.b even 2 1
528.4.b.e 8 44.c even 2 1
528.4.b.e 8 132.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 27T_{2}^{2} + 138 \) acting on \(S_{4}^{\mathrm{new}}(33, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 27 T^{2} + 138)^{2} \) Copy content Toggle raw display
$3$ \( (T^{4} - 3 T^{3} + 12 T^{2} - 81 T + 729)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} + 181 T^{2} + 4606)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 1194 T^{2} + 207552)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + 872 T^{6} + \cdots + 3138428376721 \) Copy content Toggle raw display
$13$ \( (T^{4} + 1194 T^{2} + 207552)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 15840 T^{2} + 62318592)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 30816 T^{2} + \cdots + 119549952)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 1549 T^{2} + 557326)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 12804 T^{2} + 27697152)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 205 T + 10108)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 221 T - 15446)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} - 127620 T^{2} + \cdots + 2981338752)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 129960 T^{2} + \cdots + 650883072)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 156838 T^{2} + \cdots + 971000824)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 384154 T^{2} + \cdots + 2414354656)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 111763 T^{2} + \cdots + 2268007936)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 31530 T^{2} + \cdots + 212533248)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 43 T - 31796)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 608437 T^{2} + \cdots + 77162701294)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 1400544 T^{2} + \cdots + 252967698432)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 987594 T^{2} + \cdots + 14685549312)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 131472 T^{2} + \cdots + 2499667968)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 2018707 T^{2} + \cdots + 838875084256)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 959 T - 257936)^{4} \) Copy content Toggle raw display
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