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Properties

Label 33.5.b.a
Level $33$
Weight $5$
Character orbit 33.b
Analytic conductor $3.411$
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] = [33,5,Mod(23,33)]
 
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, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("33.23");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 33 = 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 33.b (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: \(3.41120878177\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 162x^{12} + 10041x^{10} + 298396x^{8} + 4418856x^{6} + 32113344x^{4} + 102865552x^{2} + 102193344 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{9}\cdot 11^{4} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{4} q^{3} + (\beta_{2} - 7) q^{4} - \beta_{9} q^{5} - \beta_{7} q^{6} + (\beta_{4} - \beta_{3} + 6) q^{7} + (\beta_{5} - \beta_{4} - 6 \beta_1) q^{8} + (\beta_{12} - 5) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{4} q^{3} + (\beta_{2} - 7) q^{4} - \beta_{9} q^{5} - \beta_{7} q^{6} + (\beta_{4} - \beta_{3} + 6) q^{7} + (\beta_{5} - \beta_{4} - 6 \beta_1) q^{8} + (\beta_{12} - 5) q^{9} + ( - \beta_{6} - 3 \beta_{4} + \cdots - 12) q^{10}+ \cdots + ( - 3 \beta_{13} + \beta_{12} + \cdots + 274) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 5 q^{3} - 100 q^{4} - 2 q^{6} + 76 q^{7} - 67 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 5 q^{3} - 100 q^{4} - 2 q^{6} + 76 q^{7} - 67 q^{9} - 156 q^{10} - 100 q^{12} - 104 q^{13} + 151 q^{15} + 356 q^{16} - 34 q^{18} + 1072 q^{19} + 718 q^{21} + 1200 q^{24} - 1060 q^{25} - 1154 q^{27} - 1808 q^{28} - 3026 q^{30} + 3310 q^{31} - 605 q^{33} - 2304 q^{34} + 2644 q^{36} - 362 q^{37} + 4264 q^{39} + 1896 q^{40} - 7364 q^{42} - 6740 q^{43} + 3611 q^{45} - 4068 q^{46} - 2956 q^{48} + 7074 q^{49} - 7046 q^{51} + 13072 q^{52} + 20512 q^{54} + 726 q^{55} + 3876 q^{57} - 7848 q^{58} - 8416 q^{60} - 3560 q^{61} - 17662 q^{63} + 12020 q^{64} + 1210 q^{66} - 16514 q^{67} + 9833 q^{69} + 13320 q^{70} + 8160 q^{72} + 12664 q^{73} - 5386 q^{75} - 43736 q^{76} + 19096 q^{78} + 3052 q^{79} - 11611 q^{81} + 10200 q^{82} - 39184 q^{84} + 34884 q^{85} + 37068 q^{87} - 7260 q^{88} - 26686 q^{90} - 45856 q^{91} + 2719 q^{93} + 6120 q^{94} - 38368 q^{96} - 27854 q^{97} + 4235 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 162x^{12} + 10041x^{10} + 298396x^{8} + 4418856x^{6} + 32113344x^{4} + 102865552x^{2} + 102193344 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 23 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 21871 \nu^{13} + 664464 \nu^{12} + 3156768 \nu^{11} + 100588064 \nu^{10} + 162076727 \nu^{9} + \cdots - 890860271616 ) / 57508005888 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 21871 \nu^{13} - 130810 \nu^{12} + 3156768 \nu^{11} - 21010880 \nu^{10} + 162076727 \nu^{9} + \cdots - 2931692294592 ) / 57508005888 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 21871 \nu^{13} - 130810 \nu^{12} + 3156768 \nu^{11} - 21010880 \nu^{10} + 162076727 \nu^{9} + \cdots - 2931692294592 ) / 57508005888 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 65613 \nu^{13} + 2338070 \nu^{12} - 9470304 \nu^{11} + 343879424 \nu^{10} + \cdots + 31256572000320 ) / 57508005888 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 515 \nu^{13} + 1521 \nu^{12} + 82720 \nu^{11} + 226496 \nu^{10} + 4945131 \nu^{9} + \cdots + 8799490656 ) / 226409472 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 43742 \nu^{13} - 327025 \nu^{12} - 6313536 \nu^{11} - 52527200 \nu^{10} + \cdots - 7271722730592 ) / 14377001472 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 53401 \nu^{13} - 8164552 \nu^{11} - 465987745 \nu^{9} - 12206545826 \nu^{7} + \cdots - 1396072917216 \nu ) / 14377001472 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 131008 \nu^{13} - 65405 \nu^{12} - 20665512 \nu^{11} - 10505440 \nu^{10} + \cdots - 1465846147296 ) / 28754002944 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 96935 \nu^{13} + 73660 \nu^{12} - 15513224 \nu^{11} + 11023600 \nu^{10} + \cdots + 1298665846656 ) / 14377001472 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 101413 \nu^{13} - 92456 \nu^{12} - 15308004 \nu^{11} - 14123416 \nu^{10} + \cdots - 635169306240 ) / 14377001472 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 311765 \nu^{13} - 446532 \nu^{12} + 48470120 \nu^{11} - 70268592 \nu^{10} + \cdots - 7104969198720 ) / 28754002944 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 23 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} - \beta_{4} - 38\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} - \beta_{9} + 2\beta_{7} - 2\beta_{4} - 2\beta_{3} - 51\beta_{2} + 872 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -4\beta_{13} + 4\beta_{12} - 12\beta_{10} + 2\beta_{9} - 2\beta_{8} - 57\beta_{5} + 105\beta_{4} + 1644\beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 18 \beta_{13} + 18 \beta_{12} - 75 \beta_{11} + 75 \beta_{9} - 12 \beta_{8} - 186 \beta_{7} + \cdots - 37698 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 356 \beta_{13} - 304 \beta_{12} - 26 \beta_{11} + 1236 \beta_{10} - 430 \beta_{9} + 180 \beta_{8} + \cdots - 1102 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 1880 \beta_{13} - 1880 \beta_{12} + 4427 \beta_{11} - 4427 \beta_{9} + 1176 \beta_{8} + \cdots + 1695974 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 24196 \beta_{13} + 18020 \beta_{12} + 3088 \beta_{11} - 90324 \beta_{10} + 37766 \beta_{9} + \cdots + 73772 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 137542 \beta_{13} + 137542 \beta_{12} - 243291 \beta_{11} + 243291 \beta_{9} - 83988 \beta_{8} + \cdots - 77693886 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1481668 \beta_{13} - 979368 \beta_{12} - 251150 \beta_{11} + 5726868 \beta_{10} - 2551166 \beta_{9} + \cdots - 4402834 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 8733980 \beta_{13} - 8733980 \beta_{12} + 12961511 \beta_{11} - 12961511 \beta_{9} + \cdots + 3603294598 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 85753652 \beta_{13} + 50963020 \beta_{12} + 17395316 \beta_{11} - 336880932 \beta_{10} + \cdots + 248253520 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1
− 7.08422i
− 6.53421i
− 6.24203i
− 3.64872i
− 3.08869i
− 2.36054i
− 1.31515i
1.31515i
2.36054i
3.08869i
3.64872i
6.24203i
6.53421i
7.08422i
− 7.08422i 1.47928 − 8.87760i −34.1862 5.51941i −62.8909 − 10.4795i 86.8582 128.835i −76.6235 − 26.2648i 39.1007
23.2 − 6.53421i −8.21241 + 3.68190i −26.6960 12.2301i 24.0583 + 53.6616i −55.6823 69.8897i 53.8873 − 60.4745i 79.9138
23.3 − 6.24203i 7.09522 + 5.53695i −22.9629 − 42.0693i 34.5618 − 44.2886i −11.7748 43.4628i 19.6843 + 78.5718i −262.598
23.4 − 3.64872i 1.72713 + 8.83273i 2.68684 34.1690i 32.2281 − 6.30180i 45.8062 − 68.1830i −75.0341 + 30.5105i 124.673
23.5 − 3.08869i −4.49864 − 7.79501i 6.45997 − 9.71602i −24.0764 + 13.8949i −66.8890 − 69.3720i −40.5244 + 70.1340i −30.0098
23.6 − 2.36054i 7.81446 − 4.46478i 10.4279 9.92884i −10.5393 − 18.4463i −13.3317 − 62.3839i 41.1315 − 69.7797i 23.4374
23.7 − 1.31515i −7.90503 + 4.30239i 14.2704 − 39.9329i 5.65827 + 10.3963i 53.0133 − 39.8100i 43.9789 − 68.0210i −52.5176
23.8 1.31515i −7.90503 − 4.30239i 14.2704 39.9329i 5.65827 − 10.3963i 53.0133 39.8100i 43.9789 + 68.0210i −52.5176
23.9 2.36054i 7.81446 + 4.46478i 10.4279 − 9.92884i −10.5393 + 18.4463i −13.3317 62.3839i 41.1315 + 69.7797i 23.4374
23.10 3.08869i −4.49864 + 7.79501i 6.45997 9.71602i −24.0764 − 13.8949i −66.8890 69.3720i −40.5244 − 70.1340i −30.0098
23.11 3.64872i 1.72713 − 8.83273i 2.68684 − 34.1690i 32.2281 + 6.30180i 45.8062 68.1830i −75.0341 − 30.5105i 124.673
23.12 6.24203i 7.09522 − 5.53695i −22.9629 42.0693i 34.5618 + 44.2886i −11.7748 − 43.4628i 19.6843 − 78.5718i −262.598
23.13 6.53421i −8.21241 − 3.68190i −26.6960 − 12.2301i 24.0583 − 53.6616i −55.6823 − 69.8897i 53.8873 + 60.4745i 79.9138
23.14 7.08422i 1.47928 + 8.87760i −34.1862 − 5.51941i −62.8909 + 10.4795i 86.8582 − 128.835i −76.6235 + 26.2648i 39.1007
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.14
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 33.5.b.a ✓ 14
3.b odd 2 1 inner 33.5.b.a ✓ 14
4.b odd 2 1 528.5.i.d 14
12.b even 2 1 528.5.i.d 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.5.b.a ✓ 14 1.a even 1 1 trivial
33.5.b.a ✓ 14 3.b odd 2 1 inner
528.5.i.d 14 4.b odd 2 1
528.5.i.d 14 12.b even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(33, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + \cdots + 102193344 \) Copy content Toggle raw display
$3$ \( T^{14} + \cdots + 22876792454961 \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots + 13\!\cdots\!76 \) Copy content Toggle raw display
$7$ \( (T^{7} - 38 T^{6} + \cdots - 123318687104)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 1331)^{7} \) Copy content Toggle raw display
$13$ \( (T^{7} + \cdots - 81\!\cdots\!16)^{2} \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 11\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( (T^{7} + \cdots + 22\!\cdots\!04)^{2} \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 26\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 14\!\cdots\!04 \) Copy content Toggle raw display
$31$ \( (T^{7} + \cdots + 27\!\cdots\!96)^{2} \) Copy content Toggle raw display
$37$ \( (T^{7} + \cdots - 78\!\cdots\!96)^{2} \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 24\!\cdots\!16 \) Copy content Toggle raw display
$43$ \( (T^{7} + \cdots + 42\!\cdots\!88)^{2} \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 11\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 13\!\cdots\!84 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 89\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( (T^{7} + \cdots + 25\!\cdots\!04)^{2} \) Copy content Toggle raw display
$67$ \( (T^{7} + \cdots - 16\!\cdots\!36)^{2} \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 91\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( (T^{7} + \cdots + 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{7} + \cdots + 21\!\cdots\!52)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 69\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 64\!\cdots\!84 \) Copy content Toggle raw display
$97$ \( (T^{7} + \cdots + 87\!\cdots\!76)^{2} \) Copy content Toggle raw display
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