Properties

Label 33.4.e.c
Level $33$
Weight $4$
Character orbit 33.e
Analytic conductor $1.947$
Analytic rank $0$
Dimension $12$
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] = [33,4,Mod(4,33)]
 
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([0, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("33.4");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 33 = 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 33.e (of order \(5\), 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: \(1.94706303019\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(3\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3 x^{11} + 21 x^{10} - 26 x^{9} + 281 x^{8} + 486 x^{7} + 3506 x^{6} + 15102 x^{5} + 46669 x^{4} + 41850 x^{3} + 16292 x^{2} + 616 x + 1936 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} - 1) q^{2} - 3 \beta_{6} q^{3} + ( - \beta_{9} - \beta_{8} + 2 \beta_{5} - \beta_{3} + \beta_{2} - \beta_1 - 1) q^{4} + ( - \beta_{11} + 2 \beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} + 4 \beta_{6} - \beta_{4} + 4 \beta_{3} + \cdots + 4) q^{5}+ \cdots + ( - 9 \beta_{6} - 9 \beta_{5} - 9 \beta_{3} - 9) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} - 1) q^{2} - 3 \beta_{6} q^{3} + ( - \beta_{9} - \beta_{8} + 2 \beta_{5} - \beta_{3} + \beta_{2} - \beta_1 - 1) q^{4} + ( - \beta_{11} + 2 \beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} + 4 \beta_{6} - \beta_{4} + 4 \beta_{3} + \cdots + 4) q^{5}+ \cdots + ( - 9 \beta_{11} + 18 \beta_{10} - 36 \beta_{9} - 45 \beta_{8} + 81 \beta_{7} + \cdots + 27) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 9 q^{3} - 16 q^{4} + 28 q^{5} + 12 q^{7} - 112 q^{8} - 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 9 q^{3} - 16 q^{4} + 28 q^{5} + 12 q^{7} - 112 q^{8} - 27 q^{9} + 100 q^{10} - 54 q^{11} - 102 q^{12} - 18 q^{13} + 156 q^{14} + 111 q^{15} + 308 q^{16} - 80 q^{17} - 45 q^{18} - 280 q^{19} - 15 q^{20} - 6 q^{21} - 193 q^{22} - 392 q^{23} - 264 q^{24} + 77 q^{25} + 406 q^{26} + 81 q^{27} - 429 q^{28} + 13 q^{29} + 120 q^{30} + 413 q^{31} + 1314 q^{32} + 177 q^{33} + 1060 q^{34} - 1239 q^{35} - 144 q^{36} + 654 q^{37} + 912 q^{38} + 54 q^{39} - 1803 q^{40} - 1490 q^{41} + 342 q^{42} + 416 q^{43} + 695 q^{44} + 162 q^{45} - 2369 q^{46} - 150 q^{47} + 711 q^{48} - 301 q^{49} - 1878 q^{50} - 1661 q^{52} + 1359 q^{53} - 270 q^{54} + 3300 q^{55} - 858 q^{56} - 1110 q^{57} + 955 q^{58} + 1262 q^{59} + 45 q^{60} - 1044 q^{61} - 701 q^{62} + 108 q^{63} + 78 q^{64} + 4556 q^{65} + 369 q^{66} - 528 q^{67} + 703 q^{68} - 594 q^{69} + 3050 q^{70} + 558 q^{71} + 792 q^{72} - 699 q^{73} - 3224 q^{74} + 1284 q^{75} - 868 q^{76} + 390 q^{77} - 558 q^{78} - 1252 q^{79} - 1914 q^{80} - 243 q^{81} + 2987 q^{82} - 4464 q^{83} - 1443 q^{84} - 2170 q^{85} - 3209 q^{86} - 3474 q^{87} + 1302 q^{88} + 316 q^{89} - 90 q^{90} + 176 q^{91} + 4595 q^{92} - 1239 q^{93} + 1247 q^{94} + 1466 q^{95} + 1398 q^{96} + 1608 q^{97} + 2810 q^{98} - 171 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3 x^{11} + 21 x^{10} - 26 x^{9} + 281 x^{8} + 486 x^{7} + 3506 x^{6} + 15102 x^{5} + 46669 x^{4} + 41850 x^{3} + 16292 x^{2} + 616 x + 1936 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 69\!\cdots\!13 \nu^{11} + \cdots - 78\!\cdots\!98 ) / 18\!\cdots\!89 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 16\!\cdots\!34 \nu^{11} + \cdots + 11\!\cdots\!56 ) / 37\!\cdots\!78 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 46\!\cdots\!57 \nu^{11} + \cdots + 12\!\cdots\!44 ) / 74\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 12\!\cdots\!25 \nu^{11} + \cdots - 10\!\cdots\!24 ) / 14\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 66\!\cdots\!54 \nu^{11} + \cdots - 29\!\cdots\!64 ) / 74\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 83\!\cdots\!37 \nu^{11} + \cdots + 12\!\cdots\!00 ) / 74\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 11\!\cdots\!73 \nu^{11} + \cdots + 12\!\cdots\!84 ) / 74\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 53\!\cdots\!95 \nu^{11} + \cdots - 13\!\cdots\!12 ) / 14\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 55\!\cdots\!27 \nu^{11} + \cdots - 42\!\cdots\!12 ) / 74\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 18\!\cdots\!45 \nu^{11} + \cdots + 40\!\cdots\!04 ) / 14\!\cdots\!12 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} + \beta_{8} - \beta_{7} + 10\beta_{6} + \beta_{5} - 2\beta_{4} + \beta_{3} - \beta_{2} + 2\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{10} - 16\beta_{5} - 17\beta_{4} - 11\beta_{3} - \beta_{2} + \beta _1 - 11 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 16 \beta_{11} - 16 \beta_{10} - 16 \beta_{8} + 21 \beta_{7} - 162 \beta_{6} - 154 \beta_{5} - 32 \beta_{4} - 162 \beta_{3} - 32 \beta_{2} - 154 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -32\beta_{11} - 5\beta_{9} + 61\beta_{7} - 315\beta_{6} - 315\beta_{5} - 279\beta_{2} - 61\beta _1 - 429 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 29 \beta_{11} + 274 \beta_{10} - 29 \beta_{9} + 245 \beta_{8} + 807 \beta_{7} - 253 \beta_{6} + 807 \beta_{4} + 2805 \beta_{3} - 1249 \beta _1 - 253 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 168 \beta_{11} + 778 \beta_{10} - 778 \beta_{9} - 168 \beta_{8} + 5121 \beta_{7} - 2604 \beta_{6} + 7614 \beta_{5} + 7243 \beta_{4} + 7614 \beta_{3} + 5121 \beta_{2} - 7243 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1344 \beta_{10} - 4953 \beta_{9} - 4953 \beta_{8} + 64453 \beta_{5} + 28886 \beta_{4} + 13703 \beta_{3} + 19145 \beta_{2} - 19145 \beta _1 + 13703 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 4788 \beta_{11} + 4788 \beta_{10} - 13013 \beta_{8} - 98104 \beta_{7} + 62485 \beta_{6} + 238564 \beta_{5} + 60813 \beta_{4} + 62485 \beta_{3} + 60813 \beta_{2} + 238564 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 43012 \beta_{11} + 93316 \beta_{9} - 444424 \beta_{7} + 445776 \beta_{6} + 445776 \beta_{5} + 222753 \beta_{2} + 444424 \beta _1 + 1400254 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 401412 \beta_{11} - 129437 \beta_{10} + 401412 \beta_{9} + 271975 \beta_{8} - 1593097 \beta_{7} + 4013999 \beta_{6} - 1593097 \beta_{4} - 1538878 \beta_{3} + 3545004 \beta _1 + 4013999 \) Copy content Toggle raw display

Character values

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

\(n\) \(13\) \(23\)
\(\chi(n)\) \(\beta_{5}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
−1.92306 + 1.39719i
−0.654357 + 0.475418i
3.88644 2.82366i
−1.17727 + 3.62326i
0.0936861 0.288336i
1.27457 3.92271i
−1.92306 1.39719i
−0.654357 0.475418i
3.88644 + 2.82366i
−1.17727 3.62326i
0.0936861 + 0.288336i
1.27457 + 3.92271i
−2.73208 1.98497i −0.927051 + 2.85317i 1.05201 + 3.23775i −11.3314 + 8.23272i 8.19624 5.95492i 7.21292 + 22.1991i −4.79582 + 14.7600i −7.28115 5.29007i 47.2999
4.2 −1.46337 1.06320i −0.927051 + 2.85317i −1.46107 4.49672i 17.3626 12.6147i 4.39012 3.18961i −5.78498 17.8043i −7.11451 + 21.8962i −7.28115 5.29007i −38.8200
4.3 3.07742 + 2.23588i −0.927051 + 2.85317i 1.99923 + 6.15301i 2.08679 1.51614i −9.23226 + 6.70763i 0.454025 + 1.39734i 1.79887 5.53636i −7.28115 5.29007i 9.81184
16.1 −0.868251 2.67220i 2.42705 + 1.76336i 0.0853305 0.0619962i 3.76992 11.6026i 2.60475 8.01661i −5.42571 + 3.94201i −18.4246 13.3863i 2.78115 + 8.55951i −34.2777
16.2 0.402703 + 1.23939i 2.42705 + 1.76336i 5.09821 3.70407i −2.50364 + 7.70540i −1.20811 + 3.71818i −0.439746 + 0.319494i 15.0782 + 10.9549i 2.78115 + 8.55951i −10.5582
16.3 1.58358 + 4.87376i 2.42705 + 1.76336i −14.7737 + 10.7337i 4.61569 14.2056i −4.75075 + 14.6213i 9.98349 7.25343i −42.5421 30.9086i 2.78115 + 8.55951i 76.5442
25.1 −2.73208 + 1.98497i −0.927051 2.85317i 1.05201 3.23775i −11.3314 8.23272i 8.19624 + 5.95492i 7.21292 22.1991i −4.79582 14.7600i −7.28115 + 5.29007i 47.2999
25.2 −1.46337 + 1.06320i −0.927051 2.85317i −1.46107 + 4.49672i 17.3626 + 12.6147i 4.39012 + 3.18961i −5.78498 + 17.8043i −7.11451 21.8962i −7.28115 + 5.29007i −38.8200
25.3 3.07742 2.23588i −0.927051 2.85317i 1.99923 6.15301i 2.08679 + 1.51614i −9.23226 6.70763i 0.454025 1.39734i 1.79887 + 5.53636i −7.28115 + 5.29007i 9.81184
31.1 −0.868251 + 2.67220i 2.42705 1.76336i 0.0853305 + 0.0619962i 3.76992 + 11.6026i 2.60475 + 8.01661i −5.42571 3.94201i −18.4246 + 13.3863i 2.78115 8.55951i −34.2777
31.2 0.402703 1.23939i 2.42705 1.76336i 5.09821 + 3.70407i −2.50364 7.70540i −1.20811 3.71818i −0.439746 0.319494i 15.0782 10.9549i 2.78115 8.55951i −10.5582
31.3 1.58358 4.87376i 2.42705 1.76336i −14.7737 10.7337i 4.61569 + 14.2056i −4.75075 14.6213i 9.98349 + 7.25343i −42.5421 + 30.9086i 2.78115 8.55951i 76.5442
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 33.4.e.c 12
3.b odd 2 1 99.4.f.d 12
11.c even 5 1 inner 33.4.e.c 12
11.c even 5 1 363.4.a.v 6
11.d odd 10 1 363.4.a.u 6
33.f even 10 1 1089.4.a.bk 6
33.h odd 10 1 99.4.f.d 12
33.h odd 10 1 1089.4.a.bi 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.4.e.c 12 1.a even 1 1 trivial
33.4.e.c 12 11.c even 5 1 inner
99.4.f.d 12 3.b odd 2 1
99.4.f.d 12 33.h odd 10 1
363.4.a.u 6 11.d odd 10 1
363.4.a.v 6 11.c even 5 1
1089.4.a.bi 6 33.h odd 10 1
1089.4.a.bk 6 33.f even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 20 T_{2}^{10} + 64 T_{2}^{9} + 225 T_{2}^{8} + 70 T_{2}^{7} + 3191 T_{2}^{6} + 11905 T_{2}^{5} + 53975 T_{2}^{4} + 92394 T_{2}^{3} + 119980 T_{2}^{2} + 109000 T_{2} + 190096 \) acting on \(S_{4}^{\mathrm{new}}(33, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 20 T^{10} + 64 T^{9} + \cdots + 190096 \) Copy content Toggle raw display
$3$ \( (T^{4} - 3 T^{3} + 9 T^{2} - 27 T + 81)^{3} \) Copy content Toggle raw display
$5$ \( T^{12} - 28 T^{11} + \cdots + 1310363273521 \) Copy content Toggle raw display
$7$ \( T^{12} - 12 T^{11} + \cdots + 834112161 \) Copy content Toggle raw display
$11$ \( T^{12} + 54 T^{11} + \cdots + 55\!\cdots\!81 \) Copy content Toggle raw display
$13$ \( T^{12} + 18 T^{11} + \cdots + 49\!\cdots\!36 \) Copy content Toggle raw display
$17$ \( T^{12} + 80 T^{11} + \cdots + 25\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( T^{12} + 280 T^{11} + \cdots + 47\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T^{6} + 196 T^{5} - 22204 T^{4} + \cdots - 3716146836)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} - 13 T^{11} + \cdots + 95\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{12} - 413 T^{11} + \cdots + 13\!\cdots\!61 \) Copy content Toggle raw display
$37$ \( T^{12} - 654 T^{11} + \cdots + 18\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( T^{12} + 1490 T^{11} + \cdots + 40\!\cdots\!96 \) Copy content Toggle raw display
$43$ \( (T^{6} - 208 T^{5} + \cdots + 92003355120644)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + 150 T^{11} + \cdots + 61\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( T^{12} - 1359 T^{11} + \cdots + 35\!\cdots\!41 \) Copy content Toggle raw display
$59$ \( T^{12} - 1262 T^{11} + \cdots + 22\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{12} + 1044 T^{11} + \cdots + 24\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( (T^{6} + 264 T^{5} + \cdots - 66\!\cdots\!84)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} - 558 T^{11} + \cdots + 12\!\cdots\!96 \) Copy content Toggle raw display
$73$ \( T^{12} + 699 T^{11} + \cdots + 16\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( T^{12} + 1252 T^{11} + \cdots + 32\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( T^{12} + 4464 T^{11} + \cdots + 18\!\cdots\!81 \) Copy content Toggle raw display
$89$ \( (T^{6} - 158 T^{5} + \cdots - 28\!\cdots\!20)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} - 1608 T^{11} + \cdots + 11\!\cdots\!21 \) Copy content Toggle raw display
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