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Properties

Label 33.7.c.a
Level $33$
Weight $7$
Character orbit 33.c
Analytic conductor $7.592$
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] = [33,7,Mod(10,33)]
 
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([0, 1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("33.10");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 33 = 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 33.c (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: \(7.59178475946\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 486x^{10} + 82401x^{8} + 6062364x^{6} + 204706260x^{4} + 2964086784x^{2} + 15081209856 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{11} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{2} q^{3} + (\beta_{3} + \beta_{2} - 17) q^{4} + (\beta_{4} - \beta_{3} + 19) q^{5} + ( - \beta_{5} - 2 \beta_1) q^{6} + (\beta_{6} - \beta_{5} + 7 \beta_1) q^{7} + (\beta_{8} - \beta_{6} + \cdots - 19 \beta_1) q^{8}+ \cdots + 243 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{2} q^{3} + (\beta_{3} + \beta_{2} - 17) q^{4} + (\beta_{4} - \beta_{3} + 19) q^{5} + ( - \beta_{5} - 2 \beta_1) q^{6} + (\beta_{6} - \beta_{5} + 7 \beta_1) q^{7} + (\beta_{8} - \beta_{6} + \cdots - 19 \beta_1) q^{8}+ \cdots + (486 \beta_{10} - 243 \beta_{9} + \cdots - 69741) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 204 q^{4} + 224 q^{5} + 2916 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 204 q^{4} + 224 q^{5} + 2916 q^{9} - 3464 q^{11} + 1944 q^{12} - 6708 q^{14} + 1944 q^{15} + 5316 q^{16} - 44092 q^{20} + 60468 q^{22} - 15304 q^{23} + 95652 q^{25} - 76020 q^{26} - 58608 q^{31} + 4212 q^{33} + 117768 q^{34} - 49572 q^{36} - 202512 q^{37} + 29208 q^{38} - 264708 q^{42} + 434356 q^{44} + 54432 q^{45} + 516920 q^{47} - 377136 q^{48} + 157812 q^{49} - 1042192 q^{53} + 262656 q^{55} + 463020 q^{56} + 1029432 q^{58} - 461008 q^{59} - 417636 q^{60} - 725364 q^{64} + 200232 q^{66} + 364752 q^{67} + 504144 q^{69} - 1028400 q^{70} - 755176 q^{71} + 1364688 q^{75} - 102384 q^{77} + 1219212 q^{78} - 1220764 q^{80} + 708588 q^{81} - 158688 q^{82} + 248760 q^{86} - 2493252 q^{88} - 3513544 q^{89} - 702768 q^{91} + 6899300 q^{92} + 789264 q^{93} + 2370192 q^{97} - 841752 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 486x^{10} + 82401x^{8} + 6062364x^{6} + 204706260x^{4} + 2964086784x^{2} + 15081209856 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 117\nu^{10} + 56806\nu^{8} + 9541621\nu^{6} + 673291092\nu^{4} + 19439761332\nu^{2} + 162772955136 ) / 269967872 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -117\nu^{10} - 56806\nu^{8} - 9541621\nu^{6} - 673291092\nu^{4} - 19169793460\nu^{2} - 140905557504 ) / 269967872 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 40281 \nu^{10} + 18394564 \nu^{8} + 2794934961 \nu^{6} + 168462043098 \nu^{4} + \cdots + 30842537263488 ) / 7896560256 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 117 \nu^{11} - 56806 \nu^{9} - 9541621 \nu^{7} - 673291092 \nu^{5} - 19439761332 \nu^{3} - 163312890880 \nu ) / 269967872 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 139765 \nu^{11} - 65082782 \nu^{9} - 10255266645 \nu^{7} - 665103431244 \nu^{5} + \cdots - 183148581464064 \nu ) / 252689928192 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 501165 \nu^{10} + 235863262 \nu^{8} + 37684766925 \nu^{6} + 2467300644060 \nu^{4} + \cdots + 534076796284416 ) / 31586241024 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 358789 \nu^{11} - 171423614 \nu^{9} - 28117181157 \nu^{7} - 1925504355468 \nu^{5} + \cdots - 451724893747200 \nu ) / 252689928192 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 878169 \nu^{11} + 5125328 \nu^{10} - 409604982 \nu^{9} + 2413465184 \nu^{8} + \cdots + 50\!\cdots\!88 ) / 505379856384 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 292723 \nu^{11} + 136534994 \nu^{9} + 21503241747 \nu^{7} + 1376699073300 \nu^{5} + \cdots + 286275613125120 \nu ) / 84229976064 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1893075 \nu^{11} - 906040978 \nu^{9} - 148746043251 \nu^{7} - 10150508591124 \nu^{5} + \cdots - 23\!\cdots\!56 \nu ) / 252689928192 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} - 81 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} - \beta_{6} - 2\beta_{5} - 147\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -3\beta_{10} - 6\beta_{9} + 10\beta_{7} - 10\beta_{4} - 215\beta_{3} - 323\beta_{2} + 11901 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -18\beta_{11} - 22\beta_{10} - 239\beta_{8} + 191\beta_{6} + 674\beta_{5} + 26443\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 747\beta_{10} + 1494\beta_{9} - 3264\beta_{7} + 3732\beta_{4} + 42939\beta_{3} + 83549\beta_{2} - 2139957 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 7272\beta_{11} + 7656\beta_{10} + 48915\beta_{8} - 36963\beta_{6} - 177398\beta_{5} - 5117275\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 151425 \beta_{10} - 302850 \beta_{9} + 830982 \beta_{7} - 1035030 \beta_{4} - 8632887 \beta_{3} + \cdots + 413976969 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 2091294 \beta_{11} - 2048826 \beta_{10} - 9844287 \beta_{8} + 7421487 \beta_{6} + \cdots + 1024993047 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 29864367 \beta_{10} + 59728734 \beta_{9} - 194819004 \beta_{7} + 255722184 \beta_{4} + \cdots - 82894928409 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 525902724 \beta_{11} + 496986012 \beta_{10} + 1999683639 \beta_{8} - 1521853767 \beta_{6} + \cdots - 209471459367 \beta_1 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1
− 14.5918i
− 11.9419i
− 7.49503i
− 6.83773i
− 3.80817i
− 3.61107i
3.61107i
3.80817i
6.83773i
7.49503i
11.9419i
14.5918i
− 14.5918i −15.5885 −148.920 0.687766 227.463i 18.0081i 1239.14i 243.000 − 10.0357i
10.2 − 11.9419i 15.5885 −78.6078 234.218 − 186.155i − 368.050i 174.445i 243.000 − 2797.00i
10.3 − 7.49503i 15.5885 7.82458 −189.722 − 116.836i − 392.912i − 538.327i 243.000 1421.97i
10.4 − 6.83773i −15.5885 17.2454 167.968 106.590i 106.676i − 555.534i 243.000 − 1148.52i
10.5 − 3.80817i −15.5885 49.4978 −143.833 59.3636i 413.864i − 432.219i 243.000 547.741i
10.6 − 3.61107i 15.5885 50.9602 42.6809 − 56.2910i 392.632i − 415.129i 243.000 − 154.124i
10.7 3.61107i 15.5885 50.9602 42.6809 56.2910i − 392.632i 415.129i 243.000 154.124i
10.8 3.80817i −15.5885 49.4978 −143.833 − 59.3636i − 413.864i 432.219i 243.000 − 547.741i
10.9 6.83773i −15.5885 17.2454 167.968 − 106.590i − 106.676i 555.534i 243.000 1148.52i
10.10 7.49503i 15.5885 7.82458 −189.722 116.836i 392.912i 538.327i 243.000 − 1421.97i
10.11 11.9419i 15.5885 −78.6078 234.218 186.155i 368.050i − 174.445i 243.000 2797.00i
10.12 14.5918i −15.5885 −148.920 0.687766 − 227.463i − 18.0081i − 1239.14i 243.000 10.0357i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 33.7.c.a ✓ 12
3.b odd 2 1 99.7.c.d 12
4.b odd 2 1 528.7.j.c 12
11.b odd 2 1 inner 33.7.c.a ✓ 12
33.d even 2 1 99.7.c.d 12
44.c even 2 1 528.7.j.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.7.c.a ✓ 12 1.a even 1 1 trivial
33.7.c.a ✓ 12 11.b odd 2 1 inner
99.7.c.d 12 3.b odd 2 1
99.7.c.d 12 33.d even 2 1
528.7.j.c 12 4.b odd 2 1
528.7.j.c 12 44.c even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(33, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + \cdots + 15081209856 \) Copy content Toggle raw display
$3$ \( (T^{2} - 243)^{6} \) Copy content Toggle raw display
$5$ \( (T^{6} - 112 T^{5} + \cdots + 31513690000)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 20\!\cdots\!04 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 30\!\cdots\!61 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 10\!\cdots\!24 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 11\!\cdots\!36 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 11\!\cdots\!36 \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots + 11\!\cdots\!88)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 86\!\cdots\!24 \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots + 52\!\cdots\!52)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 26\!\cdots\!68)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 76\!\cdots\!76 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 61\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots - 25\!\cdots\!96)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots - 89\!\cdots\!12)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots + 59\!\cdots\!16)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 30\!\cdots\!44 \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 95\!\cdots\!32)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 79\!\cdots\!76)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 25\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 17\!\cdots\!36 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 46\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 48\!\cdots\!64)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 71\!\cdots\!28)^{2} \) Copy content Toggle raw display
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