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Properties

Label 363.3.b.n
Level $363$
Weight $3$
Character orbit 363.b
Analytic conductor $9.891$
Analytic rank $0$
Dimension $12$
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] = [363,3,Mod(122,363)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(363, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("363.122");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 363.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: \(9.89103359628\)
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: \( x^{12} - 4 x^{11} + 10 x^{10} - 12 x^{9} + 290 x^{8} + 580 x^{7} + 3656 x^{6} + 5424 x^{5} + \cdots + 48312 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + \beta_{4} q^{3} + ( - \beta_{7} - \beta_{4} + \beta_{2} - 4) q^{4} + ( - \beta_{11} - \beta_{8} + \beta_{7} + \cdots - 1) q^{5}+ \cdots + ( - \beta_{8} - 2 \beta_{7} - \beta_{4} + \cdots - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + \beta_{4} q^{3} + ( - \beta_{7} - \beta_{4} + \beta_{2} - 4) q^{4} + ( - \beta_{11} - \beta_{8} + \beta_{7} + \cdots - 1) q^{5}+ \cdots + ( - 3 \beta_{10} - 3 \beta_{9} + \cdots - \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 4 q^{3} - 44 q^{4} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 4 q^{3} - 44 q^{4} - 12 q^{9} - 80 q^{12} + 68 q^{15} + 92 q^{16} - 88 q^{25} - 232 q^{27} - 8 q^{31} + 116 q^{34} + 164 q^{36} - 244 q^{37} + 404 q^{42} - 52 q^{45} + 540 q^{48} + 100 q^{49} - 460 q^{58} + 24 q^{60} - 1276 q^{64} - 128 q^{67} + 128 q^{69} - 784 q^{70} + 684 q^{75} + 528 q^{78} + 348 q^{81} - 380 q^{82} + 120 q^{91} + 196 q^{93} - 156 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} + 10 x^{10} - 12 x^{9} + 290 x^{8} + 580 x^{7} + 3656 x^{6} + 5424 x^{5} + \cdots + 48312 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 26448881484039 \nu^{11} - 417791761524872 \nu^{10} + \cdots + 66\!\cdots\!08 ) / 33\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 11927197578380 \nu^{11} + 5134113130431 \nu^{10} + 150881956031212 \nu^{9} + \cdots - 20\!\cdots\!52 ) / 11\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 88\!\cdots\!85 \nu^{11} + \cdots + 67\!\cdots\!56 ) / 80\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 48\!\cdots\!23 \nu^{11} + \cdots - 70\!\cdots\!84 ) / 29\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 17\!\cdots\!19 \nu^{11} + \cdots + 64\!\cdots\!76 ) / 88\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 18\!\cdots\!49 \nu^{11} + \cdots - 78\!\cdots\!08 ) / 80\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 77\!\cdots\!04 \nu^{11} + \cdots - 75\!\cdots\!40 ) / 29\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 91\!\cdots\!63 \nu^{11} + \cdots + 27\!\cdots\!16 ) / 29\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 5455461050 \nu^{11} + 18803987505 \nu^{10} - 36561278785 \nu^{9} + \cdots + 54834547813980 ) / 171019613016516 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 56\!\cdots\!91 \nu^{11} + \cdots - 11\!\cdots\!20 ) / 88\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 53\!\cdots\!32 \nu^{11} + \cdots + 66\!\cdots\!36 ) / 29\!\cdots\!08 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{10} - \beta_{9} + \beta_{7} - \beta_{4} + \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} - \beta_{10} - 2\beta_{9} - 2\beta_{8} + \beta_{7} + 2\beta_{6} - 2\beta_{4} + 2\beta_{3} - \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3 \beta_{11} + 4 \beta_{9} - 6 \beta_{8} + 9 \beta_{7} + 5 \beta_{6} + 3 \beta_{5} - 7 \beta_{4} + \cdots - 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 14 \beta_{11} + 34 \beta_{10} + 60 \beta_{9} - 24 \beta_{8} + 26 \beta_{7} - 14 \beta_{6} + 8 \beta_{5} + \cdots - 120 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 33 \beta_{11} + 277 \beta_{10} + 546 \beta_{9} - 67 \beta_{8} + 10 \beta_{7} - 131 \beta_{6} + \cdots - 876 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 246 \beta_{11} + 1390 \beta_{10} + 2686 \beta_{9} + 470 \beta_{8} - 826 \beta_{7} - 1084 \beta_{6} + \cdots - 4516 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 2852 \beta_{11} + 4952 \beta_{10} + 8728 \beta_{9} + 5182 \beta_{8} - 8026 \beta_{7} - 6044 \beta_{6} + \cdots - 14538 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 18376 \beta_{11} + 5128 \beta_{10} + 6216 \beta_{9} + 33444 \beta_{8} - 43944 \beta_{7} - 20576 \beta_{6} + \cdots + 9292 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 83244 \beta_{11} - 106266 \beta_{10} - 223510 \beta_{9} + 152856 \beta_{8} - 167646 \beta_{7} + \cdots + 461822 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 205132 \beta_{11} - 1096120 \beta_{10} - 2171256 \beta_{9} + 373152 \beta_{8} - 245160 \beta_{7} + \cdots + 3889544 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 537664 \beta_{11} - 6702328 \beta_{10} - 12815952 \beta_{9} - 983848 \beta_{8} + 2755380 \beta_{7} + \cdots + 21809804 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/363\mathbb{Z}\right)^\times\).

\(n\) \(122\) \(244\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
122.1
−0.156151 + 2.44140i
−2.05018 − 2.44140i
0.293152 + 2.99036i
4.48261 − 2.99036i
−1.59440 + 0.311973i
1.02498 − 0.311973i
1.02498 + 0.311973i
−1.59440 − 0.311973i
4.48261 + 2.99036i
0.293152 − 2.99036i
−2.05018 + 2.44140i
−0.156151 − 2.44140i
− 3.83317i 1.74344 − 2.44140i −10.6932 − 0.620885i −9.35829 − 6.68291i −1.92745 25.6562i −2.92083 − 8.51286i −2.37996
122.2 − 3.83317i 1.74344 + 2.44140i −10.6932 0.620885i 9.35829 − 6.68291i 1.92745 25.6562i −2.92083 + 8.51286i 2.37996
122.3 − 2.16906i 0.240294 − 2.99036i −0.704830 5.33026i −6.48628 − 0.521212i −12.4613 − 7.14743i −8.88452 − 1.43713i 11.5617
122.4 − 2.16906i 0.240294 + 2.99036i −0.704830 − 5.33026i 6.48628 − 0.521212i 12.4613 − 7.14743i −8.88452 + 1.43713i −11.5617
122.5 − 1.89788i −2.98373 − 0.311973i 0.398042 8.25850i −0.592088 + 5.66278i −3.60566 − 8.34697i 8.80535 + 1.86169i 15.6737
122.6 − 1.89788i −2.98373 + 0.311973i 0.398042 − 8.25850i 0.592088 + 5.66278i 3.60566 − 8.34697i 8.80535 − 1.86169i −15.6737
122.7 1.89788i −2.98373 − 0.311973i 0.398042 8.25850i 0.592088 − 5.66278i 3.60566 8.34697i 8.80535 + 1.86169i −15.6737
122.8 1.89788i −2.98373 + 0.311973i 0.398042 − 8.25850i −0.592088 − 5.66278i −3.60566 8.34697i 8.80535 − 1.86169i 15.6737
122.9 2.16906i 0.240294 − 2.99036i −0.704830 5.33026i 6.48628 + 0.521212i 12.4613 7.14743i −8.88452 − 1.43713i −11.5617
122.10 2.16906i 0.240294 + 2.99036i −0.704830 − 5.33026i −6.48628 + 0.521212i −12.4613 7.14743i −8.88452 + 1.43713i 11.5617
122.11 3.83317i 1.74344 − 2.44140i −10.6932 − 0.620885i 9.35829 + 6.68291i 1.92745 − 25.6562i −2.92083 − 8.51286i 2.37996
122.12 3.83317i 1.74344 + 2.44140i −10.6932 0.620885i −9.35829 + 6.68291i −1.92745 − 25.6562i −2.92083 + 8.51286i −2.37996
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 122.12
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 363.3.b.n ✓ 12
3.b odd 2 1 inner 363.3.b.n ✓ 12
11.b odd 2 1 inner 363.3.b.n ✓ 12
11.c even 5 4 363.3.h.r 48
11.d odd 10 4 363.3.h.r 48
33.d even 2 1 inner 363.3.b.n ✓ 12
33.f even 10 4 363.3.h.r 48
33.h odd 10 4 363.3.h.r 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.3.b.n ✓ 12 1.a even 1 1 trivial
363.3.b.n ✓ 12 3.b odd 2 1 inner
363.3.b.n ✓ 12 11.b odd 2 1 inner
363.3.b.n ✓ 12 33.d even 2 1 inner
363.3.h.r 48 11.c even 5 4
363.3.h.r 48 11.d odd 10 4
363.3.h.r 48 33.f even 10 4
363.3.h.r 48 33.h odd 10 4

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(363, [\chi])\):

\( T_{2}^{6} + 23T_{2}^{4} + 139T_{2}^{2} + 249 \) Copy content Toggle raw display
\( T_{5}^{6} + 97T_{5}^{4} + 1975T_{5}^{2} + 747 \) Copy content Toggle raw display
\( T_{7}^{6} - 172T_{7}^{4} + 2644T_{7}^{2} - 7500 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} + 23 T^{4} + \cdots + 249)^{2} \) Copy content Toggle raw display
$3$ \( (T^{6} + 2 T^{5} + \cdots + 729)^{2} \) Copy content Toggle raw display
$5$ \( (T^{6} + 97 T^{4} + \cdots + 747)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} - 172 T^{4} + \cdots - 7500)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} \) Copy content Toggle raw display
$13$ \( (T^{6} - 441 T^{4} + \cdots - 648675)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + 971 T^{4} + \cdots + 14462169)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} - 916 T^{4} + \cdots - 20750700)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + 1384 T^{4} + \cdots + 43747308)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 575 T^{4} + \cdots + 3890625)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + 2 T^{2} + \cdots - 2118)^{4} \) Copy content Toggle raw display
$37$ \( (T^{3} + 61 T^{2} + \cdots + 4085)^{4} \) Copy content Toggle raw display
$41$ \( (T^{6} + 2579 T^{4} + \cdots + 579080625)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} - 8248 T^{4} + \cdots - 17046433200)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + 5728 T^{4} + \cdots + 2578766508)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 7065 T^{4} + \cdots + 3398617683)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + 6876 T^{4} + \cdots + 5445630000)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} - 6436 T^{4} + \cdots - 3248019648)^{2} \) Copy content Toggle raw display
$67$ \( (T^{3} + 32 T^{2} + \cdots - 186570)^{4} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 2030778961200)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} - 19084 T^{4} + \cdots - 31864090800)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} - 9492 T^{4} + \cdots - 21498899148)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + 17072 T^{4} + \cdots + 128420280900)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 1493575125075)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} + 39 T^{2} + \cdots - 85235)^{4} \) Copy content Toggle raw display
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