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Properties

Label 363.3.h.g
Level $363$
Weight $3$
Character orbit 363.h
Analytic conductor $9.891$
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] = [363,3,Mod(245,363)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(363, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 8]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("363.245");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 363.h (of order \(10\), degree \(4\), not 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: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{20})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-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 primitive root of unity \(\zeta_{20}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{20} q^{2} + ( - 2 \zeta_{20}^{6} - 2 \zeta_{20}^{5} + \cdots + 2) q^{3}+ \cdots + (8 \zeta_{20}^{7} + \cdots - 4 \zeta_{20}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{20} q^{2} + ( - 2 \zeta_{20}^{6} - 2 \zeta_{20}^{5} + \cdots + 2) q^{3}+ \cdots + ( - 33 \zeta_{20}^{7} + \cdots - 33 \zeta_{20}^{3}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{3} - 6 q^{4} - 10 q^{6} + 4 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{3} - 6 q^{4} - 10 q^{6} + 4 q^{7} + 2 q^{9} + 44 q^{10} - 48 q^{12} - 32 q^{13} - 50 q^{15} - 10 q^{16} - 40 q^{18} - 112 q^{19} - 88 q^{21} - 70 q^{24} - 42 q^{25} - 44 q^{27} - 78 q^{28} + 12 q^{30} - 18 q^{31} + 48 q^{34} + 6 q^{36} - 120 q^{37} + 64 q^{39} - 42 q^{40} - 70 q^{42} + 264 q^{43} + 80 q^{45} - 24 q^{46} + 20 q^{48} - 150 q^{49} - 60 q^{51} - 36 q^{52} - 136 q^{57} + 186 q^{58} + 180 q^{60} - 336 q^{61} - 4 q^{63} + 26 q^{64} - 24 q^{67} - 240 q^{69} + 42 q^{70} + 280 q^{72} + 182 q^{73} - 136 q^{75} + 264 q^{76} + 40 q^{78} + 460 q^{79} + 158 q^{81} - 72 q^{82} - 24 q^{84} + 36 q^{85} - 660 q^{87} + 16 q^{90} + 84 q^{91} + 36 q^{93} - 52 q^{94} + 330 q^{96} + 516 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(122\) \(244\)
\(\chi(n)\) \(-1\) \(-\zeta_{20}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
245.1
−0.587785 − 0.809017i
0.587785 + 0.809017i
−0.951057 − 0.309017i
0.951057 + 0.309017i
−0.951057 + 0.309017i
0.951057 − 0.309017i
−0.587785 + 0.809017i
0.587785 − 0.809017i
−0.587785 − 0.809017i 1.50859 − 2.59310i 0.927051 − 2.85317i −3.88998 + 5.35410i −2.98459 + 0.303706i 2.73607 − 8.42075i −6.65740 + 2.16312i −4.44829 − 7.82385i 6.61803
245.2 0.587785 + 0.809017i −2.74466 − 1.21113i 0.927051 − 2.85317i 3.88998 − 5.35410i −0.633446 − 2.93236i 2.73607 − 8.42075i 6.65740 − 2.16312i 6.06633 + 6.64828i 6.61803
251.1 −0.951057 − 0.309017i 2.93236 + 0.633446i −2.42705 − 1.76336i −4.16750 + 1.35410i −2.59310 − 1.50859i −1.73607 − 1.26133i 4.11450 + 5.66312i 8.19749 + 3.71499i 4.38197
251.2 0.951057 + 0.309017i 0.303706 − 2.98459i −2.42705 − 1.76336i 4.16750 − 1.35410i 1.21113 − 2.74466i −1.73607 − 1.26133i −4.11450 − 5.66312i −8.81553 − 1.81288i 4.38197
269.1 −0.951057 + 0.309017i 2.93236 − 0.633446i −2.42705 + 1.76336i −4.16750 − 1.35410i −2.59310 + 1.50859i −1.73607 + 1.26133i 4.11450 − 5.66312i 8.19749 − 3.71499i 4.38197
269.2 0.951057 − 0.309017i 0.303706 + 2.98459i −2.42705 + 1.76336i 4.16750 + 1.35410i 1.21113 + 2.74466i −1.73607 + 1.26133i −4.11450 + 5.66312i −8.81553 + 1.81288i 4.38197
323.1 −0.587785 + 0.809017i 1.50859 + 2.59310i 0.927051 + 2.85317i −3.88998 − 5.35410i −2.98459 − 0.303706i 2.73607 + 8.42075i −6.65740 − 2.16312i −4.44829 + 7.82385i 6.61803
323.2 0.587785 − 0.809017i −2.74466 + 1.21113i 0.927051 + 2.85317i 3.88998 + 5.35410i −0.633446 + 2.93236i 2.73607 + 8.42075i 6.65740 + 2.16312i 6.06633 − 6.64828i 6.61803
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 245.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.c even 5 1 inner
33.h odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 363.3.h.g 8
3.b odd 2 1 inner 363.3.h.g 8
11.b odd 2 1 33.3.h.a ✓ 8
11.c even 5 1 363.3.b.g 4
11.c even 5 1 inner 363.3.h.g 8
11.c even 5 2 363.3.h.i 8
11.d odd 10 1 33.3.h.a ✓ 8
11.d odd 10 1 363.3.b.f 4
11.d odd 10 2 363.3.h.h 8
33.d even 2 1 33.3.h.a ✓ 8
33.f even 10 1 33.3.h.a ✓ 8
33.f even 10 1 363.3.b.f 4
33.f even 10 2 363.3.h.h 8
33.h odd 10 1 363.3.b.g 4
33.h odd 10 1 inner 363.3.h.g 8
33.h odd 10 2 363.3.h.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.3.h.a ✓ 8 11.b odd 2 1
33.3.h.a ✓ 8 11.d odd 10 1
33.3.h.a ✓ 8 33.d even 2 1
33.3.h.a ✓ 8 33.f even 10 1
363.3.b.f 4 11.d odd 10 1
363.3.b.f 4 33.f even 10 1
363.3.b.g 4 11.c even 5 1
363.3.b.g 4 33.h odd 10 1
363.3.h.g 8 1.a even 1 1 trivial
363.3.h.g 8 3.b odd 2 1 inner
363.3.h.g 8 11.c even 5 1 inner
363.3.h.g 8 33.h odd 10 1 inner
363.3.h.h 8 11.d odd 10 2
363.3.h.h 8 33.f even 10 2
363.3.h.i 8 11.c even 5 2
363.3.h.i 8 33.h odd 10 2

Hecke kernels

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

\( T_{2}^{8} - T_{2}^{6} + T_{2}^{4} - T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{5}^{8} - 4T_{5}^{6} + 1446T_{5}^{4} - 49619T_{5}^{2} + 707281 \) Copy content Toggle raw display
\( T_{7}^{4} - 2T_{7}^{3} + 64T_{7}^{2} + 247T_{7} + 361 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - T^{6} + T^{4} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{8} - 4 T^{7} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( T^{8} - 4 T^{6} + \cdots + 707281 \) Copy content Toggle raw display
$7$ \( (T^{4} - 2 T^{3} + \cdots + 361)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( (T^{4} + 16 T^{3} + \cdots + 13456)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} - 36 T^{6} + \cdots + 1679616 \) Copy content Toggle raw display
$19$ \( (T^{4} + 56 T^{3} + \cdots + 80656)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 1008 T^{2} + 215296)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 2449228130001 \) Copy content Toggle raw display
$31$ \( (T^{4} + 9 T^{3} + \cdots + 36481)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 60 T^{3} + \cdots + 336400)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 3544535296 \) Copy content Toggle raw display
$43$ \( (T^{2} - 66 T + 1044)^{4} \) Copy content Toggle raw display
$47$ \( T^{8} - 496 T^{6} + \cdots + 33362176 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 3969126001 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 276281640625 \) Copy content Toggle raw display
$61$ \( (T^{4} + 168 T^{3} + \cdots + 6533136)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 6 T - 5436)^{4} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 11\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( (T^{4} - 91 T^{3} + \cdots + 866761)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 230 T^{3} + \cdots + 55428025)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 22\!\cdots\!01 \) Copy content Toggle raw display
$89$ \( (T^{4} + 9632 T^{2} + 891136)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 258 T^{3} + \cdots + 147403881)^{2} \) Copy content Toggle raw display
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