Properties

Label 121.4.c.c
Level $121$
Weight $4$
Character orbit 121.c
Analytic conductor $7.139$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [121,4,Mod(3,121)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(121, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([8]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("121.3");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 121 = 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 121.c (of order \(5\), 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: \(7.13923111069\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.324000000.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 3x^{6} + 9x^{4} + 27x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 11)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{7} + \beta_{2}) q^{2} + ( - \beta_{6} - 4 \beta_1) q^{3} + ( - 2 \beta_{7} - 2 \beta_{5} + \cdots - 2 \beta_1) q^{4}+ \cdots + (8 \beta_{7} + 22 \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{7} + \beta_{2}) q^{2} + ( - \beta_{6} - 4 \beta_1) q^{3} + ( - 2 \beta_{7} - 2 \beta_{5} + \cdots - 2 \beta_1) q^{4}+ \cdots + ( - 275 \beta_{5} - 435) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{2} + 2 q^{3} + 8 q^{4} - 2 q^{5} + 26 q^{6} - 20 q^{7} + 12 q^{8} - 44 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{2} + 2 q^{3} + 8 q^{4} - 2 q^{5} + 26 q^{6} - 20 q^{7} + 12 q^{8} - 44 q^{9} + 200 q^{10} - 160 q^{12} - 80 q^{13} + 4 q^{14} + 194 q^{15} + 8 q^{16} + 124 q^{17} - 92 q^{18} - 72 q^{19} - 88 q^{20} + 304 q^{21} - 392 q^{23} - 252 q^{24} - 136 q^{25} + 40 q^{26} + 182 q^{27} + 128 q^{28} - 144 q^{29} + 266 q^{30} + 34 q^{31} - 416 q^{32} - 208 q^{34} + 172 q^{35} + 80 q^{36} - 54 q^{37} - 432 q^{38} - 400 q^{39} + 492 q^{40} - 536 q^{41} + 140 q^{42} - 240 q^{43} + 1712 q^{45} + 314 q^{46} + 272 q^{47} - 776 q^{48} + 390 q^{49} - 232 q^{50} + 164 q^{51} + 560 q^{52} + 492 q^{53} - 440 q^{54} + 480 q^{56} + 1512 q^{57} + 192 q^{58} - 634 q^{59} - 632 q^{60} - 840 q^{61} - 134 q^{62} - 248 q^{63} - 224 q^{64} - 3520 q^{65} + 3016 q^{67} - 640 q^{68} - 962 q^{69} - 284 q^{70} + 678 q^{71} + 744 q^{72} + 400 q^{73} - 6 q^{74} + 520 q^{75} + 1728 q^{76} - 1760 q^{78} - 316 q^{79} + 1544 q^{80} + 1294 q^{81} - 512 q^{82} - 468 q^{83} + 736 q^{84} - 452 q^{85} + 156 q^{86} + 4800 q^{87} - 7368 q^{89} - 1532 q^{90} - 1280 q^{91} + 40 q^{92} + 638 q^{93} + 992 q^{94} - 2952 q^{95} + 952 q^{96} - 2194 q^{97} - 3480 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 3x^{6} + 9x^{4} + 27x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} ) / 9 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} ) / 9 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} ) / 27 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} ) / 27 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 9\beta_{4} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9\beta_{5} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 27\beta_{6} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 27\beta_{7} \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/121\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
0.535233 + 1.64728i
−0.535233 1.64728i
1.40126 1.01807i
−1.40126 + 1.01807i
1.40126 + 1.01807i
−1.40126 1.01807i
0.535233 1.64728i
−0.535233 + 1.64728i
−2.21028 + 1.60586i −2.44995 7.54017i −0.165602 + 0.509670i −12.0191 8.73238i 17.5235 + 12.7316i 0.949237 2.92145i −7.20643 22.1791i −29.0084 + 21.0759i 40.5885
3.2 0.592242 0.430289i 1.83192 + 5.63806i −2.30653 + 7.09878i 10.4011 + 7.55681i 3.51093 + 2.55084i 5.23110 16.0997i 3.49823 + 10.7664i −6.58831 + 4.78668i 9.41154
9.1 −0.226216 + 0.696222i −4.79602 + 3.48451i 6.03859 + 4.38729i −3.97285 12.2272i −1.34106 4.12734i −13.6952 9.95015i −9.15848 + 6.65403i 2.51651 7.74502i 9.41154
9.2 0.844250 2.59833i 6.41405 4.66008i 0.433551 + 0.314993i 4.59088 + 14.1293i −6.69339 20.6001i −2.48514 1.80556i 18.8667 13.7075i 11.0802 34.1015i 40.5885
27.1 −0.226216 0.696222i −4.79602 3.48451i 6.03859 4.38729i −3.97285 + 12.2272i −1.34106 + 4.12734i −13.6952 + 9.95015i −9.15848 6.65403i 2.51651 + 7.74502i 9.41154
27.2 0.844250 + 2.59833i 6.41405 + 4.66008i 0.433551 0.314993i 4.59088 14.1293i −6.69339 + 20.6001i −2.48514 + 1.80556i 18.8667 + 13.7075i 11.0802 + 34.1015i 40.5885
81.1 −2.21028 1.60586i −2.44995 + 7.54017i −0.165602 0.509670i −12.0191 + 8.73238i 17.5235 12.7316i 0.949237 + 2.92145i −7.20643 + 22.1791i −29.0084 21.0759i 40.5885
81.2 0.592242 + 0.430289i 1.83192 5.63806i −2.30653 7.09878i 10.4011 7.55681i 3.51093 2.55084i 5.23110 + 16.0997i 3.49823 10.7664i −6.58831 4.78668i 9.41154
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 121.4.c.c 8
11.b odd 2 1 121.4.c.f 8
11.c even 5 1 11.4.a.a 2
11.c even 5 3 inner 121.4.c.c 8
11.d odd 10 1 121.4.a.c 2
11.d odd 10 3 121.4.c.f 8
33.f even 10 1 1089.4.a.v 2
33.h odd 10 1 99.4.a.c 2
44.g even 10 1 1936.4.a.w 2
44.h odd 10 1 176.4.a.i 2
55.j even 10 1 275.4.a.b 2
55.k odd 20 2 275.4.b.c 4
77.j odd 10 1 539.4.a.e 2
88.l odd 10 1 704.4.a.n 2
88.o even 10 1 704.4.a.p 2
132.o even 10 1 1584.4.a.bc 2
143.n even 10 1 1859.4.a.a 2
165.o odd 10 1 2475.4.a.q 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.4.a.a 2 11.c even 5 1
99.4.a.c 2 33.h odd 10 1
121.4.a.c 2 11.d odd 10 1
121.4.c.c 8 1.a even 1 1 trivial
121.4.c.c 8 11.c even 5 3 inner
121.4.c.f 8 11.b odd 2 1
121.4.c.f 8 11.d odd 10 3
176.4.a.i 2 44.h odd 10 1
275.4.a.b 2 55.j even 10 1
275.4.b.c 4 55.k odd 20 2
539.4.a.e 2 77.j odd 10 1
704.4.a.n 2 88.l odd 10 1
704.4.a.p 2 88.o even 10 1
1089.4.a.v 2 33.f even 10 1
1584.4.a.bc 2 132.o even 10 1
1859.4.a.a 2 143.n even 10 1
1936.4.a.w 2 44.g even 10 1
2475.4.a.q 2 165.o odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 2T_{2}^{7} + 6T_{2}^{6} + 16T_{2}^{5} + 44T_{2}^{4} - 32T_{2}^{3} + 24T_{2}^{2} - 16T_{2} + 16 \) acting on \(S_{4}^{\mathrm{new}}(121, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 2 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( T^{8} - 2 T^{7} + \cdots + 4879681 \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 1330863361 \) Copy content Toggle raw display
$7$ \( T^{8} + 20 T^{7} + \cdots + 7311616 \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 25600000000 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 135530203361536 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 81\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( (T^{2} + 98 T - 1487)^{4} \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 318343244414976 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 18113272128961 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 83156680161 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 26\!\cdots\!76 \) Copy content Toggle raw display
$43$ \( (T^{2} + 60 T + 132)^{4} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 37\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 68\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 54\!\cdots\!21 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 31\!\cdots\!76 \) Copy content Toggle raw display
$67$ \( (T^{2} - 754 T + 140929)^{4} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 90\!\cdots\!01 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 14\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 25\!\cdots\!96 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 17\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( (T^{2} + 1842 T + 525489)^{4} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 16\!\cdots\!01 \) Copy content Toggle raw display
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