Properties

Label 121.5.d.c
Level $121$
Weight $5$
Character orbit 121.d
Analytic conductor $12.508$
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] = [121,5,Mod(40,121)]
 
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([7]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("121.40");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 121 = 11^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 121.d (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: \(12.5077655331\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(3\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 115x^{10} + 5030x^{8} + 102975x^{6} + 953170x^{4} + 2910655x^{2} + 73205 \) 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: $\mathrm{SU}(2)[C_{10}]$

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

\(f(q)\) \(=\) \( q + (\beta_{8} - \beta_{7} - \beta_{6} + \beta_{3} + \beta_1 - 1) q^{2} + (\beta_{11} - \beta_{9} + 2 \beta_{4} - \beta_{2} + 2) q^{3} + ( - \beta_{9} - \beta_{8} + 2 \beta_{7} + \beta_{6} + \beta_{5} - 2 \beta_{4} - 7 \beta_{3} - 2 \beta_{2} - \beta_1) q^{4} + (\beta_{11} - \beta_{9} - \beta_{8} + 2 \beta_{7} + 2 \beta_{6} - \beta_{5} + 4 \beta_{2} + 4) q^{5} + ( - 3 \beta_{11} - 2 \beta_{10} + 2 \beta_{9} + 6 \beta_{8} - 6 \beta_{7} - \beta_{5} + \cdots - 7) q^{6}+ \cdots + ( - \beta_{11} - 4 \beta_{10} + 6 \beta_{8} + 6 \beta_{7} - 9 \beta_{6} + 31 \beta_{4} + \cdots - 7) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{8} - \beta_{7} - \beta_{6} + \beta_{3} + \beta_1 - 1) q^{2} + (\beta_{11} - \beta_{9} + 2 \beta_{4} - \beta_{2} + 2) q^{3} + ( - \beta_{9} - \beta_{8} + 2 \beta_{7} + \beta_{6} + \beta_{5} - 2 \beta_{4} - 7 \beta_{3} - 2 \beta_{2} - \beta_1) q^{4} + (\beta_{11} - \beta_{9} - \beta_{8} + 2 \beta_{7} + 2 \beta_{6} - \beta_{5} + 4 \beta_{2} + 4) q^{5} + ( - 3 \beta_{11} - 2 \beta_{10} + 2 \beta_{9} + 6 \beta_{8} - 6 \beta_{7} - \beta_{5} + \cdots - 7) q^{6}+ \cdots + ( - 92 \beta_{11} - 313 \beta_{10} + 313 \beta_{9} + 39 \beta_{7} + \cdots + 630) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 15 q^{2} + 19 q^{3} + 27 q^{4} + 32 q^{5} - 80 q^{6} + 200 q^{7} - 145 q^{8} - 244 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 15 q^{2} + 19 q^{3} + 27 q^{4} + 32 q^{5} - 80 q^{6} + 200 q^{7} - 145 q^{8} - 244 q^{9} + 594 q^{12} - 250 q^{13} - 40 q^{14} - 506 q^{15} + 1167 q^{16} + 405 q^{17} + 1685 q^{18} + 530 q^{19} - 1068 q^{20} + 1684 q^{23} - 3160 q^{24} + 1707 q^{25} - 4080 q^{26} + 2563 q^{27} + 2170 q^{28} + 4230 q^{29} + 4200 q^{30} + 104 q^{31} + 2370 q^{34} - 910 q^{35} + 4761 q^{36} + 454 q^{37} + 2105 q^{38} + 1180 q^{39} + 3180 q^{40} - 5385 q^{41} + 2690 q^{42} + 5136 q^{45} + 1000 q^{46} - 4516 q^{47} + 1099 q^{48} - 4183 q^{49} - 2315 q^{50} - 3335 q^{51} + 9000 q^{52} - 4726 q^{53} - 21340 q^{56} + 9275 q^{57} - 9770 q^{58} + 11624 q^{59} - 5156 q^{60} - 5780 q^{61} - 28710 q^{62} - 20980 q^{63} + 6987 q^{64} + 12154 q^{67} + 1855 q^{68} + 268 q^{69} - 3340 q^{70} - 21646 q^{71} - 610 q^{72} + 1015 q^{73} + 1730 q^{74} - 9546 q^{75} - 42920 q^{78} + 15130 q^{79} - 12868 q^{80} + 27880 q^{81} + 13500 q^{82} - 46770 q^{83} - 21130 q^{84} - 25740 q^{85} + 31950 q^{86} + 5554 q^{89} - 5640 q^{90} - 850 q^{91} - 10246 q^{92} - 38832 q^{93} + 76630 q^{94} - 23390 q^{95} + 54645 q^{96} - 16546 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 115x^{10} + 5030x^{8} + 102975x^{6} + 953170x^{4} + 2910655x^{2} + 73205 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 13052 \nu^{11} - 117304 \nu^{10} - 1219776 \nu^{9} - 10477907 \nu^{8} - 40135806 \nu^{7} - 321264966 \nu^{6} - 549385674 \nu^{5} + \cdots - 1427754383 ) / 1339827192 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 13052 \nu^{11} - 117304 \nu^{10} + 1219776 \nu^{9} - 10477907 \nu^{8} + 40135806 \nu^{7} - 321264966 \nu^{6} + 549385674 \nu^{5} + \cdots - 1427754383 ) / 1339827192 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 31452 \nu^{11} + 117304 \nu^{10} - 2737673 \nu^{9} + 10477907 \nu^{8} - 79586472 \nu^{7} + 321264966 \nu^{6} - 827426403 \nu^{5} + \cdots + 757840787 ) / 1339827192 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 45643 \nu^{10} + 4016197 \nu^{8} + 120327945 \nu^{6} + 1386775659 \nu^{4} + 4387794733 \nu^{2} - 676101899 ) / 60901236 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 10664 \nu^{11} + 25564 \nu^{10} + 952537 \nu^{9} + 2319614 \nu^{8} + 29205906 \nu^{7} + 72240366 \nu^{6} + 350644875 \nu^{5} + 874263126 \nu^{4} + \cdots + 86861060 ) / 121802472 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 10664 \nu^{11} - 25564 \nu^{10} + 952537 \nu^{9} - 2319614 \nu^{8} + 29205906 \nu^{7} - 72240366 \nu^{6} + 350644875 \nu^{5} - 874263126 \nu^{4} + \cdots - 86861060 ) / 121802472 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 10664 \nu^{11} - 79937 \nu^{10} + 952537 \nu^{9} - 7147008 \nu^{8} + 29205906 \nu^{7} - 219213027 \nu^{6} + 350644875 \nu^{5} - 2622222042 \nu^{4} + \cdots - 209313060 ) / 121802472 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 169940 \nu^{11} - 259930 \nu^{10} - 15878131 \nu^{9} - 23961014 \nu^{8} - 530879748 \nu^{7} - 766470936 \nu^{6} - 7708194267 \nu^{5} + \cdots + 4416548158 ) / 1339827192 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 169940 \nu^{11} - 259930 \nu^{10} + 15878131 \nu^{9} - 23961014 \nu^{8} + 530879748 \nu^{7} - 766470936 \nu^{6} + 7708194267 \nu^{5} + \cdots + 4416548158 ) / 1339827192 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 380623 \nu^{11} + 502073 \nu^{10} - 35561311 \nu^{9} + 44178167 \nu^{8} - 1181014629 \nu^{7} + 1323607395 \nu^{6} - 16702393701 \nu^{5} + \cdots - 7437120889 ) / 1339827192 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} + \beta_{9} - 2\beta_{8} + \beta_{7} + \beta_{6} - \beta_{5} + \beta_{3} + \beta_{2} - \beta _1 - 19 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} - \beta_{9} - 5\beta_{7} - 5\beta_{6} - 24\beta_{4} - 9\beta_{3} - 15\beta_{2} - 25\beta _1 - 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 31 \beta_{10} - 31 \beta_{9} + 90 \beta_{8} - 51 \beta_{7} - 39 \beta_{6} + 33 \beta_{5} - 113 \beta_{3} - 113 \beta_{2} + 45 \beta _1 + 476 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 32 \beta_{11} - 33 \beta_{10} + 33 \beta_{9} + 271 \beta_{7} + 271 \beta_{6} - 16 \beta_{5} + 1300 \beta_{4} + 677 \beta_{3} + 623 \beta_{2} + 682 \beta _1 + 650 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 954 \beta_{10} + 954 \beta_{9} - 3628 \beta_{8} + 2292 \beta_{7} + 1336 \beta_{6} - 1142 \beta_{5} + 5450 \beta_{3} + 5450 \beta_{2} - 1814 \beta _1 - 12905 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 2096 \beta_{11} + 1056 \beta_{10} - 1056 \beta_{9} - 11834 \beta_{7} - 11834 \beta_{6} + 1048 \beta_{5} - 56748 \beta_{4} - 34702 \beta_{3} - 22046 \beta_{2} - 19683 \beta _1 - 28374 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 30477 \beta_{10} - 30477 \beta_{9} + 140790 \beta_{8} - 95925 \beta_{7} - 44865 \beta_{6} + 40191 \beta_{5} - 226923 \beta_{3} - 226923 \beta_{2} + 70395 \beta _1 + 369011 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 99264 \beta_{11} - 36579 \beta_{10} + 36579 \beta_{9} + 478665 \beta_{7} + 478665 \beta_{6} - 49632 \beta_{5} + 2301516 \beta_{4} + 1545459 \beta_{3} + 756057 \beta_{2} + 596465 \beta _1 + 1150758 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1012445 \beta_{10} + 1012445 \beta_{9} - 5366098 \beta_{8} + 3851627 \beta_{7} + 1514471 \beta_{6} - 1431005 \beta_{5} + 8936705 \beta_{3} + 8936705 \beta_{2} - 2683049 \beta _1 - 11069600 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 4178328 \beta_{11} + 1343903 \beta_{10} - 1343903 \beta_{9} - 18668485 \beta_{7} - 18668485 \beta_{6} + 2089164 \beta_{5} - 90111516 \beta_{4} - 64217769 \beta_{3} + \cdots - 45055758 \) Copy content Toggle raw display

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(-\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
40.1
4.56289i
2.38108i
5.04186i
5.08417i
0.159251i
6.10049i
5.08417i
0.159251i
6.10049i
4.56289i
2.38108i
5.04186i
−3.37298 + 4.64251i 3.97396 12.2306i −5.23164 16.1013i 24.4430 17.7589i 43.3765 + 59.7027i 54.3873 17.6715i 5.07527 + 1.64906i −68.2643 49.5969i 173.377i
40.2 −2.09055 + 2.87739i −4.88370 + 15.0305i 1.03528 + 3.18628i −14.3611 + 10.4339i −33.0390 45.4743i 31.7711 10.3231i −65.4537 21.2672i −136.535 99.1984i 63.1351i
40.3 2.27255 3.12789i 0.628596 1.93462i 0.325034 + 1.00035i 4.62630 3.36120i −4.62277 6.36269i 4.09081 1.32918i 62.7006 + 20.3726i 62.1828 + 45.1784i 22.1090i
94.1 −6.64435 + 2.15888i 10.8778 + 7.90316i 26.5424 19.2842i 3.84039 11.8195i −89.3377 29.0276i 14.7166 + 20.2556i −69.0217 + 95.0002i 30.8355 + 94.9021i 86.8239i
94.2 −1.65756 + 0.538574i −7.02074 5.10087i −10.4868 + 7.61913i 3.07887 9.47577i 14.3845 + 4.67381i −34.6308 47.6652i 29.6699 40.8372i −1.75841 5.41183i 17.3649i
94.3 3.99290 1.29737i 5.92412 + 4.30413i 1.31577 0.955966i −5.62746 + 17.3195i 29.2385 + 9.50015i 29.6650 + 40.8304i −35.4704 + 48.8208i −8.46066 26.0392i 76.4560i
112.1 −6.64435 2.15888i 10.8778 7.90316i 26.5424 + 19.2842i 3.84039 + 11.8195i −89.3377 + 29.0276i 14.7166 20.2556i −69.0217 95.0002i 30.8355 94.9021i 86.8239i
112.2 −1.65756 0.538574i −7.02074 + 5.10087i −10.4868 7.61913i 3.07887 + 9.47577i 14.3845 4.67381i −34.6308 + 47.6652i 29.6699 + 40.8372i −1.75841 + 5.41183i 17.3649i
112.3 3.99290 + 1.29737i 5.92412 4.30413i 1.31577 + 0.955966i −5.62746 17.3195i 29.2385 9.50015i 29.6650 40.8304i −35.4704 48.8208i −8.46066 + 26.0392i 76.4560i
118.1 −3.37298 4.64251i 3.97396 + 12.2306i −5.23164 + 16.1013i 24.4430 + 17.7589i 43.3765 59.7027i 54.3873 + 17.6715i 5.07527 1.64906i −68.2643 + 49.5969i 173.377i
118.2 −2.09055 2.87739i −4.88370 15.0305i 1.03528 3.18628i −14.3611 10.4339i −33.0390 + 45.4743i 31.7711 + 10.3231i −65.4537 + 21.2672i −136.535 + 99.1984i 63.1351i
118.3 2.27255 + 3.12789i 0.628596 + 1.93462i 0.325034 1.00035i 4.62630 + 3.36120i −4.62277 + 6.36269i 4.09081 + 1.32918i 62.7006 20.3726i 62.1828 45.1784i 22.1090i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 40.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 121.5.d.c 12
11.b odd 2 1 121.5.d.e 12
11.c even 5 1 11.5.d.a 12
11.c even 5 1 121.5.b.b 12
11.c even 5 1 121.5.d.d 12
11.c even 5 1 121.5.d.e 12
11.d odd 10 1 11.5.d.a 12
11.d odd 10 1 121.5.b.b 12
11.d odd 10 1 inner 121.5.d.c 12
11.d odd 10 1 121.5.d.d 12
33.f even 10 1 99.5.k.a 12
33.h odd 10 1 99.5.k.a 12
44.g even 10 1 176.5.n.a 12
44.h odd 10 1 176.5.n.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.5.d.a 12 11.c even 5 1
11.5.d.a 12 11.d odd 10 1
99.5.k.a 12 33.f even 10 1
99.5.k.a 12 33.h odd 10 1
121.5.b.b 12 11.c even 5 1
121.5.b.b 12 11.d odd 10 1
121.5.d.c 12 1.a even 1 1 trivial
121.5.d.c 12 11.d odd 10 1 inner
121.5.d.d 12 11.c even 5 1
121.5.d.d 12 11.d odd 10 1
121.5.d.e 12 11.b odd 2 1
121.5.d.e 12 11.c even 5 1
176.5.n.a 12 44.g even 10 1
176.5.n.a 12 44.h odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 15 T_{2}^{11} + 75 T_{2}^{10} - 5 T_{2}^{9} - 1305 T_{2}^{8} - 1950 T_{2}^{7} + 21730 T_{2}^{6} + 10180 T_{2}^{5} - 172260 T_{2}^{4} - 27560 T_{2}^{3} + 6145160 T_{2}^{2} + 18581200 T_{2} + 16272080 \) acting on \(S_{5}^{\mathrm{new}}(121, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 15 T^{11} + 75 T^{10} + \cdots + 16272080 \) Copy content Toggle raw display
$3$ \( T^{12} - 19 T^{11} + \cdots + 124779204081 \) Copy content Toggle raw display
$5$ \( T^{12} - 32 T^{11} + \cdots + 47826408560896 \) Copy content Toggle raw display
$7$ \( T^{12} - 200 T^{11} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{12} \) Copy content Toggle raw display
$13$ \( T^{12} + 250 T^{11} + \cdots + 10\!\cdots\!80 \) Copy content Toggle raw display
$17$ \( T^{12} - 405 T^{11} + \cdots + 16\!\cdots\!05 \) Copy content Toggle raw display
$19$ \( T^{12} - 530 T^{11} + \cdots + 31\!\cdots\!05 \) Copy content Toggle raw display
$23$ \( (T^{6} - 842 T^{5} + \cdots - 10\!\cdots\!96)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} - 4230 T^{11} + \cdots + 61\!\cdots\!80 \) Copy content Toggle raw display
$31$ \( T^{12} - 104 T^{11} + \cdots + 41\!\cdots\!36 \) Copy content Toggle raw display
$37$ \( T^{12} - 454 T^{11} + \cdots + 49\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( T^{12} + 5385 T^{11} + \cdots + 72\!\cdots\!05 \) Copy content Toggle raw display
$43$ \( T^{12} + 17669305 T^{10} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{12} + 4516 T^{11} + \cdots + 84\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{12} + 4726 T^{11} + \cdots + 11\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{12} - 11624 T^{11} + \cdots + 38\!\cdots\!41 \) Copy content Toggle raw display
$61$ \( T^{12} + 5780 T^{11} + \cdots + 15\!\cdots\!80 \) Copy content Toggle raw display
$67$ \( (T^{6} - 6077 T^{5} + \cdots + 15\!\cdots\!84)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + 21646 T^{11} + \cdots + 56\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( T^{12} - 1015 T^{11} + \cdots + 14\!\cdots\!05 \) Copy content Toggle raw display
$79$ \( T^{12} - 15130 T^{11} + \cdots + 36\!\cdots\!80 \) Copy content Toggle raw display
$83$ \( T^{12} + 46770 T^{11} + \cdots + 18\!\cdots\!05 \) Copy content Toggle raw display
$89$ \( (T^{6} - 2777 T^{5} + \cdots + 66\!\cdots\!44)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + 16546 T^{11} + \cdots + 18\!\cdots\!01 \) Copy content Toggle raw display
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