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Properties

Label 121.5.b.b
Level $121$
Weight $5$
Character orbit 121.b
Analytic conductor $12.508$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [121,5,Mod(120,121)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(121, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("121.120");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 121 = 11^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 121.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: \(12.5077655331\)
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: \( x^{12} + 115x^{10} + 5030x^{8} + 102975x^{6} + 953170x^{4} + 2910655x^{2} + 73205 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 11^{5} \)
Twist minimal: no (minimal twist has level 11)
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + (\beta_{3} - 2) q^{3} + (\beta_{7} - 6) q^{4} + ( - \beta_{3} + \beta_1 - 2) q^{5} + ( - \beta_{11} - \beta_{10} + \cdots + 4 \beta_{2}) q^{6}+ \cdots + (2 \beta_{9} - 3 \beta_{7} - 3 \beta_{5} + \cdots + 44) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + (\beta_{3} - 2) q^{3} + (\beta_{7} - 6) q^{4} + ( - \beta_{3} + \beta_1 - 2) q^{5} + ( - \beta_{11} - \beta_{10} + \cdots + 4 \beta_{2}) q^{6}+ \cdots + (81 \beta_{11} + 11 \beta_{10} + \cdots - 877 \beta_{2}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 26 q^{3} - 68 q^{4} - 28 q^{5} + 486 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 26 q^{3} - 68 q^{4} - 28 q^{5} + 486 q^{9} + 594 q^{12} - 1140 q^{14} - 1256 q^{15} - 1068 q^{16} + 632 q^{20} + 1684 q^{23} - 3808 q^{25} + 1180 q^{26} - 3752 q^{27} + 2064 q^{31} + 2370 q^{34} - 8494 q^{36} - 236 q^{37} - 8010 q^{38} - 8000 q^{42} + 5136 q^{45} + 8984 q^{47} - 2886 q^{48} + 6712 q^{49} + 8624 q^{53} - 21340 q^{56} + 13580 q^{58} - 3226 q^{59} + 24024 q^{60} - 8708 q^{64} + 12154 q^{67} + 17508 q^{69} + 25440 q^{70} + 8144 q^{71} + 22614 q^{75} - 42920 q^{78} + 30352 q^{80} - 13600 q^{81} + 29350 q^{82} - 43650 q^{86} + 5554 q^{89} - 13080 q^{91} + 12684 q^{92} + 3868 q^{93} - 8446 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

\(\beta_{1}\)\(=\) \( ( 6428\nu^{10} + 537997\nu^{8} + 15377352\nu^{6} + 177671499\nu^{4} + 787262006\nu^{2} + 1398748747 ) / 60901236 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 31452 \nu^{11} + 2737673 \nu^{9} + 79586472 \nu^{7} + 827426403 \nu^{5} + 1134660378 \nu^{3} - 7303391689 \nu ) / 669913596 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 66971 \nu^{10} - 5921271 \nu^{8} - 178739757 \nu^{6} - 2088065409 \nu^{4} - 6870421673 \nu^{2} + 477411429 ) / 60901236 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 86029 \nu^{11} - 7196850 \nu^{9} - 195308019 \nu^{7} - 1742959254 \nu^{5} + \cdots + 22453083312 \nu ) / 669913596 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 344\nu^{10} + 30727\nu^{8} + 942126\nu^{6} + 11311125\nu^{4} + 40042370\nu^{2} + 3293983 ) / 178596 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 6526 \nu^{11} - 609888 \nu^{9} - 20067903 \nu^{7} - 274692837 \nu^{5} - 1411940227 \nu^{3} - 2508246510 \nu ) / 30450618 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 170538 \nu^{10} - 15246553 \nu^{8} - 467631960 \nu^{6} - 5595088959 \nu^{4} - 19274817024 \nu^{2} + 669602747 ) / 60901236 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 127386 \nu^{11} + 11038024 \nu^{9} + 317228919 \nu^{7} + 3171684078 \nu^{5} + \cdots - 48594396785 \nu ) / 334956798 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 235513 \nu^{10} + 21188149 \nu^{8} + 654204021 \nu^{6} + 7872498921 \nu^{4} + 27304989541 \nu^{2} + 1997452633 ) / 60901236 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 157260 \nu^{11} - 13688365 \nu^{9} - 397932360 \nu^{7} - 4137132015 \nu^{5} + \cdots + 40201483223 \nu ) / 334956798 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 760831 \nu^{11} - 69558041 \nu^{9} - 2226629937 \nu^{7} - 29377144047 \nu^{5} + \cdots - 179167525229 \nu ) / 669913596 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{10} + 10\beta_{2} ) / 11 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{9} + \beta_{7} - 3\beta_{5} + 8\beta _1 - 205 ) / 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{11} - 38\beta_{10} - 12\beta_{8} - 2\beta_{6} + \beta_{4} - 265\beta_{2} ) / 11 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -64\beta_{9} + 45\beta_{7} + 199\beta_{5} - 22\beta_{3} - 322\beta _1 + 5502 ) / 11 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -27\beta_{11} + 1349\beta_{10} + 566\beta_{8} - 28\beta_{6} - 203\beta_{4} + 7569\beta_{2} ) / 11 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 1812\beta_{9} - 3824\beta_{7} - 8702\beta_{5} + 2068\beta_{3} + 12506\beta _1 - 160331 ) / 11 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 274\beta_{11} - 48065\beta_{10} - 23418\beta_{8} + 5288\beta_{6} + 11802\beta_{4} - 227908\beta_{2} ) / 11 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -49812\beta_{9} + 194643\beta_{7} + 348285\beta_{5} - 106854\beta_{3} - 480078\beta _1 + 4933807 ) / 11 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 15075\beta_{11} + 1734170\beta_{10} + 933246\beta_{8} - 332016\beta_{6} - 530877\beta_{4} + 7179089\beta_{2} ) / 11 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 1358338\beta_{9} - 8509153\beta_{7} - 13465683\beta_{5} + 4604160\beta_{3} + 18287710\beta _1 - 158751284 ) / 11 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 1331275 \beta_{11} - 63208837 \beta_{10} - 36432462 \beta_{8} + 16327586 \beta_{6} + \cdots - 235193531 \beta_{2} ) / 11 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
120.1
5.08417i
4.56289i
6.10049i
5.04186i
2.38108i
− 0.159251i
0.159251i
− 2.38108i
− 5.04186i
− 6.10049i
− 4.56289i
− 5.08417i
− 6.98629i −13.4457 −32.8082 12.4278 93.9353i 25.0373i 117.427i 99.7859 − 86.8239i
120.2 − 5.73846i 12.8600 −16.9299 −30.2132 − 73.7966i − 57.1862i 5.33646i 84.3794 173.377i
120.3 − 4.19838i −7.32262 −1.62639 −18.2108 30.7431i − 50.4691i − 60.3459i −27.3793 76.4560i
120.4 − 3.86629i 2.03418 1.05183 −5.71842 − 7.86472i 4.30133i − 65.9273i −76.8621 22.1090i
120.5 − 3.55665i −15.8040 3.35025 17.7513 56.2093i − 33.4062i − 68.8220i 168.766 − 63.1351i
120.6 − 1.74286i 8.67811 12.9624 9.96342 − 15.1248i − 58.9175i − 50.4775i −5.69033 − 17.3649i
120.7 1.74286i 8.67811 12.9624 9.96342 15.1248i 58.9175i 50.4775i −5.69033 17.3649i
120.8 3.55665i −15.8040 3.35025 17.7513 − 56.2093i 33.4062i 68.8220i 168.766 63.1351i
120.9 3.86629i 2.03418 1.05183 −5.71842 7.86472i − 4.30133i 65.9273i −76.8621 − 22.1090i
120.10 4.19838i −7.32262 −1.62639 −18.2108 − 30.7431i 50.4691i 60.3459i −27.3793 − 76.4560i
120.11 5.73846i 12.8600 −16.9299 −30.2132 73.7966i 57.1862i − 5.33646i 84.3794 − 173.377i
120.12 6.98629i −13.4457 −32.8082 12.4278 − 93.9353i − 25.0373i − 117.427i 99.7859 86.8239i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 120.12
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 121.5.b.b 12
11.b odd 2 1 inner 121.5.b.b 12
11.c even 5 1 11.5.d.a ✓ 12
11.c even 5 1 121.5.d.c 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.d.c 12
11.d odd 10 1 121.5.d.d 12
11.d odd 10 1 121.5.d.e 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 1.a even 1 1 trivial
121.5.b.b 12 11.b odd 2 1 inner
121.5.d.c 12 11.c even 5 1
121.5.d.c 12 11.d odd 10 1
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.c even 5 1
121.5.d.e 12 11.d odd 10 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 \( T_{2}^{12} + 130T_{2}^{10} + 6365T_{2}^{8} + 149400T_{2}^{6} + 1756840T_{2}^{4} + 9482560T_{2}^{2} + 16272080 \) acting on \(S_{5}^{\mathrm{new}}(121, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 130 T^{10} + \cdots + 16272080 \) Copy content Toggle raw display
$3$ \( (T^{6} + 13 T^{5} + \cdots - 353241)^{2} \) Copy content Toggle raw display
$5$ \( (T^{6} + 14 T^{5} + \cdots - 6915664)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{12} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 10\!\cdots\!80 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 16\!\cdots\!05 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 31\!\cdots\!05 \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots - 10\!\cdots\!96)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 61\!\cdots\!80 \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots + 64\!\cdots\!44)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots - 70\!\cdots\!96)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 72\!\cdots\!05 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots - 92\!\cdots\!36)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots - 34\!\cdots\!76)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots - 61\!\cdots\!21)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 15\!\cdots\!80 \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 15\!\cdots\!84)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 23\!\cdots\!84)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 14\!\cdots\!05 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 36\!\cdots\!80 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 18\!\cdots\!05 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 66\!\cdots\!44)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots - 13\!\cdots\!01)^{2} \) Copy content Toggle raw display
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