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Properties

Label 121.7.b.c
Level $121$
Weight $7$
Character orbit 121.b
Analytic conductor $27.837$
Analytic rank $0$
Dimension $20$
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] = [121,7,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, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("121.120");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 121 = 11^{2} \)
Weight: \( k \) \(=\) \( 7 \)
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: \(27.8365441180\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 825 x^{18} + 275175 x^{16} + 47589550 x^{14} + 4569013705 x^{12} + 245564683275 x^{10} + \cdots + 17\!\cdots\!25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{14}\cdot 5\cdot 11^{16} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{10} q^{2} + (\beta_{3} + 2) q^{3} + ( - \beta_1 - 21) q^{4} + (\beta_{7} - 9) q^{5} + ( - \beta_{16} + 4 \beta_{10}) q^{6} + (\beta_{18} - \beta_{16} + \cdots - 3 \beta_{10}) q^{7}+ \cdots + ( - \beta_{9} - \beta_{8} - 2 \beta_{7} + \cdots + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{10} q^{2} + (\beta_{3} + 2) q^{3} + ( - \beta_1 - 21) q^{4} + (\beta_{7} - 9) q^{5} + ( - \beta_{16} + 4 \beta_{10}) q^{6} + (\beta_{18} - \beta_{16} + \cdots - 3 \beta_{10}) q^{7}+ \cdots + (1955 \beta_{19} + \cdots - 4822 \beta_{10}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 46 q^{3} - 420 q^{4} - 174 q^{5} + 66 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 46 q^{3} - 420 q^{4} - 174 q^{5} + 66 q^{9} - 3006 q^{12} + 7180 q^{14} - 6260 q^{15} + 30740 q^{16} + 5384 q^{20} + 7816 q^{23} + 98014 q^{25} + 44540 q^{26} + 95428 q^{27} - 2806 q^{31} - 228190 q^{34} + 646034 q^{36} + 74594 q^{37} + 571430 q^{38} - 467120 q^{42} - 805524 q^{45} + 706194 q^{47} + 379386 q^{48} + 646590 q^{49} - 900854 q^{53} - 862620 q^{56} + 1621260 q^{58} + 741246 q^{59} + 1763880 q^{60} + 232380 q^{64} - 960956 q^{67} + 1742196 q^{69} + 1180480 q^{70} - 622350 q^{71} + 480120 q^{75} + 1703080 q^{78} - 2615744 q^{80} + 299384 q^{81} + 287430 q^{82} + 670190 q^{86} + 1111620 q^{89} - 1964640 q^{91} - 9325476 q^{92} - 3775388 q^{93} - 308266 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 825 x^{18} + 275175 x^{16} + 47589550 x^{14} + 4569013705 x^{12} + 245564683275 x^{10} + \cdots + 17\!\cdots\!25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 15\!\cdots\!72 \nu^{18} + \cdots - 10\!\cdots\!75 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 18\!\cdots\!27 \nu^{18} + \cdots + 24\!\cdots\!25 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 30\!\cdots\!81 \nu^{18} + \cdots + 99\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 30\!\cdots\!86 \nu^{18} + \cdots - 24\!\cdots\!25 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 29\!\cdots\!97 \nu^{18} + \cdots - 19\!\cdots\!50 ) / 53\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 15\!\cdots\!34 \nu^{18} + \cdots - 17\!\cdots\!25 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 22\!\cdots\!80 \nu^{18} + \cdots - 17\!\cdots\!75 ) / 19\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 29\!\cdots\!63 \nu^{18} + \cdots - 14\!\cdots\!00 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 24\!\cdots\!59 \nu^{18} + \cdots - 43\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 24\!\cdots\!89 \nu^{19} + \cdots + 41\!\cdots\!00 \nu ) / 40\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 23\!\cdots\!49 \nu^{19} + \cdots - 12\!\cdots\!00 \nu ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 62\!\cdots\!02 \nu^{19} + \cdots + 25\!\cdots\!25 \nu ) / 40\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 31\!\cdots\!57 \nu^{19} + \cdots + 10\!\cdots\!00 \nu ) / 45\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 12\!\cdots\!57 \nu^{19} + \cdots - 24\!\cdots\!50 \nu ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 58\!\cdots\!33 \nu^{19} + \cdots + 69\!\cdots\!75 \nu ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 10\!\cdots\!08 \nu^{19} + \cdots + 98\!\cdots\!25 \nu ) / 36\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 19\!\cdots\!77 \nu^{19} + \cdots - 32\!\cdots\!50 \nu ) / 36\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 50\!\cdots\!28 \nu^{19} + \cdots - 66\!\cdots\!25 \nu ) / 72\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 12\!\cdots\!83 \nu^{19} + \cdots - 51\!\cdots\!25 \nu ) / 12\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{13} - 117\beta_{10} ) / 121 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{4} - 22\beta_{2} - 121\beta _1 - 9982 ) / 121 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 55 \beta_{19} - 88 \beta_{18} - 11 \beta_{17} - 65 \beta_{15} - 187 \beta_{14} + \cdots + 18335 \beta_{10} ) / 121 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 396 \beta_{9} + 99 \beta_{8} + 627 \beta_{7} - 891 \beta_{6} + 561 \beta_{5} - 691 \beta_{4} + \cdots + 1574497 ) / 121 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 9647 \beta_{19} + 15389 \beta_{18} + 4774 \beta_{17} - 3971 \beta_{16} + 28517 \beta_{15} + \cdots - 3264695 \beta_{10} ) / 121 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 97647 \beta_{9} - 29920 \beta_{8} - 157025 \beta_{7} + 335577 \beta_{6} - 209033 \beta_{5} + \cdots - 280607293 ) / 121 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 1411619 \beta_{19} - 2619243 \beta_{18} - 1527317 \beta_{17} + 2206017 \beta_{16} + \cdots + 603084501 \beta_{10} ) / 121 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 20583398 \beta_{9} + 6740745 \beta_{8} + 36467871 \beta_{7} - 96955155 \beta_{6} + \cdots + 51924797722 ) / 121 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 180897211 \beta_{19} + 457523913 \beta_{18} + 419079848 \beta_{17} - 751189527 \beta_{16} + \cdots - 113803681904 \beta_{10} ) / 121 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 4229420085 \beta_{9} - 1348538950 \beta_{8} - 8447714825 \beta_{7} + 25224040941 \beta_{6} + \cdots - 9821668349485 ) / 121 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 17716056006 \beta_{19} - 82520811175 \beta_{18} - 105960778594 \beta_{17} + \cdots + 21855721829286 \beta_{10} ) / 121 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 867974788076 \beta_{9} + 249519741262 \beta_{8} + 1950065757530 \beta_{7} - 6198502901910 \beta_{6} + \cdots + 18\!\cdots\!37 ) / 121 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 197492262000 \beta_{19} + 15378992229810 \beta_{18} + 25532899968730 \beta_{17} + \cdots - 42\!\cdots\!15 \beta_{10} ) / 121 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 178735232282750 \beta_{9} - 43058157319210 \beta_{8} - 447084097174220 \beta_{7} + \cdots - 37\!\cdots\!10 ) / 121 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 573028171525535 \beta_{19} + \cdots + 84\!\cdots\!25 \beta_{10} ) / 121 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( - 36\!\cdots\!10 \beta_{9} + \cdots + 73\!\cdots\!65 ) / 121 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 22\!\cdots\!15 \beta_{19} + \cdots - 16\!\cdots\!55 \beta_{10} ) / 121 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( 76\!\cdots\!15 \beta_{9} + \cdots - 14\!\cdots\!85 ) / 121 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 67\!\cdots\!25 \beta_{19} + \cdots + 34\!\cdots\!15 \beta_{10} ) / 121 \) 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
13.3023i
12.4579i
14.6250i
12.6102i
6.47390i
3.41863i
6.18436i
2.50105i
4.59184i
− 0.868969i
0.868969i
− 4.59184i
− 2.50105i
− 6.18436i
− 3.41863i
− 6.47390i
− 12.6102i
− 14.6250i
− 12.4579i
− 13.3023i
− 14.4779i 25.0244 −145.610 −162.774 − 362.301i − 372.119i 1181.54i −102.779 2356.62i
120.2 − 14.3600i −11.8132 −142.211 −27.2312 169.638i 575.733i 1123.11i −589.449 391.041i
120.3 − 13.4494i −0.452917 −116.887 163.689 6.09148i 86.2142i 711.306i −728.795 − 2201.52i
120.4 − 10.7081i 12.1231 −50.6631 18.8367 − 129.815i 54.3129i − 142.813i −582.030 − 201.705i
120.5 − 7.64947i −28.7092 5.48563 101.280 219.610i 155.494i − 531.528i 95.2203 − 774.735i
120.6 − 5.32074i 44.5904 35.6897 −174.596 − 237.254i − 45.8820i − 530.423i 1259.30 928.982i
120.7 − 5.00879i −13.5381 38.9120 −180.269 67.8097i 155.034i − 515.465i −545.719 902.932i
120.8 − 4.40316i −4.82233 44.6122 219.779 21.2335i − 431.599i − 478.237i −705.745 − 967.721i
120.9 − 2.68972i −40.8764 56.7654 −141.780 109.946i − 334.732i − 324.825i 941.879 381.348i
120.10 − 0.306602i 41.4743 63.9060 96.0663 − 12.7161i 155.658i − 39.2162i 991.114 − 29.4541i
120.11 0.306602i 41.4743 63.9060 96.0663 12.7161i − 155.658i 39.2162i 991.114 29.4541i
120.12 2.68972i −40.8764 56.7654 −141.780 − 109.946i 334.732i 324.825i 941.879 − 381.348i
120.13 4.40316i −4.82233 44.6122 219.779 − 21.2335i 431.599i 478.237i −705.745 967.721i
120.14 5.00879i −13.5381 38.9120 −180.269 − 67.8097i − 155.034i 515.465i −545.719 − 902.932i
120.15 5.32074i 44.5904 35.6897 −174.596 237.254i 45.8820i 530.423i 1259.30 − 928.982i
120.16 7.64947i −28.7092 5.48563 101.280 − 219.610i − 155.494i 531.528i 95.2203 774.735i
120.17 10.7081i 12.1231 −50.6631 18.8367 129.815i − 54.3129i 142.813i −582.030 201.705i
120.18 13.4494i −0.452917 −116.887 163.689 − 6.09148i − 86.2142i − 711.306i −728.795 2201.52i
120.19 14.3600i −11.8132 −142.211 −27.2312 − 169.638i − 575.733i − 1123.11i −589.449 − 391.041i
120.20 14.4779i 25.0244 −145.610 −162.774 362.301i 372.119i − 1181.54i −102.779 − 2356.62i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 120.20
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 121.7.b.c 20
11.b odd 2 1 inner 121.7.b.c 20
11.c even 5 1 11.7.d.a ✓ 20
11.d odd 10 1 11.7.d.a ✓ 20
33.f even 10 1 99.7.k.a 20
33.h odd 10 1 99.7.k.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.7.d.a ✓ 20 11.c even 5 1
11.7.d.a ✓ 20 11.d odd 10 1
99.7.k.a 20 33.f even 10 1
99.7.k.a 20 33.h odd 10 1
121.7.b.c 20 1.a even 1 1 trivial
121.7.b.c 20 11.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} + 850 T_{2}^{18} + 292445 T_{2}^{16} + 52420200 T_{2}^{14} + 5280517640 T_{2}^{12} + \cdots + 491270438912000 \) acting on \(S_{7}^{\mathrm{new}}(121, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + \cdots + 491270438912000 \) Copy content Toggle raw display
$3$ \( (T^{10} + \cdots + 229981032405)^{2} \) Copy content Toggle raw display
$5$ \( (T^{10} + \cdots - 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 62\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{20} \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 36\!\cdots\!25 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 94\!\cdots\!25 \) Copy content Toggle raw display
$23$ \( (T^{10} + \cdots - 17\!\cdots\!80)^{2} \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 71\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots - 23\!\cdots\!96)^{2} \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots - 46\!\cdots\!20)^{2} \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 27\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 52\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( (T^{10} + \cdots + 79\!\cdots\!20)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots + 44\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{10} + \cdots - 75\!\cdots\!71)^{2} \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots + 34\!\cdots\!20)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots + 22\!\cdots\!64)^{2} \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 10\!\cdots\!25 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 15\!\cdots\!25 \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots + 92\!\cdots\!64)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots - 17\!\cdots\!05)^{2} \) Copy content Toggle raw display
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