↑

Properties

Label 130.3.o.b
Level $130$
Weight $3$
Character orbit 130.o
Analytic conductor $3.542$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [130,3,Mod(11,130)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(130, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 7]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("130.11");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 130 = 2 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 130.o (of order \(12\), degree \(4\), 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: \(3.54224343668\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} - 30 x^{14} + 64 x^{13} + 1289 x^{12} - 5316 x^{11} - 5190 x^{10} + 42576 x^{9} + \cdots + 302500 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{7} - \beta_{6} - \beta_{3} + 1) q^{2} + (\beta_{10} + \beta_{7}) q^{3} - 2 \beta_{3} q^{4} - \beta_{12} q^{5} + ( - \beta_{15} + \beta_{8} + \beta_{7} + \cdots + 1) q^{6}+ \cdots + (\beta_{14} + \beta_{13} + \cdots + \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{7} - \beta_{6} - \beta_{3} + 1) q^{2} + (\beta_{10} + \beta_{7}) q^{3} - 2 \beta_{3} q^{4} - \beta_{12} q^{5} + ( - \beta_{15} + \beta_{8} + \beta_{7} + \cdots + 1) q^{6}+ \cdots + (6 \beta_{15} - 7 \beta_{14} + \cdots + 22) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{2} + 12 q^{6} + 32 q^{7} + 32 q^{8} - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{2} + 12 q^{6} + 32 q^{7} + 32 q^{8} - 20 q^{9} + 4 q^{11} + 52 q^{13} + 8 q^{14} - 20 q^{15} + 32 q^{16} - 60 q^{17} - 40 q^{18} - 64 q^{19} + 8 q^{21} - 8 q^{22} + 24 q^{23} - 24 q^{24} + 20 q^{26} - 72 q^{27} + 64 q^{28} + 44 q^{29} - 60 q^{30} + 8 q^{31} - 32 q^{32} + 48 q^{33} - 20 q^{35} + 72 q^{36} - 192 q^{37} + 196 q^{39} - 104 q^{41} + 8 q^{42} - 60 q^{43} - 16 q^{44} - 60 q^{46} - 248 q^{47} + 84 q^{49} - 40 q^{50} - 48 q^{52} - 36 q^{54} + 60 q^{55} + 120 q^{56} + 164 q^{57} - 8 q^{58} - 76 q^{59} + 40 q^{60} - 60 q^{61} + 24 q^{62} - 340 q^{63} - 60 q^{65} - 168 q^{66} - 164 q^{67} + 408 q^{69} - 40 q^{70} - 48 q^{71} + 32 q^{72} + 224 q^{73} + 156 q^{74} + 60 q^{75} + 128 q^{76} + 120 q^{78} + 456 q^{79} + 76 q^{81} - 72 q^{82} - 256 q^{83} - 176 q^{84} + 120 q^{85} - 128 q^{86} - 32 q^{87} - 64 q^{89} - 68 q^{91} - 336 q^{92} + 248 q^{93} - 248 q^{94} + 60 q^{95} + 472 q^{97} - 16 q^{98} + 592 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} - 30 x^{14} + 64 x^{13} + 1289 x^{12} - 5316 x^{11} - 5190 x^{10} + 42576 x^{9} + \cdots + 302500 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 17\!\cdots\!69 \nu^{15} + \cdots + 72\!\cdots\!50 ) / 13\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 63\!\cdots\!54 \nu^{15} + \cdots + 17\!\cdots\!50 ) / 41\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 51\!\cdots\!54 \nu^{15} + \cdots - 17\!\cdots\!00 ) / 20\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 16\!\cdots\!00 \nu^{15} + \cdots + 14\!\cdots\!00 ) / 41\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 15\!\cdots\!51 \nu^{15} + \cdots - 23\!\cdots\!00 ) / 20\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 25\!\cdots\!62 \nu^{15} + \cdots + 54\!\cdots\!00 ) / 23\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 7404049178 \nu^{15} - 13167593277 \nu^{14} - 257320134990 \nu^{13} - 92620296358 \nu^{12} + \cdots - 679751156438750 ) / 6332413551750 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 24\!\cdots\!87 \nu^{15} + \cdots - 33\!\cdots\!00 ) / 20\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 85\!\cdots\!47 \nu^{15} + \cdots - 22\!\cdots\!50 ) / 41\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 49\!\cdots\!04 \nu^{15} + \cdots - 17\!\cdots\!00 ) / 20\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 12\!\cdots\!55 \nu^{15} + \cdots - 29\!\cdots\!00 ) / 41\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 11763720830825 \nu^{15} - 21738411048723 \nu^{14} - 407368391436513 \nu^{13} + \cdots - 12\!\cdots\!50 ) / 38\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 49\!\cdots\!27 \nu^{15} + \cdots + 49\!\cdots\!50 ) / 13\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 55\!\cdots\!88 \nu^{15} + \cdots - 77\!\cdots\!00 ) / 10\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 13\!\cdots\!06 \nu^{15} + \cdots - 23\!\cdots\!50 ) / 20\!\cdots\!50 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{14} - \beta_{12} - \beta_{11} - \beta_{4} - \beta_{3} - \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 4 \beta_{15} - 2 \beta_{14} + \beta_{13} + 2 \beta_{12} + \beta_{11} + 2 \beta_{10} + 3 \beta_{9} + \cdots + 21 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2 \beta_{15} + 13 \beta_{14} - 19 \beta_{13} - 13 \beta_{12} + 2 \beta_{10} - 27 \beta_{9} - 4 \beta_{8} + \cdots - 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 26 \beta_{15} + 29 \beta_{14} + 82 \beta_{13} - 71 \beta_{12} - 49 \beta_{11} - 4 \beta_{10} + \cdots - 90 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 190 \beta_{15} - 326 \beta_{14} - 431 \beta_{13} + 406 \beta_{12} + 495 \beta_{11} + 132 \beta_{10} + \cdots + 1969 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 1286 \beta_{15} + 2859 \beta_{14} + 743 \beta_{13} - 3623 \beta_{12} - 2454 \beta_{11} - 702 \beta_{10} + \cdots - 12233 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 7602 \beta_{15} - 12259 \beta_{14} + 3502 \beta_{13} + 11897 \beta_{12} + 10369 \beta_{11} + \cdots + 52372 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 28704 \beta_{15} + 30494 \beta_{14} - 55869 \beta_{13} - 20814 \beta_{12} - 13221 \beta_{11} + \cdots - 119839 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 25334 \beta_{15} + 102779 \beta_{14} + 396493 \beta_{13} - 212731 \beta_{12} - 148692 \beta_{11} + \cdots - 509769 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 420750 \beta_{15} - 1738731 \beta_{14} - 1884826 \beta_{13} + 2338267 \beta_{12} + 1766381 \beta_{11} + \cdots + 7313606 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 4682722 \beta_{15} + 11842708 \beta_{14} + 5694439 \beta_{13} - 13873848 \beta_{12} - 11343733 \beta_{11} + \cdots - 50404131 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 25912658 \beta_{15} - 52840897 \beta_{14} + 4710161 \beta_{13} + 54326635 \beta_{12} + \cdots + 224378651 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 100449212 \beta_{15} + 111295241 \beta_{14} - 200735900 \beta_{13} - 61388161 \beta_{12} + \cdots - 539595874 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 138930948 \beta_{15} + 539831130 \beta_{14} + 1663688651 \beta_{13} - 984932330 \beta_{12} + \cdots - 1685162309 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 1438017800 \beta_{15} - 7911936245 \beta_{14} - 8948567835 \beta_{13} + 10425764991 \beta_{12} + \cdots + 29754517717 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/130\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(41\)
\(\chi(n)\) \(1\) \(-\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
−4.39517 − 3.11490i
1.99842 + 0.576438i
2.15656 + 0.667738i
1.24019 + 0.138676i
1.77198 + 0.445706i
0.0332941 − 0.558128i
−4.01060 − 2.89287i
3.20532 + 1.27324i
−4.39517 + 3.11490i
1.99842 − 0.576438i
2.15656 − 0.667738i
1.24019 − 0.138676i
1.77198 − 0.445706i
0.0332941 + 0.558128i
−4.01060 + 2.89287i
3.20532 − 1.27324i
−0.366025 + 1.36603i −2.62448 − 4.54573i −1.73205 − 1.00000i −1.58114 − 1.58114i 7.17022 − 1.92125i 2.09397 + 7.81480i 2.00000 − 2.00000i −9.27580 + 16.0662i 2.73861 − 1.58114i
11.2 −0.366025 + 1.36603i −1.11546 − 1.93204i −1.73205 − 1.00000i 1.58114 + 1.58114i 3.04750 − 0.816574i 1.30233 + 4.86036i 2.00000 − 2.00000i 2.01149 − 3.48401i −2.73861 + 1.58114i
11.3 −0.366025 + 1.36603i −0.401420 − 0.695280i −1.73205 − 1.00000i −1.58114 − 1.58114i 1.09670 − 0.293860i −1.27078 − 4.74262i 2.00000 − 2.00000i 4.17772 − 7.23603i 2.73861 − 1.58114i
11.4 −0.366025 + 1.36603i 2.40931 + 4.17305i −1.73205 − 1.00000i 1.58114 + 1.58114i −6.58236 + 1.76374i 0.678332 + 2.53157i 2.00000 − 2.00000i −7.10957 + 12.3141i −2.73861 + 1.58114i
41.1 1.36603 + 0.366025i −1.34123 − 2.32308i 1.73205 + 1.00000i 1.58114 − 1.58114i −0.981847 − 3.66430i 3.20014 − 0.857475i 2.00000 + 2.00000i 0.902212 − 1.56268i 2.73861 − 1.58114i
41.2 1.36603 + 0.366025i −0.582857 − 1.00954i 1.73205 + 1.00000i −1.58114 + 1.58114i −0.426681 − 1.59239i 11.7519 − 3.14891i 2.00000 + 2.00000i 3.82056 − 6.61740i −2.73861 + 1.58114i
41.3 1.36603 + 0.366025i 1.62852 + 2.82067i 1.73205 + 1.00000i 1.58114 − 1.58114i 1.19216 + 4.44919i 1.23806 − 0.331737i 2.00000 + 2.00000i −0.804135 + 1.39280i 2.73861 − 1.58114i
41.4 1.36603 + 0.366025i 2.02762 + 3.51194i 1.73205 + 1.00000i −1.58114 + 1.58114i 1.48432 + 5.53956i −2.99394 + 0.802223i 2.00000 + 2.00000i −3.72248 + 6.44752i −2.73861 + 1.58114i
71.1 −0.366025 − 1.36603i −2.62448 + 4.54573i −1.73205 + 1.00000i −1.58114 + 1.58114i 7.17022 + 1.92125i 2.09397 − 7.81480i 2.00000 + 2.00000i −9.27580 − 16.0662i 2.73861 + 1.58114i
71.2 −0.366025 − 1.36603i −1.11546 + 1.93204i −1.73205 + 1.00000i 1.58114 − 1.58114i 3.04750 + 0.816574i 1.30233 − 4.86036i 2.00000 + 2.00000i 2.01149 + 3.48401i −2.73861 − 1.58114i
71.3 −0.366025 − 1.36603i −0.401420 + 0.695280i −1.73205 + 1.00000i −1.58114 + 1.58114i 1.09670 + 0.293860i −1.27078 + 4.74262i 2.00000 + 2.00000i 4.17772 + 7.23603i 2.73861 + 1.58114i
71.4 −0.366025 − 1.36603i 2.40931 − 4.17305i −1.73205 + 1.00000i 1.58114 − 1.58114i −6.58236 − 1.76374i 0.678332 − 2.53157i 2.00000 + 2.00000i −7.10957 − 12.3141i −2.73861 − 1.58114i
111.1 1.36603 − 0.366025i −1.34123 + 2.32308i 1.73205 − 1.00000i 1.58114 + 1.58114i −0.981847 + 3.66430i 3.20014 + 0.857475i 2.00000 − 2.00000i 0.902212 + 1.56268i 2.73861 + 1.58114i
111.2 1.36603 − 0.366025i −0.582857 + 1.00954i 1.73205 − 1.00000i −1.58114 − 1.58114i −0.426681 + 1.59239i 11.7519 + 3.14891i 2.00000 − 2.00000i 3.82056 + 6.61740i −2.73861 − 1.58114i
111.3 1.36603 − 0.366025i 1.62852 − 2.82067i 1.73205 − 1.00000i 1.58114 + 1.58114i 1.19216 − 4.44919i 1.23806 + 0.331737i 2.00000 − 2.00000i −0.804135 − 1.39280i 2.73861 + 1.58114i
111.4 1.36603 − 0.366025i 2.02762 − 3.51194i 1.73205 − 1.00000i −1.58114 − 1.58114i 1.48432 − 5.53956i −2.99394 − 0.802223i 2.00000 − 2.00000i −3.72248 − 6.44752i −2.73861 − 1.58114i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.f odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 130.3.o.b ✓ 16
13.f odd 12 1 inner 130.3.o.b ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.3.o.b ✓ 16 1.a even 1 1 trivial
130.3.o.b ✓ 16 13.f odd 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} + 46 T_{3}^{14} + 24 T_{3}^{13} + 1498 T_{3}^{12} + 1128 T_{3}^{11} + 23560 T_{3}^{10} + \cdots + 3500641 \) acting on \(S_{3}^{\mathrm{new}}(130, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 2 T^{3} + 2 T^{2} + \cdots + 4)^{4} \) Copy content Toggle raw display
$3$ \( T^{16} + 46 T^{14} + \cdots + 3500641 \) Copy content Toggle raw display
$5$ \( (T^{4} + 25)^{4} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 7037364321 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 2023133572161 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 66\!\cdots\!41 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 54\!\cdots\!21 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 40\!\cdots\!41 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 24\!\cdots\!41 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 3152219151601 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 14\!\cdots\!76 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 65\!\cdots\!21 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 23\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 14\!\cdots\!01 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 29\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots - 1454822525936)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 60\!\cdots\!21 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 28\!\cdots\!01 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 92\!\cdots\!61 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 24\!\cdots\!01 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots - 162220212416496)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 18\!\cdots\!61 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 17\!\cdots\!21 \) Copy content Toggle raw display
show more
show less