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Properties

Label 130.3.o.a
Level $130$
Weight $3$
Character orbit 130.o
Analytic conductor $3.542$
Analytic rank $0$
Dimension $16$
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] = [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: \( x^{16} - 8 x^{15} + 100 x^{14} - 560 x^{13} + 3632 x^{12} - 14876 x^{11} + 62910 x^{10} - 190580 x^{9} + \cdots + 404521 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{11} + \beta_{2}) q^{2} + (\beta_{14} + \beta_{12} + \cdots - \beta_{3}) q^{3}+ \cdots + ( - \beta_{15} - 3 \beta_{14} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{11} + \beta_{2}) q^{2} + (\beta_{14} + \beta_{12} + \cdots - \beta_{3}) q^{3}+ \cdots + (3 \beta_{15} + 9 \beta_{14} + \cdots - 55) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{2} - 12 q^{6} + 16 q^{7} - 32 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 8 q^{2} - 12 q^{6} + 16 q^{7} - 32 q^{8} - 4 q^{9} + 28 q^{11} - 28 q^{13} - 40 q^{14} - 20 q^{15} + 32 q^{16} - 60 q^{17} + 8 q^{18} - 56 q^{19} + 104 q^{21} + 56 q^{22} + 24 q^{23} + 24 q^{24} - 52 q^{26} + 24 q^{27} + 32 q^{28} - 36 q^{29} + 60 q^{30} - 24 q^{31} + 32 q^{32} - 64 q^{33} + 64 q^{34} - 20 q^{35} - 24 q^{36} + 320 q^{37} - 116 q^{39} - 72 q^{41} - 104 q^{42} + 36 q^{43} - 112 q^{44} + 80 q^{45} - 52 q^{46} - 184 q^{47} - 156 q^{49} + 40 q^{50} - 144 q^{52} + 352 q^{53} + 276 q^{54} + 20 q^{55} - 24 q^{56} + 100 q^{57} - 216 q^{58} - 132 q^{59} + 40 q^{60} + 20 q^{61} - 24 q^{62} - 276 q^{63} + 20 q^{65} - 152 q^{66} + 140 q^{67} - 64 q^{68} + 168 q^{69} + 40 q^{70} + 360 q^{71} + 32 q^{72} + 64 q^{73} + 4 q^{74} + 60 q^{75} + 112 q^{76} + 24 q^{78} - 248 q^{79} - 324 q^{81} + 72 q^{82} + 64 q^{83} + 208 q^{84} + 120 q^{85} - 64 q^{86} - 192 q^{87} - 176 q^{89} - 60 q^{91} + 112 q^{92} + 152 q^{93} + 184 q^{94} - 300 q^{95} + 280 q^{97} - 48 q^{98} - 448 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} + 100 x^{14} - 560 x^{13} + 3632 x^{12} - 14876 x^{11} + 62910 x^{10} - 190580 x^{9} + \cdots + 404521 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 837053 \nu^{14} + 5859371 \nu^{13} - 75891445 \nu^{12} + 379176847 \nu^{11} + \cdots - 153670175646 ) / 4127965270 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 6017906 \nu^{15} - 52603455 \nu^{14} + 132212412 \nu^{13} - 6061919023 \nu^{12} + \cdots - 16277306292520 ) / 1481939531930 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 6694 \nu^{15} + 10466 \nu^{14} + 189265 \nu^{13} + 1897389 \nu^{12} - 5104533 \nu^{11} + \cdots + 481554723 ) / 1648431070 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 6017906 \nu^{15} - 142872045 \nu^{14} + 1236116088 \nu^{13} - 11868219302 \nu^{12} + \cdots - 20713127302595 ) / 1481939531930 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2889254 \nu^{15} - 20133962 \nu^{14} + 288668453 \nu^{13} - 1515564674 \nu^{12} + \cdots - 5246108861617 ) / 621458513390 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 2889254 \nu^{15} + 23204848 \nu^{14} - 310164655 \nu^{13} + 1719545243 \nu^{12} + \cdots + 1329393016848 ) / 621458513390 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 212076073 \nu^{15} - 4250238290 \nu^{14} + 20164842969 \nu^{13} - 415035781823 \nu^{12} + \cdots - 823259391653777 ) / 19265213915090 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 212076073 \nu^{15} - 7431379385 \nu^{14} + 61606480756 \nu^{13} - 636159120831 \nu^{12} + \cdots - 976227708790711 ) / 19265213915090 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 13160968 \nu^{15} + 154281896 \nu^{14} - 1627978304 \nu^{13} + 11597913028 \nu^{12} + \cdots + 10647775872656 ) / 506979313555 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 589683658 \nu^{15} - 6582062336 \nu^{14} + 71478279857 \nu^{13} - 494026597597 \nu^{12} + \cdots - 445826666683216 ) / 19265213915090 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 26321936 \nu^{15} + 197414520 \nu^{14} - 2477911704 \nu^{13} + 13112305856 \nu^{12} + \cdots + 1501103119776 ) / 506979313555 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 1141907487 \nu^{15} - 7622078522 \nu^{14} + 101527773203 \nu^{13} - 487840787096 \nu^{12} + \cdots + 194276176299465 ) / 19265213915090 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 1759954017 \nu^{15} - 15348787626 \nu^{14} + 181112592173 \nu^{13} - 1076168027467 \nu^{12} + \cdots - 687312318730635 ) / 19265213915090 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 1849520891 \nu^{15} + 11674675451 \nu^{14} - 159973459237 \nu^{13} + \cdots - 144765876361262 ) / 19265213915090 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 338943268 \nu^{15} - 2523346198 \nu^{14} + 31736048688 \nu^{13} - 166813543443 \nu^{12} + \cdots - 9337435740620 ) / 1481939531930 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_{10} + \beta_{9} + \beta_{6} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -2\beta_{14} - 2\beta_{13} + \beta_{11} + 2\beta_{10} + 3\beta_{6} - \beta_{5} + \beta_{3} + \beta_{2} + \beta _1 - 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3 \beta_{15} - \beta_{14} - 2 \beta_{13} + 3 \beta_{12} + 18 \beta_{11} - 14 \beta_{10} - 17 \beta_{9} + \cdots - 15 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 6 \beta_{15} + 37 \beta_{14} + 35 \beta_{13} + 6 \beta_{12} + 8 \beta_{11} - 57 \beta_{10} - 7 \beta_{9} + \cdots + 138 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 69 \beta_{15} + 55 \beta_{14} + 51 \beta_{13} - 75 \beta_{12} - 487 \beta_{11} + 186 \beta_{10} + \cdots + 456 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 222 \beta_{15} - 647 \beta_{14} - 654 \beta_{13} - 240 \beta_{12} - 840 \beta_{11} + 1342 \beta_{10} + \cdots - 1952 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1217 \beta_{15} - 1751 \beta_{14} - 1428 \beta_{13} + 1547 \beta_{12} + 10090 \beta_{11} - 2054 \beta_{10} + \cdots - 11056 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 5918 \beta_{15} + 10680 \beta_{14} + 12000 \beta_{13} + 7322 \beta_{12} + 29807 \beta_{11} + \cdots + 25418 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 18560 \beta_{15} + 45186 \beta_{14} + 38806 \beta_{13} - 28164 \beta_{12} - 183394 \beta_{11} + \cdots + 246374 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 138769 \beta_{15} - 164605 \beta_{14} - 206444 \beta_{13} - 197445 \beta_{12} - 821937 \beta_{11} + \cdots - 248163 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 233238 \beta_{15} - 1055727 \beta_{14} - 978949 \beta_{13} + 433526 \beta_{12} + 2976685 \beta_{11} + \cdots - 5217972 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 3028517 \beta_{15} + 2255054 \beta_{14} + 3197852 \beta_{13} + 4899243 \beta_{12} + 20104228 \beta_{11} + \cdots - 271794 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 1670682 \beta_{15} + 23228257 \beta_{14} + 22973873 \beta_{13} - 4638348 \beta_{12} + \cdots + 105831505 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 62703355 \beta_{15} - 23374821 \beta_{14} - 40945569 \beta_{13} - 113909705 \beta_{12} + \cdots + 111149522 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 28605470 \beta_{15} - 488772233 \beta_{14} - 506129522 \beta_{13} - 14267484 \beta_{12} + \cdots - 2053305464 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(27\) \(41\)
\(\chi(n)\) \(1\) \(-\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
0.500000 + 0.248952i
0.500000 − 4.56870i
0.500000 + 2.95698i
0.500000 + 1.36277i
0.500000 − 4.40016i
0.500000 − 1.41097i
0.500000 + 2.56844i
0.500000 + 3.24268i
0.500000 − 0.248952i
0.500000 + 4.56870i
0.500000 − 2.95698i
0.500000 − 1.36277i
0.500000 + 4.40016i
0.500000 + 1.41097i
0.500000 − 2.56844i
0.500000 − 3.24268i
0.366025 − 1.36603i −2.71736 − 4.70661i −1.73205 − 1.00000i −1.58114 − 1.58114i −7.42398 + 1.98925i −0.203293 − 0.758701i −2.00000 + 2.00000i −10.2681 + 17.7849i −2.73861 + 1.58114i
11.2 0.366025 − 1.36603i −0.308537 − 0.534401i −1.73205 − 1.00000i −1.58114 − 1.58114i −0.842938 + 0.225865i −0.973520 − 3.63323i −2.00000 + 2.00000i 4.30961 − 7.46446i −2.73861 + 1.58114i
11.3 0.366025 − 1.36603i 0.248375 + 0.430197i −1.73205 − 1.00000i 1.58114 + 1.58114i 0.678572 − 0.181823i −3.32551 − 12.4110i −2.00000 + 2.00000i 4.37662 − 7.58053i 2.73861 − 1.58114i
11.4 0.366025 − 1.36603i 1.04548 + 1.81082i −1.73205 − 1.00000i 1.58114 + 1.58114i 2.85629 − 0.765341i 3.30617 + 12.3388i −2.00000 + 2.00000i 2.31396 − 4.00790i 2.73861 − 1.58114i
41.1 −1.36603 − 0.366025i −1.18833 − 2.05824i 1.73205 + 1.00000i −1.58114 + 1.58114i 0.869917 + 3.24657i −6.47546 + 1.73509i −2.00000 − 2.00000i 1.67575 − 2.90249i 2.73861 − 1.58114i
41.2 −1.36603 − 0.366025i −0.851209 − 1.47434i 1.73205 + 1.00000i 1.58114 − 1.58114i 0.623129 + 2.32555i −1.60880 + 0.431076i −2.00000 − 2.00000i 3.05088 − 5.28429i −2.73861 + 1.58114i
41.3 −1.36603 − 0.366025i 1.13850 + 1.97194i 1.73205 + 1.00000i 1.58114 − 1.58114i −0.833438 − 3.11043i 4.04700 − 1.08439i −2.00000 − 2.00000i 1.90765 − 3.30414i −2.73861 + 1.58114i
41.4 −1.36603 − 0.366025i 2.63309 + 4.56065i 1.73205 + 1.00000i −1.58114 + 1.58114i −1.92756 − 7.19374i 13.2334 − 3.54588i −2.00000 − 2.00000i −9.36633 + 16.2230i 2.73861 − 1.58114i
71.1 0.366025 + 1.36603i −2.71736 + 4.70661i −1.73205 + 1.00000i −1.58114 + 1.58114i −7.42398 − 1.98925i −0.203293 + 0.758701i −2.00000 − 2.00000i −10.2681 − 17.7849i −2.73861 − 1.58114i
71.2 0.366025 + 1.36603i −0.308537 + 0.534401i −1.73205 + 1.00000i −1.58114 + 1.58114i −0.842938 − 0.225865i −0.973520 + 3.63323i −2.00000 − 2.00000i 4.30961 + 7.46446i −2.73861 − 1.58114i
71.3 0.366025 + 1.36603i 0.248375 − 0.430197i −1.73205 + 1.00000i 1.58114 − 1.58114i 0.678572 + 0.181823i −3.32551 + 12.4110i −2.00000 − 2.00000i 4.37662 + 7.58053i 2.73861 + 1.58114i
71.4 0.366025 + 1.36603i 1.04548 − 1.81082i −1.73205 + 1.00000i 1.58114 − 1.58114i 2.85629 + 0.765341i 3.30617 − 12.3388i −2.00000 − 2.00000i 2.31396 + 4.00790i 2.73861 + 1.58114i
111.1 −1.36603 + 0.366025i −1.18833 + 2.05824i 1.73205 − 1.00000i −1.58114 − 1.58114i 0.869917 − 3.24657i −6.47546 − 1.73509i −2.00000 + 2.00000i 1.67575 + 2.90249i 2.73861 + 1.58114i
111.2 −1.36603 + 0.366025i −0.851209 + 1.47434i 1.73205 − 1.00000i 1.58114 + 1.58114i 0.623129 − 2.32555i −1.60880 − 0.431076i −2.00000 + 2.00000i 3.05088 + 5.28429i −2.73861 − 1.58114i
111.3 −1.36603 + 0.366025i 1.13850 − 1.97194i 1.73205 − 1.00000i 1.58114 + 1.58114i −0.833438 + 3.11043i 4.04700 + 1.08439i −2.00000 + 2.00000i 1.90765 + 3.30414i −2.73861 − 1.58114i
111.4 −1.36603 + 0.366025i 2.63309 − 4.56065i 1.73205 − 1.00000i −1.58114 − 1.58114i −1.92756 + 7.19374i 13.2334 + 3.54588i −2.00000 + 2.00000i −9.36633 − 16.2230i 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.a ✓ 16
13.f odd 12 1 inner 130.3.o.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.3.o.a ✓ 16 1.a even 1 1 trivial
130.3.o.a ✓ 16 13.f odd 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} + 38 T_{3}^{14} - 8 T_{3}^{13} + 1154 T_{3}^{12} - 168 T_{3}^{11} + 9752 T_{3}^{10} + \cdots + 28561 \) 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
$3$ \( T^{16} + 38 T^{14} + \cdots + 28561 \) Copy content Toggle raw display
$5$ \( (T^{4} + 25)^{4} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 96591045681 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 59\!\cdots\!61 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 66\!\cdots\!41 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 21\!\cdots\!41 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 25\!\cdots\!81 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 36\!\cdots\!81 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 48\!\cdots\!41 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 40\!\cdots\!76 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 10\!\cdots\!01 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 19\!\cdots\!41 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 13\!\cdots\!21 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 38\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots - 5651357609456)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 53\!\cdots\!01 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 13\!\cdots\!61 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 79\!\cdots\!81 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 68\!\cdots\!81 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 18\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 13\!\cdots\!44)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 19\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 21\!\cdots\!81 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 11\!\cdots\!01 \) Copy content Toggle raw display
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