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Properties

Label 120.4.k.c
Level $120$
Weight $4$
Character orbit 120.k
Analytic conductor $7.080$
Analytic rank $0$
Dimension $14$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [120,4,Mod(61,120)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(120, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("120.61");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 120 = 2^{3} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 120.k (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: \(7.08022920069\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 4 x^{13} + 7 x^{12} - 22 x^{11} + 70 x^{10} - 232 x^{9} + 1080 x^{8} - 4000 x^{7} + \cdots + 2097152 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{15} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{7} q^{2} + 3 \beta_{3} q^{3} - \beta_{2} q^{4} - 5 \beta_{3} q^{5} + 3 \beta_1 q^{6} + (\beta_{10} + 2 \beta_1 - 3) q^{7} + (\beta_{10} + \beta_{5} + 3 \beta_{3} + \cdots - 1) q^{8}+ \cdots - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{7} q^{2} + 3 \beta_{3} q^{3} - \beta_{2} q^{4} - 5 \beta_{3} q^{5} + 3 \beta_1 q^{6} + (\beta_{10} + 2 \beta_1 - 3) q^{7} + (\beta_{10} + \beta_{5} + 3 \beta_{3} + \cdots - 1) q^{8}+ \cdots + ( - 9 \beta_{12} - 9 \beta_{11} + \cdots - 9) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 2 q^{2} - 2 q^{4} + 12 q^{6} - 28 q^{7} - 8 q^{8} - 126 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 2 q^{2} - 2 q^{4} + 12 q^{6} - 28 q^{7} - 8 q^{8} - 126 q^{9} - 20 q^{10} + 12 q^{12} - 8 q^{14} + 210 q^{15} - 22 q^{16} + 204 q^{17} + 18 q^{18} - 20 q^{20} - 84 q^{22} - 328 q^{23} - 138 q^{24} - 350 q^{25} - 4 q^{26} + 68 q^{28} - 30 q^{30} + 596 q^{31} + 588 q^{32} + 264 q^{33} + 756 q^{34} + 18 q^{36} - 1144 q^{38} + 230 q^{40} - 820 q^{41} + 768 q^{42} - 2084 q^{44} - 1060 q^{46} - 104 q^{47} - 48 q^{48} + 1110 q^{49} + 50 q^{50} - 1736 q^{52} - 108 q^{54} - 440 q^{55} + 3812 q^{56} + 168 q^{57} + 2664 q^{58} - 30 q^{60} - 772 q^{62} + 252 q^{63} + 2470 q^{64} + 972 q^{66} - 2864 q^{68} - 1280 q^{70} + 1592 q^{71} + 72 q^{72} - 2260 q^{73} + 1020 q^{74} - 2468 q^{76} - 2280 q^{78} - 220 q^{79} + 80 q^{80} + 1134 q^{81} + 3444 q^{82} - 84 q^{84} - 1184 q^{86} - 1044 q^{87} - 500 q^{88} - 2492 q^{89} + 180 q^{90} - 4536 q^{92} - 4300 q^{94} - 280 q^{95} + 1542 q^{96} + 3508 q^{97} + 6246 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 4 x^{13} + 7 x^{12} - 22 x^{11} + 70 x^{10} - 232 x^{9} + 1080 x^{8} - 4000 x^{7} + \cdots + 2097152 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 405 \nu^{13} - 4300 \nu^{12} + 14835 \nu^{11} - 44822 \nu^{10} + 104462 \nu^{9} + \cdots + 484179968 ) / 231211008 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5 \nu^{13} - 67 \nu^{12} + 67 \nu^{11} + 201 \nu^{10} - 2588 \nu^{9} + 10486 \nu^{8} + \cdots + 21954560 ) / 903168 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 2369 \nu^{13} - 5956 \nu^{12} + 45401 \nu^{11} - 365410 \nu^{10} + 1260906 \nu^{9} + \cdots - 2672558080 ) / 231211008 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 355 \nu^{13} - 340 \nu^{12} - 4651 \nu^{11} + 49670 \nu^{10} - 174270 \nu^{9} + 538776 \nu^{8} + \cdots + 408682496 ) / 33030144 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 335 \nu^{13} - 1500 \nu^{12} + 4489 \nu^{11} - 9514 \nu^{10} + 17018 \nu^{9} + 5096 \nu^{8} + \cdots + 106168320 ) / 28901376 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 4265 \nu^{13} + 32700 \nu^{12} - 179455 \nu^{11} + 604462 \nu^{10} - 1808966 \nu^{9} + \cdots - 387710976 ) / 231211008 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 8399 \nu^{13} - 20508 \nu^{12} + 265655 \nu^{11} - 961694 \nu^{10} + 3013366 \nu^{9} + \cdots + 3097755648 ) / 231211008 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 37 \nu^{13} + 92 \nu^{12} - 35 \nu^{11} - 858 \nu^{10} + 4786 \nu^{9} - 12488 \nu^{8} + \cdots - 4194304 ) / 786432 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 5863 \nu^{13} + 32612 \nu^{12} - 121841 \nu^{11} + 402258 \nu^{10} - 1203674 \nu^{9} + \cdots + 2916614144 ) / 115605504 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 1965 \nu^{13} + 5800 \nu^{12} - 23979 \nu^{11} + 90890 \nu^{10} - 234278 \nu^{9} + \cdots + 1076297728 ) / 28901376 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 17995 \nu^{13} - 54100 \nu^{12} + 212909 \nu^{11} - 758410 \nu^{10} + 1905906 \nu^{9} + \cdots - 8938848256 ) / 231211008 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{13} + \beta_{12} - \beta_{7} + \beta_{3} + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{11} + \beta_{10} + \beta_{8} - 2\beta_{6} - 3\beta_{5} + \beta_{4} + \beta_{2} + 3\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 4 \beta_{12} - \beta_{11} + 5 \beta_{10} + 2 \beta_{9} + 11 \beta_{8} + 4 \beta_{7} + 4 \beta_{6} + \cdots + 30 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 8 \beta_{13} + 4 \beta_{12} + 7 \beta_{11} + 7 \beta_{10} - 6 \beta_{9} - 13 \beta_{8} - 44 \beta_{7} + \cdots - 196 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 6 \beta_{13} - 2 \beta_{12} - 35 \beta_{11} + 15 \beta_{10} - 10 \beta_{9} - 39 \beta_{8} + \cdots + 552 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 40 \beta_{13} + 28 \beta_{12} - 53 \beta_{11} + 99 \beta_{10} + 14 \beta_{9} + 35 \beta_{8} + 364 \beta_{7} + \cdots + 844 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 254 \beta_{13} - 10 \beta_{12} - 71 \beta_{11} + 263 \beta_{10} - 146 \beta_{9} - 187 \beta_{8} + \cdots - 2444 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 36 \beta_{13} + 976 \beta_{12} - 837 \beta_{11} - 233 \beta_{10} - 18 \beta_{9} - 237 \beta_{8} + \cdots + 8004 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 246 \beta_{13} + 1338 \beta_{12} - 815 \beta_{11} - 1421 \beta_{10} + 1478 \beta_{9} - 1915 \beta_{8} + \cdots - 19252 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 2336 \beta_{13} - 9596 \beta_{12} + 5995 \beta_{11} + 2083 \beta_{10} + 5814 \beta_{9} + 7307 \beta_{8} + \cdots + 99556 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 17842 \beta_{13} - 2202 \beta_{12} + 30769 \beta_{11} + 1159 \beta_{10} - 4498 \beta_{9} + 1325 \beta_{8} + \cdots + 249660 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/120\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(41\) \(61\) \(97\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
61.1
−0.594754 + 2.76519i
−0.594754 − 2.76519i
1.74774 − 2.22383i
1.74774 + 2.22383i
−2.34569 + 1.58043i
−2.34569 − 1.58043i
2.82087 − 0.206677i
2.82087 + 0.206677i
2.49456 + 1.33311i
2.49456 − 1.33311i
−2.31762 − 1.62131i
−2.31762 + 1.62131i
0.194895 + 2.82170i
0.194895 − 2.82170i
−2.76519 − 0.594754i − 3.00000i 7.29254 + 3.28921i 5.00000i −1.78426 + 8.29557i −10.3416 −18.2090 − 13.4326i −9.00000 2.97377 − 13.8259i
61.2 −2.76519 + 0.594754i 3.00000i 7.29254 − 3.28921i − 5.00000i −1.78426 − 8.29557i −10.3416 −18.2090 + 13.4326i −9.00000 2.97377 + 13.8259i
61.3 −2.22383 − 1.74774i 3.00000i 1.89082 + 7.77334i − 5.00000i 5.24321 − 6.67148i 2.94025 9.38089 − 20.5912i −9.00000 −8.73869 + 11.1191i
61.4 −2.22383 + 1.74774i − 3.00000i 1.89082 − 7.77334i 5.00000i 5.24321 + 6.67148i 2.94025 9.38089 + 20.5912i −9.00000 −8.73869 − 11.1191i
61.5 −1.58043 − 2.34569i − 3.00000i −3.00448 + 7.41439i 5.00000i −7.03706 + 4.74129i −3.68242 22.1402 − 4.67037i −9.00000 11.7284 − 7.90215i
61.6 −1.58043 + 2.34569i 3.00000i −3.00448 − 7.41439i − 5.00000i −7.03706 − 4.74129i −3.68242 22.1402 + 4.67037i −9.00000 11.7284 + 7.90215i
61.7 −0.206677 − 2.82087i 3.00000i −7.91457 + 1.16601i − 5.00000i 8.46260 − 0.620030i 31.1865 4.92492 + 22.0850i −9.00000 −14.1043 + 1.03338i
61.8 −0.206677 + 2.82087i − 3.00000i −7.91457 − 1.16601i 5.00000i 8.46260 + 0.620030i 31.1865 4.92492 − 22.0850i −9.00000 −14.1043 − 1.03338i
61.9 1.33311 − 2.49456i 3.00000i −4.44564 − 6.65104i − 5.00000i 7.48367 + 3.99933i −22.4239 −22.5179 + 2.22335i −9.00000 −12.4728 − 6.66555i
61.10 1.33311 + 2.49456i − 3.00000i −4.44564 + 6.65104i 5.00000i 7.48367 − 3.99933i −22.4239 −22.5179 − 2.22335i −9.00000 −12.4728 + 6.66555i
61.11 1.62131 − 2.31762i − 3.00000i −2.74270 − 7.51516i 5.00000i −6.95285 − 4.86393i −31.1694 −21.8640 − 5.82786i −9.00000 11.5881 + 8.10655i
61.12 1.62131 + 2.31762i 3.00000i −2.74270 + 7.51516i − 5.00000i −6.95285 + 4.86393i −31.1694 −21.8640 + 5.82786i −9.00000 11.5881 − 8.10655i
61.13 2.82170 − 0.194895i 3.00000i 7.92403 − 1.09987i − 5.00000i 0.584684 + 8.46511i 19.4906 22.1449 − 4.64786i −9.00000 −0.974474 − 14.1085i
61.14 2.82170 + 0.194895i − 3.00000i 7.92403 + 1.09987i 5.00000i 0.584684 − 8.46511i 19.4906 22.1449 + 4.64786i −9.00000 −0.974474 + 14.1085i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 61.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 120.4.k.c ✓ 14
3.b odd 2 1 360.4.k.d 14
4.b odd 2 1 480.4.k.c 14
8.b even 2 1 inner 120.4.k.c ✓ 14
8.d odd 2 1 480.4.k.c 14
12.b even 2 1 1440.4.k.d 14
24.f even 2 1 1440.4.k.d 14
24.h odd 2 1 360.4.k.d 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.4.k.c ✓ 14 1.a even 1 1 trivial
120.4.k.c ✓ 14 8.b even 2 1 inner
360.4.k.d 14 3.b odd 2 1
360.4.k.d 14 24.h odd 2 1
480.4.k.c 14 4.b odd 2 1
480.4.k.c 14 8.d odd 2 1
1440.4.k.d 14 12.b even 2 1
1440.4.k.d 14 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{7} + 14T_{7}^{6} - 1380T_{7}^{5} - 18584T_{7}^{4} + 397440T_{7}^{3} + 4875648T_{7}^{2} - 1020672T_{7} - 47570432 \) acting on \(S_{4}^{\mathrm{new}}(120, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + 2 T^{13} + \cdots + 2097152 \) Copy content Toggle raw display
$3$ \( (T^{2} + 9)^{7} \) Copy content Toggle raw display
$5$ \( (T^{2} + 25)^{7} \) Copy content Toggle raw display
$7$ \( (T^{7} + 14 T^{6} + \cdots - 47570432)^{2} \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 87\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 25\!\cdots\!44 \) Copy content Toggle raw display
$17$ \( (T^{7} + \cdots + 6976273176576)^{2} \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 10\!\cdots\!24 \) Copy content Toggle raw display
$23$ \( (T^{7} + \cdots + 94987172617216)^{2} \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 19\!\cdots\!56 \) Copy content Toggle raw display
$31$ \( (T^{7} + \cdots - 3035229478912)^{2} \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 50\!\cdots\!44 \) Copy content Toggle raw display
$41$ \( (T^{7} + \cdots + 553103116856704)^{2} \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 83\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( (T^{7} + \cdots + 12\!\cdots\!92)^{2} \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 27\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 95\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 10\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( (T^{7} + \cdots - 44\!\cdots\!24)^{2} \) Copy content Toggle raw display
$73$ \( (T^{7} + \cdots + 18\!\cdots\!36)^{2} \) Copy content Toggle raw display
$79$ \( (T^{7} + \cdots - 37\!\cdots\!60)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 29\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( (T^{7} + \cdots - 33\!\cdots\!20)^{2} \) Copy content Toggle raw display
$97$ \( (T^{7} + \cdots + 34\!\cdots\!04)^{2} \) Copy content Toggle raw display
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