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Properties

Label 119.4.a.e
Level $119$
Weight $4$
Character orbit 119.a
Self dual yes
Analytic conductor $7.021$
Analytic rank $0$
Dimension $9$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [119,4,Mod(1,119)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(119, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("119.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 119 = 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 119.a (trivial)

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: yes
Analytic conductor: \(7.02122729068\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 2x^{8} - 53x^{7} + 90x^{6} + 880x^{5} - 1087x^{4} - 4674x^{3} + 2515x^{2} + 1814x + 36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{5} + 1) q^{3} + (\beta_{2} + 4) q^{4} + (\beta_{6} + \beta_1 - 1) q^{5} + ( - \beta_{3} + \beta_1 + 1) q^{6} + 7 q^{7} + (\beta_{8} + \beta_{7} + \beta_{4} + \cdots + 2) q^{8}+ \cdots + ( - \beta_{7} - \beta_{6} + 2 \beta_{5} + \cdots + 8) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{5} + 1) q^{3} + (\beta_{2} + 4) q^{4} + (\beta_{6} + \beta_1 - 1) q^{5} + ( - \beta_{3} + \beta_1 + 1) q^{6} + 7 q^{7} + (\beta_{8} + \beta_{7} + \beta_{4} + \cdots + 2) q^{8}+ \cdots + ( - 11 \beta_{8} + 15 \beta_{7} + \cdots - 302) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 2 q^{2} + 11 q^{3} + 38 q^{4} - 3 q^{5} + 9 q^{6} + 63 q^{7} + 24 q^{8} + 74 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 2 q^{2} + 11 q^{3} + 38 q^{4} - 3 q^{5} + 9 q^{6} + 63 q^{7} + 24 q^{8} + 74 q^{9} + 134 q^{10} - 8 q^{11} + 56 q^{12} + 164 q^{13} + 14 q^{14} + 34 q^{15} + 178 q^{16} + 153 q^{17} + 98 q^{18} + 244 q^{19} - 41 q^{20} + 77 q^{21} - 80 q^{22} - 14 q^{23} + 298 q^{24} + 684 q^{25} + 326 q^{26} + 218 q^{27} + 266 q^{28} - 234 q^{29} - 335 q^{30} + 555 q^{31} - 181 q^{32} + 458 q^{33} + 34 q^{34} - 21 q^{35} - 1221 q^{36} - 364 q^{37} - 714 q^{38} - 52 q^{39} + 123 q^{40} - 45 q^{41} + 63 q^{42} - 135 q^{43} - 748 q^{44} - 844 q^{45} - 1576 q^{46} - 172 q^{47} - 949 q^{48} + 441 q^{49} - 2901 q^{50} + 187 q^{51} - 1596 q^{52} + 101 q^{53} - 1163 q^{54} + 1260 q^{55} + 168 q^{56} - 602 q^{57} + 1062 q^{58} + 280 q^{59} - 1727 q^{60} + 639 q^{61} - 1708 q^{62} + 518 q^{63} - 2390 q^{64} + 638 q^{65} - 2476 q^{66} + 35 q^{67} + 646 q^{68} + 1288 q^{69} + 938 q^{70} - 1616 q^{71} + 1335 q^{72} + 1049 q^{73} - 370 q^{74} + 1260 q^{75} + 4964 q^{76} - 56 q^{77} - 4714 q^{78} + 2304 q^{79} - 3996 q^{80} - 791 q^{81} - 215 q^{82} + 2508 q^{83} + 392 q^{84} - 51 q^{85} + 623 q^{86} + 166 q^{87} - 416 q^{88} + 2762 q^{89} + 2935 q^{90} + 1148 q^{91} - 2392 q^{92} + 2784 q^{93} - 862 q^{94} - 3462 q^{95} + 2928 q^{96} + 3107 q^{97} + 98 q^{98} - 2396 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 2x^{8} - 53x^{7} + 90x^{6} + 880x^{5} - 1087x^{4} - 4674x^{3} + 2515x^{2} + 1814x + 36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 725 \nu^{8} - 40561 \nu^{7} - 35990 \nu^{6} + 1728180 \nu^{5} + 2133388 \nu^{4} + \cdots + 2882012 ) / 3388208 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 6667 \nu^{8} - 15479 \nu^{7} + 358366 \nu^{6} + 843788 \nu^{5} - 5557212 \nu^{4} + \cdots + 22936788 ) / 3388208 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 14061 \nu^{8} + 28847 \nu^{7} + 785794 \nu^{6} - 1229500 \nu^{5} - 14101860 \nu^{4} + \cdots - 31983300 ) / 3388208 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 37659 \nu^{8} + 68585 \nu^{7} + 1904318 \nu^{6} - 2952436 \nu^{5} - 29732012 \nu^{4} + \cdots - 38173324 ) / 3388208 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 41373 \nu^{8} + 96807 \nu^{7} + 2163922 \nu^{6} - 4509364 \nu^{5} - 35178740 \nu^{4} + \cdots - 33613820 ) / 1694104 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 89413 \nu^{8} - 178135 \nu^{7} - 4686210 \nu^{6} + 8174940 \nu^{5} + 75914692 \nu^{4} + \cdots + 78172932 ) / 3388208 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} + \beta_{7} + \beta_{4} + \beta_{2} + 20\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{8} - \beta_{6} - 5\beta_{5} + 3\beta_{4} - 2\beta_{3} + 25\beta_{2} + 246 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 39\beta_{8} + 35\beta_{7} + 8\beta_{6} + \beta_{5} + 42\beta_{4} - 6\beta_{3} + 36\beta_{2} + 436\beta _1 + 47 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -41\beta_{8} - 60\beta_{6} - 153\beta_{5} + 120\beta_{4} - 76\beta_{3} + 617\beta_{2} - 29\beta _1 + 5482 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1227 \beta_{8} + 1050 \beta_{7} + 382 \beta_{6} - 15 \beta_{5} + 1334 \beta_{4} - 334 \beta_{3} + \cdots + 756 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 1246 \beta_{8} + 29 \beta_{7} - 2266 \beta_{6} - 3895 \beta_{5} + 3697 \beta_{4} - 2402 \beta_{3} + \cdots + 128026 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−5.08433
−4.05606
−2.64062
−0.411462
−0.0204472
0.923542
3.78610
4.42628
5.07700
−5.08433 −2.28838 17.8505 −18.7142 11.6349 7.00000 −50.0830 −21.7633 95.1492
1.2 −4.05606 2.41760 8.45161 20.0958 −9.80593 7.00000 −1.83177 −21.1552 −81.5097
1.3 −2.64062 8.72657 −1.02714 −19.5792 −23.0435 7.00000 23.8372 49.1531 51.7011
1.4 −0.411462 −2.90636 −7.83070 −0.206478 1.19585 7.00000 6.51373 −18.5531 0.0849579
1.5 −0.0204472 −8.34282 −7.99958 −11.9677 0.170588 7.00000 0.327147 42.6027 0.244705
1.6 0.923542 8.63083 −7.14707 12.8089 7.97094 7.00000 −13.9890 47.4913 11.8296
1.7 3.78610 5.19121 6.33458 4.98519 19.6544 7.00000 −6.30544 −0.0513763 18.8744
1.8 4.42628 −5.22352 11.5919 16.9037 −23.1207 7.00000 15.8988 0.285172 74.8205
1.9 5.07700 4.79487 17.7759 −7.32611 24.3435 7.00000 49.6323 −4.00927 −37.1947
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(-1\)
\(17\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 119.4.a.e ✓ 9
3.b odd 2 1 1071.4.a.r 9
4.b odd 2 1 1904.4.a.s 9
7.b odd 2 1 833.4.a.g 9
17.b even 2 1 2023.4.a.h 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
119.4.a.e ✓ 9 1.a even 1 1 trivial
833.4.a.g 9 7.b odd 2 1
1071.4.a.r 9 3.b odd 2 1
1904.4.a.s 9 4.b odd 2 1
2023.4.a.h 9 17.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{9} - 2T_{2}^{8} - 53T_{2}^{7} + 90T_{2}^{6} + 880T_{2}^{5} - 1087T_{2}^{4} - 4674T_{2}^{3} + 2515T_{2}^{2} + 1814T_{2} + 36 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(119))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} - 2 T^{8} + \cdots + 36 \) Copy content Toggle raw display
$3$ \( T^{9} - 11 T^{8} + \cdots - 1313648 \) Copy content Toggle raw display
$5$ \( T^{9} + 3 T^{8} + \cdots + 143880840 \) Copy content Toggle raw display
$7$ \( (T - 7)^{9} \) Copy content Toggle raw display
$11$ \( T^{9} + \cdots - 9140353044480 \) Copy content Toggle raw display
$13$ \( T^{9} + \cdots - 221595169961984 \) Copy content Toggle raw display
$17$ \( (T - 17)^{9} \) Copy content Toggle raw display
$19$ \( T^{9} + \cdots + 42\!\cdots\!80 \) Copy content Toggle raw display
$23$ \( T^{9} + \cdots - 53\!\cdots\!28 \) Copy content Toggle raw display
$29$ \( T^{9} + \cdots - 99\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{9} + \cdots + 92\!\cdots\!52 \) Copy content Toggle raw display
$37$ \( T^{9} + \cdots - 14\!\cdots\!84 \) Copy content Toggle raw display
$41$ \( T^{9} + \cdots + 64\!\cdots\!64 \) Copy content Toggle raw display
$43$ \( T^{9} + \cdots + 18\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( T^{9} + \cdots + 16\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{9} + \cdots - 53\!\cdots\!12 \) Copy content Toggle raw display
$59$ \( T^{9} + \cdots - 42\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( T^{9} + \cdots + 27\!\cdots\!92 \) Copy content Toggle raw display
$67$ \( T^{9} + \cdots - 60\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( T^{9} + \cdots + 17\!\cdots\!80 \) Copy content Toggle raw display
$73$ \( T^{9} + \cdots + 17\!\cdots\!08 \) Copy content Toggle raw display
$79$ \( T^{9} + \cdots + 87\!\cdots\!80 \) Copy content Toggle raw display
$83$ \( T^{9} + \cdots + 42\!\cdots\!92 \) Copy content Toggle raw display
$89$ \( T^{9} + \cdots - 22\!\cdots\!72 \) Copy content Toggle raw display
$97$ \( T^{9} + \cdots - 10\!\cdots\!84 \) Copy content Toggle raw display
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