↑

Properties

Label 154.4.e.a
Level $154$
Weight $4$
Character orbit 154.e
Analytic conductor $9.086$
Analytic rank $0$
Dimension $10$
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] = [154,4,Mod(23,154)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(154, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("154.23");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 154 = 2 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 154.e (of order \(3\), degree \(2\), 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: \(9.08629414088\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} + 70 x^{8} - 155 x^{7} + 3731 x^{6} - 7883 x^{5} + 88903 x^{4} - 208678 x^{3} + \cdots + 5948721 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 \beta_{3} q^{2} + (2 \beta_{3} + \beta_{2} + \beta_1 - 2) q^{3} + (4 \beta_{3} - 4) q^{4} + ( - \beta_{8} - 4 \beta_{3}) q^{5} + ( - 2 \beta_{2} + 4) q^{6} + (\beta_{9} - \beta_{6} - 4 \beta_{3} + 3) q^{7} + 8 q^{8} + ( - \beta_{8} - 5 \beta_{3} - 3 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 \beta_{3} q^{2} + (2 \beta_{3} + \beta_{2} + \beta_1 - 2) q^{3} + (4 \beta_{3} - 4) q^{4} + ( - \beta_{8} - 4 \beta_{3}) q^{5} + ( - 2 \beta_{2} + 4) q^{6} + (\beta_{9} - \beta_{6} - 4 \beta_{3} + 3) q^{7} + 8 q^{8} + ( - \beta_{8} - 5 \beta_{3} - 3 \beta_1) q^{9} + (2 \beta_{8} - 2 \beta_{4} + 8 \beta_{3} - 8) q^{10} + (11 \beta_{3} - 11) q^{11} + ( - 8 \beta_{3} - 4 \beta_1) q^{12} + (\beta_{9} - \beta_{7} + \beta_{5} + \cdots + 12) q^{13}+ \cdots + (11 \beta_{4} - 33 \beta_{2} + 55) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 10 q^{2} - 11 q^{3} - 20 q^{4} - 20 q^{5} + 44 q^{6} + 10 q^{7} + 80 q^{8} - 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 10 q^{2} - 11 q^{3} - 20 q^{4} - 20 q^{5} + 44 q^{6} + 10 q^{7} + 80 q^{8} - 28 q^{9} - 40 q^{10} - 55 q^{11} - 44 q^{12} + 122 q^{13} - 64 q^{14} + 54 q^{15} - 80 q^{16} - 185 q^{17} - 56 q^{18} - 73 q^{19} + 160 q^{20} + 4 q^{21} + 220 q^{22} - 8 q^{23} - 88 q^{24} - 201 q^{25} - 122 q^{26} + 328 q^{27} + 88 q^{28} + 42 q^{29} - 54 q^{30} - 315 q^{31} - 160 q^{32} - 121 q^{33} + 740 q^{34} - 211 q^{35} + 224 q^{36} - 6 q^{37} - 146 q^{38} + 99 q^{39} - 160 q^{40} + 750 q^{41} + 134 q^{42} + 868 q^{43} - 220 q^{44} - 807 q^{45} - 16 q^{46} - 547 q^{47} + 352 q^{48} - 662 q^{49} + 804 q^{50} - 626 q^{51} - 244 q^{52} + 601 q^{53} - 328 q^{54} + 440 q^{55} + 80 q^{56} + 344 q^{57} - 42 q^{58} - 1564 q^{59} - 108 q^{60} - 606 q^{61} + 1260 q^{62} - 264 q^{63} + 640 q^{64} - 427 q^{65} - 242 q^{66} + 820 q^{67} - 740 q^{68} - 474 q^{69} - 122 q^{70} + 2254 q^{71} - 224 q^{72} - 2018 q^{73} - 12 q^{74} - 1199 q^{75} + 584 q^{76} + 242 q^{77} - 396 q^{78} - 489 q^{79} - 320 q^{80} + 2351 q^{81} - 750 q^{82} + 2334 q^{83} - 284 q^{84} - 742 q^{85} - 868 q^{86} - 3142 q^{87} - 440 q^{88} - 973 q^{89} + 3228 q^{90} + 1601 q^{91} + 64 q^{92} - 230 q^{93} - 1094 q^{94} + 4731 q^{95} - 352 q^{96} + 2622 q^{97} + 944 q^{98} + 616 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} + 70 x^{8} - 155 x^{7} + 3731 x^{6} - 7883 x^{5} + 88903 x^{4} - 208678 x^{3} + \cdots + 5948721 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 355655814 \nu^{9} - 2759546548 \nu^{8} + 14398495266 \nu^{7} - 149899883037 \nu^{6} + \cdots - 52\!\cdots\!75 ) / 23\!\cdots\!11 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 713140894475 \nu^{9} - 1002289071257 \nu^{8} + 47676351269726 \nu^{7} + \cdots + 24\!\cdots\!51 ) / 18\!\cdots\!43 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3470858176 \nu^{9} - 36534855698 \nu^{8} + 190628038941 \nu^{7} - 2110438870821 \nu^{6} + \cdots - 61\!\cdots\!27 ) / 23\!\cdots\!11 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 4871489849290 \nu^{9} - 38301714852640 \nu^{8} + 185610203086107 \nu^{7} + \cdots - 19\!\cdots\!48 ) / 80\!\cdots\!47 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 36029470910476 \nu^{9} - 257041122723076 \nu^{8} + \cdots - 20\!\cdots\!81 ) / 56\!\cdots\!29 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 41689915460250 \nu^{9} - 297648157446767 \nu^{8} + \cdots + 23\!\cdots\!03 ) / 56\!\cdots\!29 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 17435285524994 \nu^{9} - 3882255030802 \nu^{8} + \cdots - 87\!\cdots\!50 ) / 18\!\cdots\!43 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 55471951493409 \nu^{9} - 406531795343789 \nu^{8} + \cdots - 13\!\cdots\!16 ) / 56\!\cdots\!29 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} - \beta_{4} + 28\beta_{3} - \beta_{2} - \beta _1 - 28 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{9} + 2\beta_{7} + \beta_{6} - \beta_{5} + 34\beta_{2} + 33 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7\beta_{9} - 49\beta_{8} + 2\beta_{7} - 5\beta_{6} + 5\beta_{5} - 973\beta_{3} + 61\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 71 \beta_{9} + 63 \beta_{8} - 62 \beta_{7} + 62 \beta_{6} - 71 \beta_{5} - 63 \beta_{4} + 2001 \beta_{3} + \cdots - 2001 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -436\beta_{9} + 436\beta_{7} - 97\beta_{6} - 533\beta_{5} + 2176\beta_{4} + 3181\beta_{2} + 36856 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 3387\beta_{9} - 5220\beta_{8} - 3630\beta_{7} - 7017\beta_{6} + 7017\beta_{5} - 104211\beta_{3} + 48979\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 2526 \beta_{9} + 96022 \beta_{8} - 29901 \beta_{7} + 29901 \beta_{6} + 2526 \beta_{5} + \cdots - 1471312 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 341549 \beta_{9} + 341549 \beta_{7} + 165928 \beta_{6} - 175621 \beta_{5} + 316413 \beta_{4} + \cdots + 5188830 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/154\mathbb{Z}\right)^\times\).

\(n\) \(45\) \(57\)
\(\chi(n)\) \(-\beta_{3}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1
2.98314 − 5.16695i
2.41396 − 4.18110i
1.16792 − 2.02289i
−2.66688 + 4.61917i
−3.39814 + 5.88576i
2.98314 + 5.16695i
2.41396 + 4.18110i
1.16792 + 2.02289i
−2.66688 − 4.61917i
−3.39814 − 5.88576i
−1.00000 + 1.73205i −3.98314 − 6.89900i −2.00000 − 3.46410i −8.78142 + 15.2099i 15.9326 17.1754 + 6.92862i 8.00000 −18.2308 + 31.5767i −17.5628 − 30.4197i
23.2 −1.00000 + 1.73205i −3.41396 − 5.91315i −2.00000 − 3.46410i −2.06838 + 3.58254i 13.6558 −14.1227 − 11.9812i 8.00000 −9.81026 + 16.9919i −4.13676 − 7.16507i
23.3 −1.00000 + 1.73205i −2.16792 − 3.75495i −2.00000 − 3.46410i 8.10401 − 14.0366i 8.67167 −1.57044 + 18.4536i 8.00000 4.10026 − 7.10185i 16.2080 + 28.0731i
23.4 −1.00000 + 1.73205i 1.66688 + 2.88712i −2.00000 − 3.46410i 0.442391 − 0.766243i −6.66752 11.4774 − 14.5351i 8.00000 7.94303 − 13.7577i 0.884782 + 1.53249i
23.5 −1.00000 + 1.73205i 2.39814 + 4.15371i −2.00000 − 3.46410i −7.69661 + 13.3309i −9.59257 −7.95968 + 16.7225i 8.00000 1.99782 − 3.46033i −15.3932 − 26.6618i
67.1 −1.00000 − 1.73205i −3.98314 + 6.89900i −2.00000 + 3.46410i −8.78142 − 15.2099i 15.9326 17.1754 − 6.92862i 8.00000 −18.2308 − 31.5767i −17.5628 + 30.4197i
67.2 −1.00000 − 1.73205i −3.41396 + 5.91315i −2.00000 + 3.46410i −2.06838 − 3.58254i 13.6558 −14.1227 + 11.9812i 8.00000 −9.81026 − 16.9919i −4.13676 + 7.16507i
67.3 −1.00000 − 1.73205i −2.16792 + 3.75495i −2.00000 + 3.46410i 8.10401 + 14.0366i 8.67167 −1.57044 − 18.4536i 8.00000 4.10026 + 7.10185i 16.2080 − 28.0731i
67.4 −1.00000 − 1.73205i 1.66688 − 2.88712i −2.00000 + 3.46410i 0.442391 + 0.766243i −6.66752 11.4774 + 14.5351i 8.00000 7.94303 + 13.7577i 0.884782 − 1.53249i
67.5 −1.00000 − 1.73205i 2.39814 − 4.15371i −2.00000 + 3.46410i −7.69661 − 13.3309i −9.59257 −7.95968 − 16.7225i 8.00000 1.99782 + 3.46033i −15.3932 + 26.6618i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 154.4.e.a ✓ 10
7.c even 3 1 inner 154.4.e.a ✓ 10
7.c even 3 1 1078.4.a.ba 5
7.d odd 6 1 1078.4.a.x 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.4.e.a ✓ 10 1.a even 1 1 trivial
154.4.e.a ✓ 10 7.c even 3 1 inner
1078.4.a.x 5 7.d odd 6 1
1078.4.a.ba 5 7.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} + 11 T_{3}^{9} + 142 T_{3}^{8} + 613 T_{3}^{7} + 5105 T_{3}^{6} + 13117 T_{3}^{5} + \cdots + 14220441 \) acting on \(S_{4}^{\mathrm{new}}(154, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2 T + 4)^{5} \) Copy content Toggle raw display
$3$ \( T^{10} + 11 T^{9} + \cdots + 14220441 \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 257217444 \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 4747561509943 \) Copy content Toggle raw display
$11$ \( (T^{2} + 11 T + 121)^{5} \) Copy content Toggle raw display
$13$ \( (T^{5} - 61 T^{4} + \cdots - 1235871)^{2} \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 11\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 24\!\cdots\!84 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 17\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( (T^{5} - 21 T^{4} + \cdots + 9654511473)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 45\!\cdots\!64 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 10\!\cdots\!24 \) Copy content Toggle raw display
$41$ \( (T^{5} - 375 T^{4} + \cdots + 4184213166)^{2} \) Copy content Toggle raw display
$43$ \( (T^{5} + \cdots - 1000873584272)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 20\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 43\!\cdots\!44 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 89\!\cdots\!09 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 43\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 72\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( (T^{5} - 1127 T^{4} + \cdots + 611193718962)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 20\!\cdots\!04 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 20\!\cdots\!49 \) Copy content Toggle raw display
$83$ \( (T^{5} + \cdots - 957399906264228)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 52\!\cdots\!16 \) Copy content Toggle raw display
$97$ \( (T^{5} + \cdots + 31101556401363)^{2} \) Copy content Toggle raw display
show more
show less