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Properties

Label 154.4.f.c
Level $154$
Weight $4$
Character orbit 154.f
Analytic conductor $9.086$
Analytic rank $0$
Dimension $8$
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] = [154,4,Mod(15,154)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(154, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("154.15");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 154 = 2 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 154.f (of order \(5\), 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: \(9.08629414088\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 108x^{6} - 215x^{5} + 11241x^{4} + 21415x^{3} + 1124662x^{2} + 1168333x + 119224561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 \beta_{3} q^{2} + (3 \beta_{3} + \beta_{2} + 3) q^{3} + ( - 4 \beta_{4} - 4 \beta_{3} + \cdots - 4) q^{4}+ \cdots + ( - 15 \beta_{4} - 17 \beta_{3} + 15 \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 \beta_{3} q^{2} + (3 \beta_{3} + \beta_{2} + 3) q^{3} + ( - 4 \beta_{4} - 4 \beta_{3} + \cdots - 4) q^{4}+ \cdots + (45 \beta_{7} + 64 \beta_{6} + \cdots + 202) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} + 20 q^{3} - 8 q^{4} - 21 q^{5} + 20 q^{6} + 14 q^{7} + 16 q^{8} + 94 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{2} + 20 q^{3} - 8 q^{4} - 21 q^{5} + 20 q^{6} + 14 q^{7} + 16 q^{8} + 94 q^{9} - 8 q^{10} - 55 q^{11} - 80 q^{12} + 77 q^{13} - 28 q^{14} - 70 q^{15} - 32 q^{16} - 271 q^{17} + 112 q^{18} + 259 q^{19} - 84 q^{20} + 140 q^{21} - 90 q^{22} - 80 q^{23} + 80 q^{24} + 25 q^{25} + 246 q^{26} + 290 q^{27} + 56 q^{28} + 31 q^{29} + 140 q^{30} + 725 q^{31} - 256 q^{32} + 30 q^{33} - 388 q^{34} + 147 q^{35} - 224 q^{36} + 250 q^{37} + 372 q^{38} - 215 q^{39} - 152 q^{40} - 180 q^{41} - 140 q^{42} - 1356 q^{43} + 300 q^{44} - 638 q^{45} + 160 q^{46} + 279 q^{47} + 320 q^{48} - 98 q^{49} + 50 q^{50} - 405 q^{51} - 492 q^{52} - 42 q^{53} - 560 q^{54} + 682 q^{55} + 448 q^{56} + 405 q^{57} - 62 q^{58} + 1593 q^{59} + 60 q^{60} - 1475 q^{61} + 860 q^{62} + 392 q^{63} - 128 q^{64} + 1800 q^{65} - 730 q^{66} - 2142 q^{67} - 1084 q^{68} - 600 q^{69} + 266 q^{70} + 1426 q^{71} - 752 q^{72} - 21 q^{73} - 500 q^{74} + 25 q^{75} - 584 q^{76} - 525 q^{77} + 1660 q^{78} + 859 q^{79} + 304 q^{80} + 3652 q^{81} - 360 q^{82} - 4193 q^{83} - 560 q^{84} - 2766 q^{85} - 548 q^{86} - 500 q^{87} + 480 q^{88} - 5766 q^{89} - 774 q^{90} + 861 q^{91} + 480 q^{92} + 160 q^{93} + 1122 q^{94} + 1170 q^{95} - 640 q^{96} - 64 q^{97} - 784 q^{98} + 210 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 108x^{6} - 215x^{5} + 11241x^{4} + 21415x^{3} + 1124662x^{2} + 1168333x + 119224561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 38814010115542 \nu^{7} + \cdots + 45\!\cdots\!80 ) / 16\!\cdots\!19 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 13\!\cdots\!45 \nu^{7} + 89057298007373 \nu^{6} + \cdots + 15\!\cdots\!89 ) / 16\!\cdots\!19 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 14\!\cdots\!15 \nu^{7} + \cdots - 16\!\cdots\!27 ) / 16\!\cdots\!19 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 129859326622 \nu^{7} - 13401691571339 \nu^{6} + 39597904357573 \nu^{5} + \cdots - 14\!\cdots\!55 ) / 15\!\cdots\!01 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 238969585088 \nu^{7} + 255979380157 \nu^{6} + \cdots + 30\!\cdots\!38 ) / 15\!\cdots\!01 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 13280058700688 \nu^{7} + 12790541596560 \nu^{6} + \cdots - 15\!\cdots\!85 ) / 15\!\cdots\!01 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + 105\beta_{3} - \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{6} + 105\beta_{5} - 104\beta_{4} - \beta_{2} - \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -211\beta_{7} - 2\beta_{6} - 2\beta_{5} - 10921\beta_{4} - 11131\beta_{3} + 11131\beta_{2} - 10921 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 212\beta_{7} + 11342\beta_{6} + 419\beta_{4} + 419\beta_{3} + 212\beta _1 - 21738 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 631\beta_{7} + 631\beta_{5} + 33390\beta_{3} - 1191122\beta_{2} - 32868\beta _1 + 33390 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 1224621 \beta_{7} - 1224621 \beta_{6} - 1190600 \beta_{5} - 99123 \beta_{4} - 3518026 \beta_{3} + \cdots - 1190600 \beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(45\) \(57\)
\(\chi(n)\) \(1\) \(-1 + \beta_{2} - \beta_{3} - \beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
15.1
−3.33405 + 10.2612i
3.02504 − 9.31010i
−7.83127 − 5.68975i
8.64028 + 6.27753i
−3.33405 − 10.2612i
3.02504 + 9.31010i
−7.83127 + 5.68975i
8.64028 − 6.27753i
1.61803 + 1.17557i 0.263932 − 0.812299i 1.23607 + 3.80423i −12.3467 + 8.97040i 1.38197 − 1.00406i −2.16312 − 6.65740i −2.47214 + 7.60845i 21.2533 + 15.4414i −30.5227
15.2 1.61803 + 1.17557i 0.263932 − 0.812299i 1.23607 + 3.80423i 4.30161 − 3.12530i 1.38197 − 1.00406i −2.16312 − 6.65740i −2.47214 + 7.60845i 21.2533 + 15.4414i 10.6342
71.1 −0.618034 − 1.90211i 4.73607 + 3.44095i −3.23607 + 2.35114i −4.37324 + 13.4595i 3.61803 − 11.1352i 5.66312 − 4.11450i 6.47214 + 4.70228i 2.24671 + 6.91467i 28.3042
71.2 −0.618034 − 1.90211i 4.73607 + 3.44095i −3.23607 + 2.35114i 1.91833 − 5.90401i 3.61803 − 11.1352i 5.66312 − 4.11450i 6.47214 + 4.70228i 2.24671 + 6.91467i −12.4157
113.1 1.61803 − 1.17557i 0.263932 + 0.812299i 1.23607 − 3.80423i −12.3467 − 8.97040i 1.38197 + 1.00406i −2.16312 + 6.65740i −2.47214 − 7.60845i 21.2533 − 15.4414i −30.5227
113.2 1.61803 − 1.17557i 0.263932 + 0.812299i 1.23607 − 3.80423i 4.30161 + 3.12530i 1.38197 + 1.00406i −2.16312 + 6.65740i −2.47214 − 7.60845i 21.2533 − 15.4414i 10.6342
141.1 −0.618034 + 1.90211i 4.73607 − 3.44095i −3.23607 − 2.35114i −4.37324 − 13.4595i 3.61803 + 11.1352i 5.66312 + 4.11450i 6.47214 − 4.70228i 2.24671 − 6.91467i 28.3042
141.2 −0.618034 + 1.90211i 4.73607 − 3.44095i −3.23607 − 2.35114i 1.91833 + 5.90401i 3.61803 + 11.1352i 5.66312 + 4.11450i 6.47214 − 4.70228i 2.24671 − 6.91467i −12.4157
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 15.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 154.4.f.c ✓ 8
11.c even 5 1 inner 154.4.f.c ✓ 8
11.c even 5 1 1694.4.a.v 4
11.d odd 10 1 1694.4.a.x 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.4.f.c ✓ 8 1.a even 1 1 trivial
154.4.f.c ✓ 8 11.c even 5 1 inner
1694.4.a.v 4 11.c even 5 1
1694.4.a.x 4 11.d odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 10T_{3}^{3} + 40T_{3}^{2} - 25T_{3} + 25 \) acting on \(S_{4}^{\mathrm{new}}(154, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 2 T^{3} + 4 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$3$ \( (T^{4} - 10 T^{3} + \cdots + 25)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} + 21 T^{7} + \cdots + 50822641 \) Copy content Toggle raw display
$7$ \( (T^{4} - 7 T^{3} + \cdots + 2401)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 3138428376721 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 28524689404201 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 17\!\cdots\!21 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 8310414762841 \) Copy content Toggle raw display
$23$ \( (T^{4} + 40 T^{3} + \cdots - 163259)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 132565863375625 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 22\!\cdots\!61 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 72\!\cdots\!21 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 16\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( (T^{4} + 678 T^{3} + \cdots - 950575601)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 40\!\cdots\!25 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 82\!\cdots\!81 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 94\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 60\!\cdots\!81 \) Copy content Toggle raw display
$67$ \( (T^{4} + 1071 T^{3} + \cdots - 7487660169)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 49\!\cdots\!61 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 81\!\cdots\!61 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 41\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 42\!\cdots\!21 \) Copy content Toggle raw display
$89$ \( (T^{4} + 2883 T^{3} + \cdots - 1863561605)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 24\!\cdots\!61 \) Copy content Toggle raw display
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