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Properties

Label 154.3.b.a
Level $154$
Weight $3$
Character orbit 154.b
Analytic conductor $4.196$
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] = [154,3,Mod(111,154)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(154, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("154.111");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 154 = 2 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 154.b (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: \(4.19619607115\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 92 x^{14} + 3306 x^{12} + 59516 x^{10} + 574181 x^{8} + 2964960 x^{6} + 7737300 x^{4} + \cdots + 1285956 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} - \beta_1 q^{3} + 2 q^{4} + \beta_{8} q^{5} + \beta_{9} q^{6} + ( - \beta_{5} + \beta_{4}) q^{7} + 2 \beta_{4} q^{8} + ( - \beta_{4} + \beta_{3} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{2} - \beta_1 q^{3} + 2 q^{4} + \beta_{8} q^{5} + \beta_{9} q^{6} + ( - \beta_{5} + \beta_{4}) q^{7} + 2 \beta_{4} q^{8} + ( - \beta_{4} + \beta_{3} - 3) q^{9} + \beta_{10} q^{10} - \beta_{7} q^{11} - 2 \beta_1 q^{12} + (\beta_{13} + 2 \beta_{12} + \cdots + 2 \beta_1) q^{13}+ \cdots + (2 \beta_{15} - 2 \beta_{14} + \cdots - 3 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 32 q^{4} - 4 q^{7} - 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 32 q^{4} - 4 q^{7} - 40 q^{9} + 24 q^{14} + 8 q^{15} + 64 q^{16} - 32 q^{18} + 44 q^{21} + 40 q^{23} - 136 q^{25} - 8 q^{28} + 8 q^{29} - 48 q^{30} - 12 q^{35} - 80 q^{36} + 104 q^{37} + 320 q^{39} - 120 q^{42} - 40 q^{43} + 16 q^{46} - 176 q^{49} + 192 q^{50} - 208 q^{51} - 144 q^{53} + 48 q^{56} - 392 q^{57} - 48 q^{58} + 16 q^{60} + 80 q^{63} + 128 q^{64} - 144 q^{65} - 152 q^{67} + 152 q^{70} - 8 q^{71} - 64 q^{72} - 160 q^{74} - 44 q^{77} - 240 q^{78} + 64 q^{79} + 32 q^{81} + 88 q^{84} + 736 q^{85} + 192 q^{86} + 128 q^{91} + 80 q^{92} + 712 q^{93} - 408 q^{95} - 64 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 92 x^{14} + 3306 x^{12} + 59516 x^{10} + 574181 x^{8} + 2964960 x^{6} + 7737300 x^{4} + \cdots + 1285956 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 4217 \nu^{14} - 375675 \nu^{12} - 12833287 \nu^{10} - 212618053 \nu^{8} - 1776543056 \nu^{6} + \cdots - 615087396 ) / 221487840 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 41161 \nu^{14} + 3611015 \nu^{12} + 120631431 \nu^{10} + 1932972089 \nu^{8} + \cdots + 26879861988 ) / 1064962080 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 41161 \nu^{14} + 3611015 \nu^{12} + 120631431 \nu^{10} + 1932972089 \nu^{8} + \cdots + 14100317028 ) / 1064962080 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 886249 \nu^{15} + 232356 \nu^{14} + 76856780 \nu^{13} + 19091985 \nu^{12} + \cdots + 370324893708 ) / 58306673880 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 886249 \nu^{15} - 232356 \nu^{14} + 76856780 \nu^{13} - 19091985 \nu^{12} + \cdots - 370324893708 ) / 58306673880 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 13543 \nu^{14} + 1183073 \nu^{12} + 39265995 \nu^{10} + 622584083 \nu^{8} + 4852599014 \nu^{6} + \cdots + 2953936188 ) / 180795888 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1857311 \nu^{15} + 162221650 \nu^{13} + 5382010941 \nu^{11} + 85230575524 \nu^{9} + \cdots - 214098800832 \nu ) / 58306673880 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 41161 \nu^{15} - 3611015 \nu^{13} - 120631431 \nu^{11} - 1932972089 \nu^{9} + \cdots - 14100317028 \nu ) / 1064962080 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 13732183 \nu^{15} + 1199876465 \nu^{13} + 39853529913 \nu^{11} + 633091168727 \nu^{9} + \cdots - 97825795956 \nu ) / 233226695520 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 6161 \nu^{14} - 542215 \nu^{12} - 18210966 \nu^{10} - 294496789 \nu^{8} - 2375013793 \nu^{6} + \cdots - 2013045048 ) / 34768440 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 17372927 \nu^{15} + 929424 \nu^{14} + 1516066825 \nu^{13} + 76367940 \nu^{12} + \cdots + 1481299574832 ) / 233226695520 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 31202653 \nu^{15} - 2726040635 \nu^{13} - 90506663463 \nu^{11} - 1436224635797 \nu^{9} + \cdots - 3712751886564 \nu ) / 233226695520 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 4404933 \nu^{15} + 5501660 \nu^{14} + 386798490 \nu^{13} + 480876535 \nu^{12} + \cdots + 1444180264380 ) / 38871115920 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 4404933 \nu^{15} - 5501660 \nu^{14} + 386798490 \nu^{13} - 480876535 \nu^{12} + \cdots - 1444180264380 ) / 38871115920 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{4} + \beta_{3} - 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -3\beta_{13} - 3\beta_{12} - \beta_{10} + \beta_{9} - 3\beta_{8} - \beta_{6} + 2\beta_{5} - 19\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{15} + \beta_{14} + \beta_{11} - 11 \beta_{7} + 5 \beta_{6} - 5 \beta_{5} + 55 \beta_{4} + \cdots + 247 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{15} - \beta_{14} + 110 \beta_{13} + 90 \beta_{12} + 44 \beta_{10} - 52 \beta_{9} + \cdots + 450 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 77 \beta_{15} - 77 \beta_{14} - 77 \beta_{11} + 547 \beta_{7} - 287 \beta_{6} + 287 \beta_{5} + \cdots - 6049 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 77 \beta_{15} + 77 \beta_{14} - 3520 \beta_{13} - 2476 \beta_{12} - 1702 \beta_{10} + 2070 \beta_{9} + \cdots - 11682 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 3451 \beta_{15} + 3451 \beta_{14} + 3591 \beta_{11} - 20965 \beta_{7} + 11263 \beta_{6} + \cdots + 160015 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 3731 \beta_{15} - 3731 \beta_{14} + 108714 \beta_{13} + 68814 \beta_{12} + 60080 \beta_{10} + \cdots + 318740 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 128071 \beta_{15} - 128071 \beta_{14} - 139775 \beta_{11} + 724881 \beta_{7} - 388929 \beta_{6} + \cdots - 4424131 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 151479 \beta_{15} + 151479 \beta_{14} - 3311170 \beta_{13} - 1948206 \beta_{12} - 2003832 \beta_{10} + \cdots - 8972864 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 4354231 \beta_{15} + 4354231 \beta_{14} + 4967907 \beta_{11} - 23798977 \beta_{7} + 12697335 \beta_{6} + \cdots + 125879375 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 5581583 \beta_{15} - 5581583 \beta_{14} + 100266090 \beta_{13} + 56040350 \beta_{12} + \cdots + 257999680 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 141373491 \beta_{15} - 141373491 \beta_{14} - 167484107 \beta_{11} + 759077837 \beta_{7} + \cdots - 3651860959 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 193594723 \beta_{15} + 193594723 \beta_{14} - 3029801470 \beta_{13} - 1632531386 \beta_{12} + \cdots - 7529593860 \beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(45\) \(57\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
111.1
4.69403i
3.28383i
2.16739i
1.68169i
− 1.68169i
− 2.16739i
− 3.28383i
− 4.69403i
5.50782i
4.09281i
2.09450i
0.427485i
− 0.427485i
− 2.09450i
− 4.09281i
− 5.50782i
−1.41421 − 4.69403i 2.00000 8.26329i 6.63836i −2.99810 + 6.32546i −2.82843 −13.0339 − 11.6861i
111.2 −1.41421 − 3.28383i 2.00000 − 9.79772i 4.64404i −6.91284 + 1.10118i −2.82843 −1.78357 13.8561i
111.3 −1.41421 − 2.16739i 2.00000 0.322108i 3.06516i 4.97842 + 4.92091i −2.82843 4.30241 − 0.455529i
111.4 −1.41421 − 1.68169i 2.00000 1.88686i 2.37827i −0.310114 − 6.99313i −2.82843 6.17191 − 2.66842i
111.5 −1.41421 1.68169i 2.00000 − 1.88686i − 2.37827i −0.310114 + 6.99313i −2.82843 6.17191 2.66842i
111.6 −1.41421 2.16739i 2.00000 − 0.322108i − 3.06516i 4.97842 − 4.92091i −2.82843 4.30241 0.455529i
111.7 −1.41421 3.28383i 2.00000 9.79772i − 4.64404i −6.91284 − 1.10118i −2.82843 −1.78357 − 13.8561i
111.8 −1.41421 4.69403i 2.00000 − 8.26329i − 6.63836i −2.99810 − 6.32546i −2.82843 −13.0339 11.6861i
111.9 1.41421 − 5.50782i 2.00000 1.62456i − 7.78924i 1.73923 − 6.78049i 2.82843 −21.3361 2.29748i
111.10 1.41421 − 4.09281i 2.00000 − 6.32600i − 5.78811i −1.34165 + 6.87022i 2.82843 −7.75109 − 8.94631i
111.11 1.41421 − 2.09450i 2.00000 5.95030i − 2.96207i 6.96650 + 0.684066i 2.82843 4.61308 8.41500i
111.12 1.41421 − 0.427485i 2.00000 − 4.68992i − 0.604555i −4.12143 − 5.65808i 2.82843 8.81726 − 6.63254i
111.13 1.41421 0.427485i 2.00000 4.68992i 0.604555i −4.12143 + 5.65808i 2.82843 8.81726 6.63254i
111.14 1.41421 2.09450i 2.00000 − 5.95030i 2.96207i 6.96650 − 0.684066i 2.82843 4.61308 − 8.41500i
111.15 1.41421 4.09281i 2.00000 6.32600i 5.78811i −1.34165 − 6.87022i 2.82843 −7.75109 8.94631i
111.16 1.41421 5.50782i 2.00000 − 1.62456i 7.78924i 1.73923 + 6.78049i 2.82843 −21.3361 − 2.29748i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 111.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 154.3.b.a ✓ 16
3.b odd 2 1 1386.3.b.a 16
4.b odd 2 1 1232.3.b.c 16
7.b odd 2 1 inner 154.3.b.a ✓ 16
21.c even 2 1 1386.3.b.a 16
28.d even 2 1 1232.3.b.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.3.b.a ✓ 16 1.a even 1 1 trivial
154.3.b.a ✓ 16 7.b odd 2 1 inner
1232.3.b.c 16 4.b odd 2 1
1232.3.b.c 16 28.d even 2 1
1386.3.b.a 16 3.b odd 2 1
1386.3.b.a 16 21.c even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(154, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2)^{8} \) Copy content Toggle raw display
$3$ \( T^{16} + 92 T^{14} + \cdots + 1285956 \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 199148544 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 33232930569601 \) Copy content Toggle raw display
$11$ \( (T^{2} - 11)^{8} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 33\!\cdots\!44 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 40970240640000 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 54\!\cdots\!64 \) Copy content Toggle raw display
$23$ \( (T^{8} - 20 T^{7} + \cdots + 16883500032)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} - 4 T^{7} + \cdots - 13451230656)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 63\!\cdots\!56 \) Copy content Toggle raw display
$37$ \( (T^{8} - 52 T^{7} + \cdots + 805154525092)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 35\!\cdots\!96 \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 3088482365200)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 16\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( (T^{8} + 72 T^{7} + \cdots - 551736105984)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 41\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 38\!\cdots\!64 \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 14\!\cdots\!96)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 37062756202752)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 48\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots - 1607755921856)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 29\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 35\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 47\!\cdots\!76 \) Copy content Toggle raw display
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