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Properties

Label 155.3.c.d
Level $155$
Weight $3$
Character orbit 155.c
Analytic conductor $4.223$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [155,3,Mod(154,155)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(155, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("155.154");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 155 = 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 155.c (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: \(4.22344409758\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 6 x^{19} + x^{18} + 154 x^{17} - 495 x^{16} + 2156 x^{15} - 5880 x^{14} - 11280 x^{13} + \cdots + 24211379 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{19}\cdot 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{8} q^{2} + \beta_{4} q^{3} + (\beta_{3} - 2) q^{4} + (\beta_{12} - \beta_{7} - \beta_{6} + \cdots - 2) q^{5}+ \cdots + ( - \beta_{7} - \beta_{6} - 2 \beta_{3} + 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{8} q^{2} + \beta_{4} q^{3} + (\beta_{3} - 2) q^{4} + (\beta_{12} - \beta_{7} - \beta_{6} + \cdots - 2) q^{5}+ \cdots + (3 \beta_{18} + 4 \beta_{17} + \cdots + 2 \beta_{9}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 48 q^{4} - 8 q^{5} + 112 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 48 q^{4} - 8 q^{5} + 112 q^{9} - 52 q^{10} - 8 q^{14} - 16 q^{16} - 28 q^{19} + 60 q^{20} - 164 q^{25} - 180 q^{31} + 64 q^{35} - 880 q^{36} + 392 q^{39} + 420 q^{40} + 332 q^{41} + 116 q^{45} + 260 q^{49} + 512 q^{50} - 292 q^{51} + 232 q^{56} + 60 q^{59} - 864 q^{64} + 552 q^{66} + 64 q^{69} + 496 q^{70} - 164 q^{71} - 440 q^{76} - 268 q^{80} - 532 q^{81} - 732 q^{90} + 488 q^{94} - 316 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 6 x^{19} + x^{18} + 154 x^{17} - 495 x^{16} + 2156 x^{15} - 5880 x^{14} - 11280 x^{13} + \cdots + 24211379 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 38\!\cdots\!15 \nu^{19} + \cdots - 23\!\cdots\!35 ) / 76\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 11\!\cdots\!20 \nu^{19} + \cdots - 17\!\cdots\!59 ) / 12\!\cdots\!46 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 76\!\cdots\!58 \nu^{19} + \cdots + 26\!\cdots\!03 ) / 76\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 18\!\cdots\!64 \nu^{19} + \cdots - 43\!\cdots\!49 ) / 12\!\cdots\!46 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 19\!\cdots\!04 \nu^{19} + \cdots + 10\!\cdots\!16 ) / 12\!\cdots\!46 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 26\!\cdots\!10 \nu^{19} + \cdots + 43\!\cdots\!41 ) / 15\!\cdots\!34 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 13\!\cdots\!26 \nu^{19} + \cdots - 43\!\cdots\!45 ) / 76\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 22\!\cdots\!91 \nu^{19} + \cdots + 87\!\cdots\!92 ) / 64\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 48\!\cdots\!24 \nu^{19} + \cdots - 22\!\cdots\!01 ) / 12\!\cdots\!46 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 68\!\cdots\!14 \nu^{19} + \cdots + 20\!\cdots\!11 ) / 12\!\cdots\!46 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 42\!\cdots\!40 \nu^{19} + \cdots + 43\!\cdots\!79 ) / 76\!\cdots\!70 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 91\!\cdots\!86 \nu^{19} + \cdots - 85\!\cdots\!33 ) / 12\!\cdots\!46 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 50\!\cdots\!88 \nu^{19} + \cdots - 17\!\cdots\!31 ) / 64\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 52\!\cdots\!67 \nu^{19} + \cdots - 46\!\cdots\!19 ) / 64\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 19\!\cdots\!70 \nu^{19} + \cdots - 92\!\cdots\!93 ) / 76\!\cdots\!70 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 16\!\cdots\!38 \nu^{19} + \cdots - 78\!\cdots\!06 ) / 64\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 35\!\cdots\!15 \nu^{19} + \cdots - 43\!\cdots\!22 ) / 12\!\cdots\!46 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 52\!\cdots\!14 \nu^{19} + \cdots + 37\!\cdots\!16 ) / 15\!\cdots\!34 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 59\!\cdots\!10 \nu^{19} + \cdots + 27\!\cdots\!51 ) / 76\!\cdots\!70 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{16} - 2\beta_{14} + 5\beta_{12} - 2\beta_{8} - 2\beta_{7} + 3\beta_{6} + 6\beta_{4} - 3\beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{19} - \beta_{18} - 6 \beta_{17} - 3 \beta_{16} + 2 \beta_{15} + 3 \beta_{12} + \cdots - 21 \beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 9 \beta_{19} - 9 \beta_{18} - 18 \beta_{17} - 29 \beta_{16} + 8 \beta_{14} - 60 \beta_{13} + \cdots - 366 ) / 24 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 8 \beta_{18} - 6 \beta_{17} - 16 \beta_{16} - 10 \beta_{15} + 12 \beta_{14} + 14 \beta_{13} + \cdots - 97 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 15 \beta_{19} + 435 \beta_{18} + 570 \beta_{17} + 521 \beta_{16} + 300 \beta_{15} - 104 \beta_{14} + \cdots - 16686 ) / 24 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 697 \beta_{19} + 25 \beta_{18} + 282 \beta_{17} + 2973 \beta_{16} - 338 \beta_{15} - 348 \beta_{14} + \cdots + 3930 ) / 12 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 4095 \beta_{19} + 12621 \beta_{18} + 19488 \beta_{17} + 27328 \beta_{16} + 9240 \beta_{15} + \cdots - 1794 ) / 12 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 4366 \beta_{19} + 1786 \beta_{18} + 3484 \beta_{17} + 16342 \beta_{16} + 128 \beta_{15} - 3400 \beta_{14} + \cdots + 190253 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 60561 \beta_{19} + 439245 \beta_{18} + 651150 \beta_{17} + 713341 \beta_{16} + 354456 \beta_{15} + \cdots + 10225830 ) / 24 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 79913 \beta_{19} - 581669 \beta_{18} - 855906 \beta_{17} - 896214 \beta_{16} - 485231 \beta_{15} + \cdots + 16195803 ) / 6 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 8648277 \beta_{19} - 4840077 \beta_{18} - 10696290 \beta_{17} - 36397823 \beta_{16} + \cdots + 67302666 ) / 24 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 3925613 \beta_{19} - 9849065 \beta_{18} - 15774448 \beta_{17} - 24726871 \beta_{16} - 7184310 \beta_{15} + \cdots - 68123086 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 120340896 \beta_{19} - 88879596 \beta_{18} - 178239204 \beta_{17} - 520140011 \beta_{16} + \cdots - 5460270660 ) / 12 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 91996571 \beta_{19} - 612381191 \beta_{18} - 923214006 \beta_{17} - 997386573 \beta_{16} + \cdots - 35458034196 ) / 12 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 3860105679 \beta_{19} + 10248141831 \beta_{18} + 16293280758 \beta_{17} + 24749063555 \beta_{16} + \cdots - 244692216078 ) / 24 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 4093369264 \beta_{19} + 5074353848 \beta_{18} + 8981388304 \beta_{17} + 20047825056 \beta_{16} + \cdots - 28530791119 ) / 2 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 275759833815 \beta_{19} + 503889152307 \beta_{18} + 837013645362 \beta_{17} + 1527149144293 \beta_{16} + \cdots + 7763657904234 ) / 24 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( 522909086987 \beta_{19} + 722918249747 \beta_{18} + 1254024399090 \beta_{17} + 2638261513767 \beta_{16} + \cdots + 29631432769518 ) / 12 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 41614694919 \beta_{19} + 303574448469 \beta_{18} + 451402275120 \beta_{17} + 476265166844 \beta_{16} + \cdots + 202308726293298 ) / 12 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/155\mathbb{Z}\right)^\times\).

\(n\) \(32\) \(96\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
154.1
−0.843111 − 3.44433i
4.43236 − 3.44433i
−5.01471 + 0.804536i
−0.430880 + 0.804536i
−0.297206 − 1.97112i
1.76736 − 1.97112i
0.229551 − 0.195118i
3.27578 − 0.195118i
−1.68868 + 2.75062i
1.56954 + 2.75062i
−1.68868 − 2.75062i
1.56954 − 2.75062i
0.229551 + 0.195118i
3.27578 + 0.195118i
−0.297206 + 1.97112i
1.76736 + 1.97112i
−5.01471 − 0.804536i
−0.430880 − 0.804536i
−0.843111 + 3.44433i
4.43236 + 3.44433i
− 3.76128i −5.27547 −10.1472 −1.78478 − 4.67061i 19.8425i − 4.15892i 23.1213i 18.8306 −17.5674 + 6.71303i
154.2 − 3.76128i 5.27547 −10.1472 −1.78478 − 4.67061i − 19.8425i − 4.15892i 23.1213i 18.8306 −17.5674 + 6.71303i
154.3 − 2.72570i −4.58383 −3.42943 3.48302 + 3.58728i 12.4941i 7.66847i − 1.55520i 12.0115 9.77785 − 9.49366i
154.4 − 2.72570i 4.58383 −3.42943 3.48302 + 3.58728i − 12.4941i 7.66847i − 1.55520i 12.0115 9.77785 − 9.49366i
154.5 − 2.41024i −2.06456 −1.80923 −4.63344 − 1.87915i 4.97609i 0.214886i − 5.28026i −4.73757 −4.52920 + 11.1677i
154.6 − 2.41024i 2.06456 −1.80923 −4.63344 − 1.87915i − 4.97609i 0.214886i − 5.28026i −4.73757 −4.52920 + 11.1677i
154.7 − 1.63501i −3.04623 1.32673 2.01309 − 4.57684i 4.98062i 3.42243i − 8.70928i 0.279500 −7.48319 − 3.29143i
154.8 − 1.63501i 3.04623 1.32673 2.01309 − 4.57684i − 4.98062i 3.42243i − 8.70928i 0.279500 −7.48319 − 3.29143i
154.9 − 1.39315i −3.25822 2.05912 −1.07789 + 4.88243i 4.53921i − 9.59889i − 8.44129i 1.61602 6.80198 + 1.50167i
154.10 − 1.39315i 3.25822 2.05912 −1.07789 + 4.88243i − 4.53921i − 9.59889i − 8.44129i 1.61602 6.80198 + 1.50167i
154.11 1.39315i −3.25822 2.05912 −1.07789 − 4.88243i − 4.53921i 9.59889i 8.44129i 1.61602 6.80198 − 1.50167i
154.12 1.39315i 3.25822 2.05912 −1.07789 − 4.88243i 4.53921i 9.59889i 8.44129i 1.61602 6.80198 − 1.50167i
154.13 1.63501i −3.04623 1.32673 2.01309 + 4.57684i − 4.98062i − 3.42243i 8.70928i 0.279500 −7.48319 + 3.29143i
154.14 1.63501i 3.04623 1.32673 2.01309 + 4.57684i 4.98062i − 3.42243i 8.70928i 0.279500 −7.48319 + 3.29143i
154.15 2.41024i −2.06456 −1.80923 −4.63344 + 1.87915i − 4.97609i − 0.214886i 5.28026i −4.73757 −4.52920 − 11.1677i
154.16 2.41024i 2.06456 −1.80923 −4.63344 + 1.87915i 4.97609i − 0.214886i 5.28026i −4.73757 −4.52920 − 11.1677i
154.17 2.72570i −4.58383 −3.42943 3.48302 − 3.58728i − 12.4941i − 7.66847i 1.55520i 12.0115 9.77785 + 9.49366i
154.18 2.72570i 4.58383 −3.42943 3.48302 − 3.58728i 12.4941i − 7.66847i 1.55520i 12.0115 9.77785 + 9.49366i
154.19 3.76128i −5.27547 −10.1472 −1.78478 + 4.67061i − 19.8425i 4.15892i − 23.1213i 18.8306 −17.5674 − 6.71303i
154.20 3.76128i 5.27547 −10.1472 −1.78478 + 4.67061i 19.8425i 4.15892i − 23.1213i 18.8306 −17.5674 − 6.71303i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 154.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
31.b odd 2 1 inner
155.c odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 155.3.c.d ✓ 20
5.b even 2 1 inner 155.3.c.d ✓ 20
5.c odd 4 2 775.3.d.h 20
31.b odd 2 1 inner 155.3.c.d ✓ 20
155.c odd 2 1 inner 155.3.c.d ✓ 20
155.f even 4 2 775.3.d.h 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
155.3.c.d ✓ 20 1.a even 1 1 trivial
155.3.c.d ✓ 20 5.b even 2 1 inner
155.3.c.d ✓ 20 31.b odd 2 1 inner
155.3.c.d ✓ 20 155.c odd 2 1 inner
775.3.d.h 20 5.c odd 4 2
775.3.d.h 20 155.f even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(155, [\chi])\):

\( T_{2}^{10} + 32T_{2}^{8} + 362T_{2}^{6} + 1816T_{2}^{4} + 4013T_{2}^{2} + 3168 \) Copy content Toggle raw display
\( T_{3}^{10} - 73T_{3}^{8} + 1948T_{3}^{6} - 23500T_{3}^{4} + 127704T_{3}^{2} - 245540 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{10} + 32 T^{8} + \cdots + 3168)^{2} \) Copy content Toggle raw display
$3$ \( (T^{10} - 73 T^{8} + \cdots - 245540)^{2} \) Copy content Toggle raw display
$5$ \( (T^{10} + 4 T^{9} + \cdots + 9765625)^{2} \) Copy content Toggle raw display
$7$ \( (T^{10} + 180 T^{8} + \cdots + 50688)^{2} \) Copy content Toggle raw display
$11$ \( (T^{10} + 868 T^{8} + \cdots + 63007528320)^{2} \) Copy content Toggle raw display
$13$ \( (T^{10} - 868 T^{8} + \cdots - 220986000)^{2} \) Copy content Toggle raw display
$17$ \( (T^{10} + \cdots - 371477466000)^{2} \) Copy content Toggle raw display
$19$ \( (T^{5} + 7 T^{4} + \cdots - 26872)^{4} \) Copy content Toggle raw display
$23$ \( (T^{10} + \cdots - 534927551040)^{2} \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots + 29\!\cdots\!20)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots + 819628286980801)^{2} \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots - 19\!\cdots\!40)^{2} \) Copy content Toggle raw display
$41$ \( (T^{5} - 83 T^{4} + \cdots - 78545044)^{4} \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots - 29650949030340)^{2} \) Copy content Toggle raw display
$47$ \( (T^{10} + \cdots + 338751072000000)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots - 25\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{5} - 15 T^{4} + \cdots + 16309648)^{4} \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots + 23\!\cdots\!20)^{2} \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots + 38\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{5} + 41 T^{4} + \cdots + 11242016)^{4} \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots - 20611178591760)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots + 39\!\cdots\!80)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots - 80\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots + 10\!\cdots\!80)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
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