↑

Properties

Label 165.4.c.b
Level $165$
Weight $4$
Character orbit 165.c
Analytic conductor $9.735$
Analytic rank $0$
Dimension $14$
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] = [165,4,Mod(34,165)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(165, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("165.34");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 165 = 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 165.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: \(9.73531515095\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 66x^{12} + 1705x^{10} + 22060x^{8} + 151880x^{6} + 537860x^{4} + 825344x^{2} + 262144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{4}\cdot 5 \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{6} q^{2} - \beta_{8} q^{3} + ( - \beta_{3} - 2) q^{4} + (\beta_{7} - 1) q^{5} + ( - \beta_{3} - \beta_{2} - 2) q^{6} + ( - 2 \beta_{8} + \beta_{5}) q^{7} + (\beta_{10} + 3 \beta_{8} - \beta_{7} + \beta_{5} + \beta_{4} + \beta_{2}) q^{8} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{2} - \beta_{8} q^{3} + ( - \beta_{3} - 2) q^{4} + (\beta_{7} - 1) q^{5} + ( - \beta_{3} - \beta_{2} - 2) q^{6} + ( - 2 \beta_{8} + \beta_{5}) q^{7} + (\beta_{10} + 3 \beta_{8} - \beta_{7} + \beta_{5} + \beta_{4} + \beta_{2}) q^{8} - 9 q^{9} + ( - \beta_{13} + 2 \beta_{10} + \beta_{8} - \beta_{7} - 3 \beta_{6} + \beta_{4} - \beta_{3} + 3 \beta_{2} + \cdots - 4) q^{10}+ \cdots - 99 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 26 q^{4} - 14 q^{5} - 30 q^{6} - 126 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 26 q^{4} - 14 q^{5} - 30 q^{6} - 126 q^{9} - 48 q^{10} + 154 q^{11} - 84 q^{14} - 86 q^{16} - 116 q^{19} + 442 q^{20} - 204 q^{21} + 450 q^{24} + 162 q^{25} - 400 q^{26} - 128 q^{29} + 246 q^{30} - 696 q^{31} - 412 q^{34} + 672 q^{35} + 234 q^{36} + 480 q^{39} + 1612 q^{40} - 664 q^{41} - 286 q^{44} + 126 q^{45} - 656 q^{46} + 834 q^{49} + 1908 q^{50} - 972 q^{51} + 270 q^{54} - 154 q^{55} - 3236 q^{56} + 664 q^{59} + 108 q^{60} + 44 q^{61} - 1122 q^{64} - 2328 q^{65} - 330 q^{66} + 1944 q^{69} + 1220 q^{70} - 1032 q^{71} - 3256 q^{74} + 84 q^{75} + 5588 q^{76} - 3492 q^{79} - 510 q^{80} + 1134 q^{81} - 1008 q^{84} - 1068 q^{85} + 2540 q^{86} + 4452 q^{89} + 432 q^{90} + 2144 q^{91} - 9472 q^{94} - 932 q^{95} + 450 q^{96} - 1386 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 66x^{12} + 1705x^{10} + 22060x^{8} + 151880x^{6} + 537860x^{4} + 825344x^{2} + 262144 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 13137 \nu^{12} - 712162 \nu^{10} - 13476473 \nu^{8} - 100603116 \nu^{6} - 186755016 \nu^{4} + 533460412 \nu^{2} + 542747648 ) / 37402624 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1221 \nu^{12} - 72330 \nu^{10} - 1598477 \nu^{8} - 16399708 \nu^{6} - 79067688 \nu^{4} - 150763924 \nu^{2} - 29614336 ) / 2337664 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1221 \nu^{12} - 72330 \nu^{10} - 1598477 \nu^{8} - 16399708 \nu^{6} - 79067688 \nu^{4} - 154270420 \nu^{2} - 63510464 ) / 1168832 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 241963 \nu^{13} - 2468784 \nu^{12} - 18454550 \nu^{11} - 142340960 \nu^{10} - 545049955 \nu^{9} - 3018446640 \nu^{8} - 7792773220 \nu^{7} + \cdots - 11130454016 ) / 5984419840 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 6365 \nu^{13} + 427514 \nu^{11} + 10924517 \nu^{9} + 129108220 \nu^{7} + 653010728 \nu^{5} + 723078260 \nu^{3} - 1630262272 \nu ) / 85491712 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 51749 \nu^{13} - 3102858 \nu^{11} - 69715565 \nu^{9} - 732372828 \nu^{7} - 3661312872 \nu^{5} - 7592389012 \nu^{3} - 2918278144 \nu ) / 598441984 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 737029 \nu^{13} - 1877968 \nu^{12} + 46201930 \nu^{11} - 124343200 \nu^{10} + 1093595725 \nu^{9} - 3133573200 \nu^{8} + 12139022300 \nu^{7} + \cdots - 77504233472 ) / 5984419840 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 155247 \nu^{13} + 9308574 \nu^{11} + 209146695 \nu^{9} + 2197118484 \nu^{7} + 10983938616 \nu^{5} + 22777167036 \nu^{3} + \cdots + 10550160384 \nu ) / 598441984 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1772009 \nu^{13} - 1877968 \nu^{12} + 108259090 \nu^{11} - 124343200 \nu^{10} + 2487907025 \nu^{9} - 3133573200 \nu^{8} + 26786478860 \nu^{7} + \cdots - 77504233472 ) / 5984419840 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1254519 \nu^{13} + 1858288 \nu^{12} - 77230510 \nu^{11} + 101581280 \nu^{10} - 1790751375 \nu^{9} + 1988487280 \nu^{8} - 19462750580 \nu^{7} + \cdots + 4719460352 ) / 2992209920 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 55801 \nu^{13} + 3314994 \nu^{11} + 73138529 \nu^{9} + 739391436 \nu^{7} + 3369032456 \nu^{5} + 5057154276 \nu^{3} - 2673449984 \nu ) / 85491712 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 4008547 \nu^{13} + 15002416 \nu^{12} + 238708870 \nu^{11} + 901354080 \nu^{10} + 5288248155 \nu^{9} + 20237566640 \nu^{8} + \cdots + 671570673664 ) / 5984419840 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 4008547 \nu^{13} + 10655664 \nu^{12} - 238708870 \nu^{11} + 634669920 \nu^{10} - 5288248155 \nu^{9} + 14085546800 \nu^{8} + \cdots + 582935986176 ) / 5984419840 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{8} + 3\beta_{6} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} + 2\beta_{2} - 29 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{9} - 16\beta_{8} - 3\beta_{7} - 42\beta_{6} ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 5 \beta_{13} - 3 \beta_{12} + 4 \beta_{11} - 4 \beta_{10} + 4 \beta_{9} - 4 \beta_{6} - 4 \beta_{5} + 4 \beta_{4} + 20 \beta_{3} - 47 \beta_{2} - \beta _1 + 421 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 6 \beta_{13} + 6 \beta_{12} - 15 \beta_{11} + 3 \beta_{10} - 90 \beta_{9} + 302 \beta_{8} + 93 \beta_{7} + 699 \beta_{6} + 9 \beta_{5} + 9 \beta_{4} + 3 \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 139 \beta_{13} + 105 \beta_{12} - 122 \beta_{11} + 134 \beta_{10} - 110 \beta_{9} + 12 \beta_{7} + 122 \beta_{6} + 122 \beta_{5} - 122 \beta_{4} - 354 \beta_{3} + 1013 \beta_{2} + 59 \beta _1 - 7209 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 210 \beta_{13} - 210 \beta_{12} + 537 \beta_{11} - 143 \beta_{10} + 2180 \beta_{9} - 6080 \beta_{8} - 2247 \beta_{7} - 12683 \beta_{6} - 425 \beta_{5} - 353 \beta_{4} - 143 \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 3023 \beta_{13} - 2865 \beta_{12} + 2944 \beta_{11} - 3436 \beta_{10} + 2452 \beta_{9} - 492 \beta_{7} - 2944 \beta_{6} - 2944 \beta_{5} + 2944 \beta_{4} + 5858 \beta_{3} - 21575 \beta_{2} - 1747 \beta _1 + 133657 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 5262 \beta_{13} + 5262 \beta_{12} - 14349 \beta_{11} + 4635 \beta_{10} - 49248 \beta_{9} + 126928 \beta_{8} + 49875 \beta_{7} + 240759 \beta_{6} + 13317 \beta_{5} + 9897 \beta_{4} + 4635 \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 60991 \beta_{13} + 71025 \beta_{12} - 66008 \beta_{11} + 79952 \beta_{10} - 52064 \beta_{9} + 13944 \beta_{7} + 66008 \beta_{6} + 66008 \beta_{5} - 66008 \beta_{4} - 92006 \beta_{3} + \cdots - 2578069 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 117378 \beta_{13} - 117378 \beta_{12} + 344397 \beta_{11} - 129011 \beta_{10} + 1079324 \beta_{9} - 2693260 \beta_{8} - 1067691 \beta_{7} - 4688171 \beta_{6} - 354725 \beta_{5} + \cdots - 129011 \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 399537 \beta_{13} - 557567 \beta_{12} + 478552 \beta_{11} - 592916 \beta_{10} + 364188 \beta_{9} - 114364 \beta_{7} - 478552 \beta_{6} - 478552 \beta_{5} + 478552 \beta_{4} + \cdots + 16938411 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 2496558 \beta_{13} + 2496558 \beta_{12} - 7857753 \beta_{11} + 3302799 \beta_{10} - 23294532 \beta_{9} + 57462296 \beta_{8} + 22488291 \beta_{7} + 92752251 \beta_{6} + \cdots + 3302799 \beta_{2} ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/165\mathbb{Z}\right)^\times\).

\(n\) \(46\) \(56\) \(67\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
34.1
− 4.15324i
− 2.70507i
− 4.57884i
− 1.98683i
− 3.20690i
− 2.41169i
0.647712i
− 0.647712i
2.41169i
3.20690i
1.98683i
4.57884i
2.70507i
4.15324i
− 5.15324i − 3.00000i −18.5558 −7.01850 − 8.70291i −15.4597 17.2148i 54.3968i −9.00000 −44.8482 + 36.1680i
34.2 − 3.70507i − 3.00000i −5.72754 −10.6368 + 3.44353i −11.1152 − 30.9104i − 8.41961i −9.00000 12.7585 + 39.4102i
34.3 − 3.57884i 3.00000i −4.80807 −0.925309 − 11.1420i 10.7365 7.85216i − 11.4234i −9.00000 −39.8753 + 3.31153i
34.4 − 2.98683i − 3.00000i −0.921158 7.35339 + 8.42185i −8.96049 − 6.37827i − 21.1433i −9.00000 25.1547 − 21.9633i
34.5 − 2.20690i 3.00000i 3.12958 4.62093 + 10.1807i 6.62071 1.50972i − 24.5619i −9.00000 22.4678 − 10.1979i
34.6 − 1.41169i 3.00000i 6.00714 −11.1339 + 1.01772i 4.23506 − 15.2844i − 19.7737i −9.00000 1.43670 + 15.7176i
34.7 − 0.352288i − 3.00000i 7.87589 10.7402 − 3.10602i −1.05686 − 19.8486i − 5.59288i −9.00000 −1.09421 − 3.78365i
34.8 0.352288i 3.00000i 7.87589 10.7402 + 3.10602i −1.05686 19.8486i 5.59288i −9.00000 −1.09421 + 3.78365i
34.9 1.41169i − 3.00000i 6.00714 −11.1339 − 1.01772i 4.23506 15.2844i 19.7737i −9.00000 1.43670 − 15.7176i
34.10 2.20690i − 3.00000i 3.12958 4.62093 − 10.1807i 6.62071 − 1.50972i 24.5619i −9.00000 22.4678 + 10.1979i
34.11 2.98683i 3.00000i −0.921158 7.35339 − 8.42185i −8.96049 6.37827i 21.1433i −9.00000 25.1547 + 21.9633i
34.12 3.57884i − 3.00000i −4.80807 −0.925309 + 11.1420i 10.7365 − 7.85216i 11.4234i −9.00000 −39.8753 − 3.31153i
34.13 3.70507i 3.00000i −5.72754 −10.6368 − 3.44353i −11.1152 30.9104i 8.41961i −9.00000 12.7585 − 39.4102i
34.14 5.15324i 3.00000i −18.5558 −7.01850 + 8.70291i −15.4597 − 17.2148i − 54.3968i −9.00000 −44.8482 − 36.1680i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 34.14
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 165.4.c.b ✓ 14
3.b odd 2 1 495.4.c.d 14
5.b even 2 1 inner 165.4.c.b ✓ 14
5.c odd 4 1 825.4.a.ba 7
5.c odd 4 1 825.4.a.bd 7
15.d odd 2 1 495.4.c.d 14
15.e even 4 1 2475.4.a.bo 7
15.e even 4 1 2475.4.a.bs 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.4.c.b ✓ 14 1.a even 1 1 trivial
165.4.c.b ✓ 14 5.b even 2 1 inner
495.4.c.d 14 3.b odd 2 1
495.4.c.d 14 15.d odd 2 1
825.4.a.ba 7 5.c odd 4 1
825.4.a.bd 7 5.c odd 4 1
2475.4.a.bo 7 15.e even 4 1
2475.4.a.bs 7 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} + 69T_{2}^{12} + 1798T_{2}^{10} + 22642T_{2}^{8} + 143537T_{2}^{6} + 424913T_{2}^{4} + 454864T_{2}^{2} + 50176 \) acting on \(S_{4}^{\mathrm{new}}(165, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + 69 T^{12} + 1798 T^{10} + \cdots + 50176 \) Copy content Toggle raw display
$3$ \( (T^{2} + 9)^{7} \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots + 476837158203125 \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots + 148986826592256 \) Copy content Toggle raw display
$11$ \( (T - 11)^{14} \) Copy content Toggle raw display
$13$ \( T^{14} + 17988 T^{12} + \cdots + 24\!\cdots\!76 \) Copy content Toggle raw display
$17$ \( T^{14} + 54892 T^{12} + \cdots + 46\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( (T^{7} + 58 T^{6} + \cdots - 46691778560000)^{2} \) Copy content Toggle raw display
$23$ \( T^{14} + 103772 T^{12} + \cdots + 40\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( (T^{7} + 64 T^{6} + \cdots + 38672535237120)^{2} \) Copy content Toggle raw display
$31$ \( (T^{7} + 348 T^{6} + \cdots + 197687760429056)^{2} \) Copy content Toggle raw display
$37$ \( T^{14} + 562632 T^{12} + \cdots + 94\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( (T^{7} + 332 T^{6} + \cdots + 26\!\cdots\!92)^{2} \) Copy content Toggle raw display
$43$ \( T^{14} + 1022676 T^{12} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{14} + 887148 T^{12} + \cdots + 74\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{14} + 1430852 T^{12} + \cdots + 61\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( (T^{7} - 332 T^{6} + \cdots - 30\!\cdots\!20)^{2} \) Copy content Toggle raw display
$61$ \( (T^{7} - 22 T^{6} + \cdots + 39\!\cdots\!12)^{2} \) Copy content Toggle raw display
$67$ \( T^{14} + 3775064 T^{12} + \cdots + 17\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( (T^{7} + 516 T^{6} + \cdots + 765697934131200)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} + 4480568 T^{12} + \cdots + 13\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( (T^{7} + 1746 T^{6} + \cdots + 55\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + 4133328 T^{12} + \cdots + 58\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( (T^{7} - 2226 T^{6} + \cdots - 70\!\cdots\!40)^{2} \) Copy content Toggle raw display
$97$ \( T^{14} + 3938712 T^{12} + \cdots + 40\!\cdots\!76 \) Copy content Toggle raw display
show more
show less