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Properties

Label 165.4.c.a
Level $165$
Weight $4$
Character orbit 165.c
Analytic conductor $9.735$
Analytic rank $0$
Dimension $14$
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] = [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: \( x^{14} + 97x^{12} + 3674x^{10} + 68702x^{8} + 656605x^{6} + 2988841x^{4} + 5502384x^{2} + 3385600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{4} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{8} q^{3} + (\beta_{2} - 6) q^{4} + (\beta_{6} - 1) q^{5} + (\beta_{3} + 1) q^{6} + (\beta_{10} + \beta_{9} - 3 \beta_{8} + \beta_1) q^{7} + (\beta_{13} + \beta_{12} - \beta_{11} - 5 \beta_1) q^{8} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{8} q^{3} + (\beta_{2} - 6) q^{4} + (\beta_{6} - 1) q^{5} + (\beta_{3} + 1) q^{6} + (\beta_{10} + \beta_{9} - 3 \beta_{8} + \beta_1) q^{7} + (\beta_{13} + \beta_{12} - \beta_{11} - 5 \beta_1) q^{8} - 9 q^{9} + (\beta_{12} + \beta_{9} + 4 \beta_{8} + \beta_{4} - \beta_1 - 2) q^{10} - 11 q^{11} + ( - \beta_{12} + \beta_{11} + \beta_{10} - 6 \beta_{8}) q^{12} + (\beta_{10} + 3 \beta_{9} - \beta_{8} - 5 \beta_1) q^{13} + ( - \beta_{12} - \beta_{11} + 3 \beta_{7} - 3 \beta_{6} - \beta_{5} + \beta_{4} - 6 \beta_{3} + \cdots - 20) q^{14}+ \cdots + 99 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 82 q^{4} - 14 q^{5} + 18 q^{6} - 126 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 82 q^{4} - 14 q^{5} + 18 q^{6} - 126 q^{9} - 28 q^{10} - 154 q^{11} - 284 q^{14} + 362 q^{16} - 52 q^{19} + 226 q^{20} + 300 q^{21} - 126 q^{24} - 366 q^{25} + 952 q^{26} - 1144 q^{29} - 582 q^{30} - 280 q^{31} + 1612 q^{34} - 600 q^{35} + 738 q^{36} - 144 q^{39} + 176 q^{40} + 1792 q^{41} + 902 q^{44} + 126 q^{45} - 688 q^{46} - 590 q^{49} + 388 q^{50} + 228 q^{51} - 162 q^{54} + 154 q^{55} + 3044 q^{56} - 2632 q^{59} - 1140 q^{60} - 772 q^{61} - 1738 q^{64} - 904 q^{65} - 198 q^{66} - 1368 q^{69} + 84 q^{70} + 1608 q^{71} + 1496 q^{74} - 300 q^{75} - 3396 q^{76} + 748 q^{79} - 2606 q^{80} + 1134 q^{81} - 5040 q^{84} + 2508 q^{85} - 5068 q^{86} - 1388 q^{89} + 252 q^{90} - 6752 q^{91} + 5840 q^{94} + 1724 q^{95} + 5946 q^{96} + 1386 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 97x^{12} + 3674x^{10} + 68702x^{8} + 656605x^{6} + 2988841x^{4} + 5502384x^{2} + 3385600 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 14 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 161 \nu^{12} - 10710 \nu^{10} - 199154 \nu^{8} + 151996 \nu^{6} + 31552223 \nu^{4} + 192529678 \nu^{2} + 218873720 ) / 6066280 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 404 \nu^{12} - 20281 \nu^{10} - 28756 \nu^{8} + 12274706 \nu^{6} + 207485764 \nu^{4} + 1017337739 \nu^{2} + 1028949440 ) / 6066280 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3361 \nu^{12} + 210392 \nu^{10} + 3562170 \nu^{8} - 7980844 \nu^{6} - 620131783 \nu^{4} - 3902728280 \nu^{2} - 4803256960 ) / 48530240 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 14977 \nu^{13} + 127397 \nu^{12} + 877608 \nu^{11} + 11204496 \nu^{10} + 10395194 \nu^{9} + 376493762 \nu^{8} - 209240012 \nu^{7} + \cdots + 193781027840 ) / 2232391040 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 14977 \nu^{13} - 127397 \nu^{12} + 877608 \nu^{11} - 11204496 \nu^{10} + 10395194 \nu^{9} - 376493762 \nu^{8} - 209240012 \nu^{7} + \cdots - 193781027840 ) / 2232391040 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 12225 \nu^{13} - 1215449 \nu^{11} - 46885290 \nu^{9} - 876526286 \nu^{7} - 7999028861 \nu^{5} - 30732972193 \nu^{3} + \cdots - 31841183648 \nu ) / 1116195520 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 43937 \nu^{13} + 3852489 \nu^{11} + 129337514 \nu^{9} + 2101712430 \nu^{7} + 16990191869 \nu^{5} + 60573152849 \nu^{3} + \cdots + 56499926848 \nu ) / 1116195520 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 63451 \nu^{13} - 5763816 \nu^{11} - 200137342 \nu^{9} - 3321524892 \nu^{7} - 26532139779 \nu^{5} - 87168670424 \nu^{3} + \cdots - 68876967744 \nu ) / 1116195520 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 137323 \nu^{13} - 426489 \nu^{12} - 13223110 \nu^{11} - 37166896 \nu^{10} - 492901054 \nu^{9} - 1227607658 \nu^{8} - 8921875848 \nu^{7} + \cdots - 424894743040 ) / 2232391040 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 137323 \nu^{13} - 426489 \nu^{12} + 13223110 \nu^{11} - 37166896 \nu^{10} + 492901054 \nu^{9} - 1227607658 \nu^{8} + 8921875848 \nu^{7} + \cdots - 424894743040 ) / 2232391040 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 137323 \nu^{13} - 13223110 \nu^{11} - 492901054 \nu^{9} - 8921875848 \nu^{7} - 79648655243 \nu^{5} - 306551284006 \nu^{3} + \cdots - 312070537408 \nu ) / 1116195520 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 14 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{13} + \beta_{12} - \beta_{11} - 21\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{7} + \beta_{6} - 3\beta_{5} - 3\beta_{4} + 4\beta_{3} - 29\beta_{2} + 300 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 38 \beta_{13} - 38 \beta_{12} + 38 \beta_{11} + 4 \beta_{10} - 10 \beta_{9} - 42 \beta_{8} + 12 \beta_{7} + 12 \beta_{6} + 493 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 8 \beta_{12} + 8 \beta_{11} + 22 \beta_{7} - 22 \beta_{6} + 138 \beta_{5} + 142 \beta_{4} - 206 \beta_{3} + 795 \beta_{2} - 7260 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1229 \beta_{13} + 1279 \beta_{12} - 1279 \beta_{11} - 174 \beta_{10} + 500 \beta_{9} + 2512 \beta_{8} - 612 \beta_{7} - 612 \beta_{6} - 12371 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 438 \beta_{12} - 438 \beta_{11} - 353 \beta_{7} + 353 \beta_{6} - 4861 \beta_{5} - 5185 \beta_{4} + 8150 \beta_{3} - 22251 \beta_{2} + 187798 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 38034 \beta_{13} - 41294 \beta_{12} + 41294 \beta_{11} + 6398 \beta_{10} - 18090 \beta_{9} - 106354 \beta_{8} + 23254 \beta_{7} + 23254 \beta_{6} + 326691 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 16856 \beta_{12} + 16856 \beta_{11} + 4992 \beta_{7} - 4992 \beta_{6} + 156680 \beta_{5} + 173528 \beta_{4} - 290724 \beta_{3} + 638709 \beta_{2} - 5095498 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1159309 \beta_{13} + 1305209 \beta_{12} - 1305209 \beta_{11} - 223300 \beta_{10} + 583040 \beta_{9} + 3919080 \beta_{8} - 795240 \beta_{7} - 795240 \beta_{6} - 8979273 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 571940 \beta_{12} - 571940 \beta_{11} - 70629 \beta_{7} + 70629 \beta_{6} - 4867307 \beta_{5} - 5583507 \beta_{4} + 9809816 \beta_{3} - 18700905 \beta_{2} + 143215496 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 35162906 \beta_{13} - 40750986 \beta_{12} + 40750986 \beta_{11} + 7522056 \beta_{10} - 17895570 \beta_{9} - 134478098 \beta_{8} + 25909928 \beta_{7} + 25909928 \beta_{6} + \cdots + 254428041 \beta_1 \) Copy content Toggle raw display

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

Character values

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

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

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
− 5.52667i
− 4.66025i
− 4.31207i
− 4.00699i
− 2.62783i
− 1.35250i
− 1.16334i
1.16334i
1.35250i
2.62783i
4.00699i
4.31207i
4.66025i
5.52667i
− 5.52667i 3.00000i −22.5440 −7.34773 − 8.42680i 16.5800 − 22.7119i 80.3801i −9.00000 −46.5721 + 40.6084i
34.2 − 4.66025i − 3.00000i −13.7179 0.559505 + 11.1663i −13.9808 20.4072i 26.6471i −9.00000 52.0379 − 2.60743i
34.3 − 4.31207i − 3.00000i −10.5939 7.94512 − 7.86608i −12.9362 − 9.06309i 11.1853i −9.00000 −33.9191 − 34.2599i
34.4 − 4.00699i 3.00000i −8.05595 −2.16823 + 10.9681i 12.0210 − 26.2766i 0.224208i −9.00000 43.9490 + 8.68807i
34.5 − 2.62783i 3.00000i 1.09451 −10.6820 − 3.30060i 7.88349 24.0582i − 23.8988i −9.00000 −8.67342 + 28.0706i
34.6 − 1.35250i − 3.00000i 6.17074 −4.68011 − 10.1536i −4.05750 − 5.74924i − 19.1659i −9.00000 −13.7328 + 6.32986i
34.7 − 1.16334i 3.00000i 6.64664 9.37349 − 6.09407i 3.49002 − 19.4748i − 17.0390i −9.00000 −7.08947 − 10.9046i
34.8 1.16334i − 3.00000i 6.64664 9.37349 + 6.09407i 3.49002 19.4748i 17.0390i −9.00000 −7.08947 + 10.9046i
34.9 1.35250i 3.00000i 6.17074 −4.68011 + 10.1536i −4.05750 5.74924i 19.1659i −9.00000 −13.7328 − 6.32986i
34.10 2.62783i − 3.00000i 1.09451 −10.6820 + 3.30060i 7.88349 − 24.0582i 23.8988i −9.00000 −8.67342 − 28.0706i
34.11 4.00699i − 3.00000i −8.05595 −2.16823 − 10.9681i 12.0210 26.2766i − 0.224208i −9.00000 43.9490 − 8.68807i
34.12 4.31207i 3.00000i −10.5939 7.94512 + 7.86608i −12.9362 9.06309i − 11.1853i −9.00000 −33.9191 + 34.2599i
34.13 4.66025i 3.00000i −13.7179 0.559505 − 11.1663i −13.9808 − 20.4072i − 26.6471i −9.00000 52.0379 + 2.60743i
34.14 5.52667i − 3.00000i −22.5440 −7.34773 + 8.42680i 16.5800 22.7119i − 80.3801i −9.00000 −46.5721 − 40.6084i
\(n\): 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.a ✓ 14
3.b odd 2 1 495.4.c.c 14
5.b even 2 1 inner 165.4.c.a ✓ 14
5.c odd 4 1 825.4.a.bb 7
5.c odd 4 1 825.4.a.bc 7
15.d odd 2 1 495.4.c.c 14
15.e even 4 1 2475.4.a.bq 7
15.e even 4 1 2475.4.a.br 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.4.c.a ✓ 14 1.a even 1 1 trivial
165.4.c.a ✓ 14 5.b even 2 1 inner
495.4.c.c 14 3.b odd 2 1
495.4.c.c 14 15.d odd 2 1
825.4.a.bb 7 5.c odd 4 1
825.4.a.bc 7 5.c odd 4 1
2475.4.a.bq 7 15.e even 4 1
2475.4.a.br 7 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} + 97T_{2}^{12} + 3674T_{2}^{10} + 68702T_{2}^{8} + 656605T_{2}^{6} + 2988841T_{2}^{4} + 5502384T_{2}^{2} + 3385600 \) acting on \(S_{4}^{\mathrm{new}}(165, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + 97 T^{12} + 3674 T^{10} + \cdots + 3385600 \) 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
$7$ \( T^{14} + 2696 T^{12} + \cdots + 88\!\cdots\!04 \) Copy content Toggle raw display
$11$ \( (T + 11)^{14} \) Copy content Toggle raw display
$13$ \( T^{14} + 13532 T^{12} + \cdots + 88\!\cdots\!96 \) Copy content Toggle raw display
$17$ \( T^{14} + 31700 T^{12} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{7} + 26 T^{6} + \cdots + 300905291776)^{2} \) Copy content Toggle raw display
$23$ \( T^{14} + 66284 T^{12} + \cdots + 29\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( (T^{7} + 572 T^{6} + \cdots + 95816988486144)^{2} \) Copy content Toggle raw display
$31$ \( (T^{7} + 140 T^{6} + \cdots - 75419306700800)^{2} \) Copy content Toggle raw display
$37$ \( T^{14} + 97992 T^{12} + \cdots + 17\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( (T^{7} - 896 T^{6} + \cdots - 15\!\cdots\!56)^{2} \) Copy content Toggle raw display
$43$ \( T^{14} + 207116 T^{12} + \cdots + 40\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( T^{14} + 350396 T^{12} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{14} + 911380 T^{12} + \cdots + 21\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( (T^{7} + 1316 T^{6} + \cdots + 13\!\cdots\!60)^{2} \) Copy content Toggle raw display
$61$ \( (T^{7} + 386 T^{6} + \cdots + 45\!\cdots\!20)^{2} \) Copy content Toggle raw display
$67$ \( T^{14} + 2078072 T^{12} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{7} - 804 T^{6} + \cdots + 20\!\cdots\!84)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} + 2787456 T^{12} + \cdots + 18\!\cdots\!84 \) Copy content Toggle raw display
$79$ \( (T^{7} - 374 T^{6} + \cdots + 60\!\cdots\!68)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + 3964376 T^{12} + \cdots + 79\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{7} + 694 T^{6} + \cdots + 66\!\cdots\!08)^{2} \) Copy content Toggle raw display
$97$ \( T^{14} + 5503736 T^{12} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
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