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Properties

Label 135.3.i.a
Level $135$
Weight $3$
Character orbit 135.i
Analytic conductor $3.678$
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] = [135,3,Mod(71,135)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(135, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("135.71");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 135 = 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 135.i (of order \(6\), degree \(2\), not 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: \(3.67848356886\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 48x^{14} + 912x^{12} + 8704x^{10} + 43602x^{8} + 109032x^{6} + 117844x^{4} + 36000x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{6} \)
Twist minimal: no (minimal twist has level 45)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{4} + \beta_1) q^{2} + ( - \beta_{15} + 2 \beta_{7} + \cdots + \beta_1) q^{4}+ \cdots + ( - \beta_{15} + \beta_{9} + \beta_{8} + \cdots - 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} + \beta_1) q^{2} + ( - \beta_{15} + 2 \beta_{7} + \cdots + \beta_1) q^{4}+ \cdots + (7 \beta_{15} + \beta_{14} + \beta_{13} + \cdots + 8) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 16 q^{4} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 16 q^{4} + 2 q^{7} + 18 q^{11} - 10 q^{13} + 54 q^{14} - 32 q^{16} - 52 q^{19} - 24 q^{22} + 54 q^{23} + 40 q^{25} + 32 q^{28} + 54 q^{29} + 32 q^{31} - 216 q^{32} + 54 q^{34} + 44 q^{37} - 252 q^{38} - 30 q^{40} - 144 q^{41} - 124 q^{43} - 108 q^{46} + 216 q^{47} - 54 q^{49} + 62 q^{52} + 18 q^{56} + 90 q^{58} + 486 q^{59} + 62 q^{61} + 256 q^{64} + 90 q^{65} + 14 q^{67} + 288 q^{68} - 60 q^{70} - 268 q^{73} - 540 q^{74} - 106 q^{76} - 702 q^{77} - 40 q^{79} - 204 q^{82} - 522 q^{83} + 30 q^{85} - 54 q^{86} + 144 q^{88} + 136 q^{91} + 1332 q^{92} - 150 q^{94} - 180 q^{95} - 142 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 48x^{14} + 912x^{12} + 8704x^{10} + 43602x^{8} + 109032x^{6} + 117844x^{4} + 36000x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{14} + 21\nu^{12} - 99\nu^{10} - 4607\nu^{8} - 27813\nu^{6} - 21801\nu^{4} + 63115\nu^{2} + 2223 ) / 19536 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{14} + 43 \nu^{12} + 715 \nu^{10} + 5777 \nu^{8} + 23051 \nu^{6} + 39821 \nu^{4} + 25605 \nu^{2} + \cdots + 28755 ) / 4752 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - \nu^{14} - 43 \nu^{12} - 715 \nu^{10} - 5777 \nu^{8} - 23051 \nu^{6} - 39821 \nu^{4} - 20853 \nu^{2} + \cdots - 243 ) / 4752 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 77 \nu^{15} - 9 \nu^{14} + 3245 \nu^{13} - 189 \nu^{12} + 52613 \nu^{11} + 891 \nu^{10} + \cdots - 20007 ) / 351648 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 47 \nu^{15} + 9 \nu^{14} + 2467 \nu^{13} + 189 \nu^{12} + 51883 \nu^{11} - 891 \nu^{10} + \cdots + 1074951 ) / 351648 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3 \nu^{15} + 143 \nu^{13} + 2693 \nu^{11} + 25397 \nu^{9} + 125029 \nu^{7} + 304045 \nu^{5} + \cdots + 2376 ) / 4752 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 247 \nu^{15} + 77 \nu^{14} + 11847 \nu^{13} + 3245 \nu^{12} + 225075 \nu^{11} + 52613 \nu^{10} + \cdots - 678645 ) / 351648 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 247 \nu^{15} - 77 \nu^{14} + 11847 \nu^{13} - 3245 \nu^{12} + 225075 \nu^{11} - 52613 \nu^{10} + \cdots + 678645 ) / 351648 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 277 \nu^{15} - 57 \nu^{14} - 13143 \nu^{13} - 2973 \nu^{12} - 246081 \nu^{11} - 60735 \nu^{10} + \cdots - 348489 ) / 175824 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 691 \nu^{15} + 157 \nu^{14} + 33011 \nu^{13} + 5665 \nu^{12} + 623639 \nu^{11} + 68077 \nu^{10} + \cdots + 923103 ) / 351648 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 691 \nu^{15} - 311 \nu^{14} - 33011 \nu^{13} - 14375 \nu^{12} - 623639 \nu^{11} - 258107 \nu^{10} + \cdots - 1947429 ) / 351648 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 769 \nu^{15} + 103 \nu^{14} + 36869 \nu^{13} + 4087 \nu^{12} + 700721 \nu^{11} + 62323 \nu^{10} + \cdots + 3720141 ) / 351648 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 769 \nu^{15} - 103 \nu^{14} + 36869 \nu^{13} - 4087 \nu^{12} + 700721 \nu^{11} - 62323 \nu^{10} + \cdots - 3720141 ) / 351648 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 17 \nu^{15} + 815 \nu^{13} + 15443 \nu^{11} + 146605 \nu^{9} + 727123 \nu^{7} + 1784449 \nu^{5} + \cdots - 14256 ) / 4752 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{3} - 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} - \beta_{9} - \beta_{8} - 4\beta_{7} + \beta_{6} + \beta_{5} - 10\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{9} - 3\beta_{8} - 18\beta_{4} - 12\beta_{3} + \beta_{2} + 3\beta _1 + 60 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 20 \beta_{15} - \beta_{14} - \beta_{13} + \beta_{12} + \beta_{11} - 2 \beta_{10} + 18 \beta_{9} + \cdots - 43 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 2 \beta_{15} - 4 \beta_{14} + 4 \beta_{13} - 2 \beta_{12} + 2 \beta_{11} - 59 \beta_{9} + 59 \beta_{8} + \cdots - 673 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 327 \beta_{15} + 33 \beta_{14} + 33 \beta_{13} - 27 \beta_{12} - 27 \beta_{11} + 54 \beta_{10} + \cdots + 717 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 48 \beta_{15} + 102 \beta_{14} - 102 \beta_{13} + 42 \beta_{12} - 54 \beta_{11} + 948 \beta_{9} + \cdots + 7989 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 4982 \beta_{15} - 696 \beta_{14} - 696 \beta_{13} + 540 \beta_{12} + 540 \beta_{11} - 1080 \beta_{10} + \cdots - 10894 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 792 \beta_{15} - 1902 \beta_{14} + 1902 \beta_{13} - 606 \beta_{12} + 978 \beta_{11} - 14298 \beta_{9} + \cdots - 98556 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 73279 \beta_{15} + 12398 \beta_{14} + 12398 \beta_{13} - 9518 \beta_{12} - 9518 \beta_{11} + \cdots + 158666 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 11272 \beta_{15} + 31478 \beta_{14} - 31478 \beta_{13} + 7222 \beta_{12} - 15322 \beta_{11} + \cdots + 1252244 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 1056342 \beta_{15} - 202821 \beta_{14} - 202821 \beta_{13} + 156021 \beta_{12} + 156021 \beta_{11} + \cdots - 2261853 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 149610 \beta_{15} - 490674 \beta_{14} + 490674 \beta_{13} - 73788 \beta_{12} + 225432 \beta_{11} + \cdots - 16282467 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 15042373 \beta_{15} + 3156663 \beta_{14} + 3156663 \beta_{13} - 2440557 \beta_{12} - 2440557 \beta_{11} + \cdots + 31876901 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/135\mathbb{Z}\right)^\times\).

\(n\) \(56\) \(82\)
\(\chi(n)\) \(\beta_{7}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
71.1
3.09125i
2.82877i
1.83391i
0.692902i
− 0.0476108i
− 1.39204i
− 3.27064i
− 3.73655i
− 3.09125i
− 2.82877i
− 1.83391i
− 0.692902i
0.0476108i
1.39204i
3.27064i
3.73655i
−2.67710 + 1.54563i 0 2.77793 − 4.81151i −1.93649 − 1.11803i 0 1.10296 + 1.91037i 4.80953i 0 6.91225
71.2 −2.44978 + 1.41438i 0 2.00096 − 3.46576i 1.93649 + 1.11803i 0 −3.97472 − 6.88441i 0.00541780i 0 −6.32531
71.3 −1.58822 + 0.916957i 0 −0.318381 + 0.551452i 1.93649 + 1.11803i 0 3.23398 + 5.60142i − 8.50342i 0 −4.10076
71.4 −0.600071 + 0.346451i 0 −1.75994 + 3.04831i −1.93649 − 1.11803i 0 −6.14112 − 10.6367i − 5.21055i 0 1.54938
71.5 0.0412321 − 0.0238054i 0 −1.99887 + 3.46214i −1.93649 − 1.11803i 0 1.90756 + 3.30399i 0.380778i 0 −0.106461
71.6 1.20554 − 0.696021i 0 −1.03111 + 1.78593i 1.93649 + 1.11803i 0 4.41004 + 7.63842i 8.43886i 0 3.11270
71.7 2.83245 − 1.63532i 0 3.34853 − 5.79983i 1.93649 + 1.11803i 0 −3.16931 − 5.48940i − 8.82112i 0 7.31337
71.8 3.23594 − 1.86827i 0 4.98088 − 8.62715i −1.93649 − 1.11803i 0 3.63061 + 6.28840i − 22.2764i 0 −8.35517
116.1 −2.67710 − 1.54563i 0 2.77793 + 4.81151i −1.93649 + 1.11803i 0 1.10296 − 1.91037i − 4.80953i 0 6.91225
116.2 −2.44978 − 1.41438i 0 2.00096 + 3.46576i 1.93649 − 1.11803i 0 −3.97472 + 6.88441i − 0.00541780i 0 −6.32531
116.3 −1.58822 − 0.916957i 0 −0.318381 − 0.551452i 1.93649 − 1.11803i 0 3.23398 − 5.60142i 8.50342i 0 −4.10076
116.4 −0.600071 − 0.346451i 0 −1.75994 − 3.04831i −1.93649 + 1.11803i 0 −6.14112 + 10.6367i 5.21055i 0 1.54938
116.5 0.0412321 + 0.0238054i 0 −1.99887 − 3.46214i −1.93649 + 1.11803i 0 1.90756 − 3.30399i − 0.380778i 0 −0.106461
116.6 1.20554 + 0.696021i 0 −1.03111 − 1.78593i 1.93649 − 1.11803i 0 4.41004 − 7.63842i − 8.43886i 0 3.11270
116.7 2.83245 + 1.63532i 0 3.34853 + 5.79983i 1.93649 − 1.11803i 0 −3.16931 + 5.48940i 8.82112i 0 7.31337
116.8 3.23594 + 1.86827i 0 4.98088 + 8.62715i −1.93649 + 1.11803i 0 3.63061 − 6.28840i 22.2764i 0 −8.35517
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 71.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 135.3.i.a 16
3.b odd 2 1 45.3.i.a ✓ 16
4.b odd 2 1 2160.3.bs.c 16
5.b even 2 1 675.3.j.b 16
5.c odd 4 2 675.3.i.c 32
9.c even 3 1 45.3.i.a ✓ 16
9.c even 3 1 405.3.c.a 16
9.d odd 6 1 inner 135.3.i.a 16
9.d odd 6 1 405.3.c.a 16
12.b even 2 1 720.3.bs.c 16
15.d odd 2 1 225.3.j.b 16
15.e even 4 2 225.3.i.b 32
36.f odd 6 1 720.3.bs.c 16
36.h even 6 1 2160.3.bs.c 16
45.h odd 6 1 675.3.j.b 16
45.j even 6 1 225.3.j.b 16
45.k odd 12 2 225.3.i.b 32
45.l even 12 2 675.3.i.c 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.3.i.a ✓ 16 3.b odd 2 1
45.3.i.a ✓ 16 9.c even 3 1
135.3.i.a 16 1.a even 1 1 trivial
135.3.i.a 16 9.d odd 6 1 inner
225.3.i.b 32 15.e even 4 2
225.3.i.b 32 45.k odd 12 2
225.3.j.b 16 15.d odd 2 1
225.3.j.b 16 45.j even 6 1
405.3.c.a 16 9.c even 3 1
405.3.c.a 16 9.d odd 6 1
675.3.i.c 32 5.c odd 4 2
675.3.i.c 32 45.l even 12 2
675.3.j.b 16 5.b even 2 1
675.3.j.b 16 45.h odd 6 1
720.3.bs.c 16 12.b even 2 1
720.3.bs.c 16 36.f odd 6 1
2160.3.bs.c 16 4.b odd 2 1
2160.3.bs.c 16 36.h even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(135, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 24 T^{14} + \cdots + 81 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{4} - 5 T^{2} + 25)^{4} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 4654875290256 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 112356358416 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 69380636886016 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{8} + 26 T^{7} + \cdots - 3133657244)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 64\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 21\!\cdots\!56 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 396718580736 \) Copy content Toggle raw display
$37$ \( (T^{8} - 22 T^{7} + \cdots - 143779124336)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 44\!\cdots\!61 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 36\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 28\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 25\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 18\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 39\!\cdots\!61 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots - 3873480104384)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 15\!\cdots\!16 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 23\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 17\!\cdots\!76 \) Copy content Toggle raw display
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