Properties

Label 90.3.h.a
Level $90$
Weight $3$
Character orbit 90.h
Analytic conductor $2.452$
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] = [90,3,Mod(11,90)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(90, base_ring=CyclotomicField(6))
 
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("90.11");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 90.h (of order \(6\), degree \(2\), 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: \(2.45232237924\)
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: \( x^{16} - 12x^{14} + 138x^{12} - 1040x^{10} + 5541x^{8} - 26220x^{6} + 99328x^{4} - 202728x^{2} + 181476 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{3} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - \beta_{5} q^{2} + ( - \beta_{13} + \beta_{12} - \beta_{10} + \cdots - 1) q^{3}+ \cdots + ( - 2 \beta_{13} - 2 \beta_{12} + \cdots - 2 \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{2} + ( - \beta_{13} + \beta_{12} - \beta_{10} + \cdots - 1) q^{3}+ \cdots + ( - \beta_{15} + 4 \beta_{14} + \cdots + 58) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{3} + 16 q^{4} + 16 q^{6} - 4 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{3} + 16 q^{4} + 16 q^{6} - 4 q^{7} - 4 q^{9} - 16 q^{12} + 20 q^{13} - 36 q^{14} - 20 q^{15} - 32 q^{16} + 16 q^{18} + 80 q^{19} - 44 q^{21} + 24 q^{22} + 108 q^{23} - 8 q^{24} + 40 q^{25} - 124 q^{27} - 16 q^{28} - 72 q^{29} + 20 q^{30} - 16 q^{31} - 264 q^{33} - 48 q^{34} + 32 q^{36} - 88 q^{37} - 72 q^{38} - 8 q^{39} + 108 q^{41} + 128 q^{42} + 92 q^{43} - 80 q^{45} + 24 q^{46} + 216 q^{47} - 16 q^{48} - 84 q^{49} + 168 q^{51} - 40 q^{52} + 148 q^{54} - 72 q^{56} + 28 q^{57} + 144 q^{59} + 40 q^{60} - 76 q^{61} - 104 q^{63} - 128 q^{64} + 180 q^{65} + 96 q^{66} + 56 q^{67} + 72 q^{68} - 72 q^{69} + 60 q^{70} + 64 q^{72} + 416 q^{73} - 288 q^{74} + 20 q^{75} + 80 q^{76} - 684 q^{77} + 56 q^{78} + 80 q^{79} + 320 q^{81} - 192 q^{82} + 396 q^{83} + 40 q^{84} - 60 q^{85} - 216 q^{86} + 420 q^{87} - 48 q^{88} - 160 q^{90} - 656 q^{91} + 216 q^{92} - 356 q^{93} - 84 q^{94} - 360 q^{95} - 80 q^{96} - 16 q^{97} + 816 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 12x^{14} + 138x^{12} - 1040x^{10} + 5541x^{8} - 26220x^{6} + 99328x^{4} - 202728x^{2} + 181476 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 41 \nu^{14} - 225 \nu^{12} + 1333 \nu^{10} + 31 \nu^{8} - 114072 \nu^{6} + 798716 \nu^{4} + \cdots + 15616332 ) / 4341600 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 289974 \nu^{15} - 348681 \nu^{14} + 3618395 \nu^{13} - 9875745 \nu^{12} - 23841698 \nu^{11} + \cdots - 355528882692 ) / 93092587200 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 399973 \nu^{15} - 1285313 \nu^{14} + 10390855 \nu^{13} + 16249060 \nu^{12} + \cdots + 398535722064 ) / 93092587200 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 334523 \nu^{15} + 505875 \nu^{14} - 4891765 \nu^{13} - 10514390 \nu^{12} + 54769019 \nu^{11} + \cdots - 217467572760 ) / 31030862400 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 72336 \nu^{15} - 137953 \nu^{14} - 727168 \nu^{13} + 1003727 \nu^{12} + 8534536 \nu^{11} + \cdots + 238211532 ) / 3723703488 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 389063 \nu^{15} - 2906172 \nu^{14} + 3910476 \nu^{13} + 32198997 \nu^{12} + \cdots + 254978122644 ) / 18618517440 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1043836 \nu^{15} + 64681 \nu^{14} - 7299935 \nu^{13} + 2483580 \nu^{12} + 85117948 \nu^{11} + \cdots - 1432070568 ) / 46546293600 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2122022 \nu^{15} - 3591180 \nu^{14} + 26082035 \nu^{13} + 40328710 \nu^{12} + \cdots + 527986862280 ) / 93092587200 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 215 \nu^{15} - 2296 \nu^{13} + 25907 \nu^{11} - 183272 \nu^{9} + 859816 \nu^{7} - 4023896 \nu^{5} + \cdots + 4110048 ) / 8220096 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 881988 \nu^{15} + 2003904 \nu^{14} - 8761925 \nu^{13} - 15591600 \nu^{12} + 105064844 \nu^{11} + \cdots + 1201238208 ) / 31030862400 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 2939307 \nu^{15} + 1272107 \nu^{14} + 15171655 \nu^{13} - 28863275 \nu^{12} + \cdots - 476058785436 ) / 93092587200 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 9042 \nu^{15} + 90896 \nu^{13} - 1066817 \nu^{11} + 7184007 \nu^{9} - 35214371 \nu^{7} + \cdots + 642245346 \nu ) / 232731468 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 13164 \nu^{15} + 130775 \nu^{13} - 1568132 \nu^{11} + 10603551 \nu^{9} + \cdots + 1021382172 \nu ) / 231573600 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 6407642 \nu^{15} - 1841030 \nu^{14} + 51299115 \nu^{13} + 26308340 \nu^{12} + \cdots + 264953937600 ) / 93092587200 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 48776 \nu^{15} + 2911 \nu^{14} - 509200 \nu^{13} - 15975 \nu^{12} + 5779688 \nu^{11} + \cdots + 1108759572 ) / 616507200 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( - 2 \beta_{15} + 2 \beta_{14} - 6 \beta_{13} + 2 \beta_{11} + \beta_{10} - 2 \beta_{9} - \beta_{8} + \cdots + 4 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2 \beta_{15} + 6 \beta_{14} - 3 \beta_{12} - 6 \beta_{11} - \beta_{10} - 2 \beta_{9} + 3 \beta_{8} + \cdots + 12 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 5 \beta_{15} - 4 \beta_{14} - 18 \beta_{13} + 45 \beta_{12} - 4 \beta_{11} - 11 \beta_{10} + \cdots - 38 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 4 \beta_{15} + 6 \beta_{14} - 18 \beta_{13} - 33 \beta_{12} - 6 \beta_{11} - 37 \beta_{10} + \cdots - 90 ) / 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 19 \beta_{15} - 52 \beta_{14} + 144 \beta_{13} - 27 \beta_{12} - 52 \beta_{11} - 47 \beta_{10} + \cdots - 176 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 118 \beta_{15} - 246 \beta_{14} - 90 \beta_{13} + 99 \beta_{12} + 246 \beta_{11} - 109 \beta_{10} + \cdots - 102 ) / 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 391 \beta_{15} + 290 \beta_{14} + 396 \beta_{13} - 2367 \beta_{12} + 290 \beta_{11} + 319 \beta_{10} + \cdots + 2230 ) / 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 256 \beta_{15} - 714 \beta_{14} + 1170 \beta_{13} + 3039 \beta_{12} + 714 \beta_{11} + 2747 \beta_{10} + \cdots + 5430 ) / 6 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 2417 \beta_{15} + 4028 \beta_{14} - 9216 \beta_{13} - 6435 \beta_{12} + 4028 \beta_{11} + \cdots + 13552 ) / 6 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 3614 \beta_{15} + 11814 \beta_{14} + 6534 \beta_{13} - 1815 \beta_{12} - 11814 \beta_{11} + 10925 \beta_{10} + \cdots + 24930 ) / 6 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 46367 \beta_{15} - 3310 \beta_{14} - 52128 \beta_{13} + 85239 \beta_{12} - 3310 \beta_{11} + \cdots - 80726 ) / 6 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 22112 \beta_{15} + 81330 \beta_{14} - 73674 \beta_{13} - 220923 \beta_{12} - 81330 \beta_{11} + \cdots - 212358 ) / 6 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 50989 \beta_{15} - 258820 \beta_{14} + 353016 \beta_{13} + 572463 \beta_{12} - 258820 \beta_{11} + \cdots - 856784 ) / 6 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 150718 \beta_{15} - 451686 \beta_{14} - 517374 \beta_{13} - 380793 \beta_{12} + 451686 \beta_{11} + \cdots - 2222106 ) / 6 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 3029575 \beta_{15} - 660730 \beta_{14} + 3995640 \beta_{13} - 3532959 \beta_{12} - 660730 \beta_{11} + \cdots + 2510854 ) / 6 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/90\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(37\)
\(\chi(n)\) \(1 - \beta_{9}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
2.11536 1.82514i
−2.11536 + 0.410927i
−2.11536 1.82514i
2.11536 + 0.410927i
1.42311 + 1.82514i
1.42311 0.410927i
−1.42311 0.410927i
−1.42311 + 1.82514i
2.11536 + 1.82514i
−2.11536 0.410927i
−2.11536 + 1.82514i
2.11536 0.410927i
1.42311 1.82514i
1.42311 + 0.410927i
−1.42311 + 0.410927i
−1.42311 1.82514i
−1.22474 0.707107i −2.99847 0.0956863i 1.00000 + 1.73205i 1.93649 1.11803i 3.60470 + 2.23743i −4.35057 + 7.53542i 2.82843i 8.98169 + 0.573826i −3.16228
11.2 −1.22474 0.707107i −2.68332 + 1.34156i 1.00000 + 1.73205i −1.93649 + 1.11803i 4.23501 + 0.254321i 6.95799 12.0516i 2.82843i 5.40042 7.19969i 3.16228
11.3 −1.22474 0.707107i −0.129213 2.99722i 1.00000 + 1.73205i 1.93649 1.11803i −1.96110 + 3.76219i 3.31598 5.74345i 2.82843i −8.96661 + 0.774559i −3.16228
11.4 −1.22474 0.707107i 1.13677 + 2.77628i 1.00000 + 1.73205i −1.93649 + 1.11803i 0.570872 4.20406i −3.24916 + 5.62771i 2.82843i −6.41550 + 6.31200i 3.16228
11.5 1.22474 + 0.707107i −2.43483 + 1.75260i 1.00000 + 1.73205i −1.93649 + 1.11803i −4.22132 + 0.424801i −5.39499 + 9.34440i 2.82843i 2.85680 8.53456i −3.16228
11.6 1.22474 + 0.707107i 0.605881 + 2.93818i 1.00000 + 1.73205i 1.93649 1.11803i −1.33556 + 4.02694i 0.531482 0.920554i 2.82843i −8.26582 + 3.56038i 3.16228
11.7 1.22474 + 0.707107i 1.52181 2.58536i 1.00000 + 1.73205i 1.93649 1.11803i 3.69195 2.09033i −0.496891 + 0.860641i 2.82843i −4.36821 7.86884i 3.16228
11.8 1.22474 + 0.707107i 2.98138 + 0.333742i 1.00000 + 1.73205i −1.93649 + 1.11803i 3.41544 + 2.51690i 0.686165 1.18847i 2.82843i 8.77723 + 1.99002i −3.16228
41.1 −1.22474 + 0.707107i −2.99847 + 0.0956863i 1.00000 1.73205i 1.93649 + 1.11803i 3.60470 2.23743i −4.35057 7.53542i 2.82843i 8.98169 0.573826i −3.16228
41.2 −1.22474 + 0.707107i −2.68332 1.34156i 1.00000 1.73205i −1.93649 1.11803i 4.23501 0.254321i 6.95799 + 12.0516i 2.82843i 5.40042 + 7.19969i 3.16228
41.3 −1.22474 + 0.707107i −0.129213 + 2.99722i 1.00000 1.73205i 1.93649 + 1.11803i −1.96110 3.76219i 3.31598 + 5.74345i 2.82843i −8.96661 0.774559i −3.16228
41.4 −1.22474 + 0.707107i 1.13677 2.77628i 1.00000 1.73205i −1.93649 1.11803i 0.570872 + 4.20406i −3.24916 5.62771i 2.82843i −6.41550 6.31200i 3.16228
41.5 1.22474 0.707107i −2.43483 1.75260i 1.00000 1.73205i −1.93649 1.11803i −4.22132 0.424801i −5.39499 9.34440i 2.82843i 2.85680 + 8.53456i −3.16228
41.6 1.22474 0.707107i 0.605881 2.93818i 1.00000 1.73205i 1.93649 + 1.11803i −1.33556 4.02694i 0.531482 + 0.920554i 2.82843i −8.26582 3.56038i 3.16228
41.7 1.22474 0.707107i 1.52181 + 2.58536i 1.00000 1.73205i 1.93649 + 1.11803i 3.69195 + 2.09033i −0.496891 0.860641i 2.82843i −4.36821 + 7.86884i 3.16228
41.8 1.22474 0.707107i 2.98138 0.333742i 1.00000 1.73205i −1.93649 1.11803i 3.41544 2.51690i 0.686165 + 1.18847i 2.82843i 8.77723 1.99002i −3.16228
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.8
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 90.3.h.a 16
3.b odd 2 1 270.3.h.a 16
4.b odd 2 1 720.3.bs.d 16
5.b even 2 1 450.3.i.g 16
5.c odd 4 2 450.3.k.c 32
9.c even 3 1 270.3.h.a 16
9.c even 3 1 810.3.d.c 16
9.d odd 6 1 inner 90.3.h.a 16
9.d odd 6 1 810.3.d.c 16
12.b even 2 1 2160.3.bs.d 16
15.d odd 2 1 1350.3.i.g 16
15.e even 4 2 1350.3.k.b 32
36.f odd 6 1 2160.3.bs.d 16
36.h even 6 1 720.3.bs.d 16
45.h odd 6 1 450.3.i.g 16
45.j even 6 1 1350.3.i.g 16
45.k odd 12 2 1350.3.k.b 32
45.l even 12 2 450.3.k.c 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.3.h.a 16 1.a even 1 1 trivial
90.3.h.a 16 9.d odd 6 1 inner
270.3.h.a 16 3.b odd 2 1
270.3.h.a 16 9.c even 3 1
450.3.i.g 16 5.b even 2 1
450.3.i.g 16 45.h odd 6 1
450.3.k.c 32 5.c odd 4 2
450.3.k.c 32 45.l even 12 2
720.3.bs.d 16 4.b odd 2 1
720.3.bs.d 16 36.h even 6 1
810.3.d.c 16 9.c even 3 1
810.3.d.c 16 9.d odd 6 1
1350.3.i.g 16 15.d odd 2 1
1350.3.i.g 16 45.j even 6 1
1350.3.k.b 32 15.e even 4 2
1350.3.k.b 32 45.k odd 12 2
2160.3.bs.d 16 12.b even 2 1
2160.3.bs.d 16 36.f odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(90, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 2 T^{2} + 4)^{4} \) Copy content Toggle raw display
$3$ \( T^{16} + 4 T^{15} + \cdots + 43046721 \) Copy content Toggle raw display
$5$ \( (T^{4} - 5 T^{2} + 25)^{4} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 6662640625 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 7901855928576 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 991374241321216 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{8} - 40 T^{7} + \cdots + 4403341840)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 15\!\cdots\!61 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 35\!\cdots\!61 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 12\!\cdots\!56 \) Copy content Toggle raw display
$37$ \( (T^{8} + 44 T^{7} + \cdots - 221671664)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 23\!\cdots\!81 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 82\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 64\!\cdots\!41 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 19\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 51\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 20\!\cdots\!61 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 52\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 19\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots - 638114737319936)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 11\!\cdots\!21 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 95\!\cdots\!25 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 10\!\cdots\!56 \) Copy content Toggle raw display
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