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Properties

Label 900.3.f.i
Level $900$
Weight $3$
Character orbit 900.f
Analytic conductor $24.523$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [900,3,Mod(199,900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(900, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("900.199");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 900.f (of order \(2\), degree \(1\), 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: \(24.5232237924\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 12x^{14} + 138x^{12} - 1000x^{10} + 5291x^{8} - 17800x^{6} + 39458x^{4} - 53588x^{2} + 32761 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{28} \)
Twist minimal: no (minimal twist has level 180)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{10} q^{2} + ( - \beta_{5} + 2) q^{4} - \beta_{8} q^{7} + ( - \beta_{14} - \beta_{12} + 2 \beta_{10}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{10} q^{2} + ( - \beta_{5} + 2) q^{4} - \beta_{8} q^{7} + ( - \beta_{14} - \beta_{12} + 2 \beta_{10}) q^{8} - \beta_{6} q^{11} + ( - \beta_{15} - \beta_{11} + \cdots - \beta_{8}) q^{13}+ \cdots + (16 \beta_{13} - 16 \beta_{12} + 11 \beta_{10}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 28 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 28 q^{4} - 28 q^{16} + 232 q^{34} - 368 q^{46} + 304 q^{49} + 32 q^{61} + 364 q^{64} + 768 q^{76} + 336 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 12x^{14} + 138x^{12} - 1000x^{10} + 5291x^{8} - 17800x^{6} + 39458x^{4} - 53588x^{2} + 32761 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 1564 \nu^{14} - 26035 \nu^{12} + 272717 \nu^{10} - 2294806 \nu^{8} + 12106617 \nu^{6} + \cdots - 92455767 ) / 6203680 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1517 \nu^{14} + 11182 \nu^{12} - 145387 \nu^{10} + 769817 \nu^{8} - 3409951 \nu^{6} + \cdots + 7138059 ) / 1772480 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 12301 \nu^{14} - 134882 \nu^{12} + 1585239 \nu^{10} - 10860033 \nu^{8} + 56040923 \nu^{6} + \cdots - 337443471 ) / 12407360 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 159 \nu^{14} - 1702 \nu^{12} + 19133 \nu^{10} - 128595 \nu^{8} + 614137 \nu^{6} - 1661727 \nu^{4} + \cdots - 2555965 ) / 129920 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 279 \nu^{14} + 2818 \nu^{12} - 32761 \nu^{10} + 213667 \nu^{8} - 1032517 \nu^{6} + \cdots + 4194529 ) / 129920 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 4499 \nu^{14} + 45515 \nu^{12} - 538412 \nu^{10} + 3521601 \nu^{8} - 17610972 \nu^{6} + \cdots + 90006842 ) / 886240 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 753 \nu^{14} + 7754 \nu^{12} - 90611 \nu^{10} + 594605 \nu^{8} - 2934199 \nu^{6} + \cdots + 10110515 ) / 129920 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 76 \nu^{15} + 1455 \nu^{13} - 15013 \nu^{11} + 125594 \nu^{9} - 702033 \nu^{7} + \cdots + 4520663 \nu ) / 839840 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 5664 \nu^{15} - 34121 \nu^{13} + 492575 \nu^{11} - 2166718 \nu^{9} + 9649707 \nu^{7} + \cdots - 1684841 \nu ) / 11757760 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 2233184 \nu^{15} - 21899443 \nu^{13} + 256907341 \nu^{11} - 1641213002 \nu^{9} + \cdots - 22596309219 \nu ) / 4491464320 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 2981 \nu^{15} + 30342 \nu^{13} - 360879 \nu^{11} + 2364333 \nu^{9} - 11922963 \nu^{7} + \cdots + 58773711 \nu ) / 5878880 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 85762 \nu^{15} - 867511 \nu^{13} + 10192219 \nu^{11} - 65146100 \nu^{9} + 319614791 \nu^{7} + \cdots - 195043225 \nu ) / 154878080 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 4949768 \nu^{15} + 50039697 \nu^{13} - 580754295 \nu^{11} + 3781602326 \nu^{9} + \cdots + 74953045257 \nu ) / 4491464320 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 3371926 \nu^{15} + 33490087 \nu^{13} - 394763119 \nu^{11} + 2544388208 \nu^{9} + \cdots + 54030795461 \nu ) / 2245732160 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 9430 \nu^{15} + 107187 \nu^{13} - 1220071 \nu^{11} + 8527534 \nu^{9} - 42962735 \nu^{7} + \cdots + 226052383 \nu ) / 2939440 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{14} - \beta_{11} + 2\beta_{10} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{5} + 2\beta_{4} + \beta_{3} - \beta_{2} + 6 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4\beta_{14} - 6\beta_{13} + 2\beta_{12} - 5\beta_{11} - 8\beta_{10} + 3\beta_{8} ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -4\beta_{7} - 2\beta_{6} + 16\beta_{5} + 4\beta_{4} - 13\beta_{3} - 7\beta_{2} + 12\beta _1 - 70 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 10 \beta_{15} - 15 \beta_{14} + 4 \beta_{13} + 18 \beta_{12} + 74 \beta_{11} - 64 \beta_{10} + \cdots + 5 \beta_{8} ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -\beta_{7} - 16\beta_{6} - 19\beta_{5} - 38\beta_{4} - 61\beta_{3} + 27\beta_{2} + 9\beta _1 - 65 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 42 \beta_{15} - 119 \beta_{14} + 258 \beta_{13} - 84 \beta_{12} + 477 \beta_{11} + \cdots - 252 \beta_{8} ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 192\beta_{7} + 14\beta_{6} - 632\beta_{5} - 56\beta_{4} + 597\beta_{3} + 543\beta_{2} - 920\beta _1 + 3278 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 570 \beta_{15} + 966 \beta_{14} - 772 \beta_{13} - 810 \beta_{12} - 3781 \beta_{11} + \cdots - 873 \beta_{8} ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 266 \beta_{7} + 1896 \beta_{6} + 3192 \beta_{5} + 4602 \beta_{4} + 8005 \beta_{3} - 2441 \beta_{2} + \cdots + 632 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 3190 \beta_{15} + 5751 \beta_{14} - 15672 \beta_{13} + 6458 \beta_{12} - 31096 \beta_{11} + \cdots + 13145 \beta_{8} ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 3165 \beta_{7} + 244 \beta_{6} + 9437 \beta_{5} - 122 \beta_{4} - 6613 \beta_{3} - 7989 \beta_{2} + \cdots - 54202 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 29380 \beta_{15} - 67131 \beta_{14} + 57268 \beta_{13} + 51256 \beta_{12} + 191671 \beta_{11} + \cdots + 54600 \beta_{8} ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 17696 \beta_{7} - 103684 \beta_{6} - 212746 \beta_{5} - 301226 \beta_{4} - 446003 \beta_{3} + \cdots - 17522 ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 179644 \beta_{15} - 378264 \beta_{14} + 1015030 \beta_{13} - 404910 \beta_{12} + \cdots - 729481 \beta_{8} ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
−1.97048 + 1.96040i
−1.97048 + 0.960405i
−1.97048 − 1.96040i
−1.97048 − 0.960405i
−1.36646 + 0.842371i
−1.36646 − 0.157629i
−1.36646 − 0.842371i
−1.36646 + 0.157629i
1.36646 − 0.157629i
1.36646 + 0.842371i
1.36646 + 0.157629i
1.36646 − 0.842371i
1.97048 + 0.960405i
1.97048 + 1.96040i
1.97048 − 0.960405i
1.97048 − 1.96040i
−1.97048 − 0.342371i 0 3.76556 + 1.34927i 0 0 −11.5108 −6.95801 − 3.94792i 0 0
199.2 −1.97048 − 0.342371i 0 3.76556 + 1.34927i 0 0 11.5108 −6.95801 − 3.94792i 0 0
199.3 −1.97048 + 0.342371i 0 3.76556 − 1.34927i 0 0 −11.5108 −6.95801 + 3.94792i 0 0
199.4 −1.97048 + 0.342371i 0 3.76556 − 1.34927i 0 0 11.5108 −6.95801 + 3.94792i 0 0
199.5 −1.36646 − 1.46040i 0 −0.265564 + 3.99117i 0 0 −1.87135 6.19161 − 5.06596i 0 0
199.6 −1.36646 − 1.46040i 0 −0.265564 + 3.99117i 0 0 1.87135 6.19161 − 5.06596i 0 0
199.7 −1.36646 + 1.46040i 0 −0.265564 − 3.99117i 0 0 −1.87135 6.19161 + 5.06596i 0 0
199.8 −1.36646 + 1.46040i 0 −0.265564 − 3.99117i 0 0 1.87135 6.19161 + 5.06596i 0 0
199.9 1.36646 − 1.46040i 0 −0.265564 − 3.99117i 0 0 −1.87135 −6.19161 − 5.06596i 0 0
199.10 1.36646 − 1.46040i 0 −0.265564 − 3.99117i 0 0 1.87135 −6.19161 − 5.06596i 0 0
199.11 1.36646 + 1.46040i 0 −0.265564 + 3.99117i 0 0 −1.87135 −6.19161 + 5.06596i 0 0
199.12 1.36646 + 1.46040i 0 −0.265564 + 3.99117i 0 0 1.87135 −6.19161 + 5.06596i 0 0
199.13 1.97048 − 0.342371i 0 3.76556 − 1.34927i 0 0 −11.5108 6.95801 − 3.94792i 0 0
199.14 1.97048 − 0.342371i 0 3.76556 − 1.34927i 0 0 11.5108 6.95801 − 3.94792i 0 0
199.15 1.97048 + 0.342371i 0 3.76556 + 1.34927i 0 0 −11.5108 6.95801 + 3.94792i 0 0
199.16 1.97048 + 0.342371i 0 3.76556 + 1.34927i 0 0 11.5108 6.95801 + 3.94792i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 199.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
5.b even 2 1 inner
12.b even 2 1 inner
15.d odd 2 1 inner
20.d odd 2 1 inner
60.h even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.3.f.i 16
3.b odd 2 1 inner 900.3.f.i 16
4.b odd 2 1 inner 900.3.f.i 16
5.b even 2 1 inner 900.3.f.i 16
5.c odd 4 1 180.3.c.c ✓ 8
5.c odd 4 1 900.3.c.o 8
12.b even 2 1 inner 900.3.f.i 16
15.d odd 2 1 inner 900.3.f.i 16
15.e even 4 1 180.3.c.c ✓ 8
15.e even 4 1 900.3.c.o 8
20.d odd 2 1 inner 900.3.f.i 16
20.e even 4 1 180.3.c.c ✓ 8
20.e even 4 1 900.3.c.o 8
40.i odd 4 1 2880.3.e.i 8
40.k even 4 1 2880.3.e.i 8
60.h even 2 1 inner 900.3.f.i 16
60.l odd 4 1 180.3.c.c ✓ 8
60.l odd 4 1 900.3.c.o 8
120.q odd 4 1 2880.3.e.i 8
120.w even 4 1 2880.3.e.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.3.c.c ✓ 8 5.c odd 4 1
180.3.c.c ✓ 8 15.e even 4 1
180.3.c.c ✓ 8 20.e even 4 1
180.3.c.c ✓ 8 60.l odd 4 1
900.3.c.o 8 5.c odd 4 1
900.3.c.o 8 15.e even 4 1
900.3.c.o 8 20.e even 4 1
900.3.c.o 8 60.l odd 4 1
900.3.f.i 16 1.a even 1 1 trivial
900.3.f.i 16 3.b odd 2 1 inner
900.3.f.i 16 4.b odd 2 1 inner
900.3.f.i 16 5.b even 2 1 inner
900.3.f.i 16 12.b even 2 1 inner
900.3.f.i 16 15.d odd 2 1 inner
900.3.f.i 16 20.d odd 2 1 inner
900.3.f.i 16 60.h even 2 1 inner
2880.3.e.i 8 40.i odd 4 1
2880.3.e.i 8 40.k even 4 1
2880.3.e.i 8 120.q odd 4 1
2880.3.e.i 8 120.w even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(900, [\chi])\):

\( T_{7}^{4} - 136T_{7}^{2} + 464 \) Copy content Toggle raw display
\( T_{29}^{4} - 456T_{29}^{2} + 35344 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - 7 T^{6} + \cdots + 256)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{4} - 136 T^{2} + 464)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} + 328 T^{2} + 22736)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} + 528 T^{2} + 65536)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} + 1096 T^{2} + 150544)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} + 944 T^{2} + 118784)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} - 368 T^{2} + 29696)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} - 456 T^{2} + 35344)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 1840 T^{2} + 742400)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 2832 T^{2} + 802816)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} - 832)^{8} \) Copy content Toggle raw display
$43$ \( (T^{4} - 3776 T^{2} + 1900544)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} - 5552 T^{2} + 7602176)^{4} \) Copy content Toggle raw display
$53$ \( (T^{4} + 3624 T^{2} + 21904)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} + 16488 T^{2} + 51452496)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} - 4 T - 1036)^{8} \) Copy content Toggle raw display
$67$ \( (T^{4} - 21664 T^{2} + 75732224)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 2848 T^{2} + 363776)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 3880 T^{2} + 19600)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 16816 T^{2} + 65598464)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 24608 T^{2} + 69852416)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} - 28704 T^{2} + 160985344)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 20520 T^{2} + 71571600)^{4} \) Copy content Toggle raw display
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