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Properties

Label 450.3.i.d
Level $450$
Weight $3$
Character orbit 450.i
Analytic conductor $12.262$
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] = [450,3,Mod(101,450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(450, 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("450.101");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 450.i (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: \(12.2616118962\)
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: \( x^{16} - 2 x^{15} + 7 x^{14} + 18 x^{13} - 12 x^{12} - 180 x^{11} + 819 x^{10} - 1080 x^{9} + \cdots + 43046721 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{6} \)
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.

\(f(q)\) \(=\) \( q + (\beta_{8} - \beta_{2}) q^{2} + ( - \beta_{7} + \beta_1) q^{3} + (2 \beta_{5} + 2) q^{4} + ( - \beta_{11} + \beta_{9}) q^{6} + (\beta_{11} + \beta_{8} + \beta_{6} + \cdots - 1) q^{7}+ \cdots + ( - \beta_{14} - \beta_{13} + \cdots - \beta_{5}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{8} - \beta_{2}) q^{2} + ( - \beta_{7} + \beta_1) q^{3} + (2 \beta_{5} + 2) q^{4} + ( - \beta_{11} + \beta_{9}) q^{6} + (\beta_{11} + \beta_{8} + \beta_{6} + \cdots - 1) q^{7}+ \cdots + ( - 2 \beta_{15} + 11 \beta_{14} + \cdots - 34) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 2 q^{3} + 16 q^{4} + 4 q^{6} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 2 q^{3} + 16 q^{4} + 4 q^{6} - 2 q^{7} + 2 q^{9} - 54 q^{11} - 8 q^{12} - 20 q^{13} - 36 q^{14} - 32 q^{16} + 8 q^{18} - 40 q^{19} + 34 q^{21} + 12 q^{22} + 54 q^{23} + 16 q^{24} + 88 q^{27} - 8 q^{28} + 90 q^{29} + 8 q^{31} - 72 q^{33} + 24 q^{34} + 20 q^{36} - 44 q^{37} - 36 q^{38} - 98 q^{39} + 162 q^{41} - 56 q^{42} - 44 q^{43} + 24 q^{46} + 108 q^{47} - 8 q^{48} - 6 q^{49} - 210 q^{51} + 40 q^{52} - 104 q^{54} - 72 q^{56} - 256 q^{57} - 126 q^{59} - 34 q^{61} + 458 q^{63} - 128 q^{64} - 96 q^{66} - 182 q^{67} + 36 q^{68} + 210 q^{69} + 32 q^{72} + 208 q^{73} + 360 q^{74} - 40 q^{76} + 468 q^{77} - 152 q^{78} - 22 q^{79} + 386 q^{81} - 96 q^{82} - 612 q^{83} + 172 q^{84} + 216 q^{86} + 90 q^{87} - 24 q^{88} + 76 q^{91} + 108 q^{92} - 598 q^{93} - 84 q^{94} + 16 q^{96} - 98 q^{97} - 630 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} + 7 x^{14} + 18 x^{13} - 12 x^{12} - 180 x^{11} + 819 x^{10} - 1080 x^{9} + \cdots + 43046721 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2563 \nu^{15} + 10165 \nu^{14} + 3046 \nu^{13} - 26109 \nu^{12} + 165561 \nu^{11} + \cdots + 32567235921 ) / 25708458375 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1036 \nu^{15} + 32840 \nu^{14} + 9053 \nu^{13} - 100542 \nu^{12} + 624738 \nu^{11} + \cdots + 184137929208 ) / 8569486125 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1400 \nu^{15} + 4598 \nu^{14} - 17443 \nu^{13} - 16542 \nu^{12} + 252714 \nu^{11} + \cdots + 20844178902 ) / 5141691675 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 7778 \nu^{15} - 25150 \nu^{14} - 20164 \nu^{13} + 189486 \nu^{12} - 35664 \nu^{11} + \cdots - 127662225579 ) / 25708458375 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 7712 \nu^{15} - 140015 \nu^{14} + 40381 \nu^{13} + 316161 \nu^{12} - 1861419 \nu^{11} + \cdots - 641616159474 ) / 25708458375 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1066 \nu^{15} + 8290 \nu^{14} - 5498 \nu^{13} - 6408 \nu^{12} + 156687 \nu^{11} + \cdots + 37201932882 ) / 2856495375 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 10784 \nu^{15} + 17320 \nu^{14} + 16627 \nu^{13} - 216693 \nu^{12} - 209388 \nu^{11} + \cdots + 67191148512 ) / 25708458375 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1699 \nu^{15} - 1655 \nu^{14} - 8027 \nu^{13} + 21813 \nu^{12} + 20958 \nu^{11} + \cdots - 12258749547 ) / 2856495375 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 6484 \nu^{15} - 29675 \nu^{14} - 8597 \nu^{13} + 96963 \nu^{12} - 81102 \nu^{11} + \cdots - 157134880557 ) / 8569486125 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 2171 \nu^{15} - 11890 \nu^{14} - 5518 \nu^{13} + 59457 \nu^{12} - 171888 \nu^{11} + \cdots - 63838287243 ) / 2856495375 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 26971 \nu^{15} - 54995 \nu^{14} - 119813 \nu^{13} + 588267 \nu^{12} + 745872 \nu^{11} + \cdots - 238631889348 ) / 25708458375 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 40853 \nu^{15} - 174325 \nu^{14} - 127084 \nu^{13} + 834381 \nu^{12} - 730104 \nu^{11} + \cdots - 703746926784 ) / 25708458375 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 48239 \nu^{15} - 74500 \nu^{14} - 59362 \nu^{13} + 1034298 \nu^{12} + 698583 \nu^{11} + \cdots - 290388396897 ) / 25708458375 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 64684 \nu^{15} + 183665 \nu^{14} + 63587 \nu^{13} - 1254303 \nu^{12} + 611832 \nu^{11} + \cdots + 748644656787 ) / 25708458375 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{15} + \beta_{14} + \beta_{13} - \beta_{12} + \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} - \beta_{14} + \beta_{13} + \beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} - \beta_{7} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 3 \beta_{14} - \beta_{13} + \beta_{12} + 4 \beta_{11} - 8 \beta_{10} - 7 \beta_{9} - 2 \beta_{8} + \cdots + 18 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 5 \beta_{15} - 5 \beta_{14} - 9 \beta_{13} - 7 \beta_{12} - 17 \beta_{11} - 2 \beta_{10} + 41 \beta_{9} + \cdots + 94 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 35 \beta_{15} - 13 \beta_{14} - 50 \beta_{13} + 3 \beta_{12} + 97 \beta_{11} - 41 \beta_{10} + \cdots - 46 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 159 \beta_{15} + 123 \beta_{14} + 209 \beta_{13} - 440 \beta_{12} - 260 \beta_{11} - 29 \beta_{10} + \cdots - 42 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 239 \beta_{15} - 41 \beta_{14} + 279 \beta_{13} + 218 \beta_{12} + 550 \beta_{11} + 187 \beta_{10} + \cdots - 2219 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 640 \beta_{15} - 544 \beta_{14} - 356 \beta_{13} + 129 \beta_{12} + 1033 \beta_{11} + 1138 \beta_{10} + \cdots + 13346 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 480 \beta_{15} + 2166 \beta_{14} - 2374 \beta_{13} + 964 \beta_{12} - 6056 \beta_{11} + 3652 \beta_{10} + \cdots + 16797 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1940 \beta_{15} + 6682 \beta_{14} - 9180 \beta_{13} + 1118 \beta_{12} - 7352 \beta_{11} + 5542 \beta_{10} + \cdots + 778 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 11215 \beta_{15} + 39020 \beta_{14} - 7700 \beta_{13} - 25395 \beta_{12} - 13970 \beta_{11} + \cdots - 104959 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 26445 \beta_{15} + 43170 \beta_{14} + 11270 \beta_{13} - 21500 \beta_{12} - 65555 \beta_{11} + \cdots - 915630 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 118121 \beta_{15} - 136769 \beta_{14} - 281349 \beta_{13} + 173414 \beta_{12} + 232390 \beta_{11} + \cdots - 846266 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 1495259 \beta_{15} + 66839 \beta_{14} - 1096394 \beta_{13} + 616035 \beta_{12} + 382426 \beta_{11} + \cdots + 875339 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/450\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1 + \beta_{5}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1
−2.83706 + 0.975236i
0.924116 + 2.85412i
2.92687 + 0.658348i
0.710818 − 2.91457i
−1.62909 + 2.51914i
2.23941 + 1.99626i
−2.53370 − 1.60635i
1.19864 − 2.75014i
−2.83706 − 0.975236i
0.924116 − 2.85412i
2.92687 − 0.658348i
0.710818 + 2.91457i
−1.62909 − 2.51914i
2.23941 − 1.99626i
−2.53370 + 1.60635i
1.19864 + 2.75014i
−1.22474 − 0.707107i −2.26311 − 1.96935i 1.00000 + 1.73205i 0 1.37919 + 4.01221i 1.79327 − 3.10604i − 2.82843i 1.24333 + 8.91370i 0
101.2 −1.22474 − 0.707107i −2.00968 + 2.22737i 1.00000 + 1.73205i 0 4.03634 − 1.30690i −5.80576 + 10.0559i − 2.82843i −0.922346 − 8.95261i 0
101.3 −1.22474 − 0.707107i 0.893290 + 2.86392i 1.00000 + 1.73205i 0 0.931045 − 4.13922i 2.67148 − 4.62713i − 2.82843i −7.40407 + 5.11662i 0
101.4 −1.22474 − 0.707107i 2.87950 − 0.841701i 1.00000 + 1.73205i 0 −4.12183 − 1.00525i 4.51525 − 7.82064i − 2.82843i 7.58308 − 4.84736i 0
101.5 1.22474 + 0.707107i −2.99618 − 0.151259i 1.00000 + 1.73205i 0 −3.56261 − 2.30388i 1.73734 − 3.00916i 2.82843i 8.95424 + 0.906401i 0
101.6 1.22474 + 0.707107i −0.609109 + 2.93751i 1.00000 + 1.73205i 0 −2.82314 + 3.16700i −1.74970 + 3.03056i 2.82843i −8.25797 − 3.57853i 0
101.7 1.22474 + 0.707107i 0.124284 − 2.99742i 1.00000 + 1.73205i 0 2.27172 − 3.58320i −5.26784 + 9.12416i 2.82843i −8.96911 − 0.745063i 0
101.8 1.22474 + 0.707107i 2.98101 − 0.337018i 1.00000 + 1.73205i 0 3.88928 + 1.69513i 1.10596 − 1.91558i 2.82843i 8.77284 − 2.00931i 0
401.1 −1.22474 + 0.707107i −2.26311 + 1.96935i 1.00000 − 1.73205i 0 1.37919 − 4.01221i 1.79327 + 3.10604i 2.82843i 1.24333 − 8.91370i 0
401.2 −1.22474 + 0.707107i −2.00968 − 2.22737i 1.00000 − 1.73205i 0 4.03634 + 1.30690i −5.80576 − 10.0559i 2.82843i −0.922346 + 8.95261i 0
401.3 −1.22474 + 0.707107i 0.893290 − 2.86392i 1.00000 − 1.73205i 0 0.931045 + 4.13922i 2.67148 + 4.62713i 2.82843i −7.40407 − 5.11662i 0
401.4 −1.22474 + 0.707107i 2.87950 + 0.841701i 1.00000 − 1.73205i 0 −4.12183 + 1.00525i 4.51525 + 7.82064i 2.82843i 7.58308 + 4.84736i 0
401.5 1.22474 − 0.707107i −2.99618 + 0.151259i 1.00000 − 1.73205i 0 −3.56261 + 2.30388i 1.73734 + 3.00916i − 2.82843i 8.95424 − 0.906401i 0
401.6 1.22474 − 0.707107i −0.609109 − 2.93751i 1.00000 − 1.73205i 0 −2.82314 − 3.16700i −1.74970 − 3.03056i − 2.82843i −8.25797 + 3.57853i 0
401.7 1.22474 − 0.707107i 0.124284 + 2.99742i 1.00000 − 1.73205i 0 2.27172 + 3.58320i −5.26784 − 9.12416i − 2.82843i −8.96911 + 0.745063i 0
401.8 1.22474 − 0.707107i 2.98101 + 0.337018i 1.00000 − 1.73205i 0 3.88928 − 1.69513i 1.10596 + 1.91558i − 2.82843i 8.77284 + 2.00931i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 101.8
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 450.3.i.d ✓ 16
3.b odd 2 1 1350.3.i.d 16
5.b even 2 1 450.3.i.f yes 16
5.c odd 4 2 450.3.k.b 32
9.c even 3 1 1350.3.i.d 16
9.d odd 6 1 inner 450.3.i.d ✓ 16
15.d odd 2 1 1350.3.i.f 16
15.e even 4 2 1350.3.k.c 32
45.h odd 6 1 450.3.i.f yes 16
45.j even 6 1 1350.3.i.f 16
45.k odd 12 2 1350.3.k.c 32
45.l even 12 2 450.3.k.b 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
450.3.i.d ✓ 16 1.a even 1 1 trivial
450.3.i.d ✓ 16 9.d odd 6 1 inner
450.3.i.f yes 16 5.b even 2 1
450.3.i.f yes 16 45.h odd 6 1
450.3.k.b 32 5.c odd 4 2
450.3.k.b 32 45.l even 12 2
1350.3.i.d 16 3.b odd 2 1
1350.3.i.d 16 9.c even 3 1
1350.3.i.f 16 15.d odd 2 1
1350.3.i.f 16 45.j even 6 1
1350.3.k.c 32 15.e even 4 2
1350.3.k.c 32 45.k odd 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{16} + 2 T_{7}^{15} + 201 T_{7}^{14} - 794 T_{7}^{13} + 27425 T_{7}^{12} - 125598 T_{7}^{11} + \cdots + 324187890625 \) acting on \(S_{3}^{\mathrm{new}}(450, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 2 T^{2} + 4)^{4} \) Copy content Toggle raw display
$3$ \( T^{16} + 2 T^{15} + \cdots + 43046721 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 324187890625 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 3223917207729 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 4231409447521 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 19\!\cdots\!25 \) Copy content Toggle raw display
$19$ \( (T^{8} + 20 T^{7} + \cdots - 43506935)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 87\!\cdots\!69 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 52\!\cdots\!25 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 45\!\cdots\!25 \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots - 5492582688431)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 21\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 44\!\cdots\!25 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 46\!\cdots\!49 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 65\!\cdots\!25 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 40\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 18\!\cdots\!01 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 12\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 69\!\cdots\!89 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 47602685577625)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 17\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 51\!\cdots\!29 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 76\!\cdots\!25 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 11\!\cdots\!25 \) Copy content Toggle raw display
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