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Properties

Label 450.3.i.e
Level $450$
Weight $3$
Character orbit 450.i
Analytic conductor $12.262$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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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: \( x^{16} - 2x^{14} - 230x^{12} + 96x^{10} + 25551x^{8} + 7776x^{6} - 1509030x^{4} - 1062882x^{2} + 43046721 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 90)
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.

\(f(q)\) \(=\) \( q + \beta_{9} q^{2} - \beta_{13} q^{3} + ( - 2 \beta_{5} + 2) q^{4} + ( - \beta_{3} - 1) q^{6} + ( - \beta_{15} + \beta_{13} + \cdots + \beta_1) q^{7}+ \cdots + (\beta_{10} + \beta_{5} - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{9} q^{2} - \beta_{13} q^{3} + ( - 2 \beta_{5} + 2) q^{4} + ( - \beta_{3} - 1) q^{6} + ( - \beta_{15} + \beta_{13} + \cdots + \beta_1) q^{7}+ \cdots + ( - 15 \beta_{11} + 9 \beta_{10} + \cdots - 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 16 q^{4} - 8 q^{6} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 16 q^{4} - 8 q^{6} + 16 q^{9} + 60 q^{11} - 12 q^{14} - 32 q^{16} - 144 q^{19} - 56 q^{21} - 8 q^{24} - 72 q^{29} + 28 q^{31} + 136 q^{34} + 40 q^{36} + 276 q^{39} - 180 q^{41} - 56 q^{46} - 12 q^{49} - 8 q^{51} - 260 q^{54} - 24 q^{56} + 228 q^{59} + 68 q^{61} - 128 q^{64} + 440 q^{66} + 16 q^{69} - 72 q^{74} - 144 q^{76} - 420 q^{79} - 500 q^{81} - 176 q^{84} - 48 q^{86} - 168 q^{91} + 164 q^{94} + 16 q^{96} - 268 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2x^{14} - 230x^{12} + 96x^{10} + 25551x^{8} + 7776x^{6} - 1509030x^{4} - 1062882x^{2} + 43046721 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 49 \nu^{14} - 5734 \nu^{12} + 1064 \nu^{10} + 908004 \nu^{8} - 619956 \nu^{6} + \cdots + 1545430428 ) / 271034910 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5 \nu^{14} - 55 \nu^{12} - 2032 \nu^{10} - 15657 \nu^{8} + 213588 \nu^{6} + 1070253 \nu^{4} + \cdots - 65426292 ) / 20076660 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 256 \nu^{15} - 703 \nu^{13} + 72245 \nu^{11} + 469200 \nu^{9} - 2736405 \nu^{7} + \cdots + 1856323413 \nu ) / 4878628380 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 212 \nu^{14} + 1034 \nu^{12} - 27619 \nu^{10} - 245004 \nu^{8} + 1736091 \nu^{6} + \cdots - 594151038 ) / 271034910 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 250 \nu^{14} - 769 \nu^{12} + 54935 \nu^{10} + 435540 \nu^{8} - 6064155 \nu^{6} + \cdots + 2390953059 ) / 180689940 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 475 \nu^{14} - 788 \nu^{12} - 98639 \nu^{10} + 222666 \nu^{8} + 7039071 \nu^{6} + \cdots + 1522047024 ) / 271034910 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 608 \nu^{15} + 15956 \nu^{13} - 56086 \nu^{11} - 2178771 \nu^{9} - 4310316 \nu^{7} + \cdots - 3660034167 \nu ) / 2439314190 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1225 \nu^{15} - 16666 \nu^{13} + 256559 \nu^{11} + 3638289 \nu^{9} - 19138311 \nu^{7} + \cdots + 12751926795 \nu ) / 4878628380 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 133 \nu^{14} + 1597 \nu^{12} - 26297 \nu^{10} - 280857 \nu^{8} + 1745883 \nu^{6} + \cdots - 776258154 ) / 30114990 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 2548 \nu^{14} - 4867 \nu^{12} + 472559 \nu^{10} + 1841304 \nu^{8} - 39214971 \nu^{6} + \cdots + 7589508921 ) / 542069820 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 1172 \nu^{15} - 5395 \nu^{13} - 277309 \nu^{11} - 260304 \nu^{9} + 23882841 \nu^{7} + \cdots - 2444097159 \nu ) / 1626209460 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 212 \nu^{15} - 1034 \nu^{13} + 27619 \nu^{11} + 245004 \nu^{9} - 1736091 \nu^{7} + \cdots + 594151038 \nu ) / 271034910 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 2138 \nu^{15} - 12896 \nu^{13} + 407986 \nu^{11} + 2031891 \nu^{9} - 34782714 \nu^{7} + \cdots + 5286243627 \nu ) / 2439314190 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 2365 \nu^{15} - 5503 \nu^{13} + 387998 \nu^{11} + 1278723 \nu^{9} - 31719042 \nu^{7} + \cdots + 7778170476 \nu ) / 1626209460 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{11} - \beta_{7} + \beta_{6} - 2\beta_{5} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{15} + 2\beta_{14} + 2\beta_{13} + 2\beta_{12} - \beta_{9} + 3\beta_{8} + 11\beta_{4} + 2\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{11} + 2\beta_{10} - 2\beta_{7} - 7\beta_{6} + 3\beta_{5} - 8\beta_{3} + 7\beta_{2} + 54 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -19\beta_{15} + 17\beta_{14} + 17\beta_{13} - 7\beta_{12} - 19\beta_{9} - 16\beta_{4} + 62\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -123\beta_{11} - 31\beta_{10} - 83\beta_{7} + 47\beta_{6} - 291\beta_{5} - 62\beta_{3} - 68\beta_{2} + 315 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 49 \beta_{15} + 83 \beta_{14} + 110 \beta_{13} + 116 \beta_{12} - 301 \beta_{9} + 42 \beta_{8} + \cdots + 242 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -183\beta_{11} + 122\beta_{10} - 230\beta_{7} - 265\beta_{6} - 1209\beta_{5} - 1364\beta_{3} + 427\beta_{2} + 975 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 2401 \beta_{15} + 995 \beta_{14} + 3101 \beta_{13} - 571 \beta_{12} + 551 \beta_{9} + \cdots + 2393 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 5256 \beta_{11} - 4243 \beta_{10} - 4094 \beta_{7} - 1396 \beta_{6} + 339 \beta_{5} - 8606 \beta_{3} + \cdots + 14793 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1090 \beta_{15} + 6983 \beta_{14} - 15400 \beta_{13} - 4480 \beta_{12} - 21070 \beta_{9} + \cdots + 7280 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 59370 \beta_{11} + 6440 \beta_{10} - 24860 \beta_{7} + 35030 \beta_{6} - 242760 \beta_{5} + \cdots - 71649 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 115510 \beta_{15} - 77290 \beta_{14} + 281000 \beta_{13} - 47170 \beta_{12} + 75290 \beta_{9} + \cdots + 37691 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 53931 \beta_{11} - 375340 \beta_{10} + 12289 \beta_{7} - 217309 \beta_{6} + 2862438 \beta_{5} + \cdots - 1044888 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 360511 \beta_{15} + 526802 \beta_{14} - 3202438 \beta_{13} - 961348 \beta_{12} + 1581149 \beta_{9} + \cdots - 1064458 \beta_1 \) Copy content Toggle raw display

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

Character values

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

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

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
0.416995 − 2.97088i
−2.99498 + 0.173499i
2.93235 + 0.633522i
0.870383 + 2.87096i
−0.870383 − 2.87096i
−2.93235 − 0.633522i
2.99498 − 0.173499i
−0.416995 + 2.97088i
0.416995 + 2.97088i
−2.99498 − 0.173499i
2.93235 − 0.633522i
0.870383 − 2.87096i
−0.870383 + 2.87096i
−2.93235 + 0.633522i
2.99498 + 0.173499i
−0.416995 − 2.97088i
−1.22474 − 0.707107i −2.36436 − 1.84657i 1.00000 + 1.73205i 0 1.59002 + 3.93343i 4.79375 − 8.30302i − 2.82843i 2.18038 + 8.73189i 0
101.2 −1.22474 − 0.707107i −1.34723 + 2.68048i 1.00000 + 1.73205i 0 3.54540 − 2.33026i −1.58988 + 2.75375i − 2.82843i −5.36992 − 7.22246i 0
101.3 −1.22474 − 0.707107i 2.01482 − 2.22272i 1.00000 + 1.73205i 0 −4.03934 + 1.29758i −4.38322 + 7.59195i − 2.82843i −0.881011 − 8.95678i 0
101.4 −1.22474 − 0.707107i 2.92152 + 0.681708i 1.00000 + 1.73205i 0 −3.09608 − 2.90074i 2.40409 − 4.16400i − 2.82843i 8.07055 + 3.98325i 0
101.5 1.22474 + 0.707107i −2.92152 − 0.681708i 1.00000 + 1.73205i 0 −3.09608 − 2.90074i −2.40409 + 4.16400i 2.82843i 8.07055 + 3.98325i 0
101.6 1.22474 + 0.707107i −2.01482 + 2.22272i 1.00000 + 1.73205i 0 −4.03934 + 1.29758i 4.38322 − 7.59195i 2.82843i −0.881011 − 8.95678i 0
101.7 1.22474 + 0.707107i 1.34723 − 2.68048i 1.00000 + 1.73205i 0 3.54540 − 2.33026i 1.58988 − 2.75375i 2.82843i −5.36992 − 7.22246i 0
101.8 1.22474 + 0.707107i 2.36436 + 1.84657i 1.00000 + 1.73205i 0 1.59002 + 3.93343i −4.79375 + 8.30302i 2.82843i 2.18038 + 8.73189i 0
401.1 −1.22474 + 0.707107i −2.36436 + 1.84657i 1.00000 − 1.73205i 0 1.59002 − 3.93343i 4.79375 + 8.30302i 2.82843i 2.18038 − 8.73189i 0
401.2 −1.22474 + 0.707107i −1.34723 − 2.68048i 1.00000 − 1.73205i 0 3.54540 + 2.33026i −1.58988 − 2.75375i 2.82843i −5.36992 + 7.22246i 0
401.3 −1.22474 + 0.707107i 2.01482 + 2.22272i 1.00000 − 1.73205i 0 −4.03934 − 1.29758i −4.38322 − 7.59195i 2.82843i −0.881011 + 8.95678i 0
401.4 −1.22474 + 0.707107i 2.92152 − 0.681708i 1.00000 − 1.73205i 0 −3.09608 + 2.90074i 2.40409 + 4.16400i 2.82843i 8.07055 − 3.98325i 0
401.5 1.22474 − 0.707107i −2.92152 + 0.681708i 1.00000 − 1.73205i 0 −3.09608 + 2.90074i −2.40409 − 4.16400i − 2.82843i 8.07055 − 3.98325i 0
401.6 1.22474 − 0.707107i −2.01482 − 2.22272i 1.00000 − 1.73205i 0 −4.03934 − 1.29758i 4.38322 + 7.59195i − 2.82843i −0.881011 + 8.95678i 0
401.7 1.22474 − 0.707107i 1.34723 + 2.68048i 1.00000 − 1.73205i 0 3.54540 + 2.33026i 1.58988 + 2.75375i − 2.82843i −5.36992 + 7.22246i 0
401.8 1.22474 − 0.707107i 2.36436 − 1.84657i 1.00000 − 1.73205i 0 1.59002 − 3.93343i −4.79375 − 8.30302i − 2.82843i 2.18038 − 8.73189i 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
5.b even 2 1 inner
9.d odd 6 1 inner
45.h 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.e 16
3.b odd 2 1 1350.3.i.e 16
5.b even 2 1 inner 450.3.i.e 16
5.c odd 4 2 90.3.j.b ✓ 16
9.c even 3 1 1350.3.i.e 16
9.d odd 6 1 inner 450.3.i.e 16
15.d odd 2 1 1350.3.i.e 16
15.e even 4 2 270.3.j.b 16
45.h odd 6 1 inner 450.3.i.e 16
45.j even 6 1 1350.3.i.e 16
45.k odd 12 2 270.3.j.b 16
45.k odd 12 2 810.3.b.b 16
45.l even 12 2 90.3.j.b ✓ 16
45.l even 12 2 810.3.b.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.3.j.b ✓ 16 5.c odd 4 2
90.3.j.b ✓ 16 45.l even 12 2
270.3.j.b 16 15.e even 4 2
270.3.j.b 16 45.k odd 12 2
450.3.i.e 16 1.a even 1 1 trivial
450.3.i.e 16 5.b even 2 1 inner
450.3.i.e 16 9.d odd 6 1 inner
450.3.i.e 16 45.h odd 6 1 inner
810.3.b.b 16 45.k odd 12 2
810.3.b.b 16 45.l even 12 2
1350.3.i.e 16 3.b odd 2 1
1350.3.i.e 16 9.c even 3 1
1350.3.i.e 16 15.d odd 2 1
1350.3.i.e 16 45.j even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{16} + 202 T_{7}^{14} + 27898 T_{7}^{12} + 2058640 T_{7}^{10} + 109528039 T_{7}^{8} + \cdots + 2726544000625 \) 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
$3$ \( T^{16} - 8 T^{14} + \cdots + 43046721 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 2726544000625 \) Copy content Toggle raw display
$11$ \( (T^{8} - 30 T^{7} + \cdots + 110923024)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 1991891886336 \) Copy content Toggle raw display
$17$ \( (T^{8} + 1520 T^{6} + \cdots + 640000)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 36 T^{3} + \cdots - 118800)^{4} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 26\!\cdots\!61 \) Copy content Toggle raw display
$29$ \( (T^{8} + 36 T^{7} + \cdots + 334196141409)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} - 14 T^{7} + \cdots + 27544049296)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 3016 T^{6} + \cdots + 1414963456)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 90 T^{7} + \cdots + 15399072649)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 38\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 30\!\cdots\!01 \) Copy content Toggle raw display
$53$ \( (T^{8} + 3264 T^{6} + \cdots + 81293414400)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 28666857972736)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 34 T^{7} + \cdots + 116562836569)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 31\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( (T^{8} + 11328 T^{6} + \cdots + 39661519104)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 12985356390400)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 68\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 45\!\cdots\!21 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 74\!\cdots\!00 \) Copy content Toggle raw display
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