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Properties

Label 630.4.b.a
Level $630$
Weight $4$
Character orbit 630.b
Analytic conductor $37.171$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [630,4,Mod(251,630)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(630, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("630.251");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 630.b (of order \(2\), degree \(1\), 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: \(37.1712033036\)
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: \( x^{16} + 1258 x^{14} + 600187 x^{12} + 137418288 x^{10} + 16134075531 x^{8} + 994766759010 x^{6} + \cdots + 15347730793924 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{15}\cdot 3^{8} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - \beta_{7} q^{2} - 4 q^{4} - 5 q^{5} + ( - \beta_{11} - 2) q^{7} + 4 \beta_{7} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{7} q^{2} - 4 q^{4} - 5 q^{5} + ( - \beta_{11} - 2) q^{7} + 4 \beta_{7} q^{8} + 5 \beta_{7} q^{10} - \beta_{6} q^{11} + ( - \beta_{9} + \beta_{7}) q^{13} + (\beta_{14} + 2 \beta_{7} + \beta_{2}) q^{14} + 16 q^{16} + (\beta_{5} - 2 \beta_{2}) q^{17} + (\beta_{15} + \beta_{14} + \cdots - 5 \beta_{7}) q^{19}+ \cdots + ( - 2 \beta_{14} + 2 \beta_{13} + \cdots - 6 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 64 q^{4} - 80 q^{5} - 28 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 64 q^{4} - 80 q^{5} - 28 q^{7} + 256 q^{16} + 320 q^{20} + 400 q^{25} + 48 q^{26} + 112 q^{28} + 140 q^{35} - 56 q^{37} - 336 q^{38} - 1032 q^{41} - 368 q^{43} + 240 q^{46} + 1560 q^{47} + 700 q^{49} - 624 q^{58} - 1200 q^{59} - 1200 q^{62} - 1024 q^{64} + 1664 q^{67} - 684 q^{77} + 1400 q^{79} - 1280 q^{80} - 2688 q^{83} + 2472 q^{89} + 1716 q^{91} + 72 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 1258 x^{14} + 600187 x^{12} + 137418288 x^{10} + 16134075531 x^{8} + 994766759010 x^{6} + \cdots + 15347730793924 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 330142448157 \nu^{14} + 437050528294923 \nu^{12} + \cdots - 73\!\cdots\!52 ) / 29\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 71\!\cdots\!17 \nu^{14} + \cdots + 32\!\cdots\!88 ) / 15\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 10\!\cdots\!07 \nu^{14} + \cdots + 45\!\cdots\!56 ) / 15\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 54\!\cdots\!69 \nu^{14} + \cdots + 12\!\cdots\!64 ) / 76\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 74\!\cdots\!33 \nu^{14} + \cdots + 67\!\cdots\!60 ) / 76\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 16\!\cdots\!05 \nu^{15} + \cdots + 34\!\cdots\!64 \nu ) / 29\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 28\!\cdots\!71 \nu^{15} + \cdots - 24\!\cdots\!96 \nu ) / 25\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 58048215514389 \nu^{15} + \cdots - 16\!\cdots\!80 \nu ) / 26\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 39\!\cdots\!97 \nu^{15} + \cdots - 40\!\cdots\!76 \nu ) / 14\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 19\!\cdots\!07 \nu^{15} + \cdots + 98\!\cdots\!44 ) / 59\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 19\!\cdots\!07 \nu^{15} + \cdots - 98\!\cdots\!44 ) / 59\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 82\!\cdots\!59 \nu^{15} + \cdots + 54\!\cdots\!24 ) / 14\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 82\!\cdots\!59 \nu^{15} + \cdots + 54\!\cdots\!24 ) / 14\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 18\!\cdots\!39 \nu^{15} + \cdots + 48\!\cdots\!96 \nu ) / 29\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 38\!\cdots\!32 \nu^{15} + \cdots + 23\!\cdots\!04 \nu ) / 14\!\cdots\!96 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - 3 \beta_{14} + 3 \beta_{13} - 3 \beta_{12} - 9 \beta_{11} - 9 \beta_{10} - 6 \beta_{9} + \cdots + 6 \beta_{6} ) / 36 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 15\beta_{13} + 15\beta_{12} + 27\beta_{11} - 27\beta_{10} + 120\beta_{3} - 27\beta_{2} - 97\beta _1 - 5580 ) / 36 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 438 \beta_{15} + 1425 \beta_{14} - 1425 \beta_{13} + 1425 \beta_{12} + 1809 \beta_{11} + \cdots - 1968 \beta_{6} ) / 36 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 1763 \beta_{13} - 1763 \beta_{12} - 7863 \beta_{11} + 7863 \beta_{10} - 666 \beta_{5} + \cdots + 560634 ) / 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 307770 \beta_{15} - 642075 \beta_{14} + 565935 \beta_{13} - 565935 \beta_{12} - 521595 \beta_{11} + \cdots + 650928 \beta_{6} ) / 36 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2232285 \beta_{13} + 2232285 \beta_{12} + 11408049 \beta_{11} - 11408049 \beta_{10} + 2340396 \beta_{5} + \cdots - 605269296 ) / 36 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 160373994 \beta_{15} + 272615289 \beta_{14} - 222564129 \beta_{13} + 222564129 \beta_{12} + \cdots - 229116756 \beta_{6} ) / 36 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 323167251 \beta_{13} - 323167251 \beta_{12} - 1688209119 \beta_{11} + 1688209119 \beta_{10} + \cdots + 76687579902 ) / 12 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 75460474026 \beta_{15} - 114433547547 \beta_{14} + 87559765455 \beta_{13} + \cdots + 84521742720 \beta_{6} ) / 36 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 421895186169 \beta_{13} + 421895186169 \beta_{12} + 2199183610581 \beta_{11} - 2199183610581 \beta_{10} + \cdots - 89545899221820 ) / 36 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 33903077451954 \beta_{15} + 48014466216105 \beta_{14} - 34493526638913 \beta_{13} + \cdots - 32177253371100 \beta_{6} ) / 36 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 61007904974567 \beta_{13} - 61007904974567 \beta_{12} - 315423027235035 \beta_{11} + \cdots + 11\!\cdots\!02 ) / 12 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 14\!\cdots\!22 \beta_{15} + \cdots + 12\!\cdots\!68 \beta_{6} ) / 36 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 79\!\cdots\!45 \beta_{13} + \cdots - 14\!\cdots\!00 ) / 36 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 64\!\cdots\!50 \beta_{15} + \cdots - 49\!\cdots\!36 \beta_{6} ) / 36 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/630\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(281\) \(451\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1
8.90636i
8.25157i
− 19.2871i
− 15.3027i
− 6.98019i
6.18683i
− 0.204715i
20.4299i
− 8.90636i
− 8.25157i
19.2871i
15.3027i
6.98019i
− 6.18683i
0.204715i
− 20.4299i
− 2.00000i 0 −4.00000 −5.00000 0 −18.1838 + 3.51429i 8.00000i 0 10.0000i
251.2 − 2.00000i 0 −4.00000 −5.00000 0 −17.5282 + 5.98012i 8.00000i 0 10.0000i
251.3 − 2.00000i 0 −4.00000 −5.00000 0 −11.2482 + 14.7132i 8.00000i 0 10.0000i
251.4 − 2.00000i 0 −4.00000 −5.00000 0 −10.2542 − 15.4224i 8.00000i 0 10.0000i
251.5 − 2.00000i 0 −4.00000 −5.00000 0 −1.73782 − 18.4385i 8.00000i 0 10.0000i
251.6 − 2.00000i 0 −4.00000 −5.00000 0 14.5642 − 11.4405i 8.00000i 0 10.0000i
251.7 − 2.00000i 0 −4.00000 −5.00000 0 14.6520 + 11.3278i 8.00000i 0 10.0000i
251.8 − 2.00000i 0 −4.00000 −5.00000 0 15.7361 + 9.76607i 8.00000i 0 10.0000i
251.9 2.00000i 0 −4.00000 −5.00000 0 −18.1838 − 3.51429i − 8.00000i 0 − 10.0000i
251.10 2.00000i 0 −4.00000 −5.00000 0 −17.5282 − 5.98012i − 8.00000i 0 − 10.0000i
251.11 2.00000i 0 −4.00000 −5.00000 0 −11.2482 − 14.7132i − 8.00000i 0 − 10.0000i
251.12 2.00000i 0 −4.00000 −5.00000 0 −10.2542 + 15.4224i − 8.00000i 0 − 10.0000i
251.13 2.00000i 0 −4.00000 −5.00000 0 −1.73782 + 18.4385i − 8.00000i 0 − 10.0000i
251.14 2.00000i 0 −4.00000 −5.00000 0 14.5642 + 11.4405i − 8.00000i 0 − 10.0000i
251.15 2.00000i 0 −4.00000 −5.00000 0 14.6520 − 11.3278i − 8.00000i 0 − 10.0000i
251.16 2.00000i 0 −4.00000 −5.00000 0 15.7361 − 9.76607i − 8.00000i 0 − 10.0000i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 251.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 630.4.b.a ✓ 16
3.b odd 2 1 630.4.b.b yes 16
7.b odd 2 1 630.4.b.b yes 16
21.c even 2 1 inner 630.4.b.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.4.b.a ✓ 16 1.a even 1 1 trivial
630.4.b.a ✓ 16 21.c even 2 1 inner
630.4.b.b yes 16 3.b odd 2 1
630.4.b.b yes 16 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{17}^{8} - 12570 T_{17}^{6} - 288840 T_{17}^{5} + 34659768 T_{17}^{4} + 1015861440 T_{17}^{3} + \cdots + 1897704777600 \) acting on \(S_{4}^{\mathrm{new}}(630, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 4)^{8} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T + 5)^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 19\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 50\!\cdots\!64 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 1897704777600)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 52\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 93\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 21\!\cdots\!64 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 44\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 45\!\cdots\!96)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots - 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots - 13\!\cdots\!44)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots - 83\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 53\!\cdots\!84 \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 25\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 44\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots - 56\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 29\!\cdots\!44 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots - 19\!\cdots\!52)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots - 40\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 38\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 59\!\cdots\!00 \) Copy content Toggle raw display
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