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Properties

Label 630.2.i.h
Level $630$
Weight $2$
Character orbit 630.i
Analytic conductor $5.031$
Analytic rank $0$
Dimension $12$
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] = [630,2,Mod(121,630)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(630, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("630.121");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 630.i (of order \(3\), 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: \(5.03057532734\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 5 x^{11} + 14 x^{10} - 28 x^{9} + 36 x^{8} - 24 x^{7} + 33 x^{6} + 42 x^{5} + 114 x^{4} + \cdots + 79 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + (\beta_{6} - \beta_{2}) q^{3} + q^{4} + (\beta_{2} + 1) q^{5} + (\beta_{6} - \beta_{2}) q^{6} + ( - \beta_{10} - \beta_{9} - \beta_{8} + \cdots - 1) q^{7}+ \cdots + ( - \beta_{11} + \beta_{9} + \beta_{8} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + (\beta_{6} - \beta_{2}) q^{3} + q^{4} + (\beta_{2} + 1) q^{5} + (\beta_{6} - \beta_{2}) q^{6} + ( - \beta_{10} - \beta_{9} - \beta_{8} + \cdots - 1) q^{7}+ \cdots + (2 \beta_{11} - 4 \beta_{10} + \cdots - 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{2} + 2 q^{3} + 12 q^{4} + 6 q^{5} + 2 q^{6} + 8 q^{7} + 12 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 12 q^{2} + 2 q^{3} + 12 q^{4} + 6 q^{5} + 2 q^{6} + 8 q^{7} + 12 q^{8} - 4 q^{9} + 6 q^{10} - 7 q^{11} + 2 q^{12} + 2 q^{13} + 8 q^{14} + 7 q^{15} + 12 q^{16} + 7 q^{17} - 4 q^{18} + 14 q^{19} + 6 q^{20} + 17 q^{21} - 7 q^{22} - 9 q^{23} + 2 q^{24} - 6 q^{25} + 2 q^{26} + 11 q^{27} + 8 q^{28} - 9 q^{29} + 7 q^{30} - 18 q^{31} + 12 q^{32} + 3 q^{33} + 7 q^{34} + 4 q^{35} - 4 q^{36} - 12 q^{37} + 14 q^{38} - 14 q^{39} + 6 q^{40} + q^{41} + 17 q^{42} + 7 q^{43} - 7 q^{44} - 5 q^{45} - 9 q^{46} - 14 q^{47} + 2 q^{48} - 24 q^{49} - 6 q^{50} - 3 q^{51} + 2 q^{52} + 2 q^{53} + 11 q^{54} - 14 q^{55} + 8 q^{56} - 14 q^{57} - 9 q^{58} - 58 q^{59} + 7 q^{60} + 22 q^{61} - 18 q^{62} - 13 q^{63} + 12 q^{64} + 4 q^{65} + 3 q^{66} + 44 q^{67} + 7 q^{68} - 18 q^{69} + 4 q^{70} + 10 q^{71} - 4 q^{72} + 6 q^{73} - 12 q^{74} + 5 q^{75} + 14 q^{76} - 23 q^{77} - 14 q^{78} - 2 q^{79} + 6 q^{80} - 4 q^{81} + q^{82} - 26 q^{83} + 17 q^{84} - 7 q^{85} + 7 q^{86} - 12 q^{87} - 7 q^{88} + 2 q^{89} - 5 q^{90} - 4 q^{91} - 9 q^{92} - 26 q^{93} - 14 q^{94} + 28 q^{95} + 2 q^{96} + 6 q^{97} - 24 q^{98} - 27 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 5 x^{11} + 14 x^{10} - 28 x^{9} + 36 x^{8} - 24 x^{7} + 33 x^{6} + 42 x^{5} + 114 x^{4} + \cdots + 79 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{11} - 4 \nu^{10} + 10 \nu^{9} - 18 \nu^{8} + 18 \nu^{7} - 6 \nu^{6} + 27 \nu^{5} + 69 \nu^{4} + \cdots + 407 ) / 243 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 35 \nu^{11} + 82 \nu^{10} - 1108 \nu^{9} + 4209 \nu^{8} - 10470 \nu^{7} + 18003 \nu^{6} + \cdots + 20317 ) / 5589 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 20 \nu^{11} - 124 \nu^{10} + 323 \nu^{9} - 506 \nu^{8} + 628 \nu^{7} - 608 \nu^{6} + 1063 \nu^{5} + \cdots - 2095 ) / 1863 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 21 \nu^{11} + 167 \nu^{10} - 591 \nu^{9} + 1610 \nu^{8} - 3769 \nu^{7} + 6692 \nu^{6} + \cdots + 10796 ) / 1863 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 116 \nu^{11} + 641 \nu^{10} - 1763 \nu^{9} + 3174 \nu^{8} - 3693 \nu^{7} + 2151 \nu^{6} + \cdots - 14230 ) / 5589 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 163 \nu^{11} + 1144 \nu^{10} - 4285 \nu^{9} + 11247 \nu^{8} - 21531 \nu^{7} + 29073 \nu^{6} + \cdots + 7834 ) / 5589 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 67 \nu^{11} + 397 \nu^{10} - 1327 \nu^{9} + 3450 \nu^{8} - 6828 \nu^{7} + 9774 \nu^{6} + \cdots + 8266 ) / 1863 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 83 \nu^{11} + 441 \nu^{10} - 1199 \nu^{9} + 2116 \nu^{8} - 1976 \nu^{7} - 683 \nu^{6} + \cdots - 9792 ) / 1863 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 305 \nu^{11} - 1799 \nu^{10} + 6047 \nu^{9} - 15249 \nu^{8} + 28230 \nu^{7} - 37608 \nu^{6} + \cdots - 16556 ) / 5589 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 50 \nu^{11} - 310 \nu^{10} + 1026 \nu^{9} - 2323 \nu^{8} + 3594 \nu^{7} - 3452 \nu^{6} + 2876 \nu^{5} + \cdots + 1720 ) / 621 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 670 \nu^{11} + 3349 \nu^{10} - 8923 \nu^{9} + 16284 \nu^{8} - 16530 \nu^{7} - 1827 \nu^{6} + \cdots - 92048 ) / 5589 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{11} + \beta_{10} + 2\beta_{8} + 3\beta_{6} + 2\beta_{5} - \beta_{4} + \beta_{3} - 2\beta_{2} + 2\beta _1 + 3 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 2 \beta_{11} + 2 \beta_{10} + 4 \beta_{8} + 3 \beta_{7} + 3 \beta_{6} + 4 \beta_{5} - 5 \beta_{4} + \cdots + 4 \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{10} - \beta_{9} + 2\beta_{8} + \beta_{7} + \beta_{5} - 3\beta_{4} - 2\beta_{3} + 3\beta_{2} + 2\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 11 \beta_{11} + 16 \beta_{10} - 9 \beta_{9} - \beta_{8} - 9 \beta_{6} + 5 \beta_{5} - 16 \beta_{4} + \cdots + 30 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 40 \beta_{11} + 35 \beta_{10} - 24 \beta_{9} - 38 \beta_{8} - 9 \beta_{7} - 27 \beta_{6} + 13 \beta_{5} + \cdots + 75 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 37 \beta_{11} + 19 \beta_{10} - 26 \beta_{9} - 53 \beta_{8} - 16 \beta_{7} - 34 \beta_{6} + 2 \beta_{5} + \cdots + 21 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 305 \beta_{11} + 46 \beta_{10} - 216 \beta_{9} - 502 \beta_{8} - 180 \beta_{7} - 402 \beta_{6} + \cdots - 144 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 820 \beta_{11} - 133 \beta_{10} - 444 \beta_{9} - 1442 \beta_{8} - 534 \beta_{7} - 1272 \beta_{6} + \cdots - 822 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 649 \beta_{11} - 327 \beta_{10} - 210 \beta_{9} - 1281 \beta_{8} - 468 \beta_{7} - 1090 \beta_{6} + \cdots - 963 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 3701 \beta_{11} - 4097 \beta_{10} - 288 \beta_{9} - 9100 \beta_{8} - 3483 \beta_{7} - 7119 \beta_{6} + \cdots - 9297 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 4489 \beta_{11} - 13663 \beta_{10} + 2394 \beta_{9} - 17963 \beta_{8} - 7719 \beta_{7} - 13011 \beta_{6} + \cdots - 27441 ) / 3 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(127\) \(281\) \(451\)
\(\chi(n)\) \(1\) \(\beta_{2}\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1
0.142686 + 1.50500i
−0.659665 − 0.495491i
−0.633121 + 0.576989i
0.429413 − 1.63537i
2.48293 + 0.894932i
0.737753 − 1.71208i
0.142686 − 1.50500i
−0.659665 + 0.495491i
−0.633121 − 0.576989i
0.429413 + 1.63537i
2.48293 − 0.894932i
0.737753 + 1.71208i
1.00000 −1.35158 − 1.08315i 1.00000 0.500000 + 0.866025i −1.35158 − 1.08315i −0.710533 − 2.54856i 1.00000 0.653555 + 2.92795i 0.500000 + 0.866025i
121.2 1.00000 −1.13098 − 1.31183i 1.00000 0.500000 + 0.866025i −1.13098 − 1.31183i 0.0665372 + 2.64491i 1.00000 −0.441782 + 2.96729i 0.500000 + 0.866025i
121.3 1.00000 0.0335666 + 1.73173i 1.00000 0.500000 + 0.866025i 0.0335666 + 1.73173i 2.64400 − 0.0963576i 1.00000 −2.99775 + 0.116256i 0.500000 + 0.866025i
121.4 1.00000 0.400725 − 1.68506i 1.00000 0.500000 + 0.866025i 0.400725 − 1.68506i 0.0665372 + 2.64491i 1.00000 −2.67884 − 1.35049i 0.500000 + 0.866025i
121.5 1.00000 1.31625 − 1.12583i 1.00000 0.500000 + 0.866025i 1.31625 − 1.12583i 2.64400 − 0.0963576i 1.00000 0.465015 − 2.96374i 0.500000 + 0.866025i
121.6 1.00000 1.73202 + 0.0100417i 1.00000 0.500000 + 0.866025i 1.73202 + 0.0100417i −0.710533 − 2.54856i 1.00000 2.99980 + 0.0347849i 0.500000 + 0.866025i
151.1 1.00000 −1.35158 + 1.08315i 1.00000 0.500000 − 0.866025i −1.35158 + 1.08315i −0.710533 + 2.54856i 1.00000 0.653555 − 2.92795i 0.500000 − 0.866025i
151.2 1.00000 −1.13098 + 1.31183i 1.00000 0.500000 − 0.866025i −1.13098 + 1.31183i 0.0665372 − 2.64491i 1.00000 −0.441782 − 2.96729i 0.500000 − 0.866025i
151.3 1.00000 0.0335666 − 1.73173i 1.00000 0.500000 − 0.866025i 0.0335666 − 1.73173i 2.64400 + 0.0963576i 1.00000 −2.99775 − 0.116256i 0.500000 − 0.866025i
151.4 1.00000 0.400725 + 1.68506i 1.00000 0.500000 − 0.866025i 0.400725 + 1.68506i 0.0665372 − 2.64491i 1.00000 −2.67884 + 1.35049i 0.500000 − 0.866025i
151.5 1.00000 1.31625 + 1.12583i 1.00000 0.500000 − 0.866025i 1.31625 + 1.12583i 2.64400 + 0.0963576i 1.00000 0.465015 + 2.96374i 0.500000 − 0.866025i
151.6 1.00000 1.73202 − 0.0100417i 1.00000 0.500000 − 0.866025i 1.73202 − 0.0100417i −0.710533 + 2.54856i 1.00000 2.99980 − 0.0347849i 0.500000 − 0.866025i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 121.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.h even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 630.2.i.h ✓ 12
3.b odd 2 1 1890.2.i.f 12
7.c even 3 1 630.2.l.f yes 12
9.c even 3 1 630.2.l.f yes 12
9.d odd 6 1 1890.2.l.h 12
21.h odd 6 1 1890.2.l.h 12
63.h even 3 1 inner 630.2.i.h ✓ 12
63.j odd 6 1 1890.2.i.f 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.i.h ✓ 12 1.a even 1 1 trivial
630.2.i.h ✓ 12 63.h even 3 1 inner
630.2.l.f yes 12 7.c even 3 1
630.2.l.f yes 12 9.c even 3 1
1890.2.i.f 12 3.b odd 2 1
1890.2.i.f 12 63.j odd 6 1
1890.2.l.h 12 9.d odd 6 1
1890.2.l.h 12 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(630, [\chi])\):

\( T_{11}^{12} + 7 T_{11}^{11} + 61 T_{11}^{10} + 208 T_{11}^{9} + 1306 T_{11}^{8} + 4036 T_{11}^{7} + \cdots + 131769 \) Copy content Toggle raw display
\( T_{13}^{12} - 2 T_{13}^{11} + 78 T_{13}^{10} - 140 T_{13}^{9} + 4390 T_{13}^{8} - 8067 T_{13}^{7} + \cdots + 5349969 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{12} \) Copy content Toggle raw display
$3$ \( T^{12} - 2 T^{11} + \cdots + 729 \) Copy content Toggle raw display
$5$ \( (T^{2} - T + 1)^{6} \) Copy content Toggle raw display
$7$ \( (T^{6} - 4 T^{5} + \cdots + 343)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + 7 T^{11} + \cdots + 131769 \) Copy content Toggle raw display
$13$ \( T^{12} - 2 T^{11} + \cdots + 5349969 \) Copy content Toggle raw display
$17$ \( T^{12} - 7 T^{11} + \cdots + 1046529 \) Copy content Toggle raw display
$19$ \( T^{12} - 14 T^{11} + \cdots + 62869041 \) Copy content Toggle raw display
$23$ \( T^{12} + 9 T^{11} + \cdots + 793881 \) Copy content Toggle raw display
$29$ \( T^{12} + 9 T^{11} + \cdots + 59049 \) Copy content Toggle raw display
$31$ \( (T^{6} + 9 T^{5} + \cdots - 17207)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + 12 T^{11} + \cdots + 185761 \) Copy content Toggle raw display
$41$ \( T^{12} - T^{11} + \cdots + 349281 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 5930694121 \) Copy content Toggle raw display
$47$ \( (T^{6} + 7 T^{5} + \cdots - 1203)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} - 2 T^{11} + \cdots + 41177889 \) Copy content Toggle raw display
$59$ \( (T^{6} + 29 T^{5} + \cdots - 2043)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} - 11 T^{5} + \cdots + 733)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} - 22 T^{5} + \cdots - 1863)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} - 5 T^{5} + \cdots - 43461)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 131079601 \) Copy content Toggle raw display
$79$ \( (T^{6} + T^{5} + \cdots + 99919)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 1938758266449 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 851631819921 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 212955649 \) Copy content Toggle raw display
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