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Properties

Label 1470.2.m.d
Level $1470$
Weight $2$
Character orbit 1470.m
Analytic conductor $11.738$
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] = [1470,2,Mod(97,1470)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1470, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1470.97");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1470.m (of order \(4\), 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: \(11.7380090971\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} + 12 x^{14} - 48 x^{13} + 67 x^{12} - 24 x^{11} + 118 x^{10} - 176 x^{9} + 351 x^{8} + \cdots + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 210)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{10} q^{2} - \beta_{10} q^{3} - \beta_{8} q^{4} + (\beta_{12} + \beta_{11} + \beta_{10} + \cdots + 1) q^{5}+ \cdots - \beta_{8} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{10} q^{2} - \beta_{10} q^{3} - \beta_{8} q^{4} + (\beta_{12} + \beta_{11} + \beta_{10} + \cdots + 1) q^{5}+ \cdots + (2 \beta_{11} + \beta_{10} - \beta_{8} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{10} - 8 q^{11} - 16 q^{13} + 4 q^{15} - 16 q^{16} + 24 q^{17} + 16 q^{19} - 8 q^{20} + 4 q^{22} + 8 q^{23} + 16 q^{24} + 16 q^{25} + 8 q^{30} - 4 q^{33} + 16 q^{34} - 16 q^{36} + 16 q^{37} + 8 q^{38} - 24 q^{43} - 8 q^{45} + 8 q^{46} + 24 q^{47} - 16 q^{51} + 16 q^{52} - 16 q^{53} + 16 q^{54} - 56 q^{55} - 8 q^{57} - 36 q^{58} - 16 q^{59} + 8 q^{62} - 32 q^{65} + 48 q^{67} + 24 q^{68} - 8 q^{69} - 32 q^{71} + 56 q^{73} - 16 q^{78} - 16 q^{81} + 24 q^{82} - 16 q^{83} + 8 q^{85} + 16 q^{86} + 36 q^{87} - 4 q^{88} + 32 q^{89} + 8 q^{92} - 8 q^{93} - 16 q^{94} - 24 q^{95} + 44 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} + 12 x^{14} - 48 x^{13} + 67 x^{12} - 24 x^{11} + 118 x^{10} - 176 x^{9} + 351 x^{8} + \cdots + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 1434138866353 \nu^{15} - 74224360591111 \nu^{14} + 217326371958939 \nu^{13} + \cdots - 493738919956596 ) / 51\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 55858235064187 \nu^{15} + 212060352259669 \nu^{14} - 656112655892906 \nu^{13} + \cdots + 51\!\cdots\!49 ) / 25\!\cdots\!95 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5636832106551 \nu^{15} - 21709778642242 \nu^{14} + 64379091580243 \nu^{13} + \cdots - 76799728323202 ) / 28585593772190 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 13\!\cdots\!44 \nu^{15} + \cdots + 17\!\cdots\!62 ) / 51\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 13\!\cdots\!19 \nu^{15} + \cdots - 37\!\cdots\!88 ) / 51\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 12275800546287 \nu^{15} - 43031097335949 \nu^{14} + 125528996510051 \nu^{13} + \cdots - 79799128103174 ) / 16530327389030 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 14132761546 \nu^{15} + 55431822137 \nu^{14} - 164129630598 \nu^{13} + 661673701232 \nu^{12} + \cdots + 253006704102 ) / 18195794890 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 19482487322 \nu^{15} - 71829768307 \nu^{14} + 211530176114 \nu^{13} - 869648666130 \nu^{12} + \cdots - 258618081490 ) / 18265555126 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 13\!\cdots\!97 \nu^{15} + \cdots - 16\!\cdots\!10 ) / 10\!\cdots\!78 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 22618335117691 \nu^{15} - 83577207461092 \nu^{14} + 245635592946633 \nu^{13} + \cdots - 267195717113322 ) / 16530327389030 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 78\!\cdots\!18 \nu^{15} + \cdots + 10\!\cdots\!26 ) / 51\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 816410724118240 \nu^{15} + \cdots + 44\!\cdots\!94 ) / 517399247276639 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 83\!\cdots\!81 \nu^{15} + \cdots + 11\!\cdots\!12 ) / 51\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 11\!\cdots\!77 \nu^{15} + \cdots + 16\!\cdots\!94 ) / 51\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 83\!\cdots\!09 \nu^{15} + \cdots + 98\!\cdots\!58 ) / 25\!\cdots\!95 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - \beta_{15} + \beta_{14} + \beta_{13} + \beta_{12} - \beta_{11} + 2 \beta_{6} - 2 \beta_{4} - \beta_{3} + \cdots + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{15} - \beta_{14} + \beta_{13} + \beta_{12} + \beta_{11} - 3 \beta_{10} + 3 \beta_{9} + \cdots + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{15} - \beta_{14} - 2 \beta_{13} + 3 \beta_{12} + 3 \beta_{11} + 3 \beta_{10} + 3 \beta_{9} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 9 \beta_{15} + 13 \beta_{14} + 5 \beta_{13} + 7 \beta_{12} - 3 \beta_{11} + 13 \beta_{10} - 7 \beta_{9} + \cdots + 23 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 14 \beta_{15} + 16 \beta_{14} + 48 \beta_{13} - 21 \beta_{11} - 28 \beta_{10} - 14 \beta_{9} + \cdots + 25 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 22 \beta_{15} - 111 \beta_{14} - 22 \beta_{13} + 43 \beta_{12} + 99 \beta_{11} - 46 \beta_{10} + \cdots - 91 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 66 \beta_{15} + 66 \beta_{14} - 291 \beta_{13} + 250 \beta_{12} + 285 \beta_{11} + 569 \beta_{10} + \cdots + 228 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 623 \beta_{15} + 1537 \beta_{14} + 987 \beta_{13} + 49 \beta_{12} - 865 \beta_{11} + 758 \beta_{10} + \cdots + 1699 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 318 \beta_{15} - 795 \beta_{14} + 3492 \beta_{13} - 981 \beta_{12} - 981 \beta_{11} - 4267 \beta_{10} + \cdots - 1895 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 4732 \beta_{15} - 12213 \beta_{14} - 10318 \beta_{13} + 4199 \beta_{12} + 13706 \beta_{11} + \cdots - 12701 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 9261 \beta_{15} + 27968 \beta_{14} - 20977 \beta_{13} + 15303 \beta_{12} + 9446 \beta_{11} + \cdots + 36280 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 40556 \beta_{15} + 130306 \beta_{14} + 143366 \beta_{13} - 40556 \beta_{12} - 123066 \beta_{11} + \cdots + 105726 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 155166 \beta_{15} - 341023 \beta_{14} + 134815 \beta_{13} - 99171 \beta_{12} + 87130 \beta_{11} + \cdots - 502797 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 407950 \beta_{15} - 918652 \beta_{14} - 1566704 \beta_{13} + 541254 \beta_{12} + 1452297 \beta_{11} + \cdots - 864228 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 1560582 \beta_{15} + 4825341 \beta_{14} + 529500 \beta_{12} - 1560582 \beta_{11} + 5527039 \beta_{10} + \cdots + 5340011 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1470\mathbb{Z}\right)^\times\).

\(n\) \(491\) \(1081\) \(1177\)
\(\chi(n)\) \(1\) \(-1\) \(-\beta_{8}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1
−0.709944 − 0.925217i
0.339278 − 0.0446668i
0.792206 + 1.03242i
2.69978 − 0.355433i
−0.424637 − 3.22544i
0.277956 − 0.213283i
0.117630 + 0.893490i
−1.09227 + 0.838128i
−0.709944 + 0.925217i
0.339278 + 0.0446668i
0.792206 − 1.03242i
2.69978 + 0.355433i
−0.424637 + 3.22544i
0.277956 + 0.213283i
0.117630 − 0.893490i
−1.09227 − 0.838128i
−0.707107 − 0.707107i 0.707107 + 0.707107i 1.00000i −2.17041 − 0.537883i − 1.00000i 0 0.707107 − 0.707107i 1.00000i 1.15437 + 1.91505i
97.2 −0.707107 − 0.707107i 0.707107 + 0.707107i 1.00000i −1.03078 + 1.98431i − 1.00000i 0 0.707107 − 0.707107i 1.00000i 2.13199 − 0.674247i
97.3 −0.707107 − 0.707107i 0.707107 + 0.707107i 1.00000i 1.91159 − 1.16009i − 1.00000i 0 0.707107 − 0.707107i 1.00000i −2.17201 − 0.531389i
97.4 −0.707107 − 0.707107i 0.707107 + 0.707107i 1.00000i 1.99671 + 1.00656i − 1.00000i 0 0.707107 − 0.707107i 1.00000i −0.700141 − 2.12363i
97.5 0.707107 + 0.707107i −0.707107 − 0.707107i 1.00000i −2.15899 + 0.582041i − 1.00000i 0 −0.707107 + 0.707107i 1.00000i −1.93820 − 1.11507i
97.6 0.707107 + 0.707107i −0.707107 − 0.707107i 1.00000i −1.36519 + 1.77095i − 1.00000i 0 −0.707107 + 0.707107i 1.00000i −2.21758 + 0.286912i
97.7 0.707107 + 0.707107i −0.707107 − 0.707107i 1.00000i 1.19306 + 1.89119i − 1.00000i 0 −0.707107 + 0.707107i 1.00000i −0.493652 + 2.18090i
97.8 0.707107 + 0.707107i −0.707107 − 0.707107i 1.00000i 1.62401 − 1.53707i − 1.00000i 0 −0.707107 + 0.707107i 1.00000i 2.23522 + 0.0614757i
1273.1 −0.707107 + 0.707107i 0.707107 − 0.707107i − 1.00000i −2.17041 + 0.537883i 1.00000i 0 0.707107 + 0.707107i − 1.00000i 1.15437 − 1.91505i
1273.2 −0.707107 + 0.707107i 0.707107 − 0.707107i − 1.00000i −1.03078 − 1.98431i 1.00000i 0 0.707107 + 0.707107i − 1.00000i 2.13199 + 0.674247i
1273.3 −0.707107 + 0.707107i 0.707107 − 0.707107i − 1.00000i 1.91159 + 1.16009i 1.00000i 0 0.707107 + 0.707107i − 1.00000i −2.17201 + 0.531389i
1273.4 −0.707107 + 0.707107i 0.707107 − 0.707107i − 1.00000i 1.99671 − 1.00656i 1.00000i 0 0.707107 + 0.707107i − 1.00000i −0.700141 + 2.12363i
1273.5 0.707107 − 0.707107i −0.707107 + 0.707107i − 1.00000i −2.15899 − 0.582041i 1.00000i 0 −0.707107 − 0.707107i − 1.00000i −1.93820 + 1.11507i
1273.6 0.707107 − 0.707107i −0.707107 + 0.707107i − 1.00000i −1.36519 − 1.77095i 1.00000i 0 −0.707107 − 0.707107i − 1.00000i −2.21758 − 0.286912i
1273.7 0.707107 − 0.707107i −0.707107 + 0.707107i − 1.00000i 1.19306 − 1.89119i 1.00000i 0 −0.707107 − 0.707107i − 1.00000i −0.493652 − 2.18090i
1273.8 0.707107 − 0.707107i −0.707107 + 0.707107i − 1.00000i 1.62401 + 1.53707i 1.00000i 0 −0.707107 − 0.707107i − 1.00000i 2.23522 − 0.0614757i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
35.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1470.2.m.d 16
5.c odd 4 1 1470.2.m.e 16
7.b odd 2 1 1470.2.m.e 16
7.c even 3 1 210.2.u.a ✓ 16
7.d odd 6 1 210.2.u.b yes 16
21.g even 6 1 630.2.bv.b 16
21.h odd 6 1 630.2.bv.a 16
35.f even 4 1 inner 1470.2.m.d 16
35.i odd 6 1 1050.2.bc.g 16
35.j even 6 1 1050.2.bc.h 16
35.k even 12 1 210.2.u.a ✓ 16
35.k even 12 1 1050.2.bc.h 16
35.l odd 12 1 210.2.u.b yes 16
35.l odd 12 1 1050.2.bc.g 16
105.w odd 12 1 630.2.bv.a 16
105.x even 12 1 630.2.bv.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.u.a ✓ 16 7.c even 3 1
210.2.u.a ✓ 16 35.k even 12 1
210.2.u.b yes 16 7.d odd 6 1
210.2.u.b yes 16 35.l odd 12 1
630.2.bv.a 16 21.h odd 6 1
630.2.bv.a 16 105.w odd 12 1
630.2.bv.b 16 21.g even 6 1
630.2.bv.b 16 105.x even 12 1
1050.2.bc.g 16 35.i odd 6 1
1050.2.bc.g 16 35.l odd 12 1
1050.2.bc.h 16 35.j even 6 1
1050.2.bc.h 16 35.k even 12 1
1470.2.m.d 16 1.a even 1 1 trivial
1470.2.m.d 16 35.f even 4 1 inner
1470.2.m.e 16 5.c odd 4 1
1470.2.m.e 16 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1470, [\chi])\):

\( T_{11}^{8} + 4T_{11}^{7} - 30T_{11}^{6} - 156T_{11}^{5} - 6T_{11}^{4} + 868T_{11}^{3} + 1202T_{11}^{2} + 100T_{11} - 383 \) Copy content Toggle raw display
\( T_{13}^{16} + 16 T_{13}^{15} + 128 T_{13}^{14} + 536 T_{13}^{13} + 1442 T_{13}^{12} + 4424 T_{13}^{11} + \cdots + 171295744 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 1)^{4} \) Copy content Toggle raw display
$3$ \( (T^{4} + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{16} - 8 T^{14} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{8} + 4 T^{7} + \cdots - 383)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 171295744 \) Copy content Toggle raw display
$17$ \( T^{16} - 24 T^{15} + \cdots + 16384 \) Copy content Toggle raw display
$19$ \( (T^{8} - 8 T^{7} + \cdots + 76768)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 82925569024 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 939790336 \) Copy content Toggle raw display
$31$ \( T^{16} + 404 T^{14} + \cdots + 85229824 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 1586310022144 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 729780649984 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 396169216 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 405330862336 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 7800599041 \) Copy content Toggle raw display
$59$ \( (T^{8} + 8 T^{7} + \cdots - 41408)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 4676942565376 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 13679819493376 \) Copy content Toggle raw display
$71$ \( (T^{8} + 16 T^{7} + \cdots - 17030912)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 34279432458496 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 31950847504 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 8632854701584 \) Copy content Toggle raw display
$89$ \( (T^{8} - 16 T^{7} + \cdots - 8192)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} - 44 T^{15} + \cdots + 67108864 \) Copy content Toggle raw display
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