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Properties

Label 1470.2.m.c
Level $1470$
Weight $2$
Character orbit 1470.m
Analytic conductor $11.738$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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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: \( x^{16} - 32 x^{13} + 2 x^{12} + 352 x^{10} - 288 x^{9} + 2 x^{8} - 1440 x^{7} + 8800 x^{6} + \cdots + 390625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - \beta_{6} q^{2} - \beta_{6} q^{3} - \beta_{5} q^{4} + \beta_{3} q^{5} - \beta_{5} q^{6} + \beta_{10} q^{8} - \beta_{5} q^{9} - \beta_1 q^{10} + ( - \beta_{15} - \beta_{11} + \cdots - \beta_{2}) q^{11}+ \cdots + ( - \beta_{10} - \beta_{9} + \cdots - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{5} - 8 q^{13} - 16 q^{16} + 8 q^{17} + 48 q^{19} - 8 q^{22} - 8 q^{23} - 16 q^{24} + 8 q^{25} - 8 q^{33} - 16 q^{36} + 8 q^{37} + 8 q^{38} - 16 q^{47} + 8 q^{52} + 8 q^{53} - 16 q^{54} + 8 q^{57}+ \cdots + 64 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 32 x^{13} + 2 x^{12} + 352 x^{10} - 288 x^{9} + 2 x^{8} - 1440 x^{7} + 8800 x^{6} + \cdots + 390625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 23 \nu^{15} + 151945 \nu^{14} - 513400 \nu^{13} - 30736 \nu^{12} - 3356569 \nu^{11} + \cdots + 17286250000 ) / 263500000 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 12176 \nu^{15} - 711160 \nu^{14} + 2464225 \nu^{13} + 334507 \nu^{12} + 13822768 \nu^{11} + \cdots - 88433203125 ) / 1317500000 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 7153 \nu^{15} + 234160 \nu^{14} - 230600 \nu^{13} + 200771 \nu^{12} - 4866801 \nu^{11} + \cdots + 7283671875 ) / 263500000 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 52781 \nu^{15} - 277595 \nu^{14} + 417475 \nu^{13} + 1210867 \nu^{12} + 5695603 \nu^{11} + \cdots - 6183828125 ) / 1317500000 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 10532 \nu^{15} - 94195 \nu^{14} + 91100 \nu^{13} - 196524 \nu^{12} + 2002804 \nu^{11} + \cdots - 3498281250 ) / 164687500 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5499 \nu^{15} - 36198 \nu^{14} + 69810 \nu^{13} - 110818 \nu^{12} + 799959 \nu^{11} + \cdots - 2325781250 ) / 65875000 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 124784 \nu^{15} + 289765 \nu^{14} + 543850 \nu^{13} + 2971213 \nu^{12} - 6082048 \nu^{11} + \cdots - 16529921875 ) / 1317500000 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 128033 \nu^{15} - 35880 \nu^{14} + 411075 \nu^{13} - 2683056 \nu^{12} + 1413601 \nu^{11} + \cdots - 10391250000 ) / 1317500000 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 221264 \nu^{15} + 115 \nu^{14} + 759725 \nu^{13} + 4513448 \nu^{12} - 596208 \nu^{11} + \cdots - 29524375000 ) / 1317500000 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 72414 \nu^{15} + 116510 \nu^{14} - 145225 \nu^{13} + 1495498 \nu^{12} - 2394398 \nu^{11} + \cdots + 4653906250 ) / 329375000 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 2029 \nu^{15} - 230 \nu^{14} + 12455 \nu^{13} + 40928 \nu^{12} + 1427 \nu^{11} + \cdots - 448000000 ) / 8500000 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 374136 \nu^{15} + 550015 \nu^{14} - 2860075 \nu^{13} - 7558352 \nu^{12} - 10487208 \nu^{11} + \cdots + 107188750000 ) / 1317500000 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 372809 \nu^{15} + 616665 \nu^{14} + 2116200 \nu^{13} + 7167888 \nu^{12} - 13513273 \nu^{11} + \cdots - 73231250000 ) / 1317500000 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 377912 \nu^{15} - 1106320 \nu^{14} + 575 \nu^{13} - 8294559 \nu^{12} + 23323064 \nu^{11} + \cdots - 4410859375 ) / 1317500000 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 95903 \nu^{15} + 201256 \nu^{14} - 10640 \nu^{13} + 2025921 \nu^{12} - 4520623 \nu^{11} + \cdots + 559140625 ) / 263500000 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{11} + 2\beta_{9} + \beta_{8} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{13} - 2\beta_{11} - 2\beta_{8} - \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{15} - 2\beta_{14} - 3\beta_{11} + 3\beta_{8} + 2\beta_{7} - 8\beta_{4} - 6\beta_{3} + 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{14} + \beta_{12} - 8\beta_{11} + 8\beta_{10} + 7\beta_{9} + 8\beta_{8} - 4\beta_{7} - \beta_{5} + 7\beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 2 \beta_{15} - 2 \beta_{13} - 22 \beta_{12} - 41 \beta_{11} - 2 \beta_{9} - 31 \beta_{8} + \cdots - 22 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 25 \beta_{15} - 15 \beta_{14} - \beta_{13} + 2 \beta_{11} - \beta_{10} + 2 \beta_{8} + 20 \beta_{7} + \cdots + 40 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 2 \beta_{15} + 130 \beta_{14} + 2 \beta_{13} + 38 \beta_{12} - 75 \beta_{11} + 160 \beta_{10} + \cdots + 280 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 23 \beta_{15} + \beta_{14} - 15 \beta_{13} + 15 \beta_{12} - 80 \beta_{11} - 72 \beta_{10} + 121 \beta_{9} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 510 \beta_{15} - 386 \beta_{14} + 386 \beta_{13} + 54 \beta_{12} - 279 \beta_{11} - 64 \beta_{10} + \cdots + 824 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 281 \beta_{15} + 399 \beta_{14} + 32 \beta_{13} + 281 \beta_{12} - 686 \beta_{11} + 696 \beta_{10} + \cdots - 696 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 768 \beta_{15} - 768 \beta_{14} - 130 \beta_{13} + 954 \beta_{12} - 2725 \beta_{11} + 3680 \beta_{10} + \cdots - 1088 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 2537 \beta_{15} - 2336 \beta_{14} - 1905 \beta_{13} - 2336 \beta_{12} - 456 \beta_{11} - 544 \beta_{10} + \cdots + 4239 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 12288 \beta_{15} + 5378 \beta_{14} + 7424 \beta_{12} - 1727 \beta_{11} + 10816 \beta_{10} + \cdots - 19992 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 6432 \beta_{15} + 14112 \beta_{14} + 736 \beta_{13} + 5879 \beta_{12} - 242 \beta_{11} + 19849 \beta_{10} + \cdots + 32672 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 51456 \beta_{15} - 52480 \beta_{14} - 51456 \beta_{13} + 31744 \beta_{12} + 76979 \beta_{11} + \cdots + 88640 ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(491\) \(1081\) \(1177\)
\(\chi(n)\) \(1\) \(-1\) \(-\beta_{5}\)

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
2.23272 + 0.122339i
−1.22045 − 1.87363i
−0.410538 + 2.19806i
−2.01595 + 0.967451i
−1.07534 + 1.96052i
−1.12594 − 1.93191i
2.21573 + 0.300921i
1.39977 − 1.74375i
2.23272 − 0.122339i
−1.22045 + 1.87363i
−0.410538 − 2.19806i
−2.01595 − 0.967451i
−1.07534 − 1.96052i
−1.12594 + 1.93191i
2.21573 − 0.300921i
1.39977 + 1.74375i
−0.707107 − 0.707107i −0.707107 − 0.707107i 1.00000i −2.10958 + 0.741398i 1.00000i 0 0.707107 − 0.707107i 1.00000i 2.01595 + 0.967451i
97.2 −0.707107 − 0.707107i −0.707107 − 0.707107i 1.00000i −1.84456 − 1.26397i 1.00000i 0 0.707107 − 0.707107i 1.00000i 0.410538 + 2.19806i
97.3 −0.707107 − 0.707107i −0.707107 − 0.707107i 1.00000i 0.461873 + 2.18785i 1.00000i 0 0.707107 − 0.707107i 1.00000i 1.22045 − 1.87363i
97.4 −0.707107 − 0.707107i −0.707107 − 0.707107i 1.00000i 1.49226 − 1.66528i 1.00000i 0 0.707107 − 0.707107i 1.00000i −2.23272 + 0.122339i
97.5 0.707107 + 0.707107i 0.707107 + 0.707107i 1.00000i −2.22280 − 0.243230i 1.00000i 0 −0.707107 + 0.707107i 1.00000i −1.39977 − 1.74375i
97.6 0.707107 + 0.707107i 0.707107 + 0.707107i 1.00000i −1.35397 + 1.77954i 1.00000i 0 −0.707107 + 0.707107i 1.00000i −2.21573 + 0.300921i
97.7 0.707107 + 0.707107i 0.707107 + 0.707107i 1.00000i −0.569907 − 2.16222i 1.00000i 0 −0.707107 + 0.707107i 1.00000i 1.12594 − 1.93191i
97.8 0.707107 + 0.707107i 0.707107 + 0.707107i 1.00000i 2.14668 + 0.625913i 1.00000i 0 −0.707107 + 0.707107i 1.00000i 1.07534 + 1.96052i
1273.1 −0.707107 + 0.707107i −0.707107 + 0.707107i − 1.00000i −2.10958 − 0.741398i − 1.00000i 0 0.707107 + 0.707107i − 1.00000i 2.01595 − 0.967451i
1273.2 −0.707107 + 0.707107i −0.707107 + 0.707107i − 1.00000i −1.84456 + 1.26397i − 1.00000i 0 0.707107 + 0.707107i − 1.00000i 0.410538 − 2.19806i
1273.3 −0.707107 + 0.707107i −0.707107 + 0.707107i − 1.00000i 0.461873 − 2.18785i − 1.00000i 0 0.707107 + 0.707107i − 1.00000i 1.22045 + 1.87363i
1273.4 −0.707107 + 0.707107i −0.707107 + 0.707107i − 1.00000i 1.49226 + 1.66528i − 1.00000i 0 0.707107 + 0.707107i − 1.00000i −2.23272 − 0.122339i
1273.5 0.707107 − 0.707107i 0.707107 − 0.707107i − 1.00000i −2.22280 + 0.243230i − 1.00000i 0 −0.707107 − 0.707107i − 1.00000i −1.39977 + 1.74375i
1273.6 0.707107 − 0.707107i 0.707107 − 0.707107i − 1.00000i −1.35397 − 1.77954i − 1.00000i 0 −0.707107 − 0.707107i − 1.00000i −2.21573 − 0.300921i
1273.7 0.707107 − 0.707107i 0.707107 − 0.707107i − 1.00000i −0.569907 + 2.16222i − 1.00000i 0 −0.707107 − 0.707107i − 1.00000i 1.12594 + 1.93191i
1273.8 0.707107 − 0.707107i 0.707107 − 0.707107i − 1.00000i 2.14668 − 0.625913i − 1.00000i 0 −0.707107 − 0.707107i − 1.00000i 1.07534 − 1.96052i
\(n\): 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.c ✓ 16
5.c odd 4 1 1470.2.m.f yes 16
7.b odd 2 1 1470.2.m.f yes 16
35.f even 4 1 inner 1470.2.m.c ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1470.2.m.c ✓ 16 1.a even 1 1 trivial
1470.2.m.c ✓ 16 35.f even 4 1 inner
1470.2.m.f yes 16 5.c odd 4 1
1470.2.m.f yes 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 \(S_{2}^{\mathrm{new}}(1470, [\chi])\):

\( T_{11}^{8} - 40T_{11}^{6} + 8T_{11}^{5} + 382T_{11}^{4} - 160T_{11}^{3} - 1088T_{11}^{2} + 512T_{11} + 512 \) Copy content Toggle raw display
\( T_{13}^{16} + 8 T_{13}^{15} + 32 T_{13}^{14} + 16 T_{13}^{13} + 740 T_{13}^{12} + 5632 T_{13}^{11} + \cdots + 18496 \) 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
$5$ \( T^{16} + 8 T^{15} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{8} - 40 T^{6} + \cdots + 512)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + 8 T^{15} + \cdots + 18496 \) Copy content Toggle raw display
$17$ \( T^{16} - 8 T^{15} + \cdots + 1336336 \) Copy content Toggle raw display
$19$ \( (T^{8} - 24 T^{7} + \cdots + 31744)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + 8 T^{15} + \cdots + 4194304 \) Copy content Toggle raw display
$29$ \( T^{16} + 112 T^{14} + \cdots + 64 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 303038464 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 1224191770624 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 14484603904 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 3347316736 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 1212153856 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 1176027464704 \) Copy content Toggle raw display
$59$ \( (T^{8} + 24 T^{7} + \cdots + 15023104)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 98867482624 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 4228120576 \) Copy content Toggle raw display
$71$ \( (T^{8} - 228 T^{6} + \cdots + 591872)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 36991502500096 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 148243480576 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 1401249857536 \) Copy content Toggle raw display
$89$ \( (T^{8} - 32 T^{7} + \cdots + 15376)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 244234884096256 \) Copy content Toggle raw display
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