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Properties

Label 4410.2.d.a
Level $4410$
Weight $2$
Character orbit 4410.d
Analytic conductor $35.214$
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] = [4410,2,Mod(4409,4410)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4410, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4410.4409");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4410.d (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: \(35.2140272914\)
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: \( x^{16} - 6 x^{15} + 21 x^{14} - 54 x^{13} + 113 x^{12} - 168 x^{11} + 186 x^{10} - 84 x^{9} + \cdots + 390625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 630)
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.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + \beta_{5} q^{5} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} + \beta_{5} q^{5} - q^{8} - \beta_{5} q^{10} + (\beta_{8} - \beta_{3}) q^{11} + (\beta_{14} - \beta_{10} + \cdots + \beta_{2}) q^{13}+ \cdots + ( - \beta_{14} - 2 \beta_{10} + \cdots - \beta_{2}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{2} + 16 q^{4} - 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{2} + 16 q^{4} - 16 q^{8} + 16 q^{16} - 16 q^{23} + 12 q^{25} - 16 q^{32} + 16 q^{46} - 12 q^{50} + 32 q^{53} + 16 q^{64} - 40 q^{65} - 8 q^{79} + 64 q^{85} - 16 q^{92} - 24 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 6 x^{15} + 21 x^{14} - 54 x^{13} + 113 x^{12} - 168 x^{11} + 186 x^{10} - 84 x^{9} + \cdots + 390625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 189 \nu^{15} - 7678 \nu^{14} + 25998 \nu^{13} - 49670 \nu^{12} + 103248 \nu^{11} + \cdots - 198906250 ) / 113625000 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 1553 \nu^{15} - 5487 \nu^{14} + 37592 \nu^{13} - 86901 \nu^{12} + 157080 \nu^{11} + \cdots - 346640625 ) / 340875000 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 3503 \nu^{15} + 4278 \nu^{14} - 26998 \nu^{13} + 46872 \nu^{12} - 19254 \nu^{11} + \cdots - 491250000 ) / 568125000 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 47 \nu^{15} - 6 \nu^{14} - 104 \nu^{13} - 90 \nu^{12} - 354 \nu^{11} - 918 \nu^{10} + \cdots - 3281250 ) / 3375000 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 28283 \nu^{15} + 58968 \nu^{14} - 195713 \nu^{13} + 336102 \nu^{12} - 463584 \nu^{11} + \cdots + 81562500 ) / 1704375000 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 221 \nu^{15} + 1326 \nu^{14} - 2816 \nu^{13} + 5859 \nu^{12} - 9648 \nu^{11} + 12828 \nu^{10} + \cdots + 31171875 ) / 11250000 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 72631 \nu^{15} + 152446 \nu^{14} - 816586 \nu^{13} + 1097059 \nu^{12} - 2564568 \nu^{11} + \cdots + 4389921875 ) / 3408750000 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 28999 \nu^{15} - 180524 \nu^{14} + 538634 \nu^{13} - 1227551 \nu^{12} + 2136732 \nu^{11} + \cdots - 7956484375 ) / 1136250000 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 22146 \nu^{15} + 79646 \nu^{14} - 238236 \nu^{13} + 546479 \nu^{12} - 982878 \nu^{11} + \cdots + 3061796875 ) / 852187500 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 10646 \nu^{15} - 45366 \nu^{14} + 129881 \nu^{13} - 303924 \nu^{12} + 483708 \nu^{11} + \cdots - 1730156250 ) / 340875000 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 63739 \nu^{15} - 239664 \nu^{14} + 730399 \nu^{13} - 1660236 \nu^{12} + 2476302 \nu^{11} + \cdots - 8881406250 ) / 1704375000 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 134847 \nu^{15} - 489382 \nu^{14} + 1702962 \nu^{13} - 3465913 \nu^{12} + 8035536 \nu^{11} + \cdots - 22617265625 ) / 3408750000 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 82424 \nu^{15} + 333594 \nu^{14} - 1192979 \nu^{13} + 2832846 \nu^{12} - 5451762 \nu^{11} + \cdots + 17355937500 ) / 1704375000 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 113839 \nu^{15} - 393129 \nu^{14} + 1157764 \nu^{13} - 2573376 \nu^{12} + 4791012 \nu^{11} + \cdots - 12730312500 ) / 1704375000 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 298929 \nu^{15} + 1400564 \nu^{14} - 4250574 \nu^{13} + 10377581 \nu^{12} + \cdots + 63741953125 ) / 3408750000 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} - \beta_{10} + \beta_{3} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{15} - \beta_{14} - \beta_{12} - \beta_{10} - \beta_{9} + \beta_{5} - 3 \beta_{4} - \beta_{3} + \cdots - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{14} + 2\beta_{10} + 3\beta_{9} + 2\beta_{6} - 3\beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2 \beta_{15} + 6 \beta_{14} - 4 \beta_{13} - 3 \beta_{12} - 2 \beta_{11} - 3 \beta_{10} + \beta_{9} + \cdots - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 6 \beta_{14} - 3 \beta_{13} + 12 \beta_{12} + 8 \beta_{11} + 4 \beta_{10} + 6 \beta_{9} - 18 \beta_{8} + \cdots - 19 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 14\beta_{15} - 6\beta_{13} + 16\beta_{12} + 6\beta_{11} + 14\beta_{8} + 24\beta_{7} - 38\beta_{3} - 29 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 66 \beta_{15} + 76 \beta_{14} - 104 \beta_{13} - 84 \beta_{12} - 51 \beta_{11} + 23 \beta_{10} + \cdots - 131 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 97 \beta_{15} + 171 \beta_{14} + 82 \beta_{13} - 3 \beta_{12} + 20 \beta_{11} - 63 \beta_{10} + \cdots - 205 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 45\beta_{14} + 132\beta_{10} + 345\beta_{9} - 110\beta_{6} + 190\beta_{5} - 144\beta_{4} + 209\beta_{2} - 77\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 540 \beta_{15} + 766 \beta_{14} + 408 \beta_{13} - 121 \beta_{12} - 444 \beta_{11} - 1211 \beta_{10} + \cdots - 1957 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 1008 \beta_{15} + 564 \beta_{14} + 209 \beta_{13} + 576 \beta_{12} - 2736 \beta_{11} - 2928 \beta_{10} + \cdots + 3377 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 2124 \beta_{15} + 1452 \beta_{13} + 3456 \beta_{12} + 2004 \beta_{11} + 2412 \beta_{8} - 864 \beta_{7} + \cdots + 1801 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 3708 \beta_{15} - 12624 \beta_{14} - 144 \beta_{13} - 288 \beta_{12} - 35 \beta_{11} - 61 \beta_{10} + \cdots - 8963 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 11377 \beta_{15} - 28693 \beta_{14} - 9636 \beta_{13} - 3169 \beta_{12} - 8472 \beta_{11} + \cdots - 9973 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 59 \beta_{14} + 1598 \beta_{10} - 34197 \beta_{9} + 14042 \beta_{6} - 12180 \beta_{5} + \cdots - 25031 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4410\mathbb{Z}\right)^\times\).

\(n\) \(1081\) \(2647\) \(3431\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4409.1
0.948234 − 2.02506i
0.948234 + 2.02506i
1.98669 + 1.02619i
1.98669 − 1.02619i
−0.442358 − 2.19188i
−0.442358 + 2.19188i
2.11423 − 0.728019i
2.11423 + 0.728019i
1.68760 − 1.46697i
1.68760 + 1.46697i
−2.11940 + 0.712845i
−2.11940 − 0.712845i
0.104634 − 2.23362i
0.104634 + 2.23362i
−1.27963 + 1.83372i
−1.27963 − 1.83372i
−1.00000 0 1.00000 −2.22787 − 0.191334i 0 0 −1.00000 0 2.22787 + 0.191334i
4409.2 −1.00000 0 1.00000 −2.22787 + 0.191334i 0 0 −1.00000 0 2.22787 − 0.191334i
4409.3 −1.00000 0 1.00000 −1.88205 − 1.20743i 0 0 −1.00000 0 1.88205 + 1.20743i
4409.4 −1.00000 0 1.00000 −1.88205 + 1.20743i 0 0 −1.00000 0 1.88205 − 1.20743i
4409.5 −1.00000 0 1.00000 −1.67704 − 1.47903i 0 0 −1.00000 0 1.67704 + 1.47903i
4409.6 −1.00000 0 1.00000 −1.67704 + 1.47903i 0 0 −1.00000 0 1.67704 − 1.47903i
4409.7 −1.00000 0 1.00000 −0.426635 − 2.19499i 0 0 −1.00000 0 0.426635 + 2.19499i
4409.8 −1.00000 0 1.00000 −0.426635 + 2.19499i 0 0 −1.00000 0 0.426635 − 2.19499i
4409.9 −1.00000 0 1.00000 0.426635 − 2.19499i 0 0 −1.00000 0 −0.426635 + 2.19499i
4409.10 −1.00000 0 1.00000 0.426635 + 2.19499i 0 0 −1.00000 0 −0.426635 − 2.19499i
4409.11 −1.00000 0 1.00000 1.67704 − 1.47903i 0 0 −1.00000 0 −1.67704 + 1.47903i
4409.12 −1.00000 0 1.00000 1.67704 + 1.47903i 0 0 −1.00000 0 −1.67704 − 1.47903i
4409.13 −1.00000 0 1.00000 1.88205 − 1.20743i 0 0 −1.00000 0 −1.88205 + 1.20743i
4409.14 −1.00000 0 1.00000 1.88205 + 1.20743i 0 0 −1.00000 0 −1.88205 − 1.20743i
4409.15 −1.00000 0 1.00000 2.22787 − 0.191334i 0 0 −1.00000 0 −2.22787 + 0.191334i
4409.16 −1.00000 0 1.00000 2.22787 + 0.191334i 0 0 −1.00000 0 −2.22787 − 0.191334i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4409.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
15.d odd 2 1 inner
105.g even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4410.2.d.a 16
3.b odd 2 1 4410.2.d.b 16
5.b even 2 1 4410.2.d.b 16
7.b odd 2 1 inner 4410.2.d.a 16
7.c even 3 1 630.2.bo.b yes 16
7.d odd 6 1 630.2.bo.b yes 16
15.d odd 2 1 inner 4410.2.d.a 16
21.c even 2 1 4410.2.d.b 16
21.g even 6 1 630.2.bo.a ✓ 16
21.h odd 6 1 630.2.bo.a ✓ 16
35.c odd 2 1 4410.2.d.b 16
35.i odd 6 1 630.2.bo.a ✓ 16
35.j even 6 1 630.2.bo.a ✓ 16
35.k even 12 2 3150.2.bf.f 32
35.l odd 12 2 3150.2.bf.f 32
105.g even 2 1 inner 4410.2.d.a 16
105.o odd 6 1 630.2.bo.b yes 16
105.p even 6 1 630.2.bo.b yes 16
105.w odd 12 2 3150.2.bf.f 32
105.x even 12 2 3150.2.bf.f 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.bo.a ✓ 16 21.g even 6 1
630.2.bo.a ✓ 16 21.h odd 6 1
630.2.bo.a ✓ 16 35.i odd 6 1
630.2.bo.a ✓ 16 35.j even 6 1
630.2.bo.b yes 16 7.c even 3 1
630.2.bo.b yes 16 7.d odd 6 1
630.2.bo.b yes 16 105.o odd 6 1
630.2.bo.b yes 16 105.p even 6 1
3150.2.bf.f 32 35.k even 12 2
3150.2.bf.f 32 35.l odd 12 2
3150.2.bf.f 32 105.w odd 12 2
3150.2.bf.f 32 105.x even 12 2
4410.2.d.a 16 1.a even 1 1 trivial
4410.2.d.a 16 7.b odd 2 1 inner
4410.2.d.a 16 15.d odd 2 1 inner
4410.2.d.a 16 105.g even 2 1 inner
4410.2.d.b 16 3.b odd 2 1
4410.2.d.b 16 5.b even 2 1
4410.2.d.b 16 21.c even 2 1
4410.2.d.b 16 35.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4410, [\chi])\):

\( T_{11}^{8} + 50T_{11}^{6} + 426T_{11}^{4} + 1154T_{11}^{2} + 961 \) Copy content Toggle raw display
\( T_{23}^{4} + 4T_{23}^{3} - 53T_{23}^{2} - 264T_{23} - 294 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} - 6 T^{14} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{8} + 50 T^{6} + \cdots + 961)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} - 72 T^{6} + \cdots + 1156)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + 84 T^{6} + \cdots + 61504)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 110 T^{6} + \cdots + 19044)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 4 T^{3} + \cdots - 294)^{4} \) Copy content Toggle raw display
$29$ \( (T^{8} + 122 T^{6} + \cdots + 33856)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 242 T^{6} + \cdots + 6533136)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 112 T^{6} + \cdots + 1764)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} - 244 T^{6} + \cdots + 12787776)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 148 T^{6} + \cdots + 553536)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 106 T^{6} + \cdots + 10404)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 8 T^{3} + \cdots + 831)^{4} \) Copy content Toggle raw display
$59$ \( (T^{8} - 46 T^{6} + \cdots + 324)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 420 T^{6} + \cdots + 78428736)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 184 T^{6} + \cdots + 20736)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 620 T^{6} + \cdots + 352538176)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} - 272 T^{6} + \cdots + 589824)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 2 T^{3} + \cdots + 9276)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} + 414 T^{6} + \cdots + 364816)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} - 220 T^{6} + \cdots + 553536)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} - 338 T^{6} + \cdots + 27123264)^{2} \) Copy content Toggle raw display
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