↑

Properties

Label 7200.2.d.u
Level $7200$
Weight $2$
Character orbit 7200.d
Analytic conductor $57.492$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7200,2,Mod(2449,7200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7200, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("7200.2449");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7200 = 2^{5} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7200.d (of order \(2\), degree \(1\), not 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: \(57.4922894553\)
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: \( x^{16} - 4x^{14} - 4x^{12} + 12x^{10} + 389x^{8} + 816x^{6} + 2924x^{4} + 1040x^{2} + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{31}]\)
Coefficient ring index: \( 2^{18}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 1800)
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.

\(f(q)\) \(=\) \( q + \beta_{8} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{8} q^{7} + \beta_{4} q^{11} - \beta_{3} q^{13} + ( - \beta_{7} + \beta_{5}) q^{17} + (\beta_{11} - \beta_{10}) q^{19} + \beta_{7} q^{23} + (\beta_{6} + \beta_{4}) q^{29} + (\beta_{2} - 1) q^{31} - 2 \beta_1 q^{37} + \beta_{15} q^{41} + (\beta_{3} - \beta_1) q^{43} + 2 \beta_{5} q^{47} - 2 \beta_{2} q^{49} + ( - \beta_{14} + \beta_{9}) q^{53} + 2 \beta_{6} q^{59} + ( - \beta_{11} + 2 \beta_{10}) q^{61} + ( - \beta_{3} + 3 \beta_1) q^{67} + ( - \beta_{15} - \beta_{13}) q^{71} + 2 \beta_{8} q^{73} + ( - \beta_{14} + 3 \beta_{9}) q^{77} + (2 \beta_{2} - 2) q^{79} + ( - 2 \beta_{14} - 2 \beta_{9}) q^{83} - 2 \beta_{15} q^{89} + (\beta_{11} - 3 \beta_{10}) q^{91} + ( - \beta_{12} - 5 \beta_{8}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{31} - 32 q^{79}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4x^{14} - 4x^{12} + 12x^{10} + 389x^{8} + 816x^{6} + 2924x^{4} + 1040x^{2} + 100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 23 \nu^{14} - 1936 \nu^{12} + 13426 \nu^{10} - 42572 \nu^{8} + 54683 \nu^{6} - 297796 \nu^{4} + \cdots - 6074600 ) / 1917100 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 595 \nu^{14} - 4503 \nu^{12} + 2510 \nu^{10} + 41834 \nu^{8} + 166975 \nu^{6} - 511043 \nu^{4} + \cdots - 3877090 ) / 1574040 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 191 \nu^{14} + 1662 \nu^{12} - 3442 \nu^{10} - 4576 \nu^{8} - 60311 \nu^{6} + 209482 \nu^{4} + \cdots + 1156700 ) / 296400 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 116047 \nu^{15} - 484014 \nu^{13} - 379486 \nu^{11} + 1806272 \nu^{9} + 42548887 \nu^{7} + \cdots + 184956500 \nu ) / 47221200 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 2364772 \nu^{15} + 8523213 \nu^{13} + 13435108 \nu^{11} - 21459074 \nu^{9} + \cdots - 3852390290 \nu ) / 448601400 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 255958 \nu^{15} - 949761 \nu^{13} - 1397704 \nu^{11} + 3157778 \nu^{9} + 100444018 \nu^{7} + \cdots + 413218250 \nu ) / 23610600 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 10531601 \nu^{15} + 40133094 \nu^{13} + 49733234 \nu^{11} - 113386312 \nu^{9} + \cdots - 17111451220 \nu ) / 897202800 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3015005 \nu^{14} - 12778626 \nu^{12} - 8623370 \nu^{10} + 37296688 \nu^{8} + 1152530165 \nu^{6} + \cdots + 1548946300 ) / 179440560 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 3057999 \nu^{15} + 12823458 \nu^{13} + 10065422 \nu^{11} - 39948584 \nu^{9} + \cdots - 1511895100 \nu ) / 99689200 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 67\nu^{14} - 279\nu^{12} - 226\nu^{10} + 842\nu^{8} + 26047\nu^{6} + 49981\nu^{4} + 185970\nu^{2} + 33950 ) / 1800 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 44168 \nu^{14} - 185416 \nu^{12} - 137304 \nu^{10} + 545818 \nu^{8} + 17083488 \nu^{6} + \cdots + 22699425 ) / 983775 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 1708685 \nu^{14} - 7171647 \nu^{12} - 5450390 \nu^{10} + 21883186 \nu^{8} + 659091455 \nu^{6} + \cdots + 862831150 ) / 22430070 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 51015 \nu^{15} + 214131 \nu^{13} + 165738 \nu^{11} - 668018 \nu^{9} - 19731579 \nu^{7} + \cdots - 25165110 \nu ) / 524680 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 20119258 \nu^{15} + 84307101 \nu^{13} + 64321024 \nu^{11} - 252877898 \nu^{9} + \cdots - 10088792450 \nu ) / 149533800 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 671153 \nu^{15} + 2816880 \nu^{13} + 2122346 \nu^{11} - 8445580 \nu^{9} - 259242449 \nu^{7} + \cdots - 335850040 \nu ) / 3148080 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -2\beta_{15} + 5\beta_{14} + \beta_{13} - 10\beta_{9} - 10\beta_{7} - 5\beta_{6} + 5\beta_{5} ) / 40 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -3\beta_{12} + 10\beta_{11} - 5\beta_{10} - 2\beta_{8} + 10\beta_{3} + 15\beta_{2} + 5\beta _1 + 10 ) / 20 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 6\beta_{15} - 15\beta_{14} + 17\beta_{13} - 30\beta_{9} - 10\beta_{7} + 15\beta_{6} + 45\beta_{5} - 20\beta_{4} ) / 40 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 17\beta_{12} + 10\beta_{11} - 30\beta_{10} - 37\beta_{8} + 15\beta_{3} + 10\beta_{2} + 20\beta _1 + 30 ) / 10 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 58 \beta_{15} + 85 \beta_{14} + 19 \beta_{13} - 30 \beta_{9} - 90 \beta_{7} + 85 \beta_{6} + \cdots - 280 \beta_{4} ) / 40 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 83\beta_{12} + 270\beta_{11} - 235\beta_{10} - 578\beta_{8} - 30\beta_{3} - 15\beta_{2} + 35\beta _1 + 190 ) / 20 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 144 \beta_{15} - 185 \beta_{14} - 117 \beta_{13} + 180 \beta_{9} - 920 \beta_{7} + \cdots - 1630 \beta_{4} ) / 40 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 308 \beta_{12} + 655 \beta_{11} - 840 \beta_{10} - 1288 \beta_{8} - 280 \beta_{3} - 360 \beta_{2} + \cdots - 1505 ) / 10 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 948 \beta_{15} - 1485 \beta_{14} - 1879 \beta_{13} + 5880 \beta_{9} - 2660 \beta_{7} + \cdots - 6070 \beta_{4} ) / 40 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 2417 \beta_{12} + 4550 \beta_{11} - 6275 \beta_{10} - 9222 \beta_{8} - 7450 \beta_{3} - 12135 \beta_{2} + \cdots - 26090 ) / 20 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 5506 \beta_{15} - 9015 \beta_{14} - 15683 \beta_{13} + 50970 \beta_{9} - 9850 \beta_{7} + \cdots - 17720 \beta_{4} ) / 40 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 33 \beta_{12} + 2580 \beta_{11} - 990 \beta_{10} - 4587 \beta_{8} - 24255 \beta_{3} - 38790 \beta_{2} + \cdots - 97020 ) / 10 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 48262 \beta_{15} - 76165 \beta_{14} - 92911 \beta_{13} + 293170 \beta_{9} + 26710 \beta_{7} + \cdots + 52020 \beta_{4} ) / 40 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 49167 \beta_{12} - 127270 \beta_{11} + 142235 \beta_{10} + 248522 \beta_{8} - 258570 \beta_{3} + \cdots - 1014710 ) / 20 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 179544 \beta_{15} - 281835 \beta_{14} - 432567 \beta_{13} + 1358780 \beta_{9} + \cdots + 1131470 \beta_{4} ) / 40 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7200\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(6401\) \(6751\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2449.1
2.33132 + 0.608977i
1.06152 − 1.23157i
−2.33132 − 0.608977i
−1.06152 + 1.23157i
0.0382181 − 0.438926i
0.660816 + 1.70872i
−0.0382181 + 0.438926i
−0.660816 − 1.70872i
−0.0382181 − 0.438926i
−0.660816 + 1.70872i
0.0382181 + 0.438926i
0.660816 − 1.70872i
−2.33132 + 0.608977i
−1.06152 − 1.23157i
2.33132 − 0.608977i
1.06152 + 1.23157i
0 0 0 0 0 − 3.44949i 0 0 0
2449.2 0 0 0 0 0 − 3.44949i 0 0 0
2449.3 0 0 0 0 0 − 3.44949i 0 0 0
2449.4 0 0 0 0 0 − 3.44949i 0 0 0
2449.5 0 0 0 0 0 − 1.44949i 0 0 0
2449.6 0 0 0 0 0 − 1.44949i 0 0 0
2449.7 0 0 0 0 0 − 1.44949i 0 0 0
2449.8 0 0 0 0 0 − 1.44949i 0 0 0
2449.9 0 0 0 0 0 1.44949i 0 0 0
2449.10 0 0 0 0 0 1.44949i 0 0 0
2449.11 0 0 0 0 0 1.44949i 0 0 0
2449.12 0 0 0 0 0 1.44949i 0 0 0
2449.13 0 0 0 0 0 3.44949i 0 0 0
2449.14 0 0 0 0 0 3.44949i 0 0 0
2449.15 0 0 0 0 0 3.44949i 0 0 0
2449.16 0 0 0 0 0 3.44949i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2449.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
8.b even 2 1 inner
15.d odd 2 1 inner
24.h odd 2 1 inner
40.f even 2 1 inner
120.i odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7200.2.d.u 16
3.b odd 2 1 inner 7200.2.d.u 16
4.b odd 2 1 1800.2.d.u 16
5.b even 2 1 inner 7200.2.d.u 16
5.c odd 4 1 7200.2.k.q 8
5.c odd 4 1 7200.2.k.t 8
8.b even 2 1 inner 7200.2.d.u 16
8.d odd 2 1 1800.2.d.u 16
12.b even 2 1 1800.2.d.u 16
15.d odd 2 1 inner 7200.2.d.u 16
15.e even 4 1 7200.2.k.q 8
15.e even 4 1 7200.2.k.t 8
20.d odd 2 1 1800.2.d.u 16
20.e even 4 1 1800.2.k.r ✓ 8
20.e even 4 1 1800.2.k.s yes 8
24.f even 2 1 1800.2.d.u 16
24.h odd 2 1 inner 7200.2.d.u 16
40.e odd 2 1 1800.2.d.u 16
40.f even 2 1 inner 7200.2.d.u 16
40.i odd 4 1 7200.2.k.q 8
40.i odd 4 1 7200.2.k.t 8
40.k even 4 1 1800.2.k.r ✓ 8
40.k even 4 1 1800.2.k.s yes 8
60.h even 2 1 1800.2.d.u 16
60.l odd 4 1 1800.2.k.r ✓ 8
60.l odd 4 1 1800.2.k.s yes 8
120.i odd 2 1 inner 7200.2.d.u 16
120.m even 2 1 1800.2.d.u 16
120.q odd 4 1 1800.2.k.r ✓ 8
120.q odd 4 1 1800.2.k.s yes 8
120.w even 4 1 7200.2.k.q 8
120.w even 4 1 7200.2.k.t 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1800.2.d.u 16 4.b odd 2 1
1800.2.d.u 16 8.d odd 2 1
1800.2.d.u 16 12.b even 2 1
1800.2.d.u 16 20.d odd 2 1
1800.2.d.u 16 24.f even 2 1
1800.2.d.u 16 40.e odd 2 1
1800.2.d.u 16 60.h even 2 1
1800.2.d.u 16 120.m even 2 1
1800.2.k.r ✓ 8 20.e even 4 1
1800.2.k.r ✓ 8 40.k even 4 1
1800.2.k.r ✓ 8 60.l odd 4 1
1800.2.k.r ✓ 8 120.q odd 4 1
1800.2.k.s yes 8 20.e even 4 1
1800.2.k.s yes 8 40.k even 4 1
1800.2.k.s yes 8 60.l odd 4 1
1800.2.k.s yes 8 120.q odd 4 1
7200.2.d.u 16 1.a even 1 1 trivial
7200.2.d.u 16 3.b odd 2 1 inner
7200.2.d.u 16 5.b even 2 1 inner
7200.2.d.u 16 8.b even 2 1 inner
7200.2.d.u 16 15.d odd 2 1 inner
7200.2.d.u 16 24.h odd 2 1 inner
7200.2.d.u 16 40.f even 2 1 inner
7200.2.d.u 16 120.i odd 2 1 inner
7200.2.k.q 8 5.c odd 4 1
7200.2.k.q 8 15.e even 4 1
7200.2.k.q 8 40.i odd 4 1
7200.2.k.q 8 120.w even 4 1
7200.2.k.t 8 5.c odd 4 1
7200.2.k.t 8 15.e even 4 1
7200.2.k.t 8 40.i odd 4 1
7200.2.k.t 8 120.w even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(7200, [\chi])\):

\( T_{7}^{4} + 14T_{7}^{2} + 25 \) Copy content Toggle raw display
\( T_{11}^{4} + 32T_{11}^{2} + 40 \) Copy content Toggle raw display
\( T_{13}^{2} - 15 \) Copy content Toggle raw display
\( T_{37}^{2} - 40 \) Copy content Toggle raw display
\( T_{41}^{4} - 80T_{41}^{2} + 1000 \) Copy content Toggle raw display
\( T_{53}^{4} - 80T_{53}^{2} + 1000 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{4} + 14 T^{2} + 25)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} + 32 T^{2} + 40)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} - 15)^{8} \) Copy content Toggle raw display
$17$ \( (T^{4} + 48 T^{2} + 360)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} + 50 T^{2} + 25)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} + 32 T^{2} + 40)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} + 48 T^{2} + 360)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 2 T - 5)^{8} \) Copy content Toggle raw display
$37$ \( (T^{2} - 40)^{8} \) Copy content Toggle raw display
$41$ \( (T^{4} - 80 T^{2} + 1000)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} - 50 T^{2} + 25)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} + 128 T^{2} + 2560)^{4} \) Copy content Toggle raw display
$53$ \( (T^{4} - 80 T^{2} + 1000)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} + 128 T^{2} + 2560)^{4} \) Copy content Toggle raw display
$61$ \( (T^{4} + 110 T^{2} + 625)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} - 210 T^{2} + 5625)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} - 320 T^{2} + 25000)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 56 T^{2} + 400)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 4 T - 20)^{8} \) Copy content Toggle raw display
$83$ \( (T^{4} - 192 T^{2} + 5760)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} - 320 T^{2} + 16000)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 434 T^{2} + 46225)^{4} \) Copy content Toggle raw display
show more
show less