Properties

Label 800.4.f.c
Level $800$
Weight $4$
Character orbit 800.f
Analytic conductor $47.202$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [800,4,Mod(49,800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(800, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("800.49");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 800.f (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: \(47.2015280046\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4x^{11} + 7x^{10} - 12x^{9} + 21x^{8} - 68x^{6} + 336x^{4} - 768x^{3} + 1792x^{2} - 4096x + 4096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{32} \)
Twist minimal: no (minimal twist has level 40)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{4} + 1) q^{3} + (\beta_{6} + \beta_{2}) q^{7} + ( - \beta_{7} - 2 \beta_{4} + 9) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} + 1) q^{3} + (\beta_{6} + \beta_{2}) q^{7} + ( - \beta_{7} - 2 \beta_{4} + 9) q^{9} + ( - \beta_{9} + \beta_{5} + 4 \beta_{2}) q^{11} + (\beta_{10} - \beta_{8} - 2 \beta_{4} + \beta_{3}) q^{13} + (\beta_{11} + \beta_{6} - 2 \beta_{5} - 2 \beta_{2} + 3 \beta_1) q^{17} + ( - 2 \beta_{11} - \beta_{9} - 2 \beta_{6} - \beta_{5} + 12 \beta_{2}) q^{19} + (3 \beta_{11} - \beta_{9} + 5 \beta_{6} + 4 \beta_{2} - 7 \beta_1) q^{21} + (2 \beta_{11} + \beta_{9} + 2 \beta_{6} + \beta_{5} - 21 \beta_{2} - 7 \beta_1) q^{23} + (4 \beta_{8} - 3 \beta_{7} - 11 \beta_{4} - \beta_{3} + 37) q^{27} + ( - \beta_{11} + 3 \beta_{9} + 3 \beta_{6} - 2 \beta_{5} - 5 \beta_{2} + 7 \beta_1) q^{29} + ( - 2 \beta_{10} - 2 \beta_{8} + 2 \beta_{7} + 6 \beta_{4} + 2 \beta_{3} + 22) q^{31} + (5 \beta_{11} + 3 \beta_{6} + 9 \beta_{2} - 35 \beta_1) q^{33} + ( - 2 \beta_{8} + 14 \beta_{4} + 6 \beta_{3} - 10) q^{37} + ( - \beta_{10} + 6 \beta_{8} - 6 \beta_{4} - 3 \beta_{3} + 51) q^{39} + ( - 4 \beta_{10} - \beta_{7} + 38 \beta_{4} - 4 \beta_{3} + 2) q^{41} + (\beta_{10} - 3 \beta_{7} - 26 \beta_{4} + 4 \beta_{3} - 99) q^{43} + ( - 4 \beta_{11} - 4 \beta_{9} + \beta_{6} + 8 \beta_{5} - 33 \beta_{2} - 10 \beta_1) q^{47} + ( - 2 \beta_{10} + 6 \beta_{8} - 3 \beta_{7} + 38 \beta_{4} + 6 \beta_{3} + \cdots - 105) q^{49}+ \cdots + ( - 6 \beta_{11} - 21 \beta_{9} + 30 \beta_{6} - 17 \beta_{5} + 446 \beta_{2} + 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{3} + 108 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 12 q^{3} + 108 q^{9} + 432 q^{27} + 264 q^{31} - 136 q^{37} + 600 q^{39} + 40 q^{41} - 1204 q^{43} - 1308 q^{49} - 1056 q^{53} - 2412 q^{67} + 1592 q^{71} - 824 q^{77} + 2016 q^{79} + 2508 q^{81} + 3556 q^{83} + 424 q^{89} - 2784 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4x^{11} + 7x^{10} - 12x^{9} + 21x^{8} - 68x^{6} + 336x^{4} - 768x^{3} + 1792x^{2} - 4096x + 4096 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 47 \nu^{11} - 84 \nu^{10} + 217 \nu^{9} - 156 \nu^{8} + 715 \nu^{7} + 712 \nu^{6} - 3340 \nu^{5} - 2592 \nu^{4} + 8816 \nu^{3} - 11392 \nu^{2} + 51200 \nu - 130048 ) / 15360 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 23 \nu^{11} + 60 \nu^{10} - 81 \nu^{9} + 116 \nu^{8} - 51 \nu^{7} - 480 \nu^{6} + 684 \nu^{5} + 1024 \nu^{4} - 5616 \nu^{3} + 11520 \nu^{2} - 22528 \nu + 30720 ) / 7680 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 11 \nu^{11} + 28 \nu^{10} - 349 \nu^{9} + 852 \nu^{8} - 215 \nu^{7} - 400 \nu^{6} + 220 \nu^{5} - 960 \nu^{4} + 6992 \nu^{3} + 19456 \nu^{2} - 92672 \nu + 89088 ) / 3072 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 97 \nu^{11} - 212 \nu^{10} + 215 \nu^{9} - 636 \nu^{8} + 709 \nu^{7} + 2096 \nu^{6} - 4244 \nu^{5} - 10176 \nu^{4} + 16400 \nu^{3} - 38912 \nu^{2} + 122368 \nu - 184320 ) / 15360 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 69 \nu^{11} - 36 \nu^{10} - 205 \nu^{9} - 236 \nu^{8} + 857 \nu^{7} + 2288 \nu^{6} - 2948 \nu^{5} - 12736 \nu^{4} + 16336 \nu^{3} + 44032 \nu^{2} - 16384 \nu - 131072 ) / 7680 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 211 \nu^{11} + 468 \nu^{10} - 853 \nu^{9} + 476 \nu^{8} - 319 \nu^{7} - 2584 \nu^{6} - 164 \nu^{5} + 24160 \nu^{4} - 69296 \nu^{3} + 141184 \nu^{2} - 141312 \nu + 173056 ) / 15360 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 19 \nu^{11} - 44 \nu^{10} + 85 \nu^{9} - 132 \nu^{8} + 63 \nu^{7} + 32 \nu^{6} - 1148 \nu^{5} - 192 \nu^{4} + 6320 \nu^{3} - 13824 \nu^{2} + 11776 \nu - 35840 ) / 1280 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 103 \nu^{11} + 428 \nu^{10} - 545 \nu^{9} + 1284 \nu^{8} - 2851 \nu^{7} - 1904 \nu^{6} + 5516 \nu^{5} + 9024 \nu^{4} - 18800 \nu^{3} + 48128 \nu^{2} - 143872 \nu + 376320 ) / 7680 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 67 \nu^{11} + 60 \nu^{10} - 91 \nu^{9} - 684 \nu^{8} + 1199 \nu^{7} + 1640 \nu^{6} + 484 \nu^{5} - 5856 \nu^{4} - 7376 \nu^{3} + 8320 \nu^{2} + 59392 \nu - 158720 ) / 3840 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 271 \nu^{11} + 236 \nu^{10} + 1735 \nu^{9} - 3132 \nu^{8} - 1387 \nu^{7} + 4912 \nu^{6} + 812 \nu^{5} + 38208 \nu^{4} - 51440 \nu^{3} - 154624 \nu^{2} + 482816 \nu - 122880 ) / 15360 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 165 \nu^{11} + 492 \nu^{10} - 499 \nu^{9} + 1636 \nu^{8} - 2713 \nu^{7} - 2056 \nu^{6} + 8452 \nu^{5} + 11168 \nu^{4} - 53456 \nu^{3} + 48256 \nu^{2} - 292864 \nu + 523264 ) / 7680 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{11} + \beta_{8} - \beta_{7} - \beta_{5} + 2\beta_{4} + \beta_{2} + 2\beta _1 + 11 ) / 32 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} - 2 \beta_{7} + \beta_{6} + 2 \beta_{5} - 7 \beta_{4} - 2 \beta_{3} - 13 \beta_{2} + 5 \beta _1 + 5 ) / 32 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 2 \beta_{11} + \beta_{10} + \beta_{8} + \beta_{7} - 2 \beta_{6} + 2 \beta_{5} - 8 \beta_{4} + \beta_{3} + 4 \beta_{2} - 2 \beta _1 + 22 ) / 16 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 3 \beta_{11} + \beta_{10} + 5 \beta_{9} + \beta_{8} + 2 \beta_{7} + 11 \beta_{6} - 6 \beta_{5} - 47 \beta_{4} + 4 \beta_{3} - 115 \beta_{2} + 7 \beta _1 + 39 ) / 32 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 7 \beta_{11} - 2 \beta_{10} + 8 \beta_{9} - 3 \beta_{8} - 17 \beta_{7} - 20 \beta_{6} - 7 \beta_{5} - 82 \beta_{4} - 2 \beta_{3} - 109 \beta_{2} - 46 \beta _1 - 311 ) / 32 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 11 \beta_{11} + 15 \beta_{10} + 5 \beta_{9} - 8 \beta_{8} - 8 \beta_{7} + 3 \beta_{6} + 18 \beta_{5} - 22 \beta_{4} + 15 \beta_{3} - 163 \beta_{2} + 3 \beta _1 - 3 ) / 16 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 11 \beta_{11} + 24 \beta_{10} + 44 \beta_{9} - 69 \beta_{8} + \beta_{7} - 36 \beta_{6} - 31 \beta_{5} - 174 \beta_{4} + 60 \beta_{3} + 315 \beta_{2} + 170 \beta _1 + 2397 ) / 32 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( \beta_{11} - 3 \beta_{10} - 3 \beta_{9} + 39 \beta_{8} - 138 \beta_{7} - 45 \beta_{6} - 134 \beta_{5} + 35 \beta_{4} + 74 \beta_{3} - 235 \beta_{2} + 823 \beta _1 - 1589 ) / 32 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 54 \beta_{11} - 13 \beta_{10} - 56 \beta_{9} - 5 \beta_{8} - 57 \beta_{7} - 110 \beta_{6} + 82 \beta_{5} - 436 \beta_{4} - 113 \beta_{3} - 136 \beta_{2} + 778 \beta _1 + 366 ) / 16 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 37 \beta_{11} + 235 \beta_{10} + 311 \beta_{9} + 463 \beta_{8} + 386 \beta_{7} - 455 \beta_{6} + 154 \beta_{5} - 365 \beta_{4} + 124 \beta_{3} + 4147 \beta_{2} + 1941 \beta _1 + 5001 ) / 32 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 207 \beta_{11} - 994 \beta_{10} + 844 \beta_{9} + 915 \beta_{8} + 117 \beta_{7} + 688 \beta_{6} - 1005 \beta_{5} - 2490 \beta_{4} - 638 \beta_{3} - 11795 \beta_{2} + 3698 \beta _1 + 13479 ) / 32 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/800\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(351\) \(577\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
1.23537 1.57285i
1.23537 + 1.57285i
1.71681 1.02595i
1.71681 + 1.02595i
−0.650488 1.89126i
−0.650488 + 1.89126i
−0.428316 + 1.95360i
−0.428316 1.95360i
−1.86176 + 0.730647i
−1.86176 0.730647i
1.98839 + 0.215211i
1.98839 0.215211i
0 −7.99849 0 0 0 9.93501i 0 36.9759 0
49.2 0 −7.99849 0 0 0 9.93501i 0 36.9759 0
49.3 0 −4.24443 0 0 0 14.6308i 0 −8.98481 0
49.4 0 −4.24443 0 0 0 14.6308i 0 −8.98481 0
49.5 0 0.888401 0 0 0 26.6173i 0 −26.2107 0
49.6 0 0.888401 0 0 0 26.6173i 0 −26.2107 0
49.7 0 1.51777 0 0 0 5.13620i 0 −24.6964 0
49.8 0 1.51777 0 0 0 5.13620i 0 −24.6964 0
49.9 0 6.25785 0 0 0 34.6280i 0 12.1606 0
49.10 0 6.25785 0 0 0 34.6280i 0 12.1606 0
49.11 0 9.57890 0 0 0 21.5703i 0 64.7554 0
49.12 0 9.57890 0 0 0 21.5703i 0 64.7554 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
40.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 800.4.f.c 12
4.b odd 2 1 200.4.f.b 12
5.b even 2 1 800.4.f.b 12
5.c odd 4 1 160.4.d.a 12
5.c odd 4 1 800.4.d.d 12
8.b even 2 1 800.4.f.b 12
8.d odd 2 1 200.4.f.c 12
15.e even 4 1 1440.4.k.c 12
20.d odd 2 1 200.4.f.c 12
20.e even 4 1 40.4.d.a 12
20.e even 4 1 200.4.d.b 12
40.e odd 2 1 200.4.f.b 12
40.f even 2 1 inner 800.4.f.c 12
40.i odd 4 1 160.4.d.a 12
40.i odd 4 1 800.4.d.d 12
40.k even 4 1 40.4.d.a 12
40.k even 4 1 200.4.d.b 12
60.l odd 4 1 360.4.k.c 12
80.i odd 4 1 1280.4.a.ba 6
80.j even 4 1 1280.4.a.bb 6
80.s even 4 1 1280.4.a.bc 6
80.t odd 4 1 1280.4.a.bd 6
120.q odd 4 1 360.4.k.c 12
120.w even 4 1 1440.4.k.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.4.d.a 12 20.e even 4 1
40.4.d.a 12 40.k even 4 1
160.4.d.a 12 5.c odd 4 1
160.4.d.a 12 40.i odd 4 1
200.4.d.b 12 20.e even 4 1
200.4.d.b 12 40.k even 4 1
200.4.f.b 12 4.b odd 2 1
200.4.f.b 12 40.e odd 2 1
200.4.f.c 12 8.d odd 2 1
200.4.f.c 12 20.d odd 2 1
360.4.k.c 12 60.l odd 4 1
360.4.k.c 12 120.q odd 4 1
800.4.d.d 12 5.c odd 4 1
800.4.d.d 12 40.i odd 4 1
800.4.f.b 12 5.b even 2 1
800.4.f.b 12 8.b even 2 1
800.4.f.c 12 1.a even 1 1 trivial
800.4.f.c 12 40.f even 2 1 inner
1280.4.a.ba 6 80.i odd 4 1
1280.4.a.bb 6 80.j even 4 1
1280.4.a.bc 6 80.s even 4 1
1280.4.a.bd 6 80.t odd 4 1
1440.4.k.c 12 15.e even 4 1
1440.4.k.c 12 120.w even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} - 6T_{3}^{5} - 90T_{3}^{4} + 432T_{3}^{3} + 1428T_{3}^{2} - 4632T_{3} + 2744 \) acting on \(S_{4}^{\mathrm{new}}(800, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{6} - 6 T^{5} - 90 T^{4} + 432 T^{3} + \cdots + 2744)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 220318448824384 \) Copy content Toggle raw display
$11$ \( T^{12} + 10072 T^{10} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{6} - 5572 T^{4} + 163840 T^{3} + \cdots - 157790400)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + 29000 T^{10} + \cdots + 55\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( T^{12} + 46312 T^{10} + \cdots + 17\!\cdots\!24 \) Copy content Toggle raw display
$23$ \( T^{12} + 65320 T^{10} + \cdots + 77\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( T^{12} + 92448 T^{10} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{6} - 132 T^{5} + \cdots - 1437816300032)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 68 T^{5} + \cdots + 45951464886848)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} - 20 T^{5} + \cdots - 71667547865600)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 602 T^{5} + \cdots - 508806074248)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + 562088 T^{10} + \cdots + 51\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( (T^{6} + 528 T^{5} + \cdots + 929403110278976)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + 1889384 T^{10} + \cdots + 10\!\cdots\!24 \) Copy content Toggle raw display
$61$ \( T^{12} + 627448 T^{10} + \cdots + 61\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{6} + 1206 T^{5} + \cdots - 93747278347656)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} - 796 T^{5} + \cdots - 36\!\cdots\!48)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + 1958184 T^{10} + \cdots + 14\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( (T^{6} - 1008 T^{5} + \cdots - 44\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} - 1778 T^{5} + \cdots - 28\!\cdots\!68)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} - 212 T^{5} + \cdots - 62\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + 3718376 T^{10} + \cdots + 11\!\cdots\!84 \) Copy content Toggle raw display
show more
show less