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Properties

Label 800.4.d.d
Level $800$
Weight $4$
Character orbit 800.d
Analytic conductor $47.202$
Analytic rank $0$
Dimension $12$
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] = [800,4,Mod(401,800)]
 
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, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("800.401");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 800.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: \(47.2015280046\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{32}\cdot 5^{4} \)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + ( - \beta_{4} + 2) q^{7} + (\beta_{8} - 9) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + ( - \beta_{4} + 2) q^{7} + (\beta_{8} - 9) q^{9} + (\beta_{6} - \beta_{3} + \beta_1) q^{11} + ( - \beta_{5} - \beta_1) q^{13} + ( - \beta_{4} + \beta_{2}) q^{17} + ( - 2 \beta_{10} - 3 \beta_{6} + \cdots + \beta_1) q^{19}+ \cdots + (20 \beta_{11} - 18 \beta_{10} + \cdots - 5 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 28 q^{7} - 108 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 28 q^{7} - 108 q^{9} + 604 q^{23} + 264 q^{31} + 232 q^{33} - 600 q^{39} + 40 q^{41} - 940 q^{47} + 1308 q^{49} + 680 q^{57} - 1300 q^{63} + 1592 q^{71} - 432 q^{73} - 2016 q^{79} + 2508 q^{81} - 1968 q^{87} - 424 q^{89} + 1584 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

\(\beta_{1}\)\(=\) \( ( 93 \nu^{11} - 204 \nu^{10} + 379 \nu^{9} - 388 \nu^{8} + 817 \nu^{7} + 1672 \nu^{6} - 4708 \nu^{5} + \cdots - 191488 ) / 15360 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 43 \nu^{11} - 28 \nu^{10} + 1085 \nu^{9} - 2644 \nu^{8} - 649 \nu^{7} + 3344 \nu^{6} - 2396 \nu^{5} + \cdots - 240640 ) / 5120 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 177 \nu^{11} + 228 \nu^{10} + 263 \nu^{9} - 2420 \nu^{8} + 3797 \nu^{7} + 2696 \nu^{6} + \cdots - 502784 ) / 15360 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 163 \nu^{11} - 188 \nu^{10} + 5 \nu^{9} - 564 \nu^{8} + 1231 \nu^{7} - 1456 \nu^{6} - 956 \nu^{5} + \cdots - 168960 ) / 15360 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 165 \nu^{11} + 636 \nu^{10} - 307 \nu^{9} + 3028 \nu^{8} - 5209 \nu^{7} - 2488 \nu^{6} + \cdots + 934912 ) / 15360 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 23 \nu^{11} + 60 \nu^{10} - 81 \nu^{9} + 116 \nu^{8} - 51 \nu^{7} - 480 \nu^{6} + 684 \nu^{5} + \cdots + 30720 ) / 1536 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 17 \nu^{11} - 112 \nu^{10} + 295 \nu^{9} - 576 \nu^{8} + 629 \nu^{7} + 676 \nu^{6} - 1924 \nu^{5} + \cdots - 118080 ) / 960 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 211 \nu^{11} - 476 \nu^{10} + 725 \nu^{9} - 1428 \nu^{8} + 1087 \nu^{7} + 2288 \nu^{6} + \cdots - 399360 ) / 7680 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 137 \nu^{11} - 292 \nu^{10} + 175 \nu^{9} - 876 \nu^{8} + 1229 \nu^{7} + 4096 \nu^{6} - 5044 \nu^{5} + \cdots - 261120 ) / 3840 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 271 \nu^{11} - 852 \nu^{10} + 1401 \nu^{9} - 1948 \nu^{8} + 2475 \nu^{7} + 2376 \nu^{6} + \cdots - 605184 ) / 7680 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1113 \nu^{11} - 1884 \nu^{10} + 2719 \nu^{9} - 7348 \nu^{8} + 11677 \nu^{7} + 17512 \nu^{6} + \cdots - 2219008 ) / 15360 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - \beta_{11} + 4 \beta_{10} + \beta_{9} - 2 \beta_{8} - 2 \beta_{7} + 4 \beta_{6} - 2 \beta_{5} + \cdots + 52 ) / 160 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2 \beta_{11} - 3 \beta_{10} - 3 \beta_{9} - 12 \beta_{8} - \beta_{7} - 6 \beta_{6} - 6 \beta_{5} + \cdots + 31 ) / 160 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 8 \beta_{11} - 2 \beta_{10} - 6 \beta_{9} + \beta_{8} - 2 \beta_{7} + 4 \beta_{6} - 4 \beta_{5} + \cdots + 102 ) / 80 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 2 \beta_{11} - \beta_{10} - 5 \beta_{9} - 2 \beta_{8} - \beta_{7} - 22 \beta_{6} - 6 \beta_{5} + \cdots + 35 ) / 32 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 29 \beta_{11} + 24 \beta_{10} - 21 \beta_{9} - 112 \beta_{8} + 6 \beta_{7} - 156 \beta_{6} + \cdots - 1536 ) / 160 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 32 \beta_{11} - 83 \beta_{10} + 5 \beta_{9} - 51 \beta_{8} + 16 \beta_{7} - 176 \beta_{6} + 4 \beta_{5} + \cdots - 66 ) / 80 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 73 \beta_{11} + 8 \beta_{10} - \beta_{9} - 98 \beta_{8} + 102 \beta_{7} + 452 \beta_{6} + 26 \beta_{5} + \cdots + 11928 ) / 160 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 46 \beta_{11} + 91 \beta_{10} + 31 \beta_{9} - 76 \beta_{8} - 31 \beta_{7} + 118 \beta_{6} + \cdots - 1671 ) / 32 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 108 \beta_{11} - 22 \beta_{10} - 206 \beta_{9} - 601 \beta_{8} + 110 \beta_{7} + 644 \beta_{6} + \cdots + 2310 ) / 80 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 1422 \beta_{11} - 903 \beta_{10} - 1087 \beta_{9} + 1658 \beta_{8} - 815 \beta_{7} + 5814 \beta_{6} + \cdots + 22885 ) / 160 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 1217 \beta_{11} - 412 \beta_{10} - 2099 \beta_{9} + 672 \beta_{8} - 2186 \beta_{7} - 9148 \beta_{6} + \cdots + 68196 ) / 160 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
401.1
1.98839 + 0.215211i
1.23537 + 1.57285i
−1.86176 − 0.730647i
1.71681 − 1.02595i
−0.428316 + 1.95360i
−0.650488 − 1.89126i
−0.650488 + 1.89126i
−0.428316 − 1.95360i
1.71681 + 1.02595i
−1.86176 + 0.730647i
1.23537 − 1.57285i
1.98839 − 0.215211i
0 − 9.57890i 0 0 0 21.5703 0 −64.7554 0
401.2 0 − 7.99849i 0 0 0 9.93501 0 −36.9759 0
401.3 0 − 6.25785i 0 0 0 −34.6280 0 −12.1606 0
401.4 0 − 4.24443i 0 0 0 −14.6308 0 8.98481 0
401.5 0 − 1.51777i 0 0 0 5.13620 0 24.6964 0
401.6 0 − 0.888401i 0 0 0 26.6173 0 26.2107 0
401.7 0 0.888401i 0 0 0 26.6173 0 26.2107 0
401.8 0 1.51777i 0 0 0 5.13620 0 24.6964 0
401.9 0 4.24443i 0 0 0 −14.6308 0 8.98481 0
401.10 0 6.25785i 0 0 0 −34.6280 0 −12.1606 0
401.11 0 7.99849i 0 0 0 9.93501 0 −36.9759 0
401.12 0 9.57890i 0 0 0 21.5703 0 −64.7554 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 401.12
Significant digits:
Format:

Inner twists

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(800, [\chi])\):

\( T_{3}^{12} + 216T_{3}^{10} + 16140T_{3}^{8} + 493760T_{3}^{6} + 5547312T_{3}^{4} + 13618560T_{3}^{2} + 7529536 \) Copy content Toggle raw display
\( T_{7}^{6} - 14T_{7}^{5} - 1258T_{7}^{4} + 23408T_{7}^{3} + 166612T_{7}^{2} - 4186552T_{7} + 14843128 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 216 T^{10} + \cdots + 7529536 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( (T^{6} - 14 T^{5} + \cdots + 14843128)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T^{6} - 14500 T^{4} + \cdots - 7473839808)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 17\!\cdots\!24 \) Copy content Toggle raw display
$23$ \( (T^{6} - 302 T^{5} + \cdots + 881168216)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots - 1437816300032)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 21\!\cdots\!04 \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots - 71667547865600)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 25\!\cdots\!04 \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots - 72048375466472)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 86\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 10\!\cdots\!24 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 61\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 87\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 36\!\cdots\!48)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots - 378730163491776)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots - 44\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 78\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots - 62\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots - 33\!\cdots\!28)^{2} \) Copy content Toggle raw display
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