Properties

Label 200.6.f.a
Level $200$
Weight $6$
Character orbit 200.f
Analytic conductor $32.077$
Analytic rank $0$
Dimension $8$
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] = [200,6,Mod(149,200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(200, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("200.149");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 200 = 2^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 200.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: \(32.0767639626\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.12220785438976.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 5x^{6} + 116x^{4} + 320x^{2} + 4096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{15} \)
Twist minimal: no (minimal twist has level 8)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + ( - \beta_{5} - \beta_{2}) q^{3} + ( - \beta_{6} - \beta_{4} - 5) q^{4} + ( - 2 \beta_{6} + 6 \beta_{4} + 3 \beta_{3} - 29) q^{6} + ( - 2 \beta_{7} + 2 \beta_{5} - 18 \beta_{2} - 6 \beta_1) q^{7} + (3 \beta_{7} - 7 \beta_{5} - 5 \beta_{2} - 33 \beta_1) q^{8} + (8 \beta_{6} - 8 \beta_{4} - 24 \beta_{3} + 41) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + ( - \beta_{5} - \beta_{2}) q^{3} + ( - \beta_{6} - \beta_{4} - 5) q^{4} + ( - 2 \beta_{6} + 6 \beta_{4} + 3 \beta_{3} - 29) q^{6} + ( - 2 \beta_{7} + 2 \beta_{5} - 18 \beta_{2} - 6 \beta_1) q^{7} + (3 \beta_{7} - 7 \beta_{5} - 5 \beta_{2} - 33 \beta_1) q^{8} + (8 \beta_{6} - 8 \beta_{4} - 24 \beta_{3} + 41) q^{9} + (16 \beta_{6} - 5 \beta_{4} + 16 \beta_{3}) q^{11} + ( - 5 \beta_{7} + 17 \beta_{5} - 21 \beta_{2} - 185 \beta_1) q^{12} + ( - 3 \beta_{7} + 29 \beta_{5} + 47 \beta_{2} - 3 \beta_1) q^{13} + (16 \beta_{6} + 16 \beta_{4} - 4 \beta_{3} + 596) q^{14} + ( - 4 \beta_{6} + 28 \beta_{4} - 24 \beta_{3} - 828) q^{16} + ( - 4 \beta_{7} + 4 \beta_{5} - 36 \beta_{2} - 13 \beta_1) q^{17} + (16 \beta_{7} + 48 \beta_{5} + 25 \beta_{2} + 592 \beta_1) q^{18} + (48 \beta_{6} - 7 \beta_{4} + 48 \beta_{3}) q^{19} + (24 \beta_{6} - 160 \beta_{4} + 24 \beta_{3}) q^{21} + ( - 43 \beta_{7} - 41 \beta_{5} - 21 \beta_{2} + 721 \beta_1) q^{22} + ( - 26 \beta_{7} + 26 \beta_{5} - 234 \beta_{2} + 370 \beta_1) q^{23} + (52 \beta_{6} - 44 \beta_{4} - 216 \beta_{3} + 1948) q^{24} + (28 \beta_{6} - 180 \beta_{4} - 78 \beta_{3} + 1402) q^{26} + (24 \beta_{7} - 46 \beta_{5} - 190 \beta_{2} + 24 \beta_1) q^{27} + ( - 44 \beta_{7} + 124 \beta_{5} + 596 \beta_{2} + 548 \beta_1) q^{28} + ( - 82 \beta_{6} - 256 \beta_{4} - 82 \beta_{3}) q^{29} + ( - 48 \beta_{6} + 48 \beta_{4} + 144 \beta_{3} - 3232) q^{31} + (4 \beta_{7} + 204 \beta_{5} - 796 \beta_{2} - 428 \beta_1) q^{32} + ( - 36 \beta_{7} + 36 \beta_{5} - 324 \beta_{2} - 186 \beta_1) q^{33} + (32 \beta_{6} + 32 \beta_{4} - 9 \beta_{3} + 1193) q^{34} + (183 \beta_{6} - 329 \beta_{4} + 384 \beta_{3} - 2541) q^{36} + (57 \beta_{7} + 217 \beta_{5} - 125 \beta_{2} + 57 \beta_1) q^{37} + ( - 137 \beta_{7} - 83 \beta_{5} - 55 \beta_{2} + 2059 \beta_1) q^{38} + ( - 268 \beta_{6} + 268 \beta_{4} + 804 \beta_{3} - 8776) q^{39} + ( - 272 \beta_{6} + 272 \beta_{4} + 816 \beta_{3} - 1142) q^{41} + (88 \beta_{7} - 824 \beta_{5} - 184 \beta_{2} + 3064 \beta_1) q^{42} + ( - 48 \beta_{7} - 469 \beta_{5} - 181 \beta_{2} - 48 \beta_1) q^{43} + ( - 274 \beta_{6} + 398 \beta_{4} + 1016 \beta_{3} + 7278) q^{44} + (208 \beta_{6} + 208 \beta_{4} + 396 \beta_{3} + 7300) q^{46} + ( - 108 \beta_{7} + 108 \beta_{5} - 972 \beta_{2} + 7164 \beta_1) q^{47} + (156 \beta_{7} + 532 \beta_{5} + 1852 \beta_{2} + 4044 \beta_1) q^{48} + ( - 192 \beta_{6} + 192 \beta_{4} + 576 \beta_{3} - 2457) q^{49} + (48 \beta_{6} - 322 \beta_{4} + 48 \beta_{3}) q^{51} + (202 \beta_{7} - 610 \beta_{5} + 1194 \beta_{2} + 4018 \beta_1) q^{52} + (131 \beta_{7} + 1059 \beta_{5} + 273 \beta_{2} + 131 \beta_1) q^{53} + (148 \beta_{6} + 324 \beta_{4} + 66 \beta_{3} - 5822) q^{54} + ( - 400 \beta_{6} - 1040 \beta_{4} + 352 \beta_{3} + 10192) q^{56} + ( - 76 \beta_{7} + 76 \beta_{5} - 684 \beta_{2} + 482 \beta_1) q^{57} + (502 \beta_{7} - 1198 \beta_{5} - 174 \beta_{2} - 34 \beta_1) q^{58} + ( - 256 \beta_{6} - 1427 \beta_{4} - 256 \beta_{3}) q^{59} + (1314 \beta_{6} - 2016 \beta_{4} + 1314 \beta_{3}) q^{61} + ( - 96 \beta_{7} - 288 \beta_{5} - 3136 \beta_{2} - 3552 \beta_1) q^{62} + ( - 178 \beta_{7} + 178 \beta_{5} - 1602 \beta_{2} - 18646 \beta_1) q^{63} + (1424 \beta_{6} - 240 \beta_{4} - 1056 \beta_{3} + 10480) q^{64} + (288 \beta_{6} + 288 \beta_{4} - 150 \beta_{3} + 10806) q^{66} + (792 \beta_{7} - 567 \beta_{5} - 5319 \beta_{2} + 792 \beta_1) q^{67} + ( - 87 \beta_{7} + 251 \beta_{5} + 1193 \beta_{2} + 1101 \beta_1) q^{68} + (312 \beta_{6} - 1184 \beta_{4} + 312 \beta_{3}) q^{69} + (228 \beta_{6} - 228 \beta_{4} - 684 \beta_{3} + 51672) q^{71} + ( - 421 \beta_{7} - 2431 \beta_{5} - 3053 \beta_{2} + 10775 \beta_1) q^{72} + (604 \beta_{7} - 604 \beta_{5} + 5436 \beta_{2} + 3185 \beta_1) q^{73} + (1004 \beta_{6} - 1188 \beta_{4} - 822 \beta_{3} - 4366) q^{74} + ( - 742 \beta_{6} + 1018 \beta_{4} + 2856 \beta_{3} + \cdots + 24986) q^{76}+ \cdots + ( - 3456 \beta_{6} - 1821 \beta_{4} - 3456 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 40 q^{4} - 232 q^{6} + 328 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 40 q^{4} - 232 q^{6} + 328 q^{9} + 4768 q^{14} - 6624 q^{16} + 15584 q^{24} + 11216 q^{26} - 25856 q^{31} + 9544 q^{34} - 20328 q^{36} - 70208 q^{39} - 9136 q^{41} + 58224 q^{44} + 58400 q^{46} - 19656 q^{49} - 46576 q^{54} + 81536 q^{56} + 83840 q^{64} + 86448 q^{66} + 413376 q^{71} - 34928 q^{74} + 199888 q^{76} + 495744 q^{79} + 59368 q^{81} - 393344 q^{84} - 37000 q^{86} + 169264 q^{89} + 197568 q^{94} - 231296 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 5x^{6} + 116x^{4} + 320x^{2} + 4096 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 5\nu^{7} - 7\nu^{5} + 164\nu^{3} - 832\nu ) / 2304 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + 5\nu^{5} + 116\nu^{3} + 320\nu ) / 256 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{6} + 7\nu^{4} - 164\nu^{2} + 976 ) / 144 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} - 11\nu^{4} + 100\nu^{2} - 608 ) / 48 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} - 21\nu^{5} - 68\nu^{3} - 1280\nu ) / 128 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} + 13\nu^{4} + 124\nu^{2} + 664 ) / 24 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{7} - 13\nu^{5} - 76\nu^{3} + 368\nu ) / 72 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} - \beta_{5} + \beta_{2} + \beta_1 ) / 16 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} + 3\beta_{4} + 3\beta_{3} - 10 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{7} + 9\beta_{5} + 55\beta_{2} - 73\beta_1 ) / 16 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 7\beta_{6} - 19\beta_{4} - 3\beta_{3} - 414 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -63\beta_{7} - 41\beta_{5} - 151\beta_{2} - 279\beta_1 ) / 16 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -23\beta_{6} - 125\beta_{4} - 333\beta_{3} + 1310 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 111\beta_{7} - 519\beta_{5} - 1849\beta_{2} + 9543\beta_1 ) / 16 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/200\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(151\) \(177\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1
2.10784 1.88600i
2.10784 + 1.88600i
1.51888 2.38600i
1.51888 + 2.38600i
−1.51888 2.38600i
−1.51888 + 2.38600i
−2.10784 1.88600i
−2.10784 + 1.88600i
−4.21569 3.77200i −3.25452 3.54400 + 31.8031i 0 13.7200 + 12.2760i 112.704i 105.021 147.440i −232.408 0
149.2 −4.21569 + 3.77200i −3.25452 3.54400 31.8031i 0 13.7200 12.2760i 112.704i 105.021 + 147.440i −232.408 0
149.3 −3.03776 4.77200i 23.6095 −13.5440 + 28.9924i 0 −71.7200 112.665i 160.704i 179.495 23.4400i 314.408 0
149.4 −3.03776 + 4.77200i 23.6095 −13.5440 28.9924i 0 −71.7200 + 112.665i 160.704i 179.495 + 23.4400i 314.408 0
149.5 3.03776 4.77200i −23.6095 −13.5440 28.9924i 0 −71.7200 + 112.665i 160.704i −179.495 23.4400i 314.408 0
149.6 3.03776 + 4.77200i −23.6095 −13.5440 + 28.9924i 0 −71.7200 112.665i 160.704i −179.495 + 23.4400i 314.408 0
149.7 4.21569 3.77200i 3.25452 3.54400 31.8031i 0 13.7200 12.2760i 112.704i −105.021 147.440i −232.408 0
149.8 4.21569 + 3.77200i 3.25452 3.54400 + 31.8031i 0 13.7200 + 12.2760i 112.704i −105.021 + 147.440i −232.408 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 149.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
8.b even 2 1 inner
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 200.6.f.a 8
4.b odd 2 1 800.6.f.a 8
5.b even 2 1 inner 200.6.f.a 8
5.c odd 4 1 8.6.b.a 4
5.c odd 4 1 200.6.d.a 4
8.b even 2 1 inner 200.6.f.a 8
8.d odd 2 1 800.6.f.a 8
15.e even 4 1 72.6.d.b 4
20.d odd 2 1 800.6.f.a 8
20.e even 4 1 32.6.b.a 4
20.e even 4 1 800.6.d.a 4
40.e odd 2 1 800.6.f.a 8
40.f even 2 1 inner 200.6.f.a 8
40.i odd 4 1 8.6.b.a 4
40.i odd 4 1 200.6.d.a 4
40.k even 4 1 32.6.b.a 4
40.k even 4 1 800.6.d.a 4
60.l odd 4 1 288.6.d.b 4
80.i odd 4 1 256.6.a.k 4
80.j even 4 1 256.6.a.n 4
80.s even 4 1 256.6.a.n 4
80.t odd 4 1 256.6.a.k 4
120.q odd 4 1 288.6.d.b 4
120.w even 4 1 72.6.d.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.6.b.a 4 5.c odd 4 1
8.6.b.a 4 40.i odd 4 1
32.6.b.a 4 20.e even 4 1
32.6.b.a 4 40.k even 4 1
72.6.d.b 4 15.e even 4 1
72.6.d.b 4 120.w even 4 1
200.6.d.a 4 5.c odd 4 1
200.6.d.a 4 40.i odd 4 1
200.6.f.a 8 1.a even 1 1 trivial
200.6.f.a 8 5.b even 2 1 inner
200.6.f.a 8 8.b even 2 1 inner
200.6.f.a 8 40.f even 2 1 inner
256.6.a.k 4 80.i odd 4 1
256.6.a.k 4 80.t odd 4 1
256.6.a.n 4 80.j even 4 1
256.6.a.n 4 80.s even 4 1
288.6.d.b 4 60.l odd 4 1
288.6.d.b 4 120.q odd 4 1
800.6.d.a 4 20.e even 4 1
800.6.d.a 4 40.k even 4 1
800.6.f.a 8 4.b odd 2 1
800.6.f.a 8 8.d odd 2 1
800.6.f.a 8 20.d odd 2 1
800.6.f.a 8 40.e odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 568T_{3}^{2} + 5904 \) acting on \(S_{6}^{\mathrm{new}}(200, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 20 T^{6} + 1856 T^{4} + \cdots + 1048576 \) Copy content Toggle raw display
$3$ \( (T^{4} - 568 T^{2} + 5904)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + 38528 T^{2} + \cdots + 328044544)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 347768 T^{2} + \cdots + 5520765456)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 590944 T^{2} + \cdots + 7999305984)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 154504 T^{2} + \cdots + 5220351504)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 3109816 T^{2} + \cdots + 120994976016)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 6998656 T^{2} + \cdots + 7936705990656)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 50789216 T^{2} + \cdots + 535633608132864)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 6464 T + 7754752)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} - 36113248 T^{2} + \cdots + 306881230162176)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 2284 T - 85109148)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} - 121686904 T^{2} + \cdots + 23\!\cdots\!56)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 483273216 T^{2} + \cdots + 17\!\cdots\!64)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 633629792 T^{2} + \cdots + 76\!\cdots\!44)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 1322273016 T^{2} + \cdots + 22\!\cdots\!36)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 4119483744 T^{2} + \cdots + 41\!\cdots\!16)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 4033664568 T^{2} + \cdots + 32\!\cdots\!84)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 103344 T + 2609278272)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 3608600776 T^{2} + \cdots + 25\!\cdots\!56)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 123936 T + 3701816576)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 5708307384 T^{2} + \cdots + 72\!\cdots\!56)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 42316 T - 6875717724)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 4045463048 T^{2} + \cdots + 61\!\cdots\!04)^{2} \) Copy content Toggle raw display
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