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Properties

Label 200.4.f.c
Level $200$
Weight $4$
Character orbit 200.f
Analytic conductor $11.800$
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] = [200,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("200.149");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 200 = 2^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
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: \(11.8003820011\)
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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{15} \)
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.

\(f(q)\) \(=\) \( q + (\beta_{3} + 1) q^{2} + ( - \beta_{10} + 1) q^{3} + (\beta_{8} + \beta_{3} - 1) q^{4} + ( - 2 \beta_{10} - \beta_{9} + \cdots - 2) q^{6}+ \cdots + ( - 2 \beta_{11} - \beta_{10} + \cdots + 12) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + 1) q^{2} + ( - \beta_{10} + 1) q^{3} + (\beta_{8} + \beta_{3} - 1) q^{4} + ( - 2 \beta_{10} - \beta_{9} + \cdots - 2) q^{6}+ \cdots + (12 \beta_{11} - 2 \beta_{10} + \cdots - 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{2} + 12 q^{3} - 16 q^{4} - 36 q^{6} - 24 q^{8} + 108 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{2} + 12 q^{3} - 16 q^{4} - 36 q^{6} - 24 q^{8} + 108 q^{9} - 164 q^{12} - 68 q^{14} - 56 q^{16} + 450 q^{18} + 492 q^{22} - 360 q^{24} - 308 q^{26} + 432 q^{27} + 628 q^{28} - 264 q^{31} + 856 q^{32} + 180 q^{34} + 440 q^{36} + 136 q^{37} - 1388 q^{38} - 600 q^{39} + 40 q^{41} - 2332 q^{42} - 1204 q^{43} + 472 q^{44} - 1268 q^{46} - 2536 q^{48} - 1308 q^{49} - 1272 q^{52} + 1056 q^{53} + 1512 q^{54} - 728 q^{56} + 1528 q^{58} + 2104 q^{62} + 2048 q^{64} + 2928 q^{66} - 2412 q^{67} + 960 q^{68} - 1592 q^{71} + 744 q^{72} + 420 q^{74} + 2256 q^{76} + 824 q^{77} - 160 q^{78} - 2016 q^{79} + 2508 q^{81} + 352 q^{82} + 3556 q^{83} - 1048 q^{84} - 244 q^{86} + 1008 q^{88} + 424 q^{89} + 1068 q^{92} + 2784 q^{93} - 292 q^{94} - 5920 q^{96} + 638 q^{98}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( ( - 7 \nu^{11} + 20 \nu^{10} - \nu^{9} - 100 \nu^{8} + 189 \nu^{7} - 40 \nu^{6} + 172 \nu^{5} + \cdots - 16384 ) / 7680 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 2 \nu^{11} + 5 \nu^{10} - 10 \nu^{9} + 27 \nu^{8} - 30 \nu^{7} - 55 \nu^{6} + 64 \nu^{5} + \cdots + 5888 ) / 960 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2 \nu^{11} - 5 \nu^{10} + 10 \nu^{9} - 27 \nu^{8} + 30 \nu^{7} + 55 \nu^{6} - 64 \nu^{5} - 132 \nu^{4} + \cdots - 6848 ) / 960 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 41 \nu^{11} - 60 \nu^{10} + 177 \nu^{9} + 140 \nu^{8} - 813 \nu^{7} + 480 \nu^{6} - 3084 \nu^{5} + \cdots + 27648 ) / 15360 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 47 \nu^{11} - 84 \nu^{10} + 217 \nu^{9} - 156 \nu^{8} + 715 \nu^{7} + 712 \nu^{6} - 3340 \nu^{5} + \cdots - 130048 ) / 15360 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 23 \nu^{11} - 60 \nu^{10} + 81 \nu^{9} - 116 \nu^{8} + 51 \nu^{7} + 480 \nu^{6} - 684 \nu^{5} + \cdots - 30720 ) / 7680 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 57 \nu^{11} + 204 \nu^{10} + 193 \nu^{9} - 1084 \nu^{8} - 125 \nu^{7} + 328 \nu^{6} + \cdots - 59392 ) / 15360 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 4 \nu^{11} - 3 \nu^{10} - 16 \nu^{9} + 3 \nu^{8} + 80 \nu^{7} + 89 \nu^{6} - 200 \nu^{5} - 444 \nu^{4} + \cdots - 8576 ) / 960 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 93 \nu^{11} - 84 \nu^{10} + 325 \nu^{9} - 220 \nu^{8} - 497 \nu^{7} - 1448 \nu^{6} - 604 \nu^{5} + \cdots + 48128 ) / 15360 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 97 \nu^{11} - 212 \nu^{10} + 215 \nu^{9} - 636 \nu^{8} + 709 \nu^{7} + 2096 \nu^{6} - 4244 \nu^{5} + \cdots - 184320 ) / 15360 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 245 \nu^{11} - 484 \nu^{10} + 355 \nu^{9} - 1068 \nu^{8} + 2153 \nu^{7} + 2032 \nu^{6} + \cdots - 420864 ) / 15360 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} + \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{11} - 3\beta_{10} - \beta_{9} - \beta_{6} - \beta_{5} + 2\beta_{4} + 3\beta_{3} - \beta_{2} + 6 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{11} - 9 \beta_{10} + \beta_{9} + 4 \beta_{8} + 2 \beta_{7} + 13 \beta_{6} - \beta_{5} + 7 \beta_{3} + \cdots + 6 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - \beta_{11} - 5 \beta_{10} - \beta_{9} - \beta_{8} - 2 \beta_{7} + 5 \beta_{6} - 3 \beta_{5} + \cdots - 13 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 17 \beta_{11} - 3 \beta_{10} - 5 \beta_{9} + 16 \beta_{8} - 2 \beta_{7} + 37 \beta_{6} - 3 \beta_{5} + \cdots + 14 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 3 \beta_{11} - 39 \beta_{10} + 7 \beta_{9} + 22 \beta_{8} - 12 \beta_{7} - 69 \beta_{6} + 27 \beta_{5} + \cdots + 352 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 17 \beta_{11} - 9 \beta_{10} + \beta_{9} + 6 \beta_{8} - 14 \beta_{7} + 20 \beta_{6} + 47 \beta_{5} + \cdots - 60 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 49 \beta_{11} + 11 \beta_{10} - 31 \beta_{9} - 128 \beta_{8} - 32 \beta_{7} + 73 \beta_{6} + \cdots + 66 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 47 \beta_{11} - 207 \beta_{10} - 265 \beta_{9} - 140 \beta_{8} + 102 \beta_{7} - 485 \beta_{6} + \cdots + 578 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 113 \beta_{11} - 243 \beta_{10} - 127 \beta_{9} - 59 \beta_{8} + 138 \beta_{7} + 759 \beta_{6} + \cdots + 505 ) / 2 \) Copy content Toggle raw display

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

Character values

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

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

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
−0.428316 − 1.95360i
−0.428316 + 1.95360i
−1.86176 − 0.730647i
−1.86176 + 0.730647i
1.71681 − 1.02595i
1.71681 + 1.02595i
−0.650488 − 1.89126i
−0.650488 + 1.89126i
1.98839 − 0.215211i
1.98839 + 0.215211i
1.23537 − 1.57285i
1.23537 + 1.57285i
−2.38191 − 1.52528i 1.51777 3.34703 + 7.26618i 0 −3.61520 − 2.31503i − 5.13620i 3.11063 − 22.4126i −24.6964 0
149.2 −2.38191 + 1.52528i 1.51777 3.34703 − 7.26618i 0 −3.61520 + 2.31503i 5.13620i 3.11063 + 22.4126i −24.6964 0
149.3 −1.13111 − 2.59241i 6.25785 −5.44116 + 5.86462i 0 −7.07834 − 16.2229i − 34.6280i 21.3581 + 7.47214i 12.1606 0
149.4 −1.13111 + 2.59241i 6.25785 −5.44116 − 5.86462i 0 −7.07834 + 16.2229i 34.6280i 21.3581 − 7.47214i 12.1606 0
149.5 0.690860 − 2.74276i −4.24443 −7.04543 − 3.78972i 0 −2.93231 + 11.6414i 14.6308i −15.2617 + 16.7057i −8.98481 0
149.6 0.690860 + 2.74276i −4.24443 −7.04543 + 3.78972i 0 −2.93231 − 11.6414i − 14.6308i −15.2617 − 16.7057i −8.98481 0
149.7 1.24077 − 2.54175i 0.888401 −4.92097 − 6.30746i 0 1.10230 − 2.25809i 26.6173i −22.1378 + 4.68175i −26.2107 0
149.8 1.24077 + 2.54175i 0.888401 −4.92097 + 6.30746i 0 1.10230 + 2.25809i − 26.6173i −22.1378 − 4.68175i −26.2107 0
149.9 1.77318 − 2.20360i 9.57890 −1.71169 − 7.81474i 0 16.9851 − 21.1081i − 21.5703i −20.2557 − 10.0850i 64.7554 0
149.10 1.77318 + 2.20360i 9.57890 −1.71169 + 7.81474i 0 16.9851 + 21.1081i 21.5703i −20.2557 + 10.0850i 64.7554 0
149.11 2.80822 − 0.337480i −7.99849 7.77221 − 1.89544i 0 −22.4615 + 2.69933i 9.93501i 21.1864 − 7.94578i 36.9759 0
149.12 2.80822 + 0.337480i −7.99849 7.77221 + 1.89544i 0 −22.4615 − 2.69933i − 9.93501i 21.1864 + 7.94578i 36.9759 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 149.12
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 200.4.f.c 12
4.b odd 2 1 800.4.f.b 12
5.b even 2 1 200.4.f.b 12
5.c odd 4 1 40.4.d.a ✓ 12
5.c odd 4 1 200.4.d.b 12
8.b even 2 1 200.4.f.b 12
8.d odd 2 1 800.4.f.c 12
15.e even 4 1 360.4.k.c 12
20.d odd 2 1 800.4.f.c 12
20.e even 4 1 160.4.d.a 12
20.e even 4 1 800.4.d.d 12
40.e odd 2 1 800.4.f.b 12
40.f even 2 1 inner 200.4.f.c 12
40.i odd 4 1 40.4.d.a ✓ 12
40.i odd 4 1 200.4.d.b 12
40.k even 4 1 160.4.d.a 12
40.k even 4 1 800.4.d.d 12
60.l odd 4 1 1440.4.k.c 12
80.i odd 4 1 1280.4.a.bb 6
80.j even 4 1 1280.4.a.ba 6
80.s even 4 1 1280.4.a.bd 6
80.t odd 4 1 1280.4.a.bc 6
120.q odd 4 1 1440.4.k.c 12
120.w even 4 1 360.4.k.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.4.d.a ✓ 12 5.c odd 4 1
40.4.d.a ✓ 12 40.i odd 4 1
160.4.d.a 12 20.e even 4 1
160.4.d.a 12 40.k even 4 1
200.4.d.b 12 5.c odd 4 1
200.4.d.b 12 40.i odd 4 1
200.4.f.b 12 5.b even 2 1
200.4.f.b 12 8.b even 2 1
200.4.f.c 12 1.a even 1 1 trivial
200.4.f.c 12 40.f even 2 1 inner
360.4.k.c 12 15.e even 4 1
360.4.k.c 12 120.w even 4 1
800.4.d.d 12 20.e even 4 1
800.4.d.d 12 40.k even 4 1
800.4.f.b 12 4.b odd 2 1
800.4.f.b 12 40.e odd 2 1
800.4.f.c 12 8.d odd 2 1
800.4.f.c 12 20.d odd 2 1
1280.4.a.ba 6 80.j even 4 1
1280.4.a.bb 6 80.i odd 4 1
1280.4.a.bc 6 80.t odd 4 1
1280.4.a.bd 6 80.s even 4 1
1440.4.k.c 12 60.l odd 4 1
1440.4.k.c 12 120.q odd 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 \(S_{4}^{\mathrm{new}}(200, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 6 T^{11} + \cdots + 262144 \) Copy content Toggle raw display
$3$ \( (T^{6} - 6 T^{5} + \cdots + 2744)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 220318448824384 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{6} - 5572 T^{4} + \cdots - 157790400)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 55\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 17\!\cdots\!24 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 77\!\cdots\!56 \) 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
$37$ \( (T^{6} + \cdots + 45951464886848)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots - 71667547865600)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 602 T^{5} + \cdots - 508806074248)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 51\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 929403110278976)^{2} \) 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
$67$ \( (T^{6} + \cdots - 93747278347656)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 36\!\cdots\!48)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 14\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots - 44\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots - 28\!\cdots\!68)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots - 62\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 11\!\cdots\!84 \) Copy content Toggle raw display
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