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Properties

Label 200.4.f.c.149.2
Level $200$
Weight $4$
Character 200.149
Analytic conductor $11.800$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [200,4,Mod(149,200)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://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);
 
Copy content sage: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")
 
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

Copy content comment:select newform
 
Copy content sage:traces = [12,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.8003820011\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content 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: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 149.2
Root \(-0.428316 + 1.95360i\) of defining polynomial
Character \(\chi\) \(=\) 200.149
Dual form 200.4.f.c.149.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-2.38191 + 1.52528i) q^{2} +1.51777 q^{3} +(3.34703 - 7.26618i) q^{4} +(-3.61520 + 2.31503i) q^{6} +5.13620i q^{7} +(3.11063 + 22.4126i) q^{8} -24.6964 q^{9} -31.3403i q^{11} +(5.08003 - 11.0284i) q^{12} +4.75340 q^{13} +(-7.83414 - 12.2340i) q^{14} +(-41.5948 - 48.6403i) q^{16} +108.154i q^{17} +(58.8246 - 37.6689i) q^{18} +89.8913i q^{19} +7.79558i q^{21} +(47.8028 + 74.6499i) q^{22} +68.5157i q^{23} +(4.72123 + 34.0172i) q^{24} +(-11.3222 + 7.25027i) q^{26} -78.4633 q^{27} +(37.3205 + 17.1910i) q^{28} -16.5719i q^{29} -300.523 q^{31} +(173.265 + 52.4132i) q^{32} -47.5675i q^{33} +(-164.966 - 257.614i) q^{34} +(-82.6595 + 179.448i) q^{36} -327.879 q^{37} +(-137.109 - 214.113i) q^{38} +7.21458 q^{39} -73.4968 q^{41} +(-11.8904 - 18.5684i) q^{42} -0.836008 q^{43} +(-227.724 - 104.897i) q^{44} +(-104.506 - 163.198i) q^{46} +228.335i q^{47} +(-63.1314 - 73.8249i) q^{48} +316.619 q^{49} +164.153i q^{51} +(15.9098 - 34.5391i) q^{52} -647.393 q^{53} +(186.893 - 119.679i) q^{54} +(-115.115 + 15.9768i) q^{56} +136.434i q^{57} +(25.2768 + 39.4728i) q^{58} +753.676i q^{59} -290.838i q^{61} +(715.821 - 458.383i) q^{62} -126.845i q^{63} +(-492.648 + 139.435i) q^{64} +(72.5538 + 113.302i) q^{66} -801.801 q^{67} +(785.868 + 361.995i) q^{68} +103.991i q^{69} +767.674 q^{71} +(-76.8213 - 553.509i) q^{72} +48.3194i q^{73} +(780.980 - 500.108i) q^{74} +(653.166 + 300.869i) q^{76} +160.970 q^{77} +(-17.1845 + 11.0043i) q^{78} -451.701 q^{79} +547.712 q^{81} +(175.063 - 112.103i) q^{82} +976.099 q^{83} +(56.6441 + 26.0920i) q^{84} +(1.99130 - 1.27515i) q^{86} -25.1524i q^{87} +(702.417 - 97.4881i) q^{88} +1204.25 q^{89} +24.4144i q^{91} +(497.847 + 229.324i) q^{92} -456.126 q^{93} +(-348.275 - 543.873i) q^{94} +(262.977 + 79.5513i) q^{96} -559.147i q^{97} +(-754.161 + 482.934i) q^{98} +773.992i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{2} + 12 q^{3} - 16 q^{4} - 36 q^{6} - 24 q^{8} + 108 q^{9} - 164 q^{12} - 68 q^{14} - 56 q^{16} + 450 q^{18} + 492 q^{22} - 360 q^{24} - 308 q^{26} + 432 q^{27} + 628 q^{28} - 264 q^{31} + 856 q^{32}+ \cdots + 638 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.38191 + 1.52528i −0.842134 + 0.539269i
\(3\) 1.51777 0.292095 0.146048 − 0.989278i \(-0.453345\pi\)
0.146048 + 0.989278i \(0.453345\pi\)
\(4\) 3.34703 − 7.26618i 0.418379 − 0.908273i
\(5\) 0 0
\(6\) −3.61520 + 2.31503i −0.245983 + 0.157518i
\(7\) 5.13620i 0.277328i 0.990339 + 0.138664i \(0.0442809\pi\)
−0.990339 + 0.138664i \(0.955719\pi\)
\(8\) 3.11063 + 22.4126i 0.137472 + 0.990506i
\(9\) −24.6964 −0.914680
\(10\) 0 0
\(11\) − 31.3403i − 0.859042i −0.903057 − 0.429521i \(-0.858682\pi\)
0.903057 − 0.429521i \(-0.141318\pi\)
\(12\) 5.08003 − 11.0284i 0.122207 − 0.265302i
\(13\) 4.75340 0.101412 0.0507060 − 0.998714i \(-0.483853\pi\)
0.0507060 + 0.998714i \(0.483853\pi\)
\(14\) −7.83414 − 12.2340i −0.149555 − 0.233548i
\(15\) 0 0
\(16\) −41.5948 − 48.6403i −0.649918 − 0.760004i
\(17\) 108.154i 1.54301i 0.636221 + 0.771507i \(0.280497\pi\)
−0.636221 + 0.771507i \(0.719503\pi\)
\(18\) 58.8246 − 37.6689i 0.770283 − 0.493258i
\(19\) 89.8913i 1.08539i 0.839929 + 0.542697i \(0.182597\pi\)
−0.839929 + 0.542697i \(0.817403\pi\)
\(20\) 0 0
\(21\) 7.79558i 0.0810064i
\(22\) 47.8028 + 74.6499i 0.463254 + 0.723428i
\(23\) 68.5157i 0.621152i 0.950548 + 0.310576i \(0.100522\pi\)
−0.950548 + 0.310576i \(0.899478\pi\)
\(24\) 4.72123 + 34.0172i 0.0401549 + 0.289322i
\(25\) 0 0
\(26\) −11.3222 + 7.25027i −0.0854025 + 0.0546883i
\(27\) −78.4633 −0.559269
\(28\) 37.3205 + 17.1910i 0.251890 + 0.116028i
\(29\) − 16.5719i − 0.106115i −0.998591 − 0.0530573i \(-0.983103\pi\)
0.998591 − 0.0530573i \(-0.0168966\pi\)
\(30\) 0 0
\(31\) −300.523 −1.74115 −0.870574 − 0.492037i \(-0.836252\pi\)
−0.870574 + 0.492037i \(0.836252\pi\)
\(32\) 173.265 + 52.4132i 0.957164 + 0.289545i
\(33\) − 47.5675i − 0.250922i
\(34\) −164.966 − 257.614i −0.832099 − 1.29942i
\(35\) 0 0
\(36\) −82.6595 + 179.448i −0.382683 + 0.830779i
\(37\) −327.879 −1.45684 −0.728419 − 0.685132i \(-0.759744\pi\)
−0.728419 + 0.685132i \(0.759744\pi\)
\(38\) −137.109 − 214.113i −0.585318 − 0.914046i
\(39\) 7.21458 0.0296220
\(40\) 0 0
\(41\) −73.4968 −0.279958 −0.139979 − 0.990154i \(-0.544703\pi\)
−0.139979 + 0.990154i \(0.544703\pi\)
\(42\) −11.8904 − 18.5684i −0.0436842 − 0.0682182i
\(43\) −0.836008 −0.00296489 −0.00148244 − 0.999999i \(-0.500472\pi\)
−0.00148244 + 0.999999i \(0.500472\pi\)
\(44\) −227.724 − 104.897i −0.780244 − 0.359405i
\(45\) 0 0
\(46\) −104.506 − 163.198i −0.334968 − 0.523093i
\(47\) 228.335i 0.708639i 0.935124 + 0.354319i \(0.115287\pi\)
−0.935124 + 0.354319i \(0.884713\pi\)
\(48\) −63.1314 − 73.8249i −0.189838 − 0.221994i
\(49\) 316.619 0.923089
\(50\) 0 0
\(51\) 164.153i 0.450707i
\(52\) 15.9098 − 34.5391i 0.0424286 − 0.0921097i
\(53\) −647.393 −1.67785 −0.838927 − 0.544244i \(-0.816817\pi\)
−0.838927 + 0.544244i \(0.816817\pi\)
\(54\) 186.893 − 119.679i 0.470980 − 0.301596i
\(55\) 0 0
\(56\) −115.115 + 15.9768i −0.274695 + 0.0381248i
\(57\) 136.434i 0.317038i
\(58\) 25.2768 + 39.4728i 0.0572243 + 0.0893627i
\(59\) 753.676i 1.66305i 0.555484 + 0.831527i \(0.312533\pi\)
−0.555484 + 0.831527i \(0.687467\pi\)
\(60\) 0 0
\(61\) − 290.838i − 0.610459i −0.952279 − 0.305229i \(-0.901267\pi\)
0.952279 − 0.305229i \(-0.0987331\pi\)
\(62\) 715.821 − 458.383i 1.46628 − 0.938946i
\(63\) − 126.845i − 0.253667i
\(64\) −492.648 + 139.435i −0.962203 + 0.272333i
\(65\) 0 0
\(66\) 72.5538 + 113.302i 0.135314 + 0.211310i
\(67\) −801.801 −1.46202 −0.731012 − 0.682364i \(-0.760952\pi\)
−0.731012 + 0.682364i \(0.760952\pi\)
\(68\) 785.868 + 361.995i 1.40148 + 0.645565i
\(69\) 103.991i 0.181436i
\(70\) 0 0
\(71\) 767.674 1.28318 0.641592 − 0.767046i \(-0.278274\pi\)
0.641592 + 0.767046i \(0.278274\pi\)
\(72\) −76.8213 − 553.509i −0.125743 − 0.905996i
\(73\) 48.3194i 0.0774707i 0.999250 + 0.0387353i \(0.0123329\pi\)
−0.999250 + 0.0387353i \(0.987667\pi\)
\(74\) 780.980 − 500.108i 1.22685 − 0.785627i
\(75\) 0 0
\(76\) 653.166 + 300.869i 0.985833 + 0.454106i
\(77\) 160.970 0.238237
\(78\) −17.1845 + 11.0043i −0.0249457 + 0.0159742i
\(79\) −451.701 −0.643296 −0.321648 − 0.946859i \(-0.604237\pi\)
−0.321648 + 0.946859i \(0.604237\pi\)
\(80\) 0 0
\(81\) 547.712 0.751320
\(82\) 175.063 − 112.103i 0.235762 − 0.150973i
\(83\) 976.099 1.29085 0.645427 − 0.763822i \(-0.276680\pi\)
0.645427 + 0.763822i \(0.276680\pi\)
\(84\) 56.6441 + 26.0920i 0.0735759 + 0.0338914i
\(85\) 0 0
\(86\) 1.99130 − 1.27515i 0.00249683 − 0.00159887i
\(87\) − 25.1524i − 0.0309956i
\(88\) 702.417 − 97.4881i 0.850886 − 0.118094i
\(89\) 1204.25 1.43428 0.717139 − 0.696930i \(-0.245451\pi\)
0.717139 + 0.696930i \(0.245451\pi\)
\(90\) 0 0
\(91\) 24.4144i 0.0281244i
\(92\) 497.847 + 229.324i 0.564176 + 0.259877i
\(93\) −456.126 −0.508581
\(94\) −348.275 − 543.873i −0.382147 − 0.596769i
\(95\) 0 0
\(96\) 262.977 + 79.5513i 0.279583 + 0.0845747i
\(97\) − 559.147i − 0.585287i −0.956222 − 0.292643i \(-0.905465\pi\)
0.956222 − 0.292643i \(-0.0945348\pi\)
\(98\) −754.161 + 482.934i −0.777364 + 0.497793i
\(99\) 773.992i 0.785749i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 200.4.f.c.149.2 12
4.3 odd 2 800.4.f.b.49.5 12
5.2 odd 4 200.4.d.b.101.5 12
5.3 odd 4 40.4.d.a.21.8 yes 12
5.4 even 2 200.4.f.b.149.11 12
8.3 odd 2 800.4.f.c.49.7 12
8.5 even 2 200.4.f.b.149.12 12
15.8 even 4 360.4.k.c.181.5 12
20.3 even 4 160.4.d.a.81.5 12
20.7 even 4 800.4.d.d.401.8 12
20.19 odd 2 800.4.f.c.49.8 12
40.3 even 4 160.4.d.a.81.8 12
40.13 odd 4 40.4.d.a.21.7 ✓ 12
40.19 odd 2 800.4.f.b.49.6 12
40.27 even 4 800.4.d.d.401.5 12
40.29 even 2 inner 200.4.f.c.149.1 12
40.37 odd 4 200.4.d.b.101.6 12
60.23 odd 4 1440.4.k.c.721.4 12
80.3 even 4 1280.4.a.bd.1.4 6
80.13 odd 4 1280.4.a.bb.1.3 6
80.43 even 4 1280.4.a.ba.1.3 6
80.53 odd 4 1280.4.a.bc.1.4 6
120.53 even 4 360.4.k.c.181.6 12
120.83 odd 4 1440.4.k.c.721.10 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.4.d.a.21.7 ✓ 12 40.13 odd 4
40.4.d.a.21.8 yes 12 5.3 odd 4
160.4.d.a.81.5 12 20.3 even 4
160.4.d.a.81.8 12 40.3 even 4
200.4.d.b.101.5 12 5.2 odd 4
200.4.d.b.101.6 12 40.37 odd 4
200.4.f.b.149.11 12 5.4 even 2
200.4.f.b.149.12 12 8.5 even 2
200.4.f.c.149.1 12 40.29 even 2 inner
200.4.f.c.149.2 12 1.1 even 1 trivial
360.4.k.c.181.5 12 15.8 even 4
360.4.k.c.181.6 12 120.53 even 4
800.4.d.d.401.5 12 40.27 even 4
800.4.d.d.401.8 12 20.7 even 4
800.4.f.b.49.5 12 4.3 odd 2
800.4.f.b.49.6 12 40.19 odd 2
800.4.f.c.49.7 12 8.3 odd 2
800.4.f.c.49.8 12 20.19 odd 2
1280.4.a.ba.1.3 6 80.43 even 4
1280.4.a.bb.1.3 6 80.13 odd 4
1280.4.a.bc.1.4 6 80.53 odd 4
1280.4.a.bd.1.4 6 80.3 even 4
1440.4.k.c.721.4 12 60.23 odd 4
1440.4.k.c.721.10 12 120.83 odd 4