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Properties

Label 200.4.d.b.101.5
Level $200$
Weight $4$
Character 200.101
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(101,200)] 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("200.101"); 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, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 200 = 2^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 200.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,-2] 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: \( 2^{15}\cdot 5^{4} \)
Twist minimal: no (minimal twist has level 40)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 101.5
Root \(-0.428316 + 1.95360i\) of defining polynomial
Character \(\chi\) \(=\) 200.101
Dual form 200.4.d.b.101.6

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.52528 - 2.38191i) q^{2} -1.51777i q^{3} +(-3.34703 + 7.26618i) q^{4} +(-3.61520 + 2.31503i) q^{6} -5.13620 q^{7} +(22.4126 - 3.11063i) q^{8} +24.6964 q^{9} -31.3403i q^{11} +(11.0284 + 5.08003i) q^{12} -4.75340i q^{13} +(7.83414 + 12.2340i) q^{14} +(-41.5948 - 48.6403i) q^{16} -108.154 q^{17} +(-37.6689 - 58.8246i) q^{18} -89.8913i q^{19} +7.79558i q^{21} +(-74.6499 + 47.8028i) q^{22} +68.5157 q^{23} +(-4.72123 - 34.0172i) q^{24} +(-11.3222 + 7.25027i) q^{26} -78.4633i q^{27} +(17.1910 - 37.3205i) q^{28} +16.5719i q^{29} -300.523 q^{31} +(-52.4132 + 173.265i) q^{32} -47.5675 q^{33} +(164.966 + 257.614i) q^{34} +(-82.6595 + 179.448i) q^{36} -327.879i q^{37} +(-214.113 + 137.109i) q^{38} -7.21458 q^{39} -73.4968 q^{41} +(18.5684 - 11.8904i) q^{42} +0.836008i q^{43} +(227.724 + 104.897i) q^{44} +(-104.506 - 163.198i) q^{46} -228.335 q^{47} +(-73.8249 + 63.1314i) q^{48} -316.619 q^{49} +164.153i q^{51} +(34.5391 + 15.9098i) q^{52} +647.393i q^{53} +(-186.893 + 119.679i) q^{54} +(-115.115 + 15.9768i) q^{56} -136.434 q^{57} +(39.4728 - 25.2768i) q^{58} -753.676i q^{59} -290.838i q^{61} +(458.383 + 715.821i) q^{62} -126.845 q^{63} +(492.648 - 139.435i) q^{64} +(72.5538 + 113.302i) q^{66} -801.801i q^{67} +(361.995 - 785.868i) q^{68} -103.991i q^{69} +767.674 q^{71} +(553.509 - 76.8213i) q^{72} +48.3194 q^{73} +(-780.980 + 500.108i) q^{74} +(653.166 + 300.869i) q^{76} +160.970i q^{77} +(11.0043 + 17.1845i) q^{78} +451.701 q^{79} +547.712 q^{81} +(112.103 + 175.063i) q^{82} -976.099i q^{83} +(-56.6441 - 26.0920i) q^{84} +(1.99130 - 1.27515i) q^{86} +25.1524 q^{87} +(-97.4881 - 702.417i) q^{88} -1204.25 q^{89} +24.4144i q^{91} +(-229.324 + 497.847i) q^{92} +456.126i q^{93} +(348.275 + 543.873i) q^{94} +(262.977 + 79.5513i) q^{96} +559.147 q^{97} +(482.934 + 754.161i) q^{98} -773.992i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{2} + 16 q^{4} - 36 q^{6} - 28 q^{7} + 40 q^{8} - 108 q^{9} - 188 q^{12} + 68 q^{14} - 56 q^{16} + 206 q^{18} + 164 q^{22} - 604 q^{23} + 360 q^{24} - 308 q^{26} + 436 q^{28} - 264 q^{31} - 72 q^{32}+ \cdots + 7266 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.52528 − 2.38191i −0.539269 − 0.842134i
\(3\) − 1.51777i − 0.292095i −0.989278 − 0.146048i \(-0.953345\pi\)
0.989278 − 0.146048i \(-0.0466553\pi\)
\(4\) −3.34703 + 7.26618i −0.418379 + 0.908273i
\(5\) 0 0
\(6\) −3.61520 + 2.31503i −0.245983 + 0.157518i
\(7\) −5.13620 −0.277328 −0.138664 − 0.990339i \(-0.544281\pi\)
−0.138664 + 0.990339i \(0.544281\pi\)
\(8\) 22.4126 − 3.11063i 0.990506 − 0.137472i
\(9\) 24.6964 0.914680
\(10\) 0 0
\(11\) − 31.3403i − 0.859042i −0.903057 − 0.429521i \(-0.858682\pi\)
0.903057 − 0.429521i \(-0.141318\pi\)
\(12\) 11.0284 + 5.08003i 0.265302 + 0.122207i
\(13\) − 4.75340i − 0.101412i −0.998714 − 0.0507060i \(-0.983853\pi\)
0.998714 − 0.0507060i \(-0.0161471\pi\)
\(14\) 7.83414 + 12.2340i 0.149555 + 0.233548i
\(15\) 0 0
\(16\) −41.5948 − 48.6403i −0.649918 − 0.760004i
\(17\) −108.154 −1.54301 −0.771507 − 0.636221i \(-0.780497\pi\)
−0.771507 + 0.636221i \(0.780497\pi\)
\(18\) −37.6689 − 58.8246i −0.493258 − 0.770283i
\(19\) − 89.8913i − 1.08539i −0.839929 − 0.542697i \(-0.817403\pi\)
0.839929 − 0.542697i \(-0.182597\pi\)
\(20\) 0 0
\(21\) 7.79558i 0.0810064i
\(22\) −74.6499 + 47.8028i −0.723428 + 0.463254i
\(23\) 68.5157 0.621152 0.310576 − 0.950548i \(-0.399478\pi\)
0.310576 + 0.950548i \(0.399478\pi\)
\(24\) −4.72123 − 34.0172i −0.0401549 − 0.289322i
\(25\) 0 0
\(26\) −11.3222 + 7.25027i −0.0854025 + 0.0546883i
\(27\) − 78.4633i − 0.559269i
\(28\) 17.1910 − 37.3205i 0.116028 − 0.251890i
\(29\) 16.5719i 0.106115i 0.998591 + 0.0530573i \(0.0168966\pi\)
−0.998591 + 0.0530573i \(0.983103\pi\)
\(30\) 0 0
\(31\) −300.523 −1.74115 −0.870574 − 0.492037i \(-0.836252\pi\)
−0.870574 + 0.492037i \(0.836252\pi\)
\(32\) −52.4132 + 173.265i −0.289545 + 0.957164i
\(33\) −47.5675 −0.250922
\(34\) 164.966 + 257.614i 0.832099 + 1.29942i
\(35\) 0 0
\(36\) −82.6595 + 179.448i −0.382683 + 0.830779i
\(37\) − 327.879i − 1.45684i −0.685132 − 0.728419i \(-0.740256\pi\)
0.685132 − 0.728419i \(-0.259744\pi\)
\(38\) −214.113 + 137.109i −0.914046 + 0.585318i
\(39\) −7.21458 −0.0296220
\(40\) 0 0
\(41\) −73.4968 −0.279958 −0.139979 − 0.990154i \(-0.544703\pi\)
−0.139979 + 0.990154i \(0.544703\pi\)
\(42\) 18.5684 − 11.8904i 0.0682182 − 0.0436842i
\(43\) 0.836008i 0.00296489i 0.999999 + 0.00148244i \(0.000471876\pi\)
−0.999999 + 0.00148244i \(0.999528\pi\)
\(44\) 227.724 + 104.897i 0.780244 + 0.359405i
\(45\) 0 0
\(46\) −104.506 − 163.198i −0.334968 − 0.523093i
\(47\) −228.335 −0.708639 −0.354319 − 0.935124i \(-0.615287\pi\)
−0.354319 + 0.935124i \(0.615287\pi\)
\(48\) −73.8249 + 63.1314i −0.221994 + 0.189838i
\(49\) −316.619 −0.923089
\(50\) 0 0
\(51\) 164.153i 0.450707i
\(52\) 34.5391 + 15.9098i 0.0921097 + 0.0424286i
\(53\) 647.393i 1.67785i 0.544244 + 0.838927i \(0.316817\pi\)
−0.544244 + 0.838927i \(0.683183\pi\)
\(54\) −186.893 + 119.679i −0.470980 + 0.301596i
\(55\) 0 0
\(56\) −115.115 + 15.9768i −0.274695 + 0.0381248i
\(57\) −136.434 −0.317038
\(58\) 39.4728 − 25.2768i 0.0893627 − 0.0572243i
\(59\) − 753.676i − 1.66305i −0.555484 − 0.831527i \(-0.687467\pi\)
0.555484 − 0.831527i \(-0.312533\pi\)
\(60\) 0 0
\(61\) − 290.838i − 0.610459i −0.952279 − 0.305229i \(-0.901267\pi\)
0.952279 − 0.305229i \(-0.0987331\pi\)
\(62\) 458.383 + 715.821i 0.938946 + 1.46628i
\(63\) −126.845 −0.253667
\(64\) 492.648 − 139.435i 0.962203 − 0.272333i
\(65\) 0 0
\(66\) 72.5538 + 113.302i 0.135314 + 0.211310i
\(67\) − 801.801i − 1.46202i −0.682364 − 0.731012i \(-0.739048\pi\)
0.682364 − 0.731012i \(-0.260952\pi\)
\(68\) 361.995 − 785.868i 0.645565 − 1.40148i
\(69\) − 103.991i − 0.181436i
\(70\) 0 0
\(71\) 767.674 1.28318 0.641592 − 0.767046i \(-0.278274\pi\)
0.641592 + 0.767046i \(0.278274\pi\)
\(72\) 553.509 − 76.8213i 0.905996 − 0.125743i
\(73\) 48.3194 0.0774707 0.0387353 − 0.999250i \(-0.487667\pi\)
0.0387353 + 0.999250i \(0.487667\pi\)
\(74\) −780.980 + 500.108i −1.22685 + 0.785627i
\(75\) 0 0
\(76\) 653.166 + 300.869i 0.985833 + 0.454106i
\(77\) 160.970i 0.238237i
\(78\) 11.0043 + 17.1845i 0.0159742 + 0.0249457i
\(79\) 451.701 0.643296 0.321648 − 0.946859i \(-0.395763\pi\)
0.321648 + 0.946859i \(0.395763\pi\)
\(80\) 0 0
\(81\) 547.712 0.751320
\(82\) 112.103 + 175.063i 0.150973 + 0.235762i
\(83\) − 976.099i − 1.29085i −0.763822 − 0.645427i \(-0.776680\pi\)
0.763822 − 0.645427i \(-0.223320\pi\)
\(84\) −56.6441 − 26.0920i −0.0735759 − 0.0338914i
\(85\) 0 0
\(86\) 1.99130 − 1.27515i 0.00249683 − 0.00159887i
\(87\) 25.1524 0.0309956
\(88\) −97.4881 − 702.417i −0.118094 − 0.850886i
\(89\) −1204.25 −1.43428 −0.717139 − 0.696930i \(-0.754549\pi\)
−0.717139 + 0.696930i \(0.754549\pi\)
\(90\) 0 0
\(91\) 24.4144i 0.0281244i
\(92\) −229.324 + 497.847i −0.259877 + 0.564176i
\(93\) 456.126i 0.508581i
\(94\) 348.275 + 543.873i 0.382147 + 0.596769i
\(95\) 0 0
\(96\) 262.977 + 79.5513i 0.279583 + 0.0845747i
\(97\) 559.147 0.585287 0.292643 − 0.956222i \(-0.405465\pi\)
0.292643 + 0.956222i \(0.405465\pi\)
\(98\) 482.934 + 754.161i 0.497793 + 0.777364i
\(99\) − 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.d.b.101.5 12
4.3 odd 2 800.4.d.d.401.8 12
5.2 odd 4 200.4.f.b.149.11 12
5.3 odd 4 200.4.f.c.149.2 12
5.4 even 2 40.4.d.a.21.8 yes 12
8.3 odd 2 800.4.d.d.401.5 12
8.5 even 2 inner 200.4.d.b.101.6 12
15.14 odd 2 360.4.k.c.181.5 12
20.3 even 4 800.4.f.b.49.5 12
20.7 even 4 800.4.f.c.49.8 12
20.19 odd 2 160.4.d.a.81.5 12
40.3 even 4 800.4.f.c.49.7 12
40.13 odd 4 200.4.f.b.149.12 12
40.19 odd 2 160.4.d.a.81.8 12
40.27 even 4 800.4.f.b.49.6 12
40.29 even 2 40.4.d.a.21.7 ✓ 12
40.37 odd 4 200.4.f.c.149.1 12
60.59 even 2 1440.4.k.c.721.4 12
80.19 odd 4 1280.4.a.bd.1.4 6
80.29 even 4 1280.4.a.bb.1.3 6
80.59 odd 4 1280.4.a.ba.1.3 6
80.69 even 4 1280.4.a.bc.1.4 6
120.29 odd 2 360.4.k.c.181.6 12
120.59 even 2 1440.4.k.c.721.10 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.4.d.a.21.7 ✓ 12 40.29 even 2
40.4.d.a.21.8 yes 12 5.4 even 2
160.4.d.a.81.5 12 20.19 odd 2
160.4.d.a.81.8 12 40.19 odd 2
200.4.d.b.101.5 12 1.1 even 1 trivial
200.4.d.b.101.6 12 8.5 even 2 inner
200.4.f.b.149.11 12 5.2 odd 4
200.4.f.b.149.12 12 40.13 odd 4
200.4.f.c.149.1 12 40.37 odd 4
200.4.f.c.149.2 12 5.3 odd 4
360.4.k.c.181.5 12 15.14 odd 2
360.4.k.c.181.6 12 120.29 odd 2
800.4.d.d.401.5 12 8.3 odd 2
800.4.d.d.401.8 12 4.3 odd 2
800.4.f.b.49.5 12 20.3 even 4
800.4.f.b.49.6 12 40.27 even 4
800.4.f.c.49.7 12 40.3 even 4
800.4.f.c.49.8 12 20.7 even 4
1280.4.a.ba.1.3 6 80.59 odd 4
1280.4.a.bb.1.3 6 80.29 even 4
1280.4.a.bc.1.4 6 80.69 even 4
1280.4.a.bd.1.4 6 80.19 odd 4
1440.4.k.c.721.4 12 60.59 even 2
1440.4.k.c.721.10 12 120.59 even 2