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Properties

Label 200.4.d.b.101.1
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 101.1
Root \(1.71681 - 1.02595i\) of defining polynomial
Character \(\chi\) \(=\) 200.101
Dual form 200.4.d.b.101.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-2.74276 - 0.690860i) q^{2} -4.24443i q^{3} +(7.04543 + 3.78972i) q^{4} +(-2.93231 + 11.6414i) q^{6} +14.6308 q^{7} +(-16.7057 - 15.2617i) q^{8} +8.98481 q^{9} -14.9969i q^{11} +(16.0852 - 29.9038i) q^{12} +85.6955i q^{13} +(-40.1288 - 10.1078i) q^{14} +(35.2761 + 53.4004i) q^{16} +91.9247 q^{17} +(-24.6432 - 6.20724i) q^{18} +60.3737i q^{19} -62.0995i q^{21} +(-10.3608 + 41.1330i) q^{22} -1.33730 q^{23} +(-64.7771 + 70.9063i) q^{24} +(59.2035 - 235.042i) q^{26} -152.735i q^{27} +(103.080 + 55.4467i) q^{28} -25.5118i q^{29} +73.4354 q^{31} +(-59.8615 - 170.835i) q^{32} -63.6535 q^{33} +(-252.127 - 63.5070i) q^{34} +(63.3018 + 34.0499i) q^{36} -211.259i q^{37} +(41.7098 - 165.590i) q^{38} +363.728 q^{39} +330.839 q^{41} +(-42.9020 + 170.324i) q^{42} -388.500i q^{43} +(56.8342 - 105.660i) q^{44} +(3.66788 + 0.923883i) q^{46} +550.348 q^{47} +(226.654 - 149.727i) q^{48} -128.939 q^{49} -390.168i q^{51} +(-324.762 + 603.761i) q^{52} +187.705i q^{53} +(-105.518 + 418.915i) q^{54} +(-244.419 - 223.291i) q^{56} +256.252 q^{57} +(-17.6251 + 69.9727i) q^{58} -779.090i q^{59} +358.850i q^{61} +(-201.415 - 50.7335i) q^{62} +131.455 q^{63} +(46.1625 + 509.915i) q^{64} +(174.586 + 43.9756i) q^{66} +283.674i q^{67} +(647.648 + 348.369i) q^{68} +5.67606i q^{69} -534.811 q^{71} +(-150.098 - 137.123i) q^{72} -1016.16 q^{73} +(-145.950 + 579.431i) q^{74} +(-228.799 + 425.359i) q^{76} -219.418i q^{77} +(-997.618 - 251.285i) q^{78} +1119.59 q^{79} -405.683 q^{81} +(-907.411 - 228.563i) q^{82} +1190.49i q^{83} +(235.340 - 437.518i) q^{84} +(-268.399 + 1065.56i) q^{86} -108.283 q^{87} +(-228.879 + 250.535i) q^{88} -398.940 q^{89} +1253.80i q^{91} +(-9.42182 - 5.06797i) q^{92} -311.691i q^{93} +(-1509.47 - 380.213i) q^{94} +(-725.097 + 254.078i) q^{96} -278.022 q^{97} +(353.648 + 89.0787i) q^{98} -134.745i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{2} + 16 q^{4} - 36 q^{6} - 28 q^{7} + 40 q^{8} - 108 q^{9} - 188 q^{12} + 68 q^{14} - 56 q^{16} + 206 q^{18} + 164 q^{22} - 604 q^{23} + 360 q^{24} - 308 q^{26} + 436 q^{28} - 264 q^{31} - 72 q^{32}+ \cdots + 7266 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.74276 − 0.690860i −0.969711 − 0.244256i
\(3\) − 4.24443i − 0.816841i −0.912794 − 0.408420i \(-0.866080\pi\)
0.912794 − 0.408420i \(-0.133920\pi\)
\(4\) 7.04543 + 3.78972i 0.880678 + 0.473715i
\(5\) 0 0
\(6\) −2.93231 + 11.6414i −0.199518 + 0.792100i
\(7\) 14.6308 0.789990 0.394995 − 0.918683i \(-0.370746\pi\)
0.394995 + 0.918683i \(0.370746\pi\)
\(8\) −16.7057 − 15.2617i −0.738296 − 0.674477i
\(9\) 8.98481 0.332771
\(10\) 0 0
\(11\) − 14.9969i − 0.411068i −0.978650 − 0.205534i \(-0.934107\pi\)
0.978650 − 0.205534i \(-0.0658931\pi\)
\(12\) 16.0852 − 29.9038i 0.386950 − 0.719374i
\(13\) 85.6955i 1.82828i 0.405398 + 0.914140i \(0.367133\pi\)
−0.405398 + 0.914140i \(0.632867\pi\)
\(14\) −40.1288 − 10.1078i −0.766062 − 0.192960i
\(15\) 0 0
\(16\) 35.2761 + 53.4004i 0.551188 + 0.834381i
\(17\) 91.9247 1.31147 0.655735 − 0.754991i \(-0.272359\pi\)
0.655735 + 0.754991i \(0.272359\pi\)
\(18\) −24.6432 − 6.20724i −0.322691 − 0.0812812i
\(19\) 60.3737i 0.728983i 0.931207 + 0.364492i \(0.118757\pi\)
−0.931207 + 0.364492i \(0.881243\pi\)
\(20\) 0 0
\(21\) − 62.0995i − 0.645296i
\(22\) −10.3608 + 41.1330i −0.100406 + 0.398617i
\(23\) −1.33730 −0.0121237 −0.00606186 − 0.999982i \(-0.501930\pi\)
−0.00606186 + 0.999982i \(0.501930\pi\)
\(24\) −64.7771 + 70.9063i −0.550941 + 0.603070i
\(25\) 0 0
\(26\) 59.2035 − 235.042i 0.446568 − 1.77290i
\(27\) − 152.735i − 1.08866i
\(28\) 103.080 + 55.4467i 0.695727 + 0.374230i
\(29\) − 25.5118i − 0.163360i −0.996659 − 0.0816798i \(-0.973972\pi\)
0.996659 − 0.0816798i \(-0.0260285\pi\)
\(30\) 0 0
\(31\) 73.4354 0.425464 0.212732 − 0.977111i \(-0.431764\pi\)
0.212732 + 0.977111i \(0.431764\pi\)
\(32\) −59.8615 − 170.835i −0.330691 − 0.943739i
\(33\) −63.6535 −0.335777
\(34\) −252.127 − 63.5070i −1.27175 − 0.320334i
\(35\) 0 0
\(36\) 63.3018 + 34.0499i 0.293064 + 0.157638i
\(37\) − 211.259i − 0.938668i −0.883021 − 0.469334i \(-0.844494\pi\)
0.883021 − 0.469334i \(-0.155506\pi\)
\(38\) 41.7098 − 165.590i 0.178058 − 0.706903i
\(39\) 363.728 1.49341
\(40\) 0 0
\(41\) 330.839 1.26020 0.630102 − 0.776512i \(-0.283013\pi\)
0.630102 + 0.776512i \(0.283013\pi\)
\(42\) −42.9020 + 170.324i −0.157617 + 0.625751i
\(43\) − 388.500i − 1.37781i −0.724853 − 0.688904i \(-0.758092\pi\)
0.724853 − 0.688904i \(-0.241908\pi\)
\(44\) 56.8342 − 105.660i 0.194729 − 0.362019i
\(45\) 0 0
\(46\) 3.66788 + 0.923883i 0.0117565 + 0.00296129i
\(47\) 550.348 1.70801 0.854005 − 0.520265i \(-0.174167\pi\)
0.854005 + 0.520265i \(0.174167\pi\)
\(48\) 226.654 − 149.727i 0.681556 − 0.450233i
\(49\) −128.939 −0.375915
\(50\) 0 0
\(51\) − 390.168i − 1.07126i
\(52\) −324.762 + 603.761i −0.866084 + 1.61013i
\(53\) 187.705i 0.486477i 0.969966 + 0.243239i \(0.0782098\pi\)
−0.969966 + 0.243239i \(0.921790\pi\)
\(54\) −105.518 + 418.915i −0.265912 + 1.05569i
\(55\) 0 0
\(56\) −244.419 − 223.291i −0.583246 − 0.532830i
\(57\) 256.252 0.595463
\(58\) −17.6251 + 69.9727i −0.0399015 + 0.158412i
\(59\) − 779.090i − 1.71913i −0.511023 − 0.859567i \(-0.670733\pi\)
0.511023 − 0.859567i \(-0.329267\pi\)
\(60\) 0 0
\(61\) 358.850i 0.753215i 0.926373 + 0.376607i \(0.122909\pi\)
−0.926373 + 0.376607i \(0.877091\pi\)
\(62\) −201.415 − 50.7335i −0.412577 − 0.103922i
\(63\) 131.455 0.262886
\(64\) 46.1625 + 509.915i 0.0901611 + 0.995927i
\(65\) 0 0
\(66\) 174.586 + 43.9756i 0.325607 + 0.0820155i
\(67\) 283.674i 0.517258i 0.965977 + 0.258629i \(0.0832706\pi\)
−0.965977 + 0.258629i \(0.916729\pi\)
\(68\) 647.648 + 348.369i 1.15498 + 0.621263i
\(69\) 5.67606i 0.00990315i
\(70\) 0 0
\(71\) −534.811 −0.893949 −0.446975 − 0.894547i \(-0.647499\pi\)
−0.446975 + 0.894547i \(0.647499\pi\)
\(72\) −150.098 − 137.123i −0.245683 − 0.224446i
\(73\) −1016.16 −1.62921 −0.814607 − 0.580014i \(-0.803047\pi\)
−0.814607 + 0.580014i \(0.803047\pi\)
\(74\) −145.950 + 579.431i −0.229275 + 0.910237i
\(75\) 0 0
\(76\) −228.799 + 425.359i −0.345330 + 0.642000i
\(77\) − 219.418i − 0.324740i
\(78\) −997.618 − 251.285i −1.44818 − 0.364775i
\(79\) 1119.59 1.59447 0.797237 − 0.603666i \(-0.206294\pi\)
0.797237 + 0.603666i \(0.206294\pi\)
\(80\) 0 0
\(81\) −405.683 −0.556493
\(82\) −907.411 − 228.563i −1.22203 − 0.307812i
\(83\) 1190.49i 1.57437i 0.616716 + 0.787186i \(0.288463\pi\)
−0.616716 + 0.787186i \(0.711537\pi\)
\(84\) 235.340 − 437.518i 0.305687 − 0.568299i
\(85\) 0 0
\(86\) −268.399 + 1065.56i −0.336537 + 1.33607i
\(87\) −108.283 −0.133439
\(88\) −228.879 + 250.535i −0.277256 + 0.303490i
\(89\) −398.940 −0.475141 −0.237571 − 0.971370i \(-0.576351\pi\)
−0.237571 + 0.971370i \(0.576351\pi\)
\(90\) 0 0
\(91\) 1253.80i 1.44432i
\(92\) −9.42182 − 5.06797i −0.0106771 − 0.00574318i
\(93\) − 311.691i − 0.347536i
\(94\) −1509.47 − 380.213i −1.65628 − 0.417191i
\(95\) 0 0
\(96\) −725.097 + 254.078i −0.770885 + 0.270122i
\(97\) −278.022 −0.291019 −0.145510 − 0.989357i \(-0.546482\pi\)
−0.145510 + 0.989357i \(0.546482\pi\)
\(98\) 353.648 + 89.0787i 0.364529 + 0.0918195i
\(99\) − 134.745i − 0.136791i
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.101.1 12
4.3 odd 2 800.4.d.d.401.9 12
5.2 odd 4 200.4.f.c.149.5 12
5.3 odd 4 200.4.f.b.149.8 12
5.4 even 2 40.4.d.a.21.12 yes 12
8.3 odd 2 800.4.d.d.401.4 12
8.5 even 2 inner 200.4.d.b.101.2 12
15.14 odd 2 360.4.k.c.181.1 12
20.3 even 4 800.4.f.c.49.4 12
20.7 even 4 800.4.f.b.49.9 12
20.19 odd 2 160.4.d.a.81.4 12
40.3 even 4 800.4.f.b.49.10 12
40.13 odd 4 200.4.f.c.149.6 12
40.19 odd 2 160.4.d.a.81.9 12
40.27 even 4 800.4.f.c.49.3 12
40.29 even 2 40.4.d.a.21.11 ✓ 12
40.37 odd 4 200.4.f.b.149.7 12
60.59 even 2 1440.4.k.c.721.11 12
80.19 odd 4 1280.4.a.ba.1.5 6
80.29 even 4 1280.4.a.bc.1.2 6
80.59 odd 4 1280.4.a.bd.1.2 6
80.69 even 4 1280.4.a.bb.1.5 6
120.29 odd 2 360.4.k.c.181.2 12
120.59 even 2 1440.4.k.c.721.5 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.4.d.a.21.11 ✓ 12 40.29 even 2
40.4.d.a.21.12 yes 12 5.4 even 2
160.4.d.a.81.4 12 20.19 odd 2
160.4.d.a.81.9 12 40.19 odd 2
200.4.d.b.101.1 12 1.1 even 1 trivial
200.4.d.b.101.2 12 8.5 even 2 inner
200.4.f.b.149.7 12 40.37 odd 4
200.4.f.b.149.8 12 5.3 odd 4
200.4.f.c.149.5 12 5.2 odd 4
200.4.f.c.149.6 12 40.13 odd 4
360.4.k.c.181.1 12 15.14 odd 2
360.4.k.c.181.2 12 120.29 odd 2
800.4.d.d.401.4 12 8.3 odd 2
800.4.d.d.401.9 12 4.3 odd 2
800.4.f.b.49.9 12 20.7 even 4
800.4.f.b.49.10 12 40.3 even 4
800.4.f.c.49.3 12 40.27 even 4
800.4.f.c.49.4 12 20.3 even 4
1280.4.a.ba.1.5 6 80.19 odd 4
1280.4.a.bb.1.5 6 80.69 even 4
1280.4.a.bc.1.2 6 80.29 even 4
1280.4.a.bd.1.2 6 80.59 odd 4
1440.4.k.c.721.5 12 120.59 even 2
1440.4.k.c.721.11 12 60.59 even 2