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Properties

Label 200.4.d.b.101.11
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 101.11
Root \(-1.86176 - 0.730647i\) of defining polynomial
Character \(\chi\) \(=\) 200.101
Dual form 200.4.d.b.101.12

$q$-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)
 
\(f(q)\) \(=\) \(q+(2.59241 - 1.13111i) q^{2} -6.25785i q^{3} +(5.44116 - 5.86462i) q^{4} +(-7.07834 - 16.2229i) q^{6} +34.6280 q^{7} +(7.47214 - 21.3581i) q^{8} -12.1606 q^{9} +7.91595i q^{11} +(-36.6999 - 34.0499i) q^{12} -11.0346i q^{13} +(89.7698 - 39.1682i) q^{14} +(-4.78760 - 63.8207i) q^{16} -57.6152 q^{17} +(-31.5253 + 13.7551i) q^{18} +141.133i q^{19} -216.696i q^{21} +(8.95385 + 20.5214i) q^{22} -129.328 q^{23} +(-133.655 - 46.7595i) q^{24} +(-12.4814 - 28.6062i) q^{26} -92.8625i q^{27} +(188.416 - 203.080i) q^{28} +89.1664i q^{29} -78.3307 q^{31} +(-84.5999 - 160.034i) q^{32} +49.5368 q^{33} +(-149.362 + 65.1694i) q^{34} +(-66.1679 + 71.3175i) q^{36} +249.332i q^{37} +(159.637 + 365.874i) q^{38} -69.0530 q^{39} +91.8705 q^{41} +(-245.109 - 561.766i) q^{42} +194.184i q^{43} +(46.4240 + 43.0719i) q^{44} +(-335.270 + 146.285i) q^{46} +72.6149 q^{47} +(-399.380 + 29.9600i) q^{48} +856.096 q^{49} +360.547i q^{51} +(-64.7139 - 60.0411i) q^{52} -456.782i q^{53} +(-105.038 - 240.737i) q^{54} +(258.745 - 739.586i) q^{56} +883.187 q^{57} +(100.857 + 231.156i) q^{58} -341.098i q^{59} +217.067i q^{61} +(-203.065 + 88.6011i) q^{62} -421.098 q^{63} +(-400.334 - 319.181i) q^{64} +(128.420 - 56.0318i) q^{66} +529.237i q^{67} +(-313.494 + 337.892i) q^{68} +809.314i q^{69} +381.540 q^{71} +(-90.8659 + 259.728i) q^{72} +876.902 q^{73} +(282.023 + 646.370i) q^{74} +(827.690 + 767.926i) q^{76} +274.113i q^{77} +(-179.013 + 78.1069i) q^{78} -203.950 q^{79} -909.456 q^{81} +(238.166 - 103.916i) q^{82} -996.654i q^{83} +(-1270.84 - 1179.08i) q^{84} +(219.644 + 503.403i) q^{86} +557.989 q^{87} +(169.069 + 59.1491i) q^{88} +172.456 q^{89} -382.107i q^{91} +(-703.693 + 758.459i) q^{92} +490.182i q^{93} +(188.247 - 82.1358i) q^{94} +(-1001.47 + 529.413i) q^{96} -1058.49 q^{97} +(2219.35 - 968.343i) q^{98} -96.2629i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{2} + 16 q^{4} - 36 q^{6} - 28 q^{7} + 40 q^{8} - 108 q^{9} - 188 q^{12} + 68 q^{14} - 56 q^{16} + 206 q^{18} + 164 q^{22} - 604 q^{23} + 360 q^{24} - 308 q^{26} + 436 q^{28} - 264 q^{31} - 72 q^{32}+ \cdots + 7266 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.59241 − 1.13111i 0.916555 − 0.399910i
\(3\) − 6.25785i − 1.20432i −0.798374 − 0.602161i \(-0.794306\pi\)
0.798374 − 0.602161i \(-0.205694\pi\)
\(4\) 5.44116 − 5.86462i 0.680145 − 0.733078i
\(5\) 0 0
\(6\) −7.07834 − 16.2229i −0.481620 − 1.10383i
\(7\) 34.6280 1.86973 0.934867 − 0.354998i \(-0.115518\pi\)
0.934867 + 0.354998i \(0.115518\pi\)
\(8\) 7.47214 − 21.3581i 0.330225 − 0.943902i
\(9\) −12.1606 −0.450394
\(10\) 0 0
\(11\) 7.91595i 0.216977i 0.994098 + 0.108489i \(0.0346011\pi\)
−0.994098 + 0.108489i \(0.965399\pi\)
\(12\) −36.6999 − 34.0499i −0.882862 − 0.819114i
\(13\) − 11.0346i − 0.235420i −0.993048 − 0.117710i \(-0.962445\pi\)
0.993048 − 0.117710i \(-0.0375553\pi\)
\(14\) 89.7698 − 39.1682i 1.71371 − 0.747725i
\(15\) 0 0
\(16\) −4.78760 − 63.8207i −0.0748062 − 0.997198i
\(17\) −57.6152 −0.821985 −0.410992 − 0.911639i \(-0.634818\pi\)
−0.410992 + 0.911639i \(0.634818\pi\)
\(18\) −31.5253 + 13.7551i −0.412810 + 0.180117i
\(19\) 141.133i 1.70411i 0.523453 + 0.852055i \(0.324644\pi\)
−0.523453 + 0.852055i \(0.675356\pi\)
\(20\) 0 0
\(21\) − 216.696i − 2.25176i
\(22\) 8.95385 + 20.5214i 0.0867712 + 0.198871i
\(23\) −129.328 −1.17247 −0.586233 − 0.810143i \(-0.699390\pi\)
−0.586233 + 0.810143i \(0.699390\pi\)
\(24\) −133.655 − 46.7595i −1.13676 − 0.397698i
\(25\) 0 0
\(26\) −12.4814 − 28.6062i −0.0941465 − 0.215775i
\(27\) − 92.8625i − 0.661903i
\(28\) 188.416 − 203.080i 1.27169 − 1.37066i
\(29\) 89.1664i 0.570958i 0.958385 + 0.285479i \(0.0921527\pi\)
−0.958385 + 0.285479i \(0.907847\pi\)
\(30\) 0 0
\(31\) −78.3307 −0.453826 −0.226913 − 0.973915i \(-0.572863\pi\)
−0.226913 + 0.973915i \(0.572863\pi\)
\(32\) −84.5999 − 160.034i −0.467353 − 0.884071i
\(33\) 49.5368 0.261310
\(34\) −149.362 + 65.1694i −0.753394 + 0.328720i
\(35\) 0 0
\(36\) −66.1679 + 71.3175i −0.306333 + 0.330174i
\(37\) 249.332i 1.10783i 0.832572 + 0.553917i \(0.186868\pi\)
−0.832572 + 0.553917i \(0.813132\pi\)
\(38\) 159.637 + 365.874i 0.681489 + 1.56191i
\(39\) −69.0530 −0.283521
\(40\) 0 0
\(41\) 91.8705 0.349946 0.174973 − 0.984573i \(-0.444016\pi\)
0.174973 + 0.984573i \(0.444016\pi\)
\(42\) −245.109 − 561.766i −0.900502 − 2.06386i
\(43\) 194.184i 0.688668i 0.938847 + 0.344334i \(0.111895\pi\)
−0.938847 + 0.344334i \(0.888105\pi\)
\(44\) 46.4240 + 43.0719i 0.159061 + 0.147576i
\(45\) 0 0
\(46\) −335.270 + 146.285i −1.07463 + 0.468880i
\(47\) 72.6149 0.225361 0.112680 − 0.993631i \(-0.464056\pi\)
0.112680 + 0.993631i \(0.464056\pi\)
\(48\) −399.380 + 29.9600i −1.20095 + 0.0900908i
\(49\) 856.096 2.49591
\(50\) 0 0
\(51\) 360.547i 0.989935i
\(52\) −64.7139 − 60.0411i −0.172581 − 0.160119i
\(53\) − 456.782i − 1.18385i −0.805995 − 0.591923i \(-0.798369\pi\)
0.805995 − 0.591923i \(-0.201631\pi\)
\(54\) −105.038 − 240.737i −0.264701 − 0.606671i
\(55\) 0 0
\(56\) 258.745 − 739.586i 0.617433 − 1.76485i
\(57\) 883.187 2.05230
\(58\) 100.857 + 231.156i 0.228331 + 0.523314i
\(59\) − 341.098i − 0.752664i −0.926485 − 0.376332i \(-0.877185\pi\)
0.926485 − 0.376332i \(-0.122815\pi\)
\(60\) 0 0
\(61\) 217.067i 0.455616i 0.973706 + 0.227808i \(0.0731559\pi\)
−0.973706 + 0.227808i \(0.926844\pi\)
\(62\) −203.065 + 88.6011i −0.415957 + 0.181489i
\(63\) −421.098 −0.842117
\(64\) −400.334 − 319.181i −0.781903 − 0.623400i
\(65\) 0 0
\(66\) 128.420 − 56.0318i 0.239505 − 0.104501i
\(67\) 529.237i 0.965024i 0.875889 + 0.482512i \(0.160276\pi\)
−0.875889 + 0.482512i \(0.839724\pi\)
\(68\) −313.494 + 337.892i −0.559069 + 0.602579i
\(69\) 809.314i 1.41203i
\(70\) 0 0
\(71\) 381.540 0.637754 0.318877 − 0.947796i \(-0.396694\pi\)
0.318877 + 0.947796i \(0.396694\pi\)
\(72\) −90.8659 + 259.728i −0.148731 + 0.425128i
\(73\) 876.902 1.40594 0.702970 − 0.711220i \(-0.251857\pi\)
0.702970 + 0.711220i \(0.251857\pi\)
\(74\) 282.023 + 646.370i 0.443034 + 1.01539i
\(75\) 0 0
\(76\) 827.690 + 767.926i 1.24924 + 1.15904i
\(77\) 274.113i 0.405690i
\(78\) −179.013 + 78.1069i −0.259863 + 0.113383i
\(79\) −203.950 −0.290458 −0.145229 − 0.989398i \(-0.546392\pi\)
−0.145229 + 0.989398i \(0.546392\pi\)
\(80\) 0 0
\(81\) −909.456 −1.24754
\(82\) 238.166 − 103.916i 0.320744 − 0.139947i
\(83\) − 996.654i − 1.31804i −0.752127 − 0.659018i \(-0.770972\pi\)
0.752127 − 0.659018i \(-0.229028\pi\)
\(84\) −1270.84 − 1179.08i −1.65072 − 1.53153i
\(85\) 0 0
\(86\) 219.644 + 503.403i 0.275405 + 0.631202i
\(87\) 557.989 0.687618
\(88\) 169.069 + 59.1491i 0.204805 + 0.0716513i
\(89\) 172.456 0.205396 0.102698 − 0.994713i \(-0.467252\pi\)
0.102698 + 0.994713i \(0.467252\pi\)
\(90\) 0 0
\(91\) − 382.107i − 0.440172i
\(92\) −703.693 + 758.459i −0.797447 + 0.859509i
\(93\) 490.182i 0.546553i
\(94\) 188.247 − 82.1358i 0.206556 − 0.0901240i
\(95\) 0 0
\(96\) −1001.47 + 529.413i −1.06471 + 0.562844i
\(97\) −1058.49 −1.10797 −0.553984 − 0.832527i \(-0.686893\pi\)
−0.553984 + 0.832527i \(0.686893\pi\)
\(98\) 2219.35 − 968.343i 2.28763 − 0.998137i
\(99\) − 96.2629i − 0.0977251i
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.101.11 12
4.3 odd 2 800.4.d.d.401.10 12
5.2 odd 4 200.4.f.b.149.10 12
5.3 odd 4 200.4.f.c.149.3 12
5.4 even 2 40.4.d.a.21.2 yes 12
8.3 odd 2 800.4.d.d.401.3 12
8.5 even 2 inner 200.4.d.b.101.12 12
15.14 odd 2 360.4.k.c.181.11 12
20.3 even 4 800.4.f.b.49.4 12
20.7 even 4 800.4.f.c.49.9 12
20.19 odd 2 160.4.d.a.81.3 12
40.3 even 4 800.4.f.c.49.10 12
40.13 odd 4 200.4.f.b.149.9 12
40.19 odd 2 160.4.d.a.81.10 12
40.27 even 4 800.4.f.b.49.3 12
40.29 even 2 40.4.d.a.21.1 ✓ 12
40.37 odd 4 200.4.f.c.149.4 12
60.59 even 2 1440.4.k.c.721.6 12
80.19 odd 4 1280.4.a.bd.1.5 6
80.29 even 4 1280.4.a.bb.1.2 6
80.59 odd 4 1280.4.a.ba.1.2 6
80.69 even 4 1280.4.a.bc.1.5 6
120.29 odd 2 360.4.k.c.181.12 12
120.59 even 2 1440.4.k.c.721.12 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.4.d.a.21.1 ✓ 12 40.29 even 2
40.4.d.a.21.2 yes 12 5.4 even 2
160.4.d.a.81.3 12 20.19 odd 2
160.4.d.a.81.10 12 40.19 odd 2
200.4.d.b.101.11 12 1.1 even 1 trivial
200.4.d.b.101.12 12 8.5 even 2 inner
200.4.f.b.149.9 12 40.13 odd 4
200.4.f.b.149.10 12 5.2 odd 4
200.4.f.c.149.3 12 5.3 odd 4
200.4.f.c.149.4 12 40.37 odd 4
360.4.k.c.181.11 12 15.14 odd 2
360.4.k.c.181.12 12 120.29 odd 2
800.4.d.d.401.3 12 8.3 odd 2
800.4.d.d.401.10 12 4.3 odd 2
800.4.f.b.49.3 12 40.27 even 4
800.4.f.b.49.4 12 20.3 even 4
800.4.f.c.49.9 12 20.7 even 4
800.4.f.c.49.10 12 40.3 even 4
1280.4.a.ba.1.2 6 80.59 odd 4
1280.4.a.bb.1.2 6 80.29 even 4
1280.4.a.bc.1.5 6 80.69 even 4
1280.4.a.bd.1.5 6 80.19 odd 4
1440.4.k.c.721.6 12 60.59 even 2
1440.4.k.c.721.12 12 120.59 even 2