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Properties

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

$q$-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 \(0.205694\pi\)
−0.798374 + 0.602161i \(0.794306\pi\)
\(4\) 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 \(-0.965399\pi\)
0.994098 − 0.108489i \(-0.0346011\pi\)
\(12\) −36.6999 + 34.0499i −0.882862 + 0.819114i
\(13\) 11.0346i 0.235420i 0.993048 + 0.117710i \(0.0375553\pi\)
−0.993048 + 0.117710i \(0.962445\pi\)
\(14\) 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 \(-0.675356\pi\)
0.523453 − 0.852055i \(-0.324644\pi\)
\(20\) 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 \(-0.907847\pi\)
0.958385 − 0.285479i \(-0.0921527\pi\)
\(30\) 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 \(-0.813132\pi\)
0.832572 − 0.553917i \(-0.186868\pi\)
\(38\) 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 \(-0.888105\pi\)
0.938847 − 0.344334i \(-0.111895\pi\)
\(44\) 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 \(0.201631\pi\)
−0.805995 + 0.591923i \(0.798369\pi\)
\(54\) −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 \(0.122815\pi\)
−0.926485 + 0.376332i \(0.877185\pi\)
\(60\) 0 0
\(61\) − 217.067i − 0.455616i −0.973706 − 0.227808i \(-0.926844\pi\)
0.973706 − 0.227808i \(-0.0731559\pi\)
\(62\) −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 \(-0.839724\pi\)
0.875889 − 0.482512i \(-0.160276\pi\)
\(68\) −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 \(0.229028\pi\)
−0.752127 + 0.659018i \(0.770972\pi\)
\(84\) −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.101.12 12
4.3 odd 2 800.4.d.d.401.3 12
5.2 odd 4 200.4.f.c.149.4 12
5.3 odd 4 200.4.f.b.149.9 12
5.4 even 2 40.4.d.a.21.1 ✓ 12
8.3 odd 2 800.4.d.d.401.10 12
8.5 even 2 inner 200.4.d.b.101.11 12
15.14 odd 2 360.4.k.c.181.12 12
20.3 even 4 800.4.f.c.49.10 12
20.7 even 4 800.4.f.b.49.3 12
20.19 odd 2 160.4.d.a.81.10 12
40.3 even 4 800.4.f.b.49.4 12
40.13 odd 4 200.4.f.c.149.3 12
40.19 odd 2 160.4.d.a.81.3 12
40.27 even 4 800.4.f.c.49.9 12
40.29 even 2 40.4.d.a.21.2 yes 12
40.37 odd 4 200.4.f.b.149.10 12
60.59 even 2 1440.4.k.c.721.12 12
80.19 odd 4 1280.4.a.ba.1.2 6
80.29 even 4 1280.4.a.bc.1.5 6
80.59 odd 4 1280.4.a.bd.1.5 6
80.69 even 4 1280.4.a.bb.1.2 6
120.29 odd 2 360.4.k.c.181.11 12
120.59 even 2 1440.4.k.c.721.6 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.4.d.a.21.1 ✓ 12 5.4 even 2
40.4.d.a.21.2 yes 12 40.29 even 2
160.4.d.a.81.3 12 40.19 odd 2
160.4.d.a.81.10 12 20.19 odd 2
200.4.d.b.101.11 12 8.5 even 2 inner
200.4.d.b.101.12 12 1.1 even 1 trivial
200.4.f.b.149.9 12 5.3 odd 4
200.4.f.b.149.10 12 40.37 odd 4
200.4.f.c.149.3 12 40.13 odd 4
200.4.f.c.149.4 12 5.2 odd 4
360.4.k.c.181.11 12 120.29 odd 2
360.4.k.c.181.12 12 15.14 odd 2
800.4.d.d.401.3 12 4.3 odd 2
800.4.d.d.401.10 12 8.3 odd 2
800.4.f.b.49.3 12 20.7 even 4
800.4.f.b.49.4 12 40.3 even 4
800.4.f.c.49.9 12 40.27 even 4
800.4.f.c.49.10 12 20.3 even 4
1280.4.a.ba.1.2 6 80.19 odd 4
1280.4.a.bb.1.2 6 80.69 even 4
1280.4.a.bc.1.5 6 80.29 even 4
1280.4.a.bd.1.5 6 80.59 odd 4
1440.4.k.c.721.6 12 120.59 even 2
1440.4.k.c.721.12 12 60.59 even 2