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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [50,22,Mod(49,50)] 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("50.49"); S:= CuspForms(chi, 22); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(50, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 22, names="a")
 
Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 50.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,-8388608,0,198172672] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(139.738672144\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 631153566 x^{6} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{18}\cdot 3^{4}\cdot 5^{14}\cdot 7^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.4
Root \(-16161.7i\) of defining polynomial
Character \(\chi\) \(=\) 50.49
Dual form 50.22.b.h.49.5

$q$-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)
 
\(f(q)\) \(=\) \(q-1024.00i q^{2} +185808. i q^{3} -1.04858e6 q^{4} +1.90268e8 q^{6} -4.88079e8i q^{7} +1.07374e9i q^{8} -2.40644e10 q^{9} -7.28578e10 q^{11} -1.94834e11i q^{12} +9.02768e11i q^{13} -4.99793e11 q^{14} +1.09951e12 q^{16} +6.32065e12i q^{17} +2.46419e13i q^{18} -4.36780e13 q^{19} +9.06891e13 q^{21} +7.46064e13i q^{22} -9.00682e13i q^{23} -1.99510e14 q^{24} +9.24434e14 q^{26} -2.52774e15i q^{27} +5.11788e14i q^{28} +1.34854e15 q^{29} -4.56363e14 q^{31} -1.12590e15i q^{32} -1.35376e16i q^{33} +6.47235e15 q^{34} +2.52333e16 q^{36} +2.84931e16i q^{37} +4.47263e16i q^{38} -1.67742e17 q^{39} -1.42955e17 q^{41} -9.28657e16i q^{42} +1.47393e17i q^{43} +7.63969e16 q^{44} -9.22299e16 q^{46} -1.71830e17i q^{47} +2.04298e17i q^{48} +3.20325e17 q^{49} -1.17443e18 q^{51} -9.46621e17i q^{52} +1.36482e18i q^{53} -2.58841e18 q^{54} +5.24071e17 q^{56} -8.11574e18i q^{57} -1.38090e18i q^{58} -6.60455e18 q^{59} +1.48171e18 q^{61} +4.67316e17i q^{62} +1.17453e19i q^{63} -1.15292e18 q^{64} -1.38625e19 q^{66} -2.80036e19i q^{67} -6.62769e18i q^{68} +1.67354e19 q^{69} -4.31882e19 q^{71} -2.58389e19i q^{72} +2.89648e17i q^{73} +2.91770e19 q^{74} +4.57997e19 q^{76} +3.55604e19i q^{77} +1.71768e20i q^{78} +5.22625e19 q^{79} +2.17954e20 q^{81} +1.46386e20i q^{82} -5.91809e19i q^{83} -9.50944e19 q^{84} +1.50931e20 q^{86} +2.50570e20i q^{87} -7.82305e19i q^{88} +4.79550e20 q^{89} +4.40622e20 q^{91} +9.44434e19i q^{92} -8.47961e19i q^{93} -1.75954e20 q^{94} +2.09202e20 q^{96} -8.16242e20i q^{97} -3.28013e20i q^{98} +1.75328e21 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8388608 q^{4} + 198172672 q^{6} - 47229523424 q^{9} + 175118508216 q^{11} - 333333561344 q^{14} + 8796093022208 q^{16} - 167026213839880 q^{19} - 8313944227904 q^{21} - 207799107715072 q^{24} - 200905406660608 q^{26}+ \cdots + 56\!\cdots\!52 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/50\mathbb{Z}\right)^\times\).

\(n\) \(27\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1024.00i − 0.707107i
\(3\) 185808.i 1.81674i 0.418171 + 0.908368i \(0.362671\pi\)
−0.418171 + 0.908368i \(0.637329\pi\)
\(4\) −1.04858e6 −0.500000
\(5\) 0 0
\(6\) 1.90268e8 1.28463
\(7\) − 4.88079e8i − 0.653071i −0.945185 − 0.326536i \(-0.894119\pi\)
0.945185 − 0.326536i \(-0.105881\pi\)
\(8\) 1.07374e9i 0.353553i
\(9\) −2.40644e10 −2.30053
\(10\) 0 0
\(11\) −7.28578e10 −0.846940 −0.423470 − 0.905910i \(-0.639188\pi\)
−0.423470 + 0.905910i \(0.639188\pi\)
\(12\) − 1.94834e11i − 0.908368i
\(13\) 9.02768e11i 1.81623i 0.418720 + 0.908115i \(0.362479\pi\)
−0.418720 + 0.908115i \(0.637521\pi\)
\(14\) −4.99793e11 −0.461791
\(15\) 0 0
\(16\) 1.09951e12 0.250000
\(17\) 6.32065e12i 0.760411i 0.924902 + 0.380206i \(0.124147\pi\)
−0.924902 + 0.380206i \(0.875853\pi\)
\(18\) 2.46419e13i 1.62672i
\(19\) −4.36780e13 −1.63437 −0.817184 − 0.576376i \(-0.804466\pi\)
−0.817184 + 0.576376i \(0.804466\pi\)
\(20\) 0 0
\(21\) 9.06891e13 1.18646
\(22\) 7.46064e13i 0.598877i
\(23\) − 9.00682e13i − 0.453345i −0.973971 − 0.226673i \(-0.927215\pi\)
0.973971 − 0.226673i \(-0.0727848\pi\)
\(24\) −1.99510e14 −0.642313
\(25\) 0 0
\(26\) 9.24434e14 1.28427
\(27\) − 2.52774e15i − 2.36273i
\(28\) 5.11788e14i 0.326536i
\(29\) 1.34854e15 0.595230 0.297615 − 0.954686i \(-0.403809\pi\)
0.297615 + 0.954686i \(0.403809\pi\)
\(30\) 0 0
\(31\) −4.56363e14 −0.100003 −0.0500015 − 0.998749i \(-0.515923\pi\)
−0.0500015 + 0.998749i \(0.515923\pi\)
\(32\) − 1.12590e15i − 0.176777i
\(33\) − 1.35376e16i − 1.53867i
\(34\) 6.47235e15 0.537692
\(35\) 0 0
\(36\) 2.52333e16 1.15027
\(37\) 2.84931e16i 0.974142i 0.873362 + 0.487071i \(0.161935\pi\)
−0.873362 + 0.487071i \(0.838065\pi\)
\(38\) 4.47263e16i 1.15567i
\(39\) −1.67742e17 −3.29961
\(40\) 0 0
\(41\) −1.42955e17 −1.66329 −0.831647 − 0.555304i \(-0.812602\pi\)
−0.831647 + 0.555304i \(0.812602\pi\)
\(42\) − 9.28657e16i − 0.838953i
\(43\) 1.47393e17i 1.04006i 0.854148 + 0.520030i \(0.174079\pi\)
−0.854148 + 0.520030i \(0.825921\pi\)
\(44\) 7.63969e16 0.423470
\(45\) 0 0
\(46\) −9.22299e16 −0.320564
\(47\) − 1.71830e17i − 0.476508i −0.971203 − 0.238254i \(-0.923425\pi\)
0.971203 − 0.238254i \(-0.0765751\pi\)
\(48\) 2.04298e17i 0.454184i
\(49\) 3.20325e17 0.573498
\(50\) 0 0
\(51\) −1.17443e18 −1.38147
\(52\) − 9.46621e17i − 0.908115i
\(53\) 1.36482e18i 1.07196i 0.844232 + 0.535979i \(0.180057\pi\)
−0.844232 + 0.535979i \(0.819943\pi\)
\(54\) −2.58841e18 −1.67070
\(55\) 0 0
\(56\) 5.24071e17 0.230896
\(57\) − 8.11574e18i − 2.96922i
\(58\) − 1.38090e18i − 0.420891i
\(59\) −6.60455e18 −1.68228 −0.841138 − 0.540821i \(-0.818114\pi\)
−0.841138 + 0.540821i \(0.818114\pi\)
\(60\) 0 0
\(61\) 1.48171e18 0.265950 0.132975 − 0.991119i \(-0.457547\pi\)
0.132975 + 0.991119i \(0.457547\pi\)
\(62\) 4.67316e17i 0.0707128i
\(63\) 1.17453e19i 1.50241i
\(64\) −1.15292e18 −0.125000
\(65\) 0 0
\(66\) −1.38625e19 −1.08800
\(67\) − 2.80036e19i − 1.87685i −0.345486 − 0.938424i \(-0.612286\pi\)
0.345486 − 0.938424i \(-0.387714\pi\)
\(68\) − 6.62769e18i − 0.380206i
\(69\) 1.67354e19 0.823609
\(70\) 0 0
\(71\) −4.31882e19 −1.57454 −0.787269 − 0.616610i \(-0.788505\pi\)
−0.787269 + 0.616610i \(0.788505\pi\)
\(72\) − 2.58389e19i − 0.813361i
\(73\) 2.89648e17i 0.00788825i 0.999992 + 0.00394412i \(0.00125546\pi\)
−0.999992 + 0.00394412i \(0.998745\pi\)
\(74\) 2.91770e19 0.688823
\(75\) 0 0
\(76\) 4.57997e19 0.817184
\(77\) 3.55604e19i 0.553112i
\(78\) 1.71768e20i 2.33318i
\(79\) 5.22625e19 0.621020 0.310510 − 0.950570i \(-0.399500\pi\)
0.310510 + 0.950570i \(0.399500\pi\)
\(80\) 0 0
\(81\) 2.17954e20 1.99192
\(82\) 1.46386e20i 1.17613i
\(83\) − 5.91809e19i − 0.418660i −0.977845 − 0.209330i \(-0.932872\pi\)
0.977845 − 0.209330i \(-0.0671283\pi\)
\(84\) −9.50944e19 −0.593229
\(85\) 0 0
\(86\) 1.50931e20 0.735434
\(87\) 2.50570e20i 1.08138i
\(88\) − 7.82305e19i − 0.299439i
\(89\) 4.79550e20 1.63019 0.815097 − 0.579325i \(-0.196684\pi\)
0.815097 + 0.579325i \(0.196684\pi\)
\(90\) 0 0
\(91\) 4.40622e20 1.18613
\(92\) 9.44434e19i 0.226673i
\(93\) − 8.47961e19i − 0.181679i
\(94\) −1.75954e20 −0.336942
\(95\) 0 0
\(96\) 2.09202e20 0.321157
\(97\) − 8.16242e20i − 1.12387i −0.827182 − 0.561935i \(-0.810057\pi\)
0.827182 − 0.561935i \(-0.189943\pi\)
\(98\) − 3.28013e20i − 0.405524i
\(99\) 1.75328e21 1.94841
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 50.22.b.h.49.4 8
5.2 odd 4 50.22.a.j.1.4 yes 4
5.3 odd 4 50.22.a.i.1.1 ✓ 4
5.4 even 2 inner 50.22.b.h.49.5 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
50.22.a.i.1.1 ✓ 4 5.3 odd 4
50.22.a.j.1.4 yes 4 5.2 odd 4
50.22.b.h.49.4 8 1.1 even 1 trivial
50.22.b.h.49.5 8 5.4 even 2 inner