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Properties

Label 50.4.b.b.49.2
Level $50$
Weight $4$
Character 50.49
Analytic conductor $2.950$
Analytic rank $0$
Dimension $2$
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] = [50,4,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, 4); 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, 4, names="a")
 
Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 50.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-8,0,28] 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: \(2.95009550029\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 50.49
Dual form 50.4.b.b.49.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.00000i q^{2} -7.00000i q^{3} -4.00000 q^{4} +14.0000 q^{6} -34.0000i q^{7} -8.00000i q^{8} -22.0000 q^{9} +27.0000 q^{11} +28.0000i q^{12} +28.0000i q^{13} +68.0000 q^{14} +16.0000 q^{16} +21.0000i q^{17} -44.0000i q^{18} -35.0000 q^{19} -238.000 q^{21} +54.0000i q^{22} +78.0000i q^{23} -56.0000 q^{24} -56.0000 q^{26} -35.0000i q^{27} +136.000i q^{28} +120.000 q^{29} +182.000 q^{31} +32.0000i q^{32} -189.000i q^{33} -42.0000 q^{34} +88.0000 q^{36} +146.000i q^{37} -70.0000i q^{38} +196.000 q^{39} +357.000 q^{41} -476.000i q^{42} +148.000i q^{43} -108.000 q^{44} -156.000 q^{46} -84.0000i q^{47} -112.000i q^{48} -813.000 q^{49} +147.000 q^{51} -112.000i q^{52} -702.000i q^{53} +70.0000 q^{54} -272.000 q^{56} +245.000i q^{57} +240.000i q^{58} +840.000 q^{59} -238.000 q^{61} +364.000i q^{62} +748.000i q^{63} -64.0000 q^{64} +378.000 q^{66} +461.000i q^{67} -84.0000i q^{68} +546.000 q^{69} -708.000 q^{71} +176.000i q^{72} +133.000i q^{73} -292.000 q^{74} +140.000 q^{76} -918.000i q^{77} +392.000i q^{78} -650.000 q^{79} -839.000 q^{81} +714.000i q^{82} +903.000i q^{83} +952.000 q^{84} -296.000 q^{86} -840.000i q^{87} -216.000i q^{88} -735.000 q^{89} +952.000 q^{91} -312.000i q^{92} -1274.00i q^{93} +168.000 q^{94} +224.000 q^{96} +1106.00i q^{97} -1626.00i q^{98} -594.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{4} + 28 q^{6} - 44 q^{9} + 54 q^{11} + 136 q^{14} + 32 q^{16} - 70 q^{19} - 476 q^{21} - 112 q^{24} - 112 q^{26} + 240 q^{29} + 364 q^{31} - 84 q^{34} + 176 q^{36} + 392 q^{39} + 714 q^{41} - 216 q^{44}+ \cdots - 1188 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000i 0.707107i
\(3\) − 7.00000i − 1.34715i −0.739119 − 0.673575i \(-0.764758\pi\)
0.739119 − 0.673575i \(-0.235242\pi\)
\(4\) −4.00000 −0.500000
\(5\) 0 0
\(6\) 14.0000 0.952579
\(7\) − 34.0000i − 1.83583i −0.396780 − 0.917914i \(-0.629872\pi\)
0.396780 − 0.917914i \(-0.370128\pi\)
\(8\) − 8.00000i − 0.353553i
\(9\) −22.0000 −0.814815
\(10\) 0 0
\(11\) 27.0000 0.740073 0.370037 − 0.929017i \(-0.379345\pi\)
0.370037 + 0.929017i \(0.379345\pi\)
\(12\) 28.0000i 0.673575i
\(13\) 28.0000i 0.597369i 0.954352 + 0.298685i \(0.0965479\pi\)
−0.954352 + 0.298685i \(0.903452\pi\)
\(14\) 68.0000 1.29813
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) 21.0000i 0.299603i 0.988716 + 0.149801i \(0.0478634\pi\)
−0.988716 + 0.149801i \(0.952137\pi\)
\(18\) − 44.0000i − 0.576161i
\(19\) −35.0000 −0.422608 −0.211304 − 0.977420i \(-0.567771\pi\)
−0.211304 + 0.977420i \(0.567771\pi\)
\(20\) 0 0
\(21\) −238.000 −2.47314
\(22\) 54.0000i 0.523311i
\(23\) 78.0000i 0.707136i 0.935409 + 0.353568i \(0.115032\pi\)
−0.935409 + 0.353568i \(0.884968\pi\)
\(24\) −56.0000 −0.476290
\(25\) 0 0
\(26\) −56.0000 −0.422404
\(27\) − 35.0000i − 0.249472i
\(28\) 136.000i 0.917914i
\(29\) 120.000 0.768395 0.384197 − 0.923251i \(-0.374478\pi\)
0.384197 + 0.923251i \(0.374478\pi\)
\(30\) 0 0
\(31\) 182.000 1.05446 0.527228 − 0.849724i \(-0.323231\pi\)
0.527228 + 0.849724i \(0.323231\pi\)
\(32\) 32.0000i 0.176777i
\(33\) − 189.000i − 0.996990i
\(34\) −42.0000 −0.211851
\(35\) 0 0
\(36\) 88.0000 0.407407
\(37\) 146.000i 0.648710i 0.945936 + 0.324355i \(0.105147\pi\)
−0.945936 + 0.324355i \(0.894853\pi\)
\(38\) − 70.0000i − 0.298829i
\(39\) 196.000 0.804747
\(40\) 0 0
\(41\) 357.000 1.35985 0.679927 − 0.733280i \(-0.262011\pi\)
0.679927 + 0.733280i \(0.262011\pi\)
\(42\) − 476.000i − 1.74877i
\(43\) 148.000i 0.524879i 0.964948 + 0.262439i \(0.0845270\pi\)
−0.964948 + 0.262439i \(0.915473\pi\)
\(44\) −108.000 −0.370037
\(45\) 0 0
\(46\) −156.000 −0.500021
\(47\) − 84.0000i − 0.260695i −0.991468 − 0.130347i \(-0.958391\pi\)
0.991468 − 0.130347i \(-0.0416093\pi\)
\(48\) − 112.000i − 0.336788i
\(49\) −813.000 −2.37026
\(50\) 0 0
\(51\) 147.000 0.403610
\(52\) − 112.000i − 0.298685i
\(53\) − 702.000i − 1.81938i −0.415288 − 0.909690i \(-0.636319\pi\)
0.415288 − 0.909690i \(-0.363681\pi\)
\(54\) 70.0000 0.176404
\(55\) 0 0
\(56\) −272.000 −0.649063
\(57\) 245.000i 0.569317i
\(58\) 240.000i 0.543337i
\(59\) 840.000 1.85354 0.926769 − 0.375633i \(-0.122575\pi\)
0.926769 + 0.375633i \(0.122575\pi\)
\(60\) 0 0
\(61\) −238.000 −0.499554 −0.249777 − 0.968303i \(-0.580357\pi\)
−0.249777 + 0.968303i \(0.580357\pi\)
\(62\) 364.000i 0.745614i
\(63\) 748.000i 1.49586i
\(64\) −64.0000 −0.125000
\(65\) 0 0
\(66\) 378.000 0.704979
\(67\) 461.000i 0.840599i 0.907386 + 0.420299i \(0.138075\pi\)
−0.907386 + 0.420299i \(0.861925\pi\)
\(68\) − 84.0000i − 0.149801i
\(69\) 546.000 0.952618
\(70\) 0 0
\(71\) −708.000 −1.18344 −0.591719 − 0.806144i \(-0.701551\pi\)
−0.591719 + 0.806144i \(0.701551\pi\)
\(72\) 176.000i 0.288081i
\(73\) 133.000i 0.213239i 0.994300 + 0.106620i \(0.0340027\pi\)
−0.994300 + 0.106620i \(0.965997\pi\)
\(74\) −292.000 −0.458707
\(75\) 0 0
\(76\) 140.000 0.211304
\(77\) − 918.000i − 1.35865i
\(78\) 392.000i 0.569042i
\(79\) −650.000 −0.925705 −0.462853 − 0.886435i \(-0.653174\pi\)
−0.462853 + 0.886435i \(0.653174\pi\)
\(80\) 0 0
\(81\) −839.000 −1.15089
\(82\) 714.000i 0.961562i
\(83\) 903.000i 1.19418i 0.802173 + 0.597091i \(0.203677\pi\)
−0.802173 + 0.597091i \(0.796323\pi\)
\(84\) 952.000 1.23657
\(85\) 0 0
\(86\) −296.000 −0.371145
\(87\) − 840.000i − 1.03514i
\(88\) − 216.000i − 0.261655i
\(89\) −735.000 −0.875392 −0.437696 − 0.899123i \(-0.644205\pi\)
−0.437696 + 0.899123i \(0.644205\pi\)
\(90\) 0 0
\(91\) 952.000 1.09667
\(92\) − 312.000i − 0.353568i
\(93\) − 1274.00i − 1.42051i
\(94\) 168.000 0.184339
\(95\) 0 0
\(96\) 224.000 0.238145
\(97\) 1106.00i 1.15770i 0.815433 + 0.578852i \(0.196499\pi\)
−0.815433 + 0.578852i \(0.803501\pi\)
\(98\) − 1626.00i − 1.67603i
\(99\) −594.000 −0.603023
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 50.4.b.b.49.2 2
3.2 odd 2 450.4.c.c.199.1 2
4.3 odd 2 400.4.c.d.49.2 2
5.2 odd 4 50.4.a.a.1.1 ✓ 1
5.3 odd 4 50.4.a.e.1.1 yes 1
5.4 even 2 inner 50.4.b.b.49.1 2
15.2 even 4 450.4.a.t.1.1 1
15.8 even 4 450.4.a.a.1.1 1
15.14 odd 2 450.4.c.c.199.2 2
20.3 even 4 400.4.a.d.1.1 1
20.7 even 4 400.4.a.r.1.1 1
20.19 odd 2 400.4.c.d.49.1 2
35.13 even 4 2450.4.a.y.1.1 1
35.27 even 4 2450.4.a.t.1.1 1
40.3 even 4 1600.4.a.bv.1.1 1
40.13 odd 4 1600.4.a.f.1.1 1
40.27 even 4 1600.4.a.g.1.1 1
40.37 odd 4 1600.4.a.bu.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
50.4.a.a.1.1 ✓ 1 5.2 odd 4
50.4.a.e.1.1 yes 1 5.3 odd 4
50.4.b.b.49.1 2 5.4 even 2 inner
50.4.b.b.49.2 2 1.1 even 1 trivial
400.4.a.d.1.1 1 20.3 even 4
400.4.a.r.1.1 1 20.7 even 4
400.4.c.d.49.1 2 20.19 odd 2
400.4.c.d.49.2 2 4.3 odd 2
450.4.a.a.1.1 1 15.8 even 4
450.4.a.t.1.1 1 15.2 even 4
450.4.c.c.199.1 2 3.2 odd 2
450.4.c.c.199.2 2 15.14 odd 2
1600.4.a.f.1.1 1 40.13 odd 4
1600.4.a.g.1.1 1 40.27 even 4
1600.4.a.bu.1.1 1 40.37 odd 4
1600.4.a.bv.1.1 1 40.3 even 4
2450.4.a.t.1.1 1 35.27 even 4
2450.4.a.y.1.1 1 35.13 even 4