Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [5415,2,Mod(1,5415)] 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("5415.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5415, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 5415 = 3 \cdot 5 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5415.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [5,1,5,7,5,1,2,6,5,1,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(43.2389926945\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.8797896.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 8x^{3} + 5x^{2} + 13x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 285)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.38140\) of defining polynomial
Character \(\chi\) \(=\) 5415.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.38140 q^{2} +1.00000 q^{3} -0.0917248 q^{4} +1.00000 q^{5} -1.38140 q^{6} -4.36264 q^{7} +2.88952 q^{8} +1.00000 q^{9} -1.38140 q^{10} -4.31625 q^{11} -0.0917248 q^{12} +6.36264 q^{13} +6.02657 q^{14} +1.00000 q^{15} -3.80814 q^{16} +5.71641 q^{17} -1.38140 q^{18} -0.0917248 q^{20} -4.36264 q^{21} +5.96248 q^{22} -0.579357 q^{23} +2.88952 q^{24} +1.00000 q^{25} -8.78938 q^{26} +1.00000 q^{27} +0.400163 q^{28} +3.55344 q^{29} -1.38140 q^{30} -1.18345 q^{31} -0.518458 q^{32} -4.31625 q^{33} -7.89667 q^{34} -4.36264 q^{35} -0.0917248 q^{36} +6.54609 q^{37} +6.36264 q^{39} +2.88952 q^{40} +0.762807 q^{41} +6.02657 q^{42} -6.36264 q^{43} +0.395907 q^{44} +1.00000 q^{45} +0.800326 q^{46} -2.73264 q^{47} -3.80814 q^{48} +12.0327 q^{49} -1.38140 q^{50} +5.71641 q^{51} -0.583613 q^{52} -5.13706 q^{53} -1.38140 q^{54} -4.31625 q^{55} -12.6059 q^{56} -4.90874 q^{58} -3.82968 q^{59} -0.0917248 q^{60} -12.0211 q^{61} +1.63482 q^{62} -4.36264 q^{63} +8.33247 q^{64} +6.36264 q^{65} +5.96248 q^{66} +4.00426 q^{67} -0.524337 q^{68} -0.579357 q^{69} +6.02657 q^{70} -3.07906 q^{71} +2.88952 q^{72} +8.08640 q^{73} -9.04280 q^{74} +1.00000 q^{75} +18.8303 q^{77} -8.78938 q^{78} -11.3367 q^{79} -3.80814 q^{80} +1.00000 q^{81} -1.05374 q^{82} +7.67889 q^{83} +0.400163 q^{84} +5.71641 q^{85} +8.78938 q^{86} +3.55344 q^{87} -12.4719 q^{88} -9.84186 q^{89} -1.38140 q^{90} -27.7579 q^{91} +0.0531414 q^{92} -1.18345 q^{93} +3.77487 q^{94} -0.518458 q^{96} -2.00000 q^{97} -16.6220 q^{98} -4.31625 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + q^{2} + 5 q^{3} + 7 q^{4} + 5 q^{5} + q^{6} + 2 q^{7} + 6 q^{8} + 5 q^{9} + q^{10} + 5 q^{11} + 7 q^{12} + 8 q^{13} + 4 q^{14} + 5 q^{15} + 7 q^{16} + 10 q^{17} + q^{18} + 7 q^{20} + 2 q^{21}+ \cdots + 5 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.38140 −0.976800 −0.488400 − 0.872620i \(-0.662419\pi\)
−0.488400 + 0.872620i \(0.662419\pi\)
\(3\) 1.00000 0.577350
\(4\) −0.0917248 −0.0458624
\(5\) 1.00000 0.447214
\(6\) −1.38140 −0.563956
\(7\) −4.36264 −1.64892 −0.824462 − 0.565917i \(-0.808522\pi\)
−0.824462 + 0.565917i \(0.808522\pi\)
\(8\) 2.88952 1.02160
\(9\) 1.00000 0.333333
\(10\) −1.38140 −0.436838
\(11\) −4.31625 −1.30140 −0.650699 − 0.759336i \(-0.725524\pi\)
−0.650699 + 0.759336i \(0.725524\pi\)
\(12\) −0.0917248 −0.0264787
\(13\) 6.36264 1.76468 0.882340 − 0.470613i \(-0.155967\pi\)
0.882340 + 0.470613i \(0.155967\pi\)
\(14\) 6.02657 1.61067
\(15\) 1.00000 0.258199
\(16\) −3.80814 −0.952034
\(17\) 5.71641 1.38643 0.693217 − 0.720729i \(-0.256193\pi\)
0.693217 + 0.720729i \(0.256193\pi\)
\(18\) −1.38140 −0.325600
\(19\) 0 0
\(20\) −0.0917248 −0.0205103
\(21\) −4.36264 −0.952007
\(22\) 5.96248 1.27121
\(23\) −0.579357 −0.120804 −0.0604021 − 0.998174i \(-0.519238\pi\)
−0.0604021 + 0.998174i \(0.519238\pi\)
\(24\) 2.88952 0.589820
\(25\) 1.00000 0.200000
\(26\) −8.78938 −1.72374
\(27\) 1.00000 0.192450
\(28\) 0.400163 0.0756237
\(29\) 3.55344 0.659858 0.329929 − 0.944006i \(-0.392975\pi\)
0.329929 + 0.944006i \(0.392975\pi\)
\(30\) −1.38140 −0.252209
\(31\) −1.18345 −0.212554 −0.106277 − 0.994337i \(-0.533893\pi\)
−0.106277 + 0.994337i \(0.533893\pi\)
\(32\) −0.518458 −0.0916514
\(33\) −4.31625 −0.751363
\(34\) −7.89667 −1.35427
\(35\) −4.36264 −0.737421
\(36\) −0.0917248 −0.0152875
\(37\) 6.54609 1.07617 0.538086 − 0.842890i \(-0.319148\pi\)
0.538086 + 0.842890i \(0.319148\pi\)
\(38\) 0 0
\(39\) 6.36264 1.01884
\(40\) 2.88952 0.456873
\(41\) 0.762807 0.119130 0.0595652 − 0.998224i \(-0.481029\pi\)
0.0595652 + 0.998224i \(0.481029\pi\)
\(42\) 6.02657 0.929920
\(43\) −6.36264 −0.970294 −0.485147 − 0.874433i \(-0.661234\pi\)
−0.485147 + 0.874433i \(0.661234\pi\)
\(44\) 0.395907 0.0596853
\(45\) 1.00000 0.149071
\(46\) 0.800326 0.118002
\(47\) −2.73264 −0.398596 −0.199298 − 0.979939i \(-0.563866\pi\)
−0.199298 + 0.979939i \(0.563866\pi\)
\(48\) −3.80814 −0.549657
\(49\) 12.0327 1.71895
\(50\) −1.38140 −0.195360
\(51\) 5.71641 0.800458
\(52\) −0.583613 −0.0809325
\(53\) −5.13706 −0.705629 −0.352814 − 0.935693i \(-0.614775\pi\)
−0.352814 + 0.935693i \(0.614775\pi\)
\(54\) −1.38140 −0.187985
\(55\) −4.31625 −0.582003
\(56\) −12.6059 −1.68454
\(57\) 0 0
\(58\) −4.90874 −0.644549
\(59\) −3.82968 −0.498582 −0.249291 − 0.968429i \(-0.580198\pi\)
−0.249291 + 0.968429i \(0.580198\pi\)
\(60\) −0.0917248 −0.0118416
\(61\) −12.0211 −1.53914 −0.769569 − 0.638563i \(-0.779529\pi\)
−0.769569 + 0.638563i \(0.779529\pi\)
\(62\) 1.63482 0.207623
\(63\) −4.36264 −0.549641
\(64\) 8.33247 1.04156
\(65\) 6.36264 0.789189
\(66\) 5.96248 0.733931
\(67\) 4.00426 0.489198 0.244599 − 0.969624i \(-0.421344\pi\)
0.244599 + 0.969624i \(0.421344\pi\)
\(68\) −0.524337 −0.0635852
\(69\) −0.579357 −0.0697464
\(70\) 6.02657 0.720313
\(71\) −3.07906 −0.365417 −0.182708 − 0.983167i \(-0.558486\pi\)
−0.182708 + 0.983167i \(0.558486\pi\)
\(72\) 2.88952 0.340533
\(73\) 8.08640 0.946442 0.473221 − 0.880944i \(-0.343091\pi\)
0.473221 + 0.880944i \(0.343091\pi\)
\(74\) −9.04280 −1.05120
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 18.8303 2.14591
\(78\) −8.78938 −0.995201
\(79\) −11.3367 −1.27548 −0.637741 − 0.770251i \(-0.720131\pi\)
−0.637741 + 0.770251i \(0.720131\pi\)
\(80\) −3.80814 −0.425763
\(81\) 1.00000 0.111111
\(82\) −1.05374 −0.116367
\(83\) 7.67889 0.842868 0.421434 − 0.906859i \(-0.361527\pi\)
0.421434 + 0.906859i \(0.361527\pi\)
\(84\) 0.400163 0.0436613
\(85\) 5.71641 0.620032
\(86\) 8.78938 0.947783
\(87\) 3.55344 0.380969
\(88\) −12.4719 −1.32951
\(89\) −9.84186 −1.04324 −0.521618 − 0.853179i \(-0.674671\pi\)
−0.521618 + 0.853179i \(0.674671\pi\)
\(90\) −1.38140 −0.145613
\(91\) −27.7579 −2.90982
\(92\) 0.0531414 0.00554038
\(93\) −1.18345 −0.122718
\(94\) 3.77487 0.389348
\(95\) 0 0
\(96\) −0.518458 −0.0529149
\(97\) −2.00000 −0.203069 −0.101535 − 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) −16.6220 −1.67907
\(99\) −4.31625 −0.433799
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 5415.2.a.z.1.2 5
19.8 odd 6 285.2.i.f.121.2 yes 10
19.12 odd 6 285.2.i.f.106.2 ✓ 10
19.18 odd 2 5415.2.a.y.1.4 5
57.8 even 6 855.2.k.i.406.4 10
57.50 even 6 855.2.k.i.676.4 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.i.f.106.2 ✓ 10 19.12 odd 6
285.2.i.f.121.2 yes 10 19.8 odd 6
855.2.k.i.406.4 10 57.8 even 6
855.2.k.i.676.4 10 57.50 even 6
5415.2.a.y.1.4 5 19.18 odd 2
5415.2.a.z.1.2 5 1.1 even 1 trivial