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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [90,5,Mod(71,90)] 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("90.71"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(90, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 90.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,-32,0,0,32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.30329667755\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 4x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 5^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 71.4
Root \(-1.58114 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 90.71
Dual form 90.5.d.a.71.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.82843i q^{2} -8.00000 q^{4} +11.1803i q^{5} -39.4342 q^{7} -22.6274i q^{8} -31.6228 q^{10} -62.8394i q^{11} -211.171 q^{13} -111.537i q^{14} +64.0000 q^{16} -166.505i q^{17} -468.605 q^{19} -89.4427i q^{20} +177.737 q^{22} -633.995i q^{23} -125.000 q^{25} -597.281i q^{26} +315.473 q^{28} +375.753i q^{29} +679.658 q^{31} +181.019i q^{32} +470.947 q^{34} -440.887i q^{35} -1747.85 q^{37} -1325.42i q^{38} +252.982 q^{40} +2269.66i q^{41} -945.658 q^{43} +502.715i q^{44} +1793.21 q^{46} +3667.09i q^{47} -845.947 q^{49} -353.553i q^{50} +1689.37 q^{52} +1531.37i q^{53} +702.566 q^{55} +892.293i q^{56} -1062.79 q^{58} +3971.69i q^{59} +6673.55 q^{61} +1922.36i q^{62} -512.000 q^{64} -2360.96i q^{65} -2060.74 q^{67} +1332.04i q^{68} +1247.02 q^{70} -7316.58i q^{71} -5833.18 q^{73} -4943.68i q^{74} +3748.84 q^{76} +2478.02i q^{77} +5212.97 q^{79} +715.542i q^{80} -6419.57 q^{82} -4850.89i q^{83} +1861.58 q^{85} -2674.73i q^{86} -1421.89 q^{88} +565.207i q^{89} +8327.35 q^{91} +5071.96i q^{92} -10372.1 q^{94} -5239.16i q^{95} -17434.2 q^{97} -2392.70i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 32 q^{4} + 32 q^{7} + 104 q^{13} + 256 q^{16} - 736 q^{19} - 48 q^{22} - 500 q^{25} - 256 q^{28} + 4616 q^{31} - 1152 q^{34} - 2248 q^{37} - 5680 q^{43} + 4896 q^{46} - 348 q^{49} - 832 q^{52} + 3000 q^{55}+ \cdots - 61768 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/90\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(37\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.82843i 0.707107i
\(3\) 0 0
\(4\) −8.00000 −0.500000
\(5\) 11.1803i 0.447214i
\(6\) 0 0
\(7\) −39.4342 −0.804779 −0.402389 − 0.915469i \(-0.631820\pi\)
−0.402389 + 0.915469i \(0.631820\pi\)
\(8\) − 22.6274i − 0.353553i
\(9\) 0 0
\(10\) −31.6228 −0.316228
\(11\) − 62.8394i − 0.519334i −0.965698 − 0.259667i \(-0.916387\pi\)
0.965698 − 0.259667i \(-0.0836128\pi\)
\(12\) 0 0
\(13\) −211.171 −1.24953 −0.624766 − 0.780812i \(-0.714806\pi\)
−0.624766 + 0.780812i \(0.714806\pi\)
\(14\) − 111.537i − 0.569065i
\(15\) 0 0
\(16\) 64.0000 0.250000
\(17\) − 166.505i − 0.576141i −0.957609 − 0.288071i \(-0.906986\pi\)
0.957609 − 0.288071i \(-0.0930138\pi\)
\(18\) 0 0
\(19\) −468.605 −1.29807 −0.649037 − 0.760757i \(-0.724828\pi\)
−0.649037 + 0.760757i \(0.724828\pi\)
\(20\) − 89.4427i − 0.223607i
\(21\) 0 0
\(22\) 177.737 0.367225
\(23\) − 633.995i − 1.19848i −0.800570 − 0.599240i \(-0.795470\pi\)
0.800570 − 0.599240i \(-0.204530\pi\)
\(24\) 0 0
\(25\) −125.000 −0.200000
\(26\) − 597.281i − 0.883552i
\(27\) 0 0
\(28\) 315.473 0.402389
\(29\) 375.753i 0.446793i 0.974728 + 0.223397i \(0.0717145\pi\)
−0.974728 + 0.223397i \(0.928286\pi\)
\(30\) 0 0
\(31\) 679.658 0.707241 0.353620 − 0.935389i \(-0.384950\pi\)
0.353620 + 0.935389i \(0.384950\pi\)
\(32\) 181.019i 0.176777i
\(33\) 0 0
\(34\) 470.947 0.407393
\(35\) − 440.887i − 0.359908i
\(36\) 0 0
\(37\) −1747.85 −1.27674 −0.638369 − 0.769731i \(-0.720391\pi\)
−0.638369 + 0.769731i \(0.720391\pi\)
\(38\) − 1325.42i − 0.917877i
\(39\) 0 0
\(40\) 252.982 0.158114
\(41\) 2269.66i 1.35019i 0.737733 + 0.675093i \(0.235896\pi\)
−0.737733 + 0.675093i \(0.764104\pi\)
\(42\) 0 0
\(43\) −945.658 −0.511443 −0.255722 − 0.966750i \(-0.582313\pi\)
−0.255722 + 0.966750i \(0.582313\pi\)
\(44\) 502.715i 0.259667i
\(45\) 0 0
\(46\) 1793.21 0.847453
\(47\) 3667.09i 1.66007i 0.557713 + 0.830034i \(0.311679\pi\)
−0.557713 + 0.830034i \(0.688321\pi\)
\(48\) 0 0
\(49\) −845.947 −0.352331
\(50\) − 353.553i − 0.141421i
\(51\) 0 0
\(52\) 1689.37 0.624766
\(53\) 1531.37i 0.545164i 0.962132 + 0.272582i \(0.0878777\pi\)
−0.962132 + 0.272582i \(0.912122\pi\)
\(54\) 0 0
\(55\) 702.566 0.232253
\(56\) 892.293i 0.284532i
\(57\) 0 0
\(58\) −1062.79 −0.315930
\(59\) 3971.69i 1.14096i 0.821311 + 0.570481i \(0.193243\pi\)
−0.821311 + 0.570481i \(0.806757\pi\)
\(60\) 0 0
\(61\) 6673.55 1.79348 0.896742 − 0.442554i \(-0.145928\pi\)
0.896742 + 0.442554i \(0.145928\pi\)
\(62\) 1922.36i 0.500095i
\(63\) 0 0
\(64\) −512.000 −0.125000
\(65\) − 2360.96i − 0.558807i
\(66\) 0 0
\(67\) −2060.74 −0.459064 −0.229532 − 0.973301i \(-0.573720\pi\)
−0.229532 + 0.973301i \(0.573720\pi\)
\(68\) 1332.04i 0.288071i
\(69\) 0 0
\(70\) 1247.02 0.254493
\(71\) − 7316.58i − 1.45141i −0.688004 − 0.725707i \(-0.741513\pi\)
0.688004 − 0.725707i \(-0.258487\pi\)
\(72\) 0 0
\(73\) −5833.18 −1.09461 −0.547305 − 0.836933i \(-0.684346\pi\)
−0.547305 + 0.836933i \(0.684346\pi\)
\(74\) − 4943.68i − 0.902790i
\(75\) 0 0
\(76\) 3748.84 0.649037
\(77\) 2478.02i 0.417949i
\(78\) 0 0
\(79\) 5212.97 0.835278 0.417639 − 0.908613i \(-0.362858\pi\)
0.417639 + 0.908613i \(0.362858\pi\)
\(80\) 715.542i 0.111803i
\(81\) 0 0
\(82\) −6419.57 −0.954725
\(83\) − 4850.89i − 0.704151i −0.935972 − 0.352075i \(-0.885476\pi\)
0.935972 − 0.352075i \(-0.114524\pi\)
\(84\) 0 0
\(85\) 1861.58 0.257658
\(86\) − 2674.73i − 0.361645i
\(87\) 0 0
\(88\) −1421.89 −0.183612
\(89\) 565.207i 0.0713555i 0.999363 + 0.0356778i \(0.0113590\pi\)
−0.999363 + 0.0356778i \(0.988641\pi\)
\(90\) 0 0
\(91\) 8327.35 1.00560
\(92\) 5071.96i 0.599240i
\(93\) 0 0
\(94\) −10372.1 −1.17385
\(95\) − 5239.16i − 0.580517i
\(96\) 0 0
\(97\) −17434.2 −1.85293 −0.926466 − 0.376379i \(-0.877169\pi\)
−0.926466 + 0.376379i \(0.877169\pi\)
\(98\) − 2392.70i − 0.249136i
\(99\) 0 0
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 90.5.d.a.71.4 yes 4
3.2 odd 2 inner 90.5.d.a.71.1 ✓ 4
4.3 odd 2 720.5.l.b.161.4 4
5.2 odd 4 450.5.b.d.449.2 8
5.3 odd 4 450.5.b.d.449.7 8
5.4 even 2 450.5.d.g.251.2 4
12.11 even 2 720.5.l.b.161.2 4
15.2 even 4 450.5.b.d.449.6 8
15.8 even 4 450.5.b.d.449.3 8
15.14 odd 2 450.5.d.g.251.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.5.d.a.71.1 ✓ 4 3.2 odd 2 inner
90.5.d.a.71.4 yes 4 1.1 even 1 trivial
450.5.b.d.449.2 8 5.2 odd 4
450.5.b.d.449.3 8 15.8 even 4
450.5.b.d.449.6 8 15.2 even 4
450.5.b.d.449.7 8 5.3 odd 4
450.5.d.g.251.2 4 5.4 even 2
450.5.d.g.251.4 4 15.14 odd 2
720.5.l.b.161.2 4 12.11 even 2
720.5.l.b.161.4 4 4.3 odd 2