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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [464,2,Mod(49,464)] 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("464.49"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(464, base_ring=CyclotomicField(14)) chi = DirichletCharacter(H, H._module([0, 0, 12])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 464 = 2^{4} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 464.u (of order \(7\), degree \(6\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,3,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.70505865379\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{7})\)
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: \( x^{12} - 3 x^{11} + 13 x^{10} - 9 x^{9} - 5 x^{8} + 35 x^{7} + 197 x^{6} - 140 x^{5} - 80 x^{4} + \cdots + 4096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 58)
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label 81.1
Root \(-0.260453 - 1.14112i\) of defining polynomial
Character \(\chi\) \(=\) 464.81
Dual form 464.2.u.h.401.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.260453 + 1.14112i) q^{3} +(-1.85326 + 2.32392i) q^{5} +(0.115912 - 0.507846i) q^{7} +(1.46859 + 0.707235i) q^{9} +(-0.585233 + 0.281833i) q^{11} +(-0.444717 + 0.214164i) q^{13} +(-2.16918 - 2.72007i) q^{15} -7.42032 q^{17} +(1.47532 + 6.46378i) q^{19} +(0.549323 + 0.264540i) q^{21} +(-4.74970 - 5.95594i) q^{23} +(-0.853408 - 3.73902i) q^{25} +(-3.37886 + 4.23695i) q^{27} +(-4.56917 - 2.85003i) q^{29} +(-3.72875 + 4.67571i) q^{31} +(-0.169180 - 0.741224i) q^{33} +(0.965376 + 1.21054i) q^{35} +(2.23650 + 1.07704i) q^{37} +(-0.128559 - 0.563254i) q^{39} +7.82245 q^{41} +(-0.404994 - 0.507846i) q^{43} +(-4.36524 + 2.10219i) q^{45} +(7.92488 - 3.81642i) q^{47} +(6.06231 + 2.91945i) q^{49} +(1.93264 - 8.46747i) q^{51} +(-0.717766 + 0.900050i) q^{53} +(0.429633 - 1.88234i) q^{55} -7.76020 q^{57} +5.31686 q^{59} +(-2.15779 + 9.45389i) q^{61} +(0.529394 - 0.663840i) q^{63} +(0.326477 - 1.43039i) q^{65} +(4.07320 + 1.96155i) q^{67} +(8.03351 - 3.86873i) q^{69} +(-12.7440 + 6.13719i) q^{71} +(5.44958 + 6.83356i) q^{73} +4.48894 q^{75} +(0.0752920 + 0.329876i) q^{77} +(3.71962 + 1.79127i) q^{79} +(-0.905949 - 1.13602i) q^{81} +(0.952729 + 4.17418i) q^{83} +(13.7518 - 17.2442i) q^{85} +(4.44228 - 4.47167i) q^{87} +(-0.743610 + 0.932457i) q^{89} +(0.0572142 + 0.250672i) q^{91} +(-4.36437 - 5.47275i) q^{93} +(-17.7555 - 8.55057i) q^{95} +(-2.61081 - 11.4387i) q^{97} -1.05879 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 3 q^{3} - q^{7} - 11 q^{9} + 2 q^{11} + q^{13} + 9 q^{15} - 12 q^{17} + 6 q^{19} - 13 q^{21} - 35 q^{23} - 6 q^{25} - 39 q^{27} - 14 q^{29} + 8 q^{31} + 33 q^{33} + 18 q^{35} + 31 q^{37} + 22 q^{39}+ \cdots + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/464\mathbb{Z}\right)^\times\).

\(n\) \(117\) \(175\) \(321\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{5}{7}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.260453 + 1.14112i −0.150373 + 0.658825i 0.842404 + 0.538847i \(0.181140\pi\)
−0.992776 + 0.119979i \(0.961717\pi\)
\(4\) 0 0
\(5\) −1.85326 + 2.32392i −0.828804 + 1.03929i 0.169747 + 0.985488i \(0.445705\pi\)
−0.998552 + 0.0538002i \(0.982867\pi\)
\(6\) 0 0
\(7\) 0.115912 − 0.507846i 0.0438108 − 0.191948i −0.948287 − 0.317414i \(-0.897186\pi\)
0.992098 + 0.125466i \(0.0400427\pi\)
\(8\) 0 0
\(9\) 1.46859 + 0.707235i 0.489530 + 0.235745i
\(10\) 0 0
\(11\) −0.585233 + 0.281833i −0.176454 + 0.0849759i −0.520026 − 0.854151i \(-0.674078\pi\)
0.343571 + 0.939127i \(0.388363\pi\)
\(12\) 0 0
\(13\) −0.444717 + 0.214164i −0.123342 + 0.0593985i −0.494537 − 0.869156i \(-0.664662\pi\)
0.371195 + 0.928555i \(0.378948\pi\)
\(14\) 0 0
\(15\) −2.16918 − 2.72007i −0.560080 − 0.702318i
\(16\) 0 0
\(17\) −7.42032 −1.79969 −0.899846 − 0.436208i \(-0.856321\pi\)
−0.899846 + 0.436208i \(0.856321\pi\)
\(18\) 0 0
\(19\) 1.47532 + 6.46378i 0.338461 + 1.48289i 0.802272 + 0.596959i \(0.203625\pi\)
−0.463811 + 0.885934i \(0.653518\pi\)
\(20\) 0 0
\(21\) 0.549323 + 0.264540i 0.119872 + 0.0577273i
\(22\) 0 0
\(23\) −4.74970 − 5.95594i −0.990381 − 1.24190i −0.970251 − 0.242101i \(-0.922163\pi\)
−0.0201303 − 0.999797i \(-0.506408\pi\)
\(24\) 0 0
\(25\) −0.853408 − 3.73902i −0.170682 − 0.747805i
\(26\) 0 0
\(27\) −3.37886 + 4.23695i −0.650261 + 0.815402i
\(28\) 0 0
\(29\) −4.56917 − 2.85003i −0.848474 − 0.529237i
\(30\) 0 0
\(31\) −3.72875 + 4.67571i −0.669703 + 0.839782i −0.994361 − 0.106050i \(-0.966180\pi\)
0.324657 + 0.945832i \(0.394751\pi\)
\(32\) 0 0
\(33\) −0.169180 − 0.741224i −0.0294504 − 0.129031i
\(34\) 0 0
\(35\) 0.965376 + 1.21054i 0.163178 + 0.204619i
\(36\) 0 0
\(37\) 2.23650 + 1.07704i 0.367678 + 0.177065i 0.608595 − 0.793481i \(-0.291734\pi\)
−0.240916 + 0.970546i \(0.577448\pi\)
\(38\) 0 0
\(39\) −0.128559 − 0.563254i −0.0205859 − 0.0901929i
\(40\) 0 0
\(41\) 7.82245 1.22166 0.610830 − 0.791761i \(-0.290836\pi\)
0.610830 + 0.791761i \(0.290836\pi\)
\(42\) 0 0
\(43\) −0.404994 − 0.507846i −0.0617609 − 0.0774458i 0.749991 − 0.661447i \(-0.230058\pi\)
−0.811752 + 0.584002i \(0.801486\pi\)
\(44\) 0 0
\(45\) −4.36524 + 2.10219i −0.650731 + 0.313376i
\(46\) 0 0
\(47\) 7.92488 − 3.81642i 1.15596 − 0.556682i 0.245143 − 0.969487i \(-0.421165\pi\)
0.910819 + 0.412805i \(0.135451\pi\)
\(48\) 0 0
\(49\) 6.06231 + 2.91945i 0.866044 + 0.417065i
\(50\) 0 0
\(51\) 1.93264 − 8.46747i 0.270624 − 1.18568i
\(52\) 0 0
\(53\) −0.717766 + 0.900050i −0.0985927 + 0.123631i −0.828680 − 0.559722i \(-0.810908\pi\)
0.730088 + 0.683354i \(0.239479\pi\)
\(54\) 0 0
\(55\) 0.429633 − 1.88234i 0.0579317 − 0.253815i
\(56\) 0 0
\(57\) −7.76020 −1.02786
\(58\) 0 0
\(59\) 5.31686 0.692196 0.346098 − 0.938198i \(-0.387507\pi\)
0.346098 + 0.938198i \(0.387507\pi\)
\(60\) 0 0
\(61\) −2.15779 + 9.45389i −0.276277 + 1.21045i 0.626184 + 0.779675i \(0.284616\pi\)
−0.902461 + 0.430772i \(0.858241\pi\)
\(62\) 0 0
\(63\) 0.529394 − 0.663840i 0.0666974 − 0.0836359i
\(64\) 0 0
\(65\) 0.326477 − 1.43039i 0.0404945 − 0.177418i
\(66\) 0 0
\(67\) 4.07320 + 1.96155i 0.497620 + 0.239641i 0.665821 − 0.746112i \(-0.268081\pi\)
−0.168201 + 0.985753i \(0.553796\pi\)
\(68\) 0 0
\(69\) 8.03351 − 3.86873i 0.967121 − 0.465741i
\(70\) 0 0
\(71\) −12.7440 + 6.13719i −1.51243 + 0.728350i −0.992081 − 0.125601i \(-0.959914\pi\)
−0.520353 + 0.853951i \(0.674200\pi\)
\(72\) 0 0
\(73\) 5.44958 + 6.83356i 0.637825 + 0.799808i 0.990729 − 0.135852i \(-0.0433772\pi\)
−0.352904 + 0.935660i \(0.614806\pi\)
\(74\) 0 0
\(75\) 4.48894 0.518339
\(76\) 0 0
\(77\) 0.0752920 + 0.329876i 0.00858032 + 0.0375928i
\(78\) 0 0
\(79\) 3.71962 + 1.79127i 0.418490 + 0.201534i 0.631265 − 0.775567i \(-0.282536\pi\)
−0.212775 + 0.977101i \(0.568250\pi\)
\(80\) 0 0
\(81\) −0.905949 − 1.13602i −0.100661 − 0.126225i
\(82\) 0 0
\(83\) 0.952729 + 4.17418i 0.104576 + 0.458176i 0.999918 + 0.0127974i \(0.00407365\pi\)
−0.895343 + 0.445378i \(0.853069\pi\)
\(84\) 0 0
\(85\) 13.7518 − 17.2442i 1.49159 − 1.87040i
\(86\) 0 0
\(87\) 4.44228 − 4.47167i 0.476262 − 0.479413i
\(88\) 0 0
\(89\) −0.743610 + 0.932457i −0.0788225 + 0.0988403i −0.819679 − 0.572823i \(-0.805848\pi\)
0.740857 + 0.671663i \(0.234420\pi\)
\(90\) 0 0
\(91\) 0.0572142 + 0.250672i 0.00599768 + 0.0262775i
\(92\) 0 0
\(93\) −4.36437 − 5.47275i −0.452564 − 0.567498i
\(94\) 0 0
\(95\) −17.7555 − 8.55057i −1.82167 − 0.877270i
\(96\) 0 0
\(97\) −2.61081 − 11.4387i −0.265088 − 1.16143i −0.915651 − 0.401974i \(-0.868324\pi\)
0.650563 − 0.759452i \(-0.274533\pi\)
\(98\) 0 0
\(99\) −1.05879 −0.106412
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 464.2.u.h.81.1 12
4.3 odd 2 58.2.d.b.23.2 ✓ 12
12.11 even 2 522.2.k.h.487.2 12
29.24 even 7 inner 464.2.u.h.401.1 12
116.11 even 28 1682.2.b.i.1681.4 12
116.47 even 28 1682.2.b.i.1681.9 12
116.71 odd 14 1682.2.a.q.1.4 6
116.103 odd 14 1682.2.a.t.1.3 6
116.111 odd 14 58.2.d.b.53.2 yes 12
348.227 even 14 522.2.k.h.343.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.2.d.b.23.2 ✓ 12 4.3 odd 2
58.2.d.b.53.2 yes 12 116.111 odd 14
464.2.u.h.81.1 12 1.1 even 1 trivial
464.2.u.h.401.1 12 29.24 even 7 inner
522.2.k.h.343.2 12 348.227 even 14
522.2.k.h.487.2 12 12.11 even 2
1682.2.a.q.1.4 6 116.71 odd 14
1682.2.a.t.1.3 6 116.103 odd 14
1682.2.b.i.1681.4 12 116.11 even 28
1682.2.b.i.1681.9 12 116.47 even 28