Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [464,2,Mod(33,464)] 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("464.33"); 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, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 464 = 2^{4} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 464.y (of order \(14\), degree \(6\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == 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 \(\Q(\zeta_{14})\)
Coefficient field: \(\Q(\zeta_{28})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 58)
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label 225.2
Root \(0.433884 - 0.900969i\) of defining polynomial
Character \(\chi\) \(=\) 464.225
Dual form 464.2.y.c.33.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.626980 - 1.30194i) q^{3} +(-0.136585 - 0.598418i) q^{5} +(2.59161 + 1.24805i) q^{7} +(0.568532 + 0.712916i) q^{9} +(3.39687 + 2.70891i) q^{11} +(-0.298199 + 0.373929i) q^{13} +(-0.864739 - 0.197371i) q^{15} +0.259558i q^{17} +(-3.65470 - 7.58906i) q^{19} +(3.24978 - 2.59161i) q^{21} +(-0.0317259 + 0.139000i) q^{23} +(4.16540 - 2.00595i) q^{25} +(5.51107 - 1.25786i) q^{27} +(1.07561 - 5.27665i) q^{29} +(-6.46089 + 1.47465i) q^{31} +(5.65660 - 2.72407i) q^{33} +(0.392883 - 1.72133i) q^{35} +(-7.50895 + 5.98819i) q^{37} +(0.299868 + 0.622683i) q^{39} +4.28236i q^{41} +(-3.17741 - 0.725223i) q^{43} +(0.348969 - 0.437593i) q^{45} +(3.97456 + 3.16960i) q^{47} +(0.794384 + 0.996126i) q^{49} +(0.337929 + 0.162738i) q^{51} +(-2.06111 - 9.03032i) q^{53} +(1.15710 - 2.40274i) q^{55} -12.1719 q^{57} +10.2463 q^{59} +(-4.31279 + 8.95559i) q^{61} +(0.583655 + 2.55716i) q^{63} +(0.264495 + 0.127374i) q^{65} +(1.16176 + 1.45680i) q^{67} +(0.161078 + 0.128456i) q^{69} +(-5.97581 + 7.49342i) q^{71} +(2.90704 + 0.663513i) q^{73} -6.68078i q^{75} +(5.42249 + 11.2599i) q^{77} +(-10.5977 + 8.45137i) q^{79} +(1.20895 - 5.29674i) q^{81} +(0.950401 - 0.457689i) q^{83} +(0.155324 - 0.0354518i) q^{85} +(-6.19549 - 4.70873i) q^{87} +(2.51138 - 0.573205i) q^{89} +(-1.23950 + 0.596912i) q^{91} +(-2.13094 + 9.33625i) q^{93} +(-4.04225 + 3.22359i) q^{95} +(-7.22194 - 14.9965i) q^{97} +3.96178i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{5} - 4 q^{7} - 4 q^{9} - 26 q^{13} + 14 q^{15} + 14 q^{21} + 16 q^{23} + 22 q^{25} + 18 q^{29} - 28 q^{31} - 10 q^{33} - 4 q^{35} - 28 q^{37} - 28 q^{39} + 28 q^{43} - 4 q^{45} + 14 q^{47} - 8 q^{49}+ \cdots - 28 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(117\) \(175\) \(321\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{13}{14}\right)\)

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
\(3\) 0.626980 − 1.30194i 0.361987 − 0.751674i −0.637842 − 0.770167i \(-0.720173\pi\)
0.999829 + 0.0184933i \(0.00588693\pi\)
\(4\) 0 0
\(5\) −0.136585 − 0.598418i −0.0610826 − 0.267621i 0.935160 − 0.354225i \(-0.115255\pi\)
−0.996243 + 0.0866048i \(0.972398\pi\)
\(6\) 0 0
\(7\) 2.59161 + 1.24805i 0.979537 + 0.471720i 0.853946 − 0.520362i \(-0.174203\pi\)
0.125591 + 0.992082i \(0.459917\pi\)
\(8\) 0 0
\(9\) 0.568532 + 0.712916i 0.189511 + 0.237639i
\(10\) 0 0
\(11\) 3.39687 + 2.70891i 1.02419 + 0.816767i 0.983226 − 0.182394i \(-0.0583845\pi\)
0.0409677 + 0.999160i \(0.486956\pi\)
\(12\) 0 0
\(13\) −0.298199 + 0.373929i −0.0827054 + 0.103709i −0.821463 − 0.570262i \(-0.806842\pi\)
0.738757 + 0.673972i \(0.235413\pi\)
\(14\) 0 0
\(15\) −0.864739 − 0.197371i −0.223275 − 0.0509610i
\(16\) 0 0
\(17\) 0.259558i 0.0629522i 0.999505 + 0.0314761i \(0.0100208\pi\)
−0.999505 + 0.0314761i \(0.989979\pi\)
\(18\) 0 0
\(19\) −3.65470 − 7.58906i −0.838446 − 1.74105i −0.651453 − 0.758689i \(-0.725840\pi\)
−0.186992 − 0.982361i \(-0.559874\pi\)
\(20\) 0 0
\(21\) 3.24978 − 2.59161i 0.709160 − 0.565536i
\(22\) 0 0
\(23\) −0.0317259 + 0.139000i −0.00661531 + 0.0289836i −0.978128 − 0.208005i \(-0.933303\pi\)
0.971513 + 0.236988i \(0.0761603\pi\)
\(24\) 0 0
\(25\) 4.16540 − 2.00595i 0.833079 − 0.401190i
\(26\) 0 0
\(27\) 5.51107 − 1.25786i 1.06061 − 0.242076i
\(28\) 0 0
\(29\) 1.07561 − 5.27665i 0.199736 − 0.979850i
\(30\) 0 0
\(31\) −6.46089 + 1.47465i −1.16041 + 0.264856i −0.759030 − 0.651056i \(-0.774326\pi\)
−0.401380 + 0.915912i \(0.631469\pi\)
\(32\) 0 0
\(33\) 5.65660 − 2.72407i 0.984687 − 0.474200i
\(34\) 0 0
\(35\) 0.392883 − 1.72133i 0.0664093 − 0.290958i
\(36\) 0 0
\(37\) −7.50895 + 5.98819i −1.23446 + 0.984452i −0.234541 + 0.972106i \(0.575359\pi\)
−0.999924 + 0.0123461i \(0.996070\pi\)
\(38\) 0 0
\(39\) 0.299868 + 0.622683i 0.0480173 + 0.0997090i
\(40\) 0 0
\(41\) 4.28236i 0.668792i 0.942433 + 0.334396i \(0.108532\pi\)
−0.942433 + 0.334396i \(0.891468\pi\)
\(42\) 0 0
\(43\) −3.17741 − 0.725223i −0.484550 − 0.110595i −0.0267346 − 0.999643i \(-0.508511\pi\)
−0.457816 + 0.889047i \(0.651368\pi\)
\(44\) 0 0
\(45\) 0.348969 − 0.437593i 0.0520212 − 0.0652325i
\(46\) 0 0
\(47\) 3.97456 + 3.16960i 0.579749 + 0.462334i 0.868926 − 0.494941i \(-0.164810\pi\)
−0.289178 + 0.957275i \(0.593382\pi\)
\(48\) 0 0
\(49\) 0.794384 + 0.996126i 0.113483 + 0.142304i
\(50\) 0 0
\(51\) 0.337929 + 0.162738i 0.0473195 + 0.0227879i
\(52\) 0 0
\(53\) −2.06111 − 9.03032i −0.283116 − 1.24041i −0.893774 − 0.448517i \(-0.851952\pi\)
0.610659 − 0.791894i \(-0.290905\pi\)
\(54\) 0 0
\(55\) 1.15710 − 2.40274i 0.156023 − 0.323985i
\(56\) 0 0
\(57\) −12.1719 −1.61221
\(58\) 0 0
\(59\) 10.2463 1.33395 0.666977 − 0.745078i \(-0.267588\pi\)
0.666977 + 0.745078i \(0.267588\pi\)
\(60\) 0 0
\(61\) −4.31279 + 8.95559i −0.552196 + 1.14665i 0.418917 + 0.908025i \(0.362410\pi\)
−0.971113 + 0.238622i \(0.923304\pi\)
\(62\) 0 0
\(63\) 0.583655 + 2.55716i 0.0735336 + 0.322172i
\(64\) 0 0
\(65\) 0.264495 + 0.127374i 0.0328066 + 0.0157988i
\(66\) 0 0
\(67\) 1.16176 + 1.45680i 0.141931 + 0.177976i 0.847716 − 0.530450i \(-0.177977\pi\)
−0.705785 + 0.708426i \(0.749406\pi\)
\(68\) 0 0
\(69\) 0.161078 + 0.128456i 0.0193915 + 0.0154642i
\(70\) 0 0
\(71\) −5.97581 + 7.49342i −0.709198 + 0.889306i −0.997673 − 0.0681816i \(-0.978280\pi\)
0.288475 + 0.957487i \(0.406852\pi\)
\(72\) 0 0
\(73\) 2.90704 + 0.663513i 0.340243 + 0.0776583i 0.389229 − 0.921141i \(-0.372742\pi\)
−0.0489853 + 0.998799i \(0.515599\pi\)
\(74\) 0 0
\(75\) − 6.68078i − 0.771430i
\(76\) 0 0
\(77\) 5.42249 + 11.2599i 0.617950 + 1.28319i
\(78\) 0 0
\(79\) −10.5977 + 8.45137i −1.19233 + 0.950853i −0.999538 − 0.0303860i \(-0.990326\pi\)
−0.192794 + 0.981239i \(0.561755\pi\)
\(80\) 0 0
\(81\) 1.20895 − 5.29674i 0.134327 − 0.588527i
\(82\) 0 0
\(83\) 0.950401 − 0.457689i 0.104320 − 0.0502379i −0.380996 − 0.924577i \(-0.624419\pi\)
0.485316 + 0.874339i \(0.338705\pi\)
\(84\) 0 0
\(85\) 0.155324 − 0.0354518i 0.0168473 − 0.00384528i
\(86\) 0 0
\(87\) −6.19549 − 4.70873i −0.664226 − 0.504829i
\(88\) 0 0
\(89\) 2.51138 − 0.573205i 0.266205 − 0.0607596i −0.0873347 − 0.996179i \(-0.527835\pi\)
0.353540 + 0.935419i \(0.384978\pi\)
\(90\) 0 0
\(91\) −1.23950 + 0.596912i −0.129935 + 0.0625733i
\(92\) 0 0
\(93\) −2.13094 + 9.33625i −0.220968 + 0.968124i
\(94\) 0 0
\(95\) −4.04225 + 3.22359i −0.414726 + 0.330733i
\(96\) 0 0
\(97\) −7.22194 − 14.9965i −0.733277 − 1.52267i −0.848424 − 0.529317i \(-0.822448\pi\)
0.115147 − 0.993348i \(-0.463266\pi\)
\(98\) 0 0
\(99\) 3.96178i 0.398174i
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.y.c.225.2 12
4.3 odd 2 58.2.e.a.51.2 yes 12
12.11 even 2 522.2.n.a.109.1 12
29.4 even 14 inner 464.2.y.c.33.2 12
116.27 even 28 1682.2.a.r.1.4 6
116.31 even 28 1682.2.a.s.1.4 6
116.63 odd 14 1682.2.b.j.1681.3 12
116.91 odd 14 58.2.e.a.33.2 ✓ 12
116.111 odd 14 1682.2.b.j.1681.9 12
348.323 even 14 522.2.n.a.91.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.2.e.a.33.2 ✓ 12 116.91 odd 14
58.2.e.a.51.2 yes 12 4.3 odd 2
464.2.y.c.33.2 12 29.4 even 14 inner
464.2.y.c.225.2 12 1.1 even 1 trivial
522.2.n.a.91.1 12 348.323 even 14
522.2.n.a.109.1 12 12.11 even 2
1682.2.a.r.1.4 6 116.27 even 28
1682.2.a.s.1.4 6 116.31 even 28
1682.2.b.j.1681.3 12 116.63 odd 14
1682.2.b.j.1681.9 12 116.111 odd 14