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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [522,2,Mod(89,522)] 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("522.89"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(522, base_ring=CyclotomicField(28)) chi = DirichletCharacter(H, H._module([14, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 522 = 2 \cdot 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 522.r (of order \(28\), degree \(12\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [72] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.16819098551\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(6\) over \(\Q(\zeta_{28})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{28}]$

Embedding invariants

Embedding label 143.1
Character \(\chi\) \(=\) 522.143
Dual form 522.2.r.b.449.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.993712 - 0.111964i) q^{2} +(0.974928 + 0.222521i) q^{4} +(-2.74885 - 3.44694i) q^{5} +(-0.313020 - 1.37143i) q^{7} +(-0.943883 - 0.330279i) q^{8} +(2.34563 + 3.73304i) q^{10} +(1.14582 - 0.400941i) q^{11} +(0.231462 - 0.480636i) q^{13} +(0.157500 + 1.39786i) q^{14} +(0.900969 + 0.433884i) q^{16} +(-2.44574 + 2.44574i) q^{17} +(-5.33018 + 3.34918i) q^{19} +(-1.91291 - 3.97220i) q^{20} +(-1.18351 + 0.270128i) q^{22} +(1.57772 + 1.25819i) q^{23} +(-3.21267 + 14.0756i) q^{25} +(-0.283821 + 0.451698i) q^{26} -1.40670i q^{28} +(-3.75221 + 3.86276i) q^{29} +(0.875557 - 7.77078i) q^{31} +(-0.846724 - 0.532032i) q^{32} +(2.70419 - 2.15652i) q^{34} +(-3.86680 + 4.84882i) q^{35} +(1.14443 - 3.27058i) q^{37} +(5.67166 - 2.73133i) q^{38} +(1.45614 + 4.16140i) q^{40} +(-1.44848 - 1.44848i) q^{41} +(-7.07277 + 0.796910i) q^{43} +(1.20631 - 0.135919i) q^{44} +(-1.42693 - 1.42693i) q^{46} +(-3.63307 - 10.3827i) q^{47} +(4.52394 - 2.17861i) q^{49} +(4.76843 - 13.6274i) q^{50} +(0.332610 - 0.417080i) q^{52} +(-6.62729 + 5.28509i) q^{53} +(-4.53171 - 2.84746i) q^{55} +(-0.157500 + 1.39786i) q^{56} +(4.16111 - 3.41835i) q^{58} +10.9099i q^{59} +(-2.34513 + 3.73226i) q^{61} +(-1.74010 + 7.62389i) q^{62} +(0.781831 + 0.623490i) q^{64} +(-2.29298 + 0.523357i) q^{65} +(1.30769 + 2.71545i) q^{67} +(-2.92864 + 1.84019i) q^{68} +(4.38539 - 4.38539i) q^{70} +(6.54056 + 3.14977i) q^{71} +(-0.661327 - 5.86944i) q^{73} +(-1.50342 + 3.12188i) q^{74} +(-5.94181 + 2.07913i) q^{76} +(-0.908529 - 1.44592i) q^{77} +(-10.3397 - 3.61803i) q^{79} +(-0.981052 - 4.29827i) q^{80} +(1.27719 + 1.60155i) q^{82} +(-6.29739 - 1.43734i) q^{83} +(15.1533 + 1.70736i) q^{85} +7.11752 q^{86} -1.21395 q^{88} +(13.2326 + 1.49096i) q^{89} +(-0.731611 - 0.166986i) q^{91} +(1.25819 + 1.57772i) q^{92} +(2.44773 + 10.7242i) q^{94} +(26.1963 + 9.16648i) q^{95} +(-7.05601 - 11.2296i) q^{97} +(-4.73942 + 1.65840i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 72 q - 8 q^{7} + 4 q^{10} + 12 q^{16} - 40 q^{19} - 20 q^{25} - 28 q^{31} - 12 q^{37} - 4 q^{40} - 72 q^{43} + 16 q^{46} + 52 q^{49} + 4 q^{52} + 140 q^{55} + 44 q^{58} + 24 q^{61} + 140 q^{67} - 20 q^{70}+ \cdots + 28 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/522\mathbb{Z}\right)^\times\).

\(n\) \(379\) \(407\)
\(\chi(n)\) \(e\left(\frac{15}{28}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.993712 − 0.111964i −0.702661 − 0.0791708i
\(3\) 0 0
\(4\) 0.974928 + 0.222521i 0.487464 + 0.111260i
\(5\) −2.74885 − 3.44694i −1.22932 − 1.54152i −0.745768 − 0.666206i \(-0.767917\pi\)
−0.483554 − 0.875315i \(-0.660654\pi\)
\(6\) 0 0
\(7\) −0.313020 − 1.37143i −0.118311 − 0.518352i −0.999002 − 0.0446623i \(-0.985779\pi\)
0.880692 − 0.473690i \(-0.157078\pi\)
\(8\) −0.943883 − 0.330279i −0.333713 − 0.116771i
\(9\) 0 0
\(10\) 2.34563 + 3.73304i 0.741752 + 1.18049i
\(11\) 1.14582 − 0.400941i 0.345479 − 0.120888i −0.151962 − 0.988386i \(-0.548559\pi\)
0.497440 + 0.867498i \(0.334273\pi\)
\(12\) 0 0
\(13\) 0.231462 − 0.480636i 0.0641960 − 0.133304i −0.866402 − 0.499347i \(-0.833573\pi\)
0.930598 + 0.366043i \(0.119288\pi\)
\(14\) 0.157500 + 1.39786i 0.0420938 + 0.373593i
\(15\) 0 0
\(16\) 0.900969 + 0.433884i 0.225242 + 0.108471i
\(17\) −2.44574 + 2.44574i −0.593178 + 0.593178i −0.938489 − 0.345310i \(-0.887774\pi\)
0.345310 + 0.938489i \(0.387774\pi\)
\(18\) 0 0
\(19\) −5.33018 + 3.34918i −1.22283 + 0.768354i −0.979471 − 0.201583i \(-0.935391\pi\)
−0.243357 + 0.969937i \(0.578249\pi\)
\(20\) −1.91291 − 3.97220i −0.427740 − 0.888211i
\(21\) 0 0
\(22\) −1.18351 + 0.270128i −0.252325 + 0.0575916i
\(23\) 1.57772 + 1.25819i 0.328977 + 0.262350i 0.774023 − 0.633157i \(-0.218241\pi\)
−0.445046 + 0.895508i \(0.646813\pi\)
\(24\) 0 0
\(25\) −3.21267 + 14.0756i −0.642533 + 2.81512i
\(26\) −0.283821 + 0.451698i −0.0556618 + 0.0885853i
\(27\) 0 0
\(28\) − 1.40670i − 0.265841i
\(29\) −3.75221 + 3.86276i −0.696769 + 0.717296i
\(30\) 0 0
\(31\) 0.875557 − 7.77078i 0.157255 − 1.39567i −0.628825 − 0.777547i \(-0.716464\pi\)
0.786079 − 0.618126i \(-0.212108\pi\)
\(32\) −0.846724 − 0.532032i −0.149681 − 0.0940509i
\(33\) 0 0
\(34\) 2.70419 − 2.15652i 0.463765 − 0.369840i
\(35\) −3.86680 + 4.84882i −0.653609 + 0.819600i
\(36\) 0 0
\(37\) 1.14443 − 3.27058i 0.188142 − 0.537680i −0.810717 − 0.585439i \(-0.800922\pi\)
0.998859 + 0.0477590i \(0.0152079\pi\)
\(38\) 5.67166 − 2.73133i 0.920064 − 0.443080i
\(39\) 0 0
\(40\) 1.45614 + 4.16140i 0.230235 + 0.657975i
\(41\) −1.44848 − 1.44848i −0.226214 − 0.226214i 0.584895 − 0.811109i \(-0.301136\pi\)
−0.811109 + 0.584895i \(0.801136\pi\)
\(42\) 0 0
\(43\) −7.07277 + 0.796910i −1.07859 + 0.121528i −0.633335 − 0.773878i \(-0.718315\pi\)
−0.445252 + 0.895405i \(0.646886\pi\)
\(44\) 1.20631 − 0.135919i 0.181858 − 0.0204905i
\(45\) 0 0
\(46\) −1.42693 − 1.42693i −0.210389 − 0.210389i
\(47\) −3.63307 − 10.3827i −0.529937 − 1.51447i −0.828447 − 0.560067i \(-0.810775\pi\)
0.298510 − 0.954406i \(-0.403510\pi\)
\(48\) 0 0
\(49\) 4.52394 − 2.17861i 0.646277 − 0.311231i
\(50\) 4.76843 − 13.6274i 0.674358 − 1.92720i
\(51\) 0 0
\(52\) 0.332610 − 0.417080i 0.0461247 − 0.0578386i
\(53\) −6.62729 + 5.28509i −0.910329 + 0.725963i −0.962100 − 0.272696i \(-0.912085\pi\)
0.0517715 + 0.998659i \(0.483513\pi\)
\(54\) 0 0
\(55\) −4.53171 − 2.84746i −0.611056 − 0.383952i
\(56\) −0.157500 + 1.39786i −0.0210469 + 0.186796i
\(57\) 0 0
\(58\) 4.16111 − 3.41835i 0.546381 − 0.448852i
\(59\) 10.9099i 1.42035i 0.704024 + 0.710176i \(0.251385\pi\)
−0.704024 + 0.710176i \(0.748615\pi\)
\(60\) 0 0
\(61\) −2.34513 + 3.73226i −0.300263 + 0.477867i −0.962445 − 0.271477i \(-0.912488\pi\)
0.662181 + 0.749343i \(0.269631\pi\)
\(62\) −1.74010 + 7.62389i −0.220993 + 0.968235i
\(63\) 0 0
\(64\) 0.781831 + 0.623490i 0.0977289 + 0.0779362i
\(65\) −2.29298 + 0.523357i −0.284409 + 0.0649145i
\(66\) 0 0
\(67\) 1.30769 + 2.71545i 0.159760 + 0.331745i 0.965448 − 0.260595i \(-0.0839186\pi\)
−0.805688 + 0.592340i \(0.798204\pi\)
\(68\) −2.92864 + 1.84019i −0.355150 + 0.223156i
\(69\) 0 0
\(70\) 4.38539 − 4.38539i 0.524154 − 0.524154i
\(71\) 6.54056 + 3.14977i 0.776222 + 0.373809i 0.779674 − 0.626185i \(-0.215385\pi\)
−0.00345229 + 0.999994i \(0.501099\pi\)
\(72\) 0 0
\(73\) −0.661327 − 5.86944i −0.0774025 − 0.686966i −0.971162 − 0.238420i \(-0.923371\pi\)
0.893760 − 0.448546i \(-0.148058\pi\)
\(74\) −1.50342 + 3.12188i −0.174769 + 0.362911i
\(75\) 0 0
\(76\) −5.94181 + 2.07913i −0.681572 + 0.238492i
\(77\) −0.908529 − 1.44592i −0.103536 − 0.164777i
\(78\) 0 0
\(79\) −10.3397 − 3.61803i −1.16331 − 0.407061i −0.321542 − 0.946895i \(-0.604201\pi\)
−0.841771 + 0.539835i \(0.818487\pi\)
\(80\) −0.981052 − 4.29827i −0.109685 − 0.480561i
\(81\) 0 0
\(82\) 1.27719 + 1.60155i 0.141042 + 0.176862i
\(83\) −6.29739 − 1.43734i −0.691228 − 0.157768i −0.137545 − 0.990496i \(-0.543921\pi\)
−0.553683 + 0.832727i \(0.686778\pi\)
\(84\) 0 0
\(85\) 15.1533 + 1.70736i 1.64360 + 0.185190i
\(86\) 7.11752 0.767502
\(87\) 0 0
\(88\) −1.21395 −0.129407
\(89\) 13.2326 + 1.49096i 1.40266 + 0.158042i 0.780649 − 0.624969i \(-0.214888\pi\)
0.622008 + 0.783011i \(0.286317\pi\)
\(90\) 0 0
\(91\) −0.731611 − 0.166986i −0.0766937 − 0.0175048i
\(92\) 1.25819 + 1.57772i 0.131175 + 0.164489i
\(93\) 0 0
\(94\) 2.44773 + 10.7242i 0.252464 + 1.10612i
\(95\) 26.1963 + 9.16648i 2.68768 + 0.940461i
\(96\) 0 0
\(97\) −7.05601 − 11.2296i −0.716429 − 1.14019i −0.984260 − 0.176725i \(-0.943450\pi\)
0.267831 − 0.963466i \(-0.413693\pi\)
\(98\) −4.73942 + 1.65840i −0.478754 + 0.167523i
\(99\) 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 522.2.r.b.143.1 ✓ 72
3.2 odd 2 inner 522.2.r.b.143.6 yes 72
29.14 odd 28 inner 522.2.r.b.449.6 yes 72
87.14 even 28 inner 522.2.r.b.449.1 yes 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
522.2.r.b.143.1 ✓ 72 1.1 even 1 trivial
522.2.r.b.143.6 yes 72 3.2 odd 2 inner
522.2.r.b.449.1 yes 72 87.14 even 28 inner
522.2.r.b.449.6 yes 72 29.14 odd 28 inner