Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [522,2,Mod(175,522)] 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("522.175"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(522, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 522 = 2 \cdot 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 522.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,-6,2,-6,-2] 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: \(4.16819098551\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
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: \( x^{12} - 4 x^{11} + 6 x^{10} - 10 x^{9} + 22 x^{8} - 18 x^{7} - 3 x^{6} - 54 x^{5} + 198 x^{4} + \cdots + 729 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 175.4
Root \(1.73173 - 0.0331860i\) of defining polynomial
Character \(\chi\) \(=\) 522.175
Dual form 522.2.e.h.349.4

$q$-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+(-0.500000 - 0.866025i) q^{2} +(0.837126 + 1.51632i) q^{3} +(-0.500000 + 0.866025i) q^{4} +(1.26263 - 2.18694i) q^{5} +(0.894606 - 1.48313i) q^{6} +(2.51863 + 4.36239i) q^{7} +1.00000 q^{8} +(-1.59844 + 2.53870i) q^{9} -2.52526 q^{10} +(-0.0574799 - 0.0995581i) q^{11} +(-1.73173 - 0.0331860i) q^{12} +(-1.56223 + 2.70587i) q^{13} +(2.51863 - 4.36239i) q^{14} +(4.37307 + 0.0838032i) q^{15} +(-0.500000 - 0.866025i) q^{16} -1.03726 q^{17} +(2.99780 + 0.114939i) q^{18} -2.00263 q^{19} +(1.26263 + 2.18694i) q^{20} +(-4.50636 + 7.47091i) q^{21} +(-0.0574799 + 0.0995581i) q^{22} +(1.32011 - 2.28649i) q^{23} +(0.837126 + 1.51632i) q^{24} +(-0.688457 - 1.19244i) q^{25} +3.12446 q^{26} +(-5.18757 - 0.298528i) q^{27} -5.03726 q^{28} +(0.500000 + 0.866025i) q^{29} +(-2.11396 - 3.82909i) q^{30} +(4.30332 - 7.45357i) q^{31} +(-0.500000 + 0.866025i) q^{32} +(0.102844 - 0.170500i) q^{33} +(0.518628 + 0.898290i) q^{34} +12.7204 q^{35} +(-1.39936 - 2.65364i) q^{36} +7.08514 q^{37} +(1.00132 + 1.73433i) q^{38} +(-5.41074 - 0.103689i) q^{39} +(1.26263 - 2.18694i) q^{40} +(-3.92230 + 6.79362i) q^{41} +(8.72318 + 0.167167i) q^{42} +(3.02338 + 5.23665i) q^{43} +0.114960 q^{44} +(3.53374 + 6.70111i) q^{45} -2.64021 q^{46} +(0.522065 + 0.904243i) q^{47} +(0.894606 - 1.48313i) q^{48} +(-9.18698 + 15.9123i) q^{49} +(-0.688457 + 1.19244i) q^{50} +(-0.868315 - 1.57281i) q^{51} +(-1.56223 - 2.70587i) q^{52} +6.62572 q^{53} +(2.33525 + 4.64183i) q^{54} -0.290303 q^{55} +(2.51863 + 4.36239i) q^{56} +(-1.67646 - 3.03663i) q^{57} +(0.500000 - 0.866025i) q^{58} +(3.13109 - 5.42321i) q^{59} +(-2.25911 + 3.74529i) q^{60} +(-5.74109 - 9.94386i) q^{61} -8.60664 q^{62} +(-15.1007 - 0.578976i) q^{63} +1.00000 q^{64} +(3.94504 + 6.83300i) q^{65} +(-0.199080 - 0.00381506i) q^{66} +(3.62578 - 6.28004i) q^{67} +(0.518628 - 0.898290i) q^{68} +(4.57215 + 0.0876182i) q^{69} +(-6.36018 - 11.0162i) q^{70} +10.3101 q^{71} +(-1.59844 + 2.53870i) q^{72} +4.05364 q^{73} +(-3.54257 - 6.13591i) q^{74} +(1.23180 - 2.04214i) q^{75} +(1.00132 - 1.73433i) q^{76} +(0.289541 - 0.501500i) q^{77} +(2.61557 + 4.73768i) q^{78} +(-5.91578 - 10.2464i) q^{79} -2.52526 q^{80} +(-3.88999 - 8.11591i) q^{81} +7.84459 q^{82} +(-6.47514 - 11.2153i) q^{83} +(-4.21682 - 7.63808i) q^{84} +(-1.30967 + 2.26841i) q^{85} +(3.02338 - 5.23665i) q^{86} +(-0.894606 + 1.48313i) q^{87} +(-0.0574799 - 0.0995581i) q^{88} -6.70147 q^{89} +(4.03647 - 6.41086i) q^{90} -15.7387 q^{91} +(1.32011 + 2.28649i) q^{92} +(14.9044 + 0.285620i) q^{93} +(0.522065 - 0.904243i) q^{94} +(-2.52858 + 4.37962i) q^{95} +(-1.73173 - 0.0331860i) q^{96} +(1.74508 + 3.02257i) q^{97} +18.3740 q^{98} +(0.344626 + 0.0132133i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{2} + 2 q^{3} - 6 q^{4} - 2 q^{5} + 2 q^{6} + 8 q^{7} + 12 q^{8} - 8 q^{9} + 4 q^{10} - 4 q^{12} - 6 q^{13} + 8 q^{14} + 14 q^{15} - 6 q^{16} + 32 q^{17} + 4 q^{18} + 4 q^{19} - 2 q^{20} - 10 q^{21}+ \cdots - 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(379\) \(407\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.500000 − 0.866025i −0.353553 − 0.612372i
\(3\) 0.837126 + 1.51632i 0.483315 + 0.875446i
\(4\) −0.500000 + 0.866025i −0.250000 + 0.433013i
\(5\) 1.26263 − 2.18694i 0.564664 − 0.978027i −0.432417 − 0.901674i \(-0.642339\pi\)
0.997081 − 0.0763532i \(-0.0243277\pi\)
\(6\) 0.894606 − 1.48313i 0.365222 − 0.605486i
\(7\) 2.51863 + 4.36239i 0.951952 + 1.64883i 0.741194 + 0.671291i \(0.234260\pi\)
0.210758 + 0.977538i \(0.432407\pi\)
\(8\) 1.00000 0.353553
\(9\) −1.59844 + 2.53870i −0.532813 + 0.846233i
\(10\) −2.52526 −0.798556
\(11\) −0.0574799 − 0.0995581i −0.0173308 − 0.0300179i 0.857230 − 0.514934i \(-0.172184\pi\)
−0.874561 + 0.484916i \(0.838850\pi\)
\(12\) −1.73173 − 0.0331860i −0.499908 − 0.00957998i
\(13\) −1.56223 + 2.70587i −0.433285 + 0.750472i −0.997154 − 0.0753923i \(-0.975979\pi\)
0.563869 + 0.825864i \(0.309312\pi\)
\(14\) 2.51863 − 4.36239i 0.673132 − 1.16590i
\(15\) 4.37307 + 0.0838032i 1.12912 + 0.0216379i
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) −1.03726 −0.251572 −0.125786 − 0.992057i \(-0.540145\pi\)
−0.125786 + 0.992057i \(0.540145\pi\)
\(18\) 2.99780 + 0.114939i 0.706588 + 0.0270913i
\(19\) −2.00263 −0.459435 −0.229718 − 0.973257i \(-0.573780\pi\)
−0.229718 + 0.973257i \(0.573780\pi\)
\(20\) 1.26263 + 2.18694i 0.282332 + 0.489014i
\(21\) −4.50636 + 7.47091i −0.983369 + 1.63029i
\(22\) −0.0574799 + 0.0995581i −0.0122548 + 0.0212259i
\(23\) 1.32011 − 2.28649i 0.275261 − 0.476767i −0.694940 − 0.719068i \(-0.744569\pi\)
0.970201 + 0.242301i \(0.0779023\pi\)
\(24\) 0.837126 + 1.51632i 0.170878 + 0.309517i
\(25\) −0.688457 − 1.19244i −0.137691 − 0.238488i
\(26\) 3.12446 0.612758
\(27\) −5.18757 − 0.298528i −0.998348 − 0.0574518i
\(28\) −5.03726 −0.951952
\(29\) 0.500000 + 0.866025i 0.0928477 + 0.160817i
\(30\) −2.11396 − 3.82909i −0.385954 − 0.699093i
\(31\) 4.30332 − 7.45357i 0.772899 − 1.33870i −0.163068 − 0.986615i \(-0.552139\pi\)
0.935968 − 0.352086i \(-0.114528\pi\)
\(32\) −0.500000 + 0.866025i −0.0883883 + 0.153093i
\(33\) 0.102844 − 0.170500i 0.0179028 − 0.0296803i
\(34\) 0.518628 + 0.898290i 0.0889440 + 0.154056i
\(35\) 12.7204 2.15013
\(36\) −1.39936 − 2.65364i −0.233226 − 0.442273i
\(37\) 7.08514 1.16479 0.582395 − 0.812906i \(-0.302116\pi\)
0.582395 + 0.812906i \(0.302116\pi\)
\(38\) 1.00132 + 1.73433i 0.162435 + 0.281345i
\(39\) −5.41074 − 0.103689i −0.866412 − 0.0166035i
\(40\) 1.26263 − 2.18694i 0.199639 − 0.345785i
\(41\) −3.92230 + 6.79362i −0.612560 + 1.06098i 0.378248 + 0.925704i \(0.376527\pi\)
−0.990807 + 0.135280i \(0.956807\pi\)
\(42\) 8.72318 + 0.167167i 1.34602 + 0.0257944i
\(43\) 3.02338 + 5.23665i 0.461061 + 0.798581i 0.999014 − 0.0443934i \(-0.0141355\pi\)
−0.537953 + 0.842975i \(0.680802\pi\)
\(44\) 0.114960 0.0173308
\(45\) 3.53374 + 6.70111i 0.526779 + 0.998943i
\(46\) −2.64021 −0.389278
\(47\) 0.522065 + 0.904243i 0.0761510 + 0.131897i 0.901586 − 0.432599i \(-0.142404\pi\)
−0.825435 + 0.564497i \(0.809070\pi\)
\(48\) 0.894606 − 1.48313i 0.129125 − 0.214072i
\(49\) −9.18698 + 15.9123i −1.31243 + 2.27319i
\(50\) −0.688457 + 1.19244i −0.0973625 + 0.168637i
\(51\) −0.868315 − 1.57281i −0.121588 − 0.220237i
\(52\) −1.56223 − 2.70587i −0.216643 − 0.375236i
\(53\) 6.62572 0.910112 0.455056 − 0.890463i \(-0.349619\pi\)
0.455056 + 0.890463i \(0.349619\pi\)
\(54\) 2.33525 + 4.64183i 0.317788 + 0.631673i
\(55\) −0.290303 −0.0391444
\(56\) 2.51863 + 4.36239i 0.336566 + 0.582949i
\(57\) −1.67646 − 3.03663i −0.222052 − 0.402211i
\(58\) 0.500000 − 0.866025i 0.0656532 − 0.113715i
\(59\) 3.13109 − 5.42321i 0.407633 − 0.706042i −0.586991 − 0.809594i \(-0.699687\pi\)
0.994624 + 0.103552i \(0.0330208\pi\)
\(60\) −2.25911 + 3.74529i −0.291650 + 0.483514i
\(61\) −5.74109 − 9.94386i −0.735071 − 1.27318i −0.954692 − 0.297595i \(-0.903815\pi\)
0.219621 − 0.975585i \(-0.429518\pi\)
\(62\) −8.60664 −1.09304
\(63\) −15.1007 − 0.578976i −1.90251 − 0.0729441i
\(64\) 1.00000 0.125000
\(65\) 3.94504 + 6.83300i 0.489321 + 0.847530i
\(66\) −0.199080 − 0.00381506i −0.0245050 − 0.000469601i
\(67\) 3.62578 − 6.28004i 0.442960 − 0.767228i −0.554948 − 0.831885i \(-0.687262\pi\)
0.997908 + 0.0646565i \(0.0205952\pi\)
\(68\) 0.518628 − 0.898290i 0.0628929 − 0.108934i
\(69\) 4.57215 + 0.0876182i 0.550422 + 0.0105480i
\(70\) −6.36018 − 11.0162i −0.760187 − 1.31668i
\(71\) 10.3101 1.22358 0.611790 − 0.791020i \(-0.290450\pi\)
0.611790 + 0.791020i \(0.290450\pi\)
\(72\) −1.59844 + 2.53870i −0.188378 + 0.299189i
\(73\) 4.05364 0.474442 0.237221 − 0.971456i \(-0.423763\pi\)
0.237221 + 0.971456i \(0.423763\pi\)
\(74\) −3.54257 − 6.13591i −0.411815 − 0.713285i
\(75\) 1.23180 − 2.04214i 0.142236 − 0.235807i
\(76\) 1.00132 − 1.73433i 0.114859 − 0.198941i
\(77\) 0.289541 − 0.501500i 0.0329963 − 0.0571512i
\(78\) 2.61557 + 4.73768i 0.296155 + 0.536437i
\(79\) −5.91578 − 10.2464i −0.665578 − 1.15281i −0.979128 − 0.203243i \(-0.934852\pi\)
0.313551 − 0.949572i \(-0.398481\pi\)
\(80\) −2.52526 −0.282332
\(81\) −3.88999 − 8.11591i −0.432221 − 0.901768i
\(82\) 7.84459 0.866290
\(83\) −6.47514 − 11.2153i −0.710739 − 1.23104i −0.964580 − 0.263790i \(-0.915027\pi\)
0.253841 − 0.967246i \(-0.418306\pi\)
\(84\) −4.21682 − 7.63808i −0.460093 − 0.833383i
\(85\) −1.30967 + 2.26841i −0.142054 + 0.246044i
\(86\) 3.02338 − 5.23665i 0.326020 − 0.564682i
\(87\) −0.894606 + 1.48313i −0.0959119 + 0.159008i
\(88\) −0.0574799 − 0.0995581i −0.00612738 − 0.0106129i
\(89\) −6.70147 −0.710355 −0.355177 − 0.934799i \(-0.615580\pi\)
−0.355177 + 0.934799i \(0.615580\pi\)
\(90\) 4.03647 − 6.41086i 0.425481 − 0.675764i
\(91\) −15.7387 −1.64987
\(92\) 1.32011 + 2.28649i 0.137631 + 0.238383i
\(93\) 14.9044 + 0.285620i 1.54551 + 0.0296174i
\(94\) 0.522065 − 0.904243i 0.0538469 − 0.0932656i
\(95\) −2.52858 + 4.37962i −0.259427 + 0.449340i
\(96\) −1.73173 − 0.0331860i −0.176744 − 0.00338703i
\(97\) 1.74508 + 3.02257i 0.177186 + 0.306896i 0.940916 − 0.338641i \(-0.109967\pi\)
−0.763729 + 0.645537i \(0.776634\pi\)
\(98\) 18.3740 1.85605
\(99\) 0.344626 + 0.0132133i 0.0346362 + 0.00132799i
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.e.h.175.4 ✓ 12
3.2 odd 2 1566.2.e.h.523.2 12
9.2 odd 6 1566.2.e.h.1045.2 12
9.4 even 3 4698.2.a.bh.1.2 6
9.5 odd 6 4698.2.a.be.1.5 6
9.7 even 3 inner 522.2.e.h.349.4 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
522.2.e.h.175.4 ✓ 12 1.1 even 1 trivial
522.2.e.h.349.4 yes 12 9.7 even 3 inner
1566.2.e.h.523.2 12 3.2 odd 2
1566.2.e.h.1045.2 12 9.2 odd 6
4698.2.a.be.1.5 6 9.5 odd 6
4698.2.a.bh.1.2 6 9.4 even 3