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 175.3
Root \(-1.55661 + 0.759577i\) of defining polynomial
Character \(\chi\) \(=\) 522.175
Dual form 522.2.e.h.349.3

$q$-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} +(-0.120494 - 1.72785i) q^{3} +(-0.500000 + 0.866025i) q^{4} +(-1.19682 + 2.07295i) q^{5} +(-1.43612 + 0.968278i) q^{6} +(0.279091 + 0.483399i) q^{7} +1.00000 q^{8} +(-2.97096 + 0.416391i) q^{9} +2.39363 q^{10} +(1.31563 + 2.27873i) q^{11} +(1.55661 + 0.759577i) q^{12} +(0.00438226 - 0.00759030i) q^{13} +(0.279091 - 0.483399i) q^{14} +(3.72596 + 1.81815i) q^{15} +(-0.500000 - 0.866025i) q^{16} +3.44182 q^{17} +(1.84609 + 2.36473i) q^{18} +7.57655 q^{19} +(-1.19682 - 2.07295i) q^{20} +(0.801615 - 0.540475i) q^{21} +(1.31563 - 2.27873i) q^{22} +(-2.51244 + 4.35168i) q^{23} +(-0.120494 - 1.72785i) q^{24} +(-0.364741 - 0.631750i) q^{25} -0.00876452 q^{26} +(1.07745 + 5.08322i) q^{27} -0.558181 q^{28} +(0.500000 + 0.866025i) q^{29} +(-0.288418 - 4.13585i) q^{30} +(1.96088 - 3.39634i) q^{31} +(-0.500000 + 0.866025i) q^{32} +(3.77879 - 2.54778i) q^{33} +(-1.72091 - 2.98070i) q^{34} -1.33608 q^{35} +(1.12488 - 2.78112i) q^{36} +2.34746 q^{37} +(-3.78827 - 6.56148i) q^{38} +(-0.0136430 - 0.00665732i) q^{39} +(-1.19682 + 2.07295i) q^{40} +(-2.18943 + 3.79221i) q^{41} +(-0.868872 - 0.423981i) q^{42} +(0.590334 + 1.02249i) q^{43} -2.63125 q^{44} +(2.69254 - 6.65699i) q^{45} +5.02488 q^{46} +(2.87861 + 4.98589i) q^{47} +(-1.43612 + 0.968278i) q^{48} +(3.34422 - 5.79235i) q^{49} +(-0.364741 + 0.631750i) q^{50} +(-0.414717 - 5.94696i) q^{51} +(0.00438226 + 0.00759030i) q^{52} -2.30603 q^{53} +(3.86347 - 3.47470i) q^{54} -6.29825 q^{55} +(0.279091 + 0.483399i) q^{56} +(-0.912926 - 13.0912i) q^{57} +(0.500000 - 0.866025i) q^{58} +(-2.68149 + 4.64447i) q^{59} +(-3.43754 + 2.31770i) q^{60} +(-4.96526 - 8.60009i) q^{61} -3.92176 q^{62} +(-1.03045 - 1.31995i) q^{63} +1.00000 q^{64} +(0.0104895 + 0.0181684i) q^{65} +(-4.09584 - 1.99864i) q^{66} +(-4.29704 + 7.44269i) q^{67} +(-1.72091 + 2.98070i) q^{68} +(7.82180 + 3.81678i) q^{69} +(0.668041 + 1.15708i) q^{70} -1.57370 q^{71} +(-2.97096 + 0.416391i) q^{72} +8.37970 q^{73} +(-1.17373 - 2.03296i) q^{74} +(-1.04762 + 0.706341i) q^{75} +(-3.78827 + 6.56148i) q^{76} +(-0.734357 + 1.27194i) q^{77} +(0.00105607 + 0.0151438i) q^{78} +(-0.000301254 - 0.000521786i) q^{79} +2.39363 q^{80} +(8.65324 - 2.47417i) q^{81} +4.37886 q^{82} +(3.29929 + 5.71454i) q^{83} +(0.0672573 + 0.964456i) q^{84} +(-4.11923 + 7.13471i) q^{85} +(0.590334 - 1.02249i) q^{86} +(1.43612 - 0.968278i) q^{87} +(1.31563 + 2.27873i) q^{88} +4.56307 q^{89} +(-7.11140 + 0.996688i) q^{90} +0.00489219 q^{91} +(-2.51244 - 4.35168i) q^{92} +(-6.10466 - 2.97888i) q^{93} +(2.87861 - 4.98589i) q^{94} +(-9.06774 + 15.7058i) q^{95} +(1.55661 + 0.759577i) q^{96} +(7.86908 + 13.6297i) q^{97} -6.68843 q^{98} +(-4.85752 - 6.22221i) 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
\(3\) −0.120494 − 1.72785i −0.0695671 − 0.997577i
\(4\) −0.500000 + 0.866025i −0.250000 + 0.433013i
\(5\) −1.19682 + 2.07295i −0.535233 + 0.927050i 0.463919 + 0.885877i \(0.346443\pi\)
−0.999152 + 0.0411728i \(0.986891\pi\)
\(6\) −1.43612 + 0.968278i −0.586293 + 0.395298i
\(7\) 0.279091 + 0.483399i 0.105486 + 0.182708i 0.913937 − 0.405857i \(-0.133027\pi\)
−0.808450 + 0.588564i \(0.799693\pi\)
\(8\) 1.00000 0.353553
\(9\) −2.97096 + 0.416391i −0.990321 + 0.138797i
\(10\) 2.39363 0.756933
\(11\) 1.31563 + 2.27873i 0.396676 + 0.687063i 0.993314 − 0.115448i \(-0.0368303\pi\)
−0.596638 + 0.802511i \(0.703497\pi\)
\(12\) 1.55661 + 0.759577i 0.449355 + 0.219271i
\(13\) 0.00438226 − 0.00759030i 0.00121542 − 0.00210517i −0.865417 − 0.501052i \(-0.832946\pi\)
0.866632 + 0.498947i \(0.166280\pi\)
\(14\) 0.279091 − 0.483399i 0.0745901 − 0.129194i
\(15\) 3.72596 + 1.81815i 0.962039 + 0.469444i
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) 3.44182 0.834764 0.417382 − 0.908731i \(-0.362948\pi\)
0.417382 + 0.908731i \(0.362948\pi\)
\(18\) 1.84609 + 2.36473i 0.435127 + 0.557373i
\(19\) 7.57655 1.73818 0.869089 − 0.494655i \(-0.164706\pi\)
0.869089 + 0.494655i \(0.164706\pi\)
\(20\) −1.19682 − 2.07295i −0.267616 − 0.463525i
\(21\) 0.801615 − 0.540475i 0.174927 − 0.117941i
\(22\) 1.31563 − 2.27873i 0.280492 − 0.485827i
\(23\) −2.51244 + 4.35168i −0.523880 + 0.907387i 0.475733 + 0.879590i \(0.342183\pi\)
−0.999614 + 0.0277978i \(0.991151\pi\)
\(24\) −0.120494 − 1.72785i −0.0245957 − 0.352697i
\(25\) −0.364741 − 0.631750i −0.0729482 − 0.126350i
\(26\) −0.00876452 −0.00171886
\(27\) 1.07745 + 5.08322i 0.207355 + 0.978266i
\(28\) −0.558181 −0.105486
\(29\) 0.500000 + 0.866025i 0.0928477 + 0.160817i
\(30\) −0.288418 − 4.13585i −0.0526576 − 0.755100i
\(31\) 1.96088 − 3.39634i 0.352185 − 0.610002i −0.634447 − 0.772966i \(-0.718772\pi\)
0.986632 + 0.162965i \(0.0521056\pi\)
\(32\) −0.500000 + 0.866025i −0.0883883 + 0.153093i
\(33\) 3.77879 − 2.54778i 0.657803 − 0.443512i
\(34\) −1.72091 − 2.98070i −0.295134 − 0.511186i
\(35\) −1.33608 −0.225839
\(36\) 1.12488 − 2.78112i 0.187479 − 0.463521i
\(37\) 2.34746 0.385920 0.192960 − 0.981207i \(-0.438191\pi\)
0.192960 + 0.981207i \(0.438191\pi\)
\(38\) −3.78827 − 6.56148i −0.614539 − 1.06441i
\(39\) −0.0136430 − 0.00665732i −0.00218462 − 0.00106602i
\(40\) −1.19682 + 2.07295i −0.189233 + 0.327762i
\(41\) −2.18943 + 3.79221i −0.341932 + 0.592243i −0.984791 − 0.173741i \(-0.944415\pi\)
0.642860 + 0.765984i \(0.277748\pi\)
\(42\) −0.868872 − 0.423981i −0.134070 − 0.0654218i
\(43\) 0.590334 + 1.02249i 0.0900251 + 0.155928i 0.907521 − 0.420006i \(-0.137972\pi\)
−0.817496 + 0.575934i \(0.804639\pi\)
\(44\) −2.63125 −0.396676
\(45\) 2.69254 − 6.65699i 0.401380 − 0.992366i
\(46\) 5.02488 0.740879
\(47\) 2.87861 + 4.98589i 0.419888 + 0.727267i 0.995928 − 0.0901543i \(-0.0287360\pi\)
−0.576040 + 0.817422i \(0.695403\pi\)
\(48\) −1.43612 + 0.968278i −0.207286 + 0.139759i
\(49\) 3.34422 − 5.79235i 0.477745 − 0.827479i
\(50\) −0.364741 + 0.631750i −0.0515821 + 0.0893429i
\(51\) −0.414717 − 5.94696i −0.0580721 − 0.832741i
\(52\) 0.00438226 + 0.00759030i 0.000607710 + 0.00105258i
\(53\) −2.30603 −0.316757 −0.158379 − 0.987378i \(-0.550627\pi\)
−0.158379 + 0.987378i \(0.550627\pi\)
\(54\) 3.86347 − 3.47470i 0.525752 − 0.472847i
\(55\) −6.29825 −0.849256
\(56\) 0.279091 + 0.483399i 0.0372951 + 0.0645969i
\(57\) −0.912926 − 13.0912i −0.120920 − 1.73397i
\(58\) 0.500000 − 0.866025i 0.0656532 − 0.113715i
\(59\) −2.68149 + 4.64447i −0.349100 + 0.604659i −0.986090 − 0.166213i \(-0.946846\pi\)
0.636990 + 0.770872i \(0.280179\pi\)
\(60\) −3.43754 + 2.31770i −0.443785 + 0.299214i
\(61\) −4.96526 − 8.60009i −0.635737 − 1.10113i −0.986359 − 0.164611i \(-0.947363\pi\)
0.350622 − 0.936517i \(-0.385970\pi\)
\(62\) −3.92176 −0.498064
\(63\) −1.03045 − 1.31995i −0.129825 − 0.166298i
\(64\) 1.00000 0.125000
\(65\) 0.0104895 + 0.0181684i 0.00130107 + 0.00225351i
\(66\) −4.09584 − 1.99864i −0.504163 − 0.246015i
\(67\) −4.29704 + 7.44269i −0.524967 + 0.909269i 0.474611 + 0.880196i \(0.342589\pi\)
−0.999577 + 0.0290731i \(0.990744\pi\)
\(68\) −1.72091 + 2.98070i −0.208691 + 0.361463i
\(69\) 7.82180 + 3.81678i 0.941634 + 0.459487i
\(70\) 0.668041 + 1.15708i 0.0798461 + 0.138298i
\(71\) −1.57370 −0.186764 −0.0933818 − 0.995630i \(-0.529768\pi\)
−0.0933818 + 0.995630i \(0.529768\pi\)
\(72\) −2.97096 + 0.416391i −0.350131 + 0.0490722i
\(73\) 8.37970 0.980770 0.490385 − 0.871506i \(-0.336856\pi\)
0.490385 + 0.871506i \(0.336856\pi\)
\(74\) −1.17373 − 2.03296i −0.136443 − 0.236327i
\(75\) −1.04762 + 0.706341i −0.120969 + 0.0815612i
\(76\) −3.78827 + 6.56148i −0.434545 + 0.752654i
\(77\) −0.734357 + 1.27194i −0.0836878 + 0.144952i
\(78\) 0.00105607 + 0.0151438i 0.000119576 + 0.00171470i
\(79\) −0.000301254 0 0.000521786i −3.38937e−5 0 5.87056e-5i 0.866008 − 0.500029i \(-0.166677\pi\)
−0.866042 + 0.499971i \(0.833344\pi\)
\(80\) 2.39363 0.267616
\(81\) 8.65324 − 2.47417i 0.961471 − 0.274907i
\(82\) 4.37886 0.483565
\(83\) 3.29929 + 5.71454i 0.362144 + 0.627252i 0.988314 − 0.152435i \(-0.0487114\pi\)
−0.626169 + 0.779687i \(0.715378\pi\)
\(84\) 0.0672573 + 0.964456i 0.00733838 + 0.105231i
\(85\) −4.11923 + 7.13471i −0.446793 + 0.773868i
\(86\) 0.590334 − 1.02249i 0.0636573 − 0.110258i
\(87\) 1.43612 − 0.968278i 0.153968 − 0.103810i
\(88\) 1.31563 + 2.27873i 0.140246 + 0.242913i
\(89\) 4.56307 0.483685 0.241842 − 0.970316i \(-0.422248\pi\)
0.241842 + 0.970316i \(0.422248\pi\)
\(90\) −7.11140 + 0.996688i −0.749607 + 0.105060i
\(91\) 0.00489219 0.000512841
\(92\) −2.51244 − 4.35168i −0.261940 − 0.453694i
\(93\) −6.10466 − 2.97888i −0.633024 − 0.308895i
\(94\) 2.87861 − 4.98589i 0.296906 − 0.514256i
\(95\) −9.06774 + 15.7058i −0.930330 + 1.61138i
\(96\) 1.55661 + 0.759577i 0.158871 + 0.0775240i
\(97\) 7.86908 + 13.6297i 0.798984 + 1.38388i 0.920278 + 0.391265i \(0.127962\pi\)
−0.121294 + 0.992617i \(0.538704\pi\)
\(98\) −6.68843 −0.675634
\(99\) −4.85752 − 6.22221i −0.488199 − 0.625355i
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.175.3 ✓ 12
3.2 odd 2 1566.2.e.h.523.5 12
9.2 odd 6 1566.2.e.h.1045.5 12
9.4 even 3 4698.2.a.bh.1.5 6
9.5 odd 6 4698.2.a.be.1.2 6
9.7 even 3 inner 522.2.e.h.349.3 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
522.2.e.h.175.3 ✓ 12 1.1 even 1 trivial
522.2.e.h.349.3 yes 12 9.7 even 3 inner
1566.2.e.h.523.5 12 3.2 odd 2
1566.2.e.h.1045.5 12 9.2 odd 6
4698.2.a.be.1.2 6 9.5 odd 6
4698.2.a.bh.1.5 6 9.4 even 3