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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [675,2,Mod(49,675)] 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("675.49"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(675, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([14, 9])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 675.u (of order \(18\), degree \(6\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24] 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: \(5.38990213644\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(4\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 27)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 574.3
Character \(\chi\) \(=\) 675.574
Dual form 675.2.u.b.274.3

$q$-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+(0.274138 + 0.753189i) q^{2} +(0.386327 + 1.68842i) q^{3} +(1.03995 - 0.872619i) q^{4} +(-1.16579 + 0.753837i) q^{6} +(-1.52780 + 1.82076i) q^{7} +(2.33062 + 1.34559i) q^{8} +(-2.70150 + 1.30456i) q^{9} +(-0.0434396 + 0.246358i) q^{11} +(1.87510 + 1.41875i) q^{12} +(-0.893351 + 2.45446i) q^{13} +(-1.79020 - 0.651581i) q^{14} +(0.0969067 - 0.549585i) q^{16} +(-0.254072 + 0.146688i) q^{17} +(-1.72317 - 1.67711i) q^{18} +(-1.39237 + 2.41166i) q^{19} +(-3.66443 - 1.87615i) q^{21} +(-0.197463 + 0.0348180i) q^{22} +(4.30015 + 5.12472i) q^{23} +(-1.37153 + 4.45490i) q^{24} -2.09357 q^{26} +(-3.24631 - 4.05728i) q^{27} +3.22668i q^{28} +(-0.333645 + 0.121437i) q^{29} +(2.11847 - 1.77761i) q^{31} +(5.74108 - 1.01231i) q^{32} +(-0.432738 + 0.0218307i) q^{33} +(-0.180135 - 0.151151i) q^{34} +(-1.67103 + 3.71406i) q^{36} +(-6.05558 + 3.49619i) q^{37} +(-2.19814 - 0.387591i) q^{38} +(-4.48928 - 0.560124i) q^{39} +(9.13156 + 3.32362i) q^{41} +(0.408537 - 3.27434i) q^{42} +(-0.256746 - 0.0452712i) q^{43} +(0.169802 + 0.294106i) q^{44} +(-2.68104 + 4.64370i) q^{46} +(7.34421 - 8.75249i) q^{47} +(0.965367 - 0.0487006i) q^{48} +(0.234540 + 1.33014i) q^{49} +(-0.345826 - 0.372309i) q^{51} +(1.21277 + 3.33207i) q^{52} -5.43137i q^{53} +(2.16596 - 3.55734i) q^{54} +(-6.01071 + 2.18772i) q^{56} +(-4.60980 - 1.41922i) q^{57} +(-0.182930 - 0.218007i) q^{58} +(-1.03788 - 5.88612i) q^{59} +(-9.07515 - 7.61495i) q^{61} +(1.91963 + 1.10830i) q^{62} +(1.75206 - 6.91190i) q^{63} +(1.77824 + 3.08001i) q^{64} +(-0.135073 - 0.319949i) q^{66} +(0.619160 - 1.70113i) q^{67} +(-0.136218 + 0.374256i) q^{68} +(-6.99139 + 9.24026i) q^{69} +(0.185255 + 0.320871i) q^{71} +(-8.05158 - 0.594663i) q^{72} +(4.35333 + 2.51339i) q^{73} +(-4.29336 - 3.60255i) q^{74} +(0.656467 + 3.72301i) q^{76} +(-0.382193 - 0.455479i) q^{77} +(-0.808804 - 3.53483i) q^{78} +(0.754406 - 0.274581i) q^{79} +(5.59624 - 7.04855i) q^{81} +7.78892i q^{82} +(-0.942488 - 2.58947i) q^{83} +(-5.44798 + 1.24655i) q^{84} +(-0.0362861 - 0.205789i) q^{86} +(-0.333932 - 0.516417i) q^{87} +(-0.432738 + 0.515717i) q^{88} +(5.22533 - 9.05054i) q^{89} +(-3.10412 - 5.37650i) q^{91} +(8.94385 + 1.57704i) q^{92} +(3.81976 + 2.89012i) q^{93} +(8.60560 + 3.13218i) q^{94} +(3.92713 + 9.30225i) q^{96} +(-14.6092 - 2.57600i) q^{97} +(-0.937552 + 0.541296i) q^{98} +(-0.204037 - 0.722208i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 12 q^{4} + 6 q^{11} - 30 q^{14} + 6 q^{19} - 24 q^{21} + 36 q^{24} - 60 q^{26} + 12 q^{29} + 6 q^{31} - 18 q^{34} + 36 q^{36} - 66 q^{39} + 30 q^{41} - 6 q^{44} - 6 q^{46} - 24 q^{49} - 36 q^{51}+ \cdots + 54 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/675\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\)
\(\chi(n)\) \(e\left(\frac{8}{9}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.274138 + 0.753189i 0.193845 + 0.532585i 0.998094 − 0.0617072i \(-0.0196545\pi\)
−0.804249 + 0.594292i \(0.797432\pi\)
\(3\) 0.386327 + 1.68842i 0.223046 + 0.974808i
\(4\) 1.03995 − 0.872619i 0.519974 − 0.436310i
\(5\) 0 0
\(6\) −1.16579 + 0.753837i −0.475932 + 0.307753i
\(7\) −1.52780 + 1.82076i −0.577454 + 0.688183i −0.973143 − 0.230202i \(-0.926061\pi\)
0.395689 + 0.918385i \(0.370506\pi\)
\(8\) 2.33062 + 1.34559i 0.823999 + 0.475736i
\(9\) −2.70150 + 1.30456i −0.900501 + 0.434854i
\(10\) 0 0
\(11\) −0.0434396 + 0.246358i −0.0130975 + 0.0742798i −0.990656 − 0.136385i \(-0.956452\pi\)
0.977558 + 0.210665i \(0.0675628\pi\)
\(12\) 1.87510 + 1.41875i 0.541296 + 0.409557i
\(13\) −0.893351 + 2.45446i −0.247771 + 0.680745i 0.751996 + 0.659167i \(0.229091\pi\)
−0.999767 + 0.0215777i \(0.993131\pi\)
\(14\) −1.79020 − 0.651581i −0.478452 − 0.174142i
\(15\) 0 0
\(16\) 0.0969067 − 0.549585i 0.0242267 − 0.137396i
\(17\) −0.254072 + 0.146688i −0.0616215 + 0.0355772i −0.530494 − 0.847689i \(-0.677994\pi\)
0.468873 + 0.883266i \(0.344660\pi\)
\(18\) −1.72317 − 1.67711i −0.406154 − 0.395299i
\(19\) −1.39237 + 2.41166i −0.319432 + 0.553273i −0.980370 − 0.197168i \(-0.936825\pi\)
0.660937 + 0.750441i \(0.270159\pi\)
\(20\) 0 0
\(21\) −3.66443 − 1.87615i −0.799645 − 0.409410i
\(22\) −0.197463 + 0.0348180i −0.0420992 + 0.00742323i
\(23\) 4.30015 + 5.12472i 0.896643 + 1.06858i 0.997284 + 0.0736543i \(0.0234662\pi\)
−0.100641 + 0.994923i \(0.532089\pi\)
\(24\) −1.37153 + 4.45490i −0.279962 + 0.909352i
\(25\) 0 0
\(26\) −2.09357 −0.410584
\(27\) −3.24631 − 4.05728i −0.624752 − 0.780823i
\(28\) 3.22668i 0.609786i
\(29\) −0.333645 + 0.121437i −0.0619562 + 0.0225502i −0.372812 − 0.927907i \(-0.621606\pi\)
0.310856 + 0.950457i \(0.399384\pi\)
\(30\) 0 0
\(31\) 2.11847 − 1.77761i 0.380488 − 0.319268i −0.432406 − 0.901679i \(-0.642335\pi\)
0.812894 + 0.582411i \(0.197891\pi\)
\(32\) 5.74108 − 1.01231i 1.01489 − 0.178952i
\(33\) −0.432738 + 0.0218307i −0.0753299 + 0.00380023i
\(34\) −0.180135 − 0.151151i −0.0308929 − 0.0259222i
\(35\) 0 0
\(36\) −1.67103 + 3.71406i −0.278506 + 0.619010i
\(37\) −6.05558 + 3.49619i −0.995531 + 0.574770i −0.906923 − 0.421297i \(-0.861575\pi\)
−0.0886080 + 0.996067i \(0.528242\pi\)
\(38\) −2.19814 − 0.387591i −0.356585 − 0.0628756i
\(39\) −4.48928 − 0.560124i −0.718860 − 0.0896917i
\(40\) 0 0
\(41\) 9.13156 + 3.32362i 1.42611 + 0.519062i 0.935814 − 0.352494i \(-0.114666\pi\)
0.490296 + 0.871556i \(0.336888\pi\)
\(42\) 0.408537 − 3.27434i 0.0630386 − 0.505241i
\(43\) −0.256746 − 0.0452712i −0.0391534 − 0.00690379i 0.154037 − 0.988065i \(-0.450772\pi\)
−0.193191 + 0.981161i \(0.561884\pi\)
\(44\) 0.169802 + 0.294106i 0.0255986 + 0.0443381i
\(45\) 0 0
\(46\) −2.68104 + 4.64370i −0.395298 + 0.684677i
\(47\) 7.34421 − 8.75249i 1.07126 − 1.27668i 0.112140 − 0.993692i \(-0.464230\pi\)
0.959123 − 0.282989i \(-0.0913259\pi\)
\(48\) 0.965367 − 0.0487006i 0.139339 − 0.00702933i
\(49\) 0.234540 + 1.33014i 0.0335057 + 0.190020i
\(50\) 0 0
\(51\) −0.345826 − 0.372309i −0.0484253 − 0.0521337i
\(52\) 1.21277 + 3.33207i 0.168181 + 0.462074i
\(53\) − 5.43137i − 0.746056i −0.927820 − 0.373028i \(-0.878320\pi\)
0.927820 − 0.373028i \(-0.121680\pi\)
\(54\) 2.16596 − 3.55734i 0.294750 − 0.484092i
\(55\) 0 0
\(56\) −6.01071 + 2.18772i −0.803215 + 0.292346i
\(57\) −4.60980 − 1.41922i −0.610583 − 0.187980i
\(58\) −0.182930 − 0.218007i −0.0240198 − 0.0286257i
\(59\) −1.03788 − 5.88612i −0.135121 − 0.766308i −0.974776 − 0.223188i \(-0.928354\pi\)
0.839655 − 0.543121i \(-0.182757\pi\)
\(60\) 0 0
\(61\) −9.07515 − 7.61495i −1.16195 − 0.974995i −0.162023 − 0.986787i \(-0.551802\pi\)
−0.999930 + 0.0117924i \(0.996246\pi\)
\(62\) 1.91963 + 1.10830i 0.243793 + 0.140754i
\(63\) 1.75206 − 6.91190i 0.220739 − 0.870817i
\(64\) 1.77824 + 3.08001i 0.222281 + 0.385001i
\(65\) 0 0
\(66\) −0.135073 − 0.319949i −0.0166263 − 0.0393829i
\(67\) 0.619160 − 1.70113i 0.0756424 − 0.207826i −0.896108 − 0.443835i \(-0.853617\pi\)
0.971751 + 0.236010i \(0.0758397\pi\)
\(68\) −0.136218 + 0.374256i −0.0165189 + 0.0453852i
\(69\) −6.99139 + 9.24026i −0.841665 + 1.11240i
\(70\) 0 0
\(71\) 0.185255 + 0.320871i 0.0219857 + 0.0380804i 0.876809 − 0.480839i \(-0.159668\pi\)
−0.854823 + 0.518919i \(0.826334\pi\)
\(72\) −8.05158 − 0.594663i −0.948888 − 0.0700817i
\(73\) 4.35333 + 2.51339i 0.509518 + 0.294171i 0.732636 − 0.680621i \(-0.238290\pi\)
−0.223117 + 0.974792i \(0.571623\pi\)
\(74\) −4.29336 − 3.60255i −0.499093 − 0.418789i
\(75\) 0 0
\(76\) 0.656467 + 3.72301i 0.0753019 + 0.427059i
\(77\) −0.382193 − 0.455479i −0.0435549 − 0.0519067i
\(78\) −0.808804 − 3.53483i −0.0915790 − 0.400240i
\(79\) 0.754406 − 0.274581i 0.0848773 − 0.0308928i −0.299233 − 0.954180i \(-0.596731\pi\)
0.384110 + 0.923287i \(0.374508\pi\)
\(80\) 0 0
\(81\) 5.59624 − 7.04855i 0.621804 − 0.783173i
\(82\) 7.78892i 0.860143i
\(83\) −0.942488 − 2.58947i −0.103452 − 0.284231i 0.877158 − 0.480201i \(-0.159436\pi\)
−0.980610 + 0.195971i \(0.937214\pi\)
\(84\) −5.44798 + 1.24655i −0.594424 + 0.136010i
\(85\) 0 0
\(86\) −0.0362861 − 0.205789i −0.00391283 − 0.0221908i
\(87\) −0.333932 − 0.516417i −0.0358012 − 0.0553657i
\(88\) −0.432738 + 0.515717i −0.0461300 + 0.0549756i
\(89\) 5.22533 − 9.05054i 0.553884 − 0.959356i −0.444105 − 0.895975i \(-0.646478\pi\)
0.997989 − 0.0633809i \(-0.0201883\pi\)
\(90\) 0 0
\(91\) −3.10412 − 5.37650i −0.325401 − 0.563611i
\(92\) 8.94385 + 1.57704i 0.932461 + 0.164418i
\(93\) 3.81976 + 2.89012i 0.396091 + 0.299692i
\(94\) 8.60560 + 3.13218i 0.887600 + 0.323060i
\(95\) 0 0
\(96\) 3.92713 + 9.30225i 0.400811 + 0.949407i
\(97\) −14.6092 − 2.57600i −1.48334 − 0.261553i −0.627430 − 0.778673i \(-0.715893\pi\)
−0.855913 + 0.517120i \(0.827004\pi\)
\(98\) −0.937552 + 0.541296i −0.0947070 + 0.0546791i
\(99\) −0.204037 − 0.722208i −0.0205065 − 0.0725846i
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 675.2.u.b.574.3 24
5.2 odd 4 675.2.l.c.601.1 12
5.3 odd 4 27.2.e.a.7.2 yes 12
5.4 even 2 inner 675.2.u.b.574.2 24
15.8 even 4 81.2.e.a.19.1 12
20.3 even 4 432.2.u.c.385.2 12
27.4 even 9 inner 675.2.u.b.274.2 24
45.13 odd 12 243.2.e.c.217.1 12
45.23 even 12 243.2.e.b.217.2 12
45.38 even 12 243.2.e.a.136.2 12
45.43 odd 12 243.2.e.d.136.1 12
135.4 even 18 inner 675.2.u.b.274.3 24
135.13 odd 36 243.2.e.d.109.1 12
135.23 even 36 81.2.e.a.64.1 12
135.38 even 36 729.2.c.b.244.3 12
135.43 odd 36 729.2.c.e.244.4 12
135.58 odd 36 27.2.e.a.4.2 ✓ 12
135.68 even 36 243.2.e.a.109.2 12
135.83 even 36 729.2.a.d.1.4 6
135.88 odd 36 729.2.c.e.487.4 12
135.103 odd 36 243.2.e.c.28.1 12
135.112 odd 36 675.2.l.c.301.1 12
135.113 even 36 243.2.e.b.28.2 12
135.128 even 36 729.2.c.b.487.3 12
135.133 odd 36 729.2.a.a.1.3 6
540.463 even 36 432.2.u.c.193.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.2.e.a.4.2 ✓ 12 135.58 odd 36
27.2.e.a.7.2 yes 12 5.3 odd 4
81.2.e.a.19.1 12 15.8 even 4
81.2.e.a.64.1 12 135.23 even 36
243.2.e.a.109.2 12 135.68 even 36
243.2.e.a.136.2 12 45.38 even 12
243.2.e.b.28.2 12 135.113 even 36
243.2.e.b.217.2 12 45.23 even 12
243.2.e.c.28.1 12 135.103 odd 36
243.2.e.c.217.1 12 45.13 odd 12
243.2.e.d.109.1 12 135.13 odd 36
243.2.e.d.136.1 12 45.43 odd 12
432.2.u.c.193.2 12 540.463 even 36
432.2.u.c.385.2 12 20.3 even 4
675.2.l.c.301.1 12 135.112 odd 36
675.2.l.c.601.1 12 5.2 odd 4
675.2.u.b.274.2 24 27.4 even 9 inner
675.2.u.b.274.3 24 135.4 even 18 inner
675.2.u.b.574.2 24 5.4 even 2 inner
675.2.u.b.574.3 24 1.1 even 1 trivial
729.2.a.a.1.3 6 135.133 odd 36
729.2.a.d.1.4 6 135.83 even 36
729.2.c.b.244.3 12 135.38 even 36
729.2.c.b.487.3 12 135.128 even 36
729.2.c.e.244.4 12 135.43 odd 36
729.2.c.e.487.4 12 135.88 odd 36