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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [675,2,Mod(32,675)] 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("675.32"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(675, base_ring=CyclotomicField(36)) chi = DirichletCharacter(H, H._module([10, 9])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 675.ba (of order \(36\), degree \(12\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [192] 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: \(192\)
Relative dimension: \(16\) over \(\Q(\zeta_{36})\)
Twist minimal: no (minimal twist has level 135)
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

Embedding invariants

Embedding label 257.6
Character \(\chi\) \(=\) 675.257
Dual form 675.2.ba.b.218.6

$q$-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+(-0.490069 - 0.343150i) q^{2} +(1.72553 + 0.150104i) q^{3} +(-0.561625 - 1.54305i) q^{4} +(-0.794122 - 0.665679i) q^{6} +(0.0636379 + 0.136472i) q^{7} +(-0.563947 + 2.10468i) q^{8} +(2.95494 + 0.518020i) q^{9} +(-3.54970 - 4.23037i) q^{11} +(-0.737484 - 2.74689i) q^{12} +(-1.08490 - 1.54940i) q^{13} +(0.0156434 - 0.0887180i) q^{14} +(-1.51722 + 1.27310i) q^{16} +(-1.18141 - 4.40907i) q^{17} +(-1.27036 - 1.26785i) q^{18} +(6.55178 - 3.78267i) q^{19} +(0.0893244 + 0.245039i) q^{21} +(0.287947 + 3.29125i) q^{22} +(0.883698 + 0.412075i) q^{23} +(-1.28903 + 3.54705i) q^{24} +1.13159i q^{26} +(5.02109 + 1.33741i) q^{27} +(0.174843 - 0.174843i) q^{28} +(-0.486461 - 2.75886i) q^{29} +(1.70974 - 0.622294i) q^{31} +(5.52167 - 0.483084i) q^{32} +(-5.49013 - 7.83247i) q^{33} +(-0.934001 + 2.56615i) q^{34} +(-0.860234 - 4.85055i) q^{36} +(-5.71298 + 1.53079i) q^{37} +(-4.50885 - 0.394473i) q^{38} +(-1.63946 - 2.83639i) q^{39} +(-0.682276 - 0.120304i) q^{41} +(0.0403102 - 0.150738i) q^{42} +(-0.151755 + 1.73456i) q^{43} +(-4.53407 + 7.85325i) q^{44} +(-0.291669 - 0.505186i) q^{46} +(-10.0941 + 4.70696i) q^{47} +(-2.80911 + 1.96903i) q^{48} +(4.48494 - 5.34494i) q^{49} +(-1.37674 - 7.78533i) q^{51} +(-1.78149 + 2.54423i) q^{52} +(-0.204978 - 0.204978i) q^{53} +(-2.00175 - 2.37841i) q^{54} +(-0.323118 + 0.0569745i) q^{56} +(11.8731 - 5.54368i) q^{57} +(-0.708303 + 1.51896i) q^{58} +(0.796564 + 0.668396i) q^{59} +(9.12620 + 3.32167i) q^{61} +(-1.05143 - 0.281730i) q^{62} +(0.117351 + 0.436232i) q^{63} +(0.558711 + 0.322572i) q^{64} +(0.00283170 + 5.72239i) q^{66} +(8.96216 - 6.27537i) q^{67} +(-6.13991 + 4.29921i) q^{68} +(1.46300 + 0.843697i) q^{69} +(-2.53024 - 1.46084i) q^{71} +(-2.75670 + 5.92706i) q^{72} +(9.95358 + 2.66706i) q^{73} +(3.32505 + 1.21022i) q^{74} +(-9.51649 - 7.98529i) q^{76} +(0.351431 - 0.753647i) q^{77} +(-0.169857 + 1.95261i) q^{78} +(-11.9179 + 2.10144i) q^{79} +(8.46331 + 3.06144i) q^{81} +(0.293080 + 0.293080i) q^{82} +(-4.10306 + 5.85978i) q^{83} +(0.327942 - 0.275452i) q^{84} +(0.669585 - 0.797981i) q^{86} +(-0.425289 - 4.83353i) q^{87} +(10.9054 - 5.08528i) q^{88} +(6.74192 + 11.6773i) q^{89} +(0.142409 - 0.246659i) q^{91} +(0.139546 - 1.59502i) q^{92} +(3.04362 - 0.817150i) q^{93} +(6.56201 + 1.15706i) q^{94} +(9.60035 - 0.00475069i) q^{96} +(-3.56471 - 0.311872i) q^{97} +(-4.03205 + 1.08038i) q^{98} +(-8.29773 - 14.3393i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 192 q + 12 q^{2} + 12 q^{3} - 36 q^{6} + 12 q^{7} + 18 q^{8} - 36 q^{11} + 12 q^{12} + 12 q^{13} - 24 q^{16} + 18 q^{17} + 54 q^{18} - 24 q^{21} + 12 q^{22} + 36 q^{23} + 36 q^{27} + 24 q^{28} - 24 q^{31}+ \cdots - 324 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(326\) \(352\)
\(\chi(n)\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{1}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.490069 − 0.343150i −0.346531 − 0.242644i 0.387333 − 0.921940i \(-0.373396\pi\)
−0.733864 + 0.679296i \(0.762285\pi\)
\(3\) 1.72553 + 0.150104i 0.996238 + 0.0866628i
\(4\) −0.561625 − 1.54305i −0.280812 − 0.771525i
\(5\) 0 0
\(6\) −0.794122 − 0.665679i −0.324199 − 0.271762i
\(7\) 0.0636379 + 0.136472i 0.0240529 + 0.0515816i 0.917967 − 0.396656i \(-0.129830\pi\)
−0.893914 + 0.448238i \(0.852052\pi\)
\(8\) −0.563947 + 2.10468i −0.199385 + 0.744117i
\(9\) 2.95494 + 0.518020i 0.984979 + 0.172673i
\(10\) 0 0
\(11\) −3.54970 − 4.23037i −1.07027 − 1.27550i −0.959515 − 0.281657i \(-0.909116\pi\)
−0.110760 − 0.993847i \(-0.535328\pi\)
\(12\) −0.737484 − 2.74689i −0.212893 − 0.792959i
\(13\) −1.08490 − 1.54940i −0.300897 − 0.429725i 0.640022 − 0.768356i \(-0.278925\pi\)
−0.940919 + 0.338631i \(0.890036\pi\)
\(14\) 0.0156434 − 0.0887180i 0.00418087 − 0.0237109i
\(15\) 0 0
\(16\) −1.51722 + 1.27310i −0.379305 + 0.318275i
\(17\) −1.18141 − 4.40907i −0.286533 − 1.06936i −0.947712 − 0.319127i \(-0.896610\pi\)
0.661179 − 0.750228i \(-0.270056\pi\)
\(18\) −1.27036 − 1.26785i −0.299428 − 0.298836i
\(19\) 6.55178 − 3.78267i 1.50308 − 0.867804i 0.503087 − 0.864236i \(-0.332197\pi\)
0.999994 − 0.00356839i \(-0.00113586\pi\)
\(20\) 0 0
\(21\) 0.0893244 + 0.245039i 0.0194922 + 0.0534720i
\(22\) 0.287947 + 3.29125i 0.0613906 + 0.701697i
\(23\) 0.883698 + 0.412075i 0.184264 + 0.0859236i 0.512561 − 0.858651i \(-0.328697\pi\)
−0.328297 + 0.944575i \(0.606475\pi\)
\(24\) −1.28903 + 3.54705i −0.263122 + 0.724038i
\(25\) 0 0
\(26\) 1.13159i 0.221924i
\(27\) 5.02109 + 1.33741i 0.966309 + 0.257385i
\(28\) 0.174843 − 0.174843i 0.0330421 − 0.0330421i
\(29\) −0.486461 − 2.75886i −0.0903336 − 0.512307i −0.996078 − 0.0884824i \(-0.971798\pi\)
0.905744 − 0.423825i \(-0.139313\pi\)
\(30\) 0 0
\(31\) 1.70974 − 0.622294i 0.307078 − 0.111767i −0.183885 − 0.982948i \(-0.558867\pi\)
0.490963 + 0.871181i \(0.336645\pi\)
\(32\) 5.52167 − 0.483084i 0.976103 − 0.0853979i
\(33\) −5.49013 − 7.83247i −0.955710 − 1.36346i
\(34\) −0.934001 + 2.56615i −0.160180 + 0.440090i
\(35\) 0 0
\(36\) −0.860234 − 4.85055i −0.143372 − 0.808425i
\(37\) −5.71298 + 1.53079i −0.939209 + 0.251660i −0.695777 − 0.718258i \(-0.744940\pi\)
−0.243432 + 0.969918i \(0.578273\pi\)
\(38\) −4.50885 − 0.394473i −0.731431 − 0.0639920i
\(39\) −1.63946 − 2.83639i −0.262524 − 0.454185i
\(40\) 0 0
\(41\) −0.682276 − 0.120304i −0.106554 − 0.0187883i 0.120117 − 0.992760i \(-0.461673\pi\)
−0.226671 + 0.973971i \(0.572784\pi\)
\(42\) 0.0403102 − 0.150738i 0.00621999 − 0.0232594i
\(43\) −0.151755 + 1.73456i −0.0231424 + 0.264518i 0.975805 + 0.218643i \(0.0701631\pi\)
−0.998947 + 0.0458750i \(0.985392\pi\)
\(44\) −4.53407 + 7.85325i −0.683537 + 1.18392i
\(45\) 0 0
\(46\) −0.291669 − 0.505186i −0.0430043 − 0.0744856i
\(47\) −10.0941 + 4.70696i −1.47238 + 0.686581i −0.982680 − 0.185311i \(-0.940671\pi\)
−0.489698 + 0.871892i \(0.662893\pi\)
\(48\) −2.80911 + 1.96903i −0.405460 + 0.284206i
\(49\) 4.48494 − 5.34494i 0.640705 − 0.763563i
\(50\) 0 0
\(51\) −1.37674 − 7.78533i −0.192782 − 1.09016i
\(52\) −1.78149 + 2.54423i −0.247049 + 0.352822i
\(53\) −0.204978 − 0.204978i −0.0281559 − 0.0281559i 0.692889 − 0.721045i \(-0.256338\pi\)
−0.721045 + 0.692889i \(0.756338\pi\)
\(54\) −2.00175 − 2.37841i −0.272403 − 0.323661i
\(55\) 0 0
\(56\) −0.323118 + 0.0569745i −0.0431785 + 0.00761353i
\(57\) 11.8731 − 5.54368i 1.57263 − 0.734278i
\(58\) −0.708303 + 1.51896i −0.0930047 + 0.199449i
\(59\) 0.796564 + 0.668396i 0.103704 + 0.0870178i 0.693165 − 0.720779i \(-0.256216\pi\)
−0.589462 + 0.807796i \(0.700660\pi\)
\(60\) 0 0
\(61\) 9.12620 + 3.32167i 1.16849 + 0.425296i 0.852123 − 0.523341i \(-0.175315\pi\)
0.316367 + 0.948637i \(0.397537\pi\)
\(62\) −1.05143 − 0.281730i −0.133532 − 0.0357797i
\(63\) 0.117351 + 0.436232i 0.0147848 + 0.0549601i
\(64\) 0.558711 + 0.322572i 0.0698389 + 0.0403215i
\(65\) 0 0
\(66\) 0.00283170 + 5.72239i 0.000348558 + 0.704378i
\(67\) 8.96216 − 6.27537i 1.09490 − 0.766659i 0.120780 − 0.992679i \(-0.461461\pi\)
0.974122 + 0.226021i \(0.0725717\pi\)
\(68\) −6.13991 + 4.29921i −0.744573 + 0.521356i
\(69\) 1.46300 + 0.843697i 0.176124 + 0.101569i
\(70\) 0 0
\(71\) −2.53024 − 1.46084i −0.300285 − 0.173369i 0.342286 − 0.939596i \(-0.388799\pi\)
−0.642571 + 0.766226i \(0.722132\pi\)
\(72\) −2.75670 + 5.92706i −0.324880 + 0.698511i
\(73\) 9.95358 + 2.66706i 1.16498 + 0.312155i 0.788953 − 0.614454i \(-0.210624\pi\)
0.376026 + 0.926609i \(0.377290\pi\)
\(74\) 3.32505 + 1.21022i 0.386529 + 0.140685i
\(75\) 0 0
\(76\) −9.51649 − 7.98529i −1.09162 − 0.915975i
\(77\) 0.351431 − 0.753647i 0.0400493 − 0.0858860i
\(78\) −0.169857 + 1.95261i −0.0192325 + 0.221089i
\(79\) −11.9179 + 2.10144i −1.34086 + 0.236430i −0.797628 − 0.603150i \(-0.793912\pi\)
−0.543236 + 0.839580i \(0.682801\pi\)
\(80\) 0 0
\(81\) 8.46331 + 3.06144i 0.940368 + 0.340159i
\(82\) 0.293080 + 0.293080i 0.0323653 + 0.0323653i
\(83\) −4.10306 + 5.85978i −0.450370 + 0.643194i −0.978567 − 0.205927i \(-0.933979\pi\)
0.528198 + 0.849121i \(0.322868\pi\)
\(84\) 0.327942 − 0.275452i 0.0357814 − 0.0300543i
\(85\) 0 0
\(86\) 0.669585 − 0.797981i 0.0722033 − 0.0860485i
\(87\) −0.425289 − 4.83353i −0.0455958 − 0.518208i
\(88\) 10.9054 − 5.08528i 1.16252 − 0.542092i
\(89\) 6.74192 + 11.6773i 0.714642 + 1.23780i 0.963097 + 0.269153i \(0.0867437\pi\)
−0.248455 + 0.968643i \(0.579923\pi\)
\(90\) 0 0
\(91\) 0.142409 − 0.246659i 0.0149285 − 0.0258569i
\(92\) 0.139546 − 1.59502i 0.0145487 − 0.166293i
\(93\) 3.04362 − 0.817150i 0.315609 − 0.0847345i
\(94\) 6.56201 + 1.15706i 0.676819 + 0.119341i
\(95\) 0 0
\(96\) 9.60035 − 0.00475069i 0.979831 − 0.000484865i
\(97\) −3.56471 − 0.311872i −0.361941 − 0.0316658i −0.0952651 − 0.995452i \(-0.530370\pi\)
−0.266676 + 0.963786i \(0.585925\pi\)
\(98\) −4.03205 + 1.08038i −0.407298 + 0.109135i
\(99\) −8.29773 − 14.3393i −0.833953 − 1.44115i
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.ba.b.257.6 192
5.2 odd 4 135.2.q.a.68.11 yes 192
5.3 odd 4 inner 675.2.ba.b.68.6 192
5.4 even 2 135.2.q.a.122.11 yes 192
15.2 even 4 405.2.r.a.233.6 192
15.14 odd 2 405.2.r.a.152.6 192
27.2 odd 18 inner 675.2.ba.b.407.6 192
135.2 even 36 135.2.q.a.83.11 yes 192
135.29 odd 18 135.2.q.a.2.11 ✓ 192
135.52 odd 36 405.2.r.a.8.6 192
135.79 even 18 405.2.r.a.332.6 192
135.83 even 36 inner 675.2.ba.b.218.6 192
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
135.2.q.a.2.11 ✓ 192 135.29 odd 18
135.2.q.a.68.11 yes 192 5.2 odd 4
135.2.q.a.83.11 yes 192 135.2 even 36
135.2.q.a.122.11 yes 192 5.4 even 2
405.2.r.a.8.6 192 135.52 odd 36
405.2.r.a.152.6 192 15.14 odd 2
405.2.r.a.233.6 192 15.2 even 4
405.2.r.a.332.6 192 135.79 even 18
675.2.ba.b.68.6 192 5.3 odd 4 inner
675.2.ba.b.218.6 192 135.83 even 36 inner
675.2.ba.b.257.6 192 1.1 even 1 trivial
675.2.ba.b.407.6 192 27.2 odd 18 inner