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Properties

Label 675.2.r.a.46.14
Level $675$
Weight $2$
Character 675.46
Analytic conductor $5.390$
Analytic rank $0$
Dimension $224$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [675,2,Mod(46,675)] 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("675.46"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(675, base_ring=CyclotomicField(30)) chi = DirichletCharacter(H, H._module([20, 18])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 675.r (of order \(15\), degree \(8\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.38990213644\)
Analytic rank: \(0\)
Dimension: \(224\)
Relative dimension: \(28\) over \(\Q(\zeta_{15})\)
Twist minimal: no (minimal twist has level 225)
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label 46.14
Character \(\chi\) \(=\) 675.46
Dual form 675.2.r.a.631.14

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.110043 + 0.122216i) q^{2} +(0.206230 - 1.96215i) q^{4} +(2.06082 + 0.867759i) q^{5} +(2.06663 - 3.57951i) q^{7} +(0.528597 - 0.384048i) q^{8} +(0.120726 + 0.347356i) q^{10} +(0.239690 + 0.266203i) q^{11} +(1.23537 - 1.37202i) q^{13} +(0.664892 - 0.141327i) q^{14} +(-3.75457 - 0.798059i) q^{16} +(-3.22940 + 2.34629i) q^{17} +(-0.0599095 + 0.0435268i) q^{19} +(2.12767 - 3.86468i) q^{20} +(-0.00615783 + 0.0585878i) q^{22} +(-4.28234 + 0.910239i) q^{23} +(3.49399 + 3.57660i) q^{25} +0.303627 q^{26} +(-6.59733 - 4.79324i) q^{28} +(5.93713 - 2.64338i) q^{29} +(-4.95171 - 2.20464i) q^{31} +(-0.969013 - 1.67838i) q^{32} +(-0.642128 - 0.136489i) q^{34} +(7.36512 - 5.58341i) q^{35} +(2.37480 + 7.30889i) q^{37} +(-0.0119123 - 0.00253204i) q^{38} +(1.42261 - 0.332761i) q^{40} +(-3.14813 + 3.49635i) q^{41} +(1.55061 - 2.68574i) q^{43} +(0.571760 - 0.415408i) q^{44} +(-0.582489 - 0.423203i) q^{46} +(11.8169 - 5.26122i) q^{47} +(-5.04195 - 8.73292i) q^{49} +(-0.0526256 + 0.820601i) q^{50} +(-2.43733 - 2.70693i) q^{52} +(-1.53179 - 1.11291i) q^{53} +(0.262959 + 0.756591i) q^{55} +(-0.282290 - 2.68581i) q^{56} +(0.976404 + 0.434723i) q^{58} +(-3.31117 + 3.67743i) q^{59} +(6.08932 + 6.76288i) q^{61} +(-0.275461 - 0.847783i) q^{62} +(-2.27380 + 6.99805i) q^{64} +(3.73647 - 1.75549i) q^{65} +(13.1648 + 5.86135i) q^{67} +(3.93777 + 6.82042i) q^{68} +(1.49286 + 0.285716i) q^{70} +(-10.8652 - 7.89400i) q^{71} +(4.15877 - 12.7994i) q^{73} +(-0.631930 + 1.09453i) q^{74} +(0.0730507 + 0.126528i) q^{76} +(1.44823 - 0.307831i) q^{77} +(-5.47332 + 2.43688i) q^{79} +(-7.04499 - 4.90273i) q^{80} -0.773739 q^{82} +(-0.301668 - 2.87018i) q^{83} +(-8.69124 + 2.03296i) q^{85} +(0.498874 - 0.106039i) q^{86} +(0.228934 + 0.0486615i) q^{88} +(-2.21408 + 6.81425i) q^{89} +(-2.35810 - 7.25750i) q^{91} +(0.902876 + 8.59029i) q^{92} +(1.94338 + 0.865246i) q^{94} +(-0.161234 + 0.0377140i) q^{95} +(9.65762 - 4.29985i) q^{97} +(0.512465 - 1.57721i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 224 q + 3 q^{2} + 23 q^{4} + 8 q^{5} - 8 q^{7} + 20 q^{8} - 20 q^{10} + 11 q^{11} - 3 q^{13} - q^{14} + 23 q^{16} + 24 q^{17} - 12 q^{19} - q^{20} - 11 q^{22} - q^{23} - 16 q^{25} + 136 q^{26} + 4 q^{28}+ \cdots - 146 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(326\) \(352\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{3}{5}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.110043 + 0.122216i 0.0778125 + 0.0864195i 0.780794 − 0.624789i \(-0.214815\pi\)
−0.702981 + 0.711208i \(0.748148\pi\)
\(3\) 0 0
\(4\) 0.206230 − 1.96215i 0.103115 − 0.981073i
\(5\) 2.06082 + 0.867759i 0.921628 + 0.388074i
\(6\) 0 0
\(7\) 2.06663 − 3.57951i 0.781114 − 1.35293i −0.150179 − 0.988659i \(-0.547985\pi\)
0.931293 − 0.364271i \(-0.118682\pi\)
\(8\) 0.528597 − 0.384048i 0.186887 − 0.135782i
\(9\) 0 0
\(10\) 0.120726 + 0.347356i 0.0381771 + 0.109844i
\(11\) 0.239690 + 0.266203i 0.0722693 + 0.0802632i 0.778195 − 0.628023i \(-0.216135\pi\)
−0.705926 + 0.708286i \(0.749469\pi\)
\(12\) 0 0
\(13\) 1.23537 − 1.37202i 0.342631 − 0.380530i −0.547060 − 0.837093i \(-0.684253\pi\)
0.889691 + 0.456563i \(0.150920\pi\)
\(14\) 0.664892 − 0.141327i 0.177700 − 0.0377713i
\(15\) 0 0
\(16\) −3.75457 − 0.798059i −0.938644 − 0.199515i
\(17\) −3.22940 + 2.34629i −0.783244 + 0.569060i −0.905951 − 0.423383i \(-0.860843\pi\)
0.122707 + 0.992443i \(0.460843\pi\)
\(18\) 0 0
\(19\) −0.0599095 + 0.0435268i −0.0137442 + 0.00998573i −0.594636 − 0.803995i \(-0.702704\pi\)
0.580892 + 0.813981i \(0.302704\pi\)
\(20\) 2.12767 − 3.86468i 0.475762 − 0.864168i
\(21\) 0 0
\(22\) −0.00615783 + 0.0585878i −0.00131285 + 0.0124910i
\(23\) −4.28234 + 0.910239i −0.892929 + 0.189798i −0.631455 − 0.775413i \(-0.717542\pi\)
−0.261474 + 0.965210i \(0.584209\pi\)
\(24\) 0 0
\(25\) 3.49399 + 3.57660i 0.698798 + 0.715319i
\(26\) 0.303627 0.0595462
\(27\) 0 0
\(28\) −6.59733 − 4.79324i −1.24678 − 0.905837i
\(29\) 5.93713 − 2.64338i 1.10250 − 0.490863i 0.226906 − 0.973917i \(-0.427139\pi\)
0.875591 + 0.483054i \(0.160472\pi\)
\(30\) 0 0
\(31\) −4.95171 − 2.20464i −0.889353 − 0.395966i −0.0893776 − 0.995998i \(-0.528488\pi\)
−0.799976 + 0.600032i \(0.795154\pi\)
\(32\) −0.969013 − 1.67838i −0.171299 − 0.296698i
\(33\) 0 0
\(34\) −0.642128 − 0.136489i −0.110124 − 0.0234076i
\(35\) 7.36512 − 5.58341i 1.24493 − 0.943768i
\(36\) 0 0
\(37\) 2.37480 + 7.30889i 0.390415 + 1.20157i 0.932475 + 0.361235i \(0.117645\pi\)
−0.542060 + 0.840340i \(0.682355\pi\)
\(38\) −0.0119123 − 0.00253204i −0.00193243 − 0.000410751i
\(39\) 0 0
\(40\) 1.42261 − 0.332761i 0.224934 − 0.0526141i
\(41\) −3.14813 + 3.49635i −0.491655 + 0.546038i −0.937004 − 0.349319i \(-0.886413\pi\)
0.445349 + 0.895357i \(0.353080\pi\)
\(42\) 0 0
\(43\) 1.55061 − 2.68574i 0.236466 − 0.409571i −0.723232 − 0.690605i \(-0.757344\pi\)
0.959698 + 0.281034i \(0.0906775\pi\)
\(44\) 0.571760 − 0.415408i 0.0861961 − 0.0626252i
\(45\) 0 0
\(46\) −0.582489 − 0.423203i −0.0858833 − 0.0623979i
\(47\) 11.8169 − 5.26122i 1.72367 − 0.767428i 0.726934 − 0.686707i \(-0.240945\pi\)
0.996737 − 0.0807203i \(-0.0257221\pi\)
\(48\) 0 0
\(49\) −5.04195 − 8.73292i −0.720279 − 1.24756i
\(50\) −0.0526256 + 0.820601i −0.00744238 + 0.116051i
\(51\) 0 0
\(52\) −2.43733 − 2.70693i −0.337997 − 0.375384i
\(53\) −1.53179 − 1.11291i −0.210408 − 0.152870i 0.477590 − 0.878583i \(-0.341510\pi\)
−0.687998 + 0.725712i \(0.741510\pi\)
\(54\) 0 0
\(55\) 0.262959 + 0.756591i 0.0354574 + 0.102019i
\(56\) −0.282290 − 2.68581i −0.0377225 − 0.358906i
\(57\) 0 0
\(58\) 0.976404 + 0.434723i 0.128208 + 0.0570820i
\(59\) −3.31117 + 3.67743i −0.431078 + 0.478760i −0.919074 − 0.394086i \(-0.871061\pi\)
0.487996 + 0.872846i \(0.337728\pi\)
\(60\) 0 0
\(61\) 6.08932 + 6.76288i 0.779658 + 0.865898i 0.993832 − 0.110898i \(-0.0353726\pi\)
−0.214174 + 0.976796i \(0.568706\pi\)
\(62\) −0.275461 − 0.847783i −0.0349836 − 0.107669i
\(63\) 0 0
\(64\) −2.27380 + 6.99805i −0.284226 + 0.874756i
\(65\) 3.73647 − 1.75549i 0.463452 − 0.217741i
\(66\) 0 0
\(67\) 13.1648 + 5.86135i 1.60834 + 0.716078i 0.997153 − 0.0754107i \(-0.0240268\pi\)
0.611184 + 0.791488i \(0.290693\pi\)
\(68\) 3.93777 + 6.82042i 0.477525 + 0.827098i
\(69\) 0 0
\(70\) 1.49286 + 0.285716i 0.178431 + 0.0341496i
\(71\) −10.8652 − 7.89400i −1.28946 − 0.936845i −0.289662 − 0.957129i \(-0.593543\pi\)
−0.999794 + 0.0202841i \(0.993543\pi\)
\(72\) 0 0
\(73\) 4.15877 − 12.7994i 0.486747 − 1.49805i −0.342688 − 0.939449i \(-0.611337\pi\)
0.829435 − 0.558603i \(-0.188663\pi\)
\(74\) −0.631930 + 1.09453i −0.0734603 + 0.127237i
\(75\) 0 0
\(76\) 0.0730507 + 0.126528i 0.00837950 + 0.0145137i
\(77\) 1.44823 − 0.307831i 0.165041 − 0.0350806i
\(78\) 0 0
\(79\) −5.47332 + 2.43688i −0.615797 + 0.274170i −0.690837 − 0.723010i \(-0.742758\pi\)
0.0750407 + 0.997180i \(0.476091\pi\)
\(80\) −7.04499 − 4.90273i −0.787654 − 0.548141i
\(81\) 0 0
\(82\) −0.773739 −0.0854452
\(83\) −0.301668 − 2.87018i −0.0331123 − 0.315043i −0.998524 − 0.0543124i \(-0.982703\pi\)
0.965412 − 0.260730i \(-0.0839633\pi\)
\(84\) 0 0
\(85\) −8.69124 + 2.03296i −0.942697 + 0.220506i
\(86\) 0.498874 − 0.106039i 0.0537949 − 0.0114345i
\(87\) 0 0
\(88\) 0.228934 + 0.0486615i 0.0244045 + 0.00518733i
\(89\) −2.21408 + 6.81425i −0.234692 + 0.722309i 0.762470 + 0.647024i \(0.223987\pi\)
−0.997162 + 0.0752851i \(0.976013\pi\)
\(90\) 0 0
\(91\) −2.35810 − 7.25750i −0.247196 − 0.760792i
\(92\) 0.902876 + 8.59029i 0.0941313 + 0.895600i
\(93\) 0 0
\(94\) 1.94338 + 0.865246i 0.200444 + 0.0892434i
\(95\) −0.161234 + 0.0377140i −0.0165422 + 0.00386938i
\(96\) 0 0
\(97\) 9.65762 − 4.29985i 0.980583 − 0.436584i 0.147095 − 0.989122i \(-0.453008\pi\)
0.833487 + 0.552539i \(0.186341\pi\)
\(98\) 0.512465 − 1.57721i 0.0517668 − 0.159322i
\(99\) 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 675.2.r.a.46.14 224
3.2 odd 2 225.2.q.a.196.15 yes 224
9.4 even 3 inner 675.2.r.a.496.15 224
9.5 odd 6 225.2.q.a.121.14 yes 224
25.6 even 5 inner 675.2.r.a.181.15 224
75.56 odd 10 225.2.q.a.106.14 yes 224
225.31 even 15 inner 675.2.r.a.631.14 224
225.131 odd 30 225.2.q.a.31.15 ✓ 224
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
225.2.q.a.31.15 ✓ 224 225.131 odd 30
225.2.q.a.106.14 yes 224 75.56 odd 10
225.2.q.a.121.14 yes 224 9.5 odd 6
225.2.q.a.196.15 yes 224 3.2 odd 2
675.2.r.a.46.14 224 1.1 even 1 trivial
675.2.r.a.181.15 224 25.6 even 5 inner
675.2.r.a.496.15 224 9.4 even 3 inner
675.2.r.a.631.14 224 225.31 even 15 inner