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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [925,2,Mod(99,925)] 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("925.99"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(925, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([9, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 925 = 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 925.ba (of order \(18\), degree \(6\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [72,0,0,6,0,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.38616218697\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(12\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 185)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 299.1
Character \(\chi\) \(=\) 925.299
Dual form 925.2.ba.c.99.1

$q$-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+(-2.01259 - 1.68876i) q^{2} +(0.288189 + 0.343451i) q^{3} +(0.851300 + 4.82796i) q^{4} -1.17791i q^{6} +(0.722629 - 1.98541i) q^{7} +(3.81272 - 6.60383i) q^{8} +(0.486039 - 2.75647i) q^{9} +(-0.739142 + 1.28023i) q^{11} +(-1.41283 + 1.68375i) q^{12} +(-0.605037 - 3.43134i) q^{13} +(-4.80724 + 2.77546i) q^{14} +(-9.61218 + 3.49855i) q^{16} +(0.459537 - 2.60616i) q^{17} +(-5.63322 + 4.72683i) q^{18} +(3.23599 + 3.85650i) q^{19} +(0.890144 - 0.323986i) q^{21} +(3.64960 - 1.32835i) q^{22} +(2.36639 + 4.09871i) q^{23} +(3.36688 - 0.593671i) q^{24} +(-4.57703 + 7.92764i) q^{26} +(2.25161 - 1.29997i) q^{27} +(10.2007 + 1.79865i) q^{28} +(-2.25769 - 1.30348i) q^{29} +1.85091i q^{31} +(10.9224 + 3.97544i) q^{32} +(-0.652710 + 0.115090i) q^{33} +(-5.32605 + 4.46909i) q^{34} +13.7219 q^{36} +(6.08066 - 0.159781i) q^{37} -13.2264i q^{38} +(1.00413 - 1.19668i) q^{39} +(-2.16584 - 12.2831i) q^{41} +(-2.33863 - 0.851192i) q^{42} -10.0195 q^{43} +(-6.81015 - 2.47869i) q^{44} +(2.15917 - 12.2453i) q^{46} +(-11.2904 + 6.51849i) q^{47} +(-3.97171 - 2.29307i) q^{48} +(1.94266 + 1.63009i) q^{49} +(1.02752 - 0.593240i) q^{51} +(16.0513 - 5.84220i) q^{52} +(-4.52292 - 12.4266i) q^{53} +(-6.72691 - 1.18614i) q^{54} +(-10.3561 - 12.3419i) q^{56} +(-0.391941 + 2.22281i) q^{57} +(2.34254 + 6.43607i) q^{58} +(-2.17456 - 5.97456i) q^{59} +(6.76339 - 1.19257i) q^{61} +(3.12575 - 3.72512i) q^{62} +(-5.12148 - 2.95689i) q^{63} +(-5.03975 - 8.72911i) q^{64} +(1.50800 + 0.870643i) q^{66} +(2.54937 - 7.00434i) q^{67} +12.9737 q^{68} +(-0.725735 + 1.99394i) q^{69} +(9.08889 - 7.62648i) q^{71} +(-16.3501 - 13.7194i) q^{72} -3.77705i q^{73} +(-12.5077 - 9.94723i) q^{74} +(-15.8643 + 18.9063i) q^{76} +(2.00766 + 2.39263i) q^{77} +(-4.04180 + 0.712679i) q^{78} +(3.52083 - 9.67339i) q^{79} +(-6.79520 - 2.47325i) q^{81} +(-16.3843 + 28.3784i) q^{82} +(-3.31682 - 0.584845i) q^{83} +(2.32197 + 4.02177i) q^{84} +(20.1651 + 16.9205i) q^{86} +(-0.202962 - 1.15105i) q^{87} +(5.63629 + 9.76234i) q^{88} +(1.86920 + 5.13559i) q^{89} +(-7.24982 - 1.27834i) q^{91} +(-17.7739 + 14.9141i) q^{92} +(-0.635696 + 0.533412i) q^{93} +(33.7310 + 5.94769i) q^{94} +(1.78236 + 4.89700i) q^{96} +(2.81042 + 4.86779i) q^{97} +(-1.15695 - 6.56139i) q^{98} +(3.16966 + 2.65966i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 72 q + 6 q^{4} + 6 q^{7} + 18 q^{8} + 42 q^{12} + 18 q^{13} + 18 q^{14} - 18 q^{16} - 12 q^{17} - 24 q^{18} + 18 q^{19} - 24 q^{22} + 24 q^{23} + 30 q^{24} + 48 q^{28} - 18 q^{29} - 24 q^{33} - 66 q^{34}+ \cdots + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/925\mathbb{Z}\right)^\times\).

\(n\) \(76\) \(852\)
\(\chi(n)\) \(e\left(\frac{13}{18}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.01259 − 1.68876i −1.42312 − 1.19414i −0.949645 − 0.313327i \(-0.898556\pi\)
−0.473471 − 0.880809i \(-0.656999\pi\)
\(3\) 0.288189 + 0.343451i 0.166386 + 0.198291i 0.842794 − 0.538236i \(-0.180909\pi\)
−0.676408 + 0.736527i \(0.736464\pi\)
\(4\) 0.851300 + 4.82796i 0.425650 + 2.41398i
\(5\) 0 0
\(6\) − 1.17791i − 0.480880i
\(7\) 0.722629 − 1.98541i 0.273128 − 0.750414i −0.724971 − 0.688780i \(-0.758147\pi\)
0.998099 − 0.0616338i \(-0.0196311\pi\)
\(8\) 3.81272 − 6.60383i 1.34800 − 2.33481i
\(9\) 0.486039 − 2.75647i 0.162013 − 0.918822i
\(10\) 0 0
\(11\) −0.739142 + 1.28023i −0.222860 + 0.386005i −0.955675 − 0.294423i \(-0.904873\pi\)
0.732815 + 0.680428i \(0.238206\pi\)
\(12\) −1.41283 + 1.68375i −0.407849 + 0.486056i
\(13\) −0.605037 − 3.43134i −0.167807 − 0.951682i −0.946123 − 0.323808i \(-0.895037\pi\)
0.778316 − 0.627873i \(-0.216074\pi\)
\(14\) −4.80724 + 2.77546i −1.28479 + 0.741773i
\(15\) 0 0
\(16\) −9.61218 + 3.49855i −2.40304 + 0.874637i
\(17\) 0.459537 − 2.60616i 0.111454 − 0.632087i −0.876991 − 0.480507i \(-0.840453\pi\)
0.988445 − 0.151580i \(-0.0484362\pi\)
\(18\) −5.63322 + 4.72683i −1.32776 + 1.11412i
\(19\) 3.23599 + 3.85650i 0.742387 + 0.884742i 0.996599 − 0.0824063i \(-0.0262605\pi\)
−0.254212 + 0.967149i \(0.581816\pi\)
\(20\) 0 0
\(21\) 0.890144 − 0.323986i 0.194245 − 0.0706995i
\(22\) 3.64960 − 1.32835i 0.778097 − 0.283204i
\(23\) 2.36639 + 4.09871i 0.493426 + 0.854639i 0.999971 − 0.00757431i \(-0.00241100\pi\)
−0.506545 + 0.862213i \(0.669078\pi\)
\(24\) 3.36688 − 0.593671i 0.687261 − 0.121183i
\(25\) 0 0
\(26\) −4.57703 + 7.92764i −0.897629 + 1.55474i
\(27\) 2.25161 − 1.29997i 0.433323 − 0.250179i
\(28\) 10.2007 + 1.79865i 1.92774 + 0.339913i
\(29\) −2.25769 − 1.30348i −0.419243 − 0.242050i 0.275511 − 0.961298i \(-0.411153\pi\)
−0.694753 + 0.719248i \(0.744486\pi\)
\(30\) 0 0
\(31\) 1.85091i 0.332433i 0.986089 + 0.166216i \(0.0531551\pi\)
−0.986089 + 0.166216i \(0.946845\pi\)
\(32\) 10.9224 + 3.97544i 1.93083 + 0.702766i
\(33\) −0.652710 + 0.115090i −0.113622 + 0.0200347i
\(34\) −5.32605 + 4.46909i −0.913411 + 0.766442i
\(35\) 0 0
\(36\) 13.7219 2.28698
\(37\) 6.08066 − 0.159781i 0.999655 − 0.0262679i
\(38\) − 13.2264i − 2.14560i
\(39\) 1.00413 − 1.19668i 0.160789 − 0.191621i
\(40\) 0 0
\(41\) −2.16584 − 12.2831i −0.338247 − 1.91830i −0.392467 − 0.919766i \(-0.628378\pi\)
0.0542197 − 0.998529i \(-0.482733\pi\)
\(42\) −2.33863 − 0.851192i −0.360858 − 0.131342i
\(43\) −10.0195 −1.52795 −0.763977 − 0.645244i \(-0.776756\pi\)
−0.763977 + 0.645244i \(0.776756\pi\)
\(44\) −6.81015 − 2.47869i −1.02667 − 0.373677i
\(45\) 0 0
\(46\) 2.15917 − 12.2453i 0.318353 − 1.80547i
\(47\) −11.2904 + 6.51849i −1.64687 + 0.950819i −0.668560 + 0.743658i \(0.733089\pi\)
−0.978307 + 0.207161i \(0.933578\pi\)
\(48\) −3.97171 − 2.29307i −0.573266 − 0.330975i
\(49\) 1.94266 + 1.63009i 0.277523 + 0.232869i
\(50\) 0 0
\(51\) 1.02752 − 0.593240i 0.143882 − 0.0830702i
\(52\) 16.0513 − 5.84220i 2.22592 − 0.810167i
\(53\) −4.52292 − 12.4266i −0.621271 − 1.70693i −0.703856 − 0.710343i \(-0.748540\pi\)
0.0825852 − 0.996584i \(-0.473682\pi\)
\(54\) −6.72691 − 1.18614i −0.915416 − 0.161413i
\(55\) 0 0
\(56\) −10.3561 − 12.3419i −1.38389 − 1.64926i
\(57\) −0.391941 + 2.22281i −0.0519138 + 0.294418i
\(58\) 2.34254 + 6.43607i 0.307590 + 0.845098i
\(59\) −2.17456 − 5.97456i −0.283104 − 0.777821i −0.996988 − 0.0775573i \(-0.975288\pi\)
0.713884 − 0.700264i \(-0.246934\pi\)
\(60\) 0 0
\(61\) 6.76339 − 1.19257i 0.865963 − 0.152693i 0.277017 − 0.960865i \(-0.410654\pi\)
0.588946 + 0.808172i \(0.299543\pi\)
\(62\) 3.12575 − 3.72512i 0.396970 − 0.473091i
\(63\) −5.12148 − 2.95689i −0.645246 − 0.372533i
\(64\) −5.03975 − 8.72911i −0.629969 − 1.09114i
\(65\) 0 0
\(66\) 1.50800 + 0.870643i 0.185622 + 0.107169i
\(67\) 2.54937 − 7.00434i 0.311455 − 0.855716i −0.680908 − 0.732369i \(-0.738415\pi\)
0.992364 − 0.123348i \(-0.0393630\pi\)
\(68\) 12.9737 1.57329
\(69\) −0.725735 + 1.99394i −0.0873683 + 0.240042i
\(70\) 0 0
\(71\) 9.08889 − 7.62648i 1.07865 − 0.905097i 0.0828440 − 0.996563i \(-0.473600\pi\)
0.995808 + 0.0914658i \(0.0291552\pi\)
\(72\) −16.3501 − 13.7194i −1.92688 − 1.61684i
\(73\) − 3.77705i − 0.442071i −0.975266 − 0.221035i \(-0.929056\pi\)
0.975266 − 0.221035i \(-0.0709436\pi\)
\(74\) −12.5077 − 9.94723i −1.45399 − 1.15634i
\(75\) 0 0
\(76\) −15.8643 + 18.9063i −1.81975 + 2.16870i
\(77\) 2.00766 + 2.39263i 0.228794 + 0.272666i
\(78\) −4.04180 + 0.712679i −0.457644 + 0.0806950i
\(79\) 3.52083 − 9.67339i 0.396124 − 1.08834i −0.568033 − 0.823006i \(-0.692295\pi\)
0.964156 − 0.265335i \(-0.0854826\pi\)
\(80\) 0 0
\(81\) −6.79520 − 2.47325i −0.755022 − 0.274806i
\(82\) −16.3843 + 28.3784i −1.80934 + 3.13387i
\(83\) −3.31682 − 0.584845i −0.364068 − 0.0641950i −0.0113781 − 0.999935i \(-0.503622\pi\)
−0.352690 + 0.935740i \(0.614733\pi\)
\(84\) 2.32197 + 4.02177i 0.253348 + 0.438811i
\(85\) 0 0
\(86\) 20.1651 + 16.9205i 2.17446 + 1.82459i
\(87\) −0.202962 − 1.15105i −0.0217598 − 0.123406i
\(88\) 5.63629 + 9.76234i 0.600830 + 1.04067i
\(89\) 1.86920 + 5.13559i 0.198135 + 0.544371i 0.998477 − 0.0551726i \(-0.0175709\pi\)
−0.800342 + 0.599544i \(0.795349\pi\)
\(90\) 0 0
\(91\) −7.24982 − 1.27834i −0.759988 − 0.134006i
\(92\) −17.7739 + 14.9141i −1.85306 + 1.55490i
\(93\) −0.635696 + 0.533412i −0.0659186 + 0.0553123i
\(94\) 33.7310 + 5.94769i 3.47909 + 0.613458i
\(95\) 0 0
\(96\) 1.78236 + 4.89700i 0.181912 + 0.499798i
\(97\) 2.81042 + 4.86779i 0.285355 + 0.494249i 0.972695 − 0.232086i \(-0.0745552\pi\)
−0.687340 + 0.726335i \(0.741222\pi\)
\(98\) −1.15695 − 6.56139i −0.116870 − 0.662801i
\(99\) 3.16966 + 2.65966i 0.318563 + 0.267306i
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 925.2.ba.c.299.1 72
5.2 odd 4 185.2.w.a.151.12 yes 72
5.3 odd 4 925.2.bb.b.151.1 72
5.4 even 2 925.2.ba.b.299.12 72
37.25 even 18 925.2.ba.b.99.12 72
185.32 even 36 6845.2.a.u.1.34 36
185.42 even 36 6845.2.a.t.1.3 36
185.62 odd 36 185.2.w.a.136.12 ✓ 72
185.99 even 18 inner 925.2.ba.c.99.1 72
185.173 odd 36 925.2.bb.b.876.1 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.w.a.136.12 ✓ 72 185.62 odd 36
185.2.w.a.151.12 yes 72 5.2 odd 4
925.2.ba.b.99.12 72 37.25 even 18
925.2.ba.b.299.12 72 5.4 even 2
925.2.ba.c.99.1 72 185.99 even 18 inner
925.2.ba.c.299.1 72 1.1 even 1 trivial
925.2.bb.b.151.1 72 5.3 odd 4
925.2.bb.b.876.1 72 185.173 odd 36
6845.2.a.t.1.3 36 185.42 even 36
6845.2.a.u.1.34 36 185.32 even 36