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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 324.12
Character \(\chi\) \(=\) 925.324
Dual form 925.2.ba.c.374.12

$q$-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+(0.384249 + 2.17919i) q^{2} +(-0.887108 - 0.156421i) q^{3} +(-2.72182 + 0.990661i) q^{4} -1.99328i q^{6} +(1.93805 - 2.30968i) q^{7} +(-0.991889 - 1.71800i) q^{8} +(-2.05658 - 0.748535i) q^{9} +(2.41549 + 4.18375i) q^{11} +(2.56951 - 0.453074i) q^{12} +(-4.58591 + 1.66914i) q^{13} +(5.77792 + 3.33588i) q^{14} +(-1.07498 + 0.902016i) q^{16} +(-7.56960 - 2.75511i) q^{17} +(0.840957 - 4.76930i) q^{18} +(2.22403 + 0.392156i) q^{19} +(-2.08054 + 1.74578i) q^{21} +(-8.18902 + 6.87140i) q^{22} +(-1.83171 + 3.17262i) q^{23} +(0.611181 + 1.67921i) q^{24} +(-5.39949 - 9.35220i) q^{26} +(4.04766 + 2.33692i) q^{27} +(-2.98692 + 8.20649i) q^{28} +(-7.94917 + 4.58945i) q^{29} -3.06814i q^{31} +(-5.41805 - 4.54628i) q^{32} +(-1.48837 - 4.08927i) q^{33} +(3.09528 - 17.5542i) q^{34} +6.33920 q^{36} +(-5.61221 - 2.34587i) q^{37} +4.99725i q^{38} +(4.32929 - 0.763371i) q^{39} +(3.19629 - 1.16335i) q^{41} +(-4.60384 - 3.86308i) q^{42} -5.63474 q^{43} +(-10.7192 - 8.99448i) q^{44} +(-7.61756 - 2.77257i) q^{46} +(3.03257 + 1.75085i) q^{47} +(1.09472 - 0.632036i) q^{48} +(-0.363040 - 2.05890i) q^{49} +(6.28410 + 3.62812i) q^{51} +(10.8285 - 9.08618i) q^{52} +(2.04074 + 2.43206i) q^{53} +(-3.53726 + 9.71855i) q^{54} +(-5.89037 - 1.03863i) q^{56} +(-1.91161 - 0.695769i) q^{57} +(-13.0557 - 15.5592i) q^{58} +(6.41385 + 7.64373i) q^{59} +(1.00942 + 2.77336i) q^{61} +(6.68604 - 1.17893i) q^{62} +(-5.71464 + 3.29935i) q^{63} +(6.42202 - 11.1233i) q^{64} +(8.33938 - 4.81474i) q^{66} +(-5.16145 + 6.15117i) q^{67} +23.3325 q^{68} +(2.12119 - 2.52794i) q^{69} +(1.74629 - 9.90373i) q^{71} +(0.753918 + 4.27568i) q^{72} -10.4650i q^{73} +(2.95561 - 13.1314i) q^{74} +(-6.44189 + 1.13588i) q^{76} +(14.3445 + 2.52931i) q^{77} +(3.32705 + 9.14101i) q^{78} +(3.57459 - 4.26003i) q^{79} +(1.80446 + 1.51412i) q^{81} +(3.76334 + 6.51829i) q^{82} +(-0.0655954 + 0.180222i) q^{83} +(3.93339 - 6.81283i) q^{84} +(-2.16515 - 12.2792i) q^{86} +(7.76966 - 2.82792i) q^{87} +(4.79179 - 8.29963i) q^{88} +(0.630450 + 0.751341i) q^{89} +(-5.03257 + 13.8269i) q^{91} +(1.84260 - 10.4499i) q^{92} +(-0.479921 + 2.72177i) q^{93} +(-2.65018 + 7.28130i) q^{94} +(4.09526 + 4.88054i) q^{96} +(2.36257 - 4.09209i) q^{97} +(4.34724 - 1.58226i) q^{98} +(-1.83597 - 10.4123i) 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 \(\left(\mathbb{Z}/925\mathbb{Z}\right)^\times\).

\(n\) \(76\) \(852\)
\(\chi(n)\) \(e\left(\frac{17}{18}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.384249 + 2.17919i 0.271705 + 1.54092i 0.749236 + 0.662303i \(0.230421\pi\)
−0.477531 + 0.878615i \(0.658468\pi\)
\(3\) −0.887108 − 0.156421i −0.512172 − 0.0903098i −0.0884117 − 0.996084i \(-0.528179\pi\)
−0.423761 + 0.905774i \(0.639290\pi\)
\(4\) −2.72182 + 0.990661i −1.36091 + 0.495331i
\(5\) 0 0
\(6\) − 1.99328i − 0.813753i
\(7\) 1.93805 − 2.30968i 0.732515 − 0.872977i −0.263268 − 0.964723i \(-0.584800\pi\)
0.995782 + 0.0917458i \(0.0292447\pi\)
\(8\) −0.991889 − 1.71800i −0.350686 − 0.607406i
\(9\) −2.05658 − 0.748535i −0.685528 − 0.249512i
\(10\) 0 0
\(11\) 2.41549 + 4.18375i 0.728297 + 1.26145i 0.957602 + 0.288093i \(0.0930212\pi\)
−0.229306 + 0.973354i \(0.573645\pi\)
\(12\) 2.56951 − 0.453074i 0.741753 − 0.130791i
\(13\) −4.58591 + 1.66914i −1.27190 + 0.462935i −0.887747 − 0.460332i \(-0.847730\pi\)
−0.384157 + 0.923268i \(0.625508\pi\)
\(14\) 5.77792 + 3.33588i 1.54421 + 0.891552i
\(15\) 0 0
\(16\) −1.07498 + 0.902016i −0.268745 + 0.225504i
\(17\) −7.56960 − 2.75511i −1.83590 − 0.668212i −0.991099 − 0.133127i \(-0.957498\pi\)
−0.844798 − 0.535085i \(-0.820280\pi\)
\(18\) 0.840957 − 4.76930i 0.198215 − 1.12414i
\(19\) 2.22403 + 0.392156i 0.510227 + 0.0899667i 0.422834 − 0.906207i \(-0.361035\pi\)
0.0873928 + 0.996174i \(0.472146\pi\)
\(20\) 0 0
\(21\) −2.08054 + 1.74578i −0.454012 + 0.380961i
\(22\) −8.18902 + 6.87140i −1.74590 + 1.46499i
\(23\) −1.83171 + 3.17262i −0.381938 + 0.661537i −0.991339 − 0.131326i \(-0.958077\pi\)
0.609401 + 0.792862i \(0.291410\pi\)
\(24\) 0.611181 + 1.67921i 0.124757 + 0.342767i
\(25\) 0 0
\(26\) −5.39949 − 9.35220i −1.05893 − 1.83412i
\(27\) 4.04766 + 2.33692i 0.778972 + 0.449740i
\(28\) −2.98692 + 8.20649i −0.564474 + 1.55088i
\(29\) −7.94917 + 4.58945i −1.47612 + 0.852240i −0.999637 − 0.0269420i \(-0.991423\pi\)
−0.476486 + 0.879182i \(0.658090\pi\)
\(30\) 0 0
\(31\) − 3.06814i − 0.551054i −0.961293 − 0.275527i \(-0.911148\pi\)
0.961293 − 0.275527i \(-0.0888523\pi\)
\(32\) −5.41805 − 4.54628i −0.957784 − 0.803676i
\(33\) −1.48837 − 4.08927i −0.259092 − 0.711851i
\(34\) 3.09528 − 17.5542i 0.530836 − 3.01052i
\(35\) 0 0
\(36\) 6.33920 1.05653
\(37\) −5.61221 − 2.34587i −0.922641 − 0.385659i
\(38\) 4.99725i 0.810662i
\(39\) 4.32929 − 0.763371i 0.693241 − 0.122237i
\(40\) 0 0
\(41\) 3.19629 − 1.16335i 0.499176 − 0.181685i −0.0801471 − 0.996783i \(-0.525539\pi\)
0.579324 + 0.815098i \(0.303317\pi\)
\(42\) −4.60384 − 3.86308i −0.710387 − 0.596086i
\(43\) −5.63474 −0.859290 −0.429645 − 0.902998i \(-0.641361\pi\)
−0.429645 + 0.902998i \(0.641361\pi\)
\(44\) −10.7192 − 8.99448i −1.61598 − 1.35597i
\(45\) 0 0
\(46\) −7.61756 − 2.77257i −1.12315 − 0.408792i
\(47\) 3.03257 + 1.75085i 0.442346 + 0.255388i 0.704592 − 0.709613i \(-0.251130\pi\)
−0.262246 + 0.965001i \(0.584463\pi\)
\(48\) 1.09472 − 0.632036i 0.158009 − 0.0912265i
\(49\) −0.363040 − 2.05890i −0.0518629 − 0.294129i
\(50\) 0 0
\(51\) 6.28410 + 3.62812i 0.879950 + 0.508039i
\(52\) 10.8285 − 9.08618i 1.50164 − 1.26003i
\(53\) 2.04074 + 2.43206i 0.280317 + 0.334069i 0.887770 − 0.460287i \(-0.152253\pi\)
−0.607453 + 0.794355i \(0.707809\pi\)
\(54\) −3.53726 + 9.71855i −0.481361 + 1.32253i
\(55\) 0 0
\(56\) −5.89037 − 1.03863i −0.787134 − 0.138793i
\(57\) −1.91161 − 0.695769i −0.253199 − 0.0921569i
\(58\) −13.0557 − 15.5592i −1.71430 − 2.04303i
\(59\) 6.41385 + 7.64373i 0.835012 + 0.995129i 0.999961 + 0.00882687i \(0.00280972\pi\)
−0.164949 + 0.986302i \(0.552746\pi\)
\(60\) 0 0
\(61\) 1.00942 + 2.77336i 0.129243 + 0.355093i 0.987389 − 0.158313i \(-0.0506056\pi\)
−0.858146 + 0.513406i \(0.828383\pi\)
\(62\) 6.68604 − 1.17893i 0.849128 − 0.149724i
\(63\) −5.71464 + 3.29935i −0.719977 + 0.415679i
\(64\) 6.42202 − 11.1233i 0.802753 − 1.39041i
\(65\) 0 0
\(66\) 8.33938 − 4.81474i 1.02651 − 0.592654i
\(67\) −5.16145 + 6.15117i −0.630571 + 0.751486i −0.982849 − 0.184410i \(-0.940963\pi\)
0.352278 + 0.935895i \(0.385407\pi\)
\(68\) 23.3325 2.82948
\(69\) 2.12119 − 2.52794i 0.255361 − 0.304328i
\(70\) 0 0
\(71\) 1.74629 − 9.90373i 0.207247 − 1.17536i −0.686617 − 0.727019i \(-0.740905\pi\)
0.893864 − 0.448337i \(-0.147984\pi\)
\(72\) 0.753918 + 4.27568i 0.0888501 + 0.503894i
\(73\) − 10.4650i − 1.22483i −0.790535 − 0.612417i \(-0.790197\pi\)
0.790535 − 0.612417i \(-0.209803\pi\)
\(74\) 2.95561 − 13.1314i 0.343583 − 1.52650i
\(75\) 0 0
\(76\) −6.44189 + 1.13588i −0.738936 + 0.130294i
\(77\) 14.3445 + 2.52931i 1.63470 + 0.288242i
\(78\) 3.32705 + 9.14101i 0.376715 + 1.03502i
\(79\) 3.57459 − 4.26003i 0.402173 − 0.479291i −0.526508 − 0.850170i \(-0.676499\pi\)
0.928681 + 0.370879i \(0.120944\pi\)
\(80\) 0 0
\(81\) 1.80446 + 1.51412i 0.200496 + 0.168236i
\(82\) 3.76334 + 6.51829i 0.415591 + 0.719825i
\(83\) −0.0655954 + 0.180222i −0.00720003 + 0.0197819i −0.943241 − 0.332110i \(-0.892239\pi\)
0.936041 + 0.351892i \(0.114462\pi\)
\(84\) 3.93339 − 6.81283i 0.429168 − 0.743340i
\(85\) 0 0
\(86\) −2.16515 − 12.2792i −0.233474 − 1.32410i
\(87\) 7.76966 − 2.82792i 0.832995 − 0.303185i
\(88\) 4.79179 − 8.29963i 0.510807 − 0.884743i
\(89\) 0.630450 + 0.751341i 0.0668276 + 0.0796420i 0.798423 − 0.602098i \(-0.205668\pi\)
−0.731595 + 0.681740i \(0.761224\pi\)
\(90\) 0 0
\(91\) −5.03257 + 13.8269i −0.527557 + 1.44945i
\(92\) 1.84260 − 10.4499i 0.192104 − 1.08948i
\(93\) −0.479921 + 2.72177i −0.0497655 + 0.282234i
\(94\) −2.65018 + 7.28130i −0.273345 + 0.751009i
\(95\) 0 0
\(96\) 4.09526 + 4.88054i 0.417971 + 0.498118i
\(97\) 2.36257 − 4.09209i 0.239882 − 0.415489i −0.720798 − 0.693145i \(-0.756224\pi\)
0.960680 + 0.277657i \(0.0895578\pi\)
\(98\) 4.34724 − 1.58226i 0.439137 − 0.159833i
\(99\) −1.83597 − 10.4123i −0.184522 − 1.04648i
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.324.12 72
5.2 odd 4 925.2.bb.b.176.2 72
5.3 odd 4 185.2.w.a.176.11 yes 72
5.4 even 2 925.2.ba.b.324.1 72
37.4 even 18 925.2.ba.b.374.1 72
185.4 even 18 inner 925.2.ba.c.374.12 72
185.78 odd 36 185.2.w.a.41.11 ✓ 72
185.113 even 36 6845.2.a.t.1.34 36
185.152 odd 36 925.2.bb.b.226.2 72
185.183 even 36 6845.2.a.u.1.3 36
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.w.a.41.11 ✓ 72 185.78 odd 36
185.2.w.a.176.11 yes 72 5.3 odd 4
925.2.ba.b.324.1 72 5.4 even 2
925.2.ba.b.374.1 72 37.4 even 18
925.2.ba.c.324.12 72 1.1 even 1 trivial
925.2.ba.c.374.12 72 185.4 even 18 inner
925.2.bb.b.176.2 72 5.2 odd 4
925.2.bb.b.226.2 72 185.152 odd 36
6845.2.a.t.1.34 36 185.113 even 36
6845.2.a.u.1.3 36 185.183 even 36