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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [925,2,Mod(151,925)] 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("925.151"); 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([0, 13])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 925 = 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 925.bb (of order \(18\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [72,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == 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 176.2
Character \(\chi\) \(=\) 925.176
Dual form 925.2.bb.b.226.2

$q$-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+(-2.17919 + 0.384249i) q^{2} +(-0.156421 + 0.887108i) q^{3} +(2.72182 - 0.990661i) q^{4} -1.99328i q^{6} +(2.30968 + 1.93805i) q^{7} +(-1.71800 + 0.991889i) q^{8} +(2.05658 + 0.748535i) q^{9} +(2.41549 + 4.18375i) q^{11} +(0.453074 + 2.56951i) q^{12} +(1.66914 + 4.58591i) q^{13} +(-5.77792 - 3.33588i) q^{14} +(-1.07498 + 0.902016i) q^{16} +(2.75511 - 7.56960i) q^{17} +(-4.76930 - 0.840957i) q^{18} +(-2.22403 - 0.392156i) q^{19} +(-2.08054 + 1.74578i) q^{21} +(-6.87140 - 8.18902i) q^{22} +(3.17262 + 1.83171i) q^{23} +(-0.611181 - 1.67921i) q^{24} +(-5.39949 - 9.35220i) q^{26} +(-2.33692 + 4.04766i) q^{27} +(8.20649 + 2.98692i) q^{28} +(7.94917 - 4.58945i) q^{29} -3.06814i q^{31} +(4.54628 - 5.41805i) q^{32} +(-4.08927 + 1.48837i) q^{33} +(-3.09528 + 17.5542i) q^{34} +6.33920 q^{36} +(2.34587 - 5.61221i) q^{37} +4.99725 q^{38} +(-4.32929 + 0.763371i) q^{39} +(3.19629 - 1.16335i) q^{41} +(3.86308 - 4.60384i) q^{42} +5.63474i q^{43} +(10.7192 + 8.99448i) q^{44} +(-7.61756 - 2.77257i) q^{46} +(-1.75085 + 3.03257i) q^{47} +(-0.632036 - 1.09472i) q^{48} +(0.363040 + 2.05890i) q^{49} +(6.28410 + 3.62812i) q^{51} +(9.08618 + 10.8285i) q^{52} +(2.43206 - 2.04074i) q^{53} +(3.53726 - 9.71855i) q^{54} +(-5.89037 - 1.03863i) q^{56} +(0.695769 - 1.91161i) q^{57} +(-15.5592 + 13.0557i) q^{58} +(-6.41385 - 7.64373i) q^{59} +(1.00942 + 2.77336i) q^{61} +(1.17893 + 6.68604i) q^{62} +(3.29935 + 5.71464i) q^{63} +(-6.42202 + 11.1233i) q^{64} +(8.33938 - 4.81474i) q^{66} +(-6.15117 - 5.16145i) q^{67} -23.3325i q^{68} +(-2.12119 + 2.52794i) q^{69} +(1.74629 - 9.90373i) q^{71} +(-4.27568 + 0.753918i) q^{72} -10.4650 q^{73} +(-2.95561 + 13.1314i) q^{74} +(-6.44189 + 1.13588i) q^{76} +(-2.52931 + 14.3445i) q^{77} +(9.14101 - 3.32705i) q^{78} +(-3.57459 + 4.26003i) q^{79} +(1.80446 + 1.51412i) q^{81} +(-6.51829 + 3.76334i) q^{82} +(0.180222 + 0.0655954i) q^{83} +(-3.93339 + 6.81283i) q^{84} +(-2.16515 - 12.2792i) q^{86} +(2.82792 + 7.76966i) q^{87} +(-8.29963 - 4.79179i) q^{88} +(-0.630450 - 0.751341i) q^{89} +(-5.03257 + 13.8269i) q^{91} +(10.4499 + 1.84260i) q^{92} +(2.72177 + 0.479921i) q^{93} +(2.65018 - 7.28130i) q^{94} +(4.09526 + 4.88054i) q^{96} +(4.09209 + 2.36257i) q^{97} +(-1.58226 - 4.34724i) q^{98} +(1.83597 + 10.4123i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 72 q - 6 q^{2} - 6 q^{4} + 6 q^{7} + 18 q^{8} - 30 q^{12} + 30 q^{13} - 18 q^{14} - 18 q^{16} + 30 q^{18} - 18 q^{19} - 30 q^{22} - 30 q^{24} + 24 q^{27} - 30 q^{28} + 18 q^{29} + 36 q^{32} + 12 q^{33}+ \cdots - 12 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.17919 + 0.384249i −1.54092 + 0.271705i −0.878615 − 0.477531i \(-0.841532\pi\)
−0.662303 + 0.749236i \(0.730421\pi\)
\(3\) −0.156421 + 0.887108i −0.0903098 + 0.512172i 0.905774 + 0.423761i \(0.139290\pi\)
−0.996084 + 0.0884117i \(0.971821\pi\)
\(4\) 2.72182 − 0.990661i 1.36091 − 0.495331i
\(5\) 0 0
\(6\) − 1.99328i − 0.813753i
\(7\) 2.30968 + 1.93805i 0.872977 + 0.732515i 0.964723 − 0.263268i \(-0.0848003\pi\)
−0.0917458 + 0.995782i \(0.529245\pi\)
\(8\) −1.71800 + 0.991889i −0.607406 + 0.350686i
\(9\) 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 \(0.573645\pi\)
\(12\) 0.453074 + 2.56951i 0.130791 + 0.741753i
\(13\) 1.66914 + 4.58591i 0.462935 + 1.27190i 0.923268 + 0.384157i \(0.125508\pi\)
−0.460332 + 0.887747i \(0.652270\pi\)
\(14\) −5.77792 − 3.33588i −1.54421 − 0.891552i
\(15\) 0 0
\(16\) −1.07498 + 0.902016i −0.268745 + 0.225504i
\(17\) 2.75511 − 7.56960i 0.668212 − 1.83590i 0.133127 − 0.991099i \(-0.457498\pi\)
0.535085 − 0.844798i \(-0.320280\pi\)
\(18\) −4.76930 − 0.840957i −1.12414 − 0.198215i
\(19\) −2.22403 − 0.392156i −0.510227 − 0.0899667i −0.0873928 − 0.996174i \(-0.527854\pi\)
−0.422834 + 0.906207i \(0.638965\pi\)
\(20\) 0 0
\(21\) −2.08054 + 1.74578i −0.454012 + 0.380961i
\(22\) −6.87140 − 8.18902i −1.46499 − 1.74590i
\(23\) 3.17262 + 1.83171i 0.661537 + 0.381938i 0.792862 − 0.609401i \(-0.208590\pi\)
−0.131326 + 0.991339i \(0.541923\pi\)
\(24\) −0.611181 − 1.67921i −0.124757 − 0.342767i
\(25\) 0 0
\(26\) −5.39949 − 9.35220i −1.05893 − 1.83412i
\(27\) −2.33692 + 4.04766i −0.449740 + 0.778972i
\(28\) 8.20649 + 2.98692i 1.55088 + 0.564474i
\(29\) 7.94917 − 4.58945i 1.47612 − 0.852240i 0.476486 − 0.879182i \(-0.341910\pi\)
0.999637 + 0.0269420i \(0.00857696\pi\)
\(30\) 0 0
\(31\) − 3.06814i − 0.551054i −0.961293 − 0.275527i \(-0.911148\pi\)
0.961293 − 0.275527i \(-0.0888523\pi\)
\(32\) 4.54628 − 5.41805i 0.803676 − 0.957784i
\(33\) −4.08927 + 1.48837i −0.711851 + 0.259092i
\(34\) −3.09528 + 17.5542i −0.530836 + 3.01052i
\(35\) 0 0
\(36\) 6.33920 1.05653
\(37\) 2.34587 − 5.61221i 0.385659 − 0.922641i
\(38\) 4.99725 0.810662
\(39\) −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 \(0.303317\pi\)
\(42\) 3.86308 − 4.60384i 0.596086 − 0.710387i
\(43\) 5.63474i 0.859290i 0.902998 + 0.429645i \(0.141361\pi\)
−0.902998 + 0.429645i \(0.858639\pi\)
\(44\) 10.7192 + 8.99448i 1.61598 + 1.35597i
\(45\) 0 0
\(46\) −7.61756 − 2.77257i −1.12315 − 0.408792i
\(47\) −1.75085 + 3.03257i −0.255388 + 0.442346i −0.965001 − 0.262246i \(-0.915537\pi\)
0.709613 + 0.704592i \(0.248870\pi\)
\(48\) −0.632036 − 1.09472i −0.0912265 − 0.158009i
\(49\) 0.363040 + 2.05890i 0.0518629 + 0.294129i
\(50\) 0 0
\(51\) 6.28410 + 3.62812i 0.879950 + 0.508039i
\(52\) 9.08618 + 10.8285i 1.26003 + 1.50164i
\(53\) 2.43206 − 2.04074i 0.334069 − 0.280317i −0.460287 − 0.887770i \(-0.652253\pi\)
0.794355 + 0.607453i \(0.207809\pi\)
\(54\) 3.53726 − 9.71855i 0.481361 − 1.32253i
\(55\) 0 0
\(56\) −5.89037 − 1.03863i −0.787134 − 0.138793i
\(57\) 0.695769 − 1.91161i 0.0921569 − 0.253199i
\(58\) −15.5592 + 13.0557i −2.04303 + 1.71430i
\(59\) −6.41385 − 7.64373i −0.835012 − 0.995129i −0.999961 − 0.00882687i \(-0.997190\pi\)
0.164949 − 0.986302i \(-0.447254\pi\)
\(60\) 0 0
\(61\) 1.00942 + 2.77336i 0.129243 + 0.355093i 0.987389 − 0.158313i \(-0.0506056\pi\)
−0.858146 + 0.513406i \(0.828383\pi\)
\(62\) 1.17893 + 6.68604i 0.149724 + 0.849128i
\(63\) 3.29935 + 5.71464i 0.415679 + 0.719977i
\(64\) −6.42202 + 11.1233i −0.802753 + 1.39041i
\(65\) 0 0
\(66\) 8.33938 − 4.81474i 1.02651 − 0.592654i
\(67\) −6.15117 − 5.16145i −0.751486 − 0.630571i 0.184410 − 0.982849i \(-0.440963\pi\)
−0.935895 + 0.352278i \(0.885407\pi\)
\(68\) − 23.3325i − 2.82948i
\(69\) −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 \(-0.147984\pi\)
\(72\) −4.27568 + 0.753918i −0.503894 + 0.0888501i
\(73\) −10.4650 −1.22483 −0.612417 − 0.790535i \(-0.709803\pi\)
−0.612417 + 0.790535i \(0.709803\pi\)
\(74\) −2.95561 + 13.1314i −0.343583 + 1.52650i
\(75\) 0 0
\(76\) −6.44189 + 1.13588i −0.738936 + 0.130294i
\(77\) −2.52931 + 14.3445i −0.288242 + 1.63470i
\(78\) 9.14101 − 3.32705i 1.03502 − 0.376715i
\(79\) −3.57459 + 4.26003i −0.402173 + 0.479291i −0.928681 − 0.370879i \(-0.879056\pi\)
0.526508 + 0.850170i \(0.323501\pi\)
\(80\) 0 0
\(81\) 1.80446 + 1.51412i 0.200496 + 0.168236i
\(82\) −6.51829 + 3.76334i −0.719825 + 0.415591i
\(83\) 0.180222 + 0.0655954i 0.0197819 + 0.00720003i 0.351892 − 0.936041i \(-0.385538\pi\)
−0.332110 + 0.943241i \(0.607761\pi\)
\(84\) −3.93339 + 6.81283i −0.429168 + 0.743340i
\(85\) 0 0
\(86\) −2.16515 − 12.2792i −0.233474 − 1.32410i
\(87\) 2.82792 + 7.76966i 0.303185 + 0.832995i
\(88\) −8.29963 − 4.79179i −0.884743 − 0.510807i
\(89\) −0.630450 − 0.751341i −0.0668276 − 0.0796420i 0.731595 − 0.681740i \(-0.238776\pi\)
−0.798423 + 0.602098i \(0.794332\pi\)
\(90\) 0 0
\(91\) −5.03257 + 13.8269i −0.527557 + 1.44945i
\(92\) 10.4499 + 1.84260i 1.08948 + 0.192104i
\(93\) 2.72177 + 0.479921i 0.282234 + 0.0497655i
\(94\) 2.65018 − 7.28130i 0.273345 − 0.751009i
\(95\) 0 0
\(96\) 4.09526 + 4.88054i 0.417971 + 0.498118i
\(97\) 4.09209 + 2.36257i 0.415489 + 0.239882i 0.693145 − 0.720798i \(-0.256224\pi\)
−0.277657 + 0.960680i \(0.589558\pi\)
\(98\) −1.58226 − 4.34724i −0.159833 − 0.439137i
\(99\) 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.bb.b.176.2 72
5.2 odd 4 925.2.ba.b.324.1 72
5.3 odd 4 925.2.ba.c.324.12 72
5.4 even 2 185.2.w.a.176.11 yes 72
37.4 even 18 inner 925.2.bb.b.226.2 72
185.4 even 18 185.2.w.a.41.11 ✓ 72
185.39 odd 36 6845.2.a.t.1.34 36
185.78 odd 36 925.2.ba.b.374.1 72
185.109 odd 36 6845.2.a.u.1.3 36
185.152 odd 36 925.2.ba.c.374.12 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.w.a.41.11 ✓ 72 185.4 even 18
185.2.w.a.176.11 yes 72 5.4 even 2
925.2.ba.b.324.1 72 5.2 odd 4
925.2.ba.b.374.1 72 185.78 odd 36
925.2.ba.c.324.12 72 5.3 odd 4
925.2.ba.c.374.12 72 185.152 odd 36
925.2.bb.b.176.2 72 1.1 even 1 trivial
925.2.bb.b.226.2 72 37.4 even 18 inner
6845.2.a.t.1.34 36 185.39 odd 36
6845.2.a.u.1.3 36 185.109 odd 36