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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 151.7
Character \(\chi\) \(=\) 925.151
Dual form 925.2.bb.b.876.7

$q$-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+(0.301512 - 0.359328i) q^{2} +(2.39156 - 2.00676i) q^{3} +(0.309089 + 1.75293i) q^{4} -1.46442i q^{6} +(-1.06448 - 0.387441i) q^{7} +(1.53553 + 0.886536i) q^{8} +(1.17154 - 6.64413i) q^{9} +(0.690789 - 1.19648i) q^{11} +(4.25691 + 3.57197i) q^{12} +(6.56428 - 1.15746i) q^{13} +(-0.460174 + 0.265681i) q^{14} +(-2.56372 + 0.933117i) q^{16} +(-3.42415 - 0.603770i) q^{17} +(-2.03419 - 2.42425i) q^{18} +(3.70029 + 4.40983i) q^{19} +(-3.32328 + 1.20957i) q^{21} +(-0.221648 - 0.608974i) q^{22} +(-2.26680 + 1.30874i) q^{23} +(5.45136 - 0.961222i) q^{24} +(1.56330 - 2.70772i) q^{26} +(-5.84841 - 10.1297i) q^{27} +(0.350136 - 1.98572i) q^{28} +(-6.17967 - 3.56783i) q^{29} +2.69515i q^{31} +(-1.65055 + 4.53485i) q^{32} +(-0.748985 - 4.24770i) q^{33} +(-1.24938 + 1.04835i) q^{34} +12.0088 q^{36} +(5.05847 - 3.37814i) q^{37} +2.70026 q^{38} +(13.3761 - 15.9410i) q^{39} +(1.38180 + 7.83657i) q^{41} +(-0.567375 + 1.55885i) q^{42} -10.8290i q^{43} +(2.31087 + 0.841086i) q^{44} +(-0.213202 + 1.20913i) q^{46} +(-0.933248 - 1.61643i) q^{47} +(-4.25875 + 7.37636i) q^{48} +(-4.37929 - 3.67466i) q^{49} +(-9.40068 + 5.42748i) q^{51} +(4.05790 + 11.1490i) q^{52} +(-1.92330 + 0.700025i) q^{53} +(-5.40327 - 0.952742i) q^{54} +(-1.29106 - 1.53863i) q^{56} +(17.6989 + 3.12080i) q^{57} +(-3.14527 + 1.14479i) q^{58} +(2.08921 + 5.74005i) q^{59} +(6.18979 - 1.09143i) q^{61} +(0.968445 + 0.812622i) q^{62} +(-3.82129 + 6.61867i) q^{63} +(-1.59641 - 2.76507i) q^{64} +(-1.75215 - 1.01160i) q^{66} +(-5.97892 - 2.17615i) q^{67} -6.18892i q^{68} +(-2.79487 + 7.67884i) q^{69} +(-3.10854 + 2.60837i) q^{71} +(7.68919 - 9.16362i) q^{72} +0.109760 q^{73} +(0.311329 - 2.83620i) q^{74} +(-6.58642 + 7.84939i) q^{76} +(-1.19890 + 1.00600i) q^{77} +(-1.69500 - 9.61285i) q^{78} +(-0.920146 + 2.52808i) q^{79} +(-15.2955 - 5.56709i) q^{81} +(3.23253 + 1.86630i) q^{82} +(-2.40459 + 13.6371i) q^{83} +(-3.14749 - 5.45161i) q^{84} +(-3.89117 - 3.26508i) q^{86} +(-21.9388 + 3.86841i) q^{87} +(2.12145 - 1.22482i) q^{88} +(4.36882 + 12.0032i) q^{89} +(-7.43602 - 1.31117i) q^{91} +(-2.99477 - 3.56903i) q^{92} +(5.40851 + 6.44562i) q^{93} +(-0.862216 - 0.152032i) q^{94} +(5.15295 + 14.1576i) q^{96} +(-3.43944 + 1.98576i) q^{97} +(-2.64082 + 0.465648i) q^{98} +(-7.14029 - 5.99142i) 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 \(\left(\mathbb{Z}/925\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.301512 − 0.359328i 0.213201 − 0.254084i −0.648836 − 0.760928i \(-0.724744\pi\)
0.862037 + 0.506845i \(0.169188\pi\)
\(3\) 2.39156 − 2.00676i 1.38077 − 1.15860i 0.411840 − 0.911256i \(-0.364886\pi\)
0.968927 − 0.247345i \(-0.0795581\pi\)
\(4\) 0.309089 + 1.75293i 0.154545 + 0.876466i
\(5\) 0 0
\(6\) − 1.46442i − 0.597846i
\(7\) −1.06448 − 0.387441i −0.402337 − 0.146439i 0.132922 − 0.991126i \(-0.457564\pi\)
−0.535259 + 0.844688i \(0.679786\pi\)
\(8\) 1.53553 + 0.886536i 0.542890 + 0.313438i
\(9\) 1.17154 − 6.64413i 0.390513 − 2.21471i
\(10\) 0 0
\(11\) 0.690789 − 1.19648i 0.208281 − 0.360753i −0.742892 − 0.669411i \(-0.766547\pi\)
0.951173 + 0.308658i \(0.0998799\pi\)
\(12\) 4.25691 + 3.57197i 1.22886 + 1.03114i
\(13\) 6.56428 − 1.15746i 1.82060 − 0.321022i 0.844045 − 0.536272i \(-0.180168\pi\)
0.976559 + 0.215250i \(0.0690567\pi\)
\(14\) −0.460174 + 0.265681i −0.122987 + 0.0710063i
\(15\) 0 0
\(16\) −2.56372 + 0.933117i −0.640930 + 0.233279i
\(17\) −3.42415 − 0.603770i −0.830478 − 0.146436i −0.257782 − 0.966203i \(-0.582992\pi\)
−0.572697 + 0.819767i \(0.694103\pi\)
\(18\) −2.03419 − 2.42425i −0.479463 − 0.571402i
\(19\) 3.70029 + 4.40983i 0.848905 + 1.01169i 0.999733 + 0.0231252i \(0.00736162\pi\)
−0.150828 + 0.988560i \(0.548194\pi\)
\(20\) 0 0
\(21\) −3.32328 + 1.20957i −0.725198 + 0.263951i
\(22\) −0.221648 − 0.608974i −0.0472556 − 0.129834i
\(23\) −2.26680 + 1.30874i −0.472661 + 0.272891i −0.717353 − 0.696710i \(-0.754646\pi\)
0.244692 + 0.969601i \(0.421313\pi\)
\(24\) 5.45136 − 0.961222i 1.11275 − 0.196209i
\(25\) 0 0
\(26\) 1.56330 − 2.70772i 0.306589 − 0.531028i
\(27\) −5.84841 − 10.1297i −1.12553 − 1.94947i
\(28\) 0.350136 − 1.98572i 0.0661696 − 0.375266i
\(29\) −6.17967 − 3.56783i −1.14754 − 0.662530i −0.199250 − 0.979949i \(-0.563851\pi\)
−0.948286 + 0.317418i \(0.897184\pi\)
\(30\) 0 0
\(31\) 2.69515i 0.484064i 0.970268 + 0.242032i \(0.0778139\pi\)
−0.970268 + 0.242032i \(0.922186\pi\)
\(32\) −1.65055 + 4.53485i −0.291779 + 0.801655i
\(33\) −0.748985 − 4.24770i −0.130381 − 0.739430i
\(34\) −1.24938 + 1.04835i −0.214266 + 0.179791i
\(35\) 0 0
\(36\) 12.0088 2.00147
\(37\) 5.05847 − 3.37814i 0.831608 − 0.555363i
\(38\) 2.70026 0.438040
\(39\) 13.3761 − 15.9410i 2.14189 − 2.55261i
\(40\) 0 0
\(41\) 1.38180 + 7.83657i 0.215801 + 1.22387i 0.879512 + 0.475877i \(0.157869\pi\)
−0.663711 + 0.747989i \(0.731020\pi\)
\(42\) −0.567375 + 1.55885i −0.0875478 + 0.240536i
\(43\) − 10.8290i − 1.65141i −0.564105 − 0.825703i \(-0.690779\pi\)
0.564105 − 0.825703i \(-0.309221\pi\)
\(44\) 2.31087 + 0.841086i 0.348376 + 0.126799i
\(45\) 0 0
\(46\) −0.213202 + 1.20913i −0.0314349 + 0.178276i
\(47\) −0.933248 − 1.61643i −0.136128 − 0.235781i 0.789900 − 0.613236i \(-0.210133\pi\)
−0.926028 + 0.377455i \(0.876799\pi\)
\(48\) −4.25875 + 7.37636i −0.614697 + 1.06469i
\(49\) −4.37929 − 3.67466i −0.625613 − 0.524952i
\(50\) 0 0
\(51\) −9.40068 + 5.42748i −1.31636 + 0.760000i
\(52\) 4.05790 + 11.1490i 0.562729 + 1.54609i
\(53\) −1.92330 + 0.700025i −0.264186 + 0.0961558i −0.470717 − 0.882284i \(-0.656005\pi\)
0.206531 + 0.978440i \(0.433783\pi\)
\(54\) −5.40327 − 0.952742i −0.735292 − 0.129652i
\(55\) 0 0
\(56\) −1.29106 − 1.53863i −0.172525 − 0.205608i
\(57\) 17.6989 + 3.12080i 2.34428 + 0.413360i
\(58\) −3.14527 + 1.14479i −0.412994 + 0.150318i
\(59\) 2.08921 + 5.74005i 0.271992 + 0.747291i 0.998209 + 0.0598256i \(0.0190545\pi\)
−0.726217 + 0.687465i \(0.758723\pi\)
\(60\) 0 0
\(61\) 6.18979 − 1.09143i 0.792521 − 0.139743i 0.237290 − 0.971439i \(-0.423741\pi\)
0.555231 + 0.831696i \(0.312630\pi\)
\(62\) 0.968445 + 0.812622i 0.122993 + 0.103203i
\(63\) −3.82129 + 6.61867i −0.481437 + 0.833874i
\(64\) −1.59641 − 2.76507i −0.199552 − 0.345634i
\(65\) 0 0
\(66\) −1.75215 − 1.01160i −0.215675 − 0.124520i
\(67\) −5.97892 − 2.17615i −0.730442 − 0.265859i −0.0500900 − 0.998745i \(-0.515951\pi\)
−0.680352 + 0.732886i \(0.738173\pi\)
\(68\) − 6.18892i − 0.750517i
\(69\) −2.79487 + 7.67884i −0.336463 + 0.924424i
\(70\) 0 0
\(71\) −3.10854 + 2.60837i −0.368915 + 0.309557i −0.808332 − 0.588727i \(-0.799629\pi\)
0.439417 + 0.898283i \(0.355185\pi\)
\(72\) 7.68919 − 9.16362i 0.906179 − 1.07994i
\(73\) 0.109760 0.0128464 0.00642322 − 0.999979i \(-0.497955\pi\)
0.00642322 + 0.999979i \(0.497955\pi\)
\(74\) 0.311329 − 2.83620i 0.0361913 − 0.329702i
\(75\) 0 0
\(76\) −6.58642 + 7.84939i −0.755514 + 0.900386i
\(77\) −1.19890 + 1.00600i −0.136627 + 0.114644i
\(78\) −1.69500 − 9.61285i −0.191921 − 1.08844i
\(79\) −0.920146 + 2.52808i −0.103525 + 0.284431i −0.980631 − 0.195865i \(-0.937249\pi\)
0.877106 + 0.480296i \(0.159471\pi\)
\(80\) 0 0
\(81\) −15.2955 − 5.56709i −1.69950 − 0.618566i
\(82\) 3.23253 + 1.86630i 0.356973 + 0.206099i
\(83\) −2.40459 + 13.6371i −0.263939 + 1.49687i 0.508102 + 0.861297i \(0.330347\pi\)
−0.772041 + 0.635573i \(0.780764\pi\)
\(84\) −3.14749 − 5.45161i −0.343419 − 0.594819i
\(85\) 0 0
\(86\) −3.89117 − 3.26508i −0.419595 − 0.352082i
\(87\) −21.9388 + 3.86841i −2.35209 + 0.414737i
\(88\) 2.12145 − 1.22482i 0.226147 − 0.130566i
\(89\) 4.36882 + 12.0032i 0.463093 + 1.27234i 0.923147 + 0.384448i \(0.125608\pi\)
−0.460053 + 0.887891i \(0.652170\pi\)
\(90\) 0 0
\(91\) −7.43602 − 1.31117i −0.779507 − 0.137448i
\(92\) −2.99477 − 3.56903i −0.312227 − 0.372097i
\(93\) 5.40851 + 6.44562i 0.560837 + 0.668379i
\(94\) −0.862216 − 0.152032i −0.0889308 − 0.0156809i
\(95\) 0 0
\(96\) 5.15295 + 14.1576i 0.525921 + 1.44495i
\(97\) −3.43944 + 1.98576i −0.349222 + 0.201623i −0.664343 − 0.747428i \(-0.731288\pi\)
0.315121 + 0.949052i \(0.397955\pi\)
\(98\) −2.64082 + 0.465648i −0.266763 + 0.0470376i
\(99\) −7.14029 − 5.99142i −0.717627 − 0.602160i
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.151.7 72
5.2 odd 4 925.2.ba.c.299.8 72
5.3 odd 4 925.2.ba.b.299.5 72
5.4 even 2 185.2.w.a.151.6 yes 72
37.25 even 18 inner 925.2.bb.b.876.7 72
185.62 odd 36 925.2.ba.b.99.5 72
185.69 odd 36 6845.2.a.u.1.14 36
185.79 odd 36 6845.2.a.t.1.23 36
185.99 even 18 185.2.w.a.136.6 ✓ 72
185.173 odd 36 925.2.ba.c.99.8 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.w.a.136.6 ✓ 72 185.99 even 18
185.2.w.a.151.6 yes 72 5.4 even 2
925.2.ba.b.99.5 72 185.62 odd 36
925.2.ba.b.299.5 72 5.3 odd 4
925.2.ba.c.99.8 72 185.173 odd 36
925.2.ba.c.299.8 72 5.2 odd 4
925.2.bb.b.151.7 72 1.1 even 1 trivial
925.2.bb.b.876.7 72 37.25 even 18 inner
6845.2.a.t.1.23 36 185.79 odd 36
6845.2.a.u.1.14 36 185.69 odd 36