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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 299.8
Character \(\chi\) \(=\) 925.299
Dual form 925.2.ba.c.99.8

$q$-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+(0.359328 + 0.301512i) q^{2} +(-2.00676 - 2.39156i) q^{3} +(-0.309089 - 1.75293i) q^{4} -1.46442i q^{6} +(0.387441 - 1.06448i) q^{7} +(0.886536 - 1.53553i) q^{8} +(-1.17154 + 6.64413i) q^{9} +(0.690789 - 1.19648i) q^{11} +(-3.57197 + 4.25691i) q^{12} +(-1.15746 - 6.56428i) q^{13} +(0.460174 - 0.265681i) q^{14} +(-2.56372 + 0.933117i) q^{16} +(0.603770 - 3.42415i) q^{17} +(-2.42425 + 2.03419i) q^{18} +(-3.70029 - 4.40983i) q^{19} +(-3.32328 + 1.20957i) q^{21} +(0.608974 - 0.221648i) q^{22} +(1.30874 + 2.26680i) q^{23} +(-5.45136 + 0.961222i) q^{24} +(1.56330 - 2.70772i) q^{26} +(10.1297 - 5.84841i) q^{27} +(-1.98572 - 0.350136i) q^{28} +(6.17967 + 3.56783i) q^{29} +2.69515i q^{31} +(-4.53485 - 1.65055i) q^{32} +(-4.24770 + 0.748985i) q^{33} +(1.24938 - 1.04835i) q^{34} +12.0088 q^{36} +(3.37814 + 5.05847i) q^{37} -2.70026i q^{38} +(-13.3761 + 15.9410i) q^{39} +(1.38180 + 7.83657i) q^{41} +(-1.55885 - 0.567375i) q^{42} -10.8290 q^{43} +(-2.31087 - 0.841086i) q^{44} +(-0.213202 + 1.20913i) q^{46} +(1.61643 - 0.933248i) q^{47} +(7.37636 + 4.25875i) q^{48} +(4.37929 + 3.67466i) q^{49} +(-9.40068 + 5.42748i) q^{51} +(-11.1490 + 4.05790i) q^{52} +(0.700025 + 1.92330i) q^{53} +(5.40327 + 0.952742i) q^{54} +(-1.29106 - 1.53863i) q^{56} +(-3.12080 + 17.6989i) q^{57} +(1.14479 + 3.14527i) q^{58} +(-2.08921 - 5.74005i) q^{59} +(6.18979 - 1.09143i) q^{61} +(-0.812622 + 0.968445i) q^{62} +(6.61867 + 3.82129i) q^{63} +(1.59641 + 2.76507i) q^{64} +(-1.75215 - 1.01160i) q^{66} +(2.17615 - 5.97892i) q^{67} -6.18892 q^{68} +(2.79487 - 7.67884i) q^{69} +(-3.10854 + 2.60837i) q^{71} +(9.16362 + 7.68919i) q^{72} -0.109760i q^{73} +(-0.311329 + 2.83620i) q^{74} +(-6.58642 + 7.84939i) q^{76} +(-1.00600 - 1.19890i) q^{77} +(-9.61285 + 1.69500i) q^{78} +(0.920146 - 2.52808i) q^{79} +(-15.2955 - 5.56709i) q^{81} +(-1.86630 + 3.23253i) q^{82} +(13.6371 + 2.40459i) q^{83} +(3.14749 + 5.45161i) q^{84} +(-3.89117 - 3.26508i) q^{86} +(-3.86841 - 21.9388i) q^{87} +(-1.22482 - 2.12145i) q^{88} +(-4.36882 - 12.0032i) q^{89} +(-7.43602 - 1.31117i) q^{91} +(3.56903 - 2.99477i) q^{92} +(6.44562 - 5.40851i) q^{93} +(0.862216 + 0.152032i) q^{94} +(5.15295 + 14.1576i) q^{96} +(-1.98576 - 3.43944i) q^{97} +(0.465648 + 2.64082i) q^{98} +(7.14029 + 5.99142i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.359328 + 0.301512i 0.254084 + 0.213201i 0.760928 − 0.648836i \(-0.224744\pi\)
−0.506845 + 0.862037i \(0.669188\pi\)
\(3\) −2.00676 − 2.39156i −1.15860 − 1.38077i −0.911256 − 0.411840i \(-0.864886\pi\)
−0.247345 − 0.968927i \(-0.579558\pi\)
\(4\) −0.309089 − 1.75293i −0.154545 − 0.876466i
\(5\) 0 0
\(6\) − 1.46442i − 0.597846i
\(7\) 0.387441 − 1.06448i 0.146439 − 0.402337i −0.844688 − 0.535259i \(-0.820214\pi\)
0.991126 + 0.132922i \(0.0424361\pi\)
\(8\) 0.886536 − 1.53553i 0.313438 − 0.542890i
\(9\) −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 \(0.0998799\pi\)
\(12\) −3.57197 + 4.25691i −1.03114 + 1.22886i
\(13\) −1.15746 − 6.56428i −0.321022 − 1.82060i −0.536272 − 0.844045i \(-0.680168\pi\)
0.215250 − 0.976559i \(-0.430943\pi\)
\(14\) 0.460174 − 0.265681i 0.122987 − 0.0710063i
\(15\) 0 0
\(16\) −2.56372 + 0.933117i −0.640930 + 0.233279i
\(17\) 0.603770 − 3.42415i 0.146436 − 0.830478i −0.819767 − 0.572697i \(-0.805897\pi\)
0.966203 − 0.257782i \(-0.0829916\pi\)
\(18\) −2.42425 + 2.03419i −0.571402 + 0.479463i
\(19\) −3.70029 − 4.40983i −0.848905 − 1.01169i −0.999733 − 0.0231252i \(-0.992638\pi\)
0.150828 − 0.988560i \(-0.451806\pi\)
\(20\) 0 0
\(21\) −3.32328 + 1.20957i −0.725198 + 0.263951i
\(22\) 0.608974 − 0.221648i 0.129834 − 0.0472556i
\(23\) 1.30874 + 2.26680i 0.272891 + 0.472661i 0.969601 − 0.244692i \(-0.0786869\pi\)
−0.696710 + 0.717353i \(0.745354\pi\)
\(24\) −5.45136 + 0.961222i −1.11275 + 0.196209i
\(25\) 0 0
\(26\) 1.56330 − 2.70772i 0.306589 − 0.531028i
\(27\) 10.1297 − 5.84841i 1.94947 − 1.12553i
\(28\) −1.98572 − 0.350136i −0.375266 − 0.0661696i
\(29\) 6.17967 + 3.56783i 1.14754 + 0.662530i 0.948286 − 0.317418i \(-0.102816\pi\)
0.199250 + 0.979949i \(0.436149\pi\)
\(30\) 0 0
\(31\) 2.69515i 0.484064i 0.970268 + 0.242032i \(0.0778139\pi\)
−0.970268 + 0.242032i \(0.922186\pi\)
\(32\) −4.53485 − 1.65055i −0.801655 − 0.291779i
\(33\) −4.24770 + 0.748985i −0.739430 + 0.130381i
\(34\) 1.24938 − 1.04835i 0.214266 − 0.179791i
\(35\) 0 0
\(36\) 12.0088 2.00147
\(37\) 3.37814 + 5.05847i 0.555363 + 0.831608i
\(38\) − 2.70026i − 0.438040i
\(39\) −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 \(0.731020\pi\)
\(42\) −1.55885 − 0.567375i −0.240536 − 0.0875478i
\(43\) −10.8290 −1.65141 −0.825703 − 0.564105i \(-0.809221\pi\)
−0.825703 + 0.564105i \(0.809221\pi\)
\(44\) −2.31087 − 0.841086i −0.348376 − 0.126799i
\(45\) 0 0
\(46\) −0.213202 + 1.20913i −0.0314349 + 0.178276i
\(47\) 1.61643 − 0.933248i 0.235781 − 0.136128i −0.377455 − 0.926028i \(-0.623201\pi\)
0.613236 + 0.789900i \(0.289867\pi\)
\(48\) 7.37636 + 4.25875i 1.06469 + 0.614697i
\(49\) 4.37929 + 3.67466i 0.625613 + 0.524952i
\(50\) 0 0
\(51\) −9.40068 + 5.42748i −1.31636 + 0.760000i
\(52\) −11.1490 + 4.05790i −1.54609 + 0.562729i
\(53\) 0.700025 + 1.92330i 0.0961558 + 0.264186i 0.978440 − 0.206531i \(-0.0662175\pi\)
−0.882284 + 0.470717i \(0.843995\pi\)
\(54\) 5.40327 + 0.952742i 0.735292 + 0.129652i
\(55\) 0 0
\(56\) −1.29106 − 1.53863i −0.172525 − 0.205608i
\(57\) −3.12080 + 17.6989i −0.413360 + 2.34428i
\(58\) 1.14479 + 3.14527i 0.150318 + 0.412994i
\(59\) −2.08921 − 5.74005i −0.271992 − 0.747291i −0.998209 − 0.0598256i \(-0.980946\pi\)
0.726217 − 0.687465i \(-0.241277\pi\)
\(60\) 0 0
\(61\) 6.18979 − 1.09143i 0.792521 − 0.139743i 0.237290 − 0.971439i \(-0.423741\pi\)
0.555231 + 0.831696i \(0.312630\pi\)
\(62\) −0.812622 + 0.968445i −0.103203 + 0.122993i
\(63\) 6.61867 + 3.82129i 0.833874 + 0.481437i
\(64\) 1.59641 + 2.76507i 0.199552 + 0.345634i
\(65\) 0 0
\(66\) −1.75215 − 1.01160i −0.215675 − 0.124520i
\(67\) 2.17615 − 5.97892i 0.265859 − 0.730442i −0.732886 − 0.680352i \(-0.761827\pi\)
0.998745 − 0.0500900i \(-0.0159508\pi\)
\(68\) −6.18892 −0.750517
\(69\) 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 \(0.355185\pi\)
\(72\) 9.16362 + 7.68919i 1.07994 + 0.906179i
\(73\) − 0.109760i − 0.0128464i −0.999979 − 0.00642322i \(-0.997955\pi\)
0.999979 − 0.00642322i \(-0.00204459\pi\)
\(74\) −0.311329 + 2.83620i −0.0361913 + 0.329702i
\(75\) 0 0
\(76\) −6.58642 + 7.84939i −0.755514 + 0.900386i
\(77\) −1.00600 − 1.19890i −0.114644 − 0.136627i
\(78\) −9.61285 + 1.69500i −1.08844 + 0.191921i
\(79\) 0.920146 − 2.52808i 0.103525 − 0.284431i −0.877106 − 0.480296i \(-0.840529\pi\)
0.980631 + 0.195865i \(0.0627514\pi\)
\(80\) 0 0
\(81\) −15.2955 − 5.56709i −1.69950 − 0.618566i
\(82\) −1.86630 + 3.23253i −0.206099 + 0.356973i
\(83\) 13.6371 + 2.40459i 1.49687 + 0.263939i 0.861297 − 0.508102i \(-0.169653\pi\)
0.635573 + 0.772041i \(0.280764\pi\)
\(84\) 3.14749 + 5.45161i 0.343419 + 0.594819i
\(85\) 0 0
\(86\) −3.89117 − 3.26508i −0.419595 − 0.352082i
\(87\) −3.86841 − 21.9388i −0.414737 − 2.35209i
\(88\) −1.22482 − 2.12145i −0.130566 − 0.226147i
\(89\) −4.36882 − 12.0032i −0.463093 − 1.27234i −0.923147 − 0.384448i \(-0.874392\pi\)
0.460053 − 0.887891i \(-0.347830\pi\)
\(90\) 0 0
\(91\) −7.43602 − 1.31117i −0.779507 − 0.137448i
\(92\) 3.56903 − 2.99477i 0.372097 − 0.312227i
\(93\) 6.44562 − 5.40851i 0.668379 − 0.560837i
\(94\) 0.862216 + 0.152032i 0.0889308 + 0.0156809i
\(95\) 0 0
\(96\) 5.15295 + 14.1576i 0.525921 + 1.44495i
\(97\) −1.98576 − 3.43944i −0.201623 − 0.349222i 0.747428 − 0.664343i \(-0.231288\pi\)
−0.949052 + 0.315121i \(0.897955\pi\)
\(98\) 0.465648 + 2.64082i 0.0470376 + 0.266763i
\(99\) 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.ba.c.299.8 72
5.2 odd 4 185.2.w.a.151.6 yes 72
5.3 odd 4 925.2.bb.b.151.7 72
5.4 even 2 925.2.ba.b.299.5 72
37.25 even 18 925.2.ba.b.99.5 72
185.32 even 36 6845.2.a.u.1.14 36
185.42 even 36 6845.2.a.t.1.23 36
185.62 odd 36 185.2.w.a.136.6 ✓ 72
185.99 even 18 inner 925.2.ba.c.99.8 72
185.173 odd 36 925.2.bb.b.876.7 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.w.a.136.6 ✓ 72 185.62 odd 36
185.2.w.a.151.6 yes 72 5.2 odd 4
925.2.ba.b.99.5 72 37.25 even 18
925.2.ba.b.299.5 72 5.4 even 2
925.2.ba.c.99.8 72 185.99 even 18 inner
925.2.ba.c.299.8 72 1.1 even 1 trivial
925.2.bb.b.151.7 72 5.3 odd 4
925.2.bb.b.876.7 72 185.173 odd 36
6845.2.a.t.1.23 36 185.42 even 36
6845.2.a.u.1.14 36 185.32 even 36