Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [925,2,Mod(149,925)] 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("925.149"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(925, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 925 = 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 925.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [14,0,0,-14,0,12,0,0,-20,0,32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.38616218697\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 21x^{12} + 170x^{10} + 665x^{8} + 1280x^{6} + 1087x^{4} + 311x^{2} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 149.3
Root \(-2.13289i\) of defining polynomial
Character \(\chi\) \(=\) 925.149
Dual form 925.2.b.i.149.12

$q$-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)
 
\(f(q)\) \(=\) \(q-2.13289i q^{2} +2.92598i q^{3} -2.54923 q^{4} +6.24079 q^{6} +3.01925i q^{7} +1.17144i q^{8} -5.56135 q^{9} +5.51375 q^{11} -7.45898i q^{12} -0.501630i q^{13} +6.43973 q^{14} -2.59990 q^{16} +6.61915i q^{17} +11.8618i q^{18} -4.42510 q^{19} -8.83426 q^{21} -11.7602i q^{22} -1.67307i q^{23} -3.42761 q^{24} -1.06992 q^{26} -7.49445i q^{27} -7.69675i q^{28} +1.51711 q^{29} -9.00821 q^{31} +7.88818i q^{32} +16.1331i q^{33} +14.1179 q^{34} +14.1771 q^{36} +1.00000i q^{37} +9.43826i q^{38} +1.46776 q^{39} +3.32768 q^{41} +18.8425i q^{42} +3.05695i q^{43} -14.0558 q^{44} -3.56848 q^{46} +3.80128i q^{47} -7.60725i q^{48} -2.11587 q^{49} -19.3675 q^{51} +1.27877i q^{52} +6.03086i q^{53} -15.9849 q^{54} -3.53687 q^{56} -12.9477i q^{57} -3.23583i q^{58} -13.4763 q^{59} +11.9483 q^{61} +19.2135i q^{62} -16.7911i q^{63} +11.6248 q^{64} +34.4102 q^{66} +10.8898i q^{67} -16.8737i q^{68} +4.89537 q^{69} -2.46808 q^{71} -6.51479i q^{72} +13.2652i q^{73} +2.13289 q^{74} +11.2806 q^{76} +16.6474i q^{77} -3.13057i q^{78} +11.4110 q^{79} +5.24456 q^{81} -7.09758i q^{82} -8.36622i q^{83} +22.5205 q^{84} +6.52014 q^{86} +4.43904i q^{87} +6.45903i q^{88} +7.41023 q^{89} +1.51454 q^{91} +4.26503i q^{92} -26.3578i q^{93} +8.10773 q^{94} -23.0806 q^{96} -5.75342i q^{97} +4.51291i q^{98} -30.6639 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 14 q^{4} + 12 q^{6} - 20 q^{9} + 32 q^{11} + 6 q^{14} + 6 q^{16} - 18 q^{19} + 4 q^{21} + 48 q^{26} - 6 q^{29} + 22 q^{31} + 46 q^{34} + 38 q^{36} - 4 q^{39} + 58 q^{41} - 36 q^{44} + 14 q^{46} - 10 q^{49}+ \cdots - 78 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(76\) \(852\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.13289i − 1.50818i −0.656770 − 0.754091i \(-0.728078\pi\)
0.656770 − 0.754091i \(-0.271922\pi\)
\(3\) 2.92598i 1.68931i 0.535308 + 0.844657i \(0.320196\pi\)
−0.535308 + 0.844657i \(0.679804\pi\)
\(4\) −2.54923 −1.27461
\(5\) 0 0
\(6\) 6.24079 2.54779
\(7\) 3.01925i 1.14117i 0.821239 + 0.570585i \(0.193283\pi\)
−0.821239 + 0.570585i \(0.806717\pi\)
\(8\) 1.17144i 0.414167i
\(9\) −5.56135 −1.85378
\(10\) 0 0
\(11\) 5.51375 1.66246 0.831230 − 0.555929i \(-0.187637\pi\)
0.831230 + 0.555929i \(0.187637\pi\)
\(12\) − 7.45898i − 2.15322i
\(13\) − 0.501630i − 0.139127i −0.997578 − 0.0695635i \(-0.977839\pi\)
0.997578 − 0.0695635i \(-0.0221606\pi\)
\(14\) 6.43973 1.72109
\(15\) 0 0
\(16\) −2.59990 −0.649975
\(17\) 6.61915i 1.60538i 0.596397 + 0.802690i \(0.296598\pi\)
−0.596397 + 0.802690i \(0.703402\pi\)
\(18\) 11.8618i 2.79584i
\(19\) −4.42510 −1.01519 −0.507594 − 0.861597i \(-0.669465\pi\)
−0.507594 + 0.861597i \(0.669465\pi\)
\(20\) 0 0
\(21\) −8.83426 −1.92779
\(22\) − 11.7602i − 2.50729i
\(23\) − 1.67307i − 0.348859i −0.984670 − 0.174430i \(-0.944192\pi\)
0.984670 − 0.174430i \(-0.0558081\pi\)
\(24\) −3.42761 −0.699658
\(25\) 0 0
\(26\) −1.06992 −0.209829
\(27\) − 7.49445i − 1.44231i
\(28\) − 7.69675i − 1.45455i
\(29\) 1.51711 0.281721 0.140860 − 0.990029i \(-0.455013\pi\)
0.140860 + 0.990029i \(0.455013\pi\)
\(30\) 0 0
\(31\) −9.00821 −1.61792 −0.808961 − 0.587862i \(-0.799970\pi\)
−0.808961 + 0.587862i \(0.799970\pi\)
\(32\) 7.88818i 1.39445i
\(33\) 16.1331i 2.80842i
\(34\) 14.1179 2.42120
\(35\) 0 0
\(36\) 14.1771 2.36286
\(37\) 1.00000i 0.164399i
\(38\) 9.43826i 1.53109i
\(39\) 1.46776 0.235029
\(40\) 0 0
\(41\) 3.32768 0.519696 0.259848 − 0.965649i \(-0.416327\pi\)
0.259848 + 0.965649i \(0.416327\pi\)
\(42\) 18.8425i 2.90746i
\(43\) 3.05695i 0.466180i 0.972455 + 0.233090i \(0.0748837\pi\)
−0.972455 + 0.233090i \(0.925116\pi\)
\(44\) −14.0558 −2.11899
\(45\) 0 0
\(46\) −3.56848 −0.526143
\(47\) 3.80128i 0.554474i 0.960802 + 0.277237i \(0.0894188\pi\)
−0.960802 + 0.277237i \(0.910581\pi\)
\(48\) − 7.60725i − 1.09801i
\(49\) −2.11587 −0.302267
\(50\) 0 0
\(51\) −19.3675 −2.71199
\(52\) 1.27877i 0.177333i
\(53\) 6.03086i 0.828403i 0.910185 + 0.414202i \(0.135939\pi\)
−0.910185 + 0.414202i \(0.864061\pi\)
\(54\) −15.9849 −2.17526
\(55\) 0 0
\(56\) −3.53687 −0.472634
\(57\) − 12.9477i − 1.71497i
\(58\) − 3.23583i − 0.424886i
\(59\) −13.4763 −1.75446 −0.877231 − 0.480069i \(-0.840612\pi\)
−0.877231 + 0.480069i \(0.840612\pi\)
\(60\) 0 0
\(61\) 11.9483 1.52982 0.764909 − 0.644139i \(-0.222784\pi\)
0.764909 + 0.644139i \(0.222784\pi\)
\(62\) 19.2135i 2.44012i
\(63\) − 16.7911i − 2.11548i
\(64\) 11.6248 1.45310
\(65\) 0 0
\(66\) 34.4102 4.23560
\(67\) 10.8898i 1.33040i 0.746667 + 0.665198i \(0.231653\pi\)
−0.746667 + 0.665198i \(0.768347\pi\)
\(68\) − 16.8737i − 2.04624i
\(69\) 4.89537 0.589333
\(70\) 0 0
\(71\) −2.46808 −0.292907 −0.146454 − 0.989218i \(-0.546786\pi\)
−0.146454 + 0.989218i \(0.546786\pi\)
\(72\) − 6.51479i − 0.767775i
\(73\) 13.2652i 1.55258i 0.630378 + 0.776289i \(0.282900\pi\)
−0.630378 + 0.776289i \(0.717100\pi\)
\(74\) 2.13289 0.247944
\(75\) 0 0
\(76\) 11.2806 1.29397
\(77\) 16.6474i 1.89715i
\(78\) − 3.13057i − 0.354467i
\(79\) 11.4110 1.28384 0.641918 − 0.766773i \(-0.278139\pi\)
0.641918 + 0.766773i \(0.278139\pi\)
\(80\) 0 0
\(81\) 5.24456 0.582729
\(82\) − 7.09758i − 0.783797i
\(83\) − 8.36622i − 0.918312i −0.888356 − 0.459156i \(-0.848152\pi\)
0.888356 − 0.459156i \(-0.151848\pi\)
\(84\) 22.5205 2.45719
\(85\) 0 0
\(86\) 6.52014 0.703085
\(87\) 4.43904i 0.475915i
\(88\) 6.45903i 0.688535i
\(89\) 7.41023 0.785483 0.392741 − 0.919649i \(-0.371527\pi\)
0.392741 + 0.919649i \(0.371527\pi\)
\(90\) 0 0
\(91\) 1.51454 0.158767
\(92\) 4.26503i 0.444660i
\(93\) − 26.3578i − 2.73318i
\(94\) 8.10773 0.836248
\(95\) 0 0
\(96\) −23.0806 −2.35566
\(97\) − 5.75342i − 0.584171i −0.956392 − 0.292085i \(-0.905651\pi\)
0.956392 − 0.292085i \(-0.0943492\pi\)
\(98\) 4.51291i 0.455873i
\(99\) −30.6639 −3.08184
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.b.i.149.3 14
5.2 odd 4 925.2.a.j.1.6 ✓ 7
5.3 odd 4 925.2.a.k.1.2 yes 7
5.4 even 2 inner 925.2.b.i.149.12 14
15.2 even 4 8325.2.a.cn.1.2 7
15.8 even 4 8325.2.a.cm.1.6 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
925.2.a.j.1.6 ✓ 7 5.2 odd 4
925.2.a.k.1.2 yes 7 5.3 odd 4
925.2.b.i.149.3 14 1.1 even 1 trivial
925.2.b.i.149.12 14 5.4 even 2 inner
8325.2.a.cm.1.6 7 15.8 even 4
8325.2.a.cn.1.2 7 15.2 even 4