Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [45,10,Mod(19,45)] 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("45.19"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(45, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 10, names="a")
 
Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 45.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,-1194] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(23.1766126274\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 939x^{6} + 217699x^{4} + 14559561x^{2} + 31136400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{12}\cdot 5^{4} \)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 19.4
Root \(-10.9137i\) of defining polynomial
Character \(\chi\) \(=\) 45.19
Dual form 45.10.b.c.19.5

$q$-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)
 
\(f(q)\) \(=\) \(q-14.3372i q^{2} +306.446 q^{4} +(-1380.00 + 220.717i) q^{5} +2878.61i q^{7} -11734.2i q^{8} +(3164.45 + 19785.3i) q^{10} +23286.0 q^{11} +112501. i q^{13} +41271.2 q^{14} -11335.0 q^{16} +115964. i q^{17} +213578. q^{19} +(-422896. + 67637.6i) q^{20} -333855. i q^{22} +1.83629e6i q^{23} +(1.85569e6 - 609179. i) q^{25} +1.61295e6 q^{26} +882139. i q^{28} +3.67540e6 q^{29} +8.85139e6 q^{31} -5.84540e6i q^{32} +1.66259e6 q^{34} +(-635358. - 3.97250e6i) q^{35} +9.17921e6i q^{37} -3.06210e6i q^{38} +(2.58993e6 + 1.61932e7i) q^{40} +1.17626e7 q^{41} -3.93230e7i q^{43} +7.13589e6 q^{44} +2.63272e7 q^{46} +3.26882e7i q^{47} +3.20672e7 q^{49} +(-8.73391e6 - 2.66054e7i) q^{50} +3.44754e7i q^{52} +1.04826e8i q^{53} +(-3.21348e7 + 5.13961e6i) q^{55} +3.37782e7 q^{56} -5.26948e7i q^{58} -1.02836e8 q^{59} -1.65686e8 q^{61} -1.26904e8i q^{62} -8.96099e7 q^{64} +(-2.48308e7 - 1.55252e8i) q^{65} +1.76387e8i q^{67} +3.55365e7i q^{68} +(-5.69544e7 + 9.10923e6i) q^{70} -1.30237e8 q^{71} -2.35713e8i q^{73} +1.31604e8 q^{74} +6.54500e7 q^{76} +6.70314e7i q^{77} -1.56763e8 q^{79} +(1.56423e7 - 2.50182e6i) q^{80} -1.68642e8i q^{82} +3.38241e7i q^{83} +(-2.55951e7 - 1.60030e8i) q^{85} -5.63780e8 q^{86} -2.73242e8i q^{88} +4.86766e8 q^{89} -3.23847e8 q^{91} +5.62724e8i q^{92} +4.68656e8 q^{94} +(-2.94738e8 + 4.71402e7i) q^{95} +1.40255e9i q^{97} -4.59753e8i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 1194 q^{4} + 690 q^{5} + 67090 q^{10} + 71988 q^{11} - 416364 q^{14} - 1505630 q^{16} + 851584 q^{19} - 2078100 q^{20} + 1695500 q^{25} + 877524 q^{26} + 73572 q^{29} + 474088 q^{31} - 8124388 q^{34}+ \cdots - 1698584640 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/45\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(37\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 14.3372i − 0.633619i −0.948489 − 0.316810i \(-0.897388\pi\)
0.948489 − 0.316810i \(-0.102612\pi\)
\(3\) 0 0
\(4\) 306.446 0.598526
\(5\) −1380.00 + 220.717i −0.987450 + 0.157932i
\(6\) 0 0
\(7\) 2878.61i 0.453150i 0.973994 + 0.226575i \(0.0727529\pi\)
−0.973994 + 0.226575i \(0.927247\pi\)
\(8\) − 11734.2i − 1.01286i
\(9\) 0 0
\(10\) 3164.45 + 19785.3i 0.100069 + 0.625667i
\(11\) 23286.0 0.479543 0.239772 − 0.970829i \(-0.422927\pi\)
0.239772 + 0.970829i \(0.422927\pi\)
\(12\) 0 0
\(13\) 112501.i 1.09247i 0.837630 + 0.546237i \(0.183940\pi\)
−0.837630 + 0.546237i \(0.816060\pi\)
\(14\) 41271.2 0.287125
\(15\) 0 0
\(16\) −11335.0 −0.0432396
\(17\) 115964.i 0.336745i 0.985723 + 0.168373i \(0.0538512\pi\)
−0.985723 + 0.168373i \(0.946149\pi\)
\(18\) 0 0
\(19\) 213578. 0.375980 0.187990 − 0.982171i \(-0.439803\pi\)
0.187990 + 0.982171i \(0.439803\pi\)
\(20\) −422896. + 67637.6i −0.591015 + 0.0945264i
\(21\) 0 0
\(22\) − 333855.i − 0.303848i
\(23\) 1.83629e6i 1.36825i 0.729363 + 0.684127i \(0.239816\pi\)
−0.729363 + 0.684127i \(0.760184\pi\)
\(24\) 0 0
\(25\) 1.85569e6 − 609179.i 0.950115 − 0.311900i
\(26\) 1.61295e6 0.692213
\(27\) 0 0
\(28\) 882139.i 0.271223i
\(29\) 3.67540e6 0.964969 0.482485 − 0.875904i \(-0.339734\pi\)
0.482485 + 0.875904i \(0.339734\pi\)
\(30\) 0 0
\(31\) 8.85139e6 1.72141 0.860704 − 0.509105i \(-0.170024\pi\)
0.860704 + 0.509105i \(0.170024\pi\)
\(32\) − 5.84540e6i − 0.985460i
\(33\) 0 0
\(34\) 1.66259e6 0.213368
\(35\) −635358. − 3.97250e6i −0.0715669 − 0.447463i
\(36\) 0 0
\(37\) 9.17921e6i 0.805188i 0.915379 + 0.402594i \(0.131891\pi\)
−0.915379 + 0.402594i \(0.868109\pi\)
\(38\) − 3.06210e6i − 0.238228i
\(39\) 0 0
\(40\) 2.58993e6 + 1.61932e7i 0.159963 + 1.00015i
\(41\) 1.17626e7 0.650093 0.325046 − 0.945698i \(-0.394620\pi\)
0.325046 + 0.945698i \(0.394620\pi\)
\(42\) 0 0
\(43\) − 3.93230e7i − 1.75403i −0.480459 − 0.877017i \(-0.659530\pi\)
0.480459 − 0.877017i \(-0.340470\pi\)
\(44\) 7.13589e6 0.287019
\(45\) 0 0
\(46\) 2.63272e7 0.866952
\(47\) 3.26882e7i 0.977126i 0.872529 + 0.488563i \(0.162479\pi\)
−0.872529 + 0.488563i \(0.837521\pi\)
\(48\) 0 0
\(49\) 3.20672e7 0.794655
\(50\) −8.73391e6 − 2.66054e7i −0.197626 − 0.602011i
\(51\) 0 0
\(52\) 3.44754e7i 0.653875i
\(53\) 1.04826e8i 1.82486i 0.409238 + 0.912428i \(0.365794\pi\)
−0.409238 + 0.912428i \(0.634206\pi\)
\(54\) 0 0
\(55\) −3.21348e7 + 5.13961e6i −0.473525 + 0.0757352i
\(56\) 3.37782e7 0.458977
\(57\) 0 0
\(58\) − 5.26948e7i − 0.611423i
\(59\) −1.02836e8 −1.10486 −0.552432 − 0.833558i \(-0.686300\pi\)
−0.552432 + 0.833558i \(0.686300\pi\)
\(60\) 0 0
\(61\) −1.65686e8 −1.53215 −0.766074 − 0.642752i \(-0.777792\pi\)
−0.766074 + 0.642752i \(0.777792\pi\)
\(62\) − 1.26904e8i − 1.09072i
\(63\) 0 0
\(64\) −8.96099e7 −0.667646
\(65\) −2.48308e7 − 1.55252e8i −0.172537 − 1.07876i
\(66\) 0 0
\(67\) 1.76387e8i 1.06937i 0.845051 + 0.534686i \(0.179570\pi\)
−0.845051 + 0.534686i \(0.820430\pi\)
\(68\) 3.55365e7i 0.201551i
\(69\) 0 0
\(70\) −5.69544e7 + 9.10923e6i −0.283521 + 0.0453462i
\(71\) −1.30237e8 −0.608234 −0.304117 − 0.952635i \(-0.598361\pi\)
−0.304117 + 0.952635i \(0.598361\pi\)
\(72\) 0 0
\(73\) − 2.35713e8i − 0.971474i −0.874105 − 0.485737i \(-0.838551\pi\)
0.874105 − 0.485737i \(-0.161449\pi\)
\(74\) 1.31604e8 0.510183
\(75\) 0 0
\(76\) 6.54500e7 0.225034
\(77\) 6.70314e7i 0.217305i
\(78\) 0 0
\(79\) −1.56763e8 −0.452815 −0.226407 − 0.974033i \(-0.572698\pi\)
−0.226407 + 0.974033i \(0.572698\pi\)
\(80\) 1.56423e7 − 2.50182e6i 0.0426969 − 0.00682891i
\(81\) 0 0
\(82\) − 1.68642e8i − 0.411911i
\(83\) 3.38241e7i 0.0782303i 0.999235 + 0.0391151i \(0.0124539\pi\)
−0.999235 + 0.0391151i \(0.987546\pi\)
\(84\) 0 0
\(85\) −2.55951e7 − 1.60030e8i −0.0531828 − 0.332519i
\(86\) −5.63780e8 −1.11139
\(87\) 0 0
\(88\) − 2.73242e8i − 0.485709i
\(89\) 4.86766e8 0.822366 0.411183 − 0.911553i \(-0.365116\pi\)
0.411183 + 0.911553i \(0.365116\pi\)
\(90\) 0 0
\(91\) −3.23847e8 −0.495055
\(92\) 5.62724e8i 0.818936i
\(93\) 0 0
\(94\) 4.68656e8 0.619126
\(95\) −2.94738e8 + 4.71402e7i −0.371262 + 0.0593793i
\(96\) 0 0
\(97\) 1.40255e9i 1.60859i 0.594231 + 0.804294i \(0.297456\pi\)
−0.594231 + 0.804294i \(0.702544\pi\)
\(98\) − 4.59753e8i − 0.503509i
\(99\) 0 0
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 45.10.b.c.19.4 8
3.2 odd 2 15.10.b.a.4.5 yes 8
5.2 odd 4 225.10.a.q.1.3 4
5.3 odd 4 225.10.a.u.1.2 4
5.4 even 2 inner 45.10.b.c.19.5 8
12.11 even 2 240.10.f.c.49.4 8
15.2 even 4 75.10.a.l.1.2 4
15.8 even 4 75.10.a.i.1.3 4
15.14 odd 2 15.10.b.a.4.4 ✓ 8
60.59 even 2 240.10.f.c.49.8 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.10.b.a.4.4 ✓ 8 15.14 odd 2
15.10.b.a.4.5 yes 8 3.2 odd 2
45.10.b.c.19.4 8 1.1 even 1 trivial
45.10.b.c.19.5 8 5.4 even 2 inner
75.10.a.i.1.3 4 15.8 even 4
75.10.a.l.1.2 4 15.2 even 4
225.10.a.q.1.3 4 5.2 odd 4
225.10.a.u.1.2 4 5.3 odd 4
240.10.f.c.49.4 8 12.11 even 2
240.10.f.c.49.8 8 60.59 even 2