Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [225,2,Mod(46,225)] 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("225.46"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(225, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 6])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 225.h (of order \(5\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,1] 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: \(1.79663404548\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 75)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label 91.1
Root \(0.809017 - 0.587785i\) of defining polynomial
Character \(\chi\) \(=\) 225.91
Dual form 225.2.h.a.136.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.809017 + 0.587785i) q^{2} +(-0.309017 - 0.951057i) q^{4} +(-0.690983 + 2.12663i) q^{5} +4.47214 q^{7} +(0.927051 - 2.85317i) q^{8} +(-1.80902 + 1.31433i) q^{10} +(2.61803 + 1.90211i) q^{11} +(-2.73607 + 1.98787i) q^{13} +(3.61803 + 2.62866i) q^{14} +(0.809017 - 0.587785i) q^{16} +(0.881966 - 2.71441i) q^{17} +(-1.00000 + 3.07768i) q^{19} +2.23607 q^{20} +(1.00000 + 3.07768i) q^{22} +(-3.61803 - 2.62866i) q^{23} +(-4.04508 - 2.93893i) q^{25} -3.38197 q^{26} +(-1.38197 - 4.25325i) q^{28} +(-1.35410 - 4.16750i) q^{29} +(2.23607 - 6.88191i) q^{31} -5.00000 q^{32} +(2.30902 - 1.67760i) q^{34} +(-3.09017 + 9.51057i) q^{35} +(-6.54508 + 4.75528i) q^{37} +(-2.61803 + 1.90211i) q^{38} +(5.42705 + 3.94298i) q^{40} +(-1.11803 + 0.812299i) q^{41} -5.70820 q^{43} +(1.00000 - 3.07768i) q^{44} +(-1.38197 - 4.25325i) q^{46} +(1.61803 + 4.97980i) q^{47} +13.0000 q^{49} +(-1.54508 - 4.75528i) q^{50} +(2.73607 + 1.98787i) q^{52} +(-0.427051 - 1.31433i) q^{53} +(-5.85410 + 4.25325i) q^{55} +(4.14590 - 12.7598i) q^{56} +(1.35410 - 4.16750i) q^{58} +(-3.23607 + 2.35114i) q^{59} +(0.500000 + 0.363271i) q^{61} +(5.85410 - 4.25325i) q^{62} +(-5.66312 - 4.11450i) q^{64} +(-2.33688 - 7.19218i) q^{65} +(1.61803 - 4.97980i) q^{67} -2.85410 q^{68} +(-8.09017 + 5.87785i) q^{70} +(0.236068 + 0.726543i) q^{71} +(-2.50000 - 1.81636i) q^{73} -8.09017 q^{74} +3.23607 q^{76} +(11.7082 + 8.50651i) q^{77} +(0.690983 + 2.12663i) q^{80} -1.38197 q^{82} +(-1.09017 + 3.35520i) q^{83} +(5.16312 + 3.75123i) q^{85} +(-4.61803 - 3.35520i) q^{86} +(7.85410 - 5.70634i) q^{88} +(6.16312 + 4.47777i) q^{89} +(-12.2361 + 8.89002i) q^{91} +(-1.38197 + 4.25325i) q^{92} +(-1.61803 + 4.97980i) q^{94} +(-5.85410 - 4.25325i) q^{95} +(2.73607 + 8.42075i) q^{97} +(10.5172 + 7.64121i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} + q^{4} - 5 q^{5} - 3 q^{8} - 5 q^{10} + 6 q^{11} - 2 q^{13} + 10 q^{14} + q^{16} + 8 q^{17} - 4 q^{19} + 4 q^{22} - 10 q^{23} - 5 q^{25} - 18 q^{26} - 10 q^{28} + 8 q^{29} - 20 q^{32} + 7 q^{34}+ \cdots + 13 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{5}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.809017 + 0.587785i 0.572061 + 0.415627i 0.835853 − 0.548953i \(-0.184973\pi\)
−0.263792 + 0.964580i \(0.584973\pi\)
\(3\) 0 0
\(4\) −0.309017 − 0.951057i −0.154508 − 0.475528i
\(5\) −0.690983 + 2.12663i −0.309017 + 0.951057i
\(6\) 0 0
\(7\) 4.47214 1.69031 0.845154 − 0.534522i \(-0.179509\pi\)
0.845154 + 0.534522i \(0.179509\pi\)
\(8\) 0.927051 − 2.85317i 0.327762 − 1.00875i
\(9\) 0 0
\(10\) −1.80902 + 1.31433i −0.572061 + 0.415627i
\(11\) 2.61803 + 1.90211i 0.789367 + 0.573509i 0.907776 − 0.419456i \(-0.137779\pi\)
−0.118409 + 0.992965i \(0.537779\pi\)
\(12\) 0 0
\(13\) −2.73607 + 1.98787i −0.758849 + 0.551336i −0.898557 − 0.438857i \(-0.855384\pi\)
0.139708 + 0.990193i \(0.455384\pi\)
\(14\) 3.61803 + 2.62866i 0.966960 + 0.702538i
\(15\) 0 0
\(16\) 0.809017 − 0.587785i 0.202254 − 0.146946i
\(17\) 0.881966 − 2.71441i 0.213908 − 0.658342i −0.785321 − 0.619089i \(-0.787502\pi\)
0.999229 − 0.0392530i \(-0.0124978\pi\)
\(18\) 0 0
\(19\) −1.00000 + 3.07768i −0.229416 + 0.706069i 0.768398 + 0.639973i \(0.221054\pi\)
−0.997813 + 0.0660962i \(0.978946\pi\)
\(20\) 2.23607 0.500000
\(21\) 0 0
\(22\) 1.00000 + 3.07768i 0.213201 + 0.656164i
\(23\) −3.61803 − 2.62866i −0.754412 − 0.548113i 0.142779 − 0.989755i \(-0.454396\pi\)
−0.897191 + 0.441642i \(0.854396\pi\)
\(24\) 0 0
\(25\) −4.04508 − 2.93893i −0.809017 − 0.587785i
\(26\) −3.38197 −0.663258
\(27\) 0 0
\(28\) −1.38197 − 4.25325i −0.261167 − 0.803789i
\(29\) −1.35410 − 4.16750i −0.251450 − 0.773885i −0.994508 − 0.104658i \(-0.966625\pi\)
0.743058 − 0.669227i \(-0.233375\pi\)
\(30\) 0 0
\(31\) 2.23607 − 6.88191i 0.401610 − 1.23603i −0.522083 − 0.852894i \(-0.674845\pi\)
0.923693 − 0.383133i \(-0.125155\pi\)
\(32\) −5.00000 −0.883883
\(33\) 0 0
\(34\) 2.30902 − 1.67760i 0.395993 − 0.287706i
\(35\) −3.09017 + 9.51057i −0.522334 + 1.60758i
\(36\) 0 0
\(37\) −6.54508 + 4.75528i −1.07601 + 0.781764i −0.976982 − 0.213321i \(-0.931572\pi\)
−0.0990233 + 0.995085i \(0.531572\pi\)
\(38\) −2.61803 + 1.90211i −0.424701 + 0.308563i
\(39\) 0 0
\(40\) 5.42705 + 3.94298i 0.858092 + 0.623440i
\(41\) −1.11803 + 0.812299i −0.174608 + 0.126860i −0.671657 − 0.740863i \(-0.734417\pi\)
0.497049 + 0.867722i \(0.334417\pi\)
\(42\) 0 0
\(43\) −5.70820 −0.870493 −0.435246 − 0.900311i \(-0.643339\pi\)
−0.435246 + 0.900311i \(0.643339\pi\)
\(44\) 1.00000 − 3.07768i 0.150756 − 0.463978i
\(45\) 0 0
\(46\) −1.38197 − 4.25325i −0.203760 − 0.627108i
\(47\) 1.61803 + 4.97980i 0.236015 + 0.726378i 0.996985 + 0.0775917i \(0.0247231\pi\)
−0.760971 + 0.648786i \(0.775277\pi\)
\(48\) 0 0
\(49\) 13.0000 1.85714
\(50\) −1.54508 − 4.75528i −0.218508 − 0.672499i
\(51\) 0 0
\(52\) 2.73607 + 1.98787i 0.379424 + 0.275668i
\(53\) −0.427051 − 1.31433i −0.0586600 − 0.180537i 0.917433 − 0.397890i \(-0.130258\pi\)
−0.976093 + 0.217354i \(0.930258\pi\)
\(54\) 0 0
\(55\) −5.85410 + 4.25325i −0.789367 + 0.573509i
\(56\) 4.14590 − 12.7598i 0.554019 − 1.70509i
\(57\) 0 0
\(58\) 1.35410 − 4.16750i 0.177802 − 0.547219i
\(59\) −3.23607 + 2.35114i −0.421300 + 0.306092i −0.778161 − 0.628065i \(-0.783847\pi\)
0.356861 + 0.934158i \(0.383847\pi\)
\(60\) 0 0
\(61\) 0.500000 + 0.363271i 0.0640184 + 0.0465121i 0.619334 − 0.785127i \(-0.287403\pi\)
−0.555316 + 0.831640i \(0.687403\pi\)
\(62\) 5.85410 − 4.25325i 0.743472 − 0.540164i
\(63\) 0 0
\(64\) −5.66312 − 4.11450i −0.707890 − 0.514312i
\(65\) −2.33688 − 7.19218i −0.289854 − 0.892080i
\(66\) 0 0
\(67\) 1.61803 − 4.97980i 0.197674 − 0.608379i −0.802261 − 0.596974i \(-0.796370\pi\)
0.999935 − 0.0114051i \(-0.00363042\pi\)
\(68\) −2.85410 −0.346111
\(69\) 0 0
\(70\) −8.09017 + 5.87785i −0.966960 + 0.702538i
\(71\) 0.236068 + 0.726543i 0.0280161 + 0.0862247i 0.964087 − 0.265587i \(-0.0855657\pi\)
−0.936071 + 0.351812i \(0.885566\pi\)
\(72\) 0 0
\(73\) −2.50000 − 1.81636i −0.292603 − 0.212588i 0.431793 − 0.901973i \(-0.357881\pi\)
−0.724396 + 0.689384i \(0.757881\pi\)
\(74\) −8.09017 −0.940463
\(75\) 0 0
\(76\) 3.23607 0.371202
\(77\) 11.7082 + 8.50651i 1.33427 + 0.969407i
\(78\) 0 0
\(79\) 0 0 0.951057 − 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(80\) 0.690983 + 2.12663i 0.0772542 + 0.237764i
\(81\) 0 0
\(82\) −1.38197 −0.152613
\(83\) −1.09017 + 3.35520i −0.119662 + 0.368281i −0.992891 − 0.119029i \(-0.962022\pi\)
0.873229 + 0.487310i \(0.162022\pi\)
\(84\) 0 0
\(85\) 5.16312 + 3.75123i 0.560019 + 0.406878i
\(86\) −4.61803 − 3.35520i −0.497975 − 0.361800i
\(87\) 0 0
\(88\) 7.85410 − 5.70634i 0.837250 − 0.608298i
\(89\) 6.16312 + 4.47777i 0.653289 + 0.474642i 0.864390 − 0.502822i \(-0.167705\pi\)
−0.211101 + 0.977464i \(0.567705\pi\)
\(90\) 0 0
\(91\) −12.2361 + 8.89002i −1.28269 + 0.931928i
\(92\) −1.38197 + 4.25325i −0.144080 + 0.443432i
\(93\) 0 0
\(94\) −1.61803 + 4.97980i −0.166887 + 0.513627i
\(95\) −5.85410 − 4.25325i −0.600618 − 0.436375i
\(96\) 0 0
\(97\) 2.73607 + 8.42075i 0.277806 + 0.854998i 0.988463 + 0.151460i \(0.0483974\pi\)
−0.710658 + 0.703538i \(0.751603\pi\)
\(98\) 10.5172 + 7.64121i 1.06240 + 0.771879i
\(99\) 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 225.2.h.a.91.1 4
3.2 odd 2 75.2.g.a.16.1 ✓ 4
15.2 even 4 375.2.i.a.49.2 8
15.8 even 4 375.2.i.a.49.1 8
15.14 odd 2 375.2.g.a.76.1 4
25.6 even 5 5625.2.a.a.1.2 2
25.11 even 5 inner 225.2.h.a.136.1 4
25.19 even 10 5625.2.a.h.1.1 2
75.2 even 20 375.2.i.a.199.1 8
75.8 even 20 1875.2.b.b.1249.1 4
75.11 odd 10 75.2.g.a.61.1 yes 4
75.14 odd 10 375.2.g.a.301.1 4
75.17 even 20 1875.2.b.b.1249.4 4
75.23 even 20 375.2.i.a.199.2 8
75.44 odd 10 1875.2.a.a.1.1 2
75.56 odd 10 1875.2.a.d.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.2.g.a.16.1 ✓ 4 3.2 odd 2
75.2.g.a.61.1 yes 4 75.11 odd 10
225.2.h.a.91.1 4 1.1 even 1 trivial
225.2.h.a.136.1 4 25.11 even 5 inner
375.2.g.a.76.1 4 15.14 odd 2
375.2.g.a.301.1 4 75.14 odd 10
375.2.i.a.49.1 8 15.8 even 4
375.2.i.a.49.2 8 15.2 even 4
375.2.i.a.199.1 8 75.2 even 20
375.2.i.a.199.2 8 75.23 even 20
1875.2.a.a.1.1 2 75.44 odd 10
1875.2.a.d.1.2 2 75.56 odd 10
1875.2.b.b.1249.1 4 75.8 even 20
1875.2.b.b.1249.4 4 75.17 even 20
5625.2.a.a.1.2 2 25.6 even 5
5625.2.a.h.1.1 2 25.19 even 10