Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [450,2,Mod(257,450)] 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("450.257"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(450, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([10, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 450.p (of order \(12\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,12,0,0,0,0,24,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.59326809096\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\Q(\zeta_{24})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 407.1
Root \(0.965926 - 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 450.407
Dual form 450.2.p.g.293.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.258819 - 0.965926i) q^{2} +(-1.22474 + 1.22474i) q^{3} +(-0.866025 + 0.500000i) q^{4} +(1.50000 + 0.866025i) q^{6} +(0.707107 + 0.707107i) q^{8} -3.00000i q^{9} +(3.00000 + 1.73205i) q^{11} +(0.448288 - 1.67303i) q^{12} +(-3.34607 - 0.896575i) q^{13} +(0.500000 - 0.866025i) q^{16} +(-4.24264 + 4.24264i) q^{17} +(-2.89778 + 0.776457i) q^{18} +5.00000i q^{19} +(0.896575 - 3.34607i) q^{22} +(-1.55291 + 5.79555i) q^{23} -1.73205 q^{24} +3.46410i q^{26} +(3.67423 + 3.67423i) q^{27} +(-3.46410 + 6.00000i) q^{29} +(-2.00000 - 3.46410i) q^{31} +(-0.965926 - 0.258819i) q^{32} +(-5.79555 + 1.55291i) q^{33} +(5.19615 + 3.00000i) q^{34} +(1.50000 + 2.59808i) q^{36} +(4.89898 + 4.89898i) q^{37} +(4.82963 - 1.29410i) q^{38} +(5.19615 - 3.00000i) q^{39} +(-1.50000 + 0.866025i) q^{41} +(-3.13801 - 11.7112i) q^{43} -3.46410 q^{44} +6.00000 q^{46} +(1.55291 + 5.79555i) q^{47} +(0.448288 + 1.67303i) q^{48} +(-6.06218 + 3.50000i) q^{49} -10.3923i q^{51} +(3.34607 - 0.896575i) q^{52} +(4.24264 + 4.24264i) q^{53} +(2.59808 - 4.50000i) q^{54} +(-6.12372 - 6.12372i) q^{57} +(6.69213 + 1.79315i) q^{58} +(0.866025 + 1.50000i) q^{59} +(-4.00000 + 6.92820i) q^{61} +(-2.82843 + 2.82843i) q^{62} +1.00000i q^{64} +(3.00000 + 5.19615i) q^{66} +(2.24144 - 8.36516i) q^{67} +(1.55291 - 5.79555i) q^{68} +(-5.19615 - 9.00000i) q^{69} -6.92820i q^{71} +(2.12132 - 2.12132i) q^{72} +(8.57321 - 8.57321i) q^{73} +(3.46410 - 6.00000i) q^{74} +(-2.50000 - 4.33013i) q^{76} +(-4.24264 - 4.24264i) q^{78} +(12.1244 + 7.00000i) q^{79} -9.00000 q^{81} +(1.22474 + 1.22474i) q^{82} +(8.69333 - 2.32937i) q^{83} +(-10.5000 + 6.06218i) q^{86} +(-3.10583 - 11.5911i) q^{87} +(0.896575 + 3.34607i) q^{88} -12.1244 q^{89} +(-1.55291 - 5.79555i) q^{92} +(6.69213 + 1.79315i) q^{93} +(5.19615 - 3.00000i) q^{94} +(1.50000 - 0.866025i) q^{96} +(-5.01910 + 1.34486i) q^{97} +(4.94975 + 4.94975i) q^{98} +(5.19615 - 9.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 12 q^{6} + 24 q^{11} + 4 q^{16} - 16 q^{31} + 12 q^{36} - 12 q^{41} + 48 q^{46} - 32 q^{61} + 24 q^{66} - 20 q^{76} - 72 q^{81} - 84 q^{86} + 12 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/450\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.258819 − 0.965926i −0.183013 − 0.683013i
\(3\) −1.22474 + 1.22474i −0.707107 + 0.707107i
\(4\) −0.866025 + 0.500000i −0.433013 + 0.250000i
\(5\) 0 0
\(6\) 1.50000 + 0.866025i 0.612372 + 0.353553i
\(7\) 0 0 −0.258819 − 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(8\) 0.707107 + 0.707107i 0.250000 + 0.250000i
\(9\) − 3.00000i − 1.00000i
\(10\) 0 0
\(11\) 3.00000 + 1.73205i 0.904534 + 0.522233i 0.878668 − 0.477432i \(-0.158432\pi\)
0.0258656 + 0.999665i \(0.491766\pi\)
\(12\) 0.448288 − 1.67303i 0.129410 − 0.482963i
\(13\) −3.34607 − 0.896575i −0.928032 − 0.248665i −0.237016 − 0.971506i \(-0.576170\pi\)
−0.691015 + 0.722840i \(0.742836\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0.500000 − 0.866025i 0.125000 − 0.216506i
\(17\) −4.24264 + 4.24264i −1.02899 + 1.02899i −0.0294245 + 0.999567i \(0.509367\pi\)
−0.999567 + 0.0294245i \(0.990633\pi\)
\(18\) −2.89778 + 0.776457i −0.683013 + 0.183013i
\(19\) 5.00000i 1.14708i 0.819178 + 0.573539i \(0.194430\pi\)
−0.819178 + 0.573539i \(0.805570\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0.896575 − 3.34607i 0.191151 − 0.713384i
\(23\) −1.55291 + 5.79555i −0.323805 + 1.20846i 0.591703 + 0.806156i \(0.298456\pi\)
−0.915508 + 0.402300i \(0.868211\pi\)
\(24\) −1.73205 −0.353553
\(25\) 0 0
\(26\) 3.46410i 0.679366i
\(27\) 3.67423 + 3.67423i 0.707107 + 0.707107i
\(28\) 0 0
\(29\) −3.46410 + 6.00000i −0.643268 + 1.11417i 0.341431 + 0.939907i \(0.389088\pi\)
−0.984699 + 0.174265i \(0.944245\pi\)
\(30\) 0 0
\(31\) −2.00000 − 3.46410i −0.359211 − 0.622171i 0.628619 − 0.777714i \(-0.283621\pi\)
−0.987829 + 0.155543i \(0.950287\pi\)
\(32\) −0.965926 − 0.258819i −0.170753 − 0.0457532i
\(33\) −5.79555 + 1.55291i −1.00888 + 0.270328i
\(34\) 5.19615 + 3.00000i 0.891133 + 0.514496i
\(35\) 0 0
\(36\) 1.50000 + 2.59808i 0.250000 + 0.433013i
\(37\) 4.89898 + 4.89898i 0.805387 + 0.805387i 0.983932 − 0.178545i \(-0.0571389\pi\)
−0.178545 + 0.983932i \(0.557139\pi\)
\(38\) 4.82963 − 1.29410i 0.783469 − 0.209930i
\(39\) 5.19615 − 3.00000i 0.832050 − 0.480384i
\(40\) 0 0
\(41\) −1.50000 + 0.866025i −0.234261 + 0.135250i −0.612536 − 0.790443i \(-0.709851\pi\)
0.378275 + 0.925693i \(0.376517\pi\)
\(42\) 0 0
\(43\) −3.13801 − 11.7112i −0.478543 − 1.78595i −0.607527 − 0.794299i \(-0.707838\pi\)
0.128984 − 0.991647i \(-0.458828\pi\)
\(44\) −3.46410 −0.522233
\(45\) 0 0
\(46\) 6.00000 0.884652
\(47\) 1.55291 + 5.79555i 0.226516 + 0.845369i 0.981792 + 0.189961i \(0.0608363\pi\)
−0.755276 + 0.655407i \(0.772497\pi\)
\(48\) 0.448288 + 1.67303i 0.0647048 + 0.241481i
\(49\) −6.06218 + 3.50000i −0.866025 + 0.500000i
\(50\) 0 0
\(51\) − 10.3923i − 1.45521i
\(52\) 3.34607 − 0.896575i 0.464016 − 0.124333i
\(53\) 4.24264 + 4.24264i 0.582772 + 0.582772i 0.935664 − 0.352892i \(-0.114802\pi\)
−0.352892 + 0.935664i \(0.614802\pi\)
\(54\) 2.59808 − 4.50000i 0.353553 − 0.612372i
\(55\) 0 0
\(56\) 0 0
\(57\) −6.12372 − 6.12372i −0.811107 − 0.811107i
\(58\) 6.69213 + 1.79315i 0.878720 + 0.235452i
\(59\) 0.866025 + 1.50000i 0.112747 + 0.195283i 0.916877 − 0.399170i \(-0.130702\pi\)
−0.804130 + 0.594454i \(0.797368\pi\)
\(60\) 0 0
\(61\) −4.00000 + 6.92820i −0.512148 + 0.887066i 0.487753 + 0.872982i \(0.337817\pi\)
−0.999901 + 0.0140840i \(0.995517\pi\)
\(62\) −2.82843 + 2.82843i −0.359211 + 0.359211i
\(63\) 0 0
\(64\) 1.00000i 0.125000i
\(65\) 0 0
\(66\) 3.00000 + 5.19615i 0.369274 + 0.639602i
\(67\) 2.24144 − 8.36516i 0.273835 − 1.02197i −0.682783 − 0.730622i \(-0.739230\pi\)
0.956618 − 0.291346i \(-0.0941030\pi\)
\(68\) 1.55291 − 5.79555i 0.188319 − 0.702814i
\(69\) −5.19615 − 9.00000i −0.625543 − 1.08347i
\(70\) 0 0
\(71\) − 6.92820i − 0.822226i −0.911584 − 0.411113i \(-0.865140\pi\)
0.911584 − 0.411113i \(-0.134860\pi\)
\(72\) 2.12132 − 2.12132i 0.250000 − 0.250000i
\(73\) 8.57321 − 8.57321i 1.00342 − 1.00342i 0.00342468 − 0.999994i \(-0.498910\pi\)
0.999994 − 0.00342468i \(-0.00109011\pi\)
\(74\) 3.46410 − 6.00000i 0.402694 − 0.697486i
\(75\) 0 0
\(76\) −2.50000 − 4.33013i −0.286770 − 0.496700i
\(77\) 0 0
\(78\) −4.24264 − 4.24264i −0.480384 − 0.480384i
\(79\) 12.1244 + 7.00000i 1.36410 + 0.787562i 0.990166 − 0.139895i \(-0.0446766\pi\)
0.373930 + 0.927457i \(0.378010\pi\)
\(80\) 0 0
\(81\) −9.00000 −1.00000
\(82\) 1.22474 + 1.22474i 0.135250 + 0.135250i
\(83\) 8.69333 − 2.32937i 0.954217 − 0.255682i 0.252066 − 0.967710i \(-0.418890\pi\)
0.702151 + 0.712028i \(0.252223\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −10.5000 + 6.06218i −1.13224 + 0.653701i
\(87\) −3.10583 − 11.5911i −0.332980 − 1.24270i
\(88\) 0.896575 + 3.34607i 0.0955753 + 0.356692i
\(89\) −12.1244 −1.28518 −0.642590 − 0.766211i \(-0.722140\pi\)
−0.642590 + 0.766211i \(0.722140\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −1.55291 − 5.79555i −0.161903 − 0.604228i
\(93\) 6.69213 + 1.79315i 0.693942 + 0.185941i
\(94\) 5.19615 − 3.00000i 0.535942 − 0.309426i
\(95\) 0 0
\(96\) 1.50000 − 0.866025i 0.153093 − 0.0883883i
\(97\) −5.01910 + 1.34486i −0.509612 + 0.136550i −0.504457 − 0.863437i \(-0.668307\pi\)
−0.00515471 + 0.999987i \(0.501641\pi\)
\(98\) 4.94975 + 4.94975i 0.500000 + 0.500000i
\(99\) 5.19615 − 9.00000i 0.522233 − 0.904534i
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 450.2.p.g.407.1 yes 8
3.2 odd 2 1350.2.q.c.1007.2 8
5.2 odd 4 inner 450.2.p.g.443.2 yes 8
5.3 odd 4 inner 450.2.p.g.443.1 yes 8
5.4 even 2 inner 450.2.p.g.407.2 yes 8
9.4 even 3 1350.2.q.c.557.2 8
9.5 odd 6 inner 450.2.p.g.257.1 ✓ 8
15.2 even 4 1350.2.q.c.143.1 8
15.8 even 4 1350.2.q.c.143.2 8
15.14 odd 2 1350.2.q.c.1007.1 8
45.4 even 6 1350.2.q.c.557.1 8
45.13 odd 12 1350.2.q.c.1043.2 8
45.14 odd 6 inner 450.2.p.g.257.2 yes 8
45.22 odd 12 1350.2.q.c.1043.1 8
45.23 even 12 inner 450.2.p.g.293.1 yes 8
45.32 even 12 inner 450.2.p.g.293.2 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
450.2.p.g.257.1 ✓ 8 9.5 odd 6 inner
450.2.p.g.257.2 yes 8 45.14 odd 6 inner
450.2.p.g.293.1 yes 8 45.23 even 12 inner
450.2.p.g.293.2 yes 8 45.32 even 12 inner
450.2.p.g.407.1 yes 8 1.1 even 1 trivial
450.2.p.g.407.2 yes 8 5.4 even 2 inner
450.2.p.g.443.1 yes 8 5.3 odd 4 inner
450.2.p.g.443.2 yes 8 5.2 odd 4 inner
1350.2.q.c.143.1 8 15.2 even 4
1350.2.q.c.143.2 8 15.8 even 4
1350.2.q.c.557.1 8 45.4 even 6
1350.2.q.c.557.2 8 9.4 even 3
1350.2.q.c.1007.1 8 15.14 odd 2
1350.2.q.c.1007.2 8 3.2 odd 2
1350.2.q.c.1043.1 8 45.22 odd 12
1350.2.q.c.1043.2 8 45.13 odd 12