Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1350,3,Mod(449,1350)] 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("1350.449"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1350, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([5, 3])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1350.k (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,-8,0,0,0,0,0,0,-36] 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: \(36.7848356886\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 18)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 449.1
Root \(0.258819 + 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 1350.449
Dual form 1350.3.k.a.899.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.707107 - 1.22474i) q^{2} +(-1.00000 + 1.73205i) q^{4} +(-7.22999 + 4.17423i) q^{7} +2.82843 q^{8} +(-0.825765 + 0.476756i) q^{11} +(-8.39780 - 4.84847i) q^{13} +(10.2247 + 5.90326i) q^{14} +(-2.00000 - 3.46410i) q^{16} +18.8776 q^{17} +24.6969 q^{19} +(1.16781 + 0.674235i) q^{22} +(-0.476756 + 0.825765i) q^{23} +13.7135i q^{26} -16.6969i q^{28} +(11.8485 - 6.84072i) q^{29} +(-1.52270 + 2.63740i) q^{31} +(-2.82843 + 4.89898i) q^{32} +(-13.3485 - 23.1202i) q^{34} +46.6969i q^{37} +(-17.4634 - 30.2474i) q^{38} +(9.45459 + 5.45861i) q^{41} +(-39.0105 + 22.5227i) q^{43} -1.90702i q^{44} +1.34847 q^{46} +(-22.6435 - 39.2196i) q^{47} +(10.3485 - 17.9241i) q^{49} +(16.7956 - 9.69694i) q^{52} -94.3879 q^{53} +(-20.4495 + 11.8065i) q^{56} +(-16.7563 - 9.67423i) q^{58} +(-16.2650 - 9.39063i) q^{59} +(-6.54541 - 11.3370i) q^{61} +4.30686 q^{62} +8.00000 q^{64} +(-64.9912 - 37.5227i) q^{67} +(-18.8776 + 32.6969i) q^{68} +18.0204i q^{71} +7.90918i q^{73} +(57.1918 - 33.0197i) q^{74} +(-24.6969 + 42.7764i) q^{76} +(3.98018 - 6.89388i) q^{77} +(-21.8712 - 37.8820i) q^{79} -15.4393i q^{82} +(-65.1662 - 112.871i) q^{83} +(55.1691 + 31.8519i) q^{86} +(-2.33562 + 1.34847i) q^{88} -145.300i q^{89} +80.9546 q^{91} +(-0.953512 - 1.65153i) q^{92} +(-32.0227 + 55.4650i) q^{94} +(-95.1576 + 54.9393i) q^{97} -29.2699 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{4} - 36 q^{11} + 72 q^{14} - 16 q^{16} + 80 q^{19} + 36 q^{29} + 76 q^{31} - 48 q^{34} + 252 q^{41} - 48 q^{46} + 24 q^{49} - 144 q^{56} + 252 q^{59} + 124 q^{61} + 64 q^{64} + 144 q^{74} - 80 q^{76}+ \cdots - 168 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1350\mathbb{Z}\right)^\times\).

\(n\) \(1001\) \(1027\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.707107 − 1.22474i −0.353553 − 0.612372i
\(3\) 0 0
\(4\) −1.00000 + 1.73205i −0.250000 + 0.433013i
\(5\) 0 0
\(6\) 0 0
\(7\) −7.22999 + 4.17423i −1.03286 + 0.596319i −0.917801 − 0.397040i \(-0.870037\pi\)
−0.115054 + 0.993359i \(0.536704\pi\)
\(8\) 2.82843 0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) −0.825765 + 0.476756i −0.0750696 + 0.0433414i −0.537065 − 0.843541i \(-0.680467\pi\)
0.461995 + 0.886882i \(0.347134\pi\)
\(12\) 0 0
\(13\) −8.39780 − 4.84847i −0.645984 − 0.372959i 0.140932 − 0.990019i \(-0.454990\pi\)
−0.786916 + 0.617060i \(0.788324\pi\)
\(14\) 10.2247 + 5.90326i 0.730339 + 0.421661i
\(15\) 0 0
\(16\) −2.00000 − 3.46410i −0.125000 − 0.216506i
\(17\) 18.8776 1.11045 0.555223 − 0.831701i \(-0.312633\pi\)
0.555223 + 0.831701i \(0.312633\pi\)
\(18\) 0 0
\(19\) 24.6969 1.29984 0.649919 − 0.760003i \(-0.274803\pi\)
0.649919 + 0.760003i \(0.274803\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 1.16781 + 0.674235i 0.0530822 + 0.0306470i
\(23\) −0.476756 + 0.825765i −0.0207285 + 0.0359028i −0.876204 − 0.481941i \(-0.839932\pi\)
0.855475 + 0.517844i \(0.173265\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 13.7135i 0.527444i
\(27\) 0 0
\(28\) − 16.6969i − 0.596319i
\(29\) 11.8485 − 6.84072i 0.408568 − 0.235887i −0.281606 − 0.959530i \(-0.590867\pi\)
0.690174 + 0.723643i \(0.257534\pi\)
\(30\) 0 0
\(31\) −1.52270 + 2.63740i −0.0491195 + 0.0850774i −0.889540 − 0.456858i \(-0.848975\pi\)
0.840420 + 0.541935i \(0.182308\pi\)
\(32\) −2.82843 + 4.89898i −0.0883883 + 0.153093i
\(33\) 0 0
\(34\) −13.3485 − 23.1202i −0.392602 − 0.680007i
\(35\) 0 0
\(36\) 0 0
\(37\) 46.6969i 1.26208i 0.775751 + 0.631040i \(0.217372\pi\)
−0.775751 + 0.631040i \(0.782628\pi\)
\(38\) −17.4634 − 30.2474i −0.459562 − 0.795985i
\(39\) 0 0
\(40\) 0 0
\(41\) 9.45459 + 5.45861i 0.230600 + 0.133137i 0.610849 − 0.791747i \(-0.290828\pi\)
−0.380249 + 0.924884i \(0.624162\pi\)
\(42\) 0 0
\(43\) −39.0105 + 22.5227i −0.907220 + 0.523784i −0.879536 − 0.475833i \(-0.842147\pi\)
−0.0276845 + 0.999617i \(0.508813\pi\)
\(44\) − 1.90702i − 0.0433414i
\(45\) 0 0
\(46\) 1.34847 0.0293145
\(47\) −22.6435 − 39.2196i −0.481776 − 0.834460i 0.518005 − 0.855377i \(-0.326675\pi\)
−0.999781 + 0.0209170i \(0.993341\pi\)
\(48\) 0 0
\(49\) 10.3485 − 17.9241i 0.211193 − 0.365797i
\(50\) 0 0
\(51\) 0 0
\(52\) 16.7956 − 9.69694i 0.322992 − 0.186480i
\(53\) −94.3879 −1.78090 −0.890452 − 0.455077i \(-0.849612\pi\)
−0.890452 + 0.455077i \(0.849612\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −20.4495 + 11.8065i −0.365169 + 0.210831i
\(57\) 0 0
\(58\) −16.7563 − 9.67423i −0.288901 − 0.166797i
\(59\) −16.2650 − 9.39063i −0.275679 − 0.159163i 0.355787 − 0.934567i \(-0.384213\pi\)
−0.631466 + 0.775404i \(0.717546\pi\)
\(60\) 0 0
\(61\) −6.54541 − 11.3370i −0.107302 − 0.185852i 0.807375 − 0.590039i \(-0.200888\pi\)
−0.914676 + 0.404187i \(0.867554\pi\)
\(62\) 4.30686 0.0694654
\(63\) 0 0
\(64\) 8.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −64.9912 − 37.5227i −0.970018 − 0.560040i −0.0707765 − 0.997492i \(-0.522548\pi\)
−0.899242 + 0.437452i \(0.855881\pi\)
\(68\) −18.8776 + 32.6969i −0.277612 + 0.480837i
\(69\) 0 0
\(70\) 0 0
\(71\) 18.0204i 0.253808i 0.991915 + 0.126904i \(0.0405041\pi\)
−0.991915 + 0.126904i \(0.959496\pi\)
\(72\) 0 0
\(73\) 7.90918i 0.108345i 0.998532 + 0.0541725i \(0.0172521\pi\)
−0.998532 + 0.0541725i \(0.982748\pi\)
\(74\) 57.1918 − 33.0197i 0.772863 − 0.446212i
\(75\) 0 0
\(76\) −24.6969 + 42.7764i −0.324960 + 0.562847i
\(77\) 3.98018 − 6.89388i 0.0516907 − 0.0895309i
\(78\) 0 0
\(79\) −21.8712 − 37.8820i −0.276850 − 0.479519i 0.693750 − 0.720216i \(-0.255957\pi\)
−0.970600 + 0.240697i \(0.922624\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) − 15.4393i − 0.188284i
\(83\) −65.1662 − 112.871i −0.785135 − 1.35989i −0.928918 − 0.370284i \(-0.879260\pi\)
0.143783 − 0.989609i \(-0.454073\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 55.1691 + 31.8519i 0.641502 + 0.370371i
\(87\) 0 0
\(88\) −2.33562 + 1.34847i −0.0265411 + 0.0153235i
\(89\) − 145.300i − 1.63258i −0.577642 − 0.816290i \(-0.696027\pi\)
0.577642 − 0.816290i \(-0.303973\pi\)
\(90\) 0 0
\(91\) 80.9546 0.889611
\(92\) −0.953512 − 1.65153i −0.0103643 − 0.0179514i
\(93\) 0 0
\(94\) −32.0227 + 55.4650i −0.340667 + 0.590053i
\(95\) 0 0
\(96\) 0 0
\(97\) −95.1576 + 54.9393i −0.981007 + 0.566384i −0.902574 − 0.430535i \(-0.858325\pi\)
−0.0784327 + 0.996919i \(0.524992\pi\)
\(98\) −29.2699 −0.298672
\(99\) 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 1350.3.k.a.449.1 8
3.2 odd 2 450.3.k.a.149.4 8
5.2 odd 4 1350.3.i.b.1151.2 4
5.3 odd 4 54.3.d.a.17.1 4
5.4 even 2 inner 1350.3.k.a.449.4 8
9.2 odd 6 inner 1350.3.k.a.899.4 8
9.7 even 3 450.3.k.a.299.1 8
15.2 even 4 450.3.i.b.401.1 4
15.8 even 4 18.3.d.a.5.2 ✓ 4
15.14 odd 2 450.3.k.a.149.1 8
20.3 even 4 432.3.q.d.17.1 4
40.3 even 4 1728.3.q.c.449.1 4
40.13 odd 4 1728.3.q.d.449.2 4
45.2 even 12 1350.3.i.b.251.2 4
45.7 odd 12 450.3.i.b.101.1 4
45.13 odd 12 162.3.b.a.161.1 4
45.23 even 12 162.3.b.a.161.4 4
45.29 odd 6 inner 1350.3.k.a.899.1 8
45.34 even 6 450.3.k.a.299.4 8
45.38 even 12 54.3.d.a.35.1 4
45.43 odd 12 18.3.d.a.11.2 yes 4
60.23 odd 4 144.3.q.c.113.2 4
120.53 even 4 576.3.q.f.257.2 4
120.83 odd 4 576.3.q.e.257.1 4
180.23 odd 12 1296.3.e.g.161.4 4
180.43 even 12 144.3.q.c.65.2 4
180.83 odd 12 432.3.q.d.305.1 4
180.103 even 12 1296.3.e.g.161.2 4
360.43 even 12 576.3.q.e.65.1 4
360.83 odd 12 1728.3.q.c.1601.1 4
360.133 odd 12 576.3.q.f.65.2 4
360.173 even 12 1728.3.q.d.1601.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
18.3.d.a.5.2 ✓ 4 15.8 even 4
18.3.d.a.11.2 yes 4 45.43 odd 12
54.3.d.a.17.1 4 5.3 odd 4
54.3.d.a.35.1 4 45.38 even 12
144.3.q.c.65.2 4 180.43 even 12
144.3.q.c.113.2 4 60.23 odd 4
162.3.b.a.161.1 4 45.13 odd 12
162.3.b.a.161.4 4 45.23 even 12
432.3.q.d.17.1 4 20.3 even 4
432.3.q.d.305.1 4 180.83 odd 12
450.3.i.b.101.1 4 45.7 odd 12
450.3.i.b.401.1 4 15.2 even 4
450.3.k.a.149.1 8 15.14 odd 2
450.3.k.a.149.4 8 3.2 odd 2
450.3.k.a.299.1 8 9.7 even 3
450.3.k.a.299.4 8 45.34 even 6
576.3.q.e.65.1 4 360.43 even 12
576.3.q.e.257.1 4 120.83 odd 4
576.3.q.f.65.2 4 360.133 odd 12
576.3.q.f.257.2 4 120.53 even 4
1296.3.e.g.161.2 4 180.103 even 12
1296.3.e.g.161.4 4 180.23 odd 12
1350.3.i.b.251.2 4 45.2 even 12
1350.3.i.b.1151.2 4 5.2 odd 4
1350.3.k.a.449.1 8 1.1 even 1 trivial
1350.3.k.a.449.4 8 5.4 even 2 inner
1350.3.k.a.899.1 8 45.29 odd 6 inner
1350.3.k.a.899.4 8 9.2 odd 6 inner
1728.3.q.c.449.1 4 40.3 even 4
1728.3.q.c.1601.1 4 360.83 odd 12
1728.3.q.d.449.2 4 40.13 odd 4
1728.3.q.d.1601.2 4 360.173 even 12