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Properties

Label 900.2.i.e.601.4
Level $900$
Weight $2$
Character 900.601
Analytic conductor $7.187$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [900,2,Mod(301,900)] 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("900.301"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(900, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 900.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.18653618192\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.142635249.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 3x^{6} + 3x^{5} - 11x^{4} + 6x^{3} + 12x^{2} - 24x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 601.4
Root \(0.620769 - 1.27069i\) of defining polynomial
Character \(\chi\) \(=\) 900.601
Dual form 900.2.i.e.301.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.72929 + 0.0977414i) q^{3} +(1.70089 - 2.94604i) q^{7} +(2.98089 + 0.338047i) q^{9} +(2.20089 - 3.81206i) q^{11} +(-1.03160 - 1.78679i) q^{13} -1.40179 q^{17} -6.35717 q^{19} +(3.22929 - 4.92830i) q^{21} +(-0.0539129 - 0.0933799i) q^{23} +(5.12179 + 0.875938i) q^{27} +(-4.54089 + 7.86505i) q^{29} +(-1.53160 - 2.65281i) q^{31} +(4.17858 - 6.37704i) q^{33} +1.95538 q^{37} +(-1.60930 - 3.19070i) q^{39} +(4.34788 + 7.53074i) q^{41} +(3.56320 - 6.17165i) q^{43} +(3.74179 - 6.48096i) q^{47} +(-2.28608 - 3.95961i) q^{49} +(-2.42410 - 0.137013i) q^{51} +13.0975 q^{53} +(-10.9934 - 0.621359i) q^{57} +(6.58966 + 11.4136i) q^{59} +(0.862308 - 1.49356i) q^{61} +(6.06608 - 8.20684i) q^{63} +(6.18787 + 10.7177i) q^{67} +(-0.0841040 - 0.166751i) q^{69} -7.50961 q^{71} -5.42037 q^{73} +(-7.48698 - 12.9678i) q^{77} +(-4.71806 + 8.17193i) q^{79} +(8.77145 + 2.01536i) q^{81} +(-4.48698 + 7.77167i) q^{83} +(-8.62126 + 13.1571i) q^{87} +4.01576 q^{89} -7.01858 q^{91} +(-2.38929 - 4.73718i) q^{93} +(-1.08551 + 1.88017i) q^{97} +(7.84929 - 10.6193i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{3} - q^{7} + 5 q^{9} + 3 q^{11} + 2 q^{13} + 18 q^{17} - 8 q^{19} + 13 q^{21} + 3 q^{23} + 16 q^{27} - 9 q^{29} - 2 q^{31} + 12 q^{33} + 2 q^{37} - 17 q^{39} + 9 q^{41} + 8 q^{43} - 12 q^{47}+ \cdots + 33 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.72929 + 0.0977414i 0.998406 + 0.0564310i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 1.70089 − 2.94604i 0.642878 − 1.11350i −0.341910 − 0.939733i \(-0.611074\pi\)
0.984787 − 0.173764i \(-0.0555930\pi\)
\(8\) 0 0
\(9\) 2.98089 + 0.338047i 0.993631 + 0.112682i
\(10\) 0 0
\(11\) 2.20089 − 3.81206i 0.663595 − 1.14938i −0.316070 − 0.948736i \(-0.602363\pi\)
0.979664 − 0.200644i \(-0.0643033\pi\)
\(12\) 0 0
\(13\) −1.03160 − 1.78679i −0.286115 − 0.495565i 0.686764 − 0.726880i \(-0.259031\pi\)
−0.972879 + 0.231315i \(0.925697\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.40179 −0.339984 −0.169992 − 0.985445i \(-0.554374\pi\)
−0.169992 + 0.985445i \(0.554374\pi\)
\(18\) 0 0
\(19\) −6.35717 −1.45843 −0.729217 − 0.684283i \(-0.760115\pi\)
−0.729217 + 0.684283i \(0.760115\pi\)
\(20\) 0 0
\(21\) 3.22929 − 4.92830i 0.704689 − 1.07544i
\(22\) 0 0
\(23\) −0.0539129 − 0.0933799i −0.0112416 − 0.0194711i 0.860350 − 0.509704i \(-0.170245\pi\)
−0.871591 + 0.490233i \(0.836912\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.12179 + 0.875938i 0.985689 + 0.168574i
\(28\) 0 0
\(29\) −4.54089 + 7.86505i −0.843222 + 1.46050i 0.0439339 + 0.999034i \(0.486011\pi\)
−0.887156 + 0.461469i \(0.847322\pi\)
\(30\) 0 0
\(31\) −1.53160 − 2.65281i −0.275084 − 0.476459i 0.695073 − 0.718940i \(-0.255372\pi\)
−0.970156 + 0.242481i \(0.922039\pi\)
\(32\) 0 0
\(33\) 4.17858 − 6.37704i 0.727398 − 1.11010i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 1.95538 0.321462 0.160731 − 0.986998i \(-0.448615\pi\)
0.160731 + 0.986998i \(0.448615\pi\)
\(38\) 0 0
\(39\) −1.60930 − 3.19070i −0.257694 − 0.510921i
\(40\) 0 0
\(41\) 4.34788 + 7.53074i 0.679024 + 1.17610i 0.975275 + 0.220994i \(0.0709302\pi\)
−0.296251 + 0.955110i \(0.595736\pi\)
\(42\) 0 0
\(43\) 3.56320 − 6.17165i 0.543383 − 0.941167i −0.455323 − 0.890326i \(-0.650476\pi\)
0.998707 − 0.0508414i \(-0.0161903\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.74179 − 6.48096i 0.545795 − 0.945345i −0.452761 − 0.891632i \(-0.649561\pi\)
0.998556 − 0.0537135i \(-0.0171058\pi\)
\(48\) 0 0
\(49\) −2.28608 − 3.95961i −0.326583 − 0.565659i
\(50\) 0 0
\(51\) −2.42410 − 0.137013i −0.339442 − 0.0191856i
\(52\) 0 0
\(53\) 13.0975 1.79909 0.899543 − 0.436833i \(-0.143900\pi\)
0.899543 + 0.436833i \(0.143900\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −10.9934 − 0.621359i −1.45611 − 0.0823009i
\(58\) 0 0
\(59\) 6.58966 + 11.4136i 0.857901 + 1.48593i 0.873928 + 0.486056i \(0.161565\pi\)
−0.0160267 + 0.999872i \(0.505102\pi\)
\(60\) 0 0
\(61\) 0.862308 − 1.49356i 0.110407 − 0.191231i −0.805527 − 0.592559i \(-0.798118\pi\)
0.915935 + 0.401328i \(0.131451\pi\)
\(62\) 0 0
\(63\) 6.06608 − 8.20684i 0.764255 − 1.03396i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 6.18787 + 10.7177i 0.755969 + 1.30938i 0.944891 + 0.327385i \(0.106167\pi\)
−0.188922 + 0.981992i \(0.560499\pi\)
\(68\) 0 0
\(69\) −0.0841040 − 0.166751i −0.0101249 − 0.0200744i
\(70\) 0 0
\(71\) −7.50961 −0.891227 −0.445614 − 0.895225i \(-0.647014\pi\)
−0.445614 + 0.895225i \(0.647014\pi\)
\(72\) 0 0
\(73\) −5.42037 −0.634406 −0.317203 − 0.948358i \(-0.602744\pi\)
−0.317203 + 0.948358i \(0.602744\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −7.48698 − 12.9678i −0.853220 − 1.47782i
\(78\) 0 0
\(79\) −4.71806 + 8.17193i −0.530824 + 0.919413i 0.468529 + 0.883448i \(0.344784\pi\)
−0.999353 + 0.0359656i \(0.988549\pi\)
\(80\) 0 0
\(81\) 8.77145 + 2.01536i 0.974605 + 0.223929i
\(82\) 0 0
\(83\) −4.48698 + 7.77167i −0.492510 + 0.853052i −0.999963 − 0.00862744i \(-0.997254\pi\)
0.507453 + 0.861679i \(0.330587\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −8.62126 + 13.1571i −0.924296 + 1.41059i
\(88\) 0 0
\(89\) 4.01576 0.425670 0.212835 − 0.977088i \(-0.431730\pi\)
0.212835 + 0.977088i \(0.431730\pi\)
\(90\) 0 0
\(91\) −7.01858 −0.735747
\(92\) 0 0
\(93\) −2.38929 − 4.73718i −0.247758 − 0.491223i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.08551 + 1.88017i −0.110217 + 0.190902i −0.915858 − 0.401503i \(-0.868488\pi\)
0.805641 + 0.592405i \(0.201821\pi\)
\(98\) 0 0
\(99\) 7.84929 − 10.6193i 0.788883 − 1.06728i
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 900.2.i.e.601.4 yes 8
3.2 odd 2 2700.2.i.d.1801.4 8
5.2 odd 4 900.2.s.d.349.5 16
5.3 odd 4 900.2.s.d.349.4 16
5.4 even 2 900.2.i.d.601.1 yes 8
9.2 odd 6 8100.2.a.ba.1.1 4
9.4 even 3 inner 900.2.i.e.301.4 yes 8
9.5 odd 6 2700.2.i.d.901.4 8
9.7 even 3 8100.2.a.z.1.1 4
15.2 even 4 2700.2.s.d.2449.7 16
15.8 even 4 2700.2.s.d.2449.2 16
15.14 odd 2 2700.2.i.e.1801.1 8
45.2 even 12 8100.2.d.s.649.2 8
45.4 even 6 900.2.i.d.301.1 ✓ 8
45.7 odd 12 8100.2.d.q.649.2 8
45.13 odd 12 900.2.s.d.49.5 16
45.14 odd 6 2700.2.i.e.901.1 8
45.22 odd 12 900.2.s.d.49.4 16
45.23 even 12 2700.2.s.d.1549.7 16
45.29 odd 6 8100.2.a.y.1.4 4
45.32 even 12 2700.2.s.d.1549.2 16
45.34 even 6 8100.2.a.x.1.4 4
45.38 even 12 8100.2.d.s.649.7 8
45.43 odd 12 8100.2.d.q.649.7 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
900.2.i.d.301.1 ✓ 8 45.4 even 6
900.2.i.d.601.1 yes 8 5.4 even 2
900.2.i.e.301.4 yes 8 9.4 even 3 inner
900.2.i.e.601.4 yes 8 1.1 even 1 trivial
900.2.s.d.49.4 16 45.22 odd 12
900.2.s.d.49.5 16 45.13 odd 12
900.2.s.d.349.4 16 5.3 odd 4
900.2.s.d.349.5 16 5.2 odd 4
2700.2.i.d.901.4 8 9.5 odd 6
2700.2.i.d.1801.4 8 3.2 odd 2
2700.2.i.e.901.1 8 45.14 odd 6
2700.2.i.e.1801.1 8 15.14 odd 2
2700.2.s.d.1549.2 16 45.32 even 12
2700.2.s.d.1549.7 16 45.23 even 12
2700.2.s.d.2449.2 16 15.8 even 4
2700.2.s.d.2449.7 16 15.2 even 4
8100.2.a.x.1.4 4 45.34 even 6
8100.2.a.y.1.4 4 45.29 odd 6
8100.2.a.z.1.1 4 9.7 even 3
8100.2.a.ba.1.1 4 9.2 odd 6
8100.2.d.q.649.2 8 45.7 odd 12
8100.2.d.q.649.7 8 45.43 odd 12
8100.2.d.s.649.2 8 45.2 even 12
8100.2.d.s.649.7 8 45.38 even 12