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Properties

Label 900.2.i.e.301.1
Level $900$
Weight $2$
Character 900.301
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 301.1
Root \(1.38941 + 0.263711i\) of defining polynomial
Character \(\chi\) \(=\) 900.301
Dual form 900.2.i.e.601.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.36091 + 1.07141i) q^{3} +(-0.0432397 - 0.0748933i) q^{7} +(0.704170 - 2.91619i) q^{9} +(0.456760 + 0.791132i) q^{11} +(-1.31249 + 2.27331i) q^{13} +2.08648 q^{17} +4.93847 q^{19} +(0.139087 + 0.0555960i) q^{21} +(-4.23849 + 7.34128i) q^{23} +(2.16611 + 4.72313i) q^{27} +(-1.19899 - 2.07671i) q^{29} +(-1.81249 + 3.13933i) q^{31} +(-1.46924 - 0.587286i) q^{33} -5.85199 q^{37} +(-0.649447 - 4.49999i) q^{39} +(-3.32497 + 5.75902i) q^{41} +(4.12499 + 7.14469i) q^{43} +(-1.34425 - 2.32831i) q^{47} +(3.49626 - 6.05570i) q^{49} +(-2.83952 + 2.23547i) q^{51} -5.73642 q^{53} +(-6.72083 + 5.29112i) q^{57} +(-6.16922 + 10.6854i) q^{59} +(3.16823 + 5.48753i) q^{61} +(-0.248851 + 0.0733573i) q^{63} +(-3.08274 + 5.33946i) q^{67} +(-2.09729 - 14.5320i) q^{69} -12.3905 q^{71} +5.31349 q^{73} +(0.0395003 - 0.0684166i) q^{77} +(6.72394 + 11.6462i) q^{79} +(-8.00829 - 4.10698i) q^{81} +(3.03950 + 5.26457i) q^{83} +(3.85673 + 1.54162i) q^{87} -8.13440 q^{89} +0.227007 q^{91} +(-0.896857 - 6.21428i) q^{93} +(-5.55098 - 9.61459i) q^{97} +(2.62873 - 0.774907i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{3} - q^{7} + 5 q^{9} + 3 q^{11} + 2 q^{13} + 18 q^{17} - 8 q^{19} + 13 q^{21} + 3 q^{23} + 16 q^{27} - 9 q^{29} - 2 q^{31} + 12 q^{33} + 2 q^{37} - 17 q^{39} + 9 q^{41} + 8 q^{43} - 12 q^{47}+ \cdots + 33 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(3\) −1.36091 + 1.07141i −0.785724 + 0.618578i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −0.0432397 − 0.0748933i −0.0163431 − 0.0283070i 0.857738 − 0.514087i \(-0.171869\pi\)
−0.874081 + 0.485780i \(0.838536\pi\)
\(8\) 0 0
\(9\) 0.704170 − 2.91619i 0.234723 − 0.972062i
\(10\) 0 0
\(11\) 0.456760 + 0.791132i 0.137718 + 0.238535i 0.926633 − 0.375968i \(-0.122690\pi\)
−0.788914 + 0.614503i \(0.789356\pi\)
\(12\) 0 0
\(13\) −1.31249 + 2.27331i −0.364020 + 0.630501i −0.988618 − 0.150445i \(-0.951929\pi\)
0.624598 + 0.780946i \(0.285263\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.08648 0.506046 0.253023 − 0.967460i \(-0.418575\pi\)
0.253023 + 0.967460i \(0.418575\pi\)
\(18\) 0 0
\(19\) 4.93847 1.13296 0.566482 − 0.824074i \(-0.308304\pi\)
0.566482 + 0.824074i \(0.308304\pi\)
\(20\) 0 0
\(21\) 0.139087 + 0.0555960i 0.0303512 + 0.0121320i
\(22\) 0 0
\(23\) −4.23849 + 7.34128i −0.883786 + 1.53076i −0.0366878 + 0.999327i \(0.511681\pi\)
−0.847098 + 0.531436i \(0.821653\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 2.16611 + 4.72313i 0.416868 + 0.908967i
\(28\) 0 0
\(29\) −1.19899 − 2.07671i −0.222647 − 0.385636i 0.732964 − 0.680267i \(-0.238136\pi\)
−0.955611 + 0.294632i \(0.904803\pi\)
\(30\) 0 0
\(31\) −1.81249 + 3.13933i −0.325533 + 0.563840i −0.981620 − 0.190845i \(-0.938877\pi\)
0.656087 + 0.754685i \(0.272211\pi\)
\(32\) 0 0
\(33\) −1.46924 − 0.587286i −0.255761 − 0.102233i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −5.85199 −0.962062 −0.481031 − 0.876704i \(-0.659738\pi\)
−0.481031 + 0.876704i \(0.659738\pi\)
\(38\) 0 0
\(39\) −0.649447 − 4.49999i −0.103995 − 0.720575i
\(40\) 0 0
\(41\) −3.32497 + 5.75902i −0.519273 + 0.899407i 0.480476 + 0.877008i \(0.340464\pi\)
−0.999749 + 0.0223994i \(0.992869\pi\)
\(42\) 0 0
\(43\) 4.12499 + 7.14469i 0.629055 + 1.08955i 0.987742 + 0.156097i \(0.0498911\pi\)
−0.358687 + 0.933458i \(0.616776\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1.34425 − 2.32831i −0.196079 − 0.339619i 0.751175 − 0.660103i \(-0.229488\pi\)
−0.947254 + 0.320485i \(0.896154\pi\)
\(48\) 0 0
\(49\) 3.49626 − 6.05570i 0.499466 − 0.865100i
\(50\) 0 0
\(51\) −2.83952 + 2.23547i −0.397612 + 0.313028i
\(52\) 0 0
\(53\) −5.73642 −0.787958 −0.393979 − 0.919120i \(-0.628902\pi\)
−0.393979 + 0.919120i \(0.628902\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −6.72083 + 5.29112i −0.890196 + 0.700826i
\(58\) 0 0
\(59\) −6.16922 + 10.6854i −0.803164 + 1.39112i 0.114360 + 0.993439i \(0.463518\pi\)
−0.917524 + 0.397681i \(0.869815\pi\)
\(60\) 0 0
\(61\) 3.16823 + 5.48753i 0.405650 + 0.702606i 0.994397 − 0.105711i \(-0.0337119\pi\)
−0.588747 + 0.808317i \(0.700379\pi\)
\(62\) 0 0
\(63\) −0.248851 + 0.0733573i −0.0313523 + 0.00924215i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −3.08274 + 5.33946i −0.376617 + 0.652319i −0.990568 − 0.137025i \(-0.956246\pi\)
0.613951 + 0.789344i \(0.289579\pi\)
\(68\) 0 0
\(69\) −2.09729 − 14.5320i −0.252484 − 1.74945i
\(70\) 0 0
\(71\) −12.3905 −1.47048 −0.735241 − 0.677806i \(-0.762931\pi\)
−0.735241 + 0.677806i \(0.762931\pi\)
\(72\) 0 0
\(73\) 5.31349 0.621897 0.310948 − 0.950427i \(-0.399353\pi\)
0.310948 + 0.950427i \(0.399353\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.0395003 − 0.0684166i 0.00450148 − 0.00779679i
\(78\) 0 0
\(79\) 6.72394 + 11.6462i 0.756503 + 1.31030i 0.944624 + 0.328155i \(0.106427\pi\)
−0.188121 + 0.982146i \(0.560240\pi\)
\(80\) 0 0
\(81\) −8.00829 − 4.10698i −0.889810 − 0.456331i
\(82\) 0 0
\(83\) 3.03950 + 5.26457i 0.333629 + 0.577862i 0.983220 − 0.182422i \(-0.0583936\pi\)
−0.649592 + 0.760283i \(0.725060\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 3.85673 + 1.54162i 0.413484 + 0.165279i
\(88\) 0 0
\(89\) −8.13440 −0.862244 −0.431122 − 0.902294i \(-0.641882\pi\)
−0.431122 + 0.902294i \(0.641882\pi\)
\(90\) 0 0
\(91\) 0.227007 0.0237968
\(92\) 0 0
\(93\) −0.896857 − 6.21428i −0.0929998 − 0.644390i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −5.55098 − 9.61459i −0.563617 − 0.976213i −0.997177 − 0.0750885i \(-0.976076\pi\)
0.433560 − 0.901125i \(-0.357257\pi\)
\(98\) 0 0
\(99\) 2.62873 − 0.774907i 0.264197 − 0.0778811i
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.301.1 yes 8
3.2 odd 2 2700.2.i.d.901.2 8
5.2 odd 4 900.2.s.d.49.6 16
5.3 odd 4 900.2.s.d.49.3 16
5.4 even 2 900.2.i.d.301.4 ✓ 8
9.2 odd 6 2700.2.i.d.1801.2 8
9.4 even 3 8100.2.a.z.1.3 4
9.5 odd 6 8100.2.a.ba.1.3 4
9.7 even 3 inner 900.2.i.e.601.1 yes 8
15.2 even 4 2700.2.s.d.1549.5 16
15.8 even 4 2700.2.s.d.1549.4 16
15.14 odd 2 2700.2.i.e.901.3 8
45.2 even 12 2700.2.s.d.2449.4 16
45.4 even 6 8100.2.a.x.1.2 4
45.7 odd 12 900.2.s.d.349.3 16
45.13 odd 12 8100.2.d.q.649.4 8
45.14 odd 6 8100.2.a.y.1.2 4
45.22 odd 12 8100.2.d.q.649.5 8
45.23 even 12 8100.2.d.s.649.4 8
45.29 odd 6 2700.2.i.e.1801.3 8
45.32 even 12 8100.2.d.s.649.5 8
45.34 even 6 900.2.i.d.601.4 yes 8
45.38 even 12 2700.2.s.d.2449.5 16
45.43 odd 12 900.2.s.d.349.6 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
900.2.i.d.301.4 ✓ 8 5.4 even 2
900.2.i.d.601.4 yes 8 45.34 even 6
900.2.i.e.301.1 yes 8 1.1 even 1 trivial
900.2.i.e.601.1 yes 8 9.7 even 3 inner
900.2.s.d.49.3 16 5.3 odd 4
900.2.s.d.49.6 16 5.2 odd 4
900.2.s.d.349.3 16 45.7 odd 12
900.2.s.d.349.6 16 45.43 odd 12
2700.2.i.d.901.2 8 3.2 odd 2
2700.2.i.d.1801.2 8 9.2 odd 6
2700.2.i.e.901.3 8 15.14 odd 2
2700.2.i.e.1801.3 8 45.29 odd 6
2700.2.s.d.1549.4 16 15.8 even 4
2700.2.s.d.1549.5 16 15.2 even 4
2700.2.s.d.2449.4 16 45.2 even 12
2700.2.s.d.2449.5 16 45.38 even 12
8100.2.a.x.1.2 4 45.4 even 6
8100.2.a.y.1.2 4 45.14 odd 6
8100.2.a.z.1.3 4 9.4 even 3
8100.2.a.ba.1.3 4 9.5 odd 6
8100.2.d.q.649.4 8 45.13 odd 12
8100.2.d.q.649.5 8 45.22 odd 12
8100.2.d.s.649.4 8 45.23 even 12
8100.2.d.s.649.5 8 45.32 even 12