Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [900,5,Mod(449,900)] 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("900.449"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(900, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 900.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(93.0329667755\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 30731x^{8} + 227695675x^{4} + 4228250625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{41}]\)
Coefficient ring index: \( 2^{16}\cdot 3^{16}\cdot 5^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.12
Root \(7.46303 + 7.46303i\) of defining polynomial
Character \(\chi\) \(=\) 900.449
Dual form 900.5.b.c.449.1

$q$-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)
 
\(f(q)\) \(=\) \(q+70.3259i q^{7} +218.102i q^{11} -158.792i q^{13} +76.0225 q^{17} +644.073 q^{19} +882.720 q^{23} -1176.15i q^{29} -135.186 q^{31} +626.399i q^{37} -72.2760i q^{41} +3287.18i q^{43} +2055.13 q^{47} -2544.73 q^{49} -1584.54 q^{53} +4003.49i q^{59} -3530.55 q^{61} -5196.83i q^{67} +635.751i q^{71} -5758.11i q^{73} -15338.2 q^{77} -2234.69 q^{79} +7904.44 q^{83} +12136.5i q^{89} +11167.2 q^{91} +974.647i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 756 q^{19} - 1236 q^{31} - 8256 q^{49} - 16020 q^{61} - 30720 q^{79} - 20556 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 70.3259i 1.43522i 0.696444 + 0.717611i \(0.254764\pi\)
−0.696444 + 0.717611i \(0.745236\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 218.102i 1.80250i 0.433300 + 0.901250i \(0.357349\pi\)
−0.433300 + 0.901250i \(0.642651\pi\)
\(12\) 0 0
\(13\) − 158.792i − 0.939595i −0.882774 − 0.469798i \(-0.844327\pi\)
0.882774 − 0.469798i \(-0.155673\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 76.0225 0.263053 0.131527 − 0.991313i \(-0.458012\pi\)
0.131527 + 0.991313i \(0.458012\pi\)
\(18\) 0 0
\(19\) 644.073 1.78414 0.892068 − 0.451902i \(-0.149254\pi\)
0.892068 + 0.451902i \(0.149254\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 882.720 1.66866 0.834329 − 0.551267i \(-0.185856\pi\)
0.834329 + 0.551267i \(0.185856\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 1176.15i − 1.39852i −0.714868 − 0.699259i \(-0.753513\pi\)
0.714868 − 0.699259i \(-0.246487\pi\)
\(30\) 0 0
\(31\) −135.186 −0.140672 −0.0703362 − 0.997523i \(-0.522407\pi\)
−0.0703362 + 0.997523i \(0.522407\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 626.399i 0.457559i 0.973478 + 0.228780i \(0.0734735\pi\)
−0.973478 + 0.228780i \(0.926526\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 72.2760i − 0.0429958i −0.999769 − 0.0214979i \(-0.993156\pi\)
0.999769 − 0.0214979i \(-0.00684352\pi\)
\(42\) 0 0
\(43\) 3287.18i 1.77781i 0.458088 + 0.888907i \(0.348534\pi\)
−0.458088 + 0.888907i \(0.651466\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2055.13 0.930343 0.465171 − 0.885221i \(-0.345993\pi\)
0.465171 + 0.885221i \(0.345993\pi\)
\(48\) 0 0
\(49\) −2544.73 −1.05986
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −1584.54 −0.564095 −0.282047 − 0.959400i \(-0.591014\pi\)
−0.282047 + 0.959400i \(0.591014\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4003.49i 1.15010i 0.818120 + 0.575048i \(0.195017\pi\)
−0.818120 + 0.575048i \(0.804983\pi\)
\(60\) 0 0
\(61\) −3530.55 −0.948817 −0.474408 − 0.880305i \(-0.657338\pi\)
−0.474408 + 0.880305i \(0.657338\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 5196.83i − 1.15768i −0.815440 − 0.578841i \(-0.803505\pi\)
0.815440 − 0.578841i \(-0.196495\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 635.751i 0.126116i 0.998010 + 0.0630580i \(0.0200853\pi\)
−0.998010 + 0.0630580i \(0.979915\pi\)
\(72\) 0 0
\(73\) − 5758.11i − 1.08052i −0.841497 − 0.540262i \(-0.818325\pi\)
0.841497 − 0.540262i \(-0.181675\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −15338.2 −2.58699
\(78\) 0 0
\(79\) −2234.69 −0.358066 −0.179033 − 0.983843i \(-0.557297\pi\)
−0.179033 + 0.983843i \(0.557297\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 7904.44 1.14740 0.573700 − 0.819065i \(-0.305507\pi\)
0.573700 + 0.819065i \(0.305507\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 12136.5i 1.53219i 0.642729 + 0.766093i \(0.277802\pi\)
−0.642729 + 0.766093i \(0.722198\pi\)
\(90\) 0 0
\(91\) 11167.2 1.34853
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 974.647i 0.103587i 0.998658 + 0.0517933i \(0.0164937\pi\)
−0.998658 + 0.0517933i \(0.983506\pi\)
\(98\) 0 0
\(99\) 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 900.5.b.c.449.12 12
3.2 odd 2 inner 900.5.b.c.449.11 12
5.2 odd 4 900.5.g.c.701.2 yes 6
5.3 odd 4 900.5.g.d.701.6 yes 6
5.4 even 2 inner 900.5.b.c.449.2 12
15.2 even 4 900.5.g.c.701.1 ✓ 6
15.8 even 4 900.5.g.d.701.5 yes 6
15.14 odd 2 inner 900.5.b.c.449.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
900.5.b.c.449.1 12 15.14 odd 2 inner
900.5.b.c.449.2 12 5.4 even 2 inner
900.5.b.c.449.11 12 3.2 odd 2 inner
900.5.b.c.449.12 12 1.1 even 1 trivial
900.5.g.c.701.1 ✓ 6 15.2 even 4
900.5.g.c.701.2 yes 6 5.2 odd 4
900.5.g.d.701.5 yes 6 15.8 even 4
900.5.g.d.701.6 yes 6 5.3 odd 4