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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [900,5,Mod(701,900)] 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("900.701"); 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, 0])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 900.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,0,0,-54] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(93.0329667755\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 243x^{4} + 752x^{3} + 15131x^{2} - 66318x + 98307 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{41}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 701.2
Root \(10.5543 + 1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 900.701
Dual form 900.5.g.c.701.1

$q$-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)
 
\(f(q)\) \(=\) \(q-70.3259 q^{7} +218.102i q^{11} -158.792 q^{13} +76.0225i q^{17} -644.073 q^{19} -882.720i q^{23} +1176.15i q^{29} -135.186 q^{31} -626.399 q^{37} -72.2760i q^{41} +3287.18 q^{43} +2055.13i q^{47} +2544.73 q^{49} +1584.54i q^{53} -4003.49i q^{59} -3530.55 q^{61} +5196.83 q^{67} +635.751i q^{71} -5758.11 q^{73} -15338.2i q^{77} +2234.69 q^{79} -7904.44i q^{83} -12136.5i q^{89} +11167.2 q^{91} -974.647 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 54 q^{7} - 42 q^{13} - 378 q^{19} - 618 q^{31} + 96 q^{37} + 4710 q^{43} + 4128 q^{49} - 8010 q^{61} + 8634 q^{67} + 12192 q^{73} + 15360 q^{79} - 10278 q^{91} + 12294 q^{97}+O(q^{100}) \) 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
\(7\) −70.3259 −1.43522 −0.717611 − 0.696444i \(-0.754764\pi\)
−0.717611 + 0.696444i \(0.754764\pi\)
\(8\) 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
\(13\) −158.792 −0.939595 −0.469798 − 0.882774i \(-0.655673\pi\)
−0.469798 + 0.882774i \(0.655673\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 76.0225i 0.263053i 0.991313 + 0.131527i \(0.0419879\pi\)
−0.991313 + 0.131527i \(0.958012\pi\)
\(18\) 0 0
\(19\) −644.073 −1.78414 −0.892068 − 0.451902i \(-0.850746\pi\)
−0.892068 + 0.451902i \(0.850746\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 882.720i − 1.66866i −0.551267 − 0.834329i \(-0.685856\pi\)
0.551267 − 0.834329i \(-0.314144\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1176.15i 1.39852i 0.714868 + 0.699259i \(0.246487\pi\)
−0.714868 + 0.699259i \(0.753513\pi\)
\(30\) 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
\(37\) −626.399 −0.457559 −0.228780 − 0.973478i \(-0.573474\pi\)
−0.228780 + 0.973478i \(0.573474\pi\)
\(38\) 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
\(43\) 3287.18 1.77781 0.888907 − 0.458088i \(-0.151466\pi\)
0.888907 + 0.458088i \(0.151466\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2055.13i 0.930343i 0.885221 + 0.465171i \(0.154007\pi\)
−0.885221 + 0.465171i \(0.845993\pi\)
\(48\) 0 0
\(49\) 2544.73 1.05986
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 1584.54i 0.564095i 0.959400 + 0.282047i \(0.0910135\pi\)
−0.959400 + 0.282047i \(0.908986\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 4003.49i − 1.15010i −0.818120 − 0.575048i \(-0.804983\pi\)
0.818120 − 0.575048i \(-0.195017\pi\)
\(60\) 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
\(67\) 5196.83 1.15768 0.578841 − 0.815440i \(-0.303505\pi\)
0.578841 + 0.815440i \(0.303505\pi\)
\(68\) 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
\(73\) −5758.11 −1.08052 −0.540262 − 0.841497i \(-0.681675\pi\)
−0.540262 + 0.841497i \(0.681675\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 15338.2i − 2.58699i
\(78\) 0 0
\(79\) 2234.69 0.358066 0.179033 − 0.983843i \(-0.442703\pi\)
0.179033 + 0.983843i \(0.442703\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 7904.44i − 1.14740i −0.819065 − 0.573700i \(-0.805507\pi\)
0.819065 − 0.573700i \(-0.194493\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 12136.5i − 1.53219i −0.642729 − 0.766093i \(-0.722198\pi\)
0.642729 − 0.766093i \(-0.277802\pi\)
\(90\) 0 0
\(91\) 11167.2 1.34853
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −974.647 −0.103587 −0.0517933 − 0.998658i \(-0.516494\pi\)
−0.0517933 + 0.998658i \(0.516494\pi\)
\(98\) 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.g.c.701.2 yes 6
3.2 odd 2 inner 900.5.g.c.701.1 ✓ 6
5.2 odd 4 900.5.b.c.449.2 12
5.3 odd 4 900.5.b.c.449.12 12
5.4 even 2 900.5.g.d.701.6 yes 6
15.2 even 4 900.5.b.c.449.1 12
15.8 even 4 900.5.b.c.449.11 12
15.14 odd 2 900.5.g.d.701.5 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
900.5.b.c.449.1 12 15.2 even 4
900.5.b.c.449.2 12 5.2 odd 4
900.5.b.c.449.11 12 15.8 even 4
900.5.b.c.449.12 12 5.3 odd 4
900.5.g.c.701.1 ✓ 6 3.2 odd 2 inner
900.5.g.c.701.2 yes 6 1.1 even 1 trivial
900.5.g.d.701.5 yes 6 15.14 odd 2
900.5.g.d.701.6 yes 6 5.4 even 2