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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [588,6,Mod(361,588)] 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("588.361"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(588, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 588.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,9,0,-6,0,0,0,-81,0,108] 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: \(94.3056860500\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 361.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 588.361
Dual form 588.6.i.f.373.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(4.50000 - 7.79423i) q^{3} +(-3.00000 - 5.19615i) q^{5} +(-40.5000 - 70.1481i) q^{9} +(54.0000 - 93.5307i) q^{11} -346.000 q^{13} -54.0000 q^{15} +(699.000 - 1210.70i) q^{17} +(506.000 + 876.418i) q^{19} +(768.000 + 1330.22i) q^{23} +(1544.50 - 2675.15i) q^{25} -729.000 q^{27} -3762.00 q^{29} +(368.000 - 637.395i) q^{31} +(-486.000 - 841.777i) q^{33} +(-1027.00 - 1778.82i) q^{37} +(-1557.00 + 2696.80i) q^{39} -15534.0 q^{41} +11036.0 q^{43} +(-243.000 + 420.888i) q^{45} +(-2280.00 - 3949.08i) q^{47} +(-6291.00 - 10896.3i) q^{51} +(3981.00 - 6895.29i) q^{53} -648.000 q^{55} +9108.00 q^{57} +(3510.00 - 6079.50i) q^{59} +(-13435.0 - 23270.1i) q^{61} +(1038.00 + 1797.87i) q^{65} +(-26074.0 + 45161.5i) q^{67} +13824.0 q^{69} -2544.00 q^{71} +(4883.00 - 8457.60i) q^{73} +(-13900.5 - 24076.4i) q^{75} +(-34336.0 - 59471.7i) q^{79} +(-3280.50 + 5681.99i) q^{81} -61668.0 q^{83} -8388.00 q^{85} +(-16929.0 + 29321.9i) q^{87} +(20727.0 + 35900.2i) q^{89} +(-3312.00 - 5736.55i) q^{93} +(3036.00 - 5258.51i) q^{95} -111262. q^{97} -8748.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 9 q^{3} - 6 q^{5} - 81 q^{9} + 108 q^{11} - 692 q^{13} - 108 q^{15} + 1398 q^{17} + 1012 q^{19} + 1536 q^{23} + 3089 q^{25} - 1458 q^{27} - 7524 q^{29} + 736 q^{31} - 972 q^{33} - 2054 q^{37} - 3114 q^{39}+ \cdots - 17496 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/588\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(295\) \(493\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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
\(3\) 4.50000 − 7.79423i 0.288675 − 0.500000i
\(4\) 0 0
\(5\) −3.00000 − 5.19615i −0.0536656 − 0.0929516i 0.837945 − 0.545755i \(-0.183757\pi\)
−0.891610 + 0.452804i \(0.850424\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −40.5000 − 70.1481i −0.166667 − 0.288675i
\(10\) 0 0
\(11\) 54.0000 − 93.5307i 0.134559 − 0.233063i −0.790870 − 0.611984i \(-0.790372\pi\)
0.925429 + 0.378921i \(0.123705\pi\)
\(12\) 0 0
\(13\) −346.000 −0.567829 −0.283915 − 0.958850i \(-0.591633\pi\)
−0.283915 + 0.958850i \(0.591633\pi\)
\(14\) 0 0
\(15\) −54.0000 −0.0619677
\(16\) 0 0
\(17\) 699.000 − 1210.70i 0.586617 − 1.01605i −0.408054 − 0.912958i \(-0.633793\pi\)
0.994672 − 0.103093i \(-0.0328740\pi\)
\(18\) 0 0
\(19\) 506.000 + 876.418i 0.321563 + 0.556964i 0.980811 − 0.194962i \(-0.0624584\pi\)
−0.659247 + 0.751926i \(0.729125\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 768.000 + 1330.22i 0.302720 + 0.524327i 0.976751 − 0.214376i \(-0.0687718\pi\)
−0.674031 + 0.738703i \(0.735439\pi\)
\(24\) 0 0
\(25\) 1544.50 − 2675.15i 0.494240 − 0.856049i
\(26\) 0 0
\(27\) −729.000 −0.192450
\(28\) 0 0
\(29\) −3762.00 −0.830661 −0.415330 − 0.909671i \(-0.636334\pi\)
−0.415330 + 0.909671i \(0.636334\pi\)
\(30\) 0 0
\(31\) 368.000 − 637.395i 0.0687771 − 0.119125i −0.829586 − 0.558379i \(-0.811424\pi\)
0.898363 + 0.439253i \(0.144757\pi\)
\(32\) 0 0
\(33\) −486.000 − 841.777i −0.0776875 − 0.134559i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −1027.00 − 1778.82i −0.123329 − 0.213613i 0.797749 − 0.602989i \(-0.206024\pi\)
−0.921079 + 0.389377i \(0.872690\pi\)
\(38\) 0 0
\(39\) −1557.00 + 2696.80i −0.163918 + 0.283915i
\(40\) 0 0
\(41\) −15534.0 −1.44319 −0.721595 − 0.692315i \(-0.756591\pi\)
−0.721595 + 0.692315i \(0.756591\pi\)
\(42\) 0 0
\(43\) 11036.0 0.910208 0.455104 − 0.890438i \(-0.349602\pi\)
0.455104 + 0.890438i \(0.349602\pi\)
\(44\) 0 0
\(45\) −243.000 + 420.888i −0.0178885 + 0.0309839i
\(46\) 0 0
\(47\) −2280.00 − 3949.08i −0.150553 − 0.260766i 0.780878 − 0.624684i \(-0.214772\pi\)
−0.931431 + 0.363918i \(0.881439\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −6291.00 − 10896.3i −0.338684 − 0.586617i
\(52\) 0 0
\(53\) 3981.00 − 6895.29i 0.194672 − 0.337181i −0.752121 − 0.659025i \(-0.770969\pi\)
0.946793 + 0.321844i \(0.104303\pi\)
\(54\) 0 0
\(55\) −648.000 −0.0288847
\(56\) 0 0
\(57\) 9108.00 0.371309
\(58\) 0 0
\(59\) 3510.00 − 6079.50i 0.131274 − 0.227372i −0.792894 − 0.609359i \(-0.791427\pi\)
0.924168 + 0.381987i \(0.124760\pi\)
\(60\) 0 0
\(61\) −13435.0 − 23270.1i −0.462288 − 0.800707i 0.536786 − 0.843718i \(-0.319638\pi\)
−0.999075 + 0.0430112i \(0.986305\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1038.00 + 1797.87i 0.0304729 + 0.0527806i
\(66\) 0 0
\(67\) −26074.0 + 45161.5i −0.709612 + 1.22908i 0.255390 + 0.966838i \(0.417796\pi\)
−0.965001 + 0.262245i \(0.915537\pi\)
\(68\) 0 0
\(69\) 13824.0 0.349551
\(70\) 0 0
\(71\) −2544.00 −0.0598923 −0.0299462 − 0.999552i \(-0.509534\pi\)
−0.0299462 + 0.999552i \(0.509534\pi\)
\(72\) 0 0
\(73\) 4883.00 − 8457.60i 0.107246 − 0.185755i −0.807408 − 0.589994i \(-0.799130\pi\)
0.914653 + 0.404239i \(0.132464\pi\)
\(74\) 0 0
\(75\) −13900.5 − 24076.4i −0.285350 − 0.494240i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −34336.0 − 59471.7i −0.618988 − 1.07212i −0.989671 − 0.143359i \(-0.954210\pi\)
0.370683 − 0.928759i \(-0.379124\pi\)
\(80\) 0 0
\(81\) −3280.50 + 5681.99i −0.0555556 + 0.0962250i
\(82\) 0 0
\(83\) −61668.0 −0.982573 −0.491286 − 0.870998i \(-0.663473\pi\)
−0.491286 + 0.870998i \(0.663473\pi\)
\(84\) 0 0
\(85\) −8388.00 −0.125925
\(86\) 0 0
\(87\) −16929.0 + 29321.9i −0.239791 + 0.415330i
\(88\) 0 0
\(89\) 20727.0 + 35900.2i 0.277371 + 0.480421i 0.970731 − 0.240171i \(-0.0772035\pi\)
−0.693359 + 0.720592i \(0.743870\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −3312.00 − 5736.55i −0.0397085 − 0.0687771i
\(94\) 0 0
\(95\) 3036.00 − 5258.51i 0.0345138 − 0.0597797i
\(96\) 0 0
\(97\) −111262. −1.20065 −0.600327 − 0.799755i \(-0.704963\pi\)
−0.600327 + 0.799755i \(0.704963\pi\)
\(98\) 0 0
\(99\) −8748.00 −0.0897059
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 588.6.i.f.361.1 2
7.2 even 3 inner 588.6.i.f.373.1 2
7.3 odd 6 588.6.a.e.1.1 1
7.4 even 3 84.6.a.a.1.1 ✓ 1
7.5 odd 6 588.6.i.b.373.1 2
7.6 odd 2 588.6.i.b.361.1 2
21.11 odd 6 252.6.a.b.1.1 1
28.11 odd 6 336.6.a.n.1.1 1
84.11 even 6 1008.6.a.o.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.6.a.a.1.1 ✓ 1 7.4 even 3
252.6.a.b.1.1 1 21.11 odd 6
336.6.a.n.1.1 1 28.11 odd 6
588.6.a.e.1.1 1 7.3 odd 6
588.6.i.b.361.1 2 7.6 odd 2
588.6.i.b.373.1 2 7.5 odd 6
588.6.i.f.361.1 2 1.1 even 1 trivial
588.6.i.f.373.1 2 7.2 even 3 inner
1008.6.a.o.1.1 1 84.11 even 6