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Properties

Label 700.2.r.d.149.6
Level $700$
Weight $2$
Character 700.149
Analytic conductor $5.590$
Analytic rank $0$
Dimension $12$
Inner twists $4$

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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] = [700,2,Mod(149,700)] 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("700.149"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(700, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 700.r (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,0,0,0,0,0,16,0,-8] 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: \(5.58952814149\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: 12.0.13026266817859584.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 9x^{10} + 59x^{8} - 180x^{6} + 403x^{4} - 198x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 149.6
Root \(0.617942 + 0.356769i\) of defining polynomial
Character \(\chi\) \(=\) 700.149
Dual form 700.2.r.d.249.6

$q$-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)
 
\(f(q)\) \(=\) \(q+(2.77509 - 1.60220i) q^{3} +(-2.15715 - 1.53189i) q^{7} +(3.63409 - 6.29444i) q^{9} +(-2.10220 - 3.64112i) q^{11} -0.204402i q^{13} +(-4.38537 + 2.53189i) q^{17} +(0.531894 - 0.921267i) q^{19} +(-8.44070 - 0.794959i) q^{21} +(1.85383 + 1.07031i) q^{23} -13.6770i q^{27} +7.47259 q^{29} +(4.23630 + 7.33748i) q^{31} +(-11.6676 - 6.73630i) q^{33} +(9.19130 + 5.30660i) q^{37} +(-0.327492 - 0.567233i) q^{39} +10.5494 q^{41} -8.26819i q^{43} +(-2.83033 - 1.63409i) q^{47} +(2.30660 + 6.60905i) q^{49} +(-8.11320 + 14.0525i) q^{51} +(-4.91642 + 2.83850i) q^{53} -3.40880i q^{57} +(0.602201 + 1.04304i) q^{59} +(-0.827492 + 1.43326i) q^{61} +(-17.4817 + 8.01100i) q^{63} +(-10.7463 + 6.20440i) q^{67} +6.85939 q^{69} -0.591197 q^{71} +(3.46410 - 2.00000i) q^{73} +(-1.04304 + 11.0748i) q^{77} +(3.27471 - 5.67196i) q^{79} +(-11.0110 - 19.0716i) q^{81} -3.88139i q^{83} +(20.7371 - 11.9726i) q^{87} +(-4.63409 + 8.02649i) q^{89} +(-0.313121 + 0.440925i) q^{91} +(23.5122 + 13.5748i) q^{93} +1.33198i q^{97} -30.5584 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 16 q^{9} - 8 q^{11} - 4 q^{19} - 22 q^{21} + 6 q^{31} - 28 q^{39} + 44 q^{41} - 24 q^{49} + 6 q^{51} - 10 q^{59} - 34 q^{61} + 96 q^{69} - 76 q^{71} - 2 q^{79} - 46 q^{81} - 28 q^{89} + 34 q^{91}+ \cdots - 84 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/700\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(351\) \(477\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(-1\)

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
\(3\) 2.77509 − 1.60220i 1.60220 − 0.925031i 0.611155 − 0.791511i \(-0.290705\pi\)
0.991046 − 0.133520i \(-0.0426280\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −2.15715 − 1.53189i −0.815327 − 0.579001i
\(8\) 0 0
\(9\) 3.63409 − 6.29444i 1.21136 − 2.09815i
\(10\) 0 0
\(11\) −2.10220 − 3.64112i −0.633837 − 1.09784i −0.986760 − 0.162186i \(-0.948146\pi\)
0.352923 − 0.935652i \(-0.385188\pi\)
\(12\) 0 0
\(13\) − 0.204402i − 0.0566908i −0.999598 − 0.0283454i \(-0.990976\pi\)
0.999598 − 0.0283454i \(-0.00902383\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.38537 + 2.53189i −1.06361 + 0.614074i −0.926428 − 0.376472i \(-0.877137\pi\)
−0.137180 + 0.990546i \(0.543804\pi\)
\(18\) 0 0
\(19\) 0.531894 − 0.921267i 0.122025 − 0.211353i −0.798541 − 0.601940i \(-0.794395\pi\)
0.920566 + 0.390587i \(0.127728\pi\)
\(20\) 0 0
\(21\) −8.44070 − 0.794959i −1.84191 − 0.173474i
\(22\) 0 0
\(23\) 1.85383 + 1.07031i 0.386549 + 0.223174i 0.680664 − 0.732596i \(-0.261691\pi\)
−0.294115 + 0.955770i \(0.595025\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 13.6770i − 2.63214i
\(28\) 0 0
\(29\) 7.47259 1.38763 0.693813 − 0.720156i \(-0.255930\pi\)
0.693813 + 0.720156i \(0.255930\pi\)
\(30\) 0 0
\(31\) 4.23630 + 7.33748i 0.760861 + 1.31785i 0.942407 + 0.334468i \(0.108557\pi\)
−0.181546 + 0.983382i \(0.558110\pi\)
\(32\) 0 0
\(33\) −11.6676 − 6.73630i −2.03107 − 1.17264i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 9.19130 + 5.30660i 1.51104 + 0.872400i 0.999917 + 0.0128933i \(0.00410417\pi\)
0.511124 + 0.859507i \(0.329229\pi\)
\(38\) 0 0
\(39\) −0.327492 − 0.567233i −0.0524407 − 0.0908300i
\(40\) 0 0
\(41\) 10.5494 1.64754 0.823771 − 0.566923i \(-0.191866\pi\)
0.823771 + 0.566923i \(0.191866\pi\)
\(42\) 0 0
\(43\) − 8.26819i − 1.26089i −0.776235 − 0.630444i \(-0.782873\pi\)
0.776235 − 0.630444i \(-0.217127\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2.83033 − 1.63409i −0.412847 − 0.238357i 0.279165 − 0.960243i \(-0.409942\pi\)
−0.692012 + 0.721886i \(0.743276\pi\)
\(48\) 0 0
\(49\) 2.30660 + 6.60905i 0.329515 + 0.944150i
\(50\) 0 0
\(51\) −8.11320 + 14.0525i −1.13608 + 1.96774i
\(52\) 0 0
\(53\) −4.91642 + 2.83850i −0.675322 + 0.389897i −0.798090 − 0.602538i \(-0.794156\pi\)
0.122768 + 0.992435i \(0.460823\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 3.40880i − 0.451507i
\(58\) 0 0
\(59\) 0.602201 + 1.04304i 0.0783999 + 0.135793i 0.902560 − 0.430565i \(-0.141686\pi\)
−0.824160 + 0.566357i \(0.808352\pi\)
\(60\) 0 0
\(61\) −0.827492 + 1.43326i −0.105950 + 0.183510i −0.914126 − 0.405431i \(-0.867122\pi\)
0.808176 + 0.588941i \(0.200455\pi\)
\(62\) 0 0
\(63\) −17.4817 + 8.01100i −2.20249 + 1.00929i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −10.7463 + 6.20440i −1.31287 + 0.757988i −0.982571 − 0.185887i \(-0.940484\pi\)
−0.330303 + 0.943875i \(0.607151\pi\)
\(68\) 0 0
\(69\) 6.85939 0.825773
\(70\) 0 0
\(71\) −0.591197 −0.0701622 −0.0350811 − 0.999384i \(-0.511169\pi\)
−0.0350811 + 0.999384i \(0.511169\pi\)
\(72\) 0 0
\(73\) 3.46410 − 2.00000i 0.405442 − 0.234082i −0.283387 − 0.959006i \(-0.591458\pi\)
0.688830 + 0.724923i \(0.258125\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1.04304 + 11.0748i −0.118866 + 1.26209i
\(78\) 0 0
\(79\) 3.27471 − 5.67196i 0.368433 − 0.638146i −0.620887 − 0.783900i \(-0.713228\pi\)
0.989321 + 0.145754i \(0.0465609\pi\)
\(80\) 0 0
\(81\) −11.0110 − 19.0716i −1.22344 − 2.11907i
\(82\) 0 0
\(83\) − 3.88139i − 0.426038i −0.977048 − 0.213019i \(-0.931670\pi\)
0.977048 − 0.213019i \(-0.0683297\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 20.7371 − 11.9726i 2.22325 − 1.28360i
\(88\) 0 0
\(89\) −4.63409 + 8.02649i −0.491213 + 0.850806i −0.999949 − 0.0101167i \(-0.996780\pi\)
0.508736 + 0.860923i \(0.330113\pi\)
\(90\) 0 0
\(91\) −0.313121 + 0.440925i −0.0328241 + 0.0462215i
\(92\) 0 0
\(93\) 23.5122 + 13.5748i 2.43810 + 1.40764i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1.33198i 0.135242i 0.997711 + 0.0676209i \(0.0215408\pi\)
−0.997711 + 0.0676209i \(0.978459\pi\)
\(98\) 0 0
\(99\) −30.5584 −3.07123
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 700.2.r.d.149.6 12
5.2 odd 4 700.2.i.d.401.1 ✓ 6
5.3 odd 4 700.2.i.e.401.3 yes 6
5.4 even 2 inner 700.2.r.d.149.1 12
7.2 even 3 4900.2.e.t.2549.6 6
7.4 even 3 inner 700.2.r.d.249.1 12
7.5 odd 6 4900.2.e.s.2549.1 6
35.2 odd 12 4900.2.a.bc.1.3 3
35.4 even 6 inner 700.2.r.d.249.6 12
35.9 even 6 4900.2.e.t.2549.1 6
35.12 even 12 4900.2.a.bb.1.1 3
35.18 odd 12 700.2.i.e.501.3 yes 6
35.19 odd 6 4900.2.e.s.2549.6 6
35.23 odd 12 4900.2.a.ba.1.1 3
35.32 odd 12 700.2.i.d.501.1 yes 6
35.33 even 12 4900.2.a.bd.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
700.2.i.d.401.1 ✓ 6 5.2 odd 4
700.2.i.d.501.1 yes 6 35.32 odd 12
700.2.i.e.401.3 yes 6 5.3 odd 4
700.2.i.e.501.3 yes 6 35.18 odd 12
700.2.r.d.149.1 12 5.4 even 2 inner
700.2.r.d.149.6 12 1.1 even 1 trivial
700.2.r.d.249.1 12 7.4 even 3 inner
700.2.r.d.249.6 12 35.4 even 6 inner
4900.2.a.ba.1.1 3 35.23 odd 12
4900.2.a.bb.1.1 3 35.12 even 12
4900.2.a.bc.1.3 3 35.2 odd 12
4900.2.a.bd.1.3 3 35.33 even 12
4900.2.e.s.2549.1 6 7.5 odd 6
4900.2.e.s.2549.6 6 35.19 odd 6
4900.2.e.t.2549.1 6 35.9 even 6
4900.2.e.t.2549.6 6 7.2 even 3