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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [690,2,Mod(7,690)] 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("690.7"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(690, base_ring=CyclotomicField(44)) chi = DirichletCharacter(H, H._module([0, 11, 38])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 690.w (of order \(44\), degree \(20\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [240,0,0,0,0,-24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.50967773947\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(12\) over \(\Q(\zeta_{44})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{44}]$

Embedding invariants

Embedding label 103.10
Character \(\chi\) \(=\) 690.103
Dual form 690.2.w.a.67.10

$q$-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+(0.212565 - 0.977147i) q^{2} +(-0.800541 + 0.599278i) q^{3} +(-0.909632 - 0.415415i) q^{4} +(0.475668 + 2.18489i) q^{5} +(0.415415 + 0.909632i) q^{6} +(-0.0253672 + 0.354679i) q^{7} +(-0.599278 + 0.800541i) q^{8} +(0.281733 - 0.959493i) q^{9} +(2.23607 - 0.000365709i) q^{10} +(0.177872 - 0.276775i) q^{11} +(0.977147 - 0.212565i) q^{12} +(-3.66928 + 0.262432i) q^{13} +(0.341182 + 0.100180i) q^{14} +(-1.69015 - 1.46404i) q^{15} +(0.654861 + 0.755750i) q^{16} +(-0.571079 + 1.53112i) q^{17} +(-0.877679 - 0.479249i) q^{18} +(-2.28879 + 5.01175i) q^{19} +(0.474953 - 2.18504i) q^{20} +(-0.192244 - 0.299137i) q^{21} +(-0.232640 - 0.232640i) q^{22} +(0.328862 - 4.78454i) q^{23} -1.00000i q^{24} +(-4.54748 + 2.07856i) q^{25} +(-0.523527 + 3.64121i) q^{26} +(0.349464 + 0.936950i) q^{27} +(0.170414 - 0.312090i) q^{28} +(-7.40325 + 3.38095i) q^{29} +(-1.78985 + 1.34032i) q^{30} +(1.10249 + 7.66798i) q^{31} +(0.877679 - 0.479249i) q^{32} +(0.0234708 + 0.328164i) q^{33} +(1.37474 + 0.883492i) q^{34} +(-0.787002 + 0.113285i) q^{35} +(-0.654861 + 0.755750i) q^{36} +(-2.95668 - 5.41475i) q^{37} +(4.41070 + 3.30181i) q^{38} +(2.78014 - 2.40900i) q^{39} +(-2.03415 - 0.928564i) q^{40} +(-3.28710 + 0.965180i) q^{41} +(-0.333166 + 0.124264i) q^{42} +(5.70984 + 7.62745i) q^{43} +(-0.276775 + 0.177872i) q^{44} +(2.23040 + 0.159154i) q^{45} +(-4.60530 - 1.33837i) q^{46} +(-7.08001 + 7.08001i) q^{47} +(-0.977147 - 0.212565i) q^{48} +(6.80360 + 0.978209i) q^{49} +(1.06442 + 4.88539i) q^{50} +(-0.460395 - 1.56796i) q^{51} +(3.44671 + 1.28556i) q^{52} +(-10.7666 - 0.770042i) q^{53} +(0.989821 - 0.142315i) q^{54} +(0.689330 + 0.256978i) q^{55} +(-0.268734 - 0.232859i) q^{56} +(-1.17116 - 5.38374i) q^{57} +(1.73001 + 7.95273i) q^{58} +(4.22768 + 3.66330i) q^{59} +(0.929229 + 2.03385i) q^{60} +(10.9512 - 1.57454i) q^{61} +(7.72710 + 0.552653i) q^{62} +(0.333166 + 0.124264i) q^{63} +(-0.281733 - 0.959493i) q^{64} +(-2.31874 - 7.89214i) q^{65} +(0.325654 + 0.0468219i) q^{66} +(-0.320633 - 0.0697495i) q^{67} +(1.15552 - 1.15552i) q^{68} +(2.60400 + 4.02730i) q^{69} +(-0.0565930 + 0.793097i) q^{70} +(13.0357 - 8.37754i) q^{71} +(0.599278 + 0.800541i) q^{72} +(6.56345 - 2.44804i) q^{73} +(-5.91949 + 1.73812i) q^{74} +(2.39481 - 4.38918i) q^{75} +(4.16392 - 3.60805i) q^{76} +(0.0936541 + 0.0701086i) q^{77} +(-1.76299 - 3.22868i) q^{78} +(-2.77029 + 3.19708i) q^{79} +(-1.33973 + 1.79028i) q^{80} +(-0.841254 - 0.540641i) q^{81} +(0.244399 + 3.41715i) q^{82} +(-12.4582 + 6.80271i) q^{83} +(0.0506051 + 0.351966i) q^{84} +(-3.61698 - 0.519439i) q^{85} +(8.66685 - 3.95802i) q^{86} +(3.90048 - 7.14319i) q^{87} +(0.114975 + 0.308259i) q^{88} +(0.331385 - 2.30484i) q^{89} +(0.629622 - 2.14559i) q^{90} -1.30807i q^{91} +(-2.28671 + 4.21556i) q^{92} +(-5.47784 - 5.47784i) q^{93} +(5.41324 + 8.42317i) q^{94} +(-12.0388 - 2.61683i) q^{95} +(-0.415415 + 0.909632i) q^{96} +(-8.52414 - 4.65453i) q^{97} +(2.40206 - 6.44018i) q^{98} +(-0.215451 - 0.248644i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 240 q - 24 q^{6} + 44 q^{10} - 16 q^{13} + 24 q^{16} + 44 q^{21} + 72 q^{23} + 16 q^{25} + 44 q^{28} - 16 q^{31} - 44 q^{33} - 24 q^{36} + 44 q^{37} + 88 q^{43} - 8 q^{46} + 48 q^{47} + 8 q^{50} - 16 q^{52}+ \cdots - 88 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/690\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(461\) \(511\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(1\) \(e\left(\frac{9}{22}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.212565 − 0.977147i 0.150306 − 0.690947i
\(3\) −0.800541 + 0.599278i −0.462193 + 0.345993i
\(4\) −0.909632 − 0.415415i −0.454816 − 0.207708i
\(5\) 0.475668 + 2.18489i 0.212725 + 0.977112i
\(6\) 0.415415 + 0.909632i 0.169592 + 0.371356i
\(7\) −0.0253672 + 0.354679i −0.00958789 + 0.134056i −0.999991 − 0.00414982i \(-0.998679\pi\)
0.990403 + 0.138206i \(0.0441336\pi\)
\(8\) −0.599278 + 0.800541i −0.211877 + 0.283034i
\(9\) 0.281733 − 0.959493i 0.0939109 − 0.319831i
\(10\) 2.23607 0.000365709i 0.707107 0.000115647i
\(11\) 0.177872 − 0.276775i 0.0536305 − 0.0834507i −0.813403 − 0.581701i \(-0.802387\pi\)
0.867033 + 0.498250i \(0.166024\pi\)
\(12\) 0.977147 − 0.212565i 0.282078 − 0.0613623i
\(13\) −3.66928 + 0.262432i −1.01767 + 0.0727855i −0.570196 − 0.821509i \(-0.693133\pi\)
−0.447479 + 0.894294i \(0.647678\pi\)
\(14\) 0.341182 + 0.100180i 0.0911846 + 0.0267742i
\(15\) −1.69015 − 1.46404i −0.436394 − 0.378013i
\(16\) 0.654861 + 0.755750i 0.163715 + 0.188937i
\(17\) −0.571079 + 1.53112i −0.138507 + 0.371352i −0.987621 − 0.156857i \(-0.949864\pi\)
0.849114 + 0.528209i \(0.177136\pi\)
\(18\) −0.877679 − 0.479249i −0.206871 − 0.112960i
\(19\) −2.28879 + 5.01175i −0.525085 + 1.14978i 0.442393 + 0.896821i \(0.354129\pi\)
−0.967478 + 0.252954i \(0.918598\pi\)
\(20\) 0.474953 − 2.18504i 0.106203 − 0.488591i
\(21\) −0.192244 − 0.299137i −0.0419511 − 0.0652772i
\(22\) −0.232640 − 0.232640i −0.0495990 − 0.0495990i
\(23\) 0.328862 − 4.78454i 0.0685725 − 0.997646i
\(24\) − 1.00000i − 0.204124i
\(25\) −4.54748 + 2.07856i −0.909496 + 0.415713i
\(26\) −0.523527 + 3.64121i −0.102672 + 0.714100i
\(27\) 0.349464 + 0.936950i 0.0672544 + 0.180316i
\(28\) 0.170414 − 0.312090i 0.0322052 − 0.0589794i
\(29\) −7.40325 + 3.38095i −1.37475 + 0.627827i −0.959455 − 0.281861i \(-0.909048\pi\)
−0.415293 + 0.909688i \(0.636321\pi\)
\(30\) −1.78985 + 1.34032i −0.326780 + 0.244708i
\(31\) 1.10249 + 7.66798i 0.198013 + 1.37721i 0.810040 + 0.586374i \(0.199445\pi\)
−0.612027 + 0.790837i \(0.709646\pi\)
\(32\) 0.877679 − 0.479249i 0.155153 − 0.0847201i
\(33\) 0.0234708 + 0.328164i 0.00408574 + 0.0571261i
\(34\) 1.37474 + 0.883492i 0.235766 + 0.151518i
\(35\) −0.787002 + 0.113285i −0.133028 + 0.0191487i
\(36\) −0.654861 + 0.755750i −0.109143 + 0.125958i
\(37\) −2.95668 − 5.41475i −0.486075 − 0.890179i −0.999487 − 0.0320370i \(-0.989801\pi\)
0.513412 − 0.858142i \(-0.328381\pi\)
\(38\) 4.41070 + 3.30181i 0.715510 + 0.535624i
\(39\) 2.78014 − 2.40900i 0.445179 − 0.385749i
\(40\) −2.03415 − 0.928564i −0.321627 − 0.146819i
\(41\) −3.28710 + 0.965180i −0.513359 + 0.150736i −0.528144 − 0.849155i \(-0.677112\pi\)
0.0147848 + 0.999891i \(0.495294\pi\)
\(42\) −0.333166 + 0.124264i −0.0514086 + 0.0191744i
\(43\) 5.70984 + 7.62745i 0.870742 + 1.16318i 0.985482 + 0.169780i \(0.0543057\pi\)
−0.114740 + 0.993396i \(0.536603\pi\)
\(44\) −0.276775 + 0.177872i −0.0417253 + 0.0268152i
\(45\) 2.23040 + 0.159154i 0.332488 + 0.0237253i
\(46\) −4.60530 − 1.33837i −0.679014 − 0.197333i
\(47\) −7.08001 + 7.08001i −1.03273 + 1.03273i −0.0332790 + 0.999446i \(0.510595\pi\)
−0.999446 + 0.0332790i \(0.989405\pi\)
\(48\) −0.977147 − 0.212565i −0.141039 − 0.0306812i
\(49\) 6.80360 + 0.978209i 0.971942 + 0.139744i
\(50\) 1.06442 + 4.88539i 0.150532 + 0.690898i
\(51\) −0.460395 − 1.56796i −0.0644682 − 0.219559i
\(52\) 3.44671 + 1.28556i 0.477973 + 0.178275i
\(53\) −10.7666 − 0.770042i −1.47891 − 0.105773i −0.691575 − 0.722305i \(-0.743083\pi\)
−0.787330 + 0.616531i \(0.788537\pi\)
\(54\) 0.989821 − 0.142315i 0.134698 − 0.0193666i
\(55\) 0.689330 + 0.256978i 0.0929492 + 0.0346509i
\(56\) −0.268734 − 0.232859i −0.0359110 − 0.0311171i
\(57\) −1.17116 − 5.38374i −0.155124 − 0.713094i
\(58\) 1.73001 + 7.95273i 0.227162 + 1.04425i
\(59\) 4.22768 + 3.66330i 0.550397 + 0.476921i 0.885099 − 0.465403i \(-0.154090\pi\)
−0.334703 + 0.942324i \(0.608636\pi\)
\(60\) 0.929229 + 2.03385i 0.119963 + 0.262569i
\(61\) 10.9512 − 1.57454i 1.40216 − 0.201600i 0.600614 − 0.799539i \(-0.294923\pi\)
0.801541 + 0.597939i \(0.204014\pi\)
\(62\) 7.72710 + 0.552653i 0.981342 + 0.0701870i
\(63\) 0.333166 + 0.124264i 0.0419749 + 0.0156558i
\(64\) −0.281733 − 0.959493i −0.0352166 − 0.119937i
\(65\) −2.31874 − 7.89214i −0.287605 − 0.978899i
\(66\) 0.325654 + 0.0468219i 0.0400852 + 0.00576338i
\(67\) −0.320633 − 0.0697495i −0.0391716 − 0.00852126i 0.192937 − 0.981211i \(-0.438199\pi\)
−0.232108 + 0.972690i \(0.574562\pi\)
\(68\) 1.15552 − 1.15552i 0.140128 − 0.140128i
\(69\) 2.60400 + 4.02730i 0.313485 + 0.484830i
\(70\) −0.0565930 + 0.793097i −0.00676416 + 0.0947932i
\(71\) 13.0357 − 8.37754i 1.54705 − 0.994232i 0.560999 − 0.827817i \(-0.310417\pi\)
0.986056 − 0.166415i \(-0.0532191\pi\)
\(72\) 0.599278 + 0.800541i 0.0706256 + 0.0943447i
\(73\) 6.56345 − 2.44804i 0.768194 − 0.286522i 0.0653472 − 0.997863i \(-0.479185\pi\)
0.702847 + 0.711341i \(0.251912\pi\)
\(74\) −5.91949 + 1.73812i −0.688127 + 0.202052i
\(75\) 2.39481 − 4.38918i 0.276529 − 0.506819i
\(76\) 4.16392 − 3.60805i 0.477634 − 0.413872i
\(77\) 0.0936541 + 0.0701086i 0.0106729 + 0.00798962i
\(78\) −1.76299 − 3.22868i −0.199619 − 0.365575i
\(79\) −2.77029 + 3.19708i −0.311682 + 0.359700i −0.889879 − 0.456197i \(-0.849211\pi\)
0.578197 + 0.815897i \(0.303757\pi\)
\(80\) −1.33973 + 1.79028i −0.149787 + 0.200160i
\(81\) −0.841254 − 0.540641i −0.0934726 − 0.0600712i
\(82\) 0.244399 + 3.41715i 0.0269894 + 0.377361i
\(83\) −12.4582 + 6.80271i −1.36747 + 0.746695i −0.984220 − 0.176950i \(-0.943377\pi\)
−0.383250 + 0.923645i \(0.625195\pi\)
\(84\) 0.0506051 + 0.351966i 0.00552147 + 0.0384026i
\(85\) −3.61698 − 0.519439i −0.392316 − 0.0563411i
\(86\) 8.66685 − 3.95802i 0.934571 − 0.426804i
\(87\) 3.90048 − 7.14319i 0.418175 − 0.765830i
\(88\) 0.114975 + 0.308259i 0.0122563 + 0.0328605i
\(89\) 0.331385 − 2.30484i 0.0351268 − 0.244312i −0.964691 − 0.263383i \(-0.915162\pi\)
0.999818 + 0.0190709i \(0.00607082\pi\)
\(90\) 0.629622 − 2.14559i 0.0663680 − 0.226166i
\(91\) − 1.30807i − 0.137123i
\(92\) −2.28671 + 4.21556i −0.238406 + 0.439502i
\(93\) −5.47784 − 5.47784i −0.568026 − 0.568026i
\(94\) 5.41324 + 8.42317i 0.558333 + 0.868784i
\(95\) −12.0388 − 2.61683i −1.23516 − 0.268481i
\(96\) −0.415415 + 0.909632i −0.0423981 + 0.0928389i
\(97\) −8.52414 − 4.65453i −0.865495 − 0.472596i −0.0158030 − 0.999875i \(-0.505030\pi\)
−0.849692 + 0.527279i \(0.823212\pi\)
\(98\) 2.40206 − 6.44018i 0.242645 − 0.650556i
\(99\) −0.215451 − 0.248644i −0.0216536 − 0.0249896i
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 690.2.w.a.103.10 yes 240
5.2 odd 4 inner 690.2.w.a.517.10 yes 240
23.21 odd 22 inner 690.2.w.a.343.10 yes 240
115.67 even 44 inner 690.2.w.a.67.10 ✓ 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
690.2.w.a.67.10 ✓ 240 115.67 even 44 inner
690.2.w.a.103.10 yes 240 1.1 even 1 trivial
690.2.w.a.343.10 yes 240 23.21 odd 22 inner
690.2.w.a.517.10 yes 240 5.2 odd 4 inner