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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 103.8
Character \(\chi\) \(=\) 690.103
Dual form 690.2.w.a.67.8

$q$-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} +(-1.50463 - 1.65412i) q^{5} +(0.415415 + 0.909632i) q^{6} +(0.128228 - 1.79287i) q^{7} +(-0.599278 + 0.800541i) q^{8} +(0.281733 - 0.959493i) q^{9} +(-1.93615 + 1.11863i) q^{10} +(-2.47701 + 3.85430i) q^{11} +(0.977147 - 0.212565i) q^{12} +(-0.739079 + 0.0528599i) q^{13} +(-1.72464 - 0.506400i) q^{14} +(2.19579 + 0.422501i) q^{15} +(0.654861 + 0.755750i) q^{16} +(2.00229 - 5.36835i) q^{17} +(-0.877679 - 0.479249i) q^{18} +(-2.88912 + 6.32629i) q^{19} +(0.681510 + 2.12968i) q^{20} +(0.971774 + 1.51211i) q^{21} +(3.23969 + 3.23969i) q^{22} +(-2.63754 + 4.00542i) q^{23} -1.00000i q^{24} +(-0.472207 + 4.97765i) q^{25} +(-0.105451 + 0.733425i) q^{26} +(0.349464 + 0.936950i) q^{27} +(-0.861425 + 1.57758i) q^{28} +(-3.53622 + 1.61494i) q^{29} +(0.879594 - 2.05580i) q^{30} +(-0.960321 - 6.67918i) q^{31} +(0.877679 - 0.479249i) q^{32} +(-0.326849 - 4.56994i) q^{33} +(-4.82005 - 3.09766i) q^{34} +(-3.15855 + 2.48549i) q^{35} +(-0.654861 + 0.755750i) q^{36} +(4.65449 + 8.52407i) q^{37} +(5.56759 + 4.16784i) q^{38} +(0.559985 - 0.485230i) q^{39} +(2.22588 - 0.213239i) q^{40} +(-0.551116 + 0.161822i) q^{41} +(1.68412 - 0.628144i) q^{42} +(2.66970 + 3.56631i) q^{43} +(3.85430 - 2.47701i) q^{44} +(-2.01102 + 0.977659i) q^{45} +(3.35324 + 3.42867i) q^{46} +(-9.11266 + 9.11266i) q^{47} +(-0.977147 - 0.212565i) q^{48} +(3.73081 + 0.536410i) q^{49} +(4.76352 + 1.51949i) q^{50} +(1.61422 + 5.49751i) q^{51} +(0.694248 + 0.258941i) q^{52} +(-10.2489 - 0.733017i) q^{53} +(0.989821 - 0.142315i) q^{54} +(10.1024 - 1.70202i) q^{55} +(1.35842 + 1.17708i) q^{56} +(-1.47834 - 6.79584i) q^{57} +(0.826355 + 3.79869i) q^{58} +(-9.37125 - 8.12024i) q^{59} +(-1.82185 - 1.29648i) q^{60} +(-12.6060 + 1.81246i) q^{61} +(-6.73067 - 0.481387i) q^{62} +(-1.68412 - 0.628144i) q^{63} +(-0.281733 - 0.959493i) q^{64} +(1.19947 + 1.14299i) q^{65} +(-4.53498 - 0.652032i) q^{66} +(0.0420613 + 0.00914988i) q^{67} +(-4.05144 + 4.05144i) q^{68} +(-0.288902 - 4.78712i) q^{69} +(1.75729 + 3.61470i) q^{70} +(-7.72521 + 4.96469i) q^{71} +(0.599278 + 0.800541i) q^{72} +(8.74296 - 3.26096i) q^{73} +(9.31865 - 2.73620i) q^{74} +(-2.60497 - 4.26780i) q^{75} +(5.25607 - 4.55441i) q^{76} +(6.59264 + 4.93518i) q^{77} +(-0.355107 - 0.650331i) q^{78} +(3.32560 - 3.83795i) q^{79} +(0.264778 - 2.22034i) q^{80} +(-0.841254 - 0.540641i) q^{81} +(0.0409760 + 0.572919i) q^{82} +(12.9500 - 7.07122i) q^{83} +(-0.255804 - 1.77915i) q^{84} +(-11.8926 + 4.76533i) q^{85} +(4.05229 - 1.85062i) q^{86} +(1.86310 - 3.41201i) q^{87} +(-1.60111 - 4.29275i) q^{88} +(0.625926 - 4.35341i) q^{89} +(0.527844 + 2.17287i) q^{90} +1.33185i q^{91} +(4.06310 - 2.54779i) q^{92} +(4.77146 + 4.77146i) q^{93} +(6.96737 + 10.8414i) q^{94} +(14.8115 - 4.73975i) q^{95} +(-0.415415 + 0.909632i) q^{96} +(-0.130446 - 0.0712288i) q^{97} +(1.31719 - 3.53153i) q^{98} +(3.00032 + 3.46256i) 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
\(5\) −1.50463 − 1.65412i −0.672889 − 0.739744i
\(6\) 0.415415 + 0.909632i 0.169592 + 0.371356i
\(7\) 0.128228 − 1.79287i 0.0484658 − 0.677641i −0.913910 − 0.405918i \(-0.866952\pi\)
0.962375 − 0.271723i \(-0.0875935\pi\)
\(8\) −0.599278 + 0.800541i −0.211877 + 0.283034i
\(9\) 0.281733 − 0.959493i 0.0939109 − 0.319831i
\(10\) −1.93615 + 1.11863i −0.612263 + 0.353742i
\(11\) −2.47701 + 3.85430i −0.746847 + 1.16212i 0.234918 + 0.972015i \(0.424518\pi\)
−0.981765 + 0.190101i \(0.939119\pi\)
\(12\) 0.977147 − 0.212565i 0.282078 − 0.0613623i
\(13\) −0.739079 + 0.0528599i −0.204984 + 0.0146607i −0.173453 − 0.984842i \(-0.555493\pi\)
−0.0315302 + 0.999503i \(0.510038\pi\)
\(14\) −1.72464 − 0.506400i −0.460929 − 0.135341i
\(15\) 2.19579 + 0.422501i 0.566951 + 0.109089i
\(16\) 0.654861 + 0.755750i 0.163715 + 0.188937i
\(17\) 2.00229 − 5.36835i 0.485627 − 1.30202i −0.431029 − 0.902338i \(-0.641849\pi\)
0.916656 − 0.399678i \(-0.130878\pi\)
\(18\) −0.877679 − 0.479249i −0.206871 − 0.112960i
\(19\) −2.88912 + 6.32629i −0.662810 + 1.45135i 0.217071 + 0.976156i \(0.430350\pi\)
−0.879881 + 0.475195i \(0.842378\pi\)
\(20\) 0.681510 + 2.12968i 0.152390 + 0.476211i
\(21\) 0.971774 + 1.51211i 0.212058 + 0.329969i
\(22\) 3.23969 + 3.23969i 0.690705 + 0.690705i
\(23\) −2.63754 + 4.00542i −0.549965 + 0.835188i
\(24\) − 1.00000i − 0.204124i
\(25\) −0.472207 + 4.97765i −0.0944413 + 0.995530i
\(26\) −0.105451 + 0.733425i −0.0206806 + 0.143836i
\(27\) 0.349464 + 0.936950i 0.0672544 + 0.180316i
\(28\) −0.861425 + 1.57758i −0.162794 + 0.298135i
\(29\) −3.53622 + 1.61494i −0.656660 + 0.299887i −0.715740 − 0.698367i \(-0.753911\pi\)
0.0590799 + 0.998253i \(0.481183\pi\)
\(30\) 0.879594 − 2.05580i 0.160591 − 0.375336i
\(31\) −0.960321 − 6.67918i −0.172479 − 1.19962i −0.873626 − 0.486599i \(-0.838237\pi\)
0.701147 − 0.713017i \(-0.252672\pi\)
\(32\) 0.877679 − 0.479249i 0.155153 − 0.0847201i
\(33\) −0.326849 − 4.56994i −0.0568971 − 0.795525i
\(34\) −4.82005 − 3.09766i −0.826631 − 0.531244i
\(35\) −3.15855 + 2.48549i −0.533892 + 0.420125i
\(36\) −0.654861 + 0.755750i −0.109143 + 0.125958i
\(37\) 4.65449 + 8.52407i 0.765194 + 1.40135i 0.911158 + 0.412058i \(0.135190\pi\)
−0.145964 + 0.989290i \(0.546628\pi\)
\(38\) 5.56759 + 4.16784i 0.903182 + 0.676114i
\(39\) 0.559985 − 0.485230i 0.0896694 − 0.0776990i
\(40\) 2.22588 − 0.213239i 0.351942 − 0.0337161i
\(41\) −0.551116 + 0.161822i −0.0860699 + 0.0252724i −0.324484 − 0.945891i \(-0.605191\pi\)
0.238414 + 0.971164i \(0.423372\pi\)
\(42\) 1.68412 − 0.628144i 0.259865 − 0.0969247i
\(43\) 2.66970 + 3.56631i 0.407126 + 0.543857i 0.956505 − 0.291717i \(-0.0942265\pi\)
−0.549379 + 0.835573i \(0.685136\pi\)
\(44\) 3.85430 − 2.47701i 0.581058 − 0.373423i
\(45\) −2.01102 + 0.977659i −0.299785 + 0.145741i
\(46\) 3.35324 + 3.42867i 0.494408 + 0.505531i
\(47\) −9.11266 + 9.11266i −1.32922 + 1.32922i −0.423166 + 0.906052i \(0.639081\pi\)
−0.906052 + 0.423166i \(0.860919\pi\)
\(48\) −0.977147 − 0.212565i −0.141039 − 0.0306812i
\(49\) 3.73081 + 0.536410i 0.532974 + 0.0766300i
\(50\) 4.76352 + 1.51949i 0.673664 + 0.214889i
\(51\) 1.61422 + 5.49751i 0.226035 + 0.769806i
\(52\) 0.694248 + 0.258941i 0.0962749 + 0.0359087i
\(53\) −10.2489 − 0.733017i −1.40780 − 0.100688i −0.653354 − 0.757052i \(-0.726639\pi\)
−0.754444 + 0.656365i \(0.772093\pi\)
\(54\) 0.989821 − 0.142315i 0.134698 − 0.0193666i
\(55\) 10.1024 − 1.70202i 1.36221 − 0.229500i
\(56\) 1.35842 + 1.17708i 0.181527 + 0.157294i
\(57\) −1.47834 − 6.79584i −0.195812 − 0.900131i
\(58\) 0.826355 + 3.79869i 0.108506 + 0.498793i
\(59\) −9.37125 − 8.12024i −1.22003 − 1.05716i −0.996595 − 0.0824468i \(-0.973727\pi\)
−0.223438 − 0.974718i \(-0.571728\pi\)
\(60\) −1.82185 − 1.29648i −0.235200 − 0.167375i
\(61\) −12.6060 + 1.81246i −1.61403 + 0.232062i −0.889417 − 0.457096i \(-0.848890\pi\)
−0.724611 + 0.689158i \(0.757981\pi\)
\(62\) −6.73067 − 0.481387i −0.854796 − 0.0611362i
\(63\) −1.68412 − 0.628144i −0.212179 − 0.0791387i
\(64\) −0.281733 − 0.959493i −0.0352166 − 0.119937i
\(65\) 1.19947 + 1.14299i 0.148776 + 0.141770i
\(66\) −4.53498 − 0.652032i −0.558218 − 0.0802596i
\(67\) 0.0420613 + 0.00914988i 0.00513861 + 0.00111784i 0.215134 − 0.976585i \(-0.430981\pi\)
−0.209995 + 0.977702i \(0.567345\pi\)
\(68\) −4.05144 + 4.05144i −0.491309 + 0.491309i
\(69\) −0.288902 − 4.78712i −0.0347797 − 0.576302i
\(70\) 1.75729 + 3.61470i 0.210036 + 0.432039i
\(71\) −7.72521 + 4.96469i −0.916814 + 0.589201i −0.911732 − 0.410786i \(-0.865254\pi\)
−0.00508265 + 0.999987i \(0.501618\pi\)
\(72\) 0.599278 + 0.800541i 0.0706256 + 0.0943447i
\(73\) 8.74296 − 3.26096i 1.02329 − 0.381666i 0.218884 − 0.975751i \(-0.429758\pi\)
0.804402 + 0.594085i \(0.202486\pi\)
\(74\) 9.31865 − 2.73620i 1.08327 − 0.318077i
\(75\) −2.60497 − 4.26780i −0.300797 − 0.492803i
\(76\) 5.25607 − 4.55441i 0.602913 − 0.522427i
\(77\) 6.59264 + 4.93518i 0.751300 + 0.562417i
\(78\) −0.355107 − 0.650331i −0.0402080 − 0.0736355i
\(79\) 3.32560 − 3.83795i 0.374159 − 0.431803i −0.537174 − 0.843471i \(-0.680508\pi\)
0.911334 + 0.411668i \(0.135054\pi\)
\(80\) 0.264778 − 2.22034i 0.0296031 − 0.248241i
\(81\) −0.841254 − 0.540641i −0.0934726 − 0.0600712i
\(82\) 0.0409760 + 0.572919i 0.00452504 + 0.0632684i
\(83\) 12.9500 − 7.07122i 1.42145 − 0.776168i 0.429528 − 0.903054i \(-0.358680\pi\)
0.991917 + 0.126886i \(0.0404982\pi\)
\(84\) −0.255804 − 1.77915i −0.0279105 − 0.194121i
\(85\) −11.8926 + 4.76533i −1.28993 + 0.516873i
\(86\) 4.05229 − 1.85062i 0.436970 − 0.199557i
\(87\) 1.86310 − 3.41201i 0.199745 − 0.365805i
\(88\) −1.60111 − 4.29275i −0.170679 − 0.457608i
\(89\) 0.625926 − 4.35341i 0.0663481 − 0.461461i −0.929380 − 0.369125i \(-0.879657\pi\)
0.995728 − 0.0923361i \(-0.0294334\pi\)
\(90\) 0.527844 + 2.17287i 0.0556396 + 0.229041i
\(91\) 1.33185i 0.139616i
\(92\) 4.06310 − 2.54779i 0.423607 − 0.265625i
\(93\) 4.77146 + 4.77146i 0.494777 + 0.494777i
\(94\) 6.96737 + 10.8414i 0.718629 + 1.11821i
\(95\) 14.8115 − 4.73975i 1.51962 − 0.486288i
\(96\) −0.415415 + 0.909632i −0.0423981 + 0.0928389i
\(97\) −0.130446 − 0.0712288i −0.0132448 − 0.00723219i 0.472613 − 0.881270i \(-0.343311\pi\)
−0.485858 + 0.874038i \(0.661493\pi\)
\(98\) 1.31719 − 3.53153i 0.133057 − 0.356739i
\(99\) 3.00032 + 3.46256i 0.301544 + 0.348000i
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.103.8 yes 240
5.2 odd 4 inner 690.2.w.a.517.9 yes 240
23.21 odd 22 inner 690.2.w.a.343.9 yes 240
115.67 even 44 inner 690.2.w.a.67.8 ✓ 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
690.2.w.a.67.8 ✓ 240 115.67 even 44 inner
690.2.w.a.103.8 yes 240 1.1 even 1 trivial
690.2.w.a.343.9 yes 240 23.21 odd 22 inner
690.2.w.a.517.9 yes 240 5.2 odd 4 inner