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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 217.4
Character \(\chi\) \(=\) 690.217
Dual form 690.2.w.a.283.4

$q$-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+(-0.0713392 + 0.997452i) q^{2} +(-0.977147 + 0.212565i) q^{3} +(-0.989821 - 0.142315i) q^{4} +(-0.0118400 + 2.23604i) q^{5} +(-0.142315 - 0.989821i) q^{6} +(1.21462 - 0.663233i) q^{7} +(0.212565 - 0.977147i) q^{8} +(0.909632 - 0.415415i) q^{9} +(-2.22949 - 0.171327i) q^{10} +(3.43475 + 2.97623i) q^{11} +(0.997452 - 0.0713392i) q^{12} +(-0.0572451 + 0.104837i) q^{13} +(0.574893 + 1.25884i) q^{14} +(-0.463734 - 2.18745i) q^{15} +(0.959493 + 0.281733i) q^{16} +(2.58366 + 1.93410i) q^{17} +(0.349464 + 0.936950i) q^{18} +(-0.165605 + 1.15181i) q^{19} +(0.329941 - 2.21159i) q^{20} +(-1.04588 + 0.906262i) q^{21} +(-3.21368 + 3.21368i) q^{22} +(-4.43205 + 1.83219i) q^{23} +1.00000i q^{24} +(-4.99972 - 0.0529496i) q^{25} +(-0.100486 - 0.0645782i) q^{26} +(-0.800541 + 0.599278i) q^{27} +(-1.29664 + 0.483624i) q^{28} +(2.45088 - 0.352383i) q^{29} +(2.21496 - 0.306502i) q^{30} +(-5.49019 + 3.52833i) q^{31} +(-0.349464 + 0.936950i) q^{32} +(-3.98890 - 2.17810i) q^{33} +(-2.11349 + 2.43910i) q^{34} +(1.46863 + 2.72379i) q^{35} +(-0.959493 + 0.281733i) q^{36} +(0.721730 + 0.269191i) q^{37} +(-1.13706 - 0.247352i) q^{38} +(0.0336523 - 0.114609i) q^{39} +(2.18242 + 0.486873i) q^{40} +(-1.09634 + 2.40066i) q^{41} +(-0.829340 - 1.10787i) q^{42} +(-1.03140 - 4.74127i) q^{43} +(-2.97623 - 3.43475i) q^{44} +(0.918113 + 2.03889i) q^{45} +(-1.51134 - 4.55147i) q^{46} +(-3.35604 - 3.35604i) q^{47} +(-0.997452 - 0.0713392i) q^{48} +(-2.74906 + 4.27762i) q^{49} +(0.409491 - 4.98320i) q^{50} +(-2.93574 - 1.34071i) q^{51} +(0.0715823 - 0.0956227i) q^{52} +(2.37831 + 4.35555i) q^{53} +(-0.540641 - 0.841254i) q^{54} +(-6.69562 + 7.64499i) q^{55} +(-0.389890 - 1.32784i) q^{56} +(-0.0830141 - 1.16069i) q^{57} +(0.176641 + 2.46977i) q^{58} +(-0.267799 - 0.912041i) q^{59} +(0.147707 + 2.23118i) q^{60} +(6.89732 + 10.7324i) q^{61} +(-3.12768 - 5.72791i) q^{62} +(0.829340 - 1.10787i) q^{63} +(-0.909632 - 0.415415i) q^{64} +(-0.233741 - 0.129243i) q^{65} +(2.45712 - 3.82335i) q^{66} +(-14.4218 - 1.03147i) q^{67} +(-2.28211 - 2.28211i) q^{68} +(3.94130 - 2.73242i) q^{69} +(-2.82162 + 1.27058i) q^{70} +(5.17109 + 5.96775i) q^{71} +(-0.212565 - 0.977147i) q^{72} +(-6.67685 - 8.91923i) q^{73} +(-0.319993 + 0.700687i) q^{74} +(4.89672 - 1.01103i) q^{75} +(0.327839 - 1.11652i) q^{76} +(6.14585 + 1.33695i) q^{77} +(0.111916 + 0.0417427i) q^{78} +(11.2840 - 3.31328i) q^{79} +(-0.641325 + 2.14213i) q^{80} +(0.654861 - 0.755750i) q^{81} +(-2.31633 - 1.26481i) q^{82} +(0.511355 - 1.37100i) q^{83} +(1.16421 - 0.748193i) q^{84} +(-4.35532 + 5.75426i) q^{85} +(4.80277 - 0.690534i) q^{86} +(-2.31996 + 0.865301i) q^{87} +(3.63832 - 2.72361i) q^{88} +(7.15307 + 4.59700i) q^{89} +(-2.09919 + 0.770321i) q^{90} +0.165304i q^{91} +(4.64769 - 1.18280i) q^{92} +(4.61472 - 4.61472i) q^{93} +(3.58691 - 3.10807i) q^{94} +(-2.57353 - 0.383937i) q^{95} +(0.142315 - 0.989821i) q^{96} +(4.90938 + 13.1626i) q^{97} +(-4.07061 - 3.04722i) q^{98} +(4.36073 + 1.28042i) 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 \(\left(\mathbb{Z}/690\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.0713392 + 0.997452i −0.0504444 + 0.705305i
\(3\) −0.977147 + 0.212565i −0.564156 + 0.122725i
\(4\) −0.989821 − 0.142315i −0.494911 − 0.0711574i
\(5\) −0.0118400 + 2.23604i −0.00529503 + 0.999986i
\(6\) −0.142315 − 0.989821i −0.0580998 − 0.404093i
\(7\) 1.21462 − 0.663233i 0.459083 − 0.250678i −0.233027 − 0.972470i \(-0.574863\pi\)
0.692110 + 0.721792i \(0.256681\pi\)
\(8\) 0.212565 − 0.977147i 0.0751532 − 0.345474i
\(9\) 0.909632 − 0.415415i 0.303211 − 0.138472i
\(10\) −2.22949 − 0.171327i −0.705028 − 0.0541783i
\(11\) 3.43475 + 2.97623i 1.03562 + 0.897366i 0.994805 − 0.101801i \(-0.0324604\pi\)
0.0408110 + 0.999167i \(0.487006\pi\)
\(12\) 0.997452 − 0.0713392i 0.287940 − 0.0205938i
\(13\) −0.0572451 + 0.104837i −0.0158769 + 0.0290765i −0.885497 − 0.464645i \(-0.846182\pi\)
0.869620 + 0.493721i \(0.164364\pi\)
\(14\) 0.574893 + 1.25884i 0.153647 + 0.336439i
\(15\) −0.463734 − 2.18745i −0.119736 − 0.564798i
\(16\) 0.959493 + 0.281733i 0.239873 + 0.0704331i
\(17\) 2.58366 + 1.93410i 0.626630 + 0.469089i 0.864788 − 0.502137i \(-0.167453\pi\)
−0.238158 + 0.971226i \(0.576544\pi\)
\(18\) 0.349464 + 0.936950i 0.0823695 + 0.220841i
\(19\) −0.165605 + 1.15181i −0.0379924 + 0.264243i −0.999960 − 0.00891976i \(-0.997161\pi\)
0.961968 + 0.273163i \(0.0880698\pi\)
\(20\) 0.329941 − 2.21159i 0.0737770 − 0.494527i
\(21\) −1.04588 + 0.906262i −0.228230 + 0.197763i
\(22\) −3.21368 + 3.21368i −0.685158 + 0.685158i
\(23\) −4.43205 + 1.83219i −0.924146 + 0.382038i
\(24\) 1.00000i 0.204124i
\(25\) −4.99972 − 0.0529496i −0.999944 − 0.0105899i
\(26\) −0.100486 − 0.0645782i −0.0197069 − 0.0126648i
\(27\) −0.800541 + 0.599278i −0.154064 + 0.115331i
\(28\) −1.29664 + 0.483624i −0.245043 + 0.0913963i
\(29\) 2.45088 − 0.352383i 0.455116 − 0.0654358i 0.0890552 − 0.996027i \(-0.471615\pi\)
0.366061 + 0.930591i \(0.380706\pi\)
\(30\) 2.21496 − 0.306502i 0.404395 − 0.0559593i
\(31\) −5.49019 + 3.52833i −0.986067 + 0.633707i −0.931093 − 0.364781i \(-0.881144\pi\)
−0.0549739 + 0.998488i \(0.517508\pi\)
\(32\) −0.349464 + 0.936950i −0.0617771 + 0.165631i
\(33\) −3.98890 − 2.17810i −0.694378 − 0.379159i
\(34\) −2.11349 + 2.43910i −0.362461 + 0.418302i
\(35\) 1.46863 + 2.72379i 0.248244 + 0.460404i
\(36\) −0.959493 + 0.281733i −0.159915 + 0.0469554i
\(37\) 0.721730 + 0.269191i 0.118652 + 0.0442548i 0.408089 − 0.912942i \(-0.366195\pi\)
−0.289437 + 0.957197i \(0.593468\pi\)
\(38\) −1.13706 − 0.247352i −0.184456 − 0.0401258i
\(39\) 0.0336523 − 0.114609i 0.00538868 − 0.0183521i
\(40\) 2.18242 + 0.486873i 0.345071 + 0.0769814i
\(41\) −1.09634 + 2.40066i −0.171220 + 0.374920i −0.975716 − 0.219038i \(-0.929708\pi\)
0.804496 + 0.593958i \(0.202435\pi\)
\(42\) −0.829340 − 1.10787i −0.127970 − 0.170948i
\(43\) −1.03140 − 4.74127i −0.157287 − 0.723037i −0.986454 − 0.164036i \(-0.947549\pi\)
0.829167 − 0.559000i \(-0.188815\pi\)
\(44\) −2.97623 − 3.43475i −0.448683 − 0.517808i
\(45\) 0.918113 + 2.03889i 0.136864 + 0.303940i
\(46\) −1.51134 − 4.55147i −0.222836 − 0.671077i
\(47\) −3.35604 − 3.35604i −0.489529 − 0.489529i 0.418629 − 0.908157i \(-0.362511\pi\)
−0.908157 + 0.418629i \(0.862511\pi\)
\(48\) −0.997452 − 0.0713392i −0.143970 − 0.0102969i
\(49\) −2.74906 + 4.27762i −0.392723 + 0.611089i
\(50\) 0.409491 − 4.98320i 0.0579107 − 0.704731i
\(51\) −2.93574 − 1.34071i −0.411086 − 0.187737i
\(52\) 0.0715823 − 0.0956227i 0.00992667 − 0.0132605i
\(53\) 2.37831 + 4.35555i 0.326686 + 0.598281i 0.988471 − 0.151413i \(-0.0483823\pi\)
−0.661784 + 0.749694i \(0.730201\pi\)
\(54\) −0.540641 − 0.841254i −0.0735719 − 0.114480i
\(55\) −6.69562 + 7.64499i −0.902837 + 1.03085i
\(56\) −0.389890 − 1.32784i −0.0521012 − 0.177440i
\(57\) −0.0830141 − 1.16069i −0.0109955 − 0.153737i
\(58\) 0.176641 + 2.46977i 0.0231942 + 0.324297i
\(59\) −0.267799 − 0.912041i −0.0348645 − 0.118738i 0.940222 − 0.340562i \(-0.110617\pi\)
−0.975087 + 0.221824i \(0.928799\pi\)
\(60\) 0.147707 + 2.23118i 0.0190689 + 0.288045i
\(61\) 6.89732 + 10.7324i 0.883111 + 1.37415i 0.926980 + 0.375110i \(0.122395\pi\)
−0.0438696 + 0.999037i \(0.513969\pi\)
\(62\) −3.12768 − 5.72791i −0.397215 − 0.727445i
\(63\) 0.829340 − 1.10787i 0.104487 − 0.139578i
\(64\) −0.909632 − 0.415415i −0.113704 − 0.0519269i
\(65\) −0.233741 − 0.129243i −0.0289920 − 0.0160307i
\(66\) 2.45712 − 3.82335i 0.302450 − 0.470622i
\(67\) −14.4218 − 1.03147i −1.76191 − 0.126014i −0.847497 − 0.530800i \(-0.821892\pi\)
−0.914412 + 0.404786i \(0.867346\pi\)
\(68\) −2.28211 − 2.28211i −0.276747 − 0.276747i
\(69\) 3.94130 − 2.73242i 0.474477 − 0.328945i
\(70\) −2.82162 + 1.27058i −0.337248 + 0.151863i
\(71\) 5.17109 + 5.96775i 0.613695 + 0.708242i 0.974497 − 0.224401i \(-0.0720424\pi\)
−0.360802 + 0.932642i \(0.617497\pi\)
\(72\) −0.212565 − 0.977147i −0.0250511 − 0.115158i
\(73\) −6.67685 − 8.91923i −0.781467 − 1.04392i −0.997714 − 0.0675786i \(-0.978473\pi\)
0.216247 − 0.976339i \(-0.430618\pi\)
\(74\) −0.319993 + 0.700687i −0.0371984 + 0.0814532i
\(75\) 4.89672 − 1.01103i 0.565424 − 0.116743i
\(76\) 0.327839 − 1.11652i 0.0376057 − 0.128073i
\(77\) 6.14585 + 1.33695i 0.700384 + 0.152359i
\(78\) 0.111916 + 0.0417427i 0.0126720 + 0.00472642i
\(79\) 11.2840 − 3.31328i 1.26955 − 0.372773i 0.423507 − 0.905893i \(-0.360799\pi\)
0.846040 + 0.533120i \(0.178981\pi\)
\(80\) −0.641325 + 2.14213i −0.0717023 + 0.239497i
\(81\) 0.654861 − 0.755750i 0.0727623 − 0.0839722i
\(82\) −2.31633 − 1.26481i −0.255796 − 0.139675i
\(83\) 0.511355 − 1.37100i 0.0561285 − 0.150486i −0.905859 − 0.423579i \(-0.860774\pi\)
0.961988 + 0.273092i \(0.0880465\pi\)
\(84\) 1.16421 − 0.748193i 0.127026 − 0.0816345i
\(85\) −4.35532 + 5.75426i −0.472401 + 0.624137i
\(86\) 4.80277 − 0.690534i 0.517896 − 0.0744621i
\(87\) −2.31996 + 0.865301i −0.248726 + 0.0927700i
\(88\) 3.63832 − 2.72361i 0.387846 − 0.290338i
\(89\) 7.15307 + 4.59700i 0.758224 + 0.487281i 0.861742 − 0.507347i \(-0.169374\pi\)
−0.103518 + 0.994628i \(0.533010\pi\)
\(90\) −2.09919 + 0.770321i −0.221274 + 0.0811990i
\(91\) 0.165304i 0.0173285i
\(92\) 4.64769 − 1.18280i 0.484555 − 0.123315i
\(93\) 4.61472 − 4.61472i 0.478524 − 0.478524i
\(94\) 3.58691 − 3.10807i 0.369961 − 0.320573i
\(95\) −2.57353 − 0.383937i −0.264038 − 0.0393911i
\(96\) 0.142315 − 0.989821i 0.0145249 − 0.101023i
\(97\) 4.90938 + 13.1626i 0.498472 + 1.33645i 0.906006 + 0.423265i \(0.139116\pi\)
−0.407534 + 0.913190i \(0.633611\pi\)
\(98\) −4.07061 − 3.04722i −0.411194 − 0.307816i
\(99\) 4.36073 + 1.28042i 0.438270 + 0.128688i
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.217.4 yes 240
5.3 odd 4 inner 690.2.w.a.493.12 yes 240
23.7 odd 22 inner 690.2.w.a.7.12 ✓ 240
115.53 even 44 inner 690.2.w.a.283.4 yes 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
690.2.w.a.7.12 ✓ 240 23.7 odd 22 inner
690.2.w.a.217.4 yes 240 1.1 even 1 trivial
690.2.w.a.283.4 yes 240 115.53 even 44 inner
690.2.w.a.493.12 yes 240 5.3 odd 4 inner