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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 343.3
Character \(\chi\) \(=\) 690.343
Dual form 690.2.w.a.517.3

$q$-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+(-0.977147 + 0.212565i) q^{2} +(-0.599278 + 0.800541i) q^{3} +(0.909632 - 0.415415i) q^{4} +(-1.55636 - 1.60554i) q^{5} +(0.415415 - 0.909632i) q^{6} +(0.263006 - 0.0188106i) q^{7} +(-0.800541 + 0.599278i) q^{8} +(-0.281733 - 0.959493i) q^{9} +(1.86207 + 1.23802i) q^{10} +(-1.60259 - 2.49367i) q^{11} +(-0.212565 + 0.977147i) q^{12} +(-0.319672 + 4.46959i) q^{13} +(-0.252997 + 0.0742867i) q^{14} +(2.21799 - 0.283768i) q^{15} +(0.654861 - 0.755750i) q^{16} +(0.517240 - 0.192920i) q^{17} +(0.479249 + 0.877679i) q^{18} +(0.245728 + 0.538069i) q^{19} +(-2.08268 - 0.813911i) q^{20} +(-0.142555 + 0.221820i) q^{21} +(2.09603 + 2.09603i) q^{22} +(3.12361 + 3.63910i) q^{23} -1.00000i q^{24} +(-0.155489 + 4.99758i) q^{25} +(-0.637714 - 4.43540i) q^{26} +(0.936950 + 0.349464i) q^{27} +(0.231425 - 0.126368i) q^{28} +(-1.25993 - 0.575392i) q^{29} +(-2.10698 + 0.748751i) q^{30} +(-0.953676 + 6.63296i) q^{31} +(-0.479249 + 0.877679i) q^{32} +(2.95668 + 0.211466i) q^{33} +(-0.464411 + 0.298459i) q^{34} +(-0.439534 - 0.392990i) q^{35} +(-0.654861 - 0.755750i) q^{36} +(4.00269 + 2.18563i) q^{37} +(-0.354487 - 0.473540i) q^{38} +(-3.38652 - 2.93444i) q^{39} +(2.20809 + 0.352606i) q^{40} +(5.25300 + 1.54242i) q^{41} +(0.0921461 - 0.247053i) q^{42} +(1.46922 + 1.09984i) q^{43} +(-2.49367 - 1.60259i) q^{44} +(-1.10202 + 1.94565i) q^{45} +(-3.82577 - 2.89197i) q^{46} +(-7.14989 + 7.14989i) q^{47} +(0.212565 + 0.977147i) q^{48} +(-6.85993 + 0.986309i) q^{49} +(-0.910377 - 4.91642i) q^{50} +(-0.155529 + 0.529684i) q^{51} +(1.56595 + 4.19848i) q^{52} +(0.0889742 + 1.24402i) q^{53} +(-0.989821 - 0.142315i) q^{54} +(-1.50948 + 6.45406i) q^{55} +(-0.199275 + 0.172673i) q^{56} +(-0.578006 - 0.125738i) q^{57} +(1.35345 + 0.294424i) q^{58} +(-6.47225 + 5.60824i) q^{59} +(1.89967 - 1.17951i) q^{60} +(-6.38443 - 0.917943i) q^{61} +(-0.478056 - 6.68409i) q^{62} +(-0.0921461 - 0.247053i) q^{63} +(0.281733 - 0.959493i) q^{64} +(7.67361 - 6.44305i) q^{65} +(-2.93406 + 0.421854i) q^{66} +(1.78081 + 8.18627i) q^{67} +(0.390356 - 0.390356i) q^{68} +(-4.78516 + 0.319743i) q^{69} +(0.513025 + 0.290579i) q^{70} +(1.53322 + 0.985338i) q^{71} +(0.800541 + 0.599278i) q^{72} +(2.12762 - 5.70438i) q^{73} +(-4.37581 - 1.28485i) q^{74} +(-3.90759 - 3.11941i) q^{75} +(0.447044 + 0.387366i) q^{76} +(-0.468398 - 0.625706i) q^{77} +(3.93289 + 2.14752i) q^{78} +(4.32775 + 4.99448i) q^{79} +(-2.23258 + 0.124816i) q^{80} +(-0.841254 + 0.540641i) q^{81} +(-5.46082 - 0.390565i) q^{82} +(-3.52451 + 6.45466i) q^{83} +(-0.0375253 + 0.260994i) q^{84} +(-1.11475 - 0.530193i) q^{85} +(-1.66943 - 0.762403i) q^{86} +(1.21567 - 0.663808i) q^{87} +(2.77734 + 1.03589i) q^{88} +(-0.348512 - 2.42396i) q^{89} +(0.663261 - 2.13544i) q^{90} +1.18154i q^{91} +(4.35307 + 2.01265i) q^{92} +(-4.73844 - 4.73844i) q^{93} +(5.46668 - 8.50632i) q^{94} +(0.481448 - 1.23195i) q^{95} +(-0.415415 - 0.909632i) q^{96} +(8.78010 + 16.0796i) q^{97} +(6.49351 - 2.42195i) q^{98} +(-1.94116 + 2.24022i) 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 \(\left(\mathbb{Z}/690\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(461\) \(511\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(1\) \(e\left(\frac{13}{22}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.977147 + 0.212565i −0.690947 + 0.150306i
\(3\) −0.599278 + 0.800541i −0.345993 + 0.462193i
\(4\) 0.909632 − 0.415415i 0.454816 − 0.207708i
\(5\) −1.55636 − 1.60554i −0.696025 − 0.718017i
\(6\) 0.415415 − 0.909632i 0.169592 − 0.371356i
\(7\) 0.263006 − 0.0188106i 0.0994071 − 0.00710973i −0.0215470 − 0.999768i \(-0.506859\pi\)
0.120954 + 0.992658i \(0.461405\pi\)
\(8\) −0.800541 + 0.599278i −0.283034 + 0.211877i
\(9\) −0.281733 − 0.959493i −0.0939109 − 0.319831i
\(10\) 1.86207 + 1.23802i 0.588839 + 0.391495i
\(11\) −1.60259 − 2.49367i −0.483198 − 0.751870i 0.510984 − 0.859590i \(-0.329281\pi\)
−0.994181 + 0.107720i \(0.965645\pi\)
\(12\) −0.212565 + 0.977147i −0.0613623 + 0.282078i
\(13\) −0.319672 + 4.46959i −0.0886609 + 1.23964i 0.737163 + 0.675714i \(0.236165\pi\)
−0.825824 + 0.563927i \(0.809290\pi\)
\(14\) −0.252997 + 0.0742867i −0.0676164 + 0.0198540i
\(15\) 2.21799 − 0.283768i 0.572682 − 0.0732687i
\(16\) 0.654861 − 0.755750i 0.163715 − 0.188937i
\(17\) 0.517240 − 0.192920i 0.125449 − 0.0467901i −0.285956 − 0.958243i \(-0.592311\pi\)
0.411405 + 0.911453i \(0.365038\pi\)
\(18\) 0.479249 + 0.877679i 0.112960 + 0.206871i
\(19\) 0.245728 + 0.538069i 0.0563739 + 0.123442i 0.935723 − 0.352737i \(-0.114749\pi\)
−0.879349 + 0.476178i \(0.842022\pi\)
\(20\) −2.08268 − 0.813911i −0.465701 − 0.181996i
\(21\) −0.142555 + 0.221820i −0.0311081 + 0.0484051i
\(22\) 2.09603 + 2.09603i 0.446875 + 0.446875i
\(23\) 3.12361 + 3.63910i 0.651317 + 0.758806i
\(24\) − 1.00000i − 0.204124i
\(25\) −0.155489 + 4.99758i −0.0310978 + 0.999516i
\(26\) −0.637714 − 4.43540i −0.125066 − 0.869853i
\(27\) 0.936950 + 0.349464i 0.180316 + 0.0672544i
\(28\) 0.231425 − 0.126368i 0.0437352 − 0.0238812i
\(29\) −1.25993 − 0.575392i −0.233964 − 0.106848i 0.294984 − 0.955502i \(-0.404686\pi\)
−0.528947 + 0.848655i \(0.677413\pi\)
\(30\) −2.10698 + 0.748751i −0.384680 + 0.136703i
\(31\) −0.953676 + 6.63296i −0.171285 + 1.19131i 0.704888 + 0.709319i \(0.250997\pi\)
−0.876173 + 0.481996i \(0.839912\pi\)
\(32\) −0.479249 + 0.877679i −0.0847201 + 0.155153i
\(33\) 2.95668 + 0.211466i 0.514692 + 0.0368115i
\(34\) −0.464411 + 0.298459i −0.0796458 + 0.0511852i
\(35\) −0.439534 − 0.392990i −0.0742947 − 0.0664275i
\(36\) −0.654861 − 0.755750i −0.109143 − 0.125958i
\(37\) 4.00269 + 2.18563i 0.658038 + 0.359316i 0.773319 − 0.634017i \(-0.218595\pi\)
−0.115281 + 0.993333i \(0.536777\pi\)
\(38\) −0.354487 − 0.473540i −0.0575054 − 0.0768183i
\(39\) −3.38652 − 2.93444i −0.542277 − 0.469886i
\(40\) 2.20809 + 0.352606i 0.349130 + 0.0557519i
\(41\) 5.25300 + 1.54242i 0.820381 + 0.240886i 0.664880 − 0.746950i \(-0.268483\pi\)
0.155501 + 0.987836i \(0.450301\pi\)
\(42\) 0.0921461 − 0.247053i 0.0142185 − 0.0381211i
\(43\) 1.46922 + 1.09984i 0.224054 + 0.167725i 0.705408 − 0.708802i \(-0.250764\pi\)
−0.481354 + 0.876526i \(0.659855\pi\)
\(44\) −2.49367 − 1.60259i −0.375935 − 0.241599i
\(45\) −1.10202 + 1.94565i −0.164280 + 0.290040i
\(46\) −3.82577 − 2.89197i −0.564079 − 0.426398i
\(47\) −7.14989 + 7.14989i −1.04292 + 1.04292i −0.0438827 + 0.999037i \(0.513973\pi\)
−0.999037 + 0.0438827i \(0.986027\pi\)
\(48\) 0.212565 + 0.977147i 0.0306812 + 0.141039i
\(49\) −6.85993 + 0.986309i −0.979990 + 0.140901i
\(50\) −0.910377 − 4.91642i −0.128747 − 0.695287i
\(51\) −0.155529 + 0.529684i −0.0217785 + 0.0741707i
\(52\) 1.56595 + 4.19848i 0.217158 + 0.582224i
\(53\) 0.0889742 + 1.24402i 0.0122215 + 0.170879i 0.999897 + 0.0143627i \(0.00457196\pi\)
−0.987675 + 0.156517i \(0.949973\pi\)
\(54\) −0.989821 − 0.142315i −0.134698 − 0.0193666i
\(55\) −1.50948 + 6.45406i −0.203538 + 0.870265i
\(56\) −0.199275 + 0.172673i −0.0266292 + 0.0230743i
\(57\) −0.578006 − 0.125738i −0.0765588 − 0.0166543i
\(58\) 1.35345 + 0.294424i 0.177716 + 0.0386598i
\(59\) −6.47225 + 5.60824i −0.842615 + 0.730130i −0.964971 − 0.262356i \(-0.915501\pi\)
0.122356 + 0.992486i \(0.460955\pi\)
\(60\) 1.89967 − 1.17951i 0.245247 − 0.152274i
\(61\) −6.38443 − 0.917943i −0.817443 − 0.117530i −0.279106 − 0.960260i \(-0.590038\pi\)
−0.538337 + 0.842730i \(0.680947\pi\)
\(62\) −0.478056 − 6.68409i −0.0607132 − 0.848881i
\(63\) −0.0921461 − 0.247053i −0.0116093 − 0.0311258i
\(64\) 0.281733 − 0.959493i 0.0352166 − 0.119937i
\(65\) 7.67361 − 6.44305i 0.951795 − 0.799162i
\(66\) −2.93406 + 0.421854i −0.361158 + 0.0519267i
\(67\) 1.78081 + 8.18627i 0.217561 + 1.00011i 0.948280 + 0.317434i \(0.102821\pi\)
−0.730720 + 0.682678i \(0.760815\pi\)
\(68\) 0.390356 − 0.390356i 0.0473376 − 0.0473376i
\(69\) −4.78516 + 0.319743i −0.576066 + 0.0384925i
\(70\) 0.513025 + 0.290579i 0.0613182 + 0.0347309i
\(71\) 1.53322 + 0.985338i 0.181959 + 0.116938i 0.628449 − 0.777850i \(-0.283690\pi\)
−0.446490 + 0.894789i \(0.647326\pi\)
\(72\) 0.800541 + 0.599278i 0.0943447 + 0.0706256i
\(73\) 2.12762 − 5.70438i 0.249019 − 0.667647i −0.750954 − 0.660354i \(-0.770406\pi\)
0.999973 − 0.00729227i \(-0.00232122\pi\)
\(74\) −4.37581 − 1.28485i −0.508677 − 0.149361i
\(75\) −3.90759 − 3.11941i −0.451210 − 0.360199i
\(76\) 0.447044 + 0.387366i 0.0512795 + 0.0444339i
\(77\) −0.468398 − 0.625706i −0.0533789 − 0.0713058i
\(78\) 3.93289 + 2.14752i 0.445312 + 0.243159i
\(79\) 4.32775 + 4.99448i 0.486909 + 0.561923i 0.945037 − 0.326963i \(-0.106025\pi\)
−0.458128 + 0.888886i \(0.651480\pi\)
\(80\) −2.23258 + 0.124816i −0.249610 + 0.0139548i
\(81\) −0.841254 + 0.540641i −0.0934726 + 0.0600712i
\(82\) −5.46082 − 0.390565i −0.603046 − 0.0431307i
\(83\) −3.52451 + 6.45466i −0.386866 + 0.708491i −0.996446 − 0.0842328i \(-0.973156\pi\)
0.609581 + 0.792724i \(0.291338\pi\)
\(84\) −0.0375253 + 0.260994i −0.00409435 + 0.0284768i
\(85\) −1.11475 − 0.530193i −0.120912 − 0.0575075i
\(86\) −1.66943 − 0.762403i −0.180019 − 0.0822121i
\(87\) 1.21567 − 0.663808i 0.130334 − 0.0711677i
\(88\) 2.77734 + 1.03589i 0.296065 + 0.110427i
\(89\) −0.348512 − 2.42396i −0.0369422 − 0.256939i 0.962978 − 0.269580i \(-0.0868850\pi\)
−0.999920 + 0.0126416i \(0.995976\pi\)
\(90\) 0.663261 − 2.13544i 0.0699139 − 0.225095i
\(91\) 1.18154i 0.123860i
\(92\) 4.35307 + 2.01265i 0.453839 + 0.209833i
\(93\) −4.73844 − 4.73844i −0.491354 − 0.491354i
\(94\) 5.46668 − 8.50632i 0.563845 − 0.877360i
\(95\) 0.481448 − 1.23195i 0.0493956 − 0.126396i
\(96\) −0.415415 − 0.909632i −0.0423981 − 0.0928389i
\(97\) 8.78010 + 16.0796i 0.891484 + 1.63263i 0.768153 + 0.640266i \(0.221176\pi\)
0.123331 + 0.992366i \(0.460642\pi\)
\(98\) 6.49351 − 2.42195i 0.655943 − 0.244654i
\(99\) −1.94116 + 2.24022i −0.195094 + 0.225150i
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.343.3 yes 240
5.2 odd 4 inner 690.2.w.a.67.6 ✓ 240
23.11 odd 22 inner 690.2.w.a.103.6 yes 240
115.57 even 44 inner 690.2.w.a.517.3 yes 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
690.2.w.a.67.6 ✓ 240 5.2 odd 4 inner
690.2.w.a.103.6 yes 240 23.11 odd 22 inner
690.2.w.a.343.3 yes 240 1.1 even 1 trivial
690.2.w.a.517.3 yes 240 115.57 even 44 inner