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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 103.6
Character \(\chi\) \(=\) 690.103
Dual form 690.2.w.a.67.6

$q$-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+(-0.212565 + 0.977147i) q^{2} +(0.800541 - 0.599278i) q^{3} +(-0.909632 - 0.415415i) q^{4} +(2.17731 + 0.509231i) q^{5} +(0.415415 + 0.909632i) q^{6} +(0.0188106 - 0.263006i) q^{7} +(0.599278 - 0.800541i) q^{8} +(0.281733 - 0.959493i) q^{9} +(-0.960414 + 2.01931i) q^{10} +(-1.60259 + 2.49367i) q^{11} +(-0.977147 + 0.212565i) q^{12} +(4.46959 - 0.319672i) q^{13} +(0.252997 + 0.0742867i) q^{14} +(2.04820 - 0.897154i) q^{15} +(0.654861 + 0.755750i) q^{16} +(0.192920 - 0.517240i) q^{17} +(0.877679 + 0.479249i) q^{18} +(-0.245728 + 0.538069i) q^{19} +(-1.76901 - 1.36770i) q^{20} +(-0.142555 - 0.221820i) q^{21} +(-2.09603 - 2.09603i) q^{22} +(3.63910 + 3.12361i) q^{23} -1.00000i q^{24} +(4.48137 + 2.21751i) q^{25} +(-0.637714 + 4.43540i) q^{26} +(-0.349464 - 0.936950i) q^{27} +(-0.126368 + 0.231425i) q^{28} +(1.25993 - 0.575392i) q^{29} +(0.441275 + 2.19209i) q^{30} +(-0.953676 - 6.63296i) q^{31} +(-0.877679 + 0.479249i) q^{32} +(0.211466 + 2.95668i) q^{33} +(0.464411 + 0.298459i) q^{34} +(0.174888 - 0.563068i) q^{35} +(-0.654861 + 0.755750i) q^{36} +(-2.18563 - 4.00269i) q^{37} +(-0.473540 - 0.354487i) q^{38} +(3.38652 - 2.93444i) q^{39} +(1.71247 - 1.43786i) q^{40} +(5.25300 - 1.54242i) q^{41} +(0.247053 - 0.0921461i) q^{42} +(1.09984 + 1.46922i) 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} +(0.977147 + 0.212565i) q^{48} +(6.85993 + 0.986309i) q^{49} +(-3.11941 + 3.90759i) q^{50} +(-0.155529 - 0.529684i) q^{51} +(-4.19848 - 1.56595i) q^{52} +(1.24402 + 0.0889742i) q^{53} +(0.989821 - 0.142315i) q^{54} +(-4.75918 + 4.61341i) q^{55} +(-0.199275 - 0.172673i) q^{56} +(0.125738 + 0.578006i) q^{57} +(0.294424 + 1.35345i) q^{58} +(6.47225 + 5.60824i) q^{59} +(-2.23580 - 0.0347726i) q^{60} +(-6.38443 + 0.917943i) q^{61} +(6.68409 + 0.478056i) q^{62} +(-0.247053 - 0.0921461i) q^{63} +(-0.281733 - 0.959493i) q^{64} +(9.89448 + 1.58003i) q^{65} +(-2.93406 - 0.421854i) q^{66} +(-8.18627 - 1.78081i) 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} +(-0.599278 - 0.800541i) q^{72} +(-5.70438 + 2.12762i) q^{73} +(4.37581 - 1.28485i) q^{74} +(4.91642 - 0.910377i) q^{75} +(0.447044 - 0.387366i) q^{76} +(0.625706 + 0.468398i) q^{77} +(2.14752 + 3.93289i) q^{78} +(-4.32775 + 4.99448i) q^{79} +(1.04098 + 1.97898i) q^{80} +(-0.841254 - 0.540641i) q^{81} +(0.390565 + 5.46082i) q^{82} +(6.45466 - 3.52451i) q^{83} +(0.0375253 + 0.260994i) q^{84} +(0.683442 - 1.02795i) q^{85} +(-1.66943 + 0.762403i) q^{86} +(0.663808 - 1.21567i) q^{87} +(1.03589 + 2.77734i) q^{88} +(0.348512 - 2.42396i) q^{89} +(1.66693 + 1.49042i) q^{90} -1.18154i q^{91} +(-2.01265 - 4.35307i) q^{92} +(-4.73844 - 4.73844i) q^{93} +(-5.46668 - 8.50632i) q^{94} +(-0.809028 + 1.04641i) q^{95} +(-0.415415 + 0.909632i) q^{96} +(-16.0796 - 8.78010i) q^{97} +(-2.42195 + 6.49351i) 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 \(\left(\mathbb{Z}/690\mathbb{Z}\right)^\times\).

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

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
\(5\) 2.17731 + 0.509231i 0.973723 + 0.227735i
\(6\) 0.415415 + 0.909632i 0.169592 + 0.371356i
\(7\) 0.0188106 − 0.263006i 0.00710973 − 0.0994071i −0.992658 − 0.120954i \(-0.961405\pi\)
0.999768 + 0.0215470i \(0.00685916\pi\)
\(8\) 0.599278 − 0.800541i 0.211877 − 0.283034i
\(9\) 0.281733 − 0.959493i 0.0939109 − 0.319831i
\(10\) −0.960414 + 2.01931i −0.303710 + 0.638561i
\(11\) −1.60259 + 2.49367i −0.483198 + 0.751870i −0.994181 − 0.107720i \(-0.965645\pi\)
0.510984 + 0.859590i \(0.329281\pi\)
\(12\) −0.977147 + 0.212565i −0.282078 + 0.0613623i
\(13\) 4.46959 − 0.319672i 1.23964 − 0.0886609i 0.563927 − 0.825824i \(-0.309290\pi\)
0.675714 + 0.737163i \(0.263835\pi\)
\(14\) 0.252997 + 0.0742867i 0.0676164 + 0.0198540i
\(15\) 2.04820 − 0.897154i 0.528842 − 0.231644i
\(16\) 0.654861 + 0.755750i 0.163715 + 0.188937i
\(17\) 0.192920 − 0.517240i 0.0467901 − 0.125449i −0.911453 − 0.411405i \(-0.865038\pi\)
0.958243 + 0.285956i \(0.0923110\pi\)
\(18\) 0.877679 + 0.479249i 0.206871 + 0.112960i
\(19\) −0.245728 + 0.538069i −0.0563739 + 0.123442i −0.935723 − 0.352737i \(-0.885251\pi\)
0.879349 + 0.476178i \(0.157978\pi\)
\(20\) −1.76901 − 1.36770i −0.395563 − 0.305827i
\(21\) −0.142555 − 0.221820i −0.0311081 − 0.0484051i
\(22\) −2.09603 − 2.09603i −0.446875 − 0.446875i
\(23\) 3.63910 + 3.12361i 0.758806 + 0.651317i
\(24\) − 1.00000i − 0.204124i
\(25\) 4.48137 + 2.21751i 0.896274 + 0.443502i
\(26\) −0.637714 + 4.43540i −0.125066 + 0.869853i
\(27\) −0.349464 − 0.936950i −0.0672544 − 0.180316i
\(28\) −0.126368 + 0.231425i −0.0238812 + 0.0437352i
\(29\) 1.25993 − 0.575392i 0.233964 − 0.106848i −0.294984 − 0.955502i \(-0.595314\pi\)
0.528947 + 0.848655i \(0.322587\pi\)
\(30\) 0.441275 + 2.19209i 0.0805654 + 0.400220i
\(31\) −0.953676 − 6.63296i −0.171285 − 1.19131i −0.876173 − 0.481996i \(-0.839912\pi\)
0.704888 − 0.709319i \(-0.250997\pi\)
\(32\) −0.877679 + 0.479249i −0.155153 + 0.0847201i
\(33\) 0.211466 + 2.95668i 0.0368115 + 0.514692i
\(34\) 0.464411 + 0.298459i 0.0796458 + 0.0511852i
\(35\) 0.174888 − 0.563068i 0.0295614 − 0.0951758i
\(36\) −0.654861 + 0.755750i −0.109143 + 0.125958i
\(37\) −2.18563 − 4.00269i −0.359316 − 0.658038i 0.634017 − 0.773319i \(-0.281405\pi\)
−0.993333 + 0.115281i \(0.963223\pi\)
\(38\) −0.473540 − 0.354487i −0.0768183 − 0.0575054i
\(39\) 3.38652 − 2.93444i 0.542277 − 0.469886i
\(40\) 1.71247 − 1.43786i 0.270766 − 0.227345i
\(41\) 5.25300 − 1.54242i 0.820381 − 0.240886i 0.155501 − 0.987836i \(-0.450301\pi\)
0.664880 + 0.746950i \(0.268483\pi\)
\(42\) 0.247053 − 0.0921461i 0.0381211 − 0.0142185i
\(43\) 1.09984 + 1.46922i 0.167725 + 0.224054i 0.876526 − 0.481354i \(-0.159855\pi\)
−0.708802 + 0.705408i \(0.750764\pi\)
\(44\) 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 \(0.986027\pi\)
\(48\) 0.977147 + 0.212565i 0.141039 + 0.0306812i
\(49\) 6.85993 + 0.986309i 0.979990 + 0.140901i
\(50\) −3.11941 + 3.90759i −0.441152 + 0.552617i
\(51\) −0.155529 − 0.529684i −0.0217785 − 0.0741707i
\(52\) −4.19848 − 1.56595i −0.582224 − 0.217158i
\(53\) 1.24402 + 0.0889742i 0.170879 + 0.0122215i 0.156517 − 0.987675i \(-0.449973\pi\)
0.0143627 + 0.999897i \(0.495428\pi\)
\(54\) 0.989821 − 0.142315i 0.134698 − 0.0193666i
\(55\) −4.75918 + 4.61341i −0.641728 + 0.622072i
\(56\) −0.199275 − 0.172673i −0.0266292 − 0.0230743i
\(57\) 0.125738 + 0.578006i 0.0166543 + 0.0765588i
\(58\) 0.294424 + 1.35345i 0.0386598 + 0.177716i
\(59\) 6.47225 + 5.60824i 0.842615 + 0.730130i 0.964971 − 0.262356i \(-0.0844994\pi\)
−0.122356 + 0.992486i \(0.539045\pi\)
\(60\) −2.23580 − 0.0347726i −0.288640 − 0.00448913i
\(61\) −6.38443 + 0.917943i −0.817443 + 0.117530i −0.538337 − 0.842730i \(-0.680947\pi\)
−0.279106 + 0.960260i \(0.590038\pi\)
\(62\) 6.68409 + 0.478056i 0.848881 + 0.0607132i
\(63\) −0.247053 − 0.0921461i −0.0311258 − 0.0116093i
\(64\) −0.281733 − 0.959493i −0.0352166 − 0.119937i
\(65\) 9.89448 + 1.58003i 1.22726 + 0.195979i
\(66\) −2.93406 − 0.421854i −0.361158 − 0.0519267i
\(67\) −8.18627 − 1.78081i −1.00011 − 0.217561i −0.317434 − 0.948280i \(-0.602821\pi\)
−0.682678 + 0.730720i \(0.739185\pi\)
\(68\) −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 −0.446490 − 0.894789i \(-0.647326\pi\)
0.628449 + 0.777850i \(0.283690\pi\)
\(72\) −0.599278 − 0.800541i −0.0706256 − 0.0943447i
\(73\) −5.70438 + 2.12762i −0.667647 + 0.249019i −0.660354 − 0.750954i \(-0.729594\pi\)
−0.00729227 + 0.999973i \(0.502321\pi\)
\(74\) 4.37581 − 1.28485i 0.508677 − 0.149361i
\(75\) 4.91642 − 0.910377i 0.567700 − 0.105121i
\(76\) 0.447044 − 0.387366i 0.0512795 − 0.0444339i
\(77\) 0.625706 + 0.468398i 0.0713058 + 0.0533789i
\(78\) 2.14752 + 3.93289i 0.243159 + 0.445312i
\(79\) −4.32775 + 4.99448i −0.486909 + 0.561923i −0.945037 − 0.326963i \(-0.893975\pi\)
0.458128 + 0.888886i \(0.348520\pi\)
\(80\) 1.04098 + 1.97898i 0.116386 + 0.221256i
\(81\) −0.841254 − 0.540641i −0.0934726 − 0.0600712i
\(82\) 0.390565 + 5.46082i 0.0431307 + 0.603046i
\(83\) 6.45466 − 3.52451i 0.708491 − 0.386866i −0.0842328 − 0.996446i \(-0.526844\pi\)
0.792724 + 0.609581i \(0.208662\pi\)
\(84\) 0.0375253 + 0.260994i 0.00409435 + 0.0284768i
\(85\) 0.683442 − 1.02795i 0.0741297 − 0.111497i
\(86\) −1.66943 + 0.762403i −0.180019 + 0.0822121i
\(87\) 0.663808 − 1.21567i 0.0711677 − 0.130334i
\(88\) 1.03589 + 2.77734i 0.110427 + 0.296065i
\(89\) 0.348512 − 2.42396i 0.0369422 − 0.256939i −0.962978 − 0.269580i \(-0.913115\pi\)
0.999920 + 0.0126416i \(0.00402407\pi\)
\(90\) 1.66693 + 1.49042i 0.175710 + 0.157104i
\(91\) − 1.18154i − 0.123860i
\(92\) −2.01265 − 4.35307i −0.209833 − 0.453839i
\(93\) −4.73844 − 4.73844i −0.491354 − 0.491354i
\(94\) −5.46668 − 8.50632i −0.563845 − 0.877360i
\(95\) −0.809028 + 1.04641i −0.0830045 + 0.107360i
\(96\) −0.415415 + 0.909632i −0.0423981 + 0.0928389i
\(97\) −16.0796 − 8.78010i −1.63263 − 0.891484i −0.992366 − 0.123331i \(-0.960642\pi\)
−0.640266 − 0.768153i \(-0.721176\pi\)
\(98\) −2.42195 + 6.49351i −0.244654 + 0.655943i
\(99\) 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.103.6 yes 240
5.2 odd 4 inner 690.2.w.a.517.3 yes 240
23.21 odd 22 inner 690.2.w.a.343.3 yes 240
115.67 even 44 inner 690.2.w.a.67.6 ✓ 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
690.2.w.a.67.6 ✓ 240 115.67 even 44 inner
690.2.w.a.103.6 yes 240 1.1 even 1 trivial
690.2.w.a.343.3 yes 240 23.21 odd 22 inner
690.2.w.a.517.3 yes 240 5.2 odd 4 inner