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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [575,2,Mod(26,575)] 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("575.26"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(575, base_ring=CyclotomicField(22)) chi = DirichletCharacter(H, H._module([0, 16])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 575 = 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 575.k (of order \(11\), degree \(10\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [100,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.59139811622\)
Analytic rank: \(0\)
Dimension: \(100\)
Relative dimension: \(10\) over \(\Q(\zeta_{11})\)
Twist minimal: no (minimal twist has level 115)
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label 351.3
Character \(\chi\) \(=\) 575.351
Dual form 575.2.k.g.326.3

$q$-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+(-0.683908 + 1.49755i) q^{2} +(3.06714 + 0.900592i) q^{3} +(-0.465201 - 0.536871i) q^{4} +(-3.44632 + 3.97726i) q^{6} +(0.0908739 - 0.632042i) q^{7} +(-2.03713 + 0.598154i) q^{8} +(6.07249 + 3.90255i) q^{9} +(1.69819 + 3.71852i) q^{11} +(-0.943334 - 2.06561i) q^{12} +(-0.459166 - 3.19357i) q^{13} +(0.884364 + 0.568346i) q^{14} +(0.699637 - 4.86608i) q^{16} +(0.836385 - 0.965240i) q^{17} +(-9.99729 + 6.42487i) q^{18} +(-0.307596 - 0.354985i) q^{19} +(0.847934 - 1.85672i) q^{21} -6.73007 q^{22} +(-3.57257 + 3.19949i) q^{23} -6.78684 q^{24} +(5.09656 + 1.49648i) q^{26} +(8.83052 + 10.1910i) q^{27} +(-0.381599 + 0.245239i) q^{28} +(2.61894 - 3.02242i) q^{29} +(-4.96368 + 1.45747i) q^{31} +(3.23653 + 2.07999i) q^{32} +(1.85971 + 12.9346i) q^{33} +(0.873484 + 1.91266i) q^{34} +(-0.729764 - 5.07562i) q^{36} +(-8.04195 - 5.16825i) q^{37} +(0.741974 - 0.217863i) q^{38} +(1.46778 - 10.2086i) q^{39} +(7.36949 - 4.73608i) q^{41} +(2.20062 + 2.53965i) q^{42} +(0.704508 + 0.206862i) q^{43} +(1.20636 - 2.64157i) q^{44} +(-2.34808 - 7.53826i) q^{46} +0.589562 q^{47} +(6.52824 - 14.2948i) q^{48} +(6.32523 + 1.85726i) q^{49} +(3.43459 - 2.20728i) q^{51} +(-1.50093 + 1.73217i) q^{52} +(0.955686 - 6.64695i) q^{53} +(-21.3007 + 6.25446i) q^{54} +(0.192937 + 1.34191i) q^{56} +(-0.623742 - 1.36581i) q^{57} +(2.73511 + 5.98905i) q^{58} +(0.734852 + 5.11101i) q^{59} +(-2.44333 + 0.717427i) q^{61} +(1.21207 - 8.43012i) q^{62} +(3.01841 - 3.48343i) q^{63} +(2.94303 - 1.89137i) q^{64} +(-20.6420 - 6.06105i) q^{66} +(0.998254 - 2.18587i) q^{67} -0.907297 q^{68} +(-13.8390 + 6.59584i) q^{69} +(-3.46453 + 7.58626i) q^{71} +(-14.7048 - 4.31771i) q^{72} +(0.483051 + 0.557471i) q^{73} +(13.2397 - 8.50861i) q^{74} +(-0.0474869 + 0.330279i) q^{76} +(2.50458 - 0.735411i) q^{77} +(14.2841 + 9.17984i) q^{78} +(-2.33484 - 16.2392i) q^{79} +(8.91062 + 19.5115i) q^{81} +(2.05247 + 14.2752i) q^{82} +(-8.77936 - 5.64215i) q^{83} +(-1.39128 + 0.408516i) q^{84} +(-0.791605 + 0.913560i) q^{86} +(10.7546 - 6.91157i) q^{87} +(-5.68368 - 6.55931i) q^{88} +(-5.75334 - 1.68933i) q^{89} -2.06020 q^{91} +(3.37968 + 0.429603i) q^{92} -16.5368 q^{93} +(-0.403206 + 0.882897i) q^{94} +(8.05365 + 9.29441i) q^{96} +(-5.42589 + 3.48701i) q^{97} +(-7.10721 + 8.20215i) q^{98} +(-4.19947 + 29.2080i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 100 q + 14 q^{4} - 18 q^{6} + 12 q^{9} - 26 q^{11} + 26 q^{14} - 18 q^{16} + 14 q^{19} - 22 q^{21} + 68 q^{24} - 42 q^{26} + 24 q^{29} - 12 q^{31} - 8 q^{34} - 10 q^{36} - 14 q^{39} + 8 q^{41} - 166 q^{44}+ \cdots + 60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/575\mathbb{Z}\right)^\times\).

\(n\) \(51\) \(277\)
\(\chi(n)\) \(e\left(\frac{9}{11}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.683908 + 1.49755i −0.483596 + 1.05893i 0.497863 + 0.867255i \(0.334118\pi\)
−0.981459 + 0.191671i \(0.938609\pi\)
\(3\) 3.06714 + 0.900592i 1.77081 + 0.519957i 0.993962 − 0.109729i \(-0.0349982\pi\)
0.776850 + 0.629686i \(0.216816\pi\)
\(4\) −0.465201 − 0.536871i −0.232601 − 0.268436i
\(5\) 0 0
\(6\) −3.44632 + 3.97726i −1.40695 + 1.62371i
\(7\) 0.0908739 − 0.632042i 0.0343471 − 0.238889i −0.965415 − 0.260719i \(-0.916040\pi\)
0.999762 + 0.0218302i \(0.00694933\pi\)
\(8\) −2.03713 + 0.598154i −0.720233 + 0.211480i
\(9\) 6.07249 + 3.90255i 2.02416 + 1.30085i
\(10\) 0 0
\(11\) 1.69819 + 3.71852i 0.512024 + 1.12118i 0.972372 + 0.233437i \(0.0749973\pi\)
−0.460348 + 0.887738i \(0.652275\pi\)
\(12\) −0.943334 − 2.06561i −0.272317 − 0.596291i
\(13\) −0.459166 − 3.19357i −0.127350 − 0.885738i −0.948895 − 0.315593i \(-0.897796\pi\)
0.821545 − 0.570144i \(-0.193113\pi\)
\(14\) 0.884364 + 0.568346i 0.236356 + 0.151897i
\(15\) 0 0
\(16\) 0.699637 − 4.86608i 0.174909 − 1.21652i
\(17\) 0.836385 − 0.965240i 0.202853 − 0.234105i −0.645203 − 0.764011i \(-0.723227\pi\)
0.848056 + 0.529906i \(0.177773\pi\)
\(18\) −9.99729 + 6.42487i −2.35638 + 1.51436i
\(19\) −0.307596 − 0.354985i −0.0705674 − 0.0814391i 0.719368 − 0.694629i \(-0.244431\pi\)
−0.789935 + 0.613190i \(0.789886\pi\)
\(20\) 0 0
\(21\) 0.847934 − 1.85672i 0.185034 − 0.405169i
\(22\) −6.73007 −1.43486
\(23\) −3.57257 + 3.19949i −0.744933 + 0.667140i
\(24\) −6.78684 −1.38536
\(25\) 0 0
\(26\) 5.09656 + 1.49648i 0.999517 + 0.293485i
\(27\) 8.83052 + 10.1910i 1.69943 + 1.96125i
\(28\) −0.381599 + 0.245239i −0.0721155 + 0.0463458i
\(29\) 2.61894 − 3.02242i 0.486325 − 0.561249i −0.458555 − 0.888666i \(-0.651633\pi\)
0.944880 + 0.327417i \(0.106178\pi\)
\(30\) 0 0
\(31\) −4.96368 + 1.45747i −0.891503 + 0.261769i −0.695236 − 0.718782i \(-0.744700\pi\)
−0.196267 + 0.980551i \(0.562882\pi\)
\(32\) 3.23653 + 2.07999i 0.572143 + 0.367694i
\(33\) 1.85971 + 12.9346i 0.323734 + 2.25162i
\(34\) 0.873484 + 1.91266i 0.149801 + 0.328019i
\(35\) 0 0
\(36\) −0.729764 − 5.07562i −0.121627 − 0.845936i
\(37\) −8.04195 − 5.16825i −1.32209 − 0.849655i −0.326658 − 0.945143i \(-0.605923\pi\)
−0.995431 + 0.0954879i \(0.969559\pi\)
\(38\) 0.741974 − 0.217863i 0.120364 − 0.0353421i
\(39\) 1.46778 − 10.2086i 0.235033 − 1.63469i
\(40\) 0 0
\(41\) 7.36949 − 4.73608i 1.15092 − 0.739652i 0.181099 − 0.983465i \(-0.442035\pi\)
0.969822 + 0.243813i \(0.0783983\pi\)
\(42\) 2.20062 + 2.53965i 0.339562 + 0.391876i
\(43\) 0.704508 + 0.206862i 0.107436 + 0.0315462i 0.335009 − 0.942215i \(-0.391261\pi\)
−0.227572 + 0.973761i \(0.573079\pi\)
\(44\) 1.20636 − 2.64157i 0.181866 − 0.398232i
\(45\) 0 0
\(46\) −2.34808 − 7.53826i −0.346206 − 1.11146i
\(47\) 0.589562 0.0859964 0.0429982 − 0.999075i \(-0.486309\pi\)
0.0429982 + 0.999075i \(0.486309\pi\)
\(48\) 6.52824 − 14.2948i 0.942270 − 2.06328i
\(49\) 6.32523 + 1.85726i 0.903605 + 0.265322i
\(50\) 0 0
\(51\) 3.43459 − 2.20728i 0.480939 − 0.309081i
\(52\) −1.50093 + 1.73217i −0.208142 + 0.240208i
\(53\) 0.955686 − 6.64695i 0.131274 − 0.913028i −0.812624 − 0.582789i \(-0.801961\pi\)
0.943897 − 0.330239i \(-0.107129\pi\)
\(54\) −21.3007 + 6.25446i −2.89866 + 0.851124i
\(55\) 0 0
\(56\) 0.192937 + 1.34191i 0.0257823 + 0.179320i
\(57\) −0.623742 − 1.36581i −0.0826167 − 0.180905i
\(58\) 2.73511 + 5.98905i 0.359137 + 0.786401i
\(59\) 0.734852 + 5.11101i 0.0956696 + 0.665396i 0.980068 + 0.198663i \(0.0636600\pi\)
−0.884398 + 0.466733i \(0.845431\pi\)
\(60\) 0 0
\(61\) −2.44333 + 0.717427i −0.312836 + 0.0918571i −0.434382 − 0.900729i \(-0.643033\pi\)
0.121545 + 0.992586i \(0.461215\pi\)
\(62\) 1.21207 − 8.43012i 0.153933 − 1.07063i
\(63\) 3.01841 − 3.48343i 0.380283 − 0.438871i
\(64\) 2.94303 − 1.89137i 0.367879 − 0.236422i
\(65\) 0 0
\(66\) −20.6420 − 6.06105i −2.54086 − 0.746063i
\(67\) 0.998254 − 2.18587i 0.121956 − 0.267047i −0.838801 − 0.544439i \(-0.816743\pi\)
0.960757 + 0.277392i \(0.0894700\pi\)
\(68\) −0.907297 −0.110026
\(69\) −13.8390 + 6.59584i −1.66602 + 0.794046i
\(70\) 0 0
\(71\) −3.46453 + 7.58626i −0.411164 + 0.900323i 0.584852 + 0.811140i \(0.301153\pi\)
−0.996015 + 0.0891826i \(0.971575\pi\)
\(72\) −14.7048 − 4.31771i −1.73297 − 0.508847i
\(73\) 0.483051 + 0.557471i 0.0565369 + 0.0652470i 0.783314 − 0.621627i \(-0.213528\pi\)
−0.726777 + 0.686874i \(0.758982\pi\)
\(74\) 13.2397 − 8.50861i 1.53908 − 0.989106i
\(75\) 0 0
\(76\) −0.0474869 + 0.330279i −0.00544712 + 0.0378856i
\(77\) 2.50458 − 0.735411i 0.285423 − 0.0838078i
\(78\) 14.2841 + 9.17984i 1.61736 + 1.03941i
\(79\) −2.33484 − 16.2392i −0.262691 − 1.82705i −0.512418 − 0.858736i \(-0.671250\pi\)
0.249727 − 0.968316i \(-0.419659\pi\)
\(80\) 0 0
\(81\) 8.91062 + 19.5115i 0.990069 + 2.16795i
\(82\) 2.05247 + 14.2752i 0.226657 + 1.57643i
\(83\) −8.77936 − 5.64215i −0.963660 − 0.619307i −0.0386517 − 0.999253i \(-0.512306\pi\)
−0.925009 + 0.379946i \(0.875943\pi\)
\(84\) −1.39128 + 0.408516i −0.151801 + 0.0445727i
\(85\) 0 0
\(86\) −0.791605 + 0.913560i −0.0853609 + 0.0985118i
\(87\) 10.7546 − 6.91157i 1.15302 − 0.740998i
\(88\) −5.68368 − 6.55931i −0.605882 − 0.699225i
\(89\) −5.75334 − 1.68933i −0.609853 − 0.179069i −0.0377997 − 0.999285i \(-0.512035\pi\)
−0.572054 + 0.820216i \(0.693853\pi\)
\(90\) 0 0
\(91\) −2.06020 −0.215967
\(92\) 3.37968 + 0.429603i 0.352356 + 0.0447892i
\(93\) −16.5368 −1.71479
\(94\) −0.403206 + 0.882897i −0.0415875 + 0.0910639i
\(95\) 0 0
\(96\) 8.05365 + 9.29441i 0.821972 + 0.948607i
\(97\) −5.42589 + 3.48701i −0.550916 + 0.354052i −0.786295 − 0.617852i \(-0.788003\pi\)
0.235379 + 0.971904i \(0.424367\pi\)
\(98\) −7.10721 + 8.20215i −0.717936 + 0.828543i
\(99\) −4.19947 + 29.2080i −0.422063 + 2.93551i
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 575.2.k.g.351.3 100
5.2 odd 4 115.2.j.a.29.3 yes 100
5.3 odd 4 115.2.j.a.29.8 yes 100
5.4 even 2 inner 575.2.k.g.351.8 100
23.4 even 11 inner 575.2.k.g.326.3 100
115.4 even 22 inner 575.2.k.g.326.8 100
115.27 odd 44 115.2.j.a.4.8 yes 100
115.73 odd 44 115.2.j.a.4.3 ✓ 100
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
115.2.j.a.4.3 ✓ 100 115.73 odd 44
115.2.j.a.4.8 yes 100 115.27 odd 44
115.2.j.a.29.3 yes 100 5.2 odd 4
115.2.j.a.29.8 yes 100 5.3 odd 4
575.2.k.g.326.3 100 23.4 even 11 inner
575.2.k.g.326.8 100 115.4 even 22 inner
575.2.k.g.351.3 100 1.1 even 1 trivial
575.2.k.g.351.8 100 5.4 even 2 inner