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Properties

Label 575.2.p.a.499.1
Level $575$
Weight $2$
Character 575.499
Analytic conductor $4.591$
Analytic rank $0$
Dimension $20$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [575,2,Mod(49,575)] 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("575.49"); 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([11, 16])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 575 = 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 575.p (of order \(22\), degree \(10\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.59139811622\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(2\) over \(\Q(\zeta_{22})\)
Coefficient field: \(\Q(\zeta_{44})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - x^{18} + x^{16} - x^{14} + x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 115)
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label 499.1
Root \(0.909632 + 0.415415i\) of defining polynomial
Character \(\chi\) \(=\) 575.499
Dual form 575.2.p.a.174.1

$q$-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+(-1.81219 - 1.57028i) q^{2} +(-0.540641 - 0.841254i) q^{3} +(0.533654 + 3.71165i) q^{4} +(-0.341254 + 2.37347i) q^{6} +(-0.386758 - 1.31718i) q^{7} +(2.26844 - 3.52977i) q^{8} +(0.830830 - 1.81926i) q^{9} +(1.49103 + 1.72073i) q^{11} +(2.83392 - 2.45561i) q^{12} +(0.239945 - 0.817178i) q^{13} +(-1.36745 + 2.99430i) q^{14} +(-2.45773 + 0.721655i) q^{16} +(4.04104 + 0.581014i) q^{17} +(-4.36237 + 1.99223i) q^{18} +(-0.384487 - 2.67417i) q^{19} +(-0.898983 + 1.03748i) q^{21} -5.45963i q^{22} +(4.38840 - 1.93440i) q^{23} -4.19584 q^{24} +(-1.71802 + 1.10411i) q^{26} +(-4.94911 + 0.711574i) q^{27} +(4.68251 - 2.13843i) q^{28} +(0.951362 - 6.61687i) q^{29} +(-8.68043 - 5.57857i) q^{31} +(-2.04627 - 0.934499i) q^{32} +(0.641465 - 2.18463i) q^{33} +(-6.41080 - 7.39846i) q^{34} +(7.19584 + 2.11289i) q^{36} +(-9.56379 - 4.36764i) q^{37} +(-3.50241 + 5.44986i) q^{38} +(-0.817178 + 0.239945i) q^{39} +(2.86653 + 6.27683i) q^{41} +(3.25827 - 0.468468i) q^{42} +(3.79962 + 5.91232i) q^{43} +(-5.59107 + 6.45244i) q^{44} +(-10.9902 - 3.38549i) q^{46} +8.71406i q^{47} +(1.93584 + 1.67742i) q^{48} +(4.30340 - 2.76563i) q^{49} +(-1.69597 - 3.71366i) q^{51} +(3.16113 + 0.454501i) q^{52} +(-1.92452 - 6.55432i) q^{53} +(10.0861 + 6.48195i) q^{54} +(-5.52667 - 1.62278i) q^{56} +(-2.04178 + 1.76921i) q^{57} +(-12.1144 + 10.4972i) q^{58} +(2.60598 + 0.765186i) q^{59} +(0.764538 + 0.491338i) q^{61} +(6.97073 + 23.7401i) q^{62} +(-2.71763 - 0.390736i) q^{63} +(4.36897 + 9.56672i) q^{64} +(-4.59293 + 2.95170i) q^{66} +(-7.49528 - 6.49469i) q^{67} +15.3090i q^{68} +(-3.99987 - 2.64594i) q^{69} +(-1.29740 + 1.49728i) q^{71} +(-4.53689 - 7.05954i) q^{72} +(-8.03022 + 1.15457i) q^{73} +(10.4731 + 22.9328i) q^{74} +(9.72038 - 2.85416i) q^{76} +(1.68985 - 2.62945i) q^{77} +(1.85767 + 0.848368i) q^{78} +(-14.1009 - 4.14039i) q^{79} +(-0.654861 - 0.755750i) q^{81} +(4.66164 - 15.8761i) q^{82} +(-1.59203 - 0.727055i) q^{83} +(-4.33052 - 2.78305i) q^{84} +(2.39833 - 16.6807i) q^{86} +(-6.08081 + 2.77701i) q^{87} +(9.45610 - 1.35958i) q^{88} +(-4.47262 + 2.87438i) q^{89} -1.16917 q^{91} +(9.52171 + 15.2559i) q^{92} +10.3184i q^{93} +(13.6835 - 15.7916i) q^{94} +(0.320145 + 2.22666i) q^{96} +(9.91860 - 4.52967i) q^{97} +(-12.1414 - 1.74567i) q^{98} +(4.36926 - 1.28293i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 2 q^{4} + 12 q^{6} - 4 q^{9} + 16 q^{11} + 6 q^{14} + 10 q^{16} + 26 q^{19} + 12 q^{21} + 12 q^{24} + 22 q^{26} - 4 q^{29} - 40 q^{31} - 32 q^{34} + 48 q^{36} - 10 q^{41} - 38 q^{44} - 32 q^{46}+ \cdots - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(51\) \(277\)
\(\chi(n)\) \(e\left(\frac{4}{11}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.81219 − 1.57028i −1.28142 − 1.11035i −0.988014 − 0.154363i \(-0.950667\pi\)
−0.293401 − 0.955989i \(-0.594787\pi\)
\(3\) −0.540641 − 0.841254i −0.312139 − 0.485698i 0.649369 − 0.760473i \(-0.275033\pi\)
−0.961508 + 0.274775i \(0.911397\pi\)
\(4\) 0.533654 + 3.71165i 0.266827 + 1.85582i
\(5\) 0 0
\(6\) −0.341254 + 2.37347i −0.139316 + 0.968965i
\(7\) −0.386758 − 1.31718i −0.146181 − 0.497847i 0.853551 − 0.521010i \(-0.174444\pi\)
−0.999732 + 0.0231631i \(0.992626\pi\)
\(8\) 2.26844 − 3.52977i 0.802016 − 1.24796i
\(9\) 0.830830 − 1.81926i 0.276943 − 0.606421i
\(10\) 0 0
\(11\) 1.49103 + 1.72073i 0.449561 + 0.518821i 0.934614 − 0.355664i \(-0.115745\pi\)
−0.485053 + 0.874485i \(0.661200\pi\)
\(12\) 2.83392 − 2.45561i 0.818083 − 0.708873i
\(13\) 0.239945 − 0.817178i 0.0665488 − 0.226644i −0.919503 − 0.393083i \(-0.871409\pi\)
0.986052 + 0.166438i \(0.0532267\pi\)
\(14\) −1.36745 + 2.99430i −0.365467 + 0.800261i
\(15\) 0 0
\(16\) −2.45773 + 0.721655i −0.614432 + 0.180414i
\(17\) 4.04104 + 0.581014i 0.980097 + 0.140917i 0.613698 − 0.789541i \(-0.289681\pi\)
0.366399 + 0.930458i \(0.380591\pi\)
\(18\) −4.36237 + 1.99223i −1.02822 + 0.469573i
\(19\) −0.384487 − 2.67417i −0.0882074 − 0.613496i −0.985195 − 0.171439i \(-0.945158\pi\)
0.896987 − 0.442056i \(-0.145751\pi\)
\(20\) 0 0
\(21\) −0.898983 + 1.03748i −0.196174 + 0.226397i
\(22\) − 5.45963i − 1.16400i
\(23\) 4.38840 − 1.93440i 0.915046 − 0.403351i
\(24\) −4.19584 −0.856473
\(25\) 0 0
\(26\) −1.71802 + 1.10411i −0.336932 + 0.216533i
\(27\) −4.94911 + 0.711574i −0.952456 + 0.136943i
\(28\) 4.68251 − 2.13843i 0.884911 − 0.404125i
\(29\) 0.951362 − 6.61687i 0.176664 − 1.22872i −0.687753 − 0.725945i \(-0.741403\pi\)
0.864416 − 0.502777i \(-0.167688\pi\)
\(30\) 0 0
\(31\) −8.68043 − 5.57857i −1.55905 − 1.00194i −0.982801 − 0.184668i \(-0.940879\pi\)
−0.576250 − 0.817273i \(-0.695485\pi\)
\(32\) −2.04627 − 0.934499i −0.361732 − 0.165198i
\(33\) 0.641465 − 2.18463i 0.111665 − 0.380295i
\(34\) −6.41080 − 7.39846i −1.09944 − 1.26883i
\(35\) 0 0
\(36\) 7.19584 + 2.11289i 1.19931 + 0.352148i
\(37\) −9.56379 − 4.36764i −1.57228 − 0.718035i −0.577151 − 0.816637i \(-0.695836\pi\)
−0.995127 + 0.0986020i \(0.968563\pi\)
\(38\) −3.50241 + 5.44986i −0.568166 + 0.884084i
\(39\) −0.817178 + 0.239945i −0.130853 + 0.0384220i
\(40\) 0 0
\(41\) 2.86653 + 6.27683i 0.447677 + 0.980276i 0.990125 + 0.140187i \(0.0447703\pi\)
−0.542448 + 0.840089i \(0.682502\pi\)
\(42\) 3.25827 − 0.468468i 0.502761 − 0.0722862i
\(43\) 3.79962 + 5.91232i 0.579436 + 0.901620i 0.999984 − 0.00564997i \(-0.00179845\pi\)
−0.420548 + 0.907270i \(0.638162\pi\)
\(44\) −5.59107 + 6.45244i −0.842885 + 0.972742i
\(45\) 0 0
\(46\) −10.9902 − 3.38549i −1.62041 − 0.499164i
\(47\) 8.71406i 1.27108i 0.772070 + 0.635538i \(0.219222\pi\)
−0.772070 + 0.635538i \(0.780778\pi\)
\(48\) 1.93584 + 1.67742i 0.279415 + 0.242114i
\(49\) 4.30340 − 2.76563i 0.614771 − 0.395089i
\(50\) 0 0
\(51\) −1.69597 − 3.71366i −0.237484 − 0.520016i
\(52\) 3.16113 + 0.454501i 0.438369 + 0.0630280i
\(53\) −1.92452 − 6.55432i −0.264353 − 0.900305i −0.979519 − 0.201350i \(-0.935467\pi\)
0.715166 − 0.698955i \(-0.246351\pi\)
\(54\) 10.0861 + 6.48195i 1.37255 + 0.882082i
\(55\) 0 0
\(56\) −5.52667 − 1.62278i −0.738533 − 0.216853i
\(57\) −2.04178 + 1.76921i −0.270441 + 0.234338i
\(58\) −12.1144 + 10.4972i −1.59069 + 1.37834i
\(59\) 2.60598 + 0.765186i 0.339270 + 0.0996187i 0.446930 − 0.894569i \(-0.352517\pi\)
−0.107660 + 0.994188i \(0.534336\pi\)
\(60\) 0 0
\(61\) 0.764538 + 0.491338i 0.0978890 + 0.0629094i 0.588670 − 0.808374i \(-0.299652\pi\)
−0.490781 + 0.871283i \(0.663288\pi\)
\(62\) 6.97073 + 23.7401i 0.885284 + 3.01500i
\(63\) −2.71763 − 0.390736i −0.342389 − 0.0492280i
\(64\) 4.36897 + 9.56672i 0.546122 + 1.19584i
\(65\) 0 0
\(66\) −4.59293 + 2.95170i −0.565351 + 0.363329i
\(67\) −7.49528 − 6.49469i −0.915694 − 0.793453i 0.0631624 − 0.998003i \(-0.479881\pi\)
−0.978856 + 0.204550i \(0.934427\pi\)
\(68\) 15.3090i 1.85649i
\(69\) −3.99987 − 2.64594i −0.481528 − 0.318534i
\(70\) 0 0
\(71\) −1.29740 + 1.49728i −0.153973 + 0.177694i −0.827495 − 0.561472i \(-0.810235\pi\)
0.673522 + 0.739167i \(0.264781\pi\)
\(72\) −4.53689 − 7.05954i −0.534678 − 0.831974i
\(73\) −8.03022 + 1.15457i −0.939866 + 0.135132i −0.595186 − 0.803588i \(-0.702922\pi\)
−0.344680 + 0.938720i \(0.612013\pi\)
\(74\) 10.4731 + 22.9328i 1.21747 + 2.66588i
\(75\) 0 0
\(76\) 9.72038 − 2.85416i 1.11500 − 0.327395i
\(77\) 1.68985 − 2.62945i 0.192576 − 0.299654i
\(78\) 1.85767 + 0.848368i 0.210339 + 0.0960587i
\(79\) −14.1009 − 4.14039i −1.58647 − 0.465830i −0.634730 − 0.772734i \(-0.718889\pi\)
−0.951741 + 0.306904i \(0.900707\pi\)
\(80\) 0 0
\(81\) −0.654861 − 0.755750i −0.0727623 − 0.0839722i
\(82\) 4.66164 − 15.8761i 0.514792 − 1.75322i
\(83\) −1.59203 − 0.727055i −0.174748 − 0.0798047i 0.326120 − 0.945328i \(-0.394259\pi\)
−0.500868 + 0.865524i \(0.666986\pi\)
\(84\) −4.33052 − 2.78305i −0.472498 − 0.303656i
\(85\) 0 0
\(86\) 2.39833 − 16.6807i 0.258618 − 1.79873i
\(87\) −6.08081 + 2.77701i −0.651931 + 0.297727i
\(88\) 9.45610 − 1.35958i 1.00802 − 0.144932i
\(89\) −4.47262 + 2.87438i −0.474096 + 0.304683i −0.755782 − 0.654824i \(-0.772743\pi\)
0.281685 + 0.959507i \(0.409107\pi\)
\(90\) 0 0
\(91\) −1.16917 −0.122562
\(92\) 9.52171 + 15.2559i 0.992707 + 1.59054i
\(93\) 10.3184i 1.06997i
\(94\) 13.6835 − 15.7916i 1.41134 − 1.62878i
\(95\) 0 0
\(96\) 0.320145 + 2.22666i 0.0326747 + 0.227257i
\(97\) 9.91860 − 4.52967i 1.00708 − 0.459918i 0.157580 − 0.987506i \(-0.449631\pi\)
0.849501 + 0.527588i \(0.176903\pi\)
\(98\) −12.1414 − 1.74567i −1.22647 − 0.176339i
\(99\) 4.36926 − 1.28293i 0.439127 − 0.128939i
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.p.a.499.1 20
5.2 odd 4 115.2.g.a.16.1 ✓ 10
5.3 odd 4 575.2.k.a.476.1 10
5.4 even 2 inner 575.2.p.a.499.2 20
23.13 even 11 inner 575.2.p.a.174.2 20
115.13 odd 44 575.2.k.a.151.1 10
115.17 even 44 2645.2.a.o.1.1 5
115.52 odd 44 2645.2.a.n.1.1 5
115.59 even 22 inner 575.2.p.a.174.1 20
115.82 odd 44 115.2.g.a.36.1 yes 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
115.2.g.a.16.1 ✓ 10 5.2 odd 4
115.2.g.a.36.1 yes 10 115.82 odd 44
575.2.k.a.151.1 10 115.13 odd 44
575.2.k.a.476.1 10 5.3 odd 4
575.2.p.a.174.1 20 115.59 even 22 inner
575.2.p.a.174.2 20 23.13 even 11 inner
575.2.p.a.499.1 20 1.1 even 1 trivial
575.2.p.a.499.2 20 5.4 even 2 inner
2645.2.a.n.1.1 5 115.52 odd 44
2645.2.a.o.1.1 5 115.17 even 44