↑

Properties

Label 575.2.k.a.151.1
Level $575$
Weight $2$
Character 575.151
Analytic conductor $4.591$
Analytic rank $0$
Dimension $10$
Inner twists $2$

Related objects

Downloads

Learn more

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(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 = [10,-5] 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: \(10\)
Coefficient field: \(\Q(\zeta_{22})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \) 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: $\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label 151.1
Root \(-0.841254 + 0.540641i\) of defining polynomial
Character \(\chi\) \(=\) 575.151
Dual form 575.2.k.a.476.1

$q$-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+(-1.57028 - 1.81219i) q^{2} +(0.841254 + 0.540641i) q^{3} +(-0.533654 + 3.71165i) q^{4} +(-0.341254 - 2.37347i) q^{6} +(-1.31718 - 0.386758i) q^{7} +(3.52977 - 2.26844i) q^{8} +(-0.830830 - 1.81926i) q^{9} +(1.49103 - 1.72073i) q^{11} +(-2.45561 + 2.83392i) q^{12} +(0.817178 - 0.239945i) q^{13} +(1.36745 + 2.99430i) q^{14} +(-2.45773 - 0.721655i) q^{16} +(0.581014 + 4.04104i) q^{17} +(-1.99223 + 4.36237i) q^{18} +(0.384487 - 2.67417i) q^{19} +(-0.898983 - 1.03748i) q^{21} -5.45963 q^{22} +(1.93440 - 4.38840i) q^{23} +4.19584 q^{24} +(-1.71802 - 1.10411i) q^{26} +(0.711574 - 4.94911i) q^{27} +(2.13843 - 4.68251i) q^{28} +(-0.951362 - 6.61687i) q^{29} +(-8.68043 + 5.57857i) q^{31} +(-0.934499 - 2.04627i) q^{32} +(2.18463 - 0.641465i) q^{33} +(6.41080 - 7.39846i) q^{34} +(7.19584 - 2.11289i) q^{36} +(-4.36764 - 9.56379i) q^{37} +(-5.44986 + 3.50241i) q^{38} +(0.817178 + 0.239945i) q^{39} +(2.86653 - 6.27683i) q^{41} +(-0.468468 + 3.25827i) q^{42} +(-5.91232 - 3.79962i) q^{43} +(5.59107 + 6.45244i) q^{44} +(-10.9902 + 3.38549i) q^{46} +8.71406 q^{47} +(-1.67742 - 1.93584i) q^{48} +(-4.30340 - 2.76563i) q^{49} +(-1.69597 + 3.71366i) q^{51} +(0.454501 + 3.16113i) q^{52} +(6.55432 + 1.92452i) q^{53} +(-10.0861 + 6.48195i) q^{54} +(-5.52667 + 1.62278i) q^{56} +(1.76921 - 2.04178i) q^{57} +(-10.4972 + 12.1144i) q^{58} +(-2.60598 + 0.765186i) q^{59} +(0.764538 - 0.491338i) q^{61} +(23.7401 + 6.97073i) q^{62} +(0.390736 + 2.71763i) q^{63} +(-4.36897 + 9.56672i) q^{64} +(-4.59293 - 2.95170i) q^{66} +(-6.49469 - 7.49528i) q^{67} -15.3090 q^{68} +(3.99987 - 2.64594i) q^{69} +(-1.29740 - 1.49728i) q^{71} +(-7.05954 - 4.53689i) q^{72} +(-1.15457 + 8.03022i) q^{73} +(-10.4731 + 22.9328i) q^{74} +(9.72038 + 2.85416i) q^{76} +(-2.62945 + 1.68985i) q^{77} +(-0.848368 - 1.85767i) q^{78} +(14.1009 - 4.14039i) q^{79} +(-0.654861 + 0.755750i) q^{81} +(-15.8761 + 4.66164i) q^{82} +(0.727055 + 1.59203i) q^{83} +(4.33052 - 2.78305i) q^{84} +(2.39833 + 16.6807i) q^{86} +(2.77701 - 6.08081i) q^{87} +(1.35958 - 9.45610i) q^{88} +(4.47262 + 2.87438i) q^{89} -1.16917 q^{91} +(15.2559 + 9.52171i) q^{92} -10.3184 q^{93} +(-13.6835 - 15.7916i) q^{94} +(0.320145 - 2.22666i) q^{96} +(-4.52967 + 9.91860i) q^{97} +(1.74567 + 12.1414i) q^{98} +(-4.36926 - 1.28293i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 5 q^{2} - q^{3} - q^{4} + 6 q^{6} - 5 q^{7} + 16 q^{8} + 2 q^{9} + 8 q^{11} - q^{12} - 3 q^{14} + 5 q^{16} + 23 q^{17} + 10 q^{18} - 13 q^{19} + 6 q^{21} - 4 q^{22} + 21 q^{23} - 6 q^{24} + 11 q^{26}+ \cdots + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(51\) \(277\)
\(\chi(n)\) \(e\left(\frac{7}{11}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.57028 − 1.81219i −1.11035 − 1.28142i −0.955989 − 0.293401i \(-0.905213\pi\)
−0.154363 − 0.988014i \(-0.549333\pi\)
\(3\) 0.841254 + 0.540641i 0.485698 + 0.312139i 0.760473 − 0.649369i \(-0.224967\pi\)
−0.274775 + 0.961508i \(0.588603\pi\)
\(4\) −0.533654 + 3.71165i −0.266827 + 1.85582i
\(5\) 0 0
\(6\) −0.341254 − 2.37347i −0.139316 − 0.968965i
\(7\) −1.31718 − 0.386758i −0.497847 − 0.146181i 0.0231631 − 0.999732i \(-0.492626\pi\)
−0.521010 + 0.853551i \(0.674444\pi\)
\(8\) 3.52977 − 2.26844i 1.24796 − 0.802016i
\(9\) −0.830830 − 1.81926i −0.276943 − 0.606421i
\(10\) 0 0
\(11\) 1.49103 − 1.72073i 0.449561 − 0.518821i −0.485053 − 0.874485i \(-0.661200\pi\)
0.934614 + 0.355664i \(0.115745\pi\)
\(12\) −2.45561 + 2.83392i −0.708873 + 0.818083i
\(13\) 0.817178 − 0.239945i 0.226644 − 0.0665488i −0.166438 − 0.986052i \(-0.553227\pi\)
0.393083 + 0.919503i \(0.371409\pi\)
\(14\) 1.36745 + 2.99430i 0.365467 + 0.800261i
\(15\) 0 0
\(16\) −2.45773 − 0.721655i −0.614432 − 0.180414i
\(17\) 0.581014 + 4.04104i 0.140917 + 0.980097i 0.930458 + 0.366399i \(0.119409\pi\)
−0.789541 + 0.613698i \(0.789681\pi\)
\(18\) −1.99223 + 4.36237i −0.469573 + 1.02822i
\(19\) 0.384487 − 2.67417i 0.0882074 − 0.613496i −0.896987 − 0.442056i \(-0.854249\pi\)
0.985195 − 0.171439i \(-0.0548418\pi\)
\(20\) 0 0
\(21\) −0.898983 − 1.03748i −0.196174 − 0.226397i
\(22\) −5.45963 −1.16400
\(23\) 1.93440 − 4.38840i 0.403351 − 0.915046i
\(24\) 4.19584 0.856473
\(25\) 0 0
\(26\) −1.71802 − 1.10411i −0.336932 − 0.216533i
\(27\) 0.711574 − 4.94911i 0.136943 − 0.952456i
\(28\) 2.13843 − 4.68251i 0.404125 − 0.884911i
\(29\) −0.951362 − 6.61687i −0.176664 − 1.22872i −0.864416 − 0.502777i \(-0.832312\pi\)
0.687753 − 0.725945i \(-0.258597\pi\)
\(30\) 0 0
\(31\) −8.68043 + 5.57857i −1.55905 + 1.00194i −0.576250 + 0.817273i \(0.695485\pi\)
−0.982801 + 0.184668i \(0.940879\pi\)
\(32\) −0.934499 − 2.04627i −0.165198 − 0.361732i
\(33\) 2.18463 − 0.641465i 0.380295 − 0.111665i
\(34\) 6.41080 − 7.39846i 1.09944 − 1.26883i
\(35\) 0 0
\(36\) 7.19584 − 2.11289i 1.19931 − 0.352148i
\(37\) −4.36764 − 9.56379i −0.718035 − 1.57228i −0.816637 − 0.577151i \(-0.804164\pi\)
0.0986020 − 0.995127i \(-0.468563\pi\)
\(38\) −5.44986 + 3.50241i −0.884084 + 0.568166i
\(39\) 0.817178 + 0.239945i 0.130853 + 0.0384220i
\(40\) 0 0
\(41\) 2.86653 − 6.27683i 0.447677 − 0.980276i −0.542448 − 0.840089i \(-0.682502\pi\)
0.990125 − 0.140187i \(-0.0447703\pi\)
\(42\) −0.468468 + 3.25827i −0.0722862 + 0.502761i
\(43\) −5.91232 − 3.79962i −0.901620 − 0.579436i 0.00564997 − 0.999984i \(-0.498202\pi\)
−0.907270 + 0.420548i \(0.861838\pi\)
\(44\) 5.59107 + 6.45244i 0.842885 + 0.972742i
\(45\) 0 0
\(46\) −10.9902 + 3.38549i −1.62041 + 0.499164i
\(47\) 8.71406 1.27108 0.635538 − 0.772070i \(-0.280778\pi\)
0.635538 + 0.772070i \(0.280778\pi\)
\(48\) −1.67742 − 1.93584i −0.242114 − 0.279415i
\(49\) −4.30340 − 2.76563i −0.614771 − 0.395089i
\(50\) 0 0
\(51\) −1.69597 + 3.71366i −0.237484 + 0.520016i
\(52\) 0.454501 + 3.16113i 0.0630280 + 0.438369i
\(53\) 6.55432 + 1.92452i 0.900305 + 0.264353i 0.698955 − 0.715166i \(-0.253649\pi\)
0.201350 + 0.979519i \(0.435467\pi\)
\(54\) −10.0861 + 6.48195i −1.37255 + 0.882082i
\(55\) 0 0
\(56\) −5.52667 + 1.62278i −0.738533 + 0.216853i
\(57\) 1.76921 − 2.04178i 0.234338 − 0.270441i
\(58\) −10.4972 + 12.1144i −1.37834 + 1.59069i
\(59\) −2.60598 + 0.765186i −0.339270 + 0.0996187i −0.446930 − 0.894569i \(-0.647483\pi\)
0.107660 + 0.994188i \(0.465664\pi\)
\(60\) 0 0
\(61\) 0.764538 − 0.491338i 0.0978890 − 0.0629094i −0.490781 − 0.871283i \(-0.663288\pi\)
0.588670 + 0.808374i \(0.299652\pi\)
\(62\) 23.7401 + 6.97073i 3.01500 + 0.885284i
\(63\) 0.390736 + 2.71763i 0.0492280 + 0.342389i
\(64\) −4.36897 + 9.56672i −0.546122 + 1.19584i
\(65\) 0 0
\(66\) −4.59293 − 2.95170i −0.565351 − 0.363329i
\(67\) −6.49469 − 7.49528i −0.793453 − 0.915694i 0.204550 − 0.978856i \(-0.434427\pi\)
−0.998003 + 0.0631624i \(0.979881\pi\)
\(68\) −15.3090 −1.85649
\(69\) 3.99987 − 2.64594i 0.481528 − 0.318534i
\(70\) 0 0
\(71\) −1.29740 − 1.49728i −0.153973 − 0.177694i 0.673522 − 0.739167i \(-0.264781\pi\)
−0.827495 + 0.561472i \(0.810235\pi\)
\(72\) −7.05954 − 4.53689i −0.831974 − 0.534678i
\(73\) −1.15457 + 8.03022i −0.135132 + 0.939866i 0.803588 + 0.595186i \(0.202922\pi\)
−0.938720 + 0.344680i \(0.887987\pi\)
\(74\) −10.4731 + 22.9328i −1.21747 + 2.66588i
\(75\) 0 0
\(76\) 9.72038 + 2.85416i 1.11500 + 0.327395i
\(77\) −2.62945 + 1.68985i −0.299654 + 0.192576i
\(78\) −0.848368 − 1.85767i −0.0960587 − 0.210339i
\(79\) 14.1009 − 4.14039i 1.58647 − 0.465830i 0.634730 − 0.772734i \(-0.281111\pi\)
0.951741 + 0.306904i \(0.0992929\pi\)
\(80\) 0 0
\(81\) −0.654861 + 0.755750i −0.0727623 + 0.0839722i
\(82\) −15.8761 + 4.66164i −1.75322 + 0.514792i
\(83\) 0.727055 + 1.59203i 0.0798047 + 0.174748i 0.945328 − 0.326120i \(-0.105741\pi\)
−0.865524 + 0.500868i \(0.833014\pi\)
\(84\) 4.33052 − 2.78305i 0.472498 − 0.303656i
\(85\) 0 0
\(86\) 2.39833 + 16.6807i 0.258618 + 1.79873i
\(87\) 2.77701 − 6.08081i 0.297727 − 0.651931i
\(88\) 1.35958 − 9.45610i 0.144932 − 1.00802i
\(89\) 4.47262 + 2.87438i 0.474096 + 0.304683i 0.755782 − 0.654824i \(-0.227257\pi\)
−0.281685 + 0.959507i \(0.590893\pi\)
\(90\) 0 0
\(91\) −1.16917 −0.122562
\(92\) 15.2559 + 9.52171i 1.59054 + 0.992707i
\(93\) −10.3184 −1.06997
\(94\) −13.6835 − 15.7916i −1.41134 − 1.62878i
\(95\) 0 0
\(96\) 0.320145 − 2.22666i 0.0326747 − 0.227257i
\(97\) −4.52967 + 9.91860i −0.459918 + 1.00708i 0.527588 + 0.849501i \(0.323097\pi\)
−0.987506 + 0.157580i \(0.949631\pi\)
\(98\) 1.74567 + 12.1414i 0.176339 + 1.22647i
\(99\) −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.k.a.151.1 10
5.2 odd 4 575.2.p.a.174.2 20
5.3 odd 4 575.2.p.a.174.1 20
5.4 even 2 115.2.g.a.36.1 yes 10
23.16 even 11 inner 575.2.k.a.476.1 10
115.4 even 22 2645.2.a.n.1.1 5
115.19 odd 22 2645.2.a.o.1.1 5
115.39 even 22 115.2.g.a.16.1 ✓ 10
115.62 odd 44 575.2.p.a.499.1 20
115.108 odd 44 575.2.p.a.499.2 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
115.2.g.a.16.1 ✓ 10 115.39 even 22
115.2.g.a.36.1 yes 10 5.4 even 2
575.2.k.a.151.1 10 1.1 even 1 trivial
575.2.k.a.476.1 10 23.16 even 11 inner
575.2.p.a.174.1 20 5.3 odd 4
575.2.p.a.174.2 20 5.2 odd 4
575.2.p.a.499.1 20 115.62 odd 44
575.2.p.a.499.2 20 115.108 odd 44
2645.2.a.n.1.1 5 115.4 even 22
2645.2.a.o.1.1 5 115.19 odd 22