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Properties

Label 525.3.h.b.76.9
Level $525$
Weight $3$
Character 525.76
Analytic conductor $14.305$
Analytic rank $0$
Dimension $10$
Inner twists $2$

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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] = [525,3,Mod(76,525)] 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("525.76"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(525, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 525.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,-2] 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: \(14.3052138789\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} + 13x^{8} + 2x^{7} + 118x^{6} + 8x^{5} + 403x^{4} + 299x^{3} + 931x^{2} + 186x + 36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 76.9
Root \(-1.33987 + 2.32071i\) of defining polynomial
Character \(\chi\) \(=\) 525.76
Dual form 525.3.h.b.76.10

$q$-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)
 
\(f(q)\) \(=\) \(q+2.67973 q^{2} -1.73205i q^{3} +3.18096 q^{4} -4.64143i q^{6} +(-6.16116 - 3.32267i) q^{7} -2.19482 q^{8} -3.00000 q^{9} +13.6287 q^{11} -5.50958i q^{12} -24.7132i q^{13} +(-16.5102 - 8.90386i) q^{14} -18.6053 q^{16} +11.4102i q^{17} -8.03919 q^{18} -33.2007i q^{19} +(-5.75503 + 10.6714i) q^{21} +36.5213 q^{22} -34.7650 q^{23} +3.80154i q^{24} -66.2246i q^{26} +5.19615i q^{27} +(-19.5984 - 10.5693i) q^{28} +26.5608 q^{29} -18.5429i q^{31} -41.0780 q^{32} -23.6056i q^{33} +30.5763i q^{34} -9.54287 q^{36} +24.5153 q^{37} -88.9689i q^{38} -42.8044 q^{39} -0.626750i q^{41} +(-15.4219 + 28.5966i) q^{42} -19.8109 q^{43} +43.3524 q^{44} -93.1607 q^{46} +41.9033i q^{47} +32.2254i q^{48} +(26.9197 + 40.9430i) q^{49} +19.7631 q^{51} -78.6114i q^{52} +24.7749 q^{53} +13.9243i q^{54} +(13.5226 + 7.29265i) q^{56} -57.5053 q^{57} +71.1758 q^{58} +73.3021i q^{59} +71.9661i q^{61} -49.6901i q^{62} +(18.4835 + 9.96801i) q^{63} -35.6567 q^{64} -63.2568i q^{66} +58.7724 q^{67} +36.2954i q^{68} +60.2147i q^{69} +82.1050 q^{71} +6.58445 q^{72} -97.7209i q^{73} +65.6943 q^{74} -105.610i q^{76} +(-83.9687 - 45.2837i) q^{77} -114.704 q^{78} +69.3670 q^{79} +9.00000 q^{81} -1.67952i q^{82} -27.2418i q^{83} +(-18.3065 + 33.9454i) q^{84} -53.0879 q^{86} -46.0047i q^{87} -29.9126 q^{88} -117.321i q^{89} +(-82.1136 + 152.262i) q^{91} -110.586 q^{92} -32.1173 q^{93} +112.289i q^{94} +71.1492i q^{96} -178.187i q^{97} +(72.1376 + 109.716i) q^{98} -40.8862 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 2 q^{2} + 10 q^{4} - 9 q^{7} - 28 q^{8} - 30 q^{9} + 32 q^{11} - 16 q^{14} - 54 q^{16} + 6 q^{18} - 15 q^{21} - 74 q^{22} - 32 q^{23} - 86 q^{28} - 40 q^{29} + 66 q^{32} - 30 q^{36} + 216 q^{37}+ \cdots - 96 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.67973 1.33987 0.669933 − 0.742422i \(-0.266323\pi\)
0.669933 + 0.742422i \(0.266323\pi\)
\(3\) − 1.73205i − 0.577350i
\(4\) 3.18096 0.795239
\(5\) 0 0
\(6\) − 4.64143i − 0.773572i
\(7\) −6.16116 − 3.32267i −0.880165 − 0.474667i
\(8\) −2.19482 −0.274352
\(9\) −3.00000 −0.333333
\(10\) 0 0
\(11\) 13.6287 1.23897 0.619487 − 0.785007i \(-0.287340\pi\)
0.619487 + 0.785007i \(0.287340\pi\)
\(12\) − 5.50958i − 0.459131i
\(13\) − 24.7132i − 1.90101i −0.310707 − 0.950506i \(-0.600566\pi\)
0.310707 − 0.950506i \(-0.399434\pi\)
\(14\) −16.5102 − 8.90386i −1.17930 − 0.635990i
\(15\) 0 0
\(16\) −18.6053 −1.16283
\(17\) 11.4102i 0.671190i 0.942006 + 0.335595i \(0.108937\pi\)
−0.942006 + 0.335595i \(0.891063\pi\)
\(18\) −8.03919 −0.446622
\(19\) − 33.2007i − 1.74740i −0.486461 − 0.873702i \(-0.661712\pi\)
0.486461 − 0.873702i \(-0.338288\pi\)
\(20\) 0 0
\(21\) −5.75503 + 10.6714i −0.274049 + 0.508164i
\(22\) 36.5213 1.66006
\(23\) −34.7650 −1.51152 −0.755760 − 0.654849i \(-0.772732\pi\)
−0.755760 + 0.654849i \(0.772732\pi\)
\(24\) 3.80154i 0.158397i
\(25\) 0 0
\(26\) − 66.2246i − 2.54710i
\(27\) 5.19615i 0.192450i
\(28\) −19.5984 − 10.5693i −0.699942 − 0.377474i
\(29\) 26.5608 0.915890 0.457945 − 0.888980i \(-0.348586\pi\)
0.457945 + 0.888980i \(0.348586\pi\)
\(30\) 0 0
\(31\) − 18.5429i − 0.598160i −0.954228 − 0.299080i \(-0.903320\pi\)
0.954228 − 0.299080i \(-0.0966797\pi\)
\(32\) −41.0780 −1.28369
\(33\) − 23.6056i − 0.715322i
\(34\) 30.5763i 0.899304i
\(35\) 0 0
\(36\) −9.54287 −0.265080
\(37\) 24.5153 0.662575 0.331287 − 0.943530i \(-0.392517\pi\)
0.331287 + 0.943530i \(0.392517\pi\)
\(38\) − 88.9689i − 2.34129i
\(39\) −42.8044 −1.09755
\(40\) 0 0
\(41\) − 0.626750i − 0.0152866i −0.999971 − 0.00764329i \(-0.997567\pi\)
0.999971 − 0.00764329i \(-0.00243296\pi\)
\(42\) −15.4219 + 28.5966i −0.367189 + 0.680871i
\(43\) −19.8109 −0.460718 −0.230359 − 0.973106i \(-0.573990\pi\)
−0.230359 + 0.973106i \(0.573990\pi\)
\(44\) 43.3524 0.985281
\(45\) 0 0
\(46\) −93.1607 −2.02523
\(47\) 41.9033i 0.891559i 0.895143 + 0.445779i \(0.147073\pi\)
−0.895143 + 0.445779i \(0.852927\pi\)
\(48\) 32.2254i 0.671362i
\(49\) 26.9197 + 40.9430i 0.549382 + 0.835571i
\(50\) 0 0
\(51\) 19.7631 0.387512
\(52\) − 78.6114i − 1.51176i
\(53\) 24.7749 0.467451 0.233725 − 0.972303i \(-0.424908\pi\)
0.233725 + 0.972303i \(0.424908\pi\)
\(54\) 13.9243i 0.257857i
\(55\) 0 0
\(56\) 13.5226 + 7.29265i 0.241475 + 0.130226i
\(57\) −57.5053 −1.00886
\(58\) 71.1758 1.22717
\(59\) 73.3021i 1.24241i 0.783649 + 0.621204i \(0.213356\pi\)
−0.783649 + 0.621204i \(0.786644\pi\)
\(60\) 0 0
\(61\) 71.9661i 1.17977i 0.807486 + 0.589886i \(0.200827\pi\)
−0.807486 + 0.589886i \(0.799173\pi\)
\(62\) − 49.6901i − 0.801453i
\(63\) 18.4835 + 9.96801i 0.293388 + 0.158222i
\(64\) −35.6567 −0.557136
\(65\) 0 0
\(66\) − 63.2568i − 0.958436i
\(67\) 58.7724 0.877201 0.438600 − 0.898682i \(-0.355474\pi\)
0.438600 + 0.898682i \(0.355474\pi\)
\(68\) 36.2954i 0.533756i
\(69\) 60.2147i 0.872677i
\(70\) 0 0
\(71\) 82.1050 1.15641 0.578204 − 0.815892i \(-0.303754\pi\)
0.578204 + 0.815892i \(0.303754\pi\)
\(72\) 6.58445 0.0914507
\(73\) − 97.7209i − 1.33864i −0.742973 − 0.669322i \(-0.766585\pi\)
0.742973 − 0.669322i \(-0.233415\pi\)
\(74\) 65.6943 0.887761
\(75\) 0 0
\(76\) − 105.610i − 1.38960i
\(77\) −83.9687 − 45.2837i −1.09050 − 0.588101i
\(78\) −114.704 −1.47057
\(79\) 69.3670 0.878063 0.439032 − 0.898472i \(-0.355322\pi\)
0.439032 + 0.898472i \(0.355322\pi\)
\(80\) 0 0
\(81\) 9.00000 0.111111
\(82\) − 1.67952i − 0.0204820i
\(83\) − 27.2418i − 0.328215i −0.986442 − 0.164107i \(-0.947526\pi\)
0.986442 − 0.164107i \(-0.0524743\pi\)
\(84\) −18.3065 + 33.9454i −0.217935 + 0.404112i
\(85\) 0 0
\(86\) −53.0879 −0.617301
\(87\) − 46.0047i − 0.528790i
\(88\) −29.9126 −0.339915
\(89\) − 117.321i − 1.31821i −0.752049 − 0.659107i \(-0.770934\pi\)
0.752049 − 0.659107i \(-0.229066\pi\)
\(90\) 0 0
\(91\) −82.1136 + 152.262i −0.902348 + 1.67320i
\(92\) −110.586 −1.20202
\(93\) −32.1173 −0.345348
\(94\) 112.289i 1.19457i
\(95\) 0 0
\(96\) 71.1492i 0.741138i
\(97\) − 178.187i − 1.83698i −0.395446 − 0.918489i \(-0.629410\pi\)
0.395446 − 0.918489i \(-0.370590\pi\)
\(98\) 72.1376 + 109.716i 0.736098 + 1.11955i
\(99\) −40.8862 −0.412992
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 525.3.h.b.76.9 ✓ 10
5.2 odd 4 525.3.e.b.349.17 20
5.3 odd 4 525.3.e.b.349.4 20
5.4 even 2 525.3.h.c.76.2 yes 10
7.6 odd 2 inner 525.3.h.b.76.10 yes 10
35.13 even 4 525.3.e.b.349.3 20
35.27 even 4 525.3.e.b.349.18 20
35.34 odd 2 525.3.h.c.76.1 yes 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
525.3.e.b.349.3 20 35.13 even 4
525.3.e.b.349.4 20 5.3 odd 4
525.3.e.b.349.17 20 5.2 odd 4
525.3.e.b.349.18 20 35.27 even 4
525.3.h.b.76.9 ✓ 10 1.1 even 1 trivial
525.3.h.b.76.10 yes 10 7.6 odd 2 inner
525.3.h.c.76.1 yes 10 35.34 odd 2
525.3.h.c.76.2 yes 10 5.4 even 2