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Properties

Label 525.3.e.b.349.4
Level $525$
Weight $3$
Character 525.349
Analytic conductor $14.305$
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] = [525,3,Mod(349,525)] 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("525.349"); 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, 1, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 525.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(14.3052138789\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 25 x^{18} + 409 x^{16} - 3886 x^{14} + 26830 x^{12} - 118498 x^{10} + 377533 x^{8} + \cdots + 1296 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{2}\cdot 5^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 349.4
Root \(2.32071 - 1.33987i\) of defining polynomial
Character \(\chi\) \(=\) 525.349
Dual form 525.3.e.b.349.18

$q$-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)
 
\(f(q)\) \(=\) \(q-2.67973i q^{2} +1.73205 q^{3} -3.18096 q^{4} -4.64143i q^{6} +(-3.32267 + 6.16116i) q^{7} -2.19482i q^{8} +3.00000 q^{9} +13.6287 q^{11} -5.50958 q^{12} +24.7132 q^{13} +(16.5102 + 8.90386i) q^{14} -18.6053 q^{16} +11.4102 q^{17} -8.03919i q^{18} +33.2007i q^{19} +(-5.75503 + 10.6714i) q^{21} -36.5213i q^{22} -34.7650i q^{23} -3.80154i q^{24} -66.2246i q^{26} +5.19615 q^{27} +(10.5693 - 19.5984i) q^{28} -26.5608 q^{29} -18.5429i q^{31} +41.0780i q^{32} +23.6056 q^{33} -30.5763i q^{34} -9.54287 q^{36} -24.5153i q^{37} +88.9689 q^{38} +42.8044 q^{39} -0.626750i q^{41} +(28.5966 + 15.4219i) q^{42} -19.8109i q^{43} -43.3524 q^{44} -93.1607 q^{46} +41.9033 q^{47} -32.2254 q^{48} +(-26.9197 - 40.9430i) q^{49} +19.7631 q^{51} -78.6114 q^{52} +24.7749i q^{53} -13.9243i q^{54} +(13.5226 + 7.29265i) q^{56} +57.5053i q^{57} +71.1758i q^{58} -73.3021i q^{59} +71.9661i q^{61} -49.6901 q^{62} +(-9.96801 + 18.4835i) q^{63} +35.6567 q^{64} -63.2568i q^{66} -58.7724i q^{67} -36.2954 q^{68} -60.2147i q^{69} +82.1050 q^{71} -6.58445i q^{72} +97.7209 q^{73} -65.6943 q^{74} -105.610i q^{76} +(-45.2837 + 83.9687i) q^{77} -114.704i q^{78} -69.3670 q^{79} +9.00000 q^{81} -1.67952 q^{82} +27.2418 q^{83} +(18.3065 - 33.9454i) q^{84} -53.0879 q^{86} -46.0047 q^{87} -29.9126i q^{88} +117.321i q^{89} +(-82.1136 + 152.262i) q^{91} +110.586i q^{92} -32.1173i q^{93} -112.289i q^{94} +71.1492i q^{96} -178.187 q^{97} +(-109.716 + 72.1376i) q^{98} +40.8862 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 20 q^{4} + 60 q^{9} + 64 q^{11} + 32 q^{14} - 108 q^{16} - 30 q^{21} + 80 q^{29} - 60 q^{36} - 60 q^{39} - 612 q^{44} - 452 q^{46} - 14 q^{49} - 96 q^{51} + 296 q^{56} + 336 q^{64} + 376 q^{71} + 28 q^{74}+ \cdots + 192 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.67973i − 1.33987i −0.742422 − 0.669933i \(-0.766323\pi\)
0.742422 − 0.669933i \(-0.233677\pi\)
\(3\) 1.73205 0.577350
\(4\) −3.18096 −0.795239
\(5\) 0 0
\(6\) − 4.64143i − 0.773572i
\(7\) −3.32267 + 6.16116i −0.474667 + 0.880165i
\(8\) − 2.19482i − 0.274352i
\(9\) 3.00000 0.333333
\(10\) 0 0
\(11\) 13.6287 1.23897 0.619487 − 0.785007i \(-0.287340\pi\)
0.619487 + 0.785007i \(0.287340\pi\)
\(12\) −5.50958 −0.459131
\(13\) 24.7132 1.90101 0.950506 − 0.310707i \(-0.100566\pi\)
0.950506 + 0.310707i \(0.100566\pi\)
\(14\) 16.5102 + 8.90386i 1.17930 + 0.635990i
\(15\) 0 0
\(16\) −18.6053 −1.16283
\(17\) 11.4102 0.671190 0.335595 − 0.942006i \(-0.391063\pi\)
0.335595 + 0.942006i \(0.391063\pi\)
\(18\) − 8.03919i − 0.446622i
\(19\) 33.2007i 1.74740i 0.486461 + 0.873702i \(0.338288\pi\)
−0.486461 + 0.873702i \(0.661712\pi\)
\(20\) 0 0
\(21\) −5.75503 + 10.6714i −0.274049 + 0.508164i
\(22\) − 36.5213i − 1.66006i
\(23\) − 34.7650i − 1.51152i −0.654849 − 0.755760i \(-0.727268\pi\)
0.654849 − 0.755760i \(-0.272732\pi\)
\(24\) − 3.80154i − 0.158397i
\(25\) 0 0
\(26\) − 66.2246i − 2.54710i
\(27\) 5.19615 0.192450
\(28\) 10.5693 − 19.5984i 0.377474 − 0.699942i
\(29\) −26.5608 −0.915890 −0.457945 − 0.888980i \(-0.651414\pi\)
−0.457945 + 0.888980i \(0.651414\pi\)
\(30\) 0 0
\(31\) − 18.5429i − 0.598160i −0.954228 − 0.299080i \(-0.903320\pi\)
0.954228 − 0.299080i \(-0.0966797\pi\)
\(32\) 41.0780i 1.28369i
\(33\) 23.6056 0.715322
\(34\) − 30.5763i − 0.899304i
\(35\) 0 0
\(36\) −9.54287 −0.265080
\(37\) − 24.5153i − 0.662575i −0.943530 − 0.331287i \(-0.892517\pi\)
0.943530 − 0.331287i \(-0.107483\pi\)
\(38\) 88.9689 2.34129
\(39\) 42.8044 1.09755
\(40\) 0 0
\(41\) − 0.626750i − 0.0152866i −0.999971 − 0.00764329i \(-0.997567\pi\)
0.999971 − 0.00764329i \(-0.00243296\pi\)
\(42\) 28.5966 + 15.4219i 0.680871 + 0.367189i
\(43\) − 19.8109i − 0.460718i −0.973106 − 0.230359i \(-0.926010\pi\)
0.973106 − 0.230359i \(-0.0739901\pi\)
\(44\) −43.3524 −0.985281
\(45\) 0 0
\(46\) −93.1607 −2.02523
\(47\) 41.9033 0.891559 0.445779 − 0.895143i \(-0.352927\pi\)
0.445779 + 0.895143i \(0.352927\pi\)
\(48\) −32.2254 −0.671362
\(49\) −26.9197 − 40.9430i −0.549382 − 0.835571i
\(50\) 0 0
\(51\) 19.7631 0.387512
\(52\) −78.6114 −1.51176
\(53\) 24.7749i 0.467451i 0.972303 + 0.233725i \(0.0750916\pi\)
−0.972303 + 0.233725i \(0.924908\pi\)
\(54\) − 13.9243i − 0.257857i
\(55\) 0 0
\(56\) 13.5226 + 7.29265i 0.241475 + 0.130226i
\(57\) 57.5053i 1.00886i
\(58\) 71.1758i 1.22717i
\(59\) − 73.3021i − 1.24241i −0.783649 − 0.621204i \(-0.786644\pi\)
0.783649 − 0.621204i \(-0.213356\pi\)
\(60\) 0 0
\(61\) 71.9661i 1.17977i 0.807486 + 0.589886i \(0.200827\pi\)
−0.807486 + 0.589886i \(0.799173\pi\)
\(62\) −49.6901 −0.801453
\(63\) −9.96801 + 18.4835i −0.158222 + 0.293388i
\(64\) 35.6567 0.557136
\(65\) 0 0
\(66\) − 63.2568i − 0.958436i
\(67\) − 58.7724i − 0.877201i −0.898682 − 0.438600i \(-0.855474\pi\)
0.898682 − 0.438600i \(-0.144526\pi\)
\(68\) −36.2954 −0.533756
\(69\) − 60.2147i − 0.872677i
\(70\) 0 0
\(71\) 82.1050 1.15641 0.578204 − 0.815892i \(-0.303754\pi\)
0.578204 + 0.815892i \(0.303754\pi\)
\(72\) − 6.58445i − 0.0914507i
\(73\) 97.7209 1.33864 0.669322 − 0.742973i \(-0.266585\pi\)
0.669322 + 0.742973i \(0.266585\pi\)
\(74\) −65.6943 −0.887761
\(75\) 0 0
\(76\) − 105.610i − 1.38960i
\(77\) −45.2837 + 83.9687i −0.588101 + 1.09050i
\(78\) − 114.704i − 1.47057i
\(79\) −69.3670 −0.878063 −0.439032 − 0.898472i \(-0.644678\pi\)
−0.439032 + 0.898472i \(0.644678\pi\)
\(80\) 0 0
\(81\) 9.00000 0.111111
\(82\) −1.67952 −0.0204820
\(83\) 27.2418 0.328215 0.164107 − 0.986442i \(-0.447526\pi\)
0.164107 + 0.986442i \(0.447526\pi\)
\(84\) 18.3065 − 33.9454i 0.217935 − 0.404112i
\(85\) 0 0
\(86\) −53.0879 −0.617301
\(87\) −46.0047 −0.528790
\(88\) − 29.9126i − 0.339915i
\(89\) 117.321i 1.31821i 0.752049 + 0.659107i \(0.229066\pi\)
−0.752049 + 0.659107i \(0.770934\pi\)
\(90\) 0 0
\(91\) −82.1136 + 152.262i −0.902348 + 1.67320i
\(92\) 110.586i 1.20202i
\(93\) − 32.1173i − 0.345348i
\(94\) − 112.289i − 1.19457i
\(95\) 0 0
\(96\) 71.1492i 0.741138i
\(97\) −178.187 −1.83698 −0.918489 − 0.395446i \(-0.870590\pi\)
−0.918489 + 0.395446i \(0.870590\pi\)
\(98\) −109.716 + 72.1376i −1.11955 + 0.736098i
\(99\) 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.e.b.349.4 20
5.2 odd 4 525.3.h.b.76.9 ✓ 10
5.3 odd 4 525.3.h.c.76.2 yes 10
5.4 even 2 inner 525.3.e.b.349.17 20
7.6 odd 2 inner 525.3.e.b.349.3 20
35.13 even 4 525.3.h.c.76.1 yes 10
35.27 even 4 525.3.h.b.76.10 yes 10
35.34 odd 2 inner 525.3.e.b.349.18 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
525.3.e.b.349.3 20 7.6 odd 2 inner
525.3.e.b.349.4 20 1.1 even 1 trivial
525.3.e.b.349.17 20 5.4 even 2 inner
525.3.e.b.349.18 20 35.34 odd 2 inner
525.3.h.b.76.9 ✓ 10 5.2 odd 4
525.3.h.b.76.10 yes 10 35.27 even 4
525.3.h.c.76.1 yes 10 35.13 even 4
525.3.h.c.76.2 yes 10 5.3 odd 4