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Properties

Label 525.4.d.o.274.5
Level $525$
Weight $4$
Character 525.274
Analytic conductor $30.976$
Analytic rank $0$
Dimension $8$
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] = [525,4,Mod(274,525)] 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("525.274"); S:= CuspForms(chi, 4); 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, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 525.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,-32,0,36,0,0,-72,0,114] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(30.9760027530\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 40x^{6} + 488x^{4} + 1945x^{2} + 1936 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 274.5
Root \(1.21734i\) of defining polynomial
Character \(\chi\) \(=\) 525.274
Dual form 525.4.d.o.274.4

$q$-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)
 
\(f(q)\) \(=\) \(q+0.217342i q^{2} +3.00000i q^{3} +7.95276 q^{4} -0.652027 q^{6} +7.00000i q^{7} +3.46721i q^{8} -9.00000 q^{9} -30.6085 q^{11} +23.8583i q^{12} -25.3178i q^{13} -1.52140 q^{14} +62.8685 q^{16} +72.8676i q^{17} -1.95608i q^{18} -122.711 q^{19} -21.0000 q^{21} -6.65253i q^{22} +194.258i q^{23} -10.4016 q^{24} +5.50264 q^{26} -27.0000i q^{27} +55.6693i q^{28} -48.6103 q^{29} -288.907 q^{31} +41.4017i q^{32} -91.8255i q^{33} -15.8372 q^{34} -71.5749 q^{36} +15.8251i q^{37} -26.6702i q^{38} +75.9535 q^{39} +452.905 q^{41} -4.56419i q^{42} +152.574i q^{43} -243.422 q^{44} -42.2205 q^{46} -164.435i q^{47} +188.606i q^{48} -49.0000 q^{49} -218.603 q^{51} -201.347i q^{52} +591.600i q^{53} +5.86824 q^{54} -24.2705 q^{56} -368.132i q^{57} -10.5651i q^{58} +180.823 q^{59} +115.773 q^{61} -62.7918i q^{62} -63.0000i q^{63} +493.950 q^{64} +19.9576 q^{66} -605.264i q^{67} +579.499i q^{68} -582.774 q^{69} -990.917 q^{71} -31.2049i q^{72} +863.756i q^{73} -3.43947 q^{74} -975.889 q^{76} -214.260i q^{77} +16.5079i q^{78} -965.930 q^{79} +81.0000 q^{81} +98.4355i q^{82} +160.924i q^{83} -167.008 q^{84} -33.1608 q^{86} -145.831i q^{87} -106.126i q^{88} +51.6227 q^{89} +177.225 q^{91} +1544.89i q^{92} -866.722i q^{93} +35.7387 q^{94} -124.205 q^{96} -1497.31i q^{97} -10.6498i q^{98} +275.477 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 32 q^{4} + 36 q^{6} - 72 q^{9} + 114 q^{11} + 84 q^{14} + 432 q^{16} + 24 q^{19} - 168 q^{21} - 558 q^{24} - 162 q^{26} - 756 q^{29} - 186 q^{31} - 1566 q^{34} + 288 q^{36} - 258 q^{39} - 930 q^{41}+ \cdots - 1026 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.217342i 0.0768421i 0.999262 + 0.0384211i \(0.0122328\pi\)
−0.999262 + 0.0384211i \(0.987767\pi\)
\(3\) 3.00000i 0.577350i
\(4\) 7.95276 0.994095
\(5\) 0 0
\(6\) −0.652027 −0.0443648
\(7\) 7.00000i 0.377964i
\(8\) 3.46721i 0.153231i
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) −30.6085 −0.838983 −0.419492 − 0.907759i \(-0.637792\pi\)
−0.419492 + 0.907759i \(0.637792\pi\)
\(12\) 23.8583i 0.573941i
\(13\) − 25.3178i − 0.540146i −0.962840 − 0.270073i \(-0.912952\pi\)
0.962840 − 0.270073i \(-0.0870479\pi\)
\(14\) −1.52140 −0.0290436
\(15\) 0 0
\(16\) 62.8685 0.982321
\(17\) 72.8676i 1.03959i 0.854292 + 0.519794i \(0.173991\pi\)
−0.854292 + 0.519794i \(0.826009\pi\)
\(18\) − 1.95608i − 0.0256140i
\(19\) −122.711 −1.48167 −0.740836 − 0.671686i \(-0.765570\pi\)
−0.740836 + 0.671686i \(0.765570\pi\)
\(20\) 0 0
\(21\) −21.0000 −0.218218
\(22\) − 6.65253i − 0.0644692i
\(23\) 194.258i 1.76111i 0.473943 + 0.880556i \(0.342830\pi\)
−0.473943 + 0.880556i \(0.657170\pi\)
\(24\) −10.4016 −0.0884677
\(25\) 0 0
\(26\) 5.50264 0.0415060
\(27\) − 27.0000i − 0.192450i
\(28\) 55.6693i 0.375733i
\(29\) −48.6103 −0.311266 −0.155633 − 0.987815i \(-0.549742\pi\)
−0.155633 + 0.987815i \(0.549742\pi\)
\(30\) 0 0
\(31\) −288.907 −1.67385 −0.836924 − 0.547320i \(-0.815648\pi\)
−0.836924 + 0.547320i \(0.815648\pi\)
\(32\) 41.4017i 0.228714i
\(33\) − 91.8255i − 0.484387i
\(34\) −15.8372 −0.0798841
\(35\) 0 0
\(36\) −71.5749 −0.331365
\(37\) 15.8251i 0.0703144i 0.999382 + 0.0351572i \(0.0111932\pi\)
−0.999382 + 0.0351572i \(0.988807\pi\)
\(38\) − 26.6702i − 0.113855i
\(39\) 75.9535 0.311854
\(40\) 0 0
\(41\) 452.905 1.72517 0.862584 − 0.505914i \(-0.168845\pi\)
0.862584 + 0.505914i \(0.168845\pi\)
\(42\) − 4.56419i − 0.0167683i
\(43\) 152.574i 0.541101i 0.962706 + 0.270550i \(0.0872057\pi\)
−0.962706 + 0.270550i \(0.912794\pi\)
\(44\) −243.422 −0.834029
\(45\) 0 0
\(46\) −42.2205 −0.135328
\(47\) − 164.435i − 0.510325i −0.966898 − 0.255163i \(-0.917871\pi\)
0.966898 − 0.255163i \(-0.0821290\pi\)
\(48\) 188.606i 0.567143i
\(49\) −49.0000 −0.142857
\(50\) 0 0
\(51\) −218.603 −0.600206
\(52\) − 201.347i − 0.536957i
\(53\) 591.600i 1.53326i 0.642092 + 0.766628i \(0.278067\pi\)
−0.642092 + 0.766628i \(0.721933\pi\)
\(54\) 5.86824 0.0147883
\(55\) 0 0
\(56\) −24.2705 −0.0579157
\(57\) − 368.132i − 0.855443i
\(58\) − 10.5651i − 0.0239183i
\(59\) 180.823 0.399003 0.199501 − 0.979898i \(-0.436068\pi\)
0.199501 + 0.979898i \(0.436068\pi\)
\(60\) 0 0
\(61\) 115.773 0.243004 0.121502 − 0.992591i \(-0.461229\pi\)
0.121502 + 0.992591i \(0.461229\pi\)
\(62\) − 62.7918i − 0.128622i
\(63\) − 63.0000i − 0.125988i
\(64\) 493.950 0.964746
\(65\) 0 0
\(66\) 19.9576 0.0372213
\(67\) − 605.264i − 1.10365i −0.833959 − 0.551827i \(-0.813931\pi\)
0.833959 − 0.551827i \(-0.186069\pi\)
\(68\) 579.499i 1.03345i
\(69\) −582.774 −1.01678
\(70\) 0 0
\(71\) −990.917 −1.65634 −0.828170 − 0.560477i \(-0.810618\pi\)
−0.828170 + 0.560477i \(0.810618\pi\)
\(72\) − 31.2049i − 0.0510768i
\(73\) 863.756i 1.38486i 0.721484 + 0.692431i \(0.243460\pi\)
−0.721484 + 0.692431i \(0.756540\pi\)
\(74\) −3.43947 −0.00540311
\(75\) 0 0
\(76\) −975.889 −1.47292
\(77\) − 214.260i − 0.317106i
\(78\) 16.5079i 0.0239635i
\(79\) −965.930 −1.37564 −0.687821 − 0.725881i \(-0.741432\pi\)
−0.687821 + 0.725881i \(0.741432\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 98.4355i 0.132566i
\(83\) 160.924i 0.212816i 0.994323 + 0.106408i \(0.0339350\pi\)
−0.994323 + 0.106408i \(0.966065\pi\)
\(84\) −167.008 −0.216929
\(85\) 0 0
\(86\) −33.1608 −0.0415793
\(87\) − 145.831i − 0.179709i
\(88\) − 106.126i − 0.128558i
\(89\) 51.6227 0.0614831 0.0307415 − 0.999527i \(-0.490213\pi\)
0.0307415 + 0.999527i \(0.490213\pi\)
\(90\) 0 0
\(91\) 177.225 0.204156
\(92\) 1544.89i 1.75071i
\(93\) − 866.722i − 0.966396i
\(94\) 35.7387 0.0392145
\(95\) 0 0
\(96\) −124.205 −0.132048
\(97\) − 1497.31i − 1.56731i −0.621195 − 0.783656i \(-0.713353\pi\)
0.621195 − 0.783656i \(-0.286647\pi\)
\(98\) − 10.6498i − 0.0109774i
\(99\) 275.477 0.279661
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 525.4.d.o.274.5 8
5.2 odd 4 525.4.a.v.1.2 yes 4
5.3 odd 4 525.4.a.s.1.3 ✓ 4
5.4 even 2 inner 525.4.d.o.274.4 8
15.2 even 4 1575.4.a.bf.1.3 4
15.8 even 4 1575.4.a.bm.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
525.4.a.s.1.3 ✓ 4 5.3 odd 4
525.4.a.v.1.2 yes 4 5.2 odd 4
525.4.d.o.274.4 8 5.4 even 2 inner
525.4.d.o.274.5 8 1.1 even 1 trivial
1575.4.a.bf.1.3 4 15.2 even 4
1575.4.a.bm.1.2 4 15.8 even 4