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Properties

Label 525.4.d.b.274.1
Level $525$
Weight $4$
Character 525.274
Analytic conductor $30.976$
Analytic rank $0$
Dimension $2$
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 = [2,0,0,-16,0,-24,0,0,-18,0,124] 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: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) 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 21)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 274.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 525.274
Dual form 525.4.d.b.274.2

$q$-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)
 
\(f(q)\) \(=\) \(q-4.00000i q^{2} -3.00000i q^{3} -8.00000 q^{4} -12.0000 q^{6} +7.00000i q^{7} -9.00000 q^{9} +62.0000 q^{11} +24.0000i q^{12} -62.0000i q^{13} +28.0000 q^{14} -64.0000 q^{16} -84.0000i q^{17} +36.0000i q^{18} -100.000 q^{19} +21.0000 q^{21} -248.000i q^{22} -42.0000i q^{23} -248.000 q^{26} +27.0000i q^{27} -56.0000i q^{28} +10.0000 q^{29} -48.0000 q^{31} +256.000i q^{32} -186.000i q^{33} -336.000 q^{34} +72.0000 q^{36} +246.000i q^{37} +400.000i q^{38} -186.000 q^{39} -248.000 q^{41} -84.0000i q^{42} +68.0000i q^{43} -496.000 q^{44} -168.000 q^{46} -324.000i q^{47} +192.000i q^{48} -49.0000 q^{49} -252.000 q^{51} +496.000i q^{52} +258.000i q^{53} +108.000 q^{54} +300.000i q^{57} -40.0000i q^{58} -120.000 q^{59} +622.000 q^{61} +192.000i q^{62} -63.0000i q^{63} +512.000 q^{64} -744.000 q^{66} -904.000i q^{67} +672.000i q^{68} -126.000 q^{69} -678.000 q^{71} -642.000i q^{73} +984.000 q^{74} +800.000 q^{76} +434.000i q^{77} +744.000i q^{78} -740.000 q^{79} +81.0000 q^{81} +992.000i q^{82} +468.000i q^{83} -168.000 q^{84} +272.000 q^{86} -30.0000i q^{87} -200.000 q^{89} +434.000 q^{91} +336.000i q^{92} +144.000i q^{93} -1296.00 q^{94} +768.000 q^{96} +1266.00i q^{97} +196.000i q^{98} -558.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 16 q^{4} - 24 q^{6} - 18 q^{9} + 124 q^{11} + 56 q^{14} - 128 q^{16} - 200 q^{19} + 42 q^{21} - 496 q^{26} + 20 q^{29} - 96 q^{31} - 672 q^{34} + 144 q^{36} - 372 q^{39} - 496 q^{41} - 992 q^{44}+ \cdots - 1116 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 4.00000i − 1.41421i −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 − 0.707107i \(-0.250000\pi\)
\(3\) − 3.00000i − 0.577350i
\(4\) −8.00000 −1.00000
\(5\) 0 0
\(6\) −12.0000 −0.816497
\(7\) 7.00000i 0.377964i
\(8\) 0 0
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) 62.0000 1.69943 0.849714 − 0.527244i \(-0.176775\pi\)
0.849714 + 0.527244i \(0.176775\pi\)
\(12\) 24.0000i 0.577350i
\(13\) − 62.0000i − 1.32275i −0.750057 − 0.661373i \(-0.769974\pi\)
0.750057 − 0.661373i \(-0.230026\pi\)
\(14\) 28.0000 0.534522
\(15\) 0 0
\(16\) −64.0000 −1.00000
\(17\) − 84.0000i − 1.19841i −0.800595 − 0.599206i \(-0.795483\pi\)
0.800595 − 0.599206i \(-0.204517\pi\)
\(18\) 36.0000i 0.471405i
\(19\) −100.000 −1.20745 −0.603726 − 0.797192i \(-0.706318\pi\)
−0.603726 + 0.797192i \(0.706318\pi\)
\(20\) 0 0
\(21\) 21.0000 0.218218
\(22\) − 248.000i − 2.40335i
\(23\) − 42.0000i − 0.380765i −0.981710 − 0.190383i \(-0.939027\pi\)
0.981710 − 0.190383i \(-0.0609729\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −248.000 −1.87065
\(27\) 27.0000i 0.192450i
\(28\) − 56.0000i − 0.377964i
\(29\) 10.0000 0.0640329 0.0320164 − 0.999487i \(-0.489807\pi\)
0.0320164 + 0.999487i \(0.489807\pi\)
\(30\) 0 0
\(31\) −48.0000 −0.278099 −0.139049 − 0.990285i \(-0.544405\pi\)
−0.139049 + 0.990285i \(0.544405\pi\)
\(32\) 256.000i 1.41421i
\(33\) − 186.000i − 0.981165i
\(34\) −336.000 −1.69481
\(35\) 0 0
\(36\) 72.0000 0.333333
\(37\) 246.000i 1.09303i 0.837449 + 0.546516i \(0.184046\pi\)
−0.837449 + 0.546516i \(0.815954\pi\)
\(38\) 400.000i 1.70759i
\(39\) −186.000 −0.763688
\(40\) 0 0
\(41\) −248.000 −0.944661 −0.472330 − 0.881422i \(-0.656587\pi\)
−0.472330 + 0.881422i \(0.656587\pi\)
\(42\) − 84.0000i − 0.308607i
\(43\) 68.0000i 0.241161i 0.992704 + 0.120580i \(0.0384755\pi\)
−0.992704 + 0.120580i \(0.961524\pi\)
\(44\) −496.000 −1.69943
\(45\) 0 0
\(46\) −168.000 −0.538484
\(47\) − 324.000i − 1.00554i −0.864421 − 0.502769i \(-0.832315\pi\)
0.864421 − 0.502769i \(-0.167685\pi\)
\(48\) 192.000i 0.577350i
\(49\) −49.0000 −0.142857
\(50\) 0 0
\(51\) −252.000 −0.691903
\(52\) 496.000i 1.32275i
\(53\) 258.000i 0.668661i 0.942456 + 0.334330i \(0.108510\pi\)
−0.942456 + 0.334330i \(0.891490\pi\)
\(54\) 108.000 0.272166
\(55\) 0 0
\(56\) 0 0
\(57\) 300.000i 0.697122i
\(58\) − 40.0000i − 0.0905562i
\(59\) −120.000 −0.264791 −0.132396 − 0.991197i \(-0.542267\pi\)
−0.132396 + 0.991197i \(0.542267\pi\)
\(60\) 0 0
\(61\) 622.000 1.30556 0.652778 − 0.757549i \(-0.273603\pi\)
0.652778 + 0.757549i \(0.273603\pi\)
\(62\) 192.000i 0.393291i
\(63\) − 63.0000i − 0.125988i
\(64\) 512.000 1.00000
\(65\) 0 0
\(66\) −744.000 −1.38758
\(67\) − 904.000i − 1.64838i −0.566316 − 0.824188i \(-0.691632\pi\)
0.566316 − 0.824188i \(-0.308368\pi\)
\(68\) 672.000i 1.19841i
\(69\) −126.000 −0.219835
\(70\) 0 0
\(71\) −678.000 −1.13329 −0.566646 − 0.823961i \(-0.691759\pi\)
−0.566646 + 0.823961i \(0.691759\pi\)
\(72\) 0 0
\(73\) − 642.000i − 1.02932i −0.857394 − 0.514660i \(-0.827918\pi\)
0.857394 − 0.514660i \(-0.172082\pi\)
\(74\) 984.000 1.54578
\(75\) 0 0
\(76\) 800.000 1.20745
\(77\) 434.000i 0.642323i
\(78\) 744.000i 1.08002i
\(79\) −740.000 −1.05388 −0.526940 − 0.849903i \(-0.676661\pi\)
−0.526940 + 0.849903i \(0.676661\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 992.000i 1.33595i
\(83\) 468.000i 0.618912i 0.950914 + 0.309456i \(0.100147\pi\)
−0.950914 + 0.309456i \(0.899853\pi\)
\(84\) −168.000 −0.218218
\(85\) 0 0
\(86\) 272.000 0.341052
\(87\) − 30.0000i − 0.0369694i
\(88\) 0 0
\(89\) −200.000 −0.238202 −0.119101 − 0.992882i \(-0.538001\pi\)
−0.119101 + 0.992882i \(0.538001\pi\)
\(90\) 0 0
\(91\) 434.000 0.499951
\(92\) 336.000i 0.380765i
\(93\) 144.000i 0.160560i
\(94\) −1296.00 −1.42204
\(95\) 0 0
\(96\) 768.000 0.816497
\(97\) 1266.00i 1.32518i 0.748981 + 0.662592i \(0.230544\pi\)
−0.748981 + 0.662592i \(0.769456\pi\)
\(98\) 196.000i 0.202031i
\(99\) −558.000 −0.566476
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.b.274.1 2
5.2 odd 4 21.4.a.b.1.1 ✓ 1
5.3 odd 4 525.4.a.b.1.1 1
5.4 even 2 inner 525.4.d.b.274.2 2
15.2 even 4 63.4.a.a.1.1 1
15.8 even 4 1575.4.a.k.1.1 1
20.7 even 4 336.4.a.h.1.1 1
35.2 odd 12 147.4.e.c.67.1 2
35.12 even 12 147.4.e.b.67.1 2
35.17 even 12 147.4.e.b.79.1 2
35.27 even 4 147.4.a.g.1.1 1
35.32 odd 12 147.4.e.c.79.1 2
40.27 even 4 1344.4.a.i.1.1 1
40.37 odd 4 1344.4.a.w.1.1 1
60.47 odd 4 1008.4.a.m.1.1 1
105.2 even 12 441.4.e.m.361.1 2
105.17 odd 12 441.4.e.n.226.1 2
105.32 even 12 441.4.e.m.226.1 2
105.47 odd 12 441.4.e.n.361.1 2
105.62 odd 4 441.4.a.b.1.1 1
140.27 odd 4 2352.4.a.l.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.4.a.b.1.1 ✓ 1 5.2 odd 4
63.4.a.a.1.1 1 15.2 even 4
147.4.a.g.1.1 1 35.27 even 4
147.4.e.b.67.1 2 35.12 even 12
147.4.e.b.79.1 2 35.17 even 12
147.4.e.c.67.1 2 35.2 odd 12
147.4.e.c.79.1 2 35.32 odd 12
336.4.a.h.1.1 1 20.7 even 4
441.4.a.b.1.1 1 105.62 odd 4
441.4.e.m.226.1 2 105.32 even 12
441.4.e.m.361.1 2 105.2 even 12
441.4.e.n.226.1 2 105.17 odd 12
441.4.e.n.361.1 2 105.47 odd 12
525.4.a.b.1.1 1 5.3 odd 4
525.4.d.b.274.1 2 1.1 even 1 trivial
525.4.d.b.274.2 2 5.4 even 2 inner
1008.4.a.m.1.1 1 60.47 odd 4
1344.4.a.i.1.1 1 40.27 even 4
1344.4.a.w.1.1 1 40.37 odd 4
1575.4.a.k.1.1 1 15.8 even 4
2352.4.a.l.1.1 1 140.27 odd 4