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Properties

Label 525.3.e.a.349.1
Level $525$
Weight $3$
Character 525.349
Analytic conductor $14.305$
Analytic rank $0$
Dimension $4$
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 = [4] 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: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 21)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 349.1
Root \(0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 525.349
Dual form 525.3.e.a.349.3

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.73205 q^{3} +3.00000 q^{4} +1.73205i q^{6} +(6.92820 - 1.00000i) q^{7} -7.00000i q^{8} +3.00000 q^{9} +10.0000 q^{11} -5.19615 q^{12} -6.92820 q^{13} +(-1.00000 - 6.92820i) q^{14} +5.00000 q^{16} -3.00000i q^{18} +20.7846i q^{19} +(-12.0000 + 1.73205i) q^{21} -10.0000i q^{22} -14.0000i q^{23} +12.1244i q^{24} +6.92820i q^{26} -5.19615 q^{27} +(20.7846 - 3.00000i) q^{28} +38.0000 q^{29} +27.7128i q^{31} -33.0000i q^{32} -17.3205 q^{33} +9.00000 q^{36} -26.0000i q^{37} +20.7846 q^{38} +12.0000 q^{39} -69.2820i q^{41} +(1.73205 + 12.0000i) q^{42} +26.0000i q^{43} +30.0000 q^{44} -14.0000 q^{46} +27.7128 q^{47} -8.66025 q^{48} +(47.0000 - 13.8564i) q^{49} -20.7846 q^{52} +10.0000i q^{53} +5.19615i q^{54} +(-7.00000 - 48.4974i) q^{56} -36.0000i q^{57} -38.0000i q^{58} -76.2102i q^{59} +34.6410i q^{61} +27.7128 q^{62} +(20.7846 - 3.00000i) q^{63} -13.0000 q^{64} +17.3205i q^{66} -74.0000i q^{67} +24.2487i q^{69} -62.0000 q^{71} -21.0000i q^{72} +41.5692 q^{73} -26.0000 q^{74} +62.3538i q^{76} +(69.2820 - 10.0000i) q^{77} -12.0000i q^{78} +46.0000 q^{79} +9.00000 q^{81} -69.2820 q^{82} -90.0666 q^{83} +(-36.0000 + 5.19615i) q^{84} +26.0000 q^{86} -65.8179 q^{87} -70.0000i q^{88} +41.5692i q^{89} +(-48.0000 + 6.92820i) q^{91} -42.0000i q^{92} -48.0000i q^{93} -27.7128i q^{94} +57.1577i q^{96} +55.4256 q^{97} +(-13.8564 - 47.0000i) q^{98} +30.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{4} + 12 q^{9} + 40 q^{11} - 4 q^{14} + 20 q^{16} - 48 q^{21} + 152 q^{29} + 36 q^{36} + 48 q^{39} + 120 q^{44} - 56 q^{46} + 188 q^{49} - 28 q^{56} - 52 q^{64} - 248 q^{71} - 104 q^{74} + 184 q^{79}+ \cdots + 120 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.500000i −0.968246 − 0.250000i \(-0.919569\pi\)
0.968246 − 0.250000i \(-0.0804306\pi\)
\(3\) −1.73205 −0.577350
\(4\) 3.00000 0.750000
\(5\) 0 0
\(6\) 1.73205i 0.288675i
\(7\) 6.92820 − 1.00000i 0.989743 − 0.142857i
\(8\) − 7.00000i − 0.875000i
\(9\) 3.00000 0.333333
\(10\) 0 0
\(11\) 10.0000 0.909091 0.454545 − 0.890724i \(-0.349802\pi\)
0.454545 + 0.890724i \(0.349802\pi\)
\(12\) −5.19615 −0.433013
\(13\) −6.92820 −0.532939 −0.266469 − 0.963843i \(-0.585857\pi\)
−0.266469 + 0.963843i \(0.585857\pi\)
\(14\) −1.00000 − 6.92820i −0.0714286 − 0.494872i
\(15\) 0 0
\(16\) 5.00000 0.312500
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) − 3.00000i − 0.166667i
\(19\) 20.7846i 1.09393i 0.837157 + 0.546963i \(0.184216\pi\)
−0.837157 + 0.546963i \(0.815784\pi\)
\(20\) 0 0
\(21\) −12.0000 + 1.73205i −0.571429 + 0.0824786i
\(22\) − 10.0000i − 0.454545i
\(23\) − 14.0000i − 0.608696i −0.952561 − 0.304348i \(-0.901561\pi\)
0.952561 − 0.304348i \(-0.0984385\pi\)
\(24\) 12.1244i 0.505181i
\(25\) 0 0
\(26\) 6.92820i 0.266469i
\(27\) −5.19615 −0.192450
\(28\) 20.7846 − 3.00000i 0.742307 − 0.107143i
\(29\) 38.0000 1.31034 0.655172 − 0.755479i \(-0.272596\pi\)
0.655172 + 0.755479i \(0.272596\pi\)
\(30\) 0 0
\(31\) 27.7128i 0.893962i 0.894544 + 0.446981i \(0.147501\pi\)
−0.894544 + 0.446981i \(0.852499\pi\)
\(32\) − 33.0000i − 1.03125i
\(33\) −17.3205 −0.524864
\(34\) 0 0
\(35\) 0 0
\(36\) 9.00000 0.250000
\(37\) − 26.0000i − 0.702703i −0.936244 − 0.351351i \(-0.885722\pi\)
0.936244 − 0.351351i \(-0.114278\pi\)
\(38\) 20.7846 0.546963
\(39\) 12.0000 0.307692
\(40\) 0 0
\(41\) − 69.2820i − 1.68981i −0.534920 − 0.844903i \(-0.679658\pi\)
0.534920 − 0.844903i \(-0.320342\pi\)
\(42\) 1.73205 + 12.0000i 0.0412393 + 0.285714i
\(43\) 26.0000i 0.604651i 0.953205 + 0.302326i \(0.0977630\pi\)
−0.953205 + 0.302326i \(0.902237\pi\)
\(44\) 30.0000 0.681818
\(45\) 0 0
\(46\) −14.0000 −0.304348
\(47\) 27.7128 0.589634 0.294817 − 0.955554i \(-0.404741\pi\)
0.294817 + 0.955554i \(0.404741\pi\)
\(48\) −8.66025 −0.180422
\(49\) 47.0000 − 13.8564i 0.959184 − 0.282784i
\(50\) 0 0
\(51\) 0 0
\(52\) −20.7846 −0.399704
\(53\) 10.0000i 0.188679i 0.995540 + 0.0943396i \(0.0300740\pi\)
−0.995540 + 0.0943396i \(0.969926\pi\)
\(54\) 5.19615i 0.0962250i
\(55\) 0 0
\(56\) −7.00000 − 48.4974i −0.125000 − 0.866025i
\(57\) − 36.0000i − 0.631579i
\(58\) − 38.0000i − 0.655172i
\(59\) − 76.2102i − 1.29170i −0.763465 − 0.645849i \(-0.776503\pi\)
0.763465 − 0.645849i \(-0.223497\pi\)
\(60\) 0 0
\(61\) 34.6410i 0.567886i 0.958841 + 0.283943i \(0.0916426\pi\)
−0.958841 + 0.283943i \(0.908357\pi\)
\(62\) 27.7128 0.446981
\(63\) 20.7846 − 3.00000i 0.329914 − 0.0476190i
\(64\) −13.0000 −0.203125
\(65\) 0 0
\(66\) 17.3205i 0.262432i
\(67\) − 74.0000i − 1.10448i −0.833686 − 0.552239i \(-0.813774\pi\)
0.833686 − 0.552239i \(-0.186226\pi\)
\(68\) 0 0
\(69\) 24.2487i 0.351431i
\(70\) 0 0
\(71\) −62.0000 −0.873239 −0.436620 − 0.899646i \(-0.643824\pi\)
−0.436620 + 0.899646i \(0.643824\pi\)
\(72\) − 21.0000i − 0.291667i
\(73\) 41.5692 0.569441 0.284721 − 0.958611i \(-0.408099\pi\)
0.284721 + 0.958611i \(0.408099\pi\)
\(74\) −26.0000 −0.351351
\(75\) 0 0
\(76\) 62.3538i 0.820445i
\(77\) 69.2820 − 10.0000i 0.899767 − 0.129870i
\(78\) − 12.0000i − 0.153846i
\(79\) 46.0000 0.582278 0.291139 − 0.956681i \(-0.405966\pi\)
0.291139 + 0.956681i \(0.405966\pi\)
\(80\) 0 0
\(81\) 9.00000 0.111111
\(82\) −69.2820 −0.844903
\(83\) −90.0666 −1.08514 −0.542570 − 0.840011i \(-0.682549\pi\)
−0.542570 + 0.840011i \(0.682549\pi\)
\(84\) −36.0000 + 5.19615i −0.428571 + 0.0618590i
\(85\) 0 0
\(86\) 26.0000 0.302326
\(87\) −65.8179 −0.756528
\(88\) − 70.0000i − 0.795455i
\(89\) 41.5692i 0.467070i 0.972348 + 0.233535i \(0.0750293\pi\)
−0.972348 + 0.233535i \(0.924971\pi\)
\(90\) 0 0
\(91\) −48.0000 + 6.92820i −0.527473 + 0.0761341i
\(92\) − 42.0000i − 0.456522i
\(93\) − 48.0000i − 0.516129i
\(94\) − 27.7128i − 0.294817i
\(95\) 0 0
\(96\) 57.1577i 0.595392i
\(97\) 55.4256 0.571398 0.285699 − 0.958319i \(-0.407774\pi\)
0.285699 + 0.958319i \(0.407774\pi\)
\(98\) −13.8564 − 47.0000i −0.141392 − 0.479592i
\(99\) 30.0000 0.303030
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.a.349.1 4
5.2 odd 4 21.3.d.a.13.2 yes 2
5.3 odd 4 525.3.h.a.76.1 2
5.4 even 2 inner 525.3.e.a.349.4 4
7.6 odd 2 inner 525.3.e.a.349.2 4
15.2 even 4 63.3.d.b.55.2 2
20.7 even 4 336.3.f.a.97.1 2
35.2 odd 12 147.3.f.b.31.1 2
35.12 even 12 147.3.f.d.31.1 2
35.13 even 4 525.3.h.a.76.2 2
35.17 even 12 147.3.f.b.19.1 2
35.27 even 4 21.3.d.a.13.1 ✓ 2
35.32 odd 12 147.3.f.d.19.1 2
35.34 odd 2 inner 525.3.e.a.349.3 4
40.27 even 4 1344.3.f.b.769.2 2
40.37 odd 4 1344.3.f.c.769.1 2
60.47 odd 4 1008.3.f.d.433.2 2
105.2 even 12 441.3.m.f.325.1 2
105.17 odd 12 441.3.m.f.19.1 2
105.32 even 12 441.3.m.d.19.1 2
105.47 odd 12 441.3.m.d.325.1 2
105.62 odd 4 63.3.d.b.55.1 2
140.27 odd 4 336.3.f.a.97.2 2
280.27 odd 4 1344.3.f.b.769.1 2
280.237 even 4 1344.3.f.c.769.2 2
420.167 even 4 1008.3.f.d.433.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.3.d.a.13.1 ✓ 2 35.27 even 4
21.3.d.a.13.2 yes 2 5.2 odd 4
63.3.d.b.55.1 2 105.62 odd 4
63.3.d.b.55.2 2 15.2 even 4
147.3.f.b.19.1 2 35.17 even 12
147.3.f.b.31.1 2 35.2 odd 12
147.3.f.d.19.1 2 35.32 odd 12
147.3.f.d.31.1 2 35.12 even 12
336.3.f.a.97.1 2 20.7 even 4
336.3.f.a.97.2 2 140.27 odd 4
441.3.m.d.19.1 2 105.32 even 12
441.3.m.d.325.1 2 105.47 odd 12
441.3.m.f.19.1 2 105.17 odd 12
441.3.m.f.325.1 2 105.2 even 12
525.3.e.a.349.1 4 1.1 even 1 trivial
525.3.e.a.349.2 4 7.6 odd 2 inner
525.3.e.a.349.3 4 35.34 odd 2 inner
525.3.e.a.349.4 4 5.4 even 2 inner
525.3.h.a.76.1 2 5.3 odd 4
525.3.h.a.76.2 2 35.13 even 4
1008.3.f.d.433.1 2 420.167 even 4
1008.3.f.d.433.2 2 60.47 odd 4
1344.3.f.b.769.1 2 280.27 odd 4
1344.3.f.b.769.2 2 40.27 even 4
1344.3.f.c.769.1 2 40.37 odd 4
1344.3.f.c.769.2 2 280.237 even 4