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Properties

Label 252.4.k.f.109.2
Level $252$
Weight $4$
Character 252.109
Analytic conductor $14.868$
Analytic rank $0$
Dimension $4$
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] = [252,4,Mod(37,252)] 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("252.37"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(252, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 252.k (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,11,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(14.8684813214\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{193})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 49x^{2} + 48x + 2304 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 109.2
Root \(-3.22311 - 5.58259i\) of defining polynomial
Character \(\chi\) \(=\) 252.109
Dual form 252.4.k.f.37.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(6.22311 + 10.7787i) q^{5} +(15.3924 - 10.2992i) q^{7} +(-25.5618 + 44.2743i) q^{11} +37.2311 q^{13} +(11.1076 - 19.2389i) q^{17} +(-27.1693 - 47.0587i) q^{19} +(88.4622 + 153.221i) q^{23} +(-14.9542 + 25.9015i) q^{25} -61.0916 q^{29} +(-159.962 + 277.063i) q^{31} +(206.801 + 101.818i) q^{35} +(157.540 + 272.867i) q^{37} +206.032 q^{41} +339.661 q^{43} +(71.0320 + 123.031i) q^{47} +(130.855 - 317.058i) q^{49} +(155.008 - 268.482i) q^{53} -636.295 q^{55} +(-140.825 + 243.916i) q^{59} +(271.924 + 470.987i) q^{61} +(231.693 + 401.305i) q^{65} +(239.680 - 415.137i) q^{67} -1105.63 q^{71} +(-119.675 + 207.283i) q^{73} +(62.5298 + 944.754i) q^{77} +(-580.333 - 1005.17i) q^{79} +2.93158 q^{83} +276.494 q^{85} +(-639.371 - 1107.42i) q^{89} +(573.078 - 383.449i) q^{91} +(338.156 - 585.703i) q^{95} +79.0596 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 11 q^{5} + 6 q^{7} - 5 q^{11} + 10 q^{13} + 100 q^{17} - 67 q^{19} + 76 q^{23} + 93 q^{25} - 550 q^{29} - 362 q^{31} + 466 q^{35} + 5 q^{37} + 324 q^{41} + 1442 q^{43} - 216 q^{47} + 190 q^{49} + 495 q^{53}+ \cdots + 1122 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/252\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(73\) \(127\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\)

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 0
\(3\) 0 0
\(4\) 0 0
\(5\) 6.22311 + 10.7787i 0.556612 + 0.964080i 0.997776 + 0.0666538i \(0.0212323\pi\)
−0.441164 + 0.897426i \(0.645434\pi\)
\(6\) 0 0
\(7\) 15.3924 − 10.2992i 0.831114 − 0.556102i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −25.5618 + 44.2743i −0.700651 + 1.21356i 0.267587 + 0.963534i \(0.413774\pi\)
−0.968238 + 0.250030i \(0.919559\pi\)
\(12\) 0 0
\(13\) 37.2311 0.794312 0.397156 − 0.917751i \(-0.369997\pi\)
0.397156 + 0.917751i \(0.369997\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 11.1076 − 19.2389i 0.158469 − 0.274477i −0.775848 − 0.630920i \(-0.782677\pi\)
0.934317 + 0.356444i \(0.116011\pi\)
\(18\) 0 0
\(19\) −27.1693 − 47.0587i −0.328056 − 0.568210i 0.654070 − 0.756434i \(-0.273060\pi\)
−0.982126 + 0.188224i \(0.939727\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 88.4622 + 153.221i 0.801985 + 1.38908i 0.918308 + 0.395867i \(0.129556\pi\)
−0.116323 + 0.993211i \(0.537111\pi\)
\(24\) 0 0
\(25\) −14.9542 + 25.9015i −0.119634 + 0.207212i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −61.0916 −0.391187 −0.195593 − 0.980685i \(-0.562663\pi\)
−0.195593 + 0.980685i \(0.562663\pi\)
\(30\) 0 0
\(31\) −159.962 + 277.063i −0.926776 + 1.60522i −0.138097 + 0.990419i \(0.544099\pi\)
−0.788679 + 0.614805i \(0.789235\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 206.801 + 101.818i 0.998735 + 0.491727i
\(36\) 0 0
\(37\) 157.540 + 272.867i 0.699984 + 1.21241i 0.968471 + 0.249125i \(0.0801430\pi\)
−0.268487 + 0.963283i \(0.586524\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 206.032 0.784800 0.392400 − 0.919795i \(-0.371645\pi\)
0.392400 + 0.919795i \(0.371645\pi\)
\(42\) 0 0
\(43\) 339.661 1.20460 0.602301 − 0.798269i \(-0.294251\pi\)
0.602301 + 0.798269i \(0.294251\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 71.0320 + 123.031i 0.220449 + 0.381828i 0.954944 − 0.296785i \(-0.0959146\pi\)
−0.734496 + 0.678613i \(0.762581\pi\)
\(48\) 0 0
\(49\) 130.855 − 317.058i 0.381500 − 0.924369i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 155.008 − 268.482i 0.401736 − 0.695826i −0.592200 − 0.805791i \(-0.701740\pi\)
0.993936 + 0.109965i \(0.0350738\pi\)
\(54\) 0 0
\(55\) −636.295 −1.55996
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −140.825 + 243.916i −0.310743 + 0.538223i −0.978523 − 0.206136i \(-0.933911\pi\)
0.667780 + 0.744358i \(0.267245\pi\)
\(60\) 0 0
\(61\) 271.924 + 470.987i 0.570760 + 0.988585i 0.996488 + 0.0837341i \(0.0266846\pi\)
−0.425728 + 0.904851i \(0.639982\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 231.693 + 401.305i 0.442123 + 0.765780i
\(66\) 0 0
\(67\) 239.680 − 415.137i 0.437038 − 0.756971i −0.560422 − 0.828207i \(-0.689361\pi\)
0.997459 + 0.0712360i \(0.0226943\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −1105.63 −1.84809 −0.924046 − 0.382280i \(-0.875139\pi\)
−0.924046 + 0.382280i \(0.875139\pi\)
\(72\) 0 0
\(73\) −119.675 + 207.283i −0.191876 + 0.332338i −0.945872 − 0.324541i \(-0.894790\pi\)
0.753996 + 0.656879i \(0.228124\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 62.5298 + 944.754i 0.0925445 + 1.39824i
\(78\) 0 0
\(79\) −580.333 − 1005.17i −0.826488 − 1.43152i −0.900777 − 0.434282i \(-0.857002\pi\)
0.0742888 − 0.997237i \(-0.476331\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 2.93158 0.00387690 0.00193845 − 0.999998i \(-0.499383\pi\)
0.00193845 + 0.999998i \(0.499383\pi\)
\(84\) 0 0
\(85\) 276.494 0.352824
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −639.371 − 1107.42i −0.761496 − 1.31895i −0.942079 − 0.335390i \(-0.891132\pi\)
0.180583 − 0.983560i \(-0.442202\pi\)
\(90\) 0 0
\(91\) 573.078 − 383.449i 0.660163 − 0.441719i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 338.156 − 585.703i 0.365200 − 0.632545i
\(96\) 0 0
\(97\) 79.0596 0.0827555 0.0413777 − 0.999144i \(-0.486825\pi\)
0.0413777 + 0.999144i \(0.486825\pi\)
\(98\) 0 0
\(99\) 0 0
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 252.4.k.f.109.2 4
3.2 odd 2 84.4.i.a.25.1 ✓ 4
7.2 even 3 inner 252.4.k.f.37.2 4
7.3 odd 6 1764.4.a.y.1.2 2
7.4 even 3 1764.4.a.o.1.1 2
7.5 odd 6 1764.4.k.q.1549.1 4
7.6 odd 2 1764.4.k.q.361.1 4
12.11 even 2 336.4.q.i.193.1 4
21.2 odd 6 84.4.i.a.37.1 yes 4
21.5 even 6 588.4.i.j.373.2 4
21.11 odd 6 588.4.a.i.1.2 2
21.17 even 6 588.4.a.f.1.1 2
21.20 even 2 588.4.i.j.361.2 4
84.11 even 6 2352.4.a.bt.1.2 2
84.23 even 6 336.4.q.i.289.1 4
84.59 odd 6 2352.4.a.bx.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.4.i.a.25.1 ✓ 4 3.2 odd 2
84.4.i.a.37.1 yes 4 21.2 odd 6
252.4.k.f.37.2 4 7.2 even 3 inner
252.4.k.f.109.2 4 1.1 even 1 trivial
336.4.q.i.193.1 4 12.11 even 2
336.4.q.i.289.1 4 84.23 even 6
588.4.a.f.1.1 2 21.17 even 6
588.4.a.i.1.2 2 21.11 odd 6
588.4.i.j.361.2 4 21.20 even 2
588.4.i.j.373.2 4 21.5 even 6
1764.4.a.o.1.1 2 7.4 even 3
1764.4.a.y.1.2 2 7.3 odd 6
1764.4.k.q.361.1 4 7.6 odd 2
1764.4.k.q.1549.1 4 7.5 odd 6
2352.4.a.bt.1.2 2 84.11 even 6
2352.4.a.bx.1.1 2 84.59 odd 6