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Properties

Label 675.4.a.z.1.4
Level $675$
Weight $4$
Character 675.1
Self dual yes
Analytic conductor $39.826$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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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] = [675,4,Mod(1,675)] 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("675.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(675, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 675.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,1,0,19,0,0,4] 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: yes
Analytic conductor: \(39.8262892539\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.183945.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 12x^{2} + 3x + 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{3}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-2.82516\) of defining polynomial
Character \(\chi\) \(=\) 675.1

$q$-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)
 
\(f(q)\) \(=\) \(q+4.52654 q^{2} +12.4896 q^{4} -30.8119 q^{7} +20.3223 q^{8} +6.79573 q^{11} -31.2493 q^{13} -139.471 q^{14} -7.92702 q^{16} +112.418 q^{17} -60.9161 q^{19} +30.7612 q^{22} -31.1821 q^{23} -141.451 q^{26} -384.827 q^{28} -189.228 q^{29} -343.933 q^{31} -198.460 q^{32} +508.865 q^{34} -206.503 q^{37} -275.739 q^{38} -435.018 q^{41} -60.9569 q^{43} +84.8758 q^{44} -141.147 q^{46} +251.239 q^{47} +606.370 q^{49} -390.291 q^{52} +248.620 q^{53} -626.167 q^{56} -856.547 q^{58} +571.583 q^{59} -329.038 q^{61} -1556.83 q^{62} -834.922 q^{64} +677.273 q^{67} +1404.05 q^{68} +453.668 q^{71} -1024.66 q^{73} -934.745 q^{74} -760.817 q^{76} -209.389 q^{77} +238.142 q^{79} -1969.13 q^{82} +826.853 q^{83} -275.924 q^{86} +138.105 q^{88} +1140.55 q^{89} +962.849 q^{91} -389.451 q^{92} +1137.24 q^{94} +1308.77 q^{97} +2744.76 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} + 19 q^{4} + 4 q^{7} - 15 q^{8} - 52 q^{11} - 2 q^{13} - 138 q^{14} - 5 q^{16} + 64 q^{17} - 46 q^{19} + 87 q^{22} - 90 q^{23} - 469 q^{26} - 110 q^{28} - 470 q^{29} - 262 q^{31} - 199 q^{32}+ \cdots + 4301 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.52654 1.60037 0.800187 − 0.599750i \(-0.204733\pi\)
0.800187 + 0.599750i \(0.204733\pi\)
\(3\) 0 0
\(4\) 12.4896 1.56120
\(5\) 0 0
\(6\) 0 0
\(7\) −30.8119 −1.66368 −0.831842 − 0.555013i \(-0.812713\pi\)
−0.831842 + 0.555013i \(0.812713\pi\)
\(8\) 20.3223 0.898126
\(9\) 0 0
\(10\) 0 0
\(11\) 6.79573 0.186272 0.0931359 − 0.995653i \(-0.470311\pi\)
0.0931359 + 0.995653i \(0.470311\pi\)
\(12\) 0 0
\(13\) −31.2493 −0.666692 −0.333346 − 0.942805i \(-0.608178\pi\)
−0.333346 + 0.942805i \(0.608178\pi\)
\(14\) −139.471 −2.66252
\(15\) 0 0
\(16\) −7.92702 −0.123860
\(17\) 112.418 1.60385 0.801923 − 0.597427i \(-0.203810\pi\)
0.801923 + 0.597427i \(0.203810\pi\)
\(18\) 0 0
\(19\) −60.9161 −0.735533 −0.367766 − 0.929918i \(-0.619877\pi\)
−0.367766 + 0.929918i \(0.619877\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 30.7612 0.298105
\(23\) −31.1821 −0.282692 −0.141346 − 0.989960i \(-0.545143\pi\)
−0.141346 + 0.989960i \(0.545143\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −141.451 −1.06696
\(27\) 0 0
\(28\) −384.827 −2.59734
\(29\) −189.228 −1.21168 −0.605839 − 0.795587i \(-0.707163\pi\)
−0.605839 + 0.795587i \(0.707163\pi\)
\(30\) 0 0
\(31\) −343.933 −1.99265 −0.996327 − 0.0856357i \(-0.972708\pi\)
−0.996327 + 0.0856357i \(0.972708\pi\)
\(32\) −198.460 −1.09635
\(33\) 0 0
\(34\) 508.865 2.56675
\(35\) 0 0
\(36\) 0 0
\(37\) −206.503 −0.917538 −0.458769 − 0.888556i \(-0.651709\pi\)
−0.458769 + 0.888556i \(0.651709\pi\)
\(38\) −275.739 −1.17713
\(39\) 0 0
\(40\) 0 0
\(41\) −435.018 −1.65704 −0.828518 − 0.559963i \(-0.810816\pi\)
−0.828518 + 0.559963i \(0.810816\pi\)
\(42\) 0 0
\(43\) −60.9569 −0.216183 −0.108091 − 0.994141i \(-0.534474\pi\)
−0.108091 + 0.994141i \(0.534474\pi\)
\(44\) 84.8758 0.290807
\(45\) 0 0
\(46\) −141.147 −0.452412
\(47\) 251.239 0.779721 0.389861 − 0.920874i \(-0.372523\pi\)
0.389861 + 0.920874i \(0.372523\pi\)
\(48\) 0 0
\(49\) 606.370 1.76784
\(50\) 0 0
\(51\) 0 0
\(52\) −390.291 −1.04084
\(53\) 248.620 0.644350 0.322175 − 0.946680i \(-0.395586\pi\)
0.322175 + 0.946680i \(0.395586\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −626.167 −1.49420
\(57\) 0 0
\(58\) −856.547 −1.93914
\(59\) 571.583 1.26125 0.630626 − 0.776087i \(-0.282798\pi\)
0.630626 + 0.776087i \(0.282798\pi\)
\(60\) 0 0
\(61\) −329.038 −0.690639 −0.345319 − 0.938485i \(-0.612229\pi\)
−0.345319 + 0.938485i \(0.612229\pi\)
\(62\) −1556.83 −3.18899
\(63\) 0 0
\(64\) −834.922 −1.63071
\(65\) 0 0
\(66\) 0 0
\(67\) 677.273 1.23496 0.617478 − 0.786588i \(-0.288155\pi\)
0.617478 + 0.786588i \(0.288155\pi\)
\(68\) 1404.05 2.50392
\(69\) 0 0
\(70\) 0 0
\(71\) 453.668 0.758317 0.379159 − 0.925332i \(-0.376213\pi\)
0.379159 + 0.925332i \(0.376213\pi\)
\(72\) 0 0
\(73\) −1024.66 −1.64283 −0.821417 − 0.570328i \(-0.806816\pi\)
−0.821417 + 0.570328i \(0.806816\pi\)
\(74\) −934.745 −1.46840
\(75\) 0 0
\(76\) −760.817 −1.14831
\(77\) −209.389 −0.309897
\(78\) 0 0
\(79\) 238.142 0.339153 0.169576 − 0.985517i \(-0.445760\pi\)
0.169576 + 0.985517i \(0.445760\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −1969.13 −2.65188
\(83\) 826.853 1.09348 0.546740 − 0.837302i \(-0.315869\pi\)
0.546740 + 0.837302i \(0.315869\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −275.924 −0.345973
\(87\) 0 0
\(88\) 138.105 0.167296
\(89\) 1140.55 1.35840 0.679201 − 0.733953i \(-0.262327\pi\)
0.679201 + 0.733953i \(0.262327\pi\)
\(90\) 0 0
\(91\) 962.849 1.10916
\(92\) −389.451 −0.441337
\(93\) 0 0
\(94\) 1137.24 1.24785
\(95\) 0 0
\(96\) 0 0
\(97\) 1308.77 1.36995 0.684977 − 0.728565i \(-0.259812\pi\)
0.684977 + 0.728565i \(0.259812\pi\)
\(98\) 2744.76 2.82921
\(99\) 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 675.4.a.z.1.4 yes 4
3.2 odd 2 675.4.a.v.1.1 yes 4
5.2 odd 4 675.4.b.p.649.7 8
5.3 odd 4 675.4.b.p.649.2 8
5.4 even 2 675.4.a.u.1.1 ✓ 4
15.2 even 4 675.4.b.q.649.2 8
15.8 even 4 675.4.b.q.649.7 8
15.14 odd 2 675.4.a.y.1.4 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
675.4.a.u.1.1 ✓ 4 5.4 even 2
675.4.a.v.1.1 yes 4 3.2 odd 2
675.4.a.y.1.4 yes 4 15.14 odd 2
675.4.a.z.1.4 yes 4 1.1 even 1 trivial
675.4.b.p.649.2 8 5.3 odd 4
675.4.b.p.649.7 8 5.2 odd 4
675.4.b.q.649.2 8 15.2 even 4
675.4.b.q.649.7 8 15.8 even 4