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Properties

Label 675.4.b.q.649.7
Level $675$
Weight $4$
Character 675.649
Analytic conductor $39.826$
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] = [675,4,Mod(649,675)] 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("675.649"); 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, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 675.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,-38,0,0,0,0,0,0,104,0,0,-276,0,-10,0,0,92] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(39.8262892539\)
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: \( x^{8} + 25x^{6} + 186x^{4} + 441x^{2} + 324 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 649.7
Root \(-2.82516i\) of defining polynomial
Character \(\chi\) \(=\) 675.649
Dual form 675.4.b.q.649.2

$q$-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)
 
\(f(q)\) \(=\) \(q+4.52654i q^{2} -12.4896 q^{4} +30.8119i q^{7} -20.3223i q^{8} -6.79573 q^{11} -31.2493i q^{13} -139.471 q^{14} -7.92702 q^{16} +112.418i q^{17} +60.9161 q^{19} -30.7612i q^{22} +31.1821i q^{23} +141.451 q^{26} -384.827i q^{28} -189.228 q^{29} -343.933 q^{31} -198.460i q^{32} -508.865 q^{34} +206.503i q^{37} +275.739i q^{38} +435.018 q^{41} -60.9569i q^{43} +84.8758 q^{44} -141.147 q^{46} +251.239i q^{47} -606.370 q^{49} +390.291i q^{52} -248.620i q^{53} +626.167 q^{56} -856.547i q^{58} +571.583 q^{59} -329.038 q^{61} -1556.83i q^{62} +834.922 q^{64} -677.273i q^{67} -1404.05i q^{68} -453.668 q^{71} -1024.66i q^{73} -934.745 q^{74} -760.817 q^{76} -209.389i q^{77} -238.142 q^{79} +1969.13i q^{82} -826.853i q^{83} +275.924 q^{86} +138.105i q^{88} +1140.55 q^{89} +962.849 q^{91} -389.451i q^{92} -1137.24 q^{94} -1308.77i q^{97} -2744.76i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 38 q^{4} + 104 q^{11} - 276 q^{14} - 10 q^{16} + 92 q^{19} + 938 q^{26} - 940 q^{29} - 524 q^{31} + 84 q^{34} + 1396 q^{41} - 838 q^{44} + 1074 q^{46} - 1560 q^{49} + 4068 q^{56} - 200 q^{59} + 148 q^{61}+ \cdots - 5694 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/675\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.52654i 1.60037i 0.599750 + 0.800187i \(0.295267\pi\)
−0.599750 + 0.800187i \(0.704733\pi\)
\(3\) 0 0
\(4\) −12.4896 −1.56120
\(5\) 0 0
\(6\) 0 0
\(7\) 30.8119i 1.66368i 0.555013 + 0.831842i \(0.312713\pi\)
−0.555013 + 0.831842i \(0.687287\pi\)
\(8\) − 20.3223i − 0.898126i
\(9\) 0 0
\(10\) 0 0
\(11\) −6.79573 −0.186272 −0.0931359 − 0.995653i \(-0.529689\pi\)
−0.0931359 + 0.995653i \(0.529689\pi\)
\(12\) 0 0
\(13\) − 31.2493i − 0.666692i −0.942805 − 0.333346i \(-0.891822\pi\)
0.942805 − 0.333346i \(-0.108178\pi\)
\(14\) −139.471 −2.66252
\(15\) 0 0
\(16\) −7.92702 −0.123860
\(17\) 112.418i 1.60385i 0.597427 + 0.801923i \(0.296190\pi\)
−0.597427 + 0.801923i \(0.703810\pi\)
\(18\) 0 0
\(19\) 60.9161 0.735533 0.367766 − 0.929918i \(-0.380123\pi\)
0.367766 + 0.929918i \(0.380123\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 30.7612i − 0.298105i
\(23\) 31.1821i 0.282692i 0.989960 + 0.141346i \(0.0451429\pi\)
−0.989960 + 0.141346i \(0.954857\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 141.451 1.06696
\(27\) 0 0
\(28\) − 384.827i − 2.59734i
\(29\) −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 \(0.972708\pi\)
\(32\) − 198.460i − 1.09635i
\(33\) 0 0
\(34\) −508.865 −2.56675
\(35\) 0 0
\(36\) 0 0
\(37\) 206.503i 0.917538i 0.888556 + 0.458769i \(0.151709\pi\)
−0.888556 + 0.458769i \(0.848291\pi\)
\(38\) 275.739i 1.17713i
\(39\) 0 0
\(40\) 0 0
\(41\) 435.018 1.65704 0.828518 − 0.559963i \(-0.189184\pi\)
0.828518 + 0.559963i \(0.189184\pi\)
\(42\) 0 0
\(43\) − 60.9569i − 0.216183i −0.994141 − 0.108091i \(-0.965526\pi\)
0.994141 − 0.108091i \(-0.0344739\pi\)
\(44\) 84.8758 0.290807
\(45\) 0 0
\(46\) −141.147 −0.452412
\(47\) 251.239i 0.779721i 0.920874 + 0.389861i \(0.127477\pi\)
−0.920874 + 0.389861i \(0.872523\pi\)
\(48\) 0 0
\(49\) −606.370 −1.76784
\(50\) 0 0
\(51\) 0 0
\(52\) 390.291i 1.04084i
\(53\) − 248.620i − 0.644350i −0.946680 − 0.322175i \(-0.895586\pi\)
0.946680 − 0.322175i \(-0.104414\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 626.167 1.49420
\(57\) 0 0
\(58\) − 856.547i − 1.93914i
\(59\) 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 \(0.612229\pi\)
\(62\) − 1556.83i − 3.18899i
\(63\) 0 0
\(64\) 834.922 1.63071
\(65\) 0 0
\(66\) 0 0
\(67\) − 677.273i − 1.23496i −0.786588 − 0.617478i \(-0.788155\pi\)
0.786588 − 0.617478i \(-0.211845\pi\)
\(68\) − 1404.05i − 2.50392i
\(69\) 0 0
\(70\) 0 0
\(71\) −453.668 −0.758317 −0.379159 − 0.925332i \(-0.623787\pi\)
−0.379159 + 0.925332i \(0.623787\pi\)
\(72\) 0 0
\(73\) − 1024.66i − 1.64283i −0.570328 − 0.821417i \(-0.693184\pi\)
0.570328 − 0.821417i \(-0.306816\pi\)
\(74\) −934.745 −1.46840
\(75\) 0 0
\(76\) −760.817 −1.14831
\(77\) − 209.389i − 0.309897i
\(78\) 0 0
\(79\) −238.142 −0.339153 −0.169576 − 0.985517i \(-0.554240\pi\)
−0.169576 + 0.985517i \(0.554240\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 1969.13i 2.65188i
\(83\) − 826.853i − 1.09348i −0.837302 − 0.546740i \(-0.815869\pi\)
0.837302 − 0.546740i \(-0.184131\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 275.924 0.345973
\(87\) 0 0
\(88\) 138.105i 0.167296i
\(89\) 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
\(92\) − 389.451i − 0.441337i
\(93\) 0 0
\(94\) −1137.24 −1.24785
\(95\) 0 0
\(96\) 0 0
\(97\) − 1308.77i − 1.36995i −0.728565 − 0.684977i \(-0.759812\pi\)
0.728565 − 0.684977i \(-0.240188\pi\)
\(98\) − 2744.76i − 2.82921i
\(99\) 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.b.q.649.7 8
3.2 odd 2 675.4.b.p.649.2 8
5.2 odd 4 675.4.a.v.1.1 yes 4
5.3 odd 4 675.4.a.y.1.4 yes 4
5.4 even 2 inner 675.4.b.q.649.2 8
15.2 even 4 675.4.a.z.1.4 yes 4
15.8 even 4 675.4.a.u.1.1 ✓ 4
15.14 odd 2 675.4.b.p.649.7 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
675.4.a.u.1.1 ✓ 4 15.8 even 4
675.4.a.v.1.1 yes 4 5.2 odd 4
675.4.a.y.1.4 yes 4 5.3 odd 4
675.4.a.z.1.4 yes 4 15.2 even 4
675.4.b.p.649.2 8 3.2 odd 2
675.4.b.p.649.7 8 15.14 odd 2
675.4.b.q.649.2 8 5.4 even 2 inner
675.4.b.q.649.7 8 1.1 even 1 trivial