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Properties

Label 300.8.d.h.49.1
Level $300$
Weight $8$
Character 300.49
Analytic conductor $93.716$
Analytic rank $0$
Dimension $6$
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] = [300,8,Mod(49,300)] 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("300.49"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(300, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 300.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,0,0,0,0,-4374,0,6276] 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: \(93.7155076452\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 1145x^{4} + 319204x^{2} + 15920100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{4}\cdot 5^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.1
Root \(18.3302i\) of defining polynomial
Character \(\chi\) \(=\) 300.49
Dual form 300.8.d.h.49.6

$q$-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)
 
\(f(q)\) \(=\) \(q-27.0000i q^{3} -1002.81i q^{7} -729.000 q^{9} +6311.33 q^{11} +11909.0i q^{13} +288.453i q^{17} -28007.7 q^{19} -27075.9 q^{21} +28288.8i q^{23} +19683.0i q^{27} -97664.9 q^{29} -290434. q^{31} -170406. i q^{33} +24738.0i q^{37} +321543. q^{39} +688629. q^{41} +396038. i q^{43} +920149. i q^{47} -182089. q^{49} +7788.24 q^{51} -1.64832e6i q^{53} +756208. i q^{57} -2.34227e6 q^{59} -128675. q^{61} +731050. i q^{63} +2.69575e6i q^{67} +763797. q^{69} -2.06763e6 q^{71} +2.11158e6i q^{73} -6.32907e6i q^{77} +694941. q^{79} +531441. q^{81} +4.03690e6i q^{83} +2.63695e6i q^{87} -3.43798e6 q^{89} +1.19425e7 q^{91} +7.84172e6i q^{93} +7.90999e6i q^{97} -4.60096e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 4374 q^{9} + 6276 q^{11} - 42390 q^{19} + 18954 q^{21} - 53196 q^{29} - 286926 q^{31} - 193590 q^{39} - 962640 q^{41} - 3382476 q^{49} + 282204 q^{51} - 3981516 q^{59} + 215442 q^{61} - 2183436 q^{69}+ \cdots - 4575204 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/300\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(151\) \(277\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) − 27.0000i − 0.577350i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 1002.81i − 1.10504i −0.833501 − 0.552518i \(-0.813667\pi\)
0.833501 − 0.552518i \(-0.186333\pi\)
\(8\) 0 0
\(9\) −729.000 −0.333333
\(10\) 0 0
\(11\) 6311.33 1.42970 0.714852 − 0.699276i \(-0.246494\pi\)
0.714852 + 0.699276i \(0.246494\pi\)
\(12\) 0 0
\(13\) 11909.0i 1.50340i 0.659506 + 0.751699i \(0.270765\pi\)
−0.659506 + 0.751699i \(0.729235\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 288.453i 0.0142398i 0.999975 + 0.00711991i \(0.00226636\pi\)
−0.999975 + 0.00711991i \(0.997734\pi\)
\(18\) 0 0
\(19\) −28007.7 −0.936785 −0.468392 − 0.883521i \(-0.655167\pi\)
−0.468392 + 0.883521i \(0.655167\pi\)
\(20\) 0 0
\(21\) −27075.9 −0.637993
\(22\) 0 0
\(23\) 28288.8i 0.484805i 0.970176 + 0.242402i \(0.0779354\pi\)
−0.970176 + 0.242402i \(0.922065\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 19683.0i 0.192450i
\(28\) 0 0
\(29\) −97664.9 −0.743610 −0.371805 − 0.928311i \(-0.621261\pi\)
−0.371805 + 0.928311i \(0.621261\pi\)
\(30\) 0 0
\(31\) −290434. −1.75098 −0.875491 − 0.483234i \(-0.839462\pi\)
−0.875491 + 0.483234i \(0.839462\pi\)
\(32\) 0 0
\(33\) − 170406.i − 0.825440i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 24738.0i 0.0802895i 0.999194 + 0.0401448i \(0.0127819\pi\)
−0.999194 + 0.0401448i \(0.987218\pi\)
\(38\) 0 0
\(39\) 321543. 0.867987
\(40\) 0 0
\(41\) 688629. 1.56042 0.780210 − 0.625517i \(-0.215112\pi\)
0.780210 + 0.625517i \(0.215112\pi\)
\(42\) 0 0
\(43\) 396038.i 0.759621i 0.925064 + 0.379811i \(0.124011\pi\)
−0.925064 + 0.379811i \(0.875989\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 920149.i 1.29275i 0.763018 + 0.646377i \(0.223717\pi\)
−0.763018 + 0.646377i \(0.776283\pi\)
\(48\) 0 0
\(49\) −182089. −0.221104
\(50\) 0 0
\(51\) 7788.24 0.00822136
\(52\) 0 0
\(53\) − 1.64832e6i − 1.52082i −0.649445 − 0.760409i \(-0.724999\pi\)
0.649445 − 0.760409i \(-0.275001\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 756208.i 0.540853i
\(58\) 0 0
\(59\) −2.34227e6 −1.48475 −0.742377 − 0.669983i \(-0.766301\pi\)
−0.742377 + 0.669983i \(0.766301\pi\)
\(60\) 0 0
\(61\) −128675. −0.0725839 −0.0362919 − 0.999341i \(-0.511555\pi\)
−0.0362919 + 0.999341i \(0.511555\pi\)
\(62\) 0 0
\(63\) 731050.i 0.368345i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 2.69575e6i 1.09501i 0.836802 + 0.547505i \(0.184422\pi\)
−0.836802 + 0.547505i \(0.815578\pi\)
\(68\) 0 0
\(69\) 763797. 0.279902
\(70\) 0 0
\(71\) −2.06763e6 −0.685596 −0.342798 − 0.939409i \(-0.611375\pi\)
−0.342798 + 0.939409i \(0.611375\pi\)
\(72\) 0 0
\(73\) 2.11158e6i 0.635299i 0.948208 + 0.317650i \(0.102894\pi\)
−0.948208 + 0.317650i \(0.897106\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 6.32907e6i − 1.57987i
\(78\) 0 0
\(79\) 694941. 0.158582 0.0792909 − 0.996852i \(-0.474734\pi\)
0.0792909 + 0.996852i \(0.474734\pi\)
\(80\) 0 0
\(81\) 531441. 0.111111
\(82\) 0 0
\(83\) 4.03690e6i 0.774953i 0.921880 + 0.387476i \(0.126653\pi\)
−0.921880 + 0.387476i \(0.873347\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 2.63695e6i 0.429324i
\(88\) 0 0
\(89\) −3.43798e6 −0.516938 −0.258469 − 0.966020i \(-0.583218\pi\)
−0.258469 + 0.966020i \(0.583218\pi\)
\(90\) 0 0
\(91\) 1.19425e7 1.66131
\(92\) 0 0
\(93\) 7.84172e6i 1.01093i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 7.90999e6i 0.879984i 0.898002 + 0.439992i \(0.145019\pi\)
−0.898002 + 0.439992i \(0.854981\pi\)
\(98\) 0 0
\(99\) −4.60096e6 −0.476568
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 300.8.d.h.49.1 6
5.2 odd 4 300.8.a.m.1.3 ✓ 3
5.3 odd 4 300.8.a.n.1.1 yes 3
5.4 even 2 inner 300.8.d.h.49.6 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
300.8.a.m.1.3 ✓ 3 5.2 odd 4
300.8.a.n.1.1 yes 3 5.3 odd 4
300.8.d.h.49.1 6 1.1 even 1 trivial
300.8.d.h.49.6 6 5.4 even 2 inner