Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1150,4,Mod(599,1150)] 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("1150.599"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1150, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1150.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,0,-40,0,20,0,0,-168,0,-52] 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: \(67.8521965066\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 214x^{8} + 15751x^{6} + 460323x^{4} + 4609305x^{2} + 8503056 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 599.3
Root \(-1.53483i\) of defining polynomial
Character \(\chi\) \(=\) 1150.599
Dual form 1150.4.b.r.599.8

$q$-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)
 
\(f(q)\) \(=\) \(q-2.00000i q^{2} -0.534830i q^{3} -4.00000 q^{4} -1.06966 q^{6} +28.9380i q^{7} +8.00000i q^{8} +26.7140 q^{9} +33.5234 q^{11} +2.13932i q^{12} +10.2917i q^{13} +57.8760 q^{14} +16.0000 q^{16} -56.5854i q^{17} -53.4279i q^{18} +5.26405 q^{19} +15.4769 q^{21} -67.0468i q^{22} +23.0000i q^{23} +4.27864 q^{24} +20.5834 q^{26} -28.7278i q^{27} -115.752i q^{28} +162.334 q^{29} -160.556 q^{31} -32.0000i q^{32} -17.9293i q^{33} -113.171 q^{34} -106.856 q^{36} +16.9178i q^{37} -10.5281i q^{38} +5.50431 q^{39} +22.7758 q^{41} -30.9538i q^{42} +333.620i q^{43} -134.094 q^{44} +46.0000 q^{46} -130.383i q^{47} -8.55728i q^{48} -494.408 q^{49} -30.2636 q^{51} -41.1668i q^{52} +673.206i q^{53} -57.4557 q^{54} -231.504 q^{56} -2.81537i q^{57} -324.669i q^{58} -291.958 q^{59} +454.587 q^{61} +321.113i q^{62} +773.048i q^{63} -64.0000 q^{64} -35.8586 q^{66} -132.389i q^{67} +226.342i q^{68} +12.3011 q^{69} +121.326 q^{71} +213.712i q^{72} +176.482i q^{73} +33.8356 q^{74} -21.0562 q^{76} +970.100i q^{77} -11.0086i q^{78} -563.082 q^{79} +705.912 q^{81} -45.5516i q^{82} +809.068i q^{83} -61.9076 q^{84} +667.241 q^{86} -86.8213i q^{87} +268.187i q^{88} +702.417 q^{89} -297.821 q^{91} -92.0000i q^{92} +85.8703i q^{93} -260.766 q^{94} -17.1146 q^{96} -342.945i q^{97} +988.815i q^{98} +895.543 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 40 q^{4} + 20 q^{6} - 168 q^{9} - 52 q^{11} + 12 q^{14} + 160 q^{16} - 148 q^{19} - 176 q^{21} - 80 q^{24} - 244 q^{26} - 74 q^{29} + 440 q^{31} - 924 q^{34} + 672 q^{36} - 390 q^{39} + 224 q^{41}+ \cdots - 4794 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1150\mathbb{Z}\right)^\times\).

\(n\) \(51\) \(277\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.00000i − 0.707107i
\(3\) − 0.534830i − 0.102928i −0.998675 − 0.0514640i \(-0.983611\pi\)
0.998675 − 0.0514640i \(-0.0163888\pi\)
\(4\) −4.00000 −0.500000
\(5\) 0 0
\(6\) −1.06966 −0.0727812
\(7\) 28.9380i 1.56250i 0.624215 + 0.781252i \(0.285419\pi\)
−0.624215 + 0.781252i \(0.714581\pi\)
\(8\) 8.00000i 0.353553i
\(9\) 26.7140 0.989406
\(10\) 0 0
\(11\) 33.5234 0.918880 0.459440 − 0.888209i \(-0.348050\pi\)
0.459440 + 0.888209i \(0.348050\pi\)
\(12\) 2.13932i 0.0514640i
\(13\) 10.2917i 0.219570i 0.993955 + 0.109785i \(0.0350162\pi\)
−0.993955 + 0.109785i \(0.964984\pi\)
\(14\) 57.8760 1.10486
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) − 56.5854i − 0.807293i −0.914915 − 0.403646i \(-0.867743\pi\)
0.914915 − 0.403646i \(-0.132257\pi\)
\(18\) − 53.4279i − 0.699616i
\(19\) 5.26405 0.0635608 0.0317804 − 0.999495i \(-0.489882\pi\)
0.0317804 + 0.999495i \(0.489882\pi\)
\(20\) 0 0
\(21\) 15.4769 0.160826
\(22\) − 67.0468i − 0.649747i
\(23\) 23.0000i 0.208514i
\(24\) 4.27864 0.0363906
\(25\) 0 0
\(26\) 20.5834 0.155259
\(27\) − 28.7278i − 0.204766i
\(28\) − 115.752i − 0.781252i
\(29\) 162.334 1.03947 0.519737 − 0.854327i \(-0.326030\pi\)
0.519737 + 0.854327i \(0.326030\pi\)
\(30\) 0 0
\(31\) −160.556 −0.930218 −0.465109 − 0.885253i \(-0.653985\pi\)
−0.465109 + 0.885253i \(0.653985\pi\)
\(32\) − 32.0000i − 0.176777i
\(33\) − 17.9293i − 0.0945786i
\(34\) −113.171 −0.570842
\(35\) 0 0
\(36\) −106.856 −0.494703
\(37\) 16.9178i 0.0751694i 0.999293 + 0.0375847i \(0.0119664\pi\)
−0.999293 + 0.0375847i \(0.988034\pi\)
\(38\) − 10.5281i − 0.0449443i
\(39\) 5.50431 0.0225999
\(40\) 0 0
\(41\) 22.7758 0.0867557 0.0433779 − 0.999059i \(-0.486188\pi\)
0.0433779 + 0.999059i \(0.486188\pi\)
\(42\) − 30.9538i − 0.113721i
\(43\) 333.620i 1.18318i 0.806240 + 0.591589i \(0.201499\pi\)
−0.806240 + 0.591589i \(0.798501\pi\)
\(44\) −134.094 −0.459440
\(45\) 0 0
\(46\) 46.0000 0.147442
\(47\) − 130.383i − 0.404645i −0.979319 − 0.202322i \(-0.935151\pi\)
0.979319 − 0.202322i \(-0.0648489\pi\)
\(48\) − 8.55728i − 0.0257320i
\(49\) −494.408 −1.44142
\(50\) 0 0
\(51\) −30.2636 −0.0830931
\(52\) − 41.1668i − 0.109785i
\(53\) 673.206i 1.74475i 0.488835 + 0.872376i \(0.337422\pi\)
−0.488835 + 0.872376i \(0.662578\pi\)
\(54\) −57.4557 −0.144791
\(55\) 0 0
\(56\) −231.504 −0.552429
\(57\) − 2.81537i − 0.00654219i
\(58\) − 324.669i − 0.735019i
\(59\) −291.958 −0.644233 −0.322116 − 0.946700i \(-0.604394\pi\)
−0.322116 + 0.946700i \(0.604394\pi\)
\(60\) 0 0
\(61\) 454.587 0.954162 0.477081 − 0.878859i \(-0.341695\pi\)
0.477081 + 0.878859i \(0.341695\pi\)
\(62\) 321.113i 0.657764i
\(63\) 773.048i 1.54595i
\(64\) −64.0000 −0.125000
\(65\) 0 0
\(66\) −35.8586 −0.0668772
\(67\) − 132.389i − 0.241401i −0.992689 − 0.120701i \(-0.961486\pi\)
0.992689 − 0.120701i \(-0.0385141\pi\)
\(68\) 226.342i 0.403646i
\(69\) 12.3011 0.0214620
\(70\) 0 0
\(71\) 121.326 0.202799 0.101399 − 0.994846i \(-0.467668\pi\)
0.101399 + 0.994846i \(0.467668\pi\)
\(72\) 213.712i 0.349808i
\(73\) 176.482i 0.282955i 0.989942 + 0.141477i \(0.0451852\pi\)
−0.989942 + 0.141477i \(0.954815\pi\)
\(74\) 33.8356 0.0531528
\(75\) 0 0
\(76\) −21.0562 −0.0317804
\(77\) 970.100i 1.43576i
\(78\) − 11.0086i − 0.0159805i
\(79\) −563.082 −0.801920 −0.400960 − 0.916096i \(-0.631323\pi\)
−0.400960 + 0.916096i \(0.631323\pi\)
\(80\) 0 0
\(81\) 705.912 0.968330
\(82\) − 45.5516i − 0.0613456i
\(83\) 809.068i 1.06996i 0.844864 + 0.534981i \(0.179681\pi\)
−0.844864 + 0.534981i \(0.820319\pi\)
\(84\) −61.9076 −0.0804128
\(85\) 0 0
\(86\) 667.241 0.836633
\(87\) − 86.8213i − 0.106991i
\(88\) 268.187i 0.324873i
\(89\) 702.417 0.836585 0.418293 − 0.908312i \(-0.362629\pi\)
0.418293 + 0.908312i \(0.362629\pi\)
\(90\) 0 0
\(91\) −297.821 −0.343079
\(92\) − 92.0000i − 0.104257i
\(93\) 85.8703i 0.0957456i
\(94\) −260.766 −0.286127
\(95\) 0 0
\(96\) −17.1146 −0.0181953
\(97\) − 342.945i − 0.358977i −0.983760 − 0.179489i \(-0.942556\pi\)
0.983760 − 0.179489i \(-0.0574443\pi\)
\(98\) 988.815i 1.01924i
\(99\) 895.543 0.909146
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 1150.4.b.r.599.3 10
5.2 odd 4 1150.4.a.v.1.3 yes 5
5.3 odd 4 1150.4.a.q.1.3 ✓ 5
5.4 even 2 inner 1150.4.b.r.599.8 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1150.4.a.q.1.3 ✓ 5 5.3 odd 4
1150.4.a.v.1.3 yes 5 5.2 odd 4
1150.4.b.r.599.3 10 1.1 even 1 trivial
1150.4.b.r.599.8 10 5.4 even 2 inner