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Properties

Label 230.2.e.a.137.3
Level $230$
Weight $2$
Character 230.137
Analytic conductor $1.837$
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] = [230,2,Mod(137,230)] 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("230.137"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(230, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([1, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 230.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,-4,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.83655924649\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.110166016.2
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 10x^{6} + 19x^{4} + 10x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 137.3
Root \(0.814115i\) of defining polynomial
Character \(\chi\) \(=\) 230.137
Dual form 230.2.e.a.183.3

$q$-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+(0.707107 + 0.707107i) q^{2} +(-0.868559 + 0.868559i) q^{3} +1.00000i q^{4} +(-2.22833 - 0.185885i) q^{5} -1.22833 q^{6} +(-1.38978 + 1.38978i) q^{7} +(-0.707107 + 0.707107i) q^{8} +1.49121i q^{9} +(-1.44423 - 1.70711i) q^{10} -0.642542i q^{11} +(-0.868559 - 0.868559i) q^{12} +(-2.12690 + 2.12690i) q^{13} -1.96545 q^{14} +(2.09689 - 1.77398i) q^{15} -1.00000 q^{16} +(-0.903113 + 0.903113i) q^{17} +(-1.05444 + 1.05444i) q^{18} +2.55123 q^{19} +(0.185885 - 2.22833i) q^{20} -2.41421i q^{21} +(0.454346 - 0.454346i) q^{22} +(4.74955 + 0.664664i) q^{23} -1.22833i q^{24} +(4.93089 + 0.828427i) q^{25} -3.00789 q^{26} +(-3.90088 - 3.90088i) q^{27} +(-1.38978 - 1.38978i) q^{28} +0.214016i q^{29} +(2.73712 + 0.228328i) q^{30} +6.15922 q^{31} +(-0.707107 - 0.707107i) q^{32} +(0.558086 + 0.558086i) q^{33} -1.27719 q^{34} +(3.35523 - 2.83855i) q^{35} -1.49121 q^{36} +(-7.20809 + 7.20809i) q^{37} +(1.80399 + 1.80399i) q^{38} -3.69468i q^{39} +(1.70711 - 1.44423i) q^{40} +3.33722 q^{41} +(1.70711 - 1.70711i) q^{42} +(7.85390 + 7.85390i) q^{43} +0.642542 q^{44} +(0.277194 - 3.32290i) q^{45} +(2.88845 + 3.82843i) q^{46} +(-4.15133 - 4.15133i) q^{47} +(0.868559 - 0.868559i) q^{48} +3.13702i q^{49} +(2.90088 + 4.07245i) q^{50} -1.56881i q^{51} +(-2.12690 - 2.12690i) q^{52} +(2.93732 + 2.93732i) q^{53} -5.51668i q^{54} +(-0.119439 + 1.43179i) q^{55} -1.96545i q^{56} +(-2.21590 + 2.21590i) q^{57} +(-0.151332 + 0.151332i) q^{58} -7.08223i q^{59} +(1.77398 + 2.09689i) q^{60} -5.29743i q^{61} +(4.35523 + 4.35523i) q^{62} +(-2.07245 - 2.07245i) q^{63} -1.00000i q^{64} +(5.13479 - 4.34407i) q^{65} +0.789252i q^{66} +(3.65370 - 3.65370i) q^{67} +(-0.903113 - 0.903113i) q^{68} +(-4.70257 + 3.54797i) q^{69} +(4.37966 + 0.365348i) q^{70} -1.83812 q^{71} +(-1.05444 - 1.05444i) q^{72} +(-5.20166 + 5.20166i) q^{73} -10.1938 q^{74} +(-5.00231 + 3.56323i) q^{75} +2.55123i q^{76} +(0.892992 + 0.892992i) q^{77} +(2.61253 - 2.61253i) q^{78} -6.19754 q^{79} +(2.22833 + 0.185885i) q^{80} +2.30266 q^{81} +(2.35977 + 2.35977i) q^{82} +(-6.04767 - 6.04767i) q^{83} +2.41421 q^{84} +(2.18031 - 1.84456i) q^{85} +11.1071i q^{86} +(-0.185885 - 0.185885i) q^{87} +(0.454346 + 0.454346i) q^{88} +15.7567 q^{89} +(2.54565 - 2.15364i) q^{90} -5.91185i q^{91} +(-0.664664 + 4.74955i) q^{92} +(-5.34965 + 5.34965i) q^{93} -5.87087i q^{94} +(-5.68498 - 0.474237i) q^{95} +1.22833 q^{96} +(-10.7898 + 10.7898i) q^{97} +(-2.21821 + 2.21821i) q^{98} +0.958165 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{3} - 4 q^{5} + 4 q^{6} - 4 q^{12} + 4 q^{14} - 8 q^{16} - 24 q^{17} - 8 q^{18} + 12 q^{19} + 4 q^{20} + 12 q^{22} + 16 q^{23} + 12 q^{26} + 8 q^{27} + 16 q^{30} - 4 q^{31} - 20 q^{33} + 4 q^{34}+ \cdots + 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/230\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(51\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.707107 + 0.707107i 0.500000 + 0.500000i
\(3\) −0.868559 + 0.868559i −0.501463 + 0.501463i −0.911892 − 0.410429i \(-0.865379\pi\)
0.410429 + 0.911892i \(0.365379\pi\)
\(4\) 1.00000i 0.500000i
\(5\) −2.22833 − 0.185885i −0.996539 − 0.0831305i
\(6\) −1.22833 −0.501463
\(7\) −1.38978 + 1.38978i −0.525288 + 0.525288i −0.919164 − 0.393876i \(-0.871134\pi\)
0.393876 + 0.919164i \(0.371134\pi\)
\(8\) −0.707107 + 0.707107i −0.250000 + 0.250000i
\(9\) 1.49121i 0.497070i
\(10\) −1.44423 − 1.70711i −0.456704 − 0.539835i
\(11\) − 0.642542i − 0.193734i −0.995297 − 0.0968668i \(-0.969118\pi\)
0.995297 − 0.0968668i \(-0.0308821\pi\)
\(12\) −0.868559 − 0.868559i −0.250731 − 0.250731i
\(13\) −2.12690 + 2.12690i −0.589896 + 0.589896i −0.937603 − 0.347707i \(-0.886960\pi\)
0.347707 + 0.937603i \(0.386960\pi\)
\(14\) −1.96545 −0.525288
\(15\) 2.09689 − 1.77398i 0.541414 − 0.458040i
\(16\) −1.00000 −0.250000
\(17\) −0.903113 + 0.903113i −0.219037 + 0.219037i −0.808093 − 0.589056i \(-0.799500\pi\)
0.589056 + 0.808093i \(0.299500\pi\)
\(18\) −1.05444 + 1.05444i −0.248535 + 0.248535i
\(19\) 2.55123 0.585293 0.292647 − 0.956221i \(-0.405464\pi\)
0.292647 + 0.956221i \(0.405464\pi\)
\(20\) 0.185885 − 2.22833i 0.0415652 − 0.498269i
\(21\) − 2.41421i − 0.526825i
\(22\) 0.454346 − 0.454346i 0.0968668 − 0.0968668i
\(23\) 4.74955 + 0.664664i 0.990350 + 0.138592i
\(24\) − 1.22833i − 0.250731i
\(25\) 4.93089 + 0.828427i 0.986179 + 0.165685i
\(26\) −3.00789 −0.589896
\(27\) −3.90088 − 3.90088i −0.750725 − 0.750725i
\(28\) −1.38978 − 1.38978i −0.262644 − 0.262644i
\(29\) 0.214016i 0.0397417i 0.999803 + 0.0198709i \(0.00632551\pi\)
−0.999803 + 0.0198709i \(0.993674\pi\)
\(30\) 2.73712 + 0.228328i 0.499727 + 0.0416868i
\(31\) 6.15922 1.10623 0.553114 − 0.833105i \(-0.313439\pi\)
0.553114 + 0.833105i \(0.313439\pi\)
\(32\) −0.707107 − 0.707107i −0.125000 − 0.125000i
\(33\) 0.558086 + 0.558086i 0.0971502 + 0.0971502i
\(34\) −1.27719 −0.219037
\(35\) 3.35523 − 2.83855i 0.567137 − 0.479802i
\(36\) −1.49121 −0.248535
\(37\) −7.20809 + 7.20809i −1.18500 + 1.18500i −0.206571 + 0.978432i \(0.566230\pi\)
−0.978432 + 0.206571i \(0.933770\pi\)
\(38\) 1.80399 + 1.80399i 0.292647 + 0.292647i
\(39\) − 3.69468i − 0.591622i
\(40\) 1.70711 − 1.44423i 0.269917 − 0.228352i
\(41\) 3.33722 0.521186 0.260593 − 0.965449i \(-0.416082\pi\)
0.260593 + 0.965449i \(0.416082\pi\)
\(42\) 1.70711 − 1.70711i 0.263412 − 0.263412i
\(43\) 7.85390 + 7.85390i 1.19771 + 1.19771i 0.974851 + 0.222857i \(0.0715383\pi\)
0.222857 + 0.974851i \(0.428462\pi\)
\(44\) 0.642542 0.0968668
\(45\) 0.277194 − 3.32290i 0.0413216 − 0.495349i
\(46\) 2.88845 + 3.82843i 0.425879 + 0.564471i
\(47\) −4.15133 − 4.15133i −0.605534 − 0.605534i 0.336242 − 0.941776i \(-0.390844\pi\)
−0.941776 + 0.336242i \(0.890844\pi\)
\(48\) 0.868559 − 0.868559i 0.125366 − 0.125366i
\(49\) 3.13702i 0.448146i
\(50\) 2.90088 + 4.07245i 0.410247 + 0.575932i
\(51\) − 1.56881i − 0.219678i
\(52\) −2.12690 − 2.12690i −0.294948 − 0.294948i
\(53\) 2.93732 + 2.93732i 0.403471 + 0.403471i 0.879454 − 0.475983i \(-0.157908\pi\)
−0.475983 + 0.879454i \(0.657908\pi\)
\(54\) − 5.51668i − 0.750725i
\(55\) −0.119439 + 1.43179i −0.0161052 + 0.193063i
\(56\) − 1.96545i − 0.262644i
\(57\) −2.21590 + 2.21590i −0.293503 + 0.293503i
\(58\) −0.151332 + 0.151332i −0.0198709 + 0.0198709i
\(59\) − 7.08223i − 0.922027i −0.887393 − 0.461014i \(-0.847486\pi\)
0.887393 − 0.461014i \(-0.152514\pi\)
\(60\) 1.77398 + 2.09689i 0.229020 + 0.270707i
\(61\) − 5.29743i − 0.678267i −0.940738 − 0.339134i \(-0.889866\pi\)
0.940738 − 0.339134i \(-0.110134\pi\)
\(62\) 4.35523 + 4.35523i 0.553114 + 0.553114i
\(63\) −2.07245 − 2.07245i −0.261105 − 0.261105i
\(64\) − 1.00000i − 0.125000i
\(65\) 5.13479 − 4.34407i 0.636892 − 0.538816i
\(66\) 0.789252i 0.0971502i
\(67\) 3.65370 − 3.65370i 0.446370 − 0.446370i −0.447776 − 0.894146i \(-0.647784\pi\)
0.894146 + 0.447776i \(0.147784\pi\)
\(68\) −0.903113 − 0.903113i −0.109518 − 0.109518i
\(69\) −4.70257 + 3.54797i −0.566122 + 0.427125i
\(70\) 4.37966 + 0.365348i 0.523470 + 0.0436674i
\(71\) −1.83812 −0.218144 −0.109072 − 0.994034i \(-0.534788\pi\)
−0.109072 + 0.994034i \(0.534788\pi\)
\(72\) −1.05444 − 1.05444i −0.124267 − 0.124267i
\(73\) −5.20166 + 5.20166i −0.608809 + 0.608809i −0.942635 − 0.333826i \(-0.891660\pi\)
0.333826 + 0.942635i \(0.391660\pi\)
\(74\) −10.1938 −1.18500
\(75\) −5.00231 + 3.56323i −0.577617 + 0.411447i
\(76\) 2.55123i 0.292647i
\(77\) 0.892992 + 0.892992i 0.101766 + 0.101766i
\(78\) 2.61253 − 2.61253i 0.295811 − 0.295811i
\(79\) −6.19754 −0.697277 −0.348639 − 0.937257i \(-0.613356\pi\)
−0.348639 + 0.937257i \(0.613356\pi\)
\(80\) 2.22833 + 0.185885i 0.249135 + 0.0207826i
\(81\) 2.30266 0.255852
\(82\) 2.35977 + 2.35977i 0.260593 + 0.260593i
\(83\) −6.04767 − 6.04767i −0.663818 − 0.663818i 0.292460 − 0.956278i \(-0.405526\pi\)
−0.956278 + 0.292460i \(0.905526\pi\)
\(84\) 2.41421 0.263412
\(85\) 2.18031 − 1.84456i 0.236487 − 0.200070i
\(86\) 11.1071i 1.19771i
\(87\) −0.185885 − 0.185885i −0.0199290 − 0.0199290i
\(88\) 0.454346 + 0.454346i 0.0484334 + 0.0484334i
\(89\) 15.7567 1.67020 0.835101 − 0.550096i \(-0.185409\pi\)
0.835101 + 0.550096i \(0.185409\pi\)
\(90\) 2.54565 − 2.15364i 0.268336 − 0.227014i
\(91\) − 5.91185i − 0.619730i
\(92\) −0.664664 + 4.74955i −0.0692960 + 0.495175i
\(93\) −5.34965 + 5.34965i −0.554733 + 0.554733i
\(94\) − 5.87087i − 0.605534i
\(95\) −5.68498 − 0.474237i −0.583267 − 0.0486557i
\(96\) 1.22833 0.125366
\(97\) −10.7898 + 10.7898i −1.09553 + 1.09553i −0.100608 + 0.994926i \(0.532079\pi\)
−0.994926 + 0.100608i \(0.967921\pi\)
\(98\) −2.21821 + 2.21821i −0.224073 + 0.224073i
\(99\) 0.958165 0.0962992
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 230.2.e.a.137.3 ✓ 8
5.2 odd 4 1150.2.e.b.643.2 8
5.3 odd 4 230.2.e.b.183.3 yes 8
5.4 even 2 1150.2.e.c.1057.2 8
23.22 odd 2 230.2.e.b.137.3 yes 8
115.22 even 4 1150.2.e.c.643.2 8
115.68 even 4 inner 230.2.e.a.183.3 yes 8
115.114 odd 2 1150.2.e.b.1057.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.e.a.137.3 ✓ 8 1.1 even 1 trivial
230.2.e.a.183.3 yes 8 115.68 even 4 inner
230.2.e.b.137.3 yes 8 23.22 odd 2
230.2.e.b.183.3 yes 8 5.3 odd 4
1150.2.e.b.643.2 8 5.2 odd 4
1150.2.e.b.1057.2 8 115.114 odd 2
1150.2.e.c.643.2 8 115.22 even 4
1150.2.e.c.1057.2 8 5.4 even 2