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Properties

Label 275.2.b.c.199.2
Level $275$
Weight $2$
Character 275.199
Analytic conductor $2.196$
Analytic rank $0$
Dimension $4$
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] = [275,2,Mod(199,275)] 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("275.199"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(275, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 275 = 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 275.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.19588605559\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{13})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 7x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.2
Root \(-1.30278i\) of defining polynomial
Character \(\chi\) \(=\) 275.199
Dual form 275.2.b.c.199.3

$q$-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)
 
\(f(q)\) \(=\) \(q-1.30278i q^{2} -2.30278i q^{3} +0.302776 q^{4} -3.00000 q^{6} +0.697224i q^{7} -3.00000i q^{8} -2.30278 q^{9} -1.00000 q^{11} -0.697224i q^{12} -5.00000i q^{13} +0.908327 q^{14} -3.30278 q^{16} +6.90833i q^{17} +3.00000i q^{18} +1.00000 q^{19} +1.60555 q^{21} +1.30278i q^{22} +7.30278i q^{23} -6.90833 q^{24} -6.51388 q^{26} -1.60555i q^{27} +0.211103i q^{28} -0.908327 q^{29} +10.2111 q^{31} -1.69722i q^{32} +2.30278i q^{33} +9.00000 q^{34} -0.697224 q^{36} +2.39445i q^{37} -1.30278i q^{38} -11.5139 q^{39} -5.60555 q^{41} -2.09167i q^{42} -7.21110i q^{43} -0.302776 q^{44} +9.51388 q^{46} -3.00000i q^{47} +7.60555i q^{48} +6.51388 q^{49} +15.9083 q^{51} -1.51388i q^{52} +1.30278i q^{53} -2.09167 q^{54} +2.09167 q^{56} -2.30278i q^{57} +1.18335i q^{58} +14.2111 q^{59} -7.90833 q^{61} -13.3028i q^{62} -1.60555i q^{63} -8.81665 q^{64} +3.00000 q^{66} -4.00000i q^{67} +2.09167i q^{68} +16.8167 q^{69} -2.60555 q^{71} +6.90833i q^{72} +7.90833i q^{73} +3.11943 q^{74} +0.302776 q^{76} -0.697224i q^{77} +15.0000i q^{78} +10.9083 q^{79} -10.6056 q^{81} +7.30278i q^{82} +3.51388i q^{83} +0.486122 q^{84} -9.39445 q^{86} +2.09167i q^{87} +3.00000i q^{88} -1.69722 q^{89} +3.48612 q^{91} +2.21110i q^{92} -23.5139i q^{93} -3.90833 q^{94} -3.90833 q^{96} +15.3028i q^{97} -8.48612i q^{98} +2.30278 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{4} - 12 q^{6} - 2 q^{9} - 4 q^{11} - 18 q^{14} - 6 q^{16} + 4 q^{19} - 8 q^{21} - 6 q^{24} + 10 q^{26} + 18 q^{29} + 12 q^{31} + 36 q^{34} - 10 q^{36} - 10 q^{39} - 8 q^{41} + 6 q^{44} + 2 q^{46}+ \cdots + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/275\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.30278i − 0.921201i −0.887607 − 0.460601i \(-0.847634\pi\)
0.887607 − 0.460601i \(-0.152366\pi\)
\(3\) − 2.30278i − 1.32951i −0.747062 − 0.664754i \(-0.768536\pi\)
0.747062 − 0.664754i \(-0.231464\pi\)
\(4\) 0.302776 0.151388
\(5\) 0 0
\(6\) −3.00000 −1.22474
\(7\) 0.697224i 0.263526i 0.991281 + 0.131763i \(0.0420638\pi\)
−0.991281 + 0.131763i \(0.957936\pi\)
\(8\) − 3.00000i − 1.06066i
\(9\) −2.30278 −0.767592
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) − 0.697224i − 0.201271i
\(13\) − 5.00000i − 1.38675i −0.720577 − 0.693375i \(-0.756123\pi\)
0.720577 − 0.693375i \(-0.243877\pi\)
\(14\) 0.908327 0.242761
\(15\) 0 0
\(16\) −3.30278 −0.825694
\(17\) 6.90833i 1.67552i 0.546042 + 0.837758i \(0.316134\pi\)
−0.546042 + 0.837758i \(0.683866\pi\)
\(18\) 3.00000i 0.707107i
\(19\) 1.00000 0.229416 0.114708 − 0.993399i \(-0.463407\pi\)
0.114708 + 0.993399i \(0.463407\pi\)
\(20\) 0 0
\(21\) 1.60555 0.350360
\(22\) 1.30278i 0.277753i
\(23\) 7.30278i 1.52273i 0.648321 + 0.761367i \(0.275471\pi\)
−0.648321 + 0.761367i \(0.724529\pi\)
\(24\) −6.90833 −1.41016
\(25\) 0 0
\(26\) −6.51388 −1.27748
\(27\) − 1.60555i − 0.308988i
\(28\) 0.211103i 0.0398946i
\(29\) −0.908327 −0.168672 −0.0843360 − 0.996437i \(-0.526877\pi\)
−0.0843360 + 0.996437i \(0.526877\pi\)
\(30\) 0 0
\(31\) 10.2111 1.83397 0.916984 − 0.398924i \(-0.130616\pi\)
0.916984 + 0.398924i \(0.130616\pi\)
\(32\) − 1.69722i − 0.300030i
\(33\) 2.30278i 0.400862i
\(34\) 9.00000 1.54349
\(35\) 0 0
\(36\) −0.697224 −0.116204
\(37\) 2.39445i 0.393645i 0.980439 + 0.196822i \(0.0630623\pi\)
−0.980439 + 0.196822i \(0.936938\pi\)
\(38\) − 1.30278i − 0.211338i
\(39\) −11.5139 −1.84370
\(40\) 0 0
\(41\) −5.60555 −0.875440 −0.437720 − 0.899111i \(-0.644214\pi\)
−0.437720 + 0.899111i \(0.644214\pi\)
\(42\) − 2.09167i − 0.322752i
\(43\) − 7.21110i − 1.09968i −0.835269 − 0.549841i \(-0.814688\pi\)
0.835269 − 0.549841i \(-0.185312\pi\)
\(44\) −0.302776 −0.0456451
\(45\) 0 0
\(46\) 9.51388 1.40274
\(47\) − 3.00000i − 0.437595i −0.975770 − 0.218797i \(-0.929787\pi\)
0.975770 − 0.218797i \(-0.0702134\pi\)
\(48\) 7.60555i 1.09777i
\(49\) 6.51388 0.930554
\(50\) 0 0
\(51\) 15.9083 2.22761
\(52\) − 1.51388i − 0.209937i
\(53\) 1.30278i 0.178950i 0.995989 + 0.0894750i \(0.0285189\pi\)
−0.995989 + 0.0894750i \(0.971481\pi\)
\(54\) −2.09167 −0.284641
\(55\) 0 0
\(56\) 2.09167 0.279512
\(57\) − 2.30278i − 0.305010i
\(58\) 1.18335i 0.155381i
\(59\) 14.2111 1.85013 0.925064 − 0.379811i \(-0.124011\pi\)
0.925064 + 0.379811i \(0.124011\pi\)
\(60\) 0 0
\(61\) −7.90833 −1.01256 −0.506279 − 0.862370i \(-0.668979\pi\)
−0.506279 + 0.862370i \(0.668979\pi\)
\(62\) − 13.3028i − 1.68945i
\(63\) − 1.60555i − 0.202280i
\(64\) −8.81665 −1.10208
\(65\) 0 0
\(66\) 3.00000 0.369274
\(67\) − 4.00000i − 0.488678i −0.969690 − 0.244339i \(-0.921429\pi\)
0.969690 − 0.244339i \(-0.0785709\pi\)
\(68\) 2.09167i 0.253653i
\(69\) 16.8167 2.02449
\(70\) 0 0
\(71\) −2.60555 −0.309222 −0.154611 − 0.987975i \(-0.549412\pi\)
−0.154611 + 0.987975i \(0.549412\pi\)
\(72\) 6.90833i 0.814154i
\(73\) 7.90833i 0.925600i 0.886463 + 0.462800i \(0.153155\pi\)
−0.886463 + 0.462800i \(0.846845\pi\)
\(74\) 3.11943 0.362626
\(75\) 0 0
\(76\) 0.302776 0.0347307
\(77\) − 0.697224i − 0.0794561i
\(78\) 15.0000i 1.69842i
\(79\) 10.9083 1.22728 0.613641 − 0.789585i \(-0.289704\pi\)
0.613641 + 0.789585i \(0.289704\pi\)
\(80\) 0 0
\(81\) −10.6056 −1.17839
\(82\) 7.30278i 0.806457i
\(83\) 3.51388i 0.385698i 0.981228 + 0.192849i \(0.0617728\pi\)
−0.981228 + 0.192849i \(0.938227\pi\)
\(84\) 0.486122 0.0530402
\(85\) 0 0
\(86\) −9.39445 −1.01303
\(87\) 2.09167i 0.224251i
\(88\) 3.00000i 0.319801i
\(89\) −1.69722 −0.179905 −0.0899527 − 0.995946i \(-0.528672\pi\)
−0.0899527 + 0.995946i \(0.528672\pi\)
\(90\) 0 0
\(91\) 3.48612 0.365445
\(92\) 2.21110i 0.230523i
\(93\) − 23.5139i − 2.43828i
\(94\) −3.90833 −0.403113
\(95\) 0 0
\(96\) −3.90833 −0.398892
\(97\) 15.3028i 1.55376i 0.629648 + 0.776881i \(0.283199\pi\)
−0.629648 + 0.776881i \(0.716801\pi\)
\(98\) − 8.48612i − 0.857228i
\(99\) 2.30278 0.231438
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 275.2.b.c.199.2 4
3.2 odd 2 2475.2.c.k.199.3 4
4.3 odd 2 4400.2.b.y.4049.4 4
5.2 odd 4 275.2.a.e.1.2 ✓ 2
5.3 odd 4 275.2.a.f.1.1 yes 2
5.4 even 2 inner 275.2.b.c.199.3 4
15.2 even 4 2475.2.a.t.1.1 2
15.8 even 4 2475.2.a.o.1.2 2
15.14 odd 2 2475.2.c.k.199.2 4
20.3 even 4 4400.2.a.bh.1.1 2
20.7 even 4 4400.2.a.bs.1.2 2
20.19 odd 2 4400.2.b.y.4049.1 4
55.32 even 4 3025.2.a.n.1.1 2
55.43 even 4 3025.2.a.h.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
275.2.a.e.1.2 ✓ 2 5.2 odd 4
275.2.a.f.1.1 yes 2 5.3 odd 4
275.2.b.c.199.2 4 1.1 even 1 trivial
275.2.b.c.199.3 4 5.4 even 2 inner
2475.2.a.o.1.2 2 15.8 even 4
2475.2.a.t.1.1 2 15.2 even 4
2475.2.c.k.199.2 4 15.14 odd 2
2475.2.c.k.199.3 4 3.2 odd 2
3025.2.a.h.1.2 2 55.43 even 4
3025.2.a.n.1.1 2 55.32 even 4
4400.2.a.bh.1.1 2 20.3 even 4
4400.2.a.bs.1.2 2 20.7 even 4
4400.2.b.y.4049.1 4 20.19 odd 2
4400.2.b.y.4049.4 4 4.3 odd 2