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Properties

Label 4275.2.a.bf.1.1
Level $4275$
Weight $2$
Character 4275.1
Self dual yes
Analytic conductor $34.136$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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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] = [4275,2,Mod(1,4275)] 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("4275.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4275, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4275 = 3^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4275.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,0,6,0,0,0,-3,0,0,3,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(34.1360468641\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.837.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 6x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1425)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.52892\) of defining polynomial
Character \(\chi\) \(=\) 4275.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.52892 q^{2} +4.39543 q^{4} +4.92434 q^{7} -6.05784 q^{8} +1.13349 q^{11} -4.00000 q^{13} -12.4533 q^{14} +6.52892 q^{16} -6.79085 q^{17} -1.00000 q^{19} -2.86651 q^{22} +1.92434 q^{23} +10.1157 q^{26} +21.6446 q^{28} +5.00000 q^{29} +5.13349 q^{31} -4.39543 q^{32} +17.1735 q^{34} -9.05784 q^{37} +2.52892 q^{38} -3.86651 q^{41} -4.00000 q^{43} +4.98218 q^{44} -4.86651 q^{46} +4.26698 q^{47} +17.2492 q^{49} -17.5817 q^{52} -13.1157 q^{53} -29.8309 q^{56} -12.6446 q^{58} -9.19133 q^{59} +0.733016 q^{61} -12.9822 q^{62} -1.94216 q^{64} -4.86651 q^{67} -29.8487 q^{68} -11.1913 q^{71} -11.1157 q^{73} +22.9065 q^{74} -4.39543 q^{76} +5.58170 q^{77} -13.7730 q^{79} +9.77808 q^{82} +0.866508 q^{83} +10.1157 q^{86} -6.86651 q^{88} +10.8487 q^{89} -19.6974 q^{91} +8.45831 q^{92} -10.7909 q^{94} +9.32482 q^{97} -43.6217 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 6 q^{4} - 3 q^{8} + 3 q^{11} - 12 q^{13} - 15 q^{14} + 12 q^{16} - 6 q^{17} - 3 q^{19} - 9 q^{22} - 9 q^{23} + 27 q^{28} + 15 q^{29} + 15 q^{31} - 6 q^{32} + 6 q^{34} - 12 q^{37} - 12 q^{41} - 12 q^{43}+ \cdots - 57 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.52892 −1.78822 −0.894108 − 0.447852i \(-0.852189\pi\)
−0.894108 + 0.447852i \(0.852189\pi\)
\(3\) 0 0
\(4\) 4.39543 2.19771
\(5\) 0 0
\(6\) 0 0
\(7\) 4.92434 1.86123 0.930614 − 0.366003i \(-0.119274\pi\)
0.930614 + 0.366003i \(0.119274\pi\)
\(8\) −6.05784 −2.14177
\(9\) 0 0
\(10\) 0 0
\(11\) 1.13349 0.341761 0.170880 − 0.985292i \(-0.445339\pi\)
0.170880 + 0.985292i \(0.445339\pi\)
\(12\) 0 0
\(13\) −4.00000 −1.10940 −0.554700 − 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) −12.4533 −3.32827
\(15\) 0 0
\(16\) 6.52892 1.63223
\(17\) −6.79085 −1.64702 −0.823512 − 0.567299i \(-0.807988\pi\)
−0.823512 + 0.567299i \(0.807988\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) −2.86651 −0.611142
\(23\) 1.92434 0.401253 0.200627 − 0.979668i \(-0.435702\pi\)
0.200627 + 0.979668i \(0.435702\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 10.1157 1.98385
\(27\) 0 0
\(28\) 21.6446 4.09044
\(29\) 5.00000 0.928477 0.464238 − 0.885710i \(-0.346328\pi\)
0.464238 + 0.885710i \(0.346328\pi\)
\(30\) 0 0
\(31\) 5.13349 0.922002 0.461001 − 0.887400i \(-0.347490\pi\)
0.461001 + 0.887400i \(0.347490\pi\)
\(32\) −4.39543 −0.777009
\(33\) 0 0
\(34\) 17.1735 2.94523
\(35\) 0 0
\(36\) 0 0
\(37\) −9.05784 −1.48910 −0.744550 − 0.667567i \(-0.767336\pi\)
−0.744550 + 0.667567i \(0.767336\pi\)
\(38\) 2.52892 0.410245
\(39\) 0 0
\(40\) 0 0
\(41\) −3.86651 −0.603847 −0.301924 − 0.953332i \(-0.597629\pi\)
−0.301924 + 0.953332i \(0.597629\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 4.98218 0.751092
\(45\) 0 0
\(46\) −4.86651 −0.717527
\(47\) 4.26698 0.622404 0.311202 − 0.950344i \(-0.399268\pi\)
0.311202 + 0.950344i \(0.399268\pi\)
\(48\) 0 0
\(49\) 17.2492 2.46417
\(50\) 0 0
\(51\) 0 0
\(52\) −17.5817 −2.43814
\(53\) −13.1157 −1.80158 −0.900788 − 0.434259i \(-0.857010\pi\)
−0.900788 + 0.434259i \(0.857010\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −29.8309 −3.98632
\(57\) 0 0
\(58\) −12.6446 −1.66032
\(59\) −9.19133 −1.19661 −0.598304 − 0.801269i \(-0.704159\pi\)
−0.598304 + 0.801269i \(0.704159\pi\)
\(60\) 0 0
\(61\) 0.733016 0.0938531 0.0469266 − 0.998898i \(-0.485057\pi\)
0.0469266 + 0.998898i \(0.485057\pi\)
\(62\) −12.9822 −1.64874
\(63\) 0 0
\(64\) −1.94216 −0.242771
\(65\) 0 0
\(66\) 0 0
\(67\) −4.86651 −0.594539 −0.297269 − 0.954794i \(-0.596076\pi\)
−0.297269 + 0.954794i \(0.596076\pi\)
\(68\) −29.8487 −3.61969
\(69\) 0 0
\(70\) 0 0
\(71\) −11.1913 −1.32817 −0.664083 − 0.747659i \(-0.731178\pi\)
−0.664083 + 0.747659i \(0.731178\pi\)
\(72\) 0 0
\(73\) −11.1157 −1.30099 −0.650495 − 0.759510i \(-0.725439\pi\)
−0.650495 + 0.759510i \(0.725439\pi\)
\(74\) 22.9065 2.66283
\(75\) 0 0
\(76\) −4.39543 −0.504190
\(77\) 5.58170 0.636094
\(78\) 0 0
\(79\) −13.7730 −1.54959 −0.774794 − 0.632214i \(-0.782146\pi\)
−0.774794 + 0.632214i \(0.782146\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 9.77808 1.07981
\(83\) 0.866508 0.0951116 0.0475558 − 0.998869i \(-0.484857\pi\)
0.0475558 + 0.998869i \(0.484857\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 10.1157 1.09080
\(87\) 0 0
\(88\) −6.86651 −0.731972
\(89\) 10.8487 1.14996 0.574979 − 0.818168i \(-0.305010\pi\)
0.574979 + 0.818168i \(0.305010\pi\)
\(90\) 0 0
\(91\) −19.6974 −2.06485
\(92\) 8.45831 0.881840
\(93\) 0 0
\(94\) −10.7909 −1.11299
\(95\) 0 0
\(96\) 0 0
\(97\) 9.32482 0.946792 0.473396 − 0.880850i \(-0.343028\pi\)
0.473396 + 0.880850i \(0.343028\pi\)
\(98\) −43.6217 −4.40646
\(99\) 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 4275.2.a.bf.1.1 3
3.2 odd 2 1425.2.a.t.1.3 ✓ 3
5.4 even 2 4275.2.a.bg.1.3 3
15.2 even 4 1425.2.c.o.799.6 6
15.8 even 4 1425.2.c.o.799.1 6
15.14 odd 2 1425.2.a.w.1.1 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1425.2.a.t.1.3 ✓ 3 3.2 odd 2
1425.2.a.w.1.1 yes 3 15.14 odd 2
1425.2.c.o.799.1 6 15.8 even 4
1425.2.c.o.799.6 6 15.2 even 4
4275.2.a.bf.1.1 3 1.1 even 1 trivial
4275.2.a.bg.1.3 3 5.4 even 2