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Properties

Label 245.10.a.c.1.2
Level $245$
Weight $10$
Character 245.1
Self dual yes
Analytic conductor $126.184$
Analytic rank $0$
Dimension $2$
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] = [245,10,Mod(1,245)] 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("245.1"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(245, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 10, names="a")
 
Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 245.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-24,174] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(126.183779860\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 245.1

$q$-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)
 
\(f(q)\) \(=\) \(q-9.17157 q^{2} -65.7351 q^{3} -427.882 q^{4} -625.000 q^{5} +602.894 q^{6} +8620.20 q^{8} -15361.9 q^{9} +5732.23 q^{10} +35089.6 q^{11} +28126.9 q^{12} +77401.4 q^{13} +41084.4 q^{15} +140015. q^{16} +229907. q^{17} +140893. q^{18} -16433.6 q^{19} +267426. q^{20} -321827. q^{22} -2.57284e6 q^{23} -566649. q^{24} +390625. q^{25} -709892. q^{26} +2.30368e6 q^{27} -6.62817e6 q^{29} -376809. q^{30} +8.17416e6 q^{31} -5.69770e6 q^{32} -2.30662e6 q^{33} -2.10861e6 q^{34} +6.57308e6 q^{36} +9.70272e6 q^{37} +150722. q^{38} -5.08798e6 q^{39} -5.38762e6 q^{40} -2.98108e7 q^{41} -1.95343e7 q^{43} -1.50142e7 q^{44} +9.60119e6 q^{45} +2.35970e7 q^{46} -5.93794e6 q^{47} -9.20389e6 q^{48} -3.58265e6 q^{50} -1.51130e7 q^{51} -3.31187e7 q^{52} -2.74263e7 q^{53} -2.11284e7 q^{54} -2.19310e7 q^{55} +1.08026e6 q^{57} +6.07908e7 q^{58} -5.24915e7 q^{59} -1.75793e7 q^{60} -2.23282e7 q^{61} -7.49699e7 q^{62} -1.94308e7 q^{64} -4.83759e7 q^{65} +2.11553e7 q^{66} +2.74351e8 q^{67} -9.83733e7 q^{68} +1.69126e8 q^{69} -3.63673e8 q^{71} -1.32423e8 q^{72} -2.09245e7 q^{73} -8.89892e7 q^{74} -2.56778e7 q^{75} +7.03163e6 q^{76} +4.66648e7 q^{78} -2.65896e8 q^{79} -8.75093e7 q^{80} +1.50936e8 q^{81} +2.73412e8 q^{82} +9.43764e6 q^{83} -1.43692e8 q^{85} +1.79160e8 q^{86} +4.35704e8 q^{87} +3.02479e8 q^{88} +6.64876e8 q^{89} -8.80580e7 q^{90} +1.10087e9 q^{92} -5.37329e8 q^{93} +5.44603e7 q^{94} +1.02710e7 q^{95} +3.74539e8 q^{96} +1.20731e9 q^{97} -5.39042e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 24 q^{2} + 174 q^{3} - 720 q^{4} - 1250 q^{5} - 2952 q^{6} + 20544 q^{8} + 22428 q^{9} + 15000 q^{10} + 18566 q^{11} - 41904 q^{12} + 51090 q^{13} - 108750 q^{15} + 112768 q^{16} + 373910 q^{17} - 419472 q^{18}+ \cdots - 1163466540 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −9.17157 −0.405330 −0.202665 − 0.979248i \(-0.564960\pi\)
−0.202665 + 0.979248i \(0.564960\pi\)
\(3\) −65.7351 −0.468545 −0.234273 − 0.972171i \(-0.575271\pi\)
−0.234273 + 0.972171i \(0.575271\pi\)
\(4\) −427.882 −0.835708
\(5\) −625.000 −0.447214
\(6\) 602.894 0.189915
\(7\) 0 0
\(8\) 8620.20 0.744067
\(9\) −15361.9 −0.780465
\(10\) 5732.23 0.181269
\(11\) 35089.6 0.722622 0.361311 − 0.932445i \(-0.382329\pi\)
0.361311 + 0.932445i \(0.382329\pi\)
\(12\) 28126.9 0.391567
\(13\) 77401.4 0.751629 0.375815 − 0.926695i \(-0.377363\pi\)
0.375815 + 0.926695i \(0.377363\pi\)
\(14\) 0 0
\(15\) 41084.4 0.209540
\(16\) 140015. 0.534115
\(17\) 229907. 0.667626 0.333813 − 0.942639i \(-0.391665\pi\)
0.333813 + 0.942639i \(0.391665\pi\)
\(18\) 140893. 0.316346
\(19\) −16433.6 −0.0289295 −0.0144647 − 0.999895i \(-0.504604\pi\)
−0.0144647 + 0.999895i \(0.504604\pi\)
\(20\) 267426. 0.373740
\(21\) 0 0
\(22\) −321827. −0.292900
\(23\) −2.57284e6 −1.91707 −0.958535 − 0.284975i \(-0.908015\pi\)
−0.958535 + 0.284975i \(0.908015\pi\)
\(24\) −566649. −0.348629
\(25\) 390625. 0.200000
\(26\) −709892. −0.304658
\(27\) 2.30368e6 0.834228
\(28\) 0 0
\(29\) −6.62817e6 −1.74022 −0.870108 − 0.492862i \(-0.835951\pi\)
−0.870108 + 0.492862i \(0.835951\pi\)
\(30\) −376809. −0.0849328
\(31\) 8.17416e6 1.58970 0.794851 − 0.606805i \(-0.207549\pi\)
0.794851 + 0.606805i \(0.207549\pi\)
\(32\) −5.69770e6 −0.960560
\(33\) −2.30662e6 −0.338581
\(34\) −2.10861e6 −0.270609
\(35\) 0 0
\(36\) 6.57308e6 0.652241
\(37\) 9.70272e6 0.851110 0.425555 − 0.904933i \(-0.360079\pi\)
0.425555 + 0.904933i \(0.360079\pi\)
\(38\) 150722. 0.0117260
\(39\) −5.08798e6 −0.352172
\(40\) −5.38762e6 −0.332757
\(41\) −2.98108e7 −1.64758 −0.823789 − 0.566896i \(-0.808144\pi\)
−0.823789 + 0.566896i \(0.808144\pi\)
\(42\) 0 0
\(43\) −1.95343e7 −0.871343 −0.435672 − 0.900106i \(-0.643489\pi\)
−0.435672 + 0.900106i \(0.643489\pi\)
\(44\) −1.50142e7 −0.603900
\(45\) 9.60119e6 0.349035
\(46\) 2.35970e7 0.777046
\(47\) −5.93794e6 −0.177499 −0.0887494 − 0.996054i \(-0.528287\pi\)
−0.0887494 + 0.996054i \(0.528287\pi\)
\(48\) −9.20389e6 −0.250257
\(49\) 0 0
\(50\) −3.58265e6 −0.0810660
\(51\) −1.51130e7 −0.312813
\(52\) −3.31187e7 −0.628142
\(53\) −2.74263e7 −0.477448 −0.238724 − 0.971088i \(-0.576729\pi\)
−0.238724 + 0.971088i \(0.576729\pi\)
\(54\) −2.11284e7 −0.338138
\(55\) −2.19310e7 −0.323166
\(56\) 0 0
\(57\) 1.08026e6 0.0135548
\(58\) 6.07908e7 0.705362
\(59\) −5.24915e7 −0.563969 −0.281984 − 0.959419i \(-0.590993\pi\)
−0.281984 + 0.959419i \(0.590993\pi\)
\(60\) −1.75793e7 −0.175114
\(61\) −2.23282e7 −0.206476 −0.103238 − 0.994657i \(-0.532920\pi\)
−0.103238 + 0.994657i \(0.532920\pi\)
\(62\) −7.49699e7 −0.644354
\(63\) 0 0
\(64\) −1.94308e7 −0.144771
\(65\) −4.83759e7 −0.336139
\(66\) 2.11553e7 0.137237
\(67\) 2.74351e8 1.66330 0.831649 − 0.555302i \(-0.187397\pi\)
0.831649 + 0.555302i \(0.187397\pi\)
\(68\) −9.83733e7 −0.557940
\(69\) 1.69126e8 0.898234
\(70\) 0 0
\(71\) −3.63673e8 −1.69843 −0.849216 − 0.528046i \(-0.822925\pi\)
−0.849216 + 0.528046i \(0.822925\pi\)
\(72\) −1.32423e8 −0.580719
\(73\) −2.09245e7 −0.0862387 −0.0431193 − 0.999070i \(-0.513730\pi\)
−0.0431193 + 0.999070i \(0.513730\pi\)
\(74\) −8.89892e7 −0.344980
\(75\) −2.56778e7 −0.0937090
\(76\) 7.03163e6 0.0241766
\(77\) 0 0
\(78\) 4.66648e7 0.142746
\(79\) −2.65896e8 −0.768051 −0.384025 − 0.923323i \(-0.625462\pi\)
−0.384025 + 0.923323i \(0.625462\pi\)
\(80\) −8.75093e7 −0.238863
\(81\) 1.50936e8 0.389592
\(82\) 2.73412e8 0.667813
\(83\) 9.43764e6 0.0218279 0.0109140 − 0.999940i \(-0.496526\pi\)
0.0109140 + 0.999940i \(0.496526\pi\)
\(84\) 0 0
\(85\) −1.43692e8 −0.298571
\(86\) 1.79160e8 0.353182
\(87\) 4.35704e8 0.815369
\(88\) 3.02479e8 0.537679
\(89\) 6.64876e8 1.12327 0.561637 − 0.827384i \(-0.310172\pi\)
0.561637 + 0.827384i \(0.310172\pi\)
\(90\) −8.80580e7 −0.141474
\(91\) 0 0
\(92\) 1.10087e9 1.60211
\(93\) −5.37329e8 −0.744847
\(94\) 5.44603e7 0.0719456
\(95\) 1.02710e7 0.0129377
\(96\) 3.74539e8 0.450066
\(97\) 1.20731e9 1.38467 0.692336 − 0.721575i \(-0.256582\pi\)
0.692336 + 0.721575i \(0.256582\pi\)
\(98\) 0 0
\(99\) −5.39042e8 −0.563981
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 245.10.a.c.1.2 2
7.6 odd 2 35.10.a.b.1.2 ✓ 2
21.20 even 2 315.10.a.b.1.1 2
35.13 even 4 175.10.b.c.99.3 4
35.27 even 4 175.10.b.c.99.2 4
35.34 odd 2 175.10.a.c.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.10.a.b.1.2 ✓ 2 7.6 odd 2
175.10.a.c.1.1 2 35.34 odd 2
175.10.b.c.99.2 4 35.27 even 4
175.10.b.c.99.3 4 35.13 even 4
245.10.a.c.1.2 2 1.1 even 1 trivial
315.10.a.b.1.1 2 21.20 even 2