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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [112,5,Mod(43,112)] 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("112.43"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1, 0])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 112.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.5774358654\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(48\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 43.12
Character \(\chi\) \(=\) 112.43
Dual form 112.5.k.a.99.12

$q$-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+(-2.91660 - 2.73741i) q^{2} +(0.212552 + 0.212552i) q^{3} +(1.01313 + 15.9679i) q^{4} +(14.9819 + 14.9819i) q^{5} +(-0.0380866 - 1.20177i) q^{6} +18.5203 q^{7} +(40.7558 - 49.3453i) q^{8} -80.9096i q^{9} +(-2.68458 - 84.7081i) q^{10} +(-139.313 + 139.313i) q^{11} +(-3.17866 + 3.60934i) q^{12} +(-186.037 + 186.037i) q^{13} +(-54.0162 - 50.6976i) q^{14} +6.36887i q^{15} +(-253.947 + 32.3551i) q^{16} +167.051 q^{17} +(-221.483 + 235.981i) q^{18} +(146.208 + 146.208i) q^{19} +(-224.051 + 254.409i) q^{20} +(3.93651 + 3.93651i) q^{21} +(787.680 - 24.9632i) q^{22} +367.074 q^{23} +(19.1511 - 1.82571i) q^{24} -176.083i q^{25} +(1051.86 - 33.3357i) q^{26} +(34.4141 - 34.4141i) q^{27} +(18.7634 + 295.729i) q^{28} +(-1044.05 + 1044.05i) q^{29} +(17.4342 - 18.5755i) q^{30} +1417.88i q^{31} +(829.232 + 600.791i) q^{32} -59.2225 q^{33} +(-487.222 - 457.289i) q^{34} +(277.469 + 277.469i) q^{35} +(1291.96 - 81.9720i) q^{36} +(173.533 + 173.533i) q^{37} +(-26.1987 - 826.661i) q^{38} -79.0851 q^{39} +(1349.89 - 128.688i) q^{40} +1180.37i q^{41} +(-0.705375 - 22.2571i) q^{42} +(-316.853 + 316.853i) q^{43} +(-2365.68 - 2083.40i) q^{44} +(1212.18 - 1212.18i) q^{45} +(-1070.61 - 1004.83i) q^{46} +2992.85i q^{47} +(-60.8540 - 47.0997i) q^{48} +343.000 q^{49} +(-482.012 + 513.564i) q^{50} +(35.5070 + 35.5070i) q^{51} +(-3159.11 - 2782.15i) q^{52} +(-1384.57 - 1384.57i) q^{53} +(-194.578 + 6.16660i) q^{54} -4174.37 q^{55} +(754.809 - 913.888i) q^{56} +62.1534i q^{57} +(5903.07 - 187.081i) q^{58} +(-3090.58 + 3090.58i) q^{59} +(-101.697 + 6.45250i) q^{60} +(1845.82 - 1845.82i) q^{61} +(3881.32 - 4135.38i) q^{62} -1498.47i q^{63} +(-773.925 - 4022.22i) q^{64} -5574.41 q^{65} +(172.729 + 162.117i) q^{66} +(-2028.10 - 2028.10i) q^{67} +(169.245 + 2667.46i) q^{68} +(78.0222 + 78.0222i) q^{69} +(-49.7191 - 1568.82i) q^{70} +8041.72 q^{71} +(-3992.51 - 3297.54i) q^{72} -5961.54i q^{73} +(-31.0950 - 981.159i) q^{74} +(37.4267 - 37.4267i) q^{75} +(-2186.50 + 2482.76i) q^{76} +(-2580.12 + 2580.12i) q^{77} +(230.660 + 216.489i) q^{78} -5168.83i q^{79} +(-4289.36 - 3319.88i) q^{80} -6539.05 q^{81} +(3231.16 - 3442.67i) q^{82} +(7170.62 + 7170.62i) q^{83} +(-58.8696 + 66.8460i) q^{84} +(2502.75 + 2502.75i) q^{85} +(1791.49 - 56.7761i) q^{86} -443.829 q^{87} +(1196.63 + 12552.3i) q^{88} -6317.56i q^{89} +(-6853.70 + 217.209i) q^{90} +(-3445.46 + 3445.46i) q^{91} +(371.894 + 5861.40i) q^{92} +(-301.372 + 301.372i) q^{93} +(8192.66 - 8728.94i) q^{94} +4380.95i q^{95} +(48.5554 + 303.954i) q^{96} +11147.8 q^{97} +(-1000.39 - 938.933i) q^{98} +(11271.8 + 11271.8i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 96 q + 6 q^{4} - 132 q^{6} - 200 q^{10} - 96 q^{11} + 660 q^{12} - 294 q^{14} + 642 q^{16} - 810 q^{18} + 1408 q^{19} - 2604 q^{20} - 106 q^{22} - 1152 q^{23} + 3880 q^{24} + 6828 q^{26} + 3648 q^{27} - 864 q^{29}+ \cdots - 59552 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/112\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(17\) \(85\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{1}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.91660 − 2.73741i −0.729150 − 0.684353i
\(3\) 0.212552 + 0.212552i 0.0236168 + 0.0236168i 0.718817 − 0.695200i \(-0.244684\pi\)
−0.695200 + 0.718817i \(0.744684\pi\)
\(4\) 1.01313 + 15.9679i 0.0633207 + 0.997993i
\(5\) 14.9819 + 14.9819i 0.599278 + 0.599278i 0.940120 − 0.340843i \(-0.110712\pi\)
−0.340843 + 0.940120i \(0.610712\pi\)
\(6\) −0.0380866 − 1.20177i −0.00105796 − 0.0333825i
\(7\) 18.5203 0.377964
\(8\) 40.7558 − 49.3453i 0.636810 − 0.771021i
\(9\) − 80.9096i − 0.998884i
\(10\) −2.68458 − 84.7081i −0.0268458 − 0.847081i
\(11\) −139.313 + 139.313i −1.15135 + 1.15135i −0.165067 + 0.986282i \(0.552784\pi\)
−0.986282 + 0.165067i \(0.947216\pi\)
\(12\) −3.17866 + 3.60934i −0.0220740 + 0.0250649i
\(13\) −186.037 + 186.037i −1.10081 + 1.10081i −0.106501 + 0.994313i \(0.533965\pi\)
−0.994313 + 0.106501i \(0.966035\pi\)
\(14\) −54.0162 − 50.6976i −0.275593 − 0.258661i
\(15\) 6.36887i 0.0283061i
\(16\) −253.947 + 32.3551i −0.991981 + 0.126387i
\(17\) 167.051 0.578032 0.289016 − 0.957324i \(-0.406672\pi\)
0.289016 + 0.957324i \(0.406672\pi\)
\(18\) −221.483 + 235.981i −0.683590 + 0.728337i
\(19\) 146.208 + 146.208i 0.405008 + 0.405008i 0.879993 − 0.474986i \(-0.157547\pi\)
−0.474986 + 0.879993i \(0.657547\pi\)
\(20\) −224.051 + 254.409i −0.560128 + 0.636022i
\(21\) 3.93651 + 3.93651i 0.00892633 + 0.00892633i
\(22\) 787.680 − 24.9632i 1.62744 − 0.0515770i
\(23\) 367.074 0.693902 0.346951 − 0.937883i \(-0.387217\pi\)
0.346951 + 0.937883i \(0.387217\pi\)
\(24\) 19.1511 − 1.82571i 0.0332485 − 0.00316964i
\(25\) − 176.083i − 0.281733i
\(26\) 1051.86 − 33.3357i 1.55600 − 0.0493131i
\(27\) 34.4141 − 34.4141i 0.0472073 − 0.0472073i
\(28\) 18.7634 + 295.729i 0.0239330 + 0.377206i
\(29\) −1044.05 + 1044.05i −1.24144 + 1.24144i −0.282034 + 0.959404i \(0.591009\pi\)
−0.959404 + 0.282034i \(0.908991\pi\)
\(30\) 17.4342 − 18.5755i 0.0193714 − 0.0206394i
\(31\) 1417.88i 1.47542i 0.675119 + 0.737709i \(0.264092\pi\)
−0.675119 + 0.737709i \(0.735908\pi\)
\(32\) 829.232 + 600.791i 0.809797 + 0.586710i
\(33\) −59.2225 −0.0543825
\(34\) −487.222 − 457.289i −0.421472 − 0.395578i
\(35\) 277.469 + 277.469i 0.226506 + 0.226506i
\(36\) 1291.96 − 81.9720i 0.996880 − 0.0632500i
\(37\) 173.533 + 173.533i 0.126759 + 0.126759i 0.767640 − 0.640881i \(-0.221431\pi\)
−0.640881 + 0.767640i \(0.721431\pi\)
\(38\) −26.1987 − 826.661i −0.0181431 − 0.572480i
\(39\) −79.0851 −0.0519955
\(40\) 1349.89 − 128.688i 0.843682 − 0.0804297i
\(41\) 1180.37i 0.702183i 0.936341 + 0.351091i \(0.114189\pi\)
−0.936341 + 0.351091i \(0.885811\pi\)
\(42\) −0.705375 − 22.2571i −0.000399872 − 0.0126174i
\(43\) −316.853 + 316.853i −0.171364 + 0.171364i −0.787579 − 0.616214i \(-0.788666\pi\)
0.616214 + 0.787579i \(0.288666\pi\)
\(44\) −2365.68 − 2083.40i −1.22194 − 1.07613i
\(45\) 1212.18 − 1212.18i 0.598609 − 0.598609i
\(46\) −1070.61 − 1004.83i −0.505959 − 0.474874i
\(47\) 2992.85i 1.35484i 0.735596 + 0.677421i \(0.236902\pi\)
−0.735596 + 0.677421i \(0.763098\pi\)
\(48\) −60.8540 − 47.0997i −0.0264123 − 0.0204426i
\(49\) 343.000 0.142857
\(50\) −482.012 + 513.564i −0.192805 + 0.205425i
\(51\) 35.5070 + 35.5070i 0.0136513 + 0.0136513i
\(52\) −3159.11 − 2782.15i −1.16831 − 1.02890i
\(53\) −1384.57 − 1384.57i −0.492906 − 0.492906i 0.416315 − 0.909221i \(-0.363321\pi\)
−0.909221 + 0.416315i \(0.863321\pi\)
\(54\) −194.578 + 6.16660i −0.0667277 + 0.00211474i
\(55\) −4174.37 −1.37996
\(56\) 754.809 − 913.888i 0.240691 − 0.291419i
\(57\) 62.1534i 0.0191300i
\(58\) 5903.07 − 187.081i 1.75478 − 0.0556127i
\(59\) −3090.58 + 3090.58i −0.887843 + 0.887843i −0.994316 − 0.106473i \(-0.966044\pi\)
0.106473 + 0.994316i \(0.466044\pi\)
\(60\) −101.697 + 6.45250i −0.0282493 + 0.00179236i
\(61\) 1845.82 − 1845.82i 0.496054 − 0.496054i −0.414153 − 0.910207i \(-0.635922\pi\)
0.910207 + 0.414153i \(0.135922\pi\)
\(62\) 3881.32 − 4135.38i 1.00971 − 1.07580i
\(63\) − 1498.47i − 0.377543i
\(64\) −773.925 − 4022.22i −0.188946 − 0.981987i
\(65\) −5574.41 −1.31939
\(66\) 172.729 + 162.117i 0.0396530 + 0.0372168i
\(67\) −2028.10 − 2028.10i −0.451793 − 0.451793i 0.444156 − 0.895949i \(-0.353503\pi\)
−0.895949 + 0.444156i \(0.853503\pi\)
\(68\) 169.245 + 2667.46i 0.0366014 + 0.576872i
\(69\) 78.0222 + 78.0222i 0.0163878 + 0.0163878i
\(70\) −49.7191 − 1568.82i −0.0101468 − 0.320167i
\(71\) 8041.72 1.59526 0.797632 − 0.603145i \(-0.206086\pi\)
0.797632 + 0.603145i \(0.206086\pi\)
\(72\) −3992.51 − 3297.54i −0.770161 − 0.636099i
\(73\) − 5961.54i − 1.11870i −0.828933 − 0.559348i \(-0.811051\pi\)
0.828933 − 0.559348i \(-0.188949\pi\)
\(74\) −31.0950 − 981.159i −0.00567842 − 0.179174i
\(75\) 37.4267 − 37.4267i 0.00665363 − 0.00665363i
\(76\) −2186.50 + 2482.76i −0.378550 + 0.429840i
\(77\) −2580.12 + 2580.12i −0.435169 + 0.435169i
\(78\) 230.660 + 216.489i 0.0379125 + 0.0355833i
\(79\) − 5168.83i − 0.828206i −0.910230 − 0.414103i \(-0.864095\pi\)
0.910230 − 0.414103i \(-0.135905\pi\)
\(80\) −4289.36 − 3319.88i −0.670213 − 0.518731i
\(81\) −6539.05 −0.996655
\(82\) 3231.16 − 3442.67i 0.480541 − 0.511997i
\(83\) 7170.62 + 7170.62i 1.04088 + 1.04088i 0.999128 + 0.0417514i \(0.0132938\pi\)
0.0417514 + 0.999128i \(0.486706\pi\)
\(84\) −58.8696 + 66.8460i −0.00834319 + 0.00947363i
\(85\) 2502.75 + 2502.75i 0.346402 + 0.346402i
\(86\) 1791.49 − 56.7761i 0.242224 − 0.00767660i
\(87\) −443.829 −0.0586377
\(88\) 1196.63 + 12552.3i 0.154524 + 1.62091i
\(89\) − 6317.56i − 0.797571i −0.917044 − 0.398785i \(-0.869432\pi\)
0.917044 − 0.398785i \(-0.130568\pi\)
\(90\) −6853.70 + 217.209i −0.846136 + 0.0268159i
\(91\) −3445.46 + 3445.46i −0.416068 + 0.416068i
\(92\) 371.894 + 5861.40i 0.0439383 + 0.692510i
\(93\) −301.372 + 301.372i −0.0348447 + 0.0348447i
\(94\) 8192.66 − 8728.94i 0.927191 − 0.987884i
\(95\) 4380.95i 0.485424i
\(96\) 48.5554 + 303.954i 0.00526860 + 0.0329811i
\(97\) 11147.8 1.18480 0.592402 − 0.805642i \(-0.298180\pi\)
0.592402 + 0.805642i \(0.298180\pi\)
\(98\) −1000.39 − 938.933i −0.104164 − 0.0977648i
\(99\) 11271.8 + 11271.8i 1.15007 + 1.15007i
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 112.5.k.a.43.12 ✓ 96
4.3 odd 2 448.5.k.a.15.25 96
16.3 odd 4 inner 112.5.k.a.99.12 yes 96
16.13 even 4 448.5.k.a.239.25 96
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
112.5.k.a.43.12 ✓ 96 1.1 even 1 trivial
112.5.k.a.99.12 yes 96 16.3 odd 4 inner
448.5.k.a.15.25 96 4.3 odd 2
448.5.k.a.239.25 96 16.13 even 4