LMFDB - Embedded newform 16.24.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [16,24,Mod(1,16)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(16, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 24, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("16.1");

S:= CuspForms(chi, 24);

N := Newforms(S);

Level:	$ N $	$=$	$ 16 = 2^{4} $
Weight:	$ k $	$=$	$ 24 $
Character orbit:	$[\chi]$	$=$	16.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$53.6326459752$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{144169}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 36042 $ x^2 - x - 36042
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{7}\cdot 3 $
Twist minimal:	no (minimal twist has level 1)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$190.348$ of defining polynomial
Character	$\chi$	$=$	16.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-388445. q^{3} +1.05062e8 q^{5} -3.81217e9 q^{7} +5.67462e10 q^{9} +O(q^{10})$	$q-388445. q^{3} +1.05062e8 q^{5} -3.81217e9 q^{7} +5.67462e10 q^{9} -2.52200e11 q^{11} -3.59099e12 q^{13} -4.08108e13 q^{15} +2.34190e14 q^{17} +6.23086e14 q^{19} +1.48082e15 q^{21} +3.58786e15 q^{23} -8.82898e14 q^{25} +1.45267e16 q^{27} -2.05923e16 q^{29} -1.36357e17 q^{31} +9.79656e16 q^{33} -4.00514e17 q^{35} -1.23898e18 q^{37} +1.39490e18 q^{39} +1.40074e18 q^{41} -2.18793e17 q^{43} +5.96187e18 q^{45} +8.67836e18 q^{47} -1.28361e19 q^{49} -9.09698e19 q^{51} -7.63436e19 q^{53} -2.64966e19 q^{55} -2.42034e20 q^{57} +1.01862e18 q^{59} +2.87337e20 q^{61} -2.16326e20 q^{63} -3.77277e20 q^{65} -1.47683e21 q^{67} -1.39369e21 q^{69} -7.64346e20 q^{71} -3.49433e21 q^{73} +3.42957e20 q^{75} +9.61427e20 q^{77} -1.02350e22 q^{79} -1.09851e22 q^{81} +7.71597e21 q^{83} +2.46044e22 q^{85} +7.99897e21 q^{87} +4.58518e21 q^{89} +1.36895e22 q^{91} +5.29672e22 q^{93} +6.54627e22 q^{95} -1.13703e23 q^{97} -1.43114e22 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 339480 q^{3} + 73069020 q^{5} + 1359184400 q^{7} - 34999394166 q^{9}+O(q^{10}) $ 2 * q - 339480 * q^3 + 73069020 * q^5 + 1359184400 * q^7 - 34999394166 * q^9	$ 2 q - 339480 q^{3} + 73069020 q^{5} + 1359184400 q^{7} - 34999394166 q^{9} - 856801968264 q^{11} + 4376109322060 q^{13} - 42377338985040 q^{15} + 254028147597540 q^{17} - 4260600979960 q^{19} + 17\!\cdots\!44 q^{21}+ \cdots + 41\!\cdots\!12 q^{99}+O(q^{100}) $ 2 * q - 339480 * q^3 + 73069020 * q^5 + 1359184400 * q^7 - 34999394166 * q^9 - 856801968264 * q^11 + 4376109322060 * q^13 - 42377338985040 * q^15 + 254028147597540 * q^17 - 4260600979960 * q^19 + 1734031637722944 * q^21 + 8144713079008560 * q^23 - 11780274628800850 * q^25 + 5424634982716560 * q^27 + 20818433601623340 * q^29 - 137714017177000384 * q^31 + 68361366766001760 * q^33 - 565961271250425120 * q^35 - 897721264408967780 * q^37 + 1785011473665029232 * q^39 - 2294435477168314956 * q^41 + 1750760768619855800 * q^43 + 8897092690294206540 * q^45 - 15759744217656780960 * q^47 - 13461981704376200814 * q^49 - 89998362845078292144 * q^51 - 140287253401646796420 * q^53 - 7153550955060182640 * q^55 - 272752401448627175520 * q^57 - 280872989971340771880 * q^59 - 180452892516502223636 * q^61 - 690775113933935014320 * q^63 - 632168834809440380760 * q^65 - 1754233163431557625240 * q^67 - 1170560672172404223552 * q^69 - 3055033510194143328624 * q^71 - 8063408253877606149260 * q^73 - 190631089350520885800 * q^75 - 2165184764357449665600 * q^77 - 6244916814559639980640 * q^79 - 2793528580929833975598 * q^81 - 6875994082418498976120 * q^83 + 23969743087870314902520 * q^85 + 10026640653837674384880 * q^87 + 6395093086173070004820 * q^89 + 54890178162704560146016 * q^91 + 52900811441357852079360 * q^93 + 85533361066700858502000 * q^95 - 31147288846254030500540 * q^97 + 41158245132312135981912 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−388445.	−1.26600	−0.633002	−	0.774150i	$-0.718177\pi$
$3$	−388445.	−1.26600	−0.633002	+	0.774150i	$0.718177\pi$
$4$	0	0
$5$	1.05062e8	0.962256	0.481128	−	0.876650i	$-0.340227\pi$
$5$	1.05062e8	0.962256	0.481128	+	0.876650i	$0.340227\pi$
$6$	0	0
$7$	−3.81217e9	−0.728693	−0.364346	−	0.931263i	$-0.618708\pi$
$7$	−3.81217e9	−0.728693	−0.364346	+	0.931263i	$0.618708\pi$
$8$	0	0
$9$	5.67462e10	0.602765
$10$	0	0
$11$	−2.52200e11	−0.266519	−0.133260	−	0.991081i	$-0.542544\pi$
$11$	−2.52200e11	−0.266519	−0.133260	+	0.991081i	$0.542544\pi$
$12$	0	0
$13$	−3.59099e12	−0.555733	−0.277867	−	0.960620i	$-0.589627\pi$
$13$	−3.59099e12	−0.555733	−0.277867	+	0.960620i	$0.589627\pi$
$14$	0	0
$15$	−4.08108e13	−1.21822
$16$	0	0
$17$	2.34190e14	1.65731	0.828657	−	0.559756i	$-0.189105\pi$
$17$	2.34190e14	1.65731	0.828657	+	0.559756i	$0.189105\pi$
$18$	0	0
$19$	6.23086e14	1.22710	0.613552	−	0.789654i	$-0.289740\pi$
$19$	6.23086e14	1.22710	0.613552	+	0.789654i	$0.289740\pi$
$20$	0	0
$21$	1.48082e15	0.922528
$22$	0	0
$23$	3.58786e15	0.785173	0.392587	−	0.919715i	$-0.371580\pi$
$23$	3.58786e15	0.785173	0.392587	+	0.919715i	$0.371580\pi$
$24$	0	0
$25$	−8.82898e14	−0.0740629
$26$	0	0
$27$	1.45267e16	0.502901
$28$	0	0
$29$	−2.05923e16	−0.313421	−0.156710	−	0.987645i	$-0.550089\pi$
$29$	−2.05923e16	−0.313421	−0.156710	+	0.987645i	$0.550089\pi$
$30$	0	0
$31$	−1.36357e17	−0.963870	−0.481935	−	0.876207i	$-0.660066\pi$
$31$	−1.36357e17	−0.963870	−0.481935	+	0.876207i	$0.660066\pi$
$32$	0	0
$33$	9.79656e16	0.337414
$34$	0	0
$35$	−4.00514e17	−0.701189
$36$	0	0
$37$	−1.23898e18	−1.14484	−0.572419	−	0.819961i	$-0.693995\pi$
$37$	−1.23898e18	−1.14484	−0.572419	+	0.819961i	$0.693995\pi$
$38$	0	0
$39$	1.39490e18	0.703560
$40$	0	0
$41$	1.40074e18	0.397506	0.198753	−	0.980050i	$-0.436311\pi$
$41$	1.40074e18	0.397506	0.198753	+	0.980050i	$0.436311\pi$
$42$	0	0
$43$	−2.18793e17	−0.0359043	−0.0179522	−	0.999839i	$-0.505715\pi$
$43$	−2.18793e17	−0.0359043	−0.0179522	+	0.999839i	$0.505715\pi$
$44$	0	0
$45$	5.96187e18	0.580015
$46$	0	0
$47$	8.67836e18	0.512050	0.256025	−	0.966670i	$-0.417587\pi$
$47$	8.67836e18	0.512050	0.256025	+	0.966670i	$0.417587\pi$
$48$	0	0
$49$	−1.28361e19	−0.469007
$50$	0	0
$51$	−9.09698e19	−2.09817
$52$	0	0
$53$	−7.63436e19	−1.13136	−0.565679	−	0.824626i	$-0.691386\pi$
$53$	−7.63436e19	−1.13136	−0.565679	+	0.824626i	$0.691386\pi$
$54$	0	0
$55$	−2.64966e19	−0.256460
$56$	0	0
$57$	−2.42034e20	−1.55352
$58$	0	0
$59$	1.01862e18	0.00439760	0.00219880	−	0.999998i	$-0.499300\pi$
$59$	1.01862e18	0.00439760	0.00219880	+	0.999998i	$0.499300\pi$
$60$	0	0
$61$	2.87337e20	0.845472	0.422736	−	0.906253i	$-0.361070\pi$
$61$	2.87337e20	0.845472	0.422736	+	0.906253i	$0.361070\pi$
$62$	0	0
$63$	−2.16326e20	−0.439231
$64$	0	0
$65$	−3.77277e20	−0.534758
$66$	0	0
$67$	−1.47683e21	−1.47730	−0.738652	−	0.674088i	$-0.764537\pi$
$67$	−1.47683e21	−1.47730	−0.738652	+	0.674088i	$0.764537\pi$
$68$	0	0
$69$	−1.39369e21	−0.994032
$70$	0	0
$71$	−7.64346e20	−0.392481	−0.196241	−	0.980556i	$-0.562873\pi$
$71$	−7.64346e20	−0.392481	−0.196241	+	0.980556i	$0.562873\pi$
$72$	0	0
$73$	−3.49433e21	−1.30362	−0.651810	−	0.758382i	$-0.725990\pi$
$73$	−3.49433e21	−1.30362	−0.651810	+	0.758382i	$0.725990\pi$
$74$	0	0
$75$	3.42957e20	0.0937639
$76$	0	0
$77$	9.61427e20	0.194211
$78$	0	0
$79$	−1.02350e22	−1.53948	−0.769742	−	0.638356i	$-0.779615\pi$
$79$	−1.02350e22	−1.53948	−0.769742	+	0.638356i	$0.779615\pi$
$80$	0	0
$81$	−1.09851e22	−1.23944
$82$	0	0
$83$	7.71597e21	0.657646	0.328823	−	0.944392i	$-0.393348\pi$
$83$	7.71597e21	0.657646	0.328823	+	0.944392i	$0.393348\pi$
$84$	0	0
$85$	2.46044e22	1.59476
$86$	0	0
$87$	7.99897e21	0.396792
$88$	0	0
$89$	4.58518e21	0.175134	0.0875672	−	0.996159i	$-0.472091\pi$
$89$	4.58518e21	0.175134	0.0875672	+	0.996159i	$0.472091\pi$
$90$	0	0
$91$	1.36895e22	0.404959
$92$	0	0
$93$	5.29672e22	1.22026
$94$	0	0
$95$	6.54627e22	1.18079
$96$	0	0
$97$	−1.13703e23	−1.61398	−0.806991	−	0.590564i	$-0.798905\pi$
$97$	−1.13703e23	−1.61398	−0.806991	+	0.590564i	$0.798905\pi$
$98$	0	0
$99$	−1.43114e22	−0.160648
$100$	0	0
$101$	1.36243e23	1.21512	0.607558	−	0.794275i	$-0.292149\pi$
$101$	1.36243e23	1.21512	0.607558	+	0.794275i	$0.292149\pi$
$102$	0	0
$103$	−1.41401e22	−0.100653	−0.0503264	−	0.998733i	$-0.516026\pi$
$103$	−1.41401e22	−0.100653	−0.0503264	+	0.998733i	$0.516026\pi$
$104$	0	0
$105$	1.55578e23	0.887708
$106$	0	0
$107$	−6.02971e22	−0.276938	−0.138469	−	0.990367i	$-0.544218\pi$
$107$	−6.02971e22	−0.276938	−0.138469	+	0.990367i	$0.544218\pi$
$108$	0	0
$109$	−5.58169e22	−0.207186	−0.103593	−	0.994620i	$-0.533034\pi$
$109$	−5.58169e22	−0.207186	−0.103593	+	0.994620i	$0.533034\pi$
$110$	0	0
$111$	4.81275e23	1.44937
$112$	0	0
$113$	3.51523e23	0.862089	0.431044	−	0.902331i	$-0.358145\pi$
$113$	3.51523e23	0.862089	0.431044	+	0.902331i	$0.358145\pi$
$114$	0	0
$115$	3.76948e23	0.755538
$116$	0	0
$117$	−2.03775e23	−0.334976
$118$	0	0
$119$	−8.92770e23	−1.20767
$120$	0	0
$121$	−8.31826e23	−0.928968
$122$	0	0
$123$	−5.44111e23	−0.503244
$124$	0	0
$125$	−1.34520e24	−1.03352
$126$	0	0
$127$	−2.32044e24	−1.48535	−0.742674	−	0.669653i	$-0.766443\pi$
$127$	−2.32044e24	−1.48535	−0.742674	+	0.669653i	$0.766443\pi$
$128$	0	0
$129$	8.49891e22	0.0454550
$130$	0	0
$131$	−8.70825e23	−0.390221	−0.195111	−	0.980781i	$-0.562507\pi$
$131$	−8.70825e23	−0.390221	−0.195111	+	0.980781i	$0.562507\pi$
$132$	0	0
$133$	−2.37531e24	−0.894182
$134$	0	0
$135$	1.52620e24	0.483919
$136$	0	0
$137$	4.43869e24	1.18841	0.594207	−	0.804312i	$-0.297466\pi$
$137$	4.43869e24	1.18841	0.594207	+	0.804312i	$0.297466\pi$
$138$	0	0
$139$	4.97229e23	0.112690	0.0563452	−	0.998411i	$-0.482055\pi$
$139$	4.97229e23	0.112690	0.0563452	+	0.998411i	$0.482055\pi$
$140$	0	0
$141$	−3.37106e24	−0.648257
$142$	0	0
$143$	9.05647e23	0.148114
$144$	0	0
$145$	−2.16347e24	−0.301591
$146$	0	0
$147$	4.98613e24	0.593764
$148$	0	0
$149$	1.10598e24	0.112748	0.0563738	−	0.998410i	$-0.482046\pi$
$149$	1.10598e24	0.112748	0.0563738	+	0.998410i	$0.482046\pi$
$150$	0	0
$151$	−3.76304e24	−0.329082	−0.164541	−	0.986370i	$-0.552614\pi$
$151$	−3.76304e24	−0.329082	−0.164541	+	0.986370i	$0.552614\pi$
$152$	0	0
$153$	1.32894e25	0.998972
$154$	0	0
$155$	−1.43260e25	−0.927490
$156$	0	0
$157$	1.50090e25	0.838504	0.419252	−	0.907870i	$-0.362292\pi$
$157$	1.50090e25	0.838504	0.419252	+	0.907870i	$0.362292\pi$
$158$	0	0
$159$	2.96553e25	1.43230
$160$	0	0
$161$	−1.36775e25	−0.572150
$162$	0	0
$163$	2.61170e25	0.947907	0.473953	−	0.880550i	$-0.342827\pi$
$163$	2.61170e25	0.947907	0.473953	+	0.880550i	$0.342827\pi$
$164$	0	0
$165$	1.02925e25	0.324679
$166$	0	0
$167$	−1.77408e25	−0.487230	−0.243615	−	0.969872i	$-0.578333\pi$
$167$	−1.77408e25	−0.487230	−0.243615	+	0.969872i	$0.578333\pi$
$168$	0	0
$169$	−2.88587e25	−0.691161
$170$	0	0
$171$	3.53578e25	0.739656
$172$	0	0
$173$	1.04109e26	1.90528	0.952640	−	0.304100i	$-0.0983558\pi$
$173$	1.04109e26	1.90528	0.952640	+	0.304100i	$0.0983558\pi$
$174$	0	0
$175$	3.36576e24	0.0539691
$176$	0	0
$177$	−3.95679e23	−0.00556737
$178$	0	0
$179$	1.00142e25	0.123824	0.0619122	−	0.998082i	$-0.480280\pi$
$179$	1.00142e25	0.123824	0.0619122	+	0.998082i	$0.480280\pi$
$180$	0	0
$181$	−5.17169e25	−0.562768	−0.281384	−	0.959595i	$-0.590793\pi$
$181$	−5.17169e25	−0.562768	−0.281384	+	0.959595i	$0.590793\pi$
$182$	0	0
$183$	−1.11615e26	−1.07037
$184$	0	0
$185$	−1.30170e26	−1.10163
$186$	0	0
$187$	−5.90625e25	−0.441706
$188$	0	0
$189$	−5.53780e25	−0.366460
$190$	0	0
$191$	−3.10126e26	−1.81825	−0.909127	−	0.416520i	$-0.863250\pi$
$191$	−3.10126e26	−1.81825	−0.909127	+	0.416520i	$0.863250\pi$
$192$	0	0
$193$	−1.28183e26	−0.666687	−0.333344	−	0.942805i	$-0.608177\pi$
$193$	−1.28183e26	−0.666687	−0.333344	+	0.942805i	$0.608177\pi$
$194$	0	0
$195$	1.46551e26	0.677005
$196$	0	0
$197$	−3.89967e26	−1.60201	−0.801007	−	0.598655i	$-0.795702\pi$
$197$	−3.89967e26	−1.60201	−0.801007	+	0.598655i	$0.795702\pi$
$198$	0	0
$199$	1.25611e26	0.459426	0.229713	−	0.973258i	$-0.426221\pi$
$199$	1.25611e26	0.459426	0.229713	+	0.973258i	$0.426221\pi$
$200$	0	0
$201$	5.73666e26	1.87027
$202$	0	0
$203$	7.85013e25	0.228387
$204$	0	0
$205$	1.47165e26	0.382503
$206$	0	0
$207$	2.03598e26	0.473275
$208$	0	0
$209$	−1.57142e26	−0.327047
$210$	0	0
$211$	−5.88286e26	−1.09734	−0.548669	−	0.836040i	$-0.684865\pi$
$211$	−5.88286e26	−1.09734	−0.548669	+	0.836040i	$0.684865\pi$
$212$	0	0
$213$	2.96906e26	0.496883
$214$	0	0
$215$	−2.29869e25	−0.0345491
$216$	0	0
$217$	5.19817e26	0.702365
$218$	0	0
$219$	1.35735e27	1.65039
$220$	0	0
$221$	−8.40974e26	−0.921025
$222$	0	0
$223$	−1.38821e27	−1.37072	−0.685361	−	0.728203i	$-0.740356\pi$
$223$	−1.38821e27	−1.37072	−0.685361	+	0.728203i	$0.740356\pi$
$224$	0	0
$225$	−5.01012e25	−0.0446425
$226$	0	0
$227$	2.14958e27	1.73004	0.865022	−	0.501734i	$-0.167304\pi$
$227$	2.14958e27	1.73004	0.865022	+	0.501734i	$0.167304\pi$
$228$	0	0
$229$	−6.39851e26	−0.465554	−0.232777	−	0.972530i	$-0.574781\pi$
$229$	−6.39851e26	−0.465554	−0.232777	+	0.972530i	$0.574781\pi$
$230$	0	0
$231$	−3.73461e26	−0.245871
$232$	0	0
$233$	2.19149e27	1.30661	0.653306	−	0.757094i	$-0.273381\pi$
$233$	2.19149e27	1.30661	0.653306	+	0.757094i	$0.273381\pi$
$234$	0	0
$235$	9.11766e26	0.492723
$236$	0	0
$237$	3.97572e27	1.94899
$238$	0	0
$239$	−1.09944e27	−0.489322	−0.244661	−	0.969609i	$-0.578677\pi$
$239$	−1.09944e27	−0.489322	−0.244661	+	0.969609i	$0.578677\pi$
$240$	0	0
$241$	−1.44651e27	−0.584961	−0.292480	−	0.956272i	$-0.594481\pi$
$241$	−1.44651e27	−0.584961	−0.292480	+	0.956272i	$0.594481\pi$
$242$	0	0
$243$	2.89951e27	1.06623
$244$	0	0
$245$	−1.34859e27	−0.451304
$246$	0	0
$247$	−2.23750e27	−0.681942
$248$	0	0
$249$	−2.99723e27	−0.832582
$250$	0	0
$251$	−2.24453e26	−0.0568693	−0.0284346	−	0.999596i	$-0.509052\pi$
$251$	−2.24453e26	−0.0568693	−0.0284346	+	0.999596i	$0.509052\pi$
$252$	0	0
$253$	−9.04857e26	−0.209264
$254$	0	0
$255$	−9.55747e27	−2.01897
$256$	0	0
$257$	−3.95005e27	−0.762732	−0.381366	−	0.924424i	$-0.624546\pi$
$257$	−3.95005e27	−0.762732	−0.381366	+	0.924424i	$0.624546\pi$
$258$	0	0
$259$	4.72320e27	0.834236
$260$	0	0
$261$	−1.16853e27	−0.188919
$262$	0	0
$263$	−2.01521e27	−0.298420	−0.149210	−	0.988805i	$-0.547673\pi$
$263$	−2.01521e27	−0.298420	−0.149210	+	0.988805i	$0.547673\pi$
$264$	0	0
$265$	−8.02081e27	−1.08866
$266$	0	0
$267$	−1.78109e27	−0.221721
$268$	0	0
$269$	5.72063e27	0.653571	0.326785	−	0.945099i	$-0.394035\pi$
$269$	5.72063e27	0.653571	0.326785	+	0.945099i	$0.394035\pi$
$270$	0	0
$271$	−5.18050e27	−0.543531	−0.271766	−	0.962363i	$-0.587608\pi$
$271$	−5.18050e27	−0.543531	−0.271766	+	0.962363i	$0.587608\pi$
$272$	0	0
$273$	−5.31761e27	−0.512679
$274$	0	0
$275$	2.22667e26	0.0197392
$276$	0	0
$277$	1.29611e28	1.05712	0.528560	−	0.848896i	$-0.322732\pi$
$277$	1.29611e28	1.05712	0.528560	+	0.848896i	$0.322732\pi$
$278$	0	0
$279$	−7.73776e27	−0.580987
$280$	0	0
$281$	1.24154e28	0.858691	0.429346	−	0.903140i	$-0.358744\pi$
$281$	1.24154e28	0.858691	0.429346	+	0.903140i	$0.358744\pi$
$282$	0	0
$283$	2.15031e28	1.37074	0.685372	−	0.728193i	$-0.259640\pi$
$283$	2.15031e28	1.37074	0.685372	+	0.728193i	$0.259640\pi$
$284$	0	0
$285$	−2.54286e28	−1.49488
$286$	0	0
$287$	−5.33986e27	−0.289660
$288$	0	0
$289$	3.48772e28	1.74669
$290$	0	0
$291$	4.41675e28	2.04331
$292$	0	0
$293$	−1.20307e28	−0.514416	−0.257208	−	0.966356i	$-0.582803\pi$
$293$	−1.20307e28	−0.514416	−0.257208	+	0.966356i	$0.582803\pi$
$294$	0	0
$295$	1.07019e26	0.00423161
$296$	0	0
$297$	−3.66362e27	−0.134033
$298$	0	0
$299$	−1.28840e28	−0.436347
$300$	0	0
$301$	8.34076e26	0.0261632
$302$	0	0
$303$	−5.29228e28	−1.53834
$304$	0	0
$305$	3.01883e28	0.813560
$306$	0	0
$307$	4.70428e28	1.17598	0.587992	−	0.808867i	$-0.299919\pi$
$307$	4.70428e28	1.17598	0.587992	+	0.808867i	$0.299919\pi$
$308$	0	0
$309$	5.49267e27	0.127427
$310$	0	0
$311$	8.99672e26	0.0193794	0.00968968	−	0.999953i	$-0.496916\pi$
$311$	8.99672e26	0.0193794	0.00968968	+	0.999953i	$0.496916\pi$
$312$	0	0
$313$	−1.13525e28	−0.227159	−0.113579	−	0.993529i	$-0.536232\pi$
$313$	−1.13525e28	−0.227159	−0.113579	+	0.993529i	$0.536232\pi$
$314$	0	0
$315$	−2.27277e28	−0.422653
$316$	0	0
$317$	1.37896e28	0.238436	0.119218	−	0.992868i	$-0.461961\pi$
$317$	1.37896e28	0.238436	0.119218	+	0.992868i	$0.461961\pi$
$318$	0	0
$319$	5.19337e27	0.0835326
$320$	0	0
$321$	2.34221e28	0.350605
$322$	0	0
$323$	1.45920e29	2.03370
$324$	0	0
$325$	3.17048e27	0.0411592
$326$	0	0
$327$	2.16818e28	0.262298
$328$	0	0
$329$	−3.30834e28	−0.373127
$330$	0	0
$331$	−1.62668e29	−1.71112	−0.855559	−	0.517706i	$-0.826786\pi$
$331$	−1.62668e29	−1.71112	−0.855559	+	0.517706i	$0.826786\pi$
$332$	0	0
$333$	−7.03074e28	−0.690069
$334$	0	0
$335$	−1.55159e29	−1.42154
$336$	0	0
$337$	−1.64138e29	−1.40432	−0.702160	−	0.712019i	$-0.747781\pi$
$337$	−1.64138e29	−1.40432	−0.702160	+	0.712019i	$0.747781\pi$
$338$	0	0
$339$	−1.36547e29	−1.09141
$340$	0	0
$341$	3.43892e28	0.256890
$342$	0	0
$343$	1.53268e29	1.07045
$344$	0	0
$345$	−1.46424e29	−0.956514
$346$	0	0
$347$	−1.45586e29	−0.889876	−0.444938	−	0.895561i	$-0.646774\pi$
$347$	−1.45586e29	−0.889876	−0.444938	+	0.895561i	$0.646774\pi$
$348$	0	0
$349$	2.22110e28	0.127080	0.0635398	−	0.997979i	$-0.479761\pi$
$349$	2.22110e28	0.127080	0.0635398	+	0.997979i	$0.479761\pi$
$350$	0	0
$351$	−5.21651e28	−0.279479
$352$	0	0
$353$	2.57166e28	0.129064	0.0645320	−	0.997916i	$-0.479445\pi$
$353$	2.57166e28	0.129064	0.0645320	+	0.997916i	$0.479445\pi$
$354$	0	0
$355$	−8.03037e28	−0.377668
$356$	0	0
$357$	3.46792e29	1.52892
$358$	0	0
$359$	3.34157e29	1.38154	0.690771	−	0.723074i	$-0.257271\pi$
$359$	3.34157e29	1.38154	0.690771	+	0.723074i	$0.257271\pi$
$360$	0	0
$361$	1.30406e29	0.505785
$362$	0	0
$363$	3.23118e29	1.17608
$364$	0	0
$365$	−3.67122e29	−1.25442
$366$	0	0
$367$	−1.62664e29	−0.521954	−0.260977	−	0.965345i	$-0.584045\pi$
$367$	−1.62664e29	−0.521954	−0.260977	+	0.965345i	$0.584045\pi$
$368$	0	0
$369$	7.94868e28	0.239603
$370$	0	0
$371$	2.91034e29	0.824412
$372$	0	0
$373$	1.03540e29	0.275712	0.137856	−	0.990452i	$-0.455979\pi$
$373$	1.03540e29	0.275712	0.137856	+	0.990452i	$0.455979\pi$
$374$	0	0
$375$	5.22535e29	1.30844
$376$	0	0
$377$	7.39468e28	0.174178
$378$	0	0
$379$	−6.62210e29	−1.46772	−0.733862	−	0.679298i	$-0.762284\pi$
$379$	−6.62210e29	−1.46772	−0.733862	+	0.679298i	$0.762284\pi$
$380$	0	0
$381$	9.01364e29	1.88046
$382$	0	0
$383$	−2.96585e29	−0.582591	−0.291295	−	0.956633i	$-0.594086\pi$
$383$	−2.96585e29	−0.582591	−0.291295	+	0.956633i	$0.594086\pi$
$384$	0	0
$385$	1.01010e29	0.186880
$386$	0	0
$387$	−1.24157e28	−0.0216419
$388$	0	0
$389$	−1.06609e30	−1.75134	−0.875672	−	0.482906i	$-0.839581\pi$
$389$	−1.06609e30	−1.75134	−0.875672	+	0.482906i	$0.839581\pi$
$390$	0	0
$391$	8.40240e29	1.30128
$392$	0	0
$393$	3.38267e29	0.494021
$394$	0	0
$395$	−1.07531e30	−1.48138
$396$	0	0
$397$	1.67462e28	0.0217683	0.0108842	−	0.999941i	$-0.496535\pi$
$397$	1.67462e28	0.0217683	0.0108842	+	0.999941i	$0.496535\pi$
$398$	0	0
$399$	9.22676e29	1.13204
$400$	0	0
$401$	3.87121e29	0.448421	0.224211	−	0.974541i	$-0.428020\pi$
$401$	3.87121e29	0.448421	0.224211	+	0.974541i	$0.428020\pi$
$402$	0	0
$403$	4.89658e29	0.535654
$404$	0	0
$405$	−1.15411e30	−1.19266
$406$	0	0
$407$	3.12470e29	0.305121
$408$	0	0
$409$	5.00511e29	0.461950	0.230975	−	0.972960i	$-0.425808\pi$
$409$	5.00511e29	0.461950	0.230975	+	0.972960i	$0.425808\pi$
$410$	0	0
$411$	−1.72418e30	−1.50454
$412$	0	0
$413$	−3.88316e27	−0.00320450
$414$	0	0
$415$	8.10655e29	0.632824
$416$	0	0
$417$	−1.93146e29	−0.142666
$418$	0	0
$419$	−1.70760e30	−1.19378	−0.596891	−	0.802322i	$-0.703598\pi$
$419$	−1.70760e30	−1.19378	−0.596891	+	0.802322i	$0.703598\pi$
$420$	0	0
$421$	−5.97218e29	−0.395265	−0.197633	−	0.980276i	$-0.563325\pi$
$421$	−5.97218e29	−0.395265	−0.197633	+	0.980276i	$0.563325\pi$
$422$	0	0
$423$	4.92464e29	0.308646
$424$	0	0
$425$	−2.06766e29	−0.122746
$426$	0	0
$427$	−1.09538e30	−0.616089
$428$	0	0
$429$	−3.51794e29	−0.187512
$430$	0	0
$431$	−1.91854e30	−0.969353	−0.484676	−	0.874694i	$-0.661063\pi$
$431$	−1.91854e30	−0.969353	−0.484676	+	0.874694i	$0.661063\pi$
$432$	0	0
$433$	−2.68456e30	−1.28606	−0.643032	−	0.765839i	$-0.722324\pi$
$433$	−2.68456e30	−1.28606	−0.643032	+	0.765839i	$0.722324\pi$
$434$	0	0
$435$	8.40388e29	0.381815
$436$	0	0
$437$	2.23555e30	0.963489
$438$	0	0
$439$	−2.11168e29	−0.0863549	−0.0431775	−	0.999067i	$-0.513748\pi$
$439$	−2.11168e29	−0.0863549	−0.0431775	+	0.999067i	$0.513748\pi$
$440$	0	0
$441$	−7.28401e29	−0.282701
$442$	0	0
$443$	8.12623e29	0.299396	0.149698	−	0.988732i	$-0.452170\pi$
$443$	8.12623e29	0.299396	0.149698	+	0.988732i	$0.452170\pi$
$444$	0	0
$445$	4.81728e29	0.168524
$446$	0	0
$447$	−4.29614e29	−0.142739
$448$	0	0
$449$	−3.76596e30	−1.18862	−0.594310	−	0.804236i	$-0.702575\pi$
$449$	−3.76596e30	−1.18862	−0.594310	+	0.804236i	$0.702575\pi$
$450$	0	0
$451$	−3.53267e29	−0.105943
$452$	0	0
$453$	1.46173e30	0.416619
$454$	0	0
$455$	1.43824e30	0.389674
$456$	0	0
$457$	2.14090e30	0.551518	0.275759	−	0.961227i	$-0.411071\pi$
$457$	2.14090e30	0.551518	0.275759	+	0.961227i	$0.411071\pi$
$458$	0	0
$459$	3.40199e30	0.833465
$460$	0	0
$461$	4.76379e30	1.11018	0.555088	−	0.831792i	$-0.312685\pi$
$461$	4.76379e30	1.11018	0.555088	+	0.831792i	$0.312685\pi$
$462$	0	0
$463$	7.67574e30	1.70192	0.850959	−	0.525232i	$-0.176021\pi$
$463$	7.67574e30	1.70192	0.850959	+	0.525232i	$0.176021\pi$
$464$	0	0
$465$	5.56485e30	1.17421
$466$	0	0
$467$	2.87884e30	0.578194	0.289097	−	0.957300i	$-0.406645\pi$
$467$	2.87884e30	0.578194	0.289097	+	0.957300i	$0.406645\pi$
$468$	0	0
$469$	5.62992e30	1.07650
$470$	0	0
$471$	−5.83016e30	−1.06155
$472$	0	0
$473$	5.51796e28	0.00956919
$474$	0	0
$475$	−5.50121e29	−0.0908829
$476$	0	0
$477$	−4.33221e30	−0.681943
$478$	0	0
$479$	−6.82143e30	−1.02333	−0.511666	−	0.859184i	$-0.670972\pi$
$479$	−6.82143e30	−1.02333	−0.511666	+	0.859184i	$0.670972\pi$
$480$	0	0
$481$	4.44917e30	0.636224
$482$	0	0
$483$	5.31297e30	0.724344
$484$	0	0
$485$	−1.19459e31	−1.55306
$486$	0	0
$487$	−8.59361e29	−0.106560	−0.0532798	−	0.998580i	$-0.516968\pi$
$487$	−8.59361e29	−0.106560	−0.0532798	+	0.998580i	$0.516968\pi$
$488$	0	0
$489$	−1.01450e31	−1.20005
$490$	0	0
$491$	7.15140e30	0.807149	0.403575	−	0.914947i	$-0.367768\pi$
$491$	7.15140e30	0.807149	0.403575	+	0.914947i	$0.367768\pi$
$492$	0	0
$493$	−4.82250e30	−0.519437
$494$	0	0
$495$	−1.50358e30	−0.154585
$496$	0	0
$497$	2.91382e30	0.285998
$498$	0	0
$499$	−1.77913e31	−1.66744	−0.833722	−	0.552185i	$-0.813794\pi$
$499$	−1.77913e31	−1.66744	−0.833722	+	0.552185i	$0.813794\pi$
$500$	0	0
$501$	6.89133e30	0.616835
$502$	0	0
$503$	−6.56155e30	−0.561015	−0.280507	−	0.959852i	$-0.590503\pi$
$503$	−6.56155e30	−0.561015	−0.280507	+	0.959852i	$0.590503\pi$
$504$	0	0
$505$	1.43139e31	1.16925
$506$	0	0
$507$	1.12100e31	0.875012
$508$	0	0
$509$	−1.19826e31	−0.893917	−0.446959	−	0.894555i	$-0.647493\pi$
$509$	−1.19826e31	−0.893917	−0.446959	+	0.894555i	$0.647493\pi$
$510$	0	0
$511$	1.33210e31	0.949939
$512$	0	0
$513$	9.05135e30	0.617112
$514$	0	0
$515$	−1.48559e30	−0.0968537
$516$	0	0
$517$	−2.18868e30	−0.136471
$518$	0	0
$519$	−4.04407e31	−2.41209
$520$	0	0
$521$	−2.26240e31	−1.29103	−0.645514	−	0.763749i	$-0.723357\pi$
$521$	−2.26240e31	−1.29103	−0.645514	+	0.763749i	$0.723357\pi$
$522$	0	0
$523$	3.41914e31	1.86702	0.933508	−	0.358557i	$-0.116731\pi$
$523$	3.41914e31	1.86702	0.933508	+	0.358557i	$0.116731\pi$
$524$	0	0
$525$	−1.30741e30	−0.0683251
$526$	0	0
$527$	−3.19334e31	−1.59744
$528$	0	0
$529$	−8.00772e30	−0.383503
$530$	0	0
$531$	5.78030e28	0.00265072
$532$	0	0
$533$	−5.03006e30	−0.220907
$534$	0	0
$535$	−6.33494e30	−0.266486
$536$	0	0
$537$	−3.88996e30	−0.156762
$538$	0	0
$539$	3.23726e30	0.124999
$540$	0	0
$541$	−2.75024e30	−0.101766	−0.0508829	−	0.998705i	$-0.516204\pi$
$541$	−2.75024e30	−0.101766	−0.0508829	+	0.998705i	$0.516204\pi$
$542$	0	0
$543$	2.00892e31	0.712466
$544$	0	0
$545$	−5.86424e30	−0.199366
$546$	0	0
$547$	1.03197e31	0.336367	0.168184	−	0.985756i	$-0.446210\pi$
$547$	1.03197e31	0.336367	0.168184	+	0.985756i	$0.446210\pi$
$548$	0	0
$549$	1.63053e31	0.509621
$550$	0	0
$551$	−1.28308e31	−0.384600
$552$	0	0
$553$	3.90174e31	1.12181
$554$	0	0
$555$	5.05637e31	1.39466
$556$	0	0
$557$	1.36262e31	0.360613	0.180307	−	0.983610i	$-0.442291\pi$
$557$	1.36262e31	0.360613	0.180307	+	0.983610i	$0.442291\pi$
$558$	0	0
$559$	7.85685e29	0.0199532
$560$	0	0
$561$	2.29425e31	0.559202
$562$	0	0
$563$	−3.09091e31	−0.723169	−0.361585	−	0.932339i	$-0.617764\pi$
$563$	−3.09091e31	−0.723169	−0.361585	+	0.932339i	$0.617764\pi$
$564$	0	0
$565$	3.69317e31	0.829550
$566$	0	0
$567$	4.18769e31	0.903171
$568$	0	0
$569$	−6.93051e31	−1.43540	−0.717702	−	0.696350i	$-0.754806\pi$
$569$	−6.93051e31	−1.43540	−0.717702	+	0.696350i	$0.754806\pi$
$570$	0	0
$571$	6.24961e30	0.124319	0.0621595	−	0.998066i	$-0.480201\pi$
$571$	6.24961e30	0.124319	0.0621595	+	0.998066i	$0.480201\pi$
$572$	0	0
$573$	1.20467e32	2.30192
$574$	0	0
$575$	−3.16772e30	−0.0581522
$576$	0	0
$577$	−1.59740e31	−0.281768	−0.140884	−	0.990026i	$-0.544994\pi$
$577$	−1.59740e31	−0.281768	−0.140884	+	0.990026i	$0.544994\pi$
$578$	0	0
$579$	4.97921e31	0.844029
$580$	0	0
$581$	−2.94146e31	−0.479222
$582$	0	0
$583$	1.92538e31	0.301529
$584$	0	0
$585$	−2.14091e31	−0.322333
$586$	0	0
$587$	−8.16635e31	−1.18220	−0.591099	−	0.806599i	$-0.701306\pi$
$587$	−8.16635e31	−1.18220	−0.591099	+	0.806599i	$0.701306\pi$
$588$	0	0
$589$	−8.49622e31	−1.18277
$590$	0	0
$591$	1.51481e32	2.02816
$592$	0	0
$593$	−5.61370e30	−0.0722970	−0.0361485	−	0.999346i	$-0.511509\pi$
$593$	−5.61370e30	−0.0722970	−0.0361485	+	0.999346i	$0.511509\pi$
$594$	0	0
$595$	−9.37963e31	−1.16209
$596$	0	0
$597$	−4.87928e31	−0.581635
$598$	0	0
$599$	−1.60231e32	−1.83797	−0.918985	−	0.394293i	$-0.870990\pi$
$599$	−1.60231e32	−1.83797	−0.918985	+	0.394293i	$0.870990\pi$
$600$	0	0
$601$	8.47641e31	0.935741	0.467870	−	0.883797i	$-0.345021\pi$
$601$	8.47641e31	0.935741	0.467870	+	0.883797i	$0.345021\pi$
$602$	0	0
$603$	−8.38044e31	−0.890467
$604$	0	0
$605$	−8.73933e31	−0.893905
$606$	0	0
$607$	1.63793e31	0.161296	0.0806481	−	0.996743i	$-0.474301\pi$
$607$	1.63793e31	0.161296	0.0806481	+	0.996743i	$0.474301\pi$
$608$	0	0
$609$	−3.04934e31	−0.289139
$610$	0	0
$611$	−3.11639e31	−0.284563
$612$	0	0
$613$	1.93558e31	0.170222	0.0851110	−	0.996371i	$-0.472876\pi$
$613$	1.93558e31	0.170222	0.0851110	+	0.996371i	$0.472876\pi$
$614$	0	0
$615$	−5.71654e31	−0.484250
$616$	0	0
$617$	5.25618e31	0.428934	0.214467	−	0.976731i	$-0.431199\pi$
$617$	5.25618e31	0.428934	0.214467	+	0.976731i	$0.431199\pi$
$618$	0	0
$619$	9.42009e31	0.740650	0.370325	−	0.928902i	$-0.379246\pi$
$619$	9.42009e31	0.740650	0.370325	+	0.928902i	$0.379246\pi$
$620$	0	0
$621$	5.21196e31	0.394864
$622$	0	0
$623$	−1.74795e31	−0.127619
$624$	0	0
$625$	−1.30804e32	−0.920452
$626$	0	0
$627$	6.10410e31	0.414042
$628$	0	0
$629$	−2.90156e32	−1.89736
$630$	0	0
$631$	2.22784e32	1.40458	0.702288	−	0.711892i	$-0.252162\pi$
$631$	2.22784e32	1.40458	0.702288	+	0.711892i	$0.252162\pi$
$632$	0	0
$633$	2.28517e32	1.38923
$634$	0	0
$635$	−2.43790e32	−1.42929
$636$	0	0
$637$	4.60944e31	0.260642
$638$	0	0
$639$	−4.33738e31	−0.236574
$640$	0	0
$641$	1.38602e32	0.729295	0.364648	−	0.931146i	$-0.381189\pi$
$641$	1.38602e32	0.729295	0.364648	+	0.931146i	$0.381189\pi$
$642$	0	0
$643$	−3.71401e31	−0.188546	−0.0942729	−	0.995546i	$-0.530053\pi$
$643$	−3.71401e31	−0.188546	−0.0942729	+	0.995546i	$0.530053\pi$
$644$	0	0
$645$	8.92913e30	0.0437393
$646$	0	0
$647$	3.77554e32	1.78476	0.892379	−	0.451286i	$-0.149035\pi$
$647$	3.77554e32	1.78476	0.892379	+	0.451286i	$0.149035\pi$
$648$	0	0
$649$	−2.56896e29	−0.00117204
$650$	0	0
$651$	−2.01920e32	−0.889197
$652$	0	0
$653$	−8.58042e31	−0.364760	−0.182380	−	0.983228i	$-0.558380\pi$
$653$	−8.58042e31	−0.364760	−0.182380	+	0.983228i	$0.558380\pi$
$654$	0	0
$655$	−9.14906e31	−0.375493
$656$	0	0
$657$	−1.98290e32	−0.785777
$658$	0	0
$659$	−1.46687e32	−0.561319	−0.280660	−	0.959807i	$-0.590553\pi$
$659$	−1.46687e32	−0.561319	−0.280660	+	0.959807i	$0.590553\pi$
$660$	0	0
$661$	2.02066e32	0.746752	0.373376	−	0.927680i	$-0.378200\pi$
$661$	2.02066e32	0.746752	0.373376	+	0.927680i	$0.378200\pi$
$662$	0	0
$663$	3.26672e32	1.16602
$664$	0	0
$665$	−2.49555e32	−0.860432
$666$	0	0
$667$	−7.38823e31	−0.246090
$668$	0	0
$669$	5.39243e32	1.73534
$670$	0	0
$671$	−7.24664e31	−0.225334
$672$	0	0
$673$	5.34692e32	1.60668	0.803341	−	0.595519i	$-0.203053\pi$
$673$	5.34692e32	1.60668	0.803341	+	0.595519i	$0.203053\pi$
$674$	0	0
$675$	−1.28256e31	−0.0372463
$676$	0	0
$677$	1.10049e32	0.308899	0.154449	−	0.988001i	$-0.450640\pi$
$677$	1.10049e32	0.308899	0.154449	+	0.988001i	$0.450640\pi$
$678$	0	0
$679$	4.33457e32	1.17610
$680$	0	0
$681$	−8.34995e32	−2.19024
$682$	0	0
$683$	2.97297e32	0.753967	0.376983	−	0.926220i	$-0.376961\pi$
$683$	2.97297e32	0.753967	0.376983	+	0.926220i	$0.376961\pi$
$684$	0	0
$685$	4.66337e32	1.14356
$686$	0	0
$687$	2.48547e32	0.589394
$688$	0	0
$689$	2.74149e32	0.628733
$690$	0	0
$691$	7.18838e32	1.59453	0.797266	−	0.603628i	$-0.206279\pi$
$691$	7.18838e32	1.59453	0.797266	+	0.603628i	$0.206279\pi$
$692$	0	0
$693$	5.45574e31	0.117063
$694$	0	0
$695$	5.22399e31	0.108437
$696$	0	0
$697$	3.28039e32	0.658793
$698$	0	0
$699$	−8.51274e32	−1.65418
$700$	0	0
$701$	−1.02832e33	−1.93362	−0.966810	−	0.255497i	$-0.917761\pi$
$701$	−1.02832e33	−1.93362	−0.966810	+	0.255497i	$0.917761\pi$
$702$	0	0
$703$	−7.71990e32	−1.40484
$704$	0	0
$705$	−3.54171e32	−0.623790
$706$	0	0
$707$	−5.19380e32	−0.885446
$708$	0	0
$709$	−2.50942e32	−0.414134	−0.207067	−	0.978327i	$-0.566392\pi$
$709$	−2.50942e32	−0.414134	−0.207067	+	0.978327i	$0.566392\pi$
$710$	0	0
$711$	−5.80796e32	−0.927947
$712$	0	0
$713$	−4.89231e32	−0.756805
$714$	0	0
$715$	9.51492e31	0.142523
$716$	0	0
$717$	4.27071e32	0.619483
$718$	0	0
$719$	−7.09247e32	−0.996358	−0.498179	−	0.867074i	$-0.665998\pi$
$719$	−7.09247e32	−0.996358	−0.498179	+	0.867074i	$0.665998\pi$
$720$	0	0
$721$	5.39046e31	0.0733450
$722$	0	0
$723$	5.61891e32	0.740562
$724$	0	0
$725$	1.81809e31	0.0232128
$726$	0	0
$727$	−4.63712e32	−0.573591	−0.286795	−	0.957992i	$-0.592590\pi$
$727$	−4.63712e32	−0.573591	−0.286795	+	0.957992i	$0.592590\pi$
$728$	0	0
$729$	−9.21300e31	−0.110417
$730$	0	0
$731$	−5.12391e31	−0.0595047
$732$	0	0
$733$	−1.54780e33	−1.74188	−0.870941	−	0.491387i	$-0.836490\pi$
$733$	−1.54780e33	−1.74188	−0.870941	+	0.491387i	$0.836490\pi$
$734$	0	0
$735$	5.23852e32	0.571353
$736$	0	0
$737$	3.72455e32	0.393730
$738$	0	0
$739$	1.49249e33	1.52932	0.764662	−	0.644431i	$-0.222906\pi$
$739$	1.49249e33	1.52932	0.764662	+	0.644431i	$0.222906\pi$
$740$	0	0
$741$	8.69144e32	0.863341
$742$	0	0
$743$	−1.19934e32	−0.115497	−0.0577487	−	0.998331i	$-0.518392\pi$
$743$	−1.19934e32	−0.115497	−0.0577487	+	0.998331i	$0.518392\pi$
$744$	0	0
$745$	1.16197e32	0.108492
$746$	0	0
$747$	4.37852e32	0.396406
$748$	0	0
$749$	2.29863e32	0.201803
$750$	0	0
$751$	−1.42812e33	−1.21592	−0.607959	−	0.793969i	$-0.708011\pi$
$751$	−1.42812e33	−1.21592	−0.607959	+	0.793969i	$0.708011\pi$
$752$	0	0
$753$	8.71877e31	0.0719967
$754$	0	0
$755$	−3.95352e32	−0.316661
$756$	0	0
$757$	−1.56208e33	−1.21367	−0.606835	−	0.794828i	$-0.707561\pi$
$757$	−1.56208e33	−1.21367	−0.606835	+	0.794828i	$0.707561\pi$
$758$	0	0
$759$	3.51487e32	0.264929
$760$	0	0
$761$	1.92115e33	1.40488	0.702439	−	0.711744i	$-0.252094\pi$
$761$	1.92115e33	1.40488	0.702439	+	0.711744i	$0.252094\pi$
$762$	0	0
$763$	2.12783e32	0.150975
$764$	0	0
$765$	1.39621e33	0.961267
$766$	0	0
$767$	−3.65787e30	−0.00244389
$768$	0	0
$769$	1.22615e33	0.795044	0.397522	−	0.917593i	$-0.369870\pi$
$769$	1.22615e33	0.795044	0.397522	+	0.917593i	$0.369870\pi$
$770$	0	0
$771$	1.53438e33	0.965621
$772$	0	0
$773$	−8.43914e32	−0.515507	−0.257753	−	0.966211i	$-0.582982\pi$
$773$	−8.43914e32	−0.515507	−0.257753	+	0.966211i	$0.582982\pi$
$774$	0	0
$775$	1.20390e32	0.0713870
$776$	0	0
$777$	−1.83470e33	−1.05615
$778$	0	0
$779$	8.72782e32	0.487781
$780$	0	0
$781$	1.92768e32	0.104604
$782$	0	0
$783$	−2.99137e32	−0.157620
$784$	0	0
$785$	1.57687e33	0.806856
$786$	0	0
$787$	−1.37027e33	−0.680919	−0.340460	−	0.940259i	$-0.610583\pi$
$787$	−1.37027e33	−0.680919	−0.340460	+	0.940259i	$0.610583\pi$
$788$	0	0
$789$	7.82797e32	0.377801
$790$	0	0
$791$	−1.34007e33	−0.628198
$792$	0	0
$793$	−1.03183e33	−0.469857
$794$	0	0
$795$	3.11564e33	1.37824
$796$	0	0
$797$	1.57462e33	0.676714	0.338357	−	0.941018i	$-0.390129\pi$
$797$	1.57462e33	0.676714	0.338357	+	0.941018i	$0.390129\pi$
$798$	0	0
$799$	2.03238e33	0.848628
$800$	0	0
$801$	2.60192e32	0.105565
$802$	0	0
$803$	8.81269e32	0.347440
$804$	0	0
$805$	−1.43699e33	−0.550555
$806$	0	0
$807$	−2.22215e33	−0.827423
$808$	0	0
$809$	−2.56942e33	−0.929882	−0.464941	−	0.885342i	$-0.653924\pi$
$809$	−2.56942e33	−0.929882	−0.464941	+	0.885342i	$0.653924\pi$
$810$	0	0
$811$	−1.25770e33	−0.442423	−0.221212	−	0.975226i	$-0.571001\pi$
$811$	−1.25770e33	−0.442423	−0.221212	+	0.975226i	$0.571001\pi$
$812$	0	0
$813$	2.01234e33	0.688112
$814$	0	0
$815$	2.74390e33	0.912129
$816$	0	0
$817$	−1.36327e32	−0.0440583
$818$	0	0
$819$	7.76826e32	0.244095
$820$	0	0
$821$	−2.38117e32	−0.0727519	−0.0363759	−	0.999338i	$-0.511581\pi$
$821$	−2.38117e32	−0.0727519	−0.0363759	+	0.999338i	$0.511581\pi$
$822$	0	0
$823$	−5.82453e33	−1.73047	−0.865233	−	0.501370i	$-0.832830\pi$
$823$	−5.82453e33	−1.73047	−0.865233	+	0.501370i	$0.832830\pi$
$824$	0	0
$825$	−8.64937e31	−0.0249899
$826$	0	0
$827$	3.61128e33	1.01472	0.507362	−	0.861733i	$-0.330621\pi$
$827$	3.61128e33	1.01472	0.507362	+	0.861733i	$0.330621\pi$
$828$	0	0
$829$	6.27935e33	1.71608	0.858040	−	0.513584i	$-0.171682\pi$
$829$	6.27935e33	1.71608	0.858040	+	0.513584i	$0.171682\pi$
$830$	0	0
$831$	−5.03467e33	−1.33832
$832$	0	0
$833$	−3.00609e33	−0.777292
$834$	0	0
$835$	−1.86389e33	−0.468841
$836$	0	0
$837$	−1.98081e33	−0.484731
$838$	0	0
$839$	−7.51696e32	−0.178970	−0.0894849	−	0.995988i	$-0.528522\pi$
$839$	−7.51696e32	−0.178970	−0.0894849	+	0.995988i	$0.528522\pi$
$840$	0	0
$841$	−3.89268e33	−0.901767
$842$	0	0
$843$	−4.82269e33	−1.08711
$844$	0	0
$845$	−3.03195e33	−0.665074
$846$	0	0
$847$	3.17106e33	0.676932
$848$	0	0
$849$	−8.35275e33	−1.73537
$850$	0	0
$851$	−4.44528e33	−0.898896
$852$	0	0
$853$	−9.11290e33	−1.79367	−0.896836	−	0.442364i	$-0.854140\pi$
$853$	−9.11290e33	−1.79367	−0.896836	+	0.442364i	$0.854140\pi$
$854$	0	0
$855$	3.71476e33	0.711738
$856$	0	0
$857$	6.17423e33	1.15160	0.575802	−	0.817589i	$-0.304690\pi$
$857$	6.17423e33	1.15160	0.575802	+	0.817589i	$0.304690\pi$
$858$	0	0
$859$	1.86175e33	0.338065	0.169033	−	0.985610i	$-0.445936\pi$
$859$	1.86175e33	0.338065	0.169033	+	0.985610i	$0.445936\pi$
$860$	0	0
$861$	2.07424e33	0.366710
$862$	0	0
$863$	5.22144e33	0.898806	0.449403	−	0.893329i	$-0.351637\pi$
$863$	5.22144e33	0.898806	0.449403	+	0.893329i	$0.351637\pi$
$864$	0	0
$865$	1.09379e34	1.83337
$866$	0	0
$867$	−1.35479e34	−2.21132
$868$	0	0
$869$	2.58126e33	0.410302
$870$	0	0
$871$	5.30328e33	0.820986
$872$	0	0
$873$	−6.45224e33	−0.972852
$874$	0	0
$875$	5.12811e33	0.753121
$876$	0	0
$877$	−6.73800e32	−0.0963908	−0.0481954	−	0.998838i	$-0.515347\pi$
$877$	−6.73800e32	−0.0963908	−0.0481954	+	0.998838i	$0.515347\pi$
$878$	0	0
$879$	4.67328e33	0.651253
$880$	0	0
$881$	1.36978e34	1.85963	0.929815	−	0.368028i	$-0.119967\pi$
$881$	1.36978e34	1.85963	0.929815	+	0.368028i	$0.119967\pi$
$882$	0	0
$883$	−1.49590e33	−0.197859	−0.0989294	−	0.995094i	$-0.531542\pi$
$883$	−1.49590e33	−0.197859	−0.0989294	+	0.995094i	$0.531542\pi$
$884$	0	0
$885$	−4.15708e31	−0.00535724
$886$	0	0
$887$	6.44904e33	0.809792	0.404896	−	0.914363i	$-0.367308\pi$
$887$	6.44904e33	0.809792	0.404896	+	0.914363i	$0.367308\pi$
$888$	0	0
$889$	8.84592e33	1.08236
$890$	0	0
$891$	2.77043e33	0.330334
$892$	0	0
$893$	5.40736e33	0.628339
$894$	0	0
$895$	1.05211e33	0.119151
$896$	0	0
$897$	5.00472e33	0.552417
$898$	0	0
$899$	2.80791e33	0.302097
$900$	0	0
$901$	−1.78789e34	−1.87502
$902$	0	0
$903$	−3.23993e32	−0.0331227
$904$	0	0
$905$	−5.43349e33	−0.541527
$906$	0	0
$907$	−6.42282e33	−0.624083	−0.312041	−	0.950069i	$-0.601013\pi$
$907$	−6.42282e33	−0.624083	−0.312041	+	0.950069i	$0.601013\pi$
$908$	0	0
$909$	7.73126e33	0.732430
$910$	0	0
$911$	1.48932e34	1.37571	0.687855	−	0.725848i	$-0.258553\pi$
$911$	1.48932e34	1.37571	0.687855	+	0.725848i	$0.258553\pi$
$912$	0	0
$913$	−1.94596e33	−0.175275
$914$	0	0
$915$	−1.17265e34	−1.02997
$916$	0	0
$917$	3.31973e33	0.284351
$918$	0	0
$919$	−3.10701e33	−0.259546	−0.129773	−	0.991544i	$-0.541425\pi$
$919$	−3.10701e33	−0.259546	−0.129773	+	0.991544i	$0.541425\pi$
$920$	0	0
$921$	−1.82735e34	−1.48880
$922$	0	0
$923$	2.74476e33	0.218115
$924$	0	0
$925$	1.09389e33	0.0847900
$926$	0	0
$927$	−8.02400e32	−0.0606700
$928$	0	0
$929$	1.72846e34	1.27490	0.637452	−	0.770490i	$-0.279988\pi$
$929$	1.72846e34	1.27490	0.637452	+	0.770490i	$0.279988\pi$
$930$	0	0
$931$	−7.99801e33	−0.575520
$932$	0	0
$933$	−3.49473e32	−0.0245343
$934$	0	0
$935$	−6.20523e33	−0.425035
$936$	0	0
$937$	−1.61548e34	−1.07969	−0.539843	−	0.841766i	$-0.681516\pi$
$937$	−1.61548e34	−1.07969	−0.539843	+	0.841766i	$0.681516\pi$
$938$	0	0
$939$	4.40980e33	0.287584
$940$	0	0
$941$	7.69275e33	0.489554	0.244777	−	0.969579i	$-0.421285\pi$
$941$	7.69275e33	0.489554	0.244777	+	0.969579i	$0.421285\pi$
$942$	0	0
$943$	5.02567e33	0.312111
$944$	0	0
$945$	−5.81813e33	−0.352629
$946$	0	0
$947$	2.94332e34	1.74106	0.870528	−	0.492119i	$-0.163778\pi$
$947$	2.94332e34	1.74106	0.870528	+	0.492119i	$0.163778\pi$
$948$	0	0
$949$	1.25481e34	0.724465
$950$	0	0
$951$	−5.35651e33	−0.301860
$952$	0	0
$953$	1.97292e34	1.08528	0.542641	−	0.839965i	$-0.317425\pi$
$953$	1.97292e34	1.08528	0.542641	+	0.839965i	$0.317425\pi$
$954$	0	0
$955$	−3.25825e34	−1.74963
$956$	0	0
$957$	−2.01734e33	−0.105753
$958$	0	0
$959$	−1.69210e34	−0.865989
$960$	0	0
$961$	−1.42003e33	−0.0709542
$962$	0	0
$963$	−3.42163e33	−0.166929
$964$	0	0
$965$	−1.34672e34	−0.641524
$966$	0	0
$967$	2.03266e34	0.945496	0.472748	−	0.881198i	$-0.343262\pi$
$967$	2.03266e34	0.945496	0.472748	+	0.881198i	$0.343262\pi$
$968$	0	0
$969$	−5.66820e34	−2.57467
$970$	0	0
$971$	−3.40470e34	−1.51028	−0.755141	−	0.655563i	$-0.772431\pi$
$971$	−3.40470e34	−1.51028	−0.755141	+	0.655563i	$0.772431\pi$
$972$	0	0
$973$	−1.89552e33	−0.0821166
$974$	0	0
$975$	−1.23156e33	−0.0521077
$976$	0	0
$977$	−3.68167e32	−0.0152145	−0.00760725	−	0.999971i	$-0.502421\pi$
$977$	−3.68167e32	−0.0152145	−0.00760725	+	0.999971i	$0.502421\pi$
$978$	0	0
$979$	−1.15638e33	−0.0466767
$980$	0	0
$981$	−3.16740e33	−0.124885
$982$	0	0
$983$	8.23724e33	0.317261	0.158630	−	0.987338i	$-0.449292\pi$
$983$	8.23724e33	0.317261	0.158630	+	0.987338i	$0.449292\pi$
$984$	0	0
$985$	−4.09707e34	−1.54155
$986$	0	0
$987$	1.28511e34	0.472380
$988$	0	0
$989$	−7.85000e32	−0.0281911
$990$	0	0
$991$	−2.70349e33	−0.0948590	−0.0474295	−	0.998875i	$-0.515103\pi$
$991$	−2.70349e33	−0.0948590	−0.0474295	+	0.998875i	$0.515103\pi$
$992$	0	0
$993$	6.31875e34	2.16628
$994$	0	0
$995$	1.31969e34	0.442086
$996$	0	0
$997$	−4.64692e33	−0.152115	−0.0760573	−	0.997103i	$-0.524233\pi$
$997$	−4.64692e33	−0.152115	−0.0760573	+	0.997103i	$0.524233\pi$
$998$	0	0
$999$	−1.79982e34	−0.575740

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.24.a.b.1.1		2
4.3	odd	2		1.24.a.a.1.1	✓	2
8.3	odd	2		64.24.a.d.1.1		2
8.5	even	2		64.24.a.g.1.2		2
12.11	even	2		9.24.a.b.1.2		2
20.3	even	4		25.24.b.a.24.3		4
20.7	even	4		25.24.b.a.24.2		4
20.19	odd	2		25.24.a.a.1.2		2
28.27	even	2		49.24.a.b.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1.24.a.a.1.1	✓	2	4.3	odd	2
9.24.a.b.1.2		2	12.11	even	2
16.24.a.b.1.1		2	1.1	even	1	trivial
25.24.a.a.1.2		2	20.19	odd	2
25.24.b.a.24.2		4	20.7	even	4
25.24.b.a.24.3		4	20.3	even	4
49.24.a.b.1.1		2	28.27	even	2
64.24.a.d.1.1		2	8.3	odd	2
64.24.a.g.1.2		2	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 24 \)
Character orbit:	\([\chi]\)	\(=\)	16.a (trivial)

\(f(q)\)	\(=\)	\(q-388445. q^{3} +1.05062e8 q^{5} -3.81217e9 q^{7} +5.67462e10 q^{9} +O(q^{10})\)	\(q-388445. q^{3} +1.05062e8 q^{5} -3.81217e9 q^{7} +5.67462e10 q^{9} -2.52200e11 q^{11} -3.59099e12 q^{13} -4.08108e13 q^{15} +2.34190e14 q^{17} +6.23086e14 q^{19} +1.48082e15 q^{21} +3.58786e15 q^{23} -8.82898e14 q^{25} +1.45267e16 q^{27} -2.05923e16 q^{29} -1.36357e17 q^{31} +9.79656e16 q^{33} -4.00514e17 q^{35} -1.23898e18 q^{37} +1.39490e18 q^{39} +1.40074e18 q^{41} -2.18793e17 q^{43} +5.96187e18 q^{45} +8.67836e18 q^{47} -1.28361e19 q^{49} -9.09698e19 q^{51} -7.63436e19 q^{53} -2.64966e19 q^{55} -2.42034e20 q^{57} +1.01862e18 q^{59} +2.87337e20 q^{61} -2.16326e20 q^{63} -3.77277e20 q^{65} -1.47683e21 q^{67} -1.39369e21 q^{69} -7.64346e20 q^{71} -3.49433e21 q^{73} +3.42957e20 q^{75} +9.61427e20 q^{77} -1.02350e22 q^{79} -1.09851e22 q^{81} +7.71597e21 q^{83} +2.46044e22 q^{85} +7.99897e21 q^{87} +4.58518e21 q^{89} +1.36895e22 q^{91} +5.29672e22 q^{93} +6.54627e22 q^{95} -1.13703e23 q^{97} -1.43114e22 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 339480 q^{3} + 73069020 q^{5} + 1359184400 q^{7} - 34999394166 q^{9}+O(q^{10}) \) 2 * q - 339480 * q^3 + 73069020 * q^5 + 1359184400 * q^7 - 34999394166 * q^9	\( 2 q - 339480 q^{3} + 73069020 q^{5} + 1359184400 q^{7} - 34999394166 q^{9} - 856801968264 q^{11} + 4376109322060 q^{13} - 42377338985040 q^{15} + 254028147597540 q^{17} - 4260600979960 q^{19} + 17\!\cdots\!44 q^{21}+ \cdots + 41\!\cdots\!12 q^{99}+O(q^{100}) \) 2 * q - 339480 * q^3 + 73069020 * q^5 + 1359184400 * q^7 - 34999394166 * q^9 - 856801968264 * q^11 + 4376109322060 * q^13 - 42377338985040 * q^15 + 254028147597540 * q^17 - 4260600979960 * q^19 + 1734031637722944 * q^21 + 8144713079008560 * q^23 - 11780274628800850 * q^25 + 5424634982716560 * q^27 + 20818433601623340 * q^29 - 137714017177000384 * q^31 + 68361366766001760 * q^33 - 565961271250425120 * q^35 - 897721264408967780 * q^37 + 1785011473665029232 * q^39 - 2294435477168314956 * q^41 + 1750760768619855800 * q^43 + 8897092690294206540 * q^45 - 15759744217656780960 * q^47 - 13461981704376200814 * q^49 - 89998362845078292144 * q^51 - 140287253401646796420 * q^53 - 7153550955060182640 * q^55 - 272752401448627175520 * q^57 - 280872989971340771880 * q^59 - 180452892516502223636 * q^61 - 690775113933935014320 * q^63 - 632168834809440380760 * q^65 - 1754233163431557625240 * q^67 - 1170560672172404223552 * q^69 - 3055033510194143328624 * q^71 - 8063408253877606149260 * q^73 - 190631089350520885800 * q^75 - 2165184764357449665600 * q^77 - 6244916814559639980640 * q^79 - 2793528580929833975598 * q^81 - 6875994082418498976120 * q^83 + 23969743087870314902520 * q^85 + 10026640653837674384880 * q^87 + 6395093086173070004820 * q^89 + 54890178162704560146016 * q^91 + 52900811441357852079360 * q^93 + 85533361066700858502000 * q^95 - 31147288846254030500540 * q^97 + 41158245132312135981912 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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