LMFDB - Embedded newform 80.22.a.d.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [80,22,Mod(1,80)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("80.1"); S:= CuspForms(chi, 22); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(80, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 22, names="a")

Level:	$ N $	$=$	$ 80 = 2^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 22 $
Character orbit:	$[\chi]$	$=$	80.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,126692] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$223.581875430$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{157921}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 39480 $ x^2 - x - 39480
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{4}\cdot 5\cdot 7 $
Twist minimal:	no (minimal twist has level 10)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-198.196$ of defining polynomial
Character	$\chi$	$=$	80.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+174616. q^{3} -9.76562e6 q^{5} +9.60462e8 q^{7} +2.00304e10 q^{9} -8.54661e10 q^{11} +9.74302e11 q^{13} -1.70523e12 q^{15} -1.17045e13 q^{17} -1.41128e13 q^{19} +1.67712e14 q^{21} +2.65862e13 q^{23} +9.53674e13 q^{25} +1.67108e15 q^{27} +1.45075e15 q^{29} +7.63253e15 q^{31} -1.49237e16 q^{33} -9.37951e15 q^{35} +1.09912e16 q^{37} +1.70129e17 q^{39} +7.93639e16 q^{41} +8.36751e16 q^{43} -1.95609e17 q^{45} -3.57127e17 q^{47} +3.63941e17 q^{49} -2.04379e18 q^{51} -8.17016e17 q^{53} +8.34630e17 q^{55} -2.46432e18 q^{57} -8.21011e17 q^{59} +4.53683e18 q^{61} +1.92384e19 q^{63} -9.51466e18 q^{65} -8.02826e18 q^{67} +4.64238e18 q^{69} +5.25728e19 q^{71} +9.28698e18 q^{73} +1.66527e19 q^{75} -8.20869e19 q^{77} -1.08861e19 q^{79} +8.22724e19 q^{81} +4.79994e19 q^{83} +1.14302e20 q^{85} +2.53325e20 q^{87} +2.35270e20 q^{89} +9.35779e20 q^{91} +1.33276e21 q^{93} +1.37820e20 q^{95} +7.25967e20 q^{97} -1.71192e21 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 126692 q^{3} - 19531250 q^{5} + 292598684 q^{7} + 11866737826 q^{9} + 41326831776 q^{11} + 1887051472228 q^{13} - 1237226562500 q^{15} - 3100413932364 q^{17} - 7175794652440 q^{19} + 199718603894264 q^{21}+ \cdots - 27\!\cdots\!12 q^{99}+O(q^{100}) $ 2 * q + 126692 * q^3 - 19531250 * q^5 + 292598684 * q^7 + 11866737826 * q^9 + 41326831776 * q^11 + 1887051472228 * q^13 - 1237226562500 * q^15 - 3100413932364 * q^17 - 7175794652440 * q^19 + 199718603894264 * q^21 + 357559990100052 * q^23 + 190734863281250 * q^25 + 2563616231988920 * q^27 - 2454584202185940 * q^29 + 4310677151243336 * q^31 - 21000167846983104 * q^33 - 2857409023437500 * q^35 - 32389201650205724 * q^37 + 126386038156081288 * q^39 - 16298035212712116 * q^41 + 161845702253479412 * q^43 - 115886111582031250 * q^45 - 153492993930726996 * q^47 + 251436016683357114 * q^49 - 2456129297437264344 * q^51 + 604198357317820308 * q^53 - 403582341562500000 * q^55 - 2796771718696561840 * q^57 - 3110197357974672120 * q^59 - 791142209760451676 * q^61 + 24690614624666969692 * q^63 - 18428237033476562500 * q^65 + 5111819523383561564 * q^67 - 11219196977510717208 * q^69 + 82388741313238482456 * q^71 + 10657918464992348548 * q^73 + 12082290649414062500 * q^75 - 166767237207340703808 * q^77 - 28314166125451369360 * q^79 + 124893157714537187242 * q^81 + 10427725705926651732 * q^83 + 30277479808242187500 * q^85 + 440484049640705943960 * q^87 + 478501472814113520180 * q^89 + 326187611445407942776 * q^91 + 1491957513759228855056 * q^93 + 70076119652734375000 * q^95 + 545226897779926992196 * q^97 - 2747011580657571215712 * q^99

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
$3$	174616.	1.70730	0.853652	−	0.520844i	$-0.174383\pi$
$3$	174616.	1.70730	0.853652	+	0.520844i	$0.174383\pi$
$4$	0	0
$5$	−9.76562e6	−0.447214
$5$	−9.76562e6	−0.447214
$6$	0	0
$7$	9.60462e8	1.28514	0.642570	−	0.766227i	$-0.277868\pi$
$7$	9.60462e8	1.28514	0.642570	+	0.766227i	$0.277868\pi$
$8$	0	0
$9$	2.00304e10	1.91489
$10$	0	0
$11$	−8.54661e10	−0.993507	−0.496753	−	0.867892i	$-0.665475\pi$
$11$	−8.54661e10	−0.993507	−0.496753	+	0.867892i	$0.665475\pi$
$12$	0	0
$13$	9.74302e11	1.96015	0.980073	−	0.198640i	$-0.0636523\pi$
$13$	9.74302e11	1.96015	0.980073	+	0.198640i	$0.0636523\pi$
$14$	0	0
$15$	−1.70523e12	−0.763529
$16$	0	0
$17$	−1.17045e13	−1.40812	−0.704058	−	0.710143i	$-0.748630\pi$
$17$	−1.17045e13	−1.40812	−0.704058	+	0.710143i	$0.748630\pi$
$18$	0	0
$19$	−1.41128e13	−0.528081	−0.264041	−	0.964512i	$-0.585055\pi$
$19$	−1.41128e13	−0.528081	−0.264041	+	0.964512i	$0.585055\pi$
$20$	0	0
$21$	1.67712e14	2.19413
$22$	0	0
$23$	2.65862e13	0.133818	0.0669090	−	0.997759i	$-0.478686\pi$
$23$	2.65862e13	0.133818	0.0669090	+	0.997759i	$0.478686\pi$
$24$	0	0
$25$	9.53674e13	0.200000
$26$	0	0
$27$	1.67108e15	1.56199
$28$	0	0
$29$	1.45075e15	0.640346	0.320173	−	0.947359i	$-0.396259\pi$
$29$	1.45075e15	0.640346	0.320173	+	0.947359i	$0.396259\pi$
$30$	0	0
$31$	7.63253e15	1.67252	0.836259	−	0.548335i	$-0.184738\pi$
$31$	7.63253e15	1.67252	0.836259	+	0.548335i	$0.184738\pi$
$32$	0	0
$33$	−1.49237e16	−1.69622
$34$	0	0
$35$	−9.37951e15	−0.574732
$36$	0	0
$37$	1.09912e16	0.375773	0.187886	−	0.982191i	$-0.439836\pi$
$37$	1.09912e16	0.375773	0.187886	+	0.982191i	$0.439836\pi$
$38$	0	0
$39$	1.70129e17	3.34656
$40$	0	0
$41$	7.93639e16	0.923406	0.461703	−	0.887035i	$-0.347239\pi$
$41$	7.93639e16	0.923406	0.461703	+	0.887035i	$0.347239\pi$
$42$	0	0
$43$	8.36751e16	0.590442	0.295221	−	0.955429i	$-0.404607\pi$
$43$	8.36751e16	0.590442	0.295221	+	0.955429i	$0.404607\pi$
$44$	0	0
$45$	−1.95609e17	−0.856363
$46$	0	0
$47$	−3.57127e17	−0.990365	−0.495183	−	0.868789i	$-0.664899\pi$
$47$	−3.57127e17	−0.990365	−0.495183	+	0.868789i	$0.664899\pi$
$48$	0	0
$49$	3.63941e17	0.651586
$50$	0	0
$51$	−2.04379e18	−2.40408
$52$	0	0
$53$	−8.17016e17	−0.641702	−0.320851	−	0.947130i	$-0.603969\pi$
$53$	−8.17016e17	−0.641702	−0.320851	+	0.947130i	$0.603969\pi$
$54$	0	0
$55$	8.34630e17	0.444310
$56$	0	0
$57$	−2.46432e18	−0.901595
$58$	0	0
$59$	−8.21011e17	−0.209124	−0.104562	−	0.994518i	$-0.533344\pi$
$59$	−8.21011e17	−0.209124	−0.104562	+	0.994518i	$0.533344\pi$
$60$	0	0
$61$	4.53683e18	0.814310	0.407155	−	0.913359i	$-0.366521\pi$
$61$	4.53683e18	0.814310	0.407155	+	0.913359i	$0.366521\pi$
$62$	0	0
$63$	1.92384e19	2.46090
$64$	0	0
$65$	−9.51466e18	−0.876604
$66$	0	0
$67$	−8.02826e18	−0.538066	−0.269033	−	0.963131i	$-0.586704\pi$
$67$	−8.02826e18	−0.538066	−0.269033	+	0.963131i	$0.586704\pi$
$68$	0	0
$69$	4.64238e18	0.228468
$70$	0	0
$71$	5.25728e19	1.91668	0.958338	−	0.285638i	$-0.0922056\pi$
$71$	5.25728e19	1.91668	0.958338	+	0.285638i	$0.0922056\pi$
$72$	0	0
$73$	9.28698e18	0.252921	0.126460	−	0.991972i	$-0.459638\pi$
$73$	9.28698e18	0.252921	0.126460	+	0.991972i	$0.459638\pi$
$74$	0	0
$75$	1.66527e19	0.341461
$76$	0	0
$77$	−8.20869e19	−1.27680
$78$	0	0
$79$	−1.08861e19	−0.129356	−0.0646781	−	0.997906i	$-0.520602\pi$
$79$	−1.08861e19	−0.129356	−0.0646781	+	0.997906i	$0.520602\pi$
$80$	0	0
$81$	8.22724e19	0.751902
$82$	0	0
$83$	4.79994e19	0.339560	0.169780	−	0.985482i	$-0.445694\pi$
$83$	4.79994e19	0.339560	0.169780	+	0.985482i	$0.445694\pi$
$84$	0	0
$85$	1.14302e20	0.629728
$86$	0	0
$87$	2.53325e20	1.09327
$88$	0	0
$89$	2.35270e20	0.799783	0.399891	−	0.916563i	$-0.369048\pi$
$89$	2.35270e20	0.799783	0.399891	+	0.916563i	$0.369048\pi$
$90$	0	0
$91$	9.35779e20	2.51906
$92$	0	0
$93$	1.33276e21	2.85549
$94$	0	0
$95$	1.37820e20	0.236165
$96$	0	0
$97$	7.25967e20	0.999571	0.499786	−	0.866149i	$-0.333412\pi$
$97$	7.25967e20	0.999571	0.499786	+	0.866149i	$0.333412\pi$
$98$	0	0
$99$	−1.71192e21	−1.90245
$100$	0	0
$101$	3.23940e20	0.291803	0.145902	−	0.989299i	$-0.453392\pi$
$101$	3.23940e20	0.291803	0.145902	+	0.989299i	$0.453392\pi$
$102$	0	0
$103$	−5.76074e20	−0.422364	−0.211182	−	0.977447i	$-0.567731\pi$
$103$	−5.76074e20	−0.422364	−0.211182	+	0.977447i	$0.567731\pi$
$104$	0	0
$105$	−1.63781e21	−0.981243
$106$	0	0
$107$	3.09648e21	1.52174	0.760868	−	0.648907i	$-0.224774\pi$
$107$	3.09648e21	1.52174	0.760868	+	0.648907i	$0.224774\pi$
$108$	0	0
$109$	−2.84500e21	−1.15108	−0.575539	−	0.817775i	$-0.695208\pi$
$109$	−2.84500e21	−1.15108	−0.575539	+	0.817775i	$0.695208\pi$
$110$	0	0
$111$	1.91923e21	0.641558
$112$	0	0
$113$	−5.61097e21	−1.55494	−0.777471	−	0.628919i	$-0.783498\pi$
$113$	−5.61097e21	−1.55494	−0.777471	+	0.628919i	$0.783498\pi$
$114$	0	0
$115$	−2.59631e20	−0.0598452
$116$	0	0
$117$	1.95156e22	3.75345
$118$	0	0
$119$	−1.12417e22	−1.80963
$120$	0	0
$121$	−9.57949e19	−0.0129448
$122$	0	0
$123$	1.38582e22	1.57653
$124$	0	0
$125$	−9.31323e20	−0.0894427
$126$	0	0
$127$	−1.60366e22	−1.30368	−0.651842	−	0.758355i	$-0.726003\pi$
$127$	−1.60366e22	−1.30368	−0.651842	+	0.758355i	$0.726003\pi$
$128$	0	0
$129$	1.46110e22	1.00806
$130$	0	0
$131$	1.39854e22	0.820969	0.410485	−	0.911868i	$-0.365360\pi$
$131$	1.39854e22	0.820969	0.410485	+	0.911868i	$0.365360\pi$
$132$	0	0
$133$	−1.35548e22	−0.678659
$134$	0	0
$135$	−1.63191e22	−0.698542
$136$	0	0
$137$	−3.49305e22	−1.28127	−0.640633	−	0.767847i	$-0.721328\pi$
$137$	−3.49305e22	−1.28127	−0.640633	+	0.767847i	$0.721328\pi$
$138$	0	0
$139$	1.16733e22	0.367737	0.183868	−	0.982951i	$-0.441138\pi$
$139$	1.16733e22	0.367737	0.183868	+	0.982951i	$0.441138\pi$
$140$	0	0
$141$	−6.23601e22	−1.69085
$142$	0	0
$143$	−8.32698e22	−1.94742
$144$	0	0
$145$	−1.41675e22	−0.286371
$146$	0	0
$147$	6.35499e22	1.11246
$148$	0	0
$149$	−2.99897e22	−0.455530	−0.227765	−	0.973716i	$-0.573142\pi$
$149$	−2.99897e22	−0.455530	−0.227765	+	0.973716i	$0.573142\pi$
$150$	0	0
$151$	8.80845e22	1.16317	0.581583	−	0.813487i	$-0.302434\pi$
$151$	8.80845e22	1.16317	0.581583	+	0.813487i	$0.302434\pi$
$152$	0	0
$153$	−2.34445e23	−2.69638
$154$	0	0
$155$	−7.45364e22	−0.747972
$156$	0	0
$157$	1.76504e23	1.54813	0.774067	−	0.633104i	$-0.218219\pi$
$157$	1.76504e23	1.54813	0.774067	+	0.633104i	$0.218219\pi$
$158$	0	0
$159$	−1.42664e23	−1.09558
$160$	0	0
$161$	2.55351e22	0.171975
$162$	0	0
$163$	−9.36633e22	−0.554115	−0.277057	−	0.960853i	$-0.589359\pi$
$163$	−9.36633e22	−0.554115	−0.277057	+	0.960853i	$0.589359\pi$
$164$	0	0
$165$	1.45740e23	0.758571
$166$	0	0
$167$	2.77263e23	1.27166	0.635828	−	0.771831i	$-0.280659\pi$
$167$	2.77263e23	1.27166	0.635828	+	0.771831i	$0.280659\pi$
$168$	0	0
$169$	7.02199e23	2.84217
$170$	0	0
$171$	−2.82685e23	−1.01122
$172$	0	0
$173$	−3.84494e23	−1.21732	−0.608661	−	0.793431i	$-0.708293\pi$
$173$	−3.84494e23	−1.21732	−0.608661	+	0.793431i	$0.708293\pi$
$174$	0	0
$175$	9.15968e22	0.257028
$176$	0	0
$177$	−1.43362e23	−0.357037
$178$	0	0
$179$	−4.54367e22	−0.100566	−0.0502828	−	0.998735i	$-0.516012\pi$
$179$	−4.54367e22	−0.100566	−0.0502828	+	0.998735i	$0.516012\pi$
$180$	0	0
$181$	8.61960e23	1.69770	0.848852	−	0.528630i	$-0.177294\pi$
$181$	8.61960e23	1.69770	0.848852	+	0.528630i	$0.177294\pi$
$182$	0	0
$183$	7.92204e23	1.39027
$184$	0	0
$185$	−1.07336e23	−0.168051
$186$	0	0
$187$	1.00034e24	1.39897
$188$	0	0
$189$	1.60501e24	2.00737
$190$	0	0
$191$	2.31575e23	0.259323	0.129661	−	0.991558i	$-0.458611\pi$
$191$	2.31575e23	0.259323	0.129661	+	0.991558i	$0.458611\pi$
$192$	0	0
$193$	−1.13487e24	−1.13919	−0.569595	−	0.821926i	$-0.692900\pi$
$193$	−1.13487e24	−1.13919	−0.569595	+	0.821926i	$0.692900\pi$
$194$	0	0
$195$	−1.66141e24	−1.49663
$196$	0	0
$197$	−6.29771e23	−0.509668	−0.254834	−	0.966985i	$-0.582021\pi$
$197$	−6.29771e23	−0.509668	−0.254834	+	0.966985i	$0.582021\pi$
$198$	0	0
$199$	7.13426e23	0.519268	0.259634	−	0.965707i	$-0.416398\pi$
$199$	7.13426e23	0.519268	0.259634	+	0.965707i	$0.416398\pi$
$200$	0	0
$201$	−1.40186e24	−0.918642
$202$	0	0
$203$	1.39339e24	0.822935
$204$	0	0
$205$	−7.75039e23	−0.412960
$206$	0	0
$207$	5.32532e23	0.256246
$208$	0	0
$209$	1.20617e24	0.524652
$210$	0	0
$211$	2.57567e24	1.01374	0.506868	−	0.862024i	$-0.330803\pi$
$211$	2.57567e24	1.01374	0.506868	+	0.862024i	$0.330803\pi$
$212$	0	0
$213$	9.18005e24	3.27235
$214$	0	0
$215$	−8.17140e23	−0.264054
$216$	0	0
$217$	7.33075e24	2.14942
$218$	0	0
$219$	1.62165e24	0.431812
$220$	0	0
$221$	−1.14037e25	−2.76011
$222$	0	0
$223$	7.24650e24	1.59561	0.797806	−	0.602914i	$-0.205994\pi$
$223$	7.24650e24	1.59561	0.797806	+	0.602914i	$0.205994\pi$
$224$	0	0
$225$	1.91025e24	0.382977
$226$	0	0
$227$	1.55126e24	0.283409	0.141704	−	0.989909i	$-0.454742\pi$
$227$	1.55126e24	0.283409	0.141704	+	0.989909i	$0.454742\pi$
$228$	0	0
$229$	−9.95437e24	−1.65860	−0.829299	−	0.558805i	$-0.811260\pi$
$229$	−9.95437e24	−1.65860	−0.829299	+	0.558805i	$0.811260\pi$
$230$	0	0
$231$	−1.43337e25	−2.17988
$232$	0	0
$233$	−4.62738e23	−0.0642833	−0.0321417	−	0.999483i	$-0.510233\pi$
$233$	−4.62738e23	−0.0642833	−0.0321417	+	0.999483i	$0.510233\pi$
$234$	0	0
$235$	3.48757e24	0.442905
$236$	0	0
$237$	−1.90088e24	−0.220850
$238$	0	0
$239$	−1.06208e25	−1.12975	−0.564873	−	0.825178i	$-0.691075\pi$
$239$	−1.06208e25	−1.12975	−0.564873	+	0.825178i	$0.691075\pi$
$240$	0	0
$241$	1.67701e24	0.163439	0.0817195	−	0.996655i	$-0.473959\pi$
$241$	1.67701e24	0.163439	0.0817195	+	0.996655i	$0.473959\pi$
$242$	0	0
$243$	−3.11401e24	−0.278262
$244$	0	0
$245$	−3.55411e24	−0.291398
$246$	0	0
$247$	−1.37501e25	−1.03512
$248$	0	0
$249$	8.38147e24	0.579732
$250$	0	0
$251$	2.15807e25	1.37243	0.686216	−	0.727398i	$-0.259270\pi$
$251$	2.15807e25	1.37243	0.686216	+	0.727398i	$0.259270\pi$
$252$	0	0
$253$	−2.27222e24	−0.132949
$254$	0	0
$255$	1.99589e25	1.07514
$256$	0	0
$257$	7.72467e24	0.383338	0.191669	−	0.981460i	$-0.438610\pi$
$257$	7.72467e24	0.383338	0.191669	+	0.981460i	$0.438610\pi$
$258$	0	0
$259$	1.05566e25	0.482921
$260$	0	0
$261$	2.90592e25	1.22619
$262$	0	0
$263$	−1.35616e25	−0.528174	−0.264087	−	0.964499i	$-0.585071\pi$
$263$	−1.35616e25	−0.528174	−0.264087	+	0.964499i	$0.585071\pi$
$264$	0	0
$265$	7.97867e24	0.286978
$266$	0	0
$267$	4.10819e25	1.36547
$268$	0	0
$269$	−2.23157e25	−0.685822	−0.342911	−	0.939368i	$-0.611413\pi$
$269$	−2.23157e25	−0.685822	−0.342911	+	0.939368i	$0.611413\pi$
$270$	0	0
$271$	−8.79105e24	−0.249956	−0.124978	−	0.992160i	$-0.539886\pi$
$271$	−8.79105e24	−0.249956	−0.124978	+	0.992160i	$0.539886\pi$
$272$	0	0
$273$	1.63402e26	4.30080
$274$	0	0
$275$	−8.15068e24	−0.198701
$276$	0	0
$277$	3.23947e25	0.731874	0.365937	−	0.930640i	$-0.380749\pi$
$277$	3.23947e25	0.731874	0.365937	+	0.930640i	$0.380749\pi$
$278$	0	0
$279$	1.52882e26	3.20268
$280$	0	0
$281$	1.62934e25	0.316662	0.158331	−	0.987386i	$-0.449389\pi$
$281$	1.62934e25	0.316662	0.158331	+	0.987386i	$0.449389\pi$
$282$	0	0
$283$	−7.24900e25	−1.30774	−0.653869	−	0.756608i	$-0.726855\pi$
$283$	−7.24900e25	−1.30774	−0.653869	+	0.756608i	$0.726855\pi$
$284$	0	0
$285$	2.40656e25	0.403206
$286$	0	0
$287$	7.62260e25	1.18671
$288$	0	0
$289$	6.79028e25	0.982790
$290$	0	0
$291$	1.26765e26	1.70657
$292$	0	0
$293$	−7.90881e25	−0.990835	−0.495418	−	0.868655i	$-0.664985\pi$
$293$	−7.90881e25	−0.990835	−0.495418	+	0.868655i	$0.664985\pi$
$294$	0	0
$295$	8.01769e24	0.0935229
$296$	0	0
$297$	−1.42821e26	−1.55185
$298$	0	0
$299$	2.59030e25	0.262303
$300$	0	0
$301$	8.03667e25	0.758801
$302$	0	0
$303$	5.65651e25	0.498197
$304$	0	0
$305$	−4.43050e25	−0.364170
$306$	0	0
$307$	1.19014e26	0.913363	0.456682	−	0.889630i	$-0.349038\pi$
$307$	1.19014e26	0.913363	0.456682	+	0.889630i	$0.349038\pi$
$308$	0	0
$309$	−1.00592e26	−0.721104
$310$	0	0
$311$	1.66399e26	1.11472	0.557360	−	0.830271i	$-0.311814\pi$
$311$	1.66399e26	1.11472	0.557360	+	0.830271i	$0.311814\pi$
$312$	0	0
$313$	2.24533e25	0.140626	0.0703128	−	0.997525i	$-0.477600\pi$
$313$	2.24533e25	0.140626	0.0703128	+	0.997525i	$0.477600\pi$
$314$	0	0
$315$	−1.87875e26	−1.10055
$316$	0	0
$317$	1.10755e26	0.607076	0.303538	−	0.952819i	$-0.401832\pi$
$317$	1.10755e26	0.607076	0.303538	+	0.952819i	$0.401832\pi$
$318$	0	0
$319$	−1.23990e26	−0.636188
$320$	0	0
$321$	5.40695e26	2.59806
$322$	0	0
$323$	1.65183e26	0.743599
$324$	0	0
$325$	9.29166e25	0.392029
$326$	0	0
$327$	−4.96783e26	−1.96524
$328$	0	0
$329$	−3.43007e26	−1.27276
$330$	0	0
$331$	2.18220e26	0.759803	0.379901	−	0.925027i	$-0.375958\pi$
$331$	2.18220e26	0.759803	0.379901	+	0.925027i	$0.375958\pi$
$332$	0	0
$333$	2.20157e26	0.719562
$334$	0	0
$335$	7.84009e25	0.240631
$336$	0	0
$337$	5.33804e26	1.53910	0.769551	−	0.638585i	$-0.220480\pi$
$337$	5.33804e26	1.53910	0.769551	+	0.638585i	$0.220480\pi$
$338$	0	0
$339$	−9.79765e26	−2.65476
$340$	0	0
$341$	−6.52322e26	−1.66166
$342$	0	0
$343$	−1.86911e26	−0.447761
$344$	0	0
$345$	−4.53357e25	−0.102174
$346$	0	0
$347$	4.09971e26	0.869548	0.434774	−	0.900540i	$-0.356828\pi$
$347$	4.09971e26	0.869548	0.434774	+	0.900540i	$0.356828\pi$
$348$	0	0
$349$	2.06527e26	0.412392	0.206196	−	0.978511i	$-0.433892\pi$
$349$	2.06527e26	0.412392	0.206196	+	0.978511i	$0.433892\pi$
$350$	0	0
$351$	1.62814e27	3.06172
$352$	0	0
$353$	2.63582e26	0.466961	0.233481	−	0.972361i	$-0.424988\pi$
$353$	2.63582e26	0.466961	0.233481	+	0.972361i	$0.424988\pi$
$354$	0	0
$355$	−5.13406e26	−0.857163
$356$	0	0
$357$	−1.96298e27	−3.08958
$358$	0	0
$359$	−2.72697e26	−0.404751	−0.202376	−	0.979308i	$-0.564866\pi$
$359$	−2.72697e26	−0.404751	−0.202376	+	0.979308i	$0.564866\pi$
$360$	0	0
$361$	−5.15038e26	−0.721130
$362$	0	0
$363$	−1.67273e25	−0.0221007
$364$	0	0
$365$	−9.06931e25	−0.113110
$366$	0	0
$367$	1.67539e26	0.197297	0.0986487	−	0.995122i	$-0.468548\pi$
$367$	1.67539e26	0.197297	0.0986487	+	0.995122i	$0.468548\pi$
$368$	0	0
$369$	1.58969e27	1.76822
$370$	0	0
$371$	−7.84712e26	−0.824678
$372$	0	0
$373$	9.92406e26	0.985704	0.492852	−	0.870113i	$-0.335954\pi$
$373$	9.92406e26	0.985704	0.492852	+	0.870113i	$0.335954\pi$
$374$	0	0
$375$	−1.62624e26	−0.152706
$376$	0	0
$377$	1.41347e27	1.25517
$378$	0	0
$379$	−1.41427e25	−0.0118801	−0.00594007	−	0.999982i	$-0.501891\pi$
$379$	−1.41427e25	−0.0118801	−0.00594007	+	0.999982i	$0.501891\pi$
$380$	0	0
$381$	−2.80024e27	−2.22578
$382$	0	0
$383$	−2.51013e26	−0.188847	−0.0944234	−	0.995532i	$-0.530101\pi$
$383$	−2.51013e26	−0.188847	−0.0944234	+	0.995532i	$0.530101\pi$
$384$	0	0
$385$	8.01630e26	0.571000
$386$	0	0
$387$	1.67604e27	1.13063
$388$	0	0
$389$	−2.50259e27	−1.59926	−0.799629	−	0.600495i	$-0.794970\pi$
$389$	−2.50259e27	−1.59926	−0.799629	+	0.600495i	$0.794970\pi$
$390$	0	0
$391$	−3.11178e26	−0.188431
$392$	0	0
$393$	2.44208e27	1.40164
$394$	0	0
$395$	1.06309e26	0.0578498
$396$	0	0
$397$	3.29042e26	0.169805	0.0849027	−	0.996389i	$-0.472942\pi$
$397$	3.29042e26	0.169805	0.0849027	+	0.996389i	$0.472942\pi$
$398$	0	0
$399$	−2.36689e27	−1.15868
$400$	0	0
$401$	8.45309e26	0.392645	0.196322	−	0.980539i	$-0.437100\pi$
$401$	8.45309e26	0.392645	0.196322	+	0.980539i	$0.437100\pi$
$402$	0	0
$403$	7.43638e27	3.27838
$404$	0	0
$405$	−8.03441e26	−0.336261
$406$	0	0
$407$	−9.39371e26	−0.373333
$408$	0	0
$409$	−1.58724e27	−0.599167	−0.299583	−	0.954070i	$-0.596848\pi$
$409$	−1.58724e27	−0.599167	−0.299583	+	0.954070i	$0.596848\pi$
$410$	0	0
$411$	−6.09943e27	−2.18751
$412$	0	0
$413$	−7.88550e26	−0.268753
$414$	0	0
$415$	−4.68745e26	−0.151856
$416$	0	0
$417$	2.03834e27	0.627839
$418$	0	0
$419$	1.46560e25	0.00429308	0.00214654	−	0.999998i	$-0.499317\pi$
$419$	1.46560e25	0.00429308	0.00214654	+	0.999998i	$0.499317\pi$
$420$	0	0
$421$	−2.07639e27	−0.578559	−0.289280	−	0.957245i	$-0.593416\pi$
$421$	−2.07639e27	−0.578559	−0.289280	+	0.957245i	$0.593416\pi$
$422$	0	0
$423$	−7.15340e27	−1.89644
$424$	0	0
$425$	−1.11623e27	−0.281623
$426$	0	0
$427$	4.35746e27	1.04650
$428$	0	0
$429$	−1.45402e28	−3.32483
$430$	0	0
$431$	2.88896e27	0.629115	0.314558	−	0.949238i	$-0.398144\pi$
$431$	2.88896e27	0.629115	0.314558	+	0.949238i	$0.398144\pi$
$432$	0	0
$433$	6.58569e27	1.36609	0.683044	−	0.730377i	$-0.260656\pi$
$433$	6.58569e27	1.36609	0.683044	+	0.730377i	$0.260656\pi$
$434$	0	0
$435$	−2.47387e27	−0.488923
$436$	0	0
$437$	−3.75206e26	−0.0706668
$438$	0	0
$439$	9.33293e26	0.167548	0.0837742	−	0.996485i	$-0.473303\pi$
$439$	9.33293e26	0.167548	0.0837742	+	0.996485i	$0.473303\pi$
$440$	0	0
$441$	7.28987e27	1.24771
$442$	0	0
$443$	−9.69019e27	−1.58159	−0.790793	−	0.612083i	$-0.790332\pi$
$443$	−9.69019e27	−1.58159	−0.790793	+	0.612083i	$0.790332\pi$
$444$	0	0
$445$	−2.29756e27	−0.357674
$446$	0	0
$447$	−5.23669e27	−0.777728
$448$	0	0
$449$	−9.90883e27	−1.40422	−0.702111	−	0.712067i	$-0.747759\pi$
$449$	−9.90883e27	−1.40422	−0.702111	+	0.712067i	$0.747759\pi$
$450$	0	0
$451$	−6.78293e27	−0.917410
$452$	0	0
$453$	1.53810e28	1.98588
$454$	0	0
$455$	−9.13847e27	−1.12656
$456$	0	0
$457$	−1.41062e28	−1.66069	−0.830346	−	0.557248i	$-0.811857\pi$
$457$	−1.41062e28	−1.66069	−0.830346	+	0.557248i	$0.811857\pi$
$458$	0	0
$459$	−1.95591e28	−2.19946
$460$	0	0
$461$	1.40343e26	0.0150776	0.00753880	−	0.999972i	$-0.497600\pi$
$461$	1.40343e26	0.0150776	0.00753880	+	0.999972i	$0.497600\pi$
$462$	0	0
$463$	1.54202e28	1.58303	0.791514	−	0.611150i	$-0.209293\pi$
$463$	1.54202e28	1.58303	0.791514	+	0.611150i	$0.209293\pi$
$464$	0	0
$465$	−1.30152e28	−1.27702
$466$	0	0
$467$	−1.42166e28	−1.33343	−0.666713	−	0.745314i	$-0.732299\pi$
$467$	−1.42166e28	−1.33343	−0.666713	+	0.745314i	$0.732299\pi$
$468$	0	0
$469$	−7.71083e27	−0.691491
$470$	0	0
$471$	3.08204e28	2.64313
$472$	0	0
$473$	−7.15138e27	−0.586608
$474$	0	0
$475$	−1.34590e27	−0.105616
$476$	0	0
$477$	−1.63651e28	−1.22879
$478$	0	0
$479$	−1.82659e28	−1.31256	−0.656280	−	0.754518i	$-0.727871\pi$
$479$	−1.82659e28	−1.31256	−0.656280	+	0.754518i	$0.727871\pi$
$480$	0	0
$481$	1.07087e28	0.736569
$482$	0	0
$483$	4.45883e27	0.293613
$484$	0	0
$485$	−7.08952e27	−0.447022
$486$	0	0
$487$	−3.19696e28	−1.93056	−0.965280	−	0.261218i	$-0.915876\pi$
$487$	−3.19696e28	−1.93056	−0.965280	+	0.261218i	$0.915876\pi$
$488$	0	0
$489$	−1.63551e28	−0.946042
$490$	0	0
$491$	1.06661e28	0.591086	0.295543	−	0.955329i	$-0.404499\pi$
$491$	1.06661e28	0.591086	0.295543	+	0.955329i	$0.404499\pi$
$492$	0	0
$493$	−1.69803e28	−0.901681
$494$	0	0
$495$	1.67180e28	0.850802
$496$	0	0
$497$	5.04942e28	2.46320
$498$	0	0
$499$	1.19664e28	0.559638	0.279819	−	0.960053i	$-0.409725\pi$
$499$	1.19664e28	0.559638	0.279819	+	0.960053i	$0.409725\pi$
$500$	0	0
$501$	4.84146e28	2.17110
$502$	0	0
$503$	3.75074e28	1.61307	0.806534	−	0.591188i	$-0.201341\pi$
$503$	3.75074e28	1.61307	0.806534	+	0.591188i	$0.201341\pi$
$504$	0	0
$505$	−3.16348e27	−0.130498
$506$	0	0
$507$	1.22615e29	4.85245
$508$	0	0
$509$	3.27640e28	1.24411	0.622057	−	0.782972i	$-0.286297\pi$
$509$	3.27640e28	1.24411	0.622057	+	0.782972i	$0.286297\pi$
$510$	0	0
$511$	8.91979e27	0.325039
$512$	0	0
$513$	−2.35836e28	−0.824857
$514$	0	0
$515$	5.62572e27	0.188887
$516$	0	0
$517$	3.05223e28	0.983934
$518$	0	0
$519$	−6.71388e28	−2.07834
$520$	0	0
$521$	−5.49377e27	−0.163333	−0.0816667	−	0.996660i	$-0.526024\pi$
$521$	−5.49377e27	−0.163333	−0.0816667	+	0.996660i	$0.526024\pi$
$522$	0	0
$523$	−9.60628e27	−0.274339	−0.137170	−	0.990548i	$-0.543801\pi$
$523$	−9.60628e27	−0.274339	−0.137170	+	0.990548i	$0.543801\pi$
$524$	0	0
$525$	1.59943e28	0.438825
$526$	0	0
$527$	−8.93347e28	−2.35510
$528$	0	0
$529$	−3.87648e28	−0.982093
$530$	0	0
$531$	−1.64452e28	−0.400448
$532$	0	0
$533$	7.73244e28	1.81001
$534$	0	0
$535$	−3.02391e28	−0.680541
$536$	0	0
$537$	−7.93397e27	−0.171696
$538$	0	0
$539$	−3.11046e28	−0.647355
$540$	0	0
$541$	−7.34453e28	−1.47026	−0.735128	−	0.677929i	$-0.762878\pi$
$541$	−7.34453e28	−1.47026	−0.735128	+	0.677929i	$0.762878\pi$
$542$	0	0
$543$	1.50512e29	2.89850
$544$	0	0
$545$	2.77832e28	0.514777
$546$	0	0
$547$	−4.12363e28	−0.735212	−0.367606	−	0.929982i	$-0.619822\pi$
$547$	−4.12363e28	−0.735212	−0.367606	+	0.929982i	$0.619822\pi$
$548$	0	0
$549$	9.08745e28	1.55931
$550$	0	0
$551$	−2.04742e28	−0.338155
$552$	0	0
$553$	−1.04557e28	−0.166241
$554$	0	0
$555$	−1.87425e28	−0.286914
$556$	0	0
$557$	−8.83426e27	−0.130224	−0.0651119	−	0.997878i	$-0.520740\pi$
$557$	−8.83426e27	−0.130224	−0.0651119	+	0.997878i	$0.520740\pi$
$558$	0	0
$559$	8.15248e28	1.15735
$560$	0	0
$561$	1.74675e29	2.38847
$562$	0	0
$563$	1.06900e29	1.40811	0.704057	−	0.710143i	$-0.251370\pi$
$563$	1.06900e29	1.40811	0.704057	+	0.710143i	$0.251370\pi$
$564$	0	0
$565$	5.47946e28	0.695391
$566$	0	0
$567$	7.90195e28	0.966300
$568$	0	0
$569$	−1.27688e29	−1.50477	−0.752386	−	0.658723i	$-0.771097\pi$
$569$	−1.27688e29	−1.50477	−0.752386	+	0.658723i	$0.771097\pi$
$570$	0	0
$571$	8.59333e28	0.976073	0.488037	−	0.872823i	$-0.337713\pi$
$571$	8.59333e28	0.976073	0.488037	+	0.872823i	$0.337713\pi$
$572$	0	0
$573$	4.04366e28	0.442743
$574$	0	0
$575$	2.53546e27	0.0267636
$576$	0	0
$577$	−1.28637e29	−1.30925	−0.654623	−	0.755956i	$-0.727172\pi$
$577$	−1.28637e29	−1.30925	−0.654623	+	0.755956i	$0.727172\pi$
$578$	0	0
$579$	−1.98167e29	−1.94494
$580$	0	0
$581$	4.61016e28	0.436382
$582$	0	0
$583$	6.98271e28	0.637536
$584$	0	0
$585$	−1.90582e29	−1.67860
$586$	0	0
$587$	3.48294e28	0.295969	0.147985	−	0.988990i	$-0.452721\pi$
$587$	3.48294e28	0.295969	0.147985	+	0.988990i	$0.452721\pi$
$588$	0	0
$589$	−1.07716e29	−0.883225
$590$	0	0
$591$	−1.09968e29	−0.870158
$592$	0	0
$593$	−8.77330e28	−0.670021	−0.335011	−	0.942214i	$-0.608740\pi$
$593$	−8.77330e28	−0.670021	−0.335011	+	0.942214i	$0.608740\pi$
$594$	0	0
$595$	1.09782e29	0.809290
$596$	0	0
$597$	1.24576e29	0.886548
$598$	0	0
$599$	1.00705e29	0.691944	0.345972	−	0.938245i	$-0.387549\pi$
$599$	1.00705e29	0.691944	0.345972	+	0.938245i	$0.387549\pi$
$600$	0	0
$601$	−1.06089e29	−0.703861	−0.351930	−	0.936026i	$-0.614475\pi$
$601$	−1.06089e29	−0.703861	−0.351930	+	0.936026i	$0.614475\pi$
$602$	0	0
$603$	−1.60809e29	−1.03034
$604$	0	0
$605$	9.35497e26	0.00578910
$606$	0	0
$607$	−5.05632e28	−0.302241	−0.151120	−	0.988515i	$-0.548288\pi$
$607$	−5.05632e28	−0.302241	−0.151120	+	0.988515i	$0.548288\pi$
$608$	0	0
$609$	2.43309e29	1.40500
$610$	0	0
$611$	−3.47950e29	−1.94126
$612$	0	0
$613$	1.21126e29	0.652985	0.326493	−	0.945200i	$-0.394133\pi$
$613$	1.21126e29	0.652985	0.326493	+	0.945200i	$0.394133\pi$
$614$	0	0
$615$	−1.35334e29	−0.705048
$616$	0	0
$617$	7.11151e28	0.358070	0.179035	−	0.983843i	$-0.442702\pi$
$617$	7.11151e28	0.358070	0.179035	+	0.983843i	$0.442702\pi$
$618$	0	0
$619$	1.87344e28	0.0911774	0.0455887	−	0.998960i	$-0.485484\pi$
$619$	1.87344e28	0.0911774	0.0455887	+	0.998960i	$0.485484\pi$
$620$	0	0
$621$	4.44277e28	0.209022
$622$	0	0
$623$	2.25968e29	1.02783
$624$	0	0
$625$	9.09495e27	0.0400000
$626$	0	0
$627$	2.10616e29	0.895741
$628$	0	0
$629$	−1.28646e29	−0.529131
$630$	0	0
$631$	−1.18886e29	−0.472956	−0.236478	−	0.971637i	$-0.575993\pi$
$631$	−1.18886e29	−0.472956	−0.236478	+	0.971637i	$0.575993\pi$
$632$	0	0
$633$	4.49753e29	1.73075
$634$	0	0
$635$	1.56607e29	0.583025
$636$	0	0
$637$	3.54588e29	1.27720
$638$	0	0
$639$	1.05305e30	3.67021
$640$	0	0
$641$	−4.05009e29	−1.36602	−0.683008	−	0.730411i	$-0.739329\pi$
$641$	−4.05009e29	−1.36602	−0.683008	+	0.730411i	$0.739329\pi$
$642$	0	0
$643$	−2.92040e29	−0.953296	−0.476648	−	0.879094i	$-0.658148\pi$
$643$	−2.92040e29	−0.953296	−0.476648	+	0.879094i	$0.658148\pi$
$644$	0	0
$645$	−1.42686e29	−0.450820
$646$	0	0
$647$	1.30263e29	0.398407	0.199203	−	0.979958i	$-0.436165\pi$
$647$	1.30263e29	0.398407	0.199203	+	0.979958i	$0.436165\pi$
$648$	0	0
$649$	7.01686e28	0.207766
$650$	0	0
$651$	1.28007e30	3.66971
$652$	0	0
$653$	−1.49980e29	−0.416337	−0.208169	−	0.978093i	$-0.566750\pi$
$653$	−1.49980e29	−0.416337	−0.208169	+	0.978093i	$0.566750\pi$
$654$	0	0
$655$	−1.36576e29	−0.367149
$656$	0	0
$657$	1.86022e29	0.484314
$658$	0	0
$659$	−3.09551e29	−0.780613	−0.390307	−	0.920685i	$-0.627631\pi$
$659$	−3.09551e29	−0.780613	−0.390307	+	0.920685i	$0.627631\pi$
$660$	0	0
$661$	−1.06593e29	−0.260383	−0.130191	−	0.991489i	$-0.541559\pi$
$661$	−1.06593e29	−0.260383	−0.130191	+	0.991489i	$0.541559\pi$
$662$	0	0
$663$	−1.99127e30	−4.71235
$664$	0	0
$665$	1.32371e29	0.303505
$666$	0	0
$667$	3.85701e28	0.0856898
$668$	0	0
$669$	1.26535e30	2.72419
$670$	0	0
$671$	−3.87746e29	−0.809022
$672$	0	0
$673$	−7.90406e29	−1.59842	−0.799212	−	0.601049i	$-0.794750\pi$
$673$	−7.90406e29	−1.59842	−0.799212	+	0.601049i	$0.794750\pi$
$674$	0	0
$675$	1.59367e29	0.312398
$676$	0	0
$677$	−3.48611e29	−0.662461	−0.331230	−	0.943550i	$-0.607464\pi$
$677$	−3.48611e29	−0.662461	−0.331230	+	0.943550i	$0.607464\pi$
$678$	0	0
$679$	6.97264e29	1.28459
$680$	0	0
$681$	2.70875e29	0.483865
$682$	0	0
$683$	−3.87738e28	−0.0671617	−0.0335808	−	0.999436i	$-0.510691\pi$
$683$	−3.87738e28	−0.0671617	−0.0335808	+	0.999436i	$0.510691\pi$
$684$	0	0
$685$	3.41118e29	0.572999
$686$	0	0
$687$	−1.73819e30	−2.83173
$688$	0	0
$689$	−7.96020e29	−1.25783
$690$	0	0
$691$	−7.13503e29	−1.09364	−0.546822	−	0.837249i	$-0.684163\pi$
$691$	−7.13503e29	−1.09364	−0.546822	+	0.837249i	$0.684163\pi$
$692$	0	0
$693$	−1.64423e30	−2.44492
$694$	0	0
$695$	−1.13997e29	−0.164457
$696$	0	0
$697$	−9.28913e29	−1.30026
$698$	0	0
$699$	−8.08015e28	−0.109751
$700$	0	0
$701$	−8.08128e28	−0.106522	−0.0532612	−	0.998581i	$-0.516962\pi$
$701$	−8.08128e28	−0.106522	−0.0532612	+	0.998581i	$0.516962\pi$
$702$	0	0
$703$	−1.55116e29	−0.198439
$704$	0	0
$705$	6.08986e29	0.756173
$706$	0	0
$707$	3.11132e29	0.375008
$708$	0	0
$709$	7.67692e28	0.0898259	0.0449129	−	0.998991i	$-0.485699\pi$
$709$	7.67692e28	0.0898259	0.0449129	+	0.998991i	$0.485699\pi$
$710$	0	0
$711$	−2.18052e29	−0.247702
$712$	0	0
$713$	2.02920e29	0.223813
$714$	0	0
$715$	8.13181e29	0.870911
$716$	0	0
$717$	−1.85457e30	−1.92882
$718$	0	0
$719$	−4.77523e29	−0.482326	−0.241163	−	0.970485i	$-0.577529\pi$
$719$	−4.77523e29	−0.482326	−0.241163	+	0.970485i	$0.577529\pi$
$720$	0	0
$721$	−5.53297e29	−0.542797
$722$	0	0
$723$	2.92832e29	0.279040
$724$	0	0
$725$	1.38355e29	0.128069
$726$	0	0
$727$	8.17519e29	0.735168	0.367584	−	0.929990i	$-0.380185\pi$
$727$	8.17519e29	0.735168	0.367584	+	0.929990i	$0.380185\pi$
$728$	0	0
$729$	−1.40435e30	−1.22698
$730$	0	0
$731$	−9.79373e29	−0.831411
$732$	0	0
$733$	−1.25004e30	−1.03118	−0.515588	−	0.856837i	$-0.672426\pi$
$733$	−1.25004e30	−1.03118	−0.515588	+	0.856837i	$0.672426\pi$
$734$	0	0
$735$	−6.20604e29	−0.497505
$736$	0	0
$737$	6.86144e29	0.534572
$738$	0	0
$739$	−1.41926e30	−1.07472	−0.537359	−	0.843354i	$-0.680578\pi$
$739$	−1.41926e30	−1.07472	−0.537359	+	0.843354i	$0.680578\pi$
$740$	0	0
$741$	−2.40099e30	−1.76726
$742$	0	0
$743$	4.94344e29	0.353709	0.176855	−	0.984237i	$-0.443408\pi$
$743$	4.94344e29	0.353709	0.176855	+	0.984237i	$0.443408\pi$
$744$	0	0
$745$	2.92869e29	0.203719
$746$	0	0
$747$	9.61447e29	0.650218
$748$	0	0
$749$	2.97405e30	1.95564
$750$	0	0
$751$	1.81607e30	1.16121	0.580607	−	0.814184i	$-0.302815\pi$
$751$	1.81607e30	1.16121	0.580607	+	0.814184i	$0.302815\pi$
$752$	0	0
$753$	3.76833e30	2.34316
$754$	0	0
$755$	−8.60200e29	−0.520183
$756$	0	0
$757$	−3.23429e30	−1.90227	−0.951136	−	0.308772i	$-0.900082\pi$
$757$	−3.23429e30	−1.90227	−0.951136	+	0.308772i	$0.900082\pi$
$758$	0	0
$759$	−3.96766e29	−0.226984
$760$	0	0
$761$	5.91876e29	0.329376	0.164688	−	0.986346i	$-0.447338\pi$
$761$	5.91876e29	0.329376	0.164688	+	0.986346i	$0.447338\pi$
$762$	0	0
$763$	−2.73252e30	−1.47930
$764$	0	0
$765$	2.28950e30	1.20586
$766$	0	0
$767$	−7.99912e29	−0.409912
$768$	0	0
$769$	3.65919e30	1.82456	0.912281	−	0.409565i	$-0.134320\pi$
$769$	3.65919e30	1.82456	0.912281	+	0.409565i	$0.134320\pi$
$770$	0	0
$771$	1.34885e30	0.654475
$772$	0	0
$773$	−3.12612e30	−1.47612	−0.738059	−	0.674736i	$-0.764257\pi$
$773$	−3.12612e30	−1.47612	−0.738059	+	0.674736i	$0.764257\pi$
$774$	0	0
$775$	7.27895e29	0.334503
$776$	0	0
$777$	1.84335e30	0.824492
$778$	0	0
$779$	−1.12005e30	−0.487633
$780$	0	0
$781$	−4.49319e30	−1.90423
$782$	0	0
$783$	2.42433e30	1.00021
$784$	0	0
$785$	−1.72367e30	−0.692346
$786$	0	0
$787$	−1.94010e30	−0.758735	−0.379368	−	0.925246i	$-0.623858\pi$
$787$	−1.94010e30	−0.758735	−0.379368	+	0.925246i	$0.623858\pi$
$788$	0	0
$789$	−2.36808e30	−0.901753
$790$	0	0
$791$	−5.38912e30	−1.99832
$792$	0	0
$793$	4.42024e30	1.59617
$794$	0	0
$795$	1.39320e30	0.489959
$796$	0	0
$797$	1.87837e30	0.643383	0.321691	−	0.946845i	$-0.395749\pi$
$797$	1.87837e30	0.643383	0.321691	+	0.946845i	$0.395749\pi$
$798$	0	0
$799$	4.17999e30	1.39455
$800$	0	0
$801$	4.71255e30	1.53149
$802$	0	0
$803$	−7.93722e29	−0.251278
$804$	0	0
$805$	−2.49366e29	−0.0769095
$806$	0	0
$807$	−3.89668e30	−1.17091
$808$	0	0
$809$	3.76184e30	1.10139	0.550694	−	0.834707i	$-0.314363\pi$
$809$	3.76184e30	1.10139	0.550694	+	0.834707i	$0.314363\pi$
$810$	0	0
$811$	1.39769e30	0.398741	0.199370	−	0.979924i	$-0.436110\pi$
$811$	1.39769e30	0.398741	0.199370	+	0.979924i	$0.436110\pi$
$812$	0	0
$813$	−1.53506e30	−0.426751
$814$	0	0
$815$	9.14681e29	0.247808
$816$	0	0
$817$	−1.18089e30	−0.311801
$818$	0	0
$819$	1.87440e31	4.82372
$820$	0	0
$821$	3.06130e30	0.767897	0.383948	−	0.923355i	$-0.374564\pi$
$821$	3.06130e30	0.767897	0.383948	+	0.923355i	$0.374564\pi$
$822$	0	0
$823$	2.80656e30	0.686239	0.343120	−	0.939292i	$-0.388516\pi$
$823$	2.80656e30	0.686239	0.343120	+	0.939292i	$0.388516\pi$
$824$	0	0
$825$	−1.42324e30	−0.339243
$826$	0	0
$827$	1.80807e30	0.420153	0.210077	−	0.977685i	$-0.432629\pi$
$827$	1.80807e30	0.420153	0.210077	+	0.977685i	$0.432629\pi$
$828$	0	0
$829$	6.48512e30	1.46925	0.734625	−	0.678473i	$-0.237358\pi$
$829$	6.48512e30	1.46925	0.734625	+	0.678473i	$0.237358\pi$
$830$	0	0
$831$	5.65663e30	1.24953
$832$	0	0
$833$	−4.25974e30	−0.917509
$834$	0	0
$835$	−2.70765e30	−0.568702
$836$	0	0
$837$	1.27546e31	2.61245
$838$	0	0
$839$	−5.47358e30	−1.09338	−0.546690	−	0.837335i	$-0.684112\pi$
$839$	−5.47358e30	−1.09338	−0.546690	+	0.837335i	$0.684112\pi$
$840$	0	0
$841$	−3.02816e30	−0.589957
$842$	0	0
$843$	2.84509e30	0.540639
$844$	0	0
$845$	−6.85741e30	−1.27106
$846$	0	0
$847$	−9.20073e28	−0.0166359
$848$	0	0
$849$	−1.26579e31	−2.23271
$850$	0	0
$851$	2.92213e29	0.0502851
$852$	0	0
$853$	−7.71396e30	−1.29513	−0.647564	−	0.762011i	$-0.724212\pi$
$853$	−7.71396e30	−1.29513	−0.647564	+	0.762011i	$0.724212\pi$
$854$	0	0
$855$	2.76060e30	0.452229
$856$	0	0
$857$	1.69666e29	0.0271205	0.0135602	−	0.999908i	$-0.495684\pi$
$857$	1.69666e29	0.0271205	0.0135602	+	0.999908i	$0.495684\pi$
$858$	0	0
$859$	−4.91435e30	−0.766545	−0.383273	−	0.923635i	$-0.625203\pi$
$859$	−4.91435e30	−0.766545	−0.383273	+	0.923635i	$0.625203\pi$
$860$	0	0
$861$	1.33103e31	2.02607
$862$	0	0
$863$	−1.11211e31	−1.65209	−0.826047	−	0.563602i	$-0.809415\pi$
$863$	−1.11211e31	−1.65209	−0.826047	+	0.563602i	$0.809415\pi$
$864$	0	0
$865$	3.75482e30	0.544403
$866$	0	0
$867$	1.18569e31	1.67792
$868$	0	0
$869$	9.30391e29	0.128516
$870$	0	0
$871$	−7.82194e30	−1.05469
$872$	0	0
$873$	1.45414e31	1.91406
$874$	0	0
$875$	−8.94500e29	−0.114946
$876$	0	0
$877$	−1.05245e31	−1.32041	−0.660203	−	0.751088i	$-0.729530\pi$
$877$	−1.05245e31	−1.32041	−0.660203	+	0.751088i	$0.729530\pi$
$878$	0	0
$879$	−1.38101e31	−1.69166
$880$	0	0
$881$	−8.70976e30	−1.04174	−0.520870	−	0.853636i	$-0.674392\pi$
$881$	−8.70976e30	−1.04174	−0.520870	+	0.853636i	$0.674392\pi$
$882$	0	0
$883$	−7.95603e30	−0.929199	−0.464600	−	0.885521i	$-0.653802\pi$
$883$	−7.95603e30	−0.929199	−0.464600	+	0.885521i	$0.653802\pi$
$884$	0	0
$885$	1.40002e30	0.159672
$886$	0	0
$887$	1.08949e31	1.21346	0.606729	−	0.794909i	$-0.292481\pi$
$887$	1.08949e31	1.21346	0.606729	+	0.794909i	$0.292481\pi$
$888$	0	0
$889$	−1.54025e31	−1.67542
$890$	0	0
$891$	−7.03150e30	−0.747020
$892$	0	0
$893$	5.04007e30	0.522993
$894$	0	0
$895$	4.43718e29	0.0449743
$896$	0	0
$897$	4.52308e30	0.447830
$898$	0	0
$899$	1.10729e31	1.07099
$900$	0	0
$901$	9.56274e30	0.903591
$902$	0	0
$903$	1.40333e31	1.29550
$904$	0	0
$905$	−8.41758e30	−0.759236
$906$	0	0
$907$	−7.14477e30	−0.629668	−0.314834	−	0.949147i	$-0.601949\pi$
$907$	−7.14477e30	−0.629668	−0.314834	+	0.949147i	$0.601949\pi$
$908$	0	0
$909$	6.48864e30	0.558770
$910$	0	0
$911$	9.25953e30	0.779194	0.389597	−	0.920985i	$-0.372614\pi$
$911$	9.25953e30	0.779194	0.389597	+	0.920985i	$0.372614\pi$
$912$	0	0
$913$	−4.10233e30	−0.337355
$914$	0	0
$915$	−7.73636e30	−0.621749
$916$	0	0
$917$	1.34325e31	1.05506
$918$	0	0
$919$	−2.16799e31	−1.66435	−0.832174	−	0.554515i	$-0.812904\pi$
$919$	−2.16799e31	−1.66435	−0.832174	+	0.554515i	$0.812904\pi$
$920$	0	0
$921$	2.07817e31	1.55939
$922$	0	0
$923$	5.12218e31	3.75696
$924$	0	0
$925$	1.04820e30	0.0751545
$926$	0	0
$927$	−1.15390e31	−0.808779
$928$	0	0
$929$	−1.54089e31	−1.05586	−0.527929	−	0.849289i	$-0.677031\pi$
$929$	−1.54089e31	−1.05586	−0.527929	+	0.849289i	$0.677031\pi$
$930$	0	0
$931$	−5.13623e30	−0.344091
$932$	0	0
$933$	2.90558e31	1.90316
$934$	0	0
$935$	−9.76891e30	−0.625639
$936$	0	0
$937$	−2.18719e31	−1.36969	−0.684843	−	0.728691i	$-0.740129\pi$
$937$	−2.18719e31	−1.36969	−0.684843	+	0.728691i	$0.740129\pi$
$938$	0	0
$939$	3.92070e30	0.240091
$940$	0	0
$941$	−5.71240e29	−0.0342080	−0.0171040	−	0.999854i	$-0.505445\pi$
$941$	−5.71240e29	−0.0342080	−0.0171040	+	0.999854i	$0.505445\pi$
$942$	0	0
$943$	2.10999e30	0.123568
$944$	0	0
$945$	−1.56739e31	−0.897725
$946$	0	0
$947$	−6.38251e29	−0.0357534	−0.0178767	−	0.999840i	$-0.505691\pi$
$947$	−6.38251e29	−0.0357534	−0.0178767	+	0.999840i	$0.505691\pi$
$948$	0	0
$949$	9.04832e30	0.495761
$950$	0	0
$951$	1.93397e31	1.03646
$952$	0	0
$953$	−2.06863e31	−1.08444	−0.542222	−	0.840235i	$-0.682417\pi$
$953$	−2.06863e31	−1.08444	−0.542222	+	0.840235i	$0.682417\pi$
$954$	0	0
$955$	−2.26147e30	−0.115973
$956$	0	0
$957$	−2.16507e31	−1.08617
$958$	0	0
$959$	−3.35494e31	−1.64661
$960$	0	0
$961$	3.74300e31	1.79731
$962$	0	0
$963$	6.20238e31	2.91395
$964$	0	0
$965$	1.10828e31	0.509461
$966$	0	0
$967$	−2.49286e31	−1.12129	−0.560647	−	0.828055i	$-0.689448\pi$
$967$	−2.49286e31	−1.12129	−0.560647	+	0.828055i	$0.689448\pi$
$968$	0	0
$969$	2.88436e31	1.26955
$970$	0	0
$971$	−4.80972e30	−0.207166	−0.103583	−	0.994621i	$-0.533031\pi$
$971$	−4.80972e30	−0.207166	−0.103583	+	0.994621i	$0.533031\pi$
$972$	0	0
$973$	1.12117e31	0.472594
$974$	0	0
$975$	1.62247e31	0.669313
$976$	0	0
$977$	1.25924e31	0.508411	0.254205	−	0.967150i	$-0.418186\pi$
$977$	1.25924e31	0.508411	0.254205	+	0.967150i	$0.418186\pi$
$978$	0	0
$979$	−2.01076e31	−0.794589
$980$	0	0
$981$	−5.69865e31	−2.20418
$982$	0	0
$983$	−1.94974e31	−0.738185	−0.369093	−	0.929393i	$-0.620331\pi$
$983$	−1.94974e31	−0.738185	−0.369093	+	0.929393i	$0.620331\pi$
$984$	0	0
$985$	6.15011e30	0.227930
$986$	0	0
$987$	−5.98945e31	−2.17299
$988$	0	0
$989$	2.22461e30	0.0790118
$990$	0	0
$991$	1.67935e31	0.583941	0.291971	−	0.956427i	$-0.405689\pi$
$991$	1.67935e31	0.583941	0.291971	+	0.956427i	$0.405689\pi$
$992$	0	0
$993$	3.81047e31	1.29721
$994$	0	0
$995$	−6.96705e30	−0.232224
$996$	0	0
$997$	2.24576e31	0.732932	0.366466	−	0.930431i	$-0.380568\pi$
$997$	2.24576e31	0.732932	0.366466	+	0.930431i	$0.380568\pi$
$998$	0	0
$999$	1.83671e31	0.586953

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.22.a.d.1.2		2
4.3	odd	2		10.22.a.b.1.1	✓	2
20.3	even	4		50.22.b.f.49.3		4
20.7	even	4		50.22.b.f.49.2		4
20.19	odd	2		50.22.a.f.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.22.a.b.1.1	✓	2	4.3	odd	2
50.22.a.f.1.2		2	20.19	odd	2
50.22.b.f.49.2		4	20.7	even	4
50.22.b.f.49.3		4	20.3	even	4
80.22.a.d.1.2		2	1.1	even	1	trivial

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 80 = 2^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	80.a (trivial)

\(f(q)\)	\(=\)	\(q+174616. q^{3} -9.76562e6 q^{5} +9.60462e8 q^{7} +2.00304e10 q^{9} -8.54661e10 q^{11} +9.74302e11 q^{13} -1.70523e12 q^{15} -1.17045e13 q^{17} -1.41128e13 q^{19} +1.67712e14 q^{21} +2.65862e13 q^{23} +9.53674e13 q^{25} +1.67108e15 q^{27} +1.45075e15 q^{29} +7.63253e15 q^{31} -1.49237e16 q^{33} -9.37951e15 q^{35} +1.09912e16 q^{37} +1.70129e17 q^{39} +7.93639e16 q^{41} +8.36751e16 q^{43} -1.95609e17 q^{45} -3.57127e17 q^{47} +3.63941e17 q^{49} -2.04379e18 q^{51} -8.17016e17 q^{53} +8.34630e17 q^{55} -2.46432e18 q^{57} -8.21011e17 q^{59} +4.53683e18 q^{61} +1.92384e19 q^{63} -9.51466e18 q^{65} -8.02826e18 q^{67} +4.64238e18 q^{69} +5.25728e19 q^{71} +9.28698e18 q^{73} +1.66527e19 q^{75} -8.20869e19 q^{77} -1.08861e19 q^{79} +8.22724e19 q^{81} +4.79994e19 q^{83} +1.14302e20 q^{85} +2.53325e20 q^{87} +2.35270e20 q^{89} +9.35779e20 q^{91} +1.33276e21 q^{93} +1.37820e20 q^{95} +7.25967e20 q^{97} -1.71192e21 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 126692 q^{3} - 19531250 q^{5} + 292598684 q^{7} + 11866737826 q^{9} + 41326831776 q^{11} + 1887051472228 q^{13} - 1237226562500 q^{15} - 3100413932364 q^{17} - 7175794652440 q^{19} + 199718603894264 q^{21}+ \cdots - 27\!\cdots\!12 q^{99}+O(q^{100}) \) 2 * q + 126692 * q^3 - 19531250 * q^5 + 292598684 * q^7 + 11866737826 * q^9 + 41326831776 * q^11 + 1887051472228 * q^13 - 1237226562500 * q^15 - 3100413932364 * q^17 - 7175794652440 * q^19 + 199718603894264 * q^21 + 357559990100052 * q^23 + 190734863281250 * q^25 + 2563616231988920 * q^27 - 2454584202185940 * q^29 + 4310677151243336 * q^31 - 21000167846983104 * q^33 - 2857409023437500 * q^35 - 32389201650205724 * q^37 + 126386038156081288 * q^39 - 16298035212712116 * q^41 + 161845702253479412 * q^43 - 115886111582031250 * q^45 - 153492993930726996 * q^47 + 251436016683357114 * q^49 - 2456129297437264344 * q^51 + 604198357317820308 * q^53 - 403582341562500000 * q^55 - 2796771718696561840 * q^57 - 3110197357974672120 * q^59 - 791142209760451676 * q^61 + 24690614624666969692 * q^63 - 18428237033476562500 * q^65 + 5111819523383561564 * q^67 - 11219196977510717208 * q^69 + 82388741313238482456 * q^71 + 10657918464992348548 * q^73 + 12082290649414062500 * q^75 - 166767237207340703808 * q^77 - 28314166125451369360 * q^79 + 124893157714537187242 * q^81 + 10427725705926651732 * q^83 + 30277479808242187500 * q^85 + 440484049640705943960 * q^87 + 478501472814113520180 * q^89 + 326187611445407942776 * q^91 + 1491957513759228855056 * q^93 + 70076119652734375000 * q^95 + 545226897779926992196 * q^97 - 2747011580657571215712 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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