LMFDB - Embedded newform 9968.2.a.z.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9968,2,Mod(1,9968)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9968, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("9968.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9968 = 2^{4} \cdot 7 \cdot 89 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9968.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:	$79.5948807348$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.135076.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 5x^{3} + 4x^{2} + 4x - 2 $ x^5 - x^4 - 5x^3 + 4x^2 + 4*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1246)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.420632$ of defining polynomial
Character	$\chi$	$=$	9968.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.14665 q^{3} -4.24370 q^{5} -1.00000 q^{7} -1.68519 q^{9} +O(q^{10})$	$q-1.14665 q^{3} -4.24370 q^{5} -1.00000 q^{7} -1.68519 q^{9} +5.02211 q^{11} +6.40089 q^{13} +4.86604 q^{15} -4.68519 q^{17} +7.95152 q^{19} +1.14665 q^{21} +2.44937 q^{23} +13.0090 q^{25} +5.37228 q^{27} +1.08379 q^{29} +6.68033 q^{31} -5.75859 q^{33} +4.24370 q^{35} +2.96042 q^{37} -7.33958 q^{39} -0.815317 q^{41} +6.04473 q^{43} +7.15146 q^{45} -1.21621 q^{47} +1.00000 q^{49} +5.37228 q^{51} -5.68417 q^{53} -21.3123 q^{55} -9.11761 q^{57} -6.20679 q^{59} -4.37981 q^{61} +1.68519 q^{63} -27.1635 q^{65} -8.99793 q^{67} -2.80857 q^{69} -5.88932 q^{71} -5.38983 q^{73} -14.9168 q^{75} -5.02211 q^{77} +3.84510 q^{79} -1.10453 q^{81} +10.3302 q^{83} +19.8826 q^{85} -1.24273 q^{87} -1.00000 q^{89} -6.40089 q^{91} -7.66000 q^{93} -33.7439 q^{95} +15.1229 q^{97} -8.46323 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 6 q^{3} - 10 q^{5} - 5 q^{7} + 7 q^{9}+O(q^{10}) $ 5 * q + 6 * q^3 - 10 * q^5 - 5 * q^7 + 7 * q^9	$ 5 q + 6 q^{3} - 10 q^{5} - 5 q^{7} + 7 q^{9} + 8 q^{11} - 8 q^{13} - 6 q^{15} - 8 q^{17} + 10 q^{19} - 6 q^{21} + 2 q^{23} + 17 q^{25} + 24 q^{27} - 10 q^{29} + 10 q^{31} + 10 q^{35} - 4 q^{37} - 24 q^{39} - 28 q^{41} + 10 q^{43} - 12 q^{45} + 10 q^{47} + 5 q^{49} + 24 q^{51} + 4 q^{53} - 4 q^{55} + 2 q^{57} + 10 q^{59} - 16 q^{61} - 7 q^{63} - 26 q^{65} - 2 q^{69} + 16 q^{71} + 10 q^{73} + 10 q^{75} - 8 q^{77} + 8 q^{79} + 41 q^{81} + 14 q^{83} + 18 q^{85} - 22 q^{87} - 5 q^{89} + 8 q^{91} + 22 q^{93} - 34 q^{95} + 6 q^{97} + 8 q^{99}+O(q^{100}) $ 5 * q + 6 * q^3 - 10 * q^5 - 5 * q^7 + 7 * q^9 + 8 * q^11 - 8 * q^13 - 6 * q^15 - 8 * q^17 + 10 * q^19 - 6 * q^21 + 2 * q^23 + 17 * q^25 + 24 * q^27 - 10 * q^29 + 10 * q^31 + 10 * q^35 - 4 * q^37 - 24 * q^39 - 28 * q^41 + 10 * q^43 - 12 * q^45 + 10 * q^47 + 5 * q^49 + 24 * q^51 + 4 * q^53 - 4 * q^55 + 2 * q^57 + 10 * q^59 - 16 * q^61 - 7 * q^63 - 26 * q^65 - 2 * q^69 + 16 * q^71 + 10 * q^73 + 10 * q^75 - 8 * q^77 + 8 * q^79 + 41 * q^81 + 14 * q^83 + 18 * q^85 - 22 * q^87 - 5 * q^89 + 8 * q^91 + 22 * q^93 - 34 * q^95 + 6 * q^97 + 8 * 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
$3$	−1.14665	−0.662018	−0.331009	−	0.943628i	$-0.607389\pi$
$3$	−1.14665	−0.662018	−0.331009	+	0.943628i	$0.607389\pi$
$4$	0	0
$5$	−4.24370	−1.89784	−0.948920	−	0.315516i	$-0.897822\pi$
$5$	−4.24370	−1.89784	−0.948920	+	0.315516i	$0.897822\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−1.68519	−0.561732
$10$	0	0
$11$	5.02211	1.51422	0.757111	−	0.653286i	$-0.226610\pi$
$11$	5.02211	1.51422	0.757111	+	0.653286i	$0.226610\pi$
$12$	0	0
$13$	6.40089	1.77529	0.887644	−	0.460530i	$-0.152341\pi$
$13$	6.40089	1.77529	0.887644	+	0.460530i	$0.152341\pi$
$14$	0	0
$15$	4.86604	1.25641
$16$	0	0
$17$	−4.68519	−1.13633	−0.568163	−	0.822916i	$-0.692346\pi$
$17$	−4.68519	−1.13633	−0.568163	+	0.822916i	$0.692346\pi$
$18$	0	0
$19$	7.95152	1.82420	0.912102	−	0.409963i	$-0.134458\pi$
$19$	7.95152	1.82420	0.912102	+	0.409963i	$0.134458\pi$
$20$	0	0
$21$	1.14665	0.250219
$22$	0	0
$23$	2.44937	0.510728	0.255364	−	0.966845i	$-0.417805\pi$
$23$	2.44937	0.510728	0.255364	+	0.966845i	$0.417805\pi$
$24$	0	0
$25$	13.0090	2.60180
$26$	0	0
$27$	5.37228	1.03390
$28$	0	0
$29$	1.08379	0.201255	0.100627	−	0.994924i	$-0.467915\pi$
$29$	1.08379	0.201255	0.100627	+	0.994924i	$0.467915\pi$
$30$	0	0
$31$	6.68033	1.19982	0.599911	−	0.800066i	$-0.295202\pi$
$31$	6.68033	1.19982	0.599911	+	0.800066i	$0.295202\pi$
$32$	0	0
$33$	−5.75859	−1.00244
$34$	0	0
$35$	4.24370	0.717316
$36$	0	0
$37$	2.96042	0.486690	0.243345	−	0.969940i	$-0.421755\pi$
$37$	2.96042	0.486690	0.243345	+	0.969940i	$0.421755\pi$
$38$	0	0
$39$	−7.33958	−1.17527
$40$	0	0
$41$	−0.815317	−0.127331	−0.0636655	−	0.997971i	$-0.520279\pi$
$41$	−0.815317	−0.127331	−0.0636655	+	0.997971i	$0.520279\pi$
$42$	0	0
$43$	6.04473	0.921813	0.460907	−	0.887449i	$-0.347524\pi$
$43$	6.04473	0.921813	0.460907	+	0.887449i	$0.347524\pi$
$44$	0	0
$45$	7.15146	1.06608
$46$	0	0
$47$	−1.21621	−0.177402	−0.0887011	−	0.996058i	$-0.528272\pi$
$47$	−1.21621	−0.177402	−0.0887011	+	0.996058i	$0.528272\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	5.37228	0.752269
$52$	0	0
$53$	−5.68417	−0.780781	−0.390391	−	0.920649i	$-0.627660\pi$
$53$	−5.68417	−0.780781	−0.390391	+	0.920649i	$0.627660\pi$
$54$	0	0
$55$	−21.3123	−2.87375
$56$	0	0
$57$	−9.11761	−1.20766
$58$	0	0
$59$	−6.20679	−0.808055	−0.404028	−	0.914747i	$-0.632390\pi$
$59$	−6.20679	−0.808055	−0.404028	+	0.914747i	$0.632390\pi$
$60$	0	0
$61$	−4.37981	−0.560777	−0.280389	−	0.959887i	$-0.590463\pi$
$61$	−4.37981	−0.560777	−0.280389	+	0.959887i	$0.590463\pi$
$62$	0	0
$63$	1.68519	0.212315
$64$	0	0
$65$	−27.1635	−3.36921
$66$	0	0
$67$	−8.99793	−1.09927	−0.549636	−	0.835404i	$-0.685234\pi$
$67$	−8.99793	−1.09927	−0.549636	+	0.835404i	$0.685234\pi$
$68$	0	0
$69$	−2.80857	−0.338112
$70$	0	0
$71$	−5.88932	−0.698933	−0.349467	−	0.936949i	$-0.613637\pi$
$71$	−5.88932	−0.698933	−0.349467	+	0.936949i	$0.613637\pi$
$72$	0	0
$73$	−5.38983	−0.630832	−0.315416	−	0.948954i	$-0.602144\pi$
$73$	−5.38983	−0.630832	−0.315416	+	0.948954i	$0.602144\pi$
$74$	0	0
$75$	−14.9168	−1.72244
$76$	0	0
$77$	−5.02211	−0.572322
$78$	0	0
$79$	3.84510	0.432608	0.216304	−	0.976326i	$-0.430600\pi$
$79$	3.84510	0.432608	0.216304	+	0.976326i	$0.430600\pi$
$80$	0	0
$81$	−1.10453	−0.122726
$82$	0	0
$83$	10.3302	1.13388	0.566941	−	0.823758i	$-0.308127\pi$
$83$	10.3302	1.13388	0.566941	+	0.823758i	$0.308127\pi$
$84$	0	0
$85$	19.8826	2.15657
$86$	0	0
$87$	−1.24273	−0.133234
$88$	0	0
$89$	−1.00000	−0.106000
$89$	−1.00000	−0.106000
$90$	0	0
$91$	−6.40089	−0.670996
$92$	0	0
$93$	−7.66000	−0.794305
$94$	0	0
$95$	−33.7439	−3.46205
$96$	0	0
$97$	15.1229	1.53550	0.767751	−	0.640748i	$-0.221376\pi$
$97$	15.1229	1.53550	0.767751	+	0.640748i	$0.221376\pi$
$98$	0	0
$99$	−8.46323	−0.850586
$100$	0	0
$101$	−8.51893	−0.847665	−0.423832	−	0.905741i	$-0.639315\pi$
$101$	−8.51893	−0.847665	−0.423832	+	0.905741i	$0.639315\pi$
$102$	0	0
$103$	−0.797652	−0.0785950	−0.0392975	−	0.999228i	$-0.512512\pi$
$103$	−0.797652	−0.0785950	−0.0392975	+	0.999228i	$0.512512\pi$
$104$	0	0
$105$	−4.86604	−0.474877
$106$	0	0
$107$	1.60646	0.155302	0.0776511	−	0.996981i	$-0.475258\pi$
$107$	1.60646	0.155302	0.0776511	+	0.996981i	$0.475258\pi$
$108$	0	0
$109$	2.85617	0.273571	0.136786	−	0.990601i	$-0.456323\pi$
$109$	2.85617	0.273571	0.136786	+	0.990601i	$0.456323\pi$
$110$	0	0
$111$	−3.39456	−0.322198
$112$	0	0
$113$	6.70463	0.630719	0.315359	−	0.948972i	$-0.397875\pi$
$113$	6.70463	0.630719	0.315359	+	0.948972i	$0.397875\pi$
$114$	0	0
$115$	−10.3944	−0.969281
$116$	0	0
$117$	−10.7868	−0.997235
$118$	0	0
$119$	4.68519	0.429491
$120$	0	0
$121$	14.2215	1.29287
$122$	0	0
$123$	0.934882	0.0842955
$124$	0	0
$125$	−33.9878	−3.03996
$126$	0	0
$127$	−10.7722	−0.955876	−0.477938	−	0.878394i	$-0.658616\pi$
$127$	−10.7722	−0.955876	−0.477938	+	0.878394i	$0.658616\pi$
$128$	0	0
$129$	−6.93119	−0.610257
$130$	0	0
$131$	−7.73377	−0.675703	−0.337851	−	0.941199i	$-0.609700\pi$
$131$	−7.73377	−0.675703	−0.337851	+	0.941199i	$0.609700\pi$
$132$	0	0
$133$	−7.95152	−0.689485
$134$	0	0
$135$	−22.7983	−1.96217
$136$	0	0
$137$	16.4555	1.40589	0.702946	−	0.711243i	$-0.251868\pi$
$137$	16.4555	1.40589	0.702946	+	0.711243i	$0.251868\pi$
$138$	0	0
$139$	−2.46258	−0.208873	−0.104437	−	0.994532i	$-0.533304\pi$
$139$	−2.46258	−0.208873	−0.104437	+	0.994532i	$0.533304\pi$
$140$	0	0
$141$	1.39456	0.117443
$142$	0	0
$143$	32.1460	2.68818
$144$	0	0
$145$	−4.59928	−0.381950
$146$	0	0
$147$	−1.14665	−0.0945741
$148$	0	0
$149$	22.3653	1.83224	0.916118	−	0.400909i	$-0.131305\pi$
$149$	22.3653	1.83224	0.916118	+	0.400909i	$0.131305\pi$
$150$	0	0
$151$	24.4504	1.98975	0.994873	−	0.101133i	$-0.0322467\pi$
$151$	24.4504	1.98975	0.994873	+	0.101133i	$0.0322467\pi$
$152$	0	0
$153$	7.89547	0.638311
$154$	0	0
$155$	−28.3493	−2.27707
$156$	0	0
$157$	−8.24430	−0.657967	−0.328983	−	0.944336i	$-0.606706\pi$
$157$	−8.24430	−0.657967	−0.328983	+	0.944336i	$0.606706\pi$
$158$	0	0
$159$	6.51775	0.516891
$160$	0	0
$161$	−2.44937	−0.193037
$162$	0	0
$163$	19.7295	1.54533	0.772667	−	0.634811i	$-0.218922\pi$
$163$	19.7295	1.54533	0.772667	+	0.634811i	$0.218922\pi$
$164$	0	0
$165$	24.4378	1.90248
$166$	0	0
$167$	18.2363	1.41116	0.705582	−	0.708628i	$-0.250685\pi$
$167$	18.2363	1.41116	0.705582	+	0.708628i	$0.250685\pi$
$168$	0	0
$169$	27.9714	2.15165
$170$	0	0
$171$	−13.3999	−1.02471
$172$	0	0
$173$	−8.56013	−0.650814	−0.325407	−	0.945574i	$-0.605501\pi$
$173$	−8.56013	−0.650814	−0.325407	+	0.945574i	$0.605501\pi$
$174$	0	0
$175$	−13.0090	−0.983388
$176$	0	0
$177$	7.11701	0.534947
$178$	0	0
$179$	4.67382	0.349338	0.174669	−	0.984627i	$-0.444114\pi$
$179$	4.67382	0.349338	0.174669	+	0.984627i	$0.444114\pi$
$180$	0	0
$181$	12.4948	0.928728	0.464364	−	0.885645i	$-0.346283\pi$
$181$	12.4948	0.928728	0.464364	+	0.885645i	$0.346283\pi$
$182$	0	0
$183$	5.02211	0.371245
$184$	0	0
$185$	−12.5631	−0.923660
$186$	0	0
$187$	−23.5295	−1.72065
$188$	0	0
$189$	−5.37228	−0.390776
$190$	0	0
$191$	−22.3725	−1.61881	−0.809407	−	0.587248i	$-0.800211\pi$
$191$	−22.3725	−1.61881	−0.809407	+	0.587248i	$0.800211\pi$
$192$	0	0
$193$	13.2239	0.951876	0.475938	−	0.879479i	$-0.342109\pi$
$193$	13.2239	0.951876	0.475938	+	0.879479i	$0.342109\pi$
$194$	0	0
$195$	31.1470	2.23048
$196$	0	0
$197$	−0.598736	−0.0426582	−0.0213291	−	0.999773i	$-0.506790\pi$
$197$	−0.598736	−0.0426582	−0.0213291	+	0.999773i	$0.506790\pi$
$198$	0	0
$199$	−4.00105	−0.283627	−0.141813	−	0.989893i	$-0.545293\pi$
$199$	−4.00105	−0.283627	−0.141813	+	0.989893i	$0.545293\pi$
$200$	0	0
$201$	10.3175	0.727738
$202$	0	0
$203$	−1.08379	−0.0760672
$204$	0	0
$205$	3.45996	0.241654
$206$	0	0
$207$	−4.12766	−0.286892
$208$	0	0
$209$	39.9334	2.76225
$210$	0	0
$211$	0.480701	0.0330928	0.0165464	−	0.999863i	$-0.494733\pi$
$211$	0.480701	0.0330928	0.0165464	+	0.999863i	$0.494733\pi$
$212$	0	0
$213$	6.75298	0.462707
$214$	0	0
$215$	−25.6520	−1.74945
$216$	0	0
$217$	−6.68033	−0.453490
$218$	0	0
$219$	6.18024	0.417622
$220$	0	0
$221$	−29.9894	−2.01731
$222$	0	0
$223$	15.4622	1.03543	0.517713	−	0.855554i	$-0.326784\pi$
$223$	15.4622	1.03543	0.517713	+	0.855554i	$0.326784\pi$
$224$	0	0
$225$	−21.9227	−1.46151
$226$	0	0
$227$	21.3435	1.41662	0.708310	−	0.705902i	$-0.249458\pi$
$227$	21.3435	1.41662	0.708310	+	0.705902i	$0.249458\pi$
$228$	0	0
$229$	−23.4461	−1.54936	−0.774682	−	0.632351i	$-0.782090\pi$
$229$	−23.4461	−1.54936	−0.774682	+	0.632351i	$0.782090\pi$
$230$	0	0
$231$	5.75859	0.378888
$232$	0	0
$233$	−21.8862	−1.43381	−0.716907	−	0.697169i	$-0.754443\pi$
$233$	−21.8862	−1.43381	−0.716907	+	0.697169i	$0.754443\pi$
$234$	0	0
$235$	5.16122	0.336681
$236$	0	0
$237$	−4.40899	−0.286394
$238$	0	0
$239$	24.5696	1.58928	0.794639	−	0.607082i	$-0.207660\pi$
$239$	24.5696	1.58928	0.794639	+	0.607082i	$0.207660\pi$
$240$	0	0
$241$	−30.5791	−1.96977	−0.984887	−	0.173198i	$-0.944590\pi$
$241$	−30.5791	−1.96977	−0.984887	+	0.173198i	$0.944590\pi$
$242$	0	0
$243$	−14.8503	−0.952648
$244$	0	0
$245$	−4.24370	−0.271120
$246$	0	0
$247$	50.8968	3.23849
$248$	0	0
$249$	−11.8451	−0.750651
$250$	0	0
$251$	−18.4729	−1.16600	−0.582999	−	0.812473i	$-0.698121\pi$
$251$	−18.4729	−1.16600	−0.582999	+	0.812473i	$0.698121\pi$
$252$	0	0
$253$	12.3010	0.773356
$254$	0	0
$255$	−22.7983	−1.42769
$256$	0	0
$257$	−23.7834	−1.48357	−0.741783	−	0.670640i	$-0.766020\pi$
$257$	−23.7834	−1.48357	−0.741783	+	0.670640i	$0.766020\pi$
$258$	0	0
$259$	−2.96042	−0.183952
$260$	0	0
$261$	−1.82640	−0.113051
$262$	0	0
$263$	2.88829	0.178100	0.0890499	−	0.996027i	$-0.471617\pi$
$263$	2.88829	0.178100	0.0890499	+	0.996027i	$0.471617\pi$
$264$	0	0
$265$	24.1219	1.48180
$266$	0	0
$267$	1.14665	0.0701738
$268$	0	0
$269$	−7.36605	−0.449116	−0.224558	−	0.974461i	$-0.572094\pi$
$269$	−7.36605	−0.449116	−0.224558	+	0.974461i	$0.572094\pi$
$270$	0	0
$271$	−25.8470	−1.57009	−0.785047	−	0.619436i	$-0.787361\pi$
$271$	−25.8470	−1.57009	−0.785047	+	0.619436i	$0.787361\pi$
$272$	0	0
$273$	7.33958	0.444212
$274$	0	0
$275$	65.3325	3.93970
$276$	0	0
$277$	9.63452	0.578882	0.289441	−	0.957196i	$-0.406531\pi$
$277$	9.63452	0.578882	0.289441	+	0.957196i	$0.406531\pi$
$278$	0	0
$279$	−11.2577	−0.673978
$280$	0	0
$281$	7.83466	0.467377	0.233688	−	0.972312i	$-0.424920\pi$
$281$	7.83466	0.467377	0.233688	+	0.972312i	$0.424920\pi$
$282$	0	0
$283$	10.7548	0.639308	0.319654	−	0.947534i	$-0.396433\pi$
$283$	10.7548	0.639308	0.319654	+	0.947534i	$0.396433\pi$
$284$	0	0
$285$	38.6924	2.29194
$286$	0	0
$287$	0.815317	0.0481266
$288$	0	0
$289$	4.95105	0.291238
$290$	0	0
$291$	−17.3407	−1.01653
$292$	0	0
$293$	−15.5718	−0.909717	−0.454858	−	0.890564i	$-0.650310\pi$
$293$	−15.5718	−0.909717	−0.454858	+	0.890564i	$0.650310\pi$
$294$	0	0
$295$	26.3398	1.53356
$296$	0	0
$297$	26.9801	1.56555
$298$	0	0
$299$	15.6781	0.906690
$300$	0	0
$301$	−6.04473	−0.348413
$302$	0	0
$303$	9.76822	0.561170
$304$	0	0
$305$	18.5866	1.06427
$306$	0	0
$307$	26.7317	1.52566	0.762829	−	0.646600i	$-0.223810\pi$
$307$	26.7317	1.52566	0.762829	+	0.646600i	$0.223810\pi$
$308$	0	0
$309$	0.914627	0.0520313
$310$	0	0
$311$	−31.3629	−1.77843	−0.889214	−	0.457492i	$-0.848748\pi$
$311$	−31.3629	−1.77843	−0.889214	+	0.457492i	$0.848748\pi$
$312$	0	0
$313$	−6.92411	−0.391374	−0.195687	−	0.980666i	$-0.562694\pi$
$313$	−6.92411	−0.391374	−0.195687	+	0.980666i	$0.562694\pi$
$314$	0	0
$315$	−7.15146	−0.402939
$316$	0	0
$317$	10.1850	0.572045	0.286023	−	0.958223i	$-0.407667\pi$
$317$	10.1850	0.572045	0.286023	+	0.958223i	$0.407667\pi$
$318$	0	0
$319$	5.44291	0.304745
$320$	0	0
$321$	−1.84205	−0.102813
$322$	0	0
$323$	−37.2544	−2.07289
$324$	0	0
$325$	83.2692	4.61894
$326$	0	0
$327$	−3.27502	−0.181109
$328$	0	0
$329$	1.21621	0.0670517
$330$	0	0
$331$	−21.0341	−1.15614	−0.578069	−	0.815988i	$-0.696193\pi$
$331$	−21.0341	−1.15614	−0.578069	+	0.815988i	$0.696193\pi$
$332$	0	0
$333$	−4.98889	−0.273389
$334$	0	0
$335$	38.1845	2.08624
$336$	0	0
$337$	0.358013	0.0195022	0.00975111	−	0.999952i	$-0.496896\pi$
$337$	0.358013	0.0195022	0.00975111	+	0.999952i	$0.496896\pi$
$338$	0	0
$339$	−7.68786	−0.417547
$340$	0	0
$341$	33.5493	1.81680
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	11.9187	0.641682
$346$	0	0
$347$	1.16861	0.0627340	0.0313670	−	0.999508i	$-0.490014\pi$
$347$	1.16861	0.0627340	0.0313670	+	0.999508i	$0.490014\pi$
$348$	0	0
$349$	−27.8715	−1.49193	−0.745964	−	0.665986i	$-0.768011\pi$
$349$	−27.8715	−1.49193	−0.745964	+	0.665986i	$0.768011\pi$
$350$	0	0
$351$	34.3874	1.83546
$352$	0	0
$353$	18.9349	1.00780	0.503901	−	0.863761i	$-0.331898\pi$
$353$	18.9349	1.00780	0.503901	+	0.863761i	$0.331898\pi$
$354$	0	0
$355$	24.9925	1.32646
$356$	0	0
$357$	−5.37228	−0.284331
$358$	0	0
$359$	15.0124	0.792323	0.396161	−	0.918181i	$-0.370342\pi$
$359$	15.0124	0.792323	0.396161	+	0.918181i	$0.370342\pi$
$360$	0	0
$361$	44.2267	2.32772
$362$	0	0
$363$	−16.3071	−0.855902
$364$	0	0
$365$	22.8728	1.19722
$366$	0	0
$367$	−0.580961	−0.0303259	−0.0151630	−	0.999885i	$-0.504827\pi$
$367$	−0.580961	−0.0303259	−0.0151630	+	0.999885i	$0.504827\pi$
$368$	0	0
$369$	1.37397	0.0715259
$370$	0	0
$371$	5.68417	0.295108
$372$	0	0
$373$	−30.8212	−1.59586	−0.797931	−	0.602749i	$-0.794072\pi$
$373$	−30.8212	−1.59586	−0.797931	+	0.602749i	$0.794072\pi$
$374$	0	0
$375$	38.9721	2.01251
$376$	0	0
$377$	6.93723	0.357285
$378$	0	0
$379$	6.54825	0.336361	0.168180	−	0.985756i	$-0.446211\pi$
$379$	6.54825	0.336361	0.168180	+	0.985756i	$0.446211\pi$
$380$	0	0
$381$	12.3519	0.632808
$382$	0	0
$383$	−17.1127	−0.874421	−0.437210	−	0.899359i	$-0.644033\pi$
$383$	−17.1127	−0.874421	−0.437210	+	0.899359i	$0.644033\pi$
$384$	0	0
$385$	21.3123	1.08618
$386$	0	0
$387$	−10.1866	−0.517812
$388$	0	0
$389$	24.6841	1.25153	0.625766	−	0.780011i	$-0.284787\pi$
$389$	24.6841	1.25153	0.625766	+	0.780011i	$0.284787\pi$
$390$	0	0
$391$	−11.4758	−0.580354
$392$	0	0
$393$	8.86792	0.447328
$394$	0	0
$395$	−16.3175	−0.821021
$396$	0	0
$397$	−20.0096	−1.00425	−0.502126	−	0.864794i	$-0.667449\pi$
$397$	−20.0096	−1.00425	−0.502126	+	0.864794i	$0.667449\pi$
$398$	0	0
$399$	9.11761	0.456451
$400$	0	0
$401$	−14.1679	−0.707512	−0.353756	−	0.935338i	$-0.615096\pi$
$401$	−14.1679	−0.707512	−0.353756	+	0.935338i	$0.615096\pi$
$402$	0	0
$403$	42.7601	2.13003
$404$	0	0
$405$	4.68731	0.232914
$406$	0	0
$407$	14.8675	0.736957
$408$	0	0
$409$	16.7079	0.826154	0.413077	−	0.910696i	$-0.364454\pi$
$409$	16.7079	0.826154	0.413077	+	0.910696i	$0.364454\pi$
$410$	0	0
$411$	−18.8687	−0.930726
$412$	0	0
$413$	6.20679	0.305416
$414$	0	0
$415$	−43.8381	−2.15193
$416$	0	0
$417$	2.82371	0.138278
$418$	0	0
$419$	5.58216	0.272706	0.136353	−	0.990660i	$-0.456462\pi$
$419$	5.58216	0.272706	0.136353	+	0.990660i	$0.456462\pi$
$420$	0	0
$421$	−19.0334	−0.927631	−0.463815	−	0.885932i	$-0.653520\pi$
$421$	−19.0334	−0.927631	−0.463815	+	0.885932i	$0.653520\pi$
$422$	0	0
$423$	2.04955	0.0996524
$424$	0	0
$425$	−60.9497	−2.95649
$426$	0	0
$427$	4.37981	0.211954
$428$	0	0
$429$	−36.8601	−1.77962
$430$	0	0
$431$	4.52853	0.218131	0.109066	−	0.994035i	$-0.465214\pi$
$431$	4.52853	0.218131	0.109066	+	0.994035i	$0.465214\pi$
$432$	0	0
$433$	4.63257	0.222627	0.111314	−	0.993785i	$-0.464494\pi$
$433$	4.63257	0.222627	0.111314	+	0.993785i	$0.464494\pi$
$434$	0	0
$435$	5.27377	0.252858
$436$	0	0
$437$	19.4762	0.931673
$438$	0	0
$439$	−4.87515	−0.232678	−0.116339	−	0.993210i	$-0.537116\pi$
$439$	−4.87515	−0.232678	−0.116339	+	0.993210i	$0.537116\pi$
$440$	0	0
$441$	−1.68519	−0.0802474
$442$	0	0
$443$	−7.36834	−0.350081	−0.175040	−	0.984561i	$-0.556006\pi$
$443$	−7.36834	−0.350081	−0.175040	+	0.984561i	$0.556006\pi$
$444$	0	0
$445$	4.24370	0.201171
$446$	0	0
$447$	−25.6452	−1.21297
$448$	0	0
$449$	−1.81255	−0.0855394	−0.0427697	−	0.999085i	$-0.513618\pi$
$449$	−1.81255	−0.0855394	−0.0427697	+	0.999085i	$0.513618\pi$
$450$	0	0
$451$	−4.09461	−0.192807
$452$	0	0
$453$	−28.0360	−1.31725
$454$	0	0
$455$	27.1635	1.27344
$456$	0	0
$457$	27.1581	1.27040	0.635202	−	0.772346i	$-0.280917\pi$
$457$	27.1581	1.27040	0.635202	+	0.772346i	$0.280917\pi$
$458$	0	0
$459$	−25.1702	−1.17484
$460$	0	0
$461$	33.5080	1.56062	0.780312	−	0.625391i	$-0.215060\pi$
$461$	33.5080	1.56062	0.780312	+	0.625391i	$0.215060\pi$
$462$	0	0
$463$	−2.30169	−0.106969	−0.0534844	−	0.998569i	$-0.517033\pi$
$463$	−2.30169	−0.106969	−0.0534844	+	0.998569i	$0.517033\pi$
$464$	0	0
$465$	32.5067	1.50746
$466$	0	0
$467$	19.2567	0.891093	0.445546	−	0.895259i	$-0.353009\pi$
$467$	19.2567	0.891093	0.445546	+	0.895259i	$0.353009\pi$
$468$	0	0
$469$	8.99793	0.415486
$470$	0	0
$471$	9.45332	0.435586
$472$	0	0
$473$	30.3573	1.39583
$474$	0	0
$475$	103.441	4.74621
$476$	0	0
$477$	9.57894	0.438589
$478$	0	0
$479$	−3.32488	−0.151918	−0.0759588	−	0.997111i	$-0.524202\pi$
$479$	−3.32488	−0.151918	−0.0759588	+	0.997111i	$0.524202\pi$
$480$	0	0
$481$	18.9493	0.864015
$482$	0	0
$483$	2.80857	0.127794
$484$	0	0
$485$	−64.1773	−2.91414
$486$	0	0
$487$	23.9234	1.08407	0.542037	−	0.840355i	$-0.317653\pi$
$487$	23.9234	1.08407	0.542037	+	0.840355i	$0.317653\pi$
$488$	0	0
$489$	−22.6228	−1.02304
$490$	0	0
$491$	−14.4329	−0.651349	−0.325675	−	0.945482i	$-0.605591\pi$
$491$	−14.4329	−0.651349	−0.325675	+	0.945482i	$0.605591\pi$
$492$	0	0
$493$	−5.07777	−0.228691
$494$	0	0
$495$	35.9154	1.61428
$496$	0	0
$497$	5.88932	0.264172
$498$	0	0
$499$	25.6935	1.15020	0.575100	−	0.818083i	$-0.304963\pi$
$499$	25.6935	1.15020	0.575100	+	0.818083i	$0.304963\pi$
$500$	0	0
$501$	−20.9106	−0.934217
$502$	0	0
$503$	0.687553	0.0306565	0.0153282	−	0.999883i	$-0.495121\pi$
$503$	0.687553	0.0306565	0.0153282	+	0.999883i	$0.495121\pi$
$504$	0	0
$505$	36.1518	1.60873
$506$	0	0
$507$	−32.0734	−1.42443
$508$	0	0
$509$	−33.3619	−1.47874	−0.739370	−	0.673299i	$-0.764877\pi$
$509$	−33.3619	−1.47874	−0.739370	+	0.673299i	$0.764877\pi$
$510$	0	0
$511$	5.38983	0.238432
$512$	0	0
$513$	42.7178	1.88604
$514$	0	0
$515$	3.38500	0.149161
$516$	0	0
$517$	−6.10792	−0.268626
$518$	0	0
$519$	9.81547	0.430851
$520$	0	0
$521$	5.41799	0.237367	0.118683	−	0.992932i	$-0.462133\pi$
$521$	5.41799	0.237367	0.118683	+	0.992932i	$0.462133\pi$
$522$	0	0
$523$	−24.2654	−1.06105	−0.530526	−	0.847669i	$-0.678005\pi$
$523$	−24.2654	−1.06105	−0.530526	+	0.847669i	$0.678005\pi$
$524$	0	0
$525$	14.9168	0.651021
$526$	0	0
$527$	−31.2987	−1.36339
$528$	0	0
$529$	−17.0006	−0.739157
$530$	0	0
$531$	10.4596	0.453910
$532$	0	0
$533$	−5.21875	−0.226049
$534$	0	0
$535$	−6.81733	−0.294739
$536$	0	0
$537$	−5.35923	−0.231268
$538$	0	0
$539$	5.02211	0.216317
$540$	0	0
$541$	−37.5513	−1.61446	−0.807228	−	0.590239i	$-0.799033\pi$
$541$	−37.5513	−1.61446	−0.807228	+	0.590239i	$0.799033\pi$
$542$	0	0
$543$	−14.3271	−0.614835
$544$	0	0
$545$	−12.1207	−0.519195
$546$	0	0
$547$	3.87731	0.165782	0.0828909	−	0.996559i	$-0.473585\pi$
$547$	3.87731	0.165782	0.0828909	+	0.996559i	$0.473585\pi$
$548$	0	0
$549$	7.38083	0.315006
$550$	0	0
$551$	8.61779	0.367130
$552$	0	0
$553$	−3.84510	−0.163510
$554$	0	0
$555$	14.4055	0.611480
$556$	0	0
$557$	−34.6358	−1.46756	−0.733782	−	0.679385i	$-0.762247\pi$
$557$	−34.6358	−1.46756	−0.733782	+	0.679385i	$0.762247\pi$
$558$	0	0
$559$	38.6917	1.63648
$560$	0	0
$561$	26.9801	1.13910
$562$	0	0
$563$	17.8312	0.751497	0.375749	−	0.926722i	$-0.377386\pi$
$563$	17.8312	0.751497	0.375749	+	0.926722i	$0.377386\pi$
$564$	0	0
$565$	−28.4525	−1.19700
$566$	0	0
$567$	1.10453	0.0463860
$568$	0	0
$569$	15.0167	0.629531	0.314766	−	0.949169i	$-0.398074\pi$
$569$	15.0167	0.629531	0.314766	+	0.949169i	$0.398074\pi$
$570$	0	0
$571$	27.8475	1.16538	0.582692	−	0.812693i	$-0.302000\pi$
$571$	27.8475	1.16538	0.582692	+	0.812693i	$0.302000\pi$
$572$	0	0
$573$	25.6534	1.07168
$574$	0	0
$575$	31.8638	1.32881
$576$	0	0
$577$	34.1504	1.42170	0.710849	−	0.703344i	$-0.248311\pi$
$577$	34.1504	1.42170	0.710849	+	0.703344i	$0.248311\pi$
$578$	0	0
$579$	−15.1632	−0.630160
$580$	0	0
$581$	−10.3302	−0.428567
$582$	0	0
$583$	−28.5465	−1.18228
$584$	0	0
$585$	45.7757	1.89259
$586$	0	0
$587$	14.2835	0.589545	0.294772	−	0.955567i	$-0.404756\pi$
$587$	14.2835	0.589545	0.294772	+	0.955567i	$0.404756\pi$
$588$	0	0
$589$	53.1188	2.18872
$590$	0	0
$591$	0.686540	0.0282405
$592$	0	0
$593$	37.1785	1.52674	0.763370	−	0.645962i	$-0.223544\pi$
$593$	37.1785	1.52674	0.763370	+	0.645962i	$0.223544\pi$
$594$	0	0
$595$	−19.8826	−0.815106
$596$	0	0
$597$	4.58780	0.187766
$598$	0	0
$599$	11.2640	0.460233	0.230117	−	0.973163i	$-0.426089\pi$
$599$	11.2640	0.460233	0.230117	+	0.973163i	$0.426089\pi$
$600$	0	0
$601$	−15.1947	−0.619806	−0.309903	−	0.950768i	$-0.600297\pi$
$601$	−15.1947	−0.619806	−0.309903	+	0.950768i	$0.600297\pi$
$602$	0	0
$603$	15.1633	0.617496
$604$	0	0
$605$	−60.3520	−2.45366
$606$	0	0
$607$	−13.7773	−0.559205	−0.279602	−	0.960116i	$-0.590203\pi$
$607$	−13.7773	−0.559205	−0.279602	+	0.960116i	$0.590203\pi$
$608$	0	0
$609$	1.24273	0.0503579
$610$	0	0
$611$	−7.78482	−0.314940
$612$	0	0
$613$	−20.6926	−0.835765	−0.417883	−	0.908501i	$-0.637228\pi$
$613$	−20.6926	−0.835765	−0.417883	+	0.908501i	$0.637228\pi$
$614$	0	0
$615$	−3.96736	−0.159979
$616$	0	0
$617$	−7.40383	−0.298067	−0.149033	−	0.988832i	$-0.547616\pi$
$617$	−7.40383	−0.298067	−0.149033	+	0.988832i	$0.547616\pi$
$618$	0	0
$619$	13.8751	0.557689	0.278844	−	0.960336i	$-0.410049\pi$
$619$	13.8751	0.557689	0.278844	+	0.960336i	$0.410049\pi$
$620$	0	0
$621$	13.1587	0.528040
$622$	0	0
$623$	1.00000	0.0400642
$624$	0	0
$625$	79.1890	3.16756
$626$	0	0
$627$	−45.7896	−1.82866
$628$	0	0
$629$	−13.8701	−0.553039
$630$	0	0
$631$	13.9342	0.554713	0.277357	−	0.960767i	$-0.410542\pi$
$631$	13.9342	0.554713	0.277357	+	0.960767i	$0.410542\pi$
$632$	0	0
$633$	−0.551196	−0.0219081
$634$	0	0
$635$	45.7139	1.81410
$636$	0	0
$637$	6.40089	0.253613
$638$	0	0
$639$	9.92465	0.392613
$640$	0	0
$641$	4.34255	0.171520	0.0857601	−	0.996316i	$-0.472668\pi$
$641$	4.34255	0.171520	0.0857601	+	0.996316i	$0.472668\pi$
$642$	0	0
$643$	17.4260	0.687215	0.343608	−	0.939113i	$-0.388351\pi$
$643$	17.4260	0.687215	0.343608	+	0.939113i	$0.388351\pi$
$644$	0	0
$645$	29.4139	1.15817
$646$	0	0
$647$	−12.8143	−0.503783	−0.251891	−	0.967756i	$-0.581053\pi$
$647$	−12.8143	−0.503783	−0.251891	+	0.967756i	$0.581053\pi$
$648$	0	0
$649$	−31.1711	−1.22357
$650$	0	0
$651$	7.66000	0.300219
$652$	0	0
$653$	23.0753	0.903005	0.451502	−	0.892270i	$-0.350888\pi$
$653$	23.0753	0.903005	0.451502	+	0.892270i	$0.350888\pi$
$654$	0	0
$655$	32.8198	1.28238
$656$	0	0
$657$	9.08291	0.354358
$658$	0	0
$659$	24.8537	0.968162	0.484081	−	0.875023i	$-0.339154\pi$
$659$	24.8537	0.968162	0.484081	+	0.875023i	$0.339154\pi$
$660$	0	0
$661$	−22.7052	−0.883128	−0.441564	−	0.897230i	$-0.645576\pi$
$661$	−22.7052	−0.883128	−0.441564	+	0.897230i	$0.645576\pi$
$662$	0	0
$663$	34.3874	1.33549
$664$	0	0
$665$	33.7439	1.30853
$666$	0	0
$667$	2.65460	0.102787
$668$	0	0
$669$	−17.7297	−0.685471
$670$	0	0
$671$	−21.9959	−0.849141
$672$	0	0
$673$	−32.5570	−1.25498	−0.627489	−	0.778625i	$-0.715917\pi$
$673$	−32.5570	−1.25498	−0.627489	+	0.778625i	$0.715917\pi$
$674$	0	0
$675$	69.8879	2.68999
$676$	0	0
$677$	45.6071	1.75282	0.876412	−	0.481561i	$-0.159930\pi$
$677$	45.6071	1.75282	0.876412	+	0.481561i	$0.159930\pi$
$678$	0	0
$679$	−15.1229	−0.580365
$680$	0	0
$681$	−24.4735	−0.937828
$682$	0	0
$683$	12.7957	0.489613	0.244806	−	0.969572i	$-0.421276\pi$
$683$	12.7957	0.489613	0.244806	+	0.969572i	$0.421276\pi$
$684$	0	0
$685$	−69.8324	−2.66816
$686$	0	0
$687$	26.8845	1.02571
$688$	0	0
$689$	−36.3838	−1.38611
$690$	0	0
$691$	21.7675	0.828075	0.414037	−	0.910260i	$-0.364118\pi$
$691$	21.7675	0.828075	0.414037	+	0.910260i	$0.364118\pi$
$692$	0	0
$693$	8.46323	0.321491
$694$	0	0
$695$	10.4504	0.396408
$696$	0	0
$697$	3.81992	0.144690
$698$	0	0
$699$	25.0958	0.949211
$700$	0	0
$701$	−30.3304	−1.14556	−0.572782	−	0.819708i	$-0.694136\pi$
$701$	−30.3304	−1.14556	−0.572782	+	0.819708i	$0.694136\pi$
$702$	0	0
$703$	23.5399	0.887822
$704$	0	0
$705$	−5.91811	−0.222889
$706$	0	0
$707$	8.51893	0.320387
$708$	0	0
$709$	0.840340	0.0315596	0.0157798	−	0.999875i	$-0.494977\pi$
$709$	0.840340	0.0315596	0.0157798	+	0.999875i	$0.494977\pi$
$710$	0	0
$711$	−6.47975	−0.243010
$712$	0	0
$713$	16.3626	0.612784
$714$	0	0
$715$	−136.418	−5.10174
$716$	0	0
$717$	−28.1728	−1.05213
$718$	0	0
$719$	−14.3892	−0.536628	−0.268314	−	0.963331i	$-0.586467\pi$
$719$	−14.3892	−0.536628	−0.268314	+	0.963331i	$0.586467\pi$
$720$	0	0
$721$	0.797652	0.0297061
$722$	0	0
$723$	35.0635	1.30403
$724$	0	0
$725$	14.0990	0.523625
$726$	0	0
$727$	−10.6173	−0.393773	−0.196887	−	0.980426i	$-0.563083\pi$
$727$	−10.6173	−0.393773	−0.196887	+	0.980426i	$0.563083\pi$
$728$	0	0
$729$	20.3417	0.753397
$730$	0	0
$731$	−28.3208	−1.04748
$732$	0	0
$733$	6.79142	0.250847	0.125423	−	0.992103i	$-0.459971\pi$
$733$	6.79142	0.250847	0.125423	+	0.992103i	$0.459971\pi$
$734$	0	0
$735$	4.86604	0.179486
$736$	0	0
$737$	−45.1886	−1.66454
$738$	0	0
$739$	−38.7686	−1.42612	−0.713062	−	0.701101i	$-0.752692\pi$
$739$	−38.7686	−1.42612	−0.713062	+	0.701101i	$0.752692\pi$
$740$	0	0
$741$	−58.3608	−2.14394
$742$	0	0
$743$	30.1636	1.10660	0.553298	−	0.832983i	$-0.313369\pi$
$743$	30.1636	1.10660	0.553298	+	0.832983i	$0.313369\pi$
$744$	0	0
$745$	−94.9116	−3.47729
$746$	0	0
$747$	−17.4083	−0.636938
$748$	0	0
$749$	−1.60646	−0.0586987
$750$	0	0
$751$	45.1837	1.64878	0.824388	−	0.566025i	$-0.191519\pi$
$751$	45.1837	1.64878	0.824388	+	0.566025i	$0.191519\pi$
$752$	0	0
$753$	21.1819	0.771912
$754$	0	0
$755$	−103.760	−3.77622
$756$	0	0
$757$	11.7935	0.428643	0.214321	−	0.976763i	$-0.431246\pi$
$757$	11.7935	0.428643	0.214321	+	0.976763i	$0.431246\pi$
$758$	0	0
$759$	−14.1049	−0.511976
$760$	0	0
$761$	20.2734	0.734910	0.367455	−	0.930041i	$-0.380229\pi$
$761$	20.2734	0.734910	0.367455	+	0.930041i	$0.380229\pi$
$762$	0	0
$763$	−2.85617	−0.103400
$764$	0	0
$765$	−33.5060	−1.21141
$766$	0	0
$767$	−39.7290	−1.43453
$768$	0	0
$769$	−46.2030	−1.66612	−0.833061	−	0.553182i	$-0.813414\pi$
$769$	−46.2030	−1.66612	−0.833061	+	0.553182i	$0.813414\pi$
$770$	0	0
$771$	27.2712	0.982148
$772$	0	0
$773$	−53.1268	−1.91084	−0.955419	−	0.295253i	$-0.904596\pi$
$773$	−53.1268	−1.91084	−0.955419	+	0.295253i	$0.904596\pi$
$774$	0	0
$775$	86.9044	3.12170
$776$	0	0
$777$	3.39456	0.121779
$778$	0	0
$779$	−6.48301	−0.232278
$780$	0	0
$781$	−29.5768	−1.05834
$782$	0	0
$783$	5.82242	0.208076
$784$	0	0
$785$	34.9863	1.24872
$786$	0	0
$787$	−41.3378	−1.47353	−0.736767	−	0.676147i	$-0.763649\pi$
$787$	−41.3378	−1.47353	−0.736767	+	0.676147i	$0.763649\pi$
$788$	0	0
$789$	−3.31186	−0.117905
$790$	0	0
$791$	−6.70463	−0.238389
$792$	0	0
$793$	−28.0347	−0.995541
$794$	0	0
$795$	−27.6594	−0.980978
$796$	0	0
$797$	10.8063	0.382778	0.191389	−	0.981514i	$-0.438701\pi$
$797$	10.8063	0.382778	0.191389	+	0.981514i	$0.438701\pi$
$798$	0	0
$799$	5.69817	0.201587
$800$	0	0
$801$	1.68519	0.0595434
$802$	0	0
$803$	−27.0683	−0.955219
$804$	0	0
$805$	10.3944	0.366354
$806$	0	0
$807$	8.44628	0.297323
$808$	0	0
$809$	24.5110	0.861761	0.430880	−	0.902409i	$-0.358203\pi$
$809$	24.5110	0.861761	0.430880	+	0.902409i	$0.358203\pi$
$810$	0	0
$811$	−26.8057	−0.941275	−0.470637	−	0.882327i	$-0.655976\pi$
$811$	−26.8057	−0.941275	−0.470637	+	0.882327i	$0.655976\pi$
$812$	0	0
$813$	29.6375	1.03943
$814$	0	0
$815$	−83.7261	−2.93280
$816$	0	0
$817$	48.0648	1.68158
$818$	0	0
$819$	10.7868	0.376920
$820$	0	0
$821$	−1.34905	−0.0470823	−0.0235411	−	0.999723i	$-0.507494\pi$
$821$	−1.34905	−0.0470823	−0.0235411	+	0.999723i	$0.507494\pi$
$822$	0	0
$823$	21.7395	0.757790	0.378895	−	0.925440i	$-0.376304\pi$
$823$	21.7395	0.757790	0.378895	+	0.925440i	$0.376304\pi$
$824$	0	0
$825$	−74.9135	−2.60815
$826$	0	0
$827$	4.43226	0.154125	0.0770623	−	0.997026i	$-0.475446\pi$
$827$	4.43226	0.154125	0.0770623	+	0.997026i	$0.475446\pi$
$828$	0	0
$829$	6.84137	0.237611	0.118805	−	0.992918i	$-0.462094\pi$
$829$	6.84137	0.237611	0.118805	+	0.992918i	$0.462094\pi$
$830$	0	0
$831$	−11.0474	−0.383231
$832$	0	0
$833$	−4.68519	−0.162332
$834$	0	0
$835$	−77.3893	−2.67817
$836$	0	0
$837$	35.8886	1.24049
$838$	0	0
$839$	51.2101	1.76797	0.883984	−	0.467518i	$-0.154852\pi$
$839$	51.2101	1.76797	0.883984	+	0.467518i	$0.154852\pi$
$840$	0	0
$841$	−27.8254	−0.959496
$842$	0	0
$843$	−8.98361	−0.309412
$844$	0	0
$845$	−118.702	−4.08348
$846$	0	0
$847$	−14.2215	−0.488658
$848$	0	0
$849$	−12.3320	−0.423234
$850$	0	0
$851$	7.25116	0.248566
$852$	0	0
$853$	−24.0788	−0.824443	−0.412222	−	0.911084i	$-0.635247\pi$
$853$	−24.0788	−0.824443	−0.412222	+	0.911084i	$0.635247\pi$
$854$	0	0
$855$	56.8650	1.94474
$856$	0	0
$857$	23.2395	0.793847	0.396924	−	0.917852i	$-0.370078\pi$
$857$	23.2395	0.793847	0.396924	+	0.917852i	$0.370078\pi$
$858$	0	0
$859$	−19.6148	−0.669247	−0.334623	−	0.942352i	$-0.608609\pi$
$859$	−19.6148	−0.669247	−0.334623	+	0.942352i	$0.608609\pi$
$860$	0	0
$861$	−0.934882	−0.0318607
$862$	0	0
$863$	19.6909	0.670288	0.335144	−	0.942167i	$-0.391215\pi$
$863$	19.6909	0.670288	0.335144	+	0.942167i	$0.391215\pi$
$864$	0	0
$865$	36.3266	1.23514
$866$	0	0
$867$	−5.67712	−0.192805
$868$	0	0
$869$	19.3105	0.655065
$870$	0	0
$871$	−57.5948	−1.95152
$872$	0	0
$873$	−25.4851	−0.862540
$874$	0	0
$875$	33.9878	1.14900
$876$	0	0
$877$	4.55181	0.153704	0.0768519	−	0.997043i	$-0.475513\pi$
$877$	4.55181	0.153704	0.0768519	+	0.997043i	$0.475513\pi$
$878$	0	0
$879$	17.8554	0.602249
$880$	0	0
$881$	44.0334	1.48352	0.741761	−	0.670665i	$-0.233991\pi$
$881$	44.0334	1.48352	0.741761	+	0.670665i	$0.233991\pi$
$882$	0	0
$883$	2.42526	0.0816164	0.0408082	−	0.999167i	$-0.487007\pi$
$883$	2.42526	0.0816164	0.0408082	+	0.999167i	$0.487007\pi$
$884$	0	0
$885$	−30.2025	−1.01524
$886$	0	0
$887$	51.0560	1.71429	0.857146	−	0.515073i	$-0.172235\pi$
$887$	51.0560	1.71429	0.857146	+	0.515073i	$0.172235\pi$
$888$	0	0
$889$	10.7722	0.361287
$890$	0	0
$891$	−5.54708	−0.185834
$892$	0	0
$893$	−9.67071	−0.323618
$894$	0	0
$895$	−19.8343	−0.662987
$896$	0	0
$897$	−17.9773	−0.600245
$898$	0	0
$899$	7.24008	0.241470
$900$	0	0
$901$	26.6315	0.887222
$902$	0	0
$903$	6.93119	0.230656
$904$	0	0
$905$	−53.0240	−1.76258
$906$	0	0
$907$	−51.0939	−1.69655	−0.848273	−	0.529559i	$-0.822357\pi$
$907$	−51.0939	−1.69655	−0.848273	+	0.529559i	$0.822357\pi$
$908$	0	0
$909$	14.3561	0.476160
$910$	0	0
$911$	14.3351	0.474942	0.237471	−	0.971395i	$-0.423681\pi$
$911$	14.3351	0.474942	0.237471	+	0.971395i	$0.423681\pi$
$912$	0	0
$913$	51.8792	1.71695
$914$	0	0
$915$	−21.3123	−0.704563
$916$	0	0
$917$	7.73377	0.255392
$918$	0	0
$919$	−41.6670	−1.37447	−0.687233	−	0.726437i	$-0.741175\pi$
$919$	−41.6670	−1.37447	−0.687233	+	0.726437i	$0.741175\pi$
$920$	0	0
$921$	−30.6519	−1.01001
$922$	0	0
$923$	−37.6969	−1.24081
$924$	0	0
$925$	38.5121	1.26627
$926$	0	0
$927$	1.34420	0.0441493
$928$	0	0
$929$	45.0283	1.47733	0.738666	−	0.674072i	$-0.235456\pi$
$929$	45.0283	1.47733	0.738666	+	0.674072i	$0.235456\pi$
$930$	0	0
$931$	7.95152	0.260601
$932$	0	0
$933$	35.9623	1.17735
$934$	0	0
$935$	99.8523	3.26552
$936$	0	0
$937$	28.7393	0.938873	0.469437	−	0.882966i	$-0.344457\pi$
$937$	28.7393	0.938873	0.469437	+	0.882966i	$0.344457\pi$
$938$	0	0
$939$	7.93952	0.259097
$940$	0	0
$941$	−51.1448	−1.66727	−0.833636	−	0.552315i	$-0.813745\pi$
$941$	−51.1448	−1.66727	−0.833636	+	0.552315i	$0.813745\pi$
$942$	0	0
$943$	−1.99701	−0.0650316
$944$	0	0
$945$	22.7983	0.741630
$946$	0	0
$947$	24.5729	0.798513	0.399256	−	0.916839i	$-0.369268\pi$
$947$	24.5729	0.798513	0.399256	+	0.916839i	$0.369268\pi$
$948$	0	0
$949$	−34.4997	−1.11991
$950$	0	0
$951$	−11.6786	−0.378704
$952$	0	0
$953$	48.5891	1.57396	0.786978	−	0.616981i	$-0.211644\pi$
$953$	48.5891	1.57396	0.786978	+	0.616981i	$0.211644\pi$
$954$	0	0
$955$	94.9420	3.07225
$956$	0	0
$957$	−6.24111	−0.201747
$958$	0	0
$959$	−16.4555	−0.531377
$960$	0	0
$961$	13.6268	0.439575
$962$	0	0
$963$	−2.70720	−0.0872382
$964$	0	0
$965$	−56.1182	−1.80651
$966$	0	0
$967$	−28.3555	−0.911852	−0.455926	−	0.890018i	$-0.650692\pi$
$967$	−28.3555	−0.911852	−0.455926	+	0.890018i	$0.650692\pi$
$968$	0	0
$969$	42.7178	1.37229
$970$	0	0
$971$	−48.8235	−1.56682	−0.783411	−	0.621505i	$-0.786522\pi$
$971$	−48.8235	−1.56682	−0.783411	+	0.621505i	$0.786522\pi$
$972$	0	0
$973$	2.46258	0.0789466
$974$	0	0
$975$	−95.4805	−3.05782
$976$	0	0
$977$	−7.10802	−0.227406	−0.113703	−	0.993515i	$-0.536271\pi$
$977$	−7.10802	−0.227406	−0.113703	+	0.993515i	$0.536271\pi$
$978$	0	0
$979$	−5.02211	−0.160507
$980$	0	0
$981$	−4.81320	−0.153674
$982$	0	0
$983$	−0.823967	−0.0262805	−0.0131402	−	0.999914i	$-0.504183\pi$
$983$	−0.823967	−0.0262805	−0.0131402	+	0.999914i	$0.504183\pi$
$984$	0	0
$985$	2.54086	0.0809584
$986$	0	0
$987$	−1.39456	−0.0443895
$988$	0	0
$989$	14.8058	0.470796
$990$	0	0
$991$	−20.5482	−0.652735	−0.326367	−	0.945243i	$-0.605825\pi$
$991$	−20.5482	−0.652735	−0.326367	+	0.945243i	$0.605825\pi$
$992$	0	0
$993$	24.1187	0.765384
$994$	0	0
$995$	16.9792	0.538278
$996$	0	0
$997$	9.40494	0.297858	0.148929	−	0.988848i	$-0.452417\pi$
$997$	9.40494	0.297858	0.148929	+	0.988848i	$0.452417\pi$
$998$	0	0
$999$	15.9042	0.503186

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9968.2.a.z.1.1		5
4.3	odd	2		1246.2.a.n.1.5	✓	5
28.27	even	2		8722.2.a.x.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1246.2.a.n.1.5	✓	5	4.3	odd	2
8722.2.a.x.1.1		5	28.27	even	2
9968.2.a.z.1.1		5	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 9968 = 2^{4} \cdot 7 \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9968.a (trivial)

\(f(q)\)	\(=\)	\(q-1.14665 q^{3} -4.24370 q^{5} -1.00000 q^{7} -1.68519 q^{9} +O(q^{10})\)	\(q-1.14665 q^{3} -4.24370 q^{5} -1.00000 q^{7} -1.68519 q^{9} +5.02211 q^{11} +6.40089 q^{13} +4.86604 q^{15} -4.68519 q^{17} +7.95152 q^{19} +1.14665 q^{21} +2.44937 q^{23} +13.0090 q^{25} +5.37228 q^{27} +1.08379 q^{29} +6.68033 q^{31} -5.75859 q^{33} +4.24370 q^{35} +2.96042 q^{37} -7.33958 q^{39} -0.815317 q^{41} +6.04473 q^{43} +7.15146 q^{45} -1.21621 q^{47} +1.00000 q^{49} +5.37228 q^{51} -5.68417 q^{53} -21.3123 q^{55} -9.11761 q^{57} -6.20679 q^{59} -4.37981 q^{61} +1.68519 q^{63} -27.1635 q^{65} -8.99793 q^{67} -2.80857 q^{69} -5.88932 q^{71} -5.38983 q^{73} -14.9168 q^{75} -5.02211 q^{77} +3.84510 q^{79} -1.10453 q^{81} +10.3302 q^{83} +19.8826 q^{85} -1.24273 q^{87} -1.00000 q^{89} -6.40089 q^{91} -7.66000 q^{93} -33.7439 q^{95} +15.1229 q^{97} -8.46323 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 6 q^{3} - 10 q^{5} - 5 q^{7} + 7 q^{9}+O(q^{10}) \) 5 * q + 6 * q^3 - 10 * q^5 - 5 * q^7 + 7 * q^9	\( 5 q + 6 q^{3} - 10 q^{5} - 5 q^{7} + 7 q^{9} + 8 q^{11} - 8 q^{13} - 6 q^{15} - 8 q^{17} + 10 q^{19} - 6 q^{21} + 2 q^{23} + 17 q^{25} + 24 q^{27} - 10 q^{29} + 10 q^{31} + 10 q^{35} - 4 q^{37} - 24 q^{39} - 28 q^{41} + 10 q^{43} - 12 q^{45} + 10 q^{47} + 5 q^{49} + 24 q^{51} + 4 q^{53} - 4 q^{55} + 2 q^{57} + 10 q^{59} - 16 q^{61} - 7 q^{63} - 26 q^{65} - 2 q^{69} + 16 q^{71} + 10 q^{73} + 10 q^{75} - 8 q^{77} + 8 q^{79} + 41 q^{81} + 14 q^{83} + 18 q^{85} - 22 q^{87} - 5 q^{89} + 8 q^{91} + 22 q^{93} - 34 q^{95} + 6 q^{97} + 8 q^{99}+O(q^{100}) \) 5 * q + 6 * q^3 - 10 * q^5 - 5 * q^7 + 7 * q^9 + 8 * q^11 - 8 * q^13 - 6 * q^15 - 8 * q^17 + 10 * q^19 - 6 * q^21 + 2 * q^23 + 17 * q^25 + 24 * q^27 - 10 * q^29 + 10 * q^31 + 10 * q^35 - 4 * q^37 - 24 * q^39 - 28 * q^41 + 10 * q^43 - 12 * q^45 + 10 * q^47 + 5 * q^49 + 24 * q^51 + 4 * q^53 - 4 * q^55 + 2 * q^57 + 10 * q^59 - 16 * q^61 - 7 * q^63 - 26 * q^65 - 2 * q^69 + 16 * q^71 + 10 * q^73 + 10 * q^75 - 8 * q^77 + 8 * q^79 + 41 * q^81 + 14 * q^83 + 18 * q^85 - 22 * q^87 - 5 * q^89 + 8 * q^91 + 22 * q^93 - 34 * q^95 + 6 * q^97 + 8 * q^99

Introduction

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