LMFDB - Embedded newform 9968.2.a.z.1.4

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			1.4
Root			$-1.94486$ 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)

$f(q)$	$=$	$q+2.60528 q^{3} +1.72732 q^{5} -1.00000 q^{7} +3.78749 q^{9} +O(q^{10})$	$q+2.60528 q^{3} +1.72732 q^{5} -1.00000 q^{7} +3.78749 q^{9} +5.39794 q^{11} -6.59329 q^{13} +4.50016 q^{15} +0.787490 q^{17} +1.71950 q^{19} -2.60528 q^{21} -4.31279 q^{23} -2.01636 q^{25} +2.05163 q^{27} +8.40969 q^{29} +8.32800 q^{31} +14.0631 q^{33} -1.72732 q^{35} -3.61382 q^{37} -17.1774 q^{39} -9.74714 q^{41} +11.6322 q^{43} +6.54222 q^{45} +2.84614 q^{47} +1.00000 q^{49} +2.05163 q^{51} +6.70679 q^{53} +9.32398 q^{55} +4.47978 q^{57} +2.34920 q^{59} +2.07192 q^{61} -3.78749 q^{63} -11.3887 q^{65} +7.59205 q^{67} -11.2360 q^{69} +12.1286 q^{71} +9.16898 q^{73} -5.25318 q^{75} -5.39794 q^{77} -14.9245 q^{79} -6.01739 q^{81} +15.6743 q^{83} +1.36025 q^{85} +21.9096 q^{87} -1.00000 q^{89} +6.59329 q^{91} +21.6968 q^{93} +2.97013 q^{95} +9.50409 q^{97} +20.4446 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2.60528	1.50416	0.752080	−	0.659072i	$-0.229051\pi$
$3$	2.60528	1.50416	0.752080	+	0.659072i	$0.229051\pi$
$4$	0	0
$5$	1.72732	0.772482	0.386241	−	0.922398i	$-0.373773\pi$
$5$	1.72732	0.772482	0.386241	+	0.922398i	$0.373773\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	3.78749	1.26250
$10$	0	0
$11$	5.39794	1.62754	0.813770	−	0.581188i	$-0.197412\pi$
$11$	5.39794	1.62754	0.813770	+	0.581188i	$0.197412\pi$
$12$	0	0
$13$	−6.59329	−1.82865	−0.914324	−	0.404983i	$-0.867277\pi$
$13$	−6.59329	−1.82865	−0.914324	+	0.404983i	$0.867277\pi$
$14$	0	0
$15$	4.50016	1.16194
$16$	0	0
$17$	0.787490	0.190994	0.0954972	−	0.995430i	$-0.469556\pi$
$17$	0.787490	0.190994	0.0954972	+	0.995430i	$0.469556\pi$
$18$	0	0
$19$	1.71950	0.394480	0.197240	−	0.980355i	$-0.436802\pi$
$19$	1.71950	0.394480	0.197240	+	0.980355i	$0.436802\pi$
$20$	0	0
$21$	−2.60528	−0.568519
$22$	0	0
$23$	−4.31279	−0.899278	−0.449639	−	0.893210i	$-0.648447\pi$
$23$	−4.31279	−0.899278	−0.449639	+	0.893210i	$0.648447\pi$
$24$	0	0
$25$	−2.01636	−0.403272
$26$	0	0
$27$	2.05163	0.394837
$28$	0	0
$29$	8.40969	1.56164	0.780820	−	0.624756i	$-0.214801\pi$
$29$	8.40969	1.56164	0.780820	+	0.624756i	$0.214801\pi$
$30$	0	0
$31$	8.32800	1.49575	0.747876	−	0.663838i	$-0.231074\pi$
$31$	8.32800	1.49575	0.747876	+	0.663838i	$0.231074\pi$
$32$	0	0
$33$	14.0631	2.44808
$34$	0	0
$35$	−1.72732	−0.291971
$36$	0	0
$37$	−3.61382	−0.594108	−0.297054	−	0.954861i	$-0.596004\pi$
$37$	−3.61382	−0.594108	−0.297054	+	0.954861i	$0.596004\pi$
$38$	0	0
$39$	−17.1774	−2.75058
$40$	0	0
$41$	−9.74714	−1.52225	−0.761124	−	0.648607i	$-0.775352\pi$
$41$	−9.74714	−1.52225	−0.761124	+	0.648607i	$0.775352\pi$
$42$	0	0
$43$	11.6322	1.77390	0.886951	−	0.461864i	$-0.152819\pi$
$43$	11.6322	1.77390	0.886951	+	0.461864i	$0.152819\pi$
$44$	0	0
$45$	6.54222	0.975256
$46$	0	0
$47$	2.84614	0.415153	0.207576	−	0.978219i	$-0.433442\pi$
$47$	2.84614	0.415153	0.207576	+	0.978219i	$0.433442\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	2.05163	0.287286
$52$	0	0
$53$	6.70679	0.921249	0.460624	−	0.887595i	$-0.347626\pi$
$53$	6.70679	0.921249	0.460624	+	0.887595i	$0.347626\pi$
$54$	0	0
$55$	9.32398	1.25724
$56$	0	0
$57$	4.47978	0.593361
$58$	0	0
$59$	2.34920	0.305840	0.152920	−	0.988239i	$-0.451132\pi$
$59$	2.34920	0.305840	0.152920	+	0.988239i	$0.451132\pi$
$60$	0	0
$61$	2.07192	0.265282	0.132641	−	0.991164i	$-0.457654\pi$
$61$	2.07192	0.265282	0.132641	+	0.991164i	$0.457654\pi$
$62$	0	0
$63$	−3.78749	−0.477179
$64$	0	0
$65$	−11.3887	−1.41260
$66$	0	0
$67$	7.59205	0.927517	0.463758	−	0.885962i	$-0.346501\pi$
$67$	7.59205	0.927517	0.463758	+	0.885962i	$0.346501\pi$
$68$	0	0
$69$	−11.2360	−1.35266
$70$	0	0
$71$	12.1286	1.43940	0.719702	−	0.694283i	$-0.244279\pi$
$71$	12.1286	1.43940	0.719702	+	0.694283i	$0.244279\pi$
$72$	0	0
$73$	9.16898	1.07315	0.536574	−	0.843853i	$-0.319718\pi$
$73$	9.16898	1.07315	0.536574	+	0.843853i	$0.319718\pi$
$74$	0	0
$75$	−5.25318	−0.606585
$76$	0	0
$77$	−5.39794	−0.615152
$78$	0	0
$79$	−14.9245	−1.67914	−0.839569	−	0.543253i	$-0.817192\pi$
$79$	−14.9245	−1.67914	−0.839569	+	0.543253i	$0.817192\pi$
$80$	0	0
$81$	−6.01739	−0.668599
$82$	0	0
$83$	15.6743	1.72048	0.860239	−	0.509891i	$-0.170314\pi$
$83$	15.6743	1.72048	0.860239	+	0.509891i	$0.170314\pi$
$84$	0	0
$85$	1.36025	0.147540
$86$	0	0
$87$	21.9096	2.34896
$88$	0	0
$89$	−1.00000	−0.106000
$89$	−1.00000	−0.106000
$90$	0	0
$91$	6.59329	0.691164
$92$	0	0
$93$	21.6968	2.24985
$94$	0	0
$95$	2.97013	0.304729
$96$	0	0
$97$	9.50409	0.964994	0.482497	−	0.875898i	$-0.339730\pi$
$97$	9.50409	0.964994	0.482497	+	0.875898i	$0.339730\pi$
$98$	0	0
$99$	20.4446	2.05476
$100$	0	0
$101$	−1.44635	−0.143917	−0.0719587	−	0.997408i	$-0.522925\pi$
$101$	−1.44635	−0.143917	−0.0719587	+	0.997408i	$0.522925\pi$
$102$	0	0
$103$	17.2857	1.70321	0.851605	−	0.524183i	$-0.175629\pi$
$103$	17.2857	1.70321	0.851605	+	0.524183i	$0.175629\pi$
$104$	0	0
$105$	−4.50016	−0.439171
$106$	0	0
$107$	2.50429	0.242099	0.121050	−	0.992646i	$-0.461374\pi$
$107$	2.50429	0.242099	0.121050	+	0.992646i	$0.461374\pi$
$108$	0	0
$109$	−14.3488	−1.37437	−0.687183	−	0.726484i	$-0.741153\pi$
$109$	−14.3488	−1.37437	−0.687183	+	0.726484i	$0.741153\pi$
$110$	0	0
$111$	−9.41501	−0.893633
$112$	0	0
$113$	−2.38149	−0.224032	−0.112016	−	0.993706i	$-0.535731\pi$
$113$	−2.38149	−0.224032	−0.112016	+	0.993706i	$0.535731\pi$
$114$	0	0
$115$	−7.44957	−0.694676
$116$	0	0
$117$	−24.9720	−2.30866
$118$	0	0
$119$	−0.787490	−0.0721891
$120$	0	0
$121$	18.1377	1.64888
$122$	0	0
$123$	−25.3940	−2.28970
$124$	0	0
$125$	−12.1195	−1.08400
$126$	0	0
$127$	−21.4505	−1.90343	−0.951713	−	0.306991i	$-0.900678\pi$
$127$	−21.4505	−1.90343	−0.951713	+	0.306991i	$0.900678\pi$
$128$	0	0
$129$	30.3053	2.66823
$130$	0	0
$131$	6.19003	0.540826	0.270413	−	0.962744i	$-0.412840\pi$
$131$	6.19003	0.540826	0.270413	+	0.962744i	$0.412840\pi$
$132$	0	0
$133$	−1.71950	−0.149100
$134$	0	0
$135$	3.54383	0.305004
$136$	0	0
$137$	15.6249	1.33493	0.667465	−	0.744641i	$-0.267379\pi$
$137$	15.6249	1.33493	0.667465	+	0.744641i	$0.267379\pi$
$138$	0	0
$139$	3.58153	0.303782	0.151891	−	0.988397i	$-0.451464\pi$
$139$	3.58153	0.303782	0.151891	+	0.988397i	$0.451464\pi$
$140$	0	0
$141$	7.41501	0.624456
$142$	0	0
$143$	−35.5901	−2.97620
$144$	0	0
$145$	14.5262	1.20634
$146$	0	0
$147$	2.60528	0.214880
$148$	0	0
$149$	19.8381	1.62520	0.812598	−	0.582824i	$-0.198052\pi$
$149$	19.8381	1.62520	0.812598	+	0.582824i	$0.198052\pi$
$150$	0	0
$151$	−13.7250	−1.11693	−0.558463	−	0.829530i	$-0.688609\pi$
$151$	−13.7250	−1.11693	−0.558463	+	0.829530i	$0.688609\pi$
$152$	0	0
$153$	2.98261	0.241130
$154$	0	0
$155$	14.3851	1.15544
$156$	0	0
$157$	10.3274	0.824219	0.412110	−	0.911134i	$-0.364792\pi$
$157$	10.3274	0.824219	0.412110	+	0.911134i	$0.364792\pi$
$158$	0	0
$159$	17.4731	1.38571
$160$	0	0
$161$	4.31279	0.339895
$162$	0	0
$163$	0.325343	0.0254828	0.0127414	−	0.999919i	$-0.495944\pi$
$163$	0.325343	0.0254828	0.0127414	+	0.999919i	$0.495944\pi$
$164$	0	0
$165$	24.2916	1.89110
$166$	0	0
$167$	0.713189	0.0551882	0.0275941	−	0.999619i	$-0.491215\pi$
$167$	0.713189	0.0551882	0.0275941	+	0.999619i	$0.491215\pi$
$168$	0	0
$169$	30.4714	2.34395
$170$	0	0
$171$	6.51259	0.498030
$172$	0	0
$173$	−2.37935	−0.180899	−0.0904495	−	0.995901i	$-0.528830\pi$
$173$	−2.37935	−0.180899	−0.0904495	+	0.995901i	$0.528830\pi$
$174$	0	0
$175$	2.01636	0.152422
$176$	0	0
$177$	6.12034	0.460033
$178$	0	0
$179$	16.3709	1.22362	0.611808	−	0.791007i	$-0.290443\pi$
$179$	16.3709	1.22362	0.611808	+	0.791007i	$0.290443\pi$
$180$	0	0
$181$	−11.5436	−0.858032	−0.429016	−	0.903297i	$-0.641140\pi$
$181$	−11.5436	−0.858032	−0.429016	+	0.903297i	$0.641140\pi$
$182$	0	0
$183$	5.39794	0.399027
$184$	0	0
$185$	−6.24223	−0.458938
$186$	0	0
$187$	4.25082	0.310851
$188$	0	0
$189$	−2.05163	−0.149234
$190$	0	0
$191$	13.0679	0.945560	0.472780	−	0.881181i	$-0.343251\pi$
$191$	13.0679	0.945560	0.472780	+	0.881181i	$0.343251\pi$
$192$	0	0
$193$	−18.9157	−1.36158	−0.680792	−	0.732477i	$-0.738364\pi$
$193$	−18.9157	−1.36158	−0.680792	+	0.732477i	$0.738364\pi$
$194$	0	0
$195$	−29.6708	−2.12477
$196$	0	0
$197$	16.1891	1.15343	0.576713	−	0.816947i	$-0.304335\pi$
$197$	16.1891	1.15343	0.576713	+	0.816947i	$0.304335\pi$
$198$	0	0
$199$	−13.6727	−0.969235	−0.484618	−	0.874726i	$-0.661041\pi$
$199$	−13.6727	−0.969235	−0.484618	+	0.874726i	$0.661041\pi$
$200$	0	0
$201$	19.7794	1.39513
$202$	0	0
$203$	−8.40969	−0.590245
$204$	0	0
$205$	−16.8365	−1.17591
$206$	0	0
$207$	−16.3346	−1.13534
$208$	0	0
$209$	9.28175	0.642032
$210$	0	0
$211$	−22.2288	−1.53029	−0.765145	−	0.643858i	$-0.777333\pi$
$211$	−22.2288	−1.53029	−0.765145	+	0.643858i	$0.777333\pi$
$212$	0	0
$213$	31.5985	2.16509
$214$	0	0
$215$	20.0926	1.37031
$216$	0	0
$217$	−8.32800	−0.565341
$218$	0	0
$219$	23.8878	1.61419
$220$	0	0
$221$	−5.19214	−0.349261
$222$	0	0
$223$	−20.3639	−1.36367	−0.681835	−	0.731506i	$-0.738818\pi$
$223$	−20.3639	−1.36367	−0.681835	+	0.731506i	$0.738818\pi$
$224$	0	0
$225$	−7.63694	−0.509129
$226$	0	0
$227$	2.45953	0.163245	0.0816224	−	0.996663i	$-0.473990\pi$
$227$	2.45953	0.163245	0.0816224	+	0.996663i	$0.473990\pi$
$228$	0	0
$229$	−18.1219	−1.19753	−0.598764	−	0.800925i	$-0.704341\pi$
$229$	−18.1219	−1.19753	−0.598764	+	0.800925i	$0.704341\pi$
$230$	0	0
$231$	−14.0631	−0.925287
$232$	0	0
$233$	19.6400	1.28666	0.643329	−	0.765590i	$-0.277553\pi$
$233$	19.6400	1.28666	0.643329	+	0.765590i	$0.277553\pi$
$234$	0	0
$235$	4.91621	0.320698
$236$	0	0
$237$	−38.8825	−2.52569
$238$	0	0
$239$	8.19937	0.530373	0.265187	−	0.964197i	$-0.414566\pi$
$239$	8.19937	0.530373	0.265187	+	0.964197i	$0.414566\pi$
$240$	0	0
$241$	−14.2824	−0.920008	−0.460004	−	0.887917i	$-0.652152\pi$
$241$	−14.2824	−0.920008	−0.460004	+	0.887917i	$0.652152\pi$
$242$	0	0
$243$	−21.8319	−1.40052
$244$	0	0
$245$	1.72732	0.110355
$246$	0	0
$247$	−11.3372	−0.721366
$248$	0	0
$249$	40.8360	2.58787
$250$	0	0
$251$	−10.4553	−0.659935	−0.329967	−	0.943992i	$-0.607038\pi$
$251$	−10.4553	−0.659935	−0.329967	+	0.943992i	$0.607038\pi$
$252$	0	0
$253$	−23.2801	−1.46361
$254$	0	0
$255$	3.54383	0.221923
$256$	0	0
$257$	−8.32673	−0.519407	−0.259704	−	0.965688i	$-0.583625\pi$
$257$	−8.32673	−0.519407	−0.259704	+	0.965688i	$0.583625\pi$
$258$	0	0
$259$	3.61382	0.224552
$260$	0	0
$261$	31.8516	1.97157
$262$	0	0
$263$	−22.0479	−1.35953	−0.679767	−	0.733428i	$-0.737919\pi$
$263$	−22.0479	−1.35953	−0.679767	+	0.733428i	$0.737919\pi$
$264$	0	0
$265$	11.5848	0.711648
$266$	0	0
$267$	−2.60528	−0.159441
$268$	0	0
$269$	−8.37688	−0.510748	−0.255374	−	0.966842i	$-0.582199\pi$
$269$	−8.37688	−0.510748	−0.255374	+	0.966842i	$0.582199\pi$
$270$	0	0
$271$	−7.52860	−0.457330	−0.228665	−	0.973505i	$-0.573436\pi$
$271$	−7.52860	−0.457330	−0.228665	+	0.973505i	$0.573436\pi$
$272$	0	0
$273$	17.1774	1.03962
$274$	0	0
$275$	−10.8842	−0.656340
$276$	0	0
$277$	9.03943	0.543127	0.271563	−	0.962421i	$-0.412459\pi$
$277$	9.03943	0.543127	0.271563	+	0.962421i	$0.412459\pi$
$278$	0	0
$279$	31.5422	1.88838
$280$	0	0
$281$	−22.3469	−1.33310	−0.666551	−	0.745459i	$-0.732230\pi$
$281$	−22.3469	−1.33310	−0.666551	+	0.745459i	$0.732230\pi$
$282$	0	0
$283$	−12.4648	−0.740958	−0.370479	−	0.928841i	$-0.620806\pi$
$283$	−12.4648	−0.740958	−0.370479	+	0.928841i	$0.620806\pi$
$284$	0	0
$285$	7.73802	0.458361
$286$	0	0
$287$	9.74714	0.575356
$288$	0	0
$289$	−16.3799	−0.963521
$290$	0	0
$291$	24.7608	1.45151
$292$	0	0
$293$	11.9081	0.695676	0.347838	−	0.937555i	$-0.386916\pi$
$293$	11.9081	0.695676	0.347838	+	0.937555i	$0.386916\pi$
$294$	0	0
$295$	4.05783	0.236256
$296$	0	0
$297$	11.0746	0.642612
$298$	0	0
$299$	28.4354	1.64446
$300$	0	0
$301$	−11.6322	−0.670472
$302$	0	0
$303$	−3.76815	−0.216475
$304$	0	0
$305$	3.57888	0.204926
$306$	0	0
$307$	−3.78208	−0.215855	−0.107927	−	0.994159i	$-0.534421\pi$
$307$	−3.78208	−0.215855	−0.107927	+	0.994159i	$0.534421\pi$
$308$	0	0
$309$	45.0341	2.56190
$310$	0	0
$311$	−19.1285	−1.08468	−0.542339	−	0.840160i	$-0.682461\pi$
$311$	−19.1285	−1.08468	−0.542339	+	0.840160i	$0.682461\pi$
$312$	0	0
$313$	14.8358	0.838570	0.419285	−	0.907855i	$-0.362281\pi$
$313$	14.8358	0.838570	0.419285	+	0.907855i	$0.362281\pi$
$314$	0	0
$315$	−6.54222	−0.368612
$316$	0	0
$317$	7.38678	0.414883	0.207441	−	0.978247i	$-0.433486\pi$
$317$	7.38678	0.414883	0.207441	+	0.978247i	$0.433486\pi$
$318$	0	0
$319$	45.3950	2.54163
$320$	0	0
$321$	6.52439	0.364156
$322$	0	0
$323$	1.35409	0.0753435
$324$	0	0
$325$	13.2944	0.737442
$326$	0	0
$327$	−37.3827	−2.06727
$328$	0	0
$329$	−2.84614	−0.156913
$330$	0	0
$331$	8.36052	0.459536	0.229768	−	0.973245i	$-0.426203\pi$
$331$	8.36052	0.459536	0.229768	+	0.973245i	$0.426203\pi$
$332$	0	0
$333$	−13.6873	−0.750059
$334$	0	0
$335$	13.1139	0.716490
$336$	0	0
$337$	−25.2039	−1.37294	−0.686472	−	0.727156i	$-0.740841\pi$
$337$	−25.2039	−1.37294	−0.686472	+	0.727156i	$0.740841\pi$
$338$	0	0
$339$	−6.20445	−0.336979
$340$	0	0
$341$	44.9540	2.43440
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−19.4082	−1.04490
$346$	0	0
$347$	22.7387	1.22068	0.610338	−	0.792141i	$-0.291034\pi$
$347$	22.7387	1.22068	0.610338	+	0.792141i	$0.291034\pi$
$348$	0	0
$349$	−10.5380	−0.564086	−0.282043	−	0.959402i	$-0.591012\pi$
$349$	−10.5380	−0.564086	−0.282043	+	0.959402i	$0.591012\pi$
$350$	0	0
$351$	−13.5270	−0.722018
$352$	0	0
$353$	−7.39404	−0.393545	−0.196773	−	0.980449i	$-0.563046\pi$
$353$	−7.39404	−0.393545	−0.196773	+	0.980449i	$0.563046\pi$
$354$	0	0
$355$	20.9500	1.11191
$356$	0	0
$357$	−2.05163	−0.108584
$358$	0	0
$359$	29.6289	1.56375	0.781877	−	0.623433i	$-0.214262\pi$
$359$	29.6289	1.56375	0.781877	+	0.623433i	$0.214262\pi$
$360$	0	0
$361$	−16.0433	−0.844385
$362$	0	0
$363$	47.2539	2.48019
$364$	0	0
$365$	15.8378	0.828987
$366$	0	0
$367$	−27.4871	−1.43481	−0.717406	−	0.696655i	$-0.754671\pi$
$367$	−27.4871	−1.43481	−0.717406	+	0.696655i	$0.754671\pi$
$368$	0	0
$369$	−36.9172	−1.92183
$370$	0	0
$371$	−6.70679	−0.348199
$372$	0	0
$373$	−1.21943	−0.0631397	−0.0315699	−	0.999502i	$-0.510051\pi$
$373$	−1.21943	−0.0631397	−0.0315699	+	0.999502i	$0.510051\pi$
$374$	0	0
$375$	−31.5747	−1.63051
$376$	0	0
$377$	−55.4475	−2.85569
$378$	0	0
$379$	3.60325	0.185086	0.0925432	−	0.995709i	$-0.470500\pi$
$379$	3.60325	0.185086	0.0925432	+	0.995709i	$0.470500\pi$
$380$	0	0
$381$	−55.8846	−2.86306
$382$	0	0
$383$	−10.6357	−0.543459	−0.271730	−	0.962374i	$-0.587596\pi$
$383$	−10.6357	−0.543459	−0.271730	+	0.962374i	$0.587596\pi$
$384$	0	0
$385$	−9.32398	−0.475194
$386$	0	0
$387$	44.0570	2.23954
$388$	0	0
$389$	−14.1087	−0.715340	−0.357670	−	0.933848i	$-0.616429\pi$
$389$	−14.1087	−0.715340	−0.357670	+	0.933848i	$0.616429\pi$
$390$	0	0
$391$	−3.39627	−0.171757
$392$	0	0
$393$	16.1268	0.813488
$394$	0	0
$395$	−25.7794	−1.29710
$396$	0	0
$397$	7.61647	0.382260	0.191130	−	0.981565i	$-0.438785\pi$
$397$	7.61647	0.382260	0.191130	+	0.981565i	$0.438785\pi$
$398$	0	0
$399$	−4.47978	−0.224269
$400$	0	0
$401$	−12.8388	−0.641139	−0.320569	−	0.947225i	$-0.603874\pi$
$401$	−12.8388	−0.641139	−0.320569	+	0.947225i	$0.603874\pi$
$402$	0	0
$403$	−54.9089	−2.73521
$404$	0	0
$405$	−10.3940	−0.516481
$406$	0	0
$407$	−19.5072	−0.966934
$408$	0	0
$409$	1.76378	0.0872133	0.0436066	−	0.999049i	$-0.486115\pi$
$409$	1.76378	0.0872133	0.0436066	+	0.999049i	$0.486115\pi$
$410$	0	0
$411$	40.7074	2.00795
$412$	0	0
$413$	−2.34920	−0.115597
$414$	0	0
$415$	27.0746	1.32904
$416$	0	0
$417$	9.33090	0.456936
$418$	0	0
$419$	17.2138	0.840948	0.420474	−	0.907305i	$-0.361864\pi$
$419$	17.2138	0.840948	0.420474	+	0.907305i	$0.361864\pi$
$420$	0	0
$421$	21.4089	1.04341	0.521703	−	0.853127i	$-0.325297\pi$
$421$	21.4089	1.04341	0.521703	+	0.853127i	$0.325297\pi$
$422$	0	0
$423$	10.7797	0.524129
$424$	0	0
$425$	−1.58786	−0.0770226
$426$	0	0
$427$	−2.07192	−0.100267
$428$	0	0
$429$	−92.7223	−4.47668
$430$	0	0
$431$	10.9148	0.525750	0.262875	−	0.964830i	$-0.415329\pi$
$431$	10.9148	0.525750	0.262875	+	0.964830i	$0.415329\pi$
$432$	0	0
$433$	25.7008	1.23510	0.617552	−	0.786530i	$-0.288125\pi$
$433$	25.7008	1.23510	0.617552	+	0.786530i	$0.288125\pi$
$434$	0	0
$435$	37.8450	1.81453
$436$	0	0
$437$	−7.41583	−0.354747
$438$	0	0
$439$	−27.7373	−1.32383	−0.661913	−	0.749580i	$-0.730255\pi$
$439$	−27.7373	−1.32383	−0.661913	+	0.749580i	$0.730255\pi$
$440$	0	0
$441$	3.78749	0.180357
$442$	0	0
$443$	17.4136	0.827344	0.413672	−	0.910426i	$-0.364246\pi$
$443$	17.4136	0.827344	0.413672	+	0.910426i	$0.364246\pi$
$444$	0	0
$445$	−1.72732	−0.0818829
$446$	0	0
$447$	51.6837	2.44456
$448$	0	0
$449$	−27.2812	−1.28748	−0.643739	−	0.765245i	$-0.722618\pi$
$449$	−27.2812	−1.28748	−0.643739	+	0.765245i	$0.722618\pi$
$450$	0	0
$451$	−52.6145	−2.47752
$452$	0	0
$453$	−35.7575	−1.68003
$454$	0	0
$455$	11.3887	0.533912
$456$	0	0
$457$	3.40486	0.159273	0.0796363	−	0.996824i	$-0.474624\pi$
$457$	3.40486	0.159273	0.0796363	+	0.996824i	$0.474624\pi$
$458$	0	0
$459$	1.61564	0.0754116
$460$	0	0
$461$	5.76756	0.268622	0.134311	−	0.990939i	$-0.457118\pi$
$461$	5.76756	0.268622	0.134311	+	0.990939i	$0.457118\pi$
$462$	0	0
$463$	7.62681	0.354448	0.177224	−	0.984171i	$-0.443288\pi$
$463$	7.62681	0.354448	0.177224	+	0.984171i	$0.443288\pi$
$464$	0	0
$465$	37.4773	1.73797
$466$	0	0
$467$	15.3015	0.708068	0.354034	−	0.935233i	$-0.384810\pi$
$467$	15.3015	0.708068	0.354034	+	0.935233i	$0.384810\pi$
$468$	0	0
$469$	−7.59205	−0.350568
$470$	0	0
$471$	26.9059	1.23976
$472$	0	0
$473$	62.7901	2.88709
$474$	0	0
$475$	−3.46713	−0.159083
$476$	0	0
$477$	25.4019	1.16307
$478$	0	0
$479$	−37.3553	−1.70681	−0.853405	−	0.521249i	$-0.825466\pi$
$479$	−37.3553	−1.70681	−0.853405	+	0.521249i	$0.825466\pi$
$480$	0	0
$481$	23.8269	1.08641
$482$	0	0
$483$	11.2360	0.511256
$484$	0	0
$485$	16.4166	0.745441
$486$	0	0
$487$	−20.7357	−0.939624	−0.469812	−	0.882766i	$-0.655678\pi$
$487$	−20.7357	−0.939624	−0.469812	+	0.882766i	$0.655678\pi$
$488$	0	0
$489$	0.847609	0.0383302
$490$	0	0
$491$	−11.1441	−0.502925	−0.251463	−	0.967867i	$-0.580912\pi$
$491$	−11.1441	−0.502925	−0.251463	+	0.967867i	$0.580912\pi$
$492$	0	0
$493$	6.62255	0.298264
$494$	0	0
$495$	35.3145	1.58727
$496$	0	0
$497$	−12.1286	−0.544043
$498$	0	0
$499$	25.3058	1.13284	0.566422	−	0.824116i	$-0.308327\pi$
$499$	25.3058	1.13284	0.566422	+	0.824116i	$0.308327\pi$
$500$	0	0
$501$	1.85806	0.0830119
$502$	0	0
$503$	17.9385	0.799837	0.399919	−	0.916551i	$-0.369038\pi$
$503$	17.9385	0.799837	0.399919	+	0.916551i	$0.369038\pi$
$504$	0	0
$505$	−2.49831	−0.111174
$506$	0	0
$507$	79.3866	3.52568
$508$	0	0
$509$	−1.19278	−0.0528692	−0.0264346	−	0.999651i	$-0.508415\pi$
$509$	−1.19278	−0.0528692	−0.0264346	+	0.999651i	$0.508415\pi$
$510$	0	0
$511$	−9.16898	−0.405612
$512$	0	0
$513$	3.52778	0.155755
$514$	0	0
$515$	29.8580	1.31570
$516$	0	0
$517$	15.3633	0.675678
$518$	0	0
$519$	−6.19889	−0.272101
$520$	0	0
$521$	−31.1598	−1.36514	−0.682568	−	0.730822i	$-0.739137\pi$
$521$	−31.1598	−1.36514	−0.682568	+	0.730822i	$0.739137\pi$
$522$	0	0
$523$	0.848801	0.0371155	0.0185577	−	0.999828i	$-0.494093\pi$
$523$	0.848801	0.0371155	0.0185577	+	0.999828i	$0.494093\pi$
$524$	0	0
$525$	5.25318	0.229268
$526$	0	0
$527$	6.55821	0.285680
$528$	0	0
$529$	−4.39989	−0.191299
$530$	0	0
$531$	8.89758	0.386122
$532$	0	0
$533$	64.2657	2.78366
$534$	0	0
$535$	4.32572	0.187017
$536$	0	0
$537$	42.6507	1.84051
$538$	0	0
$539$	5.39794	0.232506
$540$	0	0
$541$	−14.2762	−0.613781	−0.306890	−	0.951745i	$-0.599289\pi$
$541$	−14.2762	−0.613781	−0.306890	+	0.951745i	$0.599289\pi$
$542$	0	0
$543$	−30.0744	−1.29062
$544$	0	0
$545$	−24.7850	−1.06167
$546$	0	0
$547$	1.02486	0.0438198	0.0219099	−	0.999760i	$-0.493025\pi$
$547$	1.02486	0.0438198	0.0219099	+	0.999760i	$0.493025\pi$
$548$	0	0
$549$	7.84738	0.334918
$550$	0	0
$551$	14.4605	0.616036
$552$	0	0
$553$	14.9245	0.634654
$554$	0	0
$555$	−16.2628	−0.690315
$556$	0	0
$557$	32.5818	1.38054	0.690268	−	0.723554i	$-0.257493\pi$
$557$	32.5818	1.38054	0.690268	+	0.723554i	$0.257493\pi$
$558$	0	0
$559$	−76.6947	−3.24384
$560$	0	0
$561$	11.0746	0.467569
$562$	0	0
$563$	21.2073	0.893783	0.446891	−	0.894588i	$-0.352531\pi$
$563$	21.2073	0.893783	0.446891	+	0.894588i	$0.352531\pi$
$564$	0	0
$565$	−4.11360	−0.173060
$566$	0	0
$567$	6.01739	0.252707
$568$	0	0
$569$	27.9400	1.17131	0.585654	−	0.810561i	$-0.300838\pi$
$569$	27.9400	1.17131	0.585654	+	0.810561i	$0.300838\pi$
$570$	0	0
$571$	14.3650	0.601156	0.300578	−	0.953757i	$-0.402820\pi$
$571$	14.3650	0.601156	0.300578	+	0.953757i	$0.402820\pi$
$572$	0	0
$573$	34.0456	1.42227
$574$	0	0
$575$	8.69612	0.362653
$576$	0	0
$577$	−21.8935	−0.911439	−0.455720	−	0.890123i	$-0.650618\pi$
$577$	−21.8935	−0.911439	−0.455720	+	0.890123i	$0.650618\pi$
$578$	0	0
$579$	−49.2808	−2.04804
$580$	0	0
$581$	−15.6743	−0.650280
$582$	0	0
$583$	36.2028	1.49937
$584$	0	0
$585$	−43.1347	−1.78340
$586$	0	0
$587$	6.41544	0.264794	0.132397	−	0.991197i	$-0.457733\pi$
$587$	6.41544	0.264794	0.132397	+	0.991197i	$0.457733\pi$
$588$	0	0
$589$	14.3200	0.590045
$590$	0	0
$591$	42.1772	1.73494
$592$	0	0
$593$	15.1416	0.621792	0.310896	−	0.950444i	$-0.399371\pi$
$593$	15.1416	0.621792	0.310896	+	0.950444i	$0.399371\pi$
$594$	0	0
$595$	−1.36025	−0.0557648
$596$	0	0
$597$	−35.6214	−1.45788
$598$	0	0
$599$	−12.2190	−0.499254	−0.249627	−	0.968342i	$-0.580308\pi$
$599$	−12.2190	−0.499254	−0.249627	+	0.968342i	$0.580308\pi$
$600$	0	0
$601$	−0.909250	−0.0370891	−0.0185445	−	0.999828i	$-0.505903\pi$
$601$	−0.909250	−0.0370891	−0.0185445	+	0.999828i	$0.505903\pi$
$602$	0	0
$603$	28.7548	1.17099
$604$	0	0
$605$	31.3297	1.27373
$606$	0	0
$607$	−23.4689	−0.952575	−0.476288	−	0.879290i	$-0.658018\pi$
$607$	−23.4689	−0.952575	−0.476288	+	0.879290i	$0.658018\pi$
$608$	0	0
$609$	−21.9096	−0.887822
$610$	0	0
$611$	−18.7654	−0.759169
$612$	0	0
$613$	−37.0350	−1.49583	−0.747914	−	0.663795i	$-0.768945\pi$
$613$	−37.0350	−1.49583	−0.747914	+	0.663795i	$0.768945\pi$
$614$	0	0
$615$	−43.8637	−1.76875
$616$	0	0
$617$	−1.77912	−0.0716248	−0.0358124	−	0.999359i	$-0.511402\pi$
$617$	−1.77912	−0.0716248	−0.0358124	+	0.999359i	$0.511402\pi$
$618$	0	0
$619$	−16.1992	−0.651102	−0.325551	−	0.945524i	$-0.605550\pi$
$619$	−16.1992	−0.651102	−0.325551	+	0.945524i	$0.605550\pi$
$620$	0	0
$621$	−8.84825	−0.355068
$622$	0	0
$623$	1.00000	0.0400642
$624$	0	0
$625$	−10.8525	−0.434100
$626$	0	0
$627$	24.1816	0.965719
$628$	0	0
$629$	−2.84584	−0.113471
$630$	0	0
$631$	22.3206	0.888568	0.444284	−	0.895886i	$-0.353458\pi$
$631$	22.3206	0.888568	0.444284	+	0.895886i	$0.353458\pi$
$632$	0	0
$633$	−57.9121	−2.30180
$634$	0	0
$635$	−37.0519	−1.47036
$636$	0	0
$637$	−6.59329	−0.261235
$638$	0	0
$639$	45.9371	1.81724
$640$	0	0
$641$	−31.7228	−1.25297	−0.626487	−	0.779432i	$-0.715508\pi$
$641$	−31.7228	−1.25297	−0.626487	+	0.779432i	$0.715508\pi$
$642$	0	0
$643$	−20.2004	−0.796627	−0.398314	−	0.917249i	$-0.630404\pi$
$643$	−20.2004	−0.796627	−0.398314	+	0.917249i	$0.630404\pi$
$644$	0	0
$645$	52.3470	2.06116
$646$	0	0
$647$	−17.5813	−0.691192	−0.345596	−	0.938383i	$-0.612323\pi$
$647$	−17.5813	−0.691192	−0.345596	+	0.938383i	$0.612323\pi$
$648$	0	0
$649$	12.6809	0.497767
$650$	0	0
$651$	−21.6968	−0.850364
$652$	0	0
$653$	19.4404	0.760763	0.380382	−	0.924830i	$-0.375793\pi$
$653$	19.4404	0.760763	0.380382	+	0.924830i	$0.375793\pi$
$654$	0	0
$655$	10.6922	0.417778
$656$	0	0
$657$	34.7274	1.35485
$658$	0	0
$659$	−28.4603	−1.10865	−0.554327	−	0.832299i	$-0.687024\pi$
$659$	−28.4603	−1.10865	−0.554327	+	0.832299i	$0.687024\pi$
$660$	0	0
$661$	31.7282	1.23408	0.617041	−	0.786931i	$-0.288331\pi$
$661$	31.7282	1.23408	0.617041	+	0.786931i	$0.288331\pi$
$662$	0	0
$663$	−13.5270	−0.525345
$664$	0	0
$665$	−2.97013	−0.115177
$666$	0	0
$667$	−36.2692	−1.40435
$668$	0	0
$669$	−53.0538	−2.05118
$670$	0	0
$671$	11.1841	0.431757
$672$	0	0
$673$	−37.9989	−1.46475	−0.732376	−	0.680901i	$-0.761589\pi$
$673$	−37.9989	−1.46475	−0.732376	+	0.680901i	$0.761589\pi$
$674$	0	0
$675$	−4.13683	−0.159226
$676$	0	0
$677$	−33.7154	−1.29579	−0.647893	−	0.761731i	$-0.724350\pi$
$677$	−33.7154	−1.29579	−0.647893	+	0.761731i	$0.724350\pi$
$678$	0	0
$679$	−9.50409	−0.364734
$680$	0	0
$681$	6.40777	0.245546
$682$	0	0
$683$	29.3921	1.12466	0.562329	−	0.826913i	$-0.309905\pi$
$683$	29.3921	1.12466	0.562329	+	0.826913i	$0.309905\pi$
$684$	0	0
$685$	26.9893	1.03121
$686$	0	0
$687$	−47.2126	−1.80127
$688$	0	0
$689$	−44.2198	−1.68464
$690$	0	0
$691$	−36.7280	−1.39720	−0.698599	−	0.715513i	$-0.746193\pi$
$691$	−36.7280	−1.39720	−0.698599	+	0.715513i	$0.746193\pi$
$692$	0	0
$693$	−20.4446	−0.776627
$694$	0	0
$695$	6.18646	0.234666
$696$	0	0
$697$	−7.67577	−0.290741
$698$	0	0
$699$	51.1677	1.93534
$700$	0	0
$701$	−39.0548	−1.47508	−0.737539	−	0.675304i	$-0.764012\pi$
$701$	−39.0548	−1.47508	−0.737539	+	0.675304i	$0.764012\pi$
$702$	0	0
$703$	−6.21396	−0.234364
$704$	0	0
$705$	12.8081	0.482381
$706$	0	0
$707$	1.44635	0.0543956
$708$	0	0
$709$	−25.4921	−0.957374	−0.478687	−	0.877986i	$-0.658887\pi$
$709$	−25.4921	−0.957374	−0.478687	+	0.877986i	$0.658887\pi$
$710$	0	0
$711$	−56.5264	−2.11991
$712$	0	0
$713$	−35.9169	−1.34510
$714$	0	0
$715$	−61.4756	−2.29906
$716$	0	0
$717$	21.3617	0.797766
$718$	0	0
$719$	25.9004	0.965921	0.482961	−	0.875642i	$-0.339561\pi$
$719$	25.9004	0.965921	0.482961	+	0.875642i	$0.339561\pi$
$720$	0	0
$721$	−17.2857	−0.643753
$722$	0	0
$723$	−37.2096	−1.38384
$724$	0	0
$725$	−16.9569	−0.629765
$726$	0	0
$727$	−2.52601	−0.0936844	−0.0468422	−	0.998902i	$-0.514916\pi$
$727$	−2.52601	−0.0936844	−0.0468422	+	0.998902i	$0.514916\pi$
$728$	0	0
$729$	−38.8260	−1.43800
$730$	0	0
$731$	9.16028	0.338805
$732$	0	0
$733$	−6.96045	−0.257090	−0.128545	−	0.991704i	$-0.541031\pi$
$733$	−6.96045	−0.257090	−0.128545	+	0.991704i	$0.541031\pi$
$734$	0	0
$735$	4.50016	0.165991
$736$	0	0
$737$	40.9814	1.50957
$738$	0	0
$739$	19.9822	0.735057	0.367528	−	0.930012i	$-0.380204\pi$
$739$	19.9822	0.735057	0.367528	+	0.930012i	$0.380204\pi$
$740$	0	0
$741$	−29.5365	−1.08505
$742$	0	0
$743$	23.3542	0.856781	0.428391	−	0.903594i	$-0.359081\pi$
$743$	23.3542	0.856781	0.428391	+	0.903594i	$0.359081\pi$
$744$	0	0
$745$	34.2667	1.25544
$746$	0	0
$747$	59.3663	2.17210
$748$	0	0
$749$	−2.50429	−0.0915049
$750$	0	0
$751$	−16.1715	−0.590105	−0.295053	−	0.955481i	$-0.595337\pi$
$751$	−16.1715	−0.590105	−0.295053	+	0.955481i	$0.595337\pi$
$752$	0	0
$753$	−27.2391	−0.992647
$754$	0	0
$755$	−23.7075	−0.862805
$756$	0	0
$757$	29.2366	1.06262	0.531311	−	0.847177i	$-0.321700\pi$
$757$	29.2366	1.06262	0.531311	+	0.847177i	$0.321700\pi$
$758$	0	0
$759$	−60.6513	−2.20150
$760$	0	0
$761$	18.9785	0.687971	0.343986	−	0.938975i	$-0.388223\pi$
$761$	18.9785	0.687971	0.343986	+	0.938975i	$0.388223\pi$
$762$	0	0
$763$	14.3488	0.519462
$764$	0	0
$765$	5.15193	0.186268
$766$	0	0
$767$	−15.4890	−0.559274
$768$	0	0
$769$	−13.3541	−0.481560	−0.240780	−	0.970580i	$-0.577403\pi$
$769$	−13.3541	−0.481560	−0.240780	+	0.970580i	$0.577403\pi$
$770$	0	0
$771$	−21.6935	−0.781271
$772$	0	0
$773$	4.86222	0.174882	0.0874410	−	0.996170i	$-0.472131\pi$
$773$	4.86222	0.174882	0.0874410	+	0.996170i	$0.472131\pi$
$774$	0	0
$775$	−16.7922	−0.603195
$776$	0	0
$777$	9.41501	0.337762
$778$	0	0
$779$	−16.7602	−0.600497
$780$	0	0
$781$	65.4696	2.34269
$782$	0	0
$783$	17.2536	0.616593
$784$	0	0
$785$	17.8388	0.636694
$786$	0	0
$787$	25.2337	0.899484	0.449742	−	0.893159i	$-0.351516\pi$
$787$	25.2337	0.899484	0.449742	+	0.893159i	$0.351516\pi$
$788$	0	0
$789$	−57.4411	−2.04496
$790$	0	0
$791$	2.38149	0.0846760
$792$	0	0
$793$	−13.6608	−0.485108
$794$	0	0
$795$	30.1816	1.07043
$796$	0	0
$797$	45.0265	1.59492	0.797460	−	0.603372i	$-0.206177\pi$
$797$	45.0265	1.59492	0.797460	+	0.603372i	$0.206177\pi$
$798$	0	0
$799$	2.24131	0.0792918
$800$	0	0
$801$	−3.78749	−0.133824
$802$	0	0
$803$	49.4936	1.74659
$804$	0	0
$805$	7.44957	0.262563
$806$	0	0
$807$	−21.8241	−0.768246
$808$	0	0
$809$	−6.12302	−0.215274	−0.107637	−	0.994190i	$-0.534328\pi$
$809$	−6.12302	−0.215274	−0.107637	+	0.994190i	$0.534328\pi$
$810$	0	0
$811$	33.4113	1.17323	0.586615	−	0.809866i	$-0.300460\pi$
$811$	33.4113	1.17323	0.586615	+	0.809866i	$0.300460\pi$
$812$	0	0
$813$	−19.6141	−0.687898
$814$	0	0
$815$	0.561972	0.0196850
$816$	0	0
$817$	20.0016	0.699769
$818$	0	0
$819$	24.9720	0.872592
$820$	0	0
$821$	−52.3453	−1.82686	−0.913432	−	0.406991i	$-0.866578\pi$
$821$	−52.3453	−1.82686	−0.913432	+	0.406991i	$0.866578\pi$
$822$	0	0
$823$	42.8228	1.49271	0.746354	−	0.665549i	$-0.231802\pi$
$823$	42.8228	1.49271	0.746354	+	0.665549i	$0.231802\pi$
$824$	0	0
$825$	−28.3563	−0.987241
$826$	0	0
$827$	9.40221	0.326947	0.163473	−	0.986548i	$-0.447730\pi$
$827$	9.40221	0.326947	0.163473	+	0.986548i	$0.447730\pi$
$828$	0	0
$829$	−0.721230	−0.0250494	−0.0125247	−	0.999922i	$-0.503987\pi$
$829$	−0.721230	−0.0250494	−0.0125247	+	0.999922i	$0.503987\pi$
$830$	0	0
$831$	23.5503	0.816950
$832$	0	0
$833$	0.787490	0.0272849
$834$	0	0
$835$	1.23191	0.0426319
$836$	0	0
$837$	17.0860	0.590578
$838$	0	0
$839$	−7.04235	−0.243129	−0.121565	−	0.992584i	$-0.538791\pi$
$839$	−7.04235	−0.243129	−0.121565	+	0.992584i	$0.538791\pi$
$840$	0	0
$841$	41.7229	1.43872
$842$	0	0
$843$	−58.2199	−2.00520
$844$	0	0
$845$	52.6339	1.81066
$846$	0	0
$847$	−18.1377	−0.623220
$848$	0	0
$849$	−32.4744	−1.11452
$850$	0	0
$851$	15.5856	0.534268
$852$	0	0
$853$	−53.2082	−1.82181	−0.910907	−	0.412611i	$-0.864617\pi$
$853$	−53.2082	−1.82181	−0.910907	+	0.412611i	$0.864617\pi$
$854$	0	0
$855$	11.2493	0.384719
$856$	0	0
$857$	−2.89499	−0.0988910	−0.0494455	−	0.998777i	$-0.515745\pi$
$857$	−2.89499	−0.0988910	−0.0494455	+	0.998777i	$0.515745\pi$
$858$	0	0
$859$	−38.6140	−1.31749	−0.658747	−	0.752365i	$-0.728913\pi$
$859$	−38.6140	−1.31749	−0.658747	+	0.752365i	$0.728913\pi$
$860$	0	0
$861$	25.3940	0.865427
$862$	0	0
$863$	0.630828	0.0214736	0.0107368	−	0.999942i	$-0.496582\pi$
$863$	0.630828	0.0214736	0.0107368	+	0.999942i	$0.496582\pi$
$864$	0	0
$865$	−4.10991	−0.139741
$866$	0	0
$867$	−42.6741	−1.44929
$868$	0	0
$869$	−80.5615	−2.73286
$870$	0	0
$871$	−50.0566	−1.69610
$872$	0	0
$873$	35.9966	1.21830
$874$	0	0
$875$	12.1195	0.409714
$876$	0	0
$877$	5.13758	0.173484	0.0867419	−	0.996231i	$-0.472354\pi$
$877$	5.13758	0.173484	0.0867419	+	0.996231i	$0.472354\pi$
$878$	0	0
$879$	31.0238	1.04641
$880$	0	0
$881$	−42.1719	−1.42081	−0.710403	−	0.703795i	$-0.751487\pi$
$881$	−42.1719	−1.42081	−0.710403	+	0.703795i	$0.751487\pi$
$882$	0	0
$883$	27.2137	0.915813	0.457906	−	0.889000i	$-0.348599\pi$
$883$	27.2137	0.915813	0.457906	+	0.889000i	$0.348599\pi$
$884$	0	0
$885$	10.5718	0.355367
$886$	0	0
$887$	13.0720	0.438914	0.219457	−	0.975622i	$-0.429571\pi$
$887$	13.0720	0.438914	0.219457	+	0.975622i	$0.429571\pi$
$888$	0	0
$889$	21.4505	0.719427
$890$	0	0
$891$	−32.4815	−1.08817
$892$	0	0
$893$	4.89395	0.163770
$894$	0	0
$895$	28.2777	0.945221
$896$	0	0
$897$	74.0823	2.47354
$898$	0	0
$899$	70.0359	2.33583
$900$	0	0
$901$	5.28153	0.175953
$902$	0	0
$903$	−30.3053	−1.00850
$904$	0	0
$905$	−19.9396	−0.662814
$906$	0	0
$907$	−28.4561	−0.944869	−0.472434	−	0.881366i	$-0.656625\pi$
$907$	−28.4561	−0.944869	−0.472434	+	0.881366i	$0.656625\pi$
$908$	0	0
$909$	−5.47804	−0.181695
$910$	0	0
$911$	−47.8872	−1.58657	−0.793287	−	0.608848i	$-0.791632\pi$
$911$	−47.8872	−1.58657	−0.793287	+	0.608848i	$0.791632\pi$
$912$	0	0
$913$	84.6089	2.80015
$914$	0	0
$915$	9.32398	0.308241
$916$	0	0
$917$	−6.19003	−0.204413
$918$	0	0
$919$	23.7252	0.782624	0.391312	−	0.920258i	$-0.372021\pi$
$919$	23.7252	0.782624	0.391312	+	0.920258i	$0.372021\pi$
$920$	0	0
$921$	−9.85339	−0.324680
$922$	0	0
$923$	−79.9675	−2.63216
$924$	0	0
$925$	7.28675	0.239587
$926$	0	0
$927$	65.4694	2.15030
$928$	0	0
$929$	−9.11036	−0.298901	−0.149451	−	0.988769i	$-0.547751\pi$
$929$	−9.11036	−0.298901	−0.149451	+	0.988769i	$0.547751\pi$
$930$	0	0
$931$	1.71950	0.0563543
$932$	0	0
$933$	−49.8352	−1.63153
$934$	0	0
$935$	7.34254	0.240127
$936$	0	0
$937$	−19.5887	−0.639934	−0.319967	−	0.947429i	$-0.603672\pi$
$937$	−19.5887	−0.639934	−0.319967	+	0.947429i	$0.603672\pi$
$938$	0	0
$939$	38.6515	1.26134
$940$	0	0
$941$	−39.7681	−1.29640	−0.648202	−	0.761469i	$-0.724479\pi$
$941$	−39.7681	−1.29640	−0.648202	+	0.761469i	$0.724479\pi$
$942$	0	0
$943$	42.0373	1.36892
$944$	0	0
$945$	−3.54383	−0.115281
$946$	0	0
$947$	−38.7403	−1.25889	−0.629446	−	0.777044i	$-0.716718\pi$
$947$	−38.7403	−1.25889	−0.629446	+	0.777044i	$0.716718\pi$
$948$	0	0
$949$	−60.4537	−1.96241
$950$	0	0
$951$	19.2446	0.624050
$952$	0	0
$953$	−31.8407	−1.03142	−0.515710	−	0.856763i	$-0.672472\pi$
$953$	−31.8407	−1.03142	−0.515710	+	0.856763i	$0.672472\pi$
$954$	0	0
$955$	22.5725	0.730428
$956$	0	0
$957$	118.267	3.82302
$958$	0	0
$959$	−15.6249	−0.504556
$960$	0	0
$961$	38.3556	1.23728
$962$	0	0
$963$	9.48499	0.305649
$964$	0	0
$965$	−32.6736	−1.05180
$966$	0	0
$967$	−32.6376	−1.04955	−0.524777	−	0.851240i	$-0.675851\pi$
$967$	−32.6376	−1.04955	−0.524777	+	0.851240i	$0.675851\pi$
$968$	0	0
$969$	3.52778	0.113329
$970$	0	0
$971$	6.57104	0.210875	0.105437	−	0.994426i	$-0.466376\pi$
$971$	6.57104	0.210875	0.105437	+	0.994426i	$0.466376\pi$
$972$	0	0
$973$	−3.58153	−0.114819
$974$	0	0
$975$	34.6357	1.10923
$976$	0	0
$977$	−21.1367	−0.676223	−0.338111	−	0.941106i	$-0.609788\pi$
$977$	−21.1367	−0.676223	−0.338111	+	0.941106i	$0.609788\pi$
$978$	0	0
$979$	−5.39794	−0.172519
$980$	0	0
$981$	−54.3460	−1.73513
$982$	0	0
$983$	48.0681	1.53313	0.766566	−	0.642165i	$-0.221964\pi$
$983$	48.0681	1.53313	0.766566	+	0.642165i	$0.221964\pi$
$984$	0	0
$985$	27.9638	0.891001
$986$	0	0
$987$	−7.41501	−0.236022
$988$	0	0
$989$	−50.1674	−1.59523
$990$	0	0
$991$	30.9132	0.981990	0.490995	−	0.871162i	$-0.336633\pi$
$991$	30.9132	0.981990	0.490995	+	0.871162i	$0.336633\pi$
$992$	0	0
$993$	21.7815	0.691215
$994$	0	0
$995$	−23.6172	−0.748717
$996$	0	0
$997$	−21.6791	−0.686583	−0.343292	−	0.939229i	$-0.611542\pi$
$997$	−21.6791	−0.686583	−0.343292	+	0.939229i	$0.611542\pi$
$998$	0	0
$999$	−7.41422	−0.234576

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9968.2.a.z.1.4		5
4.3	odd	2		1246.2.a.n.1.2	✓	5
28.27	even	2		8722.2.a.x.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1246.2.a.n.1.2	✓	5	4.3	odd	2
8722.2.a.x.1.4		5	28.27	even	2
9968.2.a.z.1.4		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)

\(f(q)\)	\(=\)	\(q+2.60528 q^{3} +1.72732 q^{5} -1.00000 q^{7} +3.78749 q^{9} +O(q^{10})\)	\(q+2.60528 q^{3} +1.72732 q^{5} -1.00000 q^{7} +3.78749 q^{9} +5.39794 q^{11} -6.59329 q^{13} +4.50016 q^{15} +0.787490 q^{17} +1.71950 q^{19} -2.60528 q^{21} -4.31279 q^{23} -2.01636 q^{25} +2.05163 q^{27} +8.40969 q^{29} +8.32800 q^{31} +14.0631 q^{33} -1.72732 q^{35} -3.61382 q^{37} -17.1774 q^{39} -9.74714 q^{41} +11.6322 q^{43} +6.54222 q^{45} +2.84614 q^{47} +1.00000 q^{49} +2.05163 q^{51} +6.70679 q^{53} +9.32398 q^{55} +4.47978 q^{57} +2.34920 q^{59} +2.07192 q^{61} -3.78749 q^{63} -11.3887 q^{65} +7.59205 q^{67} -11.2360 q^{69} +12.1286 q^{71} +9.16898 q^{73} -5.25318 q^{75} -5.39794 q^{77} -14.9245 q^{79} -6.01739 q^{81} +15.6743 q^{83} +1.36025 q^{85} +21.9096 q^{87} -1.00000 q^{89} +6.59329 q^{91} +21.6968 q^{93} +2.97013 q^{95} +9.50409 q^{97} +20.4446 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 6 q^{3} - 10 q^{5} - 5 q^{7} + 7 q^{9}+O(q^{10}) \) 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

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more