LMFDB - Embedded newform 2368.2.a.bd.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2368,2,Mod(1,2368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2368, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2368.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2368 = 2^{6} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2368.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:	$18.9085751986$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.621.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 3 $ x^3 - 6*x - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1184)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.66908$ of defining polynomial
Character	$\chi$	$=$	2368.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.66908 q^{3} -1.66908 q^{5} +0.545096 q^{7} +4.12398 q^{9} +4.21417 q^{11} -0.123983 q^{13} -4.45490 q^{15} -0.909808 q^{17} +2.00000 q^{19} +1.45490 q^{21} +4.12398 q^{23} -2.21417 q^{25} +3.00000 q^{27} -1.21417 q^{29} +5.91705 q^{31} +11.2480 q^{33} -0.909808 q^{35} +1.00000 q^{37} -0.330921 q^{39} +1.33092 q^{41} +10.2480 q^{43} -6.88325 q^{45} -2.54510 q^{47} -6.70287 q^{49} -2.42835 q^{51} +6.79306 q^{53} -7.03379 q^{55} +5.33816 q^{57} +11.1578 q^{59} +2.75927 q^{61} +2.24797 q^{63} +0.206938 q^{65} +2.33092 q^{67} +11.0072 q^{69} -7.88325 q^{71} -13.5523 q^{73} -5.90981 q^{75} +2.29713 q^{77} -2.78583 q^{79} -4.36471 q^{81} -5.88325 q^{83} +1.51854 q^{85} -3.24073 q^{87} +9.09019 q^{89} -0.0675827 q^{91} +15.7931 q^{93} -3.33816 q^{95} +3.81962 q^{97} +17.3792 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{5} + 3 q^{7} + 3 q^{9} + 6 q^{11} + 9 q^{13} - 12 q^{15} + 6 q^{19} + 3 q^{21} + 3 q^{23} + 9 q^{27} + 3 q^{29} - 9 q^{31} + 15 q^{33} + 3 q^{37} - 9 q^{39} + 12 q^{41} + 12 q^{43} - 6 q^{45}+ \cdots + 6 q^{97}+O(q^{100}) $ 3 * q + 3 * q^5 + 3 * q^7 + 3 * q^9 + 6 * q^11 + 9 * q^13 - 12 * q^15 + 6 * q^19 + 3 * q^21 + 3 * q^23 + 9 * q^27 + 3 * q^29 - 9 * q^31 + 15 * q^33 + 3 * q^37 - 9 * q^39 + 12 * q^41 + 12 * q^43 - 6 * q^45 - 9 * q^47 + 6 * q^51 + 3 * q^53 - 9 * q^55 + 12 * q^59 + 3 * q^61 - 12 * q^63 + 18 * q^65 + 15 * q^67 + 9 * q^69 - 9 * q^71 - 18 * q^73 - 15 * q^75 + 27 * q^77 - 15 * q^79 - 9 * q^81 - 3 * q^83 - 6 * q^85 - 15 * q^87 + 30 * q^89 + 24 * q^91 + 30 * q^93 + 6 * q^95 + 6 * q^97

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
$3$	2.66908	1.54099	0.770497	−	0.637444i	$-0.220008\pi$
$3$	2.66908	1.54099	0.770497	+	0.637444i	$0.220008\pi$
$4$	0	0
$5$	−1.66908	−0.746435	−0.373217	−	0.927744i	$-0.621745\pi$
$5$	−1.66908	−0.746435	−0.373217	+	0.927744i	$0.621745\pi$
$6$	0	0
$7$	0.545096	0.206027	0.103013	−	0.994680i	$-0.467152\pi$
$7$	0.545096	0.206027	0.103013	+	0.994680i	$0.467152\pi$
$8$	0	0
$9$	4.12398	1.37466
$10$	0	0
$11$	4.21417	1.27062	0.635311	−	0.772257i	$-0.280872\pi$
$11$	4.21417	1.27062	0.635311	+	0.772257i	$0.280872\pi$
$12$	0	0
$13$	−0.123983	−0.0343867	−0.0171934	−	0.999852i	$-0.505473\pi$
$13$	−0.123983	−0.0343867	−0.0171934	+	0.999852i	$0.505473\pi$
$14$	0	0
$15$	−4.45490	−1.15025
$16$	0	0
$17$	−0.909808	−0.220661	−0.110330	−	0.993895i	$-0.535191\pi$
$17$	−0.909808	−0.220661	−0.110330	+	0.993895i	$0.535191\pi$
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	1.45490	0.317486
$22$	0	0
$23$	4.12398	0.859910	0.429955	−	0.902850i	$-0.358529\pi$
$23$	4.12398	0.859910	0.429955	+	0.902850i	$0.358529\pi$
$24$	0	0
$25$	−2.21417	−0.442835
$26$	0	0
$27$	3.00000	0.577350
$28$	0	0
$29$	−1.21417	−0.225467	−0.112733	−	0.993625i	$-0.535961\pi$
$29$	−1.21417	−0.225467	−0.112733	+	0.993625i	$0.535961\pi$
$30$	0	0
$31$	5.91705	1.06273	0.531366	−	0.847142i	$-0.321679\pi$
$31$	5.91705	1.06273	0.531366	+	0.847142i	$0.321679\pi$
$32$	0	0
$33$	11.2480	1.95802
$34$	0	0
$35$	−0.909808	−0.153786
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	0	0
$39$	−0.330921	−0.0529898
$40$	0	0
$41$	1.33092	0.207855	0.103927	−	0.994585i	$-0.466859\pi$
$41$	1.33092	0.207855	0.103927	+	0.994585i	$0.466859\pi$
$42$	0	0
$43$	10.2480	1.56280	0.781400	−	0.624030i	$-0.214506\pi$
$43$	10.2480	1.56280	0.781400	+	0.624030i	$0.214506\pi$
$44$	0	0
$45$	−6.88325	−1.02609
$46$	0	0
$47$	−2.54510	−0.371240	−0.185620	−	0.982622i	$-0.559429\pi$
$47$	−2.54510	−0.371240	−0.185620	+	0.982622i	$0.559429\pi$
$48$	0	0
$49$	−6.70287	−0.957553
$50$	0	0
$51$	−2.42835	−0.340037
$52$	0	0
$53$	6.79306	0.933099	0.466549	−	0.884495i	$-0.345497\pi$
$53$	6.79306	0.933099	0.466549	+	0.884495i	$0.345497\pi$
$54$	0	0
$55$	−7.03379	−0.948436
$56$	0	0
$57$	5.33816	0.707056
$58$	0	0
$59$	11.1578	1.45262	0.726309	−	0.687368i	$-0.241234\pi$
$59$	11.1578	1.45262	0.726309	+	0.687368i	$0.241234\pi$
$60$	0	0
$61$	2.75927	0.353288	0.176644	−	0.984275i	$-0.443476\pi$
$61$	2.75927	0.353288	0.176644	+	0.984275i	$0.443476\pi$
$62$	0	0
$63$	2.24797	0.283217
$64$	0	0
$65$	0.206938	0.0256675
$66$	0	0
$67$	2.33092	0.284767	0.142384	−	0.989812i	$-0.454523\pi$
$67$	2.33092	0.284767	0.142384	+	0.989812i	$0.454523\pi$
$68$	0	0
$69$	11.0072	1.32512
$70$	0	0
$71$	−7.88325	−0.935570	−0.467785	−	0.883842i	$-0.654948\pi$
$71$	−7.88325	−0.935570	−0.467785	+	0.883842i	$0.654948\pi$
$72$	0	0
$73$	−13.5523	−1.58618	−0.793090	−	0.609104i	$-0.791529\pi$
$73$	−13.5523	−1.58618	−0.793090	+	0.609104i	$0.791529\pi$
$74$	0	0
$75$	−5.90981	−0.682406
$76$	0	0
$77$	2.29713	0.261782
$78$	0	0
$79$	−2.78583	−0.313430	−0.156715	−	0.987644i	$-0.550090\pi$
$79$	−2.78583	−0.313430	−0.156715	+	0.987644i	$0.550090\pi$
$80$	0	0
$81$	−4.36471	−0.484968
$82$	0	0
$83$	−5.88325	−0.645771	−0.322886	−	0.946438i	$-0.604653\pi$
$83$	−5.88325	−0.645771	−0.322886	+	0.946438i	$0.604653\pi$
$84$	0	0
$85$	1.51854	0.164709
$86$	0	0
$87$	−3.24073	−0.347443
$88$	0	0
$89$	9.09019	0.963558	0.481779	−	0.876293i	$-0.339991\pi$
$89$	9.09019	0.963558	0.481779	+	0.876293i	$0.339991\pi$
$90$	0	0
$91$	−0.0675827	−0.00708459
$92$	0	0
$93$	15.7931	1.63766
$94$	0	0
$95$	−3.33816	−0.342488
$96$	0	0
$97$	3.81962	0.387823	0.193912	−	0.981019i	$-0.437882\pi$
$97$	3.81962	0.387823	0.193912	+	0.981019i	$0.437882\pi$
$98$	0	0
$99$	17.3792	1.74667
$100$	0	0
$101$	14.5596	1.44873	0.724366	−	0.689416i	$-0.242133\pi$
$101$	14.5596	1.44873	0.724366	+	0.689416i	$0.242133\pi$
$102$	0	0
$103$	−5.84946	−0.576365	−0.288182	−	0.957576i	$-0.593051\pi$
$103$	−5.84946	−0.576365	−0.288182	+	0.957576i	$0.593051\pi$
$104$	0	0
$105$	−2.42835	−0.236983
$106$	0	0
$107$	−7.69563	−0.743965	−0.371982	−	0.928240i	$-0.621322\pi$
$107$	−7.69563	−0.743965	−0.371982	+	0.928240i	$0.621322\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	2.66908	0.253338
$112$	0	0
$113$	6.72942	0.633051	0.316526	−	0.948584i	$-0.397484\pi$
$113$	6.72942	0.633051	0.316526	+	0.948584i	$0.397484\pi$
$114$	0	0
$115$	−6.88325	−0.641867
$116$	0	0
$117$	−0.511305	−0.0472701
$118$	0	0
$119$	−0.495933	−0.0454621
$120$	0	0
$121$	6.75927	0.614479
$122$	0	0
$123$	3.55233	0.320303
$124$	0	0
$125$	12.0410	1.07698
$126$	0	0
$127$	16.6127	1.47414	0.737068	−	0.675818i	$-0.236210\pi$
$127$	16.6127	1.47414	0.737068	+	0.675818i	$0.236210\pi$
$128$	0	0
$129$	27.3526	2.40827
$130$	0	0
$131$	−6.92428	−0.604977	−0.302489	−	0.953153i	$-0.597817\pi$
$131$	−6.92428	−0.604977	−0.302489	+	0.953153i	$0.597817\pi$
$132$	0	0
$133$	1.09019	0.0945316
$134$	0	0
$135$	−5.00724	−0.430954
$136$	0	0
$137$	−5.48870	−0.468931	−0.234465	−	0.972124i	$-0.575334\pi$
$137$	−5.48870	−0.468931	−0.234465	+	0.972124i	$0.575334\pi$
$138$	0	0
$139$	14.3719	1.21901	0.609506	−	0.792781i	$-0.291368\pi$
$139$	14.3719	1.21901	0.609506	+	0.792781i	$0.291368\pi$
$140$	0	0
$141$	−6.79306	−0.572079
$142$	0	0
$143$	−0.522487	−0.0436925
$144$	0	0
$145$	2.02655	0.168296
$146$	0	0
$147$	−17.8905	−1.47558
$148$	0	0
$149$	−12.3116	−1.00861	−0.504303	−	0.863527i	$-0.668251\pi$
$149$	−12.3116	−1.00861	−0.504303	+	0.863527i	$0.668251\pi$
$150$	0	0
$151$	−18.0145	−1.46600	−0.732999	−	0.680230i	$-0.761880\pi$
$151$	−18.0145	−1.46600	−0.732999	+	0.680230i	$0.761880\pi$
$152$	0	0
$153$	−3.75203	−0.303334
$154$	0	0
$155$	−9.87602	−0.793261
$156$	0	0
$157$	−3.15383	−0.251703	−0.125851	−	0.992049i	$-0.540166\pi$
$157$	−3.15383	−0.251703	−0.125851	+	0.992049i	$0.540166\pi$
$158$	0	0
$159$	18.1312	1.43790
$160$	0	0
$161$	2.24797	0.177165
$162$	0	0
$163$	−9.76651	−0.764972	−0.382486	−	0.923961i	$-0.624932\pi$
$163$	−9.76651	−0.764972	−0.382486	+	0.923961i	$0.624932\pi$
$164$	0	0
$165$	−18.7737	−1.46153
$166$	0	0
$167$	−6.34540	−0.491021	−0.245511	−	0.969394i	$-0.578956\pi$
$167$	−6.34540	−0.491021	−0.245511	+	0.969394i	$0.578956\pi$
$168$	0	0
$169$	−12.9846	−0.998818
$170$	0	0
$171$	8.24797	0.630738
$172$	0	0
$173$	22.7931	1.73292	0.866462	−	0.499243i	$-0.166388\pi$
$173$	22.7931	1.73292	0.866462	+	0.499243i	$0.166388\pi$
$174$	0	0
$175$	−1.20694	−0.0912359
$176$	0	0
$177$	29.7810	2.23848
$178$	0	0
$179$	−5.15777	−0.385510	−0.192755	−	0.981247i	$-0.561742\pi$
$179$	−5.15777	−0.385510	−0.192755	+	0.981247i	$0.561742\pi$
$180$	0	0
$181$	−0.297130	−0.0220855	−0.0110427	−	0.999939i	$-0.503515\pi$
$181$	−0.297130	−0.0220855	−0.0110427	+	0.999939i	$0.503515\pi$
$182$	0	0
$183$	7.36471	0.544415
$184$	0	0
$185$	−1.66908	−0.122713
$186$	0	0
$187$	−3.83409	−0.280376
$188$	0	0
$189$	1.63529	0.118950
$190$	0	0
$191$	−12.1650	−0.880229	−0.440115	−	0.897942i	$-0.645062\pi$
$191$	−12.1650	−0.880229	−0.440115	+	0.897942i	$0.645062\pi$
$192$	0	0
$193$	16.0000	1.15171	0.575853	−	0.817554i	$-0.304670\pi$
$193$	16.0000	1.15171	0.575853	+	0.817554i	$0.304670\pi$
$194$	0	0
$195$	0.552333	0.0395534
$196$	0	0
$197$	−1.93636	−0.137960	−0.0689801	−	0.997618i	$-0.521974\pi$
$197$	−1.93636	−0.137960	−0.0689801	+	0.997618i	$0.521974\pi$
$198$	0	0
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	0	0
$201$	6.22141	0.438825
$202$	0	0
$203$	−0.661842	−0.0464522
$204$	0	0
$205$	−2.22141	−0.155150
$206$	0	0
$207$	17.0072	1.18208
$208$	0	0
$209$	8.42835	0.583001
$210$	0	0
$211$	−15.9807	−1.10016	−0.550078	−	0.835113i	$-0.685402\pi$
$211$	−15.9807	−1.10016	−0.550078	+	0.835113i	$0.685402\pi$
$212$	0	0
$213$	−21.0410	−1.44171
$214$	0	0
$215$	−17.1047	−1.16653
$216$	0	0
$217$	3.22536	0.218952
$218$	0	0
$219$	−36.1722	−2.44429
$220$	0	0
$221$	0.112801	0.00758781
$222$	0	0
$223$	−6.18433	−0.414133	−0.207067	−	0.978327i	$-0.566392\pi$
$223$	−6.18433	−0.414133	−0.207067	+	0.978327i	$0.566392\pi$
$224$	0	0
$225$	−9.13122	−0.608748
$226$	0	0
$227$	−25.1578	−1.66978	−0.834890	−	0.550417i	$-0.814469\pi$
$227$	−25.1578	−1.66978	−0.834890	+	0.550417i	$0.814469\pi$
$228$	0	0
$229$	−29.7173	−1.96378	−0.981889	−	0.189459i	$-0.939327\pi$
$229$	−29.7173	−1.96378	−0.981889	+	0.189459i	$0.939327\pi$
$230$	0	0
$231$	6.13122	0.403405
$232$	0	0
$233$	−9.89049	−0.647948	−0.323974	−	0.946066i	$-0.605019\pi$
$233$	−9.89049	−0.647948	−0.323974	+	0.946066i	$0.605019\pi$
$234$	0	0
$235$	4.24797	0.277107
$236$	0	0
$237$	−7.43559	−0.482993
$238$	0	0
$239$	20.5934	1.33207	0.666037	−	0.745919i	$-0.267989\pi$
$239$	20.5934	1.33207	0.666037	+	0.745919i	$0.267989\pi$
$240$	0	0
$241$	−23.1722	−1.49266	−0.746328	−	0.665578i	$-0.768185\pi$
$241$	−23.1722	−1.49266	−0.746328	+	0.665578i	$0.768185\pi$
$242$	0	0
$243$	−20.6498	−1.32468
$244$	0	0
$245$	11.1876	0.714751
$246$	0	0
$247$	−0.247966	−0.0157777
$248$	0	0
$249$	−15.7029	−0.995129
$250$	0	0
$251$	7.63923	0.482184	0.241092	−	0.970502i	$-0.422494\pi$
$251$	7.63923	0.482184	0.241092	+	0.970502i	$0.422494\pi$
$252$	0	0
$253$	17.3792	1.09262
$254$	0	0
$255$	4.05311	0.253815
$256$	0	0
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	0	0
$259$	0.545096	0.0338706
$260$	0	0
$261$	−5.00724	−0.309940
$262$	0	0
$263$	25.9508	1.60020	0.800099	−	0.599869i	$-0.204780\pi$
$263$	25.9508	1.60020	0.800099	+	0.599869i	$0.204780\pi$
$264$	0	0
$265$	−11.3382	−0.696498
$266$	0	0
$267$	24.2624	1.48484
$268$	0	0
$269$	−16.2624	−0.991539	−0.495769	−	0.868454i	$-0.665114\pi$
$269$	−16.2624	−0.991539	−0.495769	+	0.868454i	$0.665114\pi$
$270$	0	0
$271$	−14.6127	−0.887657	−0.443829	−	0.896112i	$-0.646380\pi$
$271$	−14.6127	−0.887657	−0.443829	+	0.896112i	$0.646380\pi$
$272$	0	0
$273$	−0.180384	−0.0109173
$274$	0	0
$275$	−9.33092	−0.562676
$276$	0	0
$277$	−23.0217	−1.38324	−0.691620	−	0.722261i	$-0.743103\pi$
$277$	−23.0217	−1.38324	−0.691620	+	0.722261i	$0.743103\pi$
$278$	0	0
$279$	24.4018	1.46090
$280$	0	0
$281$	−12.8567	−0.766966	−0.383483	−	0.923548i	$-0.625276\pi$
$281$	−12.8567	−0.766966	−0.383483	+	0.923548i	$0.625276\pi$
$282$	0	0
$283$	−7.57165	−0.450088	−0.225044	−	0.974349i	$-0.572253\pi$
$283$	−7.57165	−0.450088	−0.225044	+	0.974349i	$0.572253\pi$
$284$	0	0
$285$	−8.90981	−0.527771
$286$	0	0
$287$	0.725480	0.0428237
$288$	0	0
$289$	−16.1722	−0.951309
$290$	0	0
$291$	10.1949	0.597633
$292$	0	0
$293$	22.7439	1.32871	0.664356	−	0.747416i	$-0.268706\pi$
$293$	22.7439	1.32871	0.664356	+	0.747416i	$0.268706\pi$
$294$	0	0
$295$	−18.6232	−1.08429
$296$	0	0
$297$	12.6425	0.733594
$298$	0	0
$299$	−0.511305	−0.0295695
$300$	0	0
$301$	5.58612	0.321979
$302$	0	0
$303$	38.8606	2.23249
$304$	0	0
$305$	−4.60544	−0.263707
$306$	0	0
$307$	−3.16501	−0.180637	−0.0903184	−	0.995913i	$-0.528788\pi$
$307$	−3.16501	−0.180637	−0.0903184	+	0.995913i	$0.528788\pi$
$308$	0	0
$309$	−15.6127	−0.888174
$310$	0	0
$311$	19.2962	1.09419	0.547094	−	0.837071i	$-0.315734\pi$
$311$	19.2962	1.09419	0.547094	+	0.837071i	$0.315734\pi$
$312$	0	0
$313$	22.6908	1.28256	0.641280	−	0.767307i	$-0.278404\pi$
$313$	22.6908	1.28256	0.641280	+	0.767307i	$0.278404\pi$
$314$	0	0
$315$	−3.75203	−0.211403
$316$	0	0
$317$	7.76651	0.436211	0.218105	−	0.975925i	$-0.430012\pi$
$317$	7.76651	0.436211	0.218105	+	0.975925i	$0.430012\pi$
$318$	0	0
$319$	−5.11675	−0.286483
$320$	0	0
$321$	−20.5403	−1.14645
$322$	0	0
$323$	−1.81962	−0.101246
$324$	0	0
$325$	0.274520	0.0152277
$326$	0	0
$327$	26.6908	1.47600
$328$	0	0
$329$	−1.38732	−0.0764855
$330$	0	0
$331$	5.58612	0.307041	0.153521	−	0.988145i	$-0.450939\pi$
$331$	5.58612	0.307041	0.153521	+	0.988145i	$0.450939\pi$
$332$	0	0
$333$	4.12398	0.225993
$334$	0	0
$335$	−3.89049	−0.212560
$336$	0	0
$337$	15.8003	0.860697	0.430349	−	0.902663i	$-0.358391\pi$
$337$	15.8003	0.860697	0.430349	+	0.902663i	$0.358391\pi$
$338$	0	0
$339$	17.9614	0.975528
$340$	0	0
$341$	24.9355	1.35033
$342$	0	0
$343$	−7.46938	−0.403309
$344$	0	0
$345$	−18.3719	−0.989113
$346$	0	0
$347$	−6.01447	−0.322874	−0.161437	−	0.986883i	$-0.551613\pi$
$347$	−6.01447	−0.322874	−0.161437	+	0.986883i	$0.551613\pi$
$348$	0	0
$349$	6.74390	0.360993	0.180496	−	0.983576i	$-0.442230\pi$
$349$	6.74390	0.360993	0.180496	+	0.983576i	$0.442230\pi$
$350$	0	0
$351$	−0.371950	−0.0198532
$352$	0	0
$353$	20.1949	1.07486	0.537432	−	0.843307i	$-0.319395\pi$
$353$	20.1949	1.07486	0.537432	+	0.843307i	$0.319395\pi$
$354$	0	0
$355$	13.1578	0.698342
$356$	0	0
$357$	−1.32368	−0.0700568
$358$	0	0
$359$	−3.88325	−0.204950	−0.102475	−	0.994736i	$-0.532676\pi$
$359$	−3.88325	−0.204950	−0.102475	+	0.994736i	$0.532676\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	18.0410	0.946908
$364$	0	0
$365$	22.6199	1.18398
$366$	0	0
$367$	−32.7439	−1.70922	−0.854609	−	0.519272i	$-0.826203\pi$
$367$	−32.7439	−1.70922	−0.854609	+	0.519272i	$0.826203\pi$
$368$	0	0
$369$	5.48870	0.285730
$370$	0	0
$371$	3.70287	0.192243
$372$	0	0
$373$	−31.3035	−1.62083	−0.810416	−	0.585855i	$-0.800759\pi$
$373$	−31.3035	−1.62083	−0.810416	+	0.585855i	$0.800759\pi$
$374$	0	0
$375$	32.1385	1.65962
$376$	0	0
$377$	0.150537	0.00775306
$378$	0	0
$379$	20.0072	1.02770	0.513851	−	0.857879i	$-0.328218\pi$
$379$	20.0072	1.02770	0.513851	+	0.857879i	$0.328218\pi$
$380$	0	0
$381$	44.3406	2.27164
$382$	0	0
$383$	3.17225	0.162094	0.0810472	−	0.996710i	$-0.474174\pi$
$383$	3.17225	0.162094	0.0810472	+	0.996710i	$0.474174\pi$
$384$	0	0
$385$	−3.83409	−0.195403
$386$	0	0
$387$	42.2624	2.14832
$388$	0	0
$389$	−35.5297	−1.80143	−0.900714	−	0.434412i	$-0.856956\pi$
$389$	−35.5297	−1.80143	−0.900714	+	0.434412i	$0.856956\pi$
$390$	0	0
$391$	−3.75203	−0.189748
$392$	0	0
$393$	−18.4815	−0.932266
$394$	0	0
$395$	4.64976	0.233955
$396$	0	0
$397$	2.31160	0.116016	0.0580080	−	0.998316i	$-0.481525\pi$
$397$	2.31160	0.116016	0.0580080	+	0.998316i	$0.481525\pi$
$398$	0	0
$399$	2.90981	0.145673
$400$	0	0
$401$	−32.8712	−1.64151	−0.820754	−	0.571282i	$-0.806446\pi$
$401$	−32.8712	−1.64151	−0.820754	+	0.571282i	$0.806446\pi$
$402$	0	0
$403$	−0.733614	−0.0365439
$404$	0	0
$405$	7.28505	0.361997
$406$	0	0
$407$	4.21417	0.208889
$408$	0	0
$409$	−32.9243	−1.62800	−0.814001	−	0.580864i	$-0.802715\pi$
$409$	−32.9243	−1.62800	−0.814001	+	0.580864i	$0.802715\pi$
$410$	0	0
$411$	−14.6498	−0.722619
$412$	0	0
$413$	6.08206	0.299278
$414$	0	0
$415$	9.81962	0.482026
$416$	0	0
$417$	38.3599	1.87849
$418$	0	0
$419$	18.4356	0.900637	0.450319	−	0.892868i	$-0.351310\pi$
$419$	18.4356	0.900637	0.450319	+	0.892868i	$0.351310\pi$
$420$	0	0
$421$	40.6489	1.98110	0.990552	−	0.137136i	$-0.0437897\pi$
$421$	40.6489	1.98110	0.990552	+	0.137136i	$0.0437897\pi$
$422$	0	0
$423$	−10.4959	−0.510330
$424$	0	0
$425$	2.01447	0.0977164
$426$	0	0
$427$	1.50407	0.0727869
$428$	0	0
$429$	−1.39456	−0.0673299
$430$	0	0
$431$	26.4428	1.27371	0.636853	−	0.770985i	$-0.280236\pi$
$431$	26.4428	1.27371	0.636853	+	0.770985i	$0.280236\pi$
$432$	0	0
$433$	−24.8639	−1.19488	−0.597442	−	0.801912i	$-0.703816\pi$
$433$	−24.8639	−1.19488	−0.597442	+	0.801912i	$0.703816\pi$
$434$	0	0
$435$	5.40903	0.259343
$436$	0	0
$437$	8.24797	0.394554
$438$	0	0
$439$	−6.05640	−0.289056	−0.144528	−	0.989501i	$-0.546166\pi$
$439$	−6.05640	−0.289056	−0.144528	+	0.989501i	$0.546166\pi$
$440$	0	0
$441$	−27.6425	−1.31631
$442$	0	0
$443$	−39.7472	−1.88845	−0.944223	−	0.329307i	$-0.893185\pi$
$443$	−39.7472	−1.88845	−0.944223	+	0.329307i	$0.893185\pi$
$444$	0	0
$445$	−15.1722	−0.719234
$446$	0	0
$447$	−32.8606	−1.55426
$448$	0	0
$449$	27.3526	1.29085	0.645425	−	0.763823i	$-0.276680\pi$
$449$	27.3526	1.29085	0.645425	+	0.763823i	$0.276680\pi$
$450$	0	0
$451$	5.60873	0.264105
$452$	0	0
$453$	−48.0821	−2.25909
$454$	0	0
$455$	0.112801	0.00528819
$456$	0	0
$457$	17.6006	0.823321	0.411661	−	0.911337i	$-0.364949\pi$
$457$	17.6006	0.823321	0.411661	+	0.911337i	$0.364949\pi$
$458$	0	0
$459$	−2.72942	−0.127399
$460$	0	0
$461$	−25.1722	−1.17239	−0.586194	−	0.810171i	$-0.699374\pi$
$461$	−25.1722	−1.17239	−0.586194	+	0.810171i	$0.699374\pi$
$462$	0	0
$463$	9.21417	0.428219	0.214110	−	0.976810i	$-0.431315\pi$
$463$	9.21417	0.428219	0.214110	+	0.976810i	$0.431315\pi$
$464$	0	0
$465$	−26.3599	−1.22241
$466$	0	0
$467$	−0.262441	−0.0121443	−0.00607216	−	0.999982i	$-0.501933\pi$
$467$	−0.262441	−0.0121443	−0.00607216	+	0.999982i	$0.501933\pi$
$468$	0	0
$469$	1.27058	0.0586697
$470$	0	0
$471$	−8.41782	−0.387873
$472$	0	0
$473$	43.1867	1.98573
$474$	0	0
$475$	−4.42835	−0.203187
$476$	0	0
$477$	28.0145	1.28269
$478$	0	0
$479$	−9.21417	−0.421006	−0.210503	−	0.977593i	$-0.567510\pi$
$479$	−9.21417	−0.421006	−0.210503	+	0.977593i	$0.567510\pi$
$480$	0	0
$481$	−0.123983	−0.00565315
$482$	0	0
$483$	6.00000	0.273009
$484$	0	0
$485$	−6.37524	−0.289485
$486$	0	0
$487$	−32.0289	−1.45137	−0.725685	−	0.688027i	$-0.758477\pi$
$487$	−32.0289	−1.45137	−0.725685	+	0.688027i	$0.758477\pi$
$488$	0	0
$489$	−26.0676	−1.17882
$490$	0	0
$491$	−11.9436	−0.539007	−0.269504	−	0.962999i	$-0.586860\pi$
$491$	−11.9436	−0.539007	−0.269504	+	0.962999i	$0.586860\pi$
$492$	0	0
$493$	1.10467	0.0497517
$494$	0	0
$495$	−29.0072	−1.30378
$496$	0	0
$497$	−4.29713	−0.192753
$498$	0	0
$499$	−11.4202	−0.511239	−0.255620	−	0.966777i	$-0.582279\pi$
$499$	−11.4202	−0.511239	−0.255620	+	0.966777i	$0.582279\pi$
$500$	0	0
$501$	−16.9364	−0.756661
$502$	0	0
$503$	21.5176	0.959424	0.479712	−	0.877426i	$-0.340741\pi$
$503$	21.5176	0.959424	0.479712	+	0.877426i	$0.340741\pi$
$504$	0	0
$505$	−24.3011	−1.08138
$506$	0	0
$507$	−34.6570	−1.53917
$508$	0	0
$509$	0.778588	0.0345103	0.0172551	−	0.999851i	$-0.494507\pi$
$509$	0.778588	0.0345103	0.0172551	+	0.999851i	$0.494507\pi$
$510$	0	0
$511$	−7.38732	−0.326796
$512$	0	0
$513$	6.00000	0.264906
$514$	0	0
$515$	9.76322	0.430219
$516$	0	0
$517$	−10.7255	−0.471706
$518$	0	0
$519$	60.8365	2.67043
$520$	0	0
$521$	35.6498	1.56184	0.780922	−	0.624628i	$-0.214749\pi$
$521$	35.6498	1.56184	0.780922	+	0.624628i	$0.214749\pi$
$522$	0	0
$523$	31.7810	1.38969	0.694843	−	0.719162i	$-0.255474\pi$
$523$	31.7810	1.38969	0.694843	+	0.719162i	$0.255474\pi$
$524$	0	0
$525$	−3.22141	−0.140594
$526$	0	0
$527$	−5.38338	−0.234504
$528$	0	0
$529$	−5.99276	−0.260555
$530$	0	0
$531$	46.0145	1.99686
$532$	0	0
$533$	−0.165012	−0.00714745
$534$	0	0
$535$	12.8446	0.555321
$536$	0	0
$537$	−13.7665	−0.594069
$538$	0	0
$539$	−28.2471	−1.21669
$540$	0	0
$541$	−30.8269	−1.32535	−0.662675	−	0.748907i	$-0.730579\pi$
$541$	−30.8269	−1.32535	−0.662675	+	0.748907i	$0.730579\pi$
$542$	0	0
$543$	−0.793062	−0.0340336
$544$	0	0
$545$	−16.6908	−0.714955
$546$	0	0
$547$	2.97739	0.127304	0.0636520	−	0.997972i	$-0.479725\pi$
$547$	2.97739	0.127304	0.0636520	+	0.997972i	$0.479725\pi$
$548$	0	0
$549$	11.3792	0.485652
$550$	0	0
$551$	−2.42835	−0.103451
$552$	0	0
$553$	−1.51854	−0.0645750
$554$	0	0
$555$	−4.45490	−0.189100
$556$	0	0
$557$	−41.7656	−1.76967	−0.884833	−	0.465909i	$-0.845728\pi$
$557$	−41.7656	−1.76967	−0.884833	+	0.465909i	$0.845728\pi$
$558$	0	0
$559$	−1.27058	−0.0537396
$560$	0	0
$561$	−10.2335	−0.432058
$562$	0	0
$563$	−7.45885	−0.314353	−0.157177	−	0.987571i	$-0.550239\pi$
$563$	−7.45885	−0.314353	−0.157177	+	0.987571i	$0.550239\pi$
$564$	0	0
$565$	−11.2319	−0.472531
$566$	0	0
$567$	−2.37919	−0.0999165
$568$	0	0
$569$	14.9098	0.625052	0.312526	−	0.949909i	$-0.398825\pi$
$569$	14.9098	0.625052	0.312526	+	0.949909i	$0.398825\pi$
$570$	0	0
$571$	21.0830	0.882294	0.441147	−	0.897435i	$-0.354572\pi$
$571$	21.0830	0.882294	0.441147	+	0.897435i	$0.354572\pi$
$572$	0	0
$573$	−32.4694	−1.35643
$574$	0	0
$575$	−9.13122	−0.380798
$576$	0	0
$577$	20.0145	0.833213	0.416607	−	0.909087i	$-0.363219\pi$
$577$	20.0145	0.833213	0.416607	+	0.909087i	$0.363219\pi$
$578$	0	0
$579$	42.7053	1.77477
$580$	0	0
$581$	−3.20694	−0.133046
$582$	0	0
$583$	28.6272	1.18562
$584$	0	0
$585$	0.853408	0.0352841
$586$	0	0
$587$	−9.17225	−0.378579	−0.189290	−	0.981921i	$-0.560618\pi$
$587$	−9.17225	−0.378579	−0.189290	+	0.981921i	$0.560618\pi$
$588$	0	0
$589$	11.8341	0.487615
$590$	0	0
$591$	−5.16830	−0.212596
$592$	0	0
$593$	26.8374	1.10208	0.551040	−	0.834479i	$-0.314231\pi$
$593$	26.8374	1.10208	0.551040	+	0.834479i	$0.314231\pi$
$594$	0	0
$595$	0.827751	0.0339345
$596$	0	0
$597$	−42.7053	−1.74781
$598$	0	0
$599$	−26.1988	−1.07045	−0.535227	−	0.844708i	$-0.679774\pi$
$599$	−26.1988	−1.07045	−0.535227	+	0.844708i	$0.679774\pi$
$600$	0	0
$601$	18.1795	0.741557	0.370778	−	0.928721i	$-0.379091\pi$
$601$	18.1795	0.741557	0.370778	+	0.928721i	$0.379091\pi$
$602$	0	0
$603$	9.61268	0.391459
$604$	0	0
$605$	−11.2818	−0.458669
$606$	0	0
$607$	−40.1240	−1.62858	−0.814291	−	0.580457i	$-0.802874\pi$
$607$	−40.1240	−1.62858	−0.814291	+	0.580457i	$0.802874\pi$
$608$	0	0
$609$	−1.76651	−0.0715825
$610$	0	0
$611$	0.315549	0.0127657
$612$	0	0
$613$	−5.68840	−0.229752	−0.114876	−	0.993380i	$-0.536647\pi$
$613$	−5.68840	−0.229752	−0.114876	+	0.993380i	$0.536647\pi$
$614$	0	0
$615$	−5.92913	−0.239085
$616$	0	0
$617$	5.06848	0.204049	0.102025	−	0.994782i	$-0.467468\pi$
$617$	5.06848	0.204049	0.102025	+	0.994782i	$0.467468\pi$
$618$	0	0
$619$	47.0257	1.89012	0.945060	−	0.326896i	$-0.106003\pi$
$619$	47.0257	1.89012	0.945060	+	0.326896i	$0.106003\pi$
$620$	0	0
$621$	12.3719	0.496469
$622$	0	0
$623$	4.95503	0.198519
$624$	0	0
$625$	−9.02655	−0.361062
$626$	0	0
$627$	22.4959	0.898401
$628$	0	0
$629$	−0.909808	−0.0362764
$630$	0	0
$631$	−34.1916	−1.36114	−0.680572	−	0.732681i	$-0.738269\pi$
$631$	−34.1916	−1.36114	−0.680572	+	0.732681i	$0.738269\pi$
$632$	0	0
$633$	−42.6537	−1.69533
$634$	0	0
$635$	−27.7279	−1.10035
$636$	0	0
$637$	0.831043	0.0329271
$638$	0	0
$639$	−32.5104	−1.28609
$640$	0	0
$641$	−8.48870	−0.335283	−0.167642	−	0.985848i	$-0.553615\pi$
$641$	−8.48870	−0.335283	−0.167642	+	0.985848i	$0.553615\pi$
$642$	0	0
$643$	−21.6006	−0.851844	−0.425922	−	0.904760i	$-0.640050\pi$
$643$	−21.6006	−0.851844	−0.425922	+	0.904760i	$0.640050\pi$
$644$	0	0
$645$	−45.6537	−1.79761
$646$	0	0
$647$	−1.71011	−0.0672313	−0.0336156	−	0.999435i	$-0.510702\pi$
$647$	−1.71011	−0.0672313	−0.0336156	+	0.999435i	$0.510702\pi$
$648$	0	0
$649$	47.0208	1.84573
$650$	0	0
$651$	8.60873	0.337403
$652$	0	0
$653$	−22.5668	−0.883107	−0.441554	−	0.897235i	$-0.645573\pi$
$653$	−22.5668	−0.883107	−0.441554	+	0.897235i	$0.645573\pi$
$654$	0	0
$655$	11.5572	0.451576
$656$	0	0
$657$	−55.8896	−2.18046
$658$	0	0
$659$	14.9170	0.581086	0.290543	−	0.956862i	$-0.406164\pi$
$659$	14.9170	0.581086	0.290543	+	0.956862i	$0.406164\pi$
$660$	0	0
$661$	−13.1997	−0.513409	−0.256704	−	0.966490i	$-0.582637\pi$
$661$	−13.1997	−0.513409	−0.256704	+	0.966490i	$0.582637\pi$
$662$	0	0
$663$	0.301075	0.0116928
$664$	0	0
$665$	−1.81962	−0.0705617
$666$	0	0
$667$	−5.00724	−0.193881
$668$	0	0
$669$	−16.5065	−0.638177
$670$	0	0
$671$	11.6281	0.448896
$672$	0	0
$673$	−7.16501	−0.276191	−0.138095	−	0.990419i	$-0.544098\pi$
$673$	−7.16501	−0.276191	−0.138095	+	0.990419i	$0.544098\pi$
$674$	0	0
$675$	−6.64252	−0.255671
$676$	0	0
$677$	46.2664	1.77816	0.889081	−	0.457750i	$-0.151345\pi$
$677$	46.2664	1.77816	0.889081	+	0.457750i	$0.151345\pi$
$678$	0	0
$679$	2.08206	0.0799020
$680$	0	0
$681$	−67.1481	−2.57312
$682$	0	0
$683$	49.2543	1.88466	0.942332	−	0.334680i	$-0.108628\pi$
$683$	49.2543	1.88466	0.942332	+	0.334680i	$0.108628\pi$
$684$	0	0
$685$	9.16107	0.350026
$686$	0	0
$687$	−79.3179	−3.02617
$688$	0	0
$689$	−0.842225	−0.0320862
$690$	0	0
$691$	23.5330	0.895238	0.447619	−	0.894224i	$-0.352272\pi$
$691$	23.5330	0.895238	0.447619	+	0.894224i	$0.352272\pi$
$692$	0	0
$693$	9.47332	0.359862
$694$	0	0
$695$	−23.9879	−0.909914
$696$	0	0
$697$	−1.21088	−0.0458655
$698$	0	0
$699$	−26.3985	−0.998483
$700$	0	0
$701$	13.7367	0.518827	0.259413	−	0.965766i	$-0.416471\pi$
$701$	13.7367	0.518827	0.259413	+	0.965766i	$0.416471\pi$
$702$	0	0
$703$	2.00000	0.0754314
$704$	0	0
$705$	11.3382	0.427020
$706$	0	0
$707$	7.93636	0.298478
$708$	0	0
$709$	−20.2857	−0.761846	−0.380923	−	0.924607i	$-0.624394\pi$
$709$	−20.2857	−0.761846	−0.380923	+	0.924607i	$0.624394\pi$
$710$	0	0
$711$	−11.4887	−0.430860
$712$	0	0
$713$	24.4018	0.913854
$714$	0	0
$715$	0.872072	0.0326136
$716$	0	0
$717$	54.9653	2.05272
$718$	0	0
$719$	−11.3421	−0.422989	−0.211495	−	0.977379i	$-0.567833\pi$
$719$	−11.3421	−0.422989	−0.211495	+	0.977379i	$0.567833\pi$
$720$	0	0
$721$	−3.18852	−0.118747
$722$	0	0
$723$	−61.8486	−2.30017
$724$	0	0
$725$	2.68840	0.0998445
$726$	0	0
$727$	−15.1345	−0.561308	−0.280654	−	0.959809i	$-0.590551\pi$
$727$	−15.1345	−0.561308	−0.280654	+	0.959809i	$0.590551\pi$
$728$	0	0
$729$	−42.0217	−1.55636
$730$	0	0
$731$	−9.32368	−0.344849
$732$	0	0
$733$	7.02655	0.259532	0.129766	−	0.991545i	$-0.458577\pi$
$733$	7.02655	0.259532	0.129766	+	0.991545i	$0.458577\pi$
$734$	0	0
$735$	29.8606	1.10143
$736$	0	0
$737$	9.82291	0.361831
$738$	0	0
$739$	−32.6425	−1.20077	−0.600387	−	0.799709i	$-0.704987\pi$
$739$	−32.6425	−1.20077	−0.600387	+	0.799709i	$0.704987\pi$
$740$	0	0
$741$	−0.661842	−0.0243134
$742$	0	0
$743$	−17.3487	−0.636462	−0.318231	−	0.948013i	$-0.603089\pi$
$743$	−17.3487	−0.636462	−0.318231	+	0.948013i	$0.603089\pi$
$744$	0	0
$745$	20.5490	0.752859
$746$	0	0
$747$	−24.2624	−0.887716
$748$	0	0
$749$	−4.19486	−0.153277
$750$	0	0
$751$	−31.8833	−1.16344	−0.581718	−	0.813390i	$-0.697619\pi$
$751$	−31.8833	−1.16344	−0.581718	+	0.813390i	$0.697619\pi$
$752$	0	0
$753$	20.3897	0.743043
$754$	0	0
$755$	30.0676	1.09427
$756$	0	0
$757$	−41.2021	−1.49752	−0.748758	−	0.662844i	$-0.769349\pi$
$757$	−41.2021	−1.49752	−0.748758	+	0.662844i	$0.769349\pi$
$758$	0	0
$759$	46.3864	1.68372
$760$	0	0
$761$	−31.9010	−1.15641	−0.578206	−	0.815891i	$-0.696247\pi$
$761$	−31.9010	−1.15641	−0.578206	+	0.815891i	$0.696247\pi$
$762$	0	0
$763$	5.45096	0.197338
$764$	0	0
$765$	6.26244	0.226419
$766$	0	0
$767$	−1.38338	−0.0499508
$768$	0	0
$769$	32.0821	1.15691	0.578454	−	0.815715i	$-0.303656\pi$
$769$	32.0821	1.15691	0.578454	+	0.815715i	$0.303656\pi$
$770$	0	0
$771$	16.0145	0.576747
$772$	0	0
$773$	39.0024	1.40282	0.701409	−	0.712759i	$-0.252555\pi$
$773$	39.0024	1.40282	0.701409	+	0.712759i	$0.252555\pi$
$774$	0	0
$775$	−13.1014	−0.470615
$776$	0	0
$777$	1.45490	0.0521944
$778$	0	0
$779$	2.66184	0.0953704
$780$	0	0
$781$	−33.2214	−1.18876
$782$	0	0
$783$	−3.64252	−0.130173
$784$	0	0
$785$	5.26399	0.187880
$786$	0	0
$787$	7.34210	0.261718	0.130859	−	0.991401i	$-0.458227\pi$
$787$	7.34210	0.261718	0.130859	+	0.991401i	$0.458227\pi$
$788$	0	0
$789$	69.2648	2.46589
$790$	0	0
$791$	3.66818	0.130426
$792$	0	0
$793$	−0.342103	−0.0121484
$794$	0	0
$795$	−30.2624	−1.07330
$796$	0	0
$797$	5.44767	0.192966	0.0964831	−	0.995335i	$-0.469241\pi$
$797$	5.44767	0.192966	0.0964831	+	0.995335i	$0.469241\pi$
$798$	0	0
$799$	2.31555	0.0819182
$800$	0	0
$801$	37.4878	1.32457
$802$	0	0
$803$	−57.1119	−2.01544
$804$	0	0
$805$	−3.75203	−0.132242
$806$	0	0
$807$	−43.4057	−1.52795
$808$	0	0
$809$	37.3382	1.31274	0.656370	−	0.754439i	$-0.272091\pi$
$809$	37.3382	1.31274	0.656370	+	0.754439i	$0.272091\pi$
$810$	0	0
$811$	−20.3107	−0.713205	−0.356603	−	0.934256i	$-0.616065\pi$
$811$	−20.3107	−0.713205	−0.356603	+	0.934256i	$0.616065\pi$
$812$	0	0
$813$	−39.0024	−1.36787
$814$	0	0
$815$	16.3011	0.571002
$816$	0	0
$817$	20.4959	0.717062
$818$	0	0
$819$	−0.278710	−0.00973892
$820$	0	0
$821$	4.47751	0.156266	0.0781331	−	0.996943i	$-0.475104\pi$
$821$	4.47751	0.156266	0.0781331	+	0.996943i	$0.475104\pi$
$822$	0	0
$823$	−20.1352	−0.701868	−0.350934	−	0.936400i	$-0.614136\pi$
$823$	−20.1352	−0.701868	−0.350934	+	0.936400i	$0.614136\pi$
$824$	0	0
$825$	−24.9050	−0.867080
$826$	0	0
$827$	6.66184	0.231655	0.115827	−	0.993269i	$-0.463048\pi$
$827$	6.66184	0.231655	0.115827	+	0.993269i	$0.463048\pi$
$828$	0	0
$829$	26.3044	0.913588	0.456794	−	0.889572i	$-0.348998\pi$
$829$	26.3044	0.913588	0.456794	+	0.889572i	$0.348998\pi$
$830$	0	0
$831$	−61.4468	−2.13156
$832$	0	0
$833$	6.09833	0.211294
$834$	0	0
$835$	10.5910	0.366516
$836$	0	0
$837$	17.7511	0.613569
$838$	0	0
$839$	52.3300	1.80663	0.903317	−	0.428975i	$-0.141125\pi$
$839$	52.3300	1.80663	0.903317	+	0.428975i	$0.141125\pi$
$840$	0	0
$841$	−27.5258	−0.949165
$842$	0	0
$843$	−34.3155	−1.18189
$844$	0	0
$845$	21.6724	0.745552
$846$	0	0
$847$	3.68445	0.126599
$848$	0	0
$849$	−20.2093	−0.693583
$850$	0	0
$851$	4.12398	0.141368
$852$	0	0
$853$	31.9170	1.09282	0.546409	−	0.837518i	$-0.315994\pi$
$853$	31.9170	1.09282	0.546409	+	0.837518i	$0.315994\pi$
$854$	0	0
$855$	−13.7665	−0.470805
$856$	0	0
$857$	−34.7584	−1.18732	−0.593662	−	0.804715i	$-0.702318\pi$
$857$	−34.7584	−1.18732	−0.593662	+	0.804715i	$0.702318\pi$
$858$	0	0
$859$	−37.7955	−1.28956	−0.644782	−	0.764366i	$-0.723052\pi$
$859$	−37.7955	−1.28956	−0.644782	+	0.764366i	$0.723052\pi$
$860$	0	0
$861$	1.93636	0.0659910
$862$	0	0
$863$	39.6682	1.35032	0.675160	−	0.737671i	$-0.264074\pi$
$863$	39.6682	1.35032	0.675160	+	0.737671i	$0.264074\pi$
$864$	0	0
$865$	−38.0434	−1.29352
$866$	0	0
$867$	−43.1650	−1.46596
$868$	0	0
$869$	−11.7400	−0.398251
$870$	0	0
$871$	−0.288995	−0.00979222
$872$	0	0
$873$	15.7520	0.533126
$874$	0	0
$875$	6.56352	0.221887
$876$	0	0
$877$	13.5330	0.456977	0.228489	−	0.973547i	$-0.426622\pi$
$877$	13.5330	0.456977	0.228489	+	0.973547i	$0.426622\pi$
$878$	0	0
$879$	60.7053	2.04754
$880$	0	0
$881$	−7.47422	−0.251813	−0.125906	−	0.992042i	$-0.540184\pi$
$881$	−7.47422	−0.251813	−0.125906	+	0.992042i	$0.540184\pi$
$882$	0	0
$883$	35.2995	1.18792	0.593962	−	0.804493i	$-0.297563\pi$
$883$	35.2995	1.18792	0.593962	+	0.804493i	$0.297563\pi$
$884$	0	0
$885$	−49.7068	−1.67088
$886$	0	0
$887$	−24.3792	−0.818573	−0.409286	−	0.912406i	$-0.634222\pi$
$887$	−24.3792	−0.818573	−0.409286	+	0.912406i	$0.634222\pi$
$888$	0	0
$889$	9.05550	0.303712
$890$	0	0
$891$	−18.3937	−0.616211
$892$	0	0
$893$	−5.09019	−0.170337
$894$	0	0
$895$	8.60873	0.287758
$896$	0	0
$897$	−1.36471	−0.0455664
$898$	0	0
$899$	−7.18433	−0.239611
$900$	0	0
$901$	−6.18038	−0.205898
$902$	0	0
$903$	14.9098	0.496167
$904$	0	0
$905$	0.495933	0.0164854
$906$	0	0
$907$	−5.14330	−0.170780	−0.0853902	−	0.996348i	$-0.527214\pi$
$907$	−5.14330	−0.170780	−0.0853902	+	0.996348i	$0.527214\pi$
$908$	0	0
$909$	60.0434	1.99151
$910$	0	0
$911$	−31.7665	−1.05247	−0.526236	−	0.850339i	$-0.676397\pi$
$911$	−31.7665	−1.05247	−0.526236	+	0.850339i	$0.676397\pi$
$912$	0	0
$913$	−24.7931	−0.820531
$914$	0	0
$915$	−12.2923	−0.406370
$916$	0	0
$917$	−3.77440	−0.124642
$918$	0	0
$919$	4.00000	0.131948	0.0659739	−	0.997821i	$-0.478985\pi$
$919$	4.00000	0.131948	0.0659739	+	0.997821i	$0.478985\pi$
$920$	0	0
$921$	−8.44767	−0.278360
$922$	0	0
$923$	0.977391	0.0321712
$924$	0	0
$925$	−2.21417	−0.0728016
$926$	0	0
$927$	−24.1231	−0.792306
$928$	0	0
$929$	38.8413	1.27434	0.637171	−	0.770722i	$-0.280104\pi$
$929$	38.8413	1.27434	0.637171	+	0.770722i	$0.280104\pi$
$930$	0	0
$931$	−13.4057	−0.439355
$932$	0	0
$933$	51.5032	1.68614
$934$	0	0
$935$	6.39940	0.209283
$936$	0	0
$937$	−18.7964	−0.614050	−0.307025	−	0.951701i	$-0.599334\pi$
$937$	−18.7964	−0.614050	−0.307025	+	0.951701i	$0.599334\pi$
$938$	0	0
$939$	60.5635	1.97642
$940$	0	0
$941$	53.7810	1.75321	0.876605	−	0.481211i	$-0.159803\pi$
$941$	53.7810	1.75321	0.876605	+	0.481211i	$0.159803\pi$
$942$	0	0
$943$	5.48870	0.178737
$944$	0	0
$945$	−2.72942	−0.0887882
$946$	0	0
$947$	−32.7294	−1.06356	−0.531782	−	0.846881i	$-0.678477\pi$
$947$	−32.7294	−1.06356	−0.531782	+	0.846881i	$0.678477\pi$
$948$	0	0
$949$	1.68026	0.0545436
$950$	0	0
$951$	20.7294	0.672198
$952$	0	0
$953$	−47.9952	−1.55472	−0.777358	−	0.629059i	$-0.783441\pi$
$953$	−47.9952	−1.55472	−0.777358	+	0.629059i	$0.783441\pi$
$954$	0	0
$955$	20.3044	0.657034
$956$	0	0
$957$	−13.6570	−0.441468
$958$	0	0
$959$	−2.99187	−0.0966124
$960$	0	0
$961$	4.01143	0.129401
$962$	0	0
$963$	−31.7367	−1.02270
$964$	0	0
$965$	−26.7053	−0.859673
$966$	0	0
$967$	59.1635	1.90257	0.951284	−	0.308315i	$-0.0997651\pi$
$967$	59.1635	1.90257	0.951284	+	0.308315i	$0.0997651\pi$
$968$	0	0
$969$	−4.85670	−0.156020
$970$	0	0
$971$	42.9928	1.37970	0.689852	−	0.723951i	$-0.257676\pi$
$971$	42.9928	1.37970	0.689852	+	0.723951i	$0.257676\pi$
$972$	0	0
$973$	7.83409	0.251149
$974$	0	0
$975$	0.732717	0.0234657
$976$	0	0
$977$	38.7584	1.23999	0.619995	−	0.784606i	$-0.287135\pi$
$977$	38.7584	1.23999	0.619995	+	0.784606i	$0.287135\pi$
$978$	0	0
$979$	38.3077	1.22432
$980$	0	0
$981$	41.2398	1.31669
$982$	0	0
$983$	−6.50646	−0.207524	−0.103762	−	0.994602i	$-0.533088\pi$
$983$	−6.50646	−0.207524	−0.103762	+	0.994602i	$0.533088\pi$
$984$	0	0
$985$	3.23194	0.102978
$986$	0	0
$987$	−3.70287	−0.117864
$988$	0	0
$989$	42.2624	1.34387
$990$	0	0
$991$	34.9276	1.10951	0.554755	−	0.832013i	$-0.312812\pi$
$991$	34.9276	1.10951	0.554755	+	0.832013i	$0.312812\pi$
$992$	0	0
$993$	14.9098	0.473148
$994$	0	0
$995$	26.7053	0.846614
$996$	0	0
$997$	4.41388	0.139789	0.0698944	−	0.997554i	$-0.477734\pi$
$997$	4.41388	0.139789	0.0698944	+	0.997554i	$0.477734\pi$
$998$	0	0
$999$	3.00000	0.0949158

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2368.2.a.bd.1.3		3
4.3	odd	2		2368.2.a.bc.1.1		3
8.3	odd	2		1184.2.a.l.1.3	✓	3
8.5	even	2		1184.2.a.m.1.1	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1184.2.a.l.1.3	✓	3	8.3	odd	2
1184.2.a.m.1.1	yes	3	8.5	even	2
2368.2.a.bc.1.1		3	4.3	odd	2
2368.2.a.bd.1.3		3	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:	\( N \)	\(=\)	\( 2368 = 2^{6} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2368.a (trivial)

\(f(q)\)	\(=\)	\(q+2.66908 q^{3} -1.66908 q^{5} +0.545096 q^{7} +4.12398 q^{9} +4.21417 q^{11} -0.123983 q^{13} -4.45490 q^{15} -0.909808 q^{17} +2.00000 q^{19} +1.45490 q^{21} +4.12398 q^{23} -2.21417 q^{25} +3.00000 q^{27} -1.21417 q^{29} +5.91705 q^{31} +11.2480 q^{33} -0.909808 q^{35} +1.00000 q^{37} -0.330921 q^{39} +1.33092 q^{41} +10.2480 q^{43} -6.88325 q^{45} -2.54510 q^{47} -6.70287 q^{49} -2.42835 q^{51} +6.79306 q^{53} -7.03379 q^{55} +5.33816 q^{57} +11.1578 q^{59} +2.75927 q^{61} +2.24797 q^{63} +0.206938 q^{65} +2.33092 q^{67} +11.0072 q^{69} -7.88325 q^{71} -13.5523 q^{73} -5.90981 q^{75} +2.29713 q^{77} -2.78583 q^{79} -4.36471 q^{81} -5.88325 q^{83} +1.51854 q^{85} -3.24073 q^{87} +9.09019 q^{89} -0.0675827 q^{91} +15.7931 q^{93} -3.33816 q^{95} +3.81962 q^{97} +17.3792 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{5} + 3 q^{7} + 3 q^{9} + 6 q^{11} + 9 q^{13} - 12 q^{15} + 6 q^{19} + 3 q^{21} + 3 q^{23} + 9 q^{27} + 3 q^{29} - 9 q^{31} + 15 q^{33} + 3 q^{37} - 9 q^{39} + 12 q^{41} + 12 q^{43} - 6 q^{45}+ \cdots + 6 q^{97}+O(q^{100}) \) 3 * q + 3 * q^5 + 3 * q^7 + 3 * q^9 + 6 * q^11 + 9 * q^13 - 12 * q^15 + 6 * q^19 + 3 * q^21 + 3 * q^23 + 9 * q^27 + 3 * q^29 - 9 * q^31 + 15 * q^33 + 3 * q^37 - 9 * q^39 + 12 * q^41 + 12 * q^43 - 6 * q^45 - 9 * q^47 + 6 * q^51 + 3 * q^53 - 9 * q^55 + 12 * q^59 + 3 * q^61 - 12 * q^63 + 18 * q^65 + 15 * q^67 + 9 * q^69 - 9 * q^71 - 18 * q^73 - 15 * q^75 + 27 * q^77 - 15 * q^79 - 9 * q^81 - 3 * q^83 - 6 * q^85 - 15 * q^87 + 30 * q^89 + 24 * q^91 + 30 * q^93 + 6 * q^95 + 6 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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