LMFDB - Embedded newform 968.4.a.o.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [968,4,Mod(1,968)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(968, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("968.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 968 = 2^{3} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	968.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:	$57.1138488856$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 87x^{6} + 12x^{5} + 2157x^{4} + 2939x^{3} - 5906x^{2} - 3030x + 45 $ x^8 - 2x^7 - 87x^6 + 12x^5 + 2157x^4 + 2939x^3 - 5906x^2 - 3030*x + 45
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6}\cdot 11 $
Twist minimal:	no (minimal twist has level 88)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$6.90062$ of defining polynomial
Character	$\chi$	$=$	968.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-6.24005 q^{3} +8.05999 q^{5} +25.5634 q^{7} +11.9382 q^{9} -37.7535 q^{13} -50.2947 q^{15} +40.6687 q^{17} -114.776 q^{19} -159.517 q^{21} +40.3517 q^{23} -60.0365 q^{25} +93.9864 q^{27} -95.2574 q^{29} -217.659 q^{31} +206.041 q^{35} +253.997 q^{37} +235.584 q^{39} -370.610 q^{41} +453.237 q^{43} +96.2217 q^{45} -304.697 q^{47} +310.488 q^{49} -253.775 q^{51} +263.190 q^{53} +716.207 q^{57} -26.5227 q^{59} -262.767 q^{61} +305.181 q^{63} -304.293 q^{65} +956.232 q^{67} -251.797 q^{69} -405.573 q^{71} -557.834 q^{73} +374.631 q^{75} +928.116 q^{79} -908.811 q^{81} +372.454 q^{83} +327.790 q^{85} +594.411 q^{87} -1382.64 q^{89} -965.108 q^{91} +1358.20 q^{93} -925.093 q^{95} -979.623 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{3} - 13 q^{5} + 9 q^{7} + 36 q^{9} - 7 q^{13} - 66 q^{15} - 94 q^{17} + 92 q^{19} - 17 q^{21} - 46 q^{23} + 101 q^{25} + 124 q^{27} - 241 q^{29} - 265 q^{31} + 664 q^{35} - 469 q^{37} - 788 q^{39}+ \cdots - 4702 q^{97}+O(q^{100}) $ 8 * q - 8 * q^3 - 13 * q^5 + 9 * q^7 + 36 * q^9 - 7 * q^13 - 66 * q^15 - 94 * q^17 + 92 * q^19 - 17 * q^21 - 46 * q^23 + 101 * q^25 + 124 * q^27 - 241 * q^29 - 265 * q^31 + 664 * q^35 - 469 * q^37 - 788 * q^39 + 74 * q^41 - 35 * q^43 - 566 * q^45 + 197 * q^47 + 221 * q^49 + 608 * q^51 - 247 * q^53 + 979 * q^57 - 1208 * q^59 - 2061 * q^61 + 1617 * q^63 - 837 * q^65 - 829 * q^67 - 1860 * q^69 - 2005 * q^71 - 320 * q^73 - 2323 * q^75 + 1581 * q^79 - 3752 * q^81 - 692 * q^83 - 21 * q^85 + 131 * q^87 - 3205 * q^89 - 2724 * q^91 - 722 * q^93 + 432 * q^95 - 4702 * 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
$3$	−6.24005	−1.20090	−0.600449	−	0.799663i	$-0.705011\pi$
$3$	−6.24005	−1.20090	−0.600449	+	0.799663i	$0.705011\pi$
$4$	0	0
$5$	8.05999	0.720908	0.360454	−	0.932777i	$-0.382622\pi$
$5$	8.05999	0.720908	0.360454	+	0.932777i	$0.382622\pi$
$6$	0	0
$7$	25.5634	1.38029	0.690147	−	0.723669i	$-0.257546\pi$
$7$	25.5634	1.38029	0.690147	+	0.723669i	$0.257546\pi$
$8$	0	0
$9$	11.9382	0.442155
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−37.7535	−0.805457	−0.402728	−	0.915320i	$-0.631938\pi$
$13$	−37.7535	−0.805457	−0.402728	+	0.915320i	$0.631938\pi$
$14$	0	0
$15$	−50.2947	−0.865736
$16$	0	0
$17$	40.6687	0.580213	0.290106	−	0.956994i	$-0.406309\pi$
$17$	40.6687	0.580213	0.290106	+	0.956994i	$0.406309\pi$
$18$	0	0
$19$	−114.776	−1.38586	−0.692931	−	0.721004i	$-0.743681\pi$
$19$	−114.776	−1.38586	−0.692931	+	0.721004i	$0.743681\pi$
$20$	0	0
$21$	−159.517	−1.65759
$22$	0	0
$23$	40.3517	0.365823	0.182911	−	0.983129i	$-0.441448\pi$
$23$	40.3517	0.365823	0.182911	+	0.983129i	$0.441448\pi$
$24$	0	0
$25$	−60.0365	−0.480292
$26$	0	0
$27$	93.9864	0.669915
$28$	0	0
$29$	−95.2574	−0.609960	−0.304980	−	0.952359i	$-0.598650\pi$
$29$	−95.2574	−0.609960	−0.304980	+	0.952359i	$0.598650\pi$
$30$	0	0
$31$	−217.659	−1.26106	−0.630528	−	0.776166i	$-0.717162\pi$
$31$	−217.659	−1.26106	−0.630528	+	0.776166i	$0.717162\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	206.041	0.995065
$36$	0	0
$37$	253.997	1.12856	0.564282	−	0.825582i	$-0.309153\pi$
$37$	253.997	1.12856	0.564282	+	0.825582i	$0.309153\pi$
$38$	0	0
$39$	235.584	0.967271
$40$	0	0
$41$	−370.610	−1.41170	−0.705848	−	0.708363i	$-0.749434\pi$
$41$	−370.610	−1.41170	−0.705848	+	0.708363i	$0.749434\pi$
$42$	0	0
$43$	453.237	1.60740	0.803698	−	0.595038i	$-0.202863\pi$
$43$	453.237	1.60740	0.803698	+	0.595038i	$0.202863\pi$
$44$	0	0
$45$	96.2217	0.318753
$46$	0	0
$47$	−304.697	−0.945630	−0.472815	−	0.881162i	$-0.656762\pi$
$47$	−304.697	−0.945630	−0.472815	+	0.881162i	$0.656762\pi$
$48$	0	0
$49$	310.488	0.905213
$50$	0	0
$51$	−253.775	−0.696776
$52$	0	0
$53$	263.190	0.682113	0.341057	−	0.940043i	$-0.389215\pi$
$53$	263.190	0.682113	0.341057	+	0.940043i	$0.389215\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	716.207	1.66428
$58$	0	0
$59$	−26.5227	−0.0585248	−0.0292624	−	0.999572i	$-0.509316\pi$
$59$	−26.5227	−0.0585248	−0.0292624	+	0.999572i	$0.509316\pi$
$60$	0	0
$61$	−262.767	−0.551538	−0.275769	−	0.961224i	$-0.588933\pi$
$61$	−262.767	−0.551538	−0.275769	+	0.961224i	$0.588933\pi$
$62$	0	0
$63$	305.181	0.610304
$64$	0	0
$65$	−304.293	−0.580660
$66$	0	0
$67$	956.232	1.74362	0.871808	−	0.489847i	$-0.162947\pi$
$67$	956.232	1.74362	0.871808	+	0.489847i	$0.162947\pi$
$68$	0	0
$69$	−251.797	−0.439315
$70$	0	0
$71$	−405.573	−0.677924	−0.338962	−	0.940800i	$-0.610076\pi$
$71$	−405.573	−0.677924	−0.338962	+	0.940800i	$0.610076\pi$
$72$	0	0
$73$	−557.834	−0.894377	−0.447188	−	0.894440i	$-0.647575\pi$
$73$	−557.834	−0.894377	−0.447188	+	0.894440i	$0.647575\pi$
$74$	0	0
$75$	374.631	0.576782
$76$	0	0
$77$	0	0
$78$	0	0
$79$	928.116	1.32179	0.660894	−	0.750479i	$-0.270177\pi$
$79$	928.116	1.32179	0.660894	+	0.750479i	$0.270177\pi$
$80$	0	0
$81$	−908.811	−1.24665
$82$	0	0
$83$	372.454	0.492556	0.246278	−	0.969199i	$-0.420792\pi$
$83$	372.454	0.492556	0.246278	+	0.969199i	$0.420792\pi$
$84$	0	0
$85$	327.790	0.418280
$86$	0	0
$87$	594.411	0.732500
$88$	0	0
$89$	−1382.64	−1.64673	−0.823366	−	0.567511i	$-0.807906\pi$
$89$	−1382.64	−1.64673	−0.823366	+	0.567511i	$0.807906\pi$
$90$	0	0
$91$	−965.108	−1.11177
$92$	0	0
$93$	1358.20	1.51440
$94$	0	0
$95$	−925.093	−0.999079
$96$	0	0
$97$	−979.623	−1.02542	−0.512710	−	0.858562i	$-0.671358\pi$
$97$	−979.623	−1.02542	−0.512710	+	0.858562i	$0.671358\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1842.16	−1.81487	−0.907434	−	0.420194i	$-0.861962\pi$
$101$	−1842.16	−1.81487	−0.907434	+	0.420194i	$0.861962\pi$
$102$	0	0
$103$	−1021.93	−0.977611	−0.488806	−	0.872393i	$-0.662567\pi$
$103$	−1021.93	−0.977611	−0.488806	+	0.872393i	$0.662567\pi$
$104$	0	0
$105$	−1285.71	−1.19497
$106$	0	0
$107$	1301.34	1.17575	0.587876	−	0.808951i	$-0.299964\pi$
$107$	1301.34	1.17575	0.587876	+	0.808951i	$0.299964\pi$
$108$	0	0
$109$	529.422	0.465224	0.232612	−	0.972570i	$-0.425273\pi$
$109$	529.422	0.465224	0.232612	+	0.972570i	$0.425273\pi$
$110$	0	0
$111$	−1584.96	−1.35529
$112$	0	0
$113$	−763.264	−0.635414	−0.317707	−	0.948189i	$-0.602913\pi$
$113$	−763.264	−0.635414	−0.317707	+	0.948189i	$0.602913\pi$
$114$	0	0
$115$	325.235	0.263724
$116$	0	0
$117$	−450.708	−0.356137
$118$	0	0
$119$	1039.63	0.800864
$120$	0	0
$121$	0	0
$122$	0	0
$123$	2312.62	1.69530
$124$	0	0
$125$	−1491.39	−1.06715
$126$	0	0
$127$	−1549.50	−1.08265	−0.541323	−	0.840815i	$-0.682076\pi$
$127$	−1549.50	−1.08265	−0.541323	+	0.840815i	$0.682076\pi$
$128$	0	0
$129$	−2828.22	−1.93032
$130$	0	0
$131$	−224.401	−0.149664	−0.0748322	−	0.997196i	$-0.523842\pi$
$131$	−224.401	−0.149664	−0.0748322	+	0.997196i	$0.523842\pi$
$132$	0	0
$133$	−2934.06	−1.91290
$134$	0	0
$135$	757.530	0.482947
$136$	0	0
$137$	−2089.00	−1.30274	−0.651368	−	0.758762i	$-0.725805\pi$
$137$	−2089.00	−1.30274	−0.651368	+	0.758762i	$0.725805\pi$
$138$	0	0
$139$	−1678.27	−1.02409	−0.512047	−	0.858957i	$-0.671113\pi$
$139$	−1678.27	−1.02409	−0.512047	+	0.858957i	$0.671113\pi$
$140$	0	0
$141$	1901.32	1.13560
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−767.774	−0.439725
$146$	0	0
$147$	−1937.46	−1.08707
$148$	0	0
$149$	−1169.25	−0.642877	−0.321438	−	0.946930i	$-0.604166\pi$
$149$	−1169.25	−0.642877	−0.321438	+	0.946930i	$0.604166\pi$
$150$	0	0
$151$	1902.22	1.02517	0.512586	−	0.858636i	$-0.328688\pi$
$151$	1902.22	1.02517	0.512586	+	0.858636i	$0.328688\pi$
$152$	0	0
$153$	485.511	0.256544
$154$	0	0
$155$	−1754.33	−0.909105
$156$	0	0
$157$	−3434.75	−1.74601	−0.873004	−	0.487714i	$-0.837831\pi$
$157$	−3434.75	−1.74601	−0.873004	+	0.487714i	$0.837831\pi$
$158$	0	0
$159$	−1642.32	−0.819148
$160$	0	0
$161$	1031.53	0.504943
$162$	0	0
$163$	−3142.98	−1.51029	−0.755145	−	0.655558i	$-0.772433\pi$
$163$	−3142.98	−1.51029	−0.755145	+	0.655558i	$0.772433\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3440.92	1.59441	0.797205	−	0.603709i	$-0.206311\pi$
$167$	3440.92	1.59441	0.797205	+	0.603709i	$0.206311\pi$
$168$	0	0
$169$	−771.674	−0.351240
$170$	0	0
$171$	−1370.22	−0.612766
$172$	0	0
$173$	3999.60	1.75771	0.878855	−	0.477089i	$-0.158308\pi$
$173$	3999.60	1.75771	0.878855	+	0.477089i	$0.158308\pi$
$174$	0	0
$175$	−1534.74	−0.662944
$176$	0	0
$177$	165.503	0.0702823
$178$	0	0
$179$	−2223.96	−0.928639	−0.464320	−	0.885668i	$-0.653701\pi$
$179$	−2223.96	−0.928639	−0.464320	+	0.885668i	$0.653701\pi$
$180$	0	0
$181$	903.760	0.371138	0.185569	−	0.982631i	$-0.440587\pi$
$181$	903.760	0.371138	0.185569	+	0.982631i	$0.440587\pi$
$182$	0	0
$183$	1639.68	0.662341
$184$	0	0
$185$	2047.22	0.813591
$186$	0	0
$187$	0	0
$188$	0	0
$189$	2402.61	0.924679
$190$	0	0
$191$	−2787.58	−1.05603	−0.528017	−	0.849234i	$-0.677064\pi$
$191$	−2787.58	−1.05603	−0.528017	+	0.849234i	$0.677064\pi$
$192$	0	0
$193$	−994.846	−0.371039	−0.185520	−	0.982641i	$-0.559397\pi$
$193$	−994.846	−0.371039	−0.185520	+	0.982641i	$0.559397\pi$
$194$	0	0
$195$	1898.80	0.697313
$196$	0	0
$197$	2774.75	1.00352	0.501758	−	0.865008i	$-0.332687\pi$
$197$	2774.75	1.00352	0.501758	+	0.865008i	$0.332687\pi$
$198$	0	0
$199$	2875.77	1.02441	0.512206	−	0.858863i	$-0.328828\pi$
$199$	2875.77	1.02441	0.512206	+	0.858863i	$0.328828\pi$
$200$	0	0
$201$	−5966.93	−2.09391
$202$	0	0
$203$	−2435.10	−0.841925
$204$	0	0
$205$	−2987.11	−1.01770
$206$	0	0
$207$	481.727	0.161750
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−1808.81	−0.590159	−0.295079	−	0.955473i	$-0.595346\pi$
$211$	−1808.81	−0.590159	−0.295079	+	0.955473i	$0.595346\pi$
$212$	0	0
$213$	2530.79	0.814117
$214$	0	0
$215$	3653.09	1.15878
$216$	0	0
$217$	−5564.11	−1.74063
$218$	0	0
$219$	3480.91	1.07406
$220$	0	0
$221$	−1535.39	−0.467336
$222$	0	0
$223$	2272.93	0.682542	0.341271	−	0.939965i	$-0.389143\pi$
$223$	2272.93	0.682542	0.341271	+	0.939965i	$0.389143\pi$
$224$	0	0
$225$	−716.727	−0.212364
$226$	0	0
$227$	−3706.17	−1.08364	−0.541822	−	0.840493i	$-0.682265\pi$
$227$	−3706.17	−1.08364	−0.541822	+	0.840493i	$0.682265\pi$
$228$	0	0
$229$	−2962.16	−0.854781	−0.427391	−	0.904067i	$-0.640567\pi$
$229$	−2962.16	−0.854781	−0.427391	+	0.904067i	$0.640567\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3585.02	1.00799	0.503997	−	0.863706i	$-0.331862\pi$
$233$	3585.02	1.00799	0.503997	+	0.863706i	$0.331862\pi$
$234$	0	0
$235$	−2455.85	−0.681712
$236$	0	0
$237$	−5791.49	−1.58733
$238$	0	0
$239$	−1727.50	−0.467544	−0.233772	−	0.972291i	$-0.575107\pi$
$239$	−1727.50	−0.467544	−0.233772	+	0.972291i	$0.575107\pi$
$240$	0	0
$241$	−1906.29	−0.509524	−0.254762	−	0.967004i	$-0.581997\pi$
$241$	−1906.29	−0.509524	−0.254762	+	0.967004i	$0.581997\pi$
$242$	0	0
$243$	3133.39	0.827189
$244$	0	0
$245$	2502.53	0.652575
$246$	0	0
$247$	4333.19	1.11625
$248$	0	0
$249$	−2324.13	−0.591510
$250$	0	0
$251$	−1235.30	−0.310643	−0.155322	−	0.987864i	$-0.549641\pi$
$251$	−1235.30	−0.310643	−0.155322	+	0.987864i	$0.549641\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−2045.42	−0.502311
$256$	0	0
$257$	4370.16	1.06071	0.530356	−	0.847775i	$-0.322058\pi$
$257$	4370.16	1.06071	0.530356	+	0.847775i	$0.322058\pi$
$258$	0	0
$259$	6493.04	1.55775
$260$	0	0
$261$	−1137.20	−0.269697
$262$	0	0
$263$	−2475.73	−0.580455	−0.290228	−	0.956958i	$-0.593731\pi$
$263$	−2475.73	−0.580455	−0.290228	+	0.956958i	$0.593731\pi$
$264$	0	0
$265$	2121.31	0.491741
$266$	0	0
$267$	8627.71	1.97756
$268$	0	0
$269$	6972.45	1.58036	0.790182	−	0.612873i	$-0.209986\pi$
$269$	6972.45	1.58036	0.790182	+	0.612873i	$0.209986\pi$
$270$	0	0
$271$	−1050.79	−0.235538	−0.117769	−	0.993041i	$-0.537574\pi$
$271$	−1050.79	−0.235538	−0.117769	+	0.993041i	$0.537574\pi$
$272$	0	0
$273$	6022.32	1.33512
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1601.70	0.347425	0.173712	−	0.984796i	$-0.444424\pi$
$277$	1601.70	0.347425	0.173712	+	0.984796i	$0.444424\pi$
$278$	0	0
$279$	−2598.46	−0.557582
$280$	0	0
$281$	−777.009	−0.164955	−0.0824777	−	0.996593i	$-0.526283\pi$
$281$	−777.009	−0.164955	−0.0824777	+	0.996593i	$0.526283\pi$
$282$	0	0
$283$	49.8474	0.0104704	0.00523520	−	0.999986i	$-0.498334\pi$
$283$	49.8474	0.0104704	0.00523520	+	0.999986i	$0.498334\pi$
$284$	0	0
$285$	5772.62	1.19979
$286$	0	0
$287$	−9474.05	−1.94856
$288$	0	0
$289$	−3259.05	−0.663353
$290$	0	0
$291$	6112.90	1.23142
$292$	0	0
$293$	6372.06	1.27051	0.635256	−	0.772302i	$-0.280895\pi$
$293$	6372.06	1.27051	0.635256	+	0.772302i	$0.280895\pi$
$294$	0	0
$295$	−213.773	−0.0421910
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1523.42	−0.294654
$300$	0	0
$301$	11586.3	2.21868
$302$	0	0
$303$	11495.2	2.17947
$304$	0	0
$305$	−2117.90	−0.397608
$306$	0	0
$307$	−7352.11	−1.36680	−0.683399	−	0.730045i	$-0.739499\pi$
$307$	−7352.11	−1.36680	−0.683399	+	0.730045i	$0.739499\pi$
$308$	0	0
$309$	6376.90	1.17401
$310$	0	0
$311$	−693.573	−0.126459	−0.0632297	−	0.997999i	$-0.520140\pi$
$311$	−693.573	−0.126459	−0.0632297	+	0.997999i	$0.520140\pi$
$312$	0	0
$313$	−2660.10	−0.480375	−0.240188	−	0.970726i	$-0.577209\pi$
$313$	−2660.10	−0.480375	−0.240188	+	0.970726i	$0.577209\pi$
$314$	0	0
$315$	2459.76	0.439973
$316$	0	0
$317$	−6696.52	−1.18648	−0.593240	−	0.805026i	$-0.702151\pi$
$317$	−6696.52	−1.18648	−0.593240	+	0.805026i	$0.702151\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−8120.43	−1.41196
$322$	0	0
$323$	−4667.79	−0.804095
$324$	0	0
$325$	2266.59	0.386854
$326$	0	0
$327$	−3303.62	−0.558687
$328$	0	0
$329$	−7789.09	−1.30525
$330$	0	0
$331$	9026.82	1.49897	0.749485	−	0.662021i	$-0.230301\pi$
$331$	9026.82	1.49897	0.749485	+	0.662021i	$0.230301\pi$
$332$	0	0
$333$	3032.27	0.499001
$334$	0	0
$335$	7707.22	1.25699
$336$	0	0
$337$	−5808.67	−0.938926	−0.469463	−	0.882952i	$-0.655553\pi$
$337$	−5808.67	−0.938926	−0.469463	+	0.882952i	$0.655553\pi$
$338$	0	0
$339$	4762.80	0.763068
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−831.120	−0.130835
$344$	0	0
$345$	−2029.48	−0.316706
$346$	0	0
$347$	−5901.32	−0.912967	−0.456483	−	0.889732i	$-0.650891\pi$
$347$	−5901.32	−0.912967	−0.456483	+	0.889732i	$0.650891\pi$
$348$	0	0
$349$	−1212.13	−0.185913	−0.0929566	−	0.995670i	$-0.529632\pi$
$349$	−1212.13	−0.185913	−0.0929566	+	0.995670i	$0.529632\pi$
$350$	0	0
$351$	−3548.32	−0.539587
$352$	0	0
$353$	5188.71	0.782344	0.391172	−	0.920318i	$-0.372070\pi$
$353$	5188.71	0.782344	0.391172	+	0.920318i	$0.372070\pi$
$354$	0	0
$355$	−3268.91	−0.488721
$356$	0	0
$357$	−6487.35	−0.961756
$358$	0	0
$359$	5171.56	0.760291	0.380146	−	0.924927i	$-0.375874\pi$
$359$	5171.56	0.760291	0.380146	+	0.924927i	$0.375874\pi$
$360$	0	0
$361$	6314.49	0.920614
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−4496.14	−0.644763
$366$	0	0
$367$	3672.62	0.522368	0.261184	−	0.965289i	$-0.415887\pi$
$367$	3672.62	0.522368	0.261184	+	0.965289i	$0.415887\pi$
$368$	0	0
$369$	−4424.41	−0.624189
$370$	0	0
$371$	6728.05	0.941517
$372$	0	0
$373$	3048.06	0.423116	0.211558	−	0.977365i	$-0.432146\pi$
$373$	3048.06	0.423116	0.211558	+	0.977365i	$0.432146\pi$
$374$	0	0
$375$	9306.36	1.28154
$376$	0	0
$377$	3596.30	0.491297
$378$	0	0
$379$	−10964.0	−1.48596	−0.742982	−	0.669311i	$-0.766589\pi$
$379$	−10964.0	−1.48596	−0.742982	+	0.669311i	$0.766589\pi$
$380$	0	0
$381$	9668.96	1.30015
$382$	0	0
$383$	−8661.05	−1.15551	−0.577753	−	0.816212i	$-0.696070\pi$
$383$	−8661.05	−1.15551	−0.577753	+	0.816212i	$0.696070\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	5410.83	0.710718
$388$	0	0
$389$	12109.2	1.57830	0.789152	−	0.614197i	$-0.210520\pi$
$389$	12109.2	1.57830	0.789152	+	0.614197i	$0.210520\pi$
$390$	0	0
$391$	1641.05	0.212255
$392$	0	0
$393$	1400.28	0.179732
$394$	0	0
$395$	7480.61	0.952887
$396$	0	0
$397$	−9791.79	−1.23787	−0.618937	−	0.785441i	$-0.712436\pi$
$397$	−9791.79	−1.23787	−0.618937	+	0.785441i	$0.712436\pi$
$398$	0	0
$399$	18308.7	2.29719
$400$	0	0
$401$	−12954.3	−1.61323	−0.806615	−	0.591077i	$-0.798703\pi$
$401$	−12954.3	−1.61323	−0.806615	+	0.591077i	$0.798703\pi$
$402$	0	0
$403$	8217.39	1.01573
$404$	0	0
$405$	−7325.01	−0.898723
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−1686.53	−0.203896	−0.101948	−	0.994790i	$-0.532508\pi$
$409$	−1686.53	−0.203896	−0.101948	+	0.994790i	$0.532508\pi$
$410$	0	0
$411$	13035.4	1.56445
$412$	0	0
$413$	−678.011	−0.0807815
$414$	0	0
$415$	3001.98	0.355088
$416$	0	0
$417$	10472.5	1.22983
$418$	0	0
$419$	−491.088	−0.0572583	−0.0286291	−	0.999590i	$-0.509114\pi$
$419$	−491.088	−0.0572583	−0.0286291	+	0.999590i	$0.509114\pi$
$420$	0	0
$421$	−8078.56	−0.935213	−0.467607	−	0.883937i	$-0.654884\pi$
$421$	−8078.56	−0.935213	−0.467607	+	0.883937i	$0.654884\pi$
$422$	0	0
$423$	−3637.53	−0.418115
$424$	0	0
$425$	−2441.61	−0.278671
$426$	0	0
$427$	−6717.22	−0.761285
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−6233.33	−0.696633	−0.348317	−	0.937377i	$-0.613247\pi$
$431$	−6233.33	−0.696633	−0.348317	+	0.937377i	$0.613247\pi$
$432$	0	0
$433$	−10196.2	−1.13164	−0.565820	−	0.824529i	$-0.691440\pi$
$433$	−10196.2	−1.13164	−0.565820	+	0.824529i	$0.691440\pi$
$434$	0	0
$435$	4790.95	0.528065
$436$	0	0
$437$	−4631.40	−0.506980
$438$	0	0
$439$	−1418.48	−0.154215	−0.0771073	−	0.997023i	$-0.524568\pi$
$439$	−1418.48	−0.154215	−0.0771073	+	0.997023i	$0.524568\pi$
$440$	0	0
$441$	3706.66	0.400244
$442$	0	0
$443$	9724.44	1.04294	0.521469	−	0.853270i	$-0.325384\pi$
$443$	9724.44	1.04294	0.521469	+	0.853270i	$0.325384\pi$
$444$	0	0
$445$	−11144.0	−1.18714
$446$	0	0
$447$	7296.17	0.772029
$448$	0	0
$449$	1030.78	0.108342	0.0541711	−	0.998532i	$-0.482748\pi$
$449$	1030.78	0.108342	0.0541711	+	0.998532i	$0.482748\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−11870.0	−1.23113
$454$	0	0
$455$	−7778.76	−0.801482
$456$	0	0
$457$	6659.16	0.681624	0.340812	−	0.940131i	$-0.389298\pi$
$457$	6659.16	0.681624	0.340812	+	0.940131i	$0.389298\pi$
$458$	0	0
$459$	3822.31	0.388693
$460$	0	0
$461$	−6709.39	−0.677847	−0.338924	−	0.940814i	$-0.610063\pi$
$461$	−6709.39	−0.677847	−0.338924	+	0.940814i	$0.610063\pi$
$462$	0	0
$463$	−10505.3	−1.05448	−0.527240	−	0.849716i	$-0.676773\pi$
$463$	−10505.3	−1.05448	−0.527240	+	0.849716i	$0.676773\pi$
$464$	0	0
$465$	10947.1	1.09174
$466$	0	0
$467$	−2012.33	−0.199400	−0.0996999	−	0.995018i	$-0.531788\pi$
$467$	−2012.33	−0.199400	−0.0996999	+	0.995018i	$0.531788\pi$
$468$	0	0
$469$	24444.5	2.40670
$470$	0	0
$471$	21433.0	2.09678
$472$	0	0
$473$	0	0
$474$	0	0
$475$	6890.74	0.665619
$476$	0	0
$477$	3142.02	0.301600
$478$	0	0
$479$	749.816	0.0715239	0.0357620	−	0.999360i	$-0.488614\pi$
$479$	749.816	0.0715239	0.0357620	+	0.999360i	$0.488614\pi$
$480$	0	0
$481$	−9589.28	−0.909010
$482$	0	0
$483$	−6436.78	−0.606385
$484$	0	0
$485$	−7895.76	−0.739233
$486$	0	0
$487$	−4533.33	−0.421817	−0.210908	−	0.977506i	$-0.567642\pi$
$487$	−4533.33	−0.421817	−0.210908	+	0.977506i	$0.567642\pi$
$488$	0	0
$489$	19612.3	1.81370
$490$	0	0
$491$	−10436.7	−0.959266	−0.479633	−	0.877469i	$-0.659230\pi$
$491$	−10436.7	−0.959266	−0.479633	+	0.877469i	$0.659230\pi$
$492$	0	0
$493$	−3874.00	−0.353907
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−10367.8	−0.935735
$498$	0	0
$499$	13272.7	1.19071	0.595357	−	0.803461i	$-0.297011\pi$
$499$	13272.7	1.19071	0.595357	+	0.803461i	$0.297011\pi$
$500$	0	0
$501$	−21471.5	−1.91472
$502$	0	0
$503$	13051.8	1.15696	0.578481	−	0.815696i	$-0.303646\pi$
$503$	13051.8	1.15696	0.578481	+	0.815696i	$0.303646\pi$
$504$	0	0
$505$	−14847.8	−1.30835
$506$	0	0
$507$	4815.28	0.421803
$508$	0	0
$509$	5872.85	0.511414	0.255707	−	0.966754i	$-0.417692\pi$
$509$	5872.85	0.511414	0.255707	+	0.966754i	$0.417692\pi$
$510$	0	0
$511$	−14260.1	−1.23450
$512$	0	0
$513$	−10787.4	−0.928409
$514$	0	0
$515$	−8236.77	−0.704768
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−24957.7	−2.11083
$520$	0	0
$521$	395.918	0.0332926	0.0166463	−	0.999861i	$-0.494701\pi$
$521$	395.918	0.0332926	0.0166463	+	0.999861i	$0.494701\pi$
$522$	0	0
$523$	6835.89	0.571535	0.285767	−	0.958299i	$-0.407752\pi$
$523$	6835.89	0.571535	0.285767	+	0.958299i	$0.407752\pi$
$524$	0	0
$525$	9576.83	0.796128
$526$	0	0
$527$	−8851.92	−0.731681
$528$	0	0
$529$	−10538.7	−0.866174
$530$	0	0
$531$	−316.633	−0.0258771
$532$	0	0
$533$	13991.8	1.13706
$534$	0	0
$535$	10488.8	0.847609
$536$	0	0
$537$	13877.6	1.11520
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−7770.06	−0.617488	−0.308744	−	0.951145i	$-0.599909\pi$
$541$	−7770.06	−0.617488	−0.308744	+	0.951145i	$0.599909\pi$
$542$	0	0
$543$	−5639.50	−0.445698
$544$	0	0
$545$	4267.14	0.335384
$546$	0	0
$547$	7412.71	0.579423	0.289712	−	0.957114i	$-0.406441\pi$
$547$	7412.71	0.579423	0.289712	+	0.957114i	$0.406441\pi$
$548$	0	0
$549$	−3136.96	−0.243866
$550$	0	0
$551$	10933.2	0.845321
$552$	0	0
$553$	23725.8	1.82446
$554$	0	0
$555$	−12774.7	−0.977040
$556$	0	0
$557$	−59.2615	−0.00450806	−0.00225403	−	0.999997i	$-0.500717\pi$
$557$	−59.2615	−0.00450806	−0.00225403	+	0.999997i	$0.500717\pi$
$558$	0	0
$559$	−17111.3	−1.29469
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−437.322	−0.0327370	−0.0163685	−	0.999866i	$-0.505210\pi$
$563$	−437.322	−0.0327370	−0.0163685	+	0.999866i	$0.505210\pi$
$564$	0	0
$565$	−6151.90	−0.458075
$566$	0	0
$567$	−23232.3	−1.72075
$568$	0	0
$569$	2070.16	0.152523	0.0762616	−	0.997088i	$-0.475702\pi$
$569$	2070.16	0.152523	0.0762616	+	0.997088i	$0.475702\pi$
$570$	0	0
$571$	14131.1	1.03567	0.517837	−	0.855479i	$-0.326737\pi$
$571$	14131.1	1.03567	0.517837	+	0.855479i	$0.326737\pi$
$572$	0	0
$573$	17394.6	1.26819
$574$	0	0
$575$	−2422.58	−0.175702
$576$	0	0
$577$	25774.2	1.85961	0.929803	−	0.368057i	$-0.119977\pi$
$577$	25774.2	1.85961	0.929803	+	0.368057i	$0.119977\pi$
$578$	0	0
$579$	6207.89	0.445580
$580$	0	0
$581$	9521.20	0.679873
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−3632.71	−0.256742
$586$	0	0
$587$	−4769.69	−0.335377	−0.167689	−	0.985840i	$-0.553630\pi$
$587$	−4769.69	−0.335377	−0.167689	+	0.985840i	$0.553630\pi$
$588$	0	0
$589$	24982.0	1.74765
$590$	0	0
$591$	−17314.6	−1.20512
$592$	0	0
$593$	−3093.70	−0.214238	−0.107119	−	0.994246i	$-0.534163\pi$
$593$	−3093.70	−0.214238	−0.107119	+	0.994246i	$0.534163\pi$
$594$	0	0
$595$	8379.42	0.577349
$596$	0	0
$597$	−17944.9	−1.23021
$598$	0	0
$599$	17833.1	1.21643	0.608215	−	0.793772i	$-0.291886\pi$
$599$	17833.1	1.21643	0.608215	+	0.793772i	$0.291886\pi$
$600$	0	0
$601$	17607.4	1.19504	0.597520	−	0.801854i	$-0.296153\pi$
$601$	17607.4	1.19504	0.597520	+	0.801854i	$0.296153\pi$
$602$	0	0
$603$	11415.7	0.770949
$604$	0	0
$605$	0	0
$606$	0	0
$607$	11243.6	0.751831	0.375916	−	0.926654i	$-0.377328\pi$
$607$	11243.6	0.751831	0.375916	+	0.926654i	$0.377328\pi$
$608$	0	0
$609$	15195.2	1.01107
$610$	0	0
$611$	11503.4	0.761664
$612$	0	0
$613$	−1363.56	−0.0898431	−0.0449216	−	0.998991i	$-0.514304\pi$
$613$	−1363.56	−0.0898431	−0.0449216	+	0.998991i	$0.514304\pi$
$614$	0	0
$615$	18639.7	1.22216
$616$	0	0
$617$	−9137.98	−0.596242	−0.298121	−	0.954528i	$-0.596360\pi$
$617$	−9137.98	−0.596242	−0.298121	+	0.954528i	$0.596360\pi$
$618$	0	0
$619$	6755.03	0.438623	0.219312	−	0.975655i	$-0.429619\pi$
$619$	6755.03	0.438623	0.219312	+	0.975655i	$0.429619\pi$
$620$	0	0
$621$	3792.52	0.245070
$622$	0	0
$623$	−35344.9	−2.27297
$624$	0	0
$625$	−4516.06	−0.289028
$626$	0	0
$627$	0	0
$628$	0	0
$629$	10329.7	0.654808
$630$	0	0
$631$	8959.94	0.565276	0.282638	−	0.959227i	$-0.408790\pi$
$631$	8959.94	0.565276	0.282638	+	0.959227i	$0.408790\pi$
$632$	0	0
$633$	11287.0	0.708720
$634$	0	0
$635$	−12489.0	−0.780488
$636$	0	0
$637$	−11722.0	−0.729109
$638$	0	0
$639$	−4841.80	−0.299748
$640$	0	0
$641$	1855.75	0.114349	0.0571746	−	0.998364i	$-0.481791\pi$
$641$	1855.75	0.114349	0.0571746	+	0.998364i	$0.481791\pi$
$642$	0	0
$643$	−18963.6	−1.16306	−0.581532	−	0.813524i	$-0.697546\pi$
$643$	−18963.6	−1.16306	−0.581532	+	0.813524i	$0.697546\pi$
$644$	0	0
$645$	−22795.4	−1.39158
$646$	0	0
$647$	−494.010	−0.0300178	−0.0150089	−	0.999887i	$-0.504778\pi$
$647$	−494.010	−0.0300178	−0.0150089	+	0.999887i	$0.504778\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	34720.3	2.09032
$652$	0	0
$653$	6060.45	0.363191	0.181595	−	0.983373i	$-0.441874\pi$
$653$	6060.45	0.363191	0.181595	+	0.983373i	$0.441874\pi$
$654$	0	0
$655$	−1808.67	−0.107894
$656$	0	0
$657$	−6659.52	−0.395453
$658$	0	0
$659$	22461.7	1.32774	0.663872	−	0.747846i	$-0.268912\pi$
$659$	22461.7	1.32774	0.663872	+	0.747846i	$0.268912\pi$
$660$	0	0
$661$	−10681.4	−0.628531	−0.314265	−	0.949335i	$-0.601758\pi$
$661$	−10681.4	−0.628531	−0.314265	+	0.949335i	$0.601758\pi$
$662$	0	0
$663$	9580.88	0.561223
$664$	0	0
$665$	−23648.5	−1.37902
$666$	0	0
$667$	−3843.80	−0.223137
$668$	0	0
$669$	−14183.2	−0.819663
$670$	0	0
$671$	0	0
$672$	0	0
$673$	24976.8	1.43059	0.715293	−	0.698825i	$-0.246293\pi$
$673$	24976.8	1.43059	0.715293	+	0.698825i	$0.246293\pi$
$674$	0	0
$675$	−5642.62	−0.321755
$676$	0	0
$677$	25089.8	1.42434	0.712171	−	0.702006i	$-0.247712\pi$
$677$	25089.8	1.42434	0.712171	+	0.702006i	$0.247712\pi$
$678$	0	0
$679$	−25042.5	−1.41538
$680$	0	0
$681$	23126.7	1.30135
$682$	0	0
$683$	28044.5	1.57114	0.785572	−	0.618770i	$-0.212369\pi$
$683$	28044.5	1.57114	0.785572	+	0.618770i	$0.212369\pi$
$684$	0	0
$685$	−16837.3	−0.939153
$686$	0	0
$687$	18484.0	1.02650
$688$	0	0
$689$	−9936.36	−0.549412
$690$	0	0
$691$	−10272.3	−0.565523	−0.282762	−	0.959190i	$-0.591251\pi$
$691$	−10272.3	−0.565523	−0.282762	+	0.959190i	$0.591251\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−13526.9	−0.738277
$696$	0	0
$697$	−15072.2	−0.819084
$698$	0	0
$699$	−22370.7	−1.21050
$700$	0	0
$701$	13891.9	0.748486	0.374243	−	0.927331i	$-0.377903\pi$
$701$	13891.9	0.748486	0.374243	+	0.927331i	$0.377903\pi$
$702$	0	0
$703$	−29152.8	−1.56404
$704$	0	0
$705$	15324.6	0.818666
$706$	0	0
$707$	−47091.9	−2.50505
$708$	0	0
$709$	−17336.9	−0.918339	−0.459169	−	0.888349i	$-0.651853\pi$
$709$	−17336.9	−0.918339	−0.459169	+	0.888349i	$0.651853\pi$
$710$	0	0
$711$	11080.0	0.584435
$712$	0	0
$713$	−8782.93	−0.461323
$714$	0	0
$715$	0	0
$716$	0	0
$717$	10779.7	0.561472
$718$	0	0
$719$	34184.3	1.77310	0.886550	−	0.462632i	$-0.153095\pi$
$719$	34184.3	1.77310	0.886550	+	0.462632i	$0.153095\pi$
$720$	0	0
$721$	−26124.1	−1.34939
$722$	0	0
$723$	11895.4	0.611886
$724$	0	0
$725$	5718.92	0.292959
$726$	0	0
$727$	−12623.5	−0.643987	−0.321994	−	0.946742i	$-0.604353\pi$
$727$	−12623.5	−0.643987	−0.321994	+	0.946742i	$0.604353\pi$
$728$	0	0
$729$	4985.40	0.253285
$730$	0	0
$731$	18432.6	0.932631
$732$	0	0
$733$	16017.6	0.807127	0.403563	−	0.914952i	$-0.367771\pi$
$733$	16017.6	0.807127	0.403563	+	0.914952i	$0.367771\pi$
$734$	0	0
$735$	−15615.9	−0.783675
$736$	0	0
$737$	0	0
$738$	0	0
$739$	13769.3	0.685404	0.342702	−	0.939444i	$-0.388658\pi$
$739$	13769.3	0.685404	0.342702	+	0.939444i	$0.388658\pi$
$740$	0	0
$741$	−27039.3	−1.34050
$742$	0	0
$743$	12391.0	0.611820	0.305910	−	0.952060i	$-0.401039\pi$
$743$	12391.0	0.611820	0.305910	+	0.952060i	$0.401039\pi$
$744$	0	0
$745$	−9424.14	−0.463455
$746$	0	0
$747$	4446.43	0.217786
$748$	0	0
$749$	33266.7	1.62288
$750$	0	0
$751$	−29830.3	−1.44943	−0.724715	−	0.689048i	$-0.758029\pi$
$751$	−29830.3	−1.44943	−0.724715	+	0.689048i	$0.758029\pi$
$752$	0	0
$753$	7708.33	0.373051
$754$	0	0
$755$	15331.9	0.739054
$756$	0	0
$757$	15347.1	0.736856	0.368428	−	0.929656i	$-0.379896\pi$
$757$	15347.1	0.736856	0.368428	+	0.929656i	$0.379896\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	8363.37	0.398386	0.199193	−	0.979960i	$-0.436168\pi$
$761$	8363.37	0.398386	0.199193	+	0.979960i	$0.436168\pi$
$762$	0	0
$763$	13533.8	0.642147
$764$	0	0
$765$	3913.21	0.184945
$766$	0	0
$767$	1001.33	0.0471392
$768$	0	0
$769$	−20316.4	−0.952704	−0.476352	−	0.879255i	$-0.658041\pi$
$769$	−20316.4	−0.952704	−0.476352	+	0.879255i	$0.658041\pi$
$770$	0	0
$771$	−27270.0	−1.27381
$772$	0	0
$773$	14594.1	0.679059	0.339529	−	0.940595i	$-0.389732\pi$
$773$	14594.1	0.679059	0.339529	+	0.940595i	$0.389732\pi$
$774$	0	0
$775$	13067.5	0.605675
$776$	0	0
$777$	−40516.9	−1.87070
$778$	0	0
$779$	42537.1	1.95642
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−8952.90	−0.408621
$784$	0	0
$785$	−27684.1	−1.25871
$786$	0	0
$787$	−11618.9	−0.526263	−0.263132	−	0.964760i	$-0.584755\pi$
$787$	−11618.9	−0.526263	−0.263132	+	0.964760i	$0.584755\pi$
$788$	0	0
$789$	15448.6	0.697067
$790$	0	0
$791$	−19511.6	−0.877059
$792$	0	0
$793$	9920.37	0.444240
$794$	0	0
$795$	−13237.1	−0.590530
$796$	0	0
$797$	13405.1	0.595774	0.297887	−	0.954601i	$-0.403718\pi$
$797$	13405.1	0.595774	0.297887	+	0.954601i	$0.403718\pi$
$798$	0	0
$799$	−12391.6	−0.548666
$800$	0	0
$801$	−16506.2	−0.728111
$802$	0	0
$803$	0	0
$804$	0	0
$805$	8314.11	0.364017
$806$	0	0
$807$	−43508.4	−1.89785
$808$	0	0
$809$	10020.5	0.435478	0.217739	−	0.976007i	$-0.430132\pi$
$809$	10020.5	0.435478	0.217739	+	0.976007i	$0.430132\pi$
$810$	0	0
$811$	−25543.6	−1.10599	−0.552995	−	0.833185i	$-0.686515\pi$
$811$	−25543.6	−1.10599	−0.552995	+	0.833185i	$0.686515\pi$
$812$	0	0
$813$	6556.98	0.282858
$814$	0	0
$815$	−25332.4	−1.08878
$816$	0	0
$817$	−52020.7	−2.22763
$818$	0	0
$819$	−11521.6	−0.491573
$820$	0	0
$821$	−29823.7	−1.26779	−0.633895	−	0.773419i	$-0.718545\pi$
$821$	−29823.7	−1.26779	−0.633895	+	0.773419i	$0.718545\pi$
$822$	0	0
$823$	17422.4	0.737919	0.368959	−	0.929446i	$-0.379714\pi$
$823$	17422.4	0.737919	0.368959	+	0.929446i	$0.379714\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1102.45	0.0463554	0.0231777	−	0.999731i	$-0.492622\pi$
$827$	1102.45	0.0463554	0.0231777	+	0.999731i	$0.492622\pi$
$828$	0	0
$829$	−32806.0	−1.37443	−0.687213	−	0.726456i	$-0.741166\pi$
$829$	−32806.0	−1.37443	−0.687213	+	0.726456i	$0.741166\pi$
$830$	0	0
$831$	−9994.66	−0.417221
$832$	0	0
$833$	12627.1	0.525216
$834$	0	0
$835$	27733.8	1.14942
$836$	0	0
$837$	−20457.0	−0.844800
$838$	0	0
$839$	−15448.5	−0.635689	−0.317844	−	0.948143i	$-0.602959\pi$
$839$	−15448.5	−0.635689	−0.317844	+	0.948143i	$0.602959\pi$
$840$	0	0
$841$	−15315.0	−0.627948
$842$	0	0
$843$	4848.57	0.198094
$844$	0	0
$845$	−6219.69	−0.253211
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−311.050	−0.0125739
$850$	0	0
$851$	10249.2	0.412854
$852$	0	0
$853$	−11178.9	−0.448720	−0.224360	−	0.974506i	$-0.572029\pi$
$853$	−11178.9	−0.448720	−0.224360	+	0.974506i	$0.572029\pi$
$854$	0	0
$855$	−11043.9	−0.441748
$856$	0	0
$857$	10846.6	0.432338	0.216169	−	0.976356i	$-0.430644\pi$
$857$	10846.6	0.432338	0.216169	+	0.976356i	$0.430644\pi$
$858$	0	0
$859$	−18632.5	−0.740086	−0.370043	−	0.929015i	$-0.620657\pi$
$859$	−18632.5	−0.740086	−0.370043	+	0.929015i	$0.620657\pi$
$860$	0	0
$861$	59118.5	2.34002
$862$	0	0
$863$	36583.7	1.44302	0.721508	−	0.692407i	$-0.243450\pi$
$863$	36583.7	1.44302	0.721508	+	0.692407i	$0.243450\pi$
$864$	0	0
$865$	32236.7	1.26715
$866$	0	0
$867$	20336.7	0.796619
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−36101.1	−1.40441
$872$	0	0
$873$	−11694.9	−0.453394
$874$	0	0
$875$	−38125.1	−1.47299
$876$	0	0
$877$	2744.17	0.105660	0.0528301	−	0.998604i	$-0.483176\pi$
$877$	2744.17	0.105660	0.0528301	+	0.998604i	$0.483176\pi$
$878$	0	0
$879$	−39762.0	−1.52575
$880$	0	0
$881$	−43965.1	−1.68130	−0.840649	−	0.541581i	$-0.817826\pi$
$881$	−43965.1	−1.68130	−0.840649	+	0.541581i	$0.817826\pi$
$882$	0	0
$883$	−25936.5	−0.988485	−0.494242	−	0.869324i	$-0.664554\pi$
$883$	−25936.5	−0.988485	−0.494242	+	0.869324i	$0.664554\pi$
$884$	0	0
$885$	1333.95	0.0506671
$886$	0	0
$887$	−12631.1	−0.478140	−0.239070	−	0.971002i	$-0.576843\pi$
$887$	−12631.1	−0.478140	−0.239070	+	0.971002i	$0.576843\pi$
$888$	0	0
$889$	−39610.5	−1.49437
$890$	0	0
$891$	0	0
$892$	0	0
$893$	34971.8	1.31051
$894$	0	0
$895$	−17925.1	−0.669463
$896$	0	0
$897$	9506.21	0.353850
$898$	0	0
$899$	20733.6	0.769194
$900$	0	0
$901$	10703.6	0.395771
$902$	0	0
$903$	−72299.0	−2.66441
$904$	0	0
$905$	7284.30	0.267556
$906$	0	0
$907$	−38597.3	−1.41301	−0.706506	−	0.707708i	$-0.749729\pi$
$907$	−38597.3	−1.41301	−0.706506	+	0.707708i	$0.749729\pi$
$908$	0	0
$909$	−21992.0	−0.802453
$910$	0	0
$911$	8432.65	0.306681	0.153340	−	0.988173i	$-0.450997\pi$
$911$	8432.65	0.306681	0.153340	+	0.988173i	$0.450997\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	13215.8	0.477487
$916$	0	0
$917$	−5736.47	−0.206581
$918$	0	0
$919$	−32004.2	−1.14877	−0.574385	−	0.818585i	$-0.694759\pi$
$919$	−32004.2	−1.14877	−0.574385	+	0.818585i	$0.694759\pi$
$920$	0	0
$921$	45877.5	1.64138
$922$	0	0
$923$	15311.8	0.546038
$924$	0	0
$925$	−15249.1	−0.542041
$926$	0	0
$927$	−12200.0	−0.432256
$928$	0	0
$929$	−30554.1	−1.07906	−0.539531	−	0.841966i	$-0.681398\pi$
$929$	−30554.1	−1.07906	−0.539531	+	0.841966i	$0.681398\pi$
$930$	0	0
$931$	−35636.5	−1.25450
$932$	0	0
$933$	4327.93	0.151865
$934$	0	0
$935$	0	0
$936$	0	0
$937$	50336.2	1.75498	0.877488	−	0.479599i	$-0.159218\pi$
$937$	50336.2	1.75498	0.877488	+	0.479599i	$0.159218\pi$
$938$	0	0
$939$	16599.1	0.576882
$940$	0	0
$941$	−25108.0	−0.869815	−0.434908	−	0.900475i	$-0.643219\pi$
$941$	−25108.0	−0.869815	−0.434908	+	0.900475i	$0.643219\pi$
$942$	0	0
$943$	−14954.8	−0.516430
$944$	0	0
$945$	19365.0	0.666609
$946$	0	0
$947$	−21894.4	−0.751292	−0.375646	−	0.926763i	$-0.622579\pi$
$947$	−21894.4	−0.751292	−0.375646	+	0.926763i	$0.622579\pi$
$948$	0	0
$949$	21060.2	0.720382
$950$	0	0
$951$	41786.6	1.42484
$952$	0	0
$953$	10886.3	0.370034	0.185017	−	0.982735i	$-0.440766\pi$
$953$	10886.3	0.370034	0.185017	+	0.982735i	$0.440766\pi$
$954$	0	0
$955$	−22467.9	−0.761302
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−53401.8	−1.79816
$960$	0	0
$961$	17584.5	0.590262
$962$	0	0
$963$	15535.7	0.519865
$964$	0	0
$965$	−8018.45	−0.267485
$966$	0	0
$967$	36363.5	1.20928	0.604639	−	0.796499i	$-0.293317\pi$
$967$	36363.5	1.20928	0.604639	+	0.796499i	$0.293317\pi$
$968$	0	0
$969$	29127.2	0.965635
$970$	0	0
$971$	32955.0	1.08916	0.544582	−	0.838708i	$-0.316688\pi$
$971$	32955.0	1.08916	0.544582	+	0.838708i	$0.316688\pi$
$972$	0	0
$973$	−42902.3	−1.41355
$974$	0	0
$975$	−14143.6	−0.464572
$976$	0	0
$977$	26129.3	0.855630	0.427815	−	0.903866i	$-0.359283\pi$
$977$	26129.3	0.855630	0.427815	+	0.903866i	$0.359283\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	6320.34	0.205701
$982$	0	0
$983$	−27810.2	−0.902347	−0.451174	−	0.892436i	$-0.648994\pi$
$983$	−27810.2	−0.902347	−0.451174	+	0.892436i	$0.648994\pi$
$984$	0	0
$985$	22364.5	0.723443
$986$	0	0
$987$	48604.3	1.56747
$988$	0	0
$989$	18288.9	0.588022
$990$	0	0
$991$	39402.4	1.26303	0.631513	−	0.775365i	$-0.282434\pi$
$991$	39402.4	1.26303	0.631513	+	0.775365i	$0.282434\pi$
$992$	0	0
$993$	−56327.8	−1.80011
$994$	0	0
$995$	23178.7	0.738506
$996$	0	0
$997$	−39087.9	−1.24165	−0.620826	−	0.783949i	$-0.713203\pi$
$997$	−39087.9	−1.24165	−0.620826	+	0.783949i	$0.713203\pi$
$998$	0	0
$999$	23872.3	0.756042

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	968.4.a.o.1.2		8
4.3	odd	2		1936.4.a.bv.1.7		8
11.7	odd	10		88.4.i.a.49.4	yes	16
11.8	odd	10		88.4.i.a.9.4	✓	16
11.10	odd	2		968.4.a.n.1.2		8
44.7	even	10		176.4.m.e.49.1		16
44.19	even	10		176.4.m.e.97.1		16
44.43	even	2		1936.4.a.bw.1.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
88.4.i.a.9.4	✓	16	11.8	odd	10
88.4.i.a.49.4	yes	16	11.7	odd	10
176.4.m.e.49.1		16	44.7	even	10
176.4.m.e.97.1		16	44.19	even	10
968.4.a.n.1.2		8	11.10	odd	2
968.4.a.o.1.2		8	1.1	even	1	trivial
1936.4.a.bv.1.7		8	4.3	odd	2
1936.4.a.bw.1.7		8	44.43	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 968 = 2^{3} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	968.a (trivial)

\(f(q)\)	\(=\)	\(q-6.24005 q^{3} +8.05999 q^{5} +25.5634 q^{7} +11.9382 q^{9} -37.7535 q^{13} -50.2947 q^{15} +40.6687 q^{17} -114.776 q^{19} -159.517 q^{21} +40.3517 q^{23} -60.0365 q^{25} +93.9864 q^{27} -95.2574 q^{29} -217.659 q^{31} +206.041 q^{35} +253.997 q^{37} +235.584 q^{39} -370.610 q^{41} +453.237 q^{43} +96.2217 q^{45} -304.697 q^{47} +310.488 q^{49} -253.775 q^{51} +263.190 q^{53} +716.207 q^{57} -26.5227 q^{59} -262.767 q^{61} +305.181 q^{63} -304.293 q^{65} +956.232 q^{67} -251.797 q^{69} -405.573 q^{71} -557.834 q^{73} +374.631 q^{75} +928.116 q^{79} -908.811 q^{81} +372.454 q^{83} +327.790 q^{85} +594.411 q^{87} -1382.64 q^{89} -965.108 q^{91} +1358.20 q^{93} -925.093 q^{95} -979.623 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{3} - 13 q^{5} + 9 q^{7} + 36 q^{9} - 7 q^{13} - 66 q^{15} - 94 q^{17} + 92 q^{19} - 17 q^{21} - 46 q^{23} + 101 q^{25} + 124 q^{27} - 241 q^{29} - 265 q^{31} + 664 q^{35} - 469 q^{37} - 788 q^{39}+ \cdots - 4702 q^{97}+O(q^{100}) \) 8 * q - 8 * q^3 - 13 * q^5 + 9 * q^7 + 36 * q^9 - 7 * q^13 - 66 * q^15 - 94 * q^17 + 92 * q^19 - 17 * q^21 - 46 * q^23 + 101 * q^25 + 124 * q^27 - 241 * q^29 - 265 * q^31 + 664 * q^35 - 469 * q^37 - 788 * q^39 + 74 * q^41 - 35 * q^43 - 566 * q^45 + 197 * q^47 + 221 * q^49 + 608 * q^51 - 247 * q^53 + 979 * q^57 - 1208 * q^59 - 2061 * q^61 + 1617 * q^63 - 837 * q^65 - 829 * q^67 - 1860 * q^69 - 2005 * q^71 - 320 * q^73 - 2323 * q^75 + 1581 * q^79 - 3752 * q^81 - 692 * q^83 - 21 * q^85 + 131 * q^87 - 3205 * q^89 - 2724 * q^91 - 722 * q^93 + 432 * q^95 - 4702 * q^97

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