LMFDB - Embedded newform 882.6.a.bh.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [882,6,Mod(1,882)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("882.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(882, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 882 = 2 \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	882.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,-8,0,32,53,0,0,-128,0,-212,-191] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$141.458529075$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{9601}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 2400 $ x^2 - x - 2400
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 42)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-48.4923$ of defining polynomial
Character	$\chi$	$=$	882.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-4.00000 q^{2} +16.0000 q^{4} +75.4923 q^{5} -64.0000 q^{8} -301.969 q^{10} +149.462 q^{11} +349.416 q^{13} +256.000 q^{16} +1149.85 q^{17} -2795.20 q^{19} +1207.88 q^{20} -597.847 q^{22} -1813.97 q^{23} +2574.09 q^{25} -1397.66 q^{26} +759.033 q^{29} +9031.74 q^{31} -1024.00 q^{32} -4599.39 q^{34} +7794.89 q^{37} +11180.8 q^{38} -4831.51 q^{40} -7640.49 q^{41} +12188.8 q^{43} +2391.39 q^{44} +7255.88 q^{46} -24598.8 q^{47} -10296.4 q^{50} +5590.65 q^{52} -13596.2 q^{53} +11283.2 q^{55} -3036.13 q^{58} +26358.8 q^{59} +35321.8 q^{61} -36127.0 q^{62} +4096.00 q^{64} +26378.2 q^{65} +54371.9 q^{67} +18397.6 q^{68} +70145.7 q^{71} -44468.8 q^{73} -31179.6 q^{74} -44723.2 q^{76} +61612.5 q^{79} +19326.0 q^{80} +30562.0 q^{82} +87142.0 q^{83} +86804.6 q^{85} -48755.3 q^{86} -9565.55 q^{88} -98569.4 q^{89} -29023.5 q^{92} +98395.2 q^{94} -211016. q^{95} +32342.3 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{2} + 32 q^{4} + 53 q^{5} - 128 q^{8} - 212 q^{10} - 191 q^{11} - 379 q^{13} + 512 q^{16} + 340 q^{17} - 1769 q^{19} + 848 q^{20} + 764 q^{22} - 3236 q^{23} - 45 q^{25} + 1516 q^{26} - 4459 q^{29}+ \cdots - 129423 q^{97}+O(q^{100}) $ 2 * q - 8 * q^2 + 32 * q^4 + 53 * q^5 - 128 * q^8 - 212 * q^10 - 191 * q^11 - 379 * q^13 + 512 * q^16 + 340 * q^17 - 1769 * q^19 + 848 * q^20 + 764 * q^22 - 3236 * q^23 - 45 * q^25 + 1516 * q^26 - 4459 * q^29 + 1994 * q^31 - 2048 * q^32 - 1360 * q^34 + 20587 * q^37 + 7076 * q^38 - 3392 * q^40 - 8814 * q^41 + 15853 * q^43 - 3056 * q^44 + 12944 * q^46 - 33912 * q^47 + 180 * q^50 - 6064 * q^52 - 49239 * q^53 + 18941 * q^55 + 17836 * q^58 + 56735 * q^59 + 67508 * q^61 - 7976 * q^62 + 8192 * q^64 + 42762 * q^65 + 75723 * q^67 + 5440 * q^68 + 8992 * q^71 - 3201 * q^73 - 82348 * q^74 - 28304 * q^76 + 26612 * q^79 + 13568 * q^80 + 35256 * q^82 + 949 * q^83 + 105020 * q^85 - 63412 * q^86 + 12224 * q^88 - 176562 * q^89 - 51776 * q^92 + 135648 * q^94 - 234098 * q^95 - 129423 * 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 $

$2$	−4.00000	−0.707107
$2$	−4.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	16.0000	0.500000
$5$	75.4923	1.35045	0.675224	−	0.737613i	$-0.264047\pi$
$5$	75.4923	1.35045	0.675224	+	0.737613i	$0.264047\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−64.0000	−0.353553
$9$	0	0
$10$	−301.969	−0.954911
$11$	149.462	0.372433	0.186217	−	0.982509i	$-0.440377\pi$
$11$	149.462	0.372433	0.186217	+	0.982509i	$0.440377\pi$
$12$	0	0
$13$	349.416	0.573435	0.286717	−	0.958015i	$-0.407436\pi$
$13$	349.416	0.573435	0.286717	+	0.958015i	$0.407436\pi$
$14$	0	0
$15$	0	0
$16$	256.000	0.250000
$17$	1149.85	0.964979	0.482489	−	0.875902i	$-0.339733\pi$
$17$	1149.85	0.964979	0.482489	+	0.875902i	$0.339733\pi$
$18$	0	0
$19$	−2795.20	−1.77635	−0.888176	−	0.459503i	$-0.848028\pi$
$19$	−2795.20	−1.77635	−0.888176	+	0.459503i	$0.848028\pi$
$20$	1207.88	0.675224
$21$	0	0
$22$	−597.847	−0.263350
$23$	−1813.97	−0.715007	−0.357504	−	0.933912i	$-0.616372\pi$
$23$	−1813.97	−0.715007	−0.357504	+	0.933912i	$0.616372\pi$
$24$	0	0
$25$	2574.09	0.823710
$26$	−1397.66	−0.405480
$27$	0	0
$28$	0	0
$29$	759.033	0.167597	0.0837984	−	0.996483i	$-0.473295\pi$
$29$	759.033	0.167597	0.0837984	+	0.996483i	$0.473295\pi$
$30$	0	0
$31$	9031.74	1.68798	0.843990	−	0.536359i	$-0.180201\pi$
$31$	9031.74	1.68798	0.843990	+	0.536359i	$0.180201\pi$
$32$	−1024.00	−0.176777
$33$	0	0
$34$	−4599.39	−0.682343
$35$	0	0
$36$	0	0
$37$	7794.89	0.936064	0.468032	−	0.883711i	$-0.344963\pi$
$37$	7794.89	0.936064	0.468032	+	0.883711i	$0.344963\pi$
$38$	11180.8	1.25607
$39$	0	0
$40$	−4831.51	−0.477456
$41$	−7640.49	−0.709842	−0.354921	−	0.934896i	$-0.615492\pi$
$41$	−7640.49	−0.709842	−0.354921	+	0.934896i	$0.615492\pi$
$42$	0	0
$43$	12188.8	1.00529	0.502645	−	0.864493i	$-0.332360\pi$
$43$	12188.8	1.00529	0.502645	+	0.864493i	$0.332360\pi$
$44$	2391.39	0.186217
$45$	0	0
$46$	7255.88	0.505586
$47$	−24598.8	−1.62431	−0.812156	−	0.583441i	$-0.801706\pi$
$47$	−24598.8	−1.62431	−0.812156	+	0.583441i	$0.801706\pi$
$48$	0	0
$49$	0	0
$50$	−10296.4	−0.582451
$51$	0	0
$52$	5590.65	0.286717
$53$	−13596.2	−0.664858	−0.332429	−	0.943128i	$-0.607868\pi$
$53$	−13596.2	−0.664858	−0.332429	+	0.943128i	$0.607868\pi$
$54$	0	0
$55$	11283.2	0.502952
$56$	0	0
$57$	0	0
$58$	−3036.13	−0.118509
$59$	26358.8	0.985816	0.492908	−	0.870081i	$-0.335934\pi$
$59$	26358.8	0.985816	0.492908	+	0.870081i	$0.335934\pi$
$60$	0	0
$61$	35321.8	1.21540	0.607698	−	0.794168i	$-0.292093\pi$
$61$	35321.8	1.21540	0.607698	+	0.794168i	$0.292093\pi$
$62$	−36127.0	−1.19358
$63$	0	0
$64$	4096.00	0.125000
$65$	26378.2	0.774394
$66$	0	0
$67$	54371.9	1.47975	0.739874	−	0.672746i	$-0.234885\pi$
$67$	54371.9	1.47975	0.739874	+	0.672746i	$0.234885\pi$
$68$	18397.6	0.482489
$69$	0	0
$70$	0	0
$71$	70145.7	1.65141	0.825706	−	0.564101i	$-0.190777\pi$
$71$	70145.7	1.65141	0.825706	+	0.564101i	$0.190777\pi$
$72$	0	0
$73$	−44468.8	−0.976671	−0.488335	−	0.872656i	$-0.662396\pi$
$73$	−44468.8	−0.976671	−0.488335	+	0.872656i	$0.662396\pi$
$74$	−31179.6	−0.661897
$75$	0	0
$76$	−44723.2	−0.888176
$77$	0	0
$78$	0	0
$79$	61612.5	1.11071	0.555355	−	0.831613i	$-0.312582\pi$
$79$	61612.5	1.11071	0.555355	+	0.831613i	$0.312582\pi$
$80$	19326.0	0.337612
$81$	0	0
$82$	30562.0	0.501934
$83$	87142.0	1.38846	0.694228	−	0.719755i	$-0.255746\pi$
$83$	87142.0	1.38846	0.694228	+	0.719755i	$0.255746\pi$
$84$	0	0
$85$	86804.6	1.30315
$86$	−48755.3	−0.710847
$87$	0	0
$88$	−9565.55	−0.131675
$89$	−98569.4	−1.31907	−0.659534	−	0.751675i	$-0.729246\pi$
$89$	−98569.4	−1.31907	−0.659534	+	0.751675i	$0.729246\pi$
$90$	0	0
$91$	0	0
$92$	−29023.5	−0.357504
$93$	0	0
$94$	98395.2	1.14856
$95$	−211016.	−2.39887
$96$	0	0
$97$	32342.3	0.349013	0.174507	−	0.984656i	$-0.444167\pi$
$97$	32342.3	0.349013	0.174507	+	0.984656i	$0.444167\pi$
$98$	0	0
$99$	0	0
$100$	41185.5	0.411855
$101$	−31346.4	−0.305763	−0.152881	−	0.988245i	$-0.548855\pi$
$101$	−31346.4	−0.305763	−0.152881	+	0.988245i	$0.548855\pi$
$102$	0	0
$103$	99332.9	0.922572	0.461286	−	0.887252i	$-0.347388\pi$
$103$	99332.9	0.922572	0.461286	+	0.887252i	$0.347388\pi$
$104$	−22362.6	−0.202740
$105$	0	0
$106$	54384.9	0.470125
$107$	145268.	1.22662	0.613310	−	0.789842i	$-0.289837\pi$
$107$	145268.	1.22662	0.613310	+	0.789842i	$0.289837\pi$
$108$	0	0
$109$	−180851.	−1.45799	−0.728994	−	0.684520i	$-0.760012\pi$
$109$	−180851.	−1.45799	−0.728994	+	0.684520i	$0.760012\pi$
$110$	−45132.9	−0.355640
$111$	0	0
$112$	0	0
$113$	−197832.	−1.45748	−0.728738	−	0.684793i	$-0.759893\pi$
$113$	−197832.	−1.45748	−0.728738	+	0.684793i	$0.759893\pi$
$114$	0	0
$115$	−136941.	−0.965580
$116$	12144.5	0.0837984
$117$	0	0
$118$	−105435.	−0.697077
$119$	0	0
$120$	0	0
$121$	−138712.	−0.861294
$122$	−141287.	−0.859415
$123$	0	0
$124$	144508.	0.843990
$125$	−41589.2	−0.238070
$126$	0	0
$127$	−33517.2	−0.184399	−0.0921996	−	0.995741i	$-0.529390\pi$
$127$	−33517.2	−0.184399	−0.0921996	+	0.995741i	$0.529390\pi$
$128$	−16384.0	−0.0883883
$129$	0	0
$130$	−105513.	−0.547579
$131$	10808.9	0.0550305	0.0275153	−	0.999621i	$-0.491241\pi$
$131$	10808.9	0.0550305	0.0275153	+	0.999621i	$0.491241\pi$
$132$	0	0
$133$	0	0
$134$	−217488.	−1.04634
$135$	0	0
$136$	−73590.2	−0.341171
$137$	18932.0	0.0861778	0.0430889	−	0.999071i	$-0.486280\pi$
$137$	18932.0	0.0861778	0.0430889	+	0.999071i	$0.486280\pi$
$138$	0	0
$139$	−168897.	−0.741457	−0.370729	−	0.928741i	$-0.620892\pi$
$139$	−168897.	−0.741457	−0.370729	+	0.928741i	$0.620892\pi$
$140$	0	0
$141$	0	0
$142$	−280583.	−1.16772
$143$	52224.3	0.213566
$144$	0	0
$145$	57301.2	0.226331
$146$	177875.	0.690611
$147$	0	0
$148$	124718.	0.468032
$149$	−264002.	−0.974186	−0.487093	−	0.873350i	$-0.661943\pi$
$149$	−264002.	−0.974186	−0.487093	+	0.873350i	$0.661943\pi$
$150$	0	0
$151$	279175.	0.996400	0.498200	−	0.867062i	$-0.333994\pi$
$151$	279175.	0.996400	0.498200	+	0.867062i	$0.333994\pi$
$152$	178893.	0.628035
$153$	0	0
$154$	0	0
$155$	681828.	2.27953
$156$	0	0
$157$	188602.	0.610658	0.305329	−	0.952247i	$-0.401234\pi$
$157$	188602.	0.610658	0.305329	+	0.952247i	$0.401234\pi$
$158$	−246450.	−0.785391
$159$	0	0
$160$	−77304.2	−0.238728
$161$	0	0
$162$	0	0
$163$	89717.4	0.264489	0.132244	−	0.991217i	$-0.457782\pi$
$163$	89717.4	0.264489	0.132244	+	0.991217i	$0.457782\pi$
$164$	−122248.	−0.354921
$165$	0	0
$166$	−348568.	−0.981787
$167$	−529411.	−1.46893	−0.734467	−	0.678645i	$-0.762568\pi$
$167$	−529411.	−1.46893	−0.734467	+	0.678645i	$0.762568\pi$
$168$	0	0
$169$	−249202.	−0.671172
$170$	−347219.	−0.921469
$171$	0	0
$172$	195021.	0.502645
$173$	−33838.3	−0.0859594	−0.0429797	−	0.999076i	$-0.513685\pi$
$173$	−33838.3	−0.0859594	−0.0429797	+	0.999076i	$0.513685\pi$
$174$	0	0
$175$	0	0
$176$	38262.2	0.0931083
$177$	0	0
$178$	394278.	0.932722
$179$	247744.	0.577923	0.288962	−	0.957341i	$-0.406690\pi$
$179$	247744.	0.577923	0.288962	+	0.957341i	$0.406690\pi$
$180$	0	0
$181$	369470.	0.838268	0.419134	−	0.907924i	$-0.362334\pi$
$181$	369470.	0.838268	0.419134	+	0.907924i	$0.362334\pi$
$182$	0	0
$183$	0	0
$184$	116094.	0.252793
$185$	588455.	1.26411
$186$	0	0
$187$	171858.	0.359390
$188$	−393581.	−0.812156
$189$	0	0
$190$	844065.	1.69626
$191$	485225.	0.962408	0.481204	−	0.876609i	$-0.340200\pi$
$191$	485225.	0.962408	0.481204	+	0.876609i	$0.340200\pi$
$192$	0	0
$193$	−330817.	−0.639285	−0.319643	−	0.947538i	$-0.603563\pi$
$193$	−330817.	−0.639285	−0.319643	+	0.947538i	$0.603563\pi$
$194$	−129369.	−0.246790
$195$	0	0
$196$	0	0
$197$	161963.	0.297337	0.148669	−	0.988887i	$-0.452501\pi$
$197$	161963.	0.297337	0.148669	+	0.988887i	$0.452501\pi$
$198$	0	0
$199$	55396.2	0.0991625	0.0495813	−	0.998770i	$-0.484211\pi$
$199$	55396.2	0.0991625	0.0495813	+	0.998770i	$0.484211\pi$
$200$	−164742.	−0.291226
$201$	0	0
$202$	125386.	0.216207
$203$	0	0
$204$	0	0
$205$	−576799.	−0.958605
$206$	−397332.	−0.652357
$207$	0	0
$208$	89450.4	0.143359
$209$	−417776.	−0.661572
$210$	0	0
$211$	481748.	0.744926	0.372463	−	0.928047i	$-0.378513\pi$
$211$	481748.	0.744926	0.372463	+	0.928047i	$0.378513\pi$
$212$	−217540.	−0.332429
$213$	0	0
$214$	−581072.	−0.867352
$215$	920164.	1.35759
$216$	0	0
$217$	0	0
$218$	723403.	1.03095
$219$	0	0
$220$	180531.	0.251476
$221$	401775.	0.553353
$222$	0	0
$223$	638779.	0.860178	0.430089	−	0.902787i	$-0.358482\pi$
$223$	638779.	0.860178	0.430089	+	0.902787i	$0.358482\pi$
$224$	0	0
$225$	0	0
$226$	791330.	1.03059
$227$	525682.	0.677109	0.338554	−	0.940947i	$-0.390062\pi$
$227$	525682.	0.677109	0.338554	+	0.940947i	$0.390062\pi$
$228$	0	0
$229$	932686.	1.17529	0.587647	−	0.809117i	$-0.300054\pi$
$229$	932686.	1.17529	0.587647	+	0.809117i	$0.300054\pi$
$230$	547763.	0.682768
$231$	0	0
$232$	−48578.1	−0.0592544
$233$	707793.	0.854115	0.427058	−	0.904224i	$-0.359550\pi$
$233$	707793.	0.854115	0.427058	+	0.904224i	$0.359550\pi$
$234$	0	0
$235$	−1.85702e6	−2.19355
$236$	421741.	0.492908
$237$	0	0
$238$	0	0
$239$	500614.	0.566902	0.283451	−	0.958987i	$-0.408521\pi$
$239$	500614.	0.566902	0.283451	+	0.958987i	$0.408521\pi$
$240$	0	0
$241$	1.20822e6	1.33999	0.669997	−	0.742364i	$-0.266295\pi$
$241$	1.20822e6	1.33999	0.669997	+	0.742364i	$0.266295\pi$
$242$	554849.	0.609027
$243$	0	0
$244$	565148.	0.607698
$245$	0	0
$246$	0	0
$247$	−976688.	−1.01862
$248$	−578032.	−0.596791
$249$	0	0
$250$	166357.	0.168341
$251$	−97826.7	−0.0980106	−0.0490053	−	0.998799i	$-0.515605\pi$
$251$	−97826.7	−0.0980106	−0.0490053	+	0.998799i	$0.515605\pi$
$252$	0	0
$253$	−271119.	−0.266292
$254$	134069.	0.130390
$255$	0	0
$256$	65536.0	0.0625000
$257$	1.10791e6	1.04634	0.523170	−	0.852228i	$-0.324749\pi$
$257$	1.10791e6	1.04634	0.523170	+	0.852228i	$0.324749\pi$
$258$	0	0
$259$	0	0
$260$	422052.	0.387197
$261$	0	0
$262$	−43235.6	−0.0389125
$263$	−1.50685e6	−1.34333	−0.671663	−	0.740856i	$-0.734420\pi$
$263$	−1.50685e6	−1.34333	−0.671663	+	0.740856i	$0.734420\pi$
$264$	0	0
$265$	−1.02641e6	−0.897856
$266$	0	0
$267$	0	0
$268$	869951.	0.739874
$269$	2.13502e6	1.79896	0.899481	−	0.436960i	$-0.143945\pi$
$269$	2.13502e6	1.79896	0.899481	+	0.436960i	$0.143945\pi$
$270$	0	0
$271$	1.22860e6	1.01622	0.508108	−	0.861293i	$-0.330345\pi$
$271$	1.22860e6	1.01622	0.508108	+	0.861293i	$0.330345\pi$
$272$	294361.	0.241245
$273$	0	0
$274$	−75728.1	−0.0609369
$275$	384729.	0.306777
$276$	0	0
$277$	−1.84682e6	−1.44619	−0.723097	−	0.690747i	$-0.757282\pi$
$277$	−1.84682e6	−1.44619	−0.723097	+	0.690747i	$0.757282\pi$
$278$	675590.	0.524289
$279$	0	0
$280$	0	0
$281$	2.28326e6	1.72500	0.862500	−	0.506056i	$-0.168897\pi$
$281$	2.28326e6	1.72500	0.862500	+	0.506056i	$0.168897\pi$
$282$	0	0
$283$	1.33693e6	0.992296	0.496148	−	0.868238i	$-0.334747\pi$
$283$	1.33693e6	0.992296	0.496148	+	0.868238i	$0.334747\pi$
$284$	1.12233e6	0.825706
$285$	0	0
$286$	−208897.	−0.151014
$287$	0	0
$288$	0	0
$289$	−97709.0	−0.0688161
$290$	−229205.	−0.160040
$291$	0	0
$292$	−711501.	−0.488335
$293$	2.23033e6	1.51775	0.758875	−	0.651236i	$-0.225749\pi$
$293$	2.23033e6	1.51775	0.758875	+	0.651236i	$0.225749\pi$
$294$	0	0
$295$	1.98989e6	1.33129
$296$	−498873.	−0.330949
$297$	0	0
$298$	1.05601e6	0.688853
$299$	−633830.	−0.410010
$300$	0	0
$301$	0	0
$302$	−1.11670e6	−0.704561
$303$	0	0
$304$	−715572.	−0.444088
$305$	2.66652e6	1.64133
$306$	0	0
$307$	1.77782e6	1.07657	0.538284	−	0.842763i	$-0.319073\pi$
$307$	1.77782e6	1.07657	0.538284	+	0.842763i	$0.319073\pi$
$308$	0	0
$309$	0	0
$310$	−2.72731e6	−1.61187
$311$	−1.40272e6	−0.822373	−0.411187	−	0.911551i	$-0.634886\pi$
$311$	−1.40272e6	−0.822373	−0.411187	+	0.911551i	$0.634886\pi$
$312$	0	0
$313$	1.21894e6	0.703271	0.351636	−	0.936137i	$-0.385626\pi$
$313$	1.21894e6	0.703271	0.351636	+	0.936137i	$0.385626\pi$
$314$	−754409.	−0.431800
$315$	0	0
$316$	985799.	0.555355
$317$	−2.68001e6	−1.49792	−0.748961	−	0.662615i	$-0.769447\pi$
$317$	−2.68001e6	−1.49792	−0.748961	+	0.662615i	$0.769447\pi$
$318$	0	0
$319$	113446.	0.0624186
$320$	309217.	0.168806
$321$	0	0
$322$	0	0
$323$	−3.21405e6	−1.71414
$324$	0	0
$325$	899429.	0.472344
$326$	−358869.	−0.187022
$327$	0	0
$328$	488992.	0.250967
$329$	0	0
$330$	0	0
$331$	142560.	0.0715202	0.0357601	−	0.999360i	$-0.488615\pi$
$331$	142560.	0.0715202	0.0357601	+	0.999360i	$0.488615\pi$
$332$	1.39427e6	0.694228
$333$	0	0
$334$	2.11765e6	1.03869
$335$	4.10466e6	1.99832
$336$	0	0
$337$	−1.21206e6	−0.581367	−0.290683	−	0.956819i	$-0.593883\pi$
$337$	−1.21206e6	−0.581367	−0.290683	+	0.956819i	$0.593883\pi$
$338$	996806.	0.474591
$339$	0	0
$340$	1.38887e6	0.651577
$341$	1.34990e6	0.628660
$342$	0	0
$343$	0	0
$344$	−780085.	−0.355423
$345$	0	0
$346$	135353.	0.0607825
$347$	−3.46853e6	−1.54640	−0.773199	−	0.634163i	$-0.781345\pi$
$347$	−3.46853e6	−1.54640	−0.773199	+	0.634163i	$0.781345\pi$
$348$	0	0
$349$	1.01692e6	0.446911	0.223456	−	0.974714i	$-0.428266\pi$
$349$	1.01692e6	0.446911	0.223456	+	0.974714i	$0.428266\pi$
$350$	0	0
$351$	0	0
$352$	−153049.	−0.0658375
$353$	−1.56385e6	−0.667974	−0.333987	−	0.942578i	$-0.608394\pi$
$353$	−1.56385e6	−0.667974	−0.333987	+	0.942578i	$0.608394\pi$
$354$	0	0
$355$	5.29547e6	2.23015
$356$	−1.57711e6	−0.659534
$357$	0	0
$358$	−990975.	−0.408653
$359$	−49553.1	−0.0202925	−0.0101462	−	0.999949i	$-0.503230\pi$
$359$	−49553.1	−0.0202925	−0.0101462	+	0.999949i	$0.503230\pi$
$360$	0	0
$361$	5.33705e6	2.15543
$362$	−1.47788e6	−0.592745
$363$	0	0
$364$	0	0
$365$	−3.35705e6	−1.31894
$366$	0	0
$367$	3.54831e6	1.37517	0.687585	−	0.726104i	$-0.258671\pi$
$367$	3.54831e6	1.37517	0.687585	+	0.726104i	$0.258671\pi$
$368$	−464376.	−0.178752
$369$	0	0
$370$	−2.35382e6	−0.893858
$371$	0	0
$372$	0	0
$373$	−2.25573e6	−0.839490	−0.419745	−	0.907642i	$-0.637880\pi$
$373$	−2.25573e6	−0.839490	−0.419745	+	0.907642i	$0.637880\pi$
$374$	−687432.	−0.254127
$375$	0	0
$376$	1.57432e6	0.574281
$377$	265218.	0.0961059
$378$	0	0
$379$	4.39503e6	1.57168	0.785840	−	0.618430i	$-0.212231\pi$
$379$	4.39503e6	1.57168	0.785840	+	0.618430i	$0.212231\pi$
$380$	−3.37626e6	−1.19944
$381$	0	0
$382$	−1.94090e6	−0.680525
$383$	1.22781e6	0.427694	0.213847	−	0.976867i	$-0.431401\pi$
$383$	1.22781e6	0.427694	0.213847	+	0.976867i	$0.431401\pi$
$384$	0	0
$385$	0	0
$386$	1.32327e6	0.452043
$387$	0	0
$388$	517477.	0.174507
$389$	−2.05424e6	−0.688300	−0.344150	−	0.938915i	$-0.611833\pi$
$389$	−2.05424e6	−0.688300	−0.344150	+	0.938915i	$0.611833\pi$
$390$	0	0
$391$	−2.08579e6	−0.689967
$392$	0	0
$393$	0	0
$394$	−647851.	−0.210249
$395$	4.65127e6	1.49996
$396$	0	0
$397$	−3.80367e6	−1.21123	−0.605615	−	0.795758i	$-0.707073\pi$
$397$	−3.80367e6	−1.21123	−0.605615	+	0.795758i	$0.707073\pi$
$398$	−221585.	−0.0701185
$399$	0	0
$400$	658968.	0.205928
$401$	−1.18524e6	−0.368084	−0.184042	−	0.982918i	$-0.558918\pi$
$401$	−1.18524e6	−0.368084	−0.184042	+	0.982918i	$0.558918\pi$
$402$	0	0
$403$	3.15583e6	0.967947
$404$	−501543.	−0.152881
$405$	0	0
$406$	0	0
$407$	1.16504e6	0.348621
$408$	0	0
$409$	−4.30393e6	−1.27220	−0.636102	−	0.771605i	$-0.719454\pi$
$409$	−4.30393e6	−1.27220	−0.636102	+	0.771605i	$0.719454\pi$
$410$	2.30720e6	0.677836
$411$	0	0
$412$	1.58933e6	0.461286
$413$	0	0
$414$	0	0
$415$	6.57855e6	1.87504
$416$	−357802.	−0.101370
$417$	0	0
$418$	1.67110e6	0.467802
$419$	113725.	0.0316461	0.0158230	−	0.999875i	$-0.494963\pi$
$419$	113725.	0.0316461	0.0158230	+	0.999875i	$0.494963\pi$
$420$	0	0
$421$	443417.	0.121929	0.0609645	−	0.998140i	$-0.480582\pi$
$421$	443417.	0.121929	0.0609645	+	0.998140i	$0.480582\pi$
$422$	−1.92699e6	−0.526743
$423$	0	0
$424$	870158.	0.235063
$425$	2.95981e6	0.794863
$426$	0	0
$427$	0	0
$428$	2.32429e6	0.613310
$429$	0	0
$430$	−3.68065e6	−0.959962
$431$	−4.63310e6	−1.20138	−0.600688	−	0.799484i	$-0.705107\pi$
$431$	−4.63310e6	−1.20138	−0.600688	+	0.799484i	$0.705107\pi$
$432$	0	0
$433$	6.57955e6	1.68646	0.843231	−	0.537552i	$-0.180651\pi$
$433$	6.57955e6	1.68646	0.843231	+	0.537552i	$0.180651\pi$
$434$	0	0
$435$	0	0
$436$	−2.89361e6	−0.728994
$437$	5.07041e6	1.27010
$438$	0	0
$439$	−1.82070e6	−0.450896	−0.225448	−	0.974255i	$-0.572385\pi$
$439$	−1.82070e6	−0.450896	−0.225448	+	0.974255i	$0.572385\pi$
$440$	−722126.	−0.177820
$441$	0	0
$442$	−1.60710e6	−0.391279
$443$	2.06990e6	0.501119	0.250559	−	0.968101i	$-0.419385\pi$
$443$	2.06990e6	0.501119	0.250559	+	0.968101i	$0.419385\pi$
$444$	0	0
$445$	−7.44123e6	−1.78133
$446$	−2.55511e6	−0.608238
$447$	0	0
$448$	0	0
$449$	−5.72581e6	−1.34036	−0.670180	−	0.742199i	$-0.733783\pi$
$449$	−5.72581e6	−1.34036	−0.670180	+	0.742199i	$0.733783\pi$
$450$	0	0
$451$	−1.14196e6	−0.264369
$452$	−3.16532e6	−0.728738
$453$	0	0
$454$	−2.10273e6	−0.478788
$455$	0	0
$456$	0	0
$457$	318676.	0.0713771	0.0356886	−	0.999363i	$-0.488638\pi$
$457$	318676.	0.0713771	0.0356886	+	0.999363i	$0.488638\pi$
$458$	−3.73074e6	−0.831059
$459$	0	0
$460$	−2.19105e6	−0.482790
$461$	3.42470e6	0.750535	0.375267	−	0.926917i	$-0.377551\pi$
$461$	3.42470e6	0.750535	0.375267	+	0.926917i	$0.377551\pi$
$462$	0	0
$463$	3.82945e6	0.830201	0.415101	−	0.909775i	$-0.363746\pi$
$463$	3.82945e6	0.830201	0.415101	+	0.909775i	$0.363746\pi$
$464$	194312.	0.0418992
$465$	0	0
$466$	−2.83117e6	−0.603951
$467$	−2.84462e6	−0.603576	−0.301788	−	0.953375i	$-0.597583\pi$
$467$	−2.84462e6	−0.603576	−0.301788	+	0.953375i	$0.597583\pi$
$468$	0	0
$469$	0	0
$470$	7.42809e6	1.55107
$471$	0	0
$472$	−1.68696e6	−0.348539
$473$	1.82176e6	0.374403
$474$	0	0
$475$	−7.19511e6	−1.46320
$476$	0	0
$477$	0	0
$478$	−2.00246e6	−0.400860
$479$	−1.30365e6	−0.259611	−0.129806	−	0.991539i	$-0.541435\pi$
$479$	−1.30365e6	−0.259611	−0.129806	+	0.991539i	$0.541435\pi$
$480$	0	0
$481$	2.72366e6	0.536772
$482$	−4.83287e6	−0.947519
$483$	0	0
$484$	−2.21940e6	−0.430647
$485$	2.44160e6	0.471324
$486$	0	0
$487$	6.23624e6	1.19152	0.595759	−	0.803164i	$-0.296852\pi$
$487$	6.23624e6	1.19152	0.595759	+	0.803164i	$0.296852\pi$
$488$	−2.26059e6	−0.429707
$489$	0	0
$490$	0	0
$491$	3.93928e6	0.737417	0.368709	−	0.929545i	$-0.379800\pi$
$491$	3.93928e6	0.737417	0.368709	+	0.929545i	$0.379800\pi$
$492$	0	0
$493$	872772.	0.161727
$494$	3.90675e6	0.720275
$495$	0	0
$496$	2.31213e6	0.421995
$497$	0	0
$498$	0	0
$499$	7.97571e6	1.43390	0.716948	−	0.697126i	$-0.245538\pi$
$499$	7.97571e6	1.43390	0.716948	+	0.697126i	$0.245538\pi$
$500$	−665427.	−0.119035
$501$	0	0
$502$	391307.	0.0693040
$503$	2.70777e6	0.477190	0.238595	−	0.971119i	$-0.423313\pi$
$503$	2.70777e6	0.477190	0.238595	+	0.971119i	$0.423313\pi$
$504$	0	0
$505$	−2.36641e6	−0.412917
$506$	1.08448e6	0.188297
$507$	0	0
$508$	−536276.	−0.0921996
$509$	−3.90049e6	−0.667306	−0.333653	−	0.942696i	$-0.608281\pi$
$509$	−3.90049e6	−0.667306	−0.333653	+	0.942696i	$0.608281\pi$
$510$	0	0
$511$	0	0
$512$	−262144.	−0.0441942
$513$	0	0
$514$	−4.43165e6	−0.739874
$515$	7.49887e6	1.24589
$516$	0	0
$517$	−3.67658e6	−0.604947
$518$	0	0
$519$	0	0
$520$	−1.68821e6	−0.273790
$521$	−7.00063e6	−1.12991	−0.564954	−	0.825122i	$-0.691106\pi$
$521$	−7.00063e6	−1.12991	−0.564954	+	0.825122i	$0.691106\pi$
$522$	0	0
$523$	−2.11208e6	−0.337641	−0.168821	−	0.985647i	$-0.553996\pi$
$523$	−2.11208e6	−0.337641	−0.168821	+	0.985647i	$0.553996\pi$
$524$	172943.	0.0275153
$525$	0	0
$526$	6.02741e6	0.949875
$527$	1.03851e7	1.62887
$528$	0	0
$529$	−3.14586e6	−0.488765
$530$	4.10564e6	0.634880
$531$	0	0
$532$	0	0
$533$	−2.66971e6	−0.407048
$534$	0	0
$535$	1.09666e7	1.65649
$536$	−3.47980e6	−0.523170
$537$	0	0
$538$	−8.54009e6	−1.27206
$539$	0	0
$540$	0	0
$541$	5.70567e6	0.838134	0.419067	−	0.907955i	$-0.362357\pi$
$541$	5.70567e6	0.838134	0.419067	+	0.907955i	$0.362357\pi$
$542$	−4.91439e6	−0.718574
$543$	0	0
$544$	−1.17744e6	−0.170586
$545$	−1.36528e7	−1.96894
$546$	0	0
$547$	2.28054e6	0.325888	0.162944	−	0.986635i	$-0.447901\pi$
$547$	2.28054e6	0.325888	0.162944	+	0.986635i	$0.447901\pi$
$548$	302912.	0.0430889
$549$	0	0
$550$	−1.53891e6	−0.216924
$551$	−2.12165e6	−0.297711
$552$	0	0
$553$	0	0
$554$	7.38730e6	1.02261
$555$	0	0
$556$	−2.70236e6	−0.370729
$557$	6.73941e6	0.920415	0.460208	−	0.887811i	$-0.347775\pi$
$557$	6.73941e6	0.920415	0.460208	+	0.887811i	$0.347775\pi$
$558$	0	0
$559$	4.25897e6	0.576468
$560$	0	0
$561$	0	0
$562$	−9.13304e6	−1.21976
$563$	9.66820e6	1.28551	0.642754	−	0.766073i	$-0.277792\pi$
$563$	9.66820e6	1.28551	0.642754	+	0.766073i	$0.277792\pi$
$564$	0	0
$565$	−1.49348e7	−1.96825
$566$	−5.34770e6	−0.701659
$567$	0	0
$568$	−4.48933e6	−0.583862
$569$	1.39587e7	1.80744	0.903719	−	0.428125i	$-0.140826\pi$
$569$	1.39587e7	1.80744	0.903719	+	0.428125i	$0.140826\pi$
$570$	0	0
$571$	−2.35841e6	−0.302711	−0.151355	−	0.988479i	$-0.548364\pi$
$571$	−2.35841e6	−0.302711	−0.151355	+	0.988479i	$0.548364\pi$
$572$	835589.	0.106783
$573$	0	0
$574$	0	0
$575$	−4.66933e6	−0.588959
$576$	0	0
$577$	5.01098e6	0.626589	0.313295	−	0.949656i	$-0.398567\pi$
$577$	5.01098e6	0.626589	0.313295	+	0.949656i	$0.398567\pi$
$578$	390836.	0.0486603
$579$	0	0
$580$	916819.	0.113165
$581$	0	0
$582$	0	0
$583$	−2.03211e6	−0.247615
$584$	2.84600e6	0.345305
$585$	0	0
$586$	−8.92132e6	−1.07321
$587$	−3.95106e6	−0.473280	−0.236640	−	0.971597i	$-0.576046\pi$
$587$	−3.95106e6	−0.473280	−0.236640	+	0.971597i	$0.576046\pi$
$588$	0	0
$589$	−2.52455e7	−2.99845
$590$	−7.95955e6	−0.941367
$591$	0	0
$592$	1.99549e6	0.234016
$593$	2.53680e6	0.296244	0.148122	−	0.988969i	$-0.452677\pi$
$593$	2.53680e6	0.296244	0.148122	+	0.988969i	$0.452677\pi$
$594$	0	0
$595$	0	0
$596$	−4.22403e6	−0.487093
$597$	0	0
$598$	2.53532e6	0.289921
$599$	−7.17714e6	−0.817305	−0.408652	−	0.912690i	$-0.634001\pi$
$599$	−7.17714e6	−0.817305	−0.408652	+	0.912690i	$0.634001\pi$
$600$	0	0
$601$	1.12527e7	1.27078	0.635392	−	0.772190i	$-0.280838\pi$
$601$	1.12527e7	1.27078	0.635392	+	0.772190i	$0.280838\pi$
$602$	0	0
$603$	0	0
$604$	4.46680e6	0.498200
$605$	−1.04717e7	−1.16313
$606$	0	0
$607$	−1.40370e7	−1.54633	−0.773167	−	0.634202i	$-0.781329\pi$
$607$	−1.40370e7	−1.54633	−0.773167	+	0.634202i	$0.781329\pi$
$608$	2.86229e6	0.314018
$609$	0	0
$610$	−1.06661e7	−1.16059
$611$	−8.59521e6	−0.931437
$612$	0	0
$613$	−7.62872e6	−0.819975	−0.409988	−	0.912091i	$-0.634467\pi$
$613$	−7.62872e6	−0.819975	−0.409988	+	0.912091i	$0.634467\pi$
$614$	−7.11128e6	−0.761249
$615$	0	0
$616$	0	0
$617$	4.41080e6	0.466450	0.233225	−	0.972423i	$-0.425072\pi$
$617$	4.41080e6	0.466450	0.233225	+	0.972423i	$0.425072\pi$
$618$	0	0
$619$	−9.04355e6	−0.948664	−0.474332	−	0.880346i	$-0.657310\pi$
$619$	−9.04355e6	−0.948664	−0.474332	+	0.880346i	$0.657310\pi$
$620$	1.09092e7	1.13977
$621$	0	0
$622$	5.61087e6	0.581506
$623$	0	0
$624$	0	0
$625$	−1.11837e7	−1.14521
$626$	−4.87578e6	−0.497288
$627$	0	0
$628$	3.01764e6	0.305329
$629$	8.96293e6	0.903282
$630$	0	0
$631$	−7.33039e6	−0.732916	−0.366458	−	0.930435i	$-0.619430\pi$
$631$	−7.33039e6	−0.732916	−0.366458	+	0.930435i	$0.619430\pi$
$632$	−3.94320e6	−0.392695
$633$	0	0
$634$	1.07201e7	1.05919
$635$	−2.53030e6	−0.249022
$636$	0	0
$637$	0	0
$638$	−453786.	−0.0441366
$639$	0	0
$640$	−1.23687e6	−0.119364
$641$	−3.68824e6	−0.354547	−0.177273	−	0.984162i	$-0.556728\pi$
$641$	−3.68824e6	−0.354547	−0.177273	+	0.984162i	$0.556728\pi$
$642$	0	0
$643$	−1.17584e7	−1.12155	−0.560776	−	0.827968i	$-0.689497\pi$
$643$	−1.17584e7	−1.12155	−0.560776	+	0.827968i	$0.689497\pi$
$644$	0	0
$645$	0	0
$646$	1.28562e7	1.21208
$647$	9.94050e6	0.933572	0.466786	−	0.884370i	$-0.345412\pi$
$647$	9.94050e6	0.933572	0.466786	+	0.884370i	$0.345412\pi$
$648$	0	0
$649$	3.93963e6	0.367151
$650$	−3.59772e6	−0.333998
$651$	0	0
$652$	1.43548e6	0.132244
$653$	1.02620e6	0.0941778	0.0470889	−	0.998891i	$-0.485006\pi$
$653$	1.02620e6	0.0941778	0.0470889	+	0.998891i	$0.485006\pi$
$654$	0	0
$655$	815990.	0.0743159
$656$	−1.95597e6	−0.177461
$657$	0	0
$658$	0	0
$659$	−1.00207e7	−0.898846	−0.449423	−	0.893319i	$-0.648370\pi$
$659$	−1.00207e7	−0.898846	−0.449423	+	0.893319i	$0.648370\pi$
$660$	0	0
$661$	2.62826e6	0.233972	0.116986	−	0.993134i	$-0.462677\pi$
$661$	2.62826e6	0.233972	0.116986	+	0.993134i	$0.462677\pi$
$662$	−570241.	−0.0505724
$663$	0	0
$664$	−5.57709e6	−0.490893
$665$	0	0
$666$	0	0
$667$	−1.37686e6	−0.119833
$668$	−8.47058e6	−0.734467
$669$	0	0
$670$	−1.64187e7	−1.41303
$671$	5.27925e6	0.452654
$672$	0	0
$673$	−1.50220e7	−1.27847	−0.639233	−	0.769013i	$-0.720748\pi$
$673$	−1.50220e7	−1.27847	−0.639233	+	0.769013i	$0.720748\pi$
$674$	4.84825e6	0.411088
$675$	0	0
$676$	−3.98723e6	−0.335586
$677$	6.03310e6	0.505905	0.252953	−	0.967479i	$-0.418598\pi$
$677$	6.03310e6	0.505905	0.252953	+	0.967479i	$0.418598\pi$
$678$	0	0
$679$	0	0
$680$	−5.55550e6	−0.460734
$681$	0	0
$682$	−5.39960e6	−0.444530
$683$	−1.10720e7	−0.908187	−0.454093	−	0.890954i	$-0.650037\pi$
$683$	−1.10720e7	−0.908187	−0.454093	+	0.890954i	$0.650037\pi$
$684$	0	0
$685$	1.42922e6	0.116379
$686$	0	0
$687$	0	0
$688$	3.12034e6	0.251322
$689$	−4.75073e6	−0.381253
$690$	0	0
$691$	−1.05548e7	−0.840919	−0.420460	−	0.907311i	$-0.638131\pi$
$691$	−1.05548e7	−0.840919	−0.420460	+	0.907311i	$0.638131\pi$
$692$	−541413.	−0.0429797
$693$	0	0
$694$	1.38741e7	1.09347
$695$	−1.27505e7	−1.00130
$696$	0	0
$697$	−8.78540e6	−0.684983
$698$	−4.06766e6	−0.316014
$699$	0	0
$700$	0	0
$701$	7.20675e6	0.553917	0.276958	−	0.960882i	$-0.410674\pi$
$701$	7.20675e6	0.553917	0.276958	+	0.960882i	$0.410674\pi$
$702$	0	0
$703$	−2.17883e7	−1.66278
$704$	612195.	0.0465541
$705$	0	0
$706$	6.25542e6	0.472329
$707$	0	0
$708$	0	0
$709$	2.69373e6	0.201252	0.100626	−	0.994924i	$-0.467916\pi$
$709$	2.69373e6	0.201252	0.100626	+	0.994924i	$0.467916\pi$
$710$	−2.11819e7	−1.57695
$711$	0	0
$712$	6.30844e6	0.466361
$713$	−1.63833e7	−1.20692
$714$	0	0
$715$	3.94253e6	0.288410
$716$	3.96390e6	0.288962
$717$	0	0
$718$	198213.	0.0143489
$719$	−6.92550e6	−0.499608	−0.249804	−	0.968296i	$-0.580366\pi$
$719$	−6.92550e6	−0.499608	−0.249804	+	0.968296i	$0.580366\pi$
$720$	0	0
$721$	0	0
$722$	−2.13482e7	−1.52412
$723$	0	0
$724$	5.91152e6	0.419134
$725$	1.95382e6	0.138051
$726$	0	0
$727$	4.11366e6	0.288664	0.144332	−	0.989529i	$-0.453897\pi$
$727$	4.11366e6	0.288664	0.144332	+	0.989529i	$0.453897\pi$
$728$	0	0
$729$	0	0
$730$	1.34282e7	0.932634
$731$	1.40153e7	0.970083
$732$	0	0
$733$	7.92038e6	0.544485	0.272243	−	0.962229i	$-0.412235\pi$
$733$	7.92038e6	0.544485	0.272243	+	0.962229i	$0.412235\pi$
$734$	−1.41932e7	−0.972392
$735$	0	0
$736$	1.85750e6	0.126397
$737$	8.12652e6	0.551107
$738$	0	0
$739$	1.60410e7	1.08049	0.540245	−	0.841508i	$-0.318332\pi$
$739$	1.60410e7	1.08049	0.540245	+	0.841508i	$0.318332\pi$
$740$	9.41527e6	0.632053
$741$	0	0
$742$	0	0
$743$	−1.53453e7	−1.01977	−0.509887	−	0.860241i	$-0.670313\pi$
$743$	−1.53453e7	−1.01977	−0.509887	+	0.860241i	$0.670313\pi$
$744$	0	0
$745$	−1.99301e7	−1.31559
$746$	9.02293e6	0.593609
$747$	0	0
$748$	2.74973e6	0.179695
$749$	0	0
$750$	0	0
$751$	2.24976e7	1.45558	0.727790	−	0.685800i	$-0.240547\pi$
$751$	2.24976e7	1.45558	0.727790	+	0.685800i	$0.240547\pi$
$752$	−6.29729e6	−0.406078
$753$	0	0
$754$	−1.06087e6	−0.0679571
$755$	2.10756e7	1.34559
$756$	0	0
$757$	2.30349e7	1.46099	0.730494	−	0.682919i	$-0.239290\pi$
$757$	2.30349e7	1.46099	0.730494	+	0.682919i	$0.239290\pi$
$758$	−1.75801e7	−1.11135
$759$	0	0
$760$	1.35050e7	0.848129
$761$	−5.40735e6	−0.338472	−0.169236	−	0.985576i	$-0.554130\pi$
$761$	−5.40735e6	−0.338472	−0.169236	+	0.985576i	$0.554130\pi$
$762$	0	0
$763$	0	0
$764$	7.76359e6	0.481204
$765$	0	0
$766$	−4.91123e6	−0.302426
$767$	9.21019e6	0.565301
$768$	0	0
$769$	−7.93100e6	−0.483629	−0.241814	−	0.970323i	$-0.577742\pi$
$769$	−7.93100e6	−0.483629	−0.241814	+	0.970323i	$0.577742\pi$
$770$	0	0
$771$	0	0
$772$	−5.29307e6	−0.319643
$773$	−269579.	−0.0162270	−0.00811349	−	0.999967i	$-0.502583\pi$
$773$	−269579.	−0.0162270	−0.00811349	+	0.999967i	$0.502583\pi$
$774$	0	0
$775$	2.32486e7	1.39041
$776$	−2.06991e6	−0.123395
$777$	0	0
$778$	8.21698e6	0.486702
$779$	2.13567e7	1.26093
$780$	0	0
$781$	1.04841e7	0.615041
$782$	8.34315e6	0.487880
$783$	0	0
$784$	0	0
$785$	1.42380e7	0.824662
$786$	0	0
$787$	−8.98621e6	−0.517177	−0.258589	−	0.965988i	$-0.583257\pi$
$787$	−8.98621e6	−0.517177	−0.258589	+	0.965988i	$0.583257\pi$
$788$	2.59140e6	0.148669
$789$	0	0
$790$	−1.86051e7	−1.06063
$791$	0	0
$792$	0	0
$793$	1.23420e7	0.696950
$794$	1.52147e7	0.856469
$795$	0	0
$796$	886340.	0.0495813
$797$	2.65558e7	1.48086	0.740429	−	0.672135i	$-0.234622\pi$
$797$	2.65558e7	1.48086	0.740429	+	0.672135i	$0.234622\pi$
$798$	0	0
$799$	−2.82849e7	−1.56743
$800$	−2.63587e6	−0.145613
$801$	0	0
$802$	4.74098e6	0.260275
$803$	−6.64638e6	−0.363745
$804$	0	0
$805$	0	0
$806$	−1.26233e7	−0.684442
$807$	0	0
$808$	2.00617e6	0.108103
$809$	−2.52557e7	−1.35671	−0.678356	−	0.734733i	$-0.737307\pi$
$809$	−2.52557e7	−1.35671	−0.678356	+	0.734733i	$0.737307\pi$
$810$	0	0
$811$	−4.76866e6	−0.254592	−0.127296	−	0.991865i	$-0.540630\pi$
$811$	−4.76866e6	−0.254592	−0.127296	+	0.991865i	$0.540630\pi$
$812$	0	0
$813$	0	0
$814$	−4.66015e6	−0.246513
$815$	6.77297e6	0.357179
$816$	0	0
$817$	−3.40702e7	−1.78575
$818$	1.72157e7	0.899585
$819$	0	0
$820$	−9.22878e6	−0.479303
$821$	5.95534e6	0.308353	0.154177	−	0.988043i	$-0.450727\pi$
$821$	5.95534e6	0.308353	0.154177	+	0.988043i	$0.450727\pi$
$822$	0	0
$823$	1.06659e7	0.548907	0.274453	−	0.961600i	$-0.411503\pi$
$823$	1.06659e7	0.548907	0.274453	+	0.961600i	$0.411503\pi$
$824$	−6.35731e6	−0.326178
$825$	0	0
$826$	0	0
$827$	8.52356e6	0.433369	0.216684	−	0.976242i	$-0.430476\pi$
$827$	8.52356e6	0.433369	0.216684	+	0.976242i	$0.430476\pi$
$828$	0	0
$829$	3.87458e6	0.195811	0.0979057	−	0.995196i	$-0.468786\pi$
$829$	3.87458e6	0.195811	0.0979057	+	0.995196i	$0.468786\pi$
$830$	−2.63142e7	−1.32585
$831$	0	0
$832$	1.43121e6	0.0716794
$833$	0	0
$834$	0	0
$835$	−3.99665e7	−1.98372
$836$	−6.68441e6	−0.330786
$837$	0	0
$838$	−454899.	−0.0223772
$839$	−2.13769e7	−1.04843	−0.524215	−	0.851586i	$-0.675641\pi$
$839$	−2.13769e7	−1.04843	−0.524215	+	0.851586i	$0.675641\pi$
$840$	0	0
$841$	−1.99350e7	−0.971911
$842$	−1.77367e6	−0.0862168
$843$	0	0
$844$	7.70796e6	0.372463
$845$	−1.88128e7	−0.906383
$846$	0	0
$847$	0	0
$848$	−3.48063e6	−0.166214
$849$	0	0
$850$	−1.18393e7	−0.562053
$851$	−1.41397e7	−0.669293
$852$	0	0
$853$	−85930.0	−0.00404364	−0.00202182	−	0.999998i	$-0.500644\pi$
$853$	−85930.0	−0.00404364	−0.00202182	+	0.999998i	$0.500644\pi$
$854$	0	0
$855$	0	0
$856$	−9.29715e6	−0.433676
$857$	1.85664e7	0.863528	0.431764	−	0.901987i	$-0.357891\pi$
$857$	1.85664e7	0.863528	0.431764	+	0.901987i	$0.357891\pi$
$858$	0	0
$859$	3.92215e7	1.81360	0.906799	−	0.421564i	$-0.138519\pi$
$859$	3.92215e7	1.81360	0.906799	+	0.421564i	$0.138519\pi$
$860$	1.47226e7	0.678795
$861$	0	0
$862$	1.85324e7	0.849501
$863$	−1.83642e7	−0.839352	−0.419676	−	0.907674i	$-0.637856\pi$
$863$	−1.83642e7	−0.839352	−0.419676	+	0.907674i	$0.637856\pi$
$864$	0	0
$865$	−2.55453e6	−0.116084
$866$	−2.63182e7	−1.19251
$867$	0	0
$868$	0	0
$869$	9.20870e6	0.413665
$870$	0	0
$871$	1.89984e7	0.848539
$872$	1.15744e7	0.515477
$873$	0	0
$874$	−2.02816e7	−0.898100
$875$	0	0
$876$	0	0
$877$	−9.42578e6	−0.413826	−0.206913	−	0.978359i	$-0.566342\pi$
$877$	−9.42578e6	−0.413826	−0.206913	+	0.978359i	$0.566342\pi$
$878$	7.28278e6	0.318831
$879$	0	0
$880$	2.88850e6	0.125738
$881$	−1.32820e7	−0.576534	−0.288267	−	0.957550i	$-0.593079\pi$
$881$	−1.32820e7	−0.576534	−0.288267	+	0.957550i	$0.593079\pi$
$882$	0	0
$883$	1.88897e7	0.815309	0.407655	−	0.913136i	$-0.366347\pi$
$883$	1.88897e7	0.815309	0.407655	+	0.913136i	$0.366347\pi$
$884$	6.42840e6	0.276676
$885$	0	0
$886$	−8.27961e6	−0.354345
$887$	−4.17083e7	−1.77997	−0.889987	−	0.455985i	$-0.849287\pi$
$887$	−4.17083e7	−1.77997	−0.889987	+	0.455985i	$0.849287\pi$
$888$	0	0
$889$	0	0
$890$	2.97649e7	1.25959
$891$	0	0
$892$	1.02205e7	0.430089
$893$	6.87586e7	2.88535
$894$	0	0
$895$	1.87028e7	0.780455
$896$	0	0
$897$	0	0
$898$	2.29033e7	0.947777
$899$	6.85539e6	0.282900
$900$	0	0
$901$	−1.56336e7	−0.641573
$902$	4.56785e6	0.186937
$903$	0	0
$904$	1.26613e7	0.515295
$905$	2.78922e7	1.13204
$906$	0	0
$907$	4.60801e7	1.85993	0.929963	−	0.367654i	$-0.119839\pi$
$907$	4.60801e7	1.85993	0.929963	+	0.367654i	$0.119839\pi$
$908$	8.41091e6	0.338554
$909$	0	0
$910$	0	0
$911$	−2.25527e7	−0.900334	−0.450167	−	0.892944i	$-0.648636\pi$
$911$	−2.25527e7	−0.900334	−0.450167	+	0.892944i	$0.648636\pi$
$912$	0	0
$913$	1.30244e7	0.517107
$914$	−1.27470e6	−0.0504712
$915$	0	0
$916$	1.49230e7	0.587647
$917$	0	0
$918$	0	0
$919$	−3.25898e7	−1.27290	−0.636448	−	0.771319i	$-0.719597\pi$
$919$	−3.25898e7	−1.27290	−0.636448	+	0.771319i	$0.719597\pi$
$920$	8.76421e6	0.341384
$921$	0	0
$922$	−1.36988e7	−0.530708
$923$	2.45100e7	0.946977
$924$	0	0
$925$	2.00648e7	0.771046
$926$	−1.53178e7	−0.587041
$927$	0	0
$928$	−777250.	−0.0296272
$929$	4.56149e7	1.73407	0.867036	−	0.498246i	$-0.166022\pi$
$929$	4.56149e7	1.73407	0.867036	+	0.498246i	$0.166022\pi$
$930$	0	0
$931$	0	0
$932$	1.13247e7	0.427058
$933$	0	0
$934$	1.13785e7	0.426792
$935$	1.29740e7	0.485338
$936$	0	0
$937$	3.60574e6	0.134167	0.0670835	−	0.997747i	$-0.478631\pi$
$937$	3.60574e6	0.134167	0.0670835	+	0.997747i	$0.478631\pi$
$938$	0	0
$939$	0	0
$940$	−2.97123e7	−1.09677
$941$	4.40826e6	0.162290	0.0811452	−	0.996702i	$-0.474142\pi$
$941$	4.40826e6	0.162290	0.0811452	+	0.996702i	$0.474142\pi$
$942$	0	0
$943$	1.38596e7	0.507542
$944$	6.74786e6	0.246454
$945$	0	0
$946$	−7.28706e6	−0.264743
$947$	−7.36278e6	−0.266788	−0.133394	−	0.991063i	$-0.542588\pi$
$947$	−7.36278e6	−0.266788	−0.133394	+	0.991063i	$0.542588\pi$
$948$	0	0
$949$	−1.55381e7	−0.560057
$950$	2.87804e7	1.03464
$951$	0	0
$952$	0	0
$953$	−3.73903e7	−1.33360	−0.666802	−	0.745235i	$-0.732337\pi$
$953$	−3.73903e7	−1.33360	−0.666802	+	0.745235i	$0.732337\pi$
$954$	0	0
$955$	3.66307e7	1.29968
$956$	8.00982e6	0.283451
$957$	0	0
$958$	5.21462e6	0.183573
$959$	0	0
$960$	0	0
$961$	5.29433e7	1.84928
$962$	−1.08946e7	−0.379555
$963$	0	0
$964$	1.93315e7	0.669997
$965$	−2.49742e7	−0.863322
$966$	0	0
$967$	1.30730e7	0.449582	0.224791	−	0.974407i	$-0.427830\pi$
$967$	1.30730e7	0.449582	0.224791	+	0.974407i	$0.427830\pi$
$968$	8.87758e6	0.304513
$969$	0	0
$970$	−9.76640e6	−0.333277
$971$	1.52454e7	0.518908	0.259454	−	0.965755i	$-0.416457\pi$
$971$	1.52454e7	0.518908	0.259454	+	0.965755i	$0.416457\pi$
$972$	0	0
$973$	0	0
$974$	−2.49450e7	−0.842530
$975$	0	0
$976$	9.04237e6	0.303849
$977$	−3.08690e6	−0.103463	−0.0517316	−	0.998661i	$-0.516474\pi$
$977$	−3.08690e6	−0.103463	−0.0517316	+	0.998661i	$0.516474\pi$
$978$	0	0
$979$	−1.47324e7	−0.491264
$980$	0	0
$981$	0	0
$982$	−1.57571e7	−0.521433
$983$	−1.70201e7	−0.561796	−0.280898	−	0.959738i	$-0.590632\pi$
$983$	−1.70201e7	−0.561796	−0.280898	+	0.959738i	$0.590632\pi$
$984$	0	0
$985$	1.22269e7	0.401539
$986$	−3.49109e6	−0.114359
$987$	0	0
$988$	−1.56270e7	−0.509311
$989$	−2.21102e7	−0.718789
$990$	0	0
$991$	−4.34672e7	−1.40597	−0.702987	−	0.711203i	$-0.748151\pi$
$991$	−4.34672e7	−1.40597	−0.702987	+	0.711203i	$0.748151\pi$
$992$	−9.24851e6	−0.298396
$993$	0	0
$994$	0	0
$995$	4.18199e6	0.133914
$996$	0	0
$997$	−2.46672e7	−0.785926	−0.392963	−	0.919554i	$-0.628550\pi$
$997$	−2.46672e7	−0.785926	−0.392963	+	0.919554i	$0.628550\pi$
$998$	−3.19028e7	−1.01392
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.6.a.bh.1.2		2
3.2	odd	2		294.6.a.r.1.1		2
7.2	even	3		126.6.g.h.109.1		4
7.4	even	3		126.6.g.h.37.1		4
7.6	odd	2		882.6.a.bb.1.1		2
21.2	odd	6		42.6.e.c.25.2	✓	4
21.5	even	6		294.6.e.s.67.1		4
21.11	odd	6		42.6.e.c.37.2	yes	4
21.17	even	6		294.6.e.s.79.1		4
21.20	even	2		294.6.a.w.1.2		2
84.11	even	6		336.6.q.f.289.2		4
84.23	even	6		336.6.q.f.193.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.6.e.c.25.2	✓	4	21.2	odd	6
42.6.e.c.37.2	yes	4	21.11	odd	6
126.6.g.h.37.1		4	7.4	even	3
126.6.g.h.109.1		4	7.2	even	3
294.6.a.r.1.1		2	3.2	odd	2
294.6.a.w.1.2		2	21.20	even	2
294.6.e.s.67.1		4	21.5	even	6
294.6.e.s.79.1		4	21.17	even	6
336.6.q.f.193.2		4	84.23	even	6
336.6.q.f.289.2		4	84.11	even	6
882.6.a.bb.1.1		2	7.6	odd	2
882.6.a.bh.1.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	882.a (trivial)

\(f(q)\)	\(=\)	\(q-4.00000 q^{2} +16.0000 q^{4} +75.4923 q^{5} -64.0000 q^{8} -301.969 q^{10} +149.462 q^{11} +349.416 q^{13} +256.000 q^{16} +1149.85 q^{17} -2795.20 q^{19} +1207.88 q^{20} -597.847 q^{22} -1813.97 q^{23} +2574.09 q^{25} -1397.66 q^{26} +759.033 q^{29} +9031.74 q^{31} -1024.00 q^{32} -4599.39 q^{34} +7794.89 q^{37} +11180.8 q^{38} -4831.51 q^{40} -7640.49 q^{41} +12188.8 q^{43} +2391.39 q^{44} +7255.88 q^{46} -24598.8 q^{47} -10296.4 q^{50} +5590.65 q^{52} -13596.2 q^{53} +11283.2 q^{55} -3036.13 q^{58} +26358.8 q^{59} +35321.8 q^{61} -36127.0 q^{62} +4096.00 q^{64} +26378.2 q^{65} +54371.9 q^{67} +18397.6 q^{68} +70145.7 q^{71} -44468.8 q^{73} -31179.6 q^{74} -44723.2 q^{76} +61612.5 q^{79} +19326.0 q^{80} +30562.0 q^{82} +87142.0 q^{83} +86804.6 q^{85} -48755.3 q^{86} -9565.55 q^{88} -98569.4 q^{89} -29023.5 q^{92} +98395.2 q^{94} -211016. q^{95} +32342.3 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{2} + 32 q^{4} + 53 q^{5} - 128 q^{8} - 212 q^{10} - 191 q^{11} - 379 q^{13} + 512 q^{16} + 340 q^{17} - 1769 q^{19} + 848 q^{20} + 764 q^{22} - 3236 q^{23} - 45 q^{25} + 1516 q^{26} - 4459 q^{29}+ \cdots - 129423 q^{97}+O(q^{100}) \) 2 * q - 8 * q^2 + 32 * q^4 + 53 * q^5 - 128 * q^8 - 212 * q^10 - 191 * q^11 - 379 * q^13 + 512 * q^16 + 340 * q^17 - 1769 * q^19 + 848 * q^20 + 764 * q^22 - 3236 * q^23 - 45 * q^25 + 1516 * q^26 - 4459 * q^29 + 1994 * q^31 - 2048 * q^32 - 1360 * q^34 + 20587 * q^37 + 7076 * q^38 - 3392 * q^40 - 8814 * q^41 + 15853 * q^43 - 3056 * q^44 + 12944 * q^46 - 33912 * q^47 + 180 * q^50 - 6064 * q^52 - 49239 * q^53 + 18941 * q^55 + 17836 * q^58 + 56735 * q^59 + 67508 * q^61 - 7976 * q^62 + 8192 * q^64 + 42762 * q^65 + 75723 * q^67 + 5440 * q^68 + 8992 * q^71 - 3201 * q^73 - 82348 * q^74 - 28304 * q^76 + 26612 * q^79 + 13568 * q^80 + 35256 * q^82 + 949 * q^83 + 105020 * q^85 - 63412 * q^86 + 12224 * q^88 - 176562 * q^89 - 51776 * q^92 + 135648 * q^94 - 234098 * q^95 - 129423 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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