LMFDB - Embedded newform 1028.6.a.b.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1028,6,Mod(1,1028)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1028, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("1028.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1028 = 2^{2} \cdot 257 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1028.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:	$164.874566768$
Analytic rank:	$0$
Dimension:	$57$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Character	$\chi$	$=$	1028.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-26.8472 q^{3} -27.7983 q^{5} -148.747 q^{7} +477.771 q^{9} +O(q^{10})$	$q-26.8472 q^{3} -27.7983 q^{5} -148.747 q^{7} +477.771 q^{9} -369.485 q^{11} +315.749 q^{13} +746.307 q^{15} +1189.84 q^{17} +816.166 q^{19} +3993.43 q^{21} +2048.86 q^{23} -2352.25 q^{25} -6302.95 q^{27} -3186.13 q^{29} -3402.38 q^{31} +9919.63 q^{33} +4134.91 q^{35} -4738.94 q^{37} -8476.96 q^{39} -1327.72 q^{41} -18099.2 q^{43} -13281.2 q^{45} -15291.8 q^{47} +5318.63 q^{49} -31943.9 q^{51} +13024.5 q^{53} +10271.1 q^{55} -21911.8 q^{57} +38558.7 q^{59} +8582.84 q^{61} -71067.0 q^{63} -8777.28 q^{65} +15807.1 q^{67} -55006.0 q^{69} -63031.4 q^{71} +6763.70 q^{73} +63151.4 q^{75} +54959.7 q^{77} -45605.5 q^{79} +53117.9 q^{81} -92198.4 q^{83} -33075.6 q^{85} +85538.6 q^{87} +95363.1 q^{89} -46966.6 q^{91} +91344.3 q^{93} -22688.1 q^{95} -144508. q^{97} -176529. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 57 q + 27 q^{3} + 61 q^{5} + 88 q^{7} + 5374 q^{9}+O(q^{10}) $ 57 * q + 27 * q^3 + 61 * q^5 + 88 * q^7 + 5374 * q^9	$ 57 q + 27 q^{3} + 61 q^{5} + 88 q^{7} + 5374 q^{9} + 484 q^{11} + 3162 q^{13} - 244 q^{15} + 3298 q^{17} + 2931 q^{19} + 5009 q^{21} + 3712 q^{23} + 48568 q^{25} + 8682 q^{27} + 10909 q^{29} - 3539 q^{31} + 13277 q^{33} + 13976 q^{35} + 42395 q^{37} - 14986 q^{39} - 8469 q^{41} + 69808 q^{43} + 26347 q^{45} + 12410 q^{47} + 203347 q^{49} + 24220 q^{51} + 49353 q^{53} - 34265 q^{55} + 98112 q^{57} - 9792 q^{59} + 157032 q^{61} + 24981 q^{63} + 16143 q^{65} + 20201 q^{67} - 113743 q^{69} - 229673 q^{71} - 19561 q^{73} - 482061 q^{75} + 21204 q^{77} - 10925 q^{79} + 323569 q^{81} - 62846 q^{83} + 218157 q^{85} - 169628 q^{87} + 69901 q^{89} + 56581 q^{91} + 177641 q^{93} - 192910 q^{95} + 277990 q^{97} + 424882 q^{99}+O(q^{100}) $ 57 * q + 27 * q^3 + 61 * q^5 + 88 * q^7 + 5374 * q^9 + 484 * q^11 + 3162 * q^13 - 244 * q^15 + 3298 * q^17 + 2931 * q^19 + 5009 * q^21 + 3712 * q^23 + 48568 * q^25 + 8682 * q^27 + 10909 * q^29 - 3539 * q^31 + 13277 * q^33 + 13976 * q^35 + 42395 * q^37 - 14986 * q^39 - 8469 * q^41 + 69808 * q^43 + 26347 * q^45 + 12410 * q^47 + 203347 * q^49 + 24220 * q^51 + 49353 * q^53 - 34265 * q^55 + 98112 * q^57 - 9792 * q^59 + 157032 * q^61 + 24981 * q^63 + 16143 * q^65 + 20201 * q^67 - 113743 * q^69 - 229673 * q^71 - 19561 * q^73 - 482061 * q^75 + 21204 * q^77 - 10925 * q^79 + 323569 * q^81 - 62846 * q^83 + 218157 * q^85 - 169628 * q^87 + 69901 * q^89 + 56581 * q^91 + 177641 * q^93 - 192910 * q^95 + 277990 * q^97 + 424882 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−26.8472	−1.72225	−0.861124	−	0.508395i	$-0.830239\pi$
$3$	−26.8472	−1.72225	−0.861124	+	0.508395i	$0.830239\pi$
$4$	0	0
$5$	−27.7983	−0.497272	−0.248636	−	0.968597i	$-0.579982\pi$
$5$	−27.7983	−0.497272	−0.248636	+	0.968597i	$0.579982\pi$
$6$	0	0
$7$	−148.747	−1.14737	−0.573684	−	0.819077i	$-0.694486\pi$
$7$	−148.747	−1.14737	−0.573684	+	0.819077i	$0.694486\pi$
$8$	0	0
$9$	477.771	1.96614
$10$	0	0
$11$	−369.485	−0.920694	−0.460347	−	0.887739i	$-0.652275\pi$
$11$	−369.485	−0.920694	−0.460347	+	0.887739i	$0.652275\pi$
$12$	0	0
$13$	315.749	0.518183	0.259091	−	0.965853i	$-0.416577\pi$
$13$	315.749	0.518183	0.259091	+	0.965853i	$0.416577\pi$
$14$	0	0
$15$	746.307	0.856425
$16$	0	0
$17$	1189.84	0.998542	0.499271	−	0.866446i	$-0.333601\pi$
$17$	1189.84	0.998542	0.499271	+	0.866446i	$0.333601\pi$
$18$	0	0
$19$	816.166	0.518674	0.259337	−	0.965787i	$-0.416496\pi$
$19$	816.166	0.518674	0.259337	+	0.965787i	$0.416496\pi$
$20$	0	0
$21$	3993.43	1.97605
$22$	0	0
$23$	2048.86	0.807592	0.403796	−	0.914849i	$-0.367691\pi$
$23$	2048.86	0.807592	0.403796	+	0.914849i	$0.367691\pi$
$24$	0	0
$25$	−2352.25	−0.752721
$26$	0	0
$27$	−6302.95	−1.66393
$28$	0	0
$29$	−3186.13	−0.703507	−0.351754	−	0.936093i	$-0.614414\pi$
$29$	−3186.13	−0.703507	−0.351754	+	0.936093i	$0.614414\pi$
$30$	0	0
$31$	−3402.38	−0.635885	−0.317942	−	0.948110i	$-0.602992\pi$
$31$	−3402.38	−0.635885	−0.317942	+	0.948110i	$0.602992\pi$
$32$	0	0
$33$	9919.63	1.58566
$34$	0	0
$35$	4134.91	0.570553
$36$	0	0
$37$	−4738.94	−0.569085	−0.284542	−	0.958663i	$-0.591842\pi$
$37$	−4738.94	−0.569085	−0.284542	+	0.958663i	$0.591842\pi$
$38$	0	0
$39$	−8476.96	−0.892439
$40$	0	0
$41$	−1327.72	−0.123352	−0.0616761	−	0.998096i	$-0.519645\pi$
$41$	−1327.72	−0.123352	−0.0616761	+	0.998096i	$0.519645\pi$
$42$	0	0
$43$	−18099.2	−1.49275	−0.746376	−	0.665524i	$-0.768208\pi$
$43$	−18099.2	−1.49275	−0.746376	+	0.665524i	$0.768208\pi$
$44$	0	0
$45$	−13281.2	−0.977704
$46$	0	0
$47$	−15291.8	−1.00975	−0.504874	−	0.863193i	$-0.668461\pi$
$47$	−15291.8	−1.00975	−0.504874	+	0.863193i	$0.668461\pi$
$48$	0	0
$49$	5318.63	0.316453
$50$	0	0
$51$	−31943.9	−1.71974
$52$	0	0
$53$	13024.5	0.636899	0.318450	−	0.947940i	$-0.396838\pi$
$53$	13024.5	0.636899	0.318450	+	0.947940i	$0.396838\pi$
$54$	0	0
$55$	10271.1	0.457835
$56$	0	0
$57$	−21911.8	−0.893285
$58$	0	0
$59$	38558.7	1.44209	0.721046	−	0.692888i	$-0.243662\pi$
$59$	38558.7	1.44209	0.721046	+	0.692888i	$0.243662\pi$
$60$	0	0
$61$	8582.84	0.295329	0.147665	−	0.989037i	$-0.452824\pi$
$61$	8582.84	0.295329	0.147665	+	0.989037i	$0.452824\pi$
$62$	0	0
$63$	−71067.0	−2.25588
$64$	0	0
$65$	−8777.28	−0.257678
$66$	0	0
$67$	15807.1	0.430194	0.215097	−	0.976593i	$-0.430993\pi$
$67$	15807.1	0.430194	0.215097	+	0.976593i	$0.430993\pi$
$68$	0	0
$69$	−55006.0	−1.39087
$70$	0	0
$71$	−63031.4	−1.48392	−0.741961	−	0.670443i	$-0.766104\pi$
$71$	−63031.4	−1.48392	−0.741961	+	0.670443i	$0.766104\pi$
$72$	0	0
$73$	6763.70	0.148551	0.0742757	−	0.997238i	$-0.476336\pi$
$73$	6763.70	0.148551	0.0742757	+	0.997238i	$0.476336\pi$
$74$	0	0
$75$	63151.4	1.29637
$76$	0	0
$77$	54959.7	1.05637
$78$	0	0
$79$	−45605.5	−0.822147	−0.411074	−	0.911602i	$-0.634846\pi$
$79$	−45605.5	−0.822147	−0.411074	+	0.911602i	$0.634846\pi$
$80$	0	0
$81$	53117.9	0.899557
$82$	0	0
$83$	−92198.4	−1.46902	−0.734511	−	0.678597i	$-0.762588\pi$
$83$	−92198.4	−1.46902	−0.734511	+	0.678597i	$0.762588\pi$
$84$	0	0
$85$	−33075.6	−0.496546
$86$	0	0
$87$	85538.6	1.21161
$88$	0	0
$89$	95363.1	1.27616	0.638080	−	0.769970i	$-0.279729\pi$
$89$	95363.1	1.27616	0.638080	+	0.769970i	$0.279729\pi$
$90$	0	0
$91$	−46966.6	−0.594546
$92$	0	0
$93$	91344.3	1.09515
$94$	0	0
$95$	−22688.1	−0.257922
$96$	0	0
$97$	−144508.	−1.55942	−0.779708	−	0.626144i	$-0.784632\pi$
$97$	−144508.	−1.55942	−0.779708	+	0.626144i	$0.784632\pi$
$98$	0	0
$99$	−176529.	−1.81021
$100$	0	0
$101$	−119479.	−1.16544	−0.582720	−	0.812673i	$-0.698011\pi$
$101$	−119479.	−1.16544	−0.582720	+	0.812673i	$0.698011\pi$
$102$	0	0
$103$	−134920.	−1.25309	−0.626545	−	0.779386i	$-0.715531\pi$
$103$	−134920.	−1.25309	−0.626545	+	0.779386i	$0.715531\pi$
$104$	0	0
$105$	−111011.	−0.982634
$106$	0	0
$107$	−136815.	−1.15525	−0.577623	−	0.816304i	$-0.696019\pi$
$107$	−136815.	−1.15525	−0.577623	+	0.816304i	$0.696019\pi$
$108$	0	0
$109$	−176309.	−1.42138	−0.710689	−	0.703507i	$-0.751617\pi$
$109$	−176309.	−1.42138	−0.710689	+	0.703507i	$0.751617\pi$
$110$	0	0
$111$	127227.	0.980105
$112$	0	0
$113$	−4002.18	−0.0294850	−0.0147425	−	0.999891i	$-0.504693\pi$
$113$	−4002.18	−0.0294850	−0.0147425	+	0.999891i	$0.504693\pi$
$114$	0	0
$115$	−56954.8	−0.401593
$116$	0	0
$117$	150856.	1.01882
$118$	0	0
$119$	−176985.	−1.14569
$120$	0	0
$121$	−24531.8	−0.152323
$122$	0	0
$123$	35645.5	0.212443
$124$	0	0
$125$	152258.	0.871578
$126$	0	0
$127$	−163964.	−0.902070	−0.451035	−	0.892506i	$-0.648945\pi$
$127$	−163964.	−0.902070	−0.451035	+	0.892506i	$0.648945\pi$
$128$	0	0
$129$	485912.	2.57089
$130$	0	0
$131$	34170.5	0.173969	0.0869847	−	0.996210i	$-0.472277\pi$
$131$	34170.5	0.173969	0.0869847	+	0.996210i	$0.472277\pi$
$132$	0	0
$133$	−121402.	−0.595110
$134$	0	0
$135$	175211.	0.827423
$136$	0	0
$137$	287920.	1.31060	0.655301	−	0.755367i	$-0.272542\pi$
$137$	287920.	1.31060	0.655301	+	0.755367i	$0.272542\pi$
$138$	0	0
$139$	−10055.9	−0.0441453	−0.0220726	−	0.999756i	$-0.507027\pi$
$139$	−10055.9	−0.0441453	−0.0220726	+	0.999756i	$0.507027\pi$
$140$	0	0
$141$	410540.	1.73903
$142$	0	0
$143$	−116664.	−0.477088
$144$	0	0
$145$	88569.1	0.349834
$146$	0	0
$147$	−142790.	−0.545010
$148$	0	0
$149$	1383.10	0.00510374	0.00255187	−	0.999997i	$-0.499188\pi$
$149$	1383.10	0.00510374	0.00255187	+	0.999997i	$0.499188\pi$
$150$	0	0
$151$	47502.3	0.169540	0.0847699	−	0.996401i	$-0.472984\pi$
$151$	47502.3	0.169540	0.0847699	+	0.996401i	$0.472984\pi$
$152$	0	0
$153$	568471.	1.96327
$154$	0	0
$155$	94580.4	0.316207
$156$	0	0
$157$	446041.	1.44420	0.722098	−	0.691791i	$-0.243178\pi$
$157$	446041.	1.44420	0.722098	+	0.691791i	$0.243178\pi$
$158$	0	0
$159$	−349671.	−1.09690
$160$	0	0
$161$	−304761.	−0.926605
$162$	0	0
$163$	496433.	1.46350	0.731748	−	0.681575i	$-0.238705\pi$
$163$	496433.	1.46350	0.731748	+	0.681575i	$0.238705\pi$
$164$	0	0
$165$	−275749.	−0.788505
$166$	0	0
$167$	−663646.	−1.84139	−0.920694	−	0.390284i	$-0.872377\pi$
$167$	−663646.	−1.84139	−0.920694	+	0.390284i	$0.872377\pi$
$168$	0	0
$169$	−271596.	−0.731487
$170$	0	0
$171$	389941.	1.01978
$172$	0	0
$173$	4309.74	0.0109480	0.00547401	−	0.999985i	$-0.498258\pi$
$173$	4309.74	0.0109480	0.00547401	+	0.999985i	$0.498258\pi$
$174$	0	0
$175$	349890.	0.863648
$176$	0	0
$177$	−1.03519e6	−2.48364
$178$	0	0
$179$	121125.	0.282554	0.141277	−	0.989970i	$-0.454879\pi$
$179$	121125.	0.282554	0.141277	+	0.989970i	$0.454879\pi$
$180$	0	0
$181$	435115.	0.987206	0.493603	−	0.869687i	$-0.335680\pi$
$181$	435115.	0.987206	0.493603	+	0.869687i	$0.335680\pi$
$182$	0	0
$183$	−230425.	−0.508630
$184$	0	0
$185$	131735.	0.282990
$186$	0	0
$187$	−439628.	−0.919351
$188$	0	0
$189$	937543.	1.90914
$190$	0	0
$191$	−101516.	−0.201351	−0.100675	−	0.994919i	$-0.532100\pi$
$191$	−101516.	−0.201351	−0.100675	+	0.994919i	$0.532100\pi$
$192$	0	0
$193$	191266.	0.369610	0.184805	−	0.982775i	$-0.440835\pi$
$193$	191266.	0.369610	0.184805	+	0.982775i	$0.440835\pi$
$194$	0	0
$195$	235645.	0.443785
$196$	0	0
$197$	322141.	0.591399	0.295700	−	0.955281i	$-0.404447\pi$
$197$	322141.	0.591399	0.295700	+	0.955281i	$0.404447\pi$
$198$	0	0
$199$	−474820.	−0.849956	−0.424978	−	0.905204i	$-0.639718\pi$
$199$	−474820.	−0.849956	−0.424978	+	0.905204i	$0.639718\pi$
$200$	0	0
$201$	−424375.	−0.740900
$202$	0	0
$203$	473927.	0.807181
$204$	0	0
$205$	36908.4	0.0613395
$206$	0	0
$207$	978885.	1.58784
$208$	0	0
$209$	−301561.	−0.477540
$210$	0	0
$211$	−1.12724e6	−1.74306	−0.871529	−	0.490344i	$-0.836871\pi$
$211$	−1.12724e6	−1.74306	−0.871529	+	0.490344i	$0.836871\pi$
$212$	0	0
$213$	1.69222e6	2.55568
$214$	0	0
$215$	503127.	0.742303
$216$	0	0
$217$	506093.	0.729594
$218$	0	0
$219$	−181586.	−0.255842
$220$	0	0
$221$	375690.	0.517427
$222$	0	0
$223$	661845.	0.891239	0.445620	−	0.895222i	$-0.352983\pi$
$223$	661845.	0.891239	0.445620	+	0.895222i	$0.352983\pi$
$224$	0	0
$225$	−1.12384e6	−1.47995
$226$	0	0
$227$	−627312.	−0.808014	−0.404007	−	0.914756i	$-0.632383\pi$
$227$	−627312.	−0.808014	−0.404007	+	0.914756i	$0.632383\pi$
$228$	0	0
$229$	1.25329e6	1.57929	0.789647	−	0.613562i	$-0.210264\pi$
$229$	1.25329e6	1.57929	0.789647	+	0.613562i	$0.210264\pi$
$230$	0	0
$231$	−1.47551e6	−1.81934
$232$	0	0
$233$	−382728.	−0.461850	−0.230925	−	0.972972i	$-0.574175\pi$
$233$	−382728.	−0.461850	−0.230925	+	0.972972i	$0.574175\pi$
$234$	0	0
$235$	425085.	0.502119
$236$	0	0
$237$	1.22438e6	1.41594
$238$	0	0
$239$	1.26989e6	1.43804	0.719018	−	0.694991i	$-0.244592\pi$
$239$	1.26989e6	1.43804	0.719018	+	0.694991i	$0.244592\pi$
$240$	0	0
$241$	857656.	0.951197	0.475599	−	0.879662i	$-0.342232\pi$
$241$	857656.	0.951197	0.475599	+	0.879662i	$0.342232\pi$
$242$	0	0
$243$	105549.	0.114667
$244$	0	0
$245$	−147849.	−0.157363
$246$	0	0
$247$	257703.	0.268768
$248$	0	0
$249$	2.47527e6	2.53002
$250$	0	0
$251$	−252700.	−0.253175	−0.126587	−	0.991955i	$-0.540402\pi$
$251$	−252700.	−0.253175	−0.126587	+	0.991955i	$0.540402\pi$
$252$	0	0
$253$	−757022.	−0.743545
$254$	0	0
$255$	887986.	0.855176
$256$	0	0
$257$	−66049.0	−0.0623783
$257$	−66049.0	−0.0623783
$258$	0	0
$259$	704903.	0.652950
$260$	0	0
$261$	−1.52224e6	−1.38319
$262$	0	0
$263$	922618.	0.822493	0.411247	−	0.911524i	$-0.365093\pi$
$263$	922618.	0.822493	0.411247	+	0.911524i	$0.365093\pi$
$264$	0	0
$265$	−362059.	−0.316712
$266$	0	0
$267$	−2.56023e6	−2.19786
$268$	0	0
$269$	−1.21901e6	−1.02713	−0.513566	−	0.858050i	$-0.671676\pi$
$269$	−1.21901e6	−1.02713	−0.513566	+	0.858050i	$0.671676\pi$
$270$	0	0
$271$	−204502.	−0.169151	−0.0845753	−	0.996417i	$-0.526953\pi$
$271$	−204502.	−0.169151	−0.0845753	+	0.996417i	$0.526953\pi$
$272$	0	0
$273$	1.26092e6	1.02396
$274$	0	0
$275$	869122.	0.693025
$276$	0	0
$277$	−583522.	−0.456938	−0.228469	−	0.973551i	$-0.573372\pi$
$277$	−583522.	−0.456938	−0.228469	+	0.973551i	$0.573372\pi$
$278$	0	0
$279$	−1.62556e6	−1.25024
$280$	0	0
$281$	−667829.	−0.504544	−0.252272	−	0.967656i	$-0.581178\pi$
$281$	−667829.	−0.504544	−0.252272	+	0.967656i	$0.581178\pi$
$282$	0	0
$283$	1.33138e6	0.988181	0.494090	−	0.869411i	$-0.335501\pi$
$283$	1.33138e6	0.988181	0.494090	+	0.869411i	$0.335501\pi$
$284$	0	0
$285$	609110.	0.444205
$286$	0	0
$287$	197494.	0.141530
$288$	0	0
$289$	−4137.81	−0.00291424
$290$	0	0
$291$	3.87963e6	2.68570
$292$	0	0
$293$	732354.	0.498370	0.249185	−	0.968456i	$-0.419837\pi$
$293$	732354.	0.498370	0.249185	+	0.968456i	$0.419837\pi$
$294$	0	0
$295$	−1.07187e6	−0.717111
$296$	0	0
$297$	2.32884e6	1.53197
$298$	0	0
$299$	646924.	0.418480
$300$	0	0
$301$	2.69220e6	1.71274
$302$	0	0
$303$	3.20769e6	2.00718
$304$	0	0
$305$	−238589.	−0.146859
$306$	0	0
$307$	730965.	0.442640	0.221320	−	0.975201i	$-0.428963\pi$
$307$	730965.	0.442640	0.221320	+	0.975201i	$0.428963\pi$
$308$	0	0
$309$	3.62221e6	2.15813
$310$	0	0
$311$	−810420.	−0.475126	−0.237563	−	0.971372i	$-0.576349\pi$
$311$	−810420.	−0.475126	−0.237563	+	0.971372i	$0.576349\pi$
$312$	0	0
$313$	−416022.	−0.240024	−0.120012	−	0.992772i	$-0.538293\pi$
$313$	−416022.	−0.240024	−0.120012	+	0.992772i	$0.538293\pi$
$314$	0	0
$315$	1.97554e6	1.12179
$316$	0	0
$317$	−2.00798e6	−1.12230	−0.561152	−	0.827713i	$-0.689642\pi$
$317$	−2.00798e6	−1.12230	−0.561152	+	0.827713i	$0.689642\pi$
$318$	0	0
$319$	1.17723e6	0.647715
$320$	0	0
$321$	3.67310e6	1.98962
$322$	0	0
$323$	971107.	0.517918
$324$	0	0
$325$	−742721.	−0.390047
$326$	0	0
$327$	4.73341e6	2.44796
$328$	0	0
$329$	2.27460e6	1.15855
$330$	0	0
$331$	−1.99476e6	−1.00074	−0.500368	−	0.865813i	$-0.666802\pi$
$331$	−1.99476e6	−1.00074	−0.500368	+	0.865813i	$0.666802\pi$
$332$	0	0
$333$	−2.26413e6	−1.11890
$334$	0	0
$335$	−439410.	−0.213923
$336$	0	0
$337$	−1.07888e6	−0.517485	−0.258743	−	0.965946i	$-0.583308\pi$
$337$	−1.07888e6	−0.517485	−0.258743	+	0.965946i	$0.583308\pi$
$338$	0	0
$339$	107447.	0.0507805
$340$	0	0
$341$	1.25713e6	0.585455
$342$	0	0
$343$	1.70886e6	0.784280
$344$	0	0
$345$	1.52908e6	0.691642
$346$	0	0
$347$	1.07986e6	0.481442	0.240721	−	0.970594i	$-0.422616\pi$
$347$	1.07986e6	0.481442	0.240721	+	0.970594i	$0.422616\pi$
$348$	0	0
$349$	−2.74258e6	−1.20530	−0.602650	−	0.798005i	$-0.705889\pi$
$349$	−2.74258e6	−1.20530	−0.602650	+	0.798005i	$0.705889\pi$
$350$	0	0
$351$	−1.99015e6	−0.862218
$352$	0	0
$353$	1.75444e6	0.749378	0.374689	−	0.927151i	$-0.377750\pi$
$353$	1.75444e6	0.749378	0.374689	+	0.927151i	$0.377750\pi$
$354$	0	0
$355$	1.75217e6	0.737912
$356$	0	0
$357$	4.75155e6	1.97317
$358$	0	0
$359$	4.28706e6	1.75559	0.877795	−	0.479036i	$-0.159014\pi$
$359$	4.28706e6	1.75559	0.877795	+	0.479036i	$0.159014\pi$
$360$	0	0
$361$	−1.80997e6	−0.730977
$362$	0	0
$363$	658610.	0.262338
$364$	0	0
$365$	−188019.	−0.0738704
$366$	0	0
$367$	−3.59516e6	−1.39333	−0.696664	−	0.717397i	$-0.745333\pi$
$367$	−3.59516e6	−1.39333	−0.696664	+	0.717397i	$0.745333\pi$
$368$	0	0
$369$	−634346.	−0.242527
$370$	0	0
$371$	−1.93735e6	−0.730758
$372$	0	0
$373$	2.74971e6	1.02333	0.511663	−	0.859186i	$-0.329029\pi$
$373$	2.74971e6	1.02333	0.511663	+	0.859186i	$0.329029\pi$
$374$	0	0
$375$	−4.08771e6	−1.50107
$376$	0	0
$377$	−1.00602e6	−0.364545
$378$	0	0
$379$	3.52111e6	1.25916	0.629580	−	0.776935i	$-0.283227\pi$
$379$	3.52111e6	1.25916	0.629580	+	0.776935i	$0.283227\pi$
$380$	0	0
$381$	4.40198e6	1.55359
$382$	0	0
$383$	−4.27218e6	−1.48817	−0.744085	−	0.668085i	$-0.767114\pi$
$383$	−4.27218e6	−1.48817	−0.744085	+	0.668085i	$0.767114\pi$
$384$	0	0
$385$	−1.52779e6	−0.525305
$386$	0	0
$387$	−8.64727e6	−2.93496
$388$	0	0
$389$	−4.35418e6	−1.45892	−0.729461	−	0.684022i	$-0.760229\pi$
$389$	−4.35418e6	−1.45892	−0.729461	+	0.684022i	$0.760229\pi$
$390$	0	0
$391$	2.43781e6	0.806414
$392$	0	0
$393$	−917381.	−0.299618
$394$	0	0
$395$	1.26776e6	0.408830
$396$	0	0
$397$	153022.	0.0487278	0.0243639	−	0.999703i	$-0.492244\pi$
$397$	153022.	0.0487278	0.0243639	+	0.999703i	$0.492244\pi$
$398$	0	0
$399$	3.25931e6	1.02493
$400$	0	0
$401$	−4.76073e6	−1.47847	−0.739235	−	0.673447i	$-0.764813\pi$
$401$	−4.76073e6	−1.47847	−0.739235	+	0.673447i	$0.764813\pi$
$402$	0	0
$403$	−1.07430e6	−0.329505
$404$	0	0
$405$	−1.47659e6	−0.447324
$406$	0	0
$407$	1.75097e6	0.523953
$408$	0	0
$409$	1.35257e6	0.399808	0.199904	−	0.979815i	$-0.435937\pi$
$409$	1.35257e6	0.399808	0.199904	+	0.979815i	$0.435937\pi$
$410$	0	0
$411$	−7.72985e6	−2.25718
$412$	0	0
$413$	−5.73549e6	−1.65461
$414$	0	0
$415$	2.56296e6	0.730503
$416$	0	0
$417$	269973.	0.0760291
$418$	0	0
$419$	1.55267e6	0.432061	0.216030	−	0.976387i	$-0.430689\pi$
$419$	1.55267e6	0.432061	0.216030	+	0.976387i	$0.430689\pi$
$420$	0	0
$421$	−745843.	−0.205089	−0.102544	−	0.994728i	$-0.532698\pi$
$421$	−745843.	−0.205089	−0.102544	+	0.994728i	$0.532698\pi$
$422$	0	0
$423$	−7.30596e6	−1.98530
$424$	0	0
$425$	−2.79880e6	−0.751623
$426$	0	0
$427$	−1.27667e6	−0.338851
$428$	0	0
$429$	3.13211e6	0.821663
$430$	0	0
$431$	−1.76649e6	−0.458055	−0.229027	−	0.973420i	$-0.573554\pi$
$431$	−1.76649e6	−0.458055	−0.229027	+	0.973420i	$0.573554\pi$
$432$	0	0
$433$	2.55165e6	0.654036	0.327018	−	0.945018i	$-0.393956\pi$
$433$	2.55165e6	0.654036	0.327018	+	0.945018i	$0.393956\pi$
$434$	0	0
$435$	−2.37783e6	−0.602501
$436$	0	0
$437$	1.67221e6	0.418877
$438$	0	0
$439$	−2.91636e6	−0.722236	−0.361118	−	0.932520i	$-0.617605\pi$
$439$	−2.91636e6	−0.722236	−0.361118	+	0.932520i	$0.617605\pi$
$440$	0	0
$441$	2.54109e6	0.622190
$442$	0	0
$443$	7.14554e6	1.72992	0.864959	−	0.501842i	$-0.167344\pi$
$443$	7.14554e6	1.72992	0.864959	+	0.501842i	$0.167344\pi$
$444$	0	0
$445$	−2.65093e6	−0.634599
$446$	0	0
$447$	−37132.4	−0.00878990
$448$	0	0
$449$	−1.30098e6	−0.304547	−0.152273	−	0.988338i	$-0.548659\pi$
$449$	−1.30098e6	−0.304547	−0.152273	+	0.988338i	$0.548659\pi$
$450$	0	0
$451$	490573.	0.113570
$452$	0	0
$453$	−1.27530e6	−0.291990
$454$	0	0
$455$	1.30559e6	0.295651
$456$	0	0
$457$	−4.04278e6	−0.905502	−0.452751	−	0.891637i	$-0.649557\pi$
$457$	−4.04278e6	−0.905502	−0.452751	+	0.891637i	$0.649557\pi$
$458$	0	0
$459$	−7.49950e6	−1.66150
$460$	0	0
$461$	3.47806e6	0.762228	0.381114	−	0.924528i	$-0.375541\pi$
$461$	3.47806e6	0.762228	0.381114	+	0.924528i	$0.375541\pi$
$462$	0	0
$463$	632697.	0.137165	0.0685825	−	0.997645i	$-0.478152\pi$
$463$	632697.	0.137165	0.0685825	+	0.997645i	$0.478152\pi$
$464$	0	0
$465$	−2.53922e6	−0.544588
$466$	0	0
$467$	2.26226e6	0.480010	0.240005	−	0.970772i	$-0.422851\pi$
$467$	2.26226e6	0.480010	0.240005	+	0.970772i	$0.422851\pi$
$468$	0	0
$469$	−2.35125e6	−0.493591
$470$	0	0
$471$	−1.19749e7	−2.48726
$472$	0	0
$473$	6.68738e6	1.37437
$474$	0	0
$475$	−1.91983e6	−0.390417
$476$	0	0
$477$	6.22272e6	1.25223
$478$	0	0
$479$	−5.78125e6	−1.15129	−0.575643	−	0.817701i	$-0.695248\pi$
$479$	−5.78125e6	−1.15129	−0.575643	+	0.817701i	$0.695248\pi$
$480$	0	0
$481$	−1.49631e6	−0.294890
$482$	0	0
$483$	8.18197e6	1.59584
$484$	0	0
$485$	4.01707e6	0.775453
$486$	0	0
$487$	−7.80315e6	−1.49090	−0.745448	−	0.666564i	$-0.767764\pi$
$487$	−7.80315e6	−1.49090	−0.745448	+	0.666564i	$0.767764\pi$
$488$	0	0
$489$	−1.33278e7	−2.52050
$490$	0	0
$491$	−3.94743e6	−0.738943	−0.369472	−	0.929242i	$-0.620461\pi$
$491$	−3.94743e6	−0.738943	−0.369472	+	0.929242i	$0.620461\pi$
$492$	0	0
$493$	−3.79099e6	−0.702481
$494$	0	0
$495$	4.90722e6	0.900166
$496$	0	0
$497$	9.37572e6	1.70260
$498$	0	0
$499$	−2.29772e6	−0.413091	−0.206545	−	0.978437i	$-0.566222\pi$
$499$	−2.29772e6	−0.413091	−0.206545	+	0.978437i	$0.566222\pi$
$500$	0	0
$501$	1.78170e7	3.17133
$502$	0	0
$503$	−1.12279e6	−0.197870	−0.0989350	−	0.995094i	$-0.531544\pi$
$503$	−1.12279e6	−0.197870	−0.0989350	+	0.995094i	$0.531544\pi$
$504$	0	0
$505$	3.32133e6	0.579540
$506$	0	0
$507$	7.29158e6	1.25980
$508$	0	0
$509$	928969.	0.158930	0.0794651	−	0.996838i	$-0.474679\pi$
$509$	928969.	0.158930	0.0794651	+	0.996838i	$0.474679\pi$
$510$	0	0
$511$	−1.00608e6	−0.170443
$512$	0	0
$513$	−5.14425e6	−0.863035
$514$	0	0
$515$	3.75054e6	0.623126
$516$	0	0
$517$	5.65007e6	0.929668
$518$	0	0
$519$	−115704.	−0.0188552
$520$	0	0
$521$	−784844.	−0.126674	−0.0633372	−	0.997992i	$-0.520174\pi$
$521$	−784844.	−0.126674	−0.0633372	+	0.997992i	$0.520174\pi$
$522$	0	0
$523$	−3.44049e6	−0.550005	−0.275002	−	0.961444i	$-0.588679\pi$
$523$	−3.44049e6	−0.550005	−0.275002	+	0.961444i	$0.588679\pi$
$524$	0	0
$525$	−9.39357e6	−1.48742
$526$	0	0
$527$	−4.04829e6	−0.634958
$528$	0	0
$529$	−2.23853e6	−0.347795
$530$	0	0
$531$	1.84223e7	2.83535
$532$	0	0
$533$	−419226.	−0.0639190
$534$	0	0
$535$	3.80323e6	0.574471
$536$	0	0
$537$	−3.25187e6	−0.486628
$538$	0	0
$539$	−1.96515e6	−0.291356
$540$	0	0
$541$	6.36278e6	0.934661	0.467330	−	0.884083i	$-0.345216\pi$
$541$	6.36278e6	0.934661	0.467330	+	0.884083i	$0.345216\pi$
$542$	0	0
$543$	−1.16816e7	−1.70021
$544$	0	0
$545$	4.90111e6	0.706811
$546$	0	0
$547$	1.00599e7	1.43756	0.718782	−	0.695236i	$-0.244700\pi$
$547$	1.00599e7	1.43756	0.718782	+	0.695236i	$0.244700\pi$
$548$	0	0
$549$	4.10064e6	0.580658
$550$	0	0
$551$	−2.60041e6	−0.364891
$552$	0	0
$553$	6.78368e6	0.943305
$554$	0	0
$555$	−3.53670e6	−0.487378
$556$	0	0
$557$	−3.36701e6	−0.459840	−0.229920	−	0.973210i	$-0.573846\pi$
$557$	−3.36701e6	−0.459840	−0.229920	+	0.973210i	$0.573846\pi$
$558$	0	0
$559$	−5.71479e6	−0.773519
$560$	0	0
$561$	1.18028e7	1.58335
$562$	0	0
$563$	1.07483e6	0.142912	0.0714560	−	0.997444i	$-0.477235\pi$
$563$	1.07483e6	0.142912	0.0714560	+	0.997444i	$0.477235\pi$
$564$	0	0
$565$	111254.	0.0146620
$566$	0	0
$567$	−7.90112e6	−1.03212
$568$	0	0
$569$	4.87154e6	0.630791	0.315396	−	0.948960i	$-0.397863\pi$
$569$	4.87154e6	0.630791	0.315396	+	0.948960i	$0.397863\pi$
$570$	0	0
$571$	432116.	0.0554639	0.0277320	−	0.999615i	$-0.491172\pi$
$571$	432116.	0.0554639	0.0277320	+	0.999615i	$0.491172\pi$
$572$	0	0
$573$	2.72543e6	0.346776
$574$	0	0
$575$	−4.81943e6	−0.607891
$576$	0	0
$577$	6.16019e6	0.770290	0.385145	−	0.922856i	$-0.374151\pi$
$577$	6.16019e6	0.770290	0.385145	+	0.922856i	$0.374151\pi$
$578$	0	0
$579$	−5.13495e6	−0.636561
$580$	0	0
$581$	1.37142e7	1.68551
$582$	0	0
$583$	−4.81235e6	−0.586389
$584$	0	0
$585$	−4.19353e6	−0.506629
$586$	0	0
$587$	−1.46377e7	−1.75338	−0.876690	−	0.481055i	$-0.840254\pi$
$587$	−1.46377e7	−1.75338	−0.876690	+	0.481055i	$0.840254\pi$
$588$	0	0
$589$	−2.77691e6	−0.329817
$590$	0	0
$591$	−8.64858e6	−1.01854
$592$	0	0
$593$	9.55427e6	1.11573	0.557867	−	0.829930i	$-0.311620\pi$
$593$	9.55427e6	1.11573	0.557867	+	0.829930i	$0.311620\pi$
$594$	0	0
$595$	4.91989e6	0.569721
$596$	0	0
$597$	1.27476e7	1.46383
$598$	0	0
$599$	7.92396e6	0.902350	0.451175	−	0.892436i	$-0.351005\pi$
$599$	7.92396e6	0.902350	0.451175	+	0.892436i	$0.351005\pi$
$600$	0	0
$601$	1.06645e7	1.20435	0.602177	−	0.798362i	$-0.294300\pi$
$601$	1.06645e7	1.20435	0.602177	+	0.798362i	$0.294300\pi$
$602$	0	0
$603$	7.55216e6	0.845820
$604$	0	0
$605$	681943.	0.0757460
$606$	0	0
$607$	−8.51365e6	−0.937873	−0.468937	−	0.883232i	$-0.655363\pi$
$607$	−8.51365e6	−0.937873	−0.468937	+	0.883232i	$0.655363\pi$
$608$	0	0
$609$	−1.27236e7	−1.39017
$610$	0	0
$611$	−4.82835e6	−0.523234
$612$	0	0
$613$	−468573.	−0.0503647	−0.0251824	−	0.999683i	$-0.508017\pi$
$613$	−468573.	−0.0503647	−0.0251824	+	0.999683i	$0.508017\pi$
$614$	0	0
$615$	−990887.	−0.105642
$616$	0	0
$617$	1.35050e7	1.42818	0.714088	−	0.700056i	$-0.246842\pi$
$617$	1.35050e7	1.42818	0.714088	+	0.700056i	$0.246842\pi$
$618$	0	0
$619$	−6.16135e6	−0.646323	−0.323161	−	0.946344i	$-0.604746\pi$
$619$	−6.16135e6	−0.646323	−0.323161	+	0.946344i	$0.604746\pi$
$620$	0	0
$621$	−1.29138e7	−1.34377
$622$	0	0
$623$	−1.41850e7	−1.46423
$624$	0	0
$625$	3.11826e6	0.319310
$626$	0	0
$627$	8.09607e6	0.822442
$628$	0	0
$629$	−5.63858e6	−0.568255
$630$	0	0
$631$	−6.87881e6	−0.687765	−0.343882	−	0.939013i	$-0.611742\pi$
$631$	−6.87881e6	−0.687765	−0.343882	+	0.939013i	$0.611742\pi$
$632$	0	0
$633$	3.02633e7	3.00198
$634$	0	0
$635$	4.55794e6	0.448574
$636$	0	0
$637$	1.67935e6	0.163981
$638$	0	0
$639$	−3.01146e7	−2.91759
$640$	0	0
$641$	−2.66693e6	−0.256370	−0.128185	−	0.991750i	$-0.540915\pi$
$641$	−2.66693e6	−0.256370	−0.128185	+	0.991750i	$0.540915\pi$
$642$	0	0
$643$	1.41144e7	1.34628	0.673141	−	0.739515i	$-0.264945\pi$
$643$	1.41144e7	1.34628	0.673141	+	0.739515i	$0.264945\pi$
$644$	0	0
$645$	−1.35075e7	−1.27843
$646$	0	0
$647$	−133839.	−0.0125696	−0.00628481	−	0.999980i	$-0.502001\pi$
$647$	−133839.	−0.0125696	−0.00628481	+	0.999980i	$0.502001\pi$
$648$	0	0
$649$	−1.42469e7	−1.32772
$650$	0	0
$651$	−1.35872e7	−1.25654
$652$	0	0
$653$	−6.66097e6	−0.611300	−0.305650	−	0.952144i	$-0.598874\pi$
$653$	−6.66097e6	−0.611300	−0.305650	+	0.952144i	$0.598874\pi$
$654$	0	0
$655$	−949882.	−0.0865100
$656$	0	0
$657$	3.23150e6	0.292072
$658$	0	0
$659$	6.44322e6	0.577949	0.288974	−	0.957337i	$-0.406686\pi$
$659$	6.44322e6	0.577949	0.288974	+	0.957337i	$0.406686\pi$
$660$	0	0
$661$	−2.85509e6	−0.254165	−0.127083	−	0.991892i	$-0.540561\pi$
$661$	−2.85509e6	−0.254165	−0.127083	+	0.991892i	$0.540561\pi$
$662$	0	0
$663$	−1.00862e7	−0.891138
$664$	0	0
$665$	3.37478e6	0.295931
$666$	0	0
$667$	−6.52792e6	−0.568147
$668$	0	0
$669$	−1.77687e7	−1.53494
$670$	0	0
$671$	−3.17123e6	−0.271908
$672$	0	0
$673$	6.65808e6	0.566646	0.283323	−	0.959025i	$-0.408563\pi$
$673$	6.65808e6	0.566646	0.283323	+	0.959025i	$0.408563\pi$
$674$	0	0
$675$	1.48261e7	1.25247
$676$	0	0
$677$	1.45486e6	0.121997	0.0609986	−	0.998138i	$-0.480571\pi$
$677$	1.45486e6	0.121997	0.0609986	+	0.998138i	$0.480571\pi$
$678$	0	0
$679$	2.14951e7	1.78922
$680$	0	0
$681$	1.68416e7	1.39160
$682$	0	0
$683$	1.16908e7	0.958939	0.479469	−	0.877559i	$-0.340829\pi$
$683$	1.16908e7	0.958939	0.479469	+	0.877559i	$0.340829\pi$
$684$	0	0
$685$	−8.00371e6	−0.651726
$686$	0	0
$687$	−3.36473e7	−2.71993
$688$	0	0
$689$	4.11246e6	0.330030
$690$	0	0
$691$	−1.04433e7	−0.832041	−0.416020	−	0.909355i	$-0.636576\pi$
$691$	−1.04433e7	−0.832041	−0.416020	+	0.909355i	$0.636576\pi$
$692$	0	0
$693$	2.62582e7	2.07698
$694$	0	0
$695$	279537.	0.0219522
$696$	0	0
$697$	−1.57977e6	−0.123172
$698$	0	0
$699$	1.02752e7	0.795420
$700$	0	0
$701$	1.37741e7	1.05869	0.529345	−	0.848407i	$-0.322438\pi$
$701$	1.37741e7	1.05869	0.529345	+	0.848407i	$0.322438\pi$
$702$	0	0
$703$	−3.86776e6	−0.295170
$704$	0	0
$705$	−1.14123e7	−0.864772
$706$	0	0
$707$	1.77722e7	1.33719
$708$	0	0
$709$	−5.24895e6	−0.392154	−0.196077	−	0.980588i	$-0.562820\pi$
$709$	−5.24895e6	−0.392154	−0.196077	+	0.980588i	$0.562820\pi$
$710$	0	0
$711$	−2.17890e7	−1.61645
$712$	0	0
$713$	−6.97099e6	−0.513536
$714$	0	0
$715$	3.24307e6	0.237242
$716$	0	0
$717$	−3.40929e7	−2.47665
$718$	0	0
$719$	−1.17460e7	−0.847363	−0.423682	−	0.905811i	$-0.639262\pi$
$719$	−1.17460e7	−0.847363	−0.423682	+	0.905811i	$0.639262\pi$
$720$	0	0
$721$	2.00689e7	1.43775
$722$	0	0
$723$	−2.30256e7	−1.63820
$724$	0	0
$725$	7.49458e6	0.529545
$726$	0	0
$727$	1.88496e7	1.32271	0.661357	−	0.750071i	$-0.269981\pi$
$727$	1.88496e7	1.32271	0.661357	+	0.750071i	$0.269981\pi$
$728$	0	0
$729$	−1.57414e7	−1.09704
$730$	0	0
$731$	−2.15351e7	−1.49058
$732$	0	0
$733$	−1.57924e7	−1.08564	−0.542821	−	0.839848i	$-0.682644\pi$
$733$	−1.57924e7	−1.08564	−0.542821	+	0.839848i	$0.682644\pi$
$734$	0	0
$735$	3.96933e6	0.271018
$736$	0	0
$737$	−5.84047e6	−0.396077
$738$	0	0
$739$	−2.73899e7	−1.84493	−0.922463	−	0.386085i	$-0.873827\pi$
$739$	−2.73899e7	−1.84493	−0.922463	+	0.386085i	$0.873827\pi$
$740$	0	0
$741$	−6.91861e6	−0.462885
$742$	0	0
$743$	2.33182e7	1.54962	0.774808	−	0.632197i	$-0.217847\pi$
$743$	2.33182e7	1.54962	0.774808	+	0.632197i	$0.217847\pi$
$744$	0	0
$745$	−38447.9	−0.00253794
$746$	0	0
$747$	−4.40498e7	−2.88830
$748$	0	0
$749$	2.03508e7	1.32549
$750$	0	0
$751$	3.61326e6	0.233776	0.116888	−	0.993145i	$-0.462708\pi$
$751$	3.61326e6	0.233776	0.116888	+	0.993145i	$0.462708\pi$
$752$	0	0
$753$	6.78428e6	0.436030
$754$	0	0
$755$	−1.32048e6	−0.0843074
$756$	0	0
$757$	2.24648e6	0.142483	0.0712415	−	0.997459i	$-0.477304\pi$
$757$	2.24648e6	0.142483	0.0712415	+	0.997459i	$0.477304\pi$
$758$	0	0
$759$	2.03239e7	1.28057
$760$	0	0
$761$	6.10805e6	0.382332	0.191166	−	0.981558i	$-0.438773\pi$
$761$	6.10805e6	0.382332	0.191166	+	0.981558i	$0.438773\pi$
$762$	0	0
$763$	2.62255e7	1.63084
$764$	0	0
$765$	−1.58026e7	−0.976278
$766$	0	0
$767$	1.21749e7	0.747267
$768$	0	0
$769$	−1.78303e7	−1.08728	−0.543641	−	0.839318i	$-0.682955\pi$
$769$	−1.78303e7	−1.08728	−0.543641	+	0.839318i	$0.682955\pi$
$770$	0	0
$771$	1.77323e6	0.107431
$772$	0	0
$773$	−1.02094e7	−0.614544	−0.307272	−	0.951622i	$-0.599416\pi$
$773$	−1.02094e7	−0.614544	−0.307272	+	0.951622i	$0.599416\pi$
$774$	0	0
$775$	8.00326e6	0.478644
$776$	0	0
$777$	−1.89247e7	−1.12454
$778$	0	0
$779$	−1.08364e6	−0.0639796
$780$	0	0
$781$	2.32892e7	1.36624
$782$	0	0
$783$	2.00820e7	1.17058
$784$	0	0
$785$	−1.23992e7	−0.718157
$786$	0	0
$787$	2.80624e7	1.61506	0.807530	−	0.589826i	$-0.200804\pi$
$787$	2.80624e7	1.61506	0.807530	+	0.589826i	$0.200804\pi$
$788$	0	0
$789$	−2.47697e7	−1.41654
$790$	0	0
$791$	595312.	0.0338301
$792$	0	0
$793$	2.71002e6	0.153035
$794$	0	0
$795$	9.72026e6	0.545456
$796$	0	0
$797$	−2.93306e7	−1.63559	−0.817796	−	0.575509i	$-0.804804\pi$
$797$	−2.93306e7	−1.63559	−0.817796	+	0.575509i	$0.804804\pi$
$798$	0	0
$799$	−1.81947e7	−1.00827
$800$	0	0
$801$	4.55617e7	2.50911
$802$	0	0
$803$	−2.49908e6	−0.136770
$804$	0	0
$805$	8.47185e6	0.460774
$806$	0	0
$807$	3.27269e7	1.76897
$808$	0	0
$809$	2.24344e7	1.20516	0.602579	−	0.798059i	$-0.294140\pi$
$809$	2.24344e7	1.20516	0.602579	+	0.798059i	$0.294140\pi$
$810$	0	0
$811$	2.55275e7	1.36287	0.681437	−	0.731877i	$-0.261355\pi$
$811$	2.55275e7	1.36287	0.681437	+	0.731877i	$0.261355\pi$
$812$	0	0
$813$	5.49029e6	0.291319
$814$	0	0
$815$	−1.38000e7	−0.727755
$816$	0	0
$817$	−1.47719e7	−0.774252
$818$	0	0
$819$	−2.24393e7	−1.16896
$820$	0	0
$821$	2.57101e7	1.33121	0.665604	−	0.746305i	$-0.268174\pi$
$821$	2.57101e7	1.33121	0.665604	+	0.746305i	$0.268174\pi$
$822$	0	0
$823$	−2.18552e7	−1.12475	−0.562375	−	0.826883i	$-0.690112\pi$
$823$	−2.18552e7	−1.12475	−0.562375	+	0.826883i	$0.690112\pi$
$824$	0	0
$825$	−2.33335e7	−1.19356
$826$	0	0
$827$	1.98782e7	1.01068	0.505338	−	0.862921i	$-0.331368\pi$
$827$	1.98782e7	1.01068	0.505338	+	0.862921i	$0.331368\pi$
$828$	0	0
$829$	1.82650e7	0.923066	0.461533	−	0.887123i	$-0.347300\pi$
$829$	1.82650e7	0.923066	0.461533	+	0.887123i	$0.347300\pi$
$830$	0	0
$831$	1.56659e7	0.786961
$832$	0	0
$833$	6.32831e6	0.315992
$834$	0	0
$835$	1.84483e7	0.915670
$836$	0	0
$837$	2.14450e7	1.05807
$838$	0	0
$839$	−1.80126e7	−0.883430	−0.441715	−	0.897155i	$-0.645630\pi$
$839$	−1.80126e7	−0.883430	−0.441715	+	0.897155i	$0.645630\pi$
$840$	0	0
$841$	−1.03597e7	−0.505078
$842$	0	0
$843$	1.79293e7	0.868950
$844$	0	0
$845$	7.54991e6	0.363747
$846$	0	0
$847$	3.64903e6	0.174771
$848$	0	0
$849$	−3.57438e7	−1.70189
$850$	0	0
$851$	−9.70941e6	−0.459588
$852$	0	0
$853$	2.11484e7	0.995187	0.497594	−	0.867410i	$-0.334217\pi$
$853$	2.11484e7	0.995187	0.497594	+	0.867410i	$0.334217\pi$
$854$	0	0
$855$	−1.08397e7	−0.507110
$856$	0	0
$857$	−1.41444e7	−0.657857	−0.328928	−	0.944355i	$-0.606688\pi$
$857$	−1.41444e7	−0.657857	−0.328928	+	0.944355i	$0.606688\pi$
$858$	0	0
$859$	−1.75220e6	−0.0810215	−0.0405108	−	0.999179i	$-0.512899\pi$
$859$	−1.75220e6	−0.0810215	−0.0405108	+	0.999179i	$0.512899\pi$
$860$	0	0
$861$	−5.30216e6	−0.243750
$862$	0	0
$863$	9.93037e6	0.453877	0.226939	−	0.973909i	$-0.427128\pi$
$863$	9.93037e6	0.453877	0.226939	+	0.973909i	$0.427128\pi$
$864$	0	0
$865$	−119803.	−0.00544414
$866$	0	0
$867$	111089.	0.00501905
$868$	0	0
$869$	1.68506e7	0.756946
$870$	0	0
$871$	4.99106e6	0.222919
$872$	0	0
$873$	−6.90417e7	−3.06602
$874$	0	0
$875$	−2.26480e7	−1.00002
$876$	0	0
$877$	3.97296e7	1.74428	0.872139	−	0.489258i	$-0.162733\pi$
$877$	3.97296e7	1.74428	0.872139	+	0.489258i	$0.162733\pi$
$878$	0	0
$879$	−1.96616e7	−0.858317
$880$	0	0
$881$	−3.49658e7	−1.51776	−0.758881	−	0.651230i	$-0.774253\pi$
$881$	−3.49658e7	−1.51776	−0.758881	+	0.651230i	$0.774253\pi$
$882$	0	0
$883$	1.74262e6	0.0752144	0.0376072	−	0.999293i	$-0.488026\pi$
$883$	1.74262e6	0.0752144	0.0376072	+	0.999293i	$0.488026\pi$
$884$	0	0
$885$	2.87766e7	1.23504
$886$	0	0
$887$	1.47801e6	0.0630765	0.0315383	−	0.999503i	$-0.489959\pi$
$887$	1.47801e6	0.0630765	0.0315383	+	0.999503i	$0.489959\pi$
$888$	0	0
$889$	2.43892e7	1.03501
$890$	0	0
$891$	−1.96263e7	−0.828216
$892$	0	0
$893$	−1.24806e7	−0.523730
$894$	0	0
$895$	−3.36707e6	−0.140506
$896$	0	0
$897$	−1.73681e7	−0.720727
$898$	0	0
$899$	1.08404e7	0.447350
$900$	0	0
$901$	1.54970e7	0.635971
$902$	0	0
$903$	−7.22779e7	−2.94976
$904$	0	0
$905$	−1.20955e7	−0.490909
$906$	0	0
$907$	−2.11285e7	−0.852806	−0.426403	−	0.904533i	$-0.640219\pi$
$907$	−2.11285e7	−0.852806	−0.426403	+	0.904533i	$0.640219\pi$
$908$	0	0
$909$	−5.70838e7	−2.29141
$910$	0	0
$911$	3.85191e7	1.53773	0.768865	−	0.639411i	$-0.220822\pi$
$911$	3.85191e7	1.53773	0.768865	+	0.639411i	$0.220822\pi$
$912$	0	0
$913$	3.40659e7	1.35252
$914$	0	0
$915$	6.40543e6	0.252927
$916$	0	0
$917$	−5.08275e6	−0.199607
$918$	0	0
$919$	1.69364e7	0.661502	0.330751	−	0.943718i	$-0.392698\pi$
$919$	1.69364e7	0.661502	0.330751	+	0.943718i	$0.392698\pi$
$920$	0	0
$921$	−1.96243e7	−0.762335
$922$	0	0
$923$	−1.99021e7	−0.768943
$924$	0	0
$925$	1.11472e7	0.428362
$926$	0	0
$927$	−6.44607e7	−2.46374
$928$	0	0
$929$	2.41586e6	0.0918403	0.0459201	−	0.998945i	$-0.485378\pi$
$929$	2.41586e6	0.0918403	0.0459201	+	0.998945i	$0.485378\pi$
$930$	0	0
$931$	4.34088e6	0.164136
$932$	0	0
$933$	2.17575e7	0.818285
$934$	0	0
$935$	1.22209e7	0.457167
$936$	0	0
$937$	−3.90586e7	−1.45334	−0.726671	−	0.686986i	$-0.758934\pi$
$937$	−3.90586e7	−1.45334	−0.726671	+	0.686986i	$0.758934\pi$
$938$	0	0
$939$	1.11690e7	0.413382
$940$	0	0
$941$	−3.60851e7	−1.32848	−0.664238	−	0.747521i	$-0.731244\pi$
$941$	−3.60851e7	−1.32848	−0.664238	+	0.747521i	$0.731244\pi$
$942$	0	0
$943$	−2.72031e6	−0.0996183
$944$	0	0
$945$	−2.60621e7	−0.949359
$946$	0	0
$947$	−1.60128e7	−0.580218	−0.290109	−	0.956994i	$-0.593692\pi$
$947$	−1.60128e7	−0.580218	−0.290109	+	0.956994i	$0.593692\pi$
$948$	0	0
$949$	2.13563e6	0.0769768
$950$	0	0
$951$	5.39085e7	1.93288
$952$	0	0
$953$	−7.23768e6	−0.258147	−0.129073	−	0.991635i	$-0.541200\pi$
$953$	−7.23768e6	−0.258147	−0.129073	+	0.991635i	$0.541200\pi$
$954$	0	0
$955$	2.82199e6	0.100126
$956$	0	0
$957$	−3.16052e7	−1.11552
$958$	0	0
$959$	−4.28273e7	−1.50374
$960$	0	0
$961$	−1.70530e7	−0.595651
$962$	0	0
$963$	−6.53662e7	−2.27137
$964$	0	0
$965$	−5.31687e6	−0.183797
$966$	0	0
$967$	2.26179e7	0.777833	0.388917	−	0.921273i	$-0.372849\pi$
$967$	2.26179e7	0.777833	0.388917	+	0.921273i	$0.372849\pi$
$968$	0	0
$969$	−2.60715e7	−0.891983
$970$	0	0
$971$	−3.91700e7	−1.33323	−0.666615	−	0.745402i	$-0.732257\pi$
$971$	−3.91700e7	−1.33323	−0.666615	+	0.745402i	$0.732257\pi$
$972$	0	0
$973$	1.49578e6	0.0506508
$974$	0	0
$975$	1.99400e7	0.671758
$976$	0	0
$977$	7.07036e6	0.236976	0.118488	−	0.992955i	$-0.462195\pi$
$977$	7.07036e6	0.236976	0.118488	+	0.992955i	$0.462195\pi$
$978$	0	0
$979$	−3.52352e7	−1.17495
$980$	0	0
$981$	−8.42356e7	−2.79462
$982$	0	0
$983$	7.10901e6	0.234653	0.117326	−	0.993093i	$-0.462568\pi$
$983$	7.10901e6	0.234653	0.117326	+	0.993093i	$0.462568\pi$
$984$	0	0
$985$	−8.95499e6	−0.294086
$986$	0	0
$987$	−6.10666e7	−1.99531
$988$	0	0
$989$	−3.70826e7	−1.20554
$990$	0	0
$991$	−1.30548e7	−0.422267	−0.211133	−	0.977457i	$-0.567715\pi$
$991$	−1.30548e7	−0.422267	−0.211133	+	0.977457i	$0.567715\pi$
$992$	0	0
$993$	5.35536e7	1.72352
$994$	0	0
$995$	1.31992e7	0.422659
$996$	0	0
$997$	1.24734e7	0.397419	0.198709	−	0.980058i	$-0.436325\pi$
$997$	1.24734e7	0.397419	0.198709	+	0.980058i	$0.436325\pi$
$998$	0	0
$999$	2.98693e7	0.946916

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1028.6.a.b.1.3	✓	57

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1028.6.a.b.1.3	✓	57	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:	\( N \)	\(=\)	\( 1028 = 2^{2} \cdot 257 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1028.a (trivial)

\(f(q)\)	\(=\)	\(q-26.8472 q^{3} -27.7983 q^{5} -148.747 q^{7} +477.771 q^{9} +O(q^{10})\)	\(q-26.8472 q^{3} -27.7983 q^{5} -148.747 q^{7} +477.771 q^{9} -369.485 q^{11} +315.749 q^{13} +746.307 q^{15} +1189.84 q^{17} +816.166 q^{19} +3993.43 q^{21} +2048.86 q^{23} -2352.25 q^{25} -6302.95 q^{27} -3186.13 q^{29} -3402.38 q^{31} +9919.63 q^{33} +4134.91 q^{35} -4738.94 q^{37} -8476.96 q^{39} -1327.72 q^{41} -18099.2 q^{43} -13281.2 q^{45} -15291.8 q^{47} +5318.63 q^{49} -31943.9 q^{51} +13024.5 q^{53} +10271.1 q^{55} -21911.8 q^{57} +38558.7 q^{59} +8582.84 q^{61} -71067.0 q^{63} -8777.28 q^{65} +15807.1 q^{67} -55006.0 q^{69} -63031.4 q^{71} +6763.70 q^{73} +63151.4 q^{75} +54959.7 q^{77} -45605.5 q^{79} +53117.9 q^{81} -92198.4 q^{83} -33075.6 q^{85} +85538.6 q^{87} +95363.1 q^{89} -46966.6 q^{91} +91344.3 q^{93} -22688.1 q^{95} -144508. q^{97} -176529. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 57 q + 27 q^{3} + 61 q^{5} + 88 q^{7} + 5374 q^{9}+O(q^{10}) \) 57 * q + 27 * q^3 + 61 * q^5 + 88 * q^7 + 5374 * q^9	\( 57 q + 27 q^{3} + 61 q^{5} + 88 q^{7} + 5374 q^{9} + 484 q^{11} + 3162 q^{13} - 244 q^{15} + 3298 q^{17} + 2931 q^{19} + 5009 q^{21} + 3712 q^{23} + 48568 q^{25} + 8682 q^{27} + 10909 q^{29} - 3539 q^{31} + 13277 q^{33} + 13976 q^{35} + 42395 q^{37} - 14986 q^{39} - 8469 q^{41} + 69808 q^{43} + 26347 q^{45} + 12410 q^{47} + 203347 q^{49} + 24220 q^{51} + 49353 q^{53} - 34265 q^{55} + 98112 q^{57} - 9792 q^{59} + 157032 q^{61} + 24981 q^{63} + 16143 q^{65} + 20201 q^{67} - 113743 q^{69} - 229673 q^{71} - 19561 q^{73} - 482061 q^{75} + 21204 q^{77} - 10925 q^{79} + 323569 q^{81} - 62846 q^{83} + 218157 q^{85} - 169628 q^{87} + 69901 q^{89} + 56581 q^{91} + 177641 q^{93} - 192910 q^{95} + 277990 q^{97} + 424882 q^{99}+O(q^{100}) \) 57 * q + 27 * q^3 + 61 * q^5 + 88 * q^7 + 5374 * q^9 + 484 * q^11 + 3162 * q^13 - 244 * q^15 + 3298 * q^17 + 2931 * q^19 + 5009 * q^21 + 3712 * q^23 + 48568 * q^25 + 8682 * q^27 + 10909 * q^29 - 3539 * q^31 + 13277 * q^33 + 13976 * q^35 + 42395 * q^37 - 14986 * q^39 - 8469 * q^41 + 69808 * q^43 + 26347 * q^45 + 12410 * q^47 + 203347 * q^49 + 24220 * q^51 + 49353 * q^53 - 34265 * q^55 + 98112 * q^57 - 9792 * q^59 + 157032 * q^61 + 24981 * q^63 + 16143 * q^65 + 20201 * q^67 - 113743 * q^69 - 229673 * q^71 - 19561 * q^73 - 482061 * q^75 + 21204 * q^77 - 10925 * q^79 + 323569 * q^81 - 62846 * q^83 + 218157 * q^85 - 169628 * q^87 + 69901 * q^89 + 56581 * q^91 + 177641 * q^93 - 192910 * q^95 + 277990 * q^97 + 424882 * q^99

Introduction

L-functions

Modular forms

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Database

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