LMFDB - Embedded newform 1028.6.a.b.1.9

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			1.9
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)

$f(q)$	$=$	$q-21.1395 q^{3} -58.6028 q^{5} +77.5956 q^{7} +203.880 q^{9} +O(q^{10})$	$q-21.1395 q^{3} -58.6028 q^{5} +77.5956 q^{7} +203.880 q^{9} +500.949 q^{11} +90.2740 q^{13} +1238.84 q^{15} +953.668 q^{17} +2731.10 q^{19} -1640.33 q^{21} +1245.64 q^{23} +309.287 q^{25} +826.979 q^{27} +2636.41 q^{29} +8037.67 q^{31} -10589.8 q^{33} -4547.32 q^{35} -2603.17 q^{37} -1908.35 q^{39} +20110.5 q^{41} +7215.98 q^{43} -11947.9 q^{45} -5353.35 q^{47} -10785.9 q^{49} -20160.1 q^{51} -10614.0 q^{53} -29357.0 q^{55} -57734.1 q^{57} +15677.9 q^{59} -21093.1 q^{61} +15820.2 q^{63} -5290.31 q^{65} +65628.5 q^{67} -26332.2 q^{69} +12892.9 q^{71} -34837.7 q^{73} -6538.18 q^{75} +38871.4 q^{77} -32555.6 q^{79} -67024.8 q^{81} -44683.5 q^{83} -55887.6 q^{85} -55732.4 q^{87} -6084.38 q^{89} +7004.86 q^{91} -169913. q^{93} -160050. q^{95} +107033. q^{97} +102134. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−21.1395	−1.35610	−0.678051	−	0.735015i	$-0.737175\pi$
$3$	−21.1395	−1.35610	−0.678051	+	0.735015i	$0.737175\pi$
$4$	0	0
$5$	−58.6028	−1.04832	−0.524159	−	0.851620i	$-0.675620\pi$
$5$	−58.6028	−1.04832	−0.524159	+	0.851620i	$0.675620\pi$
$6$	0	0
$7$	77.5956	0.598538	0.299269	−	0.954169i	$-0.403257\pi$
$7$	77.5956	0.598538	0.299269	+	0.954169i	$0.403257\pi$
$8$	0	0
$9$	203.880	0.839012
$10$	0	0
$11$	500.949	1.24828	0.624140	−	0.781313i	$-0.285449\pi$
$11$	500.949	1.24828	0.624140	+	0.781313i	$0.285449\pi$
$12$	0	0
$13$	90.2740	0.148151	0.0740754	−	0.997253i	$-0.476399\pi$
$13$	90.2740	0.148151	0.0740754	+	0.997253i	$0.476399\pi$
$14$	0	0
$15$	1238.84	1.42163
$16$	0	0
$17$	953.668	0.800340	0.400170	−	0.916441i	$-0.368951\pi$
$17$	953.668	0.800340	0.400170	+	0.916441i	$0.368951\pi$
$18$	0	0
$19$	2731.10	1.73561	0.867807	−	0.496901i	$-0.165529\pi$
$19$	2731.10	1.73561	0.867807	+	0.496901i	$0.165529\pi$
$20$	0	0
$21$	−1640.33	−0.811679
$22$	0	0
$23$	1245.64	0.490990	0.245495	−	0.969398i	$-0.421050\pi$
$23$	1245.64	0.490990	0.245495	+	0.969398i	$0.421050\pi$
$24$	0	0
$25$	309.287	0.0989718
$26$	0	0
$27$	826.979	0.218316
$28$	0	0
$29$	2636.41	0.582127	0.291063	−	0.956704i	$-0.405991\pi$
$29$	2636.41	0.582127	0.291063	+	0.956704i	$0.405991\pi$
$30$	0	0
$31$	8037.67	1.50219	0.751097	−	0.660192i	$-0.229525\pi$
$31$	8037.67	1.50219	0.751097	+	0.660192i	$0.229525\pi$
$32$	0	0
$33$	−10589.8	−1.69279
$34$	0	0
$35$	−4547.32	−0.627459
$36$	0	0
$37$	−2603.17	−0.312607	−0.156304	−	0.987709i	$-0.549958\pi$
$37$	−2603.17	−0.312607	−0.156304	+	0.987709i	$0.549958\pi$
$38$	0	0
$39$	−1908.35	−0.200908
$40$	0	0
$41$	20110.5	1.86837	0.934185	−	0.356788i	$-0.116128\pi$
$41$	20110.5	1.86837	0.934185	+	0.356788i	$0.116128\pi$
$42$	0	0
$43$	7215.98	0.595147	0.297574	−	0.954699i	$-0.403823\pi$
$43$	7215.98	0.595147	0.297574	+	0.954699i	$0.403823\pi$
$44$	0	0
$45$	−11947.9	−0.879552
$46$	0	0
$47$	−5353.35	−0.353493	−0.176747	−	0.984256i	$-0.556557\pi$
$47$	−5353.35	−0.353493	−0.176747	+	0.984256i	$0.556557\pi$
$48$	0	0
$49$	−10785.9	−0.641752
$50$	0	0
$51$	−20160.1	−1.08534
$52$	0	0
$53$	−10614.0	−0.519024	−0.259512	−	0.965740i	$-0.583562\pi$
$53$	−10614.0	−0.519024	−0.259512	+	0.965740i	$0.583562\pi$
$54$	0	0
$55$	−29357.0	−1.30860
$56$	0	0
$57$	−57734.1	−2.35367
$58$	0	0
$59$	15677.9	0.586350	0.293175	−	0.956059i	$-0.405288\pi$
$59$	15677.9	0.586350	0.293175	+	0.956059i	$0.405288\pi$
$60$	0	0
$61$	−21093.1	−0.725797	−0.362899	−	0.931829i	$-0.618213\pi$
$61$	−21093.1	−0.725797	−0.362899	+	0.931829i	$0.618213\pi$
$62$	0	0
$63$	15820.2	0.502181
$64$	0	0
$65$	−5290.31	−0.155309
$66$	0	0
$67$	65628.5	1.78610	0.893049	−	0.449959i	$-0.148562\pi$
$67$	65628.5	1.78610	0.893049	+	0.449959i	$0.148562\pi$
$68$	0	0
$69$	−26332.2	−0.665832
$70$	0	0
$71$	12892.9	0.303532	0.151766	−	0.988416i	$-0.451504\pi$
$71$	12892.9	0.303532	0.151766	+	0.988416i	$0.451504\pi$
$72$	0	0
$73$	−34837.7	−0.765142	−0.382571	−	0.923926i	$-0.624961\pi$
$73$	−34837.7	−0.765142	−0.382571	+	0.923926i	$0.624961\pi$
$74$	0	0
$75$	−6538.18	−0.134216
$76$	0	0
$77$	38871.4	0.747143
$78$	0	0
$79$	−32555.6	−0.586891	−0.293446	−	0.955976i	$-0.594802\pi$
$79$	−32555.6	−0.586891	−0.293446	+	0.955976i	$0.594802\pi$
$80$	0	0
$81$	−67024.8	−1.13507
$82$	0	0
$83$	−44683.5	−0.711954	−0.355977	−	0.934495i	$-0.615852\pi$
$83$	−44683.5	−0.711954	−0.355977	+	0.934495i	$0.615852\pi$
$84$	0	0
$85$	−55887.6	−0.839012
$86$	0	0
$87$	−55732.4	−0.789423
$88$	0	0
$89$	−6084.38	−0.0814219	−0.0407110	−	0.999171i	$-0.512962\pi$
$89$	−6084.38	−0.0814219	−0.0407110	+	0.999171i	$0.512962\pi$
$90$	0	0
$91$	7004.86	0.0886739
$92$	0	0
$93$	−169913.	−2.03713
$94$	0	0
$95$	−160050.	−1.81948
$96$	0	0
$97$	107033.	1.15501	0.577507	−	0.816386i	$-0.304026\pi$
$97$	107033.	1.15501	0.577507	+	0.816386i	$0.304026\pi$
$98$	0	0
$99$	102134.	1.04732
$100$	0	0
$101$	−88546.7	−0.863712	−0.431856	−	0.901943i	$-0.642141\pi$
$101$	−88546.7	−0.863712	−0.431856	+	0.901943i	$0.642141\pi$
$102$	0	0
$103$	200483.	1.86202	0.931010	−	0.364994i	$-0.118929\pi$
$103$	200483.	1.86202	0.931010	+	0.364994i	$0.118929\pi$
$104$	0	0
$105$	96128.2	0.850898
$106$	0	0
$107$	3204.05	0.0270545	0.0135273	−	0.999909i	$-0.495694\pi$
$107$	3204.05	0.0270545	0.0135273	+	0.999909i	$0.495694\pi$
$108$	0	0
$109$	−208960.	−1.68460	−0.842300	−	0.539008i	$-0.818799\pi$
$109$	−208960.	−1.68460	−0.842300	+	0.539008i	$0.818799\pi$
$110$	0	0
$111$	55029.9	0.423927
$112$	0	0
$113$	114308.	0.842129	0.421065	−	0.907031i	$-0.361657\pi$
$113$	114308.	0.842129	0.421065	+	0.907031i	$0.361657\pi$
$114$	0	0
$115$	−72997.9	−0.514714
$116$	0	0
$117$	18405.1	0.124300
$118$	0	0
$119$	74000.4	0.479034
$120$	0	0
$121$	89899.1	0.558203
$122$	0	0
$123$	−425126.	−2.53370
$124$	0	0
$125$	165009.	0.944565
$126$	0	0
$127$	37642.2	0.207093	0.103547	−	0.994625i	$-0.466981\pi$
$127$	37642.2	0.207093	0.103547	+	0.994625i	$0.466981\pi$
$128$	0	0
$129$	−152543.	−0.807080
$130$	0	0
$131$	−111892.	−0.569666	−0.284833	−	0.958577i	$-0.591938\pi$
$131$	−111892.	−0.569666	−0.284833	+	0.958577i	$0.591938\pi$
$132$	0	0
$133$	211921.	1.03883
$134$	0	0
$135$	−48463.3	−0.228865
$136$	0	0
$137$	−298300.	−1.35785	−0.678925	−	0.734208i	$-0.737554\pi$
$137$	−298300.	−1.35785	−0.678925	+	0.734208i	$0.737554\pi$
$138$	0	0
$139$	−58836.8	−0.258293	−0.129146	−	0.991626i	$-0.541224\pi$
$139$	−58836.8	−0.258293	−0.129146	+	0.991626i	$0.541224\pi$
$140$	0	0
$141$	113167.	0.479373
$142$	0	0
$143$	45222.7	0.184934
$144$	0	0
$145$	−154501.	−0.610254
$146$	0	0
$147$	228009.	0.870281
$148$	0	0
$149$	142721.	0.526650	0.263325	−	0.964707i	$-0.415181\pi$
$149$	142721.	0.526650	0.263325	+	0.964707i	$0.415181\pi$
$150$	0	0
$151$	−48685.4	−0.173763	−0.0868813	−	0.996219i	$-0.527690\pi$
$151$	−48685.4	−0.173763	−0.0868813	+	0.996219i	$0.527690\pi$
$152$	0	0
$153$	194434.	0.671495
$154$	0	0
$155$	−471030.	−1.57478
$156$	0	0
$157$	283189.	0.916910	0.458455	−	0.888718i	$-0.348403\pi$
$157$	283189.	0.916910	0.458455	+	0.888718i	$0.348403\pi$
$158$	0	0
$159$	224374.	0.703850
$160$	0	0
$161$	96656.1	0.293876
$162$	0	0
$163$	543230.	1.60145	0.800727	−	0.599029i	$-0.204447\pi$
$163$	543230.	1.60145	0.800727	+	0.599029i	$0.204447\pi$
$164$	0	0
$165$	620594.	1.77459
$166$	0	0
$167$	297909.	0.826593	0.413297	−	0.910596i	$-0.364377\pi$
$167$	297909.	0.826593	0.413297	+	0.910596i	$0.364377\pi$
$168$	0	0
$169$	−363144.	−0.978051
$170$	0	0
$171$	556816.	1.45620
$172$	0	0
$173$	115909.	0.294443	0.147221	−	0.989104i	$-0.452967\pi$
$173$	115909.	0.294443	0.147221	+	0.989104i	$0.452967\pi$
$174$	0	0
$175$	23999.3	0.0592384
$176$	0	0
$177$	−331423.	−0.795151
$178$	0	0
$179$	453920.	1.05888	0.529440	−	0.848347i	$-0.322402\pi$
$179$	453920.	1.05888	0.529440	+	0.848347i	$0.322402\pi$
$180$	0	0
$181$	380136.	0.862467	0.431234	−	0.902240i	$-0.358079\pi$
$181$	380136.	0.862467	0.431234	+	0.902240i	$0.358079\pi$
$182$	0	0
$183$	445898.	0.984255
$184$	0	0
$185$	152553.	0.327712
$186$	0	0
$187$	477739.	0.999049
$188$	0	0
$189$	64169.9	0.130670
$190$	0	0
$191$	−563745.	−1.11815	−0.559074	−	0.829117i	$-0.688843\pi$
$191$	−563745.	−1.11815	−0.559074	+	0.829117i	$0.688843\pi$
$192$	0	0
$193$	123886.	0.239403	0.119702	−	0.992810i	$-0.461806\pi$
$193$	123886.	0.239403	0.119702	+	0.992810i	$0.461806\pi$
$194$	0	0
$195$	111835.	0.210615
$196$	0	0
$197$	505819.	0.928601	0.464301	−	0.885678i	$-0.346306\pi$
$197$	505819.	0.928601	0.464301	+	0.885678i	$0.346306\pi$
$198$	0	0
$199$	−711232.	−1.27315	−0.636574	−	0.771216i	$-0.719649\pi$
$199$	−711232.	−1.27315	−0.636574	+	0.771216i	$0.719649\pi$
$200$	0	0
$201$	−1.38736e6	−2.42213
$202$	0	0
$203$	204574.	0.348425
$204$	0	0
$205$	−1.17853e6	−1.95865
$206$	0	0
$207$	253961.	0.411947
$208$	0	0
$209$	1.36814e6	2.16653
$210$	0	0
$211$	613702.	0.948968	0.474484	−	0.880264i	$-0.342635\pi$
$211$	613702.	0.948968	0.474484	+	0.880264i	$0.342635\pi$
$212$	0	0
$213$	−272550.	−0.411620
$214$	0	0
$215$	−422877.	−0.623904
$216$	0	0
$217$	623687.	0.899120
$218$	0	0
$219$	736452.	1.03761
$220$	0	0
$221$	86091.4	0.118571
$222$	0	0
$223$	−851783.	−1.14701	−0.573504	−	0.819203i	$-0.694416\pi$
$223$	−851783.	−1.14701	−0.573504	+	0.819203i	$0.694416\pi$
$224$	0	0
$225$	63057.4	0.0830386
$226$	0	0
$227$	−54753.9	−0.0705262	−0.0352631	−	0.999378i	$-0.511227\pi$
$227$	−54753.9	−0.0705262	−0.0352631	+	0.999378i	$0.511227\pi$
$228$	0	0
$229$	−604182.	−0.761341	−0.380670	−	0.924711i	$-0.624307\pi$
$229$	−604182.	−0.761341	−0.380670	+	0.924711i	$0.624307\pi$
$230$	0	0
$231$	−821724.	−1.01320
$232$	0	0
$233$	1.34395e6	1.62179	0.810895	−	0.585192i	$-0.198981\pi$
$233$	1.34395e6	1.62179	0.810895	+	0.585192i	$0.198981\pi$
$234$	0	0
$235$	313721.	0.370573
$236$	0	0
$237$	688210.	0.795884
$238$	0	0
$239$	−1.06135e6	−1.20189	−0.600944	−	0.799291i	$-0.705209\pi$
$239$	−1.06135e6	−1.20189	−0.600944	+	0.799291i	$0.705209\pi$
$240$	0	0
$241$	457925.	0.507869	0.253935	−	0.967221i	$-0.418275\pi$
$241$	457925.	0.507869	0.253935	+	0.967221i	$0.418275\pi$
$242$	0	0
$243$	1.21592e6	1.32096
$244$	0	0
$245$	632085.	0.672760
$246$	0	0
$247$	246547.	0.257133
$248$	0	0
$249$	944588.	0.965482
$250$	0	0
$251$	−1.07644e6	−1.07847	−0.539233	−	0.842157i	$-0.681286\pi$
$251$	−1.07644e6	−1.07847	−0.539233	+	0.842157i	$0.681286\pi$
$252$	0	0
$253$	624002.	0.612893
$254$	0	0
$255$	1.18144e6	1.13779
$256$	0	0
$257$	−66049.0	−0.0623783
$257$	−66049.0	−0.0623783
$258$	0	0
$259$	−201995.	−0.187107
$260$	0	0
$261$	537510.	0.488411
$262$	0	0
$263$	291813.	0.260145	0.130072	−	0.991504i	$-0.458479\pi$
$263$	291813.	0.260145	0.130072	+	0.991504i	$0.458479\pi$
$264$	0	0
$265$	622008.	0.544103
$266$	0	0
$267$	128621.	0.110416
$268$	0	0
$269$	252814.	0.213020	0.106510	−	0.994312i	$-0.466032\pi$
$269$	252814.	0.213020	0.106510	+	0.994312i	$0.466032\pi$
$270$	0	0
$271$	1.68279e6	1.39189	0.695946	−	0.718094i	$-0.254985\pi$
$271$	1.68279e6	1.39189	0.695946	+	0.718094i	$0.254985\pi$
$272$	0	0
$273$	−148080.	−0.120251
$274$	0	0
$275$	154937.	0.123545
$276$	0	0
$277$	−739898.	−0.579392	−0.289696	−	0.957119i	$-0.593554\pi$
$277$	−739898.	−0.579392	−0.289696	+	0.957119i	$0.593554\pi$
$278$	0	0
$279$	1.63872e6	1.26036
$280$	0	0
$281$	−419987.	−0.317300	−0.158650	−	0.987335i	$-0.550714\pi$
$281$	−419987.	−0.317300	−0.158650	+	0.987335i	$0.550714\pi$
$282$	0	0
$283$	1.65296e6	1.22686	0.613432	−	0.789747i	$-0.289788\pi$
$283$	1.65296e6	1.22686	0.613432	+	0.789747i	$0.289788\pi$
$284$	0	0
$285$	3.38338e6	2.46740
$286$	0	0
$287$	1.56049e6	1.11829
$288$	0	0
$289$	−510375.	−0.359455
$290$	0	0
$291$	−2.26262e6	−1.56632
$292$	0	0
$293$	−2.40736e6	−1.63822	−0.819108	−	0.573639i	$-0.805531\pi$
$293$	−2.40736e6	−1.63822	−0.819108	+	0.573639i	$0.805531\pi$
$294$	0	0
$295$	−918767.	−0.614682
$296$	0	0
$297$	414275.	0.272519
$298$	0	0
$299$	112449.	0.0727406
$300$	0	0
$301$	559929.	0.356218
$302$	0	0
$303$	1.87184e6	1.17128
$304$	0	0
$305$	1.23611e6	0.760867
$306$	0	0
$307$	−2.58353e6	−1.56447	−0.782235	−	0.622983i	$-0.785920\pi$
$307$	−2.58353e6	−1.56447	−0.782235	+	0.622983i	$0.785920\pi$
$308$	0	0
$309$	−4.23811e6	−2.52509
$310$	0	0
$311$	−1.15621e6	−0.677856	−0.338928	−	0.940812i	$-0.610064\pi$
$311$	−1.15621e6	−0.677856	−0.338928	+	0.940812i	$0.610064\pi$
$312$	0	0
$313$	489727.	0.282549	0.141274	−	0.989970i	$-0.454880\pi$
$313$	489727.	0.282549	0.141274	+	0.989970i	$0.454880\pi$
$314$	0	0
$315$	−927107.	−0.526446
$316$	0	0
$317$	2.20217e6	1.23084	0.615421	−	0.788198i	$-0.288986\pi$
$317$	2.20217e6	1.23084	0.615421	+	0.788198i	$0.288986\pi$
$318$	0	0
$319$	1.32071e6	0.726657
$320$	0	0
$321$	−67732.1	−0.0366887
$322$	0	0
$323$	2.60456e6	1.38908
$324$	0	0
$325$	27920.6	0.0146628
$326$	0	0
$327$	4.41732e6	2.28449
$328$	0	0
$329$	−415396.	−0.211579
$330$	0	0
$331$	3.56595e6	1.78898	0.894489	−	0.447090i	$-0.147539\pi$
$331$	3.56595e6	1.78898	0.894489	+	0.447090i	$0.147539\pi$
$332$	0	0
$333$	−530735.	−0.262281
$334$	0	0
$335$	−3.84601e6	−1.87240
$336$	0	0
$337$	1.00441e6	0.481765	0.240883	−	0.970554i	$-0.422563\pi$
$337$	1.00441e6	0.481765	0.240883	+	0.970554i	$0.422563\pi$
$338$	0	0
$339$	−2.41641e6	−1.14201
$340$	0	0
$341$	4.02646e6	1.87516
$342$	0	0
$343$	−2.14109e6	−0.982651
$344$	0	0
$345$	1.54314e6	0.698004
$346$	0	0
$347$	3.90327e6	1.74022	0.870111	−	0.492856i	$-0.164047\pi$
$347$	3.90327e6	1.74022	0.870111	+	0.492856i	$0.164047\pi$
$348$	0	0
$349$	−1.00839e6	−0.443167	−0.221583	−	0.975141i	$-0.571122\pi$
$349$	−1.00839e6	−0.443167	−0.221583	+	0.975141i	$0.571122\pi$
$350$	0	0
$351$	74654.7	0.0323437
$352$	0	0
$353$	−2.38370e6	−1.01816	−0.509080	−	0.860719i	$-0.670014\pi$
$353$	−2.38370e6	−1.01816	−0.509080	+	0.860719i	$0.670014\pi$
$354$	0	0
$355$	−755559.	−0.318198
$356$	0	0
$357$	−1.56433e6	−0.649619
$358$	0	0
$359$	1.27723e6	0.523040	0.261520	−	0.965198i	$-0.415776\pi$
$359$	1.27723e6	0.523040	0.261520	+	0.965198i	$0.415776\pi$
$360$	0	0
$361$	4.98280e6	2.01236
$362$	0	0
$363$	−1.90043e6	−0.756980
$364$	0	0
$365$	2.04158e6	0.802112
$366$	0	0
$367$	1.35872e6	0.526580	0.263290	−	0.964717i	$-0.415192\pi$
$367$	1.35872e6	0.526580	0.263290	+	0.964717i	$0.415192\pi$
$368$	0	0
$369$	4.10013e6	1.56759
$370$	0	0
$371$	−823596.	−0.310656
$372$	0	0
$373$	1.46565e6	0.545453	0.272727	−	0.962092i	$-0.412075\pi$
$373$	1.46565e6	0.545453	0.272727	+	0.962092i	$0.412075\pi$
$374$	0	0
$375$	−3.48821e6	−1.28093
$376$	0	0
$377$	237999.	0.0862425
$378$	0	0
$379$	746048.	0.266790	0.133395	−	0.991063i	$-0.457412\pi$
$379$	746048.	0.266790	0.133395	+	0.991063i	$0.457412\pi$
$380$	0	0
$381$	−795739.	−0.280839
$382$	0	0
$383$	−3.43983e6	−1.19823	−0.599114	−	0.800664i	$-0.704480\pi$
$383$	−3.43983e6	−1.19823	−0.599114	+	0.800664i	$0.704480\pi$
$384$	0	0
$385$	−2.27798e6	−0.783244
$386$	0	0
$387$	1.47119e6	0.499336
$388$	0	0
$389$	−514928.	−0.172533	−0.0862665	−	0.996272i	$-0.527494\pi$
$389$	−514928.	−0.172533	−0.0862665	+	0.996272i	$0.527494\pi$
$390$	0	0
$391$	1.18793e6	0.392959
$392$	0	0
$393$	2.36534e6	0.772525
$394$	0	0
$395$	1.90785e6	0.615249
$396$	0	0
$397$	−3.53672e6	−1.12622	−0.563111	−	0.826381i	$-0.690396\pi$
$397$	−3.53672e6	−1.12622	−0.563111	+	0.826381i	$0.690396\pi$
$398$	0	0
$399$	−4.47991e6	−1.40876
$400$	0	0
$401$	−3.76592e6	−1.16953	−0.584763	−	0.811204i	$-0.698813\pi$
$401$	−3.76592e6	−1.16953	−0.584763	+	0.811204i	$0.698813\pi$
$402$	0	0
$403$	725592.	0.222551
$404$	0	0
$405$	3.92784e6	1.18992
$406$	0	0
$407$	−1.30406e6	−0.390221
$408$	0	0
$409$	−4.67931e6	−1.38316	−0.691582	−	0.722298i	$-0.743086\pi$
$409$	−4.67931e6	−1.38316	−0.691582	+	0.722298i	$0.743086\pi$
$410$	0	0
$411$	6.30592e6	1.84138
$412$	0	0
$413$	1.21653e6	0.350953
$414$	0	0
$415$	2.61858e6	0.746354
$416$	0	0
$417$	1.24378e6	0.350271
$418$	0	0
$419$	−3.52772e6	−0.981655	−0.490828	−	0.871257i	$-0.663305\pi$
$419$	−3.52772e6	−0.981655	−0.490828	+	0.871257i	$0.663305\pi$
$420$	0	0
$421$	2.31159e6	0.635632	0.317816	−	0.948152i	$-0.397050\pi$
$421$	2.31159e6	0.635632	0.317816	+	0.948152i	$0.397050\pi$
$422$	0	0
$423$	−1.09144e6	−0.296585
$424$	0	0
$425$	294957.	0.0792112
$426$	0	0
$427$	−1.63673e6	−0.434417
$428$	0	0
$429$	−955986.	−0.250789
$430$	0	0
$431$	−3.04671e6	−0.790020	−0.395010	−	0.918677i	$-0.629259\pi$
$431$	−3.04671e6	−0.790020	−0.395010	+	0.918677i	$0.629259\pi$
$432$	0	0
$433$	6.35737e6	1.62951	0.814757	−	0.579803i	$-0.196871\pi$
$433$	6.35737e6	1.62951	0.814757	+	0.579803i	$0.196871\pi$
$434$	0	0
$435$	3.26607e6	0.827567
$436$	0	0
$437$	3.40196e6	0.852169
$438$	0	0
$439$	1.25016e6	0.309603	0.154802	−	0.987946i	$-0.450526\pi$
$439$	1.25016e6	0.309603	0.154802	+	0.987946i	$0.450526\pi$
$440$	0	0
$441$	−2.19903e6	−0.538438
$442$	0	0
$443$	−3.69139e6	−0.893678	−0.446839	−	0.894614i	$-0.647450\pi$
$443$	−3.69139e6	−0.893678	−0.446839	+	0.894614i	$0.647450\pi$
$444$	0	0
$445$	356562.	0.0853561
$446$	0	0
$447$	−3.01705e6	−0.714191
$448$	0	0
$449$	2.59905e6	0.608413	0.304207	−	0.952606i	$-0.401609\pi$
$449$	2.59905e6	0.608413	0.304207	+	0.952606i	$0.401609\pi$
$450$	0	0
$451$	1.00743e7	2.33225
$452$	0	0
$453$	1.02919e6	0.235640
$454$	0	0
$455$	−410504.	−0.0929585
$456$	0	0
$457$	2.15983e6	0.483759	0.241880	−	0.970306i	$-0.422236\pi$
$457$	2.15983e6	0.483759	0.241880	+	0.970306i	$0.422236\pi$
$458$	0	0
$459$	788663.	0.174727
$460$	0	0
$461$	3.00260e6	0.658029	0.329014	−	0.944325i	$-0.393284\pi$
$461$	3.00260e6	0.658029	0.329014	+	0.944325i	$0.393284\pi$
$462$	0	0
$463$	−5.91505e6	−1.28235	−0.641174	−	0.767395i	$-0.721553\pi$
$463$	−5.91505e6	−1.28235	−0.641174	+	0.767395i	$0.721553\pi$
$464$	0	0
$465$	9.95735e6	2.13556
$466$	0	0
$467$	−7.57608e6	−1.60750	−0.803752	−	0.594964i	$-0.797166\pi$
$467$	−7.57608e6	−1.60750	−0.803752	+	0.594964i	$0.797166\pi$
$468$	0	0
$469$	5.09248e6	1.06905
$470$	0	0
$471$	−5.98648e6	−1.24342
$472$	0	0
$473$	3.61484e6	0.742910
$474$	0	0
$475$	844693.	0.171777
$476$	0	0
$477$	−2.16397e6	−0.435468
$478$	0	0
$479$	−689956.	−0.137399	−0.0686993	−	0.997637i	$-0.521885\pi$
$479$	−689956.	−0.137399	−0.0686993	+	0.997637i	$0.521885\pi$
$480$	0	0
$481$	−234999.	−0.0463130
$482$	0	0
$483$	−2.04326e6	−0.398526
$484$	0	0
$485$	−6.27241e6	−1.21082
$486$	0	0
$487$	−4.42150e6	−0.844787	−0.422394	−	0.906413i	$-0.638810\pi$
$487$	−4.42150e6	−0.844787	−0.422394	+	0.906413i	$0.638810\pi$
$488$	0	0
$489$	−1.14836e7	−2.17173
$490$	0	0
$491$	859442.	0.160884	0.0804420	−	0.996759i	$-0.474367\pi$
$491$	859442.	0.160884	0.0804420	+	0.996759i	$0.474367\pi$
$492$	0	0
$493$	2.51426e6	0.465899
$494$	0	0
$495$	−5.98531e6	−1.09793
$496$	0	0
$497$	1.00043e6	0.181675
$498$	0	0
$499$	2.86744e6	0.515516	0.257758	−	0.966209i	$-0.417016\pi$
$499$	2.86744e6	0.515516	0.257758	+	0.966209i	$0.417016\pi$
$500$	0	0
$501$	−6.29765e6	−1.12094
$502$	0	0
$503$	−8.62297e6	−1.51963	−0.759813	−	0.650142i	$-0.774709\pi$
$503$	−8.62297e6	−1.51963	−0.759813	+	0.650142i	$0.774709\pi$
$504$	0	0
$505$	5.18908e6	0.905445
$506$	0	0
$507$	7.67669e6	1.32634
$508$	0	0
$509$	−1.34831e6	−0.230673	−0.115336	−	0.993326i	$-0.536795\pi$
$509$	−1.34831e6	−0.230673	−0.115336	+	0.993326i	$0.536795\pi$
$510$	0	0
$511$	−2.70325e6	−0.457967
$512$	0	0
$513$	2.25856e6	0.378912
$514$	0	0
$515$	−1.17489e7	−1.95199
$516$	0	0
$517$	−2.68176e6	−0.441258
$518$	0	0
$519$	−2.45026e6	−0.399294
$520$	0	0
$521$	6.42421e6	1.03687	0.518436	−	0.855116i	$-0.326514\pi$
$521$	6.42421e6	1.03687	0.518436	+	0.855116i	$0.326514\pi$
$522$	0	0
$523$	6.52101e6	1.04246	0.521232	−	0.853415i	$-0.325473\pi$
$523$	6.52101e6	1.04246	0.521232	+	0.853415i	$0.325473\pi$
$524$	0	0
$525$	−507334.	−0.0803333
$526$	0	0
$527$	7.66526e6	1.20227
$528$	0	0
$529$	−4.88473e6	−0.758929
$530$	0	0
$531$	3.19640e6	0.491955
$532$	0	0
$533$	1.81545e6	0.276801
$534$	0	0
$535$	−187766.	−0.0283617
$536$	0	0
$537$	−9.59566e6	−1.43595
$538$	0	0
$539$	−5.40320e6	−0.801086
$540$	0	0
$541$	7.43844e6	1.09267	0.546334	−	0.837567i	$-0.316023\pi$
$541$	7.43844e6	1.09267	0.546334	+	0.837567i	$0.316023\pi$
$542$	0	0
$543$	−8.03590e6	−1.16959
$544$	0	0
$545$	1.22456e7	1.76600
$546$	0	0
$547$	−5.41804e6	−0.774237	−0.387118	−	0.922030i	$-0.626530\pi$
$547$	−5.41804e6	−0.774237	−0.387118	+	0.922030i	$0.626530\pi$
$548$	0	0
$549$	−4.30046e6	−0.608953
$550$	0	0
$551$	7.20028e6	1.01035
$552$	0	0
$553$	−2.52617e6	−0.351277
$554$	0	0
$555$	−3.22490e6	−0.444411
$556$	0	0
$557$	931613.	0.127232	0.0636162	−	0.997974i	$-0.479737\pi$
$557$	931613.	0.127232	0.0636162	+	0.997974i	$0.479737\pi$
$558$	0	0
$559$	651415.	0.0881716
$560$	0	0
$561$	−1.00992e7	−1.35481
$562$	0	0
$563$	−8.33400e6	−1.10811	−0.554055	−	0.832480i	$-0.686920\pi$
$563$	−8.33400e6	−1.10811	−0.554055	+	0.832480i	$0.686920\pi$
$564$	0	0
$565$	−6.69874e6	−0.882820
$566$	0	0
$567$	−5.20083e6	−0.679383
$568$	0	0
$569$	−2.66836e6	−0.345513	−0.172756	−	0.984965i	$-0.555267\pi$
$569$	−2.66836e6	−0.345513	−0.172756	+	0.984965i	$0.555267\pi$
$570$	0	0
$571$	−1.17041e6	−0.150227	−0.0751137	−	0.997175i	$-0.523932\pi$
$571$	−1.17041e6	−0.150227	−0.0751137	+	0.997175i	$0.523932\pi$
$572$	0	0
$573$	1.19173e7	1.51632
$574$	0	0
$575$	385260.	0.0485942
$576$	0	0
$577$	−2.65158e6	−0.331562	−0.165781	−	0.986163i	$-0.553015\pi$
$577$	−2.65158e6	−0.331562	−0.165781	+	0.986163i	$0.553015\pi$
$578$	0	0
$579$	−2.61890e6	−0.324655
$580$	0	0
$581$	−3.46724e6	−0.426132
$582$	0	0
$583$	−5.31705e6	−0.647888
$584$	0	0
$585$	−1.07859e6	−0.130306
$586$	0	0
$587$	3.94648e6	0.472732	0.236366	−	0.971664i	$-0.424044\pi$
$587$	3.94648e6	0.472732	0.236366	+	0.971664i	$0.424044\pi$
$588$	0	0
$589$	2.19517e7	2.60723
$590$	0	0
$591$	−1.06928e7	−1.25928
$592$	0	0
$593$	6.30122e6	0.735848	0.367924	−	0.929856i	$-0.380069\pi$
$593$	6.30122e6	0.735848	0.367924	+	0.929856i	$0.380069\pi$
$594$	0	0
$595$	−4.33663e6	−0.502181
$596$	0	0
$597$	1.50351e7	1.72652
$598$	0	0
$599$	−7.44523e6	−0.847834	−0.423917	−	0.905701i	$-0.639345\pi$
$599$	−7.44523e6	−0.847834	−0.423917	+	0.905701i	$0.639345\pi$
$600$	0	0
$601$	1.37897e7	1.55728	0.778642	−	0.627468i	$-0.215909\pi$
$601$	1.37897e7	1.55728	0.778642	+	0.627468i	$0.215909\pi$
$602$	0	0
$603$	1.33803e7	1.49856
$604$	0	0
$605$	−5.26834e6	−0.585174
$606$	0	0
$607$	−6.83116e6	−0.752528	−0.376264	−	0.926512i	$-0.622791\pi$
$607$	−6.83116e6	−0.752528	−0.376264	+	0.926512i	$0.622791\pi$
$608$	0	0
$609$	−4.32459e6	−0.472500
$610$	0	0
$611$	−483268.	−0.0523703
$612$	0	0
$613$	−1.36077e7	−1.46263	−0.731314	−	0.682041i	$-0.761093\pi$
$613$	−1.36077e7	−1.46263	−0.731314	+	0.682041i	$0.761093\pi$
$614$	0	0
$615$	2.49136e7	2.65613
$616$	0	0
$617$	−3.20420e6	−0.338849	−0.169425	−	0.985543i	$-0.554191\pi$
$617$	−3.20420e6	−0.338849	−0.169425	+	0.985543i	$0.554191\pi$
$618$	0	0
$619$	−1.12598e7	−1.18115	−0.590573	−	0.806985i	$-0.701098\pi$
$619$	−1.12598e7	−1.18115	−0.590573	+	0.806985i	$0.701098\pi$
$620$	0	0
$621$	1.03012e6	0.107191
$622$	0	0
$623$	−472121.	−0.0487341
$624$	0	0
$625$	−1.06365e7	−1.08918
$626$	0	0
$627$	−2.89219e7	−2.93804
$628$	0	0
$629$	−2.48256e6	−0.250192
$630$	0	0
$631$	1.92097e7	1.92064	0.960322	−	0.278895i	$-0.0899680\pi$
$631$	1.92097e7	1.92064	0.960322	+	0.278895i	$0.0899680\pi$
$632$	0	0
$633$	−1.29734e7	−1.28690
$634$	0	0
$635$	−2.20594e6	−0.217100
$636$	0	0
$637$	−973688.	−0.0950761
$638$	0	0
$639$	2.62860e6	0.254667
$640$	0	0
$641$	−9.78058e6	−0.940199	−0.470099	−	0.882614i	$-0.655782\pi$
$641$	−9.78058e6	−0.940199	−0.470099	+	0.882614i	$0.655782\pi$
$642$	0	0
$643$	1.18533e7	1.13061	0.565305	−	0.824882i	$-0.308758\pi$
$643$	1.18533e7	1.13061	0.565305	+	0.824882i	$0.308758\pi$
$644$	0	0
$645$	8.93942e6	0.846077
$646$	0	0
$647$	−5.96635e6	−0.560335	−0.280168	−	0.959951i	$-0.590390\pi$
$647$	−5.96635e6	−0.560335	−0.280168	+	0.959951i	$0.590390\pi$
$648$	0	0
$649$	7.85382e6	0.731929
$650$	0	0
$651$	−1.31845e7	−1.21930
$652$	0	0
$653$	6.78711e6	0.622877	0.311438	−	0.950266i	$-0.399189\pi$
$653$	6.78711e6	0.622877	0.311438	+	0.950266i	$0.399189\pi$
$654$	0	0
$655$	6.55718e6	0.597192
$656$	0	0
$657$	−7.10270e6	−0.641963
$658$	0	0
$659$	551026.	0.0494263	0.0247132	−	0.999695i	$-0.492133\pi$
$659$	551026.	0.0494263	0.0247132	+	0.999695i	$0.492133\pi$
$660$	0	0
$661$	4.34487e6	0.386788	0.193394	−	0.981121i	$-0.438050\pi$
$661$	4.34487e6	0.386788	0.193394	+	0.981121i	$0.438050\pi$
$662$	0	0
$663$	−1.81993e6	−0.160794
$664$	0	0
$665$	−1.24192e7	−1.08903
$666$	0	0
$667$	3.28401e6	0.285818
$668$	0	0
$669$	1.80063e7	1.55546
$670$	0	0
$671$	−1.05666e7	−0.905998
$672$	0	0
$673$	−4.11011e6	−0.349797	−0.174899	−	0.984586i	$-0.555960\pi$
$673$	−4.11011e6	−0.349797	−0.174899	+	0.984586i	$0.555960\pi$
$674$	0	0
$675$	255774.	0.0216071
$676$	0	0
$677$	−1.10810e7	−0.929192	−0.464596	−	0.885523i	$-0.653801\pi$
$677$	−1.10810e7	−0.929192	−0.464596	+	0.885523i	$0.653801\pi$
$678$	0	0
$679$	8.30526e6	0.691320
$680$	0	0
$681$	1.15747e6	0.0956407
$682$	0	0
$683$	1.69774e7	1.39257	0.696287	−	0.717763i	$-0.254834\pi$
$683$	1.69774e7	1.39257	0.696287	+	0.717763i	$0.254834\pi$
$684$	0	0
$685$	1.74812e7	1.42346
$686$	0	0
$687$	1.27721e7	1.03246
$688$	0	0
$689$	−958164.	−0.0768939
$690$	0	0
$691$	6.27721e6	0.500117	0.250059	−	0.968231i	$-0.419550\pi$
$691$	6.27721e6	0.500117	0.250059	+	0.968231i	$0.419550\pi$
$692$	0	0
$693$	7.92511e6	0.626862
$694$	0	0
$695$	3.44800e6	0.270773
$696$	0	0
$697$	1.91787e7	1.49533
$698$	0	0
$699$	−2.84105e7	−2.19931
$700$	0	0
$701$	−1.81211e7	−1.39280	−0.696402	−	0.717652i	$-0.745217\pi$
$701$	−1.81211e7	−1.39280	−0.696402	+	0.717652i	$0.745217\pi$
$702$	0	0
$703$	−7.10952e6	−0.542565
$704$	0	0
$705$	−6.63192e6	−0.502535
$706$	0	0
$707$	−6.87083e6	−0.516965
$708$	0	0
$709$	1.10634e7	0.826556	0.413278	−	0.910605i	$-0.364384\pi$
$709$	1.10634e7	0.826556	0.413278	+	0.910605i	$0.364384\pi$
$710$	0	0
$711$	−6.63743e6	−0.492409
$712$	0	0
$713$	1.00120e7	0.737562
$714$	0	0
$715$	−2.65017e6	−0.193869
$716$	0	0
$717$	2.24365e7	1.62988
$718$	0	0
$719$	−1.67651e7	−1.20944	−0.604721	−	0.796438i	$-0.706715\pi$
$719$	−1.67651e7	−1.20944	−0.604721	+	0.796438i	$0.706715\pi$
$720$	0	0
$721$	1.55566e7	1.11449
$722$	0	0
$723$	−9.68033e6	−0.688722
$724$	0	0
$725$	815406.	0.0576141
$726$	0	0
$727$	−8.66436e6	−0.607995	−0.303998	−	0.952673i	$-0.598322\pi$
$727$	−8.66436e6	−0.607995	−0.303998	+	0.952673i	$0.598322\pi$
$728$	0	0
$729$	−9.41690e6	−0.656280
$730$	0	0
$731$	6.88165e6	0.476320
$732$	0	0
$733$	2.37193e7	1.63058	0.815291	−	0.579052i	$-0.196577\pi$
$733$	2.37193e7	1.63058	0.815291	+	0.579052i	$0.196577\pi$
$734$	0	0
$735$	−1.33620e7	−0.912332
$736$	0	0
$737$	3.28765e7	2.22955
$738$	0	0
$739$	2.88980e7	1.94651	0.973254	−	0.229732i	$-0.0737851\pi$
$739$	2.88980e7	1.94651	0.973254	+	0.229732i	$0.0737851\pi$
$740$	0	0
$741$	−5.21189e6	−0.348698
$742$	0	0
$743$	−7.63191e6	−0.507179	−0.253589	−	0.967312i	$-0.581611\pi$
$743$	−7.63191e6	−0.507179	−0.253589	+	0.967312i	$0.581611\pi$
$744$	0	0
$745$	−8.36384e6	−0.552097
$746$	0	0
$747$	−9.11007e6	−0.597338
$748$	0	0
$749$	248620.	0.0161932
$750$	0	0
$751$	−6.57025e6	−0.425091	−0.212545	−	0.977151i	$-0.568175\pi$
$751$	−6.57025e6	−0.425091	−0.212545	+	0.977151i	$0.568175\pi$
$752$	0	0
$753$	2.27555e7	1.46251
$754$	0	0
$755$	2.85310e6	0.182158
$756$	0	0
$757$	−1.54448e7	−0.979589	−0.489795	−	0.871838i	$-0.662928\pi$
$757$	−1.54448e7	−0.979589	−0.489795	+	0.871838i	$0.662928\pi$
$758$	0	0
$759$	−1.31911e7	−0.831145
$760$	0	0
$761$	−2.57244e7	−1.61022	−0.805108	−	0.593129i	$-0.797893\pi$
$761$	−2.57244e7	−1.61022	−0.805108	+	0.593129i	$0.797893\pi$
$762$	0	0
$763$	−1.62144e7	−1.00830
$764$	0	0
$765$	−1.13944e7	−0.703941
$766$	0	0
$767$	1.41530e6	0.0868683
$768$	0	0
$769$	2.20092e7	1.34211	0.671056	−	0.741407i	$-0.265841\pi$
$769$	2.20092e7	1.34211	0.671056	+	0.741407i	$0.265841\pi$
$770$	0	0
$771$	1.39625e6	0.0845913
$772$	0	0
$773$	2.57034e7	1.54719	0.773593	−	0.633682i	$-0.218457\pi$
$773$	2.57034e7	1.54719	0.773593	+	0.633682i	$0.218457\pi$
$774$	0	0
$775$	2.48595e6	0.148675
$776$	0	0
$777$	4.27007e6	0.253737
$778$	0	0
$779$	5.49237e7	3.24277
$780$	0	0
$781$	6.45868e6	0.378893
$782$	0	0
$783$	2.18025e6	0.127087
$784$	0	0
$785$	−1.65956e7	−0.961214
$786$	0	0
$787$	9.34001e6	0.537540	0.268770	−	0.963204i	$-0.413383\pi$
$787$	9.34001e6	0.537540	0.268770	+	0.963204i	$0.413383\pi$
$788$	0	0
$789$	−6.16879e6	−0.352783
$790$	0	0
$791$	8.86976e6	0.504047
$792$	0	0
$793$	−1.90416e6	−0.107527
$794$	0	0
$795$	−1.31490e7	−0.737859
$796$	0	0
$797$	3.30305e7	1.84191	0.920956	−	0.389666i	$-0.127410\pi$
$797$	3.30305e7	1.84191	0.920956	+	0.389666i	$0.127410\pi$
$798$	0	0
$799$	−5.10532e6	−0.282915
$800$	0	0
$801$	−1.24048e6	−0.0683140
$802$	0	0
$803$	−1.74519e7	−0.955111
$804$	0	0
$805$	−5.66432e6	−0.308076
$806$	0	0
$807$	−5.34437e6	−0.288877
$808$	0	0
$809$	−2.84954e7	−1.53075	−0.765373	−	0.643587i	$-0.777446\pi$
$809$	−2.84954e7	−1.53075	−0.765373	+	0.643587i	$0.777446\pi$
$810$	0	0
$811$	−2.17867e7	−1.16316	−0.581579	−	0.813490i	$-0.697565\pi$
$811$	−2.17867e7	−1.16316	−0.581579	+	0.813490i	$0.697565\pi$
$812$	0	0
$813$	−3.55733e7	−1.88755
$814$	0	0
$815$	−3.18348e7	−1.67883
$816$	0	0
$817$	1.97076e7	1.03295
$818$	0	0
$819$	1.42815e6	0.0743985
$820$	0	0
$821$	−6.93650e6	−0.359155	−0.179578	−	0.983744i	$-0.557473\pi$
$821$	−6.93650e6	−0.359155	−0.179578	+	0.983744i	$0.557473\pi$
$822$	0	0
$823$	−2.55598e7	−1.31540	−0.657699	−	0.753281i	$-0.728470\pi$
$823$	−2.55598e7	−1.31540	−0.657699	+	0.753281i	$0.728470\pi$
$824$	0	0
$825$	−3.27530e6	−0.167539
$826$	0	0
$827$	−1.40927e7	−0.716526	−0.358263	−	0.933621i	$-0.616631\pi$
$827$	−1.40927e7	−0.716526	−0.358263	+	0.933621i	$0.616631\pi$
$828$	0	0
$829$	−1.04771e7	−0.529485	−0.264743	−	0.964319i	$-0.585287\pi$
$829$	−1.04771e7	−0.529485	−0.264743	+	0.964319i	$0.585287\pi$
$830$	0	0
$831$	1.56411e7	0.785714
$832$	0	0
$833$	−1.02862e7	−0.513620
$834$	0	0
$835$	−1.74583e7	−0.866533
$836$	0	0
$837$	6.64698e6	0.327953
$838$	0	0
$839$	2.33764e7	1.14650	0.573249	−	0.819381i	$-0.305683\pi$
$839$	2.33764e7	1.14650	0.573249	+	0.819381i	$0.305683\pi$
$840$	0	0
$841$	−1.35605e7	−0.661129
$842$	0	0
$843$	8.87833e6	0.430291
$844$	0	0
$845$	2.12812e7	1.02531
$846$	0	0
$847$	6.97577e6	0.334106
$848$	0	0
$849$	−3.49428e7	−1.66375
$850$	0	0
$851$	−3.24261e6	−0.153487
$852$	0	0
$853$	−3.29898e7	−1.55241	−0.776206	−	0.630480i	$-0.782858\pi$
$853$	−3.29898e7	−1.55241	−0.776206	+	0.630480i	$0.782858\pi$
$854$	0	0
$855$	−3.26310e7	−1.52656
$856$	0	0
$857$	−1.00187e7	−0.465973	−0.232986	−	0.972480i	$-0.574850\pi$
$857$	−1.00187e7	−0.465973	−0.232986	+	0.972480i	$0.574850\pi$
$858$	0	0
$859$	−3.58552e7	−1.65794	−0.828972	−	0.559291i	$-0.811074\pi$
$859$	−3.58552e7	−1.65794	−0.828972	+	0.559291i	$0.811074\pi$
$860$	0	0
$861$	−3.29879e7	−1.51652
$862$	0	0
$863$	−1.93759e7	−0.885594	−0.442797	−	0.896622i	$-0.646014\pi$
$863$	−1.93759e7	−0.885594	−0.442797	+	0.896622i	$0.646014\pi$
$864$	0	0
$865$	−6.79257e6	−0.308670
$866$	0	0
$867$	1.07891e7	0.487458
$868$	0	0
$869$	−1.63087e7	−0.732605
$870$	0	0
$871$	5.92454e6	0.264612
$872$	0	0
$873$	2.18218e7	0.969070
$874$	0	0
$875$	1.28039e7	0.565358
$876$	0	0
$877$	−2.05350e7	−0.901563	−0.450781	−	0.892634i	$-0.648855\pi$
$877$	−2.05350e7	−0.901563	−0.450781	+	0.892634i	$0.648855\pi$
$878$	0	0
$879$	5.08904e7	2.22159
$880$	0	0
$881$	−3.57055e7	−1.54987	−0.774935	−	0.632040i	$-0.782218\pi$
$881$	−3.57055e7	−1.54987	−0.774935	+	0.632040i	$0.782218\pi$
$882$	0	0
$883$	2.15255e7	0.929078	0.464539	−	0.885553i	$-0.346220\pi$
$883$	2.15255e7	0.929078	0.464539	+	0.885553i	$0.346220\pi$
$884$	0	0
$885$	1.94223e7	0.833571
$886$	0	0
$887$	9.87963e6	0.421630	0.210815	−	0.977526i	$-0.432388\pi$
$887$	9.87963e6	0.421630	0.210815	+	0.977526i	$0.432388\pi$
$888$	0	0
$889$	2.92087e6	0.123953
$890$	0	0
$891$	−3.35760e7	−1.41689
$892$	0	0
$893$	−1.46205e7	−0.613528
$894$	0	0
$895$	−2.66010e7	−1.11004
$896$	0	0
$897$	−2.37711e6	−0.0986436
$898$	0	0
$899$	2.11906e7	0.874466
$900$	0	0
$901$	−1.01222e7	−0.415396
$902$	0	0
$903$	−1.18366e7	−0.483068
$904$	0	0
$905$	−2.22770e7	−0.904140
$906$	0	0
$907$	3.91457e7	1.58003	0.790017	−	0.613086i	$-0.210072\pi$
$907$	3.91457e7	1.58003	0.790017	+	0.613086i	$0.210072\pi$
$908$	0	0
$909$	−1.80529e7	−0.724665
$910$	0	0
$911$	4.27841e7	1.70800	0.853998	−	0.520276i	$-0.174171\pi$
$911$	4.27841e7	1.70800	0.853998	+	0.520276i	$0.174171\pi$
$912$	0	0
$913$	−2.23842e7	−0.888718
$914$	0	0
$915$	−2.61309e7	−1.03181
$916$	0	0
$917$	−8.68232e6	−0.340967
$918$	0	0
$919$	−4.20694e6	−0.164315	−0.0821575	−	0.996619i	$-0.526181\pi$
$919$	−4.20694e6	−0.164315	−0.0821575	+	0.996619i	$0.526181\pi$
$920$	0	0
$921$	5.46146e7	2.12158
$922$	0	0
$923$	1.16389e6	0.0449685
$924$	0	0
$925$	−805128.	−0.0309393
$926$	0	0
$927$	4.08744e7	1.56226
$928$	0	0
$929$	−3.92960e7	−1.49386	−0.746929	−	0.664904i	$-0.768472\pi$
$929$	−3.92960e7	−1.49386	−0.746929	+	0.664904i	$0.768472\pi$
$930$	0	0
$931$	−2.94574e7	−1.11383
$932$	0	0
$933$	2.44418e7	0.919242
$934$	0	0
$935$	−2.79968e7	−1.04732
$936$	0	0
$937$	1.19951e7	0.446329	0.223165	−	0.974781i	$-0.428361\pi$
$937$	1.19951e7	0.446329	0.223165	+	0.974781i	$0.428361\pi$
$938$	0	0
$939$	−1.03526e7	−0.383165
$940$	0	0
$941$	3.44527e6	0.126838	0.0634190	−	0.997987i	$-0.479800\pi$
$941$	3.44527e6	0.126838	0.0634190	+	0.997987i	$0.479800\pi$
$942$	0	0
$943$	2.50504e7	0.917351
$944$	0	0
$945$	−3.76054e6	−0.136984
$946$	0	0
$947$	2.81830e7	1.02121	0.510603	−	0.859817i	$-0.329422\pi$
$947$	2.81830e7	1.02121	0.510603	+	0.859817i	$0.329422\pi$
$948$	0	0
$949$	−3.14493e6	−0.113356
$950$	0	0
$951$	−4.65528e7	−1.66915
$952$	0	0
$953$	−3.50458e7	−1.24998	−0.624990	−	0.780633i	$-0.714897\pi$
$953$	−3.50458e7	−1.24998	−0.624990	+	0.780633i	$0.714897\pi$
$954$	0	0
$955$	3.30371e7	1.17218
$956$	0	0
$957$	−2.79191e7	−0.985421
$958$	0	0
$959$	−2.31468e7	−0.812725
$960$	0	0
$961$	3.59749e7	1.25658
$962$	0	0
$963$	653241.	0.0226991
$964$	0	0
$965$	−7.26009e6	−0.250971
$966$	0	0
$967$	5.47660e7	1.88341	0.941705	−	0.336441i	$-0.109223\pi$
$967$	5.47660e7	1.88341	0.941705	+	0.336441i	$0.109223\pi$
$968$	0	0
$969$	−5.50592e7	−1.88374
$970$	0	0
$971$	2.70892e6	0.0922036	0.0461018	−	0.998937i	$-0.485320\pi$
$971$	2.70892e6	0.0922036	0.0461018	+	0.998937i	$0.485320\pi$
$972$	0	0
$973$	−4.56547e6	−0.154598
$974$	0	0
$975$	−590228.	−0.0198842
$976$	0	0
$977$	2.62580e7	0.880087	0.440044	−	0.897976i	$-0.354963\pi$
$977$	2.62580e7	0.880087	0.440044	+	0.897976i	$0.354963\pi$
$978$	0	0
$979$	−3.04797e6	−0.101637
$980$	0	0
$981$	−4.26028e7	−1.41340
$982$	0	0
$983$	1.74516e7	0.576038	0.288019	−	0.957625i	$-0.407003\pi$
$983$	1.74516e7	0.576038	0.288019	+	0.957625i	$0.407003\pi$
$984$	0	0
$985$	−2.96424e7	−0.973470
$986$	0	0
$987$	8.78129e6	0.286923
$988$	0	0
$989$	8.98851e6	0.292211
$990$	0	0
$991$	−4.64909e7	−1.50378	−0.751890	−	0.659289i	$-0.770857\pi$
$991$	−4.64909e7	−1.50378	−0.751890	+	0.659289i	$0.770857\pi$
$992$	0	0
$993$	−7.53825e7	−2.42604
$994$	0	0
$995$	4.16802e7	1.33466
$996$	0	0
$997$	2.33865e7	0.745123	0.372561	−	0.928008i	$-0.378480\pi$
$997$	2.33865e7	0.745123	0.372561	+	0.928008i	$0.378480\pi$
$998$	0	0
$999$	−2.15277e6	−0.0682471

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1028.6.a.b.1.9	✓	57

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1028.6.a.b.1.9	✓	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)

\(f(q)\)	\(=\)	\(q-21.1395 q^{3} -58.6028 q^{5} +77.5956 q^{7} +203.880 q^{9} +O(q^{10})\)	\(q-21.1395 q^{3} -58.6028 q^{5} +77.5956 q^{7} +203.880 q^{9} +500.949 q^{11} +90.2740 q^{13} +1238.84 q^{15} +953.668 q^{17} +2731.10 q^{19} -1640.33 q^{21} +1245.64 q^{23} +309.287 q^{25} +826.979 q^{27} +2636.41 q^{29} +8037.67 q^{31} -10589.8 q^{33} -4547.32 q^{35} -2603.17 q^{37} -1908.35 q^{39} +20110.5 q^{41} +7215.98 q^{43} -11947.9 q^{45} -5353.35 q^{47} -10785.9 q^{49} -20160.1 q^{51} -10614.0 q^{53} -29357.0 q^{55} -57734.1 q^{57} +15677.9 q^{59} -21093.1 q^{61} +15820.2 q^{63} -5290.31 q^{65} +65628.5 q^{67} -26332.2 q^{69} +12892.9 q^{71} -34837.7 q^{73} -6538.18 q^{75} +38871.4 q^{77} -32555.6 q^{79} -67024.8 q^{81} -44683.5 q^{83} -55887.6 q^{85} -55732.4 q^{87} -6084.38 q^{89} +7004.86 q^{91} -169913. q^{93} -160050. q^{95} +107033. q^{97} +102134. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 57 q + 27 q^{3} + 61 q^{5} + 88 q^{7} + 5374 q^{9}+O(q^{10}) \) 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

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more