LMFDB - Embedded newform 528.6.a.z.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [528,6,Mod(1,528)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(528, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("528.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 528 = 2^{4} \cdot 3 \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	528.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:	$84.6826568613$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 2117x^{2} + 1518x + 1092672 $ x^4 - 2x^3 - 2117x^2 + 1518*x + 1092672
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 264)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-33.4518$ of defining polynomial
Character	$\chi$	$=$	528.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.00000 q^{3} -78.9037 q^{5} -207.206 q^{7} +81.0000 q^{9} +O(q^{10})$	$q-9.00000 q^{3} -78.9037 q^{5} -207.206 q^{7} +81.0000 q^{9} -121.000 q^{11} +680.617 q^{13} +710.133 q^{15} -744.833 q^{17} +2009.58 q^{19} +1864.85 q^{21} +4176.17 q^{23} +3100.79 q^{25} -729.000 q^{27} +4287.50 q^{29} -8936.37 q^{31} +1089.00 q^{33} +16349.3 q^{35} +5834.17 q^{37} -6125.56 q^{39} +15739.3 q^{41} -17956.9 q^{43} -6391.20 q^{45} -8258.15 q^{47} +26127.3 q^{49} +6703.49 q^{51} -8645.88 q^{53} +9547.34 q^{55} -18086.3 q^{57} -42663.7 q^{59} +39483.8 q^{61} -16783.7 q^{63} -53703.2 q^{65} -27846.5 q^{67} -37585.5 q^{69} +13818.1 q^{71} -77902.8 q^{73} -27907.1 q^{75} +25071.9 q^{77} +72444.9 q^{79} +6561.00 q^{81} +55453.5 q^{83} +58770.0 q^{85} -38587.5 q^{87} +57488.7 q^{89} -141028. q^{91} +80427.4 q^{93} -158564. q^{95} +54222.0 q^{97} -9801.00 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 36 q^{3} - 44 q^{5} - 38 q^{7} + 324 q^{9}+O(q^{10}) $ 4 * q - 36 * q^3 - 44 * q^5 - 38 * q^7 + 324 * q^9	$ 4 q - 36 q^{3} - 44 q^{5} - 38 q^{7} + 324 q^{9} - 484 q^{11} + 288 q^{13} + 396 q^{15} + 418 q^{17} + 10 q^{19} + 342 q^{21} - 576 q^{23} + 4932 q^{25} - 2916 q^{27} + 11930 q^{29} - 9968 q^{31} + 4356 q^{33} + 428 q^{35} + 13740 q^{37} - 2592 q^{39} + 29766 q^{41} - 25650 q^{43} - 3564 q^{45} + 5776 q^{47} + 39888 q^{49} - 3762 q^{51} - 7840 q^{53} + 5324 q^{55} - 90 q^{57} - 28800 q^{59} + 33932 q^{61} - 3078 q^{63} + 33216 q^{65} - 83056 q^{67} + 5184 q^{69} - 21336 q^{71} + 27044 q^{73} - 44388 q^{75} + 4598 q^{77} - 102542 q^{79} + 26244 q^{81} - 64996 q^{83} - 12132 q^{85} - 107370 q^{87} + 37888 q^{89} - 273612 q^{91} + 89712 q^{93} - 254380 q^{95} - 20996 q^{97} - 39204 q^{99}+O(q^{100}) $ 4 * q - 36 * q^3 - 44 * q^5 - 38 * q^7 + 324 * q^9 - 484 * q^11 + 288 * q^13 + 396 * q^15 + 418 * q^17 + 10 * q^19 + 342 * q^21 - 576 * q^23 + 4932 * q^25 - 2916 * q^27 + 11930 * q^29 - 9968 * q^31 + 4356 * q^33 + 428 * q^35 + 13740 * q^37 - 2592 * q^39 + 29766 * q^41 - 25650 * q^43 - 3564 * q^45 + 5776 * q^47 + 39888 * q^49 - 3762 * q^51 - 7840 * q^53 + 5324 * q^55 - 90 * q^57 - 28800 * q^59 + 33932 * q^61 - 3078 * q^63 + 33216 * q^65 - 83056 * q^67 + 5184 * q^69 - 21336 * q^71 + 27044 * q^73 - 44388 * q^75 + 4598 * q^77 - 102542 * q^79 + 26244 * q^81 - 64996 * q^83 - 12132 * q^85 - 107370 * q^87 + 37888 * q^89 - 273612 * q^91 + 89712 * q^93 - 254380 * q^95 - 20996 * q^97 - 39204 * 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
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	0	0
$5$	−78.9037	−1.41147	−0.705736	−	0.708475i	$-0.749383\pi$
$5$	−78.9037	−1.41147	−0.705736	+	0.708475i	$0.749383\pi$
$6$	0	0
$7$	−207.206	−1.59830	−0.799148	−	0.601135i	$-0.794715\pi$
$7$	−207.206	−1.59830	−0.799148	+	0.601135i	$0.794715\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	0	0
$13$	680.617	1.11698	0.558489	−	0.829512i	$-0.311381\pi$
$13$	680.617	1.11698	0.558489	+	0.829512i	$0.311381\pi$
$14$	0	0
$15$	710.133	0.814913
$16$	0	0
$17$	−744.833	−0.625081	−0.312541	−	0.949904i	$-0.601180\pi$
$17$	−744.833	−0.625081	−0.312541	+	0.949904i	$0.601180\pi$
$18$	0	0
$19$	2009.58	1.27709	0.638546	−	0.769584i	$-0.279536\pi$
$19$	2009.58	1.27709	0.638546	+	0.769584i	$0.279536\pi$
$20$	0	0
$21$	1864.85	0.922776
$22$	0	0
$23$	4176.17	1.64611	0.823055	−	0.567962i	$-0.192268\pi$
$23$	4176.17	1.64611	0.823055	+	0.567962i	$0.192268\pi$
$24$	0	0
$25$	3100.79	0.992252
$26$	0	0
$27$	−729.000	−0.192450
$28$	0	0
$29$	4287.50	0.946692	0.473346	−	0.880877i	$-0.343046\pi$
$29$	4287.50	0.946692	0.473346	+	0.880877i	$0.343046\pi$
$30$	0	0
$31$	−8936.37	−1.67016	−0.835078	−	0.550131i	$-0.814578\pi$
$31$	−8936.37	−1.67016	−0.835078	+	0.550131i	$0.814578\pi$
$32$	0	0
$33$	1089.00	0.174078
$34$	0	0
$35$	16349.3	2.25595
$36$	0	0
$37$	5834.17	0.700608	0.350304	−	0.936636i	$-0.386078\pi$
$37$	5834.17	0.700608	0.350304	+	0.936636i	$0.386078\pi$
$38$	0	0
$39$	−6125.56	−0.644887
$40$	0	0
$41$	15739.3	1.46227	0.731133	−	0.682235i	$-0.238992\pi$
$41$	15739.3	1.46227	0.731133	+	0.682235i	$0.238992\pi$
$42$	0	0
$43$	−17956.9	−1.48102	−0.740509	−	0.672047i	$-0.765415\pi$
$43$	−17956.9	−1.48102	−0.740509	+	0.672047i	$0.765415\pi$
$44$	0	0
$45$	−6391.20	−0.470491
$46$	0	0
$47$	−8258.15	−0.545303	−0.272652	−	0.962113i	$-0.587901\pi$
$47$	−8258.15	−0.545303	−0.272652	+	0.962113i	$0.587901\pi$
$48$	0	0
$49$	26127.3	1.55455
$50$	0	0
$51$	6703.49	0.360891
$52$	0	0
$53$	−8645.88	−0.422785	−0.211393	−	0.977401i	$-0.567800\pi$
$53$	−8645.88	−0.422785	−0.211393	+	0.977401i	$0.567800\pi$
$54$	0	0
$55$	9547.34	0.425575
$56$	0	0
$57$	−18086.3	−0.737329
$58$	0	0
$59$	−42663.7	−1.59562	−0.797808	−	0.602911i	$-0.794007\pi$
$59$	−42663.7	−1.59562	−0.797808	+	0.602911i	$0.794007\pi$
$60$	0	0
$61$	39483.8	1.35861	0.679303	−	0.733857i	$-0.262282\pi$
$61$	39483.8	1.35861	0.679303	+	0.733857i	$0.262282\pi$
$62$	0	0
$63$	−16783.7	−0.532765
$64$	0	0
$65$	−53703.2	−1.57658
$66$	0	0
$67$	−27846.5	−0.757850	−0.378925	−	0.925427i	$-0.623706\pi$
$67$	−27846.5	−0.757850	−0.378925	+	0.925427i	$0.623706\pi$
$68$	0	0
$69$	−37585.5	−0.950382
$70$	0	0
$71$	13818.1	0.325315	0.162657	−	0.986683i	$-0.447993\pi$
$71$	13818.1	0.325315	0.162657	+	0.986683i	$0.447993\pi$
$72$	0	0
$73$	−77902.8	−1.71098	−0.855492	−	0.517817i	$-0.826745\pi$
$73$	−77902.8	−1.71098	−0.855492	+	0.517817i	$0.826745\pi$
$74$	0	0
$75$	−27907.1	−0.572877
$76$	0	0
$77$	25071.9	0.481904
$78$	0	0
$79$	72444.9	1.30599	0.652995	−	0.757362i	$-0.273512\pi$
$79$	72444.9	1.30599	0.652995	+	0.757362i	$0.273512\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	55453.5	0.883555	0.441778	−	0.897125i	$-0.354348\pi$
$83$	55453.5	0.883555	0.441778	+	0.897125i	$0.354348\pi$
$84$	0	0
$85$	58770.0	0.882284
$86$	0	0
$87$	−38587.5	−0.546573
$88$	0	0
$89$	57488.7	0.769321	0.384661	−	0.923058i	$-0.374318\pi$
$89$	57488.7	0.769321	0.384661	+	0.923058i	$0.374318\pi$
$90$	0	0
$91$	−141028.	−1.78526
$92$	0	0
$93$	80427.4	0.964265
$94$	0	0
$95$	−158564.	−1.80258
$96$	0	0
$97$	54222.0	0.585122	0.292561	−	0.956247i	$-0.405493\pi$
$97$	54222.0	0.585122	0.292561	+	0.956247i	$0.405493\pi$
$98$	0	0
$99$	−9801.00	−0.100504
$100$	0	0
$101$	89429.5	0.872323	0.436162	−	0.899868i	$-0.356338\pi$
$101$	89429.5	0.872323	0.436162	+	0.899868i	$0.356338\pi$
$102$	0	0
$103$	18324.9	0.170195	0.0850977	−	0.996373i	$-0.472880\pi$
$103$	18324.9	0.170195	0.0850977	+	0.996373i	$0.472880\pi$
$104$	0	0
$105$	−147144.	−1.30247
$106$	0	0
$107$	−226399.	−1.91168	−0.955841	−	0.293885i	$-0.905052\pi$
$107$	−226399.	−1.91168	−0.955841	+	0.293885i	$0.905052\pi$
$108$	0	0
$109$	−73961.9	−0.596269	−0.298134	−	0.954524i	$-0.596364\pi$
$109$	−73961.9	−0.596269	−0.298134	+	0.954524i	$0.596364\pi$
$110$	0	0
$111$	−52507.6	−0.404496
$112$	0	0
$113$	89473.4	0.659170	0.329585	−	0.944126i	$-0.393091\pi$
$113$	89473.4	0.659170	0.329585	+	0.944126i	$0.393091\pi$
$114$	0	0
$115$	−329515.	−2.32344
$116$	0	0
$117$	55130.0	0.372326
$118$	0	0
$119$	154334.	0.999064
$120$	0	0
$121$	14641.0	0.0909091
$122$	0	0
$123$	−141654.	−0.844240
$124$	0	0
$125$	1910.46	0.0109361
$126$	0	0
$127$	330448.	1.81800	0.908999	−	0.416798i	$-0.136848\pi$
$127$	330448.	1.81800	0.908999	+	0.416798i	$0.136848\pi$
$128$	0	0
$129$	161612.	0.855066
$130$	0	0
$131$	248255.	1.26392	0.631960	−	0.775001i	$-0.282250\pi$
$131$	248255.	1.26392	0.631960	+	0.775001i	$0.282250\pi$
$132$	0	0
$133$	−416398.	−2.04117
$134$	0	0
$135$	57520.8	0.271638
$136$	0	0
$137$	−52952.2	−0.241037	−0.120518	−	0.992711i	$-0.538456\pi$
$137$	−52952.2	−0.241037	−0.120518	+	0.992711i	$0.538456\pi$
$138$	0	0
$139$	−203498.	−0.893354	−0.446677	−	0.894695i	$-0.647393\pi$
$139$	−203498.	−0.893354	−0.446677	+	0.894695i	$0.647393\pi$
$140$	0	0
$141$	74323.4	0.314831
$142$	0	0
$143$	−82354.7	−0.336781
$144$	0	0
$145$	−338299.	−1.33623
$146$	0	0
$147$	−235145.	−0.897518
$148$	0	0
$149$	−368523.	−1.35988	−0.679938	−	0.733270i	$-0.737993\pi$
$149$	−368523.	−1.35988	−0.679938	+	0.733270i	$0.737993\pi$
$150$	0	0
$151$	−223635.	−0.798175	−0.399088	−	0.916913i	$-0.630673\pi$
$151$	−223635.	−0.798175	−0.399088	+	0.916913i	$0.630673\pi$
$152$	0	0
$153$	−60331.4	−0.208360
$154$	0	0
$155$	705113.	2.35738
$156$	0	0
$157$	278760.	0.902570	0.451285	−	0.892380i	$-0.350966\pi$
$157$	278760.	0.902570	0.451285	+	0.892380i	$0.350966\pi$
$158$	0	0
$159$	77813.0	0.244095
$160$	0	0
$161$	−865327.	−2.63097
$162$	0	0
$163$	−458541.	−1.35179	−0.675895	−	0.736998i	$-0.736243\pi$
$163$	−458541.	−1.35179	−0.675895	+	0.736998i	$0.736243\pi$
$164$	0	0
$165$	−85926.1	−0.245706
$166$	0	0
$167$	444595.	1.23360	0.616799	−	0.787121i	$-0.288429\pi$
$167$	444595.	1.23360	0.616799	+	0.787121i	$0.288429\pi$
$168$	0	0
$169$	91946.8	0.247640
$170$	0	0
$171$	162776.	0.425697
$172$	0	0
$173$	−109766.	−0.278838	−0.139419	−	0.990233i	$-0.544523\pi$
$173$	−109766.	−0.278838	−0.139419	+	0.990233i	$0.544523\pi$
$174$	0	0
$175$	−642501.	−1.58591
$176$	0	0
$177$	383973.	0.921230
$178$	0	0
$179$	−361413.	−0.843085	−0.421542	−	0.906809i	$-0.638511\pi$
$179$	−361413.	−0.843085	−0.421542	+	0.906809i	$0.638511\pi$
$180$	0	0
$181$	−713572.	−1.61898	−0.809490	−	0.587134i	$-0.800256\pi$
$181$	−713572.	−1.61898	−0.809490	+	0.587134i	$0.800256\pi$
$182$	0	0
$183$	−355354.	−0.784392
$184$	0	0
$185$	−460338.	−0.988888
$186$	0	0
$187$	90124.7	0.188469
$188$	0	0
$189$	151053.	0.307592
$190$	0	0
$191$	−144789.	−0.287180	−0.143590	−	0.989637i	$-0.545865\pi$
$191$	−144789.	−0.287180	−0.143590	+	0.989637i	$0.545865\pi$
$192$	0	0
$193$	−176978.	−0.342000	−0.171000	−	0.985271i	$-0.554700\pi$
$193$	−176978.	−0.342000	−0.171000	+	0.985271i	$0.554700\pi$
$194$	0	0
$195$	483329.	0.910240
$196$	0	0
$197$	83671.4	0.153607	0.0768036	−	0.997046i	$-0.475529\pi$
$197$	83671.4	0.153607	0.0768036	+	0.997046i	$0.475529\pi$
$198$	0	0
$199$	19512.9	0.0349293	0.0174647	−	0.999847i	$-0.494441\pi$
$199$	19512.9	0.0349293	0.0174647	+	0.999847i	$0.494441\pi$
$200$	0	0
$201$	250618.	0.437545
$202$	0	0
$203$	−888394.	−1.51309
$204$	0	0
$205$	−1.24189e6	−2.06395
$206$	0	0
$207$	338270.	0.548703
$208$	0	0
$209$	−243160.	−0.385058
$210$	0	0
$211$	−262647.	−0.406131	−0.203066	−	0.979165i	$-0.565090\pi$
$211$	−262647.	−0.406131	−0.203066	+	0.979165i	$0.565090\pi$
$212$	0	0
$213$	−124363.	−0.187821
$214$	0	0
$215$	1.41686e6	2.09041
$216$	0	0
$217$	1.85167e6	2.66940
$218$	0	0
$219$	701125.	0.987837
$220$	0	0
$221$	−506946.	−0.698202
$222$	0	0
$223$	−389918.	−0.525063	−0.262531	−	0.964923i	$-0.584557\pi$
$223$	−389918.	−0.525063	−0.262531	+	0.964923i	$0.584557\pi$
$224$	0	0
$225$	251164.	0.330751
$226$	0	0
$227$	−283055.	−0.364592	−0.182296	−	0.983244i	$-0.558353\pi$
$227$	−283055.	−0.364592	−0.182296	+	0.983244i	$0.558353\pi$
$228$	0	0
$229$	−494501.	−0.623130	−0.311565	−	0.950225i	$-0.600853\pi$
$229$	−494501.	−0.623130	−0.311565	+	0.950225i	$0.600853\pi$
$230$	0	0
$231$	−225647.	−0.278227
$232$	0	0
$233$	148718.	0.179463	0.0897314	−	0.995966i	$-0.471399\pi$
$233$	148718.	0.179463	0.0897314	+	0.995966i	$0.471399\pi$
$234$	0	0
$235$	651598.	0.769680
$236$	0	0
$237$	−652004.	−0.754014
$238$	0	0
$239$	333573.	0.377743	0.188871	−	0.982002i	$-0.439517\pi$
$239$	333573.	0.377743	0.188871	+	0.982002i	$0.439517\pi$
$240$	0	0
$241$	1.18066e6	1.30943	0.654716	−	0.755875i	$-0.272788\pi$
$241$	1.18066e6	1.30943	0.654716	+	0.755875i	$0.272788\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	0	0
$245$	−2.06154e6	−2.19420
$246$	0	0
$247$	1.36776e6	1.42648
$248$	0	0
$249$	−499082.	−0.510121
$250$	0	0
$251$	1.78835e6	1.79171	0.895857	−	0.444343i	$-0.146563\pi$
$251$	1.78835e6	1.79171	0.895857	+	0.444343i	$0.146563\pi$
$252$	0	0
$253$	−505317.	−0.496321
$254$	0	0
$255$	−528930.	−0.509387
$256$	0	0
$257$	−262376.	−0.247795	−0.123897	−	0.992295i	$-0.539539\pi$
$257$	−262376.	−0.247795	−0.123897	+	0.992295i	$0.539539\pi$
$258$	0	0
$259$	−1.20888e6	−1.11978
$260$	0	0
$261$	347287.	0.315564
$262$	0	0
$263$	−651316.	−0.580634	−0.290317	−	0.956931i	$-0.593761\pi$
$263$	−651316.	−0.580634	−0.290317	+	0.956931i	$0.593761\pi$
$264$	0	0
$265$	682192.	0.596749
$266$	0	0
$267$	−517399.	−0.444168
$268$	0	0
$269$	281049.	0.236811	0.118405	−	0.992965i	$-0.462222\pi$
$269$	281049.	0.236811	0.118405	+	0.992965i	$0.462222\pi$
$270$	0	0
$271$	−942621.	−0.779676	−0.389838	−	0.920884i	$-0.627469\pi$
$271$	−942621.	−0.779676	−0.389838	+	0.920884i	$0.627469\pi$
$272$	0	0
$273$	1.26925e6	1.03072
$274$	0	0
$275$	−375195.	−0.299175
$276$	0	0
$277$	−623468.	−0.488219	−0.244110	−	0.969748i	$-0.578496\pi$
$277$	−623468.	−0.488219	−0.244110	+	0.969748i	$0.578496\pi$
$278$	0	0
$279$	−723846.	−0.556719
$280$	0	0
$281$	345965.	0.261376	0.130688	−	0.991424i	$-0.458281\pi$
$281$	345965.	0.261376	0.130688	+	0.991424i	$0.458281\pi$
$282$	0	0
$283$	−1.99490e6	−1.48066	−0.740329	−	0.672245i	$-0.765330\pi$
$283$	−1.99490e6	−1.48066	−0.740329	+	0.672245i	$0.765330\pi$
$284$	0	0
$285$	1.42707e6	1.04072
$286$	0	0
$287$	−3.26128e6	−2.33713
$288$	0	0
$289$	−865081.	−0.609274
$290$	0	0
$291$	−487998.	−0.337820
$292$	0	0
$293$	987681.	0.672121	0.336061	−	0.941840i	$-0.390905\pi$
$293$	987681.	0.672121	0.336061	+	0.941840i	$0.390905\pi$
$294$	0	0
$295$	3.36632e6	2.25217
$296$	0	0
$297$	88209.0	0.0580259
$298$	0	0
$299$	2.84237e6	1.83867
$300$	0	0
$301$	3.72077e6	2.36710
$302$	0	0
$303$	−804866.	−0.503636
$304$	0	0
$305$	−3.11541e6	−1.91764
$306$	0	0
$307$	374579.	0.226829	0.113414	−	0.993548i	$-0.463821\pi$
$307$	374579.	0.226829	0.113414	+	0.993548i	$0.463821\pi$
$308$	0	0
$309$	−164924.	−0.0982623
$310$	0	0
$311$	1.86191e6	1.09159	0.545793	−	0.837920i	$-0.316229\pi$
$311$	1.86191e6	1.09159	0.545793	+	0.837920i	$0.316229\pi$
$312$	0	0
$313$	−1.63895e6	−0.945592	−0.472796	−	0.881172i	$-0.656755\pi$
$313$	−1.63895e6	−0.945592	−0.472796	+	0.881172i	$0.656755\pi$
$314$	0	0
$315$	1.32429e6	0.751983
$316$	0	0
$317$	482287.	0.269561	0.134781	−	0.990875i	$-0.456967\pi$
$317$	482287.	0.269561	0.134781	+	0.990875i	$0.456967\pi$
$318$	0	0
$319$	−518787.	−0.285438
$320$	0	0
$321$	2.03759e6	1.10371
$322$	0	0
$323$	−1.49680e6	−0.798286
$324$	0	0
$325$	2.11045e6	1.10832
$326$	0	0
$327$	665657.	0.344256
$328$	0	0
$329$	1.71114e6	0.871556
$330$	0	0
$331$	−2.23220e6	−1.11986	−0.559929	−	0.828541i	$-0.689172\pi$
$331$	−2.23220e6	−1.11986	−0.559929	+	0.828541i	$0.689172\pi$
$332$	0	0
$333$	472568.	0.233536
$334$	0	0
$335$	2.19719e6	1.06968
$336$	0	0
$337$	−1.63234e6	−0.782952	−0.391476	−	0.920188i	$-0.628035\pi$
$337$	−1.63234e6	−0.782952	−0.391476	+	0.920188i	$0.628035\pi$
$338$	0	0
$339$	−805260.	−0.380572
$340$	0	0
$341$	1.08130e6	0.503571
$342$	0	0
$343$	−1.93121e6	−0.886330
$344$	0	0
$345$	2.96564e6	1.34144
$346$	0	0
$347$	3.97474e6	1.77209	0.886044	−	0.463601i	$-0.153443\pi$
$347$	3.97474e6	1.77209	0.886044	+	0.463601i	$0.153443\pi$
$348$	0	0
$349$	−506467.	−0.222581	−0.111290	−	0.993788i	$-0.535498\pi$
$349$	−506467.	−0.222581	−0.111290	+	0.993788i	$0.535498\pi$
$350$	0	0
$351$	−496170.	−0.214962
$352$	0	0
$353$	−1.82304e6	−0.778682	−0.389341	−	0.921094i	$-0.627297\pi$
$353$	−1.82304e6	−0.778682	−0.389341	+	0.921094i	$0.627297\pi$
$354$	0	0
$355$	−1.09030e6	−0.459172
$356$	0	0
$357$	−1.38900e6	−0.576810
$358$	0	0
$359$	−3.09675e6	−1.26815	−0.634075	−	0.773272i	$-0.718619\pi$
$359$	−3.09675e6	−1.26815	−0.634075	+	0.773272i	$0.718619\pi$
$360$	0	0
$361$	1.56233e6	0.630964
$362$	0	0
$363$	−131769.	−0.0524864
$364$	0	0
$365$	6.14681e6	2.41500
$366$	0	0
$367$	3.83322e6	1.48559	0.742795	−	0.669519i	$-0.233500\pi$
$367$	3.83322e6	1.48559	0.742795	+	0.669519i	$0.233500\pi$
$368$	0	0
$369$	1.27489e6	0.487422
$370$	0	0
$371$	1.79148e6	0.675735
$372$	0	0
$373$	1.09237e6	0.406535	0.203267	−	0.979123i	$-0.434844\pi$
$373$	1.09237e6	0.406535	0.203267	+	0.979123i	$0.434844\pi$
$374$	0	0
$375$	−17194.1	−0.00631395
$376$	0	0
$377$	2.91814e6	1.05743
$378$	0	0
$379$	2.14206e6	0.766008	0.383004	−	0.923747i	$-0.374890\pi$
$379$	2.14206e6	0.766008	0.383004	+	0.923747i	$0.374890\pi$
$380$	0	0
$381$	−2.97403e6	−1.04962
$382$	0	0
$383$	2.72007e6	0.947507	0.473754	−	0.880657i	$-0.342899\pi$
$383$	2.72007e6	0.947507	0.473754	+	0.880657i	$0.342899\pi$
$384$	0	0
$385$	−1.97827e6	−0.680194
$386$	0	0
$387$	−1.45451e6	−0.493672
$388$	0	0
$389$	−4.65849e6	−1.56089	−0.780443	−	0.625226i	$-0.785007\pi$
$389$	−4.65849e6	−1.56089	−0.780443	+	0.625226i	$0.785007\pi$
$390$	0	0
$391$	−3.11055e6	−1.02895
$392$	0	0
$393$	−2.23430e6	−0.729725
$394$	0	0
$395$	−5.71617e6	−1.84337
$396$	0	0
$397$	−1.38470e6	−0.440941	−0.220470	−	0.975394i	$-0.570759\pi$
$397$	−1.38470e6	−0.440941	−0.220470	+	0.975394i	$0.570759\pi$
$398$	0	0
$399$	3.74758e6	1.17847
$400$	0	0
$401$	3.27220e6	1.01620	0.508099	−	0.861299i	$-0.330348\pi$
$401$	3.27220e6	1.01620	0.508099	+	0.861299i	$0.330348\pi$
$402$	0	0
$403$	−6.08225e6	−1.86553
$404$	0	0
$405$	−517687.	−0.156830
$406$	0	0
$407$	−705935.	−0.211241
$408$	0	0
$409$	−77653.1	−0.0229536	−0.0114768	−	0.999934i	$-0.503653\pi$
$409$	−77653.1	−0.0229536	−0.0114768	+	0.999934i	$0.503653\pi$
$410$	0	0
$411$	476570.	0.139163
$412$	0	0
$413$	8.84017e6	2.55027
$414$	0	0
$415$	−4.37548e6	−1.24711
$416$	0	0
$417$	1.83148e6	0.515778
$418$	0	0
$419$	3.96236e6	1.10260	0.551301	−	0.834306i	$-0.314131\pi$
$419$	3.96236e6	1.10260	0.551301	+	0.834306i	$0.314131\pi$
$420$	0	0
$421$	−4.46374e6	−1.22742	−0.613711	−	0.789531i	$-0.710324\pi$
$421$	−4.46374e6	−1.22742	−0.613711	+	0.789531i	$0.710324\pi$
$422$	0	0
$423$	−668910.	−0.181768
$424$	0	0
$425$	−2.30957e6	−0.620238
$426$	0	0
$427$	−8.18127e6	−2.17145
$428$	0	0
$429$	741192.	0.194441
$430$	0	0
$431$	1.54839e6	0.401502	0.200751	−	0.979642i	$-0.435662\pi$
$431$	1.54839e6	0.401502	0.200751	+	0.979642i	$0.435662\pi$
$432$	0	0
$433$	−6.33554e6	−1.62392	−0.811959	−	0.583715i	$-0.801599\pi$
$433$	−6.33554e6	−1.62392	−0.811959	+	0.583715i	$0.801599\pi$
$434$	0	0
$435$	3.04469e6	0.771472
$436$	0	0
$437$	8.39237e6	2.10223
$438$	0	0
$439$	4.89838e6	1.21309	0.606543	−	0.795051i	$-0.292556\pi$
$439$	4.89838e6	1.21309	0.606543	+	0.795051i	$0.292556\pi$
$440$	0	0
$441$	2.11631e6	0.518182
$442$	0	0
$443$	5.48221e6	1.32723	0.663615	−	0.748074i	$-0.269021\pi$
$443$	5.48221e6	1.32723	0.663615	+	0.748074i	$0.269021\pi$
$444$	0	0
$445$	−4.53607e6	−1.08588
$446$	0	0
$447$	3.31671e6	0.785124
$448$	0	0
$449$	−1.87244e6	−0.438321	−0.219161	−	0.975689i	$-0.570332\pi$
$449$	−1.87244e6	−0.438321	−0.219161	+	0.975689i	$0.570332\pi$
$450$	0	0
$451$	−1.90446e6	−0.440890
$452$	0	0
$453$	2.01272e6	0.460827
$454$	0	0
$455$	1.11276e7	2.51984
$456$	0	0
$457$	2.98332e6	0.668204	0.334102	−	0.942537i	$-0.391567\pi$
$457$	2.98332e6	0.668204	0.334102	+	0.942537i	$0.391567\pi$
$458$	0	0
$459$	542983.	0.120297
$460$	0	0
$461$	4.02921e6	0.883015	0.441508	−	0.897258i	$-0.354444\pi$
$461$	4.02921e6	0.883015	0.441508	+	0.897258i	$0.354444\pi$
$462$	0	0
$463$	−5.16916e6	−1.12064	−0.560321	−	0.828275i	$-0.689322\pi$
$463$	−5.16916e6	−1.12064	−0.560321	+	0.828275i	$0.689322\pi$
$464$	0	0
$465$	−6.34601e6	−1.36103
$466$	0	0
$467$	−4.83940e6	−1.02683	−0.513416	−	0.858140i	$-0.671620\pi$
$467$	−4.83940e6	−1.02683	−0.513416	+	0.858140i	$0.671620\pi$
$468$	0	0
$469$	5.76995e6	1.21127
$470$	0	0
$471$	−2.50884e6	−0.521099
$472$	0	0
$473$	2.17278e6	0.446544
$474$	0	0
$475$	6.23129e6	1.26720
$476$	0	0
$477$	−700317.	−0.140928
$478$	0	0
$479$	2.22007e6	0.442108	0.221054	−	0.975262i	$-0.429050\pi$
$479$	2.22007e6	0.442108	0.221054	+	0.975262i	$0.429050\pi$
$480$	0	0
$481$	3.97084e6	0.782564
$482$	0	0
$483$	7.78795e6	1.51899
$484$	0	0
$485$	−4.27831e6	−0.825882
$486$	0	0
$487$	−4.24421e6	−0.810914	−0.405457	−	0.914114i	$-0.632888\pi$
$487$	−4.24421e6	−0.810914	−0.405457	+	0.914114i	$0.632888\pi$
$488$	0	0
$489$	4.12687e6	0.780457
$490$	0	0
$491$	−1.99328e6	−0.373135	−0.186567	−	0.982442i	$-0.559736\pi$
$491$	−1.99328e6	−0.373135	−0.186567	+	0.982442i	$0.559736\pi$
$492$	0	0
$493$	−3.19347e6	−0.591759
$494$	0	0
$495$	773335.	0.141858
$496$	0	0
$497$	−2.86320e6	−0.519949
$498$	0	0
$499$	−4.89841e6	−0.880651	−0.440326	−	0.897838i	$-0.645137\pi$
$499$	−4.89841e6	−0.880651	−0.440326	+	0.897838i	$0.645137\pi$
$500$	0	0
$501$	−4.00136e6	−0.712218
$502$	0	0
$503$	−78742.0	−0.0138767	−0.00693835	−	0.999976i	$-0.502209\pi$
$503$	−78742.0	−0.0138767	−0.00693835	+	0.999976i	$0.502209\pi$
$504$	0	0
$505$	−7.05632e6	−1.23126
$506$	0	0
$507$	−827521.	−0.142975
$508$	0	0
$509$	−2.36331e6	−0.404321	−0.202161	−	0.979352i	$-0.564796\pi$
$509$	−2.36331e6	−0.404321	−0.202161	+	0.979352i	$0.564796\pi$
$510$	0	0
$511$	1.61419e7	2.73466
$512$	0	0
$513$	−1.46499e6	−0.245776
$514$	0	0
$515$	−1.44590e6	−0.240226
$516$	0	0
$517$	999236.	0.164415
$518$	0	0
$519$	987892.	0.160987
$520$	0	0
$521$	−4.14846e6	−0.669564	−0.334782	−	0.942296i	$-0.608663\pi$
$521$	−4.14846e6	−0.669564	−0.334782	+	0.942296i	$0.608663\pi$
$522$	0	0
$523$	1.13161e7	1.80902	0.904511	−	0.426450i	$-0.140236\pi$
$523$	1.13161e7	1.80902	0.904511	+	0.426450i	$0.140236\pi$
$524$	0	0
$525$	5.78251e6	0.915626
$526$	0	0
$527$	6.65610e6	1.04398
$528$	0	0
$529$	1.10041e7	1.70968
$530$	0	0
$531$	−3.45576e6	−0.531872
$532$	0	0
$533$	1.07125e7	1.63332
$534$	0	0
$535$	1.78637e7	2.69828
$536$	0	0
$537$	3.25272e6	0.486755
$538$	0	0
$539$	−3.16140e6	−0.468714
$540$	0	0
$541$	−4.86131e6	−0.714102	−0.357051	−	0.934085i	$-0.616218\pi$
$541$	−4.86131e6	−0.714102	−0.357051	+	0.934085i	$0.616218\pi$
$542$	0	0
$543$	6.42215e6	0.934719
$544$	0	0
$545$	5.83587e6	0.841616
$546$	0	0
$547$	−1.10192e6	−0.157464	−0.0787321	−	0.996896i	$-0.525087\pi$
$547$	−1.10192e6	−0.157464	−0.0787321	+	0.996896i	$0.525087\pi$
$548$	0	0
$549$	3.19818e6	0.452869
$550$	0	0
$551$	8.61608e6	1.20901
$552$	0	0
$553$	−1.50110e7	−2.08736
$554$	0	0
$555$	4.14304e6	0.570935
$556$	0	0
$557$	5.73124e6	0.782727	0.391364	−	0.920236i	$-0.372003\pi$
$557$	5.73124e6	0.782727	0.391364	+	0.920236i	$0.372003\pi$
$558$	0	0
$559$	−1.22218e7	−1.65426
$560$	0	0
$561$	−811123.	−0.108813
$562$	0	0
$563$	−1.25359e7	−1.66680	−0.833401	−	0.552668i	$-0.813610\pi$
$563$	−1.25359e7	−1.66680	−0.833401	+	0.552668i	$0.813610\pi$
$564$	0	0
$565$	−7.05978e6	−0.930400
$566$	0	0
$567$	−1.35948e6	−0.177588
$568$	0	0
$569$	−871238.	−0.112812	−0.0564061	−	0.998408i	$-0.517964\pi$
$569$	−871238.	−0.112812	−0.0564061	+	0.998408i	$0.517964\pi$
$570$	0	0
$571$	−745279.	−0.0956597	−0.0478298	−	0.998855i	$-0.515231\pi$
$571$	−745279.	−0.0956597	−0.0478298	+	0.998855i	$0.515231\pi$
$572$	0	0
$573$	1.30311e6	0.165803
$574$	0	0
$575$	1.29494e7	1.63336
$576$	0	0
$577$	−8.84239e6	−1.10568	−0.552841	−	0.833287i	$-0.686456\pi$
$577$	−8.84239e6	−1.10568	−0.552841	+	0.833287i	$0.686456\pi$
$578$	0	0
$579$	1.59280e6	0.197454
$580$	0	0
$581$	−1.14903e7	−1.41218
$582$	0	0
$583$	1.04615e6	0.127475
$584$	0	0
$585$	−4.34996e6	−0.525527
$586$	0	0
$587$	−8.61341e6	−1.03176	−0.515881	−	0.856660i	$-0.672535\pi$
$587$	−8.61341e6	−1.03176	−0.515881	+	0.856660i	$0.672535\pi$
$588$	0	0
$589$	−1.79584e7	−2.13294
$590$	0	0
$591$	−753043.	−0.0886851
$592$	0	0
$593$	−1.51513e7	−1.76935	−0.884673	−	0.466212i	$-0.845618\pi$
$593$	−1.51513e7	−1.76935	−0.884673	+	0.466212i	$0.845618\pi$
$594$	0	0
$595$	−1.21775e7	−1.41015
$596$	0	0
$597$	−175617.	−0.0201665
$598$	0	0
$599$	−4.97003e6	−0.565969	−0.282984	−	0.959125i	$-0.591324\pi$
$599$	−4.97003e6	−0.565969	−0.282984	+	0.959125i	$0.591324\pi$
$600$	0	0
$601$	−1.74640e6	−0.197223	−0.0986116	−	0.995126i	$-0.531440\pi$
$601$	−1.74640e6	−0.197223	−0.0986116	+	0.995126i	$0.531440\pi$
$602$	0	0
$603$	−2.25556e6	−0.252617
$604$	0	0
$605$	−1.15523e6	−0.128316
$606$	0	0
$607$	5.19606e6	0.572403	0.286202	−	0.958169i	$-0.407607\pi$
$607$	5.19606e6	0.572403	0.286202	+	0.958169i	$0.407607\pi$
$608$	0	0
$609$	7.99555e6	0.873585
$610$	0	0
$611$	−5.62064e6	−0.609092
$612$	0	0
$613$	−1.51940e6	−0.163313	−0.0816565	−	0.996661i	$-0.526021\pi$
$613$	−1.51940e6	−0.163313	−0.0816565	+	0.996661i	$0.526021\pi$
$614$	0	0
$615$	1.11770e7	1.19162
$616$	0	0
$617$	1.03249e7	1.09188	0.545940	−	0.837824i	$-0.316173\pi$
$617$	1.03249e7	1.09188	0.545940	+	0.837824i	$0.316173\pi$
$618$	0	0
$619$	−1.65645e7	−1.73761	−0.868804	−	0.495157i	$-0.835111\pi$
$619$	−1.65645e7	−1.73761	−0.868804	+	0.495157i	$0.835111\pi$
$620$	0	0
$621$	−3.04443e6	−0.316794
$622$	0	0
$623$	−1.19120e7	−1.22960
$624$	0	0
$625$	−9.84070e6	−1.00769
$626$	0	0
$627$	2.18844e6	0.222313
$628$	0	0
$629$	−4.34548e6	−0.437937
$630$	0	0
$631$	9.17544e6	0.917388	0.458694	−	0.888594i	$-0.348317\pi$
$631$	9.17544e6	0.917388	0.458694	+	0.888594i	$0.348317\pi$
$632$	0	0
$633$	2.36382e6	0.234480
$634$	0	0
$635$	−2.60735e7	−2.56605
$636$	0	0
$637$	1.77827e7	1.73639
$638$	0	0
$639$	1.11927e6	0.108438
$640$	0	0
$641$	1.50265e7	1.44448	0.722241	−	0.691641i	$-0.243112\pi$
$641$	1.50265e7	1.44448	0.722241	+	0.691641i	$0.243112\pi$
$642$	0	0
$643$	−1.77305e7	−1.69119	−0.845596	−	0.533823i	$-0.820755\pi$
$643$	−1.77305e7	−1.69119	−0.845596	+	0.533823i	$0.820755\pi$
$644$	0	0
$645$	−1.27518e7	−1.20690
$646$	0	0
$647$	−1.35108e7	−1.26888	−0.634438	−	0.772974i	$-0.718768\pi$
$647$	−1.35108e7	−1.26888	−0.634438	+	0.772974i	$0.718768\pi$
$648$	0	0
$649$	5.16231e6	0.481097
$650$	0	0
$651$	−1.66650e7	−1.54118
$652$	0	0
$653$	−8.94301e6	−0.820731	−0.410365	−	0.911921i	$-0.634599\pi$
$653$	−8.94301e6	−0.820731	−0.410365	+	0.911921i	$0.634599\pi$
$654$	0	0
$655$	−1.95882e7	−1.78399
$656$	0	0
$657$	−6.31012e6	−0.570328
$658$	0	0
$659$	−8.39875e6	−0.753358	−0.376679	−	0.926344i	$-0.622934\pi$
$659$	−8.39875e6	−0.753358	−0.376679	+	0.926344i	$0.622934\pi$
$660$	0	0
$661$	−1.71552e7	−1.52718	−0.763592	−	0.645699i	$-0.776566\pi$
$661$	−1.71552e7	−1.52718	−0.763592	+	0.645699i	$0.776566\pi$
$662$	0	0
$663$	4.56251e6	0.403107
$664$	0	0
$665$	3.28553e7	2.88105
$666$	0	0
$667$	1.79053e7	1.55836
$668$	0	0
$669$	3.50926e6	0.303145
$670$	0	0
$671$	−4.77753e6	−0.409635
$672$	0	0
$673$	1.22119e7	1.03931	0.519657	−	0.854375i	$-0.326060\pi$
$673$	1.22119e7	1.03931	0.519657	+	0.854375i	$0.326060\pi$
$674$	0	0
$675$	−2.26047e6	−0.190959
$676$	0	0
$677$	−1.31557e7	−1.10317	−0.551583	−	0.834120i	$-0.685976\pi$
$677$	−1.31557e7	−1.10317	−0.551583	+	0.834120i	$0.685976\pi$
$678$	0	0
$679$	−1.12351e7	−0.935197
$680$	0	0
$681$	2.54750e6	0.210497
$682$	0	0
$683$	−2.33238e7	−1.91315	−0.956573	−	0.291492i	$-0.905848\pi$
$683$	−2.33238e7	−1.91315	−0.956573	+	0.291492i	$0.905848\pi$
$684$	0	0
$685$	4.17813e6	0.340216
$686$	0	0
$687$	4.45051e6	0.359764
$688$	0	0
$689$	−5.88454e6	−0.472242
$690$	0	0
$691$	1.96512e7	1.56565	0.782824	−	0.622244i	$-0.213779\pi$
$691$	1.96512e7	1.56565	0.782824	+	0.622244i	$0.213779\pi$
$692$	0	0
$693$	2.03082e6	0.160635
$694$	0	0
$695$	1.60568e7	1.26094
$696$	0	0
$697$	−1.17232e7	−0.914035
$698$	0	0
$699$	−1.33846e6	−0.103613
$700$	0	0
$701$	−7.92477e6	−0.609104	−0.304552	−	0.952496i	$-0.598507\pi$
$701$	−7.92477e6	−0.609104	−0.304552	+	0.952496i	$0.598507\pi$
$702$	0	0
$703$	1.17243e7	0.894741
$704$	0	0
$705$	−5.86438e6	−0.444375
$706$	0	0
$707$	−1.85303e7	−1.39423
$708$	0	0
$709$	1.55724e7	1.16343	0.581713	−	0.813394i	$-0.302383\pi$
$709$	1.55724e7	1.16343	0.581713	+	0.813394i	$0.302383\pi$
$710$	0	0
$711$	5.86804e6	0.435330
$712$	0	0
$713$	−3.73198e7	−2.74926
$714$	0	0
$715$	6.49809e6	0.475358
$716$	0	0
$717$	−3.00216e6	−0.218090
$718$	0	0
$719$	−4.55331e6	−0.328477	−0.164239	−	0.986421i	$-0.552517\pi$
$719$	−4.55331e6	−0.328477	−0.164239	+	0.986421i	$0.552517\pi$
$720$	0	0
$721$	−3.79702e6	−0.272022
$722$	0	0
$723$	−1.06260e7	−0.756001
$724$	0	0
$725$	1.32946e7	0.939357
$726$	0	0
$727$	−1.88184e7	−1.32052	−0.660262	−	0.751036i	$-0.729555\pi$
$727$	−1.88184e7	−1.32052	−0.660262	+	0.751036i	$0.729555\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	1.33749e7	0.925756
$732$	0	0
$733$	−2.47848e7	−1.70383	−0.851915	−	0.523680i	$-0.824559\pi$
$733$	−2.47848e7	−1.70383	−0.851915	+	0.523680i	$0.824559\pi$
$734$	0	0
$735$	1.85538e7	1.26682
$736$	0	0
$737$	3.36942e6	0.228500
$738$	0	0
$739$	−8.51943e6	−0.573852	−0.286926	−	0.957953i	$-0.592633\pi$
$739$	−8.51943e6	−0.573852	−0.286926	+	0.957953i	$0.592633\pi$
$740$	0	0
$741$	−1.23098e7	−0.823581
$742$	0	0
$743$	3.10500e6	0.206343	0.103171	−	0.994664i	$-0.467101\pi$
$743$	3.10500e6	0.206343	0.103171	+	0.994664i	$0.467101\pi$
$744$	0	0
$745$	2.90778e7	1.91943
$746$	0	0
$747$	4.49173e6	0.294518
$748$	0	0
$749$	4.69112e7	3.05543
$750$	0	0
$751$	4.41209e6	0.285459	0.142730	−	0.989762i	$-0.454412\pi$
$751$	4.41209e6	0.285459	0.142730	+	0.989762i	$0.454412\pi$
$752$	0	0
$753$	−1.60952e7	−1.03445
$754$	0	0
$755$	1.76457e7	1.12660
$756$	0	0
$757$	2.14787e6	0.136229	0.0681144	−	0.997678i	$-0.478302\pi$
$757$	2.14787e6	0.136229	0.0681144	+	0.997678i	$0.478302\pi$
$758$	0	0
$759$	4.54785e6	0.286551
$760$	0	0
$761$	1.38048e7	0.864107	0.432054	−	0.901848i	$-0.357789\pi$
$761$	1.38048e7	0.864107	0.432054	+	0.901848i	$0.357789\pi$
$762$	0	0
$763$	1.53253e7	0.953013
$764$	0	0
$765$	4.76037e6	0.294095
$766$	0	0
$767$	−2.90377e7	−1.78227
$768$	0	0
$769$	−2.59572e7	−1.58286	−0.791428	−	0.611262i	$-0.790662\pi$
$769$	−2.59572e7	−1.58286	−0.791428	+	0.611262i	$0.790662\pi$
$770$	0	0
$771$	2.36139e6	0.143064
$772$	0	0
$773$	−1.03379e7	−0.622275	−0.311137	−	0.950365i	$-0.600710\pi$
$773$	−1.03379e7	−0.622275	−0.311137	+	0.950365i	$0.600710\pi$
$774$	0	0
$775$	−2.77098e7	−1.65722
$776$	0	0
$777$	1.08799e7	0.646504
$778$	0	0
$779$	3.16295e7	1.86745
$780$	0	0
$781$	−1.67199e6	−0.0980861
$782$	0	0
$783$	−3.12558e6	−0.182191
$784$	0	0
$785$	−2.19952e7	−1.27395
$786$	0	0
$787$	2.09532e6	0.120591	0.0602954	−	0.998181i	$-0.480796\pi$
$787$	2.09532e6	0.120591	0.0602954	+	0.998181i	$0.480796\pi$
$788$	0	0
$789$	5.86185e6	0.335229
$790$	0	0
$791$	−1.85394e7	−1.05355
$792$	0	0
$793$	2.68733e7	1.51753
$794$	0	0
$795$	−6.13973e6	−0.344533
$796$	0	0
$797$	−7.20950e6	−0.402031	−0.201015	−	0.979588i	$-0.564424\pi$
$797$	−7.20950e6	−0.402031	−0.201015	+	0.979588i	$0.564424\pi$
$798$	0	0
$799$	6.15094e6	0.340859
$800$	0	0
$801$	4.65659e6	0.256440
$802$	0	0
$803$	9.42623e6	0.515881
$804$	0	0
$805$	6.82775e7	3.71354
$806$	0	0
$807$	−2.52944e6	−0.136723
$808$	0	0
$809$	4.89636e6	0.263028	0.131514	−	0.991314i	$-0.458016\pi$
$809$	4.89636e6	0.263028	0.131514	+	0.991314i	$0.458016\pi$
$810$	0	0
$811$	−2.39167e7	−1.27688	−0.638438	−	0.769673i	$-0.720419\pi$
$811$	−2.39167e7	−1.27688	−0.638438	+	0.769673i	$0.720419\pi$
$812$	0	0
$813$	8.48359e6	0.450146
$814$	0	0
$815$	3.61806e7	1.90801
$816$	0	0
$817$	−3.60859e7	−1.89140
$818$	0	0
$819$	−1.14233e7	−0.595087
$820$	0	0
$821$	2.03256e7	1.05241	0.526204	−	0.850358i	$-0.323615\pi$
$821$	2.03256e7	1.05241	0.526204	+	0.850358i	$0.323615\pi$
$822$	0	0
$823$	−1.90383e7	−0.979782	−0.489891	−	0.871784i	$-0.662964\pi$
$823$	−1.90383e7	−0.979782	−0.489891	+	0.871784i	$0.662964\pi$
$824$	0	0
$825$	3.37676e6	0.172729
$826$	0	0
$827$	−3.27003e6	−0.166260	−0.0831301	−	0.996539i	$-0.526492\pi$
$827$	−3.27003e6	−0.166260	−0.0831301	+	0.996539i	$0.526492\pi$
$828$	0	0
$829$	2.41818e7	1.22209	0.611044	−	0.791596i	$-0.290750\pi$
$829$	2.41818e7	1.22209	0.611044	+	0.791596i	$0.290750\pi$
$830$	0	0
$831$	5.61121e6	0.281873
$832$	0	0
$833$	−1.94604e7	−0.971718
$834$	0	0
$835$	−3.50802e7	−1.74119
$836$	0	0
$837$	6.51462e6	0.321422
$838$	0	0
$839$	2.29480e7	1.12549	0.562744	−	0.826631i	$-0.309746\pi$
$839$	2.29480e7	1.12549	0.562744	+	0.826631i	$0.309746\pi$
$840$	0	0
$841$	−2.12853e6	−0.103774
$842$	0	0
$843$	−3.11368e6	−0.150906
$844$	0	0
$845$	−7.25494e6	−0.349536
$846$	0	0
$847$	−3.03370e6	−0.145300
$848$	0	0
$849$	1.79541e7	0.854858
$850$	0	0
$851$	2.43645e7	1.15328
$852$	0	0
$853$	−2.24497e7	−1.05643	−0.528213	−	0.849112i	$-0.677138\pi$
$853$	−2.24497e7	−1.05643	−0.528213	+	0.849112i	$0.677138\pi$
$854$	0	0
$855$	−1.28436e7	−0.600860
$856$	0	0
$857$	3.11161e7	1.44721	0.723607	−	0.690212i	$-0.242483\pi$
$857$	3.11161e7	1.44721	0.723607	+	0.690212i	$0.242483\pi$
$858$	0	0
$859$	−5.87536e6	−0.271676	−0.135838	−	0.990731i	$-0.543373\pi$
$859$	−5.87536e6	−0.271676	−0.135838	+	0.990731i	$0.543373\pi$
$860$	0	0
$861$	2.93515e7	1.34934
$862$	0	0
$863$	3.35381e7	1.53289	0.766445	−	0.642310i	$-0.222024\pi$
$863$	3.35381e7	1.53289	0.766445	+	0.642310i	$0.222024\pi$
$864$	0	0
$865$	8.66092e6	0.393572
$866$	0	0
$867$	7.78573e6	0.351764
$868$	0	0
$869$	−8.76583e6	−0.393771
$870$	0	0
$871$	−1.89528e7	−0.846501
$872$	0	0
$873$	4.39198e6	0.195041
$874$	0	0
$875$	−395858.	−0.0174791
$876$	0	0
$877$	−3.74839e7	−1.64568	−0.822841	−	0.568272i	$-0.807612\pi$
$877$	−3.74839e7	−1.64568	−0.822841	+	0.568272i	$0.807612\pi$
$878$	0	0
$879$	−8.88913e6	−0.388049
$880$	0	0
$881$	1.19981e7	0.520801	0.260400	−	0.965501i	$-0.416145\pi$
$881$	1.19981e7	0.520801	0.260400	+	0.965501i	$0.416145\pi$
$882$	0	0
$883$	−1.24120e6	−0.0535724	−0.0267862	−	0.999641i	$-0.508527\pi$
$883$	−1.24120e6	−0.0535724	−0.0267862	+	0.999641i	$0.508527\pi$
$884$	0	0
$885$	−3.02969e7	−1.30029
$886$	0	0
$887$	−5.93499e6	−0.253286	−0.126643	−	0.991948i	$-0.540420\pi$
$887$	−5.93499e6	−0.253286	−0.126643	+	0.991948i	$0.540420\pi$
$888$	0	0
$889$	−6.84707e7	−2.90570
$890$	0	0
$891$	−793881.	−0.0335013
$892$	0	0
$893$	−1.65954e7	−0.696403
$894$	0	0
$895$	2.85168e7	1.18999
$896$	0	0
$897$	−2.55814e7	−1.06156
$898$	0	0
$899$	−3.83147e7	−1.58112
$900$	0	0
$901$	6.43974e6	0.264275
$902$	0	0
$903$	−3.34870e7	−1.36665
$904$	0	0
$905$	5.63035e7	2.28514
$906$	0	0
$907$	−3.36279e7	−1.35732	−0.678659	−	0.734454i	$-0.737439\pi$
$907$	−3.36279e7	−1.35732	−0.678659	+	0.734454i	$0.737439\pi$
$908$	0	0
$909$	7.24379e6	0.290774
$910$	0	0
$911$	1.03682e7	0.413913	0.206957	−	0.978350i	$-0.433644\pi$
$911$	1.03682e7	0.413913	0.206957	+	0.978350i	$0.433644\pi$
$912$	0	0
$913$	−6.70987e6	−0.266402
$914$	0	0
$915$	2.80387e7	1.10715
$916$	0	0
$917$	−5.14399e7	−2.02012
$918$	0	0
$919$	2.96993e6	0.116000	0.0580000	−	0.998317i	$-0.481528\pi$
$919$	2.96993e6	0.116000	0.0580000	+	0.998317i	$0.481528\pi$
$920$	0	0
$921$	−3.37121e6	−0.130960
$922$	0	0
$923$	9.40486e6	0.363369
$924$	0	0
$925$	1.80905e7	0.695180
$926$	0	0
$927$	1.48431e6	0.0567318
$928$	0	0
$929$	−1.89317e6	−0.0719696	−0.0359848	−	0.999352i	$-0.511457\pi$
$929$	−1.89317e6	−0.0719696	−0.0359848	+	0.999352i	$0.511457\pi$
$930$	0	0
$931$	5.25050e7	1.98530
$932$	0	0
$933$	−1.67572e7	−0.630227
$934$	0	0
$935$	−7.11117e6	−0.266019
$936$	0	0
$937$	−5.24929e7	−1.95322	−0.976610	−	0.215016i	$-0.931019\pi$
$937$	−5.24929e7	−1.95322	−0.976610	+	0.215016i	$0.931019\pi$
$938$	0	0
$939$	1.47505e7	0.545938
$940$	0	0
$941$	−1.75398e7	−0.645730	−0.322865	−	0.946445i	$-0.604646\pi$
$941$	−1.75398e7	−0.645730	−0.322865	+	0.946445i	$0.604646\pi$
$942$	0	0
$943$	6.57301e7	2.40705
$944$	0	0
$945$	−1.19186e7	−0.434157
$946$	0	0
$947$	−2.81654e7	−1.02056	−0.510282	−	0.860007i	$-0.670459\pi$
$947$	−2.81654e7	−1.02056	−0.510282	+	0.860007i	$0.670459\pi$
$948$	0	0
$949$	−5.30220e7	−1.91113
$950$	0	0
$951$	−4.34058e6	−0.155631
$952$	0	0
$953$	−3.62490e7	−1.29290	−0.646449	−	0.762957i	$-0.723747\pi$
$953$	−3.62490e7	−1.29290	−0.646449	+	0.762957i	$0.723747\pi$
$954$	0	0
$955$	1.14244e7	0.405346
$956$	0	0
$957$	4.66908e6	0.164798
$958$	0	0
$959$	1.09720e7	0.385248
$960$	0	0
$961$	5.12296e7	1.78942
$962$	0	0
$963$	−1.83383e7	−0.637227
$964$	0	0
$965$	1.39642e7	0.482723
$966$	0	0
$967$	−2.46567e6	−0.0847946	−0.0423973	−	0.999101i	$-0.513500\pi$
$967$	−2.46567e6	−0.0847946	−0.0423973	+	0.999101i	$0.513500\pi$
$968$	0	0
$969$	1.34712e7	0.460891
$970$	0	0
$971$	3.54087e7	1.20521	0.602603	−	0.798041i	$-0.294130\pi$
$971$	3.54087e7	1.20521	0.602603	+	0.798041i	$0.294130\pi$
$972$	0	0
$973$	4.21660e7	1.42784
$974$	0	0
$975$	−1.89940e7	−0.639891
$976$	0	0
$977$	3.29281e7	1.10365	0.551823	−	0.833961i	$-0.313932\pi$
$977$	3.29281e7	1.10365	0.551823	+	0.833961i	$0.313932\pi$
$978$	0	0
$979$	−6.95614e6	−0.231959
$980$	0	0
$981$	−5.99092e6	−0.198756
$982$	0	0
$983$	2.46982e7	0.815232	0.407616	−	0.913153i	$-0.366360\pi$
$983$	2.46982e7	0.815232	0.407616	+	0.913153i	$0.366360\pi$
$984$	0	0
$985$	−6.60198e6	−0.216812
$986$	0	0
$987$	−1.54002e7	−0.503193
$988$	0	0
$989$	−7.49911e7	−2.43792
$990$	0	0
$991$	1.51301e7	0.489393	0.244696	−	0.969600i	$-0.421312\pi$
$991$	1.51301e7	0.489393	0.244696	+	0.969600i	$0.421312\pi$
$992$	0	0
$993$	2.00898e7	0.646550
$994$	0	0
$995$	−1.53964e6	−0.0493017
$996$	0	0
$997$	4.25283e7	1.35500	0.677502	−	0.735521i	$-0.263063\pi$
$997$	4.25283e7	1.35500	0.677502	+	0.735521i	$0.263063\pi$
$998$	0	0
$999$	−4.25311e6	−0.134832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	528.6.a.z.1.1		4
4.3	odd	2		264.6.a.i.1.1	✓	4
12.11	even	2		792.6.a.m.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
264.6.a.i.1.1	✓	4	4.3	odd	2
528.6.a.z.1.1		4	1.1	even	1	trivial
792.6.a.m.1.4		4	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 528 = 2^{4} \cdot 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	528.a (trivial)

\(f(q)\)	\(=\)	\(q-9.00000 q^{3} -78.9037 q^{5} -207.206 q^{7} +81.0000 q^{9} +O(q^{10})\)	\(q-9.00000 q^{3} -78.9037 q^{5} -207.206 q^{7} +81.0000 q^{9} -121.000 q^{11} +680.617 q^{13} +710.133 q^{15} -744.833 q^{17} +2009.58 q^{19} +1864.85 q^{21} +4176.17 q^{23} +3100.79 q^{25} -729.000 q^{27} +4287.50 q^{29} -8936.37 q^{31} +1089.00 q^{33} +16349.3 q^{35} +5834.17 q^{37} -6125.56 q^{39} +15739.3 q^{41} -17956.9 q^{43} -6391.20 q^{45} -8258.15 q^{47} +26127.3 q^{49} +6703.49 q^{51} -8645.88 q^{53} +9547.34 q^{55} -18086.3 q^{57} -42663.7 q^{59} +39483.8 q^{61} -16783.7 q^{63} -53703.2 q^{65} -27846.5 q^{67} -37585.5 q^{69} +13818.1 q^{71} -77902.8 q^{73} -27907.1 q^{75} +25071.9 q^{77} +72444.9 q^{79} +6561.00 q^{81} +55453.5 q^{83} +58770.0 q^{85} -38587.5 q^{87} +57488.7 q^{89} -141028. q^{91} +80427.4 q^{93} -158564. q^{95} +54222.0 q^{97} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 36 q^{3} - 44 q^{5} - 38 q^{7} + 324 q^{9}+O(q^{10}) \) 4 * q - 36 * q^3 - 44 * q^5 - 38 * q^7 + 324 * q^9	\( 4 q - 36 q^{3} - 44 q^{5} - 38 q^{7} + 324 q^{9} - 484 q^{11} + 288 q^{13} + 396 q^{15} + 418 q^{17} + 10 q^{19} + 342 q^{21} - 576 q^{23} + 4932 q^{25} - 2916 q^{27} + 11930 q^{29} - 9968 q^{31} + 4356 q^{33} + 428 q^{35} + 13740 q^{37} - 2592 q^{39} + 29766 q^{41} - 25650 q^{43} - 3564 q^{45} + 5776 q^{47} + 39888 q^{49} - 3762 q^{51} - 7840 q^{53} + 5324 q^{55} - 90 q^{57} - 28800 q^{59} + 33932 q^{61} - 3078 q^{63} + 33216 q^{65} - 83056 q^{67} + 5184 q^{69} - 21336 q^{71} + 27044 q^{73} - 44388 q^{75} + 4598 q^{77} - 102542 q^{79} + 26244 q^{81} - 64996 q^{83} - 12132 q^{85} - 107370 q^{87} + 37888 q^{89} - 273612 q^{91} + 89712 q^{93} - 254380 q^{95} - 20996 q^{97} - 39204 q^{99}+O(q^{100}) \) 4 * q - 36 * q^3 - 44 * q^5 - 38 * q^7 + 324 * q^9 - 484 * q^11 + 288 * q^13 + 396 * q^15 + 418 * q^17 + 10 * q^19 + 342 * q^21 - 576 * q^23 + 4932 * q^25 - 2916 * q^27 + 11930 * q^29 - 9968 * q^31 + 4356 * q^33 + 428 * q^35 + 13740 * q^37 - 2592 * q^39 + 29766 * q^41 - 25650 * q^43 - 3564 * q^45 + 5776 * q^47 + 39888 * q^49 - 3762 * q^51 - 7840 * q^53 + 5324 * q^55 - 90 * q^57 - 28800 * q^59 + 33932 * q^61 - 3078 * q^63 + 33216 * q^65 - 83056 * q^67 + 5184 * q^69 - 21336 * q^71 + 27044 * q^73 - 44388 * q^75 + 4598 * q^77 - 102542 * q^79 + 26244 * q^81 - 64996 * q^83 - 12132 * q^85 - 107370 * q^87 + 37888 * q^89 - 273612 * q^91 + 89712 * q^93 - 254380 * q^95 - 20996 * q^97 - 39204 * q^99

Introduction

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