LMFDB - Embedded newform 528.6.a.r.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:	$2$
Coefficient field:	$\Q(\sqrt{409}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 102 $ x^2 - x - 102
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 264)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$10.6119$ 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)

$f(q)$	$=$	$q+9.00000 q^{3} -75.6712 q^{5} -64.4475 q^{7} +81.0000 q^{9} +O(q^{10})$	$q+9.00000 q^{3} -75.6712 q^{5} -64.4475 q^{7} +81.0000 q^{9} +121.000 q^{11} +753.041 q^{13} -681.041 q^{15} +282.621 q^{17} -1326.78 q^{19} -580.027 q^{21} +1552.76 q^{23} +2601.14 q^{25} +729.000 q^{27} +1025.25 q^{29} +940.895 q^{31} +1089.00 q^{33} +4876.82 q^{35} +5812.10 q^{37} +6777.37 q^{39} -10333.5 q^{41} +5841.39 q^{43} -6129.37 q^{45} +8213.27 q^{47} -12653.5 q^{49} +2543.59 q^{51} -29046.2 q^{53} -9156.22 q^{55} -11941.0 q^{57} -15409.5 q^{59} -2109.77 q^{61} -5220.25 q^{63} -56983.6 q^{65} -22144.1 q^{67} +13974.8 q^{69} -35135.6 q^{71} -38566.7 q^{73} +23410.2 q^{75} -7798.15 q^{77} +25093.6 q^{79} +6561.00 q^{81} +19125.5 q^{83} -21386.3 q^{85} +9227.26 q^{87} -86961.8 q^{89} -48531.6 q^{91} +8468.05 q^{93} +100399. q^{95} -112243. q^{97} +9801.00 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 18 q^{3} - 30 q^{5} - 48 q^{7} + 162 q^{9}+O(q^{10}) $ 2 * q + 18 * q^3 - 30 * q^5 - 48 * q^7 + 162 * q^9	$ 2 q + 18 q^{3} - 30 q^{5} - 48 q^{7} + 162 q^{9} + 242 q^{11} + 414 q^{13} - 270 q^{15} - 1174 q^{17} - 2694 q^{19} - 432 q^{21} - 2274 q^{23} + 1562 q^{25} + 1458 q^{27} + 1282 q^{29} + 1720 q^{31} + 2178 q^{33} + 5628 q^{35} - 2856 q^{37} + 3726 q^{39} - 7198 q^{41} + 7274 q^{43} - 2430 q^{45} + 3362 q^{47} - 29190 q^{49} - 10566 q^{51} - 44138 q^{53} - 3630 q^{55} - 24246 q^{57} + 27668 q^{59} - 25050 q^{61} - 3888 q^{63} - 72468 q^{65} - 29080 q^{67} - 20466 q^{69} + 3950 q^{71} - 52784 q^{73} + 14058 q^{75} - 5808 q^{77} + 97268 q^{79} + 13122 q^{81} + 15196 q^{83} - 87912 q^{85} + 11538 q^{87} - 227476 q^{89} - 54108 q^{91} + 15480 q^{93} + 37956 q^{95} - 96996 q^{97} + 19602 q^{99}+O(q^{100}) $ 2 * q + 18 * q^3 - 30 * q^5 - 48 * q^7 + 162 * q^9 + 242 * q^11 + 414 * q^13 - 270 * q^15 - 1174 * q^17 - 2694 * q^19 - 432 * q^21 - 2274 * q^23 + 1562 * q^25 + 1458 * q^27 + 1282 * q^29 + 1720 * q^31 + 2178 * q^33 + 5628 * q^35 - 2856 * q^37 + 3726 * q^39 - 7198 * q^41 + 7274 * q^43 - 2430 * q^45 + 3362 * q^47 - 29190 * q^49 - 10566 * q^51 - 44138 * q^53 - 3630 * q^55 - 24246 * q^57 + 27668 * q^59 - 25050 * q^61 - 3888 * q^63 - 72468 * q^65 - 29080 * q^67 - 20466 * q^69 + 3950 * q^71 - 52784 * q^73 + 14058 * q^75 - 5808 * q^77 + 97268 * q^79 + 13122 * q^81 + 15196 * q^83 - 87912 * q^85 + 11538 * q^87 - 227476 * q^89 - 54108 * q^91 + 15480 * q^93 + 37956 * q^95 - 96996 * q^97 + 19602 * 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
$5$	−75.6712	−1.35365	−0.676824	−	0.736145i	$-0.736644\pi$
$5$	−75.6712	−1.35365	−0.676824	+	0.736145i	$0.736644\pi$
$6$	0	0
$7$	−64.4475	−0.497120	−0.248560	−	0.968617i	$-0.579957\pi$
$7$	−64.4475	−0.497120	−0.248560	+	0.968617i	$0.579957\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	0	0
$13$	753.041	1.23583	0.617917	−	0.786243i	$-0.287977\pi$
$13$	753.041	1.23583	0.617917	+	0.786243i	$0.287977\pi$
$14$	0	0
$15$	−681.041	−0.781529
$16$	0	0
$17$	282.621	0.237182	0.118591	−	0.992943i	$-0.462162\pi$
$17$	282.621	0.237182	0.118591	+	0.992943i	$0.462162\pi$
$18$	0	0
$19$	−1326.78	−0.843167	−0.421584	−	0.906790i	$-0.638526\pi$
$19$	−1326.78	−0.843167	−0.421584	+	0.906790i	$0.638526\pi$
$20$	0	0
$21$	−580.027	−0.287012
$22$	0	0
$23$	1552.76	0.612046	0.306023	−	0.952024i	$-0.401001\pi$
$23$	1552.76	0.612046	0.306023	+	0.952024i	$0.401001\pi$
$24$	0	0
$25$	2601.14	0.832364
$26$	0	0
$27$	729.000	0.192450
$28$	0	0
$29$	1025.25	0.226379	0.113189	−	0.993573i	$-0.463893\pi$
$29$	1025.25	0.226379	0.113189	+	0.993573i	$0.463893\pi$
$30$	0	0
$31$	940.895	0.175848	0.0879239	−	0.996127i	$-0.471977\pi$
$31$	940.895	0.175848	0.0879239	+	0.996127i	$0.471977\pi$
$32$	0	0
$33$	1089.00	0.174078
$34$	0	0
$35$	4876.82	0.672925
$36$	0	0
$37$	5812.10	0.697957	0.348979	−	0.937131i	$-0.386529\pi$
$37$	5812.10	0.697957	0.348979	+	0.937131i	$0.386529\pi$
$38$	0	0
$39$	6777.37	0.713509
$40$	0	0
$41$	−10333.5	−0.960037	−0.480019	−	0.877258i	$-0.659370\pi$
$41$	−10333.5	−0.960037	−0.480019	+	0.877258i	$0.659370\pi$
$42$	0	0
$43$	5841.39	0.481776	0.240888	−	0.970553i	$-0.422561\pi$
$43$	5841.39	0.481776	0.240888	+	0.970553i	$0.422561\pi$
$44$	0	0
$45$	−6129.37	−0.451216
$46$	0	0
$47$	8213.27	0.542340	0.271170	−	0.962531i	$-0.412589\pi$
$47$	8213.27	0.542340	0.271170	+	0.962531i	$0.412589\pi$
$48$	0	0
$49$	−12653.5	−0.752872
$50$	0	0
$51$	2543.59	0.136937
$52$	0	0
$53$	−29046.2	−1.42036	−0.710182	−	0.704018i	$-0.751387\pi$
$53$	−29046.2	−1.42036	−0.710182	+	0.704018i	$0.751387\pi$
$54$	0	0
$55$	−9156.22	−0.408140
$56$	0	0
$57$	−11941.0	−0.486803
$58$	0	0
$59$	−15409.5	−0.576315	−0.288157	−	0.957583i	$-0.593043\pi$
$59$	−15409.5	−0.576315	−0.288157	+	0.957583i	$0.593043\pi$
$60$	0	0
$61$	−2109.77	−0.0725956	−0.0362978	−	0.999341i	$-0.511556\pi$
$61$	−2109.77	−0.0725956	−0.0362978	+	0.999341i	$0.511556\pi$
$62$	0	0
$63$	−5220.25	−0.165707
$64$	0	0
$65$	−56983.6	−1.67289
$66$	0	0
$67$	−22144.1	−0.602659	−0.301329	−	0.953520i	$-0.597430\pi$
$67$	−22144.1	−0.602659	−0.301329	+	0.953520i	$0.597430\pi$
$68$	0	0
$69$	13974.8	0.353365
$70$	0	0
$71$	−35135.6	−0.827182	−0.413591	−	0.910463i	$-0.635726\pi$
$71$	−35135.6	−0.827182	−0.413591	+	0.910463i	$0.635726\pi$
$72$	0	0
$73$	−38566.7	−0.847043	−0.423521	−	0.905886i	$-0.639206\pi$
$73$	−38566.7	−0.847043	−0.423521	+	0.905886i	$0.639206\pi$
$74$	0	0
$75$	23410.2	0.480566
$76$	0	0
$77$	−7798.15	−0.149887
$78$	0	0
$79$	25093.6	0.452371	0.226185	−	0.974084i	$-0.427374\pi$
$79$	25093.6	0.452371	0.226185	+	0.974084i	$0.427374\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	19125.5	0.304732	0.152366	−	0.988324i	$-0.451311\pi$
$83$	19125.5	0.304732	0.152366	+	0.988324i	$0.451311\pi$
$84$	0	0
$85$	−21386.3	−0.321062
$86$	0	0
$87$	9227.26	0.130700
$88$	0	0
$89$	−86961.8	−1.16373	−0.581866	−	0.813284i	$-0.697677\pi$
$89$	−86961.8	−1.16373	−0.581866	+	0.813284i	$0.697677\pi$
$90$	0	0
$91$	−48531.6	−0.614358
$92$	0	0
$93$	8468.05	0.101526
$94$	0	0
$95$	100399.	1.14135
$96$	0	0
$97$	−112243.	−1.21124	−0.605621	−	0.795753i	$-0.707075\pi$
$97$	−112243.	−1.21124	−0.605621	+	0.795753i	$0.707075\pi$
$98$	0	0
$99$	9801.00	0.100504
$100$	0	0
$101$	−136810.	−1.33449	−0.667244	−	0.744840i	$-0.732526\pi$
$101$	−136810.	−1.33449	−0.667244	+	0.744840i	$0.732526\pi$
$102$	0	0
$103$	136853.	1.27105	0.635524	−	0.772081i	$-0.280784\pi$
$103$	136853.	1.27105	0.635524	+	0.772081i	$0.280784\pi$
$104$	0	0
$105$	43891.4	0.388514
$106$	0	0
$107$	−145707.	−1.23033	−0.615165	−	0.788399i	$-0.710910\pi$
$107$	−145707.	−1.23033	−0.615165	+	0.788399i	$0.710910\pi$
$108$	0	0
$109$	−62250.6	−0.501854	−0.250927	−	0.968006i	$-0.580735\pi$
$109$	−62250.6	−0.501854	−0.250927	+	0.968006i	$0.580735\pi$
$110$	0	0
$111$	52308.9	0.402966
$112$	0	0
$113$	−24326.0	−0.179215	−0.0896074	−	0.995977i	$-0.528561\pi$
$113$	−24326.0	−0.179215	−0.0896074	+	0.995977i	$0.528561\pi$
$114$	0	0
$115$	−117499.	−0.828496
$116$	0	0
$117$	60996.3	0.411945
$118$	0	0
$119$	−18214.2	−0.117908
$120$	0	0
$121$	14641.0	0.0909091
$122$	0	0
$123$	−93001.6	−0.554278
$124$	0	0
$125$	39641.3	0.226920
$126$	0	0
$127$	275477.	1.51557	0.757786	−	0.652503i	$-0.226281\pi$
$127$	275477.	1.51557	0.757786	+	0.652503i	$0.226281\pi$
$128$	0	0
$129$	52572.5	0.278153
$130$	0	0
$131$	−140501.	−0.715319	−0.357659	−	0.933852i	$-0.616425\pi$
$131$	−140501.	−0.715319	−0.357659	+	0.933852i	$0.616425\pi$
$132$	0	0
$133$	85507.4	0.419155
$134$	0	0
$135$	−55164.3	−0.260510
$136$	0	0
$137$	106505.	0.484805	0.242403	−	0.970176i	$-0.422064\pi$
$137$	106505.	0.484805	0.242403	+	0.970176i	$0.422064\pi$
$138$	0	0
$139$	3781.45	0.0166005	0.00830025	−	0.999966i	$-0.497358\pi$
$139$	3781.45	0.0166005	0.00830025	+	0.999966i	$0.497358\pi$
$140$	0	0
$141$	73919.4	0.313120
$142$	0	0
$143$	91118.0	0.372618
$144$	0	0
$145$	−77582.0	−0.306437
$146$	0	0
$147$	−113882.	−0.434671
$148$	0	0
$149$	−469751.	−1.73341	−0.866706	−	0.498820i	$-0.833767\pi$
$149$	−469751.	−1.73341	−0.866706	+	0.498820i	$0.833767\pi$
$150$	0	0
$151$	−258100.	−0.921182	−0.460591	−	0.887613i	$-0.652362\pi$
$151$	−258100.	−0.921182	−0.460591	+	0.887613i	$0.652362\pi$
$152$	0	0
$153$	22892.3	0.0790608
$154$	0	0
$155$	−71198.7	−0.238036
$156$	0	0
$157$	562791.	1.82221	0.911104	−	0.412177i	$-0.135231\pi$
$157$	562791.	1.82221	0.911104	+	0.412177i	$0.135231\pi$
$158$	0	0
$159$	−261416.	−0.820047
$160$	0	0
$161$	−100071.	−0.304260
$162$	0	0
$163$	−151067.	−0.445349	−0.222675	−	0.974893i	$-0.571479\pi$
$163$	−151067.	−0.445349	−0.222675	+	0.974893i	$0.571479\pi$
$164$	0	0
$165$	−82406.0	−0.235640
$166$	0	0
$167$	55716.5	0.154594	0.0772970	−	0.997008i	$-0.475371\pi$
$167$	55716.5	0.154594	0.0772970	+	0.997008i	$0.475371\pi$
$168$	0	0
$169$	195778.	0.527287
$170$	0	0
$171$	−107469.	−0.281056
$172$	0	0
$173$	−202820.	−0.515224	−0.257612	−	0.966248i	$-0.582936\pi$
$173$	−202820.	−0.515224	−0.257612	+	0.966248i	$0.582936\pi$
$174$	0	0
$175$	−167637.	−0.413785
$176$	0	0
$177$	−138686.	−0.332735
$178$	0	0
$179$	−143069.	−0.333743	−0.166871	−	0.985979i	$-0.553366\pi$
$179$	−143069.	−0.333743	−0.166871	+	0.985979i	$0.553366\pi$
$180$	0	0
$181$	395910.	0.898257	0.449128	−	0.893467i	$-0.351735\pi$
$181$	395910.	0.898257	0.449128	+	0.893467i	$0.351735\pi$
$182$	0	0
$183$	−18987.9	−0.0419131
$184$	0	0
$185$	−439809.	−0.944789
$186$	0	0
$187$	34197.2	0.0715132
$188$	0	0
$189$	−46982.2	−0.0956707
$190$	0	0
$191$	301812.	0.598622	0.299311	−	0.954156i	$-0.403243\pi$
$191$	301812.	0.598622	0.299311	+	0.954156i	$0.403243\pi$
$192$	0	0
$193$	−814888.	−1.57472	−0.787362	−	0.616491i	$-0.788554\pi$
$193$	−814888.	−1.57472	−0.787362	+	0.616491i	$0.788554\pi$
$194$	0	0
$195$	−512852.	−0.965841
$196$	0	0
$197$	−670410.	−1.23076	−0.615382	−	0.788229i	$-0.710998\pi$
$197$	−670410.	−1.23076	−0.615382	+	0.788229i	$0.710998\pi$
$198$	0	0
$199$	−494158.	−0.884572	−0.442286	−	0.896874i	$-0.645832\pi$
$199$	−494158.	−0.884572	−0.442286	+	0.896874i	$0.645832\pi$
$200$	0	0
$201$	−199297.	−0.347945
$202$	0	0
$203$	−66074.9	−0.112537
$204$	0	0
$205$	781949.	1.29955
$206$	0	0
$207$	125773.	0.204015
$208$	0	0
$209$	−160540.	−0.254224
$210$	0	0
$211$	−887741.	−1.37271	−0.686357	−	0.727265i	$-0.740791\pi$
$211$	−887741.	−1.37271	−0.686357	+	0.727265i	$0.740791\pi$
$212$	0	0
$213$	−316220.	−0.477574
$214$	0	0
$215$	−442025.	−0.652155
$216$	0	0
$217$	−60638.3	−0.0874174
$218$	0	0
$219$	−347100.	−0.489040
$220$	0	0
$221$	212825.	0.293118
$222$	0	0
$223$	−1.18759e6	−1.59921	−0.799604	−	0.600528i	$-0.794957\pi$
$223$	−1.18759e6	−1.59921	−0.799604	+	0.600528i	$0.794957\pi$
$224$	0	0
$225$	210692.	0.277455
$226$	0	0
$227$	1.14107e6	1.46977	0.734884	−	0.678193i	$-0.237237\pi$
$227$	1.14107e6	1.46977	0.734884	+	0.678193i	$0.237237\pi$
$228$	0	0
$229$	−403131.	−0.507993	−0.253996	−	0.967205i	$-0.581745\pi$
$229$	−403131.	−0.507993	−0.253996	+	0.967205i	$0.581745\pi$
$230$	0	0
$231$	−70183.3	−0.0865374
$232$	0	0
$233$	609263.	0.735216	0.367608	−	0.929981i	$-0.380177\pi$
$233$	609263.	0.735216	0.367608	+	0.929981i	$0.380177\pi$
$234$	0	0
$235$	−621508.	−0.734137
$236$	0	0
$237$	225842.	0.261176
$238$	0	0
$239$	−937912.	−1.06210	−0.531052	−	0.847339i	$-0.678203\pi$
$239$	−937912.	−1.06210	−0.531052	+	0.847339i	$0.678203\pi$
$240$	0	0
$241$	−213841.	−0.237164	−0.118582	−	0.992944i	$-0.537835\pi$
$241$	−213841.	−0.237164	−0.118582	+	0.992944i	$0.537835\pi$
$242$	0	0
$243$	59049.0	0.0641500
$244$	0	0
$245$	957508.	1.01912
$246$	0	0
$247$	−999117.	−1.04202
$248$	0	0
$249$	172130.	0.175937
$250$	0	0
$251$	−245353.	−0.245814	−0.122907	−	0.992418i	$-0.539222\pi$
$251$	−245353.	−0.245814	−0.122907	+	0.992418i	$0.539222\pi$
$252$	0	0
$253$	187884.	0.184539
$254$	0	0
$255$	−192477.	−0.185365
$256$	0	0
$257$	−252288.	−0.238267	−0.119134	−	0.992878i	$-0.538012\pi$
$257$	−252288.	−0.238267	−0.119134	+	0.992878i	$0.538012\pi$
$258$	0	0
$259$	−374575.	−0.346968
$260$	0	0
$261$	83045.3	0.0754595
$262$	0	0
$263$	1.25323e6	1.11722	0.558612	−	0.829429i	$-0.311334\pi$
$263$	1.25323e6	1.11722	0.558612	+	0.829429i	$0.311334\pi$
$264$	0	0
$265$	2.19796e6	1.92267
$266$	0	0
$267$	−782656.	−0.671881
$268$	0	0
$269$	−578275.	−0.487252	−0.243626	−	0.969869i	$-0.578337\pi$
$269$	−578275.	−0.487252	−0.243626	+	0.969869i	$0.578337\pi$
$270$	0	0
$271$	1.38306e6	1.14398	0.571991	−	0.820260i	$-0.306171\pi$
$271$	1.38306e6	1.14398	0.571991	+	0.820260i	$0.306171\pi$
$272$	0	0
$273$	−436785.	−0.354700
$274$	0	0
$275$	314738.	0.250967
$276$	0	0
$277$	18488.0	0.0144774	0.00723870	−	0.999974i	$-0.497696\pi$
$277$	18488.0	0.0144774	0.00723870	+	0.999974i	$0.497696\pi$
$278$	0	0
$279$	76212.5	0.0586159
$280$	0	0
$281$	−627461.	−0.474047	−0.237023	−	0.971504i	$-0.576172\pi$
$281$	−627461.	−0.474047	−0.237023	+	0.971504i	$0.576172\pi$
$282$	0	0
$283$	1.53216e6	1.13721	0.568603	−	0.822612i	$-0.307484\pi$
$283$	1.53216e6	1.13721	0.568603	+	0.822612i	$0.307484\pi$
$284$	0	0
$285$	903589.	0.658960
$286$	0	0
$287$	665969.	0.477253
$288$	0	0
$289$	−1.33998e6	−0.943745
$290$	0	0
$291$	−1.01019e6	−0.699311
$292$	0	0
$293$	−2.30041e6	−1.56544	−0.782721	−	0.622373i	$-0.786169\pi$
$293$	−2.30041e6	−1.56544	−0.782721	+	0.622373i	$0.786169\pi$
$294$	0	0
$295$	1.16606e6	0.780127
$296$	0	0
$297$	88209.0	0.0580259
$298$	0	0
$299$	1.16929e6	0.756388
$300$	0	0
$301$	−376463.	−0.239500
$302$	0	0
$303$	−1.23129e6	−0.770466
$304$	0	0
$305$	159649.	0.0982689
$306$	0	0
$307$	1.22318e6	0.740702	0.370351	−	0.928892i	$-0.379237\pi$
$307$	1.22318e6	0.740702	0.370351	+	0.928892i	$0.379237\pi$
$308$	0	0
$309$	1.23168e6	0.733840
$310$	0	0
$311$	2.72673e6	1.59860	0.799301	−	0.600930i	$-0.205203\pi$
$311$	2.72673e6	1.59860	0.799301	+	0.600930i	$0.205203\pi$
$312$	0	0
$313$	−290463.	−0.167583	−0.0837914	−	0.996483i	$-0.526703\pi$
$313$	−290463.	−0.167583	−0.0837914	+	0.996483i	$0.526703\pi$
$314$	0	0
$315$	395023.	0.224308
$316$	0	0
$317$	−2.30520e6	−1.28843	−0.644215	−	0.764844i	$-0.722816\pi$
$317$	−2.30520e6	−1.28843	−0.644215	+	0.764844i	$0.722816\pi$
$318$	0	0
$319$	124055.	0.0682557
$320$	0	0
$321$	−1.31136e6	−0.710331
$322$	0	0
$323$	−374975.	−0.199984
$324$	0	0
$325$	1.95876e6	1.02866
$326$	0	0
$327$	−560255.	−0.289745
$328$	0	0
$329$	−529325.	−0.269608
$330$	0	0
$331$	3.65525e6	1.83378	0.916889	−	0.399141i	$-0.130692\pi$
$331$	3.65525e6	1.83378	0.916889	+	0.399141i	$0.130692\pi$
$332$	0	0
$333$	470780.	0.232652
$334$	0	0
$335$	1.67567e6	0.815788
$336$	0	0
$337$	3.31856e6	1.59175	0.795875	−	0.605461i	$-0.207011\pi$
$337$	3.31856e6	1.59175	0.795875	+	0.605461i	$0.207011\pi$
$338$	0	0
$339$	−218934.	−0.103470
$340$	0	0
$341$	113848.	0.0530201
$342$	0	0
$343$	1.89866e6	0.871387
$344$	0	0
$345$	−1.05749e6	−0.478332
$346$	0	0
$347$	−1.60553e6	−0.715803	−0.357901	−	0.933759i	$-0.616508\pi$
$347$	−1.60553e6	−0.715803	−0.357901	+	0.933759i	$0.616508\pi$
$348$	0	0
$349$	−437697.	−0.192358	−0.0961790	−	0.995364i	$-0.530662\pi$
$349$	−437697.	−0.192358	−0.0961790	+	0.995364i	$0.530662\pi$
$350$	0	0
$351$	548967.	0.237836
$352$	0	0
$353$	1.49219e6	0.637364	0.318682	−	0.947862i	$-0.396760\pi$
$353$	1.49219e6	0.637364	0.318682	+	0.947862i	$0.396760\pi$
$354$	0	0
$355$	2.65875e6	1.11971
$356$	0	0
$357$	−163928.	−0.0680742
$358$	0	0
$359$	2.74760e6	1.12517	0.562584	−	0.826740i	$-0.309807\pi$
$359$	2.74760e6	1.12517	0.562584	+	0.826740i	$0.309807\pi$
$360$	0	0
$361$	−715764.	−0.289069
$362$	0	0
$363$	131769.	0.0524864
$364$	0	0
$365$	2.91839e6	1.14660
$366$	0	0
$367$	3.62940e6	1.40660	0.703298	−	0.710895i	$-0.251710\pi$
$367$	3.62940e6	1.40660	0.703298	+	0.710895i	$0.251710\pi$
$368$	0	0
$369$	−837014.	−0.320012
$370$	0	0
$371$	1.87195e6	0.706091
$372$	0	0
$373$	−369648.	−0.137568	−0.0687839	−	0.997632i	$-0.521912\pi$
$373$	−369648.	−0.137568	−0.0687839	+	0.997632i	$0.521912\pi$
$374$	0	0
$375$	356772.	0.131012
$376$	0	0
$377$	772056.	0.279766
$378$	0	0
$379$	−2.28310e6	−0.816444	−0.408222	−	0.912883i	$-0.633851\pi$
$379$	−2.28310e6	−0.816444	−0.408222	+	0.912883i	$0.633851\pi$
$380$	0	0
$381$	2.47930e6	0.875016
$382$	0	0
$383$	−4.95846e6	−1.72723	−0.863614	−	0.504154i	$-0.831804\pi$
$383$	−4.95846e6	−1.72723	−0.863614	+	0.504154i	$0.831804\pi$
$384$	0	0
$385$	590096.	0.202895
$386$	0	0
$387$	473152.	0.160592
$388$	0	0
$389$	−4.66740e6	−1.56387	−0.781936	−	0.623359i	$-0.785768\pi$
$389$	−4.66740e6	−1.56387	−0.781936	+	0.623359i	$0.785768\pi$
$390$	0	0
$391$	438842.	0.145167
$392$	0	0
$393$	−1.26450e6	−0.412990
$394$	0	0
$395$	−1.89886e6	−0.612351
$396$	0	0
$397$	5.43505e6	1.73072	0.865361	−	0.501148i	$-0.167089\pi$
$397$	5.43505e6	1.73072	0.865361	+	0.501148i	$0.167089\pi$
$398$	0	0
$399$	769567.	0.241999
$400$	0	0
$401$	−5.76191e6	−1.78939	−0.894696	−	0.446675i	$-0.852608\pi$
$401$	−5.76191e6	−1.78939	−0.894696	+	0.446675i	$0.852608\pi$
$402$	0	0
$403$	708533.	0.217319
$404$	0	0
$405$	−496479.	−0.150405
$406$	0	0
$407$	703264.	0.210442
$408$	0	0
$409$	4.61324e6	1.36363	0.681817	−	0.731523i	$-0.261190\pi$
$409$	4.61324e6	1.36363	0.681817	+	0.731523i	$0.261190\pi$
$410$	0	0
$411$	958543.	0.279903
$412$	0	0
$413$	993106.	0.286497
$414$	0	0
$415$	−1.44725e6	−0.412500
$416$	0	0
$417$	34033.0	0.00958430
$418$	0	0
$419$	−5.62695e6	−1.56581	−0.782903	−	0.622143i	$-0.786262\pi$
$419$	−5.62695e6	−1.56581	−0.782903	+	0.622143i	$0.786262\pi$
$420$	0	0
$421$	−2.22389e6	−0.611516	−0.305758	−	0.952109i	$-0.598910\pi$
$421$	−2.22389e6	−0.611516	−0.305758	+	0.952109i	$0.598910\pi$
$422$	0	0
$423$	665275.	0.180780
$424$	0	0
$425$	735137.	0.197422
$426$	0	0
$427$	135969.	0.0360887
$428$	0	0
$429$	820062.	0.215131
$430$	0	0
$431$	−1.17803e6	−0.305466	−0.152733	−	0.988267i	$-0.548807\pi$
$431$	−1.17803e6	−0.305466	−0.152733	+	0.988267i	$0.548807\pi$
$432$	0	0
$433$	−2.50549e6	−0.642205	−0.321102	−	0.947044i	$-0.604053\pi$
$433$	−2.50549e6	−0.642205	−0.321102	+	0.947044i	$0.604053\pi$
$434$	0	0
$435$	−698238.	−0.176921
$436$	0	0
$437$	−2.06016e6	−0.516057
$438$	0	0
$439$	869760.	0.215396	0.107698	−	0.994184i	$-0.465652\pi$
$439$	869760.	0.215396	0.107698	+	0.994184i	$0.465652\pi$
$440$	0	0
$441$	−1.02494e6	−0.250957
$442$	0	0
$443$	3.71207e6	0.898683	0.449342	−	0.893360i	$-0.351659\pi$
$443$	3.71207e6	0.898683	0.449342	+	0.893360i	$0.351659\pi$
$444$	0	0
$445$	6.58050e6	1.57529
$446$	0	0
$447$	−4.22776e6	−1.00079
$448$	0	0
$449$	5.44277e6	1.27410	0.637051	−	0.770821i	$-0.280154\pi$
$449$	5.44277e6	1.27410	0.637051	+	0.770821i	$0.280154\pi$
$450$	0	0
$451$	−1.25035e6	−0.289462
$452$	0	0
$453$	−2.32290e6	−0.531845
$454$	0	0
$455$	3.67245e6	0.831624
$456$	0	0
$457$	1.61181e6	0.361013	0.180506	−	0.983574i	$-0.442226\pi$
$457$	1.61181e6	0.361013	0.180506	+	0.983574i	$0.442226\pi$
$458$	0	0
$459$	206031.	0.0456458
$460$	0	0
$461$	−7.17223e6	−1.57182	−0.785908	−	0.618343i	$-0.787804\pi$
$461$	−7.17223e6	−1.57182	−0.785908	+	0.618343i	$0.787804\pi$
$462$	0	0
$463$	5.24890e6	1.13793	0.568966	−	0.822361i	$-0.307344\pi$
$463$	5.24890e6	1.13793	0.568966	+	0.822361i	$0.307344\pi$
$464$	0	0
$465$	−640788.	−0.137430
$466$	0	0
$467$	−2.96447e6	−0.629006	−0.314503	−	0.949256i	$-0.601838\pi$
$467$	−2.96447e6	−0.629006	−0.314503	+	0.949256i	$0.601838\pi$
$468$	0	0
$469$	1.42713e6	0.299594
$470$	0	0
$471$	5.06512e6	1.05205
$472$	0	0
$473$	706808.	0.145261
$474$	0	0
$475$	−3.45113e6	−0.701822
$476$	0	0
$477$	−2.35274e6	−0.473455
$478$	0	0
$479$	2.54069e6	0.505957	0.252978	−	0.967472i	$-0.418590\pi$
$479$	2.54069e6	0.505957	0.252978	+	0.967472i	$0.418590\pi$
$480$	0	0
$481$	4.37675e6	0.862560
$482$	0	0
$483$	−900643.	−0.175665
$484$	0	0
$485$	8.49359e6	1.63960
$486$	0	0
$487$	5.58688e6	1.06745	0.533724	−	0.845658i	$-0.320792\pi$
$487$	5.58688e6	1.06745	0.533724	+	0.845658i	$0.320792\pi$
$488$	0	0
$489$	−1.35960e6	−0.257123
$490$	0	0
$491$	−813418.	−0.152268	−0.0761342	−	0.997098i	$-0.524258\pi$
$491$	−813418.	−0.152268	−0.0761342	+	0.997098i	$0.524258\pi$
$492$	0	0
$493$	289758.	0.0536930
$494$	0	0
$495$	−741654.	−0.136047
$496$	0	0
$497$	2.26440e6	0.411209
$498$	0	0
$499$	8.92488e6	1.60454	0.802271	−	0.596961i	$-0.203625\pi$
$499$	8.92488e6	1.60454	0.802271	+	0.596961i	$0.203625\pi$
$500$	0	0
$501$	501449.	0.0892549
$502$	0	0
$503$	−9.53045e6	−1.67955	−0.839776	−	0.542933i	$-0.817314\pi$
$503$	−9.53045e6	−1.67955	−0.839776	+	0.542933i	$0.817314\pi$
$504$	0	0
$505$	1.03526e7	1.80643
$506$	0	0
$507$	1.76200e6	0.304429
$508$	0	0
$509$	3.76851e6	0.644725	0.322363	−	0.946616i	$-0.395523\pi$
$509$	3.76851e6	0.644725	0.322363	+	0.946616i	$0.395523\pi$
$510$	0	0
$511$	2.48553e6	0.421082
$512$	0	0
$513$	−967220.	−0.162268
$514$	0	0
$515$	−1.03559e7	−1.72055
$516$	0	0
$517$	993806.	0.163522
$518$	0	0
$519$	−1.82538e6	−0.297465
$520$	0	0
$521$	6.79767e6	1.09715	0.548575	−	0.836101i	$-0.315170\pi$
$521$	6.79767e6	1.09715	0.548575	+	0.836101i	$0.315170\pi$
$522$	0	0
$523$	−8.66044e6	−1.38448	−0.692238	−	0.721669i	$-0.743375\pi$
$523$	−8.66044e6	−1.38448	−0.692238	+	0.721669i	$0.743375\pi$
$524$	0	0
$525$	−1.50873e6	−0.238899
$526$	0	0
$527$	265917.	0.0417080
$528$	0	0
$529$	−4.02528e6	−0.625399
$530$	0	0
$531$	−1.24817e6	−0.192105
$532$	0	0
$533$	−7.78156e6	−1.18645
$534$	0	0
$535$	1.10258e7	1.66543
$536$	0	0
$537$	−1.28762e6	−0.192686
$538$	0	0
$539$	−1.53108e6	−0.226999
$540$	0	0
$541$	165950.	0.0243772	0.0121886	−	0.999926i	$-0.496120\pi$
$541$	165950.	0.0243772	0.0121886	+	0.999926i	$0.496120\pi$
$542$	0	0
$543$	3.56319e6	0.518609
$544$	0	0
$545$	4.71058e6	0.679334
$546$	0	0
$547$	−6.44961e6	−0.921648	−0.460824	−	0.887492i	$-0.652446\pi$
$547$	−6.44961e6	−0.921648	−0.460824	+	0.887492i	$0.652446\pi$
$548$	0	0
$549$	−170891.	−0.0241985
$550$	0	0
$551$	−1.36028e6	−0.190875
$552$	0	0
$553$	−1.61722e6	−0.224882
$554$	0	0
$555$	−3.95828e6	−0.545474
$556$	0	0
$557$	−4.74451e6	−0.647968	−0.323984	−	0.946063i	$-0.605022\pi$
$557$	−4.74451e6	−0.647968	−0.323984	+	0.946063i	$0.605022\pi$
$558$	0	0
$559$	4.39881e6	0.595395
$560$	0	0
$561$	307774.	0.0412882
$562$	0	0
$563$	−1.20807e7	−1.60628	−0.803138	−	0.595793i	$-0.796838\pi$
$563$	−1.20807e7	−1.60628	−0.803138	+	0.595793i	$0.796838\pi$
$564$	0	0
$565$	1.84078e6	0.242594
$566$	0	0
$567$	−422840.	−0.0552355
$568$	0	0
$569$	7.38055e6	0.955670	0.477835	−	0.878450i	$-0.341422\pi$
$569$	7.38055e6	0.955670	0.477835	+	0.878450i	$0.341422\pi$
$570$	0	0
$571$	−2.41237e6	−0.309638	−0.154819	−	0.987943i	$-0.549479\pi$
$571$	−2.41237e6	−0.309638	−0.154819	+	0.987943i	$0.549479\pi$
$572$	0	0
$573$	2.71630e6	0.345614
$574$	0	0
$575$	4.03894e6	0.509445
$576$	0	0
$577$	−3.09401e6	−0.386886	−0.193443	−	0.981112i	$-0.561965\pi$
$577$	−3.09401e6	−0.386886	−0.193443	+	0.981112i	$0.561965\pi$
$578$	0	0
$579$	−7.33399e6	−0.909168
$580$	0	0
$581$	−1.23259e6	−0.151488
$582$	0	0
$583$	−3.51459e6	−0.428256
$584$	0	0
$585$	−4.61567e6	−0.557629
$586$	0	0
$587$	−1.13414e7	−1.35854	−0.679269	−	0.733889i	$-0.737703\pi$
$587$	−1.13414e7	−1.35854	−0.679269	+	0.733889i	$0.737703\pi$
$588$	0	0
$589$	−1.24836e6	−0.148269
$590$	0	0
$591$	−6.03369e6	−0.710582
$592$	0	0
$593$	6.53021e6	0.762588	0.381294	−	0.924454i	$-0.375478\pi$
$593$	6.53021e6	0.762588	0.381294	+	0.924454i	$0.375478\pi$
$594$	0	0
$595$	1.37829e6	0.159606
$596$	0	0
$597$	−4.44742e6	−0.510708
$598$	0	0
$599$	10374.5	0.00118141	0.000590704	−	1.00000i	$-0.499812\pi$
$599$	10374.5	0.00118141	0.000590704		1.00000i	$0.499812\pi$
$600$	0	0
$601$	−3.43855e6	−0.388320	−0.194160	−	0.980970i	$-0.562198\pi$
$601$	−3.43855e6	−0.388320	−0.194160	+	0.980970i	$0.562198\pi$
$602$	0	0
$603$	−1.79367e6	−0.200886
$604$	0	0
$605$	−1.10790e6	−0.123059
$606$	0	0
$607$	9.80221e6	1.07982	0.539911	−	0.841722i	$-0.318458\pi$
$607$	9.80221e6	1.07982	0.539911	+	0.841722i	$0.318458\pi$
$608$	0	0
$609$	−594674.	−0.0649734
$610$	0	0
$611$	6.18493e6	0.670242
$612$	0	0
$613$	376637.	0.0404829	0.0202414	−	0.999795i	$-0.493557\pi$
$613$	376637.	0.0404829	0.0202414	+	0.999795i	$0.493557\pi$
$614$	0	0
$615$	7.03754e6	0.750297
$616$	0	0
$617$	1.08771e7	1.15027	0.575134	−	0.818059i	$-0.304950\pi$
$617$	1.08771e7	1.15027	0.575134	+	0.818059i	$0.304950\pi$
$618$	0	0
$619$	−6.86300e6	−0.719925	−0.359963	−	0.932967i	$-0.617211\pi$
$619$	−6.86300e6	−0.719925	−0.359963	+	0.932967i	$0.617211\pi$
$620$	0	0
$621$	1.13196e6	0.117788
$622$	0	0
$623$	5.60447e6	0.578514
$624$	0	0
$625$	−1.11283e7	−1.13953
$626$	0	0
$627$	−1.44486e6	−0.146777
$628$	0	0
$629$	1.64262e6	0.165543
$630$	0	0
$631$	9.16578e6	0.916423	0.458211	−	0.888843i	$-0.348490\pi$
$631$	9.16578e6	0.916423	0.458211	+	0.888843i	$0.348490\pi$
$632$	0	0
$633$	−7.98967e6	−0.792537
$634$	0	0
$635$	−2.08457e7	−2.05155
$636$	0	0
$637$	−9.52862e6	−0.930425
$638$	0	0
$639$	−2.84598e6	−0.275727
$640$	0	0
$641$	−5.91461e6	−0.568566	−0.284283	−	0.958740i	$-0.591756\pi$
$641$	−5.91461e6	−0.568566	−0.284283	+	0.958740i	$0.591756\pi$
$642$	0	0
$643$	−1.02309e7	−0.975859	−0.487930	−	0.872883i	$-0.662248\pi$
$643$	−1.02309e7	−0.975859	−0.487930	+	0.872883i	$0.662248\pi$
$644$	0	0
$645$	−3.97823e6	−0.376522
$646$	0	0
$647$	7.03171e6	0.660389	0.330195	−	0.943913i	$-0.392886\pi$
$647$	7.03171e6	0.660389	0.330195	+	0.943913i	$0.392886\pi$
$648$	0	0
$649$	−1.86455e6	−0.173765
$650$	0	0
$651$	−545745.	−0.0504705
$652$	0	0
$653$	422576.	0.0387812	0.0193906	−	0.999812i	$-0.493827\pi$
$653$	422576.	0.0387812	0.0193906	+	0.999812i	$0.493827\pi$
$654$	0	0
$655$	1.06318e7	0.968290
$656$	0	0
$657$	−3.12390e6	−0.282348
$658$	0	0
$659$	505061.	0.0453033	0.0226517	−	0.999743i	$-0.492789\pi$
$659$	505061.	0.0453033	0.0226517	+	0.999743i	$0.492789\pi$
$660$	0	0
$661$	−2.95380e6	−0.262952	−0.131476	−	0.991319i	$-0.541972\pi$
$661$	−2.95380e6	−0.262952	−0.131476	+	0.991319i	$0.541972\pi$
$662$	0	0
$663$	1.91543e6	0.169232
$664$	0	0
$665$	−6.47045e6	−0.567388
$666$	0	0
$667$	1.59197e6	0.138554
$668$	0	0
$669$	−1.06883e7	−0.923303
$670$	0	0
$671$	−255282.	−0.0218884
$672$	0	0
$673$	204729.	0.0174238	0.00871188	−	0.999962i	$-0.497227\pi$
$673$	204729.	0.0174238	0.00871188	+	0.999962i	$0.497227\pi$
$674$	0	0
$675$	1.89623e6	0.160189
$676$	0	0
$677$	7.16583e6	0.600890	0.300445	−	0.953799i	$-0.402865\pi$
$677$	7.16583e6	0.600890	0.300445	+	0.953799i	$0.402865\pi$
$678$	0	0
$679$	7.23380e6	0.602132
$680$	0	0
$681$	1.02697e7	0.848571
$682$	0	0
$683$	−1.74882e7	−1.43447	−0.717236	−	0.696830i	$-0.754593\pi$
$683$	−1.74882e7	−1.43447	−0.717236	+	0.696830i	$0.754593\pi$
$684$	0	0
$685$	−8.05935e6	−0.656256
$686$	0	0
$687$	−3.62818e6	−0.293290
$688$	0	0
$689$	−2.18730e7	−1.75533
$690$	0	0
$691$	290484.	0.0231434	0.0115717	−	0.999933i	$-0.496317\pi$
$691$	290484.	0.0231434	0.0115717	+	0.999933i	$0.496317\pi$
$692$	0	0
$693$	−631650.	−0.0499624
$694$	0	0
$695$	−286147.	−0.0224712
$696$	0	0
$697$	−2.92047e6	−0.227704
$698$	0	0
$699$	5.48336e6	0.424477
$700$	0	0
$701$	1.18080e7	0.907574	0.453787	−	0.891110i	$-0.350073\pi$
$701$	1.18080e7	0.907574	0.453787	+	0.891110i	$0.350073\pi$
$702$	0	0
$703$	−7.71136e6	−0.588495
$704$	0	0
$705$	−5.59358e6	−0.423854
$706$	0	0
$707$	8.81706e6	0.663400
$708$	0	0
$709$	−8.68999e6	−0.649238	−0.324619	−	0.945845i	$-0.605236\pi$
$709$	−8.68999e6	−0.649238	−0.324619	+	0.945845i	$0.605236\pi$
$710$	0	0
$711$	2.03258e6	0.150790
$712$	0	0
$713$	1.46098e6	0.107627
$714$	0	0
$715$	−6.89501e6	−0.504394
$716$	0	0
$717$	−8.44121e6	−0.613207
$718$	0	0
$719$	−1.08515e7	−0.782829	−0.391415	−	0.920214i	$-0.628014\pi$
$719$	−1.08515e7	−0.782829	−0.391415	+	0.920214i	$0.628014\pi$
$720$	0	0
$721$	−8.81985e6	−0.631863
$722$	0	0
$723$	−1.92457e6	−0.136927
$724$	0	0
$725$	2.66682e6	0.188429
$726$	0	0
$727$	−2.04624e7	−1.43589	−0.717943	−	0.696102i	$-0.754916\pi$
$727$	−2.04624e7	−1.43589	−0.717943	+	0.696102i	$0.754916\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	1.65090e6	0.114269
$732$	0	0
$733$	5.07998e6	0.349223	0.174611	−	0.984637i	$-0.444133\pi$
$733$	5.07998e6	0.349223	0.174611	+	0.984637i	$0.444133\pi$
$734$	0	0
$735$	8.61757e6	0.588392
$736$	0	0
$737$	−2.67944e6	−0.181709
$738$	0	0
$739$	955206.	0.0643407	0.0321704	−	0.999482i	$-0.489758\pi$
$739$	955206.	0.0643407	0.0321704	+	0.999482i	$0.489758\pi$
$740$	0	0
$741$	−8.99205e6	−0.601608
$742$	0	0
$743$	2.32501e7	1.54509	0.772544	−	0.634962i	$-0.218984\pi$
$743$	2.32501e7	1.54509	0.772544	+	0.634962i	$0.218984\pi$
$744$	0	0
$745$	3.55466e7	2.34643
$746$	0	0
$747$	1.54917e6	0.101577
$748$	0	0
$749$	9.39046e6	0.611621
$750$	0	0
$751$	−682097.	−0.0441312	−0.0220656	−	0.999757i	$-0.507024\pi$
$751$	−682097.	−0.0441312	−0.0220656	+	0.999757i	$0.507024\pi$
$752$	0	0
$753$	−2.20817e6	−0.141921
$754$	0	0
$755$	1.95307e7	1.24696
$756$	0	0
$757$	−2.27991e7	−1.44603	−0.723016	−	0.690831i	$-0.757245\pi$
$757$	−2.27991e7	−1.44603	−0.723016	+	0.690831i	$0.757245\pi$
$758$	0	0
$759$	1.69095e6	0.106544
$760$	0	0
$761$	8.48003e6	0.530806	0.265403	−	0.964138i	$-0.414495\pi$
$761$	8.48003e6	0.530806	0.265403	+	0.964138i	$0.414495\pi$
$762$	0	0
$763$	4.01189e6	0.249481
$764$	0	0
$765$	−1.73229e6	−0.107021
$766$	0	0
$767$	−1.16040e7	−0.712230
$768$	0	0
$769$	1.32249e7	0.806450	0.403225	−	0.915101i	$-0.367889\pi$
$769$	1.32249e7	0.806450	0.403225	+	0.915101i	$0.367889\pi$
$770$	0	0
$771$	−2.27060e6	−0.137564
$772$	0	0
$773$	2.22097e6	0.133689	0.0668444	−	0.997763i	$-0.478707\pi$
$773$	2.22097e6	0.133689	0.0668444	+	0.997763i	$0.478707\pi$
$774$	0	0
$775$	2.44740e6	0.146369
$776$	0	0
$777$	−3.37118e6	−0.200322
$778$	0	0
$779$	1.37103e7	0.809472
$780$	0	0
$781$	−4.25140e6	−0.249405
$782$	0	0
$783$	747408.	0.0435666
$784$	0	0
$785$	−4.25871e7	−2.46663
$786$	0	0
$787$	3.14688e7	1.81111	0.905553	−	0.424234i	$-0.139457\pi$
$787$	3.14688e7	1.81111	0.905553	+	0.424234i	$0.139457\pi$
$788$	0	0
$789$	1.12790e7	0.645029
$790$	0	0
$791$	1.56775e6	0.0890912
$792$	0	0
$793$	−1.58874e6	−0.0897162
$794$	0	0
$795$	1.97817e7	1.11006
$796$	0	0
$797$	62259.3	0.00347183	0.00173592	−	0.999998i	$-0.499447\pi$
$797$	62259.3	0.00347183	0.00173592	+	0.999998i	$0.499447\pi$
$798$	0	0
$799$	2.32124e6	0.128633
$800$	0	0
$801$	−7.04390e6	−0.387911
$802$	0	0
$803$	−4.66657e6	−0.255393
$804$	0	0
$805$	7.57253e6	0.411861
$806$	0	0
$807$	−5.20447e6	−0.281315
$808$	0	0
$809$	−3.01753e7	−1.62099	−0.810496	−	0.585745i	$-0.800802\pi$
$809$	−3.01753e7	−1.62099	−0.810496	+	0.585745i	$0.800802\pi$
$810$	0	0
$811$	−5.57073e6	−0.297413	−0.148706	−	0.988881i	$-0.547511\pi$
$811$	−5.57073e6	−0.297413	−0.148706	+	0.988881i	$0.547511\pi$
$812$	0	0
$813$	1.24476e7	0.660478
$814$	0	0
$815$	1.14314e7	0.602846
$816$	0	0
$817$	−7.75022e6	−0.406218
$818$	0	0
$819$	−3.93106e6	−0.204786
$820$	0	0
$821$	−2.87212e7	−1.48711	−0.743557	−	0.668672i	$-0.766863\pi$
$821$	−2.87212e7	−1.48711	−0.743557	+	0.668672i	$0.766863\pi$
$822$	0	0
$823$	2.44210e7	1.25679	0.628396	−	0.777893i	$-0.283712\pi$
$823$	2.44210e7	1.25679	0.628396	+	0.777893i	$0.283712\pi$
$824$	0	0
$825$	2.83264e6	0.144896
$826$	0	0
$827$	−2.03111e7	−1.03269	−0.516344	−	0.856381i	$-0.672708\pi$
$827$	−2.03111e7	−1.03269	−0.516344	+	0.856381i	$0.672708\pi$
$828$	0	0
$829$	259590.	0.0131190	0.00655951	−	0.999978i	$-0.497912\pi$
$829$	259590.	0.0131190	0.00655951	+	0.999978i	$0.497912\pi$
$830$	0	0
$831$	166392.	0.00835853
$832$	0	0
$833$	−3.57615e6	−0.178568
$834$	0	0
$835$	−4.21614e6	−0.209266
$836$	0	0
$837$	685912.	0.0338419
$838$	0	0
$839$	−1.58770e7	−0.778689	−0.389345	−	0.921092i	$-0.627299\pi$
$839$	−1.58770e7	−0.778689	−0.389345	+	0.921092i	$0.627299\pi$
$840$	0	0
$841$	−1.94600e7	−0.948753
$842$	0	0
$843$	−5.64715e6	−0.273691
$844$	0	0
$845$	−1.48148e7	−0.713762
$846$	0	0
$847$	−943576.	−0.0451927
$848$	0	0
$849$	1.37895e7	0.656566
$850$	0	0
$851$	9.02479e6	0.427182
$852$	0	0
$853$	1.09646e6	0.0515964	0.0257982	−	0.999667i	$-0.491787\pi$
$853$	1.09646e6	0.0515964	0.0257982	+	0.999667i	$0.491787\pi$
$854$	0	0
$855$	8.13230e6	0.380451
$856$	0	0
$857$	2.65197e7	1.23344	0.616718	−	0.787184i	$-0.288462\pi$
$857$	2.65197e7	1.23344	0.616718	+	0.787184i	$0.288462\pi$
$858$	0	0
$859$	−1.97527e7	−0.913365	−0.456683	−	0.889630i	$-0.650963\pi$
$859$	−1.97527e7	−0.913365	−0.456683	+	0.889630i	$0.650963\pi$
$860$	0	0
$861$	5.99372e6	0.275542
$862$	0	0
$863$	−9.61293e6	−0.439368	−0.219684	−	0.975571i	$-0.570503\pi$
$863$	−9.61293e6	−0.439368	−0.219684	+	0.975571i	$0.570503\pi$
$864$	0	0
$865$	1.53477e7	0.697432
$866$	0	0
$867$	−1.20598e7	−0.544871
$868$	0	0
$869$	3.03632e6	0.136395
$870$	0	0
$871$	−1.66754e7	−0.744787
$872$	0	0
$873$	−9.09170e6	−0.403747
$874$	0	0
$875$	−2.55479e6	−0.112807
$876$	0	0
$877$	3.20329e7	1.40636	0.703180	−	0.711011i	$-0.251763\pi$
$877$	3.20329e7	1.40636	0.703180	+	0.711011i	$0.251763\pi$
$878$	0	0
$879$	−2.07037e7	−0.903808
$880$	0	0
$881$	−2.04586e7	−0.888045	−0.444023	−	0.896016i	$-0.646449\pi$
$881$	−2.04586e7	−0.888045	−0.444023	+	0.896016i	$0.646449\pi$
$882$	0	0
$883$	−3.44048e7	−1.48497	−0.742485	−	0.669863i	$-0.766353\pi$
$883$	−3.44048e7	−1.48497	−0.742485	+	0.669863i	$0.766353\pi$
$884$	0	0
$885$	1.04945e7	0.450407
$886$	0	0
$887$	−2.28170e7	−0.973755	−0.486878	−	0.873470i	$-0.661864\pi$
$887$	−2.28170e7	−0.973755	−0.486878	+	0.873470i	$0.661864\pi$
$888$	0	0
$889$	−1.77538e7	−0.753421
$890$	0	0
$891$	793881.	0.0335013
$892$	0	0
$893$	−1.08972e7	−0.457283
$894$	0	0
$895$	1.08262e7	0.451770
$896$	0	0
$897$	1.05236e7	0.436701
$898$	0	0
$899$	964654.	0.0398082
$900$	0	0
$901$	−8.20907e6	−0.336885
$902$	0	0
$903$	−3.38817e6	−0.138276
$904$	0	0
$905$	−2.99590e7	−1.21592
$906$	0	0
$907$	3.78676e6	0.152844	0.0764222	−	0.997076i	$-0.475650\pi$
$907$	3.78676e6	0.152844	0.0764222	+	0.997076i	$0.475650\pi$
$908$	0	0
$909$	−1.10816e7	−0.444829
$910$	0	0
$911$	−214848.	−0.00857699	−0.00428849	−	0.999991i	$-0.501365\pi$
$911$	−214848.	−0.00857699	−0.00428849	+	0.999991i	$0.501365\pi$
$912$	0	0
$913$	2.31419e6	0.0918802
$914$	0	0
$915$	1.43684e6	0.0567356
$916$	0	0
$917$	9.05491e6	0.355599
$918$	0	0
$919$	1.11949e7	0.437251	0.218626	−	0.975809i	$-0.429843\pi$
$919$	1.11949e7	0.437251	0.218626	+	0.975809i	$0.429843\pi$
$920$	0	0
$921$	1.10086e7	0.427644
$922$	0	0
$923$	−2.64585e7	−1.02226
$924$	0	0
$925$	1.51181e7	0.580955
$926$	0	0
$927$	1.10851e7	0.423683
$928$	0	0
$929$	2.47773e7	0.941921	0.470961	−	0.882154i	$-0.343907\pi$
$929$	2.47773e7	0.941921	0.470961	+	0.882154i	$0.343907\pi$
$930$	0	0
$931$	1.67884e7	0.634797
$932$	0	0
$933$	2.45405e7	0.922954
$934$	0	0
$935$	−2.58774e6	−0.0968037
$936$	0	0
$937$	5.15971e7	1.91989	0.959945	−	0.280189i	$-0.0903970\pi$
$937$	5.15971e7	1.91989	0.959945	+	0.280189i	$0.0903970\pi$
$938$	0	0
$939$	−2.61416e6	−0.0967540
$940$	0	0
$941$	4.78212e7	1.76054	0.880272	−	0.474470i	$-0.157360\pi$
$941$	4.78212e7	1.76054	0.880272	+	0.474470i	$0.157360\pi$
$942$	0	0
$943$	−1.60454e7	−0.587587
$944$	0	0
$945$	3.55520e6	0.129505
$946$	0	0
$947$	−2.20444e7	−0.798772	−0.399386	−	0.916783i	$-0.630777\pi$
$947$	−2.20444e7	−0.798772	−0.399386	+	0.916783i	$0.630777\pi$
$948$	0	0
$949$	−2.90423e7	−1.04680
$950$	0	0
$951$	−2.07468e7	−0.743876
$952$	0	0
$953$	−6.82257e6	−0.243341	−0.121671	−	0.992571i	$-0.538825\pi$
$953$	−6.82257e6	−0.243341	−0.121671	+	0.992571i	$0.538825\pi$
$954$	0	0
$955$	−2.28385e7	−0.810323
$956$	0	0
$957$	1.11650e6	0.0394074
$958$	0	0
$959$	−6.86396e6	−0.241006
$960$	0	0
$961$	−2.77439e7	−0.969078
$962$	0	0
$963$	−1.18023e7	−0.410110
$964$	0	0
$965$	6.16636e7	2.13162
$966$	0	0
$967$	1.96256e7	0.674927	0.337464	−	0.941339i	$-0.390431\pi$
$967$	1.96256e7	0.674927	0.337464	+	0.941339i	$0.390431\pi$
$968$	0	0
$969$	−3.37478e6	−0.115461
$970$	0	0
$971$	3.71239e7	1.26359	0.631794	−	0.775136i	$-0.282319\pi$
$971$	3.71239e7	1.26359	0.631794	+	0.775136i	$0.282319\pi$
$972$	0	0
$973$	−243705.	−0.00825244
$974$	0	0
$975$	1.76289e7	0.593900
$976$	0	0
$977$	−9.65383e6	−0.323566	−0.161783	−	0.986826i	$-0.551725\pi$
$977$	−9.65383e6	−0.323566	−0.161783	+	0.986826i	$0.551725\pi$
$978$	0	0
$979$	−1.05224e7	−0.350879
$980$	0	0
$981$	−5.04230e6	−0.167285
$982$	0	0
$983$	−3.20688e6	−0.105852	−0.0529260	−	0.998598i	$-0.516855\pi$
$983$	−3.20688e6	−0.105852	−0.0529260	+	0.998598i	$0.516855\pi$
$984$	0	0
$985$	5.07308e7	1.66602
$986$	0	0
$987$	−4.76392e6	−0.155658
$988$	0	0
$989$	9.07027e6	0.294869
$990$	0	0
$991$	−2.41830e7	−0.782213	−0.391107	−	0.920345i	$-0.627908\pi$
$991$	−2.41830e7	−0.782213	−0.391107	+	0.920345i	$0.627908\pi$
$992$	0	0
$993$	3.28972e7	1.05873
$994$	0	0
$995$	3.73935e7	1.19740
$996$	0	0
$997$	−5.35292e6	−0.170550	−0.0852752	−	0.996357i	$-0.527177\pi$
$997$	−5.35292e6	−0.170550	−0.0852752	+	0.996357i	$0.527177\pi$
$998$	0	0
$999$	4.23702e6	0.134322

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	528.6.a.r.1.1		2
4.3	odd	2		264.6.a.a.1.1	✓	2
12.11	even	2		792.6.a.d.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
264.6.a.a.1.1	✓	2	4.3	odd	2
528.6.a.r.1.1		2	1.1	even	1	trivial
792.6.a.d.1.2		2	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)

\(f(q)\)	\(=\)	\(q+9.00000 q^{3} -75.6712 q^{5} -64.4475 q^{7} +81.0000 q^{9} +O(q^{10})\)	\(q+9.00000 q^{3} -75.6712 q^{5} -64.4475 q^{7} +81.0000 q^{9} +121.000 q^{11} +753.041 q^{13} -681.041 q^{15} +282.621 q^{17} -1326.78 q^{19} -580.027 q^{21} +1552.76 q^{23} +2601.14 q^{25} +729.000 q^{27} +1025.25 q^{29} +940.895 q^{31} +1089.00 q^{33} +4876.82 q^{35} +5812.10 q^{37} +6777.37 q^{39} -10333.5 q^{41} +5841.39 q^{43} -6129.37 q^{45} +8213.27 q^{47} -12653.5 q^{49} +2543.59 q^{51} -29046.2 q^{53} -9156.22 q^{55} -11941.0 q^{57} -15409.5 q^{59} -2109.77 q^{61} -5220.25 q^{63} -56983.6 q^{65} -22144.1 q^{67} +13974.8 q^{69} -35135.6 q^{71} -38566.7 q^{73} +23410.2 q^{75} -7798.15 q^{77} +25093.6 q^{79} +6561.00 q^{81} +19125.5 q^{83} -21386.3 q^{85} +9227.26 q^{87} -86961.8 q^{89} -48531.6 q^{91} +8468.05 q^{93} +100399. q^{95} -112243. q^{97} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 18 q^{3} - 30 q^{5} - 48 q^{7} + 162 q^{9}+O(q^{10}) \) 2 * q + 18 * q^3 - 30 * q^5 - 48 * q^7 + 162 * q^9	\( 2 q + 18 q^{3} - 30 q^{5} - 48 q^{7} + 162 q^{9} + 242 q^{11} + 414 q^{13} - 270 q^{15} - 1174 q^{17} - 2694 q^{19} - 432 q^{21} - 2274 q^{23} + 1562 q^{25} + 1458 q^{27} + 1282 q^{29} + 1720 q^{31} + 2178 q^{33} + 5628 q^{35} - 2856 q^{37} + 3726 q^{39} - 7198 q^{41} + 7274 q^{43} - 2430 q^{45} + 3362 q^{47} - 29190 q^{49} - 10566 q^{51} - 44138 q^{53} - 3630 q^{55} - 24246 q^{57} + 27668 q^{59} - 25050 q^{61} - 3888 q^{63} - 72468 q^{65} - 29080 q^{67} - 20466 q^{69} + 3950 q^{71} - 52784 q^{73} + 14058 q^{75} - 5808 q^{77} + 97268 q^{79} + 13122 q^{81} + 15196 q^{83} - 87912 q^{85} + 11538 q^{87} - 227476 q^{89} - 54108 q^{91} + 15480 q^{93} + 37956 q^{95} - 96996 q^{97} + 19602 q^{99}+O(q^{100}) \) 2 * q + 18 * q^3 - 30 * q^5 - 48 * q^7 + 162 * q^9 + 242 * q^11 + 414 * q^13 - 270 * q^15 - 1174 * q^17 - 2694 * q^19 - 432 * q^21 - 2274 * q^23 + 1562 * q^25 + 1458 * q^27 + 1282 * q^29 + 1720 * q^31 + 2178 * q^33 + 5628 * q^35 - 2856 * q^37 + 3726 * q^39 - 7198 * q^41 + 7274 * q^43 - 2430 * q^45 + 3362 * q^47 - 29190 * q^49 - 10566 * q^51 - 44138 * q^53 - 3630 * q^55 - 24246 * q^57 + 27668 * q^59 - 25050 * q^61 - 3888 * q^63 - 72468 * q^65 - 29080 * q^67 - 20466 * q^69 + 3950 * q^71 - 52784 * q^73 + 14058 * q^75 - 5808 * q^77 + 97268 * q^79 + 13122 * q^81 + 15196 * q^83 - 87912 * q^85 + 11538 * q^87 - 227476 * q^89 - 54108 * q^91 + 15480 * q^93 + 37956 * q^95 - 96996 * q^97 + 19602 * q^99

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