LMFDB - Embedded newform 528.6.a.s.1.2

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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{177}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 44 $ x^2 - x - 44
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 33)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-6.15207$ 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)

$f(q)$	$=$	$q+9.00000 q^{3} +95.5207 q^{5} +209.521 q^{7} +81.0000 q^{9} +O(q^{10})$	$q+9.00000 q^{3} +95.5207 q^{5} +209.521 q^{7} +81.0000 q^{9} +121.000 q^{11} -335.779 q^{13} +859.686 q^{15} -799.124 q^{17} +658.175 q^{19} +1885.69 q^{21} +4119.66 q^{23} +5999.20 q^{25} +729.000 q^{27} +559.348 q^{29} +6052.31 q^{31} +1089.00 q^{33} +20013.6 q^{35} -14053.3 q^{37} -3022.01 q^{39} +1846.27 q^{41} -1623.49 q^{43} +7737.17 q^{45} -20728.1 q^{47} +27091.9 q^{49} -7192.12 q^{51} -7585.46 q^{53} +11558.0 q^{55} +5923.58 q^{57} -18468.5 q^{59} -16972.3 q^{61} +16971.2 q^{63} -32073.8 q^{65} +5618.62 q^{67} +37076.9 q^{69} -3703.10 q^{71} -19808.0 q^{73} +53992.8 q^{75} +25352.0 q^{77} -64009.9 q^{79} +6561.00 q^{81} +46390.2 q^{83} -76332.9 q^{85} +5034.13 q^{87} -53959.5 q^{89} -70352.5 q^{91} +54470.8 q^{93} +62869.3 q^{95} +145249. q^{97} +9801.00 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 18 q^{3} + 58 q^{5} + 286 q^{7} + 162 q^{9}+O(q^{10}) $ 2 * q + 18 * q^3 + 58 * q^5 + 286 * q^7 + 162 * q^9	$ 2 q + 18 q^{3} + 58 q^{5} + 286 q^{7} + 162 q^{9} + 242 q^{11} - 166 q^{13} + 522 q^{15} - 800 q^{17} + 1476 q^{19} + 2574 q^{21} + 3370 q^{23} + 4282 q^{25} + 1458 q^{27} + 6600 q^{29} + 7528 q^{31} + 2178 q^{33} + 17144 q^{35} - 29916 q^{37} - 1494 q^{39} - 5780 q^{41} + 16656 q^{43} + 4698 q^{45} - 7850 q^{47} + 16134 q^{49} - 7200 q^{51} + 14178 q^{53} + 7018 q^{55} + 13284 q^{57} - 17300 q^{59} - 2946 q^{61} + 23166 q^{63} - 38444 q^{65} - 31336 q^{67} + 30330 q^{69} + 33810 q^{71} + 60644 q^{73} + 38538 q^{75} + 34606 q^{77} - 1870 q^{79} + 13122 q^{81} + 58296 q^{83} - 76300 q^{85} + 59400 q^{87} + 92388 q^{89} - 57368 q^{91} + 67752 q^{93} + 32184 q^{95} + 7120 q^{97} + 19602 q^{99}+O(q^{100}) $ 2 * q + 18 * q^3 + 58 * q^5 + 286 * q^7 + 162 * q^9 + 242 * q^11 - 166 * q^13 + 522 * q^15 - 800 * q^17 + 1476 * q^19 + 2574 * q^21 + 3370 * q^23 + 4282 * q^25 + 1458 * q^27 + 6600 * q^29 + 7528 * q^31 + 2178 * q^33 + 17144 * q^35 - 29916 * q^37 - 1494 * q^39 - 5780 * q^41 + 16656 * q^43 + 4698 * q^45 - 7850 * q^47 + 16134 * q^49 - 7200 * q^51 + 14178 * q^53 + 7018 * q^55 + 13284 * q^57 - 17300 * q^59 - 2946 * q^61 + 23166 * q^63 - 38444 * q^65 - 31336 * q^67 + 30330 * q^69 + 33810 * q^71 + 60644 * q^73 + 38538 * q^75 + 34606 * q^77 - 1870 * q^79 + 13122 * q^81 + 58296 * q^83 - 76300 * q^85 + 59400 * q^87 + 92388 * q^89 - 57368 * q^91 + 67752 * q^93 + 32184 * q^95 + 7120 * 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
$5$	95.5207	1.70873	0.854363	−	0.519677i	$-0.173948\pi$
$5$	95.5207	1.70873	0.854363	+	0.519677i	$0.173948\pi$
$6$	0	0
$7$	209.521	1.61615	0.808075	−	0.589079i	$-0.200509\pi$
$7$	209.521	1.61615	0.808075	+	0.589079i	$0.200509\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	0	0
$13$	−335.779	−0.551055	−0.275527	−	0.961293i	$-0.588852\pi$
$13$	−335.779	−0.551055	−0.275527	+	0.961293i	$0.588852\pi$
$14$	0	0
$15$	859.686	0.986533
$16$	0	0
$17$	−799.124	−0.670644	−0.335322	−	0.942104i	$-0.608845\pi$
$17$	−799.124	−0.670644	−0.335322	+	0.942104i	$0.608845\pi$
$18$	0	0
$19$	658.175	0.418271	0.209135	−	0.977887i	$-0.432935\pi$
$19$	658.175	0.418271	0.209135	+	0.977887i	$0.432935\pi$
$20$	0	0
$21$	1885.69	0.933085
$22$	0	0
$23$	4119.66	1.62383	0.811917	−	0.583773i	$-0.198424\pi$
$23$	4119.66	1.62383	0.811917	+	0.583773i	$0.198424\pi$
$24$	0	0
$25$	5999.20	1.91974
$26$	0	0
$27$	729.000	0.192450
$28$	0	0
$29$	559.348	0.123506	0.0617529	−	0.998091i	$-0.480331\pi$
$29$	559.348	0.123506	0.0617529	+	0.998091i	$0.480331\pi$
$30$	0	0
$31$	6052.31	1.13114	0.565571	−	0.824700i	$-0.308656\pi$
$31$	6052.31	1.13114	0.565571	+	0.824700i	$0.308656\pi$
$32$	0	0
$33$	1089.00	0.174078
$34$	0	0
$35$	20013.6	2.76156
$36$	0	0
$37$	−14053.3	−1.68762	−0.843810	−	0.536642i	$-0.819692\pi$
$37$	−14053.3	−1.68762	−0.843810	+	0.536642i	$0.819692\pi$
$38$	0	0
$39$	−3022.01	−0.318151
$40$	0	0
$41$	1846.27	0.171528	0.0857642	−	0.996315i	$-0.472667\pi$
$41$	1846.27	0.171528	0.0857642	+	0.996315i	$0.472667\pi$
$42$	0	0
$43$	−1623.49	−0.133900	−0.0669498	−	0.997756i	$-0.521327\pi$
$43$	−1623.49	−0.133900	−0.0669498	+	0.997756i	$0.521327\pi$
$44$	0	0
$45$	7737.17	0.569575
$46$	0	0
$47$	−20728.1	−1.36872	−0.684361	−	0.729143i	$-0.739919\pi$
$47$	−20728.1	−1.36872	−0.684361	+	0.729143i	$0.739919\pi$
$48$	0	0
$49$	27091.9	1.61194
$50$	0	0
$51$	−7192.12	−0.387196
$52$	0	0
$53$	−7585.46	−0.370930	−0.185465	−	0.982651i	$-0.559379\pi$
$53$	−7585.46	−0.370930	−0.185465	+	0.982651i	$0.559379\pi$
$54$	0	0
$55$	11558.0	0.515200
$56$	0	0
$57$	5923.58	0.241489
$58$	0	0
$59$	−18468.5	−0.690717	−0.345359	−	0.938471i	$-0.612243\pi$
$59$	−18468.5	−0.690717	−0.345359	+	0.938471i	$0.612243\pi$
$60$	0	0
$61$	−16972.3	−0.584005	−0.292002	−	0.956418i	$-0.594322\pi$
$61$	−16972.3	−0.584005	−0.292002	+	0.956418i	$0.594322\pi$
$62$	0	0
$63$	16971.2	0.538717
$64$	0	0
$65$	−32073.8	−0.941601
$66$	0	0
$67$	5618.62	0.152912	0.0764561	−	0.997073i	$-0.475639\pi$
$67$	5618.62	0.152912	0.0764561	+	0.997073i	$0.475639\pi$
$68$	0	0
$69$	37076.9	0.937521
$70$	0	0
$71$	−3703.10	−0.0871807	−0.0435903	−	0.999049i	$-0.513880\pi$
$71$	−3703.10	−0.0871807	−0.0435903	+	0.999049i	$0.513880\pi$
$72$	0	0
$73$	−19808.0	−0.435044	−0.217522	−	0.976055i	$-0.569797\pi$
$73$	−19808.0	−0.435044	−0.217522	+	0.976055i	$0.569797\pi$
$74$	0	0
$75$	53992.8	1.10836
$76$	0	0
$77$	25352.0	0.487288
$78$	0	0
$79$	−64009.9	−1.15393	−0.576965	−	0.816769i	$-0.695763\pi$
$79$	−64009.9	−1.15393	−0.576965	+	0.816769i	$0.695763\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	46390.2	0.739147	0.369573	−	0.929202i	$-0.379504\pi$
$83$	46390.2	0.739147	0.369573	+	0.929202i	$0.379504\pi$
$84$	0	0
$85$	−76332.9	−1.14595
$86$	0	0
$87$	5034.13	0.0713061
$88$	0	0
$89$	−53959.5	−0.722093	−0.361046	−	0.932548i	$-0.617580\pi$
$89$	−53959.5	−0.722093	−0.361046	+	0.932548i	$0.617580\pi$
$90$	0	0
$91$	−70352.5	−0.890587
$92$	0	0
$93$	54470.8	0.653065
$94$	0	0
$95$	62869.3	0.714710
$96$	0	0
$97$	145249.	1.56741	0.783707	−	0.621130i	$-0.213326\pi$
$97$	145249.	1.56741	0.783707	+	0.621130i	$0.213326\pi$
$98$	0	0
$99$	9801.00	0.100504
$100$	0	0
$101$	−77215.1	−0.753180	−0.376590	−	0.926380i	$-0.622903\pi$
$101$	−77215.1	−0.753180	−0.376590	+	0.926380i	$0.622903\pi$
$102$	0	0
$103$	106203.	0.986374	0.493187	−	0.869923i	$-0.335832\pi$
$103$	106203.	0.986374	0.493187	+	0.869923i	$0.335832\pi$
$104$	0	0
$105$	180122.	1.59439
$106$	0	0
$107$	−33750.8	−0.284987	−0.142493	−	0.989796i	$-0.545512\pi$
$107$	−33750.8	−0.284987	−0.142493	+	0.989796i	$0.545512\pi$
$108$	0	0
$109$	−37977.1	−0.306165	−0.153083	−	0.988213i	$-0.548920\pi$
$109$	−37977.1	−0.306165	−0.153083	+	0.988213i	$0.548920\pi$
$110$	0	0
$111$	−126480.	−0.974348
$112$	0	0
$113$	239549.	1.76481	0.882407	−	0.470486i	$-0.155922\pi$
$113$	239549.	1.76481	0.882407	+	0.470486i	$0.155922\pi$
$114$	0	0
$115$	393512.	2.77469
$116$	0	0
$117$	−27198.1	−0.183685
$118$	0	0
$119$	−167433.	−1.08386
$120$	0	0
$121$	14641.0	0.0909091
$122$	0	0
$123$	16616.4	0.0990320
$124$	0	0
$125$	274545.	1.57159
$126$	0	0
$127$	−88793.2	−0.488507	−0.244253	−	0.969711i	$-0.578543\pi$
$127$	−88793.2	−0.488507	−0.244253	+	0.969711i	$0.578543\pi$
$128$	0	0
$129$	−14611.4	−0.0773070
$130$	0	0
$131$	−333640.	−1.69864	−0.849318	−	0.527881i	$-0.822987\pi$
$131$	−333640.	−1.69864	−0.849318	+	0.527881i	$0.822987\pi$
$132$	0	0
$133$	137901.	0.675988
$134$	0	0
$135$	69634.6	0.328844
$136$	0	0
$137$	−377709.	−1.71932	−0.859659	−	0.510869i	$-0.829324\pi$
$137$	−377709.	−1.71932	−0.859659	+	0.510869i	$0.829324\pi$
$138$	0	0
$139$	365308.	1.60370	0.801849	−	0.597527i	$-0.203850\pi$
$139$	365308.	1.60370	0.801849	+	0.597527i	$0.203850\pi$
$140$	0	0
$141$	−186553.	−0.790232
$142$	0	0
$143$	−40629.2	−0.166149
$144$	0	0
$145$	53429.3	0.211038
$146$	0	0
$147$	243827.	0.930655
$148$	0	0
$149$	297568.	1.09805	0.549023	−	0.835807i	$-0.315000\pi$
$149$	297568.	1.09805	0.549023	+	0.835807i	$0.315000\pi$
$150$	0	0
$151$	−318093.	−1.13530	−0.567651	−	0.823269i	$-0.692148\pi$
$151$	−318093.	−1.13530	−0.567651	+	0.823269i	$0.692148\pi$
$152$	0	0
$153$	−64729.0	−0.223548
$154$	0	0
$155$	578121.	1.93281
$156$	0	0
$157$	−34489.6	−0.111671	−0.0558353	−	0.998440i	$-0.517782\pi$
$157$	−34489.6	−0.111671	−0.0558353	+	0.998440i	$0.517782\pi$
$158$	0	0
$159$	−68269.1	−0.214157
$160$	0	0
$161$	863153.	2.62436
$162$	0	0
$163$	45865.7	0.135213	0.0676067	−	0.997712i	$-0.478464\pi$
$163$	45865.7	0.135213	0.0676067	+	0.997712i	$0.478464\pi$
$164$	0	0
$165$	104022.	0.297451
$166$	0	0
$167$	−290943.	−0.807265	−0.403633	−	0.914921i	$-0.632253\pi$
$167$	−290943.	−0.807265	−0.403633	+	0.914921i	$0.632253\pi$
$168$	0	0
$169$	−258546.	−0.696339
$170$	0	0
$171$	53312.2	0.139424
$172$	0	0
$173$	−139360.	−0.354016	−0.177008	−	0.984209i	$-0.556642\pi$
$173$	−139360.	−0.354016	−0.177008	+	0.984209i	$0.556642\pi$
$174$	0	0
$175$	1.25696e6	3.10259
$176$	0	0
$177$	−166216.	−0.398786
$178$	0	0
$179$	599556.	1.39861	0.699306	−	0.714823i	$-0.253493\pi$
$179$	599556.	1.39861	0.699306	+	0.714823i	$0.253493\pi$
$180$	0	0
$181$	130631.	0.296380	0.148190	−	0.988959i	$-0.452655\pi$
$181$	130631.	0.296380	0.148190	+	0.988959i	$0.452655\pi$
$182$	0	0
$183$	−152751.	−0.337175
$184$	0	0
$185$	−1.34238e6	−2.88368
$186$	0	0
$187$	−96694.0	−0.202207
$188$	0	0
$189$	152741.	0.311028
$190$	0	0
$191$	−338243.	−0.670882	−0.335441	−	0.942061i	$-0.608885\pi$
$191$	−338243.	−0.670882	−0.335441	+	0.942061i	$0.608885\pi$
$192$	0	0
$193$	−329094.	−0.635955	−0.317978	−	0.948098i	$-0.603004\pi$
$193$	−329094.	−0.635955	−0.317978	+	0.948098i	$0.603004\pi$
$194$	0	0
$195$	−288664.	−0.543634
$196$	0	0
$197$	517397.	0.949858	0.474929	−	0.880024i	$-0.342474\pi$
$197$	517397.	0.949858	0.474929	+	0.880024i	$0.342474\pi$
$198$	0	0
$199$	531436.	0.951302	0.475651	−	0.879634i	$-0.342212\pi$
$199$	531436.	0.951302	0.475651	+	0.879634i	$0.342212\pi$
$200$	0	0
$201$	50567.5	0.0882839
$202$	0	0
$203$	117195.	0.199604
$204$	0	0
$205$	176357.	0.293095
$206$	0	0
$207$	333692.	0.541278
$208$	0	0
$209$	79639.2	0.126113
$210$	0	0
$211$	−409913.	−0.633848	−0.316924	−	0.948451i	$-0.602650\pi$
$211$	−409913.	−0.633848	−0.316924	+	0.948451i	$0.602650\pi$
$212$	0	0
$213$	−33327.9	−0.0503338
$214$	0	0
$215$	−155077.	−0.228798
$216$	0	0
$217$	1.26808e6	1.82810
$218$	0	0
$219$	−178272.	−0.251173
$220$	0	0
$221$	268329.	0.369561
$222$	0	0
$223$	−269898.	−0.363444	−0.181722	−	0.983350i	$-0.558167\pi$
$223$	−269898.	−0.363444	−0.181722	+	0.983350i	$0.558167\pi$
$224$	0	0
$225$	485935.	0.639915
$226$	0	0
$227$	1.02146e6	1.31570	0.657852	−	0.753147i	$-0.271465\pi$
$227$	1.02146e6	1.31570	0.657852	+	0.753147i	$0.271465\pi$
$228$	0	0
$229$	169276.	0.213307	0.106654	−	0.994296i	$-0.465986\pi$
$229$	169276.	0.213307	0.106654	+	0.994296i	$0.465986\pi$
$230$	0	0
$231$	228168.	0.281336
$232$	0	0
$233$	−20445.6	−0.0246723	−0.0123361	−	0.999924i	$-0.503927\pi$
$233$	−20445.6	−0.0246723	−0.0123361	+	0.999924i	$0.503927\pi$
$234$	0	0
$235$	−1.97996e6	−2.33877
$236$	0	0
$237$	−576089.	−0.666222
$238$	0	0
$239$	933552.	1.05717	0.528583	−	0.848881i	$-0.322723\pi$
$239$	933552.	1.05717	0.528583	+	0.848881i	$0.322723\pi$
$240$	0	0
$241$	−1.30259e6	−1.44466	−0.722331	−	0.691547i	$-0.756929\pi$
$241$	−1.30259e6	−1.44466	−0.722331	+	0.691547i	$0.756929\pi$
$242$	0	0
$243$	59049.0	0.0641500
$244$	0	0
$245$	2.58784e6	2.75437
$246$	0	0
$247$	−221001.	−0.230490
$248$	0	0
$249$	417511.	0.426747
$250$	0	0
$251$	930872.	0.932622	0.466311	−	0.884621i	$-0.345583\pi$
$251$	930872.	0.932622	0.466311	+	0.884621i	$0.345583\pi$
$252$	0	0
$253$	498478.	0.489604
$254$	0	0
$255$	−686996.	−0.661612
$256$	0	0
$257$	−25313.6	−0.0239067	−0.0119534	−	0.999929i	$-0.503805\pi$
$257$	−25313.6	−0.0239067	−0.0119534	+	0.999929i	$0.503805\pi$
$258$	0	0
$259$	−2.94446e6	−2.72745
$260$	0	0
$261$	45307.2	0.0411686
$262$	0	0
$263$	1.69322e6	1.50947	0.754735	−	0.656030i	$-0.227765\pi$
$263$	1.69322e6	1.50947	0.754735	+	0.656030i	$0.227765\pi$
$264$	0	0
$265$	−724568.	−0.633818
$266$	0	0
$267$	−485636.	−0.416901
$268$	0	0
$269$	469365.	0.395485	0.197742	−	0.980254i	$-0.436639\pi$
$269$	469365.	0.395485	0.197742	+	0.980254i	$0.436639\pi$
$270$	0	0
$271$	2.18341e6	1.80598	0.902988	−	0.429666i	$-0.141369\pi$
$271$	2.18341e6	1.80598	0.902988	+	0.429666i	$0.141369\pi$
$272$	0	0
$273$	−633173.	−0.514181
$274$	0	0
$275$	725903.	0.578825
$276$	0	0
$277$	−84280.4	−0.0659974	−0.0329987	−	0.999455i	$-0.510506\pi$
$277$	−84280.4	−0.0659974	−0.0329987	+	0.999455i	$0.510506\pi$
$278$	0	0
$279$	490237.	0.377047
$280$	0	0
$281$	649515.	0.490708	0.245354	−	0.969433i	$-0.421096\pi$
$281$	649515.	0.490708	0.245354	+	0.969433i	$0.421096\pi$
$282$	0	0
$283$	−1.54893e6	−1.14965	−0.574826	−	0.818276i	$-0.694930\pi$
$283$	−1.54893e6	−1.14965	−0.574826	+	0.818276i	$0.694930\pi$
$284$	0	0
$285$	565824.	0.412638
$286$	0	0
$287$	386832.	0.277216
$288$	0	0
$289$	−781258.	−0.550237
$290$	0	0
$291$	1.30724e6	0.904947
$292$	0	0
$293$	−70276.6	−0.0478236	−0.0239118	−	0.999714i	$-0.507612\pi$
$293$	−70276.6	−0.0478236	−0.0239118	+	0.999714i	$0.507612\pi$
$294$	0	0
$295$	−1.76412e6	−1.18025
$296$	0	0
$297$	88209.0	0.0580259
$298$	0	0
$299$	−1.38329e6	−0.894821
$300$	0	0
$301$	−340155.	−0.216402
$302$	0	0
$303$	−694936.	−0.434849
$304$	0	0
$305$	−1.62121e6	−0.997904
$306$	0	0
$307$	−1.43343e6	−0.868022	−0.434011	−	0.900908i	$-0.642902\pi$
$307$	−1.43343e6	−0.868022	−0.434011	+	0.900908i	$0.642902\pi$
$308$	0	0
$309$	955823.	0.569484
$310$	0	0
$311$	227879.	0.133599	0.0667996	−	0.997766i	$-0.478721\pi$
$311$	227879.	0.133599	0.0667996	+	0.997766i	$0.478721\pi$
$312$	0	0
$313$	−723760.	−0.417574	−0.208787	−	0.977961i	$-0.566952\pi$
$313$	−723760.	−0.417574	−0.208787	+	0.977961i	$0.566952\pi$
$314$	0	0
$315$	1.62110e6	0.920519
$316$	0	0
$317$	1.26086e6	0.704723	0.352362	−	0.935864i	$-0.385379\pi$
$317$	1.26086e6	0.704723	0.352362	+	0.935864i	$0.385379\pi$
$318$	0	0
$319$	67681.1	0.0372384
$320$	0	0
$321$	−303757.	−0.164537
$322$	0	0
$323$	−525964.	−0.280511
$324$	0	0
$325$	−2.01440e6	−1.05788
$326$	0	0
$327$	−341794.	−0.176765
$328$	0	0
$329$	−4.34297e6	−2.21206
$330$	0	0
$331$	−433444.	−0.217452	−0.108726	−	0.994072i	$-0.534677\pi$
$331$	−433444.	−0.217452	−0.108726	+	0.994072i	$0.534677\pi$
$332$	0	0
$333$	−1.13832e6	−0.562540
$334$	0	0
$335$	536694.	0.261285
$336$	0	0
$337$	3.80554e6	1.82533	0.912666	−	0.408706i	$-0.134020\pi$
$337$	3.80554e6	1.82533	0.912666	+	0.408706i	$0.134020\pi$
$338$	0	0
$339$	2.15594e6	1.01892
$340$	0	0
$341$	732330.	0.341052
$342$	0	0
$343$	2.15490e6	0.988991
$344$	0	0
$345$	3.54161e6	1.60197
$346$	0	0
$347$	−2.91029e6	−1.29752	−0.648759	−	0.760994i	$-0.724712\pi$
$347$	−2.91029e6	−1.29752	−0.648759	+	0.760994i	$0.724712\pi$
$348$	0	0
$349$	−4.13500e6	−1.81724	−0.908620	−	0.417625i	$-0.862863\pi$
$349$	−4.13500e6	−1.81724	−0.908620	+	0.417625i	$0.862863\pi$
$350$	0	0
$351$	−244783.	−0.106050
$352$	0	0
$353$	499935.	0.213539	0.106769	−	0.994284i	$-0.465949\pi$
$353$	499935.	0.213539	0.106769	+	0.994284i	$0.465949\pi$
$354$	0	0
$355$	−353723.	−0.148968
$356$	0	0
$357$	−1.50690e6	−0.625768
$358$	0	0
$359$	2.96365e6	1.21364	0.606822	−	0.794838i	$-0.292444\pi$
$359$	2.96365e6	1.21364	0.606822	+	0.794838i	$0.292444\pi$
$360$	0	0
$361$	−2.04290e6	−0.825050
$362$	0	0
$363$	131769.	0.0524864
$364$	0	0
$365$	−1.89207e6	−0.743371
$366$	0	0
$367$	−257505.	−0.0997976	−0.0498988	−	0.998754i	$-0.515890\pi$
$367$	−257505.	−0.0997976	−0.0498988	+	0.998754i	$0.515890\pi$
$368$	0	0
$369$	149548.	0.0571761
$370$	0	0
$371$	−1.58931e6	−0.599479
$372$	0	0
$373$	−1.28217e6	−0.477169	−0.238584	−	0.971122i	$-0.576683\pi$
$373$	−1.28217e6	−0.477169	−0.238584	+	0.971122i	$0.576683\pi$
$374$	0	0
$375$	2.47091e6	0.907358
$376$	0	0
$377$	−187817.	−0.0680584
$378$	0	0
$379$	−3.13440e6	−1.12087	−0.560436	−	0.828198i	$-0.689366\pi$
$379$	−3.13440e6	−1.12087	−0.560436	+	0.828198i	$0.689366\pi$
$380$	0	0
$381$	−799139.	−0.282040
$382$	0	0
$383$	−875781.	−0.305069	−0.152535	−	0.988298i	$-0.548744\pi$
$383$	−875781.	−0.305069	−0.152535	+	0.988298i	$0.548744\pi$
$384$	0	0
$385$	2.42164e6	0.832641
$386$	0	0
$387$	−131503.	−0.0446332
$388$	0	0
$389$	−1.95280e6	−0.654312	−0.327156	−	0.944970i	$-0.606090\pi$
$389$	−1.95280e6	−0.654312	−0.327156	+	0.944970i	$0.606090\pi$
$390$	0	0
$391$	−3.29212e6	−1.08901
$392$	0	0
$393$	−3.00276e6	−0.980708
$394$	0	0
$395$	−6.11427e6	−1.97175
$396$	0	0
$397$	−1.70718e6	−0.543629	−0.271814	−	0.962350i	$-0.587624\pi$
$397$	−1.70718e6	−0.543629	−0.271814	+	0.962350i	$0.587624\pi$
$398$	0	0
$399$	1.24111e6	0.390282
$400$	0	0
$401$	4.51630e6	1.40256	0.701280	−	0.712886i	$-0.252612\pi$
$401$	4.51630e6	1.40256	0.701280	+	0.712886i	$0.252612\pi$
$402$	0	0
$403$	−2.03224e6	−0.623321
$404$	0	0
$405$	626711.	0.189858
$406$	0	0
$407$	−1.70045e6	−0.508836
$408$	0	0
$409$	−721013.	−0.213125	−0.106563	−	0.994306i	$-0.533984\pi$
$409$	−721013.	−0.213125	−0.106563	+	0.994306i	$0.533984\pi$
$410$	0	0
$411$	−3.39938e6	−0.992648
$412$	0	0
$413$	−3.86952e6	−1.11630
$414$	0	0
$415$	4.43122e6	1.26300
$416$	0	0
$417$	3.28777e6	0.925895
$418$	0	0
$419$	−3.89904e6	−1.08498	−0.542491	−	0.840061i	$-0.682519\pi$
$419$	−3.89904e6	−1.08498	−0.542491	+	0.840061i	$0.682519\pi$
$420$	0	0
$421$	−3.39665e6	−0.933996	−0.466998	−	0.884258i	$-0.654665\pi$
$421$	−3.39665e6	−0.933996	−0.466998	+	0.884258i	$0.654665\pi$
$422$	0	0
$423$	−1.67898e6	−0.456241
$424$	0	0
$425$	−4.79410e6	−1.28746
$426$	0	0
$427$	−3.55605e6	−0.943840
$428$	0	0
$429$	−365663.	−0.0959263
$430$	0	0
$431$	5.15196e6	1.33592	0.667958	−	0.744199i	$-0.267168\pi$
$431$	5.15196e6	1.33592	0.667958	+	0.744199i	$0.267168\pi$
$432$	0	0
$433$	−3.23450e6	−0.829063	−0.414531	−	0.910035i	$-0.636055\pi$
$433$	−3.23450e6	−0.829063	−0.414531	+	0.910035i	$0.636055\pi$
$434$	0	0
$435$	480864.	0.121843
$436$	0	0
$437$	2.71146e6	0.679202
$438$	0	0
$439$	−6.31060e6	−1.56282	−0.781411	−	0.624017i	$-0.785500\pi$
$439$	−6.31060e6	−1.56282	−0.781411	+	0.624017i	$0.785500\pi$
$440$	0	0
$441$	2.19444e6	0.537314
$442$	0	0
$443$	−2.87512e6	−0.696061	−0.348030	−	0.937483i	$-0.613149\pi$
$443$	−2.87512e6	−0.696061	−0.348030	+	0.937483i	$0.613149\pi$
$444$	0	0
$445$	−5.15425e6	−1.23386
$446$	0	0
$447$	2.67811e6	0.633957
$448$	0	0
$449$	−4.54361e6	−1.06362	−0.531809	−	0.846865i	$-0.678487\pi$
$449$	−4.54361e6	−1.06362	−0.531809	+	0.846865i	$0.678487\pi$
$450$	0	0
$451$	223399.	0.0517178
$452$	0	0
$453$	−2.86284e6	−0.655467
$454$	0	0
$455$	−6.72012e6	−1.52177
$456$	0	0
$457$	2.78485e6	0.623751	0.311876	−	0.950123i	$-0.399043\pi$
$457$	2.78485e6	0.623751	0.311876	+	0.950123i	$0.399043\pi$
$458$	0	0
$459$	−582561.	−0.129065
$460$	0	0
$461$	−2.00495e6	−0.439391	−0.219695	−	0.975569i	$-0.570506\pi$
$461$	−2.00495e6	−0.439391	−0.219695	+	0.975569i	$0.570506\pi$
$462$	0	0
$463$	4.67402e6	1.01330	0.506650	−	0.862152i	$-0.330884\pi$
$463$	4.67402e6	1.01330	0.506650	+	0.862152i	$0.330884\pi$
$464$	0	0
$465$	5.20309e6	1.11591
$466$	0	0
$467$	1.41948e6	0.301188	0.150594	−	0.988596i	$-0.451881\pi$
$467$	1.41948e6	0.301188	0.150594	+	0.988596i	$0.451881\pi$
$468$	0	0
$469$	1.17722e6	0.247129
$470$	0	0
$471$	−310406.	−0.0644731
$472$	0	0
$473$	−196443.	−0.0403722
$474$	0	0
$475$	3.94852e6	0.802973
$476$	0	0
$477$	−614422.	−0.123643
$478$	0	0
$479$	3.60077e6	0.717062	0.358531	−	0.933518i	$-0.383278\pi$
$479$	3.60077e6	0.717062	0.358531	+	0.933518i	$0.383278\pi$
$480$	0	0
$481$	4.71880e6	0.929970
$482$	0	0
$483$	7.76838e6	1.51517
$484$	0	0
$485$	1.38743e7	2.67828
$486$	0	0
$487$	5.57519e6	1.06521	0.532607	−	0.846363i	$-0.321212\pi$
$487$	5.57519e6	1.06521	0.532607	+	0.846363i	$0.321212\pi$
$488$	0	0
$489$	412792.	0.0780654
$490$	0	0
$491$	1.80948e6	0.338728	0.169364	−	0.985554i	$-0.445829\pi$
$491$	1.80948e6	0.338728	0.169364	+	0.985554i	$0.445829\pi$
$492$	0	0
$493$	−446989.	−0.0828284
$494$	0	0
$495$	936198.	0.171733
$496$	0	0
$497$	−775877.	−0.140897
$498$	0	0
$499$	5.94504e6	1.06882	0.534409	−	0.845226i	$-0.320534\pi$
$499$	5.94504e6	1.06882	0.534409	+	0.845226i	$0.320534\pi$
$500$	0	0
$501$	−2.61848e6	−0.466075
$502$	0	0
$503$	−6.57296e6	−1.15835	−0.579176	−	0.815202i	$-0.696626\pi$
$503$	−6.57296e6	−1.15835	−0.579176	+	0.815202i	$0.696626\pi$
$504$	0	0
$505$	−7.37564e6	−1.28698
$506$	0	0
$507$	−2.32691e6	−0.402031
$508$	0	0
$509$	1.76650e6	0.302218	0.151109	−	0.988517i	$-0.451716\pi$
$509$	1.76650e6	0.302218	0.151109	+	0.988517i	$0.451716\pi$
$510$	0	0
$511$	−4.15018e6	−0.703096
$512$	0	0
$513$	479810.	0.0804962
$514$	0	0
$515$	1.01445e7	1.68544
$516$	0	0
$517$	−2.50810e6	−0.412685
$518$	0	0
$519$	−1.25424e6	−0.204391
$520$	0	0
$521$	9.74164e6	1.57231	0.786154	−	0.618031i	$-0.212069\pi$
$521$	9.74164e6	1.57231	0.786154	+	0.618031i	$0.212069\pi$
$522$	0	0
$523$	−6.30069e6	−1.00724	−0.503621	−	0.863925i	$-0.667999\pi$
$523$	−6.30069e6	−1.00724	−0.503621	+	0.863925i	$0.667999\pi$
$524$	0	0
$525$	1.13126e7	1.79128
$526$	0	0
$527$	−4.83655e6	−0.758593
$528$	0	0
$529$	1.05352e7	1.63683
$530$	0	0
$531$	−1.49594e6	−0.230239
$532$	0	0
$533$	−619939.	−0.0945215
$534$	0	0
$535$	−3.22390e6	−0.486964
$536$	0	0
$537$	5.39600e6	0.807489
$538$	0	0
$539$	3.27812e6	0.486019
$540$	0	0
$541$	−1.12441e7	−1.65169	−0.825847	−	0.563894i	$-0.809303\pi$
$541$	−1.12441e7	−1.65169	−0.825847	+	0.563894i	$0.809303\pi$
$542$	0	0
$543$	1.17567e6	0.171115
$544$	0	0
$545$	−3.62760e6	−0.523153
$546$	0	0
$547$	−60699.0	−0.00867388	−0.00433694	−	0.999991i	$-0.501380\pi$
$547$	−60699.0	−0.00867388	−0.00433694	+	0.999991i	$0.501380\pi$
$548$	0	0
$549$	−1.37476e6	−0.194668
$550$	0	0
$551$	368149.	0.0516589
$552$	0	0
$553$	−1.34114e7	−1.86492
$554$	0	0
$555$	−1.20814e7	−1.66489
$556$	0	0
$557$	2.28778e6	0.312447	0.156224	−	0.987722i	$-0.450068\pi$
$557$	2.28778e6	0.312447	0.156224	+	0.987722i	$0.450068\pi$
$558$	0	0
$559$	545134.	0.0737860
$560$	0	0
$561$	−870246.	−0.116744
$562$	0	0
$563$	5.37938e6	0.715256	0.357628	−	0.933864i	$-0.383586\pi$
$563$	5.37938e6	0.715256	0.357628	+	0.933864i	$0.383586\pi$
$564$	0	0
$565$	2.28819e7	3.01558
$566$	0	0
$567$	1.37467e6	0.179572
$568$	0	0
$569$	−6.95488e6	−0.900552	−0.450276	−	0.892889i	$-0.648674\pi$
$569$	−6.95488e6	−0.900552	−0.450276	+	0.892889i	$0.648674\pi$
$570$	0	0
$571$	3.19590e6	0.410206	0.205103	−	0.978740i	$-0.434247\pi$
$571$	3.19590e6	0.410206	0.205103	+	0.978740i	$0.434247\pi$
$572$	0	0
$573$	−3.04419e6	−0.387334
$574$	0	0
$575$	2.47146e7	3.11734
$576$	0	0
$577$	2.54992e6	0.318851	0.159425	−	0.987210i	$-0.449036\pi$
$577$	2.54992e6	0.318851	0.159425	+	0.987210i	$0.449036\pi$
$578$	0	0
$579$	−2.96184e6	−0.367169
$580$	0	0
$581$	9.71970e6	1.19457
$582$	0	0
$583$	−917841.	−0.111840
$584$	0	0
$585$	−2.59798e6	−0.313867
$586$	0	0
$587$	1.96757e6	0.235687	0.117843	−	0.993032i	$-0.462402\pi$
$587$	1.96757e6	0.235687	0.117843	+	0.993032i	$0.462402\pi$
$588$	0	0
$589$	3.98348e6	0.473124
$590$	0	0
$591$	4.65658e6	0.548401
$592$	0	0
$593$	−8.41413e6	−0.982590	−0.491295	−	0.870993i	$-0.663476\pi$
$593$	−8.41413e6	−0.982590	−0.491295	+	0.870993i	$0.663476\pi$
$594$	0	0
$595$	−1.59933e7	−1.85202
$596$	0	0
$597$	4.78293e6	0.549235
$598$	0	0
$599$	1.00298e7	1.14216	0.571078	−	0.820896i	$-0.306525\pi$
$599$	1.00298e7	1.14216	0.571078	+	0.820896i	$0.306525\pi$
$600$	0	0
$601$	−1.42206e7	−1.60595	−0.802977	−	0.596010i	$-0.796752\pi$
$601$	−1.42206e7	−1.60595	−0.802977	+	0.596010i	$0.796752\pi$
$602$	0	0
$603$	455108.	0.0509708
$604$	0	0
$605$	1.39852e6	0.155339
$606$	0	0
$607$	−1.51717e7	−1.67133	−0.835665	−	0.549240i	$-0.814917\pi$
$607$	−1.51717e7	−1.67133	−0.835665	+	0.549240i	$0.814917\pi$
$608$	0	0
$609$	1.05476e6	0.115241
$610$	0	0
$611$	6.96006e6	0.754241
$612$	0	0
$613$	−1.52884e7	−1.64328	−0.821638	−	0.570010i	$-0.806939\pi$
$613$	−1.52884e7	−1.64328	−0.821638	+	0.570010i	$0.806939\pi$
$614$	0	0
$615$	1.58721e6	0.169218
$616$	0	0
$617$	−1.81989e6	−0.192457	−0.0962284	−	0.995359i	$-0.530678\pi$
$617$	−1.81989e6	−0.192457	−0.0962284	+	0.995359i	$0.530678\pi$
$618$	0	0
$619$	−1.39678e7	−1.46521	−0.732606	−	0.680653i	$-0.761696\pi$
$619$	−1.39678e7	−1.46521	−0.732606	+	0.680653i	$0.761696\pi$
$620$	0	0
$621$	3.00323e6	0.312507
$622$	0	0
$623$	−1.13056e7	−1.16701
$624$	0	0
$625$	7.47727e6	0.765672
$626$	0	0
$627$	716753.	0.0728116
$628$	0	0
$629$	1.12303e7	1.13179
$630$	0	0
$631$	1.64990e7	1.64962	0.824808	−	0.565413i	$-0.191283\pi$
$631$	1.64990e7	1.64962	0.824808	+	0.565413i	$0.191283\pi$
$632$	0	0
$633$	−3.68921e6	−0.365952
$634$	0	0
$635$	−8.48159e6	−0.834724
$636$	0	0
$637$	−9.09688e6	−0.888268
$638$	0	0
$639$	−299951.	−0.0290602
$640$	0	0
$641$	1.14852e7	1.10406	0.552030	−	0.833824i	$-0.313853\pi$
$641$	1.14852e7	1.10406	0.552030	+	0.833824i	$0.313853\pi$
$642$	0	0
$643$	5.83065e6	0.556147	0.278073	−	0.960560i	$-0.410304\pi$
$643$	5.83065e6	0.556147	0.278073	+	0.960560i	$0.410304\pi$
$644$	0	0
$645$	−1.39569e6	−0.132096
$646$	0	0
$647$	1.27091e7	1.19359	0.596795	−	0.802394i	$-0.296441\pi$
$647$	1.27091e7	1.19359	0.596795	+	0.802394i	$0.296441\pi$
$648$	0	0
$649$	−2.23468e6	−0.208259
$650$	0	0
$651$	1.14128e7	1.05545
$652$	0	0
$653$	−8.01118e6	−0.735214	−0.367607	−	0.929981i	$-0.619823\pi$
$653$	−8.01118e6	−0.735214	−0.367607	+	0.929981i	$0.619823\pi$
$654$	0	0
$655$	−3.18696e7	−2.90250
$656$	0	0
$657$	−1.60445e6	−0.145015
$658$	0	0
$659$	−1.48896e7	−1.33558	−0.667790	−	0.744350i	$-0.732759\pi$
$659$	−1.48896e7	−1.33558	−0.667790	+	0.744350i	$0.732759\pi$
$660$	0	0
$661$	−1.15205e7	−1.02558	−0.512788	−	0.858515i	$-0.671387\pi$
$661$	−1.15205e7	−1.02558	−0.512788	+	0.858515i	$0.671387\pi$
$662$	0	0
$663$	2.41496e6	0.213366
$664$	0	0
$665$	1.31724e7	1.15508
$666$	0	0
$667$	2.30432e6	0.200553
$668$	0	0
$669$	−2.42908e6	−0.209835
$670$	0	0
$671$	−2.05365e6	−0.176084
$672$	0	0
$673$	−1.68386e7	−1.43308	−0.716538	−	0.697548i	$-0.754274\pi$
$673$	−1.68386e7	−1.43308	−0.716538	+	0.697548i	$0.754274\pi$
$674$	0	0
$675$	4.37342e6	0.369455
$676$	0	0
$677$	−6.56811e6	−0.550768	−0.275384	−	0.961334i	$-0.588805\pi$
$677$	−6.56811e6	−0.550768	−0.275384	+	0.961334i	$0.588805\pi$
$678$	0	0
$679$	3.04327e7	2.53318
$680$	0	0
$681$	9.19317e6	0.759622
$682$	0	0
$683$	1.01525e7	0.832761	0.416381	−	0.909190i	$-0.363298\pi$
$683$	1.01525e7	0.832761	0.416381	+	0.909190i	$0.363298\pi$
$684$	0	0
$685$	−3.60790e7	−2.93784
$686$	0	0
$687$	1.52348e6	0.123153
$688$	0	0
$689$	2.54704e6	0.204403
$690$	0	0
$691$	1.01356e7	0.807521	0.403760	−	0.914865i	$-0.367703\pi$
$691$	1.01356e7	0.807521	0.403760	+	0.914865i	$0.367703\pi$
$692$	0	0
$693$	2.05351e6	0.162429
$694$	0	0
$695$	3.48945e7	2.74028
$696$	0	0
$697$	−1.47540e6	−0.115034
$698$	0	0
$699$	−184010.	−0.0142445
$700$	0	0
$701$	9.16091e6	0.704115	0.352058	−	0.935978i	$-0.385482\pi$
$701$	9.16091e6	0.704115	0.352058	+	0.935978i	$0.385482\pi$
$702$	0	0
$703$	−9.24955e6	−0.705882
$704$	0	0
$705$	−1.78197e7	−1.35029
$706$	0	0
$707$	−1.61782e7	−1.21725
$708$	0	0
$709$	−8.97668e6	−0.670656	−0.335328	−	0.942101i	$-0.608847\pi$
$709$	−8.97668e6	−0.670656	−0.335328	+	0.942101i	$0.608847\pi$
$710$	0	0
$711$	−5.18480e6	−0.384643
$712$	0	0
$713$	2.49334e7	1.83679
$714$	0	0
$715$	−3.88093e6	−0.283903
$716$	0	0
$717$	8.40196e6	0.610356
$718$	0	0
$719$	−7.92637e6	−0.571811	−0.285905	−	0.958258i	$-0.592294\pi$
$719$	−7.92637e6	−0.571811	−0.285905	+	0.958258i	$0.592294\pi$
$720$	0	0
$721$	2.22516e7	1.59413
$722$	0	0
$723$	−1.17233e7	−0.834076
$724$	0	0
$725$	3.35564e6	0.237099
$726$	0	0
$727$	−1.78172e7	−1.25027	−0.625135	−	0.780516i	$-0.714956\pi$
$727$	−1.78172e7	−1.25027	−0.625135	+	0.780516i	$0.714956\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	1.29737e6	0.0897989
$732$	0	0
$733$	1.81797e7	1.24976	0.624882	−	0.780719i	$-0.285147\pi$
$733$	1.81797e7	1.24976	0.624882	+	0.780719i	$0.285147\pi$
$734$	0	0
$735$	2.32905e7	1.59023
$736$	0	0
$737$	679852.	0.0461048
$738$	0	0
$739$	1.38291e7	0.931502	0.465751	−	0.884916i	$-0.345784\pi$
$739$	1.38291e7	0.931502	0.465751	+	0.884916i	$0.345784\pi$
$740$	0	0
$741$	−1.98901e6	−0.133073
$742$	0	0
$743$	7.07226e6	0.469987	0.234994	−	0.971997i	$-0.424493\pi$
$743$	7.07226e6	0.469987	0.234994	+	0.971997i	$0.424493\pi$
$744$	0	0
$745$	2.84239e7	1.87626
$746$	0	0
$747$	3.75760e6	0.246382
$748$	0	0
$749$	−7.07149e6	−0.460582
$750$	0	0
$751$	653678.	0.0422926	0.0211463	−	0.999776i	$-0.493268\pi$
$751$	653678.	0.0422926	0.0211463	+	0.999776i	$0.493268\pi$
$752$	0	0
$753$	8.37785e6	0.538449
$754$	0	0
$755$	−3.03845e7	−1.93992
$756$	0	0
$757$	2.72250e7	1.72674	0.863372	−	0.504569i	$-0.168348\pi$
$757$	2.72250e7	1.72674	0.863372	+	0.504569i	$0.168348\pi$
$758$	0	0
$759$	4.48631e6	0.282673
$760$	0	0
$761$	−5.41780e6	−0.339126	−0.169563	−	0.985519i	$-0.554236\pi$
$761$	−5.41780e6	−0.339126	−0.169563	+	0.985519i	$0.554236\pi$
$762$	0	0
$763$	−7.95700e6	−0.494809
$764$	0	0
$765$	−6.18296e6	−0.381982
$766$	0	0
$767$	6.20131e6	0.380623
$768$	0	0
$769$	2.96407e7	1.80748	0.903739	−	0.428084i	$-0.140811\pi$
$769$	2.96407e7	1.80748	0.903739	+	0.428084i	$0.140811\pi$
$770$	0	0
$771$	−227822.	−0.0138026
$772$	0	0
$773$	1.87755e7	1.13017	0.565083	−	0.825034i	$-0.308844\pi$
$773$	1.87755e7	1.13017	0.565083	+	0.825034i	$0.308844\pi$
$774$	0	0
$775$	3.63090e7	2.17150
$776$	0	0
$777$	−2.65001e7	−1.57469
$778$	0	0
$779$	1.21517e6	0.0717453
$780$	0	0
$781$	−448076.	−0.0262860
$782$	0	0
$783$	407765.	0.0237687
$784$	0	0
$785$	−3.29447e6	−0.190814
$786$	0	0
$787$	1.54112e7	0.886951	0.443475	−	0.896286i	$-0.353745\pi$
$787$	1.54112e7	0.886951	0.443475	+	0.896286i	$0.353745\pi$
$788$	0	0
$789$	1.52390e7	0.871493
$790$	0	0
$791$	5.01906e7	2.85221
$792$	0	0
$793$	5.69894e6	0.321819
$794$	0	0
$795$	−6.52111e6	−0.365935
$796$	0	0
$797$	2.10597e7	1.17437	0.587186	−	0.809452i	$-0.300236\pi$
$797$	2.10597e7	1.17437	0.587186	+	0.809452i	$0.300236\pi$
$798$	0	0
$799$	1.65643e7	0.917925
$800$	0	0
$801$	−4.37072e6	−0.240698
$802$	0	0
$803$	−2.39677e6	−0.131171
$804$	0	0
$805$	8.24490e7	4.48431
$806$	0	0
$807$	4.22428e6	0.228333
$808$	0	0
$809$	1.95659e7	1.05106	0.525532	−	0.850774i	$-0.323866\pi$
$809$	1.95659e7	1.05106	0.525532	+	0.850774i	$0.323866\pi$
$810$	0	0
$811$	−1.29945e7	−0.693758	−0.346879	−	0.937910i	$-0.612759\pi$
$811$	−1.29945e7	−0.693758	−0.346879	+	0.937910i	$0.612759\pi$
$812$	0	0
$813$	1.96507e7	1.04268
$814$	0	0
$815$	4.38113e6	0.231042
$816$	0	0
$817$	−1.06854e6	−0.0560063
$818$	0	0
$819$	−5.69856e6	−0.296862
$820$	0	0
$821$	−3.50872e7	−1.81673	−0.908367	−	0.418175i	$-0.862670\pi$
$821$	−3.50872e7	−1.81673	−0.908367	+	0.418175i	$0.862670\pi$
$822$	0	0
$823$	−1.38494e7	−0.712742	−0.356371	−	0.934345i	$-0.615986\pi$
$823$	−1.38494e7	−0.712742	−0.356371	+	0.934345i	$0.615986\pi$
$824$	0	0
$825$	6.53313e6	0.334184
$826$	0	0
$827$	−2.31031e7	−1.17464	−0.587322	−	0.809353i	$-0.699818\pi$
$827$	−2.31031e7	−1.17464	−0.587322	+	0.809353i	$0.699818\pi$
$828$	0	0
$829$	−1.28103e6	−0.0647401	−0.0323701	−	0.999476i	$-0.510306\pi$
$829$	−1.28103e6	−0.0647401	−0.0323701	+	0.999476i	$0.510306\pi$
$830$	0	0
$831$	−758523.	−0.0381036
$832$	0	0
$833$	−2.16498e7	−1.08104
$834$	0	0
$835$	−2.77910e7	−1.37939
$836$	0	0
$837$	4.41213e6	0.217688
$838$	0	0
$839$	−1.98332e7	−0.972721	−0.486361	−	0.873758i	$-0.661676\pi$
$839$	−1.98332e7	−0.972721	−0.486361	+	0.873758i	$0.661676\pi$
$840$	0	0
$841$	−2.01983e7	−0.984746
$842$	0	0
$843$	5.84564e6	0.283311
$844$	0	0
$845$	−2.46965e7	−1.18985
$846$	0	0
$847$	3.06759e6	0.146923
$848$	0	0
$849$	−1.39404e7	−0.663751
$850$	0	0
$851$	−5.78948e7	−2.74041
$852$	0	0
$853$	−2.65097e7	−1.24747	−0.623737	−	0.781634i	$-0.714386\pi$
$853$	−2.65097e7	−1.24747	−0.623737	+	0.781634i	$0.714386\pi$
$854$	0	0
$855$	5.09242e6	0.238237
$856$	0	0
$857$	−1.96831e7	−0.915465	−0.457733	−	0.889090i	$-0.651338\pi$
$857$	−1.96831e7	−0.915465	−0.457733	+	0.889090i	$0.651338\pi$
$858$	0	0
$859$	1.74273e7	0.805835	0.402917	−	0.915236i	$-0.367996\pi$
$859$	1.74273e7	0.805835	0.402917	+	0.915236i	$0.367996\pi$
$860$	0	0
$861$	3.48149e6	0.160051
$862$	0	0
$863$	6.02685e6	0.275463	0.137732	−	0.990470i	$-0.456019\pi$
$863$	6.02685e6	0.275463	0.137732	+	0.990470i	$0.456019\pi$
$864$	0	0
$865$	−1.33118e7	−0.604916
$866$	0	0
$867$	−7.03132e6	−0.317679
$868$	0	0
$869$	−7.74520e6	−0.347923
$870$	0	0
$871$	−1.88661e6	−0.0842630
$872$	0	0
$873$	1.17652e7	0.522472
$874$	0	0
$875$	5.75229e7	2.53993
$876$	0	0
$877$	−5.71518e6	−0.250917	−0.125459	−	0.992099i	$-0.540040\pi$
$877$	−5.71518e6	−0.250917	−0.125459	+	0.992099i	$0.540040\pi$
$878$	0	0
$879$	−632490.	−0.0276109
$880$	0	0
$881$	1.76619e7	0.766649	0.383324	−	0.923614i	$-0.374779\pi$
$881$	1.76619e7	0.766649	0.383324	+	0.923614i	$0.374779\pi$
$882$	0	0
$883$	−2.70117e7	−1.16587	−0.582936	−	0.812518i	$-0.698096\pi$
$883$	−2.70117e7	−1.16587	−0.582936	+	0.812518i	$0.698096\pi$
$884$	0	0
$885$	−1.58771e7	−0.681416
$886$	0	0
$887$	−1.30141e7	−0.555398	−0.277699	−	0.960668i	$-0.589572\pi$
$887$	−1.30141e7	−0.555398	−0.277699	+	0.960668i	$0.589572\pi$
$888$	0	0
$889$	−1.86040e7	−0.789500
$890$	0	0
$891$	793881.	0.0335013
$892$	0	0
$893$	−1.36427e7	−0.572496
$894$	0	0
$895$	5.72700e7	2.38984
$896$	0	0
$897$	−1.24496e7	−0.516625
$898$	0	0
$899$	3.38535e6	0.139703
$900$	0	0
$901$	6.06172e6	0.248762
$902$	0	0
$903$	−3.06140e6	−0.124940
$904$	0	0
$905$	1.24779e7	0.506431
$906$	0	0
$907$	−9.84123e6	−0.397220	−0.198610	−	0.980079i	$-0.563643\pi$
$907$	−9.84123e6	−0.397220	−0.198610	+	0.980079i	$0.563643\pi$
$908$	0	0
$909$	−6.25442e6	−0.251060
$910$	0	0
$911$	−7.41735e6	−0.296110	−0.148055	−	0.988979i	$-0.547301\pi$
$911$	−7.41735e6	−0.296110	−0.148055	+	0.988979i	$0.547301\pi$
$912$	0	0
$913$	5.61321e6	0.222861
$914$	0	0
$915$	−1.45909e7	−0.576140
$916$	0	0
$917$	−6.99046e7	−2.74525
$918$	0	0
$919$	−1.15896e7	−0.452667	−0.226333	−	0.974050i	$-0.572674\pi$
$919$	−1.15896e7	−0.452667	−0.226333	+	0.974050i	$0.572674\pi$
$920$	0	0
$921$	−1.29009e7	−0.501153
$922$	0	0
$923$	1.24342e6	0.0480413
$924$	0	0
$925$	−8.43087e7	−3.23980
$926$	0	0
$927$	8.60240e6	0.328791
$928$	0	0
$929$	1.17151e7	0.445357	0.222679	−	0.974892i	$-0.428520\pi$
$929$	1.17151e7	0.445357	0.222679	+	0.974892i	$0.428520\pi$
$930$	0	0
$931$	1.78312e7	0.674228
$932$	0	0
$933$	2.05091e6	0.0771336
$934$	0	0
$935$	−9.23628e6	−0.345516
$936$	0	0
$937$	9.18288e6	0.341688	0.170844	−	0.985298i	$-0.445351\pi$
$937$	9.18288e6	0.341688	0.170844	+	0.985298i	$0.445351\pi$
$938$	0	0
$939$	−6.51384e6	−0.241087
$940$	0	0
$941$	3.24899e6	0.119612	0.0598059	−	0.998210i	$-0.480952\pi$
$941$	3.24899e6	0.119612	0.0598059	+	0.998210i	$0.480952\pi$
$942$	0	0
$943$	7.60601e6	0.278534
$944$	0	0
$945$	1.45899e7	0.531462
$946$	0	0
$947$	−3.50906e7	−1.27150	−0.635750	−	0.771895i	$-0.719309\pi$
$947$	−3.50906e7	−1.27150	−0.635750	+	0.771895i	$0.719309\pi$
$948$	0	0
$949$	6.65109e6	0.239733
$950$	0	0
$951$	1.13477e7	0.406872
$952$	0	0
$953$	−3.27415e7	−1.16780	−0.583898	−	0.811827i	$-0.698473\pi$
$953$	−3.27415e7	−1.16780	−0.583898	+	0.811827i	$0.698473\pi$
$954$	0	0
$955$	−3.23092e7	−1.14635
$956$	0	0
$957$	609130.	0.0214996
$958$	0	0
$959$	−7.91379e7	−2.77868
$960$	0	0
$961$	8.00132e6	0.279482
$962$	0	0
$963$	−2.73382e6	−0.0949956
$964$	0	0
$965$	−3.14353e7	−1.08667
$966$	0	0
$967$	3.94890e7	1.35803	0.679016	−	0.734124i	$-0.262407\pi$
$967$	3.94890e7	1.35803	0.679016	+	0.734124i	$0.262407\pi$
$968$	0	0
$969$	−4.73367e6	−0.161953
$970$	0	0
$971$	−1.43971e7	−0.490033	−0.245017	−	0.969519i	$-0.578793\pi$
$971$	−1.43971e7	−0.490033	−0.245017	+	0.969519i	$0.578793\pi$
$972$	0	0
$973$	7.65396e7	2.59182
$974$	0	0
$975$	−1.81296e7	−0.610769
$976$	0	0
$977$	−4.55108e7	−1.52538	−0.762690	−	0.646764i	$-0.776122\pi$
$977$	−4.55108e7	−1.52538	−0.762690	+	0.646764i	$0.776122\pi$
$978$	0	0
$979$	−6.52910e6	−0.217719
$980$	0	0
$981$	−3.07615e6	−0.102055
$982$	0	0
$983$	3.34028e7	1.10255	0.551276	−	0.834323i	$-0.314141\pi$
$983$	3.34028e7	1.10255	0.551276	+	0.834323i	$0.314141\pi$
$984$	0	0
$985$	4.94222e7	1.62305
$986$	0	0
$987$	−3.90867e7	−1.27713
$988$	0	0
$989$	−6.68823e6	−0.217431
$990$	0	0
$991$	−4.23892e7	−1.37111	−0.685553	−	0.728023i	$-0.740440\pi$
$991$	−4.23892e7	−1.37111	−0.685553	+	0.728023i	$0.740440\pi$
$992$	0	0
$993$	−3.90100e6	−0.125546
$994$	0	0
$995$	5.07632e7	1.62551
$996$	0	0
$997$	3.39103e7	1.08042	0.540212	−	0.841529i	$-0.318344\pi$
$997$	3.39103e7	1.08042	0.540212	+	0.841529i	$0.318344\pi$
$998$	0	0
$999$	−1.02449e7	−0.324783

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	528.6.a.s.1.2		2
4.3	odd	2		33.6.a.c.1.1	✓	2
12.11	even	2		99.6.a.f.1.2		2
20.19	odd	2		825.6.a.e.1.2		2
44.43	even	2		363.6.a.j.1.2		2
132.131	odd	2		1089.6.a.j.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.6.a.c.1.1	✓	2	4.3	odd	2
99.6.a.f.1.2		2	12.11	even	2
363.6.a.j.1.2		2	44.43	even	2
528.6.a.s.1.2		2	1.1	even	1	trivial
825.6.a.e.1.2		2	20.19	odd	2
1089.6.a.j.1.1		2	132.131	odd	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)

\(f(q)\)	\(=\)	\(q+9.00000 q^{3} +95.5207 q^{5} +209.521 q^{7} +81.0000 q^{9} +O(q^{10})\)	\(q+9.00000 q^{3} +95.5207 q^{5} +209.521 q^{7} +81.0000 q^{9} +121.000 q^{11} -335.779 q^{13} +859.686 q^{15} -799.124 q^{17} +658.175 q^{19} +1885.69 q^{21} +4119.66 q^{23} +5999.20 q^{25} +729.000 q^{27} +559.348 q^{29} +6052.31 q^{31} +1089.00 q^{33} +20013.6 q^{35} -14053.3 q^{37} -3022.01 q^{39} +1846.27 q^{41} -1623.49 q^{43} +7737.17 q^{45} -20728.1 q^{47} +27091.9 q^{49} -7192.12 q^{51} -7585.46 q^{53} +11558.0 q^{55} +5923.58 q^{57} -18468.5 q^{59} -16972.3 q^{61} +16971.2 q^{63} -32073.8 q^{65} +5618.62 q^{67} +37076.9 q^{69} -3703.10 q^{71} -19808.0 q^{73} +53992.8 q^{75} +25352.0 q^{77} -64009.9 q^{79} +6561.00 q^{81} +46390.2 q^{83} -76332.9 q^{85} +5034.13 q^{87} -53959.5 q^{89} -70352.5 q^{91} +54470.8 q^{93} +62869.3 q^{95} +145249. q^{97} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 18 q^{3} + 58 q^{5} + 286 q^{7} + 162 q^{9}+O(q^{10}) \) 2 * q + 18 * q^3 + 58 * q^5 + 286 * q^7 + 162 * q^9	\( 2 q + 18 q^{3} + 58 q^{5} + 286 q^{7} + 162 q^{9} + 242 q^{11} - 166 q^{13} + 522 q^{15} - 800 q^{17} + 1476 q^{19} + 2574 q^{21} + 3370 q^{23} + 4282 q^{25} + 1458 q^{27} + 6600 q^{29} + 7528 q^{31} + 2178 q^{33} + 17144 q^{35} - 29916 q^{37} - 1494 q^{39} - 5780 q^{41} + 16656 q^{43} + 4698 q^{45} - 7850 q^{47} + 16134 q^{49} - 7200 q^{51} + 14178 q^{53} + 7018 q^{55} + 13284 q^{57} - 17300 q^{59} - 2946 q^{61} + 23166 q^{63} - 38444 q^{65} - 31336 q^{67} + 30330 q^{69} + 33810 q^{71} + 60644 q^{73} + 38538 q^{75} + 34606 q^{77} - 1870 q^{79} + 13122 q^{81} + 58296 q^{83} - 76300 q^{85} + 59400 q^{87} + 92388 q^{89} - 57368 q^{91} + 67752 q^{93} + 32184 q^{95} + 7120 q^{97} + 19602 q^{99}+O(q^{100}) \) 2 * q + 18 * q^3 + 58 * q^5 + 286 * q^7 + 162 * q^9 + 242 * q^11 - 166 * q^13 + 522 * q^15 - 800 * q^17 + 1476 * q^19 + 2574 * q^21 + 3370 * q^23 + 4282 * q^25 + 1458 * q^27 + 6600 * q^29 + 7528 * q^31 + 2178 * q^33 + 17144 * q^35 - 29916 * q^37 - 1494 * q^39 - 5780 * q^41 + 16656 * q^43 + 4698 * q^45 - 7850 * q^47 + 16134 * q^49 - 7200 * q^51 + 14178 * q^53 + 7018 * q^55 + 13284 * q^57 - 17300 * q^59 - 2946 * q^61 + 23166 * q^63 - 38444 * q^65 - 31336 * q^67 + 30330 * q^69 + 33810 * q^71 + 60644 * q^73 + 38538 * q^75 + 34606 * q^77 - 1870 * q^79 + 13122 * q^81 + 58296 * q^83 - 76300 * q^85 + 59400 * q^87 + 92388 * q^89 - 57368 * q^91 + 67752 * q^93 + 32184 * q^95 + 7120 * q^97 + 19602 * q^99

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