LMFDB - Embedded newform 99.6.a.f.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [99,6,Mod(1,99)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(99, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("99.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 99 = 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	99.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:	$15.8779981615$
Analytic rank:	$1$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
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	$\chi$	$=$	99.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.15207 q^{2} +51.7603 q^{4} -95.5207 q^{5} -209.521 q^{7} +180.848 q^{8} +O(q^{10})$	\(q+9.15207 q^{2} +51.7603 q^{4} -95.5207 q^{5} -209.521 q^{7} +180.848 q^{8} -874.212 q^{10} +121.000 q^{11} -335.779 q^{13} -1917.55 q^{14} -1.19832 q^{16} +799.124 q^{17} -658.175 q^{19} -4944.18 q^{20} +1107.40 q^{22} +4119.66 q^{23} +5999.20 q^{25} -3073.07 q^{26} -10844.9 q^{28} -559.348 q^{29} -6052.31 q^{31} -5798.10 q^{32} +7313.64 q^{34} +20013.6 q^{35} -14053.3 q^{37} -6023.66 q^{38} -17274.7 q^{40} -1846.27 q^{41} +1623.49 q^{43} +6263.00 q^{44} +37703.4 q^{46} -20728.1 q^{47} +27091.9 q^{49} +54905.1 q^{50} -17380.0 q^{52} +7585.46 q^{53} -11558.0 q^{55} -37891.4 q^{56} -5119.19 q^{58} -18468.5 q^{59} -16972.3 q^{61} -55391.2 q^{62} -53026.3 q^{64} +32073.8 q^{65} -5618.62 q^{67} +41362.9 q^{68} +183165. q^{70} -3703.10 q^{71} -19808.0 q^{73} -128617. q^{74} -34067.4 q^{76} -25352.0 q^{77} +64009.9 q^{79} +114.464 q^{80} -16897.2 q^{82} +46390.2 q^{83} -76332.9 q^{85} +14858.3 q^{86} +21882.6 q^{88} +53959.5 q^{89} +70352.5 q^{91} +213235. q^{92} -189705. q^{94} +62869.3 q^{95} +145249. q^{97} +247947. q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 5 q^{2} + 37 q^{4} - 58 q^{5} - 286 q^{7} + 375 q^{8}+O(q^{10}) $ 2 * q + 5 * q^2 + 37 * q^4 - 58 * q^5 - 286 * q^7 + 375 * q^8	$ 2 q + 5 q^{2} + 37 q^{4} - 58 q^{5} - 286 q^{7} + 375 q^{8} - 1030 q^{10} + 242 q^{11} - 166 q^{13} - 1600 q^{14} - 335 q^{16} + 800 q^{17} - 1476 q^{19} - 5498 q^{20} + 605 q^{22} + 3370 q^{23} + 4282 q^{25} - 3778 q^{26} - 9716 q^{28} - 6600 q^{29} - 7528 q^{31} - 10625 q^{32} + 7310 q^{34} + 17144 q^{35} - 29916 q^{37} - 2628 q^{38} - 9990 q^{40} + 5780 q^{41} - 16656 q^{43} + 4477 q^{44} + 40816 q^{46} - 7850 q^{47} + 16134 q^{49} + 62035 q^{50} - 19886 q^{52} - 14178 q^{53} - 7018 q^{55} - 52740 q^{56} + 19962 q^{58} - 17300 q^{59} - 2946 q^{61} - 49264 q^{62} - 22303 q^{64} + 38444 q^{65} + 31336 q^{67} + 41350 q^{68} + 195080 q^{70} + 33810 q^{71} + 60644 q^{73} - 62754 q^{74} - 21996 q^{76} - 34606 q^{77} + 1870 q^{79} - 12410 q^{80} - 48562 q^{82} + 58296 q^{83} - 76300 q^{85} + 90756 q^{86} + 45375 q^{88} - 92388 q^{89} + 57368 q^{91} + 224300 q^{92} - 243176 q^{94} + 32184 q^{95} + 7120 q^{97} + 293445 q^{98}+O(q^{100}) $ 2 * q + 5 * q^2 + 37 * q^4 - 58 * q^5 - 286 * q^7 + 375 * q^8 - 1030 * q^10 + 242 * q^11 - 166 * q^13 - 1600 * q^14 - 335 * q^16 + 800 * q^17 - 1476 * q^19 - 5498 * q^20 + 605 * q^22 + 3370 * q^23 + 4282 * q^25 - 3778 * q^26 - 9716 * q^28 - 6600 * q^29 - 7528 * q^31 - 10625 * q^32 + 7310 * q^34 + 17144 * q^35 - 29916 * q^37 - 2628 * q^38 - 9990 * q^40 + 5780 * q^41 - 16656 * q^43 + 4477 * q^44 + 40816 * q^46 - 7850 * q^47 + 16134 * q^49 + 62035 * q^50 - 19886 * q^52 - 14178 * q^53 - 7018 * q^55 - 52740 * q^56 + 19962 * q^58 - 17300 * q^59 - 2946 * q^61 - 49264 * q^62 - 22303 * q^64 + 38444 * q^65 + 31336 * q^67 + 41350 * q^68 + 195080 * q^70 + 33810 * q^71 + 60644 * q^73 - 62754 * q^74 - 21996 * q^76 - 34606 * q^77 + 1870 * q^79 - 12410 * q^80 - 48562 * q^82 + 58296 * q^83 - 76300 * q^85 + 90756 * q^86 + 45375 * q^88 - 92388 * q^89 + 57368 * q^91 + 224300 * q^92 - 243176 * q^94 + 32184 * q^95 + 7120 * q^97 + 293445 * q^98

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 $

$2$	9.15207	1.61787	0.808936	−	0.587897i	$-0.200044\pi$
$2$	9.15207	1.61787	0.808936	+	0.587897i	$0.200044\pi$
$3$	0	0
$3$	0	0
$4$	51.7603	1.61751
$5$	−95.5207	−1.70873	−0.854363	−	0.519677i	$-0.826052\pi$
$5$	−95.5207	−1.70873	−0.854363	+	0.519677i	$0.826052\pi$
$6$	0	0
$7$	−209.521	−1.61615	−0.808075	−	0.589079i	$-0.799491\pi$
$7$	−209.521	−1.61615	−0.808075	+	0.589079i	$0.799491\pi$
$8$	180.848	0.999053
$9$	0	0
$10$	−874.212	−2.76450
$11$	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	$0.588852\pi$
$14$	−1917.55	−2.61472
$15$	0	0
$16$	−1.19832	−0.00117023
$17$	799.124	0.670644	0.335322	−	0.942104i	$-0.391155\pi$
$17$	799.124	0.670644	0.335322	+	0.942104i	$0.391155\pi$
$18$	0	0
$19$	−658.175	−0.418271	−0.209135	−	0.977887i	$-0.567065\pi$
$19$	−658.175	−0.418271	−0.209135	+	0.977887i	$0.567065\pi$
$20$	−4944.18	−2.76388
$21$	0	0
$22$	1107.40	0.487807
$23$	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
$26$	−3073.07	−0.891536
$27$	0	0
$28$	−10844.9	−2.61414
$29$	−559.348	−0.123506	−0.0617529	−	0.998091i	$-0.519669\pi$
$29$	−559.348	−0.123506	−0.0617529	+	0.998091i	$0.519669\pi$
$30$	0	0
$31$	−6052.31	−1.13114	−0.565571	−	0.824700i	$-0.691344\pi$
$31$	−6052.31	−1.13114	−0.565571	+	0.824700i	$0.691344\pi$
$32$	−5798.10	−1.00095
$33$	0	0
$34$	7313.64	1.08502
$35$	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	$0.819692\pi$
$38$	−6023.66	−0.676709
$39$	0	0
$40$	−17274.7	−1.70711
$41$	−1846.27	−0.171528	−0.0857642	−	0.996315i	$-0.527333\pi$
$41$	−1846.27	−0.171528	−0.0857642	+	0.996315i	$0.527333\pi$
$42$	0	0
$43$	1623.49	0.133900	0.0669498	−	0.997756i	$-0.478673\pi$
$43$	1623.49	0.133900	0.0669498	+	0.997756i	$0.478673\pi$
$44$	6263.00	0.487698
$45$	0	0
$46$	37703.4	2.62715
$47$	−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
$50$	54905.1	3.10590
$51$	0	0
$52$	−17380.0	−0.891337
$53$	7585.46	0.370930	0.185465	−	0.982651i	$-0.440621\pi$
$53$	7585.46	0.370930	0.185465	+	0.982651i	$0.440621\pi$
$54$	0	0
$55$	−11558.0	−0.515200
$56$	−37891.4	−1.61462
$57$	0	0
$58$	−5119.19	−0.199817
$59$	−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	$0.594322\pi$
$62$	−55391.2	−1.83004
$63$	0	0
$64$	−53026.3	−1.61823
$65$	32073.8	0.941601
$66$	0	0
$67$	−5618.62	−0.152912	−0.0764561	−	0.997073i	$-0.524361\pi$
$67$	−5618.62	−0.152912	−0.0764561	+	0.997073i	$0.524361\pi$
$68$	41362.9	1.08477
$69$	0	0
$70$	183165.	4.46785
$71$	−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	$0.569797\pi$
$74$	−128617.	−2.73035
$75$	0	0
$76$	−34067.4	−0.676557
$77$	−25352.0	−0.487288
$78$	0	0
$79$	64009.9	1.15393	0.576965	−	0.816769i	$-0.304237\pi$
$79$	64009.9	1.15393	0.576965	+	0.816769i	$0.304237\pi$
$80$	114.464	0.00199960
$81$	0	0
$82$	−16897.2	−0.277511
$83$	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
$86$	14858.3	0.216632
$87$	0	0
$88$	21882.6	0.301226
$89$	53959.5	0.722093	0.361046	−	0.932548i	$-0.382420\pi$
$89$	53959.5	0.722093	0.361046	+	0.932548i	$0.382420\pi$
$90$	0	0
$91$	70352.5	0.890587
$92$	213235.	2.62657
$93$	0	0
$94$	−189705.	−2.21442
$95$	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	$0.213326\pi$
$98$	247947.	2.60792
$99$	0	0
$100$	310521.	3.10521
$101$	77215.1	0.753180	0.376590	−	0.926380i	$-0.377097\pi$
$101$	77215.1	0.753180	0.376590	+	0.926380i	$0.377097\pi$
$102$	0	0
$103$	−106203.	−0.986374	−0.493187	−	0.869923i	$-0.664168\pi$
$103$	−106203.	−0.986374	−0.493187	+	0.869923i	$0.664168\pi$
$104$	−60724.9	−0.550533
$105$	0	0
$106$	69422.6	0.600118
$107$	−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	$0.548920\pi$
$110$	−105780.	−0.833528
$111$	0	0
$112$	251.072	0.00189127
$113$	−239549.	−1.76481	−0.882407	−	0.470486i	$-0.844078\pi$
$113$	−239549.	−1.76481	−0.882407	+	0.470486i	$0.844078\pi$
$114$	0	0
$115$	−393512.	−2.77469
$116$	−28952.1	−0.199772
$117$	0	0
$118$	−169025.	−1.11749
$119$	−167433.	−1.08386
$120$	0	0
$121$	14641.0	0.0909091
$122$	−155332.	−0.944845
$123$	0	0
$124$	−313270.	−1.82963
$125$	−274545.	−1.57159
$126$	0	0
$127$	88793.2	0.488507	0.244253	−	0.969711i	$-0.421457\pi$
$127$	88793.2	0.488507	0.244253	+	0.969711i	$0.421457\pi$
$128$	−299761.	−1.61715
$129$	0	0
$130$	293542.	1.52339
$131$	−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
$134$	−51421.9	−0.247392
$135$	0	0
$136$	144520.	0.670009
$137$	377709.	1.71932	0.859659	−	0.510869i	$-0.170676\pi$
$137$	377709.	1.71932	0.859659	+	0.510869i	$0.170676\pi$
$138$	0	0
$139$	−365308.	−1.60370	−0.801849	−	0.597527i	$-0.796150\pi$
$139$	−365308.	−1.60370	−0.801849	+	0.597527i	$0.796150\pi$
$140$	1.03591e6	4.46685
$141$	0	0
$142$	−33891.1	−0.141047
$143$	−40629.2	−0.166149
$144$	0	0
$145$	53429.3	0.211038
$146$	−181284.	−0.703845
$147$	0	0
$148$	−727405.	−2.72974
$149$	−297568.	−1.09805	−0.549023	−	0.835807i	$-0.685000\pi$
$149$	−297568.	−1.09805	−0.549023	+	0.835807i	$0.685000\pi$
$150$	0	0
$151$	318093.	1.13530	0.567651	−	0.823269i	$-0.307852\pi$
$151$	318093.	1.13530	0.567651	+	0.823269i	$0.307852\pi$
$152$	−119030.	−0.417875
$153$	0	0
$154$	−232023.	−0.788369
$155$	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	$0.517782\pi$
$158$	585823.	1.86691
$159$	0	0
$160$	553839.	1.71034
$161$	−863153.	−2.62436
$162$	0	0
$163$	−45865.7	−0.135213	−0.0676067	−	0.997712i	$-0.521536\pi$
$163$	−45865.7	−0.135213	−0.0676067	+	0.997712i	$0.521536\pi$
$164$	−95563.7	−0.277449
$165$	0	0
$166$	424566.	1.19584
$167$	−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
$170$	−698604.	−1.85399
$171$	0	0
$172$	84032.5	0.216584
$173$	139360.	0.354016	0.177008	−	0.984209i	$-0.443358\pi$
$173$	139360.	0.354016	0.177008	+	0.984209i	$0.443358\pi$
$174$	0	0
$175$	−1.25696e6	−3.10259
$176$	−144.996	−0.000352838 0
$177$	0	0
$178$	493841.	1.16825
$179$	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	$0.452655\pi$
$182$	643871.	1.44086
$183$	0	0
$184$	745031.	1.62230
$185$	1.34238e6	2.88368
$186$	0	0
$187$	96694.0	0.202207
$188$	−1.07289e6	−2.21392
$189$	0	0
$190$	575384.	1.15631
$191$	−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	$0.603004\pi$
$194$	1.32933e6	2.53588
$195$	0	0
$196$	1.40229e6	2.60733
$197$	−517397.	−0.949858	−0.474929	−	0.880024i	$-0.657526\pi$
$197$	−517397.	−0.949858	−0.474929	+	0.880024i	$0.657526\pi$
$198$	0	0
$199$	−531436.	−0.951302	−0.475651	−	0.879634i	$-0.657788\pi$
$199$	−531436.	−0.951302	−0.475651	+	0.879634i	$0.657788\pi$
$200$	1.08494e6	1.91793
$201$	0	0
$202$	706678.	1.21855
$203$	117195.	0.199604
$204$	0	0
$205$	176357.	0.293095
$206$	−971973.	−1.59583
$207$	0	0
$208$	402.369	0.000644861 0
$209$	−79639.2	−0.126113
$210$	0	0
$211$	409913.	0.633848	0.316924	−	0.948451i	$-0.397350\pi$
$211$	409913.	0.633848	0.316924	+	0.948451i	$0.397350\pi$
$212$	392626.	0.599984
$213$	0	0
$214$	−308890.	−0.461072
$215$	−155077.	−0.228798
$216$	0	0
$217$	1.26808e6	1.82810
$218$	−347569.	−0.495336
$219$	0	0
$220$	−598246.	−0.833342
$221$	−268329.	−0.369561
$222$	0	0
$223$	269898.	0.363444	0.181722	−	0.983350i	$-0.441833\pi$
$223$	269898.	0.363444	0.181722	+	0.983350i	$0.441833\pi$
$224$	1.21482e6	1.61768
$225$	0	0
$226$	−2.19237e6	−2.85524
$227$	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	$0.465986\pi$
$230$	−3.60145e6	−4.48909
$231$	0	0
$232$	−101157.	−0.123389
$233$	20445.6	0.0246723	0.0123361	−	0.999924i	$-0.496073\pi$
$233$	20445.6	0.0246723	0.0123361	+	0.999924i	$0.496073\pi$
$234$	0	0
$235$	1.97996e6	2.33877
$236$	−955933.	−1.11724
$237$	0	0
$238$	−1.53236e6	−1.75355
$239$	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	$0.756929\pi$
$242$	133995.	0.147079
$243$	0	0
$244$	−878493.	−0.944634
$245$	−2.58784e6	−2.75437
$246$	0	0
$247$	221001.	0.230490
$248$	−1.09455e6	−1.13007
$249$	0	0
$250$	−2.51266e6	−2.54263
$251$	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
$254$	812642.	0.790342
$255$	0	0
$256$	−1.04659e6	−0.998106
$257$	25313.6	0.0239067	0.0119534	−	0.999929i	$-0.496195\pi$
$257$	25313.6	0.0239067	0.0119534	+	0.999929i	$0.496195\pi$
$258$	0	0
$259$	2.94446e6	2.72745
$260$	1.66015e6	1.52305
$261$	0	0
$262$	−3.05350e6	−2.74818
$263$	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
$266$	1.26208e6	1.09366
$267$	0	0
$268$	−290821.	−0.247337
$269$	−469365.	−0.395485	−0.197742	−	0.980254i	$-0.563361\pi$
$269$	−469365.	−0.395485	−0.197742	+	0.980254i	$0.563361\pi$
$270$	0	0
$271$	−2.18341e6	−1.80598	−0.902988	−	0.429666i	$-0.858631\pi$
$271$	−2.18341e6	−1.80598	−0.902988	+	0.429666i	$0.858631\pi$
$272$	−957.603	−0.000784808 0
$273$	0	0
$274$	3.45682e6	2.78164
$275$	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	$0.510506\pi$
$278$	−3.34333e6	−2.59458
$279$	0	0
$280$	3.61941e6	2.75894
$281$	−649515.	−0.490708	−0.245354	−	0.969433i	$-0.578904\pi$
$281$	−649515.	−0.490708	−0.245354	+	0.969433i	$0.578904\pi$
$282$	0	0
$283$	1.54893e6	1.14965	0.574826	−	0.818276i	$-0.305070\pi$
$283$	1.54893e6	1.14965	0.574826	+	0.818276i	$0.305070\pi$
$284$	−191674.	−0.141016
$285$	0	0
$286$	−371841.	−0.268808
$287$	386832.	0.277216
$288$	0	0
$289$	−781258.	−0.550237
$290$	488989.	0.341432
$291$	0	0
$292$	−1.02527e6	−0.703688
$293$	70276.6	0.0478236	0.0239118	−	0.999714i	$-0.492388\pi$
$293$	70276.6	0.0478236	0.0239118	+	0.999714i	$0.492388\pi$
$294$	0	0
$295$	1.76412e6	1.18025
$296$	−2.54151e6	−1.68602
$297$	0	0
$298$	−2.72336e6	−1.77650
$299$	−1.38329e6	−0.894821
$300$	0	0
$301$	−340155.	−0.216402
$302$	2.91121e6	1.83678
$303$	0	0
$304$	788.702	0.000489473 0
$305$	1.62121e6	0.997904
$306$	0	0
$307$	1.43343e6	0.868022	0.434011	−	0.900908i	$-0.357098\pi$
$307$	1.43343e6	0.868022	0.434011	+	0.900908i	$0.357098\pi$
$308$	−1.31223e6	−0.788193
$309$	0	0
$310$	5.29100e6	3.12704
$311$	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	$0.566952\pi$
$314$	−315651.	−0.180669
$315$	0	0
$316$	3.31317e6	1.86649
$317$	−1.26086e6	−0.704723	−0.352362	−	0.935864i	$-0.614621\pi$
$317$	−1.26086e6	−0.704723	−0.352362	+	0.935864i	$0.614621\pi$
$318$	0	0
$319$	−67681.1	−0.0372384
$320$	5.06510e6	2.76512
$321$	0	0
$322$	−7.89964e6	−4.24588
$323$	−525964.	−0.280511
$324$	0	0
$325$	−2.01440e6	−1.05788
$326$	−419766.	−0.218758
$327$	0	0
$328$	−333894.	−0.171366
$329$	4.34297e6	2.21206
$330$	0	0
$331$	433444.	0.217452	0.108726	−	0.994072i	$-0.465323\pi$
$331$	433444.	0.217452	0.108726	+	0.994072i	$0.465323\pi$
$332$	2.40117e6	1.19558
$333$	0	0
$334$	−2.66273e6	−1.30605
$335$	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	$0.134020\pi$
$338$	−2.36623e6	−1.12659
$339$	0	0
$340$	−3.95101e6	−1.85358
$341$	−732330.	−0.341052
$342$	0	0
$343$	−2.15490e6	−0.988991
$344$	293605.	0.133773
$345$	0	0
$346$	1.27543e6	0.572753
$347$	−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	$0.862863\pi$
$350$	−1.15037e7	−5.01960
$351$	0	0
$352$	−701570.	−0.301797
$353$	−499935.	−0.213539	−0.106769	−	0.994284i	$-0.534051\pi$
$353$	−499935.	−0.213539	−0.106769	+	0.994284i	$0.534051\pi$
$354$	0	0
$355$	353723.	0.148968
$356$	2.79296e6	1.16799
$357$	0	0
$358$	5.48718e6	2.26278
$359$	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
$362$	1.19554e6	0.479504
$363$	0	0
$364$	3.64147e6	1.44053
$365$	1.89207e6	0.743371
$366$	0	0
$367$	257505.	0.0997976	0.0498988	−	0.998754i	$-0.484110\pi$
$367$	257505.	0.0997976	0.0498988	+	0.998754i	$0.484110\pi$
$368$	−4936.65	−0.00190026
$369$	0	0
$370$	1.22856e7	4.66542
$371$	−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	$0.576683\pi$
$374$	884950.	0.327145
$375$	0	0
$376$	−3.74864e6	−1.36743
$377$	187817.	0.0680584
$378$	0	0
$379$	3.13440e6	1.12087	0.560436	−	0.828198i	$-0.310634\pi$
$379$	3.13440e6	1.12087	0.560436	+	0.828198i	$0.310634\pi$
$380$	3.25414e6	1.15605
$381$	0	0
$382$	−3.09563e6	−1.08540
$383$	−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
$386$	−3.01189e6	−1.02889
$387$	0	0
$388$	7.51814e6	2.53531
$389$	1.95280e6	0.654312	0.327156	−	0.944970i	$-0.393910\pi$
$389$	1.95280e6	0.654312	0.327156	+	0.944970i	$0.393910\pi$
$390$	0	0
$391$	3.29212e6	1.08901
$392$	4.89952e6	1.61042
$393$	0	0
$394$	−4.73526e6	−1.53675
$395$	−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	$0.587624\pi$
$398$	−4.86374e6	−1.53909
$399$	0	0
$400$	−7188.94	−0.00224654
$401$	−4.51630e6	−1.40256	−0.701280	−	0.712886i	$-0.747388\pi$
$401$	−4.51630e6	−1.40256	−0.701280	+	0.712886i	$0.747388\pi$
$402$	0	0
$403$	2.03224e6	0.623321
$404$	3.99668e6	1.21828
$405$	0	0
$406$	1.07258e6	0.322934
$407$	−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	$0.533984\pi$
$410$	1.61403e6	0.474190
$411$	0	0
$412$	−5.49708e6	−1.59547
$413$	3.86952e6	1.11630
$414$	0	0
$415$	−4.43122e6	−1.26300
$416$	1.94688e6	0.551576
$417$	0	0
$418$	−728863.	−0.204035
$419$	−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	$0.654665\pi$
$422$	3.75155e6	1.02549
$423$	0	0
$424$	1.37181e6	0.370579
$425$	4.79410e6	1.28746
$426$	0	0
$427$	3.55605e6	0.943840
$428$	−1.74695e6	−0.460969
$429$	0	0
$430$	−1.41928e6	−0.370165
$431$	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	$0.636055\pi$
$434$	1.16056e7	2.95762
$435$	0	0
$436$	−1.96571e6	−0.495226
$437$	−2.71146e6	−0.679202
$438$	0	0
$439$	6.31060e6	1.56282	0.781411	−	0.624017i	$-0.214500\pi$
$439$	6.31060e6	1.56282	0.781411	+	0.624017i	$0.214500\pi$
$440$	−2.09024e6	−0.514712
$441$	0	0
$442$	−2.45576e6	−0.597903
$443$	−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
$446$	2.47013e6	0.588006
$447$	0	0
$448$	1.11101e7	2.61531
$449$	4.54361e6	1.06362	0.531809	−	0.846865i	$-0.321513\pi$
$449$	4.54361e6	1.06362	0.531809	+	0.846865i	$0.321513\pi$
$450$	0	0
$451$	−223399.	−0.0517178
$452$	−1.23992e7	−2.85461
$453$	0	0
$454$	9.34850e6	2.12864
$455$	−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	$0.399043\pi$
$458$	1.54922e6	0.345104
$459$	0	0
$460$	−2.03683e7	−4.48808
$461$	2.00495e6	0.439391	0.219695	−	0.975569i	$-0.429494\pi$
$461$	2.00495e6	0.439391	0.219695	+	0.975569i	$0.429494\pi$
$462$	0	0
$463$	−4.67402e6	−1.01330	−0.506650	−	0.862152i	$-0.669116\pi$
$463$	−4.67402e6	−1.01330	−0.506650	+	0.862152i	$0.669116\pi$
$464$	670.276	0.000144530 0
$465$	0	0
$466$	187119.	0.0399166
$467$	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
$470$	1.81208e7	3.78383
$471$	0	0
$472$	−3.33998e6	−0.690063
$473$	196443.	0.0403722
$474$	0	0
$475$	−3.94852e6	−0.802973
$476$	−8.66639e6	−1.75316
$477$	0	0
$478$	8.54393e6	1.71036
$479$	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
$482$	−1.19214e7	−2.33728
$483$	0	0
$484$	757823.	0.147046
$485$	−1.38743e7	−2.67828
$486$	0	0
$487$	−5.57519e6	−1.06521	−0.532607	−	0.846363i	$-0.678788\pi$
$487$	−5.57519e6	−1.06521	−0.532607	+	0.846363i	$0.678788\pi$
$488$	−3.06941e6	−0.583452
$489$	0	0
$490$	−2.36841e7	−4.45621
$491$	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
$494$	2.02262e6	0.372903
$495$	0	0
$496$	7252.58	0.00132370
$497$	775877.	0.140897
$498$	0	0
$499$	−5.94504e6	−1.06882	−0.534409	−	0.845226i	$-0.679466\pi$
$499$	−5.94504e6	−1.06882	−0.534409	+	0.845226i	$0.679466\pi$
$500$	−1.42106e7	−2.54206
$501$	0	0
$502$	8.51940e6	1.50886
$503$	−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
$506$	4.56211e6	0.792117
$507$	0	0
$508$	4.59597e6	0.790165
$509$	−1.76650e6	−0.302218	−0.151109	−	0.988517i	$-0.548284\pi$
$509$	−1.76650e6	−0.302218	−0.151109	+	0.988517i	$0.548284\pi$
$510$	0	0
$511$	4.15018e6	0.703096
$512$	13882.8	0.00234046
$513$	0	0
$514$	231671.	0.0386780
$515$	1.01445e7	1.68544
$516$	0	0
$517$	−2.50810e6	−0.412685
$518$	2.69479e7	4.41266
$519$	0	0
$520$	5.80048e6	0.940709
$521$	−9.74164e6	−1.57231	−0.786154	−	0.618031i	$-0.787931\pi$
$521$	−9.74164e6	−1.57231	−0.786154	+	0.618031i	$0.787931\pi$
$522$	0	0
$523$	6.30069e6	1.00724	0.503621	−	0.863925i	$-0.332001\pi$
$523$	6.30069e6	1.00724	0.503621	+	0.863925i	$0.332001\pi$
$524$	−1.72693e7	−2.74756
$525$	0	0
$526$	1.54965e7	2.44213
$527$	−4.83655e6	−0.758593
$528$	0	0
$529$	1.05352e7	1.63683
$530$	−6.63130e6	−1.02544
$531$	0	0
$532$	7.13782e6	1.09342
$533$	619939.	0.0945215
$534$	0	0
$535$	3.22390e6	0.486964
$536$	−1.01611e6	−0.152767
$537$	0	0
$538$	−4.29566e6	−0.639844
$539$	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	$0.809303\pi$
$542$	−1.99827e7	−2.92184
$543$	0	0
$544$	−4.63340e6	−0.671278
$545$	3.62760e6	0.523153
$546$	0	0
$547$	60699.0	0.00867388	0.00433694	−	0.999991i	$-0.498620\pi$
$547$	60699.0	0.00867388	0.00433694	+	0.999991i	$0.498620\pi$
$548$	1.95504e7	2.78101
$549$	0	0
$550$	6.64351e6	0.936464
$551$	368149.	0.0516589
$552$	0	0
$553$	−1.34114e7	−1.86492
$554$	−771340.	−0.106775
$555$	0	0
$556$	−1.89085e7	−2.59400
$557$	−2.28778e6	−0.312447	−0.156224	−	0.987722i	$-0.549932\pi$
$557$	−2.28778e6	−0.312447	−0.156224	+	0.987722i	$0.549932\pi$
$558$	0	0
$559$	−545134.	−0.0737860
$560$	−23982.6	−0.00323166
$561$	0	0
$562$	−5.94441e6	−0.793904
$563$	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
$566$	1.41759e7	1.85999
$567$	0	0
$568$	−669699.	−0.0870981
$569$	6.95488e6	0.900552	0.450276	−	0.892889i	$-0.351326\pi$
$569$	6.95488e6	0.900552	0.450276	+	0.892889i	$0.351326\pi$
$570$	0	0
$571$	−3.19590e6	−0.410206	−0.205103	−	0.978740i	$-0.565753\pi$
$571$	−3.19590e6	−0.410206	−0.205103	+	0.978740i	$0.565753\pi$
$572$	−2.10298e6	−0.268748
$573$	0	0
$574$	3.54031e6	0.448500
$575$	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	$0.449036\pi$
$578$	−7.15012e6	−0.890213
$579$	0	0
$580$	2.76552e6	0.341355
$581$	−9.71970e6	−1.19457
$582$	0	0
$583$	917841.	0.111840
$584$	−3.58223e6	−0.434632
$585$	0	0
$586$	643177.	0.0773724
$587$	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
$590$	1.61453e7	1.90949
$591$	0	0
$592$	16840.3	0.00197490
$593$	8.41413e6	0.982590	0.491295	−	0.870993i	$-0.336524\pi$
$593$	8.41413e6	0.982590	0.491295	+	0.870993i	$0.336524\pi$
$594$	0	0
$595$	1.59933e7	1.85202
$596$	−1.54022e7	−1.77610
$597$	0	0
$598$	−1.26600e7	−1.44771
$599$	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	$0.796752\pi$
$602$	−3.11312e6	−0.350111
$603$	0	0
$604$	1.64646e7	1.83636
$605$	−1.39852e6	−0.155339
$606$	0	0
$607$	1.51717e7	1.67133	0.835665	−	0.549240i	$-0.185083\pi$
$607$	1.51717e7	1.67133	0.835665	+	0.549240i	$0.185083\pi$
$608$	3.81617e6	0.418667
$609$	0	0
$610$	1.48374e7	1.61448
$611$	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	$0.806939\pi$
$614$	1.31189e7	1.40435
$615$	0	0
$616$	−4.58486e6	−0.486826
$617$	1.81989e6	0.192457	0.0962284	−	0.995359i	$-0.469322\pi$
$617$	1.81989e6	0.192457	0.0962284	+	0.995359i	$0.469322\pi$
$618$	0	0
$619$	1.39678e7	1.46521	0.732606	−	0.680653i	$-0.238304\pi$
$619$	1.39678e7	1.46521	0.732606	+	0.680653i	$0.238304\pi$
$620$	2.99237e7	3.12634
$621$	0	0
$622$	2.08557e6	0.216147
$623$	−1.13056e7	−1.16701
$624$	0	0
$625$	7.47727e6	0.765672
$626$	−6.62390e6	−0.675582
$627$	0	0
$628$	−1.78519e6	−0.180628
$629$	−1.12303e7	−1.13179
$630$	0	0
$631$	−1.64990e7	−1.64962	−0.824808	−	0.565413i	$-0.808717\pi$
$631$	−1.64990e7	−1.64962	−0.824808	+	0.565413i	$0.808717\pi$
$632$	1.15761e7	1.15284
$633$	0	0
$634$	−1.15395e7	−1.14015
$635$	−8.48159e6	−0.834724
$636$	0	0
$637$	−9.09688e6	−0.888268
$638$	−619422.	−0.0602470
$639$	0	0
$640$	2.86333e7	2.76326
$641$	−1.14852e7	−1.10406	−0.552030	−	0.833824i	$-0.686147\pi$
$641$	−1.14852e7	−1.10406	−0.552030	+	0.833824i	$0.686147\pi$
$642$	0	0
$643$	−5.83065e6	−0.556147	−0.278073	−	0.960560i	$-0.589696\pi$
$643$	−5.83065e6	−0.556147	−0.278073	+	0.960560i	$0.589696\pi$
$644$	−4.46771e7	−4.24493
$645$	0	0
$646$	−4.81365e6	−0.453830
$647$	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
$650$	−1.84359e7	−1.71152
$651$	0	0
$652$	−2.37403e6	−0.218709
$653$	8.01118e6	0.735214	0.367607	−	0.929981i	$-0.380177\pi$
$653$	8.01118e6	0.735214	0.367607	+	0.929981i	$0.380177\pi$
$654$	0	0
$655$	3.18696e7	2.90250
$656$	2212.42	0.000200728 0
$657$	0	0
$658$	3.97472e7	3.57883
$659$	−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	$0.671387\pi$
$662$	3.96691e6	0.351809
$663$	0	0
$664$	8.38956e6	0.738447
$665$	−1.31724e7	−1.15508
$666$	0	0
$667$	−2.30432e6	−0.200553
$668$	−1.50593e7	−1.30576
$669$	0	0
$670$	4.91186e6	0.422726
$671$	−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	$0.754274\pi$
$674$	3.48286e7	2.95315
$675$	0	0
$676$	−1.33824e7	−1.12634
$677$	6.56811e6	0.550768	0.275384	−	0.961334i	$-0.411195\pi$
$677$	6.56811e6	0.550768	0.275384	+	0.961334i	$0.411195\pi$
$678$	0	0
$679$	−3.04327e7	−2.53318
$680$	−1.38046e7	−1.14486
$681$	0	0
$682$	−6.70233e6	−0.551779
$683$	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
$686$	−1.97218e7	−1.60006
$687$	0	0
$688$	−1945.46	−0.000156693 0
$689$	−2.54704e6	−0.204403
$690$	0	0
$691$	−1.01356e7	−0.807521	−0.403760	−	0.914865i	$-0.632297\pi$
$691$	−1.01356e7	−0.807521	−0.403760	+	0.914865i	$0.632297\pi$
$692$	7.21332e6	0.572625
$693$	0	0
$694$	−2.66352e7	−2.09922
$695$	3.48945e7	2.74028
$696$	0	0
$697$	−1.47540e6	−0.115034
$698$	−3.78438e7	−2.94006
$699$	0	0
$700$	−6.50605e7	−5.01848
$701$	−9.16091e6	−0.704115	−0.352058	−	0.935978i	$-0.614518\pi$
$701$	−9.16091e6	−0.704115	−0.352058	+	0.935978i	$0.614518\pi$
$702$	0	0
$703$	9.24955e6	0.705882
$704$	−6.41618e6	−0.487916
$705$	0	0
$706$	−4.57544e6	−0.345478
$707$	−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	$0.608847\pi$
$710$	3.23730e6	0.241011
$711$	0	0
$712$	9.75847e6	0.721409
$713$	−2.49334e7	−1.83679
$714$	0	0
$715$	3.88093e6	0.283903
$716$	3.10332e7	2.26227
$717$	0	0
$718$	2.71235e7	1.96352
$719$	−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
$722$	−1.86968e7	−1.33482
$723$	0	0
$724$	6.76148e6	0.479397
$725$	−3.35564e6	−0.237099
$726$	0	0
$727$	1.78172e7	1.25027	0.625135	−	0.780516i	$-0.285044\pi$
$727$	1.78172e7	1.25027	0.625135	+	0.780516i	$0.285044\pi$
$728$	1.27231e7	0.889744
$729$	0	0
$730$	1.73164e7	1.20268
$731$	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	$0.285147\pi$
$734$	2.35670e6	0.161460
$735$	0	0
$736$	−2.38862e7	−1.62537
$737$	−679852.	−0.0461048
$738$	0	0
$739$	−1.38291e7	−0.931502	−0.465751	−	0.884916i	$-0.654216\pi$
$739$	−1.38291e7	−0.931502	−0.465751	+	0.884916i	$0.654216\pi$
$740$	6.94822e7	4.66438
$741$	0	0
$742$	−1.45455e7	−0.969881
$743$	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
$746$	−1.17345e7	−0.771998
$747$	0	0
$748$	5.00491e6	0.327071
$749$	7.07149e6	0.460582
$750$	0	0
$751$	−653678.	−0.0422926	−0.0211463	−	0.999776i	$-0.506732\pi$
$751$	−653678.	−0.0422926	−0.0211463	+	0.999776i	$0.506732\pi$
$752$	24838.8	0.00160172
$753$	0	0
$754$	1.71892e6	0.110110
$755$	−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	$0.168348\pi$
$758$	2.86862e7	1.81343
$759$	0	0
$760$	1.13698e7	0.714033
$761$	5.41780e6	0.339126	0.169563	−	0.985519i	$-0.445764\pi$
$761$	5.41780e6	0.339126	0.169563	+	0.985519i	$0.445764\pi$
$762$	0	0
$763$	7.95700e6	0.494809
$764$	−1.75076e7	−1.08516
$765$	0	0
$766$	−8.01521e6	−0.493563
$767$	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	$0.140811\pi$
$770$	2.21630e7	1.34711
$771$	0	0
$772$	−1.70340e7	−1.02866
$773$	−1.87755e7	−1.13017	−0.565083	−	0.825034i	$-0.691156\pi$
$773$	−1.87755e7	−1.13017	−0.565083	+	0.825034i	$0.691156\pi$
$774$	0	0
$775$	−3.63090e7	−2.17150
$776$	2.62680e7	1.56593
$777$	0	0
$778$	1.78722e7	1.05859
$779$	1.21517e6	0.0717453
$780$	0	0
$781$	−448076.	−0.0262860
$782$	3.01297e7	1.76188
$783$	0	0
$784$	−32464.7	−0.00188634
$785$	3.29447e6	0.190814
$786$	0	0
$787$	−1.54112e7	−0.886951	−0.443475	−	0.896286i	$-0.646255\pi$
$787$	−1.54112e7	−0.886951	−0.443475	+	0.896286i	$0.646255\pi$
$788$	−2.67807e7	−1.53641
$789$	0	0
$790$	−5.59582e7	−3.19004
$791$	5.01906e7	2.85221
$792$	0	0
$793$	5.69894e6	0.321819
$794$	−1.56242e7	−0.879522
$795$	0	0
$796$	−2.75073e7	−1.53874
$797$	−2.10597e7	−1.17437	−0.587186	−	0.809452i	$-0.699764\pi$
$797$	−2.10597e7	−1.17437	−0.587186	+	0.809452i	$0.699764\pi$
$798$	0	0
$799$	−1.65643e7	−0.917925
$800$	−3.47840e7	−1.92156
$801$	0	0
$802$	−4.13334e7	−2.26916
$803$	−2.39677e6	−0.131171
$804$	0	0
$805$	8.24490e7	4.48431
$806$	1.85992e7	1.00845
$807$	0	0
$808$	1.39642e7	0.752467
$809$	−1.95659e7	−1.05106	−0.525532	−	0.850774i	$-0.676134\pi$
$809$	−1.95659e7	−1.05106	−0.525532	+	0.850774i	$0.676134\pi$
$810$	0	0
$811$	1.29945e7	0.693758	0.346879	−	0.937910i	$-0.387241\pi$
$811$	1.29945e7	0.693758	0.346879	+	0.937910i	$0.387241\pi$
$812$	6.06605e6	0.322861
$813$	0	0
$814$	−1.55626e7	−0.823232
$815$	4.38113e6	0.231042
$816$	0	0
$817$	−1.06854e6	−0.0560063
$818$	−6.59876e6	−0.344809
$819$	0	0
$820$	9.12830e6	0.474084
$821$	3.50872e7	1.81673	0.908367	−	0.418175i	$-0.137330\pi$
$821$	3.50872e7	1.81673	0.908367	+	0.418175i	$0.137330\pi$
$822$	0	0
$823$	1.38494e7	0.712742	0.356371	−	0.934345i	$-0.384014\pi$
$823$	1.38494e7	0.712742	0.356371	+	0.934345i	$0.384014\pi$
$824$	−1.92065e7	−0.985440
$825$	0	0
$826$	3.54141e7	1.80604
$827$	−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	$0.510306\pi$
$830$	−4.05548e7	−2.04337
$831$	0	0
$832$	1.78051e7	0.891735
$833$	2.16498e7	1.08104
$834$	0	0
$835$	2.77910e7	1.37939
$836$	−4.12215e6	−0.203990
$837$	0	0
$838$	−3.56843e7	−1.75536
$839$	−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
$842$	−3.10863e7	−1.51109
$843$	0	0
$844$	2.12172e7	1.02526
$845$	2.46965e7	1.18985
$846$	0	0
$847$	−3.06759e6	−0.146923
$848$	−9089.78	−0.000434074 0
$849$	0	0
$850$	4.38760e7	2.08295
$851$	−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	$0.714386\pi$
$854$	3.25452e7	1.52701
$855$	0	0
$856$	−6.10376e6	−0.284717
$857$	1.96831e7	0.915465	0.457733	−	0.889090i	$-0.348662\pi$
$857$	1.96831e7	0.915465	0.457733	+	0.889090i	$0.348662\pi$
$858$	0	0
$859$	−1.74273e7	−0.805835	−0.402917	−	0.915236i	$-0.632004\pi$
$859$	−1.74273e7	−0.805835	−0.402917	+	0.915236i	$0.632004\pi$
$860$	−8.02684e6	−0.370083
$861$	0	0
$862$	4.71511e7	2.16134
$863$	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
$866$	−2.96024e7	−1.34132
$867$	0	0
$868$	6.56365e7	2.95696
$869$	7.74520e6	0.347923
$870$	0	0
$871$	1.88661e6	0.0842630
$872$	−6.86809e6	−0.305875
$873$	0	0
$874$	−2.48154e7	−1.09886
$875$	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	$0.540040\pi$
$878$	5.77551e7	2.52845
$879$	0	0
$880$	13850.1	0.000602903 0
$881$	−1.76619e7	−0.766649	−0.383324	−	0.923614i	$-0.625221\pi$
$881$	−1.76619e7	−0.766649	−0.383324	+	0.923614i	$0.625221\pi$
$882$	0	0
$883$	2.70117e7	1.16587	0.582936	−	0.812518i	$-0.301904\pi$
$883$	2.70117e7	1.16587	0.582936	+	0.812518i	$0.301904\pi$
$884$	−1.38888e7	−0.597769
$885$	0	0
$886$	−2.63133e7	−1.12614
$887$	−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
$890$	−4.71720e7	−1.99623
$891$	0	0
$892$	1.39700e7	0.587875
$893$	1.36427e7	0.572496
$894$	0	0
$895$	−5.72700e7	−2.38984
$896$	6.28061e7	2.61355
$897$	0	0
$898$	4.15834e7	1.72080
$899$	3.38535e6	0.139703
$900$	0	0
$901$	6.06172e6	0.248762
$902$	−2.04456e6	−0.0836727
$903$	0	0
$904$	−4.33220e7	−1.76314
$905$	−1.24779e7	−0.506431
$906$	0	0
$907$	9.84123e6	0.397220	0.198610	−	0.980079i	$-0.436357\pi$
$907$	9.84123e6	0.397220	0.198610	+	0.980079i	$0.436357\pi$
$908$	5.28713e7	2.12816
$909$	0	0
$910$	−6.15030e7	−2.46203
$911$	−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
$914$	2.54871e7	1.00915
$915$	0	0
$916$	8.76177e6	0.345027
$917$	6.99046e7	2.74525
$918$	0	0
$919$	1.15896e7	0.452667	0.226333	−	0.974050i	$-0.427326\pi$
$919$	1.15896e7	0.452667	0.226333	+	0.974050i	$0.427326\pi$
$920$	−7.11659e7	−2.77206
$921$	0	0
$922$	1.83494e7	0.710878
$923$	1.24342e6	0.0480413
$924$	0	0
$925$	−8.43087e7	−3.23980
$926$	−4.27769e7	−1.63939
$927$	0	0
$928$	3.24316e6	0.123623
$929$	−1.17151e7	−0.445357	−0.222679	−	0.974892i	$-0.571480\pi$
$929$	−1.17151e7	−0.445357	−0.222679	+	0.974892i	$0.571480\pi$
$930$	0	0
$931$	−1.78312e7	−0.674228
$932$	1.05827e6	0.0399077
$933$	0	0
$934$	1.29912e7	0.487284
$935$	−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	$0.445351\pi$
$938$	1.07740e7	0.399823
$939$	0	0
$940$	1.02484e8	3.78299
$941$	−3.24899e6	−0.119612	−0.0598059	−	0.998210i	$-0.519048\pi$
$941$	−3.24899e6	−0.119612	−0.0598059	+	0.998210i	$0.519048\pi$
$942$	0	0
$943$	−7.60601e6	−0.278534
$944$	22131.0	0.000808299 0
$945$	0	0
$946$	1.79786e6	0.0653171
$947$	−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
$950$	−3.61372e7	−1.29911
$951$	0	0
$952$	−3.02799e7	−1.08283
$953$	3.27415e7	1.16780	0.583898	−	0.811827i	$-0.301527\pi$
$953$	3.27415e7	1.16780	0.583898	+	0.811827i	$0.301527\pi$
$954$	0	0
$955$	3.23092e7	1.14635
$956$	4.83209e7	1.70998
$957$	0	0
$958$	3.29545e7	1.16011
$959$	−7.91379e7	−2.77868
$960$	0	0
$961$	8.00132e6	0.279482
$962$	4.31868e7	1.50457
$963$	0	0
$964$	−6.74227e7	−2.33676
$965$	3.14353e7	1.08667
$966$	0	0
$967$	−3.94890e7	−1.35803	−0.679016	−	0.734124i	$-0.737593\pi$
$967$	−3.94890e7	−1.35803	−0.679016	+	0.734124i	$0.737593\pi$
$968$	2.64779e6	0.0908230
$969$	0	0
$970$	−1.26978e8	−4.33312
$971$	−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
$974$	−5.10245e7	−1.72338
$975$	0	0
$976$	20338.2	0.000683421 0
$977$	4.55108e7	1.52538	0.762690	−	0.646764i	$-0.223878\pi$
$977$	4.55108e7	1.52538	0.762690	+	0.646764i	$0.223878\pi$
$978$	0	0
$979$	6.52910e6	0.217719
$980$	−1.33947e8	−4.45522
$981$	0	0
$982$	1.65605e7	0.548018
$983$	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
$986$	−4.09087e6	−0.134006
$987$	0	0
$988$	1.14391e7	0.372820
$989$	6.68823e6	0.217431
$990$	0	0
$991$	4.23892e7	1.37111	0.685553	−	0.728023i	$-0.259560\pi$
$991$	4.23892e7	1.37111	0.685553	+	0.728023i	$0.259560\pi$
$992$	3.50919e7	1.13221
$993$	0	0
$994$	7.10088e6	0.227953
$995$	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	$0.318344\pi$
$998$	−5.44094e7	−1.72921
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.6.a.c.1.1	✓	2	3.2	odd	2
99.6.a.f.1.2		2	1.1	even	1	trivial
363.6.a.j.1.2		2	33.32	even	2
528.6.a.s.1.2		2	12.11	even	2
825.6.a.e.1.2		2	15.14	odd	2
1089.6.a.j.1.1		2	11.10	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:	\( N \)	\(=\)	\( 99 = 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	99.a (trivial)

\(f(q)\)	\(=\)	\(q+9.15207 q^{2} +51.7603 q^{4} -95.5207 q^{5} -209.521 q^{7} +180.848 q^{8} +O(q^{10})\)	\(q+9.15207 q^{2} +51.7603 q^{4} -95.5207 q^{5} -209.521 q^{7} +180.848 q^{8} -874.212 q^{10} +121.000 q^{11} -335.779 q^{13} -1917.55 q^{14} -1.19832 q^{16} +799.124 q^{17} -658.175 q^{19} -4944.18 q^{20} +1107.40 q^{22} +4119.66 q^{23} +5999.20 q^{25} -3073.07 q^{26} -10844.9 q^{28} -559.348 q^{29} -6052.31 q^{31} -5798.10 q^{32} +7313.64 q^{34} +20013.6 q^{35} -14053.3 q^{37} -6023.66 q^{38} -17274.7 q^{40} -1846.27 q^{41} +1623.49 q^{43} +6263.00 q^{44} +37703.4 q^{46} -20728.1 q^{47} +27091.9 q^{49} +54905.1 q^{50} -17380.0 q^{52} +7585.46 q^{53} -11558.0 q^{55} -37891.4 q^{56} -5119.19 q^{58} -18468.5 q^{59} -16972.3 q^{61} -55391.2 q^{62} -53026.3 q^{64} +32073.8 q^{65} -5618.62 q^{67} +41362.9 q^{68} +183165. q^{70} -3703.10 q^{71} -19808.0 q^{73} -128617. q^{74} -34067.4 q^{76} -25352.0 q^{77} +64009.9 q^{79} +114.464 q^{80} -16897.2 q^{82} +46390.2 q^{83} -76332.9 q^{85} +14858.3 q^{86} +21882.6 q^{88} +53959.5 q^{89} +70352.5 q^{91} +213235. q^{92} -189705. q^{94} +62869.3 q^{95} +145249. q^{97} +247947. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 5 q^{2} + 37 q^{4} - 58 q^{5} - 286 q^{7} + 375 q^{8}+O(q^{10}) \) 2 * q + 5 * q^2 + 37 * q^4 - 58 * q^5 - 286 * q^7 + 375 * q^8	\( 2 q + 5 q^{2} + 37 q^{4} - 58 q^{5} - 286 q^{7} + 375 q^{8} - 1030 q^{10} + 242 q^{11} - 166 q^{13} - 1600 q^{14} - 335 q^{16} + 800 q^{17} - 1476 q^{19} - 5498 q^{20} + 605 q^{22} + 3370 q^{23} + 4282 q^{25} - 3778 q^{26} - 9716 q^{28} - 6600 q^{29} - 7528 q^{31} - 10625 q^{32} + 7310 q^{34} + 17144 q^{35} - 29916 q^{37} - 2628 q^{38} - 9990 q^{40} + 5780 q^{41} - 16656 q^{43} + 4477 q^{44} + 40816 q^{46} - 7850 q^{47} + 16134 q^{49} + 62035 q^{50} - 19886 q^{52} - 14178 q^{53} - 7018 q^{55} - 52740 q^{56} + 19962 q^{58} - 17300 q^{59} - 2946 q^{61} - 49264 q^{62} - 22303 q^{64} + 38444 q^{65} + 31336 q^{67} + 41350 q^{68} + 195080 q^{70} + 33810 q^{71} + 60644 q^{73} - 62754 q^{74} - 21996 q^{76} - 34606 q^{77} + 1870 q^{79} - 12410 q^{80} - 48562 q^{82} + 58296 q^{83} - 76300 q^{85} + 90756 q^{86} + 45375 q^{88} - 92388 q^{89} + 57368 q^{91} + 224300 q^{92} - 243176 q^{94} + 32184 q^{95} + 7120 q^{97} + 293445 q^{98}+O(q^{100}) \) 2 * q + 5 * q^2 + 37 * q^4 - 58 * q^5 - 286 * q^7 + 375 * q^8 - 1030 * q^10 + 242 * q^11 - 166 * q^13 - 1600 * q^14 - 335 * q^16 + 800 * q^17 - 1476 * q^19 - 5498 * q^20 + 605 * q^22 + 3370 * q^23 + 4282 * q^25 - 3778 * q^26 - 9716 * q^28 - 6600 * q^29 - 7528 * q^31 - 10625 * q^32 + 7310 * q^34 + 17144 * q^35 - 29916 * q^37 - 2628 * q^38 - 9990 * q^40 + 5780 * q^41 - 16656 * q^43 + 4477 * q^44 + 40816 * q^46 - 7850 * q^47 + 16134 * q^49 + 62035 * q^50 - 19886 * q^52 - 14178 * q^53 - 7018 * q^55 - 52740 * q^56 + 19962 * q^58 - 17300 * q^59 - 2946 * q^61 - 49264 * q^62 - 22303 * q^64 + 38444 * q^65 + 31336 * q^67 + 41350 * q^68 + 195080 * q^70 + 33810 * q^71 + 60644 * q^73 - 62754 * q^74 - 21996 * q^76 - 34606 * q^77 + 1870 * q^79 - 12410 * q^80 - 48562 * q^82 + 58296 * q^83 - 76300 * q^85 + 90756 * q^86 + 45375 * q^88 - 92388 * q^89 + 57368 * q^91 + 224300 * q^92 - 243176 * q^94 + 32184 * q^95 + 7120 * q^97 + 293445 * q^98

Introduction

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