LMFDB - Embedded newform 6600.2.a.bm.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6600,2,Mod(1,6600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6600, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("6600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6600 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6600.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:	$52.7012653340$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	6600.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +4.56155 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +4.56155 q^{7} +1.00000 q^{9} +1.00000 q^{11} -2.12311 q^{13} +5.12311 q^{17} +3.00000 q^{19} +4.56155 q^{21} -6.68466 q^{23} +1.00000 q^{27} -2.68466 q^{29} +3.00000 q^{31} +1.00000 q^{33} +6.00000 q^{37} -2.12311 q^{39} -6.00000 q^{41} +6.12311 q^{43} -4.00000 q^{47} +13.8078 q^{49} +5.12311 q^{51} +1.12311 q^{53} +3.00000 q^{57} +13.3693 q^{59} +5.68466 q^{61} +4.56155 q^{63} -8.56155 q^{67} -6.68466 q^{69} +5.56155 q^{71} +8.24621 q^{73} +4.56155 q^{77} -2.24621 q^{79} +1.00000 q^{81} +2.68466 q^{83} -2.68466 q^{87} +7.56155 q^{89} -9.68466 q^{91} +3.00000 q^{93} +3.87689 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 5 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 5 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} + 5 q^{7} + 2 q^{9} + 2 q^{11} + 4 q^{13} + 2 q^{17} + 6 q^{19} + 5 q^{21} - q^{23} + 2 q^{27} + 7 q^{29} + 6 q^{31} + 2 q^{33} + 12 q^{37} + 4 q^{39} - 12 q^{41} + 4 q^{43} - 8 q^{47} + 7 q^{49} + 2 q^{51} - 6 q^{53} + 6 q^{57} + 2 q^{59} - q^{61} + 5 q^{63} - 13 q^{67} - q^{69} + 7 q^{71} + 5 q^{77} + 12 q^{79} + 2 q^{81} - 7 q^{83} + 7 q^{87} + 11 q^{89} - 7 q^{91} + 6 q^{93} + 16 q^{97} + 2 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 5 * q^7 + 2 * q^9 + 2 * q^11 + 4 * q^13 + 2 * q^17 + 6 * q^19 + 5 * q^21 - q^23 + 2 * q^27 + 7 * q^29 + 6 * q^31 + 2 * q^33 + 12 * q^37 + 4 * q^39 - 12 * q^41 + 4 * q^43 - 8 * q^47 + 7 * q^49 + 2 * q^51 - 6 * q^53 + 6 * q^57 + 2 * q^59 - q^61 + 5 * q^63 - 13 * q^67 - q^69 + 7 * q^71 + 5 * q^77 + 12 * q^79 + 2 * q^81 - 7 * q^83 + 7 * q^87 + 11 * q^89 - 7 * q^91 + 6 * q^93 + 16 * q^97 + 2 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.56155	1.72410	0.862052	−	0.506819i	$-0.169179\pi$
$7$	4.56155	1.72410	0.862052	+	0.506819i	$0.169179\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−2.12311	−0.588844	−0.294422	−	0.955676i	$-0.595127\pi$
$13$	−2.12311	−0.588844	−0.294422	+	0.955676i	$0.595127\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.12311	1.24254	0.621268	−	0.783598i	$-0.286618\pi$
$17$	5.12311	1.24254	0.621268	+	0.783598i	$0.286618\pi$
$18$	0	0
$19$	3.00000	0.688247	0.344124	−	0.938924i	$-0.388176\pi$
$19$	3.00000	0.688247	0.344124	+	0.938924i	$0.388176\pi$
$20$	0	0
$21$	4.56155	0.995412
$22$	0	0
$23$	−6.68466	−1.39385	−0.696924	−	0.717145i	$-0.745448\pi$
$23$	−6.68466	−1.39385	−0.696924	+	0.717145i	$0.745448\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−2.68466	−0.498529	−0.249264	−	0.968436i	$-0.580189\pi$
$29$	−2.68466	−0.498529	−0.249264	+	0.968436i	$0.580189\pi$
$30$	0	0
$31$	3.00000	0.538816	0.269408	−	0.963026i	$-0.413172\pi$
$31$	3.00000	0.538816	0.269408	+	0.963026i	$0.413172\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	0	0
$39$	−2.12311	−0.339969
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	6.12311	0.933765	0.466882	−	0.884319i	$-0.345377\pi$
$43$	6.12311	0.933765	0.466882	+	0.884319i	$0.345377\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−4.00000	−0.583460	−0.291730	−	0.956501i	$-0.594231\pi$
$47$	−4.00000	−0.583460	−0.291730	+	0.956501i	$0.594231\pi$
$48$	0	0
$49$	13.8078	1.97254
$50$	0	0
$51$	5.12311	0.717378
$52$	0	0
$53$	1.12311	0.154270	0.0771352	−	0.997021i	$-0.475423\pi$
$53$	1.12311	0.154270	0.0771352	+	0.997021i	$0.475423\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	3.00000	0.397360
$58$	0	0
$59$	13.3693	1.74054	0.870268	−	0.492578i	$-0.163945\pi$
$59$	13.3693	1.74054	0.870268	+	0.492578i	$0.163945\pi$
$60$	0	0
$61$	5.68466	0.727846	0.363923	−	0.931429i	$-0.381437\pi$
$61$	5.68466	0.727846	0.363923	+	0.931429i	$0.381437\pi$
$62$	0	0
$63$	4.56155	0.574702
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−8.56155	−1.04596	−0.522980	−	0.852345i	$-0.675180\pi$
$67$	−8.56155	−1.04596	−0.522980	+	0.852345i	$0.675180\pi$
$68$	0	0
$69$	−6.68466	−0.804738
$70$	0	0
$71$	5.56155	0.660035	0.330017	−	0.943975i	$-0.392945\pi$
$71$	5.56155	0.660035	0.330017	+	0.943975i	$0.392945\pi$
$72$	0	0
$73$	8.24621	0.965146	0.482573	−	0.875856i	$-0.339702\pi$
$73$	8.24621	0.965146	0.482573	+	0.875856i	$0.339702\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.56155	0.519837
$78$	0	0
$79$	−2.24621	−0.252719	−0.126359	−	0.991985i	$-0.540329\pi$
$79$	−2.24621	−0.252719	−0.126359	+	0.991985i	$0.540329\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	2.68466	0.294680	0.147340	−	0.989086i	$-0.452929\pi$
$83$	2.68466	0.294680	0.147340	+	0.989086i	$0.452929\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−2.68466	−0.287826
$88$	0	0
$89$	7.56155	0.801523	0.400761	−	0.916182i	$-0.368746\pi$
$89$	7.56155	0.801523	0.400761	+	0.916182i	$0.368746\pi$
$90$	0	0
$91$	−9.68466	−1.01523
$92$	0	0
$93$	3.00000	0.311086
$94$	0	0
$95$	0	0
$96$	0	0
$97$	3.87689	0.393639	0.196819	−	0.980440i	$-0.436939\pi$
$97$	3.87689	0.393639	0.196819	+	0.980440i	$0.436939\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	−15.8078	−1.57293	−0.786466	−	0.617634i	$-0.788091\pi$
$101$	−15.8078	−1.57293	−0.786466	+	0.617634i	$0.788091\pi$
$102$	0	0
$103$	−18.0540	−1.77891	−0.889456	−	0.457022i	$-0.848916\pi$
$103$	−18.0540	−1.77891	−0.889456	+	0.457022i	$0.848916\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−6.68466	−0.646230	−0.323115	−	0.946360i	$-0.604730\pi$
$107$	−6.68466	−0.646230	−0.323115	+	0.946360i	$0.604730\pi$
$108$	0	0
$109$	−1.87689	−0.179774	−0.0898869	−	0.995952i	$-0.528651\pi$
$109$	−1.87689	−0.179774	−0.0898869	+	0.995952i	$0.528651\pi$
$110$	0	0
$111$	6.00000	0.569495
$112$	0	0
$113$	−0.246211	−0.0231616	−0.0115808	−	0.999933i	$-0.503686\pi$
$113$	−0.246211	−0.0231616	−0.0115808	+	0.999933i	$0.503686\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−2.12311	−0.196281
$118$	0	0
$119$	23.3693	2.14226
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−6.00000	−0.541002
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−6.24621	−0.554262	−0.277131	−	0.960832i	$-0.589384\pi$
$127$	−6.24621	−0.554262	−0.277131	+	0.960832i	$0.589384\pi$
$128$	0	0
$129$	6.12311	0.539109
$130$	0	0
$131$	−4.43845	−0.387789	−0.193894	−	0.981022i	$-0.562112\pi$
$131$	−4.43845	−0.387789	−0.193894	+	0.981022i	$0.562112\pi$
$132$	0	0
$133$	13.6847	1.18661
$134$	0	0
$135$	0	0
$136$	0	0
$137$	9.56155	0.816899	0.408449	−	0.912781i	$-0.366070\pi$
$137$	9.56155	0.816899	0.408449	+	0.912781i	$0.366070\pi$
$138$	0	0
$139$	0.192236	0.0163052	0.00815262	−	0.999967i	$-0.497405\pi$
$139$	0.192236	0.0163052	0.00815262	+	0.999967i	$0.497405\pi$
$140$	0	0
$141$	−4.00000	−0.336861
$142$	0	0
$143$	−2.12311	−0.177543
$144$	0	0
$145$	0	0
$146$	0	0
$147$	13.8078	1.13885
$148$	0	0
$149$	0.246211	0.0201704	0.0100852	−	0.999949i	$-0.496790\pi$
$149$	0.246211	0.0201704	0.0100852	+	0.999949i	$0.496790\pi$
$150$	0	0
$151$	−22.5616	−1.83603	−0.918017	−	0.396542i	$-0.870210\pi$
$151$	−22.5616	−1.83603	−0.918017	+	0.396542i	$0.870210\pi$
$152$	0	0
$153$	5.12311	0.414179
$154$	0	0
$155$	0	0
$156$	0	0
$157$	16.8078	1.34141	0.670703	−	0.741726i	$-0.265993\pi$
$157$	16.8078	1.34141	0.670703	+	0.741726i	$0.265993\pi$
$158$	0	0
$159$	1.12311	0.0890681
$160$	0	0
$161$	−30.4924	−2.40314
$162$	0	0
$163$	19.6847	1.54182	0.770911	−	0.636943i	$-0.219801\pi$
$163$	19.6847	1.54182	0.770911	+	0.636943i	$0.219801\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	7.12311	0.551202	0.275601	−	0.961272i	$-0.411123\pi$
$167$	7.12311	0.551202	0.275601	+	0.961272i	$0.411123\pi$
$168$	0	0
$169$	−8.49242	−0.653263
$170$	0	0
$171$	3.00000	0.229416
$172$	0	0
$173$	22.9309	1.74340	0.871701	−	0.490038i	$-0.163017\pi$
$173$	22.9309	1.74340	0.871701	+	0.490038i	$0.163017\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	13.3693	1.00490
$178$	0	0
$179$	−17.6155	−1.31665	−0.658323	−	0.752735i	$-0.728734\pi$
$179$	−17.6155	−1.31665	−0.658323	+	0.752735i	$0.728734\pi$
$180$	0	0
$181$	12.8078	0.951994	0.475997	−	0.879447i	$-0.342087\pi$
$181$	12.8078	0.951994	0.475997	+	0.879447i	$0.342087\pi$
$182$	0	0
$183$	5.68466	0.420222
$184$	0	0
$185$	0	0
$186$	0	0
$187$	5.12311	0.374639
$188$	0	0
$189$	4.56155	0.331804
$190$	0	0
$191$	−0.192236	−0.0139097	−0.00695485	−	0.999976i	$-0.502214\pi$
$191$	−0.192236	−0.0139097	−0.00695485	+	0.999976i	$0.502214\pi$
$192$	0	0
$193$	−5.68466	−0.409191	−0.204595	−	0.978847i	$-0.565588\pi$
$193$	−5.68466	−0.409191	−0.204595	+	0.978847i	$0.565588\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−18.9309	−1.34877	−0.674384	−	0.738381i	$-0.735591\pi$
$197$	−18.9309	−1.34877	−0.674384	+	0.738381i	$0.735591\pi$
$198$	0	0
$199$	2.75379	0.195211	0.0976055	−	0.995225i	$-0.468882\pi$
$199$	2.75379	0.195211	0.0976055	+	0.995225i	$0.468882\pi$
$200$	0	0
$201$	−8.56155	−0.603885
$202$	0	0
$203$	−12.2462	−0.859516
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−6.68466	−0.464616
$208$	0	0
$209$	3.00000	0.207514
$210$	0	0
$211$	15.6847	1.07978	0.539888	−	0.841737i	$-0.318467\pi$
$211$	15.6847	1.07978	0.539888	+	0.841737i	$0.318467\pi$
$212$	0	0
$213$	5.56155	0.381071
$214$	0	0
$215$	0	0
$216$	0	0
$217$	13.6847	0.928975
$218$	0	0
$219$	8.24621	0.557227
$220$	0	0
$221$	−10.8769	−0.731659
$222$	0	0
$223$	19.0540	1.27595	0.637974	−	0.770058i	$-0.279773\pi$
$223$	19.0540	1.27595	0.637974	+	0.770058i	$0.279773\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−0.192236	−0.0127591	−0.00637957	−	0.999980i	$-0.502031\pi$
$227$	−0.192236	−0.0127591	−0.00637957	+	0.999980i	$0.502031\pi$
$228$	0	0
$229$	−21.4384	−1.41669	−0.708346	−	0.705865i	$-0.750558\pi$
$229$	−21.4384	−1.41669	−0.708346	+	0.705865i	$0.750558\pi$
$230$	0	0
$231$	4.56155	0.300128
$232$	0	0
$233$	−12.4924	−0.818406	−0.409203	−	0.912443i	$-0.634193\pi$
$233$	−12.4924	−0.818406	−0.409203	+	0.912443i	$0.634193\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−2.24621	−0.145907
$238$	0	0
$239$	10.8769	0.703568	0.351784	−	0.936081i	$-0.385575\pi$
$239$	10.8769	0.703568	0.351784	+	0.936081i	$0.385575\pi$
$240$	0	0
$241$	6.31534	0.406807	0.203403	−	0.979095i	$-0.434800\pi$
$241$	6.31534	0.406807	0.203403	+	0.979095i	$0.434800\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−6.36932	−0.405270
$248$	0	0
$249$	2.68466	0.170133
$250$	0	0
$251$	−19.3693	−1.22258	−0.611290	−	0.791407i	$-0.709349\pi$
$251$	−19.3693	−1.22258	−0.611290	+	0.791407i	$0.709349\pi$
$252$	0	0
$253$	−6.68466	−0.420261
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−0.684658	−0.0427078	−0.0213539	−	0.999772i	$-0.506798\pi$
$257$	−0.684658	−0.0427078	−0.0213539	+	0.999772i	$0.506798\pi$
$258$	0	0
$259$	27.3693	1.70065
$260$	0	0
$261$	−2.68466	−0.166176
$262$	0	0
$263$	25.1231	1.54916	0.774579	−	0.632478i	$-0.217962\pi$
$263$	25.1231	1.54916	0.774579	+	0.632478i	$0.217962\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	7.56155	0.462760
$268$	0	0
$269$	9.12311	0.556246	0.278123	−	0.960546i	$-0.410288\pi$
$269$	9.12311	0.556246	0.278123	+	0.960546i	$0.410288\pi$
$270$	0	0
$271$	−22.2462	−1.35136	−0.675681	−	0.737195i	$-0.736150\pi$
$271$	−22.2462	−1.35136	−0.675681	+	0.737195i	$0.736150\pi$
$272$	0	0
$273$	−9.68466	−0.586142
$274$	0	0
$275$	0	0
$276$	0	0
$277$	23.4384	1.40828	0.704140	−	0.710061i	$-0.251333\pi$
$277$	23.4384	1.40828	0.704140	+	0.710061i	$0.251333\pi$
$278$	0	0
$279$	3.00000	0.179605
$280$	0	0
$281$	−28.0000	−1.67034	−0.835170	−	0.549992i	$-0.814631\pi$
$281$	−28.0000	−1.67034	−0.835170	+	0.549992i	$0.814631\pi$
$282$	0	0
$283$	20.8078	1.23689	0.618447	−	0.785827i	$-0.287762\pi$
$283$	20.8078	1.23689	0.618447	+	0.785827i	$0.287762\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−27.3693	−1.61556
$288$	0	0
$289$	9.24621	0.543895
$290$	0	0
$291$	3.87689	0.227268
$292$	0	0
$293$	−20.7386	−1.21156	−0.605782	−	0.795631i	$-0.707140\pi$
$293$	−20.7386	−1.21156	−0.605782	+	0.795631i	$0.707140\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1.00000	0.0580259
$298$	0	0
$299$	14.1922	0.820758
$300$	0	0
$301$	27.9309	1.60991
$302$	0	0
$303$	−15.8078	−0.908132
$304$	0	0
$305$	0	0
$306$	0	0
$307$	20.8078	1.18756	0.593781	−	0.804627i	$-0.297635\pi$
$307$	20.8078	1.18756	0.593781	+	0.804627i	$0.297635\pi$
$308$	0	0
$309$	−18.0540	−1.02705
$310$	0	0
$311$	−31.5616	−1.78969	−0.894846	−	0.446376i	$-0.852715\pi$
$311$	−31.5616	−1.78969	−0.894846	+	0.446376i	$0.852715\pi$
$312$	0	0
$313$	−32.5616	−1.84049	−0.920244	−	0.391345i	$-0.872010\pi$
$313$	−32.5616	−1.84049	−0.920244	+	0.391345i	$0.872010\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	19.3693	1.08789	0.543945	−	0.839121i	$-0.316930\pi$
$317$	19.3693	1.08789	0.543945	+	0.839121i	$0.316930\pi$
$318$	0	0
$319$	−2.68466	−0.150312
$320$	0	0
$321$	−6.68466	−0.373101
$322$	0	0
$323$	15.3693	0.855172
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−1.87689	−0.103792
$328$	0	0
$329$	−18.2462	−1.00595
$330$	0	0
$331$	8.87689	0.487918	0.243959	−	0.969786i	$-0.421554\pi$
$331$	8.87689	0.487918	0.243959	+	0.969786i	$0.421554\pi$
$332$	0	0
$333$	6.00000	0.328798
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−20.1771	−1.09912	−0.549558	−	0.835456i	$-0.685204\pi$
$337$	−20.1771	−1.09912	−0.549558	+	0.835456i	$0.685204\pi$
$338$	0	0
$339$	−0.246211	−0.0133724
$340$	0	0
$341$	3.00000	0.162459
$342$	0	0
$343$	31.0540	1.67676
$344$	0	0
$345$	0	0
$346$	0	0
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	0	0
$349$	−7.56155	−0.404761	−0.202380	−	0.979307i	$-0.564868\pi$
$349$	−7.56155	−0.404761	−0.202380	+	0.979307i	$0.564868\pi$
$350$	0	0
$351$	−2.12311	−0.113323
$352$	0	0
$353$	0.438447	0.0233362	0.0116681	−	0.999932i	$-0.496286\pi$
$353$	0.438447	0.0233362	0.0116681	+	0.999932i	$0.496286\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	23.3693	1.23684
$358$	0	0
$359$	−0.876894	−0.0462807	−0.0231404	−	0.999732i	$-0.507366\pi$
$359$	−0.876894	−0.0462807	−0.0231404	+	0.999732i	$0.507366\pi$
$360$	0	0
$361$	−10.0000	−0.526316
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	0	0
$366$	0	0
$367$	32.3693	1.68966	0.844832	−	0.535031i	$-0.179700\pi$
$367$	32.3693	1.68966	0.844832	+	0.535031i	$0.179700\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	0	0
$371$	5.12311	0.265978
$372$	0	0
$373$	−3.43845	−0.178036	−0.0890180	−	0.996030i	$-0.528373\pi$
$373$	−3.43845	−0.178036	−0.0890180	+	0.996030i	$0.528373\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	5.69981	0.293555
$378$	0	0
$379$	25.9309	1.33198	0.665990	−	0.745961i	$-0.268009\pi$
$379$	25.9309	1.33198	0.665990	+	0.745961i	$0.268009\pi$
$380$	0	0
$381$	−6.24621	−0.320003
$382$	0	0
$383$	0.438447	0.0224036	0.0112018	−	0.999937i	$-0.496434\pi$
$383$	0.438447	0.0224036	0.0112018	+	0.999937i	$0.496434\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	6.12311	0.311255
$388$	0	0
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	0	0
$391$	−34.2462	−1.73191
$392$	0	0
$393$	−4.43845	−0.223890
$394$	0	0
$395$	0	0
$396$	0	0
$397$	15.0540	0.755537	0.377769	−	0.925900i	$-0.376691\pi$
$397$	15.0540	0.755537	0.377769	+	0.925900i	$0.376691\pi$
$398$	0	0
$399$	13.6847	0.685090
$400$	0	0
$401$	22.3002	1.11362	0.556809	−	0.830641i	$-0.312025\pi$
$401$	22.3002	1.11362	0.556809	+	0.830641i	$0.312025\pi$
$402$	0	0
$403$	−6.36932	−0.317278
$404$	0	0
$405$	0	0
$406$	0	0
$407$	6.00000	0.297409
$408$	0	0
$409$	10.8078	0.534410	0.267205	−	0.963640i	$-0.413900\pi$
$409$	10.8078	0.534410	0.267205	+	0.963640i	$0.413900\pi$
$410$	0	0
$411$	9.56155	0.471637
$412$	0	0
$413$	60.9848	3.00087
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0.192236	0.00941383
$418$	0	0
$419$	3.12311	0.152574	0.0762868	−	0.997086i	$-0.475694\pi$
$419$	3.12311	0.152574	0.0762868	+	0.997086i	$0.475694\pi$
$420$	0	0
$421$	−17.6155	−0.858528	−0.429264	−	0.903179i	$-0.641227\pi$
$421$	−17.6155	−0.858528	−0.429264	+	0.903179i	$0.641227\pi$
$422$	0	0
$423$	−4.00000	−0.194487
$424$	0	0
$425$	0	0
$426$	0	0
$427$	25.9309	1.25488
$428$	0	0
$429$	−2.12311	−0.102505
$430$	0	0
$431$	20.7386	0.998945	0.499472	−	0.866330i	$-0.333527\pi$
$431$	20.7386	0.998945	0.499472	+	0.866330i	$0.333527\pi$
$432$	0	0
$433$	−35.0000	−1.68199	−0.840996	−	0.541041i	$-0.818030\pi$
$433$	−35.0000	−1.68199	−0.840996	+	0.541041i	$0.818030\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−20.0540	−0.959312
$438$	0	0
$439$	27.9309	1.33307	0.666534	−	0.745475i	$-0.267777\pi$
$439$	27.9309	1.33307	0.666534	+	0.745475i	$0.267777\pi$
$440$	0	0
$441$	13.8078	0.657513
$442$	0	0
$443$	4.63068	0.220010	0.110005	−	0.993931i	$-0.464913\pi$
$443$	4.63068	0.220010	0.110005	+	0.993931i	$0.464913\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0.246211	0.0116454
$448$	0	0
$449$	−29.1771	−1.37695	−0.688476	−	0.725259i	$-0.741720\pi$
$449$	−29.1771	−1.37695	−0.688476	+	0.725259i	$0.741720\pi$
$450$	0	0
$451$	−6.00000	−0.282529
$452$	0	0
$453$	−22.5616	−1.06003
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−5.12311	−0.239649	−0.119824	−	0.992795i	$-0.538233\pi$
$457$	−5.12311	−0.239649	−0.119824	+	0.992795i	$0.538233\pi$
$458$	0	0
$459$	5.12311	0.239126
$460$	0	0
$461$	−3.75379	−0.174831	−0.0874157	−	0.996172i	$-0.527861\pi$
$461$	−3.75379	−0.174831	−0.0874157	+	0.996172i	$0.527861\pi$
$462$	0	0
$463$	−17.1771	−0.798287	−0.399143	−	0.916889i	$-0.630692\pi$
$463$	−17.1771	−0.798287	−0.399143	+	0.916889i	$0.630692\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	5.12311	0.237069	0.118535	−	0.992950i	$-0.462180\pi$
$467$	5.12311	0.237069	0.118535	+	0.992950i	$0.462180\pi$
$468$	0	0
$469$	−39.0540	−1.80335
$470$	0	0
$471$	16.8078	0.774461
$472$	0	0
$473$	6.12311	0.281541
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1.12311	0.0514235
$478$	0	0
$479$	−9.75379	−0.445662	−0.222831	−	0.974857i	$-0.571530\pi$
$479$	−9.75379	−0.445662	−0.222831	+	0.974857i	$0.571530\pi$
$480$	0	0
$481$	−12.7386	−0.580832
$482$	0	0
$483$	−30.4924	−1.38745
$484$	0	0
$485$	0	0
$486$	0	0
$487$	38.6155	1.74984	0.874918	−	0.484271i	$-0.160915\pi$
$487$	38.6155	1.74984	0.874918	+	0.484271i	$0.160915\pi$
$488$	0	0
$489$	19.6847	0.890171
$490$	0	0
$491$	−33.4233	−1.50837	−0.754186	−	0.656661i	$-0.771968\pi$
$491$	−33.4233	−1.50837	−0.754186	+	0.656661i	$0.771968\pi$
$492$	0	0
$493$	−13.7538	−0.619439
$494$	0	0
$495$	0	0
$496$	0	0
$497$	25.3693	1.13797
$498$	0	0
$499$	31.0540	1.39017	0.695083	−	0.718929i	$-0.255367\pi$
$499$	31.0540	1.39017	0.695083	+	0.718929i	$0.255367\pi$
$500$	0	0
$501$	7.12311	0.318237
$502$	0	0
$503$	4.63068	0.206472	0.103236	−	0.994657i	$-0.467080\pi$
$503$	4.63068	0.206472	0.103236	+	0.994657i	$0.467080\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−8.49242	−0.377162
$508$	0	0
$509$	−4.87689	−0.216165	−0.108082	−	0.994142i	$-0.534471\pi$
$509$	−4.87689	−0.216165	−0.108082	+	0.994142i	$0.534471\pi$
$510$	0	0
$511$	37.6155	1.66401
$512$	0	0
$513$	3.00000	0.132453
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−4.00000	−0.175920
$518$	0	0
$519$	22.9309	1.00655
$520$	0	0
$521$	30.3002	1.32748	0.663738	−	0.747965i	$-0.268969\pi$
$521$	30.3002	1.32748	0.663738	+	0.747965i	$0.268969\pi$
$522$	0	0
$523$	−14.1231	−0.617560	−0.308780	−	0.951133i	$-0.599921\pi$
$523$	−14.1231	−0.617560	−0.308780	+	0.951133i	$0.599921\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	15.3693	0.669498
$528$	0	0
$529$	21.6847	0.942811
$530$	0	0
$531$	13.3693	0.580179
$532$	0	0
$533$	12.7386	0.551771
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−17.6155	−0.760166
$538$	0	0
$539$	13.8078	0.594743
$540$	0	0
$541$	32.8617	1.41284	0.706418	−	0.707795i	$-0.250310\pi$
$541$	32.8617	1.41284	0.706418	+	0.707795i	$0.250310\pi$
$542$	0	0
$543$	12.8078	0.549634
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−31.4233	−1.34356	−0.671781	−	0.740749i	$-0.734471\pi$
$547$	−31.4233	−1.34356	−0.671781	+	0.740749i	$0.734471\pi$
$548$	0	0
$549$	5.68466	0.242615
$550$	0	0
$551$	−8.05398	−0.343111
$552$	0	0
$553$	−10.2462	−0.435713
$554$	0	0
$555$	0	0
$556$	0	0
$557$	18.0540	0.764971	0.382486	−	0.923961i	$-0.375068\pi$
$557$	18.0540	0.764971	0.382486	+	0.923961i	$0.375068\pi$
$558$	0	0
$559$	−13.0000	−0.549841
$560$	0	0
$561$	5.12311	0.216298
$562$	0	0
$563$	19.3693	0.816319	0.408160	−	0.912911i	$-0.366171\pi$
$563$	19.3693	0.816319	0.408160	+	0.912911i	$0.366171\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	4.56155	0.191567
$568$	0	0
$569$	−22.0000	−0.922288	−0.461144	−	0.887325i	$-0.652561\pi$
$569$	−22.0000	−0.922288	−0.461144	+	0.887325i	$0.652561\pi$
$570$	0	0
$571$	19.0000	0.795125	0.397563	−	0.917575i	$-0.369856\pi$
$571$	19.0000	0.795125	0.397563	+	0.917575i	$0.369856\pi$
$572$	0	0
$573$	−0.192236	−0.00803077
$574$	0	0
$575$	0	0
$576$	0	0
$577$	46.1771	1.92238	0.961189	−	0.275892i	$-0.0889734\pi$
$577$	46.1771	1.92238	0.961189	+	0.275892i	$0.0889734\pi$
$578$	0	0
$579$	−5.68466	−0.236246
$580$	0	0
$581$	12.2462	0.508058
$582$	0	0
$583$	1.12311	0.0465143
$584$	0	0
$585$	0	0
$586$	0	0
$587$	31.3693	1.29475	0.647375	−	0.762172i	$-0.275867\pi$
$587$	31.3693	1.29475	0.647375	+	0.762172i	$0.275867\pi$
$588$	0	0
$589$	9.00000	0.370839
$590$	0	0
$591$	−18.9309	−0.778712
$592$	0	0
$593$	−10.8769	−0.446661	−0.223330	−	0.974743i	$-0.571693\pi$
$593$	−10.8769	−0.446661	−0.223330	+	0.974743i	$0.571693\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	2.75379	0.112705
$598$	0	0
$599$	−38.7386	−1.58282	−0.791409	−	0.611287i	$-0.790652\pi$
$599$	−38.7386	−1.58282	−0.791409	+	0.611287i	$0.790652\pi$
$600$	0	0
$601$	41.7926	1.70476	0.852378	−	0.522926i	$-0.175160\pi$
$601$	41.7926	1.70476	0.852378	+	0.522926i	$0.175160\pi$
$602$	0	0
$603$	−8.56155	−0.348653
$604$	0	0
$605$	0	0
$606$	0	0
$607$	0.384472	0.0156052	0.00780262	−	0.999970i	$-0.497516\pi$
$607$	0.384472	0.0156052	0.00780262	+	0.999970i	$0.497516\pi$
$608$	0	0
$609$	−12.2462	−0.496242
$610$	0	0
$611$	8.49242	0.343567
$612$	0	0
$613$	−22.9848	−0.928349	−0.464175	−	0.885744i	$-0.653649\pi$
$613$	−22.9848	−0.928349	−0.464175	+	0.885744i	$0.653649\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	13.1771	0.530489	0.265245	−	0.964181i	$-0.414547\pi$
$617$	13.1771	0.530489	0.265245	+	0.964181i	$0.414547\pi$
$618$	0	0
$619$	25.6847	1.03235	0.516177	−	0.856482i	$-0.327355\pi$
$619$	25.6847	1.03235	0.516177	+	0.856482i	$0.327355\pi$
$620$	0	0
$621$	−6.68466	−0.268246
$622$	0	0
$623$	34.4924	1.38191
$624$	0	0
$625$	0	0
$626$	0	0
$627$	3.00000	0.119808
$628$	0	0
$629$	30.7386	1.22563
$630$	0	0
$631$	−47.5464	−1.89279	−0.946396	−	0.323008i	$-0.895306\pi$
$631$	−47.5464	−1.89279	−0.946396	+	0.323008i	$0.895306\pi$
$632$	0	0
$633$	15.6847	0.623409
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−29.3153	−1.16152
$638$	0	0
$639$	5.56155	0.220012
$640$	0	0
$641$	−39.5616	−1.56259	−0.781294	−	0.624164i	$-0.785440\pi$
$641$	−39.5616	−1.56259	−0.781294	+	0.624164i	$0.785440\pi$
$642$	0	0
$643$	−39.6155	−1.56228	−0.781142	−	0.624353i	$-0.785363\pi$
$643$	−39.6155	−1.56228	−0.781142	+	0.624353i	$0.785363\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−32.0000	−1.25805	−0.629025	−	0.777385i	$-0.716546\pi$
$647$	−32.0000	−1.25805	−0.629025	+	0.777385i	$0.716546\pi$
$648$	0	0
$649$	13.3693	0.524792
$650$	0	0
$651$	13.6847	0.536344
$652$	0	0
$653$	−13.3693	−0.523182	−0.261591	−	0.965179i	$-0.584247\pi$
$653$	−13.3693	−0.523182	−0.261591	+	0.965179i	$0.584247\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	8.24621	0.321715
$658$	0	0
$659$	−4.68466	−0.182488	−0.0912442	−	0.995829i	$-0.529084\pi$
$659$	−4.68466	−0.182488	−0.0912442	+	0.995829i	$0.529084\pi$
$660$	0	0
$661$	−7.36932	−0.286633	−0.143317	−	0.989677i	$-0.545777\pi$
$661$	−7.36932	−0.286633	−0.143317	+	0.989677i	$0.545777\pi$
$662$	0	0
$663$	−10.8769	−0.422424
$664$	0	0
$665$	0	0
$666$	0	0
$667$	17.9460	0.694873
$668$	0	0
$669$	19.0540	0.736669
$670$	0	0
$671$	5.68466	0.219454
$672$	0	0
$673$	−3.36932	−0.129878	−0.0649388	−	0.997889i	$-0.520685\pi$
$673$	−3.36932	−0.129878	−0.0649388	+	0.997889i	$0.520685\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	25.8078	0.991873	0.495936	−	0.868359i	$-0.334825\pi$
$677$	25.8078	0.991873	0.495936	+	0.868359i	$0.334825\pi$
$678$	0	0
$679$	17.6847	0.678675
$680$	0	0
$681$	−0.192236	−0.00736650
$682$	0	0
$683$	−10.4924	−0.401481	−0.200741	−	0.979644i	$-0.564335\pi$
$683$	−10.4924	−0.401481	−0.200741	+	0.979644i	$0.564335\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−21.4384	−0.817928
$688$	0	0
$689$	−2.38447	−0.0908411
$690$	0	0
$691$	48.4924	1.84474	0.922369	−	0.386309i	$-0.126250\pi$
$691$	48.4924	1.84474	0.922369	+	0.386309i	$0.126250\pi$
$692$	0	0
$693$	4.56155	0.173279
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−30.7386	−1.16431
$698$	0	0
$699$	−12.4924	−0.472507
$700$	0	0
$701$	−28.6847	−1.08340	−0.541702	−	0.840570i	$-0.682220\pi$
$701$	−28.6847	−1.08340	−0.541702	+	0.840570i	$0.682220\pi$
$702$	0	0
$703$	18.0000	0.678883
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−72.1080	−2.71190
$708$	0	0
$709$	−33.5464	−1.25986	−0.629931	−	0.776651i	$-0.716917\pi$
$709$	−33.5464	−1.25986	−0.629931	+	0.776651i	$0.716917\pi$
$710$	0	0
$711$	−2.24621	−0.0842395
$712$	0	0
$713$	−20.0540	−0.751027
$714$	0	0
$715$	0	0
$716$	0	0
$717$	10.8769	0.406205
$718$	0	0
$719$	−37.6155	−1.40282	−0.701411	−	0.712757i	$-0.747446\pi$
$719$	−37.6155	−1.40282	−0.701411	+	0.712757i	$0.747446\pi$
$720$	0	0
$721$	−82.3542	−3.06703
$722$	0	0
$723$	6.31534	0.234870
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−16.7538	−0.621364	−0.310682	−	0.950514i	$-0.600557\pi$
$727$	−16.7538	−0.621364	−0.310682	+	0.950514i	$0.600557\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	31.3693	1.16024
$732$	0	0
$733$	23.1771	0.856065	0.428033	−	0.903763i	$-0.359207\pi$
$733$	23.1771	0.856065	0.428033	+	0.903763i	$0.359207\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−8.56155	−0.315369
$738$	0	0
$739$	33.7538	1.24165	0.620827	−	0.783948i	$-0.286797\pi$
$739$	33.7538	1.24165	0.620827	+	0.783948i	$0.286797\pi$
$740$	0	0
$741$	−6.36932	−0.233983
$742$	0	0
$743$	50.1080	1.83828	0.919141	−	0.393928i	$-0.128884\pi$
$743$	50.1080	1.83828	0.919141	+	0.393928i	$0.128884\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	2.68466	0.0982265
$748$	0	0
$749$	−30.4924	−1.11417
$750$	0	0
$751$	−3.80776	−0.138947	−0.0694736	−	0.997584i	$-0.522132\pi$
$751$	−3.80776	−0.138947	−0.0694736	+	0.997584i	$0.522132\pi$
$752$	0	0
$753$	−19.3693	−0.705857
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−22.5616	−0.820014	−0.410007	−	0.912082i	$-0.634474\pi$
$757$	−22.5616	−0.820014	−0.410007	+	0.912082i	$0.634474\pi$
$758$	0	0
$759$	−6.68466	−0.242638
$760$	0	0
$761$	−53.6155	−1.94356	−0.971781	−	0.235886i	$-0.924201\pi$
$761$	−53.6155	−1.94356	−0.971781	+	0.235886i	$0.924201\pi$
$762$	0	0
$763$	−8.56155	−0.309949
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−28.3845	−1.02490
$768$	0	0
$769$	−52.1771	−1.88155	−0.940777	−	0.339026i	$-0.889902\pi$
$769$	−52.1771	−1.88155	−0.940777	+	0.339026i	$0.889902\pi$
$770$	0	0
$771$	−0.684658	−0.0246574
$772$	0	0
$773$	0.246211	0.00885560	0.00442780	−	0.999990i	$-0.498591\pi$
$773$	0.246211	0.00885560	0.00442780	+	0.999990i	$0.498591\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	27.3693	0.981869
$778$	0	0
$779$	−18.0000	−0.644917
$780$	0	0
$781$	5.56155	0.199008
$782$	0	0
$783$	−2.68466	−0.0959419
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−54.4233	−1.93998	−0.969990	−	0.243143i	$-0.921822\pi$
$787$	−54.4233	−1.93998	−0.969990	+	0.243143i	$0.921822\pi$
$788$	0	0
$789$	25.1231	0.894406
$790$	0	0
$791$	−1.12311	−0.0399330
$792$	0	0
$793$	−12.0691	−0.428587
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−20.8769	−0.739498	−0.369749	−	0.929132i	$-0.620556\pi$
$797$	−20.8769	−0.739498	−0.369749	+	0.929132i	$0.620556\pi$
$798$	0	0
$799$	−20.4924	−0.724970
$800$	0	0
$801$	7.56155	0.267174
$802$	0	0
$803$	8.24621	0.291002
$804$	0	0
$805$	0	0
$806$	0	0
$807$	9.12311	0.321149
$808$	0	0
$809$	6.63068	0.233122	0.116561	−	0.993184i	$-0.462813\pi$
$809$	6.63068	0.233122	0.116561	+	0.993184i	$0.462813\pi$
$810$	0	0
$811$	9.93087	0.348720	0.174360	−	0.984682i	$-0.444214\pi$
$811$	9.93087	0.348720	0.174360	+	0.984682i	$0.444214\pi$
$812$	0	0
$813$	−22.2462	−0.780209
$814$	0	0
$815$	0	0
$816$	0	0
$817$	18.3693	0.642661
$818$	0	0
$819$	−9.68466	−0.338409
$820$	0	0
$821$	2.93087	0.102288	0.0511440	−	0.998691i	$-0.483713\pi$
$821$	2.93087	0.102288	0.0511440	+	0.998691i	$0.483713\pi$
$822$	0	0
$823$	−40.1771	−1.40049	−0.700243	−	0.713905i	$-0.746925\pi$
$823$	−40.1771	−1.40049	−0.700243	+	0.713905i	$0.746925\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	45.4233	1.57952	0.789761	−	0.613414i	$-0.210204\pi$
$827$	45.4233	1.57952	0.789761	+	0.613414i	$0.210204\pi$
$828$	0	0
$829$	24.7386	0.859208	0.429604	−	0.903017i	$-0.358653\pi$
$829$	24.7386	0.859208	0.429604	+	0.903017i	$0.358653\pi$
$830$	0	0
$831$	23.4384	0.813071
$832$	0	0
$833$	70.7386	2.45095
$834$	0	0
$835$	0	0
$836$	0	0
$837$	3.00000	0.103695
$838$	0	0
$839$	19.3693	0.668703	0.334352	−	0.942448i	$-0.391483\pi$
$839$	19.3693	0.668703	0.334352	+	0.942448i	$0.391483\pi$
$840$	0	0
$841$	−21.7926	−0.751469
$842$	0	0
$843$	−28.0000	−0.964371
$844$	0	0
$845$	0	0
$846$	0	0
$847$	4.56155	0.156737
$848$	0	0
$849$	20.8078	0.714121
$850$	0	0
$851$	−40.1080	−1.37488
$852$	0	0
$853$	−32.4233	−1.11015	−0.555076	−	0.831800i	$-0.687311\pi$
$853$	−32.4233	−1.11015	−0.555076	+	0.831800i	$0.687311\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−3.75379	−0.128227	−0.0641135	−	0.997943i	$-0.520422\pi$
$857$	−3.75379	−0.128227	−0.0641135	+	0.997943i	$0.520422\pi$
$858$	0	0
$859$	−27.6155	−0.942230	−0.471115	−	0.882072i	$-0.656148\pi$
$859$	−27.6155	−0.942230	−0.471115	+	0.882072i	$0.656148\pi$
$860$	0	0
$861$	−27.3693	−0.932744
$862$	0	0
$863$	43.3153	1.47447	0.737236	−	0.675636i	$-0.236131\pi$
$863$	43.3153	1.47447	0.737236	+	0.675636i	$0.236131\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	9.24621	0.314018
$868$	0	0
$869$	−2.24621	−0.0761975
$870$	0	0
$871$	18.1771	0.615907
$872$	0	0
$873$	3.87689	0.131213
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−33.1080	−1.11798	−0.558988	−	0.829176i	$-0.688810\pi$
$877$	−33.1080	−1.11798	−0.558988	+	0.829176i	$0.688810\pi$
$878$	0	0
$879$	−20.7386	−0.699497
$880$	0	0
$881$	−50.7926	−1.71125	−0.855623	−	0.517599i	$-0.826826\pi$
$881$	−50.7926	−1.71125	−0.855623	+	0.517599i	$0.826826\pi$
$882$	0	0
$883$	3.05398	0.102774	0.0513872	−	0.998679i	$-0.483636\pi$
$883$	3.05398	0.102774	0.0513872	+	0.998679i	$0.483636\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−48.7386	−1.63648	−0.818242	−	0.574875i	$-0.805051\pi$
$887$	−48.7386	−1.63648	−0.818242	+	0.574875i	$0.805051\pi$
$888$	0	0
$889$	−28.4924	−0.955605
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	−12.0000	−0.401565
$894$	0	0
$895$	0	0
$896$	0	0
$897$	14.1922	0.473865
$898$	0	0
$899$	−8.05398	−0.268615
$900$	0	0
$901$	5.75379	0.191686
$902$	0	0
$903$	27.9309	0.929481
$904$	0	0
$905$	0	0
$906$	0	0
$907$	3.61553	0.120052	0.0600258	−	0.998197i	$-0.480882\pi$
$907$	3.61553	0.120052	0.0600258	+	0.998197i	$0.480882\pi$
$908$	0	0
$909$	−15.8078	−0.524310
$910$	0	0
$911$	−21.1231	−0.699840	−0.349920	−	0.936780i	$-0.613791\pi$
$911$	−21.1231	−0.699840	−0.349920	+	0.936780i	$0.613791\pi$
$912$	0	0
$913$	2.68466	0.0888492
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−20.2462	−0.668589
$918$	0	0
$919$	−15.0540	−0.496585	−0.248292	−	0.968685i	$-0.579869\pi$
$919$	−15.0540	−0.496585	−0.248292	+	0.968685i	$0.579869\pi$
$920$	0	0
$921$	20.8078	0.685639
$922$	0	0
$923$	−11.8078	−0.388657
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−18.0540	−0.592970
$928$	0	0
$929$	21.1771	0.694797	0.347399	−	0.937718i	$-0.387065\pi$
$929$	21.1771	0.694797	0.347399	+	0.937718i	$0.387065\pi$
$930$	0	0
$931$	41.4233	1.35759
$932$	0	0
$933$	−31.5616	−1.03328
$934$	0	0
$935$	0	0
$936$	0	0
$937$	53.0540	1.73320	0.866599	−	0.499005i	$-0.166301\pi$
$937$	53.0540	1.73320	0.866599	+	0.499005i	$0.166301\pi$
$938$	0	0
$939$	−32.5616	−1.06261
$940$	0	0
$941$	−1.86174	−0.0606910	−0.0303455	−	0.999539i	$-0.509661\pi$
$941$	−1.86174	−0.0606910	−0.0303455	+	0.999539i	$0.509661\pi$
$942$	0	0
$943$	40.1080	1.30609
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−48.2462	−1.56779	−0.783896	−	0.620893i	$-0.786770\pi$
$947$	−48.2462	−1.56779	−0.783896	+	0.620893i	$0.786770\pi$
$948$	0	0
$949$	−17.5076	−0.568320
$950$	0	0
$951$	19.3693	0.628093
$952$	0	0
$953$	−1.50758	−0.0488352	−0.0244176	−	0.999702i	$-0.507773\pi$
$953$	−1.50758	−0.0488352	−0.0244176	+	0.999702i	$0.507773\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−2.68466	−0.0867827
$958$	0	0
$959$	43.6155	1.40842
$960$	0	0
$961$	−22.0000	−0.709677
$962$	0	0
$963$	−6.68466	−0.215410
$964$	0	0
$965$	0	0
$966$	0	0
$967$	20.9848	0.674827	0.337414	−	0.941357i	$-0.390448\pi$
$967$	20.9848	0.674827	0.337414	+	0.941357i	$0.390448\pi$
$968$	0	0
$969$	15.3693	0.493734
$970$	0	0
$971$	3.86174	0.123929	0.0619646	−	0.998078i	$-0.480263\pi$
$971$	3.86174	0.123929	0.0619646	+	0.998078i	$0.480263\pi$
$972$	0	0
$973$	0.876894	0.0281119
$974$	0	0
$975$	0	0
$976$	0	0
$977$	6.49242	0.207711	0.103855	−	0.994592i	$-0.466882\pi$
$977$	6.49242	0.207711	0.103855	+	0.994592i	$0.466882\pi$
$978$	0	0
$979$	7.56155	0.241668
$980$	0	0
$981$	−1.87689	−0.0599246
$982$	0	0
$983$	38.9309	1.24170	0.620851	−	0.783929i	$-0.286787\pi$
$983$	38.9309	1.24170	0.620851	+	0.783929i	$0.286787\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−18.2462	−0.580783
$988$	0	0
$989$	−40.9309	−1.30153
$990$	0	0
$991$	18.4233	0.585235	0.292618	−	0.956230i	$-0.405474\pi$
$991$	18.4233	0.585235	0.292618	+	0.956230i	$0.405474\pi$
$992$	0	0
$993$	8.87689	0.281700
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−7.75379	−0.245565	−0.122782	−	0.992434i	$-0.539182\pi$
$997$	−7.75379	−0.245565	−0.122782	+	0.992434i	$0.539182\pi$
$998$	0	0
$999$	6.00000	0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6600.2.a.bm.1.2	yes	2
5.2	odd	4		6600.2.d.ba.1849.2		4
5.3	odd	4		6600.2.d.ba.1849.3		4
5.4	even	2		6600.2.a.bg.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6600.2.a.bg.1.1	✓	2	5.4	even	2
6600.2.a.bm.1.2	yes	2	1.1	even	1	trivial
6600.2.d.ba.1849.2		4	5.2	odd	4
6600.2.d.ba.1849.3		4	5.3	odd	4

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:	\( N \)	\(=\)	\( 6600 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6600.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +4.56155 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +4.56155 q^{7} +1.00000 q^{9} +1.00000 q^{11} -2.12311 q^{13} +5.12311 q^{17} +3.00000 q^{19} +4.56155 q^{21} -6.68466 q^{23} +1.00000 q^{27} -2.68466 q^{29} +3.00000 q^{31} +1.00000 q^{33} +6.00000 q^{37} -2.12311 q^{39} -6.00000 q^{41} +6.12311 q^{43} -4.00000 q^{47} +13.8078 q^{49} +5.12311 q^{51} +1.12311 q^{53} +3.00000 q^{57} +13.3693 q^{59} +5.68466 q^{61} +4.56155 q^{63} -8.56155 q^{67} -6.68466 q^{69} +5.56155 q^{71} +8.24621 q^{73} +4.56155 q^{77} -2.24621 q^{79} +1.00000 q^{81} +2.68466 q^{83} -2.68466 q^{87} +7.56155 q^{89} -9.68466 q^{91} +3.00000 q^{93} +3.87689 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 5 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 5 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} + 5 q^{7} + 2 q^{9} + 2 q^{11} + 4 q^{13} + 2 q^{17} + 6 q^{19} + 5 q^{21} - q^{23} + 2 q^{27} + 7 q^{29} + 6 q^{31} + 2 q^{33} + 12 q^{37} + 4 q^{39} - 12 q^{41} + 4 q^{43} - 8 q^{47} + 7 q^{49} + 2 q^{51} - 6 q^{53} + 6 q^{57} + 2 q^{59} - q^{61} + 5 q^{63} - 13 q^{67} - q^{69} + 7 q^{71} + 5 q^{77} + 12 q^{79} + 2 q^{81} - 7 q^{83} + 7 q^{87} + 11 q^{89} - 7 q^{91} + 6 q^{93} + 16 q^{97} + 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 5 * q^7 + 2 * q^9 + 2 * q^11 + 4 * q^13 + 2 * q^17 + 6 * q^19 + 5 * q^21 - q^23 + 2 * q^27 + 7 * q^29 + 6 * q^31 + 2 * q^33 + 12 * q^37 + 4 * q^39 - 12 * q^41 + 4 * q^43 - 8 * q^47 + 7 * q^49 + 2 * q^51 - 6 * q^53 + 6 * q^57 + 2 * q^59 - q^61 + 5 * q^63 - 13 * q^67 - q^69 + 7 * q^71 + 5 * q^77 + 12 * q^79 + 2 * q^81 - 7 * q^83 + 7 * q^87 + 11 * q^89 - 7 * q^91 + 6 * q^93 + 16 * q^97 + 2 * q^99

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