LMFDB - Embedded newform 5445.2.a.bt.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5445,2,Mod(1,5445)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5445, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("5445.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5445 = 3^{2} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5445.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:	$43.4785439006$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.725.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 3x^{2} + x + 1 $ x^4 - x^3 - 3*x^2 + x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.35567$ of defining polynomial
Character	$\chi$	$=$	5445.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.35567 q^{2} +3.54920 q^{4} -1.00000 q^{5} +0.193527 q^{7} +3.64941 q^{8} +O(q^{10})$	$q+2.35567 q^{2} +3.54920 q^{4} -1.00000 q^{5} +0.193527 q^{7} +3.64941 q^{8} -2.35567 q^{10} +0.973708 q^{13} +0.455887 q^{14} +1.49843 q^{16} +2.67571 q^{17} -5.54920 q^{19} -3.54920 q^{20} +4.80040 q^{23} +1.00000 q^{25} +2.29374 q^{26} +0.686867 q^{28} +10.1178 q^{29} -2.50533 q^{31} -3.76902 q^{32} +6.30309 q^{34} -0.193527 q^{35} +5.71217 q^{37} -13.0721 q^{38} -3.64941 q^{40} +8.27861 q^{41} +5.11353 q^{43} +11.3082 q^{46} +10.8523 q^{47} -6.96255 q^{49} +2.35567 q^{50} +3.45589 q^{52} +9.28879 q^{53} +0.706260 q^{56} +23.8342 q^{58} +11.2180 q^{59} +1.20602 q^{61} -5.90173 q^{62} -11.8754 q^{64} -0.973708 q^{65} +3.25922 q^{67} +9.49662 q^{68} -0.455887 q^{70} -5.99078 q^{71} +3.30624 q^{73} +13.4560 q^{74} -19.6952 q^{76} +10.1529 q^{79} -1.49843 q^{80} +19.5017 q^{82} -8.31508 q^{83} -2.67571 q^{85} +12.0458 q^{86} -7.34270 q^{89} +0.188439 q^{91} +17.0376 q^{92} +25.5645 q^{94} +5.54920 q^{95} -15.8523 q^{97} -16.4015 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 3 q^{2} + q^{4} - 4 q^{5} - 6 q^{7} + 3 q^{8}+O(q^{10}) $ 4 * q + 3 * q^2 + q^4 - 4 * q^5 - 6 * q^7 + 3 * q^8	$ 4 q + 3 q^{2} + q^{4} - 4 q^{5} - 6 q^{7} + 3 q^{8} - 3 q^{10} - 7 q^{13} - 3 q^{14} - q^{16} + 10 q^{17} - 9 q^{19} - q^{20} + 3 q^{23} + 4 q^{25} + 4 q^{26} + 7 q^{28} + 15 q^{29} - 13 q^{31} - 6 q^{32} - 3 q^{34} + 6 q^{35} - 3 q^{37} - 15 q^{38} - 3 q^{40} + 22 q^{41} - q^{46} + 2 q^{47} - 12 q^{49} + 3 q^{50} + 9 q^{52} - 10 q^{53} + 8 q^{56} + 39 q^{58} + 21 q^{59} - 11 q^{61} + 10 q^{62} - 3 q^{64} + 7 q^{65} + q^{67} + 3 q^{68} + 3 q^{70} + 13 q^{71} - q^{73} - 11 q^{74} - 19 q^{76} + 4 q^{79} + q^{80} + 25 q^{82} + 3 q^{83} - 10 q^{85} + 10 q^{89} + 12 q^{91} + 24 q^{92} + 35 q^{94} + 9 q^{95} - 22 q^{97} - 11 q^{98}+O(q^{100}) $ 4 * q + 3 * q^2 + q^4 - 4 * q^5 - 6 * q^7 + 3 * q^8 - 3 * q^10 - 7 * q^13 - 3 * q^14 - q^16 + 10 * q^17 - 9 * q^19 - q^20 + 3 * q^23 + 4 * q^25 + 4 * q^26 + 7 * q^28 + 15 * q^29 - 13 * q^31 - 6 * q^32 - 3 * q^34 + 6 * q^35 - 3 * q^37 - 15 * q^38 - 3 * q^40 + 22 * q^41 - q^46 + 2 * q^47 - 12 * q^49 + 3 * q^50 + 9 * q^52 - 10 * q^53 + 8 * q^56 + 39 * q^58 + 21 * q^59 - 11 * q^61 + 10 * q^62 - 3 * q^64 + 7 * q^65 + q^67 + 3 * q^68 + 3 * q^70 + 13 * q^71 - q^73 - 11 * q^74 - 19 * q^76 + 4 * q^79 + q^80 + 25 * q^82 + 3 * q^83 - 10 * q^85 + 10 * q^89 + 12 * q^91 + 24 * q^92 + 35 * q^94 + 9 * q^95 - 22 * q^97 - 11 * 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 $

$2$	2.35567	1.66571	0.832857	−	0.553489i	$-0.186704\pi$
$2$	2.35567	1.66571	0.832857	+	0.553489i	$0.186704\pi$
$3$	0	0
$3$	0	0
$4$	3.54920	1.77460
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0.193527	0.0731464	0.0365732	−	0.999331i	$-0.488356\pi$
$7$	0.193527	0.0731464	0.0365732	+	0.999331i	$0.488356\pi$
$8$	3.64941	1.29026
$9$	0	0
$10$	−2.35567	−0.744930
$11$	0	0
$11$	0	0
$12$	0	0
$13$	0.973708	0.270058	0.135029	−	0.990842i	$-0.456887\pi$
$13$	0.973708	0.270058	0.135029	+	0.990842i	$0.456887\pi$
$14$	0.455887	0.121841
$15$	0	0
$16$	1.49843	0.374607
$17$	2.67571	0.648954	0.324477	−	0.945894i	$-0.394812\pi$
$17$	2.67571	0.648954	0.324477	+	0.945894i	$0.394812\pi$
$18$	0	0
$19$	−5.54920	−1.27307	−0.636537	−	0.771246i	$-0.719634\pi$
$19$	−5.54920	−1.27307	−0.636537	+	0.771246i	$0.719634\pi$
$20$	−3.54920	−0.793626
$21$	0	0
$22$	0	0
$23$	4.80040	1.00095	0.500476	−	0.865750i	$-0.333158\pi$
$23$	4.80040	1.00095	0.500476	+	0.865750i	$0.333158\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	2.29374	0.449839
$27$	0	0
$28$	0.686867	0.129806
$29$	10.1178	1.87883	0.939414	−	0.342785i	$-0.111370\pi$
$29$	10.1178	1.87883	0.939414	+	0.342785i	$0.111370\pi$
$30$	0	0
$31$	−2.50533	−0.449970	−0.224985	−	0.974362i	$-0.572233\pi$
$31$	−2.50533	−0.449970	−0.224985	+	0.974362i	$0.572233\pi$
$32$	−3.76902	−0.666275
$33$	0	0
$34$	6.30309	1.08097
$35$	−0.193527	−0.0327120
$36$	0	0
$37$	5.71217	0.939076	0.469538	−	0.882912i	$-0.344421\pi$
$37$	5.71217	0.939076	0.469538	+	0.882912i	$0.344421\pi$
$38$	−13.0721	−2.12058
$39$	0	0
$40$	−3.64941	−0.577023
$41$	8.27861	1.29290	0.646451	−	0.762956i	$-0.276253\pi$
$41$	8.27861	1.29290	0.646451	+	0.762956i	$0.276253\pi$
$42$	0	0
$43$	5.11353	0.779807	0.389903	−	0.920856i	$-0.372508\pi$
$43$	5.11353	0.779807	0.389903	+	0.920856i	$0.372508\pi$
$44$	0	0
$45$	0	0
$46$	11.3082	1.66730
$47$	10.8523	1.58297	0.791485	−	0.611189i	$-0.209308\pi$
$47$	10.8523	1.58297	0.791485	+	0.611189i	$0.209308\pi$
$48$	0	0
$49$	−6.96255	−0.994650
$50$	2.35567	0.333143
$51$	0	0
$52$	3.45589	0.479245
$53$	9.28879	1.27591	0.637956	−	0.770072i	$-0.279780\pi$
$53$	9.28879	1.27591	0.637956	+	0.770072i	$0.279780\pi$
$54$	0	0
$55$	0	0
$56$	0.706260	0.0943780
$57$	0	0
$58$	23.8342	3.12959
$59$	11.2180	1.46046	0.730230	−	0.683201i	$-0.239413\pi$
$59$	11.2180	1.46046	0.730230	+	0.683201i	$0.239413\pi$
$60$	0	0
$61$	1.20602	0.154415	0.0772077	−	0.997015i	$-0.475400\pi$
$61$	1.20602	0.154415	0.0772077	+	0.997015i	$0.475400\pi$
$62$	−5.90173	−0.749521
$63$	0	0
$64$	−11.8754	−1.48443
$65$	−0.973708	−0.120774
$66$	0	0
$67$	3.25922	0.398176	0.199088	−	0.979982i	$-0.436202\pi$
$67$	3.25922	0.398176	0.199088	+	0.979982i	$0.436202\pi$
$68$	9.49662	1.15163
$69$	0	0
$70$	−0.455887	−0.0544889
$71$	−5.99078	−0.710975	−0.355488	−	0.934681i	$-0.615685\pi$
$71$	−5.99078	−0.710975	−0.355488	+	0.934681i	$0.615685\pi$
$72$	0	0
$73$	3.30624	0.386966	0.193483	−	0.981104i	$-0.438022\pi$
$73$	3.30624	0.386966	0.193483	+	0.981104i	$0.438022\pi$
$74$	13.4560	1.56423
$75$	0	0
$76$	−19.6952	−2.25920
$77$	0	0
$78$	0	0
$79$	10.1529	1.14229	0.571147	−	0.820848i	$-0.306499\pi$
$79$	10.1529	1.14229	0.571147	+	0.820848i	$0.306499\pi$
$80$	−1.49843	−0.167529
$81$	0	0
$82$	19.5017	2.15360
$83$	−8.31508	−0.912698	−0.456349	−	0.889801i	$-0.650843\pi$
$83$	−8.31508	−0.912698	−0.456349	+	0.889801i	$0.650843\pi$
$84$	0	0
$85$	−2.67571	−0.290221
$86$	12.0458	1.29893
$87$	0	0
$88$	0	0
$89$	−7.34270	−0.778325	−0.389163	−	0.921169i	$-0.627236\pi$
$89$	−7.34270	−0.778325	−0.389163	+	0.921169i	$0.627236\pi$
$90$	0	0
$91$	0.188439	0.0197538
$92$	17.0376	1.77629
$93$	0	0
$94$	25.5645	2.63677
$95$	5.54920	0.569336
$96$	0	0
$97$	−15.8523	−1.60956	−0.804778	−	0.593576i	$-0.797716\pi$
$97$	−15.8523	−1.60956	−0.804778	+	0.593576i	$0.797716\pi$
$98$	−16.4015	−1.65680
$99$	0	0
$100$	3.54920	0.354920
$101$	−13.1591	−1.30938	−0.654692	−	0.755896i	$-0.727201\pi$
$101$	−13.1591	−1.30938	−0.654692	+	0.755896i	$0.727201\pi$
$102$	0	0
$103$	3.99737	0.393872	0.196936	−	0.980416i	$-0.436901\pi$
$103$	3.99737	0.393872	0.196936	+	0.980416i	$0.436901\pi$
$104$	3.55346	0.348446
$105$	0	0
$106$	21.8814	2.12530
$107$	−4.88682	−0.472426	−0.236213	−	0.971701i	$-0.575906\pi$
$107$	−4.88682	−0.472426	−0.236213	+	0.971701i	$0.575906\pi$
$108$	0	0
$109$	−7.51977	−0.720263	−0.360131	−	0.932902i	$-0.617268\pi$
$109$	−7.51977	−0.720263	−0.360131	+	0.932902i	$0.617268\pi$
$110$	0	0
$111$	0	0
$112$	0.289986	0.0274011
$113$	−20.2237	−1.90249	−0.951243	−	0.308442i	$-0.900192\pi$
$113$	−20.2237	−1.90249	−0.951243	+	0.308442i	$0.900192\pi$
$114$	0	0
$115$	−4.80040	−0.447640
$116$	35.9101	3.33417
$117$	0	0
$118$	26.4260	2.43271
$119$	0.517822	0.0474686
$120$	0	0
$121$	0	0
$122$	2.84100	0.257212
$123$	0	0
$124$	−8.89190	−0.798517
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	7.10021	0.630042	0.315021	−	0.949085i	$-0.397988\pi$
$127$	7.10021	0.630042	0.315021	+	0.949085i	$0.397988\pi$
$128$	−20.4366	−1.80636
$129$	0	0
$130$	−2.29374	−0.201174
$131$	2.50024	0.218447	0.109223	−	0.994017i	$-0.465164\pi$
$131$	2.50024	0.218447	0.109223	+	0.994017i	$0.465164\pi$
$132$	0	0
$133$	−1.07392	−0.0931207
$134$	7.67765	0.663248
$135$	0	0
$136$	9.76476	0.837321
$137$	15.2944	1.30669	0.653344	−	0.757061i	$-0.273366\pi$
$137$	15.2944	1.30669	0.653344	+	0.757061i	$0.273366\pi$
$138$	0	0
$139$	−19.6586	−1.66742	−0.833711	−	0.552202i	$-0.813788\pi$
$139$	−19.6586	−1.66742	−0.833711	+	0.552202i	$0.813788\pi$
$140$	−0.686867	−0.0580508
$141$	0	0
$142$	−14.1123	−1.18428
$143$	0	0
$144$	0	0
$145$	−10.1178	−0.840237
$146$	7.78841	0.644574
$147$	0	0
$148$	20.2737	1.66648
$149$	10.1329	0.830122	0.415061	−	0.909794i	$-0.363760\pi$
$149$	10.1329	0.830122	0.415061	+	0.909794i	$0.363760\pi$
$150$	0	0
$151$	11.9632	0.973548	0.486774	−	0.873528i	$-0.338174\pi$
$151$	11.9632	0.973548	0.486774	+	0.873528i	$0.338174\pi$
$152$	−20.2513	−1.64260
$153$	0	0
$154$	0	0
$155$	2.50533	0.201233
$156$	0	0
$157$	−2.62605	−0.209582	−0.104791	−	0.994494i	$-0.533417\pi$
$157$	−2.62605	−0.209582	−0.104791	+	0.994494i	$0.533417\pi$
$158$	23.9170	1.90273
$159$	0	0
$160$	3.76902	0.297967
$161$	0.929007	0.0732160
$162$	0	0
$163$	23.7235	1.85817	0.929083	−	0.369872i	$-0.120598\pi$
$163$	23.7235	1.85817	0.929083	+	0.369872i	$0.120598\pi$
$164$	29.3825	2.29438
$165$	0	0
$166$	−19.5876	−1.52029
$167$	−3.49175	−0.270199	−0.135100	−	0.990832i	$-0.543135\pi$
$167$	−3.49175	−0.270199	−0.135100	+	0.990832i	$0.543135\pi$
$168$	0	0
$169$	−12.0519	−0.927069
$170$	−6.30309	−0.483425
$171$	0	0
$172$	18.1490	1.38385
$173$	20.3596	1.54791	0.773954	−	0.633241i	$-0.218276\pi$
$173$	20.3596	1.54791	0.773954	+	0.633241i	$0.218276\pi$
$174$	0	0
$175$	0.193527	0.0146293
$176$	0	0
$177$	0	0
$178$	−17.2970	−1.29647
$179$	−3.85117	−0.287850	−0.143925	−	0.989589i	$-0.545972\pi$
$179$	−3.85117	−0.287850	−0.143925	+	0.989589i	$0.545972\pi$
$180$	0	0
$181$	−17.6151	−1.30932	−0.654660	−	0.755923i	$-0.727188\pi$
$181$	−17.6151	−1.30932	−0.654660	+	0.755923i	$0.727188\pi$
$182$	0.443901	0.0329041
$183$	0	0
$184$	17.5186	1.29149
$185$	−5.71217	−0.419967
$186$	0	0
$187$	0	0
$188$	38.5170	2.80914
$189$	0	0
$190$	13.0721	0.948351
$191$	−0.417099	−0.0301802	−0.0150901	−	0.999886i	$-0.504804\pi$
$191$	−0.417099	−0.0301802	−0.0150901	+	0.999886i	$0.504804\pi$
$192$	0	0
$193$	11.6965	0.841935	0.420967	−	0.907076i	$-0.361691\pi$
$193$	11.6965	0.841935	0.420967	+	0.907076i	$0.361691\pi$
$194$	−37.3428	−2.68106
$195$	0	0
$196$	−24.7115	−1.76511
$197$	−21.5958	−1.53864	−0.769320	−	0.638863i	$-0.779405\pi$
$197$	−21.5958	−1.53864	−0.769320	+	0.638863i	$0.779405\pi$
$198$	0	0
$199$	7.76028	0.550111	0.275056	−	0.961428i	$-0.411304\pi$
$199$	7.76028	0.550111	0.275056	+	0.961428i	$0.411304\pi$
$200$	3.64941	0.258053
$201$	0	0
$202$	−30.9986	−2.18106
$203$	1.95807	0.137429
$204$	0	0
$205$	−8.27861	−0.578203
$206$	9.41649	0.656078
$207$	0	0
$208$	1.45903	0.101166
$209$	0	0
$210$	0	0
$211$	−12.5057	−0.860926	−0.430463	−	0.902608i	$-0.641650\pi$
$211$	−12.5057	−0.860926	−0.430463	+	0.902608i	$0.641650\pi$
$212$	32.9678	2.26424
$213$	0	0
$214$	−11.5117	−0.786927
$215$	−5.11353	−0.348740
$216$	0	0
$217$	−0.484848	−0.0329137
$218$	−17.7141	−1.19975
$219$	0	0
$220$	0	0
$221$	2.60536	0.175255
$222$	0	0
$223$	−5.37568	−0.359982	−0.179991	−	0.983668i	$-0.557607\pi$
$223$	−5.37568	−0.359982	−0.179991	+	0.983668i	$0.557607\pi$
$224$	−0.729407	−0.0487356
$225$	0	0
$226$	−47.6405	−3.16900
$227$	21.7529	1.44379	0.721895	−	0.692002i	$-0.243271\pi$
$227$	21.7529	1.44379	0.721895	+	0.692002i	$0.243271\pi$
$228$	0	0
$229$	2.67075	0.176488	0.0882441	−	0.996099i	$-0.471874\pi$
$229$	2.67075	0.176488	0.0882441	+	0.996099i	$0.471874\pi$
$230$	−11.3082	−0.745639
$231$	0	0
$232$	36.9240	2.42418
$233$	5.72699	0.375188	0.187594	−	0.982247i	$-0.439931\pi$
$233$	5.72699	0.375188	0.187594	+	0.982247i	$0.439931\pi$
$234$	0	0
$235$	−10.8523	−0.707925
$236$	39.8150	2.59173
$237$	0	0
$238$	1.21982	0.0790691
$239$	24.3680	1.57624	0.788118	−	0.615524i	$-0.211056\pi$
$239$	24.3680	1.57624	0.788118	+	0.615524i	$0.211056\pi$
$240$	0	0
$241$	−16.7082	−1.07627	−0.538135	−	0.842859i	$-0.680871\pi$
$241$	−16.7082	−1.07627	−0.538135	+	0.842859i	$0.680871\pi$
$242$	0	0
$243$	0	0
$244$	4.28042	0.274026
$245$	6.96255	0.444821
$246$	0	0
$247$	−5.40330	−0.343804
$248$	−9.14297	−0.580579
$249$	0	0
$250$	−2.35567	−0.148986
$251$	16.0984	1.01612	0.508061	−	0.861321i	$-0.330362\pi$
$251$	16.0984	1.01612	0.508061	+	0.861321i	$0.330362\pi$
$252$	0	0
$253$	0	0
$254$	16.7258	1.04947
$255$	0	0
$256$	−24.3912	−1.52445
$257$	−5.96875	−0.372321	−0.186160	−	0.982519i	$-0.559604\pi$
$257$	−5.96875	−0.372321	−0.186160	+	0.982519i	$0.559604\pi$
$258$	0	0
$259$	1.10546	0.0686900
$260$	−3.45589	−0.214325
$261$	0	0
$262$	5.88974	0.363870
$263$	0.451149	0.0278190	0.0139095	−	0.999903i	$-0.495572\pi$
$263$	0.451149	0.0278190	0.0139095	+	0.999903i	$0.495572\pi$
$264$	0	0
$265$	−9.28879	−0.570606
$266$	−2.52981	−0.155112
$267$	0	0
$268$	11.5676	0.706604
$269$	−14.7197	−0.897477	−0.448738	−	0.893663i	$-0.648127\pi$
$269$	−14.7197	−0.897477	−0.448738	+	0.893663i	$0.648127\pi$
$270$	0	0
$271$	4.42787	0.268974	0.134487	−	0.990915i	$-0.457061\pi$
$271$	4.42787	0.268974	0.134487	+	0.990915i	$0.457061\pi$
$272$	4.00935	0.243103
$273$	0	0
$274$	36.0286	2.17657
$275$	0	0
$276$	0	0
$277$	−0.0923299	−0.00554757	−0.00277378	−	0.999996i	$-0.500883\pi$
$277$	−0.0923299	−0.00554757	−0.00277378	+	0.999996i	$0.500883\pi$
$278$	−46.3093	−2.77745
$279$	0	0
$280$	−0.706260	−0.0422071
$281$	−5.18301	−0.309192	−0.154596	−	0.987978i	$-0.549408\pi$
$281$	−5.18301	−0.309192	−0.154596	+	0.987978i	$0.549408\pi$
$282$	0	0
$283$	1.56463	0.0930073	0.0465037	−	0.998918i	$-0.485192\pi$
$283$	1.56463	0.0930073	0.0465037	+	0.998918i	$0.485192\pi$
$284$	−21.2625	−1.26170
$285$	0	0
$286$	0	0
$287$	1.60214	0.0945710
$288$	0	0
$289$	−9.84060	−0.578859
$290$	−23.8342	−1.39959
$291$	0	0
$292$	11.7345	0.686709
$293$	23.1570	1.35285	0.676424	−	0.736512i	$-0.263529\pi$
$293$	23.1570	1.35285	0.676424	+	0.736512i	$0.263529\pi$
$294$	0	0
$295$	−11.2180	−0.653138
$296$	20.8461	1.21165
$297$	0	0
$298$	23.8699	1.38274
$299$	4.67419	0.270315
$300$	0	0
$301$	0.989607	0.0570400
$302$	28.1813	1.62165
$303$	0	0
$304$	−8.31508	−0.476902
$305$	−1.20602	−0.0690567
$306$	0	0
$307$	−7.72480	−0.440878	−0.220439	−	0.975401i	$-0.570749\pi$
$307$	−7.72480	−0.440878	−0.220439	+	0.975401i	$0.570749\pi$
$308$	0	0
$309$	0	0
$310$	5.90173	0.335196
$311$	19.1209	1.08424	0.542122	−	0.840300i	$-0.317621\pi$
$311$	19.1209	1.08424	0.542122	+	0.840300i	$0.317621\pi$
$312$	0	0
$313$	2.90307	0.164091	0.0820455	−	0.996629i	$-0.473855\pi$
$313$	2.90307	0.164091	0.0820455	+	0.996629i	$0.473855\pi$
$314$	−6.18612	−0.349103
$315$	0	0
$316$	36.0348	2.02712
$317$	−2.26153	−0.127020	−0.0635102	−	0.997981i	$-0.520230\pi$
$317$	−2.26153	−0.127020	−0.0635102	+	0.997981i	$0.520230\pi$
$318$	0	0
$319$	0	0
$320$	11.8754	0.663857
$321$	0	0
$322$	2.18844	0.121957
$323$	−14.8480	−0.826166
$324$	0	0
$325$	0.973708	0.0540116
$326$	55.8848	3.09517
$327$	0	0
$328$	30.2121	1.66818
$329$	2.10021	0.115788
$330$	0	0
$331$	6.02336	0.331074	0.165537	−	0.986204i	$-0.447064\pi$
$331$	6.02336	0.331074	0.165537	+	0.986204i	$0.447064\pi$
$332$	−29.5119	−1.61967
$333$	0	0
$334$	−8.22542	−0.450075
$335$	−3.25922	−0.178070
$336$	0	0
$337$	−14.5648	−0.793393	−0.396696	−	0.917950i	$-0.629843\pi$
$337$	−14.5648	−0.793393	−0.396696	+	0.917950i	$0.629843\pi$
$338$	−28.3903	−1.54423
$339$	0	0
$340$	−9.49662	−0.515026
$341$	0	0
$342$	0	0
$343$	−2.70213	−0.145901
$344$	18.6614	1.00616
$345$	0	0
$346$	47.9605	2.57837
$347$	−28.8351	−1.54795	−0.773976	−	0.633215i	$-0.781735\pi$
$347$	−28.8351	−1.54795	−0.773976	+	0.633215i	$0.781735\pi$
$348$	0	0
$349$	−1.63234	−0.0873771	−0.0436886	−	0.999045i	$-0.513911\pi$
$349$	−1.63234	−0.0873771	−0.0436886	+	0.999045i	$0.513911\pi$
$350$	0.455887	0.0243682
$351$	0	0
$352$	0	0
$353$	−24.9297	−1.32687	−0.663437	−	0.748232i	$-0.730903\pi$
$353$	−24.9297	−1.32687	−0.663437	+	0.748232i	$0.730903\pi$
$354$	0	0
$355$	5.99078	0.317958
$356$	−26.0607	−1.38122
$357$	0	0
$358$	−9.07211	−0.479476
$359$	−28.4409	−1.50106	−0.750528	−	0.660839i	$-0.770201\pi$
$359$	−28.4409	−1.50106	−0.750528	+	0.660839i	$0.770201\pi$
$360$	0	0
$361$	11.7936	0.620718
$362$	−41.4955	−2.18095
$363$	0	0
$364$	0.668808	0.0350550
$365$	−3.30624	−0.173056
$366$	0	0
$367$	−10.4413	−0.545030	−0.272515	−	0.962152i	$-0.587855\pi$
$367$	−10.4413	−0.545030	−0.272515	+	0.962152i	$0.587855\pi$
$368$	7.19305	0.374964
$369$	0	0
$370$	−13.4560	−0.699545
$371$	1.79763	0.0933284
$372$	0	0
$373$	−2.81747	−0.145883	−0.0729416	−	0.997336i	$-0.523239\pi$
$373$	−2.81747	−0.145883	−0.0729416	+	0.997336i	$0.523239\pi$
$374$	0	0
$375$	0	0
$376$	39.6045	2.04245
$377$	9.85178	0.507393
$378$	0	0
$379$	8.93319	0.458867	0.229434	−	0.973324i	$-0.426313\pi$
$379$	8.93319	0.458867	0.229434	+	0.973324i	$0.426313\pi$
$380$	19.6952	1.01034
$381$	0	0
$382$	−0.982550	−0.0502716
$383$	8.53279	0.436005	0.218003	−	0.975948i	$-0.430046\pi$
$383$	8.53279	0.436005	0.218003	+	0.975948i	$0.430046\pi$
$384$	0	0
$385$	0	0
$386$	27.5532	1.40242
$387$	0	0
$388$	−56.2630	−2.85632
$389$	−11.7757	−0.597054	−0.298527	−	0.954401i	$-0.596495\pi$
$389$	−11.7757	−0.597054	−0.298527	+	0.954401i	$0.596495\pi$
$390$	0	0
$391$	12.8445	0.649572
$392$	−25.4092	−1.28336
$393$	0	0
$394$	−50.8728	−2.56293
$395$	−10.1529	−0.510849
$396$	0	0
$397$	−1.41214	−0.0708735	−0.0354368	−	0.999372i	$-0.511282\pi$
$397$	−1.41214	−0.0708735	−0.0354368	+	0.999372i	$0.511282\pi$
$398$	18.2807	0.916328
$399$	0	0
$400$	1.49843	0.0749214
$401$	−8.45917	−0.422431	−0.211215	−	0.977440i	$-0.567742\pi$
$401$	−8.45917	−0.422431	−0.211215	+	0.977440i	$0.567742\pi$
$402$	0	0
$403$	−2.43946	−0.121518
$404$	−46.7044	−2.32363
$405$	0	0
$406$	4.61257	0.228918
$407$	0	0
$408$	0	0
$409$	10.1255	0.500675	0.250337	−	0.968159i	$-0.419458\pi$
$409$	10.1255	0.500675	0.250337	+	0.968159i	$0.419458\pi$
$410$	−19.5017	−0.963121
$411$	0	0
$412$	14.1875	0.698966
$413$	2.17099	0.106827
$414$	0	0
$415$	8.31508	0.408171
$416$	−3.66993	−0.179933
$417$	0	0
$418$	0	0
$419$	−32.8019	−1.60248	−0.801240	−	0.598343i	$-0.795826\pi$
$419$	−32.8019	−1.60248	−0.801240	+	0.598343i	$0.795826\pi$
$420$	0	0
$421$	−14.1293	−0.688619	−0.344309	−	0.938856i	$-0.611887\pi$
$421$	−14.1293	−0.688619	−0.344309	+	0.938856i	$0.611887\pi$
$422$	−29.4593	−1.43406
$423$	0	0
$424$	33.8986	1.64626
$425$	2.67571	0.129791
$426$	0	0
$427$	0.233398	0.0112949
$428$	−17.3443	−0.838368
$429$	0	0
$430$	−12.0458	−0.580901
$431$	−19.1057	−0.920291	−0.460145	−	0.887844i	$-0.652203\pi$
$431$	−19.1057	−0.920291	−0.460145	+	0.887844i	$0.652203\pi$
$432$	0	0
$433$	15.1200	0.726622	0.363311	−	0.931668i	$-0.381646\pi$
$433$	15.1200	0.726622	0.363311	+	0.931668i	$0.381646\pi$
$434$	−1.14214	−0.0548247
$435$	0	0
$436$	−26.6892	−1.27818
$437$	−26.6384	−1.27429
$438$	0	0
$439$	37.1642	1.77375	0.886876	−	0.462008i	$-0.152871\pi$
$439$	37.1642	1.77375	0.886876	+	0.462008i	$0.152871\pi$
$440$	0	0
$441$	0	0
$442$	6.13737	0.291925
$443$	2.76049	0.131155	0.0655775	−	0.997847i	$-0.479111\pi$
$443$	2.76049	0.131155	0.0655775	+	0.997847i	$0.479111\pi$
$444$	0	0
$445$	7.34270	0.348078
$446$	−12.6633	−0.599627
$447$	0	0
$448$	−2.29822	−0.108581
$449$	17.8661	0.843156	0.421578	−	0.906792i	$-0.361476\pi$
$449$	17.8661	0.843156	0.421578	+	0.906792i	$0.361476\pi$
$450$	0	0
$451$	0	0
$452$	−71.7780	−3.37615
$453$	0	0
$454$	51.2428	2.40494
$455$	−0.188439	−0.00883415
$456$	0	0
$457$	−21.1621	−0.989922	−0.494961	−	0.868915i	$-0.664818\pi$
$457$	−21.1621	−0.989922	−0.494961	+	0.868915i	$0.664818\pi$
$458$	6.29142	0.293979
$459$	0	0
$460$	−17.0376	−0.794382
$461$	34.3476	1.59973	0.799864	−	0.600181i	$-0.204905\pi$
$461$	34.3476	1.59973	0.799864	+	0.600181i	$0.204905\pi$
$462$	0	0
$463$	10.4516	0.485727	0.242864	−	0.970060i	$-0.421913\pi$
$463$	10.4516	0.485727	0.242864	+	0.970060i	$0.421913\pi$
$464$	15.1608	0.703822
$465$	0	0
$466$	13.4909	0.624955
$467$	−29.6434	−1.37173	−0.685866	−	0.727728i	$-0.740576\pi$
$467$	−29.6434	−1.37173	−0.685866	+	0.727728i	$0.740576\pi$
$468$	0	0
$469$	0.630746	0.0291252
$470$	−25.5645	−1.17920
$471$	0	0
$472$	40.9392	1.88438
$473$	0	0
$474$	0	0
$475$	−5.54920	−0.254615
$476$	1.83785	0.0842378
$477$	0	0
$478$	57.4031	2.62556
$479$	13.2186	0.603974	0.301987	−	0.953312i	$-0.402350\pi$
$479$	13.2186	0.603974	0.301987	+	0.953312i	$0.402350\pi$
$480$	0	0
$481$	5.56199	0.253605
$482$	−39.3591	−1.79276
$483$	0	0
$484$	0	0
$485$	15.8523	0.719816
$486$	0	0
$487$	−42.4976	−1.92575	−0.962875	−	0.269949i	$-0.912993\pi$
$487$	−42.4976	−1.92575	−0.962875	+	0.269949i	$0.912993\pi$
$488$	4.40128	0.199236
$489$	0	0
$490$	16.4015	0.740944
$491$	15.7457	0.710592	0.355296	−	0.934754i	$-0.384380\pi$
$491$	15.7457	0.710592	0.355296	+	0.934754i	$0.384380\pi$
$492$	0	0
$493$	27.0722	1.21927
$494$	−12.7284	−0.572679
$495$	0	0
$496$	−3.75405	−0.168562
$497$	−1.15938	−0.0520052
$498$	0	0
$499$	−33.5279	−1.50092	−0.750458	−	0.660918i	$-0.770167\pi$
$499$	−33.5279	−1.50092	−0.750458	+	0.660918i	$0.770167\pi$
$500$	−3.54920	−0.158725
$501$	0	0
$502$	37.9226	1.69257
$503$	22.8163	1.01733	0.508664	−	0.860965i	$-0.330139\pi$
$503$	22.8163	1.01733	0.508664	+	0.860965i	$0.330139\pi$
$504$	0	0
$505$	13.1591	0.585574
$506$	0	0
$507$	0	0
$508$	25.2001	1.11807
$509$	−39.7166	−1.76041	−0.880204	−	0.474596i	$-0.842594\pi$
$509$	−39.7166	−1.76041	−0.880204	+	0.474596i	$0.842594\pi$
$510$	0	0
$511$	0.639846	0.0283051
$512$	−16.5844	−0.732933
$513$	0	0
$514$	−14.0604	−0.620179
$515$	−3.99737	−0.176145
$516$	0	0
$517$	0	0
$518$	2.60410	0.114418
$519$	0	0
$520$	−3.55346	−0.155830
$521$	−25.5004	−1.11719	−0.558596	−	0.829440i	$-0.688660\pi$
$521$	−25.5004	−1.11719	−0.558596	+	0.829440i	$0.688660\pi$
$522$	0	0
$523$	−15.4304	−0.674725	−0.337363	−	0.941375i	$-0.609535\pi$
$523$	−15.4304	−0.674725	−0.337363	+	0.941375i	$0.609535\pi$
$524$	8.87385	0.387656
$525$	0	0
$526$	1.06276	0.0463385
$527$	−6.70351	−0.292010
$528$	0	0
$529$	0.0438407	0.00190612
$530$	−21.8814	−0.950465
$531$	0	0
$532$	−3.81156	−0.165252
$533$	8.06095	0.349159
$534$	0	0
$535$	4.88682	0.211276
$536$	11.8942	0.513752
$537$	0	0
$538$	−34.6749	−1.49494
$539$	0	0
$540$	0	0
$541$	27.6007	1.18665	0.593324	−	0.804964i	$-0.297815\pi$
$541$	27.6007	1.18665	0.593324	+	0.804964i	$0.297815\pi$
$542$	10.4306	0.448033
$543$	0	0
$544$	−10.0848	−0.432382
$545$	7.51977	0.322111
$546$	0	0
$547$	30.7685	1.31557	0.657784	−	0.753207i	$-0.271494\pi$
$547$	30.7685	1.31557	0.657784	+	0.753207i	$0.271494\pi$
$548$	54.2828	2.31885
$549$	0	0
$550$	0	0
$551$	−56.1457	−2.39189
$552$	0	0
$553$	1.96487	0.0835546
$554$	−0.217499	−0.00924065
$555$	0	0
$556$	−69.7723	−2.95901
$557$	13.9359	0.590482	0.295241	−	0.955423i	$-0.404600\pi$
$557$	13.9359	0.590482	0.295241	+	0.955423i	$0.404600\pi$
$558$	0	0
$559$	4.97909	0.210593
$560$	−0.289986	−0.0122542
$561$	0	0
$562$	−12.2095	−0.515026
$563$	−5.83988	−0.246122	−0.123061	−	0.992399i	$-0.539271\pi$
$563$	−5.83988	−0.246122	−0.123061	+	0.992399i	$0.539271\pi$
$564$	0	0
$565$	20.2237	0.850818
$566$	3.68575	0.154924
$567$	0	0
$568$	−21.8628	−0.917345
$569$	5.45394	0.228641	0.114321	−	0.993444i	$-0.463531\pi$
$569$	5.45394	0.228641	0.114321	+	0.993444i	$0.463531\pi$
$570$	0	0
$571$	40.2894	1.68606	0.843030	−	0.537866i	$-0.180770\pi$
$571$	40.2894	1.68606	0.843030	+	0.537866i	$0.180770\pi$
$572$	0	0
$573$	0	0
$574$	3.77411	0.157528
$575$	4.80040	0.200191
$576$	0	0
$577$	−23.8236	−0.991789	−0.495894	−	0.868383i	$-0.665160\pi$
$577$	−23.8236	−0.991789	−0.495894	+	0.868383i	$0.665160\pi$
$578$	−23.1812	−0.964213
$579$	0	0
$580$	−35.9101	−1.49109
$581$	−1.60919	−0.0667606
$582$	0	0
$583$	0	0
$584$	12.0658	0.499287
$585$	0	0
$586$	54.5504	2.25346
$587$	−19.6080	−0.809307	−0.404654	−	0.914470i	$-0.632608\pi$
$587$	−19.6080	−0.809307	−0.404654	+	0.914470i	$0.632608\pi$
$588$	0	0
$589$	13.9026	0.572845
$590$	−26.4260	−1.08794
$591$	0	0
$592$	8.55928	0.351784
$593$	−3.31095	−0.135964	−0.0679822	−	0.997687i	$-0.521656\pi$
$593$	−3.31095	−0.135964	−0.0679822	+	0.997687i	$0.521656\pi$
$594$	0	0
$595$	−0.517822	−0.0212286
$596$	35.9638	1.47313
$597$	0	0
$598$	11.0109	0.450268
$599$	−32.0710	−1.31039	−0.655193	−	0.755462i	$-0.727413\pi$
$599$	−32.0710	−1.31039	−0.655193	+	0.755462i	$0.727413\pi$
$600$	0	0
$601$	−29.2905	−1.19478	−0.597392	−	0.801950i	$-0.703796\pi$
$601$	−29.2905	−1.19478	−0.597392	+	0.801950i	$0.703796\pi$
$602$	2.33119	0.0950123
$603$	0	0
$604$	42.4596	1.72766
$605$	0	0
$606$	0	0
$607$	−39.0806	−1.58623	−0.793117	−	0.609070i	$-0.791543\pi$
$607$	−39.0806	−1.58623	−0.793117	+	0.609070i	$0.791543\pi$
$608$	20.9151	0.848217
$609$	0	0
$610$	−2.84100	−0.115029
$611$	10.5670	0.427494
$612$	0	0
$613$	−12.7324	−0.514256	−0.257128	−	0.966377i	$-0.582776\pi$
$613$	−12.7324	−0.514256	−0.257128	+	0.966377i	$0.582776\pi$
$614$	−18.1971	−0.734376
$615$	0	0
$616$	0	0
$617$	8.97789	0.361436	0.180718	−	0.983535i	$-0.442158\pi$
$617$	8.97789	0.361436	0.180718	+	0.983535i	$0.442158\pi$
$618$	0	0
$619$	−22.2121	−0.892780	−0.446390	−	0.894839i	$-0.647290\pi$
$619$	−22.2121	−0.892780	−0.446390	+	0.894839i	$0.647290\pi$
$620$	8.89190	0.357107
$621$	0	0
$622$	45.0425	1.80604
$623$	−1.42101	−0.0569316
$624$	0	0
$625$	1.00000	0.0400000
$626$	6.83868	0.273329
$627$	0	0
$628$	−9.32038	−0.371924
$629$	15.2841	0.609417
$630$	0	0
$631$	26.7672	1.06559	0.532794	−	0.846245i	$-0.321142\pi$
$631$	26.7672	1.06559	0.532794	+	0.846245i	$0.321142\pi$
$632$	37.0522	1.47386
$633$	0	0
$634$	−5.32744	−0.211580
$635$	−7.10021	−0.281763
$636$	0	0
$637$	−6.77949	−0.268613
$638$	0	0
$639$	0	0
$640$	20.4366	0.807829
$641$	−15.0726	−0.595333	−0.297666	−	0.954670i	$-0.596208\pi$
$641$	−15.0726	−0.595333	−0.297666	+	0.954670i	$0.596208\pi$
$642$	0	0
$643$	−39.2872	−1.54934	−0.774669	−	0.632367i	$-0.782083\pi$
$643$	−39.2872	−1.54934	−0.774669	+	0.632367i	$0.782083\pi$
$644$	3.29723	0.129929
$645$	0	0
$646$	−34.9771	−1.37616
$647$	−23.8136	−0.936208	−0.468104	−	0.883673i	$-0.655063\pi$
$647$	−23.8136	−0.936208	−0.468104	+	0.883673i	$0.655063\pi$
$648$	0	0
$649$	0	0
$650$	2.29374	0.0899679
$651$	0	0
$652$	84.1994	3.29750
$653$	−6.90354	−0.270156	−0.135078	−	0.990835i	$-0.543129\pi$
$653$	−6.90354	−0.270156	−0.135078	+	0.990835i	$0.543129\pi$
$654$	0	0
$655$	−2.50024	−0.0976924
$656$	12.4049	0.484330
$657$	0	0
$658$	4.94742	0.192870
$659$	20.3718	0.793571	0.396786	−	0.917911i	$-0.370126\pi$
$659$	20.3718	0.793571	0.396786	+	0.917911i	$0.370126\pi$
$660$	0	0
$661$	−19.7451	−0.767994	−0.383997	−	0.923334i	$-0.625453\pi$
$661$	−19.7451	−0.767994	−0.383997	+	0.923334i	$0.625453\pi$
$662$	14.1891	0.551474
$663$	0	0
$664$	−30.3452	−1.17762
$665$	1.07392	0.0416449
$666$	0	0
$667$	48.5695	1.88062
$668$	−12.3929	−0.479496
$669$	0	0
$670$	−7.67765	−0.296613
$671$	0	0
$672$	0	0
$673$	36.6132	1.41134	0.705669	−	0.708542i	$-0.250647\pi$
$673$	36.6132	1.41134	0.705669	+	0.708542i	$0.250647\pi$
$674$	−34.3098	−1.32157
$675$	0	0
$676$	−42.7746	−1.64518
$677$	−7.37138	−0.283305	−0.141653	−	0.989916i	$-0.545242\pi$
$677$	−7.37138	−0.283305	−0.141653	+	0.989916i	$0.545242\pi$
$678$	0	0
$679$	−3.06785	−0.117733
$680$	−9.76476	−0.374461
$681$	0	0
$682$	0	0
$683$	−24.5651	−0.939959	−0.469979	−	0.882677i	$-0.655739\pi$
$683$	−24.5651	−0.939959	−0.469979	+	0.882677i	$0.655739\pi$
$684$	0	0
$685$	−15.2944	−0.584368
$686$	−6.36534	−0.243030
$687$	0	0
$688$	7.66226	0.292121
$689$	9.04457	0.344571
$690$	0	0
$691$	29.8435	1.13530	0.567651	−	0.823269i	$-0.307852\pi$
$691$	29.8435	1.13530	0.567651	+	0.823269i	$0.307852\pi$
$692$	72.2602	2.74692
$693$	0	0
$694$	−67.9262	−2.57844
$695$	19.6586	0.745693
$696$	0	0
$697$	22.1511	0.839033
$698$	−3.84526	−0.145545
$699$	0	0
$700$	0.686867	0.0259611
$701$	14.4189	0.544595	0.272297	−	0.962213i	$-0.412217\pi$
$701$	14.4189	0.544595	0.272297	+	0.962213i	$0.412217\pi$
$702$	0	0
$703$	−31.6980	−1.19551
$704$	0	0
$705$	0	0
$706$	−58.7263	−2.21019
$707$	−2.54665	−0.0957766
$708$	0	0
$709$	−44.5088	−1.67156	−0.835781	−	0.549063i	$-0.814985\pi$
$709$	−44.5088	−1.67156	−0.835781	+	0.549063i	$0.814985\pi$
$710$	14.1123	0.529626
$711$	0	0
$712$	−26.7966	−1.00424
$713$	−12.0266	−0.450398
$714$	0	0
$715$	0	0
$716$	−13.6686	−0.510819
$717$	0	0
$718$	−66.9976	−2.50033
$719$	−3.59345	−0.134013	−0.0670066	−	0.997753i	$-0.521345\pi$
$719$	−3.59345	−0.134013	−0.0670066	+	0.997753i	$0.521345\pi$
$720$	0	0
$721$	0.773598	0.0288103
$722$	27.7820	1.03394
$723$	0	0
$724$	−62.5196	−2.32352
$725$	10.1178	0.375766
$726$	0	0
$727$	−39.0846	−1.44957	−0.724784	−	0.688976i	$-0.758060\pi$
$727$	−39.0846	−1.44957	−0.724784	+	0.688976i	$0.758060\pi$
$728$	0.687692	0.0254875
$729$	0	0
$730$	−7.78841	−0.288262
$731$	13.6823	0.506059
$732$	0	0
$733$	37.7065	1.39272	0.696360	−	0.717693i	$-0.254802\pi$
$733$	37.7065	1.39272	0.696360	+	0.717693i	$0.254802\pi$
$734$	−24.5962	−0.907863
$735$	0	0
$736$	−18.0928	−0.666910
$737$	0	0
$738$	0	0
$739$	14.9324	0.549298	0.274649	−	0.961545i	$-0.411438\pi$
$739$	14.9324	0.549298	0.274649	+	0.961545i	$0.411438\pi$
$740$	−20.2737	−0.745274
$741$	0	0
$742$	4.23463	0.155458
$743$	−13.3680	−0.490423	−0.245211	−	0.969470i	$-0.578857\pi$
$743$	−13.3680	−0.490423	−0.245211	+	0.969470i	$0.578857\pi$
$744$	0	0
$745$	−10.1329	−0.371242
$746$	−6.63705	−0.243000
$747$	0	0
$748$	0	0
$749$	−0.945731	−0.0345563
$750$	0	0
$751$	40.4655	1.47661	0.738303	−	0.674469i	$-0.235627\pi$
$751$	40.4655	1.47661	0.738303	+	0.674469i	$0.235627\pi$
$752$	16.2614	0.592991
$753$	0	0
$754$	23.2076	0.845171
$755$	−11.9632	−0.435384
$756$	0	0
$757$	−8.53757	−0.310303	−0.155152	−	0.987891i	$-0.549587\pi$
$757$	−8.53757	−0.310303	−0.155152	+	0.987891i	$0.549587\pi$
$758$	21.0437	0.764341
$759$	0	0
$760$	20.2513	0.734593
$761$	21.2805	0.771416	0.385708	−	0.922621i	$-0.373957\pi$
$761$	21.2805	0.771416	0.385708	+	0.922621i	$0.373957\pi$
$762$	0	0
$763$	−1.45528	−0.0526846
$764$	−1.48037	−0.0535579
$765$	0	0
$766$	20.1005	0.726260
$767$	10.9231	0.394409
$768$	0	0
$769$	−24.4717	−0.882471	−0.441235	−	0.897391i	$-0.645460\pi$
$769$	−24.4717	−0.882471	−0.441235	+	0.897391i	$0.645460\pi$
$770$	0	0
$771$	0	0
$772$	41.5134	1.49410
$773$	−13.6472	−0.490855	−0.245427	−	0.969415i	$-0.578928\pi$
$773$	−13.6472	−0.490855	−0.245427	+	0.969415i	$0.578928\pi$
$774$	0	0
$775$	−2.50533	−0.0899940
$776$	−57.8516	−2.07675
$777$	0	0
$778$	−27.7398	−0.994520
$779$	−45.9397	−1.64596
$780$	0	0
$781$	0	0
$782$	30.2574	1.08200
$783$	0	0
$784$	−10.4329	−0.372603
$785$	2.62605	0.0937278
$786$	0	0
$787$	−7.45709	−0.265816	−0.132908	−	0.991128i	$-0.542432\pi$
$787$	−7.45709	−0.265816	−0.132908	+	0.991128i	$0.542432\pi$
$788$	−76.6480	−2.73047
$789$	0	0
$790$	−23.9170	−0.850929
$791$	−3.91383	−0.139160
$792$	0	0
$793$	1.17431	0.0417011
$794$	−3.32655	−0.118055
$795$	0	0
$796$	27.5428	0.976228
$797$	−12.0347	−0.426289	−0.213145	−	0.977021i	$-0.568371\pi$
$797$	−12.0347	−0.426289	−0.213145	+	0.977021i	$0.568371\pi$
$798$	0	0
$799$	29.0375	1.02727
$800$	−3.76902	−0.133255
$801$	0	0
$802$	−19.9270	−0.703648
$803$	0	0
$804$	0	0
$805$	−0.929007	−0.0327432
$806$	−5.74656	−0.202414
$807$	0	0
$808$	−48.0231	−1.68945
$809$	50.0338	1.75909	0.879546	−	0.475813i	$-0.157846\pi$
$809$	50.0338	1.75909	0.879546	+	0.475813i	$0.157846\pi$
$810$	0	0
$811$	−11.2188	−0.393944	−0.196972	−	0.980409i	$-0.563111\pi$
$811$	−11.2188	−0.393944	−0.196972	+	0.980409i	$0.563111\pi$
$812$	6.94958	0.243882
$813$	0	0
$814$	0	0
$815$	−23.7235	−0.830997
$816$	0	0
$817$	−28.3760	−0.992752
$818$	23.8524	0.833981
$819$	0	0
$820$	−29.3825	−1.02608
$821$	−7.37723	−0.257467	−0.128734	−	0.991679i	$-0.541091\pi$
$821$	−7.37723	−0.257467	−0.128734	+	0.991679i	$0.541091\pi$
$822$	0	0
$823$	35.6380	1.24226	0.621132	−	0.783706i	$-0.286673\pi$
$823$	35.6380	1.24226	0.621132	+	0.783706i	$0.286673\pi$
$824$	14.5880	0.508198
$825$	0	0
$826$	5.11414	0.177944
$827$	50.5291	1.75707	0.878535	−	0.477678i	$-0.158522\pi$
$827$	50.5291	1.75707	0.878535	+	0.477678i	$0.158522\pi$
$828$	0	0
$829$	−6.14491	−0.213422	−0.106711	−	0.994290i	$-0.534032\pi$
$829$	−6.14491	−0.213422	−0.106711	+	0.994290i	$0.534032\pi$
$830$	19.5876	0.679896
$831$	0	0
$832$	−11.5632	−0.400882
$833$	−18.6297	−0.645482
$834$	0	0
$835$	3.49175	0.120837
$836$	0	0
$837$	0	0
$838$	−77.2707	−2.66927
$839$	37.5230	1.29544	0.647719	−	0.761879i	$-0.275723\pi$
$839$	37.5230	1.29544	0.647719	+	0.761879i	$0.275723\pi$
$840$	0	0
$841$	73.3698	2.52999
$842$	−33.2840	−1.14704
$843$	0	0
$844$	−44.3852	−1.52780
$845$	12.0519	0.414598
$846$	0	0
$847$	0	0
$848$	13.9186	0.477966
$849$	0	0
$850$	6.30309	0.216194
$851$	27.4207	0.939970
$852$	0	0
$853$	−27.0122	−0.924880	−0.462440	−	0.886651i	$-0.653026\pi$
$853$	−27.0122	−0.924880	−0.462440	+	0.886651i	$0.653026\pi$
$854$	0.549810	0.0188141
$855$	0	0
$856$	−17.8340	−0.609554
$857$	2.51515	0.0859159	0.0429580	−	0.999077i	$-0.486322\pi$
$857$	2.51515	0.0859159	0.0429580	+	0.999077i	$0.486322\pi$
$858$	0	0
$859$	6.70885	0.228903	0.114451	−	0.993429i	$-0.463489\pi$
$859$	6.70885	0.228903	0.114451	+	0.993429i	$0.463489\pi$
$860$	−18.1490	−0.618874
$861$	0	0
$862$	−45.0069	−1.53294
$863$	3.30744	0.112586	0.0562932	−	0.998414i	$-0.482072\pi$
$863$	3.30744	0.112586	0.0562932	+	0.998414i	$0.482072\pi$
$864$	0	0
$865$	−20.3596	−0.692246
$866$	35.6179	1.21034
$867$	0	0
$868$	−1.72082	−0.0584086
$869$	0	0
$870$	0	0
$871$	3.17352	0.107531
$872$	−27.4427	−0.929328
$873$	0	0
$874$	−62.7514	−2.12260
$875$	−0.193527	−0.00654241
$876$	0	0
$877$	32.1004	1.08395	0.541976	−	0.840394i	$-0.317676\pi$
$877$	32.1004	1.08395	0.541976	+	0.840394i	$0.317676\pi$
$878$	87.5468	2.95456
$879$	0	0
$880$	0	0
$881$	35.2547	1.18776	0.593881	−	0.804553i	$-0.297595\pi$
$881$	35.2547	1.18776	0.593881	+	0.804553i	$0.297595\pi$
$882$	0	0
$883$	−0.455852	−0.0153406	−0.00767032	−	0.999971i	$-0.502442\pi$
$883$	−0.455852	−0.0153406	−0.00767032	+	0.999971i	$0.502442\pi$
$884$	9.24694	0.311008
$885$	0	0
$886$	6.50282	0.218467
$887$	1.80160	0.0604918	0.0302459	−	0.999542i	$-0.490371\pi$
$887$	1.80160	0.0604918	0.0302459	+	0.999542i	$0.490371\pi$
$888$	0	0
$889$	1.37408	0.0460853
$890$	17.2970	0.579797
$891$	0	0
$892$	−19.0794	−0.638824
$893$	−60.2216	−2.01524
$894$	0	0
$895$	3.85117	0.128731
$896$	−3.95504	−0.132129
$897$	0	0
$898$	42.0868	1.40446
$899$	−25.3484	−0.845416
$900$	0	0
$901$	24.8541	0.828009
$902$	0	0
$903$	0	0
$904$	−73.8047	−2.45471
$905$	17.6151	0.585546
$906$	0	0
$907$	−35.3469	−1.17367	−0.586837	−	0.809705i	$-0.699627\pi$
$907$	−35.3469	−1.17367	−0.586837	+	0.809705i	$0.699627\pi$
$908$	77.2054	2.56215
$909$	0	0
$910$	−0.443901	−0.0147152
$911$	1.64877	0.0546262	0.0273131	−	0.999627i	$-0.491305\pi$
$911$	1.64877	0.0546262	0.0273131	+	0.999627i	$0.491305\pi$
$912$	0	0
$913$	0	0
$914$	−49.8511	−1.64893
$915$	0	0
$916$	9.47903	0.313196
$917$	0.483864	0.0159786
$918$	0	0
$919$	−0.318990	−0.0105225	−0.00526125	−	0.999986i	$-0.501675\pi$
$919$	−0.318990	−0.0105225	−0.00526125	+	0.999986i	$0.501675\pi$
$920$	−17.5186	−0.577573
$921$	0	0
$922$	80.9118	2.66469
$923$	−5.83327	−0.192005
$924$	0	0
$925$	5.71217	0.187815
$926$	24.6206	0.809082
$927$	0	0
$928$	−38.1342	−1.25182
$929$	−56.1509	−1.84225	−0.921125	−	0.389267i	$-0.872728\pi$
$929$	−56.1509	−1.84225	−0.921125	+	0.389267i	$0.872728\pi$
$930$	0	0
$931$	38.6366	1.26626
$932$	20.3262	0.665808
$933$	0	0
$934$	−69.8301	−2.28491
$935$	0	0
$936$	0	0
$937$	36.8424	1.20359	0.601795	−	0.798651i	$-0.294453\pi$
$937$	36.8424	1.20359	0.601795	+	0.798651i	$0.294453\pi$
$938$	1.48583	0.0485142
$939$	0	0
$940$	−38.5170	−1.25629
$941$	15.3105	0.499108	0.249554	−	0.968361i	$-0.419716\pi$
$941$	15.3105	0.499108	0.249554	+	0.968361i	$0.419716\pi$
$942$	0	0
$943$	39.7406	1.29413
$944$	16.8094	0.547099
$945$	0	0
$946$	0	0
$947$	22.6654	0.736526	0.368263	−	0.929722i	$-0.379953\pi$
$947$	22.6654	0.736526	0.368263	+	0.929722i	$0.379953\pi$
$948$	0	0
$949$	3.21931	0.104503
$950$	−13.0721	−0.424115
$951$	0	0
$952$	1.88974	0.0612470
$953$	−39.3324	−1.27410	−0.637051	−	0.770822i	$-0.719846\pi$
$953$	−39.3324	−1.27410	−0.637051	+	0.770822i	$0.719846\pi$
$954$	0	0
$955$	0.417099	0.0134970
$956$	86.4870	2.79719
$957$	0	0
$958$	31.1388	1.00605
$959$	2.95988	0.0955794
$960$	0	0
$961$	−24.7233	−0.797527
$962$	13.1022	0.422433
$963$	0	0
$964$	−59.3008	−1.90995
$965$	−11.6965	−0.376525
$966$	0	0
$967$	3.06103	0.0984360	0.0492180	−	0.998788i	$-0.484327\pi$
$967$	3.06103	0.0984360	0.0492180	+	0.998788i	$0.484327\pi$
$968$	0	0
$969$	0	0
$970$	37.3428	1.19901
$971$	−32.0385	−1.02816	−0.514082	−	0.857741i	$-0.671867\pi$
$971$	−32.0385	−1.02816	−0.514082	+	0.857741i	$0.671867\pi$
$972$	0	0
$973$	−3.80447	−0.121966
$974$	−100.110	−3.20775
$975$	0	0
$976$	1.80714	0.0578451
$977$	53.1962	1.70190	0.850948	−	0.525250i	$-0.176028\pi$
$977$	53.1962	1.70190	0.850948	+	0.525250i	$0.176028\pi$
$978$	0	0
$979$	0	0
$980$	24.7115	0.789379
$981$	0	0
$982$	37.0916	1.18364
$983$	24.5004	0.781442	0.390721	−	0.920509i	$-0.372226\pi$
$983$	24.5004	0.781442	0.390721	+	0.920509i	$0.372226\pi$
$984$	0	0
$985$	21.5958	0.688101
$986$	63.7734	2.03096
$987$	0	0
$988$	−19.1774	−0.610115
$989$	24.5470	0.780549
$990$	0	0
$991$	−6.34819	−0.201657	−0.100828	−	0.994904i	$-0.532149\pi$
$991$	−6.34819	−0.201657	−0.100828	+	0.994904i	$0.532149\pi$
$992$	9.44262	0.299804
$993$	0	0
$994$	−2.73112	−0.0866258
$995$	−7.76028	−0.246017
$996$	0	0
$997$	−25.9460	−0.821719	−0.410860	−	0.911699i	$-0.634771\pi$
$997$	−25.9460	−0.821719	−0.410860	+	0.911699i	$0.634771\pi$
$998$	−78.9808	−2.50010
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5445.2.a.bt.1.4		4
3.2	odd	2		1815.2.a.p.1.1		4
11.5	even	5		495.2.n.a.91.1		8
11.9	even	5		495.2.n.a.136.1		8
11.10	odd	2		5445.2.a.bf.1.1		4
15.14	odd	2		9075.2.a.di.1.4		4
33.5	odd	10		165.2.m.d.91.2	✓	8
33.20	odd	10		165.2.m.d.136.2	yes	8
33.32	even	2		1815.2.a.w.1.4		4
165.38	even	20		825.2.bx.f.124.1		16
165.53	even	20		825.2.bx.f.499.4		16
165.104	odd	10		825.2.n.g.751.1		8
165.119	odd	10		825.2.n.g.301.1		8
165.137	even	20		825.2.bx.f.124.4		16
165.152	even	20		825.2.bx.f.499.1		16
165.164	even	2		9075.2.a.cm.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.m.d.91.2	✓	8	33.5	odd	10
165.2.m.d.136.2	yes	8	33.20	odd	10
495.2.n.a.91.1		8	11.5	even	5
495.2.n.a.136.1		8	11.9	even	5
825.2.n.g.301.1		8	165.119	odd	10
825.2.n.g.751.1		8	165.104	odd	10
825.2.bx.f.124.1		16	165.38	even	20
825.2.bx.f.124.4		16	165.137	even	20
825.2.bx.f.499.1		16	165.152	even	20
825.2.bx.f.499.4		16	165.53	even	20
1815.2.a.p.1.1		4	3.2	odd	2
1815.2.a.w.1.4		4	33.32	even	2
5445.2.a.bf.1.1		4	11.10	odd	2
5445.2.a.bt.1.4		4	1.1	even	1	trivial
9075.2.a.cm.1.1		4	165.164	even	2
9075.2.a.di.1.4		4	15.14	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:	\( N \)	\(=\)	\( 5445 = 3^{2} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5445.a (trivial)

\(f(q)\)	\(=\)	\(q+2.35567 q^{2} +3.54920 q^{4} -1.00000 q^{5} +0.193527 q^{7} +3.64941 q^{8} +O(q^{10})\)	\(q+2.35567 q^{2} +3.54920 q^{4} -1.00000 q^{5} +0.193527 q^{7} +3.64941 q^{8} -2.35567 q^{10} +0.973708 q^{13} +0.455887 q^{14} +1.49843 q^{16} +2.67571 q^{17} -5.54920 q^{19} -3.54920 q^{20} +4.80040 q^{23} +1.00000 q^{25} +2.29374 q^{26} +0.686867 q^{28} +10.1178 q^{29} -2.50533 q^{31} -3.76902 q^{32} +6.30309 q^{34} -0.193527 q^{35} +5.71217 q^{37} -13.0721 q^{38} -3.64941 q^{40} +8.27861 q^{41} +5.11353 q^{43} +11.3082 q^{46} +10.8523 q^{47} -6.96255 q^{49} +2.35567 q^{50} +3.45589 q^{52} +9.28879 q^{53} +0.706260 q^{56} +23.8342 q^{58} +11.2180 q^{59} +1.20602 q^{61} -5.90173 q^{62} -11.8754 q^{64} -0.973708 q^{65} +3.25922 q^{67} +9.49662 q^{68} -0.455887 q^{70} -5.99078 q^{71} +3.30624 q^{73} +13.4560 q^{74} -19.6952 q^{76} +10.1529 q^{79} -1.49843 q^{80} +19.5017 q^{82} -8.31508 q^{83} -2.67571 q^{85} +12.0458 q^{86} -7.34270 q^{89} +0.188439 q^{91} +17.0376 q^{92} +25.5645 q^{94} +5.54920 q^{95} -15.8523 q^{97} -16.4015 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 3 q^{2} + q^{4} - 4 q^{5} - 6 q^{7} + 3 q^{8}+O(q^{10}) \) 4 * q + 3 * q^2 + q^4 - 4 * q^5 - 6 * q^7 + 3 * q^8	\( 4 q + 3 q^{2} + q^{4} - 4 q^{5} - 6 q^{7} + 3 q^{8} - 3 q^{10} - 7 q^{13} - 3 q^{14} - q^{16} + 10 q^{17} - 9 q^{19} - q^{20} + 3 q^{23} + 4 q^{25} + 4 q^{26} + 7 q^{28} + 15 q^{29} - 13 q^{31} - 6 q^{32} - 3 q^{34} + 6 q^{35} - 3 q^{37} - 15 q^{38} - 3 q^{40} + 22 q^{41} - q^{46} + 2 q^{47} - 12 q^{49} + 3 q^{50} + 9 q^{52} - 10 q^{53} + 8 q^{56} + 39 q^{58} + 21 q^{59} - 11 q^{61} + 10 q^{62} - 3 q^{64} + 7 q^{65} + q^{67} + 3 q^{68} + 3 q^{70} + 13 q^{71} - q^{73} - 11 q^{74} - 19 q^{76} + 4 q^{79} + q^{80} + 25 q^{82} + 3 q^{83} - 10 q^{85} + 10 q^{89} + 12 q^{91} + 24 q^{92} + 35 q^{94} + 9 q^{95} - 22 q^{97} - 11 q^{98}+O(q^{100}) \) 4 * q + 3 * q^2 + q^4 - 4 * q^5 - 6 * q^7 + 3 * q^8 - 3 * q^10 - 7 * q^13 - 3 * q^14 - q^16 + 10 * q^17 - 9 * q^19 - q^20 + 3 * q^23 + 4 * q^25 + 4 * q^26 + 7 * q^28 + 15 * q^29 - 13 * q^31 - 6 * q^32 - 3 * q^34 + 6 * q^35 - 3 * q^37 - 15 * q^38 - 3 * q^40 + 22 * q^41 - q^46 + 2 * q^47 - 12 * q^49 + 3 * q^50 + 9 * q^52 - 10 * q^53 + 8 * q^56 + 39 * q^58 + 21 * q^59 - 11 * q^61 + 10 * q^62 - 3 * q^64 + 7 * q^65 + q^67 + 3 * q^68 + 3 * q^70 + 13 * q^71 - q^73 - 11 * q^74 - 19 * q^76 + 4 * q^79 + q^80 + 25 * q^82 + 3 * q^83 - 10 * q^85 + 10 * q^89 + 12 * q^91 + 24 * q^92 + 35 * q^94 + 9 * q^95 - 22 * q^97 - 11 * q^98

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