LMFDB - Embedded newform 5445.2.a.bf.1.3

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:	$1$
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			1.3
Root			$0.737640$ 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)

$f(q)$	$=$	$q-0.262360 q^{2} -1.93117 q^{4} -1.00000 q^{5} +3.19353 q^{7} +1.03138 q^{8} +O(q^{10})$	$q-0.262360 q^{2} -1.93117 q^{4} -1.00000 q^{5} +3.19353 q^{7} +1.03138 q^{8} +0.262360 q^{10} +1.11961 q^{13} -0.837853 q^{14} +3.59174 q^{16} -0.0882264 q^{17} +0.0688326 q^{19} +1.93117 q^{20} -6.65450 q^{23} +1.00000 q^{25} -0.293740 q^{26} -6.16724 q^{28} +3.73583 q^{29} -9.58484 q^{31} -3.00509 q^{32} +0.0231471 q^{34} -3.19353 q^{35} -8.33021 q^{37} -0.0180589 q^{38} -1.03138 q^{40} -11.6657 q^{41} +11.8217 q^{43} +1.74587 q^{46} -0.908020 q^{47} +3.19862 q^{49} -0.262360 q^{50} -2.16215 q^{52} -0.872377 q^{53} +3.29374 q^{56} -0.980131 q^{58} -1.83604 q^{59} +10.0601 q^{61} +2.51468 q^{62} -6.39507 q^{64} -1.11961 q^{65} +9.53916 q^{67} +0.170380 q^{68} +0.837853 q^{70} +4.66454 q^{71} +7.16034 q^{73} +2.18551 q^{74} -0.132927 q^{76} -0.791342 q^{79} -3.59174 q^{80} +3.06060 q^{82} +0.247229 q^{83} +0.0882264 q^{85} -3.10155 q^{86} +14.5788 q^{89} +3.57549 q^{91} +12.8510 q^{92} +0.238228 q^{94} -0.0688326 q^{95} -4.09198 q^{97} -0.839188 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.262360	−0.185516	−0.0927582	−	0.995689i	$-0.529568\pi$
$2$	−0.262360	−0.185516	−0.0927582	+	0.995689i	$0.529568\pi$
$3$	0	0
$3$	0	0
$4$	−1.93117	−0.965584
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	3.19353	1.20704	0.603520	−	0.797348i	$-0.293764\pi$
$7$	3.19353	1.20704	0.603520	+	0.797348i	$0.293764\pi$
$8$	1.03138	0.364648
$9$	0	0
$10$	0.262360	0.0829654
$11$	0	0
$11$	0	0
$12$	0	0
$13$	1.11961	0.310523	0.155261	−	0.987873i	$-0.450378\pi$
$13$	1.11961	0.310523	0.155261	+	0.987873i	$0.450378\pi$
$14$	−0.837853	−0.223926
$15$	0	0
$16$	3.59174	0.897936
$17$	−0.0882264	−0.0213981	−0.0106990	−	0.999943i	$-0.503406\pi$
$17$	−0.0882264	−0.0213981	−0.0106990	+	0.999943i	$0.503406\pi$
$18$	0	0
$19$	0.0688326	0.0157913	0.00789564	−	0.999969i	$-0.497487\pi$
$19$	0.0688326	0.0157913	0.00789564	+	0.999969i	$0.497487\pi$
$20$	1.93117	0.431822
$21$	0	0
$22$	0	0
$23$	−6.65450	−1.38756	−0.693780	−	0.720187i	$-0.744056\pi$
$23$	−6.65450	−1.38756	−0.693780	+	0.720187i	$0.744056\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−0.293740	−0.0576071
$27$	0	0
$28$	−6.16724	−1.16550
$29$	3.73583	0.693726	0.346863	−	0.937916i	$-0.387247\pi$
$29$	3.73583	0.693726	0.346863	+	0.937916i	$0.387247\pi$
$30$	0	0
$31$	−9.58484	−1.72149	−0.860744	−	0.509037i	$-0.830002\pi$
$31$	−9.58484	−1.72149	−0.860744	+	0.509037i	$0.830002\pi$
$32$	−3.00509	−0.531230
$33$	0	0
$34$	0.0231471	0.00396969
$35$	−3.19353	−0.539805
$36$	0	0
$37$	−8.33021	−1.36948	−0.684739	−	0.728789i	$-0.740084\pi$
$37$	−8.33021	−1.36948	−0.684739	+	0.728789i	$0.740084\pi$
$38$	−0.0180589	−0.00292954
$39$	0	0
$40$	−1.03138	−0.163075
$41$	−11.6657	−1.82187	−0.910935	−	0.412549i	$-0.864638\pi$
$41$	−11.6657	−1.82187	−0.910935	+	0.412549i	$0.864638\pi$
$42$	0	0
$43$	11.8217	1.80280	0.901399	−	0.432989i	$-0.142541\pi$
$43$	11.8217	1.80280	0.901399	+	0.432989i	$0.142541\pi$
$44$	0	0
$45$	0	0
$46$	1.74587	0.257415
$47$	−0.908020	−0.132448	−0.0662242	−	0.997805i	$-0.521095\pi$
$47$	−0.908020	−0.132448	−0.0662242	+	0.997805i	$0.521095\pi$
$48$	0	0
$49$	3.19862	0.456945
$50$	−0.262360	−0.0371033
$51$	0	0
$52$	−2.16215	−0.299836
$53$	−0.872377	−0.119830	−0.0599151	−	0.998203i	$-0.519083\pi$
$53$	−0.872377	−0.119830	−0.0599151	+	0.998203i	$0.519083\pi$
$54$	0	0
$55$	0	0
$56$	3.29374	0.440144
$57$	0	0
$58$	−0.980131	−0.128698
$59$	−1.83604	−0.239032	−0.119516	−	0.992832i	$-0.538134\pi$
$59$	−1.83604	−0.239032	−0.119516	+	0.992832i	$0.538134\pi$
$60$	0	0
$61$	10.0601	1.28807	0.644034	−	0.764997i	$-0.277260\pi$
$61$	10.0601	1.28807	0.644034	+	0.764997i	$0.277260\pi$
$62$	2.51468	0.319364
$63$	0	0
$64$	−6.39507	−0.799384
$65$	−1.11961	−0.138870
$66$	0	0
$67$	9.53916	1.16539	0.582697	−	0.812690i	$-0.301997\pi$
$67$	9.53916	1.16539	0.582697	+	0.812690i	$0.301997\pi$
$68$	0.170380	0.0206616
$69$	0	0
$70$	0.837853	0.100143
$71$	4.66454	0.553580	0.276790	−	0.960930i	$-0.410729\pi$
$71$	4.66454	0.553580	0.276790	+	0.960930i	$0.410729\pi$
$72$	0	0
$73$	7.16034	0.838054	0.419027	−	0.907974i	$-0.362371\pi$
$73$	7.16034	0.838054	0.419027	+	0.907974i	$0.362371\pi$
$74$	2.18551	0.254060
$75$	0	0
$76$	−0.132927	−0.0152478
$77$	0	0
$78$	0	0
$79$	−0.791342	−0.0890330	−0.0445165	−	0.999009i	$-0.514175\pi$
$79$	−0.791342	−0.0890330	−0.0445165	+	0.999009i	$0.514175\pi$
$80$	−3.59174	−0.401569
$81$	0	0
$82$	3.06060	0.337987
$83$	0.247229	0.0271369	0.0135685	−	0.999908i	$-0.495681\pi$
$83$	0.247229	0.0271369	0.0135685	+	0.999908i	$0.495681\pi$
$84$	0	0
$85$	0.0882264	0.00956950
$86$	−3.10155	−0.334448
$87$	0	0
$88$	0	0
$89$	14.5788	1.54535	0.772673	−	0.634804i	$-0.218919\pi$
$89$	14.5788	1.54535	0.772673	+	0.634804i	$0.218919\pi$
$90$	0	0
$91$	3.57549	0.374814
$92$	12.8510	1.33980
$93$	0	0
$94$	0.238228	0.0245713
$95$	−0.0688326	−0.00706208
$96$	0	0
$97$	−4.09198	−0.415478	−0.207739	−	0.978184i	$-0.566610\pi$
$97$	−4.09198	−0.415478	−0.207739	+	0.978184i	$0.566610\pi$
$98$	−0.839188	−0.0847708
$99$	0	0
$100$	−1.93117	−0.193117
$101$	4.29171	0.427041	0.213521	−	0.976939i	$-0.431507\pi$
$101$	4.29171	0.427041	0.213521	+	0.976939i	$0.431507\pi$
$102$	0	0
$103$	2.09280	0.206210	0.103105	−	0.994670i	$-0.467122\pi$
$103$	2.09280	0.206210	0.103105	+	0.994670i	$0.467122\pi$
$104$	1.15474	0.113232
$105$	0	0
$106$	0.228877	0.0222305
$107$	−15.7409	−1.52173	−0.760866	−	0.648909i	$-0.775226\pi$
$107$	−15.7409	−1.52173	−0.760866	+	0.648909i	$0.775226\pi$
$108$	0	0
$109$	4.13271	0.395842	0.197921	−	0.980218i	$-0.436581\pi$
$109$	4.13271	0.395842	0.197921	+	0.980218i	$0.436581\pi$
$110$	0	0
$111$	0	0
$112$	11.4703	1.08384
$113$	−13.7550	−1.29396	−0.646981	−	0.762506i	$-0.723969\pi$
$113$	−13.7550	−1.29396	−0.646981	+	0.762506i	$0.723969\pi$
$114$	0	0
$115$	6.65450	0.620536
$116$	−7.21451	−0.669851
$117$	0	0
$118$	0.481704	0.0443444
$119$	−0.281754	−0.0258283
$120$	0	0
$121$	0	0
$122$	−2.63937	−0.238957
$123$	0	0
$124$	18.5099	1.66224
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−7.89979	−0.700993	−0.350496	−	0.936564i	$-0.613987\pi$
$127$	−7.89979	−0.700993	−0.350496	+	0.936564i	$0.613987\pi$
$128$	7.68799	0.679528
$129$	0	0
$130$	0.293740	0.0257627
$131$	−16.3539	−1.42884	−0.714422	−	0.699715i	$-0.753310\pi$
$131$	−16.3539	−1.42884	−0.714422	+	0.699715i	$0.753310\pi$
$132$	0	0
$133$	0.219819	0.0190607
$134$	−2.50269	−0.216200
$135$	0	0
$136$	−0.0909950	−0.00780275
$137$	−7.73208	−0.660596	−0.330298	−	0.943877i	$-0.607149\pi$
$137$	−7.73208	−0.660596	−0.330298	+	0.943877i	$0.607149\pi$
$138$	0	0
$139$	−20.1865	−1.71219	−0.856097	−	0.516815i	$-0.827117\pi$
$139$	−20.1865	−1.71219	−0.856097	+	0.516815i	$0.827117\pi$
$140$	6.16724	0.521227
$141$	0	0
$142$	−1.22379	−0.102698
$143$	0	0
$144$	0	0
$145$	−3.73583	−0.310244
$146$	−1.87858	−0.155473
$147$	0	0
$148$	16.0870	1.32235
$149$	9.69523	0.794264	0.397132	−	0.917761i	$-0.370005\pi$
$149$	9.69523	0.794264	0.397132	+	0.917761i	$0.370005\pi$
$150$	0	0
$151$	−15.1615	−1.23382	−0.616911	−	0.787033i	$-0.711616\pi$
$151$	−15.1615	−1.23382	−0.616911	+	0.787033i	$0.711616\pi$
$152$	0.0709926	0.00575826
$153$	0	0
$154$	0	0
$155$	9.58484	0.769873
$156$	0	0
$157$	−14.8805	−1.18760	−0.593798	−	0.804614i	$-0.702372\pi$
$157$	−14.8805	−1.18760	−0.593798	+	0.804614i	$0.702372\pi$
$158$	0.207616	0.0165171
$159$	0	0
$160$	3.00509	0.237573
$161$	−21.2513	−1.67484
$162$	0	0
$163$	3.40114	0.266398	0.133199	−	0.991089i	$-0.457475\pi$
$163$	3.40114	0.266398	0.133199	+	0.991089i	$0.457475\pi$
$164$	22.5283	1.75917
$165$	0	0
$166$	−0.0648629	−0.00503434
$167$	−22.6164	−1.75011	−0.875053	−	0.484027i	$-0.839174\pi$
$167$	−22.6164	−1.75011	−0.875053	+	0.484027i	$0.839174\pi$
$168$	0	0
$169$	−11.7465	−0.903576
$170$	−0.0231471	−0.00177530
$171$	0	0
$172$	−22.8298	−1.74075
$173$	−13.0913	−0.995312	−0.497656	−	0.867374i	$-0.665806\pi$
$173$	−13.0913	−0.995312	−0.497656	+	0.867374i	$0.665806\pi$
$174$	0	0
$175$	3.19353	0.241408
$176$	0	0
$177$	0	0
$178$	−3.82488	−0.286687
$179$	15.1774	1.13441	0.567207	−	0.823576i	$-0.308024\pi$
$179$	15.1774	1.13441	0.567207	+	0.823576i	$0.308024\pi$
$180$	0	0
$181$	1.41348	0.105063	0.0525316	−	0.998619i	$-0.483271\pi$
$181$	1.41348	0.105063	0.0525316	+	0.998619i	$0.483271\pi$
$182$	−0.938065	−0.0695340
$183$	0	0
$184$	−6.86332	−0.505971
$185$	8.33021	0.612449
$186$	0	0
$187$	0	0
$188$	1.75354	0.127890
$189$	0	0
$190$	0.0180589	0.00131013
$191$	−4.90914	−0.355213	−0.177606	−	0.984102i	$-0.556835\pi$
$191$	−4.90914	−0.355213	−0.177606	+	0.984102i	$0.556835\pi$
$192$	0	0
$193$	−9.71987	−0.699652	−0.349826	−	0.936815i	$-0.613759\pi$
$193$	−9.71987	−0.699652	−0.349826	+	0.936815i	$0.613759\pi$
$194$	1.07357	0.0770779
$195$	0	0
$196$	−6.17706	−0.441219
$197$	−8.88764	−0.633218	−0.316609	−	0.948556i	$-0.602544\pi$
$197$	−8.88764	−0.633218	−0.316609	+	0.948556i	$0.602544\pi$
$198$	0	0
$199$	18.0381	1.27869	0.639343	−	0.768921i	$-0.279206\pi$
$199$	18.0381	1.27869	0.639343	+	0.768921i	$0.279206\pi$
$200$	1.03138	0.0729296
$201$	0	0
$202$	−1.12597	−0.0792232
$203$	11.9305	0.837355
$204$	0	0
$205$	11.6657	0.814765
$206$	−0.549068	−0.0382554
$207$	0	0
$208$	4.02134	0.278830
$209$	0	0
$210$	0	0
$211$	−17.9778	−1.23764	−0.618822	−	0.785531i	$-0.712390\pi$
$211$	−17.9778	−1.23764	−0.618822	+	0.785531i	$0.712390\pi$
$212$	1.68471	0.115706
$213$	0	0
$214$	4.12978	0.282306
$215$	−11.8217	−0.806236
$216$	0	0
$217$	−30.6095	−2.07791
$218$	−1.08426	−0.0734351
$219$	0	0
$220$	0	0
$221$	−0.0987789	−0.00664459
$222$	0	0
$223$	−13.7489	−0.920697	−0.460348	−	0.887738i	$-0.652276\pi$
$223$	−13.7489	−0.920697	−0.460348	+	0.887738i	$0.652276\pi$
$224$	−9.59683	−0.641215
$225$	0	0
$226$	3.60876	0.240051
$227$	0.662735	0.0439873	0.0219936	−	0.999758i	$-0.492999\pi$
$227$	0.662735	0.0439873	0.0219936	+	0.999758i	$0.492999\pi$
$228$	0	0
$229$	−7.49041	−0.494980	−0.247490	−	0.968890i	$-0.579606\pi$
$229$	−7.49041	−0.494980	−0.247490	+	0.968890i	$0.579606\pi$
$230$	−1.74587	−0.115119
$231$	0	0
$232$	3.85306	0.252966
$233$	10.5254	0.689539	0.344770	−	0.938687i	$-0.387957\pi$
$233$	10.5254	0.689539	0.344770	+	0.938687i	$0.387957\pi$
$234$	0	0
$235$	0.908020	0.0592327
$236$	3.54571	0.230806
$237$	0	0
$238$	0.0739208	0.00479157
$239$	17.0762	1.10457	0.552284	−	0.833656i	$-0.313756\pi$
$239$	17.0762	1.10457	0.552284	+	0.833656i	$0.313756\pi$
$240$	0	0
$241$	16.7082	1.07627	0.538135	−	0.842859i	$-0.319129\pi$
$241$	16.7082	1.07627	0.538135	+	0.842859i	$0.319129\pi$
$242$	0	0
$243$	0	0
$244$	−19.4278	−1.24374
$245$	−3.19862	−0.204352
$246$	0	0
$247$	0.0770654	0.00490356
$248$	−9.88562	−0.627737
$249$	0	0
$250$	0.262360	0.0165931
$251$	5.13767	0.324287	0.162143	−	0.986767i	$-0.448159\pi$
$251$	5.13767	0.324287	0.162143	+	0.986767i	$0.448159\pi$
$252$	0	0
$253$	0	0
$254$	2.07259	0.130046
$255$	0	0
$256$	10.7731	0.673320
$257$	3.69824	0.230690	0.115345	−	0.993325i	$-0.463203\pi$
$257$	3.69824	0.230690	0.115345	+	0.993325i	$0.463203\pi$
$258$	0	0
$259$	−26.6027	−1.65301
$260$	2.16215	0.134091
$261$	0	0
$262$	4.29059	0.265074
$263$	31.6315	1.95048	0.975241	−	0.221147i	$-0.0709799\pi$
$263$	31.6315	1.95048	0.975241	+	0.221147i	$0.0709799\pi$
$264$	0	0
$265$	0.872377	0.0535897
$266$	−0.0576716	−0.00353607
$267$	0	0
$268$	−18.4217	−1.12529
$269$	14.7754	0.900875	0.450437	−	0.892808i	$-0.351268\pi$
$269$	14.7754	0.900875	0.450437	+	0.892808i	$0.351268\pi$
$270$	0	0
$271$	19.5869	1.18982	0.594910	−	0.803792i	$-0.297188\pi$
$271$	19.5869	1.18982	0.594910	+	0.803792i	$0.297188\pi$
$272$	−0.316887	−0.0192141
$273$	0	0
$274$	2.02859	0.122551
$275$	0	0
$276$	0	0
$277$	−25.7104	−1.54479	−0.772393	−	0.635145i	$-0.780940\pi$
$277$	−25.7104	−1.54479	−0.772393	+	0.635145i	$0.780940\pi$
$278$	5.29612	0.317640
$279$	0	0
$280$	−3.29374	−0.196839
$281$	−26.7109	−1.59344	−0.796719	−	0.604350i	$-0.793433\pi$
$281$	−26.7109	−1.59344	−0.796719	+	0.604350i	$0.793433\pi$
$282$	0	0
$283$	−7.23375	−0.430002	−0.215001	−	0.976614i	$-0.568975\pi$
$283$	−7.23375	−0.430002	−0.215001	+	0.976614i	$0.568975\pi$
$284$	−9.00802	−0.534527
$285$	0	0
$286$	0	0
$287$	−37.2546	−2.19907
$288$	0	0
$289$	−16.9922	−0.999542
$290$	0.980131	0.0575553
$291$	0	0
$292$	−13.8278	−0.809211
$293$	14.5947	0.852633	0.426317	−	0.904574i	$-0.359811\pi$
$293$	14.5947	0.852633	0.426317	+	0.904574i	$0.359811\pi$
$294$	0	0
$295$	1.83604	0.106899
$296$	−8.59161	−0.499377
$297$	0	0
$298$	−2.54364	−0.147349
$299$	−7.45042	−0.430869
$300$	0	0
$301$	37.7530	2.17605
$302$	3.97775	0.228894
$303$	0	0
$304$	0.247229	0.0141796
$305$	−10.0601	−0.576041
$306$	0	0
$307$	−28.5445	−1.62912	−0.814559	−	0.580080i	$-0.803021\pi$
$307$	−28.5445	−1.62912	−0.814559	+	0.580080i	$0.803021\pi$
$308$	0	0
$309$	0	0
$310$	−2.51468	−0.142824
$311$	9.94803	0.564101	0.282050	−	0.959400i	$-0.408985\pi$
$311$	9.94803	0.564101	0.282050	+	0.959400i	$0.408985\pi$
$312$	0	0
$313$	−16.4309	−0.928731	−0.464366	−	0.885644i	$-0.653718\pi$
$313$	−16.4309	−0.928731	−0.464366	+	0.885644i	$0.653718\pi$
$314$	3.90405	0.220318
$315$	0	0
$316$	1.52821	0.0859688
$317$	−14.2106	−0.798147	−0.399073	−	0.916919i	$-0.630668\pi$
$317$	−14.2106	−0.798147	−0.399073	+	0.916919i	$0.630668\pi$
$318$	0	0
$319$	0	0
$320$	6.39507	0.357495
$321$	0	0
$322$	5.57549	0.310710
$323$	−0.00607286	−0.000337903 0
$324$	0	0
$325$	1.11961	0.0621046
$326$	−0.892323	−0.0494212
$327$	0	0
$328$	−12.0317	−0.664341
$329$	−2.89979	−0.159870
$330$	0	0
$331$	−10.9119	−0.599773	−0.299886	−	0.953975i	$-0.596949\pi$
$331$	−10.9119	−0.599773	−0.299886	+	0.953975i	$0.596949\pi$
$332$	−0.477441	−0.0262030
$333$	0	0
$334$	5.93362	0.324673
$335$	−9.53916	−0.521180
$336$	0	0
$337$	5.88609	0.320636	0.160318	−	0.987065i	$-0.448748\pi$
$337$	5.88609	0.320636	0.160318	+	0.987065i	$0.448748\pi$
$338$	3.08180	0.167628
$339$	0	0
$340$	−0.170380	−0.00924015
$341$	0	0
$342$	0	0
$343$	−12.1398	−0.655489
$344$	12.1927	0.657386
$345$	0	0
$346$	3.43463	0.184647
$347$	34.5043	1.85229	0.926143	−	0.377173i	$-0.123104\pi$
$347$	34.5043	1.85229	0.926143	+	0.377173i	$0.123104\pi$
$348$	0	0
$349$	5.51356	0.295134	0.147567	−	0.989052i	$-0.452856\pi$
$349$	5.51356	0.295134	0.147567	+	0.989052i	$0.452856\pi$
$350$	−0.837853	−0.0447851
$351$	0	0
$352$	0	0
$353$	29.8740	1.59003	0.795016	−	0.606589i	$-0.207463\pi$
$353$	29.8740	1.59003	0.795016	+	0.606589i	$0.207463\pi$
$354$	0	0
$355$	−4.66454	−0.247568
$356$	−28.1540	−1.49216
$357$	0	0
$358$	−3.98194	−0.210452
$359$	−35.9131	−1.89542	−0.947710	−	0.319133i	$-0.896608\pi$
$359$	−35.9131	−1.89542	−0.947710	+	0.319133i	$0.896608\pi$
$360$	0	0
$361$	−18.9953	−0.999751
$362$	−0.370840	−0.0194909
$363$	0	0
$364$	−6.90488	−0.361914
$365$	−7.16034	−0.374789
$366$	0	0
$367$	−9.03087	−0.471408	−0.235704	−	0.971825i	$-0.575740\pi$
$367$	−9.03087	−0.471408	−0.235704	+	0.971825i	$0.575740\pi$
$368$	−23.9013	−1.24594
$369$	0	0
$370$	−2.18551	−0.113619
$371$	−2.78596	−0.144640
$372$	0	0
$373$	−17.1994	−0.890553	−0.445277	−	0.895393i	$-0.646895\pi$
$373$	−17.1994	−0.890553	−0.445277	+	0.895393i	$0.646895\pi$
$374$	0	0
$375$	0	0
$376$	−0.936514	−0.0482970
$377$	4.18266	0.215418
$378$	0	0
$379$	−8.00208	−0.411039	−0.205520	−	0.978653i	$-0.565888\pi$
$379$	−8.00208	−0.411039	−0.205520	+	0.978653i	$0.565888\pi$
$380$	0.132927	0.00681903
$381$	0	0
$382$	1.28796	0.0658978
$383$	−29.7131	−1.51827	−0.759135	−	0.650933i	$-0.774378\pi$
$383$	−29.7131	−1.51827	−0.759135	+	0.650933i	$0.774378\pi$
$384$	0	0
$385$	0	0
$386$	2.55010	0.129797
$387$	0	0
$388$	7.90230	0.401178
$389$	0.595397	0.0301878	0.0150939	−	0.999886i	$-0.495195\pi$
$389$	0.595397	0.0301878	0.0150939	+	0.999886i	$0.495195\pi$
$390$	0	0
$391$	0.587103	0.0296911
$392$	3.29899	0.166624
$393$	0	0
$394$	2.33176	0.117472
$395$	0.791342	0.0398167
$396$	0	0
$397$	1.66950	0.0837898	0.0418949	−	0.999122i	$-0.486661\pi$
$397$	1.66950	0.0837898	0.0418949	+	0.999122i	$0.486661\pi$
$398$	−4.73247	−0.237217
$399$	0	0
$400$	3.59174	0.179587
$401$	11.3690	0.567741	0.283870	−	0.958863i	$-0.408381\pi$
$401$	11.3690	0.567741	0.283870	+	0.958863i	$0.408381\pi$
$402$	0	0
$403$	−10.7313	−0.534562
$404$	−8.28802	−0.412344
$405$	0	0
$406$	−3.13008	−0.155343
$407$	0	0
$408$	0	0
$409$	8.59766	0.425127	0.212563	−	0.977147i	$-0.431819\pi$
$409$	8.59766	0.425127	0.212563	+	0.977147i	$0.431819\pi$
$410$	−3.06060	−0.151152
$411$	0	0
$412$	−4.04156	−0.199113
$413$	−5.86345	−0.288522
$414$	0	0
$415$	−0.247229	−0.0121360
$416$	−3.36452	−0.164959
$417$	0	0
$418$	0	0
$419$	−31.3915	−1.53358	−0.766789	−	0.641899i	$-0.778147\pi$
$419$	−31.3915	−1.53358	−0.766789	+	0.641899i	$0.778147\pi$
$420$	0	0
$421$	−4.57893	−0.223163	−0.111582	−	0.993755i	$-0.535592\pi$
$421$	−4.57893	−0.223163	−0.111582	+	0.993755i	$0.535592\pi$
$422$	4.71665	0.229603
$423$	0	0
$424$	−0.899752	−0.0436958
$425$	−0.0882264	−0.00427961
$426$	0	0
$427$	32.1273	1.55475
$428$	30.3983	1.46936
$429$	0	0
$430$	3.10155	0.149570
$431$	15.9074	0.766234	0.383117	−	0.923700i	$-0.374851\pi$
$431$	15.9074	0.766234	0.383117	+	0.923700i	$0.374851\pi$
$432$	0	0
$433$	15.8030	0.759441	0.379721	−	0.925101i	$-0.376020\pi$
$433$	15.8030	0.759441	0.379721	+	0.925101i	$0.376020\pi$
$434$	8.03069	0.385485
$435$	0	0
$436$	−7.98096	−0.382218
$437$	−0.458047	−0.0219113
$438$	0	0
$439$	−21.5227	−1.02722	−0.513611	−	0.858023i	$-0.671693\pi$
$439$	−21.5227	−1.02722	−0.513611	+	0.858023i	$0.671693\pi$
$440$	0	0
$441$	0	0
$442$	0.0259156	0.00123268
$443$	−10.1769	−0.483519	−0.241760	−	0.970336i	$-0.577725\pi$
$443$	−10.1769	−0.483519	−0.241760	+	0.970336i	$0.577725\pi$
$444$	0	0
$445$	−14.5788	−0.691100
$446$	3.60717	0.170804
$447$	0	0
$448$	−20.4228	−0.964888
$449$	−38.6088	−1.82206	−0.911031	−	0.412338i	$-0.864712\pi$
$449$	−38.6088	−1.82206	−0.911031	+	0.412338i	$0.864712\pi$
$450$	0	0
$451$	0	0
$452$	26.5632	1.24943
$453$	0	0
$454$	−0.173875	−0.00816035
$455$	−3.57549	−0.167622
$456$	0	0
$457$	−23.3637	−1.09291	−0.546455	−	0.837488i	$-0.684023\pi$
$457$	−23.3637	−1.09291	−0.546455	+	0.837488i	$0.684023\pi$
$458$	1.96518	0.0918269
$459$	0	0
$460$	−12.8510	−0.599179
$461$	4.93120	0.229669	0.114834	−	0.993385i	$-0.463366\pi$
$461$	4.93120	0.229669	0.114834	+	0.993385i	$0.463366\pi$
$462$	0	0
$463$	26.9648	1.25316	0.626580	−	0.779357i	$-0.284454\pi$
$463$	26.9648	1.25316	0.626580	+	0.779357i	$0.284454\pi$
$464$	13.4181	0.622922
$465$	0	0
$466$	−2.76143	−0.127921
$467$	23.6778	1.09568	0.547839	−	0.836584i	$-0.315450\pi$
$467$	23.6778	1.09568	0.547839	+	0.836584i	$0.315450\pi$
$468$	0	0
$469$	30.4636	1.40668
$470$	−0.238228	−0.0109886
$471$	0	0
$472$	−1.89366	−0.0871627
$473$	0	0
$474$	0	0
$475$	0.0688326	0.00315826
$476$	0.544113	0.0249394
$477$	0	0
$478$	−4.48011	−0.204915
$479$	−13.5240	−0.617929	−0.308964	−	0.951074i	$-0.599982\pi$
$479$	−13.5240	−0.617929	−0.308964	+	0.951074i	$0.599982\pi$
$480$	0	0
$481$	−9.32655	−0.425254
$482$	−4.38356	−0.199666
$483$	0	0
$484$	0	0
$485$	4.09198	0.185807
$486$	0	0
$487$	−8.62704	−0.390928	−0.195464	−	0.980711i	$-0.562621\pi$
$487$	−8.62704	−0.390928	−0.195464	+	0.980711i	$0.562621\pi$
$488$	10.3758	0.469691
$489$	0	0
$490$	0.839188	0.0379106
$491$	−25.9068	−1.16916	−0.584579	−	0.811337i	$-0.698740\pi$
$491$	−25.9068	−1.16916	−0.584579	+	0.811337i	$0.698740\pi$
$492$	0	0
$493$	−0.329599	−0.0148444
$494$	−0.0202189	−0.000909690 0
$495$	0	0
$496$	−34.4263	−1.54579
$497$	14.8963	0.668193
$498$	0	0
$499$	−28.0475	−1.25558	−0.627790	−	0.778383i	$-0.716040\pi$
$499$	−28.0475	−1.25558	−0.627790	+	0.778383i	$0.716040\pi$
$500$	1.93117	0.0863644
$501$	0	0
$502$	−1.34792	−0.0601604
$503$	29.3999	1.31088	0.655438	−	0.755249i	$-0.272484\pi$
$503$	29.3999	1.31088	0.655438	+	0.755249i	$0.272484\pi$
$504$	0	0
$505$	−4.29171	−0.190979
$506$	0	0
$507$	0	0
$508$	15.2558	0.676867
$509$	19.3904	0.859463	0.429731	−	0.902957i	$-0.358608\pi$
$509$	19.3904	0.859463	0.429731	+	0.902957i	$0.358608\pi$
$510$	0	0
$511$	22.8667	1.01156
$512$	−18.2024	−0.804440
$513$	0	0
$514$	−0.970270	−0.0427968
$515$	−2.09280	−0.0922200
$516$	0	0
$517$	0	0
$518$	6.97949	0.306661
$519$	0	0
$520$	−1.15474	−0.0506387
$521$	−25.0062	−1.09554	−0.547771	−	0.836628i	$-0.684523\pi$
$521$	−25.0062	−1.09554	−0.547771	+	0.836628i	$0.684523\pi$
$522$	0	0
$523$	2.18761	0.0956577	0.0478288	−	0.998856i	$-0.484770\pi$
$523$	2.18761	0.0956577	0.0478288	+	0.998856i	$0.484770\pi$
$524$	31.5820	1.37967
$525$	0	0
$526$	−8.29883	−0.361846
$527$	0.845637	0.0368365
$528$	0	0
$529$	21.2824	0.925322
$530$	−0.228877	−0.00994177
$531$	0	0
$532$	−0.424507	−0.0184047
$533$	−13.0610	−0.565733
$534$	0	0
$535$	15.7409	0.680539
$536$	9.83850	0.424958
$537$	0	0
$538$	−3.87648	−0.167127
$539$	0	0
$540$	0	0
$541$	−13.5583	−0.582918	−0.291459	−	0.956583i	$-0.594141\pi$
$541$	−13.5583	−0.582918	−0.291459	+	0.956583i	$0.594141\pi$
$542$	−5.13882	−0.220731
$543$	0	0
$544$	0.265128	0.0113673
$545$	−4.13271	−0.177026
$546$	0	0
$547$	3.71282	0.158749	0.0793743	−	0.996845i	$-0.474708\pi$
$547$	3.71282	0.158749	0.0793743	+	0.996845i	$0.474708\pi$
$548$	14.9319	0.637861
$549$	0	0
$550$	0	0
$551$	0.257147	0.0109548
$552$	0	0
$553$	−2.52717	−0.107466
$554$	6.74536	0.286583
$555$	0	0
$556$	38.9835	1.65327
$557$	6.57520	0.278600	0.139300	−	0.990250i	$-0.455515\pi$
$557$	6.57520	0.278600	0.139300	+	0.990250i	$0.455515\pi$
$558$	0	0
$559$	13.2357	0.559810
$560$	−11.4703	−0.484710
$561$	0	0
$562$	7.00786	0.295609
$563$	−6.90876	−0.291170	−0.145585	−	0.989346i	$-0.546506\pi$
$563$	−6.90876	−0.291170	−0.145585	+	0.989346i	$0.546506\pi$
$564$	0	0
$565$	13.7550	0.578678
$566$	1.89784	0.0797723
$567$	0	0
$568$	4.81092	0.201862
$569$	−6.74768	−0.282878	−0.141439	−	0.989947i	$-0.545173\pi$
$569$	−6.74768	−0.282878	−0.141439	+	0.989947i	$0.545173\pi$
$570$	0	0
$571$	−9.68928	−0.405484	−0.202742	−	0.979232i	$-0.564985\pi$
$571$	−9.68928	−0.405484	−0.202742	+	0.979232i	$0.564985\pi$
$572$	0	0
$573$	0	0
$574$	9.77411	0.407963
$575$	−6.65450	−0.277512
$576$	0	0
$577$	15.1498	0.630695	0.315348	−	0.948976i	$-0.397879\pi$
$577$	15.1498	0.630695	0.315348	+	0.948976i	$0.397879\pi$
$578$	4.45807	0.185431
$579$	0	0
$580$	7.21451	0.299566
$581$	0.789532	0.0327553
$582$	0	0
$583$	0	0
$584$	7.38503	0.305595
$585$	0	0
$586$	−3.82907	−0.158177
$587$	27.0112	1.11487	0.557436	−	0.830220i	$-0.311785\pi$
$587$	27.0112	1.11487	0.557436	+	0.830220i	$0.311785\pi$
$588$	0	0
$589$	−0.659750	−0.0271845
$590$	−0.481704	−0.0198314
$591$	0	0
$592$	−29.9200	−1.22970
$593$	−22.1863	−0.911084	−0.455542	−	0.890214i	$-0.650554\pi$
$593$	−22.1863	−0.911084	−0.455542	+	0.890214i	$0.650554\pi$
$594$	0	0
$595$	0.281754	0.0115508
$596$	−18.7231	−0.766929
$597$	0	0
$598$	1.95469	0.0799332
$599$	−11.7487	−0.480038	−0.240019	−	0.970768i	$-0.577154\pi$
$599$	−11.7487	−0.480038	−0.240019	+	0.970768i	$0.577154\pi$
$600$	0	0
$601$	−34.3806	−1.40242	−0.701208	−	0.712957i	$-0.747355\pi$
$601$	−34.3806	−1.40242	−0.701208	+	0.712957i	$0.747355\pi$
$602$	−9.90488	−0.403693
$603$	0	0
$604$	29.2793	1.19136
$605$	0	0
$606$	0	0
$607$	−13.6298	−0.553215	−0.276607	−	0.960983i	$-0.589210\pi$
$607$	−13.6298	−0.553215	−0.276607	+	0.960983i	$0.589210\pi$
$608$	−0.206848	−0.00838880
$609$	0	0
$610$	2.63937	0.106865
$611$	−1.01663	−0.0411283
$612$	0	0
$613$	6.82994	0.275858	0.137929	−	0.990442i	$-0.455955\pi$
$613$	6.82994	0.275858	0.137929	+	0.990442i	$0.455955\pi$
$614$	7.48892	0.302228
$615$	0	0
$616$	0	0
$617$	−30.3730	−1.22277	−0.611386	−	0.791333i	$-0.709388\pi$
$617$	−30.3730	−1.22277	−0.611386	+	0.791333i	$0.709388\pi$
$618$	0	0
$619$	−25.7879	−1.03650	−0.518252	−	0.855228i	$-0.673417\pi$
$619$	−25.7879	−1.03650	−0.518252	+	0.855228i	$0.673417\pi$
$620$	−18.5099	−0.743377
$621$	0	0
$622$	−2.60996	−0.104650
$623$	46.5577	1.86529
$624$	0	0
$625$	1.00000	0.0400000
$626$	4.31081	0.172295
$627$	0	0
$628$	28.7368	1.14672
$629$	0.734945	0.0293042
$630$	0	0
$631$	−25.5656	−1.01775	−0.508876	−	0.860840i	$-0.669939\pi$
$631$	−25.5656	−1.01775	−0.508876	+	0.860840i	$0.669939\pi$
$632$	−0.816174	−0.0324657
$633$	0	0
$634$	3.72829	0.148069
$635$	7.89979	0.313494
$636$	0	0
$637$	3.58119	0.141892
$638$	0	0
$639$	0	0
$640$	−7.68799	−0.303894
$641$	4.07262	0.160859	0.0804294	−	0.996760i	$-0.474371\pi$
$641$	4.07262	0.160859	0.0804294	+	0.996760i	$0.474371\pi$
$642$	0	0
$643$	11.3298	0.446804	0.223402	−	0.974726i	$-0.428284\pi$
$643$	11.3298	0.446804	0.223402	+	0.974726i	$0.428284\pi$
$644$	41.0399	1.61720
$645$	0	0
$646$	0.00159327	6.26865e−5 0
$647$	36.8480	1.44864	0.724322	−	0.689462i	$-0.242153\pi$
$647$	36.8480	1.44864	0.724322	+	0.689462i	$0.242153\pi$
$648$	0	0
$649$	0	0
$650$	−0.293740	−0.0115214
$651$	0	0
$652$	−6.56818	−0.257230
$653$	−15.2768	−0.597827	−0.298914	−	0.954280i	$-0.596624\pi$
$653$	−15.2768	−0.597827	−0.298914	+	0.954280i	$0.596624\pi$
$654$	0	0
$655$	16.3539	0.638998
$656$	−41.9001	−1.63592
$657$	0	0
$658$	0.760787	0.0296586
$659$	11.8996	0.463544	0.231772	−	0.972770i	$-0.425548\pi$
$659$	11.8996	0.463544	0.231772	+	0.972770i	$0.425548\pi$
$660$	0	0
$661$	−44.8648	−1.74504	−0.872520	−	0.488578i	$-0.837516\pi$
$661$	−44.8648	−1.74504	−0.872520	+	0.488578i	$0.837516\pi$
$662$	2.86284	0.111268
$663$	0	0
$664$	0.254987	0.00989542
$665$	−0.219819	−0.00852421
$666$	0	0
$667$	−24.8601	−0.962587
$668$	43.6760	1.68987
$669$	0	0
$670$	2.50269	0.0966874
$671$	0	0
$672$	0	0
$673$	19.1067	0.736508	0.368254	−	0.929725i	$-0.379956\pi$
$673$	19.1067	0.736508	0.368254	+	0.929725i	$0.379956\pi$
$674$	−1.54427	−0.0594832
$675$	0	0
$676$	22.6844	0.872478
$677$	46.4614	1.78566	0.892829	−	0.450396i	$-0.148717\pi$
$677$	46.4614	1.78566	0.892829	+	0.450396i	$0.148717\pi$
$678$	0	0
$679$	−13.0678	−0.501498
$680$	0.0909950	0.00348950
$681$	0	0
$682$	0	0
$683$	42.5651	1.62871	0.814355	−	0.580367i	$-0.197091\pi$
$683$	42.5651	1.62871	0.814355	+	0.580367i	$0.197091\pi$
$684$	0	0
$685$	7.73208	0.295427
$686$	3.18500	0.121604
$687$	0	0
$688$	42.4606	1.61880
$689$	−0.976719	−0.0372100
$690$	0	0
$691$	35.1352	1.33660	0.668302	−	0.743890i	$-0.267021\pi$
$691$	35.1352	1.33660	0.668302	+	0.743890i	$0.267021\pi$
$692$	25.2815	0.961057
$693$	0	0
$694$	−9.05253	−0.343629
$695$	20.1865	0.765716
$696$	0	0
$697$	1.02922	0.0389845
$698$	−1.44654	−0.0547522
$699$	0	0
$700$	−6.16724	−0.233100
$701$	−30.6713	−1.15844	−0.579219	−	0.815172i	$-0.696642\pi$
$701$	−30.6713	−1.15844	−0.579219	+	0.815172i	$0.696642\pi$
$702$	0	0
$703$	−0.573390	−0.0216258
$704$	0	0
$705$	0	0
$706$	−7.83773	−0.294977
$707$	13.7057	0.515456
$708$	0	0
$709$	15.0924	0.566806	0.283403	−	0.959001i	$-0.408537\pi$
$709$	15.0924	0.566806	0.283403	+	0.959001i	$0.408537\pi$
$710$	1.22379	0.0459280
$711$	0	0
$712$	15.0363	0.563507
$713$	63.7824	2.38867
$714$	0	0
$715$	0	0
$716$	−29.3101	−1.09537
$717$	0	0
$718$	9.42215	0.351631
$719$	6.19020	0.230856	0.115428	−	0.993316i	$-0.463176\pi$
$719$	6.19020	0.230856	0.115428	+	0.993316i	$0.463176\pi$
$720$	0	0
$721$	6.68343	0.248904
$722$	4.98359	0.185470
$723$	0	0
$724$	−2.72967	−0.101447
$725$	3.73583	0.138745
$726$	0	0
$727$	−18.1515	−0.673200	−0.336600	−	0.941648i	$-0.609277\pi$
$727$	−18.1515	−0.673200	−0.336600	+	0.941648i	$0.609277\pi$
$728$	3.68769	0.136675
$729$	0	0
$730$	1.87858	0.0695295
$731$	−1.04299	−0.0385764
$732$	0	0
$733$	28.7409	1.06157	0.530785	−	0.847507i	$-0.321897\pi$
$733$	28.7409	1.06157	0.530785	+	0.847507i	$0.321897\pi$
$734$	2.36934	0.0874538
$735$	0	0
$736$	19.9974	0.737113
$737$	0	0
$738$	0	0
$739$	46.8341	1.72282	0.861411	−	0.507909i	$-0.169581\pi$
$739$	46.8341	1.72282	0.861411	+	0.507909i	$0.169581\pi$
$740$	−16.0870	−0.591371
$741$	0	0
$742$	0.730924	0.0268331
$743$	26.6108	0.976254	0.488127	−	0.872773i	$-0.337680\pi$
$743$	26.6108	0.976254	0.488127	+	0.872773i	$0.337680\pi$
$744$	0	0
$745$	−9.69523	−0.355206
$746$	4.51244	0.165212
$747$	0	0
$748$	0	0
$749$	−50.2691	−1.83679
$750$	0	0
$751$	40.5821	1.48086	0.740431	−	0.672132i	$-0.234621\pi$
$751$	40.5821	1.48086	0.740431	+	0.672132i	$0.234621\pi$
$752$	−3.26137	−0.118930
$753$	0	0
$754$	−1.09736	−0.0399635
$755$	15.1615	0.551782
$756$	0	0
$757$	32.7179	1.18915	0.594576	−	0.804039i	$-0.297320\pi$
$757$	32.7179	1.18915	0.594576	+	0.804039i	$0.297320\pi$
$758$	2.09942	0.0762545
$759$	0	0
$760$	−0.0709926	−0.00257517
$761$	−2.62938	−0.0953148	−0.0476574	−	0.998864i	$-0.515176\pi$
$761$	−2.62938	−0.0953148	−0.0476574	+	0.998864i	$0.515176\pi$
$762$	0	0
$763$	13.1979	0.477797
$764$	9.48037	0.342988
$765$	0	0
$766$	7.79553	0.281664
$767$	−2.05565	−0.0742251
$768$	0	0
$769$	−3.23559	−0.116678	−0.0583392	−	0.998297i	$-0.518581\pi$
$769$	−3.23559	−0.116678	−0.0583392	+	0.998297i	$0.518581\pi$
$770$	0	0
$771$	0	0
$772$	18.7707	0.675572
$773$	−43.6365	−1.56950	−0.784748	−	0.619815i	$-0.787208\pi$
$773$	−43.6365	−1.56950	−0.784748	+	0.619815i	$0.787208\pi$
$774$	0	0
$775$	−9.58484	−0.344298
$776$	−4.22039	−0.151503
$777$	0	0
$778$	−0.156208	−0.00560033
$779$	−0.802978	−0.0287697
$780$	0	0
$781$	0	0
$782$	−0.154032	−0.00550818
$783$	0	0
$784$	11.4886	0.410307
$785$	14.8805	0.531109
$786$	0	0
$787$	4.56420	0.162696	0.0813480	−	0.996686i	$-0.474077\pi$
$787$	4.56420	0.162696	0.0813480	+	0.996686i	$0.474077\pi$
$788$	17.1635	0.611425
$789$	0	0
$790$	−0.207616	−0.00738666
$791$	−43.9270	−1.56186
$792$	0	0
$793$	11.2634	0.399974
$794$	−0.438009	−0.0155444
$795$	0	0
$796$	−34.8346	−1.23468
$797$	11.1806	0.396036	0.198018	−	0.980198i	$-0.436550\pi$
$797$	11.1806	0.396036	0.198018	+	0.980198i	$0.436550\pi$
$798$	0	0
$799$	0.0801114	0.00283414
$800$	−3.00509	−0.106246
$801$	0	0
$802$	−2.98277	−0.105325
$803$	0	0
$804$	0	0
$805$	21.2513	0.749011
$806$	2.81545	0.0991699
$807$	0	0
$808$	4.42639	0.155720
$809$	−45.5417	−1.60116	−0.800581	−	0.599225i	$-0.795475\pi$
$809$	−45.5417	−1.60116	−0.800581	+	0.599225i	$0.795475\pi$
$810$	0	0
$811$	−2.82363	−0.0991511	−0.0495755	−	0.998770i	$-0.515787\pi$
$811$	−2.82363	−0.0991511	−0.0495755	+	0.998770i	$0.515787\pi$
$812$	−23.0397	−0.808537
$813$	0	0
$814$	0	0
$815$	−3.40114	−0.119137
$816$	0	0
$817$	0.813721	0.0284685
$818$	−2.25568	−0.0788679
$819$	0	0
$820$	−22.5283	−0.786724
$821$	−23.4117	−0.817073	−0.408536	−	0.912742i	$-0.633961\pi$
$821$	−23.4117	−0.817073	−0.408536	+	0.912742i	$0.633961\pi$
$822$	0	0
$823$	46.1046	1.60711	0.803553	−	0.595233i	$-0.202940\pi$
$823$	46.1046	1.60711	0.803553	+	0.595233i	$0.202940\pi$
$824$	2.15848	0.0751941
$825$	0	0
$826$	1.53833	0.0535255
$827$	−43.4496	−1.51089	−0.755445	−	0.655212i	$-0.772579\pi$
$827$	−43.4496	−1.51089	−0.755445	+	0.655212i	$0.772579\pi$
$828$	0	0
$829$	15.4712	0.537335	0.268668	−	0.963233i	$-0.413417\pi$
$829$	15.4712	0.537335	0.268668	+	0.963233i	$0.413417\pi$
$830$	0.0648629	0.00225142
$831$	0	0
$832$	−7.15996	−0.248227
$833$	−0.282202	−0.00977773
$834$	0	0
$835$	22.6164	0.782671
$836$	0	0
$837$	0	0
$838$	8.23588	0.284504
$839$	−30.2181	−1.04324	−0.521622	−	0.853177i	$-0.674673\pi$
$839$	−30.2181	−1.04324	−0.521622	+	0.853177i	$0.674673\pi$
$840$	0	0
$841$	−15.0436	−0.518744
$842$	1.20133	0.0414005
$843$	0	0
$844$	34.7182	1.19505
$845$	11.7465	0.404091
$846$	0	0
$847$	0	0
$848$	−3.13335	−0.107600
$849$	0	0
$850$	0.0231471	0.000793938 0
$851$	55.4334	1.90023
$852$	0	0
$853$	2.00908	0.0687897	0.0343949	−	0.999408i	$-0.489050\pi$
$853$	2.00908	0.0687897	0.0343949	+	0.999408i	$0.489050\pi$
$854$	−8.42890	−0.288431
$855$	0	0
$856$	−16.2349	−0.554896
$857$	−33.6095	−1.14808	−0.574039	−	0.818828i	$-0.694624\pi$
$857$	−33.6095	−1.14808	−0.574039	+	0.818828i	$0.694624\pi$
$858$	0	0
$859$	−13.7301	−0.468466	−0.234233	−	0.972180i	$-0.575258\pi$
$859$	−13.7301	−0.468466	−0.234233	+	0.972180i	$0.575258\pi$
$860$	22.8298	0.778488
$861$	0	0
$862$	−4.17347	−0.142149
$863$	−8.75829	−0.298136	−0.149068	−	0.988827i	$-0.547627\pi$
$863$	−8.75829	−0.298136	−0.149068	+	0.988827i	$0.547627\pi$
$864$	0	0
$865$	13.0913	0.445117
$866$	−4.14606	−0.140889
$867$	0	0
$868$	59.1120	2.00639
$869$	0	0
$870$	0	0
$871$	10.6801	0.361881
$872$	4.26239	0.144343
$873$	0	0
$874$	0.120173	0.00406491
$875$	−3.19353	−0.107961
$876$	0	0
$877$	15.2463	0.514830	0.257415	−	0.966301i	$-0.417129\pi$
$877$	15.2463	0.514830	0.257415	+	0.966301i	$0.417129\pi$
$878$	5.64669	0.190566
$879$	0	0
$880$	0	0
$881$	15.8486	0.533953	0.266977	−	0.963703i	$-0.413975\pi$
$881$	15.8486	0.533953	0.266977	+	0.963703i	$0.413975\pi$
$882$	0	0
$883$	−32.9606	−1.10921	−0.554606	−	0.832113i	$-0.687131\pi$
$883$	−32.9606	−1.10921	−0.554606	+	0.832113i	$0.687131\pi$
$884$	0.190759	0.00641590
$885$	0	0
$886$	2.67001	0.0897007
$887$	11.2525	0.377820	0.188910	−	0.981994i	$-0.439505\pi$
$887$	11.2525	0.377820	0.188910	+	0.981994i	$0.439505\pi$
$888$	0	0
$889$	−25.2282	−0.846126
$890$	3.82488	0.128210
$891$	0	0
$892$	26.5515	0.889010
$893$	−0.0625014	−0.00209153
$894$	0	0
$895$	−15.1774	−0.507325
$896$	24.5518	0.820218
$897$	0	0
$898$	10.1294	0.338022
$899$	−35.8074	−1.19424
$900$	0	0
$901$	0.0769667	0.00256413
$902$	0	0
$903$	0	0
$904$	−14.1866	−0.471841
$905$	−1.41348	−0.0469857
$906$	0	0
$907$	−25.3745	−0.842546	−0.421273	−	0.906934i	$-0.638417\pi$
$907$	−25.3745	−0.842546	−0.421273	+	0.906934i	$0.638417\pi$
$908$	−1.27985	−0.0424734
$909$	0	0
$910$	0.938065	0.0310966
$911$	17.4070	0.576718	0.288359	−	0.957522i	$-0.406890\pi$
$911$	17.4070	0.576718	0.288359	+	0.957522i	$0.406890\pi$
$912$	0	0
$913$	0	0
$914$	6.12971	0.202753
$915$	0	0
$916$	14.4652	0.477945
$917$	−52.2265	−1.72467
$918$	0	0
$919$	32.8958	1.08513	0.542566	−	0.840013i	$-0.317453\pi$
$919$	32.8958	1.08513	0.542566	+	0.840013i	$0.317453\pi$
$920$	6.86332	0.226277
$921$	0	0
$922$	−1.29375	−0.0426073
$923$	5.22245	0.171899
$924$	0	0
$925$	−8.33021	−0.273896
$926$	−7.07448	−0.232482
$927$	0	0
$928$	−11.2265	−0.368528
$929$	−33.3131	−1.09297	−0.546484	−	0.837469i	$-0.684034\pi$
$929$	−33.3131	−1.09297	−0.546484	+	0.837469i	$0.684034\pi$
$930$	0	0
$931$	0.220169	0.00721575
$932$	−20.3262	−0.665808
$933$	0	0
$934$	−6.21211	−0.203266
$935$	0	0
$936$	0	0
$937$	23.8637	0.779593	0.389797	−	0.920901i	$-0.372545\pi$
$937$	23.8637	0.779593	0.389797	+	0.920901i	$0.372545\pi$
$938$	−7.99241	−0.260961
$939$	0	0
$940$	−1.75354	−0.0571941
$941$	−7.92557	−0.258366	−0.129183	−	0.991621i	$-0.541235\pi$
$941$	−7.92557	−0.258366	−0.129183	+	0.991621i	$0.541235\pi$
$942$	0	0
$943$	77.6292	2.52795
$944$	−6.59459	−0.214636
$945$	0	0
$946$	0	0
$947$	2.15429	0.0700050	0.0350025	−	0.999387i	$-0.488856\pi$
$947$	2.15429	0.0700050	0.0350025	+	0.999387i	$0.488856\pi$
$948$	0	0
$949$	8.01676	0.260235
$950$	−0.0180589	−0.000585908 0
$951$	0	0
$952$	−0.290595	−0.00941824
$953$	−5.99303	−0.194133	−0.0970666	−	0.995278i	$-0.530946\pi$
$953$	−5.99303	−0.194133	−0.0970666	+	0.995278i	$0.530946\pi$
$954$	0	0
$955$	4.90914	0.158856
$956$	−32.9770	−1.06655
$957$	0	0
$958$	3.54816	0.114636
$959$	−24.6926	−0.797366
$960$	0	0
$961$	60.8692	1.96352
$962$	2.44691	0.0788916
$963$	0	0
$964$	−32.2663	−1.03923
$965$	9.71987	0.312894
$966$	0	0
$967$	−22.2784	−0.716424	−0.358212	−	0.933640i	$-0.616613\pi$
$967$	−22.2784	−0.716424	−0.358212	+	0.933640i	$0.616613\pi$
$968$	0	0
$969$	0	0
$970$	−1.07357	−0.0344703
$971$	36.6909	1.17747	0.588734	−	0.808327i	$-0.299626\pi$
$971$	36.6909	1.17747	0.588734	+	0.808327i	$0.299626\pi$
$972$	0	0
$973$	−64.4660	−2.06669
$974$	2.26339	0.0725236
$975$	0	0
$976$	36.1334	1.15660
$977$	−27.9044	−0.892740	−0.446370	−	0.894848i	$-0.647283\pi$
$977$	−27.9044	−0.892740	−0.446370	+	0.894848i	$0.647283\pi$
$978$	0	0
$979$	0	0
$980$	6.17706	0.197319
$981$	0	0
$982$	6.79691	0.216898
$983$	−9.79220	−0.312323	−0.156161	−	0.987732i	$-0.549912\pi$
$983$	−9.79220	−0.312323	−0.156161	+	0.987732i	$0.549912\pi$
$984$	0	0
$985$	8.88764	0.283184
$986$	0.0864735	0.00275388
$987$	0	0
$988$	−0.148826	−0.00473479
$989$	−78.6678	−2.50149
$990$	0	0
$991$	34.9794	1.11116	0.555578	−	0.831464i	$-0.312497\pi$
$991$	34.9794	1.11116	0.555578	+	0.831464i	$0.312497\pi$
$992$	28.8033	0.914506
$993$	0	0
$994$	−3.90820	−0.123961
$995$	−18.0381	−0.571846
$996$	0	0
$997$	12.7753	0.404599	0.202299	−	0.979324i	$-0.435159\pi$
$997$	12.7753	0.404599	0.202299	+	0.979324i	$0.435159\pi$
$998$	7.35855	0.232931
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5445.2.a.bf.1.3		4
3.2	odd	2		1815.2.a.w.1.2		4
11.2	odd	10		495.2.n.a.136.2		8
11.6	odd	10		495.2.n.a.91.2		8
11.10	odd	2		5445.2.a.bt.1.2		4
15.14	odd	2		9075.2.a.cm.1.3		4
33.2	even	10		165.2.m.d.136.1	yes	8
33.17	even	10		165.2.m.d.91.1	✓	8
33.32	even	2		1815.2.a.p.1.3		4
165.2	odd	20		825.2.bx.f.499.2		16
165.17	odd	20		825.2.bx.f.124.3		16
165.68	odd	20		825.2.bx.f.499.3		16
165.83	odd	20		825.2.bx.f.124.2		16
165.134	even	10		825.2.n.g.301.2		8
165.149	even	10		825.2.n.g.751.2		8
165.164	even	2		9075.2.a.di.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.m.d.91.1	✓	8	33.17	even	10
165.2.m.d.136.1	yes	8	33.2	even	10
495.2.n.a.91.2		8	11.6	odd	10
495.2.n.a.136.2		8	11.2	odd	10
825.2.n.g.301.2		8	165.134	even	10
825.2.n.g.751.2		8	165.149	even	10
825.2.bx.f.124.2		16	165.83	odd	20
825.2.bx.f.124.3		16	165.17	odd	20
825.2.bx.f.499.2		16	165.2	odd	20
825.2.bx.f.499.3		16	165.68	odd	20
1815.2.a.p.1.3		4	33.32	even	2
1815.2.a.w.1.2		4	3.2	odd	2
5445.2.a.bf.1.3		4	1.1	even	1	trivial
5445.2.a.bt.1.2		4	11.10	odd	2
9075.2.a.cm.1.3		4	15.14	odd	2
9075.2.a.di.1.2		4	165.164	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 5445 = 3^{2} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5445.a (trivial)

\(f(q)\)	\(=\)	\(q-0.262360 q^{2} -1.93117 q^{4} -1.00000 q^{5} +3.19353 q^{7} +1.03138 q^{8} +O(q^{10})\)	\(q-0.262360 q^{2} -1.93117 q^{4} -1.00000 q^{5} +3.19353 q^{7} +1.03138 q^{8} +0.262360 q^{10} +1.11961 q^{13} -0.837853 q^{14} +3.59174 q^{16} -0.0882264 q^{17} +0.0688326 q^{19} +1.93117 q^{20} -6.65450 q^{23} +1.00000 q^{25} -0.293740 q^{26} -6.16724 q^{28} +3.73583 q^{29} -9.58484 q^{31} -3.00509 q^{32} +0.0231471 q^{34} -3.19353 q^{35} -8.33021 q^{37} -0.0180589 q^{38} -1.03138 q^{40} -11.6657 q^{41} +11.8217 q^{43} +1.74587 q^{46} -0.908020 q^{47} +3.19862 q^{49} -0.262360 q^{50} -2.16215 q^{52} -0.872377 q^{53} +3.29374 q^{56} -0.980131 q^{58} -1.83604 q^{59} +10.0601 q^{61} +2.51468 q^{62} -6.39507 q^{64} -1.11961 q^{65} +9.53916 q^{67} +0.170380 q^{68} +0.837853 q^{70} +4.66454 q^{71} +7.16034 q^{73} +2.18551 q^{74} -0.132927 q^{76} -0.791342 q^{79} -3.59174 q^{80} +3.06060 q^{82} +0.247229 q^{83} +0.0882264 q^{85} -3.10155 q^{86} +14.5788 q^{89} +3.57549 q^{91} +12.8510 q^{92} +0.238228 q^{94} -0.0688326 q^{95} -4.09198 q^{97} -0.839188 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 3 q^{2} + q^{4} - 4 q^{5} + 6 q^{7} - 3 q^{8}+O(q^{10}) \) 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

Introduction

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