LMFDB - Embedded newform 4005.2.a.n.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4005,2,Mod(1,4005)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4005, 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("4005.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4005 = 3^{2} \cdot 5 \cdot 89 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4005.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:	$31.9800860095$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.10407557.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 8x^{4} + 2x^{3} + 18x^{2} + 7x - 2 $ x^6 - x^5 - 8x^4 + 2x^3 + 18x^2 + 7x - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1335)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.43605$ of defining polynomial
Character	$\chi$	$=$	4005.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.49828 q^{2} +0.244835 q^{4} +1.00000 q^{5} -3.23110 q^{7} -2.62972 q^{8} +O(q^{10})$	$q+1.49828 q^{2} +0.244835 q^{4} +1.00000 q^{5} -3.23110 q^{7} -2.62972 q^{8} +1.49828 q^{10} +2.15654 q^{13} -4.84109 q^{14} -4.42973 q^{16} +2.54656 q^{17} +2.00000 q^{19} +0.244835 q^{20} -2.98282 q^{23} +1.00000 q^{25} +3.23110 q^{26} -0.791087 q^{28} -1.21629 q^{29} +8.46220 q^{31} -1.37751 q^{32} +3.81545 q^{34} -3.23110 q^{35} -0.546560 q^{37} +2.99655 q^{38} -2.62972 q^{40} -2.84484 q^{41} +6.19911 q^{43} -4.46909 q^{46} +9.73799 q^{47} +3.44001 q^{49} +1.49828 q^{50} +0.527997 q^{52} +11.0249 q^{53} +8.49690 q^{56} -1.82234 q^{58} -2.82628 q^{59} +2.28347 q^{61} +12.6787 q^{62} +6.79556 q^{64} +2.15654 q^{65} -1.95665 q^{67} +0.623487 q^{68} -4.84109 q^{70} +5.79380 q^{71} -12.9406 q^{73} -0.818898 q^{74} +0.489670 q^{76} +8.06481 q^{79} -4.42973 q^{80} -4.26236 q^{82} +3.47938 q^{83} +2.54656 q^{85} +9.28799 q^{86} +1.00000 q^{89} -6.96801 q^{91} -0.730299 q^{92} +14.5902 q^{94} +2.00000 q^{95} +3.30280 q^{97} +5.15410 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{2} + 8 q^{4} + 6 q^{5} + q^{7} + 9 q^{8}+O(q^{10}) $ 6 * q + 4 * q^2 + 8 * q^4 + 6 * q^5 + q^7 + 9 * q^8	$ 6 q + 4 q^{2} + 8 q^{4} + 6 q^{5} + q^{7} + 9 q^{8} + 4 q^{10} - 3 q^{13} + 5 q^{14} + 12 q^{16} + 13 q^{17} + 12 q^{19} + 8 q^{20} + 19 q^{23} + 6 q^{25} - q^{26} - 6 q^{28} + 10 q^{31} + 17 q^{32} - 2 q^{34} + q^{35} - q^{37} + 8 q^{38} + 9 q^{40} + 4 q^{41} - 7 q^{43} - 6 q^{46} + 15 q^{47} - q^{49} + 4 q^{50} + q^{52} + 27 q^{53} + 14 q^{56} - 6 q^{58} + 4 q^{59} + 8 q^{61} - 2 q^{62} - q^{64} - 3 q^{65} - 11 q^{67} + 47 q^{68} + 5 q^{70} + 16 q^{71} + q^{73} + 10 q^{74} + 16 q^{76} + 12 q^{80} + q^{82} + 17 q^{83} + 13 q^{85} + 20 q^{86} + 6 q^{89} - 18 q^{91} + 36 q^{92} + 17 q^{94} + 12 q^{95} - 29 q^{97} + 23 q^{98}+O(q^{100}) $ 6 * q + 4 * q^2 + 8 * q^4 + 6 * q^5 + q^7 + 9 * q^8 + 4 * q^10 - 3 * q^13 + 5 * q^14 + 12 * q^16 + 13 * q^17 + 12 * q^19 + 8 * q^20 + 19 * q^23 + 6 * q^25 - q^26 - 6 * q^28 + 10 * q^31 + 17 * q^32 - 2 * q^34 + q^35 - q^37 + 8 * q^38 + 9 * q^40 + 4 * q^41 - 7 * q^43 - 6 * q^46 + 15 * q^47 - q^49 + 4 * q^50 + q^52 + 27 * q^53 + 14 * q^56 - 6 * q^58 + 4 * q^59 + 8 * q^61 - 2 * q^62 - q^64 - 3 * q^65 - 11 * q^67 + 47 * q^68 + 5 * q^70 + 16 * q^71 + q^73 + 10 * q^74 + 16 * q^76 + 12 * q^80 + q^82 + 17 * q^83 + 13 * q^85 + 20 * q^86 + 6 * q^89 - 18 * q^91 + 36 * q^92 + 17 * q^94 + 12 * q^95 - 29 * q^97 + 23 * 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 $

$2$	1.49828	1.05944	0.529721	−	0.848172i	$-0.322297\pi$
$2$	1.49828	1.05944	0.529721	+	0.848172i	$0.322297\pi$
$3$	0	0
$3$	0	0
$4$	0.244835	0.122418
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−3.23110	−1.22124	−0.610621	−	0.791923i	$-0.709080\pi$
$7$	−3.23110	−1.22124	−0.610621	+	0.791923i	$0.709080\pi$
$8$	−2.62972	−0.929748
$9$	0	0
$10$	1.49828	0.473797
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	2.15654	0.598118	0.299059	−	0.954235i	$-0.403327\pi$
$13$	2.15654	0.598118	0.299059	+	0.954235i	$0.403327\pi$
$14$	−4.84109	−1.29383
$15$	0	0
$16$	−4.42973	−1.10743
$17$	2.54656	0.617631	0.308816	−	0.951122i	$-0.400067\pi$
$17$	2.54656	0.617631	0.308816	+	0.951122i	$0.400067\pi$
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0.244835	0.0547468
$21$	0	0
$22$	0	0
$23$	−2.98282	−0.621961	−0.310981	−	0.950416i	$-0.600657\pi$
$23$	−2.98282	−0.621961	−0.310981	+	0.950416i	$0.600657\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	3.23110	0.633671
$27$	0	0
$28$	−0.791087	−0.149501
$29$	−1.21629	−0.225860	−0.112930	−	0.993603i	$-0.536024\pi$
$29$	−1.21629	−0.225860	−0.112930	+	0.993603i	$0.536024\pi$
$30$	0	0
$31$	8.46220	1.51986	0.759928	−	0.650007i	$-0.225234\pi$
$31$	8.46220	1.51986	0.759928	+	0.650007i	$0.225234\pi$
$32$	−1.37751	−0.243512
$33$	0	0
$34$	3.81545	0.654345
$35$	−3.23110	−0.546156
$36$	0	0
$37$	−0.546560	−0.0898538	−0.0449269	−	0.998990i	$-0.514305\pi$
$37$	−0.546560	−0.0898538	−0.0449269	+	0.998990i	$0.514305\pi$
$38$	2.99655	0.486105
$39$	0	0
$40$	−2.62972	−0.415796
$41$	−2.84484	−0.444289	−0.222145	−	0.975014i	$-0.571306\pi$
$41$	−2.84484	−0.444289	−0.222145	+	0.975014i	$0.571306\pi$
$42$	0	0
$43$	6.19911	0.945356	0.472678	−	0.881235i	$-0.343287\pi$
$43$	6.19911	0.945356	0.472678	+	0.881235i	$0.343287\pi$
$44$	0	0
$45$	0	0
$46$	−4.46909	−0.658932
$47$	9.73799	1.42043	0.710216	−	0.703984i	$-0.248597\pi$
$47$	9.73799	1.42043	0.710216	+	0.703984i	$0.248597\pi$
$48$	0	0
$49$	3.44001	0.491431
$50$	1.49828	0.211888
$51$	0	0
$52$	0.527997	0.0732201
$53$	11.0249	1.51439	0.757193	−	0.653191i	$-0.226570\pi$
$53$	11.0249	1.51439	0.757193	+	0.653191i	$0.226570\pi$
$54$	0	0
$55$	0	0
$56$	8.49690	1.13545
$57$	0	0
$58$	−1.82234	−0.239285
$59$	−2.82628	−0.367950	−0.183975	−	0.982931i	$-0.558897\pi$
$59$	−2.82628	−0.367950	−0.183975	+	0.982931i	$0.558897\pi$
$60$	0	0
$61$	2.28347	0.292368	0.146184	−	0.989257i	$-0.453301\pi$
$61$	2.28347	0.292368	0.146184	+	0.989257i	$0.453301\pi$
$62$	12.6787	1.61020
$63$	0	0
$64$	6.79556	0.849445
$65$	2.15654	0.267486
$66$	0	0
$67$	−1.95665	−0.239043	−0.119521	−	0.992832i	$-0.538136\pi$
$67$	−1.95665	−0.239043	−0.119521	+	0.992832i	$0.538136\pi$
$68$	0.623487	0.0756089
$69$	0	0
$70$	−4.84109	−0.578620
$71$	5.79380	0.687598	0.343799	−	0.939043i	$-0.388286\pi$
$71$	5.79380	0.687598	0.343799	+	0.939043i	$0.388286\pi$
$72$	0	0
$73$	−12.9406	−1.51458	−0.757291	−	0.653078i	$-0.773478\pi$
$73$	−12.9406	−1.51458	−0.757291	+	0.653078i	$0.773478\pi$
$74$	−0.818898	−0.0951949
$75$	0	0
$76$	0.489670	0.0561690
$77$	0	0
$78$	0	0
$79$	8.06481	0.907362	0.453681	−	0.891164i	$-0.350111\pi$
$79$	8.06481	0.907362	0.453681	+	0.891164i	$0.350111\pi$
$80$	−4.42973	−0.495258
$81$	0	0
$82$	−4.26236	−0.470699
$83$	3.47938	0.381912	0.190956	−	0.981599i	$-0.438841\pi$
$83$	3.47938	0.381912	0.190956	+	0.981599i	$0.438841\pi$
$84$	0	0
$85$	2.54656	0.276213
$86$	9.28799	1.00155
$87$	0	0
$88$	0	0
$89$	1.00000	0.106000
$89$	1.00000	0.106000
$90$	0	0
$91$	−6.96801	−0.730446
$92$	−0.730299	−0.0761389
$93$	0	0
$94$	14.5902	1.50486
$95$	2.00000	0.205196
$96$	0	0
$97$	3.30280	0.335348	0.167674	−	0.985842i	$-0.446374\pi$
$97$	3.30280	0.335348	0.167674	+	0.985842i	$0.446374\pi$
$98$	5.15410	0.520642
$99$	0	0
$100$	0.244835	0.0244835
$101$	11.6600	1.16021	0.580106	−	0.814541i	$-0.303011\pi$
$101$	11.6600	1.16021	0.580106	+	0.814541i	$0.303011\pi$
$102$	0	0
$103$	9.18569	0.905093	0.452546	−	0.891741i	$-0.350516\pi$
$103$	9.18569	0.905093	0.452546	+	0.891741i	$0.350516\pi$
$104$	−5.67112	−0.556099
$105$	0	0
$106$	16.5184	1.60440
$107$	9.81908	0.949246	0.474623	−	0.880189i	$-0.342584\pi$
$107$	9.81908	0.949246	0.474623	+	0.880189i	$0.342584\pi$
$108$	0	0
$109$	−7.49468	−0.717860	−0.358930	−	0.933364i	$-0.616858\pi$
$109$	−7.49468	−0.717860	−0.358930	+	0.933364i	$0.616858\pi$
$110$	0	0
$111$	0	0
$112$	14.3129	1.35244
$113$	14.6423	1.37743	0.688714	−	0.725033i	$-0.258175\pi$
$113$	14.6423	1.37743	0.688714	+	0.725033i	$0.258175\pi$
$114$	0	0
$115$	−2.98282	−0.278149
$116$	−0.297791	−0.0276492
$117$	0	0
$118$	−4.23455	−0.389822
$119$	−8.22819	−0.754277
$120$	0	0
$121$	−11.0000	−1.00000
$122$	3.42127	0.309747
$123$	0	0
$124$	2.07184	0.186057
$125$	1.00000	0.0894427
$126$	0	0
$127$	2.21289	0.196363	0.0981813	−	0.995169i	$-0.468697\pi$
$127$	2.21289	0.196363	0.0981813	+	0.995169i	$0.468697\pi$
$128$	12.9367	1.14345
$129$	0	0
$130$	3.23110	0.283386
$131$	14.7034	1.28465	0.642323	−	0.766434i	$-0.277971\pi$
$131$	14.7034	1.28465	0.642323	+	0.766434i	$0.277971\pi$
$132$	0	0
$133$	−6.46220	−0.560344
$134$	−2.93160	−0.253252
$135$	0	0
$136$	−6.69675	−0.574241
$137$	2.17658	0.185958	0.0929790	−	0.995668i	$-0.470361\pi$
$137$	2.17658	0.185958	0.0929790	+	0.995668i	$0.470361\pi$
$138$	0	0
$139$	4.98228	0.422592	0.211296	−	0.977422i	$-0.432232\pi$
$139$	4.98228	0.422592	0.211296	+	0.977422i	$0.432232\pi$
$140$	−0.791087	−0.0668590
$141$	0	0
$142$	8.68072	0.728470
$143$	0	0
$144$	0	0
$145$	−1.21629	−0.101008
$146$	−19.3886	−1.60461
$147$	0	0
$148$	−0.133817	−0.0109997
$149$	−8.34404	−0.683570	−0.341785	−	0.939778i	$-0.611031\pi$
$149$	−8.34404	−0.683570	−0.341785	+	0.939778i	$0.611031\pi$
$150$	0	0
$151$	1.51722	0.123470	0.0617348	−	0.998093i	$-0.480337\pi$
$151$	1.51722	0.123470	0.0617348	+	0.998093i	$0.480337\pi$
$152$	−5.25945	−0.426598
$153$	0	0
$154$	0	0
$155$	8.46220	0.679700
$156$	0	0
$157$	−11.6296	−0.928141	−0.464071	−	0.885798i	$-0.653612\pi$
$157$	−11.6296	−0.928141	−0.464071	+	0.885798i	$0.653612\pi$
$158$	12.0833	0.961297
$159$	0	0
$160$	−1.37751	−0.108902
$161$	9.63780	0.759565
$162$	0	0
$163$	−7.66687	−0.600516	−0.300258	−	0.953858i	$-0.597073\pi$
$163$	−7.66687	−0.600516	−0.300258	+	0.953858i	$0.597073\pi$
$164$	−0.696516	−0.0543888
$165$	0	0
$166$	5.21308	0.404613
$167$	0.0521124	0.00403258	0.00201629	−	0.999998i	$-0.499358\pi$
$167$	0.0521124	0.00403258	0.00201629	+	0.999998i	$0.499358\pi$
$168$	0	0
$169$	−8.34932	−0.642255
$170$	3.81545	0.292632
$171$	0	0
$172$	1.51776	0.115728
$173$	1.99257	0.151492	0.0757461	−	0.997127i	$-0.475866\pi$
$173$	1.99257	0.151492	0.0757461	+	0.997127i	$0.475866\pi$
$174$	0	0
$175$	−3.23110	−0.244248
$176$	0	0
$177$	0	0
$178$	1.49828	0.112301
$179$	11.4982	0.859418	0.429709	−	0.902967i	$-0.358616\pi$
$179$	11.4982	0.859418	0.429709	+	0.902967i	$0.358616\pi$
$180$	0	0
$181$	−10.0381	−0.746126	−0.373063	−	0.927806i	$-0.621692\pi$
$181$	−10.0381	−0.746126	−0.373063	+	0.927806i	$0.621692\pi$
$182$	−10.4400	−0.773865
$183$	0	0
$184$	7.84400	0.578267
$185$	−0.546560	−0.0401839
$186$	0	0
$187$	0	0
$188$	2.38420	0.173886
$189$	0	0
$190$	2.99655	0.217393
$191$	−3.72009	−0.269176	−0.134588	−	0.990902i	$-0.542971\pi$
$191$	−3.72009	−0.269176	−0.134588	+	0.990902i	$0.542971\pi$
$192$	0	0
$193$	6.31733	0.454731	0.227366	−	0.973809i	$-0.426989\pi$
$193$	6.31733	0.454731	0.227366	+	0.973809i	$0.426989\pi$
$194$	4.94851	0.355282
$195$	0	0
$196$	0.842236	0.0601597
$197$	23.1725	1.65098	0.825488	−	0.564420i	$-0.190900\pi$
$197$	23.1725	1.65098	0.825488	+	0.564420i	$0.190900\pi$
$198$	0	0
$199$	−5.96196	−0.422632	−0.211316	−	0.977418i	$-0.567775\pi$
$199$	−5.96196	−0.422632	−0.211316	+	0.977418i	$0.567775\pi$
$200$	−2.62972	−0.185950
$201$	0	0
$202$	17.4699	1.22918
$203$	3.92996	0.275829
$204$	0	0
$205$	−2.84484	−0.198692
$206$	13.7627	0.958893
$207$	0	0
$208$	−9.55290	−0.662374
$209$	0	0
$210$	0	0
$211$	16.0794	1.10695	0.553476	−	0.832865i	$-0.313301\pi$
$211$	16.0794	1.10695	0.553476	+	0.832865i	$0.313301\pi$
$212$	2.69928	0.185387
$213$	0	0
$214$	14.7117	1.00567
$215$	6.19911	0.422776
$216$	0	0
$217$	−27.3422	−1.85611
$218$	−11.2291	−0.760531
$219$	0	0
$220$	0	0
$221$	5.49177	0.369416
$222$	0	0
$223$	17.8324	1.19415	0.597073	−	0.802187i	$-0.296330\pi$
$223$	17.8324	1.19415	0.597073	+	0.802187i	$0.296330\pi$
$224$	4.45087	0.297387
$225$	0	0
$226$	21.9382	1.45931
$227$	−12.9380	−0.858725	−0.429362	−	0.903132i	$-0.641262\pi$
$227$	−12.9380	−0.858725	−0.429362	+	0.903132i	$0.641262\pi$
$228$	0	0
$229$	−6.12854	−0.404985	−0.202493	−	0.979284i	$-0.564904\pi$
$229$	−6.12854	−0.404985	−0.202493	+	0.979284i	$0.564904\pi$
$230$	−4.46909	−0.294683
$231$	0	0
$232$	3.19851	0.209993
$233$	−15.6325	−1.02412	−0.512060	−	0.858950i	$-0.671117\pi$
$233$	−15.6325	−1.02412	−0.512060	+	0.858950i	$0.671117\pi$
$234$	0	0
$235$	9.73799	0.635236
$236$	−0.691972	−0.0450435
$237$	0	0
$238$	−12.3281	−0.799113
$239$	−14.7740	−0.955649	−0.477824	−	0.878455i	$-0.658574\pi$
$239$	−14.7740	−0.955649	−0.477824	+	0.878455i	$0.658574\pi$
$240$	0	0
$241$	−3.26281	−0.210176	−0.105088	−	0.994463i	$-0.533512\pi$
$241$	−3.26281	−0.210176	−0.105088	+	0.994463i	$0.533512\pi$
$242$	−16.4811	−1.05944
$243$	0	0
$244$	0.559073	0.0357910
$245$	3.44001	0.219774
$246$	0	0
$247$	4.31309	0.274435
$248$	−22.2533	−1.41308
$249$	0	0
$250$	1.49828	0.0947594
$251$	−20.2434	−1.27775	−0.638876	−	0.769310i	$-0.720600\pi$
$251$	−20.2434	−1.27775	−0.638876	+	0.769310i	$0.720600\pi$
$252$	0	0
$253$	0	0
$254$	3.31553	0.208035
$255$	0	0
$256$	5.79158	0.361973
$257$	1.50609	0.0939472	0.0469736	−	0.998896i	$-0.485042\pi$
$257$	1.50609	0.0939472	0.0469736	+	0.998896i	$0.485042\pi$
$258$	0	0
$259$	1.76599	0.109733
$260$	0.527997	0.0327450
$261$	0	0
$262$	22.0298	1.36101
$263$	22.0466	1.35945	0.679727	−	0.733465i	$-0.262098\pi$
$263$	22.0466	1.35945	0.679727	+	0.733465i	$0.262098\pi$
$264$	0	0
$265$	11.0249	0.677254
$266$	−9.68217	−0.593652
$267$	0	0
$268$	−0.479056	−0.0292630
$269$	−22.2582	−1.35710	−0.678552	−	0.734553i	$-0.737392\pi$
$269$	−22.2582	−1.35710	−0.678552	+	0.734553i	$0.737392\pi$
$270$	0	0
$271$	−21.3624	−1.29767	−0.648837	−	0.760928i	$-0.724744\pi$
$271$	−21.3624	−1.29767	−0.648837	+	0.760928i	$0.724744\pi$
$272$	−11.2806	−0.683984
$273$	0	0
$274$	3.26112	0.197012
$275$	0	0
$276$	0	0
$277$	13.3948	0.804817	0.402409	−	0.915460i	$-0.368173\pi$
$277$	13.3948	0.804817	0.402409	+	0.915460i	$0.368173\pi$
$278$	7.46484	0.447711
$279$	0	0
$280$	8.49690	0.507787
$281$	−10.9173	−0.651273	−0.325636	−	0.945495i	$-0.605578\pi$
$281$	−10.9173	−0.651273	−0.325636	+	0.945495i	$0.605578\pi$
$282$	0	0
$283$	−16.4451	−0.977557	−0.488779	−	0.872408i	$-0.662557\pi$
$283$	−16.4451	−0.977557	−0.488779	+	0.872408i	$0.662557\pi$
$284$	1.41853	0.0841740
$285$	0	0
$286$	0	0
$287$	9.19196	0.542584
$288$	0	0
$289$	−10.5150	−0.618531
$290$	−1.82234	−0.107012
$291$	0	0
$292$	−3.16831	−0.185411
$293$	−11.4085	−0.666493	−0.333246	−	0.942840i	$-0.608144\pi$
$293$	−11.4085	−0.666493	−0.333246	+	0.942840i	$0.608144\pi$
$294$	0	0
$295$	−2.82628	−0.164552
$296$	1.43730	0.0835414
$297$	0	0
$298$	−12.5017	−0.724203
$299$	−6.43258	−0.372006
$300$	0	0
$301$	−20.0300	−1.15451
$302$	2.27322	0.130809
$303$	0	0
$304$	−8.85945	−0.508124
$305$	2.28347	0.130751
$306$	0	0
$307$	−9.24434	−0.527602	−0.263801	−	0.964577i	$-0.584976\pi$
$307$	−9.24434	−0.527602	−0.263801	+	0.964577i	$0.584976\pi$
$308$	0	0
$309$	0	0
$310$	12.6787	0.720103
$311$	19.6701	1.11539	0.557694	−	0.830047i	$-0.311686\pi$
$311$	19.6701	1.11539	0.557694	+	0.830047i	$0.311686\pi$
$312$	0	0
$313$	−6.82421	−0.385727	−0.192864	−	0.981226i	$-0.561777\pi$
$313$	−6.82421	−0.385727	−0.192864	+	0.981226i	$0.561777\pi$
$314$	−17.4243	−0.983312
$315$	0	0
$316$	1.97455	0.111077
$317$	5.59469	0.314229	0.157114	−	0.987580i	$-0.449781\pi$
$317$	5.59469	0.314229	0.157114	+	0.987580i	$0.449781\pi$
$318$	0	0
$319$	0	0
$320$	6.79556	0.379883
$321$	0	0
$322$	14.4401	0.804715
$323$	5.09312	0.283389
$324$	0	0
$325$	2.15654	0.119624
$326$	−11.4871	−0.636212
$327$	0	0
$328$	7.48114	0.413077
$329$	−31.4644	−1.73469
$330$	0	0
$331$	27.8906	1.53301	0.766503	−	0.642241i	$-0.221995\pi$
$331$	27.8906	1.53301	0.766503	+	0.642241i	$0.221995\pi$
$332$	0.851874	0.0467527
$333$	0	0
$334$	0.0780788	0.00427228
$335$	−1.95665	−0.106903
$336$	0	0
$337$	0.472821	0.0257562	0.0128781	−	0.999917i	$-0.495901\pi$
$337$	0.472821	0.0257562	0.0128781	+	0.999917i	$0.495901\pi$
$338$	−12.5096	−0.680432
$339$	0	0
$340$	0.623487	0.0338133
$341$	0	0
$342$	0	0
$343$	11.5027	0.621086
$344$	−16.3020	−0.878943
$345$	0	0
$346$	2.98542	0.160497
$347$	−0.470376	−0.0252511	−0.0126256	−	0.999920i	$-0.504019\pi$
$347$	−0.470376	−0.0252511	−0.0126256	+	0.999920i	$0.504019\pi$
$348$	0	0
$349$	−9.95257	−0.532749	−0.266374	−	0.963870i	$-0.585826\pi$
$349$	−9.95257	−0.532749	−0.266374	+	0.963870i	$0.585826\pi$
$350$	−4.84109	−0.258767
$351$	0	0
$352$	0	0
$353$	4.51311	0.240209	0.120104	−	0.992761i	$-0.461677\pi$
$353$	4.51311	0.240209	0.120104	+	0.992761i	$0.461677\pi$
$354$	0	0
$355$	5.79380	0.307503
$356$	0.244835	0.0129762
$357$	0	0
$358$	17.2275	0.910504
$359$	8.77166	0.462951	0.231475	−	0.972841i	$-0.425645\pi$
$359$	8.77166	0.462951	0.231475	+	0.972841i	$0.425645\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	−15.0399	−0.790477
$363$	0	0
$364$	−1.70601	−0.0894194
$365$	−12.9406	−0.677342
$366$	0	0
$367$	23.2320	1.21270	0.606350	−	0.795198i	$-0.292633\pi$
$367$	23.2320	1.21270	0.606350	+	0.795198i	$0.292633\pi$
$368$	13.2131	0.688779
$369$	0	0
$370$	−0.818898	−0.0425725
$371$	−35.6226	−1.84943
$372$	0	0
$373$	3.45253	0.178765	0.0893826	−	0.995997i	$-0.471511\pi$
$373$	3.45253	0.178765	0.0893826	+	0.995997i	$0.471511\pi$
$374$	0	0
$375$	0	0
$376$	−25.6082	−1.32064
$377$	−2.62299	−0.135091
$378$	0	0
$379$	−9.71751	−0.499155	−0.249577	−	0.968355i	$-0.580292\pi$
$379$	−9.71751	−0.499155	−0.249577	+	0.968355i	$0.580292\pi$
$380$	0.489670	0.0251195
$381$	0	0
$382$	−5.57372	−0.285176
$383$	−10.5389	−0.538510	−0.269255	−	0.963069i	$-0.586778\pi$
$383$	−10.5389	−0.538510	−0.269255	+	0.963069i	$0.586778\pi$
$384$	0	0
$385$	0	0
$386$	9.46511	0.481762
$387$	0	0
$388$	0.808641	0.0410525
$389$	−8.60815	−0.436451	−0.218225	−	0.975898i	$-0.570027\pi$
$389$	−8.60815	−0.436451	−0.218225	+	0.975898i	$0.570027\pi$
$390$	0	0
$391$	−7.59593	−0.384143
$392$	−9.04629	−0.456907
$393$	0	0
$394$	34.7189	1.74911
$395$	8.06481	0.405785
$396$	0	0
$397$	11.5589	0.580127	0.290063	−	0.957007i	$-0.406324\pi$
$397$	11.5589	0.580127	0.290063	+	0.957007i	$0.406324\pi$
$398$	−8.93268	−0.447755
$399$	0	0
$400$	−4.42973	−0.221486
$401$	16.5326	0.825598	0.412799	−	0.910822i	$-0.364551\pi$
$401$	16.5326	0.825598	0.412799	+	0.910822i	$0.364551\pi$
$402$	0	0
$403$	18.2491	0.909053
$404$	2.85477	0.142030
$405$	0	0
$406$	5.88817	0.292225
$407$	0	0
$408$	0	0
$409$	−26.8607	−1.32817	−0.664087	−	0.747655i	$-0.731180\pi$
$409$	−26.8607	−1.32817	−0.664087	+	0.747655i	$0.731180\pi$
$410$	−4.26236	−0.210503
$411$	0	0
$412$	2.24898	0.110799
$413$	9.13199	0.449356
$414$	0	0
$415$	3.47938	0.170796
$416$	−2.97066	−0.145649
$417$	0	0
$418$	0	0
$419$	−14.7573	−0.720942	−0.360471	−	0.932770i	$-0.617384\pi$
$419$	−14.7573	−0.720942	−0.360471	+	0.932770i	$0.617384\pi$
$420$	0	0
$421$	−1.69539	−0.0826285	−0.0413142	−	0.999146i	$-0.513154\pi$
$421$	−1.69539	−0.0826285	−0.0413142	+	0.999146i	$0.513154\pi$
$422$	24.0914	1.17275
$423$	0	0
$424$	−28.9924	−1.40800
$425$	2.54656	0.123526
$426$	0	0
$427$	−7.37812	−0.357052
$428$	2.40405	0.116204
$429$	0	0
$430$	9.28799	0.447907
$431$	40.1685	1.93485	0.967424	−	0.253161i	$-0.0814702\pi$
$431$	40.1685	1.93485	0.967424	+	0.253161i	$0.0814702\pi$
$432$	0	0
$433$	22.4284	1.07784	0.538920	−	0.842357i	$-0.318833\pi$
$433$	22.4284	1.07784	0.538920	+	0.842357i	$0.318833\pi$
$434$	−40.9662	−1.96644
$435$	0	0
$436$	−1.83496	−0.0878786
$437$	−5.96564	−0.285375
$438$	0	0
$439$	−0.580012	−0.0276824	−0.0138412	−	0.999904i	$-0.504406\pi$
$439$	−0.580012	−0.0276824	−0.0138412	+	0.999904i	$0.504406\pi$
$440$	0	0
$441$	0	0
$442$	8.22819	0.391375
$443$	4.28962	0.203806	0.101903	−	0.994794i	$-0.467507\pi$
$443$	4.28962	0.203806	0.101903	+	0.994794i	$0.467507\pi$
$444$	0	0
$445$	1.00000	0.0474045
$446$	26.7179	1.26513
$447$	0	0
$448$	−21.9571	−1.03738
$449$	−22.9873	−1.08484	−0.542418	−	0.840109i	$-0.682491\pi$
$449$	−22.9873	−1.08484	−0.542418	+	0.840109i	$0.682491\pi$
$450$	0	0
$451$	0	0
$452$	3.58494	0.168621
$453$	0	0
$454$	−19.3847	−0.909769
$455$	−6.96801	−0.326665
$456$	0	0
$457$	−26.7097	−1.24943	−0.624714	−	0.780854i	$-0.714784\pi$
$457$	−26.7097	−1.24943	−0.624714	+	0.780854i	$0.714784\pi$
$458$	−9.18225	−0.429058
$459$	0	0
$460$	−0.730299	−0.0340504
$461$	24.1175	1.12326	0.561632	−	0.827387i	$-0.310174\pi$
$461$	24.1175	1.12326	0.561632	+	0.827387i	$0.310174\pi$
$462$	0	0
$463$	−30.8578	−1.43408	−0.717041	−	0.697031i	$-0.754504\pi$
$463$	−30.8578	−1.43408	−0.717041	+	0.697031i	$0.754504\pi$
$464$	5.38784	0.250124
$465$	0	0
$466$	−23.4218	−1.08500
$467$	30.4542	1.40925	0.704626	−	0.709579i	$-0.251115\pi$
$467$	30.4542	1.40925	0.704626	+	0.709579i	$0.251115\pi$
$468$	0	0
$469$	6.32213	0.291929
$470$	14.5902	0.672996
$471$	0	0
$472$	7.43233	0.342101
$473$	0	0
$474$	0	0
$475$	2.00000	0.0917663
$476$	−2.01455	−0.0923367
$477$	0	0
$478$	−22.1355	−1.01245
$479$	8.27076	0.377901	0.188950	−	0.981987i	$-0.439491\pi$
$479$	8.27076	0.377901	0.188950	+	0.981987i	$0.439491\pi$
$480$	0	0
$481$	−1.17868	−0.0537432
$482$	−4.88859	−0.222669
$483$	0	0
$484$	−2.69319	−0.122418
$485$	3.30280	0.149972
$486$	0	0
$487$	6.33163	0.286914	0.143457	−	0.989657i	$-0.454178\pi$
$487$	6.33163	0.286914	0.143457	+	0.989657i	$0.454178\pi$
$488$	−6.00490	−0.271829
$489$	0	0
$490$	5.15410	0.232838
$491$	43.0296	1.94190	0.970949	−	0.239288i	$-0.0769140\pi$
$491$	43.0296	1.94190	0.970949	+	0.239288i	$0.0769140\pi$
$492$	0	0
$493$	−3.09736	−0.139498
$494$	6.46220	0.290748
$495$	0	0
$496$	−37.4852	−1.68314
$497$	−18.7204	−0.839723
$498$	0	0
$499$	5.39064	0.241318	0.120659	−	0.992694i	$-0.461499\pi$
$499$	5.39064	0.241318	0.120659	+	0.992694i	$0.461499\pi$
$500$	0.244835	0.0109494
$501$	0	0
$502$	−30.3302	−1.35370
$503$	−15.9822	−0.712611	−0.356305	−	0.934370i	$-0.615964\pi$
$503$	−15.9822	−0.712611	−0.356305	+	0.934370i	$0.615964\pi$
$504$	0	0
$505$	11.6600	0.518862
$506$	0	0
$507$	0	0
$508$	0.541794	0.0240382
$509$	−25.9239	−1.14906	−0.574529	−	0.818484i	$-0.694815\pi$
$509$	−25.9239	−1.14906	−0.574529	+	0.818484i	$0.694815\pi$
$510$	0	0
$511$	41.8124	1.84967
$512$	−17.1959	−0.759960
$513$	0	0
$514$	2.25654	0.0995316
$515$	9.18569	0.404770
$516$	0	0
$517$	0	0
$518$	2.64594	0.116256
$519$	0	0
$520$	−5.67112	−0.248695
$521$	32.8737	1.44022	0.720112	−	0.693858i	$-0.244091\pi$
$521$	32.8737	1.44022	0.720112	+	0.693858i	$0.244091\pi$
$522$	0	0
$523$	36.7804	1.60830	0.804148	−	0.594430i	$-0.202622\pi$
$523$	36.7804	1.60830	0.804148	+	0.594430i	$0.202622\pi$
$524$	3.59992	0.157263
$525$	0	0
$526$	33.0320	1.44026
$527$	21.5495	0.938711
$528$	0	0
$529$	−14.1028	−0.613164
$530$	16.5184	0.717512
$531$	0	0
$532$	−1.58217	−0.0685959
$533$	−6.13502	−0.265737
$534$	0	0
$535$	9.81908	0.424516
$536$	5.14545	0.222249
$537$	0	0
$538$	−33.3489	−1.43777
$539$	0	0
$540$	0	0
$541$	14.0677	0.604818	0.302409	−	0.953178i	$-0.402209\pi$
$541$	14.0677	0.604818	0.302409	+	0.953178i	$0.402209\pi$
$542$	−32.0068	−1.37481
$543$	0	0
$544$	−3.50791	−0.150400
$545$	−7.49468	−0.321037
$546$	0	0
$547$	−43.7363	−1.87003	−0.935014	−	0.354612i	$-0.884613\pi$
$547$	−43.7363	−1.87003	−0.935014	+	0.354612i	$0.884613\pi$
$548$	0.532903	0.0227645
$549$	0	0
$550$	0	0
$551$	−2.43258	−0.103632
$552$	0	0
$553$	−26.0582	−1.10811
$554$	20.0692	0.852657
$555$	0	0
$556$	1.21984	0.0517326
$557$	−6.47140	−0.274202	−0.137101	−	0.990557i	$-0.543778\pi$
$557$	−6.47140	−0.274202	−0.137101	+	0.990557i	$0.543778\pi$
$558$	0	0
$559$	13.3687	0.565434
$560$	14.3129	0.604830
$561$	0	0
$562$	−16.3572	−0.689986
$563$	11.4507	0.482591	0.241296	−	0.970452i	$-0.422428\pi$
$563$	11.4507	0.482591	0.241296	+	0.970452i	$0.422428\pi$
$564$	0	0
$565$	14.6423	0.616005
$566$	−24.6393	−1.03567
$567$	0	0
$568$	−15.2361	−0.639292
$569$	38.6520	1.62037	0.810187	−	0.586171i	$-0.199365\pi$
$569$	38.6520	1.62037	0.810187	+	0.586171i	$0.199365\pi$
$570$	0	0
$571$	39.0392	1.63374	0.816869	−	0.576823i	$-0.195708\pi$
$571$	39.0392	1.63374	0.816869	+	0.576823i	$0.195708\pi$
$572$	0	0
$573$	0	0
$574$	13.7721	0.574837
$575$	−2.98282	−0.124392
$576$	0	0
$577$	19.7589	0.822576	0.411288	−	0.911506i	$-0.365079\pi$
$577$	19.7589	0.822576	0.411288	+	0.911506i	$0.365079\pi$
$578$	−15.7544	−0.655298
$579$	0	0
$580$	−0.297791	−0.0123651
$581$	−11.2422	−0.466406
$582$	0	0
$583$	0	0
$584$	34.0302	1.40818
$585$	0	0
$586$	−17.0931	−0.706110
$587$	−9.55716	−0.394466	−0.197233	−	0.980357i	$-0.563196\pi$
$587$	−9.55716	−0.394466	−0.197233	+	0.980357i	$0.563196\pi$
$588$	0	0
$589$	16.9244	0.697358
$590$	−4.23455	−0.174334
$591$	0	0
$592$	2.42111	0.0995070
$593$	19.2958	0.792384	0.396192	−	0.918168i	$-0.370331\pi$
$593$	19.2958	0.792384	0.396192	+	0.918168i	$0.370331\pi$
$594$	0	0
$595$	−8.22819	−0.337323
$596$	−2.04291	−0.0836809
$597$	0	0
$598$	−9.63780	−0.394119
$599$	−25.9498	−1.06028	−0.530139	−	0.847910i	$-0.677860\pi$
$599$	−25.9498	−1.06028	−0.530139	+	0.847910i	$0.677860\pi$
$600$	0	0
$601$	2.99846	0.122310	0.0611549	−	0.998128i	$-0.480522\pi$
$601$	2.99846	0.122310	0.0611549	+	0.998128i	$0.480522\pi$
$602$	−30.0104	−1.22313
$603$	0	0
$604$	0.371469	0.0151148
$605$	−11.0000	−0.447214
$606$	0	0
$607$	−27.2889	−1.10762	−0.553812	−	0.832642i	$-0.686827\pi$
$607$	−27.2889	−1.10762	−0.553812	+	0.832642i	$0.686827\pi$
$608$	−2.75502	−0.111731
$609$	0	0
$610$	3.42127	0.138523
$611$	21.0004	0.849585
$612$	0	0
$613$	−24.6000	−0.993586	−0.496793	−	0.867869i	$-0.665489\pi$
$613$	−24.6000	−0.993586	−0.496793	+	0.867869i	$0.665489\pi$
$614$	−13.8506	−0.558964
$615$	0	0
$616$	0	0
$617$	29.5894	1.19122	0.595612	−	0.803272i	$-0.296910\pi$
$617$	29.5894	1.19122	0.595612	+	0.803272i	$0.296910\pi$
$618$	0	0
$619$	−24.2403	−0.974299	−0.487150	−	0.873318i	$-0.661963\pi$
$619$	−24.2403	−0.974299	−0.487150	+	0.873318i	$0.661963\pi$
$620$	2.07184	0.0832072
$621$	0	0
$622$	29.4712	1.18169
$623$	−3.23110	−0.129451
$624$	0	0
$625$	1.00000	0.0400000
$626$	−10.2246	−0.408656
$627$	0	0
$628$	−2.84733	−0.113621
$629$	−1.39185	−0.0554966
$630$	0	0
$631$	−1.07165	−0.0426616	−0.0213308	−	0.999772i	$-0.506790\pi$
$631$	−1.07165	−0.0426616	−0.0213308	+	0.999772i	$0.506790\pi$
$632$	−21.2082	−0.843618
$633$	0	0
$634$	8.38239	0.332907
$635$	2.21289	0.0878160
$636$	0	0
$637$	7.41854	0.293933
$638$	0	0
$639$	0	0
$640$	12.9367	0.511366
$641$	31.4849	1.24358	0.621790	−	0.783184i	$-0.286406\pi$
$641$	31.4849	1.24358	0.621790	+	0.783184i	$0.286406\pi$
$642$	0	0
$643$	−16.0382	−0.632486	−0.316243	−	0.948678i	$-0.602421\pi$
$643$	−16.0382	−0.632486	−0.316243	+	0.948678i	$0.602421\pi$
$644$	2.35967	0.0929840
$645$	0	0
$646$	7.63091	0.300234
$647$	−9.45360	−0.371659	−0.185830	−	0.982582i	$-0.559497\pi$
$647$	−9.45360	−0.371659	−0.185830	+	0.982582i	$0.559497\pi$
$648$	0	0
$649$	0	0
$650$	3.23110	0.126734
$651$	0	0
$652$	−1.87712	−0.0735137
$653$	−7.23401	−0.283089	−0.141544	−	0.989932i	$-0.545207\pi$
$653$	−7.23401	−0.283089	−0.141544	+	0.989932i	$0.545207\pi$
$654$	0	0
$655$	14.7034	0.574511
$656$	12.6019	0.492020
$657$	0	0
$658$	−47.1424	−1.83780
$659$	29.3533	1.14344	0.571722	−	0.820448i	$-0.306276\pi$
$659$	29.3533	1.14344	0.571722	+	0.820448i	$0.306276\pi$
$660$	0	0
$661$	−16.8668	−0.656043	−0.328021	−	0.944670i	$-0.606382\pi$
$661$	−16.8668	−0.656043	−0.328021	+	0.944670i	$0.606382\pi$
$662$	41.7879	1.62413
$663$	0	0
$664$	−9.14981	−0.355082
$665$	−6.46220	−0.250593
$666$	0	0
$667$	3.62798	0.140476
$668$	0.0127589	0.000493658 0
$669$	0	0
$670$	−2.93160	−0.113258
$671$	0	0
$672$	0	0
$673$	49.9471	1.92532	0.962659	−	0.270716i	$-0.0872604\pi$
$673$	49.9471	1.92532	0.962659	+	0.270716i	$0.0872604\pi$
$674$	0.708416	0.0272872
$675$	0	0
$676$	−2.04421	−0.0786233
$677$	10.3125	0.396341	0.198170	−	0.980168i	$-0.436500\pi$
$677$	10.3125	0.396341	0.198170	+	0.980168i	$0.436500\pi$
$678$	0	0
$679$	−10.6717	−0.409541
$680$	−6.69675	−0.256809
$681$	0	0
$682$	0	0
$683$	29.1314	1.11468	0.557342	−	0.830283i	$-0.311821\pi$
$683$	29.1314	1.11468	0.557342	+	0.830283i	$0.311821\pi$
$684$	0	0
$685$	2.17658	0.0831629
$686$	17.2342	0.658005
$687$	0	0
$688$	−27.4604	−1.04692
$689$	23.7757	0.905781
$690$	0	0
$691$	−22.7023	−0.863635	−0.431817	−	0.901961i	$-0.642128\pi$
$691$	−22.7023	−0.863635	−0.431817	+	0.901961i	$0.642128\pi$
$692$	0.487851	0.0185453
$693$	0	0
$694$	−0.704754	−0.0267521
$695$	4.98228	0.188989
$696$	0	0
$697$	−7.24455	−0.274407
$698$	−14.9117	−0.564417
$699$	0	0
$700$	−0.791087	−0.0299003
$701$	−42.0114	−1.58675	−0.793375	−	0.608734i	$-0.791678\pi$
$701$	−42.0114	−1.58675	−0.793375	+	0.608734i	$0.791678\pi$
$702$	0	0
$703$	−1.09312	−0.0412278
$704$	0	0
$705$	0	0
$706$	6.76189	0.254487
$707$	−37.6746	−1.41690
$708$	0	0
$709$	−35.4923	−1.33294	−0.666471	−	0.745531i	$-0.732196\pi$
$709$	−35.4923	−1.33294	−0.666471	+	0.745531i	$0.732196\pi$
$710$	8.68072	0.325782
$711$	0	0
$712$	−2.62972	−0.0985531
$713$	−25.2412	−0.945292
$714$	0	0
$715$	0	0
$716$	2.81517	0.105208
$717$	0	0
$718$	13.1424	0.490470
$719$	19.8978	0.742064	0.371032	−	0.928620i	$-0.379004\pi$
$719$	19.8978	0.742064	0.371032	+	0.928620i	$0.379004\pi$
$720$	0	0
$721$	−29.6799	−1.10534
$722$	−22.4742	−0.836402
$723$	0	0
$724$	−2.45768	−0.0913389
$725$	−1.21629	−0.0451720
$726$	0	0
$727$	−20.3167	−0.753506	−0.376753	−	0.926314i	$-0.622959\pi$
$727$	−20.3167	−0.753506	−0.376753	+	0.926314i	$0.622959\pi$
$728$	18.3239	0.679131
$729$	0	0
$730$	−19.3886	−0.717604
$731$	15.7864	0.583881
$732$	0	0
$733$	−20.3832	−0.752871	−0.376435	−	0.926443i	$-0.622850\pi$
$733$	−20.3832	−0.752871	−0.376435	+	0.926443i	$0.622850\pi$
$734$	34.8079	1.28478
$735$	0	0
$736$	4.10887	0.151455
$737$	0	0
$738$	0	0
$739$	−27.2671	−1.00304	−0.501518	−	0.865147i	$-0.667225\pi$
$739$	−27.2671	−1.00304	−0.501518	+	0.865147i	$0.667225\pi$
$740$	−0.133817	−0.00491921
$741$	0	0
$742$	−53.3725	−1.95937
$743$	−24.7332	−0.907374	−0.453687	−	0.891161i	$-0.649892\pi$
$743$	−24.7332	−0.907374	−0.453687	+	0.891161i	$0.649892\pi$
$744$	0	0
$745$	−8.34404	−0.305702
$746$	5.17285	0.189391
$747$	0	0
$748$	0	0
$749$	−31.7264	−1.15926
$750$	0	0
$751$	9.20976	0.336069	0.168034	−	0.985781i	$-0.446258\pi$
$751$	9.20976	0.336069	0.168034	+	0.985781i	$0.446258\pi$
$752$	−43.1366	−1.57303
$753$	0	0
$754$	−3.92996	−0.143121
$755$	1.51722	0.0552173
$756$	0	0
$757$	−46.1220	−1.67633	−0.838166	−	0.545415i	$-0.816372\pi$
$757$	−46.1220	−1.67633	−0.838166	+	0.545415i	$0.816372\pi$
$758$	−14.5595	−0.528826
$759$	0	0
$760$	−5.25945	−0.190780
$761$	−17.2495	−0.625293	−0.312647	−	0.949869i	$-0.601216\pi$
$761$	−17.2495	−0.625293	−0.312647	+	0.949869i	$0.601216\pi$
$762$	0	0
$763$	24.2161	0.876680
$764$	−0.910808	−0.0329519
$765$	0	0
$766$	−15.7901	−0.570521
$767$	−6.09499	−0.220077
$768$	0	0
$769$	−26.5353	−0.956887	−0.478443	−	0.878118i	$-0.658799\pi$
$769$	−26.5353	−0.956887	−0.478443	+	0.878118i	$0.658799\pi$
$770$	0	0
$771$	0	0
$772$	1.54670	0.0556671
$773$	−4.11066	−0.147850	−0.0739251	−	0.997264i	$-0.523553\pi$
$773$	−4.11066	−0.147850	−0.0739251	+	0.997264i	$0.523553\pi$
$774$	0	0
$775$	8.46220	0.303971
$776$	−8.68545	−0.311790
$777$	0	0
$778$	−12.8974	−0.462394
$779$	−5.68968	−0.203854
$780$	0	0
$781$	0	0
$782$	−11.3808	−0.406977
$783$	0	0
$784$	−15.2383	−0.544226
$785$	−11.6296	−0.415077
$786$	0	0
$787$	16.2423	0.578976	0.289488	−	0.957182i	$-0.406515\pi$
$787$	16.2423	0.578976	0.289488	+	0.957182i	$0.406515\pi$
$788$	5.67345	0.202108
$789$	0	0
$790$	12.0833	0.429905
$791$	−47.3106	−1.68217
$792$	0	0
$793$	4.92440	0.174871
$794$	17.3185	0.614611
$795$	0	0
$796$	−1.45970	−0.0517376
$797$	−40.7251	−1.44256	−0.721279	−	0.692644i	$-0.756446\pi$
$797$	−40.7251	−1.44256	−0.721279	+	0.692644i	$0.756446\pi$
$798$	0	0
$799$	24.7984	0.877303
$800$	−1.37751	−0.0487023
$801$	0	0
$802$	24.7704	0.874673
$803$	0	0
$804$	0	0
$805$	9.63780	0.339688
$806$	27.3422	0.963089
$807$	0	0
$808$	−30.6625	−1.07870
$809$	18.2512	0.641678	0.320839	−	0.947134i	$-0.396035\pi$
$809$	18.2512	0.641678	0.320839	+	0.947134i	$0.396035\pi$
$810$	0	0
$811$	43.8706	1.54051	0.770253	−	0.637738i	$-0.220130\pi$
$811$	43.8706	1.54051	0.770253	+	0.637738i	$0.220130\pi$
$812$	0.962193	0.0337663
$813$	0	0
$814$	0	0
$815$	−7.66687	−0.268559
$816$	0	0
$817$	12.3982	0.433759
$818$	−40.2447	−1.40712
$819$	0	0
$820$	−0.696516	−0.0243234
$821$	14.0303	0.489661	0.244831	−	0.969566i	$-0.421268\pi$
$821$	14.0303	0.489661	0.244831	+	0.969566i	$0.421268\pi$
$822$	0	0
$823$	−5.50462	−0.191879	−0.0959394	−	0.995387i	$-0.530586\pi$
$823$	−5.50462	−0.191879	−0.0959394	+	0.995387i	$0.530586\pi$
$824$	−24.1558	−0.841508
$825$	0	0
$826$	13.6822	0.476066
$827$	−43.1206	−1.49945	−0.749726	−	0.661749i	$-0.769815\pi$
$827$	−43.1206	−1.49945	−0.749726	+	0.661749i	$0.769815\pi$
$828$	0	0
$829$	−28.8526	−1.00209	−0.501047	−	0.865420i	$-0.667052\pi$
$829$	−28.8526	−1.00209	−0.501047	+	0.865420i	$0.667052\pi$
$830$	5.21308	0.180949
$831$	0	0
$832$	14.6549	0.508068
$833$	8.76020	0.303523
$834$	0	0
$835$	0.0521124	0.00180342
$836$	0	0
$837$	0	0
$838$	−22.1106	−0.763796
$839$	−21.9011	−0.756108	−0.378054	−	0.925783i	$-0.623407\pi$
$839$	−21.9011	−0.756108	−0.378054	+	0.925783i	$0.623407\pi$
$840$	0	0
$841$	−27.5206	−0.948987
$842$	−2.54017	−0.0875401
$843$	0	0
$844$	3.93681	0.135510
$845$	−8.34932	−0.287225
$846$	0	0
$847$	35.5421	1.22124
$848$	−48.8373	−1.67708
$849$	0	0
$850$	3.81545	0.130869
$851$	1.63029	0.0558856
$852$	0	0
$853$	−1.93424	−0.0662270	−0.0331135	−	0.999452i	$-0.510542\pi$
$853$	−1.93424	−0.0662270	−0.0331135	+	0.999452i	$0.510542\pi$
$854$	−11.0545	−0.378276
$855$	0	0
$856$	−25.8215	−0.882560
$857$	58.4401	1.99627	0.998137	−	0.0610065i	$-0.0194310\pi$
$857$	58.4401	1.99627	0.998137	+	0.0610065i	$0.0194310\pi$
$858$	0	0
$859$	−28.3839	−0.968447	−0.484223	−	0.874944i	$-0.660898\pi$
$859$	−28.3839	−0.968447	−0.484223	+	0.874944i	$0.660898\pi$
$860$	1.51776	0.0517552
$861$	0	0
$862$	60.1835	2.04986
$863$	−24.2940	−0.826977	−0.413489	−	0.910509i	$-0.635690\pi$
$863$	−24.2940	−0.826977	−0.413489	+	0.910509i	$0.635690\pi$
$864$	0	0
$865$	1.99257	0.0677494
$866$	33.6039	1.14191
$867$	0	0
$868$	−6.69434	−0.227221
$869$	0	0
$870$	0	0
$871$	−4.21960	−0.142976
$872$	19.7089	0.667429
$873$	0	0
$874$	−8.93819	−0.302339
$875$	−3.23110	−0.109231
$876$	0	0
$877$	50.8099	1.71573	0.857863	−	0.513878i	$-0.171792\pi$
$877$	50.8099	1.71573	0.857863	+	0.513878i	$0.171792\pi$
$878$	−0.869018	−0.0293279
$879$	0	0
$880$	0	0
$881$	−3.05502	−0.102926	−0.0514631	−	0.998675i	$-0.516388\pi$
$881$	−3.05502	−0.102926	−0.0514631	+	0.998675i	$0.516388\pi$
$882$	0	0
$883$	−16.0936	−0.541593	−0.270797	−	0.962637i	$-0.587287\pi$
$883$	−16.0936	−0.541593	−0.270797	+	0.962637i	$0.587287\pi$
$884$	1.34458	0.0452230
$885$	0	0
$886$	6.42704	0.215920
$887$	−1.16665	−0.0391724	−0.0195862	−	0.999808i	$-0.506235\pi$
$887$	−1.16665	−0.0391724	−0.0195862	+	0.999808i	$0.506235\pi$
$888$	0	0
$889$	−7.15009	−0.239806
$890$	1.49828	0.0502224
$891$	0	0
$892$	4.36600	0.146184
$893$	19.4760	0.651739
$894$	0	0
$895$	11.4982	0.384344
$896$	−41.7996	−1.39643
$897$	0	0
$898$	−34.4413	−1.14932
$899$	−10.2925	−0.343274
$900$	0	0
$901$	28.0756	0.935333
$902$	0	0
$903$	0	0
$904$	−38.5051	−1.28066
$905$	−10.0381	−0.333678
$906$	0	0
$907$	−5.56857	−0.184901	−0.0924506	−	0.995717i	$-0.529470\pi$
$907$	−5.56857	−0.184901	−0.0924506	+	0.995717i	$0.529470\pi$
$908$	−3.16767	−0.105123
$909$	0	0
$910$	−10.4400	−0.346083
$911$	25.5924	0.847915	0.423957	−	0.905682i	$-0.360641\pi$
$911$	25.5924	0.847915	0.423957	+	0.905682i	$0.360641\pi$
$912$	0	0
$913$	0	0
$914$	−40.0186	−1.32370
$915$	0	0
$916$	−1.50048	−0.0495773
$917$	−47.5083	−1.56886
$918$	0	0
$919$	−58.7341	−1.93746	−0.968730	−	0.248118i	$-0.920188\pi$
$919$	−58.7341	−1.93746	−0.968730	+	0.248118i	$0.920188\pi$
$920$	7.84400	0.258609
$921$	0	0
$922$	36.1347	1.19003
$923$	12.4946	0.411264
$924$	0	0
$925$	−0.546560	−0.0179708
$926$	−46.2335	−1.51933
$927$	0	0
$928$	1.67545	0.0549995
$929$	29.9339	0.982101	0.491050	−	0.871131i	$-0.336613\pi$
$929$	29.9339	0.982101	0.491050	+	0.871131i	$0.336613\pi$
$930$	0	0
$931$	6.88003	0.225484
$932$	−3.82739	−0.125370
$933$	0	0
$934$	45.6288	1.49302
$935$	0	0
$936$	0	0
$937$	38.4314	1.25550	0.627750	−	0.778415i	$-0.283976\pi$
$937$	38.4314	1.25550	0.627750	+	0.778415i	$0.283976\pi$
$938$	9.47230	0.309282
$939$	0	0
$940$	2.38420	0.0777640
$941$	−38.6236	−1.25909	−0.629546	−	0.776963i	$-0.716759\pi$
$941$	−38.6236	−1.25909	−0.629546	+	0.776963i	$0.716759\pi$
$942$	0	0
$943$	8.48564	0.276331
$944$	12.5196	0.407479
$945$	0	0
$946$	0	0
$947$	8.44416	0.274398	0.137199	−	0.990543i	$-0.456190\pi$
$947$	8.44416	0.274398	0.137199	+	0.990543i	$0.456190\pi$
$948$	0	0
$949$	−27.9070	−0.905898
$950$	2.99655	0.0972211
$951$	0	0
$952$	21.6379	0.701287
$953$	40.1285	1.29989	0.649945	−	0.759981i	$-0.274792\pi$
$953$	40.1285	1.29989	0.649945	+	0.759981i	$0.274792\pi$
$954$	0	0
$955$	−3.72009	−0.120379
$956$	−3.61719	−0.116988
$957$	0	0
$958$	12.3919	0.400364
$959$	−7.03276	−0.227100
$960$	0	0
$961$	40.6089	1.30996
$962$	−1.76599	−0.0569378
$963$	0	0
$964$	−0.798850	−0.0257292
$965$	6.31733	0.203362
$966$	0	0
$967$	4.32318	0.139024	0.0695120	−	0.997581i	$-0.477856\pi$
$967$	4.32318	0.139024	0.0695120	+	0.997581i	$0.477856\pi$
$968$	28.9270	0.929748
$969$	0	0
$970$	4.94851	0.158887
$971$	3.48751	0.111919	0.0559597	−	0.998433i	$-0.482178\pi$
$971$	3.48751	0.111919	0.0559597	+	0.998433i	$0.482178\pi$
$972$	0	0
$973$	−16.0983	−0.516086
$974$	9.48654	0.303968
$975$	0	0
$976$	−10.1151	−0.323778
$977$	−19.8788	−0.635980	−0.317990	−	0.948094i	$-0.603008\pi$
$977$	−19.8788	−0.635980	−0.317990	+	0.948094i	$0.603008\pi$
$978$	0	0
$979$	0	0
$980$	0.842236	0.0269042
$981$	0	0
$982$	64.4702	2.05733
$983$	−52.3297	−1.66906	−0.834529	−	0.550963i	$-0.814260\pi$
$983$	−52.3297	−1.66906	−0.834529	+	0.550963i	$0.814260\pi$
$984$	0	0
$985$	23.1725	0.738339
$986$	−4.64071	−0.147790
$987$	0	0
$988$	1.05599	0.0335957
$989$	−18.4908	−0.587975
$990$	0	0
$991$	−16.8109	−0.534016	−0.267008	−	0.963694i	$-0.586035\pi$
$991$	−16.8109	−0.534016	−0.267008	+	0.963694i	$0.586035\pi$
$992$	−11.6568	−0.370103
$993$	0	0
$994$	−28.0483	−0.889638
$995$	−5.96196	−0.189007
$996$	0	0
$997$	−14.5992	−0.462362	−0.231181	−	0.972911i	$-0.574259\pi$
$997$	−14.5992	−0.462362	−0.231181	+	0.972911i	$0.574259\pi$
$998$	8.07667	0.255662
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4005.2.a.n.1.4		6
3.2	odd	2		1335.2.a.g.1.3	✓	6
15.14	odd	2		6675.2.a.u.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1335.2.a.g.1.3	✓	6	3.2	odd	2
4005.2.a.n.1.4		6	1.1	even	1	trivial
6675.2.a.u.1.4		6	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:	\( N \)	\(=\)	\( 4005 = 3^{2} \cdot 5 \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4005.a (trivial)

\(f(q)\)	\(=\)	\(q+1.49828 q^{2} +0.244835 q^{4} +1.00000 q^{5} -3.23110 q^{7} -2.62972 q^{8} +O(q^{10})\)	\(q+1.49828 q^{2} +0.244835 q^{4} +1.00000 q^{5} -3.23110 q^{7} -2.62972 q^{8} +1.49828 q^{10} +2.15654 q^{13} -4.84109 q^{14} -4.42973 q^{16} +2.54656 q^{17} +2.00000 q^{19} +0.244835 q^{20} -2.98282 q^{23} +1.00000 q^{25} +3.23110 q^{26} -0.791087 q^{28} -1.21629 q^{29} +8.46220 q^{31} -1.37751 q^{32} +3.81545 q^{34} -3.23110 q^{35} -0.546560 q^{37} +2.99655 q^{38} -2.62972 q^{40} -2.84484 q^{41} +6.19911 q^{43} -4.46909 q^{46} +9.73799 q^{47} +3.44001 q^{49} +1.49828 q^{50} +0.527997 q^{52} +11.0249 q^{53} +8.49690 q^{56} -1.82234 q^{58} -2.82628 q^{59} +2.28347 q^{61} +12.6787 q^{62} +6.79556 q^{64} +2.15654 q^{65} -1.95665 q^{67} +0.623487 q^{68} -4.84109 q^{70} +5.79380 q^{71} -12.9406 q^{73} -0.818898 q^{74} +0.489670 q^{76} +8.06481 q^{79} -4.42973 q^{80} -4.26236 q^{82} +3.47938 q^{83} +2.54656 q^{85} +9.28799 q^{86} +1.00000 q^{89} -6.96801 q^{91} -0.730299 q^{92} +14.5902 q^{94} +2.00000 q^{95} +3.30280 q^{97} +5.15410 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{2} + 8 q^{4} + 6 q^{5} + q^{7} + 9 q^{8}+O(q^{10}) \) 6 * q + 4 * q^2 + 8 * q^4 + 6 * q^5 + q^7 + 9 * q^8	\( 6 q + 4 q^{2} + 8 q^{4} + 6 q^{5} + q^{7} + 9 q^{8} + 4 q^{10} - 3 q^{13} + 5 q^{14} + 12 q^{16} + 13 q^{17} + 12 q^{19} + 8 q^{20} + 19 q^{23} + 6 q^{25} - q^{26} - 6 q^{28} + 10 q^{31} + 17 q^{32} - 2 q^{34} + q^{35} - q^{37} + 8 q^{38} + 9 q^{40} + 4 q^{41} - 7 q^{43} - 6 q^{46} + 15 q^{47} - q^{49} + 4 q^{50} + q^{52} + 27 q^{53} + 14 q^{56} - 6 q^{58} + 4 q^{59} + 8 q^{61} - 2 q^{62} - q^{64} - 3 q^{65} - 11 q^{67} + 47 q^{68} + 5 q^{70} + 16 q^{71} + q^{73} + 10 q^{74} + 16 q^{76} + 12 q^{80} + q^{82} + 17 q^{83} + 13 q^{85} + 20 q^{86} + 6 q^{89} - 18 q^{91} + 36 q^{92} + 17 q^{94} + 12 q^{95} - 29 q^{97} + 23 q^{98}+O(q^{100}) \) 6 * q + 4 * q^2 + 8 * q^4 + 6 * q^5 + q^7 + 9 * q^8 + 4 * q^10 - 3 * q^13 + 5 * q^14 + 12 * q^16 + 13 * q^17 + 12 * q^19 + 8 * q^20 + 19 * q^23 + 6 * q^25 - q^26 - 6 * q^28 + 10 * q^31 + 17 * q^32 - 2 * q^34 + q^35 - q^37 + 8 * q^38 + 9 * q^40 + 4 * q^41 - 7 * q^43 - 6 * q^46 + 15 * q^47 - q^49 + 4 * q^50 + q^52 + 27 * q^53 + 14 * q^56 - 6 * q^58 + 4 * q^59 + 8 * q^61 - 2 * q^62 - q^64 - 3 * q^65 - 11 * q^67 + 47 * q^68 + 5 * q^70 + 16 * q^71 + q^73 + 10 * q^74 + 16 * q^76 + 12 * q^80 + q^82 + 17 * q^83 + 13 * q^85 + 20 * q^86 + 6 * q^89 - 18 * q^91 + 36 * q^92 + 17 * q^94 + 12 * q^95 - 29 * q^97 + 23 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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