LMFDB - Embedded newform 4005.2.a.i.1.1

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:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1335)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.53209$ 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)

$f(q)$	$=$	$q-1.53209 q^{2} +0.347296 q^{4} -1.00000 q^{5} +1.34730 q^{7} +2.53209 q^{8} +O(q^{10})$	\(q-1.53209 q^{2} +0.347296 q^{4} -1.00000 q^{5} +1.34730 q^{7} +2.53209 q^{8} +1.53209 q^{10} +5.75877 q^{11} -2.87939 q^{13} -2.06418 q^{14} -4.57398 q^{16} +1.46791 q^{17} +3.06418 q^{19} -0.347296 q^{20} -8.82295 q^{22} +4.45336 q^{23} +1.00000 q^{25} +4.41147 q^{26} +0.467911 q^{28} +2.77332 q^{29} -0.610815 q^{31} +1.94356 q^{32} -2.24897 q^{34} -1.34730 q^{35} -1.14290 q^{37} -4.69459 q^{38} -2.53209 q^{40} +11.1061 q^{41} +8.29086 q^{43} +2.00000 q^{44} -6.82295 q^{46} +2.65270 q^{47} -5.18479 q^{49} -1.53209 q^{50} -1.00000 q^{52} +1.10607 q^{53} -5.75877 q^{55} +3.41147 q^{56} -4.24897 q^{58} +2.50980 q^{59} +3.06418 q^{61} +0.935822 q^{62} +6.17024 q^{64} +2.87939 q^{65} -7.43376 q^{67} +0.509800 q^{68} +2.06418 q^{70} -11.0351 q^{71} -6.45336 q^{73} +1.75103 q^{74} +1.06418 q^{76} +7.75877 q^{77} +5.47565 q^{79} +4.57398 q^{80} -17.0155 q^{82} -1.14796 q^{83} -1.46791 q^{85} -12.7023 q^{86} +14.5817 q^{88} -1.00000 q^{89} -3.87939 q^{91} +1.54664 q^{92} -4.06418 q^{94} -3.06418 q^{95} -5.43376 q^{97} +7.94356 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{5} + 3 q^{7} + 3 q^{8}+O(q^{10}) $ 3 * q - 3 * q^5 + 3 * q^7 + 3 * q^8	$ 3 q - 3 q^{5} + 3 q^{7} + 3 q^{8} + 6 q^{11} - 3 q^{13} + 3 q^{14} - 6 q^{16} + 9 q^{17} - 6 q^{22} + 3 q^{25} + 3 q^{26} + 6 q^{28} + 15 q^{29} - 6 q^{31} - 9 q^{32} + 6 q^{34} - 3 q^{35} - 3 q^{37} - 12 q^{38} - 3 q^{40} + 21 q^{41} + 9 q^{43} + 6 q^{44} + 9 q^{47} - 12 q^{49} - 3 q^{52} - 9 q^{53} - 6 q^{55} + 9 q^{59} + 12 q^{62} - 3 q^{64} + 3 q^{65} - 6 q^{67} + 3 q^{68} - 3 q^{70} + 12 q^{71} - 6 q^{73} + 18 q^{74} - 6 q^{76} + 12 q^{77} - 3 q^{79} + 6 q^{80} - 3 q^{82} + 12 q^{83} - 9 q^{85} - 12 q^{86} + 12 q^{88} - 3 q^{89} - 6 q^{91} + 18 q^{92} - 3 q^{94} + 9 q^{98}+O(q^{100}) $ 3 * q - 3 * q^5 + 3 * q^7 + 3 * q^8 + 6 * q^11 - 3 * q^13 + 3 * q^14 - 6 * q^16 + 9 * q^17 - 6 * q^22 + 3 * q^25 + 3 * q^26 + 6 * q^28 + 15 * q^29 - 6 * q^31 - 9 * q^32 + 6 * q^34 - 3 * q^35 - 3 * q^37 - 12 * q^38 - 3 * q^40 + 21 * q^41 + 9 * q^43 + 6 * q^44 + 9 * q^47 - 12 * q^49 - 3 * q^52 - 9 * q^53 - 6 * q^55 + 9 * q^59 + 12 * q^62 - 3 * q^64 + 3 * q^65 - 6 * q^67 + 3 * q^68 - 3 * q^70 + 12 * q^71 - 6 * q^73 + 18 * q^74 - 6 * q^76 + 12 * q^77 - 3 * q^79 + 6 * q^80 - 3 * q^82 + 12 * q^83 - 9 * q^85 - 12 * q^86 + 12 * q^88 - 3 * q^89 - 6 * q^91 + 18 * q^92 - 3 * q^94 + 9 * 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 $

$2$	−1.53209	−1.08335	−0.541675	−	0.840588i	$-0.682210\pi$
$2$	−1.53209	−1.08335	−0.541675	+	0.840588i	$0.682210\pi$
$3$	0	0
$3$	0	0
$4$	0.347296	0.173648
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.34730	0.509230	0.254615	−	0.967042i	$-0.418051\pi$
$7$	1.34730	0.509230	0.254615	+	0.967042i	$0.418051\pi$
$8$	2.53209	0.895229
$9$	0	0
$10$	1.53209	0.484489
$11$	5.75877	1.73633	0.868167	−	0.496272i	$-0.165298\pi$
$11$	5.75877	1.73633	0.868167	+	0.496272i	$0.165298\pi$
$12$	0	0
$13$	−2.87939	−0.798598	−0.399299	−	0.916821i	$-0.630746\pi$
$13$	−2.87939	−0.798598	−0.399299	+	0.916821i	$0.630746\pi$
$14$	−2.06418	−0.551675
$15$	0	0
$16$	−4.57398	−1.14349
$17$	1.46791	0.356021	0.178010	−	0.984029i	$-0.443034\pi$
$17$	1.46791	0.356021	0.178010	+	0.984029i	$0.443034\pi$
$18$	0	0
$19$	3.06418	0.702971	0.351485	−	0.936193i	$-0.385677\pi$
$19$	3.06418	0.702971	0.351485	+	0.936193i	$0.385677\pi$
$20$	−0.347296	−0.0776578
$21$	0	0
$22$	−8.82295	−1.88106
$23$	4.45336	0.928590	0.464295	−	0.885681i	$-0.346308\pi$
$23$	4.45336	0.928590	0.464295	+	0.885681i	$0.346308\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	4.41147	0.865161
$27$	0	0
$28$	0.467911	0.0884269
$29$	2.77332	0.514992	0.257496	−	0.966279i	$-0.417103\pi$
$29$	2.77332	0.514992	0.257496	+	0.966279i	$0.417103\pi$
$30$	0	0
$31$	−0.610815	−0.109706	−0.0548528	−	0.998494i	$-0.517469\pi$
$31$	−0.610815	−0.109706	−0.0548528	+	0.998494i	$0.517469\pi$
$32$	1.94356	0.343577
$33$	0	0
$34$	−2.24897	−0.385695
$35$	−1.34730	−0.227735
$36$	0	0
$37$	−1.14290	−0.187892	−0.0939461	−	0.995577i	$-0.529948\pi$
$37$	−1.14290	−0.187892	−0.0939461	+	0.995577i	$0.529948\pi$
$38$	−4.69459	−0.761564
$39$	0	0
$40$	−2.53209	−0.400358
$41$	11.1061	1.73448	0.867238	−	0.497894i	$-0.165893\pi$
$41$	11.1061	1.73448	0.867238	+	0.497894i	$0.165893\pi$
$42$	0	0
$43$	8.29086	1.26434	0.632172	−	0.774828i	$-0.282164\pi$
$43$	8.29086	1.26434	0.632172	+	0.774828i	$0.282164\pi$
$44$	2.00000	0.301511
$45$	0	0
$46$	−6.82295	−1.00599
$47$	2.65270	0.386937	0.193468	−	0.981107i	$-0.438026\pi$
$47$	2.65270	0.386937	0.193468	+	0.981107i	$0.438026\pi$
$48$	0	0
$49$	−5.18479	−0.740685
$50$	−1.53209	−0.216670
$51$	0	0
$52$	−1.00000	−0.138675
$53$	1.10607	0.151930	0.0759650	−	0.997110i	$-0.475796\pi$
$53$	1.10607	0.151930	0.0759650	+	0.997110i	$0.475796\pi$
$54$	0	0
$55$	−5.75877	−0.776512
$56$	3.41147	0.455877
$57$	0	0
$58$	−4.24897	−0.557917
$59$	2.50980	0.326748	0.163374	−	0.986564i	$-0.447762\pi$
$59$	2.50980	0.326748	0.163374	+	0.986564i	$0.447762\pi$
$60$	0	0
$61$	3.06418	0.392328	0.196164	−	0.980571i	$-0.437152\pi$
$61$	3.06418	0.392328	0.196164	+	0.980571i	$0.437152\pi$
$62$	0.935822	0.118850
$63$	0	0
$64$	6.17024	0.771281
$65$	2.87939	0.357144
$66$	0	0
$67$	−7.43376	−0.908179	−0.454089	−	0.890956i	$-0.650035\pi$
$67$	−7.43376	−0.908179	−0.454089	+	0.890956i	$0.650035\pi$
$68$	0.509800	0.0618224
$69$	0	0
$70$	2.06418	0.246716
$71$	−11.0351	−1.30962	−0.654812	−	0.755792i	$-0.727252\pi$
$71$	−11.0351	−1.30962	−0.654812	+	0.755792i	$0.727252\pi$
$72$	0	0
$73$	−6.45336	−0.755309	−0.377655	−	0.925947i	$-0.623269\pi$
$73$	−6.45336	−0.755309	−0.377655	+	0.925947i	$0.623269\pi$
$74$	1.75103	0.203553
$75$	0	0
$76$	1.06418	0.122070
$77$	7.75877	0.884194
$78$	0	0
$79$	5.47565	0.616059	0.308029	−	0.951377i	$-0.400330\pi$
$79$	5.47565	0.616059	0.308029	+	0.951377i	$0.400330\pi$
$80$	4.57398	0.511386
$81$	0	0
$82$	−17.0155	−1.87905
$83$	−1.14796	−0.126005	−0.0630023	−	0.998013i	$-0.520068\pi$
$83$	−1.14796	−0.126005	−0.0630023	+	0.998013i	$0.520068\pi$
$84$	0	0
$85$	−1.46791	−0.159217
$86$	−12.7023	−1.36973
$87$	0	0
$88$	14.5817	1.55442
$89$	−1.00000	−0.106000
$89$	−1.00000	−0.106000
$90$	0	0
$91$	−3.87939	−0.406670
$92$	1.54664	0.161248
$93$	0	0
$94$	−4.06418	−0.419188
$95$	−3.06418	−0.314378
$96$	0	0
$97$	−5.43376	−0.551715	−0.275858	−	0.961199i	$-0.588962\pi$
$97$	−5.43376	−0.551715	−0.275858	+	0.961199i	$0.588962\pi$
$98$	7.94356	0.802421
$99$	0	0
$100$	0.347296	0.0347296
$101$	13.0915	1.30265	0.651327	−	0.758797i	$-0.274212\pi$
$101$	13.0915	1.30265	0.651327	+	0.758797i	$0.274212\pi$
$102$	0	0
$103$	−5.09152	−0.501682	−0.250841	−	0.968028i	$-0.580707\pi$
$103$	−5.09152	−0.501682	−0.250841	+	0.968028i	$0.580707\pi$
$104$	−7.29086	−0.714928
$105$	0	0
$106$	−1.69459	−0.164593
$107$	−17.9368	−1.73401	−0.867006	−	0.498298i	$-0.833959\pi$
$107$	−17.9368	−1.73401	−0.867006	+	0.498298i	$0.833959\pi$
$108$	0	0
$109$	−11.8152	−1.13169	−0.565846	−	0.824511i	$-0.691450\pi$
$109$	−11.8152	−1.13169	−0.565846	+	0.824511i	$0.691450\pi$
$110$	8.82295	0.841235
$111$	0	0
$112$	−6.16250	−0.582302
$113$	2.85204	0.268298	0.134149	−	0.990961i	$-0.457170\pi$
$113$	2.85204	0.268298	0.134149	+	0.990961i	$0.457170\pi$
$114$	0	0
$115$	−4.45336	−0.415278
$116$	0.963163	0.0894275
$117$	0	0
$118$	−3.84524	−0.353983
$119$	1.97771	0.181296
$120$	0	0
$121$	22.1634	2.01486
$122$	−4.69459	−0.425028
$123$	0	0
$124$	−0.212134	−0.0190502
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−10.7442	−0.953396	−0.476698	−	0.879067i	$-0.658166\pi$
$127$	−10.7442	−0.953396	−0.476698	+	0.879067i	$0.658166\pi$
$128$	−13.3405	−1.17914
$129$	0	0
$130$	−4.41147	−0.386912
$131$	−3.01960	−0.263824	−0.131912	−	0.991261i	$-0.542112\pi$
$131$	−3.01960	−0.263824	−0.131912	+	0.991261i	$0.542112\pi$
$132$	0	0
$133$	4.12836	0.357974
$134$	11.3892	0.983876
$135$	0	0
$136$	3.71688	0.318720
$137$	2.69459	0.230215	0.115107	−	0.993353i	$-0.463279\pi$
$137$	2.69459	0.230215	0.115107	+	0.993353i	$0.463279\pi$
$138$	0	0
$139$	−4.15570	−0.352482	−0.176241	−	0.984347i	$-0.556394\pi$
$139$	−4.15570	−0.352482	−0.176241	+	0.984347i	$0.556394\pi$
$140$	−0.467911	−0.0395457
$141$	0	0
$142$	16.9067	1.41878
$143$	−16.5817	−1.38663
$144$	0	0
$145$	−2.77332	−0.230312
$146$	9.88713	0.818264
$147$	0	0
$148$	−0.396926	−0.0326271
$149$	−5.03508	−0.412490	−0.206245	−	0.978500i	$-0.566124\pi$
$149$	−5.03508	−0.412490	−0.206245	+	0.978500i	$0.566124\pi$
$150$	0	0
$151$	−5.84255	−0.475460	−0.237730	−	0.971331i	$-0.576403\pi$
$151$	−5.84255	−0.475460	−0.237730	+	0.971331i	$0.576403\pi$
$152$	7.75877	0.629319
$153$	0	0
$154$	−11.8871	−0.957892
$155$	0.610815	0.0490618
$156$	0	0
$157$	22.1830	1.77040	0.885200	−	0.465211i	$-0.154022\pi$
$157$	22.1830	1.77040	0.885200	+	0.465211i	$0.154022\pi$
$158$	−8.38919	−0.667408
$159$	0	0
$160$	−1.94356	−0.153652
$161$	6.00000	0.472866
$162$	0	0
$163$	22.3131	1.74770	0.873850	−	0.486195i	$-0.161616\pi$
$163$	22.3131	1.74770	0.873850	+	0.486195i	$0.161616\pi$
$164$	3.85710	0.301189
$165$	0	0
$166$	1.75877	0.136507
$167$	1.09152	0.0844643	0.0422321	−	0.999108i	$-0.486553\pi$
$167$	1.09152	0.0844643	0.0422321	+	0.999108i	$0.486553\pi$
$168$	0	0
$169$	−4.70914	−0.362242
$170$	2.24897	0.172488
$171$	0	0
$172$	2.87939	0.219551
$173$	−6.76651	−0.514448	−0.257224	−	0.966352i	$-0.582808\pi$
$173$	−6.76651	−0.514448	−0.257224	+	0.966352i	$0.582808\pi$
$174$	0	0
$175$	1.34730	0.101846
$176$	−26.3405	−1.98549
$177$	0	0
$178$	1.53209	0.114835
$179$	1.41828	0.106007	0.0530037	−	0.998594i	$-0.483121\pi$
$179$	1.41828	0.106007	0.0530037	+	0.998594i	$0.483121\pi$
$180$	0	0
$181$	−6.25671	−0.465058	−0.232529	−	0.972590i	$-0.574700\pi$
$181$	−6.25671	−0.465058	−0.232529	+	0.972590i	$0.574700\pi$
$182$	5.94356	0.440566
$183$	0	0
$184$	11.2763	0.831301
$185$	1.14290	0.0840279
$186$	0	0
$187$	8.45336	0.618171
$188$	0.921274	0.0671908
$189$	0	0
$190$	4.69459	0.340582
$191$	2.87939	0.208345	0.104173	−	0.994559i	$-0.466781\pi$
$191$	2.87939	0.208345	0.104173	+	0.994559i	$0.466781\pi$
$192$	0	0
$193$	14.5844	1.04981	0.524904	−	0.851161i	$-0.324101\pi$
$193$	14.5844	1.04981	0.524904	+	0.851161i	$0.324101\pi$
$194$	8.32501	0.597701
$195$	0	0
$196$	−1.80066	−0.128619
$197$	7.36009	0.524385	0.262192	−	0.965016i	$-0.415555\pi$
$197$	7.36009	0.524385	0.262192	+	0.965016i	$0.415555\pi$
$198$	0	0
$199$	24.3063	1.72303	0.861515	−	0.507731i	$-0.169516\pi$
$199$	24.3063	1.72303	0.861515	+	0.507731i	$0.169516\pi$
$200$	2.53209	0.179046
$201$	0	0
$202$	−20.0574	−1.41123
$203$	3.73648	0.262250
$204$	0	0
$205$	−11.1061	−0.775681
$206$	7.80066	0.543498
$207$	0	0
$208$	13.1702	0.913192
$209$	17.6459	1.22059
$210$	0	0
$211$	−2.34049	−0.161126	−0.0805630	−	0.996750i	$-0.525672\pi$
$211$	−2.34049	−0.161126	−0.0805630	+	0.996750i	$0.525672\pi$
$212$	0.384133	0.0263824
$213$	0	0
$214$	27.4807	1.87854
$215$	−8.29086	−0.565432
$216$	0	0
$217$	−0.822948	−0.0558654
$218$	18.1019	1.22602
$219$	0	0
$220$	−2.00000	−0.134840
$221$	−4.22668	−0.284317
$222$	0	0
$223$	2.28581	0.153069	0.0765345	−	0.997067i	$-0.475614\pi$
$223$	2.28581	0.153069	0.0765345	+	0.997067i	$0.475614\pi$
$224$	2.61856	0.174960
$225$	0	0
$226$	−4.36959	−0.290660
$227$	−22.9094	−1.52055	−0.760276	−	0.649601i	$-0.774936\pi$
$227$	−22.9094	−1.52055	−0.760276	+	0.649601i	$0.774936\pi$
$228$	0	0
$229$	12.4979	0.825887	0.412944	−	0.910757i	$-0.364501\pi$
$229$	12.4979	0.825887	0.412944	+	0.910757i	$0.364501\pi$
$230$	6.82295	0.449892
$231$	0	0
$232$	7.02229	0.461036
$233$	17.5449	1.14940	0.574702	−	0.818363i	$-0.305118\pi$
$233$	17.5449	1.14940	0.574702	+	0.818363i	$0.305118\pi$
$234$	0	0
$235$	−2.65270	−0.173043
$236$	0.871644	0.0567392
$237$	0	0
$238$	−3.03003	−0.196408
$239$	−0.652704	−0.0422199	−0.0211099	−	0.999777i	$-0.506720\pi$
$239$	−0.652704	−0.0422199	−0.0211099	+	0.999777i	$0.506720\pi$
$240$	0	0
$241$	27.4884	1.77069	0.885343	−	0.464938i	$-0.153923\pi$
$241$	27.4884	1.77069	0.885343	+	0.464938i	$0.153923\pi$
$242$	−33.9564	−2.18280
$243$	0	0
$244$	1.06418	0.0681270
$245$	5.18479	0.331244
$246$	0	0
$247$	−8.82295	−0.561391
$248$	−1.54664	−0.0982115
$249$	0	0
$250$	1.53209	0.0968978
$251$	21.0797	1.33054	0.665268	−	0.746605i	$-0.268317\pi$
$251$	21.0797	1.33054	0.665268	+	0.746605i	$0.268317\pi$
$252$	0	0
$253$	25.6459	1.61234
$254$	16.4611	1.03286
$255$	0	0
$256$	8.09833	0.506145
$257$	4.93851	0.308056	0.154028	−	0.988066i	$-0.450775\pi$
$257$	4.93851	0.308056	0.154028	+	0.988066i	$0.450775\pi$
$258$	0	0
$259$	−1.53983	−0.0956804
$260$	1.00000	0.0620174
$261$	0	0
$262$	4.62630	0.285814
$263$	−13.6800	−0.843548	−0.421774	−	0.906701i	$-0.638592\pi$
$263$	−13.6800	−0.843548	−0.421774	+	0.906701i	$0.638592\pi$
$264$	0	0
$265$	−1.10607	−0.0679451
$266$	−6.32501	−0.387811
$267$	0	0
$268$	−2.58172	−0.157704
$269$	−0.951304	−0.0580020	−0.0290010	−	0.999579i	$-0.509233\pi$
$269$	−0.951304	−0.0580020	−0.0290010	+	0.999579i	$0.509233\pi$
$270$	0	0
$271$	12.9932	0.789280	0.394640	−	0.918836i	$-0.370869\pi$
$271$	12.9932	0.789280	0.394640	+	0.918836i	$0.370869\pi$
$272$	−6.71419	−0.407108
$273$	0	0
$274$	−4.12836	−0.249403
$275$	5.75877	0.347267
$276$	0	0
$277$	−0.862149	−0.0518015	−0.0259008	−	0.999665i	$-0.508245\pi$
$277$	−0.862149	−0.0518015	−0.0259008	+	0.999665i	$0.508245\pi$
$278$	6.36690	0.381861
$279$	0	0
$280$	−3.41147	−0.203875
$281$	30.2226	1.80293	0.901463	−	0.432855i	$-0.142494\pi$
$281$	30.2226	1.80293	0.901463	+	0.432855i	$0.142494\pi$
$282$	0	0
$283$	6.49794	0.386262	0.193131	−	0.981173i	$-0.438136\pi$
$283$	6.49794	0.386262	0.193131	+	0.981173i	$0.438136\pi$
$284$	−3.83244	−0.227414
$285$	0	0
$286$	25.4047	1.50221
$287$	14.9632	0.883248
$288$	0	0
$289$	−14.8452	−0.873249
$290$	4.24897	0.249508
$291$	0	0
$292$	−2.24123	−0.131158
$293$	−25.7452	−1.50405	−0.752024	−	0.659136i	$-0.770922\pi$
$293$	−25.7452	−1.50405	−0.752024	+	0.659136i	$0.770922\pi$
$294$	0	0
$295$	−2.50980	−0.146126
$296$	−2.89393	−0.168206
$297$	0	0
$298$	7.71419	0.446871
$299$	−12.8229	−0.741570
$300$	0	0
$301$	11.1702	0.643842
$302$	8.95130	0.515090
$303$	0	0
$304$	−14.0155	−0.803843
$305$	−3.06418	−0.175454
$306$	0	0
$307$	−7.02498	−0.400937	−0.200468	−	0.979700i	$-0.564246\pi$
$307$	−7.02498	−0.400937	−0.200468	+	0.979700i	$0.564246\pi$
$308$	2.69459	0.153539
$309$	0	0
$310$	−0.935822	−0.0531511
$311$	29.4492	1.66991	0.834957	−	0.550316i	$-0.185493\pi$
$311$	29.4492	1.66991	0.834957	+	0.550316i	$0.185493\pi$
$312$	0	0
$313$	18.4638	1.04364	0.521818	−	0.853057i	$-0.325254\pi$
$313$	18.4638	1.04364	0.521818	+	0.853057i	$0.325254\pi$
$314$	−33.9864	−1.91796
$315$	0	0
$316$	1.90167	0.106978
$317$	7.11381	0.399551	0.199776	−	0.979842i	$-0.435979\pi$
$317$	7.11381	0.399551	0.199776	+	0.979842i	$0.435979\pi$
$318$	0	0
$319$	15.9709	0.894199
$320$	−6.17024	−0.344927
$321$	0	0
$322$	−9.19253	−0.512280
$323$	4.49794	0.250272
$324$	0	0
$325$	−2.87939	−0.159720
$326$	−34.1857	−1.89337
$327$	0	0
$328$	28.1215	1.55275
$329$	3.57398	0.197040
$330$	0	0
$331$	−24.5800	−1.35104	−0.675519	−	0.737343i	$-0.736080\pi$
$331$	−24.5800	−1.35104	−0.675519	+	0.737343i	$0.736080\pi$
$332$	−0.398681	−0.0218805
$333$	0	0
$334$	−1.67230	−0.0915044
$335$	7.43376	0.406150
$336$	0	0
$337$	−24.5398	−1.33677	−0.668385	−	0.743816i	$-0.733014\pi$
$337$	−24.5398	−1.33677	−0.668385	+	0.743816i	$0.733014\pi$
$338$	7.21482	0.392435
$339$	0	0
$340$	−0.509800	−0.0276478
$341$	−3.51754	−0.190486
$342$	0	0
$343$	−16.4165	−0.886409
$344$	20.9932	1.13188
$345$	0	0
$346$	10.3669	0.557328
$347$	6.86753	0.368668	0.184334	−	0.982864i	$-0.440987\pi$
$347$	6.86753	0.368668	0.184334	+	0.982864i	$0.440987\pi$
$348$	0	0
$349$	−21.1088	−1.12993	−0.564963	−	0.825116i	$-0.691110\pi$
$349$	−21.1088	−1.12993	−0.564963	+	0.825116i	$0.691110\pi$
$350$	−2.06418	−0.110335
$351$	0	0
$352$	11.1925	0.596564
$353$	1.14796	0.0610995	0.0305498	−	0.999533i	$-0.490274\pi$
$353$	1.14796	0.0610995	0.0305498	+	0.999533i	$0.490274\pi$
$354$	0	0
$355$	11.0351	0.585681
$356$	−0.347296	−0.0184067
$357$	0	0
$358$	−2.17293	−0.114843
$359$	12.9094	0.681333	0.340666	−	0.940184i	$-0.389347\pi$
$359$	12.9094	0.681333	0.340666	+	0.940184i	$0.389347\pi$
$360$	0	0
$361$	−9.61081	−0.505832
$362$	9.58584	0.503820
$363$	0	0
$364$	−1.34730	−0.0706175
$365$	6.45336	0.337784
$366$	0	0
$367$	−22.0547	−1.15125	−0.575623	−	0.817716i	$-0.695240\pi$
$367$	−22.0547	−1.15125	−0.575623	+	0.817716i	$0.695240\pi$
$368$	−20.3696	−1.06184
$369$	0	0
$370$	−1.75103	−0.0910317
$371$	1.49020	0.0773673
$372$	0	0
$373$	−13.2371	−0.685392	−0.342696	−	0.939446i	$-0.611340\pi$
$373$	−13.2371	−0.685392	−0.342696	+	0.939446i	$0.611340\pi$
$374$	−12.9513	−0.669696
$375$	0	0
$376$	6.71688	0.346397
$377$	−7.98545	−0.411272
$378$	0	0
$379$	6.65539	0.341865	0.170932	−	0.985283i	$-0.445322\pi$
$379$	6.65539	0.341865	0.170932	+	0.985283i	$0.445322\pi$
$380$	−1.06418	−0.0545912
$381$	0	0
$382$	−4.41147	−0.225711
$383$	3.30541	0.168898	0.0844492	−	0.996428i	$-0.473087\pi$
$383$	3.30541	0.168898	0.0844492	+	0.996428i	$0.473087\pi$
$384$	0	0
$385$	−7.75877	−0.395424
$386$	−22.3446	−1.13731
$387$	0	0
$388$	−1.88713	−0.0958043
$389$	17.9736	0.911297	0.455649	−	0.890160i	$-0.349407\pi$
$389$	17.9736	0.911297	0.455649	+	0.890160i	$0.349407\pi$
$390$	0	0
$391$	6.53714	0.330597
$392$	−13.1284	−0.663082
$393$	0	0
$394$	−11.2763	−0.568092
$395$	−5.47565	−0.275510
$396$	0	0
$397$	−3.23442	−0.162331	−0.0811655	−	0.996701i	$-0.525864\pi$
$397$	−3.23442	−0.162331	−0.0811655	+	0.996701i	$0.525864\pi$
$398$	−37.2395	−1.86665
$399$	0	0
$400$	−4.57398	−0.228699
$401$	38.7547	1.93531	0.967657	−	0.252268i	$-0.0811764\pi$
$401$	38.7547	1.93531	0.967657	+	0.252268i	$0.0811764\pi$
$402$	0	0
$403$	1.75877	0.0876106
$404$	4.54664	0.226204
$405$	0	0
$406$	−5.72462	−0.284108
$407$	−6.58172	−0.326244
$408$	0	0
$409$	−24.0966	−1.19150	−0.595749	−	0.803170i	$-0.703145\pi$
$409$	−24.0966	−1.19150	−0.595749	+	0.803170i	$0.703145\pi$
$410$	17.0155	0.840335
$411$	0	0
$412$	−1.76827	−0.0871162
$413$	3.38144	0.166390
$414$	0	0
$415$	1.14796	0.0563509
$416$	−5.59627	−0.274380
$417$	0	0
$418$	−27.0351	−1.32233
$419$	0.0395250	0.00193092	0.000965461	−	1.00000i	$-0.499693\pi$
$419$	0.0395250	0.00193092	0.000965461		1.00000i	$0.499693\pi$
$420$	0	0
$421$	25.4047	1.23815	0.619074	−	0.785333i	$-0.287508\pi$
$421$	25.4047	1.23815	0.619074	+	0.785333i	$0.287508\pi$
$422$	3.58584	0.174556
$423$	0	0
$424$	2.80066	0.136012
$425$	1.46791	0.0712041
$426$	0	0
$427$	4.12836	0.199785
$428$	−6.22937	−0.301108
$429$	0	0
$430$	12.7023	0.612561
$431$	24.0496	1.15843	0.579215	−	0.815175i	$-0.303359\pi$
$431$	24.0496	1.15843	0.579215	+	0.815175i	$0.303359\pi$
$432$	0	0
$433$	−5.85978	−0.281603	−0.140802	−	0.990038i	$-0.544968\pi$
$433$	−5.85978	−0.281603	−0.140802	+	0.990038i	$0.544968\pi$
$434$	1.26083	0.0605218
$435$	0	0
$436$	−4.10338	−0.196516
$437$	13.6459	0.652772
$438$	0	0
$439$	−9.10876	−0.434737	−0.217369	−	0.976090i	$-0.569747\pi$
$439$	−9.10876	−0.434737	−0.217369	+	0.976090i	$0.569747\pi$
$440$	−14.5817	−0.695156
$441$	0	0
$442$	6.47565	0.308015
$443$	−4.18479	−0.198825	−0.0994127	−	0.995046i	$-0.531696\pi$
$443$	−4.18479	−0.198825	−0.0994127	+	0.995046i	$0.531696\pi$
$444$	0	0
$445$	1.00000	0.0474045
$446$	−3.50206	−0.165827
$447$	0	0
$448$	8.31315	0.392759
$449$	−14.5972	−0.688884	−0.344442	−	0.938808i	$-0.611932\pi$
$449$	−14.5972	−0.688884	−0.344442	+	0.938808i	$0.611932\pi$
$450$	0	0
$451$	63.9573	3.01163
$452$	0.990505	0.0465894
$453$	0	0
$454$	35.0993	1.64729
$455$	3.87939	0.181868
$456$	0	0
$457$	4.61081	0.215685	0.107842	−	0.994168i	$-0.465606\pi$
$457$	4.61081	0.215685	0.107842	+	0.994168i	$0.465606\pi$
$458$	−19.1480	−0.894725
$459$	0	0
$460$	−1.54664	−0.0721123
$461$	41.5877	1.93693	0.968466	−	0.249145i	$-0.0801495\pi$
$461$	41.5877	1.93693	0.968466	+	0.249145i	$0.0801495\pi$
$462$	0	0
$463$	−28.1284	−1.30724	−0.653618	−	0.756825i	$-0.726750\pi$
$463$	−28.1284	−1.30724	−0.653618	+	0.756825i	$0.726750\pi$
$464$	−12.6851	−0.588891
$465$	0	0
$466$	−26.8803	−1.24521
$467$	−5.77601	−0.267282	−0.133641	−	0.991030i	$-0.542667\pi$
$467$	−5.77601	−0.267282	−0.133641	+	0.991030i	$0.542667\pi$
$468$	0	0
$469$	−10.0155	−0.462472
$470$	4.06418	0.187467
$471$	0	0
$472$	6.35504	0.292514
$473$	47.7452	2.19532
$474$	0	0
$475$	3.06418	0.140594
$476$	0.686852	0.0314818
$477$	0	0
$478$	1.00000	0.0457389
$479$	4.01548	0.183472	0.0917360	−	0.995783i	$-0.470758\pi$
$479$	4.01548	0.183472	0.0917360	+	0.995783i	$0.470758\pi$
$480$	0	0
$481$	3.29086	0.150050
$482$	−42.1147	−1.91827
$483$	0	0
$484$	7.69728	0.349876
$485$	5.43376	0.246734
$486$	0	0
$487$	−34.6364	−1.56953	−0.784763	−	0.619797i	$-0.787215\pi$
$487$	−34.6364	−1.56953	−0.784763	+	0.619797i	$0.787215\pi$
$488$	7.75877	0.351223
$489$	0	0
$490$	−7.94356	−0.358854
$491$	37.5877	1.69631	0.848155	−	0.529749i	$-0.177714\pi$
$491$	37.5877	1.69631	0.848155	+	0.529749i	$0.177714\pi$
$492$	0	0
$493$	4.07098	0.183348
$494$	13.5175	0.608183
$495$	0	0
$496$	2.79385	0.125448
$497$	−14.8675	−0.666900
$498$	0	0
$499$	17.6459	0.789939	0.394969	−	0.918694i	$-0.370755\pi$
$499$	17.6459	0.789939	0.394969	+	0.918694i	$0.370755\pi$
$500$	−0.347296	−0.0155316
$501$	0	0
$502$	−32.2959	−1.44144
$503$	−22.0993	−0.985357	−0.492679	−	0.870211i	$-0.663982\pi$
$503$	−22.0993	−0.985357	−0.492679	+	0.870211i	$0.663982\pi$
$504$	0	0
$505$	−13.0915	−0.582565
$506$	−39.2918	−1.74673
$507$	0	0
$508$	−3.73143	−0.165555
$509$	42.3560	1.87740	0.938698	−	0.344741i	$-0.112033\pi$
$509$	42.3560	1.87740	0.938698	+	0.344741i	$0.112033\pi$
$510$	0	0
$511$	−8.69459	−0.384626
$512$	14.2736	0.630811
$513$	0	0
$514$	−7.56624	−0.333732
$515$	5.09152	0.224359
$516$	0	0
$517$	15.2763	0.671851
$518$	2.35916	0.103655
$519$	0	0
$520$	7.29086	0.319725
$521$	4.49525	0.196941	0.0984703	−	0.995140i	$-0.468605\pi$
$521$	4.49525	0.196941	0.0984703	+	0.995140i	$0.468605\pi$
$522$	0	0
$523$	5.14796	0.225104	0.112552	−	0.993646i	$-0.464097\pi$
$523$	5.14796	0.225104	0.112552	+	0.993646i	$0.464097\pi$
$524$	−1.04870	−0.0458125
$525$	0	0
$526$	20.9590	0.913858
$527$	−0.896622	−0.0390574
$528$	0	0
$529$	−3.16756	−0.137720
$530$	1.69459	0.0736084
$531$	0	0
$532$	1.43376	0.0621615
$533$	−31.9786	−1.38515
$534$	0	0
$535$	17.9368	0.775474
$536$	−18.8229	−0.813028
$537$	0	0
$538$	1.45748	0.0628365
$539$	−29.8580	−1.28608
$540$	0	0
$541$	11.0506	0.475101	0.237550	−	0.971375i	$-0.423656\pi$
$541$	11.0506	0.475101	0.237550	+	0.971375i	$0.423656\pi$
$542$	−19.9067	−0.855067
$543$	0	0
$544$	2.85298	0.122320
$545$	11.8152	0.506108
$546$	0	0
$547$	34.2550	1.46464	0.732318	−	0.680963i	$-0.238438\pi$
$547$	34.2550	1.46464	0.732318	+	0.680963i	$0.238438\pi$
$548$	0.935822	0.0399763
$549$	0	0
$550$	−8.82295	−0.376212
$551$	8.49794	0.362024
$552$	0	0
$553$	7.37733	0.313716
$554$	1.32089	0.0561192
$555$	0	0
$556$	−1.44326	−0.0612078
$557$	−24.8093	−1.05121	−0.525603	−	0.850730i	$-0.676160\pi$
$557$	−24.8093	−1.05121	−0.525603	+	0.850730i	$0.676160\pi$
$558$	0	0
$559$	−23.8726	−1.00970
$560$	6.16250	0.260413
$561$	0	0
$562$	−46.3037	−1.95320
$563$	25.1343	1.05929	0.529643	−	0.848221i	$-0.322326\pi$
$563$	25.1343	1.05929	0.529643	+	0.848221i	$0.322326\pi$
$564$	0	0
$565$	−2.85204	−0.119986
$566$	−9.95542	−0.418458
$567$	0	0
$568$	−27.9418	−1.17241
$569$	37.0915	1.55496	0.777479	−	0.628909i	$-0.216498\pi$
$569$	37.0915	1.55496	0.777479	+	0.628909i	$0.216498\pi$
$570$	0	0
$571$	3.36009	0.140615	0.0703077	−	0.997525i	$-0.477602\pi$
$571$	3.36009	0.140615	0.0703077	+	0.997525i	$0.477602\pi$
$572$	−5.75877	−0.240786
$573$	0	0
$574$	−22.9249	−0.956867
$575$	4.45336	0.185718
$576$	0	0
$577$	46.4424	1.93342	0.966712	−	0.255867i	$-0.0823609\pi$
$577$	46.4424	1.93342	0.966712	+	0.255867i	$0.0823609\pi$
$578$	22.7442	0.946035
$579$	0	0
$580$	−0.963163	−0.0399932
$581$	−1.54664	−0.0641653
$582$	0	0
$583$	6.36959	0.263801
$584$	−16.3405	−0.676174
$585$	0	0
$586$	39.4439	1.62941
$587$	−26.2959	−1.08535	−0.542674	−	0.839943i	$-0.682588\pi$
$587$	−26.2959	−1.08535	−0.542674	+	0.839943i	$0.682588\pi$
$588$	0	0
$589$	−1.87164	−0.0771198
$590$	3.84524	0.158306
$591$	0	0
$592$	5.22762	0.214854
$593$	−5.14796	−0.211401	−0.105701	−	0.994398i	$-0.533709\pi$
$593$	−5.14796	−0.211401	−0.105701	+	0.994398i	$0.533709\pi$
$594$	0	0
$595$	−1.97771	−0.0810783
$596$	−1.74867	−0.0716281
$597$	0	0
$598$	19.6459	0.803380
$599$	−31.8553	−1.30157	−0.650787	−	0.759260i	$-0.725561\pi$
$599$	−31.8553	−1.30157	−0.650787	+	0.759260i	$0.725561\pi$
$600$	0	0
$601$	−8.10431	−0.330582	−0.165291	−	0.986245i	$-0.552856\pi$
$601$	−8.10431	−0.330582	−0.165291	+	0.986245i	$0.552856\pi$
$602$	−17.1138	−0.697507
$603$	0	0
$604$	−2.02910	−0.0825627
$605$	−22.1634	−0.901072
$606$	0	0
$607$	5.75877	0.233741	0.116871	−	0.993147i	$-0.462714\pi$
$607$	5.75877	0.233741	0.116871	+	0.993147i	$0.462714\pi$
$608$	5.95542	0.241524
$609$	0	0
$610$	4.69459	0.190079
$611$	−7.63816	−0.309007
$612$	0	0
$613$	23.2080	0.937363	0.468681	−	0.883367i	$-0.344729\pi$
$613$	23.2080	0.937363	0.468681	+	0.883367i	$0.344729\pi$
$614$	10.7629	0.434355
$615$	0	0
$616$	19.6459	0.791556
$617$	−34.4534	−1.38704	−0.693520	−	0.720437i	$-0.743941\pi$
$617$	−34.4534	−1.38704	−0.693520	+	0.720437i	$0.743941\pi$
$618$	0	0
$619$	33.9581	1.36489	0.682446	−	0.730936i	$-0.260916\pi$
$619$	33.9581	1.36489	0.682446	+	0.730936i	$0.260916\pi$
$620$	0.212134	0.00851949
$621$	0	0
$622$	−45.1189	−1.80910
$623$	−1.34730	−0.0539783
$624$	0	0
$625$	1.00000	0.0400000
$626$	−28.2882	−1.13062
$627$	0	0
$628$	7.70409	0.307427
$629$	−1.67768	−0.0668935
$630$	0	0
$631$	−23.7861	−0.946910	−0.473455	−	0.880818i	$-0.656993\pi$
$631$	−23.7861	−0.946910	−0.473455	+	0.880818i	$0.656993\pi$
$632$	13.8648	0.551514
$633$	0	0
$634$	−10.8990	−0.432854
$635$	10.7442	0.426371
$636$	0	0
$637$	14.9290	0.591509
$638$	−24.4688	−0.968731
$639$	0	0
$640$	13.3405	0.527329
$641$	24.5972	0.971531	0.485766	−	0.874089i	$-0.338541\pi$
$641$	24.5972	0.971531	0.485766	+	0.874089i	$0.338541\pi$
$642$	0	0
$643$	20.0838	0.792027	0.396013	−	0.918245i	$-0.370393\pi$
$643$	20.0838	0.792027	0.396013	+	0.918245i	$0.370393\pi$
$644$	2.08378	0.0821124
$645$	0	0
$646$	−6.89124	−0.271132
$647$	24.6946	0.970845	0.485422	−	0.874280i	$-0.338666\pi$
$647$	24.6946	0.970845	0.485422	+	0.874280i	$0.338666\pi$
$648$	0	0
$649$	14.4534	0.567344
$650$	4.41147	0.173032
$651$	0	0
$652$	7.74928	0.303485
$653$	38.9086	1.52261	0.761305	−	0.648393i	$-0.224559\pi$
$653$	38.9086	1.52261	0.761305	+	0.648393i	$0.224559\pi$
$654$	0	0
$655$	3.01960	0.117986
$656$	−50.7989	−1.98336
$657$	0	0
$658$	−5.47565	−0.213463
$659$	−26.0856	−1.01615	−0.508076	−	0.861312i	$-0.669643\pi$
$659$	−26.0856	−1.01615	−0.508076	+	0.861312i	$0.669643\pi$
$660$	0	0
$661$	−12.7047	−0.494155	−0.247078	−	0.968996i	$-0.579470\pi$
$661$	−12.7047	−0.494155	−0.247078	+	0.968996i	$0.579470\pi$
$662$	37.6587	1.46365
$663$	0	0
$664$	−2.90673	−0.112803
$665$	−4.12836	−0.160091
$666$	0	0
$667$	12.3506	0.478217
$668$	0.379081	0.0146671
$669$	0	0
$670$	−11.3892	−0.440003
$671$	17.6459	0.681212
$672$	0	0
$673$	37.6715	1.45213	0.726064	−	0.687627i	$-0.241347\pi$
$673$	37.6715	1.45213	0.726064	+	0.687627i	$0.241347\pi$
$674$	37.5972	1.44819
$675$	0	0
$676$	−1.63547	−0.0629026
$677$	−29.7006	−1.14149	−0.570743	−	0.821129i	$-0.693345\pi$
$677$	−29.7006	−1.14149	−0.570743	+	0.821129i	$0.693345\pi$
$678$	0	0
$679$	−7.32089	−0.280950
$680$	−3.71688	−0.142536
$681$	0	0
$682$	5.38919	0.206363
$683$	3.87702	0.148350	0.0741750	−	0.997245i	$-0.476368\pi$
$683$	3.87702	0.148350	0.0741750	+	0.997245i	$0.476368\pi$
$684$	0	0
$685$	−2.69459	−0.102955
$686$	25.1516	0.960292
$687$	0	0
$688$	−37.9222	−1.44577
$689$	−3.18479	−0.121331
$690$	0	0
$691$	18.7118	0.711832	0.355916	−	0.934518i	$-0.384169\pi$
$691$	18.7118	0.711832	0.355916	+	0.934518i	$0.384169\pi$
$692$	−2.34998	−0.0893330
$693$	0	0
$694$	−10.5217	−0.399397
$695$	4.15570	0.157635
$696$	0	0
$697$	16.3027	0.617510
$698$	32.3405	1.22411
$699$	0	0
$700$	0.467911	0.0176854
$701$	−18.0702	−0.682501	−0.341250	−	0.939972i	$-0.610850\pi$
$701$	−18.0702	−0.682501	−0.341250	+	0.939972i	$0.610850\pi$
$702$	0	0
$703$	−3.50206	−0.132083
$704$	35.5330	1.33920
$705$	0	0
$706$	−1.75877	−0.0661922
$707$	17.6382	0.663351
$708$	0	0
$709$	0.270325	0.0101523	0.00507614	−	0.999987i	$-0.498384\pi$
$709$	0.270325	0.0101523	0.00507614	+	0.999987i	$0.498384\pi$
$710$	−16.9067	−0.634498
$711$	0	0
$712$	−2.53209	−0.0948940
$713$	−2.72018	−0.101872
$714$	0	0
$715$	16.5817	0.620121
$716$	0.492564	0.0184080
$717$	0	0
$718$	−19.7784	−0.738122
$719$	24.0496	0.896900	0.448450	−	0.893808i	$-0.351976\pi$
$719$	24.0496	0.896900	0.448450	+	0.893808i	$0.351976\pi$
$720$	0	0
$721$	−6.85978	−0.255472
$722$	14.7246	0.547994
$723$	0	0
$724$	−2.17293	−0.0807564
$725$	2.77332	0.102998
$726$	0	0
$727$	−26.7543	−0.992263	−0.496132	−	0.868247i	$-0.665247\pi$
$727$	−26.7543	−0.992263	−0.496132	+	0.868247i	$0.665247\pi$
$728$	−9.82295	−0.364063
$729$	0	0
$730$	−9.88713	−0.365939
$731$	12.1702	0.450133
$732$	0	0
$733$	−43.5384	−1.60813	−0.804064	−	0.594543i	$-0.797333\pi$
$733$	−43.5384	−1.60813	−0.804064	+	0.594543i	$0.797333\pi$
$734$	33.7897	1.24720
$735$	0	0
$736$	8.65539	0.319042
$737$	−42.8093	−1.57690
$738$	0	0
$739$	25.3446	0.932316	0.466158	−	0.884701i	$-0.345638\pi$
$739$	25.3446	0.932316	0.466158	+	0.884701i	$0.345638\pi$
$740$	0.396926	0.0145913
$741$	0	0
$742$	−2.28312	−0.0838159
$743$	15.4884	0.568216	0.284108	−	0.958792i	$-0.408303\pi$
$743$	15.4884	0.568216	0.284108	+	0.958792i	$0.408303\pi$
$744$	0	0
$745$	5.03508	0.184471
$746$	20.2804	0.742519
$747$	0	0
$748$	2.93582	0.107344
$749$	−24.1661	−0.883011
$750$	0	0
$751$	22.4138	0.817893	0.408946	−	0.912558i	$-0.365896\pi$
$751$	22.4138	0.817893	0.408946	+	0.912558i	$0.365896\pi$
$752$	−12.1334	−0.442460
$753$	0	0
$754$	12.2344	0.445551
$755$	5.84255	0.212632
$756$	0	0
$757$	−40.6810	−1.47858	−0.739288	−	0.673390i	$-0.764838\pi$
$757$	−40.6810	−1.47858	−0.739288	+	0.673390i	$0.764838\pi$
$758$	−10.1967	−0.370359
$759$	0	0
$760$	−7.75877	−0.281440
$761$	−4.99050	−0.180906	−0.0904528	−	0.995901i	$-0.528831\pi$
$761$	−4.99050	−0.180906	−0.0904528	+	0.995901i	$0.528831\pi$
$762$	0	0
$763$	−15.9186	−0.576292
$764$	1.00000	0.0361787
$765$	0	0
$766$	−5.06418	−0.182976
$767$	−7.22668	−0.260940
$768$	0	0
$769$	−19.1293	−0.689820	−0.344910	−	0.938636i	$-0.612091\pi$
$769$	−19.1293	−0.689820	−0.344910	+	0.938636i	$0.612091\pi$
$770$	11.8871	0.428382
$771$	0	0
$772$	5.06511	0.182297
$773$	−32.4296	−1.16641	−0.583207	−	0.812324i	$-0.698202\pi$
$773$	−32.4296	−1.16641	−0.583207	+	0.812324i	$0.698202\pi$
$774$	0	0
$775$	−0.610815	−0.0219411
$776$	−13.7588	−0.493911
$777$	0	0
$778$	−27.5371	−0.987254
$779$	34.0310	1.21929
$780$	0	0
$781$	−63.5485	−2.27394
$782$	−10.0155	−0.358153
$783$	0	0
$784$	23.7151	0.846969
$785$	−22.1830	−0.791747
$786$	0	0
$787$	33.7134	1.20175	0.600876	−	0.799342i	$-0.294819\pi$
$787$	33.7134	1.20175	0.600876	+	0.799342i	$0.294819\pi$
$788$	2.55613	0.0910584
$789$	0	0
$790$	8.38919	0.298474
$791$	3.84255	0.136625
$792$	0	0
$793$	−8.82295	−0.313312
$794$	4.95542	0.175861
$795$	0	0
$796$	8.44150	0.299201
$797$	−21.2335	−0.752129	−0.376064	−	0.926594i	$-0.622723\pi$
$797$	−21.2335	−0.752129	−0.376064	+	0.926594i	$0.622723\pi$
$798$	0	0
$799$	3.89393	0.137757
$800$	1.94356	0.0687153
$801$	0	0
$802$	−59.3756	−2.09662
$803$	−37.1634	−1.31147
$804$	0	0
$805$	−6.00000	−0.211472
$806$	−2.69459	−0.0949130
$807$	0	0
$808$	33.1489	1.16617
$809$	−48.3952	−1.70148	−0.850742	−	0.525584i	$-0.823847\pi$
$809$	−48.3952	−1.70148	−0.850742	+	0.525584i	$0.823847\pi$
$810$	0	0
$811$	−54.8590	−1.92636	−0.963179	−	0.268860i	$-0.913353\pi$
$811$	−54.8590	−1.92636	−0.963179	+	0.268860i	$0.913353\pi$
$812$	1.29767	0.0455392
$813$	0	0
$814$	10.0838	0.353436
$815$	−22.3131	−0.781595
$816$	0	0
$817$	25.4047	0.888797
$818$	36.9181	1.29081
$819$	0	0
$820$	−3.85710	−0.134696
$821$	−20.7547	−0.724342	−0.362171	−	0.932112i	$-0.617964\pi$
$821$	−20.7547	−0.724342	−0.362171	+	0.932112i	$0.617964\pi$
$822$	0	0
$823$	−48.2431	−1.68165	−0.840824	−	0.541308i	$-0.817929\pi$
$823$	−48.2431	−1.68165	−0.840824	+	0.541308i	$0.817929\pi$
$824$	−12.8922	−0.449120
$825$	0	0
$826$	−5.18067	−0.180259
$827$	−19.0642	−0.662926	−0.331463	−	0.943468i	$-0.607542\pi$
$827$	−19.0642	−0.662926	−0.331463	+	0.943468i	$0.607542\pi$
$828$	0	0
$829$	−4.35410	−0.151224	−0.0756121	−	0.997137i	$-0.524091\pi$
$829$	−4.35410	−0.151224	−0.0756121	+	0.997137i	$0.524091\pi$
$830$	−1.75877	−0.0610478
$831$	0	0
$832$	−17.7665	−0.615943
$833$	−7.61081	−0.263699
$834$	0	0
$835$	−1.09152	−0.0377736
$836$	6.12836	0.211954
$837$	0	0
$838$	−0.0605558	−0.00209186
$839$	−24.7419	−0.854184	−0.427092	−	0.904208i	$-0.640462\pi$
$839$	−24.7419	−0.854184	−0.427092	+	0.904208i	$0.640462\pi$
$840$	0	0
$841$	−21.3087	−0.734783
$842$	−38.9222	−1.34135
$843$	0	0
$844$	−0.812843	−0.0279792
$845$	4.70914	0.161999
$846$	0	0
$847$	29.8607	1.02603
$848$	−5.05913	−0.173731
$849$	0	0
$850$	−2.24897	−0.0771390
$851$	−5.08976	−0.174475
$852$	0	0
$853$	6.60039	0.225993	0.112996	−	0.993595i	$-0.463955\pi$
$853$	6.60039	0.225993	0.112996	+	0.993595i	$0.463955\pi$
$854$	−6.32501	−0.216437
$855$	0	0
$856$	−45.4175	−1.55234
$857$	−12.2804	−0.419492	−0.209746	−	0.977756i	$-0.567264\pi$
$857$	−12.2804	−0.419492	−0.209746	+	0.977756i	$0.567264\pi$
$858$	0	0
$859$	34.3506	1.17203	0.586014	−	0.810301i	$-0.300697\pi$
$859$	34.3506	1.17203	0.586014	+	0.810301i	$0.300697\pi$
$860$	−2.87939	−0.0981862
$861$	0	0
$862$	−36.8462	−1.25499
$863$	−51.5485	−1.75473	−0.877366	−	0.479822i	$-0.840701\pi$
$863$	−51.5485	−1.75473	−0.877366	+	0.479822i	$0.840701\pi$
$864$	0	0
$865$	6.76651	0.230068
$866$	8.97771	0.305075
$867$	0	0
$868$	−0.285807	−0.00970092
$869$	31.5330	1.06968
$870$	0	0
$871$	21.4047	0.725269
$872$	−29.9172	−1.01312
$873$	0	0
$874$	−20.9067	−0.707181
$875$	−1.34730	−0.0455469
$876$	0	0
$877$	−51.9026	−1.75263	−0.876313	−	0.481742i	$-0.840004\pi$
$877$	−51.9026	−1.75263	−0.876313	+	0.481742i	$0.840004\pi$
$878$	13.9554	0.470973
$879$	0	0
$880$	26.3405	0.887938
$881$	−14.3114	−0.482163	−0.241082	−	0.970505i	$-0.577502\pi$
$881$	−14.3114	−0.482163	−0.241082	+	0.970505i	$0.577502\pi$
$882$	0	0
$883$	−50.6255	−1.70368	−0.851841	−	0.523800i	$-0.824514\pi$
$883$	−50.6255	−1.70368	−0.851841	+	0.523800i	$0.824514\pi$
$884$	−1.46791	−0.0493712
$885$	0	0
$886$	6.41147	0.215398
$887$	−1.83244	−0.0615274	−0.0307637	−	0.999527i	$-0.509794\pi$
$887$	−1.83244	−0.0615274	−0.0307637	+	0.999527i	$0.509794\pi$
$888$	0	0
$889$	−14.4757	−0.485498
$890$	−1.53209	−0.0513557
$891$	0	0
$892$	0.793852	0.0265801
$893$	8.12836	0.272005
$894$	0	0
$895$	−1.41828	−0.0474079
$896$	−17.9736	−0.600456
$897$	0	0
$898$	22.3642	0.746303
$899$	−1.69398	−0.0564975
$900$	0	0
$901$	1.62361	0.0540902
$902$	−97.9883	−3.26265
$903$	0	0
$904$	7.22163	0.240188
$905$	6.25671	0.207980
$906$	0	0
$907$	18.9959	0.630748	0.315374	−	0.948967i	$-0.397870\pi$
$907$	18.9959	0.630748	0.315374	+	0.948967i	$0.397870\pi$
$908$	−7.95636	−0.264041
$909$	0	0
$910$	−5.94356	−0.197027
$911$	4.32501	0.143294	0.0716469	−	0.997430i	$-0.477175\pi$
$911$	4.32501	0.143294	0.0716469	+	0.997430i	$0.477175\pi$
$912$	0	0
$913$	−6.61081	−0.218786
$914$	−7.06418	−0.233662
$915$	0	0
$916$	4.34049	0.143414
$917$	−4.06830	−0.134347
$918$	0	0
$919$	2.39330	0.0789478	0.0394739	−	0.999221i	$-0.487432\pi$
$919$	2.39330	0.0789478	0.0394739	+	0.999221i	$0.487432\pi$
$920$	−11.2763	−0.371769
$921$	0	0
$922$	−63.7161	−2.09838
$923$	31.7743	1.04586
$924$	0	0
$925$	−1.14290	−0.0375784
$926$	43.0951	1.41619
$927$	0	0
$928$	5.39012	0.176939
$929$	50.1248	1.64454	0.822271	−	0.569096i	$-0.192707\pi$
$929$	50.1248	1.64454	0.822271	+	0.569096i	$0.192707\pi$
$930$	0	0
$931$	−15.8871	−0.520680
$932$	6.09327	0.199592
$933$	0	0
$934$	8.84936	0.289560
$935$	−8.45336	−0.276455
$936$	0	0
$937$	21.8425	0.713565	0.356782	−	0.934187i	$-0.383874\pi$
$937$	21.8425	0.713565	0.356782	+	0.934187i	$0.383874\pi$
$938$	15.3446	0.501019
$939$	0	0
$940$	−0.921274	−0.0300487
$941$	38.7252	1.26241	0.631203	−	0.775617i	$-0.282561\pi$
$941$	38.7252	1.26241	0.631203	+	0.775617i	$0.282561\pi$
$942$	0	0
$943$	49.4593	1.61062
$944$	−11.4798	−0.373635
$945$	0	0
$946$	−73.1498	−2.37831
$947$	43.3637	1.40913	0.704566	−	0.709639i	$-0.251142\pi$
$947$	43.3637	1.40913	0.704566	+	0.709639i	$0.251142\pi$
$948$	0	0
$949$	18.5817	0.603188
$950$	−4.69459	−0.152313
$951$	0	0
$952$	5.00774	0.162302
$953$	45.2235	1.46493	0.732466	−	0.680803i	$-0.238369\pi$
$953$	45.2235	1.46493	0.732466	+	0.680803i	$0.238369\pi$
$954$	0	0
$955$	−2.87939	−0.0931747
$956$	−0.226682	−0.00733141
$957$	0	0
$958$	−6.15207	−0.198765
$959$	3.63041	0.117232
$960$	0	0
$961$	−30.6269	−0.987965
$962$	−5.04189	−0.162557
$963$	0	0
$964$	9.54664	0.307476
$965$	−14.5844	−0.469489
$966$	0	0
$967$	30.1111	0.968308	0.484154	−	0.874983i	$-0.339127\pi$
$967$	30.1111	0.968308	0.484154	+	0.874983i	$0.339127\pi$
$968$	56.1198	1.80376
$969$	0	0
$970$	−8.32501	−0.267300
$971$	4.24123	0.136108	0.0680538	−	0.997682i	$-0.478321\pi$
$971$	4.24123	0.136108	0.0680538	+	0.997682i	$0.478321\pi$
$972$	0	0
$973$	−5.59896	−0.179494
$974$	53.0660	1.70035
$975$	0	0
$976$	−14.0155	−0.448625
$977$	−37.5357	−1.20087	−0.600437	−	0.799672i	$-0.705007\pi$
$977$	−37.5357	−1.20087	−0.600437	+	0.799672i	$0.705007\pi$
$978$	0	0
$979$	−5.75877	−0.184051
$980$	1.80066	0.0575200
$981$	0	0
$982$	−57.5877	−1.83770
$983$	5.28674	0.168621	0.0843104	−	0.996440i	$-0.473131\pi$
$983$	5.28674	0.168621	0.0843104	+	0.996440i	$0.473131\pi$
$984$	0	0
$985$	−7.36009	−0.234512
$986$	−6.23711	−0.198630
$987$	0	0
$988$	−3.06418	−0.0974845
$989$	36.9222	1.17406
$990$	0	0
$991$	−25.1581	−0.799172	−0.399586	−	0.916696i	$-0.630846\pi$
$991$	−25.1581	−0.799172	−0.399586	+	0.916696i	$0.630846\pi$
$992$	−1.18716	−0.0376923
$993$	0	0
$994$	22.7784	0.722486
$995$	−24.3063	−0.770563
$996$	0	0
$997$	−42.0856	−1.33287	−0.666433	−	0.745565i	$-0.732180\pi$
$997$	−42.0856	−1.33287	−0.666433	+	0.745565i	$0.732180\pi$
$998$	−27.0351	−0.855781
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4005.2.a.i.1.1		3
3.2	odd	2		1335.2.a.e.1.3	✓	3
15.14	odd	2		6675.2.a.m.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1335.2.a.e.1.3	✓	3	3.2	odd	2
4005.2.a.i.1.1		3	1.1	even	1	trivial
6675.2.a.m.1.1		3	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)

\(f(q)\)	\(=\)	\(q-1.53209 q^{2} +0.347296 q^{4} -1.00000 q^{5} +1.34730 q^{7} +2.53209 q^{8} +O(q^{10})\)	\(q-1.53209 q^{2} +0.347296 q^{4} -1.00000 q^{5} +1.34730 q^{7} +2.53209 q^{8} +1.53209 q^{10} +5.75877 q^{11} -2.87939 q^{13} -2.06418 q^{14} -4.57398 q^{16} +1.46791 q^{17} +3.06418 q^{19} -0.347296 q^{20} -8.82295 q^{22} +4.45336 q^{23} +1.00000 q^{25} +4.41147 q^{26} +0.467911 q^{28} +2.77332 q^{29} -0.610815 q^{31} +1.94356 q^{32} -2.24897 q^{34} -1.34730 q^{35} -1.14290 q^{37} -4.69459 q^{38} -2.53209 q^{40} +11.1061 q^{41} +8.29086 q^{43} +2.00000 q^{44} -6.82295 q^{46} +2.65270 q^{47} -5.18479 q^{49} -1.53209 q^{50} -1.00000 q^{52} +1.10607 q^{53} -5.75877 q^{55} +3.41147 q^{56} -4.24897 q^{58} +2.50980 q^{59} +3.06418 q^{61} +0.935822 q^{62} +6.17024 q^{64} +2.87939 q^{65} -7.43376 q^{67} +0.509800 q^{68} +2.06418 q^{70} -11.0351 q^{71} -6.45336 q^{73} +1.75103 q^{74} +1.06418 q^{76} +7.75877 q^{77} +5.47565 q^{79} +4.57398 q^{80} -17.0155 q^{82} -1.14796 q^{83} -1.46791 q^{85} -12.7023 q^{86} +14.5817 q^{88} -1.00000 q^{89} -3.87939 q^{91} +1.54664 q^{92} -4.06418 q^{94} -3.06418 q^{95} -5.43376 q^{97} +7.94356 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{5} + 3 q^{7} + 3 q^{8}+O(q^{10}) \) 3 * q - 3 * q^5 + 3 * q^7 + 3 * q^8	\( 3 q - 3 q^{5} + 3 q^{7} + 3 q^{8} + 6 q^{11} - 3 q^{13} + 3 q^{14} - 6 q^{16} + 9 q^{17} - 6 q^{22} + 3 q^{25} + 3 q^{26} + 6 q^{28} + 15 q^{29} - 6 q^{31} - 9 q^{32} + 6 q^{34} - 3 q^{35} - 3 q^{37} - 12 q^{38} - 3 q^{40} + 21 q^{41} + 9 q^{43} + 6 q^{44} + 9 q^{47} - 12 q^{49} - 3 q^{52} - 9 q^{53} - 6 q^{55} + 9 q^{59} + 12 q^{62} - 3 q^{64} + 3 q^{65} - 6 q^{67} + 3 q^{68} - 3 q^{70} + 12 q^{71} - 6 q^{73} + 18 q^{74} - 6 q^{76} + 12 q^{77} - 3 q^{79} + 6 q^{80} - 3 q^{82} + 12 q^{83} - 9 q^{85} - 12 q^{86} + 12 q^{88} - 3 q^{89} - 6 q^{91} + 18 q^{92} - 3 q^{94} + 9 q^{98}+O(q^{100}) \) 3 * q - 3 * q^5 + 3 * q^7 + 3 * q^8 + 6 * q^11 - 3 * q^13 + 3 * q^14 - 6 * q^16 + 9 * q^17 - 6 * q^22 + 3 * q^25 + 3 * q^26 + 6 * q^28 + 15 * q^29 - 6 * q^31 - 9 * q^32 + 6 * q^34 - 3 * q^35 - 3 * q^37 - 12 * q^38 - 3 * q^40 + 21 * q^41 + 9 * q^43 + 6 * q^44 + 9 * q^47 - 12 * q^49 - 3 * q^52 - 9 * q^53 - 6 * q^55 + 9 * q^59 + 12 * q^62 - 3 * q^64 + 3 * q^65 - 6 * q^67 + 3 * q^68 - 3 * q^70 + 12 * q^71 - 6 * q^73 + 18 * q^74 - 6 * q^76 + 12 * q^77 - 3 * q^79 + 6 * q^80 - 3 * q^82 + 12 * q^83 - 9 * q^85 - 12 * q^86 + 12 * q^88 - 3 * q^89 - 6 * q^91 + 18 * q^92 - 3 * q^94 + 9 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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