LMFDB - Embedded newform 4005.2.a.n.1.2

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			1.2
Root			$-0.763968$ 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)

$f(q)$	$=$	$q-0.652386 q^{2} -1.57439 q^{4} +1.00000 q^{5} +1.82035 q^{7} +2.33188 q^{8} +O(q^{10})$	$q-0.652386 q^{2} -1.57439 q^{4} +1.00000 q^{5} +1.82035 q^{7} +2.33188 q^{8} -0.652386 q^{10} +2.79030 q^{13} -1.18757 q^{14} +1.62750 q^{16} +1.78238 q^{17} +2.00000 q^{19} -1.57439 q^{20} +4.55073 q^{23} +1.00000 q^{25} -1.82035 q^{26} -2.86595 q^{28} +8.34897 q^{29} -1.64071 q^{31} -5.72552 q^{32} -1.16280 q^{34} +1.82035 q^{35} +0.217624 q^{37} -1.30477 q^{38} +2.33188 q^{40} +1.16563 q^{41} -10.8997 q^{43} -2.96883 q^{46} +4.02366 q^{47} -3.68631 q^{49} -0.652386 q^{50} -4.39304 q^{52} +1.85182 q^{53} +4.24485 q^{56} -5.44675 q^{58} +5.34104 q^{59} -5.47661 q^{61} +1.07038 q^{62} +0.480251 q^{64} +2.79030 q^{65} +10.3032 q^{67} -2.80616 q^{68} -1.18757 q^{70} +1.67217 q^{71} -13.9234 q^{73} -0.141974 q^{74} -3.14879 q^{76} +6.85023 q^{79} +1.62750 q^{80} -0.760437 q^{82} +0.910024 q^{83} +1.78238 q^{85} +7.11081 q^{86} +1.00000 q^{89} +5.07934 q^{91} -7.16464 q^{92} -2.62498 q^{94} +2.00000 q^{95} +5.63942 q^{97} +2.40489 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.652386	−0.461306	−0.230653	−	0.973036i	$-0.574086\pi$
$2$	−0.652386	−0.461306	−0.230653	+	0.973036i	$0.574086\pi$
$3$	0	0
$3$	0	0
$4$	−1.57439	−0.787196
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.82035	0.688029	0.344015	−	0.938964i	$-0.388213\pi$
$7$	1.82035	0.688029	0.344015	+	0.938964i	$0.388213\pi$
$8$	2.33188	0.824445
$9$	0	0
$10$	−0.652386	−0.206302
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	2.79030	0.773891	0.386946	−	0.922103i	$-0.373530\pi$
$13$	2.79030	0.773891	0.386946	+	0.922103i	$0.373530\pi$
$14$	−1.18757	−0.317392
$15$	0	0
$16$	1.62750	0.406875
$17$	1.78238	0.432290	0.216145	−	0.976361i	$-0.430652\pi$
$17$	1.78238	0.432290	0.216145	+	0.976361i	$0.430652\pi$
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	−1.57439	−0.352045
$21$	0	0
$22$	0	0
$23$	4.55073	0.948893	0.474447	−	0.880284i	$-0.342648\pi$
$23$	4.55073	0.948893	0.474447	+	0.880284i	$0.342648\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−1.82035	−0.357001
$27$	0	0
$28$	−2.86595	−0.541614
$29$	8.34897	1.55036	0.775182	−	0.631738i	$-0.217658\pi$
$29$	8.34897	1.55036	0.775182	+	0.631738i	$0.217658\pi$
$30$	0	0
$31$	−1.64071	−0.294680	−0.147340	−	0.989086i	$-0.547071\pi$
$31$	−1.64071	−0.294680	−0.147340	+	0.989086i	$0.547071\pi$
$32$	−5.72552	−1.01214
$33$	0	0
$34$	−1.16280	−0.199418
$35$	1.82035	0.307696
$36$	0	0
$37$	0.217624	0.0357771	0.0178885	−	0.999840i	$-0.494306\pi$
$37$	0.217624	0.0357771	0.0178885	+	0.999840i	$0.494306\pi$
$38$	−1.30477	−0.211662
$39$	0	0
$40$	2.33188	0.368703
$41$	1.16563	0.182040	0.0910201	−	0.995849i	$-0.470987\pi$
$41$	1.16563	0.182040	0.0910201	+	0.995849i	$0.470987\pi$
$42$	0	0
$43$	−10.8997	−1.66219	−0.831094	−	0.556132i	$-0.812285\pi$
$43$	−10.8997	−1.66219	−0.831094	+	0.556132i	$0.812285\pi$
$44$	0	0
$45$	0	0
$46$	−2.96883	−0.437731
$47$	4.02366	0.586911	0.293456	−	0.955973i	$-0.405195\pi$
$47$	4.02366	0.586911	0.293456	+	0.955973i	$0.405195\pi$
$48$	0	0
$49$	−3.68631	−0.526615
$50$	−0.652386	−0.0922613
$51$	0	0
$52$	−4.39304	−0.609204
$53$	1.85182	0.254367	0.127183	−	0.991879i	$-0.459406\pi$
$53$	1.85182	0.254367	0.127183	+	0.991879i	$0.459406\pi$
$54$	0	0
$55$	0	0
$56$	4.24485	0.567242
$57$	0	0
$58$	−5.44675	−0.715193
$59$	5.34104	0.695344	0.347672	−	0.937616i	$-0.386972\pi$
$59$	5.34104	0.695344	0.347672	+	0.937616i	$0.386972\pi$
$60$	0	0
$61$	−5.47661	−0.701208	−0.350604	−	0.936524i	$-0.614024\pi$
$61$	−5.47661	−0.701208	−0.350604	+	0.936524i	$0.614024\pi$
$62$	1.07038	0.135938
$63$	0	0
$64$	0.480251	0.0600314
$65$	2.79030	0.346095
$66$	0	0
$67$	10.3032	1.25873	0.629367	−	0.777108i	$-0.283314\pi$
$67$	10.3032	1.25873	0.629367	+	0.777108i	$0.283314\pi$
$68$	−2.80616	−0.340297
$69$	0	0
$70$	−1.18757	−0.141942
$71$	1.67217	0.198450	0.0992252	−	0.995065i	$-0.468364\pi$
$71$	1.67217	0.198450	0.0992252	+	0.995065i	$0.468364\pi$
$72$	0	0
$73$	−13.9234	−1.62961	−0.814803	−	0.579737i	$-0.803155\pi$
$73$	−13.9234	−1.62961	−0.814803	+	0.579737i	$0.803155\pi$
$74$	−0.141974	−0.0165042
$75$	0	0
$76$	−3.14879	−0.361191
$77$	0	0
$78$	0	0
$79$	6.85023	0.770711	0.385356	−	0.922768i	$-0.374079\pi$
$79$	6.85023	0.770711	0.385356	+	0.922768i	$0.374079\pi$
$80$	1.62750	0.181960
$81$	0	0
$82$	−0.760437	−0.0839763
$83$	0.910024	0.0998881	0.0499440	−	0.998752i	$-0.484096\pi$
$83$	0.910024	0.0998881	0.0499440	+	0.998752i	$0.484096\pi$
$84$	0	0
$85$	1.78238	0.193326
$86$	7.11081	0.766778
$87$	0	0
$88$	0	0
$89$	1.00000	0.106000
$89$	1.00000	0.106000
$90$	0	0
$91$	5.07934	0.532460
$92$	−7.16464	−0.746966
$93$	0	0
$94$	−2.62498	−0.270746
$95$	2.00000	0.205196
$96$	0	0
$97$	5.63942	0.572596	0.286298	−	0.958141i	$-0.407575\pi$
$97$	5.63942	0.572596	0.286298	+	0.958141i	$0.407575\pi$
$98$	2.40489	0.242931
$99$	0	0
$100$	−1.57439	−0.157439
$101$	7.32955	0.729317	0.364659	−	0.931141i	$-0.381186\pi$
$101$	7.32955	0.729317	0.364659	+	0.931141i	$0.381186\pi$
$102$	0	0
$103$	−15.8036	−1.55718	−0.778589	−	0.627534i	$-0.784064\pi$
$103$	−15.8036	−1.55718	−0.778589	+	0.627534i	$0.784064\pi$
$104$	6.50666	0.638031
$105$	0	0
$106$	−1.20810	−0.117341
$107$	−2.29261	−0.221635	−0.110818	−	0.993841i	$-0.535347\pi$
$107$	−2.29261	−0.221635	−0.110818	+	0.993841i	$0.535347\pi$
$108$	0	0
$109$	6.65392	0.637330	0.318665	−	0.947867i	$-0.396766\pi$
$109$	6.65392	0.637330	0.318665	+	0.947867i	$0.396766\pi$
$110$	0	0
$111$	0	0
$112$	2.96263	0.279942
$113$	−0.623266	−0.0586320	−0.0293160	−	0.999570i	$-0.509333\pi$
$113$	−0.623266	−0.0586320	−0.0293160	+	0.999570i	$0.509333\pi$
$114$	0	0
$115$	4.55073	0.424358
$116$	−13.1446	−1.22044
$117$	0	0
$118$	−3.48442	−0.320767
$119$	3.24456	0.297428
$120$	0	0
$121$	−11.0000	−1.00000
$122$	3.57286	0.323472
$123$	0	0
$124$	2.58312	0.231971
$125$	1.00000	0.0894427
$126$	0	0
$127$	2.31939	0.205812	0.102906	−	0.994691i	$-0.467186\pi$
$127$	2.31939	0.205812	0.102906	+	0.994691i	$0.467186\pi$
$128$	11.1377	0.984446
$129$	0	0
$130$	−1.82035	−0.159656
$131$	5.35678	0.468024	0.234012	−	0.972234i	$-0.424815\pi$
$131$	5.35678	0.468024	0.234012	+	0.972234i	$0.424815\pi$
$132$	0	0
$133$	3.64071	0.315690
$134$	−6.72165	−0.580662
$135$	0	0
$136$	4.15629	0.356399
$137$	−2.72940	−0.233188	−0.116594	−	0.993180i	$-0.537198\pi$
$137$	−2.72940	−0.233188	−0.116594	+	0.993180i	$0.537198\pi$
$138$	0	0
$139$	−5.95281	−0.504911	−0.252456	−	0.967608i	$-0.581238\pi$
$139$	−5.95281	−0.504911	−0.252456	+	0.967608i	$0.581238\pi$
$140$	−2.86595	−0.242217
$141$	0	0
$142$	−1.09090	−0.0915464
$143$	0	0
$144$	0	0
$145$	8.34897	0.693344
$146$	9.08341	0.751748
$147$	0	0
$148$	−0.342625	−0.0281636
$149$	−15.8194	−1.29597	−0.647987	−	0.761652i	$-0.724389\pi$
$149$	−15.8194	−1.29597	−0.647987	+	0.761652i	$0.724389\pi$
$150$	0	0
$151$	13.7583	1.11964	0.559818	−	0.828615i	$-0.310871\pi$
$151$	13.7583	1.11964	0.559818	+	0.828615i	$0.310871\pi$
$152$	4.66377	0.378281
$153$	0	0
$154$	0	0
$155$	−1.64071	−0.131785
$156$	0	0
$157$	−4.49287	−0.358570	−0.179285	−	0.983797i	$-0.557378\pi$
$157$	−4.49287	−0.358570	−0.179285	+	0.983797i	$0.557378\pi$
$158$	−4.46899	−0.355534
$159$	0	0
$160$	−5.72552	−0.452642
$161$	8.28395	0.652867
$162$	0	0
$163$	−11.9391	−0.935142	−0.467571	−	0.883956i	$-0.654871\pi$
$163$	−11.9391	−0.935142	−0.467571	+	0.883956i	$0.654871\pi$
$164$	−1.83515	−0.143301
$165$	0	0
$166$	−0.593686	−0.0460790
$167$	21.4963	1.66344	0.831719	−	0.555197i	$-0.187357\pi$
$167$	21.4963	1.66344	0.831719	+	0.555197i	$0.187357\pi$
$168$	0	0
$169$	−5.21420	−0.401092
$170$	−1.16280	−0.0891825
$171$	0	0
$172$	17.1604	1.30847
$173$	−10.0116	−0.761170	−0.380585	−	0.924746i	$-0.624277\pi$
$173$	−10.0116	−0.761170	−0.380585	+	0.924746i	$0.624277\pi$
$174$	0	0
$175$	1.82035	0.137606
$176$	0	0
$177$	0	0
$178$	−0.652386	−0.0488984
$179$	−15.6251	−1.16788	−0.583938	−	0.811799i	$-0.698489\pi$
$179$	−15.6251	−1.16788	−0.583938	+	0.811799i	$0.698489\pi$
$180$	0	0
$181$	13.8343	1.02829	0.514147	−	0.857702i	$-0.328109\pi$
$181$	13.8343	1.02829	0.514147	+	0.857702i	$0.328109\pi$
$182$	−3.31369	−0.245627
$183$	0	0
$184$	10.6118	0.782311
$185$	0.217624	0.0160000
$186$	0	0
$187$	0	0
$188$	−6.33482	−0.462014
$189$	0	0
$190$	−1.30477	−0.0946581
$191$	1.49457	0.108143	0.0540716	−	0.998537i	$-0.482780\pi$
$191$	1.49457	0.108143	0.0540716	+	0.998537i	$0.482780\pi$
$192$	0	0
$193$	−8.86514	−0.638127	−0.319064	−	0.947733i	$-0.603368\pi$
$193$	−8.86514	−0.638127	−0.319064	+	0.947733i	$0.603368\pi$
$194$	−3.67908	−0.264142
$195$	0	0
$196$	5.80370	0.414550
$197$	12.3256	0.878163	0.439082	−	0.898447i	$-0.355304\pi$
$197$	12.3256	0.878163	0.439082	+	0.898447i	$0.355304\pi$
$198$	0	0
$199$	21.4663	1.52170	0.760852	−	0.648925i	$-0.224781\pi$
$199$	21.4663	1.52170	0.760852	+	0.648925i	$0.224781\pi$
$200$	2.33188	0.164889
$201$	0	0
$202$	−4.78169	−0.336439
$203$	15.1981	1.06670
$204$	0	0
$205$	1.16563	0.0814108
$206$	10.3101	0.718336
$207$	0	0
$208$	4.54122	0.314877
$209$	0	0
$210$	0	0
$211$	6.76086	0.465437	0.232718	−	0.972544i	$-0.425238\pi$
$211$	6.76086	0.465437	0.232718	+	0.972544i	$0.425238\pi$
$212$	−2.91549	−0.200237
$213$	0	0
$214$	1.49567	0.102242
$215$	−10.8997	−0.743353
$216$	0	0
$217$	−2.98667	−0.202749
$218$	−4.34092	−0.294004
$219$	0	0
$220$	0	0
$221$	4.97337	0.334545
$222$	0	0
$223$	19.9311	1.33469	0.667343	−	0.744750i	$-0.267432\pi$
$223$	19.9311	1.33469	0.667343	+	0.744750i	$0.267432\pi$
$224$	−10.4225	−0.696381
$225$	0	0
$226$	0.406610	0.0270473
$227$	−2.84119	−0.188577	−0.0942884	−	0.995545i	$-0.530058\pi$
$227$	−2.84119	−0.188577	−0.0942884	+	0.995545i	$0.530058\pi$
$228$	0	0
$229$	3.10370	0.205098	0.102549	−	0.994728i	$-0.467300\pi$
$229$	3.10370	0.205098	0.102549	+	0.994728i	$0.467300\pi$
$230$	−2.96883	−0.195759
$231$	0	0
$232$	19.4688	1.27819
$233$	−4.83762	−0.316923	−0.158462	−	0.987365i	$-0.550653\pi$
$233$	−4.83762	−0.316923	−0.158462	+	0.987365i	$0.550653\pi$
$234$	0	0
$235$	4.02366	0.262475
$236$	−8.40889	−0.547372
$237$	0	0
$238$	−2.11670	−0.137205
$239$	21.9339	1.41878	0.709392	−	0.704815i	$-0.248970\pi$
$239$	21.9339	1.41878	0.709392	+	0.704815i	$0.248970\pi$
$240$	0	0
$241$	11.7742	0.758442	0.379221	−	0.925306i	$-0.376192\pi$
$241$	11.7742	0.758442	0.379221	+	0.925306i	$0.376192\pi$
$242$	7.17624	0.461306
$243$	0	0
$244$	8.62234	0.551989
$245$	−3.68631	−0.235510
$246$	0	0
$247$	5.58061	0.355086
$248$	−3.82594	−0.242948
$249$	0	0
$250$	−0.652386	−0.0412605
$251$	−23.5628	−1.48727	−0.743635	−	0.668586i	$-0.766900\pi$
$251$	−23.5628	−1.48727	−0.743635	+	0.668586i	$0.766900\pi$
$252$	0	0
$253$	0	0
$254$	−1.51313	−0.0949425
$255$	0	0
$256$	−8.22660	−0.514163
$257$	21.5945	1.34703	0.673515	−	0.739173i	$-0.264784\pi$
$257$	21.5945	1.34703	0.673515	+	0.739173i	$0.264784\pi$
$258$	0	0
$259$	0.396152	0.0246157
$260$	−4.39304	−0.272445
$261$	0	0
$262$	−3.49469	−0.215902
$263$	21.4320	1.32156	0.660778	−	0.750581i	$-0.270226\pi$
$263$	21.4320	1.32156	0.660778	+	0.750581i	$0.270226\pi$
$264$	0	0
$265$	1.85182	0.113756
$266$	−2.37515	−0.145630
$267$	0	0
$268$	−16.2213	−0.990871
$269$	−10.5968	−0.646096	−0.323048	−	0.946383i	$-0.604708\pi$
$269$	−10.5968	−0.646096	−0.323048	+	0.946383i	$0.604708\pi$
$270$	0	0
$271$	−0.325304	−0.0197608	−0.00988040	−	0.999951i	$-0.503145\pi$
$271$	−0.325304	−0.0197608	−0.00988040	+	0.999951i	$0.503145\pi$
$272$	2.90082	0.175888
$273$	0	0
$274$	1.78062	0.107571
$275$	0	0
$276$	0	0
$277$	−11.1310	−0.668800	−0.334400	−	0.942431i	$-0.608534\pi$
$277$	−11.1310	−0.668800	−0.334400	+	0.942431i	$0.608534\pi$
$278$	3.88353	0.232919
$279$	0	0
$280$	4.24485	0.253679
$281$	30.5089	1.82001	0.910006	−	0.414596i	$-0.136077\pi$
$281$	30.5089	1.82001	0.910006	+	0.414596i	$0.136077\pi$
$282$	0	0
$283$	5.74938	0.341765	0.170882	−	0.985291i	$-0.445338\pi$
$283$	5.74938	0.341765	0.170882	+	0.985291i	$0.445338\pi$
$284$	−2.63266	−0.156219
$285$	0	0
$286$	0	0
$287$	2.12185	0.125249
$288$	0	0
$289$	−13.8231	−0.813126
$290$	−5.44675	−0.319844
$291$	0	0
$292$	21.9209	1.28282
$293$	23.8582	1.39381	0.696906	−	0.717163i	$-0.254560\pi$
$293$	23.8582	1.39381	0.696906	+	0.717163i	$0.254560\pi$
$294$	0	0
$295$	5.34104	0.310967
$296$	0.507472	0.0294962
$297$	0	0
$298$	10.3203	0.597841
$299$	12.6979	0.734340
$300$	0	0
$301$	−19.8413	−1.14363
$302$	−8.97574	−0.516496
$303$	0	0
$304$	3.25500	0.186687
$305$	−5.47661	−0.313590
$306$	0	0
$307$	15.0644	0.859770	0.429885	−	0.902884i	$-0.358554\pi$
$307$	15.0644	0.859770	0.429885	+	0.902884i	$0.358554\pi$
$308$	0	0
$309$	0	0
$310$	1.07038	0.0607932
$311$	−18.3987	−1.04330	−0.521648	−	0.853161i	$-0.674683\pi$
$311$	−18.3987	−1.04330	−0.521648	+	0.853161i	$0.674683\pi$
$312$	0	0
$313$	14.5426	0.821999	0.410999	−	0.911636i	$-0.365180\pi$
$313$	14.5426	0.821999	0.410999	+	0.911636i	$0.365180\pi$
$314$	2.93108	0.165411
$315$	0	0
$316$	−10.7850	−0.606701
$317$	18.5719	1.04310	0.521550	−	0.853221i	$-0.325354\pi$
$317$	18.5719	1.04310	0.521550	+	0.853221i	$0.325354\pi$
$318$	0	0
$319$	0	0
$320$	0.480251	0.0268468
$321$	0	0
$322$	−5.40433	−0.301172
$323$	3.56475	0.198348
$324$	0	0
$325$	2.79030	0.154778
$326$	7.78889	0.431387
$327$	0	0
$328$	2.71810	0.150082
$329$	7.32449	0.403812
$330$	0	0
$331$	14.5503	0.799754	0.399877	−	0.916569i	$-0.369053\pi$
$331$	14.5503	0.799754	0.399877	+	0.916569i	$0.369053\pi$
$332$	−1.43273	−0.0786315
$333$	0	0
$334$	−14.0239	−0.767354
$335$	10.3032	0.562923
$336$	0	0
$337$	30.8283	1.67932	0.839662	−	0.543109i	$-0.182753\pi$
$337$	30.8283	1.67932	0.839662	+	0.543109i	$0.182753\pi$
$338$	3.40167	0.185026
$339$	0	0
$340$	−2.80616	−0.152185
$341$	0	0
$342$	0	0
$343$	−19.4529	−1.05036
$344$	−25.4168	−1.37038
$345$	0	0
$346$	6.53144	0.351132
$347$	17.8228	0.956775	0.478388	−	0.878149i	$-0.341221\pi$
$347$	17.8228	0.956775	0.478388	+	0.878149i	$0.341221\pi$
$348$	0	0
$349$	3.69024	0.197534	0.0987671	−	0.995111i	$-0.468510\pi$
$349$	3.69024	0.197534	0.0987671	+	0.995111i	$0.468510\pi$
$350$	−1.18757	−0.0634785
$351$	0	0
$352$	0	0
$353$	7.79637	0.414959	0.207480	−	0.978239i	$-0.433474\pi$
$353$	7.79637	0.414959	0.207480	+	0.978239i	$0.433474\pi$
$354$	0	0
$355$	1.67217	0.0887497
$356$	−1.57439	−0.0834427
$357$	0	0
$358$	10.1936	0.538748
$359$	7.06971	0.373125	0.186563	−	0.982443i	$-0.440265\pi$
$359$	7.06971	0.373125	0.186563	+	0.982443i	$0.440265\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	−9.02529	−0.474359
$363$	0	0
$364$	−7.99688	−0.419151
$365$	−13.9234	−0.728782
$366$	0	0
$367$	6.84431	0.357270	0.178635	−	0.983915i	$-0.442832\pi$
$367$	6.84431	0.357270	0.178635	+	0.983915i	$0.442832\pi$
$368$	7.40631	0.386081
$369$	0	0
$370$	−0.141974	−0.00738090
$371$	3.37097	0.175012
$372$	0	0
$373$	−14.8978	−0.771381	−0.385690	−	0.922628i	$-0.626037\pi$
$373$	−14.8978	−0.771381	−0.385690	+	0.922628i	$0.626037\pi$
$374$	0	0
$375$	0	0
$376$	9.38270	0.483876
$377$	23.2962	1.19981
$378$	0	0
$379$	14.7085	0.755525	0.377762	−	0.925903i	$-0.376694\pi$
$379$	14.7085	0.755525	0.377762	+	0.925903i	$0.376694\pi$
$380$	−3.14879	−0.161529
$381$	0	0
$382$	−0.975036	−0.0498872
$383$	2.12921	0.108798	0.0543988	−	0.998519i	$-0.482676\pi$
$383$	2.12921	0.108798	0.0543988	+	0.998519i	$0.482676\pi$
$384$	0	0
$385$	0	0
$386$	5.78349	0.294372
$387$	0	0
$388$	−8.87866	−0.450746
$389$	−10.3879	−0.526687	−0.263343	−	0.964702i	$-0.584825\pi$
$389$	−10.3879	−0.526687	−0.263343	+	0.964702i	$0.584825\pi$
$390$	0	0
$391$	8.11112	0.410197
$392$	−8.59604	−0.434166
$393$	0	0
$394$	−8.04105	−0.405102
$395$	6.85023	0.344673
$396$	0	0
$397$	−7.20577	−0.361647	−0.180824	−	0.983516i	$-0.557876\pi$
$397$	−7.20577	−0.361647	−0.180824	+	0.983516i	$0.557876\pi$
$398$	−14.0043	−0.701972
$399$	0	0
$400$	1.62750	0.0813749
$401$	−14.0547	−0.701858	−0.350929	−	0.936402i	$-0.614134\pi$
$401$	−14.0547	−0.701858	−0.350929	+	0.936402i	$0.614134\pi$
$402$	0	0
$403$	−4.57808	−0.228050
$404$	−11.5396	−0.574116
$405$	0	0
$406$	−9.91501	−0.492074
$407$	0	0
$408$	0	0
$409$	−14.6725	−0.725509	−0.362755	−	0.931885i	$-0.618164\pi$
$409$	−14.6725	−0.725509	−0.362755	+	0.931885i	$0.618164\pi$
$410$	−0.760437	−0.0375553
$411$	0	0
$412$	24.8811	1.22581
$413$	9.72258	0.478417
$414$	0	0
$415$	0.910024	0.0446713
$416$	−15.9760	−0.783285
$417$	0	0
$418$	0	0
$419$	−24.6612	−1.20478	−0.602389	−	0.798202i	$-0.705785\pi$
$419$	−24.6612	−1.20478	−0.602389	+	0.798202i	$0.705785\pi$
$420$	0	0
$421$	32.4721	1.58259	0.791297	−	0.611432i	$-0.209406\pi$
$421$	32.4721	1.58259	0.791297	+	0.611432i	$0.209406\pi$
$422$	−4.41069	−0.214709
$423$	0	0
$424$	4.31822	0.209711
$425$	1.78238	0.0864580
$426$	0	0
$427$	−9.96938	−0.482452
$428$	3.60947	0.174470
$429$	0	0
$430$	7.11081	0.342914
$431$	−30.9369	−1.49018	−0.745089	−	0.666965i	$-0.767593\pi$
$431$	−30.9369	−1.49018	−0.745089	+	0.666965i	$0.767593\pi$
$432$	0	0
$433$	0.659905	0.0317130	0.0158565	−	0.999874i	$-0.494953\pi$
$433$	0.659905	0.0317130	0.0158565	+	0.999874i	$0.494953\pi$
$434$	1.94846	0.0935292
$435$	0	0
$436$	−10.4759	−0.501704
$437$	9.10147	0.435382
$438$	0	0
$439$	16.3644	0.781032	0.390516	−	0.920596i	$-0.372297\pi$
$439$	16.3644	0.781032	0.390516	+	0.920596i	$0.372297\pi$
$440$	0	0
$441$	0	0
$442$	−3.24456	−0.154328
$443$	25.3266	1.20330	0.601651	−	0.798759i	$-0.294510\pi$
$443$	25.3266	1.20330	0.601651	+	0.798759i	$0.294510\pi$
$444$	0	0
$445$	1.00000	0.0474045
$446$	−13.0028	−0.615699
$447$	0	0
$448$	0.874227	0.0413033
$449$	16.9890	0.801761	0.400880	−	0.916130i	$-0.368704\pi$
$449$	16.9890	0.801761	0.400880	+	0.916130i	$0.368704\pi$
$450$	0	0
$451$	0	0
$452$	0.981266	0.0461549
$453$	0	0
$454$	1.85355	0.0869916
$455$	5.07934	0.238123
$456$	0	0
$457$	−35.2823	−1.65043	−0.825217	−	0.564815i	$-0.808947\pi$
$457$	−35.2823	−1.65043	−0.825217	+	0.564815i	$0.808947\pi$
$458$	−2.02481	−0.0946131
$459$	0	0
$460$	−7.16464	−0.334053
$461$	−9.07343	−0.422592	−0.211296	−	0.977422i	$-0.567768\pi$
$461$	−9.07343	−0.422592	−0.211296	+	0.977422i	$0.567768\pi$
$462$	0	0
$463$	−21.3436	−0.991923	−0.495962	−	0.868344i	$-0.665184\pi$
$463$	−21.3436	−0.991923	−0.495962	+	0.868344i	$0.665184\pi$
$464$	13.5879	0.630804
$465$	0	0
$466$	3.15600	0.146199
$467$	6.70744	0.310384	0.155192	−	0.987884i	$-0.450400\pi$
$467$	6.70744	0.310384	0.155192	+	0.987884i	$0.450400\pi$
$468$	0	0
$469$	18.7555	0.866046
$470$	−2.62498	−0.121081
$471$	0	0
$472$	12.4547	0.573273
$473$	0	0
$474$	0	0
$475$	2.00000	0.0917663
$476$	−5.10821	−0.234134
$477$	0	0
$478$	−14.3093	−0.654494
$479$	−20.9346	−0.956525	−0.478262	−	0.878217i	$-0.658733\pi$
$479$	−20.9346	−0.956525	−0.478262	+	0.878217i	$0.658733\pi$
$480$	0	0
$481$	0.607236	0.0276876
$482$	−7.68131	−0.349874
$483$	0	0
$484$	17.3183	0.787196
$485$	5.63942	0.256073
$486$	0	0
$487$	7.75495	0.351410	0.175705	−	0.984443i	$-0.443779\pi$
$487$	7.75495	0.351410	0.175705	+	0.984443i	$0.443779\pi$
$488$	−12.7708	−0.578108
$489$	0	0
$490$	2.40489	0.108642
$491$	−2.44614	−0.110393	−0.0551965	−	0.998476i	$-0.517579\pi$
$491$	−2.44614	−0.110393	−0.0551965	+	0.998476i	$0.517579\pi$
$492$	0	0
$493$	14.8810	0.670207
$494$	−3.64071	−0.163803
$495$	0	0
$496$	−2.67025	−0.119898
$497$	3.04395	0.136540
$498$	0	0
$499$	−37.5082	−1.67910	−0.839549	−	0.543285i	$-0.817180\pi$
$499$	−37.5082	−1.67910	−0.839549	+	0.543285i	$0.817180\pi$
$500$	−1.57439	−0.0704090
$501$	0	0
$502$	15.3720	0.686087
$503$	−11.3670	−0.506828	−0.253414	−	0.967358i	$-0.581554\pi$
$503$	−11.3670	−0.506828	−0.253414	+	0.967358i	$0.581554\pi$
$504$	0	0
$505$	7.32955	0.326161
$506$	0	0
$507$	0	0
$508$	−3.65163	−0.162015
$509$	28.9326	1.28241	0.641207	−	0.767368i	$-0.278434\pi$
$509$	28.9326	1.28241	0.641207	+	0.767368i	$0.278434\pi$
$510$	0	0
$511$	−25.3455	−1.12122
$512$	−16.9086	−0.747260
$513$	0	0
$514$	−14.0880	−0.621394
$515$	−15.8036	−0.696391
$516$	0	0
$517$	0	0
$518$	−0.258444	−0.0113554
$519$	0	0
$520$	6.50666	0.285336
$521$	−16.7110	−0.732123	−0.366062	−	0.930591i	$-0.619294\pi$
$521$	−16.7110	−0.732123	−0.366062	+	0.930591i	$0.619294\pi$
$522$	0	0
$523$	0.158812	0.00694435	0.00347217	−	0.999994i	$-0.498895\pi$
$523$	0.158812	0.00694435	0.00347217	+	0.999994i	$0.498895\pi$
$524$	−8.43367	−0.368427
$525$	0	0
$526$	−13.9820	−0.609642
$527$	−2.92436	−0.127387
$528$	0	0
$529$	−2.29083	−0.0996012
$530$	−1.20810	−0.0524765
$531$	0	0
$532$	−5.73191	−0.248510
$533$	3.25245	0.140879
$534$	0	0
$535$	−2.29261	−0.0991182
$536$	24.0258	1.03776
$537$	0	0
$538$	6.91318	0.298048
$539$	0	0
$540$	0	0
$541$	−0.777063	−0.0334085	−0.0167043	−	0.999860i	$-0.505317\pi$
$541$	−0.777063	−0.0334085	−0.0167043	+	0.999860i	$0.505317\pi$
$542$	0.212224	0.00911578
$543$	0	0
$544$	−10.2050	−0.437537
$545$	6.65392	0.285023
$546$	0	0
$547$	17.4549	0.746317	0.373159	−	0.927768i	$-0.378275\pi$
$547$	17.4549	0.746317	0.373159	+	0.927768i	$0.378275\pi$
$548$	4.29714	0.183565
$549$	0	0
$550$	0	0
$551$	16.6979	0.711356
$552$	0	0
$553$	12.4699	0.530272
$554$	7.26174	0.308522
$555$	0	0
$556$	9.37207	0.397464
$557$	30.0347	1.27261	0.636306	−	0.771436i	$-0.280461\pi$
$557$	30.0347	1.27261	0.636306	+	0.771436i	$0.280461\pi$
$558$	0	0
$559$	−30.4135	−1.28635
$560$	2.96263	0.125194
$561$	0	0
$562$	−19.9036	−0.839583
$563$	−32.3323	−1.36264	−0.681322	−	0.731984i	$-0.738595\pi$
$563$	−32.3323	−1.36264	−0.681322	+	0.731984i	$0.738595\pi$
$564$	0	0
$565$	−0.623266	−0.0262210
$566$	−3.75081	−0.157658
$567$	0	0
$568$	3.89931	0.163611
$569$	−35.5875	−1.49191	−0.745953	−	0.665998i	$-0.768006\pi$
$569$	−35.5875	−1.49191	−0.745953	+	0.665998i	$0.768006\pi$
$570$	0	0
$571$	−7.60420	−0.318226	−0.159113	−	0.987260i	$-0.550863\pi$
$571$	−7.60420	−0.318226	−0.159113	+	0.987260i	$0.550863\pi$
$572$	0	0
$573$	0	0
$574$	−1.38427	−0.0577781
$575$	4.55073	0.189779
$576$	0	0
$577$	−21.9193	−0.912512	−0.456256	−	0.889849i	$-0.650810\pi$
$577$	−21.9193	−0.912512	−0.456256	+	0.889849i	$0.650810\pi$
$578$	9.01801	0.375100
$579$	0	0
$580$	−13.1446	−0.545798
$581$	1.65657	0.0687259
$582$	0	0
$583$	0	0
$584$	−32.4677	−1.34352
$585$	0	0
$586$	−15.5648	−0.642974
$587$	−25.1988	−1.04007	−0.520033	−	0.854146i	$-0.674080\pi$
$587$	−25.1988	−1.04007	−0.520033	+	0.854146i	$0.674080\pi$
$588$	0	0
$589$	−3.28142	−0.135209
$590$	−3.48442	−0.143451
$591$	0	0
$592$	0.354182	0.0145568
$593$	−25.9594	−1.06602	−0.533012	−	0.846107i	$-0.678940\pi$
$593$	−25.9594	−1.06602	−0.533012	+	0.846107i	$0.678940\pi$
$594$	0	0
$595$	3.24456	0.133014
$596$	24.9059	1.02019
$597$	0	0
$598$	−8.28395	−0.338756
$599$	−12.6905	−0.518520	−0.259260	−	0.965808i	$-0.583479\pi$
$599$	−12.6905	−0.518520	−0.259260	+	0.965808i	$0.583479\pi$
$600$	0	0
$601$	−25.3306	−1.03326	−0.516628	−	0.856210i	$-0.672813\pi$
$601$	−25.3306	−1.03326	−0.516628	+	0.856210i	$0.672813\pi$
$602$	12.9442	0.527566
$603$	0	0
$604$	−21.6610	−0.881374
$605$	−11.0000	−0.447214
$606$	0	0
$607$	37.5251	1.52310	0.761548	−	0.648108i	$-0.224439\pi$
$607$	37.5251	1.52310	0.761548	+	0.648108i	$0.224439\pi$
$608$	−11.4510	−0.464401
$609$	0	0
$610$	3.57286	0.144661
$611$	11.2272	0.454205
$612$	0	0
$613$	−17.8508	−0.720988	−0.360494	−	0.932761i	$-0.617392\pi$
$613$	−17.8508	−0.720988	−0.360494	+	0.932761i	$0.617392\pi$
$614$	−9.82779	−0.396617
$615$	0	0
$616$	0	0
$617$	−5.45701	−0.219691	−0.109845	−	0.993949i	$-0.535036\pi$
$617$	−5.45701	−0.219691	−0.109845	+	0.993949i	$0.535036\pi$
$618$	0	0
$619$	25.7454	1.03480	0.517398	−	0.855745i	$-0.326901\pi$
$619$	25.7454	1.03480	0.517398	+	0.855745i	$0.326901\pi$
$620$	2.58312	0.103741
$621$	0	0
$622$	12.0031	0.481279
$623$	1.82035	0.0729310
$624$	0	0
$625$	1.00000	0.0400000
$626$	−9.48741	−0.379193
$627$	0	0
$628$	7.07354	0.282265
$629$	0.387887	0.0154661
$630$	0	0
$631$	−9.01663	−0.358947	−0.179473	−	0.983763i	$-0.557439\pi$
$631$	−9.01663	−0.358947	−0.179473	+	0.983763i	$0.557439\pi$
$632$	15.9739	0.635409
$633$	0	0
$634$	−12.1160	−0.481189
$635$	2.31939	0.0920420
$636$	0	0
$637$	−10.2859	−0.407543
$638$	0	0
$639$	0	0
$640$	11.1377	0.440258
$641$	3.27591	0.129391	0.0646954	−	0.997905i	$-0.479392\pi$
$641$	3.27591	0.129391	0.0646954	+	0.997905i	$0.479392\pi$
$642$	0	0
$643$	2.97703	0.117402	0.0587012	−	0.998276i	$-0.481304\pi$
$643$	2.97703	0.117402	0.0587012	+	0.998276i	$0.481304\pi$
$644$	−13.0422	−0.513934
$645$	0	0
$646$	−2.32559	−0.0914993
$647$	13.1367	0.516457	0.258229	−	0.966084i	$-0.416861\pi$
$647$	13.1367	0.516457	0.258229	+	0.966084i	$0.416861\pi$
$648$	0	0
$649$	0	0
$650$	−1.82035	−0.0714002
$651$	0	0
$652$	18.7968	0.736140
$653$	11.4476	0.447981	0.223991	−	0.974591i	$-0.428092\pi$
$653$	11.4476	0.447981	0.223991	+	0.974591i	$0.428092\pi$
$654$	0	0
$655$	5.35678	0.209307
$656$	1.89705	0.0740675
$657$	0	0
$658$	−4.77839	−0.186281
$659$	−9.21943	−0.359138	−0.179569	−	0.983745i	$-0.557470\pi$
$659$	−9.21943	−0.359138	−0.179569	+	0.983745i	$0.557470\pi$
$660$	0	0
$661$	−29.4746	−1.14643	−0.573215	−	0.819405i	$-0.694304\pi$
$661$	−29.4746	−1.14643	−0.573215	+	0.819405i	$0.694304\pi$
$662$	−9.49238	−0.368932
$663$	0	0
$664$	2.12207	0.0823522
$665$	3.64071	0.141181
$666$	0	0
$667$	37.9939	1.47113
$668$	−33.8437	−1.30945
$669$	0	0
$670$	−6.72165	−0.259680
$671$	0	0
$672$	0	0
$673$	−24.3371	−0.938127	−0.469063	−	0.883165i	$-0.655408\pi$
$673$	−24.3371	−0.938127	−0.469063	+	0.883165i	$0.655408\pi$
$674$	−20.1119	−0.774683
$675$	0	0
$676$	8.20920	0.315738
$677$	20.8965	0.803120	0.401560	−	0.915833i	$-0.368468\pi$
$677$	20.8965	0.803120	0.401560	+	0.915833i	$0.368468\pi$
$678$	0	0
$679$	10.2657	0.393963
$680$	4.15629	0.159387
$681$	0	0
$682$	0	0
$683$	−42.9543	−1.64360	−0.821800	−	0.569777i	$-0.807030\pi$
$683$	−42.9543	−1.64360	−0.821800	+	0.569777i	$0.807030\pi$
$684$	0	0
$685$	−2.72940	−0.104285
$686$	12.6908	0.484536
$687$	0	0
$688$	−17.7392	−0.676303
$689$	5.16714	0.196852
$690$	0	0
$691$	5.65865	0.215265	0.107633	−	0.994191i	$-0.465673\pi$
$691$	5.65865	0.215265	0.107633	+	0.994191i	$0.465673\pi$
$692$	15.7622	0.599190
$693$	0	0
$694$	−11.6273	−0.441367
$695$	−5.95281	−0.225803
$696$	0	0
$697$	2.07758	0.0786941
$698$	−2.40746	−0.0911238
$699$	0	0
$700$	−2.86595	−0.108323
$701$	−21.1882	−0.800268	−0.400134	−	0.916457i	$-0.631037\pi$
$701$	−21.1882	−0.800268	−0.400134	+	0.916457i	$0.631037\pi$
$702$	0	0
$703$	0.435247	0.0164157
$704$	0	0
$705$	0	0
$706$	−5.08624	−0.191423
$707$	13.3424	0.501792
$708$	0	0
$709$	32.4198	1.21755	0.608775	−	0.793343i	$-0.291661\pi$
$709$	32.4198	1.21755	0.608775	+	0.793343i	$0.291661\pi$
$710$	−1.09090	−0.0409408
$711$	0	0
$712$	2.33188	0.0873910
$713$	−7.46643	−0.279620
$714$	0	0
$715$	0	0
$716$	24.6001	0.919347
$717$	0	0
$718$	−4.61218	−0.172125
$719$	5.94389	0.221670	0.110835	−	0.993839i	$-0.464648\pi$
$719$	5.94389	0.221670	0.110835	+	0.993839i	$0.464648\pi$
$720$	0	0
$721$	−28.7682	−1.07138
$722$	9.78579	0.364189
$723$	0	0
$724$	−21.7806	−0.809470
$725$	8.34897	0.310073
$726$	0	0
$727$	2.02688	0.0751728	0.0375864	−	0.999293i	$-0.488033\pi$
$727$	2.02688	0.0751728	0.0375864	+	0.999293i	$0.488033\pi$
$728$	11.8444	0.438984
$729$	0	0
$730$	9.08341	0.336192
$731$	−19.4274	−0.718547
$732$	0	0
$733$	0.253776	0.00937345	0.00468672	−	0.999989i	$-0.498508\pi$
$733$	0.253776	0.00937345	0.00468672	+	0.999989i	$0.498508\pi$
$734$	−4.46513	−0.164811
$735$	0	0
$736$	−26.0553	−0.960412
$737$	0	0
$738$	0	0
$739$	−17.3564	−0.638466	−0.319233	−	0.947676i	$-0.603425\pi$
$739$	−17.3564	−0.638466	−0.319233	+	0.947676i	$0.603425\pi$
$740$	−0.342625	−0.0125951
$741$	0	0
$742$	−2.19917	−0.0807341
$743$	13.1549	0.482605	0.241303	−	0.970450i	$-0.422425\pi$
$743$	13.1549	0.482605	0.241303	+	0.970450i	$0.422425\pi$
$744$	0	0
$745$	−15.8194	−0.579577
$746$	9.71914	0.355843
$747$	0	0
$748$	0	0
$749$	−4.17336	−0.152491
$750$	0	0
$751$	−19.6949	−0.718675	−0.359338	−	0.933208i	$-0.616997\pi$
$751$	−19.6949	−0.718675	−0.359338	+	0.933208i	$0.616997\pi$
$752$	6.54850	0.238799
$753$	0	0
$754$	−15.1981	−0.553481
$755$	13.7583	0.500717
$756$	0	0
$757$	19.5117	0.709165	0.354582	−	0.935025i	$-0.384623\pi$
$757$	19.5117	0.709165	0.354582	+	0.935025i	$0.384623\pi$
$758$	−9.59561	−0.348528
$759$	0	0
$760$	4.66377	0.169173
$761$	9.91619	0.359462	0.179731	−	0.983716i	$-0.442477\pi$
$761$	9.91619	0.359462	0.179731	+	0.983716i	$0.442477\pi$
$762$	0	0
$763$	12.1125	0.438502
$764$	−2.35304	−0.0851300
$765$	0	0
$766$	−1.38907	−0.0501890
$767$	14.9031	0.538121
$768$	0	0
$769$	−0.392247	−0.0141448	−0.00707239	−	0.999975i	$-0.502251\pi$
$769$	−0.392247	−0.0141448	−0.00707239	+	0.999975i	$0.502251\pi$
$770$	0	0
$771$	0	0
$772$	13.9572	0.502331
$773$	12.7537	0.458718	0.229359	−	0.973342i	$-0.426337\pi$
$773$	12.7537	0.458718	0.229359	+	0.973342i	$0.426337\pi$
$774$	0	0
$775$	−1.64071	−0.0589360
$776$	13.1505	0.472074
$777$	0	0
$778$	6.77691	0.242964
$779$	2.33125	0.0835257
$780$	0	0
$781$	0	0
$782$	−5.29158	−0.189226
$783$	0	0
$784$	−5.99946	−0.214267
$785$	−4.49287	−0.160357
$786$	0	0
$787$	−17.9336	−0.639264	−0.319632	−	0.947542i	$-0.603559\pi$
$787$	−17.9336	−0.639264	−0.319632	+	0.947542i	$0.603559\pi$
$788$	−19.4054	−0.691287
$789$	0	0
$790$	−4.46899	−0.159000
$791$	−1.13457	−0.0403405
$792$	0	0
$793$	−15.2814	−0.542659
$794$	4.70094	0.166830
$795$	0	0
$796$	−33.7964	−1.19788
$797$	24.7798	0.877747	0.438873	−	0.898549i	$-0.355378\pi$
$797$	24.7798	0.877747	0.438873	+	0.898549i	$0.355378\pi$
$798$	0	0
$799$	7.17168	0.253716
$800$	−5.72552	−0.202428
$801$	0	0
$802$	9.16908	0.323771
$803$	0	0
$804$	0	0
$805$	8.28395	0.291971
$806$	2.98667	0.105201
$807$	0	0
$808$	17.0916	0.601282
$809$	7.10308	0.249731	0.124866	−	0.992174i	$-0.460150\pi$
$809$	7.10308	0.249731	0.124866	+	0.992174i	$0.460150\pi$
$810$	0	0
$811$	24.8724	0.873387	0.436694	−	0.899610i	$-0.356149\pi$
$811$	24.8724	0.873387	0.436694	+	0.899610i	$0.356149\pi$
$812$	−23.9278	−0.839699
$813$	0	0
$814$	0	0
$815$	−11.9391	−0.418208
$816$	0	0
$817$	−21.7994	−0.762664
$818$	9.57214	0.334682
$819$	0	0
$820$	−1.83515	−0.0640863
$821$	4.62542	0.161428	0.0807141	−	0.996737i	$-0.474280\pi$
$821$	4.62542	0.161428	0.0807141	+	0.996737i	$0.474280\pi$
$822$	0	0
$823$	43.1088	1.50268	0.751339	−	0.659917i	$-0.229408\pi$
$823$	43.1088	1.50268	0.751339	+	0.659917i	$0.229408\pi$
$824$	−36.8522	−1.28381
$825$	0	0
$826$	−6.34287	−0.220697
$827$	−14.0115	−0.487228	−0.243614	−	0.969872i	$-0.578333\pi$
$827$	−14.0115	−0.487228	−0.243614	+	0.969872i	$0.578333\pi$
$828$	0	0
$829$	0.980406	0.0340509	0.0170255	−	0.999855i	$-0.494580\pi$
$829$	0.980406	0.0340509	0.0170255	+	0.999855i	$0.494580\pi$
$830$	−0.593686	−0.0206072
$831$	0	0
$832$	1.34005	0.0464577
$833$	−6.57039	−0.227651
$834$	0	0
$835$	21.4963	0.743912
$836$	0	0
$837$	0	0
$838$	16.0886	0.555772
$839$	26.5036	0.915006	0.457503	−	0.889208i	$-0.348744\pi$
$839$	26.5036	0.915006	0.457503	+	0.889208i	$0.348744\pi$
$840$	0	0
$841$	40.7052	1.40363
$842$	−21.1843	−0.730061
$843$	0	0
$844$	−10.6442	−0.366390
$845$	−5.21420	−0.179374
$846$	0	0
$847$	−20.0239	−0.688029
$848$	3.01383	0.103495
$849$	0	0
$850$	−1.16280	−0.0398836
$851$	0.990346	0.0339486
$852$	0	0
$853$	−6.72438	−0.230238	−0.115119	−	0.993352i	$-0.536725\pi$
$853$	−6.72438	−0.230238	−0.115119	+	0.993352i	$0.536725\pi$
$854$	6.50388	0.222558
$855$	0	0
$856$	−5.34610	−0.182726
$857$	−38.5694	−1.31751	−0.658753	−	0.752359i	$-0.728916\pi$
$857$	−38.5694	−1.31751	−0.658753	+	0.752359i	$0.728916\pi$
$858$	0	0
$859$	−14.2437	−0.485989	−0.242994	−	0.970028i	$-0.578130\pi$
$859$	−14.2437	−0.485989	−0.242994	+	0.970028i	$0.578130\pi$
$860$	17.1604	0.585165
$861$	0	0
$862$	20.1828	0.687429
$863$	52.7921	1.79706	0.898532	−	0.438908i	$-0.144635\pi$
$863$	52.7921	1.79706	0.898532	+	0.438908i	$0.144635\pi$
$864$	0	0
$865$	−10.0116	−0.340405
$866$	−0.430512	−0.0146294
$867$	0	0
$868$	4.70220	0.159603
$869$	0	0
$870$	0	0
$871$	28.7490	0.974123
$872$	15.5162	0.525444
$873$	0	0
$874$	−5.93767	−0.200845
$875$	1.82035	0.0615392
$876$	0	0
$877$	−54.6756	−1.84627	−0.923133	−	0.384481i	$-0.874380\pi$
$877$	−54.6756	−1.84627	−0.923133	+	0.384481i	$0.874380\pi$
$878$	−10.6759	−0.360295
$879$	0	0
$880$	0	0
$881$	−25.3990	−0.855715	−0.427858	−	0.903846i	$-0.640732\pi$
$881$	−25.3990	−0.855715	−0.427858	+	0.903846i	$0.640732\pi$
$882$	0	0
$883$	21.5793	0.726201	0.363101	−	0.931750i	$-0.381718\pi$
$883$	21.5793	0.726201	0.363101	+	0.931750i	$0.381718\pi$
$884$	−7.83004	−0.263353
$885$	0	0
$886$	−16.5227	−0.555091
$887$	25.5398	0.857541	0.428771	−	0.903413i	$-0.358947\pi$
$887$	25.5398	0.857541	0.428771	+	0.903413i	$0.358947\pi$
$888$	0	0
$889$	4.22211	0.141605
$890$	−0.652386	−0.0218680
$891$	0	0
$892$	−31.3794	−1.05066
$893$	8.04732	0.269293
$894$	0	0
$895$	−15.6251	−0.522290
$896$	20.2746	0.677328
$897$	0	0
$898$	−11.0834	−0.369857
$899$	−13.6982	−0.456861
$900$	0	0
$901$	3.30064	0.109960
$902$	0	0
$903$	0	0
$904$	−1.45338	−0.0483388
$905$	13.8343	0.459867
$906$	0	0
$907$	56.0122	1.85985	0.929927	−	0.367743i	$-0.119869\pi$
$907$	56.0122	1.85985	0.929927	+	0.367743i	$0.119869\pi$
$908$	4.47316	0.148447
$909$	0	0
$910$	−3.31369	−0.109848
$911$	−25.8303	−0.855795	−0.427897	−	0.903827i	$-0.640746\pi$
$911$	−25.8303	−0.855795	−0.427897	+	0.903827i	$0.640746\pi$
$912$	0	0
$913$	0	0
$914$	23.0176	0.761356
$915$	0	0
$916$	−4.88644	−0.161453
$917$	9.75124	0.322014
$918$	0	0
$919$	−17.1130	−0.564507	−0.282253	−	0.959340i	$-0.591082\pi$
$919$	−17.1130	−0.564507	−0.282253	+	0.959340i	$0.591082\pi$
$920$	10.6118	0.349860
$921$	0	0
$922$	5.91937	0.194944
$923$	4.66587	0.153579
$924$	0	0
$925$	0.217624	0.00715542
$926$	13.9243	0.457580
$927$	0	0
$928$	−47.8022	−1.56918
$929$	−5.83985	−0.191599	−0.0957997	−	0.995401i	$-0.530541\pi$
$929$	−5.83985	−0.191599	−0.0957997	+	0.995401i	$0.530541\pi$
$930$	0	0
$931$	−7.37262	−0.241628
$932$	7.61632	0.249481
$933$	0	0
$934$	−4.37584	−0.143182
$935$	0	0
$936$	0	0
$937$	−60.5365	−1.97764	−0.988821	−	0.149108i	$-0.952360\pi$
$937$	−60.5365	−1.97764	−0.988821	+	0.149108i	$0.952360\pi$
$938$	−12.2358	−0.399513
$939$	0	0
$940$	−6.33482	−0.206619
$941$	−35.6006	−1.16055	−0.580273	−	0.814422i	$-0.697054\pi$
$941$	−35.6006	−1.16055	−0.580273	+	0.814422i	$0.697054\pi$
$942$	0	0
$943$	5.30445	0.172737
$944$	8.69253	0.282918
$945$	0	0
$946$	0	0
$947$	5.71771	0.185801	0.0929004	−	0.995675i	$-0.470386\pi$
$947$	5.71771	0.185801	0.0929004	+	0.995675i	$0.470386\pi$
$948$	0	0
$949$	−38.8504	−1.26114
$950$	−1.30477	−0.0423324
$951$	0	0
$952$	7.56593	0.245213
$953$	5.96701	0.193291	0.0966453	−	0.995319i	$-0.469189\pi$
$953$	5.96701	0.193291	0.0966453	+	0.995319i	$0.469189\pi$
$954$	0	0
$955$	1.49457	0.0483631
$956$	−34.5325	−1.11686
$957$	0	0
$958$	13.6574	0.441251
$959$	−4.96847	−0.160440
$960$	0	0
$961$	−28.3081	−0.913164
$962$	−0.396152	−0.0127725
$963$	0	0
$964$	−18.5372	−0.597043
$965$	−8.86514	−0.285379
$966$	0	0
$967$	11.9553	0.384457	0.192228	−	0.981350i	$-0.438429\pi$
$967$	11.9553	0.384457	0.192228	+	0.981350i	$0.438429\pi$
$968$	−25.6507	−0.824445
$969$	0	0
$970$	−3.67908	−0.118128
$971$	−21.2451	−0.681789	−0.340895	−	0.940102i	$-0.610730\pi$
$971$	−21.2451	−0.681789	−0.340895	+	0.940102i	$0.610730\pi$
$972$	0	0
$973$	−10.8362	−0.347394
$974$	−5.05922	−0.162108
$975$	0	0
$976$	−8.91318	−0.285304
$977$	16.9619	0.542658	0.271329	−	0.962487i	$-0.412537\pi$
$977$	16.9619	0.542658	0.271329	+	0.962487i	$0.412537\pi$
$978$	0	0
$979$	0	0
$980$	5.80370	0.185392
$981$	0	0
$982$	1.59583	0.0509250
$983$	10.3405	0.329811	0.164906	−	0.986309i	$-0.447268\pi$
$983$	10.3405	0.329811	0.164906	+	0.986309i	$0.447268\pi$
$984$	0	0
$985$	12.3256	0.392727
$986$	−9.70815	−0.309171
$987$	0	0
$988$	−8.78607	−0.279522
$989$	−49.6016	−1.57724
$990$	0	0
$991$	−48.6759	−1.54624	−0.773121	−	0.634259i	$-0.781305\pi$
$991$	−48.6759	−1.54624	−0.773121	+	0.634259i	$0.781305\pi$
$992$	9.39392	0.298257
$993$	0	0
$994$	−1.98583	−0.0629866
$995$	21.4663	0.680527
$996$	0	0
$997$	49.7687	1.57619	0.788096	−	0.615552i	$-0.211067\pi$
$997$	49.7687	1.57619	0.788096	+	0.615552i	$0.211067\pi$
$998$	24.4698	0.774578
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4005.2.a.n.1.2		6
3.2	odd	2		1335.2.a.g.1.5	✓	6
15.14	odd	2		6675.2.a.u.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1335.2.a.g.1.5	✓	6	3.2	odd	2
4005.2.a.n.1.2		6	1.1	even	1	trivial
6675.2.a.u.1.2		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)

\(f(q)\)	\(=\)	\(q-0.652386 q^{2} -1.57439 q^{4} +1.00000 q^{5} +1.82035 q^{7} +2.33188 q^{8} +O(q^{10})\)	\(q-0.652386 q^{2} -1.57439 q^{4} +1.00000 q^{5} +1.82035 q^{7} +2.33188 q^{8} -0.652386 q^{10} +2.79030 q^{13} -1.18757 q^{14} +1.62750 q^{16} +1.78238 q^{17} +2.00000 q^{19} -1.57439 q^{20} +4.55073 q^{23} +1.00000 q^{25} -1.82035 q^{26} -2.86595 q^{28} +8.34897 q^{29} -1.64071 q^{31} -5.72552 q^{32} -1.16280 q^{34} +1.82035 q^{35} +0.217624 q^{37} -1.30477 q^{38} +2.33188 q^{40} +1.16563 q^{41} -10.8997 q^{43} -2.96883 q^{46} +4.02366 q^{47} -3.68631 q^{49} -0.652386 q^{50} -4.39304 q^{52} +1.85182 q^{53} +4.24485 q^{56} -5.44675 q^{58} +5.34104 q^{59} -5.47661 q^{61} +1.07038 q^{62} +0.480251 q^{64} +2.79030 q^{65} +10.3032 q^{67} -2.80616 q^{68} -1.18757 q^{70} +1.67217 q^{71} -13.9234 q^{73} -0.141974 q^{74} -3.14879 q^{76} +6.85023 q^{79} +1.62750 q^{80} -0.760437 q^{82} +0.910024 q^{83} +1.78238 q^{85} +7.11081 q^{86} +1.00000 q^{89} +5.07934 q^{91} -7.16464 q^{92} -2.62498 q^{94} +2.00000 q^{95} +5.63942 q^{97} +2.40489 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{2} + 8 q^{4} + 6 q^{5} + q^{7} + 9 q^{8}+O(q^{10}) \) 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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