LMFDB - Embedded newform 4005.2.a.w.1.10

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:	$17$
Coefficient field:	$\mathbb{Q}[x]/(x^{17} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{17} - 5 x^{16} - 15 x^{15} + 105 x^{14} + 45 x^{13} - 849 x^{12} + 320 x^{11} + 3371 x^{10} + \cdots + 24 $ x^17 - 5x^16 - 15x^15 + 105x^14 + 45x^13 - 849x^12 + 320x^11 + 3371x^10 - 2456x^9 - 7002x^8 + 6279x^7 + 7299x^6 - 7119x^5 - 3066x^4 + 3184x^3 + 99x^2 - 231x + 24
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$0.159847$ 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)

$f(q)$	$=$	$q-0.159847 q^{2} -1.97445 q^{4} -1.00000 q^{5} -1.46854 q^{7} +0.635302 q^{8} +O(q^{10})$	\(q-0.159847 q^{2} -1.97445 q^{4} -1.00000 q^{5} -1.46854 q^{7} +0.635302 q^{8} +0.159847 q^{10} -3.32290 q^{11} -0.911279 q^{13} +0.234742 q^{14} +3.84735 q^{16} -4.90160 q^{17} -5.74776 q^{19} +1.97445 q^{20} +0.531154 q^{22} +2.28819 q^{23} +1.00000 q^{25} +0.145665 q^{26} +2.89957 q^{28} +0.485991 q^{29} +0.644955 q^{31} -1.88559 q^{32} +0.783504 q^{34} +1.46854 q^{35} -1.19640 q^{37} +0.918760 q^{38} -0.635302 q^{40} -7.09369 q^{41} -8.51707 q^{43} +6.56090 q^{44} -0.365760 q^{46} -9.70935 q^{47} -4.84338 q^{49} -0.159847 q^{50} +1.79927 q^{52} +5.10311 q^{53} +3.32290 q^{55} -0.932969 q^{56} -0.0776841 q^{58} -5.52046 q^{59} -2.65959 q^{61} -0.103094 q^{62} -7.39329 q^{64} +0.911279 q^{65} +11.7114 q^{67} +9.67796 q^{68} -0.234742 q^{70} +7.75565 q^{71} +5.43386 q^{73} +0.191240 q^{74} +11.3487 q^{76} +4.87983 q^{77} +14.6601 q^{79} -3.84735 q^{80} +1.13390 q^{82} +3.48656 q^{83} +4.90160 q^{85} +1.36143 q^{86} -2.11105 q^{88} +1.00000 q^{89} +1.33825 q^{91} -4.51792 q^{92} +1.55201 q^{94} +5.74776 q^{95} -6.09403 q^{97} +0.774197 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 17 q - 5 q^{2} + 21 q^{4} - 17 q^{5} + 12 q^{7} - 15 q^{8}+O(q^{10}) $ 17 * q - 5 * q^2 + 21 * q^4 - 17 * q^5 + 12 * q^7 - 15 * q^8	$ 17 q - 5 q^{2} + 21 q^{4} - 17 q^{5} + 12 q^{7} - 15 q^{8} + 5 q^{10} - 2 q^{11} + 8 q^{13} - 4 q^{14} + 33 q^{16} - 10 q^{17} + 32 q^{19} - 21 q^{20} + 8 q^{22} - 15 q^{23} + 17 q^{25} + 15 q^{26} + 24 q^{28} - q^{29} + 18 q^{31} - 25 q^{32} + 14 q^{34} - 12 q^{35} + 12 q^{37} - 22 q^{38} + 15 q^{40} + 7 q^{41} + 28 q^{43} + 14 q^{44} + 4 q^{46} - 26 q^{47} + 41 q^{49} - 5 q^{50} + 10 q^{52} - 12 q^{53} + 2 q^{55} - 13 q^{56} + 16 q^{58} + 23 q^{59} + 26 q^{61} - 10 q^{62} + 59 q^{64} - 8 q^{65} + 31 q^{67} + q^{68} + 4 q^{70} + 2 q^{71} + 33 q^{73} + 10 q^{74} + 66 q^{76} - 12 q^{77} + 33 q^{79} - 33 q^{80} + 30 q^{82} - 13 q^{83} + 10 q^{85} + 20 q^{86} + 12 q^{88} + 17 q^{89} + 40 q^{91} - 16 q^{92} + 38 q^{94} - 32 q^{95} + 45 q^{97} - 2 q^{98}+O(q^{100}) $ 17 * q - 5 * q^2 + 21 * q^4 - 17 * q^5 + 12 * q^7 - 15 * q^8 + 5 * q^10 - 2 * q^11 + 8 * q^13 - 4 * q^14 + 33 * q^16 - 10 * q^17 + 32 * q^19 - 21 * q^20 + 8 * q^22 - 15 * q^23 + 17 * q^25 + 15 * q^26 + 24 * q^28 - q^29 + 18 * q^31 - 25 * q^32 + 14 * q^34 - 12 * q^35 + 12 * q^37 - 22 * q^38 + 15 * q^40 + 7 * q^41 + 28 * q^43 + 14 * q^44 + 4 * q^46 - 26 * q^47 + 41 * q^49 - 5 * q^50 + 10 * q^52 - 12 * q^53 + 2 * q^55 - 13 * q^56 + 16 * q^58 + 23 * q^59 + 26 * q^61 - 10 * q^62 + 59 * q^64 - 8 * q^65 + 31 * q^67 + q^68 + 4 * q^70 + 2 * q^71 + 33 * q^73 + 10 * q^74 + 66 * q^76 - 12 * q^77 + 33 * q^79 - 33 * q^80 + 30 * q^82 - 13 * q^83 + 10 * q^85 + 20 * q^86 + 12 * q^88 + 17 * q^89 + 40 * q^91 - 16 * q^92 + 38 * q^94 - 32 * q^95 + 45 * q^97 - 2 * 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 $

$2$	−0.159847	−0.113029	−0.0565143	−	0.998402i	$-0.517999\pi$
$2$	−0.159847	−0.113029	−0.0565143	+	0.998402i	$0.517999\pi$
$3$	0	0
$3$	0	0
$4$	−1.97445	−0.987225
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.46854	−0.555058	−0.277529	−	0.960717i	$-0.589515\pi$
$7$	−1.46854	−0.555058	−0.277529	+	0.960717i	$0.589515\pi$
$8$	0.635302	0.224613
$9$	0	0
$10$	0.159847	0.0505479
$11$	−3.32290	−1.00189	−0.500946	−	0.865479i	$-0.667014\pi$
$11$	−3.32290	−1.00189	−0.500946	+	0.865479i	$0.667014\pi$
$12$	0	0
$13$	−0.911279	−0.252743	−0.126372	−	0.991983i	$-0.540333\pi$
$13$	−0.911279	−0.252743	−0.126372	+	0.991983i	$0.540333\pi$
$14$	0.234742	0.0627374
$15$	0	0
$16$	3.84735	0.961837
$17$	−4.90160	−1.18881	−0.594406	−	0.804165i	$-0.702613\pi$
$17$	−4.90160	−1.18881	−0.594406	+	0.804165i	$0.702613\pi$
$18$	0	0
$19$	−5.74776	−1.31863	−0.659313	−	0.751868i	$-0.729153\pi$
$19$	−5.74776	−1.31863	−0.659313	+	0.751868i	$0.729153\pi$
$20$	1.97445	0.441500
$21$	0	0
$22$	0.531154	0.113242
$23$	2.28819	0.477121	0.238561	−	0.971128i	$-0.423324\pi$
$23$	2.28819	0.477121	0.238561	+	0.971128i	$0.423324\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0.145665	0.0285672
$27$	0	0
$28$	2.89957	0.547966
$29$	0.485991	0.0902463	0.0451232	−	0.998981i	$-0.485632\pi$
$29$	0.485991	0.0902463	0.0451232	+	0.998981i	$0.485632\pi$
$30$	0	0
$31$	0.644955	0.115837	0.0579186	−	0.998321i	$-0.481554\pi$
$31$	0.644955	0.115837	0.0579186	+	0.998321i	$0.481554\pi$
$32$	−1.88559	−0.333328
$33$	0	0
$34$	0.783504	0.134370
$35$	1.46854	0.248229
$36$	0	0
$37$	−1.19640	−0.196686	−0.0983432	−	0.995153i	$-0.531354\pi$
$37$	−1.19640	−0.196686	−0.0983432	+	0.995153i	$0.531354\pi$
$38$	0.918760	0.149043
$39$	0	0
$40$	−0.635302	−0.100450
$41$	−7.09369	−1.10785	−0.553924	−	0.832567i	$-0.686870\pi$
$41$	−7.09369	−1.10785	−0.553924	+	0.832567i	$0.686870\pi$
$42$	0	0
$43$	−8.51707	−1.29884	−0.649421	−	0.760429i	$-0.724989\pi$
$43$	−8.51707	−1.29884	−0.649421	+	0.760429i	$0.724989\pi$
$44$	6.56090	0.989092
$45$	0	0
$46$	−0.365760	−0.0539284
$47$	−9.70935	−1.41625	−0.708127	−	0.706085i	$-0.750460\pi$
$47$	−9.70935	−1.41625	−0.708127	+	0.706085i	$0.750460\pi$
$48$	0	0
$49$	−4.84338	−0.691911
$50$	−0.159847	−0.0226057
$51$	0	0
$52$	1.79927	0.249514
$53$	5.10311	0.700966	0.350483	−	0.936569i	$-0.386018\pi$
$53$	5.10311	0.700966	0.350483	+	0.936569i	$0.386018\pi$
$54$	0	0
$55$	3.32290	0.448060
$56$	−0.932969	−0.124673
$57$	0	0
$58$	−0.0776841	−0.0102004
$59$	−5.52046	−0.718703	−0.359351	−	0.933202i	$-0.617002\pi$
$59$	−5.52046	−0.718703	−0.359351	+	0.933202i	$0.617002\pi$
$60$	0	0
$61$	−2.65959	−0.340526	−0.170263	−	0.985399i	$-0.554462\pi$
$61$	−2.65959	−0.340526	−0.170263	+	0.985399i	$0.554462\pi$
$62$	−0.103094	−0.0130929
$63$	0	0
$64$	−7.39329	−0.924161
$65$	0.911279	0.113030
$66$	0	0
$67$	11.7114	1.43077	0.715387	−	0.698728i	$-0.246250\pi$
$67$	11.7114	1.43077	0.715387	+	0.698728i	$0.246250\pi$
$68$	9.67796	1.17363
$69$	0	0
$70$	−0.234742	−0.0280570
$71$	7.75565	0.920426	0.460213	−	0.887809i	$-0.347773\pi$
$71$	7.75565	0.920426	0.460213	+	0.887809i	$0.347773\pi$
$72$	0	0
$73$	5.43386	0.635985	0.317993	−	0.948093i	$-0.396991\pi$
$73$	5.43386	0.635985	0.317993	+	0.948093i	$0.396991\pi$
$74$	0.191240	0.0222312
$75$	0	0
$76$	11.3487	1.30178
$77$	4.87983	0.556108
$78$	0	0
$79$	14.6601	1.64939	0.824694	−	0.565580i	$-0.191348\pi$
$79$	14.6601	1.64939	0.824694	+	0.565580i	$0.191348\pi$
$80$	−3.84735	−0.430146
$81$	0	0
$82$	1.13390	0.125219
$83$	3.48656	0.382699	0.191350	−	0.981522i	$-0.438714\pi$
$83$	3.48656	0.382699	0.191350	+	0.981522i	$0.438714\pi$
$84$	0	0
$85$	4.90160	0.531653
$86$	1.36143	0.146806
$87$	0	0
$88$	−2.11105	−0.225038
$89$	1.00000	0.106000
$89$	1.00000	0.106000
$90$	0	0
$91$	1.33825	0.140287
$92$	−4.51792	−0.471026
$93$	0	0
$94$	1.55201	0.160077
$95$	5.74776	0.589708
$96$	0	0
$97$	−6.09403	−0.618755	−0.309377	−	0.950939i	$-0.600121\pi$
$97$	−6.09403	−0.618755	−0.309377	+	0.950939i	$0.600121\pi$
$98$	0.774197	0.0782057
$99$	0	0
$100$	−1.97445	−0.197445
$101$	8.19512	0.815445	0.407722	−	0.913106i	$-0.366323\pi$
$101$	8.19512	0.815445	0.407722	+	0.913106i	$0.366323\pi$
$102$	0	0
$103$	7.68497	0.757223	0.378611	−	0.925556i	$-0.376402\pi$
$103$	7.68497	0.757223	0.378611	+	0.925556i	$0.376402\pi$
$104$	−0.578938	−0.0567695
$105$	0	0
$106$	−0.815714	−0.0792292
$107$	−20.4378	−1.97580	−0.987899	−	0.155095i	$-0.950431\pi$
$107$	−20.4378	−1.97580	−0.987899	+	0.155095i	$0.950431\pi$
$108$	0	0
$109$	1.20297	0.115223	0.0576117	−	0.998339i	$-0.481651\pi$
$109$	1.20297	0.115223	0.0576117	+	0.998339i	$0.481651\pi$
$110$	−0.531154	−0.0506436
$111$	0	0
$112$	−5.65000	−0.533875
$113$	−6.95628	−0.654392	−0.327196	−	0.944957i	$-0.606104\pi$
$113$	−6.95628	−0.654392	−0.327196	+	0.944957i	$0.606104\pi$
$114$	0	0
$115$	−2.28819	−0.213375
$116$	−0.959565	−0.0890934
$117$	0	0
$118$	0.882427	0.0812340
$119$	7.19822	0.659860
$120$	0	0
$121$	0.0416632	0.00378757
$122$	0.425127	0.0384892
$123$	0	0
$124$	−1.27343	−0.114357
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	6.48359	0.575325	0.287663	−	0.957732i	$-0.407122\pi$
$127$	6.48359	0.575325	0.287663	+	0.957732i	$0.407122\pi$
$128$	4.95297	0.437785
$129$	0	0
$130$	−0.145665	−0.0127757
$131$	−8.42761	−0.736323	−0.368162	−	0.929762i	$-0.620013\pi$
$131$	−8.42761	−0.736323	−0.368162	+	0.929762i	$0.620013\pi$
$132$	0	0
$133$	8.44084	0.731914
$134$	−1.87203	−0.161718
$135$	0	0
$136$	−3.11400	−0.267023
$137$	−7.04541	−0.601930	−0.300965	−	0.953635i	$-0.597309\pi$
$137$	−7.04541	−0.601930	−0.300965	+	0.953635i	$0.597309\pi$
$138$	0	0
$139$	20.3944	1.72983	0.864916	−	0.501917i	$-0.167372\pi$
$139$	20.3944	1.72983	0.864916	+	0.501917i	$0.167372\pi$
$140$	−2.89957	−0.245058
$141$	0	0
$142$	−1.23971	−0.104034
$143$	3.02809	0.253222
$144$	0	0
$145$	−0.485991	−0.0403594
$146$	−0.868584	−0.0718845
$147$	0	0
$148$	2.36223	0.194174
$149$	−0.396584	−0.0324894	−0.0162447	−	0.999868i	$-0.505171\pi$
$149$	−0.396584	−0.0324894	−0.0162447	+	0.999868i	$0.505171\pi$
$150$	0	0
$151$	5.72691	0.466049	0.233025	−	0.972471i	$-0.425138\pi$
$151$	5.72691	0.466049	0.233025	+	0.972471i	$0.425138\pi$
$152$	−3.65156	−0.296181
$153$	0	0
$154$	−0.780024	−0.0628561
$155$	−0.644955	−0.0518040
$156$	0	0
$157$	15.5744	1.24297	0.621487	−	0.783425i	$-0.286529\pi$
$157$	15.5744	1.24297	0.621487	+	0.783425i	$0.286529\pi$
$158$	−2.34336	−0.186428
$159$	0	0
$160$	1.88559	0.149069
$161$	−3.36031	−0.264830
$162$	0	0
$163$	20.4504	1.60180	0.800901	−	0.598797i	$-0.204354\pi$
$163$	20.4504	1.60180	0.800901	+	0.598797i	$0.204354\pi$
$164$	14.0061	1.09370
$165$	0	0
$166$	−0.557314	−0.0432560
$167$	13.3551	1.03345	0.516724	−	0.856152i	$-0.327151\pi$
$167$	13.3551	1.03345	0.516724	+	0.856152i	$0.327151\pi$
$168$	0	0
$169$	−12.1696	−0.936121
$170$	−0.783504	−0.0600920
$171$	0	0
$172$	16.8165	1.28225
$173$	2.93786	0.223361	0.111681	−	0.993744i	$-0.464377\pi$
$173$	2.93786	0.223361	0.111681	+	0.993744i	$0.464377\pi$
$174$	0	0
$175$	−1.46854	−0.111012
$176$	−12.7843	−0.963657
$177$	0	0
$178$	−0.159847	−0.0119810
$179$	−16.5660	−1.23820	−0.619099	−	0.785313i	$-0.712502\pi$
$179$	−16.5660	−1.23820	−0.619099	+	0.785313i	$0.712502\pi$
$180$	0	0
$181$	4.34807	0.323190	0.161595	−	0.986857i	$-0.448336\pi$
$181$	4.34807	0.323190	0.161595	+	0.986857i	$0.448336\pi$
$182$	−0.213915	−0.0158565
$183$	0	0
$184$	1.45369	0.107168
$185$	1.19640	0.0879609
$186$	0	0
$187$	16.2875	1.19106
$188$	19.1706	1.39816
$189$	0	0
$190$	−0.918760	−0.0666538
$191$	−11.4748	−0.830284	−0.415142	−	0.909757i	$-0.636268\pi$
$191$	−11.4748	−0.830284	−0.415142	+	0.909757i	$0.636268\pi$
$192$	0	0
$193$	3.47610	0.250215	0.125108	−	0.992143i	$-0.460072\pi$
$193$	3.47610	0.250215	0.125108	+	0.992143i	$0.460072\pi$
$194$	0.974110	0.0699370
$195$	0	0
$196$	9.56300	0.683072
$197$	−0.867498	−0.0618067	−0.0309033	−	0.999522i	$-0.509838\pi$
$197$	−0.867498	−0.0618067	−0.0309033	+	0.999522i	$0.509838\pi$
$198$	0	0
$199$	0.182095	0.0129083	0.00645417	−	0.999979i	$-0.497946\pi$
$199$	0.182095	0.0129083	0.00645417	+	0.999979i	$0.497946\pi$
$200$	0.635302	0.0449226
$201$	0	0
$202$	−1.30996	−0.0921686
$203$	−0.713700	−0.0500919
$204$	0	0
$205$	7.09369	0.495445
$206$	−1.22842	−0.0855878
$207$	0	0
$208$	−3.50601	−0.243098
$209$	19.0992	1.32112
$210$	0	0
$211$	1.96428	0.135227	0.0676134	−	0.997712i	$-0.478462\pi$
$211$	1.96428	0.135227	0.0676134	+	0.997712i	$0.478462\pi$
$212$	−10.0758	−0.692010
$213$	0	0
$214$	3.26692	0.223322
$215$	8.51707	0.580860
$216$	0	0
$217$	−0.947144	−0.0642964
$218$	−0.192290	−0.0130235
$219$	0	0
$220$	−6.56090	−0.442336
$221$	4.46673	0.300465
$222$	0	0
$223$	9.05879	0.606621	0.303311	−	0.952892i	$-0.401908\pi$
$223$	9.05879	0.606621	0.303311	+	0.952892i	$0.401908\pi$
$224$	2.76907	0.185016
$225$	0	0
$226$	1.11194	0.0739650
$227$	−8.26329	−0.548454	−0.274227	−	0.961665i	$-0.588422\pi$
$227$	−8.26329	−0.548454	−0.274227	+	0.961665i	$0.588422\pi$
$228$	0	0
$229$	21.1371	1.39678	0.698389	−	0.715719i	$-0.253901\pi$
$229$	21.1371	1.39678	0.698389	+	0.715719i	$0.253901\pi$
$230$	0.365760	0.0241175
$231$	0	0
$232$	0.308751	0.0202705
$233$	−17.8116	−1.16688	−0.583439	−	0.812157i	$-0.698293\pi$
$233$	−17.8116	−1.16688	−0.583439	+	0.812157i	$0.698293\pi$
$234$	0	0
$235$	9.70935	0.633368
$236$	10.8999	0.709521
$237$	0	0
$238$	−1.15061	−0.0745830
$239$	−9.60296	−0.621164	−0.310582	−	0.950547i	$-0.600524\pi$
$239$	−9.60296	−0.621164	−0.310582	+	0.950547i	$0.600524\pi$
$240$	0	0
$241$	−3.70394	−0.238591	−0.119296	−	0.992859i	$-0.538064\pi$
$241$	−3.70394	−0.238591	−0.119296	+	0.992859i	$0.538064\pi$
$242$	−0.00665973	−0.000428103 0
$243$	0	0
$244$	5.25123	0.336176
$245$	4.84338	0.309432
$246$	0	0
$247$	5.23781	0.333274
$248$	0.409741	0.0260186
$249$	0	0
$250$	0.159847	0.0101096
$251$	22.5597	1.42396	0.711978	−	0.702202i	$-0.247800\pi$
$251$	22.5597	1.42396	0.711978	+	0.702202i	$0.247800\pi$
$252$	0	0
$253$	−7.60344	−0.478024
$254$	−1.03638	−0.0650282
$255$	0	0
$256$	13.9949	0.874679
$257$	15.6873	0.978545	0.489272	−	0.872131i	$-0.337262\pi$
$257$	15.6873	0.978545	0.489272	+	0.872131i	$0.337262\pi$
$258$	0	0
$259$	1.75696	0.109172
$260$	−1.79927	−0.111586
$261$	0	0
$262$	1.34712	0.0832256
$263$	3.13676	0.193421	0.0967105	−	0.995313i	$-0.469168\pi$
$263$	3.13676	0.193421	0.0967105	+	0.995313i	$0.469168\pi$
$264$	0	0
$265$	−5.10311	−0.313481
$266$	−1.34924	−0.0827272
$267$	0	0
$268$	−23.1235	−1.41250
$269$	20.7144	1.26298	0.631490	−	0.775384i	$-0.282444\pi$
$269$	20.7144	1.26298	0.631490	+	0.775384i	$0.282444\pi$
$270$	0	0
$271$	14.0657	0.854431	0.427216	−	0.904150i	$-0.359495\pi$
$271$	14.0657	0.854431	0.427216	+	0.904150i	$0.359495\pi$
$272$	−18.8582	−1.14344
$273$	0	0
$274$	1.12619	0.0680353
$275$	−3.32290	−0.200378
$276$	0	0
$277$	−19.5644	−1.17551	−0.587755	−	0.809039i	$-0.699988\pi$
$277$	−19.5644	−1.17551	−0.587755	+	0.809039i	$0.699988\pi$
$278$	−3.25998	−0.195520
$279$	0	0
$280$	0.932969	0.0557556
$281$	6.16242	0.367619	0.183809	−	0.982962i	$-0.441157\pi$
$281$	6.16242	0.367619	0.183809	+	0.982962i	$0.441157\pi$
$282$	0	0
$283$	1.45079	0.0862407	0.0431203	−	0.999070i	$-0.486270\pi$
$283$	1.45079	0.0862407	0.0431203	+	0.999070i	$0.486270\pi$
$284$	−15.3131	−0.908667
$285$	0	0
$286$	−0.484030	−0.0286213
$287$	10.4174	0.614920
$288$	0	0
$289$	7.02570	0.413276
$290$	0.0776841	0.00456177
$291$	0	0
$292$	−10.7289	−0.627860
$293$	15.2032	0.888181	0.444091	−	0.895982i	$-0.353527\pi$
$293$	15.2032	0.888181	0.444091	+	0.895982i	$0.353527\pi$
$294$	0	0
$295$	5.52046	0.321414
$296$	−0.760074	−0.0441784
$297$	0	0
$298$	0.0633926	0.00367224
$299$	−2.08518	−0.120589
$300$	0	0
$301$	12.5077	0.720932
$302$	−0.915427	−0.0526769
$303$	0	0
$304$	−22.1136	−1.26830
$305$	2.65959	0.152288
$306$	0	0
$307$	15.2605	0.870960	0.435480	−	0.900198i	$-0.356579\pi$
$307$	15.2605	0.870960	0.435480	+	0.900198i	$0.356579\pi$
$308$	−9.63497	−0.549003
$309$	0	0
$310$	0.103094	0.00585533
$311$	0.307673	0.0174465	0.00872327	−	0.999962i	$-0.497223\pi$
$311$	0.307673	0.0174465	0.00872327	+	0.999962i	$0.497223\pi$
$312$	0	0
$313$	0.442030	0.0249850	0.0124925	−	0.999922i	$-0.496023\pi$
$313$	0.442030	0.0249850	0.0124925	+	0.999922i	$0.496023\pi$
$314$	−2.48952	−0.140492
$315$	0	0
$316$	−28.9456	−1.62832
$317$	−21.8404	−1.22668	−0.613340	−	0.789819i	$-0.710174\pi$
$317$	−21.8404	−1.22668	−0.613340	+	0.789819i	$0.710174\pi$
$318$	0	0
$319$	−1.61490	−0.0904171
$320$	7.39329	0.413297
$321$	0	0
$322$	0.537135	0.0299333
$323$	28.1732	1.56760
$324$	0	0
$325$	−0.911279	−0.0505487
$326$	−3.26893	−0.181049
$327$	0	0
$328$	−4.50664	−0.248837
$329$	14.2586	0.786102
$330$	0	0
$331$	−13.7830	−0.757579	−0.378790	−	0.925483i	$-0.623660\pi$
$331$	−13.7830	−0.757579	−0.378790	+	0.925483i	$0.623660\pi$
$332$	−6.88403	−0.377810
$333$	0	0
$334$	−2.13476	−0.116809
$335$	−11.7114	−0.639862
$336$	0	0
$337$	−1.89359	−0.103150	−0.0515751	−	0.998669i	$-0.516424\pi$
$337$	−1.89359	−0.103150	−0.0515751	+	0.998669i	$0.516424\pi$
$338$	1.94526	0.105808
$339$	0	0
$340$	−9.67796	−0.524861
$341$	−2.14312	−0.116056
$342$	0	0
$343$	17.3925	0.939108
$344$	−5.41092	−0.291737
$345$	0	0
$346$	−0.469607	−0.0252462
$347$	−10.9822	−0.589557	−0.294778	−	0.955566i	$-0.595246\pi$
$347$	−10.9822	−0.589557	−0.294778	+	0.955566i	$0.595246\pi$
$348$	0	0
$349$	19.4071	1.03884	0.519420	−	0.854519i	$-0.326148\pi$
$349$	19.4071	1.03884	0.519420	+	0.854519i	$0.326148\pi$
$350$	0.234742	0.0125475
$351$	0	0
$352$	6.26563	0.333959
$353$	−1.27251	−0.0677287	−0.0338643	−	0.999426i	$-0.510781\pi$
$353$	−1.27251	−0.0677287	−0.0338643	+	0.999426i	$0.510781\pi$
$354$	0	0
$355$	−7.75565	−0.411627
$356$	−1.97445	−0.104646
$357$	0	0
$358$	2.64801	0.139952
$359$	25.9284	1.36845	0.684224	−	0.729272i	$-0.260141\pi$
$359$	25.9284	1.36845	0.684224	+	0.729272i	$0.260141\pi$
$360$	0	0
$361$	14.0367	0.738776
$362$	−0.695025	−0.0365297
$363$	0	0
$364$	−2.64231	−0.138495
$365$	−5.43386	−0.284421
$366$	0	0
$367$	−15.3026	−0.798790	−0.399395	−	0.916779i	$-0.630780\pi$
$367$	−15.3026	−0.798790	−0.399395	+	0.916779i	$0.630780\pi$
$368$	8.80347	0.458913
$369$	0	0
$370$	−0.191240	−0.00994209
$371$	−7.49414	−0.389076
$372$	0	0
$373$	15.4533	0.800143	0.400071	−	0.916484i	$-0.368985\pi$
$373$	15.4533	0.800143	0.400071	+	0.916484i	$0.368985\pi$
$374$	−2.60351	−0.134624
$375$	0	0
$376$	−6.16837	−0.318109
$377$	−0.442874	−0.0228092
$378$	0	0
$379$	37.2765	1.91476	0.957382	−	0.288826i	$-0.0932649\pi$
$379$	37.2765	1.91476	0.957382	+	0.288826i	$0.0932649\pi$
$380$	−11.3487	−0.582174
$381$	0	0
$382$	1.83420	0.0938459
$383$	−16.6212	−0.849302	−0.424651	−	0.905357i	$-0.639603\pi$
$383$	−16.6212	−0.849302	−0.424651	+	0.905357i	$0.639603\pi$
$384$	0	0
$385$	−4.87983	−0.248699
$386$	−0.555643	−0.0282815
$387$	0	0
$388$	12.0323	0.610850
$389$	−23.5479	−1.19393	−0.596963	−	0.802269i	$-0.703626\pi$
$389$	−23.5479	−1.19393	−0.596963	+	0.802269i	$0.703626\pi$
$390$	0	0
$391$	−11.2158	−0.567208
$392$	−3.07701	−0.155412
$393$	0	0
$394$	0.138667	0.00698592
$395$	−14.6601	−0.737628
$396$	0	0
$397$	12.4317	0.623927	0.311964	−	0.950094i	$-0.399013\pi$
$397$	12.4317	0.623927	0.311964	+	0.950094i	$0.399013\pi$
$398$	−0.0291072	−0.00145901
$399$	0	0
$400$	3.84735	0.192367
$401$	−4.23443	−0.211457	−0.105729	−	0.994395i	$-0.533718\pi$
$401$	−4.23443	−0.211457	−0.105729	+	0.994395i	$0.533718\pi$
$402$	0	0
$403$	−0.587734	−0.0292771
$404$	−16.1808	−0.805027
$405$	0	0
$406$	0.114083	0.00566182
$407$	3.97551	0.197059
$408$	0	0
$409$	13.9280	0.688697	0.344349	−	0.938842i	$-0.388100\pi$
$409$	13.9280	0.688697	0.344349	+	0.938842i	$0.388100\pi$
$410$	−1.13390	−0.0559995
$411$	0	0
$412$	−15.1736	−0.747549
$413$	8.10704	0.398921
$414$	0	0
$415$	−3.48656	−0.171148
$416$	1.71830	0.0842465
$417$	0	0
$418$	−3.05295	−0.149325
$419$	39.3047	1.92016	0.960081	−	0.279724i	$-0.0902428\pi$
$419$	39.3047	1.92016	0.960081	+	0.279724i	$0.0902428\pi$
$420$	0	0
$421$	−22.9017	−1.11616	−0.558080	−	0.829787i	$-0.688462\pi$
$421$	−22.9017	−1.11616	−0.558080	+	0.829787i	$0.688462\pi$
$422$	−0.313984	−0.0152845
$423$	0	0
$424$	3.24201	0.157446
$425$	−4.90160	−0.237763
$426$	0	0
$427$	3.90573	0.189012
$428$	40.3534	1.95056
$429$	0	0
$430$	−1.36143	−0.0656538
$431$	17.4555	0.840802	0.420401	−	0.907338i	$-0.361889\pi$
$431$	17.4555	0.840802	0.420401	+	0.907338i	$0.361889\pi$
$432$	0	0
$433$	1.38552	0.0665837	0.0332919	−	0.999446i	$-0.489401\pi$
$433$	1.38552	0.0665837	0.0332919	+	0.999446i	$0.489401\pi$
$434$	0.151398	0.00726733
$435$	0	0
$436$	−2.37520	−0.113751
$437$	−13.1520	−0.629145
$438$	0	0
$439$	−21.9081	−1.04562	−0.522809	−	0.852450i	$-0.675116\pi$
$439$	−21.9081	−1.04562	−0.522809	+	0.852450i	$0.675116\pi$
$440$	2.11105	0.100640
$441$	0	0
$442$	−0.713991	−0.0339611
$443$	−14.3486	−0.681721	−0.340861	−	0.940114i	$-0.610718\pi$
$443$	−14.3486	−0.681721	−0.340861	+	0.940114i	$0.610718\pi$
$444$	0	0
$445$	−1.00000	−0.0474045
$446$	−1.44802	−0.0685655
$447$	0	0
$448$	10.8574	0.512963
$449$	1.66811	0.0787232	0.0393616	−	0.999225i	$-0.487468\pi$
$449$	1.66811	0.0787232	0.0393616	+	0.999225i	$0.487468\pi$
$450$	0	0
$451$	23.5716	1.10994
$452$	13.7348	0.646031
$453$	0	0
$454$	1.32086	0.0619910
$455$	−1.33825	−0.0627383
$456$	0	0
$457$	−41.4544	−1.93916	−0.969578	−	0.244782i	$-0.921284\pi$
$457$	−41.4544	−1.93916	−0.969578	+	0.244782i	$0.921284\pi$
$458$	−3.37869	−0.157876
$459$	0	0
$460$	4.51792	0.210649
$461$	−28.9536	−1.34850	−0.674252	−	0.738501i	$-0.735534\pi$
$461$	−28.9536	−1.34850	−0.674252	+	0.738501i	$0.735534\pi$
$462$	0	0
$463$	−11.7216	−0.544751	−0.272375	−	0.962191i	$-0.587809\pi$
$463$	−11.7216	−0.544751	−0.272375	+	0.962191i	$0.587809\pi$
$464$	1.86978	0.0868022
$465$	0	0
$466$	2.84713	0.131891
$467$	−22.1597	−1.02543	−0.512713	−	0.858560i	$-0.671360\pi$
$467$	−22.1597	−1.02543	−0.512713	+	0.858560i	$0.671360\pi$
$468$	0	0
$469$	−17.1987	−0.794162
$470$	−1.55201	−0.0715887
$471$	0	0
$472$	−3.50716	−0.161430
$473$	28.3014	1.30130
$474$	0	0
$475$	−5.74776	−0.263725
$476$	−14.2125	−0.651430
$477$	0	0
$478$	1.53500	0.0702093
$479$	−16.1968	−0.740051	−0.370025	−	0.929022i	$-0.620651\pi$
$479$	−16.1968	−0.740051	−0.370025	+	0.929022i	$0.620651\pi$
$480$	0	0
$481$	1.09025	0.0497112
$482$	0.592061	0.0269677
$483$	0	0
$484$	−0.0822619	−0.00373918
$485$	6.09403	0.276716
$486$	0	0
$487$	−31.1084	−1.40966	−0.704829	−	0.709377i	$-0.748976\pi$
$487$	−31.1084	−1.40966	−0.704829	+	0.709377i	$0.748976\pi$
$488$	−1.68965	−0.0764866
$489$	0	0
$490$	−0.774197	−0.0349747
$491$	−10.7620	−0.485684	−0.242842	−	0.970066i	$-0.578080\pi$
$491$	−10.7620	−0.485684	−0.242842	+	0.970066i	$0.578080\pi$
$492$	0	0
$493$	−2.38214	−0.107286
$494$	−0.837247	−0.0376695
$495$	0	0
$496$	2.48136	0.111417
$497$	−11.3895	−0.510889
$498$	0	0
$499$	−15.3188	−0.685761	−0.342881	−	0.939379i	$-0.611403\pi$
$499$	−15.3188	−0.685761	−0.342881	+	0.939379i	$0.611403\pi$
$500$	1.97445	0.0883000
$501$	0	0
$502$	−3.60609	−0.160948
$503$	25.4412	1.13437	0.567183	−	0.823592i	$-0.308033\pi$
$503$	25.4412	1.13437	0.567183	+	0.823592i	$0.308033\pi$
$504$	0	0
$505$	−8.19512	−0.364678
$506$	1.21538	0.0540304
$507$	0	0
$508$	−12.8015	−0.567975
$509$	0.306166	0.0135706	0.00678528	−	0.999977i	$-0.497840\pi$
$509$	0.306166	0.0135706	0.00678528	+	0.999977i	$0.497840\pi$
$510$	0	0
$511$	−7.97986	−0.353008
$512$	−12.1430	−0.536649
$513$	0	0
$514$	−2.50756	−0.110604
$515$	−7.68497	−0.338640
$516$	0	0
$517$	32.2632	1.41893
$518$	−0.280844	−0.0123396
$519$	0	0
$520$	0.578938	0.0253881
$521$	−38.8413	−1.70167	−0.850835	−	0.525433i	$-0.823903\pi$
$521$	−38.8413	−1.70167	−0.850835	+	0.525433i	$0.823903\pi$
$522$	0	0
$523$	28.2779	1.23651	0.618254	−	0.785979i	$-0.287840\pi$
$523$	28.2779	1.23651	0.618254	+	0.785979i	$0.287840\pi$
$524$	16.6399	0.726917
$525$	0	0
$526$	−0.501401	−0.0218621
$527$	−3.16131	−0.137709
$528$	0	0
$529$	−17.7642	−0.772355
$530$	0.815714	0.0354324
$531$	0	0
$532$	−16.6660	−0.722563
$533$	6.46434	0.280001
$534$	0	0
$535$	20.4378	0.883604
$536$	7.44027	0.321371
$537$	0	0
$538$	−3.31113	−0.142753
$539$	16.0941	0.693220
$540$	0	0
$541$	16.4967	0.709249	0.354625	−	0.935009i	$-0.384609\pi$
$541$	16.4967	0.709249	0.354625	+	0.935009i	$0.384609\pi$
$542$	−2.24836	−0.0965752
$543$	0	0
$544$	9.24241	0.396265
$545$	−1.20297	−0.0515295
$546$	0	0
$547$	−8.20182	−0.350684	−0.175342	−	0.984508i	$-0.556103\pi$
$547$	−8.20182	−0.350684	−0.175342	+	0.984508i	$0.556103\pi$
$548$	13.9108	0.594240
$549$	0	0
$550$	0.531154	0.0226485
$551$	−2.79336	−0.119001
$552$	0	0
$553$	−21.5290	−0.915505
$554$	3.12730	0.132866
$555$	0	0
$556$	−40.2677	−1.70773
$557$	−19.0126	−0.805588	−0.402794	−	0.915291i	$-0.631961\pi$
$557$	−19.0126	−0.805588	−0.402794	+	0.915291i	$0.631961\pi$
$558$	0	0
$559$	7.76143	0.328274
$560$	5.65000	0.238756
$561$	0	0
$562$	−0.985041	−0.0415515
$563$	17.9023	0.754490	0.377245	−	0.926114i	$-0.376871\pi$
$563$	17.9023	0.754490	0.377245	+	0.926114i	$0.376871\pi$
$564$	0	0
$565$	6.95628	0.292653
$566$	−0.231904	−0.00974766
$567$	0	0
$568$	4.92718	0.206740
$569$	46.1274	1.93376	0.966880	−	0.255230i	$-0.0821511\pi$
$569$	46.1274	1.93376	0.966880	+	0.255230i	$0.0821511\pi$
$570$	0	0
$571$	27.2551	1.14059	0.570295	−	0.821440i	$-0.306829\pi$
$571$	27.2551	1.14059	0.570295	+	0.821440i	$0.306829\pi$
$572$	−5.97881	−0.249987
$573$	0	0
$574$	−1.66519	−0.0695035
$575$	2.28819	0.0954243
$576$	0	0
$577$	−25.4577	−1.05982	−0.529909	−	0.848054i	$-0.677774\pi$
$577$	−25.4577	−1.05982	−0.529909	+	0.848054i	$0.677774\pi$
$578$	−1.12303	−0.0467120
$579$	0	0
$580$	0.959565	0.0398438
$581$	−5.12016	−0.212420
$582$	0	0
$583$	−16.9571	−0.702292
$584$	3.45214	0.142851
$585$	0	0
$586$	−2.43018	−0.100390
$587$	−5.00058	−0.206396	−0.103198	−	0.994661i	$-0.532908\pi$
$587$	−5.00058	−0.206396	−0.103198	+	0.994661i	$0.532908\pi$
$588$	0	0
$589$	−3.70704	−0.152746
$590$	−0.882427	−0.0363289
$591$	0	0
$592$	−4.60296	−0.189180
$593$	7.82811	0.321462	0.160731	−	0.986998i	$-0.448615\pi$
$593$	7.82811	0.321462	0.160731	+	0.986998i	$0.448615\pi$
$594$	0	0
$595$	−7.19822	−0.295098
$596$	0.783035	0.0320744
$597$	0	0
$598$	0.333309	0.0136300
$599$	0.578790	0.0236487	0.0118244	−	0.999930i	$-0.496236\pi$
$599$	0.578790	0.0236487	0.0118244	+	0.999930i	$0.496236\pi$
$600$	0	0
$601$	6.87518	0.280444	0.140222	−	0.990120i	$-0.455218\pi$
$601$	6.87518	0.280444	0.140222	+	0.990120i	$0.455218\pi$
$602$	−1.99931	−0.0814859
$603$	0	0
$604$	−11.3075	−0.460095
$605$	−0.0416632	−0.00169385
$606$	0	0
$607$	−27.0820	−1.09923	−0.549613	−	0.835419i	$-0.685225\pi$
$607$	−27.0820	−1.09923	−0.549613	+	0.835419i	$0.685225\pi$
$608$	10.8379	0.439536
$609$	0	0
$610$	−0.425127	−0.0172129
$611$	8.84793	0.357949
$612$	0	0
$613$	−12.2389	−0.494325	−0.247163	−	0.968974i	$-0.579498\pi$
$613$	−12.2389	−0.494325	−0.247163	+	0.968974i	$0.579498\pi$
$614$	−2.43933	−0.0984434
$615$	0	0
$616$	3.10016	0.124909
$617$	−32.8150	−1.32108	−0.660541	−	0.750790i	$-0.729673\pi$
$617$	−32.8150	−1.32108	−0.660541	+	0.750790i	$0.729673\pi$
$618$	0	0
$619$	−41.9937	−1.68787	−0.843934	−	0.536447i	$-0.819766\pi$
$619$	−41.9937	−1.68787	−0.843934	+	0.536447i	$0.819766\pi$
$620$	1.27343	0.0511422
$621$	0	0
$622$	−0.0491805	−0.00197196
$623$	−1.46854	−0.0588360
$624$	0	0
$625$	1.00000	0.0400000
$626$	−0.0706570	−0.00282402
$627$	0	0
$628$	−30.7509	−1.22709
$629$	5.86426	0.233823
$630$	0	0
$631$	19.9565	0.794455	0.397228	−	0.917720i	$-0.369972\pi$
$631$	19.9565	0.794455	0.397228	+	0.917720i	$0.369972\pi$
$632$	9.31358	0.370474
$633$	0	0
$634$	3.49111	0.138650
$635$	−6.48359	−0.257293
$636$	0	0
$637$	4.41367	0.174876
$638$	0.258136	0.0102197
$639$	0	0
$640$	−4.95297	−0.195783
$641$	2.00575	0.0792225	0.0396113	−	0.999215i	$-0.487388\pi$
$641$	2.00575	0.0792225	0.0396113	+	0.999215i	$0.487388\pi$
$642$	0	0
$643$	−39.7658	−1.56821	−0.784106	−	0.620627i	$-0.786878\pi$
$643$	−39.7658	−1.56821	−0.784106	+	0.620627i	$0.786878\pi$
$644$	6.63477	0.261446
$645$	0	0
$646$	−4.50339	−0.177184
$647$	−18.2234	−0.716434	−0.358217	−	0.933638i	$-0.616615\pi$
$647$	−18.2234	−0.716434	−0.358217	+	0.933638i	$0.616615\pi$
$648$	0	0
$649$	18.3439	0.720063
$650$	0.145665	0.00571345
$651$	0	0
$652$	−40.3783	−1.58134
$653$	−15.3463	−0.600546	−0.300273	−	0.953853i	$-0.597078\pi$
$653$	−15.3463	−0.600546	−0.300273	+	0.953853i	$0.597078\pi$
$654$	0	0
$655$	8.42761	0.329294
$656$	−27.2919	−1.06557
$657$	0	0
$658$	−2.27919	−0.0888521
$659$	11.4638	0.446568	0.223284	−	0.974753i	$-0.428322\pi$
$659$	11.4638	0.446568	0.223284	+	0.974753i	$0.428322\pi$
$660$	0	0
$661$	−32.2572	−1.25466	−0.627329	−	0.778754i	$-0.715852\pi$
$661$	−32.2572	−1.25466	−0.627329	+	0.778754i	$0.715852\pi$
$662$	2.20316	0.0856282
$663$	0	0
$664$	2.21502	0.0859593
$665$	−8.44084	−0.327322
$666$	0	0
$667$	1.11204	0.0430584
$668$	−26.3689	−1.02024
$669$	0	0
$670$	1.87203	0.0723227
$671$	8.83756	0.341170
$672$	0	0
$673$	17.9429	0.691649	0.345825	−	0.938299i	$-0.387599\pi$
$673$	17.9429	0.691649	0.345825	+	0.938299i	$0.387599\pi$
$674$	0.302683	0.0116589
$675$	0	0
$676$	24.0282	0.924161
$677$	10.3946	0.399496	0.199748	−	0.979847i	$-0.435988\pi$
$677$	10.3946	0.399496	0.199748	+	0.979847i	$0.435988\pi$
$678$	0	0
$679$	8.94935	0.343445
$680$	3.11400	0.119416
$681$	0	0
$682$	0.342570	0.0131177
$683$	−47.2571	−1.80824	−0.904122	−	0.427274i	$-0.859474\pi$
$683$	−47.2571	−1.80824	−0.904122	+	0.427274i	$0.859474\pi$
$684$	0	0
$685$	7.04541	0.269191
$686$	−2.78014	−0.106146
$687$	0	0
$688$	−32.7681	−1.24927
$689$	−4.65035	−0.177164
$690$	0	0
$691$	19.6449	0.747327	0.373663	−	0.927564i	$-0.378102\pi$
$691$	19.6449	0.747327	0.373663	+	0.927564i	$0.378102\pi$
$692$	−5.80066	−0.220508
$693$	0	0
$694$	1.75547	0.0666368
$695$	−20.3944	−0.773604
$696$	0	0
$697$	34.7705	1.31702
$698$	−3.10216	−0.117419
$699$	0	0
$700$	2.89957	0.109593
$701$	9.67582	0.365451	0.182725	−	0.983164i	$-0.441508\pi$
$701$	9.67582	0.365451	0.182725	+	0.983164i	$0.441508\pi$
$702$	0	0
$703$	6.87660	0.259356
$704$	24.5672	0.925910
$705$	0	0
$706$	0.203406	0.00765528
$707$	−12.0349	−0.452619
$708$	0	0
$709$	−43.8556	−1.64703	−0.823515	−	0.567294i	$-0.807990\pi$
$709$	−43.8556	−1.64703	−0.823515	+	0.567294i	$0.807990\pi$
$710$	1.23971	0.0465256
$711$	0	0
$712$	0.635302	0.0238090
$713$	1.47578	0.0552684
$714$	0	0
$715$	−3.02809	−0.113244
$716$	32.7086	1.22238
$717$	0	0
$718$	−4.14456	−0.154674
$719$	40.7203	1.51861	0.759306	−	0.650734i	$-0.225538\pi$
$719$	40.7203	1.51861	0.759306	+	0.650734i	$0.225538\pi$
$720$	0	0
$721$	−11.2857	−0.420302
$722$	−2.24373	−0.0835028
$723$	0	0
$724$	−8.58505	−0.319061
$725$	0.485991	0.0180493
$726$	0	0
$727$	28.8844	1.07126	0.535631	−	0.844452i	$-0.320074\pi$
$727$	28.8844	1.07126	0.535631	+	0.844452i	$0.320074\pi$
$728$	0.850196	0.0315103
$729$	0	0
$730$	0.868584	0.0321477
$731$	41.7473	1.54408
$732$	0	0
$733$	26.4409	0.976615	0.488308	−	0.872672i	$-0.337614\pi$
$733$	26.4409	0.976615	0.488308	+	0.872672i	$0.337614\pi$
$734$	2.44607	0.0902862
$735$	0	0
$736$	−4.31459	−0.159038
$737$	−38.9158	−1.43348
$738$	0	0
$739$	−19.2035	−0.706412	−0.353206	−	0.935546i	$-0.614908\pi$
$739$	−19.2035	−0.706412	−0.353206	+	0.935546i	$0.614908\pi$
$740$	−2.36223	−0.0868371
$741$	0	0
$742$	1.19791	0.0439768
$743$	2.21771	0.0813598	0.0406799	−	0.999172i	$-0.487048\pi$
$743$	2.21771	0.0813598	0.0406799	+	0.999172i	$0.487048\pi$
$744$	0	0
$745$	0.396584	0.0145297
$746$	−2.47016	−0.0904390
$747$	0	0
$748$	−32.1589	−1.17585
$749$	30.0138	1.09668
$750$	0	0
$751$	10.3400	0.377310	0.188655	−	0.982043i	$-0.439587\pi$
$751$	10.3400	0.377310	0.188655	+	0.982043i	$0.439587\pi$
$752$	−37.3552	−1.36221
$753$	0	0
$754$	0.0707919	0.00257809
$755$	−5.72691	−0.208424
$756$	0	0
$757$	−4.25309	−0.154581	−0.0772906	−	0.997009i	$-0.524627\pi$
$757$	−4.25309	−0.154581	−0.0772906	+	0.997009i	$0.524627\pi$
$758$	−5.95852	−0.216423
$759$	0	0
$760$	3.65156	0.132456
$761$	13.5609	0.491582	0.245791	−	0.969323i	$-0.420952\pi$
$761$	13.5609	0.491582	0.245791	+	0.969323i	$0.420952\pi$
$762$	0	0
$763$	−1.76661	−0.0639556
$764$	22.6563	0.819677
$765$	0	0
$766$	2.65684	0.0959954
$767$	5.03068	0.181647
$768$	0	0
$769$	14.2040	0.512208	0.256104	−	0.966649i	$-0.417561\pi$
$769$	14.2040	0.512208	0.256104	+	0.966649i	$0.417561\pi$
$770$	0.780024	0.0281101
$771$	0	0
$772$	−6.86338	−0.247019
$773$	0.0817437	0.00294012	0.00147006	−	0.999999i	$-0.499532\pi$
$773$	0.0817437	0.00294012	0.00147006	+	0.999999i	$0.499532\pi$
$774$	0	0
$775$	0.644955	0.0231675
$776$	−3.87155	−0.138981
$777$	0	0
$778$	3.76405	0.134948
$779$	40.7728	1.46084
$780$	0	0
$781$	−25.7712	−0.922167
$782$	1.79281	0.0641107
$783$	0	0
$784$	−18.6342	−0.665506
$785$	−15.5744	−0.555875
$786$	0	0
$787$	29.7849	1.06172	0.530859	−	0.847460i	$-0.321869\pi$
$787$	29.7849	1.06172	0.530859	+	0.847460i	$0.321869\pi$
$788$	1.71283	0.0610171
$789$	0	0
$790$	2.34336	0.0833731
$791$	10.2156	0.363225
$792$	0	0
$793$	2.42363	0.0860657
$794$	−1.98716	−0.0705216
$795$	0	0
$796$	−0.359537	−0.0127434
$797$	13.6494	0.483485	0.241742	−	0.970340i	$-0.422281\pi$
$797$	13.6494	0.483485	0.241742	+	0.970340i	$0.422281\pi$
$798$	0	0
$799$	47.5913	1.68366
$800$	−1.88559	−0.0666657
$801$	0	0
$802$	0.676859	0.0239007
$803$	−18.0562	−0.637188
$804$	0	0
$805$	3.36031	0.118435
$806$	0.0939472	0.00330915
$807$	0	0
$808$	5.20638	0.183160
$809$	10.4383	0.366992	0.183496	−	0.983021i	$-0.441259\pi$
$809$	10.4383	0.366992	0.183496	+	0.983021i	$0.441259\pi$
$810$	0	0
$811$	19.8424	0.696762	0.348381	−	0.937353i	$-0.386732\pi$
$811$	19.8424	0.696762	0.348381	+	0.937353i	$0.386732\pi$
$812$	1.40916	0.0494520
$813$	0	0
$814$	−0.635471	−0.0222733
$815$	−20.4504	−0.716348
$816$	0	0
$817$	48.9541	1.71269
$818$	−2.22635	−0.0778425
$819$	0	0
$820$	−14.0061	−0.489115
$821$	36.2250	1.26426	0.632131	−	0.774861i	$-0.282180\pi$
$821$	36.2250	1.26426	0.632131	+	0.774861i	$0.282180\pi$
$822$	0	0
$823$	−27.0800	−0.943949	−0.471975	−	0.881612i	$-0.656459\pi$
$823$	−27.0800	−0.943949	−0.471975	+	0.881612i	$0.656459\pi$
$824$	4.88228	0.170082
$825$	0	0
$826$	−1.29588	−0.0450895
$827$	44.8142	1.55834	0.779170	−	0.626812i	$-0.215641\pi$
$827$	44.8142	1.55834	0.779170	+	0.626812i	$0.215641\pi$
$828$	0	0
$829$	−55.7095	−1.93487	−0.967435	−	0.253119i	$-0.918544\pi$
$829$	−55.7095	−1.93487	−0.967435	+	0.253119i	$0.918544\pi$
$830$	0.557314	0.0193447
$831$	0	0
$832$	6.73735	0.233576
$833$	23.7403	0.822553
$834$	0	0
$835$	−13.3551	−0.462172
$836$	−37.7105	−1.30424
$837$	0	0
$838$	−6.28273	−0.217033
$839$	42.8897	1.48072	0.740359	−	0.672212i	$-0.234656\pi$
$839$	42.8897	1.48072	0.740359	+	0.672212i	$0.234656\pi$
$840$	0	0
$841$	−28.7638	−0.991856
$842$	3.66075	0.126158
$843$	0	0
$844$	−3.87838	−0.133499
$845$	12.1696	0.418646
$846$	0	0
$847$	−0.0611843	−0.00210232
$848$	19.6334	0.674214
$849$	0	0
$850$	0.783504	0.0268740
$851$	−2.73759	−0.0938433
$852$	0	0
$853$	16.9153	0.579169	0.289585	−	0.957152i	$-0.406483\pi$
$853$	16.9153	0.579169	0.289585	+	0.957152i	$0.406483\pi$
$854$	−0.624318	−0.0213637
$855$	0	0
$856$	−12.9842	−0.443791
$857$	−17.1888	−0.587159	−0.293579	−	0.955935i	$-0.594847\pi$
$857$	−17.1888	−0.587159	−0.293579	+	0.955935i	$0.594847\pi$
$858$	0	0
$859$	30.4701	1.03963	0.519813	−	0.854280i	$-0.326001\pi$
$859$	30.4701	1.03963	0.519813	+	0.854280i	$0.326001\pi$
$860$	−16.8165	−0.573439
$861$	0	0
$862$	−2.79020	−0.0950347
$863$	−53.1493	−1.80922	−0.904612	−	0.426236i	$-0.859839\pi$
$863$	−53.1493	−1.80922	−0.904612	+	0.426236i	$0.859839\pi$
$864$	0	0
$865$	−2.93786	−0.0998903
$866$	−0.221470	−0.00752587
$867$	0	0
$868$	1.87009	0.0634749
$869$	−48.7140	−1.65251
$870$	0	0
$871$	−10.6723	−0.361619
$872$	0.764248	0.0258807
$873$	0	0
$874$	2.10230	0.0711114
$875$	1.46854	0.0496459
$876$	0	0
$877$	−31.4138	−1.06077	−0.530384	−	0.847757i	$-0.677952\pi$
$877$	−31.4138	−1.06077	−0.530384	+	0.847757i	$0.677952\pi$
$878$	3.50194	0.118185
$879$	0	0
$880$	12.7843	0.430960
$881$	6.80134	0.229143	0.114572	−	0.993415i	$-0.463450\pi$
$881$	6.80134	0.229143	0.114572	+	0.993415i	$0.463450\pi$
$882$	0	0
$883$	14.6729	0.493781	0.246890	−	0.969043i	$-0.420591\pi$
$883$	14.6729	0.493781	0.246890	+	0.969043i	$0.420591\pi$
$884$	−8.81933	−0.296626
$885$	0	0
$886$	2.29357	0.0770540
$887$	17.0418	0.572206	0.286103	−	0.958199i	$-0.407640\pi$
$887$	17.0418	0.572206	0.286103	+	0.958199i	$0.407640\pi$
$888$	0	0
$889$	−9.52144	−0.319339
$890$	0.159847	0.00535807
$891$	0	0
$892$	−17.8861	−0.598871
$893$	55.8070	1.86751
$894$	0	0
$895$	16.5660	0.553739
$896$	−7.27366	−0.242996
$897$	0	0
$898$	−0.266642	−0.00889797
$899$	0.313442	0.0104539
$900$	0	0
$901$	−25.0134	−0.833317
$902$	−3.76785	−0.125456
$903$	0	0
$904$	−4.41934	−0.146985
$905$	−4.34807	−0.144535
$906$	0	0
$907$	−48.2820	−1.60318	−0.801588	−	0.597877i	$-0.796011\pi$
$907$	−48.2820	−1.60318	−0.801588	+	0.597877i	$0.796011\pi$
$908$	16.3154	0.541447
$909$	0	0
$910$	0.213915	0.00709122
$911$	−9.50065	−0.314770	−0.157385	−	0.987537i	$-0.550306\pi$
$911$	−9.50065	−0.314770	−0.157385	+	0.987537i	$0.550306\pi$
$912$	0	0
$913$	−11.5855	−0.383423
$914$	6.62635	0.219180
$915$	0	0
$916$	−41.7341	−1.37893
$917$	12.3763	0.408702
$918$	0	0
$919$	5.66864	0.186991	0.0934956	−	0.995620i	$-0.470196\pi$
$919$	5.66864	0.186991	0.0934956	+	0.995620i	$0.470196\pi$
$920$	−1.45369	−0.0479269
$921$	0	0
$922$	4.62814	0.152420
$923$	−7.06756	−0.232632
$924$	0	0
$925$	−1.19640	−0.0393373
$926$	1.87366	0.0615724
$927$	0	0
$928$	−0.916380	−0.0300817
$929$	3.52744	0.115731	0.0578657	−	0.998324i	$-0.481570\pi$
$929$	3.52744	0.115731	0.0578657	+	0.998324i	$0.481570\pi$
$930$	0	0
$931$	27.8386	0.912372
$932$	35.1681	1.15197
$933$	0	0
$934$	3.54215	0.115903
$935$	−16.2875	−0.532659
$936$	0	0
$937$	31.1079	1.01625	0.508125	−	0.861284i	$-0.330339\pi$
$937$	31.1079	1.01625	0.508125	+	0.861284i	$0.330339\pi$
$938$	2.74915	0.0897630
$939$	0	0
$940$	−19.1706	−0.625276
$941$	53.3986	1.74074	0.870372	−	0.492394i	$-0.163878\pi$
$941$	53.3986	1.74074	0.870372	+	0.492394i	$0.163878\pi$
$942$	0	0
$943$	−16.2317	−0.528578
$944$	−21.2391	−0.691275
$945$	0	0
$946$	−4.52388	−0.147084
$947$	−32.9517	−1.07079	−0.535393	−	0.844603i	$-0.679837\pi$
$947$	−32.9517	−1.07079	−0.535393	+	0.844603i	$0.679837\pi$
$948$	0	0
$949$	−4.95176	−0.160741
$950$	0.918760	0.0298085
$951$	0	0
$952$	4.57304	0.148213
$953$	−27.4754	−0.890014	−0.445007	−	0.895527i	$-0.646799\pi$
$953$	−27.4754	−0.890014	−0.445007	+	0.895527i	$0.646799\pi$
$954$	0	0
$955$	11.4748	0.371314
$956$	18.9605	0.613228
$957$	0	0
$958$	2.58900	0.0836469
$959$	10.3465	0.334106
$960$	0	0
$961$	−30.5840	−0.986582
$962$	−0.174273	−0.00561879
$963$	0	0
$964$	7.31323	0.235543
$965$	−3.47610	−0.111900
$966$	0	0
$967$	6.58557	0.211778	0.105889	−	0.994378i	$-0.466231\pi$
$967$	6.58557	0.211778	0.105889	+	0.994378i	$0.466231\pi$
$968$	0.0264687	0.000850738 0
$969$	0	0
$970$	−0.974110	−0.0312768
$971$	22.7065	0.728686	0.364343	−	0.931265i	$-0.381294\pi$
$971$	22.7065	0.728686	0.364343	+	0.931265i	$0.381294\pi$
$972$	0	0
$973$	−29.9501	−0.960156
$974$	4.97258	0.159332
$975$	0	0
$976$	−10.2324	−0.327530
$977$	−57.7569	−1.84781	−0.923904	−	0.382624i	$-0.875020\pi$
$977$	−57.7569	−1.84781	−0.923904	+	0.382624i	$0.875020\pi$
$978$	0	0
$979$	−3.32290	−0.106200
$980$	−9.56300	−0.305479
$981$	0	0
$982$	1.72027	0.0548962
$983$	3.21600	0.102575	0.0512873	−	0.998684i	$-0.483668\pi$
$983$	3.21600	0.102575	0.0512873	+	0.998684i	$0.483668\pi$
$984$	0	0
$985$	0.867498	0.0276408
$986$	0.380776	0.0121264
$987$	0	0
$988$	−10.3418	−0.329016
$989$	−19.4887	−0.619705
$990$	0	0
$991$	−6.37717	−0.202577	−0.101289	−	0.994857i	$-0.532297\pi$
$991$	−6.37717	−0.202577	−0.101289	+	0.994857i	$0.532297\pi$
$992$	−1.21612	−0.0386118
$993$	0	0
$994$	1.82057	0.0577451
$995$	−0.182095	−0.00577279
$996$	0	0
$997$	−36.9006	−1.16865	−0.584327	−	0.811518i	$-0.698642\pi$
$997$	−36.9006	−1.16865	−0.584327	+	0.811518i	$0.698642\pi$
$998$	2.44865	0.0775107
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4005.2.a.w.1.10	✓	17
3.2	odd	2		4005.2.a.x.1.8	yes	17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4005.2.a.w.1.10	✓	17	1.1	even	1	trivial
4005.2.a.x.1.8	yes	17	3.2	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)

\(f(q)\)	\(=\)	\(q-0.159847 q^{2} -1.97445 q^{4} -1.00000 q^{5} -1.46854 q^{7} +0.635302 q^{8} +O(q^{10})\)	\(q-0.159847 q^{2} -1.97445 q^{4} -1.00000 q^{5} -1.46854 q^{7} +0.635302 q^{8} +0.159847 q^{10} -3.32290 q^{11} -0.911279 q^{13} +0.234742 q^{14} +3.84735 q^{16} -4.90160 q^{17} -5.74776 q^{19} +1.97445 q^{20} +0.531154 q^{22} +2.28819 q^{23} +1.00000 q^{25} +0.145665 q^{26} +2.89957 q^{28} +0.485991 q^{29} +0.644955 q^{31} -1.88559 q^{32} +0.783504 q^{34} +1.46854 q^{35} -1.19640 q^{37} +0.918760 q^{38} -0.635302 q^{40} -7.09369 q^{41} -8.51707 q^{43} +6.56090 q^{44} -0.365760 q^{46} -9.70935 q^{47} -4.84338 q^{49} -0.159847 q^{50} +1.79927 q^{52} +5.10311 q^{53} +3.32290 q^{55} -0.932969 q^{56} -0.0776841 q^{58} -5.52046 q^{59} -2.65959 q^{61} -0.103094 q^{62} -7.39329 q^{64} +0.911279 q^{65} +11.7114 q^{67} +9.67796 q^{68} -0.234742 q^{70} +7.75565 q^{71} +5.43386 q^{73} +0.191240 q^{74} +11.3487 q^{76} +4.87983 q^{77} +14.6601 q^{79} -3.84735 q^{80} +1.13390 q^{82} +3.48656 q^{83} +4.90160 q^{85} +1.36143 q^{86} -2.11105 q^{88} +1.00000 q^{89} +1.33825 q^{91} -4.51792 q^{92} +1.55201 q^{94} +5.74776 q^{95} -6.09403 q^{97} +0.774197 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17 q - 5 q^{2} + 21 q^{4} - 17 q^{5} + 12 q^{7} - 15 q^{8}+O(q^{10}) \) 17 * q - 5 * q^2 + 21 * q^4 - 17 * q^5 + 12 * q^7 - 15 * q^8	\( 17 q - 5 q^{2} + 21 q^{4} - 17 q^{5} + 12 q^{7} - 15 q^{8} + 5 q^{10} - 2 q^{11} + 8 q^{13} - 4 q^{14} + 33 q^{16} - 10 q^{17} + 32 q^{19} - 21 q^{20} + 8 q^{22} - 15 q^{23} + 17 q^{25} + 15 q^{26} + 24 q^{28} - q^{29} + 18 q^{31} - 25 q^{32} + 14 q^{34} - 12 q^{35} + 12 q^{37} - 22 q^{38} + 15 q^{40} + 7 q^{41} + 28 q^{43} + 14 q^{44} + 4 q^{46} - 26 q^{47} + 41 q^{49} - 5 q^{50} + 10 q^{52} - 12 q^{53} + 2 q^{55} - 13 q^{56} + 16 q^{58} + 23 q^{59} + 26 q^{61} - 10 q^{62} + 59 q^{64} - 8 q^{65} + 31 q^{67} + q^{68} + 4 q^{70} + 2 q^{71} + 33 q^{73} + 10 q^{74} + 66 q^{76} - 12 q^{77} + 33 q^{79} - 33 q^{80} + 30 q^{82} - 13 q^{83} + 10 q^{85} + 20 q^{86} + 12 q^{88} + 17 q^{89} + 40 q^{91} - 16 q^{92} + 38 q^{94} - 32 q^{95} + 45 q^{97} - 2 q^{98}+O(q^{100}) \) 17 * q - 5 * q^2 + 21 * q^4 - 17 * q^5 + 12 * q^7 - 15 * q^8 + 5 * q^10 - 2 * q^11 + 8 * q^13 - 4 * q^14 + 33 * q^16 - 10 * q^17 + 32 * q^19 - 21 * q^20 + 8 * q^22 - 15 * q^23 + 17 * q^25 + 15 * q^26 + 24 * q^28 - q^29 + 18 * q^31 - 25 * q^32 + 14 * q^34 - 12 * q^35 + 12 * q^37 - 22 * q^38 + 15 * q^40 + 7 * q^41 + 28 * q^43 + 14 * q^44 + 4 * q^46 - 26 * q^47 + 41 * q^49 - 5 * q^50 + 10 * q^52 - 12 * q^53 + 2 * q^55 - 13 * q^56 + 16 * q^58 + 23 * q^59 + 26 * q^61 - 10 * q^62 + 59 * q^64 - 8 * q^65 + 31 * q^67 + q^68 + 4 * q^70 + 2 * q^71 + 33 * q^73 + 10 * q^74 + 66 * q^76 - 12 * q^77 + 33 * q^79 - 33 * q^80 + 30 * q^82 - 13 * q^83 + 10 * q^85 + 20 * q^86 + 12 * q^88 + 17 * q^89 + 40 * q^91 - 16 * q^92 + 38 * q^94 - 32 * q^95 + 45 * q^97 - 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

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