LMFDB - Embedded newform 4005.2.a.k.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:	$1$
Dimension:	$4$
Coefficient field:	4.4.8069.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 5x^{2} + 5x + 1 $ x^4 - x^3 - 5x^2 + 5x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 445)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.23307$ 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)

$f(q)$	$=$	$q-1.23307 q^{2} -0.479533 q^{4} -1.00000 q^{5} +3.00000 q^{7} +3.05744 q^{8} +O(q^{10})$	\(q-1.23307 q^{2} -0.479533 q^{4} -1.00000 q^{5} +3.00000 q^{7} +3.05744 q^{8} +1.23307 q^{10} -1.82437 q^{11} +2.77005 q^{13} -3.69922 q^{14} -2.81098 q^{16} -0.344838 q^{17} +5.10150 q^{19} +0.479533 q^{20} +2.24958 q^{22} -7.77005 q^{23} +1.00000 q^{25} -3.41567 q^{26} -1.43860 q^{28} -6.60084 q^{29} -7.21326 q^{31} -2.64874 q^{32} +0.425210 q^{34} -3.00000 q^{35} -2.97245 q^{37} -6.29052 q^{38} -3.05744 q^{40} +8.01963 q^{41} -6.27713 q^{43} +0.874846 q^{44} +9.58103 q^{46} -12.4495 q^{47} +2.00000 q^{49} -1.23307 q^{50} -1.32833 q^{52} +12.3101 q^{53} +1.82437 q^{55} +9.17233 q^{56} +8.13931 q^{58} +9.60781 q^{59} +3.37474 q^{61} +8.89448 q^{62} +8.88805 q^{64} -2.77005 q^{65} +5.74712 q^{67} +0.165361 q^{68} +3.69922 q^{70} +0.418969 q^{71} -0.740151 q^{73} +3.66525 q^{74} -2.44634 q^{76} -5.47311 q^{77} -7.84088 q^{79} +2.81098 q^{80} -9.88878 q^{82} -12.4693 q^{83} +0.344838 q^{85} +7.74015 q^{86} -5.57791 q^{88} +1.00000 q^{89} +8.31014 q^{91} +3.72599 q^{92} +15.3511 q^{94} -5.10150 q^{95} +0.438599 q^{97} -2.46614 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{2} + 3 q^{4} - 4 q^{5} + 12 q^{7} + 3 q^{8}+O(q^{10}) $ 4 * q - q^2 + 3 * q^4 - 4 * q^5 + 12 * q^7 + 3 * q^8	$ 4 q - q^{2} + 3 q^{4} - 4 q^{5} + 12 q^{7} + 3 q^{8} + q^{10} - 2 q^{11} - 7 q^{13} - 3 q^{14} - 3 q^{16} - q^{17} - q^{19} - 3 q^{20} - 14 q^{22} - 13 q^{23} + 4 q^{25} - 7 q^{26} + 9 q^{28} - 14 q^{29} - 11 q^{31} - 16 q^{34} - 12 q^{35} - 5 q^{37} - 12 q^{38} - 3 q^{40} - 9 q^{41} - 9 q^{43} - 3 q^{44} + 12 q^{46} - 6 q^{47} + 8 q^{49} - q^{50} - 24 q^{52} - 5 q^{53} + 2 q^{55} + 9 q^{56} + 43 q^{58} + 18 q^{59} - 3 q^{61} - 12 q^{62} - 23 q^{64} + 7 q^{65} + 13 q^{67} - 19 q^{68} + 3 q^{70} + 28 q^{71} - q^{73} - 15 q^{74} + 12 q^{76} - 6 q^{77} - 7 q^{79} + 3 q^{80} - 17 q^{82} - 20 q^{83} + q^{85} + 29 q^{86} - 18 q^{88} + 4 q^{89} - 21 q^{91} + 9 q^{92} + 17 q^{94} + q^{95} - 13 q^{97} - 2 q^{98}+O(q^{100}) $ 4 * q - q^2 + 3 * q^4 - 4 * q^5 + 12 * q^7 + 3 * q^8 + q^10 - 2 * q^11 - 7 * q^13 - 3 * q^14 - 3 * q^16 - q^17 - q^19 - 3 * q^20 - 14 * q^22 - 13 * q^23 + 4 * q^25 - 7 * q^26 + 9 * q^28 - 14 * q^29 - 11 * q^31 - 16 * q^34 - 12 * q^35 - 5 * q^37 - 12 * q^38 - 3 * q^40 - 9 * q^41 - 9 * q^43 - 3 * q^44 + 12 * q^46 - 6 * q^47 + 8 * q^49 - q^50 - 24 * q^52 - 5 * q^53 + 2 * q^55 + 9 * q^56 + 43 * q^58 + 18 * q^59 - 3 * q^61 - 12 * q^62 - 23 * q^64 + 7 * q^65 + 13 * q^67 - 19 * q^68 + 3 * q^70 + 28 * q^71 - q^73 - 15 * q^74 + 12 * q^76 - 6 * q^77 - 7 * q^79 + 3 * q^80 - 17 * q^82 - 20 * q^83 + q^85 + 29 * q^86 - 18 * q^88 + 4 * q^89 - 21 * q^91 + 9 * q^92 + 17 * q^94 + q^95 - 13 * 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 $

$2$	−1.23307	−0.871914	−0.435957	−	0.899968i	$-0.643590\pi$
$2$	−1.23307	−0.871914	−0.435957	+	0.899968i	$0.643590\pi$
$3$	0	0
$3$	0	0
$4$	−0.479533	−0.239766
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	3.00000	1.13389	0.566947	−	0.823754i	$-0.308125\pi$
$7$	3.00000	1.13389	0.566947	+	0.823754i	$0.308125\pi$
$8$	3.05744	1.08097
$9$	0	0
$10$	1.23307	0.389932
$11$	−1.82437	−0.550069	−0.275034	−	0.961434i	$-0.588689\pi$
$11$	−1.82437	−0.550069	−0.275034	+	0.961434i	$0.588689\pi$
$12$	0	0
$13$	2.77005	0.768273	0.384137	−	0.923276i	$-0.374499\pi$
$13$	2.77005	0.768273	0.384137	+	0.923276i	$0.374499\pi$
$14$	−3.69922	−0.988657
$15$	0	0
$16$	−2.81098	−0.702746
$17$	−0.344838	−0.0836355	−0.0418178	−	0.999125i	$-0.513315\pi$
$17$	−0.344838	−0.0836355	−0.0418178	+	0.999125i	$0.513315\pi$
$18$	0	0
$19$	5.10150	1.17036	0.585182	−	0.810902i	$-0.301023\pi$
$19$	5.10150	1.17036	0.585182	+	0.810902i	$0.301023\pi$
$20$	0.479533	0.107227
$21$	0	0
$22$	2.24958	0.479612
$23$	−7.77005	−1.62017	−0.810084	−	0.586314i	$-0.800578\pi$
$23$	−7.77005	−1.62017	−0.810084	+	0.586314i	$0.800578\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−3.41567	−0.669868
$27$	0	0
$28$	−1.43860	−0.271870
$29$	−6.60084	−1.22575	−0.612873	−	0.790182i	$-0.709986\pi$
$29$	−6.60084	−1.22575	−0.612873	+	0.790182i	$0.709986\pi$
$30$	0	0
$31$	−7.21326	−1.29554	−0.647770	−	0.761836i	$-0.724298\pi$
$31$	−7.21326	−1.29554	−0.647770	+	0.761836i	$0.724298\pi$
$32$	−2.64874	−0.468236
$33$	0	0
$34$	0.425210	0.0729229
$35$	−3.00000	−0.507093
$36$	0	0
$37$	−2.97245	−0.488668	−0.244334	−	0.969691i	$-0.578569\pi$
$37$	−2.97245	−0.488668	−0.244334	+	0.969691i	$0.578569\pi$
$38$	−6.29052	−1.02046
$39$	0	0
$40$	−3.05744	−0.483424
$41$	8.01963	1.25246	0.626228	−	0.779640i	$-0.284598\pi$
$41$	8.01963	1.25246	0.626228	+	0.779640i	$0.284598\pi$
$42$	0	0
$43$	−6.27713	−0.957253	−0.478626	−	0.878019i	$-0.658865\pi$
$43$	−6.27713	−0.957253	−0.478626	+	0.878019i	$0.658865\pi$
$44$	0.874846	0.131888
$45$	0	0
$46$	9.58103	1.41265
$47$	−12.4495	−1.81594	−0.907970	−	0.419035i	$-0.862368\pi$
$47$	−12.4495	−1.81594	−0.907970	+	0.419035i	$0.862368\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	−1.23307	−0.174383
$51$	0	0
$52$	−1.32833	−0.184206
$53$	12.3101	1.69093	0.845464	−	0.534032i	$-0.179324\pi$
$53$	12.3101	1.69093	0.845464	+	0.534032i	$0.179324\pi$
$54$	0	0
$55$	1.82437	0.245998
$56$	9.17233	1.22570
$57$	0	0
$58$	8.13931	1.06874
$59$	9.60781	1.25083	0.625415	−	0.780292i	$-0.284930\pi$
$59$	9.60781	1.25083	0.625415	+	0.780292i	$0.284930\pi$
$60$	0	0
$61$	3.37474	0.432091	0.216045	−	0.976383i	$-0.430684\pi$
$61$	3.37474	0.432091	0.216045	+	0.976383i	$0.430684\pi$
$62$	8.89448	1.12960
$63$	0	0
$64$	8.88805	1.11101
$65$	−2.77005	−0.343582
$66$	0	0
$67$	5.74712	0.702122	0.351061	−	0.936353i	$-0.385821\pi$
$67$	5.74712	0.702122	0.351061	+	0.936353i	$0.385821\pi$
$68$	0.165361	0.0200530
$69$	0	0
$70$	3.69922	0.442141
$71$	0.418969	0.0497225	0.0248613	−	0.999691i	$-0.492086\pi$
$71$	0.418969	0.0497225	0.0248613	+	0.999691i	$0.492086\pi$
$72$	0	0
$73$	−0.740151	−0.0866281	−0.0433141	−	0.999062i	$-0.513792\pi$
$73$	−0.740151	−0.0866281	−0.0433141	+	0.999062i	$0.513792\pi$
$74$	3.66525	0.426077
$75$	0	0
$76$	−2.44634	−0.280614
$77$	−5.47311	−0.623719
$78$	0	0
$79$	−7.84088	−0.882168	−0.441084	−	0.897466i	$-0.645406\pi$
$79$	−7.84088	−0.882168	−0.441084	+	0.897466i	$0.645406\pi$
$80$	2.81098	0.314277
$81$	0	0
$82$	−9.88878	−1.09203
$83$	−12.4693	−1.36868	−0.684340	−	0.729163i	$-0.739909\pi$
$83$	−12.4693	−1.36868	−0.684340	+	0.729163i	$0.739909\pi$
$84$	0	0
$85$	0.344838	0.0374029
$86$	7.74015	0.834642
$87$	0	0
$88$	−5.57791	−0.594607
$89$	1.00000	0.106000
$89$	1.00000	0.106000
$90$	0	0
$91$	8.31014	0.871140
$92$	3.72599	0.388462
$93$	0	0
$94$	15.3511	1.58334
$95$	−5.10150	−0.523403
$96$	0	0
$97$	0.438599	0.0445329	0.0222665	−	0.999752i	$-0.492912\pi$
$97$	0.438599	0.0445329	0.0222665	+	0.999752i	$0.492912\pi$
$98$	−2.46614	−0.249118
$99$	0	0
$100$	−0.479533	−0.0479533
$101$	0.128275	0.0127638	0.00638190	−	0.999980i	$-0.497969\pi$
$101$	0.128275	0.0127638	0.00638190	+	0.999980i	$0.497969\pi$
$102$	0	0
$103$	4.59519	0.452777	0.226389	−	0.974037i	$-0.427308\pi$
$103$	4.59519	0.452777	0.226389	+	0.974037i	$0.427308\pi$
$104$	8.46926	0.830480
$105$	0	0
$106$	−15.1793	−1.47434
$107$	−11.4520	−1.10711	−0.553553	−	0.832814i	$-0.686728\pi$
$107$	−11.4520	−1.10711	−0.553553	+	0.832814i	$0.686728\pi$
$108$	0	0
$109$	−10.2189	−0.978795	−0.489397	−	0.872061i	$-0.662783\pi$
$109$	−10.2189	−0.978795	−0.489397	+	0.872061i	$0.662783\pi$
$110$	−2.24958	−0.214489
$111$	0	0
$112$	−8.43295	−0.796839
$113$	−0.786557	−0.0739931	−0.0369965	−	0.999315i	$-0.511779\pi$
$113$	−0.786557	−0.0739931	−0.0369965	+	0.999315i	$0.511779\pi$
$114$	0	0
$115$	7.77005	0.724561
$116$	3.16532	0.293893
$117$	0	0
$118$	−11.8471	−1.09062
$119$	−1.03451	−0.0948338
$120$	0	0
$121$	−7.67167	−0.697425
$122$	−4.16129	−0.376746
$123$	0	0
$124$	3.45900	0.310627
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	10.7644	0.955186	0.477593	−	0.878581i	$-0.341509\pi$
$127$	10.7644	0.955186	0.477593	+	0.878581i	$0.341509\pi$
$128$	−5.66213	−0.500466
$129$	0	0
$130$	3.41567	0.299574
$131$	−13.3747	−1.16856	−0.584278	−	0.811554i	$-0.698622\pi$
$131$	−13.3747	−1.16856	−0.584278	+	0.811554i	$0.698622\pi$
$132$	0	0
$133$	15.3045	1.32707
$134$	−7.08661	−0.612190
$135$	0	0
$136$	−1.05432	−0.0904074
$137$	2.58565	0.220907	0.110453	−	0.993881i	$-0.464770\pi$
$137$	2.58565	0.220907	0.110453	+	0.993881i	$0.464770\pi$
$138$	0	0
$139$	−2.16921	−0.183990	−0.0919949	−	0.995759i	$-0.529324\pi$
$139$	−2.16921	−0.183990	−0.0919949	+	0.995759i	$0.529324\pi$
$140$	1.43860	0.121584
$141$	0	0
$142$	−0.516619	−0.0433537
$143$	−5.05360	−0.422603
$144$	0	0
$145$	6.60084	0.548170
$146$	0.912660	0.0755322
$147$	0	0
$148$	1.42539	0.117166
$149$	10.8242	0.886752	0.443376	−	0.896336i	$-0.353781\pi$
$149$	10.8242	0.886752	0.443376	+	0.896336i	$0.353781\pi$
$150$	0	0
$151$	4.66525	0.379653	0.189826	−	0.981818i	$-0.439207\pi$
$151$	4.66525	0.379653	0.189826	+	0.981818i	$0.439207\pi$
$152$	15.5975	1.26513
$153$	0	0
$154$	6.74874	0.543829
$155$	7.21326	0.579383
$156$	0	0
$157$	17.5842	1.40337	0.701684	−	0.712488i	$-0.252432\pi$
$157$	17.5842	1.40337	0.701684	+	0.712488i	$0.252432\pi$
$158$	9.66837	0.769174
$159$	0	0
$160$	2.64874	0.209401
$161$	−23.3101	−1.83710
$162$	0	0
$163$	−22.6210	−1.77181	−0.885907	−	0.463862i	$-0.846463\pi$
$163$	−22.6210	−1.77181	−0.885907	+	0.463862i	$0.846463\pi$
$164$	−3.84568	−0.300297
$165$	0	0
$166$	15.3755	1.19337
$167$	19.7258	1.52643	0.763215	−	0.646145i	$-0.223620\pi$
$167$	19.7258	1.52643	0.763215	+	0.646145i	$0.223620\pi$
$168$	0	0
$169$	−5.32683	−0.409756
$170$	−0.425210	−0.0326121
$171$	0	0
$172$	3.01009	0.229517
$173$	−17.5203	−1.33204	−0.666021	−	0.745933i	$-0.732004\pi$
$173$	−17.5203	−1.33204	−0.666021	+	0.745933i	$0.732004\pi$
$174$	0	0
$175$	3.00000	0.226779
$176$	5.12827	0.386558
$177$	0	0
$178$	−1.23307	−0.0924227
$179$	2.97322	0.222229	0.111115	−	0.993808i	$-0.464558\pi$
$179$	2.97322	0.222229	0.111115	+	0.993808i	$0.464558\pi$
$180$	0	0
$181$	4.21249	0.313112	0.156556	−	0.987669i	$-0.449961\pi$
$181$	4.21249	0.313112	0.156556	+	0.987669i	$0.449961\pi$
$182$	−10.2470	−0.759559
$183$	0	0
$184$	−23.7565	−1.75135
$185$	2.97245	0.218539
$186$	0	0
$187$	0.629112	0.0460053
$188$	5.96992	0.435401
$189$	0	0
$190$	6.29052	0.456362
$191$	−10.2118	−0.738898	−0.369449	−	0.929251i	$-0.620453\pi$
$191$	−10.2118	−0.738898	−0.369449	+	0.929251i	$0.620453\pi$
$192$	0	0
$193$	−25.7170	−1.85115	−0.925577	−	0.378560i	$-0.876419\pi$
$193$	−25.7170	−1.85115	−0.925577	+	0.378560i	$0.876419\pi$
$194$	−0.540824	−0.0388289
$195$	0	0
$196$	−0.959066	−0.0685047
$197$	4.68253	0.333616	0.166808	−	0.985989i	$-0.446654\pi$
$197$	4.68253	0.333616	0.166808	+	0.985989i	$0.446654\pi$
$198$	0	0
$199$	−12.0714	−0.855720	−0.427860	−	0.903845i	$-0.640732\pi$
$199$	−12.0714	−0.855720	−0.427860	+	0.903845i	$0.640732\pi$
$200$	3.05744	0.216194
$201$	0	0
$202$	−0.158172	−0.0111289
$203$	−19.8025	−1.38986
$204$	0	0
$205$	−8.01963	−0.560115
$206$	−5.66620	−0.394783
$207$	0	0
$208$	−7.78656	−0.539901
$209$	−9.30702	−0.643780
$210$	0	0
$211$	−10.3116	−0.709883	−0.354941	−	0.934889i	$-0.615499\pi$
$211$	−10.3116	−0.709883	−0.354941	+	0.934889i	$0.615499\pi$
$212$	−5.90312	−0.405428
$213$	0	0
$214$	14.1211	0.965300
$215$	6.27713	0.428097
$216$	0	0
$217$	−21.6398	−1.46900
$218$	12.6007	0.853424
$219$	0	0
$220$	−0.874846	−0.0589821
$221$	−0.955218	−0.0642549
$222$	0	0
$223$	−12.2597	−0.820968	−0.410484	−	0.911868i	$-0.634640\pi$
$223$	−12.2597	−0.820968	−0.410484	+	0.911868i	$0.634640\pi$
$224$	−7.94623	−0.530930
$225$	0	0
$226$	0.969882	0.0645156
$227$	7.92569	0.526046	0.263023	−	0.964789i	$-0.415280\pi$
$227$	7.92569	0.526046	0.263023	+	0.964789i	$0.415280\pi$
$228$	0	0
$229$	−5.50143	−0.363545	−0.181772	−	0.983341i	$-0.558183\pi$
$229$	−5.50143	−0.363545	−0.181772	+	0.983341i	$0.558183\pi$
$230$	−9.58103	−0.631754
$231$	0	0
$232$	−20.1817	−1.32499
$233$	21.0778	1.38086	0.690428	−	0.723401i	$-0.257422\pi$
$233$	21.0778	1.38086	0.690428	+	0.723401i	$0.257422\pi$
$234$	0	0
$235$	12.4495	0.812113
$236$	−4.60726	−0.299907
$237$	0	0
$238$	1.27563	0.0826869
$239$	28.7055	1.85680	0.928401	−	0.371580i	$-0.121184\pi$
$239$	28.7055	1.85680	0.928401	+	0.371580i	$0.121184\pi$
$240$	0	0
$241$	−20.6339	−1.32914	−0.664572	−	0.747224i	$-0.731386\pi$
$241$	−20.6339	−1.32914	−0.664572	+	0.747224i	$0.731386\pi$
$242$	9.45972	0.608094
$243$	0	0
$244$	−1.61830	−0.103601
$245$	−2.00000	−0.127775
$246$	0	0
$247$	14.1314	0.899159
$248$	−22.0541	−1.40044
$249$	0	0
$250$	1.23307	0.0779863
$251$	−16.4221	−1.03655	−0.518277	−	0.855213i	$-0.673426\pi$
$251$	−16.4221	−1.03655	−0.518277	+	0.855213i	$0.673426\pi$
$252$	0	0
$253$	14.1754	0.891203
$254$	−13.2733	−0.832840
$255$	0	0
$256$	−10.7943	−0.674643
$257$	18.8981	1.17883	0.589417	−	0.807829i	$-0.299358\pi$
$257$	18.8981	1.17883	0.589417	+	0.807829i	$0.299358\pi$
$258$	0	0
$259$	−8.91736	−0.554098
$260$	1.32833	0.0823795
$261$	0	0
$262$	16.4920	1.01888
$263$	−26.8783	−1.65739	−0.828695	−	0.559701i	$-0.810916\pi$
$263$	−26.8783	−1.65739	−0.828695	+	0.559701i	$0.810916\pi$
$264$	0	0
$265$	−12.3101	−0.756206
$266$	−18.8715	−1.15709
$267$	0	0
$268$	−2.75593	−0.168345
$269$	−7.70618	−0.469854	−0.234927	−	0.972013i	$-0.575485\pi$
$269$	−7.70618	−0.469854	−0.234927	+	0.972013i	$0.575485\pi$
$270$	0	0
$271$	8.65186	0.525563	0.262782	−	0.964855i	$-0.415360\pi$
$271$	8.65186	0.525563	0.262782	+	0.964855i	$0.415360\pi$
$272$	0.969334	0.0587745
$273$	0	0
$274$	−3.18829	−0.192612
$275$	−1.82437	−0.110014
$276$	0	0
$277$	20.0714	1.20597	0.602987	−	0.797751i	$-0.293977\pi$
$277$	20.0714	1.20597	0.602987	+	0.797751i	$0.293977\pi$
$278$	2.67479	0.160423
$279$	0	0
$280$	−9.17233	−0.548152
$281$	−10.6354	−0.634452	−0.317226	−	0.948350i	$-0.602751\pi$
$281$	−10.6354	−0.634452	−0.317226	+	0.948350i	$0.602751\pi$
$282$	0	0
$283$	17.6786	1.05089	0.525443	−	0.850829i	$-0.323900\pi$
$283$	17.6786	1.05089	0.525443	+	0.850829i	$0.323900\pi$
$284$	−0.200910	−0.0119218
$285$	0	0
$286$	6.23145	0.368473
$287$	24.0589	1.42015
$288$	0	0
$289$	−16.8811	−0.993005
$290$	−8.13931	−0.477957
$291$	0	0
$292$	0.354927	0.0207705
$293$	−9.87869	−0.577119	−0.288560	−	0.957462i	$-0.593176\pi$
$293$	−9.87869	−0.577119	−0.288560	+	0.957462i	$0.593176\pi$
$294$	0	0
$295$	−9.60781	−0.559388
$296$	−9.08811	−0.528236
$297$	0	0
$298$	−13.3470	−0.773172
$299$	−21.5234	−1.24473
$300$	0	0
$301$	−18.8314	−1.08542
$302$	−5.75259	−0.331024
$303$	0	0
$304$	−14.3402	−0.822468
$305$	−3.37474	−0.193237
$306$	0	0
$307$	−2.77137	−0.158170	−0.0790851	−	0.996868i	$-0.525200\pi$
$307$	−2.77137	−0.158170	−0.0790851	+	0.996868i	$0.525200\pi$
$308$	2.62454	0.149547
$309$	0	0
$310$	−8.89448	−0.505172
$311$	−32.3823	−1.83623	−0.918117	−	0.396310i	$-0.870291\pi$
$311$	−32.3823	−1.83623	−0.918117	+	0.396310i	$0.870291\pi$
$312$	0	0
$313$	−14.7992	−0.836501	−0.418251	−	0.908332i	$-0.637357\pi$
$313$	−14.7992	−0.836501	−0.418251	+	0.908332i	$0.637357\pi$
$314$	−21.6825	−1.22362
$315$	0	0
$316$	3.75996	0.211514
$317$	−27.1446	−1.52460	−0.762298	−	0.647227i	$-0.775929\pi$
$317$	−27.1446	−1.52460	−0.762298	+	0.647227i	$0.775929\pi$
$318$	0	0
$319$	12.0424	0.674244
$320$	−8.88805	−0.496857
$321$	0	0
$322$	28.7431	1.60179
$323$	−1.75919	−0.0978840
$324$	0	0
$325$	2.77005	0.153655
$326$	27.8933	1.54487
$327$	0	0
$328$	24.5196	1.35387
$329$	−37.3484	−2.05908
$330$	0	0
$331$	−13.0961	−0.719827	−0.359914	−	0.932986i	$-0.617194\pi$
$331$	−13.0961	−0.719827	−0.359914	+	0.932986i	$0.617194\pi$
$332$	5.97942	0.328163
$333$	0	0
$334$	−24.3234	−1.33091
$335$	−5.74712	−0.313999
$336$	0	0
$337$	−22.7405	−1.23876	−0.619378	−	0.785093i	$-0.712615\pi$
$337$	−22.7405	−1.23876	−0.619378	+	0.785093i	$0.712615\pi$
$338$	6.56837	0.357272
$339$	0	0
$340$	−0.165361	−0.00896797
$341$	13.1597	0.712636
$342$	0	0
$343$	−15.0000	−0.809924
$344$	−19.1920	−1.03476
$345$	0	0
$346$	21.6038	1.16143
$347$	−4.43933	−0.238316	−0.119158	−	0.992875i	$-0.538019\pi$
$347$	−4.43933	−0.238316	−0.119158	+	0.992875i	$0.538019\pi$
$348$	0	0
$349$	15.3045	0.819231	0.409615	−	0.912258i	$-0.365663\pi$
$349$	15.3045	0.819231	0.409615	+	0.912258i	$0.365663\pi$
$350$	−3.69922	−0.197731
$351$	0	0
$352$	4.83229	0.257562
$353$	15.4500	0.822321	0.411160	−	0.911563i	$-0.365124\pi$
$353$	15.4500	0.822321	0.411160	+	0.911563i	$0.365124\pi$
$354$	0	0
$355$	−0.418969	−0.0222366
$356$	−0.479533	−0.0254152
$357$	0	0
$358$	−3.66620	−0.193765
$359$	21.8554	1.15348	0.576742	−	0.816926i	$-0.304324\pi$
$359$	21.8554	1.15348	0.576742	+	0.816926i	$0.304324\pi$
$360$	0	0
$361$	7.02528	0.369752
$362$	−5.19431	−0.273007
$363$	0	0
$364$	−3.98499	−0.208870
$365$	0.740151	0.0387413
$366$	0	0
$367$	15.0363	0.784889	0.392445	−	0.919776i	$-0.371629\pi$
$367$	15.0363	0.784889	0.392445	+	0.919776i	$0.371629\pi$
$368$	21.8415	1.13857
$369$	0	0
$370$	−3.66525	−0.190547
$371$	36.9304	1.91733
$372$	0	0
$373$	2.87390	0.148805	0.0744024	−	0.997228i	$-0.476295\pi$
$373$	2.87390	0.148805	0.0744024	+	0.997228i	$0.476295\pi$
$374$	−0.775741	−0.0401126
$375$	0	0
$376$	−38.0635	−1.96298
$377$	−18.2846	−0.941707
$378$	0	0
$379$	−29.1114	−1.49535	−0.747677	−	0.664062i	$-0.768831\pi$
$379$	−29.1114	−1.49535	−0.747677	+	0.664062i	$0.768831\pi$
$380$	2.44634	0.125494
$381$	0	0
$382$	12.5918	0.644255
$383$	11.3454	0.579724	0.289862	−	0.957068i	$-0.406391\pi$
$383$	11.3454	0.579724	0.289862	+	0.957068i	$0.406391\pi$
$384$	0	0
$385$	5.47311	0.278936
$386$	31.7110	1.61405
$387$	0	0
$388$	−0.210322	−0.0106775
$389$	−2.55421	−0.129504	−0.0647518	−	0.997901i	$-0.520626\pi$
$389$	−2.55421	−0.129504	−0.0647518	+	0.997901i	$0.520626\pi$
$390$	0	0
$391$	2.67941	0.135503
$392$	6.11489	0.308848
$393$	0	0
$394$	−5.77390	−0.290885
$395$	7.84088	0.394517
$396$	0	0
$397$	11.7660	0.590520	0.295260	−	0.955417i	$-0.404594\pi$
$397$	11.7660	0.590520	0.295260	+	0.955417i	$0.404594\pi$
$398$	14.8849	0.746114
$399$	0	0
$400$	−2.81098	−0.140549
$401$	−1.69687	−0.0847374	−0.0423687	−	0.999102i	$-0.513490\pi$
$401$	−1.69687	−0.0847374	−0.0423687	+	0.999102i	$0.513490\pi$
$402$	0	0
$403$	−19.9811	−0.995329
$404$	−0.0615119	−0.00306033
$405$	0	0
$406$	24.4179	1.21184
$407$	5.42286	0.268801
$408$	0	0
$409$	17.0501	0.843074	0.421537	−	0.906811i	$-0.361491\pi$
$409$	17.0501	0.843074	0.421537	+	0.906811i	$0.361491\pi$
$410$	9.88878	0.488372
$411$	0	0
$412$	−2.20354	−0.108561
$413$	28.8234	1.41831
$414$	0	0
$415$	12.4693	0.612092
$416$	−7.33714	−0.359733
$417$	0	0
$418$	11.4762	0.561321
$419$	−13.3488	−0.652132	−0.326066	−	0.945347i	$-0.605723\pi$
$419$	−13.3488	−0.652132	−0.326066	+	0.945347i	$0.605723\pi$
$420$	0	0
$421$	−28.4706	−1.38757	−0.693786	−	0.720182i	$-0.744058\pi$
$421$	−28.4706	−1.38757	−0.693786	+	0.720182i	$0.744058\pi$
$422$	12.7150	0.618956
$423$	0	0
$424$	37.6376	1.82784
$425$	−0.344838	−0.0167271
$426$	0	0
$427$	10.1242	0.489945
$428$	5.49160	0.265447
$429$	0	0
$430$	−7.74015	−0.373263
$431$	10.4927	0.505413	0.252707	−	0.967543i	$-0.418679\pi$
$431$	10.4927	0.505413	0.252707	+	0.967543i	$0.418679\pi$
$432$	0	0
$433$	−15.0057	−0.721128	−0.360564	−	0.932735i	$-0.617416\pi$
$433$	−15.0057	−0.721128	−0.360564	+	0.932735i	$0.617416\pi$
$434$	26.6834	1.28085
$435$	0	0
$436$	4.90031	0.234682
$437$	−39.6389	−1.89618
$438$	0	0
$439$	−3.44561	−0.164450	−0.0822250	−	0.996614i	$-0.526203\pi$
$439$	−3.44561	−0.164450	−0.0822250	+	0.996614i	$0.526203\pi$
$440$	5.57791	0.265916
$441$	0	0
$442$	1.17785	0.0560247
$443$	17.4780	0.830407	0.415203	−	0.909729i	$-0.363710\pi$
$443$	17.4780	0.830407	0.415203	+	0.909729i	$0.363710\pi$
$444$	0	0
$445$	−1.00000	−0.0474045
$446$	15.1171	0.715813
$447$	0	0
$448$	26.6642	1.25976
$449$	−34.9734	−1.65050	−0.825249	−	0.564770i	$-0.808965\pi$
$449$	−34.9734	−1.65050	−0.825249	+	0.564770i	$0.808965\pi$
$450$	0	0
$451$	−14.6308	−0.688936
$452$	0.377180	0.0177411
$453$	0	0
$454$	−9.77295	−0.458667
$455$	−8.31014	−0.389586
$456$	0	0
$457$	−17.0582	−0.797950	−0.398975	−	0.916962i	$-0.630634\pi$
$457$	−17.0582	−0.797950	−0.398975	+	0.916962i	$0.630634\pi$
$458$	6.78366	0.316980
$459$	0	0
$460$	−3.72599	−0.173725
$461$	−19.5989	−0.912810	−0.456405	−	0.889772i	$-0.650863\pi$
$461$	−19.5989	−0.912810	−0.456405	+	0.889772i	$0.650863\pi$
$462$	0	0
$463$	−27.0973	−1.25932	−0.629659	−	0.776872i	$-0.716805\pi$
$463$	−27.0973	−1.25932	−0.629659	+	0.776872i	$0.716805\pi$
$464$	18.5548	0.861387
$465$	0	0
$466$	−25.9905	−1.20399
$467$	11.2488	0.520533	0.260266	−	0.965537i	$-0.416190\pi$
$467$	11.2488	0.520533	0.260266	+	0.965537i	$0.416190\pi$
$468$	0	0
$469$	17.2414	0.796132
$470$	−15.3511	−0.708092
$471$	0	0
$472$	29.3753	1.35211
$473$	11.4518	0.526555
$474$	0	0
$475$	5.10150	0.234073
$476$	0.496084	0.0227380
$477$	0	0
$478$	−35.3959	−1.61897
$479$	31.2421	1.42749	0.713743	−	0.700407i	$-0.246998\pi$
$479$	31.2421	1.42749	0.713743	+	0.700407i	$0.246998\pi$
$480$	0	0
$481$	−8.23384	−0.375431
$482$	25.4430	1.15890
$483$	0	0
$484$	3.67882	0.167219
$485$	−0.438599	−0.0199157
$486$	0	0
$487$	0.226062	0.0102439	0.00512193	−	0.999987i	$-0.498370\pi$
$487$	0.226062	0.0102439	0.00512193	+	0.999987i	$0.498370\pi$
$488$	10.3181	0.467077
$489$	0	0
$490$	2.46614	0.111409
$491$	−13.4993	−0.609217	−0.304608	−	0.952478i	$-0.598526\pi$
$491$	−13.4993	−0.609217	−0.304608	+	0.952478i	$0.598526\pi$
$492$	0	0
$493$	2.27622	0.102516
$494$	−17.4250	−0.783989
$495$	0	0
$496$	20.2764	0.910435
$497$	1.25691	0.0563800
$498$	0	0
$499$	16.3926	0.733832	0.366916	−	0.930254i	$-0.380414\pi$
$499$	16.3926	0.733832	0.366916	+	0.930254i	$0.380414\pi$
$500$	0.479533	0.0214454
$501$	0	0
$502$	20.2496	0.903785
$503$	5.63227	0.251131	0.125565	−	0.992085i	$-0.459926\pi$
$503$	5.63227	0.251131	0.125565	+	0.992085i	$0.459926\pi$
$504$	0	0
$505$	−0.128275	−0.00570815
$506$	−17.4794	−0.777052
$507$	0	0
$508$	−5.16188	−0.229021
$509$	−27.1689	−1.20424	−0.602120	−	0.798406i	$-0.705677\pi$
$509$	−27.1689	−1.20424	−0.602120	+	0.798406i	$0.705677\pi$
$510$	0	0
$511$	−2.22045	−0.0982271
$512$	24.6344	1.08870
$513$	0	0
$514$	−23.3028	−1.02784
$515$	−4.59519	−0.202488
$516$	0	0
$517$	22.7124	0.998891
$518$	10.9958	0.483126
$519$	0	0
$520$	−8.46926	−0.371402
$521$	3.37324	0.147784	0.0738921	−	0.997266i	$-0.476458\pi$
$521$	3.37324	0.147784	0.0738921	+	0.997266i	$0.476458\pi$
$522$	0	0
$523$	2.60799	0.114039	0.0570196	−	0.998373i	$-0.481840\pi$
$523$	2.60799	0.114039	0.0570196	+	0.998373i	$0.481840\pi$
$524$	6.41363	0.280181
$525$	0	0
$526$	33.1429	1.44510
$527$	2.48741	0.108353
$528$	0	0
$529$	37.3736	1.62494
$530$	15.1793	0.659347
$531$	0	0
$532$	−7.33901	−0.318186
$533$	22.2148	0.962228
$534$	0	0
$535$	11.4520	0.495113
$536$	17.5715	0.758973
$537$	0	0
$538$	9.50228	0.409672
$539$	−3.64874	−0.157162
$540$	0	0
$541$	−11.2599	−0.484101	−0.242050	−	0.970264i	$-0.577820\pi$
$541$	−11.2599	−0.484101	−0.242050	+	0.970264i	$0.577820\pi$
$542$	−10.6684	−0.458246
$543$	0	0
$544$	0.913387	0.0391611
$545$	10.2189	0.437730
$546$	0	0
$547$	−43.1092	−1.84322	−0.921608	−	0.388122i	$-0.873124\pi$
$547$	−43.1092	−1.84322	−0.921608	+	0.388122i	$0.873124\pi$
$548$	−1.23990	−0.0529660
$549$	0	0
$550$	2.24958	0.0959225
$551$	−33.6742	−1.43457
$552$	0	0
$553$	−23.5226	−1.00028
$554$	−24.7495	−1.05151
$555$	0	0
$556$	1.04021	0.0441146
$557$	22.9424	0.972102	0.486051	−	0.873931i	$-0.338437\pi$
$557$	22.9424	0.972102	0.486051	+	0.873931i	$0.338437\pi$
$558$	0	0
$559$	−17.3879	−0.735432
$560$	8.43295	0.356357
$561$	0	0
$562$	13.1142	0.553187
$563$	−16.5064	−0.695660	−0.347830	−	0.937558i	$-0.613081\pi$
$563$	−16.5064	−0.695660	−0.347830	+	0.937558i	$0.613081\pi$
$564$	0	0
$565$	0.786557	0.0330907
$566$	−21.7990	−0.916282
$567$	0	0
$568$	1.28097	0.0537485
$569$	20.0770	0.841673	0.420836	−	0.907137i	$-0.361737\pi$
$569$	20.0770	0.841673	0.420836	+	0.907137i	$0.361737\pi$
$570$	0	0
$571$	7.23674	0.302848	0.151424	−	0.988469i	$-0.451614\pi$
$571$	7.23674	0.302848	0.151424	+	0.988469i	$0.451614\pi$
$572$	2.42337	0.101326
$573$	0	0
$574$	−29.6663	−1.23825
$575$	−7.77005	−0.324033
$576$	0	0
$577$	−5.29599	−0.220475	−0.110237	−	0.993905i	$-0.535161\pi$
$577$	−5.29599	−0.220475	−0.110237	+	0.993905i	$0.535161\pi$
$578$	20.8156	0.865815
$579$	0	0
$580$	−3.16532	−0.131433
$581$	−37.4078	−1.55194
$582$	0	0
$583$	−22.4583	−0.930126
$584$	−2.26297	−0.0936423
$585$	0	0
$586$	12.1811	0.503198
$587$	48.0955	1.98511	0.992557	−	0.121784i	$-0.0388615\pi$
$587$	48.0955	1.98511	0.992557	+	0.121784i	$0.0388615\pi$
$588$	0	0
$589$	−36.7984	−1.51625
$590$	11.8471	0.487738
$591$	0	0
$592$	8.35552	0.343410
$593$	0.320592	0.0131651	0.00658257	−	0.999978i	$-0.497905\pi$
$593$	0.320592	0.0131651	0.00658257	+	0.999978i	$0.497905\pi$
$594$	0	0
$595$	1.03451	0.0424109
$596$	−5.19056	−0.212613
$597$	0	0
$598$	26.5399	1.08530
$599$	−30.4979	−1.24611	−0.623056	−	0.782177i	$-0.714109\pi$
$599$	−30.4979	−1.24611	−0.623056	+	0.782177i	$0.714109\pi$
$600$	0	0
$601$	15.5748	0.635309	0.317654	−	0.948207i	$-0.397105\pi$
$601$	15.5748	0.635309	0.317654	+	0.948207i	$0.397105\pi$
$602$	23.2205	0.946395
$603$	0	0
$604$	−2.23714	−0.0910280
$605$	7.67167	0.311898
$606$	0	0
$607$	16.2604	0.659991	0.329995	−	0.943983i	$-0.392953\pi$
$607$	16.2604	0.659991	0.329995	+	0.943983i	$0.392953\pi$
$608$	−13.5126	−0.548006
$609$	0	0
$610$	4.16129	0.168486
$611$	−34.4856	−1.39514
$612$	0	0
$613$	34.3587	1.38773	0.693867	−	0.720103i	$-0.255906\pi$
$613$	34.3587	1.38773	0.693867	+	0.720103i	$0.255906\pi$
$614$	3.41729	0.137911
$615$	0	0
$616$	−16.7337	−0.674221
$617$	−36.8675	−1.48423	−0.742115	−	0.670272i	$-0.766177\pi$
$617$	−36.8675	−1.48423	−0.742115	+	0.670272i	$0.766177\pi$
$618$	0	0
$619$	43.1140	1.73290	0.866449	−	0.499266i	$-0.166397\pi$
$619$	43.1140	1.73290	0.866449	+	0.499266i	$0.166397\pi$
$620$	−3.45900	−0.138917
$621$	0	0
$622$	39.9298	1.60104
$623$	3.00000	0.120192
$624$	0	0
$625$	1.00000	0.0400000
$626$	18.2485	0.729357
$627$	0	0
$628$	−8.43218	−0.336481
$629$	1.02502	0.0408700
$630$	0	0
$631$	−41.4889	−1.65165	−0.825823	−	0.563929i	$-0.809289\pi$
$631$	−41.4889	−1.65165	−0.825823	+	0.563929i	$0.809289\pi$
$632$	−23.9730	−0.953596
$633$	0	0
$634$	33.4713	1.32932
$635$	−10.7644	−0.427172
$636$	0	0
$637$	5.54010	0.219507
$638$	−14.8491	−0.587882
$639$	0	0
$640$	5.66213	0.223815
$641$	−36.6519	−1.44766	−0.723832	−	0.689977i	$-0.757621\pi$
$641$	−36.6519	−1.44766	−0.723832	+	0.689977i	$0.757621\pi$
$642$	0	0
$643$	−31.0210	−1.22335	−0.611674	−	0.791110i	$-0.709504\pi$
$643$	−31.0210	−1.22335	−0.611674	+	0.791110i	$0.709504\pi$
$644$	11.1780	0.440474
$645$	0	0
$646$	2.16921	0.0853464
$647$	43.6164	1.71474	0.857370	−	0.514701i	$-0.172097\pi$
$647$	43.6164	1.71474	0.857370	+	0.514701i	$0.172097\pi$
$648$	0	0
$649$	−17.5282	−0.688042
$650$	−3.41567	−0.133974
$651$	0	0
$652$	10.8475	0.424822
$653$	−10.6109	−0.415238	−0.207619	−	0.978210i	$-0.566571\pi$
$653$	−10.6109	−0.415238	−0.207619	+	0.978210i	$0.566571\pi$
$654$	0	0
$655$	13.3747	0.522594
$656$	−22.5430	−0.880158
$657$	0	0
$658$	46.0532	1.79534
$659$	−29.2720	−1.14028	−0.570138	−	0.821549i	$-0.693110\pi$
$659$	−29.2720	−1.14028	−0.570138	+	0.821549i	$0.693110\pi$
$660$	0	0
$661$	22.2832	0.866716	0.433358	−	0.901222i	$-0.357329\pi$
$661$	22.2832	0.866716	0.433358	+	0.901222i	$0.357329\pi$
$662$	16.1485	0.627627
$663$	0	0
$664$	−38.1241	−1.47950
$665$	−15.3045	−0.593483
$666$	0	0
$667$	51.2888	1.98591
$668$	−9.45918	−0.365987
$669$	0	0
$670$	7.08661	0.273780
$671$	−6.15677	−0.237679
$672$	0	0
$673$	25.7990	0.994478	0.497239	−	0.867614i	$-0.334347\pi$
$673$	25.7990	0.994478	0.497239	+	0.867614i	$0.334347\pi$
$674$	28.0407	1.08009
$675$	0	0
$676$	2.55439	0.0982458
$677$	48.9702	1.88208	0.941038	−	0.338301i	$-0.109852\pi$
$677$	48.9702	1.88208	0.941038	+	0.338301i	$0.109852\pi$
$678$	0	0
$679$	1.31580	0.0504956
$680$	1.05432	0.0404314
$681$	0	0
$682$	−16.2268	−0.621357
$683$	−27.2541	−1.04285	−0.521424	−	0.853298i	$-0.674599\pi$
$683$	−27.2541	−1.04285	−0.521424	+	0.853298i	$0.674599\pi$
$684$	0	0
$685$	−2.58565	−0.0987925
$686$	18.4961	0.706184
$687$	0	0
$688$	17.6449	0.672705
$689$	34.0997	1.29909
$690$	0	0
$691$	−24.2323	−0.921840	−0.460920	−	0.887442i	$-0.652481\pi$
$691$	−24.2323	−0.921840	−0.460920	+	0.887442i	$0.652481\pi$
$692$	8.40155	0.319379
$693$	0	0
$694$	5.47401	0.207791
$695$	2.16921	0.0822828
$696$	0	0
$697$	−2.76547	−0.104750
$698$	−18.8715	−0.714299
$699$	0	0
$700$	−1.43860	−0.0543739
$701$	−20.2682	−0.765518	−0.382759	−	0.923848i	$-0.625026\pi$
$701$	−20.2682	−0.765518	−0.382759	+	0.923848i	$0.625026\pi$
$702$	0	0
$703$	−15.1640	−0.571920
$704$	−16.2151	−0.611130
$705$	0	0
$706$	−19.0510	−0.716993
$707$	0.384824	0.0144728
$708$	0	0
$709$	−28.6983	−1.07779	−0.538893	−	0.842374i	$-0.681157\pi$
$709$	−28.6983	−1.07779	−0.538893	+	0.842374i	$0.681157\pi$
$710$	0.516619	0.0193884
$711$	0	0
$712$	3.05744	0.114583
$713$	56.0474	2.09899
$714$	0	0
$715$	5.05360	0.188994
$716$	−1.42576	−0.0532831
$717$	0	0
$718$	−26.9493	−1.00574
$719$	0.855037	0.0318875	0.0159438	−	0.999873i	$-0.494925\pi$
$719$	0.855037	0.0318875	0.0159438	+	0.999873i	$0.494925\pi$
$720$	0	0
$721$	13.7856	0.513401
$722$	−8.66268	−0.322392
$723$	0	0
$724$	−2.02003	−0.0750738
$725$	−6.60084	−0.245149
$726$	0	0
$727$	37.7206	1.39898	0.699491	−	0.714642i	$-0.253410\pi$
$727$	37.7206	1.39898	0.699491	+	0.714642i	$0.253410\pi$
$728$	25.4078	0.941676
$729$	0	0
$730$	−0.912660	−0.0337790
$731$	2.16459	0.0800603
$732$	0	0
$733$	−43.4297	−1.60411	−0.802057	−	0.597248i	$-0.796261\pi$
$733$	−43.4297	−1.60411	−0.802057	+	0.597248i	$0.796261\pi$
$734$	−18.5409	−0.684356
$735$	0	0
$736$	20.5809	0.758620
$737$	−10.4849	−0.386215
$738$	0	0
$739$	−45.4498	−1.67190	−0.835949	−	0.548806i	$-0.815082\pi$
$739$	−45.4498	−1.67190	−0.835949	+	0.548806i	$0.815082\pi$
$740$	−1.42539	−0.0523984
$741$	0	0
$742$	−45.5379	−1.67175
$743$	−40.9622	−1.50276	−0.751378	−	0.659872i	$-0.770611\pi$
$743$	−40.9622	−1.50276	−0.751378	+	0.659872i	$0.770611\pi$
$744$	0	0
$745$	−10.8242	−0.396568
$746$	−3.54372	−0.129745
$747$	0	0
$748$	−0.301680	−0.0110305
$749$	−34.3560	−1.25534
$750$	0	0
$751$	−5.48826	−0.200270	−0.100135	−	0.994974i	$-0.531927\pi$
$751$	−5.48826	−0.200270	−0.100135	+	0.994974i	$0.531927\pi$
$752$	34.9952	1.27614
$753$	0	0
$754$	22.5463	0.821087
$755$	−4.66525	−0.169786
$756$	0	0
$757$	7.84264	0.285046	0.142523	−	0.989792i	$-0.454479\pi$
$757$	7.84264	0.285046	0.142523	+	0.989792i	$0.454479\pi$
$758$	35.8965	1.30382
$759$	0	0
$760$	−15.5975	−0.565782
$761$	33.3805	1.21004	0.605022	−	0.796209i	$-0.293164\pi$
$761$	33.3805	1.21004	0.605022	+	0.796209i	$0.293164\pi$
$762$	0	0
$763$	−30.6567	−1.10985
$764$	4.89688	0.177163
$765$	0	0
$766$	−13.9897	−0.505470
$767$	26.6141	0.960979
$768$	0	0
$769$	26.0883	0.940768	0.470384	−	0.882462i	$-0.344115\pi$
$769$	26.0883	0.940768	0.470384	+	0.882462i	$0.344115\pi$
$770$	−6.74874	−0.243208
$771$	0	0
$772$	12.3322	0.443844
$773$	−11.1366	−0.400554	−0.200277	−	0.979739i	$-0.564184\pi$
$773$	−11.1366	−0.400554	−0.200277	+	0.979739i	$0.564184\pi$
$774$	0	0
$775$	−7.21326	−0.259108
$776$	1.34099	0.0481387
$777$	0	0
$778$	3.14953	0.112916
$779$	40.9121	1.46583
$780$	0	0
$781$	−0.764355	−0.0273508
$782$	−3.30390	−0.118147
$783$	0	0
$784$	−5.62196	−0.200784
$785$	−17.5842	−0.627605
$786$	0	0
$787$	12.0097	0.428100	0.214050	−	0.976823i	$-0.431334\pi$
$787$	12.0097	0.428100	0.214050	+	0.976823i	$0.431334\pi$
$788$	−2.24543	−0.0799900
$789$	0	0
$790$	−9.66837	−0.343985
$791$	−2.35967	−0.0839003
$792$	0	0
$793$	9.34818	0.331964
$794$	−14.5084	−0.514882
$795$	0	0
$796$	5.78864	0.205173
$797$	42.3769	1.50107	0.750534	−	0.660832i	$-0.229796\pi$
$797$	42.3769	1.50107	0.750534	+	0.660832i	$0.229796\pi$
$798$	0	0
$799$	4.29305	0.151877
$800$	−2.64874	−0.0936472
$801$	0	0
$802$	2.09236	0.0738837
$803$	1.35031	0.0476514
$804$	0	0
$805$	23.3101	0.821575
$806$	24.6381	0.867841
$807$	0	0
$808$	0.392192	0.0137973
$809$	36.2788	1.27549	0.637746	−	0.770246i	$-0.279867\pi$
$809$	36.2788	1.27549	0.637746	+	0.770246i	$0.279867\pi$
$810$	0	0
$811$	12.7454	0.447550	0.223775	−	0.974641i	$-0.428162\pi$
$811$	12.7454	0.447550	0.223775	+	0.974641i	$0.428162\pi$
$812$	9.49596	0.333243
$813$	0	0
$814$	−6.68678	−0.234371
$815$	22.6210	0.792380
$816$	0	0
$817$	−32.0227	−1.12033
$818$	−21.0240	−0.735088
$819$	0	0
$820$	3.84568	0.134297
$821$	25.6666	0.895771	0.447885	−	0.894091i	$-0.352177\pi$
$821$	25.6666	0.895771	0.447885	+	0.894091i	$0.352177\pi$
$822$	0	0
$823$	33.7871	1.17775	0.588873	−	0.808226i	$-0.299572\pi$
$823$	33.7871	1.17775	0.588873	+	0.808226i	$0.299572\pi$
$824$	14.0495	0.489438
$825$	0	0
$826$	−35.5414	−1.23664
$827$	−3.49685	−0.121597	−0.0607987	−	0.998150i	$-0.519365\pi$
$827$	−3.49685	−0.121597	−0.0607987	+	0.998150i	$0.519365\pi$
$828$	0	0
$829$	11.4287	0.396935	0.198467	−	0.980107i	$-0.436404\pi$
$829$	11.4287	0.396935	0.198467	+	0.980107i	$0.436404\pi$
$830$	−15.3755	−0.533692
$831$	0	0
$832$	24.6203	0.853557
$833$	−0.689676	−0.0238959
$834$	0	0
$835$	−19.7258	−0.682640
$836$	4.46302	0.154357
$837$	0	0
$838$	16.4601	0.568603
$839$	−51.1138	−1.76464	−0.882321	−	0.470647i	$-0.844020\pi$
$839$	−51.1138	−1.76464	−0.882321	+	0.470647i	$0.844020\pi$
$840$	0	0
$841$	14.5711	0.502451
$842$	35.1063	1.20984
$843$	0	0
$844$	4.94477	0.170206
$845$	5.32683	0.183249
$846$	0	0
$847$	−23.0150	−0.790805
$848$	−34.6036	−1.18829
$849$	0	0
$850$	0.425210	0.0145846
$851$	23.0961	0.791725
$852$	0	0
$853$	32.5013	1.11282	0.556412	−	0.830906i	$-0.312178\pi$
$853$	32.5013	1.11282	0.556412	+	0.830906i	$0.312178\pi$
$854$	−12.4839	−0.427189
$855$	0	0
$856$	−35.0138	−1.19675
$857$	5.69940	0.194688	0.0973438	−	0.995251i	$-0.468965\pi$
$857$	5.69940	0.194688	0.0973438	+	0.995251i	$0.468965\pi$
$858$	0	0
$859$	12.1960	0.416123	0.208062	−	0.978116i	$-0.433285\pi$
$859$	12.1960	0.416123	0.208062	+	0.978116i	$0.433285\pi$
$860$	−3.01009	−0.102643
$861$	0	0
$862$	−12.9382	−0.440677
$863$	−55.8609	−1.90153	−0.950764	−	0.309916i	$-0.899699\pi$
$863$	−55.8609	−1.90153	−0.950764	+	0.309916i	$0.899699\pi$
$864$	0	0
$865$	17.5203	0.595708
$866$	18.5031	0.628761
$867$	0	0
$868$	10.3770	0.352218
$869$	14.3047	0.485253
$870$	0	0
$871$	15.9198	0.539422
$872$	−31.2438	−1.05805
$873$	0	0
$874$	48.8776	1.65331
$875$	−3.00000	−0.101419
$876$	0	0
$877$	13.7671	0.464882	0.232441	−	0.972610i	$-0.425329\pi$
$877$	13.7671	0.464882	0.232441	+	0.972610i	$0.425329\pi$
$878$	4.24868	0.143386
$879$	0	0
$880$	−5.12827	−0.172874
$881$	3.54056	0.119284	0.0596422	−	0.998220i	$-0.481004\pi$
$881$	3.54056	0.119284	0.0596422	+	0.998220i	$0.481004\pi$
$882$	0	0
$883$	30.7722	1.03557	0.517783	−	0.855512i	$-0.326757\pi$
$883$	30.7722	1.03557	0.517783	+	0.855512i	$0.326757\pi$
$884$	0.458058	0.0154062
$885$	0	0
$886$	−21.5517	−0.724043
$887$	0.123837	0.00415804	0.00207902	−	0.999998i	$-0.499338\pi$
$887$	0.123837	0.00415804	0.00207902	+	0.999998i	$0.499338\pi$
$888$	0	0
$889$	32.2932	1.08308
$890$	1.23307	0.0413327
$891$	0	0
$892$	5.87891	0.196841
$893$	−63.5109	−2.12531
$894$	0	0
$895$	−2.97322	−0.0993839
$896$	−16.9864	−0.567475
$897$	0	0
$898$	43.1247	1.43909
$899$	47.6136	1.58800
$900$	0	0
$901$	−4.24501	−0.141422
$902$	18.0408	0.600693
$903$	0	0
$904$	−2.40485	−0.0799842
$905$	−4.21249	−0.140028
$906$	0	0
$907$	−36.2453	−1.20351	−0.601753	−	0.798682i	$-0.705531\pi$
$907$	−36.2453	−1.20351	−0.601753	+	0.798682i	$0.705531\pi$
$908$	−3.80063	−0.126128
$909$	0	0
$910$	10.2470	0.339685
$911$	43.6730	1.44695	0.723476	−	0.690350i	$-0.242543\pi$
$911$	43.6730	1.44695	0.723476	+	0.690350i	$0.242543\pi$
$912$	0	0
$913$	22.7486	0.752868
$914$	21.0340	0.695743
$915$	0	0
$916$	2.63812	0.0871658
$917$	−40.1242	−1.32502
$918$	0	0
$919$	24.9232	0.822141	0.411070	−	0.911604i	$-0.365155\pi$
$919$	24.9232	0.822141	0.411070	+	0.911604i	$0.365155\pi$
$920$	23.7565	0.783228
$921$	0	0
$922$	24.1668	0.795891
$923$	1.16057	0.0382005
$924$	0	0
$925$	−2.97245	−0.0977337
$926$	33.4129	1.09802
$927$	0	0
$928$	17.4839	0.573938
$929$	−26.5469	−0.870975	−0.435488	−	0.900195i	$-0.643424\pi$
$929$	−26.5469	−0.870975	−0.435488	+	0.900195i	$0.643424\pi$
$930$	0	0
$931$	10.2030	0.334390
$932$	−10.1075	−0.331083
$933$	0	0
$934$	−13.8706	−0.453860
$935$	−0.629112	−0.0205742
$936$	0	0
$937$	37.0313	1.20976	0.604879	−	0.796317i	$-0.293221\pi$
$937$	37.0313	1.20976	0.604879	+	0.796317i	$0.293221\pi$
$938$	−21.2598	−0.694158
$939$	0	0
$940$	−5.96992	−0.194717
$941$	−10.5322	−0.343339	−0.171670	−	0.985155i	$-0.554916\pi$
$941$	−10.5322	−0.343339	−0.171670	+	0.985155i	$0.554916\pi$
$942$	0	0
$943$	−62.3129	−2.02919
$944$	−27.0074	−0.879015
$945$	0	0
$946$	−14.1209	−0.459110
$947$	−55.3206	−1.79768	−0.898839	−	0.438279i	$-0.855588\pi$
$947$	−55.3206	−1.79768	−0.898839	+	0.438279i	$0.855588\pi$
$948$	0	0
$949$	−2.05025	−0.0665541
$950$	−6.29052	−0.204091
$951$	0	0
$952$	−3.16297	−0.102512
$953$	43.8702	1.42109	0.710547	−	0.703650i	$-0.248448\pi$
$953$	43.8702	1.42109	0.710547	+	0.703650i	$0.248448\pi$
$954$	0	0
$955$	10.2118	0.330445
$956$	−13.7652	−0.445199
$957$	0	0
$958$	−38.5237	−1.24465
$959$	7.75694	0.250485
$960$	0	0
$961$	21.0312	0.678425
$962$	10.1529	0.327343
$963$	0	0
$964$	9.89461	0.318684
$965$	25.7170	0.827861
$966$	0	0
$967$	−39.3658	−1.26592	−0.632960	−	0.774185i	$-0.718160\pi$
$967$	−39.3658	−1.26592	−0.632960	+	0.774185i	$0.718160\pi$
$968$	−23.4557	−0.753895
$969$	0	0
$970$	0.540824	0.0173648
$971$	−22.7054	−0.728651	−0.364326	−	0.931272i	$-0.618701\pi$
$971$	−22.7054	−0.728651	−0.364326	+	0.931272i	$0.618701\pi$
$972$	0	0
$973$	−6.50763	−0.208625
$974$	−0.278751	−0.00893176
$975$	0	0
$976$	−9.48632	−0.303650
$977$	−43.4411	−1.38980	−0.694902	−	0.719105i	$-0.744552\pi$
$977$	−43.4411	−1.38980	−0.694902	+	0.719105i	$0.744552\pi$
$978$	0	0
$979$	−1.82437	−0.0583071
$980$	0.959066	0.0306362
$981$	0	0
$982$	16.6457	0.531184
$983$	53.7651	1.71484	0.857420	−	0.514618i	$-0.172066\pi$
$983$	53.7651	1.71484	0.857420	+	0.514618i	$0.172066\pi$
$984$	0	0
$985$	−4.68253	−0.149198
$986$	−2.80674	−0.0893849
$987$	0	0
$988$	−6.77647	−0.215588
$989$	48.7736	1.55091
$990$	0	0
$991$	5.47700	0.173983	0.0869914	−	0.996209i	$-0.472275\pi$
$991$	5.47700	0.173983	0.0869914	+	0.996209i	$0.472275\pi$
$992$	19.1061	0.606618
$993$	0	0
$994$	−1.54986	−0.0491585
$995$	12.0714	0.382690
$996$	0	0
$997$	−26.4856	−0.838806	−0.419403	−	0.907800i	$-0.637761\pi$
$997$	−26.4856	−0.838806	−0.419403	+	0.907800i	$0.637761\pi$
$998$	−20.2132	−0.639838
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4005.2.a.k.1.2		4
3.2	odd	2		445.2.a.e.1.3	✓	4
12.11	even	2		7120.2.a.z.1.4		4
15.14	odd	2		2225.2.a.h.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
445.2.a.e.1.3	✓	4	3.2	odd	2
2225.2.a.h.1.2		4	15.14	odd	2
4005.2.a.k.1.2		4	1.1	even	1	trivial
7120.2.a.z.1.4		4	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4005 = 3^{2} \cdot 5 \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4005.a (trivial)

\(f(q)\)	\(=\)	\(q-1.23307 q^{2} -0.479533 q^{4} -1.00000 q^{5} +3.00000 q^{7} +3.05744 q^{8} +O(q^{10})\)	\(q-1.23307 q^{2} -0.479533 q^{4} -1.00000 q^{5} +3.00000 q^{7} +3.05744 q^{8} +1.23307 q^{10} -1.82437 q^{11} +2.77005 q^{13} -3.69922 q^{14} -2.81098 q^{16} -0.344838 q^{17} +5.10150 q^{19} +0.479533 q^{20} +2.24958 q^{22} -7.77005 q^{23} +1.00000 q^{25} -3.41567 q^{26} -1.43860 q^{28} -6.60084 q^{29} -7.21326 q^{31} -2.64874 q^{32} +0.425210 q^{34} -3.00000 q^{35} -2.97245 q^{37} -6.29052 q^{38} -3.05744 q^{40} +8.01963 q^{41} -6.27713 q^{43} +0.874846 q^{44} +9.58103 q^{46} -12.4495 q^{47} +2.00000 q^{49} -1.23307 q^{50} -1.32833 q^{52} +12.3101 q^{53} +1.82437 q^{55} +9.17233 q^{56} +8.13931 q^{58} +9.60781 q^{59} +3.37474 q^{61} +8.89448 q^{62} +8.88805 q^{64} -2.77005 q^{65} +5.74712 q^{67} +0.165361 q^{68} +3.69922 q^{70} +0.418969 q^{71} -0.740151 q^{73} +3.66525 q^{74} -2.44634 q^{76} -5.47311 q^{77} -7.84088 q^{79} +2.81098 q^{80} -9.88878 q^{82} -12.4693 q^{83} +0.344838 q^{85} +7.74015 q^{86} -5.57791 q^{88} +1.00000 q^{89} +8.31014 q^{91} +3.72599 q^{92} +15.3511 q^{94} -5.10150 q^{95} +0.438599 q^{97} -2.46614 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{2} + 3 q^{4} - 4 q^{5} + 12 q^{7} + 3 q^{8}+O(q^{10}) \) 4 * q - q^2 + 3 * q^4 - 4 * q^5 + 12 * q^7 + 3 * q^8	\( 4 q - q^{2} + 3 q^{4} - 4 q^{5} + 12 q^{7} + 3 q^{8} + q^{10} - 2 q^{11} - 7 q^{13} - 3 q^{14} - 3 q^{16} - q^{17} - q^{19} - 3 q^{20} - 14 q^{22} - 13 q^{23} + 4 q^{25} - 7 q^{26} + 9 q^{28} - 14 q^{29} - 11 q^{31} - 16 q^{34} - 12 q^{35} - 5 q^{37} - 12 q^{38} - 3 q^{40} - 9 q^{41} - 9 q^{43} - 3 q^{44} + 12 q^{46} - 6 q^{47} + 8 q^{49} - q^{50} - 24 q^{52} - 5 q^{53} + 2 q^{55} + 9 q^{56} + 43 q^{58} + 18 q^{59} - 3 q^{61} - 12 q^{62} - 23 q^{64} + 7 q^{65} + 13 q^{67} - 19 q^{68} + 3 q^{70} + 28 q^{71} - q^{73} - 15 q^{74} + 12 q^{76} - 6 q^{77} - 7 q^{79} + 3 q^{80} - 17 q^{82} - 20 q^{83} + q^{85} + 29 q^{86} - 18 q^{88} + 4 q^{89} - 21 q^{91} + 9 q^{92} + 17 q^{94} + q^{95} - 13 q^{97} - 2 q^{98}+O(q^{100}) \) 4 * q - q^2 + 3 * q^4 - 4 * q^5 + 12 * q^7 + 3 * q^8 + q^10 - 2 * q^11 - 7 * q^13 - 3 * q^14 - 3 * q^16 - q^17 - q^19 - 3 * q^20 - 14 * q^22 - 13 * q^23 + 4 * q^25 - 7 * q^26 + 9 * q^28 - 14 * q^29 - 11 * q^31 - 16 * q^34 - 12 * q^35 - 5 * q^37 - 12 * q^38 - 3 * q^40 - 9 * q^41 - 9 * q^43 - 3 * q^44 + 12 * q^46 - 6 * q^47 + 8 * q^49 - q^50 - 24 * q^52 - 5 * q^53 + 2 * q^55 + 9 * q^56 + 43 * q^58 + 18 * q^59 - 3 * q^61 - 12 * q^62 - 23 * q^64 + 7 * q^65 + 13 * q^67 - 19 * q^68 + 3 * q^70 + 28 * q^71 - q^73 - 15 * q^74 + 12 * q^76 - 6 * q^77 - 7 * q^79 + 3 * q^80 - 17 * q^82 - 20 * q^83 + q^85 + 29 * q^86 - 18 * q^88 + 4 * q^89 - 21 * q^91 + 9 * q^92 + 17 * q^94 + q^95 - 13 * q^97 - 2 * q^98

Introduction

L-functions

Modular forms

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Database

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