LMFDB - Embedded newform 4005.2.a.o.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:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 8x^{5} + 6x^{4} + 19x^{3} - 10x^{2} - 12x + 6 $ x^7 - x^6 - 8x^5 + 6x^4 + 19x^3 - 10x^2 - 12*x + 6
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 445)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.96388$ 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)

$f(q)$	$=$	$q-0.856822 q^{2} -1.26586 q^{4} -1.00000 q^{5} -0.580377 q^{7} +2.79826 q^{8} +O(q^{10})$	\(q-0.856822 q^{2} -1.26586 q^{4} -1.00000 q^{5} -0.580377 q^{7} +2.79826 q^{8} +0.856822 q^{10} +4.74359 q^{11} +6.25246 q^{13} +0.497280 q^{14} +0.134107 q^{16} +5.87935 q^{17} -5.41445 q^{19} +1.26586 q^{20} -4.06441 q^{22} +8.56239 q^{23} +1.00000 q^{25} -5.35724 q^{26} +0.734675 q^{28} +3.96542 q^{29} +5.68893 q^{31} -5.71142 q^{32} -5.03755 q^{34} +0.580377 q^{35} +0.495300 q^{37} +4.63922 q^{38} -2.79826 q^{40} -11.8503 q^{41} -4.78767 q^{43} -6.00471 q^{44} -7.33644 q^{46} +3.78707 q^{47} -6.66316 q^{49} -0.856822 q^{50} -7.91471 q^{52} +7.91925 q^{53} -4.74359 q^{55} -1.62404 q^{56} -3.39766 q^{58} -10.6416 q^{59} +9.77193 q^{61} -4.87439 q^{62} +4.62545 q^{64} -6.25246 q^{65} +1.35061 q^{67} -7.44242 q^{68} -0.497280 q^{70} +10.7514 q^{71} +1.31208 q^{73} -0.424384 q^{74} +6.85392 q^{76} -2.75307 q^{77} +0.492431 q^{79} -0.134107 q^{80} +10.1536 q^{82} -1.62413 q^{83} -5.87935 q^{85} +4.10218 q^{86} +13.2738 q^{88} -1.00000 q^{89} -3.62878 q^{91} -10.8388 q^{92} -3.24485 q^{94} +5.41445 q^{95} -9.61533 q^{97} +5.70914 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 4 q^{2} + 8 q^{4} - 7 q^{5} - 16 q^{7} + 12 q^{8}+O(q^{10}) $ 7 * q + 4 * q^2 + 8 * q^4 - 7 * q^5 - 16 * q^7 + 12 * q^8	$ 7 q + 4 q^{2} + 8 q^{4} - 7 q^{5} - 16 q^{7} + 12 q^{8} - 4 q^{10} + 10 q^{11} - 7 q^{13} - 3 q^{14} + 10 q^{16} + 13 q^{17} - 7 q^{19} - 8 q^{20} + 2 q^{22} + 13 q^{23} + 7 q^{25} - q^{26} - 21 q^{28} + 4 q^{29} + q^{31} + 13 q^{32} + 10 q^{34} + 16 q^{35} - 5 q^{37} + 40 q^{38} - 12 q^{40} - 5 q^{41} - 31 q^{43} + 21 q^{44} + 16 q^{46} + 14 q^{47} + 19 q^{49} + 4 q^{50} + 13 q^{53} - 10 q^{55} + q^{56} + 17 q^{58} + 14 q^{59} + 3 q^{61} - 26 q^{62} + 14 q^{64} + 7 q^{65} + q^{67} + 35 q^{68} + 3 q^{70} + 8 q^{71} + 9 q^{73} + 35 q^{74} + 40 q^{76} - 42 q^{77} + 9 q^{79} - 10 q^{80} + 29 q^{82} + 42 q^{83} - 13 q^{85} - 35 q^{86} + 30 q^{88} - 7 q^{89} + 31 q^{91} - 19 q^{92} + 37 q^{94} + 7 q^{95} - 7 q^{97} - 9 q^{98}+O(q^{100}) $ 7 * q + 4 * q^2 + 8 * q^4 - 7 * q^5 - 16 * q^7 + 12 * q^8 - 4 * q^10 + 10 * q^11 - 7 * q^13 - 3 * q^14 + 10 * q^16 + 13 * q^17 - 7 * q^19 - 8 * q^20 + 2 * q^22 + 13 * q^23 + 7 * q^25 - q^26 - 21 * q^28 + 4 * q^29 + q^31 + 13 * q^32 + 10 * q^34 + 16 * q^35 - 5 * q^37 + 40 * q^38 - 12 * q^40 - 5 * q^41 - 31 * q^43 + 21 * q^44 + 16 * q^46 + 14 * q^47 + 19 * q^49 + 4 * q^50 + 13 * q^53 - 10 * q^55 + q^56 + 17 * q^58 + 14 * q^59 + 3 * q^61 - 26 * q^62 + 14 * q^64 + 7 * q^65 + q^67 + 35 * q^68 + 3 * q^70 + 8 * q^71 + 9 * q^73 + 35 * q^74 + 40 * q^76 - 42 * q^77 + 9 * q^79 - 10 * q^80 + 29 * q^82 + 42 * q^83 - 13 * q^85 - 35 * q^86 + 30 * q^88 - 7 * q^89 + 31 * q^91 - 19 * q^92 + 37 * q^94 + 7 * q^95 - 7 * q^97 - 9 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−0.856822	−0.605864	−0.302932	−	0.953012i	$-0.597966\pi$
$2$	−0.856822	−0.605864	−0.302932	+	0.953012i	$0.597966\pi$
$3$	0	0
$3$	0	0
$4$	−1.26586	−0.632928
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−0.580377	−0.219362	−0.109681	−	0.993967i	$-0.534983\pi$
$7$	−0.580377	−0.219362	−0.109681	+	0.993967i	$0.534983\pi$
$8$	2.79826	0.989333
$9$	0	0
$10$	0.856822	0.270951
$11$	4.74359	1.43025	0.715123	−	0.698998i	$-0.246371\pi$
$11$	4.74359	1.43025	0.715123	+	0.698998i	$0.246371\pi$
$12$	0	0
$13$	6.25246	1.73412	0.867060	−	0.498204i	$-0.166007\pi$
$13$	6.25246	1.73412	0.867060	+	0.498204i	$0.166007\pi$
$14$	0.497280	0.132904
$15$	0	0
$16$	0.134107	0.0335267
$17$	5.87935	1.42595	0.712976	−	0.701189i	$-0.247347\pi$
$17$	5.87935	1.42595	0.712976	+	0.701189i	$0.247347\pi$
$18$	0	0
$19$	−5.41445	−1.24216	−0.621080	−	0.783747i	$-0.713306\pi$
$19$	−5.41445	−1.24216	−0.621080	+	0.783747i	$0.713306\pi$
$20$	1.26586	0.283054
$21$	0	0
$22$	−4.06441	−0.866535
$23$	8.56239	1.78538	0.892691	−	0.450669i	$-0.148815\pi$
$23$	8.56239	1.78538	0.892691	+	0.450669i	$0.148815\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−5.35724	−1.05064
$27$	0	0
$28$	0.734675	0.138840
$29$	3.96542	0.736361	0.368180	−	0.929754i	$-0.379981\pi$
$29$	3.96542	0.736361	0.368180	+	0.929754i	$0.379981\pi$
$30$	0	0
$31$	5.68893	1.02176	0.510881	−	0.859652i	$-0.329319\pi$
$31$	5.68893	1.02176	0.510881	+	0.859652i	$0.329319\pi$
$32$	−5.71142	−1.00965
$33$	0	0
$34$	−5.03755	−0.863933
$35$	0.580377	0.0981017
$36$	0	0
$37$	0.495300	0.0814269	0.0407134	−	0.999171i	$-0.487037\pi$
$37$	0.495300	0.0814269	0.0407134	+	0.999171i	$0.487037\pi$
$38$	4.63922	0.752581
$39$	0	0
$40$	−2.79826	−0.442443
$41$	−11.8503	−1.85071	−0.925356	−	0.379098i	$-0.876234\pi$
$41$	−11.8503	−1.85071	−0.925356	+	0.379098i	$0.876234\pi$
$42$	0	0
$43$	−4.78767	−0.730113	−0.365056	−	0.930985i	$-0.618950\pi$
$43$	−4.78767	−0.730113	−0.365056	+	0.930985i	$0.618950\pi$
$44$	−6.00471	−0.905244
$45$	0	0
$46$	−7.33644	−1.08170
$47$	3.78707	0.552401	0.276201	−	0.961100i	$-0.410925\pi$
$47$	3.78707	0.552401	0.276201	+	0.961100i	$0.410925\pi$
$48$	0	0
$49$	−6.66316	−0.951880
$50$	−0.856822	−0.121173
$51$	0	0
$52$	−7.91471	−1.09757
$53$	7.91925	1.08779	0.543897	−	0.839152i	$-0.316948\pi$
$53$	7.91925	1.08779	0.543897	+	0.839152i	$0.316948\pi$
$54$	0	0
$55$	−4.74359	−0.639626
$56$	−1.62404	−0.217022
$57$	0	0
$58$	−3.39766	−0.446135
$59$	−10.6416	−1.38542	−0.692711	−	0.721216i	$-0.743584\pi$
$59$	−10.6416	−1.38542	−0.692711	+	0.721216i	$0.743584\pi$
$60$	0	0
$61$	9.77193	1.25117	0.625584	−	0.780157i	$-0.284861\pi$
$61$	9.77193	1.25117	0.625584	+	0.780157i	$0.284861\pi$
$62$	−4.87439	−0.619049
$63$	0	0
$64$	4.62545	0.578182
$65$	−6.25246	−0.775522
$66$	0	0
$67$	1.35061	0.165004	0.0825019	−	0.996591i	$-0.473709\pi$
$67$	1.35061	0.165004	0.0825019	+	0.996591i	$0.473709\pi$
$68$	−7.44242	−0.902525
$69$	0	0
$70$	−0.497280	−0.0594363
$71$	10.7514	1.27595	0.637976	−	0.770056i	$-0.279772\pi$
$71$	10.7514	1.27595	0.637976	+	0.770056i	$0.279772\pi$
$72$	0	0
$73$	1.31208	0.153567	0.0767834	−	0.997048i	$-0.475535\pi$
$73$	1.31208	0.153567	0.0767834	+	0.997048i	$0.475535\pi$
$74$	−0.424384	−0.0493336
$75$	0	0
$76$	6.85392	0.786198
$77$	−2.75307	−0.313742
$78$	0	0
$79$	0.492431	0.0554028	0.0277014	−	0.999616i	$-0.491181\pi$
$79$	0.492431	0.0554028	0.0277014	+	0.999616i	$0.491181\pi$
$80$	−0.134107	−0.0149936
$81$	0	0
$82$	10.1536	1.12128
$83$	−1.62413	−0.178272	−0.0891359	−	0.996019i	$-0.528411\pi$
$83$	−1.62413	−0.178272	−0.0891359	+	0.996019i	$0.528411\pi$
$84$	0	0
$85$	−5.87935	−0.637705
$86$	4.10218	0.442349
$87$	0	0
$88$	13.2738	1.41499
$89$	−1.00000	−0.106000
$89$	−1.00000	−0.106000
$90$	0	0
$91$	−3.62878	−0.380400
$92$	−10.8388	−1.13002
$93$	0	0
$94$	−3.24485	−0.334680
$95$	5.41445	0.555511
$96$	0	0
$97$	−9.61533	−0.976289	−0.488144	−	0.872763i	$-0.662326\pi$
$97$	−9.61533	−0.976289	−0.488144	+	0.872763i	$0.662326\pi$
$98$	5.70914	0.576710
$99$	0	0
$100$	−1.26586	−0.126586
$101$	−4.72415	−0.470070	−0.235035	−	0.971987i	$-0.575521\pi$
$101$	−4.72415	−0.470070	−0.235035	+	0.971987i	$0.575521\pi$
$102$	0	0
$103$	5.17345	0.509755	0.254877	−	0.966973i	$-0.417965\pi$
$103$	5.17345	0.509755	0.254877	+	0.966973i	$0.417965\pi$
$104$	17.4960	1.71562
$105$	0	0
$106$	−6.78539	−0.659055
$107$	−5.68158	−0.549259	−0.274630	−	0.961550i	$-0.588555\pi$
$107$	−5.68158	−0.549259	−0.274630	+	0.961550i	$0.588555\pi$
$108$	0	0
$109$	−6.48821	−0.621458	−0.310729	−	0.950499i	$-0.600573\pi$
$109$	−6.48821	−0.621458	−0.310729	+	0.950499i	$0.600573\pi$
$110$	4.06441	0.387526
$111$	0	0
$112$	−0.0778326	−0.00735449
$113$	−9.49005	−0.892749	−0.446375	−	0.894846i	$-0.647285\pi$
$113$	−9.49005	−0.892749	−0.446375	+	0.894846i	$0.647285\pi$
$114$	0	0
$115$	−8.56239	−0.798447
$116$	−5.01966	−0.466064
$117$	0	0
$118$	9.11797	0.839377
$119$	−3.41224	−0.312800
$120$	0	0
$121$	11.5017	1.04561
$122$	−8.37280	−0.758038
$123$	0	0
$124$	−7.20137	−0.646702
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−8.22863	−0.730173	−0.365087	−	0.930974i	$-0.618961\pi$
$127$	−8.22863	−0.730173	−0.365087	+	0.930974i	$0.618961\pi$
$128$	7.45965	0.659346
$129$	0	0
$130$	5.35724	0.469861
$131$	4.92961	0.430702	0.215351	−	0.976537i	$-0.430910\pi$
$131$	4.92961	0.430702	0.215351	+	0.976537i	$0.430910\pi$
$132$	0	0
$133$	3.14242	0.272483
$134$	−1.15723	−0.0999699
$135$	0	0
$136$	16.4519	1.41074
$137$	11.9905	1.02442	0.512211	−	0.858860i	$-0.328827\pi$
$137$	11.9905	1.02442	0.512211	+	0.858860i	$0.328827\pi$
$138$	0	0
$139$	18.0960	1.53488	0.767440	−	0.641120i	$-0.221530\pi$
$139$	18.0960	1.53488	0.767440	+	0.641120i	$0.221530\pi$
$140$	−0.734675	−0.0620913
$141$	0	0
$142$	−9.21200	−0.773054
$143$	29.6591	2.48022
$144$	0	0
$145$	−3.96542	−0.329311
$146$	−1.12421	−0.0930406
$147$	0	0
$148$	−0.626979	−0.0515374
$149$	19.9604	1.63522	0.817609	−	0.575773i	$-0.195299\pi$
$149$	19.9604	1.63522	0.817609	+	0.575773i	$0.195299\pi$
$150$	0	0
$151$	−9.01450	−0.733589	−0.366795	−	0.930302i	$-0.619545\pi$
$151$	−9.01450	−0.733589	−0.366795	+	0.930302i	$0.619545\pi$
$152$	−15.1510	−1.22891
$153$	0	0
$154$	2.35889	0.190085
$155$	−5.68893	−0.456946
$156$	0	0
$157$	−9.23610	−0.737121	−0.368561	−	0.929604i	$-0.620149\pi$
$157$	−9.23610	−0.737121	−0.368561	+	0.929604i	$0.620149\pi$
$158$	−0.421925	−0.0335666
$159$	0	0
$160$	5.71142	0.451527
$161$	−4.96942	−0.391645
$162$	0	0
$163$	−19.2838	−1.51043	−0.755213	−	0.655479i	$-0.772467\pi$
$163$	−19.2838	−1.51043	−0.755213	+	0.655479i	$0.772467\pi$
$164$	15.0008	1.17137
$165$	0	0
$166$	1.39159	0.108009
$167$	24.8161	1.92033	0.960163	−	0.279439i	$-0.0901485\pi$
$167$	24.8161	1.92033	0.960163	+	0.279439i	$0.0901485\pi$
$168$	0	0
$169$	26.0932	2.00717
$170$	5.03755	0.386363
$171$	0	0
$172$	6.06050	0.462109
$173$	17.0185	1.29389	0.646945	−	0.762537i	$-0.276046\pi$
$173$	17.0185	1.29389	0.646945	+	0.762537i	$0.276046\pi$
$174$	0	0
$175$	−0.580377	−0.0438724
$176$	0.636148	0.0479515
$177$	0	0
$178$	0.856822	0.0642215
$179$	6.10678	0.456442	0.228221	−	0.973609i	$-0.426709\pi$
$179$	6.10678	0.456442	0.228221	+	0.973609i	$0.426709\pi$
$180$	0	0
$181$	−1.02757	−0.0763785	−0.0381892	−	0.999271i	$-0.512159\pi$
$181$	−1.02757	−0.0763785	−0.0381892	+	0.999271i	$0.512159\pi$
$182$	3.10922	0.230471
$183$	0	0
$184$	23.9598	1.76634
$185$	−0.495300	−0.0364152
$186$	0	0
$187$	27.8892	2.03946
$188$	−4.79389	−0.349630
$189$	0	0
$190$	−4.63922	−0.336564
$191$	11.3023	0.817808	0.408904	−	0.912577i	$-0.365911\pi$
$191$	11.3023	0.817808	0.408904	+	0.912577i	$0.365911\pi$
$192$	0	0
$193$	−19.0759	−1.37311	−0.686555	−	0.727077i	$-0.740878\pi$
$193$	−19.0759	−1.37311	−0.686555	+	0.727077i	$0.740878\pi$
$194$	8.23862	0.591499
$195$	0	0
$196$	8.43461	0.602472
$197$	−20.1996	−1.43916	−0.719580	−	0.694409i	$-0.755666\pi$
$197$	−20.1996	−1.43916	−0.719580	+	0.694409i	$0.755666\pi$
$198$	0	0
$199$	−11.2701	−0.798915	−0.399457	−	0.916752i	$-0.630801\pi$
$199$	−11.2701	−0.798915	−0.399457	+	0.916752i	$0.630801\pi$
$200$	2.79826	0.197867
$201$	0	0
$202$	4.04775	0.284799
$203$	−2.30144	−0.161530
$204$	0	0
$205$	11.8503	0.827664
$206$	−4.43272	−0.308842
$207$	0	0
$208$	0.838497	0.0581393
$209$	−25.6839	−1.77660
$210$	0	0
$211$	6.09599	0.419665	0.209833	−	0.977737i	$-0.432708\pi$
$211$	6.09599	0.419665	0.209833	+	0.977737i	$0.432708\pi$
$212$	−10.0246	−0.688495
$213$	0	0
$214$	4.86810	0.332777
$215$	4.78767	0.326516
$216$	0	0
$217$	−3.30172	−0.224136
$218$	5.55924	0.376519
$219$	0	0
$220$	6.00471	0.404837
$221$	36.7604	2.47277
$222$	0	0
$223$	7.65809	0.512824	0.256412	−	0.966568i	$-0.417460\pi$
$223$	7.65809	0.512824	0.256412	+	0.966568i	$0.417460\pi$
$224$	3.31478	0.221478
$225$	0	0
$226$	8.13128	0.540885
$227$	29.5274	1.95980	0.979900	−	0.199490i	$-0.0639285\pi$
$227$	29.5274	1.95980	0.979900	+	0.199490i	$0.0639285\pi$
$228$	0	0
$229$	7.14724	0.472303	0.236151	−	0.971716i	$-0.424114\pi$
$229$	7.14724	0.472303	0.236151	+	0.971716i	$0.424114\pi$
$230$	7.33644	0.483751
$231$	0	0
$232$	11.0963	0.728506
$233$	10.9605	0.718045	0.359023	−	0.933329i	$-0.383110\pi$
$233$	10.9605	0.718045	0.359023	+	0.933329i	$0.383110\pi$
$234$	0	0
$235$	−3.78707	−0.247041
$236$	13.4708	0.876873
$237$	0	0
$238$	2.92368	0.189514
$239$	−19.7441	−1.27714	−0.638571	−	0.769563i	$-0.720474\pi$
$239$	−19.7441	−1.27714	−0.638571	+	0.769563i	$0.720474\pi$
$240$	0	0
$241$	−19.2285	−1.23862	−0.619309	−	0.785148i	$-0.712587\pi$
$241$	−19.2285	−1.23862	−0.619309	+	0.785148i	$0.712587\pi$
$242$	−9.85487	−0.633495
$243$	0	0
$244$	−12.3699	−0.791899
$245$	6.66316	0.425694
$246$	0	0
$247$	−33.8536	−2.15405
$248$	15.9191	1.01086
$249$	0	0
$250$	0.856822	0.0541902
$251$	2.44743	0.154481	0.0772403	−	0.997013i	$-0.475389\pi$
$251$	2.44743	0.154481	0.0772403	+	0.997013i	$0.475389\pi$
$252$	0	0
$253$	40.6165	2.55354
$254$	7.05047	0.442386
$255$	0	0
$256$	−15.6425	−0.977656
$257$	−20.5627	−1.28267	−0.641333	−	0.767263i	$-0.721618\pi$
$257$	−20.5627	−1.28267	−0.641333	+	0.767263i	$0.721618\pi$
$258$	0	0
$259$	−0.287461	−0.0178620
$260$	7.91471	0.490850
$261$	0	0
$262$	−4.22379	−0.260947
$263$	−10.4549	−0.644679	−0.322340	−	0.946624i	$-0.604469\pi$
$263$	−10.4549	−0.644679	−0.322340	+	0.946624i	$0.604469\pi$
$264$	0	0
$265$	−7.91925	−0.486476
$266$	−2.69250	−0.165088
$267$	0	0
$268$	−1.70968	−0.104436
$269$	−26.2193	−1.59862	−0.799308	−	0.600921i	$-0.794801\pi$
$269$	−26.2193	−1.59862	−0.799308	+	0.600921i	$0.794801\pi$
$270$	0	0
$271$	20.6659	1.25536	0.627681	−	0.778470i	$-0.284004\pi$
$271$	20.6659	1.25536	0.627681	+	0.778470i	$0.284004\pi$
$272$	0.788461	0.0478075
$273$	0	0
$274$	−10.2738	−0.620660
$275$	4.74359	0.286049
$276$	0	0
$277$	0.433915	0.0260714	0.0130357	−	0.999915i	$-0.495850\pi$
$277$	0.433915	0.0260714	0.0130357	+	0.999915i	$0.495850\pi$
$278$	−15.5050	−0.929930
$279$	0	0
$280$	1.62404	0.0970552
$281$	−16.7226	−0.997585	−0.498793	−	0.866721i	$-0.666223\pi$
$281$	−16.7226	−0.997585	−0.498793	+	0.866721i	$0.666223\pi$
$282$	0	0
$283$	−10.4735	−0.622587	−0.311293	−	0.950314i	$-0.600762\pi$
$283$	−10.4735	−0.622587	−0.311293	+	0.950314i	$0.600762\pi$
$284$	−13.6097	−0.807586
$285$	0	0
$286$	−25.4126	−1.50268
$287$	6.87767	0.405976
$288$	0	0
$289$	17.5668	1.03334
$290$	3.39766	0.199518
$291$	0	0
$292$	−1.66090	−0.0971968
$293$	−7.94600	−0.464210	−0.232105	−	0.972691i	$-0.574561\pi$
$293$	−7.94600	−0.464210	−0.232105	+	0.972691i	$0.574561\pi$
$294$	0	0
$295$	10.6416	0.619579
$296$	1.38598	0.0805583
$297$	0	0
$298$	−17.1025	−0.990721
$299$	53.5360	3.09607
$300$	0	0
$301$	2.77865	0.160159
$302$	7.72382	0.444456
$303$	0	0
$304$	−0.726115	−0.0416456
$305$	−9.77193	−0.559539
$306$	0	0
$307$	−2.11312	−0.120602	−0.0603011	−	0.998180i	$-0.519206\pi$
$307$	−2.11312	−0.120602	−0.0603011	+	0.998180i	$0.519206\pi$
$308$	3.48500	0.198576
$309$	0	0
$310$	4.87439	0.276847
$311$	8.66507	0.491351	0.245675	−	0.969352i	$-0.420990\pi$
$311$	8.66507	0.491351	0.245675	+	0.969352i	$0.420990\pi$
$312$	0	0
$313$	16.0660	0.908105	0.454052	−	0.890975i	$-0.349978\pi$
$313$	16.0660	0.908105	0.454052	+	0.890975i	$0.349978\pi$
$314$	7.91369	0.446595
$315$	0	0
$316$	−0.623347	−0.0350660
$317$	1.70619	0.0958291	0.0479146	−	0.998851i	$-0.484742\pi$
$317$	1.70619	0.0958291	0.0479146	+	0.998851i	$0.484742\pi$
$318$	0	0
$319$	18.8104	1.05318
$320$	−4.62545	−0.258571
$321$	0	0
$322$	4.25790	0.237284
$323$	−31.8335	−1.77126
$324$	0	0
$325$	6.25246	0.346824
$326$	16.5228	0.915113
$327$	0	0
$328$	−33.1603	−1.83097
$329$	−2.19793	−0.121176
$330$	0	0
$331$	−7.90149	−0.434305	−0.217153	−	0.976138i	$-0.569677\pi$
$331$	−7.90149	−0.434305	−0.217153	+	0.976138i	$0.569677\pi$
$332$	2.05592	0.112833
$333$	0	0
$334$	−21.2630	−1.16346
$335$	−1.35061	−0.0737919
$336$	0	0
$337$	7.99703	0.435626	0.217813	−	0.975991i	$-0.430108\pi$
$337$	7.99703	0.435626	0.217813	+	0.975991i	$0.430108\pi$
$338$	−22.3572	−1.21607
$339$	0	0
$340$	7.44242	0.403622
$341$	26.9859	1.46137
$342$	0	0
$343$	7.92979	0.428168
$344$	−13.3971	−0.722325
$345$	0	0
$346$	−14.5818	−0.783922
$347$	−6.24790	−0.335405	−0.167702	−	0.985838i	$-0.553635\pi$
$347$	−6.24790	−0.335405	−0.167702	+	0.985838i	$0.553635\pi$
$348$	0	0
$349$	7.07360	0.378641	0.189320	−	0.981915i	$-0.439372\pi$
$349$	7.07360	0.378641	0.189320	+	0.981915i	$0.439372\pi$
$350$	0.497280	0.0265807
$351$	0	0
$352$	−27.0926	−1.44404
$353$	−17.2887	−0.920184	−0.460092	−	0.887871i	$-0.652184\pi$
$353$	−17.2887	−0.920184	−0.460092	+	0.887871i	$0.652184\pi$
$354$	0	0
$355$	−10.7514	−0.570623
$356$	1.26586	0.0670903
$357$	0	0
$358$	−5.23242	−0.276542
$359$	−12.3384	−0.651193	−0.325597	−	0.945509i	$-0.605565\pi$
$359$	−12.3384	−0.651193	−0.325597	+	0.945509i	$0.605565\pi$
$360$	0	0
$361$	10.3163	0.542962
$362$	0.880442	0.0462750
$363$	0	0
$364$	4.59352	0.240766
$365$	−1.31208	−0.0686771
$366$	0	0
$367$	30.6605	1.60047	0.800233	−	0.599689i	$-0.204709\pi$
$367$	30.6605	1.60047	0.800233	+	0.599689i	$0.204709\pi$
$368$	1.14828	0.0598580
$369$	0	0
$370$	0.424384	0.0220627
$371$	−4.59616	−0.238621
$372$	0	0
$373$	−5.45234	−0.282312	−0.141156	−	0.989987i	$-0.545082\pi$
$373$	−5.45234	−0.282312	−0.141156	+	0.989987i	$0.545082\pi$
$374$	−23.8961	−1.23564
$375$	0	0
$376$	10.5972	0.546509
$377$	24.7936	1.27694
$378$	0	0
$379$	−35.5232	−1.82471	−0.912353	−	0.409405i	$-0.865736\pi$
$379$	−35.5232	−1.82471	−0.912353	+	0.409405i	$0.865736\pi$
$380$	−6.85392	−0.351599
$381$	0	0
$382$	−9.68408	−0.495481
$383$	12.8946	0.658882	0.329441	−	0.944176i	$-0.393140\pi$
$383$	12.8946	0.658882	0.329441	+	0.944176i	$0.393140\pi$
$384$	0	0
$385$	2.75307	0.140310
$386$	16.3446	0.831919
$387$	0	0
$388$	12.1716	0.617921
$389$	28.2948	1.43460	0.717300	−	0.696764i	$-0.245378\pi$
$389$	28.2948	1.43460	0.717300	+	0.696764i	$0.245378\pi$
$390$	0	0
$391$	50.3413	2.54587
$392$	−18.6452	−0.941727
$393$	0	0
$394$	17.3074	0.871936
$395$	−0.492431	−0.0247769
$396$	0	0
$397$	−32.7862	−1.64549	−0.822745	−	0.568411i	$-0.807558\pi$
$397$	−32.7862	−1.64549	−0.822745	+	0.568411i	$0.807558\pi$
$398$	9.65645	0.484034
$399$	0	0
$400$	0.134107	0.00670534
$401$	−19.8756	−0.992538	−0.496269	−	0.868169i	$-0.665297\pi$
$401$	−19.8756	−0.992538	−0.496269	+	0.868169i	$0.665297\pi$
$402$	0	0
$403$	35.5698	1.77186
$404$	5.98009	0.297521
$405$	0	0
$406$	1.97193	0.0978650
$407$	2.34950	0.116461
$408$	0	0
$409$	24.6891	1.22080	0.610398	−	0.792095i	$-0.291009\pi$
$409$	24.6891	1.22080	0.610398	+	0.792095i	$0.291009\pi$
$410$	−10.1536	−0.501452
$411$	0	0
$412$	−6.54884	−0.322638
$413$	6.17616	0.303909
$414$	0	0
$415$	1.62413	0.0797256
$416$	−35.7104	−1.75085
$417$	0	0
$418$	22.0066	1.07638
$419$	23.6692	1.15632	0.578159	−	0.815924i	$-0.303771\pi$
$419$	23.6692	1.15632	0.578159	+	0.815924i	$0.303771\pi$
$420$	0	0
$421$	15.9257	0.776169	0.388085	−	0.921624i	$-0.373137\pi$
$421$	15.9257	0.776169	0.388085	+	0.921624i	$0.373137\pi$
$422$	−5.22317	−0.254260
$423$	0	0
$424$	22.1601	1.07619
$425$	5.87935	0.285190
$426$	0	0
$427$	−5.67141	−0.274459
$428$	7.19207	0.347642
$429$	0	0
$430$	−4.10218	−0.197825
$431$	−7.78225	−0.374858	−0.187429	−	0.982278i	$-0.560015\pi$
$431$	−7.78225	−0.374858	−0.187429	+	0.982278i	$0.560015\pi$
$432$	0	0
$433$	−3.59930	−0.172971	−0.0864856	−	0.996253i	$-0.527564\pi$
$433$	−3.59930	−0.172971	−0.0864856	+	0.996253i	$0.527564\pi$
$434$	2.82899	0.135796
$435$	0	0
$436$	8.21314	0.393338
$437$	−46.3606	−2.21773
$438$	0	0
$439$	30.2840	1.44538	0.722689	−	0.691173i	$-0.242906\pi$
$439$	30.2840	1.44538	0.722689	+	0.691173i	$0.242906\pi$
$440$	−13.2738	−0.632803
$441$	0	0
$442$	−31.4971	−1.49816
$443$	−15.1094	−0.717870	−0.358935	−	0.933362i	$-0.616860\pi$
$443$	−15.1094	−0.717870	−0.358935	+	0.933362i	$0.616860\pi$
$444$	0	0
$445$	1.00000	0.0474045
$446$	−6.56162	−0.310702
$447$	0	0
$448$	−2.68451	−0.126831
$449$	−18.3101	−0.864106	−0.432053	−	0.901848i	$-0.642211\pi$
$449$	−18.3101	−0.864106	−0.432053	+	0.901848i	$0.642211\pi$
$450$	0	0
$451$	−56.2132	−2.64698
$452$	12.0130	0.565046
$453$	0	0
$454$	−25.2997	−1.18737
$455$	3.62878	0.170120
$456$	0	0
$457$	−21.0509	−0.984720	−0.492360	−	0.870392i	$-0.663866\pi$
$457$	−21.0509	−0.984720	−0.492360	+	0.870392i	$0.663866\pi$
$458$	−6.12391	−0.286151
$459$	0	0
$460$	10.8388	0.505360
$461$	38.8056	1.80735	0.903677	−	0.428214i	$-0.140857\pi$
$461$	38.8056	1.80735	0.903677	+	0.428214i	$0.140857\pi$
$462$	0	0
$463$	−12.1126	−0.562921	−0.281460	−	0.959573i	$-0.590819\pi$
$463$	−12.1126	−0.562921	−0.281460	+	0.959573i	$0.590819\pi$
$464$	0.531791	0.0246878
$465$	0	0
$466$	−9.39118	−0.435038
$467$	−2.25635	−0.104411	−0.0522056	−	0.998636i	$-0.516625\pi$
$467$	−2.25635	−0.104411	−0.0522056	+	0.998636i	$0.516625\pi$
$468$	0	0
$469$	−0.783866	−0.0361955
$470$	3.24485	0.149674
$471$	0	0
$472$	−29.7780	−1.37064
$473$	−22.7107	−1.04424
$474$	0	0
$475$	−5.41445	−0.248432
$476$	4.31941	0.197980
$477$	0	0
$478$	16.9172	0.773775
$479$	−0.237254	−0.0108404	−0.00542022	−	0.999985i	$-0.501725\pi$
$479$	−0.237254	−0.0108404	−0.00542022	+	0.999985i	$0.501725\pi$
$480$	0	0
$481$	3.09684	0.141204
$482$	16.4754	0.750434
$483$	0	0
$484$	−14.5595	−0.661793
$485$	9.61533	0.436610
$486$	0	0
$487$	1.69314	0.0767237	0.0383619	−	0.999264i	$-0.487786\pi$
$487$	1.69314	0.0767237	0.0383619	+	0.999264i	$0.487786\pi$
$488$	27.3444	1.23782
$489$	0	0
$490$	−5.70914	−0.257913
$491$	18.2064	0.821641	0.410821	−	0.911716i	$-0.365242\pi$
$491$	18.2064	0.821641	0.410821	+	0.911716i	$0.365242\pi$
$492$	0	0
$493$	23.3141	1.05002
$494$	29.0065	1.30506
$495$	0	0
$496$	0.762924	0.0342563
$497$	−6.23985	−0.279896
$498$	0	0
$499$	20.0254	0.896462	0.448231	−	0.893918i	$-0.352054\pi$
$499$	20.0254	0.896462	0.448231	+	0.893918i	$0.352054\pi$
$500$	1.26586	0.0566108
$501$	0	0
$502$	−2.09701	−0.0935943
$503$	−33.4990	−1.49365	−0.746824	−	0.665021i	$-0.768422\pi$
$503$	−33.4990	−1.49365	−0.746824	+	0.665021i	$0.768422\pi$
$504$	0	0
$505$	4.72415	0.210222
$506$	−34.8011	−1.54710
$507$	0	0
$508$	10.4163	0.462147
$509$	15.2860	0.677541	0.338770	−	0.940869i	$-0.389989\pi$
$509$	15.2860	0.677541	0.338770	+	0.940869i	$0.389989\pi$
$510$	0	0
$511$	−0.761499	−0.0336867
$512$	−1.51647	−0.0670192
$513$	0	0
$514$	17.6186	0.777122
$515$	−5.17345	−0.227969
$516$	0	0
$517$	17.9643	0.790070
$518$	0.246303	0.0108219
$519$	0	0
$520$	−17.4960	−0.767249
$521$	23.7850	1.04204	0.521021	−	0.853544i	$-0.325551\pi$
$521$	23.7850	1.04204	0.521021	+	0.853544i	$0.325551\pi$
$522$	0	0
$523$	26.9989	1.18058	0.590290	−	0.807192i	$-0.299014\pi$
$523$	26.9989	1.18058	0.590290	+	0.807192i	$0.299014\pi$
$524$	−6.24018	−0.272603
$525$	0	0
$526$	8.95802	0.390588
$527$	33.4472	1.45698
$528$	0	0
$529$	50.3145	2.18759
$530$	6.78539	0.294738
$531$	0	0
$532$	−3.97786	−0.172462
$533$	−74.0938	−3.20936
$534$	0	0
$535$	5.68158	0.245636
$536$	3.77936	0.163244
$537$	0	0
$538$	22.4652	0.968545
$539$	−31.6073	−1.36142
$540$	0	0
$541$	−21.0584	−0.905370	−0.452685	−	0.891671i	$-0.649534\pi$
$541$	−21.0584	−0.905370	−0.452685	+	0.891671i	$0.649534\pi$
$542$	−17.7070	−0.760580
$543$	0	0
$544$	−33.5794	−1.43971
$545$	6.48821	0.277924
$546$	0	0
$547$	12.6410	0.540491	0.270245	−	0.962792i	$-0.412895\pi$
$547$	12.6410	0.540491	0.270245	+	0.962792i	$0.412895\pi$
$548$	−15.1783	−0.648385
$549$	0	0
$550$	−4.06441	−0.173307
$551$	−21.4706	−0.914678
$552$	0	0
$553$	−0.285796	−0.0121533
$554$	−0.371788	−0.0157957
$555$	0	0
$556$	−22.9069	−0.971470
$557$	−7.81870	−0.331289	−0.165644	−	0.986186i	$-0.552970\pi$
$557$	−7.81870	−0.331289	−0.165644	+	0.986186i	$0.552970\pi$
$558$	0	0
$559$	−29.9347	−1.26610
$560$	0.0778326	0.00328903
$561$	0	0
$562$	14.3283	0.604401
$563$	20.7792	0.875737	0.437869	−	0.899039i	$-0.355734\pi$
$563$	20.7792	0.875737	0.437869	+	0.899039i	$0.355734\pi$
$564$	0	0
$565$	9.49005	0.399250
$566$	8.97394	0.377203
$567$	0	0
$568$	30.0851	1.26234
$569$	19.4719	0.816307	0.408153	−	0.912913i	$-0.366173\pi$
$569$	19.4719	0.816307	0.408153	+	0.912913i	$0.366173\pi$
$570$	0	0
$571$	−9.07799	−0.379902	−0.189951	−	0.981794i	$-0.560833\pi$
$571$	−9.07799	−0.379902	−0.189951	+	0.981794i	$0.560833\pi$
$572$	−37.5442	−1.56980
$573$	0	0
$574$	−5.89294	−0.245966
$575$	8.56239	0.357076
$576$	0	0
$577$	9.99714	0.416186	0.208093	−	0.978109i	$-0.433274\pi$
$577$	9.99714	0.416186	0.208093	+	0.978109i	$0.433274\pi$
$578$	−15.0516	−0.626063
$579$	0	0
$580$	5.01966	0.208430
$581$	0.942611	0.0391061
$582$	0	0
$583$	37.5657	1.55581
$584$	3.67152	0.151929
$585$	0	0
$586$	6.80830	0.281248
$587$	29.0697	1.19984	0.599918	−	0.800062i	$-0.295200\pi$
$587$	29.0697	1.19984	0.599918	+	0.800062i	$0.295200\pi$
$588$	0	0
$589$	−30.8024	−1.26919
$590$	−9.11797	−0.375381
$591$	0	0
$592$	0.0664232	0.00272998
$593$	7.26916	0.298509	0.149254	−	0.988799i	$-0.452313\pi$
$593$	7.26916	0.298509	0.149254	+	0.988799i	$0.452313\pi$
$594$	0	0
$595$	3.41224	0.139888
$596$	−25.2670	−1.03498
$597$	0	0
$598$	−45.8708	−1.87580
$599$	−15.9436	−0.651438	−0.325719	−	0.945467i	$-0.605606\pi$
$599$	−15.9436	−0.651438	−0.325719	+	0.945467i	$0.605606\pi$
$600$	0	0
$601$	−7.56369	−0.308530	−0.154265	−	0.988030i	$-0.549301\pi$
$601$	−7.56369	−0.308530	−0.154265	+	0.988030i	$0.549301\pi$
$602$	−2.38081	−0.0970346
$603$	0	0
$604$	11.4111	0.464310
$605$	−11.5017	−0.467609
$606$	0	0
$607$	−1.88969	−0.0767000	−0.0383500	−	0.999264i	$-0.512210\pi$
$607$	−1.88969	−0.0767000	−0.0383500	+	0.999264i	$0.512210\pi$
$608$	30.9242	1.25414
$609$	0	0
$610$	8.37280	0.339005
$611$	23.6785	0.957930
$612$	0	0
$613$	−12.4920	−0.504548	−0.252274	−	0.967656i	$-0.581178\pi$
$613$	−12.4920	−0.504548	−0.252274	+	0.967656i	$0.581178\pi$
$614$	1.81057	0.0730686
$615$	0	0
$616$	−7.70381	−0.310395
$617$	10.0989	0.406567	0.203284	−	0.979120i	$-0.434839\pi$
$617$	10.0989	0.406567	0.203284	+	0.979120i	$0.434839\pi$
$618$	0	0
$619$	−30.5158	−1.22653	−0.613266	−	0.789876i	$-0.710145\pi$
$619$	−30.5158	−1.22653	−0.613266	+	0.789876i	$0.710145\pi$
$620$	7.20137	0.289214
$621$	0	0
$622$	−7.42442	−0.297692
$623$	0.580377	0.0232523
$624$	0	0
$625$	1.00000	0.0400000
$626$	−13.7657	−0.550188
$627$	0	0
$628$	11.6916	0.466545
$629$	2.91204	0.116111
$630$	0	0
$631$	−15.5337	−0.618386	−0.309193	−	0.950999i	$-0.600059\pi$
$631$	−15.5337	−0.618386	−0.309193	+	0.950999i	$0.600059\pi$
$632$	1.37795	0.0548118
$633$	0	0
$634$	−1.46190	−0.0580595
$635$	8.22863	0.326543
$636$	0	0
$637$	−41.6611	−1.65067
$638$	−16.1171	−0.638083
$639$	0	0
$640$	−7.45965	−0.294869
$641$	17.5661	0.693818	0.346909	−	0.937899i	$-0.387231\pi$
$641$	17.5661	0.693818	0.346909	+	0.937899i	$0.387231\pi$
$642$	0	0
$643$	−40.7523	−1.60712	−0.803558	−	0.595227i	$-0.797062\pi$
$643$	−40.7523	−1.60712	−0.803558	+	0.595227i	$0.797062\pi$
$644$	6.29057	0.247883
$645$	0	0
$646$	27.2756	1.07314
$647$	20.1397	0.791775	0.395887	−	0.918299i	$-0.370437\pi$
$647$	20.1397	0.791775	0.395887	+	0.918299i	$0.370437\pi$
$648$	0	0
$649$	−50.4795	−1.98149
$650$	−5.35724	−0.210128
$651$	0	0
$652$	24.4106	0.955992
$653$	31.5360	1.23410	0.617050	−	0.786924i	$-0.288328\pi$
$653$	31.5360	1.23410	0.617050	+	0.786924i	$0.288328\pi$
$654$	0	0
$655$	−4.92961	−0.192616
$656$	−1.58921	−0.0620483
$657$	0	0
$658$	1.88324	0.0734162
$659$	−7.91412	−0.308290	−0.154145	−	0.988048i	$-0.549262\pi$
$659$	−7.91412	−0.308290	−0.154145	+	0.988048i	$0.549262\pi$
$660$	0	0
$661$	−39.7578	−1.54640	−0.773200	−	0.634163i	$-0.781345\pi$
$661$	−39.7578	−1.54640	−0.773200	+	0.634163i	$0.781345\pi$
$662$	6.77017	0.263130
$663$	0	0
$664$	−4.54474	−0.176370
$665$	−3.14242	−0.121858
$666$	0	0
$667$	33.9535	1.31469
$668$	−31.4136	−1.21543
$669$	0	0
$670$	1.15723	0.0447079
$671$	46.3540	1.78948
$672$	0	0
$673$	21.6297	0.833763	0.416882	−	0.908961i	$-0.363123\pi$
$673$	21.6297	0.833763	0.416882	+	0.908961i	$0.363123\pi$
$674$	−6.85202	−0.263930
$675$	0	0
$676$	−33.0303	−1.27039
$677$	16.7735	0.644658	0.322329	−	0.946628i	$-0.395534\pi$
$677$	16.7735	0.644658	0.322329	+	0.946628i	$0.395534\pi$
$678$	0	0
$679$	5.58052	0.214161
$680$	−16.4519	−0.630903
$681$	0	0
$682$	−23.1221	−0.885392
$683$	−24.7665	−0.947663	−0.473832	−	0.880615i	$-0.657129\pi$
$683$	−24.7665	−0.947663	−0.473832	+	0.880615i	$0.657129\pi$
$684$	0	0
$685$	−11.9905	−0.458135
$686$	−6.79442	−0.259412
$687$	0	0
$688$	−0.642059	−0.0244783
$689$	49.5148	1.88636
$690$	0	0
$691$	47.4194	1.80392	0.901959	−	0.431822i	$-0.142129\pi$
$691$	47.4194	1.80392	0.901959	+	0.431822i	$0.142129\pi$
$692$	−21.5429	−0.818940
$693$	0	0
$694$	5.35333	0.203210
$695$	−18.0960	−0.686420
$696$	0	0
$697$	−69.6723	−2.63903
$698$	−6.06081	−0.229405
$699$	0	0
$700$	0.734675	0.0277681
$701$	−34.9075	−1.31844	−0.659218	−	0.751952i	$-0.729113\pi$
$701$	−34.9075	−1.31844	−0.659218	+	0.751952i	$0.729113\pi$
$702$	0	0
$703$	−2.68178	−0.101145
$704$	21.9413	0.826942
$705$	0	0
$706$	14.8133	0.557507
$707$	2.74179	0.103116
$708$	0	0
$709$	48.6918	1.82866	0.914330	−	0.404970i	$-0.132718\pi$
$709$	48.6918	1.82866	0.914330	+	0.404970i	$0.132718\pi$
$710$	9.21200	0.345720
$711$	0	0
$712$	−2.79826	−0.104869
$713$	48.7108	1.82423
$714$	0	0
$715$	−29.6591	−1.10919
$716$	−7.73030	−0.288895
$717$	0	0
$718$	10.5718	0.394535
$719$	−35.5414	−1.32547	−0.662736	−	0.748853i	$-0.730605\pi$
$719$	−35.5414	−1.32547	−0.662736	+	0.748853i	$0.730605\pi$
$720$	0	0
$721$	−3.00255	−0.111821
$722$	−8.83921	−0.328961
$723$	0	0
$724$	1.30075	0.0483421
$725$	3.96542	0.147272
$726$	0	0
$727$	2.22393	0.0824812	0.0412406	−	0.999149i	$-0.486869\pi$
$727$	2.22393	0.0824812	0.0412406	+	0.999149i	$0.486869\pi$
$728$	−10.1543	−0.376342
$729$	0	0
$730$	1.12421	0.0416090
$731$	−28.1484	−1.04111
$732$	0	0
$733$	−9.92710	−0.366666	−0.183333	−	0.983051i	$-0.558689\pi$
$733$	−9.92710	−0.366666	−0.183333	+	0.983051i	$0.558689\pi$
$734$	−26.2706	−0.969665
$735$	0	0
$736$	−48.9034	−1.80260
$737$	6.40676	0.235996
$738$	0	0
$739$	−10.1642	−0.373896	−0.186948	−	0.982370i	$-0.559860\pi$
$739$	−10.1642	−0.373896	−0.186948	+	0.982370i	$0.559860\pi$
$740$	0.626979	0.0230482
$741$	0	0
$742$	3.93809	0.144572
$743$	43.3395	1.58997	0.794985	−	0.606629i	$-0.207478\pi$
$743$	43.3395	1.58997	0.794985	+	0.606629i	$0.207478\pi$
$744$	0	0
$745$	−19.9604	−0.731292
$746$	4.67168	0.171042
$747$	0	0
$748$	−35.3038	−1.29083
$749$	3.29746	0.120487
$750$	0	0
$751$	−3.92124	−0.143088	−0.0715440	−	0.997437i	$-0.522793\pi$
$751$	−3.92124	−0.143088	−0.0715440	+	0.997437i	$0.522793\pi$
$752$	0.507872	0.0185202
$753$	0	0
$754$	−21.2437	−0.773651
$755$	9.01450	0.328071
$756$	0	0
$757$	7.19956	0.261672	0.130836	−	0.991404i	$-0.458234\pi$
$757$	7.19956	0.261672	0.130836	+	0.991404i	$0.458234\pi$
$758$	30.4371	1.10552
$759$	0	0
$760$	15.1510	0.549585
$761$	−40.8629	−1.48128	−0.740639	−	0.671903i	$-0.765477\pi$
$761$	−40.8629	−1.48128	−0.740639	+	0.671903i	$0.765477\pi$
$762$	0	0
$763$	3.76561	0.136324
$764$	−14.3071	−0.517614
$765$	0	0
$766$	−11.0484	−0.399193
$767$	−66.5363	−2.40249
$768$	0	0
$769$	51.3277	1.85092	0.925462	−	0.378840i	$-0.123677\pi$
$769$	51.3277	1.85092	0.925462	+	0.378840i	$0.123677\pi$
$770$	−2.35889	−0.0850086
$771$	0	0
$772$	24.1473	0.869081
$773$	2.66946	0.0960137	0.0480069	−	0.998847i	$-0.484713\pi$
$773$	2.66946	0.0960137	0.0480069	+	0.998847i	$0.484713\pi$
$774$	0	0
$775$	5.68893	0.204352
$776$	−26.9062	−0.965875
$777$	0	0
$778$	−24.2436	−0.869173
$779$	64.1631	2.29888
$780$	0	0
$781$	51.0001	1.82493
$782$	−43.1335	−1.54245
$783$	0	0
$784$	−0.893576	−0.0319134
$785$	9.23610	0.329651
$786$	0	0
$787$	−16.3284	−0.582043	−0.291022	−	0.956716i	$-0.593995\pi$
$787$	−16.3284	−0.582043	−0.291022	+	0.956716i	$0.593995\pi$
$788$	25.5698	0.910886
$789$	0	0
$790$	0.421925	0.0150114
$791$	5.50781	0.195835
$792$	0	0
$793$	61.0986	2.16967
$794$	28.0919	0.996944
$795$	0	0
$796$	14.2663	0.505656
$797$	−34.8557	−1.23465	−0.617325	−	0.786708i	$-0.711784\pi$
$797$	−34.8557	−1.23465	−0.617325	+	0.786708i	$0.711784\pi$
$798$	0	0
$799$	22.2655	0.787698
$800$	−5.71142	−0.201929
$801$	0	0
$802$	17.0298	0.601344
$803$	6.22395	0.219638
$804$	0	0
$805$	4.96942	0.175149
$806$	−30.4769	−1.07350
$807$	0	0
$808$	−13.2194	−0.465056
$809$	41.4204	1.45626	0.728132	−	0.685437i	$-0.240389\pi$
$809$	41.4204	1.45626	0.728132	+	0.685437i	$0.240389\pi$
$810$	0	0
$811$	7.34996	0.258092	0.129046	−	0.991639i	$-0.458808\pi$
$811$	7.34996	0.258092	0.129046	+	0.991639i	$0.458808\pi$
$812$	2.91330	0.102237
$813$	0	0
$814$	−2.01310	−0.0705593
$815$	19.2838	0.675483
$816$	0	0
$817$	25.9226	0.906917
$818$	−21.1541	−0.739637
$819$	0	0
$820$	−15.0008	−0.523852
$821$	38.8242	1.35498	0.677488	−	0.735534i	$-0.263069\pi$
$821$	38.8242	1.35498	0.677488	+	0.735534i	$0.263069\pi$
$822$	0	0
$823$	−21.4901	−0.749099	−0.374550	−	0.927207i	$-0.622203\pi$
$823$	−21.4901	−0.749099	−0.374550	+	0.927207i	$0.622203\pi$
$824$	14.4766	0.504317
$825$	0	0
$826$	−5.29187	−0.184128
$827$	−19.6349	−0.682771	−0.341386	−	0.939923i	$-0.610896\pi$
$827$	−19.6349	−0.682771	−0.341386	+	0.939923i	$0.610896\pi$
$828$	0	0
$829$	−20.8272	−0.723358	−0.361679	−	0.932303i	$-0.617796\pi$
$829$	−20.8272	−0.723358	−0.361679	+	0.932303i	$0.617796\pi$
$830$	−1.39159	−0.0483029
$831$	0	0
$832$	28.9204	1.00264
$833$	−39.1751	−1.35734
$834$	0	0
$835$	−24.8161	−0.858796
$836$	32.5122	1.12446
$837$	0	0
$838$	−20.2803	−0.700572
$839$	−22.3217	−0.770630	−0.385315	−	0.922785i	$-0.625907\pi$
$839$	−22.3217	−0.770630	−0.385315	+	0.922785i	$0.625907\pi$
$840$	0	0
$841$	−13.2754	−0.457773
$842$	−13.6455	−0.470253
$843$	0	0
$844$	−7.71665	−0.265618
$845$	−26.0932	−0.897634
$846$	0	0
$847$	−6.67530	−0.229366
$848$	1.06203	0.0364701
$849$	0	0
$850$	−5.03755	−0.172787
$851$	4.24096	0.145378
$852$	0	0
$853$	−4.79245	−0.164090	−0.0820452	−	0.996629i	$-0.526145\pi$
$853$	−4.79245	−0.164090	−0.0820452	+	0.996629i	$0.526145\pi$
$854$	4.85938	0.166285
$855$	0	0
$856$	−15.8985	−0.543400
$857$	−49.1822	−1.68003	−0.840016	−	0.542561i	$-0.817455\pi$
$857$	−49.1822	−1.68003	−0.840016	+	0.542561i	$0.817455\pi$
$858$	0	0
$859$	45.6426	1.55731	0.778653	−	0.627454i	$-0.215903\pi$
$859$	45.6426	1.55731	0.778653	+	0.627454i	$0.215903\pi$
$860$	−6.06050	−0.206661
$861$	0	0
$862$	6.66800	0.227113
$863$	−16.9728	−0.577761	−0.288880	−	0.957365i	$-0.593283\pi$
$863$	−16.9728	−0.577761	−0.288880	+	0.957365i	$0.593283\pi$
$864$	0	0
$865$	−17.0185	−0.578645
$866$	3.08396	0.104797
$867$	0	0
$868$	4.17951	0.141862
$869$	2.33589	0.0792397
$870$	0	0
$871$	8.44465	0.286136
$872$	−18.1557	−0.614829
$873$	0	0
$874$	39.7228	1.34364
$875$	0.580377	0.0196203
$876$	0	0
$877$	−6.47951	−0.218797	−0.109399	−	0.993998i	$-0.534893\pi$
$877$	−6.47951	−0.218797	−0.109399	+	0.993998i	$0.534893\pi$
$878$	−25.9480	−0.875703
$879$	0	0
$880$	−0.636148	−0.0214446
$881$	13.0221	0.438724	0.219362	−	0.975644i	$-0.429602\pi$
$881$	13.0221	0.438724	0.219362	+	0.975644i	$0.429602\pi$
$882$	0	0
$883$	24.1962	0.814268	0.407134	−	0.913369i	$-0.366528\pi$
$883$	24.1962	0.814268	0.407134	+	0.913369i	$0.366528\pi$
$884$	−46.5334	−1.56509
$885$	0	0
$886$	12.9461	0.434932
$887$	18.6517	0.626264	0.313132	−	0.949710i	$-0.398622\pi$
$887$	18.6517	0.626264	0.313132	+	0.949710i	$0.398622\pi$
$888$	0	0
$889$	4.77571	0.160172
$890$	−0.856822	−0.0287207
$891$	0	0
$892$	−9.69404	−0.324581
$893$	−20.5049	−0.686171
$894$	0	0
$895$	−6.10678	−0.204127
$896$	−4.32941	−0.144636
$897$	0	0
$898$	15.6885	0.523531
$899$	22.5590	0.752385
$900$	0	0
$901$	46.5601	1.55114
$902$	48.1647	1.60371
$903$	0	0
$904$	−26.5556	−0.883226
$905$	1.02757	0.0341575
$906$	0	0
$907$	−19.1208	−0.634896	−0.317448	−	0.948276i	$-0.602826\pi$
$907$	−19.1208	−0.634896	−0.317448	+	0.948276i	$0.602826\pi$
$908$	−37.3774	−1.24041
$909$	0	0
$910$	−3.10922	−0.103070
$911$	−6.04981	−0.200439	−0.100220	−	0.994965i	$-0.531955\pi$
$911$	−6.04981	−0.200439	−0.100220	+	0.994965i	$0.531955\pi$
$912$	0	0
$913$	−7.70423	−0.254973
$914$	18.0369	0.596607
$915$	0	0
$916$	−9.04738	−0.298934
$917$	−2.86103	−0.0944796
$918$	0	0
$919$	19.9158	0.656962	0.328481	−	0.944511i	$-0.393463\pi$
$919$	19.9158	0.656962	0.328481	+	0.944511i	$0.393463\pi$
$920$	−23.9598	−0.789930
$921$	0	0
$922$	−33.2494	−1.09501
$923$	67.2224	2.21265
$924$	0	0
$925$	0.495300	0.0162854
$926$	10.3783	0.341054
$927$	0	0
$928$	−22.6482	−0.743464
$929$	−0.333759	−0.0109503	−0.00547514	−	0.999985i	$-0.501743\pi$
$929$	−0.333759	−0.0109503	−0.00547514	+	0.999985i	$0.501743\pi$
$930$	0	0
$931$	36.0774	1.18239
$932$	−13.8744	−0.454471
$933$	0	0
$934$	1.93329	0.0632590
$935$	−27.8892	−0.912075
$936$	0	0
$937$	−14.6611	−0.478957	−0.239479	−	0.970902i	$-0.576977\pi$
$937$	−14.6611	−0.478957	−0.239479	+	0.970902i	$0.576977\pi$
$938$	0.671633	0.0219296
$939$	0	0
$940$	4.79389	0.156360
$941$	4.63624	0.151137	0.0755686	−	0.997141i	$-0.475923\pi$
$941$	4.63624	0.151137	0.0755686	+	0.997141i	$0.475923\pi$
$942$	0	0
$943$	−101.467	−3.30423
$944$	−1.42711	−0.0464486
$945$	0	0
$946$	19.4591	0.632669
$947$	−10.9543	−0.355965	−0.177983	−	0.984034i	$-0.556957\pi$
$947$	−10.9543	−0.355965	−0.177983	+	0.984034i	$0.556957\pi$
$948$	0	0
$949$	8.20369	0.266303
$950$	4.63922	0.150516
$951$	0	0
$952$	−9.54833	−0.309463
$953$	26.4968	0.858315	0.429158	−	0.903230i	$-0.358810\pi$
$953$	26.4968	0.858315	0.429158	+	0.903230i	$0.358810\pi$
$954$	0	0
$955$	−11.3023	−0.365735
$956$	24.9932	0.808339
$957$	0	0
$958$	0.203285	0.00656783
$959$	−6.95904	−0.224719
$960$	0	0
$961$	1.36388	0.0439962
$962$	−2.65344	−0.0855504
$963$	0	0
$964$	24.3405	0.783956
$965$	19.0759	0.614074
$966$	0	0
$967$	−59.1812	−1.90314	−0.951569	−	0.307435i	$-0.900529\pi$
$967$	−59.1812	−1.90314	−0.951569	+	0.307435i	$0.900529\pi$
$968$	32.1846	1.03445
$969$	0	0
$970$	−8.23862	−0.264526
$971$	5.45699	0.175123	0.0875616	−	0.996159i	$-0.472093\pi$
$971$	5.45699	0.175123	0.0875616	+	0.996159i	$0.472093\pi$
$972$	0	0
$973$	−10.5025	−0.336695
$974$	−1.45072	−0.0464842
$975$	0	0
$976$	1.31048	0.0419475
$977$	−17.0837	−0.546555	−0.273278	−	0.961935i	$-0.588108\pi$
$977$	−17.0837	−0.546555	−0.273278	+	0.961935i	$0.588108\pi$
$978$	0	0
$979$	−4.74359	−0.151606
$980$	−8.43461	−0.269434
$981$	0	0
$982$	−15.5996	−0.497803
$983$	−42.1851	−1.34549	−0.672747	−	0.739872i	$-0.734886\pi$
$983$	−42.1851	−1.34549	−0.672747	+	0.739872i	$0.734886\pi$
$984$	0	0
$985$	20.1996	0.643612
$986$	−19.9760	−0.636167
$987$	0	0
$988$	42.8538	1.36336
$989$	−40.9939	−1.30353
$990$	0	0
$991$	−11.7224	−0.372375	−0.186188	−	0.982514i	$-0.559613\pi$
$991$	−11.7224	−0.372375	−0.186188	+	0.982514i	$0.559613\pi$
$992$	−32.4918	−1.03162
$993$	0	0
$994$	5.34644	0.169579
$995$	11.2701	0.357286
$996$	0	0
$997$	0.629575	0.0199388	0.00996942	−	0.999950i	$-0.496827\pi$
$997$	0.629575	0.0199388	0.00996942	+	0.999950i	$0.496827\pi$
$998$	−17.1582	−0.543134
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4005.2.a.o.1.2		7
3.2	odd	2		445.2.a.f.1.6	✓	7
12.11	even	2		7120.2.a.bj.1.4		7
15.14	odd	2		2225.2.a.k.1.2		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
445.2.a.f.1.6	✓	7	3.2	odd	2
2225.2.a.k.1.2		7	15.14	odd	2
4005.2.a.o.1.2		7	1.1	even	1	trivial
7120.2.a.bj.1.4		7	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)

\(f(q)\)	\(=\)	\(q-0.856822 q^{2} -1.26586 q^{4} -1.00000 q^{5} -0.580377 q^{7} +2.79826 q^{8} +O(q^{10})\)	\(q-0.856822 q^{2} -1.26586 q^{4} -1.00000 q^{5} -0.580377 q^{7} +2.79826 q^{8} +0.856822 q^{10} +4.74359 q^{11} +6.25246 q^{13} +0.497280 q^{14} +0.134107 q^{16} +5.87935 q^{17} -5.41445 q^{19} +1.26586 q^{20} -4.06441 q^{22} +8.56239 q^{23} +1.00000 q^{25} -5.35724 q^{26} +0.734675 q^{28} +3.96542 q^{29} +5.68893 q^{31} -5.71142 q^{32} -5.03755 q^{34} +0.580377 q^{35} +0.495300 q^{37} +4.63922 q^{38} -2.79826 q^{40} -11.8503 q^{41} -4.78767 q^{43} -6.00471 q^{44} -7.33644 q^{46} +3.78707 q^{47} -6.66316 q^{49} -0.856822 q^{50} -7.91471 q^{52} +7.91925 q^{53} -4.74359 q^{55} -1.62404 q^{56} -3.39766 q^{58} -10.6416 q^{59} +9.77193 q^{61} -4.87439 q^{62} +4.62545 q^{64} -6.25246 q^{65} +1.35061 q^{67} -7.44242 q^{68} -0.497280 q^{70} +10.7514 q^{71} +1.31208 q^{73} -0.424384 q^{74} +6.85392 q^{76} -2.75307 q^{77} +0.492431 q^{79} -0.134107 q^{80} +10.1536 q^{82} -1.62413 q^{83} -5.87935 q^{85} +4.10218 q^{86} +13.2738 q^{88} -1.00000 q^{89} -3.62878 q^{91} -10.8388 q^{92} -3.24485 q^{94} +5.41445 q^{95} -9.61533 q^{97} +5.70914 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 4 q^{2} + 8 q^{4} - 7 q^{5} - 16 q^{7} + 12 q^{8}+O(q^{10}) \) 7 * q + 4 * q^2 + 8 * q^4 - 7 * q^5 - 16 * q^7 + 12 * q^8	\( 7 q + 4 q^{2} + 8 q^{4} - 7 q^{5} - 16 q^{7} + 12 q^{8} - 4 q^{10} + 10 q^{11} - 7 q^{13} - 3 q^{14} + 10 q^{16} + 13 q^{17} - 7 q^{19} - 8 q^{20} + 2 q^{22} + 13 q^{23} + 7 q^{25} - q^{26} - 21 q^{28} + 4 q^{29} + q^{31} + 13 q^{32} + 10 q^{34} + 16 q^{35} - 5 q^{37} + 40 q^{38} - 12 q^{40} - 5 q^{41} - 31 q^{43} + 21 q^{44} + 16 q^{46} + 14 q^{47} + 19 q^{49} + 4 q^{50} + 13 q^{53} - 10 q^{55} + q^{56} + 17 q^{58} + 14 q^{59} + 3 q^{61} - 26 q^{62} + 14 q^{64} + 7 q^{65} + q^{67} + 35 q^{68} + 3 q^{70} + 8 q^{71} + 9 q^{73} + 35 q^{74} + 40 q^{76} - 42 q^{77} + 9 q^{79} - 10 q^{80} + 29 q^{82} + 42 q^{83} - 13 q^{85} - 35 q^{86} + 30 q^{88} - 7 q^{89} + 31 q^{91} - 19 q^{92} + 37 q^{94} + 7 q^{95} - 7 q^{97} - 9 q^{98}+O(q^{100}) \) 7 * q + 4 * q^2 + 8 * q^4 - 7 * q^5 - 16 * q^7 + 12 * q^8 - 4 * q^10 + 10 * q^11 - 7 * q^13 - 3 * q^14 + 10 * q^16 + 13 * q^17 - 7 * q^19 - 8 * q^20 + 2 * q^22 + 13 * q^23 + 7 * q^25 - q^26 - 21 * q^28 + 4 * q^29 + q^31 + 13 * q^32 + 10 * q^34 + 16 * q^35 - 5 * q^37 + 40 * q^38 - 12 * q^40 - 5 * q^41 - 31 * q^43 + 21 * q^44 + 16 * q^46 + 14 * q^47 + 19 * q^49 + 4 * q^50 + 13 * q^53 - 10 * q^55 + q^56 + 17 * q^58 + 14 * q^59 + 3 * q^61 - 26 * q^62 + 14 * q^64 + 7 * q^65 + q^67 + 35 * q^68 + 3 * q^70 + 8 * q^71 + 9 * q^73 + 35 * q^74 + 40 * q^76 - 42 * q^77 + 9 * q^79 - 10 * q^80 + 29 * q^82 + 42 * q^83 - 13 * q^85 - 35 * q^86 + 30 * q^88 - 7 * q^89 + 31 * q^91 - 19 * q^92 + 37 * q^94 + 7 * q^95 - 7 * q^97 - 9 * q^98

Introduction

L-functions

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