LMFDB - Embedded newform 4005.2.a.p.1.5

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:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 11x^{6} + 9x^{5} + 34x^{4} - 19x^{3} - 27x^{2} + 11x - 1 $ x^8 - x^7 - 11x^6 + 9x^5 + 34x^4 - 19x^3 - 27x^2 + 11x - 1
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			1.5
Root			$0.151894$ 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)

$f(q)$	$=$	$q-0.151894 q^{2} -1.97693 q^{4} +1.00000 q^{5} +2.88745 q^{7} +0.604071 q^{8} +O(q^{10})$	\(q-0.151894 q^{2} -1.97693 q^{4} +1.00000 q^{5} +2.88745 q^{7} +0.604071 q^{8} -0.151894 q^{10} -2.75596 q^{11} +2.78878 q^{13} -0.438585 q^{14} +3.86210 q^{16} -6.24918 q^{17} -0.721848 q^{19} -1.97693 q^{20} +0.418614 q^{22} +0.0986719 q^{23} +1.00000 q^{25} -0.423598 q^{26} -5.70828 q^{28} -6.61485 q^{29} -0.885417 q^{31} -1.79477 q^{32} +0.949212 q^{34} +2.88745 q^{35} -4.03281 q^{37} +0.109644 q^{38} +0.604071 q^{40} -9.13960 q^{41} +4.57497 q^{43} +5.44834 q^{44} -0.0149877 q^{46} +6.64268 q^{47} +1.33735 q^{49} -0.151894 q^{50} -5.51321 q^{52} -0.765878 q^{53} -2.75596 q^{55} +1.74422 q^{56} +1.00476 q^{58} -2.79467 q^{59} -8.50106 q^{61} +0.134489 q^{62} -7.45159 q^{64} +2.78878 q^{65} +4.23461 q^{67} +12.3542 q^{68} -0.438585 q^{70} +2.60032 q^{71} -9.47269 q^{73} +0.612559 q^{74} +1.42704 q^{76} -7.95770 q^{77} +13.2707 q^{79} +3.86210 q^{80} +1.38825 q^{82} +11.3012 q^{83} -6.24918 q^{85} -0.694910 q^{86} -1.66480 q^{88} -1.00000 q^{89} +8.05244 q^{91} -0.195067 q^{92} -1.00898 q^{94} -0.721848 q^{95} -11.6151 q^{97} -0.203135 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - q^{2} + 7 q^{4} + 8 q^{5} - 6 q^{7} - 3 q^{8}+O(q^{10}) $ 8 * q - q^2 + 7 * q^4 + 8 * q^5 - 6 * q^7 - 3 * q^8	$ 8 q - q^{2} + 7 q^{4} + 8 q^{5} - 6 q^{7} - 3 q^{8} - q^{10} - 14 q^{11} - 7 q^{13} - 15 q^{14} + 9 q^{16} - 17 q^{17} + 17 q^{19} + 7 q^{20} + 2 q^{22} + q^{23} + 8 q^{25} - 3 q^{26} - 29 q^{28} - 10 q^{29} + q^{31} - 2 q^{32} - 16 q^{34} - 6 q^{35} - 11 q^{37} + 30 q^{38} - 3 q^{40} - 15 q^{41} - 5 q^{43} - 7 q^{44} - 12 q^{46} - 12 q^{47} + 4 q^{49} - q^{50} - 14 q^{52} + q^{53} - 14 q^{55} - 3 q^{56} - 37 q^{58} - 26 q^{59} + 13 q^{61} - 22 q^{62} - 15 q^{64} - 7 q^{65} - 25 q^{67} - 23 q^{68} - 15 q^{70} - 28 q^{71} - 17 q^{73} + 5 q^{74} + 8 q^{76} - 7 q^{79} + 9 q^{80} + 5 q^{82} - 44 q^{83} - 17 q^{85} + 13 q^{86} - 66 q^{88} - 8 q^{89} + 27 q^{91} - 15 q^{92} - 27 q^{94} + 17 q^{95} + q^{97} + 34 q^{98}+O(q^{100}) $ 8 * q - q^2 + 7 * q^4 + 8 * q^5 - 6 * q^7 - 3 * q^8 - q^10 - 14 * q^11 - 7 * q^13 - 15 * q^14 + 9 * q^16 - 17 * q^17 + 17 * q^19 + 7 * q^20 + 2 * q^22 + q^23 + 8 * q^25 - 3 * q^26 - 29 * q^28 - 10 * q^29 + q^31 - 2 * q^32 - 16 * q^34 - 6 * q^35 - 11 * q^37 + 30 * q^38 - 3 * q^40 - 15 * q^41 - 5 * q^43 - 7 * q^44 - 12 * q^46 - 12 * q^47 + 4 * q^49 - q^50 - 14 * q^52 + q^53 - 14 * q^55 - 3 * q^56 - 37 * q^58 - 26 * q^59 + 13 * q^61 - 22 * q^62 - 15 * q^64 - 7 * q^65 - 25 * q^67 - 23 * q^68 - 15 * q^70 - 28 * q^71 - 17 * q^73 + 5 * q^74 + 8 * q^76 - 7 * q^79 + 9 * q^80 + 5 * q^82 - 44 * q^83 - 17 * q^85 + 13 * q^86 - 66 * q^88 - 8 * q^89 + 27 * q^91 - 15 * q^92 - 27 * q^94 + 17 * q^95 + q^97 + 34 * 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 $

$2$	−0.151894	−0.107405	−0.0537026	−	0.998557i	$-0.517102\pi$
$2$	−0.151894	−0.107405	−0.0537026	+	0.998557i	$0.517102\pi$
$3$	0	0
$3$	0	0
$4$	−1.97693	−0.988464
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	2.88745	1.09135	0.545676	−	0.837996i	$-0.316273\pi$
$7$	2.88745	1.09135	0.545676	+	0.837996i	$0.316273\pi$
$8$	0.604071	0.213571
$9$	0	0
$10$	−0.151894	−0.0480330
$11$	−2.75596	−0.830955	−0.415477	−	0.909604i	$-0.636385\pi$
$11$	−2.75596	−0.830955	−0.415477	+	0.909604i	$0.636385\pi$
$12$	0	0
$13$	2.78878	0.773467	0.386734	−	0.922192i	$-0.373603\pi$
$13$	2.78878	0.773467	0.386734	+	0.922192i	$0.373603\pi$
$14$	−0.438585	−0.117217
$15$	0	0
$16$	3.86210	0.965525
$17$	−6.24918	−1.51565	−0.757825	−	0.652458i	$-0.773738\pi$
$17$	−6.24918	−1.51565	−0.757825	+	0.652458i	$0.773738\pi$
$18$	0	0
$19$	−0.721848	−0.165603	−0.0828017	−	0.996566i	$-0.526387\pi$
$19$	−0.721848	−0.165603	−0.0828017	+	0.996566i	$0.526387\pi$
$20$	−1.97693	−0.442055
$21$	0	0
$22$	0.418614	0.0892488
$23$	0.0986719	0.0205745	0.0102873	−	0.999947i	$-0.496725\pi$
$23$	0.0986719	0.0205745	0.0102873	+	0.999947i	$0.496725\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−0.423598	−0.0830743
$27$	0	0
$28$	−5.70828	−1.07876
$29$	−6.61485	−1.22835	−0.614174	−	0.789171i	$-0.710511\pi$
$29$	−6.61485	−1.22835	−0.614174	+	0.789171i	$0.710511\pi$
$30$	0	0
$31$	−0.885417	−0.159026	−0.0795128	−	0.996834i	$-0.525336\pi$
$31$	−0.885417	−0.159026	−0.0795128	+	0.996834i	$0.525336\pi$
$32$	−1.79477	−0.317274
$33$	0	0
$34$	0.949212	0.162789
$35$	2.88745	0.488068
$36$	0	0
$37$	−4.03281	−0.662990	−0.331495	−	0.943457i	$-0.607553\pi$
$37$	−4.03281	−0.662990	−0.331495	+	0.943457i	$0.607553\pi$
$38$	0.109644	0.0177867
$39$	0	0
$40$	0.604071	0.0955120
$41$	−9.13960	−1.42737	−0.713683	−	0.700469i	$-0.752974\pi$
$41$	−9.13960	−1.42737	−0.713683	+	0.700469i	$0.752974\pi$
$42$	0	0
$43$	4.57497	0.697676	0.348838	−	0.937183i	$-0.386576\pi$
$43$	4.57497	0.697676	0.348838	+	0.937183i	$0.386576\pi$
$44$	5.44834	0.821369
$45$	0	0
$46$	−0.0149877	−0.00220981
$47$	6.64268	0.968934	0.484467	−	0.874809i	$-0.339014\pi$
$47$	6.64268	0.968934	0.484467	+	0.874809i	$0.339014\pi$
$48$	0	0
$49$	1.33735	0.191050
$50$	−0.151894	−0.0214810
$51$	0	0
$52$	−5.51321	−0.764544
$53$	−0.765878	−0.105201	−0.0526007	−	0.998616i	$-0.516751\pi$
$53$	−0.765878	−0.105201	−0.0526007	+	0.998616i	$0.516751\pi$
$54$	0	0
$55$	−2.75596	−0.371614
$56$	1.74422	0.233082
$57$	0	0
$58$	1.00476	0.131931
$59$	−2.79467	−0.363836	−0.181918	−	0.983314i	$-0.558230\pi$
$59$	−2.79467	−0.363836	−0.181918	+	0.983314i	$0.558230\pi$
$60$	0	0
$61$	−8.50106	−1.08845	−0.544225	−	0.838939i	$-0.683176\pi$
$61$	−8.50106	−1.08845	−0.544225	+	0.838939i	$0.683176\pi$
$62$	0.134489	0.0170802
$63$	0	0
$64$	−7.45159	−0.931449
$65$	2.78878	0.345905
$66$	0	0
$67$	4.23461	0.517339	0.258670	−	0.965966i	$-0.416716\pi$
$67$	4.23461	0.517339	0.258670	+	0.965966i	$0.416716\pi$
$68$	12.3542	1.49816
$69$	0	0
$70$	−0.438585	−0.0524210
$71$	2.60032	0.308601	0.154300	−	0.988024i	$-0.450688\pi$
$71$	2.60032	0.308601	0.154300	+	0.988024i	$0.450688\pi$
$72$	0	0
$73$	−9.47269	−1.10869	−0.554347	−	0.832285i	$-0.687032\pi$
$73$	−9.47269	−1.10869	−0.554347	+	0.832285i	$0.687032\pi$
$74$	0.612559	0.0712085
$75$	0	0
$76$	1.42704	0.163693
$77$	−7.95770	−0.906864
$78$	0	0
$79$	13.2707	1.49307	0.746536	−	0.665345i	$-0.231716\pi$
$79$	13.2707	1.49307	0.746536	+	0.665345i	$0.231716\pi$
$80$	3.86210	0.431796
$81$	0	0
$82$	1.38825	0.153306
$83$	11.3012	1.24046	0.620232	−	0.784418i	$-0.287038\pi$
$83$	11.3012	1.24046	0.620232	+	0.784418i	$0.287038\pi$
$84$	0	0
$85$	−6.24918	−0.677819
$86$	−0.694910	−0.0749340
$87$	0	0
$88$	−1.66480	−0.177468
$89$	−1.00000	−0.106000
$89$	−1.00000	−0.106000
$90$	0	0
$91$	8.05244	0.844125
$92$	−0.195067	−0.0203372
$93$	0	0
$94$	−1.00898	−0.104069
$95$	−0.721848	−0.0740601
$96$	0	0
$97$	−11.6151	−1.17933	−0.589667	−	0.807646i	$-0.700741\pi$
$97$	−11.6151	−1.17933	−0.589667	+	0.807646i	$0.700741\pi$
$98$	−0.203135	−0.0205198
$99$	0	0
$100$	−1.97693	−0.197693
$101$	−12.1795	−1.21191	−0.605953	−	0.795501i	$-0.707208\pi$
$101$	−12.1795	−1.21191	−0.605953	+	0.795501i	$0.707208\pi$
$102$	0	0
$103$	9.71335	0.957085	0.478542	−	0.878064i	$-0.341165\pi$
$103$	9.71335	0.957085	0.478542	+	0.878064i	$0.341165\pi$
$104$	1.68462	0.165190
$105$	0	0
$106$	0.116332	0.0112992
$107$	−8.02137	−0.775455	−0.387727	−	0.921774i	$-0.626740\pi$
$107$	−8.02137	−0.775455	−0.387727	+	0.921774i	$0.626740\pi$
$108$	0	0
$109$	4.86470	0.465953	0.232977	−	0.972482i	$-0.425153\pi$
$109$	4.86470	0.465953	0.232977	+	0.972482i	$0.425153\pi$
$110$	0.418614	0.0399133
$111$	0	0
$112$	11.1516	1.05373
$113$	3.12496	0.293971	0.146986	−	0.989139i	$-0.453043\pi$
$113$	3.12496	0.293971	0.146986	+	0.989139i	$0.453043\pi$
$114$	0	0
$115$	0.0986719	0.00920120
$116$	13.0771	1.21418
$117$	0	0
$118$	0.424494	0.0390778
$119$	−18.0442	−1.65411
$120$	0	0
$121$	−3.40466	−0.309514
$122$	1.29126	0.116905
$123$	0	0
$124$	1.75041	0.157191
$125$	1.00000	0.0894427
$126$	0	0
$127$	−12.6401	−1.12162	−0.560812	−	0.827943i	$-0.689511\pi$
$127$	−12.6401	−1.12162	−0.560812	+	0.827943i	$0.689511\pi$
$128$	4.72139	0.417316
$129$	0	0
$130$	−0.423598	−0.0371520
$131$	−16.2432	−1.41917	−0.709587	−	0.704618i	$-0.751119\pi$
$131$	−16.2432	−1.41917	−0.709587	+	0.704618i	$0.751119\pi$
$132$	0	0
$133$	−2.08430	−0.180732
$134$	−0.643210	−0.0555649
$135$	0	0
$136$	−3.77495	−0.323699
$137$	11.5123	0.983563	0.491781	−	0.870719i	$-0.336346\pi$
$137$	11.5123	0.983563	0.491781	+	0.870719i	$0.336346\pi$
$138$	0	0
$139$	−9.18537	−0.779093	−0.389547	−	0.921007i	$-0.627368\pi$
$139$	−9.18537	−0.779093	−0.389547	+	0.921007i	$0.627368\pi$
$140$	−5.70828	−0.482437
$141$	0	0
$142$	−0.394972	−0.0331453
$143$	−7.68577	−0.642716
$144$	0	0
$145$	−6.61485	−0.549334
$146$	1.43884	0.119080
$147$	0	0
$148$	7.97258	0.655342
$149$	3.25326	0.266517	0.133259	−	0.991081i	$-0.457456\pi$
$149$	3.25326	0.266517	0.133259	+	0.991081i	$0.457456\pi$
$150$	0	0
$151$	−11.3089	−0.920308	−0.460154	−	0.887839i	$-0.652206\pi$
$151$	−11.3089	−0.920308	−0.460154	+	0.887839i	$0.652206\pi$
$152$	−0.436048	−0.0353681
$153$	0	0
$154$	1.20873	0.0974019
$155$	−0.885417	−0.0711184
$156$	0	0
$157$	10.3585	0.826700	0.413350	−	0.910572i	$-0.364359\pi$
$157$	10.3585	0.826700	0.413350	+	0.910572i	$0.364359\pi$
$158$	−2.01574	−0.160364
$159$	0	0
$160$	−1.79477	−0.141889
$161$	0.284910	0.0224540
$162$	0	0
$163$	−8.75633	−0.685849	−0.342924	−	0.939363i	$-0.611417\pi$
$163$	−8.75633	−0.685849	−0.342924	+	0.939363i	$0.611417\pi$
$164$	18.0683	1.41090
$165$	0	0
$166$	−1.71658	−0.133232
$167$	−21.8241	−1.68880	−0.844399	−	0.535715i	$-0.820042\pi$
$167$	−21.8241	−1.68880	−0.844399	+	0.535715i	$0.820042\pi$
$168$	0	0
$169$	−5.22273	−0.401749
$170$	0.949212	0.0728012
$171$	0	0
$172$	−9.04439	−0.689628
$173$	8.11969	0.617329	0.308664	−	0.951171i	$-0.400118\pi$
$173$	8.11969	0.617329	0.308664	+	0.951171i	$0.400118\pi$
$174$	0	0
$175$	2.88745	0.218270
$176$	−10.6438	−0.802308
$177$	0	0
$178$	0.151894	0.0113849
$179$	−14.6946	−1.09832	−0.549161	−	0.835716i	$-0.685053\pi$
$179$	−14.6946	−1.09832	−0.549161	+	0.835716i	$0.685053\pi$
$180$	0	0
$181$	9.35948	0.695685	0.347842	−	0.937553i	$-0.386914\pi$
$181$	9.35948	0.695685	0.347842	+	0.937553i	$0.386914\pi$
$182$	−1.22312	−0.0906634
$183$	0	0
$184$	0.0596048	0.00439413
$185$	−4.03281	−0.296498
$186$	0	0
$187$	17.2225	1.25944
$188$	−13.1321	−0.957757
$189$	0	0
$190$	0.109644	0.00795443
$191$	−10.6961	−0.773943	−0.386971	−	0.922092i	$-0.626479\pi$
$191$	−10.6961	−0.773943	−0.386971	+	0.922092i	$0.626479\pi$
$192$	0	0
$193$	19.0333	1.37005	0.685024	−	0.728521i	$-0.259792\pi$
$193$	19.0333	1.37005	0.685024	+	0.728521i	$0.259792\pi$
$194$	1.76426	0.126667
$195$	0	0
$196$	−2.64385	−0.188846
$197$	21.9152	1.56139	0.780695	−	0.624912i	$-0.214865\pi$
$197$	21.9152	1.56139	0.780695	+	0.624912i	$0.214865\pi$
$198$	0	0
$199$	13.0013	0.921639	0.460819	−	0.887494i	$-0.347556\pi$
$199$	13.0013	0.921639	0.460819	+	0.887494i	$0.347556\pi$
$200$	0.604071	0.0427143
$201$	0	0
$202$	1.84999	0.130165
$203$	−19.1000	−1.34056
$204$	0	0
$205$	−9.13960	−0.638338
$206$	−1.47540	−0.102796
$207$	0	0
$208$	10.7705	0.746802
$209$	1.98939	0.137609
$210$	0	0
$211$	0.914119	0.0629306	0.0314653	−	0.999505i	$-0.489983\pi$
$211$	0.914119	0.0629306	0.0314653	+	0.999505i	$0.489983\pi$
$212$	1.51409	0.103988
$213$	0	0
$214$	1.21840	0.0832878
$215$	4.57497	0.312010
$216$	0	0
$217$	−2.55660	−0.173553
$218$	−0.738917	−0.0500458
$219$	0	0
$220$	5.44834	0.367327
$221$	−17.4276	−1.17230
$222$	0	0
$223$	−8.67837	−0.581147	−0.290573	−	0.956853i	$-0.593846\pi$
$223$	−8.67837	−0.581147	−0.290573	+	0.956853i	$0.593846\pi$
$224$	−5.18231	−0.346257
$225$	0	0
$226$	−0.474662	−0.0315740
$227$	−18.0379	−1.19722	−0.598608	−	0.801042i	$-0.704279\pi$
$227$	−18.0379	−1.19722	−0.598608	+	0.801042i	$0.704279\pi$
$228$	0	0
$229$	−24.4494	−1.61566	−0.807831	−	0.589414i	$-0.799359\pi$
$229$	−24.4494	−1.61566	−0.807831	+	0.589414i	$0.799359\pi$
$230$	−0.0149877	−0.000988257 0
$231$	0	0
$232$	−3.99584	−0.262340
$233$	17.4971	1.14627	0.573136	−	0.819460i	$-0.305727\pi$
$233$	17.4971	1.14627	0.573136	+	0.819460i	$0.305727\pi$
$234$	0	0
$235$	6.64268	0.433321
$236$	5.52487	0.359638
$237$	0	0
$238$	2.74080	0.177660
$239$	−7.31086	−0.472900	−0.236450	−	0.971644i	$-0.575984\pi$
$239$	−7.31086	−0.472900	−0.236450	+	0.971644i	$0.575984\pi$
$240$	0	0
$241$	−26.6186	−1.71466	−0.857328	−	0.514771i	$-0.827877\pi$
$241$	−26.6186	−1.71466	−0.857328	+	0.514771i	$0.827877\pi$
$242$	0.517147	0.0332434
$243$	0	0
$244$	16.8060	1.07589
$245$	1.33735	0.0854402
$246$	0	0
$247$	−2.01307	−0.128089
$248$	−0.534855	−0.0339633
$249$	0	0
$250$	−0.151894	−0.00960661
$251$	−21.3903	−1.35015	−0.675073	−	0.737751i	$-0.735888\pi$
$251$	−21.3903	−1.35015	−0.675073	+	0.737751i	$0.735888\pi$
$252$	0	0
$253$	−0.271936	−0.0170965
$254$	1.91995	0.120468
$255$	0	0
$256$	14.1860	0.886627
$257$	13.8639	0.864809	0.432404	−	0.901680i	$-0.357665\pi$
$257$	13.8639	0.864809	0.432404	+	0.901680i	$0.357665\pi$
$258$	0	0
$259$	−11.6445	−0.723556
$260$	−5.51321	−0.341915
$261$	0	0
$262$	2.46724	0.152427
$263$	−20.0797	−1.23817	−0.619085	−	0.785324i	$-0.712496\pi$
$263$	−20.0797	−1.23817	−0.619085	+	0.785324i	$0.712496\pi$
$264$	0	0
$265$	−0.765878	−0.0470475
$266$	0.316592	0.0194115
$267$	0	0
$268$	−8.37151	−0.511371
$269$	−10.7150	−0.653303	−0.326652	−	0.945145i	$-0.605920\pi$
$269$	−10.7150	−0.653303	−0.326652	+	0.945145i	$0.605920\pi$
$270$	0	0
$271$	−27.3435	−1.66100	−0.830500	−	0.557019i	$-0.811945\pi$
$271$	−27.3435	−1.66100	−0.830500	+	0.557019i	$0.811945\pi$
$272$	−24.1350	−1.46340
$273$	0	0
$274$	−1.74865	−0.105640
$275$	−2.75596	−0.166191
$276$	0	0
$277$	−16.5306	−0.993229	−0.496614	−	0.867971i	$-0.665424\pi$
$277$	−16.5306	−0.993229	−0.496614	+	0.867971i	$0.665424\pi$
$278$	1.39520	0.0836786
$279$	0	0
$280$	1.74422	0.104237
$281$	−6.73948	−0.402044	−0.201022	−	0.979587i	$-0.564426\pi$
$281$	−6.73948	−0.402044	−0.201022	+	0.979587i	$0.564426\pi$
$282$	0	0
$283$	−5.58139	−0.331779	−0.165890	−	0.986144i	$-0.553050\pi$
$283$	−5.58139	−0.331779	−0.165890	+	0.986144i	$0.553050\pi$
$284$	−5.14064	−0.305041
$285$	0	0
$286$	1.16742	0.0690310
$287$	−26.3901	−1.55776
$288$	0	0
$289$	22.0523	1.29719
$290$	1.00476	0.0590012
$291$	0	0
$292$	18.7268	1.09591
$293$	−7.53464	−0.440178	−0.220089	−	0.975480i	$-0.570635\pi$
$293$	−7.53464	−0.440178	−0.220089	+	0.975480i	$0.570635\pi$
$294$	0	0
$295$	−2.79467	−0.162712
$296$	−2.43610	−0.141596
$297$	0	0
$298$	−0.494149	−0.0286253
$299$	0.275174	0.0159137
$300$	0	0
$301$	13.2100	0.761411
$302$	1.71776	0.0988458
$303$	0	0
$304$	−2.78785	−0.159894
$305$	−8.50106	−0.486769
$306$	0	0
$307$	−10.4478	−0.596289	−0.298144	−	0.954521i	$-0.596368\pi$
$307$	−10.4478	−0.596289	−0.298144	+	0.954521i	$0.596368\pi$
$308$	15.7318	0.896403
$309$	0	0
$310$	0.134489	0.00763848
$311$	26.2433	1.48812	0.744061	−	0.668111i	$-0.232897\pi$
$311$	26.2433	1.48812	0.744061	+	0.668111i	$0.232897\pi$
$312$	0	0
$313$	12.4586	0.704203	0.352101	−	0.935962i	$-0.385467\pi$
$313$	12.4586	0.704203	0.352101	+	0.935962i	$0.385467\pi$
$314$	−1.57340	−0.0887918
$315$	0	0
$316$	−26.2352	−1.47585
$317$	13.7370	0.771548	0.385774	−	0.922593i	$-0.373934\pi$
$317$	13.7370	0.771548	0.385774	+	0.922593i	$0.373934\pi$
$318$	0	0
$319$	18.2303	1.02070
$320$	−7.45159	−0.416556
$321$	0	0
$322$	−0.0432761	−0.00241168
$323$	4.51096	0.250997
$324$	0	0
$325$	2.78878	0.154693
$326$	1.33003	0.0736637
$327$	0	0
$328$	−5.52097	−0.304844
$329$	19.1804	1.05745
$330$	0	0
$331$	19.4675	1.07003	0.535016	−	0.844842i	$-0.320306\pi$
$331$	19.4675	1.07003	0.535016	+	0.844842i	$0.320306\pi$
$332$	−22.3416	−1.22616
$333$	0	0
$334$	3.31494	0.181386
$335$	4.23461	0.231361
$336$	0	0
$337$	−29.0718	−1.58364	−0.791821	−	0.610753i	$-0.790867\pi$
$337$	−29.0718	−1.58364	−0.791821	+	0.610753i	$0.790867\pi$
$338$	0.793301	0.0431499
$339$	0	0
$340$	12.3542	0.670000
$341$	2.44018	0.132143
$342$	0	0
$343$	−16.3506	−0.882849
$344$	2.76361	0.149004
$345$	0	0
$346$	−1.23333	−0.0663043
$347$	5.62285	0.301850	0.150925	−	0.988545i	$-0.451775\pi$
$347$	5.62285	0.301850	0.150925	+	0.988545i	$0.451775\pi$
$348$	0	0
$349$	−2.35987	−0.126321	−0.0631605	−	0.998003i	$-0.520118\pi$
$349$	−2.35987	−0.126321	−0.0631605	+	0.998003i	$0.520118\pi$
$350$	−0.438585	−0.0234434
$351$	0	0
$352$	4.94633	0.263640
$353$	−25.8602	−1.37640	−0.688201	−	0.725520i	$-0.741599\pi$
$353$	−25.8602	−1.37640	−0.688201	+	0.725520i	$0.741599\pi$
$354$	0	0
$355$	2.60032	0.138010
$356$	1.97693	0.104777
$357$	0	0
$358$	2.23201	0.117966
$359$	22.6920	1.19764	0.598819	−	0.800884i	$-0.295637\pi$
$359$	22.6920	1.19764	0.598819	+	0.800884i	$0.295637\pi$
$360$	0	0
$361$	−18.4789	−0.972576
$362$	−1.42165	−0.0747201
$363$	0	0
$364$	−15.9191	−0.834387
$365$	−9.47269	−0.495823
$366$	0	0
$367$	−27.4660	−1.43372	−0.716858	−	0.697219i	$-0.754420\pi$
$367$	−27.4660	−1.43372	−0.716858	+	0.697219i	$0.754420\pi$
$368$	0.381081	0.0198652
$369$	0	0
$370$	0.612559	0.0318454
$371$	−2.21143	−0.114812
$372$	0	0
$373$	−3.63459	−0.188192	−0.0940958	−	0.995563i	$-0.529996\pi$
$373$	−3.63459	−0.188192	−0.0940958	+	0.995563i	$0.529996\pi$
$374$	−2.61599	−0.135270
$375$	0	0
$376$	4.01265	0.206937
$377$	−18.4473	−0.950086
$378$	0	0
$379$	−9.48226	−0.487071	−0.243536	−	0.969892i	$-0.578307\pi$
$379$	−9.48226	−0.487071	−0.243536	+	0.969892i	$0.578307\pi$
$380$	1.42704	0.0732057
$381$	0	0
$382$	1.62467	0.0831255
$383$	10.9355	0.558776	0.279388	−	0.960178i	$-0.409868\pi$
$383$	10.9355	0.558776	0.279388	+	0.960178i	$0.409868\pi$
$384$	0	0
$385$	−7.95770	−0.405562
$386$	−2.89104	−0.147150
$387$	0	0
$388$	22.9622	1.16573
$389$	11.3628	0.576117	0.288059	−	0.957613i	$-0.406990\pi$
$389$	11.3628	0.576117	0.288059	+	0.957613i	$0.406990\pi$
$390$	0	0
$391$	−0.616619	−0.0311837
$392$	0.807855	0.0408028
$393$	0	0
$394$	−3.32878	−0.167701
$395$	13.2707	0.667722
$396$	0	0
$397$	0.601588	0.0301929	0.0150964	−	0.999886i	$-0.495194\pi$
$397$	0.601588	0.0301929	0.0150964	+	0.999886i	$0.495194\pi$
$398$	−1.97482	−0.0989888
$399$	0	0
$400$	3.86210	0.193105
$401$	20.5405	1.02574	0.512872	−	0.858465i	$-0.328582\pi$
$401$	20.5405	1.02574	0.512872	+	0.858465i	$0.328582\pi$
$402$	0	0
$403$	−2.46923	−0.123001
$404$	24.0780	1.19793
$405$	0	0
$406$	2.90118	0.143983
$407$	11.1143	0.550915
$408$	0	0
$409$	29.6541	1.46630	0.733150	−	0.680067i	$-0.238049\pi$
$409$	29.6541	1.46630	0.733150	+	0.680067i	$0.238049\pi$
$410$	1.38825	0.0685607
$411$	0	0
$412$	−19.2026	−0.946044
$413$	−8.06947	−0.397073
$414$	0	0
$415$	11.3012	0.554753
$416$	−5.00521	−0.245401
$417$	0	0
$418$	−0.302176	−0.0147799
$419$	−6.46993	−0.316077	−0.158038	−	0.987433i	$-0.550517\pi$
$419$	−6.46993	−0.316077	−0.158038	+	0.987433i	$0.550517\pi$
$420$	0	0
$421$	−30.5926	−1.49099	−0.745496	−	0.666511i	$-0.767787\pi$
$421$	−30.5926	−1.49099	−0.745496	+	0.666511i	$0.767787\pi$
$422$	−0.138849	−0.00675907
$423$	0	0
$424$	−0.462645	−0.0224680
$425$	−6.24918	−0.303130
$426$	0	0
$427$	−24.5464	−1.18788
$428$	15.8577	0.766509
$429$	0	0
$430$	−0.694910	−0.0335115
$431$	15.6618	0.754402	0.377201	−	0.926131i	$-0.376887\pi$
$431$	15.6618	0.754402	0.377201	+	0.926131i	$0.376887\pi$
$432$	0	0
$433$	31.2050	1.49962	0.749809	−	0.661655i	$-0.230146\pi$
$433$	31.2050	1.49962	0.749809	+	0.661655i	$0.230146\pi$
$434$	0.388331	0.0186405
$435$	0	0
$436$	−9.61715	−0.460578
$437$	−0.0712262	−0.00340721
$438$	0	0
$439$	−17.2377	−0.822711	−0.411355	−	0.911475i	$-0.634944\pi$
$439$	−17.2377	−0.822711	−0.411355	+	0.911475i	$0.634944\pi$
$440$	−1.66480	−0.0793661
$441$	0	0
$442$	2.64714	0.125912
$443$	−1.62298	−0.0771101	−0.0385551	−	0.999256i	$-0.512275\pi$
$443$	−1.62298	−0.0771101	−0.0385551	+	0.999256i	$0.512275\pi$
$444$	0	0
$445$	−1.00000	−0.0474045
$446$	1.31819	0.0624181
$447$	0	0
$448$	−21.5161	−1.01654
$449$	37.6589	1.77723	0.888617	−	0.458650i	$-0.151667\pi$
$449$	37.6589	1.77723	0.888617	+	0.458650i	$0.151667\pi$
$450$	0	0
$451$	25.1884	1.18608
$452$	−6.17781	−0.290580
$453$	0	0
$454$	2.73984	0.128587
$455$	8.05244	0.377504
$456$	0	0
$457$	−20.4355	−0.955933	−0.477966	−	0.878378i	$-0.658626\pi$
$457$	−20.4355	−0.955933	−0.477966	+	0.878378i	$0.658626\pi$
$458$	3.71371	0.173530
$459$	0	0
$460$	−0.195067	−0.00909506
$461$	−39.5268	−1.84095	−0.920474	−	0.390803i	$-0.872197\pi$
$461$	−39.5268	−1.84095	−0.920474	+	0.390803i	$0.872197\pi$
$462$	0	0
$463$	16.4046	0.762387	0.381194	−	0.924495i	$-0.375513\pi$
$463$	16.4046	0.762387	0.381194	+	0.924495i	$0.375513\pi$
$464$	−25.5472	−1.18600
$465$	0	0
$466$	−2.65770	−0.123115
$467$	1.41022	0.0652573	0.0326286	−	0.999468i	$-0.489612\pi$
$467$	1.41022	0.0652573	0.0326286	+	0.999468i	$0.489612\pi$
$468$	0	0
$469$	12.2272	0.564600
$470$	−1.00898	−0.0465409
$471$	0	0
$472$	−1.68818	−0.0777048
$473$	−12.6085	−0.579737
$474$	0	0
$475$	−0.721848	−0.0331207
$476$	35.6721	1.63503
$477$	0	0
$478$	1.11047	0.0507919
$479$	5.09893	0.232976	0.116488	−	0.993192i	$-0.462836\pi$
$479$	5.09893	0.232976	0.116488	+	0.993192i	$0.462836\pi$
$480$	0	0
$481$	−11.2466	−0.512801
$482$	4.04320	0.184163
$483$	0	0
$484$	6.73077	0.305944
$485$	−11.6151	−0.527414
$486$	0	0
$487$	−30.3129	−1.37361	−0.686803	−	0.726843i	$-0.740987\pi$
$487$	−30.3129	−1.37361	−0.686803	+	0.726843i	$0.740987\pi$
$488$	−5.13524	−0.232462
$489$	0	0
$490$	−0.203135	−0.00917672
$491$	16.6019	0.749234	0.374617	−	0.927180i	$-0.377774\pi$
$491$	16.6019	0.749234	0.374617	+	0.927180i	$0.377774\pi$
$492$	0	0
$493$	41.3374	1.86174
$494$	0.305773	0.0137574
$495$	0	0
$496$	−3.41957	−0.153543
$497$	7.50827	0.336792
$498$	0	0
$499$	37.3084	1.67015	0.835077	−	0.550133i	$-0.185423\pi$
$499$	37.3084	1.67015	0.835077	+	0.550133i	$0.185423\pi$
$500$	−1.97693	−0.0884109
$501$	0	0
$502$	3.24906	0.145013
$503$	25.3041	1.12825	0.564127	−	0.825688i	$-0.309213\pi$
$503$	25.3041	1.12825	0.564127	+	0.825688i	$0.309213\pi$
$504$	0	0
$505$	−12.1795	−0.541981
$506$	0.0413054	0.00183625
$507$	0	0
$508$	24.9885	1.10869
$509$	2.56999	0.113913	0.0569563	−	0.998377i	$-0.481860\pi$
$509$	2.56999	0.113913	0.0569563	+	0.998377i	$0.481860\pi$
$510$	0	0
$511$	−27.3519	−1.20998
$512$	−11.5976	−0.512544
$513$	0	0
$514$	−2.10585	−0.0928849
$515$	9.71335	0.428021
$516$	0	0
$517$	−18.3070	−0.805140
$518$	1.76873	0.0777136
$519$	0	0
$520$	1.68462	0.0738754
$521$	40.9730	1.79506	0.897530	−	0.440954i	$-0.145360\pi$
$521$	40.9730	1.79506	0.897530	+	0.440954i	$0.145360\pi$
$522$	0	0
$523$	−2.35708	−0.103068	−0.0515340	−	0.998671i	$-0.516411\pi$
$523$	−2.35708	−0.103068	−0.0515340	+	0.998671i	$0.516411\pi$
$524$	32.1116	1.40280
$525$	0	0
$526$	3.04999	0.132986
$527$	5.53313	0.241027
$528$	0	0
$529$	−22.9903	−0.999577
$530$	0.116332	0.00505315
$531$	0	0
$532$	4.12051	0.178647
$533$	−25.4883	−1.10402
$534$	0	0
$535$	−8.02137	−0.346794
$536$	2.55800	0.110489
$537$	0	0
$538$	1.62754	0.0701681
$539$	−3.68569	−0.158754
$540$	0	0
$541$	−6.28140	−0.270058	−0.135029	−	0.990842i	$-0.543113\pi$
$541$	−6.28140	−0.270058	−0.135029	+	0.990842i	$0.543113\pi$
$542$	4.15331	0.178400
$543$	0	0
$544$	11.2159	0.480876
$545$	4.86470	0.208381
$546$	0	0
$547$	−37.6741	−1.61083	−0.805414	−	0.592712i	$-0.798057\pi$
$547$	−37.6741	−1.61083	−0.805414	+	0.592712i	$0.798057\pi$
$548$	−22.7590	−0.972217
$549$	0	0
$550$	0.418614	0.0178498
$551$	4.77492	0.203418
$552$	0	0
$553$	38.3185	1.62947
$554$	2.51090	0.106678
$555$	0	0
$556$	18.1588	0.770106
$557$	27.6895	1.17324	0.586622	−	0.809861i	$-0.300458\pi$
$557$	27.6895	1.17324	0.586622	+	0.809861i	$0.300458\pi$
$558$	0	0
$559$	12.7586	0.539630
$560$	11.1516	0.471242
$561$	0	0
$562$	1.02369	0.0431816
$563$	31.6924	1.33567	0.667837	−	0.744308i	$-0.267220\pi$
$563$	31.6924	1.33567	0.667837	+	0.744308i	$0.267220\pi$
$564$	0	0
$565$	3.12496	0.131468
$566$	0.847778	0.0356348
$567$	0	0
$568$	1.57077	0.0659082
$569$	9.29551	0.389688	0.194844	−	0.980834i	$-0.437580\pi$
$569$	9.29551	0.389688	0.194844	+	0.980834i	$0.437580\pi$
$570$	0	0
$571$	43.4044	1.81642	0.908208	−	0.418519i	$-0.137451\pi$
$571$	43.4044	1.81642	0.908208	+	0.418519i	$0.137451\pi$
$572$	15.1942	0.635302
$573$	0	0
$574$	4.00850	0.167311
$575$	0.0986719	0.00411490
$576$	0	0
$577$	25.8713	1.07704	0.538519	−	0.842613i	$-0.318984\pi$
$577$	25.8713	1.07704	0.538519	+	0.842613i	$0.318984\pi$
$578$	−3.34960	−0.139325
$579$	0	0
$580$	13.0771	0.542996
$581$	32.6315	1.35378
$582$	0	0
$583$	2.11073	0.0874176
$584$	−5.72218	−0.236785
$585$	0	0
$586$	1.14446	0.0472774
$587$	−35.2942	−1.45675	−0.728373	−	0.685180i	$-0.759723\pi$
$587$	−35.2942	−1.45675	−0.728373	+	0.685180i	$0.759723\pi$
$588$	0	0
$589$	0.639137	0.0263352
$590$	0.424494	0.0174761
$591$	0	0
$592$	−15.5751	−0.640134
$593$	−22.4865	−0.923411	−0.461706	−	0.887033i	$-0.652762\pi$
$593$	−22.4865	−0.923411	−0.461706	+	0.887033i	$0.652762\pi$
$594$	0	0
$595$	−18.0442	−0.739739
$596$	−6.43145	−0.263443
$597$	0	0
$598$	−0.0417972	−0.00170921
$599$	−37.4414	−1.52982	−0.764908	−	0.644140i	$-0.777215\pi$
$599$	−37.4414	−1.52982	−0.764908	+	0.644140i	$0.777215\pi$
$600$	0	0
$601$	35.1825	1.43513	0.717563	−	0.696494i	$-0.245258\pi$
$601$	35.1825	1.43513	0.717563	+	0.696494i	$0.245258\pi$
$602$	−2.00652	−0.0817794
$603$	0	0
$604$	22.3569	0.909691
$605$	−3.40466	−0.138419
$606$	0	0
$607$	−44.2600	−1.79646	−0.898230	−	0.439526i	$-0.855146\pi$
$607$	−44.2600	−1.79646	−0.898230	+	0.439526i	$0.855146\pi$
$608$	1.29555	0.0525416
$609$	0	0
$610$	1.29126	0.0522815
$611$	18.5249	0.749439
$612$	0	0
$613$	11.3225	0.457313	0.228657	−	0.973507i	$-0.426567\pi$
$613$	11.3225	0.457313	0.228657	+	0.973507i	$0.426567\pi$
$614$	1.58696	0.0640445
$615$	0	0
$616$	−4.80702	−0.193680
$617$	−13.3243	−0.536415	−0.268208	−	0.963361i	$-0.586431\pi$
$617$	−13.3243	−0.536415	−0.268208	+	0.963361i	$0.586431\pi$
$618$	0	0
$619$	13.8176	0.555378	0.277689	−	0.960671i	$-0.410432\pi$
$619$	13.8176	0.555378	0.277689	+	0.960671i	$0.410432\pi$
$620$	1.75041	0.0702980
$621$	0	0
$622$	−3.98620	−0.159832
$623$	−2.88745	−0.115683
$624$	0	0
$625$	1.00000	0.0400000
$626$	−1.89239	−0.0756350
$627$	0	0
$628$	−20.4781	−0.817163
$629$	25.2018	1.00486
$630$	0	0
$631$	45.8580	1.82558	0.912789	−	0.408431i	$-0.133924\pi$
$631$	45.8580	1.82558	0.912789	+	0.408431i	$0.133924\pi$
$632$	8.01645	0.318877
$633$	0	0
$634$	−2.08657	−0.0828682
$635$	−12.6401	−0.501606
$636$	0	0
$637$	3.72957	0.147771
$638$	−2.76907	−0.109629
$639$	0	0
$640$	4.72139	0.186629
$641$	33.8076	1.33532	0.667661	−	0.744466i	$-0.267296\pi$
$641$	33.8076	1.33532	0.667661	+	0.744466i	$0.267296\pi$
$642$	0	0
$643$	43.4445	1.71328	0.856642	−	0.515912i	$-0.172547\pi$
$643$	43.4445	1.71328	0.856642	+	0.515912i	$0.172547\pi$
$644$	−0.563247	−0.0221950
$645$	0	0
$646$	−0.685187	−0.0269583
$647$	−3.44794	−0.135553	−0.0677763	−	0.997701i	$-0.521590\pi$
$647$	−3.44794	−0.135553	−0.0677763	+	0.997701i	$0.521590\pi$
$648$	0	0
$649$	7.70202	0.302331
$650$	−0.423598	−0.0166149
$651$	0	0
$652$	17.3106	0.677937
$653$	−17.8067	−0.696831	−0.348415	−	0.937340i	$-0.613280\pi$
$653$	−17.8067	−0.696831	−0.348415	+	0.937340i	$0.613280\pi$
$654$	0	0
$655$	−16.2432	−0.634674
$656$	−35.2981	−1.37816
$657$	0	0
$658$	−2.91338	−0.113575
$659$	−16.5625	−0.645185	−0.322593	−	0.946538i	$-0.604554\pi$
$659$	−16.5625	−0.645185	−0.322593	+	0.946538i	$0.604554\pi$
$660$	0	0
$661$	2.05302	0.0798534	0.0399267	−	0.999203i	$-0.487288\pi$
$661$	2.05302	0.0798534	0.0399267	+	0.999203i	$0.487288\pi$
$662$	−2.95700	−0.114927
$663$	0	0
$664$	6.82671	0.264928
$665$	−2.08430	−0.0808257
$666$	0	0
$667$	−0.652700	−0.0252726
$668$	43.1446	1.66932
$669$	0	0
$670$	−0.643210	−0.0248494
$671$	23.4286	0.904452
$672$	0	0
$673$	−30.6267	−1.18057	−0.590286	−	0.807194i	$-0.700985\pi$
$673$	−30.6267	−1.18057	−0.590286	+	0.807194i	$0.700985\pi$
$674$	4.41583	0.170091
$675$	0	0
$676$	10.3250	0.397114
$677$	25.7858	0.991029	0.495515	−	0.868600i	$-0.334979\pi$
$677$	25.7858	0.991029	0.495515	+	0.868600i	$0.334979\pi$
$678$	0	0
$679$	−33.5380	−1.28707
$680$	−3.77495	−0.144763
$681$	0	0
$682$	−0.370648	−0.0141928
$683$	−38.0328	−1.45528	−0.727642	−	0.685957i	$-0.759384\pi$
$683$	−38.0328	−1.45528	−0.727642	+	0.685957i	$0.759384\pi$
$684$	0	0
$685$	11.5123	0.439863
$686$	2.48356	0.0948226
$687$	0	0
$688$	17.6690	0.673624
$689$	−2.13586	−0.0813698
$690$	0	0
$691$	40.2662	1.53180	0.765899	−	0.642961i	$-0.222294\pi$
$691$	40.2662	1.53180	0.765899	+	0.642961i	$0.222294\pi$
$692$	−16.0521	−0.610207
$693$	0	0
$694$	−0.854076	−0.0324203
$695$	−9.18537	−0.348421
$696$	0	0
$697$	57.1150	2.16339
$698$	0.358450	0.0135675
$699$	0	0
$700$	−5.70828	−0.215753
$701$	37.7180	1.42459	0.712295	−	0.701880i	$-0.247656\pi$
$701$	37.7180	1.42459	0.712295	+	0.701880i	$0.247656\pi$
$702$	0	0
$703$	2.91108	0.109793
$704$	20.5363	0.773992
$705$	0	0
$706$	3.92801	0.147833
$707$	−35.1677	−1.32262
$708$	0	0
$709$	38.5655	1.44836	0.724178	−	0.689613i	$-0.242219\pi$
$709$	38.5655	1.44836	0.724178	+	0.689613i	$0.242219\pi$
$710$	−0.394972	−0.0148230
$711$	0	0
$712$	−0.604071	−0.0226385
$713$	−0.0873658	−0.00327188
$714$	0	0
$715$	−7.68577	−0.287431
$716$	29.0501	1.08565
$717$	0	0
$718$	−3.44677	−0.128632
$719$	−10.0993	−0.376641	−0.188321	−	0.982108i	$-0.560304\pi$
$719$	−10.0993	−0.376641	−0.188321	+	0.982108i	$0.560304\pi$
$720$	0	0
$721$	28.0468	1.04452
$722$	2.80684	0.104460
$723$	0	0
$724$	−18.5030	−0.687660
$725$	−6.61485	−0.245669
$726$	0	0
$727$	10.8207	0.401319	0.200659	−	0.979661i	$-0.435692\pi$
$727$	10.8207	0.401319	0.200659	+	0.979661i	$0.435692\pi$
$728$	4.86424	0.180281
$729$	0	0
$730$	1.43884	0.0532540
$731$	−28.5898	−1.05743
$732$	0	0
$733$	18.8978	0.698005	0.349002	−	0.937122i	$-0.386521\pi$
$733$	18.8978	0.698005	0.349002	+	0.937122i	$0.386521\pi$
$734$	4.17192	0.153988
$735$	0	0
$736$	−0.177094	−0.00652775
$737$	−11.6704	−0.429886
$738$	0	0
$739$	32.8181	1.20723	0.603617	−	0.797275i	$-0.293726\pi$
$739$	32.8181	1.20723	0.603617	+	0.797275i	$0.293726\pi$
$740$	7.97258	0.293078
$741$	0	0
$742$	0.335903	0.0123314
$743$	22.4436	0.823374	0.411687	−	0.911325i	$-0.364940\pi$
$743$	22.4436	0.823374	0.411687	+	0.911325i	$0.364940\pi$
$744$	0	0
$745$	3.25326	0.119190
$746$	0.552071	0.0202128
$747$	0	0
$748$	−34.0477	−1.24491
$749$	−23.1613	−0.846294
$750$	0	0
$751$	11.1004	0.405058	0.202529	−	0.979276i	$-0.435084\pi$
$751$	11.1004	0.405058	0.202529	+	0.979276i	$0.435084\pi$
$752$	25.6547	0.935531
$753$	0	0
$754$	2.80204	0.102044
$755$	−11.3089	−0.411574
$756$	0	0
$757$	−17.6253	−0.640601	−0.320300	−	0.947316i	$-0.603784\pi$
$757$	−17.6253	−0.640601	−0.320300	+	0.947316i	$0.603784\pi$
$758$	1.44030	0.0523140
$759$	0	0
$760$	−0.436048	−0.0158171
$761$	8.04139	0.291500	0.145750	−	0.989321i	$-0.453440\pi$
$761$	8.04139	0.291500	0.145750	+	0.989321i	$0.453440\pi$
$762$	0	0
$763$	14.0466	0.508519
$764$	21.1454	0.765015
$765$	0	0
$766$	−1.66103	−0.0600155
$767$	−7.79372	−0.281415
$768$	0	0
$769$	13.1830	0.475392	0.237696	−	0.971340i	$-0.423608\pi$
$769$	13.1830	0.475392	0.237696	+	0.971340i	$0.423608\pi$
$770$	1.20873	0.0435595
$771$	0	0
$772$	−37.6275	−1.35424
$773$	21.9320	0.788840	0.394420	−	0.918930i	$-0.370946\pi$
$773$	21.9320	0.788840	0.394420	+	0.918930i	$0.370946\pi$
$774$	0	0
$775$	−0.885417	−0.0318051
$776$	−7.01634	−0.251872
$777$	0	0
$778$	−1.72594	−0.0618780
$779$	6.59741	0.236377
$780$	0	0
$781$	−7.16638	−0.256433
$782$	0.0936606	0.00334930
$783$	0	0
$784$	5.16498	0.184464
$785$	10.3585	0.369711
$786$	0	0
$787$	17.3072	0.616937	0.308468	−	0.951235i	$-0.400184\pi$
$787$	17.3072	0.616937	0.308468	+	0.951235i	$0.400184\pi$
$788$	−43.3247	−1.54338
$789$	0	0
$790$	−2.01574	−0.0717168
$791$	9.02315	0.320826
$792$	0	0
$793$	−23.7076	−0.841880
$794$	−0.0913776	−0.00324287
$795$	0	0
$796$	−25.7027	−0.911007
$797$	−18.0296	−0.638642	−0.319321	−	0.947647i	$-0.603455\pi$
$797$	−18.0296	−0.638642	−0.319321	+	0.947647i	$0.603455\pi$
$798$	0	0
$799$	−41.5113	−1.46856
$800$	−1.79477	−0.0634547
$801$	0	0
$802$	−3.11997	−0.110170
$803$	26.1064	0.921275
$804$	0	0
$805$	0.284910	0.0100418
$806$	0.375061	0.0132110
$807$	0	0
$808$	−7.35728	−0.258828
$809$	−10.9470	−0.384875	−0.192438	−	0.981309i	$-0.561639\pi$
$809$	−10.9470	−0.384875	−0.192438	+	0.981309i	$0.561639\pi$
$810$	0	0
$811$	−45.8898	−1.61141	−0.805705	−	0.592318i	$-0.798213\pi$
$811$	−45.8898	−1.61141	−0.805705	+	0.592318i	$0.798213\pi$
$812$	37.7594	1.32510
$813$	0	0
$814$	−1.68819	−0.0591711
$815$	−8.75633	−0.306721
$816$	0	0
$817$	−3.30244	−0.115538
$818$	−4.50427	−0.157488
$819$	0	0
$820$	18.0683	0.630974
$821$	−2.44052	−0.0851747	−0.0425874	−	0.999093i	$-0.513560\pi$
$821$	−2.44052	−0.0851747	−0.0425874	+	0.999093i	$0.513560\pi$
$822$	0	0
$823$	−24.4911	−0.853708	−0.426854	−	0.904321i	$-0.640378\pi$
$823$	−24.4911	−0.853708	−0.426854	+	0.904321i	$0.640378\pi$
$824$	5.86755	0.204406
$825$	0	0
$826$	1.22570	0.0426477
$827$	−32.8082	−1.14085	−0.570426	−	0.821349i	$-0.693222\pi$
$827$	−32.8082	−1.14085	−0.570426	+	0.821349i	$0.693222\pi$
$828$	0	0
$829$	−1.25808	−0.0436948	−0.0218474	−	0.999761i	$-0.506955\pi$
$829$	−1.25808	−0.0436948	−0.0218474	+	0.999761i	$0.506955\pi$
$830$	−1.71658	−0.0595833
$831$	0	0
$832$	−20.7808	−0.720445
$833$	−8.35735	−0.289565
$834$	0	0
$835$	−21.8241	−0.755253
$836$	−3.93288	−0.136021
$837$	0	0
$838$	0.982742	0.0339483
$839$	−31.3237	−1.08142	−0.540708	−	0.841211i	$-0.681843\pi$
$839$	−31.3237	−1.08142	−0.540708	+	0.841211i	$0.681843\pi$
$840$	0	0
$841$	14.7563	0.508837
$842$	4.64682	0.160140
$843$	0	0
$844$	−1.80715	−0.0622046
$845$	−5.22273	−0.179667
$846$	0	0
$847$	−9.83077	−0.337789
$848$	−2.95790	−0.101575
$849$	0	0
$850$	0.949212	0.0325577
$851$	−0.397925	−0.0136407
$852$	0	0
$853$	−45.1227	−1.54497	−0.772486	−	0.635032i	$-0.780987\pi$
$853$	−45.1227	−1.54497	−0.772486	+	0.635032i	$0.780987\pi$
$854$	3.72844	0.127585
$855$	0	0
$856$	−4.84547	−0.165615
$857$	−18.5362	−0.633186	−0.316593	−	0.948562i	$-0.602539\pi$
$857$	−18.5362	−0.633186	−0.316593	+	0.948562i	$0.602539\pi$
$858$	0	0
$859$	−15.7802	−0.538415	−0.269207	−	0.963082i	$-0.586762\pi$
$859$	−15.7802	−0.538415	−0.269207	+	0.963082i	$0.586762\pi$
$860$	−9.04439	−0.308411
$861$	0	0
$862$	−2.37893	−0.0810267
$863$	29.4551	1.00266	0.501332	−	0.865255i	$-0.332844\pi$
$863$	29.4551	1.00266	0.501332	+	0.865255i	$0.332844\pi$
$864$	0	0
$865$	8.11969	0.276078
$866$	−4.73985	−0.161067
$867$	0	0
$868$	5.05421	0.171551
$869$	−36.5736	−1.24067
$870$	0	0
$871$	11.8094	0.400145
$872$	2.93862	0.0995143
$873$	0	0
$874$	0.0108188	0.000365952 0
$875$	2.88745	0.0976135
$876$	0	0
$877$	−36.8048	−1.24281	−0.621405	−	0.783490i	$-0.713438\pi$
$877$	−36.8048	−1.24281	−0.621405	+	0.783490i	$0.713438\pi$
$878$	2.61830	0.0883634
$879$	0	0
$880$	−10.6438	−0.358803
$881$	−6.97606	−0.235029	−0.117515	−	0.993071i	$-0.537493\pi$
$881$	−6.97606	−0.235029	−0.117515	+	0.993071i	$0.537493\pi$
$882$	0	0
$883$	42.8921	1.44343	0.721717	−	0.692188i	$-0.243353\pi$
$883$	42.8921	1.44343	0.721717	+	0.692188i	$0.243353\pi$
$884$	34.4530	1.15878
$885$	0	0
$886$	0.246521	0.00828202
$887$	−32.4435	−1.08935	−0.544673	−	0.838649i	$-0.683346\pi$
$887$	−32.4435	−1.08935	−0.544673	+	0.838649i	$0.683346\pi$
$888$	0	0
$889$	−36.4975	−1.22409
$890$	0.151894	0.00509149
$891$	0	0
$892$	17.1565	0.574443
$893$	−4.79501	−0.160459
$894$	0	0
$895$	−14.6946	−0.491185
$896$	13.6328	0.455439
$897$	0	0
$898$	−5.72015	−0.190884
$899$	5.85690	0.195339
$900$	0	0
$901$	4.78611	0.159448
$902$	−3.82597	−0.127391
$903$	0	0
$904$	1.88769	0.0627838
$905$	9.35948	0.311120
$906$	0	0
$907$	31.4521	1.04435	0.522175	−	0.852838i	$-0.325121\pi$
$907$	31.4521	1.04435	0.522175	+	0.852838i	$0.325121\pi$
$908$	35.6596	1.18341
$909$	0	0
$910$	−1.22312	−0.0405459
$911$	12.0996	0.400877	0.200438	−	0.979706i	$-0.435763\pi$
$911$	12.0996	0.400877	0.200438	+	0.979706i	$0.435763\pi$
$912$	0	0
$913$	−31.1456	−1.03077
$914$	3.10403	0.102672
$915$	0	0
$916$	48.3347	1.59702
$917$	−46.9014	−1.54882
$918$	0	0
$919$	−31.0362	−1.02379	−0.511894	−	0.859049i	$-0.671056\pi$
$919$	−31.0362	−1.02379	−0.511894	+	0.859049i	$0.671056\pi$
$920$	0.0596048	0.00196511
$921$	0	0
$922$	6.00388	0.197727
$923$	7.25170	0.238692
$924$	0	0
$925$	−4.03281	−0.132598
$926$	−2.49176	−0.0818843
$927$	0	0
$928$	11.8721	0.389722
$929$	14.0194	0.459963	0.229982	−	0.973195i	$-0.426133\pi$
$929$	14.0194	0.459963	0.229982	+	0.973195i	$0.426133\pi$
$930$	0	0
$931$	−0.965364	−0.0316385
$932$	−34.5905	−1.13305
$933$	0	0
$934$	−0.214204	−0.00700897
$935$	17.2225	0.563237
$936$	0	0
$937$	38.5069	1.25797	0.628983	−	0.777419i	$-0.283471\pi$
$937$	38.5069	1.25797	0.628983	+	0.777419i	$0.283471\pi$
$938$	−1.85724	−0.0606409
$939$	0	0
$940$	−13.1321	−0.428322
$941$	−52.9697	−1.72676	−0.863381	−	0.504552i	$-0.831658\pi$
$941$	−52.9697	−1.72676	−0.863381	+	0.504552i	$0.831658\pi$
$942$	0	0
$943$	−0.901822	−0.0293674
$944$	−10.7933	−0.351292
$945$	0	0
$946$	1.91515	0.0622668
$947$	44.2927	1.43932	0.719659	−	0.694327i	$-0.244298\pi$
$947$	44.2927	1.43932	0.719659	+	0.694327i	$0.244298\pi$
$948$	0	0
$949$	−26.4172	−0.857539
$950$	0.109644	0.00355733
$951$	0	0
$952$	−10.9000	−0.353270
$953$	−11.7349	−0.380130	−0.190065	−	0.981772i	$-0.560870\pi$
$953$	−11.7349	−0.380130	−0.190065	+	0.981772i	$0.560870\pi$
$954$	0	0
$955$	−10.6961	−0.346118
$956$	14.4530	0.467445
$957$	0	0
$958$	−0.774496	−0.0250228
$959$	33.2412	1.07341
$960$	0	0
$961$	−30.2160	−0.974711
$962$	1.70829	0.0550775
$963$	0	0
$964$	52.6231	1.69488
$965$	19.0333	0.612704
$966$	0	0
$967$	−50.6434	−1.62858	−0.814291	−	0.580457i	$-0.802874\pi$
$967$	−50.6434	−1.62858	−0.814291	+	0.580457i	$0.802874\pi$
$968$	−2.05666	−0.0661034
$969$	0	0
$970$	1.76426	0.0566470
$971$	−14.2579	−0.457556	−0.228778	−	0.973479i	$-0.573473\pi$
$971$	−14.2579	−0.457556	−0.228778	+	0.973479i	$0.573473\pi$
$972$	0	0
$973$	−26.5223	−0.850265
$974$	4.60433	0.147532
$975$	0	0
$976$	−32.8320	−1.05093
$977$	32.2990	1.03334	0.516669	−	0.856185i	$-0.327172\pi$
$977$	32.2990	1.03334	0.516669	+	0.856185i	$0.327172\pi$
$978$	0	0
$979$	2.75596	0.0880810
$980$	−2.64385	−0.0844546
$981$	0	0
$982$	−2.52173	−0.0804716
$983$	28.7938	0.918378	0.459189	−	0.888339i	$-0.348140\pi$
$983$	28.7938	0.918378	0.459189	+	0.888339i	$0.348140\pi$
$984$	0	0
$985$	21.9152	0.698275
$986$	−6.27890	−0.199961
$987$	0	0
$988$	3.97970	0.126611
$989$	0.451421	0.0143544
$990$	0	0
$991$	46.2829	1.47023	0.735113	−	0.677945i	$-0.237129\pi$
$991$	46.2829	1.47023	0.735113	+	0.677945i	$0.237129\pi$
$992$	1.58912	0.0504546
$993$	0	0
$994$	−1.14046	−0.0361732
$995$	13.0013	0.412169
$996$	0	0
$997$	24.8140	0.785866	0.392933	−	0.919567i	$-0.371460\pi$
$997$	24.8140	0.785866	0.392933	+	0.919567i	$0.371460\pi$
$998$	−5.66692	−0.179383
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4005.2.a.p.1.5		8
3.2	odd	2		445.2.a.g.1.4	✓	8
12.11	even	2		7120.2.a.bk.1.6		8
15.14	odd	2		2225.2.a.l.1.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
445.2.a.g.1.4	✓	8	3.2	odd	2
2225.2.a.l.1.5		8	15.14	odd	2
4005.2.a.p.1.5		8	1.1	even	1	trivial
7120.2.a.bk.1.6		8	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)

\(f(q)\)	\(=\)	\(q-0.151894 q^{2} -1.97693 q^{4} +1.00000 q^{5} +2.88745 q^{7} +0.604071 q^{8} +O(q^{10})\)	\(q-0.151894 q^{2} -1.97693 q^{4} +1.00000 q^{5} +2.88745 q^{7} +0.604071 q^{8} -0.151894 q^{10} -2.75596 q^{11} +2.78878 q^{13} -0.438585 q^{14} +3.86210 q^{16} -6.24918 q^{17} -0.721848 q^{19} -1.97693 q^{20} +0.418614 q^{22} +0.0986719 q^{23} +1.00000 q^{25} -0.423598 q^{26} -5.70828 q^{28} -6.61485 q^{29} -0.885417 q^{31} -1.79477 q^{32} +0.949212 q^{34} +2.88745 q^{35} -4.03281 q^{37} +0.109644 q^{38} +0.604071 q^{40} -9.13960 q^{41} +4.57497 q^{43} +5.44834 q^{44} -0.0149877 q^{46} +6.64268 q^{47} +1.33735 q^{49} -0.151894 q^{50} -5.51321 q^{52} -0.765878 q^{53} -2.75596 q^{55} +1.74422 q^{56} +1.00476 q^{58} -2.79467 q^{59} -8.50106 q^{61} +0.134489 q^{62} -7.45159 q^{64} +2.78878 q^{65} +4.23461 q^{67} +12.3542 q^{68} -0.438585 q^{70} +2.60032 q^{71} -9.47269 q^{73} +0.612559 q^{74} +1.42704 q^{76} -7.95770 q^{77} +13.2707 q^{79} +3.86210 q^{80} +1.38825 q^{82} +11.3012 q^{83} -6.24918 q^{85} -0.694910 q^{86} -1.66480 q^{88} -1.00000 q^{89} +8.05244 q^{91} -0.195067 q^{92} -1.00898 q^{94} -0.721848 q^{95} -11.6151 q^{97} -0.203135 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - q^{2} + 7 q^{4} + 8 q^{5} - 6 q^{7} - 3 q^{8}+O(q^{10}) \) 8 * q - q^2 + 7 * q^4 + 8 * q^5 - 6 * q^7 - 3 * q^8	\( 8 q - q^{2} + 7 q^{4} + 8 q^{5} - 6 q^{7} - 3 q^{8} - q^{10} - 14 q^{11} - 7 q^{13} - 15 q^{14} + 9 q^{16} - 17 q^{17} + 17 q^{19} + 7 q^{20} + 2 q^{22} + q^{23} + 8 q^{25} - 3 q^{26} - 29 q^{28} - 10 q^{29} + q^{31} - 2 q^{32} - 16 q^{34} - 6 q^{35} - 11 q^{37} + 30 q^{38} - 3 q^{40} - 15 q^{41} - 5 q^{43} - 7 q^{44} - 12 q^{46} - 12 q^{47} + 4 q^{49} - q^{50} - 14 q^{52} + q^{53} - 14 q^{55} - 3 q^{56} - 37 q^{58} - 26 q^{59} + 13 q^{61} - 22 q^{62} - 15 q^{64} - 7 q^{65} - 25 q^{67} - 23 q^{68} - 15 q^{70} - 28 q^{71} - 17 q^{73} + 5 q^{74} + 8 q^{76} - 7 q^{79} + 9 q^{80} + 5 q^{82} - 44 q^{83} - 17 q^{85} + 13 q^{86} - 66 q^{88} - 8 q^{89} + 27 q^{91} - 15 q^{92} - 27 q^{94} + 17 q^{95} + q^{97} + 34 q^{98}+O(q^{100}) \) 8 * q - q^2 + 7 * q^4 + 8 * q^5 - 6 * q^7 - 3 * q^8 - q^10 - 14 * q^11 - 7 * q^13 - 15 * q^14 + 9 * q^16 - 17 * q^17 + 17 * q^19 + 7 * q^20 + 2 * q^22 + q^23 + 8 * q^25 - 3 * q^26 - 29 * q^28 - 10 * q^29 + q^31 - 2 * q^32 - 16 * q^34 - 6 * q^35 - 11 * q^37 + 30 * q^38 - 3 * q^40 - 15 * q^41 - 5 * q^43 - 7 * q^44 - 12 * q^46 - 12 * q^47 + 4 * q^49 - q^50 - 14 * q^52 + q^53 - 14 * q^55 - 3 * q^56 - 37 * q^58 - 26 * q^59 + 13 * q^61 - 22 * q^62 - 15 * q^64 - 7 * q^65 - 25 * q^67 - 23 * q^68 - 15 * q^70 - 28 * q^71 - 17 * q^73 + 5 * q^74 + 8 * q^76 - 7 * q^79 + 9 * q^80 + 5 * q^82 - 44 * q^83 - 17 * q^85 + 13 * q^86 - 66 * q^88 - 8 * q^89 + 27 * q^91 - 15 * q^92 - 27 * q^94 + 17 * q^95 + q^97 + 34 * q^98

Introduction

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