LMFDB - Embedded newform 4005.2.a.u.1.6

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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 13 x^{10} + 41 x^{9} + 58 x^{8} - 202 x^{7} - 95 x^{6} + 432 x^{5} + 4 x^{4} + \cdots - 27 $ x^12 - 3x^11 - 13x^10 + 41x^9 + 58x^8 - 202x^7 - 95x^6 + 432x^5 + 4x^4 - 368x^3 + 94x^2 + 77*x - 27
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$0.491103$ 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)

$f(q)$	$=$	$q-0.491103 q^{2} -1.75882 q^{4} +1.00000 q^{5} -1.39011 q^{7} +1.84597 q^{8} +O(q^{10})$	\(q-0.491103 q^{2} -1.75882 q^{4} +1.00000 q^{5} -1.39011 q^{7} +1.84597 q^{8} -0.491103 q^{10} +0.843828 q^{11} -2.66669 q^{13} +0.682689 q^{14} +2.61108 q^{16} -3.46246 q^{17} +5.51842 q^{19} -1.75882 q^{20} -0.414406 q^{22} -5.20352 q^{23} +1.00000 q^{25} +1.30962 q^{26} +2.44496 q^{28} +0.846016 q^{29} +8.90482 q^{31} -4.97424 q^{32} +1.70043 q^{34} -1.39011 q^{35} -6.81736 q^{37} -2.71011 q^{38} +1.84597 q^{40} +7.41095 q^{41} +2.31932 q^{43} -1.48414 q^{44} +2.55546 q^{46} -6.32137 q^{47} -5.06758 q^{49} -0.491103 q^{50} +4.69022 q^{52} +3.08500 q^{53} +0.843828 q^{55} -2.56610 q^{56} -0.415481 q^{58} +14.0512 q^{59} -9.19572 q^{61} -4.37318 q^{62} -2.77929 q^{64} -2.66669 q^{65} -12.3165 q^{67} +6.08984 q^{68} +0.682689 q^{70} -7.32187 q^{71} -7.96382 q^{73} +3.34803 q^{74} -9.70590 q^{76} -1.17302 q^{77} +4.56446 q^{79} +2.61108 q^{80} -3.63954 q^{82} +7.49427 q^{83} -3.46246 q^{85} -1.13902 q^{86} +1.55768 q^{88} +1.00000 q^{89} +3.70701 q^{91} +9.15204 q^{92} +3.10445 q^{94} +5.51842 q^{95} +12.2871 q^{97} +2.48871 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 3 q^{2} + 11 q^{4} + 12 q^{5} - 8 q^{7} - 9 q^{8}+O(q^{10}) $ 12 * q - 3 * q^2 + 11 * q^4 + 12 * q^5 - 8 * q^7 - 9 * q^8	$ 12 q - 3 q^{2} + 11 q^{4} + 12 q^{5} - 8 q^{7} - 9 q^{8} - 3 q^{10} - 12 q^{13} + 4 q^{14} + q^{16} - 24 q^{19} + 11 q^{20} - 16 q^{22} - 24 q^{23} + 12 q^{25} - q^{26} - 44 q^{28} + 8 q^{29} - 12 q^{31} - 31 q^{32} - 18 q^{34} - 8 q^{35} - 10 q^{37} + 2 q^{38} - 9 q^{40} - 10 q^{41} - 42 q^{43} + 42 q^{44} - 24 q^{46} - 22 q^{47} - 4 q^{49} - 3 q^{50} - 30 q^{52} + 8 q^{53} + 27 q^{56} - 12 q^{58} + 4 q^{59} - 52 q^{61} - 14 q^{62} + 7 q^{64} - 12 q^{65} - 40 q^{67} + 23 q^{68} + 4 q^{70} + 2 q^{71} - 8 q^{73} + 26 q^{74} - 46 q^{76} - 12 q^{77} - 26 q^{79} + q^{80} - 26 q^{82} - 14 q^{83} + 32 q^{86} - 60 q^{88} + 12 q^{89} - 24 q^{91} - 38 q^{92} - 26 q^{94} - 24 q^{95} - 6 q^{97} + 30 q^{98}+O(q^{100}) $ 12 * q - 3 * q^2 + 11 * q^4 + 12 * q^5 - 8 * q^7 - 9 * q^8 - 3 * q^10 - 12 * q^13 + 4 * q^14 + q^16 - 24 * q^19 + 11 * q^20 - 16 * q^22 - 24 * q^23 + 12 * q^25 - q^26 - 44 * q^28 + 8 * q^29 - 12 * q^31 - 31 * q^32 - 18 * q^34 - 8 * q^35 - 10 * q^37 + 2 * q^38 - 9 * q^40 - 10 * q^41 - 42 * q^43 + 42 * q^44 - 24 * q^46 - 22 * q^47 - 4 * q^49 - 3 * q^50 - 30 * q^52 + 8 * q^53 + 27 * q^56 - 12 * q^58 + 4 * q^59 - 52 * q^61 - 14 * q^62 + 7 * q^64 - 12 * q^65 - 40 * q^67 + 23 * q^68 + 4 * q^70 + 2 * q^71 - 8 * q^73 + 26 * q^74 - 46 * q^76 - 12 * q^77 - 26 * q^79 + q^80 - 26 * q^82 - 14 * q^83 + 32 * q^86 - 60 * q^88 + 12 * q^89 - 24 * q^91 - 38 * q^92 - 26 * q^94 - 24 * q^95 - 6 * q^97 + 30 * 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 $

$2$	−0.491103	−0.347262	−0.173631	−	0.984811i	$-0.555550\pi$
$2$	−0.491103	−0.347262	−0.173631	+	0.984811i	$0.555550\pi$
$3$	0	0
$3$	0	0
$4$	−1.75882	−0.879409
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.39011	−0.525414	−0.262707	−	0.964876i	$-0.584615\pi$
$7$	−1.39011	−0.525414	−0.262707	+	0.964876i	$0.584615\pi$
$8$	1.84597	0.652648
$9$	0	0
$10$	−0.491103	−0.155300
$11$	0.843828	0.254424	0.127212	−	0.991876i	$-0.459397\pi$
$11$	0.843828	0.254424	0.127212	+	0.991876i	$0.459397\pi$
$12$	0	0
$13$	−2.66669	−0.739607	−0.369804	−	0.929110i	$-0.620575\pi$
$13$	−2.66669	−0.739607	−0.369804	+	0.929110i	$0.620575\pi$
$14$	0.682689	0.182456
$15$	0	0
$16$	2.61108	0.652769
$17$	−3.46246	−0.839771	−0.419885	−	0.907577i	$-0.637930\pi$
$17$	−3.46246	−0.839771	−0.419885	+	0.907577i	$0.637930\pi$
$18$	0	0
$19$	5.51842	1.26601	0.633006	−	0.774147i	$-0.281821\pi$
$19$	5.51842	1.26601	0.633006	+	0.774147i	$0.281821\pi$
$20$	−1.75882	−0.393284
$21$	0	0
$22$	−0.414406	−0.0883517
$23$	−5.20352	−1.08501	−0.542504	−	0.840053i	$-0.682524\pi$
$23$	−5.20352	−1.08501	−0.542504	+	0.840053i	$0.682524\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	1.30962	0.256838
$27$	0	0
$28$	2.44496	0.462054
$29$	0.846016	0.157101	0.0785507	−	0.996910i	$-0.474971\pi$
$29$	0.846016	0.157101	0.0785507	+	0.996910i	$0.474971\pi$
$30$	0	0
$31$	8.90482	1.59935	0.799676	−	0.600432i	$-0.205005\pi$
$31$	8.90482	1.59935	0.799676	+	0.600432i	$0.205005\pi$
$32$	−4.97424	−0.879330
$33$	0	0
$34$	1.70043	0.291621
$35$	−1.39011	−0.234972
$36$	0	0
$37$	−6.81736	−1.12077	−0.560384	−	0.828233i	$-0.689346\pi$
$37$	−6.81736	−1.12077	−0.560384	+	0.828233i	$0.689346\pi$
$38$	−2.71011	−0.439638
$39$	0	0
$40$	1.84597	0.291873
$41$	7.41095	1.15740	0.578698	−	0.815542i	$-0.303561\pi$
$41$	7.41095	1.15740	0.578698	+	0.815542i	$0.303561\pi$
$42$	0	0
$43$	2.31932	0.353693	0.176846	−	0.984238i	$-0.443410\pi$
$43$	2.31932	0.353693	0.176846	+	0.984238i	$0.443410\pi$
$44$	−1.48414	−0.223742
$45$	0	0
$46$	2.55546	0.376783
$47$	−6.32137	−0.922067	−0.461033	−	0.887383i	$-0.652521\pi$
$47$	−6.32137	−0.922067	−0.461033	+	0.887383i	$0.652521\pi$
$48$	0	0
$49$	−5.06758	−0.723940
$50$	−0.491103	−0.0694525
$51$	0	0
$52$	4.69022	0.650417
$53$	3.08500	0.423757	0.211879	−	0.977296i	$-0.432042\pi$
$53$	3.08500	0.423757	0.211879	+	0.977296i	$0.432042\pi$
$54$	0	0
$55$	0.843828	0.113782
$56$	−2.56610	−0.342910
$57$	0	0
$58$	−0.415481	−0.0545554
$59$	14.0512	1.82931	0.914654	−	0.404237i	$-0.132463\pi$
$59$	14.0512	1.82931	0.914654	+	0.404237i	$0.132463\pi$
$60$	0	0
$61$	−9.19572	−1.17739	−0.588696	−	0.808355i	$-0.700358\pi$
$61$	−9.19572	−1.17739	−0.588696	+	0.808355i	$0.700358\pi$
$62$	−4.37318	−0.555395
$63$	0	0
$64$	−2.77929	−0.347411
$65$	−2.66669	−0.330762
$66$	0	0
$67$	−12.3165	−1.50470	−0.752351	−	0.658763i	$-0.771080\pi$
$67$	−12.3165	−1.50470	−0.752351	+	0.658763i	$0.771080\pi$
$68$	6.08984	0.738502
$69$	0	0
$70$	0.682689	0.0815970
$71$	−7.32187	−0.868946	−0.434473	−	0.900685i	$-0.643065\pi$
$71$	−7.32187	−0.868946	−0.434473	+	0.900685i	$0.643065\pi$
$72$	0	0
$73$	−7.96382	−0.932095	−0.466047	−	0.884760i	$-0.654322\pi$
$73$	−7.96382	−0.932095	−0.466047	+	0.884760i	$0.654322\pi$
$74$	3.34803	0.389200
$75$	0	0
$76$	−9.70590	−1.11334
$77$	−1.17302	−0.133678
$78$	0	0
$79$	4.56446	0.513542	0.256771	−	0.966472i	$-0.417341\pi$
$79$	4.56446	0.513542	0.256771	+	0.966472i	$0.417341\pi$
$80$	2.61108	0.291927
$81$	0	0
$82$	−3.63954	−0.401920
$83$	7.49427	0.822603	0.411302	−	0.911499i	$-0.365074\pi$
$83$	7.49427	0.822603	0.411302	+	0.911499i	$0.365074\pi$
$84$	0	0
$85$	−3.46246	−0.375557
$86$	−1.13902	−0.122824
$87$	0	0
$88$	1.55768	0.166049
$89$	1.00000	0.106000
$89$	1.00000	0.106000
$90$	0	0
$91$	3.70701	0.388600
$92$	9.15204	0.954166
$93$	0	0
$94$	3.10445	0.320199
$95$	5.51842	0.566178
$96$	0	0
$97$	12.2871	1.24756	0.623781	−	0.781599i	$-0.285596\pi$
$97$	12.2871	1.24756	0.623781	+	0.781599i	$0.285596\pi$
$98$	2.48871	0.251397
$99$	0	0
$100$	−1.75882	−0.175882
$101$	16.3445	1.62633	0.813167	−	0.582030i	$-0.197742\pi$
$101$	16.3445	1.62633	0.813167	+	0.582030i	$0.197742\pi$
$102$	0	0
$103$	2.89199	0.284956	0.142478	−	0.989798i	$-0.454493\pi$
$103$	2.89199	0.284956	0.142478	+	0.989798i	$0.454493\pi$
$104$	−4.92262	−0.482703
$105$	0	0
$106$	−1.51505	−0.147155
$107$	4.86649	0.470461	0.235231	−	0.971940i	$-0.424415\pi$
$107$	4.86649	0.470461	0.235231	+	0.971940i	$0.424415\pi$
$108$	0	0
$109$	−19.8869	−1.90482	−0.952409	−	0.304824i	$-0.901402\pi$
$109$	−19.8869	−1.90482	−0.952409	+	0.304824i	$0.901402\pi$
$110$	−0.414406	−0.0395121
$111$	0	0
$112$	−3.62969	−0.342974
$113$	−13.4756	−1.26768	−0.633838	−	0.773466i	$-0.718521\pi$
$113$	−13.4756	−1.26768	−0.633838	+	0.773466i	$0.718521\pi$
$114$	0	0
$115$	−5.20352	−0.485231
$116$	−1.48799	−0.138156
$117$	0	0
$118$	−6.90058	−0.635250
$119$	4.81322	0.441227
$120$	0	0
$121$	−10.2880	−0.935269
$122$	4.51605	0.408864
$123$	0	0
$124$	−15.6619	−1.40648
$125$	1.00000	0.0894427
$126$	0	0
$127$	−12.3132	−1.09262	−0.546309	−	0.837584i	$-0.683968\pi$
$127$	−12.3132	−1.09262	−0.546309	+	0.837584i	$0.683968\pi$
$128$	11.3134	0.999973
$129$	0	0
$130$	1.30962	0.114861
$131$	−18.9390	−1.65471	−0.827353	−	0.561683i	$-0.810154\pi$
$131$	−18.9390	−1.65471	−0.827353	+	0.561683i	$0.810154\pi$
$132$	0	0
$133$	−7.67124	−0.665181
$134$	6.04868	0.522526
$135$	0	0
$136$	−6.39159	−0.548075
$137$	−15.3806	−1.31405	−0.657025	−	0.753869i	$-0.728185\pi$
$137$	−15.3806	−1.31405	−0.657025	+	0.753869i	$0.728185\pi$
$138$	0	0
$139$	−8.87907	−0.753113	−0.376556	−	0.926394i	$-0.622892\pi$
$139$	−8.87907	−0.753113	−0.376556	+	0.926394i	$0.622892\pi$
$140$	2.44496	0.206637
$141$	0	0
$142$	3.59579	0.301752
$143$	−2.25023	−0.188174
$144$	0	0
$145$	0.846016	0.0702578
$146$	3.91106	0.323681
$147$	0	0
$148$	11.9905	0.985613
$149$	13.6746	1.12027	0.560135	−	0.828402i	$-0.310749\pi$
$149$	13.6746	1.12027	0.560135	+	0.828402i	$0.310749\pi$
$150$	0	0
$151$	10.0156	0.815059	0.407530	−	0.913192i	$-0.366390\pi$
$151$	10.0156	0.815059	0.407530	+	0.913192i	$0.366390\pi$
$152$	10.1868	0.826260
$153$	0	0
$154$	0.576072	0.0464212
$155$	8.90482	0.715252
$156$	0	0
$157$	11.8191	0.943263	0.471632	−	0.881796i	$-0.343665\pi$
$157$	11.8191	0.943263	0.471632	+	0.881796i	$0.343665\pi$
$158$	−2.24162	−0.178334
$159$	0	0
$160$	−4.97424	−0.393248
$161$	7.23349	0.570079
$162$	0	0
$163$	5.72799	0.448651	0.224325	−	0.974514i	$-0.427982\pi$
$163$	5.72799	0.448651	0.224325	+	0.974514i	$0.427982\pi$
$164$	−13.0345	−1.01782
$165$	0	0
$166$	−3.68046	−0.285659
$167$	−5.55230	−0.429650	−0.214825	−	0.976653i	$-0.568918\pi$
$167$	−5.55230	−0.429650	−0.214825	+	0.976653i	$0.568918\pi$
$168$	0	0
$169$	−5.88876	−0.452981
$170$	1.70043	0.130417
$171$	0	0
$172$	−4.07926	−0.311041
$173$	17.5445	1.33388	0.666941	−	0.745110i	$-0.267603\pi$
$173$	17.5445	1.33388	0.666941	+	0.745110i	$0.267603\pi$
$174$	0	0
$175$	−1.39011	−0.105083
$176$	2.20330	0.166080
$177$	0	0
$178$	−0.491103	−0.0368097
$179$	−13.3615	−0.998684	−0.499342	−	0.866405i	$-0.666425\pi$
$179$	−13.3615	−0.998684	−0.499342	+	0.866405i	$0.666425\pi$
$180$	0	0
$181$	18.4053	1.36806	0.684029	−	0.729455i	$-0.260226\pi$
$181$	18.4053	1.36806	0.684029	+	0.729455i	$0.260226\pi$
$182$	−1.82052	−0.134946
$183$	0	0
$184$	−9.60553	−0.708129
$185$	−6.81736	−0.501222
$186$	0	0
$187$	−2.92172	−0.213658
$188$	11.1181	0.810874
$189$	0	0
$190$	−2.71011	−0.196612
$191$	−26.9635	−1.95101	−0.975507	−	0.219970i	$-0.929404\pi$
$191$	−26.9635	−1.95101	−0.975507	+	0.219970i	$0.929404\pi$
$192$	0	0
$193$	−20.9722	−1.50961	−0.754807	−	0.655947i	$-0.772269\pi$
$193$	−20.9722	−1.50961	−0.754807	+	0.655947i	$0.772269\pi$
$194$	−6.03421	−0.433231
$195$	0	0
$196$	8.91295	0.636640
$197$	−15.2450	−1.08616	−0.543080	−	0.839681i	$-0.682742\pi$
$197$	−15.2450	−1.08616	−0.543080	+	0.839681i	$0.682742\pi$
$198$	0	0
$199$	−13.6672	−0.968840	−0.484420	−	0.874836i	$-0.660969\pi$
$199$	−13.6672	−0.968840	−0.484420	+	0.874836i	$0.660969\pi$
$200$	1.84597	0.130530
$201$	0	0
$202$	−8.02681	−0.564764
$203$	−1.17606	−0.0825432
$204$	0	0
$205$	7.41095	0.517603
$206$	−1.42027	−0.0989546
$207$	0	0
$208$	−6.96293	−0.482793
$209$	4.65660	0.322104
$210$	0	0
$211$	−26.5395	−1.82706	−0.913528	−	0.406775i	$-0.866653\pi$
$211$	−26.5395	−1.82706	−0.913528	+	0.406775i	$0.866653\pi$
$212$	−5.42595	−0.372656
$213$	0	0
$214$	−2.38995	−0.163374
$215$	2.31932	0.158176
$216$	0	0
$217$	−12.3787	−0.840322
$218$	9.76651	0.661471
$219$	0	0
$220$	−1.48414	−0.100061
$221$	9.23332	0.621101
$222$	0	0
$223$	8.64863	0.579155	0.289578	−	0.957155i	$-0.406485\pi$
$223$	8.64863	0.579155	0.289578	+	0.957155i	$0.406485\pi$
$224$	6.91476	0.462012
$225$	0	0
$226$	6.61790	0.440216
$227$	−9.64422	−0.640109	−0.320055	−	0.947399i	$-0.603701\pi$
$227$	−9.64422	−0.640109	−0.320055	+	0.947399i	$0.603701\pi$
$228$	0	0
$229$	−9.35677	−0.618313	−0.309156	−	0.951011i	$-0.600047\pi$
$229$	−9.35677	−0.618313	−0.309156	+	0.951011i	$0.600047\pi$
$230$	2.55546	0.168502
$231$	0	0
$232$	1.56172	0.102532
$233$	−2.94961	−0.193236	−0.0966178	−	0.995322i	$-0.530802\pi$
$233$	−2.94961	−0.193236	−0.0966178	+	0.995322i	$0.530802\pi$
$234$	0	0
$235$	−6.32137	−0.412361
$236$	−24.7135	−1.60871
$237$	0	0
$238$	−2.36379	−0.153222
$239$	−5.92878	−0.383501	−0.191751	−	0.981444i	$-0.561416\pi$
$239$	−5.92878	−0.383501	−0.191751	+	0.981444i	$0.561416\pi$
$240$	0	0
$241$	0.498640	0.0321202	0.0160601	−	0.999871i	$-0.494888\pi$
$241$	0.498640	0.0321202	0.0160601	+	0.999871i	$0.494888\pi$
$242$	5.05245	0.324784
$243$	0	0
$244$	16.1736	1.03541
$245$	−5.06758	−0.323756
$246$	0	0
$247$	−14.7159	−0.936352
$248$	16.4380	1.04381
$249$	0	0
$250$	−0.491103	−0.0310601
$251$	9.48289	0.598555	0.299277	−	0.954166i	$-0.403254\pi$
$251$	9.48289	0.598555	0.299277	+	0.954166i	$0.403254\pi$
$252$	0	0
$253$	−4.39087	−0.276052
$254$	6.04704	0.379425
$255$	0	0
$256$	0.00252865	0.000158041 0
$257$	−16.6866	−1.04088	−0.520442	−	0.853897i	$-0.674233\pi$
$257$	−16.6866	−1.04088	−0.520442	+	0.853897i	$0.674233\pi$
$258$	0	0
$259$	9.47691	0.588867
$260$	4.69022	0.290875
$261$	0	0
$262$	9.30099	0.574617
$263$	20.6328	1.27227	0.636136	−	0.771577i	$-0.280532\pi$
$263$	20.6328	1.27227	0.636136	+	0.771577i	$0.280532\pi$
$264$	0	0
$265$	3.08500	0.189510
$266$	3.76737	0.230992
$267$	0	0
$268$	21.6625	1.32325
$269$	−20.3306	−1.23958	−0.619788	−	0.784769i	$-0.712782\pi$
$269$	−20.3306	−1.23958	−0.619788	+	0.784769i	$0.712782\pi$
$270$	0	0
$271$	−17.7783	−1.07996	−0.539979	−	0.841679i	$-0.681568\pi$
$271$	−17.7783	−1.07996	−0.539979	+	0.841679i	$0.681568\pi$
$272$	−9.04076	−0.548176
$273$	0	0
$274$	7.55344	0.456320
$275$	0.843828	0.0508847
$276$	0	0
$277$	−18.5352	−1.11367	−0.556835	−	0.830623i	$-0.687984\pi$
$277$	−18.5352	−1.11367	−0.556835	+	0.830623i	$0.687984\pi$
$278$	4.36054	0.261528
$279$	0	0
$280$	−2.56610	−0.153354
$281$	−16.2926	−0.971936	−0.485968	−	0.873977i	$-0.661533\pi$
$281$	−16.2926	−0.971936	−0.485968	+	0.873977i	$0.661533\pi$
$282$	0	0
$283$	−32.7171	−1.94483	−0.972414	−	0.233260i	$-0.925061\pi$
$283$	−32.7171	−1.94483	−0.972414	+	0.233260i	$0.925061\pi$
$284$	12.8778	0.764159
$285$	0	0
$286$	1.10509	0.0653456
$287$	−10.3021	−0.608112
$288$	0	0
$289$	−5.01134	−0.294785
$290$	−0.415481	−0.0243979
$291$	0	0
$292$	14.0069	0.819693
$293$	33.3554	1.94864	0.974322	−	0.225161i	$-0.0722908\pi$
$293$	33.3554	1.94864	0.974322	+	0.225161i	$0.0722908\pi$
$294$	0	0
$295$	14.0512	0.818091
$296$	−12.5846	−0.731466
$297$	0	0
$298$	−6.71565	−0.389027
$299$	13.8762	0.802480
$300$	0	0
$301$	−3.22412	−0.185835
$302$	−4.91870	−0.283039
$303$	0	0
$304$	14.4090	0.826414
$305$	−9.19572	−0.526546
$306$	0	0
$307$	−21.9311	−1.25167	−0.625837	−	0.779954i	$-0.715242\pi$
$307$	−21.9311	−1.25167	−0.625837	+	0.779954i	$0.715242\pi$
$308$	2.06312	0.117557
$309$	0	0
$310$	−4.37318	−0.248380
$311$	28.4307	1.61216	0.806078	−	0.591810i	$-0.201586\pi$
$311$	28.4307	1.61216	0.806078	+	0.591810i	$0.201586\pi$
$312$	0	0
$313$	15.8432	0.895510	0.447755	−	0.894156i	$-0.352224\pi$
$313$	15.8432	0.895510	0.447755	+	0.894156i	$0.352224\pi$
$314$	−5.80437	−0.327560
$315$	0	0
$316$	−8.02806	−0.451614
$317$	−29.3576	−1.64888	−0.824442	−	0.565946i	$-0.808511\pi$
$317$	−29.3576	−1.64888	−0.824442	+	0.565946i	$0.808511\pi$
$318$	0	0
$319$	0.713892	0.0399703
$320$	−2.77929	−0.155367
$321$	0	0
$322$	−3.55239	−0.197967
$323$	−19.1073	−1.06316
$324$	0	0
$325$	−2.66669	−0.147921
$326$	−2.81303	−0.155800
$327$	0	0
$328$	13.6804	0.755372
$329$	8.78743	0.484467
$330$	0	0
$331$	−17.3846	−0.955542	−0.477771	−	0.878485i	$-0.658555\pi$
$331$	−17.3846	−0.955542	−0.477771	+	0.878485i	$0.658555\pi$
$332$	−13.1811	−0.723405
$333$	0	0
$334$	2.72675	0.149201
$335$	−12.3165	−0.672923
$336$	0	0
$337$	−15.6331	−0.851588	−0.425794	−	0.904820i	$-0.640005\pi$
$337$	−15.6331	−0.851588	−0.425794	+	0.904820i	$0.640005\pi$
$338$	2.89199	0.157303
$339$	0	0
$340$	6.08984	0.330268
$341$	7.51413	0.406913
$342$	0	0
$343$	16.7753	0.905782
$344$	4.28138	0.230837
$345$	0	0
$346$	−8.61615	−0.463207
$347$	25.4096	1.36406	0.682029	−	0.731325i	$-0.261098\pi$
$347$	25.4096	1.36406	0.682029	+	0.731325i	$0.261098\pi$
$348$	0	0
$349$	−3.60653	−0.193053	−0.0965266	−	0.995330i	$-0.530773\pi$
$349$	−3.60653	−0.193053	−0.0965266	+	0.995330i	$0.530773\pi$
$350$	0.682689	0.0364913
$351$	0	0
$352$	−4.19740	−0.223722
$353$	−13.6806	−0.728145	−0.364072	−	0.931371i	$-0.618614\pi$
$353$	−13.6806	−0.728145	−0.364072	+	0.931371i	$0.618614\pi$
$354$	0	0
$355$	−7.32187	−0.388604
$356$	−1.75882	−0.0932172
$357$	0	0
$358$	6.56187	0.346805
$359$	7.94331	0.419232	0.209616	−	0.977784i	$-0.432779\pi$
$359$	7.94331	0.419232	0.209616	+	0.977784i	$0.432779\pi$
$360$	0	0
$361$	11.4530	0.602788
$362$	−9.03892	−0.475075
$363$	0	0
$364$	−6.51995	−0.341738
$365$	−7.96382	−0.416845
$366$	0	0
$367$	−0.168250	−0.00878258	−0.00439129	−	0.999990i	$-0.501398\pi$
$367$	−0.168250	−0.00878258	−0.00439129	+	0.999990i	$0.501398\pi$
$368$	−13.5868	−0.708260
$369$	0	0
$370$	3.34803	0.174056
$371$	−4.28850	−0.222648
$372$	0	0
$373$	−8.38560	−0.434190	−0.217095	−	0.976150i	$-0.569658\pi$
$373$	−8.38560	−0.434190	−0.217095	+	0.976150i	$0.569658\pi$
$374$	1.43487	0.0741952
$375$	0	0
$376$	−11.6690	−0.601785
$377$	−2.25607	−0.116193
$378$	0	0
$379$	−34.4271	−1.76840	−0.884201	−	0.467107i	$-0.845296\pi$
$379$	−34.4271	−1.76840	−0.884201	+	0.467107i	$0.845296\pi$
$380$	−9.70590	−0.497902
$381$	0	0
$382$	13.2419	0.677513
$383$	31.7498	1.62234	0.811169	−	0.584812i	$-0.198832\pi$
$383$	31.7498	1.62234	0.811169	+	0.584812i	$0.198832\pi$
$384$	0	0
$385$	−1.17302	−0.0597825
$386$	10.2995	0.524232
$387$	0	0
$388$	−21.6107	−1.09712
$389$	32.9427	1.67026	0.835131	−	0.550051i	$-0.185392\pi$
$389$	32.9427	1.67026	0.835131	+	0.550051i	$0.185392\pi$
$390$	0	0
$391$	18.0170	0.911159
$392$	−9.35459	−0.472478
$393$	0	0
$394$	7.48686	0.377182
$395$	4.56446	0.229663
$396$	0	0
$397$	14.0717	0.706238	0.353119	−	0.935578i	$-0.385121\pi$
$397$	14.0717	0.706238	0.353119	+	0.935578i	$0.385121\pi$
$398$	6.71199	0.336442
$399$	0	0
$400$	2.61108	0.130554
$401$	7.27441	0.363267	0.181633	−	0.983366i	$-0.441862\pi$
$401$	7.27441	0.363267	0.181633	+	0.983366i	$0.441862\pi$
$402$	0	0
$403$	−23.7464	−1.18289
$404$	−28.7469	−1.43021
$405$	0	0
$406$	0.577566	0.0286641
$407$	−5.75268	−0.285150
$408$	0	0
$409$	27.2480	1.34733	0.673663	−	0.739039i	$-0.264720\pi$
$409$	27.2480	1.34733	0.673663	+	0.739039i	$0.264720\pi$
$410$	−3.63954	−0.179744
$411$	0	0
$412$	−5.08648	−0.250593
$413$	−19.5327	−0.961144
$414$	0	0
$415$	7.49427	0.367879
$416$	13.2648	0.650359
$417$	0	0
$418$	−2.28687	−0.111854
$419$	−13.7713	−0.672772	−0.336386	−	0.941724i	$-0.609205\pi$
$419$	−13.7713	−0.672772	−0.336386	+	0.941724i	$0.609205\pi$
$420$	0	0
$421$	−3.36210	−0.163859	−0.0819293	−	0.996638i	$-0.526108\pi$
$421$	−3.36210	−0.163859	−0.0819293	+	0.996638i	$0.526108\pi$
$422$	13.0336	0.634468
$423$	0	0
$424$	5.69481	0.276564
$425$	−3.46246	−0.167954
$426$	0	0
$427$	12.7831	0.618618
$428$	−8.55927	−0.413728
$429$	0	0
$430$	−1.13902	−0.0549286
$431$	5.92980	0.285628	0.142814	−	0.989750i	$-0.454385\pi$
$431$	5.92980	0.285628	0.142814	+	0.989750i	$0.454385\pi$
$432$	0	0
$433$	−16.5057	−0.793215	−0.396607	−	0.917988i	$-0.629813\pi$
$433$	−16.5057	−0.793215	−0.396607	+	0.917988i	$0.629813\pi$
$434$	6.07922	0.291812
$435$	0	0
$436$	34.9774	1.67511
$437$	−28.7152	−1.37364
$438$	0	0
$439$	−16.2327	−0.774747	−0.387373	−	0.921923i	$-0.626618\pi$
$439$	−16.2327	−0.774747	−0.387373	+	0.921923i	$0.626618\pi$
$440$	1.55768	0.0742594
$441$	0	0
$442$	−4.53451	−0.215685
$443$	−2.61548	−0.124265	−0.0621327	−	0.998068i	$-0.519790\pi$
$443$	−2.61548	−0.124265	−0.0621327	+	0.998068i	$0.519790\pi$
$444$	0	0
$445$	1.00000	0.0474045
$446$	−4.24737	−0.201119
$447$	0	0
$448$	3.86353	0.182534
$449$	−24.4580	−1.15424	−0.577122	−	0.816658i	$-0.695824\pi$
$449$	−24.4580	−1.15424	−0.577122	+	0.816658i	$0.695824\pi$
$450$	0	0
$451$	6.25356	0.294469
$452$	23.7011	1.11481
$453$	0	0
$454$	4.73631	0.222286
$455$	3.70701	0.173787
$456$	0	0
$457$	35.3711	1.65459	0.827294	−	0.561769i	$-0.189879\pi$
$457$	35.3711	1.65459	0.827294	+	0.561769i	$0.189879\pi$
$458$	4.59514	0.214717
$459$	0	0
$460$	9.15204	0.426716
$461$	−7.17198	−0.334033	−0.167016	−	0.985954i	$-0.553413\pi$
$461$	−7.17198	−0.334033	−0.167016	+	0.985954i	$0.553413\pi$
$462$	0	0
$463$	−29.6301	−1.37703	−0.688513	−	0.725224i	$-0.741736\pi$
$463$	−29.6301	−1.37703	−0.688513	+	0.725224i	$0.741736\pi$
$464$	2.20901	0.102551
$465$	0	0
$466$	1.44856	0.0671035
$467$	−22.3012	−1.03198	−0.515988	−	0.856596i	$-0.672575\pi$
$467$	−22.3012	−1.03198	−0.515988	+	0.856596i	$0.672575\pi$
$468$	0	0
$469$	17.1214	0.790591
$470$	3.10445	0.143197
$471$	0	0
$472$	25.9380	1.19389
$473$	1.95711	0.0899878
$474$	0	0
$475$	5.51842	0.253203
$476$	−8.46558	−0.388019
$477$	0	0
$478$	2.91164	0.133175
$479$	−4.79431	−0.219058	−0.109529	−	0.993984i	$-0.534934\pi$
$479$	−4.79431	−0.219058	−0.109529	+	0.993984i	$0.534934\pi$
$480$	0	0
$481$	18.1798	0.828928
$482$	−0.244883	−0.0111541
$483$	0	0
$484$	18.0946	0.822484
$485$	12.2871	0.557926
$486$	0	0
$487$	−13.8404	−0.627168	−0.313584	−	0.949560i	$-0.601530\pi$
$487$	−13.8404	−0.627168	−0.313584	+	0.949560i	$0.601530\pi$
$488$	−16.9750	−0.768422
$489$	0	0
$490$	2.48871	0.112428
$491$	−19.0079	−0.857815	−0.428907	−	0.903348i	$-0.641101\pi$
$491$	−19.0079	−0.857815	−0.428907	+	0.903348i	$0.641101\pi$
$492$	0	0
$493$	−2.92930	−0.131929
$494$	7.22704	0.325160
$495$	0	0
$496$	23.2511	1.04401
$497$	10.1782	0.456556
$498$	0	0
$499$	−24.8767	−1.11364	−0.556818	−	0.830635i	$-0.687978\pi$
$499$	−24.8767	−1.11364	−0.556818	+	0.830635i	$0.687978\pi$
$500$	−1.75882	−0.0786567
$501$	0	0
$502$	−4.65707	−0.207855
$503$	−7.21761	−0.321817	−0.160909	−	0.986969i	$-0.551442\pi$
$503$	−7.21761	−0.321817	−0.160909	+	0.986969i	$0.551442\pi$
$504$	0	0
$505$	16.3445	0.727319
$506$	2.15637	0.0958624
$507$	0	0
$508$	21.6566	0.960858
$509$	32.5399	1.44230	0.721152	−	0.692777i	$-0.243613\pi$
$509$	32.5399	1.44230	0.721152	+	0.692777i	$0.243613\pi$
$510$	0	0
$511$	11.0706	0.489736
$512$	−22.6280	−1.00003
$513$	0	0
$514$	8.19485	0.361459
$515$	2.89199	0.127436
$516$	0	0
$517$	−5.33415	−0.234596
$518$	−4.65414	−0.204491
$519$	0	0
$520$	−4.92262	−0.215871
$521$	−17.9351	−0.785751	−0.392875	−	0.919592i	$-0.628520\pi$
$521$	−17.9351	−0.785751	−0.392875	+	0.919592i	$0.628520\pi$
$522$	0	0
$523$	22.3547	0.977501	0.488751	−	0.872424i	$-0.337453\pi$
$523$	22.3547	0.977501	0.488751	+	0.872424i	$0.337453\pi$
$524$	33.3102	1.45516
$525$	0	0
$526$	−10.1328	−0.441812
$527$	−30.8326	−1.34309
$528$	0	0
$529$	4.07662	0.177244
$530$	−1.51505	−0.0658097
$531$	0	0
$532$	13.4923	0.584966
$533$	−19.7627	−0.856018
$534$	0	0
$535$	4.86649	0.210397
$536$	−22.7359	−0.982040
$537$	0	0
$538$	9.98440	0.430458
$539$	−4.27617	−0.184188
$540$	0	0
$541$	22.3747	0.961963	0.480981	−	0.876731i	$-0.340280\pi$
$541$	22.3747	0.961963	0.480981	+	0.876731i	$0.340280\pi$
$542$	8.73100	0.375028
$543$	0	0
$544$	17.2231	0.738436
$545$	−19.8869	−0.851860
$546$	0	0
$547$	29.0682	1.24287	0.621434	−	0.783467i	$-0.286550\pi$
$547$	29.0682	1.24287	0.621434	+	0.783467i	$0.286550\pi$
$548$	27.0516	1.15559
$549$	0	0
$550$	−0.414406	−0.0176703
$551$	4.66868	0.198892
$552$	0	0
$553$	−6.34513	−0.269822
$554$	9.10268	0.386736
$555$	0	0
$556$	15.6167	0.662294
$557$	44.6644	1.89249	0.946246	−	0.323448i	$-0.104842\pi$
$557$	44.6644	1.89249	0.946246	+	0.323448i	$0.104842\pi$
$558$	0	0
$559$	−6.18491	−0.261594
$560$	−3.62969	−0.153383
$561$	0	0
$562$	8.00135	0.337517
$563$	40.5811	1.71029	0.855144	−	0.518390i	$-0.173469\pi$
$563$	40.5811	1.71029	0.855144	+	0.518390i	$0.173469\pi$
$564$	0	0
$565$	−13.4756	−0.566922
$566$	16.0675	0.675366
$567$	0	0
$568$	−13.5159	−0.567116
$569$	14.1652	0.593837	0.296919	−	0.954903i	$-0.404041\pi$
$569$	14.1652	0.593837	0.296919	+	0.954903i	$0.404041\pi$
$570$	0	0
$571$	−26.1693	−1.09515	−0.547576	−	0.836756i	$-0.684449\pi$
$571$	−26.1693	−1.09515	−0.547576	+	0.836756i	$0.684449\pi$
$572$	3.95774	0.165481
$573$	0	0
$574$	5.05937	0.211174
$575$	−5.20352	−0.217002
$576$	0	0
$577$	−36.7700	−1.53076	−0.765378	−	0.643581i	$-0.777448\pi$
$577$	−36.7700	−1.53076	−0.765378	+	0.643581i	$0.777448\pi$
$578$	2.46109	0.102368
$579$	0	0
$580$	−1.48799	−0.0617854
$581$	−10.4179	−0.432207
$582$	0	0
$583$	2.60321	0.107814
$584$	−14.7010	−0.608330
$585$	0	0
$586$	−16.3809	−0.676690
$587$	−29.4943	−1.21736	−0.608680	−	0.793416i	$-0.708301\pi$
$587$	−29.4943	−1.21736	−0.608680	+	0.793416i	$0.708301\pi$
$588$	0	0
$589$	49.1405	2.02480
$590$	−6.90058	−0.284092
$591$	0	0
$592$	−17.8006	−0.731602
$593$	19.4743	0.799714	0.399857	−	0.916578i	$-0.369060\pi$
$593$	19.4743	0.799714	0.399857	+	0.916578i	$0.369060\pi$
$594$	0	0
$595$	4.81322	0.197323
$596$	−24.0512	−0.985175
$597$	0	0
$598$	−6.81464	−0.278671
$599$	11.3938	0.465540	0.232770	−	0.972532i	$-0.425221\pi$
$599$	11.3938	0.465540	0.232770	+	0.972532i	$0.425221\pi$
$600$	0	0
$601$	−15.9462	−0.650458	−0.325229	−	0.945635i	$-0.605441\pi$
$601$	−15.9462	−0.650458	−0.325229	+	0.945635i	$0.605441\pi$
$602$	1.58337	0.0645335
$603$	0	0
$604$	−17.6156	−0.716771
$605$	−10.2880	−0.418265
$606$	0	0
$607$	−27.0138	−1.09646	−0.548228	−	0.836329i	$-0.684697\pi$
$607$	−27.0138	−1.09646	−0.548228	+	0.836329i	$0.684697\pi$
$608$	−27.4500	−1.11324
$609$	0	0
$610$	4.51605	0.182849
$611$	16.8572	0.681967
$612$	0	0
$613$	46.7061	1.88644	0.943221	−	0.332166i	$-0.107779\pi$
$613$	46.7061	1.88644	0.943221	+	0.332166i	$0.107779\pi$
$614$	10.7704	0.434659
$615$	0	0
$616$	−2.16535	−0.0872445
$617$	11.2320	0.452183	0.226092	−	0.974106i	$-0.427405\pi$
$617$	11.2320	0.452183	0.226092	+	0.974106i	$0.427405\pi$
$618$	0	0
$619$	7.61409	0.306036	0.153018	−	0.988223i	$-0.451101\pi$
$619$	7.61409	0.306036	0.153018	+	0.988223i	$0.451101\pi$
$620$	−15.6619	−0.628999
$621$	0	0
$622$	−13.9624	−0.559841
$623$	−1.39011	−0.0556938
$624$	0	0
$625$	1.00000	0.0400000
$626$	−7.78064	−0.310977
$627$	0	0
$628$	−20.7876	−0.829514
$629$	23.6049	0.941188
$630$	0	0
$631$	−16.0786	−0.640078	−0.320039	−	0.947404i	$-0.603696\pi$
$631$	−16.0786	−0.640078	−0.320039	+	0.947404i	$0.603696\pi$
$632$	8.42585	0.335162
$633$	0	0
$634$	14.4176	0.572596
$635$	−12.3132	−0.488634
$636$	0	0
$637$	13.5137	0.535431
$638$	−0.350595	−0.0138802
$639$	0	0
$640$	11.3134	0.447201
$641$	45.1415	1.78298	0.891492	−	0.453037i	$-0.149660\pi$
$641$	45.1415	1.78298	0.891492	+	0.453037i	$0.149660\pi$
$642$	0	0
$643$	35.2129	1.38866	0.694330	−	0.719657i	$-0.255701\pi$
$643$	35.2129	1.38866	0.694330	+	0.719657i	$0.255701\pi$
$644$	−12.7224	−0.501332
$645$	0	0
$646$	9.38367	0.369196
$647$	27.4490	1.07913	0.539566	−	0.841943i	$-0.318588\pi$
$647$	27.4490	1.07913	0.539566	+	0.841943i	$0.318588\pi$
$648$	0	0
$649$	11.8568	0.465419
$650$	1.30962	0.0513675
$651$	0	0
$652$	−10.0745	−0.394548
$653$	25.6784	1.00488	0.502438	−	0.864613i	$-0.332437\pi$
$653$	25.6784	1.00488	0.502438	+	0.864613i	$0.332437\pi$
$654$	0	0
$655$	−18.9390	−0.740007
$656$	19.3505	0.755512
$657$	0	0
$658$	−4.31553	−0.168237
$659$	51.0988	1.99053	0.995263	−	0.0972197i	$-0.0309950\pi$
$659$	51.0988	1.99053	0.995263	+	0.0972197i	$0.0309950\pi$
$660$	0	0
$661$	11.9594	0.465166	0.232583	−	0.972577i	$-0.425282\pi$
$661$	11.9594	0.465166	0.232583	+	0.972577i	$0.425282\pi$
$662$	8.53761	0.331824
$663$	0	0
$664$	13.8342	0.536870
$665$	−7.67124	−0.297478
$666$	0	0
$667$	−4.40226	−0.170456
$668$	9.76548	0.377838
$669$	0	0
$670$	6.04868	0.233681
$671$	−7.75961	−0.299556
$672$	0	0
$673$	−28.6913	−1.10597	−0.552984	−	0.833192i	$-0.686511\pi$
$673$	−28.6913	−1.10597	−0.552984	+	0.833192i	$0.686511\pi$
$674$	7.67745	0.295724
$675$	0	0
$676$	10.3572	0.398356
$677$	−16.8646	−0.648160	−0.324080	−	0.946030i	$-0.605055\pi$
$677$	−16.8646	−0.648160	−0.324080	+	0.946030i	$0.605055\pi$
$678$	0	0
$679$	−17.0804	−0.655486
$680$	−6.39159	−0.245106
$681$	0	0
$682$	−3.69021	−0.141306
$683$	−34.6719	−1.32668	−0.663342	−	0.748317i	$-0.730862\pi$
$683$	−34.6719	−1.32668	−0.663342	+	0.748317i	$0.730862\pi$
$684$	0	0
$685$	−15.3806	−0.587661
$686$	−8.23841	−0.314544
$687$	0	0
$688$	6.05592	0.230880
$689$	−8.22675	−0.313414
$690$	0	0
$691$	23.8207	0.906182	0.453091	−	0.891464i	$-0.350321\pi$
$691$	23.8207	0.906182	0.453091	+	0.891464i	$0.350321\pi$
$692$	−30.8575	−1.17303
$693$	0	0
$694$	−12.4787	−0.473686
$695$	−8.87907	−0.336802
$696$	0	0
$697$	−25.6601	−0.971947
$698$	1.77118	0.0670401
$699$	0	0
$700$	2.44496	0.0924107
$701$	−44.0179	−1.66253	−0.831267	−	0.555874i	$-0.812384\pi$
$701$	−44.0179	−1.66253	−0.831267	+	0.555874i	$0.812384\pi$
$702$	0	0
$703$	−37.6211	−1.41891
$704$	−2.34524	−0.0883895
$705$	0	0
$706$	6.71858	0.252857
$707$	−22.7207	−0.854498
$708$	0	0
$709$	35.5537	1.33525	0.667623	−	0.744499i	$-0.267312\pi$
$709$	35.5537	1.33525	0.667623	+	0.744499i	$0.267312\pi$
$710$	3.59579	0.134948
$711$	0	0
$712$	1.84597	0.0691805
$713$	−46.3364	−1.73531
$714$	0	0
$715$	−2.25023	−0.0841538
$716$	23.5004	0.878252
$717$	0	0
$718$	−3.90098	−0.145583
$719$	−5.13605	−0.191542	−0.0957711	−	0.995403i	$-0.530532\pi$
$719$	−5.13605	−0.191542	−0.0957711	+	0.995403i	$0.530532\pi$
$720$	0	0
$721$	−4.02020	−0.149720
$722$	−5.62459	−0.209326
$723$	0	0
$724$	−32.3717	−1.20308
$725$	0.846016	0.0314203
$726$	0	0
$727$	12.3730	0.458888	0.229444	−	0.973322i	$-0.426309\pi$
$727$	12.3730	0.458888	0.229444	+	0.973322i	$0.426309\pi$
$728$	6.84301	0.253619
$729$	0	0
$730$	3.91106	0.144755
$731$	−8.03056	−0.297021
$732$	0	0
$733$	29.5198	1.09034	0.545169	−	0.838326i	$-0.316466\pi$
$733$	29.5198	1.09034	0.545169	+	0.838326i	$0.316466\pi$
$734$	0.0826281	0.00304986
$735$	0	0
$736$	25.8836	0.954081
$737$	−10.3930	−0.382832
$738$	0	0
$739$	−51.4459	−1.89247	−0.946234	−	0.323483i	$-0.895146\pi$
$739$	−51.4459	−1.89247	−0.946234	+	0.323483i	$0.895146\pi$
$740$	11.9905	0.440779
$741$	0	0
$742$	2.10610	0.0773173
$743$	−49.0198	−1.79836	−0.899181	−	0.437577i	$-0.855837\pi$
$743$	−49.0198	−1.79836	−0.899181	+	0.437577i	$0.855837\pi$
$744$	0	0
$745$	13.6746	0.501000
$746$	4.11819	0.150778
$747$	0	0
$748$	5.13878	0.187892
$749$	−6.76498	−0.247187
$750$	0	0
$751$	6.30801	0.230182	0.115091	−	0.993355i	$-0.463284\pi$
$751$	6.30801	0.230182	0.115091	+	0.993355i	$0.463284\pi$
$752$	−16.5056	−0.601897
$753$	0	0
$754$	1.10796	0.0403495
$755$	10.0156	0.364506
$756$	0	0
$757$	26.6669	0.969224	0.484612	−	0.874729i	$-0.338961\pi$
$757$	26.6669	0.969224	0.484612	+	0.874729i	$0.338961\pi$
$758$	16.9073	0.614099
$759$	0	0
$760$	10.1868	0.369515
$761$	−4.84411	−0.175599	−0.0877994	−	0.996138i	$-0.527983\pi$
$761$	−4.84411	−0.175599	−0.0877994	+	0.996138i	$0.527983\pi$
$762$	0	0
$763$	27.6450	1.00082
$764$	47.4240	1.71574
$765$	0	0
$766$	−15.5924	−0.563377
$767$	−37.4702	−1.35297
$768$	0	0
$769$	−18.2181	−0.656962	−0.328481	−	0.944511i	$-0.606537\pi$
$769$	−18.2181	−0.656962	−0.328481	+	0.944511i	$0.606537\pi$
$770$	0.576072	0.0207602
$771$	0	0
$772$	36.8863	1.32757
$773$	−16.8245	−0.605137	−0.302568	−	0.953128i	$-0.597844\pi$
$773$	−16.8245	−0.605137	−0.302568	+	0.953128i	$0.597844\pi$
$774$	0	0
$775$	8.90482	0.319870
$776$	22.6815	0.814218
$777$	0	0
$778$	−16.1783	−0.580019
$779$	40.8967	1.46528
$780$	0	0
$781$	−6.17840	−0.221080
$782$	−8.84820	−0.316411
$783$	0	0
$784$	−13.2318	−0.472566
$785$	11.8191	0.421840
$786$	0	0
$787$	−4.97325	−0.177277	−0.0886387	−	0.996064i	$-0.528252\pi$
$787$	−4.97325	−0.177277	−0.0886387	+	0.996064i	$0.528252\pi$
$788$	26.8132	0.955179
$789$	0	0
$790$	−2.24162	−0.0797533
$791$	18.7326	0.666054
$792$	0	0
$793$	24.5222	0.870807
$794$	−6.91065	−0.245250
$795$	0	0
$796$	24.0381	0.852006
$797$	14.5828	0.516549	0.258275	−	0.966072i	$-0.416846\pi$
$797$	14.5828	0.516549	0.258275	+	0.966072i	$0.416846\pi$
$798$	0	0
$799$	21.8875	0.774325
$800$	−4.97424	−0.175866
$801$	0	0
$802$	−3.57249	−0.126149
$803$	−6.72009	−0.237147
$804$	0	0
$805$	7.23349	0.254947
$806$	11.6619	0.410774
$807$	0	0
$808$	30.1713	1.06142
$809$	17.6121	0.619210	0.309605	−	0.950865i	$-0.399803\pi$
$809$	17.6121	0.619210	0.309605	+	0.950865i	$0.399803\pi$
$810$	0	0
$811$	22.2449	0.781124	0.390562	−	0.920577i	$-0.372281\pi$
$811$	22.2449	0.781124	0.390562	+	0.920577i	$0.372281\pi$
$812$	2.06847	0.0725892
$813$	0	0
$814$	2.82516	0.0990218
$815$	5.72799	0.200643
$816$	0	0
$817$	12.7990	0.447779
$818$	−13.3816	−0.467875
$819$	0	0
$820$	−13.0345	−0.455185
$821$	32.6129	1.13820	0.569099	−	0.822269i	$-0.307292\pi$
$821$	32.6129	1.13820	0.569099	+	0.822269i	$0.307292\pi$
$822$	0	0
$823$	38.9102	1.35633	0.678163	−	0.734912i	$-0.262776\pi$
$823$	38.9102	1.35633	0.678163	+	0.734912i	$0.262776\pi$
$824$	5.33852	0.185976
$825$	0	0
$826$	9.59259	0.333769
$827$	−30.4207	−1.05783	−0.528915	−	0.848675i	$-0.677401\pi$
$827$	−30.4207	−1.05783	−0.528915	+	0.848675i	$0.677401\pi$
$828$	0	0
$829$	17.8261	0.619125	0.309563	−	0.950879i	$-0.399817\pi$
$829$	17.8261	0.619125	0.309563	+	0.950879i	$0.399817\pi$
$830$	−3.68046	−0.127751
$831$	0	0
$832$	7.41150	0.256948
$833$	17.5463	0.607944
$834$	0	0
$835$	−5.55230	−0.192145
$836$	−8.19011	−0.283261
$837$	0	0
$838$	6.76312	0.233628
$839$	9.11047	0.314528	0.157264	−	0.987557i	$-0.449733\pi$
$839$	9.11047	0.314528	0.157264	+	0.987557i	$0.449733\pi$
$840$	0	0
$841$	−28.2843	−0.975319
$842$	1.65114	0.0569019
$843$	0	0
$844$	46.6782	1.60673
$845$	−5.88876	−0.202579
$846$	0	0
$847$	14.3014	0.491403
$848$	8.05517	0.276616
$849$	0	0
$850$	1.70043	0.0583242
$851$	35.4743	1.21604
$852$	0	0
$853$	−34.2521	−1.17277	−0.586385	−	0.810033i	$-0.699449\pi$
$853$	−34.2521	−1.17277	−0.586385	+	0.810033i	$0.699449\pi$
$854$	−6.27782	−0.214823
$855$	0	0
$856$	8.98338	0.307046
$857$	16.3278	0.557745	0.278873	−	0.960328i	$-0.410039\pi$
$857$	16.3278	0.557745	0.278873	+	0.960328i	$0.410039\pi$
$858$	0	0
$859$	−8.30761	−0.283452	−0.141726	−	0.989906i	$-0.545265\pi$
$859$	−8.30761	−0.283452	−0.141726	+	0.989906i	$0.545265\pi$
$860$	−4.07926	−0.139102
$861$	0	0
$862$	−2.91214	−0.0991879
$863$	20.7305	0.705675	0.352838	−	0.935685i	$-0.385217\pi$
$863$	20.7305	0.705675	0.352838	+	0.935685i	$0.385217\pi$
$864$	0	0
$865$	17.5445	0.596530
$866$	8.10601	0.275454
$867$	0	0
$868$	21.7719	0.738986
$869$	3.85162	0.130657
$870$	0	0
$871$	32.8443	1.11289
$872$	−36.7105	−1.24318
$873$	0	0
$874$	14.1021	0.477012
$875$	−1.39011	−0.0469944
$876$	0	0
$877$	−3.40594	−0.115010	−0.0575052	−	0.998345i	$-0.518315\pi$
$877$	−3.40594	−0.115010	−0.0575052	+	0.998345i	$0.518315\pi$
$878$	7.97195	0.269040
$879$	0	0
$880$	2.20330	0.0742732
$881$	28.2254	0.950939	0.475469	−	0.879732i	$-0.342278\pi$
$881$	28.2254	0.950939	0.475469	+	0.879732i	$0.342278\pi$
$882$	0	0
$883$	−11.8931	−0.400234	−0.200117	−	0.979772i	$-0.564132\pi$
$883$	−11.8931	−0.400234	−0.200117	+	0.979772i	$0.564132\pi$
$884$	−16.2397	−0.546201
$885$	0	0
$886$	1.28447	0.0431527
$887$	−46.2366	−1.55247	−0.776236	−	0.630442i	$-0.782874\pi$
$887$	−46.2366	−1.55247	−0.776236	+	0.630442i	$0.782874\pi$
$888$	0	0
$889$	17.1167	0.574076
$890$	−0.491103	−0.0164618
$891$	0	0
$892$	−15.2114	−0.509314
$893$	−34.8840	−1.16735
$894$	0	0
$895$	−13.3615	−0.446625
$896$	−15.7269	−0.525399
$897$	0	0
$898$	12.0114	0.400825
$899$	7.53362	0.251260
$900$	0	0
$901$	−10.6817	−0.355859
$902$	−3.07114	−0.102258
$903$	0	0
$904$	−24.8755	−0.827346
$905$	18.4053	0.611814
$906$	0	0
$907$	−9.53938	−0.316750	−0.158375	−	0.987379i	$-0.550625\pi$
$907$	−9.53938	−0.316750	−0.158375	+	0.987379i	$0.550625\pi$
$908$	16.9624	0.562918
$909$	0	0
$910$	−1.82052	−0.0603497
$911$	−25.8522	−0.856522	−0.428261	−	0.903655i	$-0.640874\pi$
$911$	−25.8522	−0.856522	−0.428261	+	0.903655i	$0.640874\pi$
$912$	0	0
$913$	6.32387	0.209290
$914$	−17.3708	−0.574576
$915$	0	0
$916$	16.4569	0.543750
$917$	26.3273	0.869405
$918$	0	0
$919$	5.58561	0.184252	0.0921261	−	0.995747i	$-0.470634\pi$
$919$	5.58561	0.184252	0.0921261	+	0.995747i	$0.470634\pi$
$920$	−9.60553	−0.316685
$921$	0	0
$922$	3.52218	0.115997
$923$	19.5252	0.642679
$924$	0	0
$925$	−6.81736	−0.224153
$926$	14.5514	0.478190
$927$	0	0
$928$	−4.20829	−0.138144
$929$	38.3430	1.25799	0.628996	−	0.777409i	$-0.283466\pi$
$929$	38.3430	1.25799	0.628996	+	0.777409i	$0.283466\pi$
$930$	0	0
$931$	−27.9651	−0.916518
$932$	5.18783	0.169933
$933$	0	0
$934$	10.9522	0.358366
$935$	−2.92172	−0.0955506
$936$	0	0
$937$	−17.6816	−0.577632	−0.288816	−	0.957385i	$-0.593262\pi$
$937$	−17.6816	−0.577632	−0.288816	+	0.957385i	$0.593262\pi$
$938$	−8.40835	−0.274542
$939$	0	0
$940$	11.1181	0.362634
$941$	0.603957	0.0196884	0.00984422	−	0.999952i	$-0.496866\pi$
$941$	0.603957	0.0196884	0.00984422	+	0.999952i	$0.496866\pi$
$942$	0	0
$943$	−38.5630	−1.25578
$944$	36.6887	1.19412
$945$	0	0
$946$	−0.961140	−0.0312494
$947$	13.2372	0.430151	0.215075	−	0.976597i	$-0.431000\pi$
$947$	13.2372	0.430151	0.215075	+	0.976597i	$0.431000\pi$
$948$	0	0
$949$	21.2371	0.689384
$950$	−2.71011	−0.0879277
$951$	0	0
$952$	8.88505	0.287966
$953$	33.2791	1.07801	0.539007	−	0.842301i	$-0.318799\pi$
$953$	33.2791	1.07801	0.539007	+	0.842301i	$0.318799\pi$
$954$	0	0
$955$	−26.9635	−0.872520
$956$	10.4277	0.337254
$957$	0	0
$958$	2.35450	0.0760705
$959$	21.3807	0.690420
$960$	0	0
$961$	48.2957	1.55793
$962$	−8.92816	−0.287855
$963$	0	0
$964$	−0.877016	−0.0282468
$965$	−20.9722	−0.675120
$966$	0	0
$967$	18.7452	0.602804	0.301402	−	0.953497i	$-0.402545\pi$
$967$	18.7452	0.602804	0.301402	+	0.953497i	$0.402545\pi$
$968$	−18.9912	−0.610401
$969$	0	0
$970$	−6.03421	−0.193747
$971$	−38.2456	−1.22736	−0.613679	−	0.789556i	$-0.710311\pi$
$971$	−38.2456	−1.22736	−0.613679	+	0.789556i	$0.710311\pi$
$972$	0	0
$973$	12.3429	0.395696
$974$	6.79706	0.217792
$975$	0	0
$976$	−24.0107	−0.768565
$977$	12.7711	0.408583	0.204292	−	0.978910i	$-0.434511\pi$
$977$	12.7711	0.408583	0.204292	+	0.978910i	$0.434511\pi$
$978$	0	0
$979$	0.843828	0.0269689
$980$	8.91295	0.284714
$981$	0	0
$982$	9.33484	0.297887
$983$	−24.6723	−0.786925	−0.393462	−	0.919341i	$-0.628723\pi$
$983$	−24.6723	−0.786925	−0.393462	+	0.919341i	$0.628723\pi$
$984$	0	0
$985$	−15.2450	−0.485746
$986$	1.43859	0.0458140
$987$	0	0
$988$	25.8826	0.823436
$989$	−12.0686	−0.383760
$990$	0	0
$991$	−3.86629	−0.122817	−0.0614084	−	0.998113i	$-0.519559\pi$
$991$	−3.86629	−0.122817	−0.0614084	+	0.998113i	$0.519559\pi$
$992$	−44.2947	−1.40636
$993$	0	0
$994$	−4.99856	−0.158545
$995$	−13.6672	−0.433278
$996$	0	0
$997$	5.57469	0.176552	0.0882761	−	0.996096i	$-0.471864\pi$
$997$	5.57469	0.176552	0.0882761	+	0.996096i	$0.471864\pi$
$998$	12.2170	0.386724
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4005.2.a.u.1.6	✓	12
3.2	odd	2		4005.2.a.v.1.7	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4005.2.a.u.1.6	✓	12	1.1	even	1	trivial
4005.2.a.v.1.7	yes	12	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-0.491103 q^{2} -1.75882 q^{4} +1.00000 q^{5} -1.39011 q^{7} +1.84597 q^{8} +O(q^{10})\)	\(q-0.491103 q^{2} -1.75882 q^{4} +1.00000 q^{5} -1.39011 q^{7} +1.84597 q^{8} -0.491103 q^{10} +0.843828 q^{11} -2.66669 q^{13} +0.682689 q^{14} +2.61108 q^{16} -3.46246 q^{17} +5.51842 q^{19} -1.75882 q^{20} -0.414406 q^{22} -5.20352 q^{23} +1.00000 q^{25} +1.30962 q^{26} +2.44496 q^{28} +0.846016 q^{29} +8.90482 q^{31} -4.97424 q^{32} +1.70043 q^{34} -1.39011 q^{35} -6.81736 q^{37} -2.71011 q^{38} +1.84597 q^{40} +7.41095 q^{41} +2.31932 q^{43} -1.48414 q^{44} +2.55546 q^{46} -6.32137 q^{47} -5.06758 q^{49} -0.491103 q^{50} +4.69022 q^{52} +3.08500 q^{53} +0.843828 q^{55} -2.56610 q^{56} -0.415481 q^{58} +14.0512 q^{59} -9.19572 q^{61} -4.37318 q^{62} -2.77929 q^{64} -2.66669 q^{65} -12.3165 q^{67} +6.08984 q^{68} +0.682689 q^{70} -7.32187 q^{71} -7.96382 q^{73} +3.34803 q^{74} -9.70590 q^{76} -1.17302 q^{77} +4.56446 q^{79} +2.61108 q^{80} -3.63954 q^{82} +7.49427 q^{83} -3.46246 q^{85} -1.13902 q^{86} +1.55768 q^{88} +1.00000 q^{89} +3.70701 q^{91} +9.15204 q^{92} +3.10445 q^{94} +5.51842 q^{95} +12.2871 q^{97} +2.48871 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 3 q^{2} + 11 q^{4} + 12 q^{5} - 8 q^{7} - 9 q^{8}+O(q^{10}) \) 12 * q - 3 * q^2 + 11 * q^4 + 12 * q^5 - 8 * q^7 - 9 * q^8	\( 12 q - 3 q^{2} + 11 q^{4} + 12 q^{5} - 8 q^{7} - 9 q^{8} - 3 q^{10} - 12 q^{13} + 4 q^{14} + q^{16} - 24 q^{19} + 11 q^{20} - 16 q^{22} - 24 q^{23} + 12 q^{25} - q^{26} - 44 q^{28} + 8 q^{29} - 12 q^{31} - 31 q^{32} - 18 q^{34} - 8 q^{35} - 10 q^{37} + 2 q^{38} - 9 q^{40} - 10 q^{41} - 42 q^{43} + 42 q^{44} - 24 q^{46} - 22 q^{47} - 4 q^{49} - 3 q^{50} - 30 q^{52} + 8 q^{53} + 27 q^{56} - 12 q^{58} + 4 q^{59} - 52 q^{61} - 14 q^{62} + 7 q^{64} - 12 q^{65} - 40 q^{67} + 23 q^{68} + 4 q^{70} + 2 q^{71} - 8 q^{73} + 26 q^{74} - 46 q^{76} - 12 q^{77} - 26 q^{79} + q^{80} - 26 q^{82} - 14 q^{83} + 32 q^{86} - 60 q^{88} + 12 q^{89} - 24 q^{91} - 38 q^{92} - 26 q^{94} - 24 q^{95} - 6 q^{97} + 30 q^{98}+O(q^{100}) \) 12 * q - 3 * q^2 + 11 * q^4 + 12 * q^5 - 8 * q^7 - 9 * q^8 - 3 * q^10 - 12 * q^13 + 4 * q^14 + q^16 - 24 * q^19 + 11 * q^20 - 16 * q^22 - 24 * q^23 + 12 * q^25 - q^26 - 44 * q^28 + 8 * q^29 - 12 * q^31 - 31 * q^32 - 18 * q^34 - 8 * q^35 - 10 * q^37 + 2 * q^38 - 9 * q^40 - 10 * q^41 - 42 * q^43 + 42 * q^44 - 24 * q^46 - 22 * q^47 - 4 * q^49 - 3 * q^50 - 30 * q^52 + 8 * q^53 + 27 * q^56 - 12 * q^58 + 4 * q^59 - 52 * q^61 - 14 * q^62 + 7 * q^64 - 12 * q^65 - 40 * q^67 + 23 * q^68 + 4 * q^70 + 2 * q^71 - 8 * q^73 + 26 * q^74 - 46 * q^76 - 12 * q^77 - 26 * q^79 + q^80 - 26 * q^82 - 14 * q^83 + 32 * q^86 - 60 * q^88 + 12 * q^89 - 24 * q^91 - 38 * q^92 - 26 * q^94 - 24 * q^95 - 6 * q^97 + 30 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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