LMFDB - Embedded newform 8046.2.a.p.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8046,2,Mod(1,8046)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8046, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8046.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8046 = 2 \cdot 3^{3} \cdot 149 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8046.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:	$64.2476334663$
Analytic rank:	$0$
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} - 5 x^{11} - 23 x^{10} + 142 x^{9} + 104 x^{8} - 1302 x^{7} + 607 x^{6} + 4323 x^{5} - 4461 x^{4} + \cdots - 553 $ x^12 - 5x^11 - 23x^10 + 142x^9 + 104x^8 - 1302x^7 + 607x^6 + 4323x^5 - 4461x^4 - 3333x^3 + 4805x^2 - 224*x - 553
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			1.10
Root			$2.74383$ of defining polynomial
Character	$\chi$	$=$	8046.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +2.74383 q^{5} +0.881706 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +2.74383 q^{5} +0.881706 q^{7} +1.00000 q^{8} +2.74383 q^{10} +0.654634 q^{11} +2.06791 q^{13} +0.881706 q^{14} +1.00000 q^{16} -5.11554 q^{17} -5.28170 q^{19} +2.74383 q^{20} +0.654634 q^{22} +3.61517 q^{23} +2.52860 q^{25} +2.06791 q^{26} +0.881706 q^{28} +1.25966 q^{29} +3.23726 q^{31} +1.00000 q^{32} -5.11554 q^{34} +2.41925 q^{35} +4.59094 q^{37} -5.28170 q^{38} +2.74383 q^{40} -0.963823 q^{41} +5.50227 q^{43} +0.654634 q^{44} +3.61517 q^{46} +12.4487 q^{47} -6.22260 q^{49} +2.52860 q^{50} +2.06791 q^{52} +6.64771 q^{53} +1.79620 q^{55} +0.881706 q^{56} +1.25966 q^{58} -9.13256 q^{59} +7.23226 q^{61} +3.23726 q^{62} +1.00000 q^{64} +5.67399 q^{65} +2.54516 q^{67} -5.11554 q^{68} +2.41925 q^{70} -8.61358 q^{71} +5.32891 q^{73} +4.59094 q^{74} -5.28170 q^{76} +0.577195 q^{77} +1.75447 q^{79} +2.74383 q^{80} -0.963823 q^{82} +16.3610 q^{83} -14.0362 q^{85} +5.50227 q^{86} +0.654634 q^{88} +10.7640 q^{89} +1.82329 q^{91} +3.61517 q^{92} +12.4487 q^{94} -14.4921 q^{95} +1.52733 q^{97} -6.22260 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{2} + 12 q^{4} + 5 q^{5} + 6 q^{7} + 12 q^{8}+O(q^{10}) $ 12 * q + 12 * q^2 + 12 * q^4 + 5 * q^5 + 6 * q^7 + 12 * q^8	$ 12 q + 12 q^{2} + 12 q^{4} + 5 q^{5} + 6 q^{7} + 12 q^{8} + 5 q^{10} + 6 q^{11} + 3 q^{13} + 6 q^{14} + 12 q^{16} + 6 q^{17} + 8 q^{19} + 5 q^{20} + 6 q^{22} + 11 q^{23} + 11 q^{25} + 3 q^{26} + 6 q^{28} + 29 q^{29} + 2 q^{31} + 12 q^{32} + 6 q^{34} + 4 q^{35} + 5 q^{37} + 8 q^{38} + 5 q^{40} + 22 q^{41} + 9 q^{43} + 6 q^{44} + 11 q^{46} + 15 q^{47} + 14 q^{49} + 11 q^{50} + 3 q^{52} + 12 q^{53} + 13 q^{55} + 6 q^{56} + 29 q^{58} + 34 q^{59} - 4 q^{61} + 2 q^{62} + 12 q^{64} + 12 q^{65} + q^{67} + 6 q^{68} + 4 q^{70} + 21 q^{71} - 2 q^{73} + 5 q^{74} + 8 q^{76} + 34 q^{77} + 9 q^{79} + 5 q^{80} + 22 q^{82} + 10 q^{83} + 5 q^{85} + 9 q^{86} + 6 q^{88} - 2 q^{89} + 17 q^{91} + 11 q^{92} + 15 q^{94} + 69 q^{95} - 13 q^{97} + 14 q^{98}+O(q^{100}) $ 12 * q + 12 * q^2 + 12 * q^4 + 5 * q^5 + 6 * q^7 + 12 * q^8 + 5 * q^10 + 6 * q^11 + 3 * q^13 + 6 * q^14 + 12 * q^16 + 6 * q^17 + 8 * q^19 + 5 * q^20 + 6 * q^22 + 11 * q^23 + 11 * q^25 + 3 * q^26 + 6 * q^28 + 29 * q^29 + 2 * q^31 + 12 * q^32 + 6 * q^34 + 4 * q^35 + 5 * q^37 + 8 * q^38 + 5 * q^40 + 22 * q^41 + 9 * q^43 + 6 * q^44 + 11 * q^46 + 15 * q^47 + 14 * q^49 + 11 * q^50 + 3 * q^52 + 12 * q^53 + 13 * q^55 + 6 * q^56 + 29 * q^58 + 34 * q^59 - 4 * q^61 + 2 * q^62 + 12 * q^64 + 12 * q^65 + q^67 + 6 * q^68 + 4 * q^70 + 21 * q^71 - 2 * q^73 + 5 * q^74 + 8 * q^76 + 34 * q^77 + 9 * q^79 + 5 * q^80 + 22 * q^82 + 10 * q^83 + 5 * q^85 + 9 * q^86 + 6 * q^88 - 2 * q^89 + 17 * q^91 + 11 * q^92 + 15 * q^94 + 69 * q^95 - 13 * q^97 + 14 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	2.74383	1.22708	0.613539	−	0.789664i	$-0.289745\pi$
$5$	2.74383	1.22708	0.613539	+	0.789664i	$0.289745\pi$
$6$	0	0
$7$	0.881706	0.333253	0.166627	−	0.986020i	$-0.446713\pi$
$7$	0.881706	0.333253	0.166627	+	0.986020i	$0.446713\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	2.74383	0.867675
$11$	0.654634	0.197380	0.0986898	−	0.995118i	$-0.468535\pi$
$11$	0.654634	0.197380	0.0986898	+	0.995118i	$0.468535\pi$
$12$	0	0
$13$	2.06791	0.573534	0.286767	−	0.958000i	$-0.407419\pi$
$13$	2.06791	0.573534	0.286767	+	0.958000i	$0.407419\pi$
$14$	0.881706	0.235646
$15$	0	0
$16$	1.00000	0.250000
$17$	−5.11554	−1.24070	−0.620351	−	0.784325i	$-0.713010\pi$
$17$	−5.11554	−1.24070	−0.620351	+	0.784325i	$0.713010\pi$
$18$	0	0
$19$	−5.28170	−1.21170	−0.605852	−	0.795577i	$-0.707168\pi$
$19$	−5.28170	−1.21170	−0.605852	+	0.795577i	$0.707168\pi$
$20$	2.74383	0.613539
$21$	0	0
$22$	0.654634	0.139568
$23$	3.61517	0.753815	0.376908	−	0.926251i	$-0.376987\pi$
$23$	3.61517	0.753815	0.376908	+	0.926251i	$0.376987\pi$
$24$	0	0
$25$	2.52860	0.505720
$26$	2.06791	0.405550
$27$	0	0
$28$	0.881706	0.166627
$29$	1.25966	0.233912	0.116956	−	0.993137i	$-0.462686\pi$
$29$	1.25966	0.233912	0.116956	+	0.993137i	$0.462686\pi$
$30$	0	0
$31$	3.23726	0.581428	0.290714	−	0.956810i	$-0.406107\pi$
$31$	3.23726	0.581428	0.290714	+	0.956810i	$0.406107\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−5.11554	−0.877308
$35$	2.41925	0.408928
$36$	0	0
$37$	4.59094	0.754746	0.377373	−	0.926061i	$-0.376827\pi$
$37$	4.59094	0.754746	0.377373	+	0.926061i	$0.376827\pi$
$38$	−5.28170	−0.856805
$39$	0	0
$40$	2.74383	0.433838
$41$	−0.963823	−0.150524	−0.0752619	−	0.997164i	$-0.523979\pi$
$41$	−0.963823	−0.150524	−0.0752619	+	0.997164i	$0.523979\pi$
$42$	0	0
$43$	5.50227	0.839088	0.419544	−	0.907735i	$-0.362190\pi$
$43$	5.50227	0.839088	0.419544	+	0.907735i	$0.362190\pi$
$44$	0.654634	0.0986898
$45$	0	0
$46$	3.61517	0.533028
$47$	12.4487	1.81583	0.907917	−	0.419149i	$-0.137672\pi$
$47$	12.4487	1.81583	0.907917	+	0.419149i	$0.137672\pi$
$48$	0	0
$49$	−6.22260	−0.888942
$50$	2.52860	0.357598
$51$	0	0
$52$	2.06791	0.286767
$53$	6.64771	0.913134	0.456567	−	0.889689i	$-0.349079\pi$
$53$	6.64771	0.913134	0.456567	+	0.889689i	$0.349079\pi$
$54$	0	0
$55$	1.79620	0.242200
$56$	0.881706	0.117823
$57$	0	0
$58$	1.25966	0.165401
$59$	−9.13256	−1.18896	−0.594479	−	0.804111i	$-0.702642\pi$
$59$	−9.13256	−1.18896	−0.594479	+	0.804111i	$0.702642\pi$
$60$	0	0
$61$	7.23226	0.925996	0.462998	−	0.886359i	$-0.346774\pi$
$61$	7.23226	0.925996	0.462998	+	0.886359i	$0.346774\pi$
$62$	3.23726	0.411132
$63$	0	0
$64$	1.00000	0.125000
$65$	5.67399	0.703771
$66$	0	0
$67$	2.54516	0.310940	0.155470	−	0.987841i	$-0.450311\pi$
$67$	2.54516	0.310940	0.155470	+	0.987841i	$0.450311\pi$
$68$	−5.11554	−0.620351
$69$	0	0
$70$	2.41925	0.289156
$71$	−8.61358	−1.02224	−0.511122	−	0.859508i	$-0.670770\pi$
$71$	−8.61358	−1.02224	−0.511122	+	0.859508i	$0.670770\pi$
$72$	0	0
$73$	5.32891	0.623702	0.311851	−	0.950131i	$-0.399051\pi$
$73$	5.32891	0.623702	0.311851	+	0.950131i	$0.399051\pi$
$74$	4.59094	0.533686
$75$	0	0
$76$	−5.28170	−0.605852
$77$	0.577195	0.0657774
$78$	0	0
$79$	1.75447	0.197394	0.0986969	−	0.995118i	$-0.468533\pi$
$79$	1.75447	0.197394	0.0986969	+	0.995118i	$0.468533\pi$
$80$	2.74383	0.306770
$81$	0	0
$82$	−0.963823	−0.106436
$83$	16.3610	1.79586	0.897928	−	0.440142i	$-0.145072\pi$
$83$	16.3610	1.79586	0.897928	+	0.440142i	$0.145072\pi$
$84$	0	0
$85$	−14.0362	−1.52244
$86$	5.50227	0.593325
$87$	0	0
$88$	0.654634	0.0697842
$89$	10.7640	1.14099	0.570493	−	0.821302i	$-0.306752\pi$
$89$	10.7640	1.14099	0.570493	+	0.821302i	$0.306752\pi$
$90$	0	0
$91$	1.82329	0.191132
$92$	3.61517	0.376908
$93$	0	0
$94$	12.4487	1.28399
$95$	−14.4921	−1.48686
$96$	0	0
$97$	1.52733	0.155077	0.0775384	−	0.996989i	$-0.475294\pi$
$97$	1.52733	0.155077	0.0775384	+	0.996989i	$0.475294\pi$
$98$	−6.22260	−0.628577
$99$	0	0
$100$	2.52860	0.252860
$101$	6.41245	0.638063	0.319031	−	0.947744i	$-0.396642\pi$
$101$	6.41245	0.638063	0.319031	+	0.947744i	$0.396642\pi$
$102$	0	0
$103$	9.90008	0.975484	0.487742	−	0.872988i	$-0.337821\pi$
$103$	9.90008	0.975484	0.487742	+	0.872988i	$0.337821\pi$
$104$	2.06791	0.202775
$105$	0	0
$106$	6.64771	0.645683
$107$	6.14489	0.594049	0.297025	−	0.954870i	$-0.404006\pi$
$107$	6.14489	0.594049	0.297025	+	0.954870i	$0.404006\pi$
$108$	0	0
$109$	7.91047	0.757686	0.378843	−	0.925461i	$-0.376322\pi$
$109$	7.91047	0.757686	0.378843	+	0.925461i	$0.376322\pi$
$110$	1.79620	0.171261
$111$	0	0
$112$	0.881706	0.0833134
$113$	14.6098	1.37438	0.687188	−	0.726480i	$-0.258845\pi$
$113$	14.6098	1.37438	0.687188	+	0.726480i	$0.258845\pi$
$114$	0	0
$115$	9.91941	0.924990
$116$	1.25966	0.116956
$117$	0	0
$118$	−9.13256	−0.840721
$119$	−4.51040	−0.413468
$120$	0	0
$121$	−10.5715	−0.961041
$122$	7.23226	0.654778
$123$	0	0
$124$	3.23726	0.290714
$125$	−6.78109	−0.606520
$126$	0	0
$127$	17.1994	1.52620	0.763101	−	0.646279i	$-0.223676\pi$
$127$	17.1994	1.52620	0.763101	+	0.646279i	$0.223676\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	5.67399	0.497642
$131$	−6.90241	−0.603067	−0.301533	−	0.953456i	$-0.597498\pi$
$131$	−6.90241	−0.603067	−0.301533	+	0.953456i	$0.597498\pi$
$132$	0	0
$133$	−4.65690	−0.403805
$134$	2.54516	0.219868
$135$	0	0
$136$	−5.11554	−0.438654
$137$	−11.6555	−0.995799	−0.497900	−	0.867235i	$-0.665895\pi$
$137$	−11.6555	−0.995799	−0.497900	+	0.867235i	$0.665895\pi$
$138$	0	0
$139$	9.96954	0.845605	0.422803	−	0.906222i	$-0.361046\pi$
$139$	9.96954	0.845605	0.422803	+	0.906222i	$0.361046\pi$
$140$	2.41925	0.204464
$141$	0	0
$142$	−8.61358	−0.722836
$143$	1.35372	0.113204
$144$	0	0
$145$	3.45628	0.287029
$146$	5.32891	0.441024
$147$	0	0
$148$	4.59094	0.377373
$149$	−1.00000	−0.0819232
$149$	−1.00000	−0.0819232
$150$	0	0
$151$	−14.8287	−1.20674	−0.603370	−	0.797461i	$-0.706176\pi$
$151$	−14.8287	−1.20674	−0.603370	+	0.797461i	$0.706176\pi$
$152$	−5.28170	−0.428402
$153$	0	0
$154$	0.577195	0.0465117
$155$	8.88248	0.713458
$156$	0	0
$157$	−7.11784	−0.568066	−0.284033	−	0.958815i	$-0.591673\pi$
$157$	−7.11784	−0.568066	−0.284033	+	0.958815i	$0.591673\pi$
$158$	1.75447	0.139578
$159$	0	0
$160$	2.74383	0.216919
$161$	3.18752	0.251212
$162$	0	0
$163$	2.33495	0.182888	0.0914439	−	0.995810i	$-0.470852\pi$
$163$	2.33495	0.182888	0.0914439	+	0.995810i	$0.470852\pi$
$164$	−0.963823	−0.0752619
$165$	0	0
$166$	16.3610	1.26986
$167$	−9.14665	−0.707789	−0.353894	−	0.935285i	$-0.615143\pi$
$167$	−9.14665	−0.707789	−0.353894	+	0.935285i	$0.615143\pi$
$168$	0	0
$169$	−8.72376	−0.671058
$170$	−14.0362	−1.07653
$171$	0	0
$172$	5.50227	0.419544
$173$	5.85264	0.444968	0.222484	−	0.974936i	$-0.428583\pi$
$173$	5.85264	0.444968	0.222484	+	0.974936i	$0.428583\pi$
$174$	0	0
$175$	2.22948	0.168533
$176$	0.654634	0.0493449
$177$	0	0
$178$	10.7640	0.806799
$179$	−21.7372	−1.62471	−0.812356	−	0.583162i	$-0.801815\pi$
$179$	−21.7372	−1.62471	−0.812356	+	0.583162i	$0.801815\pi$
$180$	0	0
$181$	−11.7960	−0.876790	−0.438395	−	0.898782i	$-0.644453\pi$
$181$	−11.7960	−0.876790	−0.438395	+	0.898782i	$0.644453\pi$
$182$	1.82329	0.135151
$183$	0	0
$184$	3.61517	0.266514
$185$	12.5968	0.926133
$186$	0	0
$187$	−3.34881	−0.244889
$188$	12.4487	0.907917
$189$	0	0
$190$	−14.4921	−1.05137
$191$	−4.40425	−0.318680	−0.159340	−	0.987224i	$-0.550937\pi$
$191$	−4.40425	−0.318680	−0.159340	+	0.987224i	$0.550937\pi$
$192$	0	0
$193$	6.02221	0.433488	0.216744	−	0.976228i	$-0.430456\pi$
$193$	6.02221	0.433488	0.216744	+	0.976228i	$0.430456\pi$
$194$	1.52733	0.109656
$195$	0	0
$196$	−6.22260	−0.444471
$197$	−6.13256	−0.436927	−0.218464	−	0.975845i	$-0.570104\pi$
$197$	−6.13256	−0.436927	−0.218464	+	0.975845i	$0.570104\pi$
$198$	0	0
$199$	0.361191	0.0256041	0.0128021	−	0.999918i	$-0.495925\pi$
$199$	0.361191	0.0256041	0.0128021	+	0.999918i	$0.495925\pi$
$200$	2.52860	0.178799
$201$	0	0
$202$	6.41245	0.451178
$203$	1.11065	0.0779521
$204$	0	0
$205$	−2.64457	−0.184704
$206$	9.90008	0.689772
$207$	0	0
$208$	2.06791	0.143384
$209$	−3.45758	−0.239166
$210$	0	0
$211$	−13.9869	−0.962897	−0.481449	−	0.876474i	$-0.659889\pi$
$211$	−13.9869	−0.962897	−0.481449	+	0.876474i	$0.659889\pi$
$212$	6.64771	0.456567
$213$	0	0
$214$	6.14489	0.420056
$215$	15.0973	1.02963
$216$	0	0
$217$	2.85431	0.193763
$218$	7.91047	0.535765
$219$	0	0
$220$	1.79620	0.121100
$221$	−10.5785	−0.711585
$222$	0	0
$223$	−17.0975	−1.14493	−0.572467	−	0.819928i	$-0.694013\pi$
$223$	−17.0975	−1.14493	−0.572467	+	0.819928i	$0.694013\pi$
$224$	0.881706	0.0589114
$225$	0	0
$226$	14.6098	0.971830
$227$	−14.5376	−0.964893	−0.482446	−	0.875926i	$-0.660252\pi$
$227$	−14.5376	−0.964893	−0.482446	+	0.875926i	$0.660252\pi$
$228$	0	0
$229$	−21.2720	−1.40569	−0.702847	−	0.711341i	$-0.748088\pi$
$229$	−21.2720	−1.40569	−0.702847	+	0.711341i	$0.748088\pi$
$230$	9.91941	0.654067
$231$	0	0
$232$	1.25966	0.0827005
$233$	10.7432	0.703807	0.351904	−	0.936036i	$-0.385534\pi$
$233$	10.7432	0.703807	0.351904	+	0.936036i	$0.385534\pi$
$234$	0	0
$235$	34.1572	2.22817
$236$	−9.13256	−0.594479
$237$	0	0
$238$	−4.51040	−0.292366
$239$	24.5694	1.58926	0.794632	−	0.607091i	$-0.207664\pi$
$239$	24.5694	1.58926	0.794632	+	0.607091i	$0.207664\pi$
$240$	0	0
$241$	1.91923	0.123628	0.0618142	−	0.998088i	$-0.480311\pi$
$241$	1.91923	0.123628	0.0618142	+	0.998088i	$0.480311\pi$
$242$	−10.5715	−0.679559
$243$	0	0
$244$	7.23226	0.462998
$245$	−17.0737	−1.09080
$246$	0	0
$247$	−10.9221	−0.694954
$248$	3.23726	0.205566
$249$	0	0
$250$	−6.78109	−0.428874
$251$	−25.4734	−1.60787	−0.803934	−	0.594719i	$-0.797263\pi$
$251$	−25.4734	−1.60787	−0.803934	+	0.594719i	$0.797263\pi$
$252$	0	0
$253$	2.36661	0.148788
$254$	17.1994	1.07919
$255$	0	0
$256$	1.00000	0.0625000
$257$	2.41342	0.150545	0.0752725	−	0.997163i	$-0.476017\pi$
$257$	2.41342	0.150545	0.0752725	+	0.997163i	$0.476017\pi$
$258$	0	0
$259$	4.04786	0.251522
$260$	5.67399	0.351886
$261$	0	0
$262$	−6.90241	−0.426433
$263$	23.8545	1.47093	0.735464	−	0.677563i	$-0.236964\pi$
$263$	23.8545	1.47093	0.735464	+	0.677563i	$0.236964\pi$
$264$	0	0
$265$	18.2402	1.12049
$266$	−4.65690	−0.285533
$267$	0	0
$268$	2.54516	0.155470
$269$	12.8156	0.781382	0.390691	−	0.920522i	$-0.372236\pi$
$269$	12.8156	0.781382	0.390691	+	0.920522i	$0.372236\pi$
$270$	0	0
$271$	−25.2316	−1.53271	−0.766355	−	0.642417i	$-0.777932\pi$
$271$	−25.2316	−1.53271	−0.766355	+	0.642417i	$0.777932\pi$
$272$	−5.11554	−0.310175
$273$	0	0
$274$	−11.6555	−0.704136
$275$	1.65531	0.0998189
$276$	0	0
$277$	−13.4492	−0.808082	−0.404041	−	0.914741i	$-0.632395\pi$
$277$	−13.4492	−0.808082	−0.404041	+	0.914741i	$0.632395\pi$
$278$	9.96954	0.597933
$279$	0	0
$280$	2.41925	0.144578
$281$	−13.8105	−0.823865	−0.411933	−	0.911214i	$-0.635146\pi$
$281$	−13.8105	−0.823865	−0.411933	+	0.911214i	$0.635146\pi$
$282$	0	0
$283$	7.49024	0.445249	0.222624	−	0.974904i	$-0.428538\pi$
$283$	7.49024	0.445249	0.222624	+	0.974904i	$0.428538\pi$
$284$	−8.61358	−0.511122
$285$	0	0
$286$	1.35372	0.0800473
$287$	−0.849808	−0.0501626
$288$	0	0
$289$	9.16877	0.539339
$290$	3.45628	0.202960
$291$	0	0
$292$	5.32891	0.311851
$293$	−32.1572	−1.87865	−0.939323	−	0.343034i	$-0.888545\pi$
$293$	−32.1572	−1.87865	−0.939323	+	0.343034i	$0.888545\pi$
$294$	0	0
$295$	−25.0582	−1.45894
$296$	4.59094	0.266843
$297$	0	0
$298$	−1.00000	−0.0579284
$299$	7.47584	0.432339
$300$	0	0
$301$	4.85138	0.279629
$302$	−14.8287	−0.853294
$303$	0	0
$304$	−5.28170	−0.302926
$305$	19.8441	1.13627
$306$	0	0
$307$	9.50165	0.542288	0.271144	−	0.962539i	$-0.412598\pi$
$307$	9.50165	0.542288	0.271144	+	0.962539i	$0.412598\pi$
$308$	0.577195	0.0328887
$309$	0	0
$310$	8.88248	0.504491
$311$	−6.21901	−0.352647	−0.176324	−	0.984332i	$-0.556421\pi$
$311$	−6.21901	−0.352647	−0.176324	+	0.984332i	$0.556421\pi$
$312$	0	0
$313$	−27.3247	−1.54448	−0.772240	−	0.635331i	$-0.780864\pi$
$313$	−27.3247	−1.54448	−0.772240	+	0.635331i	$0.780864\pi$
$314$	−7.11784	−0.401683
$315$	0	0
$316$	1.75447	0.0986969
$317$	−4.08185	−0.229260	−0.114630	−	0.993408i	$-0.536568\pi$
$317$	−4.08185	−0.229260	−0.114630	+	0.993408i	$0.536568\pi$
$318$	0	0
$319$	0.824615	0.0461696
$320$	2.74383	0.153385
$321$	0	0
$322$	3.18752	0.177633
$323$	27.0187	1.50336
$324$	0	0
$325$	5.22892	0.290048
$326$	2.33495	0.129321
$327$	0	0
$328$	−0.963823	−0.0532182
$329$	10.9761	0.605133
$330$	0	0
$331$	0.523601	0.0287797	0.0143898	−	0.999896i	$-0.495419\pi$
$331$	0.523601	0.0287797	0.0143898	+	0.999896i	$0.495419\pi$
$332$	16.3610	0.897928
$333$	0	0
$334$	−9.14665	−0.500482
$335$	6.98347	0.381548
$336$	0	0
$337$	−25.2794	−1.37706	−0.688528	−	0.725210i	$-0.741743\pi$
$337$	−25.2794	−1.37706	−0.688528	+	0.725210i	$0.741743\pi$
$338$	−8.72376	−0.474510
$339$	0	0
$340$	−14.0362	−0.761219
$341$	2.11922	0.114762
$342$	0	0
$343$	−11.6584	−0.629496
$344$	5.50227	0.296662
$345$	0	0
$346$	5.85264	0.314640
$347$	14.9548	0.802814	0.401407	−	0.915900i	$-0.368521\pi$
$347$	14.9548	0.802814	0.401407	+	0.915900i	$0.368521\pi$
$348$	0	0
$349$	−12.7902	−0.684643	−0.342322	−	0.939583i	$-0.611213\pi$
$349$	−12.7902	−0.684643	−0.342322	+	0.939583i	$0.611213\pi$
$350$	2.22948	0.119171
$351$	0	0
$352$	0.654634	0.0348921
$353$	28.3355	1.50815	0.754073	−	0.656790i	$-0.228086\pi$
$353$	28.3355	1.50815	0.754073	+	0.656790i	$0.228086\pi$
$354$	0	0
$355$	−23.6342	−1.25437
$356$	10.7640	0.570493
$357$	0	0
$358$	−21.7372	−1.14885
$359$	−0.307421	−0.0162251	−0.00811254	−	0.999967i	$-0.502582\pi$
$359$	−0.307421	−0.0162251	−0.00811254	+	0.999967i	$0.502582\pi$
$360$	0	0
$361$	8.89634	0.468228
$362$	−11.7960	−0.619984
$363$	0	0
$364$	1.82329	0.0955662
$365$	14.6216	0.765331
$366$	0	0
$367$	26.0168	1.35807	0.679034	−	0.734107i	$-0.262399\pi$
$367$	26.0168	1.35807	0.679034	+	0.734107i	$0.262399\pi$
$368$	3.61517	0.188454
$369$	0	0
$370$	12.5968	0.654875
$371$	5.86133	0.304305
$372$	0	0
$373$	−12.1658	−0.629924	−0.314962	−	0.949104i	$-0.601992\pi$
$373$	−12.1658	−0.629924	−0.314962	+	0.949104i	$0.601992\pi$
$374$	−3.34881	−0.173163
$375$	0	0
$376$	12.4487	0.641994
$377$	2.60485	0.134157
$378$	0	0
$379$	−9.40259	−0.482979	−0.241489	−	0.970403i	$-0.577636\pi$
$379$	−9.40259	−0.482979	−0.241489	+	0.970403i	$0.577636\pi$
$380$	−14.4921	−0.743428
$381$	0	0
$382$	−4.40425	−0.225341
$383$	29.2440	1.49430	0.747150	−	0.664655i	$-0.231421\pi$
$383$	29.2440	1.49430	0.747150	+	0.664655i	$0.231421\pi$
$384$	0	0
$385$	1.58372	0.0807141
$386$	6.02221	0.306522
$387$	0	0
$388$	1.52733	0.0775384
$389$	10.0683	0.510483	0.255242	−	0.966877i	$-0.417845\pi$
$389$	10.0683	0.510483	0.255242	+	0.966877i	$0.417845\pi$
$390$	0	0
$391$	−18.4936	−0.935259
$392$	−6.22260	−0.314289
$393$	0	0
$394$	−6.13256	−0.308954
$395$	4.81398	0.242218
$396$	0	0
$397$	−23.1277	−1.16075	−0.580373	−	0.814351i	$-0.697093\pi$
$397$	−23.1277	−1.16075	−0.580373	+	0.814351i	$0.697093\pi$
$398$	0.361191	0.0181049
$399$	0	0
$400$	2.52860	0.126430
$401$	16.4894	0.823439	0.411719	−	0.911311i	$-0.364928\pi$
$401$	16.4894	0.823439	0.411719	+	0.911311i	$0.364928\pi$
$402$	0	0
$403$	6.69435	0.333469
$404$	6.41245	0.319031
$405$	0	0
$406$	1.11065	0.0551205
$407$	3.00539	0.148972
$408$	0	0
$409$	−7.10611	−0.351375	−0.175687	−	0.984446i	$-0.556215\pi$
$409$	−7.10611	−0.351375	−0.175687	+	0.984446i	$0.556215\pi$
$410$	−2.64457	−0.130606
$411$	0	0
$412$	9.90008	0.487742
$413$	−8.05223	−0.396224
$414$	0	0
$415$	44.8919	2.20366
$416$	2.06791	0.101388
$417$	0	0
$418$	−3.45758	−0.169116
$419$	34.4695	1.68395	0.841973	−	0.539519i	$-0.181394\pi$
$419$	34.4695	1.68395	0.841973	+	0.539519i	$0.181394\pi$
$420$	0	0
$421$	−2.87761	−0.140246	−0.0701230	−	0.997538i	$-0.522339\pi$
$421$	−2.87761	−0.140246	−0.0701230	+	0.997538i	$0.522339\pi$
$422$	−13.9869	−0.680871
$423$	0	0
$424$	6.64771	0.322842
$425$	−12.9352	−0.627448
$426$	0	0
$427$	6.37672	0.308591
$428$	6.14489	0.297025
$429$	0	0
$430$	15.0973	0.728056
$431$	−23.7630	−1.14462	−0.572311	−	0.820036i	$-0.693953\pi$
$431$	−23.7630	−1.14462	−0.572311	+	0.820036i	$0.693953\pi$
$432$	0	0
$433$	−13.6626	−0.656585	−0.328292	−	0.944576i	$-0.606473\pi$
$433$	−13.6626	−0.656585	−0.328292	+	0.944576i	$0.606473\pi$
$434$	2.85431	0.137011
$435$	0	0
$436$	7.91047	0.378843
$437$	−19.0942	−0.913401
$438$	0	0
$439$	−5.28894	−0.252428	−0.126214	−	0.992003i	$-0.540283\pi$
$439$	−5.28894	−0.252428	−0.126214	+	0.992003i	$0.540283\pi$
$440$	1.79620	0.0856307
$441$	0	0
$442$	−10.5785	−0.503166
$443$	38.3895	1.82394	0.911969	−	0.410258i	$-0.134561\pi$
$443$	38.3895	1.82394	0.911969	+	0.410258i	$0.134561\pi$
$444$	0	0
$445$	29.5347	1.40008
$446$	−17.0975	−0.809590
$447$	0	0
$448$	0.881706	0.0416567
$449$	−22.5010	−1.06189	−0.530943	−	0.847407i	$-0.678162\pi$
$449$	−22.5010	−1.06189	−0.530943	+	0.847407i	$0.678162\pi$
$450$	0	0
$451$	−0.630951	−0.0297103
$452$	14.6098	0.687188
$453$	0	0
$454$	−14.5376	−0.682282
$455$	5.00279	0.234534
$456$	0	0
$457$	25.8987	1.21149	0.605745	−	0.795659i	$-0.292875\pi$
$457$	25.8987	1.21149	0.605745	+	0.795659i	$0.292875\pi$
$458$	−21.2720	−0.993975
$459$	0	0
$460$	9.91941	0.462495
$461$	−6.58463	−0.306677	−0.153338	−	0.988174i	$-0.549002\pi$
$461$	−6.58463	−0.306677	−0.153338	+	0.988174i	$0.549002\pi$
$462$	0	0
$463$	−5.12487	−0.238173	−0.119087	−	0.992884i	$-0.537997\pi$
$463$	−5.12487	−0.238173	−0.119087	+	0.992884i	$0.537997\pi$
$464$	1.25966	0.0584781
$465$	0	0
$466$	10.7432	0.497667
$467$	42.0808	1.94727	0.973634	−	0.228114i	$-0.0732560\pi$
$467$	42.0808	1.94727	0.973634	+	0.228114i	$0.0732560\pi$
$468$	0	0
$469$	2.24408	0.103622
$470$	34.1572	1.57555
$471$	0	0
$472$	−9.13256	−0.420360
$473$	3.60197	0.165619
$474$	0	0
$475$	−13.3553	−0.612784
$476$	−4.51040	−0.206734
$477$	0	0
$478$	24.5694	1.12378
$479$	−35.8024	−1.63585	−0.817926	−	0.575323i	$-0.804876\pi$
$479$	−35.8024	−1.63585	−0.817926	+	0.575323i	$0.804876\pi$
$480$	0	0
$481$	9.49365	0.432873
$482$	1.91923	0.0874185
$483$	0	0
$484$	−10.5715	−0.480521
$485$	4.19073	0.190291
$486$	0	0
$487$	7.56477	0.342793	0.171396	−	0.985202i	$-0.445172\pi$
$487$	7.56477	0.342793	0.171396	+	0.985202i	$0.445172\pi$
$488$	7.23226	0.327389
$489$	0	0
$490$	−17.0737	−0.771313
$491$	13.9200	0.628200	0.314100	−	0.949390i	$-0.398297\pi$
$491$	13.9200	0.628200	0.314100	+	0.949390i	$0.398297\pi$
$492$	0	0
$493$	−6.44383	−0.290215
$494$	−10.9221	−0.491407
$495$	0	0
$496$	3.23726	0.145357
$497$	−7.59464	−0.340666
$498$	0	0
$499$	25.8560	1.15747	0.578737	−	0.815514i	$-0.303546\pi$
$499$	25.8560	1.15747	0.578737	+	0.815514i	$0.303546\pi$
$500$	−6.78109	−0.303260
$501$	0	0
$502$	−25.4734	−1.13693
$503$	−5.88329	−0.262323	−0.131162	−	0.991361i	$-0.541871\pi$
$503$	−5.88329	−0.262323	−0.131162	+	0.991361i	$0.541871\pi$
$504$	0	0
$505$	17.5947	0.782953
$506$	2.36661	0.105209
$507$	0	0
$508$	17.1994	0.763101
$509$	−20.3791	−0.903288	−0.451644	−	0.892198i	$-0.649162\pi$
$509$	−20.3791	−0.903288	−0.451644	+	0.892198i	$0.649162\pi$
$510$	0	0
$511$	4.69853	0.207851
$512$	1.00000	0.0441942
$513$	0	0
$514$	2.41342	0.106451
$515$	27.1641	1.19700
$516$	0	0
$517$	8.14937	0.358409
$518$	4.04786	0.177853
$519$	0	0
$520$	5.67399	0.248821
$521$	−10.2325	−0.448295	−0.224148	−	0.974555i	$-0.571960\pi$
$521$	−10.2325	−0.448295	−0.224148	+	0.974555i	$0.571960\pi$
$522$	0	0
$523$	16.4969	0.721361	0.360680	−	0.932689i	$-0.382545\pi$
$523$	16.4969	0.721361	0.360680	+	0.932689i	$0.382545\pi$
$524$	−6.90241	−0.301533
$525$	0	0
$526$	23.8545	1.04010
$527$	−16.5603	−0.721379
$528$	0	0
$529$	−9.93054	−0.431763
$530$	18.2402	0.792303
$531$	0	0
$532$	−4.65690	−0.201902
$533$	−1.99310	−0.0863306
$534$	0	0
$535$	16.8605	0.728944
$536$	2.54516	0.109934
$537$	0	0
$538$	12.8156	0.552520
$539$	−4.07352	−0.175459
$540$	0	0
$541$	11.4996	0.494405	0.247203	−	0.968964i	$-0.420489\pi$
$541$	11.4996	0.494405	0.247203	+	0.968964i	$0.420489\pi$
$542$	−25.2316	−1.08379
$543$	0	0
$544$	−5.11554	−0.219327
$545$	21.7050	0.929739
$546$	0	0
$547$	−45.1095	−1.92874	−0.964371	−	0.264554i	$-0.914775\pi$
$547$	−45.1095	−1.92874	−0.964371	+	0.264554i	$0.914775\pi$
$548$	−11.6555	−0.497900
$549$	0	0
$550$	1.65531	0.0705826
$551$	−6.65313	−0.283433
$552$	0	0
$553$	1.54693	0.0657822
$554$	−13.4492	−0.571400
$555$	0	0
$556$	9.96954	0.422803
$557$	−6.85321	−0.290380	−0.145190	−	0.989404i	$-0.546379\pi$
$557$	−6.85321	−0.290380	−0.145190	+	0.989404i	$0.546379\pi$
$558$	0	0
$559$	11.3782	0.481246
$560$	2.41925	0.102232
$561$	0	0
$562$	−13.8105	−0.582561
$563$	28.6899	1.20914	0.604568	−	0.796554i	$-0.293346\pi$
$563$	28.6899	1.20914	0.604568	+	0.796554i	$0.293346\pi$
$564$	0	0
$565$	40.0868	1.68647
$566$	7.49024	0.314838
$567$	0	0
$568$	−8.61358	−0.361418
$569$	25.0202	1.04890	0.524452	−	0.851440i	$-0.324270\pi$
$569$	25.0202	1.04890	0.524452	+	0.851440i	$0.324270\pi$
$570$	0	0
$571$	−6.30805	−0.263984	−0.131992	−	0.991251i	$-0.542137\pi$
$571$	−6.30805	−0.263984	−0.131992	+	0.991251i	$0.542137\pi$
$572$	1.35372	0.0566020
$573$	0	0
$574$	−0.849808	−0.0354703
$575$	9.14133	0.381220
$576$	0	0
$577$	−24.1971	−1.00734	−0.503668	−	0.863897i	$-0.668017\pi$
$577$	−24.1971	−1.00734	−0.503668	+	0.863897i	$0.668017\pi$
$578$	9.16877	0.381370
$579$	0	0
$580$	3.45628	0.143514
$581$	14.4256	0.598475
$582$	0	0
$583$	4.35182	0.180234
$584$	5.32891	0.220512
$585$	0	0
$586$	−32.1572	−1.32840
$587$	9.40596	0.388225	0.194113	−	0.980979i	$-0.437817\pi$
$587$	9.40596	0.388225	0.194113	+	0.980979i	$0.437817\pi$
$588$	0	0
$589$	−17.0982	−0.704520
$590$	−25.0582	−1.03163
$591$	0	0
$592$	4.59094	0.188687
$593$	−22.5935	−0.927802	−0.463901	−	0.885887i	$-0.653551\pi$
$593$	−22.5935	−0.927802	−0.463901	+	0.885887i	$0.653551\pi$
$594$	0	0
$595$	−12.3758	−0.507357
$596$	−1.00000	−0.0409616
$597$	0	0
$598$	7.47584	0.305710
$599$	27.0858	1.10670	0.553349	−	0.832950i	$-0.313350\pi$
$599$	27.0858	1.10670	0.553349	+	0.832950i	$0.313350\pi$
$600$	0	0
$601$	−35.9213	−1.46526	−0.732631	−	0.680626i	$-0.761708\pi$
$601$	−35.9213	−1.46526	−0.732631	+	0.680626i	$0.761708\pi$
$602$	4.85138	0.197728
$603$	0	0
$604$	−14.8287	−0.603370
$605$	−29.0063	−1.17927
$606$	0	0
$607$	−31.7764	−1.28976	−0.644882	−	0.764282i	$-0.723094\pi$
$607$	−31.7764	−1.28976	−0.644882	+	0.764282i	$0.723094\pi$
$608$	−5.28170	−0.214201
$609$	0	0
$610$	19.8441	0.803463
$611$	25.7428	1.04144
$612$	0	0
$613$	−44.8926	−1.81319	−0.906597	−	0.421998i	$-0.861329\pi$
$613$	−44.8926	−1.81319	−0.906597	+	0.421998i	$0.861329\pi$
$614$	9.50165	0.383455
$615$	0	0
$616$	0.577195	0.0232558
$617$	16.7955	0.676161	0.338080	−	0.941117i	$-0.390222\pi$
$617$	16.7955	0.676161	0.338080	+	0.941117i	$0.390222\pi$
$618$	0	0
$619$	−18.8667	−0.758317	−0.379158	−	0.925332i	$-0.623786\pi$
$619$	−18.8667	−0.758317	−0.379158	+	0.925332i	$0.623786\pi$
$620$	8.88248	0.356729
$621$	0	0
$622$	−6.21901	−0.249359
$623$	9.49072	0.380238
$624$	0	0
$625$	−31.2492	−1.24997
$626$	−27.3247	−1.09211
$627$	0	0
$628$	−7.11784	−0.284033
$629$	−23.4852	−0.936415
$630$	0	0
$631$	−9.81880	−0.390880	−0.195440	−	0.980716i	$-0.562614\pi$
$631$	−9.81880	−0.390880	−0.195440	+	0.980716i	$0.562614\pi$
$632$	1.75447	0.0697892
$633$	0	0
$634$	−4.08185	−0.162111
$635$	47.1923	1.87277
$636$	0	0
$637$	−12.8678	−0.509839
$638$	0.824615	0.0326468
$639$	0	0
$640$	2.74383	0.108459
$641$	17.7828	0.702379	0.351190	−	0.936304i	$-0.385777\pi$
$641$	17.7828	0.702379	0.351190	+	0.936304i	$0.385777\pi$
$642$	0	0
$643$	6.35039	0.250435	0.125218	−	0.992129i	$-0.460037\pi$
$643$	6.35039	0.250435	0.125218	+	0.992129i	$0.460037\pi$
$644$	3.18752	0.125606
$645$	0	0
$646$	27.0187	1.06304
$647$	−32.9331	−1.29473	−0.647366	−	0.762179i	$-0.724130\pi$
$647$	−32.9331	−1.29473	−0.647366	+	0.762179i	$0.724130\pi$
$648$	0	0
$649$	−5.97849	−0.234676
$650$	5.22892	0.205095
$651$	0	0
$652$	2.33495	0.0914439
$653$	−22.1156	−0.865449	−0.432725	−	0.901526i	$-0.642448\pi$
$653$	−22.1156	−0.865449	−0.432725	+	0.901526i	$0.642448\pi$
$654$	0	0
$655$	−18.9390	−0.740010
$656$	−0.963823	−0.0376310
$657$	0	0
$658$	10.9761	0.427894
$659$	−17.3171	−0.674578	−0.337289	−	0.941401i	$-0.609510\pi$
$659$	−17.3171	−0.674578	−0.337289	+	0.941401i	$0.609510\pi$
$660$	0	0
$661$	−4.42379	−0.172066	−0.0860328	−	0.996292i	$-0.527419\pi$
$661$	−4.42379	−0.172066	−0.0860328	+	0.996292i	$0.527419\pi$
$662$	0.523601	0.0203503
$663$	0	0
$664$	16.3610	0.634931
$665$	−12.7778	−0.495500
$666$	0	0
$667$	4.55388	0.176327
$668$	−9.14665	−0.353894
$669$	0	0
$670$	6.98347	0.269795
$671$	4.73448	0.182773
$672$	0	0
$673$	26.3839	1.01702	0.508512	−	0.861055i	$-0.330196\pi$
$673$	26.3839	1.01702	0.508512	+	0.861055i	$0.330196\pi$
$674$	−25.2794	−0.973725
$675$	0	0
$676$	−8.72376	−0.335529
$677$	3.14491	0.120869	0.0604343	−	0.998172i	$-0.480751\pi$
$677$	3.14491	0.120869	0.0604343	+	0.998172i	$0.480751\pi$
$678$	0	0
$679$	1.34666	0.0516799
$680$	−14.0362	−0.538263
$681$	0	0
$682$	2.11922	0.0811491
$683$	−27.6076	−1.05638	−0.528189	−	0.849127i	$-0.677129\pi$
$683$	−27.6076	−1.05638	−0.528189	+	0.849127i	$0.677129\pi$
$684$	0	0
$685$	−31.9808	−1.22192
$686$	−11.6584	−0.445121
$687$	0	0
$688$	5.50227	0.209772
$689$	13.7469	0.523714
$690$	0	0
$691$	−25.8817	−0.984588	−0.492294	−	0.870429i	$-0.663841\pi$
$691$	−25.8817	−0.984588	−0.492294	+	0.870429i	$0.663841\pi$
$692$	5.85264	0.222484
$693$	0	0
$694$	14.9548	0.567675
$695$	27.3547	1.03762
$696$	0	0
$697$	4.93048	0.186755
$698$	−12.7902	−0.484116
$699$	0	0
$700$	2.22948	0.0842665
$701$	7.63957	0.288543	0.144271	−	0.989538i	$-0.453916\pi$
$701$	7.63957	0.288543	0.144271	+	0.989538i	$0.453916\pi$
$702$	0	0
$703$	−24.2480	−0.914530
$704$	0.654634	0.0246725
$705$	0	0
$706$	28.3355	1.06642
$707$	5.65389	0.212637
$708$	0	0
$709$	20.8901	0.784544	0.392272	−	0.919849i	$-0.371689\pi$
$709$	20.8901	0.784544	0.392272	+	0.919849i	$0.371689\pi$
$710$	−23.6342	−0.886976
$711$	0	0
$712$	10.7640	0.403400
$713$	11.7032	0.438290
$714$	0	0
$715$	3.71439	0.138910
$716$	−21.7372	−0.812356
$717$	0	0
$718$	−0.307421	−0.0114729
$719$	−38.9438	−1.45236	−0.726179	−	0.687506i	$-0.758706\pi$
$719$	−38.9438	−1.45236	−0.726179	+	0.687506i	$0.758706\pi$
$720$	0	0
$721$	8.72896	0.325083
$722$	8.89634	0.331087
$723$	0	0
$724$	−11.7960	−0.438395
$725$	3.18517	0.118294
$726$	0	0
$727$	−40.2983	−1.49458	−0.747290	−	0.664498i	$-0.768645\pi$
$727$	−40.2983	−1.49458	−0.747290	+	0.664498i	$0.768645\pi$
$728$	1.82329	0.0675755
$729$	0	0
$730$	14.6216	0.541170
$731$	−28.1471	−1.04106
$732$	0	0
$733$	13.8806	0.512693	0.256346	−	0.966585i	$-0.417481\pi$
$733$	13.8806	0.512693	0.256346	+	0.966585i	$0.417481\pi$
$734$	26.0168	0.960299
$735$	0	0
$736$	3.61517	0.133257
$737$	1.66615	0.0613733
$738$	0	0
$739$	13.1572	0.483995	0.241998	−	0.970277i	$-0.422197\pi$
$739$	13.1572	0.483995	0.241998	+	0.970277i	$0.422197\pi$
$740$	12.5968	0.463066
$741$	0	0
$742$	5.86133	0.215176
$743$	−2.41022	−0.0884223	−0.0442111	−	0.999022i	$-0.514077\pi$
$743$	−2.41022	−0.0884223	−0.0442111	+	0.999022i	$0.514077\pi$
$744$	0	0
$745$	−2.74383	−0.100526
$746$	−12.1658	−0.445423
$747$	0	0
$748$	−3.34881	−0.122445
$749$	5.41799	0.197969
$750$	0	0
$751$	27.4198	1.00056	0.500281	−	0.865863i	$-0.333230\pi$
$751$	27.4198	1.00056	0.500281	+	0.865863i	$0.333230\pi$
$752$	12.4487	0.453959
$753$	0	0
$754$	2.60485	0.0948632
$755$	−40.6874	−1.48076
$756$	0	0
$757$	5.82894	0.211856	0.105928	−	0.994374i	$-0.466219\pi$
$757$	5.82894	0.211856	0.105928	+	0.994374i	$0.466219\pi$
$758$	−9.40259	−0.341518
$759$	0	0
$760$	−14.4921	−0.525683
$761$	−2.46204	−0.0892488	−0.0446244	−	0.999004i	$-0.514209\pi$
$761$	−2.46204	−0.0892488	−0.0446244	+	0.999004i	$0.514209\pi$
$762$	0	0
$763$	6.97471	0.252501
$764$	−4.40425	−0.159340
$765$	0	0
$766$	29.2440	1.05663
$767$	−18.8853	−0.681909
$768$	0	0
$769$	−21.5369	−0.776639	−0.388319	−	0.921525i	$-0.626944\pi$
$769$	−21.5369	−0.776639	−0.388319	+	0.921525i	$0.626944\pi$
$770$	1.58372	0.0570735
$771$	0	0
$772$	6.02221	0.216744
$773$	−3.60779	−0.129763	−0.0648816	−	0.997893i	$-0.520667\pi$
$773$	−3.60779	−0.129763	−0.0648816	+	0.997893i	$0.520667\pi$
$774$	0	0
$775$	8.18573	0.294040
$776$	1.52733	0.0548280
$777$	0	0
$778$	10.0683	0.360966
$779$	5.09062	0.182390
$780$	0	0
$781$	−5.63875	−0.201770
$782$	−18.4936	−0.661328
$783$	0	0
$784$	−6.22260	−0.222236
$785$	−19.5302	−0.697061
$786$	0	0
$787$	17.5637	0.626079	0.313039	−	0.949740i	$-0.398653\pi$
$787$	17.5637	0.626079	0.313039	+	0.949740i	$0.398653\pi$
$788$	−6.13256	−0.218464
$789$	0	0
$790$	4.81398	0.171274
$791$	12.8816	0.458015
$792$	0	0
$793$	14.9556	0.531090
$794$	−23.1277	−0.820771
$795$	0	0
$796$	0.361191	0.0128021
$797$	24.2790	0.860006	0.430003	−	0.902827i	$-0.358512\pi$
$797$	24.2790	0.860006	0.430003	+	0.902827i	$0.358512\pi$
$798$	0	0
$799$	−63.6820	−2.25291
$800$	2.52860	0.0893996
$801$	0	0
$802$	16.4894	0.582259
$803$	3.48849	0.123106
$804$	0	0
$805$	8.74600	0.308256
$806$	6.69435	0.235798
$807$	0	0
$808$	6.41245	0.225589
$809$	39.1163	1.37525	0.687627	−	0.726064i	$-0.258652\pi$
$809$	39.1163	1.37525	0.687627	+	0.726064i	$0.258652\pi$
$810$	0	0
$811$	29.9556	1.05188	0.525942	−	0.850520i	$-0.323713\pi$
$811$	29.9556	1.05188	0.525942	+	0.850520i	$0.323713\pi$
$812$	1.11065	0.0389761
$813$	0	0
$814$	3.00539	0.105339
$815$	6.40672	0.224417
$816$	0	0
$817$	−29.0613	−1.01673
$818$	−7.10611	−0.248459
$819$	0	0
$820$	−2.64457	−0.0923522
$821$	−29.4060	−1.02628	−0.513139	−	0.858306i	$-0.671517\pi$
$821$	−29.4060	−1.02628	−0.513139	+	0.858306i	$0.671517\pi$
$822$	0	0
$823$	−44.6993	−1.55812	−0.779061	−	0.626949i	$-0.784304\pi$
$823$	−44.6993	−1.55812	−0.779061	+	0.626949i	$0.784304\pi$
$824$	9.90008	0.344886
$825$	0	0
$826$	−8.05223	−0.280173
$827$	20.0409	0.696891	0.348446	−	0.937329i	$-0.386710\pi$
$827$	20.0409	0.696891	0.348446	+	0.937329i	$0.386710\pi$
$828$	0	0
$829$	27.3328	0.949306	0.474653	−	0.880173i	$-0.342574\pi$
$829$	27.3328	0.949306	0.474653	+	0.880173i	$0.342574\pi$
$830$	44.8919	1.55822
$831$	0	0
$832$	2.06791	0.0716918
$833$	31.8319	1.10291
$834$	0	0
$835$	−25.0968	−0.868512
$836$	−3.45758	−0.119583
$837$	0	0
$838$	34.4695	1.19073
$839$	42.2357	1.45814	0.729068	−	0.684441i	$-0.239954\pi$
$839$	42.2357	1.45814	0.729068	+	0.684441i	$0.239954\pi$
$840$	0	0
$841$	−27.4133	−0.945285
$842$	−2.87761	−0.0991689
$843$	0	0
$844$	−13.9869	−0.481449
$845$	−23.9365	−0.823441
$846$	0	0
$847$	−9.32091	−0.320270
$848$	6.64771	0.228283
$849$	0	0
$850$	−12.9352	−0.443673
$851$	16.5970	0.568939
$852$	0	0
$853$	8.18645	0.280299	0.140149	−	0.990130i	$-0.455242\pi$
$853$	8.18645	0.280299	0.140149	+	0.990130i	$0.455242\pi$
$854$	6.37672	0.218207
$855$	0	0
$856$	6.14489	0.210028
$857$	−7.39028	−0.252447	−0.126224	−	0.992002i	$-0.540286\pi$
$857$	−7.39028	−0.252447	−0.126224	+	0.992002i	$0.540286\pi$
$858$	0	0
$859$	7.94257	0.270997	0.135498	−	0.990778i	$-0.456736\pi$
$859$	7.94257	0.270997	0.135498	+	0.990778i	$0.456736\pi$
$860$	15.0973	0.514813
$861$	0	0
$862$	−23.7630	−0.809370
$863$	23.4692	0.798900	0.399450	−	0.916755i	$-0.369201\pi$
$863$	23.4692	0.798900	0.399450	+	0.916755i	$0.369201\pi$
$864$	0	0
$865$	16.0587	0.546011
$866$	−13.6626	−0.464276
$867$	0	0
$868$	2.85431	0.0968815
$869$	1.14854	0.0389615
$870$	0	0
$871$	5.26315	0.178335
$872$	7.91047	0.267882
$873$	0	0
$874$	−19.0942	−0.645872
$875$	−5.97893	−0.202125
$876$	0	0
$877$	1.53521	0.0518404	0.0259202	−	0.999664i	$-0.491748\pi$
$877$	1.53521	0.0518404	0.0259202	+	0.999664i	$0.491748\pi$
$878$	−5.28894	−0.178493
$879$	0	0
$880$	1.79620	0.0605501
$881$	35.6573	1.20133	0.600663	−	0.799503i	$-0.294904\pi$
$881$	35.6573	1.20133	0.600663	+	0.799503i	$0.294904\pi$
$882$	0	0
$883$	−5.45880	−0.183703	−0.0918517	−	0.995773i	$-0.529279\pi$
$883$	−5.45880	−0.183703	−0.0918517	+	0.995773i	$0.529279\pi$
$884$	−10.5785	−0.355792
$885$	0	0
$886$	38.3895	1.28972
$887$	37.2684	1.25135	0.625675	−	0.780084i	$-0.284824\pi$
$887$	37.2684	1.25135	0.625675	+	0.780084i	$0.284824\pi$
$888$	0	0
$889$	15.1648	0.508612
$890$	29.5347	0.990006
$891$	0	0
$892$	−17.0975	−0.572467
$893$	−65.7505	−2.20026
$894$	0	0
$895$	−59.6431	−1.99365
$896$	0.881706	0.0294557
$897$	0	0
$898$	−22.5010	−0.750867
$899$	4.07783	0.136003
$900$	0	0
$901$	−34.0067	−1.13293
$902$	−0.630951	−0.0210084
$903$	0	0
$904$	14.6098	0.485915
$905$	−32.3662	−1.07589
$906$	0	0
$907$	−0.828857	−0.0275217	−0.0137609	−	0.999905i	$-0.504380\pi$
$907$	−0.828857	−0.0275217	−0.0137609	+	0.999905i	$0.504380\pi$
$908$	−14.5376	−0.482446
$909$	0	0
$910$	5.00279	0.165841
$911$	43.4321	1.43897	0.719484	−	0.694509i	$-0.244378\pi$
$911$	43.4321	1.43897	0.719484	+	0.694509i	$0.244378\pi$
$912$	0	0
$913$	10.7105	0.354465
$914$	25.8987	0.856652
$915$	0	0
$916$	−21.2720	−0.702847
$917$	−6.08590	−0.200974
$918$	0	0
$919$	19.3603	0.638636	0.319318	−	0.947648i	$-0.396546\pi$
$919$	19.3603	0.638636	0.319318	+	0.947648i	$0.396546\pi$
$920$	9.91941	0.327033
$921$	0	0
$922$	−6.58463	−0.216853
$923$	−17.8121	−0.586292
$924$	0	0
$925$	11.6087	0.381691
$926$	−5.12487	−0.168414
$927$	0	0
$928$	1.25966	0.0413503
$929$	21.2919	0.698566	0.349283	−	0.937017i	$-0.386425\pi$
$929$	21.2919	0.698566	0.349283	+	0.937017i	$0.386425\pi$
$930$	0	0
$931$	32.8659	1.07714
$932$	10.7432	0.351904
$933$	0	0
$934$	42.0808	1.37693
$935$	−9.18856	−0.300498
$936$	0	0
$937$	18.3462	0.599343	0.299671	−	0.954043i	$-0.403123\pi$
$937$	18.3462	0.599343	0.299671	+	0.954043i	$0.403123\pi$
$938$	2.24408	0.0732717
$939$	0	0
$940$	34.1572	1.11409
$941$	17.6325	0.574802	0.287401	−	0.957810i	$-0.407209\pi$
$941$	17.6325	0.574802	0.287401	+	0.957810i	$0.407209\pi$
$942$	0	0
$943$	−3.48438	−0.113467
$944$	−9.13256	−0.297240
$945$	0	0
$946$	3.60197	0.117110
$947$	−14.4320	−0.468977	−0.234488	−	0.972119i	$-0.575342\pi$
$947$	−14.4320	−0.468977	−0.234488	+	0.972119i	$0.575342\pi$
$948$	0	0
$949$	11.0197	0.357714
$950$	−13.3553	−0.433304
$951$	0	0
$952$	−4.51040	−0.146183
$953$	−51.4239	−1.66578	−0.832892	−	0.553436i	$-0.813316\pi$
$953$	−51.4239	−1.66578	−0.832892	+	0.553436i	$0.813316\pi$
$954$	0	0
$955$	−12.0845	−0.391045
$956$	24.5694	0.794632
$957$	0	0
$958$	−35.8024	−1.15672
$959$	−10.2767	−0.331853
$960$	0	0
$961$	−20.5202	−0.661941
$962$	9.49365	0.306087
$963$	0	0
$964$	1.91923	0.0618142
$965$	16.5239	0.531924
$966$	0	0
$967$	18.8182	0.605152	0.302576	−	0.953125i	$-0.402153\pi$
$967$	18.8182	0.605152	0.302576	+	0.953125i	$0.402153\pi$
$968$	−10.5715	−0.339779
$969$	0	0
$970$	4.19073	0.134556
$971$	−24.8210	−0.796543	−0.398271	−	0.917268i	$-0.630390\pi$
$971$	−24.8210	−0.796543	−0.398271	+	0.917268i	$0.630390\pi$
$972$	0	0
$973$	8.79020	0.281801
$974$	7.56477	0.242391
$975$	0	0
$976$	7.23226	0.231499
$977$	31.2465	0.999664	0.499832	−	0.866122i	$-0.333395\pi$
$977$	31.2465	0.999664	0.499832	+	0.866122i	$0.333395\pi$
$978$	0	0
$979$	7.04651	0.225208
$980$	−17.0737	−0.545401
$981$	0	0
$982$	13.9200	0.444204
$983$	37.6007	1.19928	0.599638	−	0.800271i	$-0.295311\pi$
$983$	37.6007	1.19928	0.599638	+	0.800271i	$0.295311\pi$
$984$	0	0
$985$	−16.8267	−0.536144
$986$	−6.44383	−0.205213
$987$	0	0
$988$	−10.9221	−0.347477
$989$	19.8916	0.632517
$990$	0	0
$991$	54.6189	1.73503	0.867514	−	0.497413i	$-0.165717\pi$
$991$	54.6189	1.73503	0.867514	+	0.497413i	$0.165717\pi$
$992$	3.23726	0.102783
$993$	0	0
$994$	−7.59464	−0.240887
$995$	0.991046	0.0314183
$996$	0	0
$997$	−2.14087	−0.0678019	−0.0339009	−	0.999425i	$-0.510793\pi$
$997$	−2.14087	−0.0678019	−0.0339009	+	0.999425i	$0.510793\pi$
$998$	25.8560	0.818458
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.p.1.10	yes	12
3.2	odd	2		8046.2.a.i.1.3	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.i.1.3	✓	12	3.2	odd	2
8046.2.a.p.1.10	yes	12	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8046 = 2 \cdot 3^{3} \cdot 149 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8046.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +2.74383 q^{5} +0.881706 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +2.74383 q^{5} +0.881706 q^{7} +1.00000 q^{8} +2.74383 q^{10} +0.654634 q^{11} +2.06791 q^{13} +0.881706 q^{14} +1.00000 q^{16} -5.11554 q^{17} -5.28170 q^{19} +2.74383 q^{20} +0.654634 q^{22} +3.61517 q^{23} +2.52860 q^{25} +2.06791 q^{26} +0.881706 q^{28} +1.25966 q^{29} +3.23726 q^{31} +1.00000 q^{32} -5.11554 q^{34} +2.41925 q^{35} +4.59094 q^{37} -5.28170 q^{38} +2.74383 q^{40} -0.963823 q^{41} +5.50227 q^{43} +0.654634 q^{44} +3.61517 q^{46} +12.4487 q^{47} -6.22260 q^{49} +2.52860 q^{50} +2.06791 q^{52} +6.64771 q^{53} +1.79620 q^{55} +0.881706 q^{56} +1.25966 q^{58} -9.13256 q^{59} +7.23226 q^{61} +3.23726 q^{62} +1.00000 q^{64} +5.67399 q^{65} +2.54516 q^{67} -5.11554 q^{68} +2.41925 q^{70} -8.61358 q^{71} +5.32891 q^{73} +4.59094 q^{74} -5.28170 q^{76} +0.577195 q^{77} +1.75447 q^{79} +2.74383 q^{80} -0.963823 q^{82} +16.3610 q^{83} -14.0362 q^{85} +5.50227 q^{86} +0.654634 q^{88} +10.7640 q^{89} +1.82329 q^{91} +3.61517 q^{92} +12.4487 q^{94} -14.4921 q^{95} +1.52733 q^{97} -6.22260 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{2} + 12 q^{4} + 5 q^{5} + 6 q^{7} + 12 q^{8}+O(q^{10}) \) 12 * q + 12 * q^2 + 12 * q^4 + 5 * q^5 + 6 * q^7 + 12 * q^8	\( 12 q + 12 q^{2} + 12 q^{4} + 5 q^{5} + 6 q^{7} + 12 q^{8} + 5 q^{10} + 6 q^{11} + 3 q^{13} + 6 q^{14} + 12 q^{16} + 6 q^{17} + 8 q^{19} + 5 q^{20} + 6 q^{22} + 11 q^{23} + 11 q^{25} + 3 q^{26} + 6 q^{28} + 29 q^{29} + 2 q^{31} + 12 q^{32} + 6 q^{34} + 4 q^{35} + 5 q^{37} + 8 q^{38} + 5 q^{40} + 22 q^{41} + 9 q^{43} + 6 q^{44} + 11 q^{46} + 15 q^{47} + 14 q^{49} + 11 q^{50} + 3 q^{52} + 12 q^{53} + 13 q^{55} + 6 q^{56} + 29 q^{58} + 34 q^{59} - 4 q^{61} + 2 q^{62} + 12 q^{64} + 12 q^{65} + q^{67} + 6 q^{68} + 4 q^{70} + 21 q^{71} - 2 q^{73} + 5 q^{74} + 8 q^{76} + 34 q^{77} + 9 q^{79} + 5 q^{80} + 22 q^{82} + 10 q^{83} + 5 q^{85} + 9 q^{86} + 6 q^{88} - 2 q^{89} + 17 q^{91} + 11 q^{92} + 15 q^{94} + 69 q^{95} - 13 q^{97} + 14 q^{98}+O(q^{100}) \) 12 * q + 12 * q^2 + 12 * q^4 + 5 * q^5 + 6 * q^7 + 12 * q^8 + 5 * q^10 + 6 * q^11 + 3 * q^13 + 6 * q^14 + 12 * q^16 + 6 * q^17 + 8 * q^19 + 5 * q^20 + 6 * q^22 + 11 * q^23 + 11 * q^25 + 3 * q^26 + 6 * q^28 + 29 * q^29 + 2 * q^31 + 12 * q^32 + 6 * q^34 + 4 * q^35 + 5 * q^37 + 8 * q^38 + 5 * q^40 + 22 * q^41 + 9 * q^43 + 6 * q^44 + 11 * q^46 + 15 * q^47 + 14 * q^49 + 11 * q^50 + 3 * q^52 + 12 * q^53 + 13 * q^55 + 6 * q^56 + 29 * q^58 + 34 * q^59 - 4 * q^61 + 2 * q^62 + 12 * q^64 + 12 * q^65 + q^67 + 6 * q^68 + 4 * q^70 + 21 * q^71 - 2 * q^73 + 5 * q^74 + 8 * q^76 + 34 * q^77 + 9 * q^79 + 5 * q^80 + 22 * q^82 + 10 * q^83 + 5 * q^85 + 9 * q^86 + 6 * q^88 - 2 * q^89 + 17 * q^91 + 11 * q^92 + 15 * q^94 + 69 * q^95 - 13 * q^97 + 14 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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