LMFDB - Embedded newform 3344.2.a.x.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3344,2,Mod(1,3344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3344, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3344.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3344 = 2^{4} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3344.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:	$26.7019744359$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 12x^{4} + 28x^{3} + 16x^{2} - 60x + 27 $ x^6 - 2x^5 - 12x^4 + 28x^3 + 16x^2 - 60*x + 27
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 836)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.658537$ of defining polynomial
Character	$\chi$	$=$	3344.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.658537 q^{3} -2.39807 q^{5} -4.45908 q^{7} -2.56633 q^{9} +O(q^{10})$	$q+0.658537 q^{3} -2.39807 q^{5} -4.45908 q^{7} -2.56633 q^{9} -1.00000 q^{11} +3.38518 q^{13} -1.57922 q^{15} -5.17423 q^{17} -1.00000 q^{19} -2.93647 q^{21} -0.851183 q^{23} +0.750745 q^{25} -3.66563 q^{27} +7.62294 q^{29} -6.34958 q^{31} -0.658537 q^{33} +10.6932 q^{35} +9.00597 q^{37} +2.22927 q^{39} -3.15490 q^{41} -5.24925 q^{43} +6.15424 q^{45} -8.77036 q^{47} +12.8834 q^{49} -3.40742 q^{51} +9.81558 q^{53} +2.39807 q^{55} -0.658537 q^{57} +1.57568 q^{59} -10.8420 q^{61} +11.4435 q^{63} -8.11791 q^{65} +3.43142 q^{67} -0.560536 q^{69} +4.02439 q^{71} +16.6117 q^{73} +0.494394 q^{75} +4.45908 q^{77} +4.15479 q^{79} +5.28503 q^{81} -6.27568 q^{83} +12.4082 q^{85} +5.01999 q^{87} +1.22927 q^{89} -15.0948 q^{91} -4.18143 q^{93} +2.39807 q^{95} -16.0254 q^{97} +2.56633 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{3} - 2 q^{7} + 10 q^{9}+O(q^{10}) $ 6 * q + 2 * q^3 - 2 * q^7 + 10 * q^9	$ 6 q + 2 q^{3} - 2 q^{7} + 10 q^{9} - 6 q^{11} + 2 q^{13} + 12 q^{15} + 12 q^{17} - 6 q^{19} + 2 q^{21} + 2 q^{23} + 26 q^{25} - 16 q^{27} + 4 q^{29} + 20 q^{31} - 2 q^{33} - 20 q^{35} + 22 q^{37} - 8 q^{39} - 2 q^{41} - 10 q^{43} - 6 q^{45} - 16 q^{47} + 48 q^{49} - 36 q^{51} + 12 q^{53} - 2 q^{57} + 14 q^{59} + 12 q^{61} + 4 q^{63} + 10 q^{65} + 12 q^{67} + 30 q^{69} + 30 q^{71} + 24 q^{73} + 50 q^{75} + 2 q^{77} + 10 q^{81} + 14 q^{83} + 12 q^{85} + 30 q^{87} - 14 q^{89} - 20 q^{91} + 14 q^{93} - 46 q^{97} - 10 q^{99}+O(q^{100}) $ 6 * q + 2 * q^3 - 2 * q^7 + 10 * q^9 - 6 * q^11 + 2 * q^13 + 12 * q^15 + 12 * q^17 - 6 * q^19 + 2 * q^21 + 2 * q^23 + 26 * q^25 - 16 * q^27 + 4 * q^29 + 20 * q^31 - 2 * q^33 - 20 * q^35 + 22 * q^37 - 8 * q^39 - 2 * q^41 - 10 * q^43 - 6 * q^45 - 16 * q^47 + 48 * q^49 - 36 * q^51 + 12 * q^53 - 2 * q^57 + 14 * q^59 + 12 * q^61 + 4 * q^63 + 10 * q^65 + 12 * q^67 + 30 * q^69 + 30 * q^71 + 24 * q^73 + 50 * q^75 + 2 * q^77 + 10 * q^81 + 14 * q^83 + 12 * q^85 + 30 * q^87 - 14 * q^89 - 20 * q^91 + 14 * q^93 - 46 * q^97 - 10 * q^99

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	0
$2$	0	0
$3$	0.658537	0.380207	0.190103	−	0.981764i	$-0.439118\pi$
$3$	0.658537	0.380207	0.190103	+	0.981764i	$0.439118\pi$
$4$	0	0
$5$	−2.39807	−1.07245	−0.536225	−	0.844075i	$-0.680150\pi$
$5$	−2.39807	−1.07245	−0.536225	+	0.844075i	$0.680150\pi$
$6$	0	0
$7$	−4.45908	−1.68537	−0.842687	−	0.538404i	$-0.819028\pi$
$7$	−4.45908	−1.68537	−0.842687	+	0.538404i	$0.819028\pi$
$8$	0	0
$9$	−2.56633	−0.855443
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	3.38518	0.938880	0.469440	−	0.882964i	$-0.344456\pi$
$13$	3.38518	0.938880	0.469440	+	0.882964i	$0.344456\pi$
$14$	0	0
$15$	−1.57922	−0.407753
$16$	0	0
$17$	−5.17423	−1.25493	−0.627467	−	0.778643i	$-0.715908\pi$
$17$	−5.17423	−1.25493	−0.627467	+	0.778643i	$0.715908\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−2.93647	−0.640790
$22$	0	0
$23$	−0.851183	−0.177484	−0.0887420	−	0.996055i	$-0.528285\pi$
$23$	−0.851183	−0.177484	−0.0887420	+	0.996055i	$0.528285\pi$
$24$	0	0
$25$	0.750745	0.150149
$26$	0	0
$27$	−3.66563	−0.705452
$28$	0	0
$29$	7.62294	1.41554	0.707772	−	0.706441i	$-0.249700\pi$
$29$	7.62294	1.41554	0.707772	+	0.706441i	$0.249700\pi$
$30$	0	0
$31$	−6.34958	−1.14042	−0.570209	−	0.821500i	$-0.693138\pi$
$31$	−6.34958	−1.14042	−0.570209	+	0.821500i	$0.693138\pi$
$32$	0	0
$33$	−0.658537	−0.114637
$34$	0	0
$35$	10.6932	1.80748
$36$	0	0
$37$	9.00597	1.48057	0.740286	−	0.672292i	$-0.234690\pi$
$37$	9.00597	1.48057	0.740286	+	0.672292i	$0.234690\pi$
$38$	0	0
$39$	2.22927	0.356968
$40$	0	0
$41$	−3.15490	−0.492712	−0.246356	−	0.969179i	$-0.579233\pi$
$41$	−3.15490	−0.492712	−0.246356	+	0.969179i	$0.579233\pi$
$42$	0	0
$43$	−5.24925	−0.800504	−0.400252	−	0.916405i	$-0.631077\pi$
$43$	−5.24925	−0.800504	−0.400252	+	0.916405i	$0.631077\pi$
$44$	0	0
$45$	6.15424	0.917420
$46$	0	0
$47$	−8.77036	−1.27929	−0.639644	−	0.768671i	$-0.720918\pi$
$47$	−8.77036	−1.27929	−0.639644	+	0.768671i	$0.720918\pi$
$48$	0	0
$49$	12.8834	1.84049
$50$	0	0
$51$	−3.40742	−0.477134
$52$	0	0
$53$	9.81558	1.34827	0.674137	−	0.738606i	$-0.264516\pi$
$53$	9.81558	1.34827	0.674137	+	0.738606i	$0.264516\pi$
$54$	0	0
$55$	2.39807	0.323356
$56$	0	0
$57$	−0.658537	−0.0872254
$58$	0	0
$59$	1.57568	0.205136	0.102568	−	0.994726i	$-0.467294\pi$
$59$	1.57568	0.205136	0.102568	+	0.994726i	$0.467294\pi$
$60$	0	0
$61$	−10.8420	−1.38818	−0.694089	−	0.719890i	$-0.744192\pi$
$61$	−10.8420	−1.38818	−0.694089	+	0.719890i	$0.744192\pi$
$62$	0	0
$63$	11.4435	1.44174
$64$	0	0
$65$	−8.11791	−1.00690
$66$	0	0
$67$	3.43142	0.419215	0.209607	−	0.977786i	$-0.432781\pi$
$67$	3.43142	0.419215	0.209607	+	0.977786i	$0.432781\pi$
$68$	0	0
$69$	−0.560536	−0.0674806
$70$	0	0
$71$	4.02439	0.477607	0.238804	−	0.971068i	$-0.423245\pi$
$71$	4.02439	0.477607	0.238804	+	0.971068i	$0.423245\pi$
$72$	0	0
$73$	16.6117	1.94426	0.972128	−	0.234452i	$-0.0753295\pi$
$73$	16.6117	1.94426	0.972128	+	0.234452i	$0.0753295\pi$
$74$	0	0
$75$	0.494394	0.0570877
$76$	0	0
$77$	4.45908	0.508159
$78$	0	0
$79$	4.15479	0.467450	0.233725	−	0.972303i	$-0.424908\pi$
$79$	4.15479	0.467450	0.233725	+	0.972303i	$0.424908\pi$
$80$	0	0
$81$	5.28503	0.587226
$82$	0	0
$83$	−6.27568	−0.688846	−0.344423	−	0.938815i	$-0.611925\pi$
$83$	−6.27568	−0.688846	−0.344423	+	0.938815i	$0.611925\pi$
$84$	0	0
$85$	12.4082	1.34585
$86$	0	0
$87$	5.01999	0.538199
$88$	0	0
$89$	1.22927	0.130302	0.0651510	−	0.997875i	$-0.479247\pi$
$89$	1.22927	0.130302	0.0651510	+	0.997875i	$0.479247\pi$
$90$	0	0
$91$	−15.0948	−1.58236
$92$	0	0
$93$	−4.18143	−0.433595
$94$	0	0
$95$	2.39807	0.246037
$96$	0	0
$97$	−16.0254	−1.62713	−0.813567	−	0.581471i	$-0.802477\pi$
$97$	−16.0254	−1.62713	−0.813567	+	0.581471i	$0.802477\pi$
$98$	0	0
$99$	2.56633	0.257926
$100$	0	0
$101$	−12.6533	−1.25905	−0.629525	−	0.776980i	$-0.716750\pi$
$101$	−12.6533	−1.25905	−0.629525	+	0.776980i	$0.716750\pi$
$102$	0	0
$103$	11.3669	1.12001	0.560006	−	0.828489i	$-0.310799\pi$
$103$	11.3669	1.12001	0.560006	+	0.828489i	$0.310799\pi$
$104$	0	0
$105$	7.04186	0.687216
$106$	0	0
$107$	17.3821	1.68039	0.840195	−	0.542284i	$-0.182440\pi$
$107$	17.3821	1.68039	0.840195	+	0.542284i	$0.182440\pi$
$108$	0	0
$109$	18.8836	1.80872	0.904359	−	0.426772i	$-0.140349\pi$
$109$	18.8836	1.80872	0.904359	+	0.426772i	$0.140349\pi$
$110$	0	0
$111$	5.93076	0.562923
$112$	0	0
$113$	−5.08147	−0.478024	−0.239012	−	0.971017i	$-0.576824\pi$
$113$	−5.08147	−0.478024	−0.239012	+	0.971017i	$0.576824\pi$
$114$	0	0
$115$	2.04120	0.190343
$116$	0	0
$117$	−8.68749	−0.803159
$118$	0	0
$119$	23.0723	2.11503
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−2.07762	−0.187332
$124$	0	0
$125$	10.1900	0.911423
$126$	0	0
$127$	9.36294	0.830827	0.415413	−	0.909633i	$-0.363637\pi$
$127$	9.36294	0.830827	0.415413	+	0.909633i	$0.363637\pi$
$128$	0	0
$129$	−3.45683	−0.304357
$130$	0	0
$131$	4.86295	0.424877	0.212439	−	0.977174i	$-0.431859\pi$
$131$	4.86295	0.424877	0.212439	+	0.977174i	$0.431859\pi$
$132$	0	0
$133$	4.45908	0.386651
$134$	0	0
$135$	8.79045	0.756562
$136$	0	0
$137$	7.38191	0.630679	0.315340	−	0.948979i	$-0.397882\pi$
$137$	7.38191	0.630679	0.315340	+	0.948979i	$0.397882\pi$
$138$	0	0
$139$	−4.02625	−0.341502	−0.170751	−	0.985314i	$-0.554619\pi$
$139$	−4.02625	−0.341502	−0.170751	+	0.985314i	$0.554619\pi$
$140$	0	0
$141$	−5.77561	−0.486394
$142$	0	0
$143$	−3.38518	−0.283083
$144$	0	0
$145$	−18.2803	−1.51810
$146$	0	0
$147$	8.48420	0.699765
$148$	0	0
$149$	12.4847	1.02278	0.511392	−	0.859348i	$-0.329130\pi$
$149$	12.4847	1.02278	0.511392	+	0.859348i	$0.329130\pi$
$150$	0	0
$151$	20.9482	1.70474	0.852372	−	0.522936i	$-0.175163\pi$
$151$	20.9482	1.70474	0.852372	+	0.522936i	$0.175163\pi$
$152$	0	0
$153$	13.2788	1.07352
$154$	0	0
$155$	15.2267	1.22304
$156$	0	0
$157$	16.4143	1.31000	0.655002	−	0.755627i	$-0.272668\pi$
$157$	16.4143	1.31000	0.655002	+	0.755627i	$0.272668\pi$
$158$	0	0
$159$	6.46393	0.512623
$160$	0	0
$161$	3.79550	0.299127
$162$	0	0
$163$	−7.46487	−0.584694	−0.292347	−	0.956312i	$-0.594436\pi$
$163$	−7.46487	−0.584694	−0.292347	+	0.956312i	$0.594436\pi$
$164$	0	0
$165$	1.57922	0.122942
$166$	0	0
$167$	−19.8734	−1.53785	−0.768924	−	0.639340i	$-0.779208\pi$
$167$	−19.8734	−1.53785	−0.768924	+	0.639340i	$0.779208\pi$
$168$	0	0
$169$	−1.54055	−0.118504
$170$	0	0
$171$	2.56633	0.196252
$172$	0	0
$173$	−8.25709	−0.627775	−0.313887	−	0.949460i	$-0.601631\pi$
$173$	−8.25709	−0.627775	−0.313887	+	0.949460i	$0.601631\pi$
$174$	0	0
$175$	−3.34763	−0.253057
$176$	0	0
$177$	1.03764	0.0779940
$178$	0	0
$179$	−15.9019	−1.18856	−0.594282	−	0.804257i	$-0.702564\pi$
$179$	−15.9019	−1.18856	−0.594282	+	0.804257i	$0.702564\pi$
$180$	0	0
$181$	−7.42992	−0.552262	−0.276131	−	0.961120i	$-0.589052\pi$
$181$	−7.42992	−0.552262	−0.276131	+	0.961120i	$0.589052\pi$
$182$	0	0
$183$	−7.13987	−0.527794
$184$	0	0
$185$	−21.5970	−1.58784
$186$	0	0
$187$	5.17423	0.378377
$188$	0	0
$189$	16.3454	1.18895
$190$	0	0
$191$	6.66247	0.482079	0.241040	−	0.970515i	$-0.422512\pi$
$191$	6.66247	0.482079	0.241040	+	0.970515i	$0.422512\pi$
$192$	0	0
$193$	−11.3323	−0.815716	−0.407858	−	0.913045i	$-0.633724\pi$
$193$	−11.3323	−0.815716	−0.407858	+	0.913045i	$0.633724\pi$
$194$	0	0
$195$	−5.34594	−0.382831
$196$	0	0
$197$	1.42331	0.101406	0.0507031	−	0.998714i	$-0.483854\pi$
$197$	1.42331	0.101406	0.0507031	+	0.998714i	$0.483854\pi$
$198$	0	0
$199$	10.6205	0.752869	0.376435	−	0.926443i	$-0.377150\pi$
$199$	10.6205	0.752869	0.376435	+	0.926443i	$0.377150\pi$
$200$	0	0
$201$	2.25972	0.159388
$202$	0	0
$203$	−33.9913	−2.38572
$204$	0	0
$205$	7.56567	0.528409
$206$	0	0
$207$	2.18442	0.151827
$208$	0	0
$209$	1.00000	0.0691714
$210$	0	0
$211$	−3.03523	−0.208954	−0.104477	−	0.994527i	$-0.533317\pi$
$211$	−3.03523	−0.208954	−0.104477	+	0.994527i	$0.533317\pi$
$212$	0	0
$213$	2.65021	0.181589
$214$	0	0
$215$	12.5881	0.858500
$216$	0	0
$217$	28.3133	1.92203
$218$	0	0
$219$	10.9394	0.739219
$220$	0	0
$221$	−17.5157	−1.17823
$222$	0	0
$223$	13.7190	0.918690	0.459345	−	0.888258i	$-0.348084\pi$
$223$	13.7190	0.918690	0.459345	+	0.888258i	$0.348084\pi$
$224$	0	0
$225$	−1.92666	−0.128444
$226$	0	0
$227$	−0.279062	−0.0185220	−0.00926098	−	0.999957i	$-0.502948\pi$
$227$	−0.279062	−0.0185220	−0.00926098	+	0.999957i	$0.502948\pi$
$228$	0	0
$229$	−17.2806	−1.14194	−0.570968	−	0.820972i	$-0.693432\pi$
$229$	−17.2806	−1.14194	−0.570968	+	0.820972i	$0.693432\pi$
$230$	0	0
$231$	2.93647	0.193206
$232$	0	0
$233$	−19.5982	−1.28392	−0.641960	−	0.766738i	$-0.721878\pi$
$233$	−19.5982	−1.28392	−0.641960	+	0.766738i	$0.721878\pi$
$234$	0	0
$235$	21.0320	1.37197
$236$	0	0
$237$	2.73608	0.177728
$238$	0	0
$239$	−27.4011	−1.77243	−0.886216	−	0.463272i	$-0.846675\pi$
$239$	−27.4011	−1.77243	−0.886216	+	0.463272i	$0.846675\pi$
$240$	0	0
$241$	−10.2204	−0.658352	−0.329176	−	0.944269i	$-0.606771\pi$
$241$	−10.2204	−0.658352	−0.329176	+	0.944269i	$0.606771\pi$
$242$	0	0
$243$	14.4773	0.928719
$244$	0	0
$245$	−30.8953	−1.97383
$246$	0	0
$247$	−3.38518	−0.215394
$248$	0	0
$249$	−4.13277	−0.261904
$250$	0	0
$251$	−4.50083	−0.284090	−0.142045	−	0.989860i	$-0.545368\pi$
$251$	−4.50083	−0.284090	−0.142045	+	0.989860i	$0.545368\pi$
$252$	0	0
$253$	0.851183	0.0535134
$254$	0	0
$255$	8.17124	0.511703
$256$	0	0
$257$	−28.5372	−1.78010	−0.890050	−	0.455863i	$-0.849331\pi$
$257$	−28.5372	−1.78010	−0.890050	+	0.455863i	$0.849331\pi$
$258$	0	0
$259$	−40.1583	−2.49532
$260$	0	0
$261$	−19.5630	−1.21092
$262$	0	0
$263$	7.12220	0.439173	0.219587	−	0.975593i	$-0.429529\pi$
$263$	7.12220	0.439173	0.219587	+	0.975593i	$0.429529\pi$
$264$	0	0
$265$	−23.5385	−1.44596
$266$	0	0
$267$	0.809518	0.0495417
$268$	0	0
$269$	21.4692	1.30900	0.654499	−	0.756063i	$-0.272879\pi$
$269$	21.4692	1.30900	0.654499	+	0.756063i	$0.272879\pi$
$270$	0	0
$271$	12.1684	0.739180	0.369590	−	0.929195i	$-0.379498\pi$
$271$	12.1684	0.739180	0.369590	+	0.929195i	$0.379498\pi$
$272$	0	0
$273$	−9.94048	−0.601625
$274$	0	0
$275$	−0.750745	−0.0452717
$276$	0	0
$277$	−10.5493	−0.633848	−0.316924	−	0.948451i	$-0.602650\pi$
$277$	−10.5493	−0.633848	−0.316924	+	0.948451i	$0.602650\pi$
$278$	0	0
$279$	16.2951	0.975563
$280$	0	0
$281$	10.4836	0.625402	0.312701	−	0.949852i	$-0.398766\pi$
$281$	10.4836	0.625402	0.312701	+	0.949852i	$0.398766\pi$
$282$	0	0
$283$	−11.9497	−0.710338	−0.355169	−	0.934802i	$-0.615577\pi$
$283$	−11.9497	−0.710338	−0.355169	+	0.934802i	$0.615577\pi$
$284$	0	0
$285$	1.57922	0.0935449
$286$	0	0
$287$	14.0679	0.830404
$288$	0	0
$289$	9.77262	0.574860
$290$	0	0
$291$	−10.5533	−0.618647
$292$	0	0
$293$	−9.96683	−0.582268	−0.291134	−	0.956682i	$-0.594033\pi$
$293$	−9.96683	−0.582268	−0.291134	+	0.956682i	$0.594033\pi$
$294$	0	0
$295$	−3.77859	−0.219998
$296$	0	0
$297$	3.66563	0.212702
$298$	0	0
$299$	−2.88141	−0.166636
$300$	0	0
$301$	23.4069	1.34915
$302$	0	0
$303$	−8.33266	−0.478699
$304$	0	0
$305$	25.9999	1.48875
$306$	0	0
$307$	−26.0470	−1.48658	−0.743290	−	0.668970i	$-0.766736\pi$
$307$	−26.0470	−1.48658	−0.743290	+	0.668970i	$0.766736\pi$
$308$	0	0
$309$	7.48551	0.425836
$310$	0	0
$311$	33.3197	1.88939	0.944693	−	0.327956i	$-0.106360\pi$
$311$	33.3197	1.88939	0.944693	+	0.327956i	$0.106360\pi$
$312$	0	0
$313$	23.1427	1.30810	0.654050	−	0.756451i	$-0.273069\pi$
$313$	23.1427	1.30810	0.654050	+	0.756451i	$0.273069\pi$
$314$	0	0
$315$	−27.4423	−1.54620
$316$	0	0
$317$	−15.9996	−0.898629	−0.449314	−	0.893374i	$-0.648332\pi$
$317$	−15.9996	−0.898629	−0.449314	+	0.893374i	$0.648332\pi$
$318$	0	0
$319$	−7.62294	−0.426803
$320$	0	0
$321$	11.4468	0.638895
$322$	0	0
$323$	5.17423	0.287902
$324$	0	0
$325$	2.54141	0.140972
$326$	0	0
$327$	12.4355	0.687687
$328$	0	0
$329$	39.1078	2.15608
$330$	0	0
$331$	4.61965	0.253919	0.126960	−	0.991908i	$-0.459478\pi$
$331$	4.61965	0.253919	0.126960	+	0.991908i	$0.459478\pi$
$332$	0	0
$333$	−23.1123	−1.26655
$334$	0	0
$335$	−8.22879	−0.449587
$336$	0	0
$337$	−4.18132	−0.227771	−0.113886	−	0.993494i	$-0.536330\pi$
$337$	−4.18132	−0.227771	−0.113886	+	0.993494i	$0.536330\pi$
$338$	0	0
$339$	−3.34634	−0.181748
$340$	0	0
$341$	6.34958	0.343849
$342$	0	0
$343$	−26.2346	−1.41653
$344$	0	0
$345$	1.34420	0.0723696
$346$	0	0
$347$	9.78259	0.525157	0.262579	−	0.964911i	$-0.415427\pi$
$347$	9.78259	0.525157	0.262579	+	0.964911i	$0.415427\pi$
$348$	0	0
$349$	7.07295	0.378606	0.189303	−	0.981919i	$-0.439377\pi$
$349$	7.07295	0.378606	0.189303	+	0.981919i	$0.439377\pi$
$350$	0	0
$351$	−12.4088	−0.662335
$352$	0	0
$353$	−7.30577	−0.388847	−0.194423	−	0.980918i	$-0.562284\pi$
$353$	−7.30577	−0.388847	−0.194423	+	0.980918i	$0.562284\pi$
$354$	0	0
$355$	−9.65077	−0.512210
$356$	0	0
$357$	15.1940	0.804150
$358$	0	0
$359$	−16.8930	−0.891580	−0.445790	−	0.895138i	$-0.647077\pi$
$359$	−16.8930	−0.891580	−0.445790	+	0.895138i	$0.647077\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0.658537	0.0345642
$364$	0	0
$365$	−39.8361	−2.08512
$366$	0	0
$367$	2.52625	0.131869	0.0659347	−	0.997824i	$-0.478997\pi$
$367$	2.52625	0.131869	0.0659347	+	0.997824i	$0.478997\pi$
$368$	0	0
$369$	8.09650	0.421487
$370$	0	0
$371$	−43.7685	−2.27235
$372$	0	0
$373$	−4.35173	−0.225324	−0.112662	−	0.993633i	$-0.535938\pi$
$373$	−4.35173	−0.225324	−0.112662	+	0.993633i	$0.535938\pi$
$374$	0	0
$375$	6.71050	0.346529
$376$	0	0
$377$	25.8050	1.32903
$378$	0	0
$379$	10.1209	0.519875	0.259938	−	0.965625i	$-0.416298\pi$
$379$	10.1209	0.519875	0.259938	+	0.965625i	$0.416298\pi$
$380$	0	0
$381$	6.16584	0.315886
$382$	0	0
$383$	−12.8072	−0.654416	−0.327208	−	0.944952i	$-0.606108\pi$
$383$	−12.8072	−0.654416	−0.327208	+	0.944952i	$0.606108\pi$
$384$	0	0
$385$	−10.6932	−0.544976
$386$	0	0
$387$	13.4713	0.684785
$388$	0	0
$389$	38.4787	1.95095	0.975475	−	0.220110i	$-0.0706418\pi$
$389$	38.4787	1.95095	0.975475	+	0.220110i	$0.0706418\pi$
$390$	0	0
$391$	4.40422	0.222731
$392$	0	0
$393$	3.20243	0.161541
$394$	0	0
$395$	−9.96347	−0.501317
$396$	0	0
$397$	22.1180	1.11007	0.555035	−	0.831827i	$-0.312705\pi$
$397$	22.1180	1.11007	0.555035	+	0.831827i	$0.312705\pi$
$398$	0	0
$399$	2.93647	0.147007
$400$	0	0
$401$	17.8505	0.891412	0.445706	−	0.895179i	$-0.352953\pi$
$401$	17.8505	0.891412	0.445706	+	0.895179i	$0.352953\pi$
$402$	0	0
$403$	−21.4945	−1.07072
$404$	0	0
$405$	−12.6739	−0.629770
$406$	0	0
$407$	−9.00597	−0.446409
$408$	0	0
$409$	−12.7440	−0.630151	−0.315075	−	0.949067i	$-0.602030\pi$
$409$	−12.7440	−0.630151	−0.315075	+	0.949067i	$0.602030\pi$
$410$	0	0
$411$	4.86126	0.239788
$412$	0	0
$413$	−7.02608	−0.345731
$414$	0	0
$415$	15.0495	0.738752
$416$	0	0
$417$	−2.65144	−0.129841
$418$	0	0
$419$	−15.6799	−0.766015	−0.383008	−	0.923745i	$-0.625112\pi$
$419$	−15.6799	−0.766015	−0.383008	+	0.923745i	$0.625112\pi$
$420$	0	0
$421$	18.1254	0.883377	0.441688	−	0.897169i	$-0.354380\pi$
$421$	18.1254	0.883377	0.441688	+	0.897169i	$0.354380\pi$
$422$	0	0
$423$	22.5076	1.09436
$424$	0	0
$425$	−3.88453	−0.188427
$426$	0	0
$427$	48.3454	2.33960
$428$	0	0
$429$	−2.22927	−0.107630
$430$	0	0
$431$	−5.55231	−0.267445	−0.133723	−	0.991019i	$-0.542693\pi$
$431$	−5.55231	−0.267445	−0.133723	+	0.991019i	$0.542693\pi$
$432$	0	0
$433$	−10.6457	−0.511600	−0.255800	−	0.966730i	$-0.582339\pi$
$433$	−10.6457	−0.511600	−0.255800	+	0.966730i	$0.582339\pi$
$434$	0	0
$435$	−12.0383	−0.577192
$436$	0	0
$437$	0.851183	0.0407176
$438$	0	0
$439$	−3.26467	−0.155814	−0.0779071	−	0.996961i	$-0.524824\pi$
$439$	−3.26467	−0.155814	−0.0779071	+	0.996961i	$0.524824\pi$
$440$	0	0
$441$	−33.0630	−1.57443
$442$	0	0
$443$	13.2828	0.631084	0.315542	−	0.948912i	$-0.397814\pi$
$443$	13.2828	0.631084	0.315542	+	0.948912i	$0.397814\pi$
$444$	0	0
$445$	−2.94787	−0.139742
$446$	0	0
$447$	8.22162	0.388869
$448$	0	0
$449$	38.0400	1.79522	0.897608	−	0.440794i	$-0.145303\pi$
$449$	38.0400	1.79522	0.897608	+	0.440794i	$0.145303\pi$
$450$	0	0
$451$	3.15490	0.148558
$452$	0	0
$453$	13.7952	0.648155
$454$	0	0
$455$	36.1984	1.69701
$456$	0	0
$457$	31.2917	1.46377	0.731883	−	0.681430i	$-0.238642\pi$
$457$	31.2917	1.46377	0.731883	+	0.681430i	$0.238642\pi$
$458$	0	0
$459$	18.9668	0.885295
$460$	0	0
$461$	−22.3830	−1.04248	−0.521241	−	0.853410i	$-0.674531\pi$
$461$	−22.3830	−1.04248	−0.521241	+	0.853410i	$0.674531\pi$
$462$	0	0
$463$	−35.5135	−1.65045	−0.825226	−	0.564802i	$-0.808952\pi$
$463$	−35.5135	−1.65045	−0.825226	+	0.564802i	$0.808952\pi$
$464$	0	0
$465$	10.0274	0.465009
$466$	0	0
$467$	15.8080	0.731507	0.365754	−	0.930712i	$-0.380811\pi$
$467$	15.8080	0.731507	0.365754	+	0.930712i	$0.380811\pi$
$468$	0	0
$469$	−15.3010	−0.706533
$470$	0	0
$471$	10.8094	0.498072
$472$	0	0
$473$	5.24925	0.241361
$474$	0	0
$475$	−0.750745	−0.0344466
$476$	0	0
$477$	−25.1900	−1.15337
$478$	0	0
$479$	18.7939	0.858716	0.429358	−	0.903134i	$-0.358740\pi$
$479$	18.7939	0.858716	0.429358	+	0.903134i	$0.358740\pi$
$480$	0	0
$481$	30.4868	1.39008
$482$	0	0
$483$	2.49947	0.113730
$484$	0	0
$485$	38.4301	1.74502
$486$	0	0
$487$	−10.2071	−0.462529	−0.231264	−	0.972891i	$-0.574286\pi$
$487$	−10.2071	−0.462529	−0.231264	+	0.972891i	$0.574286\pi$
$488$	0	0
$489$	−4.91590	−0.222305
$490$	0	0
$491$	−22.9176	−1.03426	−0.517129	−	0.855908i	$-0.672999\pi$
$491$	−22.9176	−1.03426	−0.517129	+	0.855908i	$0.672999\pi$
$492$	0	0
$493$	−39.4428	−1.77641
$494$	0	0
$495$	−6.15424	−0.276612
$496$	0	0
$497$	−17.9451	−0.804947
$498$	0	0
$499$	41.3574	1.85141	0.925706	−	0.378244i	$-0.123472\pi$
$499$	41.3574	1.85141	0.925706	+	0.378244i	$0.123472\pi$
$500$	0	0
$501$	−13.0874	−0.584700
$502$	0	0
$503$	19.5977	0.873819	0.436909	−	0.899506i	$-0.356073\pi$
$503$	19.5977	0.873819	0.436909	+	0.899506i	$0.356073\pi$
$504$	0	0
$505$	30.3435	1.35027
$506$	0	0
$507$	−1.01451	−0.0450559
$508$	0	0
$509$	−20.7913	−0.921557	−0.460778	−	0.887515i	$-0.652430\pi$
$509$	−20.7913	−0.921557	−0.460778	+	0.887515i	$0.652430\pi$
$510$	0	0
$511$	−74.0730	−3.27680
$512$	0	0
$513$	3.66563	0.161842
$514$	0	0
$515$	−27.2586	−1.20116
$516$	0	0
$517$	8.77036	0.385720
$518$	0	0
$519$	−5.43760	−0.238684
$520$	0	0
$521$	−13.9519	−0.611246	−0.305623	−	0.952153i	$-0.598865\pi$
$521$	−13.9519	−0.611246	−0.305623	+	0.952153i	$0.598865\pi$
$522$	0	0
$523$	4.81034	0.210341	0.105171	−	0.994454i	$-0.466461\pi$
$523$	4.81034	0.210341	0.105171	+	0.994454i	$0.466461\pi$
$524$	0	0
$525$	−2.20454	−0.0962141
$526$	0	0
$527$	32.8542	1.43115
$528$	0	0
$529$	−22.2755	−0.968499
$530$	0	0
$531$	−4.04371	−0.175482
$532$	0	0
$533$	−10.6799	−0.462598
$534$	0	0
$535$	−41.6835	−1.80213
$536$	0	0
$537$	−10.4720	−0.451900
$538$	0	0
$539$	−12.8834	−0.554927
$540$	0	0
$541$	10.4259	0.448245	0.224123	−	0.974561i	$-0.428048\pi$
$541$	10.4259	0.448245	0.224123	+	0.974561i	$0.428048\pi$
$542$	0	0
$543$	−4.89288	−0.209974
$544$	0	0
$545$	−45.2842	−1.93976
$546$	0	0
$547$	41.2671	1.76445	0.882226	−	0.470826i	$-0.156044\pi$
$547$	41.2671	1.76445	0.882226	+	0.470826i	$0.156044\pi$
$548$	0	0
$549$	27.8242	1.18751
$550$	0	0
$551$	−7.62294	−0.324748
$552$	0	0
$553$	−18.5265	−0.787828
$554$	0	0
$555$	−14.2224	−0.603707
$556$	0	0
$557$	−22.0908	−0.936017	−0.468008	−	0.883724i	$-0.655028\pi$
$557$	−22.0908	−0.936017	−0.468008	+	0.883724i	$0.655028\pi$
$558$	0	0
$559$	−17.7697	−0.751577
$560$	0	0
$561$	3.40742	0.143861
$562$	0	0
$563$	8.88853	0.374607	0.187303	−	0.982302i	$-0.440025\pi$
$563$	8.88853	0.374607	0.187303	+	0.982302i	$0.440025\pi$
$564$	0	0
$565$	12.1857	0.512657
$566$	0	0
$567$	−23.5664	−0.989695
$568$	0	0
$569$	12.5633	0.526679	0.263340	−	0.964703i	$-0.415176\pi$
$569$	12.5633	0.526679	0.263340	+	0.964703i	$0.415176\pi$
$570$	0	0
$571$	−15.0445	−0.629591	−0.314795	−	0.949160i	$-0.601936\pi$
$571$	−15.0445	−0.629591	−0.314795	+	0.949160i	$0.601936\pi$
$572$	0	0
$573$	4.38748	0.183290
$574$	0	0
$575$	−0.639022	−0.0266491
$576$	0	0
$577$	0.00725786	0.000302149 0	0.000151074	−	1.00000i	$-0.499952\pi$
$577$	0.00725786	0.000302149 0	0.000151074		1.00000i	$0.499952\pi$
$578$	0	0
$579$	−7.46273	−0.310141
$580$	0	0
$581$	27.9838	1.16096
$582$	0	0
$583$	−9.81558	−0.406520
$584$	0	0
$585$	20.8332	0.861347
$586$	0	0
$587$	13.5710	0.560135	0.280068	−	0.959980i	$-0.409643\pi$
$587$	13.5710	0.560135	0.280068	+	0.959980i	$0.409643\pi$
$588$	0	0
$589$	6.34958	0.261630
$590$	0	0
$591$	0.937299	0.0385553
$592$	0	0
$593$	−25.4195	−1.04385	−0.521926	−	0.852991i	$-0.674786\pi$
$593$	−25.4195	−1.04385	−0.521926	+	0.852991i	$0.674786\pi$
$594$	0	0
$595$	−55.3290	−2.26827
$596$	0	0
$597$	6.99401	0.286246
$598$	0	0
$599$	−13.3669	−0.546156	−0.273078	−	0.961992i	$-0.588042\pi$
$599$	−13.3669	−0.546156	−0.273078	+	0.961992i	$0.588042\pi$
$600$	0	0
$601$	33.3555	1.36060	0.680299	−	0.732935i	$-0.261850\pi$
$601$	33.3555	1.36060	0.680299	+	0.732935i	$0.261850\pi$
$602$	0	0
$603$	−8.80615	−0.358614
$604$	0	0
$605$	−2.39807	−0.0974955
$606$	0	0
$607$	33.1399	1.34511	0.672553	−	0.740049i	$-0.265198\pi$
$607$	33.1399	1.34511	0.672553	+	0.740049i	$0.265198\pi$
$608$	0	0
$609$	−22.3845	−0.907067
$610$	0	0
$611$	−29.6893	−1.20110
$612$	0	0
$613$	12.5401	0.506489	0.253244	−	0.967402i	$-0.418502\pi$
$613$	12.5401	0.506489	0.253244	+	0.967402i	$0.418502\pi$
$614$	0	0
$615$	4.98227	0.200905
$616$	0	0
$617$	26.8026	1.07903	0.539516	−	0.841976i	$-0.318607\pi$
$617$	26.8026	1.07903	0.539516	+	0.841976i	$0.318607\pi$
$618$	0	0
$619$	−37.6751	−1.51429	−0.757146	−	0.653246i	$-0.773407\pi$
$619$	−37.6751	−1.51429	−0.757146	+	0.653246i	$0.773407\pi$
$620$	0	0
$621$	3.12013	0.125206
$622$	0	0
$623$	−5.48140	−0.219608
$624$	0	0
$625$	−28.1901	−1.12760
$626$	0	0
$627$	0.658537	0.0262994
$628$	0	0
$629$	−46.5989	−1.85802
$630$	0	0
$631$	−44.3405	−1.76517	−0.882584	−	0.470155i	$-0.844198\pi$
$631$	−44.3405	−1.76517	−0.882584	+	0.470155i	$0.844198\pi$
$632$	0	0
$633$	−1.99881	−0.0794456
$634$	0	0
$635$	−22.4530	−0.891020
$636$	0	0
$637$	43.6127	1.72800
$638$	0	0
$639$	−10.3279	−0.408566
$640$	0	0
$641$	−0.0390296	−0.00154158	−0.000770788	−	1.00000i	$-0.500245\pi$
$641$	−0.0390296	−0.00154158	−0.000770788		1.00000i	$0.500245\pi$
$642$	0	0
$643$	38.2697	1.50921	0.754604	−	0.656180i	$-0.227829\pi$
$643$	38.2697	1.50921	0.754604	+	0.656180i	$0.227829\pi$
$644$	0	0
$645$	8.28972	0.326407
$646$	0	0
$647$	26.0701	1.02492	0.512461	−	0.858711i	$-0.328734\pi$
$647$	26.0701	1.02492	0.512461	+	0.858711i	$0.328734\pi$
$648$	0	0
$649$	−1.57568	−0.0618508
$650$	0	0
$651$	18.6454	0.730769
$652$	0	0
$653$	−33.1050	−1.29550	−0.647750	−	0.761853i	$-0.724290\pi$
$653$	−33.1050	−1.29550	−0.647750	+	0.761853i	$0.724290\pi$
$654$	0	0
$655$	−11.6617	−0.455660
$656$	0	0
$657$	−42.6312	−1.66320
$658$	0	0
$659$	24.9256	0.970963	0.485482	−	0.874247i	$-0.338644\pi$
$659$	24.9256	0.970963	0.485482	+	0.874247i	$0.338644\pi$
$660$	0	0
$661$	14.7722	0.574574	0.287287	−	0.957845i	$-0.407247\pi$
$661$	14.7722	0.574574	0.287287	+	0.957845i	$0.407247\pi$
$662$	0	0
$663$	−11.5347	−0.447972
$664$	0	0
$665$	−10.6932	−0.414664
$666$	0	0
$667$	−6.48852	−0.251236
$668$	0	0
$669$	9.03445	0.349292
$670$	0	0
$671$	10.8420	0.418551
$672$	0	0
$673$	24.5865	0.947739	0.473870	−	0.880595i	$-0.342857\pi$
$673$	24.5865	0.947739	0.473870	+	0.880595i	$0.342857\pi$
$674$	0	0
$675$	−2.75196	−0.105923
$676$	0	0
$677$	−41.2906	−1.58693	−0.793464	−	0.608617i	$-0.791725\pi$
$677$	−41.2906	−1.58693	−0.793464	+	0.608617i	$0.791725\pi$
$678$	0	0
$679$	71.4586	2.74233
$680$	0	0
$681$	−0.183772	−0.00704217
$682$	0	0
$683$	33.1446	1.26824	0.634121	−	0.773234i	$-0.281362\pi$
$683$	33.1446	1.26824	0.634121	+	0.773234i	$0.281362\pi$
$684$	0	0
$685$	−17.7024	−0.676372
$686$	0	0
$687$	−11.3799	−0.434172
$688$	0	0
$689$	33.2275	1.26587
$690$	0	0
$691$	38.8925	1.47954	0.739769	−	0.672860i	$-0.234935\pi$
$691$	38.8925	1.47954	0.739769	+	0.672860i	$0.234935\pi$
$692$	0	0
$693$	−11.4435	−0.434701
$694$	0	0
$695$	9.65524	0.366244
$696$	0	0
$697$	16.3242	0.618321
$698$	0	0
$699$	−12.9061	−0.488155
$700$	0	0
$701$	3.72084	0.140534	0.0702671	−	0.997528i	$-0.477615\pi$
$701$	3.72084	0.140534	0.0702671	+	0.997528i	$0.477615\pi$
$702$	0	0
$703$	−9.00597	−0.339667
$704$	0	0
$705$	13.8503	0.521633
$706$	0	0
$707$	56.4221	2.12197
$708$	0	0
$709$	3.32175	0.124751	0.0623755	−	0.998053i	$-0.480132\pi$
$709$	3.32175	0.124751	0.0623755	+	0.998053i	$0.480132\pi$
$710$	0	0
$711$	−10.6625	−0.399877
$712$	0	0
$713$	5.40466	0.202406
$714$	0	0
$715$	8.11791	0.303592
$716$	0	0
$717$	−18.0447	−0.673890
$718$	0	0
$719$	12.7820	0.476689	0.238344	−	0.971181i	$-0.423395\pi$
$719$	12.7820	0.476689	0.238344	+	0.971181i	$0.423395\pi$
$720$	0	0
$721$	−50.6858	−1.88764
$722$	0	0
$723$	−6.73049	−0.250310
$724$	0	0
$725$	5.72289	0.212543
$726$	0	0
$727$	42.1011	1.56144	0.780722	−	0.624878i	$-0.214851\pi$
$727$	42.1011	1.56144	0.780722	+	0.624878i	$0.214851\pi$
$728$	0	0
$729$	−6.32126	−0.234121
$730$	0	0
$731$	27.1608	1.00458
$732$	0	0
$733$	35.1283	1.29749	0.648746	−	0.761005i	$-0.275294\pi$
$733$	35.1283	1.29749	0.648746	+	0.761005i	$0.275294\pi$
$734$	0	0
$735$	−20.3457	−0.750463
$736$	0	0
$737$	−3.43142	−0.126398
$738$	0	0
$739$	−0.787459	−0.0289671	−0.0144836	−	0.999895i	$-0.504610\pi$
$739$	−0.787459	−0.0289671	−0.0144836	+	0.999895i	$0.504610\pi$
$740$	0	0
$741$	−2.22927	−0.0818942
$742$	0	0
$743$	−43.2992	−1.58849	−0.794246	−	0.607596i	$-0.792134\pi$
$743$	−43.2992	−1.58849	−0.794246	+	0.607596i	$0.792134\pi$
$744$	0	0
$745$	−29.9391	−1.09688
$746$	0	0
$747$	16.1055	0.589268
$748$	0	0
$749$	−77.5081	−2.83209
$750$	0	0
$751$	35.3019	1.28819	0.644093	−	0.764947i	$-0.277235\pi$
$751$	35.3019	1.28819	0.644093	+	0.764947i	$0.277235\pi$
$752$	0	0
$753$	−2.96396	−0.108013
$754$	0	0
$755$	−50.2354	−1.82825
$756$	0	0
$757$	−16.5618	−0.601950	−0.300975	−	0.953632i	$-0.597312\pi$
$757$	−16.5618	−0.601950	−0.300975	+	0.953632i	$0.597312\pi$
$758$	0	0
$759$	0.560536	0.0203462
$760$	0	0
$761$	25.4933	0.924130	0.462065	−	0.886846i	$-0.347109\pi$
$761$	25.4933	0.924130	0.462065	+	0.886846i	$0.347109\pi$
$762$	0	0
$763$	−84.2034	−3.04837
$764$	0	0
$765$	−31.8434	−1.15130
$766$	0	0
$767$	5.33395	0.192598
$768$	0	0
$769$	11.1304	0.401372	0.200686	−	0.979656i	$-0.435683\pi$
$769$	11.1304	0.401372	0.200686	+	0.979656i	$0.435683\pi$
$770$	0	0
$771$	−18.7928	−0.676806
$772$	0	0
$773$	7.60109	0.273392	0.136696	−	0.990613i	$-0.456352\pi$
$773$	7.60109	0.273392	0.136696	+	0.990613i	$0.456352\pi$
$774$	0	0
$775$	−4.76692	−0.171233
$776$	0	0
$777$	−26.4458	−0.948736
$778$	0	0
$779$	3.15490	0.113036
$780$	0	0
$781$	−4.02439	−0.144004
$782$	0	0
$783$	−27.9429	−0.998598
$784$	0	0
$785$	−39.3627	−1.40491
$786$	0	0
$787$	33.2096	1.18379	0.591897	−	0.806013i	$-0.298379\pi$
$787$	33.2096	1.18379	0.591897	+	0.806013i	$0.298379\pi$
$788$	0	0
$789$	4.69023	0.166977
$790$	0	0
$791$	22.6587	0.805650
$792$	0	0
$793$	−36.7022	−1.30333
$794$	0	0
$795$	−15.5010	−0.549762
$796$	0	0
$797$	−47.2995	−1.67543	−0.837716	−	0.546106i	$-0.816110\pi$
$797$	−47.2995	−1.67543	−0.837716	+	0.546106i	$0.816110\pi$
$798$	0	0
$799$	45.3798	1.60542
$800$	0	0
$801$	−3.15470	−0.111466
$802$	0	0
$803$	−16.6117	−0.586215
$804$	0	0
$805$	−9.10187	−0.320799
$806$	0	0
$807$	14.1382	0.497690
$808$	0	0
$809$	21.1977	0.745270	0.372635	−	0.927978i	$-0.378454\pi$
$809$	21.1977	0.745270	0.372635	+	0.927978i	$0.378454\pi$
$810$	0	0
$811$	−24.2019	−0.849843	−0.424921	−	0.905230i	$-0.639698\pi$
$811$	−24.2019	−0.849843	−0.424921	+	0.905230i	$0.639698\pi$
$812$	0	0
$813$	8.01337	0.281041
$814$	0	0
$815$	17.9013	0.627055
$816$	0	0
$817$	5.24925	0.183648
$818$	0	0
$819$	38.7382	1.35362
$820$	0	0
$821$	−0.557645	−0.0194620	−0.00973098	−	0.999953i	$-0.503098\pi$
$821$	−0.557645	−0.0194620	−0.00973098	+	0.999953i	$0.503098\pi$
$822$	0	0
$823$	32.4296	1.13043	0.565213	−	0.824945i	$-0.308794\pi$
$823$	32.4296	1.13043	0.565213	+	0.824945i	$0.308794\pi$
$824$	0	0
$825$	−0.494394	−0.0172126
$826$	0	0
$827$	−45.7415	−1.59059	−0.795293	−	0.606225i	$-0.792683\pi$
$827$	−45.7415	−1.59059	−0.795293	+	0.606225i	$0.792683\pi$
$828$	0	0
$829$	52.8664	1.83613	0.918063	−	0.396435i	$-0.129753\pi$
$829$	52.8664	1.83613	0.918063	+	0.396435i	$0.129753\pi$
$830$	0	0
$831$	−6.94712	−0.240993
$832$	0	0
$833$	−66.6616	−2.30969
$834$	0	0
$835$	47.6578	1.64927
$836$	0	0
$837$	23.2752	0.804510
$838$	0	0
$839$	−11.6187	−0.401123	−0.200561	−	0.979681i	$-0.564277\pi$
$839$	−11.6187	−0.401123	−0.200561	+	0.979681i	$0.564277\pi$
$840$	0	0
$841$	29.1092	1.00376
$842$	0	0
$843$	6.90387	0.237782
$844$	0	0
$845$	3.69435	0.127089
$846$	0	0
$847$	−4.45908	−0.153216
$848$	0	0
$849$	−7.86934	−0.270075
$850$	0	0
$851$	−7.66573	−0.262778
$852$	0	0
$853$	−15.3060	−0.524068	−0.262034	−	0.965059i	$-0.584393\pi$
$853$	−15.3060	−0.524068	−0.262034	+	0.965059i	$0.584393\pi$
$854$	0	0
$855$	−6.15424	−0.210471
$856$	0	0
$857$	11.4760	0.392012	0.196006	−	0.980603i	$-0.437203\pi$
$857$	11.4760	0.392012	0.196006	+	0.980603i	$0.437203\pi$
$858$	0	0
$859$	−17.6400	−0.601870	−0.300935	−	0.953645i	$-0.597299\pi$
$859$	−17.6400	−0.601870	−0.300935	+	0.953645i	$0.597299\pi$
$860$	0	0
$861$	9.26426	0.315725
$862$	0	0
$863$	−11.0895	−0.377491	−0.188745	−	0.982026i	$-0.560442\pi$
$863$	−11.0895	−0.377491	−0.188745	+	0.982026i	$0.560442\pi$
$864$	0	0
$865$	19.8011	0.673257
$866$	0	0
$867$	6.43563	0.218566
$868$	0	0
$869$	−4.15479	−0.140941
$870$	0	0
$871$	11.6160	0.393592
$872$	0	0
$873$	41.1265	1.39192
$874$	0	0
$875$	−45.4381	−1.53609
$876$	0	0
$877$	−54.4653	−1.83916	−0.919581	−	0.392900i	$-0.871472\pi$
$877$	−54.4653	−1.83916	−0.919581	+	0.392900i	$0.871472\pi$
$878$	0	0
$879$	−6.56353	−0.221382
$880$	0	0
$881$	6.95366	0.234275	0.117137	−	0.993116i	$-0.462628\pi$
$881$	6.95366	0.234275	0.117137	+	0.993116i	$0.462628\pi$
$882$	0	0
$883$	−17.2919	−0.581918	−0.290959	−	0.956736i	$-0.593974\pi$
$883$	−17.2919	−0.581918	−0.290959	+	0.956736i	$0.593974\pi$
$884$	0	0
$885$	−2.48834	−0.0836446
$886$	0	0
$887$	35.0703	1.17754	0.588772	−	0.808299i	$-0.299612\pi$
$887$	35.0703	1.17754	0.588772	+	0.808299i	$0.299612\pi$
$888$	0	0
$889$	−41.7501	−1.40025
$890$	0	0
$891$	−5.28503	−0.177055
$892$	0	0
$893$	8.77036	0.293489
$894$	0	0
$895$	38.1339	1.27467
$896$	0	0
$897$	−1.89752	−0.0633562
$898$	0	0
$899$	−48.4025	−1.61431
$900$	0	0
$901$	−50.7881	−1.69200
$902$	0	0
$903$	15.4143	0.512955
$904$	0	0
$905$	17.8175	0.592273
$906$	0	0
$907$	33.1532	1.10083	0.550417	−	0.834890i	$-0.314469\pi$
$907$	33.1532	1.10083	0.550417	+	0.834890i	$0.314469\pi$
$908$	0	0
$909$	32.4725	1.07705
$910$	0	0
$911$	3.58070	0.118634	0.0593169	−	0.998239i	$-0.481108\pi$
$911$	3.58070	0.118634	0.0593169	+	0.998239i	$0.481108\pi$
$912$	0	0
$913$	6.27568	0.207695
$914$	0	0
$915$	17.1219	0.566033
$916$	0	0
$917$	−21.6843	−0.716078
$918$	0	0
$919$	11.6466	0.384185	0.192092	−	0.981377i	$-0.438473\pi$
$919$	11.6466	0.384185	0.192092	+	0.981377i	$0.438473\pi$
$920$	0	0
$921$	−17.1529	−0.565207
$922$	0	0
$923$	13.6233	0.448416
$924$	0	0
$925$	6.76119	0.222307
$926$	0	0
$927$	−29.1711	−0.958106
$928$	0	0
$929$	−16.7802	−0.550539	−0.275270	−	0.961367i	$-0.588767\pi$
$929$	−16.7802	−0.550539	−0.275270	+	0.961367i	$0.588767\pi$
$930$	0	0
$931$	−12.8834	−0.422236
$932$	0	0
$933$	21.9423	0.718357
$934$	0	0
$935$	−12.4082	−0.405790
$936$	0	0
$937$	7.34816	0.240054	0.120027	−	0.992771i	$-0.461702\pi$
$937$	7.34816	0.240054	0.120027	+	0.992771i	$0.461702\pi$
$938$	0	0
$939$	15.2403	0.497348
$940$	0	0
$941$	−41.9997	−1.36915	−0.684576	−	0.728942i	$-0.740012\pi$
$941$	−41.9997	−1.36915	−0.684576	+	0.728942i	$0.740012\pi$
$942$	0	0
$943$	2.68540	0.0874485
$944$	0	0
$945$	−39.1973	−1.27509
$946$	0	0
$947$	−26.1607	−0.850110	−0.425055	−	0.905168i	$-0.639745\pi$
$947$	−26.1607	−0.850110	−0.425055	+	0.905168i	$0.639745\pi$
$948$	0	0
$949$	56.2337	1.82542
$950$	0	0
$951$	−10.5364	−0.341665
$952$	0	0
$953$	−24.7731	−0.802478	−0.401239	−	0.915973i	$-0.631420\pi$
$953$	−24.7731	−0.802478	−0.401239	+	0.915973i	$0.631420\pi$
$954$	0	0
$955$	−15.9771	−0.517006
$956$	0	0
$957$	−5.01999	−0.162273
$958$	0	0
$959$	−32.9165	−1.06293
$960$	0	0
$961$	9.31718	0.300554
$962$	0	0
$963$	−44.6082	−1.43748
$964$	0	0
$965$	27.1756	0.874815
$966$	0	0
$967$	−11.8644	−0.381535	−0.190767	−	0.981635i	$-0.561098\pi$
$967$	−11.8644	−0.381535	−0.190767	+	0.981635i	$0.561098\pi$
$968$	0	0
$969$	3.40742	0.109462
$970$	0	0
$971$	37.6363	1.20781	0.603904	−	0.797057i	$-0.293611\pi$
$971$	37.6363	1.20781	0.603904	+	0.797057i	$0.293611\pi$
$972$	0	0
$973$	17.9534	0.575559
$974$	0	0
$975$	1.67361	0.0535985
$976$	0	0
$977$	−4.98245	−0.159403	−0.0797013	−	0.996819i	$-0.525397\pi$
$977$	−4.98245	−0.159403	−0.0797013	+	0.996819i	$0.525397\pi$
$978$	0	0
$979$	−1.22927	−0.0392876
$980$	0	0
$981$	−48.4615	−1.54726
$982$	0	0
$983$	−14.4565	−0.461090	−0.230545	−	0.973062i	$-0.574051\pi$
$983$	−14.4565	−0.461090	−0.230545	+	0.973062i	$0.574051\pi$
$984$	0	0
$985$	−3.41319	−0.108753
$986$	0	0
$987$	25.7539	0.819756
$988$	0	0
$989$	4.46808	0.142077
$990$	0	0
$991$	34.3432	1.09095	0.545474	−	0.838128i	$-0.316350\pi$
$991$	34.3432	1.09095	0.545474	+	0.838128i	$0.316350\pi$
$992$	0	0
$993$	3.04221	0.0965418
$994$	0	0
$995$	−25.4688	−0.807415
$996$	0	0
$997$	30.1824	0.955886	0.477943	−	0.878391i	$-0.341382\pi$
$997$	30.1824	0.955886	0.477943	+	0.878391i	$0.341382\pi$
$998$	0	0
$999$	−33.0126	−1.04447

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.a.x.1.3		6
4.3	odd	2		836.2.a.d.1.4	✓	6
12.11	even	2		7524.2.a.r.1.4		6
44.43	even	2		9196.2.a.k.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
836.2.a.d.1.4	✓	6	4.3	odd	2
3344.2.a.x.1.3		6	1.1	even	1	trivial
7524.2.a.r.1.4		6	12.11	even	2
9196.2.a.k.1.4		6	44.43	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:	\( N \)	\(=\)	\( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3344.a (trivial)

\(f(q)\)	\(=\)	\(q+0.658537 q^{3} -2.39807 q^{5} -4.45908 q^{7} -2.56633 q^{9} +O(q^{10})\)	\(q+0.658537 q^{3} -2.39807 q^{5} -4.45908 q^{7} -2.56633 q^{9} -1.00000 q^{11} +3.38518 q^{13} -1.57922 q^{15} -5.17423 q^{17} -1.00000 q^{19} -2.93647 q^{21} -0.851183 q^{23} +0.750745 q^{25} -3.66563 q^{27} +7.62294 q^{29} -6.34958 q^{31} -0.658537 q^{33} +10.6932 q^{35} +9.00597 q^{37} +2.22927 q^{39} -3.15490 q^{41} -5.24925 q^{43} +6.15424 q^{45} -8.77036 q^{47} +12.8834 q^{49} -3.40742 q^{51} +9.81558 q^{53} +2.39807 q^{55} -0.658537 q^{57} +1.57568 q^{59} -10.8420 q^{61} +11.4435 q^{63} -8.11791 q^{65} +3.43142 q^{67} -0.560536 q^{69} +4.02439 q^{71} +16.6117 q^{73} +0.494394 q^{75} +4.45908 q^{77} +4.15479 q^{79} +5.28503 q^{81} -6.27568 q^{83} +12.4082 q^{85} +5.01999 q^{87} +1.22927 q^{89} -15.0948 q^{91} -4.18143 q^{93} +2.39807 q^{95} -16.0254 q^{97} +2.56633 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{3} - 2 q^{7} + 10 q^{9}+O(q^{10}) \) 6 * q + 2 * q^3 - 2 * q^7 + 10 * q^9	\( 6 q + 2 q^{3} - 2 q^{7} + 10 q^{9} - 6 q^{11} + 2 q^{13} + 12 q^{15} + 12 q^{17} - 6 q^{19} + 2 q^{21} + 2 q^{23} + 26 q^{25} - 16 q^{27} + 4 q^{29} + 20 q^{31} - 2 q^{33} - 20 q^{35} + 22 q^{37} - 8 q^{39} - 2 q^{41} - 10 q^{43} - 6 q^{45} - 16 q^{47} + 48 q^{49} - 36 q^{51} + 12 q^{53} - 2 q^{57} + 14 q^{59} + 12 q^{61} + 4 q^{63} + 10 q^{65} + 12 q^{67} + 30 q^{69} + 30 q^{71} + 24 q^{73} + 50 q^{75} + 2 q^{77} + 10 q^{81} + 14 q^{83} + 12 q^{85} + 30 q^{87} - 14 q^{89} - 20 q^{91} + 14 q^{93} - 46 q^{97} - 10 q^{99}+O(q^{100}) \) 6 * q + 2 * q^3 - 2 * q^7 + 10 * q^9 - 6 * q^11 + 2 * q^13 + 12 * q^15 + 12 * q^17 - 6 * q^19 + 2 * q^21 + 2 * q^23 + 26 * q^25 - 16 * q^27 + 4 * q^29 + 20 * q^31 - 2 * q^33 - 20 * q^35 + 22 * q^37 - 8 * q^39 - 2 * q^41 - 10 * q^43 - 6 * q^45 - 16 * q^47 + 48 * q^49 - 36 * q^51 + 12 * q^53 - 2 * q^57 + 14 * q^59 + 12 * q^61 + 4 * q^63 + 10 * q^65 + 12 * q^67 + 30 * q^69 + 30 * q^71 + 24 * q^73 + 50 * q^75 + 2 * q^77 + 10 * q^81 + 14 * q^83 + 12 * q^85 + 30 * q^87 - 14 * q^89 - 20 * q^91 + 14 * q^93 - 46 * q^97 - 10 * q^99

Introduction

L-functions

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