LMFDB - Embedded newform 4592.2.a.t.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4592,2,Mod(1,4592)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4592, 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("4592.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4592 = 2^{4} \cdot 7 \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4592.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:	$36.6673046082$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.257.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 3 $ x^3 - x^2 - 4*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 287)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.91223$ of defining polynomial
Character	$\chi$	$=$	4592.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.56885 q^{3} +2.00000 q^{5} -1.00000 q^{7} +3.59899 q^{9} +O(q^{10})$	$q-2.56885 q^{3} +2.00000 q^{5} -1.00000 q^{7} +3.59899 q^{9} +2.00000 q^{11} +2.34338 q^{13} -5.13770 q^{15} -3.22547 q^{17} -2.08777 q^{19} +2.56885 q^{21} +6.56885 q^{23} -1.00000 q^{25} -1.53871 q^{27} +7.82446 q^{29} -3.82446 q^{31} -5.13770 q^{33} -2.00000 q^{35} +4.28310 q^{37} -6.01979 q^{39} -1.00000 q^{41} -7.11021 q^{43} +7.19798 q^{45} +8.79432 q^{47} +1.00000 q^{49} +8.28575 q^{51} -2.51122 q^{53} +4.00000 q^{55} +5.36317 q^{57} +4.00000 q^{59} -4.51122 q^{61} -3.59899 q^{63} +4.68676 q^{65} -6.51122 q^{67} -16.8744 q^{69} +14.9622 q^{71} -8.45094 q^{73} +2.56885 q^{75} -2.00000 q^{77} +1.13770 q^{79} -6.84425 q^{81} +9.64892 q^{83} -6.45094 q^{85} -20.0999 q^{87} -16.9545 q^{89} -2.34338 q^{91} +9.82446 q^{93} -4.17554 q^{95} +16.3933 q^{97} +7.19798 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{3} + 6 q^{5} - 3 q^{7} + 8 q^{9}+O(q^{10}) $ 3 * q + q^3 + 6 * q^5 - 3 * q^7 + 8 * q^9	$ 3 q + q^{3} + 6 q^{5} - 3 q^{7} + 8 q^{9} + 6 q^{11} + 9 q^{13} + 2 q^{15} + q^{17} - 13 q^{19} - q^{21} + 11 q^{23} - 3 q^{25} + 10 q^{27} + 10 q^{29} + 2 q^{31} + 2 q^{33} - 6 q^{35} + 3 q^{37} + 12 q^{39} - 3 q^{41} - 9 q^{43} + 16 q^{45} + 7 q^{47} + 3 q^{49} + 26 q^{51} + 2 q^{53} + 12 q^{55} - 12 q^{57} + 12 q^{59} - 4 q^{61} - 8 q^{63} + 18 q^{65} - 10 q^{67} - 13 q^{69} + 14 q^{71} - 4 q^{73} - q^{75} - 6 q^{77} - 14 q^{79} + 23 q^{81} + 2 q^{83} + 2 q^{85} - 12 q^{87} + 5 q^{89} - 9 q^{91} + 16 q^{93} - 26 q^{95} + 27 q^{97} + 16 q^{99}+O(q^{100}) $ 3 * q + q^3 + 6 * q^5 - 3 * q^7 + 8 * q^9 + 6 * q^11 + 9 * q^13 + 2 * q^15 + q^17 - 13 * q^19 - q^21 + 11 * q^23 - 3 * q^25 + 10 * q^27 + 10 * q^29 + 2 * q^31 + 2 * q^33 - 6 * q^35 + 3 * q^37 + 12 * q^39 - 3 * q^41 - 9 * q^43 + 16 * q^45 + 7 * q^47 + 3 * q^49 + 26 * q^51 + 2 * q^53 + 12 * q^55 - 12 * q^57 + 12 * q^59 - 4 * q^61 - 8 * q^63 + 18 * q^65 - 10 * q^67 - 13 * q^69 + 14 * q^71 - 4 * q^73 - q^75 - 6 * q^77 - 14 * q^79 + 23 * q^81 + 2 * q^83 + 2 * q^85 - 12 * q^87 + 5 * q^89 - 9 * q^91 + 16 * q^93 - 26 * q^95 + 27 * q^97 + 16 * 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
$3$	−2.56885	−1.48313	−0.741563	−	0.670883i	$-0.765915\pi$
$3$	−2.56885	−1.48313	−0.741563	+	0.670883i	$0.765915\pi$
$4$	0	0
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	3.59899	1.19966
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	2.34338	0.649937	0.324968	−	0.945725i	$-0.394646\pi$
$13$	2.34338	0.649937	0.324968	+	0.945725i	$0.394646\pi$
$14$	0	0
$15$	−5.13770	−1.32655
$16$	0	0
$17$	−3.22547	−0.782291	−0.391146	−	0.920329i	$-0.627921\pi$
$17$	−3.22547	−0.782291	−0.391146	+	0.920329i	$0.627921\pi$
$18$	0	0
$19$	−2.08777	−0.478967	−0.239484	−	0.970900i	$-0.576978\pi$
$19$	−2.08777	−0.478967	−0.239484	+	0.970900i	$0.576978\pi$
$20$	0	0
$21$	2.56885	0.560569
$22$	0	0
$23$	6.56885	1.36970	0.684850	−	0.728684i	$-0.259868\pi$
$23$	6.56885	1.36970	0.684850	+	0.728684i	$0.259868\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−1.53871	−0.296125
$28$	0	0
$29$	7.82446	1.45297	0.726483	−	0.687185i	$-0.241154\pi$
$29$	7.82446	1.45297	0.726483	+	0.687185i	$0.241154\pi$
$30$	0	0
$31$	−3.82446	−0.686893	−0.343446	−	0.939172i	$-0.611594\pi$
$31$	−3.82446	−0.686893	−0.343446	+	0.939172i	$0.611594\pi$
$32$	0	0
$33$	−5.13770	−0.894359
$34$	0	0
$35$	−2.00000	−0.338062
$36$	0	0
$37$	4.28310	0.704138	0.352069	−	0.935974i	$-0.385478\pi$
$37$	4.28310	0.704138	0.352069	+	0.935974i	$0.385478\pi$
$38$	0	0
$39$	−6.01979	−0.963938
$40$	0	0
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	−7.11021	−1.08430	−0.542148	−	0.840283i	$-0.682389\pi$
$43$	−7.11021	−1.08430	−0.542148	+	0.840283i	$0.682389\pi$
$44$	0	0
$45$	7.19798	1.07301
$46$	0	0
$47$	8.79432	1.28278	0.641392	−	0.767214i	$-0.278357\pi$
$47$	8.79432	1.28278	0.641392	+	0.767214i	$0.278357\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	8.28575	1.16024
$52$	0	0
$53$	−2.51122	−0.344942	−0.172471	−	0.985015i	$-0.555175\pi$
$53$	−2.51122	−0.344942	−0.172471	+	0.985015i	$0.555175\pi$
$54$	0	0
$55$	4.00000	0.539360
$56$	0	0
$57$	5.36317	0.710369
$58$	0	0
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	−4.51122	−0.577602	−0.288801	−	0.957389i	$-0.593257\pi$
$61$	−4.51122	−0.577602	−0.288801	+	0.957389i	$0.593257\pi$
$62$	0	0
$63$	−3.59899	−0.453430
$64$	0	0
$65$	4.68676	0.581321
$66$	0	0
$67$	−6.51122	−0.795472	−0.397736	−	0.917500i	$-0.630204\pi$
$67$	−6.51122	−0.795472	−0.397736	+	0.917500i	$0.630204\pi$
$68$	0	0
$69$	−16.8744	−2.03144
$70$	0	0
$71$	14.9622	1.77568	0.887841	−	0.460151i	$-0.152205\pi$
$71$	14.9622	1.77568	0.887841	+	0.460151i	$0.152205\pi$
$72$	0	0
$73$	−8.45094	−0.989108	−0.494554	−	0.869147i	$-0.664669\pi$
$73$	−8.45094	−0.989108	−0.494554	+	0.869147i	$0.664669\pi$
$74$	0	0
$75$	2.56885	0.296625
$76$	0	0
$77$	−2.00000	−0.227921
$78$	0	0
$79$	1.13770	0.128001	0.0640006	−	0.997950i	$-0.479614\pi$
$79$	1.13770	0.128001	0.0640006	+	0.997950i	$0.479614\pi$
$80$	0	0
$81$	−6.84425	−0.760472
$82$	0	0
$83$	9.64892	1.05911	0.529553	−	0.848277i	$-0.322360\pi$
$83$	9.64892	1.05911	0.529553	+	0.848277i	$0.322360\pi$
$84$	0	0
$85$	−6.45094	−0.699703
$86$	0	0
$87$	−20.0999	−2.15493
$88$	0	0
$89$	−16.9545	−1.79717	−0.898584	−	0.438801i	$-0.855403\pi$
$89$	−16.9545	−1.79717	−0.898584	+	0.438801i	$0.855403\pi$
$90$	0	0
$91$	−2.34338	−0.245653
$92$	0	0
$93$	9.82446	1.01875
$94$	0	0
$95$	−4.17554	−0.428402
$96$	0	0
$97$	16.3933	1.66449	0.832244	−	0.554409i	$-0.187056\pi$
$97$	16.3933	1.66449	0.832244	+	0.554409i	$0.187056\pi$
$98$	0	0
$99$	7.19798	0.723424
$100$	0	0
$101$	3.37087	0.335414	0.167707	−	0.985837i	$-0.446364\pi$
$101$	3.37087	0.335414	0.167707	+	0.985837i	$0.446364\pi$
$102$	0	0
$103$	−2.35108	−0.231659	−0.115830	−	0.993269i	$-0.536953\pi$
$103$	−2.35108	−0.231659	−0.115830	+	0.993269i	$0.536953\pi$
$104$	0	0
$105$	5.13770	0.501388
$106$	0	0
$107$	−16.0121	−1.54795	−0.773973	−	0.633218i	$-0.781734\pi$
$107$	−16.0121	−1.54795	−0.773973	+	0.633218i	$0.781734\pi$
$108$	0	0
$109$	−5.70919	−0.546842	−0.273421	−	0.961895i	$-0.588155\pi$
$109$	−5.70919	−0.546842	−0.273421	+	0.961895i	$0.588155\pi$
$110$	0	0
$111$	−11.0026	−1.04432
$112$	0	0
$113$	17.5310	1.64918	0.824589	−	0.565732i	$-0.191406\pi$
$113$	17.5310	1.64918	0.824589	+	0.565732i	$0.191406\pi$
$114$	0	0
$115$	13.1377	1.22510
$116$	0	0
$117$	8.43380	0.779705
$118$	0	0
$119$	3.22547	0.295678
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	2.56885	0.231625
$124$	0	0
$125$	−12.0000	−1.07331
$126$	0	0
$127$	20.4432	1.81404	0.907022	−	0.421083i	$-0.138350\pi$
$127$	20.4432	1.81404	0.907022	+	0.421083i	$0.138350\pi$
$128$	0	0
$129$	18.2650	1.60815
$130$	0	0
$131$	−0.962158	−0.0840641	−0.0420320	−	0.999116i	$-0.513383\pi$
$131$	−0.962158	−0.0840641	−0.0420320	+	0.999116i	$0.513383\pi$
$132$	0	0
$133$	2.08777	0.181033
$134$	0	0
$135$	−3.07742	−0.264862
$136$	0	0
$137$	3.37352	0.288219	0.144110	−	0.989562i	$-0.453968\pi$
$137$	3.37352	0.288219	0.144110	+	0.989562i	$0.453968\pi$
$138$	0	0
$139$	−14.7866	−1.25418	−0.627092	−	0.778945i	$-0.715755\pi$
$139$	−14.7866	−1.25418	−0.627092	+	0.778945i	$0.715755\pi$
$140$	0	0
$141$	−22.5913	−1.90253
$142$	0	0
$143$	4.68676	0.391926
$144$	0	0
$145$	15.6489	1.29957
$146$	0	0
$147$	−2.56885	−0.211875
$148$	0	0
$149$	1.76418	0.144527	0.0722637	−	0.997386i	$-0.476978\pi$
$149$	1.76418	0.144527	0.0722637	+	0.997386i	$0.476978\pi$
$150$	0	0
$151$	1.54906	0.126061	0.0630304	−	0.998012i	$-0.479924\pi$
$151$	1.54906	0.126061	0.0630304	+	0.998012i	$0.479924\pi$
$152$	0	0
$153$	−11.6084	−0.938486
$154$	0	0
$155$	−7.64892	−0.614376
$156$	0	0
$157$	−13.8520	−1.10551	−0.552753	−	0.833345i	$-0.686423\pi$
$157$	−13.8520	−1.10551	−0.552753	+	0.833345i	$0.686423\pi$
$158$	0	0
$159$	6.45094	0.511593
$160$	0	0
$161$	−6.56885	−0.517698
$162$	0	0
$163$	24.3779	1.90942	0.954712	−	0.297531i	$-0.0961630\pi$
$163$	24.3779	1.90942	0.954712	+	0.297531i	$0.0961630\pi$
$164$	0	0
$165$	−10.2754	−0.799939
$166$	0	0
$167$	−12.1876	−0.943107	−0.471553	−	0.881837i	$-0.656307\pi$
$167$	−12.1876	−0.943107	−0.471553	+	0.881837i	$0.656307\pi$
$168$	0	0
$169$	−7.50857	−0.577582
$170$	0	0
$171$	−7.51386	−0.574599
$172$	0	0
$173$	9.31324	0.708073	0.354036	−	0.935232i	$-0.384809\pi$
$173$	9.31324	0.708073	0.354036	+	0.935232i	$0.384809\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	−10.2754	−0.772346
$178$	0	0
$179$	−12.2754	−0.917506	−0.458753	−	0.888564i	$-0.651704\pi$
$179$	−12.2754	−0.917506	−0.458753	+	0.888564i	$0.651704\pi$
$180$	0	0
$181$	13.7668	1.02328	0.511640	−	0.859200i	$-0.329038\pi$
$181$	13.7668	1.02328	0.511640	+	0.859200i	$0.329038\pi$
$182$	0	0
$183$	11.5886	0.856657
$184$	0	0
$185$	8.56620	0.629800
$186$	0	0
$187$	−6.45094	−0.471739
$188$	0	0
$189$	1.53871	0.111925
$190$	0	0
$191$	9.31324	0.673882	0.336941	−	0.941526i	$-0.390608\pi$
$191$	9.31324	0.673882	0.336941	+	0.941526i	$0.390608\pi$
$192$	0	0
$193$	7.31324	0.526419	0.263209	−	0.964739i	$-0.415219\pi$
$193$	7.31324	0.526419	0.263209	+	0.964739i	$0.415219\pi$
$194$	0	0
$195$	−12.0396	−0.862172
$196$	0	0
$197$	16.4234	1.17012	0.585061	−	0.810989i	$-0.301071\pi$
$197$	16.4234	1.17012	0.585061	+	0.810989i	$0.301071\pi$
$198$	0	0
$199$	9.13000	0.647208	0.323604	−	0.946193i	$-0.395105\pi$
$199$	9.13000	0.647208	0.323604	+	0.946193i	$0.395105\pi$
$200$	0	0
$201$	16.7263	1.17978
$202$	0	0
$203$	−7.82446	−0.549169
$204$	0	0
$205$	−2.00000	−0.139686
$206$	0	0
$207$	23.6412	1.64318
$208$	0	0
$209$	−4.17554	−0.288828
$210$	0	0
$211$	−7.53365	−0.518638	−0.259319	−	0.965792i	$-0.583498\pi$
$211$	−7.53365	−0.518638	−0.259319	+	0.965792i	$0.583498\pi$
$212$	0	0
$213$	−38.4355	−2.63356
$214$	0	0
$215$	−14.2204	−0.969824
$216$	0	0
$217$	3.82446	0.259621
$218$	0	0
$219$	21.7092	1.46697
$220$	0	0
$221$	−7.55850	−0.508440
$222$	0	0
$223$	4.57149	0.306130	0.153065	−	0.988216i	$-0.451086\pi$
$223$	4.57149	0.306130	0.153065	+	0.988216i	$0.451086\pi$
$224$	0	0
$225$	−3.59899	−0.239933
$226$	0	0
$227$	0.351083	0.0233022	0.0116511	−	0.999932i	$-0.496291\pi$
$227$	0.351083	0.0233022	0.0116511	+	0.999932i	$0.496291\pi$
$228$	0	0
$229$	−6.90453	−0.456264	−0.228132	−	0.973630i	$-0.573262\pi$
$229$	−6.90453	−0.456264	−0.228132	+	0.973630i	$0.573262\pi$
$230$	0	0
$231$	5.13770	0.338036
$232$	0	0
$233$	29.0070	1.90031	0.950157	−	0.311773i	$-0.100923\pi$
$233$	29.0070	1.90031	0.950157	+	0.311773i	$0.100923\pi$
$234$	0	0
$235$	17.5886	1.14736
$236$	0	0
$237$	−2.92258	−0.189842
$238$	0	0
$239$	28.4958	1.84324	0.921620	−	0.388093i	$-0.126866\pi$
$239$	28.4958	1.84324	0.921620	+	0.388093i	$0.126866\pi$
$240$	0	0
$241$	20.9019	1.34641	0.673204	−	0.739457i	$-0.264917\pi$
$241$	20.9019	1.34641	0.673204	+	0.739457i	$0.264917\pi$
$242$	0	0
$243$	22.1980	1.42400
$244$	0	0
$245$	2.00000	0.127775
$246$	0	0
$247$	−4.89244	−0.311298
$248$	0	0
$249$	−24.7866	−1.57079
$250$	0	0
$251$	−5.47338	−0.345476	−0.172738	−	0.984968i	$-0.555261\pi$
$251$	−5.47338	−0.345476	−0.172738	+	0.984968i	$0.555261\pi$
$252$	0	0
$253$	13.1377	0.825960
$254$	0	0
$255$	16.5715	1.03775
$256$	0	0
$257$	20.0422	1.25020	0.625100	−	0.780545i	$-0.285058\pi$
$257$	20.0422	1.25020	0.625100	+	0.780545i	$0.285058\pi$
$258$	0	0
$259$	−4.28310	−0.266139
$260$	0	0
$261$	28.1601	1.74307
$262$	0	0
$263$	8.39066	0.517390	0.258695	−	0.965959i	$-0.416707\pi$
$263$	8.39066	0.517390	0.258695	+	0.965959i	$0.416707\pi$
$264$	0	0
$265$	−5.02243	−0.308526
$266$	0	0
$267$	43.5534	2.66543
$268$	0	0
$269$	23.1826	1.41347	0.706733	−	0.707480i	$-0.250168\pi$
$269$	23.1826	1.41347	0.706733	+	0.707480i	$0.250168\pi$
$270$	0	0
$271$	20.3907	1.23864	0.619322	−	0.785137i	$-0.287407\pi$
$271$	20.3907	1.23864	0.619322	+	0.785137i	$0.287407\pi$
$272$	0	0
$273$	6.01979	0.364334
$274$	0	0
$275$	−2.00000	−0.120605
$276$	0	0
$277$	18.5534	1.11477	0.557384	−	0.830255i	$-0.311805\pi$
$277$	18.5534	1.11477	0.557384	+	0.830255i	$0.311805\pi$
$278$	0	0
$279$	−13.7642	−0.824040
$280$	0	0
$281$	−15.8847	−0.947604	−0.473802	−	0.880631i	$-0.657119\pi$
$281$	−15.8847	−0.947604	−0.473802	+	0.880631i	$0.657119\pi$
$282$	0	0
$283$	−20.6265	−1.22612	−0.613059	−	0.790037i	$-0.710061\pi$
$283$	−20.6265	−1.22612	−0.613059	+	0.790037i	$0.710061\pi$
$284$	0	0
$285$	10.7263	0.635373
$286$	0	0
$287$	1.00000	0.0590281
$288$	0	0
$289$	−6.59634	−0.388020
$290$	0	0
$291$	−42.1119	−2.46865
$292$	0	0
$293$	24.9468	1.45740	0.728702	−	0.684831i	$-0.240124\pi$
$293$	24.9468	1.45740	0.728702	+	0.684831i	$0.240124\pi$
$294$	0	0
$295$	8.00000	0.465778
$296$	0	0
$297$	−3.07742	−0.178570
$298$	0	0
$299$	15.3933	0.890218
$300$	0	0
$301$	7.11021	0.409825
$302$	0	0
$303$	−8.65927	−0.497462
$304$	0	0
$305$	−9.02243	−0.516623
$306$	0	0
$307$	10.2204	0.583310	0.291655	−	0.956524i	$-0.405794\pi$
$307$	10.2204	0.583310	0.291655	+	0.956524i	$0.405794\pi$
$308$	0	0
$309$	6.03958	0.343580
$310$	0	0
$311$	16.5387	0.937824	0.468912	−	0.883245i	$-0.344646\pi$
$311$	16.5387	0.937824	0.468912	+	0.883245i	$0.344646\pi$
$312$	0	0
$313$	16.4784	0.931416	0.465708	−	0.884938i	$-0.345800\pi$
$313$	16.4784	0.931416	0.465708	+	0.884938i	$0.345800\pi$
$314$	0	0
$315$	−7.19798	−0.405560
$316$	0	0
$317$	12.9019	0.724642	0.362321	−	0.932053i	$-0.381984\pi$
$317$	12.9019	0.724642	0.362321	+	0.932053i	$0.381984\pi$
$318$	0	0
$319$	15.6489	0.876171
$320$	0	0
$321$	41.1326	2.29580
$322$	0	0
$323$	6.73404	0.374692
$324$	0	0
$325$	−2.34338	−0.129987
$326$	0	0
$327$	14.6661	0.811035
$328$	0	0
$329$	−8.79432	−0.484847
$330$	0	0
$331$	−22.0000	−1.20923	−0.604615	−	0.796518i	$-0.706673\pi$
$331$	−22.0000	−1.20923	−0.604615	+	0.796518i	$0.706673\pi$
$332$	0	0
$333$	15.4148	0.844728
$334$	0	0
$335$	−13.0224	−0.711492
$336$	0	0
$337$	−4.39331	−0.239319	−0.119659	−	0.992815i	$-0.538180\pi$
$337$	−4.39331	−0.239319	−0.119659	+	0.992815i	$0.538180\pi$
$338$	0	0
$339$	−45.0345	−2.44594
$340$	0	0
$341$	−7.64892	−0.414212
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−33.7488	−1.81697
$346$	0	0
$347$	18.0000	0.966291	0.483145	−	0.875540i	$-0.339494\pi$
$347$	18.0000	0.966291	0.483145	+	0.875540i	$0.339494\pi$
$348$	0	0
$349$	18.9072	1.01208	0.506039	−	0.862511i	$-0.331109\pi$
$349$	18.9072	1.01208	0.506039	+	0.862511i	$0.331109\pi$
$350$	0	0
$351$	−3.60578	−0.192462
$352$	0	0
$353$	−15.3735	−0.818250	−0.409125	−	0.912478i	$-0.634166\pi$
$353$	−15.3735	−0.818250	−0.409125	+	0.912478i	$0.634166\pi$
$354$	0	0
$355$	29.9243	1.58822
$356$	0	0
$357$	−8.28575	−0.438528
$358$	0	0
$359$	11.0851	0.585051	0.292525	−	0.956258i	$-0.405504\pi$
$359$	11.0851	0.585051	0.292525	+	0.956258i	$0.405504\pi$
$360$	0	0
$361$	−14.6412	−0.770590
$362$	0	0
$363$	17.9819	0.943807
$364$	0	0
$365$	−16.9019	−0.884685
$366$	0	0
$367$	−8.57149	−0.447428	−0.223714	−	0.974655i	$-0.571818\pi$
$367$	−8.57149	−0.447428	−0.223714	+	0.974655i	$0.571818\pi$
$368$	0	0
$369$	−3.59899	−0.187356
$370$	0	0
$371$	2.51122	0.130376
$372$	0	0
$373$	−16.4234	−0.850374	−0.425187	−	0.905106i	$-0.639792\pi$
$373$	−16.4234	−0.850374	−0.425187	+	0.905106i	$0.639792\pi$
$374$	0	0
$375$	30.8262	1.59186
$376$	0	0
$377$	18.3357	0.944335
$378$	0	0
$379$	−1.43115	−0.0735133	−0.0367566	−	0.999324i	$-0.511703\pi$
$379$	−1.43115	−0.0735133	−0.0367566	+	0.999324i	$0.511703\pi$
$380$	0	0
$381$	−52.5156	−2.69046
$382$	0	0
$383$	−13.1953	−0.674250	−0.337125	−	0.941460i	$-0.609454\pi$
$383$	−13.1953	−0.674250	−0.337125	+	0.941460i	$0.609454\pi$
$384$	0	0
$385$	−4.00000	−0.203859
$386$	0	0
$387$	−25.5895	−1.30079
$388$	0	0
$389$	9.30554	0.471809	0.235905	−	0.971776i	$-0.424195\pi$
$389$	9.30554	0.471809	0.235905	+	0.971776i	$0.424195\pi$
$390$	0	0
$391$	−21.1876	−1.07150
$392$	0	0
$393$	2.47164	0.124678
$394$	0	0
$395$	2.27540	0.114488
$396$	0	0
$397$	−1.29783	−0.0651364	−0.0325682	−	0.999470i	$-0.510369\pi$
$397$	−1.29783	−0.0651364	−0.0325682	+	0.999470i	$0.510369\pi$
$398$	0	0
$399$	−5.36317	−0.268494
$400$	0	0
$401$	6.22282	0.310753	0.155377	−	0.987855i	$-0.450341\pi$
$401$	6.22282	0.310753	0.155377	+	0.987855i	$0.450341\pi$
$402$	0	0
$403$	−8.96216	−0.446437
$404$	0	0
$405$	−13.6885	−0.680187
$406$	0	0
$407$	8.56620	0.424611
$408$	0	0
$409$	33.9846	1.68043	0.840215	−	0.542253i	$-0.182429\pi$
$409$	33.9846	1.68043	0.840215	+	0.542253i	$0.182429\pi$
$410$	0	0
$411$	−8.66606	−0.427465
$412$	0	0
$413$	−4.00000	−0.196827
$414$	0	0
$415$	19.2978	0.947293
$416$	0	0
$417$	37.9846	1.86011
$418$	0	0
$419$	−33.3977	−1.63158	−0.815792	−	0.578345i	$-0.803699\pi$
$419$	−33.3977	−1.63158	−0.815792	+	0.578345i	$0.803699\pi$
$420$	0	0
$421$	7.06201	0.344182	0.172091	−	0.985081i	$-0.444948\pi$
$421$	7.06201	0.344182	0.172091	+	0.985081i	$0.444948\pi$
$422$	0	0
$423$	31.6507	1.53891
$424$	0	0
$425$	3.22547	0.156458
$426$	0	0
$427$	4.51122	0.218313
$428$	0	0
$429$	−12.0396	−0.581276
$430$	0	0
$431$	31.2978	1.50756	0.753782	−	0.657125i	$-0.228228\pi$
$431$	31.2978	1.50756	0.753782	+	0.657125i	$0.228228\pi$
$432$	0	0
$433$	−26.7712	−1.28654	−0.643271	−	0.765638i	$-0.722423\pi$
$433$	−26.7712	−1.28654	−0.643271	+	0.765638i	$0.722423\pi$
$434$	0	0
$435$	−40.1997	−1.92743
$436$	0	0
$437$	−13.7143	−0.656042
$438$	0	0
$439$	15.7065	0.749633	0.374816	−	0.927099i	$-0.377706\pi$
$439$	15.7065	0.749633	0.374816	+	0.927099i	$0.377706\pi$
$440$	0	0
$441$	3.59899	0.171380
$442$	0	0
$443$	−33.8168	−1.60668	−0.803341	−	0.595519i	$-0.796946\pi$
$443$	−33.8168	−1.60668	−0.803341	+	0.595519i	$0.796946\pi$
$444$	0	0
$445$	−33.9089	−1.60744
$446$	0	0
$447$	−4.53192	−0.214352
$448$	0	0
$449$	19.9270	0.940411	0.470206	−	0.882557i	$-0.344180\pi$
$449$	19.9270	0.940411	0.470206	+	0.882557i	$0.344180\pi$
$450$	0	0
$451$	−2.00000	−0.0941763
$452$	0	0
$453$	−3.97930	−0.186964
$454$	0	0
$455$	−4.68676	−0.219719
$456$	0	0
$457$	30.5662	1.42983	0.714913	−	0.699213i	$-0.246466\pi$
$457$	30.5662	1.42983	0.714913	+	0.699213i	$0.246466\pi$
$458$	0	0
$459$	4.96307	0.231656
$460$	0	0
$461$	−32.3203	−1.50530	−0.752652	−	0.658418i	$-0.771226\pi$
$461$	−32.3203	−1.50530	−0.752652	+	0.658418i	$0.771226\pi$
$462$	0	0
$463$	−2.16013	−0.100390	−0.0501950	−	0.998739i	$-0.515984\pi$
$463$	−2.16013	−0.100390	−0.0501950	+	0.998739i	$0.515984\pi$
$464$	0	0
$465$	19.6489	0.911197
$466$	0	0
$467$	33.9692	1.57191	0.785953	−	0.618286i	$-0.212173\pi$
$467$	33.9692	1.57191	0.785953	+	0.618286i	$0.212173\pi$
$468$	0	0
$469$	6.51122	0.300660
$470$	0	0
$471$	35.5836	1.63960
$472$	0	0
$473$	−14.2204	−0.653855
$474$	0	0
$475$	2.08777	0.0957935
$476$	0	0
$477$	−9.03784	−0.413814
$478$	0	0
$479$	12.3779	0.565561	0.282780	−	0.959185i	$-0.408743\pi$
$479$	12.3779	0.565561	0.282780	+	0.959185i	$0.408743\pi$
$480$	0	0
$481$	10.0369	0.457645
$482$	0	0
$483$	16.8744	0.767811
$484$	0	0
$485$	32.7866	1.48876
$486$	0	0
$487$	−18.4080	−0.834148	−0.417074	−	0.908873i	$-0.636944\pi$
$487$	−18.4080	−0.834148	−0.417074	+	0.908873i	$0.636944\pi$
$488$	0	0
$489$	−62.6232	−2.83192
$490$	0	0
$491$	−14.3330	−0.646841	−0.323420	−	0.946255i	$-0.604833\pi$
$491$	−14.3330	−0.646841	−0.323420	+	0.946255i	$0.604833\pi$
$492$	0	0
$493$	−25.2376	−1.13664
$494$	0	0
$495$	14.3960	0.647050
$496$	0	0
$497$	−14.9622	−0.671144
$498$	0	0
$499$	17.8245	0.797932	0.398966	−	0.916966i	$-0.369369\pi$
$499$	17.8245	0.797932	0.398966	+	0.916966i	$0.369369\pi$
$500$	0	0
$501$	31.3082	1.39875
$502$	0	0
$503$	−5.98195	−0.266722	−0.133361	−	0.991068i	$-0.542577\pi$
$503$	−5.98195	−0.266722	−0.133361	+	0.991068i	$0.542577\pi$
$504$	0	0
$505$	6.74175	0.300004
$506$	0	0
$507$	19.2884	0.856628
$508$	0	0
$509$	26.6791	1.18253	0.591264	−	0.806478i	$-0.298629\pi$
$509$	26.6791	1.18253	0.591264	+	0.806478i	$0.298629\pi$
$510$	0	0
$511$	8.45094	0.373848
$512$	0	0
$513$	3.21248	0.141834
$514$	0	0
$515$	−4.70217	−0.207202
$516$	0	0
$517$	17.5886	0.773547
$518$	0	0
$519$	−23.9243	−1.05016
$520$	0	0
$521$	−35.8115	−1.56893	−0.784464	−	0.620174i	$-0.787062\pi$
$521$	−35.8115	−1.56893	−0.784464	+	0.620174i	$0.787062\pi$
$522$	0	0
$523$	−32.2600	−1.41063	−0.705315	−	0.708894i	$-0.749195\pi$
$523$	−32.2600	−1.41063	−0.705315	+	0.708894i	$0.749195\pi$
$524$	0	0
$525$	−2.56885	−0.112114
$526$	0	0
$527$	12.3357	0.537350
$528$	0	0
$529$	20.1498	0.876078
$530$	0	0
$531$	14.3960	0.624731
$532$	0	0
$533$	−2.34338	−0.101503
$534$	0	0
$535$	−32.0242	−1.38453
$536$	0	0
$537$	31.5337	1.36078
$538$	0	0
$539$	2.00000	0.0861461
$540$	0	0
$541$	22.5233	0.968352	0.484176	−	0.874971i	$-0.339119\pi$
$541$	22.5233	0.968352	0.484176	+	0.874971i	$0.339119\pi$
$542$	0	0
$543$	−35.3649	−1.51765
$544$	0	0
$545$	−11.4184	−0.489110
$546$	0	0
$547$	−3.94501	−0.168677	−0.0843383	−	0.996437i	$-0.526878\pi$
$547$	−3.94501	−0.168677	−0.0843383	+	0.996437i	$0.526878\pi$
$548$	0	0
$549$	−16.2358	−0.692928
$550$	0	0
$551$	−16.3357	−0.695923
$552$	0	0
$553$	−1.13770	−0.0483799
$554$	0	0
$555$	−22.0053	−0.934073
$556$	0	0
$557$	−1.08271	−0.0458760	−0.0229380	−	0.999737i	$-0.507302\pi$
$557$	−1.08271	−0.0458760	−0.0229380	+	0.999737i	$0.507302\pi$
$558$	0	0
$559$	−16.6619	−0.704724
$560$	0	0
$561$	16.5715	0.699649
$562$	0	0
$563$	15.5568	0.655639	0.327820	−	0.944740i	$-0.393686\pi$
$563$	15.5568	0.655639	0.327820	+	0.944740i	$0.393686\pi$
$564$	0	0
$565$	35.0620	1.47507
$566$	0	0
$567$	6.84425	0.287431
$568$	0	0
$569$	23.7521	0.995740	0.497870	−	0.867252i	$-0.334116\pi$
$569$	23.7521	0.995740	0.497870	+	0.867252i	$0.334116\pi$
$570$	0	0
$571$	43.5886	1.82413	0.912064	−	0.410048i	$-0.134488\pi$
$571$	43.5886	1.82413	0.912064	+	0.410048i	$0.134488\pi$
$572$	0	0
$573$	−23.9243	−0.999453
$574$	0	0
$575$	−6.56885	−0.273940
$576$	0	0
$577$	36.0422	1.50046	0.750229	−	0.661178i	$-0.229943\pi$
$577$	36.0422	1.50046	0.750229	+	0.661178i	$0.229943\pi$
$578$	0	0
$579$	−18.7866	−0.780745
$580$	0	0
$581$	−9.64892	−0.400305
$582$	0	0
$583$	−5.02243	−0.208008
$584$	0	0
$585$	16.8676	0.697389
$586$	0	0
$587$	−30.9347	−1.27681	−0.638405	−	0.769701i	$-0.720406\pi$
$587$	−30.9347	−1.27681	−0.638405	+	0.769701i	$0.720406\pi$
$588$	0	0
$589$	7.98459	0.328999
$590$	0	0
$591$	−42.1894	−1.73544
$592$	0	0
$593$	20.2831	0.832927	0.416464	−	0.909152i	$-0.363269\pi$
$593$	20.2831	0.832927	0.416464	+	0.909152i	$0.363269\pi$
$594$	0	0
$595$	6.45094	0.264463
$596$	0	0
$597$	−23.4536	−0.959891
$598$	0	0
$599$	−5.71161	−0.233370	−0.116685	−	0.993169i	$-0.537227\pi$
$599$	−5.71161	−0.233370	−0.116685	+	0.993169i	$0.537227\pi$
$600$	0	0
$601$	−17.3253	−0.706715	−0.353357	−	0.935488i	$-0.614960\pi$
$601$	−17.3253	−0.706715	−0.353357	+	0.935488i	$0.614960\pi$
$602$	0	0
$603$	−23.4338	−0.954298
$604$	0	0
$605$	−14.0000	−0.569181
$606$	0	0
$607$	−20.3753	−0.827006	−0.413503	−	0.910503i	$-0.635695\pi$
$607$	−20.3753	−0.827006	−0.413503	+	0.910503i	$0.635695\pi$
$608$	0	0
$609$	20.0999	0.814487
$610$	0	0
$611$	20.6084	0.833728
$612$	0	0
$613$	3.25561	0.131493	0.0657464	−	0.997836i	$-0.479057\pi$
$613$	3.25561	0.131493	0.0657464	+	0.997836i	$0.479057\pi$
$614$	0	0
$615$	5.13770	0.207172
$616$	0	0
$617$	−13.7970	−0.555445	−0.277722	−	0.960661i	$-0.589580\pi$
$617$	−13.7970	−0.555445	−0.277722	+	0.960661i	$0.589580\pi$
$618$	0	0
$619$	−36.2600	−1.45741	−0.728706	−	0.684827i	$-0.759878\pi$
$619$	−36.2600	−1.45741	−0.728706	+	0.684827i	$0.759878\pi$
$620$	0	0
$621$	−10.1076	−0.405602
$622$	0	0
$623$	16.9545	0.679266
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	10.7263	0.428369
$628$	0	0
$629$	−13.8150	−0.550841
$630$	0	0
$631$	39.0543	1.55473	0.777364	−	0.629051i	$-0.216556\pi$
$631$	39.0543	1.55473	0.777364	+	0.629051i	$0.216556\pi$
$632$	0	0
$633$	19.3528	0.769206
$634$	0	0
$635$	40.8865	1.62253
$636$	0	0
$637$	2.34338	0.0928481
$638$	0	0
$639$	53.8486	2.13022
$640$	0	0
$641$	−3.15311	−0.124540	−0.0622701	−	0.998059i	$-0.519834\pi$
$641$	−3.15311	−0.124540	−0.0622701	+	0.998059i	$0.519834\pi$
$642$	0	0
$643$	34.3521	1.35472	0.677358	−	0.735653i	$-0.263125\pi$
$643$	34.3521	1.35472	0.677358	+	0.735653i	$0.263125\pi$
$644$	0	0
$645$	36.5301	1.43837
$646$	0	0
$647$	−37.4681	−1.47302	−0.736511	−	0.676425i	$-0.763528\pi$
$647$	−37.4681	−1.47302	−0.736511	+	0.676425i	$0.763528\pi$
$648$	0	0
$649$	8.00000	0.314027
$650$	0	0
$651$	−9.82446	−0.385051
$652$	0	0
$653$	43.9089	1.71829	0.859144	−	0.511734i	$-0.170997\pi$
$653$	43.9089	1.71829	0.859144	+	0.511734i	$0.170997\pi$
$654$	0	0
$655$	−1.92432	−0.0751892
$656$	0	0
$657$	−30.4148	−1.18660
$658$	0	0
$659$	−11.1531	−0.434463	−0.217232	−	0.976120i	$-0.569703\pi$
$659$	−11.1531	−0.434463	−0.217232	+	0.976120i	$0.569703\pi$
$660$	0	0
$661$	−20.7866	−0.808506	−0.404253	−	0.914647i	$-0.632468\pi$
$661$	−20.7866	−0.808506	−0.404253	+	0.914647i	$0.632468\pi$
$662$	0	0
$663$	19.4167	0.754080
$664$	0	0
$665$	4.17554	0.161921
$666$	0	0
$667$	51.3977	1.99013
$668$	0	0
$669$	−11.7435	−0.454029
$670$	0	0
$671$	−9.02243	−0.348307
$672$	0	0
$673$	30.1548	1.16238	0.581192	−	0.813767i	$-0.302587\pi$
$673$	30.1548	1.16238	0.581192	+	0.813767i	$0.302587\pi$
$674$	0	0
$675$	1.53871	0.0592250
$676$	0	0
$677$	21.1826	0.814112	0.407056	−	0.913403i	$-0.366555\pi$
$677$	21.1826	0.814112	0.407056	+	0.913403i	$0.366555\pi$
$678$	0	0
$679$	−16.3933	−0.629117
$680$	0	0
$681$	−0.901880	−0.0345601
$682$	0	0
$683$	49.7435	1.90338	0.951691	−	0.307058i	$-0.0993446\pi$
$683$	49.7435	1.90338	0.951691	+	0.307058i	$0.0993446\pi$
$684$	0	0
$685$	6.74704	0.257791
$686$	0	0
$687$	17.7367	0.676697
$688$	0	0
$689$	−5.88474	−0.224191
$690$	0	0
$691$	−29.8168	−1.13428	−0.567141	−	0.823620i	$-0.691951\pi$
$691$	−29.8168	−1.13428	−0.567141	+	0.823620i	$0.691951\pi$
$692$	0	0
$693$	−7.19798	−0.273429
$694$	0	0
$695$	−29.5732	−1.12178
$696$	0	0
$697$	3.22547	0.122173
$698$	0	0
$699$	−74.5147	−2.81840
$700$	0	0
$701$	−41.1696	−1.55495	−0.777477	−	0.628912i	$-0.783501\pi$
$701$	−41.1696	−1.55495	−0.777477	+	0.628912i	$0.783501\pi$
$702$	0	0
$703$	−8.94214	−0.337259
$704$	0	0
$705$	−45.1826	−1.70167
$706$	0	0
$707$	−3.37087	−0.126775
$708$	0	0
$709$	−31.8794	−1.19726	−0.598629	−	0.801027i	$-0.704288\pi$
$709$	−31.8794	−1.19726	−0.598629	+	0.801027i	$0.704288\pi$
$710$	0	0
$711$	4.09457	0.153558
$712$	0	0
$713$	−25.1223	−0.940837
$714$	0	0
$715$	9.37352	0.350550
$716$	0	0
$717$	−73.2015	−2.73376
$718$	0	0
$719$	−43.2978	−1.61474	−0.807368	−	0.590048i	$-0.799109\pi$
$719$	−43.2978	−1.61474	−0.807368	+	0.590048i	$0.799109\pi$
$720$	0	0
$721$	2.35108	0.0875589
$722$	0	0
$723$	−53.6938	−1.99689
$724$	0	0
$725$	−7.82446	−0.290593
$726$	0	0
$727$	−13.9672	−0.518015	−0.259008	−	0.965875i	$-0.583395\pi$
$727$	−13.9672	−0.518015	−0.259008	+	0.965875i	$0.583395\pi$
$728$	0	0
$729$	−36.4905	−1.35150
$730$	0	0
$731$	22.9338	0.848236
$732$	0	0
$733$	−3.38893	−0.125173	−0.0625864	−	0.998040i	$-0.519935\pi$
$733$	−3.38893	−0.125173	−0.0625864	+	0.998040i	$0.519935\pi$
$734$	0	0
$735$	−5.13770	−0.189507
$736$	0	0
$737$	−13.0224	−0.479688
$738$	0	0
$739$	−8.33809	−0.306722	−0.153361	−	0.988170i	$-0.549010\pi$
$739$	−8.33809	−0.306722	−0.153361	+	0.988170i	$0.549010\pi$
$740$	0	0
$741$	12.5679	0.461695
$742$	0	0
$743$	42.7833	1.56957	0.784783	−	0.619770i	$-0.212774\pi$
$743$	42.7833	1.56957	0.784783	+	0.619770i	$0.212774\pi$
$744$	0	0
$745$	3.52836	0.129269
$746$	0	0
$747$	34.7263	1.27057
$748$	0	0
$749$	16.0121	0.585069
$750$	0	0
$751$	−12.0396	−0.439330	−0.219665	−	0.975575i	$-0.570497\pi$
$751$	−12.0396	−0.439330	−0.219665	+	0.975575i	$0.570497\pi$
$752$	0	0
$753$	14.0603	0.512385
$754$	0	0
$755$	3.09812	0.112752
$756$	0	0
$757$	2.35108	0.0854516	0.0427258	−	0.999087i	$-0.486396\pi$
$757$	2.35108	0.0854516	0.0427258	+	0.999087i	$0.486396\pi$
$758$	0	0
$759$	−33.7488	−1.22500
$760$	0	0
$761$	35.6938	1.29390	0.646949	−	0.762533i	$-0.276044\pi$
$761$	35.6938	1.29390	0.646949	+	0.762533i	$0.276044\pi$
$762$	0	0
$763$	5.70919	0.206687
$764$	0	0
$765$	−23.2169	−0.839407
$766$	0	0
$767$	9.37352	0.338458
$768$	0	0
$769$	−2.00000	−0.0721218	−0.0360609	−	0.999350i	$-0.511481\pi$
$769$	−2.00000	−0.0721218	−0.0360609	+	0.999350i	$0.511481\pi$
$770$	0	0
$771$	−51.4855	−1.85420
$772$	0	0
$773$	25.9069	0.931808	0.465904	−	0.884835i	$-0.345729\pi$
$773$	25.9069	0.931808	0.465904	+	0.884835i	$0.345729\pi$
$774$	0	0
$775$	3.82446	0.137379
$776$	0	0
$777$	11.0026	0.394718
$778$	0	0
$779$	2.08777	0.0748022
$780$	0	0
$781$	29.9243	1.07078
$782$	0	0
$783$	−12.0396	−0.430259
$784$	0	0
$785$	−27.7039	−0.988795
$786$	0	0
$787$	−8.11526	−0.289278	−0.144639	−	0.989484i	$-0.546202\pi$
$787$	−8.11526	−0.289278	−0.144639	+	0.989484i	$0.546202\pi$
$788$	0	0
$789$	−21.5544	−0.767355
$790$	0	0
$791$	−17.5310	−0.623331
$792$	0	0
$793$	−10.5715	−0.375405
$794$	0	0
$795$	12.9019	0.457583
$796$	0	0
$797$	−46.4302	−1.64464	−0.822322	−	0.569023i	$-0.807322\pi$
$797$	−46.4302	−1.64464	−0.822322	+	0.569023i	$0.807322\pi$
$798$	0	0
$799$	−28.3658	−1.00351
$800$	0	0
$801$	−61.0189	−2.15600
$802$	0	0
$803$	−16.9019	−0.596454
$804$	0	0
$805$	−13.1377	−0.463043
$806$	0	0
$807$	−59.5525	−2.09635
$808$	0	0
$809$	28.4958	1.00186	0.500930	−	0.865488i	$-0.332992\pi$
$809$	28.4958	1.00186	0.500930	+	0.865488i	$0.332992\pi$
$810$	0	0
$811$	14.1755	0.497771	0.248885	−	0.968533i	$-0.419936\pi$
$811$	14.1755	0.497771	0.248885	+	0.968533i	$0.419936\pi$
$812$	0	0
$813$	−52.3805	−1.83707
$814$	0	0
$815$	48.7558	1.70784
$816$	0	0
$817$	14.8445	0.519343
$818$	0	0
$819$	−8.43380	−0.294701
$820$	0	0
$821$	−3.92764	−0.137075	−0.0685377	−	0.997649i	$-0.521833\pi$
$821$	−3.92764	−0.137075	−0.0685377	+	0.997649i	$0.521833\pi$
$822$	0	0
$823$	12.5662	0.438030	0.219015	−	0.975721i	$-0.429716\pi$
$823$	12.5662	0.438030	0.219015	+	0.975721i	$0.429716\pi$
$824$	0	0
$825$	5.13770	0.178872
$826$	0	0
$827$	13.5337	0.470611	0.235306	−	0.971921i	$-0.424391\pi$
$827$	13.5337	0.470611	0.235306	+	0.971921i	$0.424391\pi$
$828$	0	0
$829$	−46.3753	−1.61068	−0.805340	−	0.592814i	$-0.798017\pi$
$829$	−46.3753	−1.61068	−0.805340	+	0.592814i	$0.798017\pi$
$830$	0	0
$831$	−47.6610	−1.65334
$832$	0	0
$833$	−3.22547	−0.111756
$834$	0	0
$835$	−24.3753	−0.843540
$836$	0	0
$837$	5.88474	0.203406
$838$	0	0
$839$	−39.8218	−1.37480	−0.687401	−	0.726278i	$-0.741248\pi$
$839$	−39.8218	−1.37480	−0.687401	+	0.726278i	$0.741248\pi$
$840$	0	0
$841$	32.2221	1.11111
$842$	0	0
$843$	40.8055	1.40542
$844$	0	0
$845$	−15.0171	−0.516605
$846$	0	0
$847$	7.00000	0.240523
$848$	0	0
$849$	52.9863	1.81849
$850$	0	0
$851$	28.1351	0.964457
$852$	0	0
$853$	−27.8245	−0.952691	−0.476346	−	0.879258i	$-0.658039\pi$
$853$	−27.8245	−0.952691	−0.476346	+	0.879258i	$0.658039\pi$
$854$	0	0
$855$	−15.0277	−0.513937
$856$	0	0
$857$	9.09812	0.310786	0.155393	−	0.987853i	$-0.450336\pi$
$857$	9.09812	0.310786	0.155393	+	0.987853i	$0.450336\pi$
$858$	0	0
$859$	−6.55080	−0.223510	−0.111755	−	0.993736i	$-0.535647\pi$
$859$	−6.55080	−0.223510	−0.111755	+	0.993736i	$0.535647\pi$
$860$	0	0
$861$	−2.56885	−0.0875462
$862$	0	0
$863$	−2.04487	−0.0696082	−0.0348041	−	0.999394i	$-0.511081\pi$
$863$	−2.04487	−0.0696082	−0.0348041	+	0.999394i	$0.511081\pi$
$864$	0	0
$865$	18.6265	0.633319
$866$	0	0
$867$	16.9450	0.575483
$868$	0	0
$869$	2.27540	0.0771876
$870$	0	0
$871$	−15.2583	−0.517006
$872$	0	0
$873$	58.9993	1.99682
$874$	0	0
$875$	12.0000	0.405674
$876$	0	0
$877$	−36.3625	−1.22787	−0.613937	−	0.789355i	$-0.710415\pi$
$877$	−36.3625	−1.22787	−0.613937	+	0.789355i	$0.710415\pi$
$878$	0	0
$879$	−64.0844	−2.16151
$880$	0	0
$881$	−11.9344	−0.402081	−0.201041	−	0.979583i	$-0.564432\pi$
$881$	−11.9344	−0.402081	−0.201041	+	0.979583i	$0.564432\pi$
$882$	0	0
$883$	−27.6643	−0.930979	−0.465489	−	0.885053i	$-0.654122\pi$
$883$	−27.6643	−0.930979	−0.465489	+	0.885053i	$0.654122\pi$
$884$	0	0
$885$	−20.5508	−0.690807
$886$	0	0
$887$	−21.4811	−0.721264	−0.360632	−	0.932708i	$-0.617439\pi$
$887$	−21.4811	−0.721264	−0.360632	+	0.932708i	$0.617439\pi$
$888$	0	0
$889$	−20.4432	−0.685644
$890$	0	0
$891$	−13.6885	−0.458582
$892$	0	0
$893$	−18.3605	−0.614412
$894$	0	0
$895$	−24.5508	−0.820643
$896$	0	0
$897$	−39.5431	−1.32031
$898$	0	0
$899$	−29.9243	−0.998032
$900$	0	0
$901$	8.09986	0.269845
$902$	0	0
$903$	−18.2650	−0.607823
$904$	0	0
$905$	27.5337	0.915250
$906$	0	0
$907$	−14.3676	−0.477067	−0.238533	−	0.971134i	$-0.576667\pi$
$907$	−14.3676	−0.477067	−0.238533	+	0.971134i	$0.576667\pi$
$908$	0	0
$909$	12.1317	0.402384
$910$	0	0
$911$	−21.6161	−0.716174	−0.358087	−	0.933688i	$-0.616571\pi$
$911$	−21.6161	−0.716174	−0.358087	+	0.933688i	$0.616571\pi$
$912$	0	0
$913$	19.2978	0.638665
$914$	0	0
$915$	23.1773	0.766217
$916$	0	0
$917$	0.962158	0.0317732
$918$	0	0
$919$	27.4888	0.906771	0.453386	−	0.891314i	$-0.350216\pi$
$919$	27.4888	0.906771	0.453386	+	0.891314i	$0.350216\pi$
$920$	0	0
$921$	−26.2547	−0.865122
$922$	0	0
$923$	35.0620	1.15408
$924$	0	0
$925$	−4.28310	−0.140828
$926$	0	0
$927$	−8.46152	−0.277913
$928$	0	0
$929$	10.3176	0.338510	0.169255	−	0.985572i	$-0.445864\pi$
$929$	10.3176	0.338510	0.169255	+	0.985572i	$0.445864\pi$
$930$	0	0
$931$	−2.08777	−0.0684239
$932$	0	0
$933$	−42.4855	−1.39091
$934$	0	0
$935$	−12.9019	−0.421937
$936$	0	0
$937$	−32.8031	−1.07163	−0.535815	−	0.844335i	$-0.679996\pi$
$937$	−32.8031	−1.07163	−0.535815	+	0.844335i	$0.679996\pi$
$938$	0	0
$939$	−42.3306	−1.38141
$940$	0	0
$941$	−13.0070	−0.424017	−0.212008	−	0.977268i	$-0.568000\pi$
$941$	−13.0070	−0.424017	−0.212008	+	0.977268i	$0.568000\pi$
$942$	0	0
$943$	−6.56885	−0.213911
$944$	0	0
$945$	3.07742	0.100109
$946$	0	0
$947$	16.4985	0.536128	0.268064	−	0.963401i	$-0.413616\pi$
$947$	16.4985	0.536128	0.268064	+	0.963401i	$0.413616\pi$
$948$	0	0
$949$	−19.8038	−0.642857
$950$	0	0
$951$	−33.1430	−1.07474
$952$	0	0
$953$	−13.6258	−0.441383	−0.220692	−	0.975344i	$-0.570831\pi$
$953$	−13.6258	−0.441383	−0.220692	+	0.975344i	$0.570831\pi$
$954$	0	0
$955$	18.6265	0.602739
$956$	0	0
$957$	−40.1997	−1.29947
$958$	0	0
$959$	−3.37352	−0.108937
$960$	0	0
$961$	−16.3735	−0.528178
$962$	0	0
$963$	−57.6273	−1.85701
$964$	0	0
$965$	14.6265	0.470843
$966$	0	0
$967$	17.5182	0.563349	0.281674	−	0.959510i	$-0.409110\pi$
$967$	17.5182	0.563349	0.281674	+	0.959510i	$0.409110\pi$
$968$	0	0
$969$	−17.2987	−0.555716
$970$	0	0
$971$	−59.6852	−1.91539	−0.957694	−	0.287788i	$-0.907080\pi$
$971$	−59.6852	−1.91539	−0.957694	+	0.287788i	$0.907080\pi$
$972$	0	0
$973$	14.7866	0.474037
$974$	0	0
$975$	6.01979	0.192788
$976$	0	0
$977$	−54.0295	−1.72856	−0.864278	−	0.503015i	$-0.832224\pi$
$977$	−54.0295	−1.72856	−0.864278	+	0.503015i	$0.832224\pi$
$978$	0	0
$979$	−33.9089	−1.08373
$980$	0	0
$981$	−20.5473	−0.656026
$982$	0	0
$983$	37.5337	1.19714	0.598569	−	0.801071i	$-0.295736\pi$
$983$	37.5337	1.19714	0.598569	+	0.801071i	$0.295736\pi$
$984$	0	0
$985$	32.8469	1.04659
$986$	0	0
$987$	22.5913	0.719089
$988$	0	0
$989$	−46.7059	−1.48516
$990$	0	0
$991$	−12.0242	−0.381960	−0.190980	−	0.981594i	$-0.561167\pi$
$991$	−12.0242	−0.381960	−0.190980	+	0.981594i	$0.561167\pi$
$992$	0	0
$993$	56.5147	1.79344
$994$	0	0
$995$	18.2600	0.578881
$996$	0	0
$997$	52.0114	1.64722	0.823609	−	0.567158i	$-0.191957\pi$
$997$	52.0114	1.64722	0.823609	+	0.567158i	$0.191957\pi$
$998$	0	0
$999$	−6.59046	−0.208513

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4592.2.a.t.1.1		3
4.3	odd	2		287.2.a.c.1.1	✓	3
12.11	even	2		2583.2.a.m.1.3		3
20.19	odd	2		7175.2.a.k.1.3		3
28.27	even	2		2009.2.a.j.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.a.c.1.1	✓	3	4.3	odd	2
2009.2.a.j.1.1		3	28.27	even	2
2583.2.a.m.1.3		3	12.11	even	2
4592.2.a.t.1.1		3	1.1	even	1	trivial
7175.2.a.k.1.3		3	20.19	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:	\( N \)	\(=\)	\( 4592 = 2^{4} \cdot 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4592.a (trivial)

\(f(q)\)	\(=\)	\(q-2.56885 q^{3} +2.00000 q^{5} -1.00000 q^{7} +3.59899 q^{9} +O(q^{10})\)	\(q-2.56885 q^{3} +2.00000 q^{5} -1.00000 q^{7} +3.59899 q^{9} +2.00000 q^{11} +2.34338 q^{13} -5.13770 q^{15} -3.22547 q^{17} -2.08777 q^{19} +2.56885 q^{21} +6.56885 q^{23} -1.00000 q^{25} -1.53871 q^{27} +7.82446 q^{29} -3.82446 q^{31} -5.13770 q^{33} -2.00000 q^{35} +4.28310 q^{37} -6.01979 q^{39} -1.00000 q^{41} -7.11021 q^{43} +7.19798 q^{45} +8.79432 q^{47} +1.00000 q^{49} +8.28575 q^{51} -2.51122 q^{53} +4.00000 q^{55} +5.36317 q^{57} +4.00000 q^{59} -4.51122 q^{61} -3.59899 q^{63} +4.68676 q^{65} -6.51122 q^{67} -16.8744 q^{69} +14.9622 q^{71} -8.45094 q^{73} +2.56885 q^{75} -2.00000 q^{77} +1.13770 q^{79} -6.84425 q^{81} +9.64892 q^{83} -6.45094 q^{85} -20.0999 q^{87} -16.9545 q^{89} -2.34338 q^{91} +9.82446 q^{93} -4.17554 q^{95} +16.3933 q^{97} +7.19798 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{3} + 6 q^{5} - 3 q^{7} + 8 q^{9}+O(q^{10}) \) 3 * q + q^3 + 6 * q^5 - 3 * q^7 + 8 * q^9	\( 3 q + q^{3} + 6 q^{5} - 3 q^{7} + 8 q^{9} + 6 q^{11} + 9 q^{13} + 2 q^{15} + q^{17} - 13 q^{19} - q^{21} + 11 q^{23} - 3 q^{25} + 10 q^{27} + 10 q^{29} + 2 q^{31} + 2 q^{33} - 6 q^{35} + 3 q^{37} + 12 q^{39} - 3 q^{41} - 9 q^{43} + 16 q^{45} + 7 q^{47} + 3 q^{49} + 26 q^{51} + 2 q^{53} + 12 q^{55} - 12 q^{57} + 12 q^{59} - 4 q^{61} - 8 q^{63} + 18 q^{65} - 10 q^{67} - 13 q^{69} + 14 q^{71} - 4 q^{73} - q^{75} - 6 q^{77} - 14 q^{79} + 23 q^{81} + 2 q^{83} + 2 q^{85} - 12 q^{87} + 5 q^{89} - 9 q^{91} + 16 q^{93} - 26 q^{95} + 27 q^{97} + 16 q^{99}+O(q^{100}) \) 3 * q + q^3 + 6 * q^5 - 3 * q^7 + 8 * q^9 + 6 * q^11 + 9 * q^13 + 2 * q^15 + q^17 - 13 * q^19 - q^21 + 11 * q^23 - 3 * q^25 + 10 * q^27 + 10 * q^29 + 2 * q^31 + 2 * q^33 - 6 * q^35 + 3 * q^37 + 12 * q^39 - 3 * q^41 - 9 * q^43 + 16 * q^45 + 7 * q^47 + 3 * q^49 + 26 * q^51 + 2 * q^53 + 12 * q^55 - 12 * q^57 + 12 * q^59 - 4 * q^61 - 8 * q^63 + 18 * q^65 - 10 * q^67 - 13 * q^69 + 14 * q^71 - 4 * q^73 - q^75 - 6 * q^77 - 14 * q^79 + 23 * q^81 + 2 * q^83 + 2 * q^85 - 12 * q^87 + 5 * q^89 - 9 * q^91 + 16 * q^93 - 26 * q^95 + 27 * q^97 + 16 * q^99

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