LMFDB - Embedded newform 4368.2.a.bp.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4368,2,Mod(1,4368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4368, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4368.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4368.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:	$34.8786556029$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 2184)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.470683$ of defining polynomial
Character	$\chi$	$=$	4368.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +3.77846 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +3.77846 q^{5} -1.00000 q^{7} +1.00000 q^{9} +1.77846 q^{11} +1.00000 q^{13} -3.77846 q^{15} +4.71982 q^{17} +2.71982 q^{19} +1.00000 q^{21} +2.71982 q^{23} +9.27674 q^{25} -1.00000 q^{27} -0.719824 q^{29} -1.88273 q^{31} -1.77846 q^{33} -3.77846 q^{35} +2.83709 q^{37} -1.00000 q^{39} +1.05863 q^{41} -0.837090 q^{43} +3.77846 q^{45} -10.3810 q^{47} +1.00000 q^{49} -4.71982 q^{51} -9.11383 q^{53} +6.71982 q^{55} -2.71982 q^{57} +6.38101 q^{59} +1.16291 q^{61} -1.00000 q^{63} +3.77846 q^{65} +5.67418 q^{67} -2.71982 q^{69} +2.61555 q^{71} +13.9509 q^{73} -9.27674 q^{75} -1.77846 q^{77} +3.55691 q^{79} +1.00000 q^{81} -8.05520 q^{83} +17.8337 q^{85} +0.719824 q^{87} -2.49828 q^{89} -1.00000 q^{91} +1.88273 q^{93} +10.2767 q^{95} -13.1138 q^{97} +1.77846 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 3 q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 + 3 * q^5 - 3 * q^7 + 3 * q^9	$ 3 q - 3 q^{3} + 3 q^{5} - 3 q^{7} + 3 q^{9} - 3 q^{11} + 3 q^{13} - 3 q^{15} + 5 q^{17} - q^{19} + 3 q^{21} - q^{23} + 2 q^{25} - 3 q^{27} + 7 q^{29} - 4 q^{31} + 3 q^{33} - 3 q^{35} + q^{37} - 3 q^{39} + 4 q^{41} + 5 q^{43} + 3 q^{45} - 12 q^{47} + 3 q^{49} - 5 q^{51} + 6 q^{53} + 11 q^{55} + q^{57} + 11 q^{61} - 3 q^{63} + 3 q^{65} + 2 q^{67} + q^{69} - 8 q^{71} + q^{73} - 2 q^{75} + 3 q^{77} - 6 q^{79} + 3 q^{81} + 10 q^{83} + 11 q^{85} - 7 q^{87} + 10 q^{89} - 3 q^{91} + 4 q^{93} + 5 q^{95} - 6 q^{97} - 3 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 + 3 * q^5 - 3 * q^7 + 3 * q^9 - 3 * q^11 + 3 * q^13 - 3 * q^15 + 5 * q^17 - q^19 + 3 * q^21 - q^23 + 2 * q^25 - 3 * q^27 + 7 * q^29 - 4 * q^31 + 3 * q^33 - 3 * q^35 + q^37 - 3 * q^39 + 4 * q^41 + 5 * q^43 + 3 * q^45 - 12 * q^47 + 3 * q^49 - 5 * q^51 + 6 * q^53 + 11 * q^55 + q^57 + 11 * q^61 - 3 * q^63 + 3 * q^65 + 2 * q^67 + q^69 - 8 * q^71 + q^73 - 2 * q^75 + 3 * q^77 - 6 * q^79 + 3 * q^81 + 10 * q^83 + 11 * q^85 - 7 * q^87 + 10 * q^89 - 3 * q^91 + 4 * q^93 + 5 * q^95 - 6 * q^97 - 3 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	3.77846	1.68978	0.844889	−	0.534942i	$-0.179667\pi$
$5$	3.77846	1.68978	0.844889	+	0.534942i	$0.179667\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.77846	0.536225	0.268112	−	0.963388i	$-0.413600\pi$
$11$	1.77846	0.536225	0.268112	+	0.963388i	$0.413600\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−3.77846	−0.975593
$16$	0	0
$17$	4.71982	1.14473	0.572363	−	0.820001i	$-0.306027\pi$
$17$	4.71982	1.14473	0.572363	+	0.820001i	$0.306027\pi$
$18$	0	0
$19$	2.71982	0.623970	0.311985	−	0.950087i	$-0.399006\pi$
$19$	2.71982	0.623970	0.311985	+	0.950087i	$0.399006\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	2.71982	0.567122	0.283561	−	0.958954i	$-0.408484\pi$
$23$	2.71982	0.567122	0.283561	+	0.958954i	$0.408484\pi$
$24$	0	0
$25$	9.27674	1.85535
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−0.719824	−0.133668	−0.0668340	−	0.997764i	$-0.521290\pi$
$29$	−0.719824	−0.133668	−0.0668340	+	0.997764i	$0.521290\pi$
$30$	0	0
$31$	−1.88273	−0.338149	−0.169074	−	0.985603i	$-0.554078\pi$
$31$	−1.88273	−0.338149	−0.169074	+	0.985603i	$0.554078\pi$
$32$	0	0
$33$	−1.77846	−0.309590
$34$	0	0
$35$	−3.77846	−0.638676
$36$	0	0
$37$	2.83709	0.466415	0.233207	−	0.972427i	$-0.425078\pi$
$37$	2.83709	0.466415	0.233207	+	0.972427i	$0.425078\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	1.05863	0.165331	0.0826654	−	0.996577i	$-0.473657\pi$
$41$	1.05863	0.165331	0.0826654	+	0.996577i	$0.473657\pi$
$42$	0	0
$43$	−0.837090	−0.127655	−0.0638275	−	0.997961i	$-0.520331\pi$
$43$	−0.837090	−0.127655	−0.0638275	+	0.997961i	$0.520331\pi$
$44$	0	0
$45$	3.77846	0.563259
$46$	0	0
$47$	−10.3810	−1.51423	−0.757113	−	0.653284i	$-0.773391\pi$
$47$	−10.3810	−1.51423	−0.757113	+	0.653284i	$0.773391\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−4.71982	−0.660908
$52$	0	0
$53$	−9.11383	−1.25188	−0.625940	−	0.779871i	$-0.715285\pi$
$53$	−9.11383	−1.25188	−0.625940	+	0.779871i	$0.715285\pi$
$54$	0	0
$55$	6.71982	0.906101
$56$	0	0
$57$	−2.71982	−0.360249
$58$	0	0
$59$	6.38101	0.830737	0.415369	−	0.909653i	$-0.363653\pi$
$59$	6.38101	0.830737	0.415369	+	0.909653i	$0.363653\pi$
$60$	0	0
$61$	1.16291	0.148895	0.0744477	−	0.997225i	$-0.476281\pi$
$61$	1.16291	0.148895	0.0744477	+	0.997225i	$0.476281\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	3.77846	0.468660
$66$	0	0
$67$	5.67418	0.693211	0.346606	−	0.938011i	$-0.387334\pi$
$67$	5.67418	0.693211	0.346606	+	0.938011i	$0.387334\pi$
$68$	0	0
$69$	−2.71982	−0.327428
$70$	0	0
$71$	2.61555	0.310408	0.155204	−	0.987882i	$-0.450396\pi$
$71$	2.61555	0.310408	0.155204	+	0.987882i	$0.450396\pi$
$72$	0	0
$73$	13.9509	1.63283	0.816416	−	0.577465i	$-0.195958\pi$
$73$	13.9509	1.63283	0.816416	+	0.577465i	$0.195958\pi$
$74$	0	0
$75$	−9.27674	−1.07119
$76$	0	0
$77$	−1.77846	−0.202674
$78$	0	0
$79$	3.55691	0.400184	0.200092	−	0.979777i	$-0.435876\pi$
$79$	3.55691	0.400184	0.200092	+	0.979777i	$0.435876\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−8.05520	−0.884173	−0.442086	−	0.896973i	$-0.645761\pi$
$83$	−8.05520	−0.884173	−0.442086	+	0.896973i	$0.645761\pi$
$84$	0	0
$85$	17.8337	1.93433
$86$	0	0
$87$	0.719824	0.0771732
$88$	0	0
$89$	−2.49828	−0.264817	−0.132409	−	0.991195i	$-0.542271\pi$
$89$	−2.49828	−0.264817	−0.132409	+	0.991195i	$0.542271\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	1.88273	0.195230
$94$	0	0
$95$	10.2767	1.05437
$96$	0	0
$97$	−13.1138	−1.33151	−0.665754	−	0.746172i	$-0.731890\pi$
$97$	−13.1138	−1.33151	−0.665754	+	0.746172i	$0.731890\pi$
$98$	0	0
$99$	1.77846	0.178742
$100$	0	0
$101$	0.325819	0.0324202	0.0162101	−	0.999869i	$-0.494840\pi$
$101$	0.325819	0.0324202	0.0162101	+	0.999869i	$0.494840\pi$
$102$	0	0
$103$	12.1595	1.19811	0.599054	−	0.800708i	$-0.295543\pi$
$103$	12.1595	1.19811	0.599054	+	0.800708i	$0.295543\pi$
$104$	0	0
$105$	3.77846	0.368740
$106$	0	0
$107$	18.6707	1.80497	0.902484	−	0.430723i	$-0.141741\pi$
$107$	18.6707	1.80497	0.902484	+	0.430723i	$0.141741\pi$
$108$	0	0
$109$	−15.8337	−1.51659	−0.758294	−	0.651912i	$-0.773967\pi$
$109$	−15.8337	−1.51659	−0.758294	+	0.651912i	$0.773967\pi$
$110$	0	0
$111$	−2.83709	−0.269285
$112$	0	0
$113$	11.8827	1.11783	0.558917	−	0.829224i	$-0.311217\pi$
$113$	11.8827	1.11783	0.558917	+	0.829224i	$0.311217\pi$
$114$	0	0
$115$	10.2767	0.958311
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	−4.71982	−0.432666
$120$	0	0
$121$	−7.83709	−0.712463
$122$	0	0
$123$	−1.05863	−0.0954537
$124$	0	0
$125$	16.1595	1.44535
$126$	0	0
$127$	−3.55691	−0.315625	−0.157813	−	0.987469i	$-0.550444\pi$
$127$	−3.55691	−0.315625	−0.157813	+	0.987469i	$0.550444\pi$
$128$	0	0
$129$	0.837090	0.0737017
$130$	0	0
$131$	−19.5078	−1.70441	−0.852204	−	0.523210i	$-0.824734\pi$
$131$	−19.5078	−1.70441	−0.852204	+	0.523210i	$0.824734\pi$
$132$	0	0
$133$	−2.71982	−0.235839
$134$	0	0
$135$	−3.77846	−0.325198
$136$	0	0
$137$	0.221543	0.0189277	0.00946384	−	0.999955i	$-0.496988\pi$
$137$	0.221543	0.0189277	0.00946384	+	0.999955i	$0.496988\pi$
$138$	0	0
$139$	−4.99656	−0.423803	−0.211901	−	0.977291i	$-0.567966\pi$
$139$	−4.99656	−0.423803	−0.211901	+	0.977291i	$0.567966\pi$
$140$	0	0
$141$	10.3810	0.874239
$142$	0	0
$143$	1.77846	0.148722
$144$	0	0
$145$	−2.71982	−0.225869
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	−21.1690	−1.73423	−0.867117	−	0.498105i	$-0.834030\pi$
$149$	−21.1690	−1.73423	−0.867117	+	0.498105i	$0.834030\pi$
$150$	0	0
$151$	17.5991	1.43220	0.716098	−	0.698000i	$-0.245926\pi$
$151$	17.5991	1.43220	0.716098	+	0.698000i	$0.245926\pi$
$152$	0	0
$153$	4.71982	0.381575
$154$	0	0
$155$	−7.11383	−0.571396
$156$	0	0
$157$	12.5113	0.998508	0.499254	−	0.866456i	$-0.333607\pi$
$157$	12.5113	0.998508	0.499254	+	0.866456i	$0.333607\pi$
$158$	0	0
$159$	9.11383	0.722774
$160$	0	0
$161$	−2.71982	−0.214352
$162$	0	0
$163$	−18.4362	−1.44404	−0.722018	−	0.691875i	$-0.756785\pi$
$163$	−18.4362	−1.44404	−0.722018	+	0.691875i	$0.756785\pi$
$164$	0	0
$165$	−6.71982	−0.523138
$166$	0	0
$167$	−7.21811	−0.558554	−0.279277	−	0.960211i	$-0.590095\pi$
$167$	−7.21811	−0.558554	−0.279277	+	0.960211i	$0.590095\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	2.71982	0.207990
$172$	0	0
$173$	−1.76547	−0.134226	−0.0671130	−	0.997745i	$-0.521379\pi$
$173$	−1.76547	−0.134226	−0.0671130	+	0.997745i	$0.521379\pi$
$174$	0	0
$175$	−9.27674	−0.701255
$176$	0	0
$177$	−6.38101	−0.479626
$178$	0	0
$179$	8.00000	0.597948	0.298974	−	0.954261i	$-0.403356\pi$
$179$	8.00000	0.597948	0.298974	+	0.954261i	$0.403356\pi$
$180$	0	0
$181$	5.11383	0.380108	0.190054	−	0.981774i	$-0.439134\pi$
$181$	5.11383	0.380108	0.190054	+	0.981774i	$0.439134\pi$
$182$	0	0
$183$	−1.16291	−0.0859648
$184$	0	0
$185$	10.7198	0.788137
$186$	0	0
$187$	8.39400	0.613830
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	−10.5113	−0.760569	−0.380284	−	0.924870i	$-0.624174\pi$
$191$	−10.5113	−0.760569	−0.380284	+	0.924870i	$0.624174\pi$
$192$	0	0
$193$	−7.67418	−0.552400	−0.276200	−	0.961100i	$-0.589075\pi$
$193$	−7.67418	−0.552400	−0.276200	+	0.961100i	$0.589075\pi$
$194$	0	0
$195$	−3.77846	−0.270581
$196$	0	0
$197$	22.2897	1.58808	0.794039	−	0.607867i	$-0.207975\pi$
$197$	22.2897	1.58808	0.794039	+	0.607867i	$0.207975\pi$
$198$	0	0
$199$	−6.04221	−0.428321	−0.214160	−	0.976799i	$-0.568701\pi$
$199$	−6.04221	−0.428321	−0.214160	+	0.976799i	$0.568701\pi$
$200$	0	0
$201$	−5.67418	−0.400226
$202$	0	0
$203$	0.719824	0.0505217
$204$	0	0
$205$	4.00000	0.279372
$206$	0	0
$207$	2.71982	0.189041
$208$	0	0
$209$	4.83709	0.334589
$210$	0	0
$211$	14.9544	1.02950	0.514750	−	0.857340i	$-0.327885\pi$
$211$	14.9544	1.02950	0.514750	+	0.857340i	$0.327885\pi$
$212$	0	0
$213$	−2.61555	−0.179214
$214$	0	0
$215$	−3.16291	−0.215709
$216$	0	0
$217$	1.88273	0.127808
$218$	0	0
$219$	−13.9509	−0.942716
$220$	0	0
$221$	4.71982	0.317490
$222$	0	0
$223$	12.4431	0.833251	0.416625	−	0.909078i	$-0.363213\pi$
$223$	12.4431	0.833251	0.416625	+	0.909078i	$0.363213\pi$
$224$	0	0
$225$	9.27674	0.618449
$226$	0	0
$227$	−9.93793	−0.659604	−0.329802	−	0.944050i	$-0.606982\pi$
$227$	−9.93793	−0.659604	−0.329802	+	0.944050i	$0.606982\pi$
$228$	0	0
$229$	9.76547	0.645320	0.322660	−	0.946515i	$-0.395423\pi$
$229$	9.76547	0.645320	0.322660	+	0.946515i	$0.395423\pi$
$230$	0	0
$231$	1.77846	0.117014
$232$	0	0
$233$	12.6707	0.830088	0.415044	−	0.909801i	$-0.363766\pi$
$233$	12.6707	0.830088	0.415044	+	0.909801i	$0.363766\pi$
$234$	0	0
$235$	−39.2242	−2.55871
$236$	0	0
$237$	−3.55691	−0.231046
$238$	0	0
$239$	13.4948	0.872909	0.436454	−	0.899726i	$-0.356234\pi$
$239$	13.4948	0.872909	0.436454	+	0.899726i	$0.356234\pi$
$240$	0	0
$241$	−13.1138	−0.844736	−0.422368	−	0.906424i	$-0.638801\pi$
$241$	−13.1138	−0.844736	−0.422368	+	0.906424i	$0.638801\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	3.77846	0.241397
$246$	0	0
$247$	2.71982	0.173058
$248$	0	0
$249$	8.05520	0.510477
$250$	0	0
$251$	3.60600	0.227608	0.113804	−	0.993503i	$-0.463696\pi$
$251$	3.60600	0.227608	0.113804	+	0.993503i	$0.463696\pi$
$252$	0	0
$253$	4.83709	0.304105
$254$	0	0
$255$	−17.8337	−1.11679
$256$	0	0
$257$	14.0000	0.873296	0.436648	−	0.899632i	$-0.356166\pi$
$257$	14.0000	0.873296	0.436648	+	0.899632i	$0.356166\pi$
$258$	0	0
$259$	−2.83709	−0.176288
$260$	0	0
$261$	−0.719824	−0.0445560
$262$	0	0
$263$	17.2311	1.06251	0.531257	−	0.847210i	$-0.321720\pi$
$263$	17.2311	1.06251	0.531257	+	0.847210i	$0.321720\pi$
$264$	0	0
$265$	−34.4362	−2.11540
$266$	0	0
$267$	2.49828	0.152892
$268$	0	0
$269$	13.3484	0.813864	0.406932	−	0.913458i	$-0.366599\pi$
$269$	13.3484	0.813864	0.406932	+	0.913458i	$0.366599\pi$
$270$	0	0
$271$	5.43965	0.330435	0.165218	−	0.986257i	$-0.447167\pi$
$271$	5.43965	0.330435	0.165218	+	0.986257i	$0.447167\pi$
$272$	0	0
$273$	1.00000	0.0605228
$274$	0	0
$275$	16.4983	0.994884
$276$	0	0
$277$	−2.88617	−0.173413	−0.0867066	−	0.996234i	$-0.527634\pi$
$277$	−2.88617	−0.173413	−0.0867066	+	0.996234i	$0.527634\pi$
$278$	0	0
$279$	−1.88273	−0.112716
$280$	0	0
$281$	7.61899	0.454511	0.227255	−	0.973835i	$-0.427025\pi$
$281$	7.61899	0.454511	0.227255	+	0.973835i	$0.427025\pi$
$282$	0	0
$283$	15.7655	0.937160	0.468580	−	0.883421i	$-0.344766\pi$
$283$	15.7655	0.937160	0.468580	+	0.883421i	$0.344766\pi$
$284$	0	0
$285$	−10.2767	−0.608741
$286$	0	0
$287$	−1.05863	−0.0624891
$288$	0	0
$289$	5.27674	0.310396
$290$	0	0
$291$	13.1138	0.768746
$292$	0	0
$293$	10.4983	0.613316	0.306658	−	0.951820i	$-0.400789\pi$
$293$	10.4983	0.613316	0.306658	+	0.951820i	$0.400789\pi$
$294$	0	0
$295$	24.1104	1.40376
$296$	0	0
$297$	−1.77846	−0.103197
$298$	0	0
$299$	2.71982	0.157291
$300$	0	0
$301$	0.837090	0.0482491
$302$	0	0
$303$	−0.325819	−0.0187178
$304$	0	0
$305$	4.39400	0.251600
$306$	0	0
$307$	30.8793	1.76237	0.881187	−	0.472767i	$-0.156745\pi$
$307$	30.8793	1.76237	0.881187	+	0.472767i	$0.156745\pi$
$308$	0	0
$309$	−12.1595	−0.691728
$310$	0	0
$311$	−7.32238	−0.415214	−0.207607	−	0.978212i	$-0.566568\pi$
$311$	−7.32238	−0.415214	−0.207607	+	0.978212i	$0.566568\pi$
$312$	0	0
$313$	−14.7880	−0.835868	−0.417934	−	0.908477i	$-0.637246\pi$
$313$	−14.7880	−0.835868	−0.417934	+	0.908477i	$0.637246\pi$
$314$	0	0
$315$	−3.77846	−0.212892
$316$	0	0
$317$	32.7000	1.83661	0.918306	−	0.395871i	$-0.129557\pi$
$317$	32.7000	1.83661	0.918306	+	0.395871i	$0.129557\pi$
$318$	0	0
$319$	−1.28018	−0.0716761
$320$	0	0
$321$	−18.6707	−1.04210
$322$	0	0
$323$	12.8371	0.714275
$324$	0	0
$325$	9.27674	0.514581
$326$	0	0
$327$	15.8337	0.875603
$328$	0	0
$329$	10.3810	0.572324
$330$	0	0
$331$	23.6673	1.30087	0.650436	−	0.759561i	$-0.274586\pi$
$331$	23.6673	1.30087	0.650436	+	0.759561i	$0.274586\pi$
$332$	0	0
$333$	2.83709	0.155472
$334$	0	0
$335$	21.4396	1.17137
$336$	0	0
$337$	−25.0647	−1.36536	−0.682682	−	0.730716i	$-0.739187\pi$
$337$	−25.0647	−1.36536	−0.682682	+	0.730716i	$0.739187\pi$
$338$	0	0
$339$	−11.8827	−0.645382
$340$	0	0
$341$	−3.34836	−0.181324
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−10.2767	−0.553281
$346$	0	0
$347$	−10.8793	−0.584031	−0.292016	−	0.956414i	$-0.594326\pi$
$347$	−10.8793	−0.584031	−0.292016	+	0.956414i	$0.594326\pi$
$348$	0	0
$349$	14.1104	0.755312	0.377656	−	0.925946i	$-0.376730\pi$
$349$	14.1104	0.755312	0.377656	+	0.925946i	$0.376730\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	−15.7294	−0.837190	−0.418595	−	0.908173i	$-0.637477\pi$
$353$	−15.7294	−0.837190	−0.418595	+	0.908173i	$0.637477\pi$
$354$	0	0
$355$	9.88273	0.524521
$356$	0	0
$357$	4.71982	0.249800
$358$	0	0
$359$	−34.7259	−1.83276	−0.916382	−	0.400304i	$-0.868905\pi$
$359$	−34.7259	−1.83276	−0.916382	+	0.400304i	$0.868905\pi$
$360$	0	0
$361$	−11.6026	−0.610661
$362$	0	0
$363$	7.83709	0.411341
$364$	0	0
$365$	52.7129	2.75912
$366$	0	0
$367$	2.76891	0.144536	0.0722678	−	0.997385i	$-0.476976\pi$
$367$	2.76891	0.144536	0.0722678	+	0.997385i	$0.476976\pi$
$368$	0	0
$369$	1.05863	0.0551102
$370$	0	0
$371$	9.11383	0.473166
$372$	0	0
$373$	−22.5535	−1.16777	−0.583887	−	0.811835i	$-0.698469\pi$
$373$	−22.5535	−1.16777	−0.583887	+	0.811835i	$0.698469\pi$
$374$	0	0
$375$	−16.1595	−0.834472
$376$	0	0
$377$	−0.719824	−0.0370728
$378$	0	0
$379$	36.1104	1.85487	0.927433	−	0.373989i	$-0.122010\pi$
$379$	36.1104	1.85487	0.927433	+	0.373989i	$0.122010\pi$
$380$	0	0
$381$	3.55691	0.182226
$382$	0	0
$383$	−16.6837	−0.852499	−0.426249	−	0.904606i	$-0.640165\pi$
$383$	−16.6837	−0.852499	−0.426249	+	0.904606i	$0.640165\pi$
$384$	0	0
$385$	−6.71982	−0.342474
$386$	0	0
$387$	−0.837090	−0.0425517
$388$	0	0
$389$	−33.1138	−1.67894	−0.839469	−	0.543408i	$-0.817134\pi$
$389$	−33.1138	−1.67894	−0.839469	+	0.543408i	$0.817134\pi$
$390$	0	0
$391$	12.8371	0.649200
$392$	0	0
$393$	19.5078	0.984040
$394$	0	0
$395$	13.4396	0.676222
$396$	0	0
$397$	5.00344	0.251115	0.125558	−	0.992086i	$-0.459928\pi$
$397$	5.00344	0.251115	0.125558	+	0.992086i	$0.459928\pi$
$398$	0	0
$399$	2.71982	0.136162
$400$	0	0
$401$	−13.6121	−0.679756	−0.339878	−	0.940469i	$-0.610386\pi$
$401$	−13.6121	−0.679756	−0.339878	+	0.940469i	$0.610386\pi$
$402$	0	0
$403$	−1.88273	−0.0937856
$404$	0	0
$405$	3.77846	0.187753
$406$	0	0
$407$	5.04564	0.250103
$408$	0	0
$409$	−25.9509	−1.28319	−0.641595	−	0.767043i	$-0.721727\pi$
$409$	−25.9509	−1.28319	−0.641595	+	0.767043i	$0.721727\pi$
$410$	0	0
$411$	−0.221543	−0.0109279
$412$	0	0
$413$	−6.38101	−0.313989
$414$	0	0
$415$	−30.4362	−1.49405
$416$	0	0
$417$	4.99656	0.244683
$418$	0	0
$419$	32.1595	1.57109	0.785547	−	0.618803i	$-0.212382\pi$
$419$	32.1595	1.57109	0.785547	+	0.618803i	$0.212382\pi$
$420$	0	0
$421$	−18.9966	−0.925836	−0.462918	−	0.886401i	$-0.653198\pi$
$421$	−18.9966	−0.925836	−0.462918	+	0.886401i	$0.653198\pi$
$422$	0	0
$423$	−10.3810	−0.504742
$424$	0	0
$425$	43.7846	2.12386
$426$	0	0
$427$	−1.16291	−0.0562771
$428$	0	0
$429$	−1.77846	−0.0858647
$430$	0	0
$431$	−0.0551953	−0.00265866	−0.00132933	−	0.999999i	$-0.500423\pi$
$431$	−0.0551953	−0.00265866	−0.00132933	+	0.999999i	$0.500423\pi$
$432$	0	0
$433$	−24.5604	−1.18030	−0.590148	−	0.807295i	$-0.700930\pi$
$433$	−24.5604	−1.18030	−0.590148	+	0.807295i	$0.700930\pi$
$434$	0	0
$435$	2.71982	0.130406
$436$	0	0
$437$	7.39744	0.353868
$438$	0	0
$439$	−28.4784	−1.35920	−0.679600	−	0.733583i	$-0.737847\pi$
$439$	−28.4784	−1.35920	−0.679600	+	0.733583i	$0.737847\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−7.32238	−0.347897	−0.173948	−	0.984755i	$-0.555653\pi$
$443$	−7.32238	−0.347897	−0.173948	+	0.984755i	$0.555653\pi$
$444$	0	0
$445$	−9.43965	−0.447482
$446$	0	0
$447$	21.1690	1.00126
$448$	0	0
$449$	40.3319	1.90338	0.951691	−	0.307058i	$-0.0993446\pi$
$449$	40.3319	1.90338	0.951691	+	0.307058i	$0.0993446\pi$
$450$	0	0
$451$	1.88273	0.0886545
$452$	0	0
$453$	−17.5991	−0.826879
$454$	0	0
$455$	−3.77846	−0.177137
$456$	0	0
$457$	11.7846	0.551259	0.275629	−	0.961264i	$-0.411114\pi$
$457$	11.7846	0.551259	0.275629	+	0.961264i	$0.411114\pi$
$458$	0	0
$459$	−4.71982	−0.220303
$460$	0	0
$461$	−16.7750	−0.781291	−0.390645	−	0.920541i	$-0.627748\pi$
$461$	−16.7750	−0.781291	−0.390645	+	0.920541i	$0.627748\pi$
$462$	0	0
$463$	−24.9215	−1.15820	−0.579100	−	0.815256i	$-0.696596\pi$
$463$	−24.9215	−1.15820	−0.579100	+	0.815256i	$0.696596\pi$
$464$	0	0
$465$	7.11383	0.329896
$466$	0	0
$467$	7.84053	0.362816	0.181408	−	0.983408i	$-0.441934\pi$
$467$	7.84053	0.362816	0.181408	+	0.983408i	$0.441934\pi$
$468$	0	0
$469$	−5.67418	−0.262009
$470$	0	0
$471$	−12.5113	−0.576489
$472$	0	0
$473$	−1.48873	−0.0684518
$474$	0	0
$475$	25.2311	1.15768
$476$	0	0
$477$	−9.11383	−0.417294
$478$	0	0
$479$	40.3251	1.84250	0.921249	−	0.388972i	$-0.127170\pi$
$479$	40.3251	1.84250	0.921249	+	0.388972i	$0.127170\pi$
$480$	0	0
$481$	2.83709	0.129360
$482$	0	0
$483$	2.71982	0.123756
$484$	0	0
$485$	−49.5500	−2.24995
$486$	0	0
$487$	−40.9966	−1.85773	−0.928866	−	0.370416i	$-0.879215\pi$
$487$	−40.9966	−1.85773	−0.928866	+	0.370416i	$0.879215\pi$
$488$	0	0
$489$	18.4362	0.833714
$490$	0	0
$491$	12.0260	0.542725	0.271362	−	0.962477i	$-0.412526\pi$
$491$	12.0260	0.542725	0.271362	+	0.962477i	$0.412526\pi$
$492$	0	0
$493$	−3.39744	−0.153013
$494$	0	0
$495$	6.71982	0.302034
$496$	0	0
$497$	−2.61555	−0.117323
$498$	0	0
$499$	18.6448	0.834654	0.417327	−	0.908756i	$-0.362967\pi$
$499$	18.6448	0.834654	0.417327	+	0.908756i	$0.362967\pi$
$500$	0	0
$501$	7.21811	0.322481
$502$	0	0
$503$	18.6707	0.832487	0.416244	−	0.909253i	$-0.363346\pi$
$503$	18.6707	0.832487	0.416244	+	0.909253i	$0.363346\pi$
$504$	0	0
$505$	1.23109	0.0547830
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	43.2112	1.91530	0.957652	−	0.287928i	$-0.0929664\pi$
$509$	43.2112	1.91530	0.957652	+	0.287928i	$0.0929664\pi$
$510$	0	0
$511$	−13.9509	−0.617152
$512$	0	0
$513$	−2.71982	−0.120083
$514$	0	0
$515$	45.9440	2.02454
$516$	0	0
$517$	−18.4622	−0.811966
$518$	0	0
$519$	1.76547	0.0774954
$520$	0	0
$521$	16.2767	0.713097	0.356548	−	0.934277i	$-0.383953\pi$
$521$	16.2767	0.713097	0.356548	+	0.934277i	$0.383953\pi$
$522$	0	0
$523$	18.1173	0.792213	0.396106	−	0.918205i	$-0.370361\pi$
$523$	18.1173	0.792213	0.396106	+	0.918205i	$0.370361\pi$
$524$	0	0
$525$	9.27674	0.404870
$526$	0	0
$527$	−8.88617	−0.387088
$528$	0	0
$529$	−15.6026	−0.678372
$530$	0	0
$531$	6.38101	0.276912
$532$	0	0
$533$	1.05863	0.0458545
$534$	0	0
$535$	70.5466	3.05000
$536$	0	0
$537$	−8.00000	−0.345225
$538$	0	0
$539$	1.77846	0.0766036
$540$	0	0
$541$	28.6217	1.23054	0.615271	−	0.788316i	$-0.289047\pi$
$541$	28.6217	1.23054	0.615271	+	0.788316i	$0.289047\pi$
$542$	0	0
$543$	−5.11383	−0.219455
$544$	0	0
$545$	−59.8268	−2.56270
$546$	0	0
$547$	−4.46907	−0.191083	−0.0955417	−	0.995425i	$-0.530458\pi$
$547$	−4.46907	−0.191083	−0.0955417	+	0.995425i	$0.530458\pi$
$548$	0	0
$549$	1.16291	0.0496318
$550$	0	0
$551$	−1.95779	−0.0834048
$552$	0	0
$553$	−3.55691	−0.151255
$554$	0	0
$555$	−10.7198	−0.455031
$556$	0	0
$557$	−40.0483	−1.69690	−0.848451	−	0.529274i	$-0.822464\pi$
$557$	−40.0483	−1.69690	−0.848451	+	0.529274i	$0.822464\pi$
$558$	0	0
$559$	−0.837090	−0.0354051
$560$	0	0
$561$	−8.39400	−0.354395
$562$	0	0
$563$	31.9509	1.34657	0.673285	−	0.739383i	$-0.264883\pi$
$563$	31.9509	1.34657	0.673285	+	0.739383i	$0.264883\pi$
$564$	0	0
$565$	44.8984	1.88889
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−36.3380	−1.52337	−0.761685	−	0.647947i	$-0.775628\pi$
$569$	−36.3380	−1.52337	−0.761685	+	0.647947i	$0.775628\pi$
$570$	0	0
$571$	−2.11727	−0.0886048	−0.0443024	−	0.999018i	$-0.514107\pi$
$571$	−2.11727	−0.0886048	−0.0443024	+	0.999018i	$0.514107\pi$
$572$	0	0
$573$	10.5113	0.439115
$574$	0	0
$575$	25.2311	1.05221
$576$	0	0
$577$	13.8759	0.577660	0.288830	−	0.957380i	$-0.406734\pi$
$577$	13.8759	0.577660	0.288830	+	0.957380i	$0.406734\pi$
$578$	0	0
$579$	7.67418	0.318928
$580$	0	0
$581$	8.05520	0.334186
$582$	0	0
$583$	−16.2086	−0.671290
$584$	0	0
$585$	3.77846	0.156220
$586$	0	0
$587$	0.830976	0.0342981	0.0171490	−	0.999853i	$-0.494541\pi$
$587$	0.830976	0.0342981	0.0171490	+	0.999853i	$0.494541\pi$
$588$	0	0
$589$	−5.12070	−0.210995
$590$	0	0
$591$	−22.2897	−0.916877
$592$	0	0
$593$	38.6087	1.58547	0.792734	−	0.609568i	$-0.208657\pi$
$593$	38.6087	1.58547	0.792734	+	0.609568i	$0.208657\pi$
$594$	0	0
$595$	−17.8337	−0.731108
$596$	0	0
$597$	6.04221	0.247291
$598$	0	0
$599$	−2.71982	−0.111129	−0.0555645	−	0.998455i	$-0.517696\pi$
$599$	−2.71982	−0.111129	−0.0555645	+	0.998455i	$0.517696\pi$
$600$	0	0
$601$	−46.7880	−1.90852	−0.954261	−	0.298974i	$-0.903356\pi$
$601$	−46.7880	−1.90852	−0.954261	+	0.298974i	$0.903356\pi$
$602$	0	0
$603$	5.67418	0.231070
$604$	0	0
$605$	−29.6121	−1.20390
$606$	0	0
$607$	−14.7198	−0.597459	−0.298730	−	0.954338i	$-0.596563\pi$
$607$	−14.7198	−0.597459	−0.298730	+	0.954338i	$0.596563\pi$
$608$	0	0
$609$	−0.719824	−0.0291687
$610$	0	0
$611$	−10.3810	−0.419971
$612$	0	0
$613$	−22.8371	−0.922381	−0.461191	−	0.887301i	$-0.652578\pi$
$613$	−22.8371	−0.922381	−0.461191	+	0.887301i	$0.652578\pi$
$614$	0	0
$615$	−4.00000	−0.161296
$616$	0	0
$617$	18.7819	0.756131	0.378065	−	0.925779i	$-0.376589\pi$
$617$	18.7819	0.756131	0.378065	+	0.925779i	$0.376589\pi$
$618$	0	0
$619$	−21.8077	−0.876524	−0.438262	−	0.898847i	$-0.644406\pi$
$619$	−21.8077	−0.876524	−0.438262	+	0.898847i	$0.644406\pi$
$620$	0	0
$621$	−2.71982	−0.109143
$622$	0	0
$623$	2.49828	0.100092
$624$	0	0
$625$	14.6742	0.586967
$626$	0	0
$627$	−4.83709	−0.193175
$628$	0	0
$629$	13.3906	0.533917
$630$	0	0
$631$	−27.7424	−1.10441	−0.552203	−	0.833710i	$-0.686213\pi$
$631$	−27.7424	−1.10441	−0.552203	+	0.833710i	$0.686213\pi$
$632$	0	0
$633$	−14.9544	−0.594382
$634$	0	0
$635$	−13.4396	−0.533336
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	2.61555	0.103469
$640$	0	0
$641$	26.2208	1.03566	0.517829	−	0.855484i	$-0.326740\pi$
$641$	26.2208	1.03566	0.517829	+	0.855484i	$0.326740\pi$
$642$	0	0
$643$	8.04908	0.317425	0.158712	−	0.987325i	$-0.449266\pi$
$643$	8.04908	0.317425	0.158712	+	0.987325i	$0.449266\pi$
$644$	0	0
$645$	3.16291	0.124539
$646$	0	0
$647$	19.7655	0.777061	0.388530	−	0.921436i	$-0.372983\pi$
$647$	19.7655	0.777061	0.388530	+	0.921436i	$0.372983\pi$
$648$	0	0
$649$	11.3484	0.445462
$650$	0	0
$651$	−1.88273	−0.0737902
$652$	0	0
$653$	12.3027	0.481443	0.240721	−	0.970594i	$-0.422616\pi$
$653$	12.3027	0.481443	0.240721	+	0.970594i	$0.422616\pi$
$654$	0	0
$655$	−73.7095	−2.88007
$656$	0	0
$657$	13.9509	0.544277
$658$	0	0
$659$	−11.6673	−0.454494	−0.227247	−	0.973837i	$-0.572972\pi$
$659$	−11.6673	−0.454494	−0.227247	+	0.973837i	$0.572972\pi$
$660$	0	0
$661$	−1.00344	−0.0390292	−0.0195146	−	0.999810i	$-0.506212\pi$
$661$	−1.00344	−0.0390292	−0.0195146	+	0.999810i	$0.506212\pi$
$662$	0	0
$663$	−4.71982	−0.183303
$664$	0	0
$665$	−10.2767	−0.398515
$666$	0	0
$667$	−1.95779	−0.0758061
$668$	0	0
$669$	−12.4431	−0.481077
$670$	0	0
$671$	2.06819	0.0798414
$672$	0	0
$673$	−9.95092	−0.383580	−0.191790	−	0.981436i	$-0.561429\pi$
$673$	−9.95092	−0.383580	−0.191790	+	0.981436i	$0.561429\pi$
$674$	0	0
$675$	−9.27674	−0.357062
$676$	0	0
$677$	−44.7552	−1.72008	−0.860040	−	0.510226i	$-0.829562\pi$
$677$	−44.7552	−1.72008	−0.860040	+	0.510226i	$0.829562\pi$
$678$	0	0
$679$	13.1138	0.503263
$680$	0	0
$681$	9.93793	0.380822
$682$	0	0
$683$	3.86974	0.148072	0.0740358	−	0.997256i	$-0.476412\pi$
$683$	3.86974	0.148072	0.0740358	+	0.997256i	$0.476412\pi$
$684$	0	0
$685$	0.837090	0.0319836
$686$	0	0
$687$	−9.76547	−0.372576
$688$	0	0
$689$	−9.11383	−0.347209
$690$	0	0
$691$	−5.99312	−0.227989	−0.113995	−	0.993481i	$-0.536365\pi$
$691$	−5.99312	−0.227989	−0.113995	+	0.993481i	$0.536365\pi$
$692$	0	0
$693$	−1.77846	−0.0675580
$694$	0	0
$695$	−18.8793	−0.716133
$696$	0	0
$697$	4.99656	0.189258
$698$	0	0
$699$	−12.6707	−0.479252
$700$	0	0
$701$	−18.0000	−0.679851	−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000	−0.679851	−0.339925	+	0.940452i	$0.610402\pi$
$702$	0	0
$703$	7.71639	0.291029
$704$	0	0
$705$	39.2242	1.47727
$706$	0	0
$707$	−0.325819	−0.0122537
$708$	0	0
$709$	−2.99656	−0.112538	−0.0562691	−	0.998416i	$-0.517920\pi$
$709$	−2.99656	−0.112538	−0.0562691	+	0.998416i	$0.517920\pi$
$710$	0	0
$711$	3.55691	0.133395
$712$	0	0
$713$	−5.12070	−0.191772
$714$	0	0
$715$	6.71982	0.251307
$716$	0	0
$717$	−13.4948	−0.503974
$718$	0	0
$719$	−11.0878	−0.413507	−0.206753	−	0.978393i	$-0.566290\pi$
$719$	−11.0878	−0.413507	−0.206753	+	0.978393i	$0.566290\pi$
$720$	0	0
$721$	−12.1595	−0.452842
$722$	0	0
$723$	13.1138	0.487709
$724$	0	0
$725$	−6.67762	−0.248001
$726$	0	0
$727$	−3.95092	−0.146531	−0.0732657	−	0.997312i	$-0.523342\pi$
$727$	−3.95092	−0.146531	−0.0732657	+	0.997312i	$0.523342\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−3.95092	−0.146130
$732$	0	0
$733$	−13.6673	−0.504813	−0.252407	−	0.967621i	$-0.581222\pi$
$733$	−13.6673	−0.504813	−0.252407	+	0.967621i	$0.581222\pi$
$734$	0	0
$735$	−3.77846	−0.139370
$736$	0	0
$737$	10.0913	0.371717
$738$	0	0
$739$	−36.7880	−1.35327	−0.676634	−	0.736319i	$-0.736562\pi$
$739$	−36.7880	−1.35327	−0.676634	+	0.736319i	$0.736562\pi$
$740$	0	0
$741$	−2.71982	−0.0999152
$742$	0	0
$743$	−14.1725	−0.519937	−0.259969	−	0.965617i	$-0.583712\pi$
$743$	−14.1725	−0.519937	−0.259969	+	0.965617i	$0.583712\pi$
$744$	0	0
$745$	−79.9862	−2.93047
$746$	0	0
$747$	−8.05520	−0.294724
$748$	0	0
$749$	−18.6707	−0.682214
$750$	0	0
$751$	−35.3484	−1.28988	−0.644940	−	0.764233i	$-0.723118\pi$
$751$	−35.3484	−1.28988	−0.644940	+	0.764233i	$0.723118\pi$
$752$	0	0
$753$	−3.60600	−0.131410
$754$	0	0
$755$	66.4975	2.42009
$756$	0	0
$757$	−2.00000	−0.0726912	−0.0363456	−	0.999339i	$-0.511572\pi$
$757$	−2.00000	−0.0726912	−0.0363456	+	0.999339i	$0.511572\pi$
$758$	0	0
$759$	−4.83709	−0.175575
$760$	0	0
$761$	17.0586	0.618375	0.309187	−	0.951001i	$-0.399943\pi$
$761$	17.0586	0.618375	0.309187	+	0.951001i	$0.399943\pi$
$762$	0	0
$763$	15.8337	0.573217
$764$	0	0
$765$	17.8337	0.644777
$766$	0	0
$767$	6.38101	0.230405
$768$	0	0
$769$	−28.7198	−1.03566	−0.517832	−	0.855483i	$-0.673261\pi$
$769$	−28.7198	−1.03566	−0.517832	+	0.855483i	$0.673261\pi$
$770$	0	0
$771$	−14.0000	−0.504198
$772$	0	0
$773$	−16.2475	−0.584383	−0.292191	−	0.956360i	$-0.594384\pi$
$773$	−16.2475	−0.584383	−0.292191	+	0.956360i	$0.594384\pi$
$774$	0	0
$775$	−17.4656	−0.627384
$776$	0	0
$777$	2.83709	0.101780
$778$	0	0
$779$	2.87930	0.103161
$780$	0	0
$781$	4.65164	0.166449
$782$	0	0
$783$	0.719824	0.0257244
$784$	0	0
$785$	47.2733	1.68726
$786$	0	0
$787$	38.1786	1.36092	0.680460	−	0.732786i	$-0.261780\pi$
$787$	38.1786	1.36092	0.680460	+	0.732786i	$0.261780\pi$
$788$	0	0
$789$	−17.2311	−0.613443
$790$	0	0
$791$	−11.8827	−0.422501
$792$	0	0
$793$	1.16291	0.0412961
$794$	0	0
$795$	34.4362	1.22133
$796$	0	0
$797$	9.32238	0.330216	0.165108	−	0.986276i	$-0.447203\pi$
$797$	9.32238	0.330216	0.165108	+	0.986276i	$0.447203\pi$
$798$	0	0
$799$	−48.9966	−1.73337
$800$	0	0
$801$	−2.49828	−0.0882724
$802$	0	0
$803$	24.8111	0.875565
$804$	0	0
$805$	−10.2767	−0.362207
$806$	0	0
$807$	−13.3484	−0.469885
$808$	0	0
$809$	21.8759	0.769114	0.384557	−	0.923101i	$-0.374354\pi$
$809$	21.8759	0.769114	0.384557	+	0.923101i	$0.374354\pi$
$810$	0	0
$811$	−45.9181	−1.61240	−0.806201	−	0.591642i	$-0.798480\pi$
$811$	−45.9181	−1.61240	−0.806201	+	0.591642i	$0.798480\pi$
$812$	0	0
$813$	−5.43965	−0.190777
$814$	0	0
$815$	−69.6604	−2.44010
$816$	0	0
$817$	−2.27674	−0.0796530
$818$	0	0
$819$	−1.00000	−0.0349428
$820$	0	0
$821$	−15.8275	−0.552385	−0.276192	−	0.961102i	$-0.589073\pi$
$821$	−15.8275	−0.552385	−0.276192	+	0.961102i	$0.589073\pi$
$822$	0	0
$823$	−8.46907	−0.295213	−0.147607	−	0.989046i	$-0.547157\pi$
$823$	−8.46907	−0.295213	−0.147607	+	0.989046i	$0.547157\pi$
$824$	0	0
$825$	−16.4983	−0.574396
$826$	0	0
$827$	23.5370	0.818463	0.409232	−	0.912431i	$-0.365797\pi$
$827$	23.5370	0.818463	0.409232	+	0.912431i	$0.365797\pi$
$828$	0	0
$829$	19.6251	0.681608	0.340804	−	0.940134i	$-0.389301\pi$
$829$	19.6251	0.681608	0.340804	+	0.940134i	$0.389301\pi$
$830$	0	0
$831$	2.88617	0.100120
$832$	0	0
$833$	4.71982	0.163532
$834$	0	0
$835$	−27.2733	−0.943831
$836$	0	0
$837$	1.88273	0.0650768
$838$	0	0
$839$	−42.4914	−1.46697	−0.733483	−	0.679708i	$-0.762107\pi$
$839$	−42.4914	−1.46697	−0.733483	+	0.679708i	$0.762107\pi$
$840$	0	0
$841$	−28.4819	−0.982133
$842$	0	0
$843$	−7.61899	−0.262412
$844$	0	0
$845$	3.77846	0.129983
$846$	0	0
$847$	7.83709	0.269286
$848$	0	0
$849$	−15.7655	−0.541069
$850$	0	0
$851$	7.71639	0.264514
$852$	0	0
$853$	−43.3155	−1.48309	−0.741547	−	0.670901i	$-0.765908\pi$
$853$	−43.3155	−1.48309	−0.741547	+	0.670901i	$0.765908\pi$
$854$	0	0
$855$	10.2767	0.351457
$856$	0	0
$857$	−36.8793	−1.25977	−0.629886	−	0.776687i	$-0.716899\pi$
$857$	−36.8793	−1.25977	−0.629886	+	0.776687i	$0.716899\pi$
$858$	0	0
$859$	10.2277	0.348963	0.174482	−	0.984660i	$-0.444175\pi$
$859$	10.2277	0.348963	0.174482	+	0.984660i	$0.444175\pi$
$860$	0	0
$861$	1.05863	0.0360781
$862$	0	0
$863$	−32.9414	−1.12134	−0.560669	−	0.828040i	$-0.689456\pi$
$863$	−32.9414	−1.12134	−0.560669	+	0.828040i	$0.689456\pi$
$864$	0	0
$865$	−6.67074	−0.226812
$866$	0	0
$867$	−5.27674	−0.179207
$868$	0	0
$869$	6.32582	0.214589
$870$	0	0
$871$	5.67418	0.192262
$872$	0	0
$873$	−13.1138	−0.443836
$874$	0	0
$875$	−16.1595	−0.546290
$876$	0	0
$877$	25.7655	0.870038	0.435019	−	0.900421i	$-0.356742\pi$
$877$	25.7655	0.870038	0.435019	+	0.900421i	$0.356742\pi$
$878$	0	0
$879$	−10.4983	−0.354098
$880$	0	0
$881$	17.9509	0.604782	0.302391	−	0.953184i	$-0.402215\pi$
$881$	17.9509	0.604782	0.302391	+	0.953184i	$0.402215\pi$
$882$	0	0
$883$	8.62854	0.290373	0.145187	−	0.989404i	$-0.453622\pi$
$883$	8.62854	0.290373	0.145187	+	0.989404i	$0.453622\pi$
$884$	0	0
$885$	−24.1104	−0.810462
$886$	0	0
$887$	26.8793	0.902518	0.451259	−	0.892393i	$-0.350975\pi$
$887$	26.8793	0.902518	0.451259	+	0.892393i	$0.350975\pi$
$888$	0	0
$889$	3.55691	0.119295
$890$	0	0
$891$	1.77846	0.0595806
$892$	0	0
$893$	−28.2345	−0.944833
$894$	0	0
$895$	30.2277	1.01040
$896$	0	0
$897$	−2.71982	−0.0908123
$898$	0	0
$899$	1.35524	0.0451997
$900$	0	0
$901$	−43.0157	−1.43306
$902$	0	0
$903$	−0.837090	−0.0278566
$904$	0	0
$905$	19.3224	0.642298
$906$	0	0
$907$	−36.9966	−1.22845	−0.614225	−	0.789131i	$-0.710531\pi$
$907$	−36.9966	−1.22845	−0.614225	+	0.789131i	$0.710531\pi$
$908$	0	0
$909$	0.325819	0.0108067
$910$	0	0
$911$	−9.72326	−0.322146	−0.161073	−	0.986942i	$-0.551495\pi$
$911$	−9.72326	−0.322146	−0.161073	+	0.986942i	$0.551495\pi$
$912$	0	0
$913$	−14.3258	−0.474115
$914$	0	0
$915$	−4.39400	−0.145261
$916$	0	0
$917$	19.5078	0.644205
$918$	0	0
$919$	−32.2086	−1.06246	−0.531231	−	0.847227i	$-0.678270\pi$
$919$	−32.2086	−1.06246	−0.531231	+	0.847227i	$0.678270\pi$
$920$	0	0
$921$	−30.8793	−1.01751
$922$	0	0
$923$	2.61555	0.0860918
$924$	0	0
$925$	26.3189	0.865362
$926$	0	0
$927$	12.1595	0.399369
$928$	0	0
$929$	−47.6312	−1.56273	−0.781365	−	0.624075i	$-0.785476\pi$
$929$	−47.6312	−1.56273	−0.781365	+	0.624075i	$0.785476\pi$
$930$	0	0
$931$	2.71982	0.0891386
$932$	0	0
$933$	7.32238	0.239724
$934$	0	0
$935$	31.7164	1.03724
$936$	0	0
$937$	34.6379	1.13157	0.565785	−	0.824553i	$-0.308573\pi$
$937$	34.6379	1.13157	0.565785	+	0.824553i	$0.308573\pi$
$938$	0	0
$939$	14.7880	0.482588
$940$	0	0
$941$	−18.3741	−0.598980	−0.299490	−	0.954099i	$-0.596816\pi$
$941$	−18.3741	−0.598980	−0.299490	+	0.954099i	$0.596816\pi$
$942$	0	0
$943$	2.87930	0.0937628
$944$	0	0
$945$	3.77846	0.122913
$946$	0	0
$947$	−30.3319	−0.985655	−0.492828	−	0.870127i	$-0.664037\pi$
$947$	−30.3319	−0.985655	−0.492828	+	0.870127i	$0.664037\pi$
$948$	0	0
$949$	13.9509	0.452866
$950$	0	0
$951$	−32.7000	−1.06037
$952$	0	0
$953$	−13.3224	−0.431554	−0.215777	−	0.976443i	$-0.569228\pi$
$953$	−13.3224	−0.431554	−0.215777	+	0.976443i	$0.569228\pi$
$954$	0	0
$955$	−39.7164	−1.28519
$956$	0	0
$957$	1.28018	0.0413822
$958$	0	0
$959$	−0.221543	−0.00715399
$960$	0	0
$961$	−27.4553	−0.885655
$962$	0	0
$963$	18.6707	0.601656
$964$	0	0
$965$	−28.9966	−0.933432
$966$	0	0
$967$	−16.4922	−0.530352	−0.265176	−	0.964200i	$-0.585430\pi$
$967$	−16.4922	−0.530352	−0.265176	+	0.964200i	$0.585430\pi$
$968$	0	0
$969$	−12.8371	−0.412387
$970$	0	0
$971$	−27.5829	−0.885177	−0.442589	−	0.896725i	$-0.645940\pi$
$971$	−27.5829	−0.885177	−0.442589	+	0.896725i	$0.645940\pi$
$972$	0	0
$973$	4.99656	0.160182
$974$	0	0
$975$	−9.27674	−0.297093
$976$	0	0
$977$	−2.86631	−0.0917013	−0.0458506	−	0.998948i	$-0.514600\pi$
$977$	−2.86631	−0.0917013	−0.0458506	+	0.998948i	$0.514600\pi$
$978$	0	0
$979$	−4.44309	−0.142002
$980$	0	0
$981$	−15.8337	−0.505530
$982$	0	0
$983$	−38.1234	−1.21595	−0.607973	−	0.793957i	$-0.708017\pi$
$983$	−38.1234	−1.21595	−0.607973	+	0.793957i	$0.708017\pi$
$984$	0	0
$985$	84.2208	2.68350
$986$	0	0
$987$	−10.3810	−0.330431
$988$	0	0
$989$	−2.27674	−0.0723961
$990$	0	0
$991$	−11.4465	−0.363611	−0.181805	−	0.983335i	$-0.558194\pi$
$991$	−11.4465	−0.363611	−0.181805	+	0.983335i	$0.558194\pi$
$992$	0	0
$993$	−23.6673	−0.751059
$994$	0	0
$995$	−22.8302	−0.723766
$996$	0	0
$997$	45.3346	1.43576	0.717881	−	0.696166i	$-0.245112\pi$
$997$	45.3346	1.43576	0.717881	+	0.696166i	$0.245112\pi$
$998$	0	0
$999$	−2.83709	−0.0897616

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4368.2.a.bp.1.3		3
4.3	odd	2		2184.2.a.u.1.3	✓	3
12.11	even	2		6552.2.a.bn.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2184.2.a.u.1.3	✓	3	4.3	odd	2
4368.2.a.bp.1.3		3	1.1	even	1	trivial
6552.2.a.bn.1.1		3	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4368.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +3.77846 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +3.77846 q^{5} -1.00000 q^{7} +1.00000 q^{9} +1.77846 q^{11} +1.00000 q^{13} -3.77846 q^{15} +4.71982 q^{17} +2.71982 q^{19} +1.00000 q^{21} +2.71982 q^{23} +9.27674 q^{25} -1.00000 q^{27} -0.719824 q^{29} -1.88273 q^{31} -1.77846 q^{33} -3.77846 q^{35} +2.83709 q^{37} -1.00000 q^{39} +1.05863 q^{41} -0.837090 q^{43} +3.77846 q^{45} -10.3810 q^{47} +1.00000 q^{49} -4.71982 q^{51} -9.11383 q^{53} +6.71982 q^{55} -2.71982 q^{57} +6.38101 q^{59} +1.16291 q^{61} -1.00000 q^{63} +3.77846 q^{65} +5.67418 q^{67} -2.71982 q^{69} +2.61555 q^{71} +13.9509 q^{73} -9.27674 q^{75} -1.77846 q^{77} +3.55691 q^{79} +1.00000 q^{81} -8.05520 q^{83} +17.8337 q^{85} +0.719824 q^{87} -2.49828 q^{89} -1.00000 q^{91} +1.88273 q^{93} +10.2767 q^{95} -13.1138 q^{97} +1.77846 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 3 q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 + 3 * q^5 - 3 * q^7 + 3 * q^9	\( 3 q - 3 q^{3} + 3 q^{5} - 3 q^{7} + 3 q^{9} - 3 q^{11} + 3 q^{13} - 3 q^{15} + 5 q^{17} - q^{19} + 3 q^{21} - q^{23} + 2 q^{25} - 3 q^{27} + 7 q^{29} - 4 q^{31} + 3 q^{33} - 3 q^{35} + q^{37} - 3 q^{39} + 4 q^{41} + 5 q^{43} + 3 q^{45} - 12 q^{47} + 3 q^{49} - 5 q^{51} + 6 q^{53} + 11 q^{55} + q^{57} + 11 q^{61} - 3 q^{63} + 3 q^{65} + 2 q^{67} + q^{69} - 8 q^{71} + q^{73} - 2 q^{75} + 3 q^{77} - 6 q^{79} + 3 q^{81} + 10 q^{83} + 11 q^{85} - 7 q^{87} + 10 q^{89} - 3 q^{91} + 4 q^{93} + 5 q^{95} - 6 q^{97} - 3 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + 3 * q^5 - 3 * q^7 + 3 * q^9 - 3 * q^11 + 3 * q^13 - 3 * q^15 + 5 * q^17 - q^19 + 3 * q^21 - q^23 + 2 * q^25 - 3 * q^27 + 7 * q^29 - 4 * q^31 + 3 * q^33 - 3 * q^35 + q^37 - 3 * q^39 + 4 * q^41 + 5 * q^43 + 3 * q^45 - 12 * q^47 + 3 * q^49 - 5 * q^51 + 6 * q^53 + 11 * q^55 + q^57 + 11 * q^61 - 3 * q^63 + 3 * q^65 + 2 * q^67 + q^69 - 8 * q^71 + q^73 - 2 * q^75 + 3 * q^77 - 6 * q^79 + 3 * q^81 + 10 * q^83 + 11 * q^85 - 7 * q^87 + 10 * q^89 - 3 * q^91 + 4 * q^93 + 5 * q^95 - 6 * q^97 - 3 * q^99

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