LMFDB - Embedded newform 2368.2.a.bh.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2368,2,Mod(1,2368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2368, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2368.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2368 = 2^{6} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2368.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:	$18.9085751986$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.48389.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 8x^{2} + 3x + 10 $ x^4 - 2x^3 - 8x^2 + 3*x + 10
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 296)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.73494$ of defining polynomial
Character	$\chi$	$=$	2368.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.73494 q^{3} -0.687447 q^{5} +2.16737 q^{7} +4.47992 q^{9} +O(q^{10})$	$q-2.73494 q^{3} -0.687447 q^{5} +2.16737 q^{7} +4.47992 q^{9} +3.47992 q^{11} -2.56758 q^{13} +1.88013 q^{15} +2.00000 q^{17} +8.09499 q^{19} -5.92763 q^{21} +2.05753 q^{23} -4.52742 q^{25} -4.04750 q^{27} -6.90231 q^{29} -3.02218 q^{31} -9.51739 q^{33} -1.48995 q^{35} -1.00000 q^{37} +7.02218 q^{39} +2.22489 q^{41} +2.62511 q^{43} -3.07971 q^{45} -2.26236 q^{47} -2.30252 q^{49} -5.46989 q^{51} +9.63725 q^{53} -2.39226 q^{55} -22.1394 q^{57} -15.0548 q^{59} +1.93766 q^{61} +9.70963 q^{63} +1.76507 q^{65} +6.78244 q^{67} -5.62722 q^{69} +14.1473 q^{71} +6.73013 q^{73} +12.3822 q^{75} +7.54226 q^{77} -16.8822 q^{79} -2.37008 q^{81} +15.9720 q^{83} -1.37489 q^{85} +18.8774 q^{87} +6.42973 q^{89} -5.56488 q^{91} +8.26550 q^{93} -5.56488 q^{95} +17.2745 q^{97} +15.5898 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} - 5 q^{5} + q^{7} + 8 q^{9}+O(q^{10}) $ 4 * q + 2 * q^3 - 5 * q^5 + q^7 + 8 * q^9	$ 4 q + 2 q^{3} - 5 q^{5} + q^{7} + 8 q^{9} + 4 q^{11} - 5 q^{13} + 8 q^{17} + 2 q^{19} - q^{21} + 9 q^{23} + 7 q^{25} - q^{27} - 7 q^{29} + q^{31} + 3 q^{33} - 12 q^{35} - 4 q^{37} + 15 q^{39} + 2 q^{41} + 6 q^{43} + 29 q^{47} + 9 q^{49} + 4 q^{51} + 5 q^{53} + 5 q^{55} - 32 q^{57} - 10 q^{59} + q^{61} + 28 q^{63} - 2 q^{65} - q^{67} + 27 q^{69} + 17 q^{71} + 8 q^{73} + 19 q^{75} + 27 q^{77} - 15 q^{79} - 8 q^{81} + 15 q^{83} - 10 q^{85} + 17 q^{87} - 20 q^{89} + 34 q^{91} + 6 q^{93} + 34 q^{95} + 2 q^{97} + 44 q^{99}+O(q^{100}) $ 4 * q + 2 * q^3 - 5 * q^5 + q^7 + 8 * q^9 + 4 * q^11 - 5 * q^13 + 8 * q^17 + 2 * q^19 - q^21 + 9 * q^23 + 7 * q^25 - q^27 - 7 * q^29 + q^31 + 3 * q^33 - 12 * q^35 - 4 * q^37 + 15 * q^39 + 2 * q^41 + 6 * q^43 + 29 * q^47 + 9 * q^49 + 4 * q^51 + 5 * q^53 + 5 * q^55 - 32 * q^57 - 10 * q^59 + q^61 + 28 * q^63 - 2 * q^65 - q^67 + 27 * q^69 + 17 * q^71 + 8 * q^73 + 19 * q^75 + 27 * q^77 - 15 * q^79 - 8 * q^81 + 15 * q^83 - 10 * q^85 + 17 * q^87 - 20 * q^89 + 34 * q^91 + 6 * q^93 + 34 * q^95 + 2 * q^97 + 44 * 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
$3$	−2.73494	−1.57902	−0.789510	−	0.613737i	$-0.789665\pi$
$3$	−2.73494	−1.57902	−0.789510	+	0.613737i	$0.789665\pi$
$4$	0	0
$5$	−0.687447	−0.307436	−0.153718	−	0.988115i	$-0.549125\pi$
$5$	−0.687447	−0.307436	−0.153718	+	0.988115i	$0.549125\pi$
$6$	0	0
$7$	2.16737	0.819188	0.409594	−	0.912268i	$-0.365670\pi$
$7$	2.16737	0.819188	0.409594	+	0.912268i	$0.365670\pi$
$8$	0	0
$9$	4.47992	1.49331
$10$	0	0
$11$	3.47992	1.04924	0.524618	−	0.851338i	$-0.324208\pi$
$11$	3.47992	1.04924	0.524618	+	0.851338i	$0.324208\pi$
$12$	0	0
$13$	−2.56758	−0.712118	−0.356059	−	0.934464i	$-0.615880\pi$
$13$	−2.56758	−0.712118	−0.356059	+	0.934464i	$0.615880\pi$
$14$	0	0
$15$	1.88013	0.485447
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	8.09499	1.85712	0.928559	−	0.371184i	$-0.121048\pi$
$19$	8.09499	1.85712	0.928559	+	0.371184i	$0.121048\pi$
$20$	0	0
$21$	−5.92763	−1.29351
$22$	0	0
$23$	2.05753	0.429024	0.214512	−	0.976721i	$-0.431184\pi$
$23$	2.05753	0.429024	0.214512	+	0.976721i	$0.431184\pi$
$24$	0	0
$25$	−4.52742	−0.905483
$26$	0	0
$27$	−4.04750	−0.778941
$28$	0	0
$29$	−6.90231	−1.28173	−0.640863	−	0.767655i	$-0.721424\pi$
$29$	−6.90231	−1.28173	−0.640863	+	0.767655i	$0.721424\pi$
$30$	0	0
$31$	−3.02218	−0.542800	−0.271400	−	0.962467i	$-0.587487\pi$
$31$	−3.02218	−0.542800	−0.271400	+	0.962467i	$0.587487\pi$
$32$	0	0
$33$	−9.51739	−1.65676
$34$	0	0
$35$	−1.48995	−0.251848
$36$	0	0
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	0	0
$39$	7.02218	1.12445
$40$	0	0
$41$	2.22489	0.347470	0.173735	−	0.984792i	$-0.444416\pi$
$41$	2.22489	0.347470	0.173735	+	0.984792i	$0.444416\pi$
$42$	0	0
$43$	2.62511	0.400325	0.200162	−	0.979763i	$-0.435853\pi$
$43$	2.62511	0.400325	0.200162	+	0.979763i	$0.435853\pi$
$44$	0	0
$45$	−3.07971	−0.459096
$46$	0	0
$47$	−2.26236	−0.329999	−0.165000	−	0.986294i	$-0.552762\pi$
$47$	−2.26236	−0.329999	−0.165000	+	0.986294i	$0.552762\pi$
$48$	0	0
$49$	−2.30252	−0.328932
$50$	0	0
$51$	−5.46989	−0.765938
$52$	0	0
$53$	9.63725	1.32378	0.661889	−	0.749602i	$-0.269755\pi$
$53$	9.63725	1.32378	0.661889	+	0.749602i	$0.269755\pi$
$54$	0	0
$55$	−2.39226	−0.322572
$56$	0	0
$57$	−22.1394	−2.93243
$58$	0	0
$59$	−15.0548	−1.95997	−0.979986	−	0.199066i	$-0.936209\pi$
$59$	−15.0548	−1.95997	−0.979986	+	0.199066i	$0.936209\pi$
$60$	0	0
$61$	1.93766	0.248092	0.124046	−	0.992276i	$-0.460413\pi$
$61$	1.93766	0.248092	0.124046	+	0.992276i	$0.460413\pi$
$62$	0	0
$63$	9.70963	1.22330
$64$	0	0
$65$	1.76507	0.218931
$66$	0	0
$67$	6.78244	0.828607	0.414304	−	0.910139i	$-0.364025\pi$
$67$	6.78244	0.828607	0.414304	+	0.910139i	$0.364025\pi$
$68$	0	0
$69$	−5.62722	−0.677438
$70$	0	0
$71$	14.1473	1.67898	0.839488	−	0.543378i	$-0.182855\pi$
$71$	14.1473	1.67898	0.839488	+	0.543378i	$0.182855\pi$
$72$	0	0
$73$	6.73013	0.787702	0.393851	−	0.919174i	$-0.371143\pi$
$73$	6.73013	0.787702	0.393851	+	0.919174i	$0.371143\pi$
$74$	0	0
$75$	12.3822	1.42978
$76$	0	0
$77$	7.54226	0.859520
$78$	0	0
$79$	−16.8822	−1.89940	−0.949701	−	0.313159i	$-0.898613\pi$
$79$	−16.8822	−1.89940	−0.949701	+	0.313159i	$0.898613\pi$
$80$	0	0
$81$	−2.37008	−0.263342
$82$	0	0
$83$	15.9720	1.75315	0.876577	−	0.481262i	$-0.159821\pi$
$83$	15.9720	1.75315	0.876577	+	0.481262i	$0.159821\pi$
$84$	0	0
$85$	−1.37489	−0.149128
$86$	0	0
$87$	18.8774	2.02387
$88$	0	0
$89$	6.42973	0.681550	0.340775	−	0.940145i	$-0.389311\pi$
$89$	6.42973	0.681550	0.340775	+	0.940145i	$0.389311\pi$
$90$	0	0
$91$	−5.56488	−0.583358
$92$	0	0
$93$	8.26550	0.857092
$94$	0	0
$95$	−5.56488	−0.570945
$96$	0	0
$97$	17.2745	1.75396	0.876980	−	0.480526i	$-0.159554\pi$
$97$	17.2745	1.75396	0.876980	+	0.480526i	$0.159554\pi$
$98$	0	0
$99$	15.5898	1.56683
$100$	0	0
$101$	0.0523109	0.00520513	0.00260257	−	0.999997i	$-0.499172\pi$
$101$	0.0523109	0.00520513	0.00260257	+	0.999997i	$0.499172\pi$
$102$	0	0
$103$	−2.35271	−0.231820	−0.115910	−	0.993260i	$-0.536978\pi$
$103$	−2.35271	−0.231820	−0.115910	+	0.993260i	$0.536978\pi$
$104$	0	0
$105$	4.07493	0.397673
$106$	0	0
$107$	−4.90231	−0.473924	−0.236962	−	0.971519i	$-0.576152\pi$
$107$	−4.90231	−0.473924	−0.236962	+	0.971519i	$0.576152\pi$
$108$	0	0
$109$	−14.1900	−1.35915	−0.679577	−	0.733604i	$-0.737837\pi$
$109$	−14.1900	−1.35915	−0.679577	+	0.733604i	$0.737837\pi$
$110$	0	0
$111$	2.73494	0.259589
$112$	0	0
$113$	−8.93978	−0.840983	−0.420492	−	0.907296i	$-0.638142\pi$
$113$	−8.93978	−0.840983	−0.420492	+	0.907296i	$0.638142\pi$
$114$	0	0
$115$	−1.41444	−0.131897
$116$	0	0
$117$	−11.5025	−1.06341
$118$	0	0
$119$	4.33473	0.397364
$120$	0	0
$121$	1.10984	0.100894
$122$	0	0
$123$	−6.08496	−0.548663
$124$	0	0
$125$	6.54960	0.585814
$126$	0	0
$127$	11.3975	1.01137	0.505683	−	0.862719i	$-0.331241\pi$
$127$	11.3975	1.01137	0.505683	+	0.862719i	$0.331241\pi$
$128$	0	0
$129$	−7.17952	−0.632121
$130$	0	0
$131$	10.7201	0.936619	0.468310	−	0.883564i	$-0.344863\pi$
$131$	10.7201	0.936619	0.468310	+	0.883564i	$0.344863\pi$
$132$	0	0
$133$	17.5448	1.52133
$134$	0	0
$135$	2.78244	0.239474
$136$	0	0
$137$	−5.48787	−0.468860	−0.234430	−	0.972133i	$-0.575322\pi$
$137$	−5.48787	−0.468860	−0.234430	+	0.972133i	$0.575322\pi$
$138$	0	0
$139$	15.6521	1.32759	0.663797	−	0.747913i	$-0.268944\pi$
$139$	15.6521	1.32759	0.663797	+	0.747913i	$0.268944\pi$
$140$	0	0
$141$	6.18743	0.521076
$142$	0	0
$143$	−8.93496	−0.747179
$144$	0	0
$145$	4.74498	0.394049
$146$	0	0
$147$	6.29727	0.519390
$148$	0	0
$149$	−2.34268	−0.191920	−0.0959600	−	0.995385i	$-0.530592\pi$
$149$	−2.34268	−0.191920	−0.0959600	+	0.995385i	$0.530592\pi$
$150$	0	0
$151$	8.31467	0.676638	0.338319	−	0.941031i	$-0.390142\pi$
$151$	8.31467	0.676638	0.338319	+	0.941031i	$0.390142\pi$
$152$	0	0
$153$	8.95984	0.724360
$154$	0	0
$155$	2.07759	0.166876
$156$	0	0
$157$	22.1620	1.76872	0.884359	−	0.466807i	$-0.154596\pi$
$157$	22.1620	1.76872	0.884359	+	0.466807i	$0.154596\pi$
$158$	0	0
$159$	−26.3574	−2.09027
$160$	0	0
$161$	4.45942	0.351451
$162$	0	0
$163$	6.51005	0.509906	0.254953	−	0.966953i	$-0.417940\pi$
$163$	6.51005	0.509906	0.254953	+	0.966953i	$0.417940\pi$
$164$	0	0
$165$	6.54270	0.509349
$166$	0	0
$167$	17.3126	1.33969	0.669843	−	0.742503i	$-0.266361\pi$
$167$	17.3126	1.33969	0.669843	+	0.742503i	$0.266361\pi$
$168$	0	0
$169$	−6.40755	−0.492888
$170$	0	0
$171$	36.2649	2.77325
$172$	0	0
$173$	12.5770	0.956214	0.478107	−	0.878302i	$-0.341323\pi$
$173$	12.5770	0.956214	0.478107	+	0.878302i	$0.341323\pi$
$174$	0	0
$175$	−9.81257	−0.741761
$176$	0	0
$177$	41.1741	3.09484
$178$	0	0
$179$	1.70963	0.127784	0.0638918	−	0.997957i	$-0.479649\pi$
$179$	1.70963	0.127784	0.0638918	+	0.997957i	$0.479649\pi$
$180$	0	0
$181$	16.8168	1.24998	0.624990	−	0.780632i	$-0.285103\pi$
$181$	16.8168	1.24998	0.624990	+	0.780632i	$0.285103\pi$
$182$	0	0
$183$	−5.29939	−0.391742
$184$	0	0
$185$	0.687447	0.0505421
$186$	0	0
$187$	6.95984	0.508954
$188$	0	0
$189$	−8.77241	−0.638099
$190$	0	0
$191$	19.0875	1.38112	0.690561	−	0.723274i	$-0.257364\pi$
$191$	19.0875	1.38112	0.690561	+	0.723274i	$0.257364\pi$
$192$	0	0
$193$	1.42553	0.102612	0.0513058	−	0.998683i	$-0.483662\pi$
$193$	1.42553	0.102612	0.0513058	+	0.998683i	$0.483662\pi$
$194$	0	0
$195$	−4.82738	−0.345696
$196$	0	0
$197$	16.2423	1.15722	0.578608	−	0.815606i	$-0.303596\pi$
$197$	16.2423	1.15722	0.578608	+	0.815606i	$0.303596\pi$
$198$	0	0
$199$	−21.5849	−1.53012	−0.765058	−	0.643961i	$-0.777290\pi$
$199$	−21.5849	−1.53012	−0.765058	+	0.643961i	$0.777290\pi$
$200$	0	0
$201$	−18.5496	−1.30839
$202$	0	0
$203$	−14.9598	−1.04997
$204$	0	0
$205$	−1.52950	−0.106825
$206$	0	0
$207$	9.21756	0.640665
$208$	0	0
$209$	28.1699	1.94855
$210$	0	0
$211$	−18.0850	−1.24502	−0.622510	−	0.782612i	$-0.713887\pi$
$211$	−18.0850	−1.24502	−0.622510	+	0.782612i	$0.713887\pi$
$212$	0	0
$213$	−38.6921	−2.65114
$214$	0	0
$215$	−1.80462	−0.123074
$216$	0	0
$217$	−6.55017	−0.444655
$218$	0	0
$219$	−18.4065	−1.24380
$220$	0	0
$221$	−5.13515	−0.345428
$222$	0	0
$223$	0.907528	0.0607726	0.0303863	−	0.999538i	$-0.490326\pi$
$223$	0.907528	0.0607726	0.0303863	+	0.999538i	$0.490326\pi$
$224$	0	0
$225$	−20.2825	−1.35216
$226$	0	0
$227$	11.0645	0.734374	0.367187	−	0.930147i	$-0.380321\pi$
$227$	11.0645	0.734374	0.367187	+	0.930147i	$0.380321\pi$
$228$	0	0
$229$	−23.4419	−1.54908	−0.774541	−	0.632523i	$-0.782019\pi$
$229$	−23.4419	−1.54908	−0.774541	+	0.632523i	$0.782019\pi$
$230$	0	0
$231$	−20.6277	−1.35720
$232$	0	0
$233$	8.74289	0.572766	0.286383	−	0.958115i	$-0.407547\pi$
$233$	8.74289	0.572766	0.286383	+	0.958115i	$0.407547\pi$
$234$	0	0
$235$	1.55525	0.101454
$236$	0	0
$237$	46.1720	2.99919
$238$	0	0
$239$	15.0021	0.970406	0.485203	−	0.874401i	$-0.338746\pi$
$239$	15.0021	0.970406	0.485203	+	0.874401i	$0.338746\pi$
$240$	0	0
$241$	11.0105	0.709247	0.354623	−	0.935009i	$-0.384609\pi$
$241$	11.0105	0.709247	0.354623	+	0.935009i	$0.384609\pi$
$242$	0	0
$243$	18.6245	1.19476
$244$	0	0
$245$	1.58286	0.101125
$246$	0	0
$247$	−20.7845	−1.32249
$248$	0	0
$249$	−43.6825	−2.76827
$250$	0	0
$251$	−19.6092	−1.23772	−0.618862	−	0.785500i	$-0.712406\pi$
$251$	−19.6092	−1.23772	−0.618862	+	0.785500i	$0.712406\pi$
$252$	0	0
$253$	7.16003	0.450147
$254$	0	0
$255$	3.76026	0.235477
$256$	0	0
$257$	22.5247	1.40505	0.702527	−	0.711657i	$-0.252055\pi$
$257$	22.5247	1.40505	0.702527	+	0.711657i	$0.252055\pi$
$258$	0	0
$259$	−2.16737	−0.134674
$260$	0	0
$261$	−30.9218	−1.91401
$262$	0	0
$263$	16.1979	0.998808	0.499404	−	0.866369i	$-0.333552\pi$
$263$	16.1979	0.998808	0.499404	+	0.866369i	$0.333552\pi$
$264$	0	0
$265$	−6.62511	−0.406977
$266$	0	0
$267$	−17.5849	−1.07618
$268$	0	0
$269$	−1.76026	−0.107325	−0.0536625	−	0.998559i	$-0.517090\pi$
$269$	−1.76026	−0.107325	−0.0536625	+	0.998559i	$0.517090\pi$
$270$	0	0
$271$	26.8071	1.62842	0.814209	−	0.580572i	$-0.197171\pi$
$271$	26.8071	1.62842	0.814209	+	0.580572i	$0.197171\pi$
$272$	0	0
$273$	15.2196	0.921135
$274$	0	0
$275$	−15.7550	−0.950065
$276$	0	0
$277$	−0.687447	−0.0413047	−0.0206524	−	0.999787i	$-0.506574\pi$
$277$	−0.687447	−0.0413047	−0.0206524	+	0.999787i	$0.506574\pi$
$278$	0	0
$279$	−13.5391	−0.810566
$280$	0	0
$281$	2.09499	0.124977	0.0624884	−	0.998046i	$-0.480096\pi$
$281$	2.09499	0.124977	0.0624884	+	0.998046i	$0.480096\pi$
$282$	0	0
$283$	−18.5850	−1.10476	−0.552381	−	0.833592i	$-0.686281\pi$
$283$	−18.5850	−1.10476	−0.552381	+	0.833592i	$0.686281\pi$
$284$	0	0
$285$	15.2196	0.901534
$286$	0	0
$287$	4.82216	0.284643
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	−47.2448	−2.76954
$292$	0	0
$293$	−20.8247	−1.21659	−0.608297	−	0.793710i	$-0.708147\pi$
$293$	−20.8247	−1.21659	−0.608297	+	0.793710i	$0.708147\pi$
$294$	0	0
$295$	10.3494	0.602566
$296$	0	0
$297$	−14.0850	−0.817292
$298$	0	0
$299$	−5.28286	−0.305516
$300$	0	0
$301$	5.68957	0.327941
$302$	0	0
$303$	−0.143067	−0.00821901
$304$	0	0
$305$	−1.33204	−0.0762723
$306$	0	0
$307$	−24.8743	−1.41965	−0.709826	−	0.704377i	$-0.751226\pi$
$307$	−24.8743	−1.41965	−0.709826	+	0.704377i	$0.751226\pi$
$308$	0	0
$309$	6.43454	0.366048
$310$	0	0
$311$	15.7471	0.892936	0.446468	−	0.894800i	$-0.352682\pi$
$311$	15.7471	0.892936	0.446468	+	0.894800i	$0.352682\pi$
$312$	0	0
$313$	−2.68953	−0.152021	−0.0760106	−	0.997107i	$-0.524218\pi$
$313$	−2.68953	−0.152021	−0.0760106	+	0.997107i	$0.524218\pi$
$314$	0	0
$315$	−6.67486	−0.376086
$316$	0	0
$317$	−32.1193	−1.80400	−0.902000	−	0.431737i	$-0.857901\pi$
$317$	−32.1193	−1.80400	−0.902000	+	0.431737i	$0.857901\pi$
$318$	0	0
$319$	−24.0195	−1.34483
$320$	0	0
$321$	13.4075	0.748336
$322$	0	0
$323$	16.1900	0.900835
$324$	0	0
$325$	11.6245	0.644811
$326$	0	0
$327$	38.8088	2.14613
$328$	0	0
$329$	−4.90336	−0.270331
$330$	0	0
$331$	5.00420	0.275056	0.137528	−	0.990498i	$-0.456084\pi$
$331$	5.00420	0.275056	0.137528	+	0.990498i	$0.456084\pi$
$332$	0	0
$333$	−4.47992	−0.245498
$334$	0	0
$335$	−4.66257	−0.254743
$336$	0	0
$337$	−4.43976	−0.241849	−0.120925	−	0.992662i	$-0.538586\pi$
$337$	−4.43976	−0.241849	−0.120925	+	0.992662i	$0.538586\pi$
$338$	0	0
$339$	24.4498	1.32793
$340$	0	0
$341$	−10.5169	−0.569525
$342$	0	0
$343$	−20.1620	−1.08864
$344$	0	0
$345$	3.86842	0.208269
$346$	0	0
$347$	−14.9598	−0.803086	−0.401543	−	0.915840i	$-0.631526\pi$
$347$	−14.9598	−0.803086	−0.401543	+	0.915840i	$0.631526\pi$
$348$	0	0
$349$	2.46569	0.131985	0.0659926	−	0.997820i	$-0.478979\pi$
$349$	2.46569	0.131985	0.0659926	+	0.997820i	$0.478979\pi$
$350$	0	0
$351$	10.3923	0.554698
$352$	0	0
$353$	−26.9344	−1.43357	−0.716786	−	0.697293i	$-0.754388\pi$
$353$	−26.9344	−1.43357	−0.716786	+	0.697293i	$0.754388\pi$
$354$	0	0
$355$	−9.72553	−0.516177
$356$	0	0
$357$	−11.8553	−0.627447
$358$	0	0
$359$	3.20753	0.169287	0.0846434	−	0.996411i	$-0.473025\pi$
$359$	3.20753	0.169287	0.0846434	+	0.996411i	$0.473025\pi$
$360$	0	0
$361$	46.5289	2.44889
$362$	0	0
$363$	−3.03535	−0.159314
$364$	0	0
$365$	−4.62661	−0.242168
$366$	0	0
$367$	−7.53431	−0.393288	−0.196644	−	0.980475i	$-0.563004\pi$
$367$	−7.53431	−0.393288	−0.196644	+	0.980475i	$0.563004\pi$
$368$	0	0
$369$	9.96735	0.518879
$370$	0	0
$371$	20.8875	1.08442
$372$	0	0
$373$	−30.0469	−1.55577	−0.777885	−	0.628406i	$-0.783708\pi$
$373$	−30.0469	−1.55577	−0.777885	+	0.628406i	$0.783708\pi$
$374$	0	0
$375$	−17.9128	−0.925012
$376$	0	0
$377$	17.7222	0.912741
$378$	0	0
$379$	−3.18473	−0.163589	−0.0817944	−	0.996649i	$-0.526065\pi$
$379$	−3.18473	−0.163589	−0.0817944	+	0.996649i	$0.526065\pi$
$380$	0	0
$381$	−31.1716	−1.59697
$382$	0	0
$383$	6.97990	0.356656	0.178328	−	0.983971i	$-0.442931\pi$
$383$	6.97990	0.356656	0.178328	+	0.983971i	$0.442931\pi$
$384$	0	0
$385$	−5.18491	−0.264247
$386$	0	0
$387$	11.7603	0.597808
$388$	0	0
$389$	22.9613	1.16419	0.582093	−	0.813122i	$-0.302234\pi$
$389$	22.9613	1.16419	0.582093	+	0.813122i	$0.302234\pi$
$390$	0	0
$391$	4.11506	0.208107
$392$	0	0
$393$	−29.3189	−1.47894
$394$	0	0
$395$	11.6057	0.583944
$396$	0	0
$397$	31.1071	1.56122	0.780611	−	0.625017i	$-0.214908\pi$
$397$	31.1071	1.56122	0.780611	+	0.625017i	$0.214908\pi$
$398$	0	0
$399$	−47.9841	−2.40221
$400$	0	0
$401$	3.23015	0.161306	0.0806530	−	0.996742i	$-0.474299\pi$
$401$	3.23015	0.161306	0.0806530	+	0.996742i	$0.474299\pi$
$402$	0	0
$403$	7.75968	0.386537
$404$	0	0
$405$	1.62931	0.0809608
$406$	0	0
$407$	−3.47992	−0.172493
$408$	0	0
$409$	−12.3293	−0.609647	−0.304823	−	0.952409i	$-0.598597\pi$
$409$	−12.3293	−0.609647	−0.304823	+	0.952409i	$0.598597\pi$
$410$	0	0
$411$	15.0090	0.740340
$412$	0	0
$413$	−32.6293	−1.60558
$414$	0	0
$415$	−10.9799	−0.538982
$416$	0	0
$417$	−42.8076	−2.09630
$418$	0	0
$419$	31.2300	1.52568	0.762842	−	0.646585i	$-0.223803\pi$
$419$	31.2300	1.52568	0.762842	+	0.646585i	$0.223803\pi$
$420$	0	0
$421$	−29.7270	−1.44881	−0.724403	−	0.689376i	$-0.757885\pi$
$421$	−29.7270	−1.44881	−0.724403	+	0.689376i	$0.757885\pi$
$422$	0	0
$423$	−10.1352	−0.492790
$424$	0	0
$425$	−9.05483	−0.439224
$426$	0	0
$427$	4.19962	0.203234
$428$	0	0
$429$	24.4366	1.17981
$430$	0	0
$431$	6.93978	0.334277	0.167139	−	0.985933i	$-0.446547\pi$
$431$	6.93978	0.334277	0.167139	+	0.985933i	$0.446547\pi$
$432$	0	0
$433$	−13.7454	−0.660562	−0.330281	−	0.943883i	$-0.607144\pi$
$433$	−13.7454	−0.660562	−0.330281	+	0.943883i	$0.607144\pi$
$434$	0	0
$435$	−12.9772	−0.622211
$436$	0	0
$437$	16.6557	0.796749
$438$	0	0
$439$	9.33204	0.445394	0.222697	−	0.974888i	$-0.428514\pi$
$439$	9.33204	0.445394	0.222697	+	0.974888i	$0.428514\pi$
$440$	0	0
$441$	−10.3151	−0.491196
$442$	0	0
$443$	36.7101	1.74415	0.872074	−	0.489374i	$-0.162775\pi$
$443$	36.7101	1.74415	0.872074	+	0.489374i	$0.162775\pi$
$444$	0	0
$445$	−4.42010	−0.209533
$446$	0	0
$447$	6.40711	0.303046
$448$	0	0
$449$	7.25021	0.342159	0.171079	−	0.985257i	$-0.445275\pi$
$449$	7.25021	0.342159	0.171079	+	0.985257i	$0.445275\pi$
$450$	0	0
$451$	7.74245	0.364578
$452$	0	0
$453$	−22.7402	−1.06843
$454$	0	0
$455$	3.82556	0.179345
$456$	0	0
$457$	14.8394	0.694157	0.347079	−	0.937836i	$-0.387174\pi$
$457$	14.8394	0.694157	0.347079	+	0.937836i	$0.387174\pi$
$458$	0	0
$459$	−8.09499	−0.377842
$460$	0	0
$461$	−26.7548	−1.24610	−0.623048	−	0.782183i	$-0.714106\pi$
$461$	−26.7548	−1.24610	−0.623048	+	0.782183i	$0.714106\pi$
$462$	0	0
$463$	23.9772	1.11432	0.557158	−	0.830407i	$-0.311892\pi$
$463$	23.9772	1.11432	0.557158	+	0.830407i	$0.311892\pi$
$464$	0	0
$465$	−5.68209	−0.263501
$466$	0	0
$467$	0.725490	0.0335717	0.0167858	−	0.999859i	$-0.494657\pi$
$467$	0.725490	0.0335717	0.0167858	+	0.999859i	$0.494657\pi$
$468$	0	0
$469$	14.7000	0.678785
$470$	0	0
$471$	−60.6118	−2.79284
$472$	0	0
$473$	9.13515	0.420035
$474$	0	0
$475$	−36.6494	−1.68159
$476$	0	0
$477$	43.1741	1.97681
$478$	0	0
$479$	1.36173	0.0622190	0.0311095	−	0.999516i	$-0.490096\pi$
$479$	1.36173	0.0622190	0.0311095	+	0.999516i	$0.490096\pi$
$480$	0	0
$481$	2.56758	0.117071
$482$	0	0
$483$	−12.1963	−0.554949
$484$	0	0
$485$	−11.8753	−0.539230
$486$	0	0
$487$	−15.8553	−0.718470	−0.359235	−	0.933247i	$-0.616962\pi$
$487$	−15.8553	−0.718470	−0.359235	+	0.933247i	$0.616962\pi$
$488$	0	0
$489$	−17.8046	−0.805153
$490$	0	0
$491$	−19.0977	−0.861867	−0.430933	−	0.902384i	$-0.641816\pi$
$491$	−19.0977	−0.861867	−0.430933	+	0.902384i	$0.641816\pi$
$492$	0	0
$493$	−13.8046	−0.621729
$494$	0	0
$495$	−10.7171	−0.481700
$496$	0	0
$497$	30.6624	1.37540
$498$	0	0
$499$	−15.0548	−0.673947	−0.336973	−	0.941514i	$-0.609403\pi$
$499$	−15.0548	−0.673947	−0.336973	+	0.941514i	$0.609403\pi$
$500$	0	0
$501$	−47.3489	−2.11539
$502$	0	0
$503$	28.3770	1.26527	0.632634	−	0.774451i	$-0.281974\pi$
$503$	28.3770	1.26527	0.632634	+	0.774451i	$0.281974\pi$
$504$	0	0
$505$	−0.0359610	−0.00160024
$506$	0	0
$507$	17.5243	0.778281
$508$	0	0
$509$	42.8114	1.89758	0.948791	−	0.315906i	$-0.102308\pi$
$509$	42.8114	1.89758	0.948791	+	0.315906i	$0.102308\pi$
$510$	0	0
$511$	14.5867	0.645276
$512$	0	0
$513$	−32.7645	−1.44659
$514$	0	0
$515$	1.61737	0.0712697
$516$	0	0
$517$	−7.87283	−0.346247
$518$	0	0
$519$	−34.3975	−1.50988
$520$	0	0
$521$	−4.28242	−0.187616	−0.0938082	−	0.995590i	$-0.529904\pi$
$521$	−4.28242	−0.187616	−0.0938082	+	0.995590i	$0.529904\pi$
$522$	0	0
$523$	17.3251	0.757576	0.378788	−	0.925484i	$-0.376341\pi$
$523$	17.3251	0.757576	0.378788	+	0.925484i	$0.376341\pi$
$524$	0	0
$525$	26.8368	1.17126
$526$	0	0
$527$	−6.04436	−0.263297
$528$	0	0
$529$	−18.7666	−0.815938
$530$	0	0
$531$	−67.4444	−2.92684
$532$	0	0
$533$	−5.71259	−0.247440
$534$	0	0
$535$	3.37008	0.145701
$536$	0	0
$537$	−4.67574	−0.201773
$538$	0	0
$539$	−8.01259	−0.345127
$540$	0	0
$541$	35.2766	1.51666	0.758330	−	0.651871i	$-0.226016\pi$
$541$	35.2766	1.51666	0.758330	+	0.651871i	$0.226016\pi$
$542$	0	0
$543$	−45.9929	−1.97375
$544$	0	0
$545$	9.75487	0.417853
$546$	0	0
$547$	−5.14478	−0.219975	−0.109988	−	0.993933i	$-0.535081\pi$
$547$	−5.14478	−0.219975	−0.109988	+	0.993933i	$0.535081\pi$
$548$	0	0
$549$	8.68055	0.370477
$550$	0	0
$551$	−55.8742	−2.38032
$552$	0	0
$553$	−36.5900	−1.55597
$554$	0	0
$555$	−1.88013	−0.0798071
$556$	0	0
$557$	2.85314	0.120891	0.0604456	−	0.998171i	$-0.480748\pi$
$557$	2.85314	0.120891	0.0604456	+	0.998171i	$0.480748\pi$
$558$	0	0
$559$	−6.74016	−0.285078
$560$	0	0
$561$	−19.0348	−0.803649
$562$	0	0
$563$	−3.76026	−0.158476	−0.0792380	−	0.996856i	$-0.525249\pi$
$563$	−3.76026	−0.158476	−0.0792380	+	0.996856i	$0.525249\pi$
$564$	0	0
$565$	6.14563	0.258548
$566$	0	0
$567$	−5.13683	−0.215727
$568$	0	0
$569$	25.6155	1.07386	0.536929	−	0.843627i	$-0.319584\pi$
$569$	25.6155	1.07386	0.536929	+	0.843627i	$0.319584\pi$
$570$	0	0
$571$	−33.8195	−1.41530	−0.707650	−	0.706563i	$-0.750245\pi$
$571$	−33.8195	−1.41530	−0.707650	+	0.706563i	$0.750245\pi$
$572$	0	0
$573$	−52.2032	−2.18082
$574$	0	0
$575$	−9.31529	−0.388474
$576$	0	0
$577$	13.2544	0.551790	0.275895	−	0.961188i	$-0.411026\pi$
$577$	13.2544	0.551790	0.275895	+	0.961188i	$0.411026\pi$
$578$	0	0
$579$	−3.89874	−0.162026
$580$	0	0
$581$	34.6172	1.43616
$582$	0	0
$583$	33.5369	1.38896
$584$	0	0
$585$	7.90739	0.326930
$586$	0	0
$587$	−8.34436	−0.344409	−0.172204	−	0.985061i	$-0.555089\pi$
$587$	−8.34436	−0.344409	−0.172204	+	0.985061i	$0.555089\pi$
$588$	0	0
$589$	−24.4645	−1.00804
$590$	0	0
$591$	−44.4218	−1.82727
$592$	0	0
$593$	20.8591	0.856579	0.428289	−	0.903642i	$-0.359116\pi$
$593$	20.8591	0.856579	0.428289	+	0.903642i	$0.359116\pi$
$594$	0	0
$595$	−2.97990	−0.122164
$596$	0	0
$597$	59.0336	2.41609
$598$	0	0
$599$	15.0469	0.614799	0.307399	−	0.951581i	$-0.400541\pi$
$599$	15.0469	0.614799	0.307399	+	0.951581i	$0.400541\pi$
$600$	0	0
$601$	24.4318	0.996594	0.498297	−	0.867006i	$-0.333959\pi$
$601$	24.4318	0.996594	0.498297	+	0.867006i	$0.333959\pi$
$602$	0	0
$603$	30.3848	1.23736
$604$	0	0
$605$	−0.762956	−0.0310186
$606$	0	0
$607$	−26.2475	−1.06535	−0.532677	−	0.846319i	$-0.678814\pi$
$607$	−26.2475	−1.06535	−0.532677	+	0.846319i	$0.678814\pi$
$608$	0	0
$609$	40.9143	1.65793
$610$	0	0
$611$	5.80879	0.234998
$612$	0	0
$613$	1.41758	0.0572554	0.0286277	−	0.999590i	$-0.490886\pi$
$613$	1.41758	0.0572554	0.0286277	+	0.999590i	$0.490886\pi$
$614$	0	0
$615$	4.18309	0.168679
$616$	0	0
$617$	3.76971	0.151763	0.0758815	−	0.997117i	$-0.475823\pi$
$617$	3.76971	0.151763	0.0758815	+	0.997117i	$0.475823\pi$
$618$	0	0
$619$	33.8390	1.36010	0.680051	−	0.733165i	$-0.261958\pi$
$619$	33.8390	1.36010	0.680051	+	0.733165i	$0.261958\pi$
$620$	0	0
$621$	−8.32784	−0.334185
$622$	0	0
$623$	13.9356	0.558317
$624$	0	0
$625$	18.1346	0.725383
$626$	0	0
$627$	−77.0432	−3.07681
$628$	0	0
$629$	−2.00000	−0.0797452
$630$	0	0
$631$	−8.21278	−0.326946	−0.163473	−	0.986548i	$-0.552270\pi$
$631$	−8.21278	−0.326946	−0.163473	+	0.986548i	$0.552270\pi$
$632$	0	0
$633$	49.4614	1.96591
$634$	0	0
$635$	−7.83519	−0.310930
$636$	0	0
$637$	5.91190	0.234238
$638$	0	0
$639$	63.3788	2.50723
$640$	0	0
$641$	−3.25543	−0.128582	−0.0642908	−	0.997931i	$-0.520479\pi$
$641$	−3.25543	−0.128582	−0.0642908	+	0.997931i	$0.520479\pi$
$642$	0	0
$643$	−26.7201	−1.05374	−0.526869	−	0.849947i	$-0.676634\pi$
$643$	−26.7201	−1.05374	−0.526869	+	0.849947i	$0.676634\pi$
$644$	0	0
$645$	4.93554	0.194337
$646$	0	0
$647$	−2.24752	−0.0883589	−0.0441795	−	0.999024i	$-0.514067\pi$
$647$	−2.24752	−0.0883589	−0.0441795	+	0.999024i	$0.514067\pi$
$648$	0	0
$649$	−52.3896	−2.05647
$650$	0	0
$651$	17.9144	0.702119
$652$	0	0
$653$	40.7522	1.59476	0.797378	−	0.603480i	$-0.206220\pi$
$653$	40.7522	1.59476	0.797378	+	0.603480i	$0.206220\pi$
$654$	0	0
$655$	−7.36950	−0.287950
$656$	0	0
$657$	30.1504	1.17628
$658$	0	0
$659$	−7.70945	−0.300318	−0.150159	−	0.988662i	$-0.547979\pi$
$659$	−7.70945	−0.300318	−0.150159	+	0.988662i	$0.547979\pi$
$660$	0	0
$661$	24.5919	0.956513	0.478257	−	0.878220i	$-0.341269\pi$
$661$	24.5919	0.956513	0.478257	+	0.878220i	$0.341269\pi$
$662$	0	0
$663$	14.0444	0.545438
$664$	0	0
$665$	−12.0611	−0.467711
$666$	0	0
$667$	−14.2017	−0.549892
$668$	0	0
$669$	−2.48204	−0.0959611
$670$	0	0
$671$	6.74289	0.260307
$672$	0	0
$673$	−8.04542	−0.310128	−0.155064	−	0.987904i	$-0.549558\pi$
$673$	−8.04542	−0.310128	−0.155064	+	0.987904i	$0.549558\pi$
$674$	0	0
$675$	18.3247	0.705318
$676$	0	0
$677$	−42.7469	−1.64290	−0.821448	−	0.570283i	$-0.806833\pi$
$677$	−42.7469	−1.64290	−0.821448	+	0.570283i	$0.806833\pi$
$678$	0	0
$679$	37.4402	1.43682
$680$	0	0
$681$	−30.2607	−1.15959
$682$	0	0
$683$	41.5289	1.58906	0.794530	−	0.607225i	$-0.207717\pi$
$683$	41.5289	1.58906	0.794530	+	0.607225i	$0.207717\pi$
$684$	0	0
$685$	3.77262	0.144144
$686$	0	0
$687$	64.1122	2.44603
$688$	0	0
$689$	−24.7444	−0.942686
$690$	0	0
$691$	17.0201	0.647475	0.323738	−	0.946147i	$-0.395061\pi$
$691$	17.0201	0.647475	0.323738	+	0.946147i	$0.395061\pi$
$692$	0	0
$693$	33.7887	1.28353
$694$	0	0
$695$	−10.7600	−0.408150
$696$	0	0
$697$	4.44979	0.168548
$698$	0	0
$699$	−23.9113	−0.904409
$700$	0	0
$701$	4.52264	0.170818	0.0854089	−	0.996346i	$-0.472780\pi$
$701$	4.52264	0.170818	0.0854089	+	0.996346i	$0.472780\pi$
$702$	0	0
$703$	−8.09499	−0.305308
$704$	0	0
$705$	−4.25353	−0.160197
$706$	0	0
$707$	0.113377	0.00426398
$708$	0	0
$709$	−8.31675	−0.312342	−0.156171	−	0.987730i	$-0.549915\pi$
$709$	−8.31675	−0.312342	−0.156171	+	0.987730i	$0.549915\pi$
$710$	0	0
$711$	−75.6311	−2.83639
$712$	0	0
$713$	−6.21822	−0.232874
$714$	0	0
$715$	6.14232	0.229710
$716$	0	0
$717$	−41.0300	−1.53229
$718$	0	0
$719$	−5.36695	−0.200153	−0.100077	−	0.994980i	$-0.531909\pi$
$719$	−5.36695	−0.200153	−0.100077	+	0.994980i	$0.531909\pi$
$720$	0	0
$721$	−5.09919	−0.189904
$722$	0	0
$723$	−30.1130	−1.11992
$724$	0	0
$725$	31.2496	1.16058
$726$	0	0
$727$	−4.02222	−0.149176	−0.0745879	−	0.997214i	$-0.523764\pi$
$727$	−4.02222	−0.149176	−0.0745879	+	0.997214i	$0.523764\pi$
$728$	0	0
$729$	−43.8268	−1.62321
$730$	0	0
$731$	5.25021	0.194186
$732$	0	0
$733$	40.6817	1.50261	0.751306	−	0.659954i	$-0.229424\pi$
$733$	40.6817	1.50261	0.751306	+	0.659954i	$0.229424\pi$
$734$	0	0
$735$	−4.32904	−0.159679
$736$	0	0
$737$	23.6023	0.869404
$738$	0	0
$739$	−1.29081	−0.0474833	−0.0237416	−	0.999718i	$-0.507558\pi$
$739$	−1.29081	−0.0474833	−0.0237416	+	0.999718i	$0.507558\pi$
$740$	0	0
$741$	56.8445	2.08824
$742$	0	0
$743$	−23.9360	−0.878128	−0.439064	−	0.898456i	$-0.644690\pi$
$743$	−23.9360	−0.878128	−0.439064	+	0.898456i	$0.644690\pi$
$744$	0	0
$745$	1.61047	0.0590031
$746$	0	0
$747$	71.5532	2.61800
$748$	0	0
$749$	−10.6251	−0.388233
$750$	0	0
$751$	15.4862	0.565101	0.282550	−	0.959252i	$-0.408820\pi$
$751$	15.4862	0.565101	0.282550	+	0.959252i	$0.408820\pi$
$752$	0	0
$753$	53.6302	1.95439
$754$	0	0
$755$	−5.71590	−0.208023
$756$	0	0
$757$	14.5871	0.530176	0.265088	−	0.964224i	$-0.414599\pi$
$757$	14.5871	0.530176	0.265088	+	0.964224i	$0.414599\pi$
$758$	0	0
$759$	−19.5823	−0.710792
$760$	0	0
$761$	−23.5237	−0.852734	−0.426367	−	0.904550i	$-0.640207\pi$
$761$	−23.5237	−0.852734	−0.426367	+	0.904550i	$0.640207\pi$
$762$	0	0
$763$	−30.7549	−1.11340
$764$	0	0
$765$	−6.15942	−0.222694
$766$	0	0
$767$	38.6544	1.39573
$768$	0	0
$769$	−16.7097	−0.602566	−0.301283	−	0.953535i	$-0.597415\pi$
$769$	−16.7097	−0.602566	−0.301283	+	0.953535i	$0.597415\pi$
$770$	0	0
$771$	−61.6039	−2.21861
$772$	0	0
$773$	−38.6076	−1.38862	−0.694309	−	0.719677i	$-0.744290\pi$
$773$	−38.6076	−1.38862	−0.694309	+	0.719677i	$0.744290\pi$
$774$	0	0
$775$	13.6827	0.491496
$776$	0	0
$777$	5.92763	0.212652
$778$	0	0
$779$	18.0105	0.645293
$780$	0	0
$781$	49.2315	1.76164
$782$	0	0
$783$	27.9371	0.998390
$784$	0	0
$785$	−15.2352	−0.543767
$786$	0	0
$787$	15.2465	0.543479	0.271739	−	0.962371i	$-0.412401\pi$
$787$	15.2465	0.543479	0.271739	+	0.962371i	$0.412401\pi$
$788$	0	0
$789$	−44.3005	−1.57714
$790$	0	0
$791$	−19.3758	−0.688923
$792$	0	0
$793$	−4.97509	−0.176671
$794$	0	0
$795$	18.1193	0.642625
$796$	0	0
$797$	30.9877	1.09764	0.548820	−	0.835941i	$-0.315077\pi$
$797$	30.9877	1.09764	0.548820	+	0.835941i	$0.315077\pi$
$798$	0	0
$799$	−4.52472	−0.160073
$800$	0	0
$801$	28.8047	1.01776
$802$	0	0
$803$	23.4203	0.826485
$804$	0	0
$805$	−3.06561	−0.108049
$806$	0	0
$807$	4.81421	0.169468
$808$	0	0
$809$	−44.0042	−1.54711	−0.773553	−	0.633731i	$-0.781523\pi$
$809$	−44.0042	−1.54711	−0.773553	+	0.633731i	$0.781523\pi$
$810$	0	0
$811$	50.6984	1.78026	0.890131	−	0.455705i	$-0.150613\pi$
$811$	50.6984	1.78026	0.890131	+	0.455705i	$0.150613\pi$
$812$	0	0
$813$	−73.3160	−2.57131
$814$	0	0
$815$	−4.47532	−0.156763
$816$	0	0
$817$	21.2502	0.743451
$818$	0	0
$819$	−24.9302	−0.871132
$820$	0	0
$821$	23.3669	0.815512	0.407756	−	0.913091i	$-0.366311\pi$
$821$	23.3669	0.815512	0.407756	+	0.913091i	$0.366311\pi$
$822$	0	0
$823$	5.49958	0.191703	0.0958516	−	0.995396i	$-0.469443\pi$
$823$	5.49958	0.191703	0.0958516	+	0.995396i	$0.469443\pi$
$824$	0	0
$825$	43.0892	1.50017
$826$	0	0
$827$	49.9386	1.73653	0.868267	−	0.496097i	$-0.165234\pi$
$827$	49.9386	1.73653	0.868267	+	0.496097i	$0.165234\pi$
$828$	0	0
$829$	17.2580	0.599395	0.299697	−	0.954034i	$-0.403114\pi$
$829$	17.2580	0.599395	0.299697	+	0.954034i	$0.403114\pi$
$830$	0	0
$831$	1.88013	0.0652210
$832$	0	0
$833$	−4.60504	−0.159555
$834$	0	0
$835$	−11.9015	−0.411867
$836$	0	0
$837$	12.2323	0.422809
$838$	0	0
$839$	0.544783	0.0188080	0.00940400	−	0.999956i	$-0.497007\pi$
$839$	0.544783	0.0188080	0.00940400	+	0.999956i	$0.497007\pi$
$840$	0	0
$841$	18.6419	0.642824
$842$	0	0
$843$	−5.72969	−0.197341
$844$	0	0
$845$	4.40485	0.151531
$846$	0	0
$847$	2.40543	0.0826515
$848$	0	0
$849$	50.8289	1.74444
$850$	0	0
$851$	−2.05753	−0.0705311
$852$	0	0
$853$	−42.4624	−1.45388	−0.726942	−	0.686699i	$-0.759059\pi$
$853$	−42.4624	−1.45388	−0.726942	+	0.686699i	$0.759059\pi$
$854$	0	0
$855$	−24.9302	−0.852596
$856$	0	0
$857$	−7.52892	−0.257183	−0.128592	−	0.991698i	$-0.541046\pi$
$857$	−7.52892	−0.257183	−0.128592	+	0.991698i	$0.541046\pi$
$858$	0	0
$859$	−21.5352	−0.734771	−0.367386	−	0.930069i	$-0.619747\pi$
$859$	−21.5352	−0.734771	−0.367386	+	0.930069i	$0.619747\pi$
$860$	0	0
$861$	−13.1883	−0.449458
$862$	0	0
$863$	−22.3590	−0.761110	−0.380555	−	0.924758i	$-0.624267\pi$
$863$	−22.3590	−0.761110	−0.380555	+	0.924758i	$0.624267\pi$
$864$	0	0
$865$	−8.64605	−0.293974
$866$	0	0
$867$	35.5543	1.20749
$868$	0	0
$869$	−58.7489	−1.99292
$870$	0	0
$871$	−17.4144	−0.590066
$872$	0	0
$873$	77.3884	2.61920
$874$	0	0
$875$	14.1954	0.479891
$876$	0	0
$877$	−47.2796	−1.59652	−0.798259	−	0.602315i	$-0.794245\pi$
$877$	−47.2796	−1.59652	−0.798259	+	0.602315i	$0.794245\pi$
$878$	0	0
$879$	56.9544	1.92103
$880$	0	0
$881$	−33.1624	−1.11727	−0.558635	−	0.829414i	$-0.688675\pi$
$881$	−33.1624	−1.11727	−0.558635	+	0.829414i	$0.688675\pi$
$882$	0	0
$883$	−8.00000	−0.269221	−0.134611	−	0.990899i	$-0.542978\pi$
$883$	−8.00000	−0.269221	−0.134611	+	0.990899i	$0.542978\pi$
$884$	0	0
$885$	−28.3050	−0.951464
$886$	0	0
$887$	23.2963	0.782212	0.391106	−	0.920346i	$-0.372093\pi$
$887$	23.2963	0.782212	0.391106	+	0.920346i	$0.372093\pi$
$888$	0	0
$889$	24.7026	0.828498
$890$	0	0
$891$	−8.24769	−0.276308
$892$	0	0
$893$	−18.3138	−0.612848
$894$	0	0
$895$	−1.17528	−0.0392853
$896$	0	0
$897$	14.4483	0.482416
$898$	0	0
$899$	20.8600	0.695721
$900$	0	0
$901$	19.2745	0.642127
$902$	0	0
$903$	−15.5606	−0.517826
$904$	0	0
$905$	−11.5606	−0.384289
$906$	0	0
$907$	−52.5603	−1.74524	−0.872619	−	0.488402i	$-0.837580\pi$
$907$	−52.5603	−1.74524	−0.872619	+	0.488402i	$0.837580\pi$
$908$	0	0
$909$	0.234349	0.00777285
$910$	0	0
$911$	−16.5406	−0.548015	−0.274008	−	0.961728i	$-0.588349\pi$
$911$	−16.5406	−0.548015	−0.274008	+	0.961728i	$0.588349\pi$
$912$	0	0
$913$	55.5812	1.83947
$914$	0	0
$915$	3.64305	0.120435
$916$	0	0
$917$	23.2344	0.767267
$918$	0	0
$919$	−23.8394	−0.786388	−0.393194	−	0.919456i	$-0.628630\pi$
$919$	−23.8394	−0.786388	−0.393194	+	0.919456i	$0.628630\pi$
$920$	0	0
$921$	68.0298	2.24166
$922$	0	0
$923$	−36.3243	−1.19563
$924$	0	0
$925$	4.52742	0.148861
$926$	0	0
$927$	−10.5400	−0.346178
$928$	0	0
$929$	−27.7222	−0.909536	−0.454768	−	0.890610i	$-0.650278\pi$
$929$	−27.7222	−0.909536	−0.454768	+	0.890610i	$0.650278\pi$
$930$	0	0
$931$	−18.6389	−0.610865
$932$	0	0
$933$	−43.0674	−1.40996
$934$	0	0
$935$	−4.78452	−0.156471
$936$	0	0
$937$	35.0601	1.14536	0.572682	−	0.819778i	$-0.305903\pi$
$937$	35.0601	1.14536	0.572682	+	0.819778i	$0.305903\pi$
$938$	0	0
$939$	7.35571	0.240045
$940$	0	0
$941$	6.78452	0.221169	0.110585	−	0.993867i	$-0.464728\pi$
$941$	6.78452	0.221169	0.110585	+	0.993867i	$0.464728\pi$
$942$	0	0
$943$	4.57778	0.149073
$944$	0	0
$945$	6.03057	0.196174
$946$	0	0
$947$	−21.8637	−0.710473	−0.355237	−	0.934776i	$-0.615600\pi$
$947$	−21.8637	−0.710473	−0.355237	+	0.934776i	$0.615600\pi$
$948$	0	0
$949$	−17.2801	−0.560937
$950$	0	0
$951$	87.8445	2.84855
$952$	0	0
$953$	9.91592	0.321208	0.160604	−	0.987019i	$-0.448656\pi$
$953$	9.91592	0.321208	0.160604	+	0.987019i	$0.448656\pi$
$954$	0	0
$955$	−13.1216	−0.424606
$956$	0	0
$957$	65.6919	2.12352
$958$	0	0
$959$	−11.8942	−0.384085
$960$	0	0
$961$	−21.8664	−0.705368
$962$	0	0
$963$	−21.9620	−0.707714
$964$	0	0
$965$	−0.979975	−0.0315465
$966$	0	0
$967$	14.7720	0.475036	0.237518	−	0.971383i	$-0.423666\pi$
$967$	14.7720	0.475036	0.237518	+	0.971383i	$0.423666\pi$
$968$	0	0
$969$	−44.2787	−1.42244
$970$	0	0
$971$	−12.8921	−0.413727	−0.206864	−	0.978370i	$-0.566326\pi$
$971$	−12.8921	−0.413727	−0.206864	+	0.978370i	$0.566326\pi$
$972$	0	0
$973$	33.9238	1.08755
$974$	0	0
$975$	−31.7923	−1.01817
$976$	0	0
$977$	−26.5648	−0.849885	−0.424942	−	0.905220i	$-0.639706\pi$
$977$	−26.5648	−0.849885	−0.424942	+	0.905220i	$0.639706\pi$
$978$	0	0
$979$	22.3749	0.715106
$980$	0	0
$981$	−63.5700	−2.02963
$982$	0	0
$983$	26.3469	0.840336	0.420168	−	0.907446i	$-0.361971\pi$
$983$	26.3469	0.840336	0.420168	+	0.907446i	$0.361971\pi$
$984$	0	0
$985$	−11.1657	−0.355770
$986$	0	0
$987$	13.4104	0.426859
$988$	0	0
$989$	5.40123	0.171749
$990$	0	0
$991$	8.41320	0.267254	0.133627	−	0.991032i	$-0.457338\pi$
$991$	8.41320	0.267254	0.133627	+	0.991032i	$0.457338\pi$
$992$	0	0
$993$	−13.6862	−0.434319
$994$	0	0
$995$	14.8385	0.470412
$996$	0	0
$997$	−13.3305	−0.422182	−0.211091	−	0.977466i	$-0.567702\pi$
$997$	−13.3305	−0.422182	−0.211091	+	0.977466i	$0.567702\pi$
$998$	0	0
$999$	4.04750	0.128057

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2368.2.a.bh.1.1		4
4.3	odd	2		2368.2.a.bg.1.4		4
8.3	odd	2		296.2.a.d.1.1	✓	4
8.5	even	2		592.2.a.j.1.4		4
24.5	odd	2		5328.2.a.bp.1.3		4
24.11	even	2		2664.2.a.r.1.3		4
40.19	odd	2		7400.2.a.n.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
296.2.a.d.1.1	✓	4	8.3	odd	2
592.2.a.j.1.4		4	8.5	even	2
2368.2.a.bg.1.4		4	4.3	odd	2
2368.2.a.bh.1.1		4	1.1	even	1	trivial
2664.2.a.r.1.3		4	24.11	even	2
5328.2.a.bp.1.3		4	24.5	odd	2
7400.2.a.n.1.4		4	40.19	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:	\( N \)	\(=\)	\( 2368 = 2^{6} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2368.a (trivial)

\(f(q)\)	\(=\)	\(q-2.73494 q^{3} -0.687447 q^{5} +2.16737 q^{7} +4.47992 q^{9} +O(q^{10})\)	\(q-2.73494 q^{3} -0.687447 q^{5} +2.16737 q^{7} +4.47992 q^{9} +3.47992 q^{11} -2.56758 q^{13} +1.88013 q^{15} +2.00000 q^{17} +8.09499 q^{19} -5.92763 q^{21} +2.05753 q^{23} -4.52742 q^{25} -4.04750 q^{27} -6.90231 q^{29} -3.02218 q^{31} -9.51739 q^{33} -1.48995 q^{35} -1.00000 q^{37} +7.02218 q^{39} +2.22489 q^{41} +2.62511 q^{43} -3.07971 q^{45} -2.26236 q^{47} -2.30252 q^{49} -5.46989 q^{51} +9.63725 q^{53} -2.39226 q^{55} -22.1394 q^{57} -15.0548 q^{59} +1.93766 q^{61} +9.70963 q^{63} +1.76507 q^{65} +6.78244 q^{67} -5.62722 q^{69} +14.1473 q^{71} +6.73013 q^{73} +12.3822 q^{75} +7.54226 q^{77} -16.8822 q^{79} -2.37008 q^{81} +15.9720 q^{83} -1.37489 q^{85} +18.8774 q^{87} +6.42973 q^{89} -5.56488 q^{91} +8.26550 q^{93} -5.56488 q^{95} +17.2745 q^{97} +15.5898 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} - 5 q^{5} + q^{7} + 8 q^{9}+O(q^{10}) \) 4 * q + 2 * q^3 - 5 * q^5 + q^7 + 8 * q^9	\( 4 q + 2 q^{3} - 5 q^{5} + q^{7} + 8 q^{9} + 4 q^{11} - 5 q^{13} + 8 q^{17} + 2 q^{19} - q^{21} + 9 q^{23} + 7 q^{25} - q^{27} - 7 q^{29} + q^{31} + 3 q^{33} - 12 q^{35} - 4 q^{37} + 15 q^{39} + 2 q^{41} + 6 q^{43} + 29 q^{47} + 9 q^{49} + 4 q^{51} + 5 q^{53} + 5 q^{55} - 32 q^{57} - 10 q^{59} + q^{61} + 28 q^{63} - 2 q^{65} - q^{67} + 27 q^{69} + 17 q^{71} + 8 q^{73} + 19 q^{75} + 27 q^{77} - 15 q^{79} - 8 q^{81} + 15 q^{83} - 10 q^{85} + 17 q^{87} - 20 q^{89} + 34 q^{91} + 6 q^{93} + 34 q^{95} + 2 q^{97} + 44 q^{99}+O(q^{100}) \) 4 * q + 2 * q^3 - 5 * q^5 + q^7 + 8 * q^9 + 4 * q^11 - 5 * q^13 + 8 * q^17 + 2 * q^19 - q^21 + 9 * q^23 + 7 * q^25 - q^27 - 7 * q^29 + q^31 + 3 * q^33 - 12 * q^35 - 4 * q^37 + 15 * q^39 + 2 * q^41 + 6 * q^43 + 29 * q^47 + 9 * q^49 + 4 * q^51 + 5 * q^53 + 5 * q^55 - 32 * q^57 - 10 * q^59 + q^61 + 28 * q^63 - 2 * q^65 - q^67 + 27 * q^69 + 17 * q^71 + 8 * q^73 + 19 * q^75 + 27 * q^77 - 15 * q^79 - 8 * q^81 + 15 * q^83 - 10 * q^85 + 17 * q^87 - 20 * q^89 + 34 * q^91 + 6 * q^93 + 34 * q^95 + 2 * q^97 + 44 * q^99

Introduction

L-functions

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