LMFDB - Embedded newform 1856.4.a.z.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1856,4,Mod(1,1856)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1856, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1856.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1856 = 2^{6} \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1856.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:	$109.507544971$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 34x^{3} + 74x^{2} + 94x - 198 $ x^5 - x^4 - 34x^3 + 74x^2 + 94*x - 198
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 232)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-6.05843$ of defining polynomial
Character	$\chi$	$=$	1856.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.90002 q^{3} +14.9988 q^{5} -6.82211 q^{7} +7.81028 q^{9} +O(q^{10})$	$q+5.90002 q^{3} +14.9988 q^{5} -6.82211 q^{7} +7.81028 q^{9} +18.6870 q^{11} +39.8582 q^{13} +88.4931 q^{15} +19.3391 q^{17} -2.14572 q^{19} -40.2506 q^{21} +103.945 q^{23} +99.9629 q^{25} -113.220 q^{27} -29.0000 q^{29} +213.167 q^{31} +110.254 q^{33} -102.323 q^{35} +226.054 q^{37} +235.164 q^{39} -191.267 q^{41} -34.3923 q^{43} +117.145 q^{45} +501.173 q^{47} -296.459 q^{49} +114.101 q^{51} -259.952 q^{53} +280.282 q^{55} -12.6598 q^{57} -280.736 q^{59} -81.0752 q^{61} -53.2826 q^{63} +597.824 q^{65} +799.332 q^{67} +613.278 q^{69} -575.901 q^{71} -22.5292 q^{73} +589.783 q^{75} -127.485 q^{77} -118.878 q^{79} -878.877 q^{81} -235.673 q^{83} +290.063 q^{85} -171.101 q^{87} +1153.23 q^{89} -271.917 q^{91} +1257.69 q^{93} -32.1832 q^{95} +295.976 q^{97} +145.951 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 4 q^{3} - 10 q^{5} + 32 q^{7} + 29 q^{9}+O(q^{10}) $ 5 * q - 4 * q^3 - 10 * q^5 + 32 * q^7 + 29 * q^9	$ 5 q - 4 q^{3} - 10 q^{5} + 32 q^{7} + 29 q^{9} - 36 q^{11} - 26 q^{13} + 88 q^{15} + 82 q^{17} - 156 q^{19} - 72 q^{21} + 336 q^{23} + 151 q^{25} - 352 q^{27} - 145 q^{29} + 432 q^{31} + 108 q^{33} - 600 q^{35} + 18 q^{37} + 688 q^{39} + 82 q^{41} - 340 q^{43} + 146 q^{45} + 680 q^{47} - 115 q^{49} - 608 q^{51} + 102 q^{53} + 736 q^{55} - 576 q^{57} - 924 q^{59} + 618 q^{61} + 584 q^{63} - 704 q^{65} - 44 q^{67} + 1056 q^{69} + 1032 q^{71} - 1078 q^{73} + 468 q^{75} + 888 q^{77} + 200 q^{79} - 1843 q^{81} - 452 q^{83} + 1700 q^{85} + 116 q^{87} - 1790 q^{89} + 1128 q^{91} + 1884 q^{93} + 1024 q^{95} - 2518 q^{97} + 1500 q^{99}+O(q^{100}) $ 5 * q - 4 * q^3 - 10 * q^5 + 32 * q^7 + 29 * q^9 - 36 * q^11 - 26 * q^13 + 88 * q^15 + 82 * q^17 - 156 * q^19 - 72 * q^21 + 336 * q^23 + 151 * q^25 - 352 * q^27 - 145 * q^29 + 432 * q^31 + 108 * q^33 - 600 * q^35 + 18 * q^37 + 688 * q^39 + 82 * q^41 - 340 * q^43 + 146 * q^45 + 680 * q^47 - 115 * q^49 - 608 * q^51 + 102 * q^53 + 736 * q^55 - 576 * q^57 - 924 * q^59 + 618 * q^61 + 584 * q^63 - 704 * q^65 - 44 * q^67 + 1056 * q^69 + 1032 * q^71 - 1078 * q^73 + 468 * q^75 + 888 * q^77 + 200 * q^79 - 1843 * q^81 - 452 * q^83 + 1700 * q^85 + 116 * q^87 - 1790 * q^89 + 1128 * q^91 + 1884 * q^93 + 1024 * q^95 - 2518 * q^97 + 1500 * 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
$3$	5.90002	1.13546	0.567730	−	0.823215i	$-0.307822\pi$
$3$	5.90002	1.13546	0.567730	+	0.823215i	$0.307822\pi$
$4$	0	0
$5$	14.9988	1.34153	0.670765	−	0.741670i	$-0.265966\pi$
$5$	14.9988	1.34153	0.670765	+	0.741670i	$0.265966\pi$
$6$	0	0
$7$	−6.82211	−0.368359	−0.184180	−	0.982893i	$-0.558963\pi$
$7$	−6.82211	−0.368359	−0.184180	+	0.982893i	$0.558963\pi$
$8$	0	0
$9$	7.81028	0.289270
$10$	0	0
$11$	18.6870	0.512213	0.256107	−	0.966649i	$-0.417560\pi$
$11$	18.6870	0.512213	0.256107	+	0.966649i	$0.417560\pi$
$12$	0	0
$13$	39.8582	0.850360	0.425180	−	0.905109i	$-0.360211\pi$
$13$	39.8582	0.850360	0.425180	+	0.905109i	$0.360211\pi$
$14$	0	0
$15$	88.4931	1.52325
$16$	0	0
$17$	19.3391	0.275908	0.137954	−	0.990439i	$-0.455947\pi$
$17$	19.3391	0.275908	0.137954	+	0.990439i	$0.455947\pi$
$18$	0	0
$19$	−2.14572	−0.0259086	−0.0129543	−	0.999916i	$-0.504124\pi$
$19$	−2.14572	−0.0259086	−0.0129543	+	0.999916i	$0.504124\pi$
$20$	0	0
$21$	−40.2506	−0.418257
$22$	0	0
$23$	103.945	0.942349	0.471174	−	0.882040i	$-0.343830\pi$
$23$	103.945	0.942349	0.471174	+	0.882040i	$0.343830\pi$
$24$	0	0
$25$	99.9629	0.799703
$26$	0	0
$27$	−113.220	−0.807006
$28$	0	0
$29$	−29.0000	−0.185695
$29$	−29.0000	−0.185695
$30$	0	0
$31$	213.167	1.23503	0.617514	−	0.786560i	$-0.288140\pi$
$31$	213.167	1.23503	0.617514	+	0.786560i	$0.288140\pi$
$32$	0	0
$33$	110.254	0.581598
$34$	0	0
$35$	−102.323	−0.494165
$36$	0	0
$37$	226.054	1.00441	0.502204	−	0.864749i	$-0.332523\pi$
$37$	226.054	1.00441	0.502204	+	0.864749i	$0.332523\pi$
$38$	0	0
$39$	235.164	0.965550
$40$	0	0
$41$	−191.267	−0.728556	−0.364278	−	0.931290i	$-0.618684\pi$
$41$	−191.267	−0.728556	−0.364278	+	0.931290i	$0.618684\pi$
$42$	0	0
$43$	−34.3923	−0.121971	−0.0609857	−	0.998139i	$-0.519424\pi$
$43$	−34.3923	−0.121971	−0.0609857	+	0.998139i	$0.519424\pi$
$44$	0	0
$45$	117.145	0.388064
$46$	0	0
$47$	501.173	1.55540	0.777698	−	0.628638i	$-0.216387\pi$
$47$	501.173	1.55540	0.777698	+	0.628638i	$0.216387\pi$
$48$	0	0
$49$	−296.459	−0.864312
$50$	0	0
$51$	114.101	0.313282
$52$	0	0
$53$	−259.952	−0.673719	−0.336860	−	0.941555i	$-0.609365\pi$
$53$	−259.952	−0.673719	−0.336860	+	0.941555i	$0.609365\pi$
$54$	0	0
$55$	280.282	0.687149
$56$	0	0
$57$	−12.6598	−0.0294181
$58$	0	0
$59$	−280.736	−0.619471	−0.309735	−	0.950823i	$-0.600240\pi$
$59$	−280.736	−0.619471	−0.309735	+	0.950823i	$0.600240\pi$
$60$	0	0
$61$	−81.0752	−0.170174	−0.0850870	−	0.996374i	$-0.527117\pi$
$61$	−81.0752	−0.170174	−0.0850870	+	0.996374i	$0.527117\pi$
$62$	0	0
$63$	−53.2826	−0.106555
$64$	0	0
$65$	597.824	1.14078
$66$	0	0
$67$	799.332	1.45752	0.728761	−	0.684768i	$-0.240097\pi$
$67$	799.332	1.45752	0.728761	+	0.684768i	$0.240097\pi$
$68$	0	0
$69$	613.278	1.07000
$70$	0	0
$71$	−575.901	−0.962632	−0.481316	−	0.876547i	$-0.659841\pi$
$71$	−575.901	−0.962632	−0.481316	+	0.876547i	$0.659841\pi$
$72$	0	0
$73$	−22.5292	−0.0361211	−0.0180606	−	0.999837i	$-0.505749\pi$
$73$	−22.5292	−0.0361211	−0.0180606	+	0.999837i	$0.505749\pi$
$74$	0	0
$75$	589.783	0.908031
$76$	0	0
$77$	−127.485	−0.188678
$78$	0	0
$79$	−118.878	−0.169302	−0.0846509	−	0.996411i	$-0.526977\pi$
$79$	−118.878	−0.169302	−0.0846509	+	0.996411i	$0.526977\pi$
$80$	0	0
$81$	−878.877	−1.20559
$82$	0	0
$83$	−235.673	−0.311669	−0.155834	−	0.987783i	$-0.549807\pi$
$83$	−235.673	−0.311669	−0.155834	+	0.987783i	$0.549807\pi$
$84$	0	0
$85$	290.063	0.370138
$86$	0	0
$87$	−171.101	−0.210850
$88$	0	0
$89$	1153.23	1.37351	0.686755	−	0.726889i	$-0.259035\pi$
$89$	1153.23	1.37351	0.686755	+	0.726889i	$0.259035\pi$
$90$	0	0
$91$	−271.917	−0.313238
$92$	0	0
$93$	1257.69	1.40232
$94$	0	0
$95$	−32.1832	−0.0347571
$96$	0	0
$97$	295.976	0.309812	0.154906	−	0.987929i	$-0.450493\pi$
$97$	295.976	0.309812	0.154906	+	0.987929i	$0.450493\pi$
$98$	0	0
$99$	145.951	0.148168
$100$	0	0
$101$	1163.17	1.14594	0.572971	−	0.819575i	$-0.305791\pi$
$101$	1163.17	1.14594	0.572971	+	0.819575i	$0.305791\pi$
$102$	0	0
$103$	790.763	0.756468	0.378234	−	0.925710i	$-0.376531\pi$
$103$	790.763	0.756468	0.378234	+	0.925710i	$0.376531\pi$
$104$	0	0
$105$	−603.709	−0.561104
$106$	0	0
$107$	906.764	0.819254	0.409627	−	0.912253i	$-0.365659\pi$
$107$	906.764	0.819254	0.409627	+	0.912253i	$0.365659\pi$
$108$	0	0
$109$	897.545	0.788708	0.394354	−	0.918959i	$-0.370968\pi$
$109$	897.545	0.788708	0.394354	+	0.918959i	$0.370968\pi$
$110$	0	0
$111$	1333.73	1.14046
$112$	0	0
$113$	871.936	0.725884	0.362942	−	0.931812i	$-0.381772\pi$
$113$	871.936	0.725884	0.362942	+	0.931812i	$0.381772\pi$
$114$	0	0
$115$	1559.05	1.26419
$116$	0	0
$117$	311.304	0.245983
$118$	0	0
$119$	−131.934	−0.101633
$120$	0	0
$121$	−981.796	−0.737638
$122$	0	0
$123$	−1128.48	−0.827247
$124$	0	0
$125$	−375.526	−0.268704
$126$	0	0
$127$	1417.53	0.990433	0.495217	−	0.868770i	$-0.335089\pi$
$127$	1417.53	0.990433	0.495217	+	0.868770i	$0.335089\pi$
$128$	0	0
$129$	−202.915	−0.138494
$130$	0	0
$131$	−473.020	−0.315480	−0.157740	−	0.987481i	$-0.550421\pi$
$131$	−473.020	−0.315480	−0.157740	+	0.987481i	$0.550421\pi$
$132$	0	0
$133$	14.6384	0.00954366
$134$	0	0
$135$	−1698.16	−1.08262
$136$	0	0
$137$	1636.81	1.02075	0.510374	−	0.859953i	$-0.329507\pi$
$137$	1636.81	1.02075	0.510374	+	0.859953i	$0.329507\pi$
$138$	0	0
$139$	−1485.21	−0.906286	−0.453143	−	0.891438i	$-0.649697\pi$
$139$	−1485.21	−0.906286	−0.453143	+	0.891438i	$0.649697\pi$
$140$	0	0
$141$	2956.93	1.76609
$142$	0	0
$143$	744.831	0.435566
$144$	0	0
$145$	−434.964	−0.249116
$146$	0	0
$147$	−1749.11	−0.981391
$148$	0	0
$149$	−1370.03	−0.753269	−0.376634	−	0.926362i	$-0.622919\pi$
$149$	−1370.03	−0.753269	−0.376634	+	0.926362i	$0.622919\pi$
$150$	0	0
$151$	948.384	0.511115	0.255557	−	0.966794i	$-0.417741\pi$
$151$	948.384	0.511115	0.255557	+	0.966794i	$0.417741\pi$
$152$	0	0
$153$	151.044	0.0798117
$154$	0	0
$155$	3197.24	1.65683
$156$	0	0
$157$	1499.20	0.762098	0.381049	−	0.924555i	$-0.375563\pi$
$157$	1499.20	0.762098	0.381049	+	0.924555i	$0.375563\pi$
$158$	0	0
$159$	−1533.72	−0.764982
$160$	0	0
$161$	−709.123	−0.347123
$162$	0	0
$163$	380.266	0.182728	0.0913642	−	0.995818i	$-0.470877\pi$
$163$	380.266	0.182728	0.0913642	+	0.995818i	$0.470877\pi$
$164$	0	0
$165$	1653.67	0.780231
$166$	0	0
$167$	1418.55	0.657309	0.328655	−	0.944450i	$-0.393405\pi$
$167$	1418.55	0.657309	0.328655	+	0.944450i	$0.393405\pi$
$168$	0	0
$169$	−608.322	−0.276888
$170$	0	0
$171$	−16.7587	−0.00749456
$172$	0	0
$173$	−3637.40	−1.59853	−0.799266	−	0.600977i	$-0.794778\pi$
$173$	−3637.40	−1.59853	−0.799266	+	0.600977i	$0.794778\pi$
$174$	0	0
$175$	−681.957	−0.294578
$176$	0	0
$177$	−1656.35	−0.703384
$178$	0	0
$179$	2054.57	0.857907	0.428954	−	0.903326i	$-0.358882\pi$
$179$	2054.57	0.857907	0.428954	+	0.903326i	$0.358882\pi$
$180$	0	0
$181$	−4423.60	−1.81660	−0.908298	−	0.418324i	$-0.862618\pi$
$181$	−4423.60	−1.81660	−0.908298	+	0.418324i	$0.862618\pi$
$182$	0	0
$183$	−478.346	−0.193226
$184$	0	0
$185$	3390.53	1.34744
$186$	0	0
$187$	361.390	0.141323
$188$	0	0
$189$	772.398	0.297268
$190$	0	0
$191$	−2132.86	−0.808000	−0.404000	−	0.914759i	$-0.632380\pi$
$191$	−2132.86	−0.808000	−0.404000	+	0.914759i	$0.632380\pi$
$192$	0	0
$193$	137.560	0.0513048	0.0256524	−	0.999671i	$-0.491834\pi$
$193$	137.560	0.0513048	0.0256524	+	0.999671i	$0.491834\pi$
$194$	0	0
$195$	3527.18	1.29531
$196$	0	0
$197$	−4492.27	−1.62467	−0.812337	−	0.583189i	$-0.801805\pi$
$197$	−4492.27	−1.62467	−0.812337	+	0.583189i	$0.801805\pi$
$198$	0	0
$199$	−2435.95	−0.867738	−0.433869	−	0.900976i	$-0.642852\pi$
$199$	−2435.95	−0.867738	−0.433869	+	0.900976i	$0.642852\pi$
$200$	0	0
$201$	4716.08	1.65496
$202$	0	0
$203$	197.841	0.0684026
$204$	0	0
$205$	−2868.76	−0.977380
$206$	0	0
$207$	811.839	0.272593
$208$	0	0
$209$	−40.0971	−0.0132707
$210$	0	0
$211$	−4150.84	−1.35429	−0.677145	−	0.735849i	$-0.736783\pi$
$211$	−4150.84	−1.35429	−0.677145	+	0.735849i	$0.736783\pi$
$212$	0	0
$213$	−3397.83	−1.09303
$214$	0	0
$215$	−515.842	−0.163628
$216$	0	0
$217$	−1454.25	−0.454934
$218$	0	0
$219$	−132.923	−0.0410141
$220$	0	0
$221$	770.823	0.234621
$222$	0	0
$223$	4405.99	1.32308	0.661540	−	0.749910i	$-0.269903\pi$
$223$	4405.99	1.32308	0.661540	+	0.749910i	$0.269903\pi$
$224$	0	0
$225$	780.738	0.231330
$226$	0	0
$227$	−1996.06	−0.583627	−0.291813	−	0.956475i	$-0.594259\pi$
$227$	−1996.06	−0.583627	−0.291813	+	0.956475i	$0.594259\pi$
$228$	0	0
$229$	−4659.18	−1.34449	−0.672243	−	0.740330i	$-0.734669\pi$
$229$	−4659.18	−1.34449	−0.672243	+	0.740330i	$0.734669\pi$
$230$	0	0
$231$	−752.163	−0.214237
$232$	0	0
$233$	5821.24	1.63675	0.818374	−	0.574686i	$-0.194876\pi$
$233$	5821.24	1.63675	0.818374	+	0.574686i	$0.194876\pi$
$234$	0	0
$235$	7516.98	2.08661
$236$	0	0
$237$	−701.384	−0.192235
$238$	0	0
$239$	6555.92	1.77434	0.887170	−	0.461442i	$-0.152668\pi$
$239$	6555.92	1.77434	0.887170	+	0.461442i	$0.152668\pi$
$240$	0	0
$241$	2578.14	0.689097	0.344548	−	0.938769i	$-0.388032\pi$
$241$	2578.14	0.689097	0.344548	+	0.938769i	$0.388032\pi$
$242$	0	0
$243$	−2128.46	−0.561896
$244$	0	0
$245$	−4446.52	−1.15950
$246$	0	0
$247$	−85.5247	−0.0220316
$248$	0	0
$249$	−1390.48	−0.353887
$250$	0	0
$251$	2659.86	0.668880	0.334440	−	0.942417i	$-0.391453\pi$
$251$	2659.86	0.668880	0.334440	+	0.942417i	$0.391453\pi$
$252$	0	0
$253$	1942.42	0.482683
$254$	0	0
$255$	1711.38	0.420277
$256$	0	0
$257$	−6610.31	−1.60443	−0.802217	−	0.597032i	$-0.796346\pi$
$257$	−6610.31	−1.60443	−0.802217	+	0.597032i	$0.796346\pi$
$258$	0	0
$259$	−1542.17	−0.369983
$260$	0	0
$261$	−226.498	−0.0537160
$262$	0	0
$263$	4167.41	0.977085	0.488543	−	0.872540i	$-0.337529\pi$
$263$	4167.41	0.977085	0.488543	+	0.872540i	$0.337529\pi$
$264$	0	0
$265$	−3898.96	−0.903815
$266$	0	0
$267$	6804.10	1.55957
$268$	0	0
$269$	5180.05	1.17410	0.587050	−	0.809550i	$-0.300289\pi$
$269$	5180.05	1.17410	0.587050	+	0.809550i	$0.300289\pi$
$270$	0	0
$271$	−4805.98	−1.07728	−0.538639	−	0.842537i	$-0.681061\pi$
$271$	−4805.98	−1.07728	−0.538639	+	0.842537i	$0.681061\pi$
$272$	0	0
$273$	−1604.32	−0.355669
$274$	0	0
$275$	1868.01	0.409618
$276$	0	0
$277$	1303.90	0.282830	0.141415	−	0.989950i	$-0.454835\pi$
$277$	1303.90	0.282830	0.141415	+	0.989950i	$0.454835\pi$
$278$	0	0
$279$	1664.89	0.357256
$280$	0	0
$281$	−4084.01	−0.867016	−0.433508	−	0.901150i	$-0.642724\pi$
$281$	−4084.01	−0.867016	−0.433508	+	0.901150i	$0.642724\pi$
$282$	0	0
$283$	4493.32	0.943816	0.471908	−	0.881648i	$-0.343565\pi$
$283$	4493.32	0.943816	0.471908	+	0.881648i	$0.343565\pi$
$284$	0	0
$285$	−189.882	−0.0394653
$286$	0	0
$287$	1304.84	0.268370
$288$	0	0
$289$	−4539.00	−0.923875
$290$	0	0
$291$	1746.26	0.351779
$292$	0	0
$293$	2591.95	0.516803	0.258402	−	0.966038i	$-0.416804\pi$
$293$	2591.95	0.516803	0.258402	+	0.966038i	$0.416804\pi$
$294$	0	0
$295$	−4210.70	−0.831038
$296$	0	0
$297$	−2115.74	−0.413359
$298$	0	0
$299$	4143.06	0.801336
$300$	0	0
$301$	234.628	0.0449293
$302$	0	0
$303$	6862.76	1.30117
$304$	0	0
$305$	−1216.03	−0.228294
$306$	0	0
$307$	2355.83	0.437962	0.218981	−	0.975729i	$-0.429727\pi$
$307$	2355.83	0.437962	0.218981	+	0.975729i	$0.429727\pi$
$308$	0	0
$309$	4665.52	0.858940
$310$	0	0
$311$	3087.64	0.562971	0.281486	−	0.959565i	$-0.409173\pi$
$311$	3087.64	0.562971	0.281486	+	0.959565i	$0.409173\pi$
$312$	0	0
$313$	−1669.67	−0.301518	−0.150759	−	0.988571i	$-0.548172\pi$
$313$	−1669.67	−0.301518	−0.150759	+	0.988571i	$0.548172\pi$
$314$	0	0
$315$	−799.172	−0.142947
$316$	0	0
$317$	−6613.47	−1.17176	−0.585882	−	0.810396i	$-0.699252\pi$
$317$	−6613.47	−1.17176	−0.585882	+	0.810396i	$0.699252\pi$
$318$	0	0
$319$	−541.923	−0.0951156
$320$	0	0
$321$	5349.93	0.930230
$322$	0	0
$323$	−41.4964	−0.00714837
$324$	0	0
$325$	3984.34	0.680036
$326$	0	0
$327$	5295.54	0.895547
$328$	0	0
$329$	−3419.06	−0.572945
$330$	0	0
$331$	11636.2	1.93228	0.966138	−	0.258027i	$-0.0830722\pi$
$331$	11636.2	1.93228	0.966138	+	0.258027i	$0.0830722\pi$
$332$	0	0
$333$	1765.55	0.290545
$334$	0	0
$335$	11989.0	1.95531
$336$	0	0
$337$	−11798.2	−1.90708	−0.953542	−	0.301262i	$-0.902592\pi$
$337$	−11798.2	−1.90708	−0.953542	+	0.301262i	$0.902592\pi$
$338$	0	0
$339$	5144.44	0.824212
$340$	0	0
$341$	3983.45	0.632597
$342$	0	0
$343$	4362.46	0.686736
$344$	0	0
$345$	9198.40	1.43544
$346$	0	0
$347$	625.125	0.0967103	0.0483551	−	0.998830i	$-0.484602\pi$
$347$	625.125	0.0967103	0.0483551	+	0.998830i	$0.484602\pi$
$348$	0	0
$349$	1820.90	0.279286	0.139643	−	0.990202i	$-0.455405\pi$
$349$	1820.90	0.279286	0.139643	+	0.990202i	$0.455405\pi$
$350$	0	0
$351$	−4512.74	−0.686246
$352$	0	0
$353$	5114.36	0.771133	0.385567	−	0.922680i	$-0.374006\pi$
$353$	5114.36	0.771133	0.385567	+	0.922680i	$0.374006\pi$
$354$	0	0
$355$	−8637.80	−1.29140
$356$	0	0
$357$	−778.411	−0.115400
$358$	0	0
$359$	−10503.5	−1.54415	−0.772077	−	0.635529i	$-0.780782\pi$
$359$	−10503.5	−1.54415	−0.772077	+	0.635529i	$0.780782\pi$
$360$	0	0
$361$	−6854.40	−0.999329
$362$	0	0
$363$	−5792.62	−0.837558
$364$	0	0
$365$	−337.910	−0.0484576
$366$	0	0
$367$	5825.82	0.828625	0.414313	−	0.910135i	$-0.364022\pi$
$367$	5825.82	0.828625	0.414313	+	0.910135i	$0.364022\pi$
$368$	0	0
$369$	−1493.84	−0.210749
$370$	0	0
$371$	1773.42	0.248171
$372$	0	0
$373$	−12629.5	−1.75316	−0.876579	−	0.481257i	$-0.840180\pi$
$373$	−12629.5	−1.75316	−0.876579	+	0.481257i	$0.840180\pi$
$374$	0	0
$375$	−2215.61	−0.305103
$376$	0	0
$377$	−1155.89	−0.157908
$378$	0	0
$379$	−4152.73	−0.562826	−0.281413	−	0.959587i	$-0.590803\pi$
$379$	−4152.73	−0.562826	−0.281413	+	0.959587i	$0.590803\pi$
$380$	0	0
$381$	8363.43	1.12460
$382$	0	0
$383$	7253.54	0.967725	0.483863	−	0.875144i	$-0.339233\pi$
$383$	7253.54	0.967725	0.483863	+	0.875144i	$0.339233\pi$
$384$	0	0
$385$	−1912.11	−0.253118
$386$	0	0
$387$	−268.613	−0.0352826
$388$	0	0
$389$	−6053.84	−0.789053	−0.394527	−	0.918884i	$-0.629091\pi$
$389$	−6053.84	−0.789053	−0.394527	+	0.918884i	$0.629091\pi$
$390$	0	0
$391$	2010.20	0.260001
$392$	0	0
$393$	−2790.83	−0.358215
$394$	0	0
$395$	−1783.03	−0.227123
$396$	0	0
$397$	12932.7	1.63495	0.817473	−	0.575967i	$-0.195374\pi$
$397$	12932.7	1.63495	0.817473	+	0.575967i	$0.195374\pi$
$398$	0	0
$399$	86.3666	0.0108364
$400$	0	0
$401$	12673.6	1.57828	0.789141	−	0.614212i	$-0.210526\pi$
$401$	12673.6	1.57828	0.789141	+	0.614212i	$0.210526\pi$
$402$	0	0
$403$	8496.44	1.05022
$404$	0	0
$405$	−13182.1	−1.61734
$406$	0	0
$407$	4224.28	0.514471
$408$	0	0
$409$	−13335.1	−1.61218	−0.806088	−	0.591795i	$-0.798419\pi$
$409$	−13335.1	−1.61218	−0.806088	+	0.591795i	$0.798419\pi$
$410$	0	0
$411$	9657.23	1.15902
$412$	0	0
$413$	1915.21	0.228188
$414$	0	0
$415$	−3534.81	−0.418113
$416$	0	0
$417$	−8762.76	−1.02905
$418$	0	0
$419$	−4564.25	−0.532167	−0.266084	−	0.963950i	$-0.585730\pi$
$419$	−4564.25	−0.532167	−0.266084	+	0.963950i	$0.585730\pi$
$420$	0	0
$421$	6992.67	0.809506	0.404753	−	0.914426i	$-0.367358\pi$
$421$	6992.67	0.809506	0.404753	+	0.914426i	$0.367358\pi$
$422$	0	0
$423$	3914.30	0.449929
$424$	0	0
$425$	1933.20	0.220644
$426$	0	0
$427$	553.104	0.0626852
$428$	0	0
$429$	4394.52	0.494567
$430$	0	0
$431$	9630.58	1.07631	0.538154	−	0.842847i	$-0.319122\pi$
$431$	9630.58	1.07631	0.538154	+	0.842847i	$0.319122\pi$
$432$	0	0
$433$	13504.7	1.49883	0.749415	−	0.662100i	$-0.230335\pi$
$433$	13504.7	1.49883	0.749415	+	0.662100i	$0.230335\pi$
$434$	0	0
$435$	−2566.30	−0.282861
$436$	0	0
$437$	−223.037	−0.0244149
$438$	0	0
$439$	−2381.38	−0.258899	−0.129450	−	0.991586i	$-0.541321\pi$
$439$	−2381.38	−0.258899	−0.129450	+	0.991586i	$0.541321\pi$
$440$	0	0
$441$	−2315.43	−0.250019
$442$	0	0
$443$	−7708.83	−0.826766	−0.413383	−	0.910557i	$-0.635653\pi$
$443$	−7708.83	−0.826766	−0.413383	+	0.910557i	$0.635653\pi$
$444$	0	0
$445$	17297.1	1.84260
$446$	0	0
$447$	−8083.20	−0.855307
$448$	0	0
$449$	−2065.19	−0.217065	−0.108532	−	0.994093i	$-0.534615\pi$
$449$	−2065.19	−0.217065	−0.108532	+	0.994093i	$0.534615\pi$
$450$	0	0
$451$	−3574.20	−0.373176
$452$	0	0
$453$	5595.49	0.580351
$454$	0	0
$455$	−4078.42	−0.420218
$456$	0	0
$457$	−4959.73	−0.507673	−0.253836	−	0.967247i	$-0.581692\pi$
$457$	−4959.73	−0.507673	−0.253836	+	0.967247i	$0.581692\pi$
$458$	0	0
$459$	−2189.57	−0.222659
$460$	0	0
$461$	−16339.8	−1.65080	−0.825402	−	0.564545i	$-0.809052\pi$
$461$	−16339.8	−1.65080	−0.825402	+	0.564545i	$0.809052\pi$
$462$	0	0
$463$	7546.94	0.757529	0.378765	−	0.925493i	$-0.376349\pi$
$463$	7546.94	0.757529	0.378765	+	0.925493i	$0.376349\pi$
$464$	0	0
$465$	18863.8	1.88126
$466$	0	0
$467$	−2409.12	−0.238717	−0.119358	−	0.992851i	$-0.538084\pi$
$467$	−2409.12	−0.238717	−0.119358	+	0.992851i	$0.538084\pi$
$468$	0	0
$469$	−5453.13	−0.536891
$470$	0	0
$471$	8845.33	0.865332
$472$	0	0
$473$	−642.689	−0.0624754
$474$	0	0
$475$	−214.493	−0.0207192
$476$	0	0
$477$	−2030.30	−0.194887
$478$	0	0
$479$	17370.8	1.65698	0.828488	−	0.560007i	$-0.189202\pi$
$479$	17370.8	1.65698	0.828488	+	0.560007i	$0.189202\pi$
$480$	0	0
$481$	9010.12	0.854108
$482$	0	0
$483$	−4183.84	−0.394144
$484$	0	0
$485$	4439.27	0.415622
$486$	0	0
$487$	−6367.45	−0.592478	−0.296239	−	0.955114i	$-0.595732\pi$
$487$	−6367.45	−0.592478	−0.296239	+	0.955114i	$0.595732\pi$
$488$	0	0
$489$	2243.58	0.207481
$490$	0	0
$491$	6282.71	0.577463	0.288732	−	0.957410i	$-0.406766\pi$
$491$	6282.71	0.577463	0.288732	+	0.957410i	$0.406766\pi$
$492$	0	0
$493$	−560.835	−0.0512347
$494$	0	0
$495$	2189.08	0.198771
$496$	0	0
$497$	3928.86	0.354594
$498$	0	0
$499$	−7147.33	−0.641200	−0.320600	−	0.947215i	$-0.603884\pi$
$499$	−7147.33	−0.641200	−0.320600	+	0.947215i	$0.603884\pi$
$500$	0	0
$501$	8369.48	0.746349
$502$	0	0
$503$	−14496.5	−1.28502	−0.642512	−	0.766276i	$-0.722108\pi$
$503$	−14496.5	−1.28502	−0.642512	+	0.766276i	$0.722108\pi$
$504$	0	0
$505$	17446.2	1.53732
$506$	0	0
$507$	−3589.11	−0.314395
$508$	0	0
$509$	4185.51	0.364478	0.182239	−	0.983254i	$-0.441666\pi$
$509$	4185.51	0.364478	0.182239	+	0.983254i	$0.441666\pi$
$510$	0	0
$511$	153.697	0.0133055
$512$	0	0
$513$	242.938	0.0209084
$514$	0	0
$515$	11860.5	1.01483
$516$	0	0
$517$	9365.43	0.796695
$518$	0	0
$519$	−21460.7	−1.81507
$520$	0	0
$521$	−12392.3	−1.04206	−0.521032	−	0.853537i	$-0.674453\pi$
$521$	−12392.3	−1.04206	−0.521032	+	0.853537i	$0.674453\pi$
$522$	0	0
$523$	−18806.4	−1.57237	−0.786184	−	0.617993i	$-0.787946\pi$
$523$	−18806.4	−1.57237	−0.786184	+	0.617993i	$0.787946\pi$
$524$	0	0
$525$	−4023.56	−0.334481
$526$	0	0
$527$	4122.46	0.340753
$528$	0	0
$529$	−1362.45	−0.111979
$530$	0	0
$531$	−2192.63	−0.179194
$532$	0	0
$533$	−7623.54	−0.619535
$534$	0	0
$535$	13600.3	1.09905
$536$	0	0
$537$	12122.0	0.974120
$538$	0	0
$539$	−5539.93	−0.442712
$540$	0	0
$541$	−5080.90	−0.403780	−0.201890	−	0.979408i	$-0.564708\pi$
$541$	−5080.90	−0.403780	−0.201890	+	0.979408i	$0.564708\pi$
$542$	0	0
$543$	−26099.4	−2.06267
$544$	0	0
$545$	13462.1	1.05808
$546$	0	0
$547$	−20578.0	−1.60850	−0.804252	−	0.594288i	$-0.797434\pi$
$547$	−20578.0	−1.60850	−0.804252	+	0.594288i	$0.797434\pi$
$548$	0	0
$549$	−633.220	−0.0492262
$550$	0	0
$551$	62.2260	0.00481110
$552$	0	0
$553$	811.000	0.0623639
$554$	0	0
$555$	20004.2	1.52997
$556$	0	0
$557$	3440.76	0.261741	0.130871	−	0.991399i	$-0.458223\pi$
$557$	3440.76	0.261741	0.130871	+	0.991399i	$0.458223\pi$
$558$	0	0
$559$	−1370.82	−0.103720
$560$	0	0
$561$	2132.21	0.160467
$562$	0	0
$563$	−2647.62	−0.198195	−0.0990975	−	0.995078i	$-0.531596\pi$
$563$	−2647.62	−0.198195	−0.0990975	+	0.995078i	$0.531596\pi$
$564$	0	0
$565$	13078.0	0.973795
$566$	0	0
$567$	5995.79	0.444091
$568$	0	0
$569$	−24385.8	−1.79667	−0.898334	−	0.439313i	$-0.855222\pi$
$569$	−24385.8	−1.79667	−0.898334	+	0.439313i	$0.855222\pi$
$570$	0	0
$571$	−15785.4	−1.15692	−0.578459	−	0.815712i	$-0.696346\pi$
$571$	−15785.4	−1.15692	−0.578459	+	0.815712i	$0.696346\pi$
$572$	0	0
$573$	−12583.9	−0.917452
$574$	0	0
$575$	10390.6	0.753599
$576$	0	0
$577$	−4415.15	−0.318553	−0.159277	−	0.987234i	$-0.550916\pi$
$577$	−4415.15	−0.318553	−0.159277	+	0.987234i	$0.550916\pi$
$578$	0	0
$579$	811.610	0.0582545
$580$	0	0
$581$	1607.79	0.114806
$582$	0	0
$583$	−4857.72	−0.345088
$584$	0	0
$585$	4669.17	0.329994
$586$	0	0
$587$	−13595.4	−0.955950	−0.477975	−	0.878373i	$-0.658629\pi$
$587$	−13595.4	−0.955950	−0.477975	+	0.878373i	$0.658629\pi$
$588$	0	0
$589$	−457.397	−0.0319978
$590$	0	0
$591$	−26504.5	−1.84475
$592$	0	0
$593$	23514.8	1.62839	0.814195	−	0.580591i	$-0.197178\pi$
$593$	23514.8	1.62839	0.814195	+	0.580591i	$0.197178\pi$
$594$	0	0
$595$	−1978.84	−0.136344
$596$	0	0
$597$	−14372.2	−0.985282
$598$	0	0
$599$	−21041.9	−1.43531	−0.717654	−	0.696400i	$-0.754784\pi$
$599$	−21041.9	−1.43531	−0.717654	+	0.696400i	$0.754784\pi$
$600$	0	0
$601$	5250.69	0.356373	0.178186	−	0.983997i	$-0.442977\pi$
$601$	5250.69	0.356373	0.178186	+	0.983997i	$0.442977\pi$
$602$	0	0
$603$	6243.01	0.421617
$604$	0	0
$605$	−14725.7	−0.989563
$606$	0	0
$607$	19758.4	1.32120	0.660599	−	0.750739i	$-0.270302\pi$
$607$	19758.4	1.32120	0.660599	+	0.750739i	$0.270302\pi$
$608$	0	0
$609$	1167.27	0.0776684
$610$	0	0
$611$	19975.9	1.32265
$612$	0	0
$613$	16105.3	1.06115	0.530575	−	0.847638i	$-0.321976\pi$
$613$	16105.3	1.06115	0.530575	+	0.847638i	$0.321976\pi$
$614$	0	0
$615$	−16925.8	−1.10978
$616$	0	0
$617$	−8320.76	−0.542919	−0.271460	−	0.962450i	$-0.587506\pi$
$617$	−8320.76	−0.542919	−0.271460	+	0.962450i	$0.587506\pi$
$618$	0	0
$619$	2802.58	0.181979	0.0909896	−	0.995852i	$-0.470997\pi$
$619$	2802.58	0.181979	0.0909896	+	0.995852i	$0.470997\pi$
$620$	0	0
$621$	−11768.6	−0.760481
$622$	0	0
$623$	−7867.47	−0.505945
$624$	0	0
$625$	−18127.8	−1.16018
$626$	0	0
$627$	−236.574	−0.0150684
$628$	0	0
$629$	4371.69	0.277124
$630$	0	0
$631$	17958.7	1.13300	0.566502	−	0.824060i	$-0.308296\pi$
$631$	17958.7	1.13300	0.566502	+	0.824060i	$0.308296\pi$
$632$	0	0
$633$	−24490.0	−1.53774
$634$	0	0
$635$	21261.1	1.32870
$636$	0	0
$637$	−11816.3	−0.734976
$638$	0	0
$639$	−4497.95	−0.278460
$640$	0	0
$641$	11480.6	0.707417	0.353709	−	0.935356i	$-0.384920\pi$
$641$	11480.6	0.707417	0.353709	+	0.935356i	$0.384920\pi$
$642$	0	0
$643$	−15144.0	−0.928802	−0.464401	−	0.885625i	$-0.653730\pi$
$643$	−15144.0	−0.928802	−0.464401	+	0.885625i	$0.653730\pi$
$644$	0	0
$645$	−3043.48	−0.185793
$646$	0	0
$647$	13334.1	0.810227	0.405113	−	0.914266i	$-0.367232\pi$
$647$	13334.1	0.810227	0.405113	+	0.914266i	$0.367232\pi$
$648$	0	0
$649$	−5246.12	−0.317301
$650$	0	0
$651$	−8580.08	−0.516559
$652$	0	0
$653$	16543.2	0.991402	0.495701	−	0.868493i	$-0.334911\pi$
$653$	16543.2	0.991402	0.495701	+	0.868493i	$0.334911\pi$
$654$	0	0
$655$	−7094.71	−0.423227
$656$	0	0
$657$	−175.959	−0.0104487
$658$	0	0
$659$	9171.83	0.542161	0.271080	−	0.962557i	$-0.412619\pi$
$659$	9171.83	0.542161	0.271080	+	0.962557i	$0.412619\pi$
$660$	0	0
$661$	−16761.6	−0.986312	−0.493156	−	0.869941i	$-0.664157\pi$
$661$	−16761.6	−0.986312	−0.493156	+	0.869941i	$0.664157\pi$
$662$	0	0
$663$	4547.88	0.266403
$664$	0	0
$665$	219.557	0.0128031
$666$	0	0
$667$	−3014.40	−0.174990
$668$	0	0
$669$	25995.4	1.50230
$670$	0	0
$671$	−1515.05	−0.0871654
$672$	0	0
$673$	640.780	0.0367017	0.0183508	−	0.999832i	$-0.494158\pi$
$673$	640.780	0.0367017	0.0183508	+	0.999832i	$0.494158\pi$
$674$	0	0
$675$	−11317.8	−0.645365
$676$	0	0
$677$	30880.1	1.75305	0.876527	−	0.481353i	$-0.159855\pi$
$677$	30880.1	1.75305	0.876527	+	0.481353i	$0.159855\pi$
$678$	0	0
$679$	−2019.18	−0.114122
$680$	0	0
$681$	−11776.8	−0.662685
$682$	0	0
$683$	20491.4	1.14799	0.573997	−	0.818857i	$-0.305392\pi$
$683$	20491.4	1.14799	0.573997	+	0.818857i	$0.305392\pi$
$684$	0	0
$685$	24550.2	1.36936
$686$	0	0
$687$	−27489.3	−1.52661
$688$	0	0
$689$	−10361.2	−0.572904
$690$	0	0
$691$	−20654.5	−1.13710	−0.568549	−	0.822649i	$-0.692495\pi$
$691$	−20654.5	−1.13710	−0.568549	+	0.822649i	$0.692495\pi$
$692$	0	0
$693$	−995.691	−0.0545789
$694$	0	0
$695$	−22276.3	−1.21581
$696$	0	0
$697$	−3698.93	−0.201014
$698$	0	0
$699$	34345.5	1.85846
$700$	0	0
$701$	−11890.2	−0.640636	−0.320318	−	0.947310i	$-0.603790\pi$
$701$	−11890.2	−0.640636	−0.320318	+	0.947310i	$0.603790\pi$
$702$	0	0
$703$	−485.050	−0.0260228
$704$	0	0
$705$	44350.4	2.36926
$706$	0	0
$707$	−7935.30	−0.422118
$708$	0	0
$709$	−10009.6	−0.530209	−0.265105	−	0.964220i	$-0.585406\pi$
$709$	−10009.6	−0.530209	−0.265105	+	0.964220i	$0.585406\pi$
$710$	0	0
$711$	−928.472	−0.0489739
$712$	0	0
$713$	22157.6	1.16383
$714$	0	0
$715$	11171.5	0.584324
$716$	0	0
$717$	38680.1	2.01469
$718$	0	0
$719$	−25825.8	−1.33956	−0.669778	−	0.742561i	$-0.733611\pi$
$719$	−25825.8	−1.33956	−0.669778	+	0.742561i	$0.733611\pi$
$720$	0	0
$721$	−5394.67	−0.278652
$722$	0	0
$723$	15211.1	0.782442
$724$	0	0
$725$	−2898.92	−0.148501
$726$	0	0
$727$	−22228.4	−1.13398	−0.566991	−	0.823724i	$-0.691893\pi$
$727$	−22228.4	−1.13398	−0.566991	+	0.823724i	$0.691893\pi$
$728$	0	0
$729$	11171.7	0.567582
$730$	0	0
$731$	−665.117	−0.0336528
$732$	0	0
$733$	−13564.3	−0.683506	−0.341753	−	0.939790i	$-0.611021\pi$
$733$	−13564.3	−0.683506	−0.341753	+	0.939790i	$0.611021\pi$
$734$	0	0
$735$	−26234.5	−1.31657
$736$	0	0
$737$	14937.1	0.746562
$738$	0	0
$739$	5379.12	0.267759	0.133880	−	0.990998i	$-0.457256\pi$
$739$	5379.12	0.267759	0.133880	+	0.990998i	$0.457256\pi$
$740$	0	0
$741$	−504.598	−0.0250160
$742$	0	0
$743$	17556.8	0.866885	0.433442	−	0.901181i	$-0.357299\pi$
$743$	17556.8	0.866885	0.433442	+	0.901181i	$0.357299\pi$
$744$	0	0
$745$	−20548.7	−1.01053
$746$	0	0
$747$	−1840.67	−0.0901562
$748$	0	0
$749$	−6186.04	−0.301780
$750$	0	0
$751$	−21609.7	−1.05000	−0.525000	−	0.851102i	$-0.675935\pi$
$751$	−21609.7	−1.05000	−0.525000	+	0.851102i	$0.675935\pi$
$752$	0	0
$753$	15693.2	0.759486
$754$	0	0
$755$	14224.6	0.685676
$756$	0	0
$757$	−25756.2	−1.23662	−0.618311	−	0.785933i	$-0.712183\pi$
$757$	−25756.2	−1.23662	−0.618311	+	0.785933i	$0.712183\pi$
$758$	0	0
$759$	11460.3	0.548068
$760$	0	0
$761$	11264.3	0.536572	0.268286	−	0.963339i	$-0.413543\pi$
$761$	11264.3	0.536572	0.268286	+	0.963339i	$0.413543\pi$
$762$	0	0
$763$	−6123.15	−0.290528
$764$	0	0
$765$	2265.47	0.107070
$766$	0	0
$767$	−11189.7	−0.526773
$768$	0	0
$769$	−7851.65	−0.368189	−0.184095	−	0.982909i	$-0.558935\pi$
$769$	−7851.65	−0.368189	−0.184095	+	0.982909i	$0.558935\pi$
$770$	0	0
$771$	−39001.0	−1.82177
$772$	0	0
$773$	−1339.59	−0.0623308	−0.0311654	−	0.999514i	$-0.509922\pi$
$773$	−1339.59	−0.0623308	−0.0311654	+	0.999514i	$0.509922\pi$
$774$	0	0
$775$	21308.7	0.987655
$776$	0	0
$777$	−9098.81	−0.420101
$778$	0	0
$779$	410.405	0.0188758
$780$	0	0
$781$	−10761.9	−0.493073
$782$	0	0
$783$	3283.37	0.149857
$784$	0	0
$785$	22486.2	1.02238
$786$	0	0
$787$	4123.15	0.186753	0.0933764	−	0.995631i	$-0.470234\pi$
$787$	4123.15	0.186753	0.0933764	+	0.995631i	$0.470234\pi$
$788$	0	0
$789$	24587.8	1.10944
$790$	0	0
$791$	−5948.44	−0.267386
$792$	0	0
$793$	−3231.51	−0.144709
$794$	0	0
$795$	−23003.9	−1.02625
$796$	0	0
$797$	−27748.5	−1.23325	−0.616626	−	0.787257i	$-0.711501\pi$
$797$	−27748.5	−1.23325	−0.616626	+	0.787257i	$0.711501\pi$
$798$	0	0
$799$	9692.26	0.429146
$800$	0	0
$801$	9007.06	0.397315
$802$	0	0
$803$	−421.003	−0.0185017
$804$	0	0
$805$	−10636.0	−0.465676
$806$	0	0
$807$	30562.4	1.33314
$808$	0	0
$809$	−39614.9	−1.72162	−0.860808	−	0.508930i	$-0.830041\pi$
$809$	−39614.9	−1.72162	−0.860808	+	0.508930i	$0.830041\pi$
$810$	0	0
$811$	15565.2	0.673944	0.336972	−	0.941515i	$-0.390597\pi$
$811$	15565.2	0.673944	0.336972	+	0.941515i	$0.390597\pi$
$812$	0	0
$813$	−28355.4	−1.22321
$814$	0	0
$815$	5703.52	0.245136
$816$	0	0
$817$	73.7963	0.00316011
$818$	0	0
$819$	−2123.75	−0.0906102
$820$	0	0
$821$	−42319.2	−1.79897	−0.899483	−	0.436955i	$-0.856057\pi$
$821$	−42319.2	−1.79897	−0.899483	+	0.436955i	$0.856057\pi$
$822$	0	0
$823$	−623.931	−0.0264263	−0.0132132	−	0.999913i	$-0.504206\pi$
$823$	−623.931	−0.0264263	−0.0132132	+	0.999913i	$0.504206\pi$
$824$	0	0
$825$	11021.3	0.465105
$826$	0	0
$827$	−11493.7	−0.483283	−0.241642	−	0.970366i	$-0.577686\pi$
$827$	−11493.7	−0.483283	−0.241642	+	0.970366i	$0.577686\pi$
$828$	0	0
$829$	13934.2	0.583783	0.291892	−	0.956451i	$-0.405715\pi$
$829$	13934.2	0.583783	0.291892	+	0.956451i	$0.405715\pi$
$830$	0	0
$831$	7693.05	0.321142
$832$	0	0
$833$	−5733.26	−0.238470
$834$	0	0
$835$	21276.5	0.881800
$836$	0	0
$837$	−24134.7	−0.996675
$838$	0	0
$839$	19231.4	0.791351	0.395675	−	0.918390i	$-0.370511\pi$
$839$	19231.4	0.791351	0.395675	+	0.918390i	$0.370511\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	−24095.7	−0.984462
$844$	0	0
$845$	−9124.07	−0.371453
$846$	0	0
$847$	6697.92	0.271716
$848$	0	0
$849$	26510.7	1.07167
$850$	0	0
$851$	23497.2	0.946502
$852$	0	0
$853$	−31388.4	−1.25993	−0.629964	−	0.776624i	$-0.716930\pi$
$853$	−31388.4	−1.25993	−0.629964	+	0.776624i	$0.716930\pi$
$854$	0	0
$855$	−251.360	−0.0100542
$856$	0	0
$857$	−18862.4	−0.751839	−0.375920	−	0.926652i	$-0.622673\pi$
$857$	−18862.4	−0.751839	−0.375920	+	0.926652i	$0.622673\pi$
$858$	0	0
$859$	−41839.3	−1.66186	−0.830931	−	0.556375i	$-0.812192\pi$
$859$	−41839.3	−1.66186	−0.830931	+	0.556375i	$0.812192\pi$
$860$	0	0
$861$	7698.59	0.304724
$862$	0	0
$863$	−22898.2	−0.903204	−0.451602	−	0.892220i	$-0.649147\pi$
$863$	−22898.2	−0.903204	−0.451602	+	0.892220i	$0.649147\pi$
$864$	0	0
$865$	−54556.4	−2.14448
$866$	0	0
$867$	−26780.2	−1.04902
$868$	0	0
$869$	−2221.48	−0.0867186
$870$	0	0
$871$	31860.0	1.23942
$872$	0	0
$873$	2311.65	0.0896192
$874$	0	0
$875$	2561.88	0.0989797
$876$	0	0
$877$	15485.0	0.596226	0.298113	−	0.954531i	$-0.403643\pi$
$877$	15485.0	0.596226	0.298113	+	0.954531i	$0.403643\pi$
$878$	0	0
$879$	15292.6	0.586809
$880$	0	0
$881$	−28152.8	−1.07661	−0.538305	−	0.842750i	$-0.680935\pi$
$881$	−28152.8	−1.07661	−0.538305	+	0.842750i	$0.680935\pi$
$882$	0	0
$883$	40543.4	1.54518	0.772590	−	0.634906i	$-0.218961\pi$
$883$	40543.4	1.54518	0.772590	+	0.634906i	$0.218961\pi$
$884$	0	0
$885$	−24843.2	−0.943611
$886$	0	0
$887$	−45338.8	−1.71627	−0.858133	−	0.513428i	$-0.828375\pi$
$887$	−45338.8	−1.71627	−0.858133	+	0.513428i	$0.828375\pi$
$888$	0	0
$889$	−9670.51	−0.364835
$890$	0	0
$891$	−16423.6	−0.617520
$892$	0	0
$893$	−1075.38	−0.0402981
$894$	0	0
$895$	30815.9	1.15091
$896$	0	0
$897$	24444.2	0.909885
$898$	0	0
$899$	−6181.83	−0.229339
$900$	0	0
$901$	−5027.24	−0.185884
$902$	0	0
$903$	1384.31	0.0510154
$904$	0	0
$905$	−66348.6	−2.43702
$906$	0	0
$907$	−31106.8	−1.13879	−0.569396	−	0.822063i	$-0.692823\pi$
$907$	−31106.8	−1.13879	−0.569396	+	0.822063i	$0.692823\pi$
$908$	0	0
$909$	9084.72	0.331486
$910$	0	0
$911$	36007.5	1.30953	0.654765	−	0.755833i	$-0.272768\pi$
$911$	36007.5	1.30953	0.654765	+	0.755833i	$0.272768\pi$
$912$	0	0
$913$	−4404.03	−0.159641
$914$	0	0
$915$	−7174.59	−0.259218
$916$	0	0
$917$	3226.99	0.116210
$918$	0	0
$919$	−47066.5	−1.68942	−0.844711	−	0.535222i	$-0.820228\pi$
$919$	−47066.5	−1.68942	−0.844711	+	0.535222i	$0.820228\pi$
$920$	0	0
$921$	13899.4	0.497288
$922$	0	0
$923$	−22954.4	−0.818584
$924$	0	0
$925$	22597.0	0.803228
$926$	0	0
$927$	6176.08	0.218823
$928$	0	0
$929$	−25359.2	−0.895596	−0.447798	−	0.894135i	$-0.647792\pi$
$929$	−25359.2	−0.895596	−0.447798	+	0.894135i	$0.647792\pi$
$930$	0	0
$931$	636.119	0.0223931
$932$	0	0
$933$	18217.1	0.639231
$934$	0	0
$935$	5420.41	0.189590
$936$	0	0
$937$	12778.7	0.445530	0.222765	−	0.974872i	$-0.428492\pi$
$937$	12778.7	0.445530	0.222765	+	0.974872i	$0.428492\pi$
$938$	0	0
$939$	−9851.06	−0.342361
$940$	0	0
$941$	29785.8	1.03187	0.515935	−	0.856628i	$-0.327445\pi$
$941$	29785.8	1.03187	0.515935	+	0.856628i	$0.327445\pi$
$942$	0	0
$943$	−19881.2	−0.686554
$944$	0	0
$945$	11585.0	0.398794
$946$	0	0
$947$	3498.58	0.120051	0.0600256	−	0.998197i	$-0.480882\pi$
$947$	3498.58	0.120051	0.0600256	+	0.998197i	$0.480882\pi$
$948$	0	0
$949$	−897.974	−0.0307160
$950$	0	0
$951$	−39019.6	−1.33049
$952$	0	0
$953$	−18608.5	−0.632516	−0.316258	−	0.948673i	$-0.602427\pi$
$953$	−18608.5	−0.632516	−0.316258	+	0.948673i	$0.602427\pi$
$954$	0	0
$955$	−31990.2	−1.08396
$956$	0	0
$957$	−3197.36	−0.108000
$958$	0	0
$959$	−11166.5	−0.376002
$960$	0	0
$961$	15649.0	0.525293
$962$	0	0
$963$	7082.08	0.236985
$964$	0	0
$965$	2063.24	0.0688269
$966$	0	0
$967$	38886.4	1.29318	0.646589	−	0.762838i	$-0.276195\pi$
$967$	38886.4	1.29318	0.646589	+	0.762838i	$0.276195\pi$
$968$	0	0
$969$	−244.830	−0.00811669
$970$	0	0
$971$	−30724.6	−1.01545	−0.507724	−	0.861520i	$-0.669513\pi$
$971$	−30724.6	−1.01545	−0.507724	+	0.861520i	$0.669513\pi$
$972$	0	0
$973$	10132.2	0.333839
$974$	0	0
$975$	23507.7	0.772153
$976$	0	0
$977$	−17594.0	−0.576133	−0.288067	−	0.957610i	$-0.593012\pi$
$977$	−17594.0	−0.576133	−0.288067	+	0.957610i	$0.593012\pi$
$978$	0	0
$979$	21550.5	0.703530
$980$	0	0
$981$	7010.08	0.228149
$982$	0	0
$983$	−26914.6	−0.873290	−0.436645	−	0.899634i	$-0.643833\pi$
$983$	−26914.6	−0.873290	−0.436645	+	0.899634i	$0.643833\pi$
$984$	0	0
$985$	−67378.4	−2.17955
$986$	0	0
$987$	−20172.5	−0.650556
$988$	0	0
$989$	−3574.90	−0.114940
$990$	0	0
$991$	47353.1	1.51788	0.758940	−	0.651160i	$-0.225717\pi$
$991$	47353.1	1.51788	0.758940	+	0.651160i	$0.225717\pi$
$992$	0	0
$993$	68653.8	2.19402
$994$	0	0
$995$	−36536.2	−1.16410
$996$	0	0
$997$	20910.2	0.664227	0.332113	−	0.943240i	$-0.392238\pi$
$997$	20910.2	0.664227	0.332113	+	0.943240i	$0.392238\pi$
$998$	0	0
$999$	−25593.8	−0.810563

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1856.4.a.z.1.5		5
4.3	odd	2		1856.4.a.ba.1.1		5
8.3	odd	2		464.4.a.m.1.5		5
8.5	even	2		232.4.a.d.1.1	✓	5
24.5	odd	2		2088.4.a.f.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
232.4.a.d.1.1	✓	5	8.5	even	2
464.4.a.m.1.5		5	8.3	odd	2
1856.4.a.z.1.5		5	1.1	even	1	trivial
1856.4.a.ba.1.1		5	4.3	odd	2
2088.4.a.f.1.5		5	24.5	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:	\( N \)	\(=\)	\( 1856 = 2^{6} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1856.a (trivial)

\(f(q)\)	\(=\)	\(q+5.90002 q^{3} +14.9988 q^{5} -6.82211 q^{7} +7.81028 q^{9} +O(q^{10})\)	\(q+5.90002 q^{3} +14.9988 q^{5} -6.82211 q^{7} +7.81028 q^{9} +18.6870 q^{11} +39.8582 q^{13} +88.4931 q^{15} +19.3391 q^{17} -2.14572 q^{19} -40.2506 q^{21} +103.945 q^{23} +99.9629 q^{25} -113.220 q^{27} -29.0000 q^{29} +213.167 q^{31} +110.254 q^{33} -102.323 q^{35} +226.054 q^{37} +235.164 q^{39} -191.267 q^{41} -34.3923 q^{43} +117.145 q^{45} +501.173 q^{47} -296.459 q^{49} +114.101 q^{51} -259.952 q^{53} +280.282 q^{55} -12.6598 q^{57} -280.736 q^{59} -81.0752 q^{61} -53.2826 q^{63} +597.824 q^{65} +799.332 q^{67} +613.278 q^{69} -575.901 q^{71} -22.5292 q^{73} +589.783 q^{75} -127.485 q^{77} -118.878 q^{79} -878.877 q^{81} -235.673 q^{83} +290.063 q^{85} -171.101 q^{87} +1153.23 q^{89} -271.917 q^{91} +1257.69 q^{93} -32.1832 q^{95} +295.976 q^{97} +145.951 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 4 q^{3} - 10 q^{5} + 32 q^{7} + 29 q^{9}+O(q^{10}) \) 5 * q - 4 * q^3 - 10 * q^5 + 32 * q^7 + 29 * q^9	\( 5 q - 4 q^{3} - 10 q^{5} + 32 q^{7} + 29 q^{9} - 36 q^{11} - 26 q^{13} + 88 q^{15} + 82 q^{17} - 156 q^{19} - 72 q^{21} + 336 q^{23} + 151 q^{25} - 352 q^{27} - 145 q^{29} + 432 q^{31} + 108 q^{33} - 600 q^{35} + 18 q^{37} + 688 q^{39} + 82 q^{41} - 340 q^{43} + 146 q^{45} + 680 q^{47} - 115 q^{49} - 608 q^{51} + 102 q^{53} + 736 q^{55} - 576 q^{57} - 924 q^{59} + 618 q^{61} + 584 q^{63} - 704 q^{65} - 44 q^{67} + 1056 q^{69} + 1032 q^{71} - 1078 q^{73} + 468 q^{75} + 888 q^{77} + 200 q^{79} - 1843 q^{81} - 452 q^{83} + 1700 q^{85} + 116 q^{87} - 1790 q^{89} + 1128 q^{91} + 1884 q^{93} + 1024 q^{95} - 2518 q^{97} + 1500 q^{99}+O(q^{100}) \) 5 * q - 4 * q^3 - 10 * q^5 + 32 * q^7 + 29 * q^9 - 36 * q^11 - 26 * q^13 + 88 * q^15 + 82 * q^17 - 156 * q^19 - 72 * q^21 + 336 * q^23 + 151 * q^25 - 352 * q^27 - 145 * q^29 + 432 * q^31 + 108 * q^33 - 600 * q^35 + 18 * q^37 + 688 * q^39 + 82 * q^41 - 340 * q^43 + 146 * q^45 + 680 * q^47 - 115 * q^49 - 608 * q^51 + 102 * q^53 + 736 * q^55 - 576 * q^57 - 924 * q^59 + 618 * q^61 + 584 * q^63 - 704 * q^65 - 44 * q^67 + 1056 * q^69 + 1032 * q^71 - 1078 * q^73 + 468 * q^75 + 888 * q^77 + 200 * q^79 - 1843 * q^81 - 452 * q^83 + 1700 * q^85 + 116 * q^87 - 1790 * q^89 + 1128 * q^91 + 1884 * q^93 + 1024 * q^95 - 2518 * q^97 + 1500 * q^99

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