LMFDB - Embedded newform 6864.2.a.cd.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6864,2,Mod(1,6864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6864, 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("6864.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6864 = 2^{4} \cdot 3 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6864.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:	$54.8093159474$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.29268.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 9x^{2} + 5x + 16 $ x^4 - x^3 - 9x^2 + 5x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 3432)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.83719$ of defining polynomial
Character	$\chi$	$=$	6864.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +3.49248 q^{5} -2.46193 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +3.49248 q^{5} -2.46193 q^{7} +1.00000 q^{9} +1.00000 q^{11} +1.00000 q^{13} +3.49248 q^{15} +4.70492 q^{17} +2.46193 q^{19} -2.46193 q^{21} +1.21245 q^{23} +7.19740 q^{25} +1.00000 q^{27} +2.96945 q^{29} -6.41634 q^{31} +1.00000 q^{33} -8.59823 q^{35} +3.24948 q^{37} +1.00000 q^{39} -2.88682 q^{41} +4.64383 q^{43} +3.49248 q^{45} +6.65933 q^{47} -0.938903 q^{49} +4.70492 q^{51} +3.93890 q^{53} +3.49248 q^{55} +2.46193 q^{57} -14.3337 q^{59} +10.2344 q^{61} -2.46193 q^{63} +3.49248 q^{65} +1.00649 q^{67} +1.21245 q^{69} +1.24948 q^{71} +4.94792 q^{73} +7.19740 q^{75} -2.46193 q^{77} +4.28003 q^{79} +1.00000 q^{81} -14.3337 q^{83} +16.4318 q^{85} +2.96945 q^{87} -13.1368 q^{89} -2.46193 q^{91} -6.41634 q^{93} +8.59823 q^{95} -12.5681 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 4 q^{5} + 2 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 + 4 * q^5 + 2 * q^7 + 4 * q^9	$ 4 q + 4 q^{3} + 4 q^{5} + 2 q^{7} + 4 q^{9} + 4 q^{11} + 4 q^{13} + 4 q^{15} + 8 q^{17} - 2 q^{19} + 2 q^{21} + 4 q^{23} + 8 q^{25} + 4 q^{27} + 10 q^{29} + 8 q^{31} + 4 q^{33} + 2 q^{35} + 2 q^{37} + 4 q^{39} + 2 q^{41} + 4 q^{43} + 4 q^{45} - 6 q^{47} + 8 q^{51} + 12 q^{53} + 4 q^{55} - 2 q^{57} - 12 q^{59} + 10 q^{61} + 2 q^{63} + 4 q^{65} - 8 q^{67} + 4 q^{69} - 6 q^{71} + 10 q^{73} + 8 q^{75} + 2 q^{77} + 8 q^{79} + 4 q^{81} - 12 q^{83} + 14 q^{85} + 10 q^{87} + 10 q^{89} + 2 q^{91} + 8 q^{93} - 2 q^{95} + 26 q^{97} + 4 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 + 4 * q^5 + 2 * q^7 + 4 * q^9 + 4 * q^11 + 4 * q^13 + 4 * q^15 + 8 * q^17 - 2 * q^19 + 2 * q^21 + 4 * q^23 + 8 * q^25 + 4 * q^27 + 10 * q^29 + 8 * q^31 + 4 * q^33 + 2 * q^35 + 2 * q^37 + 4 * q^39 + 2 * q^41 + 4 * q^43 + 4 * q^45 - 6 * q^47 + 8 * q^51 + 12 * q^53 + 4 * q^55 - 2 * q^57 - 12 * q^59 + 10 * q^61 + 2 * q^63 + 4 * q^65 - 8 * q^67 + 4 * q^69 - 6 * q^71 + 10 * q^73 + 8 * q^75 + 2 * q^77 + 8 * q^79 + 4 * q^81 - 12 * q^83 + 14 * q^85 + 10 * q^87 + 10 * q^89 + 2 * q^91 + 8 * q^93 - 2 * q^95 + 26 * q^97 + 4 * 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
$5$	3.49248	1.56188	0.780942	−	0.624604i	$-0.214740\pi$
$5$	3.49248	1.56188	0.780942	+	0.624604i	$0.214740\pi$
$6$	0	0
$7$	−2.46193	−0.930522	−0.465261	−	0.885174i	$-0.654040\pi$
$7$	−2.46193	−0.930522	−0.465261	+	0.885174i	$0.654040\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	3.49248	0.901754
$16$	0	0
$17$	4.70492	1.14111	0.570556	−	0.821259i	$-0.306728\pi$
$17$	4.70492	1.14111	0.570556	+	0.821259i	$0.306728\pi$
$18$	0	0
$19$	2.46193	0.564805	0.282403	−	0.959296i	$-0.408869\pi$
$19$	2.46193	0.564805	0.282403	+	0.959296i	$0.408869\pi$
$20$	0	0
$21$	−2.46193	−0.537237
$22$	0	0
$23$	1.21245	0.252812	0.126406	−	0.991979i	$-0.459656\pi$
$23$	1.21245	0.252812	0.126406	+	0.991979i	$0.459656\pi$
$24$	0	0
$25$	7.19740	1.43948
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	2.96945	0.551413	0.275707	−	0.961242i	$-0.411088\pi$
$29$	2.96945	0.551413	0.275707	+	0.961242i	$0.411088\pi$
$30$	0	0
$31$	−6.41634	−1.15241	−0.576204	−	0.817306i	$-0.695467\pi$
$31$	−6.41634	−1.15241	−0.576204	+	0.817306i	$0.695467\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	−8.59823	−1.45337
$36$	0	0
$37$	3.24948	0.534212	0.267106	−	0.963667i	$-0.413933\pi$
$37$	3.24948	0.534212	0.267106	+	0.963667i	$0.413933\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	−2.88682	−0.450846	−0.225423	−	0.974261i	$-0.572376\pi$
$41$	−2.88682	−0.450846	−0.225423	+	0.974261i	$0.572376\pi$
$42$	0	0
$43$	4.64383	0.708177	0.354088	−	0.935212i	$-0.384791\pi$
$43$	4.64383	0.708177	0.354088	+	0.935212i	$0.384791\pi$
$44$	0	0
$45$	3.49248	0.520628
$46$	0	0
$47$	6.65933	0.971363	0.485682	−	0.874136i	$-0.338571\pi$
$47$	6.65933	0.971363	0.485682	+	0.874136i	$0.338571\pi$
$48$	0	0
$49$	−0.938903	−0.134129
$50$	0	0
$51$	4.70492	0.658821
$52$	0	0
$53$	3.93890	0.541050	0.270525	−	0.962713i	$-0.412803\pi$
$53$	3.93890	0.541050	0.270525	+	0.962713i	$0.412803\pi$
$54$	0	0
$55$	3.49248	0.470926
$56$	0	0
$57$	2.46193	0.326091
$58$	0	0
$59$	−14.3337	−1.86609	−0.933045	−	0.359760i	$-0.882858\pi$
$59$	−14.3337	−1.86609	−0.933045	+	0.359760i	$0.882858\pi$
$60$	0	0
$61$	10.2344	1.31039	0.655193	−	0.755462i	$-0.272587\pi$
$61$	10.2344	1.31039	0.655193	+	0.755462i	$0.272587\pi$
$62$	0	0
$63$	−2.46193	−0.310174
$64$	0	0
$65$	3.49248	0.433189
$66$	0	0
$67$	1.00649	0.122962	0.0614812	−	0.998108i	$-0.480418\pi$
$67$	1.00649	0.122962	0.0614812	+	0.998108i	$0.480418\pi$
$68$	0	0
$69$	1.21245	0.145961
$70$	0	0
$71$	1.24948	0.148286	0.0741432	−	0.997248i	$-0.476378\pi$
$71$	1.24948	0.148286	0.0741432	+	0.997248i	$0.476378\pi$
$72$	0	0
$73$	4.94792	0.579110	0.289555	−	0.957161i	$-0.406493\pi$
$73$	4.94792	0.579110	0.289555	+	0.957161i	$0.406493\pi$
$74$	0	0
$75$	7.19740	0.831084
$76$	0	0
$77$	−2.46193	−0.280563
$78$	0	0
$79$	4.28003	0.481541	0.240771	−	0.970582i	$-0.422600\pi$
$79$	4.28003	0.481541	0.240771	+	0.970582i	$0.422600\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−14.3337	−1.57333	−0.786664	−	0.617381i	$-0.788194\pi$
$83$	−14.3337	−1.57333	−0.786664	+	0.617381i	$0.788194\pi$
$84$	0	0
$85$	16.4318	1.78228
$86$	0	0
$87$	2.96945	0.318359
$88$	0	0
$89$	−13.1368	−1.39249	−0.696247	−	0.717802i	$-0.745148\pi$
$89$	−13.1368	−1.39249	−0.696247	+	0.717802i	$0.745148\pi$
$90$	0	0
$91$	−2.46193	−0.258080
$92$	0	0
$93$	−6.41634	−0.665343
$94$	0	0
$95$	8.59823	0.882160
$96$	0	0
$97$	−12.5681	−1.27610	−0.638051	−	0.769994i	$-0.720259\pi$
$97$	−12.5681	−1.27610	−0.638051	+	0.769994i	$0.720259\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	−5.62878	−0.560085	−0.280042	−	0.959988i	$-0.590349\pi$
$101$	−5.62878	−0.560085	−0.280042	+	0.959988i	$0.590349\pi$
$102$	0	0
$103$	7.34875	0.724094	0.362047	−	0.932160i	$-0.382078\pi$
$103$	7.34875	0.724094	0.362047	+	0.932160i	$0.382078\pi$
$104$	0	0
$105$	−8.59823	−0.839102
$106$	0	0
$107$	10.3337	0.998997	0.499499	−	0.866315i	$-0.333518\pi$
$107$	10.3337	0.998997	0.499499	+	0.866315i	$0.333518\pi$
$108$	0	0
$109$	−3.44689	−0.330152	−0.165076	−	0.986281i	$-0.552787\pi$
$109$	−3.44689	−0.330152	−0.165076	+	0.986281i	$0.552787\pi$
$110$	0	0
$111$	3.24948	0.308427
$112$	0	0
$113$	−8.33371	−0.783969	−0.391985	−	0.919972i	$-0.628211\pi$
$113$	−8.33371	−0.783969	−0.391985	+	0.919972i	$0.628211\pi$
$114$	0	0
$115$	4.23444	0.394864
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	−11.5832	−1.06183
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−2.88682	−0.260296
$124$	0	0
$125$	7.67438	0.686417
$126$	0	0
$127$	−13.9243	−1.23558	−0.617792	−	0.786342i	$-0.711973\pi$
$127$	−13.9243	−1.23558	−0.617792	+	0.786342i	$0.711973\pi$
$128$	0	0
$129$	4.64383	0.408866
$130$	0	0
$131$	16.3948	1.43242	0.716210	−	0.697885i	$-0.245875\pi$
$131$	16.3948	1.43242	0.716210	+	0.697885i	$0.245875\pi$
$132$	0	0
$133$	−6.06110	−0.525564
$134$	0	0
$135$	3.49248	0.300585
$136$	0	0
$137$	11.6528	0.995569	0.497785	−	0.867301i	$-0.334147\pi$
$137$	11.6528	0.995569	0.497785	+	0.867301i	$0.334147\pi$
$138$	0	0
$139$	14.2881	1.21190	0.605951	−	0.795502i	$-0.292793\pi$
$139$	14.2881	1.21190	0.605951	+	0.795502i	$0.292793\pi$
$140$	0	0
$141$	6.65933	0.560817
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	10.3707	0.861243
$146$	0	0
$147$	−0.938903	−0.0774394
$148$	0	0
$149$	16.4619	1.34861	0.674307	−	0.738451i	$-0.264442\pi$
$149$	16.4619	1.34861	0.674307	+	0.738451i	$0.264442\pi$
$150$	0	0
$151$	8.43184	0.686173	0.343087	−	0.939304i	$-0.388528\pi$
$151$	8.43184	0.686173	0.343087	+	0.939304i	$0.388528\pi$
$152$	0	0
$153$	4.70492	0.380371
$154$	0	0
$155$	−22.4089	−1.79993
$156$	0	0
$157$	5.15135	0.411122	0.205561	−	0.978644i	$-0.434098\pi$
$157$	5.15135	0.411122	0.205561	+	0.978644i	$0.434098\pi$
$158$	0	0
$159$	3.93890	0.312375
$160$	0	0
$161$	−2.98496	−0.235247
$162$	0	0
$163$	7.50056	0.587489	0.293745	−	0.955884i	$-0.405098\pi$
$163$	7.50056	0.587489	0.293745	+	0.955884i	$0.405098\pi$
$164$	0	0
$165$	3.49248	0.271889
$166$	0	0
$167$	9.77364	0.756307	0.378154	−	0.925743i	$-0.376559\pi$
$167$	9.77364	0.756307	0.378154	+	0.925743i	$0.376559\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	2.46193	0.188268
$172$	0	0
$173$	24.4915	1.86206	0.931029	−	0.364946i	$-0.118912\pi$
$173$	24.4915	1.86206	0.931029	+	0.364946i	$0.118912\pi$
$174$	0	0
$175$	−17.7195	−1.33947
$176$	0	0
$177$	−14.3337	−1.07739
$178$	0	0
$179$	−3.15135	−0.235543	−0.117771	−	0.993041i	$-0.537575\pi$
$179$	−3.15135	−0.235543	−0.117771	+	0.993041i	$0.537575\pi$
$180$	0	0
$181$	2.72646	0.202656	0.101328	−	0.994853i	$-0.467691\pi$
$181$	2.72646	0.202656	0.101328	+	0.994853i	$0.467691\pi$
$182$	0	0
$183$	10.2344	0.756552
$184$	0	0
$185$	11.3488	0.834377
$186$	0	0
$187$	4.70492	0.344058
$188$	0	0
$189$	−2.46193	−0.179079
$190$	0	0
$191$	−20.9549	−1.51624	−0.758120	−	0.652115i	$-0.773882\pi$
$191$	−20.9549	−1.51624	−0.758120	+	0.652115i	$0.773882\pi$
$192$	0	0
$193$	22.2185	1.59932	0.799660	−	0.600453i	$-0.205013\pi$
$193$	22.2185	1.59932	0.799660	+	0.600453i	$0.205013\pi$
$194$	0	0
$195$	3.49248	0.250102
$196$	0	0
$197$	−7.44689	−0.530569	−0.265284	−	0.964170i	$-0.585466\pi$
$197$	−7.44689	−0.530569	−0.265284	+	0.964170i	$0.585466\pi$
$198$	0	0
$199$	−5.96991	−0.423196	−0.211598	−	0.977357i	$-0.567867\pi$
$199$	−5.96991	−0.423196	−0.211598	+	0.977357i	$0.567867\pi$
$200$	0	0
$201$	1.00649	0.0709923
$202$	0	0
$203$	−7.31058	−0.513102
$204$	0	0
$205$	−10.0822	−0.704168
$206$	0	0
$207$	1.21245	0.0842708
$208$	0	0
$209$	2.46193	0.170295
$210$	0	0
$211$	3.35617	0.231048	0.115524	−	0.993305i	$-0.463145\pi$
$211$	3.35617	0.231048	0.115524	+	0.993305i	$0.463145\pi$
$212$	0	0
$213$	1.24948	0.0856132
$214$	0	0
$215$	16.2185	1.10609
$216$	0	0
$217$	15.7966	1.07234
$218$	0	0
$219$	4.94792	0.334349
$220$	0	0
$221$	4.70492	0.316487
$222$	0	0
$223$	6.41634	0.429670	0.214835	−	0.976650i	$-0.431079\pi$
$223$	6.41634	0.429670	0.214835	+	0.976650i	$0.431079\pi$
$224$	0	0
$225$	7.19740	0.479827
$226$	0	0
$227$	20.7044	1.37420	0.687101	−	0.726562i	$-0.258883\pi$
$227$	20.7044	1.37420	0.687101	+	0.726562i	$0.258883\pi$
$228$	0	0
$229$	8.65933	0.572225	0.286112	−	0.958196i	$-0.407637\pi$
$229$	8.65933	0.572225	0.286112	+	0.958196i	$0.407637\pi$
$230$	0	0
$231$	−2.46193	−0.161983
$232$	0	0
$233$	3.45544	0.226373	0.113187	−	0.993574i	$-0.463894\pi$
$233$	3.45544	0.226373	0.113187	+	0.993574i	$0.463894\pi$
$234$	0	0
$235$	23.2576	1.51716
$236$	0	0
$237$	4.28003	0.278018
$238$	0	0
$239$	−18.0069	−1.16477	−0.582386	−	0.812912i	$-0.697881\pi$
$239$	−18.0069	−1.16477	−0.582386	+	0.812912i	$0.697881\pi$
$240$	0	0
$241$	−0.788686	−0.0508037	−0.0254019	−	0.999677i	$-0.508087\pi$
$241$	−0.788686	−0.0508037	−0.0254019	+	0.999677i	$0.508087\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−3.27910	−0.209494
$246$	0	0
$247$	2.46193	0.156649
$248$	0	0
$249$	−14.3337	−0.908362
$250$	0	0
$251$	−10.4559	−0.659971	−0.329985	−	0.943986i	$-0.607044\pi$
$251$	−10.4559	−0.659971	−0.329985	+	0.943986i	$0.607044\pi$
$252$	0	0
$253$	1.21245	0.0762258
$254$	0	0
$255$	16.4318	1.02900
$256$	0	0
$257$	−8.89377	−0.554778	−0.277389	−	0.960758i	$-0.589469\pi$
$257$	−8.89377	−0.554778	−0.277389	+	0.960758i	$0.589469\pi$
$258$	0	0
$259$	−8.00000	−0.497096
$260$	0	0
$261$	2.96945	0.183804
$262$	0	0
$263$	−27.7746	−1.71265	−0.856326	−	0.516435i	$-0.827259\pi$
$263$	−27.7746	−1.71265	−0.856326	+	0.516435i	$0.827259\pi$
$264$	0	0
$265$	13.7565	0.845057
$266$	0	0
$267$	−13.1368	−0.803957
$268$	0	0
$269$	24.2856	1.48072	0.740359	−	0.672211i	$-0.234655\pi$
$269$	24.2856	1.48072	0.740359	+	0.672211i	$0.234655\pi$
$270$	0	0
$271$	−6.82572	−0.414633	−0.207317	−	0.978274i	$-0.566473\pi$
$271$	−6.82572	−0.414633	−0.207317	+	0.978274i	$0.566473\pi$
$272$	0	0
$273$	−2.46193	−0.149003
$274$	0	0
$275$	7.19740	0.434020
$276$	0	0
$277$	−1.92592	−0.115718	−0.0578588	−	0.998325i	$-0.518427\pi$
$277$	−1.92592	−0.115718	−0.0578588	+	0.998325i	$0.518427\pi$
$278$	0	0
$279$	−6.41634	−0.384136
$280$	0	0
$281$	−15.7195	−0.937746	−0.468873	−	0.883265i	$-0.655340\pi$
$281$	−15.7195	−0.937746	−0.468873	+	0.883265i	$0.655340\pi$
$282$	0	0
$283$	−1.54663	−0.0919373	−0.0459687	−	0.998943i	$-0.514637\pi$
$283$	−1.54663	−0.0919373	−0.0459687	+	0.998943i	$0.514637\pi$
$284$	0	0
$285$	8.59823	0.509315
$286$	0	0
$287$	7.10715	0.419522
$288$	0	0
$289$	5.13630	0.302136
$290$	0	0
$291$	−12.5681	−0.736758
$292$	0	0
$293$	22.5230	1.31581	0.657905	−	0.753101i	$-0.271443\pi$
$293$	22.5230	1.31581	0.657905	+	0.753101i	$0.271443\pi$
$294$	0	0
$295$	−50.0601	−2.91461
$296$	0	0
$297$	1.00000	0.0580259
$298$	0	0
$299$	1.21245	0.0701175
$300$	0	0
$301$	−11.4328	−0.658974
$302$	0	0
$303$	−5.62878	−0.323365
$304$	0	0
$305$	35.7436	2.04667
$306$	0	0
$307$	−13.2816	−0.758022	−0.379011	−	0.925392i	$-0.623736\pi$
$307$	−13.2816	−0.758022	−0.379011	+	0.925392i	$0.623736\pi$
$308$	0	0
$309$	7.34875	0.418056
$310$	0	0
$311$	−18.4089	−1.04387	−0.521937	−	0.852984i	$-0.674790\pi$
$311$	−18.4089	−1.04387	−0.521937	+	0.852984i	$0.674790\pi$
$312$	0	0
$313$	−5.63734	−0.318641	−0.159321	−	0.987227i	$-0.550930\pi$
$313$	−5.63734	−0.318641	−0.159321	+	0.987227i	$0.550930\pi$
$314$	0	0
$315$	−8.59823	−0.484456
$316$	0	0
$317$	−5.01457	−0.281646	−0.140823	−	0.990035i	$-0.544975\pi$
$317$	−5.01457	−0.281646	−0.140823	+	0.990035i	$0.544975\pi$
$318$	0	0
$319$	2.96945	0.166257
$320$	0	0
$321$	10.3337	0.576771
$322$	0	0
$323$	11.5832	0.644506
$324$	0	0
$325$	7.19740	0.399240
$326$	0	0
$327$	−3.44689	−0.190613
$328$	0	0
$329$	−16.3948	−0.903875
$330$	0	0
$331$	9.19786	0.505560	0.252780	−	0.967524i	$-0.418655\pi$
$331$	9.19786	0.505560	0.252780	+	0.967524i	$0.418655\pi$
$332$	0	0
$333$	3.24948	0.178071
$334$	0	0
$335$	3.51514	0.192053
$336$	0	0
$337$	−28.2735	−1.54016	−0.770079	−	0.637949i	$-0.779783\pi$
$337$	−28.2735	−1.54016	−0.770079	+	0.637949i	$0.779783\pi$
$338$	0	0
$339$	−8.33371	−0.452625
$340$	0	0
$341$	−6.41634	−0.347464
$342$	0	0
$343$	19.5450	1.05533
$344$	0	0
$345$	4.23444	0.227975
$346$	0	0
$347$	3.02313	0.162290	0.0811449	−	0.996702i	$-0.474142\pi$
$347$	3.02313	0.162290	0.0811449	+	0.996702i	$0.474142\pi$
$348$	0	0
$349$	8.91088	0.476988	0.238494	−	0.971144i	$-0.423346\pi$
$349$	8.91088	0.476988	0.238494	+	0.971144i	$0.423346\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	−31.4323	−1.67297	−0.836486	−	0.547988i	$-0.815394\pi$
$353$	−31.4323	−1.67297	−0.836486	+	0.547988i	$0.815394\pi$
$354$	0	0
$355$	4.36379	0.231606
$356$	0	0
$357$	−11.5832	−0.613047
$358$	0	0
$359$	−20.7526	−1.09528	−0.547639	−	0.836715i	$-0.684473\pi$
$359$	−20.7526	−1.09528	−0.547639	+	0.836715i	$0.684473\pi$
$360$	0	0
$361$	−12.9389	−0.680995
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	17.2805	0.904502
$366$	0	0
$367$	19.7866	1.03285	0.516427	−	0.856331i	$-0.327262\pi$
$367$	19.7866	1.03285	0.516427	+	0.856331i	$0.327262\pi$
$368$	0	0
$369$	−2.88682	−0.150282
$370$	0	0
$371$	−9.69730	−0.503459
$372$	0	0
$373$	3.98702	0.206440	0.103220	−	0.994659i	$-0.467085\pi$
$373$	3.98702	0.206440	0.103220	+	0.994659i	$0.467085\pi$
$374$	0	0
$375$	7.67438	0.396303
$376$	0	0
$377$	2.96945	0.152935
$378$	0	0
$379$	−10.6508	−0.547094	−0.273547	−	0.961859i	$-0.588197\pi$
$379$	−10.6508	−0.547094	−0.273547	+	0.961859i	$0.588197\pi$
$380$	0	0
$381$	−13.9243	−0.713364
$382$	0	0
$383$	−12.4379	−0.635546	−0.317773	−	0.948167i	$-0.602935\pi$
$383$	−12.4379	−0.635546	−0.317773	+	0.948167i	$0.602935\pi$
$384$	0	0
$385$	−8.59823	−0.438207
$386$	0	0
$387$	4.64383	0.236059
$388$	0	0
$389$	21.4538	1.08775	0.543876	−	0.839165i	$-0.316956\pi$
$389$	21.4538	1.08775	0.543876	+	0.839165i	$0.316956\pi$
$390$	0	0
$391$	5.70446	0.288487
$392$	0	0
$393$	16.3948	0.827008
$394$	0	0
$395$	14.9479	0.752111
$396$	0	0
$397$	−24.6903	−1.23917	−0.619586	−	0.784929i	$-0.712700\pi$
$397$	−24.6903	−1.23917	−0.619586	+	0.784929i	$0.712700\pi$
$398$	0	0
$399$	−6.06110	−0.303434
$400$	0	0
$401$	−7.44436	−0.371754	−0.185877	−	0.982573i	$-0.559513\pi$
$401$	−7.44436	−0.371754	−0.185877	+	0.982573i	$0.559513\pi$
$402$	0	0
$403$	−6.41634	−0.319620
$404$	0	0
$405$	3.49248	0.173543
$406$	0	0
$407$	3.24948	0.161071
$408$	0	0
$409$	−9.99397	−0.494170	−0.247085	−	0.968994i	$-0.579473\pi$
$409$	−9.99397	−0.494170	−0.247085	+	0.968994i	$0.579473\pi$
$410$	0	0
$411$	11.6528	0.574792
$412$	0	0
$413$	35.2886	1.73644
$414$	0	0
$415$	−50.0601	−2.45736
$416$	0	0
$417$	14.2881	0.699692
$418$	0	0
$419$	18.8949	0.923076	0.461538	−	0.887120i	$-0.347298\pi$
$419$	18.8949	0.923076	0.461538	+	0.887120i	$0.347298\pi$
$420$	0	0
$421$	−8.83175	−0.430433	−0.215217	−	0.976566i	$-0.569046\pi$
$421$	−8.83175	−0.430433	−0.215217	+	0.976566i	$0.569046\pi$
$422$	0	0
$423$	6.65933	0.323788
$424$	0	0
$425$	33.8632	1.64261
$426$	0	0
$427$	−25.1965	−1.21934
$428$	0	0
$429$	1.00000	0.0482805
$430$	0	0
$431$	20.9239	1.00787	0.503933	−	0.863743i	$-0.331886\pi$
$431$	20.9239	1.00787	0.503933	+	0.863743i	$0.331886\pi$
$432$	0	0
$433$	0.697500	0.0335197	0.0167599	−	0.999860i	$-0.494665\pi$
$433$	0.697500	0.0335197	0.0167599	+	0.999860i	$0.494665\pi$
$434$	0	0
$435$	10.3707	0.497239
$436$	0	0
$437$	2.98496	0.142790
$438$	0	0
$439$	25.8403	1.23329	0.616645	−	0.787241i	$-0.288491\pi$
$439$	25.8403	1.23329	0.616645	+	0.787241i	$0.288491\pi$
$440$	0	0
$441$	−0.938903	−0.0447097
$442$	0	0
$443$	−1.90790	−0.0906468	−0.0453234	−	0.998972i	$-0.514432\pi$
$443$	−1.90790	−0.0906468	−0.0453234	+	0.998972i	$0.514432\pi$
$444$	0	0
$445$	−45.8799	−2.17491
$446$	0	0
$447$	16.4619	0.778623
$448$	0	0
$449$	3.44436	0.162549	0.0812747	−	0.996692i	$-0.474101\pi$
$449$	3.44436	0.162549	0.0812747	+	0.996692i	$0.474101\pi$
$450$	0	0
$451$	−2.88682	−0.135935
$452$	0	0
$453$	8.43184	0.396162
$454$	0	0
$455$	−8.59823	−0.403091
$456$	0	0
$457$	−19.9218	−0.931902	−0.465951	−	0.884810i	$-0.654288\pi$
$457$	−19.9218	−0.931902	−0.465951	+	0.884810i	$0.654288\pi$
$458$	0	0
$459$	4.70492	0.219607
$460$	0	0
$461$	−6.70445	−0.312257	−0.156129	−	0.987737i	$-0.549901\pi$
$461$	−6.70445	−0.312257	−0.156129	+	0.987737i	$0.549901\pi$
$462$	0	0
$463$	23.4265	1.08872	0.544360	−	0.838851i	$-0.316772\pi$
$463$	23.4265	1.08872	0.544360	+	0.838851i	$0.316772\pi$
$464$	0	0
$465$	−22.4089	−1.03919
$466$	0	0
$467$	18.4089	0.851863	0.425931	−	0.904755i	$-0.359946\pi$
$467$	18.4089	0.851863	0.425931	+	0.904755i	$0.359946\pi$
$468$	0	0
$469$	−2.47791	−0.114419
$470$	0	0
$471$	5.15135	0.237362
$472$	0	0
$473$	4.64383	0.213523
$474$	0	0
$475$	17.7195	0.813026
$476$	0	0
$477$	3.93890	0.180350
$478$	0	0
$479$	−5.81068	−0.265497	−0.132748	−	0.991150i	$-0.542380\pi$
$479$	−5.81068	−0.265497	−0.132748	+	0.991150i	$0.542380\pi$
$480$	0	0
$481$	3.24948	0.148164
$482$	0	0
$483$	−2.98496	−0.135820
$484$	0	0
$485$	−43.8940	−1.99312
$486$	0	0
$487$	18.7299	0.848733	0.424366	−	0.905491i	$-0.360497\pi$
$487$	18.7299	0.848733	0.424366	+	0.905491i	$0.360497\pi$
$488$	0	0
$489$	7.50056	0.339187
$490$	0	0
$491$	−40.7366	−1.83842	−0.919208	−	0.393772i	$-0.871170\pi$
$491$	−40.7366	−1.83842	−0.919208	+	0.393772i	$0.871170\pi$
$492$	0	0
$493$	13.9710	0.629224
$494$	0	0
$495$	3.49248	0.156975
$496$	0	0
$497$	−3.07614	−0.137984
$498$	0	0
$499$	21.2028	0.949166	0.474583	−	0.880211i	$-0.342599\pi$
$499$	21.2028	0.949166	0.474583	+	0.880211i	$0.342599\pi$
$500$	0	0
$501$	9.77364	0.436654
$502$	0	0
$503$	−2.51608	−0.112186	−0.0560932	−	0.998426i	$-0.517864\pi$
$503$	−2.51608	−0.112186	−0.0560932	+	0.998426i	$0.517864\pi$
$504$	0	0
$505$	−19.6584	−0.874787
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	−9.30203	−0.412305	−0.206153	−	0.978520i	$-0.566094\pi$
$509$	−9.30203	−0.412305	−0.206153	+	0.978520i	$0.566094\pi$
$510$	0	0
$511$	−12.1814	−0.538874
$512$	0	0
$513$	2.46193	0.108697
$514$	0	0
$515$	25.6653	1.13095
$516$	0	0
$517$	6.65933	0.292877
$518$	0	0
$519$	24.4915	1.07506
$520$	0	0
$521$	10.7276	0.469984	0.234992	−	0.971997i	$-0.424494\pi$
$521$	10.7276	0.469984	0.234992	+	0.971997i	$0.424494\pi$
$522$	0	0
$523$	−1.17288	−0.0512866	−0.0256433	−	0.999671i	$-0.508163\pi$
$523$	−1.17288	−0.0512866	−0.0256433	+	0.999671i	$0.508163\pi$
$524$	0	0
$525$	−17.7195	−0.773342
$526$	0	0
$527$	−30.1884	−1.31503
$528$	0	0
$529$	−21.5300	−0.936086
$530$	0	0
$531$	−14.3337	−0.622030
$532$	0	0
$533$	−2.88682	−0.125042
$534$	0	0
$535$	36.0902	1.56032
$536$	0	0
$537$	−3.15135	−0.135991
$538$	0	0
$539$	−0.938903	−0.0404414
$540$	0	0
$541$	−18.0000	−0.773880	−0.386940	−	0.922105i	$-0.626468\pi$
$541$	−18.0000	−0.773880	−0.386940	+	0.922105i	$0.626468\pi$
$542$	0	0
$543$	2.72646	0.117003
$544$	0	0
$545$	−12.0382	−0.515659
$546$	0	0
$547$	11.3772	0.486455	0.243228	−	0.969969i	$-0.421794\pi$
$547$	11.3772	0.486455	0.243228	+	0.969969i	$0.421794\pi$
$548$	0	0
$549$	10.2344	0.436795
$550$	0	0
$551$	7.31058	0.311441
$552$	0	0
$553$	−10.5371	−0.448085
$554$	0	0
$555$	11.3488	0.481728
$556$	0	0
$557$	9.58206	0.406005	0.203002	−	0.979178i	$-0.434930\pi$
$557$	9.58206	0.406005	0.203002	+	0.979178i	$0.434930\pi$
$558$	0	0
$559$	4.64383	0.196413
$560$	0	0
$561$	4.70492	0.198642
$562$	0	0
$563$	−6.77158	−0.285388	−0.142694	−	0.989767i	$-0.545576\pi$
$563$	−6.77158	−0.285388	−0.142694	+	0.989767i	$0.545576\pi$
$564$	0	0
$565$	−29.1053	−1.22447
$566$	0	0
$567$	−2.46193	−0.103391
$568$	0	0
$569$	26.2629	1.10100	0.550500	−	0.834835i	$-0.314437\pi$
$569$	26.2629	1.10100	0.550500	+	0.834835i	$0.314437\pi$
$570$	0	0
$571$	−22.1588	−0.927315	−0.463658	−	0.886014i	$-0.653463\pi$
$571$	−22.1588	−0.927315	−0.463658	+	0.886014i	$0.653463\pi$
$572$	0	0
$573$	−20.9549	−0.875402
$574$	0	0
$575$	8.72646	0.363918
$576$	0	0
$577$	−33.5450	−1.39650	−0.698249	−	0.715855i	$-0.746037\pi$
$577$	−33.5450	−1.39650	−0.698249	+	0.715855i	$0.746037\pi$
$578$	0	0
$579$	22.2185	0.923368
$580$	0	0
$581$	35.2886	1.46402
$582$	0	0
$583$	3.93890	0.163133
$584$	0	0
$585$	3.49248	0.144396
$586$	0	0
$587$	−42.8187	−1.76732	−0.883659	−	0.468130i	$-0.844928\pi$
$587$	−42.8187	−1.76732	−0.883659	+	0.468130i	$0.844928\pi$
$588$	0	0
$589$	−15.7966	−0.650886
$590$	0	0
$591$	−7.44689	−0.306324
$592$	0	0
$593$	−32.7655	−1.34552	−0.672760	−	0.739861i	$-0.734891\pi$
$593$	−32.7655	−1.34552	−0.672760	+	0.739861i	$0.734891\pi$
$594$	0	0
$595$	−40.4540	−1.65845
$596$	0	0
$597$	−5.96991	−0.244332
$598$	0	0
$599$	32.1051	1.31178	0.655889	−	0.754857i	$-0.272294\pi$
$599$	32.1051	1.31178	0.655889	+	0.754857i	$0.272294\pi$
$600$	0	0
$601$	5.87781	0.239761	0.119880	−	0.992788i	$-0.461749\pi$
$601$	5.87781	0.239761	0.119880	+	0.992788i	$0.461749\pi$
$602$	0	0
$603$	1.00649	0.0409874
$604$	0	0
$605$	3.49248	0.141989
$606$	0	0
$607$	−8.68200	−0.352391	−0.176196	−	0.984355i	$-0.556379\pi$
$607$	−8.68200	−0.352391	−0.176196	+	0.984355i	$0.556379\pi$
$608$	0	0
$609$	−7.31058	−0.296240
$610$	0	0
$611$	6.65933	0.269408
$612$	0	0
$613$	−47.8578	−1.93296	−0.966480	−	0.256741i	$-0.917351\pi$
$613$	−47.8578	−1.93296	−0.966480	+	0.256741i	$0.917351\pi$
$614$	0	0
$615$	−10.0822	−0.406552
$616$	0	0
$617$	30.8292	1.24114	0.620568	−	0.784153i	$-0.286902\pi$
$617$	30.8292	1.24114	0.620568	+	0.784153i	$0.286902\pi$
$618$	0	0
$619$	−10.9836	−0.441467	−0.220733	−	0.975334i	$-0.570845\pi$
$619$	−10.9836	−0.441467	−0.220733	+	0.975334i	$0.570845\pi$
$620$	0	0
$621$	1.21245	0.0486538
$622$	0	0
$623$	32.3418	1.29575
$624$	0	0
$625$	−9.18442	−0.367377
$626$	0	0
$627$	2.46193	0.0983200
$628$	0	0
$629$	15.2886	0.609595
$630$	0	0
$631$	38.5165	1.53332	0.766659	−	0.642054i	$-0.221918\pi$
$631$	38.5165	1.53332	0.766659	+	0.642054i	$0.221918\pi$
$632$	0	0
$633$	3.35617	0.133396
$634$	0	0
$635$	−48.6304	−1.92984
$636$	0	0
$637$	−0.938903	−0.0372007
$638$	0	0
$639$	1.24948	0.0494288
$640$	0	0
$641$	−29.8917	−1.18065	−0.590326	−	0.807165i	$-0.701001\pi$
$641$	−29.8917	−1.18065	−0.590326	+	0.807165i	$0.701001\pi$
$642$	0	0
$643$	−3.09859	−0.122197	−0.0610983	−	0.998132i	$-0.519460\pi$
$643$	−3.09859	−0.122197	−0.0610983	+	0.998132i	$0.519460\pi$
$644$	0	0
$645$	16.2185	0.638601
$646$	0	0
$647$	45.7445	1.79840	0.899200	−	0.437537i	$-0.144149\pi$
$647$	45.7445	1.79840	0.899200	+	0.437537i	$0.144149\pi$
$648$	0	0
$649$	−14.3337	−0.562647
$650$	0	0
$651$	15.7966	0.619116
$652$	0	0
$653$	42.2124	1.65190	0.825950	−	0.563743i	$-0.190639\pi$
$653$	42.2124	1.65190	0.825950	+	0.563743i	$0.190639\pi$
$654$	0	0
$655$	57.2585	2.23727
$656$	0	0
$657$	4.94792	0.193037
$658$	0	0
$659$	−1.81181	−0.0705781	−0.0352891	−	0.999377i	$-0.511235\pi$
$659$	−1.81181	−0.0705781	−0.0352891	+	0.999377i	$0.511235\pi$
$660$	0	0
$661$	−36.5300	−1.42085	−0.710426	−	0.703772i	$-0.751498\pi$
$661$	−36.5300	−1.42085	−0.710426	+	0.703772i	$0.751498\pi$
$662$	0	0
$663$	4.70492	0.182724
$664$	0	0
$665$	−21.1682	−0.820869
$666$	0	0
$667$	3.60030	0.139404
$668$	0	0
$669$	6.41634	0.248070
$670$	0	0
$671$	10.2344	0.395096
$672$	0	0
$673$	−16.7024	−0.643830	−0.321915	−	0.946769i	$-0.604327\pi$
$673$	−16.7024	−0.643830	−0.321915	+	0.946769i	$0.604327\pi$
$674$	0	0
$675$	7.19740	0.277028
$676$	0	0
$677$	45.0467	1.73129	0.865643	−	0.500662i	$-0.166910\pi$
$677$	45.0467	1.73129	0.865643	+	0.500662i	$0.166910\pi$
$678$	0	0
$679$	30.9419	1.18744
$680$	0	0
$681$	20.7044	0.793396
$682$	0	0
$683$	−32.0692	−1.22709	−0.613546	−	0.789659i	$-0.710258\pi$
$683$	−32.0692	−1.22709	−0.613546	+	0.789659i	$0.710258\pi$
$684$	0	0
$685$	40.6973	1.55496
$686$	0	0
$687$	8.65933	0.330374
$688$	0	0
$689$	3.93890	0.150060
$690$	0	0
$691$	15.2441	0.579913	0.289957	−	0.957040i	$-0.406359\pi$
$691$	15.2441	0.579913	0.289957	+	0.957040i	$0.406359\pi$
$692$	0	0
$693$	−2.46193	−0.0935210
$694$	0	0
$695$	49.9009	1.89285
$696$	0	0
$697$	−13.5823	−0.514465
$698$	0	0
$699$	3.45544	0.130697
$700$	0	0
$701$	−36.3883	−1.37437	−0.687184	−	0.726484i	$-0.741153\pi$
$701$	−36.3883	−1.37437	−0.687184	+	0.726484i	$0.741153\pi$
$702$	0	0
$703$	8.00000	0.301726
$704$	0	0
$705$	23.2576	0.875931
$706$	0	0
$707$	13.8577	0.521171
$708$	0	0
$709$	39.5693	1.48605	0.743027	−	0.669261i	$-0.233389\pi$
$709$	39.5693	1.48605	0.743027	+	0.669261i	$0.233389\pi$
$710$	0	0
$711$	4.28003	0.160514
$712$	0	0
$713$	−7.77946	−0.291343
$714$	0	0
$715$	3.49248	0.130611
$716$	0	0
$717$	−18.0069	−0.672482
$718$	0	0
$719$	40.3467	1.50468	0.752339	−	0.658776i	$-0.228926\pi$
$719$	40.3467	1.50468	0.752339	+	0.658776i	$0.228926\pi$
$720$	0	0
$721$	−18.0921	−0.673785
$722$	0	0
$723$	−0.788686	−0.0293315
$724$	0	0
$725$	21.3723	0.793749
$726$	0	0
$727$	−20.0310	−0.742909	−0.371454	−	0.928451i	$-0.621141\pi$
$727$	−20.0310	−0.742909	−0.371454	+	0.928451i	$0.621141\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	21.8488	0.808109
$732$	0	0
$733$	−13.8718	−0.512366	−0.256183	−	0.966628i	$-0.582465\pi$
$733$	−13.8718	−0.512366	−0.256183	+	0.966628i	$0.582465\pi$
$734$	0	0
$735$	−3.27910	−0.120951
$736$	0	0
$737$	1.00649	0.0370745
$738$	0	0
$739$	−7.63018	−0.280681	−0.140340	−	0.990103i	$-0.544820\pi$
$739$	−7.63018	−0.280681	−0.140340	+	0.990103i	$0.544820\pi$
$740$	0	0
$741$	2.46193	0.0904412
$742$	0	0
$743$	−24.7846	−0.909257	−0.454629	−	0.890681i	$-0.650228\pi$
$743$	−24.7846	−0.909257	−0.454629	+	0.890681i	$0.650228\pi$
$744$	0	0
$745$	57.4929	2.10638
$746$	0	0
$747$	−14.3337	−0.524443
$748$	0	0
$749$	−25.4409	−0.929589
$750$	0	0
$751$	−22.2856	−0.813213	−0.406606	−	0.913603i	$-0.633288\pi$
$751$	−22.2856	−0.813213	−0.406606	+	0.913603i	$0.633288\pi$
$752$	0	0
$753$	−10.4559	−0.381034
$754$	0	0
$755$	29.4480	1.07172
$756$	0	0
$757$	−42.0150	−1.52706	−0.763531	−	0.645771i	$-0.776536\pi$
$757$	−42.0150	−1.52706	−0.763531	+	0.645771i	$0.776536\pi$
$758$	0	0
$759$	1.21245	0.0440090
$760$	0	0
$761$	46.4309	1.68312	0.841559	−	0.540165i	$-0.181638\pi$
$761$	46.4309	1.68312	0.841559	+	0.540165i	$0.181638\pi$
$762$	0	0
$763$	8.48599	0.307213
$764$	0	0
$765$	16.4318	0.594094
$766$	0	0
$767$	−14.3337	−0.517560
$768$	0	0
$769$	−18.2245	−0.657192	−0.328596	−	0.944471i	$-0.606575\pi$
$769$	−18.2245	−0.657192	−0.328596	+	0.944471i	$0.606575\pi$
$770$	0	0
$771$	−8.89377	−0.320301
$772$	0	0
$773$	26.5587	0.955249	0.477625	−	0.878564i	$-0.341498\pi$
$773$	26.5587	0.955249	0.477625	+	0.878564i	$0.341498\pi$
$774$	0	0
$775$	−46.1810	−1.65887
$776$	0	0
$777$	−8.00000	−0.286998
$778$	0	0
$779$	−7.10715	−0.254640
$780$	0	0
$781$	1.24948	0.0447100
$782$	0	0
$783$	2.96945	0.106120
$784$	0	0
$785$	17.9910	0.642125
$786$	0	0
$787$	−31.2997	−1.11571	−0.557856	−	0.829938i	$-0.688376\pi$
$787$	−31.2997	−1.11571	−0.557856	+	0.829938i	$0.688376\pi$
$788$	0	0
$789$	−27.7746	−0.988801
$790$	0	0
$791$	20.5170	0.729500
$792$	0	0
$793$	10.2344	0.363436
$794$	0	0
$795$	13.7565	0.487894
$796$	0	0
$797$	−44.5151	−1.57681	−0.788403	−	0.615159i	$-0.789092\pi$
$797$	−44.5151	−1.57681	−0.788403	+	0.615159i	$0.789092\pi$
$798$	0	0
$799$	31.3316	1.10843
$800$	0	0
$801$	−13.1368	−0.464165
$802$	0	0
$803$	4.94792	0.174608
$804$	0	0
$805$	−10.4249	−0.367429
$806$	0	0
$807$	24.2856	0.854893
$808$	0	0
$809$	3.88615	0.136630	0.0683149	−	0.997664i	$-0.478238\pi$
$809$	3.88615	0.136630	0.0683149	+	0.997664i	$0.478238\pi$
$810$	0	0
$811$	35.1775	1.23525	0.617624	−	0.786474i	$-0.288095\pi$
$811$	35.1775	1.23525	0.617624	+	0.786474i	$0.288095\pi$
$812$	0	0
$813$	−6.82572	−0.239389
$814$	0	0
$815$	26.1955	0.917590
$816$	0	0
$817$	11.4328	0.399982
$818$	0	0
$819$	−2.46193	−0.0860268
$820$	0	0
$821$	36.6434	1.27886	0.639431	−	0.768849i	$-0.279170\pi$
$821$	36.6434	1.27886	0.639431	+	0.768849i	$0.279170\pi$
$822$	0	0
$823$	−30.9270	−1.07805	−0.539024	−	0.842290i	$-0.681207\pi$
$823$	−30.9270	−1.07805	−0.539024	+	0.842290i	$0.681207\pi$
$824$	0	0
$825$	7.19740	0.250581
$826$	0	0
$827$	45.4177	1.57933	0.789664	−	0.613539i	$-0.210255\pi$
$827$	45.4177	1.57933	0.789664	+	0.613539i	$0.210255\pi$
$828$	0	0
$829$	−44.8197	−1.55665	−0.778326	−	0.627860i	$-0.783931\pi$
$829$	−44.8197	−1.55665	−0.778326	+	0.627860i	$0.783931\pi$
$830$	0	0
$831$	−1.92592	−0.0668095
$832$	0	0
$833$	−4.41747	−0.153056
$834$	0	0
$835$	34.1342	1.18126
$836$	0	0
$837$	−6.41634	−0.221781
$838$	0	0
$839$	48.0301	1.65818	0.829091	−	0.559114i	$-0.188859\pi$
$839$	48.0301	1.65818	0.829091	+	0.559114i	$0.188859\pi$
$840$	0	0
$841$	−20.1824	−0.695943
$842$	0	0
$843$	−15.7195	−0.541408
$844$	0	0
$845$	3.49248	0.120145
$846$	0	0
$847$	−2.46193	−0.0845929
$848$	0	0
$849$	−1.54663	−0.0530800
$850$	0	0
$851$	3.93982	0.135055
$852$	0	0
$853$	−29.6214	−1.01422	−0.507108	−	0.861883i	$-0.669285\pi$
$853$	−29.6214	−1.01422	−0.507108	+	0.861883i	$0.669285\pi$
$854$	0	0
$855$	8.59823	0.294053
$856$	0	0
$857$	11.8452	0.404624	0.202312	−	0.979321i	$-0.435154\pi$
$857$	11.8452	0.404624	0.202312	+	0.979321i	$0.435154\pi$
$858$	0	0
$859$	−16.7586	−0.571796	−0.285898	−	0.958260i	$-0.592292\pi$
$859$	−16.7586	−0.571796	−0.285898	+	0.958260i	$0.592292\pi$
$860$	0	0
$861$	7.10715	0.242211
$862$	0	0
$863$	−1.75653	−0.0597931	−0.0298965	−	0.999553i	$-0.509518\pi$
$863$	−1.75653	−0.0597931	−0.0298965	+	0.999553i	$0.509518\pi$
$864$	0	0
$865$	85.5362	2.90832
$866$	0	0
$867$	5.13630	0.174438
$868$	0	0
$869$	4.28003	0.145190
$870$	0	0
$871$	1.00649	0.0341036
$872$	0	0
$873$	−12.5681	−0.425367
$874$	0	0
$875$	−18.8938	−0.638726
$876$	0	0
$877$	−3.43581	−0.116019	−0.0580095	−	0.998316i	$-0.518475\pi$
$877$	−3.43581	−0.116019	−0.0580095	+	0.998316i	$0.518475\pi$
$878$	0	0
$879$	22.5230	0.759683
$880$	0	0
$881$	−30.7896	−1.03733	−0.518664	−	0.854978i	$-0.673570\pi$
$881$	−30.7896	−1.03733	−0.518664	+	0.854978i	$0.673570\pi$
$882$	0	0
$883$	−29.1201	−0.979970	−0.489985	−	0.871731i	$-0.662998\pi$
$883$	−29.1201	−0.979970	−0.489985	+	0.871731i	$0.662998\pi$
$884$	0	0
$885$	−50.0601	−1.68275
$886$	0	0
$887$	−38.2205	−1.28332	−0.641660	−	0.766990i	$-0.721754\pi$
$887$	−38.2205	−1.28332	−0.641660	+	0.766990i	$0.721754\pi$
$888$	0	0
$889$	34.2807	1.14974
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	16.3948	0.548631
$894$	0	0
$895$	−11.0060	−0.367891
$896$	0	0
$897$	1.21245	0.0404824
$898$	0	0
$899$	−19.0530	−0.635453
$900$	0	0
$901$	18.5322	0.617398
$902$	0	0
$903$	−11.4328	−0.380459
$904$	0	0
$905$	9.52209	0.316525
$906$	0	0
$907$	−29.6084	−0.983130	−0.491565	−	0.870841i	$-0.663575\pi$
$907$	−29.6084	−0.983130	−0.491565	+	0.870841i	$0.663575\pi$
$908$	0	0
$909$	−5.62878	−0.186695
$910$	0	0
$911$	59.8769	1.98381	0.991904	−	0.126987i	$-0.0405305\pi$
$911$	59.8769	1.98381	0.991904	+	0.126987i	$0.0405305\pi$
$912$	0	0
$913$	−14.3337	−0.474376
$914$	0	0
$915$	35.7436	1.18165
$916$	0	0
$917$	−40.3628	−1.33290
$918$	0	0
$919$	−58.7399	−1.93765	−0.968825	−	0.247746i	$-0.920310\pi$
$919$	−58.7399	−1.93765	−0.968825	+	0.247746i	$0.920310\pi$
$920$	0	0
$921$	−13.2816	−0.437644
$922$	0	0
$923$	1.24948	0.0411273
$924$	0	0
$925$	23.3878	0.768987
$926$	0	0
$927$	7.34875	0.241365
$928$	0	0
$929$	18.8933	0.619869	0.309934	−	0.950758i	$-0.399693\pi$
$929$	18.8933	0.619869	0.309934	+	0.950758i	$0.399693\pi$
$930$	0	0
$931$	−2.31151	−0.0757568
$932$	0	0
$933$	−18.4089	−0.602681
$934$	0	0
$935$	16.4318	0.537379
$936$	0	0
$937$	38.1312	1.24569	0.622846	−	0.782345i	$-0.285976\pi$
$937$	38.1312	1.24569	0.622846	+	0.782345i	$0.285976\pi$
$938$	0	0
$939$	−5.63734	−0.183968
$940$	0	0
$941$	−18.5823	−0.605764	−0.302882	−	0.953028i	$-0.597949\pi$
$941$	−18.5823	−0.605764	−0.302882	+	0.953028i	$0.597949\pi$
$942$	0	0
$943$	−3.50011	−0.113979
$944$	0	0
$945$	−8.59823	−0.279701
$946$	0	0
$947$	−25.3546	−0.823913	−0.411956	−	0.911204i	$-0.635154\pi$
$947$	−25.3546	−0.823913	−0.411956	+	0.911204i	$0.635154\pi$
$948$	0	0
$949$	4.94792	0.160616
$950$	0	0
$951$	−5.01457	−0.162609
$952$	0	0
$953$	−14.9523	−0.484354	−0.242177	−	0.970232i	$-0.577861\pi$
$953$	−14.9523	−0.484354	−0.242177	+	0.970232i	$0.577861\pi$
$954$	0	0
$955$	−73.1844	−2.36819
$956$	0	0
$957$	2.96945	0.0959887
$958$	0	0
$959$	−28.6885	−0.926399
$960$	0	0
$961$	10.1694	0.328044
$962$	0	0
$963$	10.3337	0.332999
$964$	0	0
$965$	77.5975	2.49795
$966$	0	0
$967$	50.5212	1.62465	0.812325	−	0.583204i	$-0.198201\pi$
$967$	50.5212	1.62465	0.812325	+	0.583204i	$0.198201\pi$
$968$	0	0
$969$	11.5832	0.372106
$970$	0	0
$971$	13.4851	0.432756	0.216378	−	0.976310i	$-0.430576\pi$
$971$	13.4851	0.432756	0.216378	+	0.976310i	$0.430576\pi$
$972$	0	0
$973$	−35.1763	−1.12770
$974$	0	0
$975$	7.19740	0.230501
$976$	0	0
$977$	−54.8192	−1.75382	−0.876911	−	0.480653i	$-0.840400\pi$
$977$	−54.8192	−1.75382	−0.876911	+	0.480653i	$0.840400\pi$
$978$	0	0
$979$	−13.1368	−0.419853
$980$	0	0
$981$	−3.44689	−0.110051
$982$	0	0
$983$	18.0944	0.577121	0.288560	−	0.957462i	$-0.406823\pi$
$983$	18.0944	0.577121	0.288560	+	0.957462i	$0.406823\pi$
$984$	0	0
$985$	−26.0081	−0.828686
$986$	0	0
$987$	−16.3948	−0.521852
$988$	0	0
$989$	5.63039	0.179036
$990$	0	0
$991$	13.8908	0.441255	0.220628	−	0.975358i	$-0.429189\pi$
$991$	13.8908	0.441255	0.220628	+	0.975358i	$0.429189\pi$
$992$	0	0
$993$	9.19786	0.291885
$994$	0	0
$995$	−20.8498	−0.660983
$996$	0	0
$997$	−1.65237	−0.0523310	−0.0261655	−	0.999658i	$-0.508330\pi$
$997$	−1.65237	−0.0523310	−0.0261655	+	0.999658i	$0.508330\pi$
$998$	0	0
$999$	3.24948	0.102809

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6864.2.a.cd.1.4		4
4.3	odd	2		3432.2.a.s.1.4	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3432.2.a.s.1.4	✓	4	4.3	odd	2
6864.2.a.cd.1.4		4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6864 = 2^{4} \cdot 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6864.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.49248 q^{5} -2.46193 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +3.49248 q^{5} -2.46193 q^{7} +1.00000 q^{9} +1.00000 q^{11} +1.00000 q^{13} +3.49248 q^{15} +4.70492 q^{17} +2.46193 q^{19} -2.46193 q^{21} +1.21245 q^{23} +7.19740 q^{25} +1.00000 q^{27} +2.96945 q^{29} -6.41634 q^{31} +1.00000 q^{33} -8.59823 q^{35} +3.24948 q^{37} +1.00000 q^{39} -2.88682 q^{41} +4.64383 q^{43} +3.49248 q^{45} +6.65933 q^{47} -0.938903 q^{49} +4.70492 q^{51} +3.93890 q^{53} +3.49248 q^{55} +2.46193 q^{57} -14.3337 q^{59} +10.2344 q^{61} -2.46193 q^{63} +3.49248 q^{65} +1.00649 q^{67} +1.21245 q^{69} +1.24948 q^{71} +4.94792 q^{73} +7.19740 q^{75} -2.46193 q^{77} +4.28003 q^{79} +1.00000 q^{81} -14.3337 q^{83} +16.4318 q^{85} +2.96945 q^{87} -13.1368 q^{89} -2.46193 q^{91} -6.41634 q^{93} +8.59823 q^{95} -12.5681 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 4 q^{5} + 2 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + 4 * q^5 + 2 * q^7 + 4 * q^9	\( 4 q + 4 q^{3} + 4 q^{5} + 2 q^{7} + 4 q^{9} + 4 q^{11} + 4 q^{13} + 4 q^{15} + 8 q^{17} - 2 q^{19} + 2 q^{21} + 4 q^{23} + 8 q^{25} + 4 q^{27} + 10 q^{29} + 8 q^{31} + 4 q^{33} + 2 q^{35} + 2 q^{37} + 4 q^{39} + 2 q^{41} + 4 q^{43} + 4 q^{45} - 6 q^{47} + 8 q^{51} + 12 q^{53} + 4 q^{55} - 2 q^{57} - 12 q^{59} + 10 q^{61} + 2 q^{63} + 4 q^{65} - 8 q^{67} + 4 q^{69} - 6 q^{71} + 10 q^{73} + 8 q^{75} + 2 q^{77} + 8 q^{79} + 4 q^{81} - 12 q^{83} + 14 q^{85} + 10 q^{87} + 10 q^{89} + 2 q^{91} + 8 q^{93} - 2 q^{95} + 26 q^{97} + 4 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 + 4 * q^5 + 2 * q^7 + 4 * q^9 + 4 * q^11 + 4 * q^13 + 4 * q^15 + 8 * q^17 - 2 * q^19 + 2 * q^21 + 4 * q^23 + 8 * q^25 + 4 * q^27 + 10 * q^29 + 8 * q^31 + 4 * q^33 + 2 * q^35 + 2 * q^37 + 4 * q^39 + 2 * q^41 + 4 * q^43 + 4 * q^45 - 6 * q^47 + 8 * q^51 + 12 * q^53 + 4 * q^55 - 2 * q^57 - 12 * q^59 + 10 * q^61 + 2 * q^63 + 4 * q^65 - 8 * q^67 + 4 * q^69 - 6 * q^71 + 10 * q^73 + 8 * q^75 + 2 * q^77 + 8 * q^79 + 4 * q^81 - 12 * q^83 + 14 * q^85 + 10 * q^87 + 10 * q^89 + 2 * q^91 + 8 * q^93 - 2 * q^95 + 26 * q^97 + 4 * q^99

Introduction

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