LMFDB - Embedded newform 6864.2.a.bq.1.1

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:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3432)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.48119$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -1.67513 q^{5} +0.806063 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.67513 q^{5} +0.806063 q^{7} +1.00000 q^{9} +1.00000 q^{11} -1.00000 q^{13} +1.67513 q^{15} +3.83146 q^{17} -5.11871 q^{19} -0.806063 q^{21} +4.54420 q^{23} -2.19394 q^{25} -1.00000 q^{27} -4.09332 q^{29} +8.63752 q^{31} -1.00000 q^{33} -1.35026 q^{35} -1.73813 q^{37} +1.00000 q^{39} +7.11871 q^{41} +3.13093 q^{43} -1.67513 q^{45} -4.70052 q^{47} -6.35026 q^{49} -3.83146 q^{51} +1.35026 q^{53} -1.67513 q^{55} +5.11871 q^{57} -6.57452 q^{59} -6.70052 q^{61} +0.806063 q^{63} +1.67513 q^{65} +10.2496 q^{67} -4.54420 q^{69} +6.18664 q^{71} +3.76845 q^{73} +2.19394 q^{75} +0.806063 q^{77} -3.44358 q^{79} +1.00000 q^{81} -1.73813 q^{83} -6.41819 q^{85} +4.09332 q^{87} -15.5247 q^{89} -0.806063 q^{91} -8.63752 q^{93} +8.57452 q^{95} +8.96239 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 2 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 + 2 * q^7 + 3 * q^9	$ 3 q - 3 q^{3} + 2 q^{7} + 3 q^{9} + 3 q^{11} - 3 q^{13} - 4 q^{17} + 6 q^{19} - 2 q^{21} + 4 q^{23} - 7 q^{25} - 3 q^{27} - 6 q^{29} + 10 q^{31} - 3 q^{33} + 6 q^{35} + 4 q^{37} + 3 q^{39} + 14 q^{43} + 6 q^{47} - 9 q^{49} + 4 q^{51} - 6 q^{53} - 6 q^{57} - 8 q^{59} + 2 q^{63} + 14 q^{67} - 4 q^{69} + 6 q^{71} + 7 q^{75} + 2 q^{77} + 6 q^{79} + 3 q^{81} + 4 q^{83} - 18 q^{85} + 6 q^{87} + 2 q^{89} - 2 q^{91} - 10 q^{93} + 14 q^{95} + 16 q^{97} + 3 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 + 2 * q^7 + 3 * q^9 + 3 * q^11 - 3 * q^13 - 4 * q^17 + 6 * q^19 - 2 * q^21 + 4 * q^23 - 7 * q^25 - 3 * q^27 - 6 * q^29 + 10 * q^31 - 3 * q^33 + 6 * q^35 + 4 * q^37 + 3 * q^39 + 14 * q^43 + 6 * q^47 - 9 * q^49 + 4 * q^51 - 6 * q^53 - 6 * q^57 - 8 * q^59 + 2 * q^63 + 14 * q^67 - 4 * q^69 + 6 * q^71 + 7 * q^75 + 2 * q^77 + 6 * q^79 + 3 * q^81 + 4 * q^83 - 18 * q^85 + 6 * q^87 + 2 * q^89 - 2 * q^91 - 10 * q^93 + 14 * q^95 + 16 * q^97 + 3 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−1.67513	−0.749141	−0.374571	−	0.927198i	$-0.622210\pi$
$5$	−1.67513	−0.749141	−0.374571	+	0.927198i	$0.622210\pi$
$6$	0	0
$7$	0.806063	0.304663	0.152332	−	0.988329i	$-0.451322\pi$
$7$	0.806063	0.304663	0.152332	+	0.988329i	$0.451322\pi$
$8$	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
$15$	1.67513	0.432517
$16$	0	0
$17$	3.83146	0.929265	0.464632	−	0.885504i	$-0.346187\pi$
$17$	3.83146	0.929265	0.464632	+	0.885504i	$0.346187\pi$
$18$	0	0
$19$	−5.11871	−1.17431	−0.587157	−	0.809473i	$-0.699753\pi$
$19$	−5.11871	−1.17431	−0.587157	+	0.809473i	$0.699753\pi$
$20$	0	0
$21$	−0.806063	−0.175897
$22$	0	0
$23$	4.54420	0.947531	0.473765	−	0.880651i	$-0.342894\pi$
$23$	4.54420	0.947531	0.473765	+	0.880651i	$0.342894\pi$
$24$	0	0
$25$	−2.19394	−0.438787
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−4.09332	−0.760111	−0.380055	−	0.924964i	$-0.624095\pi$
$29$	−4.09332	−0.760111	−0.380055	+	0.924964i	$0.624095\pi$
$30$	0	0
$31$	8.63752	1.55134	0.775672	−	0.631136i	$-0.217411\pi$
$31$	8.63752	1.55134	0.775672	+	0.631136i	$0.217411\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	−1.35026	−0.228236
$36$	0	0
$37$	−1.73813	−0.285748	−0.142874	−	0.989741i	$-0.545634\pi$
$37$	−1.73813	−0.285748	−0.142874	+	0.989741i	$0.545634\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	7.11871	1.11176	0.555878	−	0.831264i	$-0.312382\pi$
$41$	7.11871	1.11176	0.555878	+	0.831264i	$0.312382\pi$
$42$	0	0
$43$	3.13093	0.477463	0.238731	−	0.971086i	$-0.423268\pi$
$43$	3.13093	0.477463	0.238731	+	0.971086i	$0.423268\pi$
$44$	0	0
$45$	−1.67513	−0.249714
$46$	0	0
$47$	−4.70052	−0.685642	−0.342821	−	0.939401i	$-0.611382\pi$
$47$	−4.70052	−0.685642	−0.342821	+	0.939401i	$0.611382\pi$
$48$	0	0
$49$	−6.35026	−0.907180
$50$	0	0
$51$	−3.83146	−0.536511
$52$	0	0
$53$	1.35026	0.185473	0.0927364	−	0.995691i	$-0.470439\pi$
$53$	1.35026	0.185473	0.0927364	+	0.995691i	$0.470439\pi$
$54$	0	0
$55$	−1.67513	−0.225875
$56$	0	0
$57$	5.11871	0.677990
$58$	0	0
$59$	−6.57452	−0.855929	−0.427965	−	0.903796i	$-0.640769\pi$
$59$	−6.57452	−0.855929	−0.427965	+	0.903796i	$0.640769\pi$
$60$	0	0
$61$	−6.70052	−0.857914	−0.428957	−	0.903325i	$-0.641119\pi$
$61$	−6.70052	−0.857914	−0.428957	+	0.903325i	$0.641119\pi$
$62$	0	0
$63$	0.806063	0.101554
$64$	0	0
$65$	1.67513	0.207774
$66$	0	0
$67$	10.2496	1.25219	0.626097	−	0.779745i	$-0.284651\pi$
$67$	10.2496	1.25219	0.626097	+	0.779745i	$0.284651\pi$
$68$	0	0
$69$	−4.54420	−0.547057
$70$	0	0
$71$	6.18664	0.734219	0.367110	−	0.930178i	$-0.380347\pi$
$71$	6.18664	0.734219	0.367110	+	0.930178i	$0.380347\pi$
$72$	0	0
$73$	3.76845	0.441064	0.220532	−	0.975380i	$-0.429221\pi$
$73$	3.76845	0.441064	0.220532	+	0.975380i	$0.429221\pi$
$74$	0	0
$75$	2.19394	0.253334
$76$	0	0
$77$	0.806063	0.0918595
$78$	0	0
$79$	−3.44358	−0.387433	−0.193717	−	0.981058i	$-0.562054\pi$
$79$	−3.44358	−0.387433	−0.193717	+	0.981058i	$0.562054\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−1.73813	−0.190785	−0.0953925	−	0.995440i	$-0.530411\pi$
$83$	−1.73813	−0.190785	−0.0953925	+	0.995440i	$0.530411\pi$
$84$	0	0
$85$	−6.41819	−0.696150
$86$	0	0
$87$	4.09332	0.438850
$88$	0	0
$89$	−15.5247	−1.64561	−0.822807	−	0.568321i	$-0.807593\pi$
$89$	−15.5247	−1.64561	−0.822807	+	0.568321i	$0.807593\pi$
$90$	0	0
$91$	−0.806063	−0.0844984
$92$	0	0
$93$	−8.63752	−0.895669
$94$	0	0
$95$	8.57452	0.879727
$96$	0	0
$97$	8.96239	0.909993	0.454996	−	0.890493i	$-0.349641\pi$
$97$	8.96239	0.909993	0.454996	+	0.890493i	$0.349641\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	−2.99508	−0.298021	−0.149011	−	0.988836i	$-0.547609\pi$
$101$	−2.99508	−0.298021	−0.149011	+	0.988836i	$0.547609\pi$
$102$	0	0
$103$	10.3879	1.02355	0.511774	−	0.859120i	$-0.328989\pi$
$103$	10.3879	1.02355	0.511774	+	0.859120i	$0.328989\pi$
$104$	0	0
$105$	1.35026	0.131772
$106$	0	0
$107$	12.4993	1.20835	0.604176	−	0.796851i	$-0.293502\pi$
$107$	12.4993	1.20835	0.604176	+	0.796851i	$0.293502\pi$
$108$	0	0
$109$	−13.6932	−1.31157	−0.655787	−	0.754946i	$-0.727663\pi$
$109$	−13.6932	−1.31157	−0.655787	+	0.754946i	$0.727663\pi$
$110$	0	0
$111$	1.73813	0.164976
$112$	0	0
$113$	−9.66291	−0.909010	−0.454505	−	0.890744i	$-0.650184\pi$
$113$	−9.66291	−0.909010	−0.454505	+	0.890744i	$0.650184\pi$
$114$	0	0
$115$	−7.61213	−0.709834
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	3.08840	0.283113
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−7.11871	−0.641873
$124$	0	0
$125$	12.0508	1.07785
$126$	0	0
$127$	12.4812	1.10753	0.553763	−	0.832674i	$-0.313191\pi$
$127$	12.4812	1.10753	0.553763	+	0.832674i	$0.313191\pi$
$128$	0	0
$129$	−3.13093	−0.275663
$130$	0	0
$131$	−4.31265	−0.376798	−0.188399	−	0.982093i	$-0.560330\pi$
$131$	−4.31265	−0.376798	−0.188399	+	0.982093i	$0.560330\pi$
$132$	0	0
$133$	−4.12601	−0.357770
$134$	0	0
$135$	1.67513	0.144172
$136$	0	0
$137$	−6.37565	−0.544709	−0.272354	−	0.962197i	$-0.587802\pi$
$137$	−6.37565	−0.544709	−0.272354	+	0.962197i	$0.587802\pi$
$138$	0	0
$139$	18.7186	1.58769	0.793846	−	0.608118i	$-0.208075\pi$
$139$	18.7186	1.58769	0.793846	+	0.608118i	$0.208075\pi$
$140$	0	0
$141$	4.70052	0.395855
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	6.85685	0.569430
$146$	0	0
$147$	6.35026	0.523761
$148$	0	0
$149$	−10.3430	−0.847329	−0.423664	−	0.905819i	$-0.639256\pi$
$149$	−10.3430	−0.847329	−0.423664	+	0.905819i	$0.639256\pi$
$150$	0	0
$151$	−0.342968	−0.0279103	−0.0139552	−	0.999903i	$-0.504442\pi$
$151$	−0.342968	−0.0279103	−0.0139552	+	0.999903i	$0.504442\pi$
$152$	0	0
$153$	3.83146	0.309755
$154$	0	0
$155$	−14.4690	−1.16218
$156$	0	0
$157$	1.89446	0.151194	0.0755972	−	0.997138i	$-0.475914\pi$
$157$	1.89446	0.151194	0.0755972	+	0.997138i	$0.475914\pi$
$158$	0	0
$159$	−1.35026	−0.107083
$160$	0	0
$161$	3.66291	0.288678
$162$	0	0
$163$	−2.32487	−0.182098	−0.0910489	−	0.995846i	$-0.529022\pi$
$163$	−2.32487	−0.182098	−0.0910489	+	0.995846i	$0.529022\pi$
$164$	0	0
$165$	1.67513	0.130409
$166$	0	0
$167$	5.08840	0.393752	0.196876	−	0.980428i	$-0.436920\pi$
$167$	5.08840	0.393752	0.196876	+	0.980428i	$0.436920\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−5.11871	−0.391438
$172$	0	0
$173$	2.09332	0.159152	0.0795761	−	0.996829i	$-0.474643\pi$
$173$	2.09332	0.159152	0.0795761	+	0.996829i	$0.474643\pi$
$174$	0	0
$175$	−1.76845	−0.133682
$176$	0	0
$177$	6.57452	0.494171
$178$	0	0
$179$	−10.3430	−0.773070	−0.386535	−	0.922275i	$-0.626328\pi$
$179$	−10.3430	−0.773070	−0.386535	+	0.922275i	$0.626328\pi$
$180$	0	0
$181$	4.28233	0.318303	0.159152	−	0.987254i	$-0.449124\pi$
$181$	4.28233	0.318303	0.159152	+	0.987254i	$0.449124\pi$
$182$	0	0
$183$	6.70052	0.495317
$184$	0	0
$185$	2.91160	0.214065
$186$	0	0
$187$	3.83146	0.280184
$188$	0	0
$189$	−0.806063	−0.0586325
$190$	0	0
$191$	−13.5877	−0.983171	−0.491585	−	0.870829i	$-0.663582\pi$
$191$	−13.5877	−0.983171	−0.491585	+	0.870829i	$0.663582\pi$
$192$	0	0
$193$	−17.1695	−1.23589	−0.617944	−	0.786222i	$-0.712034\pi$
$193$	−17.1695	−1.23589	−0.617944	+	0.786222i	$0.712034\pi$
$194$	0	0
$195$	−1.67513	−0.119959
$196$	0	0
$197$	21.4821	1.53054	0.765270	−	0.643710i	$-0.222606\pi$
$197$	21.4821	1.53054	0.765270	+	0.643710i	$0.222606\pi$
$198$	0	0
$199$	21.0884	1.49492	0.747458	−	0.664309i	$-0.231274\pi$
$199$	21.0884	1.49492	0.747458	+	0.664309i	$0.231274\pi$
$200$	0	0
$201$	−10.2496	−0.722954
$202$	0	0
$203$	−3.29948	−0.231578
$204$	0	0
$205$	−11.9248	−0.832863
$206$	0	0
$207$	4.54420	0.315844
$208$	0	0
$209$	−5.11871	−0.354069
$210$	0	0
$211$	19.9429	1.37292	0.686462	−	0.727166i	$-0.259163\pi$
$211$	19.9429	1.37292	0.686462	+	0.727166i	$0.259163\pi$
$212$	0	0
$213$	−6.18664	−0.423902
$214$	0	0
$215$	−5.24472	−0.357687
$216$	0	0
$217$	6.96239	0.472638
$218$	0	0
$219$	−3.76845	−0.254648
$220$	0	0
$221$	−3.83146	−0.257732
$222$	0	0
$223$	2.06300	0.138149	0.0690745	−	0.997612i	$-0.477995\pi$
$223$	2.06300	0.138149	0.0690745	+	0.997612i	$0.477995\pi$
$224$	0	0
$225$	−2.19394	−0.146262
$226$	0	0
$227$	−9.95509	−0.660743	−0.330371	−	0.943851i	$-0.607174\pi$
$227$	−9.95509	−0.660743	−0.330371	+	0.943851i	$0.607174\pi$
$228$	0	0
$229$	21.9248	1.44883	0.724415	−	0.689364i	$-0.242110\pi$
$229$	21.9248	1.44883	0.724415	+	0.689364i	$0.242110\pi$
$230$	0	0
$231$	−0.806063	−0.0530351
$232$	0	0
$233$	−16.8446	−1.10353	−0.551764	−	0.834000i	$-0.686045\pi$
$233$	−16.8446	−1.10353	−0.551764	+	0.834000i	$0.686045\pi$
$234$	0	0
$235$	7.87399	0.513643
$236$	0	0
$237$	3.44358	0.223685
$238$	0	0
$239$	14.2071	0.918982	0.459491	−	0.888183i	$-0.348032\pi$
$239$	14.2071	0.918982	0.459491	+	0.888183i	$0.348032\pi$
$240$	0	0
$241$	21.7235	1.39934	0.699668	−	0.714468i	$-0.253331\pi$
$241$	21.7235	1.39934	0.699668	+	0.714468i	$0.253331\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	10.6375	0.679606
$246$	0	0
$247$	5.11871	0.325696
$248$	0	0
$249$	1.73813	0.110150
$250$	0	0
$251$	−5.73813	−0.362188	−0.181094	−	0.983466i	$-0.557964\pi$
$251$	−5.73813	−0.362188	−0.181094	+	0.983466i	$0.557964\pi$
$252$	0	0
$253$	4.54420	0.285691
$254$	0	0
$255$	6.41819	0.401923
$256$	0	0
$257$	−19.7743	−1.23349	−0.616744	−	0.787163i	$-0.711549\pi$
$257$	−19.7743	−1.23349	−0.616744	+	0.787163i	$0.711549\pi$
$258$	0	0
$259$	−1.40105	−0.0870568
$260$	0	0
$261$	−4.09332	−0.253370
$262$	0	0
$263$	6.57452	0.405402	0.202701	−	0.979241i	$-0.435028\pi$
$263$	6.57452	0.405402	0.202701	+	0.979241i	$0.435028\pi$
$264$	0	0
$265$	−2.26187	−0.138945
$266$	0	0
$267$	15.5247	0.950095
$268$	0	0
$269$	20.3634	1.24158	0.620790	−	0.783977i	$-0.286812\pi$
$269$	20.3634	1.24158	0.620790	+	0.783977i	$0.286812\pi$
$270$	0	0
$271$	12.1563	0.738444	0.369222	−	0.929341i	$-0.379624\pi$
$271$	12.1563	0.738444	0.369222	+	0.929341i	$0.379624\pi$
$272$	0	0
$273$	0.806063	0.0487852
$274$	0	0
$275$	−2.19394	−0.132299
$276$	0	0
$277$	8.38787	0.503978	0.251989	−	0.967730i	$-0.418915\pi$
$277$	8.38787	0.503978	0.251989	+	0.967730i	$0.418915\pi$
$278$	0	0
$279$	8.63752	0.517115
$280$	0	0
$281$	23.3054	1.39028	0.695140	−	0.718874i	$-0.255342\pi$
$281$	23.3054	1.39028	0.695140	+	0.718874i	$0.255342\pi$
$282$	0	0
$283$	14.9951	0.891365	0.445682	−	0.895191i	$-0.352961\pi$
$283$	14.9951	0.891365	0.445682	+	0.895191i	$0.352961\pi$
$284$	0	0
$285$	−8.57452	−0.507910
$286$	0	0
$287$	5.73813	0.338711
$288$	0	0
$289$	−2.31994	−0.136467
$290$	0	0
$291$	−8.96239	−0.525385
$292$	0	0
$293$	−13.9452	−0.814690	−0.407345	−	0.913274i	$-0.633545\pi$
$293$	−13.9452	−0.814690	−0.407345	+	0.913274i	$0.633545\pi$
$294$	0	0
$295$	11.0132	0.641212
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	−4.54420	−0.262798
$300$	0	0
$301$	2.52373	0.145465
$302$	0	0
$303$	2.99508	0.172063
$304$	0	0
$305$	11.2243	0.642699
$306$	0	0
$307$	23.0435	1.31516	0.657581	−	0.753384i	$-0.271580\pi$
$307$	23.0435	1.31516	0.657581	+	0.753384i	$0.271580\pi$
$308$	0	0
$309$	−10.3879	−0.590945
$310$	0	0
$311$	5.90431	0.334803	0.167401	−	0.985889i	$-0.446462\pi$
$311$	5.90431	0.334803	0.167401	+	0.985889i	$0.446462\pi$
$312$	0	0
$313$	26.6458	1.50611	0.753054	−	0.657959i	$-0.228580\pi$
$313$	26.6458	1.50611	0.753054	+	0.657959i	$0.228580\pi$
$314$	0	0
$315$	−1.35026	−0.0760786
$316$	0	0
$317$	28.2760	1.58814	0.794069	−	0.607828i	$-0.207959\pi$
$317$	28.2760	1.58814	0.794069	+	0.607828i	$0.207959\pi$
$318$	0	0
$319$	−4.09332	−0.229182
$320$	0	0
$321$	−12.4993	−0.697642
$322$	0	0
$323$	−19.6121	−1.09125
$324$	0	0
$325$	2.19394	0.121698
$326$	0	0
$327$	13.6932	0.757237
$328$	0	0
$329$	−3.78892	−0.208890
$330$	0	0
$331$	25.2120	1.38578	0.692889	−	0.721044i	$-0.256337\pi$
$331$	25.2120	1.38578	0.692889	+	0.721044i	$0.256337\pi$
$332$	0	0
$333$	−1.73813	−0.0952492
$334$	0	0
$335$	−17.1695	−0.938070
$336$	0	0
$337$	−9.87399	−0.537871	−0.268935	−	0.963158i	$-0.586672\pi$
$337$	−9.87399	−0.537871	−0.268935	+	0.963158i	$0.586672\pi$
$338$	0	0
$339$	9.66291	0.524817
$340$	0	0
$341$	8.63752	0.467748
$342$	0	0
$343$	−10.7612	−0.581048
$344$	0	0
$345$	7.61213	0.409823
$346$	0	0
$347$	−16.1114	−0.864906	−0.432453	−	0.901656i	$-0.642352\pi$
$347$	−16.1114	−0.864906	−0.432453	+	0.901656i	$0.642352\pi$
$348$	0	0
$349$	33.4518	1.79064	0.895318	−	0.445428i	$-0.146948\pi$
$349$	33.4518	1.79064	0.895318	+	0.445428i	$0.146948\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	13.6751	0.727854	0.363927	−	0.931428i	$-0.381436\pi$
$353$	13.6751	0.727854	0.363927	+	0.931428i	$0.381436\pi$
$354$	0	0
$355$	−10.3634	−0.550034
$356$	0	0
$357$	−3.08840	−0.163455
$358$	0	0
$359$	9.37073	0.494568	0.247284	−	0.968943i	$-0.420462\pi$
$359$	9.37073	0.494568	0.247284	+	0.968943i	$0.420462\pi$
$360$	0	0
$361$	7.20123	0.379012
$362$	0	0
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	−6.31265	−0.330419
$366$	0	0
$367$	−14.4241	−0.752930	−0.376465	−	0.926431i	$-0.622861\pi$
$367$	−14.4241	−0.752930	−0.376465	+	0.926431i	$0.622861\pi$
$368$	0	0
$369$	7.11871	0.370585
$370$	0	0
$371$	1.08840	0.0565067
$372$	0	0
$373$	17.1998	0.890573	0.445286	−	0.895388i	$-0.353102\pi$
$373$	17.1998	0.890573	0.445286	+	0.895388i	$0.353102\pi$
$374$	0	0
$375$	−12.0508	−0.622300
$376$	0	0
$377$	4.09332	0.210817
$378$	0	0
$379$	−23.8519	−1.22519	−0.612596	−	0.790397i	$-0.709875\pi$
$379$	−23.8519	−1.22519	−0.612596	+	0.790397i	$0.709875\pi$
$380$	0	0
$381$	−12.4812	−0.639431
$382$	0	0
$383$	−4.43866	−0.226805	−0.113402	−	0.993549i	$-0.536175\pi$
$383$	−4.43866	−0.226805	−0.113402	+	0.993549i	$0.536175\pi$
$384$	0	0
$385$	−1.35026	−0.0688157
$386$	0	0
$387$	3.13093	0.159154
$388$	0	0
$389$	−26.6009	−1.34872	−0.674359	−	0.738404i	$-0.735580\pi$
$389$	−26.6009	−1.34872	−0.674359	+	0.738404i	$0.735580\pi$
$390$	0	0
$391$	17.4109	0.880507
$392$	0	0
$393$	4.31265	0.217544
$394$	0	0
$395$	5.76845	0.290242
$396$	0	0
$397$	20.2882	1.01824	0.509118	−	0.860697i	$-0.329972\pi$
$397$	20.2882	1.01824	0.509118	+	0.860697i	$0.329972\pi$
$398$	0	0
$399$	4.12601	0.206559
$400$	0	0
$401$	32.1646	1.60622	0.803111	−	0.595829i	$-0.203177\pi$
$401$	32.1646	1.60622	0.803111	+	0.595829i	$0.203177\pi$
$402$	0	0
$403$	−8.63752	−0.430265
$404$	0	0
$405$	−1.67513	−0.0832379
$406$	0	0
$407$	−1.73813	−0.0861561
$408$	0	0
$409$	26.4690	1.30881	0.654403	−	0.756146i	$-0.272920\pi$
$409$	26.4690	1.30881	0.654403	+	0.756146i	$0.272920\pi$
$410$	0	0
$411$	6.37565	0.314488
$412$	0	0
$413$	−5.29948	−0.260770
$414$	0	0
$415$	2.91160	0.142925
$416$	0	0
$417$	−18.7186	−0.916655
$418$	0	0
$419$	−12.7915	−0.624904	−0.312452	−	0.949933i	$-0.601150\pi$
$419$	−12.7915	−0.624904	−0.312452	+	0.949933i	$0.601150\pi$
$420$	0	0
$421$	−17.0132	−0.829171	−0.414586	−	0.910010i	$-0.636073\pi$
$421$	−17.0132	−0.829171	−0.414586	+	0.910010i	$0.636073\pi$
$422$	0	0
$423$	−4.70052	−0.228547
$424$	0	0
$425$	−8.40597	−0.407750
$426$	0	0
$427$	−5.40105	−0.261375
$428$	0	0
$429$	1.00000	0.0482805
$430$	0	0
$431$	−28.1622	−1.35653	−0.678263	−	0.734819i	$-0.737267\pi$
$431$	−28.1622	−1.35653	−0.678263	+	0.734819i	$0.737267\pi$
$432$	0	0
$433$	3.55149	0.170674	0.0853369	−	0.996352i	$-0.472803\pi$
$433$	3.55149	0.170674	0.0853369	+	0.996352i	$0.472803\pi$
$434$	0	0
$435$	−6.85685	−0.328761
$436$	0	0
$437$	−23.2605	−1.11270
$438$	0	0
$439$	10.5056	0.501406	0.250703	−	0.968064i	$-0.419338\pi$
$439$	10.5056	0.501406	0.250703	+	0.968064i	$0.419338\pi$
$440$	0	0
$441$	−6.35026	−0.302393
$442$	0	0
$443$	20.9986	0.997673	0.498836	−	0.866696i	$-0.333761\pi$
$443$	20.9986	0.997673	0.498836	+	0.866696i	$0.333761\pi$
$444$	0	0
$445$	26.0059	1.23280
$446$	0	0
$447$	10.3430	0.489206
$448$	0	0
$449$	23.1612	1.09305	0.546523	−	0.837444i	$-0.315951\pi$
$449$	23.1612	1.09305	0.546523	+	0.837444i	$0.315951\pi$
$450$	0	0
$451$	7.11871	0.335207
$452$	0	0
$453$	0.342968	0.0161140
$454$	0	0
$455$	1.35026	0.0633012
$456$	0	0
$457$	−22.6859	−1.06120	−0.530602	−	0.847621i	$-0.678034\pi$
$457$	−22.6859	−1.06120	−0.530602	+	0.847621i	$0.678034\pi$
$458$	0	0
$459$	−3.83146	−0.178837
$460$	0	0
$461$	16.1465	0.752016	0.376008	−	0.926616i	$-0.377296\pi$
$461$	16.1465	0.752016	0.376008	+	0.926616i	$0.377296\pi$
$462$	0	0
$463$	17.9878	0.835963	0.417982	−	0.908456i	$-0.362738\pi$
$463$	17.9878	0.835963	0.417982	+	0.908456i	$0.362738\pi$
$464$	0	0
$465$	14.4690	0.670983
$466$	0	0
$467$	−18.3068	−0.847136	−0.423568	−	0.905864i	$-0.639223\pi$
$467$	−18.3068	−0.847136	−0.423568	+	0.905864i	$0.639223\pi$
$468$	0	0
$469$	8.26187	0.381497
$470$	0	0
$471$	−1.89446	−0.0872921
$472$	0	0
$473$	3.13093	0.143960
$474$	0	0
$475$	11.2301	0.515274
$476$	0	0
$477$	1.35026	0.0618242
$478$	0	0
$479$	20.9683	0.958065	0.479032	−	0.877797i	$-0.340988\pi$
$479$	20.9683	0.958065	0.479032	+	0.877797i	$0.340988\pi$
$480$	0	0
$481$	1.73813	0.0792521
$482$	0	0
$483$	−3.66291	−0.166668
$484$	0	0
$485$	−15.0132	−0.681713
$486$	0	0
$487$	25.3987	1.15092	0.575462	−	0.817829i	$-0.304822\pi$
$487$	25.3987	1.15092	0.575462	+	0.817829i	$0.304822\pi$
$488$	0	0
$489$	2.32487	0.105134
$490$	0	0
$491$	24.8265	1.12041	0.560203	−	0.828355i	$-0.310723\pi$
$491$	24.8265	1.12041	0.560203	+	0.828355i	$0.310723\pi$
$492$	0	0
$493$	−15.6834	−0.706344
$494$	0	0
$495$	−1.67513	−0.0752915
$496$	0	0
$497$	4.98683	0.223690
$498$	0	0
$499$	12.6375	0.565733	0.282867	−	0.959159i	$-0.408715\pi$
$499$	12.6375	0.565733	0.282867	+	0.959159i	$0.408715\pi$
$500$	0	0
$501$	−5.08840	−0.227333
$502$	0	0
$503$	19.1998	0.856077	0.428039	−	0.903760i	$-0.359205\pi$
$503$	19.1998	0.856077	0.428039	+	0.903760i	$0.359205\pi$
$504$	0	0
$505$	5.01714	0.223260
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	14.1236	0.626019	0.313010	−	0.949750i	$-0.398663\pi$
$509$	14.1236	0.626019	0.313010	+	0.949750i	$0.398663\pi$
$510$	0	0
$511$	3.03761	0.134376
$512$	0	0
$513$	5.11871	0.225997
$514$	0	0
$515$	−17.4010	−0.766782
$516$	0	0
$517$	−4.70052	−0.206729
$518$	0	0
$519$	−2.09332	−0.0918866
$520$	0	0
$521$	38.8481	1.70197	0.850984	−	0.525192i	$-0.176006\pi$
$521$	38.8481	1.70197	0.850984	+	0.525192i	$0.176006\pi$
$522$	0	0
$523$	−21.0068	−0.918565	−0.459282	−	0.888290i	$-0.651893\pi$
$523$	−21.0068	−0.918565	−0.459282	+	0.888290i	$0.651893\pi$
$524$	0	0
$525$	1.76845	0.0771816
$526$	0	0
$527$	33.0943	1.44161
$528$	0	0
$529$	−2.35026	−0.102185
$530$	0	0
$531$	−6.57452	−0.285310
$532$	0	0
$533$	−7.11871	−0.308346
$534$	0	0
$535$	−20.9380	−0.905227
$536$	0	0
$537$	10.3430	0.446332
$538$	0	0
$539$	−6.35026	−0.273525
$540$	0	0
$541$	7.44992	0.320297	0.160149	−	0.987093i	$-0.448803\pi$
$541$	7.44992	0.320297	0.160149	+	0.987093i	$0.448803\pi$
$542$	0	0
$543$	−4.28233	−0.183773
$544$	0	0
$545$	22.9380	0.982554
$546$	0	0
$547$	−23.8677	−1.02051	−0.510254	−	0.860024i	$-0.670448\pi$
$547$	−23.8677	−1.02051	−0.510254	+	0.860024i	$0.670448\pi$
$548$	0	0
$549$	−6.70052	−0.285971
$550$	0	0
$551$	20.9525	0.892608
$552$	0	0
$553$	−2.77575	−0.118037
$554$	0	0
$555$	−2.91160	−0.123591
$556$	0	0
$557$	−1.71767	−0.0727799	−0.0363899	−	0.999338i	$-0.511586\pi$
$557$	−1.71767	−0.0727799	−0.0363899	+	0.999338i	$0.511586\pi$
$558$	0	0
$559$	−3.13093	−0.132424
$560$	0	0
$561$	−3.83146	−0.161764
$562$	0	0
$563$	20.9380	0.882429	0.441215	−	0.897402i	$-0.354548\pi$
$563$	20.9380	0.882429	0.441215	+	0.897402i	$0.354548\pi$
$564$	0	0
$565$	16.1866	0.680977
$566$	0	0
$567$	0.806063	0.0338515
$568$	0	0
$569$	−2.53198	−0.106146	−0.0530731	−	0.998591i	$-0.516902\pi$
$569$	−2.53198	−0.106146	−0.0530731	+	0.998591i	$0.516902\pi$
$570$	0	0
$571$	10.7332	0.449171	0.224585	−	0.974454i	$-0.427897\pi$
$571$	10.7332	0.449171	0.224585	+	0.974454i	$0.427897\pi$
$572$	0	0
$573$	13.5877	0.567634
$574$	0	0
$575$	−9.96968	−0.415765
$576$	0	0
$577$	−31.9854	−1.33157	−0.665785	−	0.746144i	$-0.731903\pi$
$577$	−31.9854	−1.33157	−0.665785	+	0.746144i	$0.731903\pi$
$578$	0	0
$579$	17.1695	0.713540
$580$	0	0
$581$	−1.40105	−0.0581252
$582$	0	0
$583$	1.35026	0.0559221
$584$	0	0
$585$	1.67513	0.0692581
$586$	0	0
$587$	6.42407	0.265150	0.132575	−	0.991173i	$-0.457676\pi$
$587$	6.42407	0.265150	0.132575	+	0.991173i	$0.457676\pi$
$588$	0	0
$589$	−44.2130	−1.82176
$590$	0	0
$591$	−21.4821	−0.883658
$592$	0	0
$593$	−3.76845	−0.154752	−0.0773759	−	0.997002i	$-0.524654\pi$
$593$	−3.76845	−0.154752	−0.0773759	+	0.997002i	$0.524654\pi$
$594$	0	0
$595$	−5.17347	−0.212092
$596$	0	0
$597$	−21.0884	−0.863091
$598$	0	0
$599$	−27.4109	−1.11998	−0.559989	−	0.828500i	$-0.689195\pi$
$599$	−27.4109	−1.11998	−0.559989	+	0.828500i	$0.689195\pi$
$600$	0	0
$601$	−15.7743	−0.643448	−0.321724	−	0.946833i	$-0.604262\pi$
$601$	−15.7743	−0.643448	−0.321724	+	0.946833i	$0.604262\pi$
$602$	0	0
$603$	10.2496	0.417398
$604$	0	0
$605$	−1.67513	−0.0681038
$606$	0	0
$607$	−3.51881	−0.142824	−0.0714120	−	0.997447i	$-0.522750\pi$
$607$	−3.51881	−0.142824	−0.0714120	+	0.997447i	$0.522750\pi$
$608$	0	0
$609$	3.29948	0.133702
$610$	0	0
$611$	4.70052	0.190163
$612$	0	0
$613$	−40.8080	−1.64822	−0.824109	−	0.566431i	$-0.808324\pi$
$613$	−40.8080	−1.64822	−0.824109	+	0.566431i	$0.808324\pi$
$614$	0	0
$615$	11.9248	0.480853
$616$	0	0
$617$	9.33804	0.375935	0.187968	−	0.982175i	$-0.439810\pi$
$617$	9.33804	0.375935	0.187968	+	0.982175i	$0.439810\pi$
$618$	0	0
$619$	8.71274	0.350195	0.175097	−	0.984551i	$-0.443976\pi$
$619$	8.71274	0.350195	0.175097	+	0.984551i	$0.443976\pi$
$620$	0	0
$621$	−4.54420	−0.182352
$622$	0	0
$623$	−12.5139	−0.501358
$624$	0	0
$625$	−9.21696	−0.368678
$626$	0	0
$627$	5.11871	0.204422
$628$	0	0
$629$	−6.65959	−0.265535
$630$	0	0
$631$	30.2398	1.20383	0.601914	−	0.798561i	$-0.294405\pi$
$631$	30.2398	1.20383	0.601914	+	0.798561i	$0.294405\pi$
$632$	0	0
$633$	−19.9429	−0.792658
$634$	0	0
$635$	−20.9076	−0.829694
$636$	0	0
$637$	6.35026	0.251607
$638$	0	0
$639$	6.18664	0.244740
$640$	0	0
$641$	−29.0132	−1.14595	−0.572976	−	0.819572i	$-0.694211\pi$
$641$	−29.0132	−1.14595	−0.572976	+	0.819572i	$0.694211\pi$
$642$	0	0
$643$	5.85192	0.230777	0.115389	−	0.993320i	$-0.463189\pi$
$643$	5.85192	0.230777	0.115389	+	0.993320i	$0.463189\pi$
$644$	0	0
$645$	5.24472	0.206511
$646$	0	0
$647$	14.5139	0.570600	0.285300	−	0.958438i	$-0.407907\pi$
$647$	14.5139	0.570600	0.285300	+	0.958438i	$0.407907\pi$
$648$	0	0
$649$	−6.57452	−0.258072
$650$	0	0
$651$	−6.96239	−0.272878
$652$	0	0
$653$	35.4010	1.38535	0.692675	−	0.721250i	$-0.256432\pi$
$653$	35.4010	1.38535	0.692675	+	0.721250i	$0.256432\pi$
$654$	0	0
$655$	7.22425	0.282275
$656$	0	0
$657$	3.76845	0.147021
$658$	0	0
$659$	−4.85097	−0.188967	−0.0944835	−	0.995526i	$-0.530120\pi$
$659$	−4.85097	−0.188967	−0.0944835	+	0.995526i	$0.530120\pi$
$660$	0	0
$661$	8.36344	0.325300	0.162650	−	0.986684i	$-0.447996\pi$
$661$	8.36344	0.325300	0.162650	+	0.986684i	$0.447996\pi$
$662$	0	0
$663$	3.83146	0.148801
$664$	0	0
$665$	6.91160	0.268020
$666$	0	0
$667$	−18.6009	−0.720228
$668$	0	0
$669$	−2.06300	−0.0797603
$670$	0	0
$671$	−6.70052	−0.258671
$672$	0	0
$673$	37.1852	1.43339	0.716693	−	0.697389i	$-0.245655\pi$
$673$	37.1852	1.43339	0.716693	+	0.697389i	$0.245655\pi$
$674$	0	0
$675$	2.19394	0.0844447
$676$	0	0
$677$	−4.05712	−0.155928	−0.0779640	−	0.996956i	$-0.524842\pi$
$677$	−4.05712	−0.155928	−0.0779640	+	0.996956i	$0.524842\pi$
$678$	0	0
$679$	7.22425	0.277241
$680$	0	0
$681$	9.95509	0.381480
$682$	0	0
$683$	−19.9902	−0.764902	−0.382451	−	0.923976i	$-0.624920\pi$
$683$	−19.9902	−0.764902	−0.382451	+	0.923976i	$0.624920\pi$
$684$	0	0
$685$	10.6801	0.408064
$686$	0	0
$687$	−21.9248	−0.836482
$688$	0	0
$689$	−1.35026	−0.0514409
$690$	0	0
$691$	−22.3855	−0.851585	−0.425792	−	0.904821i	$-0.640005\pi$
$691$	−22.3855	−0.851585	−0.425792	+	0.904821i	$0.640005\pi$
$692$	0	0
$693$	0.806063	0.0306198
$694$	0	0
$695$	−31.3561	−1.18941
$696$	0	0
$697$	27.2750	1.03312
$698$	0	0
$699$	16.8446	0.637122
$700$	0	0
$701$	1.37821	0.0520542	0.0260271	−	0.999661i	$-0.491714\pi$
$701$	1.37821	0.0520542	0.0260271	+	0.999661i	$0.491714\pi$
$702$	0	0
$703$	8.89701	0.335557
$704$	0	0
$705$	−7.87399	−0.296552
$706$	0	0
$707$	−2.41422	−0.0907961
$708$	0	0
$709$	18.6399	0.700036	0.350018	−	0.936743i	$-0.386176\pi$
$709$	18.6399	0.700036	0.350018	+	0.936743i	$0.386176\pi$
$710$	0	0
$711$	−3.44358	−0.129144
$712$	0	0
$713$	39.2506	1.46995
$714$	0	0
$715$	1.67513	0.0626463
$716$	0	0
$717$	−14.2071	−0.530574
$718$	0	0
$719$	9.67276	0.360733	0.180367	−	0.983599i	$-0.442272\pi$
$719$	9.67276	0.360733	0.180367	+	0.983599i	$0.442272\pi$
$720$	0	0
$721$	8.37328	0.311837
$722$	0	0
$723$	−21.7235	−0.807907
$724$	0	0
$725$	8.98049	0.333527
$726$	0	0
$727$	16.2882	0.604096	0.302048	−	0.953293i	$-0.402330\pi$
$727$	16.2882	0.604096	0.302048	+	0.953293i	$0.402330\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	11.9960	0.443689
$732$	0	0
$733$	−13.1089	−0.484187	−0.242093	−	0.970253i	$-0.577834\pi$
$733$	−13.1089	−0.484187	−0.242093	+	0.970253i	$0.577834\pi$
$734$	0	0
$735$	−10.6375	−0.392371
$736$	0	0
$737$	10.2496	0.377551
$738$	0	0
$739$	−46.5804	−1.71349	−0.856744	−	0.515742i	$-0.827516\pi$
$739$	−46.5804	−1.71349	−0.856744	+	0.515742i	$0.827516\pi$
$740$	0	0
$741$	−5.11871	−0.188041
$742$	0	0
$743$	−27.0982	−0.994138	−0.497069	−	0.867711i	$-0.665590\pi$
$743$	−27.0982	−0.994138	−0.497069	+	0.867711i	$0.665590\pi$
$744$	0	0
$745$	17.3258	0.634769
$746$	0	0
$747$	−1.73813	−0.0635950
$748$	0	0
$749$	10.0752	0.368141
$750$	0	0
$751$	40.4993	1.47784	0.738920	−	0.673793i	$-0.235336\pi$
$751$	40.4993	1.47784	0.738920	+	0.673793i	$0.235336\pi$
$752$	0	0
$753$	5.73813	0.209109
$754$	0	0
$755$	0.574515	0.0209088
$756$	0	0
$757$	−26.0567	−0.947046	−0.473523	−	0.880782i	$-0.657018\pi$
$757$	−26.0567	−0.947046	−0.473523	+	0.880782i	$0.657018\pi$
$758$	0	0
$759$	−4.54420	−0.164944
$760$	0	0
$761$	−1.23013	−0.0445923	−0.0222961	−	0.999751i	$-0.507098\pi$
$761$	−1.23013	−0.0445923	−0.0222961	+	0.999751i	$0.507098\pi$
$762$	0	0
$763$	−11.0376	−0.399588
$764$	0	0
$765$	−6.41819	−0.232050
$766$	0	0
$767$	6.57452	0.237392
$768$	0	0
$769$	7.92478	0.285775	0.142887	−	0.989739i	$-0.454361\pi$
$769$	7.92478	0.285775	0.142887	+	0.989739i	$0.454361\pi$
$770$	0	0
$771$	19.7743	0.712155
$772$	0	0
$773$	−0.974607	−0.0350542	−0.0175271	−	0.999846i	$-0.505579\pi$
$773$	−0.974607	−0.0350542	−0.0175271	+	0.999846i	$0.505579\pi$
$774$	0	0
$775$	−18.9502	−0.680710
$776$	0	0
$777$	1.40105	0.0502623
$778$	0	0
$779$	−36.4387	−1.30555
$780$	0	0
$781$	6.18664	0.221375
$782$	0	0
$783$	4.09332	0.146283
$784$	0	0
$785$	−3.17347	−0.113266
$786$	0	0
$787$	−17.4314	−0.621361	−0.310681	−	0.950514i	$-0.600557\pi$
$787$	−17.4314	−0.621361	−0.310681	+	0.950514i	$0.600557\pi$
$788$	0	0
$789$	−6.57452	−0.234059
$790$	0	0
$791$	−7.78892	−0.276942
$792$	0	0
$793$	6.70052	0.237943
$794$	0	0
$795$	2.26187	0.0802201
$796$	0	0
$797$	23.4617	0.831055	0.415528	−	0.909581i	$-0.363597\pi$
$797$	23.4617	0.831055	0.415528	+	0.909581i	$0.363597\pi$
$798$	0	0
$799$	−18.0098	−0.637143
$800$	0	0
$801$	−15.5247	−0.548538
$802$	0	0
$803$	3.76845	0.132986
$804$	0	0
$805$	−6.13586	−0.216261
$806$	0	0
$807$	−20.3634	−0.716827
$808$	0	0
$809$	20.8954	0.734644	0.367322	−	0.930094i	$-0.380275\pi$
$809$	20.8954	0.734644	0.367322	+	0.930094i	$0.380275\pi$
$810$	0	0
$811$	11.1939	0.393072	0.196536	−	0.980497i	$-0.437031\pi$
$811$	11.1939	0.393072	0.196536	+	0.980497i	$0.437031\pi$
$812$	0	0
$813$	−12.1563	−0.426341
$814$	0	0
$815$	3.89446	0.136417
$816$	0	0
$817$	−16.0263	−0.560691
$818$	0	0
$819$	−0.806063	−0.0281661
$820$	0	0
$821$	8.75528	0.305561	0.152781	−	0.988260i	$-0.451177\pi$
$821$	8.75528	0.305561	0.152781	+	0.988260i	$0.451177\pi$
$822$	0	0
$823$	9.36485	0.326438	0.163219	−	0.986590i	$-0.447812\pi$
$823$	9.36485	0.326438	0.163219	+	0.986590i	$0.447812\pi$
$824$	0	0
$825$	2.19394	0.0763831
$826$	0	0
$827$	1.11871	0.0389015	0.0194507	−	0.999811i	$-0.493808\pi$
$827$	1.11871	0.0389015	0.0194507	+	0.999811i	$0.493808\pi$
$828$	0	0
$829$	−0.700523	−0.0243302	−0.0121651	−	0.999926i	$-0.503872\pi$
$829$	−0.700523	−0.0243302	−0.0121651	+	0.999926i	$0.503872\pi$
$830$	0	0
$831$	−8.38787	−0.290972
$832$	0	0
$833$	−24.3307	−0.843010
$834$	0	0
$835$	−8.52373	−0.294976
$836$	0	0
$837$	−8.63752	−0.298556
$838$	0	0
$839$	−40.5355	−1.39944	−0.699720	−	0.714417i	$-0.746692\pi$
$839$	−40.5355	−1.39944	−0.699720	+	0.714417i	$0.746692\pi$
$840$	0	0
$841$	−12.2447	−0.422232
$842$	0	0
$843$	−23.3054	−0.802679
$844$	0	0
$845$	−1.67513	−0.0576263
$846$	0	0
$847$	0.806063	0.0276967
$848$	0	0
$849$	−14.9951	−0.514630
$850$	0	0
$851$	−7.89843	−0.270755
$852$	0	0
$853$	56.9643	1.95042	0.975210	−	0.221280i	$-0.0710236\pi$
$853$	56.9643	1.95042	0.975210	+	0.221280i	$0.0710236\pi$
$854$	0	0
$855$	8.57452	0.293242
$856$	0	0
$857$	−35.4191	−1.20989	−0.604947	−	0.796265i	$-0.706806\pi$
$857$	−35.4191	−1.20989	−0.604947	+	0.796265i	$0.706806\pi$
$858$	0	0
$859$	−55.4636	−1.89239	−0.946197	−	0.323592i	$-0.895109\pi$
$859$	−55.4636	−1.89239	−0.946197	+	0.323592i	$0.895109\pi$
$860$	0	0
$861$	−5.73813	−0.195555
$862$	0	0
$863$	−5.63656	−0.191871	−0.0959354	−	0.995388i	$-0.530584\pi$
$863$	−5.63656	−0.191871	−0.0959354	+	0.995388i	$0.530584\pi$
$864$	0	0
$865$	−3.50659	−0.119228
$866$	0	0
$867$	2.31994	0.0787894
$868$	0	0
$869$	−3.44358	−0.116816
$870$	0	0
$871$	−10.2496	−0.347296
$872$	0	0
$873$	8.96239	0.303331
$874$	0	0
$875$	9.71370	0.328383
$876$	0	0
$877$	10.9018	0.368126	0.184063	−	0.982914i	$-0.441075\pi$
$877$	10.9018	0.368126	0.184063	+	0.982914i	$0.441075\pi$
$878$	0	0
$879$	13.9452	0.470361
$880$	0	0
$881$	30.1016	1.01415	0.507074	−	0.861903i	$-0.330727\pi$
$881$	30.1016	1.01415	0.507074	+	0.861903i	$0.330727\pi$
$882$	0	0
$883$	−16.7151	−0.562508	−0.281254	−	0.959633i	$-0.590750\pi$
$883$	−16.7151	−0.562508	−0.281254	+	0.959633i	$0.590750\pi$
$884$	0	0
$885$	−11.0132	−0.370204
$886$	0	0
$887$	−18.6253	−0.625376	−0.312688	−	0.949856i	$-0.601229\pi$
$887$	−18.6253	−0.625376	−0.312688	+	0.949856i	$0.601229\pi$
$888$	0	0
$889$	10.0606	0.337423
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	24.0606	0.805158
$894$	0	0
$895$	17.3258	0.579138
$896$	0	0
$897$	4.54420	0.151726
$898$	0	0
$899$	−35.3561	−1.17919
$900$	0	0
$901$	5.17347	0.172353
$902$	0	0
$903$	−2.52373	−0.0839845
$904$	0	0
$905$	−7.17347	−0.238454
$906$	0	0
$907$	43.9267	1.45856	0.729281	−	0.684214i	$-0.239855\pi$
$907$	43.9267	1.45856	0.729281	+	0.684214i	$0.239855\pi$
$908$	0	0
$909$	−2.99508	−0.0993404
$910$	0	0
$911$	48.6615	1.61223	0.806114	−	0.591761i	$-0.201567\pi$
$911$	48.6615	1.61223	0.806114	+	0.591761i	$0.201567\pi$
$912$	0	0
$913$	−1.73813	−0.0575239
$914$	0	0
$915$	−11.2243	−0.371062
$916$	0	0
$917$	−3.47627	−0.114797
$918$	0	0
$919$	−16.3961	−0.540858	−0.270429	−	0.962740i	$-0.587166\pi$
$919$	−16.3961	−0.540858	−0.270429	+	0.962740i	$0.587166\pi$
$920$	0	0
$921$	−23.0435	−0.759309
$922$	0	0
$923$	−6.18664	−0.203636
$924$	0	0
$925$	3.81336	0.125382
$926$	0	0
$927$	10.3879	0.341183
$928$	0	0
$929$	14.8340	0.486688	0.243344	−	0.969940i	$-0.421756\pi$
$929$	14.8340	0.486688	0.243344	+	0.969940i	$0.421756\pi$
$930$	0	0
$931$	32.5052	1.06531
$932$	0	0
$933$	−5.90431	−0.193298
$934$	0	0
$935$	−6.41819	−0.209897
$936$	0	0
$937$	−21.9003	−0.715453	−0.357726	−	0.933826i	$-0.616448\pi$
$937$	−21.9003	−0.715453	−0.357726	+	0.933826i	$0.616448\pi$
$938$	0	0
$939$	−26.6458	−0.869552
$940$	0	0
$941$	−51.2809	−1.67171	−0.835855	−	0.548950i	$-0.815028\pi$
$941$	−51.2809	−1.67171	−0.835855	+	0.548950i	$0.815028\pi$
$942$	0	0
$943$	32.3488	1.05342
$944$	0	0
$945$	1.35026	0.0439240
$946$	0	0
$947$	5.91019	0.192055	0.0960277	−	0.995379i	$-0.469386\pi$
$947$	5.91019	0.192055	0.0960277	+	0.995379i	$0.469386\pi$
$948$	0	0
$949$	−3.76845	−0.122329
$950$	0	0
$951$	−28.2760	−0.916912
$952$	0	0
$953$	−52.6288	−1.70481	−0.852407	−	0.522879i	$-0.824858\pi$
$953$	−52.6288	−1.70481	−0.852407	+	0.522879i	$0.824858\pi$
$954$	0	0
$955$	22.7612	0.736534
$956$	0	0
$957$	4.09332	0.132318
$958$	0	0
$959$	−5.13918	−0.165953
$960$	0	0
$961$	43.6067	1.40667
$962$	0	0
$963$	12.4993	0.402784
$964$	0	0
$965$	28.7612	0.925854
$966$	0	0
$967$	−15.2908	−0.491718	−0.245859	−	0.969306i	$-0.579070\pi$
$967$	−15.2908	−0.491718	−0.245859	+	0.969306i	$0.579070\pi$
$968$	0	0
$969$	19.6121	0.630032
$970$	0	0
$971$	−52.2433	−1.67657	−0.838284	−	0.545234i	$-0.816441\pi$
$971$	−52.2433	−1.67657	−0.838284	+	0.545234i	$0.816441\pi$
$972$	0	0
$973$	15.0884	0.483712
$974$	0	0
$975$	−2.19394	−0.0702622
$976$	0	0
$977$	−9.32345	−0.298284	−0.149142	−	0.988816i	$-0.547651\pi$
$977$	−9.32345	−0.298284	−0.149142	+	0.988816i	$0.547651\pi$
$978$	0	0
$979$	−15.5247	−0.496171
$980$	0	0
$981$	−13.6932	−0.437191
$982$	0	0
$983$	41.4617	1.32242	0.661211	−	0.750200i	$-0.270043\pi$
$983$	41.4617	1.32242	0.661211	+	0.750200i	$0.270043\pi$
$984$	0	0
$985$	−35.9854	−1.14659
$986$	0	0
$987$	3.78892	0.120603
$988$	0	0
$989$	14.2276	0.452411
$990$	0	0
$991$	32.0968	1.01959	0.509795	−	0.860296i	$-0.329721\pi$
$991$	32.0968	1.01959	0.509795	+	0.860296i	$0.329721\pi$
$992$	0	0
$993$	−25.2120	−0.800080
$994$	0	0
$995$	−35.3258	−1.11990
$996$	0	0
$997$	36.1524	1.14496	0.572478	−	0.819920i	$-0.305982\pi$
$997$	36.1524	1.14496	0.572478	+	0.819920i	$0.305982\pi$
$998$	0	0
$999$	1.73813	0.0549922

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6864.2.a.bq.1.1		3
4.3	odd	2		3432.2.a.p.1.1	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3432.2.a.p.1.1	✓	3	4.3	odd	2
6864.2.a.bq.1.1		3	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.67513 q^{5} +0.806063 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.67513 q^{5} +0.806063 q^{7} +1.00000 q^{9} +1.00000 q^{11} -1.00000 q^{13} +1.67513 q^{15} +3.83146 q^{17} -5.11871 q^{19} -0.806063 q^{21} +4.54420 q^{23} -2.19394 q^{25} -1.00000 q^{27} -4.09332 q^{29} +8.63752 q^{31} -1.00000 q^{33} -1.35026 q^{35} -1.73813 q^{37} +1.00000 q^{39} +7.11871 q^{41} +3.13093 q^{43} -1.67513 q^{45} -4.70052 q^{47} -6.35026 q^{49} -3.83146 q^{51} +1.35026 q^{53} -1.67513 q^{55} +5.11871 q^{57} -6.57452 q^{59} -6.70052 q^{61} +0.806063 q^{63} +1.67513 q^{65} +10.2496 q^{67} -4.54420 q^{69} +6.18664 q^{71} +3.76845 q^{73} +2.19394 q^{75} +0.806063 q^{77} -3.44358 q^{79} +1.00000 q^{81} -1.73813 q^{83} -6.41819 q^{85} +4.09332 q^{87} -15.5247 q^{89} -0.806063 q^{91} -8.63752 q^{93} +8.57452 q^{95} +8.96239 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 2 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 + 2 * q^7 + 3 * q^9	\( 3 q - 3 q^{3} + 2 q^{7} + 3 q^{9} + 3 q^{11} - 3 q^{13} - 4 q^{17} + 6 q^{19} - 2 q^{21} + 4 q^{23} - 7 q^{25} - 3 q^{27} - 6 q^{29} + 10 q^{31} - 3 q^{33} + 6 q^{35} + 4 q^{37} + 3 q^{39} + 14 q^{43} + 6 q^{47} - 9 q^{49} + 4 q^{51} - 6 q^{53} - 6 q^{57} - 8 q^{59} + 2 q^{63} + 14 q^{67} - 4 q^{69} + 6 q^{71} + 7 q^{75} + 2 q^{77} + 6 q^{79} + 3 q^{81} + 4 q^{83} - 18 q^{85} + 6 q^{87} + 2 q^{89} - 2 q^{91} - 10 q^{93} + 14 q^{95} + 16 q^{97} + 3 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + 2 * q^7 + 3 * q^9 + 3 * q^11 - 3 * q^13 - 4 * q^17 + 6 * q^19 - 2 * q^21 + 4 * q^23 - 7 * q^25 - 3 * q^27 - 6 * q^29 + 10 * q^31 - 3 * q^33 + 6 * q^35 + 4 * q^37 + 3 * q^39 + 14 * q^43 + 6 * q^47 - 9 * q^49 + 4 * q^51 - 6 * q^53 - 6 * q^57 - 8 * q^59 + 2 * q^63 + 14 * q^67 - 4 * q^69 + 6 * q^71 + 7 * q^75 + 2 * q^77 + 6 * q^79 + 3 * q^81 + 4 * q^83 - 18 * q^85 + 6 * q^87 + 2 * q^89 - 2 * q^91 - 10 * q^93 + 14 * q^95 + 16 * q^97 + 3 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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