LMFDB - Embedded newform 6864.2.a.by.1.3

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:	$1$
Dimension:	$4$
Coefficient field:	4.4.22676.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} + 6x + 2 $ x^4 - x^3 - 6x^2 + 6x + 2
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			1.3
Root			$2.31078$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +1.33970 q^{5} -0.474190 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.33970 q^{5} -0.474190 q^{7} +1.00000 q^{9} +1.00000 q^{11} -1.00000 q^{13} +1.33970 q^{15} -1.81389 q^{17} -4.62156 q^{19} -0.474190 q^{21} +6.66568 q^{23} -3.20521 q^{25} +1.00000 q^{27} -3.76976 q^{29} -4.80767 q^{31} +1.00000 q^{33} -0.635271 q^{35} -10.8709 q^{37} -1.00000 q^{39} -8.20521 q^{41} -4.90964 q^{43} +1.33970 q^{45} -9.24312 q^{47} -6.77514 q^{49} -1.81389 q^{51} -2.00000 q^{53} +1.33970 q^{55} -4.62156 q^{57} -4.14737 q^{59} +9.28724 q^{61} -0.474190 q^{63} -1.33970 q^{65} +6.85180 q^{67} +6.66568 q^{69} +7.92251 q^{71} -2.84641 q^{73} -3.20521 q^{75} -0.474190 q^{77} -3.81389 q^{79} +1.00000 q^{81} +12.2947 q^{83} -2.43006 q^{85} -3.76976 q^{87} +5.11484 q^{89} +0.474190 q^{91} -4.80767 q^{93} -6.19149 q^{95} -10.8709 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} - 3 q^{5} - 3 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 - 3 * q^5 - 3 * q^7 + 4 * q^9	$ 4 q + 4 q^{3} - 3 q^{5} - 3 q^{7} + 4 q^{9} + 4 q^{11} - 4 q^{13} - 3 q^{15} - 2 q^{19} - 3 q^{21} - 3 q^{23} + 5 q^{25} + 4 q^{27} - 21 q^{29} - 10 q^{31} + 4 q^{33} + q^{35} + 4 q^{37} - 4 q^{39} - 15 q^{41} + 3 q^{43} - 3 q^{45} - 4 q^{47} + 5 q^{49} - 8 q^{53} - 3 q^{55} - 2 q^{57} + q^{59} - 9 q^{61} - 3 q^{63} + 3 q^{65} + 5 q^{67} - 3 q^{69} - 18 q^{71} - 27 q^{73} + 5 q^{75} - 3 q^{77} - 8 q^{79} + 4 q^{81} + 14 q^{83} - 24 q^{85} - 21 q^{87} - 20 q^{89} + 3 q^{91} - 10 q^{93} + 6 q^{95} + 4 q^{97} + 4 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 - 3 * q^5 - 3 * q^7 + 4 * q^9 + 4 * q^11 - 4 * q^13 - 3 * q^15 - 2 * q^19 - 3 * q^21 - 3 * q^23 + 5 * q^25 + 4 * q^27 - 21 * q^29 - 10 * q^31 + 4 * q^33 + q^35 + 4 * q^37 - 4 * q^39 - 15 * q^41 + 3 * q^43 - 3 * q^45 - 4 * q^47 + 5 * q^49 - 8 * q^53 - 3 * q^55 - 2 * q^57 + q^59 - 9 * q^61 - 3 * q^63 + 3 * q^65 + 5 * q^67 - 3 * q^69 - 18 * q^71 - 27 * q^73 + 5 * q^75 - 3 * q^77 - 8 * q^79 + 4 * q^81 + 14 * q^83 - 24 * q^85 - 21 * q^87 - 20 * q^89 + 3 * q^91 - 10 * q^93 + 6 * q^95 + 4 * 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
$5$	1.33970	0.599131	0.299566	−	0.954076i	$-0.403158\pi$
$5$	1.33970	0.599131	0.299566	+	0.954076i	$0.403158\pi$
$6$	0	0
$7$	−0.474190	−0.179227	−0.0896134	−	0.995977i	$-0.528563\pi$
$7$	−0.474190	−0.179227	−0.0896134	+	0.995977i	$0.528563\pi$
$8$	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
$15$	1.33970	0.345909
$16$	0	0
$17$	−1.81389	−0.439932	−0.219966	−	0.975507i	$-0.570595\pi$
$17$	−1.81389	−0.439932	−0.219966	+	0.975507i	$0.570595\pi$
$18$	0	0
$19$	−4.62156	−1.06026	−0.530129	−	0.847917i	$-0.677857\pi$
$19$	−4.62156	−1.06026	−0.530129	+	0.847917i	$0.677857\pi$
$20$	0	0
$21$	−0.474190	−0.103477
$22$	0	0
$23$	6.66568	1.38989	0.694946	−	0.719062i	$-0.255428\pi$
$23$	6.66568	1.38989	0.694946	+	0.719062i	$0.255428\pi$
$24$	0	0
$25$	−3.20521	−0.641042
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−3.76976	−0.700027	−0.350014	−	0.936745i	$-0.613823\pi$
$29$	−3.76976	−0.700027	−0.350014	+	0.936745i	$0.613823\pi$
$30$	0	0
$31$	−4.80767	−0.863483	−0.431741	−	0.901997i	$-0.642101\pi$
$31$	−4.80767	−0.863483	−0.431741	+	0.901997i	$0.642101\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	−0.635271	−0.107380
$36$	0	0
$37$	−10.8709	−1.78716	−0.893582	−	0.448900i	$-0.851816\pi$
$37$	−10.8709	−1.78716	−0.893582	+	0.448900i	$0.851816\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−8.20521	−1.28144	−0.640719	−	0.767775i	$-0.721364\pi$
$41$	−8.20521	−1.28144	−0.640719	+	0.767775i	$0.721364\pi$
$42$	0	0
$43$	−4.90964	−0.748712	−0.374356	−	0.927285i	$-0.622136\pi$
$43$	−4.90964	−0.748712	−0.374356	+	0.927285i	$0.622136\pi$
$44$	0	0
$45$	1.33970	0.199710
$46$	0	0
$47$	−9.24312	−1.34825	−0.674123	−	0.738619i	$-0.735478\pi$
$47$	−9.24312	−1.34825	−0.674123	+	0.738619i	$0.735478\pi$
$48$	0	0
$49$	−6.77514	−0.967878
$50$	0	0
$51$	−1.81389	−0.253995
$52$	0	0
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	1.33970	0.180645
$56$	0	0
$57$	−4.62156	−0.612140
$58$	0	0
$59$	−4.14737	−0.539941	−0.269971	−	0.962869i	$-0.587014\pi$
$59$	−4.14737	−0.539941	−0.269971	+	0.962869i	$0.587014\pi$
$60$	0	0
$61$	9.28724	1.18911	0.594555	−	0.804055i	$-0.297328\pi$
$61$	9.28724	1.18911	0.594555	+	0.804055i	$0.297328\pi$
$62$	0	0
$63$	−0.474190	−0.0597423
$64$	0	0
$65$	−1.33970	−0.166169
$66$	0	0
$67$	6.85180	0.837080	0.418540	−	0.908198i	$-0.362542\pi$
$67$	6.85180	0.837080	0.418540	+	0.908198i	$0.362542\pi$
$68$	0	0
$69$	6.66568	0.802454
$70$	0	0
$71$	7.92251	0.940229	0.470115	−	0.882605i	$-0.344213\pi$
$71$	7.92251	0.940229	0.470115	+	0.882605i	$0.344213\pi$
$72$	0	0
$73$	−2.84641	−0.333147	−0.166574	−	0.986029i	$-0.553270\pi$
$73$	−2.84641	−0.333147	−0.166574	+	0.986029i	$0.553270\pi$
$74$	0	0
$75$	−3.20521	−0.370106
$76$	0	0
$77$	−0.474190	−0.0540389
$78$	0	0
$79$	−3.81389	−0.429096	−0.214548	−	0.976713i	$-0.568828\pi$
$79$	−3.81389	−0.429096	−0.214548	+	0.976713i	$0.568828\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	12.2947	1.34952	0.674761	−	0.738036i	$-0.264247\pi$
$83$	12.2947	1.34952	0.674761	+	0.738036i	$0.264247\pi$
$84$	0	0
$85$	−2.43006	−0.263577
$86$	0	0
$87$	−3.76976	−0.404161
$88$	0	0
$89$	5.11484	0.542172	0.271086	−	0.962555i	$-0.412617\pi$
$89$	5.11484	0.542172	0.271086	+	0.962555i	$0.412617\pi$
$90$	0	0
$91$	0.474190	0.0497086
$92$	0	0
$93$	−4.80767	−0.498532
$94$	0	0
$95$	−6.19149	−0.635234
$96$	0	0
$97$	−10.8709	−1.10377	−0.551886	−	0.833920i	$-0.686092\pi$
$97$	−10.8709	−1.10377	−0.551886	+	0.833920i	$0.686092\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	6.10862	0.607831	0.303915	−	0.952699i	$-0.401706\pi$
$101$	6.10862	0.607831	0.303915	+	0.952699i	$0.401706\pi$
$102$	0	0
$103$	0.147368	0.0145206	0.00726030	−	0.999974i	$-0.497689\pi$
$103$	0.147368	0.0145206	0.00726030	+	0.999974i	$0.497689\pi$
$104$	0	0
$105$	−0.635271	−0.0619961
$106$	0	0
$107$	9.09575	0.879319	0.439660	−	0.898165i	$-0.355099\pi$
$107$	9.09575	0.879319	0.439660	+	0.898165i	$0.355099\pi$
$108$	0	0
$109$	−9.76143	−0.934976	−0.467488	−	0.884000i	$-0.654841\pi$
$109$	−9.76143	−0.934976	−0.467488	+	0.884000i	$0.654841\pi$
$110$	0	0
$111$	−10.8709	−1.03182
$112$	0	0
$113$	5.89082	0.554162	0.277081	−	0.960847i	$-0.410633\pi$
$113$	5.89082	0.554162	0.277081	+	0.960847i	$0.410633\pi$
$114$	0	0
$115$	8.93001	0.832727
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	0.860127	0.0788477
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−8.20521	−0.739839
$124$	0	0
$125$	−10.9925	−0.983200
$126$	0	0
$127$	−0.186112	−0.0165148	−0.00825738	−	0.999966i	$-0.502628\pi$
$127$	−0.186112	−0.0165148	−0.00825738	+	0.999966i	$0.502628\pi$
$128$	0	0
$129$	−4.90964	−0.432269
$130$	0	0
$131$	−4.60784	−0.402589	−0.201295	−	0.979531i	$-0.564515\pi$
$131$	−4.60784	−0.402589	−0.201295	+	0.979531i	$0.564515\pi$
$132$	0	0
$133$	2.19149	0.190027
$134$	0	0
$135$	1.33970	0.115303
$136$	0	0
$137$	6.06322	0.518016	0.259008	−	0.965875i	$-0.416604\pi$
$137$	6.06322	0.518016	0.259008	+	0.965875i	$0.416604\pi$
$138$	0	0
$139$	−6.49329	−0.550753	−0.275377	−	0.961336i	$-0.588803\pi$
$139$	−6.49329	−0.550753	−0.275377	+	0.961336i	$0.588803\pi$
$140$	0	0
$141$	−9.24312	−0.778411
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	−5.05034	−0.419408
$146$	0	0
$147$	−6.77514	−0.558804
$148$	0	0
$149$	−17.3892	−1.42458	−0.712290	−	0.701886i	$-0.752342\pi$
$149$	−17.3892	−1.42458	−0.712290	+	0.701886i	$0.752342\pi$
$150$	0	0
$151$	1.56994	0.127760	0.0638798	−	0.997958i	$-0.479653\pi$
$151$	1.56994	0.127760	0.0638798	+	0.997958i	$0.479653\pi$
$152$	0	0
$153$	−1.81389	−0.146644
$154$	0	0
$155$	−6.44083	−0.517340
$156$	0	0
$157$	−5.67318	−0.452769	−0.226384	−	0.974038i	$-0.572691\pi$
$157$	−5.67318	−0.452769	−0.226384	+	0.974038i	$0.572691\pi$
$158$	0	0
$159$	−2.00000	−0.158610
$160$	0	0
$161$	−3.16080	−0.249106
$162$	0	0
$163$	11.4280	0.895106	0.447553	−	0.894257i	$-0.352296\pi$
$163$	11.4280	0.895106	0.447553	+	0.894257i	$0.352296\pi$
$164$	0	0
$165$	1.33970	0.104295
$166$	0	0
$167$	−17.0183	−1.31691	−0.658456	−	0.752619i	$-0.728790\pi$
$167$	−17.0183	−1.31691	−0.658456	+	0.752619i	$0.728790\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−4.62156	−0.353419
$172$	0	0
$173$	−19.8580	−1.50978	−0.754888	−	0.655853i	$-0.772309\pi$
$173$	−19.8580	−1.50978	−0.754888	+	0.655853i	$0.772309\pi$
$174$	0	0
$175$	1.51988	0.114892
$176$	0	0
$177$	−4.14737	−0.311735
$178$	0	0
$179$	−3.41663	−0.255371	−0.127686	−	0.991815i	$-0.540755\pi$
$179$	−3.41663	−0.255371	−0.127686	+	0.991815i	$0.540755\pi$
$180$	0	0
$181$	22.0562	1.63942	0.819711	−	0.572777i	$-0.194134\pi$
$181$	22.0562	1.63942	0.819711	+	0.572777i	$0.194134\pi$
$182$	0	0
$183$	9.28724	0.686533
$184$	0	0
$185$	−14.5637	−1.07075
$186$	0	0
$187$	−1.81389	−0.132645
$188$	0	0
$189$	−0.474190	−0.0344922
$190$	0	0
$191$	17.0183	1.23140	0.615699	−	0.787981i	$-0.288874\pi$
$191$	17.0183	1.23140	0.615699	+	0.787981i	$0.288874\pi$
$192$	0	0
$193$	14.9163	1.07370	0.536849	−	0.843678i	$-0.319614\pi$
$193$	14.9163	1.07370	0.536849	+	0.843678i	$0.319614\pi$
$194$	0	0
$195$	−1.33970	−0.0959378
$196$	0	0
$197$	−20.3526	−1.45006	−0.725030	−	0.688717i	$-0.758174\pi$
$197$	−20.3526	−1.45006	−0.725030	+	0.688717i	$0.758174\pi$
$198$	0	0
$199$	6.87683	0.487485	0.243743	−	0.969840i	$-0.421625\pi$
$199$	6.87683	0.487485	0.243743	+	0.969840i	$0.421625\pi$
$200$	0	0
$201$	6.85180	0.483288
$202$	0	0
$203$	1.78758	0.125464
$204$	0	0
$205$	−10.9925	−0.767750
$206$	0	0
$207$	6.66568	0.463297
$208$	0	0
$209$	−4.62156	−0.319680
$210$	0	0
$211$	3.23773	0.222895	0.111447	−	0.993770i	$-0.464451\pi$
$211$	3.23773	0.222895	0.111447	+	0.993770i	$0.464451\pi$
$212$	0	0
$213$	7.92251	0.542842
$214$	0	0
$215$	−6.57743	−0.448577
$216$	0	0
$217$	2.27975	0.154759
$218$	0	0
$219$	−2.84641	−0.192343
$220$	0	0
$221$	1.81389	0.122015
$222$	0	0
$223$	5.46131	0.365717	0.182858	−	0.983139i	$-0.441465\pi$
$223$	5.46131	0.365717	0.182858	+	0.983139i	$0.441465\pi$
$224$	0	0
$225$	−3.20521	−0.213681
$226$	0	0
$227$	3.30095	0.219092	0.109546	−	0.993982i	$-0.465060\pi$
$227$	3.30095	0.219092	0.109546	+	0.993982i	$0.465060\pi$
$228$	0	0
$229$	12.1731	0.804423	0.402211	−	0.915547i	$-0.368242\pi$
$229$	12.1731	0.804423	0.402211	+	0.915547i	$0.368242\pi$
$230$	0	0
$231$	−0.474190	−0.0311994
$232$	0	0
$233$	−10.2243	−0.669816	−0.334908	−	0.942251i	$-0.608705\pi$
$233$	−10.2243	−0.669816	−0.334908	+	0.942251i	$0.608705\pi$
$234$	0	0
$235$	−12.3830	−0.807777
$236$	0	0
$237$	−3.81389	−0.247739
$238$	0	0
$239$	25.2950	1.63620	0.818099	−	0.575077i	$-0.195028\pi$
$239$	25.2950	1.63620	0.818099	+	0.575077i	$0.195028\pi$
$240$	0	0
$241$	2.11568	0.136283	0.0681414	−	0.997676i	$-0.478293\pi$
$241$	2.11568	0.136283	0.0681414	+	0.997676i	$0.478293\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−9.07665	−0.579886
$246$	0	0
$247$	4.62156	0.294063
$248$	0	0
$249$	12.2947	0.779147
$250$	0	0
$251$	25.3314	1.59890	0.799451	−	0.600731i	$-0.205124\pi$
$251$	25.3314	1.59890	0.799451	+	0.600731i	$0.205124\pi$
$252$	0	0
$253$	6.66568	0.419068
$254$	0	0
$255$	−2.43006	−0.152176
$256$	0	0
$257$	12.8375	0.800783	0.400392	−	0.916344i	$-0.368874\pi$
$257$	12.8375	0.800783	0.400392	+	0.916344i	$0.368874\pi$
$258$	0	0
$259$	5.15486	0.320308
$260$	0	0
$261$	−3.76976	−0.233342
$262$	0	0
$263$	−5.73102	−0.353390	−0.176695	−	0.984266i	$-0.556541\pi$
$263$	−5.73102	−0.353390	−0.176695	+	0.984266i	$0.556541\pi$
$264$	0	0
$265$	−2.67940	−0.164594
$266$	0	0
$267$	5.11484	0.313023
$268$	0	0
$269$	0.795076	0.0484767	0.0242383	−	0.999706i	$-0.492284\pi$
$269$	0.795076	0.0484767	0.0242383	+	0.999706i	$0.492284\pi$
$270$	0	0
$271$	−9.56994	−0.581332	−0.290666	−	0.956825i	$-0.593877\pi$
$271$	−9.56994	−0.581332	−0.290666	+	0.956825i	$0.593877\pi$
$272$	0	0
$273$	0.474190	0.0286993
$274$	0	0
$275$	−3.20521	−0.193281
$276$	0	0
$277$	−23.3130	−1.40074	−0.700371	−	0.713779i	$-0.746982\pi$
$277$	−23.3130	−1.40074	−0.700371	+	0.713779i	$0.746982\pi$
$278$	0	0
$279$	−4.80767	−0.287828
$280$	0	0
$281$	−29.8587	−1.78122	−0.890611	−	0.454765i	$-0.849723\pi$
$281$	−29.8587	−1.78122	−0.890611	+	0.454765i	$0.849723\pi$
$282$	0	0
$283$	1.35935	0.0808048	0.0404024	−	0.999183i	$-0.487136\pi$
$283$	1.35935	0.0808048	0.0404024	+	0.999183i	$0.487136\pi$
$284$	0	0
$285$	−6.19149	−0.366752
$286$	0	0
$287$	3.89082	0.229668
$288$	0	0
$289$	−13.7098	−0.806459
$290$	0	0
$291$	−10.8709	−0.637263
$292$	0	0
$293$	−30.5441	−1.78440	−0.892202	−	0.451637i	$-0.850840\pi$
$293$	−30.5441	−1.78440	−0.892202	+	0.451637i	$0.850840\pi$
$294$	0	0
$295$	−5.55622	−0.323496
$296$	0	0
$297$	1.00000	0.0580259
$298$	0	0
$299$	−6.66568	−0.385486
$300$	0	0
$301$	2.32810	0.134189
$302$	0	0
$303$	6.10862	0.350931
$304$	0	0
$305$	12.4421	0.712433
$306$	0	0
$307$	8.08359	0.461355	0.230678	−	0.973030i	$-0.425906\pi$
$307$	8.08359	0.461355	0.230678	+	0.973030i	$0.425906\pi$
$308$	0	0
$309$	0.147368	0.00838348
$310$	0	0
$311$	−2.05784	−0.116689	−0.0583447	−	0.998296i	$-0.518582\pi$
$311$	−2.05784	−0.116689	−0.0583447	+	0.998296i	$0.518582\pi$
$312$	0	0
$313$	−9.44832	−0.534051	−0.267025	−	0.963689i	$-0.586041\pi$
$313$	−9.44832	−0.534051	−0.267025	+	0.963689i	$0.586041\pi$
$314$	0	0
$315$	−0.635271	−0.0357935
$316$	0	0
$317$	11.9142	0.669167	0.334584	−	0.942366i	$-0.391404\pi$
$317$	11.9142	0.669167	0.334584	+	0.942366i	$0.391404\pi$
$318$	0	0
$319$	−3.76976	−0.211066
$320$	0	0
$321$	9.09575	0.507675
$322$	0	0
$323$	8.38299	0.466442
$324$	0	0
$325$	3.20521	0.177793
$326$	0	0
$327$	−9.76143	−0.539808
$328$	0	0
$329$	4.38299	0.241642
$330$	0	0
$331$	−0.660301	−0.0362934	−0.0181467	−	0.999835i	$-0.505777\pi$
$331$	−0.660301	−0.0362934	−0.0181467	+	0.999835i	$0.505777\pi$
$332$	0	0
$333$	−10.8709	−0.595721
$334$	0	0
$335$	9.17934	0.501521
$336$	0	0
$337$	13.0898	0.713048	0.356524	−	0.934286i	$-0.383962\pi$
$337$	13.0898	0.713048	0.356524	+	0.934286i	$0.383962\pi$
$338$	0	0
$339$	5.89082	0.319946
$340$	0	0
$341$	−4.80767	−0.260350
$342$	0	0
$343$	6.53203	0.352696
$344$	0	0
$345$	8.93001	0.480775
$346$	0	0
$347$	10.6019	0.569140	0.284570	−	0.958655i	$-0.408149\pi$
$347$	10.6019	0.569140	0.284570	+	0.958655i	$0.408149\pi$
$348$	0	0
$349$	−31.7144	−1.69763	−0.848815	−	0.528690i	$-0.822684\pi$
$349$	−31.7144	−1.69763	−0.848815	+	0.528690i	$0.822684\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	−11.8443	−0.630409	−0.315204	−	0.949024i	$-0.602073\pi$
$353$	−11.8443	−0.630409	−0.315204	+	0.949024i	$0.602073\pi$
$354$	0	0
$355$	10.6138	0.563321
$356$	0	0
$357$	0.860127	0.0455227
$358$	0	0
$359$	−10.3709	−0.547358	−0.273679	−	0.961821i	$-0.588241\pi$
$359$	−10.3709	−0.547358	−0.273679	+	0.961821i	$0.588241\pi$
$360$	0	0
$361$	2.35879	0.124147
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	−3.81334	−0.199599
$366$	0	0
$367$	−25.7418	−1.34371	−0.671855	−	0.740683i	$-0.734502\pi$
$367$	−25.7418	−1.34371	−0.671855	+	0.740683i	$0.734502\pi$
$368$	0	0
$369$	−8.20521	−0.427146
$370$	0	0
$371$	0.948379	0.0492374
$372$	0	0
$373$	1.55790	0.0806648	0.0403324	−	0.999186i	$-0.487158\pi$
$373$	1.55790	0.0806648	0.0403324	+	0.999186i	$0.487158\pi$
$374$	0	0
$375$	−10.9925	−0.567651
$376$	0	0
$377$	3.76976	0.194153
$378$	0	0
$379$	−23.4096	−1.20247	−0.601235	−	0.799073i	$-0.705324\pi$
$379$	−23.4096	−1.20247	−0.601235	+	0.799073i	$0.705324\pi$
$380$	0	0
$381$	−0.186112	−0.00953480
$382$	0	0
$383$	−3.10168	−0.158489	−0.0792443	−	0.996855i	$-0.525251\pi$
$383$	−3.10168	−0.158489	−0.0792443	+	0.996855i	$0.525251\pi$
$384$	0	0
$385$	−0.635271	−0.0323764
$386$	0	0
$387$	−4.90964	−0.249571
$388$	0	0
$389$	−24.2297	−1.22849	−0.614247	−	0.789114i	$-0.710540\pi$
$389$	−24.2297	−1.22849	−0.614247	+	0.789114i	$0.710540\pi$
$390$	0	0
$391$	−12.0908	−0.611458
$392$	0	0
$393$	−4.60784	−0.232435
$394$	0	0
$395$	−5.10946	−0.257085
$396$	0	0
$397$	31.1721	1.56448	0.782242	−	0.622974i	$-0.214076\pi$
$397$	31.1721	1.56448	0.782242	+	0.622974i	$0.214076\pi$
$398$	0	0
$399$	2.19149	0.109712
$400$	0	0
$401$	16.5113	0.824533	0.412267	−	0.911063i	$-0.364737\pi$
$401$	16.5113	0.824533	0.412267	+	0.911063i	$0.364737\pi$
$402$	0	0
$403$	4.80767	0.239487
$404$	0	0
$405$	1.33970	0.0665702
$406$	0	0
$407$	−10.8709	−0.538850
$408$	0	0
$409$	24.0895	1.19115	0.595575	−	0.803300i	$-0.296924\pi$
$409$	24.0895	1.19115	0.595575	+	0.803300i	$0.296924\pi$
$410$	0	0
$411$	6.06322	0.299077
$412$	0	0
$413$	1.96664	0.0967720
$414$	0	0
$415$	16.4712	0.808541
$416$	0	0
$417$	−6.49329	−0.317978
$418$	0	0
$419$	11.0945	0.542000	0.271000	−	0.962579i	$-0.412646\pi$
$419$	11.0945	0.542000	0.271000	+	0.962579i	$0.412646\pi$
$420$	0	0
$421$	4.38892	0.213903	0.106952	−	0.994264i	$-0.465891\pi$
$421$	4.38892	0.213903	0.106952	+	0.994264i	$0.465891\pi$
$422$	0	0
$423$	−9.24312	−0.449416
$424$	0	0
$425$	5.81389	0.282015
$426$	0	0
$427$	−4.40391	−0.213120
$428$	0	0
$429$	−1.00000	−0.0482805
$430$	0	0
$431$	−0.653643	−0.0314849	−0.0157424	−	0.999876i	$-0.505011\pi$
$431$	−0.653643	−0.0314849	−0.0157424	+	0.999876i	$0.505011\pi$
$432$	0	0
$433$	29.4137	1.41353	0.706766	−	0.707448i	$-0.250153\pi$
$433$	29.4137	1.41353	0.706766	+	0.707448i	$0.250153\pi$
$434$	0	0
$435$	−5.05034	−0.242146
$436$	0	0
$437$	−30.8058	−1.47364
$438$	0	0
$439$	−22.3776	−1.06802	−0.534012	−	0.845477i	$-0.679316\pi$
$439$	−22.3776	−1.06802	−0.534012	+	0.845477i	$0.679316\pi$
$440$	0	0
$441$	−6.77514	−0.322626
$442$	0	0
$443$	23.4620	1.11471	0.557357	−	0.830273i	$-0.311815\pi$
$443$	23.4620	1.11471	0.557357	+	0.830273i	$0.311815\pi$
$444$	0	0
$445$	6.85235	0.324832
$446$	0	0
$447$	−17.3892	−0.822481
$448$	0	0
$449$	32.2154	1.52034	0.760170	−	0.649724i	$-0.225116\pi$
$449$	32.2154	1.52034	0.760170	+	0.649724i	$0.225116\pi$
$450$	0	0
$451$	−8.20521	−0.386368
$452$	0	0
$453$	1.56994	0.0737621
$454$	0	0
$455$	0.635271	0.0297820
$456$	0	0
$457$	−24.6718	−1.15410	−0.577049	−	0.816710i	$-0.695796\pi$
$457$	−24.6718	−1.15410	−0.577049	+	0.816710i	$0.695796\pi$
$458$	0	0
$459$	−1.81389	−0.0846650
$460$	0	0
$461$	41.4000	1.92819	0.964094	−	0.265560i	$-0.0855567\pi$
$461$	41.4000	1.92819	0.964094	+	0.265560i	$0.0855567\pi$
$462$	0	0
$463$	−27.1256	−1.26063	−0.630317	−	0.776338i	$-0.717075\pi$
$463$	−27.1256	−1.26063	−0.630317	+	0.776338i	$0.717075\pi$
$464$	0	0
$465$	−6.44083	−0.298686
$466$	0	0
$467$	−24.2219	−1.12086	−0.560428	−	0.828203i	$-0.689363\pi$
$467$	−24.2219	−1.12086	−0.560428	+	0.828203i	$0.689363\pi$
$468$	0	0
$469$	−3.24905	−0.150027
$470$	0	0
$471$	−5.67318	−0.261406
$472$	0	0
$473$	−4.90964	−0.225745
$474$	0	0
$475$	14.8131	0.679669
$476$	0	0
$477$	−2.00000	−0.0915737
$478$	0	0
$479$	−11.9362	−0.545380	−0.272690	−	0.962102i	$-0.587913\pi$
$479$	−11.9362	−0.545380	−0.272690	+	0.962102i	$0.587913\pi$
$480$	0	0
$481$	10.8709	0.495670
$482$	0	0
$483$	−3.16080	−0.143821
$484$	0	0
$485$	−14.5637	−0.661304
$486$	0	0
$487$	23.9857	1.08690	0.543449	−	0.839442i	$-0.317118\pi$
$487$	23.9857	1.08690	0.543449	+	0.839442i	$0.317118\pi$
$488$	0	0
$489$	11.4280	0.516790
$490$	0	0
$491$	−6.38892	−0.288328	−0.144164	−	0.989554i	$-0.546049\pi$
$491$	−6.38892	−0.288328	−0.144164	+	0.989554i	$0.546049\pi$
$492$	0	0
$493$	6.83793	0.307965
$494$	0	0
$495$	1.33970	0.0602150
$496$	0	0
$497$	−3.75677	−0.168514
$498$	0	0
$499$	−7.13321	−0.319327	−0.159663	−	0.987172i	$-0.551041\pi$
$499$	−7.13321	−0.319327	−0.159663	+	0.987172i	$0.551041\pi$
$500$	0	0
$501$	−17.0183	−0.760320
$502$	0	0
$503$	−19.0350	−0.848727	−0.424363	−	0.905492i	$-0.639502\pi$
$503$	−19.0350	−0.848727	−0.424363	+	0.905492i	$0.639502\pi$
$504$	0	0
$505$	8.18372	0.364171
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	−23.2455	−1.03034	−0.515170	−	0.857088i	$-0.672271\pi$
$509$	−23.2455	−1.03034	−0.515170	+	0.857088i	$0.672271\pi$
$510$	0	0
$511$	1.34974	0.0597090
$512$	0	0
$513$	−4.62156	−0.204047
$514$	0	0
$515$	0.197429	0.00869975
$516$	0	0
$517$	−9.24312	−0.406512
$518$	0	0
$519$	−19.8580	−0.871670
$520$	0	0
$521$	0.559456	0.0245102	0.0122551	−	0.999925i	$-0.496099\pi$
$521$	0.559456	0.0245102	0.0122551	+	0.999925i	$0.496099\pi$
$522$	0	0
$523$	41.0952	1.79697	0.898484	−	0.439007i	$-0.144670\pi$
$523$	41.0952	1.79697	0.898484	+	0.439007i	$0.144670\pi$
$524$	0	0
$525$	1.51988	0.0663328
$526$	0	0
$527$	8.72057	0.379874
$528$	0	0
$529$	21.4313	0.931797
$530$	0	0
$531$	−4.14737	−0.179980
$532$	0	0
$533$	8.20521	0.355407
$534$	0	0
$535$	12.1856	0.526828
$536$	0	0
$537$	−3.41663	−0.147439
$538$	0	0
$539$	−6.77514	−0.291826
$540$	0	0
$541$	39.2038	1.68550	0.842752	−	0.538302i	$-0.180934\pi$
$541$	39.2038	1.68550	0.842752	+	0.538302i	$0.180934\pi$
$542$	0	0
$543$	22.0562	0.946521
$544$	0	0
$545$	−13.0774	−0.560173
$546$	0	0
$547$	24.9730	1.06777	0.533884	−	0.845557i	$-0.320732\pi$
$547$	24.9730	1.06777	0.533884	+	0.845557i	$0.320732\pi$
$548$	0	0
$549$	9.28724	0.396370
$550$	0	0
$551$	17.4222	0.742209
$552$	0	0
$553$	1.80851	0.0769055
$554$	0	0
$555$	−14.5637	−0.618195
$556$	0	0
$557$	−10.8147	−0.458235	−0.229117	−	0.973399i	$-0.573584\pi$
$557$	−10.8147	−0.458235	−0.229117	+	0.973399i	$0.573584\pi$
$558$	0	0
$559$	4.90964	0.207655
$560$	0	0
$561$	−1.81389	−0.0765824
$562$	0	0
$563$	1.39809	0.0589225	0.0294612	−	0.999566i	$-0.490621\pi$
$563$	1.39809	0.0589225	0.0294612	+	0.999566i	$0.490621\pi$
$564$	0	0
$565$	7.89193	0.332016
$566$	0	0
$567$	−0.474190	−0.0199141
$568$	0	0
$569$	−43.1560	−1.80919	−0.904597	−	0.426267i	$-0.859828\pi$
$569$	−43.1560	−1.80919	−0.904597	+	0.426267i	$0.859828\pi$
$570$	0	0
$571$	−22.7563	−0.952323	−0.476161	−	0.879358i	$-0.657972\pi$
$571$	−22.7563	−0.952323	−0.476161	+	0.879358i	$0.657972\pi$
$572$	0	0
$573$	17.0183	0.710948
$574$	0	0
$575$	−21.3649	−0.890978
$576$	0	0
$577$	−20.5745	−0.856527	−0.428263	−	0.903654i	$-0.640874\pi$
$577$	−20.5745	−0.856527	−0.428263	+	0.903654i	$0.640874\pi$
$578$	0	0
$579$	14.9163	0.619900
$580$	0	0
$581$	−5.83004	−0.241871
$582$	0	0
$583$	−2.00000	−0.0828315
$584$	0	0
$585$	−1.33970	−0.0553897
$586$	0	0
$587$	14.3114	0.590696	0.295348	−	0.955390i	$-0.404564\pi$
$587$	14.3114	0.590696	0.295348	+	0.955390i	$0.404564\pi$
$588$	0	0
$589$	22.2189	0.915515
$590$	0	0
$591$	−20.3526	−0.837193
$592$	0	0
$593$	−17.7232	−0.727806	−0.363903	−	0.931437i	$-0.618556\pi$
$593$	−17.7232	−0.727806	−0.363903	+	0.931437i	$0.618556\pi$
$594$	0	0
$595$	1.15231	0.0472401
$596$	0	0
$597$	6.87683	0.281450
$598$	0	0
$599$	25.3130	1.03426	0.517130	−	0.855907i	$-0.327000\pi$
$599$	25.3130	1.03426	0.517130	+	0.855907i	$0.327000\pi$
$600$	0	0
$601$	6.41042	0.261486	0.130743	−	0.991416i	$-0.458264\pi$
$601$	6.41042	0.261486	0.130743	+	0.991416i	$0.458264\pi$
$602$	0	0
$603$	6.85180	0.279027
$604$	0	0
$605$	1.33970	0.0544665
$606$	0	0
$607$	−32.2625	−1.30949	−0.654747	−	0.755848i	$-0.727225\pi$
$607$	−32.2625	−1.30949	−0.654747	+	0.755848i	$0.727225\pi$
$608$	0	0
$609$	1.78758	0.0724365
$610$	0	0
$611$	9.24312	0.373936
$612$	0	0
$613$	15.6582	0.632428	0.316214	−	0.948688i	$-0.397588\pi$
$613$	15.6582	0.632428	0.316214	+	0.948688i	$0.397588\pi$
$614$	0	0
$615$	−10.9925	−0.443261
$616$	0	0
$617$	42.8841	1.72645	0.863223	−	0.504822i	$-0.168442\pi$
$617$	42.8841	1.72645	0.863223	+	0.504822i	$0.168442\pi$
$618$	0	0
$619$	6.46881	0.260003	0.130002	−	0.991514i	$-0.458502\pi$
$619$	6.46881	0.260003	0.130002	+	0.991514i	$0.458502\pi$
$620$	0	0
$621$	6.66568	0.267485
$622$	0	0
$623$	−2.42540	−0.0971718
$624$	0	0
$625$	1.29939	0.0519758
$626$	0	0
$627$	−4.62156	−0.184567
$628$	0	0
$629$	19.7186	0.786231
$630$	0	0
$631$	−13.3230	−0.530380	−0.265190	−	0.964196i	$-0.585435\pi$
$631$	−13.3230	−0.530380	−0.265190	+	0.964196i	$0.585435\pi$
$632$	0	0
$633$	3.23773	0.128688
$634$	0	0
$635$	−0.249334	−0.00989451
$636$	0	0
$637$	6.77514	0.268441
$638$	0	0
$639$	7.92251	0.313410
$640$	0	0
$641$	6.93001	0.273719	0.136859	−	0.990590i	$-0.456299\pi$
$641$	6.93001	0.273719	0.136859	+	0.990590i	$0.456299\pi$
$642$	0	0
$643$	−25.8850	−1.02081	−0.510403	−	0.859935i	$-0.670504\pi$
$643$	−25.8850	−1.02081	−0.510403	+	0.859935i	$0.670504\pi$
$644$	0	0
$645$	−6.57743	−0.258986
$646$	0	0
$647$	−29.8575	−1.17382	−0.586909	−	0.809653i	$-0.699655\pi$
$647$	−29.8575	−1.17382	−0.586909	+	0.809653i	$0.699655\pi$
$648$	0	0
$649$	−4.14737	−0.162798
$650$	0	0
$651$	2.27975	0.0893503
$652$	0	0
$653$	2.43305	0.0952126	0.0476063	−	0.998866i	$-0.484841\pi$
$653$	2.43305	0.0952126	0.0476063	+	0.998866i	$0.484841\pi$
$654$	0	0
$655$	−6.17312	−0.241204
$656$	0	0
$657$	−2.84641	−0.111049
$658$	0	0
$659$	−34.3205	−1.33694	−0.668468	−	0.743741i	$-0.733050\pi$
$659$	−34.3205	−1.33694	−0.668468	+	0.743741i	$0.733050\pi$
$660$	0	0
$661$	35.3696	1.37572	0.687858	−	0.725845i	$-0.258551\pi$
$661$	35.3696	1.37572	0.687858	+	0.725845i	$0.258551\pi$
$662$	0	0
$663$	1.81389	0.0704456
$664$	0	0
$665$	2.93594	0.113851
$666$	0	0
$667$	−25.1280	−0.972962
$668$	0	0
$669$	5.46131	0.211147
$670$	0	0
$671$	9.28724	0.358530
$672$	0	0
$673$	21.6411	0.834203	0.417101	−	0.908860i	$-0.363046\pi$
$673$	21.6411	0.834203	0.417101	+	0.908860i	$0.363046\pi$
$674$	0	0
$675$	−3.20521	−0.123369
$676$	0	0
$677$	19.7240	0.758053	0.379027	−	0.925386i	$-0.376259\pi$
$677$	19.7240	0.758053	0.379027	+	0.925386i	$0.376259\pi$
$678$	0	0
$679$	5.15486	0.197825
$680$	0	0
$681$	3.30095	0.126493
$682$	0	0
$683$	−38.5943	−1.47677	−0.738385	−	0.674379i	$-0.764411\pi$
$683$	−38.5943	−1.47677	−0.738385	+	0.674379i	$0.764411\pi$
$684$	0	0
$685$	8.12289	0.310360
$686$	0	0
$687$	12.1731	0.464434
$688$	0	0
$689$	2.00000	0.0761939
$690$	0	0
$691$	4.92335	0.187293	0.0936465	−	0.995606i	$-0.470148\pi$
$691$	4.92335	0.187293	0.0936465	+	0.995606i	$0.470148\pi$
$692$	0	0
$693$	−0.474190	−0.0180130
$694$	0	0
$695$	−8.69905	−0.329974
$696$	0	0
$697$	14.8833	0.563746
$698$	0	0
$699$	−10.2243	−0.386719
$700$	0	0
$701$	13.8563	0.523347	0.261673	−	0.965156i	$-0.415726\pi$
$701$	13.8563	0.523347	0.261673	+	0.965156i	$0.415726\pi$
$702$	0	0
$703$	50.2405	1.89485
$704$	0	0
$705$	−12.3830	−0.466370
$706$	0	0
$707$	−2.89665	−0.108940
$708$	0	0
$709$	43.1973	1.62231	0.811155	−	0.584832i	$-0.198839\pi$
$709$	43.1973	1.62231	0.811155	+	0.584832i	$0.198839\pi$
$710$	0	0
$711$	−3.81389	−0.143032
$712$	0	0
$713$	−32.0464	−1.20015
$714$	0	0
$715$	−1.33970	−0.0501019
$716$	0	0
$717$	25.2950	0.944660
$718$	0	0
$719$	−11.6320	−0.433802	−0.216901	−	0.976194i	$-0.569595\pi$
$719$	−11.6320	−0.433802	−0.216901	+	0.976194i	$0.569595\pi$
$720$	0	0
$721$	−0.0698804	−0.00260248
$722$	0	0
$723$	2.11568	0.0786830
$724$	0	0
$725$	12.0829	0.448747
$726$	0	0
$727$	−51.8558	−1.92322	−0.961612	−	0.274411i	$-0.911517\pi$
$727$	−51.8558	−1.92322	−0.961612	+	0.274411i	$0.911517\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	8.90553	0.329383
$732$	0	0
$733$	−15.0713	−0.556670	−0.278335	−	0.960484i	$-0.589783\pi$
$733$	−15.0713	−0.556670	−0.278335	+	0.960484i	$0.589783\pi$
$734$	0	0
$735$	−9.07665	−0.334797
$736$	0	0
$737$	6.85180	0.252389
$738$	0	0
$739$	−20.4533	−0.752386	−0.376193	−	0.926541i	$-0.622767\pi$
$739$	−20.4533	−0.752386	−0.376193	+	0.926541i	$0.622767\pi$
$740$	0	0
$741$	4.62156	0.169777
$742$	0	0
$743$	−15.2980	−0.561229	−0.280615	−	0.959820i	$-0.590538\pi$
$743$	−15.2980	−0.561229	−0.280615	+	0.959820i	$0.590538\pi$
$744$	0	0
$745$	−23.2963	−0.853510
$746$	0	0
$747$	12.2947	0.449841
$748$	0	0
$749$	−4.31311	−0.157598
$750$	0	0
$751$	−38.8757	−1.41860	−0.709298	−	0.704909i	$-0.750988\pi$
$751$	−38.8757	−1.41860	−0.709298	+	0.704909i	$0.750988\pi$
$752$	0	0
$753$	25.3314	0.923126
$754$	0	0
$755$	2.10324	0.0765448
$756$	0	0
$757$	−10.0562	−0.365498	−0.182749	−	0.983160i	$-0.558500\pi$
$757$	−10.0562	−0.365498	−0.182749	+	0.983160i	$0.558500\pi$
$758$	0	0
$759$	6.66568	0.241949
$760$	0	0
$761$	−29.7538	−1.07858	−0.539288	−	0.842122i	$-0.681306\pi$
$761$	−29.7538	−1.07858	−0.539288	+	0.842122i	$0.681306\pi$
$762$	0	0
$763$	4.62877	0.167573
$764$	0	0
$765$	−2.43006	−0.0878591
$766$	0	0
$767$	4.14737	0.149753
$768$	0	0
$769$	13.0941	0.472184	0.236092	−	0.971731i	$-0.424133\pi$
$769$	13.0941	0.472184	0.236092	+	0.971731i	$0.424133\pi$
$770$	0	0
$771$	12.8375	0.462332
$772$	0	0
$773$	−29.2438	−1.05183	−0.525914	−	0.850538i	$-0.676277\pi$
$773$	−29.2438	−1.05183	−0.525914	+	0.850538i	$0.676277\pi$
$774$	0	0
$775$	15.4096	0.553528
$776$	0	0
$777$	5.15486	0.184930
$778$	0	0
$779$	37.9208	1.35866
$780$	0	0
$781$	7.92251	0.283490
$782$	0	0
$783$	−3.76976	−0.134720
$784$	0	0
$785$	−7.60035	−0.271268
$786$	0	0
$787$	−44.8114	−1.59735	−0.798677	−	0.601761i	$-0.794466\pi$
$787$	−44.8114	−1.59735	−0.798677	+	0.601761i	$0.794466\pi$
$788$	0	0
$789$	−5.73102	−0.204030
$790$	0	0
$791$	−2.79337	−0.0993207
$792$	0	0
$793$	−9.28724	−0.329800
$794$	0	0
$795$	−2.67940	−0.0950284
$796$	0	0
$797$	−46.1505	−1.63474	−0.817368	−	0.576116i	$-0.804568\pi$
$797$	−46.1505	−1.63474	−0.817368	+	0.576116i	$0.804568\pi$
$798$	0	0
$799$	16.7660	0.593138
$800$	0	0
$801$	5.11484	0.180724
$802$	0	0
$803$	−2.84641	−0.100448
$804$	0	0
$805$	−4.23452	−0.149247
$806$	0	0
$807$	0.795076	0.0279880
$808$	0	0
$809$	39.1968	1.37808	0.689042	−	0.724721i	$-0.258031\pi$
$809$	39.1968	1.37808	0.689042	+	0.724721i	$0.258031\pi$
$810$	0	0
$811$	52.2482	1.83468	0.917342	−	0.398101i	$-0.130331\pi$
$811$	52.2482	1.83468	0.917342	+	0.398101i	$0.130331\pi$
$812$	0	0
$813$	−9.56994	−0.335632
$814$	0	0
$815$	15.3100	0.536286
$816$	0	0
$817$	22.6902	0.793828
$818$	0	0
$819$	0.474190	0.0165695
$820$	0	0
$821$	−27.7620	−0.968900	−0.484450	−	0.874819i	$-0.660980\pi$
$821$	−27.7620	−0.968900	−0.484450	+	0.874819i	$0.660980\pi$
$822$	0	0
$823$	44.1355	1.53847	0.769233	−	0.638968i	$-0.220638\pi$
$823$	44.1355	1.53847	0.769233	+	0.638968i	$0.220638\pi$
$824$	0	0
$825$	−3.20521	−0.111591
$826$	0	0
$827$	10.4015	0.361697	0.180848	−	0.983511i	$-0.442116\pi$
$827$	10.4015	0.361697	0.180848	+	0.983511i	$0.442116\pi$
$828$	0	0
$829$	2.43784	0.0846698	0.0423349	−	0.999103i	$-0.486520\pi$
$829$	2.43784	0.0846698	0.0423349	+	0.999103i	$0.486520\pi$
$830$	0	0
$831$	−23.3130	−0.808719
$832$	0	0
$833$	12.2894	0.425801
$834$	0	0
$835$	−22.7993	−0.789004
$836$	0	0
$837$	−4.80767	−0.166177
$838$	0	0
$839$	−0.243228	−0.00839715	−0.00419858	−	0.999991i	$-0.501336\pi$
$839$	−0.243228	−0.00839715	−0.00419858	+	0.999991i	$0.501336\pi$
$840$	0	0
$841$	−14.7889	−0.509962
$842$	0	0
$843$	−29.8587	−1.02839
$844$	0	0
$845$	1.33970	0.0460870
$846$	0	0
$847$	−0.474190	−0.0162933
$848$	0	0
$849$	1.35935	0.0466527
$850$	0	0
$851$	−72.4619	−2.48396
$852$	0	0
$853$	−24.4588	−0.837453	−0.418727	−	0.908112i	$-0.637523\pi$
$853$	−24.4588	−0.837453	−0.418727	+	0.908112i	$0.637523\pi$
$854$	0	0
$855$	−6.19149	−0.211745
$856$	0	0
$857$	−18.2469	−0.623304	−0.311652	−	0.950196i	$-0.600882\pi$
$857$	−18.2469	−0.623304	−0.311652	+	0.950196i	$0.600882\pi$
$858$	0	0
$859$	47.1113	1.60742	0.803709	−	0.595022i	$-0.202857\pi$
$859$	47.1113	1.60742	0.803709	+	0.595022i	$0.202857\pi$
$860$	0	0
$861$	3.89082	0.132599
$862$	0	0
$863$	−15.6793	−0.533729	−0.266865	−	0.963734i	$-0.585988\pi$
$863$	−15.6793	−0.533729	−0.266865	+	0.963734i	$0.585988\pi$
$864$	0	0
$865$	−26.6038	−0.904555
$866$	0	0
$867$	−13.7098	−0.465610
$868$	0	0
$869$	−3.81389	−0.129377
$870$	0	0
$871$	−6.85180	−0.232164
$872$	0	0
$873$	−10.8709	−0.367924
$874$	0	0
$875$	5.21253	0.176216
$876$	0	0
$877$	34.7934	1.17489	0.587445	−	0.809264i	$-0.300134\pi$
$877$	34.7934	1.17489	0.587445	+	0.809264i	$0.300134\pi$
$878$	0	0
$879$	−30.5441	−1.03023
$880$	0	0
$881$	−33.3006	−1.12192	−0.560962	−	0.827841i	$-0.689569\pi$
$881$	−33.3006	−1.12192	−0.560962	+	0.827841i	$0.689569\pi$
$882$	0	0
$883$	−7.30817	−0.245939	−0.122970	−	0.992410i	$-0.539242\pi$
$883$	−7.30817	−0.245939	−0.122970	+	0.992410i	$0.539242\pi$
$884$	0	0
$885$	−5.55622	−0.186770
$886$	0	0
$887$	12.3174	0.413577	0.206788	−	0.978386i	$-0.433699\pi$
$887$	12.3174	0.413577	0.206788	+	0.978386i	$0.433699\pi$
$888$	0	0
$889$	0.0882523	0.00295989
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	42.7176	1.42949
$894$	0	0
$895$	−4.57726	−0.153001
$896$	0	0
$897$	−6.66568	−0.222561
$898$	0	0
$899$	18.1238	0.604462
$900$	0	0
$901$	3.62778	0.120859
$902$	0	0
$903$	2.32810	0.0774742
$904$	0	0
$905$	29.5486	0.982229
$906$	0	0
$907$	12.5895	0.418027	0.209013	−	0.977913i	$-0.432975\pi$
$907$	12.5895	0.418027	0.209013	+	0.977913i	$0.432975\pi$
$908$	0	0
$909$	6.10862	0.202610
$910$	0	0
$911$	−34.1641	−1.13191	−0.565953	−	0.824437i	$-0.691492\pi$
$911$	−34.1641	−1.13191	−0.565953	+	0.824437i	$0.691492\pi$
$912$	0	0
$913$	12.2947	0.406896
$914$	0	0
$915$	12.4421	0.411323
$916$	0	0
$917$	2.18499	0.0721548
$918$	0	0
$919$	−2.85474	−0.0941693	−0.0470847	−	0.998891i	$-0.514993\pi$
$919$	−2.85474	−0.0941693	−0.0470847	+	0.998891i	$0.514993\pi$
$920$	0	0
$921$	8.08359	0.266363
$922$	0	0
$923$	−7.92251	−0.260773
$924$	0	0
$925$	34.8435	1.14565
$926$	0	0
$927$	0.147368	0.00484020
$928$	0	0
$929$	−1.66357	−0.0545800	−0.0272900	−	0.999628i	$-0.508688\pi$
$929$	−1.66357	−0.0545800	−0.0272900	+	0.999628i	$0.508688\pi$
$930$	0	0
$931$	31.3117	1.02620
$932$	0	0
$933$	−2.05784	−0.0673706
$934$	0	0
$935$	−2.43006	−0.0794716
$936$	0	0
$937$	38.8698	1.26982	0.634910	−	0.772586i	$-0.281037\pi$
$937$	38.8698	1.26982	0.634910	+	0.772586i	$0.281037\pi$
$938$	0	0
$939$	−9.44832	−0.308334
$940$	0	0
$941$	−32.1868	−1.04926	−0.524630	−	0.851330i	$-0.675796\pi$
$941$	−32.1868	−1.04926	−0.524630	+	0.851330i	$0.675796\pi$
$942$	0	0
$943$	−54.6933	−1.78106
$944$	0	0
$945$	−0.635271	−0.0206654
$946$	0	0
$947$	17.7966	0.578313	0.289156	−	0.957282i	$-0.406625\pi$
$947$	17.7966	0.578313	0.289156	+	0.957282i	$0.406625\pi$
$948$	0	0
$949$	2.84641	0.0923985
$950$	0	0
$951$	11.9142	0.386344
$952$	0	0
$953$	−33.7338	−1.09275	−0.546373	−	0.837542i	$-0.683992\pi$
$953$	−33.7338	−1.09275	−0.546373	+	0.837542i	$0.683992\pi$
$954$	0	0
$955$	22.7993	0.737769
$956$	0	0
$957$	−3.76976	−0.121859
$958$	0	0
$959$	−2.87512	−0.0928423
$960$	0	0
$961$	−7.88631	−0.254397
$962$	0	0
$963$	9.09575	0.293106
$964$	0	0
$965$	19.9833	0.643286
$966$	0	0
$967$	20.1676	0.648546	0.324273	−	0.945964i	$-0.394880\pi$
$967$	20.1676	0.648546	0.324273	+	0.945964i	$0.394880\pi$
$968$	0	0
$969$	8.38299	0.269300
$970$	0	0
$971$	20.8781	0.670010	0.335005	−	0.942216i	$-0.391262\pi$
$971$	20.8781	0.670010	0.335005	+	0.942216i	$0.391262\pi$
$972$	0	0
$973$	3.07905	0.0987097
$974$	0	0
$975$	3.20521	0.102649
$976$	0	0
$977$	−4.99916	−0.159937	−0.0799687	−	0.996797i	$-0.525482\pi$
$977$	−4.99916	−0.159937	−0.0799687	+	0.996797i	$0.525482\pi$
$978$	0	0
$979$	5.11484	0.163471
$980$	0	0
$981$	−9.76143	−0.311659
$982$	0	0
$983$	−51.5718	−1.64489	−0.822443	−	0.568848i	$-0.807389\pi$
$983$	−51.5718	−1.64489	−0.822443	+	0.568848i	$0.807389\pi$
$984$	0	0
$985$	−27.2663	−0.868777
$986$	0	0
$987$	4.38299	0.139512
$988$	0	0
$989$	−32.7261	−1.04063
$990$	0	0
$991$	35.3636	1.12336	0.561681	−	0.827354i	$-0.310155\pi$
$991$	35.3636	1.12336	0.561681	+	0.827354i	$0.310155\pi$
$992$	0	0
$993$	−0.660301	−0.0209540
$994$	0	0
$995$	9.21287	0.292068
$996$	0	0
$997$	52.4512	1.66115	0.830573	−	0.556910i	$-0.188013\pi$
$997$	52.4512	1.66115	0.830573	+	0.556910i	$0.188013\pi$
$998$	0	0
$999$	−10.8709	−0.343940

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6864.2.a.by.1.3		4
4.3	odd	2		3432.2.a.q.1.3	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3432.2.a.q.1.3	✓	4	4.3	odd	2
6864.2.a.by.1.3		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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.33970 q^{5} -0.474190 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.33970 q^{5} -0.474190 q^{7} +1.00000 q^{9} +1.00000 q^{11} -1.00000 q^{13} +1.33970 q^{15} -1.81389 q^{17} -4.62156 q^{19} -0.474190 q^{21} +6.66568 q^{23} -3.20521 q^{25} +1.00000 q^{27} -3.76976 q^{29} -4.80767 q^{31} +1.00000 q^{33} -0.635271 q^{35} -10.8709 q^{37} -1.00000 q^{39} -8.20521 q^{41} -4.90964 q^{43} +1.33970 q^{45} -9.24312 q^{47} -6.77514 q^{49} -1.81389 q^{51} -2.00000 q^{53} +1.33970 q^{55} -4.62156 q^{57} -4.14737 q^{59} +9.28724 q^{61} -0.474190 q^{63} -1.33970 q^{65} +6.85180 q^{67} +6.66568 q^{69} +7.92251 q^{71} -2.84641 q^{73} -3.20521 q^{75} -0.474190 q^{77} -3.81389 q^{79} +1.00000 q^{81} +12.2947 q^{83} -2.43006 q^{85} -3.76976 q^{87} +5.11484 q^{89} +0.474190 q^{91} -4.80767 q^{93} -6.19149 q^{95} -10.8709 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} - 3 q^{5} - 3 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 - 3 * q^5 - 3 * q^7 + 4 * q^9	\( 4 q + 4 q^{3} - 3 q^{5} - 3 q^{7} + 4 q^{9} + 4 q^{11} - 4 q^{13} - 3 q^{15} - 2 q^{19} - 3 q^{21} - 3 q^{23} + 5 q^{25} + 4 q^{27} - 21 q^{29} - 10 q^{31} + 4 q^{33} + q^{35} + 4 q^{37} - 4 q^{39} - 15 q^{41} + 3 q^{43} - 3 q^{45} - 4 q^{47} + 5 q^{49} - 8 q^{53} - 3 q^{55} - 2 q^{57} + q^{59} - 9 q^{61} - 3 q^{63} + 3 q^{65} + 5 q^{67} - 3 q^{69} - 18 q^{71} - 27 q^{73} + 5 q^{75} - 3 q^{77} - 8 q^{79} + 4 q^{81} + 14 q^{83} - 24 q^{85} - 21 q^{87} - 20 q^{89} + 3 q^{91} - 10 q^{93} + 6 q^{95} + 4 q^{97} + 4 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 - 3 * q^5 - 3 * q^7 + 4 * q^9 + 4 * q^11 - 4 * q^13 - 3 * q^15 - 2 * q^19 - 3 * q^21 - 3 * q^23 + 5 * q^25 + 4 * q^27 - 21 * q^29 - 10 * q^31 + 4 * q^33 + q^35 + 4 * q^37 - 4 * q^39 - 15 * q^41 + 3 * q^43 - 3 * q^45 - 4 * q^47 + 5 * q^49 - 8 * q^53 - 3 * q^55 - 2 * q^57 + q^59 - 9 * q^61 - 3 * q^63 + 3 * q^65 + 5 * q^67 - 3 * q^69 - 18 * q^71 - 27 * q^73 + 5 * q^75 - 3 * q^77 - 8 * q^79 + 4 * q^81 + 14 * q^83 - 24 * q^85 - 21 * q^87 - 20 * q^89 + 3 * q^91 - 10 * q^93 + 6 * q^95 + 4 * q^97 + 4 * q^99

Introduction

L-functions

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