LMFDB - Embedded newform 6864.2.a.bw.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:	4.4.70164.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 10x^{2} + 10x - 2 $ x^4 - x^3 - 10x^2 + 10x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
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			$3.20666$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +1.53005 q^{5} +2.90635 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.53005 q^{5} +2.90635 q^{7} +1.00000 q^{9} -1.00000 q^{11} -1.00000 q^{13} -1.53005 q^{15} +1.37630 q^{17} -6.41331 q^{19} -2.90635 q^{21} -2.90635 q^{23} -2.65894 q^{25} -1.00000 q^{27} +4.88326 q^{29} +1.78961 q^{31} +1.00000 q^{33} +4.44686 q^{35} +4.75259 q^{37} +1.00000 q^{39} -10.6729 q^{41} +0.130667 q^{43} +1.53005 q^{45} -8.00000 q^{47} +1.44686 q^{49} -1.37630 q^{51} +6.00000 q^{53} -1.53005 q^{55} +6.41331 q^{57} -9.31966 q^{59} -7.31966 q^{61} +2.90635 q^{63} -1.53005 q^{65} -2.59015 q^{67} +2.90635 q^{69} -7.06010 q^{71} -13.4716 q^{73} +2.65894 q^{75} -2.90635 q^{77} -7.37630 q^{79} +1.00000 q^{81} -11.0139 q^{83} +2.10581 q^{85} -4.88326 q^{87} -3.22432 q^{89} -2.90635 q^{91} -1.78961 q^{93} -9.81270 q^{95} +4.75259 q^{97} -1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} + q^{5} - q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 + q^5 - q^7 + 4 * q^9	$ 4 q - 4 q^{3} + q^{5} - q^{7} + 4 q^{9} - 4 q^{11} - 4 q^{13} - q^{15} - 2 q^{17} - 2 q^{19} + q^{21} + q^{23} + 17 q^{25} - 4 q^{27} + q^{29} - 24 q^{31} + 4 q^{33} + 17 q^{35} + 4 q^{37} + 4 q^{39} + 7 q^{41} - 3 q^{43} + q^{45} - 32 q^{47} + 5 q^{49} + 2 q^{51} + 24 q^{53} - q^{55} + 2 q^{57} - q^{59} + 7 q^{61} - q^{63} - q^{65} + 5 q^{67} - q^{69} - 18 q^{71} - q^{73} - 17 q^{75} + q^{77} - 22 q^{79} + 4 q^{81} - 22 q^{83} - 20 q^{85} - q^{87} - 22 q^{89} + q^{91} + 24 q^{93} - 14 q^{95} + 4 q^{97} - 4 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 + q^5 - q^7 + 4 * q^9 - 4 * q^11 - 4 * q^13 - q^15 - 2 * q^17 - 2 * q^19 + q^21 + q^23 + 17 * q^25 - 4 * q^27 + q^29 - 24 * q^31 + 4 * q^33 + 17 * q^35 + 4 * q^37 + 4 * q^39 + 7 * q^41 - 3 * q^43 + q^45 - 32 * q^47 + 5 * q^49 + 2 * q^51 + 24 * q^53 - q^55 + 2 * q^57 - q^59 + 7 * q^61 - q^63 - q^65 + 5 * q^67 - q^69 - 18 * q^71 - q^73 - 17 * q^75 + q^77 - 22 * q^79 + 4 * q^81 - 22 * q^83 - 20 * q^85 - q^87 - 22 * q^89 + q^91 + 24 * q^93 - 14 * 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
$5$	1.53005	0.684260	0.342130	−	0.939653i	$-0.388852\pi$
$5$	1.53005	0.684260	0.342130	+	0.939653i	$0.388852\pi$
$6$	0	0
$7$	2.90635	1.09850	0.549248	−	0.835659i	$-0.314914\pi$
$7$	2.90635	1.09850	0.549248	+	0.835659i	$0.314914\pi$
$8$	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
$15$	−1.53005	−0.395058
$16$	0	0
$17$	1.37630	0.333801	0.166901	−	0.985974i	$-0.446624\pi$
$17$	1.37630	0.333801	0.166901	+	0.985974i	$0.446624\pi$
$18$	0	0
$19$	−6.41331	−1.47131	−0.735657	−	0.677354i	$-0.763127\pi$
$19$	−6.41331	−1.47131	−0.735657	+	0.677354i	$0.763127\pi$
$20$	0	0
$21$	−2.90635	−0.634217
$22$	0	0
$23$	−2.90635	−0.606016	−0.303008	−	0.952988i	$-0.597991\pi$
$23$	−2.90635	−0.606016	−0.303008	+	0.952988i	$0.597991\pi$
$24$	0	0
$25$	−2.65894	−0.531789
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	4.88326	0.906799	0.453399	−	0.891307i	$-0.350211\pi$
$29$	4.88326	0.906799	0.453399	+	0.891307i	$0.350211\pi$
$30$	0	0
$31$	1.78961	0.321424	0.160712	−	0.987001i	$-0.448621\pi$
$31$	1.78961	0.321424	0.160712	+	0.987001i	$0.448621\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	4.44686	0.751657
$36$	0	0
$37$	4.75259	0.781322	0.390661	−	0.920535i	$-0.372247\pi$
$37$	4.75259	0.781322	0.390661	+	0.920535i	$0.372247\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	−10.6729	−1.66682	−0.833411	−	0.552653i	$-0.813615\pi$
$41$	−10.6729	−1.66682	−0.833411	+	0.552653i	$0.813615\pi$
$42$	0	0
$43$	0.130667	0.0199265	0.00996327	−	0.999950i	$-0.496829\pi$
$43$	0.130667	0.0199265	0.00996327	+	0.999950i	$0.496829\pi$
$44$	0	0
$45$	1.53005	0.228087
$46$	0	0
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	0	0
$49$	1.44686	0.206695
$50$	0	0
$51$	−1.37630	−0.192720
$52$	0	0
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	−1.53005	−0.206312
$56$	0	0
$57$	6.41331	0.849464
$58$	0	0
$59$	−9.31966	−1.21332	−0.606658	−	0.794963i	$-0.707490\pi$
$59$	−9.31966	−1.21332	−0.606658	+	0.794963i	$0.707490\pi$
$60$	0	0
$61$	−7.31966	−0.937187	−0.468593	−	0.883414i	$-0.655239\pi$
$61$	−7.31966	−0.937187	−0.468593	+	0.883414i	$0.655239\pi$
$62$	0	0
$63$	2.90635	0.366166
$64$	0	0
$65$	−1.53005	−0.189779
$66$	0	0
$67$	−2.59015	−0.316438	−0.158219	−	0.987404i	$-0.550575\pi$
$67$	−2.59015	−0.316438	−0.158219	+	0.987404i	$0.550575\pi$
$68$	0	0
$69$	2.90635	0.349883
$70$	0	0
$71$	−7.06010	−0.837880	−0.418940	−	0.908014i	$-0.637598\pi$
$71$	−7.06010	−0.837880	−0.418940	+	0.908014i	$0.637598\pi$
$72$	0	0
$73$	−13.4716	−1.57674	−0.788368	−	0.615204i	$-0.789074\pi$
$73$	−13.4716	−1.57674	−0.788368	+	0.615204i	$0.789074\pi$
$74$	0	0
$75$	2.65894	0.307028
$76$	0	0
$77$	−2.90635	−0.331209
$78$	0	0
$79$	−7.37630	−0.829898	−0.414949	−	0.909845i	$-0.636201\pi$
$79$	−7.37630	−0.829898	−0.414949	+	0.909845i	$0.636201\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−11.0139	−1.20894	−0.604468	−	0.796630i	$-0.706614\pi$
$83$	−11.0139	−1.20894	−0.604468	+	0.796630i	$0.706614\pi$
$84$	0	0
$85$	2.10581	0.228407
$86$	0	0
$87$	−4.88326	−0.523541
$88$	0	0
$89$	−3.22432	−0.341777	−0.170889	−	0.985290i	$-0.554664\pi$
$89$	−3.22432	−0.341777	−0.170889	+	0.985290i	$0.554664\pi$
$90$	0	0
$91$	−2.90635	−0.304668
$92$	0	0
$93$	−1.78961	−0.185574
$94$	0	0
$95$	−9.81270	−1.00676
$96$	0	0
$97$	4.75259	0.482553	0.241276	−	0.970456i	$-0.422434\pi$
$97$	4.75259	0.482553	0.241276	+	0.970456i	$0.422434\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	0	0
$101$	15.1428	1.50677	0.753384	−	0.657581i	$-0.228420\pi$
$101$	15.1428	1.50677	0.753384	+	0.657581i	$0.228420\pi$
$102$	0	0
$103$	−18.8249	−1.85487	−0.927434	−	0.373987i	$-0.877990\pi$
$103$	−18.8249	−1.85487	−0.927434	+	0.373987i	$0.877990\pi$
$104$	0	0
$105$	−4.44686	−0.433969
$106$	0	0
$107$	−5.62717	−0.543999	−0.271999	−	0.962297i	$-0.587685\pi$
$107$	−5.62717	−0.543999	−0.271999	+	0.962297i	$0.587685\pi$
$108$	0	0
$109$	13.9185	1.33315	0.666575	−	0.745438i	$-0.267759\pi$
$109$	13.9185	1.33315	0.666575	+	0.745438i	$0.267759\pi$
$110$	0	0
$111$	−4.75259	−0.451096
$112$	0	0
$113$	8.25956	0.776994	0.388497	−	0.921450i	$-0.372994\pi$
$113$	8.25956	0.776994	0.388497	+	0.921450i	$0.372994\pi$
$114$	0	0
$115$	−4.44686	−0.414672
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	4.00000	0.366679
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	10.6729	0.962340
$124$	0	0
$125$	−11.7186	−1.04814
$126$	0	0
$127$	14.3231	1.27097	0.635486	−	0.772112i	$-0.280800\pi$
$127$	14.3231	1.27097	0.635486	+	0.772112i	$0.280800\pi$
$128$	0	0
$129$	−0.130667	−0.0115046
$130$	0	0
$131$	21.5810	1.88554	0.942770	−	0.333443i	$-0.108211\pi$
$131$	21.5810	1.88554	0.942770	+	0.333443i	$0.108211\pi$
$132$	0	0
$133$	−18.6393	−1.61623
$134$	0	0
$135$	−1.53005	−0.131686
$136$	0	0
$137$	−20.6624	−1.76531	−0.882654	−	0.470023i	$-0.844246\pi$
$137$	−20.6624	−1.76531	−0.882654	+	0.470023i	$0.844246\pi$
$138$	0	0
$139$	−17.1428	−1.45404	−0.727018	−	0.686619i	$-0.759094\pi$
$139$	−17.1428	−1.45404	−0.727018	+	0.686619i	$0.759094\pi$
$140$	0	0
$141$	8.00000	0.673722
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	7.47164	0.620486
$146$	0	0
$147$	−1.44686	−0.119335
$148$	0	0
$149$	15.9925	1.31016	0.655080	−	0.755560i	$-0.272635\pi$
$149$	15.9925	1.31016	0.655080	+	0.755560i	$0.272635\pi$
$150$	0	0
$151$	8.22601	0.669423	0.334712	−	0.942321i	$-0.391361\pi$
$151$	8.22601	0.669423	0.334712	+	0.942321i	$0.391361\pi$
$152$	0	0
$153$	1.37630	0.111267
$154$	0	0
$155$	2.73820	0.219937
$156$	0	0
$157$	20.3462	1.62380	0.811902	−	0.583793i	$-0.198432\pi$
$157$	20.3462	1.62380	0.811902	+	0.583793i	$0.198432\pi$
$158$	0	0
$159$	−6.00000	−0.475831
$160$	0	0
$161$	−8.44686	−0.665706
$162$	0	0
$163$	5.29657	0.414860	0.207430	−	0.978250i	$-0.433490\pi$
$163$	5.29657	0.414860	0.207430	+	0.978250i	$0.433490\pi$
$164$	0	0
$165$	1.53005	0.119114
$166$	0	0
$167$	−13.2735	−1.02713	−0.513567	−	0.858050i	$-0.671676\pi$
$167$	−13.2735	−1.02713	−0.513567	+	0.858050i	$0.671676\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−6.41331	−0.490438
$172$	0	0
$173$	8.57575	0.652003	0.326001	−	0.945369i	$-0.394299\pi$
$173$	8.57575	0.652003	0.326001	+	0.945369i	$0.394299\pi$
$174$	0	0
$175$	−7.72782	−0.584168
$176$	0	0
$177$	9.31966	0.700509
$178$	0	0
$179$	25.0988	1.87597	0.937987	−	0.346672i	$-0.112688\pi$
$179$	25.0988	1.87597	0.937987	+	0.346672i	$0.112688\pi$
$180$	0	0
$181$	3.27918	0.243739	0.121870	−	0.992546i	$-0.461111\pi$
$181$	3.27918	0.243739	0.121870	+	0.992546i	$0.461111\pi$
$182$	0	0
$183$	7.31966	0.541085
$184$	0	0
$185$	7.27171	0.534627
$186$	0	0
$187$	−1.37630	−0.100645
$188$	0	0
$189$	−2.90635	−0.211406
$190$	0	0
$191$	0.754370	0.0545843	0.0272921	−	0.999628i	$-0.491312\pi$
$191$	0.754370	0.0545843	0.0272921	+	0.999628i	$0.491312\pi$
$192$	0	0
$193$	−19.7312	−1.42028	−0.710141	−	0.704059i	$-0.751369\pi$
$193$	−19.7312	−1.42028	−0.710141	+	0.704059i	$0.751369\pi$
$194$	0	0
$195$	1.53005	0.109569
$196$	0	0
$197$	−0.459487	−0.0327371	−0.0163685	−	0.999866i	$-0.505210\pi$
$197$	−0.459487	−0.0327371	−0.0163685	+	0.999866i	$0.505210\pi$
$198$	0	0
$199$	−2.68034	−0.190004	−0.0950021	−	0.995477i	$-0.530286\pi$
$199$	−2.68034	−0.190004	−0.0950021	+	0.995477i	$0.530286\pi$
$200$	0	0
$201$	2.59015	0.182695
$202$	0	0
$203$	14.1925	0.996115
$204$	0	0
$205$	−16.3300	−1.14054
$206$	0	0
$207$	−2.90635	−0.202005
$208$	0	0
$209$	6.41331	0.443618
$210$	0	0
$211$	−15.0688	−1.03738	−0.518689	−	0.854963i	$-0.673580\pi$
$211$	−15.0688	−1.03738	−0.518689	+	0.854963i	$0.673580\pi$
$212$	0	0
$213$	7.06010	0.483750
$214$	0	0
$215$	0.199927	0.0136349
$216$	0	0
$217$	5.20123	0.353083
$218$	0	0
$219$	13.4716	0.910329
$220$	0	0
$221$	−1.37630	−0.0925798
$222$	0	0
$223$	−10.9908	−0.736001	−0.368001	−	0.929826i	$-0.619958\pi$
$223$	−10.9908	−0.736001	−0.368001	+	0.929826i	$0.619958\pi$
$224$	0	0
$225$	−2.65894	−0.177263
$226$	0	0
$227$	10.0596	0.667681	0.333841	−	0.942630i	$-0.391655\pi$
$227$	10.0596	0.667681	0.333841	+	0.942630i	$0.391655\pi$
$228$	0	0
$229$	7.88495	0.521052	0.260526	−	0.965467i	$-0.416104\pi$
$229$	7.88495	0.521052	0.260526	+	0.965467i	$0.416104\pi$
$230$	0	0
$231$	2.90635	0.191224
$232$	0	0
$233$	12.3902	0.811711	0.405855	−	0.913937i	$-0.366974\pi$
$233$	12.3902	0.811711	0.405855	+	0.913937i	$0.366974\pi$
$234$	0	0
$235$	−12.2404	−0.798476
$236$	0	0
$237$	7.37630	0.479142
$238$	0	0
$239$	20.0405	1.29631	0.648156	−	0.761508i	$-0.275541\pi$
$239$	20.0405	1.29631	0.648156	+	0.761508i	$0.275541\pi$
$240$	0	0
$241$	−20.3318	−1.30969	−0.654844	−	0.755764i	$-0.727266\pi$
$241$	−20.3318	−1.30969	−0.654844	+	0.755764i	$0.727266\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	2.21377	0.141433
$246$	0	0
$247$	6.41331	0.408069
$248$	0	0
$249$	11.0139	0.697979
$250$	0	0
$251$	−9.81270	−0.619372	−0.309686	−	0.950839i	$-0.600224\pi$
$251$	−9.81270	−0.619372	−0.309686	+	0.950839i	$0.600224\pi$
$252$	0	0
$253$	2.90635	0.182721
$254$	0	0
$255$	−2.10581	−0.131871
$256$	0	0
$257$	−11.6272	−0.725283	−0.362641	−	0.931929i	$-0.618125\pi$
$257$	−11.6272	−0.725283	−0.362641	+	0.931929i	$0.618125\pi$
$258$	0	0
$259$	13.8127	0.858279
$260$	0	0
$261$	4.88326	0.302266
$262$	0	0
$263$	12.8728	0.793771	0.396885	−	0.917868i	$-0.370091\pi$
$263$	12.8728	0.793771	0.396885	+	0.917868i	$0.370091\pi$
$264$	0	0
$265$	9.18031	0.563942
$266$	0	0
$267$	3.22432	0.197325
$268$	0	0
$269$	−10.2613	−0.625645	−0.312822	−	0.949812i	$-0.601274\pi$
$269$	−10.2613	−0.625645	−0.312822	+	0.949812i	$0.601274\pi$
$270$	0	0
$271$	17.4272	1.05863	0.529315	−	0.848426i	$-0.322449\pi$
$271$	17.4272	1.05863	0.529315	+	0.848426i	$0.322449\pi$
$272$	0	0
$273$	2.90635	0.175900
$274$	0	0
$275$	2.65894	0.160340
$276$	0	0
$277$	−16.7787	−1.00813	−0.504067	−	0.863665i	$-0.668163\pi$
$277$	−16.7787	−1.00813	−0.504067	+	0.863665i	$0.668163\pi$
$278$	0	0
$279$	1.78961	0.107141
$280$	0	0
$281$	−25.7791	−1.53786	−0.768928	−	0.639336i	$-0.779209\pi$
$281$	−25.7791	−1.53786	−0.768928	+	0.639336i	$0.779209\pi$
$282$	0	0
$283$	−28.9537	−1.72112	−0.860561	−	0.509348i	$-0.829887\pi$
$283$	−28.9537	−1.72112	−0.860561	+	0.509348i	$0.829887\pi$
$284$	0	0
$285$	9.81270	0.581254
$286$	0	0
$287$	−31.0191	−1.83100
$288$	0	0
$289$	−15.1058	−0.888577
$290$	0	0
$291$	−4.75259	−0.278602
$292$	0	0
$293$	−26.4838	−1.54720	−0.773600	−	0.633674i	$-0.781546\pi$
$293$	−26.4838	−1.54720	−0.773600	+	0.633674i	$0.781546\pi$
$294$	0	0
$295$	−14.2596	−0.830224
$296$	0	0
$297$	1.00000	0.0580259
$298$	0	0
$299$	2.90635	0.168078
$300$	0	0
$301$	0.379764	0.0218892
$302$	0	0
$303$	−15.1428	−0.869932
$304$	0	0
$305$	−11.1995	−0.641279
$306$	0	0
$307$	−17.3601	−0.990796	−0.495398	−	0.868666i	$-0.664978\pi$
$307$	−17.3601	−0.990796	−0.495398	+	0.868666i	$0.664978\pi$
$308$	0	0
$309$	18.8249	1.07091
$310$	0	0
$311$	15.5936	0.884233	0.442117	−	0.896958i	$-0.354228\pi$
$311$	15.5936	0.884233	0.442117	+	0.896958i	$0.354228\pi$
$312$	0	0
$313$	−2.83232	−0.160092	−0.0800460	−	0.996791i	$-0.525507\pi$
$313$	−2.83232	−0.160092	−0.0800460	+	0.996791i	$0.525507\pi$
$314$	0	0
$315$	4.44686	0.250552
$316$	0	0
$317$	25.4630	1.43014	0.715071	−	0.699052i	$-0.246394\pi$
$317$	25.4630	1.43014	0.715071	+	0.699052i	$0.246394\pi$
$318$	0	0
$319$	−4.88326	−0.273410
$320$	0	0
$321$	5.62717	0.314078
$322$	0	0
$323$	−8.82663	−0.491127
$324$	0	0
$325$	2.65894	0.147492
$326$	0	0
$327$	−13.9185	−0.769695
$328$	0	0
$329$	−23.2508	−1.28186
$330$	0	0
$331$	−22.9960	−1.26397	−0.631987	−	0.774979i	$-0.717761\pi$
$331$	−22.9960	−1.26397	−0.631987	+	0.774979i	$0.717761\pi$
$332$	0	0
$333$	4.75259	0.260441
$334$	0	0
$335$	−3.96307	−0.216526
$336$	0	0
$337$	20.3780	1.11006	0.555030	−	0.831830i	$-0.312707\pi$
$337$	20.3780	1.11006	0.555030	+	0.831830i	$0.312707\pi$
$338$	0	0
$339$	−8.25956	−0.448598
$340$	0	0
$341$	−1.78961	−0.0969128
$342$	0	0
$343$	−16.1394	−0.871443
$344$	0	0
$345$	4.44686	0.239411
$346$	0	0
$347$	24.1481	1.29634	0.648168	−	0.761497i	$-0.275535\pi$
$347$	24.1481	1.29634	0.648168	+	0.761497i	$0.275535\pi$
$348$	0	0
$349$	−19.8127	−1.06055	−0.530275	−	0.847826i	$-0.677911\pi$
$349$	−19.8127	−1.06055	−0.530275	+	0.847826i	$0.677911\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	−18.5884	−0.989360	−0.494680	−	0.869075i	$-0.664715\pi$
$353$	−18.5884	−0.989360	−0.494680	+	0.869075i	$0.664715\pi$
$354$	0	0
$355$	−10.8023	−0.573328
$356$	0	0
$357$	−4.00000	−0.211702
$358$	0	0
$359$	−4.04048	−0.213248	−0.106624	−	0.994299i	$-0.534004\pi$
$359$	−4.04048	−0.213248	−0.106624	+	0.994299i	$0.534004\pi$
$360$	0	0
$361$	22.1306	1.16477
$362$	0	0
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	−20.6123	−1.07890
$366$	0	0
$367$	−2.49481	−0.130228	−0.0651140	−	0.997878i	$-0.520741\pi$
$367$	−2.49481	−0.130228	−0.0651140	+	0.997878i	$0.520741\pi$
$368$	0	0
$369$	−10.6729	−0.555608
$370$	0	0
$371$	17.4381	0.905341
$372$	0	0
$373$	−7.69427	−0.398394	−0.199197	−	0.979959i	$-0.563833\pi$
$373$	−7.69427	−0.398394	−0.199197	+	0.979959i	$0.563833\pi$
$374$	0	0
$375$	11.7186	0.605145
$376$	0	0
$377$	−4.88326	−0.251501
$378$	0	0
$379$	−19.9560	−1.02507	−0.512535	−	0.858666i	$-0.671294\pi$
$379$	−19.9560	−1.02507	−0.512535	+	0.858666i	$0.671294\pi$
$380$	0	0
$381$	−14.3231	−0.733796
$382$	0	0
$383$	3.64632	0.186318	0.0931591	−	0.995651i	$-0.470303\pi$
$383$	3.64632	0.186318	0.0931591	+	0.995651i	$0.470303\pi$
$384$	0	0
$385$	−4.44686	−0.226633
$386$	0	0
$387$	0.130667	0.00664218
$388$	0	0
$389$	−13.5792	−0.688494	−0.344247	−	0.938879i	$-0.611866\pi$
$389$	−13.5792	−0.688494	−0.344247	+	0.938879i	$0.611866\pi$
$390$	0	0
$391$	−4.00000	−0.202289
$392$	0	0
$393$	−21.5810	−1.08862
$394$	0	0
$395$	−11.2861	−0.567866
$396$	0	0
$397$	2.33359	0.117120	0.0585598	−	0.998284i	$-0.481349\pi$
$397$	2.33359	0.117120	0.0585598	+	0.998284i	$0.481349\pi$
$398$	0	0
$399$	18.6393	0.933133
$400$	0	0
$401$	26.3828	1.31749	0.658746	−	0.752365i	$-0.271087\pi$
$401$	26.3828	1.31749	0.658746	+	0.752365i	$0.271087\pi$
$402$	0	0
$403$	−1.78961	−0.0891468
$404$	0	0
$405$	1.53005	0.0760289
$406$	0	0
$407$	−4.75259	−0.235577
$408$	0	0
$409$	−6.36536	−0.314747	−0.157374	−	0.987539i	$-0.550303\pi$
$409$	−6.36536	−0.314747	−0.157374	+	0.987539i	$0.550303\pi$
$410$	0	0
$411$	20.6624	1.01920
$412$	0	0
$413$	−27.0862	−1.33282
$414$	0	0
$415$	−16.8519	−0.827226
$416$	0	0
$417$	17.1428	0.839488
$418$	0	0
$419$	3.82016	0.186627	0.0933136	−	0.995637i	$-0.470254\pi$
$419$	3.82016	0.186627	0.0933136	+	0.995637i	$0.470254\pi$
$420$	0	0
$421$	19.8850	0.969133	0.484567	−	0.874754i	$-0.338977\pi$
$421$	19.8850	0.969133	0.484567	+	0.874754i	$0.338977\pi$
$422$	0	0
$423$	−8.00000	−0.388973
$424$	0	0
$425$	−3.65950	−0.177512
$426$	0	0
$427$	−21.2735	−1.02950
$428$	0	0
$429$	−1.00000	−0.0482805
$430$	0	0
$431$	28.2116	1.35891	0.679453	−	0.733719i	$-0.262217\pi$
$431$	28.2116	1.35891	0.679453	+	0.733719i	$0.262217\pi$
$432$	0	0
$433$	21.0862	1.01334	0.506669	−	0.862141i	$-0.330877\pi$
$433$	21.0862	1.01334	0.506669	+	0.862141i	$0.330877\pi$
$434$	0	0
$435$	−7.47164	−0.358238
$436$	0	0
$437$	18.6393	0.891640
$438$	0	0
$439$	21.6410	1.03287	0.516435	−	0.856327i	$-0.327259\pi$
$439$	21.6410	1.03287	0.516435	+	0.856327i	$0.327259\pi$
$440$	0	0
$441$	1.44686	0.0688982
$442$	0	0
$443$	−15.5330	−0.737997	−0.368999	−	0.929430i	$-0.620299\pi$
$443$	−15.5330	−0.737997	−0.368999	+	0.929430i	$0.620299\pi$
$444$	0	0
$445$	−4.93337	−0.233864
$446$	0	0
$447$	−15.9925	−0.756421
$448$	0	0
$449$	−10.1711	−0.480006	−0.240003	−	0.970772i	$-0.577148\pi$
$449$	−10.1711	−0.480006	−0.240003	+	0.970772i	$0.577148\pi$
$450$	0	0
$451$	10.6729	0.502566
$452$	0	0
$453$	−8.22601	−0.386492
$454$	0	0
$455$	−4.44686	−0.208472
$456$	0	0
$457$	8.87102	0.414969	0.207485	−	0.978238i	$-0.433472\pi$
$457$	8.87102	0.414969	0.207485	+	0.978238i	$0.433472\pi$
$458$	0	0
$459$	−1.37630	−0.0642401
$460$	0	0
$461$	−28.7173	−1.33750	−0.668748	−	0.743489i	$-0.733170\pi$
$461$	−28.7173	−1.33750	−0.668748	+	0.743489i	$0.733170\pi$
$462$	0	0
$463$	−6.18946	−0.287649	−0.143824	−	0.989603i	$-0.545940\pi$
$463$	−6.18946	−0.287649	−0.143824	+	0.989603i	$0.545940\pi$
$464$	0	0
$465$	−2.73820	−0.126981
$466$	0	0
$467$	−34.9786	−1.61862	−0.809308	−	0.587384i	$-0.800158\pi$
$467$	−34.9786	−1.61862	−0.809308	+	0.587384i	$0.800158\pi$
$468$	0	0
$469$	−7.52789	−0.347606
$470$	0	0
$471$	−20.3462	−0.937504
$472$	0	0
$473$	−0.130667	−0.00600808
$474$	0	0
$475$	17.0526	0.782429
$476$	0	0
$477$	6.00000	0.274721
$478$	0	0
$479$	33.1458	1.51447	0.757236	−	0.653142i	$-0.226549\pi$
$479$	33.1458	1.51447	0.757236	+	0.653142i	$0.226549\pi$
$480$	0	0
$481$	−4.75259	−0.216700
$482$	0	0
$483$	8.44686	0.384346
$484$	0	0
$485$	7.27171	0.330191
$486$	0	0
$487$	24.4289	1.10698	0.553490	−	0.832856i	$-0.313296\pi$
$487$	24.4289	1.10698	0.553490	+	0.832856i	$0.313296\pi$
$488$	0	0
$489$	−5.29657	−0.239519
$490$	0	0
$491$	−24.5949	−1.10995	−0.554977	−	0.831866i	$-0.687273\pi$
$491$	−24.5949	−1.10995	−0.554977	+	0.831866i	$0.687273\pi$
$492$	0	0
$493$	6.72082	0.302691
$494$	0	0
$495$	−1.53005	−0.0687707
$496$	0	0
$497$	−20.5191	−0.920408
$498$	0	0
$499$	41.7024	1.86686	0.933428	−	0.358764i	$-0.116802\pi$
$499$	41.7024	1.86686	0.933428	+	0.358764i	$0.116802\pi$
$500$	0	0
$501$	13.2735	0.593016
$502$	0	0
$503$	14.7526	0.657786	0.328893	−	0.944367i	$-0.393325\pi$
$503$	14.7526	0.657786	0.328893	+	0.944367i	$0.393325\pi$
$504$	0	0
$505$	23.1693	1.03102
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	−18.9630	−0.840520	−0.420260	−	0.907404i	$-0.638061\pi$
$509$	−18.9630	−0.840520	−0.420260	+	0.907404i	$0.638061\pi$
$510$	0	0
$511$	−39.1533	−1.73204
$512$	0	0
$513$	6.41331	0.283155
$514$	0	0
$515$	−28.8030	−1.26921
$516$	0	0
$517$	8.00000	0.351840
$518$	0	0
$519$	−8.57575	−0.376434
$520$	0	0
$521$	16.4747	0.721771	0.360885	−	0.932610i	$-0.382475\pi$
$521$	16.4747	0.721771	0.360885	+	0.932610i	$0.382475\pi$
$522$	0	0
$523$	2.13244	0.0932452	0.0466226	−	0.998913i	$-0.485154\pi$
$523$	2.13244	0.0932452	0.0466226	+	0.998913i	$0.485154\pi$
$524$	0	0
$525$	7.72782	0.337270
$526$	0	0
$527$	2.46304	0.107292
$528$	0	0
$529$	−14.5531	−0.632745
$530$	0	0
$531$	−9.31966	−0.404439
$532$	0	0
$533$	10.6729	0.462293
$534$	0	0
$535$	−8.60986	−0.372237
$536$	0	0
$537$	−25.0988	−1.08309
$538$	0	0
$539$	−1.44686	−0.0623208
$540$	0	0
$541$	8.02786	0.345145	0.172572	−	0.984997i	$-0.444792\pi$
$541$	8.02786	0.345145	0.172572	+	0.984997i	$0.444792\pi$
$542$	0	0
$543$	−3.27918	−0.140723
$544$	0	0
$545$	21.2960	0.912221
$546$	0	0
$547$	9.58968	0.410025	0.205013	−	0.978759i	$-0.434276\pi$
$547$	9.58968	0.410025	0.205013	+	0.978759i	$0.434276\pi$
$548$	0	0
$549$	−7.31966	−0.312396
$550$	0	0
$551$	−31.3179	−1.33419
$552$	0	0
$553$	−21.4381	−0.911640
$554$	0	0
$555$	−7.27171	−0.308667
$556$	0	0
$557$	17.5936	0.745466	0.372733	−	0.927939i	$-0.378421\pi$
$557$	17.5936	0.745466	0.372733	+	0.927939i	$0.378421\pi$
$558$	0	0
$559$	−0.130667	−0.00552663
$560$	0	0
$561$	1.37630	0.0581073
$562$	0	0
$563$	−4.30396	−0.181390	−0.0906951	−	0.995879i	$-0.528909\pi$
$563$	−4.30396	−0.181390	−0.0906951	+	0.995879i	$0.528909\pi$
$564$	0	0
$565$	12.6375	0.531666
$566$	0	0
$567$	2.90635	0.122055
$568$	0	0
$569$	−13.0688	−0.547872	−0.273936	−	0.961748i	$-0.588326\pi$
$569$	−13.0688	−0.547872	−0.273936	+	0.961748i	$0.588326\pi$
$570$	0	0
$571$	−18.1838	−0.760967	−0.380484	−	0.924788i	$-0.624242\pi$
$571$	−18.1838	−0.760967	−0.380484	+	0.924788i	$0.624242\pi$
$572$	0	0
$573$	−0.754370	−0.0315142
$574$	0	0
$575$	7.72782	0.322272
$576$	0	0
$577$	−38.6672	−1.60974	−0.804868	−	0.593454i	$-0.797764\pi$
$577$	−38.6672	−1.60974	−0.804868	+	0.593454i	$0.797764\pi$
$578$	0	0
$579$	19.7312	0.820001
$580$	0	0
$581$	−32.0103	−1.32801
$582$	0	0
$583$	−6.00000	−0.248495
$584$	0	0
$585$	−1.53005	−0.0632598
$586$	0	0
$587$	13.0157	0.537216	0.268608	−	0.963250i	$-0.413436\pi$
$587$	13.0157	0.537216	0.268608	+	0.963250i	$0.413436\pi$
$588$	0	0
$589$	−11.4773	−0.472915
$590$	0	0
$591$	0.459487	0.0189007
$592$	0	0
$593$	14.1798	0.582296	0.291148	−	0.956678i	$-0.405963\pi$
$593$	14.1798	0.582296	0.291148	+	0.956678i	$0.405963\pi$
$594$	0	0
$595$	6.12020	0.250904
$596$	0	0
$597$	2.68034	0.109699
$598$	0	0
$599$	−29.6445	−1.21124	−0.605622	−	0.795753i	$-0.707075\pi$
$599$	−29.6445	−1.21124	−0.605622	+	0.795753i	$0.707075\pi$
$600$	0	0
$601$	−33.9329	−1.38415	−0.692076	−	0.721825i	$-0.743304\pi$
$601$	−33.9329	−1.38415	−0.692076	+	0.721825i	$0.743304\pi$
$602$	0	0
$603$	−2.59015	−0.105479
$604$	0	0
$605$	1.53005	0.0622054
$606$	0	0
$607$	−29.0757	−1.18015	−0.590074	−	0.807349i	$-0.700901\pi$
$607$	−29.0757	−1.18015	−0.590074	+	0.807349i	$0.700901\pi$
$608$	0	0
$609$	−14.1925	−0.575108
$610$	0	0
$611$	8.00000	0.323645
$612$	0	0
$613$	−41.7843	−1.68765	−0.843826	−	0.536617i	$-0.819702\pi$
$613$	−41.7843	−1.68765	−0.843826	+	0.536617i	$0.819702\pi$
$614$	0	0
$615$	16.3300	0.658491
$616$	0	0
$617$	−15.3688	−0.618726	−0.309363	−	0.950944i	$-0.600116\pi$
$617$	−15.3688	−0.618726	−0.309363	+	0.950944i	$0.600116\pi$
$618$	0	0
$619$	4.97169	0.199829	0.0999146	−	0.994996i	$-0.468143\pi$
$619$	4.97169	0.199829	0.0999146	+	0.994996i	$0.468143\pi$
$620$	0	0
$621$	2.90635	0.116628
$622$	0	0
$623$	−9.37099	−0.375441
$624$	0	0
$625$	−4.63530	−0.185412
$626$	0	0
$627$	−6.41331	−0.256123
$628$	0	0
$629$	6.54098	0.260806
$630$	0	0
$631$	−49.4672	−1.96926	−0.984628	−	0.174662i	$-0.944117\pi$
$631$	−49.4672	−1.96926	−0.984628	+	0.174662i	$0.944117\pi$
$632$	0	0
$633$	15.0688	0.598931
$634$	0	0
$635$	21.9151	0.869675
$636$	0	0
$637$	−1.44686	−0.0573268
$638$	0	0
$639$	−7.06010	−0.279293
$640$	0	0
$641$	−34.2874	−1.35427	−0.677136	−	0.735858i	$-0.736779\pi$
$641$	−34.2874	−1.35427	−0.677136	+	0.735858i	$0.736779\pi$
$642$	0	0
$643$	13.3688	0.527215	0.263608	−	0.964630i	$-0.415088\pi$
$643$	13.3688	0.527215	0.263608	+	0.964630i	$0.415088\pi$
$644$	0	0
$645$	−0.199927	−0.00787213
$646$	0	0
$647$	−9.20123	−0.361738	−0.180869	−	0.983507i	$-0.557891\pi$
$647$	−9.20123	−0.361738	−0.180869	+	0.983507i	$0.557891\pi$
$648$	0	0
$649$	9.31966	0.365829
$650$	0	0
$651$	−5.20123	−0.203852
$652$	0	0
$653$	12.0740	0.472493	0.236247	−	0.971693i	$-0.424083\pi$
$653$	12.0740	0.472493	0.236247	+	0.971693i	$0.424083\pi$
$654$	0	0
$655$	33.0200	1.29020
$656$	0	0
$657$	−13.4716	−0.525579
$658$	0	0
$659$	43.2970	1.68661	0.843305	−	0.537435i	$-0.180607\pi$
$659$	43.2970	1.68661	0.843305	+	0.537435i	$0.180607\pi$
$660$	0	0
$661$	−36.5261	−1.42070	−0.710349	−	0.703849i	$-0.751463\pi$
$661$	−36.5261	−1.42070	−0.710349	+	0.703849i	$0.751463\pi$
$662$	0	0
$663$	1.37630	0.0534510
$664$	0	0
$665$	−28.5191	−1.10592
$666$	0	0
$667$	−14.1925	−0.549534
$668$	0	0
$669$	10.9908	0.424931
$670$	0	0
$671$	7.31966	0.282572
$672$	0	0
$673$	47.8901	1.84603	0.923014	−	0.384766i	$-0.125718\pi$
$673$	47.8901	1.84603	0.923014	+	0.384766i	$0.125718\pi$
$674$	0	0
$675$	2.65894	0.102343
$676$	0	0
$677$	−8.88842	−0.341610	−0.170805	−	0.985305i	$-0.554637\pi$
$677$	−8.88842	−0.341610	−0.170805	+	0.985305i	$0.554637\pi$
$678$	0	0
$679$	13.8127	0.530083
$680$	0	0
$681$	−10.0596	−0.385486
$682$	0	0
$683$	16.7082	0.639321	0.319661	−	0.947532i	$-0.396431\pi$
$683$	16.7082	0.639321	0.319661	+	0.947532i	$0.396431\pi$
$684$	0	0
$685$	−31.6145	−1.20793
$686$	0	0
$687$	−7.88495	−0.300830
$688$	0	0
$689$	−6.00000	−0.228582
$690$	0	0
$691$	21.1284	0.803763	0.401882	−	0.915692i	$-0.368356\pi$
$691$	21.1284	0.803763	0.401882	+	0.915692i	$0.368356\pi$
$692$	0	0
$693$	−2.90635	−0.110403
$694$	0	0
$695$	−26.2294	−0.994938
$696$	0	0
$697$	−14.6890	−0.556387
$698$	0	0
$699$	−12.3902	−0.468641
$700$	0	0
$701$	32.0417	1.21020	0.605099	−	0.796150i	$-0.293133\pi$
$701$	32.0417	1.21020	0.605099	+	0.796150i	$0.293133\pi$
$702$	0	0
$703$	−30.4799	−1.14957
$704$	0	0
$705$	12.2404	0.461000
$706$	0	0
$707$	44.0103	1.65518
$708$	0	0
$709$	−20.8710	−0.783828	−0.391914	−	0.920002i	$-0.628187\pi$
$709$	−20.8710	−0.783828	−0.391914	+	0.920002i	$0.628187\pi$
$710$	0	0
$711$	−7.37630	−0.276633
$712$	0	0
$713$	−5.20123	−0.194788
$714$	0	0
$715$	1.53005	0.0572207
$716$	0	0
$717$	−20.0405	−0.748426
$718$	0	0
$719$	8.81792	0.328853	0.164426	−	0.986389i	$-0.447423\pi$
$719$	8.81792	0.328853	0.164426	+	0.986389i	$0.447423\pi$
$720$	0	0
$721$	−54.7116	−2.03757
$722$	0	0
$723$	20.3318	0.756148
$724$	0	0
$725$	−12.9843	−0.482225
$726$	0	0
$727$	22.7526	0.843847	0.421924	−	0.906631i	$-0.361355\pi$
$727$	22.7526	0.843847	0.421924	+	0.906631i	$0.361355\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0.179837	0.00665150
$732$	0	0
$733$	35.6389	1.31635	0.658176	−	0.752865i	$-0.271328\pi$
$733$	35.6389	1.31635	0.658176	+	0.752865i	$0.271328\pi$
$734$	0	0
$735$	−2.21377	−0.0816562
$736$	0	0
$737$	2.59015	0.0954095
$738$	0	0
$739$	17.7774	0.653951	0.326976	−	0.945033i	$-0.393970\pi$
$739$	17.7774	0.653951	0.326976	+	0.945033i	$0.393970\pi$
$740$	0	0
$741$	−6.41331	−0.235599
$742$	0	0
$743$	20.3798	0.747661	0.373831	−	0.927497i	$-0.378044\pi$
$743$	20.3798	0.747661	0.373831	+	0.927497i	$0.378044\pi$
$744$	0	0
$745$	24.4694	0.896489
$746$	0	0
$747$	−11.0139	−0.402979
$748$	0	0
$749$	−16.3545	−0.597581
$750$	0	0
$751$	−33.2064	−1.21172	−0.605859	−	0.795572i	$-0.707171\pi$
$751$	−33.2064	−1.21172	−0.605859	+	0.795572i	$0.707171\pi$
$752$	0	0
$753$	9.81270	0.357595
$754$	0	0
$755$	12.5862	0.458059
$756$	0	0
$757$	−24.1832	−0.878954	−0.439477	−	0.898254i	$-0.644836\pi$
$757$	−24.1832	−0.878954	−0.439477	+	0.898254i	$0.644836\pi$
$758$	0	0
$759$	−2.90635	−0.105494
$760$	0	0
$761$	−18.3480	−0.665114	−0.332557	−	0.943083i	$-0.607911\pi$
$761$	−18.3480	−0.665114	−0.332557	+	0.943083i	$0.607911\pi$
$762$	0	0
$763$	40.4520	1.46446
$764$	0	0
$765$	2.10581	0.0761356
$766$	0	0
$767$	9.31966	0.336513
$768$	0	0
$769$	44.1889	1.59349	0.796746	−	0.604314i	$-0.206553\pi$
$769$	44.1889	1.59349	0.796746	+	0.604314i	$0.206553\pi$
$770$	0	0
$771$	11.6272	0.418742
$772$	0	0
$773$	10.4960	0.377516	0.188758	−	0.982024i	$-0.439554\pi$
$773$	10.4960	0.377516	0.188758	+	0.982024i	$0.439554\pi$
$774$	0	0
$775$	−4.75847	−0.170929
$776$	0	0
$777$	−13.8127	−0.495528
$778$	0	0
$779$	68.4485	2.45242
$780$	0	0
$781$	7.06010	0.252630
$782$	0	0
$783$	−4.88326	−0.174514
$784$	0	0
$785$	31.1308	1.11110
$786$	0	0
$787$	8.73914	0.311517	0.155758	−	0.987795i	$-0.450218\pi$
$787$	8.73914	0.311517	0.155758	+	0.987795i	$0.450218\pi$
$788$	0	0
$789$	−12.8728	−0.458284
$790$	0	0
$791$	24.0052	0.853525
$792$	0	0
$793$	7.31966	0.259929
$794$	0	0
$795$	−9.18031	−0.325592
$796$	0	0
$797$	52.1479	1.84717	0.923587	−	0.383390i	$-0.125243\pi$
$797$	52.1479	1.84717	0.923587	+	0.383390i	$0.125243\pi$
$798$	0	0
$799$	−11.0104	−0.389519
$800$	0	0
$801$	−3.22432	−0.113926
$802$	0	0
$803$	13.4716	0.475404
$804$	0	0
$805$	−12.9241	−0.455516
$806$	0	0
$807$	10.2613	0.361216
$808$	0	0
$809$	−37.0688	−1.30327	−0.651635	−	0.758533i	$-0.725916\pi$
$809$	−37.0688	−1.30327	−0.651635	+	0.758533i	$0.725916\pi$
$810$	0	0
$811$	−45.2642	−1.58944	−0.794721	−	0.606974i	$-0.792383\pi$
$811$	−45.2642	−1.58944	−0.794721	+	0.606974i	$0.792383\pi$
$812$	0	0
$813$	−17.4272	−0.611200
$814$	0	0
$815$	8.10403	0.283872
$816$	0	0
$817$	−0.838009	−0.0293182
$818$	0	0
$819$	−2.90635	−0.101556
$820$	0	0
$821$	−49.2364	−1.71836	−0.859181	−	0.511672i	$-0.829026\pi$
$821$	−49.2364	−1.71836	−0.859181	+	0.511672i	$0.829026\pi$
$822$	0	0
$823$	3.34752	0.116687	0.0583436	−	0.998297i	$-0.481418\pi$
$823$	3.34752	0.116687	0.0583436	+	0.998297i	$0.481418\pi$
$824$	0	0
$825$	−2.65894	−0.0925725
$826$	0	0
$827$	18.6746	0.649381	0.324691	−	0.945820i	$-0.394740\pi$
$827$	18.6746	0.649381	0.324691	+	0.945820i	$0.394740\pi$
$828$	0	0
$829$	2.37461	0.0824735	0.0412367	−	0.999149i	$-0.486870\pi$
$829$	2.37461	0.0824735	0.0412367	+	0.999149i	$0.486870\pi$
$830$	0	0
$831$	16.7787	0.582046
$832$	0	0
$833$	1.99131	0.0689949
$834$	0	0
$835$	−20.3091	−0.702826
$836$	0	0
$837$	−1.78961	−0.0618580
$838$	0	0
$839$	5.50874	0.190183	0.0950914	−	0.995469i	$-0.469686\pi$
$839$	5.50874	0.190183	0.0950914	+	0.995469i	$0.469686\pi$
$840$	0	0
$841$	−5.15375	−0.177716
$842$	0	0
$843$	25.7791	0.887881
$844$	0	0
$845$	1.53005	0.0526354
$846$	0	0
$847$	2.90635	0.0998633
$848$	0	0
$849$	28.9537	0.993690
$850$	0	0
$851$	−13.8127	−0.473493
$852$	0	0
$853$	−32.7029	−1.11972	−0.559862	−	0.828586i	$-0.689146\pi$
$853$	−32.7029	−1.11972	−0.559862	+	0.828586i	$0.689146\pi$
$854$	0	0
$855$	−9.81270	−0.335587
$856$	0	0
$857$	−32.0192	−1.09375	−0.546877	−	0.837213i	$-0.684183\pi$
$857$	−32.0192	−1.09375	−0.546877	+	0.837213i	$0.684183\pi$
$858$	0	0
$859$	43.6566	1.48955	0.744773	−	0.667318i	$-0.232558\pi$
$859$	43.6566	1.48955	0.744773	+	0.667318i	$0.232558\pi$
$860$	0	0
$861$	31.0191	1.05713
$862$	0	0
$863$	−12.4982	−0.425443	−0.212722	−	0.977113i	$-0.568233\pi$
$863$	−12.4982	−0.425443	−0.212722	+	0.977113i	$0.568233\pi$
$864$	0	0
$865$	13.1213	0.446139
$866$	0	0
$867$	15.1058	0.513020
$868$	0	0
$869$	7.37630	0.250224
$870$	0	0
$871$	2.59015	0.0877640
$872$	0	0
$873$	4.75259	0.160851
$874$	0	0
$875$	−34.0583	−1.15138
$876$	0	0
$877$	−37.3736	−1.26202	−0.631008	−	0.775776i	$-0.717359\pi$
$877$	−37.3736	−1.26202	−0.631008	+	0.775776i	$0.717359\pi$
$878$	0	0
$879$	26.4838	0.893276
$880$	0	0
$881$	−2.07226	−0.0698161	−0.0349080	−	0.999391i	$-0.511114\pi$
$881$	−2.07226	−0.0698161	−0.0349080	+	0.999391i	$0.511114\pi$
$882$	0	0
$883$	20.8692	0.702306	0.351153	−	0.936318i	$-0.385790\pi$
$883$	20.8692	0.702306	0.351153	+	0.936318i	$0.385790\pi$
$884$	0	0
$885$	14.2596	0.479330
$886$	0	0
$887$	16.7098	0.561060	0.280530	−	0.959845i	$-0.409490\pi$
$887$	16.7098	0.561060	0.280530	+	0.959845i	$0.409490\pi$
$888$	0	0
$889$	41.6280	1.39616
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	51.3065	1.71691
$894$	0	0
$895$	38.4025	1.28365
$896$	0	0
$897$	−2.90635	−0.0970402
$898$	0	0
$899$	8.73914	0.291467
$900$	0	0
$901$	8.25778	0.275107
$902$	0	0
$903$	−0.379764	−0.0126378
$904$	0	0
$905$	5.01731	0.166781
$906$	0	0
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	0	0
$909$	15.1428	0.502256
$910$	0	0
$911$	42.1976	1.39807	0.699035	−	0.715088i	$-0.253613\pi$
$911$	42.1976	1.39807	0.699035	+	0.715088i	$0.253613\pi$
$912$	0	0
$913$	11.0139	0.364508
$914$	0	0
$915$	11.1995	0.370243
$916$	0	0
$917$	62.7219	2.07126
$918$	0	0
$919$	−35.4111	−1.16810	−0.584052	−	0.811716i	$-0.698534\pi$
$919$	−35.4111	−1.16810	−0.584052	+	0.811716i	$0.698534\pi$
$920$	0	0
$921$	17.3601	0.572036
$922$	0	0
$923$	7.06010	0.232386
$924$	0	0
$925$	−12.6369	−0.415498
$926$	0	0
$927$	−18.8249	−0.618289
$928$	0	0
$929$	31.1075	1.02060	0.510302	−	0.859995i	$-0.329534\pi$
$929$	31.1075	1.02060	0.510302	+	0.859995i	$0.329534\pi$
$930$	0	0
$931$	−9.27918	−0.304113
$932$	0	0
$933$	−15.5936	−0.510512
$934$	0	0
$935$	−2.10581	−0.0688672
$936$	0	0
$937$	60.2647	1.96876	0.984381	−	0.176050i	$-0.0563320\pi$
$937$	60.2647	1.96876	0.984381	+	0.176050i	$0.0563320\pi$
$938$	0	0
$939$	2.83232	0.0924292
$940$	0	0
$941$	8.13105	0.265065	0.132532	−	0.991179i	$-0.457689\pi$
$941$	8.13105	0.265065	0.132532	+	0.991179i	$0.457689\pi$
$942$	0	0
$943$	31.0191	1.01012
$944$	0	0
$945$	−4.44686	−0.144656
$946$	0	0
$947$	36.9190	1.19971	0.599853	−	0.800110i	$-0.295226\pi$
$947$	36.9190	1.19971	0.599853	+	0.800110i	$0.295226\pi$
$948$	0	0
$949$	13.4716	0.437308
$950$	0	0
$951$	−25.4630	−0.825693
$952$	0	0
$953$	−32.5347	−1.05390	−0.526952	−	0.849895i	$-0.676665\pi$
$953$	−32.5347	−1.05390	−0.526952	+	0.849895i	$0.676665\pi$
$954$	0	0
$955$	1.15422	0.0373498
$956$	0	0
$957$	4.88326	0.157853
$958$	0	0
$959$	−60.0522	−1.93919
$960$	0	0
$961$	−27.7973	−0.896687
$962$	0	0
$963$	−5.62717	−0.181333
$964$	0	0
$965$	−30.1897	−0.971842
$966$	0	0
$967$	36.2521	1.16579	0.582894	−	0.812548i	$-0.301920\pi$
$967$	36.2521	1.16579	0.582894	+	0.812548i	$0.301920\pi$
$968$	0	0
$969$	8.82663	0.283552
$970$	0	0
$971$	8.43163	0.270584	0.135292	−	0.990806i	$-0.456803\pi$
$971$	8.43163	0.270584	0.135292	+	0.990806i	$0.456803\pi$
$972$	0	0
$973$	−49.8230	−1.59725
$974$	0	0
$975$	−2.65894	−0.0851543
$976$	0	0
$977$	32.8069	1.04959	0.524793	−	0.851230i	$-0.324143\pi$
$977$	32.8069	1.04959	0.524793	+	0.851230i	$0.324143\pi$
$978$	0	0
$979$	3.22432	0.103050
$980$	0	0
$981$	13.9185	0.444384
$982$	0	0
$983$	−0.745662	−0.0237829	−0.0118915	−	0.999929i	$-0.503785\pi$
$983$	−0.745662	−0.0237829	−0.0118915	+	0.999929i	$0.503785\pi$
$984$	0	0
$985$	−0.703038	−0.0224006
$986$	0	0
$987$	23.2508	0.740081
$988$	0	0
$989$	−0.379764	−0.0120758
$990$	0	0
$991$	−49.3971	−1.56915	−0.784575	−	0.620034i	$-0.787119\pi$
$991$	−49.3971	−1.56915	−0.784575	+	0.620034i	$0.787119\pi$
$992$	0	0
$993$	22.9960	0.729756
$994$	0	0
$995$	−4.10105	−0.130012
$996$	0	0
$997$	−14.1220	−0.447248	−0.223624	−	0.974676i	$-0.571789\pi$
$997$	−14.1220	−0.447248	−0.223624	+	0.974676i	$0.571789\pi$
$998$	0	0
$999$	−4.75259	−0.150365

Twists

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.53005 q^{5} +2.90635 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.53005 q^{5} +2.90635 q^{7} +1.00000 q^{9} -1.00000 q^{11} -1.00000 q^{13} -1.53005 q^{15} +1.37630 q^{17} -6.41331 q^{19} -2.90635 q^{21} -2.90635 q^{23} -2.65894 q^{25} -1.00000 q^{27} +4.88326 q^{29} +1.78961 q^{31} +1.00000 q^{33} +4.44686 q^{35} +4.75259 q^{37} +1.00000 q^{39} -10.6729 q^{41} +0.130667 q^{43} +1.53005 q^{45} -8.00000 q^{47} +1.44686 q^{49} -1.37630 q^{51} +6.00000 q^{53} -1.53005 q^{55} +6.41331 q^{57} -9.31966 q^{59} -7.31966 q^{61} +2.90635 q^{63} -1.53005 q^{65} -2.59015 q^{67} +2.90635 q^{69} -7.06010 q^{71} -13.4716 q^{73} +2.65894 q^{75} -2.90635 q^{77} -7.37630 q^{79} +1.00000 q^{81} -11.0139 q^{83} +2.10581 q^{85} -4.88326 q^{87} -3.22432 q^{89} -2.90635 q^{91} -1.78961 q^{93} -9.81270 q^{95} +4.75259 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + q^{5} - q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 + q^5 - q^7 + 4 * q^9	\( 4 q - 4 q^{3} + q^{5} - q^{7} + 4 q^{9} - 4 q^{11} - 4 q^{13} - q^{15} - 2 q^{17} - 2 q^{19} + q^{21} + q^{23} + 17 q^{25} - 4 q^{27} + q^{29} - 24 q^{31} + 4 q^{33} + 17 q^{35} + 4 q^{37} + 4 q^{39} + 7 q^{41} - 3 q^{43} + q^{45} - 32 q^{47} + 5 q^{49} + 2 q^{51} + 24 q^{53} - q^{55} + 2 q^{57} - q^{59} + 7 q^{61} - q^{63} - q^{65} + 5 q^{67} - q^{69} - 18 q^{71} - q^{73} - 17 q^{75} + q^{77} - 22 q^{79} + 4 q^{81} - 22 q^{83} - 20 q^{85} - q^{87} - 22 q^{89} + q^{91} + 24 q^{93} - 14 q^{95} + 4 q^{97} - 4 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 + q^5 - q^7 + 4 * q^9 - 4 * q^11 - 4 * q^13 - q^15 - 2 * q^17 - 2 * q^19 + q^21 + q^23 + 17 * q^25 - 4 * q^27 + q^29 - 24 * q^31 + 4 * q^33 + 17 * q^35 + 4 * q^37 + 4 * q^39 + 7 * q^41 - 3 * q^43 + q^45 - 32 * q^47 + 5 * q^49 + 2 * q^51 + 24 * q^53 - q^55 + 2 * q^57 - q^59 + 7 * q^61 - q^63 - q^65 + 5 * q^67 - q^69 - 18 * q^71 - q^73 - 17 * q^75 + q^77 - 22 * q^79 + 4 * q^81 - 22 * q^83 - 20 * q^85 - q^87 - 22 * q^89 + q^91 + 24 * q^93 - 14 * q^95 + 4 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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