LMFDB - Embedded newform 6384.2.a.cd.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6384,2,Mod(1,6384)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6384, 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("6384.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6384 = 2^{4} \cdot 3 \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6384.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:	$50.9764966504$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.401584.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 7x^{3} + 8x^{2} + 12x - 4 $ x^5 - 2x^4 - 7x^3 + 8x^2 + 12x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 3192)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.06003$ of defining polynomial
Character	$\chi$	$=$	6384.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.73631 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.73631 q^{5} -1.00000 q^{7} +1.00000 q^{9} +4.12005 q^{11} -7.04948 q^{13} +1.73631 q^{15} +4.17833 q^{17} -1.00000 q^{19} +1.00000 q^{21} -2.20541 q^{23} -1.98522 q^{25} -1.00000 q^{27} +7.41435 q^{29} -3.42723 q^{31} -4.12005 q^{33} +1.73631 q^{35} +10.1843 q^{37} +7.04948 q^{39} +1.58918 q^{41} -3.37144 q^{43} -1.73631 q^{45} +6.24259 q^{47} +1.00000 q^{49} -4.17833 q^{51} +7.41435 q^{53} -7.15370 q^{55} +1.00000 q^{57} -11.3561 q^{59} -2.00000 q^{61} -1.00000 q^{63} +12.2401 q^{65} +0.236605 q^{67} +2.20541 q^{69} -11.8006 q^{71} +5.47263 q^{73} +1.98522 q^{75} -4.12005 q^{77} -2.20541 q^{79} +1.00000 q^{81} -6.41493 q^{83} -7.25489 q^{85} -7.41435 q^{87} +16.5098 q^{89} +7.04948 q^{91} +3.42723 q^{93} +1.73631 q^{95} -0.149620 q^{97} +4.12005 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{3} + 2 q^{5} - 5 q^{7} + 5 q^{9}+O(q^{10}) $ 5 * q - 5 * q^3 + 2 * q^5 - 5 * q^7 + 5 * q^9	$ 5 q - 5 q^{3} + 2 q^{5} - 5 q^{7} + 5 q^{9} - 6 q^{11} - 2 q^{15} + 6 q^{17} - 5 q^{19} + 5 q^{21} - 10 q^{23} + 7 q^{25} - 5 q^{27} + 4 q^{29} + 4 q^{31} + 6 q^{33} - 2 q^{35} + 6 q^{37} + 10 q^{41} - 4 q^{43} + 2 q^{45} - 2 q^{47} + 5 q^{49} - 6 q^{51} + 4 q^{53} - 8 q^{55} + 5 q^{57} - 12 q^{59} - 10 q^{61} - 5 q^{63} + 8 q^{65} - 2 q^{67} + 10 q^{69} - 30 q^{71} + 6 q^{73} - 7 q^{75} + 6 q^{77} - 10 q^{79} + 5 q^{81} - 14 q^{83} - 4 q^{87} + 10 q^{89} - 4 q^{93} - 2 q^{95} - 8 q^{97} - 6 q^{99}+O(q^{100}) $ 5 * q - 5 * q^3 + 2 * q^5 - 5 * q^7 + 5 * q^9 - 6 * q^11 - 2 * q^15 + 6 * q^17 - 5 * q^19 + 5 * q^21 - 10 * q^23 + 7 * q^25 - 5 * q^27 + 4 * q^29 + 4 * q^31 + 6 * q^33 - 2 * q^35 + 6 * q^37 + 10 * q^41 - 4 * q^43 + 2 * q^45 - 2 * q^47 + 5 * q^49 - 6 * q^51 + 4 * q^53 - 8 * q^55 + 5 * q^57 - 12 * q^59 - 10 * q^61 - 5 * q^63 + 8 * q^65 - 2 * q^67 + 10 * q^69 - 30 * q^71 + 6 * q^73 - 7 * q^75 + 6 * q^77 - 10 * q^79 + 5 * q^81 - 14 * q^83 - 4 * q^87 + 10 * q^89 - 4 * q^93 - 2 * q^95 - 8 * q^97 - 6 * 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
$5$	−1.73631	−0.776503	−0.388252	−	0.921553i	$-0.626921\pi$
$5$	−1.73631	−0.776503	−0.388252	+	0.921553i	$0.626921\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.12005	1.24224	0.621121	−	0.783715i	$-0.286677\pi$
$11$	4.12005	1.24224	0.621121	+	0.783715i	$0.286677\pi$
$12$	0	0
$13$	−7.04948	−1.95517	−0.977587	−	0.210534i	$-0.932480\pi$
$13$	−7.04948	−1.95517	−0.977587	+	0.210534i	$0.932480\pi$
$14$	0	0
$15$	1.73631	0.448314
$16$	0	0
$17$	4.17833	1.01339	0.506697	−	0.862124i	$-0.330866\pi$
$17$	4.17833	1.01339	0.506697	+	0.862124i	$0.330866\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	−2.20541	−0.459860	−0.229930	−	0.973207i	$-0.573850\pi$
$23$	−2.20541	−0.459860	−0.229930	+	0.973207i	$0.573850\pi$
$24$	0	0
$25$	−1.98522	−0.397043
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	7.41435	1.37681	0.688405	−	0.725326i	$-0.258311\pi$
$29$	7.41435	1.37681	0.688405	+	0.725326i	$0.258311\pi$
$30$	0	0
$31$	−3.42723	−0.615549	−0.307774	−	0.951459i	$-0.599584\pi$
$31$	−3.42723	−0.615549	−0.307774	+	0.951459i	$0.599584\pi$
$32$	0	0
$33$	−4.12005	−0.717209
$34$	0	0
$35$	1.73631	0.293491
$36$	0	0
$37$	10.1843	1.67429	0.837145	−	0.546980i	$-0.184223\pi$
$37$	10.1843	1.67429	0.837145	+	0.546980i	$0.184223\pi$
$38$	0	0
$39$	7.04948	1.12882
$40$	0	0
$41$	1.58918	0.248188	0.124094	−	0.992270i	$-0.460398\pi$
$41$	1.58918	0.248188	0.124094	+	0.992270i	$0.460398\pi$
$42$	0	0
$43$	−3.37144	−0.514140	−0.257070	−	0.966393i	$-0.582757\pi$
$43$	−3.37144	−0.514140	−0.257070	+	0.966393i	$0.582757\pi$
$44$	0	0
$45$	−1.73631	−0.258834
$46$	0	0
$47$	6.24259	0.910575	0.455288	−	0.890344i	$-0.349536\pi$
$47$	6.24259	0.910575	0.455288	+	0.890344i	$0.349536\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−4.17833	−0.585083
$52$	0	0
$53$	7.41435	1.01844	0.509220	−	0.860637i	$-0.329934\pi$
$53$	7.41435	1.01844	0.509220	+	0.860637i	$0.329934\pi$
$54$	0	0
$55$	−7.15370	−0.964605
$56$	0	0
$57$	1.00000	0.132453
$58$	0	0
$59$	−11.3561	−1.47843	−0.739217	−	0.673467i	$-0.764804\pi$
$59$	−11.3561	−1.47843	−0.739217	+	0.673467i	$0.764804\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	12.2401	1.51820
$66$	0	0
$67$	0.236605	0.0289059	0.0144529	−	0.999896i	$-0.495399\pi$
$67$	0.236605	0.0289059	0.0144529	+	0.999896i	$0.495399\pi$
$68$	0	0
$69$	2.20541	0.265500
$70$	0	0
$71$	−11.8006	−1.40047	−0.700235	−	0.713912i	$-0.746922\pi$
$71$	−11.8006	−1.40047	−0.700235	+	0.713912i	$0.746922\pi$
$72$	0	0
$73$	5.47263	0.640523	0.320261	−	0.947329i	$-0.396229\pi$
$73$	5.47263	0.640523	0.320261	+	0.947329i	$0.396229\pi$
$74$	0	0
$75$	1.98522	0.229233
$76$	0	0
$77$	−4.12005	−0.469523
$78$	0	0
$79$	−2.20541	−0.248128	−0.124064	−	0.992274i	$-0.539593\pi$
$79$	−2.20541	−0.248128	−0.124064	+	0.992274i	$0.539593\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−6.41493	−0.704130	−0.352065	−	0.935975i	$-0.614520\pi$
$83$	−6.41493	−0.704130	−0.352065	+	0.935975i	$0.614520\pi$
$84$	0	0
$85$	−7.25489	−0.786903
$86$	0	0
$87$	−7.41435	−0.794902
$88$	0	0
$89$	16.5098	1.75003	0.875016	−	0.484093i	$-0.160851\pi$
$89$	16.5098	1.75003	0.875016	+	0.484093i	$0.160851\pi$
$90$	0	0
$91$	7.04948	0.738986
$92$	0	0
$93$	3.42723	0.355387
$94$	0	0
$95$	1.73631	0.178142
$96$	0	0
$97$	−0.149620	−0.0151917	−0.00759583	−	0.999971i	$-0.502418\pi$
$97$	−0.149620	−0.0151917	−0.00759583	+	0.999971i	$0.502418\pi$
$98$	0	0
$99$	4.12005	0.414081
$100$	0	0
$101$	−17.3375	−1.72514	−0.862571	−	0.505936i	$-0.831147\pi$
$101$	−17.3375	−1.72514	−0.862571	+	0.505936i	$0.831147\pi$
$102$	0	0
$103$	−1.91464	−0.188655	−0.0943276	−	0.995541i	$-0.530070\pi$
$103$	−1.91464	−0.188655	−0.0943276	+	0.995541i	$0.530070\pi$
$104$	0	0
$105$	−1.73631	−0.169447
$106$	0	0
$107$	6.18183	0.597620	0.298810	−	0.954313i	$-0.403410\pi$
$107$	6.18183	0.597620	0.298810	+	0.954313i	$0.403410\pi$
$108$	0	0
$109$	−18.0678	−1.73058	−0.865289	−	0.501274i	$-0.832865\pi$
$109$	−18.0678	−1.73058	−0.865289	+	0.501274i	$0.832865\pi$
$110$	0	0
$111$	−10.1843	−0.966652
$112$	0	0
$113$	7.00353	0.658837	0.329418	−	0.944184i	$-0.393147\pi$
$113$	7.00353	0.658837	0.329418	+	0.944184i	$0.393147\pi$
$114$	0	0
$115$	3.82928	0.357082
$116$	0	0
$117$	−7.04948	−0.651724
$118$	0	0
$119$	−4.17833	−0.383027
$120$	0	0
$121$	5.97482	0.543166
$122$	0	0
$123$	−1.58918	−0.143292
$124$	0	0
$125$	12.1285	1.08481
$126$	0	0
$127$	12.8317	1.13863	0.569316	−	0.822119i	$-0.307208\pi$
$127$	12.8317	1.13863	0.569316	+	0.822119i	$0.307208\pi$
$128$	0	0
$129$	3.37144	0.296839
$130$	0	0
$131$	−2.05828	−0.179832	−0.0899162	−	0.995949i	$-0.528660\pi$
$131$	−2.05828	−0.179832	−0.0899162	+	0.995949i	$0.528660\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	1.73631	0.149438
$136$	0	0
$137$	13.3561	1.14109	0.570543	−	0.821268i	$-0.306733\pi$
$137$	13.3561	1.14109	0.570543	+	0.821268i	$0.306733\pi$
$138$	0	0
$139$	16.1859	1.37287	0.686437	−	0.727190i	$-0.259174\pi$
$139$	16.1859	1.37287	0.686437	+	0.727190i	$0.259174\pi$
$140$	0	0
$141$	−6.24259	−0.525721
$142$	0	0
$143$	−29.0442	−2.42880
$144$	0	0
$145$	−12.8736	−1.06910
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	−11.7981	−0.966537	−0.483269	−	0.875472i	$-0.660551\pi$
$149$	−11.7981	−0.966537	−0.483269	+	0.875472i	$0.660551\pi$
$150$	0	0
$151$	16.6233	1.35278	0.676392	−	0.736542i	$-0.263543\pi$
$151$	16.6233	1.35278	0.676392	+	0.736542i	$0.263543\pi$
$152$	0	0
$153$	4.17833	0.337798
$154$	0	0
$155$	5.95075	0.477975
$156$	0	0
$157$	−16.9830	−1.35539	−0.677695	−	0.735343i	$-0.737021\pi$
$157$	−16.9830	−1.35539	−0.677695	+	0.735343i	$0.737021\pi$
$158$	0	0
$159$	−7.41435	−0.587996
$160$	0	0
$161$	2.20541	0.173811
$162$	0	0
$163$	−15.6115	−1.22279	−0.611395	−	0.791325i	$-0.709391\pi$
$163$	−15.6115	−1.22279	−0.611395	+	0.791325i	$0.709391\pi$
$164$	0	0
$165$	7.15370	0.556915
$166$	0	0
$167$	−6.58860	−0.509841	−0.254921	−	0.966962i	$-0.582049\pi$
$167$	−6.58860	−0.509841	−0.254921	+	0.966962i	$0.582049\pi$
$168$	0	0
$169$	36.6951	2.82270
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	0	0
$173$	−5.11597	−0.388960	−0.194480	−	0.980906i	$-0.562302\pi$
$173$	−5.11597	−0.388960	−0.194480	+	0.980906i	$0.562302\pi$
$174$	0	0
$175$	1.98522	0.150068
$176$	0	0
$177$	11.3561	0.853575
$178$	0	0
$179$	−11.9122	−0.890356	−0.445178	−	0.895442i	$-0.646860\pi$
$179$	−11.9122	−0.890356	−0.445178	+	0.895442i	$0.646860\pi$
$180$	0	0
$181$	−12.9187	−0.960241	−0.480121	−	0.877203i	$-0.659407\pi$
$181$	−12.9187	−0.960241	−0.480121	+	0.877203i	$0.659407\pi$
$182$	0	0
$183$	2.00000	0.147844
$184$	0	0
$185$	−17.6832	−1.30009
$186$	0	0
$187$	17.2149	1.25888
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	11.2496	0.813994	0.406997	−	0.913430i	$-0.366576\pi$
$191$	11.2496	0.813994	0.406997	+	0.913430i	$0.366576\pi$
$192$	0	0
$193$	18.8041	1.35355	0.676775	−	0.736190i	$-0.263377\pi$
$193$	18.8041	1.35355	0.676775	+	0.736190i	$0.263377\pi$
$194$	0	0
$195$	−12.2401	−0.876532
$196$	0	0
$197$	−2.93982	−0.209453	−0.104727	−	0.994501i	$-0.533397\pi$
$197$	−2.93982	−0.209453	−0.104727	+	0.994501i	$0.533397\pi$
$198$	0	0
$199$	−25.1608	−1.78360	−0.891800	−	0.452431i	$-0.850557\pi$
$199$	−25.1608	−1.78360	−0.891800	+	0.452431i	$0.850557\pi$
$200$	0	0
$201$	−0.236605	−0.0166888
$202$	0	0
$203$	−7.41435	−0.520385
$204$	0	0
$205$	−2.75932	−0.192719
$206$	0	0
$207$	−2.20541	−0.153287
$208$	0	0
$209$	−4.12005	−0.284990
$210$	0	0
$211$	−27.1938	−1.87210	−0.936050	−	0.351866i	$-0.885547\pi$
$211$	−27.1938	−1.87210	−0.936050	+	0.351866i	$0.885547\pi$
$212$	0	0
$213$	11.8006	0.808562
$214$	0	0
$215$	5.85388	0.399231
$216$	0	0
$217$	3.42723	0.232656
$218$	0	0
$219$	−5.47263	−0.369806
$220$	0	0
$221$	−29.4550	−1.98136
$222$	0	0
$223$	18.5010	1.23892	0.619460	−	0.785029i	$-0.287352\pi$
$223$	18.5010	1.23892	0.619460	+	0.785029i	$0.287352\pi$
$224$	0	0
$225$	−1.98522	−0.132348
$226$	0	0
$227$	−2.38623	−0.158379	−0.0791897	−	0.996860i	$-0.525233\pi$
$227$	−2.38623	−0.158379	−0.0791897	+	0.996860i	$0.525233\pi$
$228$	0	0
$229$	−11.2030	−0.740312	−0.370156	−	0.928970i	$-0.620696\pi$
$229$	−11.2030	−0.740312	−0.370156	+	0.928970i	$0.620696\pi$
$230$	0	0
$231$	4.12005	0.271079
$232$	0	0
$233$	−14.8960	−0.975869	−0.487935	−	0.872880i	$-0.662250\pi$
$233$	−14.8960	−0.975869	−0.487935	+	0.872880i	$0.662250\pi$
$234$	0	0
$235$	−10.8391	−0.707064
$236$	0	0
$237$	2.20541	0.143257
$238$	0	0
$239$	−26.6315	−1.72265	−0.861323	−	0.508058i	$-0.830363\pi$
$239$	−26.6315	−1.72265	−0.861323	+	0.508058i	$0.830363\pi$
$240$	0	0
$241$	6.94169	0.447154	0.223577	−	0.974686i	$-0.428227\pi$
$241$	6.94169	0.447154	0.223577	+	0.974686i	$0.428227\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−1.73631	−0.110929
$246$	0	0
$247$	7.04948	0.448548
$248$	0	0
$249$	6.41493	0.406530
$250$	0	0
$251$	−8.96690	−0.565986	−0.282993	−	0.959122i	$-0.591327\pi$
$251$	−8.96690	−0.565986	−0.282993	+	0.959122i	$0.591327\pi$
$252$	0	0
$253$	−9.08640	−0.571257
$254$	0	0
$255$	7.25489	0.454319
$256$	0	0
$257$	5.09954	0.318100	0.159050	−	0.987270i	$-0.449157\pi$
$257$	5.09954	0.318100	0.159050	+	0.987270i	$0.449157\pi$
$258$	0	0
$259$	−10.1843	−0.632822
$260$	0	0
$261$	7.41435	0.458937
$262$	0	0
$263$	−27.9055	−1.72073	−0.860364	−	0.509680i	$-0.829764\pi$
$263$	−27.9055	−1.72073	−0.860364	+	0.509680i	$0.829764\pi$
$264$	0	0
$265$	−12.8736	−0.790821
$266$	0	0
$267$	−16.5098	−1.01038
$268$	0	0
$269$	23.1361	1.41063	0.705317	−	0.708893i	$-0.250805\pi$
$269$	23.1361	1.41063	0.705317	+	0.708893i	$0.250805\pi$
$270$	0	0
$271$	−28.4003	−1.72519	−0.862597	−	0.505891i	$-0.831164\pi$
$271$	−28.4003	−1.72519	−0.862597	+	0.505891i	$0.831164\pi$
$272$	0	0
$273$	−7.04948	−0.426654
$274$	0	0
$275$	−8.17919	−0.493224
$276$	0	0
$277$	21.9682	1.31994	0.659971	−	0.751291i	$-0.270569\pi$
$277$	21.9682	1.31994	0.659971	+	0.751291i	$0.270569\pi$
$278$	0	0
$279$	−3.42723	−0.205183
$280$	0	0
$281$	−33.0778	−1.97326	−0.986629	−	0.162983i	$-0.947888\pi$
$281$	−33.0778	−1.97326	−0.986629	+	0.162983i	$0.947888\pi$
$282$	0	0
$283$	0.294267	0.0174923	0.00874616	−	0.999962i	$-0.497216\pi$
$283$	0.294267	0.0174923	0.00874616	+	0.999962i	$0.497216\pi$
$284$	0	0
$285$	−1.73631	−0.102850
$286$	0	0
$287$	−1.58918	−0.0938063
$288$	0	0
$289$	0.458425	0.0269662
$290$	0	0
$291$	0.149620	0.00877091
$292$	0	0
$293$	−8.12361	−0.474587	−0.237293	−	0.971438i	$-0.576260\pi$
$293$	−8.12361	−0.474587	−0.237293	+	0.971438i	$0.576260\pi$
$294$	0	0
$295$	19.7177	1.14801
$296$	0	0
$297$	−4.12005	−0.239070
$298$	0	0
$299$	15.5470	0.899105
$300$	0	0
$301$	3.37144	0.194327
$302$	0	0
$303$	17.3375	0.996011
$304$	0	0
$305$	3.47263	0.198842
$306$	0	0
$307$	15.7023	0.896180	0.448090	−	0.893989i	$-0.352104\pi$
$307$	15.7023	0.896180	0.448090	+	0.893989i	$0.352104\pi$
$308$	0	0
$309$	1.91464	0.108920
$310$	0	0
$311$	−13.6786	−0.775641	−0.387821	−	0.921735i	$-0.626772\pi$
$311$	−13.6786	−0.775641	−0.387821	+	0.921735i	$0.626772\pi$
$312$	0	0
$313$	−18.3517	−1.03730	−0.518649	−	0.854987i	$-0.673565\pi$
$313$	−18.3517	−1.03730	−0.518649	+	0.854987i	$0.673565\pi$
$314$	0	0
$315$	1.73631	0.0978302
$316$	0	0
$317$	−13.5013	−0.758310	−0.379155	−	0.925333i	$-0.623785\pi$
$317$	−13.5013	−0.758310	−0.379155	+	0.925333i	$0.623785\pi$
$318$	0	0
$319$	30.5475	1.71033
$320$	0	0
$321$	−6.18183	−0.345036
$322$	0	0
$323$	−4.17833	−0.232488
$324$	0	0
$325$	13.9947	0.776288
$326$	0	0
$327$	18.0678	0.999149
$328$	0	0
$329$	−6.24259	−0.344165
$330$	0	0
$331$	−20.5428	−1.12914	−0.564568	−	0.825386i	$-0.690957\pi$
$331$	−20.5428	−1.12914	−0.564568	+	0.825386i	$0.690957\pi$
$332$	0	0
$333$	10.1843	0.558097
$334$	0	0
$335$	−0.410820	−0.0224455
$336$	0	0
$337$	7.17830	0.391027	0.195513	−	0.980701i	$-0.437363\pi$
$337$	7.17830	0.391027	0.195513	+	0.980701i	$0.437363\pi$
$338$	0	0
$339$	−7.00353	−0.380380
$340$	0	0
$341$	−14.1204	−0.764661
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−3.82928	−0.206162
$346$	0	0
$347$	−27.9236	−1.49902	−0.749508	−	0.661995i	$-0.769710\pi$
$347$	−27.9236	−1.49902	−0.749508	+	0.661995i	$0.769710\pi$
$348$	0	0
$349$	−30.9818	−1.65842	−0.829209	−	0.558938i	$-0.811209\pi$
$349$	−30.9818	−1.65842	−0.829209	+	0.558938i	$0.811209\pi$
$350$	0	0
$351$	7.04948	0.376273
$352$	0	0
$353$	−8.30685	−0.442129	−0.221065	−	0.975259i	$-0.570953\pi$
$353$	−8.30685	−0.442129	−0.221065	+	0.975259i	$0.570953\pi$
$354$	0	0
$355$	20.4895	1.08747
$356$	0	0
$357$	4.17833	0.221141
$358$	0	0
$359$	11.7228	0.618707	0.309354	−	0.950947i	$-0.399887\pi$
$359$	11.7228	0.618707	0.309354	+	0.950947i	$0.399887\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−5.97482	−0.313597
$364$	0	0
$365$	−9.50220	−0.497368
$366$	0	0
$367$	3.68107	0.192150	0.0960752	−	0.995374i	$-0.469371\pi$
$367$	3.68107	0.192150	0.0960752	+	0.995374i	$0.469371\pi$
$368$	0	0
$369$	1.58918	0.0827294
$370$	0	0
$371$	−7.41435	−0.384934
$372$	0	0
$373$	−10.8282	−0.560665	−0.280332	−	0.959903i	$-0.590445\pi$
$373$	−10.8282	−0.560665	−0.280332	+	0.959903i	$0.590445\pi$
$374$	0	0
$375$	−12.1285	−0.626314
$376$	0	0
$377$	−52.2673	−2.69190
$378$	0	0
$379$	34.2140	1.75746	0.878728	−	0.477322i	$-0.158393\pi$
$379$	34.2140	1.75746	0.878728	+	0.477322i	$0.158393\pi$
$380$	0	0
$381$	−12.8317	−0.657390
$382$	0	0
$383$	−8.85827	−0.452636	−0.226318	−	0.974053i	$-0.572669\pi$
$383$	−8.85827	−0.452636	−0.226318	+	0.974053i	$0.572669\pi$
$384$	0	0
$385$	7.15370	0.364586
$386$	0	0
$387$	−3.37144	−0.171380
$388$	0	0
$389$	9.09294	0.461030	0.230515	−	0.973069i	$-0.425959\pi$
$389$	9.09294	0.461030	0.230515	+	0.973069i	$0.425959\pi$
$390$	0	0
$391$	−9.21493	−0.466019
$392$	0	0
$393$	2.05828	0.103826
$394$	0	0
$395$	3.82928	0.192672
$396$	0	0
$397$	−9.87148	−0.495435	−0.247718	−	0.968832i	$-0.579681\pi$
$397$	−9.87148	−0.495435	−0.247718	+	0.968832i	$0.579681\pi$
$398$	0	0
$399$	−1.00000	−0.0500626
$400$	0	0
$401$	−14.6255	−0.730361	−0.365180	−	0.930937i	$-0.618993\pi$
$401$	−14.6255	−0.730361	−0.365180	+	0.930937i	$0.618993\pi$
$402$	0	0
$403$	24.1602	1.20350
$404$	0	0
$405$	−1.73631	−0.0862781
$406$	0	0
$407$	41.9599	2.07987
$408$	0	0
$409$	−18.1966	−0.899763	−0.449881	−	0.893088i	$-0.648534\pi$
$409$	−18.1966	−0.899763	−0.449881	+	0.893088i	$0.648534\pi$
$410$	0	0
$411$	−13.3561	−0.658807
$412$	0	0
$413$	11.3561	0.558796
$414$	0	0
$415$	11.1383	0.546759
$416$	0	0
$417$	−16.1859	−0.792629
$418$	0	0
$419$	−30.1654	−1.47368	−0.736838	−	0.676069i	$-0.763682\pi$
$419$	−30.1654	−1.47368	−0.736838	+	0.676069i	$0.763682\pi$
$420$	0	0
$421$	8.85283	0.431461	0.215730	−	0.976453i	$-0.430787\pi$
$421$	8.85283	0.431461	0.215730	+	0.976453i	$0.430787\pi$
$422$	0	0
$423$	6.24259	0.303525
$424$	0	0
$425$	−8.29488	−0.402361
$426$	0	0
$427$	2.00000	0.0967868
$428$	0	0
$429$	29.0442	1.40227
$430$	0	0
$431$	−24.8328	−1.19615	−0.598077	−	0.801438i	$-0.704068\pi$
$431$	−24.8328	−1.19615	−0.598077	+	0.801438i	$0.704068\pi$
$432$	0	0
$433$	37.5625	1.80514	0.902568	−	0.430547i	$-0.141679\pi$
$433$	37.5625	1.80514	0.902568	+	0.430547i	$0.141679\pi$
$434$	0	0
$435$	12.8736	0.617244
$436$	0	0
$437$	2.20541	0.105499
$438$	0	0
$439$	7.59795	0.362630	0.181315	−	0.983425i	$-0.441965\pi$
$439$	7.59795	0.362630	0.181315	+	0.983425i	$0.441965\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	8.20587	0.389873	0.194936	−	0.980816i	$-0.437550\pi$
$443$	8.20587	0.389873	0.194936	+	0.980816i	$0.437550\pi$
$444$	0	0
$445$	−28.6661	−1.35891
$446$	0	0
$447$	11.7981	0.558030
$448$	0	0
$449$	18.8082	0.887612	0.443806	−	0.896123i	$-0.353628\pi$
$449$	18.8082	0.887612	0.443806	+	0.896123i	$0.353628\pi$
$450$	0	0
$451$	6.54750	0.308310
$452$	0	0
$453$	−16.6233	−0.781030
$454$	0	0
$455$	−12.2401	−0.573825
$456$	0	0
$457$	−8.23087	−0.385024	−0.192512	−	0.981295i	$-0.561663\pi$
$457$	−8.23087	−0.385024	−0.192512	+	0.981295i	$0.561663\pi$
$458$	0	0
$459$	−4.17833	−0.195028
$460$	0	0
$461$	−19.3254	−0.900075	−0.450037	−	0.893010i	$-0.648589\pi$
$461$	−19.3254	−0.900075	−0.450037	+	0.893010i	$0.648589\pi$
$462$	0	0
$463$	−18.4299	−0.856508	−0.428254	−	0.903658i	$-0.640871\pi$
$463$	−18.4299	−0.856508	−0.428254	+	0.903658i	$0.640871\pi$
$464$	0	0
$465$	−5.95075	−0.275959
$466$	0	0
$467$	11.3590	0.525633	0.262816	−	0.964846i	$-0.415349\pi$
$467$	11.3590	0.525633	0.262816	+	0.964846i	$0.415349\pi$
$468$	0	0
$469$	−0.236605	−0.0109254
$470$	0	0
$471$	16.9830	0.782535
$472$	0	0
$473$	−13.8905	−0.638686
$474$	0	0
$475$	1.98522	0.0910879
$476$	0	0
$477$	7.41435	0.339480
$478$	0	0
$479$	−32.2508	−1.47357	−0.736787	−	0.676125i	$-0.763658\pi$
$479$	−32.2508	−1.47357	−0.736787	+	0.676125i	$0.763658\pi$
$480$	0	0
$481$	−71.7941	−3.27353
$482$	0	0
$483$	−2.20541	−0.100350
$484$	0	0
$485$	0.259788	0.0117964
$486$	0	0
$487$	−30.6434	−1.38859	−0.694293	−	0.719692i	$-0.744283\pi$
$487$	−30.6434	−1.38859	−0.694293	+	0.719692i	$0.744283\pi$
$488$	0	0
$489$	15.6115	0.705978
$490$	0	0
$491$	1.85535	0.0837310	0.0418655	−	0.999123i	$-0.486670\pi$
$491$	1.85535	0.0837310	0.0418655	+	0.999123i	$0.486670\pi$
$492$	0	0
$493$	30.9796	1.39525
$494$	0	0
$495$	−7.15370	−0.321535
$496$	0	0
$497$	11.8006	0.529328
$498$	0	0
$499$	−18.6837	−0.836399	−0.418200	−	0.908355i	$-0.637339\pi$
$499$	−18.6837	−0.836399	−0.418200	+	0.908355i	$0.637339\pi$
$500$	0	0
$501$	6.58860	0.294357
$502$	0	0
$503$	−6.54955	−0.292030	−0.146015	−	0.989282i	$-0.546645\pi$
$503$	−6.54955	−0.292030	−0.146015	+	0.989282i	$0.546645\pi$
$504$	0	0
$505$	30.1033	1.33958
$506$	0	0
$507$	−36.6951	−1.62969
$508$	0	0
$509$	−22.1166	−0.980299	−0.490149	−	0.871638i	$-0.663058\pi$
$509$	−22.1166	−0.980299	−0.490149	+	0.871638i	$0.663058\pi$
$510$	0	0
$511$	−5.47263	−0.242095
$512$	0	0
$513$	1.00000	0.0441511
$514$	0	0
$515$	3.32442	0.146491
$516$	0	0
$517$	25.7198	1.13116
$518$	0	0
$519$	5.11597	0.224566
$520$	0	0
$521$	−2.08582	−0.0913814	−0.0456907	−	0.998956i	$-0.514549\pi$
$521$	−2.08582	−0.0913814	−0.0456907	+	0.998956i	$0.514549\pi$
$522$	0	0
$523$	−9.67277	−0.422961	−0.211480	−	0.977382i	$-0.567828\pi$
$523$	−9.67277	−0.422961	−0.211480	+	0.977382i	$0.567828\pi$
$524$	0	0
$525$	−1.98522	−0.0866419
$526$	0	0
$527$	−14.3201	−0.623793
$528$	0	0
$529$	−18.1362	−0.788529
$530$	0	0
$531$	−11.3561	−0.492812
$532$	0	0
$533$	−11.2029	−0.485251
$534$	0	0
$535$	−10.7336	−0.464053
$536$	0	0
$537$	11.9122	0.514047
$538$	0	0
$539$	4.12005	0.177463
$540$	0	0
$541$	−20.3074	−0.873083	−0.436542	−	0.899684i	$-0.643797\pi$
$541$	−20.3074	−0.873083	−0.436542	+	0.899684i	$0.643797\pi$
$542$	0	0
$543$	12.9187	0.554395
$544$	0	0
$545$	31.3713	1.34380
$546$	0	0
$547$	−18.0483	−0.771689	−0.385845	−	0.922564i	$-0.626090\pi$
$547$	−18.0483	−0.771689	−0.385845	+	0.922564i	$0.626090\pi$
$548$	0	0
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	−7.41435	−0.315862
$552$	0	0
$553$	2.20541	0.0937836
$554$	0	0
$555$	17.6832	0.750608
$556$	0	0
$557$	18.3703	0.778373	0.389186	−	0.921159i	$-0.372756\pi$
$557$	18.3703	0.778373	0.389186	+	0.921159i	$0.372756\pi$
$558$	0	0
$559$	23.7669	1.00523
$560$	0	0
$561$	−17.2149	−0.726815
$562$	0	0
$563$	37.4386	1.57785	0.788924	−	0.614490i	$-0.210638\pi$
$563$	37.4386	1.57785	0.788924	+	0.614490i	$0.210638\pi$
$564$	0	0
$565$	−12.1603	−0.511589
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−3.58507	−0.150294	−0.0751469	−	0.997172i	$-0.523943\pi$
$569$	−3.58507	−0.150294	−0.0751469	+	0.997172i	$0.523943\pi$
$570$	0	0
$571$	−14.1613	−0.592634	−0.296317	−	0.955090i	$-0.595758\pi$
$571$	−14.1613	−0.592634	−0.296317	+	0.955090i	$0.595758\pi$
$572$	0	0
$573$	−11.2496	−0.469960
$574$	0	0
$575$	4.37821	0.182584
$576$	0	0
$577$	8.74288	0.363971	0.181985	−	0.983301i	$-0.441748\pi$
$577$	8.74288	0.363971	0.181985	+	0.983301i	$0.441748\pi$
$578$	0	0
$579$	−18.8041	−0.781472
$580$	0	0
$581$	6.41493	0.266136
$582$	0	0
$583$	30.5475	1.26515
$584$	0	0
$585$	12.2401	0.506066
$586$	0	0
$587$	−5.55931	−0.229457	−0.114729	−	0.993397i	$-0.536600\pi$
$587$	−5.55931	−0.229457	−0.114729	+	0.993397i	$0.536600\pi$
$588$	0	0
$589$	3.42723	0.141217
$590$	0	0
$591$	2.93982	0.120928
$592$	0	0
$593$	35.0190	1.43806	0.719029	−	0.694980i	$-0.244587\pi$
$593$	35.0190	1.43806	0.719029	+	0.694980i	$0.244587\pi$
$594$	0	0
$595$	7.25489	0.297421
$596$	0	0
$597$	25.1608	1.02976
$598$	0	0
$599$	−5.77101	−0.235797	−0.117898	−	0.993026i	$-0.537616\pi$
$599$	−5.77101	−0.235797	−0.117898	+	0.993026i	$0.537616\pi$
$600$	0	0
$601$	0.219006	0.00893344	0.00446672	−	0.999990i	$-0.498578\pi$
$601$	0.219006	0.00893344	0.00446672	+	0.999990i	$0.498578\pi$
$602$	0	0
$603$	0.236605	0.00963528
$604$	0	0
$605$	−10.3742	−0.421770
$606$	0	0
$607$	7.61331	0.309015	0.154507	−	0.987992i	$-0.450621\pi$
$607$	7.61331	0.309015	0.154507	+	0.987992i	$0.450621\pi$
$608$	0	0
$609$	7.41435	0.300445
$610$	0	0
$611$	−44.0070	−1.78033
$612$	0	0
$613$	44.3357	1.79070	0.895350	−	0.445363i	$-0.146925\pi$
$613$	44.3357	1.79070	0.895350	+	0.445363i	$0.146925\pi$
$614$	0	0
$615$	2.75932	0.111266
$616$	0	0
$617$	36.4676	1.46813	0.734065	−	0.679079i	$-0.237621\pi$
$617$	36.4676	1.46813	0.734065	+	0.679079i	$0.237621\pi$
$618$	0	0
$619$	17.8424	0.717147	0.358574	−	0.933501i	$-0.383263\pi$
$619$	17.8424	0.717147	0.358574	+	0.933501i	$0.383263\pi$
$620$	0	0
$621$	2.20541	0.0885000
$622$	0	0
$623$	−16.5098	−0.661450
$624$	0	0
$625$	−11.1328	−0.445314
$626$	0	0
$627$	4.12005	0.164539
$628$	0	0
$629$	42.5534	1.69672
$630$	0	0
$631$	30.6192	1.21893	0.609465	−	0.792813i	$-0.291384\pi$
$631$	30.6192	1.21893	0.609465	+	0.792813i	$0.291384\pi$
$632$	0	0
$633$	27.1938	1.08086
$634$	0	0
$635$	−22.2799	−0.884152
$636$	0	0
$637$	−7.04948	−0.279310
$638$	0	0
$639$	−11.8006	−0.466824
$640$	0	0
$641$	25.9291	1.02414	0.512069	−	0.858944i	$-0.328879\pi$
$641$	25.9291	1.02414	0.512069	+	0.858944i	$0.328879\pi$
$642$	0	0
$643$	19.0935	0.752973	0.376486	−	0.926422i	$-0.377132\pi$
$643$	19.0935	0.752973	0.376486	+	0.926422i	$0.377132\pi$
$644$	0	0
$645$	−5.85388	−0.230496
$646$	0	0
$647$	36.7418	1.44447	0.722235	−	0.691648i	$-0.243115\pi$
$647$	36.7418	1.44447	0.722235	+	0.691648i	$0.243115\pi$
$648$	0	0
$649$	−46.7876	−1.83657
$650$	0	0
$651$	−3.42723	−0.134324
$652$	0	0
$653$	−30.1843	−1.18120	−0.590602	−	0.806963i	$-0.701110\pi$
$653$	−30.1843	−1.18120	−0.590602	+	0.806963i	$0.701110\pi$
$654$	0	0
$655$	3.57381	0.139640
$656$	0	0
$657$	5.47263	0.213508
$658$	0	0
$659$	25.1955	0.981479	0.490740	−	0.871306i	$-0.336727\pi$
$659$	25.1955	0.981479	0.490740	+	0.871306i	$0.336727\pi$
$660$	0	0
$661$	−21.3191	−0.829219	−0.414609	−	0.909999i	$-0.636082\pi$
$661$	−21.3191	−0.829219	−0.414609	+	0.909999i	$0.636082\pi$
$662$	0	0
$663$	29.4550	1.14394
$664$	0	0
$665$	−1.73631	−0.0673313
$666$	0	0
$667$	−16.3517	−0.633140
$668$	0	0
$669$	−18.5010	−0.715290
$670$	0	0
$671$	−8.24010	−0.318106
$672$	0	0
$673$	−14.8111	−0.570926	−0.285463	−	0.958390i	$-0.592147\pi$
$673$	−14.8111	−0.570926	−0.285463	+	0.958390i	$0.592147\pi$
$674$	0	0
$675$	1.98522	0.0764110
$676$	0	0
$677$	−26.9168	−1.03450	−0.517248	−	0.855836i	$-0.673044\pi$
$677$	−26.9168	−1.03450	−0.517248	+	0.855836i	$0.673044\pi$
$678$	0	0
$679$	0.149620	0.00574191
$680$	0	0
$681$	2.38623	0.0914403
$682$	0	0
$683$	30.2676	1.15816	0.579080	−	0.815271i	$-0.303412\pi$
$683$	30.2676	1.15816	0.579080	+	0.815271i	$0.303412\pi$
$684$	0	0
$685$	−23.1903	−0.886057
$686$	0	0
$687$	11.2030	0.427419
$688$	0	0
$689$	−52.2673	−1.99123
$690$	0	0
$691$	−9.68813	−0.368554	−0.184277	−	0.982874i	$-0.558994\pi$
$691$	−9.68813	−0.368554	−0.184277	+	0.982874i	$0.558994\pi$
$692$	0	0
$693$	−4.12005	−0.156508
$694$	0	0
$695$	−28.1039	−1.06604
$696$	0	0
$697$	6.64012	0.251512
$698$	0	0
$699$	14.8960	0.563418
$700$	0	0
$701$	−5.61331	−0.212012	−0.106006	−	0.994365i	$-0.533806\pi$
$701$	−5.61331	−0.212012	−0.106006	+	0.994365i	$0.533806\pi$
$702$	0	0
$703$	−10.1843	−0.384109
$704$	0	0
$705$	10.8391	0.408224
$706$	0	0
$707$	17.3375	0.652042
$708$	0	0
$709$	4.85106	0.182186	0.0910928	−	0.995842i	$-0.470964\pi$
$709$	4.85106	0.182186	0.0910928	+	0.995842i	$0.470964\pi$
$710$	0	0
$711$	−2.20541	−0.0827093
$712$	0	0
$713$	7.55845	0.283066
$714$	0	0
$715$	50.4299	1.88597
$716$	0	0
$717$	26.6315	0.994570
$718$	0	0
$719$	−7.45254	−0.277933	−0.138966	−	0.990297i	$-0.544378\pi$
$719$	−7.45254	−0.277933	−0.138966	+	0.990297i	$0.544378\pi$
$720$	0	0
$721$	1.91464	0.0713050
$722$	0	0
$723$	−6.94169	−0.258164
$724$	0	0
$725$	−14.7191	−0.546653
$726$	0	0
$727$	36.5426	1.35529	0.677645	−	0.735389i	$-0.263001\pi$
$727$	36.5426	1.35529	0.677645	+	0.735389i	$0.263001\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−14.0870	−0.521026
$732$	0	0
$733$	−49.2974	−1.82084	−0.910421	−	0.413682i	$-0.864242\pi$
$733$	−49.2974	−1.82084	−0.910421	+	0.413682i	$0.864242\pi$
$734$	0	0
$735$	1.73631	0.0640449
$736$	0	0
$737$	0.974823	0.0359081
$738$	0	0
$739$	0.520232	0.0191370	0.00956852	−	0.999954i	$-0.496954\pi$
$739$	0.520232	0.0191370	0.00956852	+	0.999954i	$0.496954\pi$
$740$	0	0
$741$	−7.04948	−0.258969
$742$	0	0
$743$	26.1818	0.960518	0.480259	−	0.877127i	$-0.340543\pi$
$743$	26.1818	0.960518	0.480259	+	0.877127i	$0.340543\pi$
$744$	0	0
$745$	20.4852	0.750519
$746$	0	0
$747$	−6.41493	−0.234710
$748$	0	0
$749$	−6.18183	−0.225879
$750$	0	0
$751$	18.7398	0.683824	0.341912	−	0.939732i	$-0.388925\pi$
$751$	18.7398	0.683824	0.341912	+	0.939732i	$0.388925\pi$
$752$	0	0
$753$	8.96690	0.326772
$754$	0	0
$755$	−28.8632	−1.05044
$756$	0	0
$757$	−33.2831	−1.20970	−0.604848	−	0.796341i	$-0.706766\pi$
$757$	−33.2831	−1.20970	−0.604848	+	0.796341i	$0.706766\pi$
$758$	0	0
$759$	9.08640	0.329815
$760$	0	0
$761$	14.3681	0.520842	0.260421	−	0.965495i	$-0.416139\pi$
$761$	14.3681	0.520842	0.260421	+	0.965495i	$0.416139\pi$
$762$	0	0
$763$	18.0678	0.654097
$764$	0	0
$765$	−7.25489	−0.262301
$766$	0	0
$767$	80.0544	2.89060
$768$	0	0
$769$	41.6032	1.50025	0.750124	−	0.661297i	$-0.229994\pi$
$769$	41.6032	1.50025	0.750124	+	0.661297i	$0.229994\pi$
$770$	0	0
$771$	−5.09954	−0.183655
$772$	0	0
$773$	−21.2515	−0.764363	−0.382182	−	0.924087i	$-0.624827\pi$
$773$	−21.2515	−0.764363	−0.382182	+	0.924087i	$0.624827\pi$
$774$	0	0
$775$	6.80379	0.244399
$776$	0	0
$777$	10.1843	0.365360
$778$	0	0
$779$	−1.58918	−0.0569383
$780$	0	0
$781$	−48.6190	−1.73972
$782$	0	0
$783$	−7.41435	−0.264967
$784$	0	0
$785$	29.4878	1.05246
$786$	0	0
$787$	−19.7647	−0.704536	−0.352268	−	0.935899i	$-0.614590\pi$
$787$	−19.7647	−0.704536	−0.352268	+	0.935899i	$0.614590\pi$
$788$	0	0
$789$	27.9055	0.993463
$790$	0	0
$791$	−7.00353	−0.249017
$792$	0	0
$793$	14.0990	0.500669
$794$	0	0
$795$	12.8736	0.456581
$796$	0	0
$797$	13.5163	0.478770	0.239385	−	0.970925i	$-0.423054\pi$
$797$	13.5163	0.478770	0.239385	+	0.970925i	$0.423054\pi$
$798$	0	0
$799$	26.0836	0.922771
$800$	0	0
$801$	16.5098	0.583344
$802$	0	0
$803$	22.5475	0.795684
$804$	0	0
$805$	−3.82928	−0.134964
$806$	0	0
$807$	−23.1361	−0.814429
$808$	0	0
$809$	−3.34345	−0.117549	−0.0587747	−	0.998271i	$-0.518719\pi$
$809$	−3.34345	−0.117549	−0.0587747	+	0.998271i	$0.518719\pi$
$810$	0	0
$811$	−9.01580	−0.316588	−0.158294	−	0.987392i	$-0.550599\pi$
$811$	−9.01580	−0.316588	−0.158294	+	0.987392i	$0.550599\pi$
$812$	0	0
$813$	28.4003	0.996041
$814$	0	0
$815$	27.1065	0.949500
$816$	0	0
$817$	3.37144	0.117952
$818$	0	0
$819$	7.04948	0.246329
$820$	0	0
$821$	10.8025	0.377009	0.188505	−	0.982072i	$-0.439636\pi$
$821$	10.8025	0.377009	0.188505	+	0.982072i	$0.439636\pi$
$822$	0	0
$823$	19.3714	0.675246	0.337623	−	0.941281i	$-0.390377\pi$
$823$	19.3714	0.675246	0.337623	+	0.941281i	$0.390377\pi$
$824$	0	0
$825$	8.17919	0.284763
$826$	0	0
$827$	−21.7982	−0.757998	−0.378999	−	0.925397i	$-0.623732\pi$
$827$	−21.7982	−0.757998	−0.378999	+	0.925397i	$0.623732\pi$
$828$	0	0
$829$	−36.8810	−1.28093	−0.640465	−	0.767987i	$-0.721258\pi$
$829$	−36.8810	−1.28093	−0.640465	+	0.767987i	$0.721258\pi$
$830$	0	0
$831$	−21.9682	−0.762068
$832$	0	0
$833$	4.17833	0.144770
$834$	0	0
$835$	11.4399	0.395893
$836$	0	0
$837$	3.42723	0.118462
$838$	0	0
$839$	20.0070	0.690718	0.345359	−	0.938471i	$-0.387757\pi$
$839$	20.0070	0.690718	0.345359	+	0.938471i	$0.387757\pi$
$840$	0	0
$841$	25.9726	0.895607
$842$	0	0
$843$	33.0778	1.13926
$844$	0	0
$845$	−63.7143	−2.19184
$846$	0	0
$847$	−5.97482	−0.205297
$848$	0	0
$849$	−0.294267	−0.0100992
$850$	0	0
$851$	−22.4606	−0.769939
$852$	0	0
$853$	28.4014	0.972447	0.486224	−	0.873834i	$-0.338374\pi$
$853$	28.4014	0.972447	0.486224	+	0.873834i	$0.338374\pi$
$854$	0	0
$855$	1.73631	0.0593807
$856$	0	0
$857$	−13.7669	−0.470268	−0.235134	−	0.971963i	$-0.575553\pi$
$857$	−13.7669	−0.470268	−0.235134	+	0.971963i	$0.575553\pi$
$858$	0	0
$859$	23.4185	0.799028	0.399514	−	0.916727i	$-0.369179\pi$
$859$	23.4185	0.799028	0.399514	+	0.916727i	$0.369179\pi$
$860$	0	0
$861$	1.58918	0.0541591
$862$	0	0
$863$	−36.7715	−1.25172	−0.625859	−	0.779937i	$-0.715251\pi$
$863$	−36.7715	−1.25172	−0.625859	+	0.779937i	$0.715251\pi$
$864$	0	0
$865$	8.88293	0.302029
$866$	0	0
$867$	−0.458425	−0.0155689
$868$	0	0
$869$	−9.08640	−0.308235
$870$	0	0
$871$	−1.66794	−0.0565159
$872$	0	0
$873$	−0.149620	−0.00506389
$874$	0	0
$875$	−12.1285	−0.410019
$876$	0	0
$877$	4.47539	0.151123	0.0755617	−	0.997141i	$-0.475925\pi$
$877$	4.47539	0.151123	0.0755617	+	0.997141i	$0.475925\pi$
$878$	0	0
$879$	8.12361	0.274003
$880$	0	0
$881$	14.7139	0.495723	0.247861	−	0.968796i	$-0.420272\pi$
$881$	14.7139	0.495723	0.247861	+	0.968796i	$0.420272\pi$
$882$	0	0
$883$	−6.43432	−0.216532	−0.108266	−	0.994122i	$-0.534530\pi$
$883$	−6.43432	−0.216532	−0.108266	+	0.994122i	$0.534530\pi$
$884$	0	0
$885$	−19.7177	−0.662803
$886$	0	0
$887$	40.1929	1.34955	0.674773	−	0.738025i	$-0.264241\pi$
$887$	40.1929	1.34955	0.674773	+	0.738025i	$0.264241\pi$
$888$	0	0
$889$	−12.8317	−0.430363
$890$	0	0
$891$	4.12005	0.138027
$892$	0	0
$893$	−6.24259	−0.208900
$894$	0	0
$895$	20.6832	0.691364
$896$	0	0
$897$	−15.5470	−0.519099
$898$	0	0
$899$	−25.4107	−0.847494
$900$	0	0
$901$	30.9796	1.03208
$902$	0	0
$903$	−3.37144	−0.112195
$904$	0	0
$905$	22.4310	0.745630
$906$	0	0
$907$	−47.8825	−1.58991	−0.794956	−	0.606667i	$-0.792506\pi$
$907$	−47.8825	−1.58991	−0.794956	+	0.606667i	$0.792506\pi$
$908$	0	0
$909$	−17.3375	−0.575047
$910$	0	0
$911$	56.9449	1.88667	0.943334	−	0.331844i	$-0.107671\pi$
$911$	56.9449	1.88667	0.943334	+	0.331844i	$0.107671\pi$
$912$	0	0
$913$	−26.4299	−0.874700
$914$	0	0
$915$	−3.47263	−0.114802
$916$	0	0
$917$	2.05828	0.0679703
$918$	0	0
$919$	−10.4299	−0.344049	−0.172025	−	0.985093i	$-0.555031\pi$
$919$	−10.4299	−0.344049	−0.172025	+	0.985093i	$0.555031\pi$
$920$	0	0
$921$	−15.7023	−0.517410
$922$	0	0
$923$	83.1879	2.73816
$924$	0	0
$925$	−20.2181	−0.664766
$926$	0	0
$927$	−1.91464	−0.0628851
$928$	0	0
$929$	−42.6093	−1.39797	−0.698983	−	0.715139i	$-0.746364\pi$
$929$	−42.6093	−1.39797	−0.698983	+	0.715139i	$0.746364\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	13.6786	0.447817
$934$	0	0
$935$	−29.8905	−0.977524
$936$	0	0
$937$	−32.6214	−1.06569	−0.532847	−	0.846212i	$-0.678878\pi$
$937$	−32.6214	−1.06569	−0.532847	+	0.846212i	$0.678878\pi$
$938$	0	0
$939$	18.3517	0.598884
$940$	0	0
$941$	59.3033	1.93323	0.966616	−	0.256230i	$-0.0824805\pi$
$941$	59.3033	1.93323	0.966616	+	0.256230i	$0.0824805\pi$
$942$	0	0
$943$	−3.50479	−0.114132
$944$	0	0
$945$	−1.73631	−0.0564823
$946$	0	0
$947$	−31.1724	−1.01297	−0.506484	−	0.862249i	$-0.669055\pi$
$947$	−31.1724	−1.01297	−0.506484	+	0.862249i	$0.669055\pi$
$948$	0	0
$949$	−38.5792	−1.25233
$950$	0	0
$951$	13.5013	0.437811
$952$	0	0
$953$	0.879916	0.0285033	0.0142516	−	0.999898i	$-0.495463\pi$
$953$	0.879916	0.0285033	0.0142516	+	0.999898i	$0.495463\pi$
$954$	0	0
$955$	−19.5329	−0.632069
$956$	0	0
$957$	−30.5475	−0.987461
$958$	0	0
$959$	−13.3561	−0.431290
$960$	0	0
$961$	−19.2541	−0.621100
$962$	0	0
$963$	6.18183	0.199207
$964$	0	0
$965$	−32.6498	−1.05103
$966$	0	0
$967$	−17.2089	−0.553400	−0.276700	−	0.960956i	$-0.589241\pi$
$967$	−17.2089	−0.553400	−0.276700	+	0.960956i	$0.589241\pi$
$968$	0	0
$969$	4.17833	0.134227
$970$	0	0
$971$	16.0070	0.513689	0.256844	−	0.966453i	$-0.417317\pi$
$971$	16.0070	0.513689	0.256844	+	0.966453i	$0.417317\pi$
$972$	0	0
$973$	−16.1859	−0.518897
$974$	0	0
$975$	−13.9947	−0.448190
$976$	0	0
$977$	13.5485	0.433455	0.216727	−	0.976232i	$-0.430462\pi$
$977$	13.5485	0.433455	0.216727	+	0.976232i	$0.430462\pi$
$978$	0	0
$979$	68.0211	2.17396
$980$	0	0
$981$	−18.0678	−0.576859
$982$	0	0
$983$	−28.1493	−0.897824	−0.448912	−	0.893576i	$-0.648188\pi$
$983$	−28.1493	−0.897824	−0.448912	+	0.893576i	$0.648188\pi$
$984$	0	0
$985$	5.10445	0.162641
$986$	0	0
$987$	6.24259	0.198704
$988$	0	0
$989$	7.43541	0.236432
$990$	0	0
$991$	−34.3409	−1.09088	−0.545438	−	0.838151i	$-0.683637\pi$
$991$	−34.3409	−1.09088	−0.545438	+	0.838151i	$0.683637\pi$
$992$	0	0
$993$	20.5428	0.651907
$994$	0	0
$995$	43.6870	1.38497
$996$	0	0
$997$	28.9830	0.917900	0.458950	−	0.888462i	$-0.348226\pi$
$997$	28.9830	0.917900	0.458950	+	0.888462i	$0.348226\pi$
$998$	0	0
$999$	−10.1843	−0.322217

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6384.2.a.cd.1.2		5
4.3	odd	2		3192.2.a.bb.1.2	✓	5
12.11	even	2		9576.2.a.cn.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3192.2.a.bb.1.2	✓	5	4.3	odd	2
6384.2.a.cd.1.2		5	1.1	even	1	trivial
9576.2.a.cn.1.4		5	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6384 = 2^{4} \cdot 3 \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6384.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.73631 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.73631 q^{5} -1.00000 q^{7} +1.00000 q^{9} +4.12005 q^{11} -7.04948 q^{13} +1.73631 q^{15} +4.17833 q^{17} -1.00000 q^{19} +1.00000 q^{21} -2.20541 q^{23} -1.98522 q^{25} -1.00000 q^{27} +7.41435 q^{29} -3.42723 q^{31} -4.12005 q^{33} +1.73631 q^{35} +10.1843 q^{37} +7.04948 q^{39} +1.58918 q^{41} -3.37144 q^{43} -1.73631 q^{45} +6.24259 q^{47} +1.00000 q^{49} -4.17833 q^{51} +7.41435 q^{53} -7.15370 q^{55} +1.00000 q^{57} -11.3561 q^{59} -2.00000 q^{61} -1.00000 q^{63} +12.2401 q^{65} +0.236605 q^{67} +2.20541 q^{69} -11.8006 q^{71} +5.47263 q^{73} +1.98522 q^{75} -4.12005 q^{77} -2.20541 q^{79} +1.00000 q^{81} -6.41493 q^{83} -7.25489 q^{85} -7.41435 q^{87} +16.5098 q^{89} +7.04948 q^{91} +3.42723 q^{93} +1.73631 q^{95} -0.149620 q^{97} +4.12005 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{3} + 2 q^{5} - 5 q^{7} + 5 q^{9}+O(q^{10}) \) 5 * q - 5 * q^3 + 2 * q^5 - 5 * q^7 + 5 * q^9	\( 5 q - 5 q^{3} + 2 q^{5} - 5 q^{7} + 5 q^{9} - 6 q^{11} - 2 q^{15} + 6 q^{17} - 5 q^{19} + 5 q^{21} - 10 q^{23} + 7 q^{25} - 5 q^{27} + 4 q^{29} + 4 q^{31} + 6 q^{33} - 2 q^{35} + 6 q^{37} + 10 q^{41} - 4 q^{43} + 2 q^{45} - 2 q^{47} + 5 q^{49} - 6 q^{51} + 4 q^{53} - 8 q^{55} + 5 q^{57} - 12 q^{59} - 10 q^{61} - 5 q^{63} + 8 q^{65} - 2 q^{67} + 10 q^{69} - 30 q^{71} + 6 q^{73} - 7 q^{75} + 6 q^{77} - 10 q^{79} + 5 q^{81} - 14 q^{83} - 4 q^{87} + 10 q^{89} - 4 q^{93} - 2 q^{95} - 8 q^{97} - 6 q^{99}+O(q^{100}) \) 5 * q - 5 * q^3 + 2 * q^5 - 5 * q^7 + 5 * q^9 - 6 * q^11 - 2 * q^15 + 6 * q^17 - 5 * q^19 + 5 * q^21 - 10 * q^23 + 7 * q^25 - 5 * q^27 + 4 * q^29 + 4 * q^31 + 6 * q^33 - 2 * q^35 + 6 * q^37 + 10 * q^41 - 4 * q^43 + 2 * q^45 - 2 * q^47 + 5 * q^49 - 6 * q^51 + 4 * q^53 - 8 * q^55 + 5 * q^57 - 12 * q^59 - 10 * q^61 - 5 * q^63 + 8 * q^65 - 2 * q^67 + 10 * q^69 - 30 * q^71 + 6 * q^73 - 7 * q^75 + 6 * q^77 - 10 * q^79 + 5 * q^81 - 14 * q^83 - 4 * q^87 + 10 * q^89 - 4 * q^93 - 2 * q^95 - 8 * q^97 - 6 * q^99

Introduction

L-functions

Modular forms

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