LMFDB - Embedded newform 3640.2.a.q.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3640,2,Mod(1,3640)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3640, 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("3640.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3640.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:	$29.0655463357$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1573.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 7x + 2 $ x^3 - x^2 - 7*x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.277754$ of defining polynomial
Character	$\chi$	$=$	3640.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.277754 q^{3} +1.00000 q^{5} -1.00000 q^{7} -2.92285 q^{9} +O(q^{10})$	$q+0.277754 q^{3} +1.00000 q^{5} -1.00000 q^{7} -2.92285 q^{9} +4.00000 q^{11} -1.00000 q^{13} +0.277754 q^{15} +4.47836 q^{17} -2.47836 q^{19} -0.277754 q^{21} -2.00000 q^{23} +1.00000 q^{25} -1.64510 q^{27} -0.277754 q^{29} +10.1235 q^{31} +1.11102 q^{33} -1.00000 q^{35} +0.833263 q^{37} -0.277754 q^{39} -0.478361 q^{41} +4.55551 q^{43} -2.92285 q^{45} -12.4012 q^{47} +1.00000 q^{49} +1.24388 q^{51} +8.40121 q^{53} +4.00000 q^{55} -0.688376 q^{57} +8.27775 q^{59} -2.00000 q^{61} +2.92285 q^{63} -1.00000 q^{65} -7.92285 q^{67} -0.555509 q^{69} +6.95672 q^{71} +9.29020 q^{73} +0.277754 q^{75} -4.00000 q^{77} +3.92285 q^{79} +8.31162 q^{81} +15.2902 q^{83} +4.47836 q^{85} -0.0771475 q^{87} -13.0124 q^{89} +1.00000 q^{91} +2.81184 q^{93} -2.47836 q^{95} +5.44449 q^{97} -11.6914 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{3} + 3 q^{5} - 3 q^{7} + 6 q^{9}+O(q^{10}) $ 3 * q + q^3 + 3 * q^5 - 3 * q^7 + 6 * q^9	$ 3 q + q^{3} + 3 q^{5} - 3 q^{7} + 6 q^{9} + 12 q^{11} - 3 q^{13} + q^{15} - q^{17} + 7 q^{19} - q^{21} - 6 q^{23} + 3 q^{25} + 10 q^{27} - q^{29} + q^{31} + 4 q^{33} - 3 q^{35} + 3 q^{37} - q^{39} + 13 q^{41} + 14 q^{43} + 6 q^{45} - 8 q^{47} + 3 q^{49} + 18 q^{51} - 4 q^{53} + 12 q^{55} - 16 q^{57} + 25 q^{59} - 6 q^{61} - 6 q^{63} - 3 q^{65} - 9 q^{67} - 2 q^{69} - 8 q^{71} - 2 q^{73} + q^{75} - 12 q^{77} - 3 q^{79} + 11 q^{81} + 16 q^{83} - q^{85} - 15 q^{87} - 9 q^{89} + 3 q^{91} - 7 q^{93} + 7 q^{95} + 16 q^{97} + 24 q^{99}+O(q^{100}) $ 3 * q + q^3 + 3 * q^5 - 3 * q^7 + 6 * q^9 + 12 * q^11 - 3 * q^13 + q^15 - q^17 + 7 * q^19 - q^21 - 6 * q^23 + 3 * q^25 + 10 * q^27 - q^29 + q^31 + 4 * q^33 - 3 * q^35 + 3 * q^37 - q^39 + 13 * q^41 + 14 * q^43 + 6 * q^45 - 8 * q^47 + 3 * q^49 + 18 * q^51 - 4 * q^53 + 12 * q^55 - 16 * q^57 + 25 * q^59 - 6 * q^61 - 6 * q^63 - 3 * q^65 - 9 * q^67 - 2 * q^69 - 8 * q^71 - 2 * q^73 + q^75 - 12 * q^77 - 3 * q^79 + 11 * q^81 + 16 * q^83 - q^85 - 15 * q^87 - 9 * q^89 + 3 * q^91 - 7 * q^93 + 7 * q^95 + 16 * q^97 + 24 * 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
$3$	0.277754	0.160362	0.0801808	−	0.996780i	$-0.474450\pi$
$3$	0.277754	0.160362	0.0801808	+	0.996780i	$0.474450\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−2.92285	−0.974284
$10$	0	0
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0.277754	0.0717159
$16$	0	0
$17$	4.47836	1.08616	0.543081	−	0.839680i	$-0.317258\pi$
$17$	4.47836	1.08616	0.543081	+	0.839680i	$0.317258\pi$
$18$	0	0
$19$	−2.47836	−0.568575	−0.284288	−	0.958739i	$-0.591757\pi$
$19$	−2.47836	−0.568575	−0.284288	+	0.958739i	$0.591757\pi$
$20$	0	0
$21$	−0.277754	−0.0606110
$22$	0	0
$23$	−2.00000	−0.417029	−0.208514	−	0.978019i	$-0.566863\pi$
$23$	−2.00000	−0.417029	−0.208514	+	0.978019i	$0.566863\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.64510	−0.316599
$28$	0	0
$29$	−0.277754	−0.0515777	−0.0257888	−	0.999667i	$-0.508210\pi$
$29$	−0.277754	−0.0515777	−0.0257888	+	0.999667i	$0.508210\pi$
$30$	0	0
$31$	10.1235	1.81823	0.909113	−	0.416549i	$-0.136760\pi$
$31$	10.1235	1.81823	0.909113	+	0.416549i	$0.136760\pi$
$32$	0	0
$33$	1.11102	0.193403
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	0.833263	0.136988	0.0684938	−	0.997652i	$-0.478181\pi$
$37$	0.833263	0.136988	0.0684938	+	0.997652i	$0.478181\pi$
$38$	0	0
$39$	−0.277754	−0.0444763
$40$	0	0
$41$	−0.478361	−0.0747075	−0.0373537	−	0.999302i	$-0.511893\pi$
$41$	−0.478361	−0.0747075	−0.0373537	+	0.999302i	$0.511893\pi$
$42$	0	0
$43$	4.55551	0.694709	0.347354	−	0.937734i	$-0.387080\pi$
$43$	4.55551	0.694709	0.347354	+	0.937734i	$0.387080\pi$
$44$	0	0
$45$	−2.92285	−0.435713
$46$	0	0
$47$	−12.4012	−1.80890	−0.904451	−	0.426577i	$-0.859719\pi$
$47$	−12.4012	−1.80890	−0.904451	+	0.426577i	$0.859719\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	1.24388	0.174179
$52$	0	0
$53$	8.40121	1.15400	0.576998	−	0.816746i	$-0.304224\pi$
$53$	8.40121	1.15400	0.576998	+	0.816746i	$0.304224\pi$
$54$	0	0
$55$	4.00000	0.539360
$56$	0	0
$57$	−0.688376	−0.0911776
$58$	0	0
$59$	8.27775	1.07767	0.538836	−	0.842411i	$-0.318864\pi$
$59$	8.27775	1.07767	0.538836	+	0.842411i	$0.318864\pi$
$60$	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
$63$	2.92285	0.368245
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	−7.92285	−0.967930	−0.483965	−	0.875087i	$-0.660804\pi$
$67$	−7.92285	−0.967930	−0.483965	+	0.875087i	$0.660804\pi$
$68$	0	0
$69$	−0.555509	−0.0668754
$70$	0	0
$71$	6.95672	0.825611	0.412806	−	0.910819i	$-0.364549\pi$
$71$	6.95672	0.825611	0.412806	+	0.910819i	$0.364549\pi$
$72$	0	0
$73$	9.29020	1.08734	0.543668	−	0.839301i	$-0.317035\pi$
$73$	9.29020	1.08734	0.543668	+	0.839301i	$0.317035\pi$
$74$	0	0
$75$	0.277754	0.0320723
$76$	0	0
$77$	−4.00000	−0.455842
$78$	0	0
$79$	3.92285	0.441355	0.220678	−	0.975347i	$-0.429173\pi$
$79$	3.92285	0.441355	0.220678	+	0.975347i	$0.429173\pi$
$80$	0	0
$81$	8.31162	0.923514
$82$	0	0
$83$	15.2902	1.67832	0.839159	−	0.543887i	$-0.183048\pi$
$83$	15.2902	1.67832	0.839159	+	0.543887i	$0.183048\pi$
$84$	0	0
$85$	4.47836	0.485746
$86$	0	0
$87$	−0.0771475	−0.00827108
$88$	0	0
$89$	−13.0124	−1.37932	−0.689658	−	0.724135i	$-0.742239\pi$
$89$	−13.0124	−1.37932	−0.689658	+	0.724135i	$0.742239\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	2.81184	0.291574
$94$	0	0
$95$	−2.47836	−0.254274
$96$	0	0
$97$	5.44449	0.552804	0.276402	−	0.961042i	$-0.410858\pi$
$97$	5.44449	0.552804	0.276402	+	0.961042i	$0.410858\pi$
$98$	0	0
$99$	−11.6914	−1.17503
$100$	0	0
$101$	−5.29020	−0.526394	−0.263197	−	0.964742i	$-0.584777\pi$
$101$	−5.29020	−0.526394	−0.263197	+	0.964742i	$0.584777\pi$
$102$	0	0
$103$	2.96613	0.292261	0.146131	−	0.989265i	$-0.453318\pi$
$103$	2.96613	0.292261	0.146131	+	0.989265i	$0.453318\pi$
$104$	0	0
$105$	−0.277754	−0.0271061
$106$	0	0
$107$	14.7347	1.42446	0.712228	−	0.701948i	$-0.247686\pi$
$107$	14.7347	1.42446	0.712228	+	0.701948i	$0.247686\pi$
$108$	0	0
$109$	0.401214	0.0384293	0.0192147	−	0.999815i	$-0.493883\pi$
$109$	0.401214	0.0384293	0.0192147	+	0.999815i	$0.493883\pi$
$110$	0	0
$111$	0.231442	0.0219675
$112$	0	0
$113$	−10.5555	−0.992979	−0.496489	−	0.868043i	$-0.665378\pi$
$113$	−10.5555	−0.992979	−0.496489	+	0.868043i	$0.665378\pi$
$114$	0	0
$115$	−2.00000	−0.186501
$116$	0	0
$117$	2.92285	0.270218
$118$	0	0
$119$	−4.47836	−0.410531
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	−0.132867	−0.0119802
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	10.0000	0.887357	0.443678	−	0.896186i	$-0.353673\pi$
$127$	10.0000	0.887357	0.443678	+	0.896186i	$0.353673\pi$
$128$	0	0
$129$	1.26531	0.111405
$130$	0	0
$131$	15.2902	1.33591	0.667955	−	0.744201i	$-0.267170\pi$
$131$	15.2902	1.33591	0.667955	+	0.744201i	$0.267170\pi$
$132$	0	0
$133$	2.47836	0.214901
$134$	0	0
$135$	−1.64510	−0.141588
$136$	0	0
$137$	3.16674	0.270553	0.135276	−	0.990808i	$-0.456808\pi$
$137$	3.16674	0.270553	0.135276	+	0.990808i	$0.456808\pi$
$138$	0	0
$139$	11.8457	1.00474	0.502370	−	0.864653i	$-0.332462\pi$
$139$	11.8457	1.00474	0.502370	+	0.864653i	$0.332462\pi$
$140$	0	0
$141$	−3.44449	−0.290078
$142$	0	0
$143$	−4.00000	−0.334497
$144$	0	0
$145$	−0.277754	−0.0230662
$146$	0	0
$147$	0.277754	0.0229088
$148$	0	0
$149$	8.40121	0.688254	0.344127	−	0.938923i	$-0.388175\pi$
$149$	8.40121	0.688254	0.344127	+	0.938923i	$0.388175\pi$
$150$	0	0
$151$	13.8457	1.12675	0.563374	−	0.826202i	$-0.309503\pi$
$151$	13.8457	1.12675	0.563374	+	0.826202i	$0.309503\pi$
$152$	0	0
$153$	−13.0896	−1.05823
$154$	0	0
$155$	10.1235	0.813136
$156$	0	0
$157$	1.38877	0.110836	0.0554180	−	0.998463i	$-0.482351\pi$
$157$	1.38877	0.110836	0.0554180	+	0.998463i	$0.482351\pi$
$158$	0	0
$159$	2.33347	0.185057
$160$	0	0
$161$	2.00000	0.157622
$162$	0	0
$163$	−2.12346	−0.166322	−0.0831611	−	0.996536i	$-0.526502\pi$
$163$	−2.12346	−0.166322	−0.0831611	+	0.996536i	$0.526502\pi$
$164$	0	0
$165$	1.11102	0.0864926
$166$	0	0
$167$	−3.44449	−0.266543	−0.133271	−	0.991080i	$-0.542548\pi$
$167$	−3.44449	−0.266543	−0.133271	+	0.991080i	$0.542548\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	7.24388	0.553954
$172$	0	0
$173$	−22.1912	−1.68717	−0.843583	−	0.536999i	$-0.819558\pi$
$173$	−22.1912	−1.68717	−0.843583	+	0.536999i	$0.819558\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	2.29918	0.172817
$178$	0	0
$179$	−2.14489	−0.160316	−0.0801582	−	0.996782i	$-0.525543\pi$
$179$	−2.14489	−0.160316	−0.0801582	+	0.996782i	$0.525543\pi$
$180$	0	0
$181$	13.4445	0.999321	0.499661	−	0.866221i	$-0.333458\pi$
$181$	13.4445	0.999321	0.499661	+	0.866221i	$0.333458\pi$
$182$	0	0
$183$	−0.555509	−0.0410644
$184$	0	0
$185$	0.833263	0.0612627
$186$	0	0
$187$	17.9134	1.30996
$188$	0	0
$189$	1.64510	0.119663
$190$	0	0
$191$	1.01244	0.0732577	0.0366289	−	0.999329i	$-0.488338\pi$
$191$	1.01244	0.0732577	0.0366289	+	0.999329i	$0.488338\pi$
$192$	0	0
$193$	0.966130	0.0695436	0.0347718	−	0.999395i	$-0.488930\pi$
$193$	0.966130	0.0695436	0.0347718	+	0.999395i	$0.488930\pi$
$194$	0	0
$195$	−0.277754	−0.0198904
$196$	0	0
$197$	−19.4137	−1.38317	−0.691583	−	0.722297i	$-0.743086\pi$
$197$	−19.4137	−1.38317	−0.691583	+	0.722297i	$0.743086\pi$
$198$	0	0
$199$	21.3579	1.51402	0.757012	−	0.653401i	$-0.226658\pi$
$199$	21.3579	1.51402	0.757012	+	0.653401i	$0.226658\pi$
$200$	0	0
$201$	−2.20061	−0.155219
$202$	0	0
$203$	0.277754	0.0194945
$204$	0	0
$205$	−0.478361	−0.0334102
$206$	0	0
$207$	5.84571	0.406305
$208$	0	0
$209$	−9.91344	−0.685727
$210$	0	0
$211$	0.231442	0.0159332	0.00796658	−	0.999968i	$-0.497464\pi$
$211$	0.231442	0.0159332	0.00796658	+	0.999968i	$0.497464\pi$
$212$	0	0
$213$	1.93226	0.132396
$214$	0	0
$215$	4.55551	0.310683
$216$	0	0
$217$	−10.1235	−0.687225
$218$	0	0
$219$	2.58039	0.174367
$220$	0	0
$221$	−4.47836	−0.301247
$222$	0	0
$223$	−8.95672	−0.599787	−0.299893	−	0.953973i	$-0.596951\pi$
$223$	−8.95672	−0.599787	−0.299893	+	0.953973i	$0.596951\pi$
$224$	0	0
$225$	−2.92285	−0.194857
$226$	0	0
$227$	10.6232	0.705090	0.352545	−	0.935795i	$-0.385316\pi$
$227$	10.6232	0.705090	0.352545	+	0.935795i	$0.385316\pi$
$228$	0	0
$229$	−9.76856	−0.645524	−0.322762	−	0.946480i	$-0.604611\pi$
$229$	−9.76856	−0.645524	−0.322762	+	0.946480i	$0.604611\pi$
$230$	0	0
$231$	−1.11102	−0.0730996
$232$	0	0
$233$	8.17918	0.535836	0.267918	−	0.963442i	$-0.413664\pi$
$233$	8.17918	0.535836	0.267918	+	0.963442i	$0.413664\pi$
$234$	0	0
$235$	−12.4012	−0.808966
$236$	0	0
$237$	1.08959	0.0707764
$238$	0	0
$239$	−19.9134	−1.28809	−0.644047	−	0.764986i	$-0.722746\pi$
$239$	−19.9134	−1.28809	−0.644047	+	0.764986i	$0.722746\pi$
$240$	0	0
$241$	−13.4351	−0.865430	−0.432715	−	0.901531i	$-0.642444\pi$
$241$	−13.4351	−0.865430	−0.432715	+	0.901531i	$0.642444\pi$
$242$	0	0
$243$	7.24388	0.464695
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	2.47836	0.157694
$248$	0	0
$249$	4.24692	0.269138
$250$	0	0
$251$	4.55551	0.287541	0.143771	−	0.989611i	$-0.454077\pi$
$251$	4.55551	0.287541	0.143771	+	0.989611i	$0.454077\pi$
$252$	0	0
$253$	−8.00000	−0.502956
$254$	0	0
$255$	1.24388	0.0778951
$256$	0	0
$257$	−17.5122	−1.09238	−0.546191	−	0.837660i	$-0.683923\pi$
$257$	−17.5122	−1.09238	−0.546191	+	0.837660i	$0.683923\pi$
$258$	0	0
$259$	−0.833263	−0.0517764
$260$	0	0
$261$	0.811835	0.0502513
$262$	0	0
$263$	−19.9134	−1.22792	−0.613958	−	0.789339i	$-0.710424\pi$
$263$	−19.9134	−1.22792	−0.613958	+	0.789339i	$0.710424\pi$
$264$	0	0
$265$	8.40121	0.516082
$266$	0	0
$267$	−3.61426	−0.221189
$268$	0	0
$269$	6.40121	0.390289	0.195144	−	0.980775i	$-0.437482\pi$
$269$	6.40121	0.390289	0.195144	+	0.980775i	$0.437482\pi$
$270$	0	0
$271$	12.0000	0.728948	0.364474	−	0.931214i	$-0.381249\pi$
$271$	12.0000	0.728948	0.364474	+	0.931214i	$0.381249\pi$
$272$	0	0
$273$	0.277754	0.0168105
$274$	0	0
$275$	4.00000	0.241209
$276$	0	0
$277$	−8.00000	−0.480673	−0.240337	−	0.970690i	$-0.577258\pi$
$277$	−8.00000	−0.480673	−0.240337	+	0.970690i	$0.577258\pi$
$278$	0	0
$279$	−29.5894	−1.77147
$280$	0	0
$281$	−1.84571	−0.110106	−0.0550528	−	0.998483i	$-0.517533\pi$
$281$	−1.84571	−0.110106	−0.0550528	+	0.998483i	$0.517533\pi$
$282$	0	0
$283$	0.410621	0.0244089	0.0122045	−	0.999926i	$-0.496115\pi$
$283$	0.410621	0.0244089	0.0122045	+	0.999926i	$0.496115\pi$
$284$	0	0
$285$	−0.688376	−0.0407759
$286$	0	0
$287$	0.478361	0.0282368
$288$	0	0
$289$	3.05572	0.179748
$290$	0	0
$291$	1.51223	0.0886486
$292$	0	0
$293$	29.5371	1.72558	0.862788	−	0.505565i	$-0.168716\pi$
$293$	29.5371	1.72558	0.862788	+	0.505565i	$0.168716\pi$
$294$	0	0
$295$	8.27775	0.481949
$296$	0	0
$297$	−6.58039	−0.381833
$298$	0	0
$299$	2.00000	0.115663
$300$	0	0
$301$	−4.55551	−0.262575
$302$	0	0
$303$	−1.46938	−0.0844134
$304$	0	0
$305$	−2.00000	−0.114520
$306$	0	0
$307$	4.40121	0.251191	0.125595	−	0.992082i	$-0.459916\pi$
$307$	4.40121	0.251191	0.125595	+	0.992082i	$0.459916\pi$
$308$	0	0
$309$	0.823856	0.0468675
$310$	0	0
$311$	8.95672	0.507889	0.253945	−	0.967219i	$-0.418272\pi$
$311$	8.95672	0.507889	0.253945	+	0.967219i	$0.418272\pi$
$312$	0	0
$313$	12.4569	0.704107	0.352054	−	0.935980i	$-0.385483\pi$
$313$	12.4569	0.704107	0.352054	+	0.935980i	$0.385483\pi$
$314$	0	0
$315$	2.92285	0.164684
$316$	0	0
$317$	8.88898	0.499255	0.249627	−	0.968342i	$-0.419692\pi$
$317$	8.88898	0.499255	0.249627	+	0.968342i	$0.419692\pi$
$318$	0	0
$319$	−1.11102	−0.0622050
$320$	0	0
$321$	4.09262	0.228428
$322$	0	0
$323$	−11.0990	−0.617565
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	0.111439	0.00616258
$328$	0	0
$329$	12.4012	0.683701
$330$	0	0
$331$	23.0245	1.26554	0.632769	−	0.774340i	$-0.281918\pi$
$331$	23.0245	1.26554	0.632769	+	0.774340i	$0.281918\pi$
$332$	0	0
$333$	−2.43551	−0.133465
$334$	0	0
$335$	−7.92285	−0.432872
$336$	0	0
$337$	14.8024	0.806340	0.403170	−	0.915125i	$-0.367908\pi$
$337$	14.8024	0.806340	0.403170	+	0.915125i	$0.367908\pi$
$338$	0	0
$339$	−2.93184	−0.159236
$340$	0	0
$341$	40.4938	2.19286
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−0.555509	−0.0299076
$346$	0	0
$347$	2.62325	0.140823	0.0704117	−	0.997518i	$-0.477569\pi$
$347$	2.62325	0.140823	0.0704117	+	0.997518i	$0.477569\pi$
$348$	0	0
$349$	7.96916	0.426580	0.213290	−	0.976989i	$-0.431582\pi$
$349$	7.96916	0.426580	0.213290	+	0.976989i	$0.431582\pi$
$350$	0	0
$351$	1.64510	0.0878088
$352$	0	0
$353$	8.88898	0.473113	0.236556	−	0.971618i	$-0.423981\pi$
$353$	8.88898	0.473113	0.236556	+	0.971618i	$0.423981\pi$
$354$	0	0
$355$	6.95672	0.369224
$356$	0	0
$357$	−1.24388	−0.0658333
$358$	0	0
$359$	10.5555	0.557098	0.278549	−	0.960422i	$-0.410146\pi$
$359$	10.5555	0.557098	0.278549	+	0.960422i	$0.410146\pi$
$360$	0	0
$361$	−12.8577	−0.676722
$362$	0	0
$363$	1.38877	0.0728916
$364$	0	0
$365$	9.29020	0.486271
$366$	0	0
$367$	−23.0245	−1.20187	−0.600934	−	0.799299i	$-0.705204\pi$
$367$	−23.0245	−1.20187	−0.600934	+	0.799299i	$0.705204\pi$
$368$	0	0
$369$	1.39818	0.0727863
$370$	0	0
$371$	−8.40121	−0.436169
$372$	0	0
$373$	−14.0677	−0.728400	−0.364200	−	0.931321i	$-0.618658\pi$
$373$	−14.0677	−0.728400	−0.364200	+	0.931321i	$0.618658\pi$
$374$	0	0
$375$	0.277754	0.0143432
$376$	0	0
$377$	0.277754	0.0143051
$378$	0	0
$379$	12.4012	0.637008	0.318504	−	0.947922i	$-0.396820\pi$
$379$	12.4012	0.637008	0.318504	+	0.947922i	$0.396820\pi$
$380$	0	0
$381$	2.77754	0.142298
$382$	0	0
$383$	−17.7592	−0.907450	−0.453725	−	0.891142i	$-0.649905\pi$
$383$	−17.7592	−0.907450	−0.453725	+	0.891142i	$0.649905\pi$
$384$	0	0
$385$	−4.00000	−0.203859
$386$	0	0
$387$	−13.3151	−0.676844
$388$	0	0
$389$	15.7008	0.796063	0.398032	−	0.917372i	$-0.369693\pi$
$389$	15.7008	0.796063	0.398032	+	0.917372i	$0.369693\pi$
$390$	0	0
$391$	−8.95672	−0.452961
$392$	0	0
$393$	4.24692	0.214229
$394$	0	0
$395$	3.92285	0.197380
$396$	0	0
$397$	−18.9567	−0.951411	−0.475705	−	0.879605i	$-0.657807\pi$
$397$	−18.9567	−0.951411	−0.475705	+	0.879605i	$0.657807\pi$
$398$	0	0
$399$	0.688376	0.0344619
$400$	0	0
$401$	17.8457	0.891172	0.445586	−	0.895239i	$-0.352995\pi$
$401$	17.8457	0.891172	0.445586	+	0.895239i	$0.352995\pi$
$402$	0	0
$403$	−10.1235	−0.504285
$404$	0	0
$405$	8.31162	0.413008
$406$	0	0
$407$	3.33305	0.165213
$408$	0	0
$409$	−21.2036	−1.04845	−0.524226	−	0.851579i	$-0.675645\pi$
$409$	−21.2036	−1.04845	−0.524226	+	0.851579i	$0.675645\pi$
$410$	0	0
$411$	0.879575	0.0433862
$412$	0	0
$413$	−8.27775	−0.407322
$414$	0	0
$415$	15.2902	0.750566
$416$	0	0
$417$	3.29020	0.161122
$418$	0	0
$419$	−26.7347	−1.30607	−0.653037	−	0.757326i	$-0.726506\pi$
$419$	−26.7347	−1.30607	−0.653037	+	0.757326i	$0.726506\pi$
$420$	0	0
$421$	−23.2902	−1.13509	−0.567547	−	0.823341i	$-0.692108\pi$
$421$	−23.2902	−1.13509	−0.567547	+	0.823341i	$0.692108\pi$
$422$	0	0
$423$	36.2469	1.76239
$424$	0	0
$425$	4.47836	0.217232
$426$	0	0
$427$	2.00000	0.0967868
$428$	0	0
$429$	−1.11102	−0.0536404
$430$	0	0
$431$	−17.4445	−0.840272	−0.420136	−	0.907461i	$-0.638018\pi$
$431$	−17.4445	−0.840272	−0.420136	+	0.907461i	$0.638018\pi$
$432$	0	0
$433$	−28.3241	−1.36117	−0.680584	−	0.732670i	$-0.738274\pi$
$433$	−28.3241	−1.36117	−0.680584	+	0.732670i	$0.738274\pi$
$434$	0	0
$435$	−0.0771475	−0.00369894
$436$	0	0
$437$	4.95672	0.237112
$438$	0	0
$439$	22.5804	1.07770	0.538852	−	0.842401i	$-0.318858\pi$
$439$	22.5804	1.07770	0.538852	+	0.842401i	$0.318858\pi$
$440$	0	0
$441$	−2.92285	−0.139183
$442$	0	0
$443$	−2.17918	−0.103536	−0.0517680	−	0.998659i	$-0.516486\pi$
$443$	−2.17918	−0.103536	−0.0517680	+	0.998659i	$0.516486\pi$
$444$	0	0
$445$	−13.0124	−0.616849
$446$	0	0
$447$	2.33347	0.110370
$448$	0	0
$449$	19.3579	0.913557	0.456779	−	0.889580i	$-0.349003\pi$
$449$	19.3579	0.913557	0.456779	+	0.889580i	$0.349003\pi$
$450$	0	0
$451$	−1.91344	−0.0901006
$452$	0	0
$453$	3.84571	0.180687
$454$	0	0
$455$	1.00000	0.0468807
$456$	0	0
$457$	−0.0308356	−0.00144243	−0.000721214	−	1.00000i	$-0.500230\pi$
$457$	−0.0308356	−0.00144243	−0.000721214		1.00000i	$0.500230\pi$
$458$	0	0
$459$	−7.36734	−0.343878
$460$	0	0
$461$	−14.1912	−0.660950	−0.330475	−	0.943815i	$-0.607209\pi$
$461$	−14.1912	−0.660950	−0.330475	+	0.943815i	$0.607209\pi$
$462$	0	0
$463$	5.18816	0.241114	0.120557	−	0.992706i	$-0.461532\pi$
$463$	5.18816	0.241114	0.120557	+	0.992706i	$0.461532\pi$
$464$	0	0
$465$	2.81184	0.130396
$466$	0	0
$467$	20.5461	0.950760	0.475380	−	0.879781i	$-0.342311\pi$
$467$	20.5461	0.950760	0.475380	+	0.879781i	$0.342311\pi$
$468$	0	0
$469$	7.92285	0.365843
$470$	0	0
$471$	0.385737	0.0177738
$472$	0	0
$473$	18.2220	0.837850
$474$	0	0
$475$	−2.47836	−0.113715
$476$	0	0
$477$	−24.5555	−1.12432
$478$	0	0
$479$	27.9014	1.27485	0.637424	−	0.770513i	$-0.280000\pi$
$479$	27.9014	1.27485	0.637424	+	0.770513i	$0.280000\pi$
$480$	0	0
$481$	−0.833263	−0.0379935
$482$	0	0
$483$	0.555509	0.0252765
$484$	0	0
$485$	5.44449	0.247222
$486$	0	0
$487$	−5.32103	−0.241119	−0.120559	−	0.992706i	$-0.538469\pi$
$487$	−5.32103	−0.241119	−0.120559	+	0.992706i	$0.538469\pi$
$488$	0	0
$489$	−0.589800	−0.0266717
$490$	0	0
$491$	27.6914	1.24970	0.624848	−	0.780746i	$-0.285161\pi$
$491$	27.6914	1.24970	0.624848	+	0.780746i	$0.285161\pi$
$492$	0	0
$493$	−1.24388	−0.0560217
$494$	0	0
$495$	−11.6914	−0.525490
$496$	0	0
$497$	−6.95672	−0.312052
$498$	0	0
$499$	−11.6914	−0.523379	−0.261690	−	0.965152i	$-0.584280\pi$
$499$	−11.6914	−0.523379	−0.261690	+	0.965152i	$0.584280\pi$
$500$	0	0
$501$	−0.956722	−0.0427432
$502$	0	0
$503$	31.3828	1.39929	0.699645	−	0.714490i	$-0.253341\pi$
$503$	31.3828	1.39929	0.699645	+	0.714490i	$0.253341\pi$
$504$	0	0
$505$	−5.29020	−0.235411
$506$	0	0
$507$	0.277754	0.0123355
$508$	0	0
$509$	7.03387	0.311771	0.155885	−	0.987775i	$-0.450177\pi$
$509$	7.03387	0.311771	0.155885	+	0.987775i	$0.450177\pi$
$510$	0	0
$511$	−9.29020	−0.410974
$512$	0	0
$513$	4.07715	0.180010
$514$	0	0
$515$	2.96613	0.130703
$516$	0	0
$517$	−49.6049	−2.18162
$518$	0	0
$519$	−6.16370	−0.270557
$520$	0	0
$521$	19.5122	0.854846	0.427423	−	0.904052i	$-0.359421\pi$
$521$	19.5122	0.854846	0.427423	+	0.904052i	$0.359421\pi$
$522$	0	0
$523$	−27.5122	−1.20303	−0.601513	−	0.798863i	$-0.705435\pi$
$523$	−27.5122	−1.20303	−0.601513	+	0.798863i	$0.705435\pi$
$524$	0	0
$525$	−0.277754	−0.0121222
$526$	0	0
$527$	45.3365	1.97489
$528$	0	0
$529$	−19.0000	−0.826087
$530$	0	0
$531$	−24.1947	−1.04996
$532$	0	0
$533$	0.478361	0.0207201
$534$	0	0
$535$	14.7347	0.637036
$536$	0	0
$537$	−0.595752	−0.0257086
$538$	0	0
$539$	4.00000	0.172292
$540$	0	0
$541$	−17.6665	−0.759543	−0.379772	−	0.925080i	$-0.623997\pi$
$541$	−17.6665	−0.759543	−0.379772	+	0.925080i	$0.623997\pi$
$542$	0	0
$543$	3.73427	0.160253
$544$	0	0
$545$	0.401214	0.0171861
$546$	0	0
$547$	−0.401214	−0.0171547	−0.00857733	−	0.999963i	$-0.502730\pi$
$547$	−0.401214	−0.0171547	−0.00857733	+	0.999963i	$0.502730\pi$
$548$	0	0
$549$	5.84571	0.249489
$550$	0	0
$551$	0.688376	0.0293258
$552$	0	0
$553$	−3.92285	−0.166817
$554$	0	0
$555$	0.231442	0.00982419
$556$	0	0
$557$	−35.3922	−1.49962	−0.749808	−	0.661655i	$-0.769854\pi$
$557$	−35.3922	−1.49962	−0.749808	+	0.661655i	$0.769854\pi$
$558$	0	0
$559$	−4.55551	−0.192677
$560$	0	0
$561$	4.97554	0.210067
$562$	0	0
$563$	−12.1020	−0.510040	−0.255020	−	0.966936i	$-0.582082\pi$
$563$	−12.1020	−0.510040	−0.255020	+	0.966936i	$0.582082\pi$
$564$	0	0
$565$	−10.5555	−0.444074
$566$	0	0
$567$	−8.31162	−0.349055
$568$	0	0
$569$	−17.6357	−0.739327	−0.369663	−	0.929166i	$-0.620527\pi$
$569$	−17.6357	−0.739327	−0.369663	+	0.929166i	$0.620527\pi$
$570$	0	0
$571$	−20.0369	−0.838518	−0.419259	−	0.907867i	$-0.637710\pi$
$571$	−20.0369	−0.838518	−0.419259	+	0.907867i	$0.637710\pi$
$572$	0	0
$573$	0.281210	0.0117477
$574$	0	0
$575$	−2.00000	−0.0834058
$576$	0	0
$577$	−12.4878	−0.519873	−0.259936	−	0.965626i	$-0.583702\pi$
$577$	−12.4878	−0.519873	−0.259936	+	0.965626i	$0.583702\pi$
$578$	0	0
$579$	0.268347	0.0111521
$580$	0	0
$581$	−15.2902	−0.634344
$582$	0	0
$583$	33.6049	1.39177
$584$	0	0
$585$	2.92285	0.120845
$586$	0	0
$587$	15.1788	0.626494	0.313247	−	0.949672i	$-0.398583\pi$
$587$	15.1788	0.626494	0.313247	+	0.949672i	$0.398583\pi$
$588$	0	0
$589$	−25.0896	−1.03380
$590$	0	0
$591$	−5.39223	−0.221807
$592$	0	0
$593$	−5.73427	−0.235478	−0.117739	−	0.993045i	$-0.537565\pi$
$593$	−5.73427	−0.235478	−0.117739	+	0.993045i	$0.537565\pi$
$594$	0	0
$595$	−4.47836	−0.183595
$596$	0	0
$597$	5.93226	0.242791
$598$	0	0
$599$	−16.8024	−0.686529	−0.343264	−	0.939239i	$-0.611533\pi$
$599$	−16.8024	−0.686529	−0.343264	+	0.939239i	$0.611533\pi$
$600$	0	0
$601$	−16.3335	−0.666256	−0.333128	−	0.942882i	$-0.608104\pi$
$601$	−16.3335	−0.666256	−0.333128	+	0.942882i	$0.608104\pi$
$602$	0	0
$603$	23.1573	0.943039
$604$	0	0
$605$	5.00000	0.203279
$606$	0	0
$607$	−21.8149	−0.885438	−0.442719	−	0.896660i	$-0.645986\pi$
$607$	−21.8149	−0.885438	−0.442719	+	0.896660i	$0.645986\pi$
$608$	0	0
$609$	0.0771475	0.00312617
$610$	0	0
$611$	12.4012	0.501699
$612$	0	0
$613$	−22.4938	−0.908518	−0.454259	−	0.890870i	$-0.650096\pi$
$613$	−22.4938	−0.908518	−0.454259	+	0.890870i	$0.650096\pi$
$614$	0	0
$615$	−0.132867	−0.00535771
$616$	0	0
$617$	27.3922	1.10277	0.551385	−	0.834251i	$-0.314100\pi$
$617$	27.3922	1.10277	0.551385	+	0.834251i	$0.314100\pi$
$618$	0	0
$619$	−17.5002	−0.703393	−0.351696	−	0.936114i	$-0.614395\pi$
$619$	−17.5002	−0.703393	−0.351696	+	0.936114i	$0.614395\pi$
$620$	0	0
$621$	3.29020	0.132031
$622$	0	0
$623$	13.0124	0.521332
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−2.75350	−0.109964
$628$	0	0
$629$	3.73165	0.148791
$630$	0	0
$631$	30.6910	1.22179	0.610894	−	0.791712i	$-0.290810\pi$
$631$	30.6910	1.22179	0.610894	+	0.791712i	$0.290810\pi$
$632$	0	0
$633$	0.0642842	0.00255507
$634$	0	0
$635$	10.0000	0.396838
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−20.3335	−0.804380
$640$	0	0
$641$	37.8826	1.49627	0.748137	−	0.663545i	$-0.230949\pi$
$641$	37.8826	1.49627	0.748137	+	0.663545i	$0.230949\pi$
$642$	0	0
$643$	−16.7098	−0.658970	−0.329485	−	0.944161i	$-0.606875\pi$
$643$	−16.7098	−0.658970	−0.329485	+	0.944161i	$0.606875\pi$
$644$	0	0
$645$	1.26531	0.0498216
$646$	0	0
$647$	20.7655	0.816377	0.408188	−	0.912898i	$-0.366161\pi$
$647$	20.7655	0.816377	0.408188	+	0.912898i	$0.366161\pi$
$648$	0	0
$649$	33.1110	1.29972
$650$	0	0
$651$	−2.81184	−0.110205
$652$	0	0
$653$	−43.9383	−1.71944	−0.859720	−	0.510766i	$-0.829362\pi$
$653$	−43.9383	−1.71944	−0.859720	+	0.510766i	$0.829362\pi$
$654$	0	0
$655$	15.2902	0.597437
$656$	0	0
$657$	−27.1539	−1.05937
$658$	0	0
$659$	−37.5465	−1.46260	−0.731302	−	0.682053i	$-0.761087\pi$
$659$	−37.5465	−1.46260	−0.731302	+	0.682053i	$0.761087\pi$
$660$	0	0
$661$	18.8796	0.734330	0.367165	−	0.930156i	$-0.380328\pi$
$661$	18.8796	0.734330	0.367165	+	0.930156i	$0.380328\pi$
$662$	0	0
$663$	−1.24388	−0.0483085
$664$	0	0
$665$	2.47836	0.0961067
$666$	0	0
$667$	0.555509	0.0215094
$668$	0	0
$669$	−2.48777	−0.0961827
$670$	0	0
$671$	−8.00000	−0.308837
$672$	0	0
$673$	−4.93184	−0.190108	−0.0950542	−	0.995472i	$-0.530302\pi$
$673$	−4.93184	−0.190108	−0.0950542	+	0.995472i	$0.530302\pi$
$674$	0	0
$675$	−1.64510	−0.0633199
$676$	0	0
$677$	−45.6914	−1.75606	−0.878032	−	0.478602i	$-0.841144\pi$
$677$	−45.6914	−1.75606	−0.878032	+	0.478602i	$0.841144\pi$
$678$	0	0
$679$	−5.44449	−0.208940
$680$	0	0
$681$	2.95065	0.113069
$682$	0	0
$683$	−48.4167	−1.85261	−0.926307	−	0.376771i	$-0.877034\pi$
$683$	−48.4167	−1.85261	−0.926307	+	0.376771i	$0.877034\pi$
$684$	0	0
$685$	3.16674	0.120995
$686$	0	0
$687$	−2.71326	−0.103517
$688$	0	0
$689$	−8.40121	−0.320061
$690$	0	0
$691$	18.3026	0.696265	0.348133	−	0.937445i	$-0.386816\pi$
$691$	18.3026	0.696265	0.348133	+	0.937445i	$0.386816\pi$
$692$	0	0
$693$	11.6914	0.444120
$694$	0	0
$695$	11.8457	0.449333
$696$	0	0
$697$	−2.14227	−0.0811444
$698$	0	0
$699$	2.27180	0.0859275
$700$	0	0
$701$	−34.4167	−1.29990	−0.649950	−	0.759977i	$-0.725210\pi$
$701$	−34.4167	−1.29990	−0.649950	+	0.759977i	$0.725210\pi$
$702$	0	0
$703$	−2.06513	−0.0778877
$704$	0	0
$705$	−3.44449	−0.129727
$706$	0	0
$707$	5.29020	0.198958
$708$	0	0
$709$	−47.0245	−1.76604	−0.883020	−	0.469335i	$-0.844494\pi$
$709$	−47.0245	−1.76604	−0.883020	+	0.469335i	$0.844494\pi$
$710$	0	0
$711$	−11.4659	−0.430006
$712$	0	0
$713$	−20.2469	−0.758253
$714$	0	0
$715$	−4.00000	−0.149592
$716$	0	0
$717$	−5.53105	−0.206561
$718$	0	0
$719$	32.6481	1.21757	0.608785	−	0.793335i	$-0.291657\pi$
$719$	32.6481	1.21757	0.608785	+	0.793335i	$0.291657\pi$
$720$	0	0
$721$	−2.96613	−0.110464
$722$	0	0
$723$	−3.73165	−0.138782
$724$	0	0
$725$	−0.277754	−0.0103155
$726$	0	0
$727$	−20.4569	−0.758706	−0.379353	−	0.925252i	$-0.623853\pi$
$727$	−20.4569	−0.758706	−0.379353	+	0.925252i	$0.623853\pi$
$728$	0	0
$729$	−22.9229	−0.848995
$730$	0	0
$731$	20.4012	0.754566
$732$	0	0
$733$	31.9134	1.17875	0.589375	−	0.807860i	$-0.299374\pi$
$733$	31.9134	1.17875	0.589375	+	0.807860i	$0.299374\pi$
$734$	0	0
$735$	0.277754	0.0102451
$736$	0	0
$737$	−31.6914	−1.16737
$738$	0	0
$739$	−0.821243	−0.0302099	−0.0151049	−	0.999886i	$-0.504808\pi$
$739$	−0.821243	−0.0302099	−0.0151049	+	0.999886i	$0.504808\pi$
$740$	0	0
$741$	0.688376	0.0252881
$742$	0	0
$743$	5.54652	0.203482	0.101741	−	0.994811i	$-0.467559\pi$
$743$	5.54652	0.203482	0.101741	+	0.994811i	$0.467559\pi$
$744$	0	0
$745$	8.40121	0.307797
$746$	0	0
$747$	−44.6910	−1.63516
$748$	0	0
$749$	−14.7347	−0.538394
$750$	0	0
$751$	1.56795	0.0572153	0.0286077	−	0.999591i	$-0.490893\pi$
$751$	1.56795	0.0572153	0.0286077	+	0.999591i	$0.490893\pi$
$752$	0	0
$753$	1.26531	0.0461105
$754$	0	0
$755$	13.8457	0.503897
$756$	0	0
$757$	−45.5122	−1.65417	−0.827085	−	0.562077i	$-0.810003\pi$
$757$	−45.5122	−1.65417	−0.827085	+	0.562077i	$0.810003\pi$
$758$	0	0
$759$	−2.22203	−0.0806548
$760$	0	0
$761$	34.8144	1.26202	0.631011	−	0.775774i	$-0.282640\pi$
$761$	34.8144	1.26202	0.631011	+	0.775774i	$0.282640\pi$
$762$	0	0
$763$	−0.401214	−0.0145249
$764$	0	0
$765$	−13.0896	−0.473255
$766$	0	0
$767$	−8.27775	−0.298892
$768$	0	0
$769$	18.9816	0.684494	0.342247	−	0.939610i	$-0.388812\pi$
$769$	18.9816	0.684494	0.342247	+	0.939610i	$0.388812\pi$
$770$	0	0
$771$	−4.86410	−0.175176
$772$	0	0
$773$	−22.8024	−0.820146	−0.410073	−	0.912053i	$-0.634497\pi$
$773$	−22.8024	−0.820146	−0.410073	+	0.912053i	$0.634497\pi$
$774$	0	0
$775$	10.1235	0.363645
$776$	0	0
$777$	−0.231442	−0.00830295
$778$	0	0
$779$	1.18555	0.0424768
$780$	0	0
$781$	27.8269	0.995724
$782$	0	0
$783$	0.456933	0.0163295
$784$	0	0
$785$	1.38877	0.0495674
$786$	0	0
$787$	−39.3828	−1.40385	−0.701923	−	0.712253i	$-0.747675\pi$
$787$	−39.3828	−1.40385	−0.701923	+	0.712253i	$0.747675\pi$
$788$	0	0
$789$	−5.53105	−0.196910
$790$	0	0
$791$	10.5555	0.375311
$792$	0	0
$793$	2.00000	0.0710221
$794$	0	0
$795$	2.33347	0.0827598
$796$	0	0
$797$	−31.1051	−1.10180	−0.550899	−	0.834572i	$-0.685715\pi$
$797$	−31.1051	−1.10180	−0.550899	+	0.834572i	$0.685715\pi$
$798$	0	0
$799$	−55.5371	−1.96476
$800$	0	0
$801$	38.0334	1.34385
$802$	0	0
$803$	37.1608	1.31138
$804$	0	0
$805$	2.00000	0.0704907
$806$	0	0
$807$	1.77797	0.0625873
$808$	0	0
$809$	−34.7253	−1.22088	−0.610438	−	0.792064i	$-0.709007\pi$
$809$	−34.7253	−1.22088	−0.610438	+	0.792064i	$0.709007\pi$
$810$	0	0
$811$	−7.51223	−0.263790	−0.131895	−	0.991264i	$-0.542106\pi$
$811$	−7.51223	−0.263790	−0.131895	+	0.991264i	$0.542106\pi$
$812$	0	0
$813$	3.33305	0.116895
$814$	0	0
$815$	−2.12346	−0.0743815
$816$	0	0
$817$	−11.2902	−0.394994
$818$	0	0
$819$	−2.92285	−0.102133
$820$	0	0
$821$	−44.3824	−1.54896	−0.774478	−	0.632601i	$-0.781987\pi$
$821$	−44.3824	−1.54896	−0.774478	+	0.632601i	$0.781987\pi$
$822$	0	0
$823$	−28.4878	−0.993021	−0.496511	−	0.868031i	$-0.665386\pi$
$823$	−28.4878	−0.993021	−0.496511	+	0.868031i	$0.665386\pi$
$824$	0	0
$825$	1.11102	0.0386807
$826$	0	0
$827$	16.3584	0.568836	0.284418	−	0.958700i	$-0.408200\pi$
$827$	16.3584	0.568836	0.284418	+	0.958700i	$0.408200\pi$
$828$	0	0
$829$	24.6232	0.855201	0.427600	−	0.903968i	$-0.359359\pi$
$829$	24.6232	0.855201	0.427600	+	0.903968i	$0.359359\pi$
$830$	0	0
$831$	−2.22203	−0.0770815
$832$	0	0
$833$	4.47836	0.155166
$834$	0	0
$835$	−3.44449	−0.119202
$836$	0	0
$837$	−16.6541	−0.575649
$838$	0	0
$839$	−37.6049	−1.29826	−0.649132	−	0.760676i	$-0.724868\pi$
$839$	−37.6049	−1.29826	−0.649132	+	0.760676i	$0.724868\pi$
$840$	0	0
$841$	−28.9229	−0.997340
$842$	0	0
$843$	−0.512653	−0.0176567
$844$	0	0
$845$	1.00000	0.0344010
$846$	0	0
$847$	−5.00000	−0.171802
$848$	0	0
$849$	0.114052	0.00391425
$850$	0	0
$851$	−1.66653	−0.0571278
$852$	0	0
$853$	14.0926	0.482522	0.241261	−	0.970460i	$-0.422439\pi$
$853$	14.0926	0.482522	0.241261	+	0.970460i	$0.422439\pi$
$854$	0	0
$855$	7.24388	0.247736
$856$	0	0
$857$	52.9508	1.80876	0.904382	−	0.426724i	$-0.140332\pi$
$857$	52.9508	1.80876	0.904382	+	0.426724i	$0.140332\pi$
$858$	0	0
$859$	40.6481	1.38690	0.693448	−	0.720506i	$-0.256091\pi$
$859$	40.6481	1.38690	0.693448	+	0.720506i	$0.256091\pi$
$860$	0	0
$861$	0.132867	0.00452809
$862$	0	0
$863$	−17.5465	−0.597291	−0.298645	−	0.954364i	$-0.596535\pi$
$863$	−17.5465	−0.597291	−0.298645	+	0.954364i	$0.596535\pi$
$864$	0	0
$865$	−22.1912	−0.754523
$866$	0	0
$867$	0.848739	0.0288247
$868$	0	0
$869$	15.6914	0.532295
$870$	0	0
$871$	7.92285	0.268456
$872$	0	0
$873$	−15.9134	−0.538589
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−38.0771	−1.28577	−0.642887	−	0.765961i	$-0.722264\pi$
$877$	−38.0771	−1.28577	−0.642887	+	0.765961i	$0.722264\pi$
$878$	0	0
$879$	8.20406	0.276716
$880$	0	0
$881$	58.6053	1.97446	0.987231	−	0.159295i	$-0.0509222\pi$
$881$	58.6053	1.97446	0.987231	+	0.159295i	$0.0509222\pi$
$882$	0	0
$883$	−47.0245	−1.58250	−0.791250	−	0.611493i	$-0.790569\pi$
$883$	−47.0245	−1.58250	−0.791250	+	0.611493i	$0.790569\pi$
$884$	0	0
$885$	2.29918	0.0772862
$886$	0	0
$887$	−46.2589	−1.55322	−0.776612	−	0.629980i	$-0.783063\pi$
$887$	−46.2589	−1.55322	−0.776612	+	0.629980i	$0.783063\pi$
$888$	0	0
$889$	−10.0000	−0.335389
$890$	0	0
$891$	33.2465	1.11380
$892$	0	0
$893$	30.7347	1.02850
$894$	0	0
$895$	−2.14489	−0.0716957
$896$	0	0
$897$	0.555509	0.0185479
$898$	0	0
$899$	−2.81184	−0.0937799
$900$	0	0
$901$	37.6237	1.25343
$902$	0	0
$903$	−1.26531	−0.0421070
$904$	0	0
$905$	13.4445	0.446910
$906$	0	0
$907$	50.2718	1.66925	0.834624	−	0.550820i	$-0.185685\pi$
$907$	50.2718	1.66925	0.834624	+	0.550820i	$0.185685\pi$
$908$	0	0
$909$	15.4625	0.512858
$910$	0	0
$911$	−40.2812	−1.33458	−0.667288	−	0.744800i	$-0.732545\pi$
$911$	−40.2812	−1.33458	−0.667288	+	0.744800i	$0.732545\pi$
$912$	0	0
$913$	61.1608	2.02413
$914$	0	0
$915$	−0.555509	−0.0183646
$916$	0	0
$917$	−15.2902	−0.504927
$918$	0	0
$919$	−10.0120	−0.330266	−0.165133	−	0.986271i	$-0.552805\pi$
$919$	−10.0120	−0.330266	−0.165133	+	0.986271i	$0.552805\pi$
$920$	0	0
$921$	1.22246	0.0402813
$922$	0	0
$923$	−6.95672	−0.228983
$924$	0	0
$925$	0.833263	0.0273975
$926$	0	0
$927$	−8.66956	−0.284746
$928$	0	0
$929$	42.8144	1.40470	0.702348	−	0.711834i	$-0.252135\pi$
$929$	42.8144	1.40470	0.702348	+	0.711834i	$0.252135\pi$
$930$	0	0
$931$	−2.47836	−0.0812250
$932$	0	0
$933$	2.48777	0.0814459
$934$	0	0
$935$	17.9134	0.585832
$936$	0	0
$937$	−26.2375	−0.857142	−0.428571	−	0.903508i	$-0.640983\pi$
$937$	−26.2375	−0.857142	−0.428571	+	0.903508i	$0.640983\pi$
$938$	0	0
$939$	3.45997	0.112912
$940$	0	0
$941$	39.7008	1.29421	0.647105	−	0.762401i	$-0.275980\pi$
$941$	39.7008	1.29421	0.647105	+	0.762401i	$0.275980\pi$
$942$	0	0
$943$	0.956722	0.0311552
$944$	0	0
$945$	1.64510	0.0535150
$946$	0	0
$947$	17.3425	0.563554	0.281777	−	0.959480i	$-0.409076\pi$
$947$	17.3425	0.563554	0.281777	+	0.959480i	$0.409076\pi$
$948$	0	0
$949$	−9.29020	−0.301573
$950$	0	0
$951$	2.46895	0.0800613
$952$	0	0
$953$	−31.5122	−1.02078	−0.510391	−	0.859943i	$-0.670499\pi$
$953$	−31.5122	−1.02078	−0.510391	+	0.859943i	$0.670499\pi$
$954$	0	0
$955$	1.01244	0.0327619
$956$	0	0
$957$	−0.308590	−0.00997530
$958$	0	0
$959$	−3.16674	−0.102259
$960$	0	0
$961$	71.4844	2.30595
$962$	0	0
$963$	−43.0673	−1.38782
$964$	0	0
$965$	0.966130	0.0311008
$966$	0	0
$967$	−9.56795	−0.307685	−0.153842	−	0.988095i	$-0.549165\pi$
$967$	−9.56795	−0.307685	−0.153842	+	0.988095i	$0.549165\pi$
$968$	0	0
$969$	−3.08279	−0.0990336
$970$	0	0
$971$	−19.0245	−0.610524	−0.305262	−	0.952268i	$-0.598744\pi$
$971$	−19.0245	−0.610524	−0.305262	+	0.952268i	$0.598744\pi$
$972$	0	0
$973$	−11.8457	−0.379756
$974$	0	0
$975$	−0.277754	−0.00889526
$976$	0	0
$977$	2.74410	0.0877914	0.0438957	−	0.999036i	$-0.486023\pi$
$977$	2.74410	0.0877914	0.0438957	+	0.999036i	$0.486023\pi$
$978$	0	0
$979$	−52.0498	−1.66352
$980$	0	0
$981$	−1.17269	−0.0374411
$982$	0	0
$983$	36.4012	1.16102	0.580509	−	0.814254i	$-0.302854\pi$
$983$	36.4012	1.16102	0.580509	+	0.814254i	$0.302854\pi$
$984$	0	0
$985$	−19.4137	−0.618571
$986$	0	0
$987$	3.44449	0.109639
$988$	0	0
$989$	−9.11102	−0.289714
$990$	0	0
$991$	20.7253	0.658360	0.329180	−	0.944267i	$-0.393228\pi$
$991$	20.7253	0.658360	0.329180	+	0.944267i	$0.393228\pi$
$992$	0	0
$993$	6.39514	0.202944
$994$	0	0
$995$	21.3579	0.677092
$996$	0	0
$997$	−20.8118	−0.659117	−0.329559	−	0.944135i	$-0.606900\pi$
$997$	−20.8118	−0.659117	−0.329559	+	0.944135i	$0.606900\pi$
$998$	0	0
$999$	−1.37080	−0.0433702

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3640.2.a.q.1.2	✓	3
4.3	odd	2		7280.2.a.bm.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3640.2.a.q.1.2	✓	3	1.1	even	1	trivial
7280.2.a.bm.1.2		3	4.3	odd	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:	\( N \)	\(=\)	\( 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3640.a (trivial)

\(f(q)\)	\(=\)	\(q+0.277754 q^{3} +1.00000 q^{5} -1.00000 q^{7} -2.92285 q^{9} +O(q^{10})\)	\(q+0.277754 q^{3} +1.00000 q^{5} -1.00000 q^{7} -2.92285 q^{9} +4.00000 q^{11} -1.00000 q^{13} +0.277754 q^{15} +4.47836 q^{17} -2.47836 q^{19} -0.277754 q^{21} -2.00000 q^{23} +1.00000 q^{25} -1.64510 q^{27} -0.277754 q^{29} +10.1235 q^{31} +1.11102 q^{33} -1.00000 q^{35} +0.833263 q^{37} -0.277754 q^{39} -0.478361 q^{41} +4.55551 q^{43} -2.92285 q^{45} -12.4012 q^{47} +1.00000 q^{49} +1.24388 q^{51} +8.40121 q^{53} +4.00000 q^{55} -0.688376 q^{57} +8.27775 q^{59} -2.00000 q^{61} +2.92285 q^{63} -1.00000 q^{65} -7.92285 q^{67} -0.555509 q^{69} +6.95672 q^{71} +9.29020 q^{73} +0.277754 q^{75} -4.00000 q^{77} +3.92285 q^{79} +8.31162 q^{81} +15.2902 q^{83} +4.47836 q^{85} -0.0771475 q^{87} -13.0124 q^{89} +1.00000 q^{91} +2.81184 q^{93} -2.47836 q^{95} +5.44449 q^{97} -11.6914 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{3} + 3 q^{5} - 3 q^{7} + 6 q^{9}+O(q^{10}) \) 3 * q + q^3 + 3 * q^5 - 3 * q^7 + 6 * q^9	\( 3 q + q^{3} + 3 q^{5} - 3 q^{7} + 6 q^{9} + 12 q^{11} - 3 q^{13} + q^{15} - q^{17} + 7 q^{19} - q^{21} - 6 q^{23} + 3 q^{25} + 10 q^{27} - q^{29} + q^{31} + 4 q^{33} - 3 q^{35} + 3 q^{37} - q^{39} + 13 q^{41} + 14 q^{43} + 6 q^{45} - 8 q^{47} + 3 q^{49} + 18 q^{51} - 4 q^{53} + 12 q^{55} - 16 q^{57} + 25 q^{59} - 6 q^{61} - 6 q^{63} - 3 q^{65} - 9 q^{67} - 2 q^{69} - 8 q^{71} - 2 q^{73} + q^{75} - 12 q^{77} - 3 q^{79} + 11 q^{81} + 16 q^{83} - q^{85} - 15 q^{87} - 9 q^{89} + 3 q^{91} - 7 q^{93} + 7 q^{95} + 16 q^{97} + 24 q^{99}+O(q^{100}) \) 3 * q + q^3 + 3 * q^5 - 3 * q^7 + 6 * q^9 + 12 * q^11 - 3 * q^13 + q^15 - q^17 + 7 * q^19 - q^21 - 6 * q^23 + 3 * q^25 + 10 * q^27 - q^29 + q^31 + 4 * q^33 - 3 * q^35 + 3 * q^37 - q^39 + 13 * q^41 + 14 * q^43 + 6 * q^45 - 8 * q^47 + 3 * q^49 + 18 * q^51 - 4 * q^53 + 12 * q^55 - 16 * q^57 + 25 * q^59 - 6 * q^61 - 6 * q^63 - 3 * q^65 - 9 * q^67 - 2 * q^69 - 8 * q^71 - 2 * q^73 + q^75 - 12 * q^77 - 3 * q^79 + 11 * q^81 + 16 * q^83 - q^85 - 15 * q^87 - 9 * q^89 + 3 * q^91 - 7 * q^93 + 7 * q^95 + 16 * q^97 + 24 * q^99

Introduction

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