LMFDB - Embedded newform 3720.2.a.o.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3720,2,Mod(1,3720)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3720, 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("3720.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3720 = 2^{3} \cdot 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3720.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:	$29.7043495519$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.48119$ of defining polynomial
Character	$\chi$	$=$	3720.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} -2.48119 q^{7} +1.00000 q^{9} -0.387873 q^{11} +2.48119 q^{13} -1.00000 q^{15} -0.418190 q^{17} -4.00000 q^{19} -2.48119 q^{21} +5.11871 q^{23} +1.00000 q^{25} +1.00000 q^{27} -4.24965 q^{29} -1.00000 q^{31} -0.387873 q^{33} +2.48119 q^{35} +3.83146 q^{37} +2.48119 q^{39} +4.31265 q^{41} -6.00000 q^{43} -1.00000 q^{45} -10.8568 q^{47} -0.843675 q^{49} -0.418190 q^{51} -8.85685 q^{53} +0.387873 q^{55} -4.00000 q^{57} -3.86177 q^{59} +12.2374 q^{61} -2.48119 q^{63} -2.48119 q^{65} -14.2193 q^{67} +5.11871 q^{69} +7.59991 q^{71} -4.60720 q^{73} +1.00000 q^{75} +0.962389 q^{77} -5.38058 q^{79} +1.00000 q^{81} -5.89446 q^{83} +0.418190 q^{85} -4.24965 q^{87} -5.93700 q^{89} -6.15633 q^{91} -1.00000 q^{93} +4.00000 q^{95} -12.3127 q^{97} -0.387873 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} - 3 q^{5} - 2 q^{7} + 3 q^{9} - 2 q^{11} + 2 q^{13} - 3 q^{15} - 12 q^{19} - 2 q^{21} - 6 q^{23} + 3 q^{25} + 3 q^{27} + 4 q^{29} - 3 q^{31} - 2 q^{33} + 2 q^{35} - 4 q^{37} + 2 q^{39} - 8 q^{41}+ \cdots - 2 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 - 3 * q^5 - 2 * q^7 + 3 * q^9 - 2 * q^11 + 2 * q^13 - 3 * q^15 - 12 * q^19 - 2 * q^21 - 6 * q^23 + 3 * q^25 + 3 * q^27 + 4 * q^29 - 3 * q^31 - 2 * q^33 + 2 * q^35 - 4 * q^37 + 2 * q^39 - 8 * q^41 - 18 * q^43 - 3 * q^45 - 2 * q^47 - 13 * q^49 + 4 * q^53 + 2 * q^55 - 12 * q^57 + 6 * q^59 - 6 * q^61 - 2 * q^63 - 2 * q^65 - 28 * q^67 - 6 * q^69 - 4 * q^71 + 3 * q^75 - 8 * q^77 - 4 * q^79 + 3 * q^81 + 2 * q^83 + 4 * q^87 - 22 * q^89 - 8 * q^91 - 3 * q^93 + 12 * q^95 - 16 * q^97 - 2 * q^99

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
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−2.48119	−0.937803	−0.468902	−	0.883250i	$-0.655350\pi$
$7$	−2.48119	−0.937803	−0.468902	+	0.883250i	$0.655350\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.387873	−0.116948	−0.0584741	−	0.998289i	$-0.518623\pi$
$11$	−0.387873	−0.116948	−0.0584741	+	0.998289i	$0.518623\pi$
$12$	0	0
$13$	2.48119	0.688159	0.344080	−	0.938940i	$-0.388191\pi$
$13$	2.48119	0.688159	0.344080	+	0.938940i	$0.388191\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−0.418190	−0.101426	−0.0507130	−	0.998713i	$-0.516149\pi$
$17$	−0.418190	−0.101426	−0.0507130	+	0.998713i	$0.516149\pi$
$18$	0	0
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	−2.48119	−0.541441
$22$	0	0
$23$	5.11871	1.06733	0.533663	−	0.845697i	$-0.320815\pi$
$23$	5.11871	1.06733	0.533663	+	0.845697i	$0.320815\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−4.24965	−0.789140	−0.394570	−	0.918866i	$-0.629106\pi$
$29$	−4.24965	−0.789140	−0.394570	+	0.918866i	$0.629106\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	0	0
$33$	−0.387873	−0.0675200
$34$	0	0
$35$	2.48119	0.419398
$36$	0	0
$37$	3.83146	0.629887	0.314944	−	0.949110i	$-0.398014\pi$
$37$	3.83146	0.629887	0.314944	+	0.949110i	$0.398014\pi$
$38$	0	0
$39$	2.48119	0.397309
$40$	0	0
$41$	4.31265	0.673523	0.336761	−	0.941590i	$-0.390668\pi$
$41$	4.31265	0.673523	0.336761	+	0.941590i	$0.390668\pi$
$42$	0	0
$43$	−6.00000	−0.914991	−0.457496	−	0.889212i	$-0.651253\pi$
$43$	−6.00000	−0.914991	−0.457496	+	0.889212i	$0.651253\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−10.8568	−1.58363	−0.791817	−	0.610758i	$-0.790865\pi$
$47$	−10.8568	−1.58363	−0.791817	+	0.610758i	$0.790865\pi$
$48$	0	0
$49$	−0.843675	−0.120525
$50$	0	0
$51$	−0.418190	−0.0585584
$52$	0	0
$53$	−8.85685	−1.21658	−0.608291	−	0.793714i	$-0.708145\pi$
$53$	−8.85685	−1.21658	−0.608291	+	0.793714i	$0.708145\pi$
$54$	0	0
$55$	0.387873	0.0523008
$56$	0	0
$57$	−4.00000	−0.529813
$58$	0	0
$59$	−3.86177	−0.502760	−0.251380	−	0.967888i	$-0.580884\pi$
$59$	−3.86177	−0.502760	−0.251380	+	0.967888i	$0.580884\pi$
$60$	0	0
$61$	12.2374	1.56684	0.783421	−	0.621491i	$-0.213473\pi$
$61$	12.2374	1.56684	0.783421	+	0.621491i	$0.213473\pi$
$62$	0	0
$63$	−2.48119	−0.312601
$64$	0	0
$65$	−2.48119	−0.307754
$66$	0	0
$67$	−14.2193	−1.73717	−0.868584	−	0.495542i	$-0.834969\pi$
$67$	−14.2193	−1.73717	−0.868584	+	0.495542i	$0.834969\pi$
$68$	0	0
$69$	5.11871	0.616221
$70$	0	0
$71$	7.59991	0.901943	0.450972	−	0.892538i	$-0.351078\pi$
$71$	7.59991	0.901943	0.450972	+	0.892538i	$0.351078\pi$
$72$	0	0
$73$	−4.60720	−0.539232	−0.269616	−	0.962968i	$-0.586897\pi$
$73$	−4.60720	−0.539232	−0.269616	+	0.962968i	$0.586897\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	0.962389	0.109674
$78$	0	0
$79$	−5.38058	−0.605362	−0.302681	−	0.953092i	$-0.597882\pi$
$79$	−5.38058	−0.605362	−0.302681	+	0.953092i	$0.597882\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−5.89446	−0.647001	−0.323501	−	0.946228i	$-0.604860\pi$
$83$	−5.89446	−0.647001	−0.323501	+	0.946228i	$0.604860\pi$
$84$	0	0
$85$	0.418190	0.0453591
$86$	0	0
$87$	−4.24965	−0.455610
$88$	0	0
$89$	−5.93700	−0.629320	−0.314660	−	0.949204i	$-0.601891\pi$
$89$	−5.93700	−0.629320	−0.314660	+	0.949204i	$0.601891\pi$
$90$	0	0
$91$	−6.15633	−0.645358
$92$	0	0
$93$	−1.00000	−0.103695
$94$	0	0
$95$	4.00000	0.410391
$96$	0	0
$97$	−12.3127	−1.25016	−0.625080	−	0.780561i	$-0.714934\pi$
$97$	−12.3127	−1.25016	−0.625080	+	0.780561i	$0.714934\pi$
$98$	0	0
$99$	−0.387873	−0.0389827
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	7.05571	0.695220	0.347610	−	0.937639i	$-0.386993\pi$
$103$	7.05571	0.695220	0.347610	+	0.937639i	$0.386993\pi$
$104$	0	0
$105$	2.48119	0.242140
$106$	0	0
$107$	11.0435	1.06761	0.533807	−	0.845606i	$-0.320761\pi$
$107$	11.0435	1.06761	0.533807	+	0.845606i	$0.320761\pi$
$108$	0	0
$109$	−20.5950	−1.97264	−0.986321	−	0.164837i	$-0.947290\pi$
$109$	−20.5950	−1.97264	−0.986321	+	0.164837i	$0.947290\pi$
$110$	0	0
$111$	3.83146	0.363666
$112$	0	0
$113$	12.0508	1.13364	0.566821	−	0.823841i	$-0.308173\pi$
$113$	12.0508	1.13364	0.566821	+	0.823841i	$0.308173\pi$
$114$	0	0
$115$	−5.11871	−0.477323
$116$	0	0
$117$	2.48119	0.229386
$118$	0	0
$119$	1.03761	0.0951177
$120$	0	0
$121$	−10.8496	−0.986323
$122$	0	0
$123$	4.31265	0.388859
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	3.66291	0.325031	0.162515	−	0.986706i	$-0.448039\pi$
$127$	3.66291	0.325031	0.162515	+	0.986706i	$0.448039\pi$
$128$	0	0
$129$	−6.00000	−0.528271
$130$	0	0
$131$	−6.24965	−0.546034	−0.273017	−	0.962009i	$-0.588022\pi$
$131$	−6.24965	−0.546034	−0.273017	+	0.962009i	$0.588022\pi$
$132$	0	0
$133$	9.92478	0.860587
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−8.73084	−0.745926	−0.372963	−	0.927846i	$-0.621658\pi$
$137$	−8.73084	−0.745926	−0.372963	+	0.927846i	$0.621658\pi$
$138$	0	0
$139$	−11.0376	−0.936198	−0.468099	−	0.883676i	$-0.655061\pi$
$139$	−11.0376	−0.936198	−0.468099	+	0.883676i	$0.655061\pi$
$140$	0	0
$141$	−10.8568	−0.914312
$142$	0	0
$143$	−0.962389	−0.0804790
$144$	0	0
$145$	4.24965	0.352914
$146$	0	0
$147$	−0.843675	−0.0695851
$148$	0	0
$149$	−7.53690	−0.617447	−0.308724	−	0.951152i	$-0.599902\pi$
$149$	−7.53690	−0.617447	−0.308724	+	0.951152i	$0.599902\pi$
$150$	0	0
$151$	−18.9683	−1.54362	−0.771808	−	0.635856i	$-0.780647\pi$
$151$	−18.9683	−1.54362	−0.771808	+	0.635856i	$0.780647\pi$
$152$	0	0
$153$	−0.418190	−0.0338087
$154$	0	0
$155$	1.00000	0.0803219
$156$	0	0
$157$	4.77575	0.381146	0.190573	−	0.981673i	$-0.438965\pi$
$157$	4.77575	0.381146	0.190573	+	0.981673i	$0.438965\pi$
$158$	0	0
$159$	−8.85685	−0.702394
$160$	0	0
$161$	−12.7005	−1.00094
$162$	0	0
$163$	1.56959	0.122940	0.0614699	−	0.998109i	$-0.480421\pi$
$163$	1.56959	0.122940	0.0614699	+	0.998109i	$0.480421\pi$
$164$	0	0
$165$	0.387873	0.0301959
$166$	0	0
$167$	−3.87399	−0.299779	−0.149889	−	0.988703i	$-0.547892\pi$
$167$	−3.87399	−0.299779	−0.149889	+	0.988703i	$0.547892\pi$
$168$	0	0
$169$	−6.84367	−0.526437
$170$	0	0
$171$	−4.00000	−0.305888
$172$	0	0
$173$	−8.05079	−0.612090	−0.306045	−	0.952017i	$-0.599006\pi$
$173$	−8.05079	−0.612090	−0.306045	+	0.952017i	$0.599006\pi$
$174$	0	0
$175$	−2.48119	−0.187561
$176$	0	0
$177$	−3.86177	−0.290269
$178$	0	0
$179$	−10.3879	−0.776426	−0.388213	−	0.921570i	$-0.626908\pi$
$179$	−10.3879	−0.776426	−0.388213	+	0.921570i	$0.626908\pi$
$180$	0	0
$181$	2.31265	0.171898	0.0859490	−	0.996300i	$-0.472608\pi$
$181$	2.31265	0.171898	0.0859490	+	0.996300i	$0.472608\pi$
$182$	0	0
$183$	12.2374	0.904617
$184$	0	0
$185$	−3.83146	−0.281694
$186$	0	0
$187$	0.162205	0.0118616
$188$	0	0
$189$	−2.48119	−0.180480
$190$	0	0
$191$	6.18901	0.447821	0.223911	−	0.974610i	$-0.428118\pi$
$191$	6.18901	0.447821	0.223911	+	0.974610i	$0.428118\pi$
$192$	0	0
$193$	1.55149	0.111679	0.0558394	−	0.998440i	$-0.482217\pi$
$193$	1.55149	0.111679	0.0558394	+	0.998440i	$0.482217\pi$
$194$	0	0
$195$	−2.48119	−0.177682
$196$	0	0
$197$	15.1793	1.08148	0.540742	−	0.841189i	$-0.318144\pi$
$197$	15.1793	1.08148	0.540742	+	0.841189i	$0.318144\pi$
$198$	0	0
$199$	0.730841	0.0518079	0.0259040	−	0.999664i	$-0.491754\pi$
$199$	0.730841	0.0518079	0.0259040	+	0.999664i	$0.491754\pi$
$200$	0	0
$201$	−14.2193	−1.00295
$202$	0	0
$203$	10.5442	0.740058
$204$	0	0
$205$	−4.31265	−0.301209
$206$	0	0
$207$	5.11871	0.355775
$208$	0	0
$209$	1.55149	0.107319
$210$	0	0
$211$	5.27504	0.363149	0.181574	−	0.983377i	$-0.441881\pi$
$211$	5.27504	0.363149	0.181574	+	0.983377i	$0.441881\pi$
$212$	0	0
$213$	7.59991	0.520737
$214$	0	0
$215$	6.00000	0.409197
$216$	0	0
$217$	2.48119	0.168434
$218$	0	0
$219$	−4.60720	−0.311326
$220$	0	0
$221$	−1.03761	−0.0697973
$222$	0	0
$223$	−29.5125	−1.97630	−0.988150	−	0.153488i	$-0.950949\pi$
$223$	−29.5125	−1.97630	−0.988150	+	0.153488i	$0.950949\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−6.67021	−0.442717	−0.221359	−	0.975192i	$-0.571049\pi$
$227$	−6.67021	−0.442717	−0.221359	+	0.975192i	$0.571049\pi$
$228$	0	0
$229$	7.27504	0.480748	0.240374	−	0.970680i	$-0.422730\pi$
$229$	7.27504	0.480748	0.240374	+	0.970680i	$0.422730\pi$
$230$	0	0
$231$	0.962389	0.0633205
$232$	0	0
$233$	23.0943	1.51296	0.756478	−	0.654019i	$-0.226918\pi$
$233$	23.0943	1.51296	0.756478	+	0.654019i	$0.226918\pi$
$234$	0	0
$235$	10.8568	0.708223
$236$	0	0
$237$	−5.38058	−0.349506
$238$	0	0
$239$	12.8872	0.833601	0.416801	−	0.908998i	$-0.363151\pi$
$239$	12.8872	0.833601	0.416801	+	0.908998i	$0.363151\pi$
$240$	0	0
$241$	20.3634	1.31172	0.655862	−	0.754881i	$-0.272305\pi$
$241$	20.3634	1.31172	0.655862	+	0.754881i	$0.272305\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0.843675	0.0539004
$246$	0	0
$247$	−9.92478	−0.631498
$248$	0	0
$249$	−5.89446	−0.373546
$250$	0	0
$251$	−18.2374	−1.15114	−0.575568	−	0.817754i	$-0.695219\pi$
$251$	−18.2374	−1.15114	−0.575568	+	0.817754i	$0.695219\pi$
$252$	0	0
$253$	−1.98541	−0.124822
$254$	0	0
$255$	0.418190	0.0261881
$256$	0	0
$257$	−5.16950	−0.322464	−0.161232	−	0.986916i	$-0.551547\pi$
$257$	−5.16950	−0.322464	−0.161232	+	0.986916i	$0.551547\pi$
$258$	0	0
$259$	−9.50659	−0.590711
$260$	0	0
$261$	−4.24965	−0.263047
$262$	0	0
$263$	−9.58769	−0.591202	−0.295601	−	0.955311i	$-0.595520\pi$
$263$	−9.58769	−0.591202	−0.295601	+	0.955311i	$0.595520\pi$
$264$	0	0
$265$	8.85685	0.544072
$266$	0	0
$267$	−5.93700	−0.363338
$268$	0	0
$269$	27.4495	1.67362	0.836812	−	0.547491i	$-0.184417\pi$
$269$	27.4495	1.67362	0.836812	+	0.547491i	$0.184417\pi$
$270$	0	0
$271$	−10.3634	−0.629534	−0.314767	−	0.949169i	$-0.601926\pi$
$271$	−10.3634	−0.629534	−0.314767	+	0.949169i	$0.601926\pi$
$272$	0	0
$273$	−6.15633	−0.372598
$274$	0	0
$275$	−0.387873	−0.0233896
$276$	0	0
$277$	14.8691	0.893396	0.446698	−	0.894685i	$-0.352600\pi$
$277$	14.8691	0.893396	0.446698	+	0.894685i	$0.352600\pi$
$278$	0	0
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	2.71037	0.161687	0.0808436	−	0.996727i	$-0.474239\pi$
$281$	2.71037	0.161687	0.0808436	+	0.996727i	$0.474239\pi$
$282$	0	0
$283$	17.3077	1.02884	0.514419	−	0.857539i	$-0.328008\pi$
$283$	17.3077	1.02884	0.514419	+	0.857539i	$0.328008\pi$
$284$	0	0
$285$	4.00000	0.236940
$286$	0	0
$287$	−10.7005	−0.631632
$288$	0	0
$289$	−16.8251	−0.989713
$290$	0	0
$291$	−12.3127	−0.721780
$292$	0	0
$293$	18.1563	1.06070	0.530352	−	0.847778i	$-0.322060\pi$
$293$	18.1563	1.06070	0.530352	+	0.847778i	$0.322060\pi$
$294$	0	0
$295$	3.86177	0.224841
$296$	0	0
$297$	−0.387873	−0.0225067
$298$	0	0
$299$	12.7005	0.734490
$300$	0	0
$301$	14.8872	0.858082
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−12.2374	−0.700713
$306$	0	0
$307$	−14.9199	−0.851521	−0.425760	−	0.904836i	$-0.639993\pi$
$307$	−14.9199	−0.851521	−0.425760	+	0.904836i	$0.639993\pi$
$308$	0	0
$309$	7.05571	0.401385
$310$	0	0
$311$	5.86177	0.332391	0.166195	−	0.986093i	$-0.446852\pi$
$311$	5.86177	0.332391	0.166195	+	0.986093i	$0.446852\pi$
$312$	0	0
$313$	−22.5828	−1.27645	−0.638227	−	0.769849i	$-0.720332\pi$
$313$	−22.5828	−1.27645	−0.638227	+	0.769849i	$0.720332\pi$
$314$	0	0
$315$	2.48119	0.139799
$316$	0	0
$317$	29.7948	1.67344	0.836721	−	0.547629i	$-0.184469\pi$
$317$	29.7948	1.67344	0.836721	+	0.547629i	$0.184469\pi$
$318$	0	0
$319$	1.64832	0.0922884
$320$	0	0
$321$	11.0435	0.616388
$322$	0	0
$323$	1.67276	0.0930749
$324$	0	0
$325$	2.48119	0.137632
$326$	0	0
$327$	−20.5950	−1.13891
$328$	0	0
$329$	26.9380	1.48514
$330$	0	0
$331$	−7.91890	−0.435262	−0.217631	−	0.976031i	$-0.569833\pi$
$331$	−7.91890	−0.435262	−0.217631	+	0.976031i	$0.569833\pi$
$332$	0	0
$333$	3.83146	0.209962
$334$	0	0
$335$	14.2193	0.776885
$336$	0	0
$337$	13.7562	0.749350	0.374675	−	0.927156i	$-0.377754\pi$
$337$	13.7562	0.749350	0.374675	+	0.927156i	$0.377754\pi$
$338$	0	0
$339$	12.0508	0.654509
$340$	0	0
$341$	0.387873	0.0210045
$342$	0	0
$343$	19.4617	1.05083
$344$	0	0
$345$	−5.11871	−0.275582
$346$	0	0
$347$	−35.1998	−1.88963	−0.944813	−	0.327611i	$-0.893756\pi$
$347$	−35.1998	−1.88963	−0.944813	+	0.327611i	$0.893756\pi$
$348$	0	0
$349$	−18.2071	−0.974604	−0.487302	−	0.873233i	$-0.662019\pi$
$349$	−18.2071	−0.974604	−0.487302	+	0.873233i	$0.662019\pi$
$350$	0	0
$351$	2.48119	0.132436
$352$	0	0
$353$	0.478825	0.0254853	0.0127426	−	0.999919i	$-0.495944\pi$
$353$	0.478825	0.0254853	0.0127426	+	0.999919i	$0.495944\pi$
$354$	0	0
$355$	−7.59991	−0.403361
$356$	0	0
$357$	1.03761	0.0549162
$358$	0	0
$359$	20.9502	1.10571	0.552854	−	0.833278i	$-0.313539\pi$
$359$	20.9502	1.10571	0.552854	+	0.833278i	$0.313539\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	−10.8496	−0.569454
$364$	0	0
$365$	4.60720	0.241152
$366$	0	0
$367$	20.8773	1.08979	0.544894	−	0.838505i	$-0.316570\pi$
$367$	20.8773	1.08979	0.544894	+	0.838505i	$0.316570\pi$
$368$	0	0
$369$	4.31265	0.224508
$370$	0	0
$371$	21.9756	1.14091
$372$	0	0
$373$	8.20123	0.424644	0.212322	−	0.977200i	$-0.431897\pi$
$373$	8.20123	0.424644	0.212322	+	0.977200i	$0.431897\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−10.5442	−0.543054
$378$	0	0
$379$	−34.9234	−1.79389	−0.896946	−	0.442139i	$-0.854220\pi$
$379$	−34.9234	−1.79389	−0.896946	+	0.442139i	$0.854220\pi$
$380$	0	0
$381$	3.66291	0.187657
$382$	0	0
$383$	27.7586	1.41840	0.709199	−	0.705008i	$-0.249057\pi$
$383$	27.7586	1.41840	0.709199	+	0.705008i	$0.249057\pi$
$384$	0	0
$385$	−0.962389	−0.0490479
$386$	0	0
$387$	−6.00000	−0.304997
$388$	0	0
$389$	16.1989	0.821315	0.410657	−	0.911790i	$-0.365299\pi$
$389$	16.1989	0.821315	0.410657	+	0.911790i	$0.365299\pi$
$390$	0	0
$391$	−2.14060	−0.108255
$392$	0	0
$393$	−6.24965	−0.315253
$394$	0	0
$395$	5.38058	0.270726
$396$	0	0
$397$	28.2374	1.41720	0.708598	−	0.705612i	$-0.249328\pi$
$397$	28.2374	1.41720	0.708598	+	0.705612i	$0.249328\pi$
$398$	0	0
$399$	9.92478	0.496860
$400$	0	0
$401$	−7.83734	−0.391378	−0.195689	−	0.980666i	$-0.562694\pi$
$401$	−7.83734	−0.391378	−0.195689	+	0.980666i	$0.562694\pi$
$402$	0	0
$403$	−2.48119	−0.123597
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−1.48612	−0.0736642
$408$	0	0
$409$	17.0132	0.841247	0.420624	−	0.907235i	$-0.361811\pi$
$409$	17.0132	0.841247	0.420624	+	0.907235i	$0.361811\pi$
$410$	0	0
$411$	−8.73084	−0.430661
$412$	0	0
$413$	9.58181	0.471490
$414$	0	0
$415$	5.89446	0.289348
$416$	0	0
$417$	−11.0376	−0.540514
$418$	0	0
$419$	−18.1382	−0.886110	−0.443055	−	0.896494i	$-0.646105\pi$
$419$	−18.1382	−0.886110	−0.443055	+	0.896494i	$0.646105\pi$
$420$	0	0
$421$	−27.1695	−1.32416	−0.662080	−	0.749433i	$-0.730326\pi$
$421$	−27.1695	−1.32416	−0.662080	+	0.749433i	$0.730326\pi$
$422$	0	0
$423$	−10.8568	−0.527878
$424$	0	0
$425$	−0.418190	−0.0202852
$426$	0	0
$427$	−30.3634	−1.46939
$428$	0	0
$429$	−0.962389	−0.0464646
$430$	0	0
$431$	14.6375	0.705065	0.352532	−	0.935800i	$-0.385321\pi$
$431$	14.6375	0.705065	0.352532	+	0.935800i	$0.385321\pi$
$432$	0	0
$433$	23.2424	1.11696	0.558478	−	0.829519i	$-0.311385\pi$
$433$	23.2424	1.11696	0.558478	+	0.829519i	$0.311385\pi$
$434$	0	0
$435$	4.24965	0.203755
$436$	0	0
$437$	−20.4749	−0.979445
$438$	0	0
$439$	−26.5501	−1.26717	−0.633583	−	0.773675i	$-0.718417\pi$
$439$	−26.5501	−1.26717	−0.633583	+	0.773675i	$0.718417\pi$
$440$	0	0
$441$	−0.843675	−0.0401750
$442$	0	0
$443$	14.6702	0.697003	0.348501	−	0.937308i	$-0.386691\pi$
$443$	14.6702	0.697003	0.348501	+	0.937308i	$0.386691\pi$
$444$	0	0
$445$	5.93700	0.281441
$446$	0	0
$447$	−7.53690	−0.356483
$448$	0	0
$449$	−10.6883	−0.504412	−0.252206	−	0.967674i	$-0.581156\pi$
$449$	−10.6883	−0.504412	−0.252206	+	0.967674i	$0.581156\pi$
$450$	0	0
$451$	−1.67276	−0.0787672
$452$	0	0
$453$	−18.9683	−0.891207
$454$	0	0
$455$	6.15633	0.288613
$456$	0	0
$457$	−17.2931	−0.808939	−0.404469	−	0.914552i	$-0.632544\pi$
$457$	−17.2931	−0.808939	−0.404469	+	0.914552i	$0.632544\pi$
$458$	0	0
$459$	−0.418190	−0.0195195
$460$	0	0
$461$	−8.52610	−0.397100	−0.198550	−	0.980091i	$-0.563623\pi$
$461$	−8.52610	−0.397100	−0.198550	+	0.980091i	$0.563623\pi$
$462$	0	0
$463$	−9.53690	−0.443218	−0.221609	−	0.975136i	$-0.571131\pi$
$463$	−9.53690	−0.443218	−0.221609	+	0.975136i	$0.571131\pi$
$464$	0	0
$465$	1.00000	0.0463739
$466$	0	0
$467$	4.15633	0.192332	0.0961659	−	0.995365i	$-0.469342\pi$
$467$	4.15633	0.192332	0.0961659	+	0.995365i	$0.469342\pi$
$468$	0	0
$469$	35.2809	1.62912
$470$	0	0
$471$	4.77575	0.220055
$472$	0	0
$473$	2.32724	0.107007
$474$	0	0
$475$	−4.00000	−0.183533
$476$	0	0
$477$	−8.85685	−0.405527
$478$	0	0
$479$	12.5867	0.575103	0.287551	−	0.957765i	$-0.407159\pi$
$479$	12.5867	0.575103	0.287551	+	0.957765i	$0.407159\pi$
$480$	0	0
$481$	9.50659	0.433463
$482$	0	0
$483$	−12.7005	−0.577894
$484$	0	0
$485$	12.3127	0.559089
$486$	0	0
$487$	−39.3503	−1.78313	−0.891565	−	0.452892i	$-0.850392\pi$
$487$	−39.3503	−1.78313	−0.891565	+	0.452892i	$0.850392\pi$
$488$	0	0
$489$	1.56959	0.0709794
$490$	0	0
$491$	−35.0494	−1.58176	−0.790878	−	0.611974i	$-0.790376\pi$
$491$	−35.0494	−1.58176	−0.790878	+	0.611974i	$0.790376\pi$
$492$	0	0
$493$	1.77716	0.0800393
$494$	0	0
$495$	0.387873	0.0174336
$496$	0	0
$497$	−18.8568	−0.845845
$498$	0	0
$499$	−23.2243	−1.03966	−0.519830	−	0.854270i	$-0.674005\pi$
$499$	−23.2243	−1.03966	−0.519830	+	0.854270i	$0.674005\pi$
$500$	0	0
$501$	−3.87399	−0.173077
$502$	0	0
$503$	0.367405	0.0163818	0.00819089	−	0.999966i	$-0.497393\pi$
$503$	0.367405	0.0163818	0.00819089	+	0.999966i	$0.497393\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−6.84367	−0.303938
$508$	0	0
$509$	−0.0484156	−0.00214598	−0.00107299	−	0.999999i	$-0.500342\pi$
$509$	−0.0484156	−0.00214598	−0.00107299	+	0.999999i	$0.500342\pi$
$510$	0	0
$511$	11.4314	0.505694
$512$	0	0
$513$	−4.00000	−0.176604
$514$	0	0
$515$	−7.05571	−0.310912
$516$	0	0
$517$	4.21108	0.185203
$518$	0	0
$519$	−8.05079	−0.353390
$520$	0	0
$521$	−4.93795	−0.216336	−0.108168	−	0.994133i	$-0.534498\pi$
$521$	−4.93795	−0.216336	−0.108168	+	0.994133i	$0.534498\pi$
$522$	0	0
$523$	18.5745	0.812207	0.406103	−	0.913827i	$-0.366887\pi$
$523$	18.5745	0.812207	0.406103	+	0.913827i	$0.366887\pi$
$524$	0	0
$525$	−2.48119	−0.108288
$526$	0	0
$527$	0.418190	0.0182167
$528$	0	0
$529$	3.20123	0.139184
$530$	0	0
$531$	−3.86177	−0.167587
$532$	0	0
$533$	10.7005	0.463491
$534$	0	0
$535$	−11.0435	−0.477452
$536$	0	0
$537$	−10.3879	−0.448270
$538$	0	0
$539$	0.327239	0.0140952
$540$	0	0
$541$	−14.2922	−0.614469	−0.307234	−	0.951634i	$-0.599404\pi$
$541$	−14.2922	−0.614469	−0.307234	+	0.951634i	$0.599404\pi$
$542$	0	0
$543$	2.31265	0.0992453
$544$	0	0
$545$	20.5950	0.882192
$546$	0	0
$547$	11.3439	0.485031	0.242516	−	0.970148i	$-0.422027\pi$
$547$	11.3439	0.485031	0.242516	+	0.970148i	$0.422027\pi$
$548$	0	0
$549$	12.2374	0.522281
$550$	0	0
$551$	16.9986	0.724164
$552$	0	0
$553$	13.3503	0.567711
$554$	0	0
$555$	−3.83146	−0.162636
$556$	0	0
$557$	25.5672	1.08332	0.541659	−	0.840598i	$-0.317796\pi$
$557$	25.5672	1.08332	0.541659	+	0.840598i	$0.317796\pi$
$558$	0	0
$559$	−14.8872	−0.629660
$560$	0	0
$561$	0.162205	0.00684829
$562$	0	0
$563$	38.6820	1.63025	0.815125	−	0.579285i	$-0.196668\pi$
$563$	38.6820	1.63025	0.815125	+	0.579285i	$0.196668\pi$
$564$	0	0
$565$	−12.0508	−0.506980
$566$	0	0
$567$	−2.48119	−0.104200
$568$	0	0
$569$	22.0122	0.922800	0.461400	−	0.887192i	$-0.347347\pi$
$569$	22.0122	0.922800	0.461400	+	0.887192i	$0.347347\pi$
$570$	0	0
$571$	4.18664	0.175206	0.0876028	−	0.996155i	$-0.472079\pi$
$571$	4.18664	0.175206	0.0876028	+	0.996155i	$0.472079\pi$
$572$	0	0
$573$	6.18901	0.258550
$574$	0	0
$575$	5.11871	0.213465
$576$	0	0
$577$	39.8251	1.65794	0.828971	−	0.559292i	$-0.188927\pi$
$577$	39.8251	1.65794	0.828971	+	0.559292i	$0.188927\pi$
$578$	0	0
$579$	1.55149	0.0644778
$580$	0	0
$581$	14.6253	0.606760
$582$	0	0
$583$	3.43533	0.142277
$584$	0	0
$585$	−2.48119	−0.102585
$586$	0	0
$587$	−9.85940	−0.406941	−0.203471	−	0.979081i	$-0.565222\pi$
$587$	−9.85940	−0.406941	−0.203471	+	0.979081i	$0.565222\pi$
$588$	0	0
$589$	4.00000	0.164817
$590$	0	0
$591$	15.1793	0.624395
$592$	0	0
$593$	12.1114	0.497356	0.248678	−	0.968586i	$-0.420004\pi$
$593$	12.1114	0.497356	0.248678	+	0.968586i	$0.420004\pi$
$594$	0	0
$595$	−1.03761	−0.0425379
$596$	0	0
$597$	0.730841	0.0299113
$598$	0	0
$599$	19.8764	0.812126	0.406063	−	0.913845i	$-0.366901\pi$
$599$	19.8764	0.812126	0.406063	+	0.913845i	$0.366901\pi$
$600$	0	0
$601$	−28.2130	−1.15083	−0.575416	−	0.817861i	$-0.695160\pi$
$601$	−28.2130	−1.15083	−0.575416	+	0.817861i	$0.695160\pi$
$602$	0	0
$603$	−14.2193	−0.579056
$604$	0	0
$605$	10.8496	0.441097
$606$	0	0
$607$	−23.7464	−0.963836	−0.481918	−	0.876216i	$-0.660060\pi$
$607$	−23.7464	−0.963836	−0.481918	+	0.876216i	$0.660060\pi$
$608$	0	0
$609$	10.5442	0.427272
$610$	0	0
$611$	−26.9380	−1.08979
$612$	0	0
$613$	−18.9560	−0.765628	−0.382814	−	0.923826i	$-0.625045\pi$
$613$	−18.9560	−0.765628	−0.382814	+	0.923826i	$0.625045\pi$
$614$	0	0
$615$	−4.31265	−0.173903
$616$	0	0
$617$	47.7255	1.92135	0.960677	−	0.277667i	$-0.0895613\pi$
$617$	47.7255	1.92135	0.960677	+	0.277667i	$0.0895613\pi$
$618$	0	0
$619$	−15.9307	−0.640307	−0.320154	−	0.947366i	$-0.603735\pi$
$619$	−15.9307	−0.640307	−0.320154	+	0.947366i	$0.603735\pi$
$620$	0	0
$621$	5.11871	0.205407
$622$	0	0
$623$	14.7308	0.590179
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	1.55149	0.0619606
$628$	0	0
$629$	−1.60228	−0.0638870
$630$	0	0
$631$	9.92478	0.395099	0.197550	−	0.980293i	$-0.436702\pi$
$631$	9.92478	0.395099	0.197550	+	0.980293i	$0.436702\pi$
$632$	0	0
$633$	5.27504	0.209664
$634$	0	0
$635$	−3.66291	−0.145358
$636$	0	0
$637$	−2.09332	−0.0829404
$638$	0	0
$639$	7.59991	0.300648
$640$	0	0
$641$	20.6639	0.816174	0.408087	−	0.912943i	$-0.366196\pi$
$641$	20.6639	0.816174	0.408087	+	0.912943i	$0.366196\pi$
$642$	0	0
$643$	18.5745	0.732507	0.366254	−	0.930515i	$-0.380640\pi$
$643$	18.5745	0.732507	0.366254	+	0.930515i	$0.380640\pi$
$644$	0	0
$645$	6.00000	0.236250
$646$	0	0
$647$	11.7889	0.463470	0.231735	−	0.972779i	$-0.425560\pi$
$647$	11.7889	0.463470	0.231735	+	0.972779i	$0.425560\pi$
$648$	0	0
$649$	1.49788	0.0587969
$650$	0	0
$651$	2.48119	0.0972457
$652$	0	0
$653$	−10.2579	−0.401422	−0.200711	−	0.979650i	$-0.564325\pi$
$653$	−10.2579	−0.401422	−0.200711	+	0.979650i	$0.564325\pi$
$654$	0	0
$655$	6.24965	0.244194
$656$	0	0
$657$	−4.60720	−0.179744
$658$	0	0
$659$	22.5477	0.878334	0.439167	−	0.898405i	$-0.355274\pi$
$659$	22.5477	0.878334	0.439167	+	0.898405i	$0.355274\pi$
$660$	0	0
$661$	−26.3780	−1.02599	−0.512993	−	0.858393i	$-0.671463\pi$
$661$	−26.3780	−1.02599	−0.512993	+	0.858393i	$0.671463\pi$
$662$	0	0
$663$	−1.03761	−0.0402975
$664$	0	0
$665$	−9.92478	−0.384866
$666$	0	0
$667$	−21.7527	−0.842269
$668$	0	0
$669$	−29.5125	−1.14102
$670$	0	0
$671$	−4.74657	−0.183239
$672$	0	0
$673$	21.6156	0.833222	0.416611	−	0.909085i	$-0.363218\pi$
$673$	21.6156	0.833222	0.416611	+	0.909085i	$0.363218\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	9.31994	0.358195	0.179097	−	0.983831i	$-0.442682\pi$
$677$	9.31994	0.358195	0.179097	+	0.983831i	$0.442682\pi$
$678$	0	0
$679$	30.5501	1.17240
$680$	0	0
$681$	−6.67021	−0.255603
$682$	0	0
$683$	−41.5329	−1.58921	−0.794607	−	0.607124i	$-0.792323\pi$
$683$	−41.5329	−1.58921	−0.794607	+	0.607124i	$0.792323\pi$
$684$	0	0
$685$	8.73084	0.333588
$686$	0	0
$687$	7.27504	0.277560
$688$	0	0
$689$	−21.9756	−0.837202
$690$	0	0
$691$	−8.80018	−0.334775	−0.167387	−	0.985891i	$-0.553533\pi$
$691$	−8.80018	−0.334775	−0.167387	+	0.985891i	$0.553533\pi$
$692$	0	0
$693$	0.962389	0.0365581
$694$	0	0
$695$	11.0376	0.418680
$696$	0	0
$697$	−1.80351	−0.0683128
$698$	0	0
$699$	23.0943	0.873506
$700$	0	0
$701$	−24.3488	−0.919643	−0.459822	−	0.888011i	$-0.652087\pi$
$701$	−24.3488	−0.919643	−0.459822	+	0.888011i	$0.652087\pi$
$702$	0	0
$703$	−15.3258	−0.578024
$704$	0	0
$705$	10.8568	0.408893
$706$	0	0
$707$	0	0
$708$	0	0
$709$	3.40105	0.127729	0.0638645	−	0.997959i	$-0.479657\pi$
$709$	3.40105	0.127729	0.0638645	+	0.997959i	$0.479657\pi$
$710$	0	0
$711$	−5.38058	−0.201787
$712$	0	0
$713$	−5.11871	−0.191697
$714$	0	0
$715$	0.962389	0.0359913
$716$	0	0
$717$	12.8872	0.481280
$718$	0	0
$719$	−13.6385	−0.508629	−0.254315	−	0.967122i	$-0.581850\pi$
$719$	−13.6385	−0.508629	−0.254315	+	0.967122i	$0.581850\pi$
$720$	0	0
$721$	−17.5066	−0.651979
$722$	0	0
$723$	20.3634	0.757324
$724$	0	0
$725$	−4.24965	−0.157828
$726$	0	0
$727$	−8.41582	−0.312126	−0.156063	−	0.987747i	$-0.549880\pi$
$727$	−8.41582	−0.312126	−0.156063	+	0.987747i	$0.549880\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	2.50914	0.0928040
$732$	0	0
$733$	33.8496	1.25026	0.625131	−	0.780520i	$-0.285046\pi$
$733$	33.8496	1.25026	0.625131	+	0.780520i	$0.285046\pi$
$734$	0	0
$735$	0.843675	0.0311194
$736$	0	0
$737$	5.51530	0.203159
$738$	0	0
$739$	−5.86811	−0.215862	−0.107931	−	0.994158i	$-0.534423\pi$
$739$	−5.86811	−0.215862	−0.107931	+	0.994158i	$0.534423\pi$
$740$	0	0
$741$	−9.92478	−0.364596
$742$	0	0
$743$	−22.9887	−0.843375	−0.421687	−	0.906741i	$-0.638562\pi$
$743$	−22.9887	−0.843375	−0.421687	+	0.906741i	$0.638562\pi$
$744$	0	0
$745$	7.53690	0.276131
$746$	0	0
$747$	−5.89446	−0.215667
$748$	0	0
$749$	−27.4010	−1.00121
$750$	0	0
$751$	37.2144	1.35797	0.678986	−	0.734151i	$-0.262419\pi$
$751$	37.2144	1.35797	0.678986	+	0.734151i	$0.262419\pi$
$752$	0	0
$753$	−18.2374	−0.664609
$754$	0	0
$755$	18.9683	0.690326
$756$	0	0
$757$	−8.53198	−0.310100	−0.155050	−	0.987907i	$-0.549554\pi$
$757$	−8.53198	−0.310100	−0.155050	+	0.987907i	$0.549554\pi$
$758$	0	0
$759$	−1.98541	−0.0720659
$760$	0	0
$761$	−19.2266	−0.696965	−0.348482	−	0.937315i	$-0.613303\pi$
$761$	−19.2266	−0.696965	−0.348482	+	0.937315i	$0.613303\pi$
$762$	0	0
$763$	51.1002	1.84995
$764$	0	0
$765$	0.418190	0.0151197
$766$	0	0
$767$	−9.58181	−0.345979
$768$	0	0
$769$	42.2071	1.52203	0.761014	−	0.648736i	$-0.224702\pi$
$769$	42.2071	1.52203	0.761014	+	0.648736i	$0.224702\pi$
$770$	0	0
$771$	−5.16950	−0.186175
$772$	0	0
$773$	0.609572	0.0219248	0.0109624	−	0.999940i	$-0.496510\pi$
$773$	0.609572	0.0219248	0.0109624	+	0.999940i	$0.496510\pi$
$774$	0	0
$775$	−1.00000	−0.0359211
$776$	0	0
$777$	−9.50659	−0.341047
$778$	0	0
$779$	−17.2506	−0.618067
$780$	0	0
$781$	−2.94780	−0.105481
$782$	0	0
$783$	−4.24965	−0.151870
$784$	0	0
$785$	−4.77575	−0.170454
$786$	0	0
$787$	12.3225	0.439250	0.219625	−	0.975584i	$-0.429517\pi$
$787$	12.3225	0.439250	0.219625	+	0.975584i	$0.429517\pi$
$788$	0	0
$789$	−9.58769	−0.341331
$790$	0	0
$791$	−29.9003	−1.06313
$792$	0	0
$793$	30.3634	1.07824
$794$	0	0
$795$	8.85685	0.314120
$796$	0	0
$797$	−23.0289	−0.815726	−0.407863	−	0.913043i	$-0.633726\pi$
$797$	−23.0289	−0.815726	−0.407863	+	0.913043i	$0.633726\pi$
$798$	0	0
$799$	4.54023	0.160622
$800$	0	0
$801$	−5.93700	−0.209773
$802$	0	0
$803$	1.78701	0.0630622
$804$	0	0
$805$	12.7005	0.447635
$806$	0	0
$807$	27.4495	0.966267
$808$	0	0
$809$	17.4250	0.612631	0.306316	−	0.951930i	$-0.400904\pi$
$809$	17.4250	0.612631	0.306316	+	0.951930i	$0.400904\pi$
$810$	0	0
$811$	−15.8134	−0.555282	−0.277641	−	0.960685i	$-0.589553\pi$
$811$	−15.8134	−0.555282	−0.277641	+	0.960685i	$0.589553\pi$
$812$	0	0
$813$	−10.3634	−0.363462
$814$	0	0
$815$	−1.56959	−0.0549804
$816$	0	0
$817$	24.0000	0.839654
$818$	0	0
$819$	−6.15633	−0.215119
$820$	0	0
$821$	−16.0729	−0.560946	−0.280473	−	0.959862i	$-0.590491\pi$
$821$	−16.0729	−0.560946	−0.280473	+	0.959862i	$0.590491\pi$
$822$	0	0
$823$	8.11616	0.282912	0.141456	−	0.989945i	$-0.454822\pi$
$823$	8.11616	0.282912	0.141456	+	0.989945i	$0.454822\pi$
$824$	0	0
$825$	−0.387873	−0.0135040
$826$	0	0
$827$	−36.5658	−1.27152	−0.635759	−	0.771888i	$-0.719313\pi$
$827$	−36.5658	−1.27152	−0.635759	+	0.771888i	$0.719313\pi$
$828$	0	0
$829$	−40.7974	−1.41695	−0.708475	−	0.705736i	$-0.750617\pi$
$829$	−40.7974	−1.41695	−0.708475	+	0.705736i	$0.750617\pi$
$830$	0	0
$831$	14.8691	0.515802
$832$	0	0
$833$	0.352817	0.0122244
$834$	0	0
$835$	3.87399	0.134065
$836$	0	0
$837$	−1.00000	−0.0345651
$838$	0	0
$839$	36.5477	1.26177	0.630884	−	0.775878i	$-0.282693\pi$
$839$	36.5477	1.26177	0.630884	+	0.775878i	$0.282693\pi$
$840$	0	0
$841$	−10.9405	−0.377259
$842$	0	0
$843$	2.71037	0.0933502
$844$	0	0
$845$	6.84367	0.235430
$846$	0	0
$847$	26.9199	0.924977
$848$	0	0
$849$	17.3077	0.594000
$850$	0	0
$851$	19.6121	0.672295
$852$	0	0
$853$	50.3343	1.72341	0.861706	−	0.507408i	$-0.169396\pi$
$853$	50.3343	1.72341	0.861706	+	0.507408i	$0.169396\pi$
$854$	0	0
$855$	4.00000	0.136797
$856$	0	0
$857$	21.0982	0.720702	0.360351	−	0.932817i	$-0.382657\pi$
$857$	21.0982	0.720702	0.360351	+	0.932817i	$0.382657\pi$
$858$	0	0
$859$	11.0738	0.377833	0.188917	−	0.981993i	$-0.439502\pi$
$859$	11.0738	0.377833	0.188917	+	0.981993i	$0.439502\pi$
$860$	0	0
$861$	−10.7005	−0.364673
$862$	0	0
$863$	33.1002	1.12674	0.563371	−	0.826204i	$-0.309504\pi$
$863$	33.1002	1.12674	0.563371	+	0.826204i	$0.309504\pi$
$864$	0	0
$865$	8.05079	0.273735
$866$	0	0
$867$	−16.8251	−0.571411
$868$	0	0
$869$	2.08698	0.0707960
$870$	0	0
$871$	−35.2809	−1.19545
$872$	0	0
$873$	−12.3127	−0.416720
$874$	0	0
$875$	2.48119	0.0838797
$876$	0	0
$877$	17.6775	0.596927	0.298463	−	0.954421i	$-0.403526\pi$
$877$	17.6775	0.596927	0.298463	+	0.954421i	$0.403526\pi$
$878$	0	0
$879$	18.1563	0.612398
$880$	0	0
$881$	42.4772	1.43109	0.715547	−	0.698565i	$-0.246178\pi$
$881$	42.4772	1.43109	0.715547	+	0.698565i	$0.246178\pi$
$882$	0	0
$883$	−47.7499	−1.60691	−0.803456	−	0.595364i	$-0.797008\pi$
$883$	−47.7499	−1.60691	−0.803456	+	0.595364i	$0.797008\pi$
$884$	0	0
$885$	3.86177	0.129812
$886$	0	0
$887$	−19.9452	−0.669696	−0.334848	−	0.942272i	$-0.608685\pi$
$887$	−19.9452	−0.669696	−0.334848	+	0.942272i	$0.608685\pi$
$888$	0	0
$889$	−9.08840	−0.304815
$890$	0	0
$891$	−0.387873	−0.0129942
$892$	0	0
$893$	43.4274	1.45324
$894$	0	0
$895$	10.3879	0.347228
$896$	0	0
$897$	12.7005	0.424058
$898$	0	0
$899$	4.24965	0.141734
$900$	0	0
$901$	3.70385	0.123393
$902$	0	0
$903$	14.8872	0.495414
$904$	0	0
$905$	−2.31265	−0.0768751
$906$	0	0
$907$	9.79526	0.325246	0.162623	−	0.986688i	$-0.448005\pi$
$907$	9.79526	0.325246	0.162623	+	0.986688i	$0.448005\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−5.38646	−0.178461	−0.0892307	−	0.996011i	$-0.528441\pi$
$911$	−5.38646	−0.178461	−0.0892307	+	0.996011i	$0.528441\pi$
$912$	0	0
$913$	2.28630	0.0756656
$914$	0	0
$915$	−12.2374	−0.404557
$916$	0	0
$917$	15.5066	0.512073
$918$	0	0
$919$	31.3865	1.03534	0.517672	−	0.855579i	$-0.326799\pi$
$919$	31.3865	1.03534	0.517672	+	0.855579i	$0.326799\pi$
$920$	0	0
$921$	−14.9199	−0.491626
$922$	0	0
$923$	18.8568	0.620681
$924$	0	0
$925$	3.83146	0.125977
$926$	0	0
$927$	7.05571	0.231740
$928$	0	0
$929$	12.1236	0.397764	0.198882	−	0.980023i	$-0.436269\pi$
$929$	12.1236	0.397764	0.198882	+	0.980023i	$0.436269\pi$
$930$	0	0
$931$	3.37470	0.110601
$932$	0	0
$933$	5.86177	0.191906
$934$	0	0
$935$	−0.162205	−0.00530466
$936$	0	0
$937$	−43.4880	−1.42069	−0.710346	−	0.703853i	$-0.751462\pi$
$937$	−43.4880	−1.42069	−0.710346	+	0.703853i	$0.751462\pi$
$938$	0	0
$939$	−22.5828	−0.736961
$940$	0	0
$941$	9.28726	0.302756	0.151378	−	0.988476i	$-0.451629\pi$
$941$	9.28726	0.302756	0.151378	+	0.988476i	$0.451629\pi$
$942$	0	0
$943$	22.0752	0.718868
$944$	0	0
$945$	2.48119	0.0807133
$946$	0	0
$947$	−18.2071	−0.591652	−0.295826	−	0.955242i	$-0.595595\pi$
$947$	−18.2071	−0.591652	−0.295826	+	0.955242i	$0.595595\pi$
$948$	0	0
$949$	−11.4314	−0.371078
$950$	0	0
$951$	29.7948	0.966163
$952$	0	0
$953$	−47.5936	−1.54171	−0.770853	−	0.637013i	$-0.780170\pi$
$953$	−47.5936	−1.54171	−0.770853	+	0.637013i	$0.780170\pi$
$954$	0	0
$955$	−6.18901	−0.200272
$956$	0	0
$957$	1.64832	0.0532827
$958$	0	0
$959$	21.6629	0.699532
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	11.0435	0.355872
$964$	0	0
$965$	−1.55149	−0.0499443
$966$	0	0
$967$	32.0917	1.03200	0.516000	−	0.856589i	$-0.327420\pi$
$967$	32.0917	1.03200	0.516000	+	0.856589i	$0.327420\pi$
$968$	0	0
$969$	1.67276	0.0537368
$970$	0	0
$971$	48.0503	1.54201	0.771004	−	0.636830i	$-0.219755\pi$
$971$	48.0503	1.54201	0.771004	+	0.636830i	$0.219755\pi$
$972$	0	0
$973$	27.3865	0.877970
$974$	0	0
$975$	2.48119	0.0794618
$976$	0	0
$977$	12.3272	0.394383	0.197192	−	0.980365i	$-0.436818\pi$
$977$	12.3272	0.394383	0.197192	+	0.980365i	$0.436818\pi$
$978$	0	0
$979$	2.30280	0.0735978
$980$	0	0
$981$	−20.5950	−0.657547
$982$	0	0
$983$	27.0132	0.861586	0.430793	−	0.902451i	$-0.358234\pi$
$983$	27.0132	0.861586	0.430793	+	0.902451i	$0.358234\pi$
$984$	0	0
$985$	−15.1793	−0.483654
$986$	0	0
$987$	26.9380	0.857444
$988$	0	0
$989$	−30.7123	−0.976594
$990$	0	0
$991$	26.9887	0.857325	0.428663	−	0.903465i	$-0.358985\pi$
$991$	26.9887	0.857325	0.428663	+	0.903465i	$0.358985\pi$
$992$	0	0
$993$	−7.91890	−0.251299
$994$	0	0
$995$	−0.730841	−0.0231692
$996$	0	0
$997$	−4.32250	−0.136895	−0.0684475	−	0.997655i	$-0.521805\pi$
$997$	−4.32250	−0.136895	−0.0684475	+	0.997655i	$0.521805\pi$
$998$	0	0
$999$	3.83146	0.121222

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3720.2.a.o.1.1	✓	3
4.3	odd	2		7440.2.a.bo.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3720.2.a.o.1.1	✓	3	1.1	even	1	trivial
7440.2.a.bo.1.3		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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3720 = 2^{3} \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3720.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} -2.48119 q^{7} +1.00000 q^{9} -0.387873 q^{11} +2.48119 q^{13} -1.00000 q^{15} -0.418190 q^{17} -4.00000 q^{19} -2.48119 q^{21} +5.11871 q^{23} +1.00000 q^{25} +1.00000 q^{27} -4.24965 q^{29} -1.00000 q^{31} -0.387873 q^{33} +2.48119 q^{35} +3.83146 q^{37} +2.48119 q^{39} +4.31265 q^{41} -6.00000 q^{43} -1.00000 q^{45} -10.8568 q^{47} -0.843675 q^{49} -0.418190 q^{51} -8.85685 q^{53} +0.387873 q^{55} -4.00000 q^{57} -3.86177 q^{59} +12.2374 q^{61} -2.48119 q^{63} -2.48119 q^{65} -14.2193 q^{67} +5.11871 q^{69} +7.59991 q^{71} -4.60720 q^{73} +1.00000 q^{75} +0.962389 q^{77} -5.38058 q^{79} +1.00000 q^{81} -5.89446 q^{83} +0.418190 q^{85} -4.24965 q^{87} -5.93700 q^{89} -6.15633 q^{91} -1.00000 q^{93} +4.00000 q^{95} -12.3127 q^{97} -0.387873 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} - 3 q^{5} - 2 q^{7} + 3 q^{9} - 2 q^{11} + 2 q^{13} - 3 q^{15} - 12 q^{19} - 2 q^{21} - 6 q^{23} + 3 q^{25} + 3 q^{27} + 4 q^{29} - 3 q^{31} - 2 q^{33} + 2 q^{35} - 4 q^{37} + 2 q^{39} - 8 q^{41}+ \cdots - 2 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 - 3 * q^5 - 2 * q^7 + 3 * q^9 - 2 * q^11 + 2 * q^13 - 3 * q^15 - 12 * q^19 - 2 * q^21 - 6 * q^23 + 3 * q^25 + 3 * q^27 + 4 * q^29 - 3 * q^31 - 2 * q^33 + 2 * q^35 - 4 * q^37 + 2 * q^39 - 8 * q^41 - 18 * q^43 - 3 * q^45 - 2 * q^47 - 13 * q^49 + 4 * q^53 + 2 * q^55 - 12 * q^57 + 6 * q^59 - 6 * q^61 - 2 * q^63 - 2 * q^65 - 28 * q^67 - 6 * q^69 - 4 * q^71 + 3 * q^75 - 8 * q^77 - 4 * q^79 + 3 * q^81 + 2 * q^83 + 4 * q^87 - 22 * q^89 - 8 * q^91 - 3 * q^93 + 12 * q^95 - 16 * q^97 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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