LMFDB - Embedded newform 3720.2.a.i.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:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000 q^{5} -1.23607 q^{7} +1.00000 q^{9} -3.23607 q^{13} +1.00000 q^{15} +4.47214 q^{17} +6.47214 q^{19} +1.23607 q^{21} -4.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} -7.23607 q^{29} -1.00000 q^{31} +1.23607 q^{35} +4.76393 q^{37} +3.23607 q^{39} -6.00000 q^{41} +6.47214 q^{43} -1.00000 q^{45} +8.94427 q^{47} -5.47214 q^{49} -4.47214 q^{51} +8.47214 q^{53} -6.47214 q^{57} +6.76393 q^{59} -12.4721 q^{61} -1.23607 q^{63} +3.23607 q^{65} +3.70820 q^{67} +4.00000 q^{69} -9.23607 q^{71} +12.1803 q^{73} -1.00000 q^{75} -8.94427 q^{79} +1.00000 q^{81} -4.94427 q^{83} -4.47214 q^{85} +7.23607 q^{87} -7.23607 q^{89} +4.00000 q^{91} +1.00000 q^{93} -6.47214 q^{95} -8.47214 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9} - 2 q^{13} + 2 q^{15} + 4 q^{19} - 2 q^{21} - 8 q^{23} + 2 q^{25} - 2 q^{27} - 10 q^{29} - 2 q^{31} - 2 q^{35} + 14 q^{37} + 2 q^{39} - 12 q^{41} + 4 q^{43} - 2 q^{45}+ \cdots - 8 q^{97}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9 - 2 * q^13 + 2 * q^15 + 4 * q^19 - 2 * q^21 - 8 * q^23 + 2 * q^25 - 2 * q^27 - 10 * q^29 - 2 * q^31 - 2 * q^35 + 14 * q^37 + 2 * q^39 - 12 * q^41 + 4 * q^43 - 2 * q^45 - 2 * q^49 + 8 * q^53 - 4 * q^57 + 18 * q^59 - 16 * q^61 + 2 * q^63 + 2 * q^65 - 6 * q^67 + 8 * q^69 - 14 * q^71 + 2 * q^73 - 2 * q^75 + 2 * q^81 + 8 * q^83 + 10 * q^87 - 10 * q^89 + 8 * q^91 + 2 * q^93 - 4 * q^95 - 8 * q^97

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
$7$	−1.23607	−0.467190	−0.233595	−	0.972334i	$-0.575049\pi$
$7$	−1.23607	−0.467190	−0.233595	+	0.972334i	$0.575049\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	−3.23607	−0.897524	−0.448762	−	0.893651i	$-0.648135\pi$
$13$	−3.23607	−0.897524	−0.448762	+	0.893651i	$0.648135\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	4.47214	1.08465	0.542326	−	0.840168i	$-0.317544\pi$
$17$	4.47214	1.08465	0.542326	+	0.840168i	$0.317544\pi$
$18$	0	0
$19$	6.47214	1.48481	0.742405	−	0.669951i	$-0.233685\pi$
$19$	6.47214	1.48481	0.742405	+	0.669951i	$0.233685\pi$
$20$	0	0
$21$	1.23607	0.269732
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−7.23607	−1.34370	−0.671852	−	0.740685i	$-0.734501\pi$
$29$	−7.23607	−1.34370	−0.671852	+	0.740685i	$0.734501\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.23607	0.208934
$36$	0	0
$37$	4.76393	0.783186	0.391593	−	0.920139i	$-0.371924\pi$
$37$	4.76393	0.783186	0.391593	+	0.920139i	$0.371924\pi$
$38$	0	0
$39$	3.23607	0.518186
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	6.47214	0.986991	0.493496	−	0.869748i	$-0.335719\pi$
$43$	6.47214	0.986991	0.493496	+	0.869748i	$0.335719\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	8.94427	1.30466	0.652328	−	0.757937i	$-0.273792\pi$
$47$	8.94427	1.30466	0.652328	+	0.757937i	$0.273792\pi$
$48$	0	0
$49$	−5.47214	−0.781734
$50$	0	0
$51$	−4.47214	−0.626224
$52$	0	0
$53$	8.47214	1.16374	0.581869	−	0.813283i	$-0.302322\pi$
$53$	8.47214	1.16374	0.581869	+	0.813283i	$0.302322\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−6.47214	−0.857255
$58$	0	0
$59$	6.76393	0.880589	0.440294	−	0.897853i	$-0.354874\pi$
$59$	6.76393	0.880589	0.440294	+	0.897853i	$0.354874\pi$
$60$	0	0
$61$	−12.4721	−1.59689	−0.798447	−	0.602066i	$-0.794345\pi$
$61$	−12.4721	−1.59689	−0.798447	+	0.602066i	$0.794345\pi$
$62$	0	0
$63$	−1.23607	−0.155730
$64$	0	0
$65$	3.23607	0.401385
$66$	0	0
$67$	3.70820	0.453029	0.226515	−	0.974008i	$-0.427267\pi$
$67$	3.70820	0.453029	0.226515	+	0.974008i	$0.427267\pi$
$68$	0	0
$69$	4.00000	0.481543
$70$	0	0
$71$	−9.23607	−1.09612	−0.548060	−	0.836439i	$-0.684633\pi$
$71$	−9.23607	−1.09612	−0.548060	+	0.836439i	$0.684633\pi$
$72$	0	0
$73$	12.1803	1.42560	0.712800	−	0.701367i	$-0.247427\pi$
$73$	12.1803	1.42560	0.712800	+	0.701367i	$0.247427\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−8.94427	−1.00631	−0.503155	−	0.864196i	$-0.667827\pi$
$79$	−8.94427	−1.00631	−0.503155	+	0.864196i	$0.667827\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−4.94427	−0.542704	−0.271352	−	0.962480i	$-0.587471\pi$
$83$	−4.94427	−0.542704	−0.271352	+	0.962480i	$0.587471\pi$
$84$	0	0
$85$	−4.47214	−0.485071
$86$	0	0
$87$	7.23607	0.775788
$88$	0	0
$89$	−7.23607	−0.767022	−0.383511	−	0.923536i	$-0.625285\pi$
$89$	−7.23607	−0.767022	−0.383511	+	0.923536i	$0.625285\pi$
$90$	0	0
$91$	4.00000	0.419314
$92$	0	0
$93$	1.00000	0.103695
$94$	0	0
$95$	−6.47214	−0.664027
$96$	0	0
$97$	−8.47214	−0.860215	−0.430108	−	0.902778i	$-0.641524\pi$
$97$	−8.47214	−0.860215	−0.430108	+	0.902778i	$0.641524\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−10.9443	−1.08900	−0.544498	−	0.838762i	$-0.683280\pi$
$101$	−10.9443	−1.08900	−0.544498	+	0.838762i	$0.683280\pi$
$102$	0	0
$103$	−16.6525	−1.64082	−0.820409	−	0.571778i	$-0.806254\pi$
$103$	−16.6525	−1.64082	−0.820409	+	0.571778i	$0.806254\pi$
$104$	0	0
$105$	−1.23607	−0.120628
$106$	0	0
$107$	12.9443	1.25137	0.625685	−	0.780076i	$-0.284820\pi$
$107$	12.9443	1.25137	0.625685	+	0.780076i	$0.284820\pi$
$108$	0	0
$109$	−3.52786	−0.337908	−0.168954	−	0.985624i	$-0.554039\pi$
$109$	−3.52786	−0.337908	−0.168954	+	0.985624i	$0.554039\pi$
$110$	0	0
$111$	−4.76393	−0.452172
$112$	0	0
$113$	2.94427	0.276974	0.138487	−	0.990364i	$-0.455776\pi$
$113$	2.94427	0.276974	0.138487	+	0.990364i	$0.455776\pi$
$114$	0	0
$115$	4.00000	0.373002
$116$	0	0
$117$	−3.23607	−0.299175
$118$	0	0
$119$	−5.52786	−0.506738
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	6.00000	0.541002
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−11.4164	−1.01304	−0.506521	−	0.862228i	$-0.669069\pi$
$127$	−11.4164	−1.01304	−0.506521	+	0.862228i	$0.669069\pi$
$128$	0	0
$129$	−6.47214	−0.569840
$130$	0	0
$131$	−9.23607	−0.806959	−0.403480	−	0.914989i	$-0.632199\pi$
$131$	−9.23607	−0.806959	−0.403480	+	0.914989i	$0.632199\pi$
$132$	0	0
$133$	−8.00000	−0.693688
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	−14.0000	−1.19610	−0.598050	−	0.801459i	$-0.704058\pi$
$137$	−14.0000	−1.19610	−0.598050	+	0.801459i	$0.704058\pi$
$138$	0	0
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	−8.94427	−0.753244
$142$	0	0
$143$	0	0
$144$	0	0
$145$	7.23607	0.600923
$146$	0	0
$147$	5.47214	0.451334
$148$	0	0
$149$	−18.9443	−1.55198	−0.775988	−	0.630748i	$-0.782748\pi$
$149$	−18.9443	−1.55198	−0.775988	+	0.630748i	$0.782748\pi$
$150$	0	0
$151$	13.8885	1.13023	0.565117	−	0.825011i	$-0.308831\pi$
$151$	13.8885	1.13023	0.565117	+	0.825011i	$0.308831\pi$
$152$	0	0
$153$	4.47214	0.361551
$154$	0	0
$155$	1.00000	0.0803219
$156$	0	0
$157$	−5.41641	−0.432276	−0.216138	−	0.976363i	$-0.569346\pi$
$157$	−5.41641	−0.432276	−0.216138	+	0.976363i	$0.569346\pi$
$158$	0	0
$159$	−8.47214	−0.671884
$160$	0	0
$161$	4.94427	0.389663
$162$	0	0
$163$	6.76393	0.529792	0.264896	−	0.964277i	$-0.414662\pi$
$163$	6.76393	0.529792	0.264896	+	0.964277i	$0.414662\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	−2.52786	−0.194451
$170$	0	0
$171$	6.47214	0.494937
$172$	0	0
$173$	−22.0000	−1.67263	−0.836315	−	0.548250i	$-0.815294\pi$
$173$	−22.0000	−1.67263	−0.836315	+	0.548250i	$0.815294\pi$
$174$	0	0
$175$	−1.23607	−0.0934380
$176$	0	0
$177$	−6.76393	−0.508408
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	21.4164	1.59187	0.795935	−	0.605383i	$-0.206980\pi$
$181$	21.4164	1.59187	0.795935	+	0.605383i	$0.206980\pi$
$182$	0	0
$183$	12.4721	0.921967
$184$	0	0
$185$	−4.76393	−0.350251
$186$	0	0
$187$	0	0
$188$	0	0
$189$	1.23607	0.0899107
$190$	0	0
$191$	−14.7639	−1.06828	−0.534140	−	0.845396i	$-0.679365\pi$
$191$	−14.7639	−1.06828	−0.534140	+	0.845396i	$0.679365\pi$
$192$	0	0
$193$	9.41641	0.677808	0.338904	−	0.940821i	$-0.389944\pi$
$193$	9.41641	0.677808	0.338904	+	0.940821i	$0.389944\pi$
$194$	0	0
$195$	−3.23607	−0.231740
$196$	0	0
$197$	2.94427	0.209771	0.104885	−	0.994484i	$-0.466552\pi$
$197$	2.94427	0.209771	0.104885	+	0.994484i	$0.466552\pi$
$198$	0	0
$199$	13.8885	0.984533	0.492266	−	0.870445i	$-0.336169\pi$
$199$	13.8885	0.984533	0.492266	+	0.870445i	$0.336169\pi$
$200$	0	0
$201$	−3.70820	−0.261557
$202$	0	0
$203$	8.94427	0.627765
$204$	0	0
$205$	6.00000	0.419058
$206$	0	0
$207$	−4.00000	−0.278019
$208$	0	0
$209$	0	0
$210$	0	0
$211$	24.3607	1.67706	0.838529	−	0.544857i	$-0.183416\pi$
$211$	24.3607	1.67706	0.838529	+	0.544857i	$0.183416\pi$
$212$	0	0
$213$	9.23607	0.632845
$214$	0	0
$215$	−6.47214	−0.441396
$216$	0	0
$217$	1.23607	0.0839098
$218$	0	0
$219$	−12.1803	−0.823071
$220$	0	0
$221$	−14.4721	−0.973501
$222$	0	0
$223$	−4.00000	−0.267860	−0.133930	−	0.990991i	$-0.542760\pi$
$223$	−4.00000	−0.267860	−0.133930	+	0.990991i	$0.542760\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−15.4164	−1.02322	−0.511611	−	0.859217i	$-0.670951\pi$
$227$	−15.4164	−1.02322	−0.511611	+	0.859217i	$0.670951\pi$
$228$	0	0
$229$	18.9443	1.25187	0.625936	−	0.779874i	$-0.284717\pi$
$229$	18.9443	1.25187	0.625936	+	0.779874i	$0.284717\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−29.4164	−1.92713	−0.963566	−	0.267469i	$-0.913813\pi$
$233$	−29.4164	−1.92713	−0.963566	+	0.267469i	$0.913813\pi$
$234$	0	0
$235$	−8.94427	−0.583460
$236$	0	0
$237$	8.94427	0.580993
$238$	0	0
$239$	−7.41641	−0.479728	−0.239864	−	0.970807i	$-0.577103\pi$
$239$	−7.41641	−0.479728	−0.239864	+	0.970807i	$0.577103\pi$
$240$	0	0
$241$	−1.05573	−0.0680054	−0.0340027	−	0.999422i	$-0.510825\pi$
$241$	−1.05573	−0.0680054	−0.0340027	+	0.999422i	$0.510825\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	5.47214	0.349602
$246$	0	0
$247$	−20.9443	−1.33265
$248$	0	0
$249$	4.94427	0.313331
$250$	0	0
$251$	−8.00000	−0.504956	−0.252478	−	0.967603i	$-0.581245\pi$
$251$	−8.00000	−0.504956	−0.252478	+	0.967603i	$0.581245\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	4.47214	0.280056
$256$	0	0
$257$	−21.4164	−1.33592	−0.667959	−	0.744198i	$-0.732832\pi$
$257$	−21.4164	−1.33592	−0.667959	+	0.744198i	$0.732832\pi$
$258$	0	0
$259$	−5.88854	−0.365896
$260$	0	0
$261$	−7.23607	−0.447901
$262$	0	0
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	0	0
$265$	−8.47214	−0.520439
$266$	0	0
$267$	7.23607	0.442840
$268$	0	0
$269$	−30.0689	−1.83333	−0.916666	−	0.399654i	$-0.869130\pi$
$269$	−30.0689	−1.83333	−0.916666	+	0.399654i	$0.869130\pi$
$270$	0	0
$271$	−25.8885	−1.57262	−0.786309	−	0.617834i	$-0.788010\pi$
$271$	−25.8885	−1.57262	−0.786309	+	0.617834i	$0.788010\pi$
$272$	0	0
$273$	−4.00000	−0.242091
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−3.81966	−0.229501	−0.114751	−	0.993394i	$-0.536607\pi$
$277$	−3.81966	−0.229501	−0.114751	+	0.993394i	$0.536607\pi$
$278$	0	0
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	−16.4721	−0.982645	−0.491323	−	0.870978i	$-0.663486\pi$
$281$	−16.4721	−0.982645	−0.491323	+	0.870978i	$0.663486\pi$
$282$	0	0
$283$	−14.1803	−0.842934	−0.421467	−	0.906844i	$-0.638485\pi$
$283$	−14.1803	−0.842934	−0.421467	+	0.906844i	$0.638485\pi$
$284$	0	0
$285$	6.47214	0.383376
$286$	0	0
$287$	7.41641	0.437777
$288$	0	0
$289$	3.00000	0.176471
$290$	0	0
$291$	8.47214	0.496645
$292$	0	0
$293$	13.4164	0.783795	0.391897	−	0.920009i	$-0.371819\pi$
$293$	13.4164	0.783795	0.391897	+	0.920009i	$0.371819\pi$
$294$	0	0
$295$	−6.76393	−0.393811
$296$	0	0
$297$	0	0
$298$	0	0
$299$	12.9443	0.748587
$300$	0	0
$301$	−8.00000	−0.461112
$302$	0	0
$303$	10.9443	0.628732
$304$	0	0
$305$	12.4721	0.714152
$306$	0	0
$307$	−6.76393	−0.386038	−0.193019	−	0.981195i	$-0.561828\pi$
$307$	−6.76393	−0.386038	−0.193019	+	0.981195i	$0.561828\pi$
$308$	0	0
$309$	16.6525	0.947326
$310$	0	0
$311$	−14.7639	−0.837186	−0.418593	−	0.908174i	$-0.637477\pi$
$311$	−14.7639	−0.837186	−0.418593	+	0.908174i	$0.637477\pi$
$312$	0	0
$313$	−10.6525	−0.602114	−0.301057	−	0.953606i	$-0.597339\pi$
$313$	−10.6525	−0.602114	−0.301057	+	0.953606i	$0.597339\pi$
$314$	0	0
$315$	1.23607	0.0696445
$316$	0	0
$317$	−28.4721	−1.59915	−0.799577	−	0.600563i	$-0.794943\pi$
$317$	−28.4721	−1.59915	−0.799577	+	0.600563i	$0.794943\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−12.9443	−0.722479
$322$	0	0
$323$	28.9443	1.61050
$324$	0	0
$325$	−3.23607	−0.179505
$326$	0	0
$327$	3.52786	0.195091
$328$	0	0
$329$	−11.0557	−0.609522
$330$	0	0
$331$	−28.9443	−1.59092	−0.795461	−	0.606005i	$-0.792771\pi$
$331$	−28.9443	−1.59092	−0.795461	+	0.606005i	$0.792771\pi$
$332$	0	0
$333$	4.76393	0.261062
$334$	0	0
$335$	−3.70820	−0.202601
$336$	0	0
$337$	−32.1803	−1.75297	−0.876487	−	0.481425i	$-0.840119\pi$
$337$	−32.1803	−1.75297	−0.876487	+	0.481425i	$0.840119\pi$
$338$	0	0
$339$	−2.94427	−0.159911
$340$	0	0
$341$	0	0
$342$	0	0
$343$	15.4164	0.832408
$344$	0	0
$345$	−4.00000	−0.215353
$346$	0	0
$347$	−13.8885	−0.745576	−0.372788	−	0.927917i	$-0.621598\pi$
$347$	−13.8885	−0.745576	−0.372788	+	0.927917i	$0.621598\pi$
$348$	0	0
$349$	20.4721	1.09585	0.547924	−	0.836528i	$-0.315418\pi$
$349$	20.4721	1.09585	0.547924	+	0.836528i	$0.315418\pi$
$350$	0	0
$351$	3.23607	0.172729
$352$	0	0
$353$	−16.4721	−0.876723	−0.438362	−	0.898799i	$-0.644441\pi$
$353$	−16.4721	−0.876723	−0.438362	+	0.898799i	$0.644441\pi$
$354$	0	0
$355$	9.23607	0.490200
$356$	0	0
$357$	5.52786	0.292566
$358$	0	0
$359$	27.1246	1.43158	0.715791	−	0.698314i	$-0.246066\pi$
$359$	27.1246	1.43158	0.715791	+	0.698314i	$0.246066\pi$
$360$	0	0
$361$	22.8885	1.20466
$362$	0	0
$363$	11.0000	0.577350
$364$	0	0
$365$	−12.1803	−0.637548
$366$	0	0
$367$	7.05573	0.368306	0.184153	−	0.982898i	$-0.441046\pi$
$367$	7.05573	0.368306	0.184153	+	0.982898i	$0.441046\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	0	0
$371$	−10.4721	−0.543686
$372$	0	0
$373$	30.3607	1.57202	0.786008	−	0.618216i	$-0.212144\pi$
$373$	30.3607	1.57202	0.786008	+	0.618216i	$0.212144\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	23.4164	1.20601
$378$	0	0
$379$	−19.4164	−0.997354	−0.498677	−	0.866788i	$-0.666181\pi$
$379$	−19.4164	−0.997354	−0.498677	+	0.866788i	$0.666181\pi$
$380$	0	0
$381$	11.4164	0.584880
$382$	0	0
$383$	4.00000	0.204390	0.102195	−	0.994764i	$-0.467413\pi$
$383$	4.00000	0.204390	0.102195	+	0.994764i	$0.467413\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	6.47214	0.328997
$388$	0	0
$389$	−10.2918	−0.521815	−0.260907	−	0.965364i	$-0.584022\pi$
$389$	−10.2918	−0.521815	−0.260907	+	0.965364i	$0.584022\pi$
$390$	0	0
$391$	−17.8885	−0.904663
$392$	0	0
$393$	9.23607	0.465898
$394$	0	0
$395$	8.94427	0.450035
$396$	0	0
$397$	38.9443	1.95456	0.977278	−	0.211959i	$-0.0679844\pi$
$397$	38.9443	1.95456	0.977278	+	0.211959i	$0.0679844\pi$
$398$	0	0
$399$	8.00000	0.400501
$400$	0	0
$401$	−4.76393	−0.237899	−0.118950	−	0.992900i	$-0.537953\pi$
$401$	−4.76393	−0.237899	−0.118950	+	0.992900i	$0.537953\pi$
$402$	0	0
$403$	3.23607	0.161200
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	0	0
$408$	0	0
$409$	10.5836	0.523325	0.261662	−	0.965159i	$-0.415729\pi$
$409$	10.5836	0.523325	0.261662	+	0.965159i	$0.415729\pi$
$410$	0	0
$411$	14.0000	0.690569
$412$	0	0
$413$	−8.36068	−0.411402
$414$	0	0
$415$	4.94427	0.242705
$416$	0	0
$417$	−4.00000	−0.195881
$418$	0	0
$419$	−24.6525	−1.20435	−0.602176	−	0.798363i	$-0.705700\pi$
$419$	−24.6525	−1.20435	−0.602176	+	0.798363i	$0.705700\pi$
$420$	0	0
$421$	−34.3607	−1.67464	−0.837319	−	0.546715i	$-0.815878\pi$
$421$	−34.3607	−1.67464	−0.837319	+	0.546715i	$0.815878\pi$
$422$	0	0
$423$	8.94427	0.434885
$424$	0	0
$425$	4.47214	0.216930
$426$	0	0
$427$	15.4164	0.746052
$428$	0	0
$429$	0	0
$430$	0	0
$431$	35.1246	1.69189	0.845947	−	0.533268i	$-0.179036\pi$
$431$	35.1246	1.69189	0.845947	+	0.533268i	$0.179036\pi$
$432$	0	0
$433$	23.2361	1.11665	0.558327	−	0.829621i	$-0.311443\pi$
$433$	23.2361	1.11665	0.558327	+	0.829621i	$0.311443\pi$
$434$	0	0
$435$	−7.23607	−0.346943
$436$	0	0
$437$	−25.8885	−1.23842
$438$	0	0
$439$	30.8328	1.47157	0.735785	−	0.677215i	$-0.236813\pi$
$439$	30.8328	1.47157	0.735785	+	0.677215i	$0.236813\pi$
$440$	0	0
$441$	−5.47214	−0.260578
$442$	0	0
$443$	21.5279	1.02282	0.511410	−	0.859337i	$-0.329123\pi$
$443$	21.5279	1.02282	0.511410	+	0.859337i	$0.329123\pi$
$444$	0	0
$445$	7.23607	0.343023
$446$	0	0
$447$	18.9443	0.896033
$448$	0	0
$449$	−25.7082	−1.21325	−0.606623	−	0.794990i	$-0.707476\pi$
$449$	−25.7082	−1.21325	−0.606623	+	0.794990i	$0.707476\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−13.8885	−0.652541
$454$	0	0
$455$	−4.00000	−0.187523
$456$	0	0
$457$	15.2361	0.712713	0.356357	−	0.934350i	$-0.384019\pi$
$457$	15.2361	0.712713	0.356357	+	0.934350i	$0.384019\pi$
$458$	0	0
$459$	−4.47214	−0.208741
$460$	0	0
$461$	42.0689	1.95934	0.979672	−	0.200608i	$-0.0642917\pi$
$461$	42.0689	1.95934	0.979672	+	0.200608i	$0.0642917\pi$
$462$	0	0
$463$	−11.4164	−0.530565	−0.265283	−	0.964171i	$-0.585465\pi$
$463$	−11.4164	−0.530565	−0.265283	+	0.964171i	$0.585465\pi$
$464$	0	0
$465$	−1.00000	−0.0463739
$466$	0	0
$467$	4.94427	0.228794	0.114397	−	0.993435i	$-0.463506\pi$
$467$	4.94427	0.228794	0.114397	+	0.993435i	$0.463506\pi$
$468$	0	0
$469$	−4.58359	−0.211651
$470$	0	0
$471$	5.41641	0.249575
$472$	0	0
$473$	0	0
$474$	0	0
$475$	6.47214	0.296962
$476$	0	0
$477$	8.47214	0.387912
$478$	0	0
$479$	−12.2918	−0.561626	−0.280813	−	0.959762i	$-0.590604\pi$
$479$	−12.2918	−0.561626	−0.280813	+	0.959762i	$0.590604\pi$
$480$	0	0
$481$	−15.4164	−0.702928
$482$	0	0
$483$	−4.94427	−0.224972
$484$	0	0
$485$	8.47214	0.384700
$486$	0	0
$487$	9.52786	0.431749	0.215874	−	0.976421i	$-0.430740\pi$
$487$	9.52786	0.431749	0.215874	+	0.976421i	$0.430740\pi$
$488$	0	0
$489$	−6.76393	−0.305876
$490$	0	0
$491$	0.583592	0.0263371	0.0131686	−	0.999913i	$-0.495808\pi$
$491$	0.583592	0.0263371	0.0131686	+	0.999913i	$0.495808\pi$
$492$	0	0
$493$	−32.3607	−1.45745
$494$	0	0
$495$	0	0
$496$	0	0
$497$	11.4164	0.512096
$498$	0	0
$499$	32.9443	1.47479	0.737394	−	0.675463i	$-0.236056\pi$
$499$	32.9443	1.47479	0.737394	+	0.675463i	$0.236056\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−18.8328	−0.839714	−0.419857	−	0.907590i	$-0.637920\pi$
$503$	−18.8328	−0.839714	−0.419857	+	0.907590i	$0.637920\pi$
$504$	0	0
$505$	10.9443	0.487014
$506$	0	0
$507$	2.52786	0.112266
$508$	0	0
$509$	37.1246	1.64552	0.822760	−	0.568389i	$-0.192433\pi$
$509$	37.1246	1.64552	0.822760	+	0.568389i	$0.192433\pi$
$510$	0	0
$511$	−15.0557	−0.666026
$512$	0	0
$513$	−6.47214	−0.285752
$514$	0	0
$515$	16.6525	0.733796
$516$	0	0
$517$	0	0
$518$	0	0
$519$	22.0000	0.965693
$520$	0	0
$521$	−26.9443	−1.18045	−0.590225	−	0.807239i	$-0.700961\pi$
$521$	−26.9443	−1.18045	−0.590225	+	0.807239i	$0.700961\pi$
$522$	0	0
$523$	40.3607	1.76485	0.882425	−	0.470454i	$-0.155910\pi$
$523$	40.3607	1.76485	0.882425	+	0.470454i	$0.155910\pi$
$524$	0	0
$525$	1.23607	0.0539464
$526$	0	0
$527$	−4.47214	−0.194809
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	6.76393	0.293530
$532$	0	0
$533$	19.4164	0.841018
$534$	0	0
$535$	−12.9443	−0.559630
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−39.3050	−1.68985	−0.844926	−	0.534883i	$-0.820356\pi$
$541$	−39.3050	−1.68985	−0.844926	+	0.534883i	$0.820356\pi$
$542$	0	0
$543$	−21.4164	−0.919066
$544$	0	0
$545$	3.52786	0.151117
$546$	0	0
$547$	16.6525	0.712008	0.356004	−	0.934484i	$-0.384139\pi$
$547$	16.6525	0.712008	0.356004	+	0.934484i	$0.384139\pi$
$548$	0	0
$549$	−12.4721	−0.532298
$550$	0	0
$551$	−46.8328	−1.99515
$552$	0	0
$553$	11.0557	0.470137
$554$	0	0
$555$	4.76393	0.202218
$556$	0	0
$557$	−17.4164	−0.737957	−0.368978	−	0.929438i	$-0.620292\pi$
$557$	−17.4164	−0.737957	−0.368978	+	0.929438i	$0.620292\pi$
$558$	0	0
$559$	−20.9443	−0.885848
$560$	0	0
$561$	0	0
$562$	0	0
$563$	32.0000	1.34864	0.674320	−	0.738440i	$-0.264437\pi$
$563$	32.0000	1.34864	0.674320	+	0.738440i	$0.264437\pi$
$564$	0	0
$565$	−2.94427	−0.123866
$566$	0	0
$567$	−1.23607	−0.0519100
$568$	0	0
$569$	−6.65248	−0.278886	−0.139443	−	0.990230i	$-0.544531\pi$
$569$	−6.65248	−0.278886	−0.139443	+	0.990230i	$0.544531\pi$
$570$	0	0
$571$	−24.9443	−1.04389	−0.521943	−	0.852981i	$-0.674793\pi$
$571$	−24.9443	−1.04389	−0.521943	+	0.852981i	$0.674793\pi$
$572$	0	0
$573$	14.7639	0.616772
$574$	0	0
$575$	−4.00000	−0.166812
$576$	0	0
$577$	−9.05573	−0.376995	−0.188497	−	0.982074i	$-0.560362\pi$
$577$	−9.05573	−0.376995	−0.188497	+	0.982074i	$0.560362\pi$
$578$	0	0
$579$	−9.41641	−0.391333
$580$	0	0
$581$	6.11146	0.253546
$582$	0	0
$583$	0	0
$584$	0	0
$585$	3.23607	0.133795
$586$	0	0
$587$	−24.9443	−1.02956	−0.514780	−	0.857322i	$-0.672126\pi$
$587$	−24.9443	−1.02956	−0.514780	+	0.857322i	$0.672126\pi$
$588$	0	0
$589$	−6.47214	−0.266680
$590$	0	0
$591$	−2.94427	−0.121111
$592$	0	0
$593$	−8.83282	−0.362720	−0.181360	−	0.983417i	$-0.558050\pi$
$593$	−8.83282	−0.362720	−0.181360	+	0.983417i	$0.558050\pi$
$594$	0	0
$595$	5.52786	0.226620
$596$	0	0
$597$	−13.8885	−0.568420
$598$	0	0
$599$	19.1246	0.781410	0.390705	−	0.920516i	$-0.372231\pi$
$599$	19.1246	0.781410	0.390705	+	0.920516i	$0.372231\pi$
$600$	0	0
$601$	36.4721	1.48773	0.743865	−	0.668330i	$-0.232991\pi$
$601$	36.4721	1.48773	0.743865	+	0.668330i	$0.232991\pi$
$602$	0	0
$603$	3.70820	0.151010
$604$	0	0
$605$	11.0000	0.447214
$606$	0	0
$607$	−5.59675	−0.227165	−0.113582	−	0.993529i	$-0.536233\pi$
$607$	−5.59675	−0.227165	−0.113582	+	0.993529i	$0.536233\pi$
$608$	0	0
$609$	−8.94427	−0.362440
$610$	0	0
$611$	−28.9443	−1.17096
$612$	0	0
$613$	−5.70820	−0.230552	−0.115276	−	0.993333i	$-0.536775\pi$
$613$	−5.70820	−0.230552	−0.115276	+	0.993333i	$0.536775\pi$
$614$	0	0
$615$	−6.00000	−0.241943
$616$	0	0
$617$	36.8328	1.48283	0.741417	−	0.671045i	$-0.234154\pi$
$617$	36.8328	1.48283	0.741417	+	0.671045i	$0.234154\pi$
$618$	0	0
$619$	43.7771	1.75955	0.879775	−	0.475391i	$-0.157693\pi$
$619$	43.7771	1.75955	0.879775	+	0.475391i	$0.157693\pi$
$620$	0	0
$621$	4.00000	0.160514
$622$	0	0
$623$	8.94427	0.358345
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	21.3050	0.849484
$630$	0	0
$631$	−3.05573	−0.121647	−0.0608233	−	0.998149i	$-0.519373\pi$
$631$	−3.05573	−0.121647	−0.0608233	+	0.998149i	$0.519373\pi$
$632$	0	0
$633$	−24.3607	−0.968250
$634$	0	0
$635$	11.4164	0.453046
$636$	0	0
$637$	17.7082	0.701625
$638$	0	0
$639$	−9.23607	−0.365373
$640$	0	0
$641$	0.180340	0.00712300	0.00356150	−	0.999994i	$-0.498866\pi$
$641$	0.180340	0.00712300	0.00356150	+	0.999994i	$0.498866\pi$
$642$	0	0
$643$	−27.4164	−1.08120	−0.540599	−	0.841281i	$-0.681802\pi$
$643$	−27.4164	−1.08120	−0.540599	+	0.841281i	$0.681802\pi$
$644$	0	0
$645$	6.47214	0.254840
$646$	0	0
$647$	−19.0557	−0.749158	−0.374579	−	0.927195i	$-0.622213\pi$
$647$	−19.0557	−0.749158	−0.374579	+	0.927195i	$0.622213\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−1.23607	−0.0484453
$652$	0	0
$653$	29.4164	1.15115	0.575576	−	0.817748i	$-0.304778\pi$
$653$	29.4164	1.15115	0.575576	+	0.817748i	$0.304778\pi$
$654$	0	0
$655$	9.23607	0.360883
$656$	0	0
$657$	12.1803	0.475200
$658$	0	0
$659$	25.2361	0.983058	0.491529	−	0.870861i	$-0.336438\pi$
$659$	25.2361	0.983058	0.491529	+	0.870861i	$0.336438\pi$
$660$	0	0
$661$	−6.94427	−0.270101	−0.135050	−	0.990839i	$-0.543120\pi$
$661$	−6.94427	−0.270101	−0.135050	+	0.990839i	$0.543120\pi$
$662$	0	0
$663$	14.4721	0.562051
$664$	0	0
$665$	8.00000	0.310227
$666$	0	0
$667$	28.9443	1.12073
$668$	0	0
$669$	4.00000	0.154649
$670$	0	0
$671$	0	0
$672$	0	0
$673$	12.7639	0.492013	0.246007	−	0.969268i	$-0.420881\pi$
$673$	12.7639	0.492013	0.246007	+	0.969268i	$0.420881\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−33.4164	−1.28430	−0.642148	−	0.766580i	$-0.721957\pi$
$677$	−33.4164	−1.28430	−0.642148	+	0.766580i	$0.721957\pi$
$678$	0	0
$679$	10.4721	0.401884
$680$	0	0
$681$	15.4164	0.590758
$682$	0	0
$683$	−7.41641	−0.283781	−0.141890	−	0.989882i	$-0.545318\pi$
$683$	−7.41641	−0.283781	−0.141890	+	0.989882i	$0.545318\pi$
$684$	0	0
$685$	14.0000	0.534913
$686$	0	0
$687$	−18.9443	−0.722769
$688$	0	0
$689$	−27.4164	−1.04448
$690$	0	0
$691$	−12.0000	−0.456502	−0.228251	−	0.973602i	$-0.573301\pi$
$691$	−12.0000	−0.456502	−0.228251	+	0.973602i	$0.573301\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−4.00000	−0.151729
$696$	0	0
$697$	−26.8328	−1.01637
$698$	0	0
$699$	29.4164	1.11263
$700$	0	0
$701$	−35.5279	−1.34187	−0.670934	−	0.741517i	$-0.734107\pi$
$701$	−35.5279	−1.34187	−0.670934	+	0.741517i	$0.734107\pi$
$702$	0	0
$703$	30.8328	1.16288
$704$	0	0
$705$	8.94427	0.336861
$706$	0	0
$707$	13.5279	0.508768
$708$	0	0
$709$	14.0000	0.525781	0.262891	−	0.964826i	$-0.415324\pi$
$709$	14.0000	0.525781	0.262891	+	0.964826i	$0.415324\pi$
$710$	0	0
$711$	−8.94427	−0.335436
$712$	0	0
$713$	4.00000	0.149801
$714$	0	0
$715$	0	0
$716$	0	0
$717$	7.41641	0.276971
$718$	0	0
$719$	−16.0000	−0.596699	−0.298350	−	0.954457i	$-0.596436\pi$
$719$	−16.0000	−0.596699	−0.298350	+	0.954457i	$0.596436\pi$
$720$	0	0
$721$	20.5836	0.766573
$722$	0	0
$723$	1.05573	0.0392630
$724$	0	0
$725$	−7.23607	−0.268741
$726$	0	0
$727$	−2.40325	−0.0891317	−0.0445658	−	0.999006i	$-0.514190\pi$
$727$	−2.40325	−0.0891317	−0.0445658	+	0.999006i	$0.514190\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	28.9443	1.07054
$732$	0	0
$733$	−25.7771	−0.952098	−0.476049	−	0.879419i	$-0.657932\pi$
$733$	−25.7771	−0.952098	−0.476049	+	0.879419i	$0.657932\pi$
$734$	0	0
$735$	−5.47214	−0.201843
$736$	0	0
$737$	0	0
$738$	0	0
$739$	20.9443	0.770447	0.385224	−	0.922823i	$-0.374124\pi$
$739$	20.9443	0.770447	0.385224	+	0.922823i	$0.374124\pi$
$740$	0	0
$741$	20.9443	0.769407
$742$	0	0
$743$	−16.0000	−0.586983	−0.293492	−	0.955962i	$-0.594817\pi$
$743$	−16.0000	−0.586983	−0.293492	+	0.955962i	$0.594817\pi$
$744$	0	0
$745$	18.9443	0.694064
$746$	0	0
$747$	−4.94427	−0.180901
$748$	0	0
$749$	−16.0000	−0.584627
$750$	0	0
$751$	1.88854	0.0689139	0.0344570	−	0.999406i	$-0.489030\pi$
$751$	1.88854	0.0689139	0.0344570	+	0.999406i	$0.489030\pi$
$752$	0	0
$753$	8.00000	0.291536
$754$	0	0
$755$	−13.8885	−0.505456
$756$	0	0
$757$	−13.1246	−0.477022	−0.238511	−	0.971140i	$-0.576659\pi$
$757$	−13.1246	−0.477022	−0.238511	+	0.971140i	$0.576659\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	40.1803	1.45654	0.728268	−	0.685292i	$-0.240326\pi$
$761$	40.1803	1.45654	0.728268	+	0.685292i	$0.240326\pi$
$762$	0	0
$763$	4.36068	0.157867
$764$	0	0
$765$	−4.47214	−0.161690
$766$	0	0
$767$	−21.8885	−0.790350
$768$	0	0
$769$	8.47214	0.305513	0.152757	−	0.988264i	$-0.451185\pi$
$769$	8.47214	0.305513	0.152757	+	0.988264i	$0.451185\pi$
$770$	0	0
$771$	21.4164	0.771293
$772$	0	0
$773$	23.3050	0.838221	0.419110	−	0.907935i	$-0.362342\pi$
$773$	23.3050	0.838221	0.419110	+	0.907935i	$0.362342\pi$
$774$	0	0
$775$	−1.00000	−0.0359211
$776$	0	0
$777$	5.88854	0.211250
$778$	0	0
$779$	−38.8328	−1.39133
$780$	0	0
$781$	0	0
$782$	0	0
$783$	7.23607	0.258596
$784$	0	0
$785$	5.41641	0.193320
$786$	0	0
$787$	12.0000	0.427754	0.213877	−	0.976861i	$-0.431391\pi$
$787$	12.0000	0.427754	0.213877	+	0.976861i	$0.431391\pi$
$788$	0	0
$789$	−24.0000	−0.854423
$790$	0	0
$791$	−3.63932	−0.129399
$792$	0	0
$793$	40.3607	1.43325
$794$	0	0
$795$	8.47214	0.300476
$796$	0	0
$797$	28.8328	1.02131	0.510655	−	0.859785i	$-0.329403\pi$
$797$	28.8328	1.02131	0.510655	+	0.859785i	$0.329403\pi$
$798$	0	0
$799$	40.0000	1.41510
$800$	0	0
$801$	−7.23607	−0.255674
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−4.94427	−0.174263
$806$	0	0
$807$	30.0689	1.05847
$808$	0	0
$809$	−1.70820	−0.0600573	−0.0300286	−	0.999549i	$-0.509560\pi$
$809$	−1.70820	−0.0600573	−0.0300286	+	0.999549i	$0.509560\pi$
$810$	0	0
$811$	−32.3607	−1.13634	−0.568169	−	0.822912i	$-0.692348\pi$
$811$	−32.3607	−1.13634	−0.568169	+	0.822912i	$0.692348\pi$
$812$	0	0
$813$	25.8885	0.907951
$814$	0	0
$815$	−6.76393	−0.236930
$816$	0	0
$817$	41.8885	1.46549
$818$	0	0
$819$	4.00000	0.139771
$820$	0	0
$821$	−4.18034	−0.145895	−0.0729474	−	0.997336i	$-0.523241\pi$
$821$	−4.18034	−0.145895	−0.0729474	+	0.997336i	$0.523241\pi$
$822$	0	0
$823$	−8.36068	−0.291435	−0.145717	−	0.989326i	$-0.546549\pi$
$823$	−8.36068	−0.291435	−0.145717	+	0.989326i	$0.546549\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	16.0000	0.556375	0.278187	−	0.960527i	$-0.410266\pi$
$827$	16.0000	0.556375	0.278187	+	0.960527i	$0.410266\pi$
$828$	0	0
$829$	31.3050	1.08727	0.543633	−	0.839323i	$-0.317048\pi$
$829$	31.3050	1.08727	0.543633	+	0.839323i	$0.317048\pi$
$830$	0	0
$831$	3.81966	0.132503
$832$	0	0
$833$	−24.4721	−0.847909
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1.00000	0.0345651
$838$	0	0
$839$	1.23607	0.0426738	0.0213369	−	0.999772i	$-0.493208\pi$
$839$	1.23607	0.0426738	0.0213369	+	0.999772i	$0.493208\pi$
$840$	0	0
$841$	23.3607	0.805541
$842$	0	0
$843$	16.4721	0.567330
$844$	0	0
$845$	2.52786	0.0869612
$846$	0	0
$847$	13.5967	0.467190
$848$	0	0
$849$	14.1803	0.486668
$850$	0	0
$851$	−19.0557	−0.653222
$852$	0	0
$853$	37.7771	1.29346	0.646731	−	0.762718i	$-0.276135\pi$
$853$	37.7771	1.29346	0.646731	+	0.762718i	$0.276135\pi$
$854$	0	0
$855$	−6.47214	−0.221342
$856$	0	0
$857$	−2.00000	−0.0683187	−0.0341593	−	0.999416i	$-0.510875\pi$
$857$	−2.00000	−0.0683187	−0.0341593	+	0.999416i	$0.510875\pi$
$858$	0	0
$859$	53.8885	1.83865	0.919327	−	0.393495i	$-0.128734\pi$
$859$	53.8885	1.83865	0.919327	+	0.393495i	$0.128734\pi$
$860$	0	0
$861$	−7.41641	−0.252751
$862$	0	0
$863$	32.0000	1.08929	0.544646	−	0.838666i	$-0.316664\pi$
$863$	32.0000	1.08929	0.544646	+	0.838666i	$0.316664\pi$
$864$	0	0
$865$	22.0000	0.748022
$866$	0	0
$867$	−3.00000	−0.101885
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−12.0000	−0.406604
$872$	0	0
$873$	−8.47214	−0.286738
$874$	0	0
$875$	1.23607	0.0417867
$876$	0	0
$877$	14.9443	0.504632	0.252316	−	0.967645i	$-0.418808\pi$
$877$	14.9443	0.504632	0.252316	+	0.967645i	$0.418808\pi$
$878$	0	0
$879$	−13.4164	−0.452524
$880$	0	0
$881$	−14.0689	−0.473993	−0.236996	−	0.971511i	$-0.576163\pi$
$881$	−14.0689	−0.473993	−0.236996	+	0.971511i	$0.576163\pi$
$882$	0	0
$883$	−34.8328	−1.17222	−0.586109	−	0.810232i	$-0.699341\pi$
$883$	−34.8328	−1.17222	−0.586109	+	0.810232i	$0.699341\pi$
$884$	0	0
$885$	6.76393	0.227367
$886$	0	0
$887$	−19.4164	−0.651939	−0.325970	−	0.945380i	$-0.605691\pi$
$887$	−19.4164	−0.651939	−0.325970	+	0.945380i	$0.605691\pi$
$888$	0	0
$889$	14.1115	0.473283
$890$	0	0
$891$	0	0
$892$	0	0
$893$	57.8885	1.93717
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−12.9443	−0.432197
$898$	0	0
$899$	7.23607	0.241336
$900$	0	0
$901$	37.8885	1.26225
$902$	0	0
$903$	8.00000	0.266223
$904$	0	0
$905$	−21.4164	−0.711905
$906$	0	0
$907$	−50.5410	−1.67819	−0.839094	−	0.543987i	$-0.816914\pi$
$907$	−50.5410	−1.67819	−0.839094	+	0.543987i	$0.816914\pi$
$908$	0	0
$909$	−10.9443	−0.362999
$910$	0	0
$911$	−9.88854	−0.327622	−0.163811	−	0.986492i	$-0.552379\pi$
$911$	−9.88854	−0.327622	−0.163811	+	0.986492i	$0.552379\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−12.4721	−0.412316
$916$	0	0
$917$	11.4164	0.377003
$918$	0	0
$919$	−33.3050	−1.09863	−0.549314	−	0.835616i	$-0.685111\pi$
$919$	−33.3050	−1.09863	−0.549314	+	0.835616i	$0.685111\pi$
$920$	0	0
$921$	6.76393	0.222879
$922$	0	0
$923$	29.8885	0.983793
$924$	0	0
$925$	4.76393	0.156637
$926$	0	0
$927$	−16.6525	−0.546939
$928$	0	0
$929$	23.5967	0.774184	0.387092	−	0.922041i	$-0.373480\pi$
$929$	23.5967	0.774184	0.387092	+	0.922041i	$0.373480\pi$
$930$	0	0
$931$	−35.4164	−1.16073
$932$	0	0
$933$	14.7639	0.483349
$934$	0	0
$935$	0	0
$936$	0	0
$937$	2.58359	0.0844023	0.0422011	−	0.999109i	$-0.486563\pi$
$937$	2.58359	0.0844023	0.0422011	+	0.999109i	$0.486563\pi$
$938$	0	0
$939$	10.6525	0.347630
$940$	0	0
$941$	33.4853	1.09159	0.545795	−	0.837919i	$-0.316228\pi$
$941$	33.4853	1.09159	0.545795	+	0.837919i	$0.316228\pi$
$942$	0	0
$943$	24.0000	0.781548
$944$	0	0
$945$	−1.23607	−0.0402093
$946$	0	0
$947$	−4.94427	−0.160667	−0.0803336	−	0.996768i	$-0.525599\pi$
$947$	−4.94427	−0.160667	−0.0803336	+	0.996768i	$0.525599\pi$
$948$	0	0
$949$	−39.4164	−1.27951
$950$	0	0
$951$	28.4721	0.923272
$952$	0	0
$953$	−26.9443	−0.872811	−0.436405	−	0.899750i	$-0.643749\pi$
$953$	−26.9443	−0.872811	−0.436405	+	0.899750i	$0.643749\pi$
$954$	0	0
$955$	14.7639	0.477750
$956$	0	0
$957$	0	0
$958$	0	0
$959$	17.3050	0.558806
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	12.9443	0.417123
$964$	0	0
$965$	−9.41641	−0.303125
$966$	0	0
$967$	−44.0000	−1.41494	−0.707472	−	0.706741i	$-0.750165\pi$
$967$	−44.0000	−1.41494	−0.707472	+	0.706741i	$0.750165\pi$
$968$	0	0
$969$	−28.9443	−0.929824
$970$	0	0
$971$	−19.7082	−0.632466	−0.316233	−	0.948681i	$-0.602418\pi$
$971$	−19.7082	−0.632466	−0.316233	+	0.948681i	$0.602418\pi$
$972$	0	0
$973$	−4.94427	−0.158506
$974$	0	0
$975$	3.23607	0.103637
$976$	0	0
$977$	2.94427	0.0941956	0.0470978	−	0.998890i	$-0.485003\pi$
$977$	2.94427	0.0941956	0.0470978	+	0.998890i	$0.485003\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−3.52786	−0.112636
$982$	0	0
$983$	−11.7771	−0.375631	−0.187815	−	0.982204i	$-0.560141\pi$
$983$	−11.7771	−0.375631	−0.187815	+	0.982204i	$0.560141\pi$
$984$	0	0
$985$	−2.94427	−0.0938123
$986$	0	0
$987$	11.0557	0.351908
$988$	0	0
$989$	−25.8885	−0.823208
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	0	0
$993$	28.9443	0.918519
$994$	0	0
$995$	−13.8885	−0.440296
$996$	0	0
$997$	−11.5279	−0.365091	−0.182546	−	0.983197i	$-0.558434\pi$
$997$	−11.5279	−0.365091	−0.182546	+	0.983197i	$0.558434\pi$
$998$	0	0
$999$	−4.76393	−0.150724

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3720.2.a.i.1.1	✓	2
4.3	odd	2		7440.2.a.bi.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3720.2.a.i.1.1	✓	2	1.1	even	1	trivial
7440.2.a.bi.1.2		2	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:	\( N \)	\(=\)	\( 3720 = 2^{3} \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3720.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000 q^{5} -1.23607 q^{7} +1.00000 q^{9} -3.23607 q^{13} +1.00000 q^{15} +4.47214 q^{17} +6.47214 q^{19} +1.23607 q^{21} -4.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} -7.23607 q^{29} -1.00000 q^{31} +1.23607 q^{35} +4.76393 q^{37} +3.23607 q^{39} -6.00000 q^{41} +6.47214 q^{43} -1.00000 q^{45} +8.94427 q^{47} -5.47214 q^{49} -4.47214 q^{51} +8.47214 q^{53} -6.47214 q^{57} +6.76393 q^{59} -12.4721 q^{61} -1.23607 q^{63} +3.23607 q^{65} +3.70820 q^{67} +4.00000 q^{69} -9.23607 q^{71} +12.1803 q^{73} -1.00000 q^{75} -8.94427 q^{79} +1.00000 q^{81} -4.94427 q^{83} -4.47214 q^{85} +7.23607 q^{87} -7.23607 q^{89} +4.00000 q^{91} +1.00000 q^{93} -6.47214 q^{95} -8.47214 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9} - 2 q^{13} + 2 q^{15} + 4 q^{19} - 2 q^{21} - 8 q^{23} + 2 q^{25} - 2 q^{27} - 10 q^{29} - 2 q^{31} - 2 q^{35} + 14 q^{37} + 2 q^{39} - 12 q^{41} + 4 q^{43} - 2 q^{45}+ \cdots - 8 q^{97}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9 - 2 * q^13 + 2 * q^15 + 4 * q^19 - 2 * q^21 - 8 * q^23 + 2 * q^25 - 2 * q^27 - 10 * q^29 - 2 * q^31 - 2 * q^35 + 14 * q^37 + 2 * q^39 - 12 * q^41 + 4 * q^43 - 2 * q^45 - 2 * q^49 + 8 * q^53 - 4 * q^57 + 18 * q^59 - 16 * q^61 + 2 * q^63 + 2 * q^65 - 6 * q^67 + 8 * q^69 - 14 * q^71 + 2 * q^73 - 2 * q^75 + 2 * q^81 + 8 * q^83 + 10 * q^87 - 10 * q^89 + 8 * q^91 + 2 * q^93 - 4 * q^95 - 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more