LMFDB - Embedded newform 2960.2.a.v.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2960,2,Mod(1,2960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2960, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2960.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2960 = 2^{4} \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2960.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:	$23.6357189983$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.286164.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 11x^{2} - 3x + 12 $ x^4 - x^3 - 11x^2 - 3x + 12
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 740)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.24418$ of defining polynomial
Character	$\chi$	$=$	2960.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.24418 q^{3} +1.00000 q^{5} -1.24418 q^{7} +7.52471 q^{9} +O(q^{10})$	$q-3.24418 q^{3} +1.00000 q^{5} -1.24418 q^{7} +7.52471 q^{9} +6.19063 q^{11} +1.66592 q^{13} -3.24418 q^{15} +5.66592 q^{17} +4.94645 q^{19} +4.03635 q^{21} +4.48836 q^{23} +1.00000 q^{25} -14.6790 q^{27} -4.48836 q^{29} -0.385383 q^{31} -20.0835 q^{33} -1.24418 q^{35} +1.00000 q^{37} -5.40453 q^{39} -0.190629 q^{41} -4.56106 q^{43} +7.52471 q^{45} +6.08765 q^{47} -5.45201 q^{49} -18.3813 q^{51} +1.70227 q^{53} +6.19063 q^{55} -16.0472 q^{57} +6.10298 q^{59} +2.00000 q^{61} -9.36211 q^{63} +1.66592 q^{65} -13.4348 q^{67} -14.5611 q^{69} +16.6790 q^{71} -0.786097 q^{73} -3.24418 q^{75} -7.70227 q^{77} +10.9464 q^{79} +25.0472 q^{81} -13.8052 q^{83} +5.66592 q^{85} +14.5611 q^{87} +5.33183 q^{89} -2.07270 q^{91} +1.25025 q^{93} +4.94645 q^{95} -0.561064 q^{97} +46.5827 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 3 q^{3} + 4 q^{5} + 5 q^{7} + 13 q^{9}+O(q^{10}) $ 4 * q - 3 * q^3 + 4 * q^5 + 5 * q^7 + 13 * q^9	$ 4 q - 3 q^{3} + 4 q^{5} + 5 q^{7} + 13 q^{9} - 5 q^{11} - 6 q^{13} - 3 q^{15} + 10 q^{17} + 19 q^{21} - 2 q^{23} + 4 q^{25} - 9 q^{27} + 2 q^{29} + 4 q^{31} - 11 q^{33} + 5 q^{35} + 4 q^{37} - 2 q^{39} + 29 q^{41} - 4 q^{43} + 13 q^{45} + 9 q^{47} + q^{49} - 14 q^{51} - 3 q^{53} - 5 q^{55} + 8 q^{57} + 10 q^{59} + 8 q^{61} + 8 q^{63} - 6 q^{65} - 14 q^{67} - 44 q^{69} + 17 q^{71} + 7 q^{73} - 3 q^{75} - 21 q^{77} + 24 q^{79} + 28 q^{81} - 31 q^{83} + 10 q^{85} + 44 q^{87} - 4 q^{89} - 14 q^{91} + 18 q^{93} + 12 q^{97} + 22 q^{99}+O(q^{100}) $ 4 * q - 3 * q^3 + 4 * q^5 + 5 * q^7 + 13 * q^9 - 5 * q^11 - 6 * q^13 - 3 * q^15 + 10 * q^17 + 19 * q^21 - 2 * q^23 + 4 * q^25 - 9 * q^27 + 2 * q^29 + 4 * q^31 - 11 * q^33 + 5 * q^35 + 4 * q^37 - 2 * q^39 + 29 * q^41 - 4 * q^43 + 13 * q^45 + 9 * q^47 + q^49 - 14 * q^51 - 3 * q^53 - 5 * q^55 + 8 * q^57 + 10 * q^59 + 8 * q^61 + 8 * q^63 - 6 * q^65 - 14 * q^67 - 44 * q^69 + 17 * q^71 + 7 * q^73 - 3 * q^75 - 21 * q^77 + 24 * q^79 + 28 * q^81 - 31 * q^83 + 10 * q^85 + 44 * q^87 - 4 * q^89 - 14 * q^91 + 18 * q^93 + 12 * q^97 + 22 * 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
$3$	−3.24418	−1.87303	−0.936515	−	0.350629i	$-0.885968\pi$
$3$	−3.24418	−1.87303	−0.936515	+	0.350629i	$0.885968\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.24418	−0.470256	−0.235128	−	0.971964i	$-0.575551\pi$
$7$	−1.24418	−0.470256	−0.235128	+	0.971964i	$0.575551\pi$
$8$	0	0
$9$	7.52471	2.50824
$10$	0	0
$11$	6.19063	1.86654	0.933272	−	0.359169i	$-0.116940\pi$
$11$	6.19063	1.86654	0.933272	+	0.359169i	$0.116940\pi$
$12$	0	0
$13$	1.66592	0.462042	0.231021	−	0.972949i	$-0.425793\pi$
$13$	1.66592	0.462042	0.231021	+	0.972949i	$0.425793\pi$
$14$	0	0
$15$	−3.24418	−0.837644
$16$	0	0
$17$	5.66592	1.37419	0.687093	−	0.726569i	$-0.258886\pi$
$17$	5.66592	1.37419	0.687093	+	0.726569i	$0.258886\pi$
$18$	0	0
$19$	4.94645	1.13479	0.567396	−	0.823445i	$-0.307951\pi$
$19$	4.94645	1.13479	0.567396	+	0.823445i	$0.307951\pi$
$20$	0	0
$21$	4.03635	0.880804
$22$	0	0
$23$	4.48836	0.935888	0.467944	−	0.883758i	$-0.344995\pi$
$23$	4.48836	0.935888	0.467944	+	0.883758i	$0.344995\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−14.6790	−2.82497
$28$	0	0
$29$	−4.48836	−0.833468	−0.416734	−	0.909028i	$-0.636825\pi$
$29$	−4.48836	−0.833468	−0.416734	+	0.909028i	$0.636825\pi$
$30$	0	0
$31$	−0.385383	−0.0692169	−0.0346084	−	0.999401i	$-0.511018\pi$
$31$	−0.385383	−0.0692169	−0.0346084	+	0.999401i	$0.511018\pi$
$32$	0	0
$33$	−20.0835	−3.49609
$34$	0	0
$35$	−1.24418	−0.210305
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	0	0
$39$	−5.40453	−0.865418
$40$	0	0
$41$	−0.190629	−0.0297712	−0.0148856	−	0.999889i	$-0.504738\pi$
$41$	−0.190629	−0.0297712	−0.0148856	+	0.999889i	$0.504738\pi$
$42$	0	0
$43$	−4.56106	−0.695556	−0.347778	−	0.937577i	$-0.613064\pi$
$43$	−4.56106	−0.695556	−0.347778	+	0.937577i	$0.613064\pi$
$44$	0	0
$45$	7.52471	1.12172
$46$	0	0
$47$	6.08765	0.887975	0.443987	−	0.896033i	$-0.353563\pi$
$47$	6.08765	0.887975	0.443987	+	0.896033i	$0.353563\pi$
$48$	0	0
$49$	−5.45201	−0.778859
$50$	0	0
$51$	−18.3813	−2.57389
$52$	0	0
$53$	1.70227	0.233824	0.116912	−	0.993142i	$-0.462700\pi$
$53$	1.70227	0.233824	0.116912	+	0.993142i	$0.462700\pi$
$54$	0	0
$55$	6.19063	0.834744
$56$	0	0
$57$	−16.0472	−2.12550
$58$	0	0
$59$	6.10298	0.794540	0.397270	−	0.917702i	$-0.369958\pi$
$59$	6.10298	0.794540	0.397270	+	0.917702i	$0.369958\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	0	0
$63$	−9.36211	−1.17951
$64$	0	0
$65$	1.66592	0.206631
$66$	0	0
$67$	−13.4348	−1.64132	−0.820662	−	0.571414i	$-0.806395\pi$
$67$	−13.4348	−1.64132	−0.820662	+	0.571414i	$0.806395\pi$
$68$	0	0
$69$	−14.5611	−1.75295
$70$	0	0
$71$	16.6790	1.97943	0.989716	−	0.143046i	$-0.0456896\pi$
$71$	16.6790	1.97943	0.989716	+	0.143046i	$0.0456896\pi$
$72$	0	0
$73$	−0.786097	−0.0920057	−0.0460028	−	0.998941i	$-0.514648\pi$
$73$	−0.786097	−0.0920057	−0.0460028	+	0.998941i	$0.514648\pi$
$74$	0	0
$75$	−3.24418	−0.374606
$76$	0	0
$77$	−7.70227	−0.877755
$78$	0	0
$79$	10.9464	1.23157	0.615786	−	0.787914i	$-0.288839\pi$
$79$	10.9464	1.23157	0.615786	+	0.787914i	$0.288839\pi$
$80$	0	0
$81$	25.0472	2.78302
$82$	0	0
$83$	−13.8052	−1.51532	−0.757661	−	0.652648i	$-0.773658\pi$
$83$	−13.8052	−1.51532	−0.757661	+	0.652648i	$0.773658\pi$
$84$	0	0
$85$	5.66592	0.614555
$86$	0	0
$87$	14.5611	1.56111
$88$	0	0
$89$	5.33183	0.565173	0.282586	−	0.959242i	$-0.408808\pi$
$89$	5.33183	0.565173	0.282586	+	0.959242i	$0.408808\pi$
$90$	0	0
$91$	−2.07270	−0.217278
$92$	0	0
$93$	1.25025	0.129645
$94$	0	0
$95$	4.94645	0.507495
$96$	0	0
$97$	−0.561064	−0.0569674	−0.0284837	−	0.999594i	$-0.509068\pi$
$97$	−0.561064	−0.0569674	−0.0284837	+	0.999594i	$0.509068\pi$
$98$	0	0
$99$	46.5827	4.68174
$100$	0	0
$101$	10.1696	1.01191	0.505957	−	0.862559i	$-0.331139\pi$
$101$	10.1696	1.01191	0.505957	+	0.862559i	$0.331139\pi$
$102$	0	0
$103$	−2.00000	−0.197066	−0.0985329	−	0.995134i	$-0.531415\pi$
$103$	−2.00000	−0.197066	−0.0985329	+	0.995134i	$0.531415\pi$
$104$	0	0
$105$	4.03635	0.393907
$106$	0	0
$107$	−3.50751	−0.339084	−0.169542	−	0.985523i	$-0.554229\pi$
$107$	−3.50751	−0.339084	−0.169542	+	0.985523i	$0.554229\pi$
$108$	0	0
$109$	−14.4540	−1.38444	−0.692219	−	0.721687i	$-0.743367\pi$
$109$	−14.4540	−1.38444	−0.692219	+	0.721687i	$0.743367\pi$
$110$	0	0
$111$	−3.24418	−0.307924
$112$	0	0
$113$	−18.7153	−1.76059	−0.880296	−	0.474425i	$-0.842656\pi$
$113$	−18.7153	−1.76059	−0.880296	+	0.474425i	$0.842656\pi$
$114$	0	0
$115$	4.48836	0.418542
$116$	0	0
$117$	12.5355	1.15891
$118$	0	0
$119$	−7.04943	−0.646220
$120$	0	0
$121$	27.3239	2.48399
$122$	0	0
$123$	0.618435	0.0557624
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−9.80525	−0.870075	−0.435037	−	0.900412i	$-0.643265\pi$
$127$	−9.80525	−0.870075	−0.435037	+	0.900412i	$0.643265\pi$
$128$	0	0
$129$	14.7969	1.30280
$130$	0	0
$131$	−13.7899	−1.20483	−0.602415	−	0.798183i	$-0.705795\pi$
$131$	−13.7899	−1.20483	−0.602415	+	0.798183i	$0.705795\pi$
$132$	0	0
$133$	−6.15428	−0.533644
$134$	0	0
$135$	−14.6790	−1.26337
$136$	0	0
$137$	5.53779	0.473125	0.236563	−	0.971616i	$-0.423979\pi$
$137$	5.53779	0.473125	0.236563	+	0.971616i	$0.423979\pi$
$138$	0	0
$139$	−21.4307	−1.81773	−0.908863	−	0.417094i	$-0.863049\pi$
$139$	−21.4307	−1.81773	−0.908863	+	0.417094i	$0.863049\pi$
$140$	0	0
$141$	−19.7494	−1.66320
$142$	0	0
$143$	10.3131	0.862422
$144$	0	0
$145$	−4.48836	−0.372738
$146$	0	0
$147$	17.6873	1.45883
$148$	0	0
$149$	16.9787	1.39095	0.695474	−	0.718552i	$-0.255195\pi$
$149$	16.9787	1.39095	0.695474	+	0.718552i	$0.255195\pi$
$150$	0	0
$151$	6.56106	0.533932	0.266966	−	0.963706i	$-0.413979\pi$
$151$	6.56106	0.533932	0.266966	+	0.963706i	$0.413979\pi$
$152$	0	0
$153$	42.6344	3.44679
$154$	0	0
$155$	−0.385383	−0.0309547
$156$	0	0
$157$	8.19063	0.653683	0.326842	−	0.945079i	$-0.394016\pi$
$157$	8.19063	0.653683	0.326842	+	0.945079i	$0.394016\pi$
$158$	0	0
$159$	−5.52246	−0.437960
$160$	0	0
$161$	−5.58434	−0.440108
$162$	0	0
$163$	21.7131	1.70070	0.850350	−	0.526217i	$-0.176390\pi$
$163$	21.7131	1.70070	0.850350	+	0.526217i	$0.176390\pi$
$164$	0	0
$165$	−20.0835	−1.56350
$166$	0	0
$167$	17.3318	1.34118	0.670589	−	0.741829i	$-0.266042\pi$
$167$	17.3318	1.34118	0.670589	+	0.741829i	$0.266042\pi$
$168$	0	0
$169$	−10.2247	−0.786517
$170$	0	0
$171$	37.2206	2.84633
$172$	0	0
$173$	−21.8355	−1.66012	−0.830062	−	0.557671i	$-0.811695\pi$
$173$	−21.8355	−1.66012	−0.830062	+	0.557671i	$0.811695\pi$
$174$	0	0
$175$	−1.24418	−0.0940513
$176$	0	0
$177$	−19.7992	−1.48820
$178$	0	0
$179$	8.03028	0.600211	0.300106	−	0.953906i	$-0.402978\pi$
$179$	8.03028	0.600211	0.300106	+	0.953906i	$0.402978\pi$
$180$	0	0
$181$	−2.16961	−0.161266	−0.0806329	−	0.996744i	$-0.525694\pi$
$181$	−2.16961	−0.161266	−0.0806329	+	0.996744i	$0.525694\pi$
$182$	0	0
$183$	−6.48836	−0.479634
$184$	0	0
$185$	1.00000	0.0735215
$186$	0	0
$187$	35.0756	2.56498
$188$	0	0
$189$	18.2633	1.32846
$190$	0	0
$191$	26.4115	1.91107	0.955536	−	0.294875i	$-0.0952780\pi$
$191$	26.4115	1.91107	0.955536	+	0.294875i	$0.0952780\pi$
$192$	0	0
$193$	−1.40453	−0.101100	−0.0505502	−	0.998722i	$-0.516097\pi$
$193$	−1.40453	−0.101100	−0.0505502	+	0.998722i	$0.516097\pi$
$194$	0	0
$195$	−5.40453	−0.387027
$196$	0	0
$197$	20.4992	1.46051	0.730253	−	0.683177i	$-0.239402\pi$
$197$	20.4992	1.46051	0.730253	+	0.683177i	$0.239402\pi$
$198$	0	0
$199$	3.89702	0.276252	0.138126	−	0.990415i	$-0.455892\pi$
$199$	3.89702	0.276252	0.138126	+	0.990415i	$0.455892\pi$
$200$	0	0
$201$	43.5850	3.07425
$202$	0	0
$203$	5.58434	0.391944
$204$	0	0
$205$	−0.190629	−0.0133141
$206$	0	0
$207$	33.7736	2.34743
$208$	0	0
$209$	30.6216	2.11814
$210$	0	0
$211$	−6.05737	−0.417007	−0.208503	−	0.978022i	$-0.566859\pi$
$211$	−6.05737	−0.417007	−0.208503	+	0.978022i	$0.566859\pi$
$212$	0	0
$213$	−54.1097	−3.70753
$214$	0	0
$215$	−4.56106	−0.311062
$216$	0	0
$217$	0.479487	0.0325497
$218$	0	0
$219$	2.55024	0.172329
$220$	0	0
$221$	9.43894	0.634932
$222$	0	0
$223$	−8.29361	−0.555381	−0.277691	−	0.960671i	$-0.589569\pi$
$223$	−8.29361	−0.555381	−0.277691	+	0.960671i	$0.589569\pi$
$224$	0	0
$225$	7.52471	0.501648
$226$	0	0
$227$	18.3813	1.22001	0.610003	−	0.792399i	$-0.291168\pi$
$227$	18.3813	1.22001	0.610003	+	0.792399i	$0.291168\pi$
$228$	0	0
$229$	−13.1674	−0.870123	−0.435062	−	0.900401i	$-0.643273\pi$
$229$	−13.1674	−0.870123	−0.435062	+	0.900401i	$0.643273\pi$
$230$	0	0
$231$	24.9875	1.64406
$232$	0	0
$233$	−10.2480	−0.671369	−0.335684	−	0.941975i	$-0.608968\pi$
$233$	−10.2480	−0.671369	−0.335684	+	0.941975i	$0.608968\pi$
$234$	0	0
$235$	6.08765	0.397114
$236$	0	0
$237$	−35.5123	−2.30677
$238$	0	0
$239$	10.7666	0.696436	0.348218	−	0.937414i	$-0.386787\pi$
$239$	10.7666	0.696436	0.348218	+	0.937414i	$0.386787\pi$
$240$	0	0
$241$	14.8435	0.956152	0.478076	−	0.878319i	$-0.341334\pi$
$241$	14.8435	0.956152	0.478076	+	0.878319i	$0.341334\pi$
$242$	0	0
$243$	−37.2206	−2.38770
$244$	0	0
$245$	−5.45201	−0.348316
$246$	0	0
$247$	8.24036	0.524322
$248$	0	0
$249$	44.7867	2.83824
$250$	0	0
$251$	−24.4842	−1.54543	−0.772716	−	0.634752i	$-0.781102\pi$
$251$	−24.4842	−1.54543	−0.772716	+	0.634752i	$0.781102\pi$
$252$	0	0
$253$	27.7858	1.74688
$254$	0	0
$255$	−18.3813	−1.15108
$256$	0	0
$257$	9.07045	0.565799	0.282899	−	0.959150i	$-0.408704\pi$
$257$	9.07045	0.565799	0.282899	+	0.959150i	$0.408704\pi$
$258$	0	0
$259$	−1.24418	−0.0773097
$260$	0	0
$261$	−33.7736	−2.09054
$262$	0	0
$263$	25.3430	1.56272	0.781359	−	0.624081i	$-0.214527\pi$
$263$	25.3430	1.56272	0.781359	+	0.624081i	$0.214527\pi$
$264$	0	0
$265$	1.70227	0.104569
$266$	0	0
$267$	−17.2974	−1.05859
$268$	0	0
$269$	−13.1310	−0.800611	−0.400306	−	0.916382i	$-0.631096\pi$
$269$	−13.1310	−0.800611	−0.400306	+	0.916382i	$0.631096\pi$
$270$	0	0
$271$	−26.3966	−1.60348	−0.801739	−	0.597674i	$-0.796092\pi$
$271$	−26.3966	−1.60348	−0.801739	+	0.597674i	$0.796092\pi$
$272$	0	0
$273$	6.72422	0.406968
$274$	0	0
$275$	6.19063	0.373309
$276$	0	0
$277$	24.3602	1.46366	0.731832	−	0.681485i	$-0.238665\pi$
$277$	24.3602	1.46366	0.731832	+	0.681485i	$0.238665\pi$
$278$	0	0
$279$	−2.89990	−0.173612
$280$	0	0
$281$	−3.43894	−0.205150	−0.102575	−	0.994725i	$-0.532708\pi$
$281$	−3.43894	−0.205150	−0.102575	+	0.994725i	$0.532708\pi$
$282$	0	0
$283$	−26.3813	−1.56820	−0.784102	−	0.620633i	$-0.786876\pi$
$283$	−26.3813	−1.56820	−0.784102	+	0.620633i	$0.786876\pi$
$284$	0	0
$285$	−16.0472	−0.950553
$286$	0	0
$287$	0.237177	0.0140001
$288$	0	0
$289$	15.1026	0.888388
$290$	0	0
$291$	1.82019	0.106702
$292$	0	0
$293$	−25.6404	−1.49793	−0.748964	−	0.662611i	$-0.769448\pi$
$293$	−25.6404	−1.49793	−0.748964	+	0.662611i	$0.769448\pi$
$294$	0	0
$295$	6.10298	0.355329
$296$	0	0
$297$	−90.8722	−5.27294
$298$	0	0
$299$	7.47723	0.432420
$300$	0	0
$301$	5.67479	0.327090
$302$	0	0
$303$	−32.9921	−1.89534
$304$	0	0
$305$	2.00000	0.114520
$306$	0	0
$307$	3.20978	0.183192	0.0915958	−	0.995796i	$-0.470803\pi$
$307$	3.20978	0.183192	0.0915958	+	0.995796i	$0.470803\pi$
$308$	0	0
$309$	6.48836	0.369110
$310$	0	0
$311$	−31.5337	−1.78811	−0.894055	−	0.447957i	$-0.852152\pi$
$311$	−31.5337	−1.78811	−0.894055	+	0.447957i	$0.852152\pi$
$312$	0	0
$313$	1.12587	0.0636380	0.0318190	−	0.999494i	$-0.489870\pi$
$313$	1.12587	0.0636380	0.0318190	+	0.999494i	$0.489870\pi$
$314$	0	0
$315$	−9.36211	−0.527495
$316$	0	0
$317$	0.415662	0.0233459	0.0116729	−	0.999932i	$-0.496284\pi$
$317$	0.415662	0.0233459	0.0116729	+	0.999932i	$0.496284\pi$
$318$	0	0
$319$	−27.7858	−1.55571
$320$	0	0
$321$	11.3790	0.635114
$322$	0	0
$323$	28.0262	1.55942
$324$	0	0
$325$	1.66592	0.0924084
$326$	0	0
$327$	46.8913	2.59309
$328$	0	0
$329$	−7.57414	−0.417576
$330$	0	0
$331$	−30.5913	−1.68145	−0.840726	−	0.541461i	$-0.817872\pi$
$331$	−30.5913	−1.68145	−0.840726	+	0.541461i	$0.817872\pi$
$332$	0	0
$333$	7.52471	0.412352
$334$	0	0
$335$	−13.4348	−0.734022
$336$	0	0
$337$	9.52697	0.518967	0.259483	−	0.965748i	$-0.416448\pi$
$337$	9.52697	0.518967	0.259483	+	0.965748i	$0.416448\pi$
$338$	0	0
$339$	60.7160	3.29764
$340$	0	0
$341$	−2.38577	−0.129196
$342$	0	0
$343$	15.4926	0.836520
$344$	0	0
$345$	−14.5611	−0.783941
$346$	0	0
$347$	1.46509	0.0786501	0.0393250	−	0.999226i	$-0.487479\pi$
$347$	1.46509	0.0786501	0.0393250	+	0.999226i	$0.487479\pi$
$348$	0	0
$349$	24.8224	1.32872	0.664358	−	0.747415i	$-0.268705\pi$
$349$	24.8224	1.32872	0.664358	+	0.747415i	$0.268705\pi$
$350$	0	0
$351$	−24.4540	−1.30526
$352$	0	0
$353$	11.1221	0.591971	0.295986	−	0.955192i	$-0.404352\pi$
$353$	11.1221	0.591971	0.295986	+	0.955192i	$0.404352\pi$
$354$	0	0
$355$	16.6790	0.885229
$356$	0	0
$357$	22.8696	1.21039
$358$	0	0
$359$	−4.50369	−0.237696	−0.118848	−	0.992912i	$-0.537920\pi$
$359$	−4.50369	−0.237696	−0.118848	+	0.992912i	$0.537920\pi$
$360$	0	0
$361$	5.46734	0.287755
$362$	0	0
$363$	−88.6436	−4.65258
$364$	0	0
$365$	−0.786097	−0.0411462
$366$	0	0
$367$	−17.8888	−0.933786	−0.466893	−	0.884314i	$-0.654627\pi$
$367$	−17.8888	−0.933786	−0.466893	+	0.884314i	$0.654627\pi$
$368$	0	0
$369$	−1.43443	−0.0746734
$370$	0	0
$371$	−2.11793	−0.109957
$372$	0	0
$373$	30.8850	1.59916	0.799581	−	0.600558i	$-0.205055\pi$
$373$	30.8850	1.59916	0.799581	+	0.600558i	$0.205055\pi$
$374$	0	0
$375$	−3.24418	−0.167529
$376$	0	0
$377$	−7.47723	−0.385097
$378$	0	0
$379$	−20.7816	−1.06748	−0.533739	−	0.845649i	$-0.679214\pi$
$379$	−20.7816	−1.06748	−0.533739	+	0.845649i	$0.679214\pi$
$380$	0	0
$381$	31.8100	1.62968
$382$	0	0
$383$	−34.6599	−1.77104	−0.885520	−	0.464602i	$-0.846197\pi$
$383$	−34.6599	−1.77104	−0.885520	+	0.464602i	$0.846197\pi$
$384$	0	0
$385$	−7.70227	−0.392544
$386$	0	0
$387$	−34.3207	−1.74462
$388$	0	0
$389$	−8.73636	−0.442951	−0.221476	−	0.975166i	$-0.571087\pi$
$389$	−8.73636	−0.442951	−0.221476	+	0.975166i	$0.571087\pi$
$390$	0	0
$391$	25.4307	1.28609
$392$	0	0
$393$	44.7370	2.25668
$394$	0	0
$395$	10.9464	0.550776
$396$	0	0
$397$	6.29773	0.316074	0.158037	−	0.987433i	$-0.449483\pi$
$397$	6.29773	0.316074	0.158037	+	0.987433i	$0.449483\pi$
$398$	0	0
$399$	19.9656	0.999530
$400$	0	0
$401$	−30.7625	−1.53621	−0.768103	−	0.640326i	$-0.778799\pi$
$401$	−30.7625	−1.53621	−0.768103	+	0.640326i	$0.778799\pi$
$402$	0	0
$403$	−0.642016	−0.0319811
$404$	0	0
$405$	25.0472	1.24460
$406$	0	0
$407$	6.19063	0.306858
$408$	0	0
$409$	12.4884	0.617510	0.308755	−	0.951142i	$-0.400088\pi$
$409$	12.4884	0.617510	0.308755	+	0.951142i	$0.400088\pi$
$410$	0	0
$411$	−17.9656	−0.886178
$412$	0	0
$413$	−7.59321	−0.373638
$414$	0	0
$415$	−13.8052	−0.677673
$416$	0	0
$417$	69.5250	3.40466
$418$	0	0
$419$	−7.49631	−0.366219	−0.183109	−	0.983093i	$-0.558616\pi$
$419$	−7.49631	−0.366219	−0.183109	+	0.983093i	$0.558616\pi$
$420$	0	0
$421$	1.36249	0.0664038	0.0332019	−	0.999449i	$-0.489430\pi$
$421$	1.36249	0.0664038	0.0332019	+	0.999449i	$0.489430\pi$
$422$	0	0
$423$	45.8078	2.22725
$424$	0	0
$425$	5.66592	0.274837
$426$	0	0
$427$	−2.48836	−0.120420
$428$	0	0
$429$	−33.4575	−1.61534
$430$	0	0
$431$	25.1180	1.20989	0.604946	−	0.796267i	$-0.293195\pi$
$431$	25.1180	1.20989	0.604946	+	0.796267i	$0.293195\pi$
$432$	0	0
$433$	18.2168	0.875443	0.437721	−	0.899111i	$-0.355786\pi$
$433$	18.2168	0.875443	0.437721	+	0.899111i	$0.355786\pi$
$434$	0	0
$435$	14.5611	0.698150
$436$	0	0
$437$	22.2015	1.06204
$438$	0	0
$439$	13.9232	0.664517	0.332258	−	0.943188i	$-0.392189\pi$
$439$	13.9232	0.664517	0.332258	+	0.943188i	$0.392189\pi$
$440$	0	0
$441$	−41.0248	−1.95356
$442$	0	0
$443$	0.721415	0.0342754	0.0171377	−	0.999853i	$-0.494545\pi$
$443$	0.721415	0.0342754	0.0171377	+	0.999853i	$0.494545\pi$
$444$	0	0
$445$	5.33183	0.252753
$446$	0	0
$447$	−55.0819	−2.60528
$448$	0	0
$449$	−16.8696	−0.796127	−0.398063	−	0.917358i	$-0.630318\pi$
$449$	−16.8696	−0.796127	−0.398063	+	0.917358i	$0.630318\pi$
$450$	0	0
$451$	−1.18011	−0.0555694
$452$	0	0
$453$	−21.2853	−1.00007
$454$	0	0
$455$	−2.07270	−0.0971697
$456$	0	0
$457$	2.46221	0.115177	0.0575887	−	0.998340i	$-0.481659\pi$
$457$	2.46221	0.115177	0.0575887	+	0.998340i	$0.481659\pi$
$458$	0	0
$459$	−83.1699	−3.88204
$460$	0	0
$461$	14.5145	0.676008	0.338004	−	0.941145i	$-0.390248\pi$
$461$	14.5145	0.676008	0.338004	+	0.941145i	$0.390248\pi$
$462$	0	0
$463$	33.5378	1.55863	0.779317	−	0.626630i	$-0.215566\pi$
$463$	33.5378	1.55863	0.779317	+	0.626630i	$0.215566\pi$
$464$	0	0
$465$	1.25025	0.0579791
$466$	0	0
$467$	−6.00000	−0.277647	−0.138823	−	0.990317i	$-0.544332\pi$
$467$	−6.00000	−0.277647	−0.138823	+	0.990317i	$0.544332\pi$
$468$	0	0
$469$	16.7153	0.771843
$470$	0	0
$471$	−26.5719	−1.22437
$472$	0	0
$473$	−28.2359	−1.29829
$474$	0	0
$475$	4.94645	0.226959
$476$	0	0
$477$	12.8091	0.586487
$478$	0	0
$479$	−21.2550	−0.971166	−0.485583	−	0.874191i	$-0.661393\pi$
$479$	−21.2550	−0.971166	−0.485583	+	0.874191i	$0.661393\pi$
$480$	0	0
$481$	1.66592	0.0759592
$482$	0	0
$483$	18.1166	0.824334
$484$	0	0
$485$	−0.561064	−0.0254766
$486$	0	0
$487$	23.7176	1.07475	0.537373	−	0.843344i	$-0.319417\pi$
$487$	23.7176	1.07475	0.537373	+	0.843344i	$0.319417\pi$
$488$	0	0
$489$	−70.4412	−3.18546
$490$	0	0
$491$	−11.3318	−0.511398	−0.255699	−	0.966756i	$-0.582306\pi$
$491$	−11.3318	−0.511398	−0.255699	+	0.966756i	$0.582306\pi$
$492$	0	0
$493$	−25.4307	−1.14534
$494$	0	0
$495$	46.5827	2.09374
$496$	0	0
$497$	−20.7517	−0.930841
$498$	0	0
$499$	3.65666	0.163694	0.0818472	−	0.996645i	$-0.473918\pi$
$499$	3.65666	0.163694	0.0818472	+	0.996645i	$0.473918\pi$
$500$	0	0
$501$	−56.2276	−2.51206
$502$	0	0
$503$	−11.1100	−0.495370	−0.247685	−	0.968841i	$-0.579670\pi$
$503$	−11.1100	−0.495370	−0.247685	+	0.968841i	$0.579670\pi$
$504$	0	0
$505$	10.1696	0.452542
$506$	0	0
$507$	33.1709	1.47317
$508$	0	0
$509$	9.62956	0.426823	0.213411	−	0.976962i	$-0.431543\pi$
$509$	9.62956	0.426823	0.213411	+	0.976962i	$0.431543\pi$
$510$	0	0
$511$	0.978047	0.0432663
$512$	0	0
$513$	−72.6089	−3.20576
$514$	0	0
$515$	−2.00000	−0.0881305
$516$	0	0
$517$	37.6864	1.65745
$518$	0	0
$519$	70.8384	3.10946
$520$	0	0
$521$	14.6236	0.640670	0.320335	−	0.947304i	$-0.396205\pi$
$521$	14.6236	0.640670	0.320335	+	0.947304i	$0.396205\pi$
$522$	0	0
$523$	−23.4389	−1.02491	−0.512457	−	0.858713i	$-0.671264\pi$
$523$	−23.4389	−1.02491	−0.512457	+	0.858713i	$0.671264\pi$
$524$	0	0
$525$	4.03635	0.176161
$526$	0	0
$527$	−2.18355	−0.0951169
$528$	0	0
$529$	−2.85460	−0.124113
$530$	0	0
$531$	45.9232	1.99290
$532$	0	0
$533$	−0.317572	−0.0137556
$534$	0	0
$535$	−3.50751	−0.151643
$536$	0	0
$537$	−26.0517	−1.12421
$538$	0	0
$539$	−33.7514	−1.45378
$540$	0	0
$541$	21.0150	0.903506	0.451753	−	0.892143i	$-0.350799\pi$
$541$	21.0150	0.903506	0.451753	+	0.892143i	$0.350799\pi$
$542$	0	0
$543$	7.03860	0.302055
$544$	0	0
$545$	−14.4540	−0.619140
$546$	0	0
$547$	10.1753	0.435064	0.217532	−	0.976053i	$-0.430199\pi$
$547$	10.1753	0.435064	0.217532	+	0.976053i	$0.430199\pi$
$548$	0	0
$549$	15.0494	0.642294
$550$	0	0
$551$	−22.2015	−0.945814
$552$	0	0
$553$	−13.6194	−0.579154
$554$	0	0
$555$	−3.24418	−0.137708
$556$	0	0
$557$	28.7415	1.21782	0.608908	−	0.793241i	$-0.291608\pi$
$557$	28.7415	1.21782	0.608908	+	0.793241i	$0.291608\pi$
$558$	0	0
$559$	−7.59835	−0.321376
$560$	0	0
$561$	−113.792	−4.80428
$562$	0	0
$563$	−36.7281	−1.54791	−0.773953	−	0.633243i	$-0.781723\pi$
$563$	−36.7281	−1.54791	−0.773953	+	0.633243i	$0.781723\pi$
$564$	0	0
$565$	−18.7153	−0.787360
$566$	0	0
$567$	−31.1632	−1.30873
$568$	0	0
$569$	10.2480	0.429619	0.214809	−	0.976656i	$-0.431087\pi$
$569$	10.2480	0.429619	0.214809	+	0.976656i	$0.431087\pi$
$570$	0	0
$571$	21.0602	0.881344	0.440672	−	0.897668i	$-0.354740\pi$
$571$	21.0602	0.881344	0.440672	+	0.897668i	$0.354740\pi$
$572$	0	0
$573$	−85.6838	−3.57949
$574$	0	0
$575$	4.48836	0.187178
$576$	0	0
$577$	−5.14315	−0.214112	−0.107056	−	0.994253i	$-0.534142\pi$
$577$	−5.14315	−0.214112	−0.107056	+	0.994253i	$0.534142\pi$
$578$	0	0
$579$	4.55656	0.189364
$580$	0	0
$581$	17.1762	0.712590
$582$	0	0
$583$	10.5381	0.436443
$584$	0	0
$585$	12.5355	0.518281
$586$	0	0
$587$	4.86962	0.200991	0.100495	−	0.994938i	$-0.467957\pi$
$587$	4.86962	0.200991	0.100495	+	0.994938i	$0.467957\pi$
$588$	0	0
$589$	−1.90628	−0.0785468
$590$	0	0
$591$	−66.5031	−2.73557
$592$	0	0
$593$	−17.5569	−0.720974	−0.360487	−	0.932764i	$-0.617389\pi$
$593$	−17.5569	−0.720974	−0.360487	+	0.932764i	$0.617389\pi$
$594$	0	0
$595$	−7.04943	−0.288998
$596$	0	0
$597$	−12.6426	−0.517429
$598$	0	0
$599$	−45.9764	−1.87855	−0.939273	−	0.343171i	$-0.888499\pi$
$599$	−45.9764	−1.87855	−0.939273	+	0.343171i	$0.888499\pi$
$600$	0	0
$601$	−37.2764	−1.52054	−0.760268	−	0.649609i	$-0.774932\pi$
$601$	−37.2764	−1.52054	−0.760268	+	0.649609i	$0.774932\pi$
$602$	0	0
$603$	−101.093	−4.11683
$604$	0	0
$605$	27.3239	1.11087
$606$	0	0
$607$	3.96560	0.160959	0.0804793	−	0.996756i	$-0.474355\pi$
$607$	3.96560	0.160959	0.0804793	+	0.996756i	$0.474355\pi$
$608$	0	0
$609$	−18.1166	−0.734122
$610$	0	0
$611$	10.1415	0.410282
$612$	0	0
$613$	−12.3239	−0.497757	−0.248879	−	0.968535i	$-0.580062\pi$
$613$	−12.3239	−0.497757	−0.248879	+	0.968535i	$0.580062\pi$
$614$	0	0
$615$	0.618435	0.0249377
$616$	0	0
$617$	32.2551	1.29854	0.649270	−	0.760558i	$-0.275074\pi$
$617$	32.2551	1.29854	0.649270	+	0.760558i	$0.275074\pi$
$618$	0	0
$619$	−8.98755	−0.361240	−0.180620	−	0.983553i	$-0.557810\pi$
$619$	−8.98755	−0.361240	−0.180620	+	0.983553i	$0.557810\pi$
$620$	0	0
$621$	−65.8846	−2.64386
$622$	0	0
$623$	−6.63377	−0.265776
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−99.3421	−3.96734
$628$	0	0
$629$	5.66592	0.225915
$630$	0	0
$631$	10.7246	0.426940	0.213470	−	0.976950i	$-0.431524\pi$
$631$	10.7246	0.426940	0.213470	+	0.976950i	$0.431524\pi$
$632$	0	0
$633$	19.6512	0.781065
$634$	0	0
$635$	−9.80525	−0.389109
$636$	0	0
$637$	−9.08259	−0.359865
$638$	0	0
$639$	125.505	4.96489
$640$	0	0
$641$	34.9321	1.37974	0.689868	−	0.723935i	$-0.257669\pi$
$641$	34.9321	1.37974	0.689868	+	0.723935i	$0.257669\pi$
$642$	0	0
$643$	25.1483	0.991751	0.495876	−	0.868394i	$-0.334847\pi$
$643$	25.1483	0.991751	0.495876	+	0.868394i	$0.334847\pi$
$644$	0	0
$645$	14.7969	0.582628
$646$	0	0
$647$	−39.9452	−1.57041	−0.785204	−	0.619237i	$-0.787442\pi$
$647$	−39.9452	−1.57041	−0.785204	+	0.619237i	$0.787442\pi$
$648$	0	0
$649$	37.7813	1.48305
$650$	0	0
$651$	−1.55554	−0.0609665
$652$	0	0
$653$	−39.7392	−1.55512	−0.777558	−	0.628811i	$-0.783542\pi$
$653$	−39.7392	−1.55512	−0.777558	+	0.628811i	$0.783542\pi$
$654$	0	0
$655$	−13.7899	−0.538817
$656$	0	0
$657$	−5.91515	−0.230772
$658$	0	0
$659$	41.4415	1.61433	0.807166	−	0.590325i	$-0.201000\pi$
$659$	41.4415	1.61433	0.807166	+	0.590325i	$0.201000\pi$
$660$	0	0
$661$	21.2210	0.825401	0.412700	−	0.910867i	$-0.364586\pi$
$661$	21.2210	0.825401	0.412700	+	0.910867i	$0.364586\pi$
$662$	0	0
$663$	−30.6216	−1.18925
$664$	0	0
$665$	−6.15428	−0.238653
$666$	0	0
$667$	−20.1454	−0.780033
$668$	0	0
$669$	26.9060	1.04024
$670$	0	0
$671$	12.3813	0.477973
$672$	0	0
$673$	6.96590	0.268516	0.134258	−	0.990946i	$-0.457135\pi$
$673$	6.96590	0.268516	0.134258	+	0.990946i	$0.457135\pi$
$674$	0	0
$675$	−14.6790	−0.564995
$676$	0	0
$677$	16.9315	0.650730	0.325365	−	0.945588i	$-0.394513\pi$
$677$	16.9315	0.650730	0.325365	+	0.945588i	$0.394513\pi$
$678$	0	0
$679$	0.698066	0.0267893
$680$	0	0
$681$	−59.6321	−2.28511
$682$	0	0
$683$	49.3963	1.89010	0.945048	−	0.326931i	$-0.106015\pi$
$683$	49.3963	1.89010	0.945048	+	0.326931i	$0.106015\pi$
$684$	0	0
$685$	5.53779	0.211588
$686$	0	0
$687$	42.7173	1.62977
$688$	0	0
$689$	2.83583	0.108037
$690$	0	0
$691$	24.1026	0.916906	0.458453	−	0.888719i	$-0.348404\pi$
$691$	24.1026	0.916906	0.458453	+	0.888719i	$0.348404\pi$
$692$	0	0
$693$	−57.9573	−2.20162
$694$	0	0
$695$	−21.4307	−0.812912
$696$	0	0
$697$	−1.08009	−0.0409112
$698$	0	0
$699$	33.2464	1.25749
$700$	0	0
$701$	−40.3768	−1.52501	−0.762504	−	0.646983i	$-0.776030\pi$
$701$	−40.3768	−1.52501	−0.762504	+	0.646983i	$0.776030\pi$
$702$	0	0
$703$	4.94645	0.186559
$704$	0	0
$705$	−19.7494	−0.743807
$706$	0	0
$707$	−12.6528	−0.475859
$708$	0	0
$709$	−34.8435	−1.30857	−0.654287	−	0.756246i	$-0.727031\pi$
$709$	−34.8435	−1.30857	−0.654287	+	0.756246i	$0.727031\pi$
$710$	0	0
$711$	82.3689	3.08907
$712$	0	0
$713$	−1.72974	−0.0647793
$714$	0	0
$715$	10.3131	0.385687
$716$	0	0
$717$	−34.9289	−1.30445
$718$	0	0
$719$	−1.35541	−0.0505483	−0.0252742	−	0.999681i	$-0.508046\pi$
$719$	−1.35541	−0.0505483	−0.0252742	+	0.999681i	$0.508046\pi$
$720$	0	0
$721$	2.48836	0.0926715
$722$	0	0
$723$	−48.1549	−1.79090
$724$	0	0
$725$	−4.48836	−0.166694
$726$	0	0
$727$	−14.5566	−0.539873	−0.269936	−	0.962878i	$-0.587003\pi$
$727$	−14.5566	−0.539873	−0.269936	+	0.962878i	$0.587003\pi$
$728$	0	0
$729$	45.6089	1.68922
$730$	0	0
$731$	−25.8426	−0.955823
$732$	0	0
$733$	18.0415	0.666377	0.333189	−	0.942860i	$-0.391875\pi$
$733$	18.0415	0.666377	0.333189	+	0.942860i	$0.391875\pi$
$734$	0	0
$735$	17.6873	0.652407
$736$	0	0
$737$	−83.1699	−3.06360
$738$	0	0
$739$	−7.74431	−0.284879	−0.142439	−	0.989804i	$-0.545495\pi$
$739$	−7.74431	−0.284879	−0.142439	+	0.989804i	$0.545495\pi$
$740$	0	0
$741$	−26.7332	−0.982070
$742$	0	0
$743$	30.4689	1.11780	0.558898	−	0.829236i	$-0.311224\pi$
$743$	30.4689	1.11780	0.558898	+	0.829236i	$0.311224\pi$
$744$	0	0
$745$	16.9787	0.622050
$746$	0	0
$747$	−103.881	−3.80079
$748$	0	0
$749$	4.36398	0.159456
$750$	0	0
$751$	12.9270	0.471713	0.235856	−	0.971788i	$-0.424211\pi$
$751$	12.9270	0.471713	0.235856	+	0.971788i	$0.424211\pi$
$752$	0	0
$753$	79.4313	2.89464
$754$	0	0
$755$	6.56106	0.238782
$756$	0	0
$757$	14.4501	0.525197	0.262598	−	0.964905i	$-0.415421\pi$
$757$	14.4501	0.525197	0.262598	+	0.964905i	$0.415421\pi$
$758$	0	0
$759$	−90.1421	−3.27195
$760$	0	0
$761$	2.33977	0.0848168	0.0424084	−	0.999100i	$-0.486497\pi$
$761$	2.33977	0.0848168	0.0424084	+	0.999100i	$0.486497\pi$
$762$	0	0
$763$	17.9833	0.651041
$764$	0	0
$765$	42.6344	1.54145
$766$	0	0
$767$	10.1670	0.367111
$768$	0	0
$769$	25.0449	0.903143	0.451571	−	0.892235i	$-0.350864\pi$
$769$	25.0449	0.903143	0.451571	+	0.892235i	$0.350864\pi$
$770$	0	0
$771$	−29.4262	−1.05976
$772$	0	0
$773$	12.7900	0.460024	0.230012	−	0.973188i	$-0.426123\pi$
$773$	12.7900	0.460024	0.230012	+	0.973188i	$0.426123\pi$
$774$	0	0
$775$	−0.385383	−0.0138434
$776$	0	0
$777$	4.03635	0.144803
$778$	0	0
$779$	−0.942936	−0.0337842
$780$	0	0
$781$	103.253	3.69470
$782$	0	0
$783$	65.8846	2.35453
$784$	0	0
$785$	8.19063	0.292336
$786$	0	0
$787$	−32.4224	−1.15573	−0.577866	−	0.816132i	$-0.696114\pi$
$787$	−32.4224	−1.15573	−0.577866	+	0.816132i	$0.696114\pi$
$788$	0	0
$789$	−82.2174	−2.92702
$790$	0	0
$791$	23.2853	0.827929
$792$	0	0
$793$	3.33183	0.118317
$794$	0	0
$795$	−5.52246	−0.195861
$796$	0	0
$797$	18.4972	0.655206	0.327603	−	0.944816i	$-0.393759\pi$
$797$	18.4972	0.655206	0.327603	+	0.944816i	$0.393759\pi$
$798$	0	0
$799$	34.4921	1.22024
$800$	0	0
$801$	40.1205	1.41759
$802$	0	0
$803$	−4.86643	−0.171733
$804$	0	0
$805$	−5.58434	−0.196822
$806$	0	0
$807$	42.5994	1.49957
$808$	0	0
$809$	−6.38126	−0.224353	−0.112177	−	0.993688i	$-0.535782\pi$
$809$	−6.38126	−0.224353	−0.112177	+	0.993688i	$0.535782\pi$
$810$	0	0
$811$	−6.88869	−0.241895	−0.120947	−	0.992659i	$-0.538593\pi$
$811$	−6.88869	−0.241895	−0.120947	+	0.992659i	$0.538593\pi$
$812$	0	0
$813$	85.6353	3.00336
$814$	0	0
$815$	21.7131	0.760576
$816$	0	0
$817$	−22.5611	−0.789312
$818$	0	0
$819$	−15.5965	−0.544985
$820$	0	0
$821$	22.3921	0.781489	0.390745	−	0.920499i	$-0.372218\pi$
$821$	22.3921	0.781489	0.390745	+	0.920499i	$0.372218\pi$
$822$	0	0
$823$	−1.51202	−0.0527057	−0.0263528	−	0.999653i	$-0.508389\pi$
$823$	−1.51202	−0.0527057	−0.0263528	+	0.999653i	$0.508389\pi$
$824$	0	0
$825$	−20.0835	−0.699219
$826$	0	0
$827$	−7.30568	−0.254043	−0.127022	−	0.991900i	$-0.540542\pi$
$827$	−7.30568	−0.254043	−0.127022	+	0.991900i	$0.540542\pi$
$828$	0	0
$829$	−47.1438	−1.63737	−0.818685	−	0.574242i	$-0.805297\pi$
$829$	−47.1438	−1.63737	−0.818685	+	0.574242i	$0.805297\pi$
$830$	0	0
$831$	−79.0290	−2.74149
$832$	0	0
$833$	−30.8906	−1.07030
$834$	0	0
$835$	17.3318	0.599793
$836$	0	0
$837$	5.65704	0.195536
$838$	0	0
$839$	12.8313	0.442986	0.221493	−	0.975162i	$-0.428907\pi$
$839$	12.8313	0.442986	0.221493	+	0.975162i	$0.428907\pi$
$840$	0	0
$841$	−8.85460	−0.305331
$842$	0	0
$843$	11.1565	0.384251
$844$	0	0
$845$	−10.2247	−0.351741
$846$	0	0
$847$	−33.9959	−1.16811
$848$	0	0
$849$	85.5856	2.93729
$850$	0	0
$851$	4.48836	0.153859
$852$	0	0
$853$	15.6060	0.534339	0.267169	−	0.963650i	$-0.413912\pi$
$853$	15.6060	0.534339	0.267169	+	0.963650i	$0.413912\pi$
$854$	0	0
$855$	37.2206	1.27292
$856$	0	0
$857$	−24.4157	−0.834023	−0.417012	−	0.908901i	$-0.636923\pi$
$857$	−24.4157	−0.834023	−0.417012	+	0.908901i	$0.636923\pi$
$858$	0	0
$859$	46.1973	1.57623	0.788116	−	0.615526i	$-0.211057\pi$
$859$	46.1973	1.57623	0.788116	+	0.615526i	$0.211057\pi$
$860$	0	0
$861$	−0.769445	−0.0262226
$862$	0	0
$863$	−18.3089	−0.623244	−0.311622	−	0.950206i	$-0.600872\pi$
$863$	−18.3089	−0.623244	−0.311622	+	0.950206i	$0.600872\pi$
$864$	0	0
$865$	−21.8355	−0.742430
$866$	0	0
$867$	−48.9956	−1.66398
$868$	0	0
$869$	67.7654	2.29878
$870$	0	0
$871$	−22.3813	−0.758360
$872$	0	0
$873$	−4.22185	−0.142888
$874$	0	0
$875$	−1.24418	−0.0420610
$876$	0	0
$877$	−15.9963	−0.540155	−0.270078	−	0.962839i	$-0.587049\pi$
$877$	−15.9963	−0.540155	−0.270078	+	0.962839i	$0.587049\pi$
$878$	0	0
$879$	83.1821	2.80566
$880$	0	0
$881$	−43.2719	−1.45787	−0.728934	−	0.684584i	$-0.759984\pi$
$881$	−43.2719	−1.45787	−0.728934	+	0.684584i	$0.759984\pi$
$882$	0	0
$883$	47.1438	1.58651	0.793257	−	0.608887i	$-0.208384\pi$
$883$	47.1438	1.58651	0.793257	+	0.608887i	$0.208384\pi$
$884$	0	0
$885$	−19.7992	−0.665542
$886$	0	0
$887$	21.0988	0.708428	0.354214	−	0.935164i	$-0.384749\pi$
$887$	21.0988	0.708428	0.354214	+	0.935164i	$0.384749\pi$
$888$	0	0
$889$	12.1995	0.409158
$890$	0	0
$891$	155.058	5.19463
$892$	0	0
$893$	30.1122	1.00767
$894$	0	0
$895$	8.03028	0.268423
$896$	0	0
$897$	−24.2575	−0.809934
$898$	0	0
$899$	1.72974	0.0576901
$900$	0	0
$901$	9.64490	0.321318
$902$	0	0
$903$	−18.4101	−0.612648
$904$	0	0
$905$	−2.16961	−0.0721202
$906$	0	0
$907$	−37.0756	−1.23107	−0.615537	−	0.788108i	$-0.711061\pi$
$907$	−37.0756	−1.23107	−0.615537	+	0.788108i	$0.711061\pi$
$908$	0	0
$909$	76.5234	2.53812
$910$	0	0
$911$	−25.9187	−0.858724	−0.429362	−	0.903133i	$-0.641261\pi$
$911$	−25.9187	−0.858724	−0.429362	+	0.903133i	$0.641261\pi$
$912$	0	0
$913$	−85.4632	−2.82842
$914$	0	0
$915$	−6.48836	−0.214499
$916$	0	0
$917$	17.1572	0.566579
$918$	0	0
$919$	49.0332	1.61745	0.808727	−	0.588184i	$-0.200157\pi$
$919$	49.0332	1.61745	0.808727	+	0.588184i	$0.200157\pi$
$920$	0	0
$921$	−10.4131	−0.343123
$922$	0	0
$923$	27.7858	0.914580
$924$	0	0
$925$	1.00000	0.0328798
$926$	0	0
$927$	−15.0494	−0.494288
$928$	0	0
$929$	−39.2770	−1.28864	−0.644319	−	0.764757i	$-0.722859\pi$
$929$	−39.2770	−1.28864	−0.644319	+	0.764757i	$0.722859\pi$
$930$	0	0
$931$	−26.9681	−0.883844
$932$	0	0
$933$	102.301	3.34918
$934$	0	0
$935$	35.0756	1.14709
$936$	0	0
$937$	17.3855	0.567958	0.283979	−	0.958830i	$-0.408345\pi$
$937$	17.3855	0.567958	0.283979	+	0.958830i	$0.408345\pi$
$938$	0	0
$939$	−3.65253	−0.119196
$940$	0	0
$941$	−12.4628	−0.406277	−0.203138	−	0.979150i	$-0.565114\pi$
$941$	−12.4628	−0.406277	−0.203138	+	0.979150i	$0.565114\pi$
$942$	0	0
$943$	−0.855612	−0.0278626
$944$	0	0
$945$	18.2633	0.594106
$946$	0	0
$947$	34.6478	1.12590	0.562951	−	0.826490i	$-0.309666\pi$
$947$	34.6478	1.12590	0.562951	+	0.826490i	$0.309666\pi$
$948$	0	0
$949$	−1.30957	−0.0425105
$950$	0	0
$951$	−1.34848	−0.0437275
$952$	0	0
$953$	−28.0880	−0.909861	−0.454930	−	0.890527i	$-0.650336\pi$
$953$	−28.0880	−0.909861	−0.454930	+	0.890527i	$0.650336\pi$
$954$	0	0
$955$	26.4115	0.854657
$956$	0	0
$957$	90.1421	2.91388
$958$	0	0
$959$	−6.89002	−0.222490
$960$	0	0
$961$	−30.8515	−0.995209
$962$	0	0
$963$	−26.3930	−0.850503
$964$	0	0
$965$	−1.40453	−0.0452135
$966$	0	0
$967$	24.9302	0.801700	0.400850	−	0.916144i	$-0.368715\pi$
$967$	24.9302	0.801700	0.400850	+	0.916144i	$0.368715\pi$
$968$	0	0
$969$	−90.9219	−2.92083
$970$	0	0
$971$	15.7858	0.506590	0.253295	−	0.967389i	$-0.418486\pi$
$971$	15.7858	0.506590	0.253295	+	0.967389i	$0.418486\pi$
$972$	0	0
$973$	26.6637	0.854798
$974$	0	0
$975$	−5.40453	−0.173084
$976$	0	0
$977$	37.6525	1.20461	0.602306	−	0.798266i	$-0.294249\pi$
$977$	37.6525	1.20461	0.602306	+	0.798266i	$0.294249\pi$
$978$	0	0
$979$	33.0074	1.05492
$980$	0	0
$981$	−108.762	−3.47250
$982$	0	0
$983$	12.9618	0.413417	0.206708	−	0.978403i	$-0.433725\pi$
$983$	12.9618	0.413417	0.206708	+	0.978403i	$0.433725\pi$
$984$	0	0
$985$	20.4992	0.653158
$986$	0	0
$987$	24.5719	0.782132
$988$	0	0
$989$	−20.4717	−0.650963
$990$	0	0
$991$	38.7666	1.23146	0.615731	−	0.787956i	$-0.288861\pi$
$991$	38.7666	1.23146	0.615731	+	0.787956i	$0.288861\pi$
$992$	0	0
$993$	99.2439	3.14941
$994$	0	0
$995$	3.89702	0.123544
$996$	0	0
$997$	−14.6478	−0.463900	−0.231950	−	0.972728i	$-0.574511\pi$
$997$	−14.6478	−0.463900	−0.231950	+	0.972728i	$0.574511\pi$
$998$	0	0
$999$	−14.6790	−0.464423

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2960.2.a.v.1.1		4
4.3	odd	2		740.2.a.f.1.4	✓	4
12.11	even	2		6660.2.a.r.1.4		4
20.3	even	4		3700.2.d.i.149.8		8
20.7	even	4		3700.2.d.i.149.1		8
20.19	odd	2		3700.2.a.j.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
740.2.a.f.1.4	✓	4	4.3	odd	2
2960.2.a.v.1.1		4	1.1	even	1	trivial
3700.2.a.j.1.1		4	20.19	odd	2
3700.2.d.i.149.1		8	20.7	even	4
3700.2.d.i.149.8		8	20.3	even	4
6660.2.a.r.1.4		4	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2960 = 2^{4} \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2960.a (trivial)

\(f(q)\)	\(=\)	\(q-3.24418 q^{3} +1.00000 q^{5} -1.24418 q^{7} +7.52471 q^{9} +O(q^{10})\)	\(q-3.24418 q^{3} +1.00000 q^{5} -1.24418 q^{7} +7.52471 q^{9} +6.19063 q^{11} +1.66592 q^{13} -3.24418 q^{15} +5.66592 q^{17} +4.94645 q^{19} +4.03635 q^{21} +4.48836 q^{23} +1.00000 q^{25} -14.6790 q^{27} -4.48836 q^{29} -0.385383 q^{31} -20.0835 q^{33} -1.24418 q^{35} +1.00000 q^{37} -5.40453 q^{39} -0.190629 q^{41} -4.56106 q^{43} +7.52471 q^{45} +6.08765 q^{47} -5.45201 q^{49} -18.3813 q^{51} +1.70227 q^{53} +6.19063 q^{55} -16.0472 q^{57} +6.10298 q^{59} +2.00000 q^{61} -9.36211 q^{63} +1.66592 q^{65} -13.4348 q^{67} -14.5611 q^{69} +16.6790 q^{71} -0.786097 q^{73} -3.24418 q^{75} -7.70227 q^{77} +10.9464 q^{79} +25.0472 q^{81} -13.8052 q^{83} +5.66592 q^{85} +14.5611 q^{87} +5.33183 q^{89} -2.07270 q^{91} +1.25025 q^{93} +4.94645 q^{95} -0.561064 q^{97} +46.5827 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 3 q^{3} + 4 q^{5} + 5 q^{7} + 13 q^{9}+O(q^{10}) \) 4 * q - 3 * q^3 + 4 * q^5 + 5 * q^7 + 13 * q^9	\( 4 q - 3 q^{3} + 4 q^{5} + 5 q^{7} + 13 q^{9} - 5 q^{11} - 6 q^{13} - 3 q^{15} + 10 q^{17} + 19 q^{21} - 2 q^{23} + 4 q^{25} - 9 q^{27} + 2 q^{29} + 4 q^{31} - 11 q^{33} + 5 q^{35} + 4 q^{37} - 2 q^{39} + 29 q^{41} - 4 q^{43} + 13 q^{45} + 9 q^{47} + q^{49} - 14 q^{51} - 3 q^{53} - 5 q^{55} + 8 q^{57} + 10 q^{59} + 8 q^{61} + 8 q^{63} - 6 q^{65} - 14 q^{67} - 44 q^{69} + 17 q^{71} + 7 q^{73} - 3 q^{75} - 21 q^{77} + 24 q^{79} + 28 q^{81} - 31 q^{83} + 10 q^{85} + 44 q^{87} - 4 q^{89} - 14 q^{91} + 18 q^{93} + 12 q^{97} + 22 q^{99}+O(q^{100}) \) 4 * q - 3 * q^3 + 4 * q^5 + 5 * q^7 + 13 * q^9 - 5 * q^11 - 6 * q^13 - 3 * q^15 + 10 * q^17 + 19 * q^21 - 2 * q^23 + 4 * q^25 - 9 * q^27 + 2 * q^29 + 4 * q^31 - 11 * q^33 + 5 * q^35 + 4 * q^37 - 2 * q^39 + 29 * q^41 - 4 * q^43 + 13 * q^45 + 9 * q^47 + q^49 - 14 * q^51 - 3 * q^53 - 5 * q^55 + 8 * q^57 + 10 * q^59 + 8 * q^61 + 8 * q^63 - 6 * q^65 - 14 * q^67 - 44 * q^69 + 17 * q^71 + 7 * q^73 - 3 * q^75 - 21 * q^77 + 24 * q^79 + 28 * q^81 - 31 * q^83 + 10 * q^85 + 44 * q^87 - 4 * q^89 - 14 * q^91 + 18 * q^93 + 12 * q^97 + 22 * q^99

Introduction

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