LMFDB - Embedded newform 1960.2.a.w.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1960,2,Mod(1,1960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1960, 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("1960.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1960 = 2^{3} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1960.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:	$15.6506787962$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1944.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 9x - 6 $ x^3 - 9*x - 6
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 280)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.58423$ of defining polynomial
Character	$\chi$	$=$	1960.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.58423 q^{3} +1.00000 q^{5} +3.67822 q^{9} +O(q^{10})$	$q+2.58423 q^{3} +1.00000 q^{5} +3.67822 q^{9} +1.67822 q^{11} -4.84667 q^{13} +2.58423 q^{15} +2.00000 q^{17} +6.84667 q^{19} -2.26245 q^{23} +1.00000 q^{25} +1.75268 q^{27} +3.32178 q^{29} +9.16845 q^{31} +4.33690 q^{33} -2.84667 q^{37} -12.5249 q^{39} +9.52489 q^{41} +6.58423 q^{43} +3.67822 q^{45} -12.2031 q^{47} +5.16845 q^{51} +7.49023 q^{53} +1.67822 q^{55} +17.6933 q^{57} +8.00000 q^{59} -6.49023 q^{61} -4.84667 q^{65} -5.75268 q^{67} -5.84667 q^{69} -11.6933 q^{73} +2.58423 q^{75} +5.69334 q^{79} -6.50535 q^{81} -12.5842 q^{83} +2.00000 q^{85} +8.58423 q^{87} -5.84667 q^{89} +23.6933 q^{93} +6.84667 q^{95} -2.00000 q^{97} +6.17287 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{5} + 9 q^{9}+O(q^{10}) $ 3 * q + 3 * q^5 + 9 * q^9	$ 3 q + 3 q^{5} + 9 q^{9} + 3 q^{11} + 3 q^{13} + 6 q^{17} + 3 q^{19} + 3 q^{23} + 3 q^{25} - 18 q^{27} + 12 q^{29} + 12 q^{31} - 18 q^{33} + 9 q^{37} - 18 q^{39} + 9 q^{41} + 12 q^{43} + 9 q^{45} - 15 q^{47} + 9 q^{53} + 3 q^{55} + 18 q^{57} + 24 q^{59} - 6 q^{61} + 3 q^{65} + 6 q^{67} - 18 q^{79} + 27 q^{81} - 30 q^{83} + 6 q^{85} + 18 q^{87} + 36 q^{93} + 3 q^{95} - 6 q^{97} + 63 q^{99}+O(q^{100}) $ 3 * q + 3 * q^5 + 9 * q^9 + 3 * q^11 + 3 * q^13 + 6 * q^17 + 3 * q^19 + 3 * q^23 + 3 * q^25 - 18 * q^27 + 12 * q^29 + 12 * q^31 - 18 * q^33 + 9 * q^37 - 18 * q^39 + 9 * q^41 + 12 * q^43 + 9 * q^45 - 15 * q^47 + 9 * q^53 + 3 * q^55 + 18 * q^57 + 24 * q^59 - 6 * q^61 + 3 * q^65 + 6 * q^67 - 18 * q^79 + 27 * q^81 - 30 * q^83 + 6 * q^85 + 18 * q^87 + 36 * q^93 + 3 * q^95 - 6 * q^97 + 63 * 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
$3$	2.58423	1.49200	0.746002	−	0.665944i	$-0.231971\pi$
$3$	2.58423	1.49200	0.746002	+	0.665944i	$0.231971\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	3.67822	1.22607
$10$	0	0
$11$	1.67822	0.506003	0.253001	−	0.967466i	$-0.418582\pi$
$11$	1.67822	0.506003	0.253001	+	0.967466i	$0.418582\pi$
$12$	0	0
$13$	−4.84667	−1.34422	−0.672112	−	0.740449i	$-0.734613\pi$
$13$	−4.84667	−1.34422	−0.672112	+	0.740449i	$0.734613\pi$
$14$	0	0
$15$	2.58423	0.667244
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	6.84667	1.57073	0.785367	−	0.619030i	$-0.212474\pi$
$19$	6.84667	1.57073	0.785367	+	0.619030i	$0.212474\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.26245	−0.471753	−0.235876	−	0.971783i	$-0.575796\pi$
$23$	−2.26245	−0.471753	−0.235876	+	0.971783i	$0.575796\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.75268	0.337303
$28$	0	0
$29$	3.32178	0.616839	0.308419	−	0.951250i	$-0.400200\pi$
$29$	3.32178	0.616839	0.308419	+	0.951250i	$0.400200\pi$
$30$	0	0
$31$	9.16845	1.64670	0.823351	−	0.567532i	$-0.192102\pi$
$31$	9.16845	1.64670	0.823351	+	0.567532i	$0.192102\pi$
$32$	0	0
$33$	4.33690	0.754958
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−2.84667	−0.467990	−0.233995	−	0.972238i	$-0.575180\pi$
$37$	−2.84667	−0.467990	−0.233995	+	0.972238i	$0.575180\pi$
$38$	0	0
$39$	−12.5249	−2.00559
$40$	0	0
$41$	9.52489	1.48754	0.743769	−	0.668437i	$-0.233036\pi$
$41$	9.52489	1.48754	0.743769	+	0.668437i	$0.233036\pi$
$42$	0	0
$43$	6.58423	1.00408	0.502042	−	0.864843i	$-0.332582\pi$
$43$	6.58423	1.00408	0.502042	+	0.864843i	$0.332582\pi$
$44$	0	0
$45$	3.67822	0.548317
$46$	0	0
$47$	−12.2031	−1.78001	−0.890004	−	0.455954i	$-0.849298\pi$
$47$	−12.2031	−1.78001	−0.890004	+	0.455954i	$0.849298\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	5.16845	0.723728
$52$	0	0
$53$	7.49023	1.02886	0.514431	−	0.857532i	$-0.328003\pi$
$53$	7.49023	1.02886	0.514431	+	0.857532i	$0.328003\pi$
$54$	0	0
$55$	1.67822	0.226291
$56$	0	0
$57$	17.6933	2.34354
$58$	0	0
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	0	0
$61$	−6.49023	−0.830989	−0.415494	−	0.909596i	$-0.636391\pi$
$61$	−6.49023	−0.830989	−0.415494	+	0.909596i	$0.636391\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−4.84667	−0.601156
$66$	0	0
$67$	−5.75268	−0.702801	−0.351401	−	0.936225i	$-0.614294\pi$
$67$	−5.75268	−0.702801	−0.351401	+	0.936225i	$0.614294\pi$
$68$	0	0
$69$	−5.84667	−0.703857
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	−11.6933	−1.36860	−0.684301	−	0.729199i	$-0.739893\pi$
$73$	−11.6933	−1.36860	−0.684301	+	0.729199i	$0.739893\pi$
$74$	0	0
$75$	2.58423	0.298401
$76$	0	0
$77$	0	0
$78$	0	0
$79$	5.69334	0.640551	0.320276	−	0.947324i	$-0.396224\pi$
$79$	5.69334	0.640551	0.320276	+	0.947324i	$0.396224\pi$
$80$	0	0
$81$	−6.50535	−0.722817
$82$	0	0
$83$	−12.5842	−1.38130	−0.690649	−	0.723190i	$-0.742675\pi$
$83$	−12.5842	−1.38130	−0.690649	+	0.723190i	$0.742675\pi$
$84$	0	0
$85$	2.00000	0.216930
$86$	0	0
$87$	8.58423	0.920326
$88$	0	0
$89$	−5.84667	−0.619746	−0.309873	−	0.950778i	$-0.600287\pi$
$89$	−5.84667	−0.619746	−0.309873	+	0.950778i	$0.600287\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	23.6933	2.45689
$94$	0	0
$95$	6.84667	0.702454
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	0	0
$99$	6.17287	0.620397
$100$	0	0
$101$	−17.0151	−1.69307	−0.846534	−	0.532335i	$-0.821315\pi$
$101$	−17.0151	−1.69307	−0.846534	+	0.532335i	$0.821315\pi$
$102$	0	0
$103$	7.10912	0.700482	0.350241	−	0.936660i	$-0.386100\pi$
$103$	7.10912	0.700482	0.350241	+	0.936660i	$0.386100\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1.75268	−0.169438	−0.0847188	−	0.996405i	$-0.526999\pi$
$107$	−1.75268	−0.169438	−0.0847188	+	0.996405i	$0.526999\pi$
$108$	0	0
$109$	19.5400	1.87159	0.935797	−	0.352539i	$-0.114682\pi$
$109$	19.5400	1.87159	0.935797	+	0.352539i	$0.114682\pi$
$110$	0	0
$111$	−7.35644	−0.698243
$112$	0	0
$113$	10.3369	0.972414	0.486207	−	0.873844i	$-0.338380\pi$
$113$	10.3369	0.972414	0.486207	+	0.873844i	$0.338380\pi$
$114$	0	0
$115$	−2.26245	−0.210974
$116$	0	0
$117$	−17.8271	−1.64812
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−8.18357	−0.743961
$122$	0	0
$123$	24.6145	2.21941
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−19.1836	−1.70227	−0.851133	−	0.524949i	$-0.824084\pi$
$127$	−19.1836	−1.70227	−0.851133	+	0.524949i	$0.824084\pi$
$128$	0	0
$129$	17.0151	1.49810
$130$	0	0
$131$	−17.8271	−1.55756	−0.778782	−	0.627295i	$-0.784162\pi$
$131$	−17.8271	−1.55756	−0.778782	+	0.627295i	$0.784162\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	1.75268	0.150846
$136$	0	0
$137$	4.00000	0.341743	0.170872	−	0.985293i	$-0.445342\pi$
$137$	4.00000	0.341743	0.170872	+	0.985293i	$0.445342\pi$
$138$	0	0
$139$	−5.16845	−0.438382	−0.219191	−	0.975682i	$-0.570342\pi$
$139$	−5.16845	−0.438382	−0.219191	+	0.975682i	$0.570342\pi$
$140$	0	0
$141$	−31.5356	−2.65578
$142$	0	0
$143$	−8.13379	−0.680181
$144$	0	0
$145$	3.32178	0.275859
$146$	0	0
$147$	0	0
$148$	0	0
$149$	0.796886	0.0652834	0.0326417	−	0.999467i	$-0.489608\pi$
$149$	0.796886	0.0652834	0.0326417	+	0.999467i	$0.489608\pi$
$150$	0	0
$151$	−16.5249	−1.34478	−0.672388	−	0.740199i	$-0.734731\pi$
$151$	−16.5249	−1.34478	−0.672388	+	0.740199i	$0.734731\pi$
$152$	0	0
$153$	7.35644	0.594733
$154$	0	0
$155$	9.16845	0.736428
$156$	0	0
$157$	8.20311	0.654680	0.327340	−	0.944907i	$-0.393848\pi$
$157$	8.20311	0.654680	0.327340	+	0.944907i	$0.393848\pi$
$158$	0	0
$159$	19.3564	1.53507
$160$	0	0
$161$	0	0
$162$	0	0
$163$	3.69334	0.289285	0.144643	−	0.989484i	$-0.453797\pi$
$163$	3.69334	0.289285	0.144643	+	0.989484i	$0.453797\pi$
$164$	0	0
$165$	4.33690	0.337627
$166$	0	0
$167$	−0.262447	−0.0203087	−0.0101544	−	0.999948i	$-0.503232\pi$
$167$	−0.262447	−0.0203087	−0.0101544	+	0.999948i	$0.503232\pi$
$168$	0	0
$169$	10.4902	0.806941
$170$	0	0
$171$	25.1836	1.92584
$172$	0	0
$173$	−18.8467	−1.43289	−0.716443	−	0.697646i	$-0.754231\pi$
$173$	−18.8467	−1.43289	−0.716443	+	0.697646i	$0.754231\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	20.6738	1.55394
$178$	0	0
$179$	−12.0151	−0.898052	−0.449026	−	0.893519i	$-0.648229\pi$
$179$	−12.0151	−0.898052	−0.449026	+	0.893519i	$0.648229\pi$
$180$	0	0
$181$	−6.03466	−0.448553	−0.224276	−	0.974526i	$-0.572002\pi$
$181$	−6.03466	−0.448553	−0.224276	+	0.974526i	$0.572002\pi$
$182$	0	0
$183$	−16.7722	−1.23984
$184$	0	0
$185$	−2.84667	−0.209291
$186$	0	0
$187$	3.35644	0.245447
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2.83155	0.204884	0.102442	−	0.994739i	$-0.467334\pi$
$191$	2.83155	0.204884	0.102442	+	0.994739i	$0.467334\pi$
$192$	0	0
$193$	12.9805	0.934354	0.467177	−	0.884164i	$-0.345271\pi$
$193$	12.9805	0.934354	0.467177	+	0.884164i	$0.345271\pi$
$194$	0	0
$195$	−12.5249	−0.896926
$196$	0	0
$197$	−4.84667	−0.345311	−0.172656	−	0.984982i	$-0.555235\pi$
$197$	−4.84667	−0.345311	−0.172656	+	0.984982i	$0.555235\pi$
$198$	0	0
$199$	−15.3867	−1.09073	−0.545367	−	0.838198i	$-0.683610\pi$
$199$	−15.3867	−1.09073	−0.545367	+	0.838198i	$0.683610\pi$
$200$	0	0
$201$	−14.8662	−1.04858
$202$	0	0
$203$	0	0
$204$	0	0
$205$	9.52489	0.665247
$206$	0	0
$207$	−8.32178	−0.578404
$208$	0	0
$209$	11.4902	0.794796
$210$	0	0
$211$	9.18357	0.632223	0.316112	−	0.948722i	$-0.397623\pi$
$211$	9.18357	0.632223	0.316112	+	0.948722i	$0.397623\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	6.58423	0.449040
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−30.2182	−2.04196
$220$	0	0
$221$	−9.69334	−0.652045
$222$	0	0
$223$	−12.9805	−0.869236	−0.434618	−	0.900615i	$-0.643117\pi$
$223$	−12.9805	−0.869236	−0.434618	+	0.900615i	$0.643117\pi$
$224$	0	0
$225$	3.67822	0.245215
$226$	0	0
$227$	−22.0302	−1.46220	−0.731099	−	0.682271i	$-0.760992\pi$
$227$	−22.0302	−1.46220	−0.731099	+	0.682271i	$0.760992\pi$
$228$	0	0
$229$	4.30666	0.284592	0.142296	−	0.989824i	$-0.454552\pi$
$229$	4.30666	0.284592	0.142296	+	0.989824i	$0.454552\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	0	0
$235$	−12.2031	−0.796043
$236$	0	0
$237$	14.7129	0.955705
$238$	0	0
$239$	10.8618	0.702591	0.351296	−	0.936265i	$-0.385741\pi$
$239$	10.8618	0.702591	0.351296	+	0.936265i	$0.385741\pi$
$240$	0	0
$241$	0.203114	0.0130837	0.00654187	−	0.999979i	$-0.497918\pi$
$241$	0.203114	0.0130837	0.00654187	+	0.999979i	$0.497918\pi$
$242$	0	0
$243$	−22.0693	−1.41575
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−33.1836	−2.11142
$248$	0	0
$249$	−32.5205	−2.06090
$250$	0	0
$251$	5.03466	0.317785	0.158893	−	0.987296i	$-0.449208\pi$
$251$	5.03466	0.317785	0.158893	+	0.987296i	$0.449208\pi$
$252$	0	0
$253$	−3.79689	−0.238708
$254$	0	0
$255$	5.16845	0.323661
$256$	0	0
$257$	23.3867	1.45882	0.729411	−	0.684076i	$-0.239794\pi$
$257$	23.3867	1.45882	0.729411	+	0.684076i	$0.239794\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	12.2182	0.756290
$262$	0	0
$263$	27.1091	1.67162	0.835810	−	0.549019i	$-0.184999\pi$
$263$	27.1091	1.67162	0.835810	+	0.549019i	$0.184999\pi$
$264$	0	0
$265$	7.49023	0.460121
$266$	0	0
$267$	−15.1091	−0.924663
$268$	0	0
$269$	6.49023	0.395716	0.197858	−	0.980231i	$-0.436601\pi$
$269$	6.49023	0.395716	0.197858	+	0.980231i	$0.436601\pi$
$270$	0	0
$271$	−23.3867	−1.42064	−0.710320	−	0.703879i	$-0.751450\pi$
$271$	−23.3867	−1.42064	−0.710320	+	0.703879i	$0.751450\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.67822	0.101201
$276$	0	0
$277$	3.66310	0.220094	0.110047	−	0.993926i	$-0.464900\pi$
$277$	3.66310	0.220094	0.110047	+	0.993926i	$0.464900\pi$
$278$	0	0
$279$	33.7236	2.01898
$280$	0	0
$281$	13.8965	0.828993	0.414497	−	0.910051i	$-0.363958\pi$
$281$	13.8965	0.828993	0.414497	+	0.910051i	$0.363958\pi$
$282$	0	0
$283$	−20.3369	−1.20890	−0.604452	−	0.796642i	$-0.706608\pi$
$283$	−20.3369	−1.20890	−0.604452	+	0.796642i	$0.706608\pi$
$284$	0	0
$285$	17.6933	1.04806
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	−5.16845	−0.302980
$292$	0	0
$293$	18.8467	1.10103	0.550517	−	0.834824i	$-0.314431\pi$
$293$	18.8467	1.10103	0.550517	+	0.834824i	$0.314431\pi$
$294$	0	0
$295$	8.00000	0.465778
$296$	0	0
$297$	2.94138	0.170676
$298$	0	0
$299$	10.9653	0.634142
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−43.9709	−2.52606
$304$	0	0
$305$	−6.49023	−0.371629
$306$	0	0
$307$	−2.39623	−0.136760	−0.0683802	−	0.997659i	$-0.521783\pi$
$307$	−2.39623	−0.136760	−0.0683802	+	0.997659i	$0.521783\pi$
$308$	0	0
$309$	18.3716	1.04512
$310$	0	0
$311$	−5.66310	−0.321125	−0.160562	−	0.987026i	$-0.551331\pi$
$311$	−5.66310	−0.321125	−0.160562	+	0.987026i	$0.551331\pi$
$312$	0	0
$313$	29.7236	1.68008	0.840038	−	0.542527i	$-0.182532\pi$
$313$	29.7236	1.68008	0.840038	+	0.542527i	$0.182532\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	19.0498	1.06994	0.534971	−	0.844870i	$-0.320322\pi$
$317$	19.0498	1.06994	0.534971	+	0.844870i	$0.320322\pi$
$318$	0	0
$319$	5.57468	0.312122
$320$	0	0
$321$	−4.52931	−0.252801
$322$	0	0
$323$	13.6933	0.761918
$324$	0	0
$325$	−4.84667	−0.268845
$326$	0	0
$327$	50.4958	2.79242
$328$	0	0
$329$	0	0
$330$	0	0
$331$	2.32178	0.127617	0.0638083	−	0.997962i	$-0.479675\pi$
$331$	2.32178	0.127617	0.0638083	+	0.997962i	$0.479675\pi$
$332$	0	0
$333$	−10.4707	−0.573790
$334$	0	0
$335$	−5.75268	−0.314302
$336$	0	0
$337$	−16.4062	−0.893704	−0.446852	−	0.894608i	$-0.647455\pi$
$337$	−16.4062	−0.893704	−0.446852	+	0.894608i	$0.647455\pi$
$338$	0	0
$339$	26.7129	1.45084
$340$	0	0
$341$	15.3867	0.833236
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−5.84667	−0.314774
$346$	0	0
$347$	4.58423	0.246094	0.123047	−	0.992401i	$-0.460733\pi$
$347$	4.58423	0.246094	0.123047	+	0.992401i	$0.460733\pi$
$348$	0	0
$349$	12.3716	0.662235	0.331117	−	0.943590i	$-0.392574\pi$
$349$	12.3716	0.662235	0.331117	+	0.943590i	$0.392574\pi$
$350$	0	0
$351$	−8.49465	−0.453411
$352$	0	0
$353$	19.3867	1.03185	0.515925	−	0.856634i	$-0.327448\pi$
$353$	19.3867	1.03185	0.515925	+	0.856634i	$0.327448\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−9.81201	−0.517858	−0.258929	−	0.965896i	$-0.583370\pi$
$359$	−9.81201	−0.517858	−0.258929	+	0.965896i	$0.583370\pi$
$360$	0	0
$361$	27.8769	1.46721
$362$	0	0
$363$	−21.1482	−1.10999
$364$	0	0
$365$	−11.6933	−0.612058
$366$	0	0
$367$	−12.7180	−0.663875	−0.331937	−	0.943301i	$-0.607702\pi$
$367$	−12.7180	−0.663875	−0.331937	+	0.943301i	$0.607702\pi$
$368$	0	0
$369$	35.0347	1.82383
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−16.4062	−0.849482	−0.424741	−	0.905315i	$-0.639635\pi$
$373$	−16.4062	−0.849482	−0.424741	+	0.905315i	$0.639635\pi$
$374$	0	0
$375$	2.58423	0.133449
$376$	0	0
$377$	−16.0996	−0.829170
$378$	0	0
$379$	14.4707	0.743309	0.371655	−	0.928371i	$-0.378791\pi$
$379$	14.4707	0.743309	0.371655	+	0.928371i	$0.378791\pi$
$380$	0	0
$381$	−49.5747	−2.53979
$382$	0	0
$383$	−3.54956	−0.181374	−0.0906871	−	0.995879i	$-0.528906\pi$
$383$	−3.54956	−0.181374	−0.0906871	+	0.995879i	$0.528906\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	24.2182	1.23108
$388$	0	0
$389$	8.98046	0.455327	0.227664	−	0.973740i	$-0.426891\pi$
$389$	8.98046	0.455327	0.227664	+	0.973740i	$0.426891\pi$
$390$	0	0
$391$	−4.52489	−0.228834
$392$	0	0
$393$	−46.0693	−2.32389
$394$	0	0
$395$	5.69334	0.286463
$396$	0	0
$397$	−28.0996	−1.41028	−0.705139	−	0.709070i	$-0.749115\pi$
$397$	−28.0996	−1.41028	−0.705139	+	0.709070i	$0.749115\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−26.6933	−1.33300	−0.666501	−	0.745504i	$-0.732209\pi$
$401$	−26.6933	−1.33300	−0.666501	+	0.745504i	$0.732209\pi$
$402$	0	0
$403$	−44.4365	−2.21354
$404$	0	0
$405$	−6.50535	−0.323254
$406$	0	0
$407$	−4.77735	−0.236804
$408$	0	0
$409$	28.0649	1.38772	0.693860	−	0.720110i	$-0.255909\pi$
$409$	28.0649	1.38772	0.693860	+	0.720110i	$0.255909\pi$
$410$	0	0
$411$	10.3369	0.509882
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−12.5842	−0.617735
$416$	0	0
$417$	−13.3564	−0.654068
$418$	0	0
$419$	−18.8769	−0.922198	−0.461099	−	0.887349i	$-0.652545\pi$
$419$	−18.8769	−0.922198	−0.461099	+	0.887349i	$0.652545\pi$
$420$	0	0
$421$	−28.1836	−1.37358	−0.686792	−	0.726854i	$-0.740982\pi$
$421$	−28.1836	−1.37358	−0.686792	+	0.726854i	$0.740982\pi$
$422$	0	0
$423$	−44.8858	−2.18242
$424$	0	0
$425$	2.00000	0.0970143
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−21.0195	−1.01483
$430$	0	0
$431$	2.83155	0.136391	0.0681955	−	0.997672i	$-0.478276\pi$
$431$	2.83155	0.136391	0.0681955	+	0.997672i	$0.478276\pi$
$432$	0	0
$433$	−2.33690	−0.112304	−0.0561522	−	0.998422i	$-0.517883\pi$
$433$	−2.33690	−0.112304	−0.0561522	+	0.998422i	$0.517883\pi$
$434$	0	0
$435$	8.58423	0.411582
$436$	0	0
$437$	−15.4902	−0.740998
$438$	0	0
$439$	−6.06933	−0.289673	−0.144837	−	0.989456i	$-0.546266\pi$
$439$	−6.06933	−0.289673	−0.144837	+	0.989456i	$0.546266\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	11.6038	0.551312	0.275656	−	0.961256i	$-0.411105\pi$
$443$	11.6038	0.551312	0.275656	+	0.961256i	$0.411105\pi$
$444$	0	0
$445$	−5.84667	−0.277159
$446$	0	0
$447$	2.05933	0.0974031
$448$	0	0
$449$	4.13821	0.195294	0.0976470	−	0.995221i	$-0.468868\pi$
$449$	4.13821	0.195294	0.0976470	+	0.995221i	$0.468868\pi$
$450$	0	0
$451$	15.9849	0.752698
$452$	0	0
$453$	−42.7040	−2.00641
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−12.6436	−0.591441	−0.295720	−	0.955275i	$-0.595560\pi$
$457$	−12.6436	−0.591441	−0.295720	+	0.955275i	$0.595560\pi$
$458$	0	0
$459$	3.50535	0.163616
$460$	0	0
$461$	20.7129	0.964695	0.482348	−	0.875980i	$-0.339784\pi$
$461$	20.7129	0.964695	0.482348	+	0.875980i	$0.339784\pi$
$462$	0	0
$463$	−16.3811	−0.761295	−0.380647	−	0.924720i	$-0.624299\pi$
$463$	−16.3811	−0.761295	−0.380647	+	0.924720i	$0.624299\pi$
$464$	0	0
$465$	23.6933	1.09875
$466$	0	0
$467$	−1.72243	−0.0797046	−0.0398523	−	0.999206i	$-0.512689\pi$
$467$	−1.72243	−0.0797046	−0.0398523	+	0.999206i	$0.512689\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	21.1987	0.976784
$472$	0	0
$473$	11.0498	0.508070
$474$	0	0
$475$	6.84667	0.314147
$476$	0	0
$477$	27.5507	1.26146
$478$	0	0
$479$	−1.78176	−0.0814108	−0.0407054	−	0.999171i	$-0.512961\pi$
$479$	−1.78176	−0.0814108	−0.0407054	+	0.999171i	$0.512961\pi$
$480$	0	0
$481$	13.7969	0.629084
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−2.00000	−0.0908153
$486$	0	0
$487$	−11.0195	−0.499343	−0.249672	−	0.968331i	$-0.580323\pi$
$487$	−11.0195	−0.499343	−0.249672	+	0.968331i	$0.580323\pi$
$488$	0	0
$489$	9.54443	0.431614
$490$	0	0
$491$	−20.9311	−0.944608	−0.472304	−	0.881436i	$-0.656578\pi$
$491$	−20.9311	−0.944608	−0.472304	+	0.881436i	$0.656578\pi$
$492$	0	0
$493$	6.64356	0.299211
$494$	0	0
$495$	6.17287	0.277450
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−7.50535	−0.335986	−0.167993	−	0.985788i	$-0.553729\pi$
$499$	−7.50535	−0.335986	−0.167993	+	0.985788i	$0.553729\pi$
$500$	0	0
$501$	−0.678221	−0.0303007
$502$	0	0
$503$	20.5842	0.917805	0.458903	−	0.888487i	$-0.348243\pi$
$503$	20.5842	0.917805	0.458903	+	0.888487i	$0.348243\pi$
$504$	0	0
$505$	−17.0151	−0.757163
$506$	0	0
$507$	27.1091	1.20396
$508$	0	0
$509$	13.6587	0.605410	0.302705	−	0.953084i	$-0.402110\pi$
$509$	13.6587	0.605410	0.302705	+	0.953084i	$0.402110\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	12.0000	0.529813
$514$	0	0
$515$	7.10912	0.313265
$516$	0	0
$517$	−20.4795	−0.900688
$518$	0	0
$519$	−48.7040	−2.13787
$520$	0	0
$521$	−42.1640	−1.84724	−0.923620	−	0.383310i	$-0.874784\pi$
$521$	−42.1640	−1.84724	−0.923620	+	0.383310i	$0.874784\pi$
$522$	0	0
$523$	−30.6436	−1.33995	−0.669975	−	0.742384i	$-0.733695\pi$
$523$	−30.6436	−1.33995	−0.669975	+	0.742384i	$0.733695\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	18.3369	0.798768
$528$	0	0
$529$	−17.8813	−0.777449
$530$	0	0
$531$	29.4258	1.27697
$532$	0	0
$533$	−46.1640	−1.99959
$534$	0	0
$535$	−1.75268	−0.0757748
$536$	0	0
$537$	−31.0498	−1.33990
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−33.6889	−1.44840	−0.724200	−	0.689590i	$-0.757791\pi$
$541$	−33.6889	−1.44840	−0.724200	+	0.689590i	$0.757791\pi$
$542$	0	0
$543$	−15.5949	−0.669243
$544$	0	0
$545$	19.5400	0.837002
$546$	0	0
$547$	−3.03979	−0.129972	−0.0649861	−	0.997886i	$-0.520700\pi$
$547$	−3.03979	−0.129972	−0.0649861	+	0.997886i	$0.520700\pi$
$548$	0	0
$549$	−23.8725	−1.01885
$550$	0	0
$551$	22.7431	0.968890
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−7.35644	−0.312264
$556$	0	0
$557$	18.1640	0.769635	0.384817	−	0.922993i	$-0.374264\pi$
$557$	18.1640	0.769635	0.384817	+	0.922993i	$0.374264\pi$
$558$	0	0
$559$	−31.9116	−1.34972
$560$	0	0
$561$	8.67380	0.366208
$562$	0	0
$563$	−36.9905	−1.55896	−0.779481	−	0.626426i	$-0.784517\pi$
$563$	−36.9905	−1.55896	−0.779481	+	0.626426i	$0.784517\pi$
$564$	0	0
$565$	10.3369	0.434877
$566$	0	0
$567$	0	0
$568$	0	0
$569$	32.6093	1.36705	0.683527	−	0.729925i	$-0.260445\pi$
$569$	32.6093	1.36705	0.683527	+	0.729925i	$0.260445\pi$
$570$	0	0
$571$	40.4365	1.69221	0.846107	−	0.533013i	$-0.178940\pi$
$571$	40.4365	1.69221	0.846107	+	0.533013i	$0.178940\pi$
$572$	0	0
$573$	7.31736	0.305687
$574$	0	0
$575$	−2.26245	−0.0943505
$576$	0	0
$577$	−38.0302	−1.58322	−0.791610	−	0.611027i	$-0.790757\pi$
$577$	−38.0302	−1.58322	−0.791610	+	0.611027i	$0.790757\pi$
$578$	0	0
$579$	33.5444	1.39406
$580$	0	0
$581$	0	0
$582$	0	0
$583$	12.5703	0.520607
$584$	0	0
$585$	−17.8271	−0.737061
$586$	0	0
$587$	34.0302	1.40458	0.702289	−	0.711892i	$-0.252161\pi$
$587$	34.0302	1.40458	0.702289	+	0.711892i	$0.252161\pi$
$588$	0	0
$589$	62.7734	2.58653
$590$	0	0
$591$	−12.5249	−0.515205
$592$	0	0
$593$	−33.7236	−1.38486	−0.692431	−	0.721484i	$-0.743460\pi$
$593$	−33.7236	−1.38486	−0.692431	+	0.721484i	$0.743460\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−39.7627	−1.62738
$598$	0	0
$599$	6.30666	0.257683	0.128841	−	0.991665i	$-0.458874\pi$
$599$	6.30666	0.257683	0.128841	+	0.991665i	$0.458874\pi$
$600$	0	0
$601$	−11.2871	−0.460411	−0.230206	−	0.973142i	$-0.573940\pi$
$601$	−11.2871	−0.460411	−0.230206	+	0.973142i	$0.573940\pi$
$602$	0	0
$603$	−21.1596	−0.861686
$604$	0	0
$605$	−8.18357	−0.332710
$606$	0	0
$607$	−15.4611	−0.627548	−0.313774	−	0.949498i	$-0.601594\pi$
$607$	−15.4611	−0.627548	−0.313774	+	0.949498i	$0.601594\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	59.1445	2.39273
$612$	0	0
$613$	−22.1640	−0.895197	−0.447598	−	0.894235i	$-0.647721\pi$
$613$	−22.1640	−0.895197	−0.447598	+	0.894235i	$0.647721\pi$
$614$	0	0
$615$	24.6145	0.992551
$616$	0	0
$617$	−14.9805	−0.603091	−0.301545	−	0.953452i	$-0.597502\pi$
$617$	−14.9805	−0.603091	−0.301545	+	0.953452i	$0.597502\pi$
$618$	0	0
$619$	23.6391	0.950137	0.475069	−	0.879949i	$-0.342423\pi$
$619$	23.6391	0.950137	0.475069	+	0.879949i	$0.342423\pi$
$620$	0	0
$621$	−3.96534	−0.159123
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	29.6933	1.18584
$628$	0	0
$629$	−5.69334	−0.227008
$630$	0	0
$631$	−13.7818	−0.548643	−0.274322	−	0.961638i	$-0.588453\pi$
$631$	−13.7818	−0.548643	−0.274322	+	0.961638i	$0.588453\pi$
$632$	0	0
$633$	23.7324	0.943279
$634$	0	0
$635$	−19.1836	−0.761277
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	43.8618	1.73244	0.866218	−	0.499666i	$-0.166544\pi$
$641$	43.8618	1.73244	0.866218	+	0.499666i	$0.166544\pi$
$642$	0	0
$643$	7.04979	0.278016	0.139008	−	0.990291i	$-0.455609\pi$
$643$	7.04979	0.278016	0.139008	+	0.990291i	$0.455609\pi$
$644$	0	0
$645$	17.0151	0.669970
$646$	0	0
$647$	−24.4807	−0.962435	−0.481217	−	0.876601i	$-0.659805\pi$
$647$	−24.4807	−0.962435	−0.481217	+	0.876601i	$0.659805\pi$
$648$	0	0
$649$	13.4258	0.527008
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0.816426	0.0319492	0.0159746	−	0.999872i	$-0.494915\pi$
$653$	0.816426	0.0319492	0.0159746	+	0.999872i	$0.494915\pi$
$654$	0	0
$655$	−17.8271	−0.696564
$656$	0	0
$657$	−43.0107	−1.67801
$658$	0	0
$659$	21.2871	0.829228	0.414614	−	0.909997i	$-0.363917\pi$
$659$	21.2871	0.829228	0.414614	+	0.909997i	$0.363917\pi$
$660$	0	0
$661$	25.9462	1.00919	0.504596	−	0.863356i	$-0.331641\pi$
$661$	25.9462	1.00919	0.504596	+	0.863356i	$0.331641\pi$
$662$	0	0
$663$	−25.0498	−0.972853
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−7.51535	−0.290995
$668$	0	0
$669$	−33.5444	−1.29690
$670$	0	0
$671$	−10.8920	−0.420483
$672$	0	0
$673$	22.0693	0.850710	0.425355	−	0.905027i	$-0.360149\pi$
$673$	22.0693	0.850710	0.425355	+	0.905027i	$0.360149\pi$
$674$	0	0
$675$	1.75268	0.0674605
$676$	0	0
$677$	5.49023	0.211007	0.105503	−	0.994419i	$-0.466355\pi$
$677$	5.49023	0.211007	0.105503	+	0.994419i	$0.466355\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−56.9311	−2.18161
$682$	0	0
$683$	19.2278	0.735731	0.367865	−	0.929879i	$-0.380089\pi$
$683$	19.2278	0.735731	0.367865	+	0.929879i	$0.380089\pi$
$684$	0	0
$685$	4.00000	0.152832
$686$	0	0
$687$	11.1294	0.424612
$688$	0	0
$689$	−36.3027	−1.38302
$690$	0	0
$691$	14.2182	0.540887	0.270444	−	0.962736i	$-0.412830\pi$
$691$	14.2182	0.540887	0.270444	+	0.962736i	$0.412830\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−5.16845	−0.196051
$696$	0	0
$697$	19.0498	0.721562
$698$	0	0
$699$	46.5161	1.75940
$700$	0	0
$701$	14.1533	0.534564	0.267282	−	0.963618i	$-0.413875\pi$
$701$	14.1533	0.534564	0.267282	+	0.963618i	$0.413875\pi$
$702$	0	0
$703$	−19.4902	−0.735088
$704$	0	0
$705$	−31.5356	−1.18770
$706$	0	0
$707$	0	0
$708$	0	0
$709$	17.0454	0.640152	0.320076	−	0.947392i	$-0.396292\pi$
$709$	17.0454	0.640152	0.320076	+	0.947392i	$0.396292\pi$
$710$	0	0
$711$	20.9414	0.785363
$712$	0	0
$713$	−20.7431	−0.776836
$714$	0	0
$715$	−8.13379	−0.304186
$716$	0	0
$717$	28.0693	1.04827
$718$	0	0
$719$	−11.5054	−0.429077	−0.214539	−	0.976716i	$-0.568825\pi$
$719$	−11.5054	−0.429077	−0.214539	+	0.976716i	$0.568825\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0.524893	0.0195210
$724$	0	0
$725$	3.32178	0.123368
$726$	0	0
$727$	16.4114	0.608664	0.304332	−	0.952566i	$-0.401567\pi$
$727$	16.4114	0.608664	0.304332	+	0.952566i	$0.401567\pi$
$728$	0	0
$729$	−37.5161	−1.38948
$730$	0	0
$731$	13.1685	0.487053
$732$	0	0
$733$	−9.11425	−0.336642	−0.168321	−	0.985732i	$-0.553835\pi$
$733$	−9.11425	−0.336642	−0.168321	+	0.985732i	$0.553835\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−9.65426	−0.355619
$738$	0	0
$739$	−34.9956	−1.28733	−0.643667	−	0.765306i	$-0.722588\pi$
$739$	−34.9956	−1.28733	−0.643667	+	0.765306i	$0.722588\pi$
$740$	0	0
$741$	−85.7538	−3.15025
$742$	0	0
$743$	−43.5305	−1.59698	−0.798489	−	0.602009i	$-0.794367\pi$
$743$	−43.5305	−1.59698	−0.798489	+	0.602009i	$0.794367\pi$
$744$	0	0
$745$	0.796886	0.0291956
$746$	0	0
$747$	−46.2876	−1.69357
$748$	0	0
$749$	0	0
$750$	0	0
$751$	6.30666	0.230133	0.115067	−	0.993358i	$-0.463292\pi$
$751$	6.30666	0.230133	0.115067	+	0.993358i	$0.463292\pi$
$752$	0	0
$753$	13.0107	0.474136
$754$	0	0
$755$	−16.5249	−0.601402
$756$	0	0
$757$	24.3369	0.884540	0.442270	−	0.896882i	$-0.354173\pi$
$757$	24.3369	0.884540	0.442270	+	0.896882i	$0.354173\pi$
$758$	0	0
$759$	−9.81201	−0.356153
$760$	0	0
$761$	−28.2031	−1.02236	−0.511181	−	0.859473i	$-0.670792\pi$
$761$	−28.2031	−1.02236	−0.511181	+	0.859473i	$0.670792\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	7.35644	0.265973
$766$	0	0
$767$	−38.7734	−1.40003
$768$	0	0
$769$	−41.8965	−1.51082	−0.755412	−	0.655250i	$-0.772563\pi$
$769$	−41.8965	−1.51082	−0.755412	+	0.655250i	$0.772563\pi$
$770$	0	0
$771$	60.4365	2.17657
$772$	0	0
$773$	39.8574	1.43357	0.716785	−	0.697294i	$-0.245613\pi$
$773$	39.8574	1.43357	0.716785	+	0.697294i	$0.245613\pi$
$774$	0	0
$775$	9.16845	0.329340
$776$	0	0
$777$	0	0
$778$	0	0
$779$	65.2138	2.33653
$780$	0	0
$781$	0	0
$782$	0	0
$783$	5.82200	0.208061
$784$	0	0
$785$	8.20311	0.292782
$786$	0	0
$787$	35.3274	1.25928	0.629642	−	0.776885i	$-0.283202\pi$
$787$	35.3274	1.25928	0.629642	+	0.776885i	$0.283202\pi$
$788$	0	0
$789$	70.0561	2.49406
$790$	0	0
$791$	0	0
$792$	0	0
$793$	31.4560	1.11704
$794$	0	0
$795$	19.3564	0.686502
$796$	0	0
$797$	29.6933	1.05179	0.525896	−	0.850549i	$-0.323730\pi$
$797$	29.6933	1.05179	0.525896	+	0.850549i	$0.323730\pi$
$798$	0	0
$799$	−24.4062	−0.863430
$800$	0	0
$801$	−21.5054	−0.759854
$802$	0	0
$803$	−19.6240	−0.692517
$804$	0	0
$805$	0	0
$806$	0	0
$807$	16.7722	0.590410
$808$	0	0
$809$	−19.6436	−0.690631	−0.345315	−	0.938487i	$-0.612228\pi$
$809$	−19.6436	−0.690631	−0.345315	+	0.938487i	$0.612228\pi$
$810$	0	0
$811$	0.252452	0.00886479	0.00443239	−	0.999990i	$-0.498589\pi$
$811$	0.252452	0.00886479	0.00443239	+	0.999990i	$0.498589\pi$
$812$	0	0
$813$	−60.4365	−2.11960
$814$	0	0
$815$	3.69334	0.129372
$816$	0	0
$817$	45.0800	1.57715
$818$	0	0
$819$	0	0
$820$	0	0
$821$	44.7734	1.56260	0.781301	−	0.624155i	$-0.214556\pi$
$821$	44.7734	1.56260	0.781301	+	0.624155i	$0.214556\pi$
$822$	0	0
$823$	3.29711	0.114930	0.0574650	−	0.998348i	$-0.481698\pi$
$823$	3.29711	0.114930	0.0574650	+	0.998348i	$0.481698\pi$
$824$	0	0
$825$	4.33690	0.150992
$826$	0	0
$827$	9.41577	0.327419	0.163709	−	0.986509i	$-0.447654\pi$
$827$	9.41577	0.327419	0.163709	+	0.986509i	$0.447654\pi$
$828$	0	0
$829$	−4.71288	−0.163685	−0.0818426	−	0.996645i	$-0.526080\pi$
$829$	−4.71288	−0.163685	−0.0818426	+	0.996645i	$0.526080\pi$
$830$	0	0
$831$	9.46627	0.328381
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−0.262447	−0.00908234
$836$	0	0
$837$	16.0693	0.555437
$838$	0	0
$839$	22.1880	0.766015	0.383007	−	0.923745i	$-0.374888\pi$
$839$	22.1880	0.766015	0.383007	+	0.923745i	$0.374888\pi$
$840$	0	0
$841$	−17.9658	−0.619510
$842$	0	0
$843$	35.9116	1.23686
$844$	0	0
$845$	10.4902	0.360875
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−52.5551	−1.80369
$850$	0	0
$851$	6.44044	0.220776
$852$	0	0
$853$	39.9267	1.36706	0.683532	−	0.729920i	$-0.260443\pi$
$853$	39.9267	1.36706	0.683532	+	0.729920i	$0.260443\pi$
$854$	0	0
$855$	25.1836	0.861260
$856$	0	0
$857$	−1.62402	−0.0554754	−0.0277377	−	0.999615i	$-0.508830\pi$
$857$	−1.62402	−0.0554754	−0.0277377	+	0.999615i	$0.508830\pi$
$858$	0	0
$859$	8.00000	0.272956	0.136478	−	0.990643i	$-0.456422\pi$
$859$	8.00000	0.272956	0.136478	+	0.990643i	$0.456422\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	18.2234	0.620331	0.310165	−	0.950683i	$-0.399616\pi$
$863$	18.2234	0.620331	0.310165	+	0.950683i	$0.399616\pi$
$864$	0	0
$865$	−18.8467	−0.640806
$866$	0	0
$867$	−33.5949	−1.14094
$868$	0	0
$869$	9.55469	0.324121
$870$	0	0
$871$	27.8813	0.944723
$872$	0	0
$873$	−7.35644	−0.248978
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−50.8769	−1.71799	−0.858996	−	0.511983i	$-0.828911\pi$
$877$	−50.8769	−1.71799	−0.858996	+	0.511983i	$0.828911\pi$
$878$	0	0
$879$	48.7040	1.64275
$880$	0	0
$881$	−19.1187	−0.644124	−0.322062	−	0.946719i	$-0.604376\pi$
$881$	−19.1187	−0.644124	−0.322062	+	0.946719i	$0.604376\pi$
$882$	0	0
$883$	39.3174	1.32313	0.661567	−	0.749886i	$-0.269892\pi$
$883$	39.3174	1.32313	0.661567	+	0.749886i	$0.269892\pi$
$884$	0	0
$885$	20.6738	0.694942
$886$	0	0
$887$	27.0398	0.907907	0.453954	−	0.891025i	$-0.350013\pi$
$887$	27.0398	0.907907	0.453954	+	0.891025i	$0.350013\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−10.9174	−0.365747
$892$	0	0
$893$	−83.5507	−2.79592
$894$	0	0
$895$	−12.0151	−0.401621
$896$	0	0
$897$	28.3369	0.946142
$898$	0	0
$899$	30.4556	1.01575
$900$	0	0
$901$	14.9805	0.499071
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−6.03466	−0.200599
$906$	0	0
$907$	32.8025	1.08919	0.544594	−	0.838700i	$-0.316684\pi$
$907$	32.8025	1.08919	0.544594	+	0.838700i	$0.316684\pi$
$908$	0	0
$909$	−62.5854	−2.07583
$910$	0	0
$911$	6.18799	0.205017	0.102509	−	0.994732i	$-0.467313\pi$
$911$	6.18799	0.205017	0.102509	+	0.994732i	$0.467313\pi$
$912$	0	0
$913$	−21.1191	−0.698941
$914$	0	0
$915$	−16.7722	−0.554472
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−47.7929	−1.57654	−0.788271	−	0.615328i	$-0.789024\pi$
$919$	−47.7929	−1.57654	−0.788271	+	0.615328i	$0.789024\pi$
$920$	0	0
$921$	−6.19241	−0.204047
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−2.84667	−0.0935980
$926$	0	0
$927$	26.1489	0.858843
$928$	0	0
$929$	−39.8618	−1.30782	−0.653912	−	0.756571i	$-0.726873\pi$
$929$	−39.8618	−1.30782	−0.653912	+	0.756571i	$0.726873\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−14.6347	−0.479119
$934$	0	0
$935$	3.35644	0.109767
$936$	0	0
$937$	0.406229	0.0132709	0.00663546	−	0.999978i	$-0.497888\pi$
$937$	0.406229	0.0132709	0.00663546	+	0.999978i	$0.497888\pi$
$938$	0	0
$939$	76.8125	2.50668
$940$	0	0
$941$	33.6543	1.09710	0.548549	−	0.836119i	$-0.315180\pi$
$941$	33.6543	1.09710	0.548549	+	0.836119i	$0.315180\pi$
$942$	0	0
$943$	−21.5496	−0.701750
$944$	0	0
$945$	0	0
$946$	0	0
$947$	5.84110	0.189810	0.0949050	−	0.995486i	$-0.469745\pi$
$947$	5.84110	0.189810	0.0949050	+	0.995486i	$0.469745\pi$
$948$	0	0
$949$	56.6738	1.83971
$950$	0	0
$951$	49.2289	1.59636
$952$	0	0
$953$	32.0605	1.03854	0.519271	−	0.854610i	$-0.326204\pi$
$953$	32.0605	1.03854	0.519271	+	0.854610i	$0.326204\pi$
$954$	0	0
$955$	2.83155	0.0916268
$956$	0	0
$957$	14.4062	0.465687
$958$	0	0
$959$	0	0
$960$	0	0
$961$	53.0605	1.71163
$962$	0	0
$963$	−6.44673	−0.207743
$964$	0	0
$965$	12.9805	0.417856
$966$	0	0
$967$	24.0896	0.774669	0.387334	−	0.921939i	$-0.373396\pi$
$967$	24.0896	0.774669	0.387334	+	0.921939i	$0.373396\pi$
$968$	0	0
$969$	35.3867	1.13678
$970$	0	0
$971$	18.4707	0.592753	0.296376	−	0.955071i	$-0.404222\pi$
$971$	18.4707	0.592753	0.296376	+	0.955071i	$0.404222\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−12.5249	−0.401118
$976$	0	0
$977$	24.3369	0.778606	0.389303	−	0.921110i	$-0.372716\pi$
$977$	24.3369	0.778606	0.389303	+	0.921110i	$0.372716\pi$
$978$	0	0
$979$	−9.81201	−0.313593
$980$	0	0
$981$	71.8725	2.29471
$982$	0	0
$983$	8.66868	0.276488	0.138244	−	0.990398i	$-0.455854\pi$
$983$	8.66868	0.276488	0.138244	+	0.990398i	$0.455854\pi$
$984$	0	0
$985$	−4.84667	−0.154428
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−14.8965	−0.473680
$990$	0	0
$991$	−55.5054	−1.76319	−0.881593	−	0.472011i	$-0.843528\pi$
$991$	−55.5054	−1.76319	−0.881593	+	0.472011i	$0.843528\pi$
$992$	0	0
$993$	6.00000	0.190404
$994$	0	0
$995$	−15.3867	−0.487791
$996$	0	0
$997$	−35.1191	−1.11223	−0.556117	−	0.831104i	$-0.687709\pi$
$997$	−35.1191	−1.11223	−0.556117	+	0.831104i	$0.687709\pi$
$998$	0	0
$999$	−4.98929	−0.157854

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1960.2.a.w.1.3		3
4.3	odd	2		3920.2.a.cc.1.1		3
5.4	even	2		9800.2.a.ce.1.1		3
7.2	even	3		280.2.q.e.81.1	✓	6
7.3	odd	6		1960.2.q.w.961.3		6
7.4	even	3		280.2.q.e.121.1	yes	6
7.5	odd	6		1960.2.q.w.361.3		6
7.6	odd	2		1960.2.a.v.1.1		3
21.2	odd	6		2520.2.bi.q.361.1		6
21.11	odd	6		2520.2.bi.q.1801.1		6
28.11	odd	6		560.2.q.l.401.3		6
28.23	odd	6		560.2.q.l.81.3		6
28.27	even	2		3920.2.a.cb.1.3		3
35.2	odd	12		1400.2.bh.i.249.5		12
35.4	even	6		1400.2.q.j.401.3		6
35.9	even	6		1400.2.q.j.1201.3		6
35.18	odd	12		1400.2.bh.i.849.5		12
35.23	odd	12		1400.2.bh.i.249.2		12
35.32	odd	12		1400.2.bh.i.849.2		12
35.34	odd	2		9800.2.a.cf.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.2.q.e.81.1	✓	6	7.2	even	3
280.2.q.e.121.1	yes	6	7.4	even	3
560.2.q.l.81.3		6	28.23	odd	6
560.2.q.l.401.3		6	28.11	odd	6
1400.2.q.j.401.3		6	35.4	even	6
1400.2.q.j.1201.3		6	35.9	even	6
1400.2.bh.i.249.2		12	35.23	odd	12
1400.2.bh.i.249.5		12	35.2	odd	12
1400.2.bh.i.849.2		12	35.32	odd	12
1400.2.bh.i.849.5		12	35.18	odd	12
1960.2.a.v.1.1		3	7.6	odd	2
1960.2.a.w.1.3		3	1.1	even	1	trivial
1960.2.q.w.361.3		6	7.5	odd	6
1960.2.q.w.961.3		6	7.3	odd	6
2520.2.bi.q.361.1		6	21.2	odd	6
2520.2.bi.q.1801.1		6	21.11	odd	6
3920.2.a.cb.1.3		3	28.27	even	2
3920.2.a.cc.1.1		3	4.3	odd	2
9800.2.a.ce.1.1		3	5.4	even	2
9800.2.a.cf.1.3		3	35.34	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:	\( N \)	\(=\)	\( 1960 = 2^{3} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1960.a (trivial)

\(f(q)\)	\(=\)	\(q+2.58423 q^{3} +1.00000 q^{5} +3.67822 q^{9} +O(q^{10})\)	\(q+2.58423 q^{3} +1.00000 q^{5} +3.67822 q^{9} +1.67822 q^{11} -4.84667 q^{13} +2.58423 q^{15} +2.00000 q^{17} +6.84667 q^{19} -2.26245 q^{23} +1.00000 q^{25} +1.75268 q^{27} +3.32178 q^{29} +9.16845 q^{31} +4.33690 q^{33} -2.84667 q^{37} -12.5249 q^{39} +9.52489 q^{41} +6.58423 q^{43} +3.67822 q^{45} -12.2031 q^{47} +5.16845 q^{51} +7.49023 q^{53} +1.67822 q^{55} +17.6933 q^{57} +8.00000 q^{59} -6.49023 q^{61} -4.84667 q^{65} -5.75268 q^{67} -5.84667 q^{69} -11.6933 q^{73} +2.58423 q^{75} +5.69334 q^{79} -6.50535 q^{81} -12.5842 q^{83} +2.00000 q^{85} +8.58423 q^{87} -5.84667 q^{89} +23.6933 q^{93} +6.84667 q^{95} -2.00000 q^{97} +6.17287 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{5} + 9 q^{9}+O(q^{10}) \) 3 * q + 3 * q^5 + 9 * q^9	\( 3 q + 3 q^{5} + 9 q^{9} + 3 q^{11} + 3 q^{13} + 6 q^{17} + 3 q^{19} + 3 q^{23} + 3 q^{25} - 18 q^{27} + 12 q^{29} + 12 q^{31} - 18 q^{33} + 9 q^{37} - 18 q^{39} + 9 q^{41} + 12 q^{43} + 9 q^{45} - 15 q^{47} + 9 q^{53} + 3 q^{55} + 18 q^{57} + 24 q^{59} - 6 q^{61} + 3 q^{65} + 6 q^{67} - 18 q^{79} + 27 q^{81} - 30 q^{83} + 6 q^{85} + 18 q^{87} + 36 q^{93} + 3 q^{95} - 6 q^{97} + 63 q^{99}+O(q^{100}) \) 3 * q + 3 * q^5 + 9 * q^9 + 3 * q^11 + 3 * q^13 + 6 * q^17 + 3 * q^19 + 3 * q^23 + 3 * q^25 - 18 * q^27 + 12 * q^29 + 12 * q^31 - 18 * q^33 + 9 * q^37 - 18 * q^39 + 9 * q^41 + 12 * q^43 + 9 * q^45 - 15 * q^47 + 9 * q^53 + 3 * q^55 + 18 * q^57 + 24 * q^59 - 6 * q^61 + 3 * q^65 + 6 * q^67 - 18 * q^79 + 27 * q^81 - 30 * q^83 + 6 * q^85 + 18 * q^87 + 36 * q^93 + 3 * q^95 - 6 * q^97 + 63 * q^99

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