LMFDB - Embedded newform 2960.2.a.y.1.5

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:	$5$
Coefficient field:	5.5.6397264.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 10x^{3} - 2x^{2} + 14x - 4 $ x^5 - 10x^3 - 2x^2 + 14*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1480)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$3.04803$ 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)

$f(q)$	$=$	$q+3.04803 q^{3} -1.00000 q^{5} -4.39187 q^{7} +6.29050 q^{9} +O(q^{10})$	$q+3.04803 q^{3} -1.00000 q^{5} -4.39187 q^{7} +6.29050 q^{9} -2.58965 q^{11} -1.49324 q^{13} -3.04803 q^{15} +4.70085 q^{17} +6.84529 q^{19} -13.3866 q^{21} +4.09606 q^{23} +1.00000 q^{25} +10.0295 q^{27} +8.88845 q^{29} -0.295807 q^{31} -7.89332 q^{33} +4.39187 q^{35} +1.00000 q^{37} -4.55145 q^{39} -1.27732 q^{41} +2.09803 q^{43} -6.29050 q^{45} +10.3899 q^{47} +12.2885 q^{49} +14.3284 q^{51} -2.68571 q^{53} +2.58965 q^{55} +20.8647 q^{57} -4.49658 q^{59} +10.3869 q^{61} -27.6271 q^{63} +1.49324 q^{65} -4.84529 q^{67} +12.4849 q^{69} -15.8933 q^{71} +0.202743 q^{73} +3.04803 q^{75} +11.3734 q^{77} -15.0776 q^{79} +11.6989 q^{81} +14.9414 q^{83} -4.70085 q^{85} +27.0923 q^{87} +12.7140 q^{89} +6.55813 q^{91} -0.901629 q^{93} -6.84529 q^{95} +5.00487 q^{97} -16.2902 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{5} - 3 q^{7} + 5 q^{9}+O(q^{10}) $ 5 * q - 5 * q^5 - 3 * q^7 + 5 * q^9	$ 5 q - 5 q^{5} - 3 q^{7} + 5 q^{9} - 3 q^{11} + 4 q^{13} + 7 q^{17} + 4 q^{19} - 10 q^{21} - 10 q^{23} + 5 q^{25} + 6 q^{27} + 19 q^{29} - 13 q^{31} + 6 q^{33} + 3 q^{35} + 5 q^{37} - 14 q^{39} + 11 q^{41} + 13 q^{43} - 5 q^{45} + 2 q^{49} + 12 q^{51} + 27 q^{53} + 3 q^{55} + 12 q^{57} - 16 q^{59} + 27 q^{61} - 37 q^{63} - 4 q^{65} + 6 q^{67} + 40 q^{69} - 34 q^{71} + 16 q^{73} + 9 q^{77} - 16 q^{79} + 9 q^{81} + 14 q^{83} - 7 q^{85} + 42 q^{87} + 38 q^{89} + 34 q^{91} + 30 q^{93} - 4 q^{95} + 5 q^{97} - 23 q^{99}+O(q^{100}) $ 5 * q - 5 * q^5 - 3 * q^7 + 5 * q^9 - 3 * q^11 + 4 * q^13 + 7 * q^17 + 4 * q^19 - 10 * q^21 - 10 * q^23 + 5 * q^25 + 6 * q^27 + 19 * q^29 - 13 * q^31 + 6 * q^33 + 3 * q^35 + 5 * q^37 - 14 * q^39 + 11 * q^41 + 13 * q^43 - 5 * q^45 + 2 * q^49 + 12 * q^51 + 27 * q^53 + 3 * q^55 + 12 * q^57 - 16 * q^59 + 27 * q^61 - 37 * q^63 - 4 * q^65 + 6 * q^67 + 40 * q^69 - 34 * q^71 + 16 * q^73 + 9 * q^77 - 16 * q^79 + 9 * q^81 + 14 * q^83 - 7 * q^85 + 42 * q^87 + 38 * q^89 + 34 * q^91 + 30 * q^93 - 4 * q^95 + 5 * q^97 - 23 * 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
$3$	3.04803	1.75978	0.879891	−	0.475175i	$-0.157615\pi$
$3$	3.04803	1.75978	0.879891	+	0.475175i	$0.157615\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−4.39187	−1.65997	−0.829986	−	0.557785i	$-0.811651\pi$
$7$	−4.39187	−1.65997	−0.829986	+	0.557785i	$0.811651\pi$
$8$	0	0
$9$	6.29050	2.09683
$10$	0	0
$11$	−2.58965	−0.780807	−0.390404	−	0.920644i	$-0.627665\pi$
$11$	−2.58965	−0.780807	−0.390404	+	0.920644i	$0.627665\pi$
$12$	0	0
$13$	−1.49324	−0.414151	−0.207075	−	0.978325i	$-0.566395\pi$
$13$	−1.49324	−0.414151	−0.207075	+	0.978325i	$0.566395\pi$
$14$	0	0
$15$	−3.04803	−0.786998
$16$	0	0
$17$	4.70085	1.14012	0.570062	−	0.821601i	$-0.306919\pi$
$17$	4.70085	1.14012	0.570062	+	0.821601i	$0.306919\pi$
$18$	0	0
$19$	6.84529	1.57042	0.785209	−	0.619231i	$-0.212556\pi$
$19$	6.84529	1.57042	0.785209	+	0.619231i	$0.212556\pi$
$20$	0	0
$21$	−13.3866	−2.92119
$22$	0	0
$23$	4.09606	0.854088	0.427044	−	0.904231i	$-0.359555\pi$
$23$	4.09606	0.854088	0.427044	+	0.904231i	$0.359555\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	10.0295	1.93019
$28$	0	0
$29$	8.88845	1.65054	0.825272	−	0.564735i	$-0.191022\pi$
$29$	8.88845	1.65054	0.825272	+	0.564735i	$0.191022\pi$
$30$	0	0
$31$	−0.295807	−0.0531285	−0.0265642	−	0.999647i	$-0.508457\pi$
$31$	−0.295807	−0.0531285	−0.0265642	+	0.999647i	$0.508457\pi$
$32$	0	0
$33$	−7.89332	−1.37405
$34$	0	0
$35$	4.39187	0.742362
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	0	0
$39$	−4.55145	−0.728815
$40$	0	0
$41$	−1.27732	−0.199484	−0.0997422	−	0.995013i	$-0.531802\pi$
$41$	−1.27732	−0.199484	−0.0997422	+	0.995013i	$0.531802\pi$
$42$	0	0
$43$	2.09803	0.319947	0.159974	−	0.987121i	$-0.448859\pi$
$43$	2.09803	0.319947	0.159974	+	0.987121i	$0.448859\pi$
$44$	0	0
$45$	−6.29050	−0.937732
$46$	0	0
$47$	10.3899	1.51552	0.757762	−	0.652532i	$-0.226293\pi$
$47$	10.3899	1.51552	0.757762	+	0.652532i	$0.226293\pi$
$48$	0	0
$49$	12.2885	1.75550
$50$	0	0
$51$	14.3284	2.00637
$52$	0	0
$53$	−2.68571	−0.368911	−0.184455	−	0.982841i	$-0.559052\pi$
$53$	−2.68571	−0.368911	−0.184455	+	0.982841i	$0.559052\pi$
$54$	0	0
$55$	2.58965	0.349188
$56$	0	0
$57$	20.8647	2.76359
$58$	0	0
$59$	−4.49658	−0.585405	−0.292703	−	0.956204i	$-0.594555\pi$
$59$	−4.49658	−0.585405	−0.292703	+	0.956204i	$0.594555\pi$
$60$	0	0
$61$	10.3869	1.32991	0.664953	−	0.746885i	$-0.268451\pi$
$61$	10.3869	1.32991	0.664953	+	0.746885i	$0.268451\pi$
$62$	0	0
$63$	−27.6271	−3.48068
$64$	0	0
$65$	1.49324	0.185214
$66$	0	0
$67$	−4.84529	−0.591946	−0.295973	−	0.955196i	$-0.595644\pi$
$67$	−4.84529	−0.591946	−0.295973	+	0.955196i	$0.595644\pi$
$68$	0	0
$69$	12.4849	1.50301
$70$	0	0
$71$	−15.8933	−1.88619	−0.943095	−	0.332523i	$-0.892100\pi$
$71$	−15.8933	−1.88619	−0.943095	+	0.332523i	$0.892100\pi$
$72$	0	0
$73$	0.202743	0.0237293	0.0118646	−	0.999930i	$-0.496223\pi$
$73$	0.202743	0.0237293	0.0118646	+	0.999930i	$0.496223\pi$
$74$	0	0
$75$	3.04803	0.351956
$76$	0	0
$77$	11.3734	1.29612
$78$	0	0
$79$	−15.0776	−1.69636	−0.848180	−	0.529708i	$-0.822302\pi$
$79$	−15.0776	−1.69636	−0.848180	+	0.529708i	$0.822302\pi$
$80$	0	0
$81$	11.6989	1.29988
$82$	0	0
$83$	14.9414	1.64003	0.820013	−	0.572344i	$-0.193966\pi$
$83$	14.9414	1.64003	0.820013	+	0.572344i	$0.193966\pi$
$84$	0	0
$85$	−4.70085	−0.509879
$86$	0	0
$87$	27.0923	2.90460
$88$	0	0
$89$	12.7140	1.34768	0.673842	−	0.738875i	$-0.264643\pi$
$89$	12.7140	1.34768	0.673842	+	0.738875i	$0.264643\pi$
$90$	0	0
$91$	6.55813	0.687479
$92$	0	0
$93$	−0.901629	−0.0934945
$94$	0	0
$95$	−6.84529	−0.702312
$96$	0	0
$97$	5.00487	0.508167	0.254084	−	0.967182i	$-0.418226\pi$
$97$	5.00487	0.508167	0.254084	+	0.967182i	$0.418226\pi$
$98$	0	0
$99$	−16.2902	−1.63722
$100$	0	0
$101$	8.90163	0.885745	0.442873	−	0.896585i	$-0.353959\pi$
$101$	8.90163	0.885745	0.442873	+	0.896585i	$0.353959\pi$
$102$	0	0
$103$	10.9068	1.07468	0.537341	−	0.843365i	$-0.319429\pi$
$103$	10.9068	1.07468	0.537341	+	0.843365i	$0.319429\pi$
$104$	0	0
$105$	13.3866	1.30639
$106$	0	0
$107$	−12.1872	−1.17818	−0.589089	−	0.808068i	$-0.700513\pi$
$107$	−12.1872	−1.17818	−0.589089	+	0.808068i	$0.700513\pi$
$108$	0	0
$109$	−12.0611	−1.15524	−0.577620	−	0.816306i	$-0.696019\pi$
$109$	−12.0611	−1.15524	−0.577620	+	0.816306i	$0.696019\pi$
$110$	0	0
$111$	3.04803	0.289306
$112$	0	0
$113$	14.3750	1.35229	0.676144	−	0.736770i	$-0.263650\pi$
$113$	14.3750	1.35229	0.676144	+	0.736770i	$0.263650\pi$
$114$	0	0
$115$	−4.09606	−0.381960
$116$	0	0
$117$	−9.39324	−0.868405
$118$	0	0
$119$	−20.6455	−1.89257
$120$	0	0
$121$	−4.29374	−0.390340
$122$	0	0
$123$	−3.89332	−0.351049
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	4.25761	0.377802	0.188901	−	0.981996i	$-0.439507\pi$
$127$	4.25761	0.377802	0.188901	+	0.981996i	$0.439507\pi$
$128$	0	0
$129$	6.39487	0.563037
$130$	0	0
$131$	−12.5195	−1.09383	−0.546915	−	0.837188i	$-0.684198\pi$
$131$	−12.5195	−1.09383	−0.546915	+	0.837188i	$0.684198\pi$
$132$	0	0
$133$	−30.0636	−2.60685
$134$	0	0
$135$	−10.0295	−0.863206
$136$	0	0
$137$	−3.79726	−0.324422	−0.162211	−	0.986756i	$-0.551862\pi$
$137$	−3.79726	−0.324422	−0.162211	+	0.986756i	$0.551862\pi$
$138$	0	0
$139$	1.67287	0.141891	0.0709455	−	0.997480i	$-0.477398\pi$
$139$	1.67287	0.141891	0.0709455	+	0.997480i	$0.477398\pi$
$140$	0	0
$141$	31.6688	2.66699
$142$	0	0
$143$	3.86697	0.323372
$144$	0	0
$145$	−8.88845	−0.738146
$146$	0	0
$147$	37.4558	3.08931
$148$	0	0
$149$	2.18760	0.179215	0.0896075	−	0.995977i	$-0.471439\pi$
$149$	2.18760	0.179215	0.0896075	+	0.995977i	$0.471439\pi$
$150$	0	0
$151$	8.48493	0.690494	0.345247	−	0.938512i	$-0.387795\pi$
$151$	8.48493	0.690494	0.345247	+	0.938512i	$0.387795\pi$
$152$	0	0
$153$	29.5707	2.39065
$154$	0	0
$155$	0.295807	0.0237598
$156$	0	0
$157$	10.0941	0.805597	0.402798	−	0.915289i	$-0.368038\pi$
$157$	10.0941	0.805597	0.402798	+	0.915289i	$0.368038\pi$
$158$	0	0
$159$	−8.18613	−0.649202
$160$	0	0
$161$	−17.9894	−1.41776
$162$	0	0
$163$	−6.47629	−0.507262	−0.253631	−	0.967301i	$-0.581625\pi$
$163$	−6.47629	−0.507262	−0.253631	+	0.967301i	$0.581625\pi$
$164$	0	0
$165$	7.89332	0.614494
$166$	0	0
$167$	7.67416	0.593845	0.296922	−	0.954902i	$-0.404040\pi$
$167$	7.67416	0.593845	0.296922	+	0.954902i	$0.404040\pi$
$168$	0	0
$169$	−10.7702	−0.828479
$170$	0	0
$171$	43.0603	3.29290
$172$	0	0
$173$	20.2105	1.53658	0.768288	−	0.640104i	$-0.221109\pi$
$173$	20.2105	1.53658	0.768288	+	0.640104i	$0.221109\pi$
$174$	0	0
$175$	−4.39187	−0.331994
$176$	0	0
$177$	−13.7057	−1.03019
$178$	0	0
$179$	−20.6222	−1.54138	−0.770688	−	0.637213i	$-0.780087\pi$
$179$	−20.6222	−1.54138	−0.770688	+	0.637213i	$0.780087\pi$
$180$	0	0
$181$	−4.39340	−0.326559	−0.163279	−	0.986580i	$-0.552207\pi$
$181$	−4.39340	−0.326559	−0.163279	+	0.986580i	$0.552207\pi$
$182$	0	0
$183$	31.6596	2.34035
$184$	0	0
$185$	−1.00000	−0.0735215
$186$	0	0
$187$	−12.1735	−0.890218
$188$	0	0
$189$	−44.0485	−3.20406
$190$	0	0
$191$	6.61429	0.478593	0.239297	−	0.970947i	$-0.423083\pi$
$191$	6.61429	0.478593	0.239297	+	0.970947i	$0.423083\pi$
$192$	0	0
$193$	12.3586	0.889590	0.444795	−	0.895632i	$-0.353277\pi$
$193$	12.3586	0.889590	0.444795	+	0.895632i	$0.353277\pi$
$194$	0	0
$195$	4.55145	0.325936
$196$	0	0
$197$	5.19897	0.370411	0.185205	−	0.982700i	$-0.440705\pi$
$197$	5.19897	0.370411	0.185205	+	0.982700i	$0.440705\pi$
$198$	0	0
$199$	22.4825	1.59374	0.796871	−	0.604150i	$-0.206487\pi$
$199$	22.4825	1.59374	0.796871	+	0.604150i	$0.206487\pi$
$200$	0	0
$201$	−14.7686	−1.04170
$202$	0	0
$203$	−39.0369	−2.73986
$204$	0	0
$205$	1.27732	0.0892121
$206$	0	0
$207$	25.7663	1.79088
$208$	0	0
$209$	−17.7269	−1.22619
$210$	0	0
$211$	9.69922	0.667722	0.333861	−	0.942622i	$-0.391648\pi$
$211$	9.69922	0.667722	0.333861	+	0.942622i	$0.391648\pi$
$212$	0	0
$213$	−48.4434	−3.31928
$214$	0	0
$215$	−2.09803	−0.143085
$216$	0	0
$217$	1.29915	0.0881917
$218$	0	0
$219$	0.617967	0.0417583
$220$	0	0
$221$	−7.01952	−0.472184
$222$	0	0
$223$	−23.5204	−1.57504	−0.787521	−	0.616288i	$-0.788636\pi$
$223$	−23.5204	−1.57504	−0.787521	+	0.616288i	$0.788636\pi$
$224$	0	0
$225$	6.29050	0.419367
$226$	0	0
$227$	−19.0475	−1.26423	−0.632115	−	0.774874i	$-0.717813\pi$
$227$	−19.0475	−1.26423	−0.632115	+	0.774874i	$0.717813\pi$
$228$	0	0
$229$	−28.3480	−1.87329	−0.936643	−	0.350285i	$-0.886085\pi$
$229$	−28.3480	−1.87329	−0.936643	+	0.350285i	$0.886085\pi$
$230$	0	0
$231$	34.6664	2.28088
$232$	0	0
$233$	−24.4540	−1.60203	−0.801016	−	0.598642i	$-0.795707\pi$
$233$	−24.4540	−1.60203	−0.801016	+	0.598642i	$0.795707\pi$
$234$	0	0
$235$	−10.3899	−0.677762
$236$	0	0
$237$	−45.9569	−2.98522
$238$	0	0
$239$	−22.2286	−1.43785	−0.718925	−	0.695088i	$-0.755365\pi$
$239$	−22.2286	−1.43785	−0.718925	+	0.695088i	$0.755365\pi$
$240$	0	0
$241$	15.5839	1.00385	0.501924	−	0.864912i	$-0.332626\pi$
$241$	15.5839	1.00385	0.501924	+	0.864912i	$0.332626\pi$
$242$	0	0
$243$	5.56993	0.357311
$244$	0	0
$245$	−12.2885	−0.785085
$246$	0	0
$247$	−10.2217	−0.650390
$248$	0	0
$249$	45.5417	2.88609
$250$	0	0
$251$	−5.55932	−0.350901	−0.175451	−	0.984488i	$-0.556138\pi$
$251$	−5.55932	−0.350901	−0.175451	+	0.984488i	$0.556138\pi$
$252$	0	0
$253$	−10.6074	−0.666879
$254$	0	0
$255$	−14.3284	−0.897276
$256$	0	0
$257$	0.283662	0.0176943	0.00884717	−	0.999961i	$-0.497184\pi$
$257$	0.283662	0.0176943	0.00884717	+	0.999961i	$0.497184\pi$
$258$	0	0
$259$	−4.39187	−0.272898
$260$	0	0
$261$	55.9128	3.46092
$262$	0	0
$263$	−13.4779	−0.831083	−0.415542	−	0.909574i	$-0.636408\pi$
$263$	−13.4779	−0.831083	−0.415542	+	0.909574i	$0.636408\pi$
$264$	0	0
$265$	2.68571	0.164982
$266$	0	0
$267$	38.7528	2.37163
$268$	0	0
$269$	16.8883	1.02970	0.514848	−	0.857281i	$-0.327848\pi$
$269$	16.8883	1.02970	0.514848	+	0.857281i	$0.327848\pi$
$270$	0	0
$271$	−5.80719	−0.352762	−0.176381	−	0.984322i	$-0.556439\pi$
$271$	−5.80719	−0.352762	−0.176381	+	0.984322i	$0.556439\pi$
$272$	0	0
$273$	19.9894	1.20981
$274$	0	0
$275$	−2.58965	−0.156161
$276$	0	0
$277$	−15.9546	−0.958617	−0.479308	−	0.877647i	$-0.659112\pi$
$277$	−15.9546	−0.958617	−0.479308	+	0.877647i	$0.659112\pi$
$278$	0	0
$279$	−1.86077	−0.111402
$280$	0	0
$281$	−5.77023	−0.344223	−0.172111	−	0.985077i	$-0.555059\pi$
$281$	−5.77023	−0.344223	−0.172111	+	0.985077i	$0.555059\pi$
$282$	0	0
$283$	−22.5803	−1.34226	−0.671130	−	0.741339i	$-0.734191\pi$
$283$	−22.5803	−1.34226	−0.671130	+	0.741339i	$0.734191\pi$
$284$	0	0
$285$	−20.8647	−1.23592
$286$	0	0
$287$	5.60984	0.331138
$288$	0	0
$289$	5.09803	0.299884
$290$	0	0
$291$	15.2550	0.894264
$292$	0	0
$293$	14.1908	0.829038	0.414519	−	0.910041i	$-0.363950\pi$
$293$	14.1908	0.829038	0.414519	+	0.910041i	$0.363950\pi$
$294$	0	0
$295$	4.49658	0.261801
$296$	0	0
$297$	−25.9730	−1.50710
$298$	0	0
$299$	−6.11642	−0.353722
$300$	0	0
$301$	−9.21429	−0.531103
$302$	0	0
$303$	27.1325	1.55872
$304$	0	0
$305$	−10.3869	−0.594752
$306$	0	0
$307$	29.6752	1.69365	0.846826	−	0.531869i	$-0.178510\pi$
$307$	29.6752	1.69365	0.846826	+	0.531869i	$0.178510\pi$
$308$	0	0
$309$	33.2444	1.89121
$310$	0	0
$311$	−30.1522	−1.70977	−0.854886	−	0.518816i	$-0.826373\pi$
$311$	−30.1522	−1.70977	−0.854886	+	0.518816i	$0.826373\pi$
$312$	0	0
$313$	−10.1224	−0.572153	−0.286076	−	0.958207i	$-0.592351\pi$
$313$	−10.1224	−0.572153	−0.286076	+	0.958207i	$0.592351\pi$
$314$	0	0
$315$	27.6271	1.55661
$316$	0	0
$317$	−23.3931	−1.31389	−0.656943	−	0.753941i	$-0.728151\pi$
$317$	−23.3931	−1.31389	−0.656943	+	0.753941i	$0.728151\pi$
$318$	0	0
$319$	−23.0179	−1.28876
$320$	0	0
$321$	−37.1469	−2.07334
$322$	0	0
$323$	32.1787	1.79047
$324$	0	0
$325$	−1.49324	−0.0828302
$326$	0	0
$327$	−36.7625	−2.03297
$328$	0	0
$329$	−45.6311	−2.51572
$330$	0	0
$331$	−14.4195	−0.792565	−0.396282	−	0.918129i	$-0.629700\pi$
$331$	−14.4195	−0.792565	−0.396282	+	0.918129i	$0.629700\pi$
$332$	0	0
$333$	6.29050	0.344717
$334$	0	0
$335$	4.84529	0.264726
$336$	0	0
$337$	21.6899	1.18152	0.590762	−	0.806846i	$-0.298827\pi$
$337$	21.6899	1.18152	0.590762	+	0.806846i	$0.298827\pi$
$338$	0	0
$339$	43.8155	2.37973
$340$	0	0
$341$	0.766035	0.0414831
$342$	0	0
$343$	−23.2265	−1.25412
$344$	0	0
$345$	−12.4849	−0.672166
$346$	0	0
$347$	−12.1725	−0.653451	−0.326726	−	0.945119i	$-0.605945\pi$
$347$	−12.1725	−0.653451	−0.326726	+	0.945119i	$0.605945\pi$
$348$	0	0
$349$	17.8715	0.956639	0.478320	−	0.878186i	$-0.341246\pi$
$349$	17.8715	0.956639	0.478320	+	0.878186i	$0.341246\pi$
$350$	0	0
$351$	−14.9765	−0.799389
$352$	0	0
$353$	0.300776	0.0160087	0.00800434	−	0.999968i	$-0.497452\pi$
$353$	0.300776	0.0160087	0.00800434	+	0.999968i	$0.497452\pi$
$354$	0	0
$355$	15.8933	0.843530
$356$	0	0
$357$	−62.9283	−3.33052
$358$	0	0
$359$	20.8692	1.10143	0.550717	−	0.834692i	$-0.314354\pi$
$359$	20.8692	1.10143	0.550717	+	0.834692i	$0.314354\pi$
$360$	0	0
$361$	27.8580	1.46621
$362$	0	0
$363$	−13.0875	−0.686913
$364$	0	0
$365$	−0.202743	−0.0106120
$366$	0	0
$367$	31.6802	1.65369	0.826845	−	0.562429i	$-0.190133\pi$
$367$	31.6802	1.65369	0.826845	+	0.562429i	$0.190133\pi$
$368$	0	0
$369$	−8.03500	−0.418285
$370$	0	0
$371$	11.7953	0.612381
$372$	0	0
$373$	1.51507	0.0784471	0.0392235	−	0.999230i	$-0.487512\pi$
$373$	1.51507	0.0784471	0.0392235	+	0.999230i	$0.487512\pi$
$374$	0	0
$375$	−3.04803	−0.157400
$376$	0	0
$377$	−13.2726	−0.683574
$378$	0	0
$379$	24.7665	1.27217	0.636084	−	0.771620i	$-0.280553\pi$
$379$	24.7665	1.27217	0.636084	+	0.771620i	$0.280553\pi$
$380$	0	0
$381$	12.9773	0.664849
$382$	0	0
$383$	−12.1224	−0.619427	−0.309713	−	0.950830i	$-0.600233\pi$
$383$	−12.1224	−0.619427	−0.309713	+	0.950830i	$0.600233\pi$
$384$	0	0
$385$	−11.3734	−0.579641
$386$	0	0
$387$	13.1977	0.670875
$388$	0	0
$389$	18.4724	0.936585	0.468293	−	0.883573i	$-0.344869\pi$
$389$	18.4724	0.936585	0.468293	+	0.883573i	$0.344869\pi$
$390$	0	0
$391$	19.2550	0.973767
$392$	0	0
$393$	−38.1597	−1.92490
$394$	0	0
$395$	15.0776	0.758635
$396$	0	0
$397$	0.714031	0.0358362	0.0179181	−	0.999839i	$-0.494296\pi$
$397$	0.714031	0.0358362	0.0179181	+	0.999839i	$0.494296\pi$
$398$	0	0
$399$	−91.6349	−4.58748
$400$	0	0
$401$	36.1979	1.80764	0.903819	−	0.427915i	$-0.140752\pi$
$401$	36.1979	1.80764	0.903819	+	0.427915i	$0.140752\pi$
$402$	0	0
$403$	0.441711	0.0220032
$404$	0	0
$405$	−11.6989	−0.581322
$406$	0	0
$407$	−2.58965	−0.128364
$408$	0	0
$409$	−19.5748	−0.967913	−0.483957	−	0.875092i	$-0.660801\pi$
$409$	−19.5748	−0.967913	−0.483957	+	0.875092i	$0.660801\pi$
$410$	0	0
$411$	−11.5742	−0.570911
$412$	0	0
$413$	19.7484	0.971755
$414$	0	0
$415$	−14.9414	−0.733442
$416$	0	0
$417$	5.09896	0.249697
$418$	0	0
$419$	−10.8435	−0.529740	−0.264870	−	0.964284i	$-0.585329\pi$
$419$	−10.8435	−0.529740	−0.264870	+	0.964284i	$0.585329\pi$
$420$	0	0
$421$	19.5978	0.955137	0.477569	−	0.878594i	$-0.341518\pi$
$421$	19.5978	0.955137	0.477569	+	0.878594i	$0.341518\pi$
$422$	0	0
$423$	65.3577	3.17780
$424$	0	0
$425$	4.70085	0.228025
$426$	0	0
$427$	−45.6179	−2.20761
$428$	0	0
$429$	11.7866	0.569064
$430$	0	0
$431$	−25.7199	−1.23889	−0.619443	−	0.785042i	$-0.712641\pi$
$431$	−25.7199	−1.23889	−0.619443	+	0.785042i	$0.712641\pi$
$432$	0	0
$433$	−27.8827	−1.33996	−0.669979	−	0.742380i	$-0.733697\pi$
$433$	−27.8827	−1.33996	−0.669979	+	0.742380i	$0.733697\pi$
$434$	0	0
$435$	−27.0923	−1.29898
$436$	0	0
$437$	28.0387	1.34127
$438$	0	0
$439$	31.4070	1.49898	0.749488	−	0.662018i	$-0.230300\pi$
$439$	31.4070	1.49898	0.749488	+	0.662018i	$0.230300\pi$
$440$	0	0
$441$	77.3010	3.68100
$442$	0	0
$443$	25.8653	1.22890	0.614448	−	0.788957i	$-0.289379\pi$
$443$	25.8653	1.22890	0.614448	+	0.788957i	$0.289379\pi$
$444$	0	0
$445$	−12.7140	−0.602703
$446$	0	0
$447$	6.66787	0.315379
$448$	0	0
$449$	−23.8827	−1.12709	−0.563547	−	0.826084i	$-0.690564\pi$
$449$	−23.8827	−1.12709	−0.563547	+	0.826084i	$0.690564\pi$
$450$	0	0
$451$	3.30781	0.155759
$452$	0	0
$453$	25.8624	1.21512
$454$	0	0
$455$	−6.55813	−0.307450
$456$	0	0
$457$	17.1970	0.804442	0.402221	−	0.915543i	$-0.368238\pi$
$457$	17.1970	0.804442	0.402221	+	0.915543i	$0.368238\pi$
$458$	0	0
$459$	47.1474	2.20065
$460$	0	0
$461$	−0.413455	−0.0192565	−0.00962826	−	0.999954i	$-0.503065\pi$
$461$	−0.413455	−0.0192565	−0.00962826	+	0.999954i	$0.503065\pi$
$462$	0	0
$463$	−15.8762	−0.737831	−0.368915	−	0.929463i	$-0.620271\pi$
$463$	−15.8762	−0.737831	−0.368915	+	0.929463i	$0.620271\pi$
$464$	0	0
$465$	0.901629	0.0418120
$466$	0	0
$467$	16.5693	0.766735	0.383368	−	0.923596i	$-0.374764\pi$
$467$	16.5693	0.766735	0.383368	+	0.923596i	$0.374764\pi$
$468$	0	0
$469$	21.2799	0.982614
$470$	0	0
$471$	30.7671	1.41767
$472$	0	0
$473$	−5.43316	−0.249817
$474$	0	0
$475$	6.84529	0.314083
$476$	0	0
$477$	−16.8945	−0.773544
$478$	0	0
$479$	14.7977	0.676124	0.338062	−	0.941124i	$-0.390229\pi$
$479$	14.7977	0.676124	0.338062	+	0.941124i	$0.390229\pi$
$480$	0	0
$481$	−1.49324	−0.0680860
$482$	0	0
$483$	−54.8322	−2.49495
$484$	0	0
$485$	−5.00487	−0.227259
$486$	0	0
$487$	−32.4043	−1.46838	−0.734190	−	0.678944i	$-0.762438\pi$
$487$	−32.4043	−1.46838	−0.734190	+	0.678944i	$0.762438\pi$
$488$	0	0
$489$	−19.7399	−0.892671
$490$	0	0
$491$	27.4705	1.23973	0.619864	−	0.784709i	$-0.287188\pi$
$491$	27.4705	1.23973	0.619864	+	0.784709i	$0.287188\pi$
$492$	0	0
$493$	41.7833	1.88183
$494$	0	0
$495$	16.2902	0.732188
$496$	0	0
$497$	69.8014	3.13102
$498$	0	0
$499$	−22.0114	−0.985368	−0.492684	−	0.870208i	$-0.663984\pi$
$499$	−22.0114	−0.985368	−0.492684	+	0.870208i	$0.663984\pi$
$500$	0	0
$501$	23.3911	1.04504
$502$	0	0
$503$	21.2844	0.949025	0.474513	−	0.880249i	$-0.342624\pi$
$503$	21.2844	0.949025	0.474513	+	0.880249i	$0.342624\pi$
$504$	0	0
$505$	−8.90163	−0.396117
$506$	0	0
$507$	−32.8280	−1.45794
$508$	0	0
$509$	0.620707	0.0275124	0.0137562	−	0.999905i	$-0.495621\pi$
$509$	0.620707	0.0275124	0.0137562	+	0.999905i	$0.495621\pi$
$510$	0	0
$511$	−0.890421	−0.0393899
$512$	0	0
$513$	68.6552	3.03120
$514$	0	0
$515$	−10.9068	−0.480613
$516$	0	0
$517$	−26.9062	−1.18333
$518$	0	0
$519$	61.6023	2.70404
$520$	0	0
$521$	−7.29179	−0.319459	−0.159730	−	0.987161i	$-0.551062\pi$
$521$	−7.29179	−0.319459	−0.159730	+	0.987161i	$0.551062\pi$
$522$	0	0
$523$	35.1456	1.53681	0.768405	−	0.639964i	$-0.221051\pi$
$523$	35.1456	1.53681	0.768405	+	0.639964i	$0.221051\pi$
$524$	0	0
$525$	−13.3866	−0.584238
$526$	0	0
$527$	−1.39054	−0.0605731
$528$	0	0
$529$	−6.22226	−0.270533
$530$	0	0
$531$	−28.2857	−1.22750
$532$	0	0
$533$	1.90735	0.0826166
$534$	0	0
$535$	12.1872	0.526897
$536$	0	0
$537$	−62.8571	−2.71248
$538$	0	0
$539$	−31.8229	−1.37071
$540$	0	0
$541$	−7.55755	−0.324924	−0.162462	−	0.986715i	$-0.551944\pi$
$541$	−7.55755	−0.324924	−0.162462	+	0.986715i	$0.551944\pi$
$542$	0	0
$543$	−13.3912	−0.574673
$544$	0	0
$545$	12.0611	0.516639
$546$	0	0
$547$	26.7652	1.14440	0.572198	−	0.820116i	$-0.306091\pi$
$547$	26.7652	1.14440	0.572198	+	0.820116i	$0.306091\pi$
$548$	0	0
$549$	65.3388	2.78859
$550$	0	0
$551$	60.8440	2.59204
$552$	0	0
$553$	66.2188	2.81591
$554$	0	0
$555$	−3.04803	−0.129382
$556$	0	0
$557$	22.9277	0.971478	0.485739	−	0.874104i	$-0.338551\pi$
$557$	22.9277	0.971478	0.485739	+	0.874104i	$0.338551\pi$
$558$	0	0
$559$	−3.13287	−0.132506
$560$	0	0
$561$	−37.1054	−1.56659
$562$	0	0
$563$	−10.9418	−0.461143	−0.230572	−	0.973055i	$-0.574060\pi$
$563$	−10.9418	−0.461143	−0.230572	+	0.973055i	$0.574060\pi$
$564$	0	0
$565$	−14.3750	−0.604762
$566$	0	0
$567$	−51.3800	−2.15776
$568$	0	0
$569$	−28.1185	−1.17879	−0.589394	−	0.807846i	$-0.700633\pi$
$569$	−28.1185	−1.17879	−0.589394	+	0.807846i	$0.700633\pi$
$570$	0	0
$571$	−6.32319	−0.264617	−0.132309	−	0.991209i	$-0.542239\pi$
$571$	−6.32319	−0.264617	−0.132309	+	0.991209i	$0.542239\pi$
$572$	0	0
$573$	20.1606	0.842220
$574$	0	0
$575$	4.09606	0.170818
$576$	0	0
$577$	32.0703	1.33510	0.667552	−	0.744563i	$-0.267342\pi$
$577$	32.0703	1.33510	0.667552	+	0.744563i	$0.267342\pi$
$578$	0	0
$579$	37.6693	1.56548
$580$	0	0
$581$	−65.6205	−2.72240
$582$	0	0
$583$	6.95503	0.288048
$584$	0	0
$585$	9.39324	0.388363
$586$	0	0
$587$	28.3326	1.16941	0.584707	−	0.811245i	$-0.301210\pi$
$587$	28.3326	1.16941	0.584707	+	0.811245i	$0.301210\pi$
$588$	0	0
$589$	−2.02488	−0.0834339
$590$	0	0
$591$	15.8466	0.651843
$592$	0	0
$593$	−32.7165	−1.34350	−0.671752	−	0.740776i	$-0.734458\pi$
$593$	−32.7165	−1.34350	−0.671752	+	0.740776i	$0.734458\pi$
$594$	0	0
$595$	20.6455	0.846385
$596$	0	0
$597$	68.5273	2.80464
$598$	0	0
$599$	−26.1855	−1.06991	−0.534954	−	0.844881i	$-0.679671\pi$
$599$	−26.1855	−1.06991	−0.534954	+	0.844881i	$0.679671\pi$
$600$	0	0
$601$	−14.1190	−0.575925	−0.287963	−	0.957642i	$-0.592978\pi$
$601$	−14.1190	−0.575925	−0.287963	+	0.957642i	$0.592978\pi$
$602$	0	0
$603$	−30.4793	−1.24121
$604$	0	0
$605$	4.29374	0.174565
$606$	0	0
$607$	9.84023	0.399402	0.199701	−	0.979857i	$-0.436003\pi$
$607$	9.84023	0.399402	0.199701	+	0.979857i	$0.436003\pi$
$608$	0	0
$609$	−118.986	−4.82155
$610$	0	0
$611$	−15.5146	−0.627655
$612$	0	0
$613$	−13.1510	−0.531163	−0.265581	−	0.964088i	$-0.585564\pi$
$613$	−13.1510	−0.531163	−0.265581	+	0.964088i	$0.585564\pi$
$614$	0	0
$615$	3.89332	0.156994
$616$	0	0
$617$	−2.85278	−0.114848	−0.0574242	−	0.998350i	$-0.518289\pi$
$617$	−2.85278	−0.114848	−0.0574242	+	0.998350i	$0.518289\pi$
$618$	0	0
$619$	−4.31051	−0.173254	−0.0866271	−	0.996241i	$-0.527609\pi$
$619$	−4.31051	−0.173254	−0.0866271	+	0.996241i	$0.527609\pi$
$620$	0	0
$621$	41.0817	1.64855
$622$	0	0
$623$	−55.8384	−2.23712
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−54.0321	−2.15783
$628$	0	0
$629$	4.70085	0.187435
$630$	0	0
$631$	25.5070	1.01542	0.507709	−	0.861528i	$-0.330492\pi$
$631$	25.5070	1.01542	0.507709	+	0.861528i	$0.330492\pi$
$632$	0	0
$633$	29.5635	1.17505
$634$	0	0
$635$	−4.25761	−0.168958
$636$	0	0
$637$	−18.3498	−0.727044
$638$	0	0
$639$	−99.9769	−3.95503
$640$	0	0
$641$	−38.5572	−1.52292	−0.761459	−	0.648213i	$-0.775517\pi$
$641$	−38.5572	−1.52292	−0.761459	+	0.648213i	$0.775517\pi$
$642$	0	0
$643$	−16.6626	−0.657109	−0.328555	−	0.944485i	$-0.606562\pi$
$643$	−16.6626	−0.657109	−0.328555	+	0.944485i	$0.606562\pi$
$644$	0	0
$645$	−6.39487	−0.251798
$646$	0	0
$647$	−20.6777	−0.812926	−0.406463	−	0.913667i	$-0.633238\pi$
$647$	−20.6777	−0.812926	−0.406463	+	0.913667i	$0.633238\pi$
$648$	0	0
$649$	11.6445	0.457089
$650$	0	0
$651$	3.95984	0.155198
$652$	0	0
$653$	−19.1754	−0.750390	−0.375195	−	0.926946i	$-0.622424\pi$
$653$	−19.1754	−0.750390	−0.375195	+	0.926946i	$0.622424\pi$
$654$	0	0
$655$	12.5195	0.489175
$656$	0	0
$657$	1.27535	0.0497563
$658$	0	0
$659$	18.2346	0.710320	0.355160	−	0.934806i	$-0.384426\pi$
$659$	18.2346	0.710320	0.355160	+	0.934806i	$0.384426\pi$
$660$	0	0
$661$	25.4602	0.990287	0.495143	−	0.868811i	$-0.335116\pi$
$661$	25.4602	0.990287	0.495143	+	0.868811i	$0.335116\pi$
$662$	0	0
$663$	−21.3957	−0.830940
$664$	0	0
$665$	30.0636	1.16582
$666$	0	0
$667$	36.4077	1.40971
$668$	0	0
$669$	−71.6909	−2.77173
$670$	0	0
$671$	−26.8984	−1.03840
$672$	0	0
$673$	7.10929	0.274043	0.137022	−	0.990568i	$-0.456247\pi$
$673$	7.10929	0.274043	0.137022	+	0.990568i	$0.456247\pi$
$674$	0	0
$675$	10.0295	0.386037
$676$	0	0
$677$	−12.1988	−0.468838	−0.234419	−	0.972136i	$-0.575319\pi$
$677$	−12.1988	−0.468838	−0.234419	+	0.972136i	$0.575319\pi$
$678$	0	0
$679$	−21.9807	−0.843543
$680$	0	0
$681$	−58.0575	−2.22477
$682$	0	0
$683$	3.70522	0.141776	0.0708882	−	0.997484i	$-0.477417\pi$
$683$	3.70522	0.141776	0.0708882	+	0.997484i	$0.477417\pi$
$684$	0	0
$685$	3.79726	0.145086
$686$	0	0
$687$	−86.4055	−3.29658
$688$	0	0
$689$	4.01042	0.152785
$690$	0	0
$691$	−27.0503	−1.02904	−0.514521	−	0.857478i	$-0.672030\pi$
$691$	−27.0503	−1.02904	−0.514521	+	0.857478i	$0.672030\pi$
$692$	0	0
$693$	71.5443	2.71774
$694$	0	0
$695$	−1.67287	−0.0634556
$696$	0	0
$697$	−6.00451	−0.227437
$698$	0	0
$699$	−74.5365	−2.81923
$700$	0	0
$701$	−39.4042	−1.48827	−0.744137	−	0.668027i	$-0.767139\pi$
$701$	−39.4042	−1.48827	−0.744137	+	0.668027i	$0.767139\pi$
$702$	0	0
$703$	6.84529	0.258175
$704$	0	0
$705$	−31.6688	−1.19271
$706$	0	0
$707$	−39.0948	−1.47031
$708$	0	0
$709$	−1.43997	−0.0540792	−0.0270396	−	0.999634i	$-0.508608\pi$
$709$	−1.43997	−0.0540792	−0.0270396	+	0.999634i	$0.508608\pi$
$710$	0	0
$711$	−94.8455	−3.55698
$712$	0	0
$713$	−1.21164	−0.0453764
$714$	0	0
$715$	−3.86697	−0.144616
$716$	0	0
$717$	−67.7535	−2.53030
$718$	0	0
$719$	−23.1423	−0.863063	−0.431531	−	0.902098i	$-0.642027\pi$
$719$	−23.1423	−0.863063	−0.431531	+	0.902098i	$0.642027\pi$
$720$	0	0
$721$	−47.9014	−1.78394
$722$	0	0
$723$	47.5002	1.76655
$724$	0	0
$725$	8.88845	0.330109
$726$	0	0
$727$	20.7368	0.769087	0.384543	−	0.923107i	$-0.374359\pi$
$727$	20.7368	0.769087	0.384543	+	0.923107i	$0.374359\pi$
$728$	0	0
$729$	−18.1193	−0.671086
$730$	0	0
$731$	9.86255	0.364779
$732$	0	0
$733$	33.9170	1.25275	0.626377	−	0.779520i	$-0.284537\pi$
$733$	33.9170	1.25275	0.626377	+	0.779520i	$0.284537\pi$
$734$	0	0
$735$	−37.4558	−1.38158
$736$	0	0
$737$	12.5476	0.462196
$738$	0	0
$739$	50.3386	1.85174	0.925868	−	0.377846i	$-0.123335\pi$
$739$	50.3386	1.85174	0.925868	+	0.377846i	$0.123335\pi$
$740$	0	0
$741$	−31.1560	−1.14454
$742$	0	0
$743$	19.1045	0.700878	0.350439	−	0.936586i	$-0.386032\pi$
$743$	19.1045	0.700878	0.350439	+	0.936586i	$0.386032\pi$
$744$	0	0
$745$	−2.18760	−0.0801474
$746$	0	0
$747$	93.9886	3.43886
$748$	0	0
$749$	53.5244	1.95574
$750$	0	0
$751$	32.3704	1.18121	0.590606	−	0.806960i	$-0.298889\pi$
$751$	32.3704	1.18121	0.590606	+	0.806960i	$0.298889\pi$
$752$	0	0
$753$	−16.9450	−0.617510
$754$	0	0
$755$	−8.48493	−0.308798
$756$	0	0
$757$	−29.7853	−1.08256	−0.541282	−	0.840841i	$-0.682061\pi$
$757$	−29.7853	−1.08256	−0.541282	+	0.840841i	$0.682061\pi$
$758$	0	0
$759$	−32.3316	−1.17356
$760$	0	0
$761$	−47.2924	−1.71435	−0.857174	−	0.515027i	$-0.827782\pi$
$761$	−47.2924	−1.71435	−0.857174	+	0.515027i	$0.827782\pi$
$762$	0	0
$763$	52.9706	1.91767
$764$	0	0
$765$	−29.5707	−1.06913
$766$	0	0
$767$	6.71449	0.242446
$768$	0	0
$769$	9.92361	0.357855	0.178927	−	0.983862i	$-0.442737\pi$
$769$	9.92361	0.357855	0.178927	+	0.983862i	$0.442737\pi$
$770$	0	0
$771$	0.864610	0.0311382
$772$	0	0
$773$	28.3068	1.01812	0.509062	−	0.860730i	$-0.329992\pi$
$773$	28.3068	1.01812	0.509062	+	0.860730i	$0.329992\pi$
$774$	0	0
$775$	−0.295807	−0.0106257
$776$	0	0
$777$	−13.3866	−0.480240
$778$	0	0
$779$	−8.74364	−0.313274
$780$	0	0
$781$	41.1581	1.47275
$782$	0	0
$783$	89.1472	3.18586
$784$	0	0
$785$	−10.0941	−0.360274
$786$	0	0
$787$	0.296422	0.0105663	0.00528315	−	0.999986i	$-0.498318\pi$
$787$	0.296422	0.0105663	0.00528315	+	0.999986i	$0.498318\pi$
$788$	0	0
$789$	−41.0811	−1.46253
$790$	0	0
$791$	−63.1332	−2.24476
$792$	0	0
$793$	−15.5102	−0.550782
$794$	0	0
$795$	8.18613	0.290332
$796$	0	0
$797$	−35.2856	−1.24988	−0.624940	−	0.780673i	$-0.714877\pi$
$797$	−35.2856	−1.24988	−0.624940	+	0.780673i	$0.714877\pi$
$798$	0	0
$799$	48.8414	1.72789
$800$	0	0
$801$	79.9776	2.82587
$802$	0	0
$803$	−0.525032	−0.0185280
$804$	0	0
$805$	17.9894	0.634043
$806$	0	0
$807$	51.4760	1.81204
$808$	0	0
$809$	20.8837	0.734233	0.367117	−	0.930175i	$-0.380345\pi$
$809$	20.8837	0.734233	0.367117	+	0.930175i	$0.380345\pi$
$810$	0	0
$811$	−22.6704	−0.796065	−0.398032	−	0.917371i	$-0.630307\pi$
$811$	−22.6704	−0.796065	−0.398032	+	0.917371i	$0.630307\pi$
$812$	0	0
$813$	−17.7005	−0.620784
$814$	0	0
$815$	6.47629	0.226855
$816$	0	0
$817$	14.3616	0.502450
$818$	0	0
$819$	41.2539	1.44153
$820$	0	0
$821$	−6.17619	−0.215551	−0.107775	−	0.994175i	$-0.534373\pi$
$821$	−6.17619	−0.215551	−0.107775	+	0.994175i	$0.534373\pi$
$822$	0	0
$823$	−48.1296	−1.67769	−0.838847	−	0.544367i	$-0.816770\pi$
$823$	−48.1296	−1.67769	−0.838847	+	0.544367i	$0.816770\pi$
$824$	0	0
$825$	−7.89332	−0.274810
$826$	0	0
$827$	−41.9170	−1.45760	−0.728799	−	0.684728i	$-0.759921\pi$
$827$	−41.9170	−1.45760	−0.728799	+	0.684728i	$0.759921\pi$
$828$	0	0
$829$	−41.4722	−1.44039	−0.720194	−	0.693772i	$-0.755947\pi$
$829$	−41.4722	−1.44039	−0.720194	+	0.693772i	$0.755947\pi$
$830$	0	0
$831$	−48.6300	−1.68696
$832$	0	0
$833$	57.7666	2.00149
$834$	0	0
$835$	−7.67416	−0.265575
$836$	0	0
$837$	−2.96681	−0.102548
$838$	0	0
$839$	−33.3740	−1.15220	−0.576099	−	0.817380i	$-0.695426\pi$
$839$	−33.3740	−1.15220	−0.576099	+	0.817380i	$0.695426\pi$
$840$	0	0
$841$	50.0046	1.72430
$842$	0	0
$843$	−17.5878	−0.605757
$844$	0	0
$845$	10.7702	0.370507
$846$	0	0
$847$	18.8575	0.647953
$848$	0	0
$849$	−68.8255	−2.36209
$850$	0	0
$851$	4.09606	0.140411
$852$	0	0
$853$	−12.1166	−0.414865	−0.207433	−	0.978249i	$-0.566511\pi$
$853$	−12.1166	−0.414865	−0.207433	+	0.978249i	$0.566511\pi$
$854$	0	0
$855$	−43.0603	−1.47263
$856$	0	0
$857$	−9.19441	−0.314075	−0.157038	−	0.987593i	$-0.550194\pi$
$857$	−9.19441	−0.314075	−0.157038	+	0.987593i	$0.550194\pi$
$858$	0	0
$859$	−2.39010	−0.0815492	−0.0407746	−	0.999168i	$-0.512983\pi$
$859$	−2.39010	−0.0815492	−0.0407746	+	0.999168i	$0.512983\pi$
$860$	0	0
$861$	17.0990	0.582731
$862$	0	0
$863$	−44.9320	−1.52950	−0.764752	−	0.644325i	$-0.777139\pi$
$863$	−44.9320	−1.52950	−0.764752	+	0.644325i	$0.777139\pi$
$864$	0	0
$865$	−20.2105	−0.687178
$866$	0	0
$867$	15.5390	0.527731
$868$	0	0
$869$	39.0456	1.32453
$870$	0	0
$871$	7.23519	0.245155
$872$	0	0
$873$	31.4831	1.06554
$874$	0	0
$875$	4.39187	0.148472
$876$	0	0
$877$	48.1128	1.62465	0.812327	−	0.583202i	$-0.198200\pi$
$877$	48.1128	1.62465	0.812327	+	0.583202i	$0.198200\pi$
$878$	0	0
$879$	43.2541	1.45893
$880$	0	0
$881$	−44.6632	−1.50474	−0.752371	−	0.658740i	$-0.771090\pi$
$881$	−44.6632	−1.50474	−0.752371	+	0.658740i	$0.771090\pi$
$882$	0	0
$883$	40.2662	1.35507	0.677533	−	0.735493i	$-0.263049\pi$
$883$	40.2662	1.35507	0.677533	+	0.735493i	$0.263049\pi$
$884$	0	0
$885$	13.7057	0.460713
$886$	0	0
$887$	24.4446	0.820769	0.410384	−	0.911913i	$-0.365395\pi$
$887$	24.4446	0.820769	0.410384	+	0.911913i	$0.365395\pi$
$888$	0	0
$889$	−18.6989	−0.627141
$890$	0	0
$891$	−30.2960	−1.01495
$892$	0	0
$893$	71.1219	2.38000
$894$	0	0
$895$	20.6222	0.689324
$896$	0	0
$897$	−18.6430	−0.622473
$898$	0	0
$899$	−2.62927	−0.0876909
$900$	0	0
$901$	−12.6251	−0.420604
$902$	0	0
$903$	−28.0854	−0.934625
$904$	0	0
$905$	4.39340	0.146042
$906$	0	0
$907$	49.1980	1.63359	0.816796	−	0.576926i	$-0.195748\pi$
$907$	49.1980	1.63359	0.816796	+	0.576926i	$0.195748\pi$
$908$	0	0
$909$	55.9957	1.85726
$910$	0	0
$911$	−36.2029	−1.19946	−0.599728	−	0.800204i	$-0.704724\pi$
$911$	−36.2029	−1.19946	−0.599728	+	0.800204i	$0.704724\pi$
$912$	0	0
$913$	−38.6928	−1.28054
$914$	0	0
$915$	−31.6596	−1.04663
$916$	0	0
$917$	54.9838	1.81573
$918$	0	0
$919$	−44.0029	−1.45152	−0.725761	−	0.687947i	$-0.758512\pi$
$919$	−44.0029	−1.45152	−0.725761	+	0.687947i	$0.758512\pi$
$920$	0	0
$921$	90.4509	2.98046
$922$	0	0
$923$	23.7326	0.781167
$924$	0	0
$925$	1.00000	0.0328798
$926$	0	0
$927$	68.6094	2.25343
$928$	0	0
$929$	−40.2361	−1.32010	−0.660052	−	0.751220i	$-0.729466\pi$
$929$	−40.2361	−1.32010	−0.660052	+	0.751220i	$0.729466\pi$
$930$	0	0
$931$	84.1185	2.75687
$932$	0	0
$933$	−91.9048	−3.00883
$934$	0	0
$935$	12.1735	0.398117
$936$	0	0
$937$	10.3942	0.339564	0.169782	−	0.985482i	$-0.445694\pi$
$937$	10.3942	0.339564	0.169782	+	0.985482i	$0.445694\pi$
$938$	0	0
$939$	−30.8535	−1.00686
$940$	0	0
$941$	−35.1903	−1.14717	−0.573585	−	0.819146i	$-0.694448\pi$
$941$	−35.1903	−1.14717	−0.573585	+	0.819146i	$0.694448\pi$
$942$	0	0
$943$	−5.23200	−0.170377
$944$	0	0
$945$	44.0485	1.43290
$946$	0	0
$947$	15.3097	0.497498	0.248749	−	0.968568i	$-0.419981\pi$
$947$	15.3097	0.497498	0.248749	+	0.968568i	$0.419981\pi$
$948$	0	0
$949$	−0.302744	−0.00982750
$950$	0	0
$951$	−71.3028	−2.31215
$952$	0	0
$953$	−9.73013	−0.315190	−0.157595	−	0.987504i	$-0.550374\pi$
$953$	−9.73013	−0.315190	−0.157595	+	0.987504i	$0.550374\pi$
$954$	0	0
$955$	−6.61429	−0.214033
$956$	0	0
$957$	−70.1594	−2.26793
$958$	0	0
$959$	16.6771	0.538530
$960$	0	0
$961$	−30.9125	−0.997177
$962$	0	0
$963$	−76.6633	−2.47044
$964$	0	0
$965$	−12.3586	−0.397837
$966$	0	0
$967$	−45.5301	−1.46415	−0.732075	−	0.681224i	$-0.761448\pi$
$967$	−45.5301	−1.46415	−0.732075	+	0.681224i	$0.761448\pi$
$968$	0	0
$969$	98.0817	3.15084
$970$	0	0
$971$	30.7614	0.987181	0.493590	−	0.869694i	$-0.335684\pi$
$971$	30.7614	0.987181	0.493590	+	0.869694i	$0.335684\pi$
$972$	0	0
$973$	−7.34703	−0.235535
$974$	0	0
$975$	−4.55145	−0.145763
$976$	0	0
$977$	−21.1970	−0.678152	−0.339076	−	0.940759i	$-0.610114\pi$
$977$	−21.1970	−0.678152	−0.339076	+	0.940759i	$0.610114\pi$
$978$	0	0
$979$	−32.9248	−1.05228
$980$	0	0
$981$	−75.8701	−2.42235
$982$	0	0
$983$	39.8068	1.26964	0.634820	−	0.772660i	$-0.281074\pi$
$983$	39.8068	1.26964	0.634820	+	0.772660i	$0.281074\pi$
$984$	0	0
$985$	−5.19897	−0.165653
$986$	0	0
$987$	−139.085	−4.42713
$988$	0	0
$989$	8.59368	0.273263
$990$	0	0
$991$	−3.92409	−0.124653	−0.0623265	−	0.998056i	$-0.519852\pi$
$991$	−3.92409	−0.124653	−0.0623265	+	0.998056i	$0.519852\pi$
$992$	0	0
$993$	−43.9509	−1.39474
$994$	0	0
$995$	−22.4825	−0.712743
$996$	0	0
$997$	47.1040	1.49180	0.745899	−	0.666059i	$-0.232020\pi$
$997$	47.1040	1.49180	0.745899	+	0.666059i	$0.232020\pi$
$998$	0	0
$999$	10.0295	0.317321

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2960.2.a.y.1.5		5
4.3	odd	2		1480.2.a.i.1.1	✓	5
20.19	odd	2		7400.2.a.p.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1480.2.a.i.1.1	✓	5	4.3	odd	2
2960.2.a.y.1.5		5	1.1	even	1	trivial
7400.2.a.p.1.5		5	20.19	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:	\( N \)	\(=\)	\( 2960 = 2^{4} \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2960.a (trivial)

\(f(q)\)	\(=\)	\(q+3.04803 q^{3} -1.00000 q^{5} -4.39187 q^{7} +6.29050 q^{9} +O(q^{10})\)	\(q+3.04803 q^{3} -1.00000 q^{5} -4.39187 q^{7} +6.29050 q^{9} -2.58965 q^{11} -1.49324 q^{13} -3.04803 q^{15} +4.70085 q^{17} +6.84529 q^{19} -13.3866 q^{21} +4.09606 q^{23} +1.00000 q^{25} +10.0295 q^{27} +8.88845 q^{29} -0.295807 q^{31} -7.89332 q^{33} +4.39187 q^{35} +1.00000 q^{37} -4.55145 q^{39} -1.27732 q^{41} +2.09803 q^{43} -6.29050 q^{45} +10.3899 q^{47} +12.2885 q^{49} +14.3284 q^{51} -2.68571 q^{53} +2.58965 q^{55} +20.8647 q^{57} -4.49658 q^{59} +10.3869 q^{61} -27.6271 q^{63} +1.49324 q^{65} -4.84529 q^{67} +12.4849 q^{69} -15.8933 q^{71} +0.202743 q^{73} +3.04803 q^{75} +11.3734 q^{77} -15.0776 q^{79} +11.6989 q^{81} +14.9414 q^{83} -4.70085 q^{85} +27.0923 q^{87} +12.7140 q^{89} +6.55813 q^{91} -0.901629 q^{93} -6.84529 q^{95} +5.00487 q^{97} -16.2902 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{5} - 3 q^{7} + 5 q^{9}+O(q^{10}) \) 5 * q - 5 * q^5 - 3 * q^7 + 5 * q^9	\( 5 q - 5 q^{5} - 3 q^{7} + 5 q^{9} - 3 q^{11} + 4 q^{13} + 7 q^{17} + 4 q^{19} - 10 q^{21} - 10 q^{23} + 5 q^{25} + 6 q^{27} + 19 q^{29} - 13 q^{31} + 6 q^{33} + 3 q^{35} + 5 q^{37} - 14 q^{39} + 11 q^{41} + 13 q^{43} - 5 q^{45} + 2 q^{49} + 12 q^{51} + 27 q^{53} + 3 q^{55} + 12 q^{57} - 16 q^{59} + 27 q^{61} - 37 q^{63} - 4 q^{65} + 6 q^{67} + 40 q^{69} - 34 q^{71} + 16 q^{73} + 9 q^{77} - 16 q^{79} + 9 q^{81} + 14 q^{83} - 7 q^{85} + 42 q^{87} + 38 q^{89} + 34 q^{91} + 30 q^{93} - 4 q^{95} + 5 q^{97} - 23 q^{99}+O(q^{100}) \) 5 * q - 5 * q^5 - 3 * q^7 + 5 * q^9 - 3 * q^11 + 4 * q^13 + 7 * q^17 + 4 * q^19 - 10 * q^21 - 10 * q^23 + 5 * q^25 + 6 * q^27 + 19 * q^29 - 13 * q^31 + 6 * q^33 + 3 * q^35 + 5 * q^37 - 14 * q^39 + 11 * q^41 + 13 * q^43 - 5 * q^45 + 2 * q^49 + 12 * q^51 + 27 * q^53 + 3 * q^55 + 12 * q^57 - 16 * q^59 + 27 * q^61 - 37 * q^63 - 4 * q^65 + 6 * q^67 + 40 * q^69 - 34 * q^71 + 16 * q^73 + 9 * q^77 - 16 * q^79 + 9 * q^81 + 14 * q^83 - 7 * q^85 + 42 * q^87 + 38 * q^89 + 34 * q^91 + 30 * q^93 - 4 * q^95 + 5 * q^97 - 23 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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