LMFDB - Embedded newform 3330.2.a.bk.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3330,2,Mod(1,3330)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3330, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3330.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3330.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:	$26.5901838731$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.23544108.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 20x^{3} + 39x^{2} + 9x - 36 $ x^5 - x^4 - 20x^3 + 39x^2 + 9*x - 36
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$3.56575$ of defining polynomial
Character	$\chi$	$=$	3330.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +3.21441 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +3.21441 q^{7} -1.00000 q^{8} +1.00000 q^{10} -6.35809 q^{11} +6.24735 q^{13} -3.21441 q^{14} +1.00000 q^{16} +1.21441 q^{17} -6.95004 q^{19} -1.00000 q^{20} +6.35809 q^{22} +4.24735 q^{23} +1.00000 q^{25} -6.24735 q^{26} +3.21441 q^{28} -3.03295 q^{29} +4.59195 q^{31} -1.00000 q^{32} -1.21441 q^{34} -3.21441 q^{35} +1.00000 q^{37} +6.95004 q^{38} +1.00000 q^{40} +3.47395 q^{41} -1.47395 q^{43} -6.35809 q^{44} -4.24735 q^{46} +8.01349 q^{47} +3.33241 q^{49} -1.00000 q^{50} +6.24735 q^{52} +0.773406 q^{53} +6.35809 q^{55} -3.21441 q^{56} +3.03295 q^{58} +12.7162 q^{59} -10.1523 q^{61} -4.59195 q^{62} +1.00000 q^{64} -6.24735 q^{65} +6.28737 q^{67} +1.21441 q^{68} +3.21441 q^{70} -0.882000 q^{71} -14.0815 q^{73} -1.00000 q^{74} -6.95004 q^{76} -20.4375 q^{77} +2.88200 q^{79} -1.00000 q^{80} -3.47395 q^{82} +16.9635 q^{83} -1.21441 q^{85} +1.47395 q^{86} +6.35809 q^{88} +6.61027 q^{89} +20.0815 q^{91} +4.24735 q^{92} -8.01349 q^{94} +6.95004 q^{95} -2.53955 q^{97} -3.33241 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{2} + 5 q^{4} - 5 q^{5} + 3 q^{7} - 5 q^{8}+O(q^{10}) $ 5 * q - 5 * q^2 + 5 * q^4 - 5 * q^5 + 3 * q^7 - 5 * q^8	$ 5 q - 5 q^{2} + 5 q^{4} - 5 q^{5} + 3 q^{7} - 5 q^{8} + 5 q^{10} - 5 q^{11} + 5 q^{13} - 3 q^{14} + 5 q^{16} - 7 q^{17} - q^{19} - 5 q^{20} + 5 q^{22} - 5 q^{23} + 5 q^{25} - 5 q^{26} + 3 q^{28} - 2 q^{29} + 16 q^{31} - 5 q^{32} + 7 q^{34} - 3 q^{35} + 5 q^{37} + q^{38} + 5 q^{40} - 2 q^{41} + 12 q^{43} - 5 q^{44} + 5 q^{46} - 6 q^{47} + 16 q^{49} - 5 q^{50} + 5 q^{52} - 3 q^{53} + 5 q^{55} - 3 q^{56} + 2 q^{58} + 10 q^{59} + 16 q^{61} - 16 q^{62} + 5 q^{64} - 5 q^{65} + 4 q^{67} - 7 q^{68} + 3 q^{70} + 8 q^{71} - 3 q^{73} - 5 q^{74} - q^{76} - 3 q^{77} + 2 q^{79} - 5 q^{80} + 2 q^{82} + 5 q^{83} + 7 q^{85} - 12 q^{86} + 5 q^{88} + 7 q^{89} + 33 q^{91} - 5 q^{92} + 6 q^{94} + q^{95} + 14 q^{97} - 16 q^{98}+O(q^{100}) $ 5 * q - 5 * q^2 + 5 * q^4 - 5 * q^5 + 3 * q^7 - 5 * q^8 + 5 * q^10 - 5 * q^11 + 5 * q^13 - 3 * q^14 + 5 * q^16 - 7 * q^17 - q^19 - 5 * q^20 + 5 * q^22 - 5 * q^23 + 5 * q^25 - 5 * q^26 + 3 * q^28 - 2 * q^29 + 16 * q^31 - 5 * q^32 + 7 * q^34 - 3 * q^35 + 5 * q^37 + q^38 + 5 * q^40 - 2 * q^41 + 12 * q^43 - 5 * q^44 + 5 * q^46 - 6 * q^47 + 16 * q^49 - 5 * q^50 + 5 * q^52 - 3 * q^53 + 5 * q^55 - 3 * q^56 + 2 * q^58 + 10 * q^59 + 16 * q^61 - 16 * q^62 + 5 * q^64 - 5 * q^65 + 4 * q^67 - 7 * q^68 + 3 * q^70 + 8 * q^71 - 3 * q^73 - 5 * q^74 - q^76 - 3 * q^77 + 2 * q^79 - 5 * q^80 + 2 * q^82 + 5 * q^83 + 7 * q^85 - 12 * q^86 + 5 * q^88 + 7 * q^89 + 33 * q^91 - 5 * q^92 + 6 * q^94 + q^95 + 14 * q^97 - 16 * q^98

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	3.21441	1.21493	0.607466	−	0.794346i	$-0.292186\pi$
$7$	3.21441	1.21493	0.607466	+	0.794346i	$0.292186\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	−6.35809	−1.91704	−0.958518	−	0.285032i	$-0.907996\pi$
$11$	−6.35809	−1.91704	−0.958518	+	0.285032i	$0.907996\pi$
$12$	0	0
$13$	6.24735	1.73270	0.866352	−	0.499434i	$-0.166459\pi$
$13$	6.24735	1.73270	0.866352	+	0.499434i	$0.166459\pi$
$14$	−3.21441	−0.859086
$15$	0	0
$16$	1.00000	0.250000
$17$	1.21441	0.294537	0.147268	−	0.989097i	$-0.452952\pi$
$17$	1.21441	0.294537	0.147268	+	0.989097i	$0.452952\pi$
$18$	0	0
$19$	−6.95004	−1.59445	−0.797224	−	0.603684i	$-0.793699\pi$
$19$	−6.95004	−1.59445	−0.797224	+	0.603684i	$0.793699\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	6.35809	1.35555
$23$	4.24735	0.885634	0.442817	−	0.896612i	$-0.353979\pi$
$23$	4.24735	0.885634	0.442817	+	0.896612i	$0.353979\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−6.24735	−1.22521
$27$	0	0
$28$	3.21441	0.607466
$29$	−3.03295	−0.563204	−0.281602	−	0.959531i	$-0.590866\pi$
$29$	−3.03295	−0.563204	−0.281602	+	0.959531i	$0.590866\pi$
$30$	0	0
$31$	4.59195	0.824738	0.412369	−	0.911017i	$-0.364701\pi$
$31$	4.59195	0.824738	0.412369	+	0.911017i	$0.364701\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−1.21441	−0.208269
$35$	−3.21441	−0.543334
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	6.95004	1.12744
$39$	0	0
$40$	1.00000	0.158114
$41$	3.47395	0.542539	0.271270	−	0.962503i	$-0.412556\pi$
$41$	3.47395	0.542539	0.271270	+	0.962503i	$0.412556\pi$
$42$	0	0
$43$	−1.47395	−0.224775	−0.112387	−	0.993664i	$-0.535850\pi$
$43$	−1.47395	−0.224775	−0.112387	+	0.993664i	$0.535850\pi$
$44$	−6.35809	−0.958518
$45$	0	0
$46$	−4.24735	−0.626238
$47$	8.01349	1.16889	0.584444	−	0.811434i	$-0.301313\pi$
$47$	8.01349	1.16889	0.584444	+	0.811434i	$0.301313\pi$
$48$	0	0
$49$	3.33241	0.476058
$50$	−1.00000	−0.141421
$51$	0	0
$52$	6.24735	0.866352
$53$	0.773406	0.106235	0.0531177	−	0.998588i	$-0.483084\pi$
$53$	0.773406	0.106235	0.0531177	+	0.998588i	$0.483084\pi$
$54$	0	0
$55$	6.35809	0.857324
$56$	−3.21441	−0.429543
$57$	0	0
$58$	3.03295	0.398246
$59$	12.7162	1.65551	0.827753	−	0.561093i	$-0.189619\pi$
$59$	12.7162	1.65551	0.827753	+	0.561093i	$0.189619\pi$
$60$	0	0
$61$	−10.1523	−1.29986	−0.649931	−	0.759993i	$-0.725202\pi$
$61$	−10.1523	−1.29986	−0.649931	+	0.759993i	$0.725202\pi$
$62$	−4.59195	−0.583178
$63$	0	0
$64$	1.00000	0.125000
$65$	−6.24735	−0.774889
$66$	0	0
$67$	6.28737	0.768124	0.384062	−	0.923307i	$-0.374525\pi$
$67$	6.28737	0.768124	0.384062	+	0.923307i	$0.374525\pi$
$68$	1.21441	0.147268
$69$	0	0
$70$	3.21441	0.384195
$71$	−0.882000	−0.104674	−0.0523371	−	0.998629i	$-0.516667\pi$
$71$	−0.882000	−0.104674	−0.0523371	+	0.998629i	$0.516667\pi$
$72$	0	0
$73$	−14.0815	−1.64812	−0.824059	−	0.566504i	$-0.808296\pi$
$73$	−14.0815	−1.64812	−0.824059	+	0.566504i	$0.808296\pi$
$74$	−1.00000	−0.116248
$75$	0	0
$76$	−6.95004	−0.797224
$77$	−20.4375	−2.32907
$78$	0	0
$79$	2.88200	0.324250	0.162125	−	0.986770i	$-0.448165\pi$
$79$	2.88200	0.324250	0.162125	+	0.986770i	$0.448165\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	−3.47395	−0.383633
$83$	16.9635	1.86199	0.930995	−	0.365033i	$-0.118942\pi$
$83$	16.9635	1.86199	0.930995	+	0.365033i	$0.118942\pi$
$84$	0	0
$85$	−1.21441	−0.131721
$86$	1.47395	0.158940
$87$	0	0
$88$	6.35809	0.677775
$89$	6.61027	0.700687	0.350344	−	0.936621i	$-0.386065\pi$
$89$	6.61027	0.700687	0.350344	+	0.936621i	$0.386065\pi$
$90$	0	0
$91$	20.0815	2.10512
$92$	4.24735	0.442817
$93$	0	0
$94$	−8.01349	−0.826529
$95$	6.95004	0.713059
$96$	0	0
$97$	−2.53955	−0.257852	−0.128926	−	0.991654i	$-0.541153\pi$
$97$	−2.53955	−0.257852	−0.128926	+	0.991654i	$0.541153\pi$
$98$	−3.33241	−0.336624
$99$	0	0
$100$	1.00000	0.100000
$101$	6.46668	0.643459	0.321729	−	0.946832i	$-0.395736\pi$
$101$	6.46668	0.643459	0.321729	+	0.946832i	$0.395736\pi$
$102$	0	0
$103$	10.0135	0.986659	0.493329	−	0.869843i	$-0.335780\pi$
$103$	10.0135	0.986659	0.493329	+	0.869843i	$0.335780\pi$
$104$	−6.24735	−0.612603
$105$	0	0
$106$	−0.773406	−0.0751198
$107$	10.3132	0.997019	0.498510	−	0.866884i	$-0.333881\pi$
$107$	10.3132	0.997019	0.498510	+	0.866884i	$0.333881\pi$
$108$	0	0
$109$	−5.24009	−0.501909	−0.250955	−	0.967999i	$-0.580745\pi$
$109$	−5.24009	−0.501909	−0.250955	+	0.967999i	$0.580745\pi$
$110$	−6.35809	−0.606220
$111$	0	0
$112$	3.21441	0.303733
$113$	−9.68323	−0.910922	−0.455461	−	0.890256i	$-0.650525\pi$
$113$	−9.68323	−0.910922	−0.455461	+	0.890256i	$0.650525\pi$
$114$	0	0
$115$	−4.24735	−0.396068
$116$	−3.03295	−0.281602
$117$	0	0
$118$	−12.7162	−1.17062
$119$	3.90359	0.357842
$120$	0	0
$121$	29.4253	2.67503
$122$	10.1523	0.919142
$123$	0	0
$124$	4.59195	0.412369
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−15.8601	−1.40735	−0.703676	−	0.710521i	$-0.748460\pi$
$127$	−15.8601	−1.40735	−0.703676	+	0.710521i	$0.748460\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	6.24735	0.547929
$131$	−17.5738	−1.53543	−0.767715	−	0.640791i	$-0.778606\pi$
$131$	−17.5738	−1.53543	−0.767715	+	0.640791i	$0.778606\pi$
$132$	0	0
$133$	−22.3402	−1.93714
$134$	−6.28737	−0.543146
$135$	0	0
$136$	−1.21441	−0.104134
$137$	7.13149	0.609285	0.304642	−	0.952467i	$-0.401463\pi$
$137$	7.13149	0.609285	0.304642	+	0.952467i	$0.401463\pi$
$138$	0	0
$139$	−6.80123	−0.576873	−0.288436	−	0.957499i	$-0.593135\pi$
$139$	−6.80123	−0.576873	−0.288436	+	0.957499i	$0.593135\pi$
$140$	−3.21441	−0.271667
$141$	0	0
$142$	0.882000	0.0740158
$143$	−39.7212	−3.32166
$144$	0	0
$145$	3.03295	0.251873
$146$	14.0815	1.16540
$147$	0	0
$148$	1.00000	0.0821995
$149$	−14.7442	−1.20789	−0.603946	−	0.797025i	$-0.706406\pi$
$149$	−14.7442	−1.20789	−0.603946	+	0.797025i	$0.706406\pi$
$150$	0	0
$151$	17.4312	1.41853	0.709267	−	0.704940i	$-0.249026\pi$
$151$	17.4312	1.41853	0.709267	+	0.704940i	$0.249026\pi$
$152$	6.95004	0.563722
$153$	0	0
$154$	20.4375	1.64690
$155$	−4.59195	−0.368834
$156$	0	0
$157$	17.8150	1.42179	0.710897	−	0.703296i	$-0.248289\pi$
$157$	17.8150	1.42179	0.710897	+	0.703296i	$0.248289\pi$
$158$	−2.88200	−0.229280
$159$	0	0
$160$	1.00000	0.0790569
$161$	13.6527	1.07598
$162$	0	0
$163$	23.1023	1.80951	0.904756	−	0.425931i	$-0.140053\pi$
$163$	23.1023	1.80951	0.904756	+	0.425931i	$0.140053\pi$
$164$	3.47395	0.271270
$165$	0	0
$166$	−16.9635	−1.31663
$167$	−4.24735	−0.328670	−0.164335	−	0.986405i	$-0.552548\pi$
$167$	−4.24735	−0.328670	−0.164335	+	0.986405i	$0.552548\pi$
$168$	0	0
$169$	26.0294	2.00226
$170$	1.21441	0.0931407
$171$	0	0
$172$	−1.47395	−0.112387
$173$	−12.9052	−0.981164	−0.490582	−	0.871395i	$-0.663216\pi$
$173$	−12.9052	−0.981164	−0.490582	+	0.871395i	$0.663216\pi$
$174$	0	0
$175$	3.21441	0.242986
$176$	−6.35809	−0.479259
$177$	0	0
$178$	−6.61027	−0.495461
$179$	21.3767	1.59777	0.798885	−	0.601484i	$-0.205424\pi$
$179$	21.3767	1.59777	0.798885	+	0.601484i	$0.205424\pi$
$180$	0	0
$181$	3.71263	0.275958	0.137979	−	0.990435i	$-0.455939\pi$
$181$	3.71263	0.275958	0.137979	+	0.990435i	$0.455939\pi$
$182$	−20.0815	−1.48854
$183$	0	0
$184$	−4.24735	−0.313119
$185$	−1.00000	−0.0735215
$186$	0	0
$187$	−7.72130	−0.564637
$188$	8.01349	0.584444
$189$	0	0
$190$	−6.95004	−0.504209
$191$	−2.16230	−0.156459	−0.0782293	−	0.996935i	$-0.524927\pi$
$191$	−2.16230	−0.156459	−0.0782293	+	0.996935i	$0.524927\pi$
$192$	0	0
$193$	16.6638	1.19948	0.599742	−	0.800193i	$-0.295270\pi$
$193$	16.6638	1.19948	0.599742	+	0.800193i	$0.295270\pi$
$194$	2.53955	0.182329
$195$	0	0
$196$	3.33241	0.238029
$197$	−6.04001	−0.430333	−0.215167	−	0.976577i	$-0.569029\pi$
$197$	−6.04001	−0.430333	−0.215167	+	0.976577i	$0.569029\pi$
$198$	0	0
$199$	25.6397	1.81755	0.908775	−	0.417287i	$-0.137019\pi$
$199$	25.6397	1.81755	0.908775	+	0.417287i	$0.137019\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	−6.46668	−0.454994
$203$	−9.74912	−0.684254
$204$	0	0
$205$	−3.47395	−0.242631
$206$	−10.0135	−0.697673
$207$	0	0
$208$	6.24735	0.433176
$209$	44.1889	3.05661
$210$	0	0
$211$	−18.9330	−1.30340	−0.651701	−	0.758476i	$-0.725944\pi$
$211$	−18.9330	−1.30340	−0.651701	+	0.758476i	$0.725944\pi$
$212$	0.773406	0.0531177
$213$	0	0
$214$	−10.3132	−0.704999
$215$	1.47395	0.100522
$216$	0	0
$217$	14.7604	1.00200
$218$	5.24009	0.354904
$219$	0	0
$220$	6.35809	0.428662
$221$	7.58682	0.510345
$222$	0	0
$223$	3.48614	0.233449	0.116724	−	0.993164i	$-0.462761\pi$
$223$	3.48614	0.233449	0.116724	+	0.993164i	$0.462761\pi$
$224$	−3.21441	−0.214772
$225$	0	0
$226$	9.68323	0.644119
$227$	−1.25442	−0.0832586	−0.0416293	−	0.999133i	$-0.513255\pi$
$227$	−1.25442	−0.0832586	−0.0416293	+	0.999133i	$0.513255\pi$
$228$	0	0
$229$	−6.82098	−0.450743	−0.225371	−	0.974273i	$-0.572360\pi$
$229$	−6.82098	−0.450743	−0.225371	+	0.974273i	$0.572360\pi$
$230$	4.24735	0.280062
$231$	0	0
$232$	3.03295	0.199123
$233$	3.79202	0.248424	0.124212	−	0.992256i	$-0.460360\pi$
$233$	3.79202	0.248424	0.124212	+	0.992256i	$0.460360\pi$
$234$	0	0
$235$	−8.01349	−0.522743
$236$	12.7162	0.827753
$237$	0	0
$238$	−3.90359	−0.253032
$239$	−13.7247	−0.887780	−0.443890	−	0.896081i	$-0.646402\pi$
$239$	−13.7247	−0.887780	−0.443890	+	0.896081i	$0.646402\pi$
$240$	0	0
$241$	1.86821	0.120342	0.0601710	−	0.998188i	$-0.480835\pi$
$241$	1.86821	0.120342	0.0601710	+	0.998188i	$0.480835\pi$
$242$	−29.4253	−1.89153
$243$	0	0
$244$	−10.1523	−0.649931
$245$	−3.33241	−0.212900
$246$	0	0
$247$	−43.4193	−2.76271
$248$	−4.59195	−0.291589
$249$	0	0
$250$	1.00000	0.0632456
$251$	19.6641	1.24119	0.620593	−	0.784133i	$-0.286892\pi$
$251$	19.6641	1.24119	0.620593	+	0.784133i	$0.286892\pi$
$252$	0	0
$253$	−27.0050	−1.69779
$254$	15.8601	0.995149
$255$	0	0
$256$	1.00000	0.0625000
$257$	16.0670	1.00223	0.501116	−	0.865380i	$-0.332923\pi$
$257$	16.0670	1.00223	0.501116	+	0.865380i	$0.332923\pi$
$258$	0	0
$259$	3.21441	0.199733
$260$	−6.24735	−0.387444
$261$	0	0
$262$	17.5738	1.08571
$263$	28.3737	1.74960	0.874799	−	0.484485i	$-0.160993\pi$
$263$	28.3737	1.74960	0.874799	+	0.484485i	$0.160993\pi$
$264$	0	0
$265$	−0.773406	−0.0475100
$266$	22.3402	1.36977
$267$	0	0
$268$	6.28737	0.384062
$269$	19.1574	1.16805	0.584023	−	0.811737i	$-0.301478\pi$
$269$	19.1574	1.16805	0.584023	+	0.811737i	$0.301478\pi$
$270$	0	0
$271$	24.0270	1.45954	0.729768	−	0.683695i	$-0.239628\pi$
$271$	24.0270	1.45954	0.729768	+	0.683695i	$0.239628\pi$
$272$	1.21441	0.0736342
$273$	0	0
$274$	−7.13149	−0.430829
$275$	−6.35809	−0.383407
$276$	0	0
$277$	−11.3264	−0.680540	−0.340270	−	0.940328i	$-0.610519\pi$
$277$	−11.3264	−0.680540	−0.340270	+	0.940328i	$0.610519\pi$
$278$	6.80123	0.407911
$279$	0	0
$280$	3.21441	0.192098
$281$	0.870647	0.0519384	0.0259692	−	0.999663i	$-0.491733\pi$
$281$	0.870647	0.0519384	0.0259692	+	0.999663i	$0.491733\pi$
$282$	0	0
$283$	24.1474	1.43542	0.717708	−	0.696344i	$-0.245191\pi$
$283$	24.1474	1.43542	0.717708	+	0.696344i	$0.245191\pi$
$284$	−0.882000	−0.0523371
$285$	0	0
$286$	39.7212	2.34877
$287$	11.1667	0.659148
$288$	0	0
$289$	−15.5252	−0.913248
$290$	−3.03295	−0.178101
$291$	0	0
$292$	−14.0815	−0.824059
$293$	21.9843	1.28434	0.642168	−	0.766564i	$-0.278035\pi$
$293$	21.9843	1.28434	0.642168	+	0.766564i	$0.278035\pi$
$294$	0	0
$295$	−12.7162	−0.740364
$296$	−1.00000	−0.0581238
$297$	0	0
$298$	14.7442	0.854109
$299$	26.5347	1.53454
$300$	0	0
$301$	−4.73787	−0.273086
$302$	−17.4312	−1.00306
$303$	0	0
$304$	−6.95004	−0.398612
$305$	10.1523	0.581316
$306$	0	0
$307$	−16.7162	−0.954043	−0.477021	−	0.878892i	$-0.658284\pi$
$307$	−16.7162	−0.954043	−0.477021	+	0.878892i	$0.658284\pi$
$308$	−20.4375	−1.16453
$309$	0	0
$310$	4.59195	0.260805
$311$	7.99050	0.453100	0.226550	−	0.974000i	$-0.427255\pi$
$311$	7.99050	0.453100	0.226550	+	0.974000i	$0.427255\pi$
$312$	0	0
$313$	23.9379	1.35305	0.676526	−	0.736419i	$-0.263485\pi$
$313$	23.9379	1.35305	0.676526	+	0.736419i	$0.263485\pi$
$314$	−17.8150	−1.00536
$315$	0	0
$316$	2.88200	0.162125
$317$	−33.7126	−1.89349	−0.946743	−	0.321990i	$-0.895648\pi$
$317$	−33.7126	−1.89349	−0.946743	+	0.321990i	$0.895648\pi$
$318$	0	0
$319$	19.2837	1.07968
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	−13.6527	−0.760836
$323$	−8.44017	−0.469623
$324$	0	0
$325$	6.24735	0.346541
$326$	−23.1023	−1.27952
$327$	0	0
$328$	−3.47395	−0.191817
$329$	25.7586	1.42012
$330$	0	0
$331$	−0.612412	−0.0336612	−0.0168306	−	0.999858i	$-0.505358\pi$
$331$	−0.612412	−0.0336612	−0.0168306	+	0.999858i	$0.505358\pi$
$332$	16.9635	0.930995
$333$	0	0
$334$	4.24735	0.232405
$335$	−6.28737	−0.343515
$336$	0	0
$337$	−31.3168	−1.70593	−0.852967	−	0.521965i	$-0.825199\pi$
$337$	−31.3168	−1.70593	−0.852967	+	0.521965i	$0.825199\pi$
$338$	−26.0294	−1.41581
$339$	0	0
$340$	−1.21441	−0.0658604
$341$	−29.1960	−1.58105
$342$	0	0
$343$	−11.7891	−0.636554
$344$	1.47395	0.0794699
$345$	0	0
$346$	12.9052	0.693788
$347$	−3.63708	−0.195249	−0.0976244	−	0.995223i	$-0.531124\pi$
$347$	−3.63708	−0.195249	−0.0976244	+	0.995223i	$0.531124\pi$
$348$	0	0
$349$	−7.54253	−0.403742	−0.201871	−	0.979412i	$-0.564702\pi$
$349$	−7.54253	−0.403742	−0.201871	+	0.979412i	$0.564702\pi$
$350$	−3.21441	−0.171817
$351$	0	0
$352$	6.35809	0.338887
$353$	8.38267	0.446164	0.223082	−	0.974800i	$-0.428388\pi$
$353$	8.38267	0.446164	0.223082	+	0.974800i	$0.428388\pi$
$354$	0	0
$355$	0.882000	0.0468117
$356$	6.61027	0.350344
$357$	0	0
$358$	−21.3767	−1.12979
$359$	31.3665	1.65546	0.827729	−	0.561128i	$-0.189632\pi$
$359$	31.3665	1.65546	0.827729	+	0.561128i	$0.189632\pi$
$360$	0	0
$361$	29.3030	1.54226
$362$	−3.71263	−0.195132
$363$	0	0
$364$	20.0815	1.05256
$365$	14.0815	0.737061
$366$	0	0
$367$	35.6751	1.86222	0.931112	−	0.364733i	$-0.118840\pi$
$367$	35.6751	1.86222	0.931112	+	0.364733i	$0.118840\pi$
$368$	4.24735	0.221409
$369$	0	0
$370$	1.00000	0.0519875
$371$	2.48604	0.129069
$372$	0	0
$373$	−0.674658	−0.0349325	−0.0174662	−	0.999847i	$-0.505560\pi$
$373$	−0.674658	−0.0349325	−0.0174662	+	0.999847i	$0.505560\pi$
$374$	7.72130	0.399259
$375$	0	0
$376$	−8.01349	−0.413264
$377$	−18.9479	−0.975866
$378$	0	0
$379$	−23.5840	−1.21143	−0.605716	−	0.795681i	$-0.707113\pi$
$379$	−23.5840	−1.21143	−0.605716	+	0.795681i	$0.707113\pi$
$380$	6.95004	0.356529
$381$	0	0
$382$	2.16230	0.110633
$383$	2.06439	0.105485	0.0527426	−	0.998608i	$-0.483204\pi$
$383$	2.06439	0.105485	0.0527426	+	0.998608i	$0.483204\pi$
$384$	0	0
$385$	20.4375	1.04159
$386$	−16.6638	−0.848164
$387$	0	0
$388$	−2.53955	−0.128926
$389$	−23.1960	−1.17608	−0.588042	−	0.808830i	$-0.700101\pi$
$389$	−23.1960	−1.17608	−0.588042	+	0.808830i	$0.700101\pi$
$390$	0	0
$391$	5.15801	0.260852
$392$	−3.33241	−0.168312
$393$	0	0
$394$	6.04001	0.304291
$395$	−2.88200	−0.145009
$396$	0	0
$397$	16.3948	0.822830	0.411415	−	0.911448i	$-0.365035\pi$
$397$	16.3948	0.822830	0.411415	+	0.911448i	$0.365035\pi$
$398$	−25.6397	−1.28520
$399$	0	0
$400$	1.00000	0.0500000
$401$	34.7177	1.73372	0.866859	−	0.498553i	$-0.166135\pi$
$401$	34.7177	1.73372	0.866859	+	0.498553i	$0.166135\pi$
$402$	0	0
$403$	28.6875	1.42903
$404$	6.46668	0.321729
$405$	0	0
$406$	9.74912	0.483841
$407$	−6.35809	−0.315159
$408$	0	0
$409$	1.16937	0.0578214	0.0289107	−	0.999582i	$-0.490796\pi$
$409$	1.16937	0.0578214	0.0289107	+	0.999582i	$0.490796\pi$
$410$	3.47395	0.171566
$411$	0	0
$412$	10.0135	0.493329
$413$	40.8750	2.01133
$414$	0	0
$415$	−16.9635	−0.832707
$416$	−6.24735	−0.306302
$417$	0	0
$418$	−44.1889	−2.16135
$419$	12.2095	0.596472	0.298236	−	0.954492i	$-0.403602\pi$
$419$	12.2095	0.596472	0.298236	+	0.954492i	$0.403602\pi$
$420$	0	0
$421$	6.86851	0.334750	0.167375	−	0.985893i	$-0.446471\pi$
$421$	6.86851	0.334750	0.167375	+	0.985893i	$0.446471\pi$
$422$	18.9330	0.921645
$423$	0	0
$424$	−0.773406	−0.0375599
$425$	1.21441	0.0589073
$426$	0	0
$427$	−32.6335	−1.57924
$428$	10.3132	0.498510
$429$	0	0
$430$	−1.47395	−0.0710801
$431$	−26.4955	−1.27624	−0.638121	−	0.769936i	$-0.720288\pi$
$431$	−26.4955	−1.27624	−0.638121	+	0.769936i	$0.720288\pi$
$432$	0	0
$433$	−8.18146	−0.393176	−0.196588	−	0.980486i	$-0.562986\pi$
$433$	−8.18146	−0.393176	−0.196588	+	0.980486i	$0.562986\pi$
$434$	−14.7604	−0.708521
$435$	0	0
$436$	−5.24009	−0.250955
$437$	−29.5193	−1.41210
$438$	0	0
$439$	−28.6990	−1.36973	−0.684863	−	0.728672i	$-0.740138\pi$
$439$	−28.6990	−1.36973	−0.684863	+	0.728672i	$0.740138\pi$
$440$	−6.35809	−0.303110
$441$	0	0
$442$	−7.58682	−0.360868
$443$	19.8299	0.942147	0.471073	−	0.882094i	$-0.343867\pi$
$443$	19.8299	0.942147	0.471073	+	0.882094i	$0.343867\pi$
$444$	0	0
$445$	−6.61027	−0.313357
$446$	−3.48614	−0.165073
$447$	0	0
$448$	3.21441	0.151866
$449$	−15.2109	−0.717846	−0.358923	−	0.933367i	$-0.616856\pi$
$449$	−15.2109	−0.717846	−0.358923	+	0.933367i	$0.616856\pi$
$450$	0	0
$451$	−22.0877	−1.04007
$452$	−9.68323	−0.455461
$453$	0	0
$454$	1.25442	0.0588727
$455$	−20.0815	−0.941437
$456$	0	0
$457$	32.9343	1.54060	0.770301	−	0.637680i	$-0.220106\pi$
$457$	32.9343	1.54060	0.770301	+	0.637680i	$0.220106\pi$
$458$	6.82098	0.318723
$459$	0	0
$460$	−4.24735	−0.198034
$461$	24.0266	1.11903	0.559516	−	0.828819i	$-0.310987\pi$
$461$	24.0266	1.11903	0.559516	+	0.828819i	$0.310987\pi$
$462$	0	0
$463$	−5.42911	−0.252312	−0.126156	−	0.992010i	$-0.540264\pi$
$463$	−5.42911	−0.252312	−0.126156	+	0.992010i	$0.540264\pi$
$464$	−3.03295	−0.140801
$465$	0	0
$466$	−3.79202	−0.175662
$467$	−6.44369	−0.298178	−0.149089	−	0.988824i	$-0.547634\pi$
$467$	−6.44369	−0.298178	−0.149089	+	0.988824i	$0.547634\pi$
$468$	0	0
$469$	20.2101	0.933218
$470$	8.01349	0.369635
$471$	0	0
$472$	−12.7162	−0.585309
$473$	9.37149	0.430902
$474$	0	0
$475$	−6.95004	−0.318890
$476$	3.90359	0.178921
$477$	0	0
$478$	13.7247	0.627755
$479$	4.60766	0.210529	0.105265	−	0.994444i	$-0.466431\pi$
$479$	4.60766	0.210529	0.105265	+	0.994444i	$0.466431\pi$
$480$	0	0
$481$	6.24735	0.284855
$482$	−1.86821	−0.0850946
$483$	0	0
$484$	29.4253	1.33751
$485$	2.53955	0.115315
$486$	0	0
$487$	5.87205	0.266088	0.133044	−	0.991110i	$-0.457525\pi$
$487$	5.87205	0.266088	0.133044	+	0.991110i	$0.457525\pi$
$488$	10.1523	0.459571
$489$	0	0
$490$	3.33241	0.150543
$491$	−16.0584	−0.724704	−0.362352	−	0.932041i	$-0.618026\pi$
$491$	−16.0584	−0.724704	−0.362352	+	0.932041i	$0.618026\pi$
$492$	0	0
$493$	−3.68323	−0.165884
$494$	43.4193	1.95353
$495$	0	0
$496$	4.59195	0.206185
$497$	−2.83511	−0.127172
$498$	0	0
$499$	−10.2095	−0.457039	−0.228520	−	0.973539i	$-0.573389\pi$
$499$	−10.2095	−0.457039	−0.228520	+	0.973539i	$0.573389\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	−19.6641	−0.877650
$503$	−5.82005	−0.259503	−0.129752	−	0.991547i	$-0.541418\pi$
$503$	−5.82005	−0.259503	−0.129752	+	0.991547i	$0.541418\pi$
$504$	0	0
$505$	−6.46668	−0.287764
$506$	27.0050	1.20052
$507$	0	0
$508$	−15.8601	−0.703676
$509$	−31.6273	−1.40186	−0.700928	−	0.713232i	$-0.747231\pi$
$509$	−31.6273	−1.40186	−0.700928	+	0.713232i	$0.747231\pi$
$510$	0	0
$511$	−45.2638	−2.00235
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−16.0670	−0.708685
$515$	−10.0135	−0.441247
$516$	0	0
$517$	−50.9505	−2.24080
$518$	−3.21441	−0.141233
$519$	0	0
$520$	6.24735	0.273965
$521$	−1.12556	−0.0493117	−0.0246558	−	0.999696i	$-0.507849\pi$
$521$	−1.12556	−0.0493117	−0.0246558	+	0.999696i	$0.507849\pi$
$522$	0	0
$523$	43.4597	1.90036	0.950181	−	0.311699i	$-0.100898\pi$
$523$	43.4597	1.90036	0.950181	+	0.311699i	$0.100898\pi$
$524$	−17.5738	−0.767715
$525$	0	0
$526$	−28.3737	−1.23715
$527$	5.57649	0.242916
$528$	0	0
$529$	−4.95999	−0.215652
$530$	0.773406	0.0335946
$531$	0	0
$532$	−22.3402	−0.968572
$533$	21.7030	0.940060
$534$	0	0
$535$	−10.3132	−0.445881
$536$	−6.28737	−0.271573
$537$	0	0
$538$	−19.1574	−0.825933
$539$	−21.1877	−0.912620
$540$	0	0
$541$	2.12906	0.0915354	0.0457677	−	0.998952i	$-0.485427\pi$
$541$	2.12906	0.0915354	0.0457677	+	0.998952i	$0.485427\pi$
$542$	−24.0270	−1.03205
$543$	0	0
$544$	−1.21441	−0.0520672
$545$	5.24009	0.224461
$546$	0	0
$547$	−10.2404	−0.437847	−0.218924	−	0.975742i	$-0.570255\pi$
$547$	−10.2404	−0.437847	−0.218924	+	0.975742i	$0.570255\pi$
$548$	7.13149	0.304642
$549$	0	0
$550$	6.35809	0.271110
$551$	21.0791	0.898000
$552$	0	0
$553$	9.26392	0.393942
$554$	11.3264	0.481215
$555$	0	0
$556$	−6.80123	−0.288436
$557$	−4.98621	−0.211273	−0.105636	−	0.994405i	$-0.533688\pi$
$557$	−4.98621	−0.211273	−0.105636	+	0.994405i	$0.533688\pi$
$558$	0	0
$559$	−9.20827	−0.389468
$560$	−3.21441	−0.135833
$561$	0	0
$562$	−0.870647	−0.0367260
$563$	41.0356	1.72944	0.864721	−	0.502252i	$-0.167495\pi$
$563$	41.0356	1.72944	0.864721	+	0.502252i	$0.167495\pi$
$564$	0	0
$565$	9.68323	0.407377
$566$	−24.1474	−1.01499
$567$	0	0
$568$	0.882000	0.0370079
$569$	−16.9781	−0.711757	−0.355879	−	0.934532i	$-0.615818\pi$
$569$	−16.9781	−0.711757	−0.355879	+	0.934532i	$0.615818\pi$
$570$	0	0
$571$	−37.1770	−1.55581	−0.777905	−	0.628382i	$-0.783717\pi$
$571$	−37.1770	−1.55581	−0.777905	+	0.628382i	$0.783717\pi$
$572$	−39.7212	−1.66083
$573$	0	0
$574$	−11.1667	−0.466088
$575$	4.24735	0.177127
$576$	0	0
$577$	7.25841	0.302172	0.151086	−	0.988521i	$-0.451723\pi$
$577$	7.25841	0.302172	0.151086	+	0.988521i	$0.451723\pi$
$578$	15.5252	0.645764
$579$	0	0
$580$	3.03295	0.125936
$581$	54.5277	2.26219
$582$	0	0
$583$	−4.91738	−0.203657
$584$	14.0815	0.582698
$585$	0	0
$586$	−21.9843	−0.908163
$587$	−11.5422	−0.476397	−0.238198	−	0.971217i	$-0.576557\pi$
$587$	−11.5422	−0.476397	−0.238198	+	0.971217i	$0.576557\pi$
$588$	0	0
$589$	−31.9142	−1.31500
$590$	12.7162	0.523517
$591$	0	0
$592$	1.00000	0.0410997
$593$	−15.1272	−0.621200	−0.310600	−	0.950541i	$-0.600530\pi$
$593$	−15.1272	−0.621200	−0.310600	+	0.950541i	$0.600530\pi$
$594$	0	0
$595$	−3.90359	−0.160032
$596$	−14.7442	−0.603946
$597$	0	0
$598$	−26.5347	−1.08509
$599$	10.2874	0.420330	0.210165	−	0.977666i	$-0.432600\pi$
$599$	10.2874	0.420330	0.210165	+	0.977666i	$0.432600\pi$
$600$	0	0
$601$	4.99294	0.203666	0.101833	−	0.994802i	$-0.467529\pi$
$601$	4.99294	0.203666	0.101833	+	0.994802i	$0.467529\pi$
$602$	4.73787	0.193101
$603$	0	0
$604$	17.4312	0.709267
$605$	−29.4253	−1.19631
$606$	0	0
$607$	−27.6651	−1.12289	−0.561446	−	0.827513i	$-0.689755\pi$
$607$	−27.6651	−1.12289	−0.561446	+	0.827513i	$0.689755\pi$
$608$	6.95004	0.281861
$609$	0	0
$610$	−10.1523	−0.411053
$611$	50.0631	2.02534
$612$	0	0
$613$	3.42024	0.138142	0.0690711	−	0.997612i	$-0.477996\pi$
$613$	3.42024	0.138142	0.0690711	+	0.997612i	$0.477996\pi$
$614$	16.7162	0.674610
$615$	0	0
$616$	20.4375	0.823449
$617$	−41.7574	−1.68109	−0.840545	−	0.541742i	$-0.817765\pi$
$617$	−41.7574	−1.68109	−0.840545	+	0.541742i	$0.817765\pi$
$618$	0	0
$619$	−17.8388	−0.717002	−0.358501	−	0.933529i	$-0.616712\pi$
$619$	−17.8388	−0.717002	−0.358501	+	0.933529i	$0.616712\pi$
$620$	−4.59195	−0.184417
$621$	0	0
$622$	−7.99050	−0.320390
$623$	21.2481	0.851287
$624$	0	0
$625$	1.00000	0.0400000
$626$	−23.9379	−0.956753
$627$	0	0
$628$	17.8150	0.710897
$629$	1.21441	0.0484215
$630$	0	0
$631$	18.7749	0.747418	0.373709	−	0.927546i	$-0.378086\pi$
$631$	18.7749	0.747418	0.373709	+	0.927546i	$0.378086\pi$
$632$	−2.88200	−0.114640
$633$	0	0
$634$	33.7126	1.33890
$635$	15.8601	0.629387
$636$	0	0
$637$	20.8187	0.824868
$638$	−19.2837	−0.763451
$639$	0	0
$640$	1.00000	0.0395285
$641$	−27.1423	−1.07206	−0.536028	−	0.844200i	$-0.680076\pi$
$641$	−27.1423	−1.07206	−0.536028	+	0.844200i	$0.680076\pi$
$642$	0	0
$643$	29.0024	1.14374	0.571871	−	0.820344i	$-0.306218\pi$
$643$	29.0024	1.14374	0.571871	+	0.820344i	$0.306218\pi$
$644$	13.6527	0.537992
$645$	0	0
$646$	8.44017	0.332074
$647$	−19.0538	−0.749082	−0.374541	−	0.927210i	$-0.622200\pi$
$647$	−19.0538	−0.749082	−0.374541	+	0.927210i	$0.622200\pi$
$648$	0	0
$649$	−80.8506	−3.17366
$650$	−6.24735	−0.245041
$651$	0	0
$652$	23.1023	0.904756
$653$	−29.9373	−1.17154	−0.585769	−	0.810478i	$-0.699207\pi$
$653$	−29.9373	−1.17154	−0.585769	+	0.810478i	$0.699207\pi$
$654$	0	0
$655$	17.5738	0.686665
$656$	3.47395	0.135635
$657$	0	0
$658$	−25.7586	−1.00418
$659$	30.3813	1.18349	0.591743	−	0.806126i	$-0.298440\pi$
$659$	30.3813	1.18349	0.591743	+	0.806126i	$0.298440\pi$
$660$	0	0
$661$	21.8709	0.850678	0.425339	−	0.905034i	$-0.360155\pi$
$661$	21.8709	0.850678	0.425339	+	0.905034i	$0.360155\pi$
$662$	0.612412	0.0238021
$663$	0	0
$664$	−16.9635	−0.658313
$665$	22.3402	0.866317
$666$	0	0
$667$	−12.8820	−0.498793
$668$	−4.24735	−0.164335
$669$	0	0
$670$	6.28737	0.242902
$671$	64.5489	2.49188
$672$	0	0
$673$	30.4714	1.17459	0.587294	−	0.809374i	$-0.300193\pi$
$673$	30.4714	1.17459	0.587294	+	0.809374i	$0.300193\pi$
$674$	31.3168	1.20628
$675$	0	0
$676$	26.0294	1.00113
$677$	−28.0869	−1.07947	−0.539734	−	0.841836i	$-0.681475\pi$
$677$	−28.0869	−1.07947	−0.539734	+	0.841836i	$0.681475\pi$
$678$	0	0
$679$	−8.16314	−0.313272
$680$	1.21441	0.0465703
$681$	0	0
$682$	29.1960	1.11797
$683$	−1.12263	−0.0429562	−0.0214781	−	0.999769i	$-0.506837\pi$
$683$	−1.12263	−0.0429562	−0.0214781	+	0.999769i	$0.506837\pi$
$684$	0	0
$685$	−7.13149	−0.272480
$686$	11.7891	0.450111
$687$	0	0
$688$	−1.47395	−0.0561937
$689$	4.83174	0.184075
$690$	0	0
$691$	−20.4852	−0.779294	−0.389647	−	0.920964i	$-0.627403\pi$
$691$	−20.4852	−0.779294	−0.389647	+	0.920964i	$0.627403\pi$
$692$	−12.9052	−0.490582
$693$	0	0
$694$	3.63708	0.138062
$695$	6.80123	0.257985
$696$	0	0
$697$	4.21878	0.159798
$698$	7.54253	0.285489
$699$	0	0
$700$	3.21441	0.121493
$701$	11.0694	0.418087	0.209043	−	0.977906i	$-0.432965\pi$
$701$	11.0694	0.418087	0.209043	+	0.977906i	$0.432965\pi$
$702$	0	0
$703$	−6.95004	−0.262126
$704$	−6.35809	−0.239629
$705$	0	0
$706$	−8.38267	−0.315486
$707$	20.7865	0.781758
$708$	0	0
$709$	40.4925	1.52073	0.760364	−	0.649498i	$-0.225021\pi$
$709$	40.4925	1.52073	0.760364	+	0.649498i	$0.225021\pi$
$710$	−0.882000	−0.0331009
$711$	0	0
$712$	−6.61027	−0.247730
$713$	19.5036	0.730416
$714$	0	0
$715$	39.7212	1.48549
$716$	21.3767	0.798885
$717$	0	0
$718$	−31.3665	−1.17059
$719$	−47.1136	−1.75704	−0.878520	−	0.477706i	$-0.841468\pi$
$719$	−47.1136	−1.75704	−0.878520	+	0.477706i	$0.841468\pi$
$720$	0	0
$721$	32.1874	1.19872
$722$	−29.3030	−1.09054
$723$	0	0
$724$	3.71263	0.137979
$725$	−3.03295	−0.112641
$726$	0	0
$727$	−16.0837	−0.596510	−0.298255	−	0.954486i	$-0.596405\pi$
$727$	−16.0837	−0.596510	−0.298255	+	0.954486i	$0.596405\pi$
$728$	−20.0815	−0.744271
$729$	0	0
$730$	−14.0815	−0.521181
$731$	−1.78997	−0.0662045
$732$	0	0
$733$	22.5068	0.831308	0.415654	−	0.909523i	$-0.363553\pi$
$733$	22.5068	0.831308	0.415654	+	0.909523i	$0.363553\pi$
$734$	−35.6751	−1.31679
$735$	0	0
$736$	−4.24735	−0.156560
$737$	−39.9756	−1.47252
$738$	0	0
$739$	−38.4653	−1.41497	−0.707484	−	0.706729i	$-0.750170\pi$
$739$	−38.4653	−1.41497	−0.707484	+	0.706729i	$0.750170\pi$
$740$	−1.00000	−0.0367607
$741$	0	0
$742$	−2.48604	−0.0912654
$743$	15.0732	0.552981	0.276490	−	0.961017i	$-0.410829\pi$
$743$	15.0732	0.552981	0.276490	+	0.961017i	$0.410829\pi$
$744$	0	0
$745$	14.7442	0.540186
$746$	0.674658	0.0247010
$747$	0	0
$748$	−7.72130	−0.282319
$749$	33.1510	1.21131
$750$	0	0
$751$	−29.3810	−1.07213	−0.536064	−	0.844177i	$-0.680089\pi$
$751$	−29.3810	−1.07213	−0.536064	+	0.844177i	$0.680089\pi$
$752$	8.01349	0.292222
$753$	0	0
$754$	18.9479	0.690042
$755$	−17.4312	−0.634388
$756$	0	0
$757$	−15.4966	−0.563232	−0.281616	−	0.959527i	$-0.590870\pi$
$757$	−15.4966	−0.563232	−0.281616	+	0.959527i	$0.590870\pi$
$758$	23.5840	0.856611
$759$	0	0
$760$	−6.95004	−0.252104
$761$	−6.06858	−0.219986	−0.109993	−	0.993932i	$-0.535083\pi$
$761$	−6.06858	−0.219986	−0.109993	+	0.993932i	$0.535083\pi$
$762$	0	0
$763$	−16.8438	−0.609785
$764$	−2.16230	−0.0782293
$765$	0	0
$766$	−2.06439	−0.0745894
$767$	79.4425	2.86850
$768$	0	0
$769$	−5.06235	−0.182553	−0.0912766	−	0.995826i	$-0.529095\pi$
$769$	−5.06235	−0.182553	−0.0912766	+	0.995826i	$0.529095\pi$
$770$	−20.4375	−0.736516
$771$	0	0
$772$	16.6638	0.599742
$773$	−39.8043	−1.43166	−0.715831	−	0.698273i	$-0.753952\pi$
$773$	−39.8043	−1.43166	−0.715831	+	0.698273i	$0.753952\pi$
$774$	0	0
$775$	4.59195	0.164948
$776$	2.53955	0.0911644
$777$	0	0
$778$	23.1960	0.831617
$779$	−24.1441	−0.865051
$780$	0	0
$781$	5.60783	0.200664
$782$	−5.15801	−0.184450
$783$	0	0
$784$	3.33241	0.119015
$785$	−17.8150	−0.635845
$786$	0	0
$787$	−7.01320	−0.249994	−0.124997	−	0.992157i	$-0.539892\pi$
$787$	−7.01320	−0.249994	−0.124997	+	0.992157i	$0.539892\pi$
$788$	−6.04001	−0.215167
$789$	0	0
$790$	2.88200	0.102537
$791$	−31.1258	−1.10671
$792$	0	0
$793$	−63.4247	−2.25228
$794$	−16.3948	−0.581829
$795$	0	0
$796$	25.6397	0.908775
$797$	−25.3665	−0.898526	−0.449263	−	0.893400i	$-0.648313\pi$
$797$	−25.3665	−0.898526	−0.449263	+	0.893400i	$0.648313\pi$
$798$	0	0
$799$	9.73164	0.344281
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	−34.7177	−1.22592
$803$	89.5316	3.15950
$804$	0	0
$805$	−13.6527	−0.481195
$806$	−28.6875	−1.01047
$807$	0	0
$808$	−6.46668	−0.227497
$809$	−43.0462	−1.51342	−0.756711	−	0.653749i	$-0.773195\pi$
$809$	−43.0462	−1.51342	−0.756711	+	0.653749i	$0.773195\pi$
$810$	0	0
$811$	−25.4571	−0.893921	−0.446960	−	0.894554i	$-0.647494\pi$
$811$	−25.4571	−0.893921	−0.446960	+	0.894554i	$0.647494\pi$
$812$	−9.74912	−0.342127
$813$	0	0
$814$	6.35809	0.222851
$815$	−23.1023	−0.809238
$816$	0	0
$817$	10.2440	0.358392
$818$	−1.16937	−0.0408859
$819$	0	0
$820$	−3.47395	−0.121316
$821$	−53.1185	−1.85385	−0.926924	−	0.375250i	$-0.877557\pi$
$821$	−53.1185	−1.85385	−0.926924	+	0.375250i	$0.877557\pi$
$822$	0	0
$823$	−29.1606	−1.01648	−0.508238	−	0.861217i	$-0.669703\pi$
$823$	−29.1606	−1.01648	−0.508238	+	0.861217i	$0.669703\pi$
$824$	−10.0135	−0.348837
$825$	0	0
$826$	−40.8750	−1.42222
$827$	−10.8256	−0.376443	−0.188222	−	0.982127i	$-0.560272\pi$
$827$	−10.8256	−0.376443	−0.188222	+	0.982127i	$0.560272\pi$
$828$	0	0
$829$	−31.3304	−1.08815	−0.544074	−	0.839037i	$-0.683119\pi$
$829$	−31.3304	−1.08815	−0.544074	+	0.839037i	$0.683119\pi$
$830$	16.9635	0.588813
$831$	0	0
$832$	6.24735	0.216588
$833$	4.04689	0.140217
$834$	0	0
$835$	4.24735	0.146986
$836$	44.1889	1.52831
$837$	0	0
$838$	−12.2095	−0.421770
$839$	32.3002	1.11513	0.557564	−	0.830134i	$-0.311736\pi$
$839$	32.3002	1.11513	0.557564	+	0.830134i	$0.311736\pi$
$840$	0	0
$841$	−19.8012	−0.682801
$842$	−6.86851	−0.236704
$843$	0	0
$844$	−18.9330	−0.651701
$845$	−26.0294	−0.895439
$846$	0	0
$847$	94.5848	3.24997
$848$	0.773406	0.0265589
$849$	0	0
$850$	−1.21441	−0.0416538
$851$	4.24735	0.145597
$852$	0	0
$853$	−13.0050	−0.445284	−0.222642	−	0.974900i	$-0.571468\pi$
$853$	−13.0050	−0.445284	−0.222642	+	0.974900i	$0.571468\pi$
$854$	32.6335	1.11669
$855$	0	0
$856$	−10.3132	−0.352500
$857$	−39.8550	−1.36142	−0.680711	−	0.732552i	$-0.738329\pi$
$857$	−39.8550	−1.36142	−0.680711	+	0.732552i	$0.738329\pi$
$858$	0	0
$859$	20.7971	0.709587	0.354794	−	0.934945i	$-0.384551\pi$
$859$	20.7971	0.709587	0.354794	+	0.934945i	$0.384551\pi$
$860$	1.47395	0.0502612
$861$	0	0
$862$	26.4955	0.902439
$863$	0.237652	0.00808977	0.00404489	−	0.999992i	$-0.498712\pi$
$863$	0.237652	0.00808977	0.00404489	+	0.999992i	$0.498712\pi$
$864$	0	0
$865$	12.9052	0.438790
$866$	8.18146	0.278017
$867$	0	0
$868$	14.7604	0.501000
$869$	−18.3240	−0.621600
$870$	0	0
$871$	39.2794	1.33093
$872$	5.24009	0.177452
$873$	0	0
$874$	29.5193	0.998504
$875$	−3.21441	−0.108667
$876$	0	0
$877$	19.0015	0.641636	0.320818	−	0.947141i	$-0.396042\pi$
$877$	19.0015	0.641636	0.320818	+	0.947141i	$0.396042\pi$
$878$	28.6990	0.968543
$879$	0	0
$880$	6.35809	0.214331
$881$	15.9956	0.538907	0.269453	−	0.963013i	$-0.413157\pi$
$881$	15.9956	0.538907	0.269453	+	0.963013i	$0.413157\pi$
$882$	0	0
$883$	−23.2437	−0.782214	−0.391107	−	0.920345i	$-0.627908\pi$
$883$	−23.2437	−0.782214	−0.391107	+	0.920345i	$0.627908\pi$
$884$	7.58682	0.255172
$885$	0	0
$886$	−19.8299	−0.666198
$887$	−46.5428	−1.56275	−0.781376	−	0.624060i	$-0.785482\pi$
$887$	−46.5428	−1.56275	−0.781376	+	0.624060i	$0.785482\pi$
$888$	0	0
$889$	−50.9807	−1.70984
$890$	6.61027	0.221577
$891$	0	0
$892$	3.48614	0.116724
$893$	−55.6941	−1.86373
$894$	0	0
$895$	−21.3767	−0.714545
$896$	−3.21441	−0.107386
$897$	0	0
$898$	15.2109	0.507594
$899$	−13.9271	−0.464496
$900$	0	0
$901$	0.939229	0.0312903
$902$	22.0877	0.735439
$903$	0	0
$904$	9.68323	0.322059
$905$	−3.71263	−0.123412
$906$	0	0
$907$	−4.64213	−0.154139	−0.0770697	−	0.997026i	$-0.524556\pi$
$907$	−4.64213	−0.154139	−0.0770697	+	0.997026i	$0.524556\pi$
$908$	−1.25442	−0.0416293
$909$	0	0
$910$	20.0815	0.665696
$911$	−18.6216	−0.616962	−0.308481	−	0.951231i	$-0.599821\pi$
$911$	−18.6216	−0.616962	−0.308481	+	0.951231i	$0.599821\pi$
$912$	0	0
$913$	−107.856	−3.56950
$914$	−32.9343	−1.08937
$915$	0	0
$916$	−6.82098	−0.225371
$917$	−56.4893	−1.86544
$918$	0	0
$919$	−22.7284	−0.749742	−0.374871	−	0.927077i	$-0.622313\pi$
$919$	−22.7284	−0.749742	−0.374871	+	0.927077i	$0.622313\pi$
$920$	4.24735	0.140031
$921$	0	0
$922$	−24.0266	−0.791275
$923$	−5.51017	−0.181369
$924$	0	0
$925$	1.00000	0.0328798
$926$	5.42911	0.178412
$927$	0	0
$928$	3.03295	0.0995614
$929$	42.9575	1.40939	0.704695	−	0.709511i	$-0.251084\pi$
$929$	42.9575	1.40939	0.704695	+	0.709511i	$0.251084\pi$
$930$	0	0
$931$	−23.1603	−0.759050
$932$	3.79202	0.124212
$933$	0	0
$934$	6.44369	0.210844
$935$	7.72130	0.252514
$936$	0	0
$937$	−36.7724	−1.20130	−0.600651	−	0.799511i	$-0.705092\pi$
$937$	−36.7724	−1.20130	−0.600651	+	0.799511i	$0.705092\pi$
$938$	−20.2101	−0.659885
$939$	0	0
$940$	−8.01349	−0.261371
$941$	19.2115	0.626278	0.313139	−	0.949707i	$-0.398619\pi$
$941$	19.2115	0.626278	0.313139	+	0.949707i	$0.398619\pi$
$942$	0	0
$943$	14.7551	0.480492
$944$	12.7162	0.413876
$945$	0	0
$946$	−9.37149	−0.304693
$947$	−1.12263	−0.0364805	−0.0182402	−	0.999834i	$-0.505806\pi$
$947$	−1.12263	−0.0364805	−0.0182402	+	0.999834i	$0.505806\pi$
$948$	0	0
$949$	−87.9723	−2.85570
$950$	6.95004	0.225489
$951$	0	0
$952$	−3.90359	−0.126516
$953$	−18.4153	−0.596531	−0.298265	−	0.954483i	$-0.596408\pi$
$953$	−18.4153	−0.596531	−0.298265	+	0.954483i	$0.596408\pi$
$954$	0	0
$955$	2.16230	0.0699704
$956$	−13.7247	−0.443890
$957$	0	0
$958$	−4.60766	−0.148867
$959$	22.9235	0.740239
$960$	0	0
$961$	−9.91402	−0.319807
$962$	−6.24735	−0.201423
$963$	0	0
$964$	1.86821	0.0601710
$965$	−16.6638	−0.536426
$966$	0	0
$967$	−9.65058	−0.310342	−0.155171	−	0.987888i	$-0.549593\pi$
$967$	−9.65058	−0.310342	−0.155171	+	0.987888i	$0.549593\pi$
$968$	−29.4253	−0.945765
$969$	0	0
$970$	−2.53955	−0.0815399
$971$	3.29724	0.105814	0.0529068	−	0.998599i	$-0.483151\pi$
$971$	3.29724	0.105814	0.0529068	+	0.998599i	$0.483151\pi$
$972$	0	0
$973$	−21.8619	−0.700861
$974$	−5.87205	−0.188153
$975$	0	0
$976$	−10.1523	−0.324966
$977$	−17.1614	−0.549041	−0.274520	−	0.961581i	$-0.588519\pi$
$977$	−17.1614	−0.549041	−0.274520	+	0.961581i	$0.588519\pi$
$978$	0	0
$979$	−42.0287	−1.34324
$980$	−3.33241	−0.106450
$981$	0	0
$982$	16.0584	0.512443
$983$	52.4255	1.67211	0.836057	−	0.548643i	$-0.184855\pi$
$983$	52.4255	1.67211	0.836057	+	0.548643i	$0.184855\pi$
$984$	0	0
$985$	6.04001	0.192451
$986$	3.68323	0.117298
$987$	0	0
$988$	−43.4193	−1.38135
$989$	−6.26038	−0.199068
$990$	0	0
$991$	28.1512	0.894253	0.447126	−	0.894471i	$-0.352447\pi$
$991$	28.1512	0.894253	0.447126	+	0.894471i	$0.352447\pi$
$992$	−4.59195	−0.145794
$993$	0	0
$994$	2.83511	0.0899241
$995$	−25.6397	−0.812833
$996$	0	0
$997$	2.21996	0.0703069	0.0351535	−	0.999382i	$-0.488808\pi$
$997$	2.21996	0.0703069	0.0351535	+	0.999382i	$0.488808\pi$
$998$	10.2095	0.323176
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3330.2.a.bk.1.4	✓	5
3.2	odd	2		3330.2.a.bl.1.4	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3330.2.a.bk.1.4	✓	5	1.1	even	1	trivial
3330.2.a.bl.1.4	yes	5	3.2	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:	\( N \)	\(=\)	\( 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3330.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +3.21441 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +3.21441 q^{7} -1.00000 q^{8} +1.00000 q^{10} -6.35809 q^{11} +6.24735 q^{13} -3.21441 q^{14} +1.00000 q^{16} +1.21441 q^{17} -6.95004 q^{19} -1.00000 q^{20} +6.35809 q^{22} +4.24735 q^{23} +1.00000 q^{25} -6.24735 q^{26} +3.21441 q^{28} -3.03295 q^{29} +4.59195 q^{31} -1.00000 q^{32} -1.21441 q^{34} -3.21441 q^{35} +1.00000 q^{37} +6.95004 q^{38} +1.00000 q^{40} +3.47395 q^{41} -1.47395 q^{43} -6.35809 q^{44} -4.24735 q^{46} +8.01349 q^{47} +3.33241 q^{49} -1.00000 q^{50} +6.24735 q^{52} +0.773406 q^{53} +6.35809 q^{55} -3.21441 q^{56} +3.03295 q^{58} +12.7162 q^{59} -10.1523 q^{61} -4.59195 q^{62} +1.00000 q^{64} -6.24735 q^{65} +6.28737 q^{67} +1.21441 q^{68} +3.21441 q^{70} -0.882000 q^{71} -14.0815 q^{73} -1.00000 q^{74} -6.95004 q^{76} -20.4375 q^{77} +2.88200 q^{79} -1.00000 q^{80} -3.47395 q^{82} +16.9635 q^{83} -1.21441 q^{85} +1.47395 q^{86} +6.35809 q^{88} +6.61027 q^{89} +20.0815 q^{91} +4.24735 q^{92} -8.01349 q^{94} +6.95004 q^{95} -2.53955 q^{97} -3.33241 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{2} + 5 q^{4} - 5 q^{5} + 3 q^{7} - 5 q^{8}+O(q^{10}) \) 5 * q - 5 * q^2 + 5 * q^4 - 5 * q^5 + 3 * q^7 - 5 * q^8	\( 5 q - 5 q^{2} + 5 q^{4} - 5 q^{5} + 3 q^{7} - 5 q^{8} + 5 q^{10} - 5 q^{11} + 5 q^{13} - 3 q^{14} + 5 q^{16} - 7 q^{17} - q^{19} - 5 q^{20} + 5 q^{22} - 5 q^{23} + 5 q^{25} - 5 q^{26} + 3 q^{28} - 2 q^{29} + 16 q^{31} - 5 q^{32} + 7 q^{34} - 3 q^{35} + 5 q^{37} + q^{38} + 5 q^{40} - 2 q^{41} + 12 q^{43} - 5 q^{44} + 5 q^{46} - 6 q^{47} + 16 q^{49} - 5 q^{50} + 5 q^{52} - 3 q^{53} + 5 q^{55} - 3 q^{56} + 2 q^{58} + 10 q^{59} + 16 q^{61} - 16 q^{62} + 5 q^{64} - 5 q^{65} + 4 q^{67} - 7 q^{68} + 3 q^{70} + 8 q^{71} - 3 q^{73} - 5 q^{74} - q^{76} - 3 q^{77} + 2 q^{79} - 5 q^{80} + 2 q^{82} + 5 q^{83} + 7 q^{85} - 12 q^{86} + 5 q^{88} + 7 q^{89} + 33 q^{91} - 5 q^{92} + 6 q^{94} + q^{95} + 14 q^{97} - 16 q^{98}+O(q^{100}) \) 5 * q - 5 * q^2 + 5 * q^4 - 5 * q^5 + 3 * q^7 - 5 * q^8 + 5 * q^10 - 5 * q^11 + 5 * q^13 - 3 * q^14 + 5 * q^16 - 7 * q^17 - q^19 - 5 * q^20 + 5 * q^22 - 5 * q^23 + 5 * q^25 - 5 * q^26 + 3 * q^28 - 2 * q^29 + 16 * q^31 - 5 * q^32 + 7 * q^34 - 3 * q^35 + 5 * q^37 + q^38 + 5 * q^40 - 2 * q^41 + 12 * q^43 - 5 * q^44 + 5 * q^46 - 6 * q^47 + 16 * q^49 - 5 * q^50 + 5 * q^52 - 3 * q^53 + 5 * q^55 - 3 * q^56 + 2 * q^58 + 10 * q^59 + 16 * q^61 - 16 * q^62 + 5 * q^64 - 5 * q^65 + 4 * q^67 - 7 * q^68 + 3 * q^70 + 8 * q^71 - 3 * q^73 - 5 * q^74 - q^76 - 3 * q^77 + 2 * q^79 - 5 * q^80 + 2 * q^82 + 5 * q^83 + 7 * q^85 - 12 * q^86 + 5 * q^88 + 7 * q^89 + 33 * q^91 - 5 * q^92 + 6 * q^94 + q^95 + 14 * q^97 - 16 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more