LMFDB - Embedded newform 3381.2.a.w.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3381,2,Mod(1,3381)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3381, 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("3381.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3381 = 3 \cdot 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3381.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:	$26.9974209234$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.24197.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} - x + 2 $ x^4 - 6*x^2 - x + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 483)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-2.27460$ of defining polynomial
Character	$\chi$	$=$	3381.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.27460 q^{2} +1.00000 q^{3} +3.17380 q^{4} -2.39532 q^{5} +2.27460 q^{6} +2.66992 q^{8} +1.00000 q^{9} -5.44840 q^{10} -4.56912 q^{11} +3.17380 q^{12} -5.05307 q^{13} -2.39532 q^{15} -0.274599 q^{16} +1.44840 q^{17} +2.27460 q^{18} -1.98008 q^{19} -7.60227 q^{20} -10.3929 q^{22} -1.00000 q^{23} +2.66992 q^{24} +0.737570 q^{25} -11.4937 q^{26} +1.00000 q^{27} +3.68985 q^{29} -5.44840 q^{30} +3.44840 q^{31} -5.96444 q^{32} -4.56912 q^{33} +3.29452 q^{34} +3.17380 q^{36} -3.99759 q^{37} -4.50388 q^{38} -5.05307 q^{39} -6.39532 q^{40} -1.77848 q^{41} -1.25467 q^{43} -14.5015 q^{44} -2.39532 q^{45} -2.27460 q^{46} -2.87928 q^{47} -0.274599 q^{48} +1.67768 q^{50} +1.44840 q^{51} -16.0374 q^{52} -10.4128 q^{53} +2.27460 q^{54} +10.9445 q^{55} -1.98008 q^{57} +8.39292 q^{58} +14.1315 q^{59} -7.60227 q^{60} -8.37540 q^{61} +7.84372 q^{62} -13.0175 q^{64} +12.1037 q^{65} -10.3929 q^{66} -7.73517 q^{67} +4.59692 q^{68} -1.00000 q^{69} -0.946925 q^{71} +2.66992 q^{72} -3.86711 q^{73} -9.09292 q^{74} +0.737570 q^{75} -6.28436 q^{76} -11.4937 q^{78} +4.23904 q^{79} +0.657752 q^{80} +1.00000 q^{81} -4.04532 q^{82} -10.8281 q^{83} -3.46938 q^{85} -2.85388 q^{86} +3.68985 q^{87} -12.1992 q^{88} -0.262430 q^{89} -5.44840 q^{90} -3.17380 q^{92} +3.44840 q^{93} -6.54920 q^{94} +4.74292 q^{95} -5.96444 q^{96} +7.20695 q^{97} -4.56912 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 4 q^{4} - 5 q^{5} - 3 q^{8} + 4 q^{9} - 4 q^{10} - 5 q^{11} + 4 q^{12} - 7 q^{13} - 5 q^{15} + 8 q^{16} - 12 q^{17} - 3 q^{19} + q^{20} - q^{22} - 4 q^{23} - 3 q^{24} + 7 q^{25} - 5 q^{26}+ \cdots - 5 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 + 4 * q^4 - 5 * q^5 - 3 * q^8 + 4 * q^9 - 4 * q^10 - 5 * q^11 + 4 * q^12 - 7 * q^13 - 5 * q^15 + 8 * q^16 - 12 * q^17 - 3 * q^19 + q^20 - q^22 - 4 * q^23 - 3 * q^24 + 7 * q^25 - 5 * q^26 + 4 * q^27 + 6 * q^29 - 4 * q^30 - 4 * q^31 - 6 * q^32 - 5 * q^33 + 9 * q^34 + 4 * q^36 + 20 * q^37 - 23 * q^38 - 7 * q^39 - 21 * q^40 - 3 * q^41 + 9 * q^43 - 27 * q^44 - 5 * q^45 - 7 * q^47 + 8 * q^48 + 3 * q^50 - 12 * q^51 - 38 * q^52 - 6 * q^53 + 21 * q^55 - 3 * q^57 - 7 * q^58 + 2 * q^59 + q^60 - 24 * q^61 + 9 * q^62 - 21 * q^64 - 14 * q^65 - q^66 + q^67 + 13 * q^68 - 4 * q^69 - 17 * q^71 - 3 * q^72 - 16 * q^73 - 33 * q^74 + 7 * q^75 + 25 * q^76 - 5 * q^78 - 10 * q^79 - 6 * q^80 + 4 * q^81 + 7 * q^82 - 8 * q^83 + 17 * q^85 - 35 * q^86 + 6 * q^87 - 12 * q^88 + 3 * q^89 - 4 * q^90 - 4 * q^92 - 4 * q^93 - 8 * q^94 - 3 * q^95 - 6 * q^96 + 2 * q^97 - 5 * q^99

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 $

$2$	2.27460	1.60838	0.804192	−	0.594370i	$-0.202598\pi$
$2$	2.27460	1.60838	0.804192	+	0.594370i	$0.202598\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	3.17380	1.58690
$5$	−2.39532	−1.07122	−0.535610	−	0.844465i	$-0.679918\pi$
$5$	−2.39532	−1.07122	−0.535610	+	0.844465i	$0.679918\pi$
$6$	2.27460	0.928601
$7$	0	0
$7$	0	0
$8$	2.66992	0.943960
$9$	1.00000	0.333333
$10$	−5.44840	−1.72293
$11$	−4.56912	−1.37764	−0.688821	−	0.724931i	$-0.741871\pi$
$11$	−4.56912	−1.37764	−0.688821	+	0.724931i	$0.741871\pi$
$12$	3.17380	0.916197
$13$	−5.05307	−1.40147	−0.700735	−	0.713421i	$-0.747145\pi$
$13$	−5.05307	−1.40147	−0.700735	+	0.713421i	$0.747145\pi$
$14$	0	0
$15$	−2.39532	−0.618470
$16$	−0.274599	−0.0686497
$17$	1.44840	0.351288	0.175644	−	0.984454i	$-0.443799\pi$
$17$	1.44840	0.351288	0.175644	+	0.984454i	$0.443799\pi$
$18$	2.27460	0.536128
$19$	−1.98008	−0.454261	−0.227130	−	0.973864i	$-0.572934\pi$
$19$	−1.98008	−0.454261	−0.227130	+	0.973864i	$0.572934\pi$
$20$	−7.60227	−1.69992
$21$	0	0
$22$	−10.3929	−2.21578
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	2.66992	0.544995
$25$	0.737570	0.147514
$26$	−11.4937	−2.25410
$27$	1.00000	0.192450
$28$	0	0
$29$	3.68985	0.685187	0.342594	−	0.939484i	$-0.388695\pi$
$29$	3.68985	0.685187	0.342594	+	0.939484i	$0.388695\pi$
$30$	−5.44840	−0.994737
$31$	3.44840	0.619350	0.309675	−	0.950842i	$-0.399780\pi$
$31$	3.44840	0.619350	0.309675	+	0.950842i	$0.399780\pi$
$32$	−5.96444	−1.05437
$33$	−4.56912	−0.795382
$34$	3.29452	0.565006
$35$	0	0
$36$	3.17380	0.528966
$37$	−3.99759	−0.657201	−0.328600	−	0.944469i	$-0.606577\pi$
$37$	−3.99759	−0.657201	−0.328600	+	0.944469i	$0.606577\pi$
$38$	−4.50388	−0.730625
$39$	−5.05307	−0.809140
$40$	−6.39532	−1.01119
$41$	−1.77848	−0.277751	−0.138876	−	0.990310i	$-0.544349\pi$
$41$	−1.77848	−0.277751	−0.138876	+	0.990310i	$0.544349\pi$
$42$	0	0
$43$	−1.25467	−0.191336	−0.0956680	−	0.995413i	$-0.530499\pi$
$43$	−1.25467	−0.191336	−0.0956680	+	0.995413i	$0.530499\pi$
$44$	−14.5015	−2.18618
$45$	−2.39532	−0.357074
$46$	−2.27460	−0.335371
$47$	−2.87928	−0.419986	−0.209993	−	0.977703i	$-0.567344\pi$
$47$	−2.87928	−0.419986	−0.209993	+	0.977703i	$0.567344\pi$
$48$	−0.274599	−0.0396349
$49$	0	0
$50$	1.67768	0.237259
$51$	1.44840	0.202816
$52$	−16.0374	−2.22399
$53$	−10.4128	−1.43031	−0.715157	−	0.698964i	$-0.753645\pi$
$53$	−10.4128	−1.43031	−0.715157	+	0.698964i	$0.753645\pi$
$54$	2.27460	0.309534
$55$	10.9445	1.47576
$56$	0	0
$57$	−1.98008	−0.262267
$58$	8.39292	1.10204
$59$	14.1315	1.83977	0.919885	−	0.392188i	$-0.128282\pi$
$59$	14.1315	1.83977	0.919885	+	0.392188i	$0.128282\pi$
$60$	−7.60227	−0.981449
$61$	−8.37540	−1.07236	−0.536180	−	0.844104i	$-0.680133\pi$
$61$	−8.37540	−1.07236	−0.536180	+	0.844104i	$0.680133\pi$
$62$	7.84372	0.996153
$63$	0	0
$64$	−13.0175	−1.62719
$65$	12.1037	1.50128
$66$	−10.3929	−1.27928
$67$	−7.73517	−0.945001	−0.472500	−	0.881330i	$-0.656648\pi$
$67$	−7.73517	−0.945001	−0.472500	+	0.881330i	$0.656648\pi$
$68$	4.59692	0.557459
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−0.946925	−0.112379	−0.0561897	−	0.998420i	$-0.517895\pi$
$71$	−0.946925	−0.112379	−0.0561897	+	0.998420i	$0.517895\pi$
$72$	2.66992	0.314653
$73$	−3.86711	−0.452611	−0.226305	−	0.974056i	$-0.572665\pi$
$73$	−3.86711	−0.452611	−0.226305	+	0.974056i	$0.572665\pi$
$74$	−9.09292	−1.05703
$75$	0.737570	0.0851673
$76$	−6.28436	−0.720866
$77$	0	0
$78$	−11.4937	−1.30141
$79$	4.23904	0.476930	0.238465	−	0.971151i	$-0.423356\pi$
$79$	4.23904	0.476930	0.238465	+	0.971151i	$0.423356\pi$
$80$	0.657752	0.0735389
$81$	1.00000	0.111111
$82$	−4.04532	−0.446731
$83$	−10.8281	−1.18854	−0.594269	−	0.804267i	$-0.702558\pi$
$83$	−10.8281	−1.18854	−0.594269	+	0.804267i	$0.702558\pi$
$84$	0	0
$85$	−3.46938	−0.376307
$86$	−2.85388	−0.307742
$87$	3.68985	0.395593
$88$	−12.1992	−1.30044
$89$	−0.262430	−0.0278175	−0.0139087	−	0.999903i	$-0.504427\pi$
$89$	−0.262430	−0.0278175	−0.0139087	+	0.999903i	$0.504427\pi$
$90$	−5.44840	−0.574312
$91$	0	0
$92$	−3.17380	−0.330891
$93$	3.44840	0.357582
$94$	−6.54920	−0.675498
$95$	4.74292	0.486613
$96$	−5.96444	−0.608744
$97$	7.20695	0.731755	0.365877	−	0.930663i	$-0.380769\pi$
$97$	7.20695	0.731755	0.365877	+	0.930663i	$0.380769\pi$
$98$	0	0
$99$	−4.56912	−0.459214
$100$	2.34090	0.234090
$101$	8.26684	0.822582	0.411291	−	0.911504i	$-0.365078\pi$
$101$	8.26684	0.822582	0.411291	+	0.911504i	$0.365078\pi$
$102$	3.29452	0.326206
$103$	19.2619	1.89793	0.948966	−	0.315378i	$-0.102131\pi$
$103$	19.2619	1.89793	0.948966	+	0.315378i	$0.102131\pi$
$104$	−13.4913	−1.32293
$105$	0	0
$106$	−23.6850	−2.30049
$107$	−11.0531	−1.06854	−0.534271	−	0.845314i	$-0.679414\pi$
$107$	−11.0531	−1.06854	−0.534271	+	0.845314i	$0.679414\pi$
$108$	3.17380	0.305399
$109$	12.9047	1.23604	0.618022	−	0.786161i	$-0.287934\pi$
$109$	12.9047	1.23604	0.618022	+	0.786161i	$0.287934\pi$
$110$	24.8944	2.37359
$111$	−3.99759	−0.379435
$112$	0	0
$113$	−1.25708	−0.118256	−0.0591281	−	0.998250i	$-0.518832\pi$
$113$	−1.25708	−0.118256	−0.0591281	+	0.998250i	$0.518832\pi$
$114$	−4.50388	−0.421827
$115$	2.39532	0.223365
$116$	11.7108	1.08732
$117$	−5.05307	−0.467157
$118$	32.1436	2.95906
$119$	0	0
$120$	−6.39532	−0.583810
$121$	9.87687	0.897897
$122$	−19.0507	−1.72477
$123$	−1.77848	−0.160360
$124$	10.9445	0.982847
$125$	10.2099	0.913201
$126$	0	0
$127$	19.3385	1.71601	0.858007	−	0.513638i	$-0.171703\pi$
$127$	19.3385	1.71601	0.858007	+	0.513638i	$0.171703\pi$
$128$	−17.6807	−1.56277
$129$	−1.25467	−0.110468
$130$	27.5312	2.41464
$131$	−19.4659	−1.70075	−0.850373	−	0.526181i	$-0.823623\pi$
$131$	−19.4659	−1.70075	−0.850373	+	0.526181i	$0.823623\pi$
$132$	−14.5015	−1.26219
$133$	0	0
$134$	−17.5944	−1.51992
$135$	−2.39532	−0.206157
$136$	3.86711	0.331602
$137$	20.4980	1.75126	0.875632	−	0.482980i	$-0.160446\pi$
$137$	20.4980	1.75126	0.875632	+	0.482980i	$0.160446\pi$
$138$	−2.27460	−0.193627
$139$	−21.0507	−1.78549	−0.892747	−	0.450558i	$-0.851225\pi$
$139$	−21.0507	−1.78549	−0.892747	+	0.450558i	$0.851225\pi$
$140$	0	0
$141$	−2.87928	−0.242479
$142$	−2.15387	−0.180749
$143$	23.0881	1.93072
$144$	−0.274599	−0.0228832
$145$	−8.83837	−0.733987
$146$	−8.79612	−0.727972
$147$	0	0
$148$	−12.6876	−1.04291
$149$	−19.1159	−1.56604	−0.783018	−	0.621999i	$-0.786321\pi$
$149$	−19.1159	−1.56604	−0.783018	+	0.621999i	$0.786321\pi$
$150$	1.67768	0.136982
$151$	18.2542	1.48550	0.742751	−	0.669568i	$-0.233521\pi$
$151$	18.2542	1.48550	0.742751	+	0.669568i	$0.233521\pi$
$152$	−5.28665	−0.428804
$153$	1.44840	0.117096
$154$	0	0
$155$	−8.26002	−0.663461
$156$	−16.0374	−1.28402
$157$	−6.63783	−0.529756	−0.264878	−	0.964282i	$-0.585332\pi$
$157$	−6.63783	−0.529756	−0.264878	+	0.964282i	$0.585332\pi$
$158$	9.64212	0.767086
$159$	−10.4128	−0.825792
$160$	14.2868	1.12947
$161$	0	0
$162$	2.27460	0.178709
$163$	21.1080	1.65331	0.826655	−	0.562710i	$-0.190241\pi$
$163$	21.1080	1.65331	0.826655	+	0.562710i	$0.190241\pi$
$164$	−5.64453	−0.440763
$165$	10.9445	0.852030
$166$	−24.6296	−1.91162
$167$	1.01217	0.0783240	0.0391620	−	0.999233i	$-0.487531\pi$
$167$	1.01217	0.0783240	0.0391620	+	0.999233i	$0.487531\pi$
$168$	0	0
$169$	12.5336	0.964120
$170$	−7.89144	−0.605246
$171$	−1.98008	−0.151420
$172$	−3.98208	−0.303631
$173$	−15.2069	−1.15616	−0.578081	−	0.815979i	$-0.696198\pi$
$173$	−15.2069	−1.15616	−0.578081	+	0.815979i	$0.696198\pi$
$174$	8.39292	0.636265
$175$	0	0
$176$	1.25467	0.0945746
$177$	14.1315	1.06219
$178$	−0.596922	−0.0447412
$179$	18.1491	1.35652	0.678262	−	0.734820i	$-0.262733\pi$
$179$	18.1491	1.35652	0.678262	+	0.734820i	$0.262733\pi$
$180$	−7.60227	−0.566640
$181$	−10.5866	−0.786899	−0.393449	−	0.919346i	$-0.628718\pi$
$181$	−10.5866	−0.786899	−0.393449	+	0.919346i	$0.628718\pi$
$182$	0	0
$183$	−8.37540	−0.619127
$184$	−2.66992	−0.196829
$185$	9.57553	0.704007
$186$	7.84372	0.575129
$187$	−6.61790	−0.483949
$188$	−9.13824	−0.666475
$189$	0	0
$190$	10.7882	0.782661
$191$	−19.8004	−1.43271	−0.716354	−	0.697737i	$-0.754190\pi$
$191$	−19.8004	−1.43271	−0.716354	+	0.697737i	$0.754190\pi$
$192$	−13.0175	−0.939459
$193$	22.4956	1.61927	0.809635	−	0.586934i	$-0.199665\pi$
$193$	22.4956	1.61927	0.809635	+	0.586934i	$0.199665\pi$
$194$	16.3929	1.17694
$195$	12.1037	0.866767
$196$	0	0
$197$	1.67338	0.119224	0.0596118	−	0.998222i	$-0.481014\pi$
$197$	1.67338	0.119224	0.0596118	+	0.998222i	$0.481014\pi$
$198$	−10.3929	−0.738592
$199$	−5.82380	−0.412838	−0.206419	−	0.978464i	$-0.566181\pi$
$199$	−5.82380	−0.412838	−0.206419	+	0.978464i	$0.566181\pi$
$200$	1.96925	0.139247
$201$	−7.73517	−0.545596
$202$	18.8038	1.32303
$203$	0	0
$204$	4.59692	0.321849
$205$	4.26002	0.297533
$206$	43.8131	3.05260
$207$	−1.00000	−0.0695048
$208$	1.38757	0.0962105
$209$	9.04721	0.625808
$210$	0	0
$211$	−14.2809	−0.983138	−0.491569	−	0.870839i	$-0.663576\pi$
$211$	−14.2809	−0.983138	−0.491569	+	0.870839i	$0.663576\pi$
$212$	−33.0483	−2.26976
$213$	−0.946925	−0.0648822
$214$	−25.1413	−1.71862
$215$	3.00535	0.204963
$216$	2.66992	0.181665
$217$	0	0
$218$	29.3529	1.98803
$219$	−3.86711	−0.261315
$220$	34.7357	2.34188
$221$	−7.31886	−0.492320
$222$	−9.09292	−0.610277
$223$	20.0031	1.33950	0.669752	−	0.742585i	$-0.266400\pi$
$223$	20.0031	1.33950	0.669752	+	0.742585i	$0.266400\pi$
$224$	0	0
$225$	0.737570	0.0491714
$226$	−2.85935	−0.190201
$227$	−18.4547	−1.22488	−0.612441	−	0.790517i	$-0.709812\pi$
$227$	−18.4547	−1.22488	−0.612441	+	0.790517i	$0.709812\pi$
$228$	−6.28436	−0.416192
$229$	17.2299	1.13859	0.569293	−	0.822135i	$-0.307217\pi$
$229$	17.2299	1.13859	0.569293	+	0.822135i	$0.307217\pi$
$230$	5.44840	0.359257
$231$	0	0
$232$	9.85160	0.646789
$233$	−16.2844	−1.06682	−0.533412	−	0.845856i	$-0.679090\pi$
$233$	−16.2844	−1.06682	−0.533412	+	0.845856i	$0.679090\pi$
$234$	−11.4937	−0.751368
$235$	6.89679	0.449897
$236$	44.8507	2.91953
$237$	4.23904	0.275355
$238$	0	0
$239$	−26.1539	−1.69175	−0.845877	−	0.533378i	$-0.820922\pi$
$239$	−26.1539	−1.69175	−0.845877	+	0.533378i	$0.820922\pi$
$240$	0.657752	0.0424577
$241$	−8.67527	−0.558823	−0.279412	−	0.960171i	$-0.590139\pi$
$241$	−8.67527	−0.558823	−0.279412	+	0.960171i	$0.590139\pi$
$242$	22.4659	1.44416
$243$	1.00000	0.0641500
$244$	−26.5818	−1.70173
$245$	0	0
$246$	−4.04532	−0.257920
$247$	10.0055	0.636633
$248$	9.20695	0.584642
$249$	−10.8281	−0.686202
$250$	23.2234	1.46878
$251$	−9.22888	−0.582522	−0.291261	−	0.956644i	$-0.594075\pi$
$251$	−9.22888	−0.582522	−0.291261	+	0.956644i	$0.594075\pi$
$252$	0	0
$253$	4.56912	0.287258
$254$	43.9873	2.76001
$255$	−3.46938	−0.217261
$256$	−14.1816	−0.886347
$257$	−17.7606	−1.10787	−0.553937	−	0.832559i	$-0.686875\pi$
$257$	−17.7606	−1.10787	−0.553937	+	0.832559i	$0.686875\pi$
$258$	−2.85388	−0.177675
$259$	0	0
$260$	38.4148	2.38239
$261$	3.68985	0.228396
$262$	−44.2771	−2.73545
$263$	−10.2721	−0.633403	−0.316702	−	0.948525i	$-0.602575\pi$
$263$	−10.2721	−0.633403	−0.316702	+	0.948525i	$0.602575\pi$
$264$	−12.1992	−0.750809
$265$	24.9421	1.53218
$266$	0	0
$267$	−0.262430	−0.0160604
$268$	−24.5499	−1.49962
$269$	17.5312	1.06889	0.534447	−	0.845202i	$-0.320520\pi$
$269$	17.5312	1.06889	0.534447	+	0.845202i	$0.320520\pi$
$270$	−5.44840	−0.331579
$271$	−21.1602	−1.28539	−0.642695	−	0.766123i	$-0.722184\pi$
$271$	−21.1602	−1.28539	−0.642695	+	0.766123i	$0.722184\pi$
$272$	−0.397728	−0.0241158
$273$	0	0
$274$	46.6247	2.81670
$275$	−3.37005	−0.203222
$276$	−3.17380	−0.191040
$277$	2.92941	0.176011	0.0880055	−	0.996120i	$-0.471951\pi$
$277$	2.92941	0.176011	0.0880055	+	0.996120i	$0.471951\pi$
$278$	−47.8818	−2.87176
$279$	3.44840	0.206450
$280$	0	0
$281$	−26.8655	−1.60266	−0.801332	−	0.598220i	$-0.795875\pi$
$281$	−26.8655	−1.60266	−0.801332	+	0.598220i	$0.795875\pi$
$282$	−6.54920	−0.389999
$283$	22.0584	1.31124	0.655619	−	0.755092i	$-0.272408\pi$
$283$	22.0584	1.31124	0.655619	+	0.755092i	$0.272408\pi$
$284$	−3.00535	−0.178335
$285$	4.74292	0.280946
$286$	52.5162	3.10535
$287$	0	0
$288$	−5.96444	−0.351458
$289$	−14.9021	−0.876597
$290$	−20.1037	−1.18053
$291$	7.20695	0.422479
$292$	−12.2734	−0.718248
$293$	−23.7537	−1.38771	−0.693854	−	0.720116i	$-0.744089\pi$
$293$	−23.7537	−1.38771	−0.693854	+	0.720116i	$0.744089\pi$
$294$	0	0
$295$	−33.8496	−1.97080
$296$	−10.6733	−0.620371
$297$	−4.56912	−0.265127
$298$	−43.4810	−2.51879
$299$	5.05307	0.292227
$300$	2.34090	0.135152
$301$	0	0
$302$	41.5209	2.38926
$303$	8.26684	0.474918
$304$	0.543726	0.0311848
$305$	20.0618	1.14873
$306$	3.29452	0.188335
$307$	−9.61015	−0.548480	−0.274240	−	0.961661i	$-0.588426\pi$
$307$	−9.61015	−0.548480	−0.274240	+	0.961661i	$0.588426\pi$
$308$	0	0
$309$	19.2619	1.09577
$310$	−18.7882	−1.06710
$311$	−13.5642	−0.769155	−0.384577	−	0.923093i	$-0.625653\pi$
$311$	−13.5642	−0.769155	−0.384577	+	0.923093i	$0.625653\pi$
$312$	−13.4913	−0.763795
$313$	−1.23168	−0.0696189	−0.0348095	−	0.999394i	$-0.511082\pi$
$313$	−1.23168	−0.0696189	−0.0348095	+	0.999394i	$0.511082\pi$
$314$	−15.0984	−0.852052
$315$	0	0
$316$	13.4539	0.756839
$317$	16.0584	0.901931	0.450965	−	0.892541i	$-0.351080\pi$
$317$	16.0584	0.901931	0.450965	+	0.892541i	$0.351080\pi$
$318$	−23.6850	−1.32819
$319$	−16.8594	−0.943942
$320$	31.1812	1.74308
$321$	−11.0531	−0.616922
$322$	0	0
$323$	−2.86794	−0.159576
$324$	3.17380	0.176322
$325$	−3.72700	−0.206737
$326$	48.0123	2.65916
$327$	12.9047	0.713630
$328$	−4.74839	−0.262186
$329$	0	0
$330$	24.8944	1.37039
$331$	−12.5424	−0.689391	−0.344696	−	0.938714i	$-0.612018\pi$
$331$	−12.5424	−0.689391	−0.344696	+	0.938714i	$0.612018\pi$
$332$	−34.3662	−1.88609
$333$	−3.99759	−0.219067
$334$	2.30228	0.125975
$335$	18.5282	1.01230
$336$	0	0
$337$	9.62661	0.524395	0.262197	−	0.965014i	$-0.415553\pi$
$337$	9.62661	0.524395	0.262197	+	0.965014i	$0.415553\pi$
$338$	28.5088	1.55068
$339$	−1.25708	−0.0682752
$340$	−11.0111	−0.597161
$341$	−15.7561	−0.853243
$342$	−4.50388	−0.243542
$343$	0	0
$344$	−3.34988	−0.180614
$345$	2.39532	0.128960
$346$	−34.5897	−1.85955
$347$	−9.94199	−0.533714	−0.266857	−	0.963736i	$-0.585985\pi$
$347$	−9.94199	−0.533714	−0.266857	+	0.963736i	$0.585985\pi$
$348$	11.7108	0.627766
$349$	12.7274	0.681283	0.340641	−	0.940193i	$-0.389356\pi$
$349$	12.7274	0.681283	0.340641	+	0.940193i	$0.389356\pi$
$350$	0	0
$351$	−5.05307	−0.269713
$352$	27.2523	1.45255
$353$	1.67433	0.0891159	0.0445579	−	0.999007i	$-0.485812\pi$
$353$	1.67433	0.0891159	0.0445579	+	0.999007i	$0.485812\pi$
$354$	32.1436	1.70841
$355$	2.26819	0.120383
$356$	−0.832899	−0.0441436
$357$	0	0
$358$	41.2818	2.18181
$359$	−36.9662	−1.95100	−0.975500	−	0.220001i	$-0.929394\pi$
$359$	−36.9662	−1.95100	−0.975500	+	0.220001i	$0.929394\pi$
$360$	−6.39532	−0.337063
$361$	−15.0793	−0.793647
$362$	−24.0804	−1.26564
$363$	9.87687	0.518401
$364$	0	0
$365$	9.26297	0.484846
$366$	−19.0507	−0.995794
$367$	22.5080	1.17491	0.587454	−	0.809258i	$-0.300130\pi$
$367$	22.5080	1.17491	0.587454	+	0.809258i	$0.300130\pi$
$368$	0.274599	0.0143144
$369$	−1.77848	−0.0925838
$370$	21.7805	1.13231
$371$	0	0
$372$	10.9445	0.567447
$373$	23.5896	1.22142	0.610711	−	0.791853i	$-0.290884\pi$
$373$	23.5896	1.22142	0.610711	+	0.791853i	$0.290884\pi$
$374$	−15.0531	−0.778376
$375$	10.2099	0.527237
$376$	−7.68744	−0.396449
$377$	−18.6451	−0.960270
$378$	0	0
$379$	6.93664	0.356311	0.178156	−	0.984002i	$-0.442987\pi$
$379$	6.93664	0.356311	0.178156	+	0.984002i	$0.442987\pi$
$380$	15.0531	0.772206
$381$	19.3385	0.990741
$382$	−45.0380	−2.30434
$383$	22.8281	1.16646	0.583230	−	0.812307i	$-0.301788\pi$
$383$	22.8281	1.16646	0.583230	+	0.812307i	$0.301788\pi$
$384$	−17.6807	−0.902267
$385$	0	0
$386$	51.1685	2.60441
$387$	−1.25467	−0.0637787
$388$	22.8734	1.16122
$389$	16.1350	0.818077	0.409039	−	0.912517i	$-0.365864\pi$
$389$	16.1350	0.818077	0.409039	+	0.912517i	$0.365864\pi$
$390$	27.5312	1.39409
$391$	−1.44840	−0.0732486
$392$	0	0
$393$	−19.4659	−0.981926
$394$	3.80628	0.191757
$395$	−10.1539	−0.510897
$396$	−14.5015	−0.728726
$397$	−18.5687	−0.931938	−0.465969	−	0.884801i	$-0.654294\pi$
$397$	−18.5687	−0.931938	−0.465969	+	0.884801i	$0.654294\pi$
$398$	−13.2468	−0.664002
$399$	0	0
$400$	−0.202536	−0.0101268
$401$	−7.22928	−0.361013	−0.180506	−	0.983574i	$-0.557774\pi$
$401$	−7.22928	−0.361013	−0.180506	+	0.983574i	$0.557774\pi$
$402$	−17.5944	−0.877529
$403$	−17.4250	−0.868002
$404$	26.2373	1.30535
$405$	−2.39532	−0.119025
$406$	0	0
$407$	18.2655	0.905387
$408$	3.86711	0.191450
$409$	−17.7912	−0.879717	−0.439859	−	0.898067i	$-0.644971\pi$
$409$	−17.7912	−0.879717	−0.439859	+	0.898067i	$0.644971\pi$
$410$	9.68985	0.478547
$411$	20.4980	1.01109
$412$	61.1334	3.01183
$413$	0	0
$414$	−2.27460	−0.111790
$415$	25.9368	1.27319
$416$	30.1388	1.47768
$417$	−21.0507	−1.03086
$418$	20.5788	1.00654
$419$	16.7937	0.820426	0.410213	−	0.911990i	$-0.365454\pi$
$419$	16.7937	0.820426	0.410213	+	0.911990i	$0.365454\pi$
$420$	0	0
$421$	−11.1194	−0.541925	−0.270963	−	0.962590i	$-0.587342\pi$
$421$	−11.1194	−0.541925	−0.270963	+	0.962590i	$0.587342\pi$
$422$	−32.4833	−1.58126
$423$	−2.87928	−0.139995
$424$	−27.8015	−1.35016
$425$	1.06829	0.0518199
$426$	−2.15387	−0.104356
$427$	0	0
$428$	−35.0802	−1.69567
$429$	23.0881	1.11470
$430$	6.83596	0.329659
$431$	35.8516	1.72691	0.863456	−	0.504424i	$-0.168295\pi$
$431$	35.8516	1.72691	0.863456	+	0.504424i	$0.168295\pi$
$432$	−0.274599	−0.0132116
$433$	−13.2199	−0.635310	−0.317655	−	0.948206i	$-0.602895\pi$
$433$	−13.2199	−0.635310	−0.317655	+	0.948206i	$0.602895\pi$
$434$	0	0
$435$	−8.83837	−0.423767
$436$	40.9568	1.96148
$437$	1.98008	0.0947199
$438$	−8.79612	−0.420295
$439$	28.2898	1.35020	0.675100	−	0.737726i	$-0.264100\pi$
$439$	28.2898	1.35020	0.675100	+	0.737726i	$0.264100\pi$
$440$	29.2210	1.39306
$441$	0	0
$442$	−16.6475	−0.791839
$443$	23.7781	1.12973	0.564865	−	0.825183i	$-0.308928\pi$
$443$	23.7781	1.12973	0.564865	+	0.825183i	$0.308928\pi$
$444$	−12.6876	−0.602125
$445$	0.628604	0.0297987
$446$	45.4989	2.15444
$447$	−19.1159	−0.904152
$448$	0	0
$449$	−2.97578	−0.140436	−0.0702179	−	0.997532i	$-0.522369\pi$
$449$	−2.97578	−0.140436	−0.0702179	+	0.997532i	$0.522369\pi$
$450$	1.67768	0.0790864
$451$	8.12607	0.382642
$452$	−3.98972	−0.187661
$453$	18.2542	0.857655
$454$	−41.9770	−1.97008
$455$	0	0
$456$	−5.28665	−0.247570
$457$	−19.4381	−0.909277	−0.454638	−	0.890676i	$-0.650231\pi$
$457$	−19.4381	−0.909277	−0.454638	+	0.890676i	$0.650231\pi$
$458$	39.1912	1.83128
$459$	1.44840	0.0676054
$460$	7.60227	0.354458
$461$	4.92877	0.229556	0.114778	−	0.993391i	$-0.463384\pi$
$461$	4.92877	0.229556	0.114778	+	0.993391i	$0.463384\pi$
$462$	0	0
$463$	3.27983	0.152427	0.0762133	−	0.997092i	$-0.475717\pi$
$463$	3.27983	0.152427	0.0762133	+	0.997092i	$0.475717\pi$
$464$	−1.01323	−0.0470379
$465$	−8.26002	−0.383049
$466$	−37.0404	−1.71586
$467$	−3.04520	−0.140915	−0.0704575	−	0.997515i	$-0.522446\pi$
$467$	−3.04520	−0.140915	−0.0704575	+	0.997515i	$0.522446\pi$
$468$	−16.0374	−0.741331
$469$	0	0
$470$	15.6874	0.723608
$471$	−6.63783	−0.305855
$472$	37.7301	1.73667
$473$	5.73276	0.263593
$474$	9.64212	0.442877
$475$	−1.46045	−0.0670098
$476$	0	0
$477$	−10.4128	−0.476771
$478$	−59.4896	−2.72099
$479$	−3.10080	−0.141679	−0.0708396	−	0.997488i	$-0.522568\pi$
$479$	−3.10080	−0.141679	−0.0708396	+	0.997488i	$0.522568\pi$
$480$	14.2868	0.652099
$481$	20.2001	0.921047
$482$	−19.7328	−0.898803
$483$	0	0
$484$	31.3472	1.42487
$485$	−17.2630	−0.783871
$486$	2.27460	0.103178
$487$	17.5634	0.795872	0.397936	−	0.917413i	$-0.369727\pi$
$487$	17.5634	0.795872	0.397936	+	0.917413i	$0.369727\pi$
$488$	−22.3617	−1.01226
$489$	21.1080	0.954538
$490$	0	0
$491$	32.6646	1.47413	0.737066	−	0.675821i	$-0.236211\pi$
$491$	32.6646	1.47413	0.737066	+	0.675821i	$0.236211\pi$
$492$	−5.64453	−0.254475
$493$	5.34436	0.240698
$494$	22.7584	1.02395
$495$	10.9445	0.491920
$496$	−0.946925	−0.0425182
$497$	0	0
$498$	−24.6296	−1.10368
$499$	−16.9084	−0.756926	−0.378463	−	0.925616i	$-0.623547\pi$
$499$	−16.9084	−0.756926	−0.378463	+	0.925616i	$0.623547\pi$
$500$	32.4041	1.44916
$501$	1.01217	0.0452204
$502$	−20.9920	−0.936919
$503$	9.22582	0.411359	0.205679	−	0.978619i	$-0.434060\pi$
$503$	9.22582	0.411359	0.205679	+	0.978619i	$0.434060\pi$
$504$	0	0
$505$	−19.8018	−0.881167
$506$	10.3929	0.462022
$507$	12.5336	0.556635
$508$	61.3765	2.72314
$509$	12.4362	0.551226	0.275613	−	0.961269i	$-0.411119\pi$
$509$	12.4362	0.551226	0.275613	+	0.961269i	$0.411119\pi$
$510$	−7.89144	−0.349439
$511$	0	0
$512$	3.10414	0.137185
$513$	−1.98008	−0.0874225
$514$	−40.3981	−1.78189
$515$	−46.1385	−2.03310
$516$	−3.98208	−0.175301
$517$	13.1558	0.578590
$518$	0	0
$519$	−15.2069	−0.667511
$520$	32.3160	1.41715
$521$	26.2123	1.14838	0.574191	−	0.818721i	$-0.305317\pi$
$521$	26.2123	1.14838	0.574191	+	0.818721i	$0.305317\pi$
$522$	8.39292	0.367348
$523$	−25.3309	−1.10764	−0.553822	−	0.832635i	$-0.686831\pi$
$523$	−25.3309	−1.10764	−0.553822	+	0.832635i	$0.686831\pi$
$524$	−61.7809	−2.69891
$525$	0	0
$526$	−23.3648	−1.01876
$527$	4.99465	0.217570
$528$	1.25467	0.0546027
$529$	1.00000	0.0434783
$530$	56.7333	2.46434
$531$	14.1315	0.613257
$532$	0	0
$533$	8.98677	0.389260
$534$	−0.596922	−0.0258313
$535$	26.4757	1.14464
$536$	−20.6523	−0.892043
$537$	18.1491	0.783190
$538$	39.8764	1.71919
$539$	0	0
$540$	−7.60227	−0.327150
$541$	−16.2940	−0.700534	−0.350267	−	0.936650i	$-0.613909\pi$
$541$	−16.2940	−0.700534	−0.350267	+	0.936650i	$0.613909\pi$
$542$	−48.1309	−2.06740
$543$	−10.5866	−0.454316
$544$	−8.63889	−0.370389
$545$	−30.9109	−1.32408
$546$	0	0
$547$	13.1461	0.562087	0.281044	−	0.959695i	$-0.409319\pi$
$547$	13.1461	0.562087	0.281044	+	0.959695i	$0.409319\pi$
$548$	65.0566	2.77908
$549$	−8.37540	−0.357453
$550$	−7.66551	−0.326858
$551$	−7.30617	−0.311253
$552$	−2.66992	−0.113639
$553$	0	0
$554$	6.66322	0.283093
$555$	9.57553	0.406459
$556$	−66.8106	−2.83340
$557$	1.28424	0.0544150	0.0272075	−	0.999630i	$-0.491339\pi$
$557$	1.28424	0.0544150	0.0272075	+	0.999630i	$0.491339\pi$
$558$	7.84372	0.332051
$559$	6.33996	0.268152
$560$	0	0
$561$	−6.61790	−0.279408
$562$	−61.1083	−2.57770
$563$	1.02181	0.0430642	0.0215321	−	0.999768i	$-0.493146\pi$
$563$	1.02181	0.0430642	0.0215321	+	0.999768i	$0.493146\pi$
$564$	−9.13824	−0.384789
$565$	3.01111	0.126678
$566$	50.1741	2.10897
$567$	0	0
$568$	−2.52822	−0.106082
$569$	−14.9943	−0.628592	−0.314296	−	0.949325i	$-0.601768\pi$
$569$	−14.9943	−0.628592	−0.314296	+	0.949325i	$0.601768\pi$
$570$	10.7882	0.451870
$571$	−15.3179	−0.641035	−0.320517	−	0.947243i	$-0.603857\pi$
$571$	−15.3179	−0.641035	−0.320517	+	0.947243i	$0.603857\pi$
$572$	73.2770	3.06387
$573$	−19.8004	−0.827174
$574$	0	0
$575$	−0.737570	−0.0307588
$576$	−13.0175	−0.542397
$577$	36.4091	1.51573	0.757865	−	0.652411i	$-0.226243\pi$
$577$	36.4091	1.51573	0.757865	+	0.652411i	$0.226243\pi$
$578$	−33.8964	−1.40990
$579$	22.4956	0.934885
$580$	−28.0512	−1.16476
$581$	0	0
$582$	16.3929	0.679508
$583$	47.5775	1.97046
$584$	−10.3249	−0.427246
$585$	12.1037	0.500428
$586$	−54.0302	−2.23197
$587$	32.1484	1.32691	0.663454	−	0.748217i	$-0.269090\pi$
$587$	32.1484	1.32691	0.663454	+	0.748217i	$0.269090\pi$
$588$	0	0
$589$	−6.82809	−0.281346
$590$	−76.9943	−3.16980
$591$	1.67338	0.0688338
$592$	1.09773	0.0451166
$593$	−31.6719	−1.30061	−0.650305	−	0.759673i	$-0.725359\pi$
$593$	−31.6719	−1.30061	−0.650305	+	0.759673i	$0.725359\pi$
$594$	−10.3929	−0.426427
$595$	0	0
$596$	−60.6701	−2.48514
$597$	−5.82380	−0.238352
$598$	11.4937	0.470013
$599$	−36.0114	−1.47138	−0.735692	−	0.677316i	$-0.763143\pi$
$599$	−36.0114	−1.47138	−0.735692	+	0.677316i	$0.763143\pi$
$600$	1.96925	0.0803945
$601$	33.4295	1.36362	0.681810	−	0.731530i	$-0.261193\pi$
$601$	33.4295	1.36362	0.681810	+	0.731530i	$0.261193\pi$
$602$	0	0
$603$	−7.73517	−0.315000
$604$	57.9350	2.35734
$605$	−23.6583	−0.961846
$606$	18.8038	0.763850
$607$	−0.510803	−0.0207329	−0.0103664	−	0.999946i	$-0.503300\pi$
$607$	−0.510803	−0.0207329	−0.0103664	+	0.999946i	$0.503300\pi$
$608$	11.8101	0.478961
$609$	0	0
$610$	45.6325	1.84761
$611$	14.5492	0.588598
$612$	4.59692	0.185820
$613$	−29.7444	−1.20137	−0.600683	−	0.799488i	$-0.705104\pi$
$613$	−29.7444	−1.20137	−0.600683	+	0.799488i	$0.705104\pi$
$614$	−21.8592	−0.882167
$615$	4.26002	0.171781
$616$	0	0
$617$	−18.2312	−0.733959	−0.366980	−	0.930229i	$-0.619608\pi$
$617$	−18.2312	−0.733959	−0.366980	+	0.930229i	$0.619608\pi$
$618$	43.8131	1.76242
$619$	−31.3889	−1.26163	−0.630814	−	0.775934i	$-0.717279\pi$
$619$	−31.3889	−1.26163	−0.630814	+	0.775934i	$0.717279\pi$
$620$	−26.2157	−1.05285
$621$	−1.00000	−0.0401286
$622$	−30.8531	−1.23710
$623$	0	0
$624$	1.38757	0.0555471
$625$	−28.1438	−1.12575
$626$	−2.80159	−0.111974
$627$	9.04721	0.361311
$628$	−21.0671	−0.840670
$629$	−5.79011	−0.230867
$630$	0	0
$631$	42.9030	1.70794	0.853970	−	0.520322i	$-0.174188\pi$
$631$	42.9030	1.70794	0.853970	+	0.520322i	$0.174188\pi$
$632$	11.3179	0.450202
$633$	−14.2809	−0.567615
$634$	36.5265	1.45065
$635$	−46.3219	−1.83823
$636$	−33.0483	−1.31045
$637$	0	0
$638$	−38.3483	−1.51822
$639$	−0.946925	−0.0374598
$640$	42.3511	1.67407
$641$	−19.8505	−0.784049	−0.392025	−	0.919955i	$-0.628225\pi$
$641$	−19.8505	−0.784049	−0.392025	+	0.919955i	$0.628225\pi$
$642$	−25.1413	−0.992248
$643$	−26.5457	−1.04686	−0.523431	−	0.852068i	$-0.675348\pi$
$643$	−26.5457	−1.04686	−0.523431	+	0.852068i	$0.675348\pi$
$644$	0	0
$645$	3.00535	0.118336
$646$	−6.52340	−0.256660
$647$	−30.0117	−1.17988	−0.589940	−	0.807447i	$-0.700849\pi$
$647$	−30.0117	−1.17988	−0.589940	+	0.807447i	$0.700849\pi$
$648$	2.66992	0.104884
$649$	−64.5687	−2.53454
$650$	−8.47742	−0.332512
$651$	0	0
$652$	66.9927	2.62364
$653$	4.41083	0.172609	0.0863046	−	0.996269i	$-0.472494\pi$
$653$	4.41083	0.172609	0.0863046	+	0.996269i	$0.472494\pi$
$654$	29.3529	1.14779
$655$	46.6271	1.82187
$656$	0.488367	0.0190675
$657$	−3.86711	−0.150870
$658$	0	0
$659$	−30.2147	−1.17700	−0.588499	−	0.808498i	$-0.700281\pi$
$659$	−30.2147	−1.17700	−0.588499	+	0.808498i	$0.700281\pi$
$660$	34.7357	1.35209
$661$	−18.8322	−0.732488	−0.366244	−	0.930519i	$-0.619356\pi$
$661$	−18.8322	−0.732488	−0.366244	+	0.930519i	$0.619356\pi$
$662$	−28.5289	−1.10881
$663$	−7.31886	−0.284241
$664$	−28.9101	−1.12193
$665$	0	0
$666$	−9.09292	−0.352344
$667$	−3.68985	−0.142871
$668$	3.21242	0.124292
$669$	20.0031	0.773363
$670$	42.1443	1.62817
$671$	38.2682	1.47733
$672$	0	0
$673$	−17.0208	−0.656102	−0.328051	−	0.944660i	$-0.606392\pi$
$673$	−17.0208	−0.656102	−0.328051	+	0.944660i	$0.606392\pi$
$674$	21.8967	0.843428
$675$	0.737570	0.0283891
$676$	39.7790	1.52996
$677$	−33.4116	−1.28411	−0.642056	−	0.766657i	$-0.721918\pi$
$677$	−33.4116	−1.28411	−0.642056	+	0.766657i	$0.721918\pi$
$678$	−2.85935	−0.109813
$679$	0	0
$680$	−9.26297	−0.355219
$681$	−18.4547	−0.707186
$682$	−35.8389	−1.37234
$683$	−9.95082	−0.380758	−0.190379	−	0.981711i	$-0.560972\pi$
$683$	−9.95082	−0.380758	−0.190379	+	0.981711i	$0.560972\pi$
$684$	−6.28436	−0.240289
$685$	−49.0993	−1.87599
$686$	0	0
$687$	17.2299	0.657363
$688$	0.344532	0.0131352
$689$	52.6169	2.00454
$690$	5.44840	0.207417
$691$	−21.7795	−0.828533	−0.414266	−	0.910156i	$-0.635962\pi$
$691$	−21.7795	−0.828533	−0.414266	+	0.910156i	$0.635962\pi$
$692$	−48.2638	−1.83471
$693$	0	0
$694$	−22.6140	−0.858417
$695$	50.4231	1.91266
$696$	9.85160	0.373424
$697$	−2.57594	−0.0975707
$698$	28.9497	1.09576
$699$	−16.2844	−0.615931
$700$	0	0
$701$	17.4292	0.658291	0.329146	−	0.944279i	$-0.393239\pi$
$701$	17.4292	0.658291	0.329146	+	0.944279i	$0.393239\pi$
$702$	−11.4937	−0.433802
$703$	7.91554	0.298540
$704$	59.4786	2.24169
$705$	6.89679	0.259748
$706$	3.80844	0.143333
$707$	0	0
$708$	44.8507	1.68559
$709$	29.6266	1.11265	0.556325	−	0.830965i	$-0.312211\pi$
$709$	29.6266	1.11265	0.556325	+	0.830965i	$0.312211\pi$
$710$	5.15922	0.193622
$711$	4.23904	0.158977
$712$	−0.700667	−0.0262586
$713$	−3.44840	−0.129143
$714$	0	0
$715$	−55.3035	−2.06823
$716$	57.6015	2.15267
$717$	−26.1539	−0.976734
$718$	−84.0832	−3.13796
$719$	17.1492	0.639558	0.319779	−	0.947492i	$-0.396391\pi$
$719$	17.1492	0.639558	0.319779	+	0.947492i	$0.396391\pi$
$720$	0.657752	0.0245130
$721$	0	0
$722$	−34.2994	−1.27649
$723$	−8.67527	−0.322637
$724$	−33.5999	−1.24873
$725$	2.72152	0.101075
$726$	22.4659	0.833788
$727$	−48.6199	−1.80321	−0.901606	−	0.432557i	$-0.857611\pi$
$727$	−48.6199	−1.80321	−0.901606	+	0.432557i	$0.857611\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	21.0695	0.779819
$731$	−1.81727	−0.0672141
$732$	−26.5818	−0.982493
$733$	39.3216	1.45238	0.726189	−	0.687495i	$-0.241290\pi$
$733$	39.3216	1.45238	0.726189	+	0.687495i	$0.241290\pi$
$734$	51.1967	1.88970
$735$	0	0
$736$	5.96444	0.219852
$737$	35.3429	1.30187
$738$	−4.04532	−0.148910
$739$	20.5818	0.757115	0.378557	−	0.925578i	$-0.376420\pi$
$739$	20.5818	0.757115	0.378557	+	0.925578i	$0.376420\pi$
$740$	30.3908	1.11719
$741$	10.0055	0.367560
$742$	0	0
$743$	−38.0313	−1.39523	−0.697616	−	0.716472i	$-0.745756\pi$
$743$	−38.0313	−1.39523	−0.697616	+	0.716472i	$0.745756\pi$
$744$	9.20695	0.337543
$745$	45.7888	1.67757
$746$	53.6568	1.96452
$747$	−10.8281	−0.396179
$748$	−21.0039	−0.767978
$749$	0	0
$750$	23.2234	0.847999
$751$	−38.3146	−1.39812	−0.699060	−	0.715063i	$-0.746398\pi$
$751$	−38.3146	−1.39812	−0.699060	+	0.715063i	$0.746398\pi$
$752$	0.790645	0.0288319
$753$	−9.22888	−0.336319
$754$	−42.4100	−1.54448
$755$	−43.7246	−1.59130
$756$	0	0
$757$	−25.4278	−0.924190	−0.462095	−	0.886830i	$-0.652902\pi$
$757$	−25.4278	−0.924190	−0.462095	+	0.886830i	$0.652902\pi$
$758$	15.7781	0.573086
$759$	4.56912	0.165849
$760$	12.6632	0.459343
$761$	−46.3205	−1.67912	−0.839558	−	0.543271i	$-0.817186\pi$
$761$	−46.3205	−1.67912	−0.839558	+	0.543271i	$0.817186\pi$
$762$	43.9873	1.59349
$763$	0	0
$764$	−62.8425	−2.27356
$765$	−3.46938	−0.125436
$766$	51.9247	1.87612
$767$	−71.4078	−2.57838
$768$	−14.1816	−0.511733
$769$	−2.92806	−0.105588	−0.0527942	−	0.998605i	$-0.516813\pi$
$769$	−2.92806	−0.105588	−0.0527942	+	0.998605i	$0.516813\pi$
$770$	0	0
$771$	−17.7606	−0.639631
$772$	71.3965	2.56962
$773$	−0.848652	−0.0305239	−0.0152619	−	0.999884i	$-0.504858\pi$
$773$	−0.848652	−0.0305239	−0.0152619	+	0.999884i	$0.504858\pi$
$774$	−2.85388	−0.102581
$775$	2.54344	0.0913629
$776$	19.2420	0.690747
$777$	0	0
$778$	36.7007	1.31578
$779$	3.52152	0.126171
$780$	38.4148	1.37547
$781$	4.32662	0.154818
$782$	−3.29452	−0.117812
$783$	3.68985	0.131864
$784$	0	0
$785$	15.8997	0.567486
$786$	−44.2771	−1.57931
$787$	−31.3118	−1.11614	−0.558072	−	0.829793i	$-0.688459\pi$
$787$	−31.3118	−1.11614	−0.558072	+	0.829793i	$0.688459\pi$
$788$	5.31098	0.189196
$789$	−10.2721	−0.365695
$790$	−23.0960	−0.821718
$791$	0	0
$792$	−12.1992	−0.433479
$793$	42.3215	1.50288
$794$	−42.2364	−1.49891
$795$	24.9421	0.884606
$796$	−18.4836	−0.655132
$797$	−4.66393	−0.165205	−0.0826025	−	0.996583i	$-0.526323\pi$
$797$	−4.66393	−0.165205	−0.0826025	+	0.996583i	$0.526323\pi$
$798$	0	0
$799$	−4.17034	−0.147536
$800$	−4.39920	−0.155535
$801$	−0.262430	−0.00927250
$802$	−16.4437	−0.580648
$803$	17.6693	0.623535
$804$	−24.5499	−0.865807
$805$	0	0
$806$	−39.6349	−1.39608
$807$	17.5312	0.617126
$808$	22.0718	0.776484
$809$	26.2106	0.921516	0.460758	−	0.887526i	$-0.347578\pi$
$809$	26.2106	0.921516	0.460758	+	0.887526i	$0.347578\pi$
$810$	−5.44840	−0.191437
$811$	35.2107	1.23642	0.618208	−	0.786015i	$-0.287859\pi$
$811$	35.2107	1.23642	0.618208	+	0.786015i	$0.287859\pi$
$812$	0	0
$813$	−21.1602	−0.742120
$814$	41.5467	1.45621
$815$	−50.5606	−1.77106
$816$	−0.397728	−0.0139233
$817$	2.48435	0.0869164
$818$	−40.4678	−1.41492
$819$	0	0
$820$	13.5205	0.472155
$821$	30.9369	1.07970	0.539852	−	0.841760i	$-0.318480\pi$
$821$	30.9369	1.07970	0.539852	+	0.841760i	$0.318480\pi$
$822$	46.6247	1.62622
$823$	−17.7770	−0.619668	−0.309834	−	0.950791i	$-0.600273\pi$
$823$	−17.7770	−0.619668	−0.309834	+	0.950791i	$0.600273\pi$
$824$	51.4278	1.79157
$825$	−3.37005	−0.117330
$826$	0	0
$827$	−3.40509	−0.118406	−0.0592032	−	0.998246i	$-0.518856\pi$
$827$	−3.40509	−0.118406	−0.0592032	+	0.998246i	$0.518856\pi$
$828$	−3.17380	−0.110297
$829$	−0.475680	−0.0165210	−0.00826052	−	0.999966i	$-0.502629\pi$
$829$	−0.475680	−0.0165210	−0.00826052	+	0.999966i	$0.502629\pi$
$830$	58.9957	2.04777
$831$	2.92941	0.101620
$832$	65.7785	2.28046
$833$	0	0
$834$	−47.8818	−1.65801
$835$	−2.42447	−0.0839023
$836$	28.7140	0.993095
$837$	3.44840	0.119194
$838$	38.1990	1.31956
$839$	15.4804	0.534442	0.267221	−	0.963635i	$-0.413895\pi$
$839$	15.4804	0.534442	0.267221	+	0.963635i	$0.413895\pi$
$840$	0	0
$841$	−15.3850	−0.530519
$842$	−25.2921	−0.871624
$843$	−26.8655	−0.925298
$844$	−45.3247	−1.56014
$845$	−30.0219	−1.03279
$846$	−6.54920	−0.225166
$847$	0	0
$848$	2.85935	0.0981905
$849$	22.0584	0.757043
$850$	2.42994	0.0833463
$851$	3.99759	0.137036
$852$	−3.00535	−0.102962
$853$	−23.6583	−0.810044	−0.405022	−	0.914307i	$-0.632736\pi$
$853$	−23.6583	−0.810044	−0.405022	+	0.914307i	$0.632736\pi$
$854$	0	0
$855$	4.74292	0.162204
$856$	−29.5108	−1.00866
$857$	10.2346	0.349608	0.174804	−	0.984603i	$-0.444071\pi$
$857$	10.2346	0.349608	0.174804	+	0.984603i	$0.444071\pi$
$858$	52.5162	1.79287
$859$	38.4043	1.31034	0.655169	−	0.755483i	$-0.272598\pi$
$859$	38.4043	1.31034	0.655169	+	0.755483i	$0.272598\pi$
$860$	9.53838	0.325256
$861$	0	0
$862$	81.5480	2.77754
$863$	33.5209	1.14106	0.570532	−	0.821275i	$-0.306737\pi$
$863$	33.5209	1.14106	0.570532	+	0.821275i	$0.306737\pi$
$864$	−5.96444	−0.202915
$865$	36.4255	1.23851
$866$	−30.0701	−1.02182
$867$	−14.9021	−0.506103
$868$	0	0
$869$	−19.3687	−0.657038
$870$	−20.1037	−0.681581
$871$	39.0864	1.32439
$872$	34.4545	1.16678
$873$	7.20695	0.243918
$874$	4.50388	0.152346
$875$	0	0
$876$	−12.2734	−0.414680
$877$	−52.7392	−1.78088	−0.890438	−	0.455105i	$-0.849602\pi$
$877$	−52.7392	−1.78088	−0.890438	+	0.455105i	$0.849602\pi$
$878$	64.3480	2.17164
$879$	−23.7537	−0.801194
$880$	−3.00535	−0.101310
$881$	−19.1890	−0.646495	−0.323247	−	0.946314i	$-0.604775\pi$
$881$	−19.1890	−0.646495	−0.323247	+	0.946314i	$0.604775\pi$
$882$	0	0
$883$	−33.2985	−1.12059	−0.560293	−	0.828295i	$-0.689311\pi$
$883$	−33.2985	−1.12059	−0.560293	+	0.828295i	$0.689311\pi$
$884$	−23.2286	−0.781262
$885$	−33.8496	−1.13784
$886$	54.0856	1.81704
$887$	17.8750	0.600183	0.300092	−	0.953910i	$-0.402983\pi$
$887$	17.8750	0.600183	0.300092	+	0.953910i	$0.402983\pi$
$888$	−10.6733	−0.358171
$889$	0	0
$890$	1.42982	0.0479277
$891$	−4.56912	−0.153071
$892$	63.4857	2.12566
$893$	5.70118	0.190783
$894$	−43.4810	−1.45422
$895$	−43.4729	−1.45314
$896$	0	0
$897$	5.05307	0.168717
$898$	−6.76871	−0.225875
$899$	12.7241	0.424371
$900$	2.34090	0.0780300
$901$	−15.0819	−0.502452
$902$	18.4836	0.615435
$903$	0	0
$904$	−3.35630	−0.111629
$905$	25.3584	0.842942
$906$	41.5209	1.37944
$907$	6.71166	0.222857	0.111428	−	0.993772i	$-0.464457\pi$
$907$	6.71166	0.222857	0.111428	+	0.993772i	$0.464457\pi$
$908$	−58.5715	−1.94376
$909$	8.26684	0.274194
$910$	0	0
$911$	−14.5404	−0.481744	−0.240872	−	0.970557i	$-0.577433\pi$
$911$	−14.5404	−0.481744	−0.240872	+	0.970557i	$0.577433\pi$
$912$	0.543726	0.0180046
$913$	49.4749	1.63738
$914$	−44.2139	−1.46247
$915$	20.0618	0.663222
$916$	54.6844	1.80682
$917$	0	0
$918$	3.29452	0.108735
$919$	25.3532	0.836325	0.418162	−	0.908372i	$-0.362674\pi$
$919$	25.3532	0.836325	0.418162	+	0.908372i	$0.362674\pi$
$920$	6.39532	0.210848
$921$	−9.61015	−0.316665
$922$	11.2110	0.369214
$923$	4.78488	0.157496
$924$	0	0
$925$	−2.94851	−0.0969463
$926$	7.46029	0.245160
$927$	19.2619	0.632644
$928$	−22.0079	−0.722444
$929$	−46.6378	−1.53014	−0.765069	−	0.643948i	$-0.777295\pi$
$929$	−46.6378	−1.53014	−0.765069	+	0.643948i	$0.777295\pi$
$930$	−18.7882	−0.616091
$931$	0	0
$932$	−51.6833	−1.69294
$933$	−13.5642	−0.444072
$934$	−6.92660	−0.226645
$935$	15.8520	0.518416
$936$	−13.4913	−0.440977
$937$	34.8699	1.13915	0.569576	−	0.821939i	$-0.307108\pi$
$937$	34.8699	1.13915	0.569576	+	0.821939i	$0.307108\pi$
$938$	0	0
$939$	−1.23168	−0.0401945
$940$	21.8890	0.713942
$941$	12.0243	0.391982	0.195991	−	0.980606i	$-0.437208\pi$
$941$	12.0243	0.391982	0.195991	+	0.980606i	$0.437208\pi$
$942$	−15.0984	−0.491932
$943$	1.77848	0.0579152
$944$	−3.88050	−0.126300
$945$	0	0
$946$	13.0397	0.423958
$947$	−18.4626	−0.599953	−0.299977	−	0.953947i	$-0.596979\pi$
$947$	−18.4626	−0.599953	−0.299977	+	0.953947i	$0.596979\pi$
$948$	13.4539	0.436961
$949$	19.5408	0.634321
$950$	−3.32193	−0.107778
$951$	16.0584	0.520730
$952$	0	0
$953$	8.91601	0.288818	0.144409	−	0.989518i	$-0.453872\pi$
$953$	8.91601	0.288818	0.144409	+	0.989518i	$0.453872\pi$
$954$	−23.6850	−0.766831
$955$	47.4284	1.53475
$956$	−83.0071	−2.68464
$957$	−16.8594	−0.544985
$958$	−7.05307	−0.227875
$959$	0	0
$960$	31.1812	1.00637
$961$	−19.1086	−0.616405
$962$	45.9472	1.48140
$963$	−11.0531	−0.356180
$964$	−27.5336	−0.886796
$965$	−53.8842	−1.73459
$966$	0	0
$967$	−11.1091	−0.357244	−0.178622	−	0.983918i	$-0.557164\pi$
$967$	−11.1091	−0.357244	−0.178622	+	0.983918i	$0.557164\pi$
$968$	26.3705	0.847579
$969$	−2.86794	−0.0921314
$970$	−39.2663	−1.26077
$971$	−18.9226	−0.607255	−0.303627	−	0.952791i	$-0.598198\pi$
$971$	−18.9226	−0.607255	−0.303627	+	0.952791i	$0.598198\pi$
$972$	3.17380	0.101800
$973$	0	0
$974$	39.9496	1.28007
$975$	−3.72700	−0.119359
$976$	2.29987	0.0736171
$977$	17.0219	0.544579	0.272290	−	0.962215i	$-0.412219\pi$
$977$	17.0219	0.544579	0.272290	+	0.962215i	$0.412219\pi$
$978$	48.0123	1.53526
$979$	1.19907	0.0383225
$980$	0	0
$981$	12.9047	0.412014
$982$	74.2988	2.37097
$983$	−37.7249	−1.20324	−0.601618	−	0.798784i	$-0.705477\pi$
$983$	−37.7249	−1.20324	−0.601618	+	0.798784i	$0.705477\pi$
$984$	−4.74839	−0.151373
$985$	−4.00829	−0.127715
$986$	12.1563	0.387135
$987$	0	0
$988$	31.7554	1.01027
$989$	1.25467	0.0398963
$990$	24.8944	0.791196
$991$	−51.0190	−1.62067	−0.810336	−	0.585965i	$-0.800715\pi$
$991$	−51.0190	−1.62067	−0.810336	+	0.585965i	$0.800715\pi$
$992$	−20.5678	−0.653027
$993$	−12.5424	−0.398020
$994$	0	0
$995$	13.9499	0.442241
$996$	−34.3662	−1.08893
$997$	43.6823	1.38343	0.691717	−	0.722169i	$-0.256855\pi$
$997$	43.6823	1.38343	0.691717	+	0.722169i	$0.256855\pi$
$998$	−38.4599	−1.21743
$999$	−3.99759	−0.126478

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3381.2.a.w.1.4		4
7.6	odd	2		483.2.a.i.1.4	✓	4
21.20	even	2		1449.2.a.p.1.1		4
28.27	even	2		7728.2.a.cd.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.a.i.1.4	✓	4	7.6	odd	2
1449.2.a.p.1.1		4	21.20	even	2
3381.2.a.w.1.4		4	1.1	even	1	trivial
7728.2.a.cd.1.3		4	28.27	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3381 = 3 \cdot 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3381.a (trivial)

\(f(q)\)	\(=\)	\(q+2.27460 q^{2} +1.00000 q^{3} +3.17380 q^{4} -2.39532 q^{5} +2.27460 q^{6} +2.66992 q^{8} +1.00000 q^{9} -5.44840 q^{10} -4.56912 q^{11} +3.17380 q^{12} -5.05307 q^{13} -2.39532 q^{15} -0.274599 q^{16} +1.44840 q^{17} +2.27460 q^{18} -1.98008 q^{19} -7.60227 q^{20} -10.3929 q^{22} -1.00000 q^{23} +2.66992 q^{24} +0.737570 q^{25} -11.4937 q^{26} +1.00000 q^{27} +3.68985 q^{29} -5.44840 q^{30} +3.44840 q^{31} -5.96444 q^{32} -4.56912 q^{33} +3.29452 q^{34} +3.17380 q^{36} -3.99759 q^{37} -4.50388 q^{38} -5.05307 q^{39} -6.39532 q^{40} -1.77848 q^{41} -1.25467 q^{43} -14.5015 q^{44} -2.39532 q^{45} -2.27460 q^{46} -2.87928 q^{47} -0.274599 q^{48} +1.67768 q^{50} +1.44840 q^{51} -16.0374 q^{52} -10.4128 q^{53} +2.27460 q^{54} +10.9445 q^{55} -1.98008 q^{57} +8.39292 q^{58} +14.1315 q^{59} -7.60227 q^{60} -8.37540 q^{61} +7.84372 q^{62} -13.0175 q^{64} +12.1037 q^{65} -10.3929 q^{66} -7.73517 q^{67} +4.59692 q^{68} -1.00000 q^{69} -0.946925 q^{71} +2.66992 q^{72} -3.86711 q^{73} -9.09292 q^{74} +0.737570 q^{75} -6.28436 q^{76} -11.4937 q^{78} +4.23904 q^{79} +0.657752 q^{80} +1.00000 q^{81} -4.04532 q^{82} -10.8281 q^{83} -3.46938 q^{85} -2.85388 q^{86} +3.68985 q^{87} -12.1992 q^{88} -0.262430 q^{89} -5.44840 q^{90} -3.17380 q^{92} +3.44840 q^{93} -6.54920 q^{94} +4.74292 q^{95} -5.96444 q^{96} +7.20695 q^{97} -4.56912 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 4 q^{4} - 5 q^{5} - 3 q^{8} + 4 q^{9} - 4 q^{10} - 5 q^{11} + 4 q^{12} - 7 q^{13} - 5 q^{15} + 8 q^{16} - 12 q^{17} - 3 q^{19} + q^{20} - q^{22} - 4 q^{23} - 3 q^{24} + 7 q^{25} - 5 q^{26}+ \cdots - 5 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 + 4 * q^4 - 5 * q^5 - 3 * q^8 + 4 * q^9 - 4 * q^10 - 5 * q^11 + 4 * q^12 - 7 * q^13 - 5 * q^15 + 8 * q^16 - 12 * q^17 - 3 * q^19 + q^20 - q^22 - 4 * q^23 - 3 * q^24 + 7 * q^25 - 5 * q^26 + 4 * q^27 + 6 * q^29 - 4 * q^30 - 4 * q^31 - 6 * q^32 - 5 * q^33 + 9 * q^34 + 4 * q^36 + 20 * q^37 - 23 * q^38 - 7 * q^39 - 21 * q^40 - 3 * q^41 + 9 * q^43 - 27 * q^44 - 5 * q^45 - 7 * q^47 + 8 * q^48 + 3 * q^50 - 12 * q^51 - 38 * q^52 - 6 * q^53 + 21 * q^55 - 3 * q^57 - 7 * q^58 + 2 * q^59 + q^60 - 24 * q^61 + 9 * q^62 - 21 * q^64 - 14 * q^65 - q^66 + q^67 + 13 * q^68 - 4 * q^69 - 17 * q^71 - 3 * q^72 - 16 * q^73 - 33 * q^74 + 7 * q^75 + 25 * q^76 - 5 * q^78 - 10 * q^79 - 6 * q^80 + 4 * q^81 + 7 * q^82 - 8 * q^83 + 17 * q^85 - 35 * q^86 + 6 * q^87 - 12 * q^88 + 3 * q^89 - 4 * q^90 - 4 * q^92 - 4 * q^93 - 8 * q^94 - 3 * q^95 - 6 * q^96 + 2 * q^97 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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