LMFDB - Embedded newform 9904.2.a.n.1.28

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9904,2,Mod(1,9904)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9904, 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("9904.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9904 = 2^{4} \cdot 619 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9904.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:	$79.0838381619$
Analytic rank:	$0$
Dimension:	$30$
Twist minimal:	no (minimal twist has level 619)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.28
Character	$\chi$	$=$	9904.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.02694 q^{3} -0.0323024 q^{5} +3.37580 q^{7} +6.16239 q^{9} +O(q^{10})$	$q+3.02694 q^{3} -0.0323024 q^{5} +3.37580 q^{7} +6.16239 q^{9} -0.333322 q^{11} -1.65481 q^{13} -0.0977777 q^{15} -1.50147 q^{17} +5.02748 q^{19} +10.2184 q^{21} +5.35036 q^{23} -4.99896 q^{25} +9.57238 q^{27} +3.38998 q^{29} +5.45120 q^{31} -1.00895 q^{33} -0.109047 q^{35} -4.83972 q^{37} -5.00902 q^{39} +0.0826437 q^{41} -1.78583 q^{43} -0.199060 q^{45} -7.93095 q^{47} +4.39602 q^{49} -4.54487 q^{51} +11.6068 q^{53} +0.0107671 q^{55} +15.2179 q^{57} +12.7123 q^{59} -1.27687 q^{61} +20.8030 q^{63} +0.0534544 q^{65} -11.1702 q^{67} +16.1952 q^{69} -9.44501 q^{71} +16.0226 q^{73} -15.1316 q^{75} -1.12523 q^{77} -2.17453 q^{79} +10.4879 q^{81} -0.893308 q^{83} +0.0485011 q^{85} +10.2613 q^{87} +11.3541 q^{89} -5.58631 q^{91} +16.5005 q^{93} -0.162400 q^{95} -8.98396 q^{97} -2.05406 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q - q^{3} + 21 q^{5} - 2 q^{7} + 43 q^{9}+O(q^{10}) $ 30 * q - q^3 + 21 * q^5 - 2 * q^7 + 43 * q^9	$ 30 q - q^{3} + 21 q^{5} - 2 q^{7} + 43 q^{9} - 23 q^{11} + 9 q^{13} + 2 q^{15} + 4 q^{17} + q^{19} + 30 q^{21} - 4 q^{23} + 35 q^{25} + 5 q^{27} + 90 q^{29} - 2 q^{31} - 6 q^{33} - 9 q^{35} + 19 q^{37} - 32 q^{39} + 59 q^{41} + 4 q^{43} + 30 q^{45} - 4 q^{47} + 30 q^{49} + 34 q^{53} + 17 q^{55} - 8 q^{57} - 13 q^{59} + 16 q^{61} + 40 q^{63} + 31 q^{65} + 11 q^{67} + 6 q^{69} - 42 q^{71} - 4 q^{73} + 52 q^{75} + 29 q^{77} - 3 q^{79} + 30 q^{81} + 11 q^{83} + 19 q^{85} + 20 q^{87} + 58 q^{89} + 39 q^{91} - 15 q^{93} - 23 q^{95} - 9 q^{97} + 8 q^{99}+O(q^{100}) $ 30 * q - q^3 + 21 * q^5 - 2 * q^7 + 43 * q^9 - 23 * q^11 + 9 * q^13 + 2 * q^15 + 4 * q^17 + q^19 + 30 * q^21 - 4 * q^23 + 35 * q^25 + 5 * q^27 + 90 * q^29 - 2 * q^31 - 6 * q^33 - 9 * q^35 + 19 * q^37 - 32 * q^39 + 59 * q^41 + 4 * q^43 + 30 * q^45 - 4 * q^47 + 30 * q^49 + 34 * q^53 + 17 * q^55 - 8 * q^57 - 13 * q^59 + 16 * q^61 + 40 * q^63 + 31 * q^65 + 11 * q^67 + 6 * q^69 - 42 * q^71 - 4 * q^73 + 52 * q^75 + 29 * q^77 - 3 * q^79 + 30 * q^81 + 11 * q^83 + 19 * q^85 + 20 * q^87 + 58 * q^89 + 39 * q^91 - 15 * q^93 - 23 * q^95 - 9 * q^97 + 8 * 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
$3$	3.02694	1.74761	0.873804	−	0.486279i	$-0.161646\pi$
$3$	3.02694	1.74761	0.873804	+	0.486279i	$0.161646\pi$
$4$	0	0
$5$	−0.0323024	−0.0144461	−0.00722304	−	0.999974i	$-0.502299\pi$
$5$	−0.0323024	−0.0144461	−0.00722304	+	0.999974i	$0.502299\pi$
$6$	0	0
$7$	3.37580	1.27593	0.637966	−	0.770064i	$-0.279776\pi$
$7$	3.37580	1.27593	0.637966	+	0.770064i	$0.279776\pi$
$8$	0	0
$9$	6.16239	2.05413
$10$	0	0
$11$	−0.333322	−0.100500	−0.0502502	−	0.998737i	$-0.516002\pi$
$11$	−0.333322	−0.100500	−0.0502502	+	0.998737i	$0.516002\pi$
$12$	0	0
$13$	−1.65481	−0.458962	−0.229481	−	0.973313i	$-0.573703\pi$
$13$	−1.65481	−0.458962	−0.229481	+	0.973313i	$0.573703\pi$
$14$	0	0
$15$	−0.0977777	−0.0252461
$16$	0	0
$17$	−1.50147	−0.364160	−0.182080	−	0.983284i	$-0.558283\pi$
$17$	−1.50147	−0.364160	−0.182080	+	0.983284i	$0.558283\pi$
$18$	0	0
$19$	5.02748	1.15338	0.576691	−	0.816962i	$-0.304344\pi$
$19$	5.02748	1.15338	0.576691	+	0.816962i	$0.304344\pi$
$20$	0	0
$21$	10.2184	2.22983
$22$	0	0
$23$	5.35036	1.11563	0.557814	−	0.829966i	$-0.311640\pi$
$23$	5.35036	1.11563	0.557814	+	0.829966i	$0.311640\pi$
$24$	0	0
$25$	−4.99896	−0.999791
$26$	0	0
$27$	9.57238	1.84221
$28$	0	0
$29$	3.38998	0.629503	0.314752	−	0.949174i	$-0.398079\pi$
$29$	3.38998	0.629503	0.314752	+	0.949174i	$0.398079\pi$
$30$	0	0
$31$	5.45120	0.979065	0.489532	−	0.871985i	$-0.337168\pi$
$31$	5.45120	0.979065	0.489532	+	0.871985i	$0.337168\pi$
$32$	0	0
$33$	−1.00895	−0.175635
$34$	0	0
$35$	−0.109047	−0.0184322
$36$	0	0
$37$	−4.83972	−0.795644	−0.397822	−	0.917463i	$-0.630234\pi$
$37$	−4.83972	−0.795644	−0.397822	+	0.917463i	$0.630234\pi$
$38$	0	0
$39$	−5.00902	−0.802085
$40$	0	0
$41$	0.0826437	0.0129068	0.00645339	−	0.999979i	$-0.497946\pi$
$41$	0.0826437	0.0129068	0.00645339	+	0.999979i	$0.497946\pi$
$42$	0	0
$43$	−1.78583	−0.272337	−0.136168	−	0.990686i	$-0.543479\pi$
$43$	−1.78583	−0.272337	−0.136168	+	0.990686i	$0.543479\pi$
$44$	0	0
$45$	−0.199060	−0.0296742
$46$	0	0
$47$	−7.93095	−1.15685	−0.578424	−	0.815736i	$-0.696332\pi$
$47$	−7.93095	−1.15685	−0.578424	+	0.815736i	$0.696332\pi$
$48$	0	0
$49$	4.39602	0.628003
$50$	0	0
$51$	−4.54487	−0.636409
$52$	0	0
$53$	11.6068	1.59432	0.797161	−	0.603767i	$-0.206334\pi$
$53$	11.6068	1.59432	0.797161	+	0.603767i	$0.206334\pi$
$54$	0	0
$55$	0.0107671	0.00145184
$56$	0	0
$57$	15.2179	2.01566
$58$	0	0
$59$	12.7123	1.65501	0.827503	−	0.561461i	$-0.189760\pi$
$59$	12.7123	1.65501	0.827503	+	0.561461i	$0.189760\pi$
$60$	0	0
$61$	−1.27687	−0.163486	−0.0817432	−	0.996653i	$-0.526049\pi$
$61$	−1.27687	−0.163486	−0.0817432	+	0.996653i	$0.526049\pi$
$62$	0	0
$63$	20.8030	2.62093
$64$	0	0
$65$	0.0534544	0.00663020
$66$	0	0
$67$	−11.1702	−1.36466	−0.682329	−	0.731045i	$-0.739033\pi$
$67$	−11.1702	−1.36466	−0.682329	+	0.731045i	$0.739033\pi$
$68$	0	0
$69$	16.1952	1.94968
$70$	0	0
$71$	−9.44501	−1.12092	−0.560458	−	0.828183i	$-0.689375\pi$
$71$	−9.44501	−1.12092	−0.560458	+	0.828183i	$0.689375\pi$
$72$	0	0
$73$	16.0226	1.87530	0.937651	−	0.347579i	$-0.112996\pi$
$73$	16.0226	1.87530	0.937651	+	0.347579i	$0.112996\pi$
$74$	0	0
$75$	−15.1316	−1.74724
$76$	0	0
$77$	−1.12523	−0.128232
$78$	0	0
$79$	−2.17453	−0.244654	−0.122327	−	0.992490i	$-0.539036\pi$
$79$	−2.17453	−0.244654	−0.122327	+	0.992490i	$0.539036\pi$
$80$	0	0
$81$	10.4879	1.16532
$82$	0	0
$83$	−0.893308	−0.0980533	−0.0490266	−	0.998797i	$-0.515612\pi$
$83$	−0.893308	−0.0980533	−0.0490266	+	0.998797i	$0.515612\pi$
$84$	0	0
$85$	0.0485011	0.00526069
$86$	0	0
$87$	10.2613	1.10012
$88$	0	0
$89$	11.3541	1.20353	0.601767	−	0.798672i	$-0.294464\pi$
$89$	11.3541	1.20353	0.601767	+	0.798672i	$0.294464\pi$
$90$	0	0
$91$	−5.58631	−0.585604
$92$	0	0
$93$	16.5005	1.71102
$94$	0	0
$95$	−0.162400	−0.0166619
$96$	0	0
$97$	−8.98396	−0.912183	−0.456091	−	0.889933i	$-0.650751\pi$
$97$	−8.98396	−0.912183	−0.456091	+	0.889933i	$0.650751\pi$
$98$	0	0
$99$	−2.05406	−0.206441
$100$	0	0
$101$	−19.1212	−1.90263	−0.951315	−	0.308221i	$-0.900266\pi$
$101$	−19.1212	−1.90263	−0.951315	+	0.308221i	$0.900266\pi$
$102$	0	0
$103$	19.7952	1.95048	0.975241	−	0.221143i	$-0.0709787\pi$
$103$	19.7952	1.95048	0.975241	+	0.221143i	$0.0709787\pi$
$104$	0	0
$105$	−0.330078	−0.0322123
$106$	0	0
$107$	0.880564	0.0851273	0.0425637	−	0.999094i	$-0.486447\pi$
$107$	0.880564	0.0851273	0.0425637	+	0.999094i	$0.486447\pi$
$108$	0	0
$109$	10.0304	0.960736	0.480368	−	0.877067i	$-0.340503\pi$
$109$	10.0304	0.960736	0.480368	+	0.877067i	$0.340503\pi$
$110$	0	0
$111$	−14.6495	−1.39047
$112$	0	0
$113$	7.05601	0.663774	0.331887	−	0.943319i	$-0.392315\pi$
$113$	7.05601	0.663774	0.331887	+	0.943319i	$0.392315\pi$
$114$	0	0
$115$	−0.172830	−0.0161165
$116$	0	0
$117$	−10.1976	−0.942767
$118$	0	0
$119$	−5.06866	−0.464644
$120$	0	0
$121$	−10.8889	−0.989900
$122$	0	0
$123$	0.250158	0.0225560
$124$	0	0
$125$	0.322991	0.0288892
$126$	0	0
$127$	3.23199	0.286793	0.143396	−	0.989665i	$-0.454198\pi$
$127$	3.23199	0.286793	0.143396	+	0.989665i	$0.454198\pi$
$128$	0	0
$129$	−5.40561	−0.475938
$130$	0	0
$131$	−10.2444	−0.895054	−0.447527	−	0.894270i	$-0.647695\pi$
$131$	−10.2444	−0.895054	−0.447527	+	0.894270i	$0.647695\pi$
$132$	0	0
$133$	16.9717	1.47164
$134$	0	0
$135$	−0.309211	−0.0266127
$136$	0	0
$137$	−3.81470	−0.325912	−0.162956	−	0.986633i	$-0.552103\pi$
$137$	−3.81470	−0.325912	−0.162956	+	0.986633i	$0.552103\pi$
$138$	0	0
$139$	−14.8581	−1.26025	−0.630123	−	0.776496i	$-0.716995\pi$
$139$	−14.8581	−1.26025	−0.630123	+	0.776496i	$0.716995\pi$
$140$	0	0
$141$	−24.0065	−2.02172
$142$	0	0
$143$	0.551585	0.0461258
$144$	0	0
$145$	−0.109505	−0.00909386
$146$	0	0
$147$	13.3065	1.09750
$148$	0	0
$149$	−4.00235	−0.327885	−0.163943	−	0.986470i	$-0.552421\pi$
$149$	−4.00235	−0.327885	−0.163943	+	0.986470i	$0.552421\pi$
$150$	0	0
$151$	−4.75933	−0.387309	−0.193654	−	0.981070i	$-0.562034\pi$
$151$	−4.75933	−0.387309	−0.193654	+	0.981070i	$0.562034\pi$
$152$	0	0
$153$	−9.25265	−0.748032
$154$	0	0
$155$	−0.176087	−0.0141437
$156$	0	0
$157$	19.7896	1.57938	0.789690	−	0.613506i	$-0.210241\pi$
$157$	19.7896	1.57938	0.789690	+	0.613506i	$0.210241\pi$
$158$	0	0
$159$	35.1333	2.78625
$160$	0	0
$161$	18.0617	1.42346
$162$	0	0
$163$	6.82013	0.534194	0.267097	−	0.963670i	$-0.413936\pi$
$163$	6.82013	0.534194	0.267097	+	0.963670i	$0.413936\pi$
$164$	0	0
$165$	0.0325915	0.00253724
$166$	0	0
$167$	−13.2644	−1.02643	−0.513215	−	0.858260i	$-0.671546\pi$
$167$	−13.2644	−1.02643	−0.513215	+	0.858260i	$0.671546\pi$
$168$	0	0
$169$	−10.2616	−0.789354
$170$	0	0
$171$	30.9813	2.36920
$172$	0	0
$173$	14.2557	1.08384	0.541921	−	0.840429i	$-0.317697\pi$
$173$	14.2557	1.08384	0.541921	+	0.840429i	$0.317697\pi$
$174$	0	0
$175$	−16.8755	−1.27567
$176$	0	0
$177$	38.4796	2.89230
$178$	0	0
$179$	−12.2721	−0.917260	−0.458630	−	0.888627i	$-0.651660\pi$
$179$	−12.2721	−0.917260	−0.458630	+	0.888627i	$0.651660\pi$
$180$	0	0
$181$	5.03653	0.374362	0.187181	−	0.982325i	$-0.440065\pi$
$181$	5.03653	0.374362	0.187181	+	0.982325i	$0.440065\pi$
$182$	0	0
$183$	−3.86501	−0.285710
$184$	0	0
$185$	0.156335	0.0114939
$186$	0	0
$187$	0.500473	0.0365982
$188$	0	0
$189$	32.3144	2.35053
$190$	0	0
$191$	−6.68323	−0.483581	−0.241791	−	0.970328i	$-0.577735\pi$
$191$	−6.68323	−0.483581	−0.241791	+	0.970328i	$0.577735\pi$
$192$	0	0
$193$	−9.66090	−0.695407	−0.347703	−	0.937605i	$-0.613038\pi$
$193$	−9.66090	−0.695407	−0.347703	+	0.937605i	$0.613038\pi$
$194$	0	0
$195$	0.161804	0.0115870
$196$	0	0
$197$	22.6726	1.61536	0.807678	−	0.589624i	$-0.200724\pi$
$197$	22.6726	1.61536	0.807678	+	0.589624i	$0.200724\pi$
$198$	0	0
$199$	10.8711	0.770633	0.385316	−	0.922785i	$-0.374092\pi$
$199$	10.8711	0.770633	0.385316	+	0.922785i	$0.374092\pi$
$200$	0	0
$201$	−33.8116	−2.38489
$202$	0	0
$203$	11.4439	0.803203
$204$	0	0
$205$	−0.00266959	−0.000186452 0
$206$	0	0
$207$	32.9710	2.29164
$208$	0	0
$209$	−1.67577	−0.115915
$210$	0	0
$211$	−22.4482	−1.54540	−0.772699	−	0.634773i	$-0.781094\pi$
$211$	−22.4482	−1.54540	−0.772699	+	0.634773i	$0.781094\pi$
$212$	0	0
$213$	−28.5895	−1.95892
$214$	0	0
$215$	0.0576867	0.00393420
$216$	0	0
$217$	18.4022	1.24922
$218$	0	0
$219$	48.4995	3.27729
$220$	0	0
$221$	2.48465	0.167136
$222$	0	0
$223$	−9.86150	−0.660375	−0.330187	−	0.943915i	$-0.607112\pi$
$223$	−9.86150	−0.660375	−0.330187	+	0.943915i	$0.607112\pi$
$224$	0	0
$225$	−30.8055	−2.05370
$226$	0	0
$227$	9.95777	0.660920	0.330460	−	0.943820i	$-0.392796\pi$
$227$	9.95777	0.660920	0.330460	+	0.943820i	$0.392796\pi$
$228$	0	0
$229$	12.5539	0.829585	0.414792	−	0.909916i	$-0.363854\pi$
$229$	12.5539	0.829585	0.414792	+	0.909916i	$0.363854\pi$
$230$	0	0
$231$	−3.40600	−0.224099
$232$	0	0
$233$	−6.22700	−0.407944	−0.203972	−	0.978977i	$-0.565385\pi$
$233$	−6.22700	−0.407944	−0.203972	+	0.978977i	$0.565385\pi$
$234$	0	0
$235$	0.256189	0.0167119
$236$	0	0
$237$	−6.58218	−0.427559
$238$	0	0
$239$	−2.13865	−0.138338	−0.0691688	−	0.997605i	$-0.522035\pi$
$239$	−2.13865	−0.138338	−0.0691688	+	0.997605i	$0.522035\pi$
$240$	0	0
$241$	23.6818	1.52548	0.762739	−	0.646706i	$-0.223854\pi$
$241$	23.6818	1.52548	0.762739	+	0.646706i	$0.223854\pi$
$242$	0	0
$243$	3.02913	0.194319
$244$	0	0
$245$	−0.142002	−0.00907219
$246$	0	0
$247$	−8.31952	−0.529358
$248$	0	0
$249$	−2.70399	−0.171359
$250$	0	0
$251$	20.3919	1.28713	0.643563	−	0.765393i	$-0.277455\pi$
$251$	20.3919	1.28713	0.643563	+	0.765393i	$0.277455\pi$
$252$	0	0
$253$	−1.78339	−0.112121
$254$	0	0
$255$	0.146810	0.00919362
$256$	0	0
$257$	−30.6145	−1.90968	−0.954839	−	0.297125i	$-0.903972\pi$
$257$	−30.6145	−1.90968	−0.954839	+	0.297125i	$0.903972\pi$
$258$	0	0
$259$	−16.3379	−1.01519
$260$	0	0
$261$	20.8904	1.29308
$262$	0	0
$263$	24.3347	1.50054	0.750270	−	0.661132i	$-0.229924\pi$
$263$	24.3347	1.50054	0.750270	+	0.661132i	$0.229924\pi$
$264$	0	0
$265$	−0.374929	−0.0230317
$266$	0	0
$267$	34.3683	2.10331
$268$	0	0
$269$	12.3260	0.751528	0.375764	−	0.926715i	$-0.377380\pi$
$269$	12.3260	0.751528	0.375764	+	0.926715i	$0.377380\pi$
$270$	0	0
$271$	13.9583	0.847906	0.423953	−	0.905684i	$-0.360642\pi$
$271$	13.9583	0.847906	0.423953	+	0.905684i	$0.360642\pi$
$272$	0	0
$273$	−16.9094	−1.02341
$274$	0	0
$275$	1.66626	0.100479
$276$	0	0
$277$	1.93138	0.116046	0.0580228	−	0.998315i	$-0.481520\pi$
$277$	1.93138	0.116046	0.0580228	+	0.998315i	$0.481520\pi$
$278$	0	0
$279$	33.5924	2.01113
$280$	0	0
$281$	16.4329	0.980304	0.490152	−	0.871637i	$-0.336941\pi$
$281$	16.4329	0.980304	0.490152	+	0.871637i	$0.336941\pi$
$282$	0	0
$283$	29.2492	1.73868	0.869342	−	0.494210i	$-0.164543\pi$
$283$	29.2492	1.73868	0.869342	+	0.494210i	$0.164543\pi$
$284$	0	0
$285$	−0.491575	−0.0291184
$286$	0	0
$287$	0.278988	0.0164682
$288$	0	0
$289$	−14.7456	−0.867387
$290$	0	0
$291$	−27.1939	−1.59414
$292$	0	0
$293$	9.62737	0.562437	0.281218	−	0.959644i	$-0.409261\pi$
$293$	9.62737	0.562437	0.281218	+	0.959644i	$0.409261\pi$
$294$	0	0
$295$	−0.410640	−0.0239084
$296$	0	0
$297$	−3.19069	−0.185142
$298$	0	0
$299$	−8.85383	−0.512030
$300$	0	0
$301$	−6.02861	−0.347483
$302$	0	0
$303$	−57.8788	−3.32505
$304$	0	0
$305$	0.0412460	0.00236174
$306$	0	0
$307$	−31.4085	−1.79258	−0.896288	−	0.443473i	$-0.853746\pi$
$307$	−31.4085	−1.79258	−0.896288	+	0.443473i	$0.853746\pi$
$308$	0	0
$309$	59.9191	3.40868
$310$	0	0
$311$	9.93124	0.563149	0.281575	−	0.959539i	$-0.409143\pi$
$311$	9.93124	0.563149	0.281575	+	0.959539i	$0.409143\pi$
$312$	0	0
$313$	−17.6550	−0.997917	−0.498959	−	0.866626i	$-0.666284\pi$
$313$	−17.6550	−0.997917	−0.498959	+	0.866626i	$0.666284\pi$
$314$	0	0
$315$	−0.671988	−0.0378622
$316$	0	0
$317$	−22.9524	−1.28914	−0.644568	−	0.764547i	$-0.722963\pi$
$317$	−22.9524	−1.28914	−0.644568	+	0.764547i	$0.722963\pi$
$318$	0	0
$319$	−1.12995	−0.0632653
$320$	0	0
$321$	2.66542	0.148769
$322$	0	0
$323$	−7.54860	−0.420016
$324$	0	0
$325$	8.27232	0.458866
$326$	0	0
$327$	30.3614	1.67899
$328$	0	0
$329$	−26.7733	−1.47606
$330$	0	0
$331$	−12.6516	−0.695396	−0.347698	−	0.937606i	$-0.613037\pi$
$331$	−12.6516	−0.695396	−0.347698	+	0.937606i	$0.613037\pi$
$332$	0	0
$333$	−29.8242	−1.63436
$334$	0	0
$335$	0.360825	0.0197140
$336$	0	0
$337$	3.30965	0.180288	0.0901440	−	0.995929i	$-0.471267\pi$
$337$	3.30965	0.180288	0.0901440	+	0.995929i	$0.471267\pi$
$338$	0	0
$339$	21.3582	1.16002
$340$	0	0
$341$	−1.81701	−0.0983964
$342$	0	0
$343$	−8.79051	−0.474643
$344$	0	0
$345$	−0.523146	−0.0281652
$346$	0	0
$347$	−17.1461	−0.920450	−0.460225	−	0.887802i	$-0.652231\pi$
$347$	−17.1461	−0.920450	−0.460225	+	0.887802i	$0.652231\pi$
$348$	0	0
$349$	−13.5765	−0.726734	−0.363367	−	0.931646i	$-0.618373\pi$
$349$	−13.5765	−0.726734	−0.363367	+	0.931646i	$0.618373\pi$
$350$	0	0
$351$	−15.8405	−0.845502
$352$	0	0
$353$	−35.8042	−1.90566	−0.952832	−	0.303499i	$-0.901845\pi$
$353$	−35.8042	−1.90566	−0.952832	+	0.303499i	$0.901845\pi$
$354$	0	0
$355$	0.305097	0.0161929
$356$	0	0
$357$	−15.3426	−0.812014
$358$	0	0
$359$	−31.2324	−1.64838	−0.824191	−	0.566312i	$-0.808370\pi$
$359$	−31.2324	−1.64838	−0.824191	+	0.566312i	$0.808370\pi$
$360$	0	0
$361$	6.27551	0.330290
$362$	0	0
$363$	−32.9601	−1.72996
$364$	0	0
$365$	−0.517568	−0.0270908
$366$	0	0
$367$	−32.6644	−1.70507	−0.852534	−	0.522672i	$-0.824935\pi$
$367$	−32.6644	−1.70507	−0.852534	+	0.522672i	$0.824935\pi$
$368$	0	0
$369$	0.509283	0.0265122
$370$	0	0
$371$	39.1824	2.03425
$372$	0	0
$373$	19.1389	0.990975	0.495487	−	0.868615i	$-0.334989\pi$
$373$	19.1389	0.990975	0.495487	+	0.868615i	$0.334989\pi$
$374$	0	0
$375$	0.977675	0.0504869
$376$	0	0
$377$	−5.60977	−0.288918
$378$	0	0
$379$	−22.0104	−1.13060	−0.565299	−	0.824886i	$-0.691239\pi$
$379$	−22.0104	−1.13060	−0.565299	+	0.824886i	$0.691239\pi$
$380$	0	0
$381$	9.78306	0.501201
$382$	0	0
$383$	12.3848	0.632833	0.316416	−	0.948620i	$-0.397520\pi$
$383$	12.3848	0.632833	0.316416	+	0.948620i	$0.397520\pi$
$384$	0	0
$385$	0.0363476	0.00185245
$386$	0	0
$387$	−11.0050	−0.559415
$388$	0	0
$389$	16.1516	0.818919	0.409460	−	0.912328i	$-0.365717\pi$
$389$	16.1516	0.818919	0.409460	+	0.912328i	$0.365717\pi$
$390$	0	0
$391$	−8.03341	−0.406267
$392$	0	0
$393$	−31.0091	−1.56420
$394$	0	0
$395$	0.0702426	0.00353429
$396$	0	0
$397$	−23.0393	−1.15631	−0.578155	−	0.815927i	$-0.696227\pi$
$397$	−23.0393	−1.15631	−0.578155	+	0.815927i	$0.696227\pi$
$398$	0	0
$399$	51.3725	2.57184
$400$	0	0
$401$	−6.98825	−0.348977	−0.174488	−	0.984659i	$-0.555827\pi$
$401$	−6.98825	−0.348977	−0.174488	+	0.984659i	$0.555827\pi$
$402$	0	0
$403$	−9.02070	−0.449353
$404$	0	0
$405$	−0.338785	−0.0168343
$406$	0	0
$407$	1.61318	0.0799626
$408$	0	0
$409$	−16.6359	−0.822591	−0.411296	−	0.911502i	$-0.634924\pi$
$409$	−16.6359	−0.822591	−0.411296	+	0.911502i	$0.634924\pi$
$410$	0	0
$411$	−11.5469	−0.569566
$412$	0	0
$413$	42.9143	2.11168
$414$	0	0
$415$	0.0288560	0.00141649
$416$	0	0
$417$	−44.9746	−2.20241
$418$	0	0
$419$	−16.2490	−0.793815	−0.396907	−	0.917859i	$-0.629917\pi$
$419$	−16.2490	−0.793815	−0.396907	+	0.917859i	$0.629917\pi$
$420$	0	0
$421$	−21.2631	−1.03630	−0.518150	−	0.855290i	$-0.673379\pi$
$421$	−21.2631	−1.03630	−0.518150	+	0.855290i	$0.673379\pi$
$422$	0	0
$423$	−48.8736	−2.37632
$424$	0	0
$425$	7.50578	0.364084
$426$	0	0
$427$	−4.31045	−0.208598
$428$	0	0
$429$	1.66962	0.0806098
$430$	0	0
$431$	12.5978	0.606814	0.303407	−	0.952861i	$-0.401876\pi$
$431$	12.5978	0.606814	0.303407	+	0.952861i	$0.401876\pi$
$432$	0	0
$433$	9.83400	0.472592	0.236296	−	0.971681i	$-0.424067\pi$
$433$	9.83400	0.472592	0.236296	+	0.971681i	$0.424067\pi$
$434$	0	0
$435$	−0.331464	−0.0158925
$436$	0	0
$437$	26.8988	1.28674
$438$	0	0
$439$	10.9426	0.522259	0.261130	−	0.965304i	$-0.415905\pi$
$439$	10.9426	0.522259	0.261130	+	0.965304i	$0.415905\pi$
$440$	0	0
$441$	27.0900	1.29000
$442$	0	0
$443$	9.86639	0.468766	0.234383	−	0.972144i	$-0.424693\pi$
$443$	9.86639	0.468766	0.234383	+	0.972144i	$0.424693\pi$
$444$	0	0
$445$	−0.366766	−0.0173864
$446$	0	0
$447$	−12.1149	−0.573015
$448$	0	0
$449$	35.9120	1.69479	0.847397	−	0.530961i	$-0.178169\pi$
$449$	35.9120	1.69479	0.847397	+	0.530961i	$0.178169\pi$
$450$	0	0
$451$	−0.0275470	−0.00129714
$452$	0	0
$453$	−14.4062	−0.676864
$454$	0	0
$455$	0.180451	0.00845969
$456$	0	0
$457$	−24.8431	−1.16211	−0.581056	−	0.813864i	$-0.697360\pi$
$457$	−24.8431	−1.16211	−0.581056	+	0.813864i	$0.697360\pi$
$458$	0	0
$459$	−14.3726	−0.670858
$460$	0	0
$461$	32.2525	1.50215	0.751073	−	0.660219i	$-0.229536\pi$
$461$	32.2525	1.50215	0.751073	+	0.660219i	$0.229536\pi$
$462$	0	0
$463$	10.2424	0.476005	0.238002	−	0.971265i	$-0.423507\pi$
$463$	10.2424	0.476005	0.238002	+	0.971265i	$0.423507\pi$
$464$	0	0
$465$	−0.533006	−0.0247176
$466$	0	0
$467$	12.5826	0.582253	0.291127	−	0.956685i	$-0.405970\pi$
$467$	12.5826	0.582253	0.291127	+	0.956685i	$0.405970\pi$
$468$	0	0
$469$	−37.7084	−1.74121
$470$	0	0
$471$	59.9020	2.76014
$472$	0	0
$473$	0.595257	0.0273700
$474$	0	0
$475$	−25.1321	−1.15314
$476$	0	0
$477$	71.5259	3.27495
$478$	0	0
$479$	5.02267	0.229492	0.114746	−	0.993395i	$-0.463395\pi$
$479$	5.02267	0.229492	0.114746	+	0.993395i	$0.463395\pi$
$480$	0	0
$481$	8.00881	0.365170
$482$	0	0
$483$	54.6719	2.48766
$484$	0	0
$485$	0.290204	0.0131775
$486$	0	0
$487$	−14.1461	−0.641019	−0.320510	−	0.947245i	$-0.603854\pi$
$487$	−14.1461	−0.641019	−0.320510	+	0.947245i	$0.603854\pi$
$488$	0	0
$489$	20.6442	0.933561
$490$	0	0
$491$	2.94129	0.132739	0.0663693	−	0.997795i	$-0.478858\pi$
$491$	2.94129	0.132739	0.0663693	+	0.997795i	$0.478858\pi$
$492$	0	0
$493$	−5.08995	−0.229240
$494$	0	0
$495$	0.0663512	0.00298226
$496$	0	0
$497$	−31.8844	−1.43021
$498$	0	0
$499$	−26.8514	−1.20204	−0.601018	−	0.799236i	$-0.705238\pi$
$499$	−26.8514	−1.20204	−0.601018	+	0.799236i	$0.705238\pi$
$500$	0	0
$501$	−40.1506	−1.79380
$502$	0	0
$503$	22.6589	1.01031	0.505155	−	0.863029i	$-0.331435\pi$
$503$	22.6589	1.01031	0.505155	+	0.863029i	$0.331435\pi$
$504$	0	0
$505$	0.617661	0.0274856
$506$	0	0
$507$	−31.0613	−1.37948
$508$	0	0
$509$	4.59675	0.203747	0.101874	−	0.994797i	$-0.467516\pi$
$509$	4.59675	0.203747	0.101874	+	0.994797i	$0.467516\pi$
$510$	0	0
$511$	54.0890	2.39276
$512$	0	0
$513$	48.1249	2.12477
$514$	0	0
$515$	−0.639434	−0.0281768
$516$	0	0
$517$	2.64356	0.116264
$518$	0	0
$519$	43.1513	1.89413
$520$	0	0
$521$	−20.6840	−0.906182	−0.453091	−	0.891464i	$-0.649679\pi$
$521$	−20.6840	−0.906182	−0.453091	+	0.891464i	$0.649679\pi$
$522$	0	0
$523$	33.1967	1.45159	0.725794	−	0.687912i	$-0.241472\pi$
$523$	33.1967	1.45159	0.725794	+	0.687912i	$0.241472\pi$
$524$	0	0
$525$	−51.0811	−2.22936
$526$	0	0
$527$	−8.18482	−0.356536
$528$	0	0
$529$	5.62636	0.244624
$530$	0	0
$531$	78.3385	3.39960
$532$	0	0
$533$	−0.136760	−0.00592372
$534$	0	0
$535$	−0.0284444	−0.00122976
$536$	0	0
$537$	−37.1470	−1.60301
$538$	0	0
$539$	−1.46529	−0.0631146
$540$	0	0
$541$	32.4624	1.39567	0.697833	−	0.716261i	$-0.254148\pi$
$541$	32.4624	1.39567	0.697833	+	0.716261i	$0.254148\pi$
$542$	0	0
$543$	15.2453	0.654238
$544$	0	0
$545$	−0.324006	−0.0138789
$546$	0	0
$547$	−7.64787	−0.326999	−0.163500	−	0.986543i	$-0.552278\pi$
$547$	−7.64787	−0.326999	−0.163500	+	0.986543i	$0.552278\pi$
$548$	0	0
$549$	−7.86857	−0.335822
$550$	0	0
$551$	17.0430	0.726058
$552$	0	0
$553$	−7.34078	−0.312162
$554$	0	0
$555$	0.473216	0.0200869
$556$	0	0
$557$	11.5304	0.488559	0.244279	−	0.969705i	$-0.421449\pi$
$557$	11.5304	0.488559	0.244279	+	0.969705i	$0.421449\pi$
$558$	0	0
$559$	2.95521	0.124992
$560$	0	0
$561$	1.51490	0.0639593
$562$	0	0
$563$	15.3958	0.648858	0.324429	−	0.945910i	$-0.394828\pi$
$563$	15.3958	0.648858	0.324429	+	0.945910i	$0.394828\pi$
$564$	0	0
$565$	−0.227926	−0.00958894
$566$	0	0
$567$	35.4050	1.48687
$568$	0	0
$569$	−44.9142	−1.88290	−0.941450	−	0.337153i	$-0.890536\pi$
$569$	−44.9142	−1.88290	−0.941450	+	0.337153i	$0.890536\pi$
$570$	0	0
$571$	13.6202	0.569988	0.284994	−	0.958529i	$-0.408008\pi$
$571$	13.6202	0.569988	0.284994	+	0.958529i	$0.408008\pi$
$572$	0	0
$573$	−20.2298	−0.845110
$574$	0	0
$575$	−26.7462	−1.11539
$576$	0	0
$577$	25.4563	1.05976	0.529879	−	0.848073i	$-0.322237\pi$
$577$	25.4563	1.05976	0.529879	+	0.848073i	$0.322237\pi$
$578$	0	0
$579$	−29.2430	−1.21530
$580$	0	0
$581$	−3.01563	−0.125109
$582$	0	0
$583$	−3.86882	−0.160230
$584$	0	0
$585$	0.329407	0.0136193
$586$	0	0
$587$	0.0745162	0.00307561	0.00153781	−	0.999999i	$-0.499511\pi$
$587$	0.0745162	0.00307561	0.00153781	+	0.999999i	$0.499511\pi$
$588$	0	0
$589$	27.4058	1.12924
$590$	0	0
$591$	68.6287	2.82301
$592$	0	0
$593$	18.1883	0.746903	0.373452	−	0.927650i	$-0.378174\pi$
$593$	18.1883	0.746903	0.373452	+	0.927650i	$0.378174\pi$
$594$	0	0
$595$	0.163730	0.00671228
$596$	0	0
$597$	32.9062	1.34676
$598$	0	0
$599$	−10.8427	−0.443021	−0.221510	−	0.975158i	$-0.571099\pi$
$599$	−10.8427	−0.443021	−0.221510	+	0.975158i	$0.571099\pi$
$600$	0	0
$601$	−10.7179	−0.437193	−0.218597	−	0.975815i	$-0.570148\pi$
$601$	−10.7179	−0.437193	−0.218597	+	0.975815i	$0.570148\pi$
$602$	0	0
$603$	−68.8352	−2.80318
$604$	0	0
$605$	0.351738	0.0143002
$606$	0	0
$607$	−40.9798	−1.66332	−0.831660	−	0.555285i	$-0.812609\pi$
$607$	−40.9798	−1.66332	−0.831660	+	0.555285i	$0.812609\pi$
$608$	0	0
$609$	34.6400	1.40368
$610$	0	0
$611$	13.1242	0.530949
$612$	0	0
$613$	−33.5595	−1.35546	−0.677728	−	0.735313i	$-0.737035\pi$
$613$	−33.5595	−1.35546	−0.677728	+	0.735313i	$0.737035\pi$
$614$	0	0
$615$	−0.00808071	−0.000325846 0
$616$	0	0
$617$	37.5593	1.51208	0.756040	−	0.654526i	$-0.227132\pi$
$617$	37.5593	1.51208	0.756040	+	0.654526i	$0.227132\pi$
$618$	0	0
$619$	−1.00000	−0.0401934
$619$	−1.00000	−0.0401934
$620$	0	0
$621$	51.2157	2.05522
$622$	0	0
$623$	38.3292	1.53563
$624$	0	0
$625$	24.9843	0.999374
$626$	0	0
$627$	−5.07246	−0.202574
$628$	0	0
$629$	7.26669	0.289742
$630$	0	0
$631$	33.2887	1.32520	0.662601	−	0.748973i	$-0.269453\pi$
$631$	33.2887	1.32520	0.662601	+	0.748973i	$0.269453\pi$
$632$	0	0
$633$	−67.9494	−2.70075
$634$	0	0
$635$	−0.104401	−0.00414304
$636$	0	0
$637$	−7.27458	−0.288229
$638$	0	0
$639$	−58.2038	−2.30251
$640$	0	0
$641$	−43.8778	−1.73307	−0.866535	−	0.499116i	$-0.833658\pi$
$641$	−43.8778	−1.73307	−0.866535	+	0.499116i	$0.833658\pi$
$642$	0	0
$643$	16.4937	0.650448	0.325224	−	0.945637i	$-0.394560\pi$
$643$	16.4937	0.650448	0.325224	+	0.945637i	$0.394560\pi$
$644$	0	0
$645$	0.174614	0.00687544
$646$	0	0
$647$	−16.1560	−0.635160	−0.317580	−	0.948232i	$-0.602870\pi$
$647$	−16.1560	−0.635160	−0.317580	+	0.948232i	$0.602870\pi$
$648$	0	0
$649$	−4.23731	−0.166329
$650$	0	0
$651$	55.7023	2.18315
$652$	0	0
$653$	−14.2100	−0.556079	−0.278039	−	0.960570i	$-0.589685\pi$
$653$	−14.2100	−0.556079	−0.278039	+	0.960570i	$0.589685\pi$
$654$	0	0
$655$	0.330918	0.0129300
$656$	0	0
$657$	98.7374	3.85211
$658$	0	0
$659$	−23.0308	−0.897152	−0.448576	−	0.893745i	$-0.648069\pi$
$659$	−23.0308	−0.897152	−0.448576	+	0.893745i	$0.648069\pi$
$660$	0	0
$661$	−30.3656	−1.18109	−0.590543	−	0.807006i	$-0.701087\pi$
$661$	−30.3656	−1.18109	−0.590543	+	0.807006i	$0.701087\pi$
$662$	0	0
$663$	7.52089	0.292087
$664$	0	0
$665$	−0.548229	−0.0212594
$666$	0	0
$667$	18.1376	0.702291
$668$	0	0
$669$	−29.8502	−1.15408
$670$	0	0
$671$	0.425609	0.0164304
$672$	0	0
$673$	−10.7264	−0.413474	−0.206737	−	0.978397i	$-0.566284\pi$
$673$	−10.7264	−0.413474	−0.206737	+	0.978397i	$0.566284\pi$
$674$	0	0
$675$	−47.8519	−1.84182
$676$	0	0
$677$	39.4288	1.51537	0.757687	−	0.652618i	$-0.226329\pi$
$677$	39.4288	1.51537	0.757687	+	0.652618i	$0.226329\pi$
$678$	0	0
$679$	−30.3280	−1.16388
$680$	0	0
$681$	30.1416	1.15503
$682$	0	0
$683$	−13.3888	−0.512308	−0.256154	−	0.966636i	$-0.582455\pi$
$683$	−13.3888	−0.512308	−0.256154	+	0.966636i	$0.582455\pi$
$684$	0	0
$685$	0.123224	0.00470815
$686$	0	0
$687$	38.0000	1.44979
$688$	0	0
$689$	−19.2071	−0.731733
$690$	0	0
$691$	10.3514	0.393787	0.196893	−	0.980425i	$-0.436915\pi$
$691$	10.3514	0.393787	0.196893	+	0.980425i	$0.436915\pi$
$692$	0	0
$693$	−6.93410	−0.263405
$694$	0	0
$695$	0.479952	0.0182056
$696$	0	0
$697$	−0.124087	−0.00470013
$698$	0	0
$699$	−18.8488	−0.712926
$700$	0	0
$701$	1.46168	0.0552070	0.0276035	−	0.999619i	$-0.491212\pi$
$701$	1.46168	0.0552070	0.0276035	+	0.999619i	$0.491212\pi$
$702$	0	0
$703$	−24.3315	−0.917682
$704$	0	0
$705$	0.775470	0.0292059
$706$	0	0
$707$	−64.5493	−2.42763
$708$	0	0
$709$	−2.10668	−0.0791180	−0.0395590	−	0.999217i	$-0.512595\pi$
$709$	−2.10668	−0.0791180	−0.0395590	+	0.999217i	$0.512595\pi$
$710$	0	0
$711$	−13.4003	−0.502551
$712$	0	0
$713$	29.1659	1.09227
$714$	0	0
$715$	−0.0178175	−0.000666338 0
$716$	0	0
$717$	−6.47356	−0.241760
$718$	0	0
$719$	21.1806	0.789903	0.394952	−	0.918702i	$-0.370761\pi$
$719$	21.1806	0.789903	0.394952	+	0.918702i	$0.370761\pi$
$720$	0	0
$721$	66.8248	2.48868
$722$	0	0
$723$	71.6835	2.66594
$724$	0	0
$725$	−16.9464	−0.629372
$726$	0	0
$727$	−39.1371	−1.45151	−0.725757	−	0.687951i	$-0.758511\pi$
$727$	−39.1371	−1.45151	−0.725757	+	0.687951i	$0.758511\pi$
$728$	0	0
$729$	−22.2947	−0.825729
$730$	0	0
$731$	2.68137	0.0991742
$732$	0	0
$733$	−19.7887	−0.730914	−0.365457	−	0.930828i	$-0.619087\pi$
$733$	−19.7887	−0.730914	−0.365457	+	0.930828i	$0.619087\pi$
$734$	0	0
$735$	−0.429833	−0.0158546
$736$	0	0
$737$	3.72328	0.137149
$738$	0	0
$739$	−34.3551	−1.26377	−0.631886	−	0.775061i	$-0.717719\pi$
$739$	−34.3551	−1.26377	−0.631886	+	0.775061i	$0.717719\pi$
$740$	0	0
$741$	−25.1827	−0.925110
$742$	0	0
$743$	−8.25712	−0.302924	−0.151462	−	0.988463i	$-0.548398\pi$
$743$	−8.25712	−0.302924	−0.151462	+	0.988463i	$0.548398\pi$
$744$	0	0
$745$	0.129286	0.00473666
$746$	0	0
$747$	−5.50491	−0.201414
$748$	0	0
$749$	2.97261	0.108617
$750$	0	0
$751$	−12.9342	−0.471975	−0.235988	−	0.971756i	$-0.575833\pi$
$751$	−12.9342	−0.471975	−0.235988	+	0.971756i	$0.575833\pi$
$752$	0	0
$753$	61.7252	2.24939
$754$	0	0
$755$	0.153738	0.00559510
$756$	0	0
$757$	−9.35561	−0.340035	−0.170018	−	0.985441i	$-0.554382\pi$
$757$	−9.35561	−0.340035	−0.170018	+	0.985441i	$0.554382\pi$
$758$	0	0
$759$	−5.39823	−0.195943
$760$	0	0
$761$	−33.5186	−1.21505	−0.607524	−	0.794301i	$-0.707837\pi$
$761$	−33.5186	−1.21505	−0.607524	+	0.794301i	$0.707837\pi$
$762$	0	0
$763$	33.8606	1.22583
$764$	0	0
$765$	0.298883	0.0108061
$766$	0	0
$767$	−21.0365	−0.759585
$768$	0	0
$769$	−11.1247	−0.401166	−0.200583	−	0.979677i	$-0.564284\pi$
$769$	−11.1247	−0.401166	−0.200583	+	0.979677i	$0.564284\pi$
$770$	0	0
$771$	−92.6682	−3.33737
$772$	0	0
$773$	16.8751	0.606953	0.303477	−	0.952839i	$-0.401853\pi$
$773$	16.8751	0.606953	0.303477	+	0.952839i	$0.401853\pi$
$774$	0	0
$775$	−27.2503	−0.978860
$776$	0	0
$777$	−49.4539	−1.77415
$778$	0	0
$779$	0.415489	0.0148864
$780$	0	0
$781$	3.14823	0.112652
$782$	0	0
$783$	32.4502	1.15967
$784$	0	0
$785$	−0.639252	−0.0228159
$786$	0	0
$787$	−10.5636	−0.376553	−0.188277	−	0.982116i	$-0.560290\pi$
$787$	−10.5636	−0.376553	−0.188277	+	0.982116i	$0.560290\pi$
$788$	0	0
$789$	73.6596	2.62235
$790$	0	0
$791$	23.8197	0.846930
$792$	0	0
$793$	2.11298	0.0750340
$794$	0	0
$795$	−1.13489	−0.0402504
$796$	0	0
$797$	27.0776	0.959137	0.479568	−	0.877504i	$-0.340793\pi$
$797$	27.0776	0.959137	0.479568	+	0.877504i	$0.340793\pi$
$798$	0	0
$799$	11.9081	0.421278
$800$	0	0
$801$	69.9685	2.47222
$802$	0	0
$803$	−5.34068	−0.188469
$804$	0	0
$805$	−0.583438	−0.0205635
$806$	0	0
$807$	37.3100	1.31338
$808$	0	0
$809$	5.49991	0.193367	0.0966833	−	0.995315i	$-0.469177\pi$
$809$	5.49991	0.193367	0.0966833	+	0.995315i	$0.469177\pi$
$810$	0	0
$811$	−40.2347	−1.41283	−0.706415	−	0.707798i	$-0.749689\pi$
$811$	−40.2347	−1.41283	−0.706415	+	0.707798i	$0.749689\pi$
$812$	0	0
$813$	42.2510	1.48181
$814$	0	0
$815$	−0.220307	−0.00771701
$816$	0	0
$817$	−8.97822	−0.314108
$818$	0	0
$819$	−34.4250	−1.20291
$820$	0	0
$821$	22.0387	0.769157	0.384579	−	0.923092i	$-0.374347\pi$
$821$	22.0387	0.769157	0.384579	+	0.923092i	$0.374347\pi$
$822$	0	0
$823$	−1.67762	−0.0584781	−0.0292390	−	0.999572i	$-0.509308\pi$
$823$	−1.67762	−0.0584781	−0.0292390	+	0.999572i	$0.509308\pi$
$824$	0	0
$825$	5.04368	0.175599
$826$	0	0
$827$	3.15787	0.109810	0.0549049	−	0.998492i	$-0.482514\pi$
$827$	3.15787	0.109810	0.0549049	+	0.998492i	$0.482514\pi$
$828$	0	0
$829$	24.1320	0.838138	0.419069	−	0.907954i	$-0.362357\pi$
$829$	24.1320	0.838138	0.419069	+	0.907954i	$0.362357\pi$
$830$	0	0
$831$	5.84619	0.202802
$832$	0	0
$833$	−6.60050	−0.228694
$834$	0	0
$835$	0.428472	0.0148279
$836$	0	0
$837$	52.1810	1.80364
$838$	0	0
$839$	54.9414	1.89679	0.948394	−	0.317095i	$-0.102707\pi$
$839$	54.9414	1.89679	0.948394	+	0.317095i	$0.102707\pi$
$840$	0	0
$841$	−17.5080	−0.603726
$842$	0	0
$843$	49.7414	1.71319
$844$	0	0
$845$	0.331475	0.0114031
$846$	0	0
$847$	−36.7587	−1.26304
$848$	0	0
$849$	88.5357	3.03854
$850$	0	0
$851$	−25.8942	−0.887642
$852$	0	0
$853$	−50.7022	−1.73601	−0.868006	−	0.496554i	$-0.834598\pi$
$853$	−50.7022	−1.73601	−0.868006	+	0.496554i	$0.834598\pi$
$854$	0	0
$855$	−1.00077	−0.0342256
$856$	0	0
$857$	50.1517	1.71315	0.856575	−	0.516023i	$-0.172588\pi$
$857$	50.1517	1.71315	0.856575	+	0.516023i	$0.172588\pi$
$858$	0	0
$859$	31.2237	1.06534	0.532669	−	0.846324i	$-0.321189\pi$
$859$	31.2237	1.06534	0.532669	+	0.846324i	$0.321189\pi$
$860$	0	0
$861$	0.844483	0.0287799
$862$	0	0
$863$	−55.4201	−1.88652	−0.943261	−	0.332053i	$-0.892259\pi$
$863$	−55.4201	−1.88652	−0.943261	+	0.332053i	$0.892259\pi$
$864$	0	0
$865$	−0.460495	−0.0156573
$866$	0	0
$867$	−44.6341	−1.51585
$868$	0	0
$869$	0.724819	0.0245878
$870$	0	0
$871$	18.4846	0.626326
$872$	0	0
$873$	−55.3627	−1.87374
$874$	0	0
$875$	1.09035	0.0368606
$876$	0	0
$877$	−49.2248	−1.66220	−0.831102	−	0.556120i	$-0.812289\pi$
$877$	−49.2248	−1.66220	−0.831102	+	0.556120i	$0.812289\pi$
$878$	0	0
$879$	29.1415	0.982918
$880$	0	0
$881$	14.1734	0.477515	0.238758	−	0.971079i	$-0.423260\pi$
$881$	14.1734	0.477515	0.238758	+	0.971079i	$0.423260\pi$
$882$	0	0
$883$	44.5572	1.49947	0.749734	−	0.661739i	$-0.230181\pi$
$883$	44.5572	1.49947	0.749734	+	0.661739i	$0.230181\pi$
$884$	0	0
$885$	−1.24298	−0.0417824
$886$	0	0
$887$	−43.2413	−1.45190	−0.725950	−	0.687748i	$-0.758599\pi$
$887$	−43.2413	−1.45190	−0.725950	+	0.687748i	$0.758599\pi$
$888$	0	0
$889$	10.9106	0.365928
$890$	0	0
$891$	−3.49585	−0.117115
$892$	0	0
$893$	−39.8726	−1.33429
$894$	0	0
$895$	0.396419	0.0132508
$896$	0	0
$897$	−26.8001	−0.894828
$898$	0	0
$899$	18.4795	0.616324
$900$	0	0
$901$	−17.4273	−0.580588
$902$	0	0
$903$	−18.2483	−0.607264
$904$	0	0
$905$	−0.162692	−0.00540807
$906$	0	0
$907$	−0.447171	−0.0148481	−0.00742403	−	0.999972i	$-0.502363\pi$
$907$	−0.447171	−0.0148481	−0.00742403	+	0.999972i	$0.502363\pi$
$908$	0	0
$909$	−117.832	−3.90825
$910$	0	0
$911$	−10.6948	−0.354335	−0.177167	−	0.984181i	$-0.556693\pi$
$911$	−10.6948	−0.354335	−0.177167	+	0.984181i	$0.556693\pi$
$912$	0	0
$913$	0.297759	0.00985439
$914$	0	0
$915$	0.124849	0.00412739
$916$	0	0
$917$	−34.5829	−1.14203
$918$	0	0
$919$	−30.0350	−0.990762	−0.495381	−	0.868676i	$-0.664972\pi$
$919$	−30.0350	−0.990762	−0.495381	+	0.868676i	$0.664972\pi$
$920$	0	0
$921$	−95.0716	−3.13272
$922$	0	0
$923$	15.6297	0.514458
$924$	0	0
$925$	24.1935	0.795478
$926$	0	0
$927$	121.986	4.00655
$928$	0	0
$929$	57.2823	1.87937	0.939686	−	0.342038i	$-0.111117\pi$
$929$	57.2823	1.87937	0.939686	+	0.342038i	$0.111117\pi$
$930$	0	0
$931$	22.1009	0.724327
$932$	0	0
$933$	30.0613	0.984163
$934$	0	0
$935$	−0.0161665	−0.000528701 0
$936$	0	0
$937$	1.22719	0.0400907	0.0200453	−	0.999799i	$-0.493619\pi$
$937$	1.22719	0.0400907	0.0200453	+	0.999799i	$0.493619\pi$
$938$	0	0
$939$	−53.4406	−1.74397
$940$	0	0
$941$	−38.6797	−1.26092	−0.630460	−	0.776221i	$-0.717134\pi$
$941$	−38.6797	−1.26092	−0.630460	+	0.776221i	$0.717134\pi$
$942$	0	0
$943$	0.442173	0.0143991
$944$	0	0
$945$	−1.04384	−0.0339560
$946$	0	0
$947$	−26.2148	−0.851867	−0.425934	−	0.904754i	$-0.640054\pi$
$947$	−26.2148	−0.851867	−0.425934	+	0.904754i	$0.640054\pi$
$948$	0	0
$949$	−26.5143	−0.860692
$950$	0	0
$951$	−69.4757	−2.25290
$952$	0	0
$953$	−0.823051	−0.0266613	−0.0133306	−	0.999911i	$-0.504243\pi$
$953$	−0.823051	−0.0266613	−0.0133306	+	0.999911i	$0.504243\pi$
$954$	0	0
$955$	0.215884	0.00698586
$956$	0	0
$957$	−3.42031	−0.110563
$958$	0	0
$959$	−12.8777	−0.415842
$960$	0	0
$961$	−1.28441	−0.0414325
$962$	0	0
$963$	5.42638	0.174863
$964$	0	0
$965$	0.312071	0.0100459
$966$	0	0
$967$	12.7952	0.411466	0.205733	−	0.978608i	$-0.434042\pi$
$967$	12.7952	0.411466	0.205733	+	0.978608i	$0.434042\pi$
$968$	0	0
$969$	−22.8492	−0.734022
$970$	0	0
$971$	25.3492	0.813494	0.406747	−	0.913541i	$-0.366663\pi$
$971$	25.3492	0.813494	0.406747	+	0.913541i	$0.366663\pi$
$972$	0	0
$973$	−50.1579	−1.60799
$974$	0	0
$975$	25.0399	0.801917
$976$	0	0
$977$	15.0049	0.480050	0.240025	−	0.970767i	$-0.422844\pi$
$977$	15.0049	0.480050	0.240025	+	0.970767i	$0.422844\pi$
$978$	0	0
$979$	−3.78458	−0.120956
$980$	0	0
$981$	61.8112	1.97348
$982$	0	0
$983$	12.4298	0.396451	0.198225	−	0.980156i	$-0.436482\pi$
$983$	12.4298	0.396451	0.198225	+	0.980156i	$0.436482\pi$
$984$	0	0
$985$	−0.732380	−0.0233356
$986$	0	0
$987$	−81.0413	−2.57957
$988$	0	0
$989$	−9.55484	−0.303826
$990$	0	0
$991$	26.4538	0.840331	0.420166	−	0.907447i	$-0.361972\pi$
$991$	26.4538	0.840331	0.420166	+	0.907447i	$0.361972\pi$
$992$	0	0
$993$	−38.2958	−1.21528
$994$	0	0
$995$	−0.351163	−0.0111326
$996$	0	0
$997$	−33.4142	−1.05824	−0.529119	−	0.848547i	$-0.677478\pi$
$997$	−33.4142	−1.05824	−0.529119	+	0.848547i	$0.677478\pi$
$998$	0	0
$999$	−46.3276	−1.46574

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9904.2.a.n.1.28		30
4.3	odd	2		619.2.a.b.1.2	✓	30
12.11	even	2		5571.2.a.g.1.29		30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
619.2.a.b.1.2	✓	30	4.3	odd	2
5571.2.a.g.1.29		30	12.11	even	2
9904.2.a.n.1.28		30	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 9904 = 2^{4} \cdot 619 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9904.a (trivial)

\(f(q)\)	\(=\)	\(q+3.02694 q^{3} -0.0323024 q^{5} +3.37580 q^{7} +6.16239 q^{9} +O(q^{10})\)	\(q+3.02694 q^{3} -0.0323024 q^{5} +3.37580 q^{7} +6.16239 q^{9} -0.333322 q^{11} -1.65481 q^{13} -0.0977777 q^{15} -1.50147 q^{17} +5.02748 q^{19} +10.2184 q^{21} +5.35036 q^{23} -4.99896 q^{25} +9.57238 q^{27} +3.38998 q^{29} +5.45120 q^{31} -1.00895 q^{33} -0.109047 q^{35} -4.83972 q^{37} -5.00902 q^{39} +0.0826437 q^{41} -1.78583 q^{43} -0.199060 q^{45} -7.93095 q^{47} +4.39602 q^{49} -4.54487 q^{51} +11.6068 q^{53} +0.0107671 q^{55} +15.2179 q^{57} +12.7123 q^{59} -1.27687 q^{61} +20.8030 q^{63} +0.0534544 q^{65} -11.1702 q^{67} +16.1952 q^{69} -9.44501 q^{71} +16.0226 q^{73} -15.1316 q^{75} -1.12523 q^{77} -2.17453 q^{79} +10.4879 q^{81} -0.893308 q^{83} +0.0485011 q^{85} +10.2613 q^{87} +11.3541 q^{89} -5.58631 q^{91} +16.5005 q^{93} -0.162400 q^{95} -8.98396 q^{97} -2.05406 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - q^{3} + 21 q^{5} - 2 q^{7} + 43 q^{9}+O(q^{10}) \) 30 * q - q^3 + 21 * q^5 - 2 * q^7 + 43 * q^9	\( 30 q - q^{3} + 21 q^{5} - 2 q^{7} + 43 q^{9} - 23 q^{11} + 9 q^{13} + 2 q^{15} + 4 q^{17} + q^{19} + 30 q^{21} - 4 q^{23} + 35 q^{25} + 5 q^{27} + 90 q^{29} - 2 q^{31} - 6 q^{33} - 9 q^{35} + 19 q^{37} - 32 q^{39} + 59 q^{41} + 4 q^{43} + 30 q^{45} - 4 q^{47} + 30 q^{49} + 34 q^{53} + 17 q^{55} - 8 q^{57} - 13 q^{59} + 16 q^{61} + 40 q^{63} + 31 q^{65} + 11 q^{67} + 6 q^{69} - 42 q^{71} - 4 q^{73} + 52 q^{75} + 29 q^{77} - 3 q^{79} + 30 q^{81} + 11 q^{83} + 19 q^{85} + 20 q^{87} + 58 q^{89} + 39 q^{91} - 15 q^{93} - 23 q^{95} - 9 q^{97} + 8 q^{99}+O(q^{100}) \) 30 * q - q^3 + 21 * q^5 - 2 * q^7 + 43 * q^9 - 23 * q^11 + 9 * q^13 + 2 * q^15 + 4 * q^17 + q^19 + 30 * q^21 - 4 * q^23 + 35 * q^25 + 5 * q^27 + 90 * q^29 - 2 * q^31 - 6 * q^33 - 9 * q^35 + 19 * q^37 - 32 * q^39 + 59 * q^41 + 4 * q^43 + 30 * q^45 - 4 * q^47 + 30 * q^49 + 34 * q^53 + 17 * q^55 - 8 * q^57 - 13 * q^59 + 16 * q^61 + 40 * q^63 + 31 * q^65 + 11 * q^67 + 6 * q^69 - 42 * q^71 - 4 * q^73 + 52 * q^75 + 29 * q^77 - 3 * q^79 + 30 * q^81 + 11 * q^83 + 19 * q^85 + 20 * q^87 + 58 * q^89 + 39 * q^91 - 15 * q^93 - 23 * q^95 - 9 * q^97 + 8 * q^99

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