LMFDB - Embedded newform 6004.2.a.f.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6004,2,Mod(1,6004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6004, 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("6004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6004 = 2^{2} \cdot 19 \cdot 79 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6004.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:	$47.9421813736$
Analytic rank:	$1$
Dimension:	$25$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Character	$\chi$	$=$	6004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.53952 q^{3} +0.492772 q^{5} -1.61008 q^{7} -0.629866 q^{9} +O(q^{10})$	$q+1.53952 q^{3} +0.492772 q^{5} -1.61008 q^{7} -0.629866 q^{9} +4.80774 q^{11} -0.407594 q^{13} +0.758635 q^{15} -2.43674 q^{17} -1.00000 q^{19} -2.47875 q^{21} -1.48873 q^{23} -4.75718 q^{25} -5.58827 q^{27} +3.50177 q^{29} -5.52173 q^{31} +7.40162 q^{33} -0.793401 q^{35} -3.84260 q^{37} -0.627500 q^{39} -4.91114 q^{41} -8.50157 q^{43} -0.310381 q^{45} -4.77373 q^{47} -4.40765 q^{49} -3.75142 q^{51} +1.77383 q^{53} +2.36912 q^{55} -1.53952 q^{57} +0.295569 q^{59} +0.405339 q^{61} +1.01413 q^{63} -0.200851 q^{65} +6.04899 q^{67} -2.29194 q^{69} +2.60154 q^{71} -5.03698 q^{73} -7.32378 q^{75} -7.74082 q^{77} +1.00000 q^{79} -6.71367 q^{81} +5.78347 q^{83} -1.20076 q^{85} +5.39106 q^{87} +5.38410 q^{89} +0.656257 q^{91} -8.50083 q^{93} -0.492772 q^{95} +11.8079 q^{97} -3.02823 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q + 4 q^{3} - 8 q^{5} + 2 q^{7} + 13 q^{9}+O(q^{10}) $ 25 * q + 4 * q^3 - 8 * q^5 + 2 * q^7 + 13 * q^9	$ 25 q + 4 q^{3} - 8 q^{5} + 2 q^{7} + 13 q^{9} - 3 q^{11} + q^{13} - 5 q^{15} - 13 q^{17} - 25 q^{19} - 24 q^{21} - 31 q^{23} + 21 q^{25} + 7 q^{27} - 19 q^{29} - 7 q^{31} - 30 q^{33} - q^{35} - 29 q^{37} - 26 q^{39} - 40 q^{41} - 40 q^{45} - 8 q^{47} - 9 q^{49} + 12 q^{51} - 38 q^{53} - 29 q^{55} - 4 q^{57} + 18 q^{59} - 26 q^{61} - 40 q^{63} - 70 q^{65} - 13 q^{67} + q^{69} - 47 q^{71} - 8 q^{73} + 7 q^{75} - 19 q^{77} + 25 q^{79} - 19 q^{81} - 8 q^{83} - 33 q^{85} - 50 q^{87} - 54 q^{89} - 12 q^{91} - 24 q^{93} + 8 q^{95} - 4 q^{97} - 39 q^{99}+O(q^{100}) $ 25 * q + 4 * q^3 - 8 * q^5 + 2 * q^7 + 13 * q^9 - 3 * q^11 + q^13 - 5 * q^15 - 13 * q^17 - 25 * q^19 - 24 * q^21 - 31 * q^23 + 21 * q^25 + 7 * q^27 - 19 * q^29 - 7 * q^31 - 30 * q^33 - q^35 - 29 * q^37 - 26 * q^39 - 40 * q^41 - 40 * q^45 - 8 * q^47 - 9 * q^49 + 12 * q^51 - 38 * q^53 - 29 * q^55 - 4 * q^57 + 18 * q^59 - 26 * q^61 - 40 * q^63 - 70 * q^65 - 13 * q^67 + q^69 - 47 * q^71 - 8 * q^73 + 7 * q^75 - 19 * q^77 + 25 * q^79 - 19 * q^81 - 8 * q^83 - 33 * q^85 - 50 * q^87 - 54 * q^89 - 12 * q^91 - 24 * q^93 + 8 * q^95 - 4 * q^97 - 39 * 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
$3$	1.53952	0.888845	0.444422	−	0.895817i	$-0.353409\pi$
$3$	1.53952	0.888845	0.444422	+	0.895817i	$0.353409\pi$
$4$	0	0
$5$	0.492772	0.220375	0.110187	−	0.993911i	$-0.464855\pi$
$5$	0.492772	0.220375	0.110187	+	0.993911i	$0.464855\pi$
$6$	0	0
$7$	−1.61008	−0.608552	−0.304276	−	0.952584i	$-0.598414\pi$
$7$	−1.61008	−0.608552	−0.304276	+	0.952584i	$0.598414\pi$
$8$	0	0
$9$	−0.629866	−0.209955
$10$	0	0
$11$	4.80774	1.44959	0.724793	−	0.688966i	$-0.241935\pi$
$11$	4.80774	1.44959	0.724793	+	0.688966i	$0.241935\pi$
$12$	0	0
$13$	−0.407594	−0.113046	−0.0565231	−	0.998401i	$-0.518001\pi$
$13$	−0.407594	−0.113046	−0.0565231	+	0.998401i	$0.518001\pi$
$14$	0	0
$15$	0.758635	0.195879
$16$	0	0
$17$	−2.43674	−0.590997	−0.295498	−	0.955343i	$-0.595486\pi$
$17$	−2.43674	−0.590997	−0.295498	+	0.955343i	$0.595486\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−2.47875	−0.540908
$22$	0	0
$23$	−1.48873	−0.310422	−0.155211	−	0.987881i	$-0.549606\pi$
$23$	−1.48873	−0.310422	−0.155211	+	0.987881i	$0.549606\pi$
$24$	0	0
$25$	−4.75718	−0.951435
$26$	0	0
$27$	−5.58827	−1.07546
$28$	0	0
$29$	3.50177	0.650263	0.325131	−	0.945669i	$-0.394591\pi$
$29$	3.50177	0.650263	0.325131	+	0.945669i	$0.394591\pi$
$30$	0	0
$31$	−5.52173	−0.991732	−0.495866	−	0.868399i	$-0.665149\pi$
$31$	−5.52173	−0.991732	−0.495866	+	0.868399i	$0.665149\pi$
$32$	0	0
$33$	7.40162	1.28846
$34$	0	0
$35$	−0.793401	−0.134109
$36$	0	0
$37$	−3.84260	−0.631719	−0.315859	−	0.948806i	$-0.602293\pi$
$37$	−3.84260	−0.631719	−0.315859	+	0.948806i	$0.602293\pi$
$38$	0	0
$39$	−0.627500	−0.100480
$40$	0	0
$41$	−4.91114	−0.766991	−0.383495	−	0.923543i	$-0.625280\pi$
$41$	−4.91114	−0.766991	−0.383495	+	0.923543i	$0.625280\pi$
$42$	0	0
$43$	−8.50157	−1.29648	−0.648238	−	0.761438i	$-0.724494\pi$
$43$	−8.50157	−1.29648	−0.648238	+	0.761438i	$0.724494\pi$
$44$	0	0
$45$	−0.310381	−0.0462688
$46$	0	0
$47$	−4.77373	−0.696320	−0.348160	−	0.937435i	$-0.613193\pi$
$47$	−4.77373	−0.696320	−0.348160	+	0.937435i	$0.613193\pi$
$48$	0	0
$49$	−4.40765	−0.629665
$50$	0	0
$51$	−3.75142	−0.525304
$52$	0	0
$53$	1.77383	0.243654	0.121827	−	0.992551i	$-0.461125\pi$
$53$	1.77383	0.243654	0.121827	+	0.992551i	$0.461125\pi$
$54$	0	0
$55$	2.36912	0.319452
$56$	0	0
$57$	−1.53952	−0.203915
$58$	0	0
$59$	0.295569	0.0384798	0.0192399	−	0.999815i	$-0.493875\pi$
$59$	0.295569	0.0384798	0.0192399	+	0.999815i	$0.493875\pi$
$60$	0	0
$61$	0.405339	0.0518984	0.0259492	−	0.999663i	$-0.491739\pi$
$61$	0.405339	0.0518984	0.0259492	+	0.999663i	$0.491739\pi$
$62$	0	0
$63$	1.01413	0.127769
$64$	0	0
$65$	−0.200851	−0.0249125
$66$	0	0
$67$	6.04899	0.739002	0.369501	−	0.929230i	$-0.379529\pi$
$67$	6.04899	0.739002	0.369501	+	0.929230i	$0.379529\pi$
$68$	0	0
$69$	−2.29194	−0.275917
$70$	0	0
$71$	2.60154	0.308746	0.154373	−	0.988013i	$-0.450664\pi$
$71$	2.60154	0.308746	0.154373	+	0.988013i	$0.450664\pi$
$72$	0	0
$73$	−5.03698	−0.589534	−0.294767	−	0.955569i	$-0.595242\pi$
$73$	−5.03698	−0.589534	−0.294767	+	0.955569i	$0.595242\pi$
$74$	0	0
$75$	−7.32378	−0.845678
$76$	0	0
$77$	−7.74082	−0.882148
$78$	0	0
$79$	1.00000	0.112509
$79$	1.00000	0.112509
$80$	0	0
$81$	−6.71367	−0.745963
$82$	0	0
$83$	5.78347	0.634819	0.317409	−	0.948289i	$-0.397187\pi$
$83$	5.78347	0.634819	0.317409	+	0.948289i	$0.397187\pi$
$84$	0	0
$85$	−1.20076	−0.130241
$86$	0	0
$87$	5.39106	0.577983
$88$	0	0
$89$	5.38410	0.570714	0.285357	−	0.958421i	$-0.407888\pi$
$89$	5.38410	0.570714	0.285357	+	0.958421i	$0.407888\pi$
$90$	0	0
$91$	0.656257	0.0687944
$92$	0	0
$93$	−8.50083	−0.881495
$94$	0	0
$95$	−0.492772	−0.0505574
$96$	0	0
$97$	11.8079	1.19891	0.599456	−	0.800407i	$-0.295383\pi$
$97$	11.8079	1.19891	0.599456	+	0.800407i	$0.295383\pi$
$98$	0	0
$99$	−3.02823	−0.304349
$100$	0	0
$101$	10.3553	1.03039	0.515194	−	0.857073i	$-0.327720\pi$
$101$	10.3553	1.03039	0.515194	+	0.857073i	$0.327720\pi$
$102$	0	0
$103$	−11.4183	−1.12508	−0.562540	−	0.826770i	$-0.690176\pi$
$103$	−11.4183	−1.12508	−0.562540	+	0.826770i	$0.690176\pi$
$104$	0	0
$105$	−1.22146	−0.119202
$106$	0	0
$107$	−8.34233	−0.806483	−0.403242	−	0.915094i	$-0.632117\pi$
$107$	−8.34233	−0.806483	−0.403242	+	0.915094i	$0.632117\pi$
$108$	0	0
$109$	−8.03555	−0.769666	−0.384833	−	0.922986i	$-0.625741\pi$
$109$	−8.03555	−0.769666	−0.384833	+	0.922986i	$0.625741\pi$
$110$	0	0
$111$	−5.91577	−0.561500
$112$	0	0
$113$	−2.65027	−0.249316	−0.124658	−	0.992200i	$-0.539783\pi$
$113$	−2.65027	−0.249316	−0.124658	+	0.992200i	$0.539783\pi$
$114$	0	0
$115$	−0.733607	−0.0684092
$116$	0	0
$117$	0.256730	0.0237347
$118$	0	0
$119$	3.92334	0.359652
$120$	0	0
$121$	12.1143	1.10130
$122$	0	0
$123$	−7.56081	−0.681735
$124$	0	0
$125$	−4.80807	−0.430047
$126$	0	0
$127$	1.43594	0.127419	0.0637096	−	0.997968i	$-0.479707\pi$
$127$	1.43594	0.127419	0.0637096	+	0.997968i	$0.479707\pi$
$128$	0	0
$129$	−13.0884	−1.15237
$130$	0	0
$131$	7.22327	0.631100	0.315550	−	0.948909i	$-0.397811\pi$
$131$	7.22327	0.631100	0.315550	+	0.948909i	$0.397811\pi$
$132$	0	0
$133$	1.61008	0.139611
$134$	0	0
$135$	−2.75374	−0.237004
$136$	0	0
$137$	19.0829	1.63036	0.815182	−	0.579205i	$-0.196637\pi$
$137$	19.0829	1.63036	0.815182	+	0.579205i	$0.196637\pi$
$138$	0	0
$139$	−18.2549	−1.54836	−0.774179	−	0.632967i	$-0.781837\pi$
$139$	−18.2549	−1.54836	−0.774179	+	0.632967i	$0.781837\pi$
$140$	0	0
$141$	−7.34927	−0.618920
$142$	0	0
$143$	−1.95960	−0.163870
$144$	0	0
$145$	1.72558	0.143301
$146$	0	0
$147$	−6.78569	−0.559674
$148$	0	0
$149$	11.6668	0.955779	0.477889	−	0.878420i	$-0.341402\pi$
$149$	11.6668	0.955779	0.477889	+	0.878420i	$0.341402\pi$
$150$	0	0
$151$	−20.7991	−1.69261	−0.846304	−	0.532700i	$-0.821178\pi$
$151$	−20.7991	−1.69261	−0.846304	+	0.532700i	$0.821178\pi$
$152$	0	0
$153$	1.53482	0.124083
$154$	0	0
$155$	−2.72095	−0.218552
$156$	0	0
$157$	7.09809	0.566490	0.283245	−	0.959048i	$-0.408589\pi$
$157$	7.09809	0.566490	0.283245	+	0.959048i	$0.408589\pi$
$158$	0	0
$159$	2.73085	0.216571
$160$	0	0
$161$	2.39697	0.188908
$162$	0	0
$163$	18.5698	1.45450	0.727249	−	0.686374i	$-0.240799\pi$
$163$	18.5698	1.45450	0.727249	+	0.686374i	$0.240799\pi$
$164$	0	0
$165$	3.64732	0.283943
$166$	0	0
$167$	−4.31738	−0.334089	−0.167044	−	0.985949i	$-0.553422\pi$
$167$	−4.31738	−0.334089	−0.167044	+	0.985949i	$0.553422\pi$
$168$	0	0
$169$	−12.8339	−0.987221
$170$	0	0
$171$	0.629866	0.0481671
$172$	0	0
$173$	1.29282	0.0982910	0.0491455	−	0.998792i	$-0.484350\pi$
$173$	1.29282	0.0982910	0.0491455	+	0.998792i	$0.484350\pi$
$174$	0	0
$175$	7.65942	0.578997
$176$	0	0
$177$	0.455036	0.0342026
$178$	0	0
$179$	−20.7907	−1.55397	−0.776984	−	0.629520i	$-0.783252\pi$
$179$	−20.7907	−1.55397	−0.776984	+	0.629520i	$0.783252\pi$
$180$	0	0
$181$	−15.8367	−1.17713	−0.588567	−	0.808449i	$-0.700308\pi$
$181$	−15.8367	−1.17713	−0.588567	+	0.808449i	$0.700308\pi$
$182$	0	0
$183$	0.624029	0.0461296
$184$	0	0
$185$	−1.89353	−0.139215
$186$	0	0
$187$	−11.7152	−0.856701
$188$	0	0
$189$	8.99753	0.654474
$190$	0	0
$191$	14.3120	1.03558	0.517790	−	0.855508i	$-0.326755\pi$
$191$	14.3120	1.03558	0.517790	+	0.855508i	$0.326755\pi$
$192$	0	0
$193$	0.866486	0.0623710	0.0311855	−	0.999514i	$-0.490072\pi$
$193$	0.866486	0.0623710	0.0311855	+	0.999514i	$0.490072\pi$
$194$	0	0
$195$	−0.309215	−0.0221433
$196$	0	0
$197$	−11.6493	−0.829982	−0.414991	−	0.909826i	$-0.636215\pi$
$197$	−11.6493	−0.829982	−0.414991	+	0.909826i	$0.636215\pi$
$198$	0	0
$199$	−13.3791	−0.948419	−0.474210	−	0.880412i	$-0.657266\pi$
$199$	−13.3791	−0.948419	−0.474210	+	0.880412i	$0.657266\pi$
$200$	0	0
$201$	9.31256	0.656857
$202$	0	0
$203$	−5.63812	−0.395719
$204$	0	0
$205$	−2.42007	−0.169025
$206$	0	0
$207$	0.937703	0.0651749
$208$	0	0
$209$	−4.80774	−0.332558
$210$	0	0
$211$	−12.1841	−0.838789	−0.419395	−	0.907804i	$-0.637758\pi$
$211$	−12.1841	−0.838789	−0.419395	+	0.907804i	$0.637758\pi$
$212$	0	0
$213$	4.00513	0.274427
$214$	0	0
$215$	−4.18934	−0.285710
$216$	0	0
$217$	8.89040	0.603520
$218$	0	0
$219$	−7.75455	−0.524004
$220$	0	0
$221$	0.993200	0.0668099
$222$	0	0
$223$	16.2966	1.09130	0.545652	−	0.838012i	$-0.316282\pi$
$223$	16.2966	1.09130	0.545652	+	0.838012i	$0.316282\pi$
$224$	0	0
$225$	2.99638	0.199759
$226$	0	0
$227$	16.1531	1.07212	0.536058	−	0.844181i	$-0.319913\pi$
$227$	16.1531	1.07212	0.536058	+	0.844181i	$0.319913\pi$
$228$	0	0
$229$	−24.1450	−1.59555	−0.797773	−	0.602958i	$-0.793989\pi$
$229$	−24.1450	−1.59555	−0.797773	+	0.602958i	$0.793989\pi$
$230$	0	0
$231$	−11.9172	−0.784093
$232$	0	0
$233$	−8.95044	−0.586363	−0.293181	−	0.956057i	$-0.594714\pi$
$233$	−8.95044	−0.586363	−0.293181	+	0.956057i	$0.594714\pi$
$234$	0	0
$235$	−2.35236	−0.153451
$236$	0	0
$237$	1.53952	0.100003
$238$	0	0
$239$	−11.3997	−0.737386	−0.368693	−	0.929551i	$-0.620195\pi$
$239$	−11.3997	−0.737386	−0.368693	+	0.929551i	$0.620195\pi$
$240$	0	0
$241$	4.13582	0.266411	0.133206	−	0.991088i	$-0.457473\pi$
$241$	4.13582	0.266411	0.133206	+	0.991088i	$0.457473\pi$
$242$	0	0
$243$	6.42894	0.412417
$244$	0	0
$245$	−2.17197	−0.138762
$246$	0	0
$247$	0.407594	0.0259346
$248$	0	0
$249$	8.90380	0.564255
$250$	0	0
$251$	11.2865	0.712400	0.356200	−	0.934410i	$-0.384072\pi$
$251$	11.2865	0.712400	0.356200	+	0.934410i	$0.384072\pi$
$252$	0	0
$253$	−7.15744	−0.449984
$254$	0	0
$255$	−1.84860	−0.115764
$256$	0	0
$257$	−29.2910	−1.82713	−0.913563	−	0.406698i	$-0.866680\pi$
$257$	−29.2910	−1.82713	−0.913563	+	0.406698i	$0.866680\pi$
$258$	0	0
$259$	6.18687	0.384434
$260$	0	0
$261$	−2.20565	−0.136526
$262$	0	0
$263$	−7.31442	−0.451027	−0.225513	−	0.974240i	$-0.572406\pi$
$263$	−7.31442	−0.451027	−0.225513	+	0.974240i	$0.572406\pi$
$264$	0	0
$265$	0.874095	0.0536952
$266$	0	0
$267$	8.28896	0.507276
$268$	0	0
$269$	−28.0009	−1.70724	−0.853622	−	0.520893i	$-0.825599\pi$
$269$	−28.0009	−1.70724	−0.853622	+	0.520893i	$0.825599\pi$
$270$	0	0
$271$	−11.1710	−0.678593	−0.339296	−	0.940679i	$-0.610189\pi$
$271$	−11.1710	−0.678593	−0.339296	+	0.940679i	$0.610189\pi$
$272$	0	0
$273$	1.01032	0.0611475
$274$	0	0
$275$	−22.8712	−1.37919
$276$	0	0
$277$	−23.9277	−1.43768	−0.718839	−	0.695177i	$-0.755326\pi$
$277$	−23.9277	−1.43768	−0.718839	+	0.695177i	$0.755326\pi$
$278$	0	0
$279$	3.47795	0.208219
$280$	0	0
$281$	12.5468	0.748481	0.374241	−	0.927332i	$-0.377903\pi$
$281$	12.5468	0.748481	0.374241	+	0.927332i	$0.377903\pi$
$282$	0	0
$283$	−15.4948	−0.921068	−0.460534	−	0.887642i	$-0.652342\pi$
$283$	−15.4948	−0.921068	−0.460534	+	0.887642i	$0.652342\pi$
$284$	0	0
$285$	−0.758635	−0.0449376
$286$	0	0
$287$	7.90730	0.466753
$288$	0	0
$289$	−11.0623	−0.650723
$290$	0	0
$291$	18.1786	1.06565
$292$	0	0
$293$	12.2029	0.712902	0.356451	−	0.934314i	$-0.383987\pi$
$293$	12.2029	0.712902	0.356451	+	0.934314i	$0.383987\pi$
$294$	0	0
$295$	0.145648	0.00847997
$296$	0	0
$297$	−26.8669	−1.55898
$298$	0	0
$299$	0.606798	0.0350921
$300$	0	0
$301$	13.6882	0.788973
$302$	0	0
$303$	15.9422	0.915855
$304$	0	0
$305$	0.199740	0.0114371
$306$	0	0
$307$	−22.2996	−1.27271	−0.636354	−	0.771397i	$-0.719558\pi$
$307$	−22.2996	−1.27271	−0.636354	+	0.771397i	$0.719558\pi$
$308$	0	0
$309$	−17.5788	−1.00002
$310$	0	0
$311$	21.1007	1.19651	0.598256	−	0.801305i	$-0.295860\pi$
$311$	21.1007	1.19651	0.598256	+	0.801305i	$0.295860\pi$
$312$	0	0
$313$	−18.4994	−1.04565	−0.522825	−	0.852440i	$-0.675122\pi$
$313$	−18.4994	−1.04565	−0.522825	+	0.852440i	$0.675122\pi$
$314$	0	0
$315$	0.499737	0.0281570
$316$	0	0
$317$	−27.9169	−1.56797	−0.783985	−	0.620780i	$-0.786816\pi$
$317$	−27.9169	−1.56797	−0.783985	+	0.620780i	$0.786816\pi$
$318$	0	0
$319$	16.8356	0.942613
$320$	0	0
$321$	−12.8432	−0.716838
$322$	0	0
$323$	2.43674	0.135584
$324$	0	0
$325$	1.93899	0.107556
$326$	0	0
$327$	−12.3709	−0.684113
$328$	0	0
$329$	7.68607	0.423747
$330$	0	0
$331$	7.90799	0.434662	0.217331	−	0.976098i	$-0.430265\pi$
$331$	7.90799	0.434662	0.217331	+	0.976098i	$0.430265\pi$
$332$	0	0
$333$	2.42032	0.132633
$334$	0	0
$335$	2.98077	0.162857
$336$	0	0
$337$	14.9155	0.812501	0.406250	−	0.913762i	$-0.366836\pi$
$337$	14.9155	0.812501	0.406250	+	0.913762i	$0.366836\pi$
$338$	0	0
$339$	−4.08015	−0.221603
$340$	0	0
$341$	−26.5470	−1.43760
$342$	0	0
$343$	18.3672	0.991735
$344$	0	0
$345$	−1.12941	−0.0608051
$346$	0	0
$347$	28.4070	1.52497	0.762483	−	0.647009i	$-0.223980\pi$
$347$	28.4070	1.52497	0.762483	+	0.647009i	$0.223980\pi$
$348$	0	0
$349$	−12.3166	−0.659291	−0.329645	−	0.944105i	$-0.606929\pi$
$349$	−12.3166	−0.659291	−0.329645	+	0.944105i	$0.606929\pi$
$350$	0	0
$351$	2.27774	0.121577
$352$	0	0
$353$	13.9646	0.743261	0.371631	−	0.928381i	$-0.378799\pi$
$353$	13.9646	0.743261	0.371631	+	0.928381i	$0.378799\pi$
$354$	0	0
$355$	1.28197	0.0680398
$356$	0	0
$357$	6.04007	0.319675
$358$	0	0
$359$	−26.0353	−1.37409	−0.687045	−	0.726615i	$-0.741092\pi$
$359$	−26.0353	−1.37409	−0.687045	+	0.726615i	$0.741092\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	18.6503	0.978886
$364$	0	0
$365$	−2.48208	−0.129918
$366$	0	0
$367$	14.2003	0.741251	0.370625	−	0.928782i	$-0.379143\pi$
$367$	14.2003	0.741251	0.370625	+	0.928782i	$0.379143\pi$
$368$	0	0
$369$	3.09336	0.161034
$370$	0	0
$371$	−2.85600	−0.148276
$372$	0	0
$373$	25.4904	1.31984	0.659921	−	0.751335i	$-0.270590\pi$
$373$	25.4904	1.31984	0.659921	+	0.751335i	$0.270590\pi$
$374$	0	0
$375$	−7.40213	−0.382245
$376$	0	0
$377$	−1.42730	−0.0735097
$378$	0	0
$379$	11.9490	0.613780	0.306890	−	0.951745i	$-0.400712\pi$
$379$	11.9490	0.613780	0.306890	+	0.951745i	$0.400712\pi$
$380$	0	0
$381$	2.21067	0.113256
$382$	0	0
$383$	6.48025	0.331125	0.165563	−	0.986199i	$-0.447056\pi$
$383$	6.48025	0.331125	0.165563	+	0.986199i	$0.447056\pi$
$384$	0	0
$385$	−3.81446	−0.194403
$386$	0	0
$387$	5.35485	0.272202
$388$	0	0
$389$	−30.2247	−1.53245	−0.766226	−	0.642572i	$-0.777868\pi$
$389$	−30.2247	−1.53245	−0.766226	+	0.642572i	$0.777868\pi$
$390$	0	0
$391$	3.62766	0.183459
$392$	0	0
$393$	11.1204	0.560950
$394$	0	0
$395$	0.492772	0.0247941
$396$	0	0
$397$	27.7157	1.39101	0.695507	−	0.718520i	$-0.255180\pi$
$397$	27.7157	1.39101	0.695507	+	0.718520i	$0.255180\pi$
$398$	0	0
$399$	2.47875	0.124093
$400$	0	0
$401$	36.5795	1.82669	0.913345	−	0.407186i	$-0.133490\pi$
$401$	36.5795	1.82669	0.913345	+	0.407186i	$0.133490\pi$
$402$	0	0
$403$	2.25062	0.112111
$404$	0	0
$405$	−3.30831	−0.164391
$406$	0	0
$407$	−18.4742	−0.915731
$408$	0	0
$409$	31.4163	1.55343	0.776717	−	0.629850i	$-0.216884\pi$
$409$	31.4163	1.55343	0.776717	+	0.629850i	$0.216884\pi$
$410$	0	0
$411$	29.3786	1.44914
$412$	0	0
$413$	−0.475889	−0.0234170
$414$	0	0
$415$	2.84994	0.139898
$416$	0	0
$417$	−28.1038	−1.37625
$418$	0	0
$419$	32.3443	1.58012	0.790061	−	0.613028i	$-0.210049\pi$
$419$	32.3443	1.58012	0.790061	+	0.613028i	$0.210049\pi$
$420$	0	0
$421$	−4.81119	−0.234483	−0.117242	−	0.993103i	$-0.537405\pi$
$421$	−4.81119	−0.234483	−0.117242	+	0.993103i	$0.537405\pi$
$422$	0	0
$423$	3.00681	0.146196
$424$	0	0
$425$	11.5920	0.562295
$426$	0	0
$427$	−0.652627	−0.0315828
$428$	0	0
$429$	−3.01685	−0.145655
$430$	0	0
$431$	−5.04130	−0.242831	−0.121415	−	0.992602i	$-0.538743\pi$
$431$	−5.04130	−0.242831	−0.121415	+	0.992602i	$0.538743\pi$
$432$	0	0
$433$	6.97831	0.335356	0.167678	−	0.985842i	$-0.446373\pi$
$433$	6.97831	0.335356	0.167678	+	0.985842i	$0.446373\pi$
$434$	0	0
$435$	2.65657	0.127373
$436$	0	0
$437$	1.48873	0.0712158
$438$	0	0
$439$	17.6810	0.843870	0.421935	−	0.906626i	$-0.361351\pi$
$439$	17.6810	0.843870	0.421935	+	0.906626i	$0.361351\pi$
$440$	0	0
$441$	2.77623	0.132202
$442$	0	0
$443$	21.7921	1.03538	0.517688	−	0.855570i	$-0.326793\pi$
$443$	21.7921	1.03538	0.517688	+	0.855570i	$0.326793\pi$
$444$	0	0
$445$	2.65314	0.125771
$446$	0	0
$447$	17.9613	0.849539
$448$	0	0
$449$	−1.93892	−0.0915032	−0.0457516	−	0.998953i	$-0.514568\pi$
$449$	−1.93892	−0.0915032	−0.0457516	+	0.998953i	$0.514568\pi$
$450$	0	0
$451$	−23.6114	−1.11182
$452$	0	0
$453$	−32.0207	−1.50447
$454$	0	0
$455$	0.323385	0.0151605
$456$	0	0
$457$	25.8756	1.21041	0.605205	−	0.796070i	$-0.293091\pi$
$457$	25.8756	1.21041	0.605205	+	0.796070i	$0.293091\pi$
$458$	0	0
$459$	13.6172	0.635594
$460$	0	0
$461$	−7.07899	−0.329702	−0.164851	−	0.986319i	$-0.552714\pi$
$461$	−7.07899	−0.329702	−0.164851	+	0.986319i	$0.552714\pi$
$462$	0	0
$463$	−18.2991	−0.850432	−0.425216	−	0.905092i	$-0.639802\pi$
$463$	−18.2991	−0.850432	−0.425216	+	0.905092i	$0.639802\pi$
$464$	0	0
$465$	−4.18897	−0.194259
$466$	0	0
$467$	−13.7207	−0.634918	−0.317459	−	0.948272i	$-0.602830\pi$
$467$	−13.7207	−0.634918	−0.317459	+	0.948272i	$0.602830\pi$
$468$	0	0
$469$	−9.73933	−0.449721
$470$	0	0
$471$	10.9277	0.503521
$472$	0	0
$473$	−40.8733	−1.87936
$474$	0	0
$475$	4.75718	0.218274
$476$	0	0
$477$	−1.11728	−0.0511566
$478$	0	0
$479$	−26.2189	−1.19797	−0.598987	−	0.800759i	$-0.704430\pi$
$479$	−26.2189	−1.19797	−0.598987	+	0.800759i	$0.704430\pi$
$480$	0	0
$481$	1.56622	0.0714134
$482$	0	0
$483$	3.69020	0.167910
$484$	0	0
$485$	5.81862	0.264210
$486$	0	0
$487$	−2.31106	−0.104724	−0.0523620	−	0.998628i	$-0.516675\pi$
$487$	−2.31106	−0.104724	−0.0523620	+	0.998628i	$0.516675\pi$
$488$	0	0
$489$	28.5886	1.29282
$490$	0	0
$491$	−15.5510	−0.701807	−0.350903	−	0.936412i	$-0.614125\pi$
$491$	−15.5510	−0.701807	−0.350903	+	0.936412i	$0.614125\pi$
$492$	0	0
$493$	−8.53291	−0.384303
$494$	0	0
$495$	−1.49223	−0.0670707
$496$	0	0
$497$	−4.18868	−0.187888
$498$	0	0
$499$	−19.5403	−0.874744	−0.437372	−	0.899281i	$-0.644091\pi$
$499$	−19.5403	−0.874744	−0.437372	+	0.899281i	$0.644091\pi$
$500$	0	0
$501$	−6.64671	−0.296953
$502$	0	0
$503$	29.6101	1.32025	0.660124	−	0.751157i	$-0.270504\pi$
$503$	29.6101	1.32025	0.660124	+	0.751157i	$0.270504\pi$
$504$	0	0
$505$	5.10280	0.227071
$506$	0	0
$507$	−19.7580	−0.877486
$508$	0	0
$509$	26.2164	1.16202	0.581011	−	0.813896i	$-0.302657\pi$
$509$	26.2164	1.16202	0.581011	+	0.813896i	$0.302657\pi$
$510$	0	0
$511$	8.10992	0.358762
$512$	0	0
$513$	5.58827	0.246728
$514$	0	0
$515$	−5.62663	−0.247939
$516$	0	0
$517$	−22.9508	−1.00938
$518$	0	0
$519$	1.99032	0.0873654
$520$	0	0
$521$	24.2115	1.06073	0.530363	−	0.847771i	$-0.322056\pi$
$521$	24.2115	1.06073	0.530363	+	0.847771i	$0.322056\pi$
$522$	0	0
$523$	−0.389079	−0.0170132	−0.00850661	−	0.999964i	$-0.502708\pi$
$523$	−0.389079	−0.0170132	−0.00850661	+	0.999964i	$0.502708\pi$
$524$	0	0
$525$	11.7919	0.514639
$526$	0	0
$527$	13.4550	0.586110
$528$	0	0
$529$	−20.7837	−0.903638
$530$	0	0
$531$	−0.186169	−0.00807905
$532$	0	0
$533$	2.00175	0.0867053
$534$	0	0
$535$	−4.11087	−0.177728
$536$	0	0
$537$	−32.0078	−1.38124
$538$	0	0
$539$	−21.1908	−0.912754
$540$	0	0
$541$	−16.3622	−0.703464	−0.351732	−	0.936101i	$-0.614407\pi$
$541$	−16.3622	−0.703464	−0.351732	+	0.936101i	$0.614407\pi$
$542$	0	0
$543$	−24.3810	−1.04629
$544$	0	0
$545$	−3.95969	−0.169615
$546$	0	0
$547$	1.47912	0.0632426	0.0316213	−	0.999500i	$-0.489933\pi$
$547$	1.47912	0.0632426	0.0316213	+	0.999500i	$0.489933\pi$
$548$	0	0
$549$	−0.255310	−0.0108963
$550$	0	0
$551$	−3.50177	−0.149181
$552$	0	0
$553$	−1.61008	−0.0684674
$554$	0	0
$555$	−2.91513	−0.123740
$556$	0	0
$557$	−18.5548	−0.786194	−0.393097	−	0.919497i	$-0.628596\pi$
$557$	−18.5548	−0.786194	−0.393097	+	0.919497i	$0.628596\pi$
$558$	0	0
$559$	3.46518	0.146562
$560$	0	0
$561$	−18.0358	−0.761474
$562$	0	0
$563$	16.5094	0.695790	0.347895	−	0.937534i	$-0.386897\pi$
$563$	16.5094	0.695790	0.347895	+	0.937534i	$0.386897\pi$
$564$	0	0
$565$	−1.30598	−0.0549429
$566$	0	0
$567$	10.8095	0.453957
$568$	0	0
$569$	36.0343	1.51064	0.755318	−	0.655358i	$-0.227482\pi$
$569$	36.0343	1.51064	0.755318	+	0.655358i	$0.227482\pi$
$570$	0	0
$571$	41.9698	1.75638	0.878191	−	0.478310i	$-0.158750\pi$
$571$	41.9698	1.75638	0.878191	+	0.478310i	$0.158750\pi$
$572$	0	0
$573$	22.0337	0.920470
$574$	0	0
$575$	7.08217	0.295347
$576$	0	0
$577$	36.2918	1.51085	0.755424	−	0.655236i	$-0.227431\pi$
$577$	36.2918	1.51085	0.755424	+	0.655236i	$0.227431\pi$
$578$	0	0
$579$	1.33398	0.0554381
$580$	0	0
$581$	−9.31183	−0.386320
$582$	0	0
$583$	8.52811	0.353198
$584$	0	0
$585$	0.126509	0.00523051
$586$	0	0
$587$	32.2040	1.32920	0.664601	−	0.747198i	$-0.268601\pi$
$587$	32.2040	1.32920	0.664601	+	0.747198i	$0.268601\pi$
$588$	0	0
$589$	5.52173	0.227519
$590$	0	0
$591$	−17.9345	−0.737725
$592$	0	0
$593$	48.3027	1.98356	0.991778	−	0.127971i	$-0.0408464\pi$
$593$	48.3027	1.98356	0.991778	+	0.127971i	$0.0408464\pi$
$594$	0	0
$595$	1.93331	0.0792581
$596$	0	0
$597$	−20.5974	−0.842997
$598$	0	0
$599$	−32.7019	−1.33617	−0.668083	−	0.744087i	$-0.732885\pi$
$599$	−32.7019	−1.33617	−0.668083	+	0.744087i	$0.732885\pi$
$600$	0	0
$601$	−14.8332	−0.605058	−0.302529	−	0.953140i	$-0.597831\pi$
$601$	−14.8332	−0.605058	−0.302529	+	0.953140i	$0.597831\pi$
$602$	0	0
$603$	−3.81005	−0.155157
$604$	0	0
$605$	5.96960	0.242699
$606$	0	0
$607$	−7.49946	−0.304394	−0.152197	−	0.988350i	$-0.548635\pi$
$607$	−7.49946	−0.304394	−0.152197	+	0.988350i	$0.548635\pi$
$608$	0	0
$609$	−8.68002	−0.351732
$610$	0	0
$611$	1.94574	0.0787163
$612$	0	0
$613$	47.8193	1.93140	0.965702	−	0.259654i	$-0.0836086\pi$
$613$	47.8193	1.93140	0.965702	+	0.259654i	$0.0836086\pi$
$614$	0	0
$615$	−3.72576	−0.150237
$616$	0	0
$617$	−16.0085	−0.644479	−0.322239	−	0.946658i	$-0.604436\pi$
$617$	−16.0085	−0.644479	−0.322239	+	0.946658i	$0.604436\pi$
$618$	0	0
$619$	−9.70892	−0.390234	−0.195117	−	0.980780i	$-0.562509\pi$
$619$	−9.70892	−0.390234	−0.195117	+	0.980780i	$0.562509\pi$
$620$	0	0
$621$	8.31944	0.333848
$622$	0	0
$623$	−8.66882	−0.347309
$624$	0	0
$625$	21.4166	0.856664
$626$	0	0
$627$	−7.40162	−0.295592
$628$	0	0
$629$	9.36341	0.373344
$630$	0	0
$631$	−12.8017	−0.509629	−0.254814	−	0.966990i	$-0.582014\pi$
$631$	−12.8017	−0.509629	−0.254814	+	0.966990i	$0.582014\pi$
$632$	0	0
$633$	−18.7577	−0.745553
$634$	0	0
$635$	0.707592	0.0280799
$636$	0	0
$637$	1.79653	0.0711812
$638$	0	0
$639$	−1.63862	−0.0648229
$640$	0	0
$641$	−43.6124	−1.72259	−0.861293	−	0.508109i	$-0.830345\pi$
$641$	−43.6124	−1.72259	−0.861293	+	0.508109i	$0.830345\pi$
$642$	0	0
$643$	18.2879	0.721204	0.360602	−	0.932720i	$-0.382571\pi$
$643$	18.2879	0.721204	0.360602	+	0.932720i	$0.382571\pi$
$644$	0	0
$645$	−6.44958	−0.253952
$646$	0	0
$647$	28.3032	1.11271	0.556357	−	0.830943i	$-0.312199\pi$
$647$	28.3032	1.11271	0.556357	+	0.830943i	$0.312199\pi$
$648$	0	0
$649$	1.42102	0.0557798
$650$	0	0
$651$	13.6870	0.536435
$652$	0	0
$653$	−21.0007	−0.821820	−0.410910	−	0.911676i	$-0.634789\pi$
$653$	−21.0007	−0.821820	−0.410910	+	0.911676i	$0.634789\pi$
$654$	0	0
$655$	3.55943	0.139078
$656$	0	0
$657$	3.17262	0.123776
$658$	0	0
$659$	10.5456	0.410800	0.205400	−	0.978678i	$-0.434150\pi$
$659$	10.5456	0.410800	0.205400	+	0.978678i	$0.434150\pi$
$660$	0	0
$661$	−44.1025	−1.71539	−0.857694	−	0.514161i	$-0.828103\pi$
$661$	−44.1025	−1.71539	−0.857694	+	0.514161i	$0.828103\pi$
$662$	0	0
$663$	1.52906	0.0593836
$664$	0	0
$665$	0.793401	0.0307668
$666$	0	0
$667$	−5.21321	−0.201856
$668$	0	0
$669$	25.0891	0.970000
$670$	0	0
$671$	1.94876	0.0752312
$672$	0	0
$673$	−7.44605	−0.287024	−0.143512	−	0.989649i	$-0.545840\pi$
$673$	−7.44605	−0.287024	−0.143512	+	0.989649i	$0.545840\pi$
$674$	0	0
$675$	26.5844	1.02323
$676$	0	0
$677$	−41.7267	−1.60369	−0.801844	−	0.597533i	$-0.796148\pi$
$677$	−41.7267	−1.60369	−0.801844	+	0.597533i	$0.796148\pi$
$678$	0	0
$679$	−19.0117	−0.729600
$680$	0	0
$681$	24.8680	0.952944
$682$	0	0
$683$	−15.8325	−0.605816	−0.302908	−	0.953020i	$-0.597957\pi$
$683$	−15.8325	−0.605816	−0.302908	+	0.953020i	$0.597957\pi$
$684$	0	0
$685$	9.40354	0.359291
$686$	0	0
$687$	−37.1718	−1.41819
$688$	0	0
$689$	−0.723002	−0.0275442
$690$	0	0
$691$	−11.0877	−0.421797	−0.210899	−	0.977508i	$-0.567639\pi$
$691$	−11.0877	−0.421797	−0.210899	+	0.977508i	$0.567639\pi$
$692$	0	0
$693$	4.87568	0.185212
$694$	0	0
$695$	−8.99549	−0.341219
$696$	0	0
$697$	11.9672	0.453289
$698$	0	0
$699$	−13.7794	−0.521185
$700$	0	0
$701$	−12.2378	−0.462214	−0.231107	−	0.972928i	$-0.574235\pi$
$701$	−12.2378	−0.462214	−0.231107	+	0.972928i	$0.574235\pi$
$702$	0	0
$703$	3.84260	0.144926
$704$	0	0
$705$	−3.62152	−0.136394
$706$	0	0
$707$	−16.6728	−0.627045
$708$	0	0
$709$	−39.3681	−1.47850	−0.739250	−	0.673431i	$-0.764820\pi$
$709$	−39.3681	−1.47850	−0.739250	+	0.673431i	$0.764820\pi$
$710$	0	0
$711$	−0.629866	−0.0236218
$712$	0	0
$713$	8.22038	0.307856
$714$	0	0
$715$	−0.965638	−0.0361128
$716$	0	0
$717$	−17.5501	−0.655422
$718$	0	0
$719$	14.5441	0.542403	0.271202	−	0.962523i	$-0.412579\pi$
$719$	14.5441	0.542403	0.271202	+	0.962523i	$0.412579\pi$
$720$	0	0
$721$	18.3844	0.684669
$722$	0	0
$723$	6.36719	0.236798
$724$	0	0
$725$	−16.6585	−0.618683
$726$	0	0
$727$	26.4195	0.979845	0.489923	−	0.871766i	$-0.337025\pi$
$727$	26.4195	0.979845	0.489923	+	0.871766i	$0.337025\pi$
$728$	0	0
$729$	30.0385	1.11254
$730$	0	0
$731$	20.7161	0.766213
$732$	0	0
$733$	38.1457	1.40894	0.704472	−	0.709732i	$-0.251184\pi$
$733$	38.1457	1.40894	0.704472	+	0.709732i	$0.251184\pi$
$734$	0	0
$735$	−3.34380	−0.123338
$736$	0	0
$737$	29.0819	1.07125
$738$	0	0
$739$	0.840015	0.0309005	0.0154502	−	0.999881i	$-0.495082\pi$
$739$	0.840015	0.0309005	0.0154502	+	0.999881i	$0.495082\pi$
$740$	0	0
$741$	0.627500	0.0230518
$742$	0	0
$743$	24.6717	0.905117	0.452558	−	0.891735i	$-0.350511\pi$
$743$	24.6717	0.905117	0.452558	+	0.891735i	$0.350511\pi$
$744$	0	0
$745$	5.74906	0.210629
$746$	0	0
$747$	−3.64282	−0.133284
$748$	0	0
$749$	13.4318	0.490787
$750$	0	0
$751$	−16.8209	−0.613803	−0.306902	−	0.951741i	$-0.599292\pi$
$751$	−16.8209	−0.613803	−0.306902	+	0.951741i	$0.599292\pi$
$752$	0	0
$753$	17.3759	0.633213
$754$	0	0
$755$	−10.2492	−0.373008
$756$	0	0
$757$	−19.0018	−0.690633	−0.345317	−	0.938486i	$-0.612228\pi$
$757$	−19.0018	−0.690633	−0.345317	+	0.938486i	$0.612228\pi$
$758$	0	0
$759$	−11.0190	−0.399966
$760$	0	0
$761$	−8.60254	−0.311842	−0.155921	−	0.987770i	$-0.549834\pi$
$761$	−8.60254	−0.311842	−0.155921	+	0.987770i	$0.549834\pi$
$762$	0	0
$763$	12.9378	0.468381
$764$	0	0
$765$	0.756318	0.0273447
$766$	0	0
$767$	−0.120472	−0.00434999
$768$	0	0
$769$	−17.2864	−0.623363	−0.311681	−	0.950187i	$-0.600892\pi$
$769$	−17.2864	−0.623363	−0.311681	+	0.950187i	$0.600892\pi$
$770$	0	0
$771$	−45.0943	−1.62403
$772$	0	0
$773$	−25.7981	−0.927892	−0.463946	−	0.885863i	$-0.653567\pi$
$773$	−25.7981	−0.927892	−0.463946	+	0.885863i	$0.653567\pi$
$774$	0	0
$775$	26.2678	0.943568
$776$	0	0
$777$	9.52484	0.341702
$778$	0	0
$779$	4.91114	0.175960
$780$	0	0
$781$	12.5075	0.447554
$782$	0	0
$783$	−19.5688	−0.699333
$784$	0	0
$785$	3.49774	0.124840
$786$	0	0
$787$	10.2223	0.364386	0.182193	−	0.983263i	$-0.441680\pi$
$787$	10.2223	0.364386	0.182193	+	0.983263i	$0.441680\pi$
$788$	0	0
$789$	−11.2607	−0.400893
$790$	0	0
$791$	4.26713	0.151722
$792$	0	0
$793$	−0.165214	−0.00586691
$794$	0	0
$795$	1.34569	0.0477267
$796$	0	0
$797$	−25.4212	−0.900467	−0.450233	−	0.892911i	$-0.648659\pi$
$797$	−25.4212	−0.900467	−0.450233	+	0.892911i	$0.648659\pi$
$798$	0	0
$799$	11.6323	0.411523
$800$	0	0
$801$	−3.39127	−0.119824
$802$	0	0
$803$	−24.2165	−0.854580
$804$	0	0
$805$	1.18116	0.0416305
$806$	0	0
$807$	−43.1080	−1.51747
$808$	0	0
$809$	−6.40827	−0.225303	−0.112651	−	0.993635i	$-0.535934\pi$
$809$	−6.40827	−0.225303	−0.112651	+	0.993635i	$0.535934\pi$
$810$	0	0
$811$	−34.3894	−1.20757	−0.603787	−	0.797146i	$-0.706342\pi$
$811$	−34.3894	−1.20757	−0.603787	+	0.797146i	$0.706342\pi$
$812$	0	0
$813$	−17.1981	−0.603164
$814$	0	0
$815$	9.15068	0.320534
$816$	0	0
$817$	8.50157	0.297432
$818$	0	0
$819$	−0.413354	−0.0144438
$820$	0	0
$821$	13.7070	0.478379	0.239190	−	0.970973i	$-0.423118\pi$
$821$	13.7070	0.478379	0.239190	+	0.970973i	$0.423118\pi$
$822$	0	0
$823$	39.8960	1.39069	0.695344	−	0.718677i	$-0.255252\pi$
$823$	39.8960	1.39069	0.695344	+	0.718677i	$0.255252\pi$
$824$	0	0
$825$	−35.2108	−1.22588
$826$	0	0
$827$	8.83512	0.307227	0.153614	−	0.988131i	$-0.450909\pi$
$827$	8.83512	0.307227	0.153614	+	0.988131i	$0.450909\pi$
$828$	0	0
$829$	31.0279	1.07764	0.538821	−	0.842420i	$-0.318870\pi$
$829$	31.0279	1.07764	0.538821	+	0.842420i	$0.318870\pi$
$830$	0	0
$831$	−36.8373	−1.27787
$832$	0	0
$833$	10.7403	0.372130
$834$	0	0
$835$	−2.12749	−0.0736247
$836$	0	0
$837$	30.8569	1.06657
$838$	0	0
$839$	31.3699	1.08301	0.541504	−	0.840698i	$-0.317855\pi$
$839$	31.3699	1.08301	0.541504	+	0.840698i	$0.317855\pi$
$840$	0	0
$841$	−16.7376	−0.577158
$842$	0	0
$843$	19.3162	0.665284
$844$	0	0
$845$	−6.32418	−0.217558
$846$	0	0
$847$	−19.5050	−0.670199
$848$	0	0
$849$	−23.8546	−0.818686
$850$	0	0
$851$	5.72060	0.196100
$852$	0	0
$853$	−20.4320	−0.699579	−0.349789	−	0.936828i	$-0.613747\pi$
$853$	−20.4320	−0.699579	−0.349789	+	0.936828i	$0.613747\pi$
$854$	0	0
$855$	0.310381	0.0106148
$856$	0	0
$857$	9.58014	0.327251	0.163626	−	0.986522i	$-0.447681\pi$
$857$	9.58014	0.327251	0.163626	+	0.986522i	$0.447681\pi$
$858$	0	0
$859$	−14.3723	−0.490376	−0.245188	−	0.969475i	$-0.578850\pi$
$859$	−14.3723	−0.490376	−0.245188	+	0.969475i	$0.578850\pi$
$860$	0	0
$861$	12.1735	0.414871
$862$	0	0
$863$	30.6002	1.04164	0.520821	−	0.853666i	$-0.325626\pi$
$863$	30.6002	1.04164	0.520821	+	0.853666i	$0.325626\pi$
$864$	0	0
$865$	0.637064	0.0216608
$866$	0	0
$867$	−17.0307	−0.578392
$868$	0	0
$869$	4.80774	0.163091
$870$	0	0
$871$	−2.46553	−0.0835413
$872$	0	0
$873$	−7.43741	−0.251718
$874$	0	0
$875$	7.74135	0.261706
$876$	0	0
$877$	−39.6767	−1.33979	−0.669894	−	0.742457i	$-0.733660\pi$
$877$	−39.6767	−1.33979	−0.669894	+	0.742457i	$0.733660\pi$
$878$	0	0
$879$	18.7867	0.633659
$880$	0	0
$881$	−51.4059	−1.73191	−0.865954	−	0.500124i	$-0.833288\pi$
$881$	−51.4059	−1.73191	−0.865954	+	0.500124i	$0.833288\pi$
$882$	0	0
$883$	18.4341	0.620358	0.310179	−	0.950678i	$-0.399611\pi$
$883$	18.4341	0.620358	0.310179	+	0.950678i	$0.399611\pi$
$884$	0	0
$885$	0.224229	0.00753737
$886$	0	0
$887$	−22.4931	−0.755245	−0.377622	−	0.925960i	$-0.623258\pi$
$887$	−22.4931	−0.755245	−0.377622	+	0.925960i	$0.623258\pi$
$888$	0	0
$889$	−2.31198	−0.0775412
$890$	0	0
$891$	−32.2775	−1.08134
$892$	0	0
$893$	4.77373	0.159747
$894$	0	0
$895$	−10.2451	−0.342455
$896$	0	0
$897$	0.934181	0.0311914
$898$	0	0
$899$	−19.3358	−0.644886
$900$	0	0
$901$	−4.32237	−0.143999
$902$	0	0
$903$	21.0733	0.701274
$904$	0	0
$905$	−7.80390	−0.259410
$906$	0	0
$907$	−15.9207	−0.528640	−0.264320	−	0.964435i	$-0.585147\pi$
$907$	−15.9207	−0.528640	−0.264320	+	0.964435i	$0.585147\pi$
$908$	0	0
$909$	−6.52244	−0.216336
$910$	0	0
$911$	37.0938	1.22897	0.614486	−	0.788927i	$-0.289363\pi$
$911$	37.0938	1.22897	0.614486	+	0.788927i	$0.289363\pi$
$912$	0	0
$913$	27.8054	0.920225
$914$	0	0
$915$	0.307505	0.0101658
$916$	0	0
$917$	−11.6300	−0.384057
$918$	0	0
$919$	−35.6784	−1.17692	−0.588461	−	0.808526i	$-0.700266\pi$
$919$	−35.6784	−1.17692	−0.588461	+	0.808526i	$0.700266\pi$
$920$	0	0
$921$	−34.3308	−1.13124
$922$	0	0
$923$	−1.06037	−0.0349026
$924$	0	0
$925$	18.2799	0.601040
$926$	0	0
$927$	7.19201	0.236217
$928$	0	0
$929$	−30.0290	−0.985219	−0.492609	−	0.870251i	$-0.663957\pi$
$929$	−30.0290	−0.985219	−0.492609	+	0.870251i	$0.663957\pi$
$930$	0	0
$931$	4.40765	0.144455
$932$	0	0
$933$	32.4851	1.06351
$934$	0	0
$935$	−5.77293	−0.188795
$936$	0	0
$937$	45.6323	1.49074	0.745372	−	0.666649i	$-0.232272\pi$
$937$	45.6323	1.49074	0.745372	+	0.666649i	$0.232272\pi$
$938$	0	0
$939$	−28.4803	−0.929419
$940$	0	0
$941$	16.3069	0.531590	0.265795	−	0.964030i	$-0.414366\pi$
$941$	16.3069	0.531590	0.265795	+	0.964030i	$0.414366\pi$
$942$	0	0
$943$	7.31137	0.238091
$944$	0	0
$945$	4.43374	0.144229
$946$	0	0
$947$	8.42906	0.273907	0.136954	−	0.990577i	$-0.456269\pi$
$947$	8.42906	0.273907	0.136954	+	0.990577i	$0.456269\pi$
$948$	0	0
$949$	2.05304	0.0666445
$950$	0	0
$951$	−42.9787	−1.39368
$952$	0	0
$953$	−22.0783	−0.715186	−0.357593	−	0.933877i	$-0.616403\pi$
$953$	−22.0783	−0.715186	−0.357593	+	0.933877i	$0.616403\pi$
$954$	0	0
$955$	7.05256	0.228215
$956$	0	0
$957$	25.9188	0.837836
$958$	0	0
$959$	−30.7250	−0.992161
$960$	0	0
$961$	−0.510524	−0.0164685
$962$	0	0
$963$	5.25455	0.169326
$964$	0	0
$965$	0.426980	0.0137450
$966$	0	0
$967$	21.6823	0.697254	0.348627	−	0.937261i	$-0.386648\pi$
$967$	21.6823	0.697254	0.348627	+	0.937261i	$0.386648\pi$
$968$	0	0
$969$	3.75142	0.120513
$970$	0	0
$971$	14.1954	0.455553	0.227776	−	0.973713i	$-0.426854\pi$
$971$	14.1954	0.455553	0.227776	+	0.973713i	$0.426854\pi$
$972$	0	0
$973$	29.3917	0.942255
$974$	0	0
$975$	2.98513	0.0956006
$976$	0	0
$977$	20.1124	0.643454	0.321727	−	0.946832i	$-0.395737\pi$
$977$	20.1124	0.643454	0.321727	+	0.946832i	$0.395737\pi$
$978$	0	0
$979$	25.8853	0.827299
$980$	0	0
$981$	5.06132	0.161595
$982$	0	0
$983$	−16.7602	−0.534566	−0.267283	−	0.963618i	$-0.586126\pi$
$983$	−16.7602	−0.534566	−0.267283	+	0.963618i	$0.586126\pi$
$984$	0	0
$985$	−5.74048	−0.182907
$986$	0	0
$987$	11.8329	0.376645
$988$	0	0
$989$	12.6566	0.402455
$990$	0	0
$991$	−54.5006	−1.73127	−0.865634	−	0.500677i	$-0.833084\pi$
$991$	−54.5006	−1.73127	−0.865634	+	0.500677i	$0.833084\pi$
$992$	0	0
$993$	12.1745	0.386347
$994$	0	0
$995$	−6.59285	−0.209007
$996$	0	0
$997$	−6.86527	−0.217425	−0.108713	−	0.994073i	$-0.534673\pi$
$997$	−6.86527	−0.217425	−0.108713	+	0.994073i	$0.534673\pi$
$998$	0	0
$999$	21.4734	0.679390

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6004.2.a.f.1.18	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6004.2.a.f.1.18	✓	25	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:	\( N \)	\(=\)	\( 6004 = 2^{2} \cdot 19 \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6004.a (trivial)

\(f(q)\)	\(=\)	\(q+1.53952 q^{3} +0.492772 q^{5} -1.61008 q^{7} -0.629866 q^{9} +O(q^{10})\)	\(q+1.53952 q^{3} +0.492772 q^{5} -1.61008 q^{7} -0.629866 q^{9} +4.80774 q^{11} -0.407594 q^{13} +0.758635 q^{15} -2.43674 q^{17} -1.00000 q^{19} -2.47875 q^{21} -1.48873 q^{23} -4.75718 q^{25} -5.58827 q^{27} +3.50177 q^{29} -5.52173 q^{31} +7.40162 q^{33} -0.793401 q^{35} -3.84260 q^{37} -0.627500 q^{39} -4.91114 q^{41} -8.50157 q^{43} -0.310381 q^{45} -4.77373 q^{47} -4.40765 q^{49} -3.75142 q^{51} +1.77383 q^{53} +2.36912 q^{55} -1.53952 q^{57} +0.295569 q^{59} +0.405339 q^{61} +1.01413 q^{63} -0.200851 q^{65} +6.04899 q^{67} -2.29194 q^{69} +2.60154 q^{71} -5.03698 q^{73} -7.32378 q^{75} -7.74082 q^{77} +1.00000 q^{79} -6.71367 q^{81} +5.78347 q^{83} -1.20076 q^{85} +5.39106 q^{87} +5.38410 q^{89} +0.656257 q^{91} -8.50083 q^{93} -0.492772 q^{95} +11.8079 q^{97} -3.02823 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q + 4 q^{3} - 8 q^{5} + 2 q^{7} + 13 q^{9}+O(q^{10}) \) 25 * q + 4 * q^3 - 8 * q^5 + 2 * q^7 + 13 * q^9	\( 25 q + 4 q^{3} - 8 q^{5} + 2 q^{7} + 13 q^{9} - 3 q^{11} + q^{13} - 5 q^{15} - 13 q^{17} - 25 q^{19} - 24 q^{21} - 31 q^{23} + 21 q^{25} + 7 q^{27} - 19 q^{29} - 7 q^{31} - 30 q^{33} - q^{35} - 29 q^{37} - 26 q^{39} - 40 q^{41} - 40 q^{45} - 8 q^{47} - 9 q^{49} + 12 q^{51} - 38 q^{53} - 29 q^{55} - 4 q^{57} + 18 q^{59} - 26 q^{61} - 40 q^{63} - 70 q^{65} - 13 q^{67} + q^{69} - 47 q^{71} - 8 q^{73} + 7 q^{75} - 19 q^{77} + 25 q^{79} - 19 q^{81} - 8 q^{83} - 33 q^{85} - 50 q^{87} - 54 q^{89} - 12 q^{91} - 24 q^{93} + 8 q^{95} - 4 q^{97} - 39 q^{99}+O(q^{100}) \) 25 * q + 4 * q^3 - 8 * q^5 + 2 * q^7 + 13 * q^9 - 3 * q^11 + q^13 - 5 * q^15 - 13 * q^17 - 25 * q^19 - 24 * q^21 - 31 * q^23 + 21 * q^25 + 7 * q^27 - 19 * q^29 - 7 * q^31 - 30 * q^33 - q^35 - 29 * q^37 - 26 * q^39 - 40 * q^41 - 40 * q^45 - 8 * q^47 - 9 * q^49 + 12 * q^51 - 38 * q^53 - 29 * q^55 - 4 * q^57 + 18 * q^59 - 26 * q^61 - 40 * q^63 - 70 * q^65 - 13 * q^67 + q^69 - 47 * q^71 - 8 * q^73 + 7 * q^75 - 19 * q^77 + 25 * q^79 - 19 * q^81 - 8 * q^83 - 33 * q^85 - 50 * q^87 - 54 * q^89 - 12 * q^91 - 24 * q^93 + 8 * q^95 - 4 * q^97 - 39 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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