LMFDB - Embedded newform 6004.2.a.g.1.2

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:	$27$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
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)

$f(q)$	$=$	$q-2.81775 q^{3} -0.243389 q^{5} +0.929492 q^{7} +4.93969 q^{9} +O(q^{10})$	$q-2.81775 q^{3} -0.243389 q^{5} +0.929492 q^{7} +4.93969 q^{9} +4.38796 q^{11} -1.17004 q^{13} +0.685809 q^{15} +6.41492 q^{17} +1.00000 q^{19} -2.61907 q^{21} -6.17766 q^{23} -4.94076 q^{25} -5.46555 q^{27} -5.16503 q^{29} -8.29647 q^{31} -12.3642 q^{33} -0.226229 q^{35} -11.1767 q^{37} +3.29689 q^{39} +3.11040 q^{41} +5.53876 q^{43} -1.20227 q^{45} +8.81464 q^{47} -6.13604 q^{49} -18.0756 q^{51} +10.2900 q^{53} -1.06798 q^{55} -2.81775 q^{57} -11.0099 q^{59} +0.358578 q^{61} +4.59140 q^{63} +0.284776 q^{65} +11.8751 q^{67} +17.4071 q^{69} +7.27001 q^{71} -14.0340 q^{73} +13.9218 q^{75} +4.07858 q^{77} -1.00000 q^{79} +0.581451 q^{81} +6.31006 q^{83} -1.56132 q^{85} +14.5537 q^{87} +5.87396 q^{89} -1.08755 q^{91} +23.3773 q^{93} -0.243389 q^{95} -10.4974 q^{97} +21.6752 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 27 q - 4 q^{3} - 10 q^{5} - 8 q^{7} + 19 q^{9}+O(q^{10}) $ 27 * q - 4 * q^3 - 10 * q^5 - 8 * q^7 + 19 * q^9	$ 27 q - 4 q^{3} - 10 q^{5} - 8 q^{7} + 19 q^{9} + 3 q^{11} - 5 q^{13} - 11 q^{15} - 17 q^{17} + 27 q^{19} - 28 q^{21} - 11 q^{23} + 13 q^{25} - 7 q^{27} - 39 q^{29} - 27 q^{31} - 18 q^{33} - 5 q^{35} - q^{37} - 22 q^{39} - 36 q^{41} - 2 q^{43} - 18 q^{45} - 12 q^{47} + 15 q^{49} + 4 q^{51} - 28 q^{53} + 5 q^{55} - 4 q^{57} - 30 q^{59} - 6 q^{61} - 4 q^{63} - 32 q^{65} + 13 q^{67} - 27 q^{69} - 59 q^{71} - 30 q^{73} - 21 q^{75} - 39 q^{77} - 27 q^{79} - 5 q^{81} + 4 q^{83} - 3 q^{85} + 22 q^{87} - 56 q^{89} - 8 q^{91} - 38 q^{93} - 10 q^{95} - 30 q^{97} + 31 q^{99}+O(q^{100}) $ 27 * q - 4 * q^3 - 10 * q^5 - 8 * q^7 + 19 * q^9 + 3 * q^11 - 5 * q^13 - 11 * q^15 - 17 * q^17 + 27 * q^19 - 28 * q^21 - 11 * q^23 + 13 * q^25 - 7 * q^27 - 39 * q^29 - 27 * q^31 - 18 * q^33 - 5 * q^35 - q^37 - 22 * q^39 - 36 * q^41 - 2 * q^43 - 18 * q^45 - 12 * q^47 + 15 * q^49 + 4 * q^51 - 28 * q^53 + 5 * q^55 - 4 * q^57 - 30 * q^59 - 6 * q^61 - 4 * q^63 - 32 * q^65 + 13 * q^67 - 27 * q^69 - 59 * q^71 - 30 * q^73 - 21 * q^75 - 39 * q^77 - 27 * q^79 - 5 * q^81 + 4 * q^83 - 3 * q^85 + 22 * q^87 - 56 * q^89 - 8 * q^91 - 38 * q^93 - 10 * q^95 - 30 * q^97 + 31 * 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
$3$	−2.81775	−1.62683	−0.813413	−	0.581687i	$-0.802393\pi$
$3$	−2.81775	−1.62683	−0.813413	+	0.581687i	$0.802393\pi$
$4$	0	0
$5$	−0.243389	−0.108847	−0.0544235	−	0.998518i	$-0.517332\pi$
$5$	−0.243389	−0.108847	−0.0544235	+	0.998518i	$0.517332\pi$
$6$	0	0
$7$	0.929492	0.351315	0.175658	−	0.984451i	$-0.443795\pi$
$7$	0.929492	0.351315	0.175658	+	0.984451i	$0.443795\pi$
$8$	0	0
$9$	4.93969	1.64656
$10$	0	0
$11$	4.38796	1.32302	0.661510	−	0.749936i	$-0.269916\pi$
$11$	4.38796	1.32302	0.661510	+	0.749936i	$0.269916\pi$
$12$	0	0
$13$	−1.17004	−0.324512	−0.162256	−	0.986749i	$-0.551877\pi$
$13$	−1.17004	−0.324512	−0.162256	+	0.986749i	$0.551877\pi$
$14$	0	0
$15$	0.685809	0.177075
$16$	0	0
$17$	6.41492	1.55585	0.777923	−	0.628360i	$-0.216273\pi$
$17$	6.41492	1.55585	0.777923	+	0.628360i	$0.216273\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−2.61907	−0.571528
$22$	0	0
$23$	−6.17766	−1.28813	−0.644066	−	0.764970i	$-0.722754\pi$
$23$	−6.17766	−1.28813	−0.644066	+	0.764970i	$0.722754\pi$
$24$	0	0
$25$	−4.94076	−0.988152
$26$	0	0
$27$	−5.46555	−1.05184
$28$	0	0
$29$	−5.16503	−0.959122	−0.479561	−	0.877508i	$-0.659204\pi$
$29$	−5.16503	−0.959122	−0.479561	+	0.877508i	$0.659204\pi$
$30$	0	0
$31$	−8.29647	−1.49009	−0.745045	−	0.667015i	$-0.767572\pi$
$31$	−8.29647	−1.49009	−0.745045	+	0.667015i	$0.767572\pi$
$32$	0	0
$33$	−12.3642	−2.15232
$34$	0	0
$35$	−0.226229	−0.0382396
$36$	0	0
$37$	−11.1767	−1.83744	−0.918718	−	0.394914i	$-0.870775\pi$
$37$	−11.1767	−1.83744	−0.918718	+	0.394914i	$0.870775\pi$
$38$	0	0
$39$	3.29689	0.527925
$40$	0	0
$41$	3.11040	0.485763	0.242881	−	0.970056i	$-0.421907\pi$
$41$	3.11040	0.485763	0.242881	+	0.970056i	$0.421907\pi$
$42$	0	0
$43$	5.53876	0.844653	0.422326	−	0.906444i	$-0.361214\pi$
$43$	5.53876	0.844653	0.422326	+	0.906444i	$0.361214\pi$
$44$	0	0
$45$	−1.20227	−0.179223
$46$	0	0
$47$	8.81464	1.28575	0.642874	−	0.765972i	$-0.277742\pi$
$47$	8.81464	1.28575	0.642874	+	0.765972i	$0.277742\pi$
$48$	0	0
$49$	−6.13604	−0.876578
$50$	0	0
$51$	−18.0756	−2.53109
$52$	0	0
$53$	10.2900	1.41344	0.706720	−	0.707494i	$-0.250174\pi$
$53$	10.2900	1.41344	0.706720	+	0.707494i	$0.250174\pi$
$54$	0	0
$55$	−1.06798	−0.144007
$56$	0	0
$57$	−2.81775	−0.373219
$58$	0	0
$59$	−11.0099	−1.43336	−0.716682	−	0.697400i	$-0.754340\pi$
$59$	−11.0099	−1.43336	−0.716682	+	0.697400i	$0.754340\pi$
$60$	0	0
$61$	0.358578	0.0459113	0.0229556	−	0.999736i	$-0.492692\pi$
$61$	0.358578	0.0459113	0.0229556	+	0.999736i	$0.492692\pi$
$62$	0	0
$63$	4.59140	0.578462
$64$	0	0
$65$	0.284776	0.0353222
$66$	0	0
$67$	11.8751	1.45077	0.725385	−	0.688344i	$-0.241662\pi$
$67$	11.8751	1.45077	0.725385	+	0.688344i	$0.241662\pi$
$68$	0	0
$69$	17.4071	2.09557
$70$	0	0
$71$	7.27001	0.862792	0.431396	−	0.902163i	$-0.358021\pi$
$71$	7.27001	0.862792	0.431396	+	0.902163i	$0.358021\pi$
$72$	0	0
$73$	−14.0340	−1.64255	−0.821276	−	0.570530i	$-0.806738\pi$
$73$	−14.0340	−1.64255	−0.821276	+	0.570530i	$0.806738\pi$
$74$	0	0
$75$	13.9218	1.60755
$76$	0	0
$77$	4.07858	0.464797
$78$	0	0
$79$	−1.00000	−0.112509
$79$	−1.00000	−0.112509
$80$	0	0
$81$	0.581451	0.0646056
$82$	0	0
$83$	6.31006	0.692619	0.346309	−	0.938120i	$-0.387435\pi$
$83$	6.31006	0.692619	0.346309	+	0.938120i	$0.387435\pi$
$84$	0	0
$85$	−1.56132	−0.169349
$86$	0	0
$87$	14.5537	1.56033
$88$	0	0
$89$	5.87396	0.622639	0.311319	−	0.950305i	$-0.399229\pi$
$89$	5.87396	0.622639	0.311319	+	0.950305i	$0.399229\pi$
$90$	0	0
$91$	−1.08755	−0.114006
$92$	0	0
$93$	23.3773	2.42412
$94$	0	0
$95$	−0.243389	−0.0249712
$96$	0	0
$97$	−10.4974	−1.06585	−0.532926	−	0.846162i	$-0.678908\pi$
$97$	−10.4974	−1.06585	−0.532926	+	0.846162i	$0.678908\pi$
$98$	0	0
$99$	21.6752	2.17844
$100$	0	0
$101$	1.14271	0.113704	0.0568519	−	0.998383i	$-0.481894\pi$
$101$	1.14271	0.113704	0.0568519	+	0.998383i	$0.481894\pi$
$102$	0	0
$103$	−14.8972	−1.46787	−0.733934	−	0.679221i	$-0.762318\pi$
$103$	−14.8972	−1.46787	−0.733934	+	0.679221i	$0.762318\pi$
$104$	0	0
$105$	0.637454	0.0622092
$106$	0	0
$107$	−12.8657	−1.24378	−0.621888	−	0.783106i	$-0.713634\pi$
$107$	−12.8657	−1.24378	−0.621888	+	0.783106i	$0.713634\pi$
$108$	0	0
$109$	1.90646	0.182606	0.0913030	−	0.995823i	$-0.470897\pi$
$109$	1.90646	0.182606	0.0913030	+	0.995823i	$0.470897\pi$
$110$	0	0
$111$	31.4931	2.98919
$112$	0	0
$113$	10.3402	0.972723	0.486362	−	0.873758i	$-0.338324\pi$
$113$	10.3402	0.972723	0.486362	+	0.873758i	$0.338324\pi$
$114$	0	0
$115$	1.50358	0.140209
$116$	0	0
$117$	−5.77966	−0.534329
$118$	0	0
$119$	5.96262	0.546592
$120$	0	0
$121$	8.25422	0.750384
$122$	0	0
$123$	−8.76431	−0.790251
$124$	0	0
$125$	2.41948	0.216404
$126$	0	0
$127$	6.81705	0.604915	0.302458	−	0.953163i	$-0.402193\pi$
$127$	6.81705	0.604915	0.302458	+	0.953163i	$0.402193\pi$
$128$	0	0
$129$	−15.6068	−1.37410
$130$	0	0
$131$	8.88813	0.776559	0.388280	−	0.921542i	$-0.373069\pi$
$131$	8.88813	0.776559	0.388280	+	0.921542i	$0.373069\pi$
$132$	0	0
$133$	0.929492	0.0805972
$134$	0	0
$135$	1.33026	0.114490
$136$	0	0
$137$	−3.19350	−0.272839	−0.136420	−	0.990651i	$-0.543560\pi$
$137$	−3.19350	−0.272839	−0.136420	+	0.990651i	$0.543560\pi$
$138$	0	0
$139$	10.6744	0.905392	0.452696	−	0.891665i	$-0.350462\pi$
$139$	10.6744	0.905392	0.452696	+	0.891665i	$0.350462\pi$
$140$	0	0
$141$	−24.8374	−2.09169
$142$	0	0
$143$	−5.13411	−0.429336
$144$	0	0
$145$	1.25711	0.104398
$146$	0	0
$147$	17.2898	1.42604
$148$	0	0
$149$	−6.99354	−0.572933	−0.286467	−	0.958090i	$-0.592481\pi$
$149$	−6.99354	−0.572933	−0.286467	+	0.958090i	$0.592481\pi$
$150$	0	0
$151$	20.9826	1.70754	0.853771	−	0.520648i	$-0.174310\pi$
$151$	20.9826	1.70754	0.853771	+	0.520648i	$0.174310\pi$
$152$	0	0
$153$	31.6877	2.56180
$154$	0	0
$155$	2.01927	0.162192
$156$	0	0
$157$	7.91794	0.631921	0.315960	−	0.948772i	$-0.397673\pi$
$157$	7.91794	0.631921	0.315960	+	0.948772i	$0.397673\pi$
$158$	0	0
$159$	−28.9946	−2.29942
$160$	0	0
$161$	−5.74209	−0.452540
$162$	0	0
$163$	6.71896	0.526270	0.263135	−	0.964759i	$-0.415244\pi$
$163$	6.71896	0.526270	0.263135	+	0.964759i	$0.415244\pi$
$164$	0	0
$165$	3.00931	0.234274
$166$	0	0
$167$	−10.9171	−0.844790	−0.422395	−	0.906412i	$-0.638810\pi$
$167$	−10.9171	−0.844790	−0.422395	+	0.906412i	$0.638810\pi$
$168$	0	0
$169$	−11.6310	−0.894692
$170$	0	0
$171$	4.93969	0.377747
$172$	0	0
$173$	−24.2211	−1.84149	−0.920747	−	0.390159i	$-0.872420\pi$
$173$	−24.2211	−1.84149	−0.920747	+	0.390159i	$0.872420\pi$
$174$	0	0
$175$	−4.59240	−0.347153
$176$	0	0
$177$	31.0230	2.33183
$178$	0	0
$179$	−21.3740	−1.59757	−0.798783	−	0.601619i	$-0.794522\pi$
$179$	−21.3740	−1.59757	−0.798783	+	0.601619i	$0.794522\pi$
$180$	0	0
$181$	−16.2771	−1.20987	−0.604933	−	0.796277i	$-0.706800\pi$
$181$	−16.2771	−1.20987	−0.604933	+	0.796277i	$0.706800\pi$
$182$	0	0
$183$	−1.01038	−0.0746896
$184$	0	0
$185$	2.72029	0.200000
$186$	0	0
$187$	28.1484	2.05842
$188$	0	0
$189$	−5.08018	−0.369529
$190$	0	0
$191$	−19.8455	−1.43597	−0.717984	−	0.696060i	$-0.754935\pi$
$191$	−19.8455	−1.43597	−0.717984	+	0.696060i	$0.754935\pi$
$192$	0	0
$193$	−18.2372	−1.31274	−0.656371	−	0.754438i	$-0.727910\pi$
$193$	−18.2372	−1.31274	−0.656371	+	0.754438i	$0.727910\pi$
$194$	0	0
$195$	−0.802427	−0.0574630
$196$	0	0
$197$	−10.7045	−0.762666	−0.381333	−	0.924438i	$-0.624535\pi$
$197$	−10.7045	−0.762666	−0.381333	+	0.924438i	$0.624535\pi$
$198$	0	0
$199$	−7.75540	−0.549766	−0.274883	−	0.961478i	$-0.588639\pi$
$199$	−7.75540	−0.549766	−0.274883	+	0.961478i	$0.588639\pi$
$200$	0	0
$201$	−33.4609	−2.36015
$202$	0	0
$203$	−4.80086	−0.336954
$204$	0	0
$205$	−0.757038	−0.0528738
$206$	0	0
$207$	−30.5157	−2.12099
$208$	0	0
$209$	4.38796	0.303522
$210$	0	0
$211$	−26.6690	−1.83597	−0.917985	−	0.396615i	$-0.870185\pi$
$211$	−26.6690	−1.83597	−0.917985	+	0.396615i	$0.870185\pi$
$212$	0	0
$213$	−20.4850	−1.40361
$214$	0	0
$215$	−1.34808	−0.0919380
$216$	0	0
$217$	−7.71150	−0.523491
$218$	0	0
$219$	39.5442	2.67215
$220$	0	0
$221$	−7.50574	−0.504891
$222$	0	0
$223$	12.3667	0.828135	0.414067	−	0.910246i	$-0.364108\pi$
$223$	12.3667	0.828135	0.414067	+	0.910246i	$0.364108\pi$
$224$	0	0
$225$	−24.4058	−1.62705
$226$	0	0
$227$	16.9197	1.12300	0.561499	−	0.827477i	$-0.310225\pi$
$227$	16.9197	1.12300	0.561499	+	0.827477i	$0.310225\pi$
$228$	0	0
$229$	16.3415	1.07988	0.539939	−	0.841704i	$-0.318447\pi$
$229$	16.3415	1.07988	0.539939	+	0.841704i	$0.318447\pi$
$230$	0	0
$231$	−11.4924	−0.756144
$232$	0	0
$233$	−8.98952	−0.588923	−0.294461	−	0.955663i	$-0.595140\pi$
$233$	−8.98952	−0.588923	−0.294461	+	0.955663i	$0.595140\pi$
$234$	0	0
$235$	−2.14539	−0.139950
$236$	0	0
$237$	2.81775	0.183032
$238$	0	0
$239$	19.1449	1.23838	0.619191	−	0.785240i	$-0.287461\pi$
$239$	19.1449	1.23838	0.619191	+	0.785240i	$0.287461\pi$
$240$	0	0
$241$	−10.0048	−0.644466	−0.322233	−	0.946660i	$-0.604433\pi$
$241$	−10.0048	−0.644466	−0.322233	+	0.946660i	$0.604433\pi$
$242$	0	0
$243$	14.7583	0.946743
$244$	0	0
$245$	1.49345	0.0954129
$246$	0	0
$247$	−1.17004	−0.0744482
$248$	0	0
$249$	−17.7801	−1.12677
$250$	0	0
$251$	4.33666	0.273727	0.136864	−	0.990590i	$-0.456298\pi$
$251$	4.33666	0.273727	0.136864	+	0.990590i	$0.456298\pi$
$252$	0	0
$253$	−27.1074	−1.70422
$254$	0	0
$255$	4.39941	0.275502
$256$	0	0
$257$	28.4324	1.77356	0.886782	−	0.462187i	$-0.152935\pi$
$257$	28.4324	1.77356	0.886782	+	0.462187i	$0.152935\pi$
$258$	0	0
$259$	−10.3886	−0.645519
$260$	0	0
$261$	−25.5136	−1.57926
$262$	0	0
$263$	−17.4259	−1.07453	−0.537263	−	0.843415i	$-0.680542\pi$
$263$	−17.4259	−1.07453	−0.537263	+	0.843415i	$0.680542\pi$
$264$	0	0
$265$	−2.50447	−0.153849
$266$	0	0
$267$	−16.5513	−1.01292
$268$	0	0
$269$	−25.8170	−1.57409	−0.787045	−	0.616895i	$-0.788390\pi$
$269$	−25.8170	−1.57409	−0.787045	+	0.616895i	$0.788390\pi$
$270$	0	0
$271$	−23.2688	−1.41348	−0.706739	−	0.707475i	$-0.749834\pi$
$271$	−23.2688	−1.41348	−0.706739	+	0.707475i	$0.749834\pi$
$272$	0	0
$273$	3.06443	0.185468
$274$	0	0
$275$	−21.6799	−1.30735
$276$	0	0
$277$	11.8827	0.713962	0.356981	−	0.934112i	$-0.383806\pi$
$277$	11.8827	0.713962	0.356981	+	0.934112i	$0.383806\pi$
$278$	0	0
$279$	−40.9820	−2.45353
$280$	0	0
$281$	−10.6975	−0.638157	−0.319079	−	0.947728i	$-0.603373\pi$
$281$	−10.6975	−0.638157	−0.319079	+	0.947728i	$0.603373\pi$
$282$	0	0
$283$	9.06296	0.538737	0.269368	−	0.963037i	$-0.413185\pi$
$283$	9.06296	0.538737	0.269368	+	0.963037i	$0.413185\pi$
$284$	0	0
$285$	0.685809	0.0406238
$286$	0	0
$287$	2.89109	0.170656
$288$	0	0
$289$	24.1512	1.42066
$290$	0	0
$291$	29.5791	1.73396
$292$	0	0
$293$	28.2216	1.64872	0.824361	−	0.566064i	$-0.191535\pi$
$293$	28.2216	1.64872	0.824361	+	0.566064i	$0.191535\pi$
$294$	0	0
$295$	2.67969	0.156017
$296$	0	0
$297$	−23.9826	−1.39161
$298$	0	0
$299$	7.22814	0.418014
$300$	0	0
$301$	5.14823	0.296739
$302$	0	0
$303$	−3.21986	−0.184976
$304$	0	0
$305$	−0.0872742	−0.00499731
$306$	0	0
$307$	30.9355	1.76558	0.882791	−	0.469765i	$-0.155661\pi$
$307$	30.9355	1.76558	0.882791	+	0.469765i	$0.155661\pi$
$308$	0	0
$309$	41.9766	2.38796
$310$	0	0
$311$	−3.88612	−0.220361	−0.110181	−	0.993912i	$-0.535143\pi$
$311$	−3.88612	−0.220361	−0.110181	+	0.993912i	$0.535143\pi$
$312$	0	0
$313$	−5.03801	−0.284765	−0.142383	−	0.989812i	$-0.545476\pi$
$313$	−5.03801	−0.284765	−0.142383	+	0.989812i	$0.545476\pi$
$314$	0	0
$315$	−1.11750	−0.0629639
$316$	0	0
$317$	14.8736	0.835386	0.417693	−	0.908588i	$-0.362839\pi$
$317$	14.8736	0.835386	0.417693	+	0.908588i	$0.362839\pi$
$318$	0	0
$319$	−22.6640	−1.26894
$320$	0	0
$321$	36.2523	2.02341
$322$	0	0
$323$	6.41492	0.356936
$324$	0	0
$325$	5.78091	0.320667
$326$	0	0
$327$	−5.37193	−0.297068
$328$	0	0
$329$	8.19314	0.451703
$330$	0	0
$331$	28.8329	1.58480	0.792399	−	0.610003i	$-0.208832\pi$
$331$	28.8329	1.58480	0.792399	+	0.610003i	$0.208832\pi$
$332$	0	0
$333$	−55.2094	−3.02545
$334$	0	0
$335$	−2.89026	−0.157912
$336$	0	0
$337$	−30.1520	−1.64249	−0.821243	−	0.570579i	$-0.806719\pi$
$337$	−30.1520	−1.64249	−0.821243	+	0.570579i	$0.806719\pi$
$338$	0	0
$339$	−29.1360	−1.58245
$340$	0	0
$341$	−36.4046	−1.97142
$342$	0	0
$343$	−12.2099	−0.659270
$344$	0	0
$345$	−4.23670	−0.228096
$346$	0	0
$347$	14.0836	0.756045	0.378023	−	0.925796i	$-0.376604\pi$
$347$	14.0836	0.756045	0.378023	+	0.925796i	$0.376604\pi$
$348$	0	0
$349$	−2.47266	−0.132358	−0.0661792	−	0.997808i	$-0.521081\pi$
$349$	−2.47266	−0.132358	−0.0661792	+	0.997808i	$0.521081\pi$
$350$	0	0
$351$	6.39493	0.341336
$352$	0	0
$353$	8.95360	0.476552	0.238276	−	0.971197i	$-0.423418\pi$
$353$	8.95360	0.476552	0.238276	+	0.971197i	$0.423418\pi$
$354$	0	0
$355$	−1.76944	−0.0939123
$356$	0	0
$357$	−16.8011	−0.889210
$358$	0	0
$359$	7.84896	0.414252	0.207126	−	0.978314i	$-0.433589\pi$
$359$	7.84896	0.414252	0.207126	+	0.978314i	$0.433589\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−23.2583	−1.22074
$364$	0	0
$365$	3.41572	0.178787
$366$	0	0
$367$	−26.8354	−1.40079	−0.700397	−	0.713753i	$-0.746994\pi$
$367$	−26.8354	−1.40079	−0.700397	+	0.713753i	$0.746994\pi$
$368$	0	0
$369$	15.3644	0.799839
$370$	0	0
$371$	9.56447	0.496563
$372$	0	0
$373$	−19.5460	−1.01205	−0.506027	−	0.862517i	$-0.668887\pi$
$373$	−19.5460	−1.01205	−0.506027	+	0.862517i	$0.668887\pi$
$374$	0	0
$375$	−6.81747	−0.352052
$376$	0	0
$377$	6.04332	0.311247
$378$	0	0
$379$	8.31232	0.426975	0.213488	−	0.976946i	$-0.431518\pi$
$379$	8.31232	0.426975	0.213488	+	0.976946i	$0.431518\pi$
$380$	0	0
$381$	−19.2087	−0.984092
$382$	0	0
$383$	−4.08070	−0.208514	−0.104257	−	0.994550i	$-0.533246\pi$
$383$	−4.08070	−0.208514	−0.104257	+	0.994550i	$0.533246\pi$
$384$	0	0
$385$	−0.992682	−0.0505918
$386$	0	0
$387$	27.3597	1.39077
$388$	0	0
$389$	−18.2795	−0.926808	−0.463404	−	0.886147i	$-0.653372\pi$
$389$	−18.2795	−0.926808	−0.463404	+	0.886147i	$0.653372\pi$
$390$	0	0
$391$	−39.6292	−2.00413
$392$	0	0
$393$	−25.0445	−1.26333
$394$	0	0
$395$	0.243389	0.0122462
$396$	0	0
$397$	5.50119	0.276097	0.138048	−	0.990426i	$-0.455917\pi$
$397$	5.50119	0.276097	0.138048	+	0.990426i	$0.455917\pi$
$398$	0	0
$399$	−2.61907	−0.131118
$400$	0	0
$401$	−4.46487	−0.222965	−0.111482	−	0.993766i	$-0.535560\pi$
$401$	−4.46487	−0.222965	−0.111482	+	0.993766i	$0.535560\pi$
$402$	0	0
$403$	9.70724	0.483552
$404$	0	0
$405$	−0.141519	−0.00703213
$406$	0	0
$407$	−49.0429	−2.43097
$408$	0	0
$409$	13.5789	0.671432	0.335716	−	0.941963i	$-0.391022\pi$
$409$	13.5789	0.671432	0.335716	+	0.941963i	$0.391022\pi$
$410$	0	0
$411$	8.99847	0.443862
$412$	0	0
$413$	−10.2336	−0.503562
$414$	0	0
$415$	−1.53580	−0.0753895
$416$	0	0
$417$	−30.0778	−1.47292
$418$	0	0
$419$	30.8314	1.50621	0.753106	−	0.657899i	$-0.228555\pi$
$419$	30.8314	1.50621	0.753106	+	0.657899i	$0.228555\pi$
$420$	0	0
$421$	−16.4084	−0.799697	−0.399849	−	0.916581i	$-0.630937\pi$
$421$	−16.4084	−0.799697	−0.399849	+	0.916581i	$0.630937\pi$
$422$	0	0
$423$	43.5416	2.11706
$424$	0	0
$425$	−31.6946	−1.53741
$426$	0	0
$427$	0.333296	0.0161293
$428$	0	0
$429$	14.4666	0.698455
$430$	0	0
$431$	−40.2586	−1.93919	−0.969594	−	0.244718i	$-0.921305\pi$
$431$	−40.2586	−1.93919	−0.969594	+	0.244718i	$0.921305\pi$
$432$	0	0
$433$	0.477039	0.0229250	0.0114625	−	0.999934i	$-0.496351\pi$
$433$	0.477039	0.0229250	0.0114625	+	0.999934i	$0.496351\pi$
$434$	0	0
$435$	−3.54223	−0.169837
$436$	0	0
$437$	−6.17766	−0.295518
$438$	0	0
$439$	−26.8587	−1.28190	−0.640948	−	0.767584i	$-0.721459\pi$
$439$	−26.8587	−1.28190	−0.640948	+	0.767584i	$0.721459\pi$
$440$	0	0
$441$	−30.3101	−1.44334
$442$	0	0
$443$	−24.0326	−1.14182	−0.570912	−	0.821011i	$-0.693410\pi$
$443$	−24.0326	−1.14182	−0.570912	+	0.821011i	$0.693410\pi$
$444$	0	0
$445$	−1.42966	−0.0677724
$446$	0	0
$447$	19.7060	0.932063
$448$	0	0
$449$	−40.9547	−1.93277	−0.966386	−	0.257097i	$-0.917234\pi$
$449$	−40.9547	−1.93277	−0.966386	+	0.257097i	$0.917234\pi$
$450$	0	0
$451$	13.6483	0.642674
$452$	0	0
$453$	−59.1237	−2.77787
$454$	0	0
$455$	0.264698	0.0124092
$456$	0	0
$457$	2.32008	0.108529	0.0542644	−	0.998527i	$-0.482719\pi$
$457$	2.32008	0.108529	0.0542644	+	0.998527i	$0.482719\pi$
$458$	0	0
$459$	−35.0610	−1.63651
$460$	0	0
$461$	4.92454	0.229359	0.114679	−	0.993403i	$-0.463416\pi$
$461$	4.92454	0.229359	0.114679	+	0.993403i	$0.463416\pi$
$462$	0	0
$463$	−13.1476	−0.611021	−0.305511	−	0.952189i	$-0.598827\pi$
$463$	−13.1476	−0.611021	−0.305511	+	0.952189i	$0.598827\pi$
$464$	0	0
$465$	−5.68979	−0.263858
$466$	0	0
$467$	−16.5271	−0.764784	−0.382392	−	0.924000i	$-0.624900\pi$
$467$	−16.5271	−0.764784	−0.382392	+	0.924000i	$0.624900\pi$
$468$	0	0
$469$	11.0378	0.509677
$470$	0	0
$471$	−22.3107	−1.02802
$472$	0	0
$473$	24.3039	1.11749
$474$	0	0
$475$	−4.94076	−0.226698
$476$	0	0
$477$	50.8294	2.32732
$478$	0	0
$479$	20.2636	0.925869	0.462934	−	0.886393i	$-0.346797\pi$
$479$	20.2636	0.925869	0.462934	+	0.886393i	$0.346797\pi$
$480$	0	0
$481$	13.0772	0.596270
$482$	0	0
$483$	16.1797	0.736204
$484$	0	0
$485$	2.55496	0.116015
$486$	0	0
$487$	27.7595	1.25791	0.628953	−	0.777444i	$-0.283484\pi$
$487$	27.7595	1.25791	0.628953	+	0.777444i	$0.283484\pi$
$488$	0	0
$489$	−18.9323	−0.856149
$490$	0	0
$491$	−20.9007	−0.943235	−0.471617	−	0.881803i	$-0.656330\pi$
$491$	−20.9007	−0.943235	−0.471617	+	0.881803i	$0.656330\pi$
$492$	0	0
$493$	−33.1333	−1.49225
$494$	0	0
$495$	−5.27550	−0.237116
$496$	0	0
$497$	6.75742	0.303112
$498$	0	0
$499$	18.9602	0.848775	0.424387	−	0.905481i	$-0.360490\pi$
$499$	18.9602	0.848775	0.424387	+	0.905481i	$0.360490\pi$
$500$	0	0
$501$	30.7616	1.37433
$502$	0	0
$503$	9.88996	0.440972	0.220486	−	0.975390i	$-0.429236\pi$
$503$	9.88996	0.440972	0.220486	+	0.975390i	$0.429236\pi$
$504$	0	0
$505$	−0.278123	−0.0123763
$506$	0	0
$507$	32.7732	1.45551
$508$	0	0
$509$	−5.67539	−0.251557	−0.125779	−	0.992058i	$-0.540143\pi$
$509$	−5.67539	−0.251557	−0.125779	+	0.992058i	$0.540143\pi$
$510$	0	0
$511$	−13.0445	−0.577054
$512$	0	0
$513$	−5.46555	−0.241310
$514$	0	0
$515$	3.62583	0.159773
$516$	0	0
$517$	38.6783	1.70107
$518$	0	0
$519$	68.2488	2.99579
$520$	0	0
$521$	−31.6584	−1.38698	−0.693490	−	0.720466i	$-0.743928\pi$
$521$	−31.6584	−1.38698	−0.693490	+	0.720466i	$0.743928\pi$
$522$	0	0
$523$	−1.52863	−0.0668423	−0.0334212	−	0.999441i	$-0.510640\pi$
$523$	−1.52863	−0.0668423	−0.0334212	+	0.999441i	$0.510640\pi$
$524$	0	0
$525$	12.9402	0.564757
$526$	0	0
$527$	−53.2211	−2.31835
$528$	0	0
$529$	15.1635	0.659282
$530$	0	0
$531$	−54.3854	−2.36012
$532$	0	0
$533$	−3.63931	−0.157636
$534$	0	0
$535$	3.13138	0.135381
$536$	0	0
$537$	60.2264	2.59896
$538$	0	0
$539$	−26.9247	−1.15973
$540$	0	0
$541$	−28.5133	−1.22588	−0.612942	−	0.790128i	$-0.710014\pi$
$541$	−28.5133	−1.22588	−0.612942	+	0.790128i	$0.710014\pi$
$542$	0	0
$543$	45.8646	1.96824
$544$	0	0
$545$	−0.464013	−0.0198761
$546$	0	0
$547$	−5.90948	−0.252671	−0.126336	−	0.991988i	$-0.540322\pi$
$547$	−5.90948	−0.252671	−0.126336	+	0.991988i	$0.540322\pi$
$548$	0	0
$549$	1.77127	0.0755958
$550$	0	0
$551$	−5.16503	−0.220038
$552$	0	0
$553$	−0.929492	−0.0395260
$554$	0	0
$555$	−7.66508	−0.325364
$556$	0	0
$557$	−22.9039	−0.970467	−0.485234	−	0.874385i	$-0.661265\pi$
$557$	−22.9039	−0.970467	−0.485234	+	0.874385i	$0.661265\pi$
$558$	0	0
$559$	−6.48060	−0.274100
$560$	0	0
$561$	−79.3151	−3.34869
$562$	0	0
$563$	−11.6582	−0.491333	−0.245667	−	0.969354i	$-0.579007\pi$
$563$	−11.6582	−0.491333	−0.245667	+	0.969354i	$0.579007\pi$
$564$	0	0
$565$	−2.51669	−0.105878
$566$	0	0
$567$	0.540454	0.0226969
$568$	0	0
$569$	1.57711	0.0661160	0.0330580	−	0.999453i	$-0.489475\pi$
$569$	1.57711	0.0661160	0.0330580	+	0.999453i	$0.489475\pi$
$570$	0	0
$571$	30.3873	1.27167	0.635834	−	0.771826i	$-0.280656\pi$
$571$	30.3873	1.27167	0.635834	+	0.771826i	$0.280656\pi$
$572$	0	0
$573$	55.9195	2.33607
$574$	0	0
$575$	30.5224	1.27287
$576$	0	0
$577$	−24.6605	−1.02663	−0.513315	−	0.858200i	$-0.671583\pi$
$577$	−24.6605	−1.02663	−0.513315	+	0.858200i	$0.671583\pi$
$578$	0	0
$579$	51.3878	2.13560
$580$	0	0
$581$	5.86515	0.243327
$582$	0	0
$583$	45.1521	1.87001
$584$	0	0
$585$	1.40671	0.0581602
$586$	0	0
$587$	−26.9821	−1.11367	−0.556835	−	0.830623i	$-0.687985\pi$
$587$	−26.9821	−1.11367	−0.556835	+	0.830623i	$0.687985\pi$
$588$	0	0
$589$	−8.29647	−0.341850
$590$	0	0
$591$	30.1626	1.24073
$592$	0	0
$593$	−15.7153	−0.645351	−0.322676	−	0.946510i	$-0.604582\pi$
$593$	−15.7153	−0.645351	−0.322676	+	0.946510i	$0.604582\pi$
$594$	0	0
$595$	−1.45124	−0.0594949
$596$	0	0
$597$	21.8527	0.894373
$598$	0	0
$599$	17.2265	0.703856	0.351928	−	0.936027i	$-0.385526\pi$
$599$	17.2265	0.703856	0.351928	+	0.936027i	$0.385526\pi$
$600$	0	0
$601$	−36.2061	−1.47688	−0.738438	−	0.674321i	$-0.764436\pi$
$601$	−36.2061	−1.47688	−0.738438	+	0.674321i	$0.764436\pi$
$602$	0	0
$603$	58.6591	2.38878
$604$	0	0
$605$	−2.00899	−0.0816770
$606$	0	0
$607$	10.7548	0.436525	0.218263	−	0.975890i	$-0.429961\pi$
$607$	10.7548	0.436525	0.218263	+	0.975890i	$0.429961\pi$
$608$	0	0
$609$	13.5276	0.548166
$610$	0	0
$611$	−10.3135	−0.417241
$612$	0	0
$613$	−5.77027	−0.233059	−0.116530	−	0.993187i	$-0.537177\pi$
$613$	−5.77027	−0.233059	−0.116530	+	0.993187i	$0.537177\pi$
$614$	0	0
$615$	2.13314	0.0860165
$616$	0	0
$617$	−1.48806	−0.0599069	−0.0299535	−	0.999551i	$-0.509536\pi$
$617$	−1.48806	−0.0599069	−0.0299535	+	0.999551i	$0.509536\pi$
$618$	0	0
$619$	−38.1127	−1.53188	−0.765940	−	0.642913i	$-0.777726\pi$
$619$	−38.1127	−1.53188	−0.765940	+	0.642913i	$0.777726\pi$
$620$	0	0
$621$	33.7643	1.35491
$622$	0	0
$623$	5.45980	0.218742
$624$	0	0
$625$	24.1149	0.964597
$626$	0	0
$627$	−12.3642	−0.493777
$628$	0	0
$629$	−71.6975	−2.85877
$630$	0	0
$631$	12.2918	0.489329	0.244664	−	0.969608i	$-0.421322\pi$
$631$	12.2918	0.489329	0.244664	+	0.969608i	$0.421322\pi$
$632$	0	0
$633$	75.1465	2.98680
$634$	0	0
$635$	−1.65920	−0.0658432
$636$	0	0
$637$	7.17945	0.284460
$638$	0	0
$639$	35.9116	1.42064
$640$	0	0
$641$	−36.1875	−1.42932	−0.714659	−	0.699473i	$-0.753418\pi$
$641$	−36.1875	−1.42932	−0.714659	+	0.699473i	$0.753418\pi$
$642$	0	0
$643$	−19.1619	−0.755672	−0.377836	−	0.925872i	$-0.623332\pi$
$643$	−19.1619	−0.755672	−0.377836	+	0.925872i	$0.623332\pi$
$644$	0	0
$645$	3.79853	0.149567
$646$	0	0
$647$	41.0331	1.61318	0.806589	−	0.591113i	$-0.201311\pi$
$647$	41.0331	1.61318	0.806589	+	0.591113i	$0.201311\pi$
$648$	0	0
$649$	−48.3109	−1.89637
$650$	0	0
$651$	21.7290	0.851628
$652$	0	0
$653$	9.30259	0.364038	0.182019	−	0.983295i	$-0.441737\pi$
$653$	9.30259	0.364038	0.182019	+	0.983295i	$0.441737\pi$
$654$	0	0
$655$	−2.16328	−0.0845262
$656$	0	0
$657$	−69.3235	−2.70457
$658$	0	0
$659$	11.9866	0.466931	0.233466	−	0.972365i	$-0.424993\pi$
$659$	11.9866	0.466931	0.233466	+	0.972365i	$0.424993\pi$
$660$	0	0
$661$	−15.0263	−0.584455	−0.292228	−	0.956349i	$-0.594396\pi$
$661$	−15.0263	−0.584455	−0.292228	+	0.956349i	$0.594396\pi$
$662$	0	0
$663$	21.1493	0.821369
$664$	0	0
$665$	−0.226229	−0.00877277
$666$	0	0
$667$	31.9078	1.23548
$668$	0	0
$669$	−34.8462	−1.34723
$670$	0	0
$671$	1.57343	0.0607416
$672$	0	0
$673$	−0.447914	−0.0172658	−0.00863291	−	0.999963i	$-0.502748\pi$
$673$	−0.447914	−0.0172658	−0.00863291	+	0.999963i	$0.502748\pi$
$674$	0	0
$675$	27.0040	1.03938
$676$	0	0
$677$	−15.7839	−0.606624	−0.303312	−	0.952891i	$-0.598092\pi$
$677$	−15.7839	−0.606624	−0.303312	+	0.952891i	$0.598092\pi$
$678$	0	0
$679$	−9.75728	−0.374450
$680$	0	0
$681$	−47.6753	−1.82692
$682$	0	0
$683$	41.9490	1.60513	0.802566	−	0.596563i	$-0.203467\pi$
$683$	41.9490	1.60513	0.802566	+	0.596563i	$0.203467\pi$
$684$	0	0
$685$	0.777264	0.0296977
$686$	0	0
$687$	−46.0462	−1.75677
$688$	0	0
$689$	−12.0398	−0.458678
$690$	0	0
$691$	12.9636	0.493160	0.246580	−	0.969122i	$-0.420693\pi$
$691$	12.9636	0.493160	0.246580	+	0.969122i	$0.420693\pi$
$692$	0	0
$693$	20.1469	0.765318
$694$	0	0
$695$	−2.59804	−0.0985492
$696$	0	0
$697$	19.9529	0.755772
$698$	0	0
$699$	25.3302	0.958075
$700$	0	0
$701$	−41.7159	−1.57559	−0.787795	−	0.615938i	$-0.788777\pi$
$701$	−41.7159	−1.57559	−0.787795	+	0.615938i	$0.788777\pi$
$702$	0	0
$703$	−11.1767	−0.421537
$704$	0	0
$705$	6.04516	0.227674
$706$	0	0
$707$	1.06214	0.0399458
$708$	0	0
$709$	−17.9243	−0.673160	−0.336580	−	0.941655i	$-0.609270\pi$
$709$	−17.9243	−0.673160	−0.336580	+	0.941655i	$0.609270\pi$
$710$	0	0
$711$	−4.93969	−0.185253
$712$	0	0
$713$	51.2528	1.91943
$714$	0	0
$715$	1.24959	0.0467320
$716$	0	0
$717$	−53.9455	−2.01463
$718$	0	0
$719$	−25.7196	−0.959180	−0.479590	−	0.877493i	$-0.659215\pi$
$719$	−25.7196	−0.959180	−0.479590	+	0.877493i	$0.659215\pi$
$720$	0	0
$721$	−13.8469	−0.515684
$722$	0	0
$723$	28.1910	1.04843
$724$	0	0
$725$	25.5192	0.947759
$726$	0	0
$727$	−39.8235	−1.47697	−0.738485	−	0.674270i	$-0.764459\pi$
$727$	−39.8235	−1.47697	−0.738485	+	0.674270i	$0.764459\pi$
$728$	0	0
$729$	−43.3294	−1.60479
$730$	0	0
$731$	35.5307	1.31415
$732$	0	0
$733$	32.2629	1.19166	0.595829	−	0.803111i	$-0.296823\pi$
$733$	32.2629	1.19166	0.595829	+	0.803111i	$0.296823\pi$
$734$	0	0
$735$	−4.20816	−0.155220
$736$	0	0
$737$	52.1073	1.91940
$738$	0	0
$739$	−51.8038	−1.90563	−0.952816	−	0.303548i	$-0.901829\pi$
$739$	−51.8038	−1.90563	−0.952816	+	0.303548i	$0.901829\pi$
$740$	0	0
$741$	3.29689	0.121114
$742$	0	0
$743$	−14.9802	−0.549570	−0.274785	−	0.961506i	$-0.588607\pi$
$743$	−14.9802	−0.549570	−0.274785	+	0.961506i	$0.588607\pi$
$744$	0	0
$745$	1.70215	0.0623621
$746$	0	0
$747$	31.1697	1.14044
$748$	0	0
$749$	−11.9586	−0.436957
$750$	0	0
$751$	−12.6191	−0.460477	−0.230238	−	0.973134i	$-0.573951\pi$
$751$	−12.6191	−0.460477	−0.230238	+	0.973134i	$0.573951\pi$
$752$	0	0
$753$	−12.2196	−0.445307
$754$	0	0
$755$	−5.10695	−0.185861
$756$	0	0
$757$	−10.6477	−0.386997	−0.193499	−	0.981101i	$-0.561984\pi$
$757$	−10.6477	−0.386997	−0.193499	+	0.981101i	$0.561984\pi$
$758$	0	0
$759$	76.3816	2.77248
$760$	0	0
$761$	26.9322	0.976293	0.488146	−	0.872762i	$-0.337673\pi$
$761$	26.9322	0.976293	0.488146	+	0.872762i	$0.337673\pi$
$762$	0	0
$763$	1.77204	0.0641523
$764$	0	0
$765$	−7.71245	−0.278844
$766$	0	0
$767$	12.8821	0.465144
$768$	0	0
$769$	21.3676	0.770536	0.385268	−	0.922805i	$-0.374109\pi$
$769$	21.3676	0.770536	0.385268	+	0.922805i	$0.374109\pi$
$770$	0	0
$771$	−80.1153	−2.88528
$772$	0	0
$773$	−10.1433	−0.364829	−0.182415	−	0.983222i	$-0.558391\pi$
$773$	−10.1433	−0.364829	−0.182415	+	0.983222i	$0.558391\pi$
$774$	0	0
$775$	40.9909	1.47244
$776$	0	0
$777$	29.2726	1.05015
$778$	0	0
$779$	3.11040	0.111442
$780$	0	0
$781$	31.9005	1.14149
$782$	0	0
$783$	28.2297	1.00885
$784$	0	0
$785$	−1.92714	−0.0687827
$786$	0	0
$787$	28.2597	1.00735	0.503675	−	0.863893i	$-0.331981\pi$
$787$	28.2597	1.00735	0.503675	+	0.863893i	$0.331981\pi$
$788$	0	0
$789$	49.1017	1.74807
$790$	0	0
$791$	9.61113	0.341732
$792$	0	0
$793$	−0.419553	−0.0148988
$794$	0	0
$795$	7.05697	0.250285
$796$	0	0
$797$	53.3864	1.89104	0.945522	−	0.325557i	$-0.105552\pi$
$797$	53.3864	1.89104	0.945522	+	0.325557i	$0.105552\pi$
$798$	0	0
$799$	56.5452	2.00043
$800$	0	0
$801$	29.0155	1.02521
$802$	0	0
$803$	−61.5806	−2.17313
$804$	0	0
$805$	1.39756	0.0492576
$806$	0	0
$807$	72.7457	2.56077
$808$	0	0
$809$	−1.13616	−0.0399453	−0.0199726	−	0.999801i	$-0.506358\pi$
$809$	−1.13616	−0.0399453	−0.0199726	+	0.999801i	$0.506358\pi$
$810$	0	0
$811$	−39.5890	−1.39016	−0.695078	−	0.718934i	$-0.744630\pi$
$811$	−39.5890	−1.39016	−0.695078	+	0.718934i	$0.744630\pi$
$812$	0	0
$813$	65.5655	2.29948
$814$	0	0
$815$	−1.63532	−0.0572829
$816$	0	0
$817$	5.53876	0.193777
$818$	0	0
$819$	−5.37215	−0.187718
$820$	0	0
$821$	−11.1082	−0.387677	−0.193839	−	0.981033i	$-0.562094\pi$
$821$	−11.1082	−0.387677	−0.193839	+	0.981033i	$0.562094\pi$
$822$	0	0
$823$	−8.81783	−0.307370	−0.153685	−	0.988120i	$-0.549114\pi$
$823$	−8.81783	−0.307370	−0.153685	+	0.988120i	$0.549114\pi$
$824$	0	0
$825$	61.0884	2.12682
$826$	0	0
$827$	48.3086	1.67985	0.839927	−	0.542699i	$-0.182598\pi$
$827$	48.3086	1.67985	0.839927	+	0.542699i	$0.182598\pi$
$828$	0	0
$829$	−20.9899	−0.729008	−0.364504	−	0.931202i	$-0.618761\pi$
$829$	−20.9899	−0.729008	−0.364504	+	0.931202i	$0.618761\pi$
$830$	0	0
$831$	−33.4824	−1.16149
$832$	0	0
$833$	−39.3622	−1.36382
$834$	0	0
$835$	2.65710	0.0919528
$836$	0	0
$837$	45.3447	1.56734
$838$	0	0
$839$	−12.6224	−0.435774	−0.217887	−	0.975974i	$-0.569916\pi$
$839$	−12.6224	−0.435774	−0.217887	+	0.975974i	$0.569916\pi$
$840$	0	0
$841$	−2.32244	−0.0800841
$842$	0	0
$843$	30.1427	1.03817
$844$	0	0
$845$	2.83086	0.0973846
$846$	0	0
$847$	7.67224	0.263621
$848$	0	0
$849$	−25.5371	−0.876431
$850$	0	0
$851$	69.0458	2.36686
$852$	0	0
$853$	28.2597	0.967592	0.483796	−	0.875181i	$-0.339258\pi$
$853$	28.2597	0.967592	0.483796	+	0.875181i	$0.339258\pi$
$854$	0	0
$855$	−1.20227	−0.0411167
$856$	0	0
$857$	7.10377	0.242660	0.121330	−	0.992612i	$-0.461284\pi$
$857$	7.10377	0.242660	0.121330	+	0.992612i	$0.461284\pi$
$858$	0	0
$859$	25.6812	0.876232	0.438116	−	0.898918i	$-0.355646\pi$
$859$	25.6812	0.876232	0.438116	+	0.898918i	$0.355646\pi$
$860$	0	0
$861$	−8.14636	−0.277627
$862$	0	0
$863$	−52.1636	−1.77567	−0.887835	−	0.460161i	$-0.847792\pi$
$863$	−52.1636	−1.77567	−0.887835	+	0.460161i	$0.847792\pi$
$864$	0	0
$865$	5.89515	0.200441
$866$	0	0
$867$	−68.0518	−2.31116
$868$	0	0
$869$	−4.38796	−0.148851
$870$	0	0
$871$	−13.8944	−0.470792
$872$	0	0
$873$	−51.8540	−1.75499
$874$	0	0
$875$	2.24888	0.0760262
$876$	0	0
$877$	−25.2518	−0.852692	−0.426346	−	0.904560i	$-0.640199\pi$
$877$	−25.2518	−0.852692	−0.426346	+	0.904560i	$0.640199\pi$
$878$	0	0
$879$	−79.5212	−2.68218
$880$	0	0
$881$	−53.7053	−1.80938	−0.904689	−	0.426072i	$-0.859897\pi$
$881$	−53.7053	−1.80938	−0.904689	+	0.426072i	$0.859897\pi$
$882$	0	0
$883$	−7.33046	−0.246690	−0.123345	−	0.992364i	$-0.539362\pi$
$883$	−7.33046	−0.246690	−0.123345	+	0.992364i	$0.539362\pi$
$884$	0	0
$885$	−7.55067	−0.253813
$886$	0	0
$887$	46.5751	1.56384	0.781920	−	0.623379i	$-0.214241\pi$
$887$	46.5751	1.56384	0.781920	+	0.623379i	$0.214241\pi$
$888$	0	0
$889$	6.33639	0.212516
$890$	0	0
$891$	2.55138	0.0854746
$892$	0	0
$893$	8.81464	0.294971
$894$	0	0
$895$	5.20220	0.173890
$896$	0	0
$897$	−20.3671	−0.680036
$898$	0	0
$899$	42.8515	1.42918
$900$	0	0
$901$	66.0095	2.19909
$902$	0	0
$903$	−14.5064	−0.482743
$904$	0	0
$905$	3.96167	0.131690
$906$	0	0
$907$	52.2496	1.73492	0.867460	−	0.497508i	$-0.165751\pi$
$907$	52.2496	1.73492	0.867460	+	0.497508i	$0.165751\pi$
$908$	0	0
$909$	5.64462	0.187220
$910$	0	0
$911$	−39.5725	−1.31109	−0.655547	−	0.755154i	$-0.727562\pi$
$911$	−39.5725	−1.31109	−0.655547	+	0.755154i	$0.727562\pi$
$912$	0	0
$913$	27.6883	0.916349
$914$	0	0
$915$	0.245916	0.00812975
$916$	0	0
$917$	8.26145	0.272817
$918$	0	0
$919$	−20.1585	−0.664968	−0.332484	−	0.943109i	$-0.607887\pi$
$919$	−20.1585	−0.664968	−0.332484	+	0.943109i	$0.607887\pi$
$920$	0	0
$921$	−87.1684	−2.87230
$922$	0	0
$923$	−8.50624	−0.279986
$924$	0	0
$925$	55.2214	1.81567
$926$	0	0
$927$	−73.5876	−2.41694
$928$	0	0
$929$	11.9692	0.392696	0.196348	−	0.980534i	$-0.437092\pi$
$929$	11.9692	0.392696	0.196348	+	0.980534i	$0.437092\pi$
$930$	0	0
$931$	−6.13604	−0.201101
$932$	0	0
$933$	10.9501	0.358490
$934$	0	0
$935$	−6.85103	−0.224053
$936$	0	0
$937$	14.8914	0.486480	0.243240	−	0.969966i	$-0.421790\pi$
$937$	14.8914	0.486480	0.243240	+	0.969966i	$0.421790\pi$
$938$	0	0
$939$	14.1958	0.463263
$940$	0	0
$941$	−13.8992	−0.453103	−0.226551	−	0.973999i	$-0.572745\pi$
$941$	−13.8992	−0.453103	−0.226551	+	0.973999i	$0.572745\pi$
$942$	0	0
$943$	−19.2150	−0.625726
$944$	0	0
$945$	1.23646	0.0402221
$946$	0	0
$947$	−44.8171	−1.45636	−0.728180	−	0.685386i	$-0.759633\pi$
$947$	−44.8171	−1.45636	−0.728180	+	0.685386i	$0.759633\pi$
$948$	0	0
$949$	16.4204	0.533028
$950$	0	0
$951$	−41.9101	−1.35903
$952$	0	0
$953$	34.2210	1.10853	0.554264	−	0.832341i	$-0.313000\pi$
$953$	34.2210	1.10853	0.554264	+	0.832341i	$0.313000\pi$
$954$	0	0
$955$	4.83018	0.156301
$956$	0	0
$957$	63.8613	2.06434
$958$	0	0
$959$	−2.96833	−0.0958525
$960$	0	0
$961$	37.8313	1.22037
$962$	0	0
$963$	−63.5526	−2.04796
$964$	0	0
$965$	4.43874	0.142888
$966$	0	0
$967$	48.2353	1.55114	0.775571	−	0.631260i	$-0.217462\pi$
$967$	48.2353	1.55114	0.775571	+	0.631260i	$0.217462\pi$
$968$	0	0
$969$	−18.0756	−0.580672
$970$	0	0
$971$	35.7510	1.14730	0.573652	−	0.819099i	$-0.305526\pi$
$971$	35.7510	1.14730	0.573652	+	0.819099i	$0.305526\pi$
$972$	0	0
$973$	9.92179	0.318078
$974$	0	0
$975$	−16.2891	−0.521670
$976$	0	0
$977$	−2.06258	−0.0659876	−0.0329938	−	0.999456i	$-0.510504\pi$
$977$	−2.06258	−0.0659876	−0.0329938	+	0.999456i	$0.510504\pi$
$978$	0	0
$979$	25.7747	0.823764
$980$	0	0
$981$	9.41733	0.300672
$982$	0	0
$983$	−23.1827	−0.739413	−0.369707	−	0.929149i	$-0.620542\pi$
$983$	−23.1827	−0.739413	−0.369707	+	0.929149i	$0.620542\pi$
$984$	0	0
$985$	2.60537	0.0830140
$986$	0	0
$987$	−23.0862	−0.734841
$988$	0	0
$989$	−34.2166	−1.08802
$990$	0	0
$991$	−12.8141	−0.407053	−0.203526	−	0.979069i	$-0.565240\pi$
$991$	−12.8141	−0.407053	−0.203526	+	0.979069i	$0.565240\pi$
$992$	0	0
$993$	−81.2437	−2.57819
$994$	0	0
$995$	1.88758	0.0598404
$996$	0	0
$997$	30.1866	0.956019	0.478010	−	0.878355i	$-0.341358\pi$
$997$	30.1866	0.956019	0.478010	+	0.878355i	$0.341358\pi$
$998$	0	0
$999$	61.0867	1.93270

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6004.2.a.g.1.2	✓	27

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6004.2.a.g.1.2	✓	27	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)

\(f(q)\)	\(=\)	\(q-2.81775 q^{3} -0.243389 q^{5} +0.929492 q^{7} +4.93969 q^{9} +O(q^{10})\)	\(q-2.81775 q^{3} -0.243389 q^{5} +0.929492 q^{7} +4.93969 q^{9} +4.38796 q^{11} -1.17004 q^{13} +0.685809 q^{15} +6.41492 q^{17} +1.00000 q^{19} -2.61907 q^{21} -6.17766 q^{23} -4.94076 q^{25} -5.46555 q^{27} -5.16503 q^{29} -8.29647 q^{31} -12.3642 q^{33} -0.226229 q^{35} -11.1767 q^{37} +3.29689 q^{39} +3.11040 q^{41} +5.53876 q^{43} -1.20227 q^{45} +8.81464 q^{47} -6.13604 q^{49} -18.0756 q^{51} +10.2900 q^{53} -1.06798 q^{55} -2.81775 q^{57} -11.0099 q^{59} +0.358578 q^{61} +4.59140 q^{63} +0.284776 q^{65} +11.8751 q^{67} +17.4071 q^{69} +7.27001 q^{71} -14.0340 q^{73} +13.9218 q^{75} +4.07858 q^{77} -1.00000 q^{79} +0.581451 q^{81} +6.31006 q^{83} -1.56132 q^{85} +14.5537 q^{87} +5.87396 q^{89} -1.08755 q^{91} +23.3773 q^{93} -0.243389 q^{95} -10.4974 q^{97} +21.6752 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 27 q - 4 q^{3} - 10 q^{5} - 8 q^{7} + 19 q^{9}+O(q^{10}) \) 27 * q - 4 * q^3 - 10 * q^5 - 8 * q^7 + 19 * q^9	\( 27 q - 4 q^{3} - 10 q^{5} - 8 q^{7} + 19 q^{9} + 3 q^{11} - 5 q^{13} - 11 q^{15} - 17 q^{17} + 27 q^{19} - 28 q^{21} - 11 q^{23} + 13 q^{25} - 7 q^{27} - 39 q^{29} - 27 q^{31} - 18 q^{33} - 5 q^{35} - q^{37} - 22 q^{39} - 36 q^{41} - 2 q^{43} - 18 q^{45} - 12 q^{47} + 15 q^{49} + 4 q^{51} - 28 q^{53} + 5 q^{55} - 4 q^{57} - 30 q^{59} - 6 q^{61} - 4 q^{63} - 32 q^{65} + 13 q^{67} - 27 q^{69} - 59 q^{71} - 30 q^{73} - 21 q^{75} - 39 q^{77} - 27 q^{79} - 5 q^{81} + 4 q^{83} - 3 q^{85} + 22 q^{87} - 56 q^{89} - 8 q^{91} - 38 q^{93} - 10 q^{95} - 30 q^{97} + 31 q^{99}+O(q^{100}) \) 27 * q - 4 * q^3 - 10 * q^5 - 8 * q^7 + 19 * q^9 + 3 * q^11 - 5 * q^13 - 11 * q^15 - 17 * q^17 + 27 * q^19 - 28 * q^21 - 11 * q^23 + 13 * q^25 - 7 * q^27 - 39 * q^29 - 27 * q^31 - 18 * q^33 - 5 * q^35 - q^37 - 22 * q^39 - 36 * q^41 - 2 * q^43 - 18 * q^45 - 12 * q^47 + 15 * q^49 + 4 * q^51 - 28 * q^53 + 5 * q^55 - 4 * q^57 - 30 * q^59 - 6 * q^61 - 4 * q^63 - 32 * q^65 + 13 * q^67 - 27 * q^69 - 59 * q^71 - 30 * q^73 - 21 * q^75 - 39 * q^77 - 27 * q^79 - 5 * q^81 + 4 * q^83 - 3 * q^85 + 22 * q^87 - 56 * q^89 - 8 * q^91 - 38 * q^93 - 10 * q^95 - 30 * q^97 + 31 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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