LMFDB - Embedded newform 6004.2.a.g.1.5

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			1.5
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)

$f(q)$	$=$	$q-2.14389 q^{3} +2.12568 q^{5} -4.35707 q^{7} +1.59626 q^{9} +O(q^{10})$	$q-2.14389 q^{3} +2.12568 q^{5} -4.35707 q^{7} +1.59626 q^{9} +2.62997 q^{11} +1.60555 q^{13} -4.55722 q^{15} -7.43738 q^{17} +1.00000 q^{19} +9.34107 q^{21} +6.34749 q^{23} -0.481487 q^{25} +3.00946 q^{27} -6.78691 q^{29} +5.19656 q^{31} -5.63836 q^{33} -9.26173 q^{35} +9.25264 q^{37} -3.44212 q^{39} -10.6249 q^{41} +1.30088 q^{43} +3.39314 q^{45} +13.2589 q^{47} +11.9840 q^{49} +15.9449 q^{51} -4.96110 q^{53} +5.59047 q^{55} -2.14389 q^{57} -9.67374 q^{59} +0.0771850 q^{61} -6.95502 q^{63} +3.41289 q^{65} +8.62707 q^{67} -13.6083 q^{69} -10.4767 q^{71} -6.36465 q^{73} +1.03225 q^{75} -11.4589 q^{77} -1.00000 q^{79} -11.2407 q^{81} -0.269206 q^{83} -15.8095 q^{85} +14.5504 q^{87} +13.0122 q^{89} -6.99549 q^{91} -11.1408 q^{93} +2.12568 q^{95} -5.91056 q^{97} +4.19812 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.14389	−1.23778	−0.618888	−	0.785480i	$-0.712416\pi$
$3$	−2.14389	−1.23778	−0.618888	+	0.785480i	$0.712416\pi$
$4$	0	0
$5$	2.12568	0.950633	0.475316	−	0.879815i	$-0.342334\pi$
$5$	2.12568	0.950633	0.475316	+	0.879815i	$0.342334\pi$
$6$	0	0
$7$	−4.35707	−1.64682	−0.823408	−	0.567449i	$-0.807930\pi$
$7$	−4.35707	−1.64682	−0.823408	+	0.567449i	$0.807930\pi$
$8$	0	0
$9$	1.59626	0.532088
$10$	0	0
$11$	2.62997	0.792965	0.396483	−	0.918042i	$-0.370231\pi$
$11$	2.62997	0.792965	0.396483	+	0.918042i	$0.370231\pi$
$12$	0	0
$13$	1.60555	0.445300	0.222650	−	0.974898i	$-0.428529\pi$
$13$	1.60555	0.445300	0.222650	+	0.974898i	$0.428529\pi$
$14$	0	0
$15$	−4.55722	−1.17667
$16$	0	0
$17$	−7.43738	−1.80383	−0.901914	−	0.431915i	$-0.857838\pi$
$17$	−7.43738	−1.80383	−0.901914	+	0.431915i	$0.857838\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	9.34107	2.03839
$22$	0	0
$23$	6.34749	1.32354	0.661771	−	0.749706i	$-0.269805\pi$
$23$	6.34749	1.32354	0.661771	+	0.749706i	$0.269805\pi$
$24$	0	0
$25$	−0.481487	−0.0962973
$26$	0	0
$27$	3.00946	0.579170
$28$	0	0
$29$	−6.78691	−1.26030	−0.630149	−	0.776475i	$-0.717006\pi$
$29$	−6.78691	−1.26030	−0.630149	+	0.776475i	$0.717006\pi$
$30$	0	0
$31$	5.19656	0.933329	0.466664	−	0.884434i	$-0.345456\pi$
$31$	5.19656	0.933329	0.466664	+	0.884434i	$0.345456\pi$
$32$	0	0
$33$	−5.63836	−0.981513
$34$	0	0
$35$	−9.26173	−1.56552
$36$	0	0
$37$	9.25264	1.52112	0.760562	−	0.649265i	$-0.224923\pi$
$37$	9.25264	1.52112	0.760562	+	0.649265i	$0.224923\pi$
$38$	0	0
$39$	−3.44212	−0.551181
$40$	0	0
$41$	−10.6249	−1.65933	−0.829667	−	0.558259i	$-0.811470\pi$
$41$	−10.6249	−1.65933	−0.829667	+	0.558259i	$0.811470\pi$
$42$	0	0
$43$	1.30088	0.198383	0.0991913	−	0.995068i	$-0.468374\pi$
$43$	1.30088	0.198383	0.0991913	+	0.995068i	$0.468374\pi$
$44$	0	0
$45$	3.39314	0.505820
$46$	0	0
$47$	13.2589	1.93401	0.967003	−	0.254766i	$-0.0819985\pi$
$47$	13.2589	1.93401	0.967003	+	0.254766i	$0.0819985\pi$
$48$	0	0
$49$	11.9840	1.71200
$50$	0	0
$51$	15.9449	2.23273
$52$	0	0
$53$	−4.96110	−0.681460	−0.340730	−	0.940161i	$-0.610674\pi$
$53$	−4.96110	−0.681460	−0.340730	+	0.940161i	$0.610674\pi$
$54$	0	0
$55$	5.59047	0.753819
$56$	0	0
$57$	−2.14389	−0.283965
$58$	0	0
$59$	−9.67374	−1.25941	−0.629707	−	0.776833i	$-0.716825\pi$
$59$	−9.67374	−1.25941	−0.629707	+	0.776833i	$0.716825\pi$
$60$	0	0
$61$	0.0771850	0.00988252	0.00494126	−	0.999988i	$-0.498427\pi$
$61$	0.0771850	0.00988252	0.00494126	+	0.999988i	$0.498427\pi$
$62$	0	0
$63$	−6.95502	−0.876251
$64$	0	0
$65$	3.41289	0.423316
$66$	0	0
$67$	8.62707	1.05396	0.526982	−	0.849876i	$-0.323323\pi$
$67$	8.62707	1.05396	0.526982	+	0.849876i	$0.323323\pi$
$68$	0	0
$69$	−13.6083	−1.63825
$70$	0	0
$71$	−10.4767	−1.24336	−0.621679	−	0.783273i	$-0.713549\pi$
$71$	−10.4767	−1.24336	−0.621679	+	0.783273i	$0.713549\pi$
$72$	0	0
$73$	−6.36465	−0.744926	−0.372463	−	0.928047i	$-0.621487\pi$
$73$	−6.36465	−0.744926	−0.372463	+	0.928047i	$0.621487\pi$
$74$	0	0
$75$	1.03225	0.119194
$76$	0	0
$77$	−11.4589	−1.30587
$78$	0	0
$79$	−1.00000	−0.112509
$79$	−1.00000	−0.112509
$80$	0	0
$81$	−11.2407	−1.24897
$82$	0	0
$83$	−0.269206	−0.0295492	−0.0147746	−	0.999891i	$-0.504703\pi$
$83$	−0.269206	−0.0295492	−0.0147746	+	0.999891i	$0.504703\pi$
$84$	0	0
$85$	−15.8095	−1.71478
$86$	0	0
$87$	14.5504	1.55996
$88$	0	0
$89$	13.0122	1.37929	0.689644	−	0.724149i	$-0.257767\pi$
$89$	13.0122	1.37929	0.689644	+	0.724149i	$0.257767\pi$
$90$	0	0
$91$	−6.99549	−0.733327
$92$	0	0
$93$	−11.1408	−1.15525
$94$	0	0
$95$	2.12568	0.218090
$96$	0	0
$97$	−5.91056	−0.600126	−0.300063	−	0.953919i	$-0.597008\pi$
$97$	−5.91056	−0.600126	−0.300063	+	0.953919i	$0.597008\pi$
$98$	0	0
$99$	4.19812	0.421927
$100$	0	0
$101$	15.3724	1.52961	0.764803	−	0.644264i	$-0.222836\pi$
$101$	15.3724	1.52961	0.764803	+	0.644264i	$0.222836\pi$
$102$	0	0
$103$	15.3785	1.51529	0.757643	−	0.652669i	$-0.226351\pi$
$103$	15.3785	1.51529	0.757643	+	0.652669i	$0.226351\pi$
$104$	0	0
$105$	19.8561	1.93776
$106$	0	0
$107$	−12.5103	−1.20942	−0.604709	−	0.796446i	$-0.706711\pi$
$107$	−12.5103	−1.20942	−0.604709	+	0.796446i	$0.706711\pi$
$108$	0	0
$109$	8.72744	0.835937	0.417969	−	0.908462i	$-0.362742\pi$
$109$	8.72744	0.835937	0.417969	+	0.908462i	$0.362742\pi$
$110$	0	0
$111$	−19.8366	−1.88281
$112$	0	0
$113$	−2.86294	−0.269323	−0.134661	−	0.990892i	$-0.542995\pi$
$113$	−2.86294	−0.269323	−0.134661	+	0.990892i	$0.542995\pi$
$114$	0	0
$115$	13.4927	1.25820
$116$	0	0
$117$	2.56288	0.236938
$118$	0	0
$119$	32.4051	2.97057
$120$	0	0
$121$	−4.08326	−0.371206
$122$	0	0
$123$	22.7787	2.05388
$124$	0	0
$125$	−11.6519	−1.04218
$126$	0	0
$127$	13.0048	1.15399	0.576994	−	0.816749i	$-0.304226\pi$
$127$	13.0048	1.15399	0.576994	+	0.816749i	$0.304226\pi$
$128$	0	0
$129$	−2.78895	−0.245553
$130$	0	0
$131$	−10.0451	−0.877641	−0.438821	−	0.898575i	$-0.644604\pi$
$131$	−10.0451	−0.877641	−0.438821	+	0.898575i	$0.644604\pi$
$132$	0	0
$133$	−4.35707	−0.377806
$134$	0	0
$135$	6.39714	0.550578
$136$	0	0
$137$	7.67431	0.655661	0.327830	−	0.944737i	$-0.393683\pi$
$137$	7.67431	0.655661	0.327830	+	0.944737i	$0.393683\pi$
$138$	0	0
$139$	−9.24532	−0.784178	−0.392089	−	0.919927i	$-0.628247\pi$
$139$	−9.24532	−0.784178	−0.392089	+	0.919927i	$0.628247\pi$
$140$	0	0
$141$	−28.4256	−2.39386
$142$	0	0
$143$	4.22255	0.353107
$144$	0	0
$145$	−14.4268	−1.19808
$146$	0	0
$147$	−25.6924	−2.11908
$148$	0	0
$149$	−12.3267	−1.00985	−0.504923	−	0.863165i	$-0.668479\pi$
$149$	−12.3267	−1.00985	−0.504923	+	0.863165i	$0.668479\pi$
$150$	0	0
$151$	5.46737	0.444928	0.222464	−	0.974941i	$-0.428590\pi$
$151$	5.46737	0.444928	0.222464	+	0.974941i	$0.428590\pi$
$152$	0	0
$153$	−11.8720	−0.959795
$154$	0	0
$155$	11.0462	0.887253
$156$	0	0
$157$	−5.28258	−0.421596	−0.210798	−	0.977530i	$-0.567606\pi$
$157$	−5.28258	−0.421596	−0.210798	+	0.977530i	$0.567606\pi$
$158$	0	0
$159$	10.6361	0.843494
$160$	0	0
$161$	−27.6564	−2.17963
$162$	0	0
$163$	−22.5062	−1.76282	−0.881410	−	0.472352i	$-0.843405\pi$
$163$	−22.5062	−1.76282	−0.881410	+	0.472352i	$0.843405\pi$
$164$	0	0
$165$	−11.9854	−0.933058
$166$	0	0
$167$	−24.5736	−1.90156	−0.950782	−	0.309860i	$-0.899718\pi$
$167$	−24.5736	−1.90156	−0.950782	+	0.309860i	$0.899718\pi$
$168$	0	0
$169$	−10.4222	−0.801708
$170$	0	0
$171$	1.59626	0.122069
$172$	0	0
$173$	12.8035	0.973434	0.486717	−	0.873560i	$-0.338194\pi$
$173$	12.8035	0.973434	0.486717	+	0.873560i	$0.338194\pi$
$174$	0	0
$175$	2.09787	0.158584
$176$	0	0
$177$	20.7394	1.55887
$178$	0	0
$179$	−2.11092	−0.157777	−0.0788887	−	0.996883i	$-0.525137\pi$
$179$	−2.11092	−0.157777	−0.0788887	+	0.996883i	$0.525137\pi$
$180$	0	0
$181$	2.71295	0.201652	0.100826	−	0.994904i	$-0.467851\pi$
$181$	2.71295	0.201652	0.100826	+	0.994904i	$0.467851\pi$
$182$	0	0
$183$	−0.165476	−0.0122323
$184$	0	0
$185$	19.6681	1.44603
$186$	0	0
$187$	−19.5601	−1.43037
$188$	0	0
$189$	−13.1124	−0.953787
$190$	0	0
$191$	15.8457	1.14655	0.573277	−	0.819362i	$-0.305672\pi$
$191$	15.8457	1.14655	0.573277	+	0.819362i	$0.305672\pi$
$192$	0	0
$193$	4.80511	0.345879	0.172940	−	0.984932i	$-0.444673\pi$
$193$	4.80511	0.345879	0.172940	+	0.984932i	$0.444673\pi$
$194$	0	0
$195$	−7.31685	−0.523971
$196$	0	0
$197$	−10.2230	−0.728358	−0.364179	−	0.931329i	$-0.618650\pi$
$197$	−10.2230	−0.728358	−0.364179	+	0.931329i	$0.618650\pi$
$198$	0	0
$199$	−2.55409	−0.181055	−0.0905273	−	0.995894i	$-0.528855\pi$
$199$	−2.55409	−0.181055	−0.0905273	+	0.995894i	$0.528855\pi$
$200$	0	0
$201$	−18.4955	−1.30457
$202$	0	0
$203$	29.5710	2.07548
$204$	0	0
$205$	−22.5852	−1.57742
$206$	0	0
$207$	10.1323	0.704241
$208$	0	0
$209$	2.62997	0.181919
$210$	0	0
$211$	−24.8781	−1.71268	−0.856338	−	0.516415i	$-0.827266\pi$
$211$	−24.8781	−1.71268	−0.856338	+	0.516415i	$0.827266\pi$
$212$	0	0
$213$	22.4609	1.53900
$214$	0	0
$215$	2.76526	0.188589
$216$	0	0
$217$	−22.6417	−1.53702
$218$	0	0
$219$	13.6451	0.922051
$220$	0	0
$221$	−11.9411	−0.803244
$222$	0	0
$223$	18.8472	1.26210	0.631050	−	0.775742i	$-0.282624\pi$
$223$	18.8472	1.26210	0.631050	+	0.775742i	$0.282624\pi$
$224$	0	0
$225$	−0.768579	−0.0512386
$226$	0	0
$227$	13.0581	0.866695	0.433347	−	0.901227i	$-0.357332\pi$
$227$	13.0581	0.866695	0.433347	+	0.901227i	$0.357332\pi$
$228$	0	0
$229$	3.20186	0.211585	0.105792	−	0.994388i	$-0.466262\pi$
$229$	3.20186	0.211585	0.105792	+	0.994388i	$0.466262\pi$
$230$	0	0
$231$	24.5667	1.61637
$232$	0	0
$233$	7.72987	0.506400	0.253200	−	0.967414i	$-0.418517\pi$
$233$	7.72987	0.506400	0.253200	+	0.967414i	$0.418517\pi$
$234$	0	0
$235$	28.1841	1.83853
$236$	0	0
$237$	2.14389	0.139261
$238$	0	0
$239$	17.0828	1.10499	0.552497	−	0.833515i	$-0.313675\pi$
$239$	17.0828	1.10499	0.552497	+	0.833515i	$0.313675\pi$
$240$	0	0
$241$	−24.4878	−1.57740	−0.788700	−	0.614778i	$-0.789246\pi$
$241$	−24.4878	−1.57740	−0.788700	+	0.614778i	$0.789246\pi$
$242$	0	0
$243$	15.0705	0.966774
$244$	0	0
$245$	25.4742	1.62749
$246$	0	0
$247$	1.60555	0.102159
$248$	0	0
$249$	0.577147	0.0365752
$250$	0	0
$251$	−16.6351	−1.05000	−0.524999	−	0.851103i	$-0.675934\pi$
$251$	−16.6351	−1.05000	−0.524999	+	0.851103i	$0.675934\pi$
$252$	0	0
$253$	16.6937	1.04952
$254$	0	0
$255$	33.8938	2.12251
$256$	0	0
$257$	−32.0134	−1.99694	−0.998472	−	0.0552644i	$-0.982400\pi$
$257$	−32.0134	−1.99694	−0.998472	+	0.0552644i	$0.982400\pi$
$258$	0	0
$259$	−40.3144	−2.50501
$260$	0	0
$261$	−10.8337	−0.670588
$262$	0	0
$263$	−10.5677	−0.651635	−0.325817	−	0.945433i	$-0.605639\pi$
$263$	−10.5677	−0.651635	−0.325817	+	0.945433i	$0.605639\pi$
$264$	0	0
$265$	−10.5457	−0.647818
$266$	0	0
$267$	−27.8967	−1.70725
$268$	0	0
$269$	−25.8886	−1.57846	−0.789228	−	0.614100i	$-0.789519\pi$
$269$	−25.8886	−1.57846	−0.789228	+	0.614100i	$0.789519\pi$
$270$	0	0
$271$	−9.75807	−0.592761	−0.296380	−	0.955070i	$-0.595780\pi$
$271$	−9.75807	−0.592761	−0.296380	+	0.955070i	$0.595780\pi$
$272$	0	0
$273$	14.9976	0.907694
$274$	0	0
$275$	−1.26629	−0.0763604
$276$	0	0
$277$	7.32901	0.440358	0.220179	−	0.975460i	$-0.429336\pi$
$277$	7.32901	0.440358	0.220179	+	0.975460i	$0.429336\pi$
$278$	0	0
$279$	8.29507	0.496613
$280$	0	0
$281$	0.0135689	0.000809454 0	0.000404727	−	1.00000i	$-0.499871\pi$
$281$	0.0135689	0.000809454 0	0.000404727		1.00000i	$0.499871\pi$
$282$	0	0
$283$	−11.3812	−0.676542	−0.338271	−	0.941049i	$-0.609842\pi$
$283$	−11.3812	−0.676542	−0.338271	+	0.941049i	$0.609842\pi$
$284$	0	0
$285$	−4.55722	−0.269947
$286$	0	0
$287$	46.2935	2.73262
$288$	0	0
$289$	38.3146	2.25380
$290$	0	0
$291$	12.6716	0.742821
$292$	0	0
$293$	16.6159	0.970714	0.485357	−	0.874316i	$-0.338690\pi$
$293$	16.6159	0.970714	0.485357	+	0.874316i	$0.338690\pi$
$294$	0	0
$295$	−20.5633	−1.19724
$296$	0	0
$297$	7.91478	0.459262
$298$	0	0
$299$	10.1912	0.589373
$300$	0	0
$301$	−5.66803	−0.326700
$302$	0	0
$303$	−32.9566	−1.89331
$304$	0	0
$305$	0.164070	0.00939465
$306$	0	0
$307$	5.25463	0.299897	0.149949	−	0.988694i	$-0.452089\pi$
$307$	5.25463	0.299897	0.149949	+	0.988694i	$0.452089\pi$
$308$	0	0
$309$	−32.9698	−1.87558
$310$	0	0
$311$	30.5038	1.72971	0.864857	−	0.502018i	$-0.167409\pi$
$311$	30.5038	1.72971	0.864857	+	0.502018i	$0.167409\pi$
$312$	0	0
$313$	14.1889	0.802006	0.401003	−	0.916077i	$-0.368662\pi$
$313$	14.1889	0.802006	0.401003	+	0.916077i	$0.368662\pi$
$314$	0	0
$315$	−14.7842	−0.832993
$316$	0	0
$317$	−34.8894	−1.95958	−0.979792	−	0.200020i	$-0.935899\pi$
$317$	−34.8894	−1.95958	−0.979792	+	0.200020i	$0.935899\pi$
$318$	0	0
$319$	−17.8494	−0.999372
$320$	0	0
$321$	26.8208	1.49699
$322$	0	0
$323$	−7.43738	−0.413827
$324$	0	0
$325$	−0.773051	−0.0428812
$326$	0	0
$327$	−18.7107	−1.03470
$328$	0	0
$329$	−57.7698	−3.18495
$330$	0	0
$331$	−22.4092	−1.23172	−0.615860	−	0.787856i	$-0.711191\pi$
$331$	−22.4092	−1.23172	−0.615860	+	0.787856i	$0.711191\pi$
$332$	0	0
$333$	14.7696	0.809371
$334$	0	0
$335$	18.3384	1.00193
$336$	0	0
$337$	−24.8371	−1.35296	−0.676481	−	0.736460i	$-0.736496\pi$
$337$	−24.8371	−1.35296	−0.676481	+	0.736460i	$0.736496\pi$
$338$	0	0
$339$	6.13783	0.333361
$340$	0	0
$341$	13.6668	0.740098
$342$	0	0
$343$	−21.7158	−1.17254
$344$	0	0
$345$	−28.9269	−1.55737
$346$	0	0
$347$	−1.89664	−0.101817	−0.0509086	−	0.998703i	$-0.516212\pi$
$347$	−1.89664	−0.101817	−0.0509086	+	0.998703i	$0.516212\pi$
$348$	0	0
$349$	−12.1821	−0.652095	−0.326048	−	0.945353i	$-0.605717\pi$
$349$	−12.1821	−0.652095	−0.326048	+	0.945353i	$0.605717\pi$
$350$	0	0
$351$	4.83184	0.257904
$352$	0	0
$353$	2.72479	0.145026	0.0725130	−	0.997367i	$-0.476898\pi$
$353$	2.72479	0.145026	0.0725130	+	0.997367i	$0.476898\pi$
$354$	0	0
$355$	−22.2701	−1.18198
$356$	0	0
$357$	−69.4731	−3.67690
$358$	0	0
$359$	−30.6161	−1.61586	−0.807929	−	0.589280i	$-0.799412\pi$
$359$	−30.6161	−1.61586	−0.807929	+	0.589280i	$0.799412\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	8.75407	0.459469
$364$	0	0
$365$	−13.5292	−0.708151
$366$	0	0
$367$	−31.4506	−1.64171	−0.820855	−	0.571137i	$-0.806503\pi$
$367$	−31.4506	−1.64171	−0.820855	+	0.571137i	$0.806503\pi$
$368$	0	0
$369$	−16.9602	−0.882911
$370$	0	0
$371$	21.6159	1.12224
$372$	0	0
$373$	−18.2606	−0.945500	−0.472750	−	0.881196i	$-0.656739\pi$
$373$	−18.2606	−0.945500	−0.472750	+	0.881196i	$0.656739\pi$
$374$	0	0
$375$	24.9804	1.28998
$376$	0	0
$377$	−10.8967	−0.561210
$378$	0	0
$379$	18.2307	0.936448	0.468224	−	0.883610i	$-0.344894\pi$
$379$	18.2307	0.936448	0.468224	+	0.883610i	$0.344894\pi$
$380$	0	0
$381$	−27.8808	−1.42838
$382$	0	0
$383$	7.22982	0.369426	0.184713	−	0.982792i	$-0.440864\pi$
$383$	7.22982	0.369426	0.184713	+	0.982792i	$0.440864\pi$
$384$	0	0
$385$	−24.3581	−1.24140
$386$	0	0
$387$	2.07655	0.105557
$388$	0	0
$389$	17.9081	0.907975	0.453987	−	0.891008i	$-0.350001\pi$
$389$	17.9081	0.907975	0.453987	+	0.891008i	$0.350001\pi$
$390$	0	0
$391$	−47.2086	−2.38744
$392$	0	0
$393$	21.5355	1.08632
$394$	0	0
$395$	−2.12568	−0.106955
$396$	0	0
$397$	−5.98507	−0.300382	−0.150191	−	0.988657i	$-0.547989\pi$
$397$	−5.98507	−0.300382	−0.150191	+	0.988657i	$0.547989\pi$
$398$	0	0
$399$	9.34107	0.467638
$400$	0	0
$401$	−8.00498	−0.399750	−0.199875	−	0.979821i	$-0.564054\pi$
$401$	−8.00498	−0.399750	−0.199875	+	0.979821i	$0.564054\pi$
$402$	0	0
$403$	8.34333	0.415611
$404$	0	0
$405$	−23.8942	−1.18731
$406$	0	0
$407$	24.3341	1.20620
$408$	0	0
$409$	−17.5025	−0.865445	−0.432722	−	0.901527i	$-0.642447\pi$
$409$	−17.5025	−0.865445	−0.432722	+	0.901527i	$0.642447\pi$
$410$	0	0
$411$	−16.4529	−0.811560
$412$	0	0
$413$	42.1491	2.07402
$414$	0	0
$415$	−0.572245	−0.0280904
$416$	0	0
$417$	19.8209	0.970636
$418$	0	0
$419$	22.1126	1.08027	0.540136	−	0.841578i	$-0.318373\pi$
$419$	22.1126	1.08027	0.540136	+	0.841578i	$0.318373\pi$
$420$	0	0
$421$	13.3413	0.650213	0.325106	−	0.945677i	$-0.394600\pi$
$421$	13.3413	0.650213	0.325106	+	0.945677i	$0.394600\pi$
$422$	0	0
$423$	21.1646	1.02906
$424$	0	0
$425$	3.58100	0.173704
$426$	0	0
$427$	−0.336300	−0.0162747
$428$	0	0
$429$	−9.05268	−0.437067
$430$	0	0
$431$	7.71926	0.371824	0.185912	−	0.982566i	$-0.440476\pi$
$431$	7.71926	0.371824	0.185912	+	0.982566i	$0.440476\pi$
$432$	0	0
$433$	−7.36989	−0.354174	−0.177087	−	0.984195i	$-0.556667\pi$
$433$	−7.36989	−0.354174	−0.177087	+	0.984195i	$0.556667\pi$
$434$	0	0
$435$	30.9294	1.48295
$436$	0	0
$437$	6.34749	0.303641
$438$	0	0
$439$	3.26857	0.156000	0.0780001	−	0.996953i	$-0.475147\pi$
$439$	3.26857	0.156000	0.0780001	+	0.996953i	$0.475147\pi$
$440$	0	0
$441$	19.1297	0.910937
$442$	0	0
$443$	−37.7130	−1.79180	−0.895900	−	0.444256i	$-0.853468\pi$
$443$	−37.7130	−1.79180	−0.895900	+	0.444256i	$0.853468\pi$
$444$	0	0
$445$	27.6597	1.31120
$446$	0	0
$447$	26.4271	1.24996
$448$	0	0
$449$	−40.1857	−1.89648	−0.948240	−	0.317555i	$-0.897138\pi$
$449$	−40.1857	−1.89648	−0.948240	+	0.317555i	$0.897138\pi$
$450$	0	0
$451$	−27.9432	−1.31579
$452$	0	0
$453$	−11.7214	−0.550721
$454$	0	0
$455$	−14.8702	−0.697125
$456$	0	0
$457$	16.1353	0.754776	0.377388	−	0.926055i	$-0.376822\pi$
$457$	16.1353	0.754776	0.377388	+	0.926055i	$0.376822\pi$
$458$	0	0
$459$	−22.3825	−1.04472
$460$	0	0
$461$	−38.2154	−1.77987	−0.889934	−	0.456090i	$-0.849250\pi$
$461$	−38.2154	−1.77987	−0.889934	+	0.456090i	$0.849250\pi$
$462$	0	0
$463$	−24.8069	−1.15288	−0.576438	−	0.817141i	$-0.695558\pi$
$463$	−24.8069	−1.15288	−0.576438	+	0.817141i	$0.695558\pi$
$464$	0	0
$465$	−23.6819	−1.09822
$466$	0	0
$467$	6.85118	0.317035	0.158517	−	0.987356i	$-0.449329\pi$
$467$	6.85118	0.317035	0.158517	+	0.987356i	$0.449329\pi$
$468$	0	0
$469$	−37.5887	−1.73569
$470$	0	0
$471$	11.3253	0.521841
$472$	0	0
$473$	3.42128	0.157311
$474$	0	0
$475$	−0.481487	−0.0220921
$476$	0	0
$477$	−7.91922	−0.362596
$478$	0	0
$479$	−24.1668	−1.10421	−0.552105	−	0.833774i	$-0.686175\pi$
$479$	−24.1668	−1.10421	−0.552105	+	0.833774i	$0.686175\pi$
$480$	0	0
$481$	14.8556	0.677356
$482$	0	0
$483$	59.2923	2.69789
$484$	0	0
$485$	−12.5640	−0.570500
$486$	0	0
$487$	−31.0689	−1.40787	−0.703934	−	0.710265i	$-0.748575\pi$
$487$	−31.0689	−1.40787	−0.703934	+	0.710265i	$0.748575\pi$
$488$	0	0
$489$	48.2508	2.18197
$490$	0	0
$491$	−0.377765	−0.0170483	−0.00852414	−	0.999964i	$-0.502713\pi$
$491$	−0.377765	−0.0170483	−0.00852414	+	0.999964i	$0.502713\pi$
$492$	0	0
$493$	50.4768	2.27336
$494$	0	0
$495$	8.92386	0.401098
$496$	0	0
$497$	45.6477	2.04758
$498$	0	0
$499$	−22.0949	−0.989101	−0.494551	−	0.869149i	$-0.664667\pi$
$499$	−22.0949	−0.989101	−0.494551	+	0.869149i	$0.664667\pi$
$500$	0	0
$501$	52.6832	2.35371
$502$	0	0
$503$	24.9086	1.11062	0.555309	−	0.831644i	$-0.312600\pi$
$503$	24.9086	1.11062	0.555309	+	0.831644i	$0.312600\pi$
$504$	0	0
$505$	32.6767	1.45409
$506$	0	0
$507$	22.3441	0.992335
$508$	0	0
$509$	−1.71529	−0.0760290	−0.0380145	−	0.999277i	$-0.512103\pi$
$509$	−1.71529	−0.0760290	−0.0380145	+	0.999277i	$0.512103\pi$
$510$	0	0
$511$	27.7312	1.22676
$512$	0	0
$513$	3.00946	0.132871
$514$	0	0
$515$	32.6897	1.44048
$516$	0	0
$517$	34.8704	1.53360
$518$	0	0
$519$	−27.4493	−1.20489
$520$	0	0
$521$	−16.6626	−0.730002	−0.365001	−	0.931007i	$-0.618931\pi$
$521$	−16.6626	−0.730002	−0.365001	+	0.931007i	$0.618931\pi$
$522$	0	0
$523$	−2.13944	−0.0935512	−0.0467756	−	0.998905i	$-0.514895\pi$
$523$	−2.13944	−0.0935512	−0.0467756	+	0.998905i	$0.514895\pi$
$524$	0	0
$525$	−4.49760	−0.196291
$526$	0	0
$527$	−38.6487	−1.68357
$528$	0	0
$529$	17.2906	0.751765
$530$	0	0
$531$	−15.4418	−0.670118
$532$	0	0
$533$	−17.0588	−0.738901
$534$	0	0
$535$	−26.5929	−1.14971
$536$	0	0
$537$	4.52557	0.195293
$538$	0	0
$539$	31.5176	1.35756
$540$	0	0
$541$	26.5841	1.14294	0.571469	−	0.820624i	$-0.306374\pi$
$541$	26.5841	1.14294	0.571469	+	0.820624i	$0.306374\pi$
$542$	0	0
$543$	−5.81626	−0.249600
$544$	0	0
$545$	18.5517	0.794669
$546$	0	0
$547$	−19.5487	−0.835844	−0.417922	−	0.908483i	$-0.637241\pi$
$547$	−19.5487	−0.835844	−0.417922	+	0.908483i	$0.637241\pi$
$548$	0	0
$549$	0.123207	0.00525837
$550$	0	0
$551$	−6.78691	−0.289132
$552$	0	0
$553$	4.35707	0.185281
$554$	0	0
$555$	−42.1663	−1.78986
$556$	0	0
$557$	−15.5728	−0.659839	−0.329919	−	0.944009i	$-0.607022\pi$
$557$	−15.5728	−0.659839	−0.329919	+	0.944009i	$0.607022\pi$
$558$	0	0
$559$	2.08863	0.0883398
$560$	0	0
$561$	41.9346	1.77048
$562$	0	0
$563$	−14.7846	−0.623098	−0.311549	−	0.950230i	$-0.600848\pi$
$563$	−14.7846	−0.623098	−0.311549	+	0.950230i	$0.600848\pi$
$564$	0	0
$565$	−6.08569	−0.256027
$566$	0	0
$567$	48.9766	2.05683
$568$	0	0
$569$	−21.5807	−0.904710	−0.452355	−	0.891838i	$-0.649416\pi$
$569$	−21.5807	−0.904710	−0.452355	+	0.891838i	$0.649416\pi$
$570$	0	0
$571$	15.2869	0.639739	0.319869	−	0.947462i	$-0.396361\pi$
$571$	15.2869	0.639739	0.319869	+	0.947462i	$0.396361\pi$
$572$	0	0
$573$	−33.9714	−1.41918
$574$	0	0
$575$	−3.05623	−0.127454
$576$	0	0
$577$	−23.4394	−0.975797	−0.487898	−	0.872900i	$-0.662236\pi$
$577$	−23.4394	−0.975797	−0.487898	+	0.872900i	$0.662236\pi$
$578$	0	0
$579$	−10.3016	−0.428121
$580$	0	0
$581$	1.17295	0.0486620
$582$	0	0
$583$	−13.0475	−0.540374
$584$	0	0
$585$	5.44786	0.225241
$586$	0	0
$587$	−20.0445	−0.827327	−0.413663	−	0.910430i	$-0.635751\pi$
$587$	−20.0445	−0.827327	−0.413663	+	0.910430i	$0.635751\pi$
$588$	0	0
$589$	5.19656	0.214120
$590$	0	0
$591$	21.9170	0.901543
$592$	0	0
$593$	15.9108	0.653377	0.326689	−	0.945132i	$-0.394067\pi$
$593$	15.9108	0.653377	0.326689	+	0.945132i	$0.394067\pi$
$594$	0	0
$595$	68.8830	2.82393
$596$	0	0
$597$	5.47568	0.224105
$598$	0	0
$599$	−41.0020	−1.67530	−0.837649	−	0.546209i	$-0.816070\pi$
$599$	−41.0020	−1.67530	−0.837649	+	0.546209i	$0.816070\pi$
$600$	0	0
$601$	−25.3062	−1.03226	−0.516131	−	0.856509i	$-0.672628\pi$
$601$	−25.3062	−1.03226	−0.516131	+	0.856509i	$0.672628\pi$
$602$	0	0
$603$	13.7711	0.560802
$604$	0	0
$605$	−8.67971	−0.352880
$606$	0	0
$607$	23.8276	0.967134	0.483567	−	0.875307i	$-0.339341\pi$
$607$	23.8276	0.967134	0.483567	+	0.875307i	$0.339341\pi$
$608$	0	0
$609$	−63.3970	−2.56898
$610$	0	0
$611$	21.2878	0.861212
$612$	0	0
$613$	20.2971	0.819792	0.409896	−	0.912132i	$-0.365565\pi$
$613$	20.2971	0.819792	0.409896	+	0.912132i	$0.365565\pi$
$614$	0	0
$615$	48.4201	1.95249
$616$	0	0
$617$	23.9811	0.965442	0.482721	−	0.875774i	$-0.339649\pi$
$617$	23.9811	0.965442	0.482721	+	0.875774i	$0.339649\pi$
$618$	0	0
$619$	10.7967	0.433958	0.216979	−	0.976176i	$-0.430380\pi$
$619$	10.7967	0.433958	0.216979	+	0.976176i	$0.430380\pi$
$620$	0	0
$621$	19.1025	0.766557
$622$	0	0
$623$	−56.6949	−2.27143
$624$	0	0
$625$	−22.3607	−0.894429
$626$	0	0
$627$	−5.63836	−0.225175
$628$	0	0
$629$	−68.8154	−2.74385
$630$	0	0
$631$	−29.4905	−1.17400	−0.587000	−	0.809587i	$-0.699691\pi$
$631$	−29.4905	−1.17400	−0.587000	+	0.809587i	$0.699691\pi$
$632$	0	0
$633$	53.3358	2.11991
$634$	0	0
$635$	27.6440	1.09702
$636$	0	0
$637$	19.2410	0.762355
$638$	0	0
$639$	−16.7236	−0.661575
$640$	0	0
$641$	−11.8446	−0.467833	−0.233916	−	0.972257i	$-0.575154\pi$
$641$	−11.8446	−0.467833	−0.233916	+	0.972257i	$0.575154\pi$
$642$	0	0
$643$	21.3846	0.843327	0.421663	−	0.906752i	$-0.361446\pi$
$643$	21.3846	0.843327	0.421663	+	0.906752i	$0.361446\pi$
$644$	0	0
$645$	−5.92841	−0.233431
$646$	0	0
$647$	37.5431	1.47597	0.737985	−	0.674817i	$-0.235777\pi$
$647$	37.5431	1.47597	0.737985	+	0.674817i	$0.235777\pi$
$648$	0	0
$649$	−25.4416	−0.998671
$650$	0	0
$651$	48.5414	1.90249
$652$	0	0
$653$	−35.7235	−1.39797	−0.698984	−	0.715137i	$-0.746364\pi$
$653$	−35.7235	−1.39797	−0.698984	+	0.715137i	$0.746364\pi$
$654$	0	0
$655$	−21.3526	−0.834314
$656$	0	0
$657$	−10.1597	−0.396366
$658$	0	0
$659$	−32.7924	−1.27741	−0.638706	−	0.769451i	$-0.720530\pi$
$659$	−32.7924	−1.27741	−0.638706	+	0.769451i	$0.720530\pi$
$660$	0	0
$661$	−15.8731	−0.617391	−0.308696	−	0.951161i	$-0.599892\pi$
$661$	−15.8731	−0.617391	−0.308696	+	0.951161i	$0.599892\pi$
$662$	0	0
$663$	25.6004	0.994236
$664$	0	0
$665$	−9.26173	−0.359154
$666$	0	0
$667$	−43.0798	−1.66806
$668$	0	0
$669$	−40.4063	−1.56220
$670$	0	0
$671$	0.202994	0.00783650
$672$	0	0
$673$	16.0180	0.617448	0.308724	−	0.951152i	$-0.400098\pi$
$673$	16.0180	0.617448	0.308724	+	0.951152i	$0.400098\pi$
$674$	0	0
$675$	−1.44901	−0.0557726
$676$	0	0
$677$	28.6912	1.10269	0.551346	−	0.834277i	$-0.314114\pi$
$677$	28.6912	1.10269	0.551346	+	0.834277i	$0.314114\pi$
$678$	0	0
$679$	25.7527	0.988298
$680$	0	0
$681$	−27.9951	−1.07277
$682$	0	0
$683$	38.7196	1.48157	0.740783	−	0.671744i	$-0.234455\pi$
$683$	38.7196	1.48157	0.740783	+	0.671744i	$0.234455\pi$
$684$	0	0
$685$	16.3131	0.623292
$686$	0	0
$687$	−6.86443	−0.261894
$688$	0	0
$689$	−7.96530	−0.303454
$690$	0	0
$691$	−40.1229	−1.52635	−0.763174	−	0.646193i	$-0.776360\pi$
$691$	−40.1229	−1.52635	−0.763174	+	0.646193i	$0.776360\pi$
$692$	0	0
$693$	−18.2915	−0.694836
$694$	0	0
$695$	−19.6526	−0.745465
$696$	0	0
$697$	79.0215	2.99315
$698$	0	0
$699$	−16.5720	−0.626810
$700$	0	0
$701$	−34.4368	−1.30066	−0.650329	−	0.759652i	$-0.725369\pi$
$701$	−34.4368	−1.30066	−0.650329	+	0.759652i	$0.725369\pi$
$702$	0	0
$703$	9.25264	0.348970
$704$	0	0
$705$	−60.4236	−2.27569
$706$	0	0
$707$	−66.9784	−2.51898
$708$	0	0
$709$	−5.82146	−0.218630	−0.109315	−	0.994007i	$-0.534866\pi$
$709$	−5.82146	−0.218630	−0.109315	+	0.994007i	$0.534866\pi$
$710$	0	0
$711$	−1.59626	−0.0598645
$712$	0	0
$713$	32.9851	1.23530
$714$	0	0
$715$	8.97578	0.335675
$716$	0	0
$717$	−36.6236	−1.36773
$718$	0	0
$719$	30.6248	1.14211	0.571055	−	0.820911i	$-0.306534\pi$
$719$	30.6248	1.14211	0.571055	+	0.820911i	$0.306534\pi$
$720$	0	0
$721$	−67.0050	−2.49540
$722$	0	0
$723$	52.4992	1.95247
$724$	0	0
$725$	3.26781	0.121363
$726$	0	0
$727$	1.31811	0.0488860	0.0244430	−	0.999701i	$-0.492219\pi$
$727$	1.31811	0.0488860	0.0244430	+	0.999701i	$0.492219\pi$
$728$	0	0
$729$	1.41267	0.0523211
$730$	0	0
$731$	−9.67515	−0.357848
$732$	0	0
$733$	4.24944	0.156957	0.0784783	−	0.996916i	$-0.474994\pi$
$733$	4.24944	0.156957	0.0784783	+	0.996916i	$0.474994\pi$
$734$	0	0
$735$	−54.6139	−2.01446
$736$	0	0
$737$	22.6889	0.835758
$738$	0	0
$739$	38.8400	1.42875	0.714375	−	0.699763i	$-0.246711\pi$
$739$	38.8400	1.42875	0.714375	+	0.699763i	$0.246711\pi$
$740$	0	0
$741$	−3.44212	−0.126450
$742$	0	0
$743$	33.1921	1.21770	0.608850	−	0.793285i	$-0.291631\pi$
$743$	33.1921	1.21770	0.608850	+	0.793285i	$0.291631\pi$
$744$	0	0
$745$	−26.2027	−0.959992
$746$	0	0
$747$	−0.429723	−0.0157227
$748$	0	0
$749$	54.5083	1.99169
$750$	0	0
$751$	−12.0409	−0.439380	−0.219690	−	0.975570i	$-0.570505\pi$
$751$	−12.0409	−0.439380	−0.219690	+	0.975570i	$0.570505\pi$
$752$	0	0
$753$	35.6638	1.29966
$754$	0	0
$755$	11.6219	0.422963
$756$	0	0
$757$	−4.07712	−0.148185	−0.0740927	−	0.997251i	$-0.523606\pi$
$757$	−4.07712	−0.148185	−0.0740927	+	0.997251i	$0.523606\pi$
$758$	0	0
$759$	−35.7894	−1.29907
$760$	0	0
$761$	21.7769	0.789411	0.394706	−	0.918808i	$-0.370847\pi$
$761$	21.7769	0.789411	0.394706	+	0.918808i	$0.370847\pi$
$762$	0	0
$763$	−38.0260	−1.37663
$764$	0	0
$765$	−25.2361	−0.912412
$766$	0	0
$767$	−15.5317	−0.560816
$768$	0	0
$769$	46.6151	1.68098	0.840491	−	0.541825i	$-0.182267\pi$
$769$	46.6151	1.68098	0.840491	+	0.541825i	$0.182267\pi$
$770$	0	0
$771$	68.6333	2.47177
$772$	0	0
$773$	12.6364	0.454498	0.227249	−	0.973837i	$-0.427027\pi$
$773$	12.6364	0.454498	0.227249	+	0.973837i	$0.427027\pi$
$774$	0	0
$775$	−2.50207	−0.0898771
$776$	0	0
$777$	86.4296	3.10064
$778$	0	0
$779$	−10.6249	−0.380677
$780$	0	0
$781$	−27.5534	−0.985939
$782$	0	0
$783$	−20.4249	−0.729927
$784$	0	0
$785$	−11.2291	−0.400783
$786$	0	0
$787$	−6.47695	−0.230878	−0.115439	−	0.993315i	$-0.536828\pi$
$787$	−6.47695	−0.230878	−0.115439	+	0.993315i	$0.536828\pi$
$788$	0	0
$789$	22.6561	0.806577
$790$	0	0
$791$	12.4740	0.443525
$792$	0	0
$793$	0.123924	0.00440068
$794$	0	0
$795$	22.6088	0.801853
$796$	0	0
$797$	−31.0359	−1.09935	−0.549675	−	0.835379i	$-0.685248\pi$
$797$	−31.0359	−1.09935	−0.549675	+	0.835379i	$0.685248\pi$
$798$	0	0
$799$	−98.6112	−3.48861
$800$	0	0
$801$	20.7708	0.733902
$802$	0	0
$803$	−16.7388	−0.590701
$804$	0	0
$805$	−58.7887	−2.07203
$806$	0	0
$807$	55.5023	1.95377
$808$	0	0
$809$	−27.6555	−0.972316	−0.486158	−	0.873871i	$-0.661602\pi$
$809$	−27.6555	−0.972316	−0.486158	+	0.873871i	$0.661602\pi$
$810$	0	0
$811$	12.3852	0.434905	0.217452	−	0.976071i	$-0.430225\pi$
$811$	12.3852	0.434905	0.217452	+	0.976071i	$0.430225\pi$
$812$	0	0
$813$	20.9202	0.733704
$814$	0	0
$815$	−47.8409	−1.67579
$816$	0	0
$817$	1.30088	0.0455121
$818$	0	0
$819$	−11.1666	−0.390194
$820$	0	0
$821$	−42.6582	−1.48878	−0.744391	−	0.667744i	$-0.767260\pi$
$821$	−42.6582	−1.48878	−0.744391	+	0.667744i	$0.767260\pi$
$822$	0	0
$823$	−31.8548	−1.11039	−0.555194	−	0.831721i	$-0.687356\pi$
$823$	−31.8548	−1.11039	−0.555194	+	0.831721i	$0.687356\pi$
$824$	0	0
$825$	2.71480	0.0945171
$826$	0	0
$827$	−19.4568	−0.676578	−0.338289	−	0.941042i	$-0.609848\pi$
$827$	−19.4568	−0.676578	−0.338289	+	0.941042i	$0.609848\pi$
$828$	0	0
$829$	8.11597	0.281879	0.140940	−	0.990018i	$-0.454988\pi$
$829$	8.11597	0.281879	0.140940	+	0.990018i	$0.454988\pi$
$830$	0	0
$831$	−15.7126	−0.545064
$832$	0	0
$833$	−89.1298	−3.08816
$834$	0	0
$835$	−52.2357	−1.80769
$836$	0	0
$837$	15.6388	0.540556
$838$	0	0
$839$	−28.4402	−0.981864	−0.490932	−	0.871198i	$-0.663344\pi$
$839$	−28.4402	−0.981864	−0.490932	+	0.871198i	$0.663344\pi$
$840$	0	0
$841$	17.0621	0.588349
$842$	0	0
$843$	−0.0290903	−0.00100192
$844$	0	0
$845$	−22.1543	−0.762130
$846$	0	0
$847$	17.7911	0.611308
$848$	0	0
$849$	24.4000	0.837407
$850$	0	0
$851$	58.7310	2.01327
$852$	0	0
$853$	55.4942	1.90008	0.950042	−	0.312122i	$-0.101040\pi$
$853$	55.4942	1.90008	0.950042	+	0.312122i	$0.101040\pi$
$854$	0	0
$855$	3.39314	0.116043
$856$	0	0
$857$	28.3573	0.968667	0.484334	−	0.874883i	$-0.339062\pi$
$857$	28.3573	0.968667	0.484334	+	0.874883i	$0.339062\pi$
$858$	0	0
$859$	34.2990	1.17027	0.585133	−	0.810937i	$-0.301042\pi$
$859$	34.2990	1.17027	0.585133	+	0.810937i	$0.301042\pi$
$860$	0	0
$861$	−99.2481	−3.38237
$862$	0	0
$863$	23.1331	0.787459	0.393729	−	0.919226i	$-0.371185\pi$
$863$	23.1331	0.787459	0.393729	+	0.919226i	$0.371185\pi$
$864$	0	0
$865$	27.2162	0.925378
$866$	0	0
$867$	−82.1422	−2.78969
$868$	0	0
$869$	−2.62997	−0.0892156
$870$	0	0
$871$	13.8512	0.469330
$872$	0	0
$873$	−9.43480	−0.319320
$874$	0	0
$875$	50.7680	1.71627
$876$	0	0
$877$	17.7369	0.598933	0.299467	−	0.954107i	$-0.403191\pi$
$877$	17.7369	0.598933	0.299467	+	0.954107i	$0.403191\pi$
$878$	0	0
$879$	−35.6227	−1.20153
$880$	0	0
$881$	10.9644	0.369401	0.184700	−	0.982795i	$-0.440869\pi$
$881$	10.9644	0.369401	0.184700	+	0.982795i	$0.440869\pi$
$882$	0	0
$883$	49.5814	1.66855	0.834274	−	0.551350i	$-0.185887\pi$
$883$	49.5814	1.66855	0.834274	+	0.551350i	$0.185887\pi$
$884$	0	0
$885$	44.0854	1.48191
$886$	0	0
$887$	−26.6259	−0.894011	−0.447005	−	0.894531i	$-0.647510\pi$
$887$	−26.6259	−0.894011	−0.447005	+	0.894531i	$0.647510\pi$
$888$	0	0
$889$	−56.6627	−1.90041
$890$	0	0
$891$	−29.5628	−0.990390
$892$	0	0
$893$	13.2589	0.443691
$894$	0	0
$895$	−4.48713	−0.149988
$896$	0	0
$897$	−21.8488	−0.729511
$898$	0	0
$899$	−35.2685	−1.17627
$900$	0	0
$901$	36.8976	1.22924
$902$	0	0
$903$	12.1516	0.404381
$904$	0	0
$905$	5.76686	0.191697
$906$	0	0
$907$	−34.6956	−1.15205	−0.576025	−	0.817432i	$-0.695397\pi$
$907$	−34.6956	−1.15205	−0.576025	+	0.817432i	$0.695397\pi$
$908$	0	0
$909$	24.5383	0.813884
$910$	0	0
$911$	−31.3442	−1.03848	−0.519240	−	0.854629i	$-0.673785\pi$
$911$	−31.3442	−1.03848	−0.519240	+	0.854629i	$0.673785\pi$
$912$	0	0
$913$	−0.708002	−0.0234315
$914$	0	0
$915$	−0.351749	−0.0116285
$916$	0	0
$917$	43.7670	1.44531
$918$	0	0
$919$	−5.00119	−0.164974	−0.0824871	−	0.996592i	$-0.526286\pi$
$919$	−5.00119	−0.164974	−0.0824871	+	0.996592i	$0.526286\pi$
$920$	0	0
$921$	−11.2653	−0.371206
$922$	0	0
$923$	−16.8209	−0.553666
$924$	0	0
$925$	−4.45502	−0.146480
$926$	0	0
$927$	24.5481	0.806265
$928$	0	0
$929$	19.5074	0.640018	0.320009	−	0.947414i	$-0.396314\pi$
$929$	19.5074	0.640018	0.320009	+	0.947414i	$0.396314\pi$
$930$	0	0
$931$	11.9840	0.392761
$932$	0	0
$933$	−65.3969	−2.14100
$934$	0	0
$935$	−41.5784	−1.35976
$936$	0	0
$937$	4.20144	0.137255	0.0686275	−	0.997642i	$-0.478138\pi$
$937$	4.20144	0.137255	0.0686275	+	0.997642i	$0.478138\pi$
$938$	0	0
$939$	−30.4195	−0.992704
$940$	0	0
$941$	−11.0142	−0.359054	−0.179527	−	0.983753i	$-0.557457\pi$
$941$	−11.0142	−0.359054	−0.179527	+	0.983753i	$0.557457\pi$
$942$	0	0
$943$	−67.4415	−2.19620
$944$	0	0
$945$	−27.8728	−0.906701
$946$	0	0
$947$	26.2525	0.853091	0.426546	−	0.904466i	$-0.359730\pi$
$947$	26.2525	0.853091	0.426546	+	0.904466i	$0.359730\pi$
$948$	0	0
$949$	−10.2188	−0.331715
$950$	0	0
$951$	74.7990	2.42552
$952$	0	0
$953$	20.3400	0.658879	0.329439	−	0.944177i	$-0.393140\pi$
$953$	20.3400	0.658879	0.329439	+	0.944177i	$0.393140\pi$
$954$	0	0
$955$	33.6829	1.08995
$956$	0	0
$957$	38.2670	1.23700
$958$	0	0
$959$	−33.4375	−1.07975
$960$	0	0
$961$	−3.99581	−0.128897
$962$	0	0
$963$	−19.9698	−0.643517
$964$	0	0
$965$	10.2141	0.328804
$966$	0	0
$967$	27.8742	0.896375	0.448188	−	0.893940i	$-0.352070\pi$
$967$	27.8742	0.896375	0.448188	+	0.893940i	$0.352070\pi$
$968$	0	0
$969$	15.9449	0.512224
$970$	0	0
$971$	36.8990	1.18415	0.592073	−	0.805885i	$-0.298310\pi$
$971$	36.8990	1.18415	0.592073	+	0.805885i	$0.298310\pi$
$972$	0	0
$973$	40.2825	1.29140
$974$	0	0
$975$	1.65734	0.0530773
$976$	0	0
$977$	12.0080	0.384169	0.192084	−	0.981378i	$-0.438475\pi$
$977$	12.0080	0.384169	0.192084	+	0.981378i	$0.438475\pi$
$978$	0	0
$979$	34.2216	1.09373
$980$	0	0
$981$	13.9313	0.444792
$982$	0	0
$983$	10.3993	0.331687	0.165844	−	0.986152i	$-0.446965\pi$
$983$	10.3993	0.331687	0.165844	+	0.986152i	$0.446965\pi$
$984$	0	0
$985$	−21.7308	−0.692401
$986$	0	0
$987$	123.852	3.94225
$988$	0	0
$989$	8.25733	0.262568
$990$	0	0
$991$	−23.7903	−0.755724	−0.377862	−	0.925862i	$-0.623341\pi$
$991$	−23.7903	−0.755724	−0.377862	+	0.925862i	$0.623341\pi$
$992$	0	0
$993$	48.0428	1.52459
$994$	0	0
$995$	−5.42917	−0.172116
$996$	0	0
$997$	−15.9782	−0.506035	−0.253018	−	0.967462i	$-0.581423\pi$
$997$	−15.9782	−0.506035	−0.253018	+	0.967462i	$0.581423\pi$
$998$	0	0
$999$	27.8454	0.880990

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6004.2.a.g.1.5	✓	27

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6004.2.a.g.1.5	✓	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)

\(f(q)\)	\(=\)	\(q-2.14389 q^{3} +2.12568 q^{5} -4.35707 q^{7} +1.59626 q^{9} +O(q^{10})\)	\(q-2.14389 q^{3} +2.12568 q^{5} -4.35707 q^{7} +1.59626 q^{9} +2.62997 q^{11} +1.60555 q^{13} -4.55722 q^{15} -7.43738 q^{17} +1.00000 q^{19} +9.34107 q^{21} +6.34749 q^{23} -0.481487 q^{25} +3.00946 q^{27} -6.78691 q^{29} +5.19656 q^{31} -5.63836 q^{33} -9.26173 q^{35} +9.25264 q^{37} -3.44212 q^{39} -10.6249 q^{41} +1.30088 q^{43} +3.39314 q^{45} +13.2589 q^{47} +11.9840 q^{49} +15.9449 q^{51} -4.96110 q^{53} +5.59047 q^{55} -2.14389 q^{57} -9.67374 q^{59} +0.0771850 q^{61} -6.95502 q^{63} +3.41289 q^{65} +8.62707 q^{67} -13.6083 q^{69} -10.4767 q^{71} -6.36465 q^{73} +1.03225 q^{75} -11.4589 q^{77} -1.00000 q^{79} -11.2407 q^{81} -0.269206 q^{83} -15.8095 q^{85} +14.5504 q^{87} +13.0122 q^{89} -6.99549 q^{91} -11.1408 q^{93} +2.12568 q^{95} -5.91056 q^{97} +4.19812 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 27 q - 4 q^{3} - 10 q^{5} - 8 q^{7} + 19 q^{9}+O(q^{10}) \) 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

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