LMFDB - Embedded newform 6004.2.a.g.1.3

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			1.3
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)

$f(q)$	$=$	$q-2.74782 q^{3} -0.322226 q^{5} -1.10048 q^{7} +4.55049 q^{9} +O(q^{10})$	$q-2.74782 q^{3} -0.322226 q^{5} -1.10048 q^{7} +4.55049 q^{9} -1.90455 q^{11} -3.46075 q^{13} +0.885416 q^{15} +0.596613 q^{17} +1.00000 q^{19} +3.02391 q^{21} +3.97163 q^{23} -4.89617 q^{25} -4.26046 q^{27} +4.48417 q^{29} -0.142441 q^{31} +5.23334 q^{33} +0.354603 q^{35} +1.73291 q^{37} +9.50951 q^{39} -2.38206 q^{41} +5.85585 q^{43} -1.46628 q^{45} -1.46562 q^{47} -5.78895 q^{49} -1.63938 q^{51} -2.82437 q^{53} +0.613693 q^{55} -2.74782 q^{57} +10.6456 q^{59} -3.06513 q^{61} -5.00772 q^{63} +1.11514 q^{65} +3.30641 q^{67} -10.9133 q^{69} +9.93768 q^{71} +1.31446 q^{73} +13.4538 q^{75} +2.09591 q^{77} -1.00000 q^{79} -1.94452 q^{81} +12.5240 q^{83} -0.192244 q^{85} -12.3217 q^{87} -16.0319 q^{89} +3.80849 q^{91} +0.391402 q^{93} -0.322226 q^{95} +14.2275 q^{97} -8.66661 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.74782	−1.58645	−0.793226	−	0.608927i	$-0.791600\pi$
$3$	−2.74782	−1.58645	−0.793226	+	0.608927i	$0.791600\pi$
$4$	0	0
$5$	−0.322226	−0.144104	−0.0720518	−	0.997401i	$-0.522955\pi$
$5$	−0.322226	−0.144104	−0.0720518	+	0.997401i	$0.522955\pi$
$6$	0	0
$7$	−1.10048	−0.415942	−0.207971	−	0.978135i	$-0.566686\pi$
$7$	−1.10048	−0.415942	−0.207971	+	0.978135i	$0.566686\pi$
$8$	0	0
$9$	4.55049	1.51683
$10$	0	0
$11$	−1.90455	−0.574242	−0.287121	−	0.957894i	$-0.592698\pi$
$11$	−1.90455	−0.574242	−0.287121	+	0.957894i	$0.592698\pi$
$12$	0	0
$13$	−3.46075	−0.959840	−0.479920	−	0.877312i	$-0.659334\pi$
$13$	−3.46075	−0.959840	−0.479920	+	0.877312i	$0.659334\pi$
$14$	0	0
$15$	0.885416	0.228614
$16$	0	0
$17$	0.596613	0.144700	0.0723499	−	0.997379i	$-0.476950\pi$
$17$	0.596613	0.144700	0.0723499	+	0.997379i	$0.476950\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	3.02391	0.659872
$22$	0	0
$23$	3.97163	0.828143	0.414071	−	0.910244i	$-0.364106\pi$
$23$	3.97163	0.828143	0.414071	+	0.910244i	$0.364106\pi$
$24$	0	0
$25$	−4.89617	−0.979234
$26$	0	0
$27$	−4.26046	−0.819925
$28$	0	0
$29$	4.48417	0.832690	0.416345	−	0.909207i	$-0.363311\pi$
$29$	4.48417	0.832690	0.416345	+	0.909207i	$0.363311\pi$
$30$	0	0
$31$	−0.142441	−0.0255832	−0.0127916	−	0.999918i	$-0.504072\pi$
$31$	−0.142441	−0.0255832	−0.0127916	+	0.999918i	$0.504072\pi$
$32$	0	0
$33$	5.23334	0.911007
$34$	0	0
$35$	0.354603	0.0599388
$36$	0	0
$37$	1.73291	0.284889	0.142444	−	0.989803i	$-0.454504\pi$
$37$	1.73291	0.284889	0.142444	+	0.989803i	$0.454504\pi$
$38$	0	0
$39$	9.50951	1.52274
$40$	0	0
$41$	−2.38206	−0.372015	−0.186007	−	0.982548i	$-0.559555\pi$
$41$	−2.38206	−0.372015	−0.186007	+	0.982548i	$0.559555\pi$
$42$	0	0
$43$	5.85585	0.893009	0.446504	−	0.894781i	$-0.352669\pi$
$43$	5.85585	0.893009	0.446504	+	0.894781i	$0.352669\pi$
$44$	0	0
$45$	−1.46628	−0.218581
$46$	0	0
$47$	−1.46562	−0.213782	−0.106891	−	0.994271i	$-0.534090\pi$
$47$	−1.46562	−0.213782	−0.106891	+	0.994271i	$0.534090\pi$
$48$	0	0
$49$	−5.78895	−0.826992
$50$	0	0
$51$	−1.63938	−0.229559
$52$	0	0
$53$	−2.82437	−0.387957	−0.193979	−	0.981006i	$-0.562139\pi$
$53$	−2.82437	−0.387957	−0.193979	+	0.981006i	$0.562139\pi$
$54$	0	0
$55$	0.613693	0.0827504
$56$	0	0
$57$	−2.74782	−0.363957
$58$	0	0
$59$	10.6456	1.38594	0.692968	−	0.720969i	$-0.256303\pi$
$59$	10.6456	1.38594	0.692968	+	0.720969i	$0.256303\pi$
$60$	0	0
$61$	−3.06513	−0.392449	−0.196225	−	0.980559i	$-0.562868\pi$
$61$	−3.06513	−0.392449	−0.196225	+	0.980559i	$0.562868\pi$
$62$	0	0
$63$	−5.00772	−0.630913
$64$	0	0
$65$	1.11514	0.138317
$66$	0	0
$67$	3.30641	0.403943	0.201971	−	0.979391i	$-0.435265\pi$
$67$	3.30641	0.403943	0.201971	+	0.979391i	$0.435265\pi$
$68$	0	0
$69$	−10.9133	−1.31381
$70$	0	0
$71$	9.93768	1.17939	0.589693	−	0.807628i	$-0.299249\pi$
$71$	9.93768	1.17939	0.589693	+	0.807628i	$0.299249\pi$
$72$	0	0
$73$	1.31446	0.153846	0.0769228	−	0.997037i	$-0.475490\pi$
$73$	1.31446	0.153846	0.0769228	+	0.997037i	$0.475490\pi$
$74$	0	0
$75$	13.4538	1.55351
$76$	0	0
$77$	2.09591	0.238851
$78$	0	0
$79$	−1.00000	−0.112509
$79$	−1.00000	−0.112509
$80$	0	0
$81$	−1.94452	−0.216058
$82$	0	0
$83$	12.5240	1.37468	0.687341	−	0.726334i	$-0.258778\pi$
$83$	12.5240	1.37468	0.687341	+	0.726334i	$0.258778\pi$
$84$	0	0
$85$	−0.192244	−0.0208518
$86$	0	0
$87$	−12.3217	−1.32102
$88$	0	0
$89$	−16.0319	−1.69938	−0.849688	−	0.527287i	$-0.823209\pi$
$89$	−16.0319	−1.69938	−0.849688	+	0.527287i	$0.823209\pi$
$90$	0	0
$91$	3.80849	0.399238
$92$	0	0
$93$	0.391402	0.0405865
$94$	0	0
$95$	−0.322226	−0.0330597
$96$	0	0
$97$	14.2275	1.44459	0.722294	−	0.691586i	$-0.243088\pi$
$97$	14.2275	1.44459	0.722294	+	0.691586i	$0.243088\pi$
$98$	0	0
$99$	−8.66661	−0.871027
$100$	0	0
$101$	9.75462	0.970621	0.485311	−	0.874342i	$-0.338707\pi$
$101$	9.75462	0.970621	0.485311	+	0.874342i	$0.338707\pi$
$102$	0	0
$103$	8.82932	0.869978	0.434989	−	0.900436i	$-0.356752\pi$
$103$	8.82932	0.869978	0.434989	+	0.900436i	$0.356752\pi$
$104$	0	0
$105$	−0.974382	−0.0950900
$106$	0	0
$107$	1.59907	0.154588	0.0772941	−	0.997008i	$-0.475372\pi$
$107$	1.59907	0.154588	0.0772941	+	0.997008i	$0.475372\pi$
$108$	0	0
$109$	−18.1592	−1.73934	−0.869670	−	0.493634i	$-0.835668\pi$
$109$	−18.1592	−1.73934	−0.869670	+	0.493634i	$0.835668\pi$
$110$	0	0
$111$	−4.76172	−0.451962
$112$	0	0
$113$	−10.6686	−1.00362	−0.501809	−	0.864978i	$-0.667332\pi$
$113$	−10.6686	−1.00362	−0.501809	+	0.864978i	$0.667332\pi$
$114$	0	0
$115$	−1.27976	−0.119338
$116$	0	0
$117$	−15.7481	−1.45591
$118$	0	0
$119$	−0.656560	−0.0601867
$120$	0	0
$121$	−7.37271	−0.670246
$122$	0	0
$123$	6.54545	0.590183
$124$	0	0
$125$	3.18880	0.285215
$126$	0	0
$127$	−4.33340	−0.384527	−0.192263	−	0.981343i	$-0.561583\pi$
$127$	−4.33340	−0.384527	−0.192263	+	0.981343i	$0.561583\pi$
$128$	0	0
$129$	−16.0908	−1.41672
$130$	0	0
$131$	0.0882175	0.00770760	0.00385380	−	0.999993i	$-0.498773\pi$
$131$	0.0882175	0.00770760	0.00385380	+	0.999993i	$0.498773\pi$
$132$	0	0
$133$	−1.10048	−0.0954236
$134$	0	0
$135$	1.37283	0.118154
$136$	0	0
$137$	19.5661	1.67165	0.835823	−	0.548999i	$-0.184991\pi$
$137$	19.5661	1.67165	0.835823	+	0.548999i	$0.184991\pi$
$138$	0	0
$139$	8.38056	0.710830	0.355415	−	0.934709i	$-0.384340\pi$
$139$	8.38056	0.710830	0.355415	+	0.934709i	$0.384340\pi$
$140$	0	0
$141$	4.02724	0.339155
$142$	0	0
$143$	6.59116	0.551181
$144$	0	0
$145$	−1.44491	−0.119994
$146$	0	0
$147$	15.9070	1.31198
$148$	0	0
$149$	6.43262	0.526981	0.263490	−	0.964662i	$-0.415126\pi$
$149$	6.43262	0.526981	0.263490	+	0.964662i	$0.415126\pi$
$150$	0	0
$151$	5.40288	0.439680	0.219840	−	0.975536i	$-0.429446\pi$
$151$	5.40288	0.439680	0.219840	+	0.975536i	$0.429446\pi$
$152$	0	0
$153$	2.71488	0.219485
$154$	0	0
$155$	0.0458982	0.00368663
$156$	0	0
$157$	11.7271	0.935928	0.467964	−	0.883748i	$-0.344988\pi$
$157$	11.7271	0.935928	0.467964	+	0.883748i	$0.344988\pi$
$158$	0	0
$159$	7.76085	0.615476
$160$	0	0
$161$	−4.37070	−0.344459
$162$	0	0
$163$	−12.0793	−0.946121	−0.473061	−	0.881030i	$-0.656851\pi$
$163$	−12.0793	−0.946121	−0.473061	+	0.881030i	$0.656851\pi$
$164$	0	0
$165$	−1.68632	−0.131279
$166$	0	0
$167$	−15.6644	−1.21215	−0.606073	−	0.795409i	$-0.707256\pi$
$167$	−15.6644	−1.21215	−0.606073	+	0.795409i	$0.707256\pi$
$168$	0	0
$169$	−1.02318	−0.0787064
$170$	0	0
$171$	4.55049	0.347985
$172$	0	0
$173$	−4.05671	−0.308426	−0.154213	−	0.988038i	$-0.549284\pi$
$173$	−4.05671	−0.308426	−0.154213	+	0.988038i	$0.549284\pi$
$174$	0	0
$175$	5.38813	0.407305
$176$	0	0
$177$	−29.2521	−2.19872
$178$	0	0
$179$	−18.4769	−1.38103	−0.690513	−	0.723320i	$-0.742615\pi$
$179$	−18.4769	−1.38103	−0.690513	+	0.723320i	$0.742615\pi$
$180$	0	0
$181$	17.7692	1.32077	0.660385	−	0.750927i	$-0.270393\pi$
$181$	17.7692	1.32077	0.660385	+	0.750927i	$0.270393\pi$
$182$	0	0
$183$	8.42240	0.622602
$184$	0	0
$185$	−0.558388	−0.0410535
$186$	0	0
$187$	−1.13628	−0.0830927
$188$	0	0
$189$	4.68854	0.341041
$190$	0	0
$191$	11.2665	0.815217	0.407609	−	0.913157i	$-0.366363\pi$
$191$	11.2665	0.815217	0.407609	+	0.913157i	$0.366363\pi$
$192$	0	0
$193$	10.2092	0.734871	0.367436	−	0.930049i	$-0.380236\pi$
$193$	10.2092	0.734871	0.367436	+	0.930049i	$0.380236\pi$
$194$	0	0
$195$	−3.06421	−0.219433
$196$	0	0
$197$	2.20613	0.157180	0.0785900	−	0.996907i	$-0.474958\pi$
$197$	2.20613	0.157180	0.0785900	+	0.996907i	$0.474958\pi$
$198$	0	0
$199$	8.02285	0.568725	0.284363	−	0.958717i	$-0.408218\pi$
$199$	8.02285	0.568725	0.284363	+	0.958717i	$0.408218\pi$
$200$	0	0
$201$	−9.08541	−0.640836
$202$	0	0
$203$	−4.93474	−0.346351
$204$	0	0
$205$	0.767559	0.0536087
$206$	0	0
$207$	18.0729	1.25615
$208$	0	0
$209$	−1.90455	−0.131740
$210$	0	0
$211$	−26.9577	−1.85584	−0.927922	−	0.372775i	$-0.878406\pi$
$211$	−26.9577	−1.85584	−0.927922	+	0.372775i	$0.878406\pi$
$212$	0	0
$213$	−27.3069	−1.87104
$214$	0	0
$215$	−1.88690	−0.128686
$216$	0	0
$217$	0.156754	0.0106411
$218$	0	0
$219$	−3.61189	−0.244069
$220$	0	0
$221$	−2.06473	−0.138889
$222$	0	0
$223$	−17.5112	−1.17264	−0.586319	−	0.810080i	$-0.699423\pi$
$223$	−17.5112	−1.17264	−0.586319	+	0.810080i	$0.699423\pi$
$224$	0	0
$225$	−22.2800	−1.48533
$226$	0	0
$227$	15.9225	1.05681	0.528406	−	0.848992i	$-0.322790\pi$
$227$	15.9225	1.05681	0.528406	+	0.848992i	$0.322790\pi$
$228$	0	0
$229$	−10.2773	−0.679141	−0.339571	−	0.940581i	$-0.610282\pi$
$229$	−10.2773	−0.679141	−0.339571	+	0.940581i	$0.610282\pi$
$230$	0	0
$231$	−5.75918	−0.378926
$232$	0	0
$233$	−19.2565	−1.26154	−0.630768	−	0.775971i	$-0.717260\pi$
$233$	−19.2565	−1.26154	−0.630768	+	0.775971i	$0.717260\pi$
$234$	0	0
$235$	0.472259	0.0308068
$236$	0	0
$237$	2.74782	0.178490
$238$	0	0
$239$	−9.33202	−0.603638	−0.301819	−	0.953365i	$-0.597594\pi$
$239$	−9.33202	−0.603638	−0.301819	+	0.953365i	$0.597594\pi$
$240$	0	0
$241$	−4.26126	−0.274492	−0.137246	−	0.990537i	$-0.543825\pi$
$241$	−4.26126	−0.274492	−0.137246	+	0.990537i	$0.543825\pi$
$242$	0	0
$243$	18.1245	1.16269
$244$	0	0
$245$	1.86535	0.119173
$246$	0	0
$247$	−3.46075	−0.220202
$248$	0	0
$249$	−34.4135	−2.18087
$250$	0	0
$251$	3.99243	0.252000	0.126000	−	0.992030i	$-0.459786\pi$
$251$	3.99243	0.252000	0.126000	+	0.992030i	$0.459786\pi$
$252$	0	0
$253$	−7.56416	−0.475554
$254$	0	0
$255$	0.528251	0.0330803
$256$	0	0
$257$	6.04628	0.377157	0.188578	−	0.982058i	$-0.439612\pi$
$257$	6.04628	0.377157	0.188578	+	0.982058i	$0.439612\pi$
$258$	0	0
$259$	−1.90703	−0.118497
$260$	0	0
$261$	20.4052	1.26305
$262$	0	0
$263$	−28.4841	−1.75640	−0.878201	−	0.478291i	$-0.841256\pi$
$263$	−28.4841	−1.75640	−0.878201	+	0.478291i	$0.841256\pi$
$264$	0	0
$265$	0.910085	0.0559061
$266$	0	0
$267$	44.0526	2.69598
$268$	0	0
$269$	25.6804	1.56576	0.782879	−	0.622173i	$-0.213750\pi$
$269$	25.6804	1.56576	0.782879	+	0.622173i	$0.213750\pi$
$270$	0	0
$271$	5.47566	0.332623	0.166311	−	0.986073i	$-0.446814\pi$
$271$	5.47566	0.332623	0.166311	+	0.986073i	$0.446814\pi$
$272$	0	0
$273$	−10.4650	−0.633372
$274$	0	0
$275$	9.32498	0.562317
$276$	0	0
$277$	6.23158	0.374419	0.187210	−	0.982320i	$-0.440056\pi$
$277$	6.23158	0.374419	0.187210	+	0.982320i	$0.440056\pi$
$278$	0	0
$279$	−0.648177	−0.0388054
$280$	0	0
$281$	−12.5009	−0.745739	−0.372870	−	0.927884i	$-0.621626\pi$
$281$	−12.5009	−0.745739	−0.372870	+	0.927884i	$0.621626\pi$
$282$	0	0
$283$	23.8462	1.41751	0.708753	−	0.705456i	$-0.249258\pi$
$283$	23.8462	1.41751	0.708753	+	0.705456i	$0.249258\pi$
$284$	0	0
$285$	0.885416	0.0524475
$286$	0	0
$287$	2.62140	0.154736
$288$	0	0
$289$	−16.6441	−0.979062
$290$	0	0
$291$	−39.0947	−2.29177
$292$	0	0
$293$	−6.44855	−0.376728	−0.188364	−	0.982099i	$-0.560319\pi$
$293$	−6.44855	−0.376728	−0.188364	+	0.982099i	$0.560319\pi$
$294$	0	0
$295$	−3.43028	−0.199718
$296$	0	0
$297$	8.11423	0.470836
$298$	0	0
$299$	−13.7448	−0.794885
$300$	0	0
$301$	−6.44424	−0.371440
$302$	0	0
$303$	−26.8039	−1.53984
$304$	0	0
$305$	0.987663	0.0565534
$306$	0	0
$307$	−24.6078	−1.40444	−0.702221	−	0.711959i	$-0.747808\pi$
$307$	−24.6078	−1.40444	−0.702221	+	0.711959i	$0.747808\pi$
$308$	0	0
$309$	−24.2613	−1.38018
$310$	0	0
$311$	−33.4895	−1.89901	−0.949506	−	0.313748i	$-0.898415\pi$
$311$	−33.4895	−1.89901	−0.949506	+	0.313748i	$0.898415\pi$
$312$	0	0
$313$	−17.4341	−0.985436	−0.492718	−	0.870189i	$-0.663997\pi$
$313$	−17.4341	−0.985436	−0.492718	+	0.870189i	$0.663997\pi$
$314$	0	0
$315$	1.61361	0.0909169
$316$	0	0
$317$	−8.81607	−0.495160	−0.247580	−	0.968867i	$-0.579635\pi$
$317$	−8.81607	−0.495160	−0.247580	+	0.968867i	$0.579635\pi$
$318$	0	0
$319$	−8.54031	−0.478165
$320$	0	0
$321$	−4.39396	−0.245247
$322$	0	0
$323$	0.596613	0.0331964
$324$	0	0
$325$	16.9444	0.939908
$326$	0	0
$327$	49.8982	2.75938
$328$	0	0
$329$	1.61288	0.0889209
$330$	0	0
$331$	−12.4641	−0.685089	−0.342544	−	0.939502i	$-0.611289\pi$
$331$	−12.4641	−0.685089	−0.342544	+	0.939502i	$0.611289\pi$
$332$	0	0
$333$	7.88559	0.432128
$334$	0	0
$335$	−1.06541	−0.0582096
$336$	0	0
$337$	−32.5786	−1.77467	−0.887334	−	0.461127i	$-0.847445\pi$
$337$	−32.5786	−1.77467	−0.887334	+	0.461127i	$0.847445\pi$
$338$	0	0
$339$	29.3154	1.59219
$340$	0	0
$341$	0.271286	0.0146909
$342$	0	0
$343$	14.0740	0.759923
$344$	0	0
$345$	3.51655	0.189325
$346$	0	0
$347$	9.25713	0.496949	0.248474	−	0.968638i	$-0.420071\pi$
$347$	9.25713	0.496949	0.248474	+	0.968638i	$0.420071\pi$
$348$	0	0
$349$	3.00026	0.160600	0.0803001	−	0.996771i	$-0.474412\pi$
$349$	3.00026	0.160600	0.0803001	+	0.996771i	$0.474412\pi$
$350$	0	0
$351$	14.7444	0.786997
$352$	0	0
$353$	−2.16535	−0.115250	−0.0576250	−	0.998338i	$-0.518353\pi$
$353$	−2.16535	−0.115250	−0.0576250	+	0.998338i	$0.518353\pi$
$354$	0	0
$355$	−3.20218	−0.169954
$356$	0	0
$357$	1.80410	0.0954834
$358$	0	0
$359$	−31.0025	−1.63625	−0.818125	−	0.575041i	$-0.804986\pi$
$359$	−31.0025	−1.63625	−0.818125	+	0.575041i	$0.804986\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	20.2588	1.06331
$364$	0	0
$365$	−0.423552	−0.0221697
$366$	0	0
$367$	30.4518	1.58957	0.794784	−	0.606892i	$-0.207584\pi$
$367$	30.4518	1.58957	0.794784	+	0.606892i	$0.207584\pi$
$368$	0	0
$369$	−10.8395	−0.564283
$370$	0	0
$371$	3.10816	0.161368
$372$	0	0
$373$	−21.3117	−1.10348	−0.551740	−	0.834016i	$-0.686036\pi$
$373$	−21.3117	−1.10348	−0.551740	+	0.834016i	$0.686036\pi$
$374$	0	0
$375$	−8.76223	−0.452480
$376$	0	0
$377$	−15.5186	−0.799249
$378$	0	0
$379$	10.8481	0.557228	0.278614	−	0.960403i	$-0.410125\pi$
$379$	10.8481	0.557228	0.278614	+	0.960403i	$0.410125\pi$
$380$	0	0
$381$	11.9074	0.610034
$382$	0	0
$383$	−6.10185	−0.311790	−0.155895	−	0.987774i	$-0.549826\pi$
$383$	−6.10185	−0.311790	−0.155895	+	0.987774i	$0.549826\pi$
$384$	0	0
$385$	−0.675357	−0.0344194
$386$	0	0
$387$	26.6470	1.35454
$388$	0	0
$389$	−13.7729	−0.698312	−0.349156	−	0.937065i	$-0.613532\pi$
$389$	−13.7729	−0.698312	−0.349156	+	0.937065i	$0.613532\pi$
$390$	0	0
$391$	2.36953	0.119832
$392$	0	0
$393$	−0.242405	−0.0122277
$394$	0	0
$395$	0.322226	0.0162129
$396$	0	0
$397$	−18.0140	−0.904095	−0.452047	−	0.891994i	$-0.649306\pi$
$397$	−18.0140	−0.904095	−0.452047	+	0.891994i	$0.649306\pi$
$398$	0	0
$399$	3.02391	0.151385
$400$	0	0
$401$	−26.4039	−1.31855	−0.659274	−	0.751903i	$-0.729136\pi$
$401$	−26.4039	−1.31855	−0.659274	+	0.751903i	$0.729136\pi$
$402$	0	0
$403$	0.492954	0.0245558
$404$	0	0
$405$	0.626574	0.0311347
$406$	0	0
$407$	−3.30041	−0.163595
$408$	0	0
$409$	−25.3726	−1.25459	−0.627297	−	0.778780i	$-0.715839\pi$
$409$	−25.3726	−1.25459	−0.627297	+	0.778780i	$0.715839\pi$
$410$	0	0
$411$	−53.7641	−2.65199
$412$	0	0
$413$	−11.7152	−0.576469
$414$	0	0
$415$	−4.03554	−0.198097
$416$	0	0
$417$	−23.0282	−1.12770
$418$	0	0
$419$	−17.5022	−0.855039	−0.427519	−	0.904006i	$-0.640612\pi$
$419$	−17.5022	−0.855039	−0.427519	+	0.904006i	$0.640612\pi$
$420$	0	0
$421$	−16.5524	−0.806715	−0.403358	−	0.915042i	$-0.632157\pi$
$421$	−16.5524	−0.806715	−0.403358	+	0.915042i	$0.632157\pi$
$422$	0	0
$423$	−6.66927	−0.324271
$424$	0	0
$425$	−2.92112	−0.141695
$426$	0	0
$427$	3.37311	0.163236
$428$	0	0
$429$	−18.1113	−0.874422
$430$	0	0
$431$	−19.6764	−0.947778	−0.473889	−	0.880585i	$-0.657150\pi$
$431$	−19.6764	−0.947778	−0.473889	+	0.880585i	$0.657150\pi$
$432$	0	0
$433$	−10.3389	−0.496857	−0.248429	−	0.968650i	$-0.579914\pi$
$433$	−10.3389	−0.496857	−0.248429	+	0.968650i	$0.579914\pi$
$434$	0	0
$435$	3.97036	0.190364
$436$	0	0
$437$	3.97163	0.189989
$438$	0	0
$439$	−35.8829	−1.71260	−0.856298	−	0.516482i	$-0.827241\pi$
$439$	−35.8829	−1.71260	−0.856298	+	0.516482i	$0.827241\pi$
$440$	0	0
$441$	−26.3425	−1.25441
$442$	0	0
$443$	23.9914	1.13987	0.569934	−	0.821691i	$-0.306969\pi$
$443$	23.9914	1.13987	0.569934	+	0.821691i	$0.306969\pi$
$444$	0	0
$445$	5.16588	0.244886
$446$	0	0
$447$	−17.6757	−0.836030
$448$	0	0
$449$	−1.95556	−0.0922887	−0.0461443	−	0.998935i	$-0.514693\pi$
$449$	−1.95556	−0.0922887	−0.0461443	+	0.998935i	$0.514693\pi$
$450$	0	0
$451$	4.53673	0.213626
$452$	0	0
$453$	−14.8461	−0.697531
$454$	0	0
$455$	−1.22719	−0.0575317
$456$	0	0
$457$	30.9199	1.44637	0.723186	−	0.690653i	$-0.242677\pi$
$457$	30.9199	1.44637	0.723186	+	0.690653i	$0.242677\pi$
$458$	0	0
$459$	−2.54184	−0.118643
$460$	0	0
$461$	−32.9900	−1.53650	−0.768248	−	0.640153i	$-0.778871\pi$
$461$	−32.9900	−1.53650	−0.768248	+	0.640153i	$0.778871\pi$
$462$	0	0
$463$	35.6325	1.65599	0.827993	−	0.560739i	$-0.189483\pi$
$463$	35.6325	1.65599	0.827993	+	0.560739i	$0.189483\pi$
$464$	0	0
$465$	−0.126120	−0.00584867
$466$	0	0
$467$	26.8577	1.24283	0.621414	−	0.783483i	$-0.286559\pi$
$467$	26.8577	1.24283	0.621414	+	0.783483i	$0.286559\pi$
$468$	0	0
$469$	−3.63864	−0.168017
$470$	0	0
$471$	−32.2240	−1.48480
$472$	0	0
$473$	−11.1527	−0.512803
$474$	0	0
$475$	−4.89617	−0.224652
$476$	0	0
$477$	−12.8523	−0.588465
$478$	0	0
$479$	10.6099	0.484776	0.242388	−	0.970179i	$-0.422069\pi$
$479$	10.6099	0.484776	0.242388	+	0.970179i	$0.422069\pi$
$480$	0	0
$481$	−5.99718	−0.273448
$482$	0	0
$483$	12.0099	0.546468
$484$	0	0
$485$	−4.58448	−0.208170
$486$	0	0
$487$	−15.1683	−0.687341	−0.343670	−	0.939090i	$-0.611670\pi$
$487$	−15.1683	−0.687341	−0.343670	+	0.939090i	$0.611670\pi$
$488$	0	0
$489$	33.1916	1.50098
$490$	0	0
$491$	−36.0083	−1.62503	−0.812516	−	0.582938i	$-0.801903\pi$
$491$	−36.0083	−1.62503	−0.812516	+	0.582938i	$0.801903\pi$
$492$	0	0
$493$	2.67531	0.120490
$494$	0	0
$495$	2.79260	0.125518
$496$	0	0
$497$	−10.9362	−0.490556
$498$	0	0
$499$	−0.380587	−0.0170374	−0.00851870	−	0.999964i	$-0.502712\pi$
$499$	−0.380587	−0.0170374	−0.00851870	+	0.999964i	$0.502712\pi$
$500$	0	0
$501$	43.0428	1.92301
$502$	0	0
$503$	−40.0370	−1.78516	−0.892581	−	0.450886i	$-0.851108\pi$
$503$	−40.0370	−1.78516	−0.892581	+	0.450886i	$0.851108\pi$
$504$	0	0
$505$	−3.14319	−0.139870
$506$	0	0
$507$	2.81152	0.124864
$508$	0	0
$509$	−33.3739	−1.47927	−0.739636	−	0.673007i	$-0.765002\pi$
$509$	−33.3739	−1.47927	−0.739636	+	0.673007i	$0.765002\pi$
$510$	0	0
$511$	−1.44653	−0.0639909
$512$	0	0
$513$	−4.26046	−0.188104
$514$	0	0
$515$	−2.84503	−0.125367
$516$	0	0
$517$	2.79133	0.122763
$518$	0	0
$519$	11.1471	0.489303
$520$	0	0
$521$	21.2022	0.928886	0.464443	−	0.885603i	$-0.346255\pi$
$521$	21.2022	0.928886	0.464443	+	0.885603i	$0.346255\pi$
$522$	0	0
$523$	3.24994	0.142110	0.0710549	−	0.997472i	$-0.477363\pi$
$523$	3.24994	0.142110	0.0710549	+	0.997472i	$0.477363\pi$
$524$	0	0
$525$	−14.8056	−0.646169
$526$	0	0
$527$	−0.0849823	−0.00370189
$528$	0	0
$529$	−7.22612	−0.314179
$530$	0	0
$531$	48.4426	2.10223
$532$	0	0
$533$	8.24371	0.357075
$534$	0	0
$535$	−0.515262	−0.0222767
$536$	0	0
$537$	50.7710	2.19093
$538$	0	0
$539$	11.0253	0.474894
$540$	0	0
$541$	31.1466	1.33910	0.669550	−	0.742767i	$-0.266487\pi$
$541$	31.1466	1.33910	0.669550	+	0.742767i	$0.266487\pi$
$542$	0	0
$543$	−48.8264	−2.09534
$544$	0	0
$545$	5.85137	0.250645
$546$	0	0
$547$	−4.03247	−0.172416	−0.0862079	−	0.996277i	$-0.527475\pi$
$547$	−4.03247	−0.172416	−0.0862079	+	0.996277i	$0.527475\pi$
$548$	0	0
$549$	−13.9478	−0.595279
$550$	0	0
$551$	4.48417	0.191032
$552$	0	0
$553$	1.10048	0.0467971
$554$	0	0
$555$	1.53435	0.0651295
$556$	0	0
$557$	8.20095	0.347485	0.173743	−	0.984791i	$-0.444414\pi$
$557$	8.20095	0.347485	0.173743	+	0.984791i	$0.444414\pi$
$558$	0	0
$559$	−20.2657	−0.857146
$560$	0	0
$561$	3.12228	0.131823
$562$	0	0
$563$	11.6848	0.492458	0.246229	−	0.969212i	$-0.420809\pi$
$563$	11.6848	0.492458	0.246229	+	0.969212i	$0.420809\pi$
$564$	0	0
$565$	3.43770	0.144625
$566$	0	0
$567$	2.13990	0.0898674
$568$	0	0
$569$	−24.0710	−1.00911	−0.504555	−	0.863380i	$-0.668343\pi$
$569$	−24.0710	−1.00911	−0.504555	+	0.863380i	$0.668343\pi$
$570$	0	0
$571$	31.8220	1.33171	0.665856	−	0.746081i	$-0.268067\pi$
$571$	31.8220	1.33171	0.665856	+	0.746081i	$0.268067\pi$
$572$	0	0
$573$	−30.9583	−1.29330
$574$	0	0
$575$	−19.4458	−0.810946
$576$	0	0
$577$	8.11051	0.337645	0.168823	−	0.985646i	$-0.446004\pi$
$577$	8.11051	0.337645	0.168823	+	0.985646i	$0.446004\pi$
$578$	0	0
$579$	−28.0529	−1.16584
$580$	0	0
$581$	−13.7824	−0.571788
$582$	0	0
$583$	5.37915	0.222781
$584$	0	0
$585$	5.07445	0.209803
$586$	0	0
$587$	−31.3012	−1.29194	−0.645969	−	0.763364i	$-0.723546\pi$
$587$	−31.3012	−1.29194	−0.645969	+	0.763364i	$0.723546\pi$
$588$	0	0
$589$	−0.142441	−0.00586919
$590$	0	0
$591$	−6.06203	−0.249359
$592$	0	0
$593$	18.2853	0.750886	0.375443	−	0.926845i	$-0.377491\pi$
$593$	18.2853	0.750886	0.375443	+	0.926845i	$0.377491\pi$
$594$	0	0
$595$	0.211560	0.00867313
$596$	0	0
$597$	−22.0453	−0.902255
$598$	0	0
$599$	41.7951	1.70770	0.853852	−	0.520517i	$-0.174261\pi$
$599$	41.7951	1.70770	0.853852	+	0.520517i	$0.174261\pi$
$600$	0	0
$601$	−26.9296	−1.09848	−0.549240	−	0.835665i	$-0.685083\pi$
$601$	−26.9296	−1.09848	−0.549240	+	0.835665i	$0.685083\pi$
$602$	0	0
$603$	15.0458	0.612712
$604$	0	0
$605$	2.37568	0.0965849
$606$	0	0
$607$	−5.20348	−0.211203	−0.105601	−	0.994409i	$-0.533677\pi$
$607$	−5.20348	−0.211203	−0.105601	+	0.994409i	$0.533677\pi$
$608$	0	0
$609$	13.5597	0.549469
$610$	0	0
$611$	5.07213	0.205197
$612$	0	0
$613$	−37.5556	−1.51685	−0.758427	−	0.651758i	$-0.774032\pi$
$613$	−37.5556	−1.51685	−0.758427	+	0.651758i	$0.774032\pi$
$614$	0	0
$615$	−2.10911	−0.0850476
$616$	0	0
$617$	1.11484	0.0448818	0.0224409	−	0.999748i	$-0.492856\pi$
$617$	1.11484	0.0448818	0.0224409	+	0.999748i	$0.492856\pi$
$618$	0	0
$619$	32.3564	1.30051	0.650256	−	0.759715i	$-0.274661\pi$
$619$	32.3564	1.30051	0.650256	+	0.759715i	$0.274661\pi$
$620$	0	0
$621$	−16.9210	−0.679015
$622$	0	0
$623$	17.6427	0.706841
$624$	0	0
$625$	23.4533	0.938134
$626$	0	0
$627$	5.23334	0.208999
$628$	0	0
$629$	1.03388	0.0412234
$630$	0	0
$631$	−21.2398	−0.845542	−0.422771	−	0.906237i	$-0.638942\pi$
$631$	−21.2398	−0.845542	−0.422771	+	0.906237i	$0.638942\pi$
$632$	0	0
$633$	74.0747	2.94421
$634$	0	0
$635$	1.39633	0.0554118
$636$	0	0
$637$	20.0341	0.793781
$638$	0	0
$639$	45.2213	1.78893
$640$	0	0
$641$	37.6080	1.48542	0.742712	−	0.669610i	$-0.233539\pi$
$641$	37.6080	1.48542	0.742712	+	0.669610i	$0.233539\pi$
$642$	0	0
$643$	25.8329	1.01875	0.509376	−	0.860544i	$-0.329876\pi$
$643$	25.8329	1.01875	0.509376	+	0.860544i	$0.329876\pi$
$644$	0	0
$645$	5.18487	0.204154
$646$	0	0
$647$	−10.4698	−0.411609	−0.205804	−	0.978593i	$-0.565981\pi$
$647$	−10.4698	−0.411609	−0.205804	+	0.978593i	$0.565981\pi$
$648$	0	0
$649$	−20.2750	−0.795862
$650$	0	0
$651$	−0.430730	−0.0168816
$652$	0	0
$653$	6.18617	0.242083	0.121042	−	0.992647i	$-0.461377\pi$
$653$	6.18617	0.242083	0.121042	+	0.992647i	$0.461377\pi$
$654$	0	0
$655$	−0.0284259	−0.00111069
$656$	0	0
$657$	5.98143	0.233358
$658$	0	0
$659$	−12.7354	−0.496100	−0.248050	−	0.968747i	$-0.579790\pi$
$659$	−12.7354	−0.496100	−0.248050	+	0.968747i	$0.579790\pi$
$660$	0	0
$661$	41.4179	1.61097	0.805485	−	0.592616i	$-0.201905\pi$
$661$	41.4179	1.61097	0.805485	+	0.592616i	$0.201905\pi$
$662$	0	0
$663$	5.67350	0.220340
$664$	0	0
$665$	0.354603	0.0137509
$666$	0	0
$667$	17.8095	0.689586
$668$	0	0
$669$	48.1176	1.86033
$670$	0	0
$671$	5.83767	0.225361
$672$	0	0
$673$	−15.7266	−0.606216	−0.303108	−	0.952956i	$-0.598024\pi$
$673$	−15.7266	−0.606216	−0.303108	+	0.952956i	$0.598024\pi$
$674$	0	0
$675$	20.8599	0.802899
$676$	0	0
$677$	−23.9547	−0.920653	−0.460327	−	0.887750i	$-0.652268\pi$
$677$	−23.9547	−0.920653	−0.460327	+	0.887750i	$0.652268\pi$
$678$	0	0
$679$	−15.6571	−0.600865
$680$	0	0
$681$	−43.7520	−1.67658
$682$	0	0
$683$	−5.49575	−0.210289	−0.105144	−	0.994457i	$-0.533530\pi$
$683$	−5.49575	−0.210289	−0.105144	+	0.994457i	$0.533530\pi$
$684$	0	0
$685$	−6.30470	−0.240890
$686$	0	0
$687$	28.2400	1.07742
$688$	0	0
$689$	9.77446	0.372377
$690$	0	0
$691$	−41.4625	−1.57731	−0.788653	−	0.614838i	$-0.789221\pi$
$691$	−41.4625	−1.57731	−0.788653	+	0.614838i	$0.789221\pi$
$692$	0	0
$693$	9.53742	0.362297
$694$	0	0
$695$	−2.70043	−0.102433
$696$	0	0
$697$	−1.42116	−0.0538304
$698$	0	0
$699$	52.9134	2.00137
$700$	0	0
$701$	34.4107	1.29967	0.649836	−	0.760074i	$-0.274837\pi$
$701$	34.4107	1.29967	0.649836	+	0.760074i	$0.274837\pi$
$702$	0	0
$703$	1.73291	0.0653580
$704$	0	0
$705$	−1.29768	−0.0488735
$706$	0	0
$707$	−10.7348	−0.403722
$708$	0	0
$709$	1.23435	0.0463568	0.0231784	−	0.999731i	$-0.492621\pi$
$709$	1.23435	0.0463568	0.0231784	+	0.999731i	$0.492621\pi$
$710$	0	0
$711$	−4.55049	−0.170657
$712$	0	0
$713$	−0.565725	−0.0211866
$714$	0	0
$715$	−2.12384	−0.0794272
$716$	0	0
$717$	25.6427	0.957643
$718$	0	0
$719$	−28.9956	−1.08135	−0.540677	−	0.841230i	$-0.681832\pi$
$719$	−28.9956	−1.08135	−0.540677	+	0.841230i	$0.681832\pi$
$720$	0	0
$721$	−9.71648	−0.361860
$722$	0	0
$723$	11.7092	0.435468
$724$	0	0
$725$	−21.9553	−0.815398
$726$	0	0
$727$	23.0888	0.856315	0.428157	−	0.903704i	$-0.359163\pi$
$727$	23.0888	0.856315	0.428157	+	0.903704i	$0.359163\pi$
$728$	0	0
$729$	−43.9694	−1.62849
$730$	0	0
$731$	3.49367	0.129218
$732$	0	0
$733$	51.6398	1.90736	0.953680	−	0.300824i	$-0.0972617\pi$
$733$	51.6398	1.90736	0.953680	+	0.300824i	$0.0972617\pi$
$734$	0	0
$735$	−5.12563	−0.189062
$736$	0	0
$737$	−6.29721	−0.231961
$738$	0	0
$739$	−7.46711	−0.274682	−0.137341	−	0.990524i	$-0.543856\pi$
$739$	−7.46711	−0.274682	−0.137341	+	0.990524i	$0.543856\pi$
$740$	0	0
$741$	9.50951	0.349341
$742$	0	0
$743$	12.9801	0.476195	0.238098	−	0.971241i	$-0.423476\pi$
$743$	12.9801	0.476195	0.238098	+	0.971241i	$0.423476\pi$
$744$	0	0
$745$	−2.07276	−0.0759399
$746$	0	0
$747$	56.9901	2.08516
$748$	0	0
$749$	−1.75975	−0.0642997
$750$	0	0
$751$	6.29746	0.229798	0.114899	−	0.993377i	$-0.463346\pi$
$751$	6.29746	0.229798	0.114899	+	0.993377i	$0.463346\pi$
$752$	0	0
$753$	−10.9705	−0.399786
$754$	0	0
$755$	−1.74095	−0.0633595
$756$	0	0
$757$	15.3741	0.558780	0.279390	−	0.960178i	$-0.409868\pi$
$757$	15.3741	0.558780	0.279390	+	0.960178i	$0.409868\pi$
$758$	0	0
$759$	20.7849	0.754444
$760$	0	0
$761$	22.5592	0.817770	0.408885	−	0.912586i	$-0.365918\pi$
$761$	22.5592	0.817770	0.408885	+	0.912586i	$0.365918\pi$
$762$	0	0
$763$	19.9839	0.723465
$764$	0	0
$765$	−0.874804	−0.0316286
$766$	0	0
$767$	−36.8417	−1.33028
$768$	0	0
$769$	39.2098	1.41394	0.706971	−	0.707243i	$-0.250061\pi$
$769$	39.2098	1.41394	0.706971	+	0.707243i	$0.250061\pi$
$770$	0	0
$771$	−16.6141	−0.598341
$772$	0	0
$773$	4.23161	0.152200	0.0761002	−	0.997100i	$-0.475753\pi$
$773$	4.23161	0.152200	0.0761002	+	0.997100i	$0.475753\pi$
$774$	0	0
$775$	0.697417	0.0250519
$776$	0	0
$777$	5.24017	0.187990
$778$	0	0
$779$	−2.38206	−0.0853460
$780$	0	0
$781$	−18.9268	−0.677253
$782$	0	0
$783$	−19.1046	−0.682743
$784$	0	0
$785$	−3.77879	−0.134871
$786$	0	0
$787$	22.4768	0.801212	0.400606	−	0.916251i	$-0.368800\pi$
$787$	22.4768	0.801212	0.400606	+	0.916251i	$0.368800\pi$
$788$	0	0
$789$	78.2690	2.78645
$790$	0	0
$791$	11.7406	0.417447
$792$	0	0
$793$	10.6077	0.376689
$794$	0	0
$795$	−2.50075	−0.0886923
$796$	0	0
$797$	−38.8015	−1.37442	−0.687209	−	0.726459i	$-0.741164\pi$
$797$	−38.8015	−1.37442	−0.687209	+	0.726459i	$0.741164\pi$
$798$	0	0
$799$	−0.874405	−0.0309342
$800$	0	0
$801$	−72.9529	−2.57766
$802$	0	0
$803$	−2.50344	−0.0883446
$804$	0	0
$805$	1.40835	0.0496379
$806$	0	0
$807$	−70.5649	−2.48400
$808$	0	0
$809$	30.9845	1.08936	0.544678	−	0.838645i	$-0.316652\pi$
$809$	30.9845	1.08936	0.544678	+	0.838645i	$0.316652\pi$
$810$	0	0
$811$	−13.8246	−0.485448	−0.242724	−	0.970095i	$-0.578041\pi$
$811$	−13.8246	−0.485448	−0.242724	+	0.970095i	$0.578041\pi$
$812$	0	0
$813$	−15.0461	−0.527690
$814$	0	0
$815$	3.89225	0.136340
$816$	0	0
$817$	5.85585	0.204870
$818$	0	0
$819$	17.3305	0.605576
$820$	0	0
$821$	−48.7769	−1.70233	−0.851163	−	0.524901i	$-0.824102\pi$
$821$	−48.7769	−1.70233	−0.851163	+	0.524901i	$0.824102\pi$
$822$	0	0
$823$	16.7394	0.583497	0.291749	−	0.956495i	$-0.405763\pi$
$823$	16.7394	0.583497	0.291749	+	0.956495i	$0.405763\pi$
$824$	0	0
$825$	−25.6233	−0.892089
$826$	0	0
$827$	12.6269	0.439081	0.219540	−	0.975603i	$-0.429544\pi$
$827$	12.6269	0.439081	0.219540	+	0.975603i	$0.429544\pi$
$828$	0	0
$829$	10.2351	0.355479	0.177739	−	0.984078i	$-0.443122\pi$
$829$	10.2351	0.355479	0.177739	+	0.984078i	$0.443122\pi$
$830$	0	0
$831$	−17.1232	−0.593998
$832$	0	0
$833$	−3.45376	−0.119666
$834$	0	0
$835$	5.04747	0.174675
$836$	0	0
$837$	0.606865	0.0209763
$838$	0	0
$839$	−7.64201	−0.263832	−0.131916	−	0.991261i	$-0.542113\pi$
$839$	−7.64201	−0.263832	−0.131916	+	0.991261i	$0.542113\pi$
$840$	0	0
$841$	−8.89221	−0.306628
$842$	0	0
$843$	34.3501	1.18308
$844$	0	0
$845$	0.329696	0.0113419
$846$	0	0
$847$	8.11351	0.278784
$848$	0	0
$849$	−65.5248	−2.24881
$850$	0	0
$851$	6.88249	0.235929
$852$	0	0
$853$	−34.8680	−1.19386	−0.596929	−	0.802294i	$-0.703613\pi$
$853$	−34.8680	−1.19386	−0.596929	+	0.802294i	$0.703613\pi$
$854$	0	0
$855$	−1.46628	−0.0501459
$856$	0	0
$857$	−45.3258	−1.54830	−0.774149	−	0.633003i	$-0.781822\pi$
$857$	−45.3258	−1.54830	−0.774149	+	0.633003i	$0.781822\pi$
$858$	0	0
$859$	16.9737	0.579135	0.289568	−	0.957158i	$-0.406488\pi$
$859$	16.9737	0.579135	0.289568	+	0.957158i	$0.406488\pi$
$860$	0	0
$861$	−7.20313	−0.245482
$862$	0	0
$863$	−16.6412	−0.566472	−0.283236	−	0.959050i	$-0.591408\pi$
$863$	−16.6412	−0.566472	−0.283236	+	0.959050i	$0.591408\pi$
$864$	0	0
$865$	1.30718	0.0444453
$866$	0	0
$867$	45.7348	1.55323
$868$	0	0
$869$	1.90455	0.0646073
$870$	0	0
$871$	−11.4427	−0.387721
$872$	0	0
$873$	64.7423	2.19119
$874$	0	0
$875$	−3.50921	−0.118633
$876$	0	0
$877$	22.8014	0.769948	0.384974	−	0.922927i	$-0.374210\pi$
$877$	22.8014	0.769948	0.384974	+	0.922927i	$0.374210\pi$
$878$	0	0
$879$	17.7194	0.597661
$880$	0	0
$881$	30.1721	1.01653	0.508263	−	0.861202i	$-0.330288\pi$
$881$	30.1721	1.01653	0.508263	+	0.861202i	$0.330288\pi$
$882$	0	0
$883$	−42.7120	−1.43737	−0.718686	−	0.695334i	$-0.755256\pi$
$883$	−42.7120	−1.43737	−0.718686	+	0.695334i	$0.755256\pi$
$884$	0	0
$885$	9.42576	0.316844
$886$	0	0
$887$	−17.9941	−0.604182	−0.302091	−	0.953279i	$-0.597685\pi$
$887$	−17.9941	−0.604182	−0.302091	+	0.953279i	$0.597685\pi$
$888$	0	0
$889$	4.76881	0.159941
$890$	0	0
$891$	3.70342	0.124069
$892$	0	0
$893$	−1.46562	−0.0490449
$894$	0	0
$895$	5.95372	0.199011
$896$	0	0
$897$	37.7683	1.26105
$898$	0	0
$899$	−0.638731	−0.0213029
$900$	0	0
$901$	−1.68506	−0.0561374
$902$	0	0
$903$	17.7076	0.589271
$904$	0	0
$905$	−5.72568	−0.190328
$906$	0	0
$907$	−34.6271	−1.14978	−0.574888	−	0.818232i	$-0.694954\pi$
$907$	−34.6271	−1.14978	−0.574888	+	0.818232i	$0.694954\pi$
$908$	0	0
$909$	44.3883	1.47227
$910$	0	0
$911$	−44.5789	−1.47697	−0.738483	−	0.674272i	$-0.764457\pi$
$911$	−44.5789	−1.47697	−0.738483	+	0.674272i	$0.764457\pi$
$912$	0	0
$913$	−23.8524	−0.789401
$914$	0	0
$915$	−2.71391	−0.0897192
$916$	0	0
$917$	−0.0970815	−0.00320591
$918$	0	0
$919$	42.3848	1.39815	0.699073	−	0.715050i	$-0.253596\pi$
$919$	42.3848	1.39815	0.699073	+	0.715050i	$0.253596\pi$
$920$	0	0
$921$	67.6177	2.22808
$922$	0	0
$923$	−34.3919	−1.13202
$924$	0	0
$925$	−8.48463	−0.278973
$926$	0	0
$927$	40.1777	1.31961
$928$	0	0
$929$	−1.53086	−0.0502260	−0.0251130	−	0.999685i	$-0.507995\pi$
$929$	−1.53086	−0.0502260	−0.0251130	+	0.999685i	$0.507995\pi$
$930$	0	0
$931$	−5.78895	−0.189725
$932$	0	0
$933$	92.0228	3.01269
$934$	0	0
$935$	0.366137	0.0119740
$936$	0	0
$937$	16.3179	0.533081	0.266541	−	0.963824i	$-0.414119\pi$
$937$	16.3179	0.533081	0.266541	+	0.963824i	$0.414119\pi$
$938$	0	0
$939$	47.9058	1.56335
$940$	0	0
$941$	−3.40682	−0.111059	−0.0555296	−	0.998457i	$-0.517685\pi$
$941$	−3.40682	−0.111059	−0.0555296	+	0.998457i	$0.517685\pi$
$942$	0	0
$943$	−9.46065	−0.308081
$944$	0	0
$945$	−1.51077	−0.0491453
$946$	0	0
$947$	−16.3858	−0.532468	−0.266234	−	0.963908i	$-0.585779\pi$
$947$	−16.3858	−0.532468	−0.266234	+	0.963908i	$0.585779\pi$
$948$	0	0
$949$	−4.54902	−0.147667
$950$	0	0
$951$	24.2249	0.785548
$952$	0	0
$953$	2.25457	0.0730327	0.0365164	−	0.999333i	$-0.488374\pi$
$953$	2.25457	0.0730327	0.0365164	+	0.999333i	$0.488374\pi$
$954$	0	0
$955$	−3.63036	−0.117476
$956$	0	0
$957$	23.4672	0.758586
$958$	0	0
$959$	−21.5321	−0.695308
$960$	0	0
$961$	−30.9797	−0.999345
$962$	0	0
$963$	7.27656	0.234484
$964$	0	0
$965$	−3.28965	−0.105898
$966$	0	0
$967$	35.3853	1.13792	0.568958	−	0.822367i	$-0.307347\pi$
$967$	35.3853	1.13792	0.568958	+	0.822367i	$0.307347\pi$
$968$	0	0
$969$	−1.63938	−0.0526645
$970$	0	0
$971$	−23.8146	−0.764245	−0.382123	−	0.924112i	$-0.624807\pi$
$971$	−23.8146	−0.764245	−0.382123	+	0.924112i	$0.624807\pi$
$972$	0	0
$973$	−9.22263	−0.295664
$974$	0	0
$975$	−46.5602	−1.49112
$976$	0	0
$977$	37.9373	1.21372	0.606861	−	0.794808i	$-0.292428\pi$
$977$	37.9373	1.21372	0.606861	+	0.794808i	$0.292428\pi$
$978$	0	0
$979$	30.5334	0.975852
$980$	0	0
$981$	−82.6334	−2.63828
$982$	0	0
$983$	6.03830	0.192592	0.0962959	−	0.995353i	$-0.469300\pi$
$983$	6.03830	0.192592	0.0962959	+	0.995353i	$0.469300\pi$
$984$	0	0
$985$	−0.710871	−0.0226502
$986$	0	0
$987$	−4.43189	−0.141069
$988$	0	0
$989$	23.2573	0.739539
$990$	0	0
$991$	22.2803	0.707757	0.353879	−	0.935291i	$-0.384863\pi$
$991$	22.2803	0.707757	0.353879	+	0.935291i	$0.384863\pi$
$992$	0	0
$993$	34.2490	1.08686
$994$	0	0
$995$	−2.58517	−0.0819554
$996$	0	0
$997$	33.8602	1.07236	0.536181	−	0.844103i	$-0.319866\pi$
$997$	33.8602	1.07236	0.536181	+	0.844103i	$0.319866\pi$
$998$	0	0
$999$	−7.38299	−0.233588

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6004.2.a.g.1.3	✓	27

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6004.2.a.g.1.3	✓	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)

\(f(q)\)	\(=\)	\(q-2.74782 q^{3} -0.322226 q^{5} -1.10048 q^{7} +4.55049 q^{9} +O(q^{10})\)	\(q-2.74782 q^{3} -0.322226 q^{5} -1.10048 q^{7} +4.55049 q^{9} -1.90455 q^{11} -3.46075 q^{13} +0.885416 q^{15} +0.596613 q^{17} +1.00000 q^{19} +3.02391 q^{21} +3.97163 q^{23} -4.89617 q^{25} -4.26046 q^{27} +4.48417 q^{29} -0.142441 q^{31} +5.23334 q^{33} +0.354603 q^{35} +1.73291 q^{37} +9.50951 q^{39} -2.38206 q^{41} +5.85585 q^{43} -1.46628 q^{45} -1.46562 q^{47} -5.78895 q^{49} -1.63938 q^{51} -2.82437 q^{53} +0.613693 q^{55} -2.74782 q^{57} +10.6456 q^{59} -3.06513 q^{61} -5.00772 q^{63} +1.11514 q^{65} +3.30641 q^{67} -10.9133 q^{69} +9.93768 q^{71} +1.31446 q^{73} +13.4538 q^{75} +2.09591 q^{77} -1.00000 q^{79} -1.94452 q^{81} +12.5240 q^{83} -0.192244 q^{85} -12.3217 q^{87} -16.0319 q^{89} +3.80849 q^{91} +0.391402 q^{93} -0.322226 q^{95} +14.2275 q^{97} -8.66661 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 27 q - 4 q^{3} - 10 q^{5} - 8 q^{7} + 19 q^{9}+O(q^{10}) \) 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

Related objects

Downloads

Learn more