LMFDB - Embedded newform 6036.2.a.h.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6036,2,Mod(1,6036)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6036, 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("6036.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6036 = 2^{2} \cdot 3 \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6036.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:	$48.1977026600$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Character	$\chi$	$=$	6036.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.40769 q^{5} +3.41112 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.40769 q^{5} +3.41112 q^{7} +1.00000 q^{9} +2.64296 q^{11} +0.477768 q^{13} -1.40769 q^{15} +4.32494 q^{17} +5.99732 q^{19} +3.41112 q^{21} -6.32739 q^{23} -3.01841 q^{25} +1.00000 q^{27} +9.36681 q^{29} -0.871081 q^{31} +2.64296 q^{33} -4.80179 q^{35} -2.54899 q^{37} +0.477768 q^{39} -5.56833 q^{41} -0.111128 q^{43} -1.40769 q^{45} +9.54229 q^{47} +4.63571 q^{49} +4.32494 q^{51} +6.08525 q^{53} -3.72046 q^{55} +5.99732 q^{57} +14.5591 q^{59} -8.51162 q^{61} +3.41112 q^{63} -0.672549 q^{65} -1.75582 q^{67} -6.32739 q^{69} +8.75598 q^{71} -6.16070 q^{73} -3.01841 q^{75} +9.01544 q^{77} -5.60445 q^{79} +1.00000 q^{81} -14.6857 q^{83} -6.08817 q^{85} +9.36681 q^{87} +9.59659 q^{89} +1.62972 q^{91} -0.871081 q^{93} -8.44235 q^{95} +0.679094 q^{97} +2.64296 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 24 q^{3} + 18 q^{5} + 9 q^{7} + 24 q^{9}+O(q^{10}) $ 24 * q + 24 * q^3 + 18 * q^5 + 9 * q^7 + 24 * q^9	$ 24 q + 24 q^{3} + 18 q^{5} + 9 q^{7} + 24 q^{9} + 9 q^{11} + 7 q^{13} + 18 q^{15} + 12 q^{17} - q^{19} + 9 q^{21} + 22 q^{23} + 48 q^{25} + 24 q^{27} + 44 q^{29} + 15 q^{31} + 9 q^{33} + 3 q^{35} + 18 q^{37} + 7 q^{39} + 37 q^{41} - 8 q^{43} + 18 q^{45} + 24 q^{47} + 63 q^{49} + 12 q^{51} + 59 q^{53} + 20 q^{55} - q^{57} + 44 q^{59} + 33 q^{61} + 9 q^{63} + 15 q^{65} + 13 q^{67} + 22 q^{69} + 47 q^{71} + 28 q^{73} + 48 q^{75} + 19 q^{77} + 12 q^{79} + 24 q^{81} + 17 q^{83} + 31 q^{85} + 44 q^{87} + 41 q^{89} - q^{91} + 15 q^{93} + 58 q^{95} + 51 q^{97} + 9 q^{99}+O(q^{100}) $ 24 * q + 24 * q^3 + 18 * q^5 + 9 * q^7 + 24 * q^9 + 9 * q^11 + 7 * q^13 + 18 * q^15 + 12 * q^17 - q^19 + 9 * q^21 + 22 * q^23 + 48 * q^25 + 24 * q^27 + 44 * q^29 + 15 * q^31 + 9 * q^33 + 3 * q^35 + 18 * q^37 + 7 * q^39 + 37 * q^41 - 8 * q^43 + 18 * q^45 + 24 * q^47 + 63 * q^49 + 12 * q^51 + 59 * q^53 + 20 * q^55 - q^57 + 44 * q^59 + 33 * q^61 + 9 * q^63 + 15 * q^65 + 13 * q^67 + 22 * q^69 + 47 * q^71 + 28 * q^73 + 48 * q^75 + 19 * q^77 + 12 * q^79 + 24 * q^81 + 17 * q^83 + 31 * q^85 + 44 * q^87 + 41 * q^89 - q^91 + 15 * q^93 + 58 * q^95 + 51 * q^97 + 9 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−1.40769	−0.629537	−0.314769	−	0.949168i	$-0.601927\pi$
$5$	−1.40769	−0.629537	−0.314769	+	0.949168i	$0.601927\pi$
$6$	0	0
$7$	3.41112	1.28928	0.644640	−	0.764486i	$-0.277007\pi$
$7$	3.41112	1.28928	0.644640	+	0.764486i	$0.277007\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.64296	0.796882	0.398441	−	0.917194i	$-0.369551\pi$
$11$	2.64296	0.796882	0.398441	+	0.917194i	$0.369551\pi$
$12$	0	0
$13$	0.477768	0.132509	0.0662545	−	0.997803i	$-0.478895\pi$
$13$	0.477768	0.132509	0.0662545	+	0.997803i	$0.478895\pi$
$14$	0	0
$15$	−1.40769	−0.363464
$16$	0	0
$17$	4.32494	1.04895	0.524476	−	0.851425i	$-0.324261\pi$
$17$	4.32494	1.04895	0.524476	+	0.851425i	$0.324261\pi$
$18$	0	0
$19$	5.99732	1.37588	0.687939	−	0.725768i	$-0.258516\pi$
$19$	5.99732	1.37588	0.687939	+	0.725768i	$0.258516\pi$
$20$	0	0
$21$	3.41112	0.744366
$22$	0	0
$23$	−6.32739	−1.31935	−0.659676	−	0.751550i	$-0.729307\pi$
$23$	−6.32739	−1.31935	−0.659676	+	0.751550i	$0.729307\pi$
$24$	0	0
$25$	−3.01841	−0.603683
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	9.36681	1.73937	0.869686	−	0.493605i	$-0.164321\pi$
$29$	9.36681	1.73937	0.869686	+	0.493605i	$0.164321\pi$
$30$	0	0
$31$	−0.871081	−0.156451	−0.0782254	−	0.996936i	$-0.524925\pi$
$31$	−0.871081	−0.156451	−0.0782254	+	0.996936i	$0.524925\pi$
$32$	0	0
$33$	2.64296	0.460080
$34$	0	0
$35$	−4.80179	−0.811650
$36$	0	0
$37$	−2.54899	−0.419052	−0.209526	−	0.977803i	$-0.567192\pi$
$37$	−2.54899	−0.419052	−0.209526	+	0.977803i	$0.567192\pi$
$38$	0	0
$39$	0.477768	0.0765041
$40$	0	0
$41$	−5.56833	−0.869628	−0.434814	−	0.900520i	$-0.643186\pi$
$41$	−5.56833	−0.869628	−0.434814	+	0.900520i	$0.643186\pi$
$42$	0	0
$43$	−0.111128	−0.0169468	−0.00847340	−	0.999964i	$-0.502697\pi$
$43$	−0.111128	−0.0169468	−0.00847340	+	0.999964i	$0.502697\pi$
$44$	0	0
$45$	−1.40769	−0.209846
$46$	0	0
$47$	9.54229	1.39189	0.695943	−	0.718097i	$-0.254987\pi$
$47$	9.54229	1.39189	0.695943	+	0.718097i	$0.254987\pi$
$48$	0	0
$49$	4.63571	0.662244
$50$	0	0
$51$	4.32494	0.605613
$52$	0	0
$53$	6.08525	0.835873	0.417936	−	0.908476i	$-0.362753\pi$
$53$	6.08525	0.835873	0.417936	+	0.908476i	$0.362753\pi$
$54$	0	0
$55$	−3.72046	−0.501667
$56$	0	0
$57$	5.99732	0.794364
$58$	0	0
$59$	14.5591	1.89544	0.947719	−	0.319107i	$-0.103383\pi$
$59$	14.5591	1.89544	0.947719	+	0.319107i	$0.103383\pi$
$60$	0	0
$61$	−8.51162	−1.08980	−0.544901	−	0.838501i	$-0.683433\pi$
$61$	−8.51162	−1.08980	−0.544901	+	0.838501i	$0.683433\pi$
$62$	0	0
$63$	3.41112	0.429760
$64$	0	0
$65$	−0.672549	−0.0834194
$66$	0	0
$67$	−1.75582	−0.214508	−0.107254	−	0.994232i	$-0.534206\pi$
$67$	−1.75582	−0.214508	−0.107254	+	0.994232i	$0.534206\pi$
$68$	0	0
$69$	−6.32739	−0.761728
$70$	0	0
$71$	8.75598	1.03914	0.519572	−	0.854427i	$-0.326092\pi$
$71$	8.75598	1.03914	0.519572	+	0.854427i	$0.326092\pi$
$72$	0	0
$73$	−6.16070	−0.721056	−0.360528	−	0.932748i	$-0.617403\pi$
$73$	−6.16070	−0.721056	−0.360528	+	0.932748i	$0.617403\pi$
$74$	0	0
$75$	−3.01841	−0.348536
$76$	0	0
$77$	9.01544	1.02740
$78$	0	0
$79$	−5.60445	−0.630550	−0.315275	−	0.949000i	$-0.602097\pi$
$79$	−5.60445	−0.630550	−0.315275	+	0.949000i	$0.602097\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−14.6857	−1.61196	−0.805980	−	0.591943i	$-0.798361\pi$
$83$	−14.6857	−1.61196	−0.805980	+	0.591943i	$0.798361\pi$
$84$	0	0
$85$	−6.08817	−0.660355
$86$	0	0
$87$	9.36681	1.00423
$88$	0	0
$89$	9.59659	1.01724	0.508618	−	0.860992i	$-0.330156\pi$
$89$	9.59659	1.01724	0.508618	+	0.860992i	$0.330156\pi$
$90$	0	0
$91$	1.62972	0.170841
$92$	0	0
$93$	−0.871081	−0.0903269
$94$	0	0
$95$	−8.44235	−0.866167
$96$	0	0
$97$	0.679094	0.0689516	0.0344758	−	0.999406i	$-0.489024\pi$
$97$	0.679094	0.0689516	0.0344758	+	0.999406i	$0.489024\pi$
$98$	0	0
$99$	2.64296	0.265627
$100$	0	0
$101$	1.18567	0.117979	0.0589894	−	0.998259i	$-0.481212\pi$
$101$	1.18567	0.117979	0.0589894	+	0.998259i	$0.481212\pi$
$102$	0	0
$103$	−15.5890	−1.53603	−0.768014	−	0.640433i	$-0.778755\pi$
$103$	−15.5890	−1.53603	−0.768014	+	0.640433i	$0.778755\pi$
$104$	0	0
$105$	−4.80179	−0.468606
$106$	0	0
$107$	−11.7649	−1.13735	−0.568677	−	0.822561i	$-0.692545\pi$
$107$	−11.7649	−1.13735	−0.568677	+	0.822561i	$0.692545\pi$
$108$	0	0
$109$	−3.18315	−0.304890	−0.152445	−	0.988312i	$-0.548715\pi$
$109$	−3.18315	−0.304890	−0.152445	+	0.988312i	$0.548715\pi$
$110$	0	0
$111$	−2.54899	−0.241940
$112$	0	0
$113$	−12.9658	−1.21972	−0.609862	−	0.792508i	$-0.708775\pi$
$113$	−12.9658	−1.21972	−0.609862	+	0.792508i	$0.708775\pi$
$114$	0	0
$115$	8.90699	0.830582
$116$	0	0
$117$	0.477768	0.0441697
$118$	0	0
$119$	14.7529	1.35239
$120$	0	0
$121$	−4.01476	−0.364979
$122$	0	0
$123$	−5.56833	−0.502080
$124$	0	0
$125$	11.2874	1.00958
$126$	0	0
$127$	19.1751	1.70152	0.850758	−	0.525557i	$-0.176143\pi$
$127$	19.1751	1.70152	0.850758	+	0.525557i	$0.176143\pi$
$128$	0	0
$129$	−0.111128	−0.00978424
$130$	0	0
$131$	−2.11716	−0.184977	−0.0924885	−	0.995714i	$-0.529482\pi$
$131$	−2.11716	−0.184977	−0.0924885	+	0.995714i	$0.529482\pi$
$132$	0	0
$133$	20.4575	1.77389
$134$	0	0
$135$	−1.40769	−0.121155
$136$	0	0
$137$	6.23587	0.532766	0.266383	−	0.963867i	$-0.414171\pi$
$137$	6.23587	0.532766	0.266383	+	0.963867i	$0.414171\pi$
$138$	0	0
$139$	−6.29369	−0.533824	−0.266912	−	0.963721i	$-0.586003\pi$
$139$	−6.29369	−0.533824	−0.266912	+	0.963721i	$0.586003\pi$
$140$	0	0
$141$	9.54229	0.803606
$142$	0	0
$143$	1.26272	0.105594
$144$	0	0
$145$	−13.1855	−1.09500
$146$	0	0
$147$	4.63571	0.382347
$148$	0	0
$149$	18.5660	1.52099	0.760493	−	0.649346i	$-0.224957\pi$
$149$	18.5660	1.52099	0.760493	+	0.649346i	$0.224957\pi$
$150$	0	0
$151$	10.7315	0.873315	0.436658	−	0.899628i	$-0.356162\pi$
$151$	10.7315	0.873315	0.436658	+	0.899628i	$0.356162\pi$
$152$	0	0
$153$	4.32494	0.349651
$154$	0	0
$155$	1.22621	0.0984917
$156$	0	0
$157$	−5.42969	−0.433336	−0.216668	−	0.976245i	$-0.569519\pi$
$157$	−5.42969	−0.433336	−0.216668	+	0.976245i	$0.569519\pi$
$158$	0	0
$159$	6.08525	0.482591
$160$	0	0
$161$	−21.5835	−1.70101
$162$	0	0
$163$	18.0618	1.41471	0.707357	−	0.706857i	$-0.249888\pi$
$163$	18.0618	1.41471	0.707357	+	0.706857i	$0.249888\pi$
$164$	0	0
$165$	−3.72046	−0.289638
$166$	0	0
$167$	12.7548	0.986997	0.493499	−	0.869747i	$-0.335718\pi$
$167$	12.7548	0.986997	0.493499	+	0.869747i	$0.335718\pi$
$168$	0	0
$169$	−12.7717	−0.982441
$170$	0	0
$171$	5.99732	0.458626
$172$	0	0
$173$	−20.9486	−1.59270	−0.796348	−	0.604839i	$-0.793238\pi$
$173$	−20.9486	−1.59270	−0.796348	+	0.604839i	$0.793238\pi$
$174$	0	0
$175$	−10.2962	−0.778316
$176$	0	0
$177$	14.5591	1.09433
$178$	0	0
$179$	−1.96802	−0.147097	−0.0735483	−	0.997292i	$-0.523432\pi$
$179$	−1.96802	−0.147097	−0.0735483	+	0.997292i	$0.523432\pi$
$180$	0	0
$181$	0.327913	0.0243736	0.0121868	−	0.999926i	$-0.496121\pi$
$181$	0.327913	0.0243736	0.0121868	+	0.999926i	$0.496121\pi$
$182$	0	0
$183$	−8.51162	−0.629197
$184$	0	0
$185$	3.58819	0.263809
$186$	0	0
$187$	11.4306	0.835892
$188$	0	0
$189$	3.41112	0.248122
$190$	0	0
$191$	11.8291	0.855925	0.427963	−	0.903796i	$-0.359232\pi$
$191$	11.8291	0.855925	0.427963	+	0.903796i	$0.359232\pi$
$192$	0	0
$193$	−23.8607	−1.71753	−0.858766	−	0.512368i	$-0.828768\pi$
$193$	−23.8607	−1.71753	−0.858766	+	0.512368i	$0.828768\pi$
$194$	0	0
$195$	−0.672549	−0.0481622
$196$	0	0
$197$	−1.95528	−0.139308	−0.0696538	−	0.997571i	$-0.522189\pi$
$197$	−1.95528	−0.139308	−0.0696538	+	0.997571i	$0.522189\pi$
$198$	0	0
$199$	15.1464	1.07370	0.536851	−	0.843677i	$-0.319614\pi$
$199$	15.1464	1.07370	0.536851	+	0.843677i	$0.319614\pi$
$200$	0	0
$201$	−1.75582	−0.123846
$202$	0	0
$203$	31.9513	2.24254
$204$	0	0
$205$	7.83848	0.547463
$206$	0	0
$207$	−6.32739	−0.439784
$208$	0	0
$209$	15.8507	1.09641
$210$	0	0
$211$	−0.603036	−0.0415147	−0.0207573	−	0.999785i	$-0.506608\pi$
$211$	−0.603036	−0.0415147	−0.0207573	+	0.999785i	$0.506608\pi$
$212$	0	0
$213$	8.75598	0.599950
$214$	0	0
$215$	0.156433	0.0106686
$216$	0	0
$217$	−2.97136	−0.201709
$218$	0	0
$219$	−6.16070	−0.416302
$220$	0	0
$221$	2.06632	0.138996
$222$	0	0
$223$	7.43382	0.497805	0.248903	−	0.968529i	$-0.419930\pi$
$223$	7.43382	0.497805	0.248903	+	0.968529i	$0.419930\pi$
$224$	0	0
$225$	−3.01841	−0.201228
$226$	0	0
$227$	6.33199	0.420269	0.210135	−	0.977672i	$-0.432610\pi$
$227$	6.33199	0.420269	0.210135	+	0.977672i	$0.432610\pi$
$228$	0	0
$229$	−21.1295	−1.39628	−0.698138	−	0.715964i	$-0.745988\pi$
$229$	−21.1295	−1.39628	−0.698138	+	0.715964i	$0.745988\pi$
$230$	0	0
$231$	9.01544	0.593172
$232$	0	0
$233$	−0.910860	−0.0596724	−0.0298362	−	0.999555i	$-0.509499\pi$
$233$	−0.910860	−0.0596724	−0.0298362	+	0.999555i	$0.509499\pi$
$234$	0	0
$235$	−13.4326	−0.876245
$236$	0	0
$237$	−5.60445	−0.364048
$238$	0	0
$239$	23.4231	1.51511	0.757557	−	0.652769i	$-0.226393\pi$
$239$	23.4231	1.51511	0.757557	+	0.652769i	$0.226393\pi$
$240$	0	0
$241$	17.0657	1.09930	0.549648	−	0.835396i	$-0.314762\pi$
$241$	17.0657	1.09930	0.549648	+	0.835396i	$0.314762\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−6.52563	−0.416907
$246$	0	0
$247$	2.86533	0.182316
$248$	0	0
$249$	−14.6857	−0.930666
$250$	0	0
$251$	−5.25254	−0.331538	−0.165769	−	0.986165i	$-0.553011\pi$
$251$	−5.25254	−0.331538	−0.165769	+	0.986165i	$0.553011\pi$
$252$	0	0
$253$	−16.7230	−1.05137
$254$	0	0
$255$	−6.08817	−0.381256
$256$	0	0
$257$	−24.0457	−1.49993	−0.749964	−	0.661479i	$-0.769929\pi$
$257$	−24.0457	−1.49993	−0.749964	+	0.661479i	$0.769929\pi$
$258$	0	0
$259$	−8.69491	−0.540275
$260$	0	0
$261$	9.36681	0.579791
$262$	0	0
$263$	4.73773	0.292141	0.146070	−	0.989274i	$-0.453337\pi$
$263$	4.73773	0.292141	0.146070	+	0.989274i	$0.453337\pi$
$264$	0	0
$265$	−8.56613	−0.526213
$266$	0	0
$267$	9.59659	0.587302
$268$	0	0
$269$	27.6327	1.68479	0.842396	−	0.538859i	$-0.181144\pi$
$269$	27.6327	1.68479	0.842396	+	0.538859i	$0.181144\pi$
$270$	0	0
$271$	8.29964	0.504167	0.252084	−	0.967705i	$-0.418884\pi$
$271$	8.29964	0.504167	0.252084	+	0.967705i	$0.418884\pi$
$272$	0	0
$273$	1.62972	0.0986352
$274$	0	0
$275$	−7.97754	−0.481064
$276$	0	0
$277$	−5.89610	−0.354262	−0.177131	−	0.984187i	$-0.556682\pi$
$277$	−5.89610	−0.354262	−0.177131	+	0.984187i	$0.556682\pi$
$278$	0	0
$279$	−0.871081	−0.0521503
$280$	0	0
$281$	6.47156	0.386061	0.193030	−	0.981193i	$-0.438168\pi$
$281$	6.47156	0.386061	0.193030	+	0.981193i	$0.438168\pi$
$282$	0	0
$283$	2.91714	0.173406	0.0867031	−	0.996234i	$-0.472367\pi$
$283$	2.91714	0.173406	0.0867031	+	0.996234i	$0.472367\pi$
$284$	0	0
$285$	−8.44235	−0.500082
$286$	0	0
$287$	−18.9942	−1.12119
$288$	0	0
$289$	1.70512	0.100301
$290$	0	0
$291$	0.679094	0.0398092
$292$	0	0
$293$	29.5099	1.72399	0.861994	−	0.506919i	$-0.169216\pi$
$293$	29.5099	1.72399	0.861994	+	0.506919i	$0.169216\pi$
$294$	0	0
$295$	−20.4947	−1.19325
$296$	0	0
$297$	2.64296	0.153360
$298$	0	0
$299$	−3.02303	−0.174826
$300$	0	0
$301$	−0.379069	−0.0218492
$302$	0	0
$303$	1.18567	0.0681151
$304$	0	0
$305$	11.9817	0.686071
$306$	0	0
$307$	19.7270	1.12588	0.562941	−	0.826497i	$-0.309670\pi$
$307$	19.7270	1.12588	0.562941	+	0.826497i	$0.309670\pi$
$308$	0	0
$309$	−15.5890	−0.886826
$310$	0	0
$311$	15.9534	0.904633	0.452316	−	0.891857i	$-0.350598\pi$
$311$	15.9534	0.904633	0.452316	+	0.891857i	$0.350598\pi$
$312$	0	0
$313$	−6.94029	−0.392288	−0.196144	−	0.980575i	$-0.562842\pi$
$313$	−6.94029	−0.392288	−0.196144	+	0.980575i	$0.562842\pi$
$314$	0	0
$315$	−4.80179	−0.270550
$316$	0	0
$317$	23.2018	1.30314	0.651571	−	0.758587i	$-0.274110\pi$
$317$	23.2018	1.30314	0.651571	+	0.758587i	$0.274110\pi$
$318$	0	0
$319$	24.7561	1.38608
$320$	0	0
$321$	−11.7649	−0.656652
$322$	0	0
$323$	25.9380	1.44323
$324$	0	0
$325$	−1.44210	−0.0799934
$326$	0	0
$327$	−3.18315	−0.176028
$328$	0	0
$329$	32.5499	1.79453
$330$	0	0
$331$	26.3978	1.45096	0.725479	−	0.688245i	$-0.241619\pi$
$331$	26.3978	1.45096	0.725479	+	0.688245i	$0.241619\pi$
$332$	0	0
$333$	−2.54899	−0.139684
$334$	0	0
$335$	2.47165	0.135041
$336$	0	0
$337$	18.1707	0.989819	0.494909	−	0.868945i	$-0.335201\pi$
$337$	18.1707	0.989819	0.494909	+	0.868945i	$0.335201\pi$
$338$	0	0
$339$	−12.9658	−0.704208
$340$	0	0
$341$	−2.30223	−0.124673
$342$	0	0
$343$	−8.06488	−0.435462
$344$	0	0
$345$	8.90699	0.479536
$346$	0	0
$347$	−5.36100	−0.287793	−0.143897	−	0.989593i	$-0.545963\pi$
$347$	−5.36100	−0.287793	−0.143897	+	0.989593i	$0.545963\pi$
$348$	0	0
$349$	25.7512	1.37843	0.689215	−	0.724557i	$-0.257956\pi$
$349$	25.7512	1.37843	0.689215	+	0.724557i	$0.257956\pi$
$350$	0	0
$351$	0.477768	0.0255014
$352$	0	0
$353$	−18.4592	−0.982482	−0.491241	−	0.871024i	$-0.663457\pi$
$353$	−18.4592	−0.982482	−0.491241	+	0.871024i	$0.663457\pi$
$354$	0	0
$355$	−12.3257	−0.654180
$356$	0	0
$357$	14.7529	0.780805
$358$	0	0
$359$	−14.1864	−0.748727	−0.374364	−	0.927282i	$-0.622139\pi$
$359$	−14.1864	−0.748727	−0.374364	+	0.927282i	$0.622139\pi$
$360$	0	0
$361$	16.9678	0.893042
$362$	0	0
$363$	−4.01476	−0.210721
$364$	0	0
$365$	8.67235	0.453932
$366$	0	0
$367$	−19.2676	−1.00576	−0.502880	−	0.864356i	$-0.667726\pi$
$367$	−19.2676	−1.00576	−0.502880	+	0.864356i	$0.667726\pi$
$368$	0	0
$369$	−5.56833	−0.289876
$370$	0	0
$371$	20.7575	1.07767
$372$	0	0
$373$	−13.8158	−0.715355	−0.357678	−	0.933845i	$-0.616431\pi$
$373$	−13.8158	−0.715355	−0.357678	+	0.933845i	$0.616431\pi$
$374$	0	0
$375$	11.2874	0.582880
$376$	0	0
$377$	4.47516	0.230483
$378$	0	0
$379$	5.84656	0.300317	0.150159	−	0.988662i	$-0.452022\pi$
$379$	5.84656	0.300317	0.150159	+	0.988662i	$0.452022\pi$
$380$	0	0
$381$	19.1751	0.982371
$382$	0	0
$383$	−31.4795	−1.60853	−0.804265	−	0.594271i	$-0.797441\pi$
$383$	−31.4795	−1.60853	−0.804265	+	0.594271i	$0.797441\pi$
$384$	0	0
$385$	−12.6909	−0.646790
$386$	0	0
$387$	−0.111128	−0.00564893
$388$	0	0
$389$	−5.68941	−0.288465	−0.144232	−	0.989544i	$-0.546071\pi$
$389$	−5.68941	−0.288465	−0.144232	+	0.989544i	$0.546071\pi$
$390$	0	0
$391$	−27.3656	−1.38394
$392$	0	0
$393$	−2.11716	−0.106796
$394$	0	0
$395$	7.88932	0.396955
$396$	0	0
$397$	20.4453	1.02612	0.513060	−	0.858353i	$-0.328512\pi$
$397$	20.4453	1.02612	0.513060	+	0.858353i	$0.328512\pi$
$398$	0	0
$399$	20.4575	1.02416
$400$	0	0
$401$	−15.7593	−0.786981	−0.393490	−	0.919329i	$-0.628733\pi$
$401$	−15.7593	−0.786981	−0.393490	+	0.919329i	$0.628733\pi$
$402$	0	0
$403$	−0.416175	−0.0207311
$404$	0	0
$405$	−1.40769	−0.0699486
$406$	0	0
$407$	−6.73688	−0.333935
$408$	0	0
$409$	7.81737	0.386544	0.193272	−	0.981145i	$-0.438090\pi$
$409$	7.81737	0.386544	0.193272	+	0.981145i	$0.438090\pi$
$410$	0	0
$411$	6.23587	0.307593
$412$	0	0
$413$	49.6629	2.44375
$414$	0	0
$415$	20.6728	1.01479
$416$	0	0
$417$	−6.29369	−0.308204
$418$	0	0
$419$	−31.8685	−1.55688	−0.778438	−	0.627721i	$-0.783988\pi$
$419$	−31.8685	−1.55688	−0.778438	+	0.627721i	$0.783988\pi$
$420$	0	0
$421$	−33.0375	−1.61015	−0.805074	−	0.593174i	$-0.797875\pi$
$421$	−33.0375	−1.61015	−0.805074	+	0.593174i	$0.797875\pi$
$422$	0	0
$423$	9.54229	0.463962
$424$	0	0
$425$	−13.0545	−0.633234
$426$	0	0
$427$	−29.0341	−1.40506
$428$	0	0
$429$	1.26272	0.0609648
$430$	0	0
$431$	−21.0377	−1.01335	−0.506674	−	0.862138i	$-0.669125\pi$
$431$	−21.0377	−1.01335	−0.506674	+	0.862138i	$0.669125\pi$
$432$	0	0
$433$	37.9448	1.82351	0.911756	−	0.410733i	$-0.134727\pi$
$433$	37.9448	1.82351	0.911756	+	0.410733i	$0.134727\pi$
$434$	0	0
$435$	−13.1855	−0.632199
$436$	0	0
$437$	−37.9474	−1.81527
$438$	0	0
$439$	3.48653	0.166403	0.0832014	−	0.996533i	$-0.473486\pi$
$439$	3.48653	0.166403	0.0832014	+	0.996533i	$0.473486\pi$
$440$	0	0
$441$	4.63571	0.220748
$442$	0	0
$443$	15.3266	0.728189	0.364095	−	0.931362i	$-0.381378\pi$
$443$	15.3266	0.728189	0.364095	+	0.931362i	$0.381378\pi$
$444$	0	0
$445$	−13.5090	−0.640388
$446$	0	0
$447$	18.5660	0.878141
$448$	0	0
$449$	−15.6967	−0.740775	−0.370388	−	0.928877i	$-0.620775\pi$
$449$	−15.6967	−0.740775	−0.370388	+	0.928877i	$0.620775\pi$
$450$	0	0
$451$	−14.7169	−0.692991
$452$	0	0
$453$	10.7315	0.504209
$454$	0	0
$455$	−2.29414	−0.107551
$456$	0	0
$457$	11.9013	0.556718	0.278359	−	0.960477i	$-0.410210\pi$
$457$	11.9013	0.556718	0.278359	+	0.960477i	$0.410210\pi$
$458$	0	0
$459$	4.32494	0.201871
$460$	0	0
$461$	−1.14715	−0.0534283	−0.0267141	−	0.999643i	$-0.508504\pi$
$461$	−1.14715	−0.0534283	−0.0267141	+	0.999643i	$0.508504\pi$
$462$	0	0
$463$	−23.3419	−1.08479	−0.542394	−	0.840124i	$-0.682482\pi$
$463$	−23.3419	−1.08479	−0.542394	+	0.840124i	$0.682482\pi$
$464$	0	0
$465$	1.22621	0.0568642
$466$	0	0
$467$	−3.93430	−0.182058	−0.0910288	−	0.995848i	$-0.529016\pi$
$467$	−3.93430	−0.182058	−0.0910288	+	0.995848i	$0.529016\pi$
$468$	0	0
$469$	−5.98932	−0.276561
$470$	0	0
$471$	−5.42969	−0.250187
$472$	0	0
$473$	−0.293706	−0.0135046
$474$	0	0
$475$	−18.1024	−0.830594
$476$	0	0
$477$	6.08525	0.278624
$478$	0	0
$479$	4.78311	0.218546	0.109273	−	0.994012i	$-0.465148\pi$
$479$	4.78311	0.218546	0.109273	+	0.994012i	$0.465148\pi$
$480$	0	0
$481$	−1.21783	−0.0555281
$482$	0	0
$483$	−21.5835	−0.982081
$484$	0	0
$485$	−0.955953	−0.0434076
$486$	0	0
$487$	32.0811	1.45373	0.726866	−	0.686779i	$-0.240976\pi$
$487$	32.0811	1.45373	0.726866	+	0.686779i	$0.240976\pi$
$488$	0	0
$489$	18.0618	0.816785
$490$	0	0
$491$	27.2879	1.23148	0.615742	−	0.787948i	$-0.288856\pi$
$491$	27.2879	1.23148	0.615742	+	0.787948i	$0.288856\pi$
$492$	0	0
$493$	40.5109	1.82452
$494$	0	0
$495$	−3.72046	−0.167222
$496$	0	0
$497$	29.8676	1.33975
$498$	0	0
$499$	−0.803528	−0.0359709	−0.0179854	−	0.999838i	$-0.505725\pi$
$499$	−0.803528	−0.0359709	−0.0179854	+	0.999838i	$0.505725\pi$
$500$	0	0
$501$	12.7548	0.569843
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	−1.66906	−0.0742721
$506$	0	0
$507$	−12.7717	−0.567213
$508$	0	0
$509$	10.2114	0.452614	0.226307	−	0.974056i	$-0.427335\pi$
$509$	10.2114	0.452614	0.226307	+	0.974056i	$0.427335\pi$
$510$	0	0
$511$	−21.0149	−0.929643
$512$	0	0
$513$	5.99732	0.264788
$514$	0	0
$515$	21.9444	0.966987
$516$	0	0
$517$	25.2199	1.10917
$518$	0	0
$519$	−20.9486	−0.919544
$520$	0	0
$521$	−1.94149	−0.0850584	−0.0425292	−	0.999095i	$-0.513542\pi$
$521$	−1.94149	−0.0850584	−0.0425292	+	0.999095i	$0.513542\pi$
$522$	0	0
$523$	−35.1335	−1.53628	−0.768140	−	0.640281i	$-0.778818\pi$
$523$	−35.1335	−1.53628	−0.768140	+	0.640281i	$0.778818\pi$
$524$	0	0
$525$	−10.2962	−0.449361
$526$	0	0
$527$	−3.76738	−0.164109
$528$	0	0
$529$	17.0359	0.740690
$530$	0	0
$531$	14.5591	0.631813
$532$	0	0
$533$	−2.66037	−0.115234
$534$	0	0
$535$	16.5613	0.716007
$536$	0	0
$537$	−1.96802	−0.0849263
$538$	0	0
$539$	12.2520	0.527730
$540$	0	0
$541$	−26.5418	−1.14112	−0.570561	−	0.821255i	$-0.693274\pi$
$541$	−26.5418	−1.14112	−0.570561	+	0.821255i	$0.693274\pi$
$542$	0	0
$543$	0.327913	0.0140721
$544$	0	0
$545$	4.48088	0.191940
$546$	0	0
$547$	−12.1414	−0.519129	−0.259564	−	0.965726i	$-0.583579\pi$
$547$	−12.1414	−0.519129	−0.259564	+	0.965726i	$0.583579\pi$
$548$	0	0
$549$	−8.51162	−0.363267
$550$	0	0
$551$	56.1757	2.39317
$552$	0	0
$553$	−19.1174	−0.812956
$554$	0	0
$555$	3.58819	0.152310
$556$	0	0
$557$	−14.2696	−0.604622	−0.302311	−	0.953209i	$-0.597758\pi$
$557$	−14.2696	−0.604622	−0.302311	+	0.953209i	$0.597758\pi$
$558$	0	0
$559$	−0.0530932	−0.00224560
$560$	0	0
$561$	11.4306	0.482602
$562$	0	0
$563$	−26.7493	−1.12735	−0.563674	−	0.825998i	$-0.690612\pi$
$563$	−26.7493	−1.12735	−0.563674	+	0.825998i	$0.690612\pi$
$564$	0	0
$565$	18.2519	0.767862
$566$	0	0
$567$	3.41112	0.143253
$568$	0	0
$569$	13.8180	0.579283	0.289641	−	0.957135i	$-0.406464\pi$
$569$	13.8180	0.579283	0.289641	+	0.957135i	$0.406464\pi$
$570$	0	0
$571$	−13.9875	−0.585360	−0.292680	−	0.956210i	$-0.594547\pi$
$571$	−13.9875	−0.585360	−0.292680	+	0.956210i	$0.594547\pi$
$572$	0	0
$573$	11.8291	0.494169
$574$	0	0
$575$	19.0987	0.796470
$576$	0	0
$577$	41.7739	1.73907	0.869534	−	0.493873i	$-0.164419\pi$
$577$	41.7739	1.73907	0.869534	+	0.493873i	$0.164419\pi$
$578$	0	0
$579$	−23.8607	−0.991617
$580$	0	0
$581$	−50.0945	−2.07827
$582$	0	0
$583$	16.0831	0.666092
$584$	0	0
$585$	−0.672549	−0.0278065
$586$	0	0
$587$	−17.7279	−0.731710	−0.365855	−	0.930672i	$-0.619223\pi$
$587$	−17.7279	−0.731710	−0.365855	+	0.930672i	$0.619223\pi$
$588$	0	0
$589$	−5.22415	−0.215257
$590$	0	0
$591$	−1.95528	−0.0804293
$592$	0	0
$593$	4.75500	0.195264	0.0976321	−	0.995223i	$-0.468873\pi$
$593$	4.75500	0.195264	0.0976321	+	0.995223i	$0.468873\pi$
$594$	0	0
$595$	−20.7675	−0.851382
$596$	0	0
$597$	15.1464	0.619902
$598$	0	0
$599$	−12.6441	−0.516623	−0.258311	−	0.966062i	$-0.583166\pi$
$599$	−12.6441	−0.516623	−0.258311	+	0.966062i	$0.583166\pi$
$600$	0	0
$601$	−38.8311	−1.58395	−0.791977	−	0.610550i	$-0.790948\pi$
$601$	−38.8311	−1.58395	−0.791977	+	0.610550i	$0.790948\pi$
$602$	0	0
$603$	−1.75582	−0.0715026
$604$	0	0
$605$	5.65154	0.229768
$606$	0	0
$607$	−37.5519	−1.52418	−0.762091	−	0.647469i	$-0.775827\pi$
$607$	−37.5519	−1.52418	−0.762091	+	0.647469i	$0.775827\pi$
$608$	0	0
$609$	31.9513	1.29473
$610$	0	0
$611$	4.55900	0.184437
$612$	0	0
$613$	−20.5574	−0.830307	−0.415154	−	0.909751i	$-0.636272\pi$
$613$	−20.5574	−0.830307	−0.415154	+	0.909751i	$0.636272\pi$
$614$	0	0
$615$	7.83848	0.316078
$616$	0	0
$617$	−4.82442	−0.194224	−0.0971119	−	0.995273i	$-0.530960\pi$
$617$	−4.82442	−0.194224	−0.0971119	+	0.995273i	$0.530960\pi$
$618$	0	0
$619$	−2.18601	−0.0878630	−0.0439315	−	0.999035i	$-0.513988\pi$
$619$	−2.18601	−0.0878630	−0.0439315	+	0.999035i	$0.513988\pi$
$620$	0	0
$621$	−6.32739	−0.253909
$622$	0	0
$623$	32.7351	1.31150
$624$	0	0
$625$	−0.797116	−0.0318846
$626$	0	0
$627$	15.8507	0.633014
$628$	0	0
$629$	−11.0242	−0.439565
$630$	0	0
$631$	36.2963	1.44493	0.722466	−	0.691406i	$-0.243009\pi$
$631$	36.2963	1.44493	0.722466	+	0.691406i	$0.243009\pi$
$632$	0	0
$633$	−0.603036	−0.0239685
$634$	0	0
$635$	−26.9926	−1.07117
$636$	0	0
$637$	2.21479	0.0877533
$638$	0	0
$639$	8.75598	0.346381
$640$	0	0
$641$	−34.2239	−1.35176	−0.675882	−	0.737010i	$-0.736237\pi$
$641$	−34.2239	−1.35176	−0.675882	+	0.737010i	$0.736237\pi$
$642$	0	0
$643$	−39.8063	−1.56981	−0.784903	−	0.619618i	$-0.787287\pi$
$643$	−39.8063	−1.56981	−0.784903	+	0.619618i	$0.787287\pi$
$644$	0	0
$645$	0.156433	0.00615955
$646$	0	0
$647$	16.1837	0.636247	0.318123	−	0.948049i	$-0.396947\pi$
$647$	16.1837	0.636247	0.318123	+	0.948049i	$0.396947\pi$
$648$	0	0
$649$	38.4792	1.51044
$650$	0	0
$651$	−2.97136	−0.116457
$652$	0	0
$653$	1.29413	0.0506431	0.0253216	−	0.999679i	$-0.491939\pi$
$653$	1.29413	0.0506431	0.0253216	+	0.999679i	$0.491939\pi$
$654$	0	0
$655$	2.98030	0.116450
$656$	0	0
$657$	−6.16070	−0.240352
$658$	0	0
$659$	16.0187	0.624001	0.312001	−	0.950082i	$-0.399001\pi$
$659$	16.0187	0.624001	0.312001	+	0.950082i	$0.399001\pi$
$660$	0	0
$661$	30.7238	1.19502	0.597509	−	0.801862i	$-0.296157\pi$
$661$	30.7238	1.19502	0.597509	+	0.801862i	$0.296157\pi$
$662$	0	0
$663$	2.06632	0.0802492
$664$	0	0
$665$	−28.7978	−1.11673
$666$	0	0
$667$	−59.2675	−2.29485
$668$	0	0
$669$	7.43382	0.287408
$670$	0	0
$671$	−22.4959	−0.868443
$672$	0	0
$673$	28.3765	1.09383	0.546917	−	0.837187i	$-0.315801\pi$
$673$	28.3765	1.09383	0.546917	+	0.837187i	$0.315801\pi$
$674$	0	0
$675$	−3.01841	−0.116179
$676$	0	0
$677$	−19.7474	−0.758956	−0.379478	−	0.925201i	$-0.623896\pi$
$677$	−19.7474	−0.758956	−0.379478	+	0.925201i	$0.623896\pi$
$678$	0	0
$679$	2.31647	0.0888979
$680$	0	0
$681$	6.33199	0.242643
$682$	0	0
$683$	−35.3534	−1.35276	−0.676379	−	0.736554i	$-0.736452\pi$
$683$	−35.3534	−1.35276	−0.676379	+	0.736554i	$0.736452\pi$
$684$	0	0
$685$	−8.77816	−0.335396
$686$	0	0
$687$	−21.1295	−0.806140
$688$	0	0
$689$	2.90734	0.110761
$690$	0	0
$691$	−0.150378	−0.00572066	−0.00286033	−	0.999996i	$-0.500910\pi$
$691$	−0.150378	−0.00572066	−0.00286033	+	0.999996i	$0.500910\pi$
$692$	0	0
$693$	9.01544	0.342468
$694$	0	0
$695$	8.85956	0.336062
$696$	0	0
$697$	−24.0827	−0.912198
$698$	0	0
$699$	−0.910860	−0.0344519
$700$	0	0
$701$	18.9340	0.715128	0.357564	−	0.933889i	$-0.383607\pi$
$701$	18.9340	0.715128	0.357564	+	0.933889i	$0.383607\pi$
$702$	0	0
$703$	−15.2871	−0.576564
$704$	0	0
$705$	−13.4326	−0.505900
$706$	0	0
$707$	4.04447	0.152108
$708$	0	0
$709$	14.3060	0.537271	0.268636	−	0.963242i	$-0.413427\pi$
$709$	14.3060	0.537271	0.268636	+	0.963242i	$0.413427\pi$
$710$	0	0
$711$	−5.60445	−0.210183
$712$	0	0
$713$	5.51167	0.206414
$714$	0	0
$715$	−1.77752	−0.0664754
$716$	0	0
$717$	23.4231	0.874752
$718$	0	0
$719$	32.3766	1.20744	0.603721	−	0.797195i	$-0.293684\pi$
$719$	32.3766	1.20744	0.603721	+	0.797195i	$0.293684\pi$
$720$	0	0
$721$	−53.1758	−1.98037
$722$	0	0
$723$	17.0657	0.634679
$724$	0	0
$725$	−28.2729	−1.05003
$726$	0	0
$727$	−15.6849	−0.581720	−0.290860	−	0.956766i	$-0.593941\pi$
$727$	−15.6849	−0.581720	−0.290860	+	0.956766i	$0.593941\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−0.480620	−0.0177764
$732$	0	0
$733$	−14.7555	−0.545006	−0.272503	−	0.962155i	$-0.587851\pi$
$733$	−14.7555	−0.545006	−0.272503	+	0.962155i	$0.587851\pi$
$734$	0	0
$735$	−6.52563	−0.240702
$736$	0	0
$737$	−4.64057	−0.170938
$738$	0	0
$739$	10.0757	0.370642	0.185321	−	0.982678i	$-0.440668\pi$
$739$	10.0757	0.370642	0.185321	+	0.982678i	$0.440668\pi$
$740$	0	0
$741$	2.86533	0.105260
$742$	0	0
$743$	−3.97514	−0.145834	−0.0729169	−	0.997338i	$-0.523231\pi$
$743$	−3.97514	−0.145834	−0.0729169	+	0.997338i	$0.523231\pi$
$744$	0	0
$745$	−26.1351	−0.957517
$746$	0	0
$747$	−14.6857	−0.537320
$748$	0	0
$749$	−40.1314	−1.46637
$750$	0	0
$751$	−23.0002	−0.839291	−0.419646	−	0.907688i	$-0.637846\pi$
$751$	−23.0002	−0.839291	−0.419646	+	0.907688i	$0.637846\pi$
$752$	0	0
$753$	−5.25254	−0.191413
$754$	0	0
$755$	−15.1066	−0.549784
$756$	0	0
$757$	−22.3187	−0.811186	−0.405593	−	0.914054i	$-0.632935\pi$
$757$	−22.3187	−0.811186	−0.405593	+	0.914054i	$0.632935\pi$
$758$	0	0
$759$	−16.7230	−0.607008
$760$	0	0
$761$	21.5645	0.781714	0.390857	−	0.920451i	$-0.372179\pi$
$761$	21.5645	0.781714	0.390857	+	0.920451i	$0.372179\pi$
$762$	0	0
$763$	−10.8581	−0.393089
$764$	0	0
$765$	−6.08817	−0.220118
$766$	0	0
$767$	6.95589	0.251163
$768$	0	0
$769$	−22.0918	−0.796650	−0.398325	−	0.917244i	$-0.630408\pi$
$769$	−22.0918	−0.796650	−0.398325	+	0.917244i	$0.630408\pi$
$770$	0	0
$771$	−24.0457	−0.865983
$772$	0	0
$773$	−50.4144	−1.81328	−0.906641	−	0.421903i	$-0.861362\pi$
$773$	−50.4144	−1.81328	−0.906641	+	0.421903i	$0.861362\pi$
$774$	0	0
$775$	2.62928	0.0944467
$776$	0	0
$777$	−8.69491	−0.311928
$778$	0	0
$779$	−33.3951	−1.19650
$780$	0	0
$781$	23.1417	0.828075
$782$	0	0
$783$	9.36681	0.334742
$784$	0	0
$785$	7.64331	0.272801
$786$	0	0
$787$	−44.8556	−1.59893	−0.799465	−	0.600712i	$-0.794884\pi$
$787$	−44.8556	−1.59893	−0.799465	+	0.600712i	$0.794884\pi$
$788$	0	0
$789$	4.73773	0.168668
$790$	0	0
$791$	−44.2280	−1.57257
$792$	0	0
$793$	−4.06658	−0.144409
$794$	0	0
$795$	−8.56613	−0.303809
$796$	0	0
$797$	−13.4590	−0.476741	−0.238371	−	0.971174i	$-0.576613\pi$
$797$	−13.4590	−0.476741	−0.238371	+	0.971174i	$0.576613\pi$
$798$	0	0
$799$	41.2699	1.46002
$800$	0	0
$801$	9.59659	0.339079
$802$	0	0
$803$	−16.2825	−0.574596
$804$	0	0
$805$	30.3828	1.07085
$806$	0	0
$807$	27.6327	0.972715
$808$	0	0
$809$	40.4779	1.42313	0.711564	−	0.702621i	$-0.247987\pi$
$809$	40.4779	1.42313	0.711564	+	0.702621i	$0.247987\pi$
$810$	0	0
$811$	−40.5012	−1.42219	−0.711094	−	0.703097i	$-0.751800\pi$
$811$	−40.5012	−1.42219	−0.711094	+	0.703097i	$0.751800\pi$
$812$	0	0
$813$	8.29964	0.291081
$814$	0	0
$815$	−25.4255	−0.890615
$816$	0	0
$817$	−0.666467	−0.0233167
$818$	0	0
$819$	1.62972	0.0569471
$820$	0	0
$821$	−38.2325	−1.33432	−0.667161	−	0.744913i	$-0.732491\pi$
$821$	−38.2325	−1.33432	−0.667161	+	0.744913i	$0.732491\pi$
$822$	0	0
$823$	9.39516	0.327495	0.163747	−	0.986502i	$-0.447642\pi$
$823$	9.39516	0.327495	0.163747	+	0.986502i	$0.447642\pi$
$824$	0	0
$825$	−7.97754	−0.277742
$826$	0	0
$827$	22.2739	0.774540	0.387270	−	0.921966i	$-0.373418\pi$
$827$	22.2739	0.774540	0.387270	+	0.921966i	$0.373418\pi$
$828$	0	0
$829$	25.5089	0.885962	0.442981	−	0.896531i	$-0.353921\pi$
$829$	25.5089	0.885962	0.442981	+	0.896531i	$0.353921\pi$
$830$	0	0
$831$	−5.89610	−0.204533
$832$	0	0
$833$	20.0492	0.694662
$834$	0	0
$835$	−17.9548	−0.621352
$836$	0	0
$837$	−0.871081	−0.0301090
$838$	0	0
$839$	−45.4880	−1.57042	−0.785211	−	0.619228i	$-0.787446\pi$
$839$	−45.4880	−1.57042	−0.785211	+	0.619228i	$0.787446\pi$
$840$	0	0
$841$	58.7371	2.02542
$842$	0	0
$843$	6.47156	0.222892
$844$	0	0
$845$	17.9786	0.618484
$846$	0	0
$847$	−13.6948	−0.470560
$848$	0	0
$849$	2.91714	0.100116
$850$	0	0
$851$	16.1285	0.552877
$852$	0	0
$853$	39.6498	1.35758	0.678792	−	0.734331i	$-0.262504\pi$
$853$	39.6498	1.35758	0.678792	+	0.734331i	$0.262504\pi$
$854$	0	0
$855$	−8.44235	−0.288722
$856$	0	0
$857$	−16.6688	−0.569395	−0.284697	−	0.958617i	$-0.591893\pi$
$857$	−16.6688	−0.569395	−0.284697	+	0.958617i	$0.591893\pi$
$858$	0	0
$859$	−34.2891	−1.16993	−0.584965	−	0.811059i	$-0.698892\pi$
$859$	−34.2891	−1.16993	−0.584965	+	0.811059i	$0.698892\pi$
$860$	0	0
$861$	−18.9942	−0.647322
$862$	0	0
$863$	33.4514	1.13870	0.569349	−	0.822096i	$-0.307195\pi$
$863$	33.4514	1.13870	0.569349	+	0.822096i	$0.307195\pi$
$864$	0	0
$865$	29.4892	1.00266
$866$	0	0
$867$	1.70512	0.0579089
$868$	0	0
$869$	−14.8123	−0.502474
$870$	0	0
$871$	−0.838876	−0.0284242
$872$	0	0
$873$	0.679094	0.0229839
$874$	0	0
$875$	38.5027	1.30163
$876$	0	0
$877$	34.5797	1.16767	0.583836	−	0.811871i	$-0.301551\pi$
$877$	34.5797	1.16767	0.583836	+	0.811871i	$0.301551\pi$
$878$	0	0
$879$	29.5099	0.995345
$880$	0	0
$881$	−40.9375	−1.37922	−0.689610	−	0.724181i	$-0.742218\pi$
$881$	−40.9375	−1.37922	−0.689610	+	0.724181i	$0.742218\pi$
$882$	0	0
$883$	−44.7943	−1.50745	−0.753724	−	0.657191i	$-0.771744\pi$
$883$	−44.7943	−1.50745	−0.753724	+	0.657191i	$0.771744\pi$
$884$	0	0
$885$	−20.4947	−0.688923
$886$	0	0
$887$	27.7758	0.932621	0.466310	−	0.884621i	$-0.345583\pi$
$887$	27.7758	0.932621	0.466310	+	0.884621i	$0.345583\pi$
$888$	0	0
$889$	65.4085	2.19373
$890$	0	0
$891$	2.64296	0.0885425
$892$	0	0
$893$	57.2281	1.91507
$894$	0	0
$895$	2.77036	0.0926028
$896$	0	0
$897$	−3.02303	−0.100936
$898$	0	0
$899$	−8.15925	−0.272126
$900$	0	0
$901$	26.3183	0.876791
$902$	0	0
$903$	−0.379069	−0.0126146
$904$	0	0
$905$	−0.461599	−0.0153441
$906$	0	0
$907$	−35.6640	−1.18420	−0.592101	−	0.805864i	$-0.701701\pi$
$907$	−35.6640	−1.18420	−0.592101	+	0.805864i	$0.701701\pi$
$908$	0	0
$909$	1.18567	0.0393263
$910$	0	0
$911$	−20.3045	−0.672717	−0.336359	−	0.941734i	$-0.609195\pi$
$911$	−20.3045	−0.672717	−0.336359	+	0.941734i	$0.609195\pi$
$912$	0	0
$913$	−38.8136	−1.28454
$914$	0	0
$915$	11.9817	0.396103
$916$	0	0
$917$	−7.22187	−0.238487
$918$	0	0
$919$	23.5599	0.777171	0.388585	−	0.921413i	$-0.372964\pi$
$919$	23.5599	0.777171	0.388585	+	0.921413i	$0.372964\pi$
$920$	0	0
$921$	19.7270	0.650028
$922$	0	0
$923$	4.18333	0.137696
$924$	0	0
$925$	7.69391	0.252974
$926$	0	0
$927$	−15.5890	−0.512009
$928$	0	0
$929$	−2.01942	−0.0662552	−0.0331276	−	0.999451i	$-0.510547\pi$
$929$	−2.01942	−0.0662552	−0.0331276	+	0.999451i	$0.510547\pi$
$930$	0	0
$931$	27.8018	0.911167
$932$	0	0
$933$	15.9534	0.522290
$934$	0	0
$935$	−16.0908	−0.526225
$936$	0	0
$937$	37.9562	1.23997	0.619987	−	0.784612i	$-0.287138\pi$
$937$	37.9562	1.23997	0.619987	+	0.784612i	$0.287138\pi$
$938$	0	0
$939$	−6.94029	−0.226488
$940$	0	0
$941$	0.155534	0.00507028	0.00253514	−	0.999997i	$-0.499193\pi$
$941$	0.155534	0.00507028	0.00253514	+	0.999997i	$0.499193\pi$
$942$	0	0
$943$	35.2330	1.14735
$944$	0	0
$945$	−4.80179	−0.156202
$946$	0	0
$947$	−46.5864	−1.51385	−0.756927	−	0.653500i	$-0.773300\pi$
$947$	−46.5864	−1.51385	−0.756927	+	0.653500i	$0.773300\pi$
$948$	0	0
$949$	−2.94339	−0.0955464
$950$	0	0
$951$	23.2018	0.752370
$952$	0	0
$953$	38.0816	1.23358	0.616792	−	0.787126i	$-0.288432\pi$
$953$	38.0816	1.23358	0.616792	+	0.787126i	$0.288432\pi$
$954$	0	0
$955$	−16.6517	−0.538837
$956$	0	0
$957$	24.7561	0.800251
$958$	0	0
$959$	21.2713	0.686885
$960$	0	0
$961$	−30.2412	−0.975523
$962$	0	0
$963$	−11.7649	−0.379118
$964$	0	0
$965$	33.5885	1.08125
$966$	0	0
$967$	−30.9145	−0.994143	−0.497072	−	0.867709i	$-0.665591\pi$
$967$	−30.9145	−0.994143	−0.497072	+	0.867709i	$0.665591\pi$
$968$	0	0
$969$	25.9380	0.833250
$970$	0	0
$971$	−6.97191	−0.223739	−0.111870	−	0.993723i	$-0.535684\pi$
$971$	−6.97191	−0.223739	−0.111870	+	0.993723i	$0.535684\pi$
$972$	0	0
$973$	−21.4685	−0.688249
$974$	0	0
$975$	−1.44210	−0.0461842
$976$	0	0
$977$	−30.6567	−0.980794	−0.490397	−	0.871499i	$-0.663148\pi$
$977$	−30.6567	−0.980794	−0.490397	+	0.871499i	$0.663148\pi$
$978$	0	0
$979$	25.3634	0.810618
$980$	0	0
$981$	−3.18315	−0.101630
$982$	0	0
$983$	−10.9586	−0.349526	−0.174763	−	0.984611i	$-0.555916\pi$
$983$	−10.9586	−0.349526	−0.174763	+	0.984611i	$0.555916\pi$
$984$	0	0
$985$	2.75242	0.0876994
$986$	0	0
$987$	32.5499	1.03607
$988$	0	0
$989$	0.703148	0.0223588
$990$	0	0
$991$	34.2296	1.08734	0.543669	−	0.839300i	$-0.317035\pi$
$991$	34.2296	1.08734	0.543669	+	0.839300i	$0.317035\pi$
$992$	0	0
$993$	26.3978	0.837710
$994$	0	0
$995$	−21.3214	−0.675935
$996$	0	0
$997$	−7.03376	−0.222761	−0.111381	−	0.993778i	$-0.535527\pi$
$997$	−7.03376	−0.222761	−0.111381	+	0.993778i	$0.535527\pi$
$998$	0	0
$999$	−2.54899	−0.0806465

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6036.2.a.h.1.7	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6036.2.a.h.1.7	✓	24	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:	\( N \)	\(=\)	\( 6036 = 2^{2} \cdot 3 \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6036.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.40769 q^{5} +3.41112 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.40769 q^{5} +3.41112 q^{7} +1.00000 q^{9} +2.64296 q^{11} +0.477768 q^{13} -1.40769 q^{15} +4.32494 q^{17} +5.99732 q^{19} +3.41112 q^{21} -6.32739 q^{23} -3.01841 q^{25} +1.00000 q^{27} +9.36681 q^{29} -0.871081 q^{31} +2.64296 q^{33} -4.80179 q^{35} -2.54899 q^{37} +0.477768 q^{39} -5.56833 q^{41} -0.111128 q^{43} -1.40769 q^{45} +9.54229 q^{47} +4.63571 q^{49} +4.32494 q^{51} +6.08525 q^{53} -3.72046 q^{55} +5.99732 q^{57} +14.5591 q^{59} -8.51162 q^{61} +3.41112 q^{63} -0.672549 q^{65} -1.75582 q^{67} -6.32739 q^{69} +8.75598 q^{71} -6.16070 q^{73} -3.01841 q^{75} +9.01544 q^{77} -5.60445 q^{79} +1.00000 q^{81} -14.6857 q^{83} -6.08817 q^{85} +9.36681 q^{87} +9.59659 q^{89} +1.62972 q^{91} -0.871081 q^{93} -8.44235 q^{95} +0.679094 q^{97} +2.64296 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 24 q^{3} + 18 q^{5} + 9 q^{7} + 24 q^{9}+O(q^{10}) \) 24 * q + 24 * q^3 + 18 * q^5 + 9 * q^7 + 24 * q^9	\( 24 q + 24 q^{3} + 18 q^{5} + 9 q^{7} + 24 q^{9} + 9 q^{11} + 7 q^{13} + 18 q^{15} + 12 q^{17} - q^{19} + 9 q^{21} + 22 q^{23} + 48 q^{25} + 24 q^{27} + 44 q^{29} + 15 q^{31} + 9 q^{33} + 3 q^{35} + 18 q^{37} + 7 q^{39} + 37 q^{41} - 8 q^{43} + 18 q^{45} + 24 q^{47} + 63 q^{49} + 12 q^{51} + 59 q^{53} + 20 q^{55} - q^{57} + 44 q^{59} + 33 q^{61} + 9 q^{63} + 15 q^{65} + 13 q^{67} + 22 q^{69} + 47 q^{71} + 28 q^{73} + 48 q^{75} + 19 q^{77} + 12 q^{79} + 24 q^{81} + 17 q^{83} + 31 q^{85} + 44 q^{87} + 41 q^{89} - q^{91} + 15 q^{93} + 58 q^{95} + 51 q^{97} + 9 q^{99}+O(q^{100}) \) 24 * q + 24 * q^3 + 18 * q^5 + 9 * q^7 + 24 * q^9 + 9 * q^11 + 7 * q^13 + 18 * q^15 + 12 * q^17 - q^19 + 9 * q^21 + 22 * q^23 + 48 * q^25 + 24 * q^27 + 44 * q^29 + 15 * q^31 + 9 * q^33 + 3 * q^35 + 18 * q^37 + 7 * q^39 + 37 * q^41 - 8 * q^43 + 18 * q^45 + 24 * q^47 + 63 * q^49 + 12 * q^51 + 59 * q^53 + 20 * q^55 - q^57 + 44 * q^59 + 33 * q^61 + 9 * q^63 + 15 * q^65 + 13 * q^67 + 22 * q^69 + 47 * q^71 + 28 * q^73 + 48 * q^75 + 19 * q^77 + 12 * q^79 + 24 * q^81 + 17 * q^83 + 31 * q^85 + 44 * q^87 + 41 * q^89 - q^91 + 15 * q^93 + 58 * q^95 + 51 * q^97 + 9 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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