LMFDB - Embedded newform 6036.2.a.h.1.11

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			1.11
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)

$f(q)$	$=$	$q+1.00000 q^{3} +0.357952 q^{5} -4.62593 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +0.357952 q^{5} -4.62593 q^{7} +1.00000 q^{9} -3.69957 q^{11} -1.83240 q^{13} +0.357952 q^{15} -6.88058 q^{17} +4.94026 q^{19} -4.62593 q^{21} -3.59653 q^{23} -4.87187 q^{25} +1.00000 q^{27} +7.34256 q^{29} -6.14270 q^{31} -3.69957 q^{33} -1.65586 q^{35} +3.98715 q^{37} -1.83240 q^{39} +9.17046 q^{41} -0.816130 q^{43} +0.357952 q^{45} +11.4358 q^{47} +14.3993 q^{49} -6.88058 q^{51} -8.10482 q^{53} -1.32427 q^{55} +4.94026 q^{57} +6.88856 q^{59} -0.313062 q^{61} -4.62593 q^{63} -0.655911 q^{65} +5.40636 q^{67} -3.59653 q^{69} +14.3460 q^{71} +4.67033 q^{73} -4.87187 q^{75} +17.1140 q^{77} +13.0253 q^{79} +1.00000 q^{81} -12.5449 q^{83} -2.46292 q^{85} +7.34256 q^{87} -10.2026 q^{89} +8.47654 q^{91} -6.14270 q^{93} +1.76838 q^{95} +10.4047 q^{97} -3.69957 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	0.357952	0.160081	0.0800406	−	0.996792i	$-0.474495\pi$
$5$	0.357952	0.160081	0.0800406	+	0.996792i	$0.474495\pi$
$6$	0	0
$7$	−4.62593	−1.74844	−0.874219	−	0.485531i	$-0.838626\pi$
$7$	−4.62593	−1.74844	−0.874219	+	0.485531i	$0.838626\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.69957	−1.11546	−0.557732	−	0.830021i	$-0.688328\pi$
$11$	−3.69957	−1.11546	−0.557732	+	0.830021i	$0.688328\pi$
$12$	0	0
$13$	−1.83240	−0.508215	−0.254108	−	0.967176i	$-0.581782\pi$
$13$	−1.83240	−0.508215	−0.254108	+	0.967176i	$0.581782\pi$
$14$	0	0
$15$	0.357952	0.0924229
$16$	0	0
$17$	−6.88058	−1.66879	−0.834393	−	0.551170i	$-0.814182\pi$
$17$	−6.88058	−1.66879	−0.834393	+	0.551170i	$0.814182\pi$
$18$	0	0
$19$	4.94026	1.13337	0.566687	−	0.823933i	$-0.308225\pi$
$19$	4.94026	1.13337	0.566687	+	0.823933i	$0.308225\pi$
$20$	0	0
$21$	−4.62593	−1.00946
$22$	0	0
$23$	−3.59653	−0.749929	−0.374964	−	0.927039i	$-0.622345\pi$
$23$	−3.59653	−0.749929	−0.374964	+	0.927039i	$0.622345\pi$
$24$	0	0
$25$	−4.87187	−0.974374
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	7.34256	1.36348	0.681740	−	0.731595i	$-0.261224\pi$
$29$	7.34256	1.36348	0.681740	+	0.731595i	$0.261224\pi$
$30$	0	0
$31$	−6.14270	−1.10326	−0.551630	−	0.834089i	$-0.685994\pi$
$31$	−6.14270	−1.10326	−0.551630	+	0.834089i	$0.685994\pi$
$32$	0	0
$33$	−3.69957	−0.644013
$34$	0	0
$35$	−1.65586	−0.279892
$36$	0	0
$37$	3.98715	0.655484	0.327742	−	0.944767i	$-0.393712\pi$
$37$	3.98715	0.655484	0.327742	+	0.944767i	$0.393712\pi$
$38$	0	0
$39$	−1.83240	−0.293418
$40$	0	0
$41$	9.17046	1.43218	0.716092	−	0.698005i	$-0.245929\pi$
$41$	9.17046	1.43218	0.716092	+	0.698005i	$0.245929\pi$
$42$	0	0
$43$	−0.816130	−0.124459	−0.0622293	−	0.998062i	$-0.519821\pi$
$43$	−0.816130	−0.124459	−0.0622293	+	0.998062i	$0.519821\pi$
$44$	0	0
$45$	0.357952	0.0533604
$46$	0	0
$47$	11.4358	1.66808	0.834040	−	0.551704i	$-0.186022\pi$
$47$	11.4358	1.66808	0.834040	+	0.551704i	$0.186022\pi$
$48$	0	0
$49$	14.3993	2.05704
$50$	0	0
$51$	−6.88058	−0.963474
$52$	0	0
$53$	−8.10482	−1.11328	−0.556642	−	0.830753i	$-0.687910\pi$
$53$	−8.10482	−1.11328	−0.556642	+	0.830753i	$0.687910\pi$
$54$	0	0
$55$	−1.32427	−0.178565
$56$	0	0
$57$	4.94026	0.654353
$58$	0	0
$59$	6.88856	0.896814	0.448407	−	0.893829i	$-0.351991\pi$
$59$	6.88856	0.896814	0.448407	+	0.893829i	$0.351991\pi$
$60$	0	0
$61$	−0.313062	−0.0400835	−0.0200418	−	0.999799i	$-0.506380\pi$
$61$	−0.313062	−0.0400835	−0.0200418	+	0.999799i	$0.506380\pi$
$62$	0	0
$63$	−4.62593	−0.582813
$64$	0	0
$65$	−0.655911	−0.0813557
$66$	0	0
$67$	5.40636	0.660493	0.330246	−	0.943895i	$-0.392868\pi$
$67$	5.40636	0.660493	0.330246	+	0.943895i	$0.392868\pi$
$68$	0	0
$69$	−3.59653	−0.432972
$70$	0	0
$71$	14.3460	1.70255	0.851277	−	0.524716i	$-0.175828\pi$
$71$	14.3460	1.70255	0.851277	+	0.524716i	$0.175828\pi$
$72$	0	0
$73$	4.67033	0.546621	0.273311	−	0.961926i	$-0.411881\pi$
$73$	4.67033	0.546621	0.273311	+	0.961926i	$0.411881\pi$
$74$	0	0
$75$	−4.87187	−0.562555
$76$	0	0
$77$	17.1140	1.95032
$78$	0	0
$79$	13.0253	1.46546	0.732731	−	0.680519i	$-0.238245\pi$
$79$	13.0253	1.46546	0.732731	+	0.680519i	$0.238245\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−12.5449	−1.37698	−0.688490	−	0.725246i	$-0.741726\pi$
$83$	−12.5449	−1.37698	−0.688490	+	0.725246i	$0.741726\pi$
$84$	0	0
$85$	−2.46292	−0.267141
$86$	0	0
$87$	7.34256	0.787205
$88$	0	0
$89$	−10.2026	−1.08147	−0.540735	−	0.841193i	$-0.681854\pi$
$89$	−10.2026	−1.08147	−0.540735	+	0.841193i	$0.681854\pi$
$90$	0	0
$91$	8.47654	0.888583
$92$	0	0
$93$	−6.14270	−0.636968
$94$	0	0
$95$	1.76838	0.181432
$96$	0	0
$97$	10.4047	1.05644	0.528220	−	0.849107i	$-0.322859\pi$
$97$	10.4047	1.05644	0.528220	+	0.849107i	$0.322859\pi$
$98$	0	0
$99$	−3.69957	−0.371821
$100$	0	0
$101$	12.6069	1.25444	0.627219	−	0.778843i	$-0.284193\pi$
$101$	12.6069	1.25444	0.627219	+	0.778843i	$0.284193\pi$
$102$	0	0
$103$	−10.1459	−0.999706	−0.499853	−	0.866110i	$-0.666613\pi$
$103$	−10.1459	−0.999706	−0.499853	+	0.866110i	$0.666613\pi$
$104$	0	0
$105$	−1.65586	−0.161596
$106$	0	0
$107$	1.15833	0.111980	0.0559902	−	0.998431i	$-0.482168\pi$
$107$	1.15833	0.111980	0.0559902	+	0.998431i	$0.482168\pi$
$108$	0	0
$109$	12.9868	1.24391	0.621955	−	0.783053i	$-0.286339\pi$
$109$	12.9868	1.24391	0.621955	+	0.783053i	$0.286339\pi$
$110$	0	0
$111$	3.98715	0.378444
$112$	0	0
$113$	−0.579579	−0.0545222	−0.0272611	−	0.999628i	$-0.508679\pi$
$113$	−0.579579	−0.0545222	−0.0272611	+	0.999628i	$0.508679\pi$
$114$	0	0
$115$	−1.28739	−0.120050
$116$	0	0
$117$	−1.83240	−0.169405
$118$	0	0
$119$	31.8291	2.91777
$120$	0	0
$121$	2.68684	0.244258
$122$	0	0
$123$	9.17046	0.826872
$124$	0	0
$125$	−3.53366	−0.316060
$126$	0	0
$127$	−18.3761	−1.63062	−0.815308	−	0.579028i	$-0.803432\pi$
$127$	−18.3761	−1.63062	−0.815308	+	0.579028i	$0.803432\pi$
$128$	0	0
$129$	−0.816130	−0.0718562
$130$	0	0
$131$	−8.18581	−0.715197	−0.357599	−	0.933875i	$-0.616404\pi$
$131$	−8.18581	−0.715197	−0.357599	+	0.933875i	$0.616404\pi$
$132$	0	0
$133$	−22.8533	−1.98163
$134$	0	0
$135$	0.357952	0.0308076
$136$	0	0
$137$	−4.12290	−0.352243	−0.176122	−	0.984368i	$-0.556355\pi$
$137$	−4.12290	−0.352243	−0.176122	+	0.984368i	$0.556355\pi$
$138$	0	0
$139$	5.67821	0.481620	0.240810	−	0.970572i	$-0.422587\pi$
$139$	5.67821	0.481620	0.240810	+	0.970572i	$0.422587\pi$
$140$	0	0
$141$	11.4358	0.963067
$142$	0	0
$143$	6.77908	0.566895
$144$	0	0
$145$	2.62829	0.218267
$146$	0	0
$147$	14.3993	1.18763
$148$	0	0
$149$	12.0354	0.985976	0.492988	−	0.870036i	$-0.335905\pi$
$149$	12.0354	0.985976	0.492988	+	0.870036i	$0.335905\pi$
$150$	0	0
$151$	−14.3806	−1.17028	−0.585138	−	0.810934i	$-0.698960\pi$
$151$	−14.3806	−1.17028	−0.585138	+	0.810934i	$0.698960\pi$
$152$	0	0
$153$	−6.88058	−0.556262
$154$	0	0
$155$	−2.19879	−0.176611
$156$	0	0
$157$	16.2965	1.30060	0.650302	−	0.759675i	$-0.274642\pi$
$157$	16.2965	1.30060	0.650302	+	0.759675i	$0.274642\pi$
$158$	0	0
$159$	−8.10482	−0.642754
$160$	0	0
$161$	16.6373	1.31120
$162$	0	0
$163$	8.53340	0.668387	0.334194	−	0.942504i	$-0.391536\pi$
$163$	8.53340	0.668387	0.334194	+	0.942504i	$0.391536\pi$
$164$	0	0
$165$	−1.32427	−0.103094
$166$	0	0
$167$	−20.4586	−1.58313	−0.791566	−	0.611083i	$-0.790734\pi$
$167$	−20.4586	−1.58313	−0.791566	+	0.611083i	$0.790734\pi$
$168$	0	0
$169$	−9.64233	−0.741717
$170$	0	0
$171$	4.94026	0.377791
$172$	0	0
$173$	16.3590	1.24376	0.621878	−	0.783114i	$-0.286370\pi$
$173$	16.3590	1.24376	0.621878	+	0.783114i	$0.286370\pi$
$174$	0	0
$175$	22.5369	1.70363
$176$	0	0
$177$	6.88856	0.517776
$178$	0	0
$179$	−5.72835	−0.428157	−0.214078	−	0.976816i	$-0.568675\pi$
$179$	−5.72835	−0.428157	−0.214078	+	0.976816i	$0.568675\pi$
$180$	0	0
$181$	10.6224	0.789556	0.394778	−	0.918777i	$-0.370822\pi$
$181$	10.6224	0.789556	0.394778	+	0.918777i	$0.370822\pi$
$182$	0	0
$183$	−0.313062	−0.0231422
$184$	0	0
$185$	1.42721	0.104931
$186$	0	0
$187$	25.4552	1.86147
$188$	0	0
$189$	−4.62593	−0.336487
$190$	0	0
$191$	−17.7558	−1.28477	−0.642384	−	0.766383i	$-0.722054\pi$
$191$	−17.7558	−1.28477	−0.642384	+	0.766383i	$0.722054\pi$
$192$	0	0
$193$	−2.77225	−0.199551	−0.0997755	−	0.995010i	$-0.531812\pi$
$193$	−2.77225	−0.199551	−0.0997755	+	0.995010i	$0.531812\pi$
$194$	0	0
$195$	−0.655911	−0.0469707
$196$	0	0
$197$	8.38198	0.597191	0.298596	−	0.954380i	$-0.403482\pi$
$197$	8.38198	0.597191	0.298596	+	0.954380i	$0.403482\pi$
$198$	0	0
$199$	7.14553	0.506533	0.253266	−	0.967397i	$-0.418495\pi$
$199$	7.14553	0.506533	0.253266	+	0.967397i	$0.418495\pi$
$200$	0	0
$201$	5.40636	0.381336
$202$	0	0
$203$	−33.9662	−2.38396
$204$	0	0
$205$	3.28259	0.229266
$206$	0	0
$207$	−3.59653	−0.249976
$208$	0	0
$209$	−18.2768	−1.26424
$210$	0	0
$211$	18.1448	1.24914	0.624569	−	0.780969i	$-0.285275\pi$
$211$	18.1448	1.24914	0.624569	+	0.780969i	$0.285275\pi$
$212$	0	0
$213$	14.3460	0.982971
$214$	0	0
$215$	−0.292136	−0.0199235
$216$	0	0
$217$	28.4157	1.92898
$218$	0	0
$219$	4.67033	0.315592
$220$	0	0
$221$	12.6079	0.848102
$222$	0	0
$223$	−21.7141	−1.45408	−0.727041	−	0.686594i	$-0.759105\pi$
$223$	−21.7141	−1.45408	−0.727041	+	0.686594i	$0.759105\pi$
$224$	0	0
$225$	−4.87187	−0.324791
$226$	0	0
$227$	−0.390774	−0.0259366	−0.0129683	−	0.999916i	$-0.504128\pi$
$227$	−0.390774	−0.0259366	−0.0129683	+	0.999916i	$0.504128\pi$
$228$	0	0
$229$	8.47381	0.559965	0.279983	−	0.960005i	$-0.409671\pi$
$229$	8.47381	0.559965	0.279983	+	0.960005i	$0.409671\pi$
$230$	0	0
$231$	17.1140	1.12602
$232$	0	0
$233$	−7.71474	−0.505409	−0.252705	−	0.967543i	$-0.581320\pi$
$233$	−7.71474	−0.505409	−0.252705	+	0.967543i	$0.581320\pi$
$234$	0	0
$235$	4.09347	0.267028
$236$	0	0
$237$	13.0253	0.846085
$238$	0	0
$239$	9.50740	0.614983	0.307491	−	0.951551i	$-0.400511\pi$
$239$	9.50740	0.614983	0.307491	+	0.951551i	$0.400511\pi$
$240$	0	0
$241$	−4.76188	−0.306740	−0.153370	−	0.988169i	$-0.549013\pi$
$241$	−4.76188	−0.306740	−0.153370	+	0.988169i	$0.549013\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	5.15425	0.329293
$246$	0	0
$247$	−9.05251	−0.575997
$248$	0	0
$249$	−12.5449	−0.794999
$250$	0	0
$251$	−14.5880	−0.920784	−0.460392	−	0.887716i	$-0.652291\pi$
$251$	−14.5880	−0.920784	−0.460392	+	0.887716i	$0.652291\pi$
$252$	0	0
$253$	13.3056	0.836518
$254$	0	0
$255$	−2.46292	−0.154234
$256$	0	0
$257$	16.9474	1.05715	0.528576	−	0.848886i	$-0.322726\pi$
$257$	16.9474	1.05715	0.528576	+	0.848886i	$0.322726\pi$
$258$	0	0
$259$	−18.4443	−1.14607
$260$	0	0
$261$	7.34256	0.454493
$262$	0	0
$263$	25.3617	1.56387	0.781934	−	0.623361i	$-0.214233\pi$
$263$	25.3617	1.56387	0.781934	+	0.623361i	$0.214233\pi$
$264$	0	0
$265$	−2.90114	−0.178216
$266$	0	0
$267$	−10.2026	−0.624387
$268$	0	0
$269$	−1.95686	−0.119312	−0.0596558	−	0.998219i	$-0.519000\pi$
$269$	−1.95686	−0.119312	−0.0596558	+	0.998219i	$0.519000\pi$
$270$	0	0
$271$	−2.18664	−0.132829	−0.0664146	−	0.997792i	$-0.521156\pi$
$271$	−2.18664	−0.132829	−0.0664146	+	0.997792i	$0.521156\pi$
$272$	0	0
$273$	8.47654	0.513024
$274$	0	0
$275$	18.0238	1.08688
$276$	0	0
$277$	−12.8288	−0.770808	−0.385404	−	0.922748i	$-0.625938\pi$
$277$	−12.8288	−0.770808	−0.385404	+	0.922748i	$0.625938\pi$
$278$	0	0
$279$	−6.14270	−0.367754
$280$	0	0
$281$	3.70278	0.220889	0.110445	−	0.993882i	$-0.464772\pi$
$281$	3.70278	0.220889	0.110445	+	0.993882i	$0.464772\pi$
$282$	0	0
$283$	−4.45606	−0.264885	−0.132443	−	0.991191i	$-0.542282\pi$
$283$	−4.45606	−0.264885	−0.132443	+	0.991191i	$0.542282\pi$
$284$	0	0
$285$	1.76838	0.104750
$286$	0	0
$287$	−42.4219	−2.50409
$288$	0	0
$289$	30.3424	1.78485
$290$	0	0
$291$	10.4047	0.609936
$292$	0	0
$293$	19.7872	1.15598	0.577990	−	0.816044i	$-0.303837\pi$
$293$	19.7872	1.15598	0.577990	+	0.816044i	$0.303837\pi$
$294$	0	0
$295$	2.46578	0.143563
$296$	0	0
$297$	−3.69957	−0.214671
$298$	0	0
$299$	6.59027	0.381125
$300$	0	0
$301$	3.77536	0.217608
$302$	0	0
$303$	12.6069	0.724250
$304$	0	0
$305$	−0.112061	−0.00641662
$306$	0	0
$307$	−26.7657	−1.52760	−0.763800	−	0.645453i	$-0.776669\pi$
$307$	−26.7657	−1.52760	−0.763800	+	0.645453i	$0.776669\pi$
$308$	0	0
$309$	−10.1459	−0.577180
$310$	0	0
$311$	−1.46623	−0.0831423	−0.0415711	−	0.999136i	$-0.513236\pi$
$311$	−1.46623	−0.0831423	−0.0415711	+	0.999136i	$0.513236\pi$
$312$	0	0
$313$	−29.3389	−1.65833	−0.829167	−	0.559001i	$-0.811185\pi$
$313$	−29.3389	−1.65833	−0.829167	+	0.559001i	$0.811185\pi$
$314$	0	0
$315$	−1.65586	−0.0932974
$316$	0	0
$317$	7.17204	0.402822	0.201411	−	0.979507i	$-0.435447\pi$
$317$	7.17204	0.402822	0.201411	+	0.979507i	$0.435447\pi$
$318$	0	0
$319$	−27.1643	−1.52091
$320$	0	0
$321$	1.15833	0.0646519
$322$	0	0
$323$	−33.9919	−1.89136
$324$	0	0
$325$	8.92719	0.495192
$326$	0	0
$327$	12.9868	0.718172
$328$	0	0
$329$	−52.9012	−2.91654
$330$	0	0
$331$	13.2043	0.725774	0.362887	−	0.931833i	$-0.381791\pi$
$331$	13.2043	0.725774	0.362887	+	0.931833i	$0.381791\pi$
$332$	0	0
$333$	3.98715	0.218495
$334$	0	0
$335$	1.93522	0.105732
$336$	0	0
$337$	−23.0164	−1.25378	−0.626890	−	0.779108i	$-0.715673\pi$
$337$	−23.0164	−1.25378	−0.626890	+	0.779108i	$0.715673\pi$
$338$	0	0
$339$	−0.579579	−0.0314784
$340$	0	0
$341$	22.7254	1.23065
$342$	0	0
$343$	−34.2285	−1.84816
$344$	0	0
$345$	−1.28739	−0.0693106
$346$	0	0
$347$	−1.13265	−0.0608037	−0.0304019	−	0.999538i	$-0.509679\pi$
$347$	−1.13265	−0.0608037	−0.0304019	+	0.999538i	$0.509679\pi$
$348$	0	0
$349$	28.2031	1.50968	0.754838	−	0.655911i	$-0.227715\pi$
$349$	28.2031	1.50968	0.754838	+	0.655911i	$0.227715\pi$
$350$	0	0
$351$	−1.83240	−0.0978061
$352$	0	0
$353$	1.92492	0.102453	0.0512265	−	0.998687i	$-0.483687\pi$
$353$	1.92492	0.102453	0.0512265	+	0.998687i	$0.483687\pi$
$354$	0	0
$355$	5.13518	0.272547
$356$	0	0
$357$	31.8291	1.68458
$358$	0	0
$359$	−24.5556	−1.29599	−0.647997	−	0.761643i	$-0.724393\pi$
$359$	−24.5556	−1.29599	−0.647997	+	0.761643i	$0.724393\pi$
$360$	0	0
$361$	5.40616	0.284535
$362$	0	0
$363$	2.68684	0.141023
$364$	0	0
$365$	1.67176	0.0875038
$366$	0	0
$367$	14.6262	0.763482	0.381741	−	0.924269i	$-0.375325\pi$
$367$	14.6262	0.763482	0.381741	+	0.924269i	$0.375325\pi$
$368$	0	0
$369$	9.17046	0.477395
$370$	0	0
$371$	37.4924	1.94651
$372$	0	0
$373$	20.1213	1.04184	0.520922	−	0.853604i	$-0.325588\pi$
$373$	20.1213	1.04184	0.520922	+	0.853604i	$0.325588\pi$
$374$	0	0
$375$	−3.53366	−0.182477
$376$	0	0
$377$	−13.4545	−0.692941
$378$	0	0
$379$	−32.7592	−1.68273	−0.841364	−	0.540469i	$-0.818247\pi$
$379$	−32.7592	−1.68273	−0.841364	+	0.540469i	$0.818247\pi$
$380$	0	0
$381$	−18.3761	−0.941436
$382$	0	0
$383$	22.4948	1.14943	0.574714	−	0.818354i	$-0.305113\pi$
$383$	22.4948	1.14943	0.574714	+	0.818354i	$0.305113\pi$
$384$	0	0
$385$	6.12599	0.312209
$386$	0	0
$387$	−0.816130	−0.0414862
$388$	0	0
$389$	20.0468	1.01641	0.508206	−	0.861236i	$-0.330309\pi$
$389$	20.0468	1.01641	0.508206	+	0.861236i	$0.330309\pi$
$390$	0	0
$391$	24.7462	1.25147
$392$	0	0
$393$	−8.18581	−0.412919
$394$	0	0
$395$	4.66244	0.234593
$396$	0	0
$397$	29.8722	1.49924	0.749622	−	0.661866i	$-0.230235\pi$
$397$	29.8722	1.49924	0.749622	+	0.661866i	$0.230235\pi$
$398$	0	0
$399$	−22.8533	−1.14410
$400$	0	0
$401$	21.1679	1.05707	0.528536	−	0.848911i	$-0.322741\pi$
$401$	21.1679	1.05707	0.528536	+	0.848911i	$0.322741\pi$
$402$	0	0
$403$	11.2559	0.560694
$404$	0	0
$405$	0.357952	0.0177868
$406$	0	0
$407$	−14.7508	−0.731168
$408$	0	0
$409$	−35.3067	−1.74580	−0.872901	−	0.487897i	$-0.837764\pi$
$409$	−35.3067	−1.74580	−0.872901	+	0.487897i	$0.837764\pi$
$410$	0	0
$411$	−4.12290	−0.203368
$412$	0	0
$413$	−31.8660	−1.56802
$414$	0	0
$415$	−4.49047	−0.220429
$416$	0	0
$417$	5.67821	0.278063
$418$	0	0
$419$	23.6725	1.15648	0.578239	−	0.815867i	$-0.303740\pi$
$419$	23.6725	1.15648	0.578239	+	0.815867i	$0.303740\pi$
$420$	0	0
$421$	−5.92326	−0.288682	−0.144341	−	0.989528i	$-0.546106\pi$
$421$	−5.92326	−0.288682	−0.144341	+	0.989528i	$0.546106\pi$
$422$	0	0
$423$	11.4358	0.556027
$424$	0	0
$425$	33.5213	1.62602
$426$	0	0
$427$	1.44821	0.0700836
$428$	0	0
$429$	6.77908	0.327297
$430$	0	0
$431$	−14.1287	−0.680556	−0.340278	−	0.940325i	$-0.610521\pi$
$431$	−14.1287	−0.680556	−0.340278	+	0.940325i	$0.610521\pi$
$432$	0	0
$433$	−0.457456	−0.0219839	−0.0109920	−	0.999940i	$-0.503499\pi$
$433$	−0.457456	−0.0219839	−0.0109920	+	0.999940i	$0.503499\pi$
$434$	0	0
$435$	2.62829	0.126017
$436$	0	0
$437$	−17.7678	−0.849949
$438$	0	0
$439$	−32.9422	−1.57224	−0.786122	−	0.618071i	$-0.787914\pi$
$439$	−32.9422	−1.57224	−0.786122	+	0.618071i	$0.787914\pi$
$440$	0	0
$441$	14.3993	0.685679
$442$	0	0
$443$	22.9584	1.09079	0.545393	−	0.838181i	$-0.316381\pi$
$443$	22.9584	1.09079	0.545393	+	0.838181i	$0.316381\pi$
$444$	0	0
$445$	−3.65203	−0.173123
$446$	0	0
$447$	12.0354	0.569254
$448$	0	0
$449$	8.26833	0.390206	0.195103	−	0.980783i	$-0.437496\pi$
$449$	8.26833	0.390206	0.195103	+	0.980783i	$0.437496\pi$
$450$	0	0
$451$	−33.9268	−1.59755
$452$	0	0
$453$	−14.3806	−0.675659
$454$	0	0
$455$	3.03420	0.142245
$456$	0	0
$457$	15.8098	0.739549	0.369775	−	0.929121i	$-0.379435\pi$
$457$	15.8098	0.739549	0.369775	+	0.929121i	$0.379435\pi$
$458$	0	0
$459$	−6.88058	−0.321158
$460$	0	0
$461$	7.05773	0.328711	0.164356	−	0.986401i	$-0.447446\pi$
$461$	7.05773	0.328711	0.164356	+	0.986401i	$0.447446\pi$
$462$	0	0
$463$	−8.44409	−0.392430	−0.196215	−	0.980561i	$-0.562865\pi$
$463$	−8.44409	−0.392430	−0.196215	+	0.980561i	$0.562865\pi$
$464$	0	0
$465$	−2.19879	−0.101967
$466$	0	0
$467$	21.4584	0.992977	0.496488	−	0.868043i	$-0.334623\pi$
$467$	21.4584	0.992977	0.496488	+	0.868043i	$0.334623\pi$
$468$	0	0
$469$	−25.0095	−1.15483
$470$	0	0
$471$	16.2965	0.750904
$472$	0	0
$473$	3.01933	0.138829
$474$	0	0
$475$	−24.0683	−1.10433
$476$	0	0
$477$	−8.10482	−0.371094
$478$	0	0
$479$	−2.72550	−0.124531	−0.0622656	−	0.998060i	$-0.519833\pi$
$479$	−2.72550	−0.124531	−0.0622656	+	0.998060i	$0.519833\pi$
$480$	0	0
$481$	−7.30604	−0.333127
$482$	0	0
$483$	16.6373	0.757024
$484$	0	0
$485$	3.72440	0.169116
$486$	0	0
$487$	−20.7205	−0.938935	−0.469468	−	0.882950i	$-0.655554\pi$
$487$	−20.7205	−0.938935	−0.469468	+	0.882950i	$0.655554\pi$
$488$	0	0
$489$	8.53340	0.385894
$490$	0	0
$491$	18.9575	0.855541	0.427771	−	0.903887i	$-0.359299\pi$
$491$	18.9575	0.855541	0.427771	+	0.903887i	$0.359299\pi$
$492$	0	0
$493$	−50.5211	−2.27535
$494$	0	0
$495$	−1.32427	−0.0595216
$496$	0	0
$497$	−66.3636	−2.97681
$498$	0	0
$499$	29.8081	1.33440	0.667198	−	0.744881i	$-0.267494\pi$
$499$	29.8081	1.33440	0.667198	+	0.744881i	$0.267494\pi$
$500$	0	0
$501$	−20.4586	−0.914022
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	4.51268	0.200812
$506$	0	0
$507$	−9.64233	−0.428231
$508$	0	0
$509$	5.42532	0.240473	0.120237	−	0.992745i	$-0.461635\pi$
$509$	5.42532	0.240473	0.120237	+	0.992745i	$0.461635\pi$
$510$	0	0
$511$	−21.6046	−0.955733
$512$	0	0
$513$	4.94026	0.218118
$514$	0	0
$515$	−3.63175	−0.160034
$516$	0	0
$517$	−42.3075	−1.86068
$518$	0	0
$519$	16.3590	0.718082
$520$	0	0
$521$	22.0616	0.966538	0.483269	−	0.875472i	$-0.339449\pi$
$521$	22.0616	0.966538	0.483269	+	0.875472i	$0.339449\pi$
$522$	0	0
$523$	−32.5729	−1.42432	−0.712158	−	0.702019i	$-0.752282\pi$
$523$	−32.5729	−1.42432	−0.712158	+	0.702019i	$0.752282\pi$
$524$	0	0
$525$	22.5369	0.983593
$526$	0	0
$527$	42.2653	1.84111
$528$	0	0
$529$	−10.0650	−0.437607
$530$	0	0
$531$	6.88856	0.298938
$532$	0	0
$533$	−16.8039	−0.727858
$534$	0	0
$535$	0.414629	0.0179260
$536$	0	0
$537$	−5.72835	−0.247196
$538$	0	0
$539$	−53.2711	−2.29455
$540$	0	0
$541$	−5.70470	−0.245264	−0.122632	−	0.992452i	$-0.539134\pi$
$541$	−5.70470	−0.245264	−0.122632	+	0.992452i	$0.539134\pi$
$542$	0	0
$543$	10.6224	0.455850
$544$	0	0
$545$	4.64866	0.199127
$546$	0	0
$547$	40.6173	1.73667	0.868335	−	0.495978i	$-0.165190\pi$
$547$	40.6173	1.73667	0.868335	+	0.495978i	$0.165190\pi$
$548$	0	0
$549$	−0.313062	−0.0133612
$550$	0	0
$551$	36.2741	1.54533
$552$	0	0
$553$	−60.2542	−2.56227
$554$	0	0
$555$	1.42721	0.0605817
$556$	0	0
$557$	1.47909	0.0626712	0.0313356	−	0.999509i	$-0.490024\pi$
$557$	1.47909	0.0626712	0.0313356	+	0.999509i	$0.490024\pi$
$558$	0	0
$559$	1.49547	0.0632517
$560$	0	0
$561$	25.4552	1.07472
$562$	0	0
$563$	30.5135	1.28599	0.642994	−	0.765871i	$-0.277692\pi$
$563$	30.5135	1.28599	0.642994	+	0.765871i	$0.277692\pi$
$564$	0	0
$565$	−0.207462	−0.00872798
$566$	0	0
$567$	−4.62593	−0.194271
$568$	0	0
$569$	28.4007	1.19062	0.595309	−	0.803497i	$-0.297030\pi$
$569$	28.4007	1.19062	0.595309	+	0.803497i	$0.297030\pi$
$570$	0	0
$571$	−25.0005	−1.04624	−0.523119	−	0.852260i	$-0.675232\pi$
$571$	−25.0005	−1.04624	−0.523119	+	0.852260i	$0.675232\pi$
$572$	0	0
$573$	−17.7558	−0.741761
$574$	0	0
$575$	17.5218	0.730711
$576$	0	0
$577$	−19.7960	−0.824118	−0.412059	−	0.911157i	$-0.635190\pi$
$577$	−19.7960	−0.824118	−0.412059	+	0.911157i	$0.635190\pi$
$578$	0	0
$579$	−2.77225	−0.115211
$580$	0	0
$581$	58.0318	2.40756
$582$	0	0
$583$	29.9844	1.24183
$584$	0	0
$585$	−0.655911	−0.0271186
$586$	0	0
$587$	25.3936	1.04811	0.524054	−	0.851685i	$-0.324419\pi$
$587$	25.3936	1.04811	0.524054	+	0.851685i	$0.324419\pi$
$588$	0	0
$589$	−30.3465	−1.25041
$590$	0	0
$591$	8.38198	0.344788
$592$	0	0
$593$	−21.5401	−0.884547	−0.442273	−	0.896880i	$-0.645828\pi$
$593$	−21.5401	−0.884547	−0.442273	+	0.896880i	$0.645828\pi$
$594$	0	0
$595$	11.3933	0.467080
$596$	0	0
$597$	7.14553	0.292447
$598$	0	0
$599$	10.2870	0.420317	0.210158	−	0.977667i	$-0.432602\pi$
$599$	10.2870	0.420317	0.210158	+	0.977667i	$0.432602\pi$
$600$	0	0
$601$	35.0935	1.43149	0.715747	−	0.698360i	$-0.246086\pi$
$601$	35.0935	1.43149	0.715747	+	0.698360i	$0.246086\pi$
$602$	0	0
$603$	5.40636	0.220164
$604$	0	0
$605$	0.961761	0.0391012
$606$	0	0
$607$	−10.8136	−0.438912	−0.219456	−	0.975622i	$-0.570428\pi$
$607$	−10.8136	−0.438912	−0.219456	+	0.975622i	$0.570428\pi$
$608$	0	0
$609$	−33.9662	−1.37638
$610$	0	0
$611$	−20.9549	−0.847744
$612$	0	0
$613$	−2.84066	−0.114733	−0.0573665	−	0.998353i	$-0.518270\pi$
$613$	−2.84066	−0.114733	−0.0573665	+	0.998353i	$0.518270\pi$
$614$	0	0
$615$	3.28259	0.132367
$616$	0	0
$617$	−17.7255	−0.713600	−0.356800	−	0.934181i	$-0.616132\pi$
$617$	−17.7255	−0.713600	−0.356800	+	0.934181i	$0.616132\pi$
$618$	0	0
$619$	−4.72444	−0.189891	−0.0949456	−	0.995482i	$-0.530268\pi$
$619$	−4.72444	−0.189891	−0.0949456	+	0.995482i	$0.530268\pi$
$620$	0	0
$621$	−3.59653	−0.144324
$622$	0	0
$623$	47.1964	1.89088
$624$	0	0
$625$	23.0945	0.923779
$626$	0	0
$627$	−18.2768	−0.729907
$628$	0	0
$629$	−27.4339	−1.09386
$630$	0	0
$631$	−8.44879	−0.336341	−0.168171	−	0.985758i	$-0.553786\pi$
$631$	−8.44879	−0.336341	−0.168171	+	0.985758i	$0.553786\pi$
$632$	0	0
$633$	18.1448	0.721190
$634$	0	0
$635$	−6.57777	−0.261031
$636$	0	0
$637$	−26.3851	−1.04542
$638$	0	0
$639$	14.3460	0.567518
$640$	0	0
$641$	33.9140	1.33952	0.669760	−	0.742577i	$-0.266397\pi$
$641$	33.9140	1.33952	0.669760	+	0.742577i	$0.266397\pi$
$642$	0	0
$643$	−22.7937	−0.898897	−0.449448	−	0.893306i	$-0.648379\pi$
$643$	−22.7937	−0.898897	−0.449448	+	0.893306i	$0.648379\pi$
$644$	0	0
$645$	−0.292136	−0.0115028
$646$	0	0
$647$	45.6908	1.79629	0.898145	−	0.439699i	$-0.144915\pi$
$647$	45.6908	1.79629	0.898145	+	0.439699i	$0.144915\pi$
$648$	0	0
$649$	−25.4847	−1.00036
$650$	0	0
$651$	28.4157	1.11370
$652$	0	0
$653$	5.16214	0.202010	0.101005	−	0.994886i	$-0.467794\pi$
$653$	5.16214	0.202010	0.101005	+	0.994886i	$0.467794\pi$
$654$	0	0
$655$	−2.93013	−0.114490
$656$	0	0
$657$	4.67033	0.182207
$658$	0	0
$659$	4.75114	0.185078	0.0925392	−	0.995709i	$-0.470502\pi$
$659$	4.75114	0.185078	0.0925392	+	0.995709i	$0.470502\pi$
$660$	0	0
$661$	−9.21171	−0.358294	−0.179147	−	0.983822i	$-0.557334\pi$
$661$	−9.21171	−0.358294	−0.179147	+	0.983822i	$0.557334\pi$
$662$	0	0
$663$	12.6079	0.489652
$664$	0	0
$665$	−8.18040	−0.317222
$666$	0	0
$667$	−26.4078	−1.02251
$668$	0	0
$669$	−21.7141	−0.839515
$670$	0	0
$671$	1.15820	0.0447117
$672$	0	0
$673$	−10.7161	−0.413074	−0.206537	−	0.978439i	$-0.566219\pi$
$673$	−10.7161	−0.413074	−0.206537	+	0.978439i	$0.566219\pi$
$674$	0	0
$675$	−4.87187	−0.187518
$676$	0	0
$677$	−35.5452	−1.36611	−0.683057	−	0.730366i	$-0.739350\pi$
$677$	−35.5452	−1.36611	−0.683057	+	0.730366i	$0.739350\pi$
$678$	0	0
$679$	−48.1316	−1.84712
$680$	0	0
$681$	−0.390774	−0.0149745
$682$	0	0
$683$	49.4463	1.89201	0.946005	−	0.324151i	$-0.105079\pi$
$683$	49.4463	1.89201	0.946005	+	0.324151i	$0.105079\pi$
$684$	0	0
$685$	−1.47580	−0.0563875
$686$	0	0
$687$	8.47381	0.323296
$688$	0	0
$689$	14.8512	0.565787
$690$	0	0
$691$	43.7125	1.66290	0.831452	−	0.555597i	$-0.187510\pi$
$691$	43.7125	1.66290	0.831452	+	0.555597i	$0.187510\pi$
$692$	0	0
$693$	17.1140	0.650106
$694$	0	0
$695$	2.03253	0.0770983
$696$	0	0
$697$	−63.0981	−2.39001
$698$	0	0
$699$	−7.71474	−0.291798
$700$	0	0
$701$	17.0767	0.644977	0.322489	−	0.946573i	$-0.395481\pi$
$701$	17.0767	0.644977	0.322489	+	0.946573i	$0.395481\pi$
$702$	0	0
$703$	19.6976	0.742908
$704$	0	0
$705$	4.09347	0.154169
$706$	0	0
$707$	−58.3189	−2.19331
$708$	0	0
$709$	48.4858	1.82092	0.910461	−	0.413595i	$-0.135727\pi$
$709$	48.4858	1.82092	0.910461	+	0.413595i	$0.135727\pi$
$710$	0	0
$711$	13.0253	0.488487
$712$	0	0
$713$	22.0924	0.827367
$714$	0	0
$715$	2.42659	0.0907493
$716$	0	0
$717$	9.50740	0.355060
$718$	0	0
$719$	1.42290	0.0530653	0.0265326	−	0.999648i	$-0.491553\pi$
$719$	1.42290	0.0530653	0.0265326	+	0.999648i	$0.491553\pi$
$720$	0	0
$721$	46.9343	1.74792
$722$	0	0
$723$	−4.76188	−0.177096
$724$	0	0
$725$	−35.7720	−1.32854
$726$	0	0
$727$	−32.4443	−1.20329	−0.601645	−	0.798763i	$-0.705488\pi$
$727$	−32.4443	−1.20329	−0.601645	+	0.798763i	$0.705488\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	5.61545	0.207695
$732$	0	0
$733$	45.3249	1.67411	0.837056	−	0.547117i	$-0.184275\pi$
$733$	45.3249	1.67411	0.837056	+	0.547117i	$0.184275\pi$
$734$	0	0
$735$	5.15425	0.190117
$736$	0	0
$737$	−20.0012	−0.736755
$738$	0	0
$739$	−7.56616	−0.278326	−0.139163	−	0.990270i	$-0.544441\pi$
$739$	−7.56616	−0.278326	−0.139163	+	0.990270i	$0.544441\pi$
$740$	0	0
$741$	−9.05251	−0.332552
$742$	0	0
$743$	−12.7260	−0.466872	−0.233436	−	0.972372i	$-0.574997\pi$
$743$	−12.7260	−0.466872	−0.233436	+	0.972372i	$0.574997\pi$
$744$	0	0
$745$	4.30809	0.157836
$746$	0	0
$747$	−12.5449	−0.458993
$748$	0	0
$749$	−5.35838	−0.195791
$750$	0	0
$751$	47.2386	1.72376	0.861880	−	0.507112i	$-0.169287\pi$
$751$	47.2386	1.72376	0.861880	+	0.507112i	$0.169287\pi$
$752$	0	0
$753$	−14.5880	−0.531615
$754$	0	0
$755$	−5.14757	−0.187339
$756$	0	0
$757$	22.8717	0.831286	0.415643	−	0.909528i	$-0.363557\pi$
$757$	22.8717	0.831286	0.415643	+	0.909528i	$0.363557\pi$
$758$	0	0
$759$	13.3056	0.482964
$760$	0	0
$761$	−4.97520	−0.180351	−0.0901755	−	0.995926i	$-0.528743\pi$
$761$	−4.97520	−0.180351	−0.0901755	+	0.995926i	$0.528743\pi$
$762$	0	0
$763$	−60.0761	−2.17490
$764$	0	0
$765$	−2.46292	−0.0890471
$766$	0	0
$767$	−12.6226	−0.455775
$768$	0	0
$769$	32.8681	1.18525	0.592627	−	0.805477i	$-0.298091\pi$
$769$	32.8681	1.18525	0.592627	+	0.805477i	$0.298091\pi$
$770$	0	0
$771$	16.9474	0.610347
$772$	0	0
$773$	−8.47052	−0.304663	−0.152332	−	0.988329i	$-0.548678\pi$
$773$	−8.47052	−0.304663	−0.152332	+	0.988329i	$0.548678\pi$
$774$	0	0
$775$	29.9264	1.07499
$776$	0	0
$777$	−18.4443	−0.661686
$778$	0	0
$779$	45.3044	1.62320
$780$	0	0
$781$	−53.0740	−1.89914
$782$	0	0
$783$	7.34256	0.262402
$784$	0	0
$785$	5.83338	0.208202
$786$	0	0
$787$	26.1698	0.932852	0.466426	−	0.884560i	$-0.345541\pi$
$787$	26.1698	0.932852	0.466426	+	0.884560i	$0.345541\pi$
$788$	0	0
$789$	25.3617	0.902900
$790$	0	0
$791$	2.68109	0.0953287
$792$	0	0
$793$	0.573654	0.0203711
$794$	0	0
$795$	−2.90114	−0.102893
$796$	0	0
$797$	−14.2915	−0.506232	−0.253116	−	0.967436i	$-0.581455\pi$
$797$	−14.2915	−0.506232	−0.253116	+	0.967436i	$0.581455\pi$
$798$	0	0
$799$	−78.6848	−2.78367
$800$	0	0
$801$	−10.2026	−0.360490
$802$	0	0
$803$	−17.2782	−0.609736
$804$	0	0
$805$	5.95537	0.209899
$806$	0	0
$807$	−1.95686	−0.0688846
$808$	0	0
$809$	−3.39905	−0.119504	−0.0597521	−	0.998213i	$-0.519031\pi$
$809$	−3.39905	−0.119504	−0.0597521	+	0.998213i	$0.519031\pi$
$810$	0	0
$811$	−24.1409	−0.847702	−0.423851	−	0.905732i	$-0.639322\pi$
$811$	−24.1409	−0.847702	−0.423851	+	0.905732i	$0.639322\pi$
$812$	0	0
$813$	−2.18664	−0.0766890
$814$	0	0
$815$	3.05455	0.106996
$816$	0	0
$817$	−4.03189	−0.141058
$818$	0	0
$819$	8.47654	0.296194
$820$	0	0
$821$	0.683500	0.0238543	0.0119272	−	0.999929i	$-0.496203\pi$
$821$	0.683500	0.0238543	0.0119272	+	0.999929i	$0.496203\pi$
$822$	0	0
$823$	−30.7430	−1.07163	−0.535817	−	0.844334i	$-0.679996\pi$
$823$	−30.7430	−1.07163	−0.535817	+	0.844334i	$0.679996\pi$
$824$	0	0
$825$	18.0238	0.627510
$826$	0	0
$827$	7.11513	0.247417	0.123709	−	0.992319i	$-0.460521\pi$
$827$	7.11513	0.247417	0.123709	+	0.992319i	$0.460521\pi$
$828$	0	0
$829$	6.01691	0.208976	0.104488	−	0.994526i	$-0.466680\pi$
$829$	6.01691	0.208976	0.104488	+	0.994526i	$0.466680\pi$
$830$	0	0
$831$	−12.8288	−0.445026
$832$	0	0
$833$	−99.0753	−3.43276
$834$	0	0
$835$	−7.32320	−0.253430
$836$	0	0
$837$	−6.14270	−0.212323
$838$	0	0
$839$	19.3118	0.666717	0.333358	−	0.942800i	$-0.391818\pi$
$839$	19.3118	0.666717	0.333358	+	0.942800i	$0.391818\pi$
$840$	0	0
$841$	24.9132	0.859075
$842$	0	0
$843$	3.70278	0.127531
$844$	0	0
$845$	−3.45149	−0.118735
$846$	0	0
$847$	−12.4291	−0.427071
$848$	0	0
$849$	−4.45606	−0.152932
$850$	0	0
$851$	−14.3399	−0.491566
$852$	0	0
$853$	42.8885	1.46847	0.734236	−	0.678894i	$-0.237540\pi$
$853$	42.8885	1.46847	0.734236	+	0.678894i	$0.237540\pi$
$854$	0	0
$855$	1.76838	0.0604772
$856$	0	0
$857$	−20.3062	−0.693645	−0.346823	−	0.937931i	$-0.612739\pi$
$857$	−20.3062	−0.693645	−0.346823	+	0.937931i	$0.612739\pi$
$858$	0	0
$859$	−14.4014	−0.491370	−0.245685	−	0.969350i	$-0.579013\pi$
$859$	−14.4014	−0.491370	−0.245685	+	0.969350i	$0.579013\pi$
$860$	0	0
$861$	−42.4219	−1.44574
$862$	0	0
$863$	46.7934	1.59287	0.796433	−	0.604727i	$-0.206718\pi$
$863$	46.7934	1.59287	0.796433	+	0.604727i	$0.206718\pi$
$864$	0	0
$865$	5.85576	0.199102
$866$	0	0
$867$	30.3424	1.03048
$868$	0	0
$869$	−48.1881	−1.63467
$870$	0	0
$871$	−9.90660	−0.335672
$872$	0	0
$873$	10.4047	0.352147
$874$	0	0
$875$	16.3465	0.552612
$876$	0	0
$877$	2.45406	0.0828677	0.0414339	−	0.999141i	$-0.486807\pi$
$877$	2.45406	0.0828677	0.0414339	+	0.999141i	$0.486807\pi$
$878$	0	0
$879$	19.7872	0.667405
$880$	0	0
$881$	−44.1666	−1.48801	−0.744005	−	0.668174i	$-0.767076\pi$
$881$	−44.1666	−1.48801	−0.744005	+	0.668174i	$0.767076\pi$
$882$	0	0
$883$	−23.1266	−0.778271	−0.389136	−	0.921180i	$-0.627226\pi$
$883$	−23.1266	−0.778271	−0.389136	+	0.921180i	$0.627226\pi$
$884$	0	0
$885$	2.46578	0.0828862
$886$	0	0
$887$	46.2903	1.55428	0.777138	−	0.629330i	$-0.216670\pi$
$887$	46.2903	1.55428	0.777138	+	0.629330i	$0.216670\pi$
$888$	0	0
$889$	85.0066	2.85103
$890$	0	0
$891$	−3.69957	−0.123940
$892$	0	0
$893$	56.4957	1.89056
$894$	0	0
$895$	−2.05048	−0.0685398
$896$	0	0
$897$	6.59027	0.220043
$898$	0	0
$899$	−45.1031	−1.50427
$900$	0	0
$901$	55.7659	1.85783
$902$	0	0
$903$	3.77536	0.125636
$904$	0	0
$905$	3.80231	0.126393
$906$	0	0
$907$	−37.9111	−1.25882	−0.629409	−	0.777074i	$-0.716703\pi$
$907$	−37.9111	−1.25882	−0.629409	+	0.777074i	$0.716703\pi$
$908$	0	0
$909$	12.6069	0.418146
$910$	0	0
$911$	49.5349	1.64117	0.820583	−	0.571527i	$-0.193649\pi$
$911$	49.5349	1.64117	0.820583	+	0.571527i	$0.193649\pi$
$912$	0	0
$913$	46.4107	1.53597
$914$	0	0
$915$	−0.112061	−0.00370464
$916$	0	0
$917$	37.8670	1.25048
$918$	0	0
$919$	−25.8662	−0.853246	−0.426623	−	0.904429i	$-0.640297\pi$
$919$	−25.8662	−0.853246	−0.426623	+	0.904429i	$0.640297\pi$
$920$	0	0
$921$	−26.7657	−0.881960
$922$	0	0
$923$	−26.2875	−0.865264
$924$	0	0
$925$	−19.4249	−0.638686
$926$	0	0
$927$	−10.1459	−0.333235
$928$	0	0
$929$	15.0891	0.495058	0.247529	−	0.968881i	$-0.420382\pi$
$929$	15.0891	0.495058	0.247529	+	0.968881i	$0.420382\pi$
$930$	0	0
$931$	71.1361	2.33139
$932$	0	0
$933$	−1.46623	−0.0480022
$934$	0	0
$935$	9.11176	0.297986
$936$	0	0
$937$	−50.8818	−1.66223	−0.831117	−	0.556097i	$-0.812298\pi$
$937$	−50.8818	−1.66223	−0.831117	+	0.556097i	$0.812298\pi$
$938$	0	0
$939$	−29.3389	−0.957440
$940$	0	0
$941$	8.66605	0.282505	0.141253	−	0.989974i	$-0.454887\pi$
$941$	8.66605	0.282505	0.141253	+	0.989974i	$0.454887\pi$
$942$	0	0
$943$	−32.9819	−1.07404
$944$	0	0
$945$	−1.65586	−0.0538653
$946$	0	0
$947$	33.0615	1.07435	0.537177	−	0.843470i	$-0.319491\pi$
$947$	33.0615	1.07435	0.537177	+	0.843470i	$0.319491\pi$
$948$	0	0
$949$	−8.55790	−0.277801
$950$	0	0
$951$	7.17204	0.232569
$952$	0	0
$953$	−0.562967	−0.0182363	−0.00911814	−	0.999958i	$-0.502902\pi$
$953$	−0.562967	−0.0182363	−0.00911814	+	0.999958i	$0.502902\pi$
$954$	0	0
$955$	−6.35575	−0.205667
$956$	0	0
$957$	−27.1643	−0.878098
$958$	0	0
$959$	19.0723	0.615876
$960$	0	0
$961$	6.73271	0.217184
$962$	0	0
$963$	1.15833	0.0373268
$964$	0	0
$965$	−0.992334	−0.0319444
$966$	0	0
$967$	33.8460	1.08842	0.544208	−	0.838951i	$-0.316830\pi$
$967$	33.8460	1.08842	0.544208	+	0.838951i	$0.316830\pi$
$968$	0	0
$969$	−33.9919	−1.09198
$970$	0	0
$971$	−25.8293	−0.828902	−0.414451	−	0.910072i	$-0.636026\pi$
$971$	−25.8293	−0.828902	−0.414451	+	0.910072i	$0.636026\pi$
$972$	0	0
$973$	−26.2670	−0.842082
$974$	0	0
$975$	8.92719	0.285899
$976$	0	0
$977$	50.6825	1.62148	0.810739	−	0.585408i	$-0.199066\pi$
$977$	50.6825	1.62148	0.810739	+	0.585408i	$0.199066\pi$
$978$	0	0
$979$	37.7451	1.20634
$980$	0	0
$981$	12.9868	0.414637
$982$	0	0
$983$	−5.10548	−0.162840	−0.0814198	−	0.996680i	$-0.525945\pi$
$983$	−5.10548	−0.162840	−0.0814198	+	0.996680i	$0.525945\pi$
$984$	0	0
$985$	3.00035	0.0955991
$986$	0	0
$987$	−52.9012	−1.68386
$988$	0	0
$989$	2.93524	0.0933351
$990$	0	0
$991$	−2.92436	−0.0928955	−0.0464478	−	0.998921i	$-0.514790\pi$
$991$	−2.92436	−0.0928955	−0.0464478	+	0.998921i	$0.514790\pi$
$992$	0	0
$993$	13.2043	0.419026
$994$	0	0
$995$	2.55776	0.0810864
$996$	0	0
$997$	−3.91989	−0.124144	−0.0620722	−	0.998072i	$-0.519771\pi$
$997$	−3.91989	−0.124144	−0.0620722	+	0.998072i	$0.519771\pi$
$998$	0	0
$999$	3.98715	0.126148

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6036.2.a.h.1.11	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6036.2.a.h.1.11	✓	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +0.357952 q^{5} -4.62593 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +0.357952 q^{5} -4.62593 q^{7} +1.00000 q^{9} -3.69957 q^{11} -1.83240 q^{13} +0.357952 q^{15} -6.88058 q^{17} +4.94026 q^{19} -4.62593 q^{21} -3.59653 q^{23} -4.87187 q^{25} +1.00000 q^{27} +7.34256 q^{29} -6.14270 q^{31} -3.69957 q^{33} -1.65586 q^{35} +3.98715 q^{37} -1.83240 q^{39} +9.17046 q^{41} -0.816130 q^{43} +0.357952 q^{45} +11.4358 q^{47} +14.3993 q^{49} -6.88058 q^{51} -8.10482 q^{53} -1.32427 q^{55} +4.94026 q^{57} +6.88856 q^{59} -0.313062 q^{61} -4.62593 q^{63} -0.655911 q^{65} +5.40636 q^{67} -3.59653 q^{69} +14.3460 q^{71} +4.67033 q^{73} -4.87187 q^{75} +17.1140 q^{77} +13.0253 q^{79} +1.00000 q^{81} -12.5449 q^{83} -2.46292 q^{85} +7.34256 q^{87} -10.2026 q^{89} +8.47654 q^{91} -6.14270 q^{93} +1.76838 q^{95} +10.4047 q^{97} -3.69957 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 24 q^{3} + 18 q^{5} + 9 q^{7} + 24 q^{9}+O(q^{10}) \) 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

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Varieties

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Database

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