LMFDB - Embedded newform 1840.3.c.b.1151.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,3,Mod(1151,1840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0, 0]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1840.1151");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1840 = 2^{4} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1840.c (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$50.1363686423$
Analytic rank:	$0$
Dimension:	$56$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1151.10
Character	$\chi$	$=$	1840.1151
Dual form			1840.3.c.b.1151.47

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.65039i q^{3} -2.23607 q^{5} -6.59183i q^{7} -4.32537 q^{9} +O(q^{10})$	$q-3.65039i q^{3} -2.23607 q^{5} -6.59183i q^{7} -4.32537 q^{9} +0.493827i q^{11} +16.5461 q^{13} +8.16253i q^{15} +13.0387 q^{17} -7.99615i q^{19} -24.0628 q^{21} -4.79583i q^{23} +5.00000 q^{25} -17.0642i q^{27} -10.0847 q^{29} -47.1009i q^{31} +1.80266 q^{33} +14.7398i q^{35} +50.1172 q^{37} -60.3997i q^{39} +47.2666 q^{41} -1.05969i q^{43} +9.67182 q^{45} +26.4962i q^{47} +5.54780 q^{49} -47.5964i q^{51} +43.7983 q^{53} -1.10423i q^{55} -29.1891 q^{57} -6.97073i q^{59} -17.5860 q^{61} +28.5121i q^{63} -36.9982 q^{65} -15.7988i q^{67} -17.5067 q^{69} +11.8764i q^{71} -64.4116 q^{73} -18.2520i q^{75} +3.25522 q^{77} -111.028i q^{79} -101.220 q^{81} -66.2700i q^{83} -29.1555 q^{85} +36.8131i q^{87} -85.2746 q^{89} -109.069i q^{91} -171.937 q^{93} +17.8799i q^{95} -88.5769 q^{97} -2.13599i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q - 120 q^{9}+O(q^{10}) $ 56 * q - 120 * q^9	$ 56 q - 120 q^{9} - 56 q^{13} - 96 q^{17} + 104 q^{21} + 280 q^{25} - 76 q^{29} + 240 q^{33} - 88 q^{37} - 76 q^{41} - 356 q^{49} - 88 q^{53} - 256 q^{57} + 376 q^{61} + 120 q^{65} + 192 q^{73} - 168 q^{77} - 392 q^{81} - 60 q^{85} + 368 q^{89} + 216 q^{93} + 264 q^{97}+O(q^{100}) $ 56 * q - 120 * q^9 - 56 * q^13 - 96 * q^17 + 104 * q^21 + 280 * q^25 - 76 * q^29 + 240 * q^33 - 88 * q^37 - 76 * q^41 - 356 * q^49 - 88 * q^53 - 256 * q^57 + 376 * q^61 + 120 * q^65 + 192 * q^73 - 168 * q^77 - 392 * q^81 - 60 * q^85 + 368 * q^89 + 216 * q^93 + 264 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times$.

$n$	$737$	$1151$	$1201$	$1381$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$	− 3.65039i	− 1.21680i	−0.793631	−	0.608399i	$-0.791812\pi$
$3$	− 3.65039i	− 1.21680i	0.793631	−	0.608399i	$-0.208188\pi$
$4$	0	0
$5$	−2.23607	−0.447214
$5$	−2.23607	−0.447214
$6$	0	0
$7$	− 6.59183i	− 0.941690i	−0.882216	−	0.470845i	$-0.843949\pi$
$7$	− 6.59183i	− 0.941690i	0.882216	−	0.470845i	$-0.156051\pi$
$8$	0	0
$9$	−4.32537	−0.480597
$10$	0	0
$11$	0.493827i	0.0448934i	0.999748	+	0.0224467i	$0.00714560\pi$
$11$	0.493827i	0.0448934i	−0.999748	+	0.0224467i	$0.992854\pi$
$12$	0	0
$13$	16.5461	1.27278	0.636388	−	0.771369i	$-0.280428\pi$
$13$	16.5461	1.27278	0.636388	+	0.771369i	$0.280428\pi$
$14$	0	0
$15$	8.16253i	0.544168i
$16$	0	0
$17$	13.0387	0.766983	0.383492	−	0.923544i	$-0.374721\pi$
$17$	13.0387	0.766983	0.383492	+	0.923544i	$0.374721\pi$
$18$	0	0
$19$	− 7.99615i	− 0.420850i	−0.977610	−	0.210425i	$-0.932515\pi$
$19$	− 7.99615i	− 0.420850i	0.977610	−	0.210425i	$-0.0674848\pi$
$20$	0	0
$21$	−24.0628	−1.14585
$22$	0	0
$23$	− 4.79583i	− 0.208514i
$23$	− 4.79583i	− 0.208514i
$24$	0	0
$25$	5.00000	0.200000
$26$	0	0
$27$	− 17.0642i	− 0.632009i
$28$	0	0
$29$	−10.0847	−0.347748	−0.173874	−	0.984768i	$-0.555629\pi$
$29$	−10.0847	−0.347748	−0.173874	+	0.984768i	$0.555629\pi$
$30$	0	0
$31$	− 47.1009i	− 1.51938i	−0.650283	−	0.759692i	$-0.725350\pi$
$31$	− 47.1009i	− 1.51938i	0.650283	−	0.759692i	$-0.274650\pi$
$32$	0	0
$33$	1.80266	0.0546262
$34$	0	0
$35$	14.7398i	0.421136i
$36$	0	0
$37$	50.1172	1.35452	0.677260	−	0.735744i	$-0.263167\pi$
$37$	50.1172	1.35452	0.677260	+	0.735744i	$0.263167\pi$
$38$	0	0
$39$	− 60.3997i	− 1.54871i
$40$	0	0
$41$	47.2666	1.15284	0.576422	−	0.817152i	$-0.304448\pi$
$41$	47.2666	1.15284	0.576422	+	0.817152i	$0.304448\pi$
$42$	0	0
$43$	− 1.05969i	− 0.0246439i	−0.999924	−	0.0123219i	$-0.996078\pi$
$43$	− 1.05969i	− 0.0246439i	0.999924	−	0.0123219i	$-0.00392229\pi$
$44$	0	0
$45$	9.67182	0.214929
$46$	0	0
$47$	26.4962i	0.563749i	0.959451	+	0.281874i	$0.0909561\pi$
$47$	26.4962i	0.563749i	−0.959451	+	0.281874i	$0.909044\pi$
$48$	0	0
$49$	5.54780	0.113220
$50$	0	0
$51$	− 47.5964i	− 0.933264i
$52$	0	0
$53$	43.7983	0.826383	0.413191	−	0.910644i	$-0.364414\pi$
$53$	43.7983	0.826383	0.413191	+	0.910644i	$0.364414\pi$
$54$	0	0
$55$	− 1.10423i	− 0.0200769i
$56$	0	0
$57$	−29.1891	−0.512089
$58$	0	0
$59$	− 6.97073i	− 0.118148i	−0.998254	−	0.0590739i	$-0.981185\pi$
$59$	− 6.97073i	− 0.118148i	0.998254	−	0.0590739i	$-0.0188148\pi$
$60$	0	0
$61$	−17.5860	−0.288295	−0.144147	−	0.989556i	$-0.546044\pi$
$61$	−17.5860	−0.288295	−0.144147	+	0.989556i	$0.546044\pi$
$62$	0	0
$63$	28.5121i	0.452573i
$64$	0	0
$65$	−36.9982	−0.569203
$66$	0	0
$67$	− 15.7988i	− 0.235803i	−0.993025	−	0.117902i	$-0.962383\pi$
$67$	− 15.7988i	− 0.235803i	0.993025	−	0.117902i	$-0.0376167\pi$
$68$	0	0
$69$	−17.5067	−0.253720
$70$	0	0
$71$	11.8764i	0.167273i	0.996496	+	0.0836367i	$0.0266535\pi$
$71$	11.8764i	0.167273i	−0.996496	+	0.0836367i	$0.973346\pi$
$72$	0	0
$73$	−64.4116	−0.882350	−0.441175	−	0.897421i	$-0.645438\pi$
$73$	−64.4116	−0.882350	−0.441175	+	0.897421i	$0.645438\pi$
$74$	0	0
$75$	− 18.2520i	− 0.243360i
$76$	0	0
$77$	3.25522	0.0422756
$78$	0	0
$79$	− 111.028i	− 1.40542i	−0.711475	−	0.702712i	$-0.751972\pi$
$79$	− 111.028i	− 1.40542i	0.711475	−	0.702712i	$-0.248028\pi$
$80$	0	0
$81$	−101.220	−1.24962
$82$	0	0
$83$	− 66.2700i	− 0.798434i	−0.916856	−	0.399217i	$-0.869282\pi$
$83$	− 66.2700i	− 0.798434i	0.916856	−	0.399217i	$-0.130718\pi$
$84$	0	0
$85$	−29.1555	−0.343005
$86$	0	0
$87$	36.8131i	0.423139i
$88$	0	0
$89$	−85.2746	−0.958142	−0.479071	−	0.877776i	$-0.659026\pi$
$89$	−85.2746	−0.958142	−0.479071	+	0.877776i	$0.659026\pi$
$90$	0	0
$91$	− 109.069i	− 1.19856i
$92$	0	0
$93$	−171.937	−1.84878
$94$	0	0
$95$	17.8799i	0.188210i
$96$	0	0
$97$	−88.5769	−0.913164	−0.456582	−	0.889681i	$-0.650926\pi$
$97$	−88.5769	−0.913164	−0.456582	+	0.889681i	$0.650926\pi$
$98$	0	0
$99$	− 2.13599i	− 0.0215756i
$100$	0	0
$101$	66.2131	0.655575	0.327787	−	0.944752i	$-0.393697\pi$
$101$	66.2131	0.655575	0.327787	+	0.944752i	$0.393697\pi$
$102$	0	0
$103$	119.639i	1.16155i	0.814065	+	0.580774i	$0.197250\pi$
$103$	119.639i	1.16155i	−0.814065	+	0.580774i	$0.802750\pi$
$104$	0	0
$105$	53.8060	0.512438
$106$	0	0
$107$	116.358i	1.08746i	0.839260	+	0.543731i	$0.182989\pi$
$107$	116.358i	1.08746i	−0.839260	+	0.543731i	$0.817011\pi$
$108$	0	0
$109$	−98.5593	−0.904214	−0.452107	−	0.891964i	$-0.649327\pi$
$109$	−98.5593	−0.904214	−0.452107	+	0.891964i	$0.649327\pi$
$110$	0	0
$111$	− 182.948i	− 1.64818i
$112$	0	0
$113$	−21.6650	−0.191726	−0.0958628	−	0.995395i	$-0.530561\pi$
$113$	−21.6650	−0.191726	−0.0958628	+	0.995395i	$0.530561\pi$
$114$	0	0
$115$	10.7238i	0.0932505i
$116$	0	0
$117$	−71.5680	−0.611692
$118$	0	0
$119$	− 85.9490i	− 0.722260i
$120$	0	0
$121$	120.756	0.997985
$122$	0	0
$123$	− 172.542i	− 1.40278i
$124$	0	0
$125$	−11.1803	−0.0894427
$126$	0	0
$127$	− 47.5208i	− 0.374179i	−0.982343	−	0.187090i	$-0.940095\pi$
$127$	− 47.5208i	− 0.374179i	0.982343	−	0.187090i	$-0.0599055\pi$
$128$	0	0
$129$	−3.86827	−0.0299866
$130$	0	0
$131$	221.679i	1.69221i	0.533017	+	0.846104i	$0.321058\pi$
$131$	221.679i	1.69221i	−0.533017	+	0.846104i	$0.678942\pi$
$132$	0	0
$133$	−52.7092	−0.396310
$134$	0	0
$135$	38.1568i	0.282643i
$136$	0	0
$137$	−119.742	−0.874030	−0.437015	−	0.899454i	$-0.643964\pi$
$137$	−119.742	−0.874030	−0.437015	+	0.899454i	$0.643964\pi$
$138$	0	0
$139$	− 65.2939i	− 0.469740i	−0.972027	−	0.234870i	$-0.924534\pi$
$139$	− 65.2939i	− 0.469740i	0.972027	−	0.234870i	$-0.0754665\pi$
$140$	0	0
$141$	96.7215	0.685968
$142$	0	0
$143$	8.17091i	0.0571392i
$144$	0	0
$145$	22.5501	0.155518
$146$	0	0
$147$	− 20.2517i	− 0.137766i
$148$	0	0
$149$	−56.2403	−0.377452	−0.188726	−	0.982030i	$-0.560436\pi$
$149$	−56.2403	−0.377452	−0.188726	+	0.982030i	$0.560436\pi$
$150$	0	0
$151$	215.558i	1.42753i	0.700383	+	0.713767i	$0.253012\pi$
$151$	215.558i	1.42753i	−0.700383	+	0.713767i	$0.746988\pi$
$152$	0	0
$153$	−56.3973	−0.368610
$154$	0	0
$155$	105.321i	0.679489i
$156$	0	0
$157$	−222.277	−1.41577	−0.707887	−	0.706326i	$-0.750351\pi$
$157$	−222.277	−1.41577	−0.707887	+	0.706326i	$0.750351\pi$
$158$	0	0
$159$	− 159.881i	− 1.00554i
$160$	0	0
$161$	−31.6133	−0.196356
$162$	0	0
$163$	− 3.74020i	− 0.0229460i	−0.999934	−	0.0114730i	$-0.996348\pi$
$163$	− 3.74020i	− 0.0229460i	0.999934	−	0.0114730i	$-0.00365205\pi$
$164$	0	0
$165$	−4.03088	−0.0244296
$166$	0	0
$167$	− 189.464i	− 1.13451i	−0.823541	−	0.567256i	$-0.808005\pi$
$167$	− 189.464i	− 1.13451i	0.823541	−	0.567256i	$-0.191995\pi$
$168$	0	0
$169$	104.773	0.619959
$170$	0	0
$171$	34.5863i	0.202259i
$172$	0	0
$173$	268.944	1.55459	0.777295	−	0.629136i	$-0.216591\pi$
$173$	268.944	1.55459	0.777295	+	0.629136i	$0.216591\pi$
$174$	0	0
$175$	− 32.9591i	− 0.188338i
$176$	0	0
$177$	−25.4459	−0.143762
$178$	0	0
$179$	4.41735i	0.0246779i	0.999924	+	0.0123390i	$0.00392771\pi$
$179$	4.41735i	0.0246779i	−0.999924	+	0.0123390i	$0.996072\pi$
$180$	0	0
$181$	−8.05447	−0.0444998	−0.0222499	−	0.999752i	$-0.507083\pi$
$181$	−8.05447	−0.0444998	−0.0222499	+	0.999752i	$0.507083\pi$
$182$	0	0
$183$	64.1957i	0.350796i
$184$	0	0
$185$	−112.066	−0.605760
$186$	0	0
$187$	6.43887i	0.0344325i
$188$	0	0
$189$	−112.485	−0.595156
$190$	0	0
$191$	− 200.481i	− 1.04964i	−0.851214	−	0.524818i	$-0.824133\pi$
$191$	− 200.481i	− 1.04964i	0.851214	−	0.524818i	$-0.175867\pi$
$192$	0	0
$193$	−187.075	−0.969299	−0.484649	−	0.874708i	$-0.661053\pi$
$193$	−187.075	−0.969299	−0.484649	+	0.874708i	$0.661053\pi$
$194$	0	0
$195$	135.058i	0.692605i
$196$	0	0
$197$	−221.408	−1.12390	−0.561948	−	0.827172i	$-0.689948\pi$
$197$	−221.408	−1.12390	−0.561948	+	0.827172i	$0.689948\pi$
$198$	0	0
$199$	− 42.7149i	− 0.214648i	−0.994224	−	0.107324i	$-0.965772\pi$
$199$	− 42.7149i	− 0.214648i	0.994224	−	0.107324i	$-0.0342282\pi$
$200$	0	0
$201$	−57.6719	−0.286925
$202$	0	0
$203$	66.4766i	0.327471i
$204$	0	0
$205$	−105.691	−0.515568
$206$	0	0
$207$	20.7437i	0.100211i
$208$	0	0
$209$	3.94872	0.0188934
$210$	0	0
$211$	− 75.9653i	− 0.360025i	−0.983664	−	0.180013i	$-0.942386\pi$
$211$	− 75.9653i	− 0.360025i	0.983664	−	0.180013i	$-0.0576138\pi$
$212$	0	0
$213$	43.3536	0.203538
$214$	0	0
$215$	2.36953i	0.0110211i
$216$	0	0
$217$	−310.481	−1.43079
$218$	0	0
$219$	235.128i	1.07364i
$220$	0	0
$221$	215.740	0.976198
$222$	0	0
$223$	− 234.944i	− 1.05356i	−0.850002	−	0.526779i	$-0.823399\pi$
$223$	− 234.944i	− 1.05356i	0.850002	−	0.526779i	$-0.176601\pi$
$224$	0	0
$225$	−21.6268	−0.0961193
$226$	0	0
$227$	383.582i	1.68979i	0.534933	+	0.844894i	$0.320337\pi$
$227$	383.582i	1.68979i	−0.534933	+	0.844894i	$0.679663\pi$
$228$	0	0
$229$	404.959	1.76838	0.884190	−	0.467127i	$-0.154711\pi$
$229$	404.959	1.76838	0.884190	+	0.467127i	$0.154711\pi$
$230$	0	0
$231$	− 11.8828i	− 0.0514409i
$232$	0	0
$233$	−271.408	−1.16484	−0.582420	−	0.812888i	$-0.697894\pi$
$233$	−271.408	−1.16484	−0.582420	+	0.812888i	$0.697894\pi$
$234$	0	0
$235$	− 59.2473i	− 0.252116i
$236$	0	0
$237$	−405.298	−1.71012
$238$	0	0
$239$	− 341.600i	− 1.42929i	−0.699488	−	0.714644i	$-0.746589\pi$
$239$	− 341.600i	− 1.42929i	0.699488	−	0.714644i	$-0.253411\pi$
$240$	0	0
$241$	−261.561	−1.08532	−0.542659	−	0.839953i	$-0.682582\pi$
$241$	−261.561	−1.08532	−0.542659	+	0.839953i	$0.682582\pi$
$242$	0	0
$243$	215.913i	0.888530i
$244$	0	0
$245$	−12.4053	−0.0506337
$246$	0	0
$247$	− 132.305i	− 0.535648i
$248$	0	0
$249$	−241.912	−0.971533
$250$	0	0
$251$	52.7497i	0.210158i	0.994464	+	0.105079i	$0.0335096\pi$
$251$	52.7497i	0.210158i	−0.994464	+	0.105079i	$0.966490\pi$
$252$	0	0
$253$	2.36831	0.00936092
$254$	0	0
$255$	106.429i	0.417368i
$256$	0	0
$257$	−15.0884	−0.0587099	−0.0293550	−	0.999569i	$-0.509345\pi$
$257$	−15.0884	−0.0587099	−0.0293550	+	0.999569i	$0.509345\pi$
$258$	0	0
$259$	− 330.364i	− 1.27554i
$260$	0	0
$261$	43.6200	0.167127
$262$	0	0
$263$	71.1836i	0.270660i	0.990801	+	0.135330i	$0.0432095\pi$
$263$	71.1836i	0.270660i	−0.990801	+	0.135330i	$0.956791\pi$
$264$	0	0
$265$	−97.9359	−0.369570
$266$	0	0
$267$	311.286i	1.16586i
$268$	0	0
$269$	106.114	0.394475	0.197237	−	0.980356i	$-0.436803\pi$
$269$	106.114	0.394475	0.197237	+	0.980356i	$0.436803\pi$
$270$	0	0
$271$	− 47.4410i	− 0.175059i	−0.996162	−	0.0875294i	$-0.972103\pi$
$271$	− 47.4410i	− 0.175059i	0.996162	−	0.0875294i	$-0.0278972\pi$
$272$	0	0
$273$	−398.145	−1.45841
$274$	0	0
$275$	2.46914i	0.00897868i
$276$	0	0
$277$	228.259	0.824038	0.412019	−	0.911175i	$-0.364824\pi$
$277$	228.259	0.824038	0.412019	+	0.911175i	$0.364824\pi$
$278$	0	0
$279$	203.729i	0.730211i
$280$	0	0
$281$	−167.318	−0.595437	−0.297719	−	0.954654i	$-0.596226\pi$
$281$	−167.318	−0.595437	−0.297719	+	0.954654i	$0.596226\pi$
$282$	0	0
$283$	− 38.3760i	− 0.135604i	−0.997699	−	0.0678021i	$-0.978401\pi$
$283$	− 38.3760i	− 0.135604i	0.997699	−	0.0678021i	$-0.0215986\pi$
$284$	0	0
$285$	65.2688	0.229013
$286$	0	0
$287$	− 311.574i	− 1.08562i
$288$	0	0
$289$	−118.992	−0.411737
$290$	0	0
$291$	323.341i	1.11114i
$292$	0	0
$293$	376.306	1.28432	0.642161	−	0.766570i	$-0.278038\pi$
$293$	376.306	1.28432	0.642161	+	0.766570i	$0.278038\pi$
$294$	0	0
$295$	15.5870i	0.0528373i
$296$	0	0
$297$	8.42678	0.0283730
$298$	0	0
$299$	− 79.3523i	− 0.265392i
$300$	0	0
$301$	−6.98527	−0.0232069
$302$	0	0
$303$	− 241.704i	− 0.797702i
$304$	0	0
$305$	39.3234	0.128929
$306$	0	0
$307$	228.953i	0.745776i	0.927876	+	0.372888i	$0.121632\pi$
$307$	228.953i	0.745776i	−0.927876	+	0.372888i	$0.878368\pi$
$308$	0	0
$309$	436.731	1.41337
$310$	0	0
$311$	380.953i	1.22493i	0.790498	+	0.612464i	$0.209822\pi$
$311$	380.953i	1.22493i	−0.790498	+	0.612464i	$0.790178\pi$
$312$	0	0
$313$	0.893804	0.00285560	0.00142780	−	0.999999i	$-0.499546\pi$
$313$	0.893804	0.00285560	0.00142780	+	0.999999i	$0.499546\pi$
$314$	0	0
$315$	− 63.7550i	− 0.202397i
$316$	0	0
$317$	−188.068	−0.593273	−0.296637	−	0.954990i	$-0.595865\pi$
$317$	−188.068	−0.593273	−0.296637	+	0.954990i	$0.595865\pi$
$318$	0	0
$319$	− 4.98010i	− 0.0156116i
$320$	0	0
$321$	424.754	1.32322
$322$	0	0
$323$	− 104.260i	− 0.322785i
$324$	0	0
$325$	82.7305	0.254555
$326$	0	0
$327$	359.780i	1.10025i
$328$	0	0
$329$	174.658	0.530876
$330$	0	0
$331$	42.0186i	0.126945i	0.997984	+	0.0634723i	$0.0202174\pi$
$331$	42.0186i	0.126945i	−0.997984	+	0.0634723i	$0.979783\pi$
$332$	0	0
$333$	−216.776	−0.650978
$334$	0	0
$335$	35.3272i	0.105454i
$336$	0	0
$337$	−22.7736	−0.0675774	−0.0337887	−	0.999429i	$-0.510757\pi$
$337$	−22.7736	−0.0675774	−0.0337887	+	0.999429i	$0.510757\pi$
$338$	0	0
$339$	79.0857i	0.233291i
$340$	0	0
$341$	23.2597	0.0682103
$342$	0	0
$343$	− 359.570i	− 1.04831i
$344$	0	0
$345$	39.1461	0.113467
$346$	0	0
$347$	331.793i	0.956175i	0.878312	+	0.478087i	$0.158670\pi$
$347$	331.793i	0.956175i	−0.878312	+	0.478087i	$0.841330\pi$
$348$	0	0
$349$	83.7240	0.239897	0.119948	−	0.992780i	$-0.461727\pi$
$349$	83.7240	0.239897	0.119948	+	0.992780i	$0.461727\pi$
$350$	0	0
$351$	− 282.346i	− 0.804406i
$352$	0	0
$353$	361.191	1.02320	0.511601	−	0.859223i	$-0.329052\pi$
$353$	361.191	1.02320	0.511601	+	0.859223i	$0.329052\pi$
$354$	0	0
$355$	− 26.5565i	− 0.0748070i
$356$	0	0
$357$	−313.748	−0.878845
$358$	0	0
$359$	− 159.832i	− 0.445214i	−0.974908	−	0.222607i	$-0.928543\pi$
$359$	− 159.832i	− 0.445214i	0.974908	−	0.222607i	$-0.0714566\pi$
$360$	0	0
$361$	297.062	0.822885
$362$	0	0
$363$	− 440.807i	− 1.21435i
$364$	0	0
$365$	144.029	0.394599
$366$	0	0
$367$	− 207.884i	− 0.566443i	−0.959055	−	0.283221i	$-0.908597\pi$
$367$	− 207.884i	− 0.566443i	0.959055	−	0.283221i	$-0.0914031\pi$
$368$	0	0
$369$	−204.446	−0.554053
$370$	0	0
$371$	− 288.711i	− 0.778196i
$372$	0	0
$373$	339.246	0.909505	0.454753	−	0.890618i	$-0.349728\pi$
$373$	339.246	0.909505	0.454753	+	0.890618i	$0.349728\pi$
$374$	0	0
$375$	40.8126i	0.108834i
$376$	0	0
$377$	−166.862	−0.442606
$378$	0	0
$379$	− 427.407i	− 1.12772i	−0.825869	−	0.563862i	$-0.809315\pi$
$379$	− 427.407i	− 1.12772i	0.825869	−	0.563862i	$-0.190685\pi$
$380$	0	0
$381$	−173.469	−0.455300
$382$	0	0
$383$	278.470i	0.727076i	0.931579	+	0.363538i	$0.118431\pi$
$383$	278.470i	0.727076i	−0.931579	+	0.363538i	$0.881569\pi$
$384$	0	0
$385$	−7.27890	−0.0189062
$386$	0	0
$387$	4.58354i	0.0118438i
$388$	0	0
$389$	255.909	0.657864	0.328932	−	0.944354i	$-0.393311\pi$
$389$	255.909	0.657864	0.328932	+	0.944354i	$0.393311\pi$
$390$	0	0
$391$	− 62.5315i	− 0.159927i
$392$	0	0
$393$	809.217	2.05908
$394$	0	0
$395$	248.267i	0.628525i
$396$	0	0
$397$	−527.215	−1.32800	−0.663998	−	0.747734i	$-0.731142\pi$
$397$	−527.215	−1.32800	−0.663998	+	0.747734i	$0.731142\pi$
$398$	0	0
$399$	192.409i	0.482229i
$400$	0	0
$401$	−65.5098	−0.163366	−0.0816831	−	0.996658i	$-0.526030\pi$
$401$	−65.5098	−0.163366	−0.0816831	+	0.996658i	$0.526030\pi$
$402$	0	0
$403$	− 779.336i	− 1.93384i
$404$	0	0
$405$	226.334	0.558849
$406$	0	0
$407$	24.7493i	0.0608090i
$408$	0	0
$409$	−690.565	−1.68842	−0.844212	−	0.536010i	$-0.819931\pi$
$409$	−690.565	−1.68842	−0.844212	+	0.536010i	$0.819931\pi$
$410$	0	0
$411$	437.106i	1.06352i
$412$	0	0
$413$	−45.9498	−0.111259
$414$	0	0
$415$	148.184i	0.357071i
$416$	0	0
$417$	−238.348	−0.571579
$418$	0	0
$419$	329.841i	0.787211i	0.919279	+	0.393605i	$0.128772\pi$
$419$	329.841i	0.787211i	−0.919279	+	0.393605i	$0.871228\pi$
$420$	0	0
$421$	421.115	1.00027	0.500136	−	0.865947i	$-0.333283\pi$
$421$	421.115	1.00027	0.500136	+	0.865947i	$0.333283\pi$
$422$	0	0
$423$	− 114.606i	− 0.270936i
$424$	0	0
$425$	65.1936	0.153397
$426$	0	0
$427$	115.924i	0.271484i
$428$	0	0
$429$	29.8270	0.0695269
$430$	0	0
$431$	− 591.965i	− 1.37347i	−0.726909	−	0.686734i	$-0.759044\pi$
$431$	− 591.965i	− 1.37347i	0.726909	−	0.686734i	$-0.240956\pi$
$432$	0	0
$433$	−498.906	−1.15221	−0.576103	−	0.817377i	$-0.695427\pi$
$433$	−498.906	−1.15221	−0.576103	+	0.817377i	$0.695427\pi$
$434$	0	0
$435$	− 82.3166i	− 0.189234i
$436$	0	0
$437$	−38.3482	−0.0877533
$438$	0	0
$439$	− 357.599i	− 0.814577i	−0.913300	−	0.407289i	$-0.866474\pi$
$439$	− 357.599i	− 0.814577i	0.913300	−	0.407289i	$-0.133526\pi$
$440$	0	0
$441$	−23.9963	−0.0544134
$442$	0	0
$443$	− 418.187i	− 0.943990i	−0.881601	−	0.471995i	$-0.843534\pi$
$443$	− 418.187i	− 0.943990i	0.881601	−	0.471995i	$-0.156466\pi$
$444$	0	0
$445$	190.680	0.428494
$446$	0	0
$447$	205.299i	0.459282i
$448$	0	0
$449$	69.7377	0.155318	0.0776589	−	0.996980i	$-0.475255\pi$
$449$	69.7377	0.155318	0.0776589	+	0.996980i	$0.475255\pi$
$450$	0	0
$451$	23.3416i	0.0517551i
$452$	0	0
$453$	786.870	1.73702
$454$	0	0
$455$	243.886i	0.536012i
$456$	0	0
$457$	625.090	1.36781	0.683906	−	0.729570i	$-0.260280\pi$
$457$	625.090	1.36781	0.683906	+	0.729570i	$0.260280\pi$
$458$	0	0
$459$	− 222.496i	− 0.484740i
$460$	0	0
$461$	533.609	1.15750	0.578752	−	0.815504i	$-0.303540\pi$
$461$	533.609	1.15750	0.578752	+	0.815504i	$0.303540\pi$
$462$	0	0
$463$	440.282i	0.950932i	0.879734	+	0.475466i	$0.157721\pi$
$463$	440.282i	0.950932i	−0.879734	+	0.475466i	$0.842279\pi$
$464$	0	0
$465$	384.462	0.826801
$466$	0	0
$467$	− 342.994i	− 0.734462i	−0.930130	−	0.367231i	$-0.880306\pi$
$467$	− 342.994i	− 0.734462i	0.930130	−	0.367231i	$-0.119694\pi$
$468$	0	0
$469$	−104.143	−0.222053
$470$	0	0
$471$	811.397i	1.72271i
$472$	0	0
$473$	0.523302	0.00110635
$474$	0	0
$475$	− 39.9807i	− 0.0841700i
$476$	0	0
$477$	−189.444	−0.397157
$478$	0	0
$479$	− 29.3388i	− 0.0612502i	−0.999531	−	0.0306251i	$-0.990250\pi$
$479$	− 29.3388i	− 0.0612502i	0.999531	−	0.0306251i	$-0.00974980\pi$
$480$	0	0
$481$	829.244	1.72400
$482$	0	0
$483$	115.401i	0.238925i
$484$	0	0
$485$	198.064	0.408379
$486$	0	0
$487$	− 662.369i	− 1.36010i	−0.733166	−	0.680050i	$-0.761958\pi$
$487$	− 662.369i	− 1.36010i	0.733166	−	0.680050i	$-0.238042\pi$
$488$	0	0
$489$	−13.6532	−0.0279206
$490$	0	0
$491$	305.596i	0.622395i	0.950345	+	0.311198i	$0.100730\pi$
$491$	305.596i	0.622395i	−0.950345	+	0.311198i	$0.899270\pi$
$492$	0	0
$493$	−131.492	−0.266717
$494$	0	0
$495$	4.77621i	0.00964891i
$496$	0	0
$497$	78.2873	0.157520
$498$	0	0
$499$	− 199.873i	− 0.400548i	−0.979740	−	0.200274i	$-0.935817\pi$
$499$	− 199.873i	− 0.400548i	0.979740	−	0.200274i	$-0.0641832\pi$
$500$	0	0
$501$	−691.617	−1.38047
$502$	0	0
$503$	− 875.522i	− 1.74060i	−0.492522	−	0.870300i	$-0.663925\pi$
$503$	− 875.522i	− 1.74060i	0.492522	−	0.870300i	$-0.336075\pi$
$504$	0	0
$505$	−148.057	−0.293182
$506$	0	0
$507$	− 382.463i	− 0.754365i
$508$	0	0
$509$	−70.2582	−0.138032	−0.0690159	−	0.997616i	$-0.521986\pi$
$509$	−70.2582	−0.138032	−0.0690159	+	0.997616i	$0.521986\pi$
$510$	0	0
$511$	424.590i	0.830900i
$512$	0	0
$513$	−136.448	−0.265981
$514$	0	0
$515$	− 267.522i	− 0.519460i
$516$	0	0
$517$	−13.0845	−0.0253086
$518$	0	0
$519$	− 981.752i	− 1.89162i
$520$	0	0
$521$	783.558	1.50395	0.751975	−	0.659191i	$-0.229101\pi$
$521$	783.558	1.50395	0.751975	+	0.659191i	$0.229101\pi$
$522$	0	0
$523$	605.261i	1.15729i	0.815580	+	0.578644i	$0.196418\pi$
$523$	605.261i	1.15729i	−0.815580	+	0.578644i	$0.803582\pi$
$524$	0	0
$525$	−120.314	−0.229169
$526$	0	0
$527$	− 614.135i	− 1.16534i
$528$	0	0
$529$	−23.0000	−0.0434783
$530$	0	0
$531$	30.1510i	0.0567815i
$532$	0	0
$533$	782.078	1.46731
$534$	0	0
$535$	− 260.185i	− 0.486328i
$536$	0	0
$537$	16.1251	0.0300280
$538$	0	0
$539$	2.73966i	0.00508285i
$540$	0	0
$541$	886.941	1.63945	0.819724	−	0.572759i	$-0.194127\pi$
$541$	886.941	1.63945	0.819724	+	0.572759i	$0.194127\pi$
$542$	0	0
$543$	29.4020i	0.0541473i
$544$	0	0
$545$	220.385	0.404377
$546$	0	0
$547$	97.7131i	0.178635i	0.996003	+	0.0893173i	$0.0284685\pi$
$547$	97.7131i	0.178635i	−0.996003	+	0.0893173i	$0.971531\pi$
$548$	0	0
$549$	76.0658	0.138553
$550$	0	0
$551$	80.6388i	0.146350i
$552$	0	0
$553$	−731.881	−1.32347
$554$	0	0
$555$	409.083i	0.737087i
$556$	0	0
$557$	−17.9507	−0.0322275	−0.0161138	−	0.999870i	$-0.505129\pi$
$557$	−17.9507	−0.0322275	−0.0161138	+	0.999870i	$0.505129\pi$
$558$	0	0
$559$	− 17.5337i	− 0.0313661i
$560$	0	0
$561$	23.5044	0.0418974
$562$	0	0
$563$	244.141i	0.433643i	0.976211	+	0.216822i	$0.0695690\pi$
$563$	244.141i	0.433643i	−0.976211	+	0.216822i	$0.930431\pi$
$564$	0	0
$565$	48.4444	0.0857423
$566$	0	0
$567$	667.222i	1.17676i
$568$	0	0
$569$	−126.109	−0.221634	−0.110817	−	0.993841i	$-0.535347\pi$
$569$	−126.109	−0.221634	−0.110817	+	0.993841i	$0.535347\pi$
$570$	0	0
$571$	485.359i	0.850016i	0.905190	+	0.425008i	$0.139729\pi$
$571$	485.359i	0.850016i	−0.905190	+	0.425008i	$0.860271\pi$
$572$	0	0
$573$	−731.833	−1.27720
$574$	0	0
$575$	− 23.9792i	− 0.0417029i
$576$	0	0
$577$	1037.65	1.79836	0.899178	−	0.437584i	$-0.144166\pi$
$577$	1037.65	1.79836	0.899178	+	0.437584i	$0.144166\pi$
$578$	0	0
$579$	682.896i	1.17944i
$580$	0	0
$581$	−436.841	−0.751877
$582$	0	0
$583$	21.6288i	0.0370991i
$584$	0	0
$585$	160.031	0.273557
$586$	0	0
$587$	311.741i	0.531075i	0.964100	+	0.265538i	$0.0855495\pi$
$587$	311.741i	0.531075i	−0.964100	+	0.265538i	$0.914451\pi$
$588$	0	0
$589$	−376.626	−0.639432
$590$	0	0
$591$	808.225i	1.36755i
$592$	0	0
$593$	185.268	0.312426	0.156213	−	0.987723i	$-0.450071\pi$
$593$	185.268	0.312426	0.156213	+	0.987723i	$0.450071\pi$
$594$	0	0
$595$	192.188i	0.323005i
$596$	0	0
$597$	−155.926	−0.261183
$598$	0	0
$599$	− 1046.83i	− 1.74763i	−0.486258	−	0.873815i	$-0.661639\pi$
$599$	− 1046.83i	− 1.74763i	0.486258	−	0.873815i	$-0.338361\pi$
$600$	0	0
$601$	686.631	1.14248	0.571240	−	0.820783i	$-0.306463\pi$
$601$	686.631	1.14248	0.571240	+	0.820783i	$0.306463\pi$
$602$	0	0
$603$	68.3357i	0.113326i
$604$	0	0
$605$	−270.019	−0.446312
$606$	0	0
$607$	705.270i	1.16189i	0.813941	+	0.580947i	$0.197318\pi$
$607$	705.270i	1.16189i	−0.813941	+	0.580947i	$0.802682\pi$
$608$	0	0
$609$	242.666	0.398466
$610$	0	0
$611$	438.408i	0.717526i
$612$	0	0
$613$	−440.737	−0.718984	−0.359492	−	0.933148i	$-0.617050\pi$
$613$	−440.737	−0.718984	−0.359492	+	0.933148i	$0.617050\pi$
$614$	0	0
$615$	385.815i	0.627342i
$616$	0	0
$617$	−733.966	−1.18957	−0.594786	−	0.803884i	$-0.702763\pi$
$617$	−733.966	−1.18957	−0.594786	+	0.803884i	$0.702763\pi$
$618$	0	0
$619$	262.328i	0.423793i	0.977292	+	0.211897i	$0.0679640\pi$
$619$	262.328i	0.423793i	−0.977292	+	0.211897i	$0.932036\pi$
$620$	0	0
$621$	−81.8372	−0.131783
$622$	0	0
$623$	562.116i	0.902272i
$624$	0	0
$625$	25.0000	0.0400000
$626$	0	0
$627$	− 14.4144i	− 0.0229894i
$628$	0	0
$629$	653.465	1.03889
$630$	0	0
$631$	46.2868i	0.0733546i	0.999327	+	0.0366773i	$0.0116774\pi$
$631$	46.2868i	0.0733546i	−0.999327	+	0.0366773i	$0.988323\pi$
$632$	0	0
$633$	−277.303	−0.438078
$634$	0	0
$635$	106.260i	0.167338i
$636$	0	0
$637$	91.7945	0.144104
$638$	0	0
$639$	− 51.3699i	− 0.0803911i
$640$	0	0
$641$	698.294	1.08938	0.544691	−	0.838637i	$-0.316647\pi$
$641$	698.294	1.08938	0.544691	+	0.838637i	$0.316647\pi$
$642$	0	0
$643$	698.744i	1.08669i	0.839508	+	0.543347i	$0.182843\pi$
$643$	698.744i	1.08669i	−0.839508	+	0.543347i	$0.817157\pi$
$644$	0	0
$645$	8.64972	0.0134104
$646$	0	0
$647$	708.758i	1.09545i	0.836658	+	0.547726i	$0.184506\pi$
$647$	708.758i	1.09545i	−0.836658	+	0.547726i	$0.815494\pi$
$648$	0	0
$649$	3.44233	0.00530406
$650$	0	0
$651$	1133.38i	1.74098i
$652$	0	0
$653$	1098.89	1.68283	0.841417	−	0.540386i	$-0.181722\pi$
$653$	1098.89	1.68283	0.841417	+	0.540386i	$0.181722\pi$
$654$	0	0
$655$	− 495.690i	− 0.756779i
$656$	0	0
$657$	278.604	0.424055
$658$	0	0
$659$	591.005i	0.896822i	0.893828	+	0.448411i	$0.148010\pi$
$659$	591.005i	0.896822i	−0.893828	+	0.448411i	$0.851990\pi$
$660$	0	0
$661$	78.6893	0.119046	0.0595229	−	0.998227i	$-0.481042\pi$
$661$	78.6893	0.119046	0.0595229	+	0.998227i	$0.481042\pi$
$662$	0	0
$663$	− 787.535i	− 1.18784i
$664$	0	0
$665$	117.861	0.177235
$666$	0	0
$667$	48.3645i	0.0725105i
$668$	0	0
$669$	−857.637	−1.28197
$670$	0	0
$671$	− 8.68443i	− 0.0129425i
$672$	0	0
$673$	−78.3203	−0.116375	−0.0581874	−	0.998306i	$-0.518532\pi$
$673$	−78.3203	−0.116375	−0.0581874	+	0.998306i	$0.518532\pi$
$674$	0	0
$675$	− 85.3212i	− 0.126402i
$676$	0	0
$677$	171.386	0.253156	0.126578	−	0.991957i	$-0.459601\pi$
$677$	171.386	0.253156	0.126578	+	0.991957i	$0.459601\pi$
$678$	0	0
$679$	583.884i	0.859917i
$680$	0	0
$681$	1400.23	2.05613
$682$	0	0
$683$	939.259i	1.37520i	0.726091	+	0.687598i	$0.241335\pi$
$683$	939.259i	1.37520i	−0.726091	+	0.687598i	$0.758665\pi$
$684$	0	0
$685$	267.751	0.390878
$686$	0	0
$687$	− 1478.26i	− 2.15176i
$688$	0	0
$689$	724.690	1.05180
$690$	0	0
$691$	− 41.7396i	− 0.0604046i	−0.999544	−	0.0302023i	$-0.990385\pi$
$691$	− 41.7396i	− 0.0604046i	0.999544	−	0.0302023i	$-0.00961515\pi$
$692$	0	0
$693$	−14.0800	−0.0203175
$694$	0	0
$695$	146.002i	0.210074i
$696$	0	0
$697$	616.296	0.884213
$698$	0	0
$699$	990.745i	1.41738i
$700$	0	0
$701$	290.290	0.414109	0.207054	−	0.978329i	$-0.433612\pi$
$701$	290.290	0.414109	0.207054	+	0.978329i	$0.433612\pi$
$702$	0	0
$703$	− 400.745i	− 0.570050i
$704$	0	0
$705$	−216.276	−0.306774
$706$	0	0
$707$	− 436.465i	− 0.617348i
$708$	0	0
$709$	−1305.90	−1.84188	−0.920942	−	0.389700i	$-0.872579\pi$
$709$	−1305.90	−1.84188	−0.920942	+	0.389700i	$0.872579\pi$
$710$	0	0
$711$	480.239i	0.675442i
$712$	0	0
$713$	−225.888	−0.316813
$714$	0	0
$715$	− 18.2707i	− 0.0255534i
$716$	0	0
$717$	−1246.97	−1.73916
$718$	0	0
$719$	710.211i	0.987776i	0.869526	+	0.493888i	$0.164425\pi$
$719$	710.211i	0.987776i	−0.869526	+	0.493888i	$0.835575\pi$
$720$	0	0
$721$	788.642	1.09382
$722$	0	0
$723$	954.802i	1.32061i
$724$	0	0
$725$	−50.4235	−0.0695496
$726$	0	0
$727$	454.006i	0.624492i	0.950001	+	0.312246i	$0.101081\pi$
$727$	454.006i	0.624492i	−0.950001	+	0.312246i	$0.898919\pi$
$728$	0	0
$729$	−122.809	−0.168462
$730$	0	0
$731$	− 13.8170i	− 0.0189014i
$732$	0	0
$733$	449.775	0.613609	0.306804	−	0.951773i	$-0.400740\pi$
$733$	449.775	0.613609	0.306804	+	0.951773i	$0.400740\pi$
$734$	0	0
$735$	45.2841i	0.0616110i
$736$	0	0
$737$	7.80188	0.0105860
$738$	0	0
$739$	946.045i	1.28017i	0.768304	+	0.640085i	$0.221101\pi$
$739$	946.045i	1.28017i	−0.768304	+	0.640085i	$0.778899\pi$
$740$	0	0
$741$	−482.965	−0.651775
$742$	0	0
$743$	970.587i	1.30631i	0.757225	+	0.653154i	$0.226555\pi$
$743$	970.587i	1.30631i	−0.757225	+	0.653154i	$0.773445\pi$
$744$	0	0
$745$	125.757	0.168801
$746$	0	0
$747$	286.642i	0.383725i
$748$	0	0
$749$	767.015	1.02405
$750$	0	0
$751$	− 35.5808i	− 0.0473779i	−0.999719	−	0.0236889i	$-0.992459\pi$
$751$	− 35.5808i	− 0.0473779i	0.999719	−	0.0236889i	$-0.00754113\pi$
$752$	0	0
$753$	192.557	0.255720
$754$	0	0
$755$	− 482.001i	− 0.638412i
$756$	0	0
$757$	−1004.24	−1.32661	−0.663303	−	0.748351i	$-0.730846\pi$
$757$	−1004.24	−1.32661	−0.663303	+	0.748351i	$0.730846\pi$
$758$	0	0
$759$	− 8.64527i	− 0.0113903i
$760$	0	0
$761$	989.038	1.29966	0.649828	−	0.760082i	$-0.274841\pi$
$761$	989.038	1.29966	0.649828	+	0.760082i	$0.274841\pi$
$762$	0	0
$763$	649.686i	0.851489i
$764$	0	0
$765$	126.108	0.164847
$766$	0	0
$767$	− 115.338i	− 0.150376i
$768$	0	0
$769$	−182.602	−0.237454	−0.118727	−	0.992927i	$-0.537881\pi$
$769$	−182.602	−0.237454	−0.118727	+	0.992927i	$0.537881\pi$
$770$	0	0
$771$	55.0788i	0.0714381i
$772$	0	0
$773$	−658.437	−0.851794	−0.425897	−	0.904772i	$-0.640041\pi$
$773$	−658.437	−0.851794	−0.425897	+	0.904772i	$0.640041\pi$
$774$	0	0
$775$	− 235.504i	− 0.303877i
$776$	0	0
$777$	−1205.96	−1.55207
$778$	0	0
$779$	− 377.951i	− 0.485175i
$780$	0	0
$781$	−5.86490	−0.00750947
$782$	0	0
$783$	172.088i	0.219780i
$784$	0	0
$785$	497.026	0.633154
$786$	0	0
$787$	− 1038.17i	− 1.31915i	−0.751640	−	0.659573i	$-0.770737\pi$
$787$	− 1038.17i	− 1.31915i	0.751640	−	0.659573i	$-0.229263\pi$
$788$	0	0
$789$	259.848	0.329339
$790$	0	0
$791$	142.812i	0.180546i
$792$	0	0
$793$	−290.979	−0.366934
$794$	0	0
$795$	357.505i	0.449691i
$796$	0	0
$797$	−202.727	−0.254362	−0.127181	−	0.991880i	$-0.540593\pi$
$797$	−202.727	−0.254362	−0.127181	+	0.991880i	$0.540593\pi$
$798$	0	0
$799$	345.476i	0.432386i
$800$	0	0
$801$	368.844	0.460480
$802$	0	0
$803$	− 31.8082i	− 0.0396117i
$804$	0	0
$805$	70.6895	0.0878130
$806$	0	0
$807$	− 387.357i	− 0.479996i
$808$	0	0
$809$	−535.482	−0.661907	−0.330953	−	0.943647i	$-0.607370\pi$
$809$	−535.482	−0.661907	−0.330953	+	0.943647i	$0.607370\pi$
$810$	0	0
$811$	− 1098.55i	− 1.35456i	−0.735725	−	0.677280i	$-0.763159\pi$
$811$	− 1098.55i	− 1.35456i	0.735725	−	0.677280i	$-0.236841\pi$
$812$	0	0
$813$	−173.178	−0.213011
$814$	0	0
$815$	8.36334i	0.0102618i
$816$	0	0
$817$	−8.47342	−0.0103714
$818$	0	0
$819$	471.764i	0.576024i
$820$	0	0
$821$	806.201	0.981975	0.490987	−	0.871167i	$-0.336636\pi$
$821$	806.201	0.981975	0.490987	+	0.871167i	$0.336636\pi$
$822$	0	0
$823$	− 779.969i	− 0.947715i	−0.880602	−	0.473857i	$-0.842861\pi$
$823$	− 779.969i	− 0.947715i	0.880602	−	0.473857i	$-0.157139\pi$
$824$	0	0
$825$	9.01332	0.0109252
$826$	0	0
$827$	− 785.480i	− 0.949794i	−0.880041	−	0.474897i	$-0.842485\pi$
$827$	− 785.480i	− 0.949794i	0.880041	−	0.474897i	$-0.157515\pi$
$828$	0	0
$829$	−1092.29	−1.31760	−0.658801	−	0.752317i	$-0.728936\pi$
$829$	−1092.29	−1.31760	−0.658801	+	0.752317i	$0.728936\pi$
$830$	0	0
$831$	− 833.234i	− 1.00269i
$832$	0	0
$833$	72.3362	0.0868382
$834$	0	0
$835$	423.653i	0.507369i
$836$	0	0
$837$	−803.741	−0.960264
$838$	0	0
$839$	239.389i	0.285326i	0.989771	+	0.142663i	$0.0455666\pi$
$839$	239.389i	0.285326i	−0.989771	+	0.142663i	$0.954433\pi$
$840$	0	0
$841$	−739.299	−0.879071
$842$	0	0
$843$	610.776i	0.724527i
$844$	0	0
$845$	−234.280	−0.277254
$846$	0	0
$847$	− 796.004i	− 0.939792i
$848$	0	0
$849$	−140.087	−0.165003
$850$	0	0
$851$	− 240.354i	− 0.282437i
$852$	0	0
$853$	369.902	0.433648	0.216824	−	0.976211i	$-0.430430\pi$
$853$	369.902	0.433648	0.216824	+	0.976211i	$0.430430\pi$
$854$	0	0
$855$	− 77.3373i	− 0.0904530i
$856$	0	0
$857$	−84.1672	−0.0982114	−0.0491057	−	0.998794i	$-0.515637\pi$
$857$	−84.1672	−0.0982114	−0.0491057	+	0.998794i	$0.515637\pi$
$858$	0	0
$859$	345.829i	0.402594i	0.979530	+	0.201297i	$0.0645157\pi$
$859$	345.829i	0.402594i	−0.979530	+	0.201297i	$0.935484\pi$
$860$	0	0
$861$	−1137.37	−1.32098
$862$	0	0
$863$	− 943.438i	− 1.09321i	−0.837392	−	0.546604i	$-0.815920\pi$
$863$	− 943.438i	− 1.09321i	0.837392	−	0.546604i	$-0.184080\pi$
$864$	0	0
$865$	−601.377	−0.695234
$866$	0	0
$867$	434.367i	0.501000i
$868$	0	0
$869$	54.8289	0.0630942
$870$	0	0
$871$	− 261.409i	− 0.300125i
$872$	0	0
$873$	383.128	0.438864
$874$	0	0
$875$	73.6989i	0.0842273i
$876$	0	0
$877$	930.352	1.06083	0.530417	−	0.847737i	$-0.322035\pi$
$877$	930.352	1.06083	0.530417	+	0.847737i	$0.322035\pi$
$878$	0	0
$879$	− 1373.67i	− 1.56276i
$880$	0	0
$881$	710.690	0.806686	0.403343	−	0.915049i	$-0.367848\pi$
$881$	710.690	0.806686	0.403343	+	0.915049i	$0.367848\pi$
$882$	0	0
$883$	361.804i	0.409744i	0.978789	+	0.204872i	$0.0656778\pi$
$883$	361.804i	0.409744i	−0.978789	+	0.204872i	$0.934322\pi$
$884$	0	0
$885$	56.8987	0.0642924
$886$	0	0
$887$	− 1390.54i	− 1.56769i	−0.620960	−	0.783843i	$-0.713257\pi$
$887$	− 1390.54i	− 1.56769i	0.620960	−	0.783843i	$-0.286743\pi$
$888$	0	0
$889$	−313.249	−0.352361
$890$	0	0
$891$	− 49.9849i	− 0.0560998i
$892$	0	0
$893$	211.868	0.237254
$894$	0	0
$895$	− 9.87749i	− 0.0110363i
$896$	0	0
$897$	−289.667	−0.322929
$898$	0	0
$899$	474.998i	0.528363i
$900$	0	0
$901$	571.073	0.633822
$902$	0	0
$903$	25.4990i	0.0282381i
$904$	0	0
$905$	18.0103	0.0199009
$906$	0	0
$907$	− 1189.28i	− 1.31122i	−0.755100	−	0.655610i	$-0.772412\pi$
$907$	− 1189.28i	− 1.31122i	0.755100	−	0.655610i	$-0.227588\pi$
$908$	0	0
$909$	−286.396	−0.315067
$910$	0	0
$911$	201.667i	0.221369i	0.993856	+	0.110684i	$0.0353042\pi$
$911$	201.667i	0.221369i	−0.993856	+	0.110684i	$0.964696\pi$
$912$	0	0
$913$	32.7259	0.0358444
$914$	0	0
$915$	− 143.546i	− 0.156881i
$916$	0	0
$917$	1461.27	1.59354
$918$	0	0
$919$	− 60.9869i	− 0.0663623i	−0.999449	−	0.0331811i	$-0.989436\pi$
$919$	− 60.9869i	− 0.0663623i	0.999449	−	0.0331811i	$-0.0105638\pi$
$920$	0	0
$921$	835.769	0.907458
$922$	0	0
$923$	196.508i	0.212902i
$924$	0	0
$925$	250.586	0.270904
$926$	0	0
$927$	− 517.485i	− 0.558236i
$928$	0	0
$929$	1054.63	1.13524	0.567618	−	0.823292i	$-0.307865\pi$
$929$	1054.63	1.13524	0.567618	+	0.823292i	$0.307865\pi$
$930$	0	0
$931$	− 44.3611i	− 0.0476488i
$932$	0	0
$933$	1390.63	1.49049
$934$	0	0
$935$	− 14.3978i	− 0.0153987i
$936$	0	0
$937$	−694.929	−0.741653	−0.370827	−	0.928702i	$-0.620926\pi$
$937$	−694.929	−0.741653	−0.370827	+	0.928702i	$0.620926\pi$
$938$	0	0
$939$	− 3.26274i	− 0.00347469i
$940$	0	0
$941$	−1559.80	−1.65760	−0.828800	−	0.559545i	$-0.810976\pi$
$941$	−1559.80	−1.65760	−0.828800	+	0.559545i	$0.810976\pi$
$942$	0	0
$943$	− 226.683i	− 0.240385i
$944$	0	0
$945$	251.523	0.266162
$946$	0	0
$947$	− 302.041i	− 0.318946i	−0.987202	−	0.159473i	$-0.949021\pi$
$947$	− 302.041i	− 0.318946i	0.987202	−	0.159473i	$-0.0509794\pi$
$948$	0	0
$949$	−1065.76	−1.12303
$950$	0	0
$951$	686.521i	0.721893i
$952$	0	0
$953$	−1498.32	−1.57222	−0.786109	−	0.618088i	$-0.787908\pi$
$953$	−1498.32	−1.57222	−0.786109	+	0.618088i	$0.787908\pi$
$954$	0	0
$955$	448.288i	0.469412i
$956$	0	0
$957$	−18.1793	−0.0189962
$958$	0	0
$959$	789.319i	0.823065i
$960$	0	0
$961$	−1257.49	−1.30853
$962$	0	0
$963$	− 503.293i	− 0.522630i
$964$	0	0
$965$	418.312	0.433484
$966$	0	0
$967$	− 362.202i	− 0.374563i	−0.982306	−	0.187281i	$-0.940032\pi$
$967$	− 362.202i	− 0.374563i	0.982306	−	0.187281i	$-0.0599676\pi$
$968$	0	0
$969$	−380.588	−0.392764
$970$	0	0
$971$	1761.46i	1.81407i	0.421057	+	0.907034i	$0.361659\pi$
$971$	1761.46i	1.81407i	−0.421057	+	0.907034i	$0.638341\pi$
$972$	0	0
$973$	−430.406	−0.442349
$974$	0	0
$975$	− 301.999i	− 0.309742i
$976$	0	0
$977$	1288.25	1.31858	0.659289	−	0.751890i	$-0.270857\pi$
$977$	1288.25	1.31858	0.659289	+	0.751890i	$0.270857\pi$
$978$	0	0
$979$	− 42.1109i	− 0.0430142i
$980$	0	0
$981$	426.305	0.434562
$982$	0	0
$983$	1665.11i	1.69391i	0.531664	+	0.846955i	$0.321567\pi$
$983$	1665.11i	1.69391i	−0.531664	+	0.846955i	$0.678433\pi$
$984$	0	0
$985$	495.082	0.502622
$986$	0	0
$987$	− 637.572i	− 0.645969i
$988$	0	0
$989$	−5.08208	−0.00513860
$990$	0	0
$991$	191.431i	0.193169i	0.995325	+	0.0965846i	$0.0307918\pi$
$991$	191.431i	0.193169i	−0.995325	+	0.0965846i	$0.969208\pi$
$992$	0	0
$993$	153.385	0.154466
$994$	0	0
$995$	95.5134i	0.0959934i
$996$	0	0
$997$	−1606.70	−1.61153	−0.805765	−	0.592235i	$-0.798246\pi$
$997$	−1606.70	−1.61153	−0.805765	+	0.592235i	$0.798246\pi$
$998$	0	0
$999$	− 855.213i	− 0.856069i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.3.c.b.1151.10	✓	56
4.3	odd	2	inner	1840.3.c.b.1151.47	yes	56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1840.3.c.b.1151.10	✓	56	1.1	even	1	trivial
1840.3.c.b.1151.47	yes	56	4.3	odd	2	inner

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:	\( N \)	\(=\)	\( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1840.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.65039i q^{3} -2.23607 q^{5} -6.59183i q^{7} -4.32537 q^{9} +O(q^{10})\)	\(q-3.65039i q^{3} -2.23607 q^{5} -6.59183i q^{7} -4.32537 q^{9} +0.493827i q^{11} +16.5461 q^{13} +8.16253i q^{15} +13.0387 q^{17} -7.99615i q^{19} -24.0628 q^{21} -4.79583i q^{23} +5.00000 q^{25} -17.0642i q^{27} -10.0847 q^{29} -47.1009i q^{31} +1.80266 q^{33} +14.7398i q^{35} +50.1172 q^{37} -60.3997i q^{39} +47.2666 q^{41} -1.05969i q^{43} +9.67182 q^{45} +26.4962i q^{47} +5.54780 q^{49} -47.5964i q^{51} +43.7983 q^{53} -1.10423i q^{55} -29.1891 q^{57} -6.97073i q^{59} -17.5860 q^{61} +28.5121i q^{63} -36.9982 q^{65} -15.7988i q^{67} -17.5067 q^{69} +11.8764i q^{71} -64.4116 q^{73} -18.2520i q^{75} +3.25522 q^{77} -111.028i q^{79} -101.220 q^{81} -66.2700i q^{83} -29.1555 q^{85} +36.8131i q^{87} -85.2746 q^{89} -109.069i q^{91} -171.937 q^{93} +17.8799i q^{95} -88.5769 q^{97} -2.13599i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 120 q^{9}+O(q^{10}) \) 56 * q - 120 * q^9	\( 56 q - 120 q^{9} - 56 q^{13} - 96 q^{17} + 104 q^{21} + 280 q^{25} - 76 q^{29} + 240 q^{33} - 88 q^{37} - 76 q^{41} - 356 q^{49} - 88 q^{53} - 256 q^{57} + 376 q^{61} + 120 q^{65} + 192 q^{73} - 168 q^{77} - 392 q^{81} - 60 q^{85} + 368 q^{89} + 216 q^{93} + 264 q^{97}+O(q^{100}) \) 56 * q - 120 * q^9 - 56 * q^13 - 96 * q^17 + 104 * q^21 + 280 * q^25 - 76 * q^29 + 240 * q^33 - 88 * q^37 - 76 * q^41 - 356 * q^49 - 88 * q^53 - 256 * q^57 + 376 * q^61 + 120 * q^65 + 192 * q^73 - 168 * q^77 - 392 * q^81 - 60 * q^85 + 368 * q^89 + 216 * q^93 + 264 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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