LMFDB - Embedded newform 6040.2.a.s.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6040,2,Mod(1,6040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6040, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6040.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6040 = 2^{3} \cdot 5 \cdot 151 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6040.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:	$48.2296428209$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Character	$\chi$	$=$	6040.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.08470 q^{3} +1.00000 q^{5} +4.94762 q^{7} -1.82343 q^{9} +O(q^{10})$	$q-1.08470 q^{3} +1.00000 q^{5} +4.94762 q^{7} -1.82343 q^{9} +5.13009 q^{11} -2.50012 q^{13} -1.08470 q^{15} -7.14595 q^{17} +5.89977 q^{19} -5.36666 q^{21} -4.09995 q^{23} +1.00000 q^{25} +5.23196 q^{27} +6.60869 q^{29} +4.11277 q^{31} -5.56460 q^{33} +4.94762 q^{35} -10.1353 q^{37} +2.71187 q^{39} -3.11447 q^{41} +5.51843 q^{43} -1.82343 q^{45} -8.11320 q^{47} +17.4789 q^{49} +7.75119 q^{51} +12.1731 q^{53} +5.13009 q^{55} -6.39946 q^{57} +12.3336 q^{59} +3.41094 q^{61} -9.02165 q^{63} -2.50012 q^{65} -0.762500 q^{67} +4.44720 q^{69} +8.10672 q^{71} -5.82769 q^{73} -1.08470 q^{75} +25.3817 q^{77} -7.14489 q^{79} -0.204792 q^{81} +4.08603 q^{83} -7.14595 q^{85} -7.16843 q^{87} +13.6706 q^{89} -12.3696 q^{91} -4.46111 q^{93} +5.89977 q^{95} -9.95070 q^{97} -9.35438 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 2 q^{3} + 24 q^{5} + 3 q^{7} + 40 q^{9}+O(q^{10}) $ 24 * q + 2 * q^3 + 24 * q^5 + 3 * q^7 + 40 * q^9	$ 24 q + 2 q^{3} + 24 q^{5} + 3 q^{7} + 40 q^{9} + 17 q^{11} + 16 q^{13} + 2 q^{15} + 22 q^{17} + 16 q^{19} - q^{21} + 7 q^{23} + 24 q^{25} - 4 q^{27} + 25 q^{29} + 28 q^{31} + 11 q^{33} + 3 q^{35} + 26 q^{37} + 13 q^{39} + 38 q^{41} - 13 q^{43} + 40 q^{45} + 12 q^{47} + 61 q^{49} + 53 q^{53} + 17 q^{55} + 30 q^{57} + 35 q^{59} + 44 q^{61} - 9 q^{63} + 16 q^{65} - 15 q^{67} + 9 q^{69} + 22 q^{71} + 31 q^{73} + 2 q^{75} + 26 q^{77} + 20 q^{79} + 88 q^{81} - 14 q^{83} + 22 q^{85} - 18 q^{87} + 37 q^{89} - 26 q^{91} + 13 q^{93} + 16 q^{95} + 21 q^{97} - 12 q^{99}+O(q^{100}) $ 24 * q + 2 * q^3 + 24 * q^5 + 3 * q^7 + 40 * q^9 + 17 * q^11 + 16 * q^13 + 2 * q^15 + 22 * q^17 + 16 * q^19 - q^21 + 7 * q^23 + 24 * q^25 - 4 * q^27 + 25 * q^29 + 28 * q^31 + 11 * q^33 + 3 * q^35 + 26 * q^37 + 13 * q^39 + 38 * q^41 - 13 * q^43 + 40 * q^45 + 12 * q^47 + 61 * q^49 + 53 * q^53 + 17 * q^55 + 30 * q^57 + 35 * q^59 + 44 * q^61 - 9 * q^63 + 16 * q^65 - 15 * q^67 + 9 * q^69 + 22 * q^71 + 31 * q^73 + 2 * q^75 + 26 * q^77 + 20 * q^79 + 88 * q^81 - 14 * q^83 + 22 * q^85 - 18 * q^87 + 37 * q^89 - 26 * q^91 + 13 * q^93 + 16 * q^95 + 21 * q^97 - 12 * 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
$3$	−1.08470	−0.626250	−0.313125	−	0.949712i	$-0.601376\pi$
$3$	−1.08470	−0.626250	−0.313125	+	0.949712i	$0.601376\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	4.94762	1.87002	0.935011	−	0.354618i	$-0.115389\pi$
$7$	4.94762	1.87002	0.935011	+	0.354618i	$0.115389\pi$
$8$	0	0
$9$	−1.82343	−0.607811
$10$	0	0
$11$	5.13009	1.54678	0.773391	−	0.633930i	$-0.218559\pi$
$11$	5.13009	1.54678	0.773391	+	0.633930i	$0.218559\pi$
$12$	0	0
$13$	−2.50012	−0.693407	−0.346704	−	0.937975i	$-0.612699\pi$
$13$	−2.50012	−0.693407	−0.346704	+	0.937975i	$0.612699\pi$
$14$	0	0
$15$	−1.08470	−0.280067
$16$	0	0
$17$	−7.14595	−1.73315	−0.866574	−	0.499049i	$-0.833683\pi$
$17$	−7.14595	−1.73315	−0.866574	+	0.499049i	$0.833683\pi$
$18$	0	0
$19$	5.89977	1.35350	0.676750	−	0.736213i	$-0.263388\pi$
$19$	5.89977	1.35350	0.676750	+	0.736213i	$0.263388\pi$
$20$	0	0
$21$	−5.36666	−1.17110
$22$	0	0
$23$	−4.09995	−0.854899	−0.427449	−	0.904039i	$-0.640588\pi$
$23$	−4.09995	−0.854899	−0.427449	+	0.904039i	$0.640588\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.23196	1.00689
$28$	0	0
$29$	6.60869	1.22720	0.613602	−	0.789616i	$-0.289720\pi$
$29$	6.60869	1.22720	0.613602	+	0.789616i	$0.289720\pi$
$30$	0	0
$31$	4.11277	0.738675	0.369338	−	0.929295i	$-0.379585\pi$
$31$	4.11277	0.738675	0.369338	+	0.929295i	$0.379585\pi$
$32$	0	0
$33$	−5.56460	−0.968672
$34$	0	0
$35$	4.94762	0.836300
$36$	0	0
$37$	−10.1353	−1.66624	−0.833118	−	0.553095i	$-0.813447\pi$
$37$	−10.1353	−1.66624	−0.833118	+	0.553095i	$0.813447\pi$
$38$	0	0
$39$	2.71187	0.434246
$40$	0	0
$41$	−3.11447	−0.486398	−0.243199	−	0.969976i	$-0.578197\pi$
$41$	−3.11447	−0.486398	−0.243199	+	0.969976i	$0.578197\pi$
$42$	0	0
$43$	5.51843	0.841553	0.420777	−	0.907164i	$-0.361758\pi$
$43$	5.51843	0.841553	0.420777	+	0.907164i	$0.361758\pi$
$44$	0	0
$45$	−1.82343	−0.271821
$46$	0	0
$47$	−8.11320	−1.18343	−0.591716	−	0.806146i	$-0.701549\pi$
$47$	−8.11320	−1.18343	−0.591716	+	0.806146i	$0.701549\pi$
$48$	0	0
$49$	17.4789	2.49699
$50$	0	0
$51$	7.75119	1.08538
$52$	0	0
$53$	12.1731	1.67210	0.836052	−	0.548650i	$-0.184858\pi$
$53$	12.1731	1.67210	0.836052	+	0.548650i	$0.184858\pi$
$54$	0	0
$55$	5.13009	0.691742
$56$	0	0
$57$	−6.39946	−0.847629
$58$	0	0
$59$	12.3336	1.60570	0.802849	−	0.596182i	$-0.203316\pi$
$59$	12.3336	1.60570	0.802849	+	0.596182i	$0.203316\pi$
$60$	0	0
$61$	3.41094	0.436726	0.218363	−	0.975868i	$-0.429928\pi$
$61$	3.41094	0.436726	0.218363	+	0.975868i	$0.429928\pi$
$62$	0	0
$63$	−9.02165	−1.13662
$64$	0	0
$65$	−2.50012	−0.310101
$66$	0	0
$67$	−0.762500	−0.0931542	−0.0465771	−	0.998915i	$-0.514831\pi$
$67$	−0.762500	−0.0931542	−0.0465771	+	0.998915i	$0.514831\pi$
$68$	0	0
$69$	4.44720	0.535380
$70$	0	0
$71$	8.10672	0.962091	0.481045	−	0.876696i	$-0.340257\pi$
$71$	8.10672	0.962091	0.481045	+	0.876696i	$0.340257\pi$
$72$	0	0
$73$	−5.82769	−0.682080	−0.341040	−	0.940049i	$-0.610779\pi$
$73$	−5.82769	−0.682080	−0.341040	+	0.940049i	$0.610779\pi$
$74$	0	0
$75$	−1.08470	−0.125250
$76$	0	0
$77$	25.3817	2.89252
$78$	0	0
$79$	−7.14489	−0.803863	−0.401931	−	0.915670i	$-0.631661\pi$
$79$	−7.14489	−0.803863	−0.401931	+	0.915670i	$0.631661\pi$
$80$	0	0
$81$	−0.204792	−0.0227547
$82$	0	0
$83$	4.08603	0.448501	0.224250	−	0.974532i	$-0.428007\pi$
$83$	4.08603	0.448501	0.224250	+	0.974532i	$0.428007\pi$
$84$	0	0
$85$	−7.14595	−0.775087
$86$	0	0
$87$	−7.16843	−0.768536
$88$	0	0
$89$	13.6706	1.44908	0.724542	−	0.689230i	$-0.242051\pi$
$89$	13.6706	1.44908	0.724542	+	0.689230i	$0.242051\pi$
$90$	0	0
$91$	−12.3696	−1.29669
$92$	0	0
$93$	−4.46111	−0.462595
$94$	0	0
$95$	5.89977	0.605303
$96$	0	0
$97$	−9.95070	−1.01034	−0.505170	−	0.863020i	$-0.668570\pi$
$97$	−9.95070	−1.01034	−0.505170	+	0.863020i	$0.668570\pi$
$98$	0	0
$99$	−9.35438	−0.940151
$100$	0	0
$101$	−9.03685	−0.899200	−0.449600	−	0.893230i	$-0.648433\pi$
$101$	−9.03685	−0.899200	−0.449600	+	0.893230i	$0.648433\pi$
$102$	0	0
$103$	19.1151	1.88346	0.941731	−	0.336367i	$-0.109198\pi$
$103$	19.1151	1.88346	0.941731	+	0.336367i	$0.109198\pi$
$104$	0	0
$105$	−5.36666	−0.523733
$106$	0	0
$107$	−6.15670	−0.595191	−0.297595	−	0.954692i	$-0.596185\pi$
$107$	−6.15670	−0.595191	−0.297595	+	0.954692i	$0.596185\pi$
$108$	0	0
$109$	−16.8015	−1.60929	−0.804644	−	0.593757i	$-0.797644\pi$
$109$	−16.8015	−1.60929	−0.804644	+	0.593757i	$0.797644\pi$
$110$	0	0
$111$	10.9937	1.04348
$112$	0	0
$113$	14.3009	1.34532	0.672659	−	0.739953i	$-0.265152\pi$
$113$	14.3009	1.34532	0.672659	+	0.739953i	$0.265152\pi$
$114$	0	0
$115$	−4.09995	−0.382322
$116$	0	0
$117$	4.55879	0.421461
$118$	0	0
$119$	−35.3554	−3.24103
$120$	0	0
$121$	15.3179	1.39253
$122$	0	0
$123$	3.37825	0.304607
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−16.5698	−1.47033	−0.735167	−	0.677886i	$-0.762896\pi$
$127$	−16.5698	−1.47033	−0.735167	+	0.677886i	$0.762896\pi$
$128$	0	0
$129$	−5.98583	−0.527023
$130$	0	0
$131$	11.6657	1.01923	0.509617	−	0.860401i	$-0.329787\pi$
$131$	11.6657	1.01923	0.509617	+	0.860401i	$0.329787\pi$
$132$	0	0
$133$	29.1898	2.53108
$134$	0	0
$135$	5.23196	0.450296
$136$	0	0
$137$	16.3064	1.39315	0.696574	−	0.717485i	$-0.254707\pi$
$137$	16.3064	1.39315	0.696574	+	0.717485i	$0.254707\pi$
$138$	0	0
$139$	9.20843	0.781049	0.390524	−	0.920593i	$-0.372294\pi$
$139$	9.20843	0.781049	0.390524	+	0.920593i	$0.372294\pi$
$140$	0	0
$141$	8.80037	0.741124
$142$	0	0
$143$	−12.8258	−1.07255
$144$	0	0
$145$	6.60869	0.548822
$146$	0	0
$147$	−18.9593	−1.56374
$148$	0	0
$149$	−3.68722	−0.302069	−0.151034	−	0.988529i	$-0.548260\pi$
$149$	−3.68722	−0.302069	−0.151034	+	0.988529i	$0.548260\pi$
$150$	0	0
$151$	1.00000	0.0813788
$151$	1.00000	0.0813788
$152$	0	0
$153$	13.0302	1.05343
$154$	0	0
$155$	4.11277	0.330346
$156$	0	0
$157$	−11.2583	−0.898513	−0.449257	−	0.893403i	$-0.648311\pi$
$157$	−11.2583	−0.898513	−0.449257	+	0.893403i	$0.648311\pi$
$158$	0	0
$159$	−13.2041	−1.04716
$160$	0	0
$161$	−20.2850	−1.59868
$162$	0	0
$163$	9.28188	0.727013	0.363506	−	0.931592i	$-0.381579\pi$
$163$	9.28188	0.727013	0.363506	+	0.931592i	$0.381579\pi$
$164$	0	0
$165$	−5.56460	−0.433203
$166$	0	0
$167$	12.3140	0.952885	0.476442	−	0.879206i	$-0.341926\pi$
$167$	12.3140	0.952885	0.476442	+	0.879206i	$0.341926\pi$
$168$	0	0
$169$	−6.74942	−0.519186
$170$	0	0
$171$	−10.7578	−0.822672
$172$	0	0
$173$	17.6144	1.33920	0.669600	−	0.742722i	$-0.266466\pi$
$173$	17.6144	1.33920	0.669600	+	0.742722i	$0.266466\pi$
$174$	0	0
$175$	4.94762	0.374005
$176$	0	0
$177$	−13.3782	−1.00557
$178$	0	0
$179$	−26.3482	−1.96936	−0.984678	−	0.174384i	$-0.944206\pi$
$179$	−26.3482	−1.96936	−0.984678	+	0.174384i	$0.944206\pi$
$180$	0	0
$181$	−25.1878	−1.87219	−0.936096	−	0.351745i	$-0.885588\pi$
$181$	−25.1878	−1.87219	−0.936096	+	0.351745i	$0.885588\pi$
$182$	0	0
$183$	−3.69984	−0.273500
$184$	0	0
$185$	−10.1353	−0.745163
$186$	0	0
$187$	−36.6594	−2.68080
$188$	0	0
$189$	25.8857	1.88291
$190$	0	0
$191$	16.9165	1.22403	0.612016	−	0.790845i	$-0.290359\pi$
$191$	16.9165	1.22403	0.612016	+	0.790845i	$0.290359\pi$
$192$	0	0
$193$	−12.5035	−0.900018	−0.450009	−	0.893024i	$-0.648579\pi$
$193$	−12.5035	−0.900018	−0.450009	+	0.893024i	$0.648579\pi$
$194$	0	0
$195$	2.71187	0.194201
$196$	0	0
$197$	−9.91164	−0.706175	−0.353088	−	0.935590i	$-0.614868\pi$
$197$	−9.91164	−0.706175	−0.353088	+	0.935590i	$0.614868\pi$
$198$	0	0
$199$	−8.14803	−0.577598	−0.288799	−	0.957390i	$-0.593256\pi$
$199$	−8.14803	−0.577598	−0.288799	+	0.957390i	$0.593256\pi$
$200$	0	0
$201$	0.827081	0.0583378
$202$	0	0
$203$	32.6973	2.29490
$204$	0	0
$205$	−3.11447	−0.217524
$206$	0	0
$207$	7.47599	0.519617
$208$	0	0
$209$	30.2664	2.09357
$210$	0	0
$211$	−3.56455	−0.245393	−0.122697	−	0.992444i	$-0.539154\pi$
$211$	−3.56455	−0.245393	−0.122697	+	0.992444i	$0.539154\pi$
$212$	0	0
$213$	−8.79333	−0.602509
$214$	0	0
$215$	5.51843	0.376354
$216$	0	0
$217$	20.3484	1.38134
$218$	0	0
$219$	6.32128	0.427153
$220$	0	0
$221$	17.8657	1.20178
$222$	0	0
$223$	−10.0190	−0.670920	−0.335460	−	0.942054i	$-0.608892\pi$
$223$	−10.0190	−0.670920	−0.335460	+	0.942054i	$0.608892\pi$
$224$	0	0
$225$	−1.82343	−0.121562
$226$	0	0
$227$	−14.3309	−0.951173	−0.475586	−	0.879669i	$-0.657764\pi$
$227$	−14.3309	−0.951173	−0.475586	+	0.879669i	$0.657764\pi$
$228$	0	0
$229$	5.97598	0.394904	0.197452	−	0.980313i	$-0.436733\pi$
$229$	5.97598	0.394904	0.197452	+	0.980313i	$0.436733\pi$
$230$	0	0
$231$	−27.5315	−1.81144
$232$	0	0
$233$	7.94330	0.520383	0.260191	−	0.965557i	$-0.416214\pi$
$233$	7.94330	0.520383	0.260191	+	0.965557i	$0.416214\pi$
$234$	0	0
$235$	−8.11320	−0.529247
$236$	0	0
$237$	7.75003	0.503419
$238$	0	0
$239$	17.2553	1.11615	0.558077	−	0.829789i	$-0.311539\pi$
$239$	17.2553	1.11615	0.558077	+	0.829789i	$0.311539\pi$
$240$	0	0
$241$	11.6050	0.747546	0.373773	−	0.927520i	$-0.378064\pi$
$241$	11.6050	0.747546	0.373773	+	0.927520i	$0.378064\pi$
$242$	0	0
$243$	−15.4737	−0.992641
$244$	0	0
$245$	17.4789	1.11669
$246$	0	0
$247$	−14.7501	−0.938527
$248$	0	0
$249$	−4.43211	−0.280873
$250$	0	0
$251$	−13.4261	−0.847446	−0.423723	−	0.905792i	$-0.639277\pi$
$251$	−13.4261	−0.847446	−0.423723	+	0.905792i	$0.639277\pi$
$252$	0	0
$253$	−21.0331	−1.32234
$254$	0	0
$255$	7.75119	0.485398
$256$	0	0
$257$	−28.4227	−1.77296	−0.886480	−	0.462767i	$-0.846857\pi$
$257$	−28.4227	−1.77296	−0.886480	+	0.462767i	$0.846857\pi$
$258$	0	0
$259$	−50.1457	−3.11590
$260$	0	0
$261$	−12.0505	−0.745908
$262$	0	0
$263$	7.19032	0.443374	0.221687	−	0.975118i	$-0.428844\pi$
$263$	7.19032	0.443374	0.221687	+	0.975118i	$0.428844\pi$
$264$	0	0
$265$	12.1731	0.747788
$266$	0	0
$267$	−14.8285	−0.907489
$268$	0	0
$269$	4.07055	0.248186	0.124093	−	0.992271i	$-0.460398\pi$
$269$	4.07055	0.248186	0.124093	+	0.992271i	$0.460398\pi$
$270$	0	0
$271$	10.1325	0.615504	0.307752	−	0.951467i	$-0.400423\pi$
$271$	10.1325	0.615504	0.307752	+	0.951467i	$0.400423\pi$
$272$	0	0
$273$	13.4173	0.812051
$274$	0	0
$275$	5.13009	0.309356
$276$	0	0
$277$	18.0521	1.08465	0.542323	−	0.840170i	$-0.317545\pi$
$277$	18.0521	1.08465	0.542323	+	0.840170i	$0.317545\pi$
$278$	0	0
$279$	−7.49936	−0.448975
$280$	0	0
$281$	30.7765	1.83597	0.917986	−	0.396612i	$-0.129814\pi$
$281$	30.7765	1.83597	0.917986	+	0.396612i	$0.129814\pi$
$282$	0	0
$283$	21.0648	1.25217	0.626085	−	0.779755i	$-0.284656\pi$
$283$	21.0648	1.25217	0.626085	+	0.779755i	$0.284656\pi$
$284$	0	0
$285$	−6.39946	−0.379071
$286$	0	0
$287$	−15.4092	−0.909575
$288$	0	0
$289$	34.0646	2.00380
$290$	0	0
$291$	10.7935	0.632726
$292$	0	0
$293$	26.2217	1.53189	0.765945	−	0.642906i	$-0.222271\pi$
$293$	26.2217	1.53189	0.765945	+	0.642906i	$0.222271\pi$
$294$	0	0
$295$	12.3336	0.718090
$296$	0	0
$297$	26.8405	1.55744
$298$	0	0
$299$	10.2504	0.592793
$300$	0	0
$301$	27.3031	1.57372
$302$	0	0
$303$	9.80224	0.563124
$304$	0	0
$305$	3.41094	0.195310
$306$	0	0
$307$	8.59551	0.490572	0.245286	−	0.969451i	$-0.421118\pi$
$307$	8.59551	0.490572	0.245286	+	0.969451i	$0.421118\pi$
$308$	0	0
$309$	−20.7340	−1.17952
$310$	0	0
$311$	−9.87369	−0.559886	−0.279943	−	0.960017i	$-0.590315\pi$
$311$	−9.87369	−0.559886	−0.279943	+	0.960017i	$0.590315\pi$
$312$	0	0
$313$	−1.66755	−0.0942555	−0.0471278	−	0.998889i	$-0.515007\pi$
$313$	−1.66755	−0.0942555	−0.0471278	+	0.998889i	$0.515007\pi$
$314$	0	0
$315$	−9.02165	−0.508312
$316$	0	0
$317$	3.48532	0.195755	0.0978775	−	0.995198i	$-0.468795\pi$
$317$	3.48532	0.195755	0.0978775	+	0.995198i	$0.468795\pi$
$318$	0	0
$319$	33.9032	1.89822
$320$	0	0
$321$	6.67815	0.372738
$322$	0	0
$323$	−42.1595	−2.34582
$324$	0	0
$325$	−2.50012	−0.138681
$326$	0	0
$327$	18.2245	1.00782
$328$	0	0
$329$	−40.1410	−2.21305
$330$	0	0
$331$	12.6062	0.692901	0.346450	−	0.938068i	$-0.387387\pi$
$331$	12.6062	0.692901	0.346450	+	0.938068i	$0.387387\pi$
$332$	0	0
$333$	18.4811	1.01276
$334$	0	0
$335$	−0.762500	−0.0416598
$336$	0	0
$337$	16.4473	0.895942	0.447971	−	0.894048i	$-0.352147\pi$
$337$	16.4473	0.895942	0.447971	+	0.894048i	$0.352147\pi$
$338$	0	0
$339$	−15.5122	−0.842505
$340$	0	0
$341$	21.0989	1.14257
$342$	0	0
$343$	51.8455	2.79940
$344$	0	0
$345$	4.44720	0.239429
$346$	0	0
$347$	3.32924	0.178723	0.0893614	−	0.995999i	$-0.471517\pi$
$347$	3.32924	0.178723	0.0893614	+	0.995999i	$0.471517\pi$
$348$	0	0
$349$	−5.65830	−0.302882	−0.151441	−	0.988466i	$-0.548391\pi$
$349$	−5.65830	−0.302882	−0.151441	+	0.988466i	$0.548391\pi$
$350$	0	0
$351$	−13.0805	−0.698186
$352$	0	0
$353$	−21.5263	−1.14573	−0.572863	−	0.819651i	$-0.694167\pi$
$353$	−21.5263	−1.14573	−0.572863	+	0.819651i	$0.694167\pi$
$354$	0	0
$355$	8.10672	0.430260
$356$	0	0
$357$	38.3499	2.02969
$358$	0	0
$359$	34.4874	1.82018	0.910089	−	0.414414i	$-0.136013\pi$
$359$	34.4874	1.82018	0.910089	+	0.414414i	$0.136013\pi$
$360$	0	0
$361$	15.8073	0.831961
$362$	0	0
$363$	−16.6152	−0.872074
$364$	0	0
$365$	−5.82769	−0.305035
$366$	0	0
$367$	8.14522	0.425177	0.212588	−	0.977142i	$-0.431811\pi$
$367$	8.14522	0.425177	0.212588	+	0.977142i	$0.431811\pi$
$368$	0	0
$369$	5.67902	0.295638
$370$	0	0
$371$	60.2278	3.12687
$372$	0	0
$373$	1.05619	0.0546875	0.0273438	−	0.999626i	$-0.491295\pi$
$373$	1.05619	0.0546875	0.0273438	+	0.999626i	$0.491295\pi$
$374$	0	0
$375$	−1.08470	−0.0560135
$376$	0	0
$377$	−16.5225	−0.850952
$378$	0	0
$379$	7.76128	0.398670	0.199335	−	0.979931i	$-0.436122\pi$
$379$	7.76128	0.398670	0.199335	+	0.979931i	$0.436122\pi$
$380$	0	0
$381$	17.9732	0.920797
$382$	0	0
$383$	3.38532	0.172982	0.0864908	−	0.996253i	$-0.472435\pi$
$383$	3.38532	0.172982	0.0864908	+	0.996253i	$0.472435\pi$
$384$	0	0
$385$	25.3817	1.29357
$386$	0	0
$387$	−10.0625	−0.511505
$388$	0	0
$389$	4.46696	0.226484	0.113242	−	0.993567i	$-0.463876\pi$
$389$	4.46696	0.226484	0.113242	+	0.993567i	$0.463876\pi$
$390$	0	0
$391$	29.2981	1.48167
$392$	0	0
$393$	−12.6537	−0.638296
$394$	0	0
$395$	−7.14489	−0.359498
$396$	0	0
$397$	19.4989	0.978620	0.489310	−	0.872110i	$-0.337249\pi$
$397$	19.4989	0.978620	0.489310	+	0.872110i	$0.337249\pi$
$398$	0	0
$399$	−31.6621	−1.58509
$400$	0	0
$401$	−6.07592	−0.303417	−0.151708	−	0.988425i	$-0.548477\pi$
$401$	−6.07592	−0.303417	−0.151708	+	0.988425i	$0.548477\pi$
$402$	0	0
$403$	−10.2824	−0.512203
$404$	0	0
$405$	−0.204792	−0.0101762
$406$	0	0
$407$	−51.9951	−2.57730
$408$	0	0
$409$	−12.3493	−0.610632	−0.305316	−	0.952251i	$-0.598762\pi$
$409$	−12.3493	−0.610632	−0.305316	+	0.952251i	$0.598762\pi$
$410$	0	0
$411$	−17.6875	−0.872459
$412$	0	0
$413$	61.0219	3.00269
$414$	0	0
$415$	4.08603	0.200576
$416$	0	0
$417$	−9.98835	−0.489132
$418$	0	0
$419$	10.8973	0.532369	0.266185	−	0.963922i	$-0.414237\pi$
$419$	10.8973	0.532369	0.266185	+	0.963922i	$0.414237\pi$
$420$	0	0
$421$	−3.39295	−0.165362	−0.0826812	−	0.996576i	$-0.526348\pi$
$421$	−3.39295	−0.165362	−0.0826812	+	0.996576i	$0.526348\pi$
$422$	0	0
$423$	14.7939	0.719303
$424$	0	0
$425$	−7.14595	−0.346630
$426$	0	0
$427$	16.8760	0.816688
$428$	0	0
$429$	13.9121	0.671684
$430$	0	0
$431$	−11.8008	−0.568424	−0.284212	−	0.958762i	$-0.591732\pi$
$431$	−11.8008	−0.568424	−0.284212	+	0.958762i	$0.591732\pi$
$432$	0	0
$433$	−11.2690	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$433$	−11.2690	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$434$	0	0
$435$	−7.16843	−0.343700
$436$	0	0
$437$	−24.1888	−1.15711
$438$	0	0
$439$	−3.08365	−0.147175	−0.0735873	−	0.997289i	$-0.523445\pi$
$439$	−3.08365	−0.147175	−0.0735873	+	0.997289i	$0.523445\pi$
$440$	0	0
$441$	−31.8716	−1.51770
$442$	0	0
$443$	−29.7294	−1.41249	−0.706243	−	0.707970i	$-0.749611\pi$
$443$	−29.7294	−1.41249	−0.706243	+	0.707970i	$0.749611\pi$
$444$	0	0
$445$	13.6706	0.648050
$446$	0	0
$447$	3.99951	0.189170
$448$	0	0
$449$	24.6622	1.16388	0.581941	−	0.813231i	$-0.302294\pi$
$449$	24.6622	1.16388	0.581941	+	0.813231i	$0.302294\pi$
$450$	0	0
$451$	−15.9775	−0.752351
$452$	0	0
$453$	−1.08470	−0.0509635
$454$	0	0
$455$	−12.3696	−0.579896
$456$	0	0
$457$	8.69831	0.406890	0.203445	−	0.979086i	$-0.434786\pi$
$457$	8.69831	0.406890	0.203445	+	0.979086i	$0.434786\pi$
$458$	0	0
$459$	−37.3874	−1.74509
$460$	0	0
$461$	−23.9332	−1.11468	−0.557339	−	0.830285i	$-0.688178\pi$
$461$	−23.9332	−1.11468	−0.557339	+	0.830285i	$0.688178\pi$
$462$	0	0
$463$	2.90875	0.135181	0.0675906	−	0.997713i	$-0.478469\pi$
$463$	2.90875	0.135181	0.0675906	+	0.997713i	$0.478469\pi$
$464$	0	0
$465$	−4.46111	−0.206879
$466$	0	0
$467$	22.1993	1.02726	0.513630	−	0.858012i	$-0.328300\pi$
$467$	22.1993	1.02726	0.513630	+	0.858012i	$0.328300\pi$
$468$	0	0
$469$	−3.77256	−0.174200
$470$	0	0
$471$	12.2119	0.562694
$472$	0	0
$473$	28.3101	1.30170
$474$	0	0
$475$	5.89977	0.270700
$476$	0	0
$477$	−22.1968	−1.01632
$478$	0	0
$479$	39.5352	1.80641	0.903204	−	0.429211i	$-0.141208\pi$
$479$	39.5352	1.80641	0.903204	+	0.429211i	$0.141208\pi$
$480$	0	0
$481$	25.3395	1.15538
$482$	0	0
$483$	22.0030	1.00117
$484$	0	0
$485$	−9.95070	−0.451838
$486$	0	0
$487$	29.4468	1.33436	0.667180	−	0.744896i	$-0.267501\pi$
$487$	29.4468	1.33436	0.667180	+	0.744896i	$0.267501\pi$
$488$	0	0
$489$	−10.0680	−0.455292
$490$	0	0
$491$	−26.8019	−1.20955	−0.604776	−	0.796396i	$-0.706737\pi$
$491$	−26.8019	−1.20955	−0.604776	+	0.796396i	$0.706737\pi$
$492$	0	0
$493$	−47.2254	−2.12692
$494$	0	0
$495$	−9.35438	−0.420448
$496$	0	0
$497$	40.1089	1.79913
$498$	0	0
$499$	−17.2925	−0.774118	−0.387059	−	0.922055i	$-0.626509\pi$
$499$	−17.2925	−0.774118	−0.387059	+	0.922055i	$0.626509\pi$
$500$	0	0
$501$	−13.3569	−0.596744
$502$	0	0
$503$	−3.24324	−0.144609	−0.0723044	−	0.997383i	$-0.523035\pi$
$503$	−3.24324	−0.144609	−0.0723044	+	0.997383i	$0.523035\pi$
$504$	0	0
$505$	−9.03685	−0.402135
$506$	0	0
$507$	7.32107	0.325140
$508$	0	0
$509$	8.55236	0.379076	0.189538	−	0.981873i	$-0.439301\pi$
$509$	8.55236	0.379076	0.189538	+	0.981873i	$0.439301\pi$
$510$	0	0
$511$	−28.8332	−1.27551
$512$	0	0
$513$	30.8674	1.36283
$514$	0	0
$515$	19.1151	0.842310
$516$	0	0
$517$	−41.6215	−1.83051
$518$	0	0
$519$	−19.1063	−0.838673
$520$	0	0
$521$	−20.8839	−0.914939	−0.457469	−	0.889225i	$-0.651244\pi$
$521$	−20.8839	−0.914939	−0.457469	+	0.889225i	$0.651244\pi$
$522$	0	0
$523$	−8.02243	−0.350797	−0.175398	−	0.984498i	$-0.556121\pi$
$523$	−8.02243	−0.350797	−0.175398	+	0.984498i	$0.556121\pi$
$524$	0	0
$525$	−5.36666	−0.234220
$526$	0	0
$527$	−29.3897	−1.28023
$528$	0	0
$529$	−6.19040	−0.269148
$530$	0	0
$531$	−22.4895	−0.975961
$532$	0	0
$533$	7.78653	0.337272
$534$	0	0
$535$	−6.15670	−0.266177
$536$	0	0
$537$	28.5798	1.23331
$538$	0	0
$539$	89.6684	3.86229
$540$	0	0
$541$	−2.29290	−0.0985795	−0.0492898	−	0.998785i	$-0.515696\pi$
$541$	−2.29290	−0.0985795	−0.0492898	+	0.998785i	$0.515696\pi$
$542$	0	0
$543$	27.3211	1.17246
$544$	0	0
$545$	−16.8015	−0.719696
$546$	0	0
$547$	7.28912	0.311660	0.155830	−	0.987784i	$-0.450195\pi$
$547$	7.28912	0.311660	0.155830	+	0.987784i	$0.450195\pi$
$548$	0	0
$549$	−6.21962	−0.265447
$550$	0	0
$551$	38.9897	1.66102
$552$	0	0
$553$	−35.3501	−1.50324
$554$	0	0
$555$	10.9937	0.466659
$556$	0	0
$557$	−16.2125	−0.686944	−0.343472	−	0.939163i	$-0.611603\pi$
$557$	−16.2125	−0.686944	−0.343472	+	0.939163i	$0.611603\pi$
$558$	0	0
$559$	−13.7967	−0.583539
$560$	0	0
$561$	39.7643	1.67885
$562$	0	0
$563$	26.0030	1.09589	0.547947	−	0.836513i	$-0.315410\pi$
$563$	26.0030	1.09589	0.547947	+	0.836513i	$0.315410\pi$
$564$	0	0
$565$	14.3009	0.601644
$566$	0	0
$567$	−1.01323	−0.0425517
$568$	0	0
$569$	39.4604	1.65427	0.827133	−	0.562006i	$-0.189970\pi$
$569$	39.4604	1.65427	0.827133	+	0.562006i	$0.189970\pi$
$570$	0	0
$571$	17.0485	0.713459	0.356730	−	0.934208i	$-0.383892\pi$
$571$	17.0485	0.713459	0.356730	+	0.934208i	$0.383892\pi$
$572$	0	0
$573$	−18.3492	−0.766550
$574$	0	0
$575$	−4.09995	−0.170980
$576$	0	0
$577$	−0.159675	−0.00664736	−0.00332368	−	0.999994i	$-0.501058\pi$
$577$	−0.159675	−0.00664736	−0.00332368	+	0.999994i	$0.501058\pi$
$578$	0	0
$579$	13.5625	0.563636
$580$	0	0
$581$	20.2161	0.838706
$582$	0	0
$583$	62.4492	2.58638
$584$	0	0
$585$	4.55879	0.188483
$586$	0	0
$587$	−11.8158	−0.487692	−0.243846	−	0.969814i	$-0.578409\pi$
$587$	−11.8158	−0.487692	−0.243846	+	0.969814i	$0.578409\pi$
$588$	0	0
$589$	24.2644	0.999797
$590$	0	0
$591$	10.7511	0.442242
$592$	0	0
$593$	−14.8279	−0.608907	−0.304453	−	0.952527i	$-0.598474\pi$
$593$	−14.8279	−0.608907	−0.304453	+	0.952527i	$0.598474\pi$
$594$	0	0
$595$	−35.3554	−1.44943
$596$	0	0
$597$	8.83814	0.361721
$598$	0	0
$599$	−13.2752	−0.542410	−0.271205	−	0.962522i	$-0.587422\pi$
$599$	−13.2752	−0.542410	−0.271205	+	0.962522i	$0.587422\pi$
$600$	0	0
$601$	−35.4424	−1.44572	−0.722862	−	0.690992i	$-0.757174\pi$
$601$	−35.4424	−1.44572	−0.722862	+	0.690992i	$0.757174\pi$
$602$	0	0
$603$	1.39037	0.0566201
$604$	0	0
$605$	15.3179	0.622760
$606$	0	0
$607$	−7.69283	−0.312242	−0.156121	−	0.987738i	$-0.549899\pi$
$607$	−7.69283	−0.312242	−0.156121	+	0.987738i	$0.549899\pi$
$608$	0	0
$609$	−35.4666	−1.43718
$610$	0	0
$611$	20.2840	0.820601
$612$	0	0
$613$	−22.2230	−0.897578	−0.448789	−	0.893638i	$-0.648144\pi$
$613$	−22.2230	−0.897578	−0.448789	+	0.893638i	$0.648144\pi$
$614$	0	0
$615$	3.37825	0.136224
$616$	0	0
$617$	15.7857	0.635509	0.317755	−	0.948173i	$-0.397071\pi$
$617$	15.7857	0.635509	0.317755	+	0.948173i	$0.397071\pi$
$618$	0	0
$619$	15.2229	0.611860	0.305930	−	0.952054i	$-0.401033\pi$
$619$	15.2229	0.611860	0.305930	+	0.952054i	$0.401033\pi$
$620$	0	0
$621$	−21.4508	−0.860790
$622$	0	0
$623$	67.6371	2.70982
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−32.8298	−1.31110
$628$	0	0
$629$	72.4265	2.88783
$630$	0	0
$631$	24.3360	0.968803	0.484401	−	0.874846i	$-0.339037\pi$
$631$	24.3360	0.968803	0.484401	+	0.874846i	$0.339037\pi$
$632$	0	0
$633$	3.86645	0.153678
$634$	0	0
$635$	−16.5698	−0.657554
$636$	0	0
$637$	−43.6993	−1.73143
$638$	0	0
$639$	−14.7821	−0.584769
$640$	0	0
$641$	−22.4879	−0.888219	−0.444110	−	0.895972i	$-0.646480\pi$
$641$	−22.4879	−0.888219	−0.444110	+	0.895972i	$0.646480\pi$
$642$	0	0
$643$	6.00624	0.236863	0.118432	−	0.992962i	$-0.462213\pi$
$643$	6.00624	0.236863	0.118432	+	0.992962i	$0.462213\pi$
$644$	0	0
$645$	−5.98583	−0.235692
$646$	0	0
$647$	−24.6133	−0.967647	−0.483823	−	0.875166i	$-0.660752\pi$
$647$	−24.6133	−0.967647	−0.483823	+	0.875166i	$0.660752\pi$
$648$	0	0
$649$	63.2726	2.48367
$650$	0	0
$651$	−22.0719	−0.865064
$652$	0	0
$653$	41.3867	1.61959	0.809793	−	0.586716i	$-0.199580\pi$
$653$	41.3867	1.61959	0.809793	+	0.586716i	$0.199580\pi$
$654$	0	0
$655$	11.6657	0.455816
$656$	0	0
$657$	10.6264	0.414576
$658$	0	0
$659$	28.6986	1.11794	0.558969	−	0.829188i	$-0.311197\pi$
$659$	28.6986	1.11794	0.558969	+	0.829188i	$0.311197\pi$
$660$	0	0
$661$	−34.7440	−1.35138	−0.675692	−	0.737184i	$-0.736155\pi$
$661$	−34.7440	−1.35138	−0.675692	+	0.737184i	$0.736155\pi$
$662$	0	0
$663$	−19.3789	−0.752613
$664$	0	0
$665$	29.1898	1.13193
$666$	0	0
$667$	−27.0953	−1.04913
$668$	0	0
$669$	10.8675	0.420164
$670$	0	0
$671$	17.4985	0.675520
$672$	0	0
$673$	41.8548	1.61339	0.806693	−	0.590971i	$-0.201255\pi$
$673$	41.8548	1.61339	0.806693	+	0.590971i	$0.201255\pi$
$674$	0	0
$675$	5.23196	0.201378
$676$	0	0
$677$	−36.8755	−1.41724	−0.708620	−	0.705590i	$-0.750682\pi$
$677$	−36.8755	−1.41724	−0.708620	+	0.705590i	$0.750682\pi$
$678$	0	0
$679$	−49.2323	−1.88936
$680$	0	0
$681$	15.5446	0.595672
$682$	0	0
$683$	−34.2889	−1.31203	−0.656013	−	0.754749i	$-0.727758\pi$
$683$	−34.2889	−1.31203	−0.656013	+	0.754749i	$0.727758\pi$
$684$	0	0
$685$	16.3064	0.623035
$686$	0	0
$687$	−6.48213	−0.247309
$688$	0	0
$689$	−30.4342	−1.15945
$690$	0	0
$691$	−36.9082	−1.40406	−0.702028	−	0.712150i	$-0.747722\pi$
$691$	−36.9082	−1.40406	−0.702028	+	0.712150i	$0.747722\pi$
$692$	0	0
$693$	−46.2819	−1.75810
$694$	0	0
$695$	9.20843	0.349296
$696$	0	0
$697$	22.2558	0.842999
$698$	0	0
$699$	−8.61607	−0.325890
$700$	0	0
$701$	−26.5693	−1.00351	−0.501755	−	0.865010i	$-0.667312\pi$
$701$	−26.5693	−1.00351	−0.501755	+	0.865010i	$0.667312\pi$
$702$	0	0
$703$	−59.7960	−2.25525
$704$	0	0
$705$	8.80037	0.331441
$706$	0	0
$707$	−44.7109	−1.68152
$708$	0	0
$709$	−15.0719	−0.566037	−0.283019	−	0.959114i	$-0.591336\pi$
$709$	−15.0719	−0.566037	−0.283019	+	0.959114i	$0.591336\pi$
$710$	0	0
$711$	13.0282	0.488597
$712$	0	0
$713$	−16.8622	−0.631493
$714$	0	0
$715$	−12.8258	−0.479659
$716$	0	0
$717$	−18.7168	−0.698992
$718$	0	0
$719$	2.01801	0.0752592	0.0376296	−	0.999292i	$-0.488019\pi$
$719$	2.01801	0.0752592	0.0376296	+	0.999292i	$0.488019\pi$
$720$	0	0
$721$	94.5739	3.52212
$722$	0	0
$723$	−12.5879	−0.468151
$724$	0	0
$725$	6.60869	0.245441
$726$	0	0
$727$	36.9416	1.37009	0.685045	−	0.728501i	$-0.259783\pi$
$727$	36.9416	1.37009	0.685045	+	0.728501i	$0.259783\pi$
$728$	0	0
$729$	17.3987	0.644396
$730$	0	0
$731$	−39.4345	−1.45854
$732$	0	0
$733$	−31.7937	−1.17433	−0.587163	−	0.809469i	$-0.699755\pi$
$733$	−31.7937	−1.17433	−0.587163	+	0.809469i	$0.699755\pi$
$734$	0	0
$735$	−18.9593	−0.699324
$736$	0	0
$737$	−3.91170	−0.144089
$738$	0	0
$739$	−25.7661	−0.947822	−0.473911	−	0.880573i	$-0.657158\pi$
$739$	−25.7661	−0.947822	−0.473911	+	0.880573i	$0.657158\pi$
$740$	0	0
$741$	15.9994	0.587752
$742$	0	0
$743$	−23.1777	−0.850309	−0.425154	−	0.905121i	$-0.639780\pi$
$743$	−23.1777	−0.850309	−0.425154	+	0.905121i	$0.639780\pi$
$744$	0	0
$745$	−3.68722	−0.135089
$746$	0	0
$747$	−7.45061	−0.272604
$748$	0	0
$749$	−30.4610	−1.11302
$750$	0	0
$751$	0.617998	0.0225511	0.0112755	−	0.999936i	$-0.496411\pi$
$751$	0.617998	0.0225511	0.0112755	+	0.999936i	$0.496411\pi$
$752$	0	0
$753$	14.5632	0.530713
$754$	0	0
$755$	1.00000	0.0363937
$756$	0	0
$757$	−47.4993	−1.72639	−0.863196	−	0.504868i	$-0.831541\pi$
$757$	−47.4993	−1.72639	−0.863196	+	0.504868i	$0.831541\pi$
$758$	0	0
$759$	22.8146	0.828116
$760$	0	0
$761$	−14.4914	−0.525314	−0.262657	−	0.964889i	$-0.584599\pi$
$761$	−14.4914	−0.525314	−0.262657	+	0.964889i	$0.584599\pi$
$762$	0	0
$763$	−83.1272	−3.00941
$764$	0	0
$765$	13.0302	0.471107
$766$	0	0
$767$	−30.8354	−1.11340
$768$	0	0
$769$	−48.4391	−1.74676	−0.873380	−	0.487039i	$-0.838077\pi$
$769$	−48.4391	−1.74676	−0.873380	+	0.487039i	$0.838077\pi$
$770$	0	0
$771$	30.8300	1.11032
$772$	0	0
$773$	24.0391	0.864626	0.432313	−	0.901724i	$-0.357698\pi$
$773$	24.0391	0.864626	0.432313	+	0.901724i	$0.357698\pi$
$774$	0	0
$775$	4.11277	0.147735
$776$	0	0
$777$	54.3928	1.95133
$778$	0	0
$779$	−18.3746	−0.658339
$780$	0	0
$781$	41.5882	1.48814
$782$	0	0
$783$	34.5764	1.23566
$784$	0	0
$785$	−11.2583	−0.401827
$786$	0	0
$787$	−0.289763	−0.0103289	−0.00516446	−	0.999987i	$-0.501644\pi$
$787$	−0.289763	−0.0103289	−0.00516446	+	0.999987i	$0.501644\pi$
$788$	0	0
$789$	−7.79931	−0.277663
$790$	0	0
$791$	70.7555	2.51577
$792$	0	0
$793$	−8.52775	−0.302829
$794$	0	0
$795$	−13.2041	−0.468302
$796$	0	0
$797$	45.7942	1.62211	0.811056	−	0.584968i	$-0.198893\pi$
$797$	45.7942	1.62211	0.811056	+	0.584968i	$0.198893\pi$
$798$	0	0
$799$	57.9766	2.05106
$800$	0	0
$801$	−24.9275	−0.880770
$802$	0	0
$803$	−29.8966	−1.05503
$804$	0	0
$805$	−20.2850	−0.714952
$806$	0	0
$807$	−4.41531	−0.155426
$808$	0	0
$809$	−1.37939	−0.0484967	−0.0242483	−	0.999706i	$-0.507719\pi$
$809$	−1.37939	−0.0484967	−0.0242483	+	0.999706i	$0.507719\pi$
$810$	0	0
$811$	−5.06900	−0.177997	−0.0889984	−	0.996032i	$-0.528367\pi$
$811$	−5.06900	−0.177997	−0.0889984	+	0.996032i	$0.528367\pi$
$812$	0	0
$813$	−10.9907	−0.385459
$814$	0	0
$815$	9.28188	0.325130
$816$	0	0
$817$	32.5575	1.13904
$818$	0	0
$819$	22.5552	0.788141
$820$	0	0
$821$	18.9625	0.661797	0.330898	−	0.943666i	$-0.392648\pi$
$821$	18.9625	0.661797	0.330898	+	0.943666i	$0.392648\pi$
$822$	0	0
$823$	−0.357756	−0.0124706	−0.00623530	−	0.999981i	$-0.501985\pi$
$823$	−0.357756	−0.0124706	−0.00623530	+	0.999981i	$0.501985\pi$
$824$	0	0
$825$	−5.56460	−0.193734
$826$	0	0
$827$	12.7286	0.442617	0.221309	−	0.975204i	$-0.428967\pi$
$827$	12.7286	0.442617	0.221309	+	0.975204i	$0.428967\pi$
$828$	0	0
$829$	−3.66411	−0.127260	−0.0636299	−	0.997974i	$-0.520268\pi$
$829$	−3.66411	−0.127260	−0.0636299	+	0.997974i	$0.520268\pi$
$830$	0	0
$831$	−19.5811	−0.679259
$832$	0	0
$833$	−124.903	−4.32765
$834$	0	0
$835$	12.3140	0.426143
$836$	0	0
$837$	21.5179	0.743766
$838$	0	0
$839$	9.58262	0.330829	0.165414	−	0.986224i	$-0.447104\pi$
$839$	9.58262	0.330829	0.165414	+	0.986224i	$0.447104\pi$
$840$	0	0
$841$	14.6748	0.506028
$842$	0	0
$843$	−33.3832	−1.14978
$844$	0	0
$845$	−6.74942	−0.232187
$846$	0	0
$847$	75.7869	2.60407
$848$	0	0
$849$	−22.8489	−0.784172
$850$	0	0
$851$	41.5543	1.42446
$852$	0	0
$853$	−27.1987	−0.931266	−0.465633	−	0.884978i	$-0.654173\pi$
$853$	−27.1987	−0.931266	−0.465633	+	0.884978i	$0.654173\pi$
$854$	0	0
$855$	−10.7578	−0.367910
$856$	0	0
$857$	−34.5810	−1.18126	−0.590632	−	0.806941i	$-0.701122\pi$
$857$	−34.5810	−1.18126	−0.590632	+	0.806941i	$0.701122\pi$
$858$	0	0
$859$	20.2575	0.691176	0.345588	−	0.938386i	$-0.387679\pi$
$859$	20.2575	0.691176	0.345588	+	0.938386i	$0.387679\pi$
$860$	0	0
$861$	16.7143	0.569621
$862$	0	0
$863$	19.0594	0.648790	0.324395	−	0.945922i	$-0.394839\pi$
$863$	19.0594	0.648790	0.324395	+	0.945922i	$0.394839\pi$
$864$	0	0
$865$	17.6144	0.598908
$866$	0	0
$867$	−36.9498	−1.25488
$868$	0	0
$869$	−36.6539	−1.24340
$870$	0	0
$871$	1.90634	0.0645938
$872$	0	0
$873$	18.1444	0.614096
$874$	0	0
$875$	4.94762	0.167260
$876$	0	0
$877$	4.78964	0.161734	0.0808672	−	0.996725i	$-0.474231\pi$
$877$	4.78964	0.161734	0.0808672	+	0.996725i	$0.474231\pi$
$878$	0	0
$879$	−28.4426	−0.959346
$880$	0	0
$881$	−19.7675	−0.665986	−0.332993	−	0.942929i	$-0.608058\pi$
$881$	−19.7675	−0.665986	−0.332993	+	0.942929i	$0.608058\pi$
$882$	0	0
$883$	−34.5940	−1.16418	−0.582091	−	0.813124i	$-0.697765\pi$
$883$	−34.5940	−1.16418	−0.582091	+	0.813124i	$0.697765\pi$
$884$	0	0
$885$	−13.3782	−0.449704
$886$	0	0
$887$	37.1594	1.24769	0.623845	−	0.781548i	$-0.285570\pi$
$887$	37.1594	1.24769	0.623845	+	0.781548i	$0.285570\pi$
$888$	0	0
$889$	−81.9812	−2.74956
$890$	0	0
$891$	−1.05060	−0.0351965
$892$	0	0
$893$	−47.8660	−1.60178
$894$	0	0
$895$	−26.3482	−0.880722
$896$	0	0
$897$	−11.1185	−0.371237
$898$	0	0
$899$	27.1800	0.906505
$900$	0	0
$901$	−86.9884	−2.89800
$902$	0	0
$903$	−29.6156	−0.985544
$904$	0	0
$905$	−25.1878	−0.837270
$906$	0	0
$907$	−6.30105	−0.209223	−0.104611	−	0.994513i	$-0.533360\pi$
$907$	−6.30105	−0.209223	−0.104611	+	0.994513i	$0.533360\pi$
$908$	0	0
$909$	16.4781	0.546544
$910$	0	0
$911$	−29.6864	−0.983554	−0.491777	−	0.870721i	$-0.663652\pi$
$911$	−29.6864	−0.983554	−0.491777	+	0.870721i	$0.663652\pi$
$912$	0	0
$913$	20.9617	0.693732
$914$	0	0
$915$	−3.69984	−0.122313
$916$	0	0
$917$	57.7173	1.90599
$918$	0	0
$919$	−16.9974	−0.560693	−0.280347	−	0.959899i	$-0.590449\pi$
$919$	−16.9974	−0.560693	−0.280347	+	0.959899i	$0.590449\pi$
$920$	0	0
$921$	−9.32352	−0.307220
$922$	0	0
$923$	−20.2677	−0.667121
$924$	0	0
$925$	−10.1353	−0.333247
$926$	0	0
$927$	−34.8550	−1.14479
$928$	0	0
$929$	−17.0336	−0.558853	−0.279427	−	0.960167i	$-0.590144\pi$
$929$	−17.0336	−0.558853	−0.279427	+	0.960167i	$0.590144\pi$
$930$	0	0
$931$	103.121	3.37967
$932$	0	0
$933$	10.7100	0.350628
$934$	0	0
$935$	−36.6594	−1.19889
$936$	0	0
$937$	13.7922	0.450570	0.225285	−	0.974293i	$-0.427669\pi$
$937$	13.7922	0.450570	0.225285	+	0.974293i	$0.427669\pi$
$938$	0	0
$939$	1.80879	0.0590275
$940$	0	0
$941$	13.5611	0.442080	0.221040	−	0.975265i	$-0.429055\pi$
$941$	13.5611	0.442080	0.221040	+	0.975265i	$0.429055\pi$
$942$	0	0
$943$	12.7692	0.415821
$944$	0	0
$945$	25.8857	0.842063
$946$	0	0
$947$	−21.6418	−0.703265	−0.351633	−	0.936138i	$-0.614373\pi$
$947$	−21.6418	−0.703265	−0.351633	+	0.936138i	$0.614373\pi$
$948$	0	0
$949$	14.5699	0.472959
$950$	0	0
$951$	−3.78052	−0.122592
$952$	0	0
$953$	−46.3775	−1.50231	−0.751157	−	0.660124i	$-0.770504\pi$
$953$	−46.3775	−1.50231	−0.751157	+	0.660124i	$0.770504\pi$
$954$	0	0
$955$	16.9165	0.547404
$956$	0	0
$957$	−36.7747	−1.18876
$958$	0	0
$959$	80.6777	2.60522
$960$	0	0
$961$	−14.0851	−0.454359
$962$	0	0
$963$	11.2263	0.361763
$964$	0	0
$965$	−12.5035	−0.402500
$966$	0	0
$967$	−14.4215	−0.463763	−0.231882	−	0.972744i	$-0.574488\pi$
$967$	−14.4215	−0.463763	−0.231882	+	0.972744i	$0.574488\pi$
$968$	0	0
$969$	45.7302	1.46907
$970$	0	0
$971$	7.66997	0.246141	0.123071	−	0.992398i	$-0.460726\pi$
$971$	7.66997	0.246141	0.123071	+	0.992398i	$0.460726\pi$
$972$	0	0
$973$	45.5598	1.46058
$974$	0	0
$975$	2.71187	0.0868493
$976$	0	0
$977$	42.5118	1.36007	0.680036	−	0.733179i	$-0.261964\pi$
$977$	42.5118	1.36007	0.680036	+	0.733179i	$0.261964\pi$
$978$	0	0
$979$	70.1317	2.24142
$980$	0	0
$981$	30.6364	0.978144
$982$	0	0
$983$	−52.6414	−1.67900	−0.839501	−	0.543359i	$-0.817152\pi$
$983$	−52.6414	−1.67900	−0.839501	+	0.543359i	$0.817152\pi$
$984$	0	0
$985$	−9.91164	−0.315811
$986$	0	0
$987$	43.5408	1.38592
$988$	0	0
$989$	−22.6253	−0.719443
$990$	0	0
$991$	17.8093	0.565730	0.282865	−	0.959160i	$-0.408715\pi$
$991$	17.8093	0.565730	0.282865	+	0.959160i	$0.408715\pi$
$992$	0	0
$993$	−13.6739	−0.433929
$994$	0	0
$995$	−8.14803	−0.258310
$996$	0	0
$997$	−41.6526	−1.31915	−0.659576	−	0.751638i	$-0.729264\pi$
$997$	−41.6526	−1.31915	−0.659576	+	0.751638i	$0.729264\pi$
$998$	0	0
$999$	−53.0276	−1.67772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6040.2.a.s.1.8	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6040.2.a.s.1.8	✓	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:	\( N \)	\(=\)	\( 6040 = 2^{3} \cdot 5 \cdot 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6040.a (trivial)

\(f(q)\)	\(=\)	\(q-1.08470 q^{3} +1.00000 q^{5} +4.94762 q^{7} -1.82343 q^{9} +O(q^{10})\)	\(q-1.08470 q^{3} +1.00000 q^{5} +4.94762 q^{7} -1.82343 q^{9} +5.13009 q^{11} -2.50012 q^{13} -1.08470 q^{15} -7.14595 q^{17} +5.89977 q^{19} -5.36666 q^{21} -4.09995 q^{23} +1.00000 q^{25} +5.23196 q^{27} +6.60869 q^{29} +4.11277 q^{31} -5.56460 q^{33} +4.94762 q^{35} -10.1353 q^{37} +2.71187 q^{39} -3.11447 q^{41} +5.51843 q^{43} -1.82343 q^{45} -8.11320 q^{47} +17.4789 q^{49} +7.75119 q^{51} +12.1731 q^{53} +5.13009 q^{55} -6.39946 q^{57} +12.3336 q^{59} +3.41094 q^{61} -9.02165 q^{63} -2.50012 q^{65} -0.762500 q^{67} +4.44720 q^{69} +8.10672 q^{71} -5.82769 q^{73} -1.08470 q^{75} +25.3817 q^{77} -7.14489 q^{79} -0.204792 q^{81} +4.08603 q^{83} -7.14595 q^{85} -7.16843 q^{87} +13.6706 q^{89} -12.3696 q^{91} -4.46111 q^{93} +5.89977 q^{95} -9.95070 q^{97} -9.35438 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 2 q^{3} + 24 q^{5} + 3 q^{7} + 40 q^{9}+O(q^{10}) \) 24 * q + 2 * q^3 + 24 * q^5 + 3 * q^7 + 40 * q^9	\( 24 q + 2 q^{3} + 24 q^{5} + 3 q^{7} + 40 q^{9} + 17 q^{11} + 16 q^{13} + 2 q^{15} + 22 q^{17} + 16 q^{19} - q^{21} + 7 q^{23} + 24 q^{25} - 4 q^{27} + 25 q^{29} + 28 q^{31} + 11 q^{33} + 3 q^{35} + 26 q^{37} + 13 q^{39} + 38 q^{41} - 13 q^{43} + 40 q^{45} + 12 q^{47} + 61 q^{49} + 53 q^{53} + 17 q^{55} + 30 q^{57} + 35 q^{59} + 44 q^{61} - 9 q^{63} + 16 q^{65} - 15 q^{67} + 9 q^{69} + 22 q^{71} + 31 q^{73} + 2 q^{75} + 26 q^{77} + 20 q^{79} + 88 q^{81} - 14 q^{83} + 22 q^{85} - 18 q^{87} + 37 q^{89} - 26 q^{91} + 13 q^{93} + 16 q^{95} + 21 q^{97} - 12 q^{99}+O(q^{100}) \) 24 * q + 2 * q^3 + 24 * q^5 + 3 * q^7 + 40 * q^9 + 17 * q^11 + 16 * q^13 + 2 * q^15 + 22 * q^17 + 16 * q^19 - q^21 + 7 * q^23 + 24 * q^25 - 4 * q^27 + 25 * q^29 + 28 * q^31 + 11 * q^33 + 3 * q^35 + 26 * q^37 + 13 * q^39 + 38 * q^41 - 13 * q^43 + 40 * q^45 + 12 * q^47 + 61 * q^49 + 53 * q^53 + 17 * q^55 + 30 * q^57 + 35 * q^59 + 44 * q^61 - 9 * q^63 + 16 * q^65 - 15 * q^67 + 9 * q^69 + 22 * q^71 + 31 * q^73 + 2 * q^75 + 26 * q^77 + 20 * q^79 + 88 * q^81 - 14 * q^83 + 22 * q^85 - 18 * q^87 + 37 * q^89 - 26 * q^91 + 13 * q^93 + 16 * q^95 + 21 * q^97 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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