LMFDB - Embedded newform 6020.2.a.g.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6020,2,Mod(1,6020)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6020, 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("6020.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6020 = 2^{2} \cdot 5 \cdot 7 \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6020.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:	$48.0699420168$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 16x^{7} + 83x^{5} - 9x^{4} - 160x^{3} + 32x^{2} + 77x - 2 $ x^9 - 16x^7 + 83x^5 - 9x^4 - 160x^3 + 32x^2 + 77x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$1.79476$ of defining polynomial
Character	$\chi$	$=$	6020.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.79476 q^{3} +1.00000 q^{5} -1.00000 q^{7} +0.221173 q^{9} +O(q^{10})$	$q+1.79476 q^{3} +1.00000 q^{5} -1.00000 q^{7} +0.221173 q^{9} +1.19036 q^{11} -5.93103 q^{13} +1.79476 q^{15} -0.643045 q^{17} -2.17219 q^{19} -1.79476 q^{21} +6.38895 q^{23} +1.00000 q^{25} -4.98734 q^{27} -4.54493 q^{29} -6.26984 q^{31} +2.13641 q^{33} -1.00000 q^{35} +7.62257 q^{37} -10.6448 q^{39} -1.55660 q^{41} -1.00000 q^{43} +0.221173 q^{45} +9.34336 q^{47} +1.00000 q^{49} -1.15411 q^{51} -8.12607 q^{53} +1.19036 q^{55} -3.89856 q^{57} -9.70055 q^{59} +2.24881 q^{61} -0.221173 q^{63} -5.93103 q^{65} +0.590435 q^{67} +11.4666 q^{69} -14.5136 q^{71} -7.83496 q^{73} +1.79476 q^{75} -1.19036 q^{77} -9.04520 q^{79} -9.61460 q^{81} -13.4133 q^{83} -0.643045 q^{85} -8.15706 q^{87} +16.1756 q^{89} +5.93103 q^{91} -11.2529 q^{93} -2.17219 q^{95} +8.22492 q^{97} +0.263275 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 9 q^{5} - 9 q^{7} + 5 q^{9}+O(q^{10}) $ 9 * q + 9 * q^5 - 9 * q^7 + 5 * q^9	$ 9 q + 9 q^{5} - 9 q^{7} + 5 q^{9} + q^{11} - 14 q^{13} - 11 q^{17} - 2 q^{19} - 6 q^{23} + 9 q^{25} - 6 q^{29} + 6 q^{31} - 14 q^{33} - 9 q^{35} - 20 q^{37} - 6 q^{39} - 6 q^{41} - 9 q^{43} + 5 q^{45} + 9 q^{49} - 8 q^{51} - 31 q^{53} + q^{55} - 16 q^{57} + 2 q^{59} - 13 q^{61} - 5 q^{63} - 14 q^{65} - 10 q^{67} - 18 q^{69} + 12 q^{71} - 32 q^{73} - q^{77} + q^{79} - 27 q^{81} - 10 q^{83} - 11 q^{85} - 5 q^{87} - q^{89} + 14 q^{91} - 49 q^{93} - 2 q^{95} - 28 q^{97} - 14 q^{99}+O(q^{100}) $ 9 * q + 9 * q^5 - 9 * q^7 + 5 * q^9 + q^11 - 14 * q^13 - 11 * q^17 - 2 * q^19 - 6 * q^23 + 9 * q^25 - 6 * q^29 + 6 * q^31 - 14 * q^33 - 9 * q^35 - 20 * q^37 - 6 * q^39 - 6 * q^41 - 9 * q^43 + 5 * q^45 + 9 * q^49 - 8 * q^51 - 31 * q^53 + q^55 - 16 * q^57 + 2 * q^59 - 13 * q^61 - 5 * q^63 - 14 * q^65 - 10 * q^67 - 18 * q^69 + 12 * q^71 - 32 * q^73 - q^77 + q^79 - 27 * q^81 - 10 * q^83 - 11 * q^85 - 5 * q^87 - q^89 + 14 * q^91 - 49 * q^93 - 2 * q^95 - 28 * q^97 - 14 * 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
$3$	1.79476	1.03621	0.518103	−	0.855318i	$-0.326638\pi$
$3$	1.79476	1.03621	0.518103	+	0.855318i	$0.326638\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0.221173	0.0737242
$10$	0	0
$11$	1.19036	0.358907	0.179454	−	0.983766i	$-0.442567\pi$
$11$	1.19036	0.358907	0.179454	+	0.983766i	$0.442567\pi$
$12$	0	0
$13$	−5.93103	−1.64497	−0.822486	−	0.568785i	$-0.807414\pi$
$13$	−5.93103	−1.64497	−0.822486	+	0.568785i	$0.807414\pi$
$14$	0	0
$15$	1.79476	0.463406
$16$	0	0
$17$	−0.643045	−0.155961	−0.0779807	−	0.996955i	$-0.524847\pi$
$17$	−0.643045	−0.155961	−0.0779807	+	0.996955i	$0.524847\pi$
$18$	0	0
$19$	−2.17219	−0.498334	−0.249167	−	0.968460i	$-0.580157\pi$
$19$	−2.17219	−0.498334	−0.249167	+	0.968460i	$0.580157\pi$
$20$	0	0
$21$	−1.79476	−0.391649
$22$	0	0
$23$	6.38895	1.33219	0.666094	−	0.745868i	$-0.267965\pi$
$23$	6.38895	1.33219	0.666094	+	0.745868i	$0.267965\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−4.98734	−0.959813
$28$	0	0
$29$	−4.54493	−0.843972	−0.421986	−	0.906602i	$-0.638667\pi$
$29$	−4.54493	−0.843972	−0.421986	+	0.906602i	$0.638667\pi$
$30$	0	0
$31$	−6.26984	−1.12610	−0.563048	−	0.826424i	$-0.690371\pi$
$31$	−6.26984	−1.12610	−0.563048	+	0.826424i	$0.690371\pi$
$32$	0	0
$33$	2.13641	0.371902
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	7.62257	1.25314	0.626571	−	0.779364i	$-0.284458\pi$
$37$	7.62257	1.25314	0.626571	+	0.779364i	$0.284458\pi$
$38$	0	0
$39$	−10.6448	−1.70453
$40$	0	0
$41$	−1.55660	−0.243101	−0.121550	−	0.992585i	$-0.538787\pi$
$41$	−1.55660	−0.243101	−0.121550	+	0.992585i	$0.538787\pi$
$42$	0	0
$43$	−1.00000	−0.152499
$43$	−1.00000	−0.152499
$44$	0	0
$45$	0.221173	0.0329705
$46$	0	0
$47$	9.34336	1.36287	0.681435	−	0.731879i	$-0.261356\pi$
$47$	9.34336	1.36287	0.681435	+	0.731879i	$0.261356\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−1.15411	−0.161608
$52$	0	0
$53$	−8.12607	−1.11620	−0.558101	−	0.829773i	$-0.688470\pi$
$53$	−8.12607	−1.11620	−0.558101	+	0.829773i	$0.688470\pi$
$54$	0	0
$55$	1.19036	0.160508
$56$	0	0
$57$	−3.89856	−0.516377
$58$	0	0
$59$	−9.70055	−1.26290	−0.631452	−	0.775415i	$-0.717541\pi$
$59$	−9.70055	−1.26290	−0.631452	+	0.775415i	$0.717541\pi$
$60$	0	0
$61$	2.24881	0.287930	0.143965	−	0.989583i	$-0.454015\pi$
$61$	2.24881	0.287930	0.143965	+	0.989583i	$0.454015\pi$
$62$	0	0
$63$	−0.221173	−0.0278651
$64$	0	0
$65$	−5.93103	−0.735654
$66$	0	0
$67$	0.590435	0.0721331	0.0360665	−	0.999349i	$-0.488517\pi$
$67$	0.590435	0.0721331	0.0360665	+	0.999349i	$0.488517\pi$
$68$	0	0
$69$	11.4666	1.38042
$70$	0	0
$71$	−14.5136	−1.72244	−0.861222	−	0.508229i	$-0.830301\pi$
$71$	−14.5136	−1.72244	−0.861222	+	0.508229i	$0.830301\pi$
$72$	0	0
$73$	−7.83496	−0.917013	−0.458507	−	0.888691i	$-0.651615\pi$
$73$	−7.83496	−0.917013	−0.458507	+	0.888691i	$0.651615\pi$
$74$	0	0
$75$	1.79476	0.207241
$76$	0	0
$77$	−1.19036	−0.135654
$78$	0	0
$79$	−9.04520	−1.01767	−0.508833	−	0.860866i	$-0.669923\pi$
$79$	−9.04520	−1.01767	−0.508833	+	0.860866i	$0.669923\pi$
$80$	0	0
$81$	−9.61460	−1.06829
$82$	0	0
$83$	−13.4133	−1.47230	−0.736149	−	0.676819i	$-0.763358\pi$
$83$	−13.4133	−1.47230	−0.736149	+	0.676819i	$0.763358\pi$
$84$	0	0
$85$	−0.643045	−0.0697481
$86$	0	0
$87$	−8.15706	−0.874529
$88$	0	0
$89$	16.1756	1.71461	0.857307	−	0.514805i	$-0.172136\pi$
$89$	16.1756	1.71461	0.857307	+	0.514805i	$0.172136\pi$
$90$	0	0
$91$	5.93103	0.621741
$92$	0	0
$93$	−11.2529	−1.16687
$94$	0	0
$95$	−2.17219	−0.222862
$96$	0	0
$97$	8.22492	0.835114	0.417557	−	0.908651i	$-0.362886\pi$
$97$	8.22492	0.835114	0.417557	+	0.908651i	$0.362886\pi$
$98$	0	0
$99$	0.263275	0.0264601
$100$	0	0
$101$	−2.84472	−0.283060	−0.141530	−	0.989934i	$-0.545202\pi$
$101$	−2.84472	−0.283060	−0.141530	+	0.989934i	$0.545202\pi$
$102$	0	0
$103$	10.8895	1.07297	0.536486	−	0.843909i	$-0.319751\pi$
$103$	10.8895	1.07297	0.536486	+	0.843909i	$0.319751\pi$
$104$	0	0
$105$	−1.79476	−0.175151
$106$	0	0
$107$	−16.3768	−1.58321	−0.791603	−	0.611036i	$-0.790753\pi$
$107$	−16.3768	−1.58321	−0.791603	+	0.611036i	$0.790753\pi$
$108$	0	0
$109$	−5.07631	−0.486223	−0.243111	−	0.969998i	$-0.578168\pi$
$109$	−5.07631	−0.486223	−0.243111	+	0.969998i	$0.578168\pi$
$110$	0	0
$111$	13.6807	1.29852
$112$	0	0
$113$	−10.2068	−0.960179	−0.480090	−	0.877219i	$-0.659396\pi$
$113$	−10.2068	−0.960179	−0.480090	+	0.877219i	$0.659396\pi$
$114$	0	0
$115$	6.38895	0.595773
$116$	0	0
$117$	−1.31178	−0.121274
$118$	0	0
$119$	0.643045	0.0589479
$120$	0	0
$121$	−9.58304	−0.871186
$122$	0	0
$123$	−2.79373	−0.251903
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−5.98381	−0.530978	−0.265489	−	0.964114i	$-0.585533\pi$
$127$	−5.98381	−0.530978	−0.265489	+	0.964114i	$0.585533\pi$
$128$	0	0
$129$	−1.79476	−0.158020
$130$	0	0
$131$	−6.17081	−0.539146	−0.269573	−	0.962980i	$-0.586883\pi$
$131$	−6.17081	−0.539146	−0.269573	+	0.962980i	$0.586883\pi$
$132$	0	0
$133$	2.17219	0.188353
$134$	0	0
$135$	−4.98734	−0.429241
$136$	0	0
$137$	14.6210	1.24916	0.624579	−	0.780961i	$-0.285270\pi$
$137$	14.6210	1.24916	0.624579	+	0.780961i	$0.285270\pi$
$138$	0	0
$139$	5.23783	0.444267	0.222134	−	0.975016i	$-0.428698\pi$
$139$	5.23783	0.444267	0.222134	+	0.975016i	$0.428698\pi$
$140$	0	0
$141$	16.7691	1.41221
$142$	0	0
$143$	−7.06007	−0.590392
$144$	0	0
$145$	−4.54493	−0.377436
$146$	0	0
$147$	1.79476	0.148030
$148$	0	0
$149$	−6.03441	−0.494358	−0.247179	−	0.968970i	$-0.579504\pi$
$149$	−6.03441	−0.494358	−0.247179	+	0.968970i	$0.579504\pi$
$150$	0	0
$151$	−16.0630	−1.30719	−0.653595	−	0.756844i	$-0.726740\pi$
$151$	−16.0630	−1.30719	−0.653595	+	0.756844i	$0.726740\pi$
$152$	0	0
$153$	−0.142224	−0.0114981
$154$	0	0
$155$	−6.26984	−0.503605
$156$	0	0
$157$	−5.92482	−0.472852	−0.236426	−	0.971650i	$-0.575976\pi$
$157$	−5.92482	−0.472852	−0.236426	+	0.971650i	$0.575976\pi$
$158$	0	0
$159$	−14.5844	−1.15662
$160$	0	0
$161$	−6.38895	−0.503520
$162$	0	0
$163$	−14.8624	−1.16411	−0.582057	−	0.813148i	$-0.697752\pi$
$163$	−14.8624	−1.16411	−0.582057	+	0.813148i	$0.697752\pi$
$164$	0	0
$165$	2.13641	0.166320
$166$	0	0
$167$	−3.40622	−0.263581	−0.131791	−	0.991278i	$-0.542073\pi$
$167$	−3.40622	−0.263581	−0.131791	+	0.991278i	$0.542073\pi$
$168$	0	0
$169$	22.1772	1.70594
$170$	0	0
$171$	−0.480429	−0.0367393
$172$	0	0
$173$	−14.5520	−1.10637	−0.553183	−	0.833060i	$-0.686587\pi$
$173$	−14.5520	−1.10637	−0.553183	+	0.833060i	$0.686587\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	−17.4102	−1.30863
$178$	0	0
$179$	−0.0545803	−0.00407952	−0.00203976	−	0.999998i	$-0.500649\pi$
$179$	−0.0545803	−0.00407952	−0.00203976	+	0.999998i	$0.500649\pi$
$180$	0	0
$181$	6.06426	0.450753	0.225377	−	0.974272i	$-0.427639\pi$
$181$	6.06426	0.450753	0.225377	+	0.974272i	$0.427639\pi$
$182$	0	0
$183$	4.03607	0.298355
$184$	0	0
$185$	7.62257	0.560423
$186$	0	0
$187$	−0.765456	−0.0559757
$188$	0	0
$189$	4.98734	0.362775
$190$	0	0
$191$	3.59902	0.260416	0.130208	−	0.991487i	$-0.458435\pi$
$191$	3.59902	0.260416	0.130208	+	0.991487i	$0.458435\pi$
$192$	0	0
$193$	−7.60481	−0.547406	−0.273703	−	0.961814i	$-0.588249\pi$
$193$	−7.60481	−0.547406	−0.273703	+	0.961814i	$0.588249\pi$
$194$	0	0
$195$	−10.6448	−0.762290
$196$	0	0
$197$	−26.2236	−1.86835	−0.934176	−	0.356811i	$-0.883864\pi$
$197$	−26.2236	−1.86835	−0.934176	+	0.356811i	$0.883864\pi$
$198$	0	0
$199$	27.4204	1.94378	0.971889	−	0.235440i	$-0.0756531\pi$
$199$	27.4204	1.94378	0.971889	+	0.235440i	$0.0756531\pi$
$200$	0	0
$201$	1.05969	0.0747448
$202$	0	0
$203$	4.54493	0.318991
$204$	0	0
$205$	−1.55660	−0.108718
$206$	0	0
$207$	1.41306	0.0982145
$208$	0	0
$209$	−2.58569	−0.178856
$210$	0	0
$211$	18.8061	1.29467	0.647333	−	0.762207i	$-0.275884\pi$
$211$	18.8061	1.29467	0.647333	+	0.762207i	$0.275884\pi$
$212$	0	0
$213$	−26.0484	−1.78481
$214$	0	0
$215$	−1.00000	−0.0681994
$216$	0	0
$217$	6.26984	0.425624
$218$	0	0
$219$	−14.0619	−0.950215
$220$	0	0
$221$	3.81392	0.256552
$222$	0	0
$223$	11.4812	0.768837	0.384418	−	0.923159i	$-0.374402\pi$
$223$	11.4812	0.768837	0.384418	+	0.923159i	$0.374402\pi$
$224$	0	0
$225$	0.221173	0.0147448
$226$	0	0
$227$	−19.1815	−1.27312	−0.636561	−	0.771226i	$-0.719644\pi$
$227$	−19.1815	−1.27312	−0.636561	+	0.771226i	$0.719644\pi$
$228$	0	0
$229$	7.75372	0.512380	0.256190	−	0.966626i	$-0.417533\pi$
$229$	7.75372	0.512380	0.256190	+	0.966626i	$0.417533\pi$
$230$	0	0
$231$	−2.13641	−0.140566
$232$	0	0
$233$	1.89529	0.124164	0.0620822	−	0.998071i	$-0.480226\pi$
$233$	1.89529	0.124164	0.0620822	+	0.998071i	$0.480226\pi$
$234$	0	0
$235$	9.34336	0.609494
$236$	0	0
$237$	−16.2340	−1.05451
$238$	0	0
$239$	16.5311	1.06931	0.534653	−	0.845072i	$-0.320442\pi$
$239$	16.5311	1.06931	0.534653	+	0.845072i	$0.320442\pi$
$240$	0	0
$241$	4.03409	0.259858	0.129929	−	0.991523i	$-0.458525\pi$
$241$	4.03409	0.259858	0.129929	+	0.991523i	$0.458525\pi$
$242$	0	0
$243$	−2.29392	−0.147155
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	12.8833	0.819747
$248$	0	0
$249$	−24.0736	−1.52561
$250$	0	0
$251$	−16.4631	−1.03914	−0.519570	−	0.854428i	$-0.673908\pi$
$251$	−16.4631	−1.03914	−0.519570	+	0.854428i	$0.673908\pi$
$252$	0	0
$253$	7.60515	0.478132
$254$	0	0
$255$	−1.15411	−0.0722734
$256$	0	0
$257$	9.69151	0.604540	0.302270	−	0.953222i	$-0.402256\pi$
$257$	9.69151	0.604540	0.302270	+	0.953222i	$0.402256\pi$
$258$	0	0
$259$	−7.62257	−0.473644
$260$	0	0
$261$	−1.00521	−0.0622211
$262$	0	0
$263$	11.3433	0.699460	0.349730	−	0.936851i	$-0.386273\pi$
$263$	11.3433	0.699460	0.349730	+	0.936851i	$0.386273\pi$
$264$	0	0
$265$	−8.12607	−0.499181
$266$	0	0
$267$	29.0314	1.77669
$268$	0	0
$269$	28.7709	1.75419	0.877097	−	0.480314i	$-0.159477\pi$
$269$	28.7709	1.75419	0.877097	+	0.480314i	$0.159477\pi$
$270$	0	0
$271$	24.0696	1.46212	0.731061	−	0.682312i	$-0.239025\pi$
$271$	24.0696	1.46212	0.731061	+	0.682312i	$0.239025\pi$
$272$	0	0
$273$	10.6448	0.644252
$274$	0	0
$275$	1.19036	0.0717814
$276$	0	0
$277$	−6.59145	−0.396042	−0.198021	−	0.980198i	$-0.563451\pi$
$277$	−6.59145	−0.396042	−0.198021	+	0.980198i	$0.563451\pi$
$278$	0	0
$279$	−1.38672	−0.0830205
$280$	0	0
$281$	8.34502	0.497822	0.248911	−	0.968526i	$-0.419927\pi$
$281$	8.34502	0.497822	0.248911	+	0.968526i	$0.419927\pi$
$282$	0	0
$283$	−2.52210	−0.149923	−0.0749617	−	0.997186i	$-0.523883\pi$
$283$	−2.52210	−0.149923	−0.0749617	+	0.997186i	$0.523883\pi$
$284$	0	0
$285$	−3.89856	−0.230931
$286$	0	0
$287$	1.55660	0.0918834
$288$	0	0
$289$	−16.5865	−0.975676
$290$	0	0
$291$	14.7618	0.865350
$292$	0	0
$293$	−14.6140	−0.853757	−0.426878	−	0.904309i	$-0.640387\pi$
$293$	−14.6140	−0.853757	−0.426878	+	0.904309i	$0.640387\pi$
$294$	0	0
$295$	−9.70055	−0.564788
$296$	0	0
$297$	−5.93672	−0.344484
$298$	0	0
$299$	−37.8931	−2.19141
$300$	0	0
$301$	1.00000	0.0576390
$302$	0	0
$303$	−5.10559	−0.293309
$304$	0	0
$305$	2.24881	0.128766
$306$	0	0
$307$	18.0325	1.02917	0.514584	−	0.857440i	$-0.327946\pi$
$307$	18.0325	1.02917	0.514584	+	0.857440i	$0.327946\pi$
$308$	0	0
$309$	19.5440	1.11182
$310$	0	0
$311$	−5.09085	−0.288676	−0.144338	−	0.989528i	$-0.546105\pi$
$311$	−5.09085	−0.288676	−0.144338	+	0.989528i	$0.546105\pi$
$312$	0	0
$313$	6.10254	0.344936	0.172468	−	0.985015i	$-0.444826\pi$
$313$	6.10254	0.344936	0.172468	+	0.985015i	$0.444826\pi$
$314$	0	0
$315$	−0.221173	−0.0124617
$316$	0	0
$317$	18.2614	1.02566	0.512832	−	0.858489i	$-0.328596\pi$
$317$	18.2614	1.02566	0.512832	+	0.858489i	$0.328596\pi$
$318$	0	0
$319$	−5.41010	−0.302907
$320$	0	0
$321$	−29.3925	−1.64053
$322$	0	0
$323$	1.39682	0.0777209
$324$	0	0
$325$	−5.93103	−0.328995
$326$	0	0
$327$	−9.11077	−0.503827
$328$	0	0
$329$	−9.34336	−0.515116
$330$	0	0
$331$	−13.0401	−0.716747	−0.358374	−	0.933578i	$-0.616669\pi$
$331$	−13.0401	−0.716747	−0.358374	+	0.933578i	$0.616669\pi$
$332$	0	0
$333$	1.68590	0.0923870
$334$	0	0
$335$	0.590435	0.0322589
$336$	0	0
$337$	−14.2881	−0.778323	−0.389162	−	0.921169i	$-0.627235\pi$
$337$	−14.2881	−0.778323	−0.389162	+	0.921169i	$0.627235\pi$
$338$	0	0
$339$	−18.3189	−0.994944
$340$	0	0
$341$	−7.46336	−0.404164
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	11.4666	0.617344
$346$	0	0
$347$	33.9087	1.82031	0.910156	−	0.414266i	$-0.135962\pi$
$347$	33.9087	1.82031	0.910156	+	0.414266i	$0.135962\pi$
$348$	0	0
$349$	16.9563	0.907648	0.453824	−	0.891091i	$-0.350060\pi$
$349$	16.9563	0.907648	0.453824	+	0.891091i	$0.350060\pi$
$350$	0	0
$351$	29.5801	1.57887
$352$	0	0
$353$	16.4849	0.877401	0.438701	−	0.898633i	$-0.355439\pi$
$353$	16.4849	0.877401	0.438701	+	0.898633i	$0.355439\pi$
$354$	0	0
$355$	−14.5136	−0.770301
$356$	0	0
$357$	1.15411	0.0610822
$358$	0	0
$359$	17.8693	0.943106	0.471553	−	0.881838i	$-0.343694\pi$
$359$	17.8693	0.943106	0.471553	+	0.881838i	$0.343694\pi$
$360$	0	0
$361$	−14.2816	−0.751663
$362$	0	0
$363$	−17.1993	−0.902728
$364$	0	0
$365$	−7.83496	−0.410101
$366$	0	0
$367$	32.5248	1.69778	0.848889	−	0.528570i	$-0.177272\pi$
$367$	32.5248	1.69778	0.848889	+	0.528570i	$0.177272\pi$
$368$	0	0
$369$	−0.344278	−0.0179224
$370$	0	0
$371$	8.12607	0.421885
$372$	0	0
$373$	−13.3477	−0.691119	−0.345559	−	0.938397i	$-0.612311\pi$
$373$	−13.3477	−0.691119	−0.345559	+	0.938397i	$0.612311\pi$
$374$	0	0
$375$	1.79476	0.0926811
$376$	0	0
$377$	26.9561	1.38831
$378$	0	0
$379$	−36.7609	−1.88828	−0.944140	−	0.329546i	$-0.893104\pi$
$379$	−36.7609	−1.88828	−0.944140	+	0.329546i	$0.893104\pi$
$380$	0	0
$381$	−10.7395	−0.550203
$382$	0	0
$383$	−14.1575	−0.723414	−0.361707	−	0.932292i	$-0.617806\pi$
$383$	−14.1575	−0.723414	−0.361707	+	0.932292i	$0.617806\pi$
$384$	0	0
$385$	−1.19036	−0.0606664
$386$	0	0
$387$	−0.221173	−0.0112428
$388$	0	0
$389$	−4.31141	−0.218597	−0.109299	−	0.994009i	$-0.534860\pi$
$389$	−4.31141	−0.218597	−0.109299	+	0.994009i	$0.534860\pi$
$390$	0	0
$391$	−4.10839	−0.207770
$392$	0	0
$393$	−11.0751	−0.558667
$394$	0	0
$395$	−9.04520	−0.455114
$396$	0	0
$397$	9.71490	0.487577	0.243789	−	0.969828i	$-0.421610\pi$
$397$	9.71490	0.487577	0.243789	+	0.969828i	$0.421610\pi$
$398$	0	0
$399$	3.89856	0.195172
$400$	0	0
$401$	−22.0654	−1.10189	−0.550946	−	0.834541i	$-0.685733\pi$
$401$	−22.0654	−1.10189	−0.550946	+	0.834541i	$0.685733\pi$
$402$	0	0
$403$	37.1866	1.85240
$404$	0	0
$405$	−9.61460	−0.477753
$406$	0	0
$407$	9.07360	0.449762
$408$	0	0
$409$	−32.0196	−1.58327	−0.791633	−	0.610997i	$-0.790769\pi$
$409$	−32.0196	−1.58327	−0.791633	+	0.610997i	$0.790769\pi$
$410$	0	0
$411$	26.2413	1.29439
$412$	0	0
$413$	9.70055	0.477333
$414$	0	0
$415$	−13.4133	−0.658432
$416$	0	0
$417$	9.40066	0.460352
$418$	0	0
$419$	32.2597	1.57599	0.787995	−	0.615681i	$-0.211119\pi$
$419$	32.2597	1.57599	0.787995	+	0.615681i	$0.211119\pi$
$420$	0	0
$421$	−22.3731	−1.09040	−0.545200	−	0.838306i	$-0.683546\pi$
$421$	−22.3731	−1.09040	−0.545200	+	0.838306i	$0.683546\pi$
$422$	0	0
$423$	2.06650	0.100476
$424$	0	0
$425$	−0.643045	−0.0311923
$426$	0	0
$427$	−2.24881	−0.108827
$428$	0	0
$429$	−12.6711	−0.611768
$430$	0	0
$431$	−8.35682	−0.402534	−0.201267	−	0.979536i	$-0.564506\pi$
$431$	−8.35682	−0.402534	−0.201267	+	0.979536i	$0.564506\pi$
$432$	0	0
$433$	−31.2201	−1.50034	−0.750172	−	0.661243i	$-0.770029\pi$
$433$	−31.2201	−1.50034	−0.750172	+	0.661243i	$0.770029\pi$
$434$	0	0
$435$	−8.15706	−0.391101
$436$	0	0
$437$	−13.8780	−0.663875
$438$	0	0
$439$	32.4506	1.54878	0.774392	−	0.632706i	$-0.218056\pi$
$439$	32.4506	1.54878	0.774392	+	0.632706i	$0.218056\pi$
$440$	0	0
$441$	0.221173	0.0105320
$442$	0	0
$443$	−37.1734	−1.76616	−0.883080	−	0.469222i	$-0.844535\pi$
$443$	−37.1734	−1.76616	−0.883080	+	0.469222i	$0.844535\pi$
$444$	0	0
$445$	16.1756	0.766799
$446$	0	0
$447$	−10.8303	−0.512257
$448$	0	0
$449$	23.5248	1.11020	0.555102	−	0.831783i	$-0.312679\pi$
$449$	23.5248	1.11020	0.555102	+	0.831783i	$0.312679\pi$
$450$	0	0
$451$	−1.85292	−0.0872506
$452$	0	0
$453$	−28.8293	−1.35452
$454$	0	0
$455$	5.93103	0.278051
$456$	0	0
$457$	7.36293	0.344423	0.172212	−	0.985060i	$-0.444909\pi$
$457$	7.36293	0.344423	0.172212	+	0.985060i	$0.444909\pi$
$458$	0	0
$459$	3.20708	0.149694
$460$	0	0
$461$	−22.7336	−1.05881	−0.529405	−	0.848369i	$-0.677585\pi$
$461$	−22.7336	−1.05881	−0.529405	+	0.848369i	$0.677585\pi$
$462$	0	0
$463$	19.8866	0.924206	0.462103	−	0.886826i	$-0.347095\pi$
$463$	19.8866	0.924206	0.462103	+	0.886826i	$0.347095\pi$
$464$	0	0
$465$	−11.2529	−0.521839
$466$	0	0
$467$	−19.0227	−0.880264	−0.440132	−	0.897933i	$-0.645068\pi$
$467$	−19.0227	−0.880264	−0.440132	+	0.897933i	$0.645068\pi$
$468$	0	0
$469$	−0.590435	−0.0272637
$470$	0	0
$471$	−10.6336	−0.489972
$472$	0	0
$473$	−1.19036	−0.0547328
$474$	0	0
$475$	−2.17219	−0.0996669
$476$	0	0
$477$	−1.79726	−0.0822911
$478$	0	0
$479$	17.2712	0.789142	0.394571	−	0.918865i	$-0.370893\pi$
$479$	17.2712	0.789142	0.394571	+	0.918865i	$0.370893\pi$
$480$	0	0
$481$	−45.2097	−2.06139
$482$	0	0
$483$	−11.4666	−0.521751
$484$	0	0
$485$	8.22492	0.373474
$486$	0	0
$487$	4.07359	0.184592	0.0922959	−	0.995732i	$-0.470579\pi$
$487$	4.07359	0.184592	0.0922959	+	0.995732i	$0.470579\pi$
$488$	0	0
$489$	−26.6745	−1.20626
$490$	0	0
$491$	34.3373	1.54962	0.774810	−	0.632194i	$-0.217845\pi$
$491$	34.3373	1.54962	0.774810	+	0.632194i	$0.217845\pi$
$492$	0	0
$493$	2.92259	0.131627
$494$	0	0
$495$	0.263275	0.0118333
$496$	0	0
$497$	14.5136	0.651023
$498$	0	0
$499$	11.6428	0.521201	0.260601	−	0.965447i	$-0.416079\pi$
$499$	11.6428	0.521201	0.260601	+	0.965447i	$0.416079\pi$
$500$	0	0
$501$	−6.11336	−0.273125
$502$	0	0
$503$	−9.06287	−0.404093	−0.202047	−	0.979376i	$-0.564759\pi$
$503$	−9.06287	−0.404093	−0.202047	+	0.979376i	$0.564759\pi$
$504$	0	0
$505$	−2.84472	−0.126588
$506$	0	0
$507$	39.8027	1.76770
$508$	0	0
$509$	33.3998	1.48042	0.740211	−	0.672375i	$-0.234726\pi$
$509$	33.3998	1.48042	0.740211	+	0.672375i	$0.234726\pi$
$510$	0	0
$511$	7.83496	0.346598
$512$	0	0
$513$	10.8334	0.478308
$514$	0	0
$515$	10.8895	0.479848
$516$	0	0
$517$	11.1220	0.489144
$518$	0	0
$519$	−26.1173	−1.14642
$520$	0	0
$521$	−20.4092	−0.894141	−0.447071	−	0.894499i	$-0.647533\pi$
$521$	−20.4092	−0.894141	−0.447071	+	0.894499i	$0.647533\pi$
$522$	0	0
$523$	44.3000	1.93710	0.968551	−	0.248815i	$-0.0800410\pi$
$523$	44.3000	1.93710	0.968551	+	0.248815i	$0.0800410\pi$
$524$	0	0
$525$	−1.79476	−0.0783299
$526$	0	0
$527$	4.03179	0.175628
$528$	0	0
$529$	17.8187	0.774725
$530$	0	0
$531$	−2.14550	−0.0931066
$532$	0	0
$533$	9.23227	0.399894
$534$	0	0
$535$	−16.3768	−0.708031
$536$	0	0
$537$	−0.0979586	−0.00422723
$538$	0	0
$539$	1.19036	0.0512724
$540$	0	0
$541$	34.0043	1.46196	0.730980	−	0.682399i	$-0.239063\pi$
$541$	34.0043	1.46196	0.730980	+	0.682399i	$0.239063\pi$
$542$	0	0
$543$	10.8839	0.467073
$544$	0	0
$545$	−5.07631	−0.217445
$546$	0	0
$547$	−33.3809	−1.42726	−0.713632	−	0.700521i	$-0.752951\pi$
$547$	−33.3809	−1.42726	−0.713632	+	0.700521i	$0.752951\pi$
$548$	0	0
$549$	0.497374	0.0212274
$550$	0	0
$551$	9.87244	0.420580
$552$	0	0
$553$	9.04520	0.384641
$554$	0	0
$555$	13.6807	0.580714
$556$	0	0
$557$	15.5607	0.659326	0.329663	−	0.944099i	$-0.393065\pi$
$557$	15.5607	0.659326	0.329663	+	0.944099i	$0.393065\pi$
$558$	0	0
$559$	5.93103	0.250856
$560$	0	0
$561$	−1.37381	−0.0580023
$562$	0	0
$563$	−3.36969	−0.142016	−0.0710078	−	0.997476i	$-0.522622\pi$
$563$	−3.36969	−0.142016	−0.0710078	+	0.997476i	$0.522622\pi$
$564$	0	0
$565$	−10.2068	−0.429405
$566$	0	0
$567$	9.61460	0.403775
$568$	0	0
$569$	33.2806	1.39519	0.697597	−	0.716491i	$-0.254253\pi$
$569$	33.2806	1.39519	0.697597	+	0.716491i	$0.254253\pi$
$570$	0	0
$571$	−8.24255	−0.344940	−0.172470	−	0.985015i	$-0.555175\pi$
$571$	−8.24255	−0.344940	−0.172470	+	0.985015i	$0.555175\pi$
$572$	0	0
$573$	6.45939	0.269845
$574$	0	0
$575$	6.38895	0.266438
$576$	0	0
$577$	0.672868	0.0280119	0.0140059	−	0.999902i	$-0.495542\pi$
$577$	0.672868	0.0280119	0.0140059	+	0.999902i	$0.495542\pi$
$578$	0	0
$579$	−13.6488	−0.567226
$580$	0	0
$581$	13.4133	0.556476
$582$	0	0
$583$	−9.67295	−0.400613
$584$	0	0
$585$	−1.31178	−0.0542355
$586$	0	0
$587$	−0.388256	−0.0160250	−0.00801252	−	0.999968i	$-0.502550\pi$
$587$	−0.388256	−0.0160250	−0.00801252	+	0.999968i	$0.502550\pi$
$588$	0	0
$589$	13.6193	0.561172
$590$	0	0
$591$	−47.0651	−1.93600
$592$	0	0
$593$	−5.18982	−0.213121	−0.106560	−	0.994306i	$-0.533984\pi$
$593$	−5.18982	−0.213121	−0.106560	+	0.994306i	$0.533984\pi$
$594$	0	0
$595$	0.643045	0.0263623
$596$	0	0
$597$	49.2130	2.01416
$598$	0	0
$599$	8.77892	0.358697	0.179348	−	0.983786i	$-0.442601\pi$
$599$	8.77892	0.358697	0.179348	+	0.983786i	$0.442601\pi$
$600$	0	0
$601$	−14.0944	−0.574921	−0.287461	−	0.957792i	$-0.592811\pi$
$601$	−14.0944	−0.574921	−0.287461	+	0.957792i	$0.592811\pi$
$602$	0	0
$603$	0.130588	0.00531795
$604$	0	0
$605$	−9.58304	−0.389606
$606$	0	0
$607$	38.9628	1.58145	0.790725	−	0.612171i	$-0.209704\pi$
$607$	38.9628	1.58145	0.790725	+	0.612171i	$0.209704\pi$
$608$	0	0
$609$	8.15706	0.330541
$610$	0	0
$611$	−55.4158	−2.24188
$612$	0	0
$613$	−12.3733	−0.499754	−0.249877	−	0.968278i	$-0.580390\pi$
$613$	−12.3733	−0.499754	−0.249877	+	0.968278i	$0.580390\pi$
$614$	0	0
$615$	−2.79373	−0.112654
$616$	0	0
$617$	17.3966	0.700360	0.350180	−	0.936683i	$-0.386120\pi$
$617$	17.3966	0.700360	0.350180	+	0.936683i	$0.386120\pi$
$618$	0	0
$619$	−37.7920	−1.51899	−0.759495	−	0.650513i	$-0.774554\pi$
$619$	−37.7920	−1.51899	−0.759495	+	0.650513i	$0.774554\pi$
$620$	0	0
$621$	−31.8638	−1.27865
$622$	0	0
$623$	−16.1756	−0.648063
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−4.64070	−0.185332
$628$	0	0
$629$	−4.90166	−0.195442
$630$	0	0
$631$	3.44018	0.136951	0.0684757	−	0.997653i	$-0.478186\pi$
$631$	3.44018	0.136951	0.0684757	+	0.997653i	$0.478186\pi$
$632$	0	0
$633$	33.7525	1.34154
$634$	0	0
$635$	−5.98381	−0.237460
$636$	0	0
$637$	−5.93103	−0.234996
$638$	0	0
$639$	−3.21000	−0.126986
$640$	0	0
$641$	37.4653	1.47979	0.739896	−	0.672722i	$-0.234875\pi$
$641$	37.4653	1.47979	0.739896	+	0.672722i	$0.234875\pi$
$642$	0	0
$643$	15.3230	0.604280	0.302140	−	0.953264i	$-0.402299\pi$
$643$	15.3230	0.604280	0.302140	+	0.953264i	$0.402299\pi$
$644$	0	0
$645$	−1.79476	−0.0706687
$646$	0	0
$647$	−38.0547	−1.49609	−0.748043	−	0.663650i	$-0.769006\pi$
$647$	−38.0547	−1.49609	−0.748043	+	0.663650i	$0.769006\pi$
$648$	0	0
$649$	−11.5471	−0.453265
$650$	0	0
$651$	11.2529	0.441035
$652$	0	0
$653$	−7.85567	−0.307416	−0.153708	−	0.988116i	$-0.549122\pi$
$653$	−7.85567	−0.307416	−0.153708	+	0.988116i	$0.549122\pi$
$654$	0	0
$655$	−6.17081	−0.241114
$656$	0	0
$657$	−1.73288	−0.0676061
$658$	0	0
$659$	2.24857	0.0875917	0.0437959	−	0.999041i	$-0.486055\pi$
$659$	2.24857	0.0875917	0.0437959	+	0.999041i	$0.486055\pi$
$660$	0	0
$661$	19.1892	0.746373	0.373187	−	0.927756i	$-0.378265\pi$
$661$	19.1892	0.746373	0.373187	+	0.927756i	$0.378265\pi$
$662$	0	0
$663$	6.84509	0.265841
$664$	0	0
$665$	2.17219	0.0842339
$666$	0	0
$667$	−29.0373	−1.12433
$668$	0	0
$669$	20.6060	0.796673
$670$	0	0
$671$	2.67689	0.103340
$672$	0	0
$673$	−6.09182	−0.234823	−0.117411	−	0.993083i	$-0.537460\pi$
$673$	−6.09182	−0.234823	−0.117411	+	0.993083i	$0.537460\pi$
$674$	0	0
$675$	−4.98734	−0.191963
$676$	0	0
$677$	30.5918	1.17574	0.587869	−	0.808956i	$-0.299967\pi$
$677$	30.5918	1.17574	0.587869	+	0.808956i	$0.299967\pi$
$678$	0	0
$679$	−8.22492	−0.315643
$680$	0	0
$681$	−34.4263	−1.31922
$682$	0	0
$683$	8.20479	0.313948	0.156974	−	0.987603i	$-0.449826\pi$
$683$	8.20479	0.313948	0.156974	+	0.987603i	$0.449826\pi$
$684$	0	0
$685$	14.6210	0.558641
$686$	0	0
$687$	13.9161	0.530932
$688$	0	0
$689$	48.1960	1.83612
$690$	0	0
$691$	−23.1115	−0.879204	−0.439602	−	0.898193i	$-0.644881\pi$
$691$	−23.1115	−0.879204	−0.439602	+	0.898193i	$0.644881\pi$
$692$	0	0
$693$	−0.263275	−0.0100010
$694$	0	0
$695$	5.23783	0.198682
$696$	0	0
$697$	1.00097	0.0379143
$698$	0	0
$699$	3.40159	0.128660
$700$	0	0
$701$	1.02647	0.0387693	0.0193847	−	0.999812i	$-0.493829\pi$
$701$	1.02647	0.0387693	0.0193847	+	0.999812i	$0.493829\pi$
$702$	0	0
$703$	−16.5577	−0.624484
$704$	0	0
$705$	16.7691	0.631562
$706$	0	0
$707$	2.84472	0.106987
$708$	0	0
$709$	−14.6081	−0.548620	−0.274310	−	0.961641i	$-0.588449\pi$
$709$	−14.6081	−0.548620	−0.274310	+	0.961641i	$0.588449\pi$
$710$	0	0
$711$	−2.00055	−0.0750265
$712$	0	0
$713$	−40.0577	−1.50017
$714$	0	0
$715$	−7.06007	−0.264031
$716$	0	0
$717$	29.6693	1.10802
$718$	0	0
$719$	19.3533	0.721757	0.360878	−	0.932613i	$-0.382477\pi$
$719$	19.3533	0.721757	0.360878	+	0.932613i	$0.382477\pi$
$720$	0	0
$721$	−10.8895	−0.405546
$722$	0	0
$723$	7.24023	0.269267
$724$	0	0
$725$	−4.54493	−0.168794
$726$	0	0
$727$	−31.7758	−1.17850	−0.589250	−	0.807950i	$-0.700577\pi$
$727$	−31.7758	−1.17850	−0.589250	+	0.807950i	$0.700577\pi$
$728$	0	0
$729$	24.7268	0.915806
$730$	0	0
$731$	0.643045	0.0237839
$732$	0	0
$733$	−4.08324	−0.150818	−0.0754089	−	0.997153i	$-0.524026\pi$
$733$	−4.08324	−0.150818	−0.0754089	+	0.997153i	$0.524026\pi$
$734$	0	0
$735$	1.79476	0.0662008
$736$	0	0
$737$	0.702830	0.0258891
$738$	0	0
$739$	4.86752	0.179055	0.0895274	−	0.995984i	$-0.471464\pi$
$739$	4.86752	0.179055	0.0895274	+	0.995984i	$0.471464\pi$
$740$	0	0
$741$	23.1225	0.849427
$742$	0	0
$743$	−37.0400	−1.35887	−0.679433	−	0.733737i	$-0.737774\pi$
$743$	−37.0400	−1.35887	−0.679433	+	0.733737i	$0.737774\pi$
$744$	0	0
$745$	−6.03441	−0.221084
$746$	0	0
$747$	−2.96665	−0.108544
$748$	0	0
$749$	16.3768	0.598396
$750$	0	0
$751$	1.31377	0.0479400	0.0239700	−	0.999713i	$-0.492369\pi$
$751$	1.31377	0.0479400	0.0239700	+	0.999713i	$0.492369\pi$
$752$	0	0
$753$	−29.5473	−1.07676
$754$	0	0
$755$	−16.0630	−0.584593
$756$	0	0
$757$	−30.2669	−1.10007	−0.550034	−	0.835142i	$-0.685385\pi$
$757$	−30.2669	−1.10007	−0.550034	+	0.835142i	$0.685385\pi$
$758$	0	0
$759$	13.6494	0.495443
$760$	0	0
$761$	7.83053	0.283856	0.141928	−	0.989877i	$-0.454670\pi$
$761$	7.83053	0.283856	0.141928	+	0.989877i	$0.454670\pi$
$762$	0	0
$763$	5.07631	0.183775
$764$	0	0
$765$	−0.142224	−0.00514212
$766$	0	0
$767$	57.5343	2.07744
$768$	0	0
$769$	−29.5736	−1.06645	−0.533225	−	0.845973i	$-0.679020\pi$
$769$	−29.5736	−1.06645	−0.533225	+	0.845973i	$0.679020\pi$
$770$	0	0
$771$	17.3940	0.626428
$772$	0	0
$773$	−52.5963	−1.89176	−0.945879	−	0.324521i	$-0.894797\pi$
$773$	−52.5963	−1.89176	−0.945879	+	0.324521i	$0.894797\pi$
$774$	0	0
$775$	−6.26984	−0.225219
$776$	0	0
$777$	−13.6807	−0.490793
$778$	0	0
$779$	3.38124	0.121145
$780$	0	0
$781$	−17.2764	−0.618198
$782$	0	0
$783$	22.6671	0.810055
$784$	0	0
$785$	−5.92482	−0.211466
$786$	0	0
$787$	11.5256	0.410845	0.205422	−	0.978673i	$-0.434143\pi$
$787$	11.5256	0.410845	0.205422	+	0.978673i	$0.434143\pi$
$788$	0	0
$789$	20.3586	0.724785
$790$	0	0
$791$	10.2068	0.362914
$792$	0	0
$793$	−13.3377	−0.473637
$794$	0	0
$795$	−14.5844	−0.517254
$796$	0	0
$797$	17.8616	0.632689	0.316345	−	0.948644i	$-0.397544\pi$
$797$	17.8616	0.632689	0.316345	+	0.948644i	$0.397544\pi$
$798$	0	0
$799$	−6.00821	−0.212555
$800$	0	0
$801$	3.57761	0.126409
$802$	0	0
$803$	−9.32643	−0.329123
$804$	0	0
$805$	−6.38895	−0.225181
$806$	0	0
$807$	51.6370	1.81771
$808$	0	0
$809$	−21.1748	−0.744465	−0.372233	−	0.928139i	$-0.621408\pi$
$809$	−21.1748	−0.744465	−0.372233	+	0.928139i	$0.621408\pi$
$810$	0	0
$811$	45.3742	1.59330	0.796652	−	0.604438i	$-0.206602\pi$
$811$	45.3742	1.59330	0.796652	+	0.604438i	$0.206602\pi$
$812$	0	0
$813$	43.1992	1.51506
$814$	0	0
$815$	−14.8624	−0.520607
$816$	0	0
$817$	2.17219	0.0759953
$818$	0	0
$819$	1.31178	0.0458374
$820$	0	0
$821$	−5.58675	−0.194979	−0.0974894	−	0.995237i	$-0.531081\pi$
$821$	−5.58675	−0.194979	−0.0974894	+	0.995237i	$0.531081\pi$
$822$	0	0
$823$	−47.4665	−1.65458	−0.827289	−	0.561776i	$-0.810118\pi$
$823$	−47.4665	−1.65458	−0.827289	+	0.561776i	$0.810118\pi$
$824$	0	0
$825$	2.13641	0.0743804
$826$	0	0
$827$	−27.9351	−0.971400	−0.485700	−	0.874126i	$-0.661435\pi$
$827$	−27.9351	−0.971400	−0.485700	+	0.874126i	$0.661435\pi$
$828$	0	0
$829$	27.0202	0.938451	0.469225	−	0.883078i	$-0.344533\pi$
$829$	27.0202	0.938451	0.469225	+	0.883078i	$0.344533\pi$
$830$	0	0
$831$	−11.8301	−0.410381
$832$	0	0
$833$	−0.643045	−0.0222802
$834$	0	0
$835$	−3.40622	−0.117877
$836$	0	0
$837$	31.2698	1.08084
$838$	0	0
$839$	12.9416	0.446793	0.223397	−	0.974728i	$-0.428285\pi$
$839$	12.9416	0.446793	0.223397	+	0.974728i	$0.428285\pi$
$840$	0	0
$841$	−8.34364	−0.287712
$842$	0	0
$843$	14.9773	0.515847
$844$	0	0
$845$	22.1772	0.762917
$846$	0	0
$847$	9.58304	0.329277
$848$	0	0
$849$	−4.52658	−0.155352
$850$	0	0
$851$	48.7002	1.66942
$852$	0	0
$853$	−23.2202	−0.795046	−0.397523	−	0.917592i	$-0.630130\pi$
$853$	−23.2202	−0.795046	−0.397523	+	0.917592i	$0.630130\pi$
$854$	0	0
$855$	−0.480429	−0.0164303
$856$	0	0
$857$	−6.14560	−0.209930	−0.104965	−	0.994476i	$-0.533473\pi$
$857$	−6.14560	−0.209930	−0.104965	+	0.994476i	$0.533473\pi$
$858$	0	0
$859$	−48.5182	−1.65542	−0.827709	−	0.561157i	$-0.810356\pi$
$859$	−48.5182	−1.65542	−0.827709	+	0.561157i	$0.810356\pi$
$860$	0	0
$861$	2.79373	0.0952102
$862$	0	0
$863$	−53.8288	−1.83235	−0.916176	−	0.400776i	$-0.868741\pi$
$863$	−53.8288	−1.83235	−0.916176	+	0.400776i	$0.868741\pi$
$864$	0	0
$865$	−14.5520	−0.494782
$866$	0	0
$867$	−29.7688	−1.01100
$868$	0	0
$869$	−10.7671	−0.365247
$870$	0	0
$871$	−3.50189	−0.118657
$872$	0	0
$873$	1.81913	0.0615681
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−35.2102	−1.18896	−0.594482	−	0.804109i	$-0.702643\pi$
$877$	−35.2102	−1.18896	−0.594482	+	0.804109i	$0.702643\pi$
$878$	0	0
$879$	−26.2286	−0.884669
$880$	0	0
$881$	5.57878	0.187954	0.0939769	−	0.995574i	$-0.470042\pi$
$881$	5.57878	0.187954	0.0939769	+	0.995574i	$0.470042\pi$
$882$	0	0
$883$	15.9552	0.536935	0.268468	−	0.963289i	$-0.413483\pi$
$883$	15.9552	0.536935	0.268468	+	0.963289i	$0.413483\pi$
$884$	0	0
$885$	−17.4102	−0.585237
$886$	0	0
$887$	44.4477	1.49241	0.746204	−	0.665717i	$-0.231874\pi$
$887$	44.4477	1.49241	0.746204	+	0.665717i	$0.231874\pi$
$888$	0	0
$889$	5.98381	0.200691
$890$	0	0
$891$	−11.4448	−0.383416
$892$	0	0
$893$	−20.2956	−0.679165
$894$	0	0
$895$	−0.0545803	−0.00182442
$896$	0	0
$897$	−68.0091	−2.27076
$898$	0	0
$899$	28.4959	0.950393
$900$	0	0
$901$	5.22543	0.174084
$902$	0	0
$903$	1.79476	0.0597260
$904$	0	0
$905$	6.06426	0.201583
$906$	0	0
$907$	−7.28999	−0.242060	−0.121030	−	0.992649i	$-0.538620\pi$
$907$	−7.28999	−0.242060	−0.121030	+	0.992649i	$0.538620\pi$
$908$	0	0
$909$	−0.629173	−0.0208684
$910$	0	0
$911$	−39.9108	−1.32230	−0.661152	−	0.750252i	$-0.729932\pi$
$911$	−39.9108	−1.32230	−0.661152	+	0.750252i	$0.729932\pi$
$912$	0	0
$913$	−15.9666	−0.528418
$914$	0	0
$915$	4.03607	0.133428
$916$	0	0
$917$	6.17081	0.203778
$918$	0	0
$919$	−35.0157	−1.15506	−0.577531	−	0.816369i	$-0.695984\pi$
$919$	−35.0157	−1.15506	−0.577531	+	0.816369i	$0.695984\pi$
$920$	0	0
$921$	32.3640	1.06643
$922$	0	0
$923$	86.0805	2.83337
$924$	0	0
$925$	7.62257	0.250629
$926$	0	0
$927$	2.40846	0.0791041
$928$	0	0
$929$	−52.5765	−1.72498	−0.862490	−	0.506075i	$-0.831096\pi$
$929$	−52.5765	−1.72498	−0.862490	+	0.506075i	$0.831096\pi$
$930$	0	0
$931$	−2.17219	−0.0711906
$932$	0	0
$933$	−9.13686	−0.299128
$934$	0	0
$935$	−0.765456	−0.0250331
$936$	0	0
$937$	−32.8339	−1.07264	−0.536318	−	0.844016i	$-0.680185\pi$
$937$	−32.8339	−1.07264	−0.536318	+	0.844016i	$0.680185\pi$
$938$	0	0
$939$	10.9526	0.357425
$940$	0	0
$941$	55.2645	1.80157	0.900786	−	0.434263i	$-0.142991\pi$
$941$	55.2645	1.80157	0.900786	+	0.434263i	$0.142991\pi$
$942$	0	0
$943$	−9.94506	−0.323856
$944$	0	0
$945$	4.98734	0.162238
$946$	0	0
$947$	22.4742	0.730314	0.365157	−	0.930946i	$-0.381015\pi$
$947$	22.4742	0.730314	0.365157	+	0.930946i	$0.381015\pi$
$948$	0	0
$949$	46.4694	1.50846
$950$	0	0
$951$	32.7749	1.06280
$952$	0	0
$953$	−28.0942	−0.910061	−0.455031	−	0.890476i	$-0.650372\pi$
$953$	−28.0942	−0.910061	−0.455031	+	0.890476i	$0.650372\pi$
$954$	0	0
$955$	3.59902	0.116462
$956$	0	0
$957$	−9.70984	−0.313875
$958$	0	0
$959$	−14.6210	−0.472138
$960$	0	0
$961$	8.31086	0.268092
$962$	0	0
$963$	−3.62210	−0.116721
$964$	0	0
$965$	−7.60481	−0.244808
$966$	0	0
$967$	−35.2294	−1.13290	−0.566451	−	0.824096i	$-0.691684\pi$
$967$	−35.2294	−1.13290	−0.566451	+	0.824096i	$0.691684\pi$
$968$	0	0
$969$	2.50695	0.0805350
$970$	0	0
$971$	26.0969	0.837490	0.418745	−	0.908104i	$-0.362470\pi$
$971$	26.0969	0.837490	0.418745	+	0.908104i	$0.362470\pi$
$972$	0	0
$973$	−5.23783	−0.167917
$974$	0	0
$975$	−10.6448	−0.340906
$976$	0	0
$977$	−28.0347	−0.896909	−0.448455	−	0.893806i	$-0.648025\pi$
$977$	−28.0347	−0.896909	−0.448455	+	0.893806i	$0.648025\pi$
$978$	0	0
$979$	19.2548	0.615387
$980$	0	0
$981$	−1.12274	−0.0358464
$982$	0	0
$983$	2.73097	0.0871045	0.0435522	−	0.999051i	$-0.486133\pi$
$983$	2.73097	0.0871045	0.0435522	+	0.999051i	$0.486133\pi$
$984$	0	0
$985$	−26.2236	−0.835553
$986$	0	0
$987$	−16.7691	−0.533767
$988$	0	0
$989$	−6.38895	−0.203157
$990$	0	0
$991$	−19.3028	−0.613174	−0.306587	−	0.951843i	$-0.599187\pi$
$991$	−19.3028	−0.613174	−0.306587	+	0.951843i	$0.599187\pi$
$992$	0	0
$993$	−23.4038	−0.742698
$994$	0	0
$995$	27.4204	0.869284
$996$	0	0
$997$	46.0638	1.45885	0.729427	−	0.684058i	$-0.239787\pi$
$997$	46.0638	1.45885	0.729427	+	0.684058i	$0.239787\pi$
$998$	0	0
$999$	−38.0163	−1.20278

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6020.2.a.g.1.8	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6020.2.a.g.1.8	✓	9	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:	\( N \)	\(=\)	\( 6020 = 2^{2} \cdot 5 \cdot 7 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6020.a (trivial)

\(f(q)\)	\(=\)	\(q+1.79476 q^{3} +1.00000 q^{5} -1.00000 q^{7} +0.221173 q^{9} +O(q^{10})\)	\(q+1.79476 q^{3} +1.00000 q^{5} -1.00000 q^{7} +0.221173 q^{9} +1.19036 q^{11} -5.93103 q^{13} +1.79476 q^{15} -0.643045 q^{17} -2.17219 q^{19} -1.79476 q^{21} +6.38895 q^{23} +1.00000 q^{25} -4.98734 q^{27} -4.54493 q^{29} -6.26984 q^{31} +2.13641 q^{33} -1.00000 q^{35} +7.62257 q^{37} -10.6448 q^{39} -1.55660 q^{41} -1.00000 q^{43} +0.221173 q^{45} +9.34336 q^{47} +1.00000 q^{49} -1.15411 q^{51} -8.12607 q^{53} +1.19036 q^{55} -3.89856 q^{57} -9.70055 q^{59} +2.24881 q^{61} -0.221173 q^{63} -5.93103 q^{65} +0.590435 q^{67} +11.4666 q^{69} -14.5136 q^{71} -7.83496 q^{73} +1.79476 q^{75} -1.19036 q^{77} -9.04520 q^{79} -9.61460 q^{81} -13.4133 q^{83} -0.643045 q^{85} -8.15706 q^{87} +16.1756 q^{89} +5.93103 q^{91} -11.2529 q^{93} -2.17219 q^{95} +8.22492 q^{97} +0.263275 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{5} - 9 q^{7} + 5 q^{9}+O(q^{10}) \) 9 * q + 9 * q^5 - 9 * q^7 + 5 * q^9	\( 9 q + 9 q^{5} - 9 q^{7} + 5 q^{9} + q^{11} - 14 q^{13} - 11 q^{17} - 2 q^{19} - 6 q^{23} + 9 q^{25} - 6 q^{29} + 6 q^{31} - 14 q^{33} - 9 q^{35} - 20 q^{37} - 6 q^{39} - 6 q^{41} - 9 q^{43} + 5 q^{45} + 9 q^{49} - 8 q^{51} - 31 q^{53} + q^{55} - 16 q^{57} + 2 q^{59} - 13 q^{61} - 5 q^{63} - 14 q^{65} - 10 q^{67} - 18 q^{69} + 12 q^{71} - 32 q^{73} - q^{77} + q^{79} - 27 q^{81} - 10 q^{83} - 11 q^{85} - 5 q^{87} - q^{89} + 14 q^{91} - 49 q^{93} - 2 q^{95} - 28 q^{97} - 14 q^{99}+O(q^{100}) \) 9 * q + 9 * q^5 - 9 * q^7 + 5 * q^9 + q^11 - 14 * q^13 - 11 * q^17 - 2 * q^19 - 6 * q^23 + 9 * q^25 - 6 * q^29 + 6 * q^31 - 14 * q^33 - 9 * q^35 - 20 * q^37 - 6 * q^39 - 6 * q^41 - 9 * q^43 + 5 * q^45 + 9 * q^49 - 8 * q^51 - 31 * q^53 + q^55 - 16 * q^57 + 2 * q^59 - 13 * q^61 - 5 * q^63 - 14 * q^65 - 10 * q^67 - 18 * q^69 + 12 * q^71 - 32 * q^73 - q^77 + q^79 - 27 * q^81 - 10 * q^83 - 11 * q^85 - 5 * q^87 - q^89 + 14 * q^91 - 49 * q^93 - 2 * q^95 - 28 * q^97 - 14 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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