LMFDB - Embedded newform 320.6.a.r.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [320,6,Mod(1,320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(320, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("320.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 320 = 2^{6} \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	320.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:	$51.3228223402$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{70}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 70 $ x^2 - 70
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 160)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-8.36660$ of defining polynomial
Character	$\chi$	$=$	320.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-20.7332 q^{3} -25.0000 q^{5} +35.2668 q^{7} +186.866 q^{9} +O(q^{10})$	$q-20.7332 q^{3} -25.0000 q^{5} +35.2668 q^{7} +186.866 q^{9} +7.33201 q^{11} -619.328 q^{13} +518.330 q^{15} +959.328 q^{17} +309.328 q^{19} -731.194 q^{21} +2467.12 q^{23} +625.000 q^{25} +1163.85 q^{27} +1284.66 q^{29} +7095.32 q^{31} -152.016 q^{33} -881.670 q^{35} +6100.59 q^{37} +12840.7 q^{39} -18830.4 q^{41} -3147.48 q^{43} -4671.64 q^{45} +20556.8 q^{47} -15563.3 q^{49} -19889.9 q^{51} -33741.9 q^{53} -183.300 q^{55} -6413.36 q^{57} -15065.3 q^{59} +7542.11 q^{61} +6590.15 q^{63} +15483.2 q^{65} -25574.9 q^{67} -51151.3 q^{69} -56232.3 q^{71} +58657.5 q^{73} -12958.3 q^{75} +258.576 q^{77} +32238.7 q^{79} -69538.6 q^{81} -31173.4 q^{83} -23983.2 q^{85} -26635.0 q^{87} -75258.9 q^{89} -21841.7 q^{91} -147109. q^{93} -7733.20 q^{95} -176059. q^{97} +1370.10 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{3} - 50 q^{5} + 104 q^{7} + 106 q^{9}+O(q^{10}) $ 2 * q - 8 * q^3 - 50 * q^5 + 104 * q^7 + 106 * q^9	$ 2 q - 8 q^{3} - 50 q^{5} + 104 q^{7} + 106 q^{9} - 320 q^{11} + 100 q^{13} + 200 q^{15} + 580 q^{17} - 720 q^{19} + 144 q^{21} + 1688 q^{23} + 1250 q^{25} - 2960 q^{27} - 108 q^{29} + 9840 q^{31} - 4320 q^{33} - 2600 q^{35} - 6540 q^{37} + 22000 q^{39} - 10620 q^{41} - 25672 q^{43} - 2650 q^{45} + 28296 q^{47} - 27646 q^{49} - 24720 q^{51} - 31340 q^{53} + 8000 q^{55} - 19520 q^{57} - 30800 q^{59} - 24540 q^{61} + 1032 q^{63} - 2500 q^{65} - 34584 q^{67} - 61072 q^{69} - 12400 q^{71} - 7180 q^{73} - 5000 q^{75} - 22240 q^{77} + 71840 q^{79} - 102398 q^{81} + 31928 q^{83} - 14500 q^{85} - 44368 q^{87} - 40748 q^{89} + 27600 q^{91} - 112160 q^{93} + 18000 q^{95} - 190140 q^{97} + 27840 q^{99}+O(q^{100}) $ 2 * q - 8 * q^3 - 50 * q^5 + 104 * q^7 + 106 * q^9 - 320 * q^11 + 100 * q^13 + 200 * q^15 + 580 * q^17 - 720 * q^19 + 144 * q^21 + 1688 * q^23 + 1250 * q^25 - 2960 * q^27 - 108 * q^29 + 9840 * q^31 - 4320 * q^33 - 2600 * q^35 - 6540 * q^37 + 22000 * q^39 - 10620 * q^41 - 25672 * q^43 - 2650 * q^45 + 28296 * q^47 - 27646 * q^49 - 24720 * q^51 - 31340 * q^53 + 8000 * q^55 - 19520 * q^57 - 30800 * q^59 - 24540 * q^61 + 1032 * q^63 - 2500 * q^65 - 34584 * q^67 - 61072 * q^69 - 12400 * q^71 - 7180 * q^73 - 5000 * q^75 - 22240 * q^77 + 71840 * q^79 - 102398 * q^81 + 31928 * q^83 - 14500 * q^85 - 44368 * q^87 - 40748 * q^89 + 27600 * q^91 - 112160 * q^93 + 18000 * q^95 - 190140 * q^97 + 27840 * 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
$3$	−20.7332	−1.33004	−0.665018	−	0.746828i	$-0.731576\pi$
$3$	−20.7332	−1.33004	−0.665018	+	0.746828i	$0.731576\pi$
$4$	0	0
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	0	0
$7$	35.2668	0.272033	0.136016	−	0.990707i	$-0.456570\pi$
$7$	35.2668	0.272033	0.136016	+	0.990707i	$0.456570\pi$
$8$	0	0
$9$	186.866	0.768994
$10$	0	0
$11$	7.33201	0.0182701	0.00913505	−	0.999958i	$-0.497092\pi$
$11$	7.33201	0.0182701	0.00913505	+	0.999958i	$0.497092\pi$
$12$	0	0
$13$	−619.328	−1.01639	−0.508197	−	0.861241i	$-0.669688\pi$
$13$	−619.328	−1.01639	−0.508197	+	0.861241i	$0.669688\pi$
$14$	0	0
$15$	518.330	0.594810
$16$	0	0
$17$	959.328	0.805091	0.402545	−	0.915400i	$-0.368126\pi$
$17$	959.328	0.805091	0.402545	+	0.915400i	$0.368126\pi$
$18$	0	0
$19$	309.328	0.196578	0.0982891	−	0.995158i	$-0.468663\pi$
$19$	309.328	0.196578	0.0982891	+	0.995158i	$0.468663\pi$
$20$	0	0
$21$	−731.194	−0.361813
$22$	0	0
$23$	2467.12	0.972458	0.486229	−	0.873831i	$-0.338372\pi$
$23$	2467.12	0.972458	0.486229	+	0.873831i	$0.338372\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	1163.85	0.307246
$28$	0	0
$29$	1284.66	0.283656	0.141828	−	0.989891i	$-0.454702\pi$
$29$	1284.66	0.283656	0.141828	+	0.989891i	$0.454702\pi$
$30$	0	0
$31$	7095.32	1.32607	0.663037	−	0.748587i	$-0.269267\pi$
$31$	7095.32	1.32607	0.663037	+	0.748587i	$0.269267\pi$
$32$	0	0
$33$	−152.016	−0.0242999
$34$	0	0
$35$	−881.670	−0.121657
$36$	0	0
$37$	6100.59	0.732601	0.366301	−	0.930497i	$-0.380624\pi$
$37$	6100.59	0.732601	0.366301	+	0.930497i	$0.380624\pi$
$38$	0	0
$39$	12840.7	1.35184
$40$	0	0
$41$	−18830.4	−1.74945	−0.874723	−	0.484623i	$-0.838957\pi$
$41$	−18830.4	−1.74945	−0.874723	+	0.484623i	$0.838957\pi$
$42$	0	0
$43$	−3147.48	−0.259592	−0.129796	−	0.991541i	$-0.541432\pi$
$43$	−3147.48	−0.259592	−0.129796	+	0.991541i	$0.541432\pi$
$44$	0	0
$45$	−4671.64	−0.343905
$46$	0	0
$47$	20556.8	1.35741	0.678705	−	0.734411i	$-0.262541\pi$
$47$	20556.8	1.35741	0.678705	+	0.734411i	$0.262541\pi$
$48$	0	0
$49$	−15563.3	−0.925998
$50$	0	0
$51$	−19889.9	−1.07080
$52$	0	0
$53$	−33741.9	−1.64998	−0.824991	−	0.565146i	$-0.808820\pi$
$53$	−33741.9	−1.64998	−0.824991	+	0.565146i	$0.808820\pi$
$54$	0	0
$55$	−183.300	−0.00817064
$56$	0	0
$57$	−6413.36	−0.261456
$58$	0	0
$59$	−15065.3	−0.563441	−0.281721	−	0.959496i	$-0.590905\pi$
$59$	−15065.3	−0.563441	−0.281721	+	0.959496i	$0.590905\pi$
$60$	0	0
$61$	7542.11	0.259518	0.129759	−	0.991546i	$-0.458580\pi$
$61$	7542.11	0.259518	0.129759	+	0.991546i	$0.458580\pi$
$62$	0	0
$63$	6590.15	0.209192
$64$	0	0
$65$	15483.2	0.454545
$66$	0	0
$67$	−25574.9	−0.696029	−0.348015	−	0.937489i	$-0.613144\pi$
$67$	−25574.9	−0.696029	−0.348015	+	0.937489i	$0.613144\pi$
$68$	0	0
$69$	−51151.3	−1.29340
$70$	0	0
$71$	−56232.3	−1.32385	−0.661926	−	0.749569i	$-0.730261\pi$
$71$	−56232.3	−1.32385	−0.661926	+	0.749569i	$0.730261\pi$
$72$	0	0
$73$	58657.5	1.28830	0.644149	−	0.764900i	$-0.277212\pi$
$73$	58657.5	1.28830	0.644149	+	0.764900i	$0.277212\pi$
$74$	0	0
$75$	−12958.3	−0.266007
$76$	0	0
$77$	258.576	0.00497006
$78$	0	0
$79$	32238.7	0.581179	0.290589	−	0.956848i	$-0.406149\pi$
$79$	32238.7	0.581179	0.290589	+	0.956848i	$0.406149\pi$
$80$	0	0
$81$	−69538.6	−1.17764
$82$	0	0
$83$	−31173.4	−0.496694	−0.248347	−	0.968671i	$-0.579887\pi$
$83$	−31173.4	−0.496694	−0.248347	+	0.968671i	$0.579887\pi$
$84$	0	0
$85$	−23983.2	−0.360048
$86$	0	0
$87$	−26635.0	−0.377272
$88$	0	0
$89$	−75258.9	−1.00712	−0.503562	−	0.863959i	$-0.667977\pi$
$89$	−75258.9	−1.00712	−0.503562	+	0.863959i	$0.667977\pi$
$90$	0	0
$91$	−21841.7	−0.276492
$92$	0	0
$93$	−147109.	−1.76372
$94$	0	0
$95$	−7733.20	−0.0879124
$96$	0	0
$97$	−176059.	−1.89989	−0.949944	−	0.312419i	$-0.898861\pi$
$97$	−176059.	−1.89989	−0.949944	+	0.312419i	$0.898861\pi$
$98$	0	0
$99$	1370.10	0.0140496
$100$	0	0
$101$	102454.	0.999370	0.499685	−	0.866207i	$-0.333449\pi$
$101$	102454.	0.999370	0.499685	+	0.866207i	$0.333449\pi$
$102$	0	0
$103$	−150320.	−1.39612	−0.698062	−	0.716038i	$-0.745954\pi$
$103$	−150320.	−1.39612	−0.698062	+	0.716038i	$0.745954\pi$
$104$	0	0
$105$	18279.8	0.161808
$106$	0	0
$107$	220857.	1.86488	0.932441	−	0.361321i	$-0.117674\pi$
$107$	220857.	1.86488	0.932441	+	0.361321i	$0.117674\pi$
$108$	0	0
$109$	24621.0	0.198490	0.0992450	−	0.995063i	$-0.468357\pi$
$109$	24621.0	0.198490	0.0992450	+	0.995063i	$0.468357\pi$
$110$	0	0
$111$	−126485.	−0.974386
$112$	0	0
$113$	−64568.7	−0.475692	−0.237846	−	0.971303i	$-0.576441\pi$
$113$	−64568.7	−0.475692	−0.237846	+	0.971303i	$0.576441\pi$
$114$	0	0
$115$	−61678.0	−0.434896
$116$	0	0
$117$	−115731.	−0.781602
$118$	0	0
$119$	33832.4	0.219011
$120$	0	0
$121$	−160997.	−0.999666
$122$	0	0
$123$	390415.	2.32682
$124$	0	0
$125$	−15625.0	−0.0894427
$126$	0	0
$127$	159599.	0.878054	0.439027	−	0.898474i	$-0.355323\pi$
$127$	159599.	0.878054	0.439027	+	0.898474i	$0.355323\pi$
$128$	0	0
$129$	65257.3	0.345267
$130$	0	0
$131$	−116112.	−0.591153	−0.295577	−	0.955319i	$-0.595512\pi$
$131$	−116112.	−0.591153	−0.295577	+	0.955319i	$0.595512\pi$
$132$	0	0
$133$	10909.0	0.0534757
$134$	0	0
$135$	−29096.1	−0.137405
$136$	0	0
$137$	−1446.13	−0.00658272	−0.00329136	−	0.999995i	$-0.501048\pi$
$137$	−1446.13	−0.00658272	−0.00329136	+	0.999995i	$0.501048\pi$
$138$	0	0
$139$	135055.	0.592891	0.296446	−	0.955050i	$-0.404199\pi$
$139$	135055.	0.592891	0.296446	+	0.955050i	$0.404199\pi$
$140$	0	0
$141$	−426209.	−1.80540
$142$	0	0
$143$	−4540.92	−0.0185696
$144$	0	0
$145$	−32116.4	−0.126855
$146$	0	0
$147$	322676.	1.23161
$148$	0	0
$149$	−16097.8	−0.0594019	−0.0297010	−	0.999559i	$-0.509455\pi$
$149$	−16097.8	−0.0594019	−0.0297010	+	0.999559i	$0.509455\pi$
$150$	0	0
$151$	−467353.	−1.66802	−0.834012	−	0.551746i	$-0.813962\pi$
$151$	−467353.	−1.66802	−0.834012	+	0.551746i	$0.813962\pi$
$152$	0	0
$153$	179265.	0.619110
$154$	0	0
$155$	−177383.	−0.593038
$156$	0	0
$157$	−210478.	−0.681487	−0.340743	−	0.940156i	$-0.610679\pi$
$157$	−210478.	−0.681487	−0.340743	+	0.940156i	$0.610679\pi$
$158$	0	0
$159$	699577.	2.19454
$160$	0	0
$161$	87007.4	0.264540
$162$	0	0
$163$	136230.	0.401610	0.200805	−	0.979631i	$-0.435644\pi$
$163$	136230.	0.401610	0.200805	+	0.979631i	$0.435644\pi$
$164$	0	0
$165$	3800.40	0.0108672
$166$	0	0
$167$	246713.	0.684542	0.342271	−	0.939601i	$-0.388804\pi$
$167$	246713.	0.684542	0.342271	+	0.939601i	$0.388804\pi$
$168$	0	0
$169$	12274.2	0.0330580
$170$	0	0
$171$	57802.8	0.151167
$172$	0	0
$173$	652180.	1.65673	0.828366	−	0.560187i	$-0.189271\pi$
$173$	652180.	1.65673	0.828366	+	0.560187i	$0.189271\pi$
$174$	0	0
$175$	22041.7	0.0544065
$176$	0	0
$177$	312353.	0.749397
$178$	0	0
$179$	−786542.	−1.83480	−0.917402	−	0.397962i	$-0.869718\pi$
$179$	−786542.	−1.83480	−0.917402	+	0.397962i	$0.869718\pi$
$180$	0	0
$181$	974.027	0.00220991	0.00110495	−	0.999999i	$-0.499648\pi$
$181$	974.027	0.00220991	0.00110495	+	0.999999i	$0.499648\pi$
$182$	0	0
$183$	−156372.	−0.345169
$184$	0	0
$185$	−152515.	−0.327629
$186$	0	0
$187$	7033.80	0.0147091
$188$	0	0
$189$	41045.1	0.0835809
$190$	0	0
$191$	−469675.	−0.931567	−0.465783	−	0.884899i	$-0.654227\pi$
$191$	−469675.	−0.931567	−0.465783	+	0.884899i	$0.654227\pi$
$192$	0	0
$193$	−644811.	−1.24606	−0.623030	−	0.782198i	$-0.714099\pi$
$193$	−644811.	−1.24606	−0.623030	+	0.782198i	$0.714099\pi$
$194$	0	0
$195$	−321016.	−0.604562
$196$	0	0
$197$	521778.	0.957899	0.478950	−	0.877842i	$-0.341018\pi$
$197$	521778.	0.957899	0.478950	+	0.877842i	$0.341018\pi$
$198$	0	0
$199$	394042.	0.705358	0.352679	−	0.935744i	$-0.385271\pi$
$199$	394042.	0.705358	0.352679	+	0.935744i	$0.385271\pi$
$200$	0	0
$201$	530250.	0.925744
$202$	0	0
$203$	45305.7	0.0771637
$204$	0	0
$205$	470761.	0.782376
$206$	0	0
$207$	461020.	0.747814
$208$	0	0
$209$	2267.99	0.00359150
$210$	0	0
$211$	−750041.	−1.15979	−0.579894	−	0.814692i	$-0.696906\pi$
$211$	−750041.	−1.15979	−0.579894	+	0.814692i	$0.696906\pi$
$212$	0	0
$213$	1.16587e6	1.76077
$214$	0	0
$215$	78686.9	0.116093
$216$	0	0
$217$	250229.	0.360735
$218$	0	0
$219$	−1.21616e6	−1.71348
$220$	0	0
$221$	−594139.	−0.818290
$222$	0	0
$223$	−1.00328e6	−1.35102	−0.675509	−	0.737351i	$-0.736076\pi$
$223$	−1.00328e6	−1.35102	−0.675509	+	0.737351i	$0.736076\pi$
$224$	0	0
$225$	116791.	0.153799
$226$	0	0
$227$	−385367.	−0.496375	−0.248188	−	0.968712i	$-0.579835\pi$
$227$	−385367.	−0.496375	−0.248188	+	0.968712i	$0.579835\pi$
$228$	0	0
$229$	16601.6	0.0209200	0.0104600	−	0.999945i	$-0.496670\pi$
$229$	16601.6	0.0209200	0.0104600	+	0.999945i	$0.496670\pi$
$230$	0	0
$231$	−5361.12	−0.00661036
$232$	0	0
$233$	933464.	1.12644	0.563219	−	0.826307i	$-0.309563\pi$
$233$	933464.	1.12644	0.563219	+	0.826307i	$0.309563\pi$
$234$	0	0
$235$	−513920.	−0.607052
$236$	0	0
$237$	−668411.	−0.772988
$238$	0	0
$239$	1.34120e6	1.51880	0.759398	−	0.650626i	$-0.225493\pi$
$239$	1.34120e6	1.51880	0.759398	+	0.650626i	$0.225493\pi$
$240$	0	0
$241$	390139.	0.432690	0.216345	−	0.976317i	$-0.430586\pi$
$241$	390139.	0.432690	0.216345	+	0.976317i	$0.430586\pi$
$242$	0	0
$243$	1.15894e6	1.25906
$244$	0	0
$245$	389081.	0.414119
$246$	0	0
$247$	−191576.	−0.199801
$248$	0	0
$249$	646325.	0.660621
$250$	0	0
$251$	−805856.	−0.807371	−0.403686	−	0.914898i	$-0.632271\pi$
$251$	−805856.	−0.807371	−0.403686	+	0.914898i	$0.632271\pi$
$252$	0	0
$253$	18088.9	0.0177669
$254$	0	0
$255$	497249.	0.478876
$256$	0	0
$257$	153621.	0.145084	0.0725419	−	0.997365i	$-0.476889\pi$
$257$	153621.	0.145084	0.0725419	+	0.997365i	$0.476889\pi$
$258$	0	0
$259$	215148.	0.199291
$260$	0	0
$261$	240058.	0.218130
$262$	0	0
$263$	−28007.0	−0.0249676	−0.0124838	−	0.999922i	$-0.503974\pi$
$263$	−28007.0	−0.0249676	−0.0124838	+	0.999922i	$0.503974\pi$
$264$	0	0
$265$	843546.	0.737895
$266$	0	0
$267$	1.56036e6	1.33951
$268$	0	0
$269$	−715607.	−0.602968	−0.301484	−	0.953471i	$-0.597482\pi$
$269$	−715607.	−0.602968	−0.301484	+	0.953471i	$0.597482\pi$
$270$	0	0
$271$	−1.35674e6	−1.12221	−0.561104	−	0.827745i	$-0.689623\pi$
$271$	−1.35674e6	−1.12221	−0.561104	+	0.827745i	$0.689623\pi$
$272$	0	0
$273$	452849.	0.367745
$274$	0	0
$275$	4582.50	0.00365402
$276$	0	0
$277$	791037.	0.619437	0.309719	−	0.950828i	$-0.399765\pi$
$277$	791037.	0.619437	0.309719	+	0.950828i	$0.399765\pi$
$278$	0	0
$279$	1.32587e6	1.01974
$280$	0	0
$281$	−191136.	−0.144403	−0.0722016	−	0.997390i	$-0.523002\pi$
$281$	−191136.	−0.144403	−0.0722016	+	0.997390i	$0.523002\pi$
$282$	0	0
$283$	1.16924e6	0.867834	0.433917	−	0.900953i	$-0.357131\pi$
$283$	1.16924e6	0.867834	0.433917	+	0.900953i	$0.357131\pi$
$284$	0	0
$285$	160334.	0.116927
$286$	0	0
$287$	−664089.	−0.475906
$288$	0	0
$289$	−499547.	−0.351829
$290$	0	0
$291$	3.65026e6	2.52692
$292$	0	0
$293$	932811.	0.634782	0.317391	−	0.948295i	$-0.397193\pi$
$293$	932811.	0.634782	0.317391	+	0.948295i	$0.397193\pi$
$294$	0	0
$295$	376633.	0.251979
$296$	0	0
$297$	8533.32	0.00561341
$298$	0	0
$299$	−1.52796e6	−0.988401
$300$	0	0
$301$	−111001.	−0.0706175
$302$	0	0
$303$	−2.12420e6	−1.32920
$304$	0	0
$305$	−188553.	−0.116060
$306$	0	0
$307$	678740.	0.411015	0.205507	−	0.978656i	$-0.434116\pi$
$307$	678740.	0.411015	0.205507	+	0.978656i	$0.434116\pi$
$308$	0	0
$309$	3.11662e6	1.85689
$310$	0	0
$311$	1.25645e6	0.736621	0.368311	−	0.929703i	$-0.379936\pi$
$311$	1.25645e6	0.736621	0.368311	+	0.929703i	$0.379936\pi$
$312$	0	0
$313$	3.13670e6	1.80972	0.904861	−	0.425708i	$-0.139975\pi$
$313$	3.13670e6	1.80972	0.904861	+	0.425708i	$0.139975\pi$
$314$	0	0
$315$	−164754.	−0.0935533
$316$	0	0
$317$	−353375.	−0.197510	−0.0987548	−	0.995112i	$-0.531486\pi$
$317$	−353375.	−0.197510	−0.0987548	+	0.995112i	$0.531486\pi$
$318$	0	0
$319$	9419.10	0.00518242
$320$	0	0
$321$	−4.57907e6	−2.48036
$322$	0	0
$323$	296747.	0.158263
$324$	0	0
$325$	−387080.	−0.203279
$326$	0	0
$327$	−510471.	−0.263999
$328$	0	0
$329$	724973.	0.369260
$330$	0	0
$331$	2.91561e6	1.46271	0.731356	−	0.681996i	$-0.238888\pi$
$331$	2.91561e6	1.46271	0.731356	+	0.681996i	$0.238888\pi$
$332$	0	0
$333$	1.13999e6	0.563366
$334$	0	0
$335$	639373.	0.311274
$336$	0	0
$337$	−3.11496e6	−1.49409	−0.747046	−	0.664772i	$-0.768529\pi$
$337$	−3.11496e6	−1.49409	−0.747046	+	0.664772i	$0.768529\pi$
$338$	0	0
$339$	1.33871e6	0.632687
$340$	0	0
$341$	52022.9	0.0242275
$342$	0	0
$343$	−1.14160e6	−0.523934
$344$	0	0
$345$	1.27878e6	0.578428
$346$	0	0
$347$	−4.09755e6	−1.82684	−0.913419	−	0.407020i	$-0.866568\pi$
$347$	−4.09755e6	−1.82684	−0.913419	+	0.407020i	$0.866568\pi$
$348$	0	0
$349$	−2.46309e6	−1.08247	−0.541237	−	0.840870i	$-0.682044\pi$
$349$	−2.46309e6	−1.08247	−0.541237	+	0.840870i	$0.682044\pi$
$350$	0	0
$351$	−720802.	−0.312283
$352$	0	0
$353$	3.56682e6	1.52350	0.761752	−	0.647868i	$-0.224339\pi$
$353$	3.56682e6	1.52350	0.761752	+	0.647868i	$0.224339\pi$
$354$	0	0
$355$	1.40581e6	0.592045
$356$	0	0
$357$	−701455.	−0.291292
$358$	0	0
$359$	−648318.	−0.265492	−0.132746	−	0.991150i	$-0.542380\pi$
$359$	−648318.	−0.265492	−0.132746	+	0.991150i	$0.542380\pi$
$360$	0	0
$361$	−2.38042e6	−0.961357
$362$	0	0
$363$	3.33799e6	1.32959
$364$	0	0
$365$	−1.46644e6	−0.576145
$366$	0	0
$367$	−3.84494e6	−1.49013	−0.745065	−	0.666992i	$-0.767581\pi$
$367$	−3.84494e6	−1.49013	−0.745065	+	0.666992i	$0.767581\pi$
$368$	0	0
$369$	−3.51876e6	−1.34531
$370$	0	0
$371$	−1.18997e6	−0.448849
$372$	0	0
$373$	−2.40550e6	−0.895228	−0.447614	−	0.894227i	$-0.647726\pi$
$373$	−2.40550e6	−0.895228	−0.447614	+	0.894227i	$0.647726\pi$
$374$	0	0
$375$	323956.	0.118962
$376$	0	0
$377$	−795623.	−0.288306
$378$	0	0
$379$	−5.03689e6	−1.80121	−0.900605	−	0.434638i	$-0.856876\pi$
$379$	−5.03689e6	−1.80121	−0.900605	+	0.434638i	$0.856876\pi$
$380$	0	0
$381$	−3.30900e6	−1.16784
$382$	0	0
$383$	−1.61371e6	−0.562121	−0.281060	−	0.959690i	$-0.590686\pi$
$383$	−1.61371e6	−0.562121	−0.281060	+	0.959690i	$0.590686\pi$
$384$	0	0
$385$	−6464.41	−0.00222268
$386$	0	0
$387$	−588155.	−0.199625
$388$	0	0
$389$	−1.99508e6	−0.668477	−0.334238	−	0.942489i	$-0.608479\pi$
$389$	−1.99508e6	−0.668477	−0.334238	+	0.942489i	$0.608479\pi$
$390$	0	0
$391$	2.36678e6	0.782917
$392$	0	0
$393$	2.40738e6	0.786255
$394$	0	0
$395$	−805967.	−0.259911
$396$	0	0
$397$	−3.61674e6	−1.15170	−0.575852	−	0.817554i	$-0.695329\pi$
$397$	−3.61674e6	−1.15170	−0.575852	+	0.817554i	$0.695329\pi$
$398$	0	0
$399$	−226179.	−0.0711245
$400$	0	0
$401$	−2.64534e6	−0.821525	−0.410763	−	0.911742i	$-0.634738\pi$
$401$	−2.64534e6	−0.821525	−0.410763	+	0.911742i	$0.634738\pi$
$402$	0	0
$403$	−4.39433e6	−1.34781
$404$	0	0
$405$	1.73846e6	0.526658
$406$	0	0
$407$	44729.6	0.0133847
$408$	0	0
$409$	2.86360e6	0.846456	0.423228	−	0.906023i	$-0.360897\pi$
$409$	2.86360e6	0.846456	0.423228	+	0.906023i	$0.360897\pi$
$410$	0	0
$411$	29982.9	0.00875525
$412$	0	0
$413$	−531306.	−0.153274
$414$	0	0
$415$	779336.	0.222129
$416$	0	0
$417$	−2.80013e6	−0.788566
$418$	0	0
$419$	−1.09856e6	−0.305695	−0.152847	−	0.988250i	$-0.548844\pi$
$419$	−1.09856e6	−0.305695	−0.152847	+	0.988250i	$0.548844\pi$
$420$	0	0
$421$	−2.31987e6	−0.637909	−0.318955	−	0.947770i	$-0.603332\pi$
$421$	−2.31987e6	−0.637909	−0.318955	+	0.947770i	$0.603332\pi$
$422$	0	0
$423$	3.84136e6	1.04384
$424$	0	0
$425$	599580.	0.161018
$426$	0	0
$427$	265986.	0.0705975
$428$	0	0
$429$	94147.7	0.0246983
$430$	0	0
$431$	5.05072e6	1.30967	0.654833	−	0.755774i	$-0.272739\pi$
$431$	5.05072e6	1.30967	0.654833	+	0.755774i	$0.272739\pi$
$432$	0	0
$433$	−1.01025e6	−0.258946	−0.129473	−	0.991583i	$-0.541329\pi$
$433$	−1.01025e6	−0.258946	−0.129473	+	0.991583i	$0.541329\pi$
$434$	0	0
$435$	665876.	0.168721
$436$	0	0
$437$	763149.	0.191164
$438$	0	0
$439$	−5.14635e6	−1.27450	−0.637248	−	0.770659i	$-0.719927\pi$
$439$	−5.14635e6	−1.27450	−0.637248	+	0.770659i	$0.719927\pi$
$440$	0	0
$441$	−2.90824e6	−0.712087
$442$	0	0
$443$	−5.25603e6	−1.27247	−0.636236	−	0.771494i	$-0.719510\pi$
$443$	−5.25603e6	−1.27247	−0.636236	+	0.771494i	$0.719510\pi$
$444$	0	0
$445$	1.88147e6	0.450399
$446$	0	0
$447$	333759.	0.0790067
$448$	0	0
$449$	−4.10261e6	−0.960384	−0.480192	−	0.877163i	$-0.659433\pi$
$449$	−4.10261e6	−0.960384	−0.480192	+	0.877163i	$0.659433\pi$
$450$	0	0
$451$	−138065.	−0.0319626
$452$	0	0
$453$	9.68972e6	2.21853
$454$	0	0
$455$	546043.	0.123651
$456$	0	0
$457$	−4.01053e6	−0.898278	−0.449139	−	0.893462i	$-0.648269\pi$
$457$	−4.01053e6	−0.898278	−0.449139	+	0.893462i	$0.648269\pi$
$458$	0	0
$459$	1.11651e6	0.247361
$460$	0	0
$461$	−3.83280e6	−0.839971	−0.419986	−	0.907531i	$-0.637965\pi$
$461$	−3.83280e6	−0.839971	−0.419986	+	0.907531i	$0.637965\pi$
$462$	0	0
$463$	1.14398e6	0.248009	0.124004	−	0.992282i	$-0.460426\pi$
$463$	1.14398e6	0.248009	0.124004	+	0.992282i	$0.460426\pi$
$464$	0	0
$465$	3.67772e6	0.788762
$466$	0	0
$467$	−7.12577e6	−1.51196	−0.755979	−	0.654596i	$-0.772839\pi$
$467$	−7.12577e6	−1.51196	−0.755979	+	0.654596i	$0.772839\pi$
$468$	0	0
$469$	−901946.	−0.189343
$470$	0	0
$471$	4.36388e6	0.906401
$472$	0	0
$473$	−23077.3	−0.00474277
$474$	0	0
$475$	193330.	0.0393156
$476$	0	0
$477$	−6.30519e6	−1.26883
$478$	0	0
$479$	6.45039e6	1.28454	0.642269	−	0.766479i	$-0.277993\pi$
$479$	6.45039e6	1.28454	0.642269	+	0.766479i	$0.277993\pi$
$480$	0	0
$481$	−3.77827e6	−0.744612
$482$	0	0
$483$	−1.80394e6	−0.351848
$484$	0	0
$485$	4.40147e6	0.849656
$486$	0	0
$487$	−1.46947e6	−0.280762	−0.140381	−	0.990098i	$-0.544833\pi$
$487$	−1.46947e6	−0.280762	−0.140381	+	0.990098i	$0.544833\pi$
$488$	0	0
$489$	−2.82449e6	−0.534155
$490$	0	0
$491$	5.19488e6	0.972460	0.486230	−	0.873831i	$-0.338372\pi$
$491$	5.19488e6	0.972460	0.486230	+	0.873831i	$0.338372\pi$
$492$	0	0
$493$	1.23241e6	0.228369
$494$	0	0
$495$	−34252.5	−0.00628317
$496$	0	0
$497$	−1.98313e6	−0.360131
$498$	0	0
$499$	7.07689e6	1.27230	0.636152	−	0.771564i	$-0.280525\pi$
$499$	7.07689e6	1.27230	0.636152	+	0.771564i	$0.280525\pi$
$500$	0	0
$501$	−5.11514e6	−0.910465
$502$	0	0
$503$	8.92098e6	1.57215	0.786073	−	0.618134i	$-0.212111\pi$
$503$	8.92098e6	1.57215	0.786073	+	0.618134i	$0.212111\pi$
$504$	0	0
$505$	−2.56135e6	−0.446932
$506$	0	0
$507$	−254483.	−0.0439683
$508$	0	0
$509$	−5.79714e6	−0.991789	−0.495895	−	0.868383i	$-0.665160\pi$
$509$	−5.79714e6	−0.991789	−0.495895	+	0.868383i	$0.665160\pi$
$510$	0	0
$511$	2.06866e6	0.350459
$512$	0	0
$513$	360010.	0.0603978
$514$	0	0
$515$	3.75800e6	0.624365
$516$	0	0
$517$	150723.	0.0248000
$518$	0	0
$519$	−1.35218e7	−2.20351
$520$	0	0
$521$	7.45721e6	1.20360	0.601800	−	0.798647i	$-0.294450\pi$
$521$	7.45721e6	1.20360	0.601800	+	0.798647i	$0.294450\pi$
$522$	0	0
$523$	1.01015e7	1.61485	0.807424	−	0.589971i	$-0.200861\pi$
$523$	1.01015e7	1.61485	0.807424	+	0.589971i	$0.200861\pi$
$524$	0	0
$525$	−456996.	−0.0723626
$526$	0	0
$527$	6.80674e6	1.06761
$528$	0	0
$529$	−349660.	−0.0543258
$530$	0	0
$531$	−2.81519e6	−0.433283
$532$	0	0
$533$	1.16622e7	1.77813
$534$	0	0
$535$	−5.52142e6	−0.834001
$536$	0	0
$537$	1.63075e7	2.44035
$538$	0	0
$539$	−114110.	−0.0169181
$540$	0	0
$541$	−1.23616e7	−1.81585	−0.907925	−	0.419132i	$-0.862334\pi$
$541$	−1.23616e7	−1.81585	−0.907925	+	0.419132i	$0.862334\pi$
$542$	0	0
$543$	−20194.7	−0.00293926
$544$	0	0
$545$	−615524.	−0.0887675
$546$	0	0
$547$	3.72735e6	0.532637	0.266319	−	0.963885i	$-0.414193\pi$
$547$	3.72735e6	0.532637	0.266319	+	0.963885i	$0.414193\pi$
$548$	0	0
$549$	1.40936e6	0.199568
$550$	0	0
$551$	397380.	0.0557606
$552$	0	0
$553$	1.13696e6	0.158100
$554$	0	0
$555$	3.16212e6	0.435759
$556$	0	0
$557$	1.68294e6	0.229842	0.114921	−	0.993375i	$-0.463338\pi$
$557$	1.68294e6	0.229842	0.114921	+	0.993375i	$0.463338\pi$
$558$	0	0
$559$	1.94932e6	0.263848
$560$	0	0
$561$	−145833.	−0.0195636
$562$	0	0
$563$	−1.31501e7	−1.74848	−0.874238	−	0.485498i	$-0.838638\pi$
$563$	−1.31501e7	−1.74848	−0.874238	+	0.485498i	$0.838638\pi$
$564$	0	0
$565$	1.61422e6	0.212736
$566$	0	0
$567$	−2.45240e6	−0.320357
$568$	0	0
$569$	6.91087e6	0.894854	0.447427	−	0.894320i	$-0.352340\pi$
$569$	6.91087e6	0.894854	0.447427	+	0.894320i	$0.352340\pi$
$570$	0	0
$571$	−1.49642e6	−0.192072	−0.0960361	−	0.995378i	$-0.530616\pi$
$571$	−1.49642e6	−0.192072	−0.0960361	+	0.995378i	$0.530616\pi$
$572$	0	0
$573$	9.73787e6	1.23902
$574$	0	0
$575$	1.54195e6	0.194492
$576$	0	0
$577$	−2.81857e6	−0.352444	−0.176222	−	0.984350i	$-0.556388\pi$
$577$	−2.81857e6	−0.352444	−0.176222	+	0.984350i	$0.556388\pi$
$578$	0	0
$579$	1.33690e7	1.65730
$580$	0	0
$581$	−1.09939e6	−0.135117
$582$	0	0
$583$	−247395.	−0.0301454
$584$	0	0
$585$	2.89328e6	0.349543
$586$	0	0
$587$	9.61995e6	1.15233	0.576166	−	0.817333i	$-0.304548\pi$
$587$	9.61995e6	1.15233	0.576166	+	0.817333i	$0.304548\pi$
$588$	0	0
$589$	2.19478e6	0.260677
$590$	0	0
$591$	−1.08181e7	−1.27404
$592$	0	0
$593$	7.39724e6	0.863839	0.431920	−	0.901912i	$-0.357836\pi$
$593$	7.39724e6	0.863839	0.431920	+	0.901912i	$0.357836\pi$
$594$	0	0
$595$	−845811.	−0.0979447
$596$	0	0
$597$	−8.16975e6	−0.938152
$598$	0	0
$599$	−1.14938e7	−1.30887	−0.654434	−	0.756119i	$-0.727093\pi$
$599$	−1.14938e7	−1.30887	−0.654434	+	0.756119i	$0.727093\pi$
$600$	0	0
$601$	−7.58526e6	−0.856612	−0.428306	−	0.903634i	$-0.640889\pi$
$601$	−7.58526e6	−0.856612	−0.428306	+	0.903634i	$0.640889\pi$
$602$	0	0
$603$	−4.77908e6	−0.535243
$604$	0	0
$605$	4.02493e6	0.447064
$606$	0	0
$607$	−2.36915e6	−0.260988	−0.130494	−	0.991449i	$-0.541656\pi$
$607$	−2.36915e6	−0.260988	−0.130494	+	0.991449i	$0.541656\pi$
$608$	0	0
$609$	−939332.	−0.102630
$610$	0	0
$611$	−1.27314e7	−1.37966
$612$	0	0
$613$	1.05992e7	1.13926	0.569629	−	0.821902i	$-0.307087\pi$
$613$	1.05992e7	1.13926	0.569629	+	0.821902i	$0.307087\pi$
$614$	0	0
$615$	−9.76037e6	−1.04059
$616$	0	0
$617$	−1.22009e7	−1.29026	−0.645131	−	0.764072i	$-0.723197\pi$
$617$	−1.22009e7	−1.29026	−0.645131	+	0.764072i	$0.723197\pi$
$618$	0	0
$619$	1.46019e7	1.53173	0.765865	−	0.643001i	$-0.222311\pi$
$619$	1.46019e7	1.53173	0.765865	+	0.643001i	$0.222311\pi$
$620$	0	0
$621$	2.87135e6	0.298784
$622$	0	0
$623$	−2.65414e6	−0.273970
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	−47022.8	−0.00477683
$628$	0	0
$629$	5.85247e6	0.589811
$630$	0	0
$631$	−9.40997e6	−0.940838	−0.470419	−	0.882443i	$-0.655897\pi$
$631$	−9.40997e6	−0.940838	−0.470419	+	0.882443i	$0.655897\pi$
$632$	0	0
$633$	1.55507e7	1.54256
$634$	0	0
$635$	−3.98998e6	−0.392678
$636$	0	0
$637$	9.63876e6	0.941180
$638$	0	0
$639$	−1.05079e7	−1.01804
$640$	0	0
$641$	−8.44166e6	−0.811489	−0.405744	−	0.913987i	$-0.632988\pi$
$641$	−8.44166e6	−0.811489	−0.405744	+	0.913987i	$0.632988\pi$
$642$	0	0
$643$	−1.03157e7	−0.983942	−0.491971	−	0.870612i	$-0.663723\pi$
$643$	−1.03157e7	−0.983942	−0.491971	+	0.870612i	$0.663723\pi$
$644$	0	0
$645$	−1.63143e6	−0.154408
$646$	0	0
$647$	3.52118e6	0.330695	0.165348	−	0.986235i	$-0.447125\pi$
$647$	3.52118e6	0.330695	0.165348	+	0.986235i	$0.447125\pi$
$648$	0	0
$649$	−110459.	−0.0102941
$650$	0	0
$651$	−5.18805e6	−0.479791
$652$	0	0
$653$	2.20492e6	0.202354	0.101177	−	0.994868i	$-0.467739\pi$
$653$	2.20492e6	0.202354	0.101177	+	0.994868i	$0.467739\pi$
$654$	0	0
$655$	2.90281e6	0.264372
$656$	0	0
$657$	1.09611e7	0.990694
$658$	0	0
$659$	1.26903e7	1.13830	0.569152	−	0.822232i	$-0.307271\pi$
$659$	1.26903e7	1.13830	0.569152	+	0.822232i	$0.307271\pi$
$660$	0	0
$661$	−2.20351e6	−0.196160	−0.0980801	−	0.995179i	$-0.531270\pi$
$661$	−2.20351e6	−0.196160	−0.0980801	+	0.995179i	$0.531270\pi$
$662$	0	0
$663$	1.23184e7	1.08835
$664$	0	0
$665$	−272725.	−0.0239150
$666$	0	0
$667$	3.16940e6	0.275843
$668$	0	0
$669$	2.08013e7	1.79690
$670$	0	0
$671$	55298.8	0.00474143
$672$	0	0
$673$	2.02905e6	0.172685	0.0863424	−	0.996266i	$-0.472482\pi$
$673$	2.02905e6	0.172685	0.0863424	+	0.996266i	$0.472482\pi$
$674$	0	0
$675$	727404.	0.0614492
$676$	0	0
$677$	1.33080e7	1.11594	0.557971	−	0.829860i	$-0.311580\pi$
$677$	1.33080e7	1.11594	0.557971	+	0.829860i	$0.311580\pi$
$678$	0	0
$679$	−6.20903e6	−0.516832
$680$	0	0
$681$	7.98989e6	0.660196
$682$	0	0
$683$	5.95063e6	0.488102	0.244051	−	0.969762i	$-0.421524\pi$
$683$	5.95063e6	0.488102	0.244051	+	0.969762i	$0.421524\pi$
$684$	0	0
$685$	36153.2	0.00294388
$686$	0	0
$687$	−344204.	−0.0278243
$688$	0	0
$689$	2.08973e7	1.67703
$690$	0	0
$691$	−1.95555e7	−1.55802	−0.779010	−	0.627011i	$-0.784278\pi$
$691$	−1.95555e7	−1.55802	−0.779010	+	0.627011i	$0.784278\pi$
$692$	0	0
$693$	48319.0	0.00382195
$694$	0	0
$695$	−3.37639e6	−0.265149
$696$	0	0
$697$	−1.80646e7	−1.40846
$698$	0	0
$699$	−1.93537e7	−1.49820
$700$	0	0
$701$	1.70838e7	1.31308	0.656538	−	0.754293i	$-0.272020\pi$
$701$	1.70838e7	1.31308	0.656538	+	0.754293i	$0.272020\pi$
$702$	0	0
$703$	1.88708e6	0.144013
$704$	0	0
$705$	1.06552e7	0.807401
$706$	0	0
$707$	3.61323e6	0.271861
$708$	0	0
$709$	−2.47096e7	−1.84608	−0.923038	−	0.384710i	$-0.874301\pi$
$709$	−2.47096e7	−1.84608	−0.923038	+	0.384710i	$0.874301\pi$
$710$	0	0
$711$	6.02430e6	0.446923
$712$	0	0
$713$	1.75050e7	1.28955
$714$	0	0
$715$	113523.	0.00830459
$716$	0	0
$717$	−2.78074e7	−2.02005
$718$	0	0
$719$	2.02894e7	1.46369	0.731843	−	0.681474i	$-0.238661\pi$
$719$	2.02894e7	1.46369	0.731843	+	0.681474i	$0.238661\pi$
$720$	0	0
$721$	−5.30131e6	−0.379791
$722$	0	0
$723$	−8.08882e6	−0.575492
$724$	0	0
$725$	802910.	0.0567312
$726$	0	0
$727$	2.59786e7	1.82297	0.911486	−	0.411331i	$-0.134936\pi$
$727$	2.59786e7	1.82297	0.911486	+	0.411331i	$0.134936\pi$
$728$	0	0
$729$	−7.13072e6	−0.496952
$730$	0	0
$731$	−3.01946e6	−0.208995
$732$	0	0
$733$	−1.75705e7	−1.20788	−0.603940	−	0.797030i	$-0.706403\pi$
$733$	−1.75705e7	−1.20788	−0.603940	+	0.797030i	$0.706403\pi$
$734$	0	0
$735$	−8.06690e6	−0.550793
$736$	0	0
$737$	−187516.	−0.0127165
$738$	0	0
$739$	2.26188e7	1.52356	0.761779	−	0.647837i	$-0.224326\pi$
$739$	2.26188e7	1.52356	0.761779	+	0.647837i	$0.224326\pi$
$740$	0	0
$741$	3.97197e6	0.265742
$742$	0	0
$743$	1.75610e7	1.16701	0.583507	−	0.812108i	$-0.301680\pi$
$743$	1.75610e7	1.16701	0.583507	+	0.812108i	$0.301680\pi$
$744$	0	0
$745$	402445.	0.0265653
$746$	0	0
$747$	−5.82524e6	−0.381955
$748$	0	0
$749$	7.78892e6	0.507309
$750$	0	0
$751$	1.36472e6	0.0882967	0.0441483	−	0.999025i	$-0.485943\pi$
$751$	1.36472e6	0.0882967	0.0441483	+	0.999025i	$0.485943\pi$
$752$	0	0
$753$	1.67080e7	1.07383
$754$	0	0
$755$	1.16838e7	0.745963
$756$	0	0
$757$	2.89968e6	0.183912	0.0919560	−	0.995763i	$-0.470688\pi$
$757$	2.89968e6	0.183912	0.0919560	+	0.995763i	$0.470688\pi$
$758$	0	0
$759$	−375042.	−0.0236306
$760$	0	0
$761$	−9.33991e6	−0.584630	−0.292315	−	0.956322i	$-0.594426\pi$
$761$	−9.33991e6	−0.584630	−0.292315	+	0.956322i	$0.594426\pi$
$762$	0	0
$763$	868303.	0.0539958
$764$	0	0
$765$	−4.48164e6	−0.276874
$766$	0	0
$767$	9.33038e6	0.572679
$768$	0	0
$769$	−1.86030e7	−1.13440	−0.567200	−	0.823580i	$-0.691974\pi$
$769$	−1.86030e7	−1.13440	−0.567200	+	0.823580i	$0.691974\pi$
$770$	0	0
$771$	−3.18506e6	−0.192967
$772$	0	0
$773$	−2.71314e7	−1.63314	−0.816570	−	0.577247i	$-0.804127\pi$
$773$	−2.71314e7	−1.63314	−0.816570	+	0.577247i	$0.804127\pi$
$774$	0	0
$775$	4.43457e6	0.265215
$776$	0	0
$777$	−4.46071e6	−0.265065
$778$	0	0
$779$	−5.82478e6	−0.343903
$780$	0	0
$781$	−412295.	−0.0241869
$782$	0	0
$783$	1.49514e6	0.0871521
$784$	0	0
$785$	5.26195e6	0.304770
$786$	0	0
$787$	9.16693e6	0.527579	0.263789	−	0.964580i	$-0.415028\pi$
$787$	9.16693e6	0.527579	0.263789	+	0.964580i	$0.415028\pi$
$788$	0	0
$789$	580674.	0.0332078
$790$	0	0
$791$	−2.27713e6	−0.129404
$792$	0	0
$793$	−4.67104e6	−0.263773
$794$	0	0
$795$	−1.74894e7	−0.981426
$796$	0	0
$797$	1.90109e7	1.06012	0.530062	−	0.847959i	$-0.322169\pi$
$797$	1.90109e7	1.06012	0.530062	+	0.847959i	$0.322169\pi$
$798$	0	0
$799$	1.97207e7	1.09284
$800$	0	0
$801$	−1.40633e7	−0.774472
$802$	0	0
$803$	430077.	0.0235373
$804$	0	0
$805$	−2.17519e6	−0.118306
$806$	0	0
$807$	1.48368e7	0.801969
$808$	0	0
$809$	−4.04747e6	−0.217426	−0.108713	−	0.994073i	$-0.534673\pi$
$809$	−4.04747e6	−0.217426	−0.108713	+	0.994073i	$0.534673\pi$
$810$	0	0
$811$	−984765.	−0.0525752	−0.0262876	−	0.999654i	$-0.508369\pi$
$811$	−984765.	−0.0525752	−0.0262876	+	0.999654i	$0.508369\pi$
$812$	0	0
$813$	2.81296e7	1.49258
$814$	0	0
$815$	−3.40575e6	−0.179605
$816$	0	0
$817$	−973603.	−0.0510301
$818$	0	0
$819$	−4.08147e6	−0.212621
$820$	0	0
$821$	−3.53595e6	−0.183083	−0.0915415	−	0.995801i	$-0.529179\pi$
$821$	−3.53595e6	−0.183083	−0.0915415	+	0.995801i	$0.529179\pi$
$822$	0	0
$823$	9.27174e6	0.477157	0.238579	−	0.971123i	$-0.423319\pi$
$823$	9.27174e6	0.477157	0.238579	+	0.971123i	$0.423319\pi$
$824$	0	0
$825$	−95010.0	−0.00485998
$826$	0	0
$827$	−3.88822e6	−0.197691	−0.0988456	−	0.995103i	$-0.531515\pi$
$827$	−3.88822e6	−0.197691	−0.0988456	+	0.995103i	$0.531515\pi$
$828$	0	0
$829$	−2.20046e7	−1.11206	−0.556028	−	0.831163i	$-0.687675\pi$
$829$	−2.20046e7	−1.11206	−0.556028	+	0.831163i	$0.687675\pi$
$830$	0	0
$831$	−1.64007e7	−0.823874
$832$	0	0
$833$	−1.49303e7	−0.745513
$834$	0	0
$835$	−6.16781e6	−0.306136
$836$	0	0
$837$	8.25785e6	0.407430
$838$	0	0
$839$	888779.	0.0435902	0.0217951	−	0.999762i	$-0.493062\pi$
$839$	888779.	0.0435902	0.0217951	+	0.999762i	$0.493062\pi$
$840$	0	0
$841$	−1.88608e7	−0.919539
$842$	0	0
$843$	3.96286e6	0.192061
$844$	0	0
$845$	−306855.	−0.0147840
$846$	0	0
$847$	−5.67786e6	−0.271942
$848$	0	0
$849$	−2.42420e7	−1.15425
$850$	0	0
$851$	1.50509e7	0.712424
$852$	0	0
$853$	−2.03265e7	−0.956511	−0.478255	−	0.878221i	$-0.658731\pi$
$853$	−2.03265e7	−0.956511	−0.478255	+	0.878221i	$0.658731\pi$
$854$	0	0
$855$	−1.44507e6	−0.0676041
$856$	0	0
$857$	1.96142e6	0.0912258	0.0456129	−	0.998959i	$-0.485476\pi$
$857$	1.96142e6	0.0912258	0.0456129	+	0.998959i	$0.485476\pi$
$858$	0	0
$859$	−1.54216e7	−0.713095	−0.356548	−	0.934277i	$-0.616046\pi$
$859$	−1.54216e7	−0.713095	−0.356548	+	0.934277i	$0.616046\pi$
$860$	0	0
$861$	1.37687e7	0.632972
$862$	0	0
$863$	−2.79115e7	−1.27572	−0.637860	−	0.770152i	$-0.720180\pi$
$863$	−2.79115e7	−1.27572	−0.637860	+	0.770152i	$0.720180\pi$
$864$	0	0
$865$	−1.63045e7	−0.740913
$866$	0	0
$867$	1.03572e7	0.467945
$868$	0	0
$869$	236374.	0.0106182
$870$	0	0
$871$	1.58393e7	0.707440
$872$	0	0
$873$	−3.28993e7	−1.46100
$874$	0	0
$875$	−551044.	−0.0243313
$876$	0	0
$877$	−1.72191e7	−0.755980	−0.377990	−	0.925810i	$-0.623385\pi$
$877$	−1.72191e7	−0.755980	−0.377990	+	0.925810i	$0.623385\pi$
$878$	0	0
$879$	−1.93402e7	−0.844282
$880$	0	0
$881$	3.57166e7	1.55035	0.775177	−	0.631745i	$-0.217661\pi$
$881$	3.57166e7	1.55035	0.775177	+	0.631745i	$0.217661\pi$
$882$	0	0
$883$	−3.78278e6	−0.163271	−0.0816356	−	0.996662i	$-0.526014\pi$
$883$	−3.78278e6	−0.163271	−0.0816356	+	0.996662i	$0.526014\pi$
$884$	0	0
$885$	−7.80882e6	−0.335141
$886$	0	0
$887$	−2.45768e7	−1.04886	−0.524428	−	0.851455i	$-0.675721\pi$
$887$	−2.45768e7	−1.04886	−0.524428	+	0.851455i	$0.675721\pi$
$888$	0	0
$889$	5.62855e6	0.238859
$890$	0	0
$891$	−509857.	−0.0215156
$892$	0	0
$893$	6.35880e6	0.266837
$894$	0	0
$895$	1.96636e7	0.820549
$896$	0	0
$897$	3.16794e7	1.31461
$898$	0	0
$899$	9.11504e6	0.376149
$900$	0	0
$901$	−3.23695e7	−1.32839
$902$	0	0
$903$	2.30142e6	0.0939238
$904$	0	0
$905$	−24350.7	−0.000988301 0
$906$	0	0
$907$	1.47776e6	0.0596467	0.0298234	−	0.999555i	$-0.490506\pi$
$907$	1.47776e6	0.0596467	0.0298234	+	0.999555i	$0.490506\pi$
$908$	0	0
$909$	1.91452e7	0.768510
$910$	0	0
$911$	−2.62169e7	−1.04661	−0.523305	−	0.852146i	$-0.675301\pi$
$911$	−2.62169e7	−1.04661	−0.523305	+	0.852146i	$0.675301\pi$
$912$	0	0
$913$	−228564.	−0.00907466
$914$	0	0
$915$	3.90930e6	0.154364
$916$	0	0
$917$	−4.09491e6	−0.160813
$918$	0	0
$919$	1.10734e6	0.0432507	0.0216254	−	0.999766i	$-0.493116\pi$
$919$	1.10734e6	0.0432507	0.0216254	+	0.999766i	$0.493116\pi$
$920$	0	0
$921$	−1.40724e7	−0.546664
$922$	0	0
$923$	3.48262e7	1.34556
$924$	0	0
$925$	3.81287e6	0.146520
$926$	0	0
$927$	−2.80896e7	−1.07361
$928$	0	0
$929$	2.36543e7	0.899229	0.449615	−	0.893223i	$-0.351561\pi$
$929$	2.36543e7	0.899229	0.449615	+	0.893223i	$0.351561\pi$
$930$	0	0
$931$	−4.81415e6	−0.182031
$932$	0	0
$933$	−2.60502e7	−0.979732
$934$	0	0
$935$	−175845.	−0.00657811
$936$	0	0
$937$	−3.28168e7	−1.22109	−0.610544	−	0.791983i	$-0.709049\pi$
$937$	−3.28168e7	−1.22109	−0.610544	+	0.791983i	$0.709049\pi$
$938$	0	0
$939$	−6.50338e7	−2.40699
$940$	0	0
$941$	−3.66896e7	−1.35073	−0.675366	−	0.737483i	$-0.736014\pi$
$941$	−3.66896e7	−1.35073	−0.675366	+	0.737483i	$0.736014\pi$
$942$	0	0
$943$	−4.64569e7	−1.70126
$944$	0	0
$945$	−1.02613e6	−0.0373785
$946$	0	0
$947$	−8.63334e6	−0.312827	−0.156413	−	0.987692i	$-0.549993\pi$
$947$	−8.63334e6	−0.312827	−0.156413	+	0.987692i	$0.549993\pi$
$948$	0	0
$949$	−3.63282e7	−1.30942
$950$	0	0
$951$	7.32660e6	0.262695
$952$	0	0
$953$	−1.76257e7	−0.628659	−0.314329	−	0.949314i	$-0.601780\pi$
$953$	−1.76257e7	−0.628659	−0.314329	+	0.949314i	$0.601780\pi$
$954$	0	0
$955$	1.17419e7	0.416609
$956$	0	0
$957$	−195288.	−0.00689281
$958$	0	0
$959$	−51000.3	−0.00179071
$960$	0	0
$961$	2.17144e7	0.758470
$962$	0	0
$963$	4.12706e7	1.43408
$964$	0	0
$965$	1.61203e7	0.557255
$966$	0	0
$967$	−4.87856e7	−1.67774	−0.838871	−	0.544330i	$-0.816784\pi$
$967$	−4.87856e7	−1.67774	−0.838871	+	0.544330i	$0.816784\pi$
$968$	0	0
$969$	−6.15252e6	−0.210496
$970$	0	0
$971$	3.81830e7	1.29964	0.649818	−	0.760090i	$-0.274845\pi$
$971$	3.81830e7	1.29964	0.649818	+	0.760090i	$0.274845\pi$
$972$	0	0
$973$	4.76297e6	0.161286
$974$	0	0
$975$	8.02541e6	0.270368
$976$	0	0
$977$	3.64067e7	1.22024	0.610120	−	0.792309i	$-0.291121\pi$
$977$	3.64067e7	1.22024	0.610120	+	0.792309i	$0.291121\pi$
$978$	0	0
$979$	−551799.	−0.0184003
$980$	0	0
$981$	4.60081e6	0.152638
$982$	0	0
$983$	−4.61248e7	−1.52248	−0.761239	−	0.648472i	$-0.775408\pi$
$983$	−4.61248e7	−1.52248	−0.761239	+	0.648472i	$0.775408\pi$
$984$	0	0
$985$	−1.30444e7	−0.428386
$986$	0	0
$987$	−1.50310e7	−0.491129
$988$	0	0
$989$	−7.76520e6	−0.252442
$990$	0	0
$991$	2.77541e7	0.897725	0.448862	−	0.893601i	$-0.351829\pi$
$991$	2.77541e7	0.897725	0.448862	+	0.893601i	$0.351829\pi$
$992$	0	0
$993$	−6.04499e7	−1.94546
$994$	0	0
$995$	−9.85105e6	−0.315446
$996$	0	0
$997$	−2.50394e7	−0.797784	−0.398892	−	0.916998i	$-0.630605\pi$
$997$	−2.50394e7	−0.797784	−0.398892	+	0.916998i	$0.630605\pi$
$998$	0	0
$999$	7.10015e6	0.225089

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.6.a.r.1.1		2
4.3	odd	2		320.6.a.v.1.2		2
8.3	odd	2		160.6.a.a.1.1	✓	2
8.5	even	2		160.6.a.e.1.2	yes	2
40.3	even	4		800.6.c.f.449.1		4
40.13	odd	4		800.6.c.g.449.4		4
40.19	odd	2		800.6.a.l.1.2		2
40.27	even	4		800.6.c.f.449.4		4
40.29	even	2		800.6.a.g.1.1		2
40.37	odd	4		800.6.c.g.449.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.6.a.a.1.1	✓	2	8.3	odd	2
160.6.a.e.1.2	yes	2	8.5	even	2
320.6.a.r.1.1		2	1.1	even	1	trivial
320.6.a.v.1.2		2	4.3	odd	2
800.6.a.g.1.1		2	40.29	even	2
800.6.a.l.1.2		2	40.19	odd	2
800.6.c.f.449.1		4	40.3	even	4
800.6.c.f.449.4		4	40.27	even	4
800.6.c.g.449.1		4	40.37	odd	4
800.6.c.g.449.4		4	40.13	odd	4

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:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	320.a (trivial)

\(f(q)\)	\(=\)	\(q-20.7332 q^{3} -25.0000 q^{5} +35.2668 q^{7} +186.866 q^{9} +O(q^{10})\)	\(q-20.7332 q^{3} -25.0000 q^{5} +35.2668 q^{7} +186.866 q^{9} +7.33201 q^{11} -619.328 q^{13} +518.330 q^{15} +959.328 q^{17} +309.328 q^{19} -731.194 q^{21} +2467.12 q^{23} +625.000 q^{25} +1163.85 q^{27} +1284.66 q^{29} +7095.32 q^{31} -152.016 q^{33} -881.670 q^{35} +6100.59 q^{37} +12840.7 q^{39} -18830.4 q^{41} -3147.48 q^{43} -4671.64 q^{45} +20556.8 q^{47} -15563.3 q^{49} -19889.9 q^{51} -33741.9 q^{53} -183.300 q^{55} -6413.36 q^{57} -15065.3 q^{59} +7542.11 q^{61} +6590.15 q^{63} +15483.2 q^{65} -25574.9 q^{67} -51151.3 q^{69} -56232.3 q^{71} +58657.5 q^{73} -12958.3 q^{75} +258.576 q^{77} +32238.7 q^{79} -69538.6 q^{81} -31173.4 q^{83} -23983.2 q^{85} -26635.0 q^{87} -75258.9 q^{89} -21841.7 q^{91} -147109. q^{93} -7733.20 q^{95} -176059. q^{97} +1370.10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{3} - 50 q^{5} + 104 q^{7} + 106 q^{9}+O(q^{10}) \) 2 * q - 8 * q^3 - 50 * q^5 + 104 * q^7 + 106 * q^9	\( 2 q - 8 q^{3} - 50 q^{5} + 104 q^{7} + 106 q^{9} - 320 q^{11} + 100 q^{13} + 200 q^{15} + 580 q^{17} - 720 q^{19} + 144 q^{21} + 1688 q^{23} + 1250 q^{25} - 2960 q^{27} - 108 q^{29} + 9840 q^{31} - 4320 q^{33} - 2600 q^{35} - 6540 q^{37} + 22000 q^{39} - 10620 q^{41} - 25672 q^{43} - 2650 q^{45} + 28296 q^{47} - 27646 q^{49} - 24720 q^{51} - 31340 q^{53} + 8000 q^{55} - 19520 q^{57} - 30800 q^{59} - 24540 q^{61} + 1032 q^{63} - 2500 q^{65} - 34584 q^{67} - 61072 q^{69} - 12400 q^{71} - 7180 q^{73} - 5000 q^{75} - 22240 q^{77} + 71840 q^{79} - 102398 q^{81} + 31928 q^{83} - 14500 q^{85} - 44368 q^{87} - 40748 q^{89} + 27600 q^{91} - 112160 q^{93} + 18000 q^{95} - 190140 q^{97} + 27840 q^{99}+O(q^{100}) \) 2 * q - 8 * q^3 - 50 * q^5 + 104 * q^7 + 106 * q^9 - 320 * q^11 + 100 * q^13 + 200 * q^15 + 580 * q^17 - 720 * q^19 + 144 * q^21 + 1688 * q^23 + 1250 * q^25 - 2960 * q^27 - 108 * q^29 + 9840 * q^31 - 4320 * q^33 - 2600 * q^35 - 6540 * q^37 + 22000 * q^39 - 10620 * q^41 - 25672 * q^43 - 2650 * q^45 + 28296 * q^47 - 27646 * q^49 - 24720 * q^51 - 31340 * q^53 + 8000 * q^55 - 19520 * q^57 - 30800 * q^59 - 24540 * q^61 + 1032 * q^63 - 2500 * q^65 - 34584 * q^67 - 61072 * q^69 - 12400 * q^71 - 7180 * q^73 - 5000 * q^75 - 22240 * q^77 + 71840 * q^79 - 102398 * q^81 + 31928 * q^83 - 14500 * q^85 - 44368 * q^87 - 40748 * q^89 + 27600 * q^91 - 112160 * q^93 + 18000 * q^95 - 190140 * q^97 + 27840 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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