LMFDB - Embedded newform 400.6.a.o.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,6,Mod(1,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, 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("400.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	400.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:	$64.1535279252$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{241}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 60 $ x^2 - x - 60
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 25)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-7.26209$ of defining polynomial
Character	$\chi$	$=$	400.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.52417 q^{3} -68.9517 q^{7} -212.483 q^{9} +O(q^{10})$	$q+5.52417 q^{3} -68.9517 q^{7} -212.483 q^{9} +486.104 q^{11} -428.387 q^{13} +1800.64 q^{17} +1046.65 q^{19} -380.901 q^{21} -686.855 q^{23} -2516.17 q^{27} -1339.03 q^{29} -7990.30 q^{31} +2685.33 q^{33} -1970.64 q^{37} -2366.48 q^{39} +10772.2 q^{41} -15017.7 q^{43} -895.337 q^{47} -12052.7 q^{49} +9947.07 q^{51} -19327.1 q^{53} +5781.90 q^{57} -21193.7 q^{59} -27722.2 q^{61} +14651.1 q^{63} +7719.33 q^{67} -3794.31 q^{69} +51410.1 q^{71} -43776.4 q^{73} -33517.7 q^{77} +6225.68 q^{79} +37733.7 q^{81} -52949.9 q^{83} -7397.05 q^{87} +44631.2 q^{89} +29538.0 q^{91} -44139.8 q^{93} -148018. q^{97} -103289. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 20 q^{3} - 200 q^{7} + 196 q^{9}+O(q^{10}) $ 2 * q - 20 * q^3 - 200 * q^7 + 196 * q^9	$ 2 q - 20 q^{3} - 200 q^{7} + 196 q^{9} + 196 q^{11} - 360 q^{13} + 1490 q^{17} + 3180 q^{19} + 2964 q^{21} - 1560 q^{23} - 6740 q^{27} - 3920 q^{29} + 1096 q^{31} + 10090 q^{33} + 2020 q^{37} - 4112 q^{39} + 27754 q^{41} + 3000 q^{43} - 25760 q^{47} - 11686 q^{49} + 17876 q^{51} - 26980 q^{53} - 48670 q^{57} - 11960 q^{59} - 24396 q^{61} - 38880 q^{63} + 40060 q^{67} + 18492 q^{69} + 87296 q^{71} - 70290 q^{73} + 4500 q^{77} - 65480 q^{79} + 46282 q^{81} - 92580 q^{83} + 58480 q^{87} - 72810 q^{89} + 20576 q^{91} - 276060 q^{93} - 126140 q^{97} - 221792 q^{99}+O(q^{100}) $ 2 * q - 20 * q^3 - 200 * q^7 + 196 * q^9 + 196 * q^11 - 360 * q^13 + 1490 * q^17 + 3180 * q^19 + 2964 * q^21 - 1560 * q^23 - 6740 * q^27 - 3920 * q^29 + 1096 * q^31 + 10090 * q^33 + 2020 * q^37 - 4112 * q^39 + 27754 * q^41 + 3000 * q^43 - 25760 * q^47 - 11686 * q^49 + 17876 * q^51 - 26980 * q^53 - 48670 * q^57 - 11960 * q^59 - 24396 * q^61 - 38880 * q^63 + 40060 * q^67 + 18492 * q^69 + 87296 * q^71 - 70290 * q^73 + 4500 * q^77 - 65480 * q^79 + 46282 * q^81 - 92580 * q^83 + 58480 * q^87 - 72810 * q^89 + 20576 * q^91 - 276060 * q^93 - 126140 * q^97 - 221792 * 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
$3$	5.52417	0.354376	0.177188	−	0.984177i	$-0.443300\pi$
$3$	5.52417	0.354376	0.177188	+	0.984177i	$0.443300\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−68.9517	−0.531863	−0.265931	−	0.963992i	$-0.585679\pi$
$7$	−68.9517	−0.531863	−0.265931	+	0.963992i	$0.585679\pi$
$8$	0	0
$9$	−212.483	−0.874418
$10$	0	0
$11$	486.104	1.21129	0.605645	−	0.795735i	$-0.292915\pi$
$11$	486.104	1.21129	0.605645	+	0.795735i	$0.292915\pi$
$12$	0	0
$13$	−428.387	−0.703036	−0.351518	−	0.936181i	$-0.614334\pi$
$13$	−428.387	−0.703036	−0.351518	+	0.936181i	$0.614334\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1800.64	1.51114	0.755571	−	0.655066i	$-0.227359\pi$
$17$	1800.64	1.51114	0.755571	+	0.655066i	$0.227359\pi$
$18$	0	0
$19$	1046.65	0.665149	0.332575	−	0.943077i	$-0.392083\pi$
$19$	1046.65	0.665149	0.332575	+	0.943077i	$0.392083\pi$
$20$	0	0
$21$	−380.901	−0.188479
$22$	0	0
$23$	−686.855	−0.270736	−0.135368	−	0.990795i	$-0.543222\pi$
$23$	−686.855	−0.270736	−0.135368	+	0.990795i	$0.543222\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−2516.17	−0.664249
$28$	0	0
$29$	−1339.03	−0.295663	−0.147831	−	0.989013i	$-0.547229\pi$
$29$	−1339.03	−0.295663	−0.147831	+	0.989013i	$0.547229\pi$
$30$	0	0
$31$	−7990.30	−1.49334	−0.746670	−	0.665195i	$-0.768349\pi$
$31$	−7990.30	−1.49334	−0.746670	+	0.665195i	$0.768349\pi$
$32$	0	0
$33$	2685.33	0.429252
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−1970.64	−0.236648	−0.118324	−	0.992975i	$-0.537752\pi$
$37$	−1970.64	−0.236648	−0.118324	+	0.992975i	$0.537752\pi$
$38$	0	0
$39$	−2366.48	−0.249139
$40$	0	0
$41$	10772.2	1.00079	0.500395	−	0.865797i	$-0.333188\pi$
$41$	10772.2	1.00079	0.500395	+	0.865797i	$0.333188\pi$
$42$	0	0
$43$	−15017.7	−1.23861	−0.619303	−	0.785152i	$-0.712585\pi$
$43$	−15017.7	−1.23861	−0.619303	+	0.785152i	$0.712585\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−895.337	−0.0591210	−0.0295605	−	0.999563i	$-0.509411\pi$
$47$	−895.337	−0.0591210	−0.0295605	+	0.999563i	$0.509411\pi$
$48$	0	0
$49$	−12052.7	−0.717122
$50$	0	0
$51$	9947.07	0.535513
$52$	0	0
$53$	−19327.1	−0.945098	−0.472549	−	0.881304i	$-0.656666\pi$
$53$	−19327.1	−0.945098	−0.472549	+	0.881304i	$0.656666\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	5781.90	0.235713
$58$	0	0
$59$	−21193.7	−0.792641	−0.396321	−	0.918112i	$-0.629713\pi$
$59$	−21193.7	−0.792641	−0.396321	+	0.918112i	$0.629713\pi$
$60$	0	0
$61$	−27722.2	−0.953900	−0.476950	−	0.878931i	$-0.658258\pi$
$61$	−27722.2	−0.953900	−0.476950	+	0.878931i	$0.658258\pi$
$62$	0	0
$63$	14651.1	0.465070
$64$	0	0
$65$	0	0
$66$	0	0
$67$	7719.33	0.210084	0.105042	−	0.994468i	$-0.466502\pi$
$67$	7719.33	0.210084	0.105042	+	0.994468i	$0.466502\pi$
$68$	0	0
$69$	−3794.31	−0.0959422
$70$	0	0
$71$	51410.1	1.21033	0.605163	−	0.796101i	$-0.293108\pi$
$71$	51410.1	1.21033	0.605163	+	0.796101i	$0.293108\pi$
$72$	0	0
$73$	−43776.4	−0.961465	−0.480732	−	0.876867i	$-0.659629\pi$
$73$	−43776.4	−0.961465	−0.480732	+	0.876867i	$0.659629\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−33517.7	−0.644240
$78$	0	0
$79$	6225.68	0.112233	0.0561163	−	0.998424i	$-0.482128\pi$
$79$	6225.68	0.112233	0.0561163	+	0.998424i	$0.482128\pi$
$80$	0	0
$81$	37733.7	0.639024
$82$	0	0
$83$	−52949.9	−0.843664	−0.421832	−	0.906674i	$-0.638613\pi$
$83$	−52949.9	−0.843664	−0.421832	+	0.906674i	$0.638613\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−7397.05	−0.104776
$88$	0	0
$89$	44631.2	0.597260	0.298630	−	0.954369i	$-0.403470\pi$
$89$	44631.2	0.597260	0.298630	+	0.954369i	$0.403470\pi$
$90$	0	0
$91$	29538.0	0.373919
$92$	0	0
$93$	−44139.8	−0.529204
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−148018.	−1.59730	−0.798649	−	0.601797i	$-0.794452\pi$
$97$	−148018.	−1.59730	−0.798649	+	0.601797i	$0.794452\pi$
$98$	0	0
$99$	−103289.	−1.05917
$100$	0	0
$101$	148476.	1.44828	0.724141	−	0.689652i	$-0.242237\pi$
$101$	148476.	1.44828	0.724141	+	0.689652i	$0.242237\pi$
$102$	0	0
$103$	−188391.	−1.74972	−0.874859	−	0.484378i	$-0.839046\pi$
$103$	−188391.	−1.74972	−0.874859	+	0.484378i	$0.839046\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−67887.7	−0.573234	−0.286617	−	0.958045i	$-0.592531\pi$
$107$	−67887.7	−0.573234	−0.286617	+	0.958045i	$0.592531\pi$
$108$	0	0
$109$	−219292.	−1.76790	−0.883949	−	0.467582i	$-0.845125\pi$
$109$	−219292.	−1.76790	−0.883949	+	0.467582i	$0.845125\pi$
$110$	0	0
$111$	−10886.2	−0.0838625
$112$	0	0
$113$	−80783.9	−0.595153	−0.297577	−	0.954698i	$-0.596178\pi$
$113$	−80783.9	−0.595153	−0.297577	+	0.954698i	$0.596178\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	91025.1	0.614747
$118$	0	0
$119$	−124157.	−0.803721
$120$	0	0
$121$	75246.5	0.467221
$122$	0	0
$123$	59507.3	0.354656
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−161301.	−0.887417	−0.443708	−	0.896171i	$-0.646337\pi$
$127$	−161301.	−0.887417	−0.443708	+	0.896171i	$0.646337\pi$
$128$	0	0
$129$	−82960.5	−0.438932
$130$	0	0
$131$	193006.	0.982636	0.491318	−	0.870980i	$-0.336515\pi$
$131$	193006.	0.982636	0.491318	+	0.870980i	$0.336515\pi$
$132$	0	0
$133$	−72168.5	−0.353768
$134$	0	0
$135$	0	0
$136$	0	0
$137$	250340.	1.13954	0.569768	−	0.821806i	$-0.307033\pi$
$137$	250340.	1.13954	0.569768	+	0.821806i	$0.307033\pi$
$138$	0	0
$139$	218650.	0.959871	0.479935	−	0.877304i	$-0.340660\pi$
$139$	218650.	0.959871	0.479935	+	0.877304i	$0.340660\pi$
$140$	0	0
$141$	−4946.00	−0.0209511
$142$	0	0
$143$	−208241.	−0.851580
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−66581.1	−0.254131
$148$	0	0
$149$	−38740.0	−0.142953	−0.0714766	−	0.997442i	$-0.522771\pi$
$149$	−38740.0	−0.142953	−0.0714766	+	0.997442i	$0.522771\pi$
$150$	0	0
$151$	154945.	0.553013	0.276507	−	0.961012i	$-0.410823\pi$
$151$	154945.	0.553013	0.276507	+	0.961012i	$0.410823\pi$
$152$	0	0
$153$	−382607.	−1.32137
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−344442.	−1.11523	−0.557617	−	0.830098i	$-0.688284\pi$
$157$	−344442.	−1.11523	−0.557617	+	0.830098i	$0.688284\pi$
$158$	0	0
$159$	−106766.	−0.334920
$160$	0	0
$161$	47359.8	0.143994
$162$	0	0
$163$	−366203.	−1.07957	−0.539787	−	0.841801i	$-0.681495\pi$
$163$	−366203.	−1.07957	−0.539787	+	0.841801i	$0.681495\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−249272.	−0.691644	−0.345822	−	0.938300i	$-0.612400\pi$
$167$	−249272.	−0.691644	−0.345822	+	0.938300i	$0.612400\pi$
$168$	0	0
$169$	−187778.	−0.505740
$170$	0	0
$171$	−222397.	−0.581618
$172$	0	0
$173$	−61460.1	−0.156127	−0.0780635	−	0.996948i	$-0.524874\pi$
$173$	−61460.1	−0.156127	−0.0780635	+	0.996948i	$0.524874\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−117078.	−0.280893
$178$	0	0
$179$	−606803.	−1.41552	−0.707759	−	0.706454i	$-0.750294\pi$
$179$	−606803.	−1.41552	−0.707759	+	0.706454i	$0.750294\pi$
$180$	0	0
$181$	153684.	0.348685	0.174343	−	0.984685i	$-0.444220\pi$
$181$	153684.	0.348685	0.174343	+	0.984685i	$0.444220\pi$
$182$	0	0
$183$	−153142.	−0.338039
$184$	0	0
$185$	0	0
$186$	0	0
$187$	875301.	1.83043
$188$	0	0
$189$	173494.	0.353289
$190$	0	0
$191$	−182315.	−0.361608	−0.180804	−	0.983519i	$-0.557870\pi$
$191$	−182315.	−0.361608	−0.180804	+	0.983519i	$0.557870\pi$
$192$	0	0
$193$	−102080.	−0.197265	−0.0986323	−	0.995124i	$-0.531447\pi$
$193$	−102080.	−0.197265	−0.0986323	+	0.995124i	$0.531447\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−404656.	−0.742882	−0.371441	−	0.928456i	$-0.621136\pi$
$197$	−404656.	−0.742882	−0.371441	+	0.928456i	$0.621136\pi$
$198$	0	0
$199$	167297.	0.299472	0.149736	−	0.988726i	$-0.452158\pi$
$199$	167297.	0.299472	0.149736	+	0.988726i	$0.452158\pi$
$200$	0	0
$201$	42642.9	0.0744487
$202$	0	0
$203$	92328.5	0.157252
$204$	0	0
$205$	0	0
$206$	0	0
$207$	145945.	0.236736
$208$	0	0
$209$	508783.	0.805688
$210$	0	0
$211$	460778.	0.712502	0.356251	−	0.934390i	$-0.384055\pi$
$211$	460778.	0.712502	0.356251	+	0.934390i	$0.384055\pi$
$212$	0	0
$213$	283998.	0.428911
$214$	0	0
$215$	0	0
$216$	0	0
$217$	550944.	0.794252
$218$	0	0
$219$	−241829.	−0.340720
$220$	0	0
$221$	−771372.	−1.06239
$222$	0	0
$223$	1.08298e6	1.45834	0.729172	−	0.684330i	$-0.239905\pi$
$223$	1.08298e6	1.45834	0.729172	+	0.684330i	$0.239905\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−412201.	−0.530938	−0.265469	−	0.964119i	$-0.585527\pi$
$227$	−412201.	−0.530938	−0.265469	+	0.964119i	$0.585527\pi$
$228$	0	0
$229$	−433163.	−0.545836	−0.272918	−	0.962037i	$-0.587989\pi$
$229$	−433163.	−0.545836	−0.272918	+	0.962037i	$0.587989\pi$
$230$	0	0
$231$	−185158.	−0.228303
$232$	0	0
$233$	760097.	0.917232	0.458616	−	0.888635i	$-0.348345\pi$
$233$	760097.	0.917232	0.458616	+	0.888635i	$0.348345\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	34391.7	0.0397725
$238$	0	0
$239$	988624.	1.11953	0.559766	−	0.828651i	$-0.310891\pi$
$239$	988624.	1.11953	0.559766	+	0.828651i	$0.310891\pi$
$240$	0	0
$241$	−358878.	−0.398020	−0.199010	−	0.979997i	$-0.563773\pi$
$241$	−358878.	−0.398020	−0.199010	+	0.979997i	$0.563773\pi$
$242$	0	0
$243$	819877.	0.890703
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−448373.	−0.467624
$248$	0	0
$249$	−292504.	−0.298974
$250$	0	0
$251$	851049.	0.852649	0.426324	−	0.904570i	$-0.359808\pi$
$251$	851049.	0.852649	0.426324	+	0.904570i	$0.359808\pi$
$252$	0	0
$253$	−333883.	−0.327939
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−76358.4	−0.0721147	−0.0360574	−	0.999350i	$-0.511480\pi$
$257$	−76358.4	−0.0721147	−0.0360574	+	0.999350i	$0.511480\pi$
$258$	0	0
$259$	135879.	0.125864
$260$	0	0
$261$	284522.	0.258533
$262$	0	0
$263$	−1.19420e6	−1.06460	−0.532301	−	0.846555i	$-0.678673\pi$
$263$	−1.19420e6	−1.06460	−0.532301	+	0.846555i	$0.678673\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	246551.	0.211655
$268$	0	0
$269$	1.02930e6	0.867286	0.433643	−	0.901085i	$-0.357228\pi$
$269$	1.02930e6	0.867286	0.433643	+	0.901085i	$0.357228\pi$
$270$	0	0
$271$	−2.12144e6	−1.75472	−0.877359	−	0.479834i	$-0.840697\pi$
$271$	−2.12144e6	−1.75472	−0.877359	+	0.479834i	$0.840697\pi$
$272$	0	0
$273$	163173.	0.132508
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1.85145e6	1.44982	0.724908	−	0.688845i	$-0.241882\pi$
$277$	1.85145e6	1.44982	0.724908	+	0.688845i	$0.241882\pi$
$278$	0	0
$279$	1.69781e6	1.30580
$280$	0	0
$281$	90653.2	0.0684884	0.0342442	−	0.999413i	$-0.489098\pi$
$281$	90653.2	0.0684884	0.0342442	+	0.999413i	$0.489098\pi$
$282$	0	0
$283$	929308.	0.689753	0.344877	−	0.938648i	$-0.387921\pi$
$283$	929308.	0.689753	0.344877	+	0.938648i	$0.387921\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−742759.	−0.532283
$288$	0	0
$289$	1.82246e6	1.28355
$290$	0	0
$291$	−817679.	−0.566044
$292$	0	0
$293$	2.72733e6	1.85596	0.927979	−	0.372632i	$-0.121545\pi$
$293$	2.72733e6	1.85596	0.927979	+	0.372632i	$0.121545\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−1.22312e6	−0.804597
$298$	0	0
$299$	294240.	0.190337
$300$	0	0
$301$	1.03550e6	0.658768
$302$	0	0
$303$	820208.	0.513236
$304$	0	0
$305$	0	0
$306$	0	0
$307$	2.29648e6	1.39064	0.695322	−	0.718698i	$-0.255262\pi$
$307$	2.29648e6	1.39064	0.695322	+	0.718698i	$0.255262\pi$
$308$	0	0
$309$	−1.04071e6	−0.620058
$310$	0	0
$311$	−984847.	−0.577388	−0.288694	−	0.957421i	$-0.593221\pi$
$311$	−984847.	−0.577388	−0.288694	+	0.957421i	$0.593221\pi$
$312$	0	0
$313$	2.06650e6	1.19227	0.596135	−	0.802884i	$-0.296702\pi$
$313$	2.06650e6	1.19227	0.596135	+	0.802884i	$0.296702\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.14349e6	0.639125	0.319563	−	0.947565i	$-0.396464\pi$
$317$	1.14349e6	0.639125	0.319563	+	0.947565i	$0.396464\pi$
$318$	0	0
$319$	−650910.	−0.358133
$320$	0	0
$321$	−375024.	−0.203140
$322$	0	0
$323$	1.88465e6	1.00514
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−1.21141e6	−0.626501
$328$	0	0
$329$	61735.0	0.0314443
$330$	0	0
$331$	−205230.	−0.102961	−0.0514804	−	0.998674i	$-0.516394\pi$
$331$	−205230.	−0.102961	−0.0514804	+	0.998674i	$0.516394\pi$
$332$	0	0
$333$	418729.	0.206929
$334$	0	0
$335$	0	0
$336$	0	0
$337$	488213.	0.234172	0.117086	−	0.993122i	$-0.462645\pi$
$337$	488213.	0.234172	0.117086	+	0.993122i	$0.462645\pi$
$338$	0	0
$339$	−446265.	−0.210908
$340$	0	0
$341$	−3.88412e6	−1.80887
$342$	0	0
$343$	1.98992e6	0.913273
$344$	0	0
$345$	0	0
$346$	0	0
$347$	3.82809e6	1.70670	0.853351	−	0.521336i	$-0.174566\pi$
$347$	3.82809e6	1.70670	0.853351	+	0.521336i	$0.174566\pi$
$348$	0	0
$349$	1.45476e6	0.639333	0.319667	−	0.947530i	$-0.396429\pi$
$349$	1.45476e6	0.639333	0.319667	+	0.947530i	$0.396429\pi$
$350$	0	0
$351$	1.07789e6	0.466991
$352$	0	0
$353$	−778492.	−0.332520	−0.166260	−	0.986082i	$-0.553169\pi$
$353$	−778492.	−0.332520	−0.166260	+	0.986082i	$0.553169\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−685867.	−0.284819
$358$	0	0
$359$	2.12510e6	0.870247	0.435124	−	0.900371i	$-0.356705\pi$
$359$	2.12510e6	0.870247	0.435124	+	0.900371i	$0.356705\pi$
$360$	0	0
$361$	−1.38061e6	−0.557577
$362$	0	0
$363$	415675.	0.165572
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−4.10801e6	−1.59208	−0.796042	−	0.605242i	$-0.793077\pi$
$367$	−4.10801e6	−1.59208	−0.796042	+	0.605242i	$0.793077\pi$
$368$	0	0
$369$	−2.28891e6	−0.875109
$370$	0	0
$371$	1.33263e6	0.502662
$372$	0	0
$373$	−4.54570e6	−1.69172	−0.845860	−	0.533405i	$-0.820912\pi$
$373$	−4.54570e6	−1.69172	−0.845860	+	0.533405i	$0.820912\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	573624.	0.207861
$378$	0	0
$379$	−1.40554e6	−0.502626	−0.251313	−	0.967906i	$-0.580862\pi$
$379$	−1.40554e6	−0.502626	−0.251313	+	0.967906i	$0.580862\pi$
$380$	0	0
$381$	−891054.	−0.314479
$382$	0	0
$383$	−4.64417e6	−1.61775	−0.808874	−	0.587982i	$-0.799922\pi$
$383$	−4.64417e6	−1.61775	−0.808874	+	0.587982i	$0.799922\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3.19102e6	1.08306
$388$	0	0
$389$	3.53606e6	1.18480	0.592400	−	0.805644i	$-0.298181\pi$
$389$	3.53606e6	1.18480	0.592400	+	0.805644i	$0.298181\pi$
$390$	0	0
$391$	−1.23678e6	−0.409120
$392$	0	0
$393$	1.06620e6	0.348223
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−2.95611e6	−0.941336	−0.470668	−	0.882310i	$-0.655987\pi$
$397$	−2.95611e6	−0.941336	−0.470668	+	0.882310i	$0.655987\pi$
$398$	0	0
$399$	−398671.	−0.125367
$400$	0	0
$401$	−799254.	−0.248213	−0.124106	−	0.992269i	$-0.539606\pi$
$401$	−799254.	−0.248213	−0.124106	+	0.992269i	$0.539606\pi$
$402$	0	0
$403$	3.42294e6	1.04987
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−957937.	−0.286649
$408$	0	0
$409$	898422.	0.265566	0.132783	−	0.991145i	$-0.457609\pi$
$409$	898422.	0.265566	0.132783	+	0.991145i	$0.457609\pi$
$410$	0	0
$411$	1.38292e6	0.403824
$412$	0	0
$413$	1.46134e6	0.421576
$414$	0	0
$415$	0	0
$416$	0	0
$417$	1.20786e6	0.340155
$418$	0	0
$419$	−2.31259e6	−0.643523	−0.321761	−	0.946821i	$-0.604275\pi$
$419$	−2.31259e6	−0.643523	−0.321761	+	0.946821i	$0.604275\pi$
$420$	0	0
$421$	4.43296e6	1.21896	0.609478	−	0.792803i	$-0.291379\pi$
$421$	4.43296e6	1.21896	0.609478	+	0.792803i	$0.291379\pi$
$422$	0	0
$423$	190244.	0.0516965
$424$	0	0
$425$	0	0
$426$	0	0
$427$	1.91149e6	0.507344
$428$	0	0
$429$	−1.15036e6	−0.301780
$430$	0	0
$431$	−5.97999e6	−1.55063	−0.775314	−	0.631576i	$-0.782408\pi$
$431$	−5.97999e6	−1.55063	−0.775314	+	0.631576i	$0.782408\pi$
$432$	0	0
$433$	2.06419e6	0.529089	0.264545	−	0.964373i	$-0.414778\pi$
$433$	2.06419e6	0.529089	0.264545	+	0.964373i	$0.414778\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−718899.	−0.180080
$438$	0	0
$439$	4.09148e6	1.01326	0.506628	−	0.862165i	$-0.330892\pi$
$439$	4.09148e6	1.01326	0.506628	+	0.862165i	$0.330892\pi$
$440$	0	0
$441$	2.56099e6	0.627064
$442$	0	0
$443$	−2.75822e6	−0.667759	−0.333879	−	0.942616i	$-0.608358\pi$
$443$	−2.75822e6	−0.667759	−0.333879	+	0.942616i	$0.608358\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−214006.	−0.0506592
$448$	0	0
$449$	−3.76648e6	−0.881698	−0.440849	−	0.897581i	$-0.645323\pi$
$449$	−3.76648e6	−0.881698	−0.440849	+	0.897581i	$0.645323\pi$
$450$	0	0
$451$	5.23640e6	1.21225
$452$	0	0
$453$	855944.	0.195975
$454$	0	0
$455$	0	0
$456$	0	0
$457$	480604.	0.107646	0.0538229	−	0.998550i	$-0.482859\pi$
$457$	480604.	0.107646	0.0538229	+	0.998550i	$0.482859\pi$
$458$	0	0
$459$	−4.53073e6	−1.00377
$460$	0	0
$461$	4.52514e6	0.991699	0.495849	−	0.868409i	$-0.334857\pi$
$461$	4.52514e6	0.991699	0.495849	+	0.868409i	$0.334857\pi$
$462$	0	0
$463$	−7.39975e6	−1.60422	−0.802111	−	0.597175i	$-0.796290\pi$
$463$	−7.39975e6	−1.60422	−0.802111	+	0.597175i	$0.796290\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−1.84711e6	−0.391923	−0.195962	−	0.980612i	$-0.562783\pi$
$467$	−1.84711e6	−0.391923	−0.195962	+	0.980612i	$0.562783\pi$
$468$	0	0
$469$	−532261.	−0.111736
$470$	0	0
$471$	−1.90276e6	−0.395212
$472$	0	0
$473$	−7.30018e6	−1.50031
$474$	0	0
$475$	0	0
$476$	0	0
$477$	4.10669e6	0.826410
$478$	0	0
$479$	3.05088e6	0.607555	0.303778	−	0.952743i	$-0.401752\pi$
$479$	3.05088e6	0.607555	0.303778	+	0.952743i	$0.401752\pi$
$480$	0	0
$481$	844197.	0.166372
$482$	0	0
$483$	261624.	0.0510281
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−7.28136e6	−1.39120	−0.695601	−	0.718429i	$-0.744862\pi$
$487$	−7.28136e6	−1.39120	−0.695601	+	0.718429i	$0.744862\pi$
$488$	0	0
$489$	−2.02297e6	−0.382575
$490$	0	0
$491$	6.60475e6	1.23638	0.618191	−	0.786028i	$-0.287866\pi$
$491$	6.60475e6	1.23638	0.618191	+	0.786028i	$0.287866\pi$
$492$	0	0
$493$	−2.41112e6	−0.446788
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−3.54481e6	−0.643727
$498$	0	0
$499$	−4.87006e6	−0.875555	−0.437777	−	0.899083i	$-0.644234\pi$
$499$	−4.87006e6	−0.875555	−0.437777	+	0.899083i	$0.644234\pi$
$500$	0	0
$501$	−1.37702e6	−0.245102
$502$	0	0
$503$	1.16752e6	0.205753	0.102876	−	0.994694i	$-0.467195\pi$
$503$	1.16752e6	0.205753	0.102876	+	0.994694i	$0.467195\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−1.03732e6	−0.179222
$508$	0	0
$509$	−7.41468e6	−1.26852	−0.634261	−	0.773119i	$-0.718695\pi$
$509$	−7.41468e6	−1.26852	−0.634261	+	0.773119i	$0.718695\pi$
$510$	0	0
$511$	3.01846e6	0.511367
$512$	0	0
$513$	−2.63356e6	−0.441824
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−435227.	−0.0716127
$518$	0	0
$519$	−339516.	−0.0553277
$520$	0	0
$521$	−811897.	−0.131041	−0.0655204	−	0.997851i	$-0.520871\pi$
$521$	−811897.	−0.131041	−0.0655204	+	0.997851i	$0.520871\pi$
$522$	0	0
$523$	−5.06828e6	−0.810226	−0.405113	−	0.914267i	$-0.632768\pi$
$523$	−5.06828e6	−0.810226	−0.405113	+	0.914267i	$0.632768\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.43877e7	−2.25665
$528$	0	0
$529$	−5.96457e6	−0.926702
$530$	0	0
$531$	4.50331e6	0.693099
$532$	0	0
$533$	−4.61465e6	−0.703592
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−3.35209e6	−0.501626
$538$	0	0
$539$	−5.85886e6	−0.868642
$540$	0	0
$541$	1.52830e6	0.224499	0.112250	−	0.993680i	$-0.464194\pi$
$541$	1.52830e6	0.224499	0.112250	+	0.993680i	$0.464194\pi$
$542$	0	0
$543$	848980.	0.123566
$544$	0	0
$545$	0	0
$546$	0	0
$547$	1.23234e7	1.76101	0.880506	−	0.474036i	$-0.157203\pi$
$547$	1.23234e7	1.76101	0.880506	+	0.474036i	$0.157203\pi$
$548$	0	0
$549$	5.89050e6	0.834107
$550$	0	0
$551$	−1.40150e6	−0.196660
$552$	0	0
$553$	−429271.	−0.0596923
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−4.08606e6	−0.558042	−0.279021	−	0.960285i	$-0.590010\pi$
$557$	−4.08606e6	−0.558042	−0.279021	+	0.960285i	$0.590010\pi$
$558$	0	0
$559$	6.43339e6	0.870784
$560$	0	0
$561$	4.83531e6	0.648661
$562$	0	0
$563$	−24160.3	−0.00321241	−0.00160621	−	0.999999i	$-0.500511\pi$
$563$	−24160.3	−0.00321241	−0.00160621	+	0.999999i	$0.500511\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−2.60180e6	−0.339873
$568$	0	0
$569$	1.42000e7	1.83869	0.919344	−	0.393454i	$-0.128720\pi$
$569$	1.42000e7	1.83869	0.919344	+	0.393454i	$0.128720\pi$
$570$	0	0
$571$	767642.	0.0985300	0.0492650	−	0.998786i	$-0.484312\pi$
$571$	767642.	0.0985300	0.0492650	+	0.998786i	$0.484312\pi$
$572$	0	0
$573$	−1.00714e6	−0.128145
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−1.51488e6	−0.189426	−0.0947129	−	0.995505i	$-0.530193\pi$
$577$	−1.51488e6	−0.189426	−0.0947129	+	0.995505i	$0.530193\pi$
$578$	0	0
$579$	−563910.	−0.0699059
$580$	0	0
$581$	3.65098e6	0.448714
$582$	0	0
$583$	−9.39498e6	−1.14479
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.28973e7	1.54491	0.772455	−	0.635070i	$-0.219029\pi$
$587$	1.28973e7	1.54491	0.772455	+	0.635070i	$0.219029\pi$
$588$	0	0
$589$	−8.36307e6	−0.993294
$590$	0	0
$591$	−2.23539e6	−0.263260
$592$	0	0
$593$	−5.43125e6	−0.634254	−0.317127	−	0.948383i	$-0.602718\pi$
$593$	−5.43125e6	−0.634254	−0.317127	+	0.948383i	$0.602718\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	924180.	0.106126
$598$	0	0
$599$	−3.92217e6	−0.446642	−0.223321	−	0.974745i	$-0.571690\pi$
$599$	−3.92217e6	−0.446642	−0.223321	+	0.974745i	$0.571690\pi$
$600$	0	0
$601$	−5.64824e6	−0.637863	−0.318931	−	0.947778i	$-0.603324\pi$
$601$	−5.64824e6	−0.637863	−0.318931	+	0.947778i	$0.603324\pi$
$602$	0	0
$603$	−1.64023e6	−0.183701
$604$	0	0
$605$	0	0
$606$	0	0
$607$	1.07148e7	1.18035	0.590177	−	0.807274i	$-0.299058\pi$
$607$	1.07148e7	1.18035	0.590177	+	0.807274i	$0.299058\pi$
$608$	0	0
$609$	510039.	0.0557263
$610$	0	0
$611$	383551.	0.0415642
$612$	0	0
$613$	4.08748e6	0.439344	0.219672	−	0.975574i	$-0.429501\pi$
$613$	4.08748e6	0.439344	0.219672	+	0.975574i	$0.429501\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−7.83395e6	−0.828453	−0.414227	−	0.910174i	$-0.635948\pi$
$617$	−7.83395e6	−0.828453	−0.414227	+	0.910174i	$0.635948\pi$
$618$	0	0
$619$	−1.23423e7	−1.29470	−0.647352	−	0.762191i	$-0.724124\pi$
$619$	−1.23423e7	−1.29470	−0.647352	+	0.762191i	$0.724124\pi$
$620$	0	0
$621$	1.72824e6	0.179836
$622$	0	0
$623$	−3.07739e6	−0.317660
$624$	0	0
$625$	0	0
$626$	0	0
$627$	2.81061e6	0.285516
$628$	0	0
$629$	−3.54842e6	−0.357609
$630$	0	0
$631$	1.31578e6	0.131556	0.0657780	−	0.997834i	$-0.479047\pi$
$631$	1.31578e6	0.131556	0.0657780	+	0.997834i	$0.479047\pi$
$632$	0	0
$633$	2.54542e6	0.252494
$634$	0	0
$635$	0	0
$636$	0	0
$637$	5.16320e6	0.504163
$638$	0	0
$639$	−1.09238e7	−1.05833
$640$	0	0
$641$	6.55744e6	0.630360	0.315180	−	0.949032i	$-0.397935\pi$
$641$	6.55744e6	0.630360	0.315180	+	0.949032i	$0.397935\pi$
$642$	0	0
$643$	4.69954e6	0.448258	0.224129	−	0.974559i	$-0.428046\pi$
$643$	4.69954e6	0.448258	0.224129	+	0.974559i	$0.428046\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−2.05827e7	−1.93305	−0.966523	−	0.256580i	$-0.917404\pi$
$647$	−2.05827e7	−1.93305	−0.966523	+	0.256580i	$0.917404\pi$
$648$	0	0
$649$	−1.03023e7	−0.960117
$650$	0	0
$651$	3.04351e6	0.281464
$652$	0	0
$653$	−1.42466e7	−1.30746	−0.653731	−	0.756727i	$-0.726797\pi$
$653$	−1.42466e7	−1.30746	−0.653731	+	0.756727i	$0.726797\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	9.30177e6	0.840722
$658$	0	0
$659$	1.35369e7	1.21425	0.607123	−	0.794608i	$-0.292324\pi$
$659$	1.35369e7	1.21425	0.607123	+	0.794608i	$0.292324\pi$
$660$	0	0
$661$	−1.30443e7	−1.16122	−0.580612	−	0.814180i	$-0.697187\pi$
$661$	−1.30443e7	−1.16122	−0.580612	+	0.814180i	$0.697187\pi$
$662$	0	0
$663$	−4.26119e6	−0.376485
$664$	0	0
$665$	0	0
$666$	0	0
$667$	919721.	0.0800464
$668$	0	0
$669$	5.98260e6	0.516802
$670$	0	0
$671$	−1.34759e7	−1.15545
$672$	0	0
$673$	4.75951e6	0.405065	0.202532	−	0.979276i	$-0.435083\pi$
$673$	4.75951e6	0.405065	0.202532	+	0.979276i	$0.435083\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−1.51397e7	−1.26954	−0.634770	−	0.772701i	$-0.718905\pi$
$677$	−1.51397e7	−1.26954	−0.634770	+	0.772701i	$0.718905\pi$
$678$	0	0
$679$	1.02061e7	0.849543
$680$	0	0
$681$	−2.27707e6	−0.188152
$682$	0	0
$683$	2.34145e7	1.92058	0.960292	−	0.278998i	$-0.0900024\pi$
$683$	2.34145e7	1.92058	0.960292	+	0.278998i	$0.0900024\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−2.39287e6	−0.193431
$688$	0	0
$689$	8.27947e6	0.664438
$690$	0	0
$691$	1.62194e7	1.29223	0.646113	−	0.763242i	$-0.276394\pi$
$691$	1.62194e7	1.29223	0.646113	+	0.763242i	$0.276394\pi$
$692$	0	0
$693$	7.12196e6	0.563334
$694$	0	0
$695$	0	0
$696$	0	0
$697$	1.93968e7	1.51234
$698$	0	0
$699$	4.19891e6	0.325045
$700$	0	0
$701$	−1.89605e7	−1.45732	−0.728659	−	0.684876i	$-0.759856\pi$
$701$	−1.89605e7	−1.45732	−0.728659	+	0.684876i	$0.759856\pi$
$702$	0	0
$703$	−2.06258e6	−0.157406
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1.02377e7	−0.770287
$708$	0	0
$709$	128325.	0.00958732	0.00479366	−	0.999989i	$-0.498474\pi$
$709$	128325.	0.00958732	0.00479366	+	0.999989i	$0.498474\pi$
$710$	0	0
$711$	−1.32285e6	−0.0981382
$712$	0	0
$713$	5.48817e6	0.404300
$714$	0	0
$715$	0	0
$716$	0	0
$717$	5.46133e6	0.396735
$718$	0	0
$719$	2.41874e7	1.74489	0.872444	−	0.488714i	$-0.162534\pi$
$719$	2.41874e7	1.74489	0.872444	+	0.488714i	$0.162534\pi$
$720$	0	0
$721$	1.29899e7	0.930610
$722$	0	0
$723$	−1.98251e6	−0.141049
$724$	0	0
$725$	0	0
$726$	0	0
$727$	513307.	0.0360198	0.0180099	−	0.999838i	$-0.494267\pi$
$727$	513307.	0.0360198	0.0180099	+	0.999838i	$0.494267\pi$
$728$	0	0
$729$	−4.64015e6	−0.323380
$730$	0	0
$731$	−2.70416e7	−1.87171
$732$	0	0
$733$	1.64153e7	1.12847	0.564234	−	0.825615i	$-0.309172\pi$
$733$	1.64153e7	1.12847	0.564234	+	0.825615i	$0.309172\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3.75240e6	0.254472
$738$	0	0
$739$	−1.16112e7	−0.782109	−0.391054	−	0.920368i	$-0.627890\pi$
$739$	−1.16112e7	−0.782109	−0.391054	+	0.920368i	$0.627890\pi$
$740$	0	0
$741$	−2.47689e6	−0.165715
$742$	0	0
$743$	5.72590e6	0.380515	0.190257	−	0.981734i	$-0.439068\pi$
$743$	5.72590e6	0.380515	0.190257	+	0.981734i	$0.439068\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	1.12510e7	0.737715
$748$	0	0
$749$	4.68097e6	0.304882
$750$	0	0
$751$	−1.15324e7	−0.746137	−0.373069	−	0.927804i	$-0.621694\pi$
$751$	−1.15324e7	−0.746137	−0.373069	+	0.927804i	$0.621694\pi$
$752$	0	0
$753$	4.70134e6	0.302158
$754$	0	0
$755$	0	0
$756$	0	0
$757$	8.63293e6	0.547544	0.273772	−	0.961795i	$-0.411729\pi$
$757$	8.63293e6	0.547544	0.273772	+	0.961795i	$0.411729\pi$
$758$	0	0
$759$	−1.84443e6	−0.116214
$760$	0	0
$761$	−3.52622e6	−0.220723	−0.110361	−	0.993892i	$-0.535201\pi$
$761$	−3.52622e6	−0.220723	−0.110361	+	0.993892i	$0.535201\pi$
$762$	0	0
$763$	1.51206e7	0.940280
$764$	0	0
$765$	0	0
$766$	0	0
$767$	9.07910e6	0.557255
$768$	0	0
$769$	1.40471e7	0.856585	0.428293	−	0.903640i	$-0.359115\pi$
$769$	1.40471e7	0.856585	0.428293	+	0.903640i	$0.359115\pi$
$770$	0	0
$771$	−421817.	−0.0255557
$772$	0	0
$773$	2.44760e7	1.47330	0.736651	−	0.676274i	$-0.236406\pi$
$773$	2.44760e7	1.47330	0.736651	+	0.676274i	$0.236406\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	750619.	0.0446033
$778$	0	0
$779$	1.12747e7	0.665675
$780$	0	0
$781$	2.49907e7	1.46606
$782$	0	0
$783$	3.36924e6	0.196393
$784$	0	0
$785$	0	0
$786$	0	0
$787$	4.35977e6	0.250915	0.125458	−	0.992099i	$-0.459960\pi$
$787$	4.35977e6	0.250915	0.125458	+	0.992099i	$0.459960\pi$
$788$	0	0
$789$	−6.59697e6	−0.377270
$790$	0	0
$791$	5.57019e6	0.316540
$792$	0	0
$793$	1.18758e7	0.670626
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−1.06887e7	−0.596044	−0.298022	−	0.954559i	$-0.596327\pi$
$797$	−1.06887e7	−0.596044	−0.298022	+	0.954559i	$0.596327\pi$
$798$	0	0
$799$	−1.61218e6	−0.0893403
$800$	0	0
$801$	−9.48339e6	−0.522255
$802$	0	0
$803$	−2.12799e7	−1.16461
$804$	0	0
$805$	0	0
$806$	0	0
$807$	5.68605e6	0.307345
$808$	0	0
$809$	9.12014e6	0.489926	0.244963	−	0.969532i	$-0.421224\pi$
$809$	9.12014e6	0.489926	0.244963	+	0.969532i	$0.421224\pi$
$810$	0	0
$811$	−5.22575e6	−0.278995	−0.139497	−	0.990222i	$-0.544549\pi$
$811$	−5.22575e6	−0.278995	−0.139497	+	0.990222i	$0.544549\pi$
$812$	0	0
$813$	−1.17192e7	−0.621830
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−1.57184e7	−0.823857
$818$	0	0
$819$	−6.27633e6	−0.326961
$820$	0	0
$821$	9.00437e6	0.466225	0.233112	−	0.972450i	$-0.425109\pi$
$821$	9.00437e6	0.466225	0.233112	+	0.972450i	$0.425109\pi$
$822$	0	0
$823$	2.78867e7	1.43515	0.717574	−	0.696482i	$-0.245252\pi$
$823$	2.78867e7	1.43515	0.717574	+	0.696482i	$0.245252\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	6.64309e6	0.337758	0.168879	−	0.985637i	$-0.445985\pi$
$827$	6.64309e6	0.337758	0.168879	+	0.985637i	$0.445985\pi$
$828$	0	0
$829$	2.17030e7	1.09682	0.548408	−	0.836211i	$-0.315234\pi$
$829$	2.17030e7	1.09682	0.548408	+	0.836211i	$0.315234\pi$
$830$	0	0
$831$	1.02277e7	0.513780
$832$	0	0
$833$	−2.17026e7	−1.08367
$834$	0	0
$835$	0	0
$836$	0	0
$837$	2.01049e7	0.991949
$838$	0	0
$839$	1.01238e7	0.496520	0.248260	−	0.968693i	$-0.420141\pi$
$839$	1.01238e7	0.496520	0.248260	+	0.968693i	$0.420141\pi$
$840$	0	0
$841$	−1.87181e7	−0.912584
$842$	0	0
$843$	500784.	0.0242707
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−5.18837e6	−0.248498
$848$	0	0
$849$	5.13366e6	0.244432
$850$	0	0
$851$	1.35354e6	0.0640691
$852$	0	0
$853$	1.46326e7	0.688573	0.344286	−	0.938865i	$-0.388121\pi$
$853$	1.46326e7	0.688573	0.344286	+	0.938865i	$0.388121\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−5.52218e6	−0.256838	−0.128419	−	0.991720i	$-0.540990\pi$
$857$	−5.52218e6	−0.256838	−0.128419	+	0.991720i	$0.540990\pi$
$858$	0	0
$859$	3.02260e6	0.139765	0.0698824	−	0.997555i	$-0.477738\pi$
$859$	3.02260e6	0.139765	0.0698824	+	0.997555i	$0.477738\pi$
$860$	0	0
$861$	−4.10313e6	−0.188628
$862$	0	0
$863$	3.06818e7	1.40234	0.701172	−	0.712992i	$-0.252661\pi$
$863$	3.06818e7	1.40234	0.701172	+	0.712992i	$0.252661\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	1.00676e7	0.454860
$868$	0	0
$869$	3.02633e6	0.135946
$870$	0	0
$871$	−3.30686e6	−0.147697
$872$	0	0
$873$	3.14514e7	1.39671
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−5.17607e6	−0.227249	−0.113624	−	0.993524i	$-0.536246\pi$
$877$	−5.17607e6	−0.227249	−0.113624	+	0.993524i	$0.536246\pi$
$878$	0	0
$879$	1.50662e7	0.657707
$880$	0	0
$881$	−4.25937e7	−1.84887	−0.924433	−	0.381345i	$-0.875461\pi$
$881$	−4.25937e7	−1.84887	−0.924433	+	0.381345i	$0.875461\pi$
$882$	0	0
$883$	1.72076e7	0.742709	0.371354	−	0.928491i	$-0.378893\pi$
$883$	1.72076e7	0.742709	0.371354	+	0.928491i	$0.378893\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−2.53773e6	−0.108302	−0.0541510	−	0.998533i	$-0.517245\pi$
$887$	−2.53773e6	−0.108302	−0.0541510	+	0.998533i	$0.517245\pi$
$888$	0	0
$889$	1.11220e7	0.471984
$890$	0	0
$891$	1.83425e7	0.774043
$892$	0	0
$893$	−937108.	−0.0393243
$894$	0	0
$895$	0	0
$896$	0	0
$897$	1.62543e6	0.0674508
$898$	0	0
$899$	1.06993e7	0.441525
$900$	0	0
$901$	−3.48012e7	−1.42818
$902$	0	0
$903$	5.72026e6	0.233452
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−2.60899e7	−1.05306	−0.526531	−	0.850156i	$-0.676508\pi$
$907$	−2.60899e7	−1.05306	−0.526531	+	0.850156i	$0.676508\pi$
$908$	0	0
$909$	−3.15487e7	−1.26640
$910$	0	0
$911$	−1.44818e7	−0.578130	−0.289065	−	0.957309i	$-0.593344\pi$
$911$	−1.44818e7	−0.578130	−0.289065	+	0.957309i	$0.593344\pi$
$912$	0	0
$913$	−2.57392e7	−1.02192
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1.33081e7	−0.522627
$918$	0	0
$919$	−4.61041e7	−1.80074	−0.900369	−	0.435127i	$-0.856704\pi$
$919$	−4.61041e7	−1.80074	−0.900369	+	0.435127i	$0.856704\pi$
$920$	0	0
$921$	1.26861e7	0.492811
$922$	0	0
$923$	−2.20234e7	−0.850903
$924$	0	0
$925$	0	0
$926$	0	0
$927$	4.00301e7	1.52998
$928$	0	0
$929$	−1.81557e7	−0.690197	−0.345098	−	0.938567i	$-0.612154\pi$
$929$	−1.81557e7	−0.690197	−0.345098	+	0.938567i	$0.612154\pi$
$930$	0	0
$931$	−1.26150e7	−0.476993
$932$	0	0
$933$	−5.44047e6	−0.204612
$934$	0	0
$935$	0	0
$936$	0	0
$937$	1.78946e7	0.665844	0.332922	−	0.942954i	$-0.391965\pi$
$937$	1.78946e7	0.665844	0.332922	+	0.942954i	$0.391965\pi$
$938$	0	0
$939$	1.14157e7	0.422512
$940$	0	0
$941$	−3.04463e6	−0.112088	−0.0560441	−	0.998428i	$-0.517849\pi$
$941$	−3.04463e6	−0.112088	−0.0560441	+	0.998428i	$0.517849\pi$
$942$	0	0
$943$	−7.39891e6	−0.270950
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−3.17110e7	−1.14904	−0.574519	−	0.818491i	$-0.694811\pi$
$947$	−3.17110e7	−1.14904	−0.574519	+	0.818491i	$0.694811\pi$
$948$	0	0
$949$	1.87532e7	0.675944
$950$	0	0
$951$	6.31686e6	0.226491
$952$	0	0
$953$	1.01913e7	0.363494	0.181747	−	0.983345i	$-0.441825\pi$
$953$	1.01913e7	0.363494	0.181747	+	0.983345i	$0.441825\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−3.59574e6	−0.126914
$958$	0	0
$959$	−1.72613e7	−0.606077
$960$	0	0
$961$	3.52157e7	1.23006
$962$	0	0
$963$	1.44250e7	0.501246
$964$	0	0
$965$	0	0
$966$	0	0
$967$	3.21125e7	1.10435	0.552177	−	0.833727i	$-0.313797\pi$
$967$	3.21125e7	1.10435	0.552177	+	0.833727i	$0.313797\pi$
$968$	0	0
$969$	1.04111e7	0.356196
$970$	0	0
$971$	−2.29867e7	−0.782399	−0.391200	−	0.920306i	$-0.627940\pi$
$971$	−2.29867e7	−0.782399	−0.391200	+	0.920306i	$0.627940\pi$
$972$	0	0
$973$	−1.50763e7	−0.510519
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−2.47331e7	−0.828978	−0.414489	−	0.910054i	$-0.636040\pi$
$977$	−2.47331e7	−0.828978	−0.414489	+	0.910054i	$0.636040\pi$
$978$	0	0
$979$	2.16954e7	0.723455
$980$	0	0
$981$	4.65960e7	1.54588
$982$	0	0
$983$	−5.57031e7	−1.83863	−0.919317	−	0.393518i	$-0.871258\pi$
$983$	−5.57031e7	−1.83863	−0.919317	+	0.393518i	$0.871258\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	341035.	0.0111431
$988$	0	0
$989$	1.03150e7	0.335335
$990$	0	0
$991$	6.86029e6	0.221901	0.110950	−	0.993826i	$-0.464611\pi$
$991$	6.86029e6	0.221901	0.110950	+	0.993826i	$0.464611\pi$
$992$	0	0
$993$	−1.13373e6	−0.0364868
$994$	0	0
$995$	0	0
$996$	0	0
$997$	6.03725e7	1.92354	0.961771	−	0.273856i	$-0.0882993\pi$
$997$	6.03725e7	1.92354	0.961771	+	0.273856i	$0.0882993\pi$
$998$	0	0
$999$	4.95847e6	0.157193

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.6.a.o.1.2		2
4.3	odd	2		25.6.a.d.1.2	yes	2
5.2	odd	4		400.6.c.n.49.2		4
5.3	odd	4		400.6.c.n.49.3		4
5.4	even	2		400.6.a.w.1.1		2
12.11	even	2		225.6.a.l.1.1		2
20.3	even	4		25.6.b.b.24.1		4
20.7	even	4		25.6.b.b.24.4		4
20.19	odd	2		25.6.a.b.1.1	✓	2
60.23	odd	4		225.6.b.i.199.4		4
60.47	odd	4		225.6.b.i.199.1		4
60.59	even	2		225.6.a.s.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.6.a.b.1.1	✓	2	20.19	odd	2
25.6.a.d.1.2	yes	2	4.3	odd	2
25.6.b.b.24.1		4	20.3	even	4
25.6.b.b.24.4		4	20.7	even	4
225.6.a.l.1.1		2	12.11	even	2
225.6.a.s.1.2		2	60.59	even	2
225.6.b.i.199.1		4	60.47	odd	4
225.6.b.i.199.4		4	60.23	odd	4
400.6.a.o.1.2		2	1.1	even	1	trivial
400.6.a.w.1.1		2	5.4	even	2
400.6.c.n.49.2		4	5.2	odd	4
400.6.c.n.49.3		4	5.3	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:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	400.a (trivial)

\(f(q)\)	\(=\)	\(q+5.52417 q^{3} -68.9517 q^{7} -212.483 q^{9} +O(q^{10})\)	\(q+5.52417 q^{3} -68.9517 q^{7} -212.483 q^{9} +486.104 q^{11} -428.387 q^{13} +1800.64 q^{17} +1046.65 q^{19} -380.901 q^{21} -686.855 q^{23} -2516.17 q^{27} -1339.03 q^{29} -7990.30 q^{31} +2685.33 q^{33} -1970.64 q^{37} -2366.48 q^{39} +10772.2 q^{41} -15017.7 q^{43} -895.337 q^{47} -12052.7 q^{49} +9947.07 q^{51} -19327.1 q^{53} +5781.90 q^{57} -21193.7 q^{59} -27722.2 q^{61} +14651.1 q^{63} +7719.33 q^{67} -3794.31 q^{69} +51410.1 q^{71} -43776.4 q^{73} -33517.7 q^{77} +6225.68 q^{79} +37733.7 q^{81} -52949.9 q^{83} -7397.05 q^{87} +44631.2 q^{89} +29538.0 q^{91} -44139.8 q^{93} -148018. q^{97} -103289. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 20 q^{3} - 200 q^{7} + 196 q^{9}+O(q^{10}) \) 2 * q - 20 * q^3 - 200 * q^7 + 196 * q^9	\( 2 q - 20 q^{3} - 200 q^{7} + 196 q^{9} + 196 q^{11} - 360 q^{13} + 1490 q^{17} + 3180 q^{19} + 2964 q^{21} - 1560 q^{23} - 6740 q^{27} - 3920 q^{29} + 1096 q^{31} + 10090 q^{33} + 2020 q^{37} - 4112 q^{39} + 27754 q^{41} + 3000 q^{43} - 25760 q^{47} - 11686 q^{49} + 17876 q^{51} - 26980 q^{53} - 48670 q^{57} - 11960 q^{59} - 24396 q^{61} - 38880 q^{63} + 40060 q^{67} + 18492 q^{69} + 87296 q^{71} - 70290 q^{73} + 4500 q^{77} - 65480 q^{79} + 46282 q^{81} - 92580 q^{83} + 58480 q^{87} - 72810 q^{89} + 20576 q^{91} - 276060 q^{93} - 126140 q^{97} - 221792 q^{99}+O(q^{100}) \) 2 * q - 20 * q^3 - 200 * q^7 + 196 * q^9 + 196 * q^11 - 360 * q^13 + 1490 * q^17 + 3180 * q^19 + 2964 * q^21 - 1560 * q^23 - 6740 * q^27 - 3920 * q^29 + 1096 * q^31 + 10090 * q^33 + 2020 * q^37 - 4112 * q^39 + 27754 * q^41 + 3000 * q^43 - 25760 * q^47 - 11686 * q^49 + 17876 * q^51 - 26980 * q^53 - 48670 * q^57 - 11960 * q^59 - 24396 * q^61 - 38880 * q^63 + 40060 * q^67 + 18492 * q^69 + 87296 * q^71 - 70290 * q^73 + 4500 * q^77 - 65480 * q^79 + 46282 * q^81 - 92580 * q^83 + 58480 * q^87 - 72810 * q^89 + 20576 * q^91 - 276060 * q^93 - 126140 * q^97 - 221792 * q^99

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