LMFDB - Embedded newform 800.6.a.n.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,6,Mod(1,800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(800, 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("800.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	800.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:	$128.307055850$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.39180.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 36x - 24 $ x^3 - x^2 - 36*x - 24
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 160)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$6.80681$ of defining polynomial
Character	$\chi$	$=$	800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-29.2272 q^{3} -44.5253 q^{7} +611.232 q^{9} +O(q^{10})$	$q-29.2272 q^{3} -44.5253 q^{7} +611.232 q^{9} -349.258 q^{11} -255.040 q^{13} -1505.12 q^{17} +2431.11 q^{19} +1301.35 q^{21} -1435.50 q^{23} -10762.4 q^{27} +2872.04 q^{29} -8940.96 q^{31} +10207.8 q^{33} +14536.6 q^{37} +7454.11 q^{39} +7504.01 q^{41} -13490.7 q^{43} -8449.69 q^{47} -14824.5 q^{49} +43990.5 q^{51} +28317.3 q^{53} -71054.8 q^{57} -3530.13 q^{59} +45644.9 q^{61} -27215.3 q^{63} +69849.6 q^{67} +41955.6 q^{69} +60090.2 q^{71} +49204.6 q^{73} +15550.8 q^{77} -981.509 q^{79} +166026. q^{81} +38372.3 q^{83} -83941.9 q^{87} +39418.6 q^{89} +11355.7 q^{91} +261320. q^{93} -51340.9 q^{97} -213477. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 10 q^{3} + 6 q^{7} + 467 q^{9}+O(q^{10}) $ 3 * q - 10 * q^3 + 6 * q^7 + 467 * q^9	$ 3 q - 10 q^{3} + 6 q^{7} + 467 q^{9} - 396 q^{11} + 354 q^{13} - 1158 q^{17} + 3192 q^{19} - 820 q^{21} - 6126 q^{23} - 13804 q^{27} + 426 q^{29} + 3276 q^{31} + 2408 q^{33} + 11562 q^{37} + 21348 q^{39} + 12450 q^{41} - 26346 q^{43} - 36762 q^{47} - 3849 q^{49} + 71444 q^{51} + 21162 q^{53} - 69136 q^{57} + 35040 q^{59} - 24138 q^{61} - 80986 q^{63} + 9570 q^{67} - 27036 q^{69} + 88092 q^{71} - 66750 q^{73} + 136488 q^{77} - 92952 q^{79} + 151391 q^{81} - 30258 q^{83} + 26228 q^{87} + 172686 q^{89} - 106812 q^{91} + 318232 q^{93} - 170910 q^{97} - 351436 q^{99}+O(q^{100}) $ 3 * q - 10 * q^3 + 6 * q^7 + 467 * q^9 - 396 * q^11 + 354 * q^13 - 1158 * q^17 + 3192 * q^19 - 820 * q^21 - 6126 * q^23 - 13804 * q^27 + 426 * q^29 + 3276 * q^31 + 2408 * q^33 + 11562 * q^37 + 21348 * q^39 + 12450 * q^41 - 26346 * q^43 - 36762 * q^47 - 3849 * q^49 + 71444 * q^51 + 21162 * q^53 - 69136 * q^57 + 35040 * q^59 - 24138 * q^61 - 80986 * q^63 + 9570 * q^67 - 27036 * q^69 + 88092 * q^71 - 66750 * q^73 + 136488 * q^77 - 92952 * q^79 + 151391 * q^81 - 30258 * q^83 + 26228 * q^87 + 172686 * q^89 - 106812 * q^91 + 318232 * q^93 - 170910 * q^97 - 351436 * 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
$3$	−29.2272	−1.87493	−0.937464	−	0.348081i	$-0.886833\pi$
$3$	−29.2272	−1.87493	−0.937464	+	0.348081i	$0.886833\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−44.5253	−0.343449	−0.171724	−	0.985145i	$-0.554934\pi$
$7$	−44.5253	−0.343449	−0.171724	+	0.985145i	$0.554934\pi$
$8$	0	0
$9$	611.232	2.51536
$10$	0	0
$11$	−349.258	−0.870290	−0.435145	−	0.900360i	$-0.643303\pi$
$11$	−349.258	−0.870290	−0.435145	+	0.900360i	$0.643303\pi$
$12$	0	0
$13$	−255.040	−0.418552	−0.209276	−	0.977857i	$-0.567111\pi$
$13$	−255.040	−0.418552	−0.209276	+	0.977857i	$0.567111\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1505.12	−1.26313	−0.631566	−	0.775322i	$-0.717588\pi$
$17$	−1505.12	−1.26313	−0.631566	+	0.775322i	$0.717588\pi$
$18$	0	0
$19$	2431.11	1.54498	0.772488	−	0.635030i	$-0.219012\pi$
$19$	2431.11	1.54498	0.772488	+	0.635030i	$0.219012\pi$
$20$	0	0
$21$	1301.35	0.643942
$22$	0	0
$23$	−1435.50	−0.565825	−0.282913	−	0.959146i	$-0.591301\pi$
$23$	−1435.50	−0.565825	−0.282913	+	0.959146i	$0.591301\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−10762.4	−2.84119
$28$	0	0
$29$	2872.04	0.634156	0.317078	−	0.948400i	$-0.397298\pi$
$29$	2872.04	0.634156	0.317078	+	0.948400i	$0.397298\pi$
$30$	0	0
$31$	−8940.96	−1.67101	−0.835507	−	0.549480i	$-0.814826\pi$
$31$	−8940.96	−1.67101	−0.835507	+	0.549480i	$0.814826\pi$
$32$	0	0
$33$	10207.8	1.63173
$34$	0	0
$35$	0	0
$36$	0	0
$37$	14536.6	1.74566	0.872830	−	0.488024i	$-0.162282\pi$
$37$	14536.6	1.74566	0.872830	+	0.488024i	$0.162282\pi$
$38$	0	0
$39$	7454.11	0.784756
$40$	0	0
$41$	7504.01	0.697162	0.348581	−	0.937279i	$-0.386664\pi$
$41$	7504.01	0.697162	0.348581	+	0.937279i	$0.386664\pi$
$42$	0	0
$43$	−13490.7	−1.11266	−0.556331	−	0.830961i	$-0.687791\pi$
$43$	−13490.7	−1.11266	−0.556331	+	0.830961i	$0.687791\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−8449.69	−0.557951	−0.278975	−	0.960298i	$-0.589995\pi$
$47$	−8449.69	−0.557951	−0.278975	+	0.960298i	$0.589995\pi$
$48$	0	0
$49$	−14824.5	−0.882043
$50$	0	0
$51$	43990.5	2.36828
$52$	0	0
$53$	28317.3	1.38472	0.692360	−	0.721553i	$-0.256571\pi$
$53$	28317.3	1.38472	0.692360	+	0.721553i	$0.256571\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−71054.8	−2.89672
$58$	0	0
$59$	−3530.13	−0.132026	−0.0660131	−	0.997819i	$-0.521028\pi$
$59$	−3530.13	−0.132026	−0.0660131	+	0.997819i	$0.521028\pi$
$60$	0	0
$61$	45644.9	1.57061	0.785304	−	0.619110i	$-0.212506\pi$
$61$	45644.9	1.57061	0.785304	+	0.619110i	$0.212506\pi$
$62$	0	0
$63$	−27215.3	−0.863896
$64$	0	0
$65$	0	0
$66$	0	0
$67$	69849.6	1.90098	0.950489	−	0.310759i	$-0.100583\pi$
$67$	69849.6	1.90098	0.950489	+	0.310759i	$0.100583\pi$
$68$	0	0
$69$	41955.6	1.06088
$70$	0	0
$71$	60090.2	1.41468	0.707339	−	0.706874i	$-0.249895\pi$
$71$	60090.2	1.41468	0.707339	+	0.706874i	$0.249895\pi$
$72$	0	0
$73$	49204.6	1.08068	0.540342	−	0.841445i	$-0.318295\pi$
$73$	49204.6	1.08068	0.540342	+	0.841445i	$0.318295\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	15550.8	0.298900
$78$	0	0
$79$	−981.509	−0.0176940	−0.00884702	−	0.999961i	$-0.502816\pi$
$79$	−981.509	−0.0176940	−0.00884702	+	0.999961i	$0.502816\pi$
$80$	0	0
$81$	166026.	2.81167
$82$	0	0
$83$	38372.3	0.611396	0.305698	−	0.952129i	$-0.401110\pi$
$83$	38372.3	0.611396	0.305698	+	0.952129i	$0.401110\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−83941.9	−1.18900
$88$	0	0
$89$	39418.6	0.527504	0.263752	−	0.964591i	$-0.415040\pi$
$89$	39418.6	0.527504	0.263752	+	0.964591i	$0.415040\pi$
$90$	0	0
$91$	11355.7	0.143751
$92$	0	0
$93$	261320.	3.13303
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−51340.9	−0.554031	−0.277016	−	0.960865i	$-0.589345\pi$
$97$	−51340.9	−0.554031	−0.277016	+	0.960865i	$0.589345\pi$
$98$	0	0
$99$	−213477.	−2.18909
$100$	0	0
$101$	−51557.0	−0.502903	−0.251451	−	0.967870i	$-0.580908\pi$
$101$	−51557.0	−0.502903	−0.251451	+	0.967870i	$0.580908\pi$
$102$	0	0
$103$	−6370.92	−0.0591710	−0.0295855	−	0.999562i	$-0.509419\pi$
$103$	−6370.92	−0.0591710	−0.0295855	+	0.999562i	$0.509419\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5654.13	0.0477426	0.0238713	−	0.999715i	$-0.492401\pi$
$107$	5654.13	0.0477426	0.0238713	+	0.999715i	$0.492401\pi$
$108$	0	0
$109$	−217953.	−1.75710	−0.878550	−	0.477650i	$-0.841489\pi$
$109$	−217953.	−1.75710	−0.878550	+	0.477650i	$0.841489\pi$
$110$	0	0
$111$	−424866.	−3.27299
$112$	0	0
$113$	129389.	0.953240	0.476620	−	0.879109i	$-0.341862\pi$
$113$	129389.	0.953240	0.476620	+	0.879109i	$0.341862\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−155889.	−1.05281
$118$	0	0
$119$	67015.9	0.433821
$120$	0	0
$121$	−39070.1	−0.242595
$122$	0	0
$123$	−219322.	−1.30713
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−106258.	−0.584594	−0.292297	−	0.956328i	$-0.594420\pi$
$127$	−106258.	−0.584594	−0.292297	+	0.956328i	$0.594420\pi$
$128$	0	0
$129$	394296.	2.08616
$130$	0	0
$131$	132808.	0.676155	0.338077	−	0.941118i	$-0.390223\pi$
$131$	132808.	0.676155	0.338077	+	0.941118i	$0.390223\pi$
$132$	0	0
$133$	−108246.	−0.530619
$134$	0	0
$135$	0	0
$136$	0	0
$137$	301171.	1.37092	0.685460	−	0.728110i	$-0.259601\pi$
$137$	301171.	1.37092	0.685460	+	0.728110i	$0.259601\pi$
$138$	0	0
$139$	−139021.	−0.610298	−0.305149	−	0.952305i	$-0.598706\pi$
$139$	−139021.	−0.610298	−0.305149	+	0.952305i	$0.598706\pi$
$140$	0	0
$141$	246961.	1.04612
$142$	0	0
$143$	89074.6	0.364262
$144$	0	0
$145$	0	0
$146$	0	0
$147$	433279.	1.65377
$148$	0	0
$149$	−195083.	−0.719869	−0.359934	−	0.932978i	$-0.617201\pi$
$149$	−195083.	−0.719869	−0.359934	+	0.932978i	$0.617201\pi$
$150$	0	0
$151$	−161597.	−0.576755	−0.288378	−	0.957517i	$-0.593116\pi$
$151$	−161597.	−0.576755	−0.288378	+	0.957517i	$0.593116\pi$
$152$	0	0
$153$	−919977.	−3.17723
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−108055.	−0.349860	−0.174930	−	0.984581i	$-0.555970\pi$
$157$	−108055.	−0.349860	−0.174930	+	0.984581i	$0.555970\pi$
$158$	0	0
$159$	−827636.	−2.59625
$160$	0	0
$161$	63915.9	0.194332
$162$	0	0
$163$	35513.9	0.104696	0.0523479	−	0.998629i	$-0.483330\pi$
$163$	35513.9	0.104696	0.0523479	+	0.998629i	$0.483330\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−377413.	−1.04719	−0.523595	−	0.851967i	$-0.675409\pi$
$167$	−377413.	−1.04719	−0.523595	+	0.851967i	$0.675409\pi$
$168$	0	0
$169$	−306248.	−0.824814
$170$	0	0
$171$	1.48598e6	3.88617
$172$	0	0
$173$	142405.	0.361752	0.180876	−	0.983506i	$-0.442107\pi$
$173$	142405.	0.361752	0.180876	+	0.983506i	$0.442107\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	103176.	0.247540
$178$	0	0
$179$	−457275.	−1.06671	−0.533353	−	0.845893i	$-0.679068\pi$
$179$	−457275.	−1.06671	−0.533353	+	0.845893i	$0.679068\pi$
$180$	0	0
$181$	−871860.	−1.97811	−0.989055	−	0.147549i	$-0.952862\pi$
$181$	−871860.	−1.97811	−0.989055	+	0.147549i	$0.952862\pi$
$182$	0	0
$183$	−1.33408e6	−2.94478
$184$	0	0
$185$	0	0
$186$	0	0
$187$	525675.	1.09929
$188$	0	0
$189$	479199.	0.975802
$190$	0	0
$191$	330154.	0.654837	0.327418	−	0.944879i	$-0.393821\pi$
$191$	330154.	0.654837	0.327418	+	0.944879i	$0.393821\pi$
$192$	0	0
$193$	152914.	0.295498	0.147749	−	0.989025i	$-0.452797\pi$
$193$	152914.	0.295498	0.147749	+	0.989025i	$0.452797\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	714867.	1.31238	0.656190	−	0.754596i	$-0.272167\pi$
$197$	714867.	1.31238	0.656190	+	0.754596i	$0.272167\pi$
$198$	0	0
$199$	292872.	0.524258	0.262129	−	0.965033i	$-0.415575\pi$
$199$	292872.	0.524258	0.262129	+	0.965033i	$0.415575\pi$
$200$	0	0
$201$	−2.04151e6	−3.56420
$202$	0	0
$203$	−127879.	−0.217800
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−877421.	−1.42325
$208$	0	0
$209$	−849085.	−1.34458
$210$	0	0
$211$	365716.	0.565507	0.282754	−	0.959193i	$-0.408752\pi$
$211$	365716.	0.565507	0.282754	+	0.959193i	$0.408752\pi$
$212$	0	0
$213$	−1.75627e6	−2.65242
$214$	0	0
$215$	0	0
$216$	0	0
$217$	398099.	0.573907
$218$	0	0
$219$	−1.43812e6	−2.02621
$220$	0	0
$221$	383866.	0.528687
$222$	0	0
$223$	−748539.	−1.00798	−0.503990	−	0.863709i	$-0.668135\pi$
$223$	−748539.	−1.00798	−0.503990	+	0.863709i	$0.668135\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−424142.	−0.546319	−0.273159	−	0.961969i	$-0.588069\pi$
$227$	−424142.	−0.546319	−0.273159	+	0.961969i	$0.588069\pi$
$228$	0	0
$229$	1.02868e6	1.29625	0.648126	−	0.761533i	$-0.275553\pi$
$229$	1.02868e6	1.29625	0.648126	+	0.761533i	$0.275553\pi$
$230$	0	0
$231$	−454507.	−0.560416
$232$	0	0
$233$	−614128.	−0.741087	−0.370544	−	0.928815i	$-0.620829\pi$
$233$	−614128.	−0.741087	−0.370544	+	0.928815i	$0.620829\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	28686.8	0.0331750
$238$	0	0
$239$	1.36204e6	1.54239	0.771197	−	0.636597i	$-0.219658\pi$
$239$	1.36204e6	1.54239	0.771197	+	0.636597i	$0.219658\pi$
$240$	0	0
$241$	354377.	0.393027	0.196514	−	0.980501i	$-0.437038\pi$
$241$	354377.	0.393027	0.196514	+	0.980501i	$0.437038\pi$
$242$	0	0
$243$	−2.23722e6	−2.43049
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−620031.	−0.646653
$248$	0	0
$249$	−1.12152e6	−1.14632
$250$	0	0
$251$	−826955.	−0.828509	−0.414255	−	0.910161i	$-0.635958\pi$
$251$	−826955.	−0.828509	−0.414255	+	0.910161i	$0.635958\pi$
$252$	0	0
$253$	501358.	0.492432
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1.47307e6	−1.39121	−0.695603	−	0.718426i	$-0.744863\pi$
$257$	−1.47307e6	−1.39121	−0.695603	+	0.718426i	$0.744863\pi$
$258$	0	0
$259$	−647248.	−0.599545
$260$	0	0
$261$	1.75548e6	1.59513
$262$	0	0
$263$	−669755.	−0.597072	−0.298536	−	0.954398i	$-0.596498\pi$
$263$	−669755.	−0.597072	−0.298536	+	0.954398i	$0.596498\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−1.15210e6	−0.989032
$268$	0	0
$269$	−161730.	−0.136273	−0.0681367	−	0.997676i	$-0.521705\pi$
$269$	−161730.	−0.136273	−0.0681367	+	0.997676i	$0.521705\pi$
$270$	0	0
$271$	−709813.	−0.587112	−0.293556	−	0.955942i	$-0.594839\pi$
$271$	−709813.	−0.587112	−0.293556	+	0.955942i	$0.594839\pi$
$272$	0	0
$273$	−331897.	−0.269523
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−1.25136e6	−0.979904	−0.489952	−	0.871749i	$-0.662986\pi$
$277$	−1.25136e6	−0.979904	−0.489952	+	0.871749i	$0.662986\pi$
$278$	0	0
$279$	−5.46500e6	−4.20320
$280$	0	0
$281$	−1.14718e6	−0.866692	−0.433346	−	0.901228i	$-0.642667\pi$
$281$	−1.14718e6	−0.866692	−0.433346	+	0.901228i	$0.642667\pi$
$282$	0	0
$283$	1.01151e6	0.750766	0.375383	−	0.926870i	$-0.377511\pi$
$283$	1.01151e6	0.750766	0.375383	+	0.926870i	$0.377511\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−334118.	−0.239439
$288$	0	0
$289$	845528.	0.595502
$290$	0	0
$291$	1.50055e6	1.03877
$292$	0	0
$293$	161999.	0.110241	0.0551204	−	0.998480i	$-0.482446\pi$
$293$	161999.	0.110241	0.0551204	+	0.998480i	$0.482446\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	3.75885e6	2.47266
$298$	0	0
$299$	366109.	0.236828
$300$	0	0
$301$	600677.	0.382142
$302$	0	0
$303$	1.50687e6	0.942907
$304$	0	0
$305$	0	0
$306$	0	0
$307$	1.20761e6	0.731276	0.365638	−	0.930757i	$-0.380851\pi$
$307$	1.20761e6	0.731276	0.365638	+	0.930757i	$0.380851\pi$
$308$	0	0
$309$	186204.	0.110941
$310$	0	0
$311$	2.90780e6	1.70476	0.852382	−	0.522920i	$-0.175157\pi$
$311$	2.90780e6	1.70476	0.852382	+	0.522920i	$0.175157\pi$
$312$	0	0
$313$	1.27324e6	0.734597	0.367298	−	0.930103i	$-0.380283\pi$
$313$	1.27324e6	0.734597	0.367298	+	0.930103i	$0.380283\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1.34305e6	−0.750659	−0.375330	−	0.926891i	$-0.622470\pi$
$317$	−1.34305e6	−0.750659	−0.375330	+	0.926891i	$0.622470\pi$
$318$	0	0
$319$	−1.00308e6	−0.551899
$320$	0	0
$321$	−165255.	−0.0895140
$322$	0	0
$323$	−3.65912e6	−1.95151
$324$	0	0
$325$	0	0
$326$	0	0
$327$	6.37016e6	3.29444
$328$	0	0
$329$	376225.	0.191627
$330$	0	0
$331$	−1.59250e6	−0.798929	−0.399464	−	0.916749i	$-0.630804\pi$
$331$	−1.59250e6	−0.798929	−0.399464	+	0.916749i	$0.630804\pi$
$332$	0	0
$333$	8.88526e6	4.39096
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−1.34715e6	−0.646163	−0.323082	−	0.946371i	$-0.604719\pi$
$337$	−1.34715e6	−0.646163	−0.323082	+	0.946371i	$0.604719\pi$
$338$	0	0
$339$	−3.78169e6	−1.78726
$340$	0	0
$341$	3.12270e6	1.45427
$342$	0	0
$343$	1.40840e6	0.646385
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.61456e6	0.719832	0.359916	−	0.932985i	$-0.382805\pi$
$347$	1.61456e6	0.719832	0.359916	+	0.932985i	$0.382805\pi$
$348$	0	0
$349$	−1.78417e6	−0.784103	−0.392052	−	0.919943i	$-0.628235\pi$
$349$	−1.78417e6	−0.784103	−0.392052	+	0.919943i	$0.628235\pi$
$350$	0	0
$351$	2.74484e6	1.18919
$352$	0	0
$353$	−1.02848e6	−0.439299	−0.219650	−	0.975579i	$-0.570491\pi$
$353$	−1.02848e6	−0.439299	−0.219650	+	0.975579i	$0.570491\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−1.95869e6	−0.813383
$358$	0	0
$359$	−719588.	−0.294678	−0.147339	−	0.989086i	$-0.547071\pi$
$359$	−719588.	−0.294678	−0.147339	+	0.989086i	$0.547071\pi$
$360$	0	0
$361$	3.43422e6	1.38695
$362$	0	0
$363$	1.14191e6	0.454848
$364$	0	0
$365$	0	0
$366$	0	0
$367$	412535.	0.159881	0.0799403	−	0.996800i	$-0.474527\pi$
$367$	412535.	0.159881	0.0799403	+	0.996800i	$0.474527\pi$
$368$	0	0
$369$	4.58669e6	1.75361
$370$	0	0
$371$	−1.26083e6	−0.475580
$372$	0	0
$373$	4.01227e6	1.49320	0.746600	−	0.665273i	$-0.231685\pi$
$373$	4.01227e6	1.49320	0.746600	+	0.665273i	$0.231685\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−732485.	−0.265427
$378$	0	0
$379$	725543.	0.259457	0.129728	−	0.991550i	$-0.458589\pi$
$379$	725543.	0.259457	0.129728	+	0.991550i	$0.458589\pi$
$380$	0	0
$381$	3.10564e6	1.09607
$382$	0	0
$383$	−4.16948e6	−1.45240	−0.726199	−	0.687485i	$-0.758715\pi$
$383$	−4.16948e6	−1.45240	−0.726199	+	0.687485i	$0.758715\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−8.24594e6	−2.79874
$388$	0	0
$389$	710805.	0.238164	0.119082	−	0.992884i	$-0.462005\pi$
$389$	710805.	0.238164	0.119082	+	0.992884i	$0.462005\pi$
$390$	0	0
$391$	2.16059e6	0.714712
$392$	0	0
$393$	−3.88161e6	−1.26774
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−2.60119e6	−0.828315	−0.414157	−	0.910205i	$-0.635924\pi$
$397$	−2.60119e6	−0.828315	−0.414157	+	0.910205i	$0.635924\pi$
$398$	0	0
$399$	3.16374e6	0.994874
$400$	0	0
$401$	1.82517e6	0.566817	0.283409	−	0.958999i	$-0.408535\pi$
$401$	1.82517e6	0.566817	0.283409	+	0.958999i	$0.408535\pi$
$402$	0	0
$403$	2.28030e6	0.699407
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−5.07703e6	−1.51923
$408$	0	0
$409$	−527537.	−0.155935	−0.0779677	−	0.996956i	$-0.524843\pi$
$409$	−527537.	−0.155935	−0.0779677	+	0.996956i	$0.524843\pi$
$410$	0	0
$411$	−8.80241e6	−2.57038
$412$	0	0
$413$	157180.	0.0453442
$414$	0	0
$415$	0	0
$416$	0	0
$417$	4.06319e6	1.14427
$418$	0	0
$419$	2.11729e6	0.589175	0.294587	−	0.955625i	$-0.404818\pi$
$419$	2.11729e6	0.589175	0.294587	+	0.955625i	$0.404818\pi$
$420$	0	0
$421$	1.67556e6	0.460738	0.230369	−	0.973103i	$-0.426007\pi$
$421$	1.67556e6	0.460738	0.230369	+	0.973103i	$0.426007\pi$
$422$	0	0
$423$	−5.16472e6	−1.40345
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−2.03235e6	−0.539423
$428$	0	0
$429$	−2.60341e6	−0.682965
$430$	0	0
$431$	1.44216e6	0.373957	0.186978	−	0.982364i	$-0.440131\pi$
$431$	1.44216e6	0.373957	0.186978	+	0.982364i	$0.440131\pi$
$432$	0	0
$433$	−5.49118e6	−1.40749	−0.703746	−	0.710452i	$-0.748491\pi$
$433$	−5.49118e6	−1.40749	−0.703746	+	0.710452i	$0.748491\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−3.48986e6	−0.874186
$438$	0	0
$439$	−3.57088e6	−0.884329	−0.442165	−	0.896934i	$-0.645789\pi$
$439$	−3.57088e6	−0.884329	−0.442165	+	0.896934i	$0.645789\pi$
$440$	0	0
$441$	−9.06121e6	−2.21865
$442$	0	0
$443$	2.15711e6	0.522231	0.261116	−	0.965308i	$-0.415910\pi$
$443$	2.15711e6	0.522231	0.261116	+	0.965308i	$0.415910\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	5.70174e6	1.34970
$448$	0	0
$449$	−4.15945e6	−0.973689	−0.486845	−	0.873489i	$-0.661852\pi$
$449$	−4.15945e6	−0.973689	−0.486845	+	0.873489i	$0.661852\pi$
$450$	0	0
$451$	−2.62083e6	−0.606733
$452$	0	0
$453$	4.72304e6	1.08138
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−6.32204e6	−1.41601	−0.708006	−	0.706207i	$-0.750405\pi$
$457$	−6.32204e6	−1.41601	−0.708006	+	0.706207i	$0.750405\pi$
$458$	0	0
$459$	1.61987e7	3.58880
$460$	0	0
$461$	−8.12007e6	−1.77954	−0.889769	−	0.456411i	$-0.849135\pi$
$461$	−8.12007e6	−1.77954	−0.889769	+	0.456411i	$0.849135\pi$
$462$	0	0
$463$	5.51746e6	1.19615	0.598076	−	0.801439i	$-0.295932\pi$
$463$	5.51746e6	1.19615	0.598076	+	0.801439i	$0.295932\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−7.98287e6	−1.69382	−0.846909	−	0.531737i	$-0.821539\pi$
$467$	−7.98287e6	−1.69382	−0.846909	+	0.531737i	$0.821539\pi$
$468$	0	0
$469$	−3.11007e6	−0.652888
$470$	0	0
$471$	3.15814e6	0.655963
$472$	0	0
$473$	4.71172e6	0.968338
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1.73084e7	3.48307
$478$	0	0
$479$	7.21796e6	1.43739	0.718697	−	0.695323i	$-0.244739\pi$
$479$	7.21796e6	1.43739	0.718697	+	0.695323i	$0.244739\pi$
$480$	0	0
$481$	−3.70742e6	−0.730650
$482$	0	0
$483$	−1.86809e6	−0.364358
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−1.74447e6	−0.333305	−0.166652	−	0.986016i	$-0.553296\pi$
$487$	−1.74447e6	−0.333305	−0.166652	+	0.986016i	$0.553296\pi$
$488$	0	0
$489$	−1.03797e6	−0.196297
$490$	0	0
$491$	9.10988e6	1.70533	0.852666	−	0.522457i	$-0.174985\pi$
$491$	9.10988e6	1.70533	0.852666	+	0.522457i	$0.174985\pi$
$492$	0	0
$493$	−4.32277e6	−0.801022
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−2.67553e6	−0.485869
$498$	0	0
$499$	−5.94465e6	−1.06875	−0.534374	−	0.845248i	$-0.679452\pi$
$499$	−5.94465e6	−1.06875	−0.534374	+	0.845248i	$0.679452\pi$
$500$	0	0
$501$	1.10307e7	1.96341
$502$	0	0
$503$	2.53195e6	0.446207	0.223103	−	0.974795i	$-0.428381\pi$
$503$	2.53195e6	0.446207	0.223103	+	0.974795i	$0.428381\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	8.95078e6	1.54647
$508$	0	0
$509$	4.91009e6	0.840030	0.420015	−	0.907517i	$-0.362025\pi$
$509$	4.91009e6	0.840030	0.420015	+	0.907517i	$0.362025\pi$
$510$	0	0
$511$	−2.19085e6	−0.371160
$512$	0	0
$513$	−2.61647e7	−4.38957
$514$	0	0
$515$	0	0
$516$	0	0
$517$	2.95112e6	0.485579
$518$	0	0
$519$	−4.16211e6	−0.678259
$520$	0	0
$521$	−5.90160e6	−0.952522	−0.476261	−	0.879304i	$-0.658008\pi$
$521$	−5.90160e6	−0.952522	−0.476261	+	0.879304i	$0.658008\pi$
$522$	0	0
$523$	−6.73830e6	−1.07720	−0.538600	−	0.842562i	$-0.681047\pi$
$523$	−6.73830e6	−1.07720	−0.538600	+	0.842562i	$0.681047\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.34572e7	2.11071
$528$	0	0
$529$	−4.37569e6	−0.679842
$530$	0	0
$531$	−2.15773e6	−0.332093
$532$	0	0
$533$	−1.91382e6	−0.291799
$534$	0	0
$535$	0	0
$536$	0	0
$537$	1.33649e7	2.00000
$538$	0	0
$539$	5.17757e6	0.767634
$540$	0	0
$541$	−6.56711e6	−0.964675	−0.482337	−	0.875986i	$-0.660212\pi$
$541$	−6.56711e6	−0.964675	−0.482337	+	0.875986i	$0.660212\pi$
$542$	0	0
$543$	2.54821e7	3.70881
$544$	0	0
$545$	0	0
$546$	0	0
$547$	1.27471e7	1.82156	0.910778	−	0.412896i	$-0.135483\pi$
$547$	1.27471e7	1.82156	0.910778	+	0.412896i	$0.135483\pi$
$548$	0	0
$549$	2.78997e7	3.95064
$550$	0	0
$551$	6.98226e6	0.979755
$552$	0	0
$553$	43702.0	0.00607699
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−1.22479e7	−1.67273	−0.836364	−	0.548175i	$-0.815323\pi$
$557$	−1.22479e7	−1.67273	−0.836364	+	0.548175i	$0.815323\pi$
$558$	0	0
$559$	3.44066e6	0.465707
$560$	0	0
$561$	−1.53640e7	−2.06109
$562$	0	0
$563$	2.23945e6	0.297763	0.148882	−	0.988855i	$-0.452433\pi$
$563$	2.23945e6	0.297763	0.148882	+	0.988855i	$0.452433\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−7.39237e6	−0.965664
$568$	0	0
$569$	−3.95223e6	−0.511755	−0.255877	−	0.966709i	$-0.582364\pi$
$569$	−3.95223e6	−0.511755	−0.255877	+	0.966709i	$0.582364\pi$
$570$	0	0
$571$	−6.99947e6	−0.898411	−0.449206	−	0.893428i	$-0.648293\pi$
$571$	−6.99947e6	−0.898411	−0.449206	+	0.893428i	$0.648293\pi$
$572$	0	0
$573$	−9.64949e6	−1.22777
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−3.20338e6	−0.400561	−0.200280	−	0.979739i	$-0.564185\pi$
$577$	−3.20338e6	−0.400561	−0.200280	+	0.979739i	$0.564185\pi$
$578$	0	0
$579$	−4.46927e6	−0.554039
$580$	0	0
$581$	−1.70854e6	−0.209983
$582$	0	0
$583$	−9.89002e6	−1.20511
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−1.35926e7	−1.62820	−0.814102	−	0.580722i	$-0.802770\pi$
$587$	−1.35926e7	−1.62820	−0.814102	+	0.580722i	$0.802770\pi$
$588$	0	0
$589$	−2.17365e7	−2.58168
$590$	0	0
$591$	−2.08936e7	−2.46062
$592$	0	0
$593$	−713635.	−0.0833373	−0.0416687	−	0.999131i	$-0.513267\pi$
$593$	−713635.	−0.0833373	−0.0416687	+	0.999131i	$0.513267\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−8.55984e6	−0.982946
$598$	0	0
$599$	5.09641e6	0.580359	0.290180	−	0.956972i	$-0.406285\pi$
$599$	5.09641e6	0.580359	0.290180	+	0.956972i	$0.406285\pi$
$600$	0	0
$601$	136310.	0.0153937	0.00769683	−	0.999970i	$-0.497550\pi$
$601$	136310.	0.0153937	0.00769683	+	0.999970i	$0.497550\pi$
$602$	0	0
$603$	4.26943e7	4.78164
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−6.23096e6	−0.686409	−0.343204	−	0.939261i	$-0.611512\pi$
$607$	−6.23096e6	−0.686409	−0.343204	+	0.939261i	$0.611512\pi$
$608$	0	0
$609$	3.73754e6	0.408359
$610$	0	0
$611$	2.15501e6	0.233532
$612$	0	0
$613$	1.37774e7	1.48087	0.740435	−	0.672128i	$-0.234620\pi$
$613$	1.37774e7	1.48087	0.740435	+	0.672128i	$0.234620\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	8.85147e6	0.936058	0.468029	−	0.883713i	$-0.344964\pi$
$617$	8.85147e6	0.936058	0.468029	+	0.883713i	$0.344964\pi$
$618$	0	0
$619$	1.22847e7	1.28866	0.644329	−	0.764748i	$-0.277137\pi$
$619$	1.22847e7	1.28866	0.644329	+	0.764748i	$0.277137\pi$
$620$	0	0
$621$	1.54494e7	1.60762
$622$	0	0
$623$	−1.75512e6	−0.181170
$624$	0	0
$625$	0	0
$626$	0	0
$627$	2.48164e7	2.52099
$628$	0	0
$629$	−2.18794e7	−2.20500
$630$	0	0
$631$	−1.75131e7	−1.75101	−0.875506	−	0.483207i	$-0.839472\pi$
$631$	−1.75131e7	−1.75101	−0.875506	+	0.483207i	$0.839472\pi$
$632$	0	0
$633$	−1.06889e7	−1.06029
$634$	0	0
$635$	0	0
$636$	0	0
$637$	3.78084e6	0.369181
$638$	0	0
$639$	3.67291e7	3.55842
$640$	0	0
$641$	−1.24879e7	−1.20045	−0.600223	−	0.799832i	$-0.704922\pi$
$641$	−1.24879e7	−1.20045	−0.600223	+	0.799832i	$0.704922\pi$
$642$	0	0
$643$	−5.50652e6	−0.525231	−0.262615	−	0.964901i	$-0.584585\pi$
$643$	−5.50652e6	−0.525231	−0.262615	+	0.964901i	$0.584585\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−2.74623e6	−0.257915	−0.128957	−	0.991650i	$-0.541163\pi$
$647$	−2.74623e6	−0.257915	−0.128957	+	0.991650i	$0.541163\pi$
$648$	0	0
$649$	1.23292e6	0.114901
$650$	0	0
$651$	−1.16353e7	−1.07604
$652$	0	0
$653$	1.24705e7	1.14446	0.572232	−	0.820092i	$-0.306078\pi$
$653$	1.24705e7	1.14446	0.572232	+	0.820092i	$0.306078\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	3.00755e7	2.71831
$658$	0	0
$659$	−1.75360e7	−1.57296	−0.786478	−	0.617618i	$-0.788098\pi$
$659$	−1.75360e7	−1.57296	−0.786478	+	0.617618i	$0.788098\pi$
$660$	0	0
$661$	9.04528e6	0.805227	0.402613	−	0.915370i	$-0.368102\pi$
$661$	9.04528e6	0.805227	0.402613	+	0.915370i	$0.368102\pi$
$662$	0	0
$663$	−1.12193e7	−0.991250
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−4.12281e6	−0.358821
$668$	0	0
$669$	2.18777e7	1.88989
$670$	0	0
$671$	−1.59418e7	−1.36689
$672$	0	0
$673$	9.09208e6	0.773794	0.386897	−	0.922123i	$-0.373547\pi$
$673$	9.09208e6	0.773794	0.386897	+	0.922123i	$0.373547\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−1.55584e7	−1.30465	−0.652325	−	0.757940i	$-0.726206\pi$
$677$	−1.55584e7	−1.30465	−0.652325	+	0.757940i	$0.726206\pi$
$678$	0	0
$679$	2.28597e6	0.190281
$680$	0	0
$681$	1.23965e7	1.02431
$682$	0	0
$683$	−5.22623e6	−0.428684	−0.214342	−	0.976759i	$-0.568761\pi$
$683$	−5.22623e6	−0.428684	−0.214342	+	0.976759i	$0.568761\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−3.00653e7	−2.43038
$688$	0	0
$689$	−7.22203e6	−0.579577
$690$	0	0
$691$	2.01073e7	1.60199	0.800994	−	0.598672i	$-0.204305\pi$
$691$	2.01073e7	1.60199	0.800994	+	0.598672i	$0.204305\pi$
$692$	0	0
$693$	9.50515e6	0.751840
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−1.12944e7	−0.880608
$698$	0	0
$699$	1.79493e7	1.38949
$700$	0	0
$701$	1.57869e7	1.21339	0.606695	−	0.794934i	$-0.292495\pi$
$701$	1.57869e7	1.21339	0.606695	+	0.794934i	$0.292495\pi$
$702$	0	0
$703$	3.53402e7	2.69700
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2.29559e6	0.172721
$708$	0	0
$709$	−5.13552e6	−0.383680	−0.191840	−	0.981426i	$-0.561445\pi$
$709$	−5.13552e6	−0.383680	−0.191840	+	0.981426i	$0.561445\pi$
$710$	0	0
$711$	−599930.	−0.0445068
$712$	0	0
$713$	1.28347e7	0.945502
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−3.98087e7	−2.89188
$718$	0	0
$719$	2.90785e6	0.209773	0.104886	−	0.994484i	$-0.466552\pi$
$719$	2.90785e6	0.209773	0.104886	+	0.994484i	$0.466552\pi$
$720$	0	0
$721$	283667.	0.0203222
$722$	0	0
$723$	−1.03575e7	−0.736898
$724$	0	0
$725$	0	0
$726$	0	0
$727$	7.85123e6	0.550937	0.275468	−	0.961310i	$-0.411167\pi$
$727$	7.85123e6	0.550937	0.275468	+	0.961310i	$0.411167\pi$
$728$	0	0
$729$	2.50435e7	1.74533
$730$	0	0
$731$	2.03051e7	1.40544
$732$	0	0
$733$	1.41173e7	0.970491	0.485245	−	0.874378i	$-0.338730\pi$
$733$	1.41173e7	0.970491	0.485245	+	0.874378i	$0.338730\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−2.43955e7	−1.65440
$738$	0	0
$739$	−4.20049e6	−0.282937	−0.141468	−	0.989943i	$-0.545182\pi$
$739$	−4.20049e6	−0.282937	−0.141468	+	0.989943i	$0.545182\pi$
$740$	0	0
$741$	1.81218e7	1.21243
$742$	0	0
$743$	−1.18490e6	−0.0787425	−0.0393712	−	0.999225i	$-0.512535\pi$
$743$	−1.18490e6	−0.0787425	−0.0393712	+	0.999225i	$0.512535\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	2.34544e7	1.53788
$748$	0	0
$749$	−251752.	−0.0163971
$750$	0	0
$751$	−3.94605e6	−0.255307	−0.127653	−	0.991819i	$-0.540745\pi$
$751$	−3.94605e6	−0.255307	−0.127653	+	0.991819i	$0.540745\pi$
$752$	0	0
$753$	2.41696e7	1.55340
$754$	0	0
$755$	0	0
$756$	0	0
$757$	1.22109e6	0.0774479	0.0387239	−	0.999250i	$-0.487671\pi$
$757$	1.22109e6	0.0774479	0.0387239	+	0.999250i	$0.487671\pi$
$758$	0	0
$759$	−1.46533e7	−0.923276
$760$	0	0
$761$	1.91652e7	1.19964	0.599822	−	0.800134i	$-0.295238\pi$
$761$	1.91652e7	1.19964	0.599822	+	0.800134i	$0.295238\pi$
$762$	0	0
$763$	9.70442e6	0.603473
$764$	0	0
$765$	0	0
$766$	0	0
$767$	900323.	0.0552599
$768$	0	0
$769$	9.50951e6	0.579886	0.289943	−	0.957044i	$-0.406364\pi$
$769$	9.50951e6	0.579886	0.289943	+	0.957044i	$0.406364\pi$
$770$	0	0
$771$	4.30539e7	2.60841
$772$	0	0
$773$	−526263.	−0.0316777	−0.0158389	−	0.999875i	$-0.505042\pi$
$773$	−526263.	−0.0316777	−0.0158389	+	0.999875i	$0.505042\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	1.89173e7	1.12410
$778$	0	0
$779$	1.82431e7	1.07710
$780$	0	0
$781$	−2.09870e7	−1.23118
$782$	0	0
$783$	−3.09101e7	−1.80176
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−2.05915e7	−1.18509	−0.592544	−	0.805538i	$-0.701877\pi$
$787$	−2.05915e7	−1.18509	−0.592544	+	0.805538i	$0.701877\pi$
$788$	0	0
$789$	1.95751e7	1.11947
$790$	0	0
$791$	−5.76109e6	−0.327389
$792$	0	0
$793$	−1.16413e7	−0.657382
$794$	0	0
$795$	0	0
$796$	0	0
$797$	2.08952e7	1.16520	0.582599	−	0.812760i	$-0.302036\pi$
$797$	2.08952e7	1.16520	0.582599	+	0.812760i	$0.302036\pi$
$798$	0	0
$799$	1.27178e7	0.704766
$800$	0	0
$801$	2.40939e7	1.32686
$802$	0	0
$803$	−1.71851e7	−0.940509
$804$	0	0
$805$	0	0
$806$	0	0
$807$	4.72694e6	0.255503
$808$	0	0
$809$	−3.38823e7	−1.82013	−0.910063	−	0.414470i	$-0.863967\pi$
$809$	−3.38823e7	−1.82013	−0.910063	+	0.414470i	$0.863967\pi$
$810$	0	0
$811$	8.25428e6	0.440684	0.220342	−	0.975423i	$-0.429283\pi$
$811$	8.25428e6	0.440684	0.220342	+	0.975423i	$0.429283\pi$
$812$	0	0
$813$	2.07459e7	1.10079
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−3.27974e7	−1.71903
$818$	0	0
$819$	6.94098e6	0.361586
$820$	0	0
$821$	1.99898e6	0.103503	0.0517513	−	0.998660i	$-0.483520\pi$
$821$	1.99898e6	0.103503	0.0517513	+	0.998660i	$0.483520\pi$
$822$	0	0
$823$	−1.28086e7	−0.659176	−0.329588	−	0.944125i	$-0.606910\pi$
$823$	−1.28086e7	−0.659176	−0.329588	+	0.944125i	$0.606910\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	2.83308e7	1.44044	0.720221	−	0.693745i	$-0.244040\pi$
$827$	2.83308e7	1.44044	0.720221	+	0.693745i	$0.244040\pi$
$828$	0	0
$829$	7.91663e6	0.400087	0.200043	−	0.979787i	$-0.435892\pi$
$829$	7.91663e6	0.400087	0.200043	+	0.979787i	$0.435892\pi$
$830$	0	0
$831$	3.65739e7	1.83725
$832$	0	0
$833$	2.23126e7	1.11414
$834$	0	0
$835$	0	0
$836$	0	0
$837$	9.62263e7	4.74767
$838$	0	0
$839$	−2.18676e7	−1.07250	−0.536249	−	0.844060i	$-0.680159\pi$
$839$	−2.18676e7	−1.07250	−0.536249	+	0.844060i	$0.680159\pi$
$840$	0	0
$841$	−1.22625e7	−0.597847
$842$	0	0
$843$	3.35289e7	1.62499
$844$	0	0
$845$	0	0
$846$	0	0
$847$	1.73961e6	0.0833188
$848$	0	0
$849$	−2.95637e7	−1.40763
$850$	0	0
$851$	−2.08673e7	−0.987739
$852$	0	0
$853$	6.73771e6	0.317059	0.158529	−	0.987354i	$-0.449325\pi$
$853$	6.73771e6	0.317059	0.158529	+	0.987354i	$0.449325\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−2.01368e7	−0.936565	−0.468283	−	0.883579i	$-0.655127\pi$
$857$	−2.01368e7	−0.936565	−0.468283	+	0.883579i	$0.655127\pi$
$858$	0	0
$859$	−2.31020e7	−1.06823	−0.534117	−	0.845410i	$-0.679356\pi$
$859$	−2.31020e7	−1.06823	−0.534117	+	0.845410i	$0.679356\pi$
$860$	0	0
$861$	9.76535e6	0.448932
$862$	0	0
$863$	2.79582e7	1.27785	0.638927	−	0.769267i	$-0.279378\pi$
$863$	2.79582e7	1.27785	0.638927	+	0.769267i	$0.279378\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−2.47125e7	−1.11652
$868$	0	0
$869$	342800.	0.0153989
$870$	0	0
$871$	−1.78144e7	−0.795658
$872$	0	0
$873$	−3.13812e7	−1.39359
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−3.75058e7	−1.64664	−0.823322	−	0.567574i	$-0.807882\pi$
$877$	−3.75058e7	−1.64664	−0.823322	+	0.567574i	$0.807882\pi$
$878$	0	0
$879$	−4.73477e6	−0.206693
$880$	0	0
$881$	−1.94116e6	−0.0842601	−0.0421300	−	0.999112i	$-0.513414\pi$
$881$	−1.94116e6	−0.0842601	−0.0421300	+	0.999112i	$0.513414\pi$
$882$	0	0
$883$	−3.90752e7	−1.68655	−0.843276	−	0.537481i	$-0.819376\pi$
$883$	−3.90752e7	−1.68655	−0.843276	+	0.537481i	$0.819376\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−1.87662e7	−0.800878	−0.400439	−	0.916323i	$-0.631142\pi$
$887$	−1.87662e7	−0.800878	−0.400439	+	0.916323i	$0.631142\pi$
$888$	0	0
$889$	4.73119e6	0.200778
$890$	0	0
$891$	−5.79859e7	−2.44697
$892$	0	0
$893$	−2.05422e7	−0.862020
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−1.07004e7	−0.444035
$898$	0	0
$899$	−2.56788e7	−1.05968
$900$	0	0
$901$	−4.26209e7	−1.74908
$902$	0	0
$903$	−1.75561e7	−0.716489
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−2.79416e7	−1.12780	−0.563901	−	0.825842i	$-0.690700\pi$
$907$	−2.79416e7	−1.12780	−0.563901	+	0.825842i	$0.690700\pi$
$908$	0	0
$909$	−3.15133e7	−1.26498
$910$	0	0
$911$	−1.32374e7	−0.528452	−0.264226	−	0.964461i	$-0.585116\pi$
$911$	−1.32374e7	−0.528452	−0.264226	+	0.964461i	$0.585116\pi$
$912$	0	0
$913$	−1.34018e7	−0.532092
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−5.91332e6	−0.232224
$918$	0	0
$919$	2.80433e7	1.09532	0.547660	−	0.836701i	$-0.315519\pi$
$919$	2.80433e7	1.09532	0.547660	+	0.836701i	$0.315519\pi$
$920$	0	0
$921$	−3.52951e7	−1.37109
$922$	0	0
$923$	−1.53254e7	−0.592117
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−3.89411e6	−0.148836
$928$	0	0
$929$	1.96341e7	0.746400	0.373200	−	0.927751i	$-0.378261\pi$
$929$	1.96341e7	0.746400	0.373200	+	0.927751i	$0.378261\pi$
$930$	0	0
$931$	−3.60401e7	−1.36273
$932$	0	0
$933$	−8.49871e7	−3.19631
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−2.71944e7	−1.01188	−0.505942	−	0.862567i	$-0.668855\pi$
$937$	−2.71944e7	−1.01188	−0.505942	+	0.862567i	$0.668855\pi$
$938$	0	0
$939$	−3.72133e7	−1.37732
$940$	0	0
$941$	2.74326e7	1.00993	0.504966	−	0.863139i	$-0.331505\pi$
$941$	2.74326e7	1.00993	0.504966	+	0.863139i	$0.331505\pi$
$942$	0	0
$943$	−1.07720e7	−0.394472
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.42413e7	0.516029	0.258014	−	0.966141i	$-0.416932\pi$
$947$	1.42413e7	0.516029	0.258014	+	0.966141i	$0.416932\pi$
$948$	0	0
$949$	−1.25491e7	−0.452323
$950$	0	0
$951$	3.92535e7	1.40743
$952$	0	0
$953$	−3.25786e7	−1.16198	−0.580992	−	0.813910i	$-0.697335\pi$
$953$	−3.25786e7	−1.16198	−0.580992	+	0.813910i	$0.697335\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	2.93173e7	1.03477
$958$	0	0
$959$	−1.34097e7	−0.470841
$960$	0	0
$961$	5.13117e7	1.79229
$962$	0	0
$963$	3.45598e6	0.120090
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−4.03932e6	−0.138913	−0.0694564	−	0.997585i	$-0.522126\pi$
$967$	−4.03932e6	−0.138913	−0.0694564	+	0.997585i	$0.522126\pi$
$968$	0	0
$969$	1.06946e8	3.65894
$970$	0	0
$971$	−2.98023e7	−1.01438	−0.507191	−	0.861834i	$-0.669316\pi$
$971$	−2.98023e7	−1.01438	−0.507191	+	0.861834i	$0.669316\pi$
$972$	0	0
$973$	6.18993e6	0.209606
$974$	0	0
$975$	0	0
$976$	0	0
$977$	2.70486e7	0.906586	0.453293	−	0.891362i	$-0.350249\pi$
$977$	2.70486e7	0.906586	0.453293	+	0.891362i	$0.350249\pi$
$978$	0	0
$979$	−1.37672e7	−0.459082
$980$	0	0
$981$	−1.33220e8	−4.41974
$982$	0	0
$983$	3.15761e7	1.04226	0.521128	−	0.853479i	$-0.325511\pi$
$983$	3.15761e7	1.04226	0.521128	+	0.853479i	$0.325511\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−1.09960e7	−0.359288
$988$	0	0
$989$	1.93658e7	0.629572
$990$	0	0
$991$	6.49963e6	0.210235	0.105117	−	0.994460i	$-0.466478\pi$
$991$	6.49963e6	0.210235	0.105117	+	0.994460i	$0.466478\pi$
$992$	0	0
$993$	4.65443e7	1.49794
$994$	0	0
$995$	0	0
$996$	0	0
$997$	1.34950e7	0.429966	0.214983	−	0.976618i	$-0.431030\pi$
$997$	1.34950e7	0.429966	0.214983	+	0.976618i	$0.431030\pi$
$998$	0	0
$999$	−1.56449e8	−4.95975

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.6.a.n.1.1		3
4.3	odd	2		800.6.a.o.1.3		3
5.2	odd	4		800.6.c.j.449.6		6
5.3	odd	4		800.6.c.j.449.1		6
5.4	even	2		160.6.a.g.1.3	yes	3
20.3	even	4		800.6.c.k.449.6		6
20.7	even	4		800.6.c.k.449.1		6
20.19	odd	2		160.6.a.f.1.1	✓	3
40.19	odd	2		320.6.a.y.1.3		3
40.29	even	2		320.6.a.x.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.6.a.f.1.1	✓	3	20.19	odd	2
160.6.a.g.1.3	yes	3	5.4	even	2
320.6.a.x.1.1		3	40.29	even	2
320.6.a.y.1.3		3	40.19	odd	2
800.6.a.n.1.1		3	1.1	even	1	trivial
800.6.a.o.1.3		3	4.3	odd	2
800.6.c.j.449.1		6	5.3	odd	4
800.6.c.j.449.6		6	5.2	odd	4
800.6.c.k.449.1		6	20.7	even	4
800.6.c.k.449.6		6	20.3	even	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:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	800.a (trivial)

\(f(q)\)	\(=\)	\(q-29.2272 q^{3} -44.5253 q^{7} +611.232 q^{9} +O(q^{10})\)	\(q-29.2272 q^{3} -44.5253 q^{7} +611.232 q^{9} -349.258 q^{11} -255.040 q^{13} -1505.12 q^{17} +2431.11 q^{19} +1301.35 q^{21} -1435.50 q^{23} -10762.4 q^{27} +2872.04 q^{29} -8940.96 q^{31} +10207.8 q^{33} +14536.6 q^{37} +7454.11 q^{39} +7504.01 q^{41} -13490.7 q^{43} -8449.69 q^{47} -14824.5 q^{49} +43990.5 q^{51} +28317.3 q^{53} -71054.8 q^{57} -3530.13 q^{59} +45644.9 q^{61} -27215.3 q^{63} +69849.6 q^{67} +41955.6 q^{69} +60090.2 q^{71} +49204.6 q^{73} +15550.8 q^{77} -981.509 q^{79} +166026. q^{81} +38372.3 q^{83} -83941.9 q^{87} +39418.6 q^{89} +11355.7 q^{91} +261320. q^{93} -51340.9 q^{97} -213477. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 10 q^{3} + 6 q^{7} + 467 q^{9}+O(q^{10}) \) 3 * q - 10 * q^3 + 6 * q^7 + 467 * q^9	\( 3 q - 10 q^{3} + 6 q^{7} + 467 q^{9} - 396 q^{11} + 354 q^{13} - 1158 q^{17} + 3192 q^{19} - 820 q^{21} - 6126 q^{23} - 13804 q^{27} + 426 q^{29} + 3276 q^{31} + 2408 q^{33} + 11562 q^{37} + 21348 q^{39} + 12450 q^{41} - 26346 q^{43} - 36762 q^{47} - 3849 q^{49} + 71444 q^{51} + 21162 q^{53} - 69136 q^{57} + 35040 q^{59} - 24138 q^{61} - 80986 q^{63} + 9570 q^{67} - 27036 q^{69} + 88092 q^{71} - 66750 q^{73} + 136488 q^{77} - 92952 q^{79} + 151391 q^{81} - 30258 q^{83} + 26228 q^{87} + 172686 q^{89} - 106812 q^{91} + 318232 q^{93} - 170910 q^{97} - 351436 q^{99}+O(q^{100}) \) 3 * q - 10 * q^3 + 6 * q^7 + 467 * q^9 - 396 * q^11 + 354 * q^13 - 1158 * q^17 + 3192 * q^19 - 820 * q^21 - 6126 * q^23 - 13804 * q^27 + 426 * q^29 + 3276 * q^31 + 2408 * q^33 + 11562 * q^37 + 21348 * q^39 + 12450 * q^41 - 26346 * q^43 - 36762 * q^47 - 3849 * q^49 + 71444 * q^51 + 21162 * q^53 - 69136 * q^57 + 35040 * q^59 - 24138 * q^61 - 80986 * q^63 + 9570 * q^67 - 27036 * q^69 + 88092 * q^71 - 66750 * q^73 + 136488 * q^77 - 92952 * q^79 + 151391 * q^81 - 30258 * q^83 + 26228 * q^87 + 172686 * q^89 - 106812 * q^91 + 318232 * q^93 - 170910 * q^97 - 351436 * q^99

Introduction

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