LMFDB - Embedded newform 800.6.c.j.449.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,6,Mod(449,800)]

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, 1]))

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

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

chi := DirichletCharacter("800.449");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	800.c (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$128.307055850$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.6140289600.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 32x^{4} + 116x^{3} + 256x^{2} + 2778x + 7605 $ x^6 - 2x^5 + 32x^4 + 116x^3 + 256x^2 + 2778*x + 7605
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{16} $
Twist minimal:	no (minimal twist has level 160)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.6
Root			$-2.90341 - 0.978064i$ of defining polynomial
Character	$\chi$	$=$	800.449
Dual form			800.6.c.j.449.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+29.2272i q^{3} -44.5253i q^{7} -611.232 q^{9} +O(q^{10})$	$q+29.2272i q^{3} -44.5253i q^{7} -611.232 q^{9} -349.258 q^{11} +255.040i q^{13} -1505.12i q^{17} -2431.11 q^{19} +1301.35 q^{21} +1435.50i q^{23} -10762.4i q^{27} -2872.04 q^{29} -8940.96 q^{31} -10207.8i q^{33} +14536.6i q^{37} -7454.11 q^{39} +7504.01 q^{41} +13490.7i q^{43} -8449.69i q^{47} +14824.5 q^{49} +43990.5 q^{51} -28317.3i q^{53} -71054.8i q^{57} +3530.13 q^{59} +45644.9 q^{61} +27215.3i q^{63} +69849.6i q^{67} -41955.6 q^{69} +60090.2 q^{71} -49204.6i q^{73} +15550.8i q^{77} +981.509 q^{79} +166026. q^{81} -38372.3i q^{83} -83941.9i q^{87} -39418.6 q^{89} +11355.7 q^{91} -261320. i q^{93} -51340.9i q^{97} +213477. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 934 q^{9}+O(q^{10}) $ 6 * q - 934 * q^9	$ 6 q - 934 q^{9} - 792 q^{11} - 6384 q^{19} - 1640 q^{21} - 852 q^{29} + 6552 q^{31} - 42696 q^{39} + 24900 q^{41} + 7698 q^{49} + 142888 q^{51} - 70080 q^{59} - 48276 q^{61} + 54072 q^{69} + 176184 q^{71} + 185904 q^{79} + 302782 q^{81} - 345372 q^{89} - 213624 q^{91} + 702872 q^{99}+O(q^{100}) $ 6 * q - 934 * q^9 - 792 * q^11 - 6384 * q^19 - 1640 * q^21 - 852 * q^29 + 6552 * q^31 - 42696 * q^39 + 24900 * q^41 + 7698 * q^49 + 142888 * q^51 - 70080 * q^59 - 48276 * q^61 + 54072 * q^69 + 176184 * q^71 + 185904 * q^79 + 302782 * q^81 - 345372 * q^89 - 213624 * q^91 + 702872 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/800\mathbb{Z}\right)^\times$.

$n$	$101$	$351$	$577$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	29.2272i	1.87493i	0.348081	+	0.937464i	$0.386833\pi$
$3$	29.2272i	1.87493i	−0.348081	+	0.937464i	$0.613167\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 44.5253i	− 0.343449i	−0.985145	−	0.171724i	$-0.945066\pi$
$7$	− 44.5253i	− 0.343449i	0.985145	−	0.171724i	$-0.0549338\pi$
$8$	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
$13$	255.040i	0.418552i	0.977857	+	0.209276i	$0.0671108\pi$
$13$	255.040i	0.418552i	−0.977857	+	0.209276i	$0.932889\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 1505.12i	− 1.26313i	−0.775322	−	0.631566i	$-0.782412\pi$
$17$	− 1505.12i	− 1.26313i	0.775322	−	0.631566i	$-0.217588\pi$
$18$	0	0
$19$	−2431.11	−1.54498	−0.772488	−	0.635030i	$-0.780988\pi$
$19$	−2431.11	−1.54498	−0.772488	+	0.635030i	$0.780988\pi$
$20$	0	0
$21$	1301.35	0.643942
$22$	0	0
$23$	1435.50i	0.565825i	0.959146	+	0.282913i	$0.0913007\pi$
$23$	1435.50i	0.565825i	−0.959146	+	0.282913i	$0.908699\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 10762.4i	− 2.84119i
$28$	0	0
$29$	−2872.04	−0.634156	−0.317078	−	0.948400i	$-0.602702\pi$
$29$	−2872.04	−0.634156	−0.317078	+	0.948400i	$0.602702\pi$
$30$	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
$33$	− 10207.8i	− 1.63173i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	14536.6i	1.74566i	0.488024	+	0.872830i	$0.337718\pi$
$37$	14536.6i	1.74566i	−0.488024	+	0.872830i	$0.662282\pi$
$38$	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
$43$	13490.7i	1.11266i	0.830961	+	0.556331i	$0.187791\pi$
$43$	13490.7i	1.11266i	−0.830961	+	0.556331i	$0.812209\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 8449.69i	− 0.557951i	−0.960298	−	0.278975i	$-0.910005\pi$
$47$	− 8449.69i	− 0.557951i	0.960298	−	0.278975i	$-0.0899948\pi$
$48$	0	0
$49$	14824.5	0.882043
$50$	0	0
$51$	43990.5	2.36828
$52$	0	0
$53$	− 28317.3i	− 1.38472i	−0.721553	−	0.692360i	$-0.756571\pi$
$53$	− 28317.3i	− 1.38472i	0.721553	−	0.692360i	$-0.243429\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 71054.8i	− 2.89672i
$58$	0	0
$59$	3530.13	0.132026	0.0660131	−	0.997819i	$-0.478972\pi$
$59$	3530.13	0.132026	0.0660131	+	0.997819i	$0.478972\pi$
$60$	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
$63$	27215.3i	0.863896i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	69849.6i	1.90098i	0.310759	+	0.950489i	$0.399417\pi$
$67$	69849.6i	1.90098i	−0.310759	+	0.950489i	$0.600583\pi$
$68$	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
$73$	− 49204.6i	− 1.08068i	−0.841445	−	0.540342i	$-0.818295\pi$
$73$	− 49204.6i	− 1.08068i	0.841445	−	0.540342i	$-0.181705\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	15550.8i	0.298900i
$78$	0	0
$79$	981.509	0.0176940	0.00884702	−	0.999961i	$-0.497184\pi$
$79$	981.509	0.0176940	0.00884702	+	0.999961i	$0.497184\pi$
$80$	0	0
$81$	166026.	2.81167
$82$	0	0
$83$	− 38372.3i	− 0.611396i	−0.952129	−	0.305698i	$-0.901110\pi$
$83$	− 38372.3i	− 0.611396i	0.952129	−	0.305698i	$-0.0988898\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 83941.9i	− 1.18900i
$88$	0	0
$89$	−39418.6	−0.527504	−0.263752	−	0.964591i	$-0.584960\pi$
$89$	−39418.6	−0.527504	−0.263752	+	0.964591i	$0.584960\pi$
$90$	0	0
$91$	11355.7	0.143751
$92$	0	0
$93$	− 261320.i	− 3.13303i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 51340.9i	− 0.554031i	−0.960865	−	0.277016i	$-0.910655\pi$
$97$	− 51340.9i	− 0.554031i	0.960865	−	0.277016i	$-0.0893454\pi$
$98$	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
$103$	6370.92i	0.0591710i	0.999562	+	0.0295855i	$0.00941873\pi$
$103$	6370.92i	0.0591710i	−0.999562	+	0.0295855i	$0.990581\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5654.13i	0.0477426i	0.999715	+	0.0238713i	$0.00759919\pi$
$107$	5654.13i	0.0477426i	−0.999715	+	0.0238713i	$0.992401\pi$
$108$	0	0
$109$	217953.	1.75710	0.878550	−	0.477650i	$-0.158511\pi$
$109$	217953.	1.75710	0.878550	+	0.477650i	$0.158511\pi$
$110$	0	0
$111$	−424866.	−3.27299
$112$	0	0
$113$	− 129389.i	− 0.953240i	−0.879109	−	0.476620i	$-0.841862\pi$
$113$	− 129389.i	− 0.953240i	0.879109	−	0.476620i	$-0.158138\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 155889.i	− 1.05281i
$118$	0	0
$119$	−67015.9	−0.433821
$120$	0	0
$121$	−39070.1	−0.242595
$122$	0	0
$123$	219322.i	1.30713i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 106258.i	− 0.584594i	−0.956328	−	0.292297i	$-0.905580\pi$
$127$	− 106258.i	− 0.584594i	0.956328	−	0.292297i	$-0.0944195\pi$
$128$	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.i	0.530619i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	301171.i	1.37092i	0.728110	+	0.685460i	$0.240399\pi$
$137$	301171.i	1.37092i	−0.728110	+	0.685460i	$0.759601\pi$
$138$	0	0
$139$	139021.	0.610298	0.305149	−	0.952305i	$-0.401294\pi$
$139$	139021.	0.610298	0.305149	+	0.952305i	$0.401294\pi$
$140$	0	0
$141$	246961.	1.04612
$142$	0	0
$143$	− 89074.6i	− 0.364262i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	433279.i	1.65377i
$148$	0	0
$149$	195083.	0.719869	0.359934	−	0.932978i	$-0.382799\pi$
$149$	195083.	0.719869	0.359934	+	0.932978i	$0.382799\pi$
$150$	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.i	3.17723i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 108055.i	− 0.349860i	−0.984581	−	0.174930i	$-0.944030\pi$
$157$	− 108055.i	− 0.349860i	0.984581	−	0.174930i	$-0.0559700\pi$
$158$	0	0
$159$	827636.	2.59625
$160$	0	0
$161$	63915.9	0.194332
$162$	0	0
$163$	− 35513.9i	− 0.104696i	−0.998629	−	0.0523479i	$-0.983330\pi$
$163$	− 35513.9i	− 0.104696i	0.998629	−	0.0523479i	$-0.0166705\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 377413.i	− 1.04719i	−0.851967	−	0.523595i	$-0.824591\pi$
$167$	− 377413.i	− 1.04719i	0.851967	−	0.523595i	$-0.175409\pi$
$168$	0	0
$169$	306248.	0.824814
$170$	0	0
$171$	1.48598e6	3.88617
$172$	0	0
$173$	− 142405.i	− 0.361752i	−0.983506	−	0.180876i	$-0.942107\pi$
$173$	− 142405.i	− 0.361752i	0.983506	−	0.180876i	$-0.0578932\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	103176.i	0.247540i
$178$	0	0
$179$	457275.	1.06671	0.533353	−	0.845893i	$-0.320932\pi$
$179$	457275.	1.06671	0.533353	+	0.845893i	$0.320932\pi$
$180$	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
$183$	1.33408e6i	2.94478i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	525675.i	1.09929i
$188$	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.i	− 0.295498i	−0.989025	−	0.147749i	$-0.952797\pi$
$193$	− 152914.i	− 0.295498i	0.989025	−	0.147749i	$-0.0472029\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	714867.i	1.31238i	0.754596	+	0.656190i	$0.227833\pi$
$197$	714867.i	1.31238i	−0.754596	+	0.656190i	$0.772167\pi$
$198$	0	0
$199$	−292872.	−0.524258	−0.262129	−	0.965033i	$-0.584425\pi$
$199$	−292872.	−0.524258	−0.262129	+	0.965033i	$0.584425\pi$
$200$	0	0
$201$	−2.04151e6	−3.56420
$202$	0	0
$203$	127879.i	0.217800i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 877421.i	− 1.42325i
$208$	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
$213$	1.75627e6i	2.65242i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	398099.i	0.573907i
$218$	0	0
$219$	1.43812e6	2.02621
$220$	0	0
$221$	383866.	0.528687
$222$	0	0
$223$	748539.i	1.00798i	0.863709	+	0.503990i	$0.168135\pi$
$223$	748539.i	1.00798i	−0.863709	+	0.503990i	$0.831865\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 424142.i	− 0.546319i	−0.961969	−	0.273159i	$-0.911931\pi$
$227$	− 424142.i	− 0.546319i	0.961969	−	0.273159i	$-0.0880687\pi$
$228$	0	0
$229$	−1.02868e6	−1.29625	−0.648126	−	0.761533i	$-0.724447\pi$
$229$	−1.02868e6	−1.29625	−0.648126	+	0.761533i	$0.724447\pi$
$230$	0	0
$231$	−454507.	−0.560416
$232$	0	0
$233$	614128.i	0.741087i	0.928815	+	0.370544i	$0.120829\pi$
$233$	614128.i	0.741087i	−0.928815	+	0.370544i	$0.879171\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	28686.8i	0.0331750i
$238$	0	0
$239$	−1.36204e6	−1.54239	−0.771197	−	0.636597i	$-0.780342\pi$
$239$	−1.36204e6	−1.54239	−0.771197	+	0.636597i	$0.780342\pi$
$240$	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
$243$	2.23722e6i	2.43049i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 620031.i	− 0.646653i
$248$	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.i	− 0.492432i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 1.47307e6i	− 1.39121i	−0.718426	−	0.695603i	$-0.755137\pi$
$257$	− 1.47307e6i	− 1.39121i	0.718426	−	0.695603i	$-0.244863\pi$
$258$	0	0
$259$	647248.	0.599545
$260$	0	0
$261$	1.75548e6	1.59513
$262$	0	0
$263$	669755.i	0.597072i	0.954398	+	0.298536i	$0.0964982\pi$
$263$	669755.i	0.597072i	−0.954398	+	0.298536i	$0.903502\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 1.15210e6i	− 0.989032i
$268$	0	0
$269$	161730.	0.136273	0.0681367	−	0.997676i	$-0.478295\pi$
$269$	161730.	0.136273	0.0681367	+	0.997676i	$0.478295\pi$
$270$	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.i	0.269523i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 1.25136e6i	− 0.979904i	−0.871749	−	0.489952i	$-0.837014\pi$
$277$	− 1.25136e6i	− 0.979904i	0.871749	−	0.489952i	$-0.162986\pi$
$278$	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
$283$	− 1.01151e6i	− 0.750766i	−0.926870	−	0.375383i	$-0.877511\pi$
$283$	− 1.01151e6i	− 0.750766i	0.926870	−	0.375383i	$-0.122489\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 334118.i	− 0.239439i
$288$	0	0
$289$	−845528.	−0.595502
$290$	0	0
$291$	1.50055e6	1.03877
$292$	0	0
$293$	− 161999.i	− 0.110241i	−0.998480	−	0.0551204i	$-0.982446\pi$
$293$	− 161999.i	− 0.110241i	0.998480	−	0.0551204i	$-0.0175542\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	3.75885e6i	2.47266i
$298$	0	0
$299$	−366109.	−0.236828
$300$	0	0
$301$	600677.	0.382142
$302$	0	0
$303$	− 1.50687e6i	− 0.942907i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	1.20761e6i	0.731276i	0.930757	+	0.365638i	$0.119149\pi$
$307$	1.20761e6i	0.731276i	−0.930757	+	0.365638i	$0.880851\pi$
$308$	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
$313$	− 1.27324e6i	− 0.734597i	−0.930103	−	0.367298i	$-0.880283\pi$
$313$	− 1.27324e6i	− 0.734597i	0.930103	−	0.367298i	$-0.119717\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 1.34305e6i	− 0.750659i	−0.926891	−	0.375330i	$-0.877530\pi$
$317$	− 1.34305e6i	− 0.750659i	0.926891	−	0.375330i	$-0.122470\pi$
$318$	0	0
$319$	1.00308e6	0.551899
$320$	0	0
$321$	−165255.	−0.0895140
$322$	0	0
$323$	3.65912e6i	1.95151i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	6.37016e6i	3.29444i
$328$	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
$333$	− 8.88526e6i	− 4.39096i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 1.34715e6i	− 0.646163i	−0.946371	−	0.323082i	$-0.895281\pi$
$337$	− 1.34715e6i	− 0.646163i	0.946371	−	0.323082i	$-0.104719\pi$
$338$	0	0
$339$	3.78169e6	1.78726
$340$	0	0
$341$	3.12270e6	1.45427
$342$	0	0
$343$	− 1.40840e6i	− 0.646385i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.61456e6i	0.719832i	0.932985	+	0.359916i	$0.117195\pi$
$347$	1.61456e6i	0.719832i	−0.932985	+	0.359916i	$0.882805\pi$
$348$	0	0
$349$	1.78417e6	0.784103	0.392052	−	0.919943i	$-0.371765\pi$
$349$	1.78417e6	0.784103	0.392052	+	0.919943i	$0.371765\pi$
$350$	0	0
$351$	2.74484e6	1.18919
$352$	0	0
$353$	1.02848e6i	0.439299i	0.975579	+	0.219650i	$0.0704914\pi$
$353$	1.02848e6i	0.439299i	−0.975579	+	0.219650i	$0.929509\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 1.95869e6i	− 0.813383i
$358$	0	0
$359$	719588.	0.294678	0.147339	−	0.989086i	$-0.452929\pi$
$359$	719588.	0.294678	0.147339	+	0.989086i	$0.452929\pi$
$360$	0	0
$361$	3.43422e6	1.38695
$362$	0	0
$363$	− 1.14191e6i	− 0.454848i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	412535.i	0.159881i	0.996800	+	0.0799403i	$0.0254730\pi$
$367$	412535.i	0.159881i	−0.996800	+	0.0799403i	$0.974527\pi$
$368$	0	0
$369$	−4.58669e6	−1.75361
$370$	0	0
$371$	−1.26083e6	−0.475580
$372$	0	0
$373$	− 4.01227e6i	− 1.49320i	−0.665273	−	0.746600i	$-0.731685\pi$
$373$	− 4.01227e6i	− 1.49320i	0.665273	−	0.746600i	$-0.268315\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 732485.i	− 0.265427i
$378$	0	0
$379$	−725543.	−0.259457	−0.129728	−	0.991550i	$-0.541411\pi$
$379$	−725543.	−0.259457	−0.129728	+	0.991550i	$0.541411\pi$
$380$	0	0
$381$	3.10564e6	1.09607
$382$	0	0
$383$	4.16948e6i	1.45240i	0.687485	+	0.726199i	$0.258715\pi$
$383$	4.16948e6i	1.45240i	−0.687485	+	0.726199i	$0.741285\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 8.24594e6i	− 2.79874i
$388$	0	0
$389$	−710805.	−0.238164	−0.119082	−	0.992884i	$-0.537995\pi$
$389$	−710805.	−0.238164	−0.119082	+	0.992884i	$0.537995\pi$
$390$	0	0
$391$	2.16059e6	0.714712
$392$	0	0
$393$	3.88161e6i	1.26774i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 2.60119e6i	− 0.828315i	−0.910205	−	0.414157i	$-0.864076\pi$
$397$	− 2.60119e6i	− 0.828315i	0.910205	−	0.414157i	$-0.135924\pi$
$398$	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
$403$	− 2.28030e6i	− 0.699407i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 5.07703e6i	− 1.51923i
$408$	0	0
$409$	527537.	0.155935	0.0779677	−	0.996956i	$-0.475157\pi$
$409$	527537.	0.155935	0.0779677	+	0.996956i	$0.475157\pi$
$410$	0	0
$411$	−8.80241e6	−2.57038
$412$	0	0
$413$	− 157180.i	− 0.0453442i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	4.06319e6i	1.14427i
$418$	0	0
$419$	−2.11729e6	−0.589175	−0.294587	−	0.955625i	$-0.595182\pi$
$419$	−2.11729e6	−0.589175	−0.294587	+	0.955625i	$0.595182\pi$
$420$	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
$423$	5.16472e6i	1.40345i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 2.03235e6i	− 0.539423i
$428$	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
$433$	5.49118e6i	1.40749i	0.710452	+	0.703746i	$0.248491\pi$
$433$	5.49118e6i	1.40749i	−0.710452	+	0.703746i	$0.751509\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 3.48986e6i	− 0.874186i
$438$	0	0
$439$	3.57088e6	0.884329	0.442165	−	0.896934i	$-0.354211\pi$
$439$	3.57088e6	0.884329	0.442165	+	0.896934i	$0.354211\pi$
$440$	0	0
$441$	−9.06121e6	−2.21865
$442$	0	0
$443$	− 2.15711e6i	− 0.522231i	−0.965308	−	0.261116i	$-0.915910\pi$
$443$	− 2.15711e6i	− 0.522231i	0.965308	−	0.261116i	$-0.0840904\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	5.70174e6i	1.34970i
$448$	0	0
$449$	4.15945e6	0.973689	0.486845	−	0.873489i	$-0.338148\pi$
$449$	4.15945e6	0.973689	0.486845	+	0.873489i	$0.338148\pi$
$450$	0	0
$451$	−2.62083e6	−0.606733
$452$	0	0
$453$	− 4.72304e6i	− 1.08138i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 6.32204e6i	− 1.41601i	−0.706207	−	0.708006i	$-0.749595\pi$
$457$	− 6.32204e6i	− 1.41601i	0.706207	−	0.708006i	$-0.250405\pi$
$458$	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
$463$	− 5.51746e6i	− 1.19615i	−0.801439	−	0.598076i	$-0.795932\pi$
$463$	− 5.51746e6i	− 1.19615i	0.801439	−	0.598076i	$-0.204068\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 7.98287e6i	− 1.69382i	−0.531737	−	0.846909i	$-0.678461\pi$
$467$	− 7.98287e6i	− 1.69382i	0.531737	−	0.846909i	$-0.321539\pi$
$468$	0	0
$469$	3.11007e6	0.652888
$470$	0	0
$471$	3.15814e6	0.655963
$472$	0	0
$473$	− 4.71172e6i	− 0.968338i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1.73084e7i	3.48307i
$478$	0	0
$479$	−7.21796e6	−1.43739	−0.718697	−	0.695323i	$-0.755261\pi$
$479$	−7.21796e6	−1.43739	−0.718697	+	0.695323i	$0.755261\pi$
$480$	0	0
$481$	−3.70742e6	−0.730650
$482$	0	0
$483$	1.86809e6i	0.364358i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 1.74447e6i	− 0.333305i	−0.986016	−	0.166652i	$-0.946704\pi$
$487$	− 1.74447e6i	− 0.333305i	0.986016	−	0.166652i	$-0.0532958\pi$
$488$	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
$493$	4.32277e6i	0.801022i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 2.67553e6i	− 0.485869i
$498$	0	0
$499$	5.94465e6	1.06875	0.534374	−	0.845248i	$-0.320548\pi$
$499$	5.94465e6	1.06875	0.534374	+	0.845248i	$0.320548\pi$
$500$	0	0
$501$	1.10307e7	1.96341
$502$	0	0
$503$	− 2.53195e6i	− 0.446207i	−0.974795	−	0.223103i	$-0.928381\pi$
$503$	− 2.53195e6i	− 0.446207i	0.974795	−	0.223103i	$-0.0716187\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	8.95078e6i	1.54647i
$508$	0	0
$509$	−4.91009e6	−0.840030	−0.420015	−	0.907517i	$-0.637975\pi$
$509$	−4.91009e6	−0.840030	−0.420015	+	0.907517i	$0.637975\pi$
$510$	0	0
$511$	−2.19085e6	−0.371160
$512$	0	0
$513$	2.61647e7i	4.38957i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	2.95112e6i	0.485579i
$518$	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
$523$	6.73830e6i	1.07720i	0.842562	+	0.538600i	$0.181047\pi$
$523$	6.73830e6i	1.07720i	−0.842562	+	0.538600i	$0.818953\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.34572e7i	2.11071i
$528$	0	0
$529$	4.37569e6	0.679842
$530$	0	0
$531$	−2.15773e6	−0.332093
$532$	0	0
$533$	1.91382e6i	0.291799i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	1.33649e7i	2.00000i
$538$	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
$543$	− 2.54821e7i	− 3.70881i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	1.27471e7i	1.82156i	0.412896	+	0.910778i	$0.364517\pi$
$547$	1.27471e7i	1.82156i	−0.412896	+	0.910778i	$0.635483\pi$
$548$	0	0
$549$	−2.78997e7	−3.95064
$550$	0	0
$551$	6.98226e6	0.979755
$552$	0	0
$553$	− 43702.0i	− 0.00607699i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 1.22479e7i	− 1.67273i	−0.548175	−	0.836364i	$-0.684677\pi$
$557$	− 1.22479e7i	− 1.67273i	0.548175	−	0.836364i	$-0.315323\pi$
$558$	0	0
$559$	−3.44066e6	−0.465707
$560$	0	0
$561$	−1.53640e7	−2.06109
$562$	0	0
$563$	− 2.23945e6i	− 0.297763i	−0.988855	−	0.148882i	$-0.952433\pi$
$563$	− 2.23945e6i	− 0.297763i	0.988855	−	0.148882i	$-0.0475673\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 7.39237e6i	− 0.965664i
$568$	0	0
$569$	3.95223e6	0.511755	0.255877	−	0.966709i	$-0.417636\pi$
$569$	3.95223e6	0.511755	0.255877	+	0.966709i	$0.417636\pi$
$570$	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
$573$	9.64949e6i	1.22777i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 3.20338e6i	− 0.400561i	−0.979739	−	0.200280i	$-0.935815\pi$
$577$	− 3.20338e6i	− 0.400561i	0.979739	−	0.200280i	$-0.0641853\pi$
$578$	0	0
$579$	4.46927e6	0.554039
$580$	0	0
$581$	−1.70854e6	−0.209983
$582$	0	0
$583$	9.89002e6i	1.20511i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 1.35926e7i	− 1.62820i	−0.580722	−	0.814102i	$-0.697230\pi$
$587$	− 1.35926e7i	− 1.62820i	0.580722	−	0.814102i	$-0.302770\pi$
$588$	0	0
$589$	2.17365e7	2.58168
$590$	0	0
$591$	−2.08936e7	−2.46062
$592$	0	0
$593$	713635.i	0.0833373i	0.999131	+	0.0416687i	$0.0132674\pi$
$593$	713635.i	0.0833373i	−0.999131	+	0.0416687i	$0.986733\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 8.55984e6i	− 0.982946i
$598$	0	0
$599$	−5.09641e6	−0.580359	−0.290180	−	0.956972i	$-0.593715\pi$
$599$	−5.09641e6	−0.580359	−0.290180	+	0.956972i	$0.593715\pi$
$600$	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
$603$	− 4.26943e7i	− 4.78164i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 6.23096e6i	− 0.686409i	−0.939261	−	0.343204i	$-0.888488\pi$
$607$	− 6.23096e6i	− 0.686409i	0.939261	−	0.343204i	$-0.111512\pi$
$608$	0	0
$609$	−3.73754e6	−0.408359
$610$	0	0
$611$	2.15501e6	0.233532
$612$	0	0
$613$	− 1.37774e7i	− 1.48087i	−0.672128	−	0.740435i	$-0.734620\pi$
$613$	− 1.37774e7i	− 1.48087i	0.672128	−	0.740435i	$-0.265380\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	8.85147e6i	0.936058i	0.883713	+	0.468029i	$0.155036\pi$
$617$	8.85147e6i	0.936058i	−0.883713	+	0.468029i	$0.844964\pi$
$618$	0	0
$619$	−1.22847e7	−1.28866	−0.644329	−	0.764748i	$-0.722863\pi$
$619$	−1.22847e7	−1.28866	−0.644329	+	0.764748i	$0.722863\pi$
$620$	0	0
$621$	1.54494e7	1.60762
$622$	0	0
$623$	1.75512e6i	0.181170i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	2.48164e7i	2.52099i
$628$	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
$633$	1.06889e7i	1.06029i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	3.78084e6i	0.369181i
$638$	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
$643$	5.50652e6i	0.525231i	0.964901	+	0.262615i	$0.0845850\pi$
$643$	5.50652e6i	0.525231i	−0.964901	+	0.262615i	$0.915415\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 2.74623e6i	− 0.257915i	−0.991650	−	0.128957i	$-0.958837\pi$
$647$	− 2.74623e6i	− 0.257915i	0.991650	−	0.128957i	$-0.0411630\pi$
$648$	0	0
$649$	−1.23292e6	−0.114901
$650$	0	0
$651$	−1.16353e7	−1.07604
$652$	0	0
$653$	− 1.24705e7i	− 1.14446i	−0.820092	−	0.572232i	$-0.806078\pi$
$653$	− 1.24705e7i	− 1.14446i	0.820092	−	0.572232i	$-0.193922\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	3.00755e7i	2.71831i
$658$	0	0
$659$	1.75360e7	1.57296	0.786478	−	0.617618i	$-0.211902\pi$
$659$	1.75360e7	1.57296	0.786478	+	0.617618i	$0.211902\pi$
$660$	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
$663$	1.12193e7i	0.991250i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 4.12281e6i	− 0.358821i
$668$	0	0
$669$	−2.18777e7	−1.88989
$670$	0	0
$671$	−1.59418e7	−1.36689
$672$	0	0
$673$	− 9.09208e6i	− 0.773794i	−0.922123	−	0.386897i	$-0.873547\pi$
$673$	− 9.09208e6i	− 0.773794i	0.922123	−	0.386897i	$-0.126453\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 1.55584e7i	− 1.30465i	−0.757940	−	0.652325i	$-0.773794\pi$
$677$	− 1.55584e7i	− 1.30465i	0.757940	−	0.652325i	$-0.226206\pi$
$678$	0	0
$679$	−2.28597e6	−0.190281
$680$	0	0
$681$	1.23965e7	1.02431
$682$	0	0
$683$	5.22623e6i	0.428684i	0.976759	+	0.214342i	$0.0687607\pi$
$683$	5.22623e6i	0.428684i	−0.976759	+	0.214342i	$0.931239\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 3.00653e7i	− 2.43038i
$688$	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
$693$	− 9.50515e6i	− 0.751840i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 1.12944e7i	− 0.880608i
$698$	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
$703$	− 3.53402e7i	− 2.69700i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2.29559e6i	0.172721i
$708$	0	0
$709$	5.13552e6	0.383680	0.191840	−	0.981426i	$-0.438555\pi$
$709$	5.13552e6	0.383680	0.191840	+	0.981426i	$0.438555\pi$
$710$	0	0
$711$	−599930.	−0.0445068
$712$	0	0
$713$	− 1.28347e7i	− 0.945502i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 3.98087e7i	− 2.89188i
$718$	0	0
$719$	−2.90785e6	−0.209773	−0.104886	−	0.994484i	$-0.533448\pi$
$719$	−2.90785e6	−0.209773	−0.104886	+	0.994484i	$0.533448\pi$
$720$	0	0
$721$	283667.	0.0203222
$722$	0	0
$723$	1.03575e7i	0.736898i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	7.85123e6i	0.550937i	0.961310	+	0.275468i	$0.0888329\pi$
$727$	7.85123e6i	0.550937i	−0.961310	+	0.275468i	$0.911167\pi$
$728$	0	0
$729$	−2.50435e7	−1.74533
$730$	0	0
$731$	2.03051e7	1.40544
$732$	0	0
$733$	− 1.41173e7i	− 0.970491i	−0.874378	−	0.485245i	$-0.838730\pi$
$733$	− 1.41173e7i	− 0.970491i	0.874378	−	0.485245i	$-0.161270\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 2.43955e7i	− 1.65440i
$738$	0	0
$739$	4.20049e6	0.282937	0.141468	−	0.989943i	$-0.454818\pi$
$739$	4.20049e6	0.282937	0.141468	+	0.989943i	$0.454818\pi$
$740$	0	0
$741$	1.81218e7	1.21243
$742$	0	0
$743$	1.18490e6i	0.0787425i	0.999225	+	0.0393712i	$0.0125355\pi$
$743$	1.18490e6i	0.0787425i	−0.999225	+	0.0393712i	$0.987465\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	2.34544e7i	1.53788i
$748$	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
$753$	− 2.41696e7i	− 1.55340i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	1.22109e6i	0.0774479i	0.999250	+	0.0387239i	$0.0123293\pi$
$757$	1.22109e6i	0.0774479i	−0.999250	+	0.0387239i	$0.987671\pi$
$758$	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
$763$	− 9.70442e6i	− 0.603473i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	900323.i	0.0552599i
$768$	0	0
$769$	−9.50951e6	−0.579886	−0.289943	−	0.957044i	$-0.593636\pi$
$769$	−9.50951e6	−0.579886	−0.289943	+	0.957044i	$0.593636\pi$
$770$	0	0
$771$	4.30539e7	2.60841
$772$	0	0
$773$	526263.i	0.0316777i	0.999875	+	0.0158389i	$0.00504187\pi$
$773$	526263.i	0.0316777i	−0.999875	+	0.0158389i	$0.994958\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	1.89173e7i	1.12410i
$778$	0	0
$779$	−1.82431e7	−1.07710
$780$	0	0
$781$	−2.09870e7	−1.23118
$782$	0	0
$783$	3.09101e7i	1.80176i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 2.05915e7i	− 1.18509i	−0.805538	−	0.592544i	$-0.798123\pi$
$787$	− 2.05915e7i	− 1.18509i	0.805538	−	0.592544i	$-0.201877\pi$
$788$	0	0
$789$	−1.95751e7	−1.11947
$790$	0	0
$791$	−5.76109e6	−0.327389
$792$	0	0
$793$	1.16413e7i	0.657382i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	2.08952e7i	1.16520i	0.812760	+	0.582599i	$0.197964\pi$
$797$	2.08952e7i	1.16520i	−0.812760	+	0.582599i	$0.802036\pi$
$798$	0	0
$799$	−1.27178e7	−0.704766
$800$	0	0
$801$	2.40939e7	1.32686
$802$	0	0
$803$	1.71851e7i	0.940509i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	4.72694e6i	0.255503i
$808$	0	0
$809$	3.38823e7	1.82013	0.910063	−	0.414470i	$-0.136033\pi$
$809$	3.38823e7	1.82013	0.910063	+	0.414470i	$0.136033\pi$
$810$	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
$813$	− 2.07459e7i	− 1.10079i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 3.27974e7i	− 1.71903i
$818$	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
$823$	1.28086e7i	0.659176i	0.944125	+	0.329588i	$0.106910\pi$
$823$	1.28086e7i	0.659176i	−0.944125	+	0.329588i	$0.893090\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	2.83308e7i	1.44044i	0.693745	+	0.720221i	$0.255960\pi$
$827$	2.83308e7i	1.44044i	−0.693745	+	0.720221i	$0.744040\pi$
$828$	0	0
$829$	−7.91663e6	−0.400087	−0.200043	−	0.979787i	$-0.564108\pi$
$829$	−7.91663e6	−0.400087	−0.200043	+	0.979787i	$0.564108\pi$
$830$	0	0
$831$	3.65739e7	1.83725
$832$	0	0
$833$	− 2.23126e7i	− 1.11414i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	9.62263e7i	4.74767i
$838$	0	0
$839$	2.18676e7	1.07250	0.536249	−	0.844060i	$-0.319841\pi$
$839$	2.18676e7	1.07250	0.536249	+	0.844060i	$0.319841\pi$
$840$	0	0
$841$	−1.22625e7	−0.597847
$842$	0	0
$843$	− 3.35289e7i	− 1.62499i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	1.73961e6i	0.0833188i
$848$	0	0
$849$	2.95637e7	1.40763
$850$	0	0
$851$	−2.08673e7	−0.987739
$852$	0	0
$853$	− 6.73771e6i	− 0.317059i	−0.987354	−	0.158529i	$-0.949325\pi$
$853$	− 6.73771e6i	− 0.317059i	0.987354	−	0.158529i	$-0.0506753\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 2.01368e7i	− 0.936565i	−0.883579	−	0.468283i	$-0.844873\pi$
$857$	− 2.01368e7i	− 0.936565i	0.883579	−	0.468283i	$-0.155127\pi$
$858$	0	0
$859$	2.31020e7	1.06823	0.534117	−	0.845410i	$-0.320644\pi$
$859$	2.31020e7	1.06823	0.534117	+	0.845410i	$0.320644\pi$
$860$	0	0
$861$	9.76535e6	0.448932
$862$	0	0
$863$	− 2.79582e7i	− 1.27785i	−0.769267	−	0.638927i	$-0.779378\pi$
$863$	− 2.79582e7i	− 1.27785i	0.769267	−	0.638927i	$-0.220622\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 2.47125e7i	− 1.11652i
$868$	0	0
$869$	−342800.	−0.0153989
$870$	0	0
$871$	−1.78144e7	−0.795658
$872$	0	0
$873$	3.13812e7i	1.39359i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 3.75058e7i	− 1.64664i	−0.567574	−	0.823322i	$-0.692118\pi$
$877$	− 3.75058e7i	− 1.64664i	0.567574	−	0.823322i	$-0.307882\pi$
$878$	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
$883$	3.90752e7i	1.68655i	0.537481	+	0.843276i	$0.319376\pi$
$883$	3.90752e7i	1.68655i	−0.537481	+	0.843276i	$0.680624\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1.87662e7i	− 0.800878i	−0.916323	−	0.400439i	$-0.868858\pi$
$887$	− 1.87662e7i	− 0.800878i	0.916323	−	0.400439i	$-0.131142\pi$
$888$	0	0
$889$	−4.73119e6	−0.200778
$890$	0	0
$891$	−5.79859e7	−2.44697
$892$	0	0
$893$	2.05422e7i	0.862020i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 1.07004e7i	− 0.444035i
$898$	0	0
$899$	2.56788e7	1.05968
$900$	0	0
$901$	−4.26209e7	−1.74908
$902$	0	0
$903$	1.75561e7i	0.716489i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 2.79416e7i	− 1.12780i	−0.825842	−	0.563901i	$-0.809300\pi$
$907$	− 2.79416e7i	− 1.12780i	0.825842	−	0.563901i	$-0.190700\pi$
$908$	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
$913$	1.34018e7i	0.532092i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 5.91332e6i	− 0.232224i
$918$	0	0
$919$	−2.80433e7	−1.09532	−0.547660	−	0.836701i	$-0.684481\pi$
$919$	−2.80433e7	−1.09532	−0.547660	+	0.836701i	$0.684481\pi$
$920$	0	0
$921$	−3.52951e7	−1.37109
$922$	0	0
$923$	1.53254e7i	0.592117i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 3.89411e6i	− 0.148836i
$928$	0	0
$929$	−1.96341e7	−0.746400	−0.373200	−	0.927751i	$-0.621739\pi$
$929$	−1.96341e7	−0.746400	−0.373200	+	0.927751i	$0.621739\pi$
$930$	0	0
$931$	−3.60401e7	−1.36273
$932$	0	0
$933$	8.49871e7i	3.19631i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 2.71944e7i	− 1.01188i	−0.862567	−	0.505942i	$-0.831145\pi$
$937$	− 2.71944e7i	− 1.01188i	0.862567	−	0.505942i	$-0.168855\pi$
$938$	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
$943$	1.07720e7i	0.394472i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.42413e7i	0.516029i	0.966141	+	0.258014i	$0.0830682\pi$
$947$	1.42413e7i	0.516029i	−0.966141	+	0.258014i	$0.916932\pi$
$948$	0	0
$949$	1.25491e7	0.452323
$950$	0	0
$951$	3.92535e7	1.40743
$952$	0	0
$953$	3.25786e7i	1.16198i	0.813910	+	0.580992i	$0.197335\pi$
$953$	3.25786e7i	1.16198i	−0.813910	+	0.580992i	$0.802665\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	2.93173e7i	1.03477i
$958$	0	0
$959$	1.34097e7	0.470841
$960$	0	0
$961$	5.13117e7	1.79229
$962$	0	0
$963$	− 3.45598e6i	− 0.120090i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 4.03932e6i	− 0.138913i	−0.997585	−	0.0694564i	$-0.977874\pi$
$967$	− 4.03932e6i	− 0.138913i	0.997585	−	0.0694564i	$-0.0221265\pi$
$968$	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
$973$	− 6.18993e6i	− 0.209606i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	2.70486e7i	0.906586i	0.891362	+	0.453293i	$0.149751\pi$
$977$	2.70486e7i	0.906586i	−0.891362	+	0.453293i	$0.850249\pi$
$978$	0	0
$979$	1.37672e7	0.459082
$980$	0	0
$981$	−1.33220e8	−4.41974
$982$	0	0
$983$	− 3.15761e7i	− 1.04226i	−0.853479	−	0.521128i	$-0.825511\pi$
$983$	− 3.15761e7i	− 1.04226i	0.853479	−	0.521128i	$-0.174489\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 1.09960e7i	− 0.359288i
$988$	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
$993$	− 4.65443e7i	− 1.49794i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	1.34950e7i	0.429966i	0.976618	+	0.214983i	$0.0689695\pi$
$997$	1.34950e7i	0.429966i	−0.976618	+	0.214983i	$0.931030\pi$
$998$	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.c.j.449.6		6
4.3	odd	2		800.6.c.k.449.1		6
5.2	odd	4		160.6.a.g.1.3	yes	3
5.3	odd	4		800.6.a.n.1.1		3
5.4	even	2	inner	800.6.c.j.449.1		6
20.3	even	4		800.6.a.o.1.3		3
20.7	even	4		160.6.a.f.1.1	✓	3
20.19	odd	2		800.6.c.k.449.6		6
40.27	even	4		320.6.a.y.1.3		3
40.37	odd	4		320.6.a.x.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.6.a.f.1.1	✓	3	20.7	even	4
160.6.a.g.1.3	yes	3	5.2	odd	4
320.6.a.x.1.1		3	40.37	odd	4
320.6.a.y.1.3		3	40.27	even	4
800.6.a.n.1.1		3	5.3	odd	4
800.6.a.o.1.3		3	20.3	even	4
800.6.c.j.449.1		6	5.4	even	2	inner
800.6.c.j.449.6		6	1.1	even	1	trivial
800.6.c.k.449.1		6	4.3	odd	2
800.6.c.k.449.6		6	20.19	odd	2

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.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+29.2272i q^{3} -44.5253i q^{7} -611.232 q^{9} +O(q^{10})\)	\(q+29.2272i q^{3} -44.5253i q^{7} -611.232 q^{9} -349.258 q^{11} +255.040i q^{13} -1505.12i q^{17} -2431.11 q^{19} +1301.35 q^{21} +1435.50i q^{23} -10762.4i q^{27} -2872.04 q^{29} -8940.96 q^{31} -10207.8i q^{33} +14536.6i q^{37} -7454.11 q^{39} +7504.01 q^{41} +13490.7i q^{43} -8449.69i q^{47} +14824.5 q^{49} +43990.5 q^{51} -28317.3i q^{53} -71054.8i q^{57} +3530.13 q^{59} +45644.9 q^{61} +27215.3i q^{63} +69849.6i q^{67} -41955.6 q^{69} +60090.2 q^{71} -49204.6i q^{73} +15550.8i q^{77} +981.509 q^{79} +166026. q^{81} -38372.3i q^{83} -83941.9i q^{87} -39418.6 q^{89} +11355.7 q^{91} -261320. i q^{93} -51340.9i q^{97} +213477. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 934 q^{9}+O(q^{10}) \) 6 * q - 934 * q^9	\( 6 q - 934 q^{9} - 792 q^{11} - 6384 q^{19} - 1640 q^{21} - 852 q^{29} + 6552 q^{31} - 42696 q^{39} + 24900 q^{41} + 7698 q^{49} + 142888 q^{51} - 70080 q^{59} - 48276 q^{61} + 54072 q^{69} + 176184 q^{71} + 185904 q^{79} + 302782 q^{81} - 345372 q^{89} - 213624 q^{91} + 702872 q^{99}+O(q^{100}) \) 6 * q - 934 * q^9 - 792 * q^11 - 6384 * q^19 - 1640 * q^21 - 852 * q^29 + 6552 * q^31 - 42696 * q^39 + 24900 * q^41 + 7698 * q^49 + 142888 * q^51 - 70080 * q^59 - 48276 * q^61 + 54072 * q^69 + 176184 * q^71 + 185904 * q^79 + 302782 * q^81 - 345372 * q^89 - 213624 * q^91 + 702872 * q^99

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