LMFDB - Embedded newform 320.3.b.b.191.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [320,3,Mod(191,320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

chi = DirichletCharacter(H, H._module([1, 0, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("320.191");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 320 = 2^{6} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	320.b (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:	$8.71936845953$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 160)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			191.1
Root			$-1.61803i$ of defining polynomial
Character	$\chi$	$=$	320.191
Dual form			320.3.b.b.191.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.23607i q^{3} -2.23607 q^{5} +10.1803i q^{7} -18.4164 q^{9} +O(q^{10})$	$q-5.23607i q^{3} -2.23607 q^{5} +10.1803i q^{7} -18.4164 q^{9} +14.4721i q^{11} -11.5279 q^{13} +11.7082i q^{15} -18.9443 q^{17} -12.0000i q^{19} +53.3050 q^{21} +17.5967i q^{23} +5.00000 q^{25} +49.3050i q^{27} -8.83282 q^{29} -0.583592i q^{31} +75.7771 q^{33} -22.7639i q^{35} -32.4721 q^{37} +60.3607i q^{39} +71.3050 q^{41} +4.65248i q^{43} +41.1803 q^{45} -22.5410i q^{47} -54.6393 q^{49} +99.1935i q^{51} -63.3050 q^{53} -32.3607i q^{55} -62.8328 q^{57} +30.6099i q^{59} -65.1935 q^{61} -187.485i q^{63} +25.7771 q^{65} -92.2067i q^{67} +92.1378 q^{69} +41.7508i q^{71} -136.164 q^{73} -26.1803i q^{75} -147.331 q^{77} +81.1672i q^{79} +92.4164 q^{81} +86.5410i q^{83} +42.3607 q^{85} +46.2492i q^{87} -30.0000 q^{89} -117.358i q^{91} -3.05573 q^{93} +26.8328i q^{95} +119.666 q^{97} -266.525i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 20 q^{9}+O(q^{10}) $ 4 * q - 20 * q^9	$ 4 q - 20 q^{9} - 64 q^{13} - 40 q^{17} + 88 q^{21} + 20 q^{25} + 72 q^{29} + 160 q^{33} - 112 q^{37} + 160 q^{41} + 120 q^{45} - 308 q^{49} - 128 q^{53} - 144 q^{57} - 64 q^{61} - 40 q^{65} + 136 q^{69} - 8 q^{73} - 160 q^{77} + 316 q^{81} + 80 q^{85} - 120 q^{89} - 48 q^{93} + 264 q^{97}+O(q^{100}) $ 4 * q - 20 * q^9 - 64 * q^13 - 40 * q^17 + 88 * q^21 + 20 * q^25 + 72 * q^29 + 160 * q^33 - 112 * q^37 + 160 * q^41 + 120 * q^45 - 308 * q^49 - 128 * q^53 - 144 * q^57 - 64 * q^61 - 40 * q^65 + 136 * q^69 - 8 * q^73 - 160 * q^77 + 316 * q^81 + 80 * q^85 - 120 * q^89 - 48 * q^93 + 264 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/320\mathbb{Z}\right)^\times$.

$n$	$191$	$257$	$261$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	− 5.23607i	− 1.74536i	−0.488296	−	0.872678i	$-0.662381\pi$
$3$	− 5.23607i	− 1.74536i	0.488296	−	0.872678i	$-0.337619\pi$
$4$	0	0
$5$	−2.23607	−0.447214
$5$	−2.23607	−0.447214
$6$	0	0
$7$	10.1803i	1.45433i	0.686460	+	0.727167i	$0.259163\pi$
$7$	10.1803i	1.45433i	−0.686460	+	0.727167i	$0.740837\pi$
$8$	0	0
$9$	−18.4164	−2.04627
$10$	0	0
$11$	14.4721i	1.31565i	0.753171	+	0.657824i	$0.228523\pi$
$11$	14.4721i	1.31565i	−0.753171	+	0.657824i	$0.771477\pi$
$12$	0	0
$13$	−11.5279	−0.886759	−0.443379	−	0.896334i	$-0.646221\pi$
$13$	−11.5279	−0.886759	−0.443379	+	0.896334i	$0.646221\pi$
$14$	0	0
$15$	11.7082i	0.780547i
$16$	0	0
$17$	−18.9443	−1.11437	−0.557184	−	0.830389i	$-0.688118\pi$
$17$	−18.9443	−1.11437	−0.557184	+	0.830389i	$0.688118\pi$
$18$	0	0
$19$	− 12.0000i	− 0.631579i	−0.948829	−	0.315789i	$-0.897731\pi$
$19$	− 12.0000i	− 0.631579i	0.948829	−	0.315789i	$-0.102269\pi$
$20$	0	0
$21$	53.3050	2.53833
$22$	0	0
$23$	17.5967i	0.765076i	0.923940	+	0.382538i	$0.124950\pi$
$23$	17.5967i	0.765076i	−0.923940	+	0.382538i	$0.875050\pi$
$24$	0	0
$25$	5.00000	0.200000
$26$	0	0
$27$	49.3050i	1.82611i
$28$	0	0
$29$	−8.83282	−0.304580	−0.152290	−	0.988336i	$-0.548665\pi$
$29$	−8.83282	−0.304580	−0.152290	+	0.988336i	$0.548665\pi$
$30$	0	0
$31$	− 0.583592i	− 0.0188256i	−0.999956	−	0.00941278i	$-0.997004\pi$
$31$	− 0.583592i	− 0.0188256i	0.999956	−	0.00941278i	$-0.00299622\pi$
$32$	0	0
$33$	75.7771	2.29628
$34$	0	0
$35$	− 22.7639i	− 0.650398i
$36$	0	0
$37$	−32.4721	−0.877625	−0.438813	−	0.898579i	$-0.644601\pi$
$37$	−32.4721	−0.877625	−0.438813	+	0.898579i	$0.644601\pi$
$38$	0	0
$39$	60.3607i	1.54771i
$40$	0	0
$41$	71.3050	1.73915	0.869573	−	0.493805i	$-0.164394\pi$
$41$	71.3050	1.73915	0.869573	+	0.493805i	$0.164394\pi$
$42$	0	0
$43$	4.65248i	0.108197i	0.998536	+	0.0540986i	$0.0172285\pi$
$43$	4.65248i	0.108197i	−0.998536	+	0.0540986i	$0.982771\pi$
$44$	0	0
$45$	41.1803	0.915119
$46$	0	0
$47$	− 22.5410i	− 0.479596i	−0.970823	−	0.239798i	$-0.922919\pi$
$47$	− 22.5410i	− 0.479596i	0.970823	−	0.239798i	$-0.0770812\pi$
$48$	0	0
$49$	−54.6393	−1.11509
$50$	0	0
$51$	99.1935i	1.94497i
$52$	0	0
$53$	−63.3050	−1.19443	−0.597217	−	0.802080i	$-0.703727\pi$
$53$	−63.3050	−1.19443	−0.597217	+	0.802080i	$0.703727\pi$
$54$	0	0
$55$	− 32.3607i	− 0.588376i
$56$	0	0
$57$	−62.8328	−1.10233
$58$	0	0
$59$	30.6099i	0.518812i	0.965768	+	0.259406i	$0.0835268\pi$
$59$	30.6099i	0.518812i	−0.965768	+	0.259406i	$0.916473\pi$
$60$	0	0
$61$	−65.1935	−1.06875	−0.534373	−	0.845249i	$-0.679452\pi$
$61$	−65.1935	−1.06875	−0.534373	+	0.845249i	$0.679452\pi$
$62$	0	0
$63$	− 187.485i	− 2.97596i
$64$	0	0
$65$	25.7771	0.396571
$66$	0	0
$67$	− 92.2067i	− 1.37622i	−0.725607	−	0.688109i	$-0.758441\pi$
$67$	− 92.2067i	− 1.37622i	0.725607	−	0.688109i	$-0.241559\pi$
$68$	0	0
$69$	92.1378	1.33533
$70$	0	0
$71$	41.7508i	0.588039i	0.955799	+	0.294020i	$0.0949931\pi$
$71$	41.7508i	0.588039i	−0.955799	+	0.294020i	$0.905007\pi$
$72$	0	0
$73$	−136.164	−1.86526	−0.932631	−	0.360832i	$-0.882493\pi$
$73$	−136.164	−1.86526	−0.932631	+	0.360832i	$0.882493\pi$
$74$	0	0
$75$	− 26.1803i	− 0.349071i
$76$	0	0
$77$	−147.331	−1.91339
$78$	0	0
$79$	81.1672i	1.02743i	0.857960	+	0.513716i	$0.171732\pi$
$79$	81.1672i	1.02743i	−0.857960	+	0.513716i	$0.828268\pi$
$80$	0	0
$81$	92.4164	1.14094
$82$	0	0
$83$	86.5410i	1.04266i	0.853354	+	0.521331i	$0.174564\pi$
$83$	86.5410i	1.04266i	−0.853354	+	0.521331i	$0.825436\pi$
$84$	0	0
$85$	42.3607	0.498361
$86$	0	0
$87$	46.2492i	0.531600i
$88$	0	0
$89$	−30.0000	−0.337079	−0.168539	−	0.985695i	$-0.553905\pi$
$89$	−30.0000	−0.337079	−0.168539	+	0.985695i	$0.553905\pi$
$90$	0	0
$91$	− 117.358i	− 1.28964i
$92$	0	0
$93$	−3.05573	−0.0328573
$94$	0	0
$95$	26.8328i	0.282451i
$96$	0	0
$97$	119.666	1.23367	0.616833	−	0.787094i	$-0.288415\pi$
$97$	119.666	1.23367	0.616833	+	0.787094i	$0.288415\pi$
$98$	0	0
$99$	− 266.525i	− 2.69217i
$100$	0	0
$101$	37.5542	0.371824	0.185912	−	0.982566i	$-0.440476\pi$
$101$	37.5542	0.371824	0.185912	+	0.982566i	$0.440476\pi$
$102$	0	0
$103$	− 177.013i	− 1.71857i	−0.511493	−	0.859287i	$-0.670908\pi$
$103$	− 177.013i	− 1.71857i	0.511493	−	0.859287i	$-0.329092\pi$
$104$	0	0
$105$	−119.193	−1.13518
$106$	0	0
$107$	63.1246i	0.589950i	0.955505	+	0.294975i	$0.0953113\pi$
$107$	63.1246i	0.589950i	−0.955505	+	0.294975i	$0.904689\pi$
$108$	0	0
$109$	44.4721	0.408001	0.204001	−	0.978971i	$-0.434606\pi$
$109$	44.4721	0.408001	0.204001	+	0.978971i	$0.434606\pi$
$110$	0	0
$111$	170.026i	1.53177i
$112$	0	0
$113$	18.7214	0.165676	0.0828379	−	0.996563i	$-0.473602\pi$
$113$	18.7214	0.165676	0.0828379	+	0.996563i	$0.473602\pi$
$114$	0	0
$115$	− 39.3475i	− 0.342152i
$116$	0	0
$117$	212.302	1.81455
$118$	0	0
$119$	− 192.859i	− 1.62066i
$120$	0	0
$121$	−88.4427	−0.730932
$122$	0	0
$123$	− 373.358i	− 3.03543i
$124$	0	0
$125$	−11.1803	−0.0894427
$126$	0	0
$127$	81.4590i	0.641409i	0.947179	+	0.320705i	$0.103920\pi$
$127$	81.4590i	0.641409i	−0.947179	+	0.320705i	$0.896080\pi$
$128$	0	0
$129$	24.3607	0.188842
$130$	0	0
$131$	169.082i	1.29070i	0.763886	+	0.645351i	$0.223289\pi$
$131$	169.082i	1.29070i	−0.763886	+	0.645351i	$0.776711\pi$
$132$	0	0
$133$	122.164	0.918527
$134$	0	0
$135$	− 110.249i	− 0.816661i
$136$	0	0
$137$	−24.6099	−0.179634	−0.0898172	−	0.995958i	$-0.528628\pi$
$137$	−24.6099	−0.179634	−0.0898172	+	0.995958i	$0.528628\pi$
$138$	0	0
$139$	73.3901i	0.527986i	0.964525	+	0.263993i	$0.0850396\pi$
$139$	73.3901i	0.527986i	−0.964525	+	0.263993i	$0.914960\pi$
$140$	0	0
$141$	−118.026	−0.837066
$142$	0	0
$143$	− 166.833i	− 1.16666i
$144$	0	0
$145$	19.7508	0.136212
$146$	0	0
$147$	286.095i	1.94623i
$148$	0	0
$149$	68.9180	0.462537	0.231268	−	0.972890i	$-0.425712\pi$
$149$	68.9180	0.462537	0.231268	+	0.972890i	$0.425712\pi$
$150$	0	0
$151$	− 109.803i	− 0.727175i	−0.931560	−	0.363587i	$-0.881552\pi$
$151$	− 109.803i	− 0.727175i	0.931560	−	0.363587i	$-0.118448\pi$
$152$	0	0
$153$	348.885	2.28030
$154$	0	0
$155$	1.30495i	0.00841904i
$156$	0	0
$157$	279.082	1.77759	0.888796	−	0.458302i	$-0.151542\pi$
$157$	279.082	1.77759	0.888796	+	0.458302i	$0.151542\pi$
$158$	0	0
$159$	331.469i	2.08471i
$160$	0	0
$161$	−179.141	−1.11268
$162$	0	0
$163$	− 82.6262i	− 0.506909i	−0.967347	−	0.253454i	$-0.918433\pi$
$163$	− 82.6262i	− 0.506909i	0.967347	−	0.253454i	$-0.0815668\pi$
$164$	0	0
$165$	−169.443	−1.02693
$166$	0	0
$167$	93.2361i	0.558300i	0.960248	+	0.279150i	$0.0900526\pi$
$167$	93.2361i	0.558300i	−0.960248	+	0.279150i	$0.909947\pi$
$168$	0	0
$169$	−36.1084	−0.213659
$170$	0	0
$171$	220.997i	1.29238i
$172$	0	0
$173$	−159.580	−0.922431	−0.461215	−	0.887288i	$-0.652586\pi$
$173$	−159.580	−0.922431	−0.461215	+	0.887288i	$0.652586\pi$
$174$	0	0
$175$	50.9017i	0.290867i
$176$	0	0
$177$	160.276	0.905511
$178$	0	0
$179$	171.830i	0.959943i	0.877284	+	0.479971i	$0.159353\pi$
$179$	171.830i	0.959943i	−0.877284	+	0.479971i	$0.840647\pi$
$180$	0	0
$181$	−237.331	−1.31122	−0.655611	−	0.755099i	$-0.727589\pi$
$181$	−237.331	−1.31122	−0.655611	+	0.755099i	$0.727589\pi$
$182$	0	0
$183$	341.358i	1.86534i
$184$	0	0
$185$	72.6099	0.392486
$186$	0	0
$187$	− 274.164i	− 1.46612i
$188$	0	0
$189$	−501.941	−2.65577
$190$	0	0
$191$	− 90.7477i	− 0.475119i	−0.971373	−	0.237559i	$-0.923653\pi$
$191$	− 90.7477i	− 0.475119i	0.971373	−	0.237559i	$-0.0763474\pi$
$192$	0	0
$193$	−49.5016	−0.256485	−0.128242	−	0.991743i	$-0.540934\pi$
$193$	−49.5016	−0.256485	−0.128242	+	0.991743i	$0.540934\pi$
$194$	0	0
$195$	− 134.971i	− 0.692157i
$196$	0	0
$197$	−377.023	−1.91382	−0.956912	−	0.290379i	$-0.906219\pi$
$197$	−377.023	−1.91382	−0.956912	+	0.290379i	$0.906219\pi$
$198$	0	0
$199$	− 391.108i	− 1.96537i	−0.185288	−	0.982684i	$-0.559322\pi$
$199$	− 391.108i	− 1.96537i	0.185288	−	0.982684i	$-0.440678\pi$
$200$	0	0
$201$	−482.800	−2.40199
$202$	0	0
$203$	− 89.9211i	− 0.442961i
$204$	0	0
$205$	−159.443	−0.777769
$206$	0	0
$207$	− 324.069i	− 1.56555i
$208$	0	0
$209$	173.666	0.830936
$210$	0	0
$211$	267.416i	1.26738i	0.773589	+	0.633688i	$0.218460\pi$
$211$	267.416i	1.26738i	−0.773589	+	0.633688i	$0.781540\pi$
$212$	0	0
$213$	218.610	1.02634
$214$	0	0
$215$	− 10.4033i	− 0.0483872i
$216$	0	0
$217$	5.94117	0.0273786
$218$	0	0
$219$	712.964i	3.25555i
$220$	0	0
$221$	218.387	0.988176
$222$	0	0
$223$	31.7082i	0.142189i	0.997470	+	0.0710946i	$0.0226492\pi$
$223$	31.7082i	0.142189i	−0.997470	+	0.0710946i	$0.977351\pi$
$224$	0	0
$225$	−92.0820	−0.409254
$226$	0	0
$227$	162.456i	0.715665i	0.933786	+	0.357832i	$0.116484\pi$
$227$	162.456i	0.715665i	−0.933786	+	0.357832i	$0.883516\pi$
$228$	0	0
$229$	425.161	1.85660	0.928299	−	0.371834i	$-0.121271\pi$
$229$	425.161	1.85660	0.928299	+	0.371834i	$0.121271\pi$
$230$	0	0
$231$	771.437i	3.33955i
$232$	0	0
$233$	−177.331	−0.761078	−0.380539	−	0.924765i	$-0.624262\pi$
$233$	−177.331	−0.761078	−0.380539	+	0.924765i	$0.624262\pi$
$234$	0	0
$235$	50.4033i	0.214482i
$236$	0	0
$237$	424.997	1.79324
$238$	0	0
$239$	85.4953i	0.357721i	0.983874	+	0.178861i	$0.0572411\pi$
$239$	85.4953i	0.357721i	−0.983874	+	0.178861i	$0.942759\pi$
$240$	0	0
$241$	50.0851	0.207822	0.103911	−	0.994587i	$-0.466864\pi$
$241$	50.0851	0.207822	0.103911	+	0.994587i	$0.466864\pi$
$242$	0	0
$243$	− 40.1540i	− 0.165243i
$244$	0	0
$245$	122.177	0.498683
$246$	0	0
$247$	138.334i	0.560058i
$248$	0	0
$249$	453.135	1.81982
$250$	0	0
$251$	− 54.1966i	− 0.215923i	−0.994155	−	0.107961i	$-0.965568\pi$
$251$	− 54.1966i	− 0.215923i	0.994155	−	0.107961i	$-0.0344323\pi$
$252$	0	0
$253$	−254.663	−1.00657
$254$	0	0
$255$	− 221.803i	− 0.869817i
$256$	0	0
$257$	285.502	1.11090	0.555450	−	0.831550i	$-0.312546\pi$
$257$	285.502	1.11090	0.555450	+	0.831550i	$0.312546\pi$
$258$	0	0
$259$	− 330.577i	− 1.27636i
$260$	0	0
$261$	162.669	0.623252
$262$	0	0
$263$	123.039i	0.467831i	0.972257	+	0.233915i	$0.0751539\pi$
$263$	123.039i	0.467831i	−0.972257	+	0.233915i	$0.924846\pi$
$264$	0	0
$265$	141.554	0.534167
$266$	0	0
$267$	157.082i	0.588322i
$268$	0	0
$269$	184.354	0.685333	0.342666	−	0.939457i	$-0.388670\pi$
$269$	184.354	0.685333	0.342666	+	0.939457i	$0.388670\pi$
$270$	0	0
$271$	− 184.413i	− 0.680492i	−0.940336	−	0.340246i	$-0.889490\pi$
$271$	− 184.413i	− 0.680492i	0.940336	−	0.340246i	$-0.110510\pi$
$272$	0	0
$273$	−614.492	−2.25089
$274$	0	0
$275$	72.3607i	0.263130i
$276$	0	0
$277$	−20.4195	−0.0737167	−0.0368583	−	0.999321i	$-0.511735\pi$
$277$	−20.4195	−0.0737167	−0.0368583	+	0.999321i	$0.511735\pi$
$278$	0	0
$279$	10.7477i	0.0385221i
$280$	0	0
$281$	463.410	1.64915	0.824573	−	0.565755i	$-0.191415\pi$
$281$	463.410	1.64915	0.824573	+	0.565755i	$0.191415\pi$
$282$	0	0
$283$	− 316.895i	− 1.11977i	−0.828570	−	0.559886i	$-0.810845\pi$
$283$	− 316.895i	− 1.11977i	0.828570	−	0.559886i	$-0.189155\pi$
$284$	0	0
$285$	140.498	0.492977
$286$	0	0
$287$	725.909i	2.52930i
$288$	0	0
$289$	69.8854	0.241818
$290$	0	0
$291$	− 626.577i	− 2.15319i
$292$	0	0
$293$	407.082	1.38936	0.694679	−	0.719320i	$-0.255546\pi$
$293$	407.082	1.38936	0.694679	+	0.719320i	$0.255546\pi$
$294$	0	0
$295$	− 68.4458i	− 0.232020i
$296$	0	0
$297$	−713.548	−2.40252
$298$	0	0
$299$	− 202.853i	− 0.678438i
$300$	0	0
$301$	−47.3638	−0.157355
$302$	0	0
$303$	− 196.636i	− 0.648964i
$304$	0	0
$305$	145.777	0.477958
$306$	0	0
$307$	− 318.541i	− 1.03759i	−0.854898	−	0.518796i	$-0.826380\pi$
$307$	− 318.541i	− 1.03759i	0.854898	−	0.518796i	$-0.173620\pi$
$308$	0	0
$309$	−926.853	−2.99952
$310$	0	0
$311$	156.912i	0.504539i	0.967657	+	0.252270i	$0.0811770\pi$
$311$	156.912i	0.504539i	−0.967657	+	0.252270i	$0.918823\pi$
$312$	0	0
$313$	−184.780	−0.590352	−0.295176	−	0.955443i	$-0.595378\pi$
$313$	−184.780	−0.590352	−0.295176	+	0.955443i	$0.595378\pi$
$314$	0	0
$315$	419.230i	1.33089i
$316$	0	0
$317$	−14.5836	−0.0460050	−0.0230025	−	0.999735i	$-0.507323\pi$
$317$	−14.5836	−0.0460050	−0.0230025	+	0.999735i	$0.507323\pi$
$318$	0	0
$319$	− 127.830i	− 0.400720i
$320$	0	0
$321$	330.525	1.02967
$322$	0	0
$323$	227.331i	0.703812i
$324$	0	0
$325$	−57.6393	−0.177352
$326$	0	0
$327$	− 232.859i	− 0.712107i
$328$	0	0
$329$	229.475	0.697493
$330$	0	0
$331$	− 76.5836i	− 0.231370i	−0.993286	−	0.115685i	$-0.963094\pi$
$331$	− 76.5836i	− 0.231370i	0.993286	−	0.115685i	$-0.0369064\pi$
$332$	0	0
$333$	598.020	1.79586
$334$	0	0
$335$	206.180i	0.615464i
$336$	0	0
$337$	−214.997	−0.637973	−0.318986	−	0.947759i	$-0.603342\pi$
$337$	−214.997	−0.637973	−0.318986	+	0.947759i	$0.603342\pi$
$338$	0	0
$339$	− 98.0263i	− 0.289163i
$340$	0	0
$341$	8.44582	0.0247678
$342$	0	0
$343$	− 57.4102i	− 0.167377i
$344$	0	0
$345$	−206.026	−0.597178
$346$	0	0
$347$	86.5735i	0.249491i	0.992189	+	0.124746i	$0.0398115\pi$
$347$	86.5735i	0.249491i	−0.992189	+	0.124746i	$0.960188\pi$
$348$	0	0
$349$	116.833	0.334765	0.167382	−	0.985892i	$-0.446469\pi$
$349$	116.833	0.334765	0.167382	+	0.985892i	$0.446469\pi$
$350$	0	0
$351$	− 568.381i	− 1.61932i
$352$	0	0
$353$	−330.774	−0.937037	−0.468518	−	0.883454i	$-0.655212\pi$
$353$	−330.774	−0.937037	−0.468518	+	0.883454i	$0.655212\pi$
$354$	0	0
$355$	− 93.3576i	− 0.262979i
$356$	0	0
$357$	−1009.82	−2.82864
$358$	0	0
$359$	− 432.105i	− 1.20364i	−0.798633	−	0.601818i	$-0.794443\pi$
$359$	− 432.105i	− 1.20364i	0.798633	−	0.601818i	$-0.205557\pi$
$360$	0	0
$361$	217.000	0.601108
$362$	0	0
$363$	463.092i	1.27574i
$364$	0	0
$365$	304.472	0.834170
$366$	0	0
$367$	− 288.259i	− 0.785448i	−0.919656	−	0.392724i	$-0.871533\pi$
$367$	− 288.259i	− 0.785448i	0.919656	−	0.392724i	$-0.128467\pi$
$368$	0	0
$369$	−1313.18	−3.55876
$370$	0	0
$371$	− 644.466i	− 1.73710i
$372$	0	0
$373$	−253.076	−0.678487	−0.339244	−	0.940699i	$-0.610171\pi$
$373$	−253.076	−0.678487	−0.339244	+	0.940699i	$0.610171\pi$
$374$	0	0
$375$	58.5410i	0.156109i
$376$	0	0
$377$	101.823	0.270089
$378$	0	0
$379$	42.2167i	0.111390i	0.998448	+	0.0556949i	$0.0177374\pi$
$379$	42.2167i	0.111390i	−0.998448	+	0.0556949i	$0.982263\pi$
$380$	0	0
$381$	426.525	1.11949
$382$	0	0
$383$	− 179.177i	− 0.467826i	−0.972258	−	0.233913i	$-0.924847\pi$
$383$	− 179.177i	− 0.467826i	0.972258	−	0.233913i	$-0.0751530\pi$
$384$	0	0
$385$	329.443	0.855695
$386$	0	0
$387$	− 85.6819i	− 0.221400i
$388$	0	0
$389$	75.9211	0.195170	0.0975849	−	0.995227i	$-0.468888\pi$
$389$	75.9211	0.195170	0.0975849	+	0.995227i	$0.468888\pi$
$390$	0	0
$391$	− 333.358i	− 0.852577i
$392$	0	0
$393$	885.325	2.25274
$394$	0	0
$395$	− 181.495i	− 0.459482i
$396$	0	0
$397$	−480.407	−1.21009	−0.605047	−	0.796190i	$-0.706846\pi$
$397$	−480.407	−1.21009	−0.605047	+	0.796190i	$0.706846\pi$
$398$	0	0
$399$	− 639.659i	− 1.60316i
$400$	0	0
$401$	−514.328	−1.28261	−0.641307	−	0.767284i	$-0.721607\pi$
$401$	−514.328	−1.28261	−0.641307	+	0.767284i	$0.721607\pi$
$402$	0	0
$403$	6.72757i	0.0166937i
$404$	0	0
$405$	−206.649	−0.510245
$406$	0	0
$407$	− 469.941i	− 1.15465i
$408$	0	0
$409$	610.466	1.49258	0.746291	−	0.665620i	$-0.231833\pi$
$409$	610.466	1.49258	0.746291	+	0.665620i	$0.231833\pi$
$410$	0	0
$411$	128.859i	0.313526i
$412$	0	0
$413$	−311.619	−0.754526
$414$	0	0
$415$	− 193.512i	− 0.466293i
$416$	0	0
$417$	384.276	0.921524
$418$	0	0
$419$	556.381i	1.32788i	0.747787	+	0.663939i	$0.231117\pi$
$419$	556.381i	1.32788i	−0.747787	+	0.663939i	$0.768883\pi$
$420$	0	0
$421$	201.246	0.478019	0.239010	−	0.971017i	$-0.423177\pi$
$421$	201.246	0.478019	0.239010	+	0.971017i	$0.423177\pi$
$422$	0	0
$423$	415.125i	0.981382i
$424$	0	0
$425$	−94.7214	−0.222874
$426$	0	0
$427$	− 663.692i	− 1.55431i
$428$	0	0
$429$	−873.548	−2.03624
$430$	0	0
$431$	749.909i	1.73993i	0.493116	+	0.869964i	$0.335858\pi$
$431$	749.909i	1.73993i	−0.493116	+	0.869964i	$0.664142\pi$
$432$	0	0
$433$	663.325	1.53193	0.765964	−	0.642883i	$-0.222262\pi$
$433$	663.325	1.53193	0.765964	+	0.642883i	$0.222262\pi$
$434$	0	0
$435$	− 103.416i	− 0.237739i
$436$	0	0
$437$	211.161	0.483206
$438$	0	0
$439$	717.220i	1.63376i	0.576809	+	0.816879i	$0.304298\pi$
$439$	717.220i	1.63376i	−0.576809	+	0.816879i	$0.695702\pi$
$440$	0	0
$441$	1006.26	2.28177
$442$	0	0
$443$	− 182.508i	− 0.411983i	−0.978554	−	0.205992i	$-0.933958\pi$
$443$	− 182.508i	− 0.411983i	0.978554	−	0.205992i	$-0.0660419\pi$
$444$	0	0
$445$	67.0820	0.150746
$446$	0	0
$447$	− 360.859i	− 0.807291i
$448$	0	0
$449$	−416.971	−0.928665	−0.464332	−	0.885661i	$-0.653706\pi$
$449$	−416.971	−0.928665	−0.464332	+	0.885661i	$0.653706\pi$
$450$	0	0
$451$	1031.93i	2.28810i
$452$	0	0
$453$	−574.938	−1.26918
$454$	0	0
$455$	262.420i	0.576746i
$456$	0	0
$457$	−550.997	−1.20568	−0.602841	−	0.797861i	$-0.705965\pi$
$457$	−550.997	−1.20568	−0.602841	+	0.797861i	$0.705965\pi$
$458$	0	0
$459$	− 934.046i	− 2.03496i
$460$	0	0
$461$	330.774	0.717514	0.358757	−	0.933431i	$-0.383201\pi$
$461$	330.774	0.717514	0.358757	+	0.933431i	$0.383201\pi$
$462$	0	0
$463$	789.925i	1.70610i	0.521828	+	0.853051i	$0.325250\pi$
$463$	789.925i	1.70610i	−0.521828	+	0.853051i	$0.674750\pi$
$464$	0	0
$465$	6.83282	0.0146942
$466$	0	0
$467$	− 30.3707i	− 0.0650337i	−0.999471	−	0.0325168i	$-0.989648\pi$
$467$	− 30.3707i	− 0.0650337i	0.999471	−	0.0325168i	$-0.0103523\pi$
$468$	0	0
$469$	938.695	2.00148
$470$	0	0
$471$	− 1461.29i	− 3.10253i
$472$	0	0
$473$	−67.3313	−0.142349
$474$	0	0
$475$	− 60.0000i	− 0.126316i
$476$	0	0
$477$	1165.85	2.44413
$478$	0	0
$479$	774.597i	1.61711i	0.588418	+	0.808557i	$0.299751\pi$
$479$	774.597i	1.61711i	−0.588418	+	0.808557i	$0.700249\pi$
$480$	0	0
$481$	374.334	0.778242
$482$	0	0
$483$	937.994i	1.94202i
$484$	0	0
$485$	−267.580	−0.551712
$486$	0	0
$487$	− 578.869i	− 1.18864i	−0.804228	−	0.594322i	$-0.797421\pi$
$487$	− 578.869i	− 1.18864i	0.804228	−	0.594322i	$-0.202579\pi$
$488$	0	0
$489$	−432.636	−0.884737
$490$	0	0
$491$	− 347.416i	− 0.707569i	−0.935327	−	0.353785i	$-0.884895\pi$
$491$	− 347.416i	− 0.707569i	0.935327	−	0.353785i	$-0.115105\pi$
$492$	0	0
$493$	167.331	0.339414
$494$	0	0
$495$	595.967i	1.20397i
$496$	0	0
$497$	−425.037	−0.855205
$498$	0	0
$499$	224.774i	0.450449i	0.974307	+	0.225224i	$0.0723115\pi$
$499$	224.774i	0.450449i	−0.974307	+	0.225224i	$0.927688\pi$
$500$	0	0
$501$	488.190	0.974432
$502$	0	0
$503$	192.089i	0.381887i	0.981601	+	0.190943i	$0.0611547\pi$
$503$	192.089i	0.381887i	−0.981601	+	0.190943i	$0.938845\pi$
$504$	0	0
$505$	−83.9737	−0.166285
$506$	0	0
$507$	189.066i	0.372911i
$508$	0	0
$509$	367.823	0.722639	0.361320	−	0.932442i	$-0.382326\pi$
$509$	367.823	0.722639	0.361320	+	0.932442i	$0.382326\pi$
$510$	0	0
$511$	− 1386.20i	− 2.71271i
$512$	0	0
$513$	591.659	1.15333
$514$	0	0
$515$	395.813i	0.768570i
$516$	0	0
$517$	326.217	0.630980
$518$	0	0
$519$	835.574i	1.60997i
$520$	0	0
$521$	−858.984	−1.64872	−0.824361	−	0.566064i	$-0.808466\pi$
$521$	−858.984	−1.64872	−0.824361	+	0.566064i	$0.808466\pi$
$522$	0	0
$523$	609.872i	1.16610i	0.812435	+	0.583052i	$0.198142\pi$
$523$	609.872i	1.16610i	−0.812435	+	0.583052i	$0.801858\pi$
$524$	0	0
$525$	266.525	0.507666
$526$	0	0
$527$	11.0557i	0.0209786i
$528$	0	0
$529$	219.354	0.414659
$530$	0	0
$531$	− 563.724i	− 1.06163i
$532$	0	0
$533$	−821.994	−1.54220
$534$	0	0
$535$	− 141.151i	− 0.263834i
$536$	0	0
$537$	899.712	1.67544
$538$	0	0
$539$	− 790.748i	− 1.46706i
$540$	0	0
$541$	−1061.44	−1.96199	−0.980995	−	0.194033i	$-0.937843\pi$
$541$	−1061.44	−1.96199	−0.980995	+	0.194033i	$0.937843\pi$
$542$	0	0
$543$	1242.68i	2.28855i
$544$	0	0
$545$	−99.4427	−0.182464
$546$	0	0
$547$	235.453i	0.430444i	0.976565	+	0.215222i	$0.0690475\pi$
$547$	235.453i	0.430444i	−0.976565	+	0.215222i	$0.930953\pi$
$548$	0	0
$549$	1200.63	2.18694
$550$	0	0
$551$	105.994i	0.192366i
$552$	0	0
$553$	−826.310	−1.49423
$554$	0	0
$555$	− 380.190i	− 0.685028i
$556$	0	0
$557$	589.574	1.05848	0.529241	−	0.848472i	$-0.322477\pi$
$557$	589.574	1.05848	0.529241	+	0.848472i	$0.322477\pi$
$558$	0	0
$559$	− 53.6331i	− 0.0959447i
$560$	0	0
$561$	−1435.54	−2.55890
$562$	0	0
$563$	− 688.778i	− 1.22341i	−0.791087	−	0.611703i	$-0.790485\pi$
$563$	− 688.778i	− 1.22341i	0.791087	−	0.611703i	$-0.209515\pi$
$564$	0	0
$565$	−41.8622	−0.0740924
$566$	0	0
$567$	940.830i	1.65931i
$568$	0	0
$569$	492.906	0.866266	0.433133	−	0.901330i	$-0.357408\pi$
$569$	492.906	0.866266	0.433133	+	0.901330i	$0.357408\pi$
$570$	0	0
$571$	− 517.240i	− 0.905849i	−0.891549	−	0.452925i	$-0.850381\pi$
$571$	− 517.240i	− 0.905849i	0.891549	−	0.452925i	$-0.149619\pi$
$572$	0	0
$573$	−475.161	−0.829251
$574$	0	0
$575$	87.9837i	0.153015i
$576$	0	0
$577$	−302.170	−0.523692	−0.261846	−	0.965110i	$-0.584331\pi$
$577$	−302.170	−0.523692	−0.261846	+	0.965110i	$0.584331\pi$
$578$	0	0
$579$	259.193i	0.447657i
$580$	0	0
$581$	−881.017	−1.51638
$582$	0	0
$583$	− 916.158i	− 1.57145i
$584$	0	0
$585$	−474.721	−0.811490
$586$	0	0
$587$	1086.61i	1.85113i	0.378588	+	0.925565i	$0.376410\pi$
$587$	1086.61i	1.85113i	−0.378588	+	0.925565i	$0.623590\pi$
$588$	0	0
$589$	−7.00311	−0.0118898
$590$	0	0
$591$	1974.12i	3.34030i
$592$	0	0
$593$	−439.548	−0.741228	−0.370614	−	0.928787i	$-0.620853\pi$
$593$	−439.548	−0.741228	−0.370614	+	0.928787i	$0.620853\pi$
$594$	0	0
$595$	431.246i	0.724783i
$596$	0	0
$597$	−2047.87	−3.43027
$598$	0	0
$599$	− 489.718i	− 0.817560i	−0.912633	−	0.408780i	$-0.865954\pi$
$599$	− 489.718i	− 0.817560i	0.912633	−	0.408780i	$-0.134046\pi$
$600$	0	0
$601$	23.3050	0.0387770	0.0193885	−	0.999812i	$-0.493828\pi$
$601$	23.3050	0.0387770	0.0193885	+	0.999812i	$0.493828\pi$
$602$	0	0
$603$	1698.12i	2.81611i
$604$	0	0
$605$	197.764	0.326883
$606$	0	0
$607$	1110.44i	1.82940i	0.404138	+	0.914698i	$0.367572\pi$
$607$	1110.44i	1.82940i	−0.404138	+	0.914698i	$0.632428\pi$
$608$	0	0
$609$	−470.833	−0.773124
$610$	0	0
$611$	259.850i	0.425286i
$612$	0	0
$613$	−149.587	−0.244024	−0.122012	−	0.992529i	$-0.538935\pi$
$613$	−149.587	−0.244024	−0.122012	+	0.992529i	$0.538935\pi$
$614$	0	0
$615$	834.853i	1.35748i
$616$	0	0
$617$	247.325	0.400851	0.200425	−	0.979709i	$-0.435768\pi$
$617$	247.325	0.400851	0.200425	+	0.979709i	$0.435768\pi$
$618$	0	0
$619$	− 61.3375i	− 0.0990912i	−0.998772	−	0.0495456i	$-0.984223\pi$
$619$	− 61.3375i	− 0.0990912i	0.998772	−	0.0495456i	$-0.0157773\pi$
$620$	0	0
$621$	−867.607	−1.39711
$622$	0	0
$623$	− 305.410i	− 0.490225i
$624$	0	0
$625$	25.0000	0.0400000
$626$	0	0
$627$	− 909.325i	− 1.45028i
$628$	0	0
$629$	615.161	0.977998
$630$	0	0
$631$	172.636i	0.273591i	0.990599	+	0.136796i	$0.0436804\pi$
$631$	172.636i	0.273591i	−0.990599	+	0.136796i	$0.956320\pi$
$632$	0	0
$633$	1400.21	2.21202
$634$	0	0
$635$	− 182.148i	− 0.286847i
$636$	0	0
$637$	629.875	0.988814
$638$	0	0
$639$	− 768.899i	− 1.20329i
$640$	0	0
$641$	−646.020	−1.00783	−0.503916	−	0.863753i	$-0.668108\pi$
$641$	−646.020	−1.00783	−0.503916	+	0.863753i	$0.668108\pi$
$642$	0	0
$643$	296.810i	0.461602i	0.973001	+	0.230801i	$0.0741347\pi$
$643$	296.810i	0.461602i	−0.973001	+	0.230801i	$0.925865\pi$
$644$	0	0
$645$	−54.4721	−0.0844529
$646$	0	0
$647$	204.450i	0.315996i	0.987439	+	0.157998i	$0.0505040\pi$
$647$	204.450i	0.315996i	−0.987439	+	0.157998i	$0.949496\pi$
$648$	0	0
$649$	−442.991	−0.682574
$650$	0	0
$651$	− 31.1084i	− 0.0477855i
$652$	0	0
$653$	653.639	1.00098	0.500490	−	0.865743i	$-0.333153\pi$
$653$	653.639	1.00098	0.500490	+	0.865743i	$0.333153\pi$
$654$	0	0
$655$	− 378.079i	− 0.577220i
$656$	0	0
$657$	2507.65	3.81682
$658$	0	0
$659$	− 1245.72i	− 1.89032i	−0.326613	−	0.945158i	$-0.605908\pi$
$659$	− 1245.72i	− 1.89032i	0.326613	−	0.945158i	$-0.394092\pi$
$660$	0	0
$661$	−1005.42	−1.52105	−0.760527	−	0.649307i	$-0.775059\pi$
$661$	−1005.42	−1.52105	−0.760527	+	0.649307i	$0.775059\pi$
$662$	0	0
$663$	− 1143.49i	− 1.72472i
$664$	0	0
$665$	−273.167	−0.410778
$666$	0	0
$667$	− 155.429i	− 0.233027i
$668$	0	0
$669$	166.026	0.248171
$670$	0	0
$671$	− 943.489i	− 1.40609i
$672$	0	0
$673$	−897.331	−1.33333	−0.666665	−	0.745357i	$-0.732279\pi$
$673$	−897.331	−1.33333	−0.666665	+	0.745357i	$0.732279\pi$
$674$	0	0
$675$	246.525i	0.365222i
$676$	0	0
$677$	796.067	1.17587	0.587937	−	0.808907i	$-0.299940\pi$
$677$	796.067	1.17587	0.587937	+	0.808907i	$0.299940\pi$
$678$	0	0
$679$	1218.24i	1.79416i
$680$	0	0
$681$	850.630	1.24909
$682$	0	0
$683$	1124.31i	1.64614i	0.567942	+	0.823069i	$0.307740\pi$
$683$	1124.31i	1.64614i	−0.567942	+	0.823069i	$0.692260\pi$
$684$	0	0
$685$	55.0294	0.0803349
$686$	0	0
$687$	− 2226.17i	− 3.24042i
$688$	0	0
$689$	729.771	1.05917
$690$	0	0
$691$	188.584i	0.272914i	0.990646	+	0.136457i	$0.0435716\pi$
$691$	188.584i	0.272914i	−0.990646	+	0.136457i	$0.956428\pi$
$692$	0	0
$693$	2713.31	3.91531
$694$	0	0
$695$	− 164.105i	− 0.236123i
$696$	0	0
$697$	−1350.82	−1.93805
$698$	0	0
$699$	928.519i	1.32835i
$700$	0	0
$701$	−177.404	−0.253073	−0.126536	−	0.991962i	$-0.540386\pi$
$701$	−177.404	−0.253073	−0.126536	+	0.991962i	$0.540386\pi$
$702$	0	0
$703$	389.666i	0.554290i
$704$	0	0
$705$	263.915	0.374347
$706$	0	0
$707$	382.314i	0.540756i
$708$	0	0
$709$	−1025.15	−1.44591	−0.722954	−	0.690896i	$-0.757216\pi$
$709$	−1025.15	−1.44591	−0.722954	+	0.690896i	$0.757216\pi$
$710$	0	0
$711$	− 1494.81i	− 2.10240i
$712$	0	0
$713$	10.2693	0.0144030
$714$	0	0
$715$	373.050i	0.521748i
$716$	0	0
$717$	447.659	0.624351
$718$	0	0
$719$	− 1215.38i	− 1.69038i	−0.534465	−	0.845190i	$-0.679487\pi$
$719$	− 1215.38i	− 1.69038i	0.534465	−	0.845190i	$-0.320513\pi$
$720$	0	0
$721$	1802.05	2.49938
$722$	0	0
$723$	− 262.249i	− 0.362724i
$724$	0	0
$725$	−44.1641	−0.0609160
$726$	0	0
$727$	753.735i	1.03677i	0.855146	+	0.518387i	$0.173467\pi$
$727$	753.735i	1.03677i	−0.855146	+	0.518387i	$0.826533\pi$
$728$	0	0
$729$	621.498	0.852536
$730$	0	0
$731$	− 88.1378i	− 0.120572i
$732$	0	0
$733$	−901.141	−1.22939	−0.614694	−	0.788766i	$-0.710720\pi$
$733$	−901.141	−1.22939	−0.614694	+	0.788766i	$0.710720\pi$
$734$	0	0
$735$	− 639.728i	− 0.870379i
$736$	0	0
$737$	1334.43	1.81062
$738$	0	0
$739$	415.947i	0.562852i	0.959583	+	0.281426i	$0.0908073\pi$
$739$	415.947i	0.562852i	−0.959583	+	0.281426i	$0.909193\pi$
$740$	0	0
$741$	724.328	0.977501
$742$	0	0
$743$	492.207i	0.662458i	0.943550	+	0.331229i	$0.107463\pi$
$743$	492.207i	0.662458i	−0.943550	+	0.331229i	$0.892537\pi$
$744$	0	0
$745$	−154.105	−0.206853
$746$	0	0
$747$	− 1593.77i	− 2.13357i
$748$	0	0
$749$	−642.630	−0.857984
$750$	0	0
$751$	994.512i	1.32425i	0.749393	+	0.662125i	$0.230345\pi$
$751$	994.512i	1.32425i	−0.749393	+	0.662125i	$0.769655\pi$
$752$	0	0
$753$	−283.777	−0.376862
$754$	0	0
$755$	245.528i	0.325202i
$756$	0	0
$757$	615.463	0.813029	0.406514	−	0.913644i	$-0.366744\pi$
$757$	615.463	0.813029	0.406514	+	0.913644i	$0.366744\pi$
$758$	0	0
$759$	1333.43i	1.75683i
$760$	0	0
$761$	−1083.88	−1.42428	−0.712139	−	0.702038i	$-0.752274\pi$
$761$	−1083.88	−1.42428	−0.712139	+	0.702038i	$0.752274\pi$
$762$	0	0
$763$	452.741i	0.593370i
$764$	0	0
$765$	−780.132	−1.01978
$766$	0	0
$767$	− 352.867i	− 0.460061i
$768$	0	0
$769$	−411.115	−0.534609	−0.267305	−	0.963612i	$-0.586133\pi$
$769$	−411.115	−0.534609	−0.267305	+	0.963612i	$0.586133\pi$
$770$	0	0
$771$	− 1494.91i	− 1.93892i
$772$	0	0
$773$	−905.574	−1.17151	−0.585753	−	0.810490i	$-0.699201\pi$
$773$	−905.574	−1.17151	−0.585753	+	0.810490i	$0.699201\pi$
$774$	0	0
$775$	− 2.91796i	− 0.00376511i
$776$	0	0
$777$	−1730.93	−2.22770
$778$	0	0
$779$	− 855.659i	− 1.09841i
$780$	0	0
$781$	−604.223	−0.773653
$782$	0	0
$783$	− 435.502i	− 0.556196i
$784$	0	0
$785$	−624.046	−0.794964
$786$	0	0
$787$	− 583.262i	− 0.741121i	−0.928808	−	0.370561i	$-0.879166\pi$
$787$	− 583.262i	− 0.741121i	0.928808	−	0.370561i	$-0.120834\pi$
$788$	0	0
$789$	644.243	0.816531
$790$	0	0
$791$	190.590i	0.240948i
$792$	0	0
$793$	751.542	0.947720
$794$	0	0
$795$	− 741.187i	− 0.932311i
$796$	0	0
$797$	−1061.80	−1.33224	−0.666121	−	0.745844i	$-0.732047\pi$
$797$	−1061.80	−1.33224	−0.666121	+	0.745844i	$0.732047\pi$
$798$	0	0
$799$	427.023i	0.534447i
$800$	0	0
$801$	552.492	0.689753
$802$	0	0
$803$	− 1970.59i	− 2.45403i
$804$	0	0
$805$	400.571	0.497604
$806$	0	0
$807$	− 965.293i	− 1.19615i
$808$	0	0
$809$	909.672	1.12444	0.562220	−	0.826988i	$-0.309947\pi$
$809$	909.672	1.12444	0.562220	+	0.826988i	$0.309947\pi$
$810$	0	0
$811$	604.689i	0.745609i	0.927910	+	0.372804i	$0.121604\pi$
$811$	604.689i	0.745609i	−0.927910	+	0.372804i	$0.878396\pi$
$812$	0	0
$813$	−965.601	−1.18770
$814$	0	0
$815$	184.758i	0.226697i
$816$	0	0
$817$	55.8297	0.0683350
$818$	0	0
$819$	2161.30i	2.63896i
$820$	0	0
$821$	674.584	0.821661	0.410830	−	0.911712i	$-0.365239\pi$
$821$	674.584	0.821661	0.410830	+	0.911712i	$0.365239\pi$
$822$	0	0
$823$	499.039i	0.606366i	0.952932	+	0.303183i	$0.0980494\pi$
$823$	499.039i	0.606366i	−0.952932	+	0.303183i	$0.901951\pi$
$824$	0	0
$825$	378.885	0.459255
$826$	0	0
$827$	1146.90i	1.38682i	0.720542	+	0.693411i	$0.243893\pi$
$827$	1146.90i	1.38682i	−0.720542	+	0.693411i	$0.756107\pi$
$828$	0	0
$829$	−1196.74	−1.44359	−0.721794	−	0.692107i	$-0.756682\pi$
$829$	−1196.74	−1.44359	−0.721794	+	0.692107i	$0.756682\pi$
$830$	0	0
$831$	106.918i	0.128662i
$832$	0	0
$833$	1035.10	1.24262
$834$	0	0
$835$	− 208.482i	− 0.249679i
$836$	0	0
$837$	28.7740	0.0343775
$838$	0	0
$839$	− 1121.82i	− 1.33710i	−0.743669	−	0.668548i	$-0.766916\pi$
$839$	− 1121.82i	− 1.33710i	0.743669	−	0.668548i	$-0.233084\pi$
$840$	0	0
$841$	−762.981	−0.907231
$842$	0	0
$843$	− 2426.45i	− 2.87835i
$844$	0	0
$845$	80.7407	0.0955512
$846$	0	0
$847$	− 900.377i	− 1.06302i
$848$	0	0
$849$	−1659.29	−1.95440
$850$	0	0
$851$	− 571.404i	− 0.671450i
$852$	0	0
$853$	1278.40	1.49871	0.749356	−	0.662168i	$-0.230363\pi$
$853$	1278.40	1.49871	0.749356	+	0.662168i	$0.230363\pi$
$854$	0	0
$855$	− 494.164i	− 0.577970i
$856$	0	0
$857$	−485.384	−0.566376	−0.283188	−	0.959064i	$-0.591392\pi$
$857$	−485.384	−0.566376	−0.283188	+	0.959064i	$0.591392\pi$
$858$	0	0
$859$	− 781.483i	− 0.909759i	−0.890553	−	0.454879i	$-0.849682\pi$
$859$	− 781.483i	− 0.909759i	0.890553	−	0.454879i	$-0.150318\pi$
$860$	0	0
$861$	3800.91	4.41453
$862$	0	0
$863$	154.659i	0.179211i	0.995977	+	0.0896053i	$0.0285605\pi$
$863$	154.659i	0.179211i	−0.995977	+	0.0896053i	$0.971439\pi$
$864$	0	0
$865$	356.833	0.412523
$866$	0	0
$867$	− 365.925i	− 0.422059i
$868$	0	0
$869$	−1174.66	−1.35174
$870$	0	0
$871$	1062.95i	1.22037i
$872$	0	0
$873$	−2203.81	−2.52441
$874$	0	0
$875$	− 113.820i	− 0.130080i
$876$	0	0
$877$	146.203	0.166708	0.0833539	−	0.996520i	$-0.473437\pi$
$877$	146.203	0.166708	0.0833539	+	0.996520i	$0.473437\pi$
$878$	0	0
$879$	− 2131.51i	− 2.42493i
$880$	0	0
$881$	−39.2523	−0.0445543	−0.0222771	−	0.999752i	$-0.507092\pi$
$881$	−39.2523	−0.0445543	−0.0222771	+	0.999752i	$0.507092\pi$
$882$	0	0
$883$	− 493.131i	− 0.558472i	−0.960222	−	0.279236i	$-0.909919\pi$
$883$	− 493.131i	− 0.558472i	0.960222	−	0.279236i	$-0.0900812\pi$
$884$	0	0
$885$	−358.387	−0.404957
$886$	0	0
$887$	865.183i	0.975404i	0.873010	+	0.487702i	$0.162165\pi$
$887$	865.183i	0.975404i	−0.873010	+	0.487702i	$0.837835\pi$
$888$	0	0
$889$	−829.280	−0.932824
$890$	0	0
$891$	1337.46i	1.50108i
$892$	0	0
$893$	−270.492	−0.302903
$894$	0	0
$895$	− 384.223i	− 0.429299i
$896$	0	0
$897$	−1062.15	−1.18412
$898$	0	0
$899$	5.15476i	0.00573388i
$900$	0	0
$901$	1199.27	1.33104
$902$	0	0
$903$	248.000i	0.274640i
$904$	0	0
$905$	530.689	0.586397
$906$	0	0
$907$	833.183i	0.918615i	0.888277	+	0.459307i	$0.151902\pi$
$907$	833.183i	0.918615i	−0.888277	+	0.459307i	$0.848098\pi$
$908$	0	0
$909$	−691.613	−0.760850
$910$	0	0
$911$	− 541.227i	− 0.594103i	−0.954862	−	0.297051i	$-0.903997\pi$
$911$	− 541.227i	− 0.594103i	0.954862	−	0.297051i	$-0.0960032\pi$
$912$	0	0
$913$	−1252.43	−1.37178
$914$	0	0
$915$	− 763.299i	− 0.834206i
$916$	0	0
$917$	−1721.31	−1.87711
$918$	0	0
$919$	330.610i	0.359750i	0.983690	+	0.179875i	$0.0575693\pi$
$919$	330.610i	0.359750i	−0.983690	+	0.179875i	$0.942431\pi$
$920$	0	0
$921$	−1667.90	−1.81097
$922$	0	0
$923$	− 481.297i	− 0.521449i
$924$	0	0
$925$	−162.361	−0.175525
$926$	0	0
$927$	3259.95i	3.51666i
$928$	0	0
$929$	632.237	0.680556	0.340278	−	0.940325i	$-0.389479\pi$
$929$	632.237	0.680556	0.340278	+	0.940325i	$0.389479\pi$
$930$	0	0
$931$	655.672i	0.704266i
$932$	0	0
$933$	821.601	0.880601
$934$	0	0
$935$	613.050i	0.655668i
$936$	0	0
$937$	−147.666	−0.157594	−0.0787970	−	0.996891i	$-0.525108\pi$
$937$	−147.666	−0.157594	−0.0787970	+	0.996891i	$0.525108\pi$
$938$	0	0
$939$	967.522i	1.03037i
$940$	0	0
$941$	−804.885	−0.855351	−0.427676	−	0.903932i	$-0.640667\pi$
$941$	−804.885	−0.855351	−0.427676	+	0.903932i	$0.640667\pi$
$942$	0	0
$943$	1254.74i	1.33058i
$944$	0	0
$945$	1122.37	1.18770
$946$	0	0
$947$	1365.45i	1.44187i	0.693005	+	0.720933i	$0.256286\pi$
$947$	1365.45i	1.44187i	−0.693005	+	0.720933i	$0.743714\pi$
$948$	0	0
$949$	1569.68	1.65404
$950$	0	0
$951$	76.3607i	0.0802951i
$952$	0	0
$953$	1075.99	1.12906	0.564530	−	0.825413i	$-0.309058\pi$
$953$	1075.99	1.12906	0.564530	+	0.825413i	$0.309058\pi$
$954$	0	0
$955$	202.918i	0.212480i
$956$	0	0
$957$	−669.325	−0.699399
$958$	0	0
$959$	− 250.537i	− 0.261248i
$960$	0	0
$961$	960.659	0.999646
$962$	0	0
$963$	− 1162.53i	− 1.20719i
$964$	0	0
$965$	110.689	0.114703
$966$	0	0
$967$	− 1057.05i	− 1.09312i	−0.837420	−	0.546559i	$-0.815937\pi$
$967$	− 1057.05i	− 1.09312i	0.837420	−	0.546559i	$-0.184063\pi$
$968$	0	0
$969$	1190.32	1.22840
$970$	0	0
$971$	112.190i	0.115541i	0.998330	+	0.0577705i	$0.0183992\pi$
$971$	112.190i	0.115541i	−0.998330	+	0.0577705i	$0.981601\pi$
$972$	0	0
$973$	−747.136	−0.767869
$974$	0	0
$975$	301.803i	0.309542i
$976$	0	0
$977$	−1627.22	−1.66553	−0.832763	−	0.553629i	$-0.813243\pi$
$977$	−1627.22	−1.66553	−0.832763	+	0.553629i	$0.813243\pi$
$978$	0	0
$979$	− 434.164i	− 0.443477i
$980$	0	0
$981$	−819.017	−0.834880
$982$	0	0
$983$	− 1451.62i	− 1.47673i	−0.674403	−	0.738364i	$-0.735599\pi$
$983$	− 1451.62i	− 1.47673i	0.674403	−	0.738364i	$-0.264401\pi$
$984$	0	0
$985$	843.050	0.855888
$986$	0	0
$987$	− 1201.55i	− 1.21737i
$988$	0	0
$989$	−81.8684	−0.0827790
$990$	0	0
$991$	− 1179.44i	− 1.19016i	−0.803668	−	0.595078i	$-0.797121\pi$
$991$	− 1179.44i	− 1.19016i	0.803668	−	0.595078i	$-0.202879\pi$
$992$	0	0
$993$	−400.997	−0.403824
$994$	0	0
$995$	874.545i	0.878940i
$996$	0	0
$997$	987.896	0.990869	0.495434	−	0.868645i	$-0.335009\pi$
$997$	987.896	0.990869	0.495434	+	0.868645i	$0.335009\pi$
$998$	0	0
$999$	− 1601.04i	− 1.60264i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.3.b.b.191.1		4
3.2	odd	2		2880.3.e.a.2431.4		4
4.3	odd	2	inner	320.3.b.b.191.4		4
5.2	odd	4		1600.3.h.d.1599.2		4
5.3	odd	4		1600.3.h.m.1599.4		4
5.4	even	2		1600.3.b.n.1151.4		4
8.3	odd	2		160.3.b.a.31.1	✓	4
8.5	even	2		160.3.b.a.31.4	yes	4
12.11	even	2		2880.3.e.a.2431.3		4
16.3	odd	4		1280.3.g.a.1151.2		4
16.5	even	4		1280.3.g.a.1151.1		4
16.11	odd	4		1280.3.g.d.1151.3		4
16.13	even	4		1280.3.g.d.1151.4		4
20.3	even	4		1600.3.h.d.1599.1		4
20.7	even	4		1600.3.h.m.1599.3		4
20.19	odd	2		1600.3.b.n.1151.1		4
24.5	odd	2		1440.3.e.b.991.2		4
24.11	even	2		1440.3.e.b.991.1		4
40.3	even	4		800.3.h.j.799.4		4
40.13	odd	4		800.3.h.c.799.1		4
40.19	odd	2		800.3.b.d.351.4		4
40.27	even	4		800.3.h.c.799.2		4
40.29	even	2		800.3.b.d.351.1		4
40.37	odd	4		800.3.h.j.799.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.3.b.a.31.1	✓	4	8.3	odd	2
160.3.b.a.31.4	yes	4	8.5	even	2
320.3.b.b.191.1		4	1.1	even	1	trivial
320.3.b.b.191.4		4	4.3	odd	2	inner
800.3.b.d.351.1		4	40.29	even	2
800.3.b.d.351.4		4	40.19	odd	2
800.3.h.c.799.1		4	40.13	odd	4
800.3.h.c.799.2		4	40.27	even	4
800.3.h.j.799.3		4	40.37	odd	4
800.3.h.j.799.4		4	40.3	even	4
1280.3.g.a.1151.1		4	16.5	even	4
1280.3.g.a.1151.2		4	16.3	odd	4
1280.3.g.d.1151.3		4	16.11	odd	4
1280.3.g.d.1151.4		4	16.13	even	4
1440.3.e.b.991.1		4	24.11	even	2
1440.3.e.b.991.2		4	24.5	odd	2
1600.3.b.n.1151.1		4	20.19	odd	2
1600.3.b.n.1151.4		4	5.4	even	2
1600.3.h.d.1599.1		4	20.3	even	4
1600.3.h.d.1599.2		4	5.2	odd	4
1600.3.h.m.1599.3		4	20.7	even	4
1600.3.h.m.1599.4		4	5.3	odd	4
2880.3.e.a.2431.3		4	12.11	even	2
2880.3.e.a.2431.4		4	3.2	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:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	320.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-5.23607i q^{3} -2.23607 q^{5} +10.1803i q^{7} -18.4164 q^{9} +O(q^{10})\)	\(q-5.23607i q^{3} -2.23607 q^{5} +10.1803i q^{7} -18.4164 q^{9} +14.4721i q^{11} -11.5279 q^{13} +11.7082i q^{15} -18.9443 q^{17} -12.0000i q^{19} +53.3050 q^{21} +17.5967i q^{23} +5.00000 q^{25} +49.3050i q^{27} -8.83282 q^{29} -0.583592i q^{31} +75.7771 q^{33} -22.7639i q^{35} -32.4721 q^{37} +60.3607i q^{39} +71.3050 q^{41} +4.65248i q^{43} +41.1803 q^{45} -22.5410i q^{47} -54.6393 q^{49} +99.1935i q^{51} -63.3050 q^{53} -32.3607i q^{55} -62.8328 q^{57} +30.6099i q^{59} -65.1935 q^{61} -187.485i q^{63} +25.7771 q^{65} -92.2067i q^{67} +92.1378 q^{69} +41.7508i q^{71} -136.164 q^{73} -26.1803i q^{75} -147.331 q^{77} +81.1672i q^{79} +92.4164 q^{81} +86.5410i q^{83} +42.3607 q^{85} +46.2492i q^{87} -30.0000 q^{89} -117.358i q^{91} -3.05573 q^{93} +26.8328i q^{95} +119.666 q^{97} -266.525i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 20 q^{9}+O(q^{10}) \) 4 * q - 20 * q^9	\( 4 q - 20 q^{9} - 64 q^{13} - 40 q^{17} + 88 q^{21} + 20 q^{25} + 72 q^{29} + 160 q^{33} - 112 q^{37} + 160 q^{41} + 120 q^{45} - 308 q^{49} - 128 q^{53} - 144 q^{57} - 64 q^{61} - 40 q^{65} + 136 q^{69} - 8 q^{73} - 160 q^{77} + 316 q^{81} + 80 q^{85} - 120 q^{89} - 48 q^{93} + 264 q^{97}+O(q^{100}) \) 4 * q - 20 * q^9 - 64 * q^13 - 40 * q^17 + 88 * q^21 + 20 * q^25 + 72 * q^29 + 160 * q^33 - 112 * q^37 + 160 * q^41 + 120 * q^45 - 308 * q^49 - 128 * q^53 - 144 * q^57 - 64 * q^61 - 40 * q^65 + 136 * q^69 - 8 * q^73 - 160 * q^77 + 316 * q^81 + 80 * q^85 - 120 * q^89 - 48 * q^93 + 264 * q^97

Introduction

L-functions

Modular forms

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