LMFDB - Embedded newform 320.6.f.d.289.27

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [320,6,Mod(289,320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("320.289");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 320 = 2^{6} \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	320.f (of order $2$, degree $1$, 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:	$51.3228223402$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			289.27
Character	$\chi$	$=$	320.289
Dual form			320.6.f.d.289.26

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+19.3064 q^{3} +(19.7709 - 52.2887i) q^{5} -226.751i q^{7} +129.737 q^{9} +O(q^{10})$	$q+19.3064 q^{3} +(19.7709 - 52.2887i) q^{5} -226.751i q^{7} +129.737 q^{9} -533.326i q^{11} +540.576 q^{13} +(381.706 - 1009.51i) q^{15} -300.043i q^{17} +2825.45i q^{19} -4377.74i q^{21} -1511.21i q^{23} +(-2343.22 - 2067.59i) q^{25} -2186.69 q^{27} +5353.99i q^{29} -2542.00 q^{31} -10296.6i q^{33} +(-11856.5 - 4483.07i) q^{35} -2944.93 q^{37} +10436.6 q^{39} -1971.77 q^{41} -9176.59 q^{43} +(2565.03 - 6783.81i) q^{45} +8141.80i q^{47} -34608.9 q^{49} -5792.75i q^{51} +25644.1 q^{53} +(-27886.9 - 10544.3i) q^{55} +54549.3i q^{57} -39333.8i q^{59} -11682.5i q^{61} -29418.1i q^{63} +(10687.7 - 28266.0i) q^{65} +69680.1 q^{67} -29176.1i q^{69} +74876.9 q^{71} +57405.0i q^{73} +(-45239.2 - 39917.8i) q^{75} -120932. q^{77} +42125.5 q^{79} -73743.4 q^{81} +2655.02 q^{83} +(-15688.9 - 5932.13i) q^{85} +103366. i q^{87} +14466.5 q^{89} -122576. i q^{91} -49076.9 q^{93} +(147739. + 55861.8i) q^{95} +97239.7i q^{97} -69192.3i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 2944 q^{9}+O(q^{10}) $ 32 * q + 2944 * q^9	$ 32 q + 2944 q^{9} - 46528 q^{25} - 36768 q^{41} - 113920 q^{49} + 25376 q^{65} + 633472 q^{81} + 968704 q^{89}+O(q^{100}) $ 32 * q + 2944 * q^9 - 46528 * q^25 - 36768 * q^41 - 113920 * q^49 + 25376 * q^65 + 633472 * q^81 + 968704 * q^89

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
$3$	19.3064			1.23851			0.619253	−	0.785191i	$-0.287435\pi$
$3$	19.3064			1.23851			0.619253	+	0.785191i	$0.287435\pi$
$4$		0			0
$5$	19.7709	−	52.2887i	0.353673	−	0.935369i
$5$	19.7709	−	52.2887i	0.353673	−	0.935369i
$6$		0			0
$7$		−	226.751i		−	1.74906i	−0.484976	−	0.874528i	$-0.661172\pi$
$7$		−	226.751i		−	1.74906i	0.484976	−	0.874528i	$-0.338828\pi$
$8$		0			0
$9$	129.737			0.533899
$10$		0			0
$11$		−	533.326i		−	1.32896i	−0.747307	−	0.664478i	$-0.768654\pi$
$11$		−	533.326i		−	1.32896i	0.747307	−	0.664478i	$-0.231346\pi$
$12$		0			0
$13$	540.576			0.887153			0.443577	−	0.896236i	$-0.353709\pi$
$13$	540.576			0.887153			0.443577	+	0.896236i	$0.353709\pi$
$14$		0			0
$15$	381.706	−	1009.51i	0.438026	−	1.15846i
$16$		0			0
$17$		−	300.043i		−	0.251803i	−0.992043	−	0.125902i	$-0.959818\pi$
$17$		−	300.043i		−	0.251803i	0.992043	−	0.125902i	$-0.0401824\pi$
$18$		0			0
$19$			2825.45i			1.79558i	0.440427	+	0.897788i	$0.354827\pi$
$19$			2825.45i			1.79558i	−0.440427	+	0.897788i	$0.645173\pi$
$20$		0			0
$21$		−	4377.74i		−	2.16622i
$22$		0			0
$23$		−	1511.21i		−	0.595671i	−0.954617	−	0.297836i	$-0.903735\pi$
$23$		−	1511.21i		−	0.595671i	0.954617	−	0.297836i	$-0.0962647\pi$
$24$		0			0
$25$	−2343.22	−	2067.59i	−0.749831	−	0.661630i
$26$		0			0
$27$	−2186.69			−0.577269
$28$		0			0
$29$			5353.99i			1.18218i	0.806607	+	0.591088i	$0.201302\pi$
$29$			5353.99i			1.18218i	−0.806607	+	0.591088i	$0.798698\pi$
$30$		0			0
$31$	−2542.00			−0.475085			−0.237543	−	0.971377i	$-0.576342\pi$
$31$	−2542.00			−0.475085			−0.237543	+	0.971377i	$0.576342\pi$
$32$		0			0
$33$		−	10296.6i		−	1.64592i
$34$		0			0
$35$	−11856.5	−	4483.07i	−1.63601	−	0.618594i
$36$		0			0
$37$	−2944.93			−0.353648			−0.176824	−	0.984242i	$-0.556582\pi$
$37$	−2944.93			−0.353648			−0.176824	+	0.984242i	$0.556582\pi$
$38$		0			0
$39$	10436.6			1.09875
$40$		0			0
$41$	−1971.77			−0.183188			−0.0915939	−	0.995796i	$-0.529196\pi$
$41$	−1971.77			−0.183188			−0.0915939	+	0.995796i	$0.529196\pi$
$42$		0			0
$43$	−9176.59			−0.756850			−0.378425	−	0.925632i	$-0.623534\pi$
$43$	−9176.59			−0.756850			−0.378425	+	0.925632i	$0.623534\pi$
$44$		0			0
$45$	2565.03	−	6783.81i	0.188826	−	0.499393i
$46$		0			0
$47$			8141.80i			0.537620i	0.963193	+	0.268810i	$0.0866304\pi$
$47$			8141.80i			0.537620i	−0.963193	+	0.268810i	$0.913370\pi$
$48$		0			0
$49$	−34608.9			−2.05919
$50$		0			0
$51$		−	5792.75i		−	0.311860i
$52$		0			0
$53$	25644.1			1.25400			0.627002	−	0.779018i	$-0.284282\pi$
$53$	25644.1			1.25400			0.627002	+	0.779018i	$0.284282\pi$
$54$		0			0
$55$	−27886.9	−	10544.3i	−1.24307	−	0.470016i
$56$		0			0
$57$			54549.3i			2.22383i
$58$		0			0
$59$		−	39333.8i		−	1.47108i	−0.677482	−	0.735539i	$-0.736929\pi$
$59$		−	39333.8i		−	1.47108i	0.677482	−	0.735539i	$-0.263071\pi$
$60$		0			0
$61$		−	11682.5i		−	0.401985i	−0.979593	−	0.200993i	$-0.935583\pi$
$61$		−	11682.5i		−	0.401985i	0.979593	−	0.200993i	$-0.0644167\pi$
$62$		0			0
$63$		−	29418.1i		−	0.933819i
$64$		0			0
$65$	10687.7	−	28266.0i	0.313762	−	0.829816i
$66$		0			0
$67$	69680.1			1.89636			0.948181	−	0.317729i	$-0.102920\pi$
$67$	69680.1			1.89636			0.948181	+	0.317729i	$0.102920\pi$
$68$		0			0
$69$		−	29176.1i		−	0.737743i
$70$		0			0
$71$	74876.9			1.76280			0.881398	−	0.472374i	$-0.156603\pi$
$71$	74876.9			1.76280			0.881398	+	0.472374i	$0.156603\pi$
$72$		0			0
$73$			57405.0i			1.26079i	0.776275	+	0.630394i	$0.217107\pi$
$73$			57405.0i			1.26079i	−0.776275	+	0.630394i	$0.782893\pi$
$74$		0			0
$75$	−45239.2	−	39917.8i	−0.928670	−	0.819433i
$76$		0			0
$77$	−120932.			−2.32442
$78$		0			0
$79$	42125.5			0.759411			0.379706	−	0.925107i	$-0.376025\pi$
$79$	42125.5			0.759411			0.379706	+	0.925107i	$0.376025\pi$
$80$		0			0
$81$	−73743.4			−1.24885
$82$		0			0
$83$	2655.02			0.0423031			0.0211516	−	0.999776i	$-0.493267\pi$
$83$	2655.02			0.0423031			0.0211516	+	0.999776i	$0.493267\pi$
$84$		0			0
$85$	−15688.9	−	5932.13i	−0.235529	−	0.0890560i
$86$		0			0
$87$			103366.i			1.46413i
$88$		0			0
$89$	14466.5			0.193593			0.0967964	−	0.995304i	$-0.469140\pi$
$89$	14466.5			0.193593			0.0967964	+	0.995304i	$0.469140\pi$
$90$		0			0
$91$		−	122576.i		−	1.55168i
$92$		0			0
$93$	−49076.9			−0.588396
$94$		0			0
$95$	147739.	+	55861.8i	1.67953	+	0.635047i
$96$		0			0
$97$			97239.7i			1.04934i	0.851307	+	0.524668i	$0.175810\pi$
$97$			97239.7i			1.04934i	−0.851307	+	0.524668i	$0.824190\pi$
$98$		0			0
$99$		−	69192.3i		−	0.709529i
$100$		0			0
$101$		−	84471.8i		−	0.823964i	−0.911192	−	0.411982i	$-0.864837\pi$
$101$		−	84471.8i		−	0.823964i	0.911192	−	0.411982i	$-0.135163\pi$
$102$		0			0
$103$		−	180020.i		−	1.67196i	−0.548756	−	0.835982i	$-0.684899\pi$
$103$		−	180020.i		−	1.67196i	0.548756	−	0.835982i	$-0.315101\pi$
$104$		0			0
$105$	−228907.	−	86552.0i	−2.02621	−	0.766133i
$106$		0			0
$107$	−36894.9			−0.311535			−0.155768	−	0.987794i	$-0.549785\pi$
$107$	−36894.9			−0.311535			−0.155768	+	0.987794i	$0.549785\pi$
$108$		0			0
$109$		−	126450.i		−	1.01942i	−0.860348	−	0.509708i	$-0.829754\pi$
$109$		−	126450.i		−	1.01942i	0.860348	−	0.509708i	$-0.170246\pi$
$110$		0			0
$111$	−56856.1			−0.437995
$112$		0			0
$113$			155890.i			1.14848i	0.818688	+	0.574238i	$0.194702\pi$
$113$			155890.i			1.14848i	−0.818688	+	0.574238i	$0.805298\pi$
$114$		0			0
$115$	−79019.5	−	29878.1i	−0.557172	−	0.210673i
$116$		0			0
$117$	70133.0			0.473650
$118$		0			0
$119$	−68035.0			−0.440418
$120$		0			0
$121$	−123385.			−0.766126
$122$		0			0
$123$	−38067.8			−0.226879
$124$		0			0
$125$	−154439.	+	81645.8i	−0.884063	+	0.467368i
$126$		0			0
$127$		−	145038.i		−	0.797943i	−0.916963	−	0.398972i	$-0.869367\pi$
$127$		−	145038.i		−	0.797943i	0.916963	−	0.398972i	$-0.130633\pi$
$128$		0			0
$129$	−177167.			−0.937364
$130$		0			0
$131$		−	144244.i		−	0.734380i	−0.930146	−	0.367190i	$-0.880320\pi$
$131$		−	144244.i		−	0.734380i	0.930146	−	0.367190i	$-0.119680\pi$
$132$		0			0
$133$	640673.			3.14056
$134$		0			0
$135$	−43232.9	+	114339.i	−0.204165	+	0.539960i
$136$		0			0
$137$		−	317515.i		−	1.44532i	−0.691205	−	0.722658i	$-0.742920\pi$
$137$		−	317515.i		−	1.44532i	0.691205	−	0.722658i	$-0.257080\pi$
$138$		0			0
$139$		−	188691.i		−	0.828351i	−0.910197	−	0.414176i	$-0.864070\pi$
$139$		−	188691.i		−	0.828351i	0.910197	−	0.414176i	$-0.135930\pi$
$140$		0			0
$141$			157189.i			0.665846i
$142$		0			0
$143$		−	288303.i		−	1.17899i
$144$		0			0
$145$	279953.	+	105853.i	1.10577	+	0.418104i
$146$		0			0
$147$	−668173.			−2.55033
$148$		0			0
$149$		−	332812.i		−	1.22810i	−0.789268	−	0.614049i	$-0.789540\pi$
$149$		−	332812.i		−	1.22810i	0.789268	−	0.614049i	$-0.210460\pi$
$150$		0			0
$151$	−58016.7			−0.207067			−0.103533	−	0.994626i	$-0.533015\pi$
$151$	−58016.7			−0.207067			−0.103533	+	0.994626i	$0.533015\pi$
$152$		0			0
$153$		−	38926.8i		−	0.134437i
$154$		0			0
$155$	−50257.7	+	132918.i	−0.168025	+	0.444380i
$156$		0			0
$157$	−42076.6			−0.136236			−0.0681179	−	0.997677i	$-0.521699\pi$
$157$	−42076.6			−0.136236			−0.0681179	+	0.997677i	$0.521699\pi$
$158$		0			0
$159$	495096.			1.55309
$160$		0			0
$161$	−342669.			−1.04186
$162$		0			0
$163$	522575.			1.54056			0.770282	−	0.637703i	$-0.220115\pi$
$163$	522575.			1.54056			0.770282	+	0.637703i	$0.220115\pi$
$164$		0			0
$165$	−538396.	−	203573.i	−1.53954	−	0.582118i
$166$		0			0
$167$			99237.7i			0.275350i	0.990477	+	0.137675i	$0.0439630\pi$
$167$			99237.7i			0.275350i	−0.990477	+	0.137675i	$0.956037\pi$
$168$		0			0
$169$	−79070.3			−0.212959
$170$		0			0
$171$			366567.i			0.958657i
$172$		0			0
$173$	−371065.			−0.942617			−0.471308	−	0.881969i	$-0.656218\pi$
$173$	−371065.			−0.942617			−0.471308	+	0.881969i	$0.656218\pi$
$174$		0			0
$175$	−468828.	+	531327.i	−1.15723	+	1.31150i
$176$		0			0
$177$		−	759394.i		−	1.82194i
$178$		0			0
$179$			550467.i			1.28410i	0.766662	+	0.642050i	$0.221916\pi$
$179$			550467.i			1.28410i	−0.766662	+	0.642050i	$0.778084\pi$
$180$		0			0
$181$			151958.i			0.344768i	0.985030	+	0.172384i	$0.0551470\pi$
$181$			151958.i			0.344768i	−0.985030	+	0.172384i	$0.944853\pi$
$182$		0			0
$183$		−	225547.i		−	0.497861i
$184$		0			0
$185$	−58224.0	+	153987.i	−0.125076	+	0.330791i
$186$		0			0
$187$	−160021.			−0.334636
$188$		0			0
$189$			495834.i			1.00968i
$190$		0			0
$191$	361650.			0.717308			0.358654	−	0.933471i	$-0.383236\pi$
$191$	361650.			0.717308			0.358654	+	0.933471i	$0.383236\pi$
$192$		0			0
$193$			721956.i			1.39514i	0.716518	+	0.697569i	$0.245735\pi$
$193$			721956.i			1.39514i	−0.716518	+	0.697569i	$0.754265\pi$
$194$		0			0
$195$	206341.	−	545716.i	0.388597	−	1.02773i
$196$		0			0
$197$	−225516.			−0.414011			−0.207006	−	0.978340i	$-0.566372\pi$
$197$	−225516.			−0.414011			−0.207006	+	0.978340i	$0.566372\pi$
$198$		0			0
$199$	39098.4			0.0699884			0.0349942	−	0.999388i	$-0.488859\pi$
$199$	39098.4			0.0699884			0.0349942	+	0.999388i	$0.488859\pi$
$200$		0			0
$201$	1.34527e6			2.34866
$202$		0			0
$203$	1.21402e6			2.06769
$204$		0			0
$205$	−38983.7	+	103101.i	−0.0647886	+	0.171348i
$206$		0			0
$207$		−	196061.i		−	0.318028i
$208$		0			0
$209$	1.50689e6			2.38624
$210$		0			0
$211$		−	218246.i		−	0.337474i	−0.985661	−	0.168737i	$-0.946031\pi$
$211$		−	218246.i		−	0.337474i	0.985661	−	0.168737i	$-0.0539688\pi$
$212$		0			0
$213$	1.44560e6			2.18324
$214$		0			0
$215$	−181430.	+	479832.i	−0.267678	+	0.707934i
$216$		0			0
$217$			576401.i			0.830951i
$218$		0			0
$219$			1.10828e6i			1.56150i
$220$		0			0
$221$		−	162196.i		−	0.223388i
$222$		0			0
$223$		−	169634.i		−	0.228429i	−0.993456	−	0.114214i	$-0.963565\pi$
$223$		−	169634.i		−	0.228429i	0.993456	−	0.114214i	$-0.0364350\pi$
$224$		0			0
$225$	−304004.	−	268244.i	−0.400334	−	0.353243i
$226$		0			0
$227$	1.00025e6			1.28839			0.644193	−	0.764863i	$-0.277193\pi$
$227$	1.00025e6			1.28839			0.644193	+	0.764863i	$0.277193\pi$
$228$		0			0
$229$			642993.i			0.810247i	0.914262	+	0.405124i	$0.132772\pi$
$229$			642993.i			0.810247i	−0.914262	+	0.405124i	$0.867228\pi$
$230$		0			0
$231$	−2.33476e6			−2.87881
$232$		0			0
$233$			853190.i			1.02957i	0.857319	+	0.514785i	$0.172128\pi$
$233$			853190.i			1.02957i	−0.857319	+	0.514785i	$0.827872\pi$
$234$		0			0
$235$	425724.	+	160971.i	0.502873	+	0.190142i
$236$		0			0
$237$	813292.			0.940536
$238$		0			0
$239$	138181.			0.156478			0.0782389	−	0.996935i	$-0.475070\pi$
$239$	138181.			0.156478			0.0782389	+	0.996935i	$0.475070\pi$
$240$		0			0
$241$	−208470.			−0.231207			−0.115604	−	0.993295i	$-0.536880\pi$
$241$	−208470.			−0.231207			−0.115604	+	0.993295i	$0.536880\pi$
$242$		0			0
$243$	−892354.			−0.969441
$244$		0			0
$245$	−684250.	+	1.80965e6i	−0.728282	+	1.92611i
$246$		0			0
$247$			1.52737e6i			1.59295i
$248$		0			0
$249$	51258.9			0.0523927
$250$		0			0
$251$			537488.i			0.538499i	0.963071	+	0.269249i	$0.0867756\pi$
$251$			537488.i			0.538499i	−0.963071	+	0.269249i	$0.913224\pi$
$252$		0			0
$253$	−805970.			−0.791621
$254$		0			0
$255$	−302896.	−	114528.i	−0.291704	−	0.110296i
$256$		0			0
$257$		−	516235.i		−	0.487545i	−0.969832	−	0.243772i	$-0.921615\pi$
$257$		−	516235.i		−	0.487545i	0.969832	−	0.243772i	$-0.0783850\pi$
$258$		0			0
$259$			667766.i			0.618550i
$260$		0			0
$261$			694613.i			0.631163i
$262$		0			0
$263$			43960.1i			0.0391894i	0.999808	+	0.0195947i	$0.00623759\pi$
$263$			43960.1i			0.0391894i	−0.999808	+	0.0195947i	$0.993762\pi$
$264$		0			0
$265$	507009.	−	1.34090e6i	0.443507	−	1.17296i
$266$		0			0
$267$	279297.			0.239766
$268$		0			0
$269$		−	2.17299e6i		−	1.83095i	−0.402376	−	0.915475i	$-0.631815\pi$
$269$		−	2.17299e6i		−	1.83095i	0.402376	−	0.915475i	$-0.368185\pi$
$270$		0			0
$271$	1.99442e6			1.64965			0.824826	−	0.565386i	$-0.191273\pi$
$271$	1.99442e6			1.64965			0.824826	+	0.565386i	$0.191273\pi$
$272$		0			0
$273$		−	2.36650e6i		−	1.92177i
$274$		0			0
$275$	−1.10270e6	+	1.24970e6i	−0.879277	+	0.996493i
$276$		0			0
$277$	−2.13008e6			−1.66800			−0.834002	−	0.551761i	$-0.813956\pi$
$277$	−2.13008e6			−1.66800			−0.834002	+	0.551761i	$0.813956\pi$
$278$		0			0
$279$	−329793.			−0.253648
$280$		0			0
$281$	541626.			0.409198			0.204599	−	0.978846i	$-0.434411\pi$
$281$	541626.			0.409198			0.204599	+	0.978846i	$0.434411\pi$
$282$		0			0
$283$	−1.41618e6			−1.05112			−0.525560	−	0.850757i	$-0.676144\pi$
$283$	−1.41618e6			−1.05112			−0.525560	+	0.850757i	$0.676144\pi$
$284$		0			0
$285$	2.85232e6	+	1.07849e6i	2.08011	+	0.786510i
$286$		0			0
$287$			447100.i			0.320406i
$288$		0			0
$289$	1.32983e6			0.936595
$290$		0			0
$291$			1.87735e6i			1.29961i
$292$		0			0
$293$	546813.			0.372109			0.186054	−	0.982539i	$-0.440430\pi$
$293$	546813.			0.372109			0.186054	+	0.982539i	$0.440430\pi$
$294$		0			0
$295$	−2.05671e6	−	777665.i	−1.37600	−	0.520281i
$296$		0			0
$297$			1.16622e6i			0.767166i
$298$		0			0
$299$		−	816927.i		−	0.528451i
$300$		0			0
$301$			2.08080e6i			1.32377i
$302$		0			0
$303$		−	1.63085e6i		−	1.02048i
$304$		0			0
$305$	−610861.	−	230973.i	−0.376005	−	0.142171i
$306$		0			0
$307$	1.16103e6			0.703067			0.351533	−	0.936175i	$-0.385660\pi$
$307$	1.16103e6			0.703067			0.351533	+	0.936175i	$0.385660\pi$
$308$		0			0
$309$		−	3.47554e6i		−	2.07074i
$310$		0			0
$311$	−1.56419e6			−0.917041			−0.458521	−	0.888684i	$-0.651620\pi$
$311$	−1.56419e6			−0.917041			−0.458521	+	0.888684i	$0.651620\pi$
$312$		0			0
$313$		−	2.79472e6i		−	1.61242i	−0.591630	−	0.806209i	$-0.701515\pi$
$313$		−	2.79472e6i		−	1.61242i	0.591630	−	0.806209i	$-0.298485\pi$
$314$		0			0
$315$	−1.53823e6	−	581622.i	−0.873465	−	0.330267i
$316$		0			0
$317$	2.09586e6			1.17142			0.585712	−	0.810519i	$-0.300815\pi$
$317$	2.09586e6			1.17142			0.585712	+	0.810519i	$0.300815\pi$
$318$		0			0
$319$	2.85542e6			1.57106
$320$		0			0
$321$	−712308.			−0.385838
$322$		0			0
$323$	847757.			0.452132
$324$		0			0
$325$	−1.26669e6	−	1.11769e6i	−0.665215	−	0.586967i
$326$		0			0
$327$		−	2.44129e6i		−	1.26255i
$328$		0			0
$329$	1.84616e6			0.940328
$330$		0			0
$331$		−	748106.i		−	0.375312i	−0.982235	−	0.187656i	$-0.939911\pi$
$331$		−	748106.i		−	0.375312i	0.982235	−	0.187656i	$-0.0600891\pi$
$332$		0			0
$333$	−382068.			−0.188812
$334$		0			0
$335$	1.37764e6	−	3.64348e6i	0.670692	−	1.77380i
$336$		0			0
$337$			1.65588e6i			0.794244i	0.917766	+	0.397122i	$0.129991\pi$
$337$			1.65588e6i			0.794244i	−0.917766	+	0.397122i	$0.870009\pi$
$338$		0			0
$339$			3.00968e6i			1.42240i
$340$		0			0
$341$			1.35572e6i			0.631368i
$342$		0			0
$343$			4.03659e6i			1.85259i
$344$		0			0
$345$	−1.52558e6	−	576839.i	−0.690062	−	0.260920i
$346$		0			0
$347$	1.38459e6			0.617301			0.308651	−	0.951175i	$-0.400123\pi$
$347$	1.38459e6			0.617301			0.308651	+	0.951175i	$0.400123\pi$
$348$		0			0
$349$		−	1.99067e6i		−	0.874853i	−0.899254	−	0.437426i	$-0.855890\pi$
$349$		−	1.99067e6i		−	0.874853i	0.899254	−	0.437426i	$-0.144110\pi$
$350$		0			0
$351$	−1.18207e6			−0.512126
$352$		0			0
$353$			2.93425e6i			1.25332i	0.779294	+	0.626658i	$0.215578\pi$
$353$			2.93425e6i			1.25332i	−0.779294	+	0.626658i	$0.784422\pi$
$354$		0			0
$355$	1.48039e6	−	3.91522e6i	0.623454	−	1.64887i
$356$		0			0
$357$	−1.31351e6			−0.545460
$358$		0			0
$359$	−89539.5			−0.0366673			−0.0183336	−	0.999832i	$-0.505836\pi$
$359$	−89539.5			−0.0366673			−0.0183336	+	0.999832i	$0.505836\pi$
$360$		0			0
$361$	−5.50708e6			−2.22410
$362$		0			0
$363$	−2.38213e6			−0.948853
$364$		0			0
$365$	3.00163e6	+	1.13495e6i	1.17930	+	0.445907i
$366$		0			0
$367$		−	52253.3i		−	0.0202511i	−0.999949	−	0.0101256i	$-0.996777\pi$
$367$		−	52253.3i		−	0.0202511i	0.999949	−	0.0101256i	$-0.00322312\pi$
$368$		0			0
$369$	−255812.			−0.0978038
$370$		0			0
$371$		−	5.81483e6i		−	2.19332i
$372$		0			0
$373$	−2.64065e6			−0.982740			−0.491370	−	0.870951i	$-0.663504\pi$
$373$	−2.64065e6			−0.982740			−0.491370	+	0.870951i	$0.663504\pi$
$374$		0			0
$375$	−2.98167e6	+	1.57629e6i	−1.09492	+	0.578838i
$376$		0			0
$377$			2.89424e6i			1.04877i
$378$		0			0
$379$			848415.i			0.303396i	0.988427	+	0.151698i	$0.0484742\pi$
$379$			848415.i			0.303396i	−0.988427	+	0.151698i	$0.951526\pi$
$380$		0			0
$381$		−	2.80016e6i		−	0.988258i
$382$		0			0
$383$			4.92606e6i			1.71594i	0.513698	+	0.857971i	$0.328275\pi$
$383$			4.92606e6i			1.71594i	−0.513698	+	0.857971i	$0.671725\pi$
$384$		0			0
$385$	−2.39094e6	+	6.32338e6i	−0.822084	+	2.17419i
$386$		0			0
$387$	−1.19055e6			−0.404082
$388$		0			0
$389$			2.26748e6i			0.759749i	0.925038	+	0.379874i	$0.124033\pi$
$389$			2.26748e6i			0.759749i	−0.925038	+	0.379874i	$0.875967\pi$
$390$		0			0
$391$	−453429.			−0.149992
$392$		0			0
$393$		−	2.78484e6i		−	0.909535i
$394$		0			0
$395$	832860.	−	2.20269e6i	0.268583	−	0.710330i
$396$		0			0
$397$	184998.			0.0589103			0.0294552	−	0.999566i	$-0.490623\pi$
$397$	184998.			0.0589103			0.0294552	+	0.999566i	$0.490623\pi$
$398$		0			0
$399$	1.23691e7			3.88961
$400$		0			0
$401$	3.28234e6			1.01935			0.509675	−	0.860367i	$-0.329766\pi$
$401$	3.28234e6			1.01935			0.509675	+	0.860367i	$0.329766\pi$
$402$		0			0
$403$	−1.37415e6			−0.421473
$404$		0			0
$405$	−1.45798e6	+	3.85595e6i	−0.441685	+	1.16814i
$406$		0			0
$407$			1.57061e6i			0.469983i
$408$		0			0
$409$	20416.4			0.00603491			0.00301745	−	0.999995i	$-0.499040\pi$
$409$	20416.4			0.00603491			0.00301745	+	0.999995i	$0.499040\pi$
$410$		0			0
$411$		−	6.13008e6i		−	1.79003i
$412$		0			0
$413$	−8.91897e6			−2.57300
$414$		0			0
$415$	52492.2	−	138828.i	0.0149615	−	0.0395690i
$416$		0			0
$417$		−	3.64295e6i		−	1.02592i
$418$		0			0
$419$		−	1.02946e6i		−	0.286468i	−0.989689	−	0.143234i	$-0.954250\pi$
$419$		−	1.02946e6i		−	0.286468i	0.989689	−	0.143234i	$-0.0457502\pi$
$420$		0			0
$421$		−	205645.i		−	0.0565474i	−0.999600	−	0.0282737i	$-0.990999\pi$
$421$		−	205645.i		−	0.0565474i	0.999600	−	0.0282737i	$-0.00900100\pi$
$422$		0			0
$423$			1.05630e6i			0.287035i
$424$		0			0
$425$	−620367.	+	703067.i	−0.166600	+	0.188810i
$426$		0			0
$427$	−2.64901e6			−0.703094
$428$		0			0
$429$		−	5.56610e6i		−	1.46018i
$430$		0			0
$431$	−2.60103e6			−0.674455			−0.337227	−	0.941423i	$-0.609489\pi$
$431$	−2.60103e6			−0.674455			−0.337227	+	0.941423i	$0.609489\pi$
$432$		0			0
$433$			2.78929e6i			0.714948i	0.933923	+	0.357474i	$0.116362\pi$
$433$			2.78929e6i			0.714948i	−0.933923	+	0.357474i	$0.883638\pi$
$434$		0			0
$435$	5.40489e6	+	2.04365e6i	1.36951	+	0.517825i
$436$		0			0
$437$	4.26986e6			1.06957
$438$		0			0
$439$	3.77670e6			0.935301			0.467651	−	0.883913i	$-0.345101\pi$
$439$	3.77670e6			0.935301			0.467651	+	0.883913i	$0.345101\pi$
$440$		0			0
$441$	−4.49007e6			−1.09940
$442$		0			0
$443$	2.17639e6			0.526899			0.263449	−	0.964673i	$-0.415140\pi$
$443$	2.17639e6			0.526899			0.263449	+	0.964673i	$0.415140\pi$
$444$		0			0
$445$	286017.	−	756436.i	0.0684686	−	0.181081i
$446$		0			0
$447$		−	6.42540e6i		−	1.52101i
$448$		0			0
$449$	−7.28951e6			−1.70641			−0.853204	−	0.521578i	$-0.825344\pi$
$449$	−7.28951e6			−1.70641			−0.853204	+	0.521578i	$0.825344\pi$
$450$		0			0
$451$			1.05160e6i			0.243449i
$452$		0			0
$453$	−1.12009e6			−0.256454
$454$		0			0
$455$	−6.40935e6	−	2.42344e6i	−1.45139	−	0.548787i
$456$		0			0
$457$			2.29557e6i			0.514162i	0.966390	+	0.257081i	$0.0827608\pi$
$457$			2.29557e6i			0.514162i	−0.966390	+	0.257081i	$0.917239\pi$
$458$		0			0
$459$			656102.i			0.145358i
$460$		0			0
$461$			728492.i			0.159651i	0.996809	+	0.0798256i	$0.0254363\pi$
$461$			728492.i			0.159651i	−0.996809	+	0.0798256i	$0.974564\pi$
$462$		0			0
$463$		−	5.59971e6i		−	1.21398i	−0.794708	−	0.606992i	$-0.792376\pi$
$463$		−	5.59971e6i		−	1.21398i	0.794708	−	0.606992i	$-0.207624\pi$
$464$		0			0
$465$	−970296.	+	2.56617e6i	−0.208100	+	0.550368i
$466$		0			0
$467$	−4.92150e6			−1.04425			−0.522126	−	0.852868i	$-0.674861\pi$
$467$	−4.92150e6			−1.04425			−0.522126	+	0.852868i	$0.674861\pi$
$468$		0			0
$469$		−	1.58000e7i		−	3.31684i
$470$		0			0
$471$	−812347.			−0.168729
$472$		0			0
$473$			4.89411e6i			1.00582i
$474$		0			0
$475$	5.84188e6	−	6.62066e6i	1.18801	−	1.34638i
$476$		0			0
$477$	3.32701e6			0.669511
$478$		0			0
$479$	8.31194e6			1.65525			0.827625	−	0.561281i	$-0.189691\pi$
$479$	8.31194e6			1.65525			0.827625	+	0.561281i	$0.189691\pi$
$480$		0			0
$481$	−1.59196e6			−0.313740
$482$		0			0
$483$	−6.61571e6			−1.29035
$484$		0			0
$485$	5.08454e6	+	1.92252e6i	0.981516	+	0.371122i
$486$		0			0
$487$		−	7.86871e6i		−	1.50342i	−0.659493	−	0.751711i	$-0.729229\pi$
$487$		−	7.86871e6i		−	1.50342i	0.659493	−	0.751711i	$-0.270771\pi$
$488$		0			0
$489$	1.00891e7			1.90800
$490$		0			0
$491$			8.53923e6i			1.59851i	0.600993	+	0.799254i	$0.294772\pi$
$491$			8.53923e6i			1.59851i	−0.600993	+	0.799254i	$0.705228\pi$
$492$		0			0
$493$	1.60643e6			0.297676
$494$		0			0
$495$	−3.61798e6	−	1.36800e6i	−0.663671	−	0.250941i
$496$		0			0
$497$		−	1.69784e7i		−	3.08323i
$498$		0			0
$499$		−	6.12241e6i		−	1.10070i	−0.834933	−	0.550352i	$-0.814493\pi$
$499$		−	6.12241e6i		−	1.10070i	0.834933	−	0.550352i	$-0.185507\pi$
$500$		0			0
$501$			1.91592e6i			0.341023i
$502$		0			0
$503$			8.42810e6i			1.48528i	0.669689	+	0.742642i	$0.266428\pi$
$503$			8.42810e6i			1.48528i	−0.669689	+	0.742642i	$0.733572\pi$
$504$		0			0
$505$	−4.41692e6	−	1.67008e6i	−0.770710	−	0.291414i
$506$		0			0
$507$	−1.52656e6			−0.263752
$508$		0			0
$509$		−	1.64869e6i		−	0.282062i	−0.990005	−	0.141031i	$-0.954958\pi$
$509$		−	1.64869e6i		−	0.282062i	0.990005	−	0.141031i	$-0.0450417\pi$
$510$		0			0
$511$	1.30166e7			2.20519
$512$		0			0
$513$		−	6.17840e6i		−	1.03653i
$514$		0			0
$515$	−9.41300e6	−	3.55916e6i	−1.56390	−	0.591329i
$516$		0			0
$517$	4.34223e6			0.714474
$518$		0			0
$519$	−7.16394e6			−1.16744
$520$		0			0
$521$	−3.34886e6			−0.540509			−0.270255	−	0.962789i	$-0.587108\pi$
$521$	−3.34886e6			−0.540509			−0.270255	+	0.962789i	$0.587108\pi$
$522$		0			0
$523$	−4.84532e6			−0.774583			−0.387292	−	0.921957i	$-0.626589\pi$
$523$	−4.84532e6			−0.774583			−0.387292	+	0.921957i	$0.626589\pi$
$524$		0			0
$525$	−9.05139e6	+	1.02580e7i	−1.43323	+	1.62430i
$526$		0			0
$527$			762710.i			0.119628i
$528$		0			0
$529$	4.15257e6			0.645176
$530$		0			0
$531$		−	5.10307e6i		−	0.785407i
$532$		0			0
$533$	−1.06589e6			−0.162516
$534$		0			0
$535$	−729446.	+	1.92919e6i	−0.110182	+	0.291400i
$536$		0			0
$537$			1.06276e7i			1.59037i
$538$		0			0
$539$			1.84578e7i			2.73658i
$540$		0			0
$541$			1.00573e7i			1.47737i	0.674053	+	0.738683i	$0.264552\pi$
$541$			1.00573e7i			1.47737i	−0.674053	+	0.738683i	$0.735448\pi$
$542$		0			0
$543$			2.93376e6i			0.426998i
$544$		0			0
$545$	−6.61189e6	−	2.50003e6i	−0.953530	−	0.360540i
$546$		0			0
$547$	−2.20875e6			−0.315630			−0.157815	−	0.987469i	$-0.550445\pi$
$547$	−2.20875e6			−0.315630			−0.157815	+	0.987469i	$0.550445\pi$
$548$		0			0
$549$		−	1.51565e6i		−	0.214620i
$550$		0			0
$551$	−1.51274e7			−2.12269
$552$		0			0
$553$		−	9.55198e6i		−	1.32825i
$554$		0			0
$555$	−1.12410e6	+	2.97293e6i	−0.154907	+	0.409687i
$556$		0			0
$557$	−9.18396e6			−1.25427			−0.627136	−	0.778909i	$-0.715773\pi$
$557$	−9.18396e6			−1.25427			−0.627136	+	0.778909i	$0.715773\pi$
$558$		0			0
$559$	−4.96065e6			−0.671442
$560$		0			0
$561$	−3.08942e6			−0.414448
$562$		0			0
$563$	1.10566e7			1.47011			0.735054	−	0.678009i	$-0.237157\pi$
$563$	1.10566e7			1.47011			0.735054	+	0.678009i	$0.237157\pi$
$564$		0			0
$565$	8.15129e6	+	3.08209e6i	1.07425	+	0.406185i
$566$		0			0
$567$			1.67214e7i			2.18431i
$568$		0			0
$569$	1.00197e7			1.29740			0.648699	−	0.761045i	$-0.275314\pi$
$569$	1.00197e7			1.29740			0.648699	+	0.761045i	$0.275314\pi$
$570$		0			0
$571$		−	5.44283e6i		−	0.698609i	−0.937009	−	0.349305i	$-0.886418\pi$
$571$		−	5.44283e6i		−	0.698609i	0.937009	−	0.349305i	$-0.113582\pi$
$572$		0			0
$573$	6.98217e6			0.888390
$574$		0			0
$575$	−3.12458e6	+	3.54111e6i	−0.394114	+	0.446652i
$576$		0			0
$577$			1.03799e6i			0.129793i	0.997892	+	0.0648967i	$0.0206718\pi$
$577$			1.03799e6i			0.129793i	−0.997892	+	0.0648967i	$0.979328\pi$
$578$		0			0
$579$			1.39384e7i			1.72789i
$580$		0			0
$581$		−	602028.i		−	0.0739905i
$582$		0			0
$583$		−	1.36767e7i		−	1.66652i
$584$		0			0
$585$	1.38659e6	−	3.66716e6i	0.167517	−	0.443038i
$586$		0			0
$587$	4.01891e6			0.481407			0.240704	−	0.970599i	$-0.422622\pi$
$587$	4.01891e6			0.481407			0.240704	+	0.970599i	$0.422622\pi$
$588$		0			0
$589$		−	7.18230e6i		−	0.853052i
$590$		0			0
$591$	−4.35391e6			−0.512755
$592$		0			0
$593$			9.73063e6i			1.13633i	0.822915	+	0.568165i	$0.192346\pi$
$593$			9.73063e6i			1.13633i	−0.822915	+	0.568165i	$0.807654\pi$
$594$		0			0
$595$	−1.34511e6	+	3.55746e6i	−0.155764	+	0.411953i
$596$		0			0
$597$	754849.			0.0866811
$598$		0			0
$599$	1.50621e7			1.71522			0.857609	−	0.514301i	$-0.171949\pi$
$599$	1.50621e7			1.71522			0.857609	+	0.514301i	$0.171949\pi$
$600$		0			0
$601$	6.76656e6			0.764156			0.382078	−	0.924130i	$-0.375209\pi$
$601$	6.76656e6			0.764156			0.382078	+	0.924130i	$0.375209\pi$
$602$		0			0
$603$	9.04011e6			1.01247
$604$		0			0
$605$	−2.43944e6	+	6.45167e6i	−0.270958	+	0.716611i
$606$		0			0
$607$			787439.i			0.0867451i	0.999059	+	0.0433726i	$0.0138102\pi$
$607$			787439.i			0.0867451i	−0.999059	+	0.0433726i	$0.986190\pi$
$608$		0			0
$609$	2.34384e7			2.56085
$610$		0			0
$611$			4.40126e6i			0.476952i
$612$		0			0
$613$	1.14982e7			1.23588			0.617942	−	0.786224i	$-0.287967\pi$
$613$	1.14982e7			1.23588			0.617942	+	0.786224i	$0.287967\pi$
$614$		0			0
$615$	−752636.	+	1.99052e6i	−0.0802411	+	0.212216i
$616$		0			0
$617$			1.15286e7i			1.21917i	0.792720	+	0.609586i	$0.208664\pi$
$617$			1.15286e7i			1.21917i	−0.792720	+	0.609586i	$0.791336\pi$
$618$		0			0
$619$		−	1.23282e7i		−	1.29323i	−0.762819	−	0.646613i	$-0.776185\pi$
$619$		−	1.23282e7i		−	1.29323i	0.762819	−	0.646613i	$-0.223815\pi$
$620$		0			0
$621$			3.30456e6i			0.343863i
$622$		0			0
$623$		−	3.28029e6i		−	0.338605i
$624$		0			0
$625$	1.21574e6	+	9.68965e6i	0.124492	+	0.992221i
$626$		0			0
$627$	2.90926e7			2.95538
$628$		0			0
$629$			883606.i			0.0890497i
$630$		0			0
$631$	−1.55376e7			−1.55350			−0.776749	−	0.629810i	$-0.783133\pi$
$631$	−1.55376e7			−1.55350			−0.776749	+	0.629810i	$0.783133\pi$
$632$		0			0
$633$		−	4.21355e6i		−	0.417964i
$634$		0			0
$635$	−7.58384e6	−	2.86753e6i	−0.746371	−	0.282211i
$636$		0			0
$637$	−1.87087e7			−1.82682
$638$		0			0
$639$	9.71434e6			0.941155
$640$		0			0
$641$	1.69129e7			1.62582			0.812910	−	0.582389i	$-0.197882\pi$
$641$	1.69129e7			1.62582			0.812910	+	0.582389i	$0.197882\pi$
$642$		0			0
$643$	451136.			0.0430309			0.0215154	−	0.999769i	$-0.493151\pi$
$643$	451136.			0.0430309			0.0215154	+	0.999769i	$0.493151\pi$
$644$		0			0
$645$	−3.50275e6	+	9.26383e6i	−0.331521	+	0.876782i
$646$		0			0
$647$			2.24615e6i			0.210949i	0.994422	+	0.105475i	$0.0336362\pi$
$647$			2.24615e6i			0.210949i	−0.994422	+	0.105475i	$0.966364\pi$
$648$		0			0
$649$	−2.09777e7			−1.95500
$650$		0			0
$651$			1.11282e7i			1.02914i
$652$		0			0
$653$	1.66966e7			1.53230			0.766151	−	0.642660i	$-0.222169\pi$
$653$	1.66966e7			1.53230			0.766151	+	0.642660i	$0.222169\pi$
$654$		0			0
$655$	−7.54236e6	−	2.85185e6i	−0.686916	−	0.259730i
$656$		0			0
$657$			7.44757e6i			0.673134i
$658$		0			0
$659$			1.16732e7i			1.04708i	0.852002	+	0.523538i	$0.175388\pi$
$659$			1.16732e7i			1.04708i	−0.852002	+	0.523538i	$0.824612\pi$
$660$		0			0
$661$			699962.i			0.0623118i	0.999515	+	0.0311559i	$0.00991884\pi$
$661$			699962.i			0.0623118i	−0.999515	+	0.0311559i	$0.990081\pi$
$662$		0			0
$663$		−	3.13142e6i		−	0.276668i
$664$		0			0
$665$	1.26667e7	−	3.35000e7i	1.11073	−	2.93759i
$666$		0			0
$667$	8.09102e6			0.704188
$668$		0			0
$669$		−	3.27502e6i		−	0.282910i
$670$		0			0
$671$	−6.23056e6			−0.534221
$672$		0			0
$673$			7.73374e6i			0.658191i	0.944297	+	0.329096i	$0.106744\pi$
$673$			7.73374e6i			0.658191i	−0.944297	+	0.329096i	$0.893256\pi$
$674$		0			0
$675$	5.12391e6	+	4.52119e6i	0.432854	+	0.381938i
$676$		0			0
$677$	−1.12616e6			−0.0944337			−0.0472169	−	0.998885i	$-0.515035\pi$
$677$	−1.12616e6			−0.0944337			−0.0472169	+	0.998885i	$0.515035\pi$
$678$		0			0
$679$	2.20492e7			1.83535
$680$		0			0
$681$	1.93113e7			1.59567
$682$		0			0
$683$	1.30937e7			1.07402			0.537009	−	0.843577i	$-0.319554\pi$
$683$	1.30937e7			1.07402			0.537009	+	0.843577i	$0.319554\pi$
$684$		0			0
$685$	−1.66025e7	−	6.27757e6i	−1.35190	−	0.511170i
$686$		0			0
$687$			1.24139e7i			1.00350i
$688$		0			0
$689$	1.38626e7			1.11249
$690$		0			0
$691$		−	1.11322e6i		−	0.0886923i	−0.999016	−	0.0443461i	$-0.985880\pi$
$691$		−	1.11322e6i		−	0.0886923i	0.999016	−	0.0443461i	$-0.0141204\pi$
$692$		0			0
$693$	−1.56894e7			−1.24100
$694$		0			0
$695$	−9.86642e6	−	3.73060e6i	−0.774814	−	0.292966i
$696$		0			0
$697$			591616.i			0.0461273i
$698$		0			0
$699$			1.64720e7i			1.27513i
$700$		0			0
$701$			1.36127e7i			1.04629i	0.852245	+	0.523143i	$0.175241\pi$
$701$			1.36127e7i			1.04629i	−0.852245	+	0.523143i	$0.824759\pi$
$702$		0			0
$703$		−	8.32077e6i		−	0.635002i
$704$		0			0
$705$	8.21921e6	+	3.10777e6i	0.622812	+	0.235492i
$706$		0			0
$707$	−1.91540e7			−1.44116
$708$		0			0
$709$		−	2.49344e6i		−	0.186288i	−0.995653	−	0.0931438i	$-0.970308\pi$
$709$		−	2.49344e6i		−	0.186288i	0.995653	−	0.0931438i	$-0.0296916\pi$
$710$		0			0
$711$	5.46525e6			0.405449
$712$		0			0
$713$			3.84151e6i			0.282995i
$714$		0			0
$715$	−1.50750e7	−	5.70002e6i	−1.10279	−	0.416976i
$716$		0			0
$717$	2.66777e6			0.193799
$718$		0			0
$719$	−8.87181e6			−0.640015			−0.320008	−	0.947415i	$-0.603685\pi$
$719$	−8.87181e6			−0.640015			−0.320008	+	0.947415i	$0.603685\pi$
$720$		0			0
$721$	−4.08196e7			−2.92436
$722$		0			0
$723$	−4.02481e6			−0.286352
$724$		0			0
$725$	1.10699e7	−	1.25456e7i	0.782163	−	0.886432i
$726$		0			0
$727$			2.19342e7i			1.53917i	0.638545	+	0.769585i	$0.279537\pi$
$727$			2.19342e7i			1.53917i	−0.638545	+	0.769585i	$0.720463\pi$
$728$		0			0
$729$	691498.			0.0481917
$730$		0			0
$731$			2.75337e6i			0.190577i
$732$		0			0
$733$	2.46781e7			1.69649			0.848246	−	0.529602i	$-0.177659\pi$
$733$	2.46781e7			1.69649			0.848246	+	0.529602i	$0.177659\pi$
$734$		0			0
$735$	−1.32104e7	+	3.49379e7i	−0.901982	+	2.38550i
$736$		0			0
$737$		−	3.71622e7i		−	2.52018i
$738$		0			0
$739$			1.01741e7i			0.685310i	0.939461	+	0.342655i	$0.111326\pi$
$739$			1.01741e7i			0.685310i	−0.939461	+	0.342655i	$0.888674\pi$
$740$		0			0
$741$			2.94881e7i			1.97288i
$742$		0			0
$743$		−	1.62520e7i		−	1.08003i	−0.841657	−	0.540013i	$-0.818419\pi$
$743$		−	1.62520e7i		−	1.08003i	0.841657	−	0.540013i	$-0.181581\pi$
$744$		0			0
$745$	−1.74023e7	−	6.57999e6i	−1.14872	−	0.434345i
$746$		0			0
$747$	344455.			0.0225856
$748$		0			0
$749$			8.36595e6i			0.544892i
$750$		0			0
$751$	−4.07295e6			−0.263517			−0.131759	−	0.991282i	$-0.542062\pi$
$751$	−4.07295e6			−0.263517			−0.131759	+	0.991282i	$0.542062\pi$
$752$		0			0
$753$			1.03770e7i			0.666934i
$754$		0			0
$755$	−1.14704e6	+	3.03362e6i	−0.0732339	+	0.193684i
$756$		0			0
$757$	2.59518e6			0.164599			0.0822996	−	0.996608i	$-0.473774\pi$
$757$	2.59518e6			0.164599			0.0822996	+	0.996608i	$0.473774\pi$
$758$		0			0
$759$	−1.55604e7			−0.980428
$760$		0			0
$761$	−3.27320e6			−0.204885			−0.102443	−	0.994739i	$-0.532666\pi$
$761$	−3.27320e6			−0.204885			−0.102443	+	0.994739i	$0.532666\pi$
$762$		0			0
$763$	−2.86725e7			−1.78301
$764$		0			0
$765$	−2.03543e6	−	769619.i	−0.125749	−	0.0475469i
$766$		0			0
$767$		−	2.12629e7i		−	1.30507i
$768$		0			0
$769$	−3.06637e6			−0.186986			−0.0934930	−	0.995620i	$-0.529803\pi$
$769$	−3.06637e6			−0.186986			−0.0934930	+	0.995620i	$0.529803\pi$
$770$		0			0
$771$		−	9.96664e6i		−	0.603827i
$772$		0			0
$773$	−8.35882e6			−0.503149			−0.251574	−	0.967838i	$-0.580948\pi$
$773$	−8.35882e6			−0.503149			−0.251574	+	0.967838i	$0.580948\pi$
$774$		0			0
$775$	5.95647e6	+	5.25582e6i	0.356234	+	0.314331i
$776$		0			0
$777$			1.28922e7i			0.766078i
$778$		0			0
$779$		−	5.57114e6i		−	0.328928i
$780$		0			0
$781$		−	3.99338e7i		−	2.34268i
$782$		0			0
$783$		−	1.17075e7i		−	0.682434i
$784$		0			0
$785$	−831893.	+	2.20013e6i	−0.0481829	+	0.127431i
$786$		0			0
$787$	−2.90786e7			−1.67354			−0.836771	−	0.547552i	$-0.815560\pi$
$787$	−2.90786e7			−1.67354			−0.836771	+	0.547552i	$0.815560\pi$
$788$		0			0
$789$			848711.i			0.0485364i
$790$		0			0
$791$	3.53482e7			2.00875
$792$		0			0
$793$		−	6.31527e6i		−	0.356622i
$794$		0			0
$795$	9.78851e6	−	2.58880e7i	0.549287	−	1.45271i
$796$		0			0
$797$	1.76998e7			0.987014			0.493507	−	0.869742i	$-0.335715\pi$
$797$	1.76998e7			0.987014			0.493507	+	0.869742i	$0.335715\pi$
$798$		0			0
$799$	2.44289e6			0.135375
$800$		0			0
$801$	1.87685e6			0.103359
$802$		0			0
$803$	3.06155e7			1.67553
$804$		0			0
$805$	−6.77488e6	+	1.79177e7i	−0.368478	+	0.974525i
$806$		0			0
$807$		−	4.19526e7i		−	2.26764i
$808$		0			0
$809$	1.59612e7			0.857422			0.428711	−	0.903442i	$-0.358968\pi$
$809$	1.59612e7			0.857422			0.428711	+	0.903442i	$0.358968\pi$
$810$		0			0
$811$		−	1.63310e7i		−	0.871886i	−0.899974	−	0.435943i	$-0.856415\pi$
$811$		−	1.63310e7i		−	0.871886i	0.899974	−	0.435943i	$-0.143585\pi$
$812$		0			0
$813$	3.85050e7			2.04311
$814$		0			0
$815$	1.03318e7	−	2.73248e7i	0.544856	−	1.44100i
$816$		0			0
$817$		−	2.59280e7i		−	1.35898i
$818$		0			0
$819$		−	1.59027e7i		−	0.828440i
$820$		0			0
$821$		−	3.07399e7i		−	1.59164i	−0.605535	−	0.795819i	$-0.707041\pi$
$821$		−	3.07399e7i		−	1.59164i	0.605535	−	0.795819i	$-0.292959\pi$
$822$		0			0
$823$		−	1.30101e7i		−	0.669546i	−0.942299	−	0.334773i	$-0.891340\pi$
$823$		−	1.30101e7i		−	0.669546i	0.942299	−	0.334773i	$-0.108660\pi$
$824$		0			0
$825$	−2.12892e7	+	2.41272e7i	−1.08899	+	1.23416i
$826$		0			0
$827$	−3.19344e7			−1.62366			−0.811831	−	0.583893i	$-0.801529\pi$
$827$	−3.19344e7			−1.62366			−0.811831	+	0.583893i	$0.801529\pi$
$828$		0			0
$829$		−	1.18291e7i		−	0.597813i	−0.954282	−	0.298907i	$-0.903378\pi$
$829$		−	1.18291e7i		−	0.597813i	0.954282	−	0.298907i	$-0.0966219\pi$
$830$		0			0
$831$	−4.11243e7			−2.06584
$832$		0			0
$833$			1.03842e7i			0.518512i
$834$		0			0
$835$	5.18901e6	+	1.96202e6i	0.257554	+	0.0973839i
$836$		0			0
$837$	5.55858e6			0.274252
$838$		0			0
$839$	−1.24709e7			−0.611638			−0.305819	−	0.952090i	$-0.598930\pi$
$839$	−1.24709e7			−0.611638			−0.305819	+	0.952090i	$0.598930\pi$
$840$		0			0
$841$	−8.15404e6			−0.397542
$842$		0			0
$843$	1.04568e7			0.506794
$844$		0			0
$845$	−1.56329e6	+	4.13448e6i	−0.0753180	+	0.199196i
$846$		0			0
$847$			2.79777e7i			1.34000i
$848$		0			0
$849$	−2.73413e7			−1.30182
$850$		0			0
$851$			4.45042e6i			0.210658i
$852$		0			0
$853$	−2.87080e7			−1.35092			−0.675462	−	0.737395i	$-0.736056\pi$
$853$	−2.87080e7			−1.35092			−0.675462	+	0.737395i	$0.736056\pi$
$854$		0			0
$855$	1.91673e7	+	7.24737e6i	0.896698	+	0.339051i
$856$		0			0
$857$			3.98090e6i			0.185153i	0.995706	+	0.0925763i	$0.0295102\pi$
$857$			3.98090e6i			0.185153i	−0.995706	+	0.0925763i	$0.970490\pi$
$858$		0			0
$859$		−	3.20253e7i		−	1.48085i	−0.672140	−	0.740424i	$-0.734625\pi$
$859$		−	3.20253e7i		−	1.48085i	0.672140	−	0.740424i	$-0.265375\pi$
$860$		0			0
$861$			8.63190e6i			0.396825i
$862$		0			0
$863$			2.67858e7i			1.22427i	0.790752	+	0.612136i	$0.209690\pi$
$863$			2.67858e7i			1.22427i	−0.790752	+	0.612136i	$0.790310\pi$
$864$		0			0
$865$	−7.33630e6	+	1.94025e7i	−0.333378	+	0.881695i
$866$		0			0
$867$	2.56743e7			1.15998
$868$		0			0
$869$		−	2.24666e7i		−	1.00923i
$870$		0			0
$871$	3.76674e7			1.68236
$872$		0			0
$873$			1.26156e7i			0.560239i
$874$		0			0
$875$	1.85132e7	+	3.50193e7i	0.817452	+	1.54627i
$876$		0			0
$877$	−1.84339e7			−0.809314			−0.404657	−	0.914469i	$-0.632609\pi$
$877$	−1.84339e7			−0.809314			−0.404657	+	0.914469i	$0.632609\pi$
$878$		0			0
$879$	1.05570e7			0.460859
$880$		0			0
$881$	5.79005e6			0.251329			0.125664	−	0.992073i	$-0.459894\pi$
$881$	5.79005e6			0.251329			0.125664	+	0.992073i	$0.459894\pi$
$882$		0			0
$883$	−2.88548e7			−1.24542			−0.622711	−	0.782452i	$-0.713969\pi$
$883$	−2.88548e7			−1.24542			−0.622711	+	0.782452i	$0.713969\pi$
$884$		0			0
$885$	−3.97078e7	−	1.50139e7i	−1.70419	−	0.644371i
$886$		0			0
$887$		−	1.45668e7i		−	0.621662i	−0.950465	−	0.310831i	$-0.899393\pi$
$887$		−	1.45668e7i		−	0.621662i	0.950465	−	0.310831i	$-0.100607\pi$
$888$		0			0
$889$	−3.28874e7			−1.39565
$890$		0			0
$891$			3.93293e7i			1.65967i
$892$		0			0
$893$	−2.30043e7			−0.965339
$894$		0			0
$895$	2.87832e7	+	1.08833e7i	1.20111	+	0.454152i
$896$		0			0
$897$		−	1.57719e7i		−	0.654491i
$898$		0			0
$899$		−	1.36098e7i		−	0.561635i
$900$		0			0
$901$		−	7.69435e6i		−	0.315762i
$902$		0			0
$903$			4.01727e7i			1.63950i
$904$		0			0
$905$	7.94569e6	+	3.00435e6i	0.322486	+	0.121935i
$906$		0			0
$907$	−1.84235e7			−0.743624			−0.371812	−	0.928308i	$-0.621263\pi$
$907$	−1.84235e7			−0.743624			−0.371812	+	0.928308i	$0.621263\pi$
$908$		0			0
$909$		−	1.09592e7i		−	0.439913i
$910$		0			0
$911$	8.98672e6			0.358761			0.179381	−	0.983780i	$-0.442591\pi$
$911$	8.98672e6			0.358761			0.179381	+	0.983780i	$0.442591\pi$
$912$		0			0
$913$		−	1.41599e6i		−	0.0562190i
$914$		0			0
$915$	−1.17935e7	−	4.45926e6i	−0.465684	−	0.176080i
$916$		0			0
$917$	−3.27075e7			−1.28447
$918$		0			0
$919$	3.72490e7			1.45488			0.727438	−	0.686173i	$-0.240711\pi$
$919$	3.72490e7			1.45488			0.727438	+	0.686173i	$0.240711\pi$
$920$		0			0
$921$	2.24153e7			0.870753
$922$		0			0
$923$	4.04767e7			1.56387
$924$		0			0
$925$	6.90063e6	+	6.08892e6i	0.265176	+	0.233984i
$926$		0			0
$927$		−	2.33553e7i		−	0.892660i
$928$		0			0
$929$	−3.06146e7			−1.16383			−0.581914	−	0.813251i	$-0.697696\pi$
$929$	−3.06146e7			−1.16383			−0.581914	+	0.813251i	$0.697696\pi$
$930$		0			0
$931$		−	9.77857e7i		−	3.69744i
$932$		0			0
$933$	−3.01989e7			−1.13576
$934$		0			0
$935$	−3.16376e6	+	8.36728e6i	−0.118352	+	0.313008i
$936$		0			0
$937$			3.65796e7i			1.36110i	0.732701	+	0.680550i	$0.238259\pi$
$937$			3.65796e7i			1.36110i	−0.732701	+	0.680550i	$0.761741\pi$
$938$		0			0
$939$		−	5.39561e7i		−	1.99699i
$940$		0			0
$941$			1.02605e7i			0.377742i	0.982002	+	0.188871i	$0.0604829\pi$
$941$			1.02605e7i			0.377742i	−0.982002	+	0.188871i	$0.939517\pi$
$942$		0			0
$943$			2.97977e6i			0.109120i
$944$		0			0
$945$	2.59265e7	+	9.80310e6i	0.944420	+	0.357095i
$946$		0			0
$947$	−1.63044e7			−0.590785			−0.295393	−	0.955376i	$-0.595450\pi$
$947$	−1.63044e7			−0.590785			−0.295393	+	0.955376i	$0.595450\pi$
$948$		0			0
$949$			3.10318e7i			1.11851i
$950$		0			0
$951$	4.04635e7			1.45082
$952$		0			0
$953$		−	1.65678e6i		−	0.0590924i	−0.999563	−	0.0295462i	$-0.990594\pi$
$953$		−	1.65678e6i		−	0.0590924i	0.999563	−	0.0295462i	$-0.00940621\pi$
$954$		0			0
$955$	7.15016e6	−	1.89102e7i	0.253692	−	0.670947i
$956$		0			0
$957$	5.51279e7			1.94577
$958$		0			0
$959$	−7.19968e7			−2.52794
$960$		0			0
$961$	−2.21674e7			−0.774294
$962$		0			0
$963$	−4.78665e6			−0.166328
$964$		0			0
$965$	3.77501e7	+	1.42737e7i	1.30497	+	0.493423i
$966$		0			0
$967$		−	7.85647e6i		−	0.270185i	−0.990833	−	0.135093i	$-0.956867\pi$
$967$		−	7.85647e6i		−	0.270185i	0.990833	−	0.135093i	$-0.0431332\pi$
$968$		0			0
$969$	1.63671e7			0.559968
$970$		0			0
$971$			2.58774e7i			0.880792i	0.897804	+	0.440396i	$0.145162\pi$
$971$			2.58774e7i			0.880792i	−0.897804	+	0.440396i	$0.854838\pi$
$972$		0			0
$973$	−4.27859e7			−1.44883
$974$		0			0
$975$	−2.44552e7	−	2.15786e7i	−0.823873	−	0.726962i
$976$		0			0
$977$			3.25226e7i			1.09006i	0.838418	+	0.545028i	$0.183481\pi$
$977$			3.25226e7i			1.09006i	−0.838418	+	0.545028i	$0.816519\pi$
$978$		0			0
$979$		−	7.71537e6i		−	0.257276i
$980$		0			0
$981$		−	1.64052e7i		−	0.544265i
$982$		0			0
$983$		−	8.88835e6i		−	0.293385i	−0.989182	−	0.146692i	$-0.953137\pi$
$983$		−	8.88835e6i		−	0.293385i	0.989182	−	0.146692i	$-0.0468627\pi$
$984$		0			0
$985$	−4.45866e6	+	1.17919e7i	−0.146425	+	0.387253i
$986$		0			0
$987$	3.56427e7			1.16460
$988$		0			0
$989$			1.38678e7i			0.450834i
$990$		0			0
$991$	−7.76877e6			−0.251286			−0.125643	−	0.992076i	$-0.540099\pi$
$991$	−7.76877e6			−0.251286			−0.125643	+	0.992076i	$0.540099\pi$
$992$		0			0
$993$		−	1.44432e7i		−	0.464827i
$994$		0			0
$995$	773011.	−	2.04440e6i	0.0247530	−	0.0654650i
$996$		0			0
$997$	1.13163e7			0.360551			0.180275	−	0.983616i	$-0.442301\pi$
$997$	1.13163e7			0.360551			0.180275	+	0.983616i	$0.442301\pi$
$998$		0			0
$999$	6.43966e6			0.204150

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.6.f.d.289.27	yes	32
4.3	odd	2	inner	320.6.f.d.289.6	yes	32
5.4	even	2	inner	320.6.f.d.289.5	✓	32
8.3	odd	2	inner	320.6.f.d.289.25	yes	32
8.5	even	2	inner	320.6.f.d.289.8	yes	32
20.19	odd	2	inner	320.6.f.d.289.28	yes	32
40.19	odd	2	inner	320.6.f.d.289.7	yes	32
40.29	even	2	inner	320.6.f.d.289.26	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.6.f.d.289.5	✓	32	5.4	even	2	inner
320.6.f.d.289.6	yes	32	4.3	odd	2	inner
320.6.f.d.289.7	yes	32	40.19	odd	2	inner
320.6.f.d.289.8	yes	32	8.5	even	2	inner
320.6.f.d.289.25	yes	32	8.3	odd	2	inner
320.6.f.d.289.26	yes	32	40.29	even	2	inner
320.6.f.d.289.27	yes	32	1.1	even	1	trivial
320.6.f.d.289.28	yes	32	20.19	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	320.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+19.3064 q^{3} +(19.7709 - 52.2887i) q^{5} -226.751i q^{7} +129.737 q^{9} +O(q^{10})\)	\(q+19.3064 q^{3} +(19.7709 - 52.2887i) q^{5} -226.751i q^{7} +129.737 q^{9} -533.326i q^{11} +540.576 q^{13} +(381.706 - 1009.51i) q^{15} -300.043i q^{17} +2825.45i q^{19} -4377.74i q^{21} -1511.21i q^{23} +(-2343.22 - 2067.59i) q^{25} -2186.69 q^{27} +5353.99i q^{29} -2542.00 q^{31} -10296.6i q^{33} +(-11856.5 - 4483.07i) q^{35} -2944.93 q^{37} +10436.6 q^{39} -1971.77 q^{41} -9176.59 q^{43} +(2565.03 - 6783.81i) q^{45} +8141.80i q^{47} -34608.9 q^{49} -5792.75i q^{51} +25644.1 q^{53} +(-27886.9 - 10544.3i) q^{55} +54549.3i q^{57} -39333.8i q^{59} -11682.5i q^{61} -29418.1i q^{63} +(10687.7 - 28266.0i) q^{65} +69680.1 q^{67} -29176.1i q^{69} +74876.9 q^{71} +57405.0i q^{73} +(-45239.2 - 39917.8i) q^{75} -120932. q^{77} +42125.5 q^{79} -73743.4 q^{81} +2655.02 q^{83} +(-15688.9 - 5932.13i) q^{85} +103366. i q^{87} +14466.5 q^{89} -122576. i q^{91} -49076.9 q^{93} +(147739. + 55861.8i) q^{95} +97239.7i q^{97} -69192.3i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 2944 q^{9}+O(q^{10}) \) 32 * q + 2944 * q^9	\( 32 q + 2944 q^{9} - 46528 q^{25} - 36768 q^{41} - 113920 q^{49} + 25376 q^{65} + 633472 q^{81} + 968704 q^{89}+O(q^{100}) \) 32 * q + 2944 * q^9 - 46528 * q^25 - 36768 * q^41 - 113920 * q^49 + 25376 * q^65 + 633472 * q^81 + 968704 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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