LMFDB - Embedded newform 320.6.f.d.289.31

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			289.31
Character	$\chi$	$=$	320.289
Dual form			320.6.f.d.289.30

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+28.5405 q^{3} +(-5.81388 - 55.5985i) q^{5} +92.3750i q^{7} +571.559 q^{9} +O(q^{10})$	$q+28.5405 q^{3} +(-5.81388 - 55.5985i) q^{5} +92.3750i q^{7} +571.559 q^{9} +541.547i q^{11} +782.810 q^{13} +(-165.931 - 1586.81i) q^{15} +1556.72i q^{17} +484.475i q^{19} +2636.43i q^{21} -1096.90i q^{23} +(-3057.40 + 646.487i) q^{25} +9377.22 q^{27} +597.650i q^{29} -8869.25 q^{31} +15456.0i q^{33} +(5135.92 - 537.058i) q^{35} +13624.1 q^{37} +22341.8 q^{39} +14540.4 q^{41} +1828.39 q^{43} +(-3322.98 - 31777.8i) q^{45} -17951.8i q^{47} +8273.86 q^{49} +44429.6i q^{51} -1650.99 q^{53} +(30109.3 - 3148.49i) q^{55} +13827.1i q^{57} -14453.6i q^{59} +24574.4i q^{61} +52797.7i q^{63} +(-4551.17 - 43523.1i) q^{65} -26842.8 q^{67} -31306.0i q^{69} -48050.6 q^{71} -60646.8i q^{73} +(-87259.6 + 18451.0i) q^{75} -50025.5 q^{77} +28175.9 q^{79} +128742. q^{81} -24525.8 q^{83} +(86551.6 - 9050.61i) q^{85} +17057.2i q^{87} +117015. q^{89} +72312.1i q^{91} -253133. q^{93} +(26936.1 - 2816.68i) q^{95} -37703.4i q^{97} +309526. i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	28.5405			1.83087			0.915436	−	0.402463i	$-0.131846\pi$
$3$	28.5405			1.83087			0.915436	+	0.402463i	$0.131846\pi$
$4$		0			0
$5$	−5.81388	−	55.5985i	−0.104002	−	0.994577i
$5$	−5.81388	−	55.5985i	−0.104002	−	0.994577i
$6$		0			0
$7$			92.3750i			0.712540i	0.934383	+	0.356270i	$0.115952\pi$
$7$			92.3750i			0.712540i	−0.934383	+	0.356270i	$0.884048\pi$
$8$		0			0
$9$	571.559			2.35209
$10$		0			0
$11$			541.547i			1.34944i	0.738072	+	0.674722i	$0.235736\pi$
$11$			541.547i			1.34944i	−0.738072	+	0.674722i	$0.764264\pi$
$12$		0			0
$13$	782.810			1.28469			0.642345	−	0.766416i	$-0.277962\pi$
$13$	782.810			1.28469			0.642345	+	0.766416i	$0.277962\pi$
$14$		0			0
$15$	−165.931	−	1586.81i	−0.190414	−	1.82094i
$16$		0			0
$17$			1556.72i			1.30644i	0.757169	+	0.653219i	$0.226582\pi$
$17$			1556.72i			1.30644i	−0.757169	+	0.653219i	$0.773418\pi$
$18$		0			0
$19$			484.475i			0.307884i	0.988080	+	0.153942i	$0.0491969\pi$
$19$			484.475i			0.307884i	−0.988080	+	0.153942i	$0.950803\pi$
$20$		0			0
$21$			2636.43i			1.30457i
$22$		0			0
$23$		−	1096.90i		−	0.432361i	−0.976353	−	0.216181i	$-0.930640\pi$
$23$		−	1096.90i		−	0.432361i	0.976353	−	0.216181i	$-0.0693600\pi$
$24$		0			0
$25$	−3057.40	+	646.487i	−0.978367	+	0.206876i
$26$		0			0
$27$	9377.22			2.47551
$28$		0			0
$29$			597.650i			0.131963i	0.997821	+	0.0659814i	$0.0210178\pi$
$29$			597.650i			0.131963i	−0.997821	+	0.0659814i	$0.978982\pi$
$30$		0			0
$31$	−8869.25			−1.65761			−0.828805	−	0.559537i	$-0.810979\pi$
$31$	−8869.25			−1.65761			−0.828805	+	0.559537i	$0.810979\pi$
$32$		0			0
$33$			15456.0i			2.47066i
$34$		0			0
$35$	5135.92	−	537.058i	0.708676	−	0.0741056i
$36$		0			0
$37$	13624.1			1.63608			0.818039	−	0.575163i	$-0.195061\pi$
$37$	13624.1			1.63608			0.818039	+	0.575163i	$0.195061\pi$
$38$		0			0
$39$	22341.8			2.35210
$40$		0			0
$41$	14540.4			1.35088			0.675441	−	0.737414i	$-0.263953\pi$
$41$	14540.4			1.35088			0.675441	+	0.737414i	$0.263953\pi$
$42$		0			0
$43$	1828.39			0.150799			0.0753993	−	0.997153i	$-0.475977\pi$
$43$	1828.39			0.150799			0.0753993	+	0.997153i	$0.475977\pi$
$44$		0			0
$45$	−3322.98	−	31777.8i	−0.244622	−	2.33934i
$46$		0			0
$47$		−	17951.8i		−	1.18539i	−0.805426	−	0.592697i	$-0.798063\pi$
$47$		−	17951.8i		−	1.18539i	0.805426	−	0.592697i	$-0.201937\pi$
$48$		0			0
$49$	8273.86			0.492286
$50$		0			0
$51$			44429.6i			2.39192i
$52$		0			0
$53$	−1650.99			−0.0807337			−0.0403669	−	0.999185i	$-0.512853\pi$
$53$	−1650.99			−0.0807337			−0.0403669	+	0.999185i	$0.512853\pi$
$54$		0			0
$55$	30109.3	−	3148.49i	1.34213	−	0.140345i
$56$		0			0
$57$			13827.1i			0.563697i
$58$		0			0
$59$		−	14453.6i		−	0.540564i	−0.962781	−	0.270282i	$-0.912883\pi$
$59$		−	14453.6i		−	0.540564i	0.962781	−	0.270282i	$-0.0871169\pi$
$60$		0			0
$61$			24574.4i			0.845588i	0.906226	+	0.422794i	$0.138951\pi$
$61$			24574.4i			0.845588i	−0.906226	+	0.422794i	$0.861049\pi$
$62$		0			0
$63$			52797.7i			1.67596i
$64$		0			0
$65$	−4551.17	−	43523.1i	−0.133610	−	1.27772i
$66$		0			0
$67$	−26842.8			−0.730535			−0.365268	−	0.930903i	$-0.619022\pi$
$67$	−26842.8			−0.730535			−0.365268	+	0.930903i	$0.619022\pi$
$68$		0			0
$69$		−	31306.0i		−	0.791598i
$70$		0			0
$71$	−48050.6			−1.13124			−0.565618	−	0.824668i	$-0.691362\pi$
$71$	−48050.6			−1.13124			−0.565618	+	0.824668i	$0.691362\pi$
$72$		0			0
$73$		−	60646.8i		−	1.33199i	−0.745957	−	0.665994i	$-0.768007\pi$
$73$		−	60646.8i		−	1.33199i	0.745957	−	0.665994i	$-0.231993\pi$
$74$		0			0
$75$	−87259.6	+	18451.0i	−1.79127	+	0.378763i
$76$		0			0
$77$	−50025.5			−0.961533
$78$		0			0
$79$	28175.9			0.507937			0.253969	−	0.967212i	$-0.418264\pi$
$79$	28175.9			0.507937			0.253969	+	0.967212i	$0.418264\pi$
$80$		0			0
$81$	128742.			2.18025
$82$		0			0
$83$	−24525.8			−0.390777			−0.195388	−	0.980726i	$-0.562597\pi$
$83$	−24525.8			−0.390777			−0.195388	+	0.980726i	$0.562597\pi$
$84$		0			0
$85$	86551.6	−	9050.61i	1.29935	−	0.135872i
$86$		0			0
$87$			17057.2i			0.241607i
$88$		0			0
$89$	117015.			1.56591			0.782953	−	0.622081i	$-0.213713\pi$
$89$	117015.			1.56591			0.782953	+	0.622081i	$0.213713\pi$
$90$		0			0
$91$			72312.1i			0.915393i
$92$		0			0
$93$	−253133.			−3.03487
$94$		0			0
$95$	26936.1	−	2816.68i	0.306215	−	0.0320205i
$96$		0			0
$97$		−	37703.4i		−	0.406865i	−0.979089	−	0.203433i	$-0.934790\pi$
$97$		−	37703.4i		−	0.406865i	0.979089	−	0.203433i	$-0.0652098\pi$
$98$		0			0
$99$			309526.i			3.17402i
$100$		0			0
$101$		−	106605.i		−	1.03985i	−0.854211	−	0.519927i	$-0.825959\pi$
$101$		−	106605.i		−	1.03985i	0.854211	−	0.519927i	$-0.174041\pi$
$102$		0			0
$103$			140551.i			1.30539i	0.757619	+	0.652697i	$0.226362\pi$
$103$			140551.i			1.30539i	−0.757619	+	0.652697i	$0.773638\pi$
$104$		0			0
$105$	146581.	−	15327.9i	1.29750	−	0.135678i
$106$		0			0
$107$	−78552.8			−0.663289			−0.331644	−	0.943405i	$-0.607603\pi$
$107$	−78552.8			−0.663289			−0.331644	+	0.943405i	$0.607603\pi$
$108$		0			0
$109$		−	149209.i		−	1.20290i	−0.798910	−	0.601450i	$-0.794590\pi$
$109$		−	149209.i		−	1.20290i	0.798910	−	0.601450i	$-0.205410\pi$
$110$		0			0
$111$	388839.			2.99545
$112$		0			0
$113$			190766.i			1.40541i	0.711479	+	0.702707i	$0.248026\pi$
$113$			190766.i			1.40541i	−0.711479	+	0.702707i	$0.751974\pi$
$114$		0			0
$115$	−60986.0	+	6377.24i	−0.430017	+	0.0449664i
$116$		0			0
$117$	447422.			3.02171
$118$		0			0
$119$	−143802.			−0.930890
$120$		0			0
$121$	−132223.			−0.820999
$122$		0			0
$123$	414991.			2.47329
$124$		0			0
$125$	53719.1	+	166228.i	0.307506	+	0.951546i
$126$		0			0
$127$		−	257744.i		−	1.41801i	−0.705205	−	0.709004i	$-0.749145\pi$
$127$		−	257744.i		−	1.41801i	0.705205	−	0.709004i	$-0.250855\pi$
$128$		0			0
$129$	52183.1			0.276093
$130$		0			0
$131$		−	165038.i		−	0.840242i	−0.907468	−	0.420121i	$-0.861988\pi$
$131$		−	165038.i		−	0.840242i	0.907468	−	0.420121i	$-0.138012\pi$
$132$		0			0
$133$	−44753.4			−0.219380
$134$		0			0
$135$	−54518.1	−	521360.i	−0.257458	−	2.46209i
$136$		0			0
$137$			17244.2i			0.0784948i	0.999230	+	0.0392474i	$0.0124960\pi$
$137$			17244.2i			0.0784948i	−0.999230	+	0.0392474i	$0.987504\pi$
$138$		0			0
$139$		−	140468.i		−	0.616654i	−0.951280	−	0.308327i	$-0.900231\pi$
$139$		−	140468.i		−	0.616654i	0.951280	−	0.308327i	$-0.0997692\pi$
$140$		0			0
$141$		−	512352.i		−	2.17031i
$142$		0			0
$143$			423929.i			1.73362i
$144$		0			0
$145$	33228.5	−	3474.67i	0.131247	−	0.0137244i
$146$		0			0
$147$	236140.			0.901314
$148$		0			0
$149$			314120.i			1.15913i	0.814928	+	0.579563i	$0.196777\pi$
$149$			314120.i			1.15913i	−0.814928	+	0.579563i	$0.803223\pi$
$150$		0			0
$151$	242427.			0.865242			0.432621	−	0.901576i	$-0.357589\pi$
$151$	242427.			0.865242			0.432621	+	0.901576i	$0.357589\pi$
$152$		0			0
$153$			889759.i			3.07287i
$154$		0			0
$155$	51564.8	+	493117.i	0.172395	+	1.64862i
$156$		0			0
$157$	−343720.			−1.11290			−0.556450	−	0.830881i	$-0.687837\pi$
$157$	−343720.			−1.11290			−0.556450	+	0.830881i	$0.687837\pi$
$158$		0			0
$159$	−47120.1			−0.147813
$160$		0			0
$161$	101326.			0.308075
$162$		0			0
$163$	154899.			0.456647			0.228324	−	0.973585i	$-0.426676\pi$
$163$	154899.			0.456647			0.228324	+	0.973585i	$0.426676\pi$
$164$		0			0
$165$	859332.	−	89859.5i	2.45726	−	0.256953i
$166$		0			0
$167$			41199.1i			0.114313i	0.998365	+	0.0571567i	$0.0182035\pi$
$167$			41199.1i			0.114313i	−0.998365	+	0.0571567i	$0.981797\pi$
$168$		0			0
$169$	241499.			0.650427
$170$		0			0
$171$			276906.i			0.724172i
$172$		0			0
$173$	−707204.			−1.79651			−0.898255	−	0.439475i	$-0.855164\pi$
$173$	−707204.			−1.79651			−0.898255	+	0.439475i	$0.855164\pi$
$174$		0			0
$175$	−59719.2	−	282427.i	−0.147407	−	0.697126i
$176$		0			0
$177$		−	412513.i		−	0.989703i
$178$		0			0
$179$			385923.i			0.900259i	0.892963	+	0.450130i	$0.148622\pi$
$179$			385923.i			0.900259i	−0.892963	+	0.450130i	$0.851378\pi$
$180$		0			0
$181$			283701.i			0.643672i	0.946795	+	0.321836i	$0.104300\pi$
$181$			283701.i			0.643672i	−0.946795	+	0.321836i	$0.895700\pi$
$182$		0			0
$183$			701366.i			1.54816i
$184$		0			0
$185$	−79209.0	−	757481.i	−0.170155	−	1.62721i
$186$		0			0
$187$	−843040.			−1.76297
$188$		0			0
$189$			866221.i			1.76390i
$190$		0			0
$191$	−749997.			−1.48756			−0.743782	−	0.668422i	$-0.766970\pi$
$191$	−749997.			−1.48756			−0.743782	+	0.668422i	$0.766970\pi$
$192$		0			0
$193$		−	232810.i		−	0.449892i	−0.974371	−	0.224946i	$-0.927779\pi$
$193$		−	232810.i		−	0.449892i	0.974371	−	0.224946i	$-0.0722206\pi$
$194$		0			0
$195$	−129893.	−	1.24217e6i	−0.244623	−	2.33935i
$196$		0			0
$197$	88921.6			0.163246			0.0816228	−	0.996663i	$-0.473990\pi$
$197$	88921.6			0.163246			0.0816228	+	0.996663i	$0.473990\pi$
$198$		0			0
$199$	−194624.			−0.348388			−0.174194	−	0.984711i	$-0.555732\pi$
$199$	−194624.			−0.348388			−0.174194	+	0.984711i	$0.555732\pi$
$200$		0			0
$201$	−766107.			−1.33752
$202$		0			0
$203$	−55207.9			−0.0940288
$204$		0			0
$205$	−84536.4	−	808427.i	−0.140494	−	1.34356i
$206$		0			0
$207$		−	626942.i		−	1.01695i
$208$		0			0
$209$	−262366.			−0.415472
$210$		0			0
$211$		−	932681.i		−	1.44220i	−0.692828	−	0.721102i	$-0.743636\pi$
$211$		−	932681.i		−	1.44220i	0.692828	−	0.721102i	$-0.256364\pi$
$212$		0			0
$213$	−1.37139e6			−2.07115
$214$		0			0
$215$	−10630.0	−	101656.i	−0.0156833	−	0.149981i
$216$		0			0
$217$		−	819297.i		−	1.18111i
$218$		0			0
$219$		−	1.73089e6i		−	2.43870i
$220$		0			0
$221$			1.21862e6i			1.67837i
$222$		0			0
$223$			1.37279e6i			1.84860i	0.381666	+	0.924300i	$0.375351\pi$
$223$			1.37279e6i			1.84860i	−0.381666	+	0.924300i	$0.624649\pi$
$224$		0			0
$225$	−1.74748e6	+	369505.i	−2.30121	+	0.486591i
$226$		0			0
$227$	−1.38224e6			−1.78040			−0.890200	−	0.455569i	$-0.849436\pi$
$227$	−1.38224e6			−1.78040			−0.890200	+	0.455569i	$0.849436\pi$
$228$		0			0
$229$		−	330003.i		−	0.415843i	−0.978146	−	0.207922i	$-0.933330\pi$
$229$		−	330003.i		−	0.415843i	0.978146	−	0.207922i	$-0.0666699\pi$
$230$		0			0
$231$	−1.42775e6			−1.76044
$232$		0			0
$233$		−	1.14254e6i		−	1.37874i	−0.724409	−	0.689370i	$-0.757887\pi$
$233$		−	1.14254e6i		−	1.37874i	0.724409	−	0.689370i	$-0.242113\pi$
$234$		0			0
$235$	−998093.	+	104370.i	−1.17897	+	0.123283i
$236$		0			0
$237$	804154.			0.929969
$238$		0			0
$239$	441560.			0.500029			0.250015	−	0.968242i	$-0.419565\pi$
$239$	441560.			0.500029			0.250015	+	0.968242i	$0.419565\pi$
$240$		0			0
$241$	509548.			0.565122			0.282561	−	0.959249i	$-0.408816\pi$
$241$	509548.			0.565122			0.282561	+	0.959249i	$0.408816\pi$
$242$		0			0
$243$	1.39568e6			1.51625
$244$		0			0
$245$	−48103.3	−	460015.i	−0.0511987	−	0.489617i
$246$		0			0
$247$			379252.i			0.395536i
$248$		0			0
$249$	−699979.			−0.715462
$250$		0			0
$251$			253562.i			0.254039i	0.991900	+	0.127019i	$0.0405411\pi$
$251$			253562.i			0.254039i	−0.991900	+	0.127019i	$0.959459\pi$
$252$		0			0
$253$	594023.			0.583447
$254$		0			0
$255$	2.47022e6	−	258309.i	2.37895	−	0.248765i
$256$		0			0
$257$		−	203987.i		−	0.192650i	−0.995350	−	0.0963252i	$-0.969291\pi$
$257$		−	203987.i		−	0.192650i	0.995350	−	0.0963252i	$-0.0307089\pi$
$258$		0			0
$259$			1.25853e6i			1.16577i
$260$		0			0
$261$			341592.i			0.310389i
$262$		0			0
$263$		−	1.49608e6i		−	1.33372i	−0.745181	−	0.666862i	$-0.767637\pi$
$263$		−	1.49608e6i		−	1.33372i	0.745181	−	0.666862i	$-0.232363\pi$
$264$		0			0
$265$	9598.67	+	91792.7i	0.00839646	+	0.0802959i
$266$		0			0
$267$	3.33966e6			2.86697
$268$		0			0
$269$			522744.i			0.440462i	0.975448	+	0.220231i	$0.0706811\pi$
$269$			522744.i			0.440462i	−0.975448	+	0.220231i	$0.929319\pi$
$270$		0			0
$271$	1.43490e6			1.18686			0.593429	−	0.804886i	$-0.297774\pi$
$271$	1.43490e6			1.18686			0.593429	+	0.804886i	$0.297774\pi$
$272$		0			0
$273$			2.06382e6i			1.67597i
$274$		0			0
$275$	−350103.	−	1.65573e6i	−0.279167	−	1.32025i
$276$		0			0
$277$	−360298.			−0.282139			−0.141069	−	0.990000i	$-0.545054\pi$
$277$	−360298.			−0.282139			−0.141069	+	0.990000i	$0.545054\pi$
$278$		0			0
$279$	−5.06930e6			−3.89886
$280$		0			0
$281$	−982971.			−0.742634			−0.371317	−	0.928506i	$-0.621094\pi$
$281$	−982971.			−0.742634			−0.371317	+	0.928506i	$0.621094\pi$
$282$		0			0
$283$	−597953.			−0.443814			−0.221907	−	0.975068i	$-0.571228\pi$
$283$	−597953.			−0.443814			−0.221907	+	0.975068i	$0.571228\pi$
$284$		0			0
$285$	768769.	−	80389.4i	0.560640	−	0.0586255i
$286$		0			0
$287$			1.34317e6i			0.962558i
$288$		0			0
$289$	−1.00353e6			−0.706783
$290$		0			0
$291$		−	1.07607e6i		−	0.744919i
$292$		0			0
$293$	−1.14992e6			−0.782529			−0.391264	−	0.920278i	$-0.627962\pi$
$293$	−1.14992e6			−0.782529			−0.391264	+	0.920278i	$0.627962\pi$
$294$		0			0
$295$	−803601.	+	84031.7i	−0.537632	+	0.0562197i
$296$		0			0
$297$			5.07821e6i			3.34056i
$298$		0			0
$299$		−	858663.i		−	0.555450i
$300$		0			0
$301$			168897.i			0.107450i
$302$		0			0
$303$		−	3.04255e6i		−	1.90384i
$304$		0			0
$305$	1.36630e6	−	142873.i	0.841003	−	0.0879428i
$306$		0			0
$307$	1.78536e6			1.08114			0.540568	−	0.841300i	$-0.318209\pi$
$307$	1.78536e6			1.08114			0.540568	+	0.841300i	$0.318209\pi$
$308$		0			0
$309$			4.01140e6i			2.39001i
$310$		0			0
$311$	−669162.			−0.392311			−0.196155	−	0.980573i	$-0.562846\pi$
$311$	−669162.			−0.392311			−0.196155	+	0.980573i	$0.562846\pi$
$312$		0			0
$313$		−	1.22019e6i		−	0.703989i	−0.936002	−	0.351994i	$-0.885504\pi$
$313$		−	1.22019e6i		−	0.703989i	0.936002	−	0.351994i	$-0.114496\pi$
$314$		0			0
$315$	2.93548e6	−	306960.i	1.66687	−	0.174303i
$316$		0			0
$317$	1.63087e6			0.911529			0.455764	−	0.890101i	$-0.349366\pi$
$317$	1.63087e6			0.911529			0.455764	+	0.890101i	$0.349366\pi$
$318$		0			0
$319$	−323656.			−0.178076
$320$		0			0
$321$	−2.24194e6			−1.21440
$322$		0			0
$323$	−754194.			−0.402232
$324$		0			0
$325$	−2.39336e6	+	506077.i	−1.25690	+	0.265771i
$326$		0			0
$327$		−	4.25851e6i		−	2.20236i
$328$		0			0
$329$	1.65830e6			0.844641
$330$		0			0
$331$		−	1.47491e6i		−	0.739936i	−0.929044	−	0.369968i	$-0.879369\pi$
$331$		−	1.47491e6i		−	0.739936i	0.929044	−	0.369968i	$-0.120631\pi$
$332$		0			0
$333$	7.78698e6			3.84821
$334$		0			0
$335$	156061.	+	1.49242e6i	0.0759771	+	0.726573i
$336$		0			0
$337$		−	3.04284e6i		−	1.45950i	−0.683713	−	0.729751i	$-0.739636\pi$
$337$		−	3.04284e6i		−	1.45950i	0.683713	−	0.729751i	$-0.260364\pi$
$338$		0			0
$339$			5.44455e6i			2.57314i
$340$		0			0
$341$		−	4.80312e6i		−	2.23685i
$342$		0			0
$343$			2.31684e6i			1.06331i
$344$		0			0
$345$	−1.74057e6	+	182009.i	−0.787306	+	0.0823277i
$346$		0			0
$347$	−3.03217e6			−1.35185			−0.675927	−	0.736968i	$-0.736257\pi$
$347$	−3.03217e6			−1.35185			−0.675927	+	0.736968i	$0.736257\pi$
$348$		0			0
$349$			142845.i			0.0627773i	0.999507	+	0.0313887i	$0.00999296\pi$
$349$			142845.i			0.0627773i	−0.999507	+	0.0313887i	$0.990007\pi$
$350$		0			0
$351$	7.34059e6			3.18026
$352$		0			0
$353$		−	955673.i		−	0.408200i	−0.978950	−	0.204100i	$-0.934573\pi$
$353$		−	955673.i		−	0.408200i	0.978950	−	0.204100i	$-0.0654267\pi$
$354$		0			0
$355$	279361.	+	2.67154e6i	0.117651	+	1.12510i
$356$		0			0
$357$	−4.10419e6			−1.70434
$358$		0			0
$359$	−1.33046e6			−0.544837			−0.272418	−	0.962179i	$-0.587823\pi$
$359$	−1.33046e6			−0.544837			−0.272418	+	0.962179i	$0.587823\pi$
$360$		0			0
$361$	2.24138e6			0.905207
$362$		0			0
$363$	−3.77370e6			−1.50314
$364$		0			0
$365$	−3.37187e6	+	352593.i	−1.32477	+	0.138529i
$366$		0			0
$367$			1.97269e6i			0.764527i	0.924053	+	0.382263i	$0.124855\pi$
$367$			1.97269e6i			0.764527i	−0.924053	+	0.382263i	$0.875145\pi$
$368$		0			0
$369$	8.31071e6			3.17740
$370$		0			0
$371$		−	152510.i		−	0.0575260i
$372$		0			0
$373$	−386597.			−0.143875			−0.0719377	−	0.997409i	$-0.522918\pi$
$373$	−386597.			−0.143875			−0.0719377	+	0.997409i	$0.522918\pi$
$374$		0			0
$375$	1.53317e6	+	4.74423e6i	0.563004	+	1.74216i
$376$		0			0
$377$			467846.i			0.169531i
$378$		0			0
$379$			3.21349e6i			1.14915i	0.818450	+	0.574577i	$0.194833\pi$
$379$			3.21349e6i			1.14915i	−0.818450	+	0.574577i	$0.805167\pi$
$380$		0			0
$381$		−	7.35612e6i		−	2.59619i
$382$		0			0
$383$		−	2.09419e6i		−	0.729488i	−0.931108	−	0.364744i	$-0.881156\pi$
$383$		−	2.09419e6i		−	0.729488i	0.931108	−	0.364744i	$-0.118844\pi$
$384$		0			0
$385$	290842.	+	2.78134e6i	0.100001	+	0.956319i
$386$		0			0
$387$	1.04503e6			0.354692
$388$		0			0
$389$			944090.i			0.316329i	0.987413	+	0.158165i	$0.0505577\pi$
$389$			944090.i			0.316329i	−0.987413	+	0.158165i	$0.949442\pi$
$390$		0			0
$391$	1.70757e6			0.564854
$392$		0			0
$393$		−	4.71025e6i		−	1.53838i
$394$		0			0
$395$	−163811.	−	1.56654e6i	−0.0528265	−	0.505183i
$396$		0			0
$397$	−3.62777e6			−1.15522			−0.577608	−	0.816314i	$-0.696014\pi$
$397$	−3.62777e6			−1.15522			−0.577608	+	0.816314i	$0.696014\pi$
$398$		0			0
$399$	−1.27728e6			−0.401657
$400$		0			0
$401$	−3.56035e6			−1.10569			−0.552843	−	0.833285i	$-0.686457\pi$
$401$	−3.56035e6			−1.10569			−0.552843	+	0.833285i	$0.686457\pi$
$402$		0			0
$403$	−6.94294e6			−2.12952
$404$		0			0
$405$	−748489.	−	7.15785e6i	−0.226750	−	2.16843i
$406$		0			0
$407$			7.37811e6i			2.20780i
$408$		0			0
$409$	4.04814e6			1.19660			0.598298	−	0.801273i	$-0.295844\pi$
$409$	4.04814e6			1.19660			0.598298	+	0.801273i	$0.295844\pi$
$410$		0			0
$411$			492157.i			0.143714i
$412$		0			0
$413$	1.33515e6			0.385173
$414$		0			0
$415$	142590.	+	1.36360e6i	0.0406415	+	0.388657i
$416$		0			0
$417$		−	4.00904e6i		−	1.12902i
$418$		0			0
$419$		−	4.13360e6i		−	1.15025i	−0.818064	−	0.575127i	$-0.804953\pi$
$419$		−	4.13360e6i		−	1.15025i	0.818064	−	0.575127i	$-0.195047\pi$
$420$		0			0
$421$			1.98891e6i			0.546902i	0.961886	+	0.273451i	$0.0881651\pi$
$421$			1.98891e6i			0.546902i	−0.961886	+	0.273451i	$0.911835\pi$
$422$		0			0
$423$		−	1.02605e7i		−	2.78816i
$424$		0			0
$425$	−1.00640e6	−	4.75952e6i	−0.270271	−	1.27818i
$426$		0			0
$427$	−2.27006e6			−0.602516
$428$		0			0
$429$			1.20991e7i			3.17403i
$430$		0			0
$431$	2.17585e6			0.564203			0.282101	−	0.959385i	$-0.408969\pi$
$431$	2.17585e6			0.564203			0.282101	+	0.959385i	$0.408969\pi$
$432$		0			0
$433$		−	3.97546e6i		−	1.01899i	−0.860475	−	0.509493i	$-0.829833\pi$
$433$		−	3.97546e6i		−	1.01899i	0.860475	−	0.509493i	$-0.170167\pi$
$434$		0			0
$435$	948356.	−	99168.6i	0.240297	−	0.0251276i
$436$		0			0
$437$	531420.			0.133117
$438$		0			0
$439$	−2.62877e6			−0.651017			−0.325508	−	0.945539i	$-0.605535\pi$
$439$	−2.62877e6			−0.651017			−0.325508	+	0.945539i	$0.605535\pi$
$440$		0			0
$441$	4.72900e6			1.15790
$442$		0			0
$443$	3.63413e6			0.879814			0.439907	−	0.898043i	$-0.355011\pi$
$443$	3.63413e6			0.879814			0.439907	+	0.898043i	$0.355011\pi$
$444$		0			0
$445$	−680310.	−	6.50585e6i	−0.162857	−	1.55741i
$446$		0			0
$447$			8.96515e6i			2.12221i
$448$		0			0
$449$	−1.40882e6			−0.329792			−0.164896	−	0.986311i	$-0.552729\pi$
$449$	−1.40882e6			−0.329792			−0.164896	+	0.986311i	$0.552729\pi$
$450$		0			0
$451$			7.87433e6i			1.82294i
$452$		0			0
$453$	6.91897e6			1.58415
$454$		0			0
$455$	4.02045e6	−	420414.i	0.910429	−	0.0952026i
$456$		0			0
$457$			7.07341e6i			1.58430i	0.610325	+	0.792151i	$0.291039\pi$
$457$			7.07341e6i			1.58430i	−0.610325	+	0.792151i	$0.708961\pi$
$458$		0			0
$459$			1.45977e7i			3.23410i
$460$		0			0
$461$			1.57690e6i			0.345582i	0.984958	+	0.172791i	$0.0552786\pi$
$461$			1.57690e6i			0.345582i	−0.984958	+	0.172791i	$0.944721\pi$
$462$		0			0
$463$		−	5.38641e6i		−	1.16774i	−0.811847	−	0.583871i	$-0.801537\pi$
$463$		−	5.38641e6i		−	1.16774i	0.811847	−	0.583871i	$-0.198463\pi$
$464$		0			0
$465$	1.47168e6	+	1.40738e7i	0.315633	+	3.01842i
$466$		0			0
$467$	−1.22575e6			−0.260082			−0.130041	−	0.991509i	$-0.541511\pi$
$467$	−1.22575e6			−0.260082			−0.130041	+	0.991509i	$0.541511\pi$
$468$		0			0
$469$		−	2.47961e6i		−	0.520536i
$470$		0			0
$471$	−9.80994e6			−2.03758
$472$		0			0
$473$			990159.i			0.203494i
$474$		0			0
$475$	−313207.	−	1.48123e6i	−0.0636938	−	0.301224i
$476$		0			0
$477$	−943638.			−0.189893
$478$		0			0
$479$	2.58563e6			0.514906			0.257453	−	0.966291i	$-0.417117\pi$
$479$	2.58563e6			0.514906			0.257453	+	0.966291i	$0.417117\pi$
$480$		0			0
$481$	1.06651e7			2.10185
$482$		0			0
$483$	2.89189e6			0.564046
$484$		0			0
$485$	−2.09625e6	+	219203.i	−0.404659	+	0.0423148i
$486$		0			0
$487$		−	8.96610e6i		−	1.71309i	−0.516069	−	0.856547i	$-0.672605\pi$
$487$		−	8.96610e6i		−	1.71309i	0.516069	−	0.856547i	$-0.327395\pi$
$488$		0			0
$489$	4.42090e6			0.836063
$490$		0			0
$491$			2.48071e6i			0.464378i	0.972671	+	0.232189i	$0.0745889\pi$
$491$			2.48071e6i			0.464378i	−0.972671	+	0.232189i	$0.925411\pi$
$492$		0			0
$493$	−930375.			−0.172401
$494$		0			0
$495$	1.72092e7	−	1.79955e6i	3.15681	−	0.330104i
$496$		0			0
$497$		−	4.43867e6i		−	0.806051i
$498$		0			0
$499$		−	2.82421e6i		−	0.507744i	−0.967238	−	0.253872i	$-0.918296\pi$
$499$		−	2.82421e6i		−	0.507744i	0.967238	−	0.253872i	$-0.0817043\pi$
$500$		0			0
$501$			1.17584e6i			0.209293i
$502$		0			0
$503$			1.47261e6i			0.259517i	0.991546	+	0.129759i	$0.0414203\pi$
$503$			1.47261e6i			0.259517i	−0.991546	+	0.129759i	$0.958580\pi$
$504$		0			0
$505$	−5.92706e6	+	619787.i	−1.03422	+	0.108147i
$506$		0			0
$507$	6.89250e6			1.19085
$508$		0			0
$509$		−	5.12155e6i		−	0.876207i	−0.898924	−	0.438104i	$-0.855650\pi$
$509$		−	5.12155e6i		−	0.876207i	0.898924	−	0.438104i	$-0.144350\pi$
$510$		0			0
$511$	5.60225e6			0.949096
$512$		0			0
$513$			4.54303e6i			0.762171i
$514$		0			0
$515$	7.81444e6	−	817148.i	1.29831	−	0.135763i
$516$		0			0
$517$	9.72174e6			1.59962
$518$		0			0
$519$	−2.01839e7			−3.28918
$520$		0			0
$521$	−2.85592e6			−0.460948			−0.230474	−	0.973078i	$-0.574028\pi$
$521$	−2.85592e6			−0.460948			−0.230474	+	0.973078i	$0.574028\pi$
$522$		0			0
$523$	5.00538e6			0.800171			0.400086	−	0.916478i	$-0.368980\pi$
$523$	5.00538e6			0.800171			0.400086	+	0.916478i	$0.368980\pi$
$524$		0			0
$525$	−1.70442e6	−	8.06060e6i	−0.269884	−	1.27635i
$526$		0			0
$527$		−	1.38070e7i		−	2.16557i
$528$		0			0
$529$	5.23316e6			0.813064
$530$		0			0
$531$		−	8.26110e6i		−	1.27146i
$532$		0			0
$533$	1.13824e7			1.73547
$534$		0			0
$535$	456697.	+	4.36742e6i	0.0689833	+	0.659692i
$536$		0			0
$537$			1.10144e7i			1.64826i
$538$		0			0
$539$			4.48069e6i			0.664313i
$540$		0			0
$541$			6.29808e6i			0.925157i	0.886578	+	0.462578i	$0.153076\pi$
$541$			6.29808e6i			0.925157i	−0.886578	+	0.462578i	$0.846924\pi$
$542$		0			0
$543$			8.09696e6i			1.17848i
$544$		0			0
$545$	−8.29582e6	+	867486.i	−1.19638	+	0.125104i
$546$		0			0
$547$	1.56773e6			0.224029			0.112014	−	0.993707i	$-0.464270\pi$
$547$	1.56773e6			0.224029			0.112014	+	0.993707i	$0.464270\pi$
$548$		0			0
$549$			1.40457e7i			1.98890i
$550$		0			0
$551$	−289546.			−0.0406293
$552$		0			0
$553$			2.60275e6i			0.361926i
$554$		0			0
$555$	−2.26066e6	−	2.16189e7i	−0.311533	−	2.97921i
$556$		0			0
$557$	−8.52468e6			−1.16423			−0.582117	−	0.813105i	$-0.697775\pi$
$557$	−8.52468e6			−1.16423			−0.582117	+	0.813105i	$0.697775\pi$
$558$		0			0
$559$	1.43128e6			0.193729
$560$		0			0
$561$	−2.40608e7			−3.22777
$562$		0			0
$563$	935676.			0.124410			0.0622049	−	0.998063i	$-0.480187\pi$
$563$	935676.			0.124410			0.0622049	+	0.998063i	$0.480187\pi$
$564$		0			0
$565$	1.06063e7	−	1.10909e6i	1.39779	−	0.146166i
$566$		0			0
$567$			1.18925e7i			1.55352i
$568$		0			0
$569$	−9.09876e6			−1.17815			−0.589076	−	0.808077i	$-0.700508\pi$
$569$	−9.09876e6			−1.17815			−0.589076	+	0.808077i	$0.700508\pi$
$570$		0			0
$571$		−	1.41349e6i		−	0.181427i	−0.995877	−	0.0907136i	$-0.971085\pi$
$571$		−	1.41349e6i		−	0.181427i	0.995877	−	0.0907136i	$-0.0289148\pi$
$572$		0			0
$573$	−2.14053e7			−2.72354
$574$		0			0
$575$	709131.	+	3.35365e6i	0.0894451	+	0.423008i
$576$		0			0
$577$		−	1.18400e7i		−	1.48052i	−0.672322	−	0.740259i	$-0.734703\pi$
$577$		−	1.18400e7i		−	1.48052i	0.672322	−	0.740259i	$-0.265297\pi$
$578$		0			0
$579$		−	6.64451e6i		−	0.823695i
$580$		0			0
$581$		−	2.26557e6i		−	0.278444i
$582$		0			0
$583$		−	894090.i		−	0.108946i
$584$		0			0
$585$	−2.60126e6	−	2.48760e7i	−0.314264	−	3.00532i
$586$		0			0
$587$	−3.37505e6			−0.404283			−0.202141	−	0.979356i	$-0.564790\pi$
$587$	−3.37505e6			−0.404283			−0.202141	+	0.979356i	$0.564790\pi$
$588$		0			0
$589$		−	4.29693e6i		−	0.510352i
$590$		0			0
$591$	2.53786e6			0.298882
$592$		0			0
$593$			1.32900e7i			1.55199i	0.630737	+	0.775997i	$0.282753\pi$
$593$			1.32900e7i			1.55199i	−0.630737	+	0.775997i	$0.717247\pi$
$594$		0			0
$595$	836050.	+	7.99520e6i	0.0968144	+	0.925842i
$596$		0			0
$597$	−5.55466e6			−0.637855
$598$		0			0
$599$	−8.21563e6			−0.935565			−0.467783	−	0.883844i	$-0.654947\pi$
$599$	−8.21563e6			−0.935565			−0.467783	+	0.883844i	$0.654947\pi$
$600$		0			0
$601$	4.66795e6			0.527157			0.263579	−	0.964638i	$-0.415097\pi$
$601$	4.66795e6			0.527157			0.263579	+	0.964638i	$0.415097\pi$
$602$		0			0
$603$	−1.53422e7			−1.71829
$604$		0			0
$605$	768727.	+	7.35139e6i	0.0853855	+	0.816547i
$606$		0			0
$607$		−	280377.i		−	0.0308867i	−0.999881	−	0.0154433i	$-0.995084\pi$
$607$		−	280377.i		−	0.0308867i	0.999881	−	0.0154433i	$-0.00491596\pi$
$608$		0			0
$609$	−1.57566e6			−0.172155
$610$		0			0
$611$		−	1.40528e7i		−	1.52286i
$612$		0			0
$613$	−963624.			−0.103575			−0.0517877	−	0.998658i	$-0.516492\pi$
$613$	−963624.			−0.103575			−0.0517877	+	0.998658i	$0.516492\pi$
$614$		0			0
$615$	−2.41271e6	−	2.30729e7i	−0.257227	−	2.45988i
$616$		0			0
$617$			1.00959e7i			1.06766i	0.845593	+	0.533829i	$0.179247\pi$
$617$			1.00959e7i			1.06766i	−0.845593	+	0.533829i	$0.820753\pi$
$618$		0			0
$619$			1.57741e7i			1.65469i	0.561691	+	0.827347i	$0.310151\pi$
$619$			1.57741e7i			1.65469i	−0.561691	+	0.827347i	$0.689849\pi$
$620$		0			0
$621$		−	1.02859e7i		−	1.07031i
$622$		0			0
$623$			1.08092e7i			1.11577i
$624$		0			0
$625$	8.92973e6	−	3.95314e6i	0.914405	−	0.404801i
$626$		0			0
$627$	−7.48806e6			−0.760677
$628$		0			0
$629$			2.12090e7i			2.13744i
$630$		0			0
$631$	1.60364e7			1.60337			0.801685	−	0.597747i	$-0.203937\pi$
$631$	1.60364e7			1.60337			0.801685	+	0.597747i	$0.203937\pi$
$632$		0			0
$633$		−	2.66192e7i		−	2.64049i
$634$		0			0
$635$	−1.43302e7	+	1.49849e6i	−1.41032	+	0.147475i
$636$		0			0
$637$	6.47686e6			0.632435
$638$		0			0
$639$	−2.74637e7			−2.66077
$640$		0			0
$641$	9.42961e6			0.906460			0.453230	−	0.891394i	$-0.350272\pi$
$641$	9.42961e6			0.906460			0.453230	+	0.891394i	$0.350272\pi$
$642$		0			0
$643$	−102240.			−0.00975200			−0.00487600	−	0.999988i	$-0.501552\pi$
$643$	−102240.			−0.00975200			−0.00487600	+	0.999988i	$0.501552\pi$
$644$		0			0
$645$	−303386.	−	2.90130e6i	−0.0287142	−	0.274596i
$646$		0			0
$647$		−	7.52613e6i		−	0.706824i	−0.935468	−	0.353412i	$-0.885021\pi$
$647$		−	7.52613e6i		−	0.706824i	0.935468	−	0.353412i	$-0.114979\pi$
$648$		0			0
$649$	7.82733e6			0.729460
$650$		0			0
$651$		−	2.33831e7i		−	2.16247i
$652$		0			0
$653$	−1.11241e7			−1.02090			−0.510450	−	0.859907i	$-0.670521\pi$
$653$	−1.11241e7			−1.02090			−0.510450	+	0.859907i	$0.670521\pi$
$654$		0			0
$655$	−9.17585e6	+	959509.i	−0.835686	+	0.0873868i
$656$		0			0
$657$		−	3.46632e7i		−	3.13296i
$658$		0			0
$659$		−	9.90902e6i		−	0.888827i	−0.895822	−	0.444413i	$-0.853412\pi$
$659$		−	9.90902e6i		−	0.888827i	0.895822	−	0.444413i	$-0.146588\pi$
$660$		0			0
$661$		−	1.94571e7i		−	1.73211i	−0.499950	−	0.866054i	$-0.666648\pi$
$661$		−	1.94571e7i		−	1.73211i	0.499950	−	0.866054i	$-0.333352\pi$
$662$		0			0
$663$			3.47800e7i			3.07288i
$664$		0			0
$665$	260191.	+	2.48822e6i	0.0228159	+	0.218190i
$666$		0			0
$667$	655561.			0.0570556
$668$		0			0
$669$			3.91802e7i			3.38455i
$670$		0			0
$671$	−1.33082e7			−1.14107
$672$		0			0
$673$			1.66792e7i			1.41951i	0.704450	+	0.709754i	$0.251194\pi$
$673$			1.66792e7i			1.41951i	−0.704450	+	0.709754i	$0.748806\pi$
$674$		0			0
$675$	−2.86699e7	+	6.06225e6i	−2.42196	+	0.512123i
$676$		0			0
$677$	1.43143e7			1.20033			0.600163	−	0.799878i	$-0.295102\pi$
$677$	1.43143e7			1.20033			0.600163	+	0.799878i	$0.295102\pi$
$678$		0			0
$679$	3.48285e6			0.289908
$680$		0			0
$681$	−3.94497e7			−3.25969
$682$		0			0
$683$	2.35057e7			1.92806			0.964032	−	0.265786i	$-0.0856314\pi$
$683$	2.35057e7			1.92806			0.964032	+	0.265786i	$0.0856314\pi$
$684$		0			0
$685$	958751.	−	100256.i	0.0780691	−	0.00816361i
$686$		0			0
$687$		−	9.41845e6i		−	0.761355i
$688$		0			0
$689$	−1.29241e6			−0.103718
$690$		0			0
$691$			9.02428e6i			0.718981i	0.933149	+	0.359490i	$0.117049\pi$
$691$			9.02428e6i			0.718981i	−0.933149	+	0.359490i	$0.882951\pi$
$692$		0			0
$693$	−2.85925e7			−2.26162
$694$		0			0
$695$	−7.80984e6	+	816667.i	−0.613310	+	0.0641332i
$696$		0			0
$697$			2.26354e7i			1.76485i
$698$		0			0
$699$		−	3.26087e7i		−	2.52430i
$700$		0			0
$701$			2.56364e6i			0.197044i	0.995135	+	0.0985218i	$0.0314114\pi$
$701$			2.56364e6i			0.197044i	−0.995135	+	0.0985218i	$0.968589\pi$
$702$		0			0
$703$			6.60054e6i			0.503723i
$704$		0			0
$705$	−2.84860e7	+	2.97876e6i	−2.15854	+	0.225716i
$706$		0			0
$707$	9.84760e6			0.740938
$708$		0			0
$709$			1.83762e6i			0.137290i	0.997641	+	0.0686450i	$0.0218676\pi$
$709$			1.83762e6i			0.137290i	−0.997641	+	0.0686450i	$0.978132\pi$
$710$		0			0
$711$	1.61042e7			1.19472
$712$		0			0
$713$			9.72866e6i			0.716687i
$714$		0			0
$715$	2.35698e7	−	2.46467e6i	1.72422	−	0.180299i
$716$		0			0
$717$	1.26023e7			0.915489
$718$		0			0
$719$	−9.48236e6			−0.684060			−0.342030	−	0.939689i	$-0.611115\pi$
$719$	−9.48236e6			−0.684060			−0.342030	+	0.939689i	$0.611115\pi$
$720$		0			0
$721$	−1.29834e7			−0.930145
$722$		0			0
$723$	1.45427e7			1.03467
$724$		0			0
$725$	−386373.	−	1.82725e6i	−0.0272999	−	0.129108i
$726$		0			0
$727$		−	2.49674e6i		−	0.175201i	−0.996156	−	0.0876007i	$-0.972080\pi$
$727$		−	2.49674e6i		−	0.175201i	0.996156	−	0.0876007i	$-0.0279200\pi$
$728$		0			0
$729$	8.54920e6			0.595809
$730$		0			0
$731$			2.84629e6i			0.197009i
$732$		0			0
$733$	−2.33505e7			−1.60522			−0.802612	−	0.596502i	$-0.796557\pi$
$733$	−2.33505e7			−1.60522			−0.802612	+	0.596502i	$0.796557\pi$
$734$		0			0
$735$	−1.37289e6	−	1.31290e7i	−0.0937384	−	0.896426i
$736$		0			0
$737$		−	1.45367e7i		−	0.985816i
$738$		0			0
$739$			2.42153e7i			1.63109i	0.578692	+	0.815546i	$0.303563\pi$
$739$			2.42153e7i			1.63109i	−0.578692	+	0.815546i	$0.696437\pi$
$740$		0			0
$741$			1.08240e7i			0.724175i
$742$		0			0
$743$		−	1.99899e7i		−	1.32843i	−0.747542	−	0.664214i	$-0.768766\pi$
$743$		−	1.99899e7i		−	1.32843i	0.747542	−	0.664214i	$-0.231234\pi$
$744$		0			0
$745$	1.74646e7	−	1.82626e6i	1.15284	−	0.120551i
$746$		0			0
$747$	−1.40180e7			−0.919143
$748$		0			0
$749$		−	7.25632e6i		−	0.472620i
$750$		0			0
$751$	4.33613e6			0.280545			0.140272	−	0.990113i	$-0.455202\pi$
$751$	4.33613e6			0.280545			0.140272	+	0.990113i	$0.455202\pi$
$752$		0			0
$753$			7.23679e6i			0.465113i
$754$		0			0
$755$	−1.40944e6	−	1.34786e7i	−0.0899869	−	0.860550i
$756$		0			0
$757$	1.47258e7			0.933984			0.466992	−	0.884262i	$-0.345338\pi$
$757$	1.47258e7			0.933984			0.466992	+	0.884262i	$0.345338\pi$
$758$		0			0
$759$	1.69537e7			1.06822
$760$		0			0
$761$	−366653.			−0.0229506			−0.0114753	−	0.999934i	$-0.503653\pi$
$761$	−366653.			−0.0229506			−0.0114753	+	0.999934i	$0.503653\pi$
$762$		0			0
$763$	1.37832e7			0.857115
$764$		0			0
$765$	4.94693e7	−	5.17295e6i	3.05620	−	0.319584i
$766$		0			0
$767$		−	1.13145e7i		−	0.694457i
$768$		0			0
$769$	5.25862e6			0.320668			0.160334	−	0.987063i	$-0.448743\pi$
$769$	5.25862e6			0.320668			0.160334	+	0.987063i	$0.448743\pi$
$770$		0			0
$771$		−	5.82189e6i		−	0.352718i
$772$		0			0
$773$	249509.			0.0150189			0.00750943	−	0.999972i	$-0.497610\pi$
$773$	249509.			0.0150189			0.00750943	+	0.999972i	$0.497610\pi$
$774$		0			0
$775$	2.71168e7	−	5.73385e6i	1.62175	−	0.342920i
$776$		0			0
$777$			3.59190e7i			2.13438i
$778$		0			0
$779$			7.04448e6i			0.415915i
$780$		0			0
$781$		−	2.60217e7i		−	1.52654i
$782$		0			0
$783$			5.60429e6i			0.326675i
$784$		0			0
$785$	1.99835e6	+	1.91104e7i	0.115744	+	1.10686i
$786$		0			0
$787$	1.94586e6			0.111989			0.0559945	−	0.998431i	$-0.482167\pi$
$787$	1.94586e6			0.111989			0.0559945	+	0.998431i	$0.482167\pi$
$788$		0			0
$789$		−	4.26989e7i		−	2.44188i
$790$		0			0
$791$	−1.76220e7			−1.00141
$792$		0			0
$793$			1.92371e7i			1.08632i
$794$		0			0
$795$	273951.	+	2.61981e6i	0.0153729	+	0.147012i
$796$		0			0
$797$	1.32215e7			0.737286			0.368643	−	0.929571i	$-0.379823\pi$
$797$	1.32215e7			0.737286			0.368643	+	0.929571i	$0.379823\pi$
$798$		0			0
$799$	2.79459e7			1.54864
$800$		0			0
$801$	6.68808e7			3.68316
$802$		0			0
$803$	3.28431e7			1.79744
$804$		0			0
$805$	−589098.	−	5.63358e6i	−0.0320404	−	0.306404i
$806$		0			0
$807$			1.49194e7i			0.806430i
$808$		0			0
$809$	−4.37501e6			−0.235022			−0.117511	−	0.993072i	$-0.537491\pi$
$809$	−4.37501e6			−0.235022			−0.117511	+	0.993072i	$0.537491\pi$
$810$		0			0
$811$			3.36426e7i			1.79613i	0.439863	+	0.898065i	$0.355027\pi$
$811$			3.36426e7i			1.79613i	−0.439863	+	0.898065i	$0.644973\pi$
$812$		0			0
$813$	4.09528e7			2.17299
$814$		0			0
$815$	−900568.	−	8.61219e6i	−0.0474922	−	0.454171i
$816$		0			0
$817$			885808.i			0.0464285i
$818$		0			0
$819$			4.13306e7i			2.15309i
$820$		0			0
$821$			3.51821e7i			1.82164i	0.412799	+	0.910822i	$0.364551\pi$
$821$			3.51821e7i			1.82164i	−0.412799	+	0.910822i	$0.635449\pi$
$822$		0			0
$823$			8.11861e6i			0.417813i	0.977936	+	0.208907i	$0.0669905\pi$
$823$			8.11861e6i			0.417813i	−0.977936	+	0.208907i	$0.933010\pi$
$824$		0			0
$825$	−9.99212e6	−	4.72552e7i	−0.511120	−	2.41721i
$826$		0			0
$827$	1.82572e7			0.928263			0.464132	−	0.885766i	$-0.346366\pi$
$827$	1.82572e7			0.928263			0.464132	+	0.885766i	$0.346366\pi$
$828$		0			0
$829$		−	3.65919e6i		−	0.184926i	−0.995716	−	0.0924632i	$-0.970526\pi$
$829$		−	3.65919e6i		−	0.184926i	0.995716	−	0.0924632i	$-0.0294740\pi$
$830$		0			0
$831$	−1.02831e7			−0.516560
$832$		0			0
$833$			1.28801e7i			0.643142i
$834$		0			0
$835$	2.29061e6	−	239527.i	0.113693	−	0.0118888i
$836$		0			0
$837$	−8.31689e7			−4.10343
$838$		0			0
$839$	−86068.5			−0.00422123			−0.00211062	−	0.999998i	$-0.500672\pi$
$839$	−86068.5			−0.00422123			−0.00211062	+	0.999998i	$0.500672\pi$
$840$		0			0
$841$	2.01540e7			0.982586
$842$		0			0
$843$	−2.80545e7			−1.35967
$844$		0			0
$845$	−1.40405e6	−	1.34270e7i	−0.0676457	−	0.646900i
$846$		0			0
$847$		−	1.22141e7i		−	0.584995i
$848$		0			0
$849$	−1.70659e7			−0.812567
$850$		0			0
$851$		−	1.49443e7i		−	0.707377i
$852$		0			0
$853$	−3.70986e6			−0.174576			−0.0872880	−	0.996183i	$-0.527820\pi$
$853$	−3.70986e6			−0.174576			−0.0872880	+	0.996183i	$0.527820\pi$
$854$		0			0
$855$	1.53956e7	−	1.60990e6i	0.720245	−	0.0753153i
$856$		0			0
$857$		−	2.77130e7i		−	1.28893i	−0.764632	−	0.644467i	$-0.777079\pi$
$857$		−	2.77130e7i		−	1.28893i	0.764632	−	0.644467i	$-0.222921\pi$
$858$		0			0
$859$		−	1.79861e7i		−	0.831677i	−0.909439	−	0.415838i	$-0.863488\pi$
$859$		−	1.79861e7i		−	0.831677i	0.909439	−	0.415838i	$-0.136512\pi$
$860$		0			0
$861$			3.83348e7i			1.76232i
$862$		0			0
$863$			1.53944e7i			0.703615i	0.936072	+	0.351807i	$0.114433\pi$
$863$			1.53944e7i			0.703615i	−0.936072	+	0.351807i	$0.885567\pi$
$864$		0			0
$865$	4.11160e6	+	3.93195e7i	0.186840	+	1.78677i
$866$		0			0
$867$	−2.86412e7			−1.29403
$868$		0			0
$869$			1.52586e7i			0.685433i
$870$		0			0
$871$	−2.10128e7			−0.938511
$872$		0			0
$873$		−	2.15497e7i		−	0.956985i
$874$		0			0
$875$	−1.53553e7	+	4.96230e6i	−0.678015	+	0.219110i
$876$		0			0
$877$	2.20053e7			0.966114			0.483057	−	0.875589i	$-0.339526\pi$
$877$	2.20053e7			0.966114			0.483057	+	0.875589i	$0.339526\pi$
$878$		0			0
$879$	−3.28194e7			−1.43271
$880$		0			0
$881$	−3.35282e7			−1.45536			−0.727680	−	0.685916i	$-0.759402\pi$
$881$	−3.35282e7			−1.45536			−0.727680	+	0.685916i	$0.759402\pi$
$882$		0			0
$883$	−2.51274e7			−1.08454			−0.542271	−	0.840203i	$-0.682435\pi$
$883$	−2.51274e7			−1.08454			−0.542271	+	0.840203i	$0.682435\pi$
$884$		0			0
$885$	−2.29351e7	+	2.39831e6i	−0.984336	+	0.102931i
$886$		0			0
$887$		−	2.68037e7i		−	1.14389i	−0.820291	−	0.571947i	$-0.806188\pi$
$887$		−	2.68037e7i		−	1.14389i	0.820291	−	0.571947i	$-0.193812\pi$
$888$		0			0
$889$	2.38091e7			1.01039
$890$		0			0
$891$			6.97197e7i			2.94213i
$892$		0			0
$893$	8.69719e6			0.364964
$894$		0			0
$895$	2.14567e7	−	2.24371e6i	0.895377	−	0.0936287i
$896$		0			0
$897$		−	2.45067e7i		−	1.01696i
$898$		0			0
$899$		−	5.30070e6i		−	0.218743i
$900$		0			0
$901$		−	2.57014e6i		−	0.105474i
$902$		0			0
$903$			4.82041e6i			0.196727i
$904$		0			0
$905$	1.57734e7	−	1.64941e6i	0.640181	−	0.0669431i
$906$		0			0
$907$	−2.72736e7			−1.10084			−0.550420	−	0.834888i	$-0.685532\pi$
$907$	−2.72736e7			−1.10084			−0.550420	+	0.834888i	$0.685532\pi$
$908$		0			0
$909$		−	6.09308e7i		−	2.44584i
$910$		0			0
$911$	5.24856e6			0.209529			0.104765	−	0.994497i	$-0.466591\pi$
$911$	5.24856e6			0.209529			0.104765	+	0.994497i	$0.466591\pi$
$912$		0			0
$913$		−	1.32819e7i		−	0.527331i
$914$		0			0
$915$	3.89949e7	−	4.07766e6i	1.53977	−	0.161012i
$916$		0			0
$917$	1.52453e7			0.598706
$918$		0			0
$919$	2.33248e7			0.911024			0.455512	−	0.890230i	$-0.349456\pi$
$919$	2.33248e7			0.911024			0.455512	+	0.890230i	$0.349456\pi$
$920$		0			0
$921$	5.09551e7			1.97942
$922$		0			0
$923$	−3.76145e7			−1.45329
$924$		0			0
$925$	−4.16543e7	+	8.80782e6i	−1.60069	+	0.338465i
$926$		0			0
$927$			8.03333e7i			3.07041i
$928$		0			0
$929$	3.98540e6			0.151507			0.0757536	−	0.997127i	$-0.475864\pi$
$929$	3.98540e6			0.151507			0.0757536	+	0.997127i	$0.475864\pi$
$930$		0			0
$931$			4.00848e6i			0.151567i
$932$		0			0
$933$	−1.90982e7			−0.718271
$934$		0			0
$935$	4.90133e6	+	4.68718e7i	0.183352	+	1.75341i
$936$		0			0
$937$		−	2.33906e7i		−	0.870347i	−0.900347	−	0.435174i	$-0.856687\pi$
$937$		−	2.33906e7i		−	0.870347i	0.900347	−	0.435174i	$-0.143313\pi$
$938$		0			0
$939$		−	3.48247e7i		−	1.28891i
$940$		0			0
$941$			1.77697e7i			0.654192i	0.944991	+	0.327096i	$0.106070\pi$
$941$			1.77697e7i			0.654192i	−0.944991	+	0.327096i	$0.893930\pi$
$942$		0			0
$943$		−	1.59494e7i		−	0.584069i
$944$		0			0
$945$	4.81606e7	−	5.03611e6i	1.75434	−	0.183449i
$946$		0			0
$947$	−1.60461e7			−0.581426			−0.290713	−	0.956810i	$-0.593893\pi$
$947$	−1.60461e7			−0.581426			−0.290713	+	0.956810i	$0.593893\pi$
$948$		0			0
$949$		−	4.74749e7i		−	1.71119i
$950$		0			0
$951$	4.65457e7			1.66889
$952$		0			0
$953$			8.96733e6i			0.319839i	0.987130	+	0.159919i	$0.0511234\pi$
$953$			8.96733e6i			0.319839i	−0.987130	+	0.159919i	$0.948877\pi$
$954$		0			0
$955$	4.36039e6	+	4.16987e7i	0.154710	+	1.47950i
$956$		0			0
$957$	−9.23729e6			−0.326035
$958$		0			0
$959$	−1.59293e6			−0.0559307
$960$		0			0
$961$	5.00344e7			1.74767
$962$		0			0
$963$	−4.48976e7			−1.56012
$964$		0			0
$965$	−1.29439e7	+	1.35353e6i	−0.447453	+	0.0467897i
$966$		0			0
$967$			1.42347e7i			0.489532i	0.969582	+	0.244766i	$0.0787111\pi$
$967$			1.42347e7i			0.489532i	−0.969582	+	0.244766i	$0.921289\pi$
$968$		0			0
$969$	−2.15250e7			−0.736435
$970$		0			0
$971$			1.83592e7i			0.624893i	0.949935	+	0.312447i	$0.101149\pi$
$971$			1.83592e7i			0.624893i	−0.949935	+	0.312447i	$0.898851\pi$
$972$		0			0
$973$	1.29758e7			0.439391
$974$		0			0
$975$	−6.83077e7	+	1.44437e7i	−2.30122	+	0.486593i
$976$		0			0
$977$		−	2.23259e7i		−	0.748294i	−0.927370	−	0.374147i	$-0.877936\pi$
$977$		−	2.23259e7i		−	0.748294i	0.927370	−	0.374147i	$-0.122064\pi$
$978$		0			0
$979$			6.33690e7i			2.11310i
$980$		0			0
$981$		−	8.52819e7i		−	2.82933i
$982$		0			0
$983$		−	2.90678e7i		−	0.959463i	−0.877415	−	0.479732i	$-0.840734\pi$
$983$		−	2.90678e7i		−	0.959463i	0.877415	−	0.479732i	$-0.159266\pi$
$984$		0			0
$985$	−516980.	−	4.94391e6i	−0.0169779	−	0.162360i
$986$		0			0
$987$	4.73285e7			1.54643
$988$		0			0
$989$		−	2.00556e6i		−	0.0651995i
$990$		0			0
$991$	−3.44761e7			−1.11515			−0.557575	−	0.830126i	$-0.688268\pi$
$991$	−3.44761e7			−1.11515			−0.557575	+	0.830126i	$0.688268\pi$
$992$		0			0
$993$		−	4.20945e7i		−	1.35473i
$994$		0			0
$995$	1.13152e6	+	1.08208e7i	0.0362331	+	0.346499i
$996$		0			0
$997$	2.87402e7			0.915696			0.457848	−	0.889031i	$-0.348620\pi$
$997$	2.87402e7			0.915696			0.457848	+	0.889031i	$0.348620\pi$
$998$		0			0
$999$	1.27756e8			4.05013

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.6.f.d.289.31	yes	32
4.3	odd	2	inner	320.6.f.d.289.2	yes	32
5.4	even	2	inner	320.6.f.d.289.4	yes	32
8.3	odd	2	inner	320.6.f.d.289.32	yes	32
8.5	even	2	inner	320.6.f.d.289.1	✓	32
20.19	odd	2	inner	320.6.f.d.289.29	yes	32
40.19	odd	2	inner	320.6.f.d.289.3	yes	32
40.29	even	2	inner	320.6.f.d.289.30	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.6.f.d.289.1	✓	32	8.5	even	2	inner
320.6.f.d.289.2	yes	32	4.3	odd	2	inner
320.6.f.d.289.3	yes	32	40.19	odd	2	inner
320.6.f.d.289.4	yes	32	5.4	even	2	inner
320.6.f.d.289.29	yes	32	20.19	odd	2	inner
320.6.f.d.289.30	yes	32	40.29	even	2	inner
320.6.f.d.289.31	yes	32	1.1	even	1	trivial
320.6.f.d.289.32	yes	32	8.3	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)

\(f(q)\)	\(=\)	\(q+28.5405 q^{3} +(-5.81388 - 55.5985i) q^{5} +92.3750i q^{7} +571.559 q^{9} +O(q^{10})\)	\(q+28.5405 q^{3} +(-5.81388 - 55.5985i) q^{5} +92.3750i q^{7} +571.559 q^{9} +541.547i q^{11} +782.810 q^{13} +(-165.931 - 1586.81i) q^{15} +1556.72i q^{17} +484.475i q^{19} +2636.43i q^{21} -1096.90i q^{23} +(-3057.40 + 646.487i) q^{25} +9377.22 q^{27} +597.650i q^{29} -8869.25 q^{31} +15456.0i q^{33} +(5135.92 - 537.058i) q^{35} +13624.1 q^{37} +22341.8 q^{39} +14540.4 q^{41} +1828.39 q^{43} +(-3322.98 - 31777.8i) q^{45} -17951.8i q^{47} +8273.86 q^{49} +44429.6i q^{51} -1650.99 q^{53} +(30109.3 - 3148.49i) q^{55} +13827.1i q^{57} -14453.6i q^{59} +24574.4i q^{61} +52797.7i q^{63} +(-4551.17 - 43523.1i) q^{65} -26842.8 q^{67} -31306.0i q^{69} -48050.6 q^{71} -60646.8i q^{73} +(-87259.6 + 18451.0i) q^{75} -50025.5 q^{77} +28175.9 q^{79} +128742. q^{81} -24525.8 q^{83} +(86551.6 - 9050.61i) q^{85} +17057.2i q^{87} +117015. q^{89} +72312.1i q^{91} -253133. q^{93} +(26936.1 - 2816.68i) q^{95} -37703.4i q^{97} +309526. i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 2944 q^{9}+O(q^{10}) \) 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

Related objects

Downloads

Learn more