LMFDB - Embedded newform 60.6.d.a.49.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [60,6,Mod(49,60)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(60, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("60.49");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 60 = 2^{2} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	60.d (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:	$9.62302918878$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 373x^{4} + 33732x^{2} + 186624 $ x^6 + 373x^4 + 33732x^2 + 186624
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{4}\cdot 5^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.6
Root			$-11.7229i$ of defining polynomial
Character	$\chi$	$=$	60.49
Dual form			60.6.d.a.49.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.00000i q^{3} +(54.4661 + 12.5876i) q^{5} +12.5817i q^{7} -81.0000 q^{9} +O(q^{10})$	$q+9.00000i q^{3} +(54.4661 + 12.5876i) q^{5} +12.5817i q^{7} -81.0000 q^{9} +164.713 q^{11} +849.118i q^{13} +(-113.288 + 490.195i) q^{15} +1608.11i q^{17} +446.747 q^{19} -113.235 q^{21} +802.223i q^{23} +(2808.10 + 1371.20i) q^{25} -729.000i q^{27} -4477.98 q^{29} +5200.90 q^{31} +1482.41i q^{33} +(-158.373 + 685.275i) q^{35} -8248.56i q^{37} -7642.07 q^{39} -6250.10 q^{41} -11314.2i q^{43} +(-4411.75 - 1019.60i) q^{45} -29842.0i q^{47} +16648.7 q^{49} -14473.0 q^{51} +9195.04i q^{53} +(8971.25 + 2073.34i) q^{55} +4020.72i q^{57} -9948.84 q^{59} +41550.9 q^{61} -1019.12i q^{63} +(-10688.4 + 46248.1i) q^{65} -49074.0i q^{67} -7220.00 q^{69} -44977.6 q^{71} +65643.2i q^{73} +(-12340.8 + 25272.9i) q^{75} +2072.36i q^{77} +67157.8 q^{79} +6561.00 q^{81} +54883.8i q^{83} +(-20242.3 + 87587.6i) q^{85} -40301.8i q^{87} +53984.2 q^{89} -10683.3 q^{91} +46808.1i q^{93} +(24332.5 + 5623.47i) q^{95} -100814. i q^{97} -13341.7 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 38 q^{5} - 486 q^{9}+O(q^{10}) $ 6 * q - 38 * q^5 - 486 * q^9	$ 6 q - 38 q^{5} - 486 q^{9} + 296 q^{11} + 396 q^{15} - 6000 q^{19} + 1584 q^{21} + 6054 q^{25} - 15924 q^{29} + 264 q^{31} + 20096 q^{35} + 16340 q^{41} + 3078 q^{45} - 27654 q^{49} + 15624 q^{51} - 26088 q^{55} + 92456 q^{59} + 6252 q^{61} - 52440 q^{65} + 18936 q^{69} - 160800 q^{71} - 29448 q^{75} + 128952 q^{79} + 39366 q^{81} - 177864 q^{85} + 76060 q^{89} - 98400 q^{91} + 232800 q^{95} - 23976 q^{99}+O(q^{100}) $ 6 * q - 38 * q^5 - 486 * q^9 + 296 * q^11 + 396 * q^15 - 6000 * q^19 + 1584 * q^21 + 6054 * q^25 - 15924 * q^29 + 264 * q^31 + 20096 * q^35 + 16340 * q^41 + 3078 * q^45 - 27654 * q^49 + 15624 * q^51 - 26088 * q^55 + 92456 * q^59 + 6252 * q^61 - 52440 * q^65 + 18936 * q^69 - 160800 * q^71 - 29448 * q^75 + 128952 * q^79 + 39366 * q^81 - 177864 * q^85 + 76060 * q^89 - 98400 * q^91 + 232800 * q^95 - 23976 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/60\mathbb{Z}\right)^\times$.

$n$	$31$	$37$	$41$
$\chi(n)$	$1$	$-1$	$1$

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
$3$			9.00000i			0.577350i
$3$			9.00000i			0.577350i
$4$		0			0
$5$	54.4661	+	12.5876i	0.974319	+	0.225174i
$5$	54.4661	+	12.5876i	0.974319	+	0.225174i
$6$		0			0
$7$			12.5817i			0.0970496i	0.998822	+	0.0485248i	$0.0154520\pi$
$7$			12.5817i			0.0970496i	−0.998822	+	0.0485248i	$0.984548\pi$
$8$		0			0
$9$	−81.0000			−0.333333
$10$		0			0
$11$	164.713			0.410436			0.205218	−	0.978716i	$-0.434210\pi$
$11$	164.713			0.410436			0.205218	+	0.978716i	$0.434210\pi$
$12$		0			0
$13$			849.118i			1.39351i	0.717310	+	0.696755i	$0.245373\pi$
$13$			849.118i			1.39351i	−0.717310	+	0.696755i	$0.754627\pi$
$14$		0			0
$15$	−113.288	+	490.195i	−0.130004	+	0.562523i
$16$		0			0
$17$			1608.11i			1.34957i	0.738016	+	0.674783i	$0.235763\pi$
$17$			1608.11i			1.34957i	−0.738016	+	0.674783i	$0.764237\pi$
$18$		0			0
$19$	446.747			0.283908			0.141954	−	0.989873i	$-0.454662\pi$
$19$	446.747			0.283908			0.141954	+	0.989873i	$0.454662\pi$
$20$		0			0
$21$	−113.235			−0.0560316
$22$		0			0
$23$			802.223i			0.316210i	0.987422	+	0.158105i	$0.0505384\pi$
$23$			802.223i			0.316210i	−0.987422	+	0.158105i	$0.949462\pi$
$24$		0			0
$25$	2808.10	+	1371.20i	0.898593	+	0.438782i
$26$		0			0
$27$		−	729.000i		−	0.192450i
$28$		0			0
$29$	−4477.98			−0.988752			−0.494376	−	0.869248i	$-0.664603\pi$
$29$	−4477.98			−0.988752			−0.494376	+	0.869248i	$0.664603\pi$
$30$		0			0
$31$	5200.90			0.972019			0.486009	−	0.873954i	$-0.338452\pi$
$31$	5200.90			0.972019			0.486009	+	0.873954i	$0.338452\pi$
$32$		0			0
$33$			1482.41i			0.236965i
$34$		0			0
$35$	−158.373	+	685.275i	−0.0218531	+	0.0945573i
$36$		0			0
$37$		−	8248.56i		−	0.990544i	−0.868738	−	0.495272i	$-0.835068\pi$
$37$		−	8248.56i		−	0.990544i	0.868738	−	0.495272i	$-0.164932\pi$
$38$		0			0
$39$	−7642.07			−0.804543
$40$		0			0
$41$	−6250.10			−0.580667			−0.290334	−	0.956925i	$-0.593766\pi$
$41$	−6250.10			−0.580667			−0.290334	+	0.956925i	$0.593766\pi$
$42$		0			0
$43$		−	11314.2i		−	0.933152i	−0.884481	−	0.466576i	$-0.845487\pi$
$43$		−	11314.2i		−	0.933152i	0.884481	−	0.466576i	$-0.154513\pi$
$44$		0			0
$45$	−4411.75	−	1019.60i	−0.324773	−	0.0750580i
$46$		0			0
$47$		−	29842.0i		−	1.97053i	−0.171035	−	0.985265i	$-0.554711\pi$
$47$		−	29842.0i		−	1.97053i	0.171035	−	0.985265i	$-0.445289\pi$
$48$		0			0
$49$	16648.7			0.990581
$50$		0			0
$51$	−14473.0			−0.779172
$52$		0			0
$53$			9195.04i			0.449639i	0.974400	+	0.224820i	$0.0721793\pi$
$53$			9195.04i			0.449639i	−0.974400	+	0.224820i	$0.927821\pi$
$54$		0			0
$55$	8971.25	+	2073.34i	0.399895	+	0.0924195i
$56$		0			0
$57$			4020.72i			0.163914i
$58$		0			0
$59$	−9948.84			−0.372085			−0.186043	−	0.982542i	$-0.559566\pi$
$59$	−9948.84			−0.372085			−0.186043	+	0.982542i	$0.559566\pi$
$60$		0			0
$61$	41550.9			1.42974			0.714868	−	0.699259i	$-0.246487\pi$
$61$	41550.9			1.42974			0.714868	+	0.699259i	$0.246487\pi$
$62$		0			0
$63$		−	1019.12i		−	0.0323499i
$64$		0			0
$65$	−10688.4	+	46248.1i	−0.313782	+	1.35772i
$66$		0			0
$67$		−	49074.0i		−	1.33556i	−0.744357	−	0.667781i	$-0.767244\pi$
$67$		−	49074.0i		−	1.33556i	0.744357	−	0.667781i	$-0.232756\pi$
$68$		0			0
$69$	−7220.00			−0.182564
$70$		0			0
$71$	−44977.6			−1.05889			−0.529445	−	0.848344i	$-0.677600\pi$
$71$	−44977.6			−1.05889			−0.529445	+	0.848344i	$0.677600\pi$
$72$		0			0
$73$			65643.2i			1.44173i	0.693077	+	0.720863i	$0.256254\pi$
$73$			65643.2i			1.44173i	−0.693077	+	0.720863i	$0.743746\pi$
$74$		0			0
$75$	−12340.8	+	25272.9i	−0.253331	+	0.518803i
$76$		0			0
$77$			2072.36i			0.0398327i
$78$		0			0
$79$	67157.8			1.21068			0.605339	−	0.795968i	$-0.293037\pi$
$79$	67157.8			1.21068			0.605339	+	0.795968i	$0.293037\pi$
$80$		0			0
$81$	6561.00			0.111111
$82$		0			0
$83$			54883.8i			0.874478i	0.899345	+	0.437239i	$0.144044\pi$
$83$			54883.8i			0.874478i	−0.899345	+	0.437239i	$0.855956\pi$
$84$		0			0
$85$	−20242.3	+	87587.6i	−0.303887	+	1.31491i
$86$		0			0
$87$		−	40301.8i		−	0.570856i
$88$		0			0
$89$	53984.2			0.722423			0.361212	−	0.932484i	$-0.382363\pi$
$89$	53984.2			0.722423			0.361212	+	0.932484i	$0.382363\pi$
$90$		0			0
$91$	−10683.3			−0.135240
$92$		0			0
$93$			46808.1i			0.561195i
$94$		0			0
$95$	24332.5	+	5623.47i	0.276617	+	0.0639287i
$96$		0			0
$97$		−	100814.i		−	1.08791i	−0.839114	−	0.543956i	$-0.816926\pi$
$97$		−	100814.i		−	1.08791i	0.839114	−	0.543956i	$-0.183074\pi$
$98$		0			0
$99$	−13341.7			−0.136812
$100$		0			0
$101$	35164.5			0.343005			0.171503	−	0.985184i	$-0.445138\pi$
$101$	35164.5			0.343005			0.171503	+	0.985184i	$0.445138\pi$
$102$		0			0
$103$		−	26256.9i		−	0.243865i	−0.992538	−	0.121933i	$-0.961091\pi$
$103$		−	26256.9i		−	0.243865i	0.992538	−	0.121933i	$-0.0389092\pi$
$104$		0			0
$105$	−6167.48	−	1425.36i	−0.0545927	−	0.0126169i
$106$		0			0
$107$		−	192949.i		−	1.62923i	−0.580003	−	0.814615i	$-0.696948\pi$
$107$		−	192949.i		−	1.62923i	0.580003	−	0.814615i	$-0.303052\pi$
$108$		0			0
$109$	−210504.			−1.69705			−0.848523	−	0.529159i	$-0.822508\pi$
$109$	−210504.			−1.69705			−0.848523	+	0.529159i	$0.822508\pi$
$110$		0			0
$111$	74237.1			0.571891
$112$		0			0
$113$		−	69072.9i		−	0.508875i	−0.967089	−	0.254438i	$-0.918110\pi$
$113$		−	69072.9i		−	0.508875i	0.967089	−	0.254438i	$-0.0818904\pi$
$114$		0			0
$115$	−10098.1	+	43693.9i	−0.0712022	+	0.308089i
$116$		0			0
$117$		−	68778.6i		−	0.464503i
$118$		0			0
$119$	−20232.8			−0.130975
$120$		0			0
$121$	−133921.			−0.831542
$122$		0			0
$123$		−	56250.9i		−	0.335248i
$124$		0			0
$125$	135686.	+	110031.i	0.776714	+	0.629854i
$126$		0			0
$127$		−	137071.i		−	0.754115i	−0.926190	−	0.377057i	$-0.876936\pi$
$127$		−	137071.i		−	0.754115i	0.926190	−	0.377057i	$-0.123064\pi$
$128$		0			0
$129$	101828.			0.538755
$130$		0			0
$131$	−161384.			−0.821644			−0.410822	−	0.911716i	$-0.634758\pi$
$131$	−161384.			−0.821644			−0.410822	+	0.911716i	$0.634758\pi$
$132$		0			0
$133$			5620.83i			0.0275532i
$134$		0			0
$135$	9176.37	−	39705.8i	0.0433348	−	0.187508i
$136$		0			0
$137$		−	236746.i		−	1.07766i	−0.842415	−	0.538829i	$-0.818867\pi$
$137$		−	236746.i		−	1.07766i	0.842415	−	0.538829i	$-0.181133\pi$
$138$		0			0
$139$	−139568.			−0.612703			−0.306352	−	0.951918i	$-0.599108\pi$
$139$	−139568.			−0.612703			−0.306352	+	0.951918i	$0.599108\pi$
$140$		0			0
$141$	268578.			1.13769
$142$		0			0
$143$			139861.i			0.571946i
$144$		0			0
$145$	−243898.	−	56367.1i	−0.963359	−	0.222641i
$146$		0			0
$147$			149838.i			0.571912i
$148$		0			0
$149$	169441.			0.625249			0.312624	−	0.949877i	$-0.398792\pi$
$149$	169441.			0.625249			0.312624	+	0.949877i	$0.398792\pi$
$150$		0			0
$151$	141483.			0.504966			0.252483	−	0.967601i	$-0.418753\pi$
$151$	141483.			0.504966			0.252483	+	0.967601i	$0.418753\pi$
$152$		0			0
$153$		−	130257.i		−	0.449855i
$154$		0			0
$155$	283273.	+	65467.0i	0.947056	+	0.218873i
$156$		0			0
$157$			356278.i			1.15356i	0.816900	+	0.576780i	$0.195691\pi$
$157$			356278.i			1.15356i	−0.816900	+	0.576780i	$0.804309\pi$
$158$		0			0
$159$	−82755.4			−0.259599
$160$		0			0
$161$	−10093.3			−0.0306881
$162$		0			0
$163$			492593.i			1.45218i	0.687602	+	0.726088i	$0.258663\pi$
$163$			492593.i			1.45218i	−0.687602	+	0.726088i	$0.741337\pi$
$164$		0			0
$165$	−18660.1	+	80741.3i	−0.0533584	+	0.230880i
$166$		0			0
$167$			558396.i			1.54936i	0.632356	+	0.774678i	$0.282088\pi$
$167$			558396.i			1.54936i	−0.632356	+	0.774678i	$0.717912\pi$
$168$		0			0
$169$	−349709.			−0.941868
$170$		0			0
$171$	−36186.5			−0.0946359
$172$		0			0
$173$		−	104434.i		−	0.265294i	−0.991163	−	0.132647i	$-0.957652\pi$
$173$		−	104434.i		−	0.265294i	0.991163	−	0.132647i	$-0.0423477\pi$
$174$		0			0
$175$	−17252.0	+	35330.7i	−0.0425837	+	0.0872082i
$176$		0			0
$177$		−	89539.6i		−	0.214823i
$178$		0			0
$179$	439589.			1.02545			0.512725	−	0.858553i	$-0.328636\pi$
$179$	439589.			1.02545			0.512725	+	0.858553i	$0.328636\pi$
$180$		0			0
$181$	66771.5			0.151494			0.0757469	−	0.997127i	$-0.475866\pi$
$181$	66771.5			0.151494			0.0757469	+	0.997127i	$0.475866\pi$
$182$		0			0
$183$			373958.i			0.825459i
$184$		0			0
$185$	103830.	−	449267.i	0.223045	−	0.965106i
$186$		0			0
$187$			264877.i			0.553911i
$188$		0			0
$189$	9172.05			0.0186772
$190$		0			0
$191$	−907090.			−1.79915			−0.899574	−	0.436768i	$-0.856123\pi$
$191$	−907090.			−1.79915			−0.899574	+	0.436768i	$0.856123\pi$
$192$		0			0
$193$		−	992517.i		−	1.91798i	−0.283436	−	0.958991i	$-0.591474\pi$
$193$		−	992517.i		−	1.91798i	0.283436	−	0.958991i	$-0.408526\pi$
$194$		0			0
$195$	−416233.	−	96195.3i	−0.783881	−	0.181162i
$196$		0			0
$197$			520497.i			0.955548i	0.878483	+	0.477774i	$0.158556\pi$
$197$			520497.i			0.955548i	−0.878483	+	0.477774i	$0.841444\pi$
$198$		0			0
$199$	861410.			1.54198			0.770988	−	0.636850i	$-0.219763\pi$
$199$	861410.			1.54198			0.770988	+	0.636850i	$0.219763\pi$
$200$		0			0
$201$	441666.			0.771088
$202$		0			0
$203$		−	56340.6i		−	0.0959580i
$204$		0			0
$205$	−340418.	−	78673.8i	−0.565755	−	0.130751i
$206$		0			0
$207$		−	64980.0i		−	0.105403i
$208$		0			0
$209$	73584.9			0.116526
$210$		0			0
$211$	1.08168e6			1.67261			0.836304	−	0.548266i	$-0.184712\pi$
$211$	1.08168e6			1.67261			0.836304	+	0.548266i	$0.184712\pi$
$212$		0			0
$213$		−	404799.i		−	0.611350i
$214$		0			0
$215$	142419.	−	616239.i	0.210122	−	0.909187i
$216$		0			0
$217$			65436.2i			0.0943341i
$218$		0			0
$219$	−590789.			−0.832381
$220$		0			0
$221$	−1.36548e6			−1.88063
$222$		0			0
$223$		−	294324.i		−	0.396336i	−0.980168	−	0.198168i	$-0.936501\pi$
$223$		−	294324.i		−	0.396336i	0.980168	−	0.198168i	$-0.0634992\pi$
$224$		0			0
$225$	−227456.	−	111067.i	−0.299531	−	0.146261i
$226$		0			0
$227$			223019.i			0.287261i	0.989631	+	0.143631i	$0.0458777\pi$
$227$			223019.i			0.287261i	−0.989631	+	0.143631i	$0.954122\pi$
$228$		0			0
$229$	75342.8			0.0949408			0.0474704	−	0.998873i	$-0.484884\pi$
$229$	75342.8			0.0949408			0.0474704	+	0.998873i	$0.484884\pi$
$230$		0			0
$231$	−18651.3			−0.0229974
$232$		0			0
$233$		−	1.04814e6i		−	1.26482i	−0.774633	−	0.632411i	$-0.782065\pi$
$233$		−	1.04814e6i		−	1.26482i	0.774633	−	0.632411i	$-0.217935\pi$
$234$		0			0
$235$	375639.	−	1.62538e6i	0.443712	−	1.91992i
$236$		0			0
$237$			604420.i			0.698986i
$238$		0			0
$239$	−871642.			−0.987060			−0.493530	−	0.869729i	$-0.664294\pi$
$239$	−871642.			−0.987060			−0.493530	+	0.869729i	$0.664294\pi$
$240$		0			0
$241$	228926.			0.253894			0.126947	−	0.991910i	$-0.459482\pi$
$241$	228926.			0.253894			0.126947	+	0.991910i	$0.459482\pi$
$242$		0			0
$243$			59049.0i			0.0641500i
$244$		0			0
$245$	906789.	+	209567.i	0.965142	+	0.223053i
$246$		0			0
$247$			379341.i			0.395628i
$248$		0			0
$249$	−493954.			−0.504880
$250$		0			0
$251$	−594245.			−0.595362			−0.297681	−	0.954665i	$-0.596213\pi$
$251$	−594245.			−0.595362			−0.297681	+	0.954665i	$0.596213\pi$
$252$		0			0
$253$			132136.i			0.129784i
$254$		0			0
$255$	−788288.	−	182181.i	−0.759162	−	0.175449i
$256$		0			0
$257$			499436.i			0.471679i	0.971792	+	0.235840i	$0.0757840\pi$
$257$			499436.i			0.471679i	−0.971792	+	0.235840i	$0.924216\pi$
$258$		0			0
$259$	103781.			0.0961320
$260$		0			0
$261$	362717.			0.329584
$262$		0			0
$263$			760007.i			0.677530i	0.940871	+	0.338765i	$0.110009\pi$
$263$			760007.i			0.677530i	−0.940871	+	0.338765i	$0.889991\pi$
$264$		0			0
$265$	−115744.	+	500818.i	−0.101247	+	0.438092i
$266$		0			0
$267$			485858.i			0.417091i
$268$		0			0
$269$	1.88976e6			1.59230			0.796150	−	0.605099i	$-0.206866\pi$
$269$	1.88976e6			1.59230			0.796150	+	0.605099i	$0.206866\pi$
$270$		0			0
$271$	2.23595e6			1.84943			0.924717	−	0.380655i	$-0.124302\pi$
$271$	2.23595e6			1.84943			0.924717	+	0.380655i	$0.124302\pi$
$272$		0			0
$273$		−	96150.1i		−	0.0780806i
$274$		0			0
$275$	462530.	+	225853.i	0.368815	+	0.180092i
$276$		0			0
$277$			1.82310e6i			1.42761i	0.700343	+	0.713806i	$0.253030\pi$
$277$			1.82310e6i			1.42761i	−0.700343	+	0.713806i	$0.746970\pi$
$278$		0			0
$279$	−421273.			−0.324006
$280$		0			0
$281$	−1.28839e6			−0.973381			−0.486690	−	0.873575i	$-0.661796\pi$
$281$	−1.28839e6			−0.973381			−0.486690	+	0.873575i	$0.661796\pi$
$282$		0			0
$283$			497215.i			0.369044i	0.982828	+	0.184522i	$0.0590736\pi$
$283$			497215.i			0.369044i	−0.982828	+	0.184522i	$0.940926\pi$
$284$		0			0
$285$	−50611.3	+	218993.i	−0.0369092	+	0.159705i
$286$		0			0
$287$		−	78636.8i		−	0.0563535i
$288$		0			0
$289$	−1.16617e6			−0.821329
$290$		0			0
$291$	907330.			0.628106
$292$		0			0
$293$			445745.i			0.303332i	0.988432	+	0.151666i	$0.0484637\pi$
$293$			445745.i			0.303332i	−0.988432	+	0.151666i	$0.951536\pi$
$294$		0			0
$295$	−541874.	−	125232.i	−0.362530	−	0.0837839i
$296$		0			0
$297$		−	120076.i		−	0.0789884i
$298$		0			0
$299$	−681182.			−0.440641
$300$		0			0
$301$	142352.			0.0905620
$302$		0			0
$303$			316480.i			0.198034i
$304$		0			0
$305$	2.26311e6	+	523027.i	1.39302	+	0.321939i
$306$		0			0
$307$		−	947164.i		−	0.573560i	−0.957996	−	0.286780i	$-0.907415\pi$
$307$		−	947164.i		−	0.573560i	0.957996	−	0.286780i	$-0.0925849\pi$
$308$		0			0
$309$	236312.			0.140796
$310$		0			0
$311$	−2.79602e6			−1.63923			−0.819615	−	0.572914i	$-0.805813\pi$
$311$	−2.79602e6			−1.63923			−0.819615	+	0.572914i	$0.805813\pi$
$312$		0			0
$313$		−	559111.i		−	0.322580i	−0.986907	−	0.161290i	$-0.948435\pi$
$313$		−	559111.i		−	0.322580i	0.986907	−	0.161290i	$-0.0515654\pi$
$314$		0			0
$315$	12828.2	−	55507.3i	0.00728435	−	0.0315191i
$316$		0			0
$317$			1.91082e6i			1.06800i	0.845485	+	0.534000i	$0.179312\pi$
$317$			1.91082e6i			1.06800i	−0.845485	+	0.534000i	$0.820688\pi$
$318$		0			0
$319$	−737581.			−0.405819
$320$		0			0
$321$	1.73654e6			0.940636
$322$		0			0
$323$			718419.i			0.383152i
$324$		0			0
$325$	−1.16431e6	+	2.38441e6i	−0.611447	+	1.25220i
$326$		0			0
$327$		−	1.89453e6i		−	0.979790i
$328$		0			0
$329$	375463.			0.191239
$330$		0			0
$331$	−564319.			−0.283110			−0.141555	−	0.989930i	$-0.545210\pi$
$331$	−564319.			−0.283110			−0.141555	+	0.989930i	$0.545210\pi$
$332$		0			0
$333$			668133.i			0.330181i
$334$		0			0
$335$	617724.	−	2.67287e6i	0.300734	−	1.30126i
$336$		0			0
$337$		−	1.79332e6i		−	0.860166i	−0.902789	−	0.430083i	$-0.858484\pi$
$337$		−	1.79332e6i		−	0.860166i	0.902789	−	0.430083i	$-0.141516\pi$
$338$		0			0
$339$	621656.			0.293799
$340$		0			0
$341$	856655.			0.398951
$342$		0			0
$343$			420929.i			0.193185i
$344$		0			0
$345$	−393245.	−	90882.6i	−0.177875	−	0.0411086i
$346$		0			0
$347$		−	4.22394e6i		−	1.88319i	−0.336748	−	0.941595i	$-0.609327\pi$
$347$		−	4.22394e6i		−	1.88319i	0.336748	−	0.941595i	$-0.390673\pi$
$348$		0			0
$349$	−1.31539e6			−0.578084			−0.289042	−	0.957316i	$-0.593337\pi$
$349$	−1.31539e6			−0.578084			−0.289042	+	0.957316i	$0.593337\pi$
$350$		0			0
$351$	619007.			0.268181
$352$		0			0
$353$		−	3.39771e6i		−	1.45128i	−0.688077	−	0.725638i	$-0.741545\pi$
$353$		−	3.39771e6i		−	1.45128i	0.688077	−	0.725638i	$-0.258455\pi$
$354$		0			0
$355$	−2.44975e6	−	566161.i	−1.03170	−	0.238434i
$356$		0			0
$357$		−	182095.i		−	0.0756184i
$358$		0			0
$359$	4.61103e6			1.88826			0.944131	−	0.329570i	$-0.106904\pi$
$359$	4.61103e6			1.88826			0.944131	+	0.329570i	$0.106904\pi$
$360$		0			0
$361$	−2.27652e6			−0.919396
$362$		0			0
$363$		−	1.20529e6i		−	0.480091i
$364$		0			0
$365$	−826291.	+	3.57533e6i	−0.324639	+	1.40470i
$366$		0			0
$367$		−	4.16806e6i		−	1.61536i	−0.589623	−	0.807679i	$-0.700723\pi$
$367$		−	4.16806e6i		−	1.61536i	0.589623	−	0.807679i	$-0.299277\pi$
$368$		0			0
$369$	506258.			0.193556
$370$		0			0
$371$	−115689.			−0.0436373
$372$		0			0
$373$			873693.i			0.325152i	0.986696	+	0.162576i	$0.0519803\pi$
$373$			873693.i			0.325152i	−0.986696	+	0.162576i	$0.948020\pi$
$374$		0			0
$375$	−990278.	+	1.22118e6i	−0.363646	+	0.448436i
$376$		0			0
$377$		−	3.80234e6i		−	1.37784i
$378$		0			0
$379$	−2.00655e6			−0.717551			−0.358775	−	0.933424i	$-0.616806\pi$
$379$	−2.00655e6			−0.717551			−0.358775	+	0.933424i	$0.616806\pi$
$380$		0			0
$381$	1.23364e6			0.435388
$382$		0			0
$383$			3.34193e6i			1.16413i	0.813143	+	0.582064i	$0.197755\pi$
$383$			3.34193e6i			1.16413i	−0.813143	+	0.582064i	$0.802245\pi$
$384$		0			0
$385$	−26086.1	+	112874.i	−0.00896928	+	0.0388097i
$386$		0			0
$387$			916449.i			0.311051i
$388$		0			0
$389$	1.41873e6			0.475364			0.237682	−	0.971343i	$-0.423612\pi$
$389$	1.41873e6			0.475364			0.237682	+	0.971343i	$0.423612\pi$
$390$		0			0
$391$	−1.29006e6			−0.426746
$392$		0			0
$393$		−	1.45246e6i		−	0.474376i
$394$		0			0
$395$	3.65782e6	+	845356.i	1.17959	+	0.272613i
$396$		0			0
$397$			1.31507e6i			0.418766i	0.977834	+	0.209383i	$0.0671455\pi$
$397$			1.31507e6i			0.418766i	−0.977834	+	0.209383i	$0.932854\pi$
$398$		0			0
$399$	−50587.5			−0.0159078
$400$		0			0
$401$	−5.01811e6			−1.55840			−0.779200	−	0.626775i	$-0.784374\pi$
$401$	−5.01811e6			−1.55840			−0.779200	+	0.626775i	$0.784374\pi$
$402$		0			0
$403$			4.41618e6i			1.35452i
$404$		0			0
$405$	357352.	+	82587.3i	0.108258	+	0.0250193i
$406$		0			0
$407$		−	1.35864e6i		−	0.406555i
$408$		0			0
$409$	314814.			0.0930563			0.0465282	−	0.998917i	$-0.485184\pi$
$409$	314814.			0.0930563			0.0465282	+	0.998917i	$0.485184\pi$
$410$		0			0
$411$	2.13071e6			0.622187
$412$		0			0
$413$		−	125173.i		−	0.0361107i
$414$		0			0
$415$	−690856.	+	2.98931e6i	−0.196910	+	0.852021i
$416$		0			0
$417$		−	1.25612e6i		−	0.353744i
$418$		0			0
$419$	−4.54817e6			−1.26561			−0.632807	−	0.774310i	$-0.718097\pi$
$419$	−4.54817e6			−1.26561			−0.632807	+	0.774310i	$0.718097\pi$
$420$		0			0
$421$	4.05291e6			1.11445			0.557227	−	0.830360i	$-0.311865\pi$
$421$	4.05291e6			1.11445			0.557227	+	0.830360i	$0.311865\pi$
$422$		0			0
$423$			2.41720e6i			0.656843i
$424$		0			0
$425$	−2.20504e6	+	4.51575e6i	−0.592166	+	1.21271i
$426$		0			0
$427$			522781.i			0.138755i
$428$		0			0
$429$	−1.25875e6			−0.330213
$430$		0			0
$431$	5.02007e6			1.30172			0.650858	−	0.759200i	$-0.274409\pi$
$431$	5.02007e6			1.30172			0.650858	+	0.759200i	$0.274409\pi$
$432$		0			0
$433$		−	4.54995e6i		−	1.16624i	−0.812387	−	0.583118i	$-0.801832\pi$
$433$		−	4.54995e6i		−	1.16624i	0.812387	−	0.583118i	$-0.198168\pi$
$434$		0			0
$435$	507304.	−	2.19508e6i	0.128542	−	0.556196i
$436$		0			0
$437$			358390.i			0.0897744i
$438$		0			0
$439$	3.75909e6			0.930938			0.465469	−	0.885064i	$-0.345886\pi$
$439$	3.75909e6			0.930938			0.465469	+	0.885064i	$0.345886\pi$
$440$		0			0
$441$	−1.34854e6			−0.330194
$442$		0			0
$443$			6.67476e6i			1.61594i	0.589220	+	0.807972i	$0.299435\pi$
$443$			6.67476e6i			1.61594i	−0.589220	+	0.807972i	$0.700565\pi$
$444$		0			0
$445$	2.94031e6	+	679532.i	0.703871	+	0.162671i
$446$		0			0
$447$			1.52497e6i			0.360988i
$448$		0			0
$449$	2.57947e6			0.603831			0.301916	−	0.953335i	$-0.402374\pi$
$449$	2.57947e6			0.603831			0.301916	+	0.953335i	$0.402374\pi$
$450$		0			0
$451$	−1.02947e6			−0.238327
$452$		0			0
$453$			1.27335e6i			0.291542i
$454$		0			0
$455$	−581880.	−	134478.i	−0.131766	−	0.0304524i
$456$		0			0
$457$		−	1.78248e6i		−	0.399240i	−0.979873	−	0.199620i	$-0.936029\pi$
$457$		−	1.78248e6i		−	0.399240i	0.979873	−	0.199620i	$-0.0639707\pi$
$458$		0			0
$459$	1.17231e6			0.259724
$460$		0			0
$461$	−2.57073e6			−0.563383			−0.281691	−	0.959505i	$-0.590895\pi$
$461$	−2.57073e6			−0.563383			−0.281691	+	0.959505i	$0.590895\pi$
$462$		0			0
$463$			5.50764e6i			1.19402i	0.802232	+	0.597012i	$0.203646\pi$
$463$			5.50764e6i			1.19402i	−0.802232	+	0.597012i	$0.796354\pi$
$464$		0			0
$465$	−589203.	+	2.54946e6i	−0.126367	+	0.546783i
$466$		0			0
$467$			1.10049e6i			0.233504i	0.993161	+	0.116752i	$0.0372483\pi$
$467$			1.10049e6i			0.233504i	−0.993161	+	0.116752i	$0.962752\pi$
$468$		0			0
$469$	617434.			0.129616
$470$		0			0
$471$	−3.20650e6			−0.666008
$472$		0			0
$473$		−	1.86359e6i		−	0.382999i
$474$		0			0
$475$	1.25451e6	+	612577.i	0.255118	+	0.124574i
$476$		0			0
$477$		−	744798.i		−	0.149880i
$478$		0			0
$479$	−2.85993e6			−0.569530			−0.284765	−	0.958597i	$-0.591916\pi$
$479$	−2.85993e6			−0.569530			−0.284765	+	0.958597i	$0.591916\pi$
$480$		0			0
$481$	7.00401e6			1.38033
$482$		0			0
$483$		−	90839.9i		−	0.0177178i
$484$		0			0
$485$	1.26901e6	−	5.49097e6i	0.244969	−	1.05997i
$486$		0			0
$487$		−	988304.i		−	0.188829i	−0.995533	−	0.0944144i	$-0.969902\pi$
$487$		−	988304.i		−	0.188829i	0.995533	−	0.0944144i	$-0.0300979\pi$
$488$		0			0
$489$	−4.43334e6			−0.838414
$490$		0			0
$491$	−2.57116e6			−0.481310			−0.240655	−	0.970611i	$-0.577362\pi$
$491$	−2.57116e6			−0.481310			−0.240655	+	0.970611i	$0.577362\pi$
$492$		0			0
$493$		−	7.20110e6i		−	1.33439i
$494$		0			0
$495$	−726671.	−	167940.i	−0.133298	−	0.0308065i
$496$		0			0
$497$		−	565895.i		−	0.102765i
$498$		0			0
$499$	−3.15722e6			−0.567614			−0.283807	−	0.958881i	$-0.591598\pi$
$499$	−3.15722e6			−0.567614			−0.283807	+	0.958881i	$0.591598\pi$
$500$		0			0
$501$	−5.02556e6			−0.894521
$502$		0			0
$503$		−	6.90234e6i		−	1.21640i	−0.793784	−	0.608200i	$-0.791892\pi$
$503$		−	6.90234e6i		−	1.21640i	0.793784	−	0.608200i	$-0.208108\pi$
$504$		0			0
$505$	1.91527e6	+	442637.i	0.334196	+	0.0772359i
$506$		0			0
$507$		−	3.14738e6i		−	0.543788i
$508$		0			0
$509$	−8.99803e6			−1.53941			−0.769703	−	0.638402i	$-0.779596\pi$
$509$	−8.99803e6			−1.53941			−0.769703	+	0.638402i	$0.779596\pi$
$510$		0			0
$511$	−825903.			−0.139919
$512$		0			0
$513$		−	325678.i		−	0.0546381i
$514$		0			0
$515$	330511.	−	1.43011e6i	0.0549121	−	0.237602i
$516$		0			0
$517$		−	4.91535e6i		−	0.808776i
$518$		0			0
$519$	939909.			0.153168
$520$		0			0
$521$	8.75002e6			1.41226			0.706130	−	0.708082i	$-0.250439\pi$
$521$	8.75002e6			1.41226			0.706130	+	0.708082i	$0.250439\pi$
$522$		0			0
$523$		−	2.22239e6i		−	0.355276i	−0.984096	−	0.177638i	$-0.943154\pi$
$523$		−	2.22239e6i		−	0.355276i	0.984096	−	0.177638i	$-0.0568456\pi$
$524$		0			0
$525$	−317976.	−	155268.i	−0.0503497	−	0.0245857i
$526$		0			0
$527$			8.36364e6i			1.31180i
$528$		0			0
$529$	5.79278e6			0.900011
$530$		0			0
$531$	805856.			0.124028
$532$		0			0
$533$		−	5.30707e6i		−	0.809165i
$534$		0			0
$535$	2.42876e6	−	1.05092e7i	0.366860	−	1.58739i
$536$		0			0
$537$			3.95630e6i			0.592044i
$538$		0			0
$539$	2.74225e6			0.406570
$540$		0			0
$541$	2.93950e6			0.431798			0.215899	−	0.976416i	$-0.430732\pi$
$541$	2.93950e6			0.431798			0.215899	+	0.976416i	$0.430732\pi$
$542$		0			0
$543$			600944.i			0.0874650i
$544$		0			0
$545$	−1.14653e7	−	2.64974e6i	−1.65346	−	0.382130i
$546$		0			0
$547$			6.12529e6i			0.875303i	0.899145	+	0.437651i	$0.144190\pi$
$547$			6.12529e6i			0.875303i	−0.899145	+	0.437651i	$0.855810\pi$
$548$		0			0
$549$	−3.36562e6			−0.476579
$550$		0			0
$551$	−2.00052e6			−0.280714
$552$		0			0
$553$			844959.i			0.117496i
$554$		0			0
$555$	4.04340e6	+	934467.i	0.557204	+	0.128775i
$556$		0			0
$557$			2.19245e6i			0.299428i	0.988729	+	0.149714i	$0.0478352\pi$
$557$			2.19245e6i			0.299428i	−0.988729	+	0.149714i	$0.952165\pi$
$558$		0			0
$559$	9.60709e6			1.30036
$560$		0			0
$561$	−2.38389e6			−0.319800
$562$		0			0
$563$		−	44998.2i		−	0.00598307i	−0.999996	−	0.00299154i	$-0.999048\pi$
$563$		−	44998.2i		−	0.00598307i	0.999996	−	0.00299154i	$-0.000952237\pi$
$564$		0			0
$565$	869463.	−	3.76213e6i	0.114586	−	0.495807i
$566$		0			0
$567$			82548.5i			0.0107833i
$568$		0			0
$569$	1.09576e6			0.141884			0.0709420	−	0.997480i	$-0.477399\pi$
$569$	1.09576e6			0.141884			0.0709420	+	0.997480i	$0.477399\pi$
$570$		0			0
$571$	−1.11639e7			−1.43294			−0.716468	−	0.697620i	$-0.754242\pi$
$571$	−1.11639e7			−1.43294			−0.716468	+	0.697620i	$0.754242\pi$
$572$		0			0
$573$		−	8.16381e6i		−	1.03874i
$574$		0			0
$575$	−1.10000e6	+	2.25272e6i	−0.138747	+	0.284144i
$576$		0			0
$577$			8.62132e6i			1.07804i	0.842294	+	0.539019i	$0.181205\pi$
$577$			8.62132e6i			1.07804i	−0.842294	+	0.539019i	$0.818795\pi$
$578$		0			0
$579$	8.93265e6			1.10735
$580$		0			0
$581$	−690531.			−0.0848678
$582$		0			0
$583$			1.51454e6i			0.184548i
$584$		0			0
$585$	865758.	−	3.74610e6i	0.104594	−	0.452574i
$586$		0			0
$587$		−	1.34006e6i		−	0.160520i	−0.996774	−	0.0802599i	$-0.974425\pi$
$587$		−	1.34006e6i		−	0.160520i	0.996774	−	0.0802599i	$-0.0255750\pi$
$588$		0			0
$589$	2.32349e6			0.275964
$590$		0			0
$591$	−4.68447e6			−0.551686
$592$		0			0
$593$			1.08044e7i			1.26172i	0.775897	+	0.630860i	$0.217298\pi$
$593$			1.08044e7i			1.26172i	−0.775897	+	0.630860i	$0.782702\pi$
$594$		0			0
$595$	−1.10200e6	−	254682.i	−0.127611	−	0.0294922i
$596$		0			0
$597$			7.75269e6i			0.890260i
$598$		0			0
$599$	−6.92132e6			−0.788174			−0.394087	−	0.919073i	$-0.628939\pi$
$599$	−6.92132e6			−0.788174			−0.394087	+	0.919073i	$0.628939\pi$
$600$		0			0
$601$	−6.16706e6			−0.696453			−0.348226	−	0.937410i	$-0.613216\pi$
$601$	−6.16706e6			−0.696453			−0.348226	+	0.937410i	$0.613216\pi$
$602$		0			0
$603$			3.97499e6i			0.445188i
$604$		0			0
$605$	−7.29413e6	−	1.68574e6i	−0.810187	−	0.187242i
$606$		0			0
$607$			1.18721e7i			1.30785i	0.756560	+	0.653924i	$0.226878\pi$
$607$			1.18721e7i			1.30785i	−0.756560	+	0.653924i	$0.773122\pi$
$608$		0			0
$609$	507065.			0.0554014
$610$		0			0
$611$	2.53394e7			2.74595
$612$		0			0
$613$		−	2.27563e6i		−	0.244597i	−0.992493	−	0.122298i	$-0.960974\pi$
$613$		−	2.27563e6i		−	0.244597i	0.992493	−	0.122298i	$-0.0390265\pi$
$614$		0			0
$615$	708064.	−	3.06376e6i	0.0754892	−	0.326639i
$616$		0			0
$617$			1.34715e7i			1.42463i	0.701860	+	0.712315i	$0.252353\pi$
$617$			1.34715e7i			1.42463i	−0.701860	+	0.712315i	$0.747647\pi$
$618$		0			0
$619$	−1.36512e7			−1.43200			−0.716000	−	0.698100i	$-0.754029\pi$
$619$	−1.36512e7			−1.43200			−0.716000	+	0.698100i	$0.754029\pi$
$620$		0			0
$621$	584820.			0.0608546
$622$		0			0
$623$			679213.i			0.0701109i
$624$		0			0
$625$	6.00527e6	+	7.70092e6i	0.614940	+	0.788574i
$626$		0			0
$627$			662264.i			0.0672763i
$628$		0			0
$629$	1.32646e7			1.33681
$630$		0			0
$631$	−6.51822e6			−0.651711			−0.325856	−	0.945420i	$-0.605652\pi$
$631$	−6.51822e6			−0.651711			−0.325856	+	0.945420i	$0.605652\pi$
$632$		0			0
$633$			9.73515e6i			0.965681i
$634$		0			0
$635$	1.72540e6	−	7.46574e6i	0.169807	−	0.734748i
$636$		0			0
$637$			1.41367e7i			1.38038i
$638$		0			0
$639$	3.64319e6			0.352963
$640$		0			0
$641$	8.25104e6			0.793165			0.396583	−	0.917999i	$-0.370196\pi$
$641$	8.25104e6			0.793165			0.396583	+	0.917999i	$0.370196\pi$
$642$		0			0
$643$		−	8.10283e6i		−	0.772875i	−0.922316	−	0.386437i	$-0.873706\pi$
$643$		−	8.10283e6i		−	0.772875i	0.922316	−	0.386437i	$-0.126294\pi$
$644$		0			0
$645$	5.54615e6	+	1.28177e6i	0.524919	+	0.121314i
$646$		0			0
$647$		−	5.23767e6i		−	0.491901i	−0.969282	−	0.245950i	$-0.920900\pi$
$647$		−	5.23767e6i		−	0.491901i	0.969282	−	0.245950i	$-0.0791000\pi$
$648$		0			0
$649$	−1.63870e6			−0.152717
$650$		0			0
$651$	−588926.			−0.0544638
$652$		0			0
$653$		−	3.53455e6i		−	0.324377i	−0.986760	−	0.162189i	$-0.948145\pi$
$653$		−	3.53455e6i		−	0.324377i	0.986760	−	0.162189i	$-0.0518553\pi$
$654$		0			0
$655$	−8.78998e6	−	2.03144e6i	−0.800543	−	0.185013i
$656$		0			0
$657$		−	5.31710e6i		−	0.480575i
$658$		0			0
$659$	−2.78813e6			−0.250092			−0.125046	−	0.992151i	$-0.539908\pi$
$659$	−2.78813e6			−0.250092			−0.125046	+	0.992151i	$0.539908\pi$
$660$		0			0
$661$	−7.98219e6			−0.710589			−0.355294	−	0.934754i	$-0.615619\pi$
$661$	−7.98219e6			−0.710589			−0.355294	+	0.934754i	$0.615619\pi$
$662$		0			0
$663$		−	1.22893e7i		−	1.08578i
$664$		0			0
$665$	−70752.8	+	306144.i	−0.00620425	+	0.0268455i
$666$		0			0
$667$		−	3.59234e6i		−	0.312653i
$668$		0			0
$669$	2.64892e6			0.228825
$670$		0			0
$671$	6.84396e6			0.586815
$672$		0			0
$673$			1.01108e7i			0.860496i	0.902711	+	0.430248i	$0.141574\pi$
$673$			1.01108e7i			0.860496i	−0.902711	+	0.430248i	$0.858426\pi$
$674$		0			0
$675$	999601.	−	2.04711e6i	0.0844437	−	0.172934i
$676$		0			0
$677$			6.25054e6i			0.524138i	0.965049	+	0.262069i	$0.0844048\pi$
$677$			6.25054e6i			0.524138i	−0.965049	+	0.262069i	$0.915595\pi$
$678$		0			0
$679$	1.26842e6			0.105581
$680$		0			0
$681$	−2.00717e6			−0.165850
$682$		0			0
$683$		−	1.86733e7i		−	1.53169i	−0.643027	−	0.765843i	$-0.722322\pi$
$683$		−	1.86733e7i		−	1.53169i	0.643027	−	0.765843i	$-0.277678\pi$
$684$		0			0
$685$	2.98007e6	−	1.28946e7i	0.242661	−	1.04998i
$686$		0			0
$687$			678085.i			0.0548141i
$688$		0			0
$689$	−7.80768e6			−0.626576
$690$		0			0
$691$	−1.93615e7			−1.54257			−0.771285	−	0.636490i	$-0.780386\pi$
$691$	−1.93615e7			−1.54257			−0.771285	+	0.636490i	$0.780386\pi$
$692$		0			0
$693$		−	167862.i		−	0.0132776i
$694$		0			0
$695$	−7.60174e6	−	1.75683e6i	−0.596968	−	0.137965i
$696$		0			0
$697$		−	1.00509e7i		−	0.783649i
$698$		0			0
$699$	9.43326e6			0.730246
$700$		0			0
$701$	6.67337e6			0.512921			0.256460	−	0.966555i	$-0.417444\pi$
$701$	6.67337e6			0.512921			0.256460	+	0.966555i	$0.417444\pi$
$702$		0			0
$703$		−	3.68502e6i		−	0.281223i
$704$		0			0
$705$	1.46284e7	+	3.38075e6i	1.10847	+	0.256177i
$706$		0			0
$707$			442429.i			0.0332885i
$708$		0			0
$709$	−4.97370e6			−0.371590			−0.185795	−	0.982589i	$-0.559486\pi$
$709$	−4.97370e6			−0.371590			−0.185795	+	0.982589i	$0.559486\pi$
$710$		0			0
$711$	−5.43978e6			−0.403559
$712$		0			0
$713$			4.17228e6i			0.307362i
$714$		0			0
$715$	−1.76051e6	+	7.61766e6i	−0.128787	+	0.557258i
$716$		0			0
$717$		−	7.84478e6i		−	0.569879i
$718$		0			0
$719$	−1.76815e7			−1.27555			−0.637773	−	0.770225i	$-0.720144\pi$
$719$	−1.76815e7			−1.27555			−0.637773	+	0.770225i	$0.720144\pi$
$720$		0			0
$721$	330356.			0.0236670
$722$		0			0
$723$			2.06033e6i			0.146586i
$724$		0			0
$725$	−1.25746e7	−	6.14019e6i	−0.888486	−	0.433847i
$726$		0			0
$727$		−	1.84948e7i		−	1.29782i	−0.760867	−	0.648908i	$-0.775226\pi$
$727$		−	1.84948e7i		−	1.29782i	0.760867	−	0.648908i	$-0.224774\pi$
$728$		0			0
$729$	−531441.			−0.0370370
$730$		0			0
$731$	1.81945e7			1.25935
$732$		0			0
$733$			1.93313e6i			0.132893i	0.997790	+	0.0664464i	$0.0211662\pi$
$733$			1.93313e6i			0.132893i	−0.997790	+	0.0664464i	$0.978834\pi$
$734$		0			0
$735$	−1.88611e6	+	8.16110e6i	−0.128780	+	0.557225i
$736$		0			0
$737$		−	8.08311e6i		−	0.548163i
$738$		0			0
$739$	−1.17689e7			−0.792730			−0.396365	−	0.918093i	$-0.629729\pi$
$739$	−1.17689e7			−0.792730			−0.396365	+	0.918093i	$0.629729\pi$
$740$		0			0
$741$	−3.41407e6			−0.228416
$742$		0			0
$743$			1.34755e6i			0.0895516i	0.998997	+	0.0447758i	$0.0142573\pi$
$743$			1.34755e6i			0.0895516i	−0.998997	+	0.0447758i	$0.985743\pi$
$744$		0			0
$745$	9.22879e6	+	2.13286e6i	0.609192	+	0.140790i
$746$		0			0
$747$		−	4.44559e6i		−	0.291493i
$748$		0			0
$749$	2.42762e6			0.158116
$750$		0			0
$751$	1.32950e7			0.860179			0.430089	−	0.902786i	$-0.358482\pi$
$751$	1.32950e7			0.860179			0.430089	+	0.902786i	$0.358482\pi$
$752$		0			0
$753$		−	5.34821e6i		−	0.343733i
$754$		0			0
$755$	7.70603e6	+	1.78093e6i	0.491998	+	0.113705i
$756$		0			0
$757$		−	1.12779e7i		−	0.715302i	−0.933855	−	0.357651i	$-0.883578\pi$
$757$		−	1.12779e7i		−	0.715302i	0.933855	−	0.357651i	$-0.116422\pi$
$758$		0			0
$759$	−1.18923e6			−0.0749307
$760$		0			0
$761$	833469.			0.0521708			0.0260854	−	0.999660i	$-0.491696\pi$
$761$	833469.			0.0521708			0.0260854	+	0.999660i	$0.491696\pi$
$762$		0			0
$763$		−	2.64849e6i		−	0.164698i
$764$		0			0
$765$	1.63963e6	−	7.09459e6i	0.101296	−	0.438303i
$766$		0			0
$767$		−	8.44774e6i		−	0.518504i
$768$		0			0
$769$	−1.29320e7			−0.788590			−0.394295	−	0.918984i	$-0.629011\pi$
$769$	−1.29320e7			−0.788590			−0.394295	+	0.918984i	$0.629011\pi$
$770$		0			0
$771$	−4.49492e6			−0.272324
$772$		0			0
$773$		−	160225.i		−	0.00964457i	−0.999988	−	0.00482228i	$-0.998465\pi$
$773$		−	160225.i		−	0.00964457i	0.999988	−	0.00482228i	$-0.00153499\pi$
$774$		0			0
$775$	1.46047e7	+	7.13145e6i	0.873450	+	0.426505i
$776$		0			0
$777$			934028.i			0.0555018i
$778$		0			0
$779$	−2.79221e6			−0.164856
$780$		0			0
$781$	−7.40839e6			−0.434606
$782$		0			0
$783$			3.26445e6i			0.190285i
$784$		0			0
$785$	−4.48469e6	+	1.94051e7i	−0.259752	+	1.12393i
$786$		0			0
$787$			1.19076e7i			0.685309i	0.939462	+	0.342654i	$0.111326\pi$
$787$			1.19076e7i			0.685309i	−0.939462	+	0.342654i	$0.888674\pi$
$788$		0			0
$789$	−6.84006e6			−0.391172
$790$		0			0
$791$	869054.			0.0493862
$792$		0			0
$793$			3.52816e7i			1.99235i
$794$		0			0
$795$	−4.50736e6	−	1.04169e6i	−0.252932	−	0.0584550i
$796$		0			0
$797$		−	2.19475e7i		−	1.22388i	−0.790903	−	0.611941i	$-0.790389\pi$
$797$		−	2.19475e7i		−	1.22388i	0.790903	−	0.611941i	$-0.209611\pi$
$798$		0			0
$799$	4.79893e7			2.65936
$800$		0			0
$801$	−4.37272e6			−0.240808
$802$		0			0
$803$			1.08123e7i			0.591736i
$804$		0			0
$805$	−549743.	−	127051.i	−0.0298999	−	0.00691015i
$806$		0			0
$807$			1.70078e7i			0.919315i
$808$		0			0
$809$	2.12570e7			1.14191			0.570953	−	0.820983i	$-0.306574\pi$
$809$	2.12570e7			1.14191			0.570953	+	0.820983i	$0.306574\pi$
$810$		0			0
$811$	−4.03970e6			−0.215674			−0.107837	−	0.994169i	$-0.534392\pi$
$811$	−4.03970e6			−0.215674			−0.107837	+	0.994169i	$0.534392\pi$
$812$		0			0
$813$			2.01235e7i			1.06777i
$814$		0			0
$815$	−6.20057e6	+	2.68296e7i	−0.326992	+	1.41488i
$816$		0			0
$817$		−	5.05458e6i		−	0.264929i
$818$		0			0
$819$	865351.			0.0450799
$820$		0			0
$821$	7.62547e6			0.394829			0.197414	−	0.980320i	$-0.436746\pi$
$821$	7.62547e6			0.394829			0.197414	+	0.980320i	$0.436746\pi$
$822$		0			0
$823$		−	2.50958e7i		−	1.29152i	−0.763541	−	0.645760i	$-0.776541\pi$
$823$		−	2.50958e7i		−	1.29152i	0.763541	−	0.645760i	$-0.223459\pi$
$824$		0			0
$825$	−2.03268e6	+	4.16277e6i	−0.103976	+	0.212935i
$826$		0			0
$827$			9.79557e6i			0.498042i	0.968498	+	0.249021i	$0.0801088\pi$
$827$			9.79557e6i			0.498042i	−0.968498	+	0.249021i	$0.919891\pi$
$828$		0			0
$829$	1.01425e7			0.512578			0.256289	−	0.966600i	$-0.417500\pi$
$829$	1.01425e7			0.512578			0.256289	+	0.966600i	$0.417500\pi$
$830$		0			0
$831$	−1.64079e7			−0.824232
$832$		0			0
$833$			2.67730e7i			1.33686i
$834$		0			0
$835$	−7.02887e6	+	3.04136e7i	−0.348875	+	1.50957i
$836$		0			0
$837$		−	3.79146e6i		−	0.187065i
$838$		0			0
$839$	−3.72767e7			−1.82824			−0.914119	−	0.405447i	$-0.867116\pi$
$839$	−3.72767e7			−1.82824			−0.914119	+	0.405447i	$0.867116\pi$
$840$		0			0
$841$	−458827.			−0.0223696
$842$		0			0
$843$		−	1.15955e7i		−	0.561982i
$844$		0			0
$845$	−1.90473e7	−	4.40200e6i	−0.917680	−	0.212084i
$846$		0			0
$847$		−	1.68495e6i		−	0.0807009i
$848$		0			0
$849$	−4.47493e6			−0.213067
$850$		0			0
$851$	6.61718e6			0.313220
$852$		0			0
$853$		−	1.56786e7i		−	0.737794i	−0.929470	−	0.368897i	$-0.879735\pi$
$853$		−	1.56786e7i		−	0.737794i	0.929470	−	0.368897i	$-0.120265\pi$
$854$		0			0
$855$	−1.97094e6	−	455501.i	−0.0922055	−	0.0213096i
$856$		0			0
$857$		−	2.90275e7i		−	1.35007i	−0.737784	−	0.675037i	$-0.764128\pi$
$857$		−	2.90275e7i		−	1.35007i	0.737784	−	0.675037i	$-0.235872\pi$
$858$		0			0
$859$	1.72480e7			0.797546			0.398773	−	0.917050i	$-0.369436\pi$
$859$	1.72480e7			0.797546			0.398773	+	0.917050i	$0.369436\pi$
$860$		0			0
$861$	707732.			0.0325357
$862$		0			0
$863$			2.50082e7i			1.14302i	0.820594	+	0.571511i	$0.193643\pi$
$863$			2.50082e7i			1.14302i	−0.820594	+	0.571511i	$0.806357\pi$
$864$		0			0
$865$	1.31458e6	−	5.68813e6i	0.0597374	−	0.258481i
$866$		0			0
$867$		−	1.04955e7i		−	0.474195i
$868$		0			0
$869$	1.10617e7			0.496906
$870$		0			0
$871$	4.16696e7			1.86112
$872$		0			0
$873$			8.16597e6i			0.362637i
$874$		0			0
$875$	−1.38438e6	+	1.70716e6i	−0.0611271	+	0.0753798i
$876$		0			0
$877$		−	1.21258e7i		−	0.532366i	−0.963923	−	0.266183i	$-0.914238\pi$
$877$		−	1.21258e7i		−	0.532366i	0.963923	−	0.266183i	$-0.0857625\pi$
$878$		0			0
$879$	−4.01171e6			−0.175129
$880$		0			0
$881$	−2.32403e7			−1.00879			−0.504397	−	0.863472i	$-0.668285\pi$
$881$	−2.32403e7			−1.00879			−0.504397	+	0.863472i	$0.668285\pi$
$882$		0			0
$883$			2.03793e7i			0.879604i	0.898095	+	0.439802i	$0.144951\pi$
$883$			2.03793e7i			0.879604i	−0.898095	+	0.439802i	$0.855049\pi$
$884$		0			0
$885$	1.12709e6	−	4.87687e6i	0.0483727	−	0.209307i
$886$		0			0
$887$		−	7.73639e6i		−	0.330164i	−0.986280	−	0.165082i	$-0.947211\pi$
$887$		−	7.73639e6i		−	0.330164i	0.986280	−	0.165082i	$-0.0527888\pi$
$888$		0			0
$889$	1.72459e6			0.0731866
$890$		0			0
$891$	1.08068e6			0.0456040
$892$		0			0
$893$		−	1.33318e7i		−	0.559449i
$894$		0			0
$895$	2.39427e7	+	5.53338e6i	0.999115	+	0.230905i
$896$		0			0
$897$		−	6.13064e6i		−	0.254404i
$898$		0			0
$899$	−2.32896e7			−0.961085
$900$		0			0
$901$	−1.47867e7			−0.606818
$902$		0			0
$903$			1.28116e6i			0.0522860i
$904$		0			0
$905$	3.63678e6	+	840494.i	0.147603	+	0.0341125i
$906$		0			0
$907$		−	3.25270e7i		−	1.31288i	−0.754377	−	0.656442i	$-0.772061\pi$
$907$		−	3.25270e7i		−	1.31288i	0.754377	−	0.656442i	$-0.227939\pi$
$908$		0			0
$909$	−2.84832e6			−0.114335
$910$		0			0
$911$	−1.69652e7			−0.677270			−0.338635	−	0.940918i	$-0.609965\pi$
$911$	−1.69652e7			−0.677270			−0.338635	+	0.940918i	$0.609965\pi$
$912$		0			0
$913$			9.04006e6i			0.358917i
$914$		0			0
$915$	−4.70724e6	+	2.03680e7i	−0.185872	+	0.804260i
$916$		0			0
$917$		−	2.03049e6i		−	0.0797402i
$918$		0			0
$919$	2.34135e7			0.914485			0.457243	−	0.889342i	$-0.348837\pi$
$919$	2.34135e7			0.914485			0.457243	+	0.889342i	$0.348837\pi$
$920$		0			0
$921$	8.52447e6			0.331145
$922$		0			0
$923$		−	3.81913e7i		−	1.47557i
$924$		0			0
$925$	1.13104e7	−	2.31628e7i	0.434633	−	0.890096i
$926$		0			0
$927$			2.12681e6i			0.0812884i
$928$		0			0
$929$	3.30115e7			1.25495			0.627475	−	0.778637i	$-0.284089\pi$
$929$	3.30115e7			1.25495			0.627475	+	0.778637i	$0.284089\pi$
$930$		0			0
$931$	7.43775e6			0.281234
$932$		0			0
$933$		−	2.51642e7i		−	0.946410i
$934$		0			0
$935$	−3.33416e6	+	1.44268e7i	−0.124726	+	0.539685i
$936$		0			0
$937$		−	3.91088e7i		−	1.45521i	−0.685997	−	0.727604i	$-0.740634\pi$
$937$		−	3.91088e7i		−	1.45521i	0.685997	−	0.727604i	$-0.259366\pi$
$938$		0			0
$939$	5.03200e6			0.186241
$940$		0			0
$941$	−6.31958e6			−0.232656			−0.116328	−	0.993211i	$-0.537112\pi$
$941$	−6.31958e6			−0.232656			−0.116328	+	0.993211i	$0.537112\pi$
$942$		0			0
$943$		−	5.01397e6i		−	0.183613i
$944$		0			0
$945$	499566.	+	115454.i	0.0181976	+	0.00420562i
$946$		0			0
$947$			3.24853e7i			1.17710i	0.808462	+	0.588548i	$0.200300\pi$
$947$			3.24853e7i			1.17710i	−0.808462	+	0.588548i	$0.799700\pi$
$948$		0			0
$949$	−5.57389e7			−2.00906
$950$		0			0
$951$	−1.71974e7			−0.616610
$952$		0			0
$953$		−	4.86404e6i		−	0.173486i	−0.996231	−	0.0867431i	$-0.972354\pi$
$953$		−	4.86404e6i		−	0.173486i	0.996231	−	0.0867431i	$-0.0276459\pi$
$954$		0			0
$955$	−4.94056e7	−	1.14181e7i	−1.75294	−	0.405121i
$956$		0			0
$957$		−	6.63822e6i		−	0.234300i
$958$		0			0
$959$	2.97867e6			0.104586
$960$		0			0
$961$	−1.57974e6			−0.0551794
$962$		0			0
$963$			1.56288e7i			0.543076i
$964$		0			0
$965$	1.24934e7	−	5.40585e7i	0.431880	−	1.86873i
$966$		0			0
$967$			1.42868e7i			0.491324i	0.969356	+	0.245662i	$0.0790054\pi$
$967$			1.42868e7i			0.491324i	−0.969356	+	0.245662i	$0.920995\pi$
$968$		0			0
$969$	−6.46577e6			−0.221213
$970$		0			0
$971$	−1.36593e7			−0.464924			−0.232462	−	0.972606i	$-0.574678\pi$
$971$	−1.36593e7			−0.464924			−0.232462	+	0.972606i	$0.574678\pi$
$972$		0			0
$973$		−	1.75601e6i		−	0.0594626i
$974$		0			0
$975$	−2.14597e7	−	1.04788e7i	−0.722957	−	0.353019i
$976$		0			0
$977$		−	2.24945e7i		−	0.753945i	−0.926225	−	0.376972i	$-0.876965\pi$
$977$		−	2.24945e7i		−	0.753945i	0.926225	−	0.376972i	$-0.123035\pi$
$978$		0			0
$979$	8.89189e6			0.296509
$980$		0			0
$981$	1.70508e7			0.565682
$982$		0			0
$983$		−	2.82995e7i		−	0.934103i	−0.884230	−	0.467051i	$-0.845316\pi$
$983$		−	2.82995e7i		−	0.934103i	0.884230	−	0.467051i	$-0.154684\pi$
$984$		0			0
$985$	−6.55181e6	+	2.83494e7i	−0.215165	+	0.931008i
$986$		0			0
$987$			3.37916e6i			0.110412i
$988$		0			0
$989$	9.07650e6			0.295072
$990$		0			0
$991$	4.01846e6			0.129980			0.0649898	−	0.997886i	$-0.479299\pi$
$991$	4.01846e6			0.129980			0.0649898	+	0.997886i	$0.479299\pi$
$992$		0			0
$993$		−	5.07887e6i		−	0.163453i
$994$		0			0
$995$	4.69176e7	+	1.08431e7i	1.50238	+	0.347213i
$996$		0			0
$997$			3.39350e7i			1.08121i	0.841277	+	0.540604i	$0.181804\pi$
$997$			3.39350e7i			1.08121i	−0.841277	+	0.540604i	$0.818196\pi$
$998$		0			0
$999$	−6.01320e6			−0.190630

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	60.6.d.a.49.6	yes	6
3.2	odd	2		180.6.d.d.109.1		6
4.3	odd	2		240.6.f.d.49.3		6
5.2	odd	4		300.6.a.j.1.2		3
5.3	odd	4		300.6.a.i.1.2		3
5.4	even	2	inner	60.6.d.a.49.3	✓	6
12.11	even	2		720.6.f.m.289.1		6
15.2	even	4		900.6.a.x.1.2		3
15.8	even	4		900.6.a.w.1.2		3
15.14	odd	2		180.6.d.d.109.2		6
20.19	odd	2		240.6.f.d.49.6		6
60.59	even	2		720.6.f.m.289.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.6.d.a.49.3	✓	6	5.4	even	2	inner
60.6.d.a.49.6	yes	6	1.1	even	1	trivial
180.6.d.d.109.1		6	3.2	odd	2
180.6.d.d.109.2		6	15.14	odd	2
240.6.f.d.49.3		6	4.3	odd	2
240.6.f.d.49.6		6	20.19	odd	2
300.6.a.i.1.2		3	5.3	odd	4
300.6.a.j.1.2		3	5.2	odd	4
720.6.f.m.289.1		6	12.11	even	2
720.6.f.m.289.2		6	60.59	even	2
900.6.a.w.1.2		3	15.8	even	4
900.6.a.x.1.2		3	15.2	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	60.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+9.00000i q^{3} +(54.4661 + 12.5876i) q^{5} +12.5817i q^{7} -81.0000 q^{9} +O(q^{10})\)	\(q+9.00000i q^{3} +(54.4661 + 12.5876i) q^{5} +12.5817i q^{7} -81.0000 q^{9} +164.713 q^{11} +849.118i q^{13} +(-113.288 + 490.195i) q^{15} +1608.11i q^{17} +446.747 q^{19} -113.235 q^{21} +802.223i q^{23} +(2808.10 + 1371.20i) q^{25} -729.000i q^{27} -4477.98 q^{29} +5200.90 q^{31} +1482.41i q^{33} +(-158.373 + 685.275i) q^{35} -8248.56i q^{37} -7642.07 q^{39} -6250.10 q^{41} -11314.2i q^{43} +(-4411.75 - 1019.60i) q^{45} -29842.0i q^{47} +16648.7 q^{49} -14473.0 q^{51} +9195.04i q^{53} +(8971.25 + 2073.34i) q^{55} +4020.72i q^{57} -9948.84 q^{59} +41550.9 q^{61} -1019.12i q^{63} +(-10688.4 + 46248.1i) q^{65} -49074.0i q^{67} -7220.00 q^{69} -44977.6 q^{71} +65643.2i q^{73} +(-12340.8 + 25272.9i) q^{75} +2072.36i q^{77} +67157.8 q^{79} +6561.00 q^{81} +54883.8i q^{83} +(-20242.3 + 87587.6i) q^{85} -40301.8i q^{87} +53984.2 q^{89} -10683.3 q^{91} +46808.1i q^{93} +(24332.5 + 5623.47i) q^{95} -100814. i q^{97} -13341.7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 38 q^{5} - 486 q^{9}+O(q^{10}) \) 6 * q - 38 * q^5 - 486 * q^9	\( 6 q - 38 q^{5} - 486 q^{9} + 296 q^{11} + 396 q^{15} - 6000 q^{19} + 1584 q^{21} + 6054 q^{25} - 15924 q^{29} + 264 q^{31} + 20096 q^{35} + 16340 q^{41} + 3078 q^{45} - 27654 q^{49} + 15624 q^{51} - 26088 q^{55} + 92456 q^{59} + 6252 q^{61} - 52440 q^{65} + 18936 q^{69} - 160800 q^{71} - 29448 q^{75} + 128952 q^{79} + 39366 q^{81} - 177864 q^{85} + 76060 q^{89} - 98400 q^{91} + 232800 q^{95} - 23976 q^{99}+O(q^{100}) \) 6 * q - 38 * q^5 - 486 * q^9 + 296 * q^11 + 396 * q^15 - 6000 * q^19 + 1584 * q^21 + 6054 * q^25 - 15924 * q^29 + 264 * q^31 + 20096 * q^35 + 16340 * q^41 + 3078 * q^45 - 27654 * q^49 + 15624 * q^51 - 26088 * q^55 + 92456 * q^59 + 6252 * q^61 - 52440 * q^65 + 18936 * q^69 - 160800 * q^71 - 29448 * q^75 + 128952 * q^79 + 39366 * q^81 - 177864 * q^85 + 76060 * q^89 - 98400 * q^91 + 232800 * q^95 - 23976 * q^99

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