LMFDB - Embedded newform 333.6.c.d.73.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [333,6,Mod(73,333)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(333, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("333.73");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 333 = 3^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	333.c (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:	$53.4078119977$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 390 x^{14} + 60701 x^{12} + 4799932 x^{10} + 203487156 x^{8} + 4519465040 x^{6} + \cdots + 178006118400 $ x^16 + 390x^14 + 60701x^12 + 4799932x^10 + 203487156x^8 + 4519465040x^6 + 48993644736x^4 + 211923220224*x^2 + 178006118400
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{5} $
Twist minimal:	no (minimal twist has level 37)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			73.9
Root			$1.04262i$ of defining polynomial
Character	$\chi$	$=$	333.73
Dual form			333.6.c.d.73.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.04262i q^{2} +30.9129 q^{4} -86.4713i q^{5} -87.1072 q^{7} +65.5944i q^{8} +O(q^{10})$	\(q+1.04262i q^{2} +30.9129 q^{4} -86.4713i q^{5} -87.1072 q^{7} +65.5944i q^{8} +90.1568 q^{10} -153.498 q^{11} -27.6086i q^{13} -90.8199i q^{14} +920.824 q^{16} +926.725i q^{17} -2749.94i q^{19} -2673.08i q^{20} -160.041i q^{22} +1336.01i q^{23} -4352.28 q^{25} +28.7854 q^{26} -2692.74 q^{28} -7147.31i q^{29} -4026.24i q^{31} +3059.09i q^{32} -966.224 q^{34} +7532.28i q^{35} +(-7808.85 + 2892.37i) q^{37} +2867.14 q^{38} +5672.03 q^{40} -11398.2 q^{41} +8183.21i q^{43} -4745.08 q^{44} -1392.95 q^{46} +13313.5 q^{47} -9219.33 q^{49} -4537.78i q^{50} -853.464i q^{52} -27617.4 q^{53} +13273.2i q^{55} -5713.75i q^{56} +7451.94 q^{58} +16008.1i q^{59} +44974.5i q^{61} +4197.84 q^{62} +26276.9 q^{64} -2387.35 q^{65} -45824.8 q^{67} +28647.8i q^{68} -7853.31 q^{70} +18749.2 q^{71} -6614.81 q^{73} +(-3015.65 - 8141.68i) q^{74} -85008.6i q^{76} +13370.8 q^{77} -69916.6i q^{79} -79624.8i q^{80} -11884.0i q^{82} -81375.6 q^{83} +80135.1 q^{85} -8531.99 q^{86} -10068.6i q^{88} -120861. i q^{89} +2404.91i q^{91} +41300.0i q^{92} +13881.0i q^{94} -237791. q^{95} -5970.34i q^{97} -9612.27i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 268 q^{4} + 190 q^{7}+O(q^{10}) $ 16 * q - 268 * q^4 + 190 * q^7	$ 16 q - 268 q^{4} + 190 q^{7} - 74 q^{10} + 1110 q^{11} + 2900 q^{16} - 12052 q^{25} - 4902 q^{26} - 16824 q^{28} + 20556 q^{34} - 11400 q^{37} - 12108 q^{38} + 16966 q^{40} - 3918 q^{41} - 125394 q^{44} + 17470 q^{46} - 3822 q^{47} - 32618 q^{49} + 24126 q^{53} - 164718 q^{58} + 81426 q^{62} + 158076 q^{64} - 98976 q^{65} + 23560 q^{67} - 222404 q^{70} + 50046 q^{71} - 196274 q^{73} - 141216 q^{74} + 239574 q^{77} + 215814 q^{83} - 346472 q^{85} - 197640 q^{86} + 132504 q^{95}+O(q^{100}) $ 16 * q - 268 * q^4 + 190 * q^7 - 74 * q^10 + 1110 * q^11 + 2900 * q^16 - 12052 * q^25 - 4902 * q^26 - 16824 * q^28 + 20556 * q^34 - 11400 * q^37 - 12108 * q^38 + 16966 * q^40 - 3918 * q^41 - 125394 * q^44 + 17470 * q^46 - 3822 * q^47 - 32618 * q^49 + 24126 * q^53 - 164718 * q^58 + 81426 * q^62 + 158076 * q^64 - 98976 * q^65 + 23560 * q^67 - 222404 * q^70 + 50046 * q^71 - 196274 * q^73 - 141216 * q^74 + 239574 * q^77 + 215814 * q^83 - 346472 * q^85 - 197640 * q^86 + 132504 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/333\mathbb{Z}\right)^\times$.

$n$	$38$	$298$
$\chi(n)$	$1$	$-1$

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 $

$2$			1.04262i			0.184311i	0.995745	+	0.0921556i	$0.0293757\pi$
$2$			1.04262i			0.184311i	−0.995745	+	0.0921556i	$0.970624\pi$
$3$		0			0
$3$		0			0
$4$	30.9129			0.966029
$5$		−	86.4713i		−	1.54685i	−0.633891	−	0.773423i	$-0.718543\pi$
$5$		−	86.4713i		−	1.54685i	0.633891	−	0.773423i	$-0.281457\pi$
$6$		0			0
$7$	−87.1072			−0.671907			−0.335954	−	0.941879i	$-0.609059\pi$
$7$	−87.1072			−0.671907			−0.335954	+	0.941879i	$0.609059\pi$
$8$			65.5944i			0.362361i
$9$		0			0
$10$	90.1568			0.285101
$11$	−153.498			−0.382491			−0.191246	−	0.981542i	$-0.561253\pi$
$11$	−153.498			−0.382491			−0.191246	+	0.981542i	$0.561253\pi$
$12$		0			0
$13$		−	27.6086i		−	0.0453092i	−0.999743	−	0.0226546i	$-0.992788\pi$
$13$		−	27.6086i		−	0.0453092i	0.999743	−	0.0226546i	$-0.00721181\pi$
$14$		−	90.8199i		−	0.123840i
$15$		0			0
$16$	920.824			0.899242
$17$			926.725i			0.777730i	0.921295	+	0.388865i	$0.127133\pi$
$17$			926.725i			0.777730i	−0.921295	+	0.388865i	$0.872867\pi$
$18$		0			0
$19$		−	2749.94i		−	1.74759i	−0.486298	−	0.873793i	$-0.661653\pi$
$19$		−	2749.94i		−	1.74759i	0.486298	−	0.873793i	$-0.338347\pi$
$20$		−	2673.08i		−	1.49430i
$21$		0			0
$22$		−	160.041i		−	0.0704974i
$23$			1336.01i			0.526611i	0.964713	+	0.263305i	$0.0848127\pi$
$23$			1336.01i			0.526611i	−0.964713	+	0.263305i	$0.915187\pi$
$24$		0			0
$25$	−4352.28			−1.39273
$26$	28.7854			0.00835100
$27$		0			0
$28$	−2692.74			−0.649082
$29$		−	7147.31i		−	1.57815i	−0.614299	−	0.789073i	$-0.710561\pi$
$29$		−	7147.31i		−	1.57815i	0.614299	−	0.789073i	$-0.289439\pi$
$30$		0			0
$31$		−	4026.24i		−	0.752481i	−0.926522	−	0.376240i	$-0.877217\pi$
$31$		−	4026.24i		−	0.752481i	0.926522	−	0.376240i	$-0.122783\pi$
$32$			3059.09i			0.528102i
$33$		0			0
$34$	−966.224			−0.143344
$35$			7532.28i			1.03934i
$36$		0			0
$37$	−7808.85	+	2892.37i	−0.937741	+	0.347336i
$37$	−7808.85	+	2892.37i	−0.937741	+	0.347336i
$38$	2867.14			0.322100
$39$		0			0
$40$	5672.03			0.560517
$41$	−11398.2			−1.05896			−0.529478	−	0.848324i	$-0.677612\pi$
$41$	−11398.2			−1.05896			−0.529478	+	0.848324i	$0.677612\pi$
$42$		0			0
$43$			8183.21i			0.674920i	0.941340	+	0.337460i	$0.109568\pi$
$43$			8183.21i			0.674920i	−0.941340	+	0.337460i	$0.890432\pi$
$44$	−4745.08			−0.369498
$45$		0			0
$46$	−1392.95			−0.0970603
$47$	13313.5			0.879122			0.439561	−	0.898213i	$-0.355134\pi$
$47$	13313.5			0.879122			0.439561	+	0.898213i	$0.355134\pi$
$48$		0			0
$49$	−9219.33			−0.548541
$50$		−	4537.78i		−	0.256696i
$51$		0			0
$52$		−	853.464i		−	0.0437701i
$53$	−27617.4			−1.35050			−0.675248	−	0.737591i	$-0.735963\pi$
$53$	−27617.4			−1.35050			−0.675248	+	0.737591i	$0.735963\pi$
$54$		0			0
$55$			13273.2i			0.591655i
$56$		−	5713.75i		−	0.243473i
$57$		0			0
$58$	7451.94			0.290870
$59$			16008.1i			0.598700i	0.954143	+	0.299350i	$0.0967699\pi$
$59$			16008.1i			0.598700i	−0.954143	+	0.299350i	$0.903230\pi$
$60$		0			0
$61$			44974.5i			1.54754i	0.633466	+	0.773770i	$0.281631\pi$
$61$			44974.5i			1.54754i	−0.633466	+	0.773770i	$0.718369\pi$
$62$	4197.84			0.138691
$63$		0			0
$64$	26276.9			0.801907
$65$	−2387.35			−0.0700864
$66$		0			0
$67$	−45824.8			−1.24713			−0.623567	−	0.781770i	$-0.714317\pi$
$67$	−45824.8			−1.24713			−0.623567	+	0.781770i	$0.714317\pi$
$68$			28647.8i			0.751310i
$69$		0			0
$70$	−7853.31			−0.191561
$71$	18749.2			0.441404			0.220702	−	0.975341i	$-0.429165\pi$
$71$	18749.2			0.441404			0.220702	+	0.975341i	$0.429165\pi$
$72$		0			0
$73$	−6614.81			−0.145281			−0.0726407	−	0.997358i	$-0.523143\pi$
$73$	−6614.81			−0.145281			−0.0726407	+	0.997358i	$0.523143\pi$
$74$	−3015.65	−	8141.68i	−0.0640179	−	0.172836i
$75$		0			0
$76$		−	85008.6i		−	1.68822i
$77$	13370.8			0.256999
$78$		0			0
$79$		−	69916.6i		−	1.26041i	−0.776428	−	0.630206i	$-0.782970\pi$
$79$		−	69916.6i		−	1.26041i	0.776428	−	0.630206i	$-0.217030\pi$
$80$		−	79624.8i		−	1.39099i
$81$		0			0
$82$		−	11884.0i		−	0.195177i
$83$	−81375.6			−1.29658			−0.648289	−	0.761394i	$-0.724515\pi$
$83$	−81375.6			−1.29658			−0.648289	+	0.761394i	$0.724515\pi$
$84$		0			0
$85$	80135.1			1.20303
$86$	−8531.99			−0.124395
$87$		0			0
$88$		−	10068.6i		−	0.138600i
$89$		−	120861.i		−	1.61738i	−0.588235	−	0.808690i	$-0.700177\pi$
$89$		−	120861.i		−	1.61738i	0.588235	−	0.808690i	$-0.299823\pi$
$90$		0			0
$91$			2404.91i			0.0304436i
$92$			41300.0i			0.508722i
$93$		0			0
$94$			13881.0i			0.162032i
$95$	−237791.			−2.70325
$96$		0			0
$97$		−	5970.34i		−	0.0644273i	−0.999481	−	0.0322137i	$-0.989744\pi$
$97$		−	5970.34i		−	0.0644273i	0.999481	−	0.0322137i	$-0.0102557\pi$
$98$		−	9612.27i		−	0.101102i
$99$		0			0
$100$	−134542.			−1.34542
$101$	−102083.			−0.995754			−0.497877	−	0.867248i	$-0.665887\pi$
$101$	−102083.			−0.995754			−0.497877	+	0.867248i	$0.665887\pi$
$102$		0			0
$103$			193144.i			1.79386i	0.442172	+	0.896930i	$0.354208\pi$
$103$			193144.i			1.79386i	−0.442172	+	0.896930i	$0.645792\pi$
$104$	1810.97			0.0164183
$105$		0			0
$106$		−	28794.5i		−	0.248911i
$107$	−120881.			−1.02070			−0.510351	−	0.859966i	$-0.670485\pi$
$107$	−120881.			−1.02070			−0.510351	+	0.859966i	$0.670485\pi$
$108$		0			0
$109$		−	153803.i		−	1.23993i	−0.784629	−	0.619966i	$-0.787147\pi$
$109$		−	153803.i		−	1.23993i	0.784629	−	0.619966i	$-0.212853\pi$
$110$	−13838.9			−0.109049
$111$		0			0
$112$	−80210.4			−0.604207
$113$		−	19713.0i		−	0.145230i	−0.997360	−	0.0726150i	$-0.976866\pi$
$113$		−	19713.0i		−	0.145230i	0.997360	−	0.0726150i	$-0.0231344\pi$
$114$		0			0
$115$	115526.			0.814586
$116$		−	220944.i		−	1.52454i
$117$		0			0
$118$	−16690.4			−0.110347
$119$		−	80724.5i		−	0.522562i
$120$		0			0
$121$	−137489.			−0.853700
$122$	−46891.4			−0.285229
$123$		0			0
$124$		−	124463.i		−	0.726918i
$125$			106125.i			0.607493i
$126$		0			0
$127$	−41588.5			−0.228804			−0.114402	−	0.993435i	$-0.536495\pi$
$127$	−41588.5			−0.228804			−0.114402	+	0.993435i	$0.536495\pi$
$128$			125288.i			0.675902i
$129$		0			0
$130$		−	2489.11i		−	0.0129177i
$131$			167404.i			0.852289i	0.904655	+	0.426145i	$0.140129\pi$
$131$			167404.i			0.852289i	−0.904655	+	0.426145i	$0.859871\pi$
$132$		0			0
$133$			239539.i			1.17422i
$134$		−	47777.9i		−	0.229861i
$135$		0			0
$136$	−60788.0			−0.281819
$137$	326960.			1.48831			0.744155	−	0.668007i	$-0.232852\pi$
$137$	326960.			1.48831			0.744155	+	0.668007i	$0.232852\pi$
$138$		0			0
$139$	62658.9			0.275071			0.137536	−	0.990497i	$-0.456082\pi$
$139$	62658.9			0.275071			0.137536	+	0.990497i	$0.456082\pi$
$140$			232845.i			1.00403i
$141$		0			0
$142$			19548.3i			0.0813557i
$143$			4237.88i			0.0173304i
$144$		0			0
$145$	−618037.			−2.44115
$146$		−	6896.74i		−	0.0267770i
$147$		0			0
$148$	−241395.	+	89411.8i	−0.905885	+	0.335537i
$149$	65982.0			0.243478			0.121739	−	0.992562i	$-0.461153\pi$
$149$	65982.0			0.243478			0.121739	+	0.992562i	$0.461153\pi$
$150$		0			0
$151$	95460.7			0.340708			0.170354	−	0.985383i	$-0.445509\pi$
$151$	95460.7			0.340708			0.170354	+	0.985383i	$0.445509\pi$
$152$	180380.			0.633258
$153$		0			0
$154$			13940.7i			0.0473677i
$155$	−348154.			−1.16397
$156$		0			0
$157$	290753.			0.941403			0.470702	−	0.882292i	$-0.344001\pi$
$157$	290753.			0.941403			0.470702	+	0.882292i	$0.344001\pi$
$158$	72896.6			0.232308
$159$		0			0
$160$	264524.			0.816892
$161$		−	116376.i		−	0.353834i
$162$		0			0
$163$		−	522500.i		−	1.54034i	−0.637837	−	0.770171i	$-0.720171\pi$
$163$		−	522500.i		−	1.54034i	0.637837	−	0.770171i	$-0.279829\pi$
$164$	−352353.			−1.02298
$165$		0			0
$166$		−	84843.9i		−	0.238974i
$167$		−	320029.i		−	0.887968i	−0.896035	−	0.443984i	$-0.853565\pi$
$167$		−	320029.i		−	0.887968i	0.896035	−	0.443984i	$-0.146435\pi$
$168$		0			0
$169$	370531.			0.997947
$170$			83550.6i			0.221731i
$171$		0			0
$172$			252967.i			0.651993i
$173$	386890.			0.982816			0.491408	−	0.870929i	$-0.336482\pi$
$173$	386890.			0.982816			0.491408	+	0.870929i	$0.336482\pi$
$174$		0			0
$175$	379115.			0.935785
$176$	−141345.			−0.343952
$177$		0			0
$178$	126013.			0.298101
$179$		−	772439.i		−	1.80190i	−0.433918	−	0.900952i	$-0.642869\pi$
$179$		−	772439.i		−	1.80190i	0.433918	−	0.900952i	$-0.357131\pi$
$180$		0			0
$181$	−219471.			−0.497944			−0.248972	−	0.968511i	$-0.580093\pi$
$181$	−219471.			−0.497944			−0.248972	+	0.968511i	$0.580093\pi$
$182$	−2507.41			−0.00561110
$183$		0			0
$184$	−87634.7			−0.190823
$185$	250107.	+	675241.i	0.537275	+	1.45054i
$186$		0			0
$187$		−	142251.i		−	0.297475i
$188$	411561.			0.849257
$189$		0			0
$190$		−	247926.i		−	0.498238i
$191$			119268.i			0.236559i	0.992980	+	0.118279i	$0.0377378\pi$
$191$			119268.i			0.236559i	−0.992980	+	0.118279i	$0.962262\pi$
$192$		0			0
$193$		−	606131.i		−	1.17131i	−0.810559	−	0.585657i	$-0.800836\pi$
$193$		−	606131.i		−	1.17131i	0.810559	−	0.585657i	$-0.199164\pi$
$194$	6224.81			0.0118747
$195$		0			0
$196$	−284997.			−0.529907
$197$	48998.3			0.0899529			0.0449765	−	0.998988i	$-0.485679\pi$
$197$	48998.3			0.0899529			0.0449765	+	0.998988i	$0.485679\pi$
$198$		0			0
$199$			401960.i			0.719532i	0.933043	+	0.359766i	$0.117143\pi$
$199$			401960.i			0.719532i	−0.933043	+	0.359766i	$0.882857\pi$
$200$		−	285485.i		−	0.504671i
$201$		0			0
$202$		−	106434.i		−	0.183529i
$203$			622582.i			1.06037i
$204$		0			0
$205$			985619.i			1.63804i
$206$	−201376.			−0.330629
$207$		0			0
$208$		−	25422.7i		−	0.0407440i
$209$			422110.i			0.668436i
$210$		0			0
$211$	981075.			1.51704			0.758518	−	0.651652i	$-0.225924\pi$
$211$	981075.			1.51704			0.758518	+	0.651652i	$0.225924\pi$
$212$	−853735.			−1.30462
$213$		0			0
$214$		−	126033.i		−	0.188127i
$215$	707613.			1.04400
$216$		0			0
$217$			350715.i			0.505597i
$218$	160358.			0.228533
$219$		0			0
$220$			410313.i			0.571556i
$221$	25585.6			0.0352383
$222$		0			0
$223$	−988479.			−1.33108			−0.665542	−	0.746361i	$-0.731799\pi$
$223$	−988479.			−1.33108			−0.665542	+	0.746361i	$0.731799\pi$
$224$		−	266469.i		−	0.354835i
$225$		0			0
$226$	20553.2			0.0267675
$227$		−	1.06424e6i		−	1.37080i	−0.728168	−	0.685398i	$-0.759628\pi$
$227$		−	1.06424e6i		−	1.37080i	0.728168	−	0.685398i	$-0.240372\pi$
$228$		0			0
$229$	437725.			0.551586			0.275793	−	0.961217i	$-0.411060\pi$
$229$	437725.			0.551586			0.275793	+	0.961217i	$0.411060\pi$
$230$			120450.i			0.150137i
$231$		0			0
$232$	468823.			0.571859
$233$	1.01255e6			1.22188			0.610940	−	0.791677i	$-0.290792\pi$
$233$	1.01255e6			1.22188			0.610940	+	0.791677i	$0.290792\pi$
$234$		0			0
$235$		−	1.15124e6i		−	1.35987i
$236$			494857.i			0.578362i
$237$		0			0
$238$	84165.1			0.0963140
$239$			286370.i			0.324290i	0.986767	+	0.162145i	$0.0518412\pi$
$239$			286370.i			0.324290i	−0.986767	+	0.162145i	$0.948159\pi$
$240$		0			0
$241$			372657.i			0.413302i	0.978415	+	0.206651i	$0.0662564\pi$
$241$			372657.i			0.413302i	−0.978415	+	0.206651i	$0.933744\pi$
$242$		−	143349.i		−	0.157347i
$243$		0			0
$244$			1.39029e6i			1.49497i
$245$			797207.i			0.848508i
$246$		0			0
$247$	−75922.0			−0.0791818
$248$	264099.			0.272670
$249$		0			0
$250$	−110648.			−0.111968
$251$		−	807096.i		−	0.808613i	−0.914624	−	0.404306i	$-0.867513\pi$
$251$		−	807096.i		−	0.808613i	0.914624	−	0.404306i	$-0.132487\pi$
$252$		0			0
$253$		−	205075.i		−	0.201424i
$254$		−	43361.1i		−	0.0421712i
$255$		0			0
$256$	710233.			0.677331
$257$			1.29694e6i			1.22486i	0.790524	+	0.612432i	$0.209809\pi$
$257$			1.29694e6i			1.22486i	−0.790524	+	0.612432i	$0.790191\pi$
$258$		0			0
$259$	680207.	−	251947.i	0.630075	−	0.233378i
$260$	−73800.2			−0.0677055
$261$		0			0
$262$	−174539.			−0.157086
$263$	358188.			0.319317			0.159658	−	0.987172i	$-0.448961\pi$
$263$	358188.			0.319317			0.159658	+	0.987172i	$0.448961\pi$
$264$		0			0
$265$			2.38811e6i			2.08901i
$266$	−249749.			−0.216421
$267$		0			0
$268$	−1.41658e6			−1.20477
$269$	−2.26507e6			−1.90853			−0.954267	−	0.298955i	$-0.903362\pi$
$269$	−2.26507e6			−1.90853			−0.954267	+	0.298955i	$0.903362\pi$
$270$		0			0
$271$	1.06501e6			0.880910			0.440455	−	0.897775i	$-0.354817\pi$
$271$	1.06501e6			0.880910			0.440455	+	0.897775i	$0.354817\pi$
$272$			853351.i			0.699367i
$273$		0			0
$274$			340896.i			0.274312i
$275$	668067.			0.532707
$276$		0			0
$277$		−	1.07621e6i		−	0.842751i	−0.906886	−	0.421375i	$-0.861547\pi$
$277$		−	1.07621e6i		−	0.842751i	0.906886	−	0.421375i	$-0.138453\pi$
$278$			65329.5i			0.0506988i
$279$		0			0
$280$	−494075.			−0.376615
$281$			540316.i			0.408208i	0.978949	+	0.204104i	$0.0654281\pi$
$281$			540316.i			0.408208i	−0.978949	+	0.204104i	$0.934572\pi$
$282$		0			0
$283$		−	669160.i		−	0.496665i	−0.968675	−	0.248333i	$-0.920117\pi$
$283$		−	669160.i		−	0.496665i	0.968675	−	0.248333i	$-0.0798826\pi$
$284$	579592.			0.426409
$285$		0			0
$286$	−4418.50			−0.00319418
$287$	992868.			0.711520
$288$		0			0
$289$	561038.			0.395137
$290$		−	644378.i		−	0.449931i
$291$		0			0
$292$	−204483.			−0.140346
$293$	2.14251e6			1.45799			0.728994	−	0.684521i	$-0.239988\pi$
$293$	2.14251e6			1.45799			0.728994	+	0.684521i	$0.239988\pi$
$294$		0			0
$295$	1.38424e6			0.926097
$296$	−189723.	−	512217.i	−0.125861	−	0.339801i
$297$		0			0
$298$			68794.2i			0.0448757i
$299$	36885.4			0.0238603
$300$		0			0
$301$		−	712817.i		−	0.453484i
$302$			99529.4i			0.0627963i
$303$		0			0
$304$		−	2.53221e6i		−	1.57150i
$305$	3.88900e6			2.39381
$306$		0			0
$307$	−1.35256e6			−0.819047			−0.409524	−	0.912299i	$-0.634305\pi$
$307$	−1.35256e6			−0.819047			−0.409524	+	0.912299i	$0.634305\pi$
$308$	413331.			0.248268
$309$		0			0
$310$		−	362993.i		−	0.214533i
$311$		−	371845.i		−	0.218002i	−0.994042	−	0.109001i	$-0.965235\pi$
$311$		−	371845.i		−	0.218002i	0.994042	−	0.109001i	$-0.0347652\pi$
$312$		0			0
$313$			1.27822e6i			0.737470i	0.929535	+	0.368735i	$0.120209\pi$
$313$			1.27822e6i			0.737470i	−0.929535	+	0.368735i	$0.879791\pi$
$314$			303146.i			0.173511i
$315$		0			0
$316$		−	2.16133e6i		−	1.21760i
$317$	888017.			0.496333			0.248166	−	0.968717i	$-0.420172\pi$
$317$	888017.			0.496333			0.248166	+	0.968717i	$0.420172\pi$
$318$		0			0
$319$			1.09710e6i			0.603627i
$320$		−	2.27220e6i		−	1.24043i
$321$		0			0
$322$	121336.			0.0652155
$323$	2.54843e6			1.35915
$324$		0			0
$325$			120161.i			0.0631035i
$326$	544770.			0.283902
$327$		0			0
$328$		−	747660.i		−	0.383724i
$329$	−1.15971e6			−0.590688
$330$		0			0
$331$			1.13045e6i			0.567129i	0.958953	+	0.283565i	$0.0915171\pi$
$331$			1.13045e6i			0.567129i	−0.958953	+	0.283565i	$0.908483\pi$
$332$	−2.51556e6			−1.25253
$333$		0			0
$334$	333669.			0.163663
$335$			3.96253e6i			1.92912i
$336$		0			0
$337$	−2.75184e6			−1.31992			−0.659961	−	0.751300i	$-0.729427\pi$
$337$	−2.75184e6			−1.31992			−0.659961	+	0.751300i	$0.729427\pi$
$338$			386323.i			0.183933i
$339$		0			0
$340$	2.47721e6			1.16216
$341$			618020.i			0.287817i
$342$		0			0
$343$	2.26708e6			1.04048
$344$	−536773.			−0.244565
$345$		0			0
$346$			403380.i			0.181144i
$347$		−	883744.i		−	0.394006i	−0.980403	−	0.197003i	$-0.936879\pi$
$347$		−	883744.i		−	0.394006i	0.980403	−	0.197003i	$-0.0631208\pi$
$348$		0			0
$349$	−55889.7			−0.0245623			−0.0122811	−	0.999925i	$-0.503909\pi$
$349$	−55889.7			−0.0245623			−0.0122811	+	0.999925i	$0.503909\pi$
$350$			395274.i			0.172476i
$351$		0			0
$352$		−	469565.i		−	0.201994i
$353$			388952.i			0.166134i	0.996544	+	0.0830672i	$0.0264716\pi$
$353$			388952.i			0.166134i	−0.996544	+	0.0830672i	$0.973528\pi$
$354$		0			0
$355$		−	1.62127e6i		−	0.682784i
$356$		−	3.73618e6i		−	1.56244i
$357$		0			0
$358$	805362.			0.332111
$359$	−3.13551e6			−1.28402			−0.642010	−	0.766697i	$-0.721899\pi$
$359$	−3.13551e6			−1.28402			−0.642010	+	0.766697i	$0.721899\pi$
$360$		0			0
$361$	−5.08605e6			−2.05406
$362$		−	228825.i		−	0.0917767i
$363$		0			0
$364$			74342.9i			0.0294094i
$365$			571991.i			0.224728i
$366$		0			0
$367$	−160181.			−0.0620793			−0.0310396	−	0.999518i	$-0.509882\pi$
$367$	−160181.			−0.0620793			−0.0310396	+	0.999518i	$0.509882\pi$
$368$			1.23023e6i			0.473551i
$369$		0			0
$370$	−704021.	+	260767.i	−0.267351	+	0.0990259i
$371$	2.40568e6			0.907407
$372$		0			0
$373$	3.26910e6			1.21662			0.608311	−	0.793699i	$-0.291847\pi$
$373$	3.26910e6			1.21662			0.608311	+	0.793699i	$0.291847\pi$
$374$	148314.			0.0548279
$375$		0			0
$376$			873294.i			0.318560i
$377$	−197327.			−0.0715046
$378$		0			0
$379$	−756174.			−0.270411			−0.135205	−	0.990818i	$-0.543169\pi$
$379$	−756174.			−0.270411			−0.135205	+	0.990818i	$0.543169\pi$
$380$	−7.35080e6			−2.61141
$381$		0			0
$382$	−124351.			−0.0436004
$383$		−	3.40102e6i		−	1.18471i	−0.805677	−	0.592355i	$-0.798198\pi$
$383$		−	3.40102e6i		−	1.18471i	0.805677	−	0.592355i	$-0.201802\pi$
$384$		0			0
$385$		−	1.15619e6i		−	0.397537i
$386$	631966.			0.215886
$387$		0			0
$388$		−	184561.i		−	0.0622387i
$389$		−	3.01289e6i		−	1.00951i	−0.863263	−	0.504754i	$-0.831583\pi$
$389$		−	3.01289e6i		−	1.00951i	0.863263	−	0.504754i	$-0.168417\pi$
$390$		0			0
$391$	−1.23811e6			−0.409561
$392$		−	604736.i		−	0.198770i
$393$		0			0
$394$			51086.7i			0.0165793i
$395$	−6.04578e6			−1.94966
$396$		0			0
$397$	2.29374e6			0.730413			0.365207	−	0.930926i	$-0.380998\pi$
$397$	2.29374e6			0.730413			0.365207	+	0.930926i	$0.380998\pi$
$398$	−419092.			−0.132618
$399$		0			0
$400$	−4.00769e6			−1.25240
$401$			3.73275e6i			1.15922i	0.814892	+	0.579612i	$0.196796\pi$
$401$			3.73275e6i			1.15922i	−0.814892	+	0.579612i	$0.803204\pi$
$402$		0			0
$403$	−111159.			−0.0340943
$404$	−3.15570e6			−0.961927
$405$		0			0
$406$	−649118.			−0.195438
$407$	1.19864e6	−	443974.i	0.358678	−	0.132853i
$408$		0			0
$409$			4.34834e6i			1.28533i	0.766146	+	0.642667i	$0.222172\pi$
$409$			4.34834e6i			1.28533i	−0.766146	+	0.642667i	$0.777828\pi$
$410$	−1.02763e6			−0.301909
$411$		0			0
$412$			5.97066e6i			1.73292i
$413$		−	1.39442e6i		−	0.402271i
$414$		0			0
$415$			7.03665e6i			2.00561i
$416$	84457.4			0.0239279
$417$		0			0
$418$	−440101.			−0.123200
$419$	−3.92059e6			−1.09098			−0.545490	−	0.838117i	$-0.683656\pi$
$419$	−3.92059e6			−1.09098			−0.545490	+	0.838117i	$0.683656\pi$
$420$		0			0
$421$		−	252803.i		−	0.0695146i	−0.999396	−	0.0347573i	$-0.988934\pi$
$421$		−	252803.i		−	0.0695146i	0.999396	−	0.0347573i	$-0.0110658\pi$
$422$			1.02289e6i			0.279607i
$423$		0			0
$424$		−	1.81155e6i		−	0.489367i
$425$		−	4.03337e6i		−	1.08317i
$426$		0			0
$427$		−	3.91761e6i		−	1.03980i
$428$	−3.73679e6			−0.986029
$429$		0			0
$430$			737772.i			0.192420i
$431$			5.98928e6i			1.55304i	0.630095	+	0.776518i	$0.283016\pi$
$431$			5.98928e6i			1.55304i	−0.630095	+	0.776518i	$0.716984\pi$
$432$		0			0
$433$	2.64435e6			0.677795			0.338898	−	0.940823i	$-0.389946\pi$
$433$	2.64435e6			0.677795			0.338898	+	0.940823i	$0.389946\pi$
$434$	−365663.			−0.0931872
$435$		0			0
$436$		−	4.75449e6i		−	1.19781i
$437$	3.67394e6			0.920298
$438$		0			0
$439$		−	6.50479e6i		−	1.61091i	−0.592655	−	0.805457i	$-0.701920\pi$
$439$		−	6.50479e6i		−	1.61091i	0.592655	−	0.805457i	$-0.298080\pi$
$440$	−870646.			−0.214393
$441$		0			0
$442$			26676.1i			0.00649482i
$443$	−1.83180e6			−0.443475			−0.221738	−	0.975106i	$-0.571173\pi$
$443$	−1.83180e6			−0.443475			−0.221738	+	0.975106i	$0.571173\pi$
$444$		0			0
$445$	−1.04510e7			−2.50184
$446$		−	1.03061e6i		−	0.245334i
$447$		0			0
$448$	−2.28891e6			−0.538807
$449$			3.59888e6i			0.842465i	0.906953	+	0.421233i	$0.138402\pi$
$449$			3.59888e6i			0.842465i	−0.906953	+	0.421233i	$0.861598\pi$
$450$		0			0
$451$	1.74961e6			0.405041
$452$		−	609386.i		−	0.140296i
$453$		0			0
$454$	1.10959e6			0.252653
$455$	207956.			0.0470915
$456$		0			0
$457$		−	2.96825e6i		−	0.664829i	−0.943133	−	0.332415i	$-0.892137\pi$
$457$		−	2.96825e6i		−	0.664829i	0.943133	−	0.332415i	$-0.107863\pi$
$458$			456382.i			0.101663i
$459$		0			0
$460$	3.57126e6			0.786914
$461$		−	271586.i		−	0.0595190i	−0.999557	−	0.0297595i	$-0.990526\pi$
$461$		−	271586.i		−	0.0595190i	0.999557	−	0.0297595i	$-0.00947415\pi$
$462$		0			0
$463$		−	5.47928e6i		−	1.18788i	−0.804511	−	0.593938i	$-0.797573\pi$
$463$		−	5.47928e6i		−	1.18788i	0.804511	−	0.593938i	$-0.202427\pi$
$464$		−	6.58141e6i		−	1.41914i
$465$		0			0
$466$			1.05571e6i			0.225206i
$467$			361304.i			0.0766621i	0.999265	+	0.0383311i	$0.0122041\pi$
$467$			361304.i			0.0766621i	−0.999265	+	0.0383311i	$0.987796\pi$
$468$		0			0
$469$	3.99167e6			0.837959
$470$	1.20031e6			0.250638
$471$		0			0
$472$	−1.05004e6			−0.216946
$473$		−	1.25611e6i		−	0.258151i
$474$		0			0
$475$			1.19685e7i			2.43392i
$476$		−	2.49543e6i		−	0.504810i
$477$		0			0
$478$	−298576.			−0.0597703
$479$		−	6.39822e6i		−	1.27415i	−0.770802	−	0.637074i	$-0.780144\pi$
$479$		−	6.39822e6i		−	1.27415i	0.770802	−	0.637074i	$-0.219856\pi$
$480$		0			0
$481$	79854.5	+	215592.i	0.0157375	+	0.0424883i
$482$	−388541.			−0.0761761
$483$		0			0
$484$	−4.25020e6			−0.824700
$485$	−516263.			−0.0996591
$486$		0			0
$487$		−	5.35237e6i		−	1.02264i	−0.859390	−	0.511321i	$-0.829156\pi$
$487$		−	5.35237e6i		−	1.02264i	0.859390	−	0.511321i	$-0.170844\pi$
$488$	−2.95008e6			−0.560769
$489$		0			0
$490$	−831185.			−0.156390
$491$	6.11966e6			1.14557			0.572787	−	0.819704i	$-0.305862\pi$
$491$	6.11966e6			1.14557			0.572787	+	0.819704i	$0.305862\pi$
$492$		0			0
$493$	6.62359e6			1.22737
$494$		−	79157.9i		−	0.0145941i
$495$		0			0
$496$		−	3.70746e6i		−	0.676662i
$497$	−1.63319e6			−0.296582
$498$		0			0
$499$		−	9.16295e6i		−	1.64734i	−0.567067	−	0.823672i	$-0.691922\pi$
$499$		−	9.16295e6i		−	1.64734i	0.567067	−	0.823672i	$-0.308078\pi$
$500$			3.28063e6i			0.586856i
$501$		0			0
$502$	841495.			0.149036
$503$			434586.i			0.0765871i	0.999267	+	0.0382935i	$0.0121922\pi$
$503$			434586.i			0.0765871i	−0.999267	+	0.0382935i	$0.987808\pi$
$504$		0			0
$505$			8.82729e6i			1.54028i
$506$	213816.			0.0371247
$507$		0			0
$508$	−1.28562e6			−0.221032
$509$	−770690.			−0.131852			−0.0659258	−	0.997825i	$-0.521000\pi$
$509$	−770690.			−0.131852			−0.0659258	+	0.997825i	$0.521000\pi$
$510$		0			0
$511$	576198.			0.0976156
$512$			4.74971e6i			0.800742i
$513$		0			0
$514$	−1.35222e6			−0.225756
$515$	1.67014e7			2.77482
$516$		0			0
$517$	−2.04360e6			−0.336256
$518$	262685.	+	709199.i	0.0430141	+	0.116130i
$519$		0			0
$520$		−	156597.i		−	0.0253966i
$521$	−401386.			−0.0647841			−0.0323920	−	0.999475i	$-0.510313\pi$
$521$	−401386.			−0.0647841			−0.0323920	+	0.999475i	$0.510313\pi$
$522$		0			0
$523$		−	2.01101e6i		−	0.321485i	−0.986996	−	0.160743i	$-0.948611\pi$
$523$		−	2.01101e6i		−	0.321485i	0.986996	−	0.160743i	$-0.0513889\pi$
$524$			5.17494e6i			0.823336i
$525$		0			0
$526$			373455.i			0.0588537i
$527$	3.73122e6			0.585226
$528$		0			0
$529$	4.65142e6			0.722681
$530$	−2.48990e6			−0.385027
$531$		0			0
$532$			7.40487e6i			1.13433i
$533$			314690.i			0.0479805i
$534$		0			0
$535$			1.04528e7i			1.57887i
$536$		−	3.00585e6i		−	0.451913i
$537$		0			0
$538$		−	2.36161e6i		−	0.351764i
$539$	1.41515e6			0.209812
$540$		0			0
$541$		−	1.06718e7i		−	1.56764i	−0.620990	−	0.783818i	$-0.713269\pi$
$541$		−	1.06718e7i		−	1.56764i	0.620990	−	0.783818i	$-0.286731\pi$
$542$			1.11041e6i			0.162362i
$543$		0			0
$544$	−2.83494e6			−0.410720
$545$	−1.32995e7			−1.91798
$546$		0			0
$547$		−	8.72544e6i		−	1.24686i	−0.781878	−	0.623432i	$-0.785738\pi$
$547$		−	8.72544e6i		−	1.24686i	0.781878	−	0.623432i	$-0.214262\pi$
$548$	1.01073e7			1.43775
$549$		0			0
$550$			696541.i			0.0981839i
$551$	−1.96546e7			−2.75795
$552$		0			0
$553$			6.09024e6i			0.846880i
$554$	1.12208e6			0.155328
$555$		0			0
$556$	1.93697e6			0.265727
$557$			258227.i			0.0352667i	0.999845	+	0.0176333i	$0.00561316\pi$
$557$			258227.i			0.0352667i	−0.999845	+	0.0176333i	$0.994387\pi$
$558$		0			0
$559$	225927.			0.0305801
$560$			6.93590e6i			0.934615i
$561$		0			0
$562$	−563345.			−0.0752374
$563$			1.26414e6i			0.168083i	0.996462	+	0.0840413i	$0.0267828\pi$
$563$			1.26414e6i			0.168083i	−0.996462	+	0.0840413i	$0.973217\pi$
$564$		0			0
$565$	−1.70461e6			−0.224648
$566$	697681.			0.0915410
$567$		0			0
$568$			1.22984e6i			0.159948i
$569$		−	2.30163e6i		−	0.298027i	−0.988835	−	0.149014i	$-0.952390\pi$
$569$		−	2.30163e6i		−	0.298027i	0.988835	−	0.149014i	$-0.0476098\pi$
$570$		0			0
$571$	1.11302e6			0.142861			0.0714305	−	0.997446i	$-0.477244\pi$
$571$	1.11302e6			0.142861			0.0714305	+	0.997446i	$0.477244\pi$
$572$			131005.i			0.0167417i
$573$		0			0
$574$			1.03519e6i			0.131141i
$575$		−	5.81469e6i		−	0.733427i
$576$		0			0
$577$			5.49305e6i			0.686869i	0.939177	+	0.343434i	$0.111590\pi$
$577$			5.49305e6i			0.686869i	−0.939177	+	0.343434i	$0.888410\pi$
$578$			584950.i			0.0728281i
$579$		0			0
$580$	−1.91053e7			−2.35822
$581$	7.08840e6			0.871180
$582$		0			0
$583$	4.23922e6			0.516553
$584$		−	433894.i		−	0.0526443i
$585$		0			0
$586$			2.23383e6i			0.268723i
$587$		−	4.66139e6i		−	0.558367i	−0.960238	−	0.279184i	$-0.909936\pi$
$587$		−	4.66139e6i		−	0.558367i	0.960238	−	0.279184i	$-0.0900638\pi$
$588$		0			0
$589$	−1.10719e7			−1.31502
$590$			1.44324e6i			0.170690i
$591$		0			0
$592$	−7.19058e6	+	2.66337e6i	−0.843256	+	0.312339i
$593$	8.26345e6			0.964994			0.482497	−	0.875898i	$-0.339730\pi$
$593$	8.26345e6			0.964994			0.482497	+	0.875898i	$0.339730\pi$
$594$		0			0
$595$	−6.98035e6			−0.808323
$596$	2.03970e6			0.235207
$597$		0			0
$598$			38457.5i			0.00439773i
$599$	3.84507e6			0.437862			0.218931	−	0.975740i	$-0.429743\pi$
$599$	3.84507e6			0.437862			0.218931	+	0.975740i	$0.429743\pi$
$600$		0			0
$601$	1.18174e7			1.33456			0.667278	−	0.744809i	$-0.267459\pi$
$601$	1.18174e7			1.33456			0.667278	+	0.744809i	$0.267459\pi$
$602$	743198.			0.0835821
$603$		0			0
$604$	2.95097e6			0.329134
$605$			1.18889e7i			1.32054i
$606$		0			0
$607$		−	8.13937e6i		−	0.896642i	−0.893873	−	0.448321i	$-0.852022\pi$
$607$		−	8.13937e6i		−	0.896642i	0.893873	−	0.448321i	$-0.147978\pi$
$608$	8.41231e6			0.922903
$609$		0			0
$610$			4.05476e6i			0.441205i
$611$		−	367569.i		−	0.0398323i
$612$		0			0
$613$	6.23345e6			0.670003			0.335002	−	0.942218i	$-0.391263\pi$
$613$	6.23345e6			0.670003			0.335002	+	0.942218i	$0.391263\pi$
$614$		−	1.41020e6i		−	0.150960i
$615$		0			0
$616$			877050.i			0.0931263i
$617$	−1.61232e6			−0.170506			−0.0852530	−	0.996359i	$-0.527170\pi$
$617$	−1.61232e6			−0.170506			−0.0852530	+	0.996359i	$0.527170\pi$
$618$		0			0
$619$	1.33080e7			1.39600			0.697999	−	0.716099i	$-0.254074\pi$
$619$	1.33080e7			1.39600			0.697999	+	0.716099i	$0.254074\pi$
$620$	−1.07625e7			−1.12443
$621$		0			0
$622$	387694.			0.0401803
$623$			1.05279e7i			1.08673i
$624$		0			0
$625$	−4.42415e6			−0.453033
$626$	−1.33270e6			−0.135924
$627$		0			0
$628$	8.98805e6			0.909423
$629$	−2.68043e6	−	7.23666e6i	−0.270134	−	0.729309i
$630$		0			0
$631$			1.16451e7i			1.16431i	0.813078	+	0.582154i	$0.197790\pi$
$631$			1.16451e7i			1.16431i	−0.813078	+	0.582154i	$0.802210\pi$
$632$	4.58614e6			0.456725
$633$		0			0
$634$			925866.i			0.0914797i
$635$			3.59621e6i			0.353925i
$636$		0			0
$637$			254533.i			0.0248540i
$638$	−1.14386e6			−0.111255
$639$		0			0
$640$	1.08338e7			1.04552
$641$	−1.89809e6			−0.182462			−0.0912309	−	0.995830i	$-0.529080\pi$
$641$	−1.89809e6			−0.182462			−0.0912309	+	0.995830i	$0.529080\pi$
$642$		0			0
$643$			3.21597e6i			0.306750i	0.988168	+	0.153375i	$0.0490143\pi$
$643$			3.21597e6i			0.306750i	−0.988168	+	0.153375i	$0.950986\pi$
$644$		−	3.59753e6i		−	0.341814i
$645$		0			0
$646$			2.65705e6i			0.250506i
$647$		−	9.14794e6i		−	0.859137i	−0.903034	−	0.429568i	$-0.858666\pi$
$647$		−	9.14794e6i		−	0.859137i	0.903034	−	0.429568i	$-0.141334\pi$
$648$		0			0
$649$		−	2.45721e6i		−	0.228998i
$650$	−125282.			−0.0116307
$651$		0			0
$652$		−	1.61520e7i		−	1.48802i
$653$			2.31944e6i			0.212863i	0.994320	+	0.106432i	$0.0339425\pi$
$653$			2.31944e6i			0.212863i	−0.994320	+	0.106432i	$0.966058\pi$
$654$		0			0
$655$	1.44756e7			1.31836
$656$	−1.04958e7			−0.952257
$657$		0			0
$658$		−	1.20913e6i		−	0.108870i
$659$	−916434.			−0.0822030			−0.0411015	−	0.999155i	$-0.513087\pi$
$659$	−916434.			−0.0822030			−0.0411015	+	0.999155i	$0.513087\pi$
$660$		0			0
$661$		−	2.37664e6i		−	0.211572i	−0.994389	−	0.105786i	$-0.966264\pi$
$661$		−	2.37664e6i		−	0.211572i	0.994389	−	0.105786i	$-0.0337359\pi$
$662$	−1.17863e6			−0.104528
$663$		0			0
$664$		−	5.33778e6i		−	0.469830i
$665$	2.07133e7			1.81633
$666$		0			0
$667$	9.54887e6			0.831069
$668$		−	9.89302e6i		−	0.857804i
$669$		0			0
$670$	−4.13142e6			−0.355559
$671$		−	6.90351e6i		−	0.591921i
$672$		0			0
$673$	5.10059e6			0.434093			0.217047	−	0.976161i	$-0.430358\pi$
$673$	5.10059e6			0.434093			0.217047	+	0.976161i	$0.430358\pi$
$674$		−	2.86912e6i		−	0.243276i
$675$		0			0
$676$	1.14542e7			0.964046
$677$	1.83738e7			1.54073			0.770365	−	0.637603i	$-0.220074\pi$
$677$	1.83738e7			1.54073			0.770365	+	0.637603i	$0.220074\pi$
$678$		0			0
$679$			520060.i			0.0432892i
$680$			5.25641e6i			0.435930i
$681$		0			0
$682$	−644361.			−0.0530479
$683$			1.34013e7i			1.09925i	0.835413	+	0.549623i	$0.185229\pi$
$683$			1.34013e7i			1.09925i	−0.835413	+	0.549623i	$0.814771\pi$
$684$		0			0
$685$		−	2.82727e7i		−	2.30219i
$686$			2.36371e6i			0.191771i
$687$		0			0
$688$			7.53530e6i			0.606917i
$689$			762479.i			0.0611899i
$690$		0			0
$691$	−5.66245e6			−0.451138			−0.225569	−	0.974227i	$-0.572424\pi$
$691$	−5.66245e6			−0.451138			−0.225569	+	0.974227i	$0.572424\pi$
$692$	1.19599e7			0.949430
$693$		0			0
$694$	921410.			0.0726197
$695$		−	5.41819e6i		−	0.425493i
$696$		0			0
$697$		−	1.05630e7i		−	0.823581i
$698$		−	58271.9i		−	0.00452710i
$699$		0			0
$700$	1.17196e7			0.903996
$701$			4.15435e6i			0.319306i	0.987173	+	0.159653i	$0.0510376\pi$
$701$			4.15435e6i			0.319306i	−0.987173	+	0.159653i	$0.948962\pi$
$702$		0			0
$703$	7.95384e6	+	2.14738e7i	0.607000	+	1.63878i
$704$	−4.03345e6			−0.306722
$705$		0			0
$706$	−405530.			−0.0306204
$707$	8.89221e6			0.669054
$708$		0			0
$709$			2.41883e7i			1.80713i	0.428449	+	0.903566i	$0.359060\pi$
$709$			2.41883e7i			1.80713i	−0.428449	+	0.903566i	$0.640940\pi$
$710$	1.69037e6			0.125845
$711$		0			0
$712$	7.92782e6			0.586076
$713$	5.37909e6			0.396265
$714$		0			0
$715$	366455.			0.0268074
$716$		−	2.38784e7i		−	1.74069i
$717$		0			0
$718$		−	3.26915e6i		−	0.236659i
$719$	1.98486e6			0.143188			0.0715941	−	0.997434i	$-0.477191\pi$
$719$	1.98486e6			0.143188			0.0715941	+	0.997434i	$0.477191\pi$
$720$		0			0
$721$		−	1.68243e7i		−	1.20531i
$722$		−	5.30283e6i		−	0.378586i
$723$		0			0
$724$	−6.78449e6			−0.481029
$725$			3.11071e7i			2.19793i
$726$		0			0
$727$			2.20799e7i			1.54939i	0.632333	+	0.774697i	$0.282098\pi$
$727$			2.20799e7i			1.54939i	−0.632333	+	0.774697i	$0.717902\pi$
$728$	−157749.			−0.0110316
$729$		0			0
$730$	−596370.			−0.0414199
$731$	−7.58359e6			−0.524906
$732$		0			0
$733$	−1.72118e7			−1.18322			−0.591612	−	0.806223i	$-0.701508\pi$
$733$	−1.72118e7			−1.18322			−0.591612	+	0.806223i	$0.701508\pi$
$734$		−	167008.i		−	0.0114419i
$735$		0			0
$736$	−4.08697e6			−0.278104
$737$	7.03402e6			0.477018
$738$		0			0
$739$	−2.51693e7			−1.69535			−0.847675	−	0.530515i	$-0.821998\pi$
$739$	−2.51693e7			−1.69535			−0.847675	+	0.530515i	$0.821998\pi$
$740$	7.73155e6	+	2.08737e7i	0.519024	+	1.40126i
$741$		0			0
$742$			2.50821e6i			0.167245i
$743$	−2.36324e7			−1.57049			−0.785245	−	0.619185i	$-0.787463\pi$
$743$	−2.36324e7			−1.57049			−0.785245	+	0.619185i	$0.787463\pi$
$744$		0			0
$745$		−	5.70555e6i		−	0.376623i
$746$			3.40843e6i			0.224237i
$747$		0			0
$748$		−	4.39738e6i		−	0.287369i
$749$	1.05296e7			0.685817
$750$		0			0
$751$	761070.			0.0492408			0.0246204	−	0.999697i	$-0.492162\pi$
$751$	761070.			0.0492408			0.0246204	+	0.999697i	$0.492162\pi$
$752$	1.22594e7			0.790543
$753$		0			0
$754$		−	205738.i		−	0.0131791i
$755$		−	8.25461e6i		−	0.527022i
$756$		0			0
$757$			2.12851e7i			1.35001i	0.737814	+	0.675004i	$0.235858\pi$
$757$			2.12851e7i			1.35001i	−0.737814	+	0.675004i	$0.764142\pi$
$758$		−	788403.i		−	0.0498397i
$759$		0			0
$760$		−	1.55977e7i		−	0.979551i
$761$	1.99836e7			1.25087			0.625435	−	0.780277i	$-0.284922\pi$
$761$	1.99836e7			1.25087			0.625435	+	0.780277i	$0.284922\pi$
$762$		0			0
$763$			1.33973e7i			0.833118i
$764$			3.68691e6i			0.228523i
$765$		0			0
$766$	3.54597e6			0.218355
$767$	441962.			0.0271267
$768$		0			0
$769$		−	1.94814e7i		−	1.18797i	−0.804477	−	0.593984i	$-0.797554\pi$
$769$		−	1.94814e7i		−	1.18797i	0.804477	−	0.593984i	$-0.202446\pi$
$770$	1.20547e6			0.0732705
$771$		0			0
$772$		−	1.87373e7i		−	1.13152i
$773$	−1.13110e7			−0.680849			−0.340425	−	0.940272i	$-0.610571\pi$
$773$	−1.13110e7			−0.680849			−0.340425	+	0.940272i	$0.610571\pi$
$774$		0			0
$775$			1.75233e7i			1.04800i
$776$	391621.			0.0233460
$777$		0			0
$778$	3.14131e6			0.186064
$779$			3.13444e7i			1.85062i
$780$		0			0
$781$	−2.87796e6			−0.168833
$782$		−	1.29088e6i		−	0.0754867i
$783$		0			0
$784$	−8.48938e6			−0.493271
$785$		−	2.51418e7i		−	1.45621i
$786$		0			0
$787$	−7.02612e6			−0.404370			−0.202185	−	0.979347i	$-0.564804\pi$
$787$	−7.02612e6			−0.404370			−0.202185	+	0.979347i	$0.564804\pi$
$788$	1.51468e6			0.0868972
$789$		0			0
$790$		−	6.30346e6i		−	0.359345i
$791$			1.71714e6i			0.0975810i
$792$		0			0
$793$	1.24169e6			0.0701179
$794$			2.39151e6i			0.134623i
$795$		0			0
$796$			1.24258e7i			0.695089i
$797$			2.61247e7i			1.45682i	0.685143	+	0.728409i	$0.259740\pi$
$797$			2.61247e7i			1.45682i	−0.685143	+	0.728409i	$0.740260\pi$
$798$		0			0
$799$			1.23380e7i			0.683719i
$800$		−	1.33140e7i		−	0.735503i
$801$		0			0
$802$	−3.89184e6			−0.213658
$803$	1.01536e6			0.0555689
$804$		0			0
$805$	−1.00632e7			−0.547326
$806$		−	115897.i		−	0.00628397i
$807$		0			0
$808$		−	6.69610e6i		−	0.360823i
$809$		−	2.91750e7i		−	1.56725i	−0.621233	−	0.783626i	$-0.713368\pi$
$809$		−	2.91750e7i		−	1.56725i	0.621233	−	0.783626i	$-0.286632\pi$
$810$		0			0
$811$	3.56190e6			0.190164			0.0950822	−	0.995469i	$-0.469689\pi$
$811$	3.56190e6			0.190164			0.0950822	+	0.995469i	$0.469689\pi$
$812$			1.92458e7i			1.02435i
$813$		0			0
$814$	462897.	+	1.24973e6i	0.0244863	+	0.0661083i
$815$	−4.51812e7			−2.38267
$816$		0			0
$817$	2.25033e7			1.17948
$818$	−4.53368e6			−0.236901
$819$		0			0
$820$			3.04684e7i			1.58239i
$821$	1.48511e7			0.768954			0.384477	−	0.923135i	$-0.374382\pi$
$821$	1.48511e7			0.768954			0.384477	+	0.923135i	$0.374382\pi$
$822$		0			0
$823$	4.45575e6			0.229309			0.114654	−	0.993405i	$-0.463424\pi$
$823$	4.45575e6			0.229309			0.114654	+	0.993405i	$0.463424\pi$
$824$	−1.26692e7			−0.650026
$825$		0			0
$826$	1.45385e6			0.0741431
$827$		−	9.37369e6i		−	0.476592i	−0.971193	−	0.238296i	$-0.923411\pi$
$827$		−	9.37369e6i		−	0.476592i	0.971193	−	0.238296i	$-0.0765889\pi$
$828$		0			0
$829$			4.05029e6i			0.204692i	0.994749	+	0.102346i	$0.0326348\pi$
$829$			4.05029e6i			0.204692i	−0.994749	+	0.102346i	$0.967365\pi$
$830$	−7.33656e6			−0.369656
$831$		0			0
$832$		−	725469.i		−	0.0363338i
$833$		−	8.54378e6i		−	0.426617i
$834$		0			0
$835$	−2.76733e7			−1.37355
$836$			1.30487e7i			0.645729i
$837$		0			0
$838$		−	4.08769e6i		−	0.201080i
$839$	−2.21153e7			−1.08465			−0.542323	−	0.840170i	$-0.682455\pi$
$839$	−2.21153e7			−1.08465			−0.542323	+	0.840170i	$0.682455\pi$
$840$		0			0
$841$	−3.05728e7			−1.49055
$842$	263577.			0.0128123
$843$		0			0
$844$	3.03279e7			1.46550
$845$		−	3.20403e7i		−	1.54367i
$846$		0			0
$847$	1.19763e7			0.573607
$848$	−2.54308e7			−1.21442
$849$		0			0
$850$	4.20528e6			0.199640
$851$	−3.86424e6	−	1.04327e7i	−0.182911	−	0.493824i
$852$		0			0
$853$			1.41295e7i			0.664898i	0.943121	+	0.332449i	$0.107875\pi$
$853$			1.41295e7i			0.664898i	−0.943121	+	0.332449i	$0.892125\pi$
$854$	4.08458e6			0.191647
$855$		0			0
$856$		−	7.92913e6i		−	0.369863i
$857$			2.53580e7i			1.17940i	0.807621	+	0.589702i	$0.200755\pi$
$857$			2.53580e7i			1.17940i	−0.807621	+	0.589702i	$0.799245\pi$
$858$		0			0
$859$		−	2.22970e7i		−	1.03101i	−0.856886	−	0.515506i	$-0.827604\pi$
$859$		−	2.22970e7i		−	1.03101i	0.856886	−	0.515506i	$-0.172396\pi$
$860$	2.18744e7			1.00853
$861$		0			0
$862$	−6.24456e6			−0.286242
$863$	125737.			0.00574694			0.00287347	−	0.999996i	$-0.499085\pi$
$863$	125737.			0.00574694			0.00287347	+	0.999996i	$0.499085\pi$
$864$		0			0
$865$		−	3.34549e7i		−	1.52026i
$866$			2.75705e6i			0.124925i
$867$		0			0
$868$			1.08416e7i			0.488422i
$869$			1.07321e7i			0.482097i
$870$		0			0
$871$			1.26516e6i			0.0565067i
$872$	1.00886e7			0.449303
$873$		0			0
$874$			3.83053e6i			0.169621i
$875$		−	9.24423e6i		−	0.408179i
$876$		0			0
$877$	−6.73604e6			−0.295737			−0.147869	−	0.989007i	$-0.547241\pi$
$877$	−6.73604e6			−0.295737			−0.147869	+	0.989007i	$0.547241\pi$
$878$	6.78204e6			0.296909
$879$		0			0
$880$			1.22223e7i			0.532041i
$881$	2.67472e7			1.16102			0.580509	−	0.814254i	$-0.302854\pi$
$881$	2.67472e7			1.16102			0.580509	+	0.814254i	$0.302854\pi$
$882$		0			0
$883$		−	1.30359e7i		−	0.562652i	−0.959612	−	0.281326i	$-0.909226\pi$
$883$		−	1.30359e7i		−	0.562652i	0.959612	−	0.281326i	$-0.0907742\pi$
$884$	790927.			0.0340413
$885$		0			0
$886$		−	1.90988e6i		−	0.0817375i
$887$	−3.76423e7			−1.60645			−0.803225	−	0.595675i	$-0.796885\pi$
$887$	−3.76423e7			−1.60645			−0.803225	+	0.595675i	$0.796885\pi$
$888$		0			0
$889$	3.62266e6			0.153735
$890$		−	1.08965e7i		−	0.461117i
$891$		0			0
$892$	−3.05568e7			−1.28587
$893$		−	3.66114e7i		−	1.53634i
$894$		0			0
$895$	−6.67938e7			−2.78727
$896$		−	1.09135e7i		−	0.454143i
$897$		0			0
$898$	−3.75227e6			−0.155276
$899$	−2.87768e7			−1.18752
$900$		0			0
$901$		−	2.55937e7i		−	1.05032i
$902$			1.82418e6i			0.0746536i
$903$		0			0
$904$	1.29306e6			0.0526257
$905$			1.89779e7i			0.770242i
$906$		0			0
$907$		−	3.29270e7i		−	1.32903i	−0.747275	−	0.664514i	$-0.768638\pi$
$907$		−	3.29270e7i		−	1.32903i	0.747275	−	0.664514i	$-0.231362\pi$
$908$		−	3.28986e7i		−	1.32423i
$909$		0			0
$910$			216819.i			0.00867950i
$911$			2.83567e7i			1.13203i	0.824394	+	0.566017i	$0.191516\pi$
$911$			2.83567e7i			1.13203i	−0.824394	+	0.566017i	$0.808484\pi$
$912$		0			0
$913$	1.24910e7			0.495930
$914$	3.09476e6			0.122535
$915$		0			0
$916$	1.35314e7			0.532848
$917$		−	1.45821e7i		−	0.572659i
$918$		0			0
$919$			4.60773e6i			0.179969i	0.995943	+	0.0899846i	$0.0286818\pi$
$919$			4.60773e6i			0.179969i	−0.995943	+	0.0899846i	$0.971318\pi$
$920$			7.57789e6i			0.295174i
$921$		0			0
$922$	283162.			0.0109700
$923$		−	517639.i		−	0.0199997i
$924$		0			0
$925$	3.39863e7	−	1.25884e7i	1.30602	−	0.483746i
$926$	5.71281e6			0.218939
$927$		0			0
$928$	2.18643e7			0.833422
$929$	−2.22109e7			−0.844360			−0.422180	−	0.906512i	$-0.638735\pi$
$929$	−2.22109e7			−0.844360			−0.422180	+	0.906512i	$0.638735\pi$
$930$		0			0
$931$			2.53526e7i			0.958623i
$932$	3.13010e7			1.18037
$933$		0			0
$934$	−376704.			−0.0141297
$935$	−1.23006e7			−0.460147
$936$		0			0
$937$	−2.95907e7			−1.10105			−0.550523	−	0.834820i	$-0.685572\pi$
$937$	−2.95907e7			−1.10105			−0.550523	+	0.834820i	$0.685572\pi$
$938$			4.16180e6i			0.154445i
$939$		0			0
$940$		−	3.55882e7i		−	1.31367i
$941$	−2.05391e7			−0.756148			−0.378074	−	0.925775i	$-0.623414\pi$
$941$	−2.05391e7			−0.756148			−0.378074	+	0.925775i	$0.623414\pi$
$942$		0			0
$943$		−	1.52281e7i		−	0.557657i
$944$			1.47406e7i			0.538377i
$945$		0			0
$946$	1.30965e6			0.0475801
$947$		−	4.21333e7i		−	1.52669i	−0.645991	−	0.763345i	$-0.723556\pi$
$947$		−	4.21333e7i		−	1.52669i	0.645991	−	0.763345i	$-0.276444\pi$
$948$		0			0
$949$			182626.i			0.00658259i
$950$	−1.24786e7			−0.448598
$951$		0			0
$952$	5.29507e6			0.189356
$953$	−4.48612e7			−1.60007			−0.800034	−	0.599954i	$-0.795185\pi$
$953$	−4.48612e7			−1.60007			−0.800034	+	0.599954i	$0.795185\pi$
$954$		0			0
$955$	1.03132e7			0.365920
$956$			8.85255e6i			0.313274i
$957$		0			0
$958$	6.67092e6			0.234840
$959$	−2.84806e7			−1.00001
$960$		0			0
$961$	1.24185e7			0.433773
$962$	−224781.	+	83258.0i	−0.00783107	+	0.00290060i
$963$		0			0
$964$			1.15199e7i			0.399262i
$965$	−5.24130e7			−1.81184
$966$		0			0
$967$		−	1.77799e7i		−	0.611453i	−0.952119	−	0.305727i	$-0.901101\pi$
$967$		−	1.77799e7i		−	0.611453i	0.952119	−	0.305727i	$-0.0988994\pi$
$968$		−	9.01853e6i		−	0.309348i
$969$		0			0
$970$		−	538267.i		−	0.0183683i
$971$	−1.29448e7			−0.440603			−0.220302	−	0.975432i	$-0.570704\pi$
$971$	−1.29448e7			−0.440603			−0.220302	+	0.975432i	$0.570704\pi$
$972$		0			0
$973$	−5.45804e6			−0.184822
$974$	5.58049e6			0.188484
$975$		0			0
$976$			4.14136e7i			1.39161i
$977$		−	4.56506e7i		−	1.53007i	−0.643990	−	0.765034i	$-0.722722\pi$
$977$		−	4.56506e7i		−	1.53007i	0.643990	−	0.765034i	$-0.277278\pi$
$978$		0			0
$979$			1.85520e7i			0.618634i
$980$			2.46440e7i			0.819684i
$981$		0			0
$982$			6.38049e6i			0.211142i
$983$	5.14288e7			1.69755			0.848775	−	0.528755i	$-0.177341\pi$
$983$	5.14288e7			1.69755			0.848775	+	0.528755i	$0.177341\pi$
$984$		0			0
$985$		−	4.23694e6i		−	0.139143i
$986$			6.90590e6i			0.226218i
$987$		0			0
$988$	−2.34697e6			−0.0764919
$989$	−1.09328e7			−0.355420
$990$		0			0
$991$		−	2.27943e6i		−	0.0737295i	−0.999320	−	0.0368648i	$-0.988263\pi$
$991$		−	2.27943e6i		−	0.0737295i	0.999320	−	0.0368648i	$-0.0117371\pi$
$992$	1.23166e7			0.397386
$993$		0			0
$994$		−	1.70280e6i		−	0.0546635i
$995$	3.47580e7			1.11300
$996$		0			0
$997$			2.09171e7i			0.666444i	0.942848	+	0.333222i	$0.108136\pi$
$997$			2.09171e7i			0.666444i	−0.942848	+	0.333222i	$0.891864\pi$
$998$	9.55349e6			0.303624
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	333.6.c.d.73.9		16
3.2	odd	2		37.6.b.a.36.8	✓	16
12.11	even	2		592.6.g.c.369.4		16
37.36	even	2	inner	333.6.c.d.73.8		16
111.110	odd	2		37.6.b.a.36.9	yes	16
444.443	even	2		592.6.g.c.369.3		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.6.b.a.36.8	✓	16	3.2	odd	2
37.6.b.a.36.9	yes	16	111.110	odd	2
333.6.c.d.73.8		16	37.36	even	2	inner
333.6.c.d.73.9		16	1.1	even	1	trivial
592.6.g.c.369.3		16	444.443	even	2
592.6.g.c.369.4		16	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 333 = 3^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	333.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.04262i q^{2} +30.9129 q^{4} -86.4713i q^{5} -87.1072 q^{7} +65.5944i q^{8} +O(q^{10})\)	\(q+1.04262i q^{2} +30.9129 q^{4} -86.4713i q^{5} -87.1072 q^{7} +65.5944i q^{8} +90.1568 q^{10} -153.498 q^{11} -27.6086i q^{13} -90.8199i q^{14} +920.824 q^{16} +926.725i q^{17} -2749.94i q^{19} -2673.08i q^{20} -160.041i q^{22} +1336.01i q^{23} -4352.28 q^{25} +28.7854 q^{26} -2692.74 q^{28} -7147.31i q^{29} -4026.24i q^{31} +3059.09i q^{32} -966.224 q^{34} +7532.28i q^{35} +(-7808.85 + 2892.37i) q^{37} +2867.14 q^{38} +5672.03 q^{40} -11398.2 q^{41} +8183.21i q^{43} -4745.08 q^{44} -1392.95 q^{46} +13313.5 q^{47} -9219.33 q^{49} -4537.78i q^{50} -853.464i q^{52} -27617.4 q^{53} +13273.2i q^{55} -5713.75i q^{56} +7451.94 q^{58} +16008.1i q^{59} +44974.5i q^{61} +4197.84 q^{62} +26276.9 q^{64} -2387.35 q^{65} -45824.8 q^{67} +28647.8i q^{68} -7853.31 q^{70} +18749.2 q^{71} -6614.81 q^{73} +(-3015.65 - 8141.68i) q^{74} -85008.6i q^{76} +13370.8 q^{77} -69916.6i q^{79} -79624.8i q^{80} -11884.0i q^{82} -81375.6 q^{83} +80135.1 q^{85} -8531.99 q^{86} -10068.6i q^{88} -120861. i q^{89} +2404.91i q^{91} +41300.0i q^{92} +13881.0i q^{94} -237791. q^{95} -5970.34i q^{97} -9612.27i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 268 q^{4} + 190 q^{7}+O(q^{10}) \) 16 * q - 268 * q^4 + 190 * q^7	\( 16 q - 268 q^{4} + 190 q^{7} - 74 q^{10} + 1110 q^{11} + 2900 q^{16} - 12052 q^{25} - 4902 q^{26} - 16824 q^{28} + 20556 q^{34} - 11400 q^{37} - 12108 q^{38} + 16966 q^{40} - 3918 q^{41} - 125394 q^{44} + 17470 q^{46} - 3822 q^{47} - 32618 q^{49} + 24126 q^{53} - 164718 q^{58} + 81426 q^{62} + 158076 q^{64} - 98976 q^{65} + 23560 q^{67} - 222404 q^{70} + 50046 q^{71} - 196274 q^{73} - 141216 q^{74} + 239574 q^{77} + 215814 q^{83} - 346472 q^{85} - 197640 q^{86} + 132504 q^{95}+O(q^{100}) \) 16 * q - 268 * q^4 + 190 * q^7 - 74 * q^10 + 1110 * q^11 + 2900 * q^16 - 12052 * q^25 - 4902 * q^26 - 16824 * q^28 + 20556 * q^34 - 11400 * q^37 - 12108 * q^38 + 16966 * q^40 - 3918 * q^41 - 125394 * q^44 + 17470 * q^46 - 3822 * q^47 - 32618 * q^49 + 24126 * q^53 - 164718 * q^58 + 81426 * q^62 + 158076 * q^64 - 98976 * q^65 + 23560 * q^67 - 222404 * q^70 + 50046 * q^71 - 196274 * q^73 - 141216 * q^74 + 239574 * q^77 + 215814 * q^83 - 346472 * q^85 - 197640 * q^86 + 132504 * q^95

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