LMFDB - Embedded newform 207.6.c.a.206.22

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [207,6,Mod(206,207)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(207, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("207.206");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 207 = 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	207.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:	$33.1994507013$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			206.22
Character	$\chi$	$=$	207.206
Dual form			207.6.c.a.206.21

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.37708i q^{2} +12.8412 q^{4} +13.0959 q^{5} +213.282i q^{7} +196.273i q^{8} +O(q^{10})$	\(q+4.37708i q^{2} +12.8412 q^{4} +13.0959 q^{5} +213.282i q^{7} +196.273i q^{8} +57.3218i q^{10} -174.086 q^{11} -394.801 q^{13} -933.553 q^{14} -448.184 q^{16} +1403.13 q^{17} +100.345i q^{19} +168.168 q^{20} -761.986i q^{22} +(1457.37 - 2076.64i) q^{23} -2953.50 q^{25} -1728.08i q^{26} +2738.81i q^{28} +5002.82i q^{29} -7046.64 q^{31} +4319.01i q^{32} +6141.63i q^{34} +2793.13i q^{35} +4980.90i q^{37} -439.219 q^{38} +2570.38i q^{40} -1934.58i q^{41} +2118.06i q^{43} -2235.47 q^{44} +(9089.59 + 6379.02i) q^{46} -6979.17i q^{47} -28682.4 q^{49} -12927.7i q^{50} -5069.73 q^{52} +5358.84 q^{53} -2279.81 q^{55} -41861.7 q^{56} -21897.7 q^{58} -21952.4i q^{59} -26926.6i q^{61} -30843.7i q^{62} -33246.5 q^{64} -5170.29 q^{65} +54081.7i q^{67} +18018.0 q^{68} -12225.7 q^{70} -23805.0i q^{71} -2165.67 q^{73} -21801.8 q^{74} +1288.55i q^{76} -37129.4i q^{77} +103875. i q^{79} -5869.39 q^{80} +8467.79 q^{82} -83380.3 q^{83} +18375.3 q^{85} -9270.89 q^{86} -34168.4i q^{88} -60517.8 q^{89} -84204.2i q^{91} +(18714.4 - 26666.5i) q^{92} +30548.3 q^{94} +1314.11i q^{95} +22107.1i q^{97} -125545. i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 600 q^{4}+O(q^{10}) $ 40 * q - 600 * q^4	$ 40 q - 600 q^{4} - 1048 q^{13} + 9728 q^{16} + 14704 q^{25} + 4640 q^{31} - 91864 q^{46} - 8192 q^{49} + 150360 q^{52} + 134592 q^{55} - 195704 q^{58} - 183416 q^{64} - 257448 q^{70} + 31088 q^{73} - 77096 q^{82} - 368760 q^{85} - 123512 q^{94}+O(q^{100}) $ 40 * q - 600 * q^4 - 1048 * q^13 + 9728 * q^16 + 14704 * q^25 + 4640 * q^31 - 91864 * q^46 - 8192 * q^49 + 150360 * q^52 + 134592 * q^55 - 195704 * q^58 - 183416 * q^64 - 257448 * q^70 + 31088 * q^73 - 77096 * q^82 - 368760 * q^85 - 123512 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/207\mathbb{Z}\right)^\times$.

$n$	$28$	$47$
$\chi(n)$	$-1$	$-1$

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 $

$2$			4.37708i			0.773765i	0.922129	+	0.386882i	$0.126448\pi$
$2$			4.37708i			0.773765i	−0.922129	+	0.386882i	$0.873552\pi$
$3$		0			0
$3$		0			0
$4$	12.8412			0.401288
$5$	13.0959			0.234267			0.117133	−	0.993116i	$-0.462629\pi$
$5$	13.0959			0.234267			0.117133	+	0.993116i	$0.462629\pi$
$6$		0			0
$7$			213.282i			1.64517i	0.568644	+	0.822584i	$0.307468\pi$
$7$			213.282i			1.64517i	−0.568644	+	0.822584i	$0.692532\pi$
$8$			196.273i			1.08427i
$9$		0			0
$10$			57.3218i			0.181268i
$11$	−174.086			−0.433792			−0.216896	−	0.976195i	$-0.569593\pi$
$11$	−174.086			−0.433792			−0.216896	+	0.976195i	$0.569593\pi$
$12$		0			0
$13$	−394.801			−0.647919			−0.323959	−	0.946071i	$-0.605014\pi$
$13$	−394.801			−0.647919			−0.323959	+	0.946071i	$0.605014\pi$
$14$	−933.553			−1.27297
$15$		0			0
$16$	−448.184			−0.437680
$17$	1403.13			1.17754			0.588772	−	0.808299i	$-0.299612\pi$
$17$	1403.13			1.17754			0.588772	+	0.808299i	$0.299612\pi$
$18$		0			0
$19$			100.345i			0.0637695i	0.999492	+	0.0318847i	$0.0101509\pi$
$19$			100.345i			0.0637695i	−0.999492	+	0.0318847i	$0.989849\pi$
$20$	168.168			0.0940085
$21$		0			0
$22$		−	761.986i		−	0.335653i
$23$	1457.37	−	2076.64i	0.574447	−	0.818542i
$23$	1457.37	−	2076.64i	0.574447	−	0.818542i
$24$		0			0
$25$	−2953.50			−0.945119
$26$		−	1728.08i		−	0.501337i
$27$		0			0
$28$			2738.81i			0.660186i
$29$			5002.82i			1.10464i	0.833633	+	0.552319i	$0.186257\pi$
$29$			5002.82i			1.10464i	−0.833633	+	0.552319i	$0.813743\pi$
$30$		0			0
$31$	−7046.64			−1.31698			−0.658488	−	0.752591i	$-0.728804\pi$
$31$	−7046.64			−1.31698			−0.658488	+	0.752591i	$0.728804\pi$
$32$			4319.01i			0.745606i
$33$		0			0
$34$			6141.63i			0.911142i
$35$			2793.13i			0.385408i
$36$		0			0
$37$			4980.90i			0.598141i	0.954231	+	0.299071i	$0.0966766\pi$
$37$			4980.90i			0.598141i	−0.954231	+	0.299071i	$0.903323\pi$
$38$	−439.219			−0.0493426
$39$		0			0
$40$			2570.38i			0.254008i
$41$		−	1934.58i		−	0.179733i	−0.995954	−	0.0898663i	$-0.971356\pi$
$41$		−	1934.58i		−	0.179733i	0.995954	−	0.0898663i	$-0.0286440\pi$
$42$		0			0
$43$			2118.06i			0.174689i	0.996178	+	0.0873446i	$0.0278381\pi$
$43$			2118.06i			0.174689i	−0.996178	+	0.0873446i	$0.972162\pi$
$44$	−2235.47			−0.174075
$45$		0			0
$46$	9089.59	+	6379.02i	0.633359	+	0.444487i
$47$		−	6979.17i		−	0.460849i	−0.973090	−	0.230425i	$-0.925988\pi$
$47$		−	6979.17i		−	0.460849i	0.973090	−	0.230425i	$-0.0740115\pi$
$48$		0			0
$49$	−28682.4			−1.70658
$50$		−	12927.7i		−	0.731300i
$51$		0			0
$52$	−5069.73			−0.260002
$53$	5358.84			0.262048			0.131024	−	0.991379i	$-0.458173\pi$
$53$	5358.84			0.262048			0.131024	+	0.991379i	$0.458173\pi$
$54$		0			0
$55$	−2279.81			−0.101623
$56$	−41861.7			−1.78380
$57$		0			0
$58$	−21897.7			−0.854730
$59$		−	21952.4i		−	0.821017i	−0.911857	−	0.410509i	$-0.865351\pi$
$59$		−	21952.4i		−	0.821017i	0.911857	−	0.410509i	$-0.134649\pi$
$60$		0			0
$61$		−	26926.6i		−	0.926526i	−0.886221	−	0.463263i	$-0.846679\pi$
$61$		−	26926.6i		−	0.926526i	0.886221	−	0.463263i	$-0.153321\pi$
$62$		−	30843.7i		−	1.01903i
$63$		0			0
$64$	−33246.5			−1.01460
$65$	−5170.29			−0.151786
$66$		0			0
$67$			54081.7i			1.47185i	0.677064	+	0.735924i	$0.263252\pi$
$67$			54081.7i			1.47185i	−0.677064	+	0.735924i	$0.736748\pi$
$68$	18018.0			0.472534
$69$		0			0
$70$	−12225.7			−0.298215
$71$		−	23805.0i		−	0.560432i	−0.959937	−	0.280216i	$-0.909594\pi$
$71$		−	23805.0i		−	0.560432i	0.959937	−	0.280216i	$-0.0904061\pi$
$72$		0			0
$73$	−2165.67			−0.0475647			−0.0237823	−	0.999717i	$-0.507571\pi$
$73$	−2165.67			−0.0475647			−0.0237823	+	0.999717i	$0.507571\pi$
$74$	−21801.8			−0.462821
$75$		0			0
$76$			1288.55i			0.0255899i
$77$		−	37129.4i		−	0.713660i
$78$		0			0
$79$			103875.i			1.87259i	0.351221	+	0.936293i	$0.385767\pi$
$79$			103875.i			1.87259i	−0.351221	+	0.936293i	$0.614233\pi$
$80$	−5869.39			−0.102534
$81$		0			0
$82$	8467.79			0.139071
$83$	−83380.3			−1.32852			−0.664260	−	0.747501i	$-0.731253\pi$
$83$	−83380.3			−1.32852			−0.664260	+	0.747501i	$0.731253\pi$
$84$		0			0
$85$	18375.3			0.275860
$86$	−9270.89			−0.135168
$87$		0			0
$88$		−	34168.4i		−	0.470346i
$89$	−60517.8			−0.809856			−0.404928	−	0.914349i	$-0.632703\pi$
$89$	−60517.8			−0.809856			−0.404928	+	0.914349i	$0.632703\pi$
$90$		0			0
$91$		−	84204.2i		−	1.06593i
$92$	18714.4	−	26666.5i	0.230519	−	0.328471i
$93$		0			0
$94$	30548.3			0.356589
$95$			1314.11i			0.0149391i
$96$		0			0
$97$			22107.1i			0.238563i	0.992861	+	0.119281i	$0.0380590\pi$
$97$			22107.1i			0.238563i	−0.992861	+	0.119281i	$0.961941\pi$
$98$		−	125545.i		−	1.32049i
$99$		0			0
$100$	−37926.5			−0.379265
$101$			98505.8i			0.960856i	0.877034	+	0.480428i	$0.159519\pi$
$101$			98505.8i			0.960856i	−0.877034	+	0.480428i	$0.840481\pi$
$102$		0			0
$103$			124557.i			1.15684i	0.815737	+	0.578422i	$0.196331\pi$
$103$			124557.i			1.15684i	−0.815737	+	0.578422i	$0.803669\pi$
$104$		−	77489.0i		−	0.702517i
$105$		0			0
$106$			23456.1i			0.202764i
$107$	162601.			1.37298			0.686488	−	0.727142i	$-0.259152\pi$
$107$	162601.			1.37298			0.686488	+	0.727142i	$0.259152\pi$
$108$		0			0
$109$		−	195809.i		−	1.57858i	−0.614022	−	0.789289i	$-0.710449\pi$
$109$		−	195809.i		−	1.57858i	0.614022	−	0.789289i	$-0.289551\pi$
$110$		−	9978.91i		−	0.0786324i
$111$		0			0
$112$		−	95589.9i		−	0.720057i
$113$	61196.8			0.450851			0.225425	−	0.974260i	$-0.427623\pi$
$113$	61196.8			0.450851			0.225425	+	0.974260i	$0.427623\pi$
$114$		0			0
$115$	19085.6	−	27195.5i	0.134574	−	0.191757i
$116$			64242.3i			0.443278i
$117$		0			0
$118$	96087.4			0.635274
$119$			299264.i			1.93726i
$120$		0			0
$121$	−130745.			−0.811825
$122$	117860.			0.716913
$123$		0			0
$124$	−90487.5			−0.528487
$125$	−79603.5			−0.455677
$126$		0			0
$127$	−52147.5			−0.286896			−0.143448	−	0.989658i	$-0.545819\pi$
$127$	−52147.5			−0.286896			−0.143448	+	0.989658i	$0.545819\pi$
$128$		−	7314.23i		−	0.0394588i
$129$		0			0
$130$		−	22630.7i		−	0.117447i
$131$			294583.i			1.49978i	0.661560	+	0.749892i	$0.269895\pi$
$131$			294583.i			1.49978i	−0.661560	+	0.749892i	$0.730105\pi$
$132$		0			0
$133$	−21401.9			−0.104911
$134$	−236719.			−1.13886
$135$		0			0
$136$			275398.i			1.27677i
$137$	302964.			1.37908			0.689540	−	0.724248i	$-0.257813\pi$
$137$	302964.			1.37908			0.689540	+	0.724248i	$0.257813\pi$
$138$		0			0
$139$	419258.			1.84053			0.920267	−	0.391290i	$-0.127971\pi$
$139$	419258.			1.84053			0.920267	+	0.391290i	$0.127971\pi$
$140$			35867.2i			0.154660i
$141$		0			0
$142$	104196.			0.433643
$143$	68729.3			0.281062
$144$		0			0
$145$			65516.6i			0.258780i
$146$		−	9479.28i		−	0.0368039i
$147$		0			0
$148$			63960.8i			0.240027i
$149$	317591.			1.17193			0.585966	−	0.810336i	$-0.300715\pi$
$149$	317591.			1.17193			0.585966	+	0.810336i	$0.300715\pi$
$150$		0			0
$151$	101652.			0.362806			0.181403	−	0.983409i	$-0.441936\pi$
$151$	101652.			0.362806			0.181403	+	0.983409i	$0.441936\pi$
$152$	−19695.1			−0.0691431
$153$		0			0
$154$	162518.			0.552205
$155$	−92282.3			−0.308524
$156$		0			0
$157$		−	367020.i		−	1.18834i	−0.804339	−	0.594170i	$-0.797480\pi$
$157$		−	367020.i		−	1.18834i	0.804339	−	0.594170i	$-0.202520\pi$
$158$	−454667.			−1.44894
$159$		0			0
$160$			56561.4i			0.174671i
$161$	442910.	+	310831.i	1.34664	+	0.945062i
$162$		0			0
$163$	199252.			0.587399			0.293700	−	0.955898i	$-0.405113\pi$
$163$	199252.			0.587399			0.293700	+	0.955898i	$0.405113\pi$
$164$		−	24842.3i		−	0.0721245i
$165$		0			0
$166$		−	364962.i		−	1.02796i
$167$		−	124469.i		−	0.345357i	−0.984978	−	0.172679i	$-0.944758\pi$
$167$		−	124469.i		−	0.345357i	0.984978	−	0.172679i	$-0.0552422\pi$
$168$		0			0
$169$	−215425.			−0.580202
$170$			80430.3i			0.213451i
$171$		0			0
$172$			27198.4i			0.0701007i
$173$		−	28739.1i		−	0.0730059i	−0.999334	−	0.0365029i	$-0.988378\pi$
$173$		−	28739.1i		−	0.0730059i	0.999334	−	0.0365029i	$-0.0116218\pi$
$174$		0			0
$175$		−	629929.i		−	1.55488i
$176$	78022.5			0.189862
$177$		0			0
$178$		−	264891.i		−	0.626638i
$179$			309761.i			0.722593i	0.932451	+	0.361297i	$0.117666\pi$
$179$			309761.i			0.722593i	−0.932451	+	0.361297i	$0.882334\pi$
$180$		0			0
$181$		−	245024.i		−	0.555919i	−0.960593	−	0.277959i	$-0.910342\pi$
$181$		−	245024.i		−	0.555919i	0.960593	−	0.277959i	$-0.0896581\pi$
$182$	368568.			0.824783
$183$		0			0
$184$	407588.	+	286043.i	0.887518	+	0.622854i
$185$			65229.5i			0.140125i
$186$		0			0
$187$	−244266.			−0.510809
$188$		−	89621.0i		−	0.184933i
$189$		0			0
$190$	−5751.97			−0.0115593
$191$	593684.			1.17753			0.588765	−	0.808304i	$-0.299614\pi$
$191$	593684.			1.17753			0.588765	+	0.808304i	$0.299614\pi$
$192$		0			0
$193$	916986.			1.77202			0.886012	−	0.463663i	$-0.153465\pi$
$193$	916986.			1.77202			0.886012	+	0.463663i	$0.153465\pi$
$194$	−96764.4			−0.184591
$195$		0			0
$196$	−368317.			−0.684828
$197$		−	14077.2i		−	0.0258435i	−0.999917	−	0.0129217i	$-0.995887\pi$
$197$		−	14077.2i		−	0.0258435i	0.999917	−	0.0129217i	$-0.00411323\pi$
$198$		0			0
$199$			690535.i			1.23610i	0.786139	+	0.618049i	$0.212077\pi$
$199$			690535.i			1.23610i	−0.786139	+	0.618049i	$0.787923\pi$
$200$		−	579693.i		−	1.02476i
$201$		0			0
$202$	−431167.			−0.743477
$203$	−1.06701e6			−1.81731
$204$		0			0
$205$		−	25335.1i		−	0.0421054i
$206$	−545195.			−0.895126
$207$		0			0
$208$	176944.			0.283581
$209$		−	17468.7i		−	0.0276627i
$210$		0			0
$211$	1.08126e6			1.67195			0.835977	−	0.548765i	$-0.184902\pi$
$211$	1.08126e6			1.67195			0.835977	+	0.548765i	$0.184902\pi$
$212$	68814.1			0.105157
$213$		0			0
$214$			711715.i			1.06236i
$215$			27737.9i			0.0409239i
$216$		0			0
$217$		−	1.50293e6i		−	2.16665i
$218$	857070.			1.22145
$219$		0			0
$220$	−29275.6			−0.0407801
$221$	−553960.			−0.762953
$222$		0			0
$223$	555700.			0.748305			0.374152	−	0.927367i	$-0.377934\pi$
$223$	555700.			0.748305			0.374152	+	0.927367i	$0.377934\pi$
$224$	−921169.			−1.22665
$225$		0			0
$226$			267863.i			0.348852i
$227$	−756275.			−0.974126			−0.487063	−	0.873367i	$-0.661932\pi$
$227$	−756275.			−0.974126			−0.487063	+	0.873367i	$0.661932\pi$
$228$		0			0
$229$			387715.i			0.488567i	0.969704	+	0.244284i	$0.0785528\pi$
$229$			387715.i			0.488567i	−0.969704	+	0.244284i	$0.921447\pi$
$230$	119037.	+	83539.1i	0.148375	+	0.104129i
$231$		0			0
$232$	−981921.			−1.19772
$233$		−	642988.i		−	0.775913i	−0.921678	−	0.387957i	$-0.873181\pi$
$233$		−	642988.i		−	0.775913i	0.921678	−	0.387957i	$-0.126819\pi$
$234$		0			0
$235$		−	91398.6i		−	0.107962i
$236$		−	281896.i		−	0.329464i
$237$		0			0
$238$	−1.30990e6			−1.49898
$239$			230335.i			0.260835i	0.991459	+	0.130417i	$0.0416317\pi$
$239$			230335.i			0.260835i	−0.991459	+	0.130417i	$0.958368\pi$
$240$		0			0
$241$			1.49830e6i			1.66172i	0.556483	+	0.830859i	$0.312150\pi$
$241$			1.49830e6i			1.66172i	−0.556483	+	0.830859i	$0.687850\pi$
$242$		−	572282.i		−	0.628161i
$243$		0			0
$244$		−	345771.i		−	0.371804i
$245$	−375623.			−0.399794
$246$		0			0
$247$		−	39616.4i		−	0.0413174i
$248$		−	1.38307e6i		−	1.42795i
$249$		0			0
$250$		−	348431.i		−	0.352587i
$251$	1.73712e6			1.74039			0.870193	−	0.492712i	$-0.163994\pi$
$251$	1.73712e6			1.74039			0.870193	+	0.492712i	$0.163994\pi$
$252$		0			0
$253$	−253707.	+	361513.i	−0.249190	+	0.355077i
$254$		−	228254.i		−	0.221990i
$255$		0			0
$256$	−1.03187e6			−0.984072
$257$		−	233130.i		−	0.220174i	−0.993922	−	0.110087i	$-0.964887\pi$
$257$		−	233130.i		−	0.220174i	0.993922	−	0.110087i	$-0.0351129\pi$
$258$		0			0
$259$	−1.06234e6			−0.984043
$260$	−66392.8			−0.0609099
$261$		0			0
$262$	−1.28941e6			−1.16048
$263$	937684.			0.835925			0.417962	−	0.908464i	$-0.362744\pi$
$263$	937684.			0.835925			0.417962	+	0.908464i	$0.362744\pi$
$264$		0			0
$265$	70179.0			0.0613893
$266$		−	93677.6i		−	0.0811768i
$267$		0			0
$268$			694474.i			0.590635i
$269$			722713.i			0.608955i	0.952520	+	0.304477i	$0.0984818\pi$
$269$			722713.i			0.608955i	−0.952520	+	0.304477i	$0.901518\pi$
$270$		0			0
$271$	283478.			0.234475			0.117237	−	0.993104i	$-0.462596\pi$
$271$	283478.			0.234475			0.117237	+	0.993104i	$0.462596\pi$
$272$	−628863.			−0.515388
$273$		0			0
$274$			1.32610e6i			1.06708i
$275$	514161.			0.409985
$276$		0			0
$277$	−1.09368e6			−0.856430			−0.428215	−	0.903677i	$-0.640857\pi$
$277$	−1.09368e6			−0.856430			−0.428215	+	0.903677i	$0.640857\pi$
$278$			1.83512e6i			1.42414i
$279$		0			0
$280$	−548217.			−0.417886
$281$	−538747.			−0.407023			−0.203512	−	0.979073i	$-0.565235\pi$
$281$	−538747.			−0.407023			−0.203512	+	0.979073i	$0.565235\pi$
$282$		0			0
$283$		−	53011.9i		−	0.0393466i	−0.999806	−	0.0196733i	$-0.993737\pi$
$283$		−	53011.9i		−	0.0393466i	0.999806	−	0.0196733i	$-0.00626261\pi$
$284$		−	305686.i		−	0.224895i
$285$		0			0
$286$			300833.i			0.217476i
$287$	412612.			0.295690
$288$		0			0
$289$	548931.			0.386610
$290$	−286771.			−0.200235
$291$		0			0
$292$	−27809.8			−0.0190871
$293$	−1.87451e6			−1.27561			−0.637807	−	0.770197i	$-0.720158\pi$
$293$	−1.87451e6			−1.27561			−0.637807	+	0.770197i	$0.720158\pi$
$294$		0			0
$295$		−	287487.i		−	0.192337i
$296$	−977619.			−0.648545
$297$		0			0
$298$			1.39012e6i			0.906800i
$299$	−575372.	+	819859.i	−0.372195	+	0.530348i
$300$		0			0
$301$	−451744.			−0.287393
$302$			444940.i			0.280727i
$303$		0			0
$304$		−	44973.2i		−	0.0279106i
$305$		−	352629.i		−	0.217054i
$306$		0			0
$307$	1.01140e6			0.612458			0.306229	−	0.951958i	$-0.400933\pi$
$307$	1.01140e6			0.612458			0.306229	+	0.951958i	$0.400933\pi$
$308$		−	476787.i		−	0.286383i
$309$		0			0
$310$		−	403927.i		−	0.238725i
$311$		−	2.21149e6i		−	1.29653i	−0.761414	−	0.648266i	$-0.775495\pi$
$311$		−	2.21149e6i		−	1.29653i	0.761414	−	0.648266i	$-0.224505\pi$
$312$		0			0
$313$			114774.i			0.0662189i	0.999452	+	0.0331094i	$0.0105410\pi$
$313$			114774.i			0.0662189i	−0.999452	+	0.0331094i	$0.989459\pi$
$314$	1.60648e6			0.919496
$315$		0			0
$316$			1.33388e6i			0.751446i
$317$			857671.i			0.479372i	0.970851	+	0.239686i	$0.0770444\pi$
$317$			857671.i			0.479372i	−0.970851	+	0.239686i	$0.922956\pi$
$318$		0			0
$319$		−	870919.i		−	0.479183i
$320$	−435394.			−0.237688
$321$		0			0
$322$	−1.36053e6	+	1.93865e6i	−0.731255	+	1.04198i
$323$			140798.i			0.0750913i
$324$		0			0
$325$	1.16604e6			0.612360
$326$			872141.i			0.454509i
$327$		0			0
$328$	379706.			0.194878
$329$	1.48853e6			0.758174
$330$		0			0
$331$	−2.00888e6			−1.00782			−0.503912	−	0.863755i	$-0.668107\pi$
$331$	−2.00888e6			−1.00782			−0.503912	+	0.863755i	$0.668107\pi$
$332$	−1.07070e6			−0.533119
$333$		0			0
$334$	544809.			0.267225
$335$			708249.i			0.344805i
$336$		0			0
$337$		−	1.50990e6i		−	0.724223i	−0.932135	−	0.362112i	$-0.882056\pi$
$337$		−	1.50990e6i		−	0.724223i	0.932135	−	0.362112i	$-0.117944\pi$
$338$		−	942930.i		−	0.448940i
$339$		0			0
$340$	235962.			0.110699
$341$	1.22672e6			0.571294
$342$		0			0
$343$		−	2.53282e6i		−	1.16243i
$344$	−415718.			−0.189410
$345$		0			0
$346$	125793.			0.0564894
$347$		−	3.05477e6i		−	1.36193i	−0.732316	−	0.680965i	$-0.761561\pi$
$347$		−	3.05477e6i		−	1.36193i	0.732316	−	0.680965i	$-0.238439\pi$
$348$		0			0
$349$	804245.			0.353447			0.176724	−	0.984260i	$-0.443450\pi$
$349$	804245.			0.353447			0.176724	+	0.984260i	$0.443450\pi$
$350$	2.75725e6			1.20311
$351$		0			0
$352$		−	751878.i		−	0.323438i
$353$			2.37720e6i			1.01538i	0.861540	+	0.507690i	$0.169500\pi$
$353$			2.37720e6i			1.01538i	−0.861540	+	0.507690i	$0.830500\pi$
$354$		0			0
$355$		−	311749.i		−	0.131291i
$356$	−777122.			−0.324986
$357$		0			0
$358$	−1.35585e6			−0.559117
$359$	−146469.			−0.0599805			−0.0299902	−	0.999550i	$-0.509548\pi$
$359$	−146469.			−0.0599805			−0.0299902	+	0.999550i	$0.509548\pi$
$360$		0			0
$361$	2.46603e6			0.995933
$362$	1.07249e6			0.430151
$363$		0			0
$364$		−	1.08128e6i		−	0.427747i
$365$	−28361.4			−0.0111428
$366$		0			0
$367$		−	4.16885e6i		−	1.61566i	−0.589413	−	0.807832i	$-0.700641\pi$
$367$		−	4.16885e6i		−	1.61566i	0.589413	−	0.807832i	$-0.299359\pi$
$368$	−653170.	+	930716.i	−0.251424	+	0.358259i
$369$		0			0
$370$	−285515.			−0.108424
$371$			1.14295e6i			0.431113i
$372$		0			0
$373$			1.86966e6i			0.695809i	0.937530	+	0.347905i	$0.113107\pi$
$373$			1.86966e6i			0.695809i	−0.937530	+	0.347905i	$0.886893\pi$
$374$		−	1.06917e6i		−	0.395246i
$375$		0			0
$376$	1.36982e6			0.499684
$377$		−	1.97512e6i		−	0.715716i
$378$		0			0
$379$			1.55889e6i			0.557463i	0.960369	+	0.278732i	$0.0899140\pi$
$379$			1.55889e6i			0.557463i	−0.960369	+	0.278732i	$0.910086\pi$
$380$			16874.8i			0.00599487i
$381$		0			0
$382$			2.59860e6i			0.911131i
$383$	516004.			0.179745			0.0898724	−	0.995953i	$-0.471354\pi$
$383$	516004.			0.179745			0.0898724	+	0.995953i	$0.471354\pi$
$384$		0			0
$385$		−	486244.i		−	0.167187i
$386$			4.01372e6i			1.37113i
$387$		0			0
$388$			283882.i			0.0957323i
$389$	−3.17699e6			−1.06449			−0.532245	−	0.846590i	$-0.678651\pi$
$389$	−3.17699e6			−1.06449			−0.532245	+	0.846590i	$0.678651\pi$
$390$		0			0
$391$	2.04489e6	−	2.91380e6i	0.676437	−	0.963869i
$392$		−	5.62959e6i		−	1.85038i
$393$		0			0
$394$	61617.0			0.0199968
$395$			1.36033e6i			0.438685i
$396$		0			0
$397$	4.10721e6			1.30789			0.653944	−	0.756543i	$-0.273113\pi$
$397$	4.10721e6			1.30789			0.653944	+	0.756543i	$0.273113\pi$
$398$	−3.02252e6			−0.956450
$399$		0			0
$400$	1.32371e6			0.413660
$401$	−660961.			−0.205265			−0.102633	−	0.994719i	$-0.532727\pi$
$401$	−660961.			−0.205265			−0.102633	+	0.994719i	$0.532727\pi$
$402$		0			0
$403$	2.78203e6			0.853294
$404$			1.26493e6i			0.385580i
$405$		0			0
$406$		−	4.67040e6i		−	1.40617i
$407$		−	867104.i		−	0.259469i
$408$		0			0
$409$	4.08213e6			1.20664			0.603322	−	0.797498i	$-0.293843\pi$
$409$	4.08213e6			1.20664			0.603322	+	0.797498i	$0.293843\pi$
$410$	110894.			0.0325797
$411$		0			0
$412$			1.59946e6i			0.464228i
$413$	4.68207e6			1.35071
$414$		0			0
$415$	−1.09194e6			−0.311228
$416$		−	1.70515e6i		−	0.483092i
$417$		0			0
$418$	76461.7			0.0214044
$419$	−4.99912e6			−1.39110			−0.695551	−	0.718477i	$-0.744840\pi$
$419$	−4.99912e6			−1.39110			−0.695551	+	0.718477i	$0.744840\pi$
$420$		0			0
$421$			58306.3i			0.0160328i	0.999968	+	0.00801642i	$0.00255173\pi$
$421$			58306.3i			0.0160328i	−0.999968	+	0.00801642i	$0.997448\pi$
$422$			4.73276e6i			1.29370i
$423$		0			0
$424$			1.05180e6i			0.284130i
$425$	−4.14415e6			−1.11292
$426$		0			0
$427$	5.74298e6			1.52429
$428$	2.08799e6			0.550958
$429$		0			0
$430$	−121411.			−0.0316655
$431$	6.18474e6			1.60372			0.801860	−	0.597512i	$-0.203844\pi$
$431$	6.18474e6			1.60372			0.801860	+	0.597512i	$0.203844\pi$
$432$		0			0
$433$		−	206352.i		−	0.0528919i	−0.999650	−	0.0264459i	$-0.991581\pi$
$433$		−	206352.i		−	0.0528919i	0.999650	−	0.0264459i	$-0.00841899\pi$
$434$	6.57842e6			1.67648
$435$		0			0
$436$		−	2.51442e6i		−	0.633464i
$437$	208381.	+	146240.i	0.0521980	+	0.0366322i
$438$		0			0
$439$	−1.75544e6			−0.434736			−0.217368	−	0.976090i	$-0.569747\pi$
$439$	−1.75544e6			−0.434736			−0.217368	+	0.976090i	$0.569747\pi$
$440$		−	447466.i		−	0.110187i
$441$		0			0
$442$		−	2.42472e6i		−	0.590346i
$443$			7.46113e6i			1.80632i	0.429301	+	0.903161i	$0.358760\pi$
$443$			7.46113e6i			1.80632i	−0.429301	+	0.903161i	$0.641240\pi$
$444$		0			0
$445$	−792536.			−0.189723
$446$			2.43234e6i			0.579012i
$447$		0			0
$448$		−	7.09090e6i		−	1.66919i
$449$			5.64567e6i			1.32160i	0.750563	+	0.660799i	$0.229782\pi$
$449$			5.64567e6i			1.32160i	−0.750563	+	0.660799i	$0.770218\pi$
$450$		0			0
$451$			336782.i			0.0779665i
$452$	785841.			0.180921
$453$		0			0
$454$		−	3.31027e6i		−	0.753745i
$455$		−	1.10273e6i		−	0.249713i
$456$		0			0
$457$			7.55756e6i			1.69274i	0.532593	+	0.846371i	$0.321217\pi$
$457$			7.55756e6i			1.69274i	−0.532593	+	0.846371i	$0.678783\pi$
$458$	−1.69706e6			−0.378036
$459$		0			0
$460$	245082.	−	349223.i	0.0540029	−	0.0769499i
$461$		−	5.85167e6i		−	1.28241i	−0.767369	−	0.641206i	$-0.778434\pi$
$461$		−	5.85167e6i		−	1.28241i	0.767369	−	0.641206i	$-0.221566\pi$
$462$		0			0
$463$	−5.15533e6			−1.11764			−0.558822	−	0.829287i	$-0.688747\pi$
$463$	−5.15533e6			−1.11764			−0.558822	+	0.829287i	$0.688747\pi$
$464$		−	2.24219e6i		−	0.483478i
$465$		0			0
$466$	2.81441e6			0.600375
$467$	208721.			0.0442868			0.0221434	−	0.999755i	$-0.492951\pi$
$467$	208721.			0.0442868			0.0221434	+	0.999755i	$0.492951\pi$
$468$		0			0
$469$	−1.15347e7			−2.42144
$470$	400059.			0.0835370
$471$		0			0
$472$	4.30867e6			0.890202
$473$		−	368723.i		−	0.0757788i
$474$		0			0
$475$		−	296369.i		−	0.0602697i
$476$			3.84291e6i			0.777398i
$477$		0			0
$478$	−1.00819e6			−0.201825
$479$	9.23003e6			1.83808			0.919040	−	0.394165i	$-0.128966\pi$
$479$	9.23003e6			1.83808			0.919040	+	0.394165i	$0.128966\pi$
$480$		0			0
$481$		−	1.96647e6i		−	0.387547i
$482$	−6.55819e6			−1.28578
$483$		0			0
$484$	−1.67893e6			−0.325775
$485$			289513.i			0.0558873i
$486$		0			0
$487$	1.09708e6			0.209612			0.104806	−	0.994493i	$-0.466578\pi$
$487$	1.09708e6			0.209612			0.104806	+	0.994493i	$0.466578\pi$
$488$	5.28498e6			1.00460
$489$		0			0
$490$		−	1.64413e6i		−	0.309347i
$491$			3.40794e6i			0.637952i	0.947763	+	0.318976i	$0.103339\pi$
$491$			3.40794e6i			0.637952i	−0.947763	+	0.318976i	$0.896661\pi$
$492$		0			0
$493$			7.01963e6i			1.30076i
$494$	173404.			0.0319700
$495$		0			0
$496$	3.15820e6			0.576414
$497$	5.07720e6			0.922004
$498$		0			0
$499$	586822.			0.105501			0.0527503	−	0.998608i	$-0.483201\pi$
$499$	586822.			0.105501			0.0527503	+	0.998608i	$0.483201\pi$
$500$	−1.02221e6			−0.182858
$501$		0			0
$502$			7.60350e6i			1.34665i
$503$	629638.			0.110961			0.0554806	−	0.998460i	$-0.482331\pi$
$503$	629638.			0.110961			0.0554806	+	0.998460i	$0.482331\pi$
$504$		0			0
$505$			1.29002e6i			0.225097i
$506$	−1.58237e6	−	1.11049e6i	−0.274746	−	0.192815i
$507$		0			0
$508$	−669637.			−0.115128
$509$		−	1.90466e6i		−	0.325855i	−0.986638	−	0.162927i	$-0.947906\pi$
$509$		−	1.90466e6i		−	0.325855i	0.986638	−	0.162927i	$-0.0520936\pi$
$510$		0			0
$511$		−	461899.i		−	0.0782518i
$512$		−	4.75065e6i		−	0.800899i
$513$		0			0
$514$	1.02043e6			0.170363
$515$			1.63119e6i			0.271010i
$516$		0			0
$517$			1.21497e6i			0.199913i
$518$		−	4.64994e6i		−	0.761418i
$519$		0			0
$520$		−	1.01479e6i		−	0.164577i
$521$	−9.78506e6			−1.57932			−0.789658	−	0.613547i	$-0.789742\pi$
$521$	−9.78506e6			−1.57932			−0.789658	+	0.613547i	$0.789742\pi$
$522$		0			0
$523$			6.91407e6i			1.10530i	0.833414	+	0.552649i	$0.186383\pi$
$523$			6.91407e6i			1.10530i	−0.833414	+	0.552649i	$0.813617\pi$
$524$			3.78280e6i			0.601845i
$525$		0			0
$526$			4.10431e6i			0.646809i
$527$	−9.88739e6			−1.55080
$528$		0			0
$529$	−2.18849e6	−	6.05285e6i	−0.340021	−	0.940418i
$530$			307179.i			0.0475009i
$531$		0			0
$532$	−274826.			−0.0420997
$533$			763774.i			0.116452i
$534$		0			0
$535$	2.12940e6			0.321643
$536$	−1.06148e7			−1.59588
$537$		0			0
$538$	−3.16337e6			−0.471188
$539$	4.99320e6			0.740298
$540$		0			0
$541$	−1.16376e7			−1.70951			−0.854753	−	0.519034i	$-0.826292\pi$
$541$	−1.16376e7			−1.70951			−0.854753	+	0.519034i	$0.826292\pi$
$542$			1.24080e6i			0.181428i
$543$		0			0
$544$			6.06016e6i			0.877984i
$545$		−	2.56430e6i		−	0.369809i
$546$		0			0
$547$	3.94107e6			0.563178			0.281589	−	0.959535i	$-0.409139\pi$
$547$	3.94107e6			0.563178			0.281589	+	0.959535i	$0.409139\pi$
$548$	3.89042e6			0.553408
$549$		0			0
$550$			2.25052e6i			0.317232i
$551$	−502009.			−0.0704422
$552$		0			0
$553$	−2.21546e7			−3.08072
$554$		−	4.78713e6i		−	0.662675i
$555$		0			0
$556$	5.38378e6			0.738584
$557$	−2.75090e6			−0.375696			−0.187848	−	0.982198i	$-0.560151\pi$
$557$	−2.75090e6			−0.375696			−0.187848	+	0.982198i	$0.560151\pi$
$558$		0			0
$559$		−	836212.i		−	0.113184i
$560$		−	1.25184e6i		−	0.168686i
$561$		0			0
$562$		−	2.35814e6i		−	0.314940i
$563$	−2.52208e6			−0.335342			−0.167671	−	0.985843i	$-0.553625\pi$
$563$	−2.52208e6			−0.335342			−0.167671	+	0.985843i	$0.553625\pi$
$564$		0			0
$565$	801429.			0.105619
$566$	232037.			0.0304450
$567$		0			0
$568$	4.67229e6			0.607658
$569$	1.00871e7			1.30613			0.653066	−	0.757301i	$-0.273482\pi$
$569$	1.00871e7			1.30613			0.653066	+	0.757301i	$0.273482\pi$
$570$		0			0
$571$			6.62870e6i			0.850820i	0.905001	+	0.425410i	$0.139870\pi$
$571$			6.62870e6i			0.850820i	−0.905001	+	0.425410i	$0.860130\pi$
$572$	882567.			0.112787
$573$		0			0
$574$			1.80603e6i			0.228795i
$575$	−4.30434e6	+	6.13334e6i	−0.542921	+	0.773619i
$576$		0			0
$577$	−6.94310e6			−0.868188			−0.434094	−	0.900868i	$-0.642931\pi$
$577$	−6.94310e6			−0.868188			−0.434094	+	0.900868i	$0.642931\pi$
$578$			2.40271e6i			0.299145i
$579$		0			0
$580$			841312.i			0.103845i
$581$		−	1.77836e7i		−	2.18564i
$582$		0			0
$583$	−932898.			−0.113674
$584$		−	425063.i		−	0.0515728i
$585$		0			0
$586$		−	8.20488e6i		−	0.987025i
$587$		−	4.03546e6i		−	0.483390i	−0.970352	−	0.241695i	$-0.922297\pi$
$587$		−	4.03546e6i		−	0.483390i	0.970352	−	0.241695i	$-0.0777033\pi$
$588$		0			0
$589$		−	707097.i		−	0.0839829i
$590$	1.25835e6			0.148824
$591$		0			0
$592$		−	2.23236e6i		−	0.261795i
$593$		−	7.93784e6i		−	0.926970i	−0.886105	−	0.463485i	$-0.846599\pi$
$593$		−	7.93784e6i		−	0.926970i	0.886105	−	0.463485i	$-0.153401\pi$
$594$		0			0
$595$			3.91914e6i			0.453835i
$596$	4.07825e6			0.470282
$597$		0			0
$598$	−3.58858e6	−	2.51844e6i	−0.410365	−	0.287991i
$599$		−	6.14111e6i		−	0.699326i	−0.936876	−	0.349663i	$-0.886296\pi$
$599$		−	6.14111e6i		−	0.699326i	0.936876	−	0.349663i	$-0.113704\pi$
$600$		0			0
$601$	−4.86684e6			−0.549618			−0.274809	−	0.961499i	$-0.588615\pi$
$601$	−4.86684e6			−0.549618			−0.274809	+	0.961499i	$0.588615\pi$
$602$		−	1.97732e6i		−	0.222375i
$603$		0			0
$604$	1.30534e6			0.145590
$605$	−1.71223e6			−0.190184
$606$		0			0
$607$	−1.24930e7			−1.37624			−0.688122	−	0.725595i	$-0.741565\pi$
$607$	−1.24930e7			−1.37624			−0.688122	+	0.725595i	$0.741565\pi$
$608$	−433392.			−0.0475469
$609$		0			0
$610$	1.54348e6			0.167949
$611$			2.75539e6i			0.298593i
$612$		0			0
$613$			2.76801e6i			0.297521i	0.988873	+	0.148760i	$0.0475283\pi$
$613$			2.76801e6i			0.297521i	−0.988873	+	0.148760i	$0.952472\pi$
$614$			4.42696e6i			0.473898i
$615$		0			0
$616$	7.28751e6			0.773798
$617$	−1.52359e7			−1.61122			−0.805612	−	0.592444i	$-0.798163\pi$
$617$	−1.52359e7			−1.61122			−0.805612	+	0.592444i	$0.798163\pi$
$618$		0			0
$619$		−	1.22425e7i		−	1.28423i	−0.766607	−	0.642116i	$-0.778057\pi$
$619$		−	1.22425e7i		−	1.28423i	0.766607	−	0.642116i	$-0.221943\pi$
$620$	−1.18502e6			−0.123807
$621$		0			0
$622$	9.67984e6			1.00321
$623$		−	1.29074e7i		−	1.33235i
$624$		0			0
$625$	8.18720e6			0.838369
$626$	−502373.			−0.0512378
$627$		0			0
$628$		−	4.71299e6i		−	0.476867i
$629$			6.98888e6i			0.704338i
$630$		0			0
$631$		−	4.28921e6i		−	0.428848i	−0.976741	−	0.214424i	$-0.931213\pi$
$631$		−	4.28921e6i		−	0.428848i	0.976741	−	0.214424i	$-0.0687874\pi$
$632$	−2.03878e7			−2.03038
$633$		0			0
$634$	−3.75409e6			−0.370921
$635$	−682920.			−0.0672102
$636$		0			0
$637$	1.13239e7			1.10572
$638$	3.81208e6			0.370775
$639$		0			0
$640$		−	95786.6i		−	0.00924389i
$641$	6.94501e6			0.667617			0.333809	−	0.942641i	$-0.391666\pi$
$641$	6.94501e6			0.667617			0.333809	+	0.942641i	$0.391666\pi$
$642$		0			0
$643$			1.77243e7i			1.69060i	0.534291	+	0.845301i	$0.320579\pi$
$643$			1.77243e7i			1.69060i	−0.534291	+	0.845301i	$0.679421\pi$
$644$	5.68750e6	+	3.99145e6i	0.540390	+	0.379242i
$645$		0			0
$646$	−616283.			−0.0581030
$647$		−	3.63111e6i		−	0.341019i	−0.985356	−	0.170509i	$-0.945459\pi$
$647$		−	3.63111e6i		−	0.341019i	0.985356	−	0.170509i	$-0.0545413\pi$
$648$		0			0
$649$			3.82160e6i			0.356150i
$650$			5.10387e6i			0.473823i
$651$		0			0
$652$	2.55864e6			0.235716
$653$		−	1.71301e7i		−	1.57209i	−0.618169	−	0.786046i	$-0.712125\pi$
$653$		−	1.71301e7i		−	1.57209i	0.618169	−	0.786046i	$-0.287875\pi$
$654$		0			0
$655$			3.85783e6i			0.351350i
$656$			867048.i			0.0786653i
$657$		0			0
$658$			6.51543e6i			0.586649i
$659$	−1.67386e7			−1.50143			−0.750716	−	0.660625i	$-0.770291\pi$
$659$	−1.67386e7			−1.50143			−0.750716	+	0.660625i	$0.770291\pi$
$660$		0			0
$661$			1.16254e7i			1.03492i	0.855709	+	0.517458i	$0.173122\pi$
$661$			1.16254e7i			1.03492i	−0.855709	+	0.517458i	$0.826878\pi$
$662$		−	8.79304e6i		−	0.779820i
$663$		0			0
$664$		−	1.63653e7i		−	1.44047i
$665$	−280277.			−0.0245773
$666$		0			0
$667$	1.03890e7	+	7.29096e6i	0.904192	+	0.634556i
$668$		−	1.59833e6i		−	0.138588i
$669$		0			0
$670$	−3.10006e6			−0.266798
$671$			4.68754e6i			0.401919i
$672$		0			0
$673$	−1.08990e7			−0.927577			−0.463788	−	0.885946i	$-0.653510\pi$
$673$	−1.08990e7			−0.927577			−0.463788	+	0.885946i	$0.653510\pi$
$674$	6.60893e6			0.560378
$675$		0			0
$676$	−2.76632e6			−0.232828
$677$	4.00143e6			0.335539			0.167770	−	0.985826i	$-0.446344\pi$
$677$	4.00143e6			0.335539			0.167770	+	0.985826i	$0.446344\pi$
$678$		0			0
$679$	−4.71505e6			−0.392475
$680$			3.60659e6i			0.299106i
$681$		0			0
$682$			5.36944e6i			0.442047i
$683$			2.18521e7i			1.79243i	0.443624	+	0.896213i	$0.353692\pi$
$683$			2.18521e7i			1.79243i	−0.443624	+	0.896213i	$0.646308\pi$
$684$		0			0
$685$	3.96759e6			0.323073
$686$	1.10863e7			0.899451
$687$		0			0
$688$		−	949280.i		−	0.0764580i
$689$	−2.11568e6			−0.169786
$690$		0			0
$691$	8.13830e6			0.648394			0.324197	−	0.945990i	$-0.394906\pi$
$691$	8.13830e6			0.648394			0.324197	+	0.945990i	$0.394906\pi$
$692$		−	369045.i		−	0.0292964i
$693$		0			0
$694$	1.33710e7			1.05381
$695$	5.49057e6			0.431176
$696$		0			0
$697$		−	2.71447e6i		−	0.211643i
$698$			3.52024e6i			0.273485i
$699$		0			0
$700$		−	8.08905e6i		−	0.623954i
$701$	2.09752e7			1.61217			0.806084	−	0.591801i	$-0.201583\pi$
$701$	2.09752e7			1.61217			0.806084	+	0.591801i	$0.201583\pi$
$702$		0			0
$703$	−499810.			−0.0381432
$704$	5.78774e6			0.440127
$705$		0			0
$706$	−1.04052e7			−0.785665
$707$	−2.10096e7			−1.58077
$708$		0			0
$709$		−	4.96485e6i		−	0.370929i	−0.982651	−	0.185465i	$-0.940621\pi$
$709$		−	4.96485e6i		−	0.370929i	0.982651	−	0.185465i	$-0.0593790\pi$
$710$	1.36455e6			0.101588
$711$		0			0
$712$		−	1.18780e7i		−	0.878101i
$713$	−1.02696e7	+	1.46333e7i	−0.756533	+	1.07800i
$714$		0			0
$715$	900073.			0.0658435
$716$			3.97770e6i			0.289968i
$717$		0			0
$718$		−	641106.i		−	0.0464108i
$719$			9.76377e6i			0.704361i	0.935932	+	0.352180i	$0.114560\pi$
$719$			9.76377e6i			0.704361i	−0.935932	+	0.352180i	$0.885440\pi$
$720$		0			0
$721$	−2.65658e7			−1.90320
$722$			1.07940e7i			0.770618i
$723$		0			0
$724$		−	3.14640e6i		−	0.223084i
$725$		−	1.47758e7i		−	1.04401i
$726$		0			0
$727$		−	1.09982e7i		−	0.771765i	−0.922548	−	0.385883i	$-0.873897\pi$
$727$		−	1.09982e7i		−	0.771765i	0.922548	−	0.385883i	$-0.126103\pi$
$728$	1.65270e7			1.15576
$729$		0			0
$730$		−	124140.i		−	0.00862193i
$731$			2.97192e6i			0.205704i
$732$		0			0
$733$		−	9.29717e6i		−	0.639132i	−0.947564	−	0.319566i	$-0.896463\pi$
$733$		−	9.29717e6i		−	0.639132i	0.947564	−	0.319566i	$-0.103537\pi$
$734$	1.82474e7			1.25014
$735$		0			0
$736$	8.96901e6	+	6.29439e6i	0.610310	+	0.428311i
$737$		−	9.41484e6i		−	0.638475i
$738$		0			0
$739$	−1.56062e7			−1.05120			−0.525600	−	0.850732i	$-0.676159\pi$
$739$	−1.56062e7			−1.05120			−0.525600	+	0.850732i	$0.676159\pi$
$740$			837626.i			0.0562304i
$741$		0			0
$742$	−5.00277e6			−0.333580
$743$	2.61252e7			1.73615			0.868074	−	0.496435i	$-0.165358\pi$
$743$	2.61252e7			1.73615			0.868074	+	0.496435i	$0.165358\pi$
$744$		0			0
$745$	4.15914e6			0.274545
$746$	−8.18363e6			−0.538393
$747$		0			0
$748$	−3.13667e6			−0.204981
$749$			3.46799e7i			2.25877i
$750$		0			0
$751$		−	1.09372e7i		−	0.707629i	−0.935316	−	0.353814i	$-0.884884\pi$
$751$		−	1.09372e7i		−	0.707629i	0.935316	−	0.353814i	$-0.115116\pi$
$752$			3.12795e6i			0.201705i
$753$		0			0
$754$	8.64526e6			0.553796
$755$	1.33123e6			0.0849935
$756$		0			0
$757$		−	2.50193e6i		−	0.158685i	−0.996847	−	0.0793424i	$-0.974718\pi$
$757$		−	2.50193e6i		−	0.158685i	0.996847	−	0.0793424i	$-0.0252820\pi$
$758$	−6.82336e6			−0.431346
$759$		0			0
$760$	−257925.			−0.0161980
$761$		−	2.93461e7i		−	1.83691i	−0.395521	−	0.918457i	$-0.629436\pi$
$761$		−	2.93461e7i		−	1.83691i	0.395521	−	0.918457i	$-0.370564\pi$
$762$		0			0
$763$	4.17626e7			2.59702
$764$	7.62362e6			0.472528
$765$		0			0
$766$			2.25859e6i			0.139080i
$767$			8.66685e6i			0.531952i
$768$		0			0
$769$			2.19492e7i			1.33845i	0.743059	+	0.669226i	$0.233374\pi$
$769$			2.19492e7i			1.33845i	−0.743059	+	0.669226i	$0.766626\pi$
$770$	2.12833e6			0.129363
$771$		0			0
$772$	1.17752e7			0.711092
$773$	−2.39338e7			−1.44066			−0.720332	−	0.693629i	$-0.756011\pi$
$773$	−2.39338e7			−1.44066			−0.720332	+	0.693629i	$0.756011\pi$
$774$		0			0
$775$	2.08122e7			1.24470
$776$	−4.33903e6			−0.258666
$777$		0			0
$778$		−	1.39059e7i		−	0.823665i
$779$	194126.			0.0114614
$780$		0			0
$781$			4.14411e6i			0.243111i
$782$	1.27539e7	+	8.95062e6i	0.745808	+	0.523403i
$783$		0			0
$784$	1.28550e7			0.746934
$785$		−	4.80647e6i		−	0.278389i
$786$		0			0
$787$		−	1.61006e7i		−	0.926625i	−0.886195	−	0.463313i	$-0.846661\pi$
$787$		−	1.61006e7i		−	0.926625i	0.886195	−	0.463313i	$-0.153339\pi$
$788$		−	180768.i		−	0.0103707i
$789$		0			0
$790$	−5.95428e6			−0.339439
$791$			1.30522e7i			0.741725i
$792$		0			0
$793$			1.06307e7i			0.600313i
$794$			1.79776e7i			1.01200i
$795$		0			0
$796$			8.86731e6i			0.496031i
$797$	2.42931e7			1.35468			0.677341	−	0.735669i	$-0.263132\pi$
$797$	2.42931e7			1.35468			0.677341	+	0.735669i	$0.263132\pi$
$798$		0			0
$799$		−	9.79271e6i		−	0.542670i
$800$		−	1.27562e7i		−	0.704686i
$801$		0			0
$802$		−	2.89308e6i		−	0.158827i
$803$	377011.			0.0206332
$804$		0			0
$805$	5.80032e6	+	4.07062e6i	0.315473	+	0.221397i
$806$			1.21771e7i			0.660249i
$807$		0			0
$808$	−1.93341e7			−1.04183
$809$			1.41926e7i			0.762414i	0.924490	+	0.381207i	$0.124492\pi$
$809$			1.41926e7i			0.762414i	−0.924490	+	0.381207i	$0.875508\pi$
$810$		0			0
$811$	3.08355e6			0.164626			0.0823132	−	0.996607i	$-0.473769\pi$
$811$	3.08355e6			0.164626			0.0823132	+	0.996607i	$0.473769\pi$
$812$	−1.37018e7			−0.729266
$813$		0			0
$814$	3.79538e6			0.200768
$815$	2.60939e6			0.137608
$816$		0			0
$817$	−212537.			−0.0111398
$818$			1.78678e7i			0.933658i
$819$		0			0
$820$		−	325333.i		−	0.0168964i
$821$			7.13047e6i			0.369199i	0.982814	+	0.184599i	$0.0590987\pi$
$821$			7.13047e6i			0.369199i	−0.982814	+	0.184599i	$0.940901\pi$
$822$		0			0
$823$	1.93439e7			0.995506			0.497753	−	0.867319i	$-0.334159\pi$
$823$	1.93439e7			0.995506			0.497753	+	0.867319i	$0.334159\pi$
$824$	−2.44472e7			−1.25433
$825$		0			0
$826$			2.04938e7i			1.04513i
$827$	1.78153e7			0.905793			0.452897	−	0.891563i	$-0.350391\pi$
$827$	1.78153e7			0.905793			0.452897	+	0.891563i	$0.350391\pi$
$828$		0			0
$829$	−1.54420e7			−0.780401			−0.390201	−	0.920730i	$-0.627594\pi$
$829$	−1.54420e7			−0.780401			−0.390201	+	0.920730i	$0.627594\pi$
$830$		−	4.77951e6i		−	0.240818i
$831$		0			0
$832$	1.31258e7			0.657381
$833$	−4.02453e7			−2.00957
$834$		0			0
$835$		−	1.63003e6i		−	0.0809059i
$836$		−	224319.i		−	0.0111007i
$837$		0			0
$838$		−	2.18815e7i		−	1.07639i
$839$	3.70477e7			1.81700			0.908502	−	0.417879i	$-0.137227\pi$
$839$	3.70477e7			1.81700			0.908502	+	0.417879i	$0.137227\pi$
$840$		0			0
$841$	−4.51708e6			−0.220226
$842$	−255211.			−0.0124056
$843$		0			0
$844$	1.38847e7			0.670935
$845$	−2.82119e6			−0.135922
$846$		0			0
$847$		−	2.78857e7i		−	1.33559i
$848$	−2.40175e6			−0.114693
$849$		0			0
$850$		−	1.81393e7i		−	0.861138i
$851$	1.03435e7	+	7.25902e6i	0.489604	+	0.343601i
$852$		0			0
$853$	3.43956e6			0.161856			0.0809282	−	0.996720i	$-0.474212\pi$
$853$	3.43956e6			0.161856			0.0809282	+	0.996720i	$0.474212\pi$
$854$			2.51375e7i			1.17944i
$855$		0			0
$856$			3.19142e7i			1.48867i
$857$		−	4.05994e7i		−	1.88829i	−0.329534	−	0.944144i	$-0.606892\pi$
$857$		−	4.05994e7i		−	1.88829i	0.329534	−	0.944144i	$-0.393108\pi$
$858$		0			0
$859$	1.78738e7			0.826483			0.413241	−	0.910622i	$-0.364397\pi$
$859$	1.78738e7			0.826483			0.413241	+	0.910622i	$0.364397\pi$
$860$			356188.i			0.0164223i
$861$		0			0
$862$			2.70711e7i			1.24090i
$863$		−	2.15087e7i		−	0.983075i	−0.870856	−	0.491538i	$-0.836435\pi$
$863$		−	2.15087e7i		−	0.983075i	0.870856	−	0.491538i	$-0.163565\pi$
$864$		0			0
$865$		−	376365.i		−	0.0171029i
$866$	903219.			0.0409259
$867$		0			0
$868$		−	1.92994e7i		−	0.869449i
$869$		−	1.80831e7i		−	0.812312i
$870$		0			0
$871$		−	2.13515e7i		−	0.953638i
$872$	3.84320e7			1.71160
$873$		0			0
$874$	−640104.	+	912097.i	−0.0283447	+	0.0403890i
$875$		−	1.69780e7i		−	0.749665i
$876$		0			0
$877$	−2.33588e7			−1.02554			−0.512768	−	0.858527i	$-0.671380\pi$
$877$	−2.33588e7			−1.02554			−0.512768	+	0.858527i	$0.671380\pi$
$878$		−	7.68371e6i		−	0.336384i
$879$		0			0
$880$	1.02178e6			0.0444784
$881$	3.62203e6			0.157222			0.0786108	−	0.996905i	$-0.474952\pi$
$881$	3.62203e6			0.157222			0.0786108	+	0.996905i	$0.474952\pi$
$882$		0			0
$883$	−2.05445e7			−0.886734			−0.443367	−	0.896340i	$-0.646216\pi$
$883$	−2.05445e7			−0.886734			−0.443367	+	0.896340i	$0.646216\pi$
$884$	−7.11352e6			−0.306164
$885$		0			0
$886$	−3.26579e7			−1.39767
$887$		−	1.06827e7i		−	0.455904i	−0.973672	−	0.227952i	$-0.926797\pi$
$887$		−	1.06827e7i		−	0.455904i	0.973672	−	0.227952i	$-0.0732028\pi$
$888$		0			0
$889$		−	1.11221e7i		−	0.471992i
$890$		−	3.46899e6i		−	0.146801i
$891$		0			0
$892$	7.13586e6			0.300286
$893$	700326.			0.0293881
$894$		0			0
$895$			4.05660e6i			0.169280i
$896$	1.56000e6			0.0649163
$897$		0			0
$898$	−2.47115e7			−1.02261
$899$		−	3.52531e7i		−	1.45478i
$900$		0			0
$901$	7.51918e6			0.308573
$902$	−1.47412e6			−0.0603277
$903$		0			0
$904$			1.20113e7i			0.488843i
$905$		−	3.20881e6i		−	0.130233i
$906$		0			0
$907$			3.34819e6i			0.135142i	0.997714	+	0.0675712i	$0.0215250\pi$
$907$			3.34819e6i			0.135142i	−0.997714	+	0.0675712i	$0.978475\pi$
$908$	−9.71149e6			−0.390905
$909$		0			0
$910$	4.82674e6			0.193219
$911$	−1.56830e7			−0.626085			−0.313042	−	0.949739i	$-0.601348\pi$
$911$	−1.56830e7			−0.626085			−0.313042	+	0.949739i	$0.601348\pi$
$912$		0			0
$913$	1.45153e7			0.576301
$914$	−3.30800e7			−1.30978
$915$		0			0
$916$			4.97873e6i			0.196056i
$917$	−6.28293e7			−2.46740
$918$		0			0
$919$			9.02567e6i			0.352526i	0.984343	+	0.176263i	$0.0564009\pi$
$919$			9.02567e6i			0.352526i	−0.984343	+	0.176263i	$0.943599\pi$
$920$	5.33775e6	+	3.74599e6i	0.207916	+	0.145914i
$921$		0			0
$922$	2.56132e7			0.992286
$923$			9.39826e6i			0.363114i
$924$		0			0
$925$		−	1.47111e7i		−	0.565315i
$926$		−	2.25653e7i		−	0.864794i
$927$		0			0
$928$	−2.16072e7			−0.823625
$929$			2.92952e7i			1.11367i	0.830623	+	0.556836i	$0.187985\pi$
$929$			2.92952e7i			1.11367i	−0.830623	+	0.556836i	$0.812015\pi$
$930$		0			0
$931$		−	2.87814e6i		−	0.108827i
$932$		−	8.25675e6i		−	0.311365i
$933$		0			0
$934$			913588.i			0.0342676i
$935$	−3.19888e6			−0.119666
$936$		0			0
$937$		−	2.01618e7i		−	0.750204i	−0.926983	−	0.375102i	$-0.877608\pi$
$937$		−	2.01618e7i		−	0.750204i	0.926983	−	0.375102i	$-0.122392\pi$
$938$		−	5.04881e7i		−	1.87362i
$939$		0			0
$940$		−	1.17367e6i		−	0.0433238i
$941$	−4.07629e7			−1.50069			−0.750345	−	0.661047i	$-0.770112\pi$
$941$	−4.07629e7			−1.50069			−0.750345	+	0.661047i	$0.770112\pi$
$942$		0			0
$943$	−4.01741e6	−	2.81940e6i	−0.147119	−	0.103247i
$944$			9.83873e6i			0.359343i
$945$		0			0
$946$	1.61393e6			0.0586349
$947$		−	2.71301e7i		−	0.983051i	−0.870863	−	0.491525i	$-0.836439\pi$
$947$		−	2.71301e7i		−	0.983051i	0.870863	−	0.491525i	$-0.163561\pi$
$948$		0			0
$949$	855008.			0.0308180
$950$	1.29723e6			0.0466346
$951$		0			0
$952$	−5.87376e7			−2.10050
$953$	−4.23008e7			−1.50875			−0.754373	−	0.656446i	$-0.772059\pi$
$953$	−4.23008e7			−1.50875			−0.754373	+	0.656446i	$0.772059\pi$
$954$		0			0
$955$	7.77484e6			0.275856
$956$			2.95778e6i			0.104670i
$957$		0			0
$958$			4.04005e7i			1.42224i
$959$			6.46169e7i			2.26882i
$960$		0			0
$961$	2.10260e7			0.734427
$962$	8.60738e6			0.299870
$963$		0			0
$964$			1.92400e7i			0.666827i
$965$	1.20088e7			0.415127
$966$		0			0
$967$	1.10872e7			0.381289			0.190644	−	0.981659i	$-0.438942\pi$
$967$	1.10872e7			0.381289			0.190644	+	0.981659i	$0.438942\pi$
$968$		−	2.56618e7i		−	0.880235i
$969$		0			0
$970$	−1.26722e6			−0.0432436
$971$	3.31161e7			1.12717			0.563587	−	0.826057i	$-0.309421\pi$
$971$	3.31161e7			1.12717			0.563587	+	0.826057i	$0.309421\pi$
$972$		0			0
$973$			8.94203e7i			3.02799i
$974$			4.80200e6i			0.162190i
$975$		0			0
$976$			1.20681e7i			0.405522i
$977$	4.70544e7			1.57712			0.788559	−	0.614959i	$-0.210828\pi$
$977$	4.70544e7			1.57712			0.788559	+	0.614959i	$0.210828\pi$
$978$		0			0
$979$	1.05353e7			0.351309
$980$	−4.82345e6			−0.160433
$981$		0			0
$982$	−1.49168e7			−0.493625
$983$	1.96698e7			0.649255			0.324628	−	0.945842i	$-0.394761\pi$
$983$	1.96698e7			0.649255			0.324628	+	0.945842i	$0.394761\pi$
$984$		0			0
$985$		−	184354.i		−	0.00605427i
$986$	−3.07255e7			−1.00648
$987$		0			0
$988$		−	508723.i		−	0.0165802i
$989$	4.39843e6	+	3.08679e6i	0.142990	+	0.100350i
$990$		0			0
$991$	−2.33031e7			−0.753753			−0.376876	−	0.926264i	$-0.623002\pi$
$991$	−2.33031e7			−0.753753			−0.376876	+	0.926264i	$0.623002\pi$
$992$		−	3.04345e7i		−	0.981946i
$993$		0			0
$994$			2.22233e7i			0.713415i
$995$			9.04319e6i			0.289577i
$996$		0			0
$997$	−1.98732e7			−0.633183			−0.316591	−	0.948562i	$-0.602538\pi$
$997$	−1.98732e7			−0.633183			−0.316591	+	0.948562i	$0.602538\pi$
$998$			2.56856e6i			0.0816327i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.6.c.a.206.22	yes	40
3.2	odd	2	inner	207.6.c.a.206.19	✓	40
23.22	odd	2	inner	207.6.c.a.206.20	yes	40
69.68	even	2	inner	207.6.c.a.206.21	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
207.6.c.a.206.19	✓	40	3.2	odd	2	inner
207.6.c.a.206.20	yes	40	23.22	odd	2	inner
207.6.c.a.206.21	yes	40	69.68	even	2	inner
207.6.c.a.206.22	yes	40	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 207 = 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	207.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+4.37708i q^{2} +12.8412 q^{4} +13.0959 q^{5} +213.282i q^{7} +196.273i q^{8} +O(q^{10})\)	\(q+4.37708i q^{2} +12.8412 q^{4} +13.0959 q^{5} +213.282i q^{7} +196.273i q^{8} +57.3218i q^{10} -174.086 q^{11} -394.801 q^{13} -933.553 q^{14} -448.184 q^{16} +1403.13 q^{17} +100.345i q^{19} +168.168 q^{20} -761.986i q^{22} +(1457.37 - 2076.64i) q^{23} -2953.50 q^{25} -1728.08i q^{26} +2738.81i q^{28} +5002.82i q^{29} -7046.64 q^{31} +4319.01i q^{32} +6141.63i q^{34} +2793.13i q^{35} +4980.90i q^{37} -439.219 q^{38} +2570.38i q^{40} -1934.58i q^{41} +2118.06i q^{43} -2235.47 q^{44} +(9089.59 + 6379.02i) q^{46} -6979.17i q^{47} -28682.4 q^{49} -12927.7i q^{50} -5069.73 q^{52} +5358.84 q^{53} -2279.81 q^{55} -41861.7 q^{56} -21897.7 q^{58} -21952.4i q^{59} -26926.6i q^{61} -30843.7i q^{62} -33246.5 q^{64} -5170.29 q^{65} +54081.7i q^{67} +18018.0 q^{68} -12225.7 q^{70} -23805.0i q^{71} -2165.67 q^{73} -21801.8 q^{74} +1288.55i q^{76} -37129.4i q^{77} +103875. i q^{79} -5869.39 q^{80} +8467.79 q^{82} -83380.3 q^{83} +18375.3 q^{85} -9270.89 q^{86} -34168.4i q^{88} -60517.8 q^{89} -84204.2i q^{91} +(18714.4 - 26666.5i) q^{92} +30548.3 q^{94} +1314.11i q^{95} +22107.1i q^{97} -125545. i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 600 q^{4}+O(q^{10}) \) 40 * q - 600 * q^4	\( 40 q - 600 q^{4} - 1048 q^{13} + 9728 q^{16} + 14704 q^{25} + 4640 q^{31} - 91864 q^{46} - 8192 q^{49} + 150360 q^{52} + 134592 q^{55} - 195704 q^{58} - 183416 q^{64} - 257448 q^{70} + 31088 q^{73} - 77096 q^{82} - 368760 q^{85} - 123512 q^{94}+O(q^{100}) \) 40 * q - 600 * q^4 - 1048 * q^13 + 9728 * q^16 + 14704 * q^25 + 4640 * q^31 - 91864 * q^46 - 8192 * q^49 + 150360 * q^52 + 134592 * q^55 - 195704 * q^58 - 183416 * q^64 - 257448 * q^70 + 31088 * q^73 - 77096 * q^82 - 368760 * q^85 - 123512 * q^94

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