LMFDB - Embedded newform 207.6.c.a.206.12

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			206.12
Character	$\chi$	$=$	207.206
Dual form			207.6.c.a.206.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+10.4590i q^{2} -77.3916 q^{4} +44.1133 q^{5} +173.403i q^{7} -474.753i q^{8} +O(q^{10})$	\(q+10.4590i q^{2} -77.3916 q^{4} +44.1133 q^{5} +173.403i q^{7} -474.753i q^{8} +461.383i q^{10} +517.890 q^{11} -1131.71 q^{13} -1813.63 q^{14} +2488.93 q^{16} -1731.06 q^{17} -2751.29i q^{19} -3414.00 q^{20} +5416.64i q^{22} +(-2392.69 + 843.420i) q^{23} -1179.02 q^{25} -11836.6i q^{26} -13419.9i q^{28} +3332.27i q^{29} +6810.10 q^{31} +10839.7i q^{32} -18105.3i q^{34} +7649.36i q^{35} +5314.45i q^{37} +28775.9 q^{38} -20942.9i q^{40} -8223.85i q^{41} -5226.89i q^{43} -40080.4 q^{44} +(-8821.36 - 25025.3i) q^{46} -2036.00i q^{47} -13261.4 q^{49} -12331.4i q^{50} +87585.1 q^{52} -19718.6 q^{53} +22845.8 q^{55} +82323.3 q^{56} -34852.4 q^{58} -48019.1i q^{59} +11193.7i q^{61} +71227.2i q^{62} -33727.4 q^{64} -49923.6 q^{65} -5007.80i q^{67} +133970. q^{68} -80005.0 q^{70} +41134.1i q^{71} +5100.58 q^{73} -55584.0 q^{74} +212927. i q^{76} +89803.5i q^{77} +78609.5i q^{79} +109795. q^{80} +86013.6 q^{82} +37862.0 q^{83} -76363.0 q^{85} +54668.3 q^{86} -245870. i q^{88} -58343.1 q^{89} -196242. i q^{91} +(185174. - 65273.6i) q^{92} +21294.6 q^{94} -121369. i q^{95} -61829.5i q^{97} -138702. i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$			10.4590i			1.84892i	0.381285	+	0.924458i	$0.375482\pi$
$2$			10.4590i			1.84892i	−0.381285	+	0.924458i	$0.624518\pi$
$3$		0			0
$3$		0			0
$4$	−77.3916			−2.41849
$5$	44.1133			0.789123			0.394561	−	0.918870i	$-0.370897\pi$
$5$	44.1133			0.789123			0.394561	+	0.918870i	$0.370897\pi$
$6$		0			0
$7$			173.403i			1.33755i	0.743464	+	0.668776i	$0.233181\pi$
$7$			173.403i			1.33755i	−0.743464	+	0.668776i	$0.766819\pi$
$8$		−	474.753i		−	2.62266i
$9$		0			0
$10$			461.383i			1.45902i
$11$	517.890			1.29049			0.645247	−	0.763974i	$-0.276754\pi$
$11$	517.890			1.29049			0.645247	+	0.763974i	$0.276754\pi$
$12$		0			0
$13$	−1131.71			−1.85728			−0.928641	−	0.370980i	$-0.879022\pi$
$13$	−1131.71			−1.85728			−0.928641	+	0.370980i	$0.879022\pi$
$14$	−1813.63			−2.47302
$15$		0			0
$16$	2488.93			2.43059
$17$	−1731.06			−1.45275			−0.726375	−	0.687298i	$-0.758796\pi$
$17$	−1731.06			−1.45275			−0.726375	+	0.687298i	$0.758796\pi$
$18$		0			0
$19$		−	2751.29i		−	1.74845i	−0.485523	−	0.874224i	$-0.661371\pi$
$19$		−	2751.29i		−	1.74845i	0.485523	−	0.874224i	$-0.338629\pi$
$20$	−3414.00			−1.90848
$21$		0			0
$22$			5416.64i			2.38601i
$23$	−2392.69	+	843.420i	−0.943121	+	0.332448i
$23$	−2392.69	+	843.420i	−0.943121	+	0.332448i
$24$		0			0
$25$	−1179.02			−0.377286
$26$		−	11836.6i		−	3.43396i
$27$		0			0
$28$		−	13419.9i		−	3.23485i
$29$			3332.27i			0.735776i	0.929870	+	0.367888i	$0.119919\pi$
$29$			3332.27i			0.735776i	−0.929870	+	0.367888i	$0.880081\pi$
$30$		0			0
$31$	6810.10			1.27277			0.636384	−	0.771372i	$-0.280429\pi$
$31$	6810.10			1.27277			0.636384	+	0.771372i	$0.280429\pi$
$32$			10839.7i			1.87130i
$33$		0			0
$34$		−	18105.3i		−	2.68601i
$35$			7649.36i			1.05549i
$36$		0			0
$37$			5314.45i			0.638195i	0.947722	+	0.319098i	$0.103380\pi$
$37$			5314.45i			0.638195i	−0.947722	+	0.319098i	$0.896620\pi$
$38$	28775.9			3.23273
$39$		0			0
$40$		−	20942.9i		−	2.06960i
$41$		−	8223.85i		−	0.764039i	−0.924154	−	0.382019i	$-0.875229\pi$
$41$		−	8223.85i		−	0.764039i	0.924154	−	0.382019i	$-0.124771\pi$
$42$		0			0
$43$		−	5226.89i		−	0.431094i	−0.976493	−	0.215547i	$-0.930847\pi$
$43$		−	5226.89i		−	0.431094i	0.976493	−	0.215547i	$-0.0691535\pi$
$44$	−40080.4			−3.12104
$45$		0			0
$46$	−8821.36	−	25025.3i	−0.614669	−	1.74375i
$47$		−	2036.00i		−	0.134441i	−0.997738	−	0.0672206i	$-0.978587\pi$
$47$		−	2036.00i		−	0.134441i	0.997738	−	0.0672206i	$-0.0214131\pi$
$48$		0			0
$49$	−13261.4			−0.789043
$50$		−	12331.4i		−	0.697569i
$51$		0			0
$52$	87585.1			4.49181
$53$	−19718.6			−0.964244			−0.482122	−	0.876104i	$-0.660134\pi$
$53$	−19718.6			−0.964244			−0.482122	+	0.876104i	$0.660134\pi$
$54$		0			0
$55$	22845.8			1.01836
$56$	82323.3			3.50795
$57$		0			0
$58$	−34852.4			−1.36039
$59$		−	48019.1i		−	1.79591i	−0.440091	−	0.897953i	$-0.645054\pi$
$59$		−	48019.1i		−	1.79591i	0.440091	−	0.897953i	$-0.354946\pi$
$60$		0			0
$61$			11193.7i			0.385167i	0.981281	+	0.192584i	$0.0616866\pi$
$61$			11193.7i			0.385167i	−0.981281	+	0.192584i	$0.938313\pi$
$62$			71227.2i			2.35324i
$63$		0			0
$64$	−33727.4			−1.02928
$65$	−49923.6			−1.46562
$66$		0			0
$67$		−	5007.80i		−	0.136289i	−0.997675	−	0.0681444i	$-0.978292\pi$
$67$		−	5007.80i		−	0.136289i	0.997675	−	0.0681444i	$-0.0217079\pi$
$68$	133970.			3.51346
$69$		0			0
$70$	−80005.0			−1.95151
$71$			41134.1i			0.968402i	0.874957	+	0.484201i	$0.160890\pi$
$71$			41134.1i			0.968402i	−0.874957	+	0.484201i	$0.839110\pi$
$72$		0			0
$73$	5100.58			0.112024			0.0560122	−	0.998430i	$-0.482161\pi$
$73$	5100.58			0.112024			0.0560122	+	0.998430i	$0.482161\pi$
$74$	−55584.0			−1.17997
$75$		0			0
$76$			212927.i			4.22860i
$77$			89803.5i			1.72610i
$78$		0			0
$79$			78609.5i			1.41712i	0.705650	+	0.708561i	$0.250655\pi$
$79$			78609.5i			1.41712i	−0.705650	+	0.708561i	$0.749345\pi$
$80$	109795.			1.91804
$81$		0			0
$82$	86013.6			1.41264
$83$	37862.0			0.603265			0.301633	−	0.953424i	$-0.402468\pi$
$83$	37862.0			0.603265			0.301633	+	0.953424i	$0.402468\pi$
$84$		0			0
$85$	−76363.0			−1.14640
$86$	54668.3			0.797057
$87$		0			0
$88$		−	245870.i		−	3.38453i
$89$	−58343.1			−0.780755			−0.390377	−	0.920655i	$-0.627655\pi$
$89$	−58343.1			−0.780755			−0.390377	+	0.920655i	$0.627655\pi$
$90$		0			0
$91$		−	196242.i		−	2.48421i
$92$	185174.	−	65273.6i	2.28093	−	0.804022i
$93$		0			0
$94$	21294.6			0.248570
$95$		−	121369.i		−	1.37974i
$96$		0			0
$97$		−	61829.5i		−	0.667216i	−0.942712	−	0.333608i	$-0.891734\pi$
$97$		−	61829.5i		−	0.667216i	0.942712	−	0.333608i	$-0.108266\pi$
$98$		−	138702.i		−	1.45887i
$99$		0			0
$100$	91246.1			0.912461
$101$		−	76854.6i		−	0.749664i	−0.927093	−	0.374832i	$-0.877700\pi$
$101$		−	76854.6i		−	0.749664i	0.927093	−	0.374832i	$-0.122300\pi$
$102$		0			0
$103$		−	100330.i		−	0.931830i	−0.884830	−	0.465915i	$-0.845725\pi$
$103$		−	100330.i		−	0.931830i	0.884830	−	0.465915i	$-0.154275\pi$
$104$			537284.i			4.87102i
$105$		0			0
$106$		−	206238.i		−	1.78281i
$107$	−117832.			−0.994954			−0.497477	−	0.867477i	$-0.665740\pi$
$107$	−117832.			−0.994954			−0.497477	+	0.867477i	$0.665740\pi$
$108$		0			0
$109$			90155.4i			0.726817i	0.931630	+	0.363409i	$0.118387\pi$
$109$			90155.4i			0.726817i	−0.931630	+	0.363409i	$0.881613\pi$
$110$			238946.i			1.88286i
$111$		0			0
$112$			431587.i			3.25104i
$113$	−18081.2			−0.133209			−0.0666043	−	0.997779i	$-0.521217\pi$
$113$	−18081.2			−0.133209			−0.0666043	+	0.997779i	$0.521217\pi$
$114$		0			0
$115$	−105550.	+	37206.0i	−0.744238	+	0.262342i
$116$		−	257890.i		−	1.77946i
$117$		0			0
$118$	502234.			3.32048
$119$		−	300171.i		−	1.94313i
$120$		0			0
$121$	107159.			0.665376
$122$	−117075.			−0.712141
$123$		0			0
$124$	−527045.			−3.07818
$125$	−189864.			−1.08685
$126$		0			0
$127$	−86093.6			−0.473655			−0.236827	−	0.971552i	$-0.576108\pi$
$127$	−86093.6			−0.473655			−0.236827	+	0.971552i	$0.576108\pi$
$128$		−	5885.61i		−	0.0317517i
$129$		0			0
$130$		−	522153.i		−	2.70981i
$131$		−	26526.8i		−	0.135054i	−0.997717	−	0.0675270i	$-0.978489\pi$
$131$		−	26526.8i		−	0.135054i	0.997717	−	0.0675270i	$-0.0215109\pi$
$132$		0			0
$133$	477081.			2.33864
$134$	52376.8			0.251986
$135$		0			0
$136$			821828.i			3.81007i
$137$	−128955.			−0.586997			−0.293498	−	0.955960i	$-0.594820\pi$
$137$	−128955.			−0.586997			−0.293498	+	0.955960i	$0.594820\pi$
$138$		0			0
$139$	−176015.			−0.772701			−0.386351	−	0.922352i	$-0.626265\pi$
$139$	−176015.			−0.772701			−0.386351	+	0.922352i	$0.626265\pi$
$140$		−	591996.i		−	2.55269i
$141$		0			0
$142$	−430223.			−1.79049
$143$	−586103.			−2.39681
$144$		0			0
$145$			146998.i			0.580617i
$146$			53347.2i			0.207124i
$147$		0			0
$148$		−	411293.i		−	1.54347i
$149$	−101289.			−0.373764			−0.186882	−	0.982382i	$-0.559838\pi$
$149$	−101289.			−0.373764			−0.186882	+	0.982382i	$0.559838\pi$
$150$		0			0
$151$	198135.			0.707162			0.353581	−	0.935404i	$-0.384964\pi$
$151$	198135.			0.707162			0.353581	+	0.935404i	$0.384964\pi$
$152$	−1.30618e6			−4.58559
$153$		0			0
$154$	−939259.			−3.19142
$155$	300416.			1.00437
$156$		0			0
$157$			65451.1i			0.211918i	0.994371	+	0.105959i	$0.0337912\pi$
$157$			65451.1i			0.211918i	−0.994371	+	0.105959i	$0.966209\pi$
$158$	−822180.			−2.62014
$159$		0			0
$160$			478176.i			1.47668i
$161$	−146251.	−	414899.i	−0.444667	−	1.26147i
$162$		0			0
$163$	−170460.			−0.502520			−0.251260	−	0.967920i	$-0.580845\pi$
$163$	−170460.			−0.502520			−0.251260	+	0.967920i	$0.580845\pi$
$164$			636457.i			1.84782i
$165$		0			0
$166$			396000.i			1.11539i
$167$			503704.i			1.39760i	0.715315	+	0.698802i	$0.246283\pi$
$167$			503704.i			1.39760i	−0.715315	+	0.698802i	$0.753717\pi$
$168$		0			0
$169$	909481.			2.44950
$170$		−	798683.i		−	2.11959i
$171$		0			0
$172$			404517.i			1.04260i
$173$			243566.i			0.618731i	0.950943	+	0.309366i	$0.100117\pi$
$173$			243566.i			0.618731i	−0.950943	+	0.309366i	$0.899883\pi$
$174$		0			0
$175$		−	204445.i		−	0.504639i
$176$	1.28899e6			3.13667
$177$		0			0
$178$		−	610213.i		−	1.44355i
$179$			231089.i			0.539073i	0.962990	+	0.269537i	$0.0868705\pi$
$179$			231089.i			0.539073i	−0.962990	+	0.269537i	$0.913130\pi$
$180$		0			0
$181$		−	517085.i		−	1.17318i	−0.809883	−	0.586591i	$-0.800470\pi$
$181$		−	517085.i		−	1.17318i	0.809883	−	0.586591i	$-0.199530\pi$
$182$	2.05250e6			4.59309
$183$		0			0
$184$	400416.	+	1.13594e6i	0.871900	+	2.47349i
$185$			234438.i			0.503614i
$186$		0			0
$187$	−896502.			−1.87477
$188$			157569.i			0.325144i
$189$		0			0
$190$	1.26940e6			2.55102
$191$	−358901.			−0.711855			−0.355928	−	0.934514i	$-0.615835\pi$
$191$	−358901.			−0.711855			−0.355928	+	0.934514i	$0.615835\pi$
$192$		0			0
$193$	−333477.			−0.644425			−0.322212	−	0.946667i	$-0.604427\pi$
$193$	−333477.			−0.644425			−0.322212	+	0.946667i	$0.604427\pi$
$194$	646678.			1.23363
$195$		0			0
$196$	1.02632e6			1.90829
$197$			336020.i			0.616878i	0.951244	+	0.308439i	$0.0998066\pi$
$197$			336020.i			0.616878i	−0.951244	+	0.308439i	$0.900193\pi$
$198$		0			0
$199$			626490.i			1.12145i	0.828000	+	0.560727i	$0.189478\pi$
$199$			626490.i			1.12145i	−0.828000	+	0.560727i	$0.810522\pi$
$200$			559742.i			0.989493i
$201$		0			0
$202$	803826.			1.38607
$203$	−577825.			−0.984138
$204$		0			0
$205$		−	362781.i		−	0.602920i
$206$	1.04935e6			1.72287
$207$		0			0
$208$	−2.81675e6			−4.51430
$209$		−	1.42487e6i		−	2.25636i
$210$		0			0
$211$	−1.26731e6			−1.95963			−0.979817	−	0.199894i	$-0.935940\pi$
$211$	−1.26731e6			−1.95963			−0.979817	+	0.199894i	$0.935940\pi$
$212$	1.52606e6			2.33201
$213$		0			0
$214$		−	1.23241e6i		−	1.83959i
$215$		−	230575.i		−	0.340186i
$216$		0			0
$217$			1.18089e6i			1.70239i
$218$	−942939.			−1.34382
$219$		0			0
$220$	−1.76808e6			−2.46289
$221$	1.95907e6			2.69817
$222$		0			0
$223$	860931.			1.15933			0.579664	−	0.814856i	$-0.303184\pi$
$223$	860931.			1.15933			0.579664	+	0.814856i	$0.303184\pi$
$224$	−1.87964e6			−2.50296
$225$		0			0
$226$		−	189113.i		−	0.246291i
$227$	−755694.			−0.973377			−0.486689	−	0.873576i	$-0.661795\pi$
$227$	−755694.			−0.973377			−0.486689	+	0.873576i	$0.661795\pi$
$228$		0			0
$229$		−	562240.i		−	0.708489i	−0.935153	−	0.354244i	$-0.884738\pi$
$229$		−	562240.i		−	0.708489i	0.935153	−	0.354244i	$-0.115262\pi$
$230$	−389139.	−	1.10395e6i	−0.485049	−	1.37603i
$231$		0			0
$232$	1.58201e6			1.92969
$233$		−	250842.i		−	0.302698i	−0.988480	−	0.151349i	$-0.951638\pi$
$233$		−	250842.i		−	0.302698i	0.988480	−	0.151349i	$-0.0483618\pi$
$234$		0			0
$235$		−	89814.5i		−	0.106091i
$236$			3.71627e6i			4.34338i
$237$		0			0
$238$	3.13950e6			3.59268
$239$			509820.i			0.577328i	0.957431	+	0.288664i	$0.0932110\pi$
$239$			509820.i			0.577328i	−0.957431	+	0.288664i	$0.906789\pi$
$240$		0			0
$241$			786704.i			0.872507i	0.899824	+	0.436253i	$0.143695\pi$
$241$			786704.i			0.872507i	−0.899824	+	0.436253i	$0.856305\pi$
$242$			1.12078e6i			1.23022i
$243$		0			0
$244$		−	866298.i		−	0.931522i
$245$	−585006.			−0.622652
$246$		0			0
$247$			3.11367e6i			3.24736i
$248$		−	3.23312e6i		−	3.33804i
$249$		0			0
$250$		−	1.98580e6i		−	2.00949i
$251$	−1.45085e6			−1.45358			−0.726788	−	0.686862i	$-0.758988\pi$
$251$	−1.45085e6			−1.45358			−0.726788	+	0.686862i	$0.758988\pi$
$252$		0			0
$253$	−1.23915e6	+	436799.i	−1.21709	+	0.429023i
$254$		−	900457.i		−	0.875747i
$255$		0			0
$256$	−1.01772e6			−0.970574
$257$		−	650857.i		−	0.614686i	−0.951599	−	0.307343i	$-0.900560\pi$
$257$		−	650857.i		−	0.614686i	0.951599	−	0.307343i	$-0.0994398\pi$
$258$		0			0
$259$	−921538.			−0.853619
$260$	3.86367e6			3.54459
$261$		0			0
$262$	277445.			0.249703
$263$	1.17033e6			1.04332			0.521661	−	0.853153i	$-0.325312\pi$
$263$	1.17033e6			1.04332			0.521661	+	0.853153i	$0.325312\pi$
$264$		0			0
$265$	−869854.			−0.760907
$266$			4.98981e6i			4.32394i
$267$		0			0
$268$			387562.i			0.329613i
$269$			615328.i			0.518473i	0.965814	+	0.259237i	$0.0834709\pi$
$269$			615328.i			0.518473i	−0.965814	+	0.259237i	$0.916529\pi$
$270$		0			0
$271$	1.37516e6			1.13745			0.568723	−	0.822529i	$-0.307438\pi$
$271$	1.37516e6			1.13745			0.568723	+	0.822529i	$0.307438\pi$
$272$	−4.30850e6			−3.53105
$273$		0			0
$274$		−	1.34874e6i		−	1.08531i
$275$	−610602.			−0.486885
$276$		0			0
$277$	−1.86615e6			−1.46133			−0.730663	−	0.682739i	$-0.760789\pi$
$277$	−1.86615e6			−1.46133			−0.730663	+	0.682739i	$0.760789\pi$
$278$		−	1.84094e6i		−	1.42866i
$279$		0			0
$280$	3.63155e6			2.76820
$281$	13753.1			0.0103904			0.00519521	−	0.999987i	$-0.498346\pi$
$281$	13753.1			0.0103904			0.00519521	+	0.999987i	$0.498346\pi$
$282$		0			0
$283$		−	130122.i		−	0.0965796i	−0.998833	−	0.0482898i	$-0.984623\pi$
$283$		−	130122.i		−	0.0965796i	0.998833	−	0.0482898i	$-0.0153771\pi$
$284$		−	3.18343e6i		−	2.34207i
$285$		0			0
$286$		−	6.13008e6i		−	4.43150i
$287$	1.42604e6			1.02194
$288$		0			0
$289$	1.57673e6			1.11048
$290$	−1.53745e6			−1.07351
$291$		0			0
$292$	−394742.			−0.270930
$293$	−2.43582e6			−1.65758			−0.828792	−	0.559557i	$-0.810971\pi$
$293$	−2.43582e6			−1.65758			−0.828792	+	0.559557i	$0.810971\pi$
$294$		0			0
$295$		−	2.11828e6i		−	1.41719i
$296$	2.52305e6			1.67377
$297$		0			0
$298$		−	1.05939e6i		−	0.691057i
$299$	2.70784e6	−	954509.i	1.75164	−	0.617450i
$300$		0			0
$301$	906356.			0.576611
$302$			2.07230e6i			1.30748i
$303$		0			0
$304$		−	6.84777e6i		−	4.24977i
$305$			493791.i			0.303944i
$306$		0			0
$307$	−154914.			−0.0938089			−0.0469044	−	0.998899i	$-0.514936\pi$
$307$	−154914.			−0.0938089			−0.0469044	+	0.998899i	$0.514936\pi$
$308$		−	6.95004e6i		−	4.17456i
$309$		0			0
$310$			3.14207e6i			1.85700i
$311$			546429.i			0.320356i	0.987088	+	0.160178i	$0.0512068\pi$
$311$			546429.i			0.320356i	−0.987088	+	0.160178i	$0.948793\pi$
$312$		0			0
$313$			2.87208e6i			1.65705i	0.559953	+	0.828524i	$0.310819\pi$
$313$			2.87208e6i			1.65705i	−0.559953	+	0.828524i	$0.689181\pi$
$314$	−684555.			−0.391818
$315$		0			0
$316$		−	6.08371e6i		−	3.42729i
$317$			143647.i			0.0802875i	0.999194	+	0.0401437i	$0.0127816\pi$
$317$			143647.i			0.0802875i	−0.999194	+	0.0401437i	$0.987218\pi$
$318$		0			0
$319$			1.72575e6i			0.949515i
$320$	−1.48783e6			−0.812228
$321$		0			0
$322$	4.33945e6	−	1.52965e6i	2.33236	−	0.822151i
$323$			4.76266e6i			2.54006i
$324$		0			0
$325$	1.33431e6			0.700726
$326$		−	1.78285e6i		−	0.929117i
$327$		0			0
$328$	−3.90429e6			−2.00382
$329$	353047.			0.179822
$330$		0			0
$331$	1.55592e6			0.780582			0.390291	−	0.920692i	$-0.372374\pi$
$331$	1.55592e6			0.780582			0.390291	+	0.920692i	$0.372374\pi$
$332$	−2.93020e6			−1.45899
$333$		0			0
$334$	−5.26826e6			−2.58405
$335$		−	220911.i		−	0.107549i
$336$		0			0
$337$			2.28296e6i			1.09502i	0.836798	+	0.547512i	$0.184425\pi$
$337$			2.28296e6i			1.09502i	−0.836798	+	0.547512i	$0.815575\pi$
$338$			9.51230e6i			4.52891i
$339$		0			0
$340$	5.90985e6			2.77255
$341$	3.52689e6			1.64250
$342$		0			0
$343$			614808.i			0.282165i
$344$	−2.48148e6			−1.13061
$345$		0			0
$346$	−2.54747e6			−1.14398
$347$		−	3.53356e6i		−	1.57539i	−0.616064	−	0.787696i	$-0.711274\pi$
$347$		−	3.53356e6i		−	1.57539i	0.616064	−	0.787696i	$-0.288726\pi$
$348$		0			0
$349$	1.76621e6			0.776210			0.388105	−	0.921615i	$-0.373130\pi$
$349$	1.76621e6			0.776210			0.388105	+	0.921615i	$0.373130\pi$
$350$	2.13830e6			0.933035
$351$		0			0
$352$			5.61379e6i			2.41490i
$353$			710193.i			0.303347i	0.988431	+	0.151673i	$0.0484662\pi$
$353$			710193.i			0.303347i	−0.988431	+	0.151673i	$0.951534\pi$
$354$		0			0
$355$			1.81456e6i			0.764188i
$356$	4.51527e6			1.88825
$357$		0			0
$358$	−2.41698e6			−0.996700
$359$	1.05118e6			0.430468			0.215234	−	0.976563i	$-0.430949\pi$
$359$	1.05118e6			0.430468			0.215234	+	0.976563i	$0.430949\pi$
$360$		0			0
$361$	−5.09351e6			−2.05707
$362$	5.40821e6			2.16911
$363$		0			0
$364$			1.51875e7i			6.00803i
$365$	225004.			0.0884010
$366$		0			0
$367$			1.52284e6i			0.590186i	0.955469	+	0.295093i	$0.0953506\pi$
$367$			1.52284e6i			0.590186i	−0.955469	+	0.295093i	$0.904649\pi$
$368$	−5.95524e6	+	2.09921e6i	−2.29235	+	0.808047i
$369$		0			0
$370$	−2.45199e6			−0.931140
$371$		−	3.41926e6i		−	1.28973i
$372$		0			0
$373$		−	2.05333e6i		−	0.764163i	−0.924129	−	0.382082i	$-0.875207\pi$
$373$		−	2.05333e6i		−	0.764163i	0.924129	−	0.382082i	$-0.124793\pi$
$374$		−	9.37655e6i		−	3.46628i
$375$		0			0
$376$	−966595.			−0.352594
$377$		−	3.77118e6i		−	1.36654i
$378$		0			0
$379$			2.26784e6i			0.810989i	0.914098	+	0.405494i	$0.132901\pi$
$379$			2.26784e6i			0.810989i	−0.914098	+	0.405494i	$0.867099\pi$
$380$			9.39290e6i			3.33688i
$381$		0			0
$382$		−	3.75376e6i		−	1.31616i
$383$	−1.88984e6			−0.658307			−0.329153	−	0.944276i	$-0.606763\pi$
$383$	−1.88984e6			−0.658307			−0.329153	+	0.944276i	$0.606763\pi$
$384$		0			0
$385$			3.96153e6i			1.36211i
$386$		−	3.48785e6i		−	1.19149i
$387$		0			0
$388$			4.78509e6i			1.61365i
$389$	−5.14489e6			−1.72386			−0.861930	−	0.507028i	$-0.830744\pi$
$389$	−5.14489e6			−1.72386			−0.861930	+	0.507028i	$0.830744\pi$
$390$		0			0
$391$	4.14191e6	−	1.46001e6i	1.37012	−	0.482964i
$392$			6.29591e6i			2.06939i
$393$		0			0
$394$	−3.51445e6			−1.14056
$395$			3.46772e6i			1.11828i
$396$		0			0
$397$	−3.65434e6			−1.16368			−0.581838	−	0.813305i	$-0.697666\pi$
$397$	−3.65434e6			−1.16368			−0.581838	+	0.813305i	$0.697666\pi$
$398$	−6.55249e6			−2.07347
$399$		0			0
$400$	−2.93449e6			−0.917028
$401$	−1.73549e6			−0.538966			−0.269483	−	0.963005i	$-0.586853\pi$
$401$	−1.73549e6			−0.538966			−0.269483	+	0.963005i	$0.586853\pi$
$402$		0			0
$403$	−7.70708e6			−2.36389
$404$			5.94790e6i			1.81305i
$405$		0			0
$406$		−	6.04349e6i		−	1.81959i
$407$			2.75230e6i			0.823587i
$408$		0			0
$409$	773910.			0.228761			0.114381	−	0.993437i	$-0.463512\pi$
$409$	773910.			0.228761			0.114381	+	0.993437i	$0.463512\pi$
$410$	3.79434e6			1.11475
$411$		0			0
$412$			7.76468e6i			2.25362i
$413$	8.32663e6			2.40212
$414$		0			0
$415$	1.67022e6			0.476050
$416$		−	1.22675e7i		−	3.47553i
$417$		0			0
$418$	1.49028e7			4.17182
$419$	912124.			0.253816			0.126908	−	0.991914i	$-0.459495\pi$
$419$	912124.			0.253816			0.126908	+	0.991914i	$0.459495\pi$
$420$		0			0
$421$		−	5.15333e6i		−	1.41704i	−0.705690	−	0.708521i	$-0.749363\pi$
$421$		−	5.15333e6i		−	1.41704i	0.705690	−	0.708521i	$-0.250637\pi$
$422$		−	1.32548e7i		−	3.62320i
$423$		0			0
$424$			9.36147e6i			2.52889i
$425$	2.04096e6			0.548102
$426$		0			0
$427$	−1.94102e6			−0.515181
$428$	9.11919e6			2.40628
$429$		0			0
$430$	2.41160e6			0.628975
$431$	−4.34258e6			−1.12604			−0.563021	−	0.826443i	$-0.690361\pi$
$431$	−4.34258e6			−1.12604			−0.563021	+	0.826443i	$0.690361\pi$
$432$		0			0
$433$		−	7.16361e6i		−	1.83617i	−0.396386	−	0.918084i	$-0.629736\pi$
$433$		−	7.16361e6i		−	1.83617i	0.396386	−	0.918084i	$-0.370264\pi$
$434$	−1.23510e7			−3.14758
$435$		0			0
$436$		−	6.97727e6i		−	1.75780i
$437$	2.32049e6	+	6.58300e6i	0.581268	+	1.64900i
$438$		0			0
$439$	1.93694e6			0.479683			0.239841	−	0.970812i	$-0.422905\pi$
$439$	1.93694e6			0.479683			0.239841	+	0.970812i	$0.422905\pi$
$440$		−	1.08461e7i		−	2.67081i
$441$		0			0
$442$			2.04900e7i			4.98868i
$443$			3.21582e6i			0.778544i	0.921123	+	0.389272i	$0.127273\pi$
$443$			3.21582e6i			0.778544i	−0.921123	+	0.389272i	$0.872727\pi$
$444$		0			0
$445$	−2.57371e6			−0.616111
$446$			9.00451e6i			2.14350i
$447$		0			0
$448$		−	5.84842e6i		−	1.37671i
$449$			3.92589e6i			0.919014i	0.888174	+	0.459507i	$0.151974\pi$
$449$			3.92589e6i			0.919014i	−0.888174	+	0.459507i	$0.848026\pi$
$450$		0			0
$451$		−	4.25905e6i		−	0.985988i
$452$	1.39934e6			0.322163
$453$		0			0
$454$		−	7.90383e6i		−	1.79969i
$455$		−	8.65688e6i		−	1.96035i
$456$		0			0
$457$			969241.i			0.217091i	0.994092	+	0.108545i	$0.0346193\pi$
$457$			969241.i			0.217091i	−0.994092	+	0.108545i	$0.965381\pi$
$458$	5.88049e6			1.30994
$459$		0			0
$460$	8.16865e6	−	2.87943e6i	1.79993	−	0.634472i
$461$		−	1.67188e6i		−	0.366397i	−0.983076	−	0.183198i	$-0.941355\pi$
$461$		−	1.67188e6i		−	0.366397i	0.983076	−	0.183198i	$-0.0586451\pi$
$462$		0			0
$463$	844202.			0.183018			0.0915090	−	0.995804i	$-0.470831\pi$
$463$	844202.			0.183018			0.0915090	+	0.995804i	$0.470831\pi$
$464$			8.29379e6i			1.78837i
$465$		0			0
$466$	2.62356e6			0.559663
$467$	−1.10340e6			−0.234121			−0.117061	−	0.993125i	$-0.537347\pi$
$467$	−1.10340e6			−0.234121			−0.117061	+	0.993125i	$0.537347\pi$
$468$		0			0
$469$	868366.			0.182293
$470$	939374.			0.196153
$471$		0			0
$472$	−2.27972e7			−4.71006
$473$		−	2.70696e6i		−	0.556325i
$474$		0			0
$475$			3.24382e6i			0.659664i
$476$			2.32307e7i			4.69943i
$477$		0			0
$478$	−5.33223e6			−1.06743
$479$	902832.			0.179791			0.0898955	−	0.995951i	$-0.471347\pi$
$479$	902832.			0.179791			0.0898955	+	0.995951i	$0.471347\pi$
$480$		0			0
$481$		−	6.01443e6i		−	1.18531i
$482$	−8.22817e6			−1.61319
$483$		0			0
$484$	−8.29324e6			−1.60920
$485$		−	2.72750e6i		−	0.526515i
$486$		0			0
$487$	7.41295e6			1.41634			0.708172	−	0.706040i	$-0.249520\pi$
$487$	7.41295e6			1.41634			0.708172	+	0.706040i	$0.249520\pi$
$488$	5.31424e6			1.01016
$489$		0			0
$490$		−	6.11861e6i		−	1.15123i
$491$		−	5.33114e6i		−	0.997968i	−0.866611	−	0.498984i	$-0.833707\pi$
$491$		−	5.33114e6i		−	0.997968i	0.866611	−	0.498984i	$-0.166293\pi$
$492$		0			0
$493$		−	5.76838e6i		−	1.06890i
$494$	−3.25660e7			−6.00409
$495$		0			0
$496$	1.69499e7			3.09358
$497$	−7.13275e6			−1.29529
$498$		0			0
$499$	9.49878e6			1.70772			0.853860	−	0.520503i	$-0.174255\pi$
$499$	9.49878e6			1.70772			0.853860	+	0.520503i	$0.174255\pi$
$500$	1.46939e7			2.62853
$501$		0			0
$502$		−	1.51745e7i		−	2.68754i
$503$	−4.96896e6			−0.875681			−0.437841	−	0.899053i	$-0.644257\pi$
$503$	−4.96896e6			−0.875681			−0.437841	+	0.899053i	$0.644257\pi$
$504$		0			0
$505$		−	3.39031e6i		−	0.591577i
$506$	−4.56850e6	−	1.29604e7i	−0.793226	−	2.25030i
$507$		0			0
$508$	6.66292e6			1.14553
$509$		−	408548.i		−	0.0698954i	−0.999389	−	0.0349477i	$-0.988874\pi$
$509$		−	408548.i		−	0.0698954i	0.999389	−	0.0349477i	$-0.0111265\pi$
$510$		0			0
$511$			884454.i			0.149838i
$512$		−	1.08327e7i		−	1.82626i
$513$		0			0
$514$	6.80735e6			1.13650
$515$		−	4.42587e6i		−	0.735328i
$516$		0			0
$517$		−	1.05442e6i		−	0.173496i
$518$		−	9.63841e6i		−	1.57827i
$519$		0			0
$520$			2.37014e7i			3.84384i
$521$	−244432.			−0.0394515			−0.0197258	−	0.999805i	$-0.506279\pi$
$521$	−244432.			−0.0394515			−0.0197258	+	0.999805i	$0.506279\pi$
$522$		0			0
$523$			4.18847e6i			0.669579i	0.942293	+	0.334789i	$0.108665\pi$
$523$			4.18847e6i			0.669579i	−0.942293	+	0.334789i	$0.891335\pi$
$524$			2.05296e6i			0.326626i
$525$		0			0
$526$			1.22405e7i			1.92902i
$527$	−1.17887e7			−1.84902
$528$		0			0
$529$	5.01363e6	−	4.03609e6i	0.778956	−	0.627078i
$530$		−	9.09784e6i		−	1.40685i
$531$		0			0
$532$	−3.69221e7			−5.65597
$533$			9.30703e6i			1.41904i
$534$		0			0
$535$	−5.19795e6			−0.785141
$536$	−2.37747e6			−0.357440
$537$		0			0
$538$	−6.43575e6			−0.958613
$539$	−6.86798e6			−1.01826
$540$		0			0
$541$	8.72415e6			1.28153			0.640767	−	0.767735i	$-0.278616\pi$
$541$	8.72415e6			1.28153			0.640767	+	0.767735i	$0.278616\pi$
$542$			1.43829e7i			2.10304i
$543$		0			0
$544$		−	1.87643e7i		−	2.71853i
$545$			3.97705e6i			0.573548i
$546$		0			0
$547$	−3.58856e6			−0.512804			−0.256402	−	0.966570i	$-0.582537\pi$
$547$	−3.58856e6			−0.512804			−0.256402	+	0.966570i	$0.582537\pi$
$548$	9.98001e6			1.41964
$549$		0			0
$550$		−	6.38631e6i		−	0.900209i
$551$	9.16806e6			1.28647
$552$		0			0
$553$	−1.36311e7			−1.89547
$554$		−	1.95181e7i		−	2.70187i
$555$		0			0
$556$	1.36220e7			1.86877
$557$	−7.31648e6			−0.999228			−0.499614	−	0.866248i	$-0.666525\pi$
$557$	−7.31648e6			−0.999228			−0.499614	+	0.866248i	$0.666525\pi$
$558$		0			0
$559$			5.91534e6i			0.800663i
$560$			1.90387e7i			2.56547i
$561$		0			0
$562$			143844.i			0.0192110i
$563$	4.49544e6			0.597725			0.298862	−	0.954296i	$-0.403393\pi$
$563$	4.49544e6			0.597725			0.298862	+	0.954296i	$0.403393\pi$
$564$		0			0
$565$	−797623.			−0.105118
$566$	1.36095e6			0.178567
$567$		0			0
$568$	1.95285e7			2.53979
$569$	859174.			0.111250			0.0556251	−	0.998452i	$-0.482285\pi$
$569$	859174.			0.111250			0.0556251	+	0.998452i	$0.482285\pi$
$570$		0			0
$571$			6.65401e6i			0.854069i	0.904235	+	0.427035i	$0.140442\pi$
$571$			6.65401e6i			0.854069i	−0.904235	+	0.427035i	$0.859558\pi$
$572$	4.53595e7			5.79666
$573$		0			0
$574$			1.49150e7i			1.88948i
$575$	2.82103e6	−	994407.i	0.355826	−	0.125428i
$576$		0			0
$577$	1.43698e7			1.79684			0.898422	−	0.439134i	$-0.144715\pi$
$577$	1.43698e7			1.79684			0.898422	+	0.439134i	$0.144715\pi$
$578$			1.64911e7i			2.05319i
$579$		0			0
$580$		−	1.13764e7i		−	1.40422i
$581$			6.56537e6i			0.806898i
$582$		0			0
$583$	−1.02121e7			−1.24435
$584$		−	2.42152e6i		−	0.293802i
$585$		0			0
$586$		−	2.54763e7i		−	3.06473i
$587$			182057.i			0.0218078i	0.999941	+	0.0109039i	$0.00347089\pi$
$587$			182057.i			0.0218078i	−0.999941	+	0.0109039i	$0.996529\pi$
$588$		0			0
$589$		−	1.87366e7i		−	2.22537i
$590$	2.21552e7			2.62026
$591$		0			0
$592$			1.32273e7i			1.55119i
$593$			1.63564e6i			0.191008i	0.995429	+	0.0955040i	$0.0304463\pi$
$593$			1.63564e6i			0.191008i	−0.995429	+	0.0955040i	$0.969554\pi$
$594$		0			0
$595$		−	1.32415e7i		−	1.53337i
$596$	7.83893e6			0.903943
$597$		0			0
$598$	9.98325e6	+	2.83214e7i	1.14161	+	3.23864i
$599$		−	2.82294e6i		−	0.321465i	−0.986998	−	0.160733i	$-0.948614\pi$
$599$		−	2.82294e6i		−	0.321465i	0.986998	−	0.160733i	$-0.0513857\pi$
$600$		0			0
$601$	1.25163e7			1.41348			0.706739	−	0.707474i	$-0.250166\pi$
$601$	1.25163e7			1.41348			0.706739	+	0.707474i	$0.250166\pi$
$602$			9.47962e6i			1.06610i
$603$		0			0
$604$	−1.53340e7			−1.71026
$605$	4.72715e6			0.525063
$606$		0			0
$607$	−1.05369e7			−1.16075			−0.580377	−	0.814348i	$-0.697095\pi$
$607$	−1.05369e7			−1.16075			−0.580377	+	0.814348i	$0.697095\pi$
$608$	2.98232e7			3.27187
$609$		0			0
$610$	−5.16458e6			−0.561967
$611$			2.30416e6i			0.249695i
$612$		0			0
$613$			3.08564e6i			0.331661i	0.986154	+	0.165830i	$0.0530304\pi$
$613$			3.08564e6i			0.331661i	−0.986154	+	0.165830i	$0.946970\pi$
$614$		−	1.62025e6i		−	0.173445i
$615$		0			0
$616$	4.26345e7			4.52698
$617$	−3.51365e6			−0.371574			−0.185787	−	0.982590i	$-0.559483\pi$
$617$	−3.51365e6			−0.371574			−0.185787	+	0.982590i	$0.559483\pi$
$618$		0			0
$619$		−	2.54370e6i		−	0.266832i	−0.991060	−	0.133416i	$-0.957405\pi$
$619$		−	2.54370e6i		−	0.266832i	0.991060	−	0.133416i	$-0.0425947\pi$
$620$	−2.32497e7			−2.42906
$621$		0			0
$622$	−5.71512e6			−0.592311
$623$		−	1.01168e7i		−	1.04430i
$624$		0			0
$625$	−4.69111e6			−0.480370
$626$	−3.00392e7			−3.06374
$627$		0			0
$628$		−	5.06536e6i		−	0.512521i
$629$		−	9.19965e6i		−	0.927138i
$630$		0			0
$631$		−	194710.i		−	0.0194677i	−0.999953	−	0.00973387i	$-0.996902\pi$
$631$		−	194710.i		−	0.0194677i	0.999953	−	0.00973387i	$-0.00309844\pi$
$632$	3.73201e7			3.71663
$633$		0			0
$634$	−1.50241e6			−0.148445
$635$	−3.79787e6			−0.373771
$636$		0			0
$637$	1.50082e7			1.46548
$638$	−1.80497e7			−1.75557
$639$		0			0
$640$		−	259634.i		−	0.0250560i
$641$	1.29594e7			1.24577			0.622887	−	0.782312i	$-0.285960\pi$
$641$	1.29594e7			1.24577			0.622887	+	0.782312i	$0.285960\pi$
$642$		0			0
$643$		−	1.33034e7i		−	1.26893i	−0.772953	−	0.634463i	$-0.781221\pi$
$643$		−	1.33034e7i		−	1.26893i	0.772953	−	0.634463i	$-0.218779\pi$
$644$	1.13186e7	+	3.21097e7i	1.07542	+	3.05086i
$645$		0			0
$646$	−4.98129e7			−4.69635
$647$			3.02518e6i			0.284112i	0.989859	+	0.142056i	$0.0453713\pi$
$647$			3.02518e6i			0.284112i	−0.989859	+	0.142056i	$0.954629\pi$
$648$		0			0
$649$		−	2.48686e7i		−	2.31761i
$650$			1.39556e7i			1.29558i
$651$		0			0
$652$	1.31922e7			1.21534
$653$		−	125503.i		−	0.0115178i	−0.999983	−	0.00575890i	$-0.998167\pi$
$653$		−	125503.i		−	0.0115178i	0.999983	−	0.00575890i	$-0.00183313\pi$
$654$		0			0
$655$		−	1.17019e6i		−	0.106574i
$656$		−	2.04686e7i		−	1.85707i
$657$		0			0
$658$			3.69253e6i			0.332476i
$659$	−4.03251e6			−0.361711			−0.180856	−	0.983510i	$-0.557887\pi$
$659$	−4.03251e6			−0.361711			−0.180856	+	0.983510i	$0.557887\pi$
$660$		0			0
$661$			2.15378e7i			1.91733i	0.284532	+	0.958666i	$0.408162\pi$
$661$			2.15378e7i			1.91733i	−0.284532	+	0.958666i	$0.591838\pi$
$662$			1.62735e7i			1.44323i
$663$		0			0
$664$		−	1.79751e7i		−	1.58216i
$665$	2.10456e7			1.84547
$666$		0			0
$667$	−2.81050e6	−	7.97311e6i	−0.244607	−	0.693926i
$668$		−	3.89825e7i		−	3.38009i
$669$		0			0
$670$	2.31051e6			0.198848
$671$			5.79711e6i			0.497056i
$672$		0			0
$673$	1.28568e7			1.09419			0.547097	−	0.837069i	$-0.315733\pi$
$673$	1.28568e7			1.09419			0.547097	+	0.837069i	$0.315733\pi$
$674$	−2.38776e7			−2.02461
$675$		0			0
$676$	−7.03862e7			−5.92408
$677$	−2.89355e6			−0.242638			−0.121319	−	0.992614i	$-0.538712\pi$
$677$	−2.89355e6			−0.242638			−0.121319	+	0.992614i	$0.538712\pi$
$678$		0			0
$679$	1.07214e7			0.892436
$680$			3.62535e7i			3.00662i
$681$		0			0
$682$			3.68879e7i			3.03684i
$683$		−	1.96624e7i		−	1.61282i	−0.591359	−	0.806409i	$-0.701408\pi$
$683$		−	1.96624e7i		−	1.61282i	0.591359	−	0.806409i	$-0.298592\pi$
$684$		0			0
$685$	−5.68862e6			−0.463212
$686$	−6.43030e6			−0.521700
$687$		0			0
$688$		−	1.30094e7i		−	1.04781i
$689$	2.23158e7			1.79087
$690$		0			0
$691$	−6.34782e6			−0.505742			−0.252871	−	0.967500i	$-0.581375\pi$
$691$	−6.34782e6			−0.505742			−0.252871	+	0.967500i	$0.581375\pi$
$692$		−	1.88500e7i		−	1.49639i
$693$		0			0
$694$	3.69576e7			2.91277
$695$	−7.76458e6			−0.609756
$696$		0			0
$697$			1.42360e7i			1.10996i
$698$			1.84729e7i			1.43515i
$699$		0			0
$700$			1.58223e7i			1.22046i
$701$	−2.18395e6			−0.167860			−0.0839299	−	0.996472i	$-0.526747\pi$
$701$	−2.18395e6			−0.167860			−0.0839299	+	0.996472i	$0.526747\pi$
$702$		0			0
$703$	1.46216e7			1.11585
$704$	−1.74671e7			−1.32828
$705$		0			0
$706$	−7.42794e6			−0.560863
$707$	1.33268e7			1.00271
$708$		0			0
$709$			1.99365e7i			1.48947i	0.667359	+	0.744736i	$0.267425\pi$
$709$			1.99365e7i			1.48947i	−0.667359	+	0.744736i	$0.732575\pi$
$710$	−1.89785e7			−1.41292
$711$		0			0
$712$			2.76986e7i			2.04766i
$713$	−1.62945e7	+	5.74378e6i	−1.20038	+	0.423130i
$714$		0			0
$715$	−2.58549e7			−1.89138
$716$		−	1.78844e7i		−	1.30374i
$717$		0			0
$718$			1.09943e7i			0.795898i
$719$			1.54837e6i			0.111700i	0.998439	+	0.0558499i	$0.0177868\pi$
$719$			1.54837e6i			0.111700i	−0.998439	+	0.0558499i	$0.982213\pi$
$720$		0			0
$721$	1.73974e7			1.24637
$722$		−	5.32732e7i		−	3.80335i
$723$		0			0
$724$			4.00180e7i			2.83733i
$725$		−	3.92881e6i		−	0.277598i
$726$		0			0
$727$			1.59775e7i			1.12118i	0.828095	+	0.560588i	$0.189425\pi$
$727$			1.59775e7i			1.12118i	−0.828095	+	0.560588i	$0.810575\pi$
$728$	−9.31664e7			−6.51524
$729$		0			0
$730$			2.35332e6i			0.163446i
$731$			9.04808e6i			0.626272i
$732$		0			0
$733$		−	1.49482e6i		−	0.102761i	−0.998679	−	0.0513805i	$-0.983638\pi$
$733$		−	1.49482e6i		−	0.102761i	0.998679	−	0.0513805i	$-0.0163621\pi$
$734$	−1.59274e7			−1.09120
$735$		0			0
$736$	−9.14244e6	−	2.59361e7i	−0.622110	−	1.76486i
$737$		−	2.59349e6i		−	0.175880i
$738$		0			0
$739$	−1.29707e7			−0.873680			−0.436840	−	0.899539i	$-0.643902\pi$
$739$	−1.29707e7			−0.873680			−0.436840	+	0.899539i	$0.643902\pi$
$740$		−	1.81435e7i		−	1.21798i
$741$		0			0
$742$	3.57622e7			2.38459
$743$	9.56253e6			0.635478			0.317739	−	0.948178i	$-0.397076\pi$
$743$	9.56253e6			0.635478			0.317739	+	0.948178i	$0.397076\pi$
$744$		0			0
$745$	−4.46820e6			−0.294945
$746$	2.14758e7			1.41287
$747$		0			0
$748$	6.93817e7			4.53410
$749$		−	2.04323e7i		−	1.33080i
$750$		0			0
$751$		−	8.43657e6i		−	0.545841i	−0.962037	−	0.272920i	$-0.912010\pi$
$751$		−	8.43657e6i		−	0.545841i	0.962037	−	0.272920i	$-0.0879896\pi$
$752$		−	5.06745e6i		−	0.326772i
$753$		0			0
$754$	3.94429e7			2.52662
$755$	8.74039e6			0.558038
$756$		0			0
$757$		−	2.72144e7i		−	1.72607i	−0.505144	−	0.863035i	$-0.668560\pi$
$757$		−	2.72144e7i		−	1.72607i	0.505144	−	0.863035i	$-0.331440\pi$
$758$	−2.37195e7			−1.49945
$759$		0			0
$760$	−5.76200e7			−3.61859
$761$			2.54108e7i			1.59058i	0.606226	+	0.795292i	$0.292683\pi$
$761$			2.54108e7i			1.59058i	−0.606226	+	0.795292i	$0.707317\pi$
$762$		0			0
$763$	−1.56332e7			−0.972155
$764$	2.77759e7			1.72161
$765$		0			0
$766$		−	1.97659e7i		−	1.21715i
$767$			5.43438e7i			3.33550i
$768$		0			0
$769$		−	2.94827e7i		−	1.79784i	−0.438108	−	0.898922i	$-0.644351\pi$
$769$		−	2.94827e7i		−	1.79784i	0.438108	−	0.898922i	$-0.355649\pi$
$770$	−4.14338e7			−2.51842
$771$		0			0
$772$	2.58083e7			1.55853
$773$	−2.04418e7			−1.23047			−0.615233	−	0.788346i	$-0.710938\pi$
$773$	−2.04418e7			−1.23047			−0.615233	+	0.788346i	$0.710938\pi$
$774$		0			0
$775$	−8.02923e6			−0.480197
$776$	−2.93537e7			−1.74988
$777$		0			0
$778$		−	5.38106e7i		−	3.18727i
$779$	−2.26262e7			−1.33588
$780$		0			0
$781$			2.13029e7i			1.24972i
$782$	1.52703e7	+	4.33204e7i	0.892960	+	2.53324i
$783$		0			0
$784$	−3.30068e7			−1.91784
$785$			2.88726e6i			0.167229i
$786$		0			0
$787$			2.11163e7i			1.21529i	0.794208	+	0.607645i	$0.207886\pi$
$787$			2.11163e7i			1.21529i	−0.794208	+	0.607645i	$0.792114\pi$
$788$		−	2.60051e7i		−	1.49191i
$789$		0			0
$790$	−3.62691e7			−2.06761
$791$		−	3.13533e6i		−	0.178173i
$792$		0			0
$793$		−	1.26681e7i		−	0.715364i
$794$		−	3.82209e7i		−	2.15154i
$795$		0			0
$796$		−	4.84851e7i		−	2.71222i
$797$	2.89970e7			1.61699			0.808495	−	0.588503i	$-0.200283\pi$
$797$	2.89970e7			1.61699			0.808495	+	0.588503i	$0.200283\pi$
$798$		0			0
$799$			3.52444e6i			0.195310i
$800$		−	1.27802e7i		−	0.706014i
$801$		0			0
$802$		−	1.81516e7i		−	0.996502i
$803$	2.64154e6			0.144567
$804$		0			0
$805$	−6.45162e6	−	1.83026e7i	−0.350896	−	0.995457i
$806$		−	8.06087e7i		−	4.37063i
$807$		0			0
$808$	−3.64869e7			−1.96612
$809$		−	2.40945e7i		−	1.29433i	−0.762348	−	0.647167i	$-0.775954\pi$
$809$		−	2.40945e7i		−	1.29433i	0.762348	−	0.647167i	$-0.224046\pi$
$810$		0			0
$811$	2.11124e7			1.12716			0.563580	−	0.826061i	$-0.309424\pi$
$811$	2.11124e7			1.12716			0.563580	+	0.826061i	$0.309424\pi$
$812$	4.47188e7			2.38013
$813$		0			0
$814$	−2.87864e7			−1.52274
$815$	−7.51955e6			−0.396550
$816$		0			0
$817$	−1.43807e7			−0.753746
$818$			8.09436e6i			0.422960i
$819$		0			0
$820$			2.80762e7i			1.45816i
$821$			3.42065e7i			1.77113i	0.464512	+	0.885567i	$0.346230\pi$
$821$			3.42065e7i			1.77113i	−0.464512	+	0.885567i	$0.653770\pi$
$822$		0			0
$823$	2.86621e7			1.47506			0.737529	−	0.675316i	$-0.235993\pi$
$823$	2.86621e7			1.47506			0.737529	+	0.675316i	$0.235993\pi$
$824$	−4.76318e7			−2.44388
$825$		0			0
$826$			8.70886e7i			4.44131i
$827$	1.29229e7			0.657044			0.328522	−	0.944496i	$-0.393449\pi$
$827$	1.29229e7			0.657044			0.328522	+	0.944496i	$0.393449\pi$
$828$		0			0
$829$	−2.75684e7			−1.39324			−0.696620	−	0.717440i	$-0.745314\pi$
$829$	−2.75684e7			−1.39324			−0.696620	+	0.717440i	$0.745314\pi$
$830$			1.74689e7i			0.880177i
$831$		0			0
$832$	3.81698e7			1.91166
$833$	2.29564e7			1.14628
$834$		0			0
$835$			2.22201e7i			1.10288i
$836$			1.10273e8i			5.45698i
$837$		0			0
$838$			9.53995e6i			0.469284i
$839$	2.22366e7			1.09060			0.545298	−	0.838242i	$-0.316416\pi$
$839$	2.22366e7			1.09060			0.545298	+	0.838242i	$0.316416\pi$
$840$		0			0
$841$	9.40710e6			0.458634
$842$	5.38989e7			2.61999
$843$		0			0
$844$	9.80788e7			4.73935
$845$	4.01202e7			1.93295
$846$		0			0
$847$			1.85817e7i			0.889974i
$848$	−4.90783e7			−2.34369
$849$		0			0
$850$			2.13464e7i			1.01339i
$851$	−4.48231e6	−	1.27158e7i	−0.212167	−	0.601896i
$852$		0			0
$853$	−3.22072e7			−1.51558			−0.757792	−	0.652496i	$-0.773722\pi$
$853$	−3.22072e7			−1.51558			−0.757792	+	0.652496i	$0.773722\pi$
$854$		−	2.03012e7i		−	0.952525i
$855$		0			0
$856$			5.59410e7i			2.60943i
$857$		−	1.99180e7i		−	0.926388i	−0.886257	−	0.463194i	$-0.846703\pi$
$857$		−	1.99180e7i		−	0.926388i	0.886257	−	0.463194i	$-0.153297\pi$
$858$		0			0
$859$	1.23144e7			0.569419			0.284709	−	0.958614i	$-0.408103\pi$
$859$	1.23144e7			0.569419			0.284709	+	0.958614i	$0.408103\pi$
$860$			1.78446e7i			0.822736i
$861$		0			0
$862$		−	4.54192e7i		−	2.08196i
$863$			2.72897e6i			0.124730i	0.998053	+	0.0623652i	$0.0198643\pi$
$863$			2.72897e6i			0.124730i	−0.998053	+	0.0623652i	$0.980136\pi$
$864$		0			0
$865$			1.07445e7i			0.488255i
$866$	7.49245e7			3.39492
$867$		0			0
$868$		−	9.13909e7i		−	4.11722i
$869$			4.07111e7i			1.82879i
$870$		0			0
$871$			5.66739e6i			0.253127i
$872$	4.28015e7			1.90620
$873$		0			0
$874$	−6.88519e7	+	2.42701e7i	−3.04886	+	1.07472i
$875$		−	3.29230e7i		−	1.45371i
$876$		0			0
$877$	−1.07440e7			−0.471700			−0.235850	−	0.971789i	$-0.575787\pi$
$877$	−1.07440e7			−0.471700			−0.235850	+	0.971789i	$0.575787\pi$
$878$			2.02585e7i			0.886893i
$879$		0			0
$880$	5.68617e7			2.47522
$881$	2.93111e7			1.27231			0.636155	−	0.771561i	$-0.280524\pi$
$881$	2.93111e7			1.27231			0.636155	+	0.771561i	$0.280524\pi$
$882$		0			0
$883$	129541.			0.00559123			0.00279561	−	0.999996i	$-0.499110\pi$
$883$	129541.			0.00559123			0.00279561	+	0.999996i	$0.499110\pi$
$884$	−1.51615e8			−6.52548
$885$		0			0
$886$	−3.36344e7			−1.43946
$887$			1.76523e7i			0.753341i	0.926347	+	0.376670i	$0.122931\pi$
$887$			1.76523e7i			0.753341i	−0.926347	+	0.376670i	$0.877069\pi$
$888$		0			0
$889$		−	1.49289e7i		−	0.633537i
$890$		−	2.69185e7i		−	1.13914i
$891$		0			0
$892$	−6.66288e7			−2.80382
$893$	−5.60162e6			−0.235063
$894$		0			0
$895$			1.01941e7i			0.425395i
$896$	1.02058e6			0.0424695
$897$		0			0
$898$	−4.10610e7			−1.69918
$899$			2.26931e7i			0.936473i
$900$		0			0
$901$	3.41342e7			1.40081
$902$	4.45456e7			1.82301
$903$		0			0
$904$			8.58412e6i			0.349361i
$905$		−	2.28103e7i		−	0.925784i
$906$		0			0
$907$			4.43455e7i			1.78991i	0.446154	+	0.894956i	$0.352793\pi$
$907$			4.43455e7i			1.78991i	−0.446154	+	0.894956i	$0.647207\pi$
$908$	5.84843e7			2.35410
$909$		0			0
$910$	9.05426e7			3.62451
$911$	−3.02130e7			−1.20614			−0.603071	−	0.797688i	$-0.706056\pi$
$911$	−3.02130e7			−1.20614			−0.603071	+	0.797688i	$0.706056\pi$
$912$		0			0
$913$	1.96084e7			0.778511
$914$	−1.01373e7			−0.401382
$915$		0			0
$916$			4.35127e7i			1.71347i
$917$	4.59982e6			0.180642
$918$		0			0
$919$			3.46670e7i			1.35403i	0.735970	+	0.677014i	$0.236726\pi$
$919$			3.46670e7i			1.35403i	−0.735970	+	0.677014i	$0.763274\pi$
$920$	1.76637e7	+	5.01100e7i	0.688036	+	1.95189i
$921$		0			0
$922$	1.74862e7			0.677436
$923$		−	4.65519e7i		−	1.79860i
$924$		0			0
$925$		−	6.26582e6i		−	0.240782i
$926$			8.82954e6i			0.338385i
$927$		0			0
$928$	−3.61209e7			−1.37686
$929$			2.40828e7i			0.915521i	0.889076	+	0.457760i	$0.151348\pi$
$929$			2.40828e7i			0.915521i	−0.889076	+	0.457760i	$0.848652\pi$
$930$		0			0
$931$			3.64861e7i			1.37960i
$932$			1.94130e7i			0.732072i
$933$		0			0
$934$		−	1.15405e7i		−	0.432870i
$935$	−3.95476e7			−1.47942
$936$		0			0
$937$		−	4.49222e7i		−	1.67152i	−0.549095	−	0.835760i	$-0.685027\pi$
$937$		−	4.49222e7i		−	1.67152i	0.549095	−	0.835760i	$-0.314973\pi$
$938$			9.08228e6i			0.337045i
$939$		0			0
$940$			6.95089e6i			0.256579i
$941$	−4.18049e7			−1.53905			−0.769525	−	0.638616i	$-0.779507\pi$
$941$	−4.18049e7			−1.53905			−0.769525	+	0.638616i	$0.779507\pi$
$942$		0			0
$943$	6.93615e6	+	1.96772e7i	0.254003	+	0.720581i
$944$		−	1.19516e8i		−	4.36512i
$945$		0			0
$946$	2.83122e7			1.02860
$947$		−	1.31722e7i		−	0.477290i	−0.971107	−	0.238645i	$-0.923297\pi$
$947$		−	1.31722e7i		−	0.477290i	0.971107	−	0.238645i	$-0.0767032\pi$
$948$		0			0
$949$	−5.77240e6			−0.208061
$950$	−3.39273e7			−1.21966
$951$		0			0
$952$	−1.42507e8			−5.09617
$953$	3.82766e7			1.36521			0.682607	−	0.730785i	$-0.260846\pi$
$953$	3.82766e7			1.36521			0.682607	+	0.730785i	$0.260846\pi$
$954$		0			0
$955$	−1.58323e7			−0.561741
$956$		−	3.94558e7i		−	1.39626i
$957$		0			0
$958$			9.44275e6i			0.332418i
$959$		−	2.23611e7i		−	0.785138i
$960$		0			0
$961$	1.77484e7			0.619940
$962$	6.29051e7			2.19154
$963$		0			0
$964$		−	6.08843e7i		−	2.11015i
$965$	−1.47108e7			−0.508530
$966$		0			0
$967$	5.49242e7			1.88885			0.944426	−	0.328725i	$-0.106619\pi$
$967$	5.49242e7			1.88885			0.944426	+	0.328725i	$0.106619\pi$
$968$		−	5.08742e7i		−	1.74506i
$969$		0			0
$970$	2.85271e7			0.973482
$971$	−3.32006e7			−1.13005			−0.565025	−	0.825073i	$-0.691134\pi$
$971$	−3.32006e7			−1.13005			−0.565025	+	0.825073i	$0.691134\pi$
$972$		0			0
$973$		−	3.05214e7i		−	1.03353i
$974$			7.75324e7i			2.61870i
$975$		0			0
$976$			2.78603e7i			0.936185i
$977$	3.36673e7			1.12842			0.564212	−	0.825630i	$-0.309180\pi$
$977$	3.36673e7			1.12842			0.564212	+	0.825630i	$0.309180\pi$
$978$		0			0
$979$	−3.02153e7			−1.00756
$980$	4.52746e7			1.50588
$981$		0			0
$982$	5.57587e7			1.84516
$983$	2.62406e7			0.866143			0.433072	−	0.901360i	$-0.357430\pi$
$983$	2.62406e7			0.866143			0.433072	+	0.901360i	$0.357430\pi$
$984$		0			0
$985$			1.48229e7i			0.486792i
$986$	6.03317e7			1.97630
$987$		0			0
$988$		−	2.40972e8i		−	7.85370i
$989$	4.40846e6	+	1.25063e7i	0.143317	+	0.406574i
$990$		0			0
$991$	−6.06145e7			−1.96062			−0.980308	−	0.197475i	$-0.936726\pi$
$991$	−6.06145e7			−1.96062			−0.980308	+	0.197475i	$0.936726\pi$
$992$			7.38197e7i			2.38173i
$993$		0			0
$994$		−	7.46018e7i		−	2.39488i
$995$			2.76366e7i			0.884965i
$996$		0			0
$997$	−1.57678e7			−0.502380			−0.251190	−	0.967938i	$-0.580822\pi$
$997$	−1.57678e7			−0.502380			−0.251190	+	0.967938i	$0.580822\pi$
$998$			9.93482e7i			3.15743i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.6.c.a.206.12	yes	40
3.2	odd	2	inner	207.6.c.a.206.29	yes	40
23.22	odd	2	inner	207.6.c.a.206.30	yes	40
69.68	even	2	inner	207.6.c.a.206.11	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
207.6.c.a.206.11	✓	40	69.68	even	2	inner
207.6.c.a.206.12	yes	40	1.1	even	1	trivial
207.6.c.a.206.29	yes	40	3.2	odd	2	inner
207.6.c.a.206.30	yes	40	23.22	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+10.4590i q^{2} -77.3916 q^{4} +44.1133 q^{5} +173.403i q^{7} -474.753i q^{8} +O(q^{10})\)	\(q+10.4590i q^{2} -77.3916 q^{4} +44.1133 q^{5} +173.403i q^{7} -474.753i q^{8} +461.383i q^{10} +517.890 q^{11} -1131.71 q^{13} -1813.63 q^{14} +2488.93 q^{16} -1731.06 q^{17} -2751.29i q^{19} -3414.00 q^{20} +5416.64i q^{22} +(-2392.69 + 843.420i) q^{23} -1179.02 q^{25} -11836.6i q^{26} -13419.9i q^{28} +3332.27i q^{29} +6810.10 q^{31} +10839.7i q^{32} -18105.3i q^{34} +7649.36i q^{35} +5314.45i q^{37} +28775.9 q^{38} -20942.9i q^{40} -8223.85i q^{41} -5226.89i q^{43} -40080.4 q^{44} +(-8821.36 - 25025.3i) q^{46} -2036.00i q^{47} -13261.4 q^{49} -12331.4i q^{50} +87585.1 q^{52} -19718.6 q^{53} +22845.8 q^{55} +82323.3 q^{56} -34852.4 q^{58} -48019.1i q^{59} +11193.7i q^{61} +71227.2i q^{62} -33727.4 q^{64} -49923.6 q^{65} -5007.80i q^{67} +133970. q^{68} -80005.0 q^{70} +41134.1i q^{71} +5100.58 q^{73} -55584.0 q^{74} +212927. i q^{76} +89803.5i q^{77} +78609.5i q^{79} +109795. q^{80} +86013.6 q^{82} +37862.0 q^{83} -76363.0 q^{85} +54668.3 q^{86} -245870. i q^{88} -58343.1 q^{89} -196242. i q^{91} +(185174. - 65273.6i) q^{92} +21294.6 q^{94} -121369. i q^{95} -61829.5i q^{97} -138702. i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 600 q^{4}+O(q^{10}) \) 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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