LMFDB - Embedded newform 414.3.b.c.91.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [414,3,Mod(91,414)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(414, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("414.91");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 414 = 2 \cdot 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	414.b (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:	$11.2806829445$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.1358954496.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 8x^{6} + 20x^{4} + 16x^{2} + 1 $ x^8 + 8x^6 + 20x^4 + 16*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 138)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			91.5
Root			$1.58671i$ of defining polynomial
Character	$\chi$	$=$	414.91
Dual form			414.3.b.c.91.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{2} +2.00000 q^{4} -1.69484i q^{5} -4.18388i q^{7} +2.82843 q^{8} +O(q^{10})$	\(q+1.41421 q^{2} +2.00000 q^{4} -1.69484i q^{5} -4.18388i q^{7} +2.82843 q^{8} -2.39686i q^{10} -20.2970i q^{11} -11.6410 q^{13} -5.91690i q^{14} +4.00000 q^{16} -12.5377i q^{17} +27.1971i q^{19} -3.38968i q^{20} -28.7043i q^{22} +(-4.46926 - 22.5616i) q^{23} +22.1275 q^{25} -16.4629 q^{26} -8.36776i q^{28} +50.9389 q^{29} -11.5732 q^{31} +5.65685 q^{32} -17.7310i q^{34} -7.09100 q^{35} -52.7319i q^{37} +38.4625i q^{38} -4.79373i q^{40} -42.4390 q^{41} +2.10282i q^{43} -40.5940i q^{44} +(-6.32049 - 31.9069i) q^{46} -20.0260 q^{47} +31.4951 q^{49} +31.2930 q^{50} -23.2820 q^{52} +49.6304i q^{53} -34.4001 q^{55} -11.8338i q^{56} +72.0384 q^{58} +39.9319 q^{59} +70.9635i q^{61} -16.3669 q^{62} +8.00000 q^{64} +19.7296i q^{65} +4.38918i q^{67} -25.0755i q^{68} -10.0282 q^{70} +46.5698 q^{71} -3.71985 q^{73} -74.5741i q^{74} +54.3941i q^{76} -84.9203 q^{77} +26.4542i q^{79} -6.77935i q^{80} -60.0178 q^{82} -106.806i q^{83} -21.2494 q^{85} +2.97384i q^{86} -57.4086i q^{88} +137.125i q^{89} +48.7046i q^{91} +(-8.93852 - 45.1232i) q^{92} -28.3210 q^{94} +46.0946 q^{95} +119.514i q^{97} +44.5408 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{4}+O(q^{10}) $ 8 * q + 16 * q^4	$ 8 q + 16 q^{4} + 16 q^{13} + 32 q^{16} - 16 q^{23} + 72 q^{25} - 32 q^{26} + 144 q^{29} - 128 q^{31} + 112 q^{35} + 16 q^{41} - 80 q^{46} + 112 q^{47} + 40 q^{49} + 160 q^{50} + 32 q^{52} - 64 q^{55} + 128 q^{58} - 80 q^{59} + 96 q^{62} + 64 q^{64} - 144 q^{70} - 32 q^{71} + 64 q^{73} - 224 q^{77} + 48 q^{85} - 32 q^{92} - 16 q^{94} - 112 q^{95} - 224 q^{98}+O(q^{100}) $ 8 * q + 16 * q^4 + 16 * q^13 + 32 * q^16 - 16 * q^23 + 72 * q^25 - 32 * q^26 + 144 * q^29 - 128 * q^31 + 112 * q^35 + 16 * q^41 - 80 * q^46 + 112 * q^47 + 40 * q^49 + 160 * q^50 + 32 * q^52 - 64 * q^55 + 128 * q^58 - 80 * q^59 + 96 * q^62 + 64 * q^64 - 144 * q^70 - 32 * q^71 + 64 * q^73 - 224 * q^77 + 48 * q^85 - 32 * q^92 - 16 * q^94 - 112 * q^95 - 224 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/414\mathbb{Z}\right)^\times$.

$n$	$47$	$235$
$\chi(n)$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	1.41421			0.707107
$2$	1.41421			0.707107
$3$		0			0
$3$		0			0
$4$	2.00000			0.500000
$5$		−	1.69484i		−	0.338968i	−0.985533	−	0.169484i	$-0.945790\pi$
$5$		−	1.69484i		−	0.338968i	0.985533	−	0.169484i	$-0.0542100\pi$
$6$		0			0
$7$		−	4.18388i		−	0.597697i	−0.954300	−	0.298849i	$-0.903397\pi$
$7$		−	4.18388i		−	0.597697i	0.954300	−	0.298849i	$-0.0966026\pi$
$8$	2.82843			0.353553
$9$		0			0
$10$		−	2.39686i		−	0.239686i
$11$		−	20.2970i		−	1.84518i	−0.385779	−	0.922591i	$-0.626067\pi$
$11$		−	20.2970i		−	1.84518i	0.385779	−	0.922591i	$-0.373933\pi$
$12$		0			0
$13$	−11.6410			−0.895461			−0.447731	−	0.894168i	$-0.647768\pi$
$13$	−11.6410			−0.895461			−0.447731	+	0.894168i	$0.647768\pi$
$14$		−	5.91690i		−	0.422636i
$15$		0			0
$16$	4.00000			0.250000
$17$		−	12.5377i		−	0.737514i	−0.929526	−	0.368757i	$-0.879783\pi$
$17$		−	12.5377i		−	0.737514i	0.929526	−	0.368757i	$-0.120217\pi$
$18$		0			0
$19$			27.1971i			1.43142i	0.698395	+	0.715712i	$0.253898\pi$
$19$			27.1971i			1.43142i	−0.698395	+	0.715712i	$0.746102\pi$
$20$		−	3.38968i		−	0.169484i
$21$		0			0
$22$		−	28.7043i		−	1.30474i
$23$	−4.46926	−	22.5616i	−0.194316	−	0.980939i
$23$	−4.46926	−	22.5616i	−0.194316	−	0.980939i
$24$		0			0
$25$	22.1275			0.885101
$26$	−16.4629			−0.633187
$27$		0			0
$28$		−	8.36776i		−	0.298849i
$29$	50.9389			1.75651			0.878256	−	0.478191i	$-0.158707\pi$
$29$	50.9389			1.75651			0.878256	+	0.478191i	$0.158707\pi$
$30$		0			0
$31$	−11.5732			−0.373328			−0.186664	−	0.982424i	$-0.559768\pi$
$31$	−11.5732			−0.373328			−0.186664	+	0.982424i	$0.559768\pi$
$32$	5.65685			0.176777
$33$		0			0
$34$		−	17.7310i		−	0.521501i
$35$	−7.09100			−0.202600
$36$		0			0
$37$		−	52.7319i		−	1.42519i	−0.701578	−	0.712593i	$-0.747521\pi$
$37$		−	52.7319i		−	1.42519i	0.701578	−	0.712593i	$-0.252479\pi$
$38$			38.4625i			1.01217i
$39$		0			0
$40$		−	4.79373i		−	0.119843i
$41$	−42.4390			−1.03510			−0.517548	−	0.855654i	$-0.673155\pi$
$41$	−42.4390			−1.03510			−0.517548	+	0.855654i	$0.673155\pi$
$42$		0			0
$43$			2.10282i			0.0489029i	0.999701	+	0.0244514i	$0.00778391\pi$
$43$			2.10282i			0.0489029i	−0.999701	+	0.0244514i	$0.992216\pi$
$44$		−	40.5940i		−	0.922591i
$45$		0			0
$46$	−6.32049	−	31.9069i	−0.137402	−	0.693629i
$47$	−20.0260			−0.426085			−0.213043	−	0.977043i	$-0.568337\pi$
$47$	−20.0260			−0.426085			−0.213043	+	0.977043i	$0.568337\pi$
$48$		0			0
$49$	31.4951			0.642758
$50$	31.2930			0.625861
$51$		0			0
$52$	−23.2820			−0.447731
$53$			49.6304i			0.936423i	0.883616	+	0.468212i	$0.155102\pi$
$53$			49.6304i			0.936423i	−0.883616	+	0.468212i	$0.844898\pi$
$54$		0			0
$55$	−34.4001			−0.625457
$56$		−	11.8338i		−	0.211318i
$57$		0			0
$58$	72.0384			1.24204
$59$	39.9319			0.676811			0.338406	−	0.941000i	$-0.390112\pi$
$59$	39.9319			0.676811			0.338406	+	0.941000i	$0.390112\pi$
$60$		0			0
$61$			70.9635i			1.16334i	0.813426	+	0.581668i	$0.197600\pi$
$61$			70.9635i			1.16334i	−0.813426	+	0.581668i	$0.802400\pi$
$62$	−16.3669			−0.263983
$63$		0			0
$64$	8.00000			0.125000
$65$			19.7296i			0.303532i
$66$		0			0
$67$			4.38918i			0.0655101i	0.999463	+	0.0327551i	$0.0104281\pi$
$67$			4.38918i			0.0655101i	−0.999463	+	0.0327551i	$0.989572\pi$
$68$		−	25.0755i		−	0.368757i
$69$		0			0
$70$	−10.0282			−0.143260
$71$	46.5698			0.655912			0.327956	−	0.944693i	$-0.393640\pi$
$71$	46.5698			0.655912			0.327956	+	0.944693i	$0.393640\pi$
$72$		0			0
$73$	−3.71985			−0.0509569			−0.0254784	−	0.999675i	$-0.508111\pi$
$73$	−3.71985			−0.0509569			−0.0254784	+	0.999675i	$0.508111\pi$
$74$		−	74.5741i		−	1.00776i
$75$		0			0
$76$			54.3941i			0.715712i
$77$	−84.9203			−1.10286
$78$		0			0
$79$			26.4542i			0.334864i	0.985884	+	0.167432i	$0.0535474\pi$
$79$			26.4542i			0.334864i	−0.985884	+	0.167432i	$0.946453\pi$
$80$		−	6.77935i		−	0.0847419i
$81$		0			0
$82$	−60.0178			−0.731924
$83$		−	106.806i		−	1.28681i	−0.765524	−	0.643407i	$-0.777520\pi$
$83$		−	106.806i		−	1.28681i	0.765524	−	0.643407i	$-0.222480\pi$
$84$		0			0
$85$	−21.2494			−0.249993
$86$			2.97384i			0.0345795i
$87$		0			0
$88$		−	57.4086i		−	0.652370i
$89$			137.125i			1.54073i	0.637602	+	0.770366i	$0.279926\pi$
$89$			137.125i			1.54073i	−0.637602	+	0.770366i	$0.720074\pi$
$90$		0			0
$91$			48.7046i			0.535215i
$92$	−8.93852	−	45.1232i	−0.0971579	−	0.490470i
$93$		0			0
$94$	−28.3210			−0.301288
$95$	46.0946			0.485207
$96$		0			0
$97$			119.514i			1.23211i	0.787705	+	0.616053i	$0.211269\pi$
$97$			119.514i			1.23211i	−0.787705	+	0.616053i	$0.788731\pi$
$98$	44.5408			0.454498
$99$		0			0
$100$	44.2550			0.442550
$101$	−24.5512			−0.243082			−0.121541	−	0.992586i	$-0.538784\pi$
$101$	−24.5512			−0.243082			−0.121541	+	0.992586i	$0.538784\pi$
$102$		0			0
$103$		−	7.77997i		−	0.0755337i	−0.999287	−	0.0377668i	$-0.987976\pi$
$103$		−	7.77997i		−	0.0755337i	0.999287	−	0.0377668i	$-0.0120244\pi$
$104$	−32.9257			−0.316593
$105$		0			0
$106$			70.1880i			0.662151i
$107$			42.8558i			0.400522i	0.979743	+	0.200261i	$0.0641790\pi$
$107$			42.8558i			0.400522i	−0.979743	+	0.200261i	$0.935821\pi$
$108$		0			0
$109$		−	1.36851i		−	0.0125552i	−0.999980	−	0.00627759i	$-0.998002\pi$
$109$		−	1.36851i		−	0.0125552i	0.999980	−	0.00627759i	$-0.00199823\pi$
$110$	−48.6491			−0.442265
$111$		0			0
$112$		−	16.7355i		−	0.149424i
$113$			121.497i			1.07520i	0.843201	+	0.537598i	$0.180668\pi$
$113$			121.497i			1.07520i	−0.843201	+	0.537598i	$0.819332\pi$
$114$		0			0
$115$	−38.2383	+	7.57467i	−0.332507	+	0.0658667i
$116$	101.878			0.878256
$117$		0			0
$118$	56.4722			0.478578
$119$	−52.4564			−0.440810
$120$		0			0
$121$	−290.968			−2.40470
$122$			100.358i			0.822603i
$123$		0			0
$124$	−23.1463			−0.186664
$125$		−	79.8735i		−	0.638988i
$126$		0			0
$127$	223.757			1.76186			0.880932	−	0.473243i	$-0.156917\pi$
$127$	223.757			1.76186			0.880932	+	0.473243i	$0.156917\pi$
$128$	11.3137			0.0883883
$129$		0			0
$130$			27.9019i			0.214630i
$131$	−49.2839			−0.376213			−0.188106	−	0.982149i	$-0.560235\pi$
$131$	−49.2839			−0.376213			−0.188106	+	0.982149i	$0.560235\pi$
$132$		0			0
$133$	113.789			0.855559
$134$			6.20724i			0.0463227i
$135$		0			0
$136$		−	35.4621i		−	0.260751i
$137$			97.8172i			0.713994i	0.934105	+	0.356997i	$0.116199\pi$
$137$			97.8172i			0.713994i	−0.934105	+	0.356997i	$0.883801\pi$
$138$		0			0
$139$	174.434			1.25492			0.627459	−	0.778650i	$-0.284095\pi$
$139$	174.434			1.25492			0.627459	+	0.778650i	$0.284095\pi$
$140$	−14.1820			−0.101300
$141$		0			0
$142$	65.8596			0.463800
$143$			236.277i			1.65229i
$144$		0			0
$145$		−	86.3331i		−	0.595401i
$146$	−5.26067			−0.0360320
$147$		0			0
$148$		−	105.464i		−	0.712593i
$149$			8.72531i			0.0585591i	0.999571	+	0.0292796i	$0.00932131\pi$
$149$			8.72531i			0.0585591i	−0.999571	+	0.0292796i	$0.990679\pi$
$150$		0			0
$151$	−86.1741			−0.570689			−0.285345	−	0.958425i	$-0.592108\pi$
$151$	−86.1741			−0.570689			−0.285345	+	0.958425i	$0.592108\pi$
$152$			76.9249i			0.506085i
$153$		0			0
$154$	−120.095			−0.779840
$155$			19.6146i			0.126546i
$156$		0			0
$157$		−	177.857i		−	1.13285i	−0.824114	−	0.566424i	$-0.808327\pi$
$157$		−	177.857i		−	1.13285i	0.824114	−	0.566424i	$-0.191673\pi$
$158$			37.4119i			0.236784i
$159$		0			0
$160$		−	9.58745i		−	0.0599216i
$161$	−94.3951	+	18.6989i	−0.586305	+	0.116142i
$162$		0			0
$163$	−116.103			−0.712291			−0.356146	−	0.934430i	$-0.615909\pi$
$163$	−116.103			−0.712291			−0.356146	+	0.934430i	$0.615909\pi$
$164$	−84.8780			−0.517548
$165$		0			0
$166$		−	151.046i		−	0.909915i
$167$	281.959			1.68838			0.844189	−	0.536046i	$-0.180083\pi$
$167$	281.959			1.68838			0.844189	+	0.536046i	$0.180083\pi$
$168$		0			0
$169$	−33.4872			−0.198149
$170$	−30.0512			−0.176772
$171$		0			0
$172$			4.20565i			0.0244514i
$173$	−115.730			−0.668962			−0.334481	−	0.942403i	$-0.608561\pi$
$173$	−115.730			−0.668962			−0.334481	+	0.942403i	$0.608561\pi$
$174$		0			0
$175$		−	92.5790i		−	0.529023i
$176$		−	81.1880i		−	0.461296i
$177$		0			0
$178$			193.924i			1.08946i
$179$	−184.904			−1.03299			−0.516493	−	0.856292i	$-0.672763\pi$
$179$	−184.904			−1.03299			−0.516493	+	0.856292i	$0.672763\pi$
$180$		0			0
$181$			349.157i			1.92904i	0.264001	+	0.964522i	$0.414958\pi$
$181$			349.157i			1.92904i	−0.264001	+	0.964522i	$0.585042\pi$
$182$			68.8787i			0.378454i
$183$		0			0
$184$	−12.6410	−	63.8138i	−0.0687010	−	0.346814i
$185$	−89.3720			−0.483092
$186$		0			0
$187$	−254.478			−1.36085
$188$	−40.0520			−0.213043
$189$		0			0
$190$	65.1877			0.343093
$191$		−	319.163i		−	1.67101i	−0.549483	−	0.835505i	$-0.685175\pi$
$191$		−	319.163i		−	1.67101i	0.549483	−	0.835505i	$-0.314825\pi$
$192$		0			0
$193$	371.496			1.92485			0.962425	−	0.271546i	$-0.0875349\pi$
$193$	371.496			1.92485			0.962425	+	0.271546i	$0.0875349\pi$
$194$			169.019i			0.871231i
$195$		0			0
$196$	62.9903			0.321379
$197$	190.805			0.968552			0.484276	−	0.874915i	$-0.339083\pi$
$197$	190.805			0.968552			0.484276	+	0.874915i	$0.339083\pi$
$198$		0			0
$199$		−	165.204i		−	0.830171i	−0.909782	−	0.415085i	$-0.863752\pi$
$199$		−	165.204i		−	0.830171i	0.909782	−	0.415085i	$-0.136248\pi$
$200$	62.5861			0.312930
$201$		0			0
$202$	−34.7207			−0.171885
$203$		−	213.122i		−	1.04986i
$204$		0			0
$205$			71.9272i			0.350864i
$206$		−	11.0025i		−	0.0534104i
$207$		0			0
$208$	−46.5640			−0.223865
$209$	552.019			2.64124
$210$		0			0
$211$	217.886			1.03263			0.516317	−	0.856397i	$-0.327303\pi$
$211$	217.886			1.03263			0.516317	+	0.856397i	$0.327303\pi$
$212$			99.2609i			0.468212i
$213$		0			0
$214$			60.6073i			0.283212i
$215$	3.56394			0.0165765
$216$		0			0
$217$			48.4208i			0.223137i
$218$		−	1.93537i		−	0.00887785i
$219$		0			0
$220$	−68.8003			−0.312729
$221$			145.952i			0.660415i
$222$		0			0
$223$	−229.084			−1.02728			−0.513641	−	0.858005i	$-0.671704\pi$
$223$	−229.084			−1.02728			−0.513641	+	0.858005i	$0.671704\pi$
$224$		−	23.6676i		−	0.105659i
$225$		0			0
$226$			171.823i			0.760279i
$227$		−	272.590i		−	1.20084i	−0.799686	−	0.600418i	$-0.795001\pi$
$227$		−	272.590i		−	1.20084i	0.799686	−	0.600418i	$-0.204999\pi$
$228$		0			0
$229$		−	155.398i		−	0.678593i	−0.940679	−	0.339297i	$-0.889811\pi$
$229$		−	155.398i		−	0.678593i	0.940679	−	0.339297i	$-0.110189\pi$
$230$	−54.0771	+	10.7122i	−0.235118	+	0.0465748i
$231$		0			0
$232$	144.077			0.621021
$233$	−200.272			−0.859537			−0.429768	−	0.902939i	$-0.641405\pi$
$233$	−200.272			−0.859537			−0.429768	+	0.902939i	$0.641405\pi$
$234$		0			0
$235$			33.9408i			0.144429i
$236$	79.8637			0.338406
$237$		0			0
$238$	−74.1846			−0.311700
$239$	−440.599			−1.84351			−0.921755	−	0.387772i	$-0.873245\pi$
$239$	−440.599			−1.84351			−0.921755	+	0.387772i	$0.873245\pi$
$240$		0			0
$241$			331.186i			1.37421i	0.726556	+	0.687107i	$0.241120\pi$
$241$			331.186i			1.37421i	−0.726556	+	0.687107i	$0.758880\pi$
$242$	−411.491			−1.70038
$243$		0			0
$244$			141.927i			0.581668i
$245$		−	53.3791i		−	0.217874i
$246$		0			0
$247$		−	316.601i		−	1.28179i
$248$	−32.7339			−0.131991
$249$		0			0
$250$		−	112.958i		−	0.451833i
$251$		−	163.892i		−	0.652957i	−0.945205	−	0.326479i	$-0.894138\pi$
$251$		−	163.892i		−	0.652957i	0.945205	−	0.326479i	$-0.105862\pi$
$252$		0			0
$253$	−457.933	+	90.7126i	−1.81001	+	0.358548i
$254$	316.440			1.24583
$255$		0			0
$256$	16.0000			0.0625000
$257$	210.371			0.818565			0.409283	−	0.912408i	$-0.365779\pi$
$257$	210.371			0.818565			0.409283	+	0.912408i	$0.365779\pi$
$258$		0			0
$259$	−220.624			−0.851830
$260$			39.4592i			0.151766i
$261$		0			0
$262$	−69.6979			−0.266023
$263$			257.455i			0.978916i	0.872027	+	0.489458i	$0.162805\pi$
$263$			257.455i			0.978916i	−0.872027	+	0.489458i	$0.837195\pi$
$264$		0			0
$265$	84.1156			0.317417
$266$	160.922			0.604972
$267$		0			0
$268$			8.77836i			0.0327551i
$269$	−126.614			−0.470683			−0.235342	−	0.971913i	$-0.575621\pi$
$269$	−126.614			−0.470683			−0.235342	+	0.971913i	$0.575621\pi$
$270$		0			0
$271$	452.242			1.66879			0.834394	−	0.551168i	$-0.185818\pi$
$271$	452.242			1.66879			0.834394	+	0.551168i	$0.185818\pi$
$272$		−	50.1509i		−	0.184378i
$273$		0			0
$274$			138.334i			0.504870i
$275$		−	449.122i		−	1.63317i
$276$		0			0
$277$	97.8176			0.353132			0.176566	−	0.984289i	$-0.443501\pi$
$277$	97.8176			0.353132			0.176566	+	0.984289i	$0.443501\pi$
$278$	246.686			0.887361
$279$		0			0
$280$	−20.0564			−0.0716299
$281$		−	411.158i		−	1.46320i	−0.681736	−	0.731598i	$-0.738775\pi$
$281$		−	411.158i		−	1.46320i	0.681736	−	0.731598i	$-0.261225\pi$
$282$		0			0
$283$		−	167.168i		−	0.590701i	−0.955389	−	0.295350i	$-0.904564\pi$
$283$		−	167.168i		−	0.590701i	0.955389	−	0.295350i	$-0.0954364\pi$
$284$	93.1395			0.327956
$285$		0			0
$286$			334.147i			1.16835i
$287$			177.560i			0.618675i
$288$		0			0
$289$	131.805			0.456073
$290$		−	122.093i		−	0.421012i
$291$		0			0
$292$	−7.43970			−0.0254784
$293$		−	359.823i		−	1.22807i	−0.789280	−	0.614033i	$-0.789546\pi$
$293$		−	359.823i		−	1.22807i	0.789280	−	0.614033i	$-0.210454\pi$
$294$		0			0
$295$		−	67.6780i		−	0.229417i
$296$		−	149.148i		−	0.503879i
$297$		0			0
$298$			12.3395i			0.0414076i
$299$	52.0267	+	262.640i	0.174002	+	0.878393i
$300$		0			0
$301$	8.79796			0.0292291
$302$	−121.869			−0.403538
$303$		0			0
$304$			108.788i			0.357856i
$305$	120.272			0.394333
$306$		0			0
$307$	159.380			0.519152			0.259576	−	0.965723i	$-0.416417\pi$
$307$	159.380			0.519152			0.259576	+	0.965723i	$0.416417\pi$
$308$	−169.841			−0.551430
$309$		0			0
$310$			27.7393i			0.0894816i
$311$	212.603			0.683610			0.341805	−	0.939771i	$-0.388962\pi$
$311$	212.603			0.683610			0.341805	+	0.939771i	$0.388962\pi$
$312$		0			0
$313$		−	6.75351i		−	0.0215767i	−0.999942	−	0.0107884i	$-0.996566\pi$
$313$		−	6.75351i		−	0.0215767i	0.999942	−	0.0107884i	$-0.00343411\pi$
$314$		−	251.528i		−	0.801044i
$315$		0			0
$316$			52.9085i			0.167432i
$317$	−262.878			−0.829268			−0.414634	−	0.909988i	$-0.636090\pi$
$317$	−262.878			−0.829268			−0.414634	+	0.909988i	$0.636090\pi$
$318$		0			0
$319$		−	1033.91i		−	3.24108i
$320$		−	13.5587i		−	0.0423710i
$321$		0			0
$322$	−133.495	+	26.4442i	−0.414580	+	0.0821248i
$323$	340.990			1.05570
$324$		0			0
$325$	−257.586			−0.792574
$326$	−164.195			−0.503666
$327$		0			0
$328$	−120.036			−0.365962
$329$			83.7865i			0.254670i
$330$		0			0
$331$	353.071			1.06668			0.533339	−	0.845901i	$-0.320937\pi$
$331$	353.071			1.06668			0.533339	+	0.845901i	$0.320937\pi$
$332$		−	213.611i		−	0.643407i
$333$		0			0
$334$	398.750			1.19386
$335$	7.43895			0.0222058
$336$		0			0
$337$		−	95.3746i		−	0.283011i	−0.989937	−	0.141505i	$-0.954806\pi$
$337$		−	95.3746i		−	0.283011i	0.989937	−	0.141505i	$-0.0451942\pi$
$338$	−47.3580			−0.140112
$339$		0			0
$340$	−42.4989			−0.124997
$341$			234.901i			0.688858i
$342$		0			0
$343$		−	336.782i		−	0.981872i
$344$			5.94768i			0.0172898i
$345$		0			0
$346$	−163.668			−0.473027
$347$	348.533			1.00442			0.502209	−	0.864746i	$-0.332521\pi$
$347$	348.533			1.00442			0.502209	+	0.864746i	$0.332521\pi$
$348$		0			0
$349$	458.963			1.31508			0.657541	−	0.753419i	$-0.271597\pi$
$349$	458.963			1.31508			0.657541	+	0.753419i	$0.271597\pi$
$350$		−	130.926i		−	0.374075i
$351$		0			0
$352$		−	114.817i		−	0.326185i
$353$	−225.020			−0.637452			−0.318726	−	0.947847i	$-0.603255\pi$
$353$	−225.020			−0.637452			−0.318726	+	0.947847i	$0.603255\pi$
$354$		0			0
$355$		−	78.9282i		−	0.222333i
$356$			274.250i			0.770366i
$357$		0			0
$358$	−261.494			−0.730431
$359$			318.972i			0.888502i	0.895902	+	0.444251i	$0.146530\pi$
$359$			318.972i			0.888502i	−0.895902	+	0.444251i	$0.853470\pi$
$360$		0			0
$361$	−378.681			−1.04898
$362$			493.783i			1.36404i
$363$		0			0
$364$			97.4091i			0.267608i
$365$			6.30455i			0.0172727i
$366$		0			0
$367$			10.8441i			0.0295479i	0.999891	+	0.0147740i	$0.00470287\pi$
$367$			10.8441i			0.0295479i	−0.999891	+	0.0147740i	$0.995297\pi$
$368$	−17.8770	−	90.2464i	−0.0485789	−	0.245235i
$369$		0			0
$370$	−126.391			−0.341598
$371$	207.648			0.559698
$372$		0			0
$373$		−	252.337i		−	0.676507i	−0.941055	−	0.338254i	$-0.890164\pi$
$373$		−	252.337i		−	0.676507i	0.941055	−	0.338254i	$-0.109836\pi$
$374$	−359.887			−0.962264
$375$		0			0
$376$	−56.6421			−0.150644
$377$	−592.979			−1.57289
$378$		0			0
$379$		−	675.838i		−	1.78321i	−0.452811	−	0.891606i	$-0.649579\pi$
$379$		−	675.838i		−	1.78321i	0.452811	−	0.891606i	$-0.350421\pi$
$380$	92.1893			0.242603
$381$		0			0
$382$		−	451.364i		−	1.18158i
$383$		−	285.533i		−	0.745516i	−0.927929	−	0.372758i	$-0.878412\pi$
$383$		−	285.533i		−	0.745516i	0.927929	−	0.372758i	$-0.121588\pi$
$384$		0			0
$385$			143.926i			0.373834i
$386$	525.375			1.36108
$387$		0			0
$388$			239.029i			0.616053i
$389$			119.403i			0.306949i	0.988153	+	0.153475i	$0.0490463\pi$
$389$			119.403i			0.306949i	−0.988153	+	0.153475i	$0.950954\pi$
$390$		0			0
$391$	−282.871	+	56.0344i	−0.723456	+	0.143311i
$392$	89.0817			0.227249
$393$		0			0
$394$	269.839			0.684869
$395$	44.8356			0.113508
$396$		0			0
$397$	−135.547			−0.341429			−0.170715	−	0.985321i	$-0.554608\pi$
$397$	−135.547			−0.341429			−0.170715	+	0.985321i	$0.554608\pi$
$398$		−	233.634i		−	0.587020i
$399$		0			0
$400$	88.5101			0.221275
$401$			135.478i			0.337850i	0.985629	+	0.168925i	$0.0540296\pi$
$401$			135.478i			0.337850i	−0.985629	+	0.168925i	$0.945970\pi$
$402$		0			0
$403$	134.723			0.334301
$404$	−49.1025			−0.121541
$405$		0			0
$406$		−	301.400i		−	0.742365i
$407$	−1070.30			−2.62973
$408$		0			0
$409$	−271.448			−0.663686			−0.331843	−	0.943335i	$-0.607670\pi$
$409$	−271.448			−0.663686			−0.331843	+	0.943335i	$0.607670\pi$
$410$			101.720i			0.248099i
$411$		0			0
$412$		−	15.5599i		−	0.0377668i
$413$		−	167.070i		−	0.404528i
$414$		0			0
$415$	−181.018			−0.436188
$416$	−65.8514			−0.158297
$417$		0			0
$418$	780.673			1.86764
$419$			214.387i			0.511664i	0.966721	+	0.255832i	$0.0823494\pi$
$419$			214.387i			0.511664i	−0.966721	+	0.255832i	$0.917651\pi$
$420$		0			0
$421$			569.380i			1.35245i	0.736696	+	0.676224i	$0.236385\pi$
$421$			569.380i			1.35245i	−0.736696	+	0.676224i	$0.763615\pi$
$422$	308.137			0.730183
$423$		0			0
$424$			140.376i			0.331076i
$425$		−	277.429i		−	0.652774i
$426$		0			0
$427$	296.903			0.695323
$428$			85.7117i			0.200261i
$429$		0			0
$430$	5.04018			0.0117213
$431$			183.203i			0.425064i	0.977154	+	0.212532i	$0.0681710\pi$
$431$			183.203i			0.425064i	−0.977154	+	0.212532i	$0.931829\pi$
$432$		0			0
$433$			749.313i			1.73052i	0.501327	+	0.865258i	$0.332845\pi$
$433$			749.313i			1.73052i	−0.501327	+	0.865258i	$0.667155\pi$
$434$			68.4773i			0.157782i
$435$		0			0
$436$		−	2.73703i		−	0.00627759i
$437$	613.609	−	121.551i	1.40414	−	0.278148i
$438$		0			0
$439$	91.4979			0.208423			0.104212	−	0.994555i	$-0.466768\pi$
$439$	91.4979			0.208423			0.104212	+	0.994555i	$0.466768\pi$
$440$	−97.2983			−0.221132
$441$		0			0
$442$			206.407i			0.466984i
$443$	366.672			0.827701			0.413851	−	0.910345i	$-0.364184\pi$
$443$	366.672			0.827701			0.413851	+	0.910345i	$0.364184\pi$
$444$		0			0
$445$	232.405			0.522258
$446$	−323.974			−0.726398
$447$		0			0
$448$		−	33.4711i		−	0.0747122i
$449$	410.543			0.914349			0.457175	−	0.889377i	$-0.348861\pi$
$449$	410.543			0.914349			0.457175	+	0.889377i	$0.348861\pi$
$450$		0			0
$451$			861.384i			1.90994i
$452$			242.994i			0.537598i
$453$		0			0
$454$		−	385.500i		−	0.849120i
$455$	82.5464			0.181421
$456$		0			0
$457$			840.088i			1.83827i	0.393947	+	0.919133i	$0.371109\pi$
$457$			840.088i			1.83827i	−0.393947	+	0.919133i	$0.628891\pi$
$458$		−	219.766i		−	0.479838i
$459$		0			0
$460$	−76.4765	+	15.1493i	−0.166253	+	0.0329334i
$461$	−35.9604			−0.0780052			−0.0390026	−	0.999239i	$-0.512418\pi$
$461$	−35.9604			−0.0780052			−0.0390026	+	0.999239i	$0.512418\pi$
$462$		0			0
$463$	−615.180			−1.32868			−0.664341	−	0.747430i	$-0.731288\pi$
$463$	−615.180			−1.32868			−0.664341	+	0.747430i	$0.731288\pi$
$464$	203.755			0.439128
$465$		0			0
$466$	−283.228			−0.607784
$467$			504.679i			1.08068i	0.841446	+	0.540342i	$0.181705\pi$
$467$			504.679i			1.08068i	−0.841446	+	0.540342i	$0.818295\pi$
$468$		0			0
$469$	18.3638			0.0391552
$470$			47.9996i			0.102127i
$471$		0			0
$472$	112.944			0.239289
$473$	42.6810			0.0902347
$474$		0			0
$475$			601.804i			1.26696i
$476$	−104.913			−0.220405
$477$		0			0
$478$	−623.101			−1.30356
$479$			50.1143i			0.104623i	0.998631	+	0.0523114i	$0.0166588\pi$
$479$			50.1143i			0.104623i	−0.998631	+	0.0523114i	$0.983341\pi$
$480$		0			0
$481$			613.852i			1.27620i
$482$			468.367i			0.971716i
$483$		0			0
$484$	−581.937			−1.20235
$485$	202.557			0.417644
$486$		0			0
$487$	359.536			0.738268			0.369134	−	0.929376i	$-0.379654\pi$
$487$	359.536			0.738268			0.369134	+	0.929376i	$0.379654\pi$
$488$			200.715i			0.411301i
$489$		0			0
$490$		−	75.4895i		−	0.154060i
$491$	−516.399			−1.05173			−0.525864	−	0.850569i	$-0.676258\pi$
$491$	−516.399			−1.05173			−0.525864	+	0.850569i	$0.676258\pi$
$492$		0			0
$493$		−	638.658i		−	1.29545i
$494$		−	447.742i		−	0.906359i
$495$		0			0
$496$	−46.2927			−0.0933320
$497$		−	194.842i		−	0.392037i
$498$		0			0
$499$	−61.0247			−0.122294			−0.0611470	−	0.998129i	$-0.519476\pi$
$499$	−61.0247			−0.122294			−0.0611470	+	0.998129i	$0.519476\pi$
$500$		−	159.747i		−	0.319494i
$501$		0			0
$502$		−	231.779i		−	0.461710i
$503$			54.2794i			0.107911i	0.998543	+	0.0539557i	$0.0171830\pi$
$503$			54.2794i			0.107911i	−0.998543	+	0.0539557i	$0.982817\pi$
$504$		0			0
$505$			41.6104i			0.0823968i
$506$	−647.615	+	128.287i	−1.27987	+	0.253532i
$507$		0			0
$508$	447.513			0.880932
$509$	−515.585			−1.01294			−0.506468	−	0.862258i	$-0.669049\pi$
$509$	−515.585			−1.01294			−0.506468	+	0.862258i	$0.669049\pi$
$510$		0			0
$511$			15.5634i			0.0304568i
$512$	22.6274			0.0441942
$513$		0			0
$514$	297.510			0.578813
$515$	−13.1858			−0.0256035
$516$		0			0
$517$			406.468i			0.786205i
$518$	−312.009			−0.602335
$519$		0			0
$520$			55.8038i			0.107315i
$521$			950.381i			1.82415i	0.410027	+	0.912073i	$0.365519\pi$
$521$			950.381i			1.82415i	−0.410027	+	0.912073i	$0.634481\pi$
$522$		0			0
$523$			632.161i			1.20872i	0.796711	+	0.604360i	$0.206571\pi$
$523$			632.161i			1.20872i	−0.796711	+	0.604360i	$0.793429\pi$
$524$	−98.5678			−0.188106
$525$		0			0
$526$			364.096i			0.692198i
$527$			145.101i			0.275335i
$528$		0			0
$529$	−489.051	+	201.667i	−0.924483	+	0.381224i
$530$	118.957			0.224448
$531$		0			0
$532$	227.579			0.427780
$533$	494.032			0.926889
$534$		0			0
$535$	72.6337			0.135764
$536$			12.4145i			0.0231613i
$537$		0			0
$538$	−179.059			−0.332823
$539$		−	639.257i		−	1.18601i
$540$		0			0
$541$	42.8880			0.0792755			0.0396377	−	0.999214i	$-0.487380\pi$
$541$	42.8880			0.0792755			0.0396377	+	0.999214i	$0.487380\pi$
$542$	639.566			1.18001
$543$		0			0
$544$		−	70.9241i		−	0.130375i
$545$	−2.31941			−0.00425580
$546$		0			0
$547$	−597.669			−1.09263			−0.546315	−	0.837580i	$-0.683970\pi$
$547$	−597.669			−1.09263			−0.546315	+	0.837580i	$0.683970\pi$
$548$			195.634i			0.356997i
$549$		0			0
$550$		−	635.155i		−	1.15483i
$551$			1385.39i			2.51432i
$552$		0			0
$553$	110.681			0.200147
$554$	138.335			0.249702
$555$		0			0
$556$	348.867			0.627459
$557$			39.1372i			0.0702642i	0.999383	+	0.0351321i	$0.0111852\pi$
$557$			39.1372i			0.0702642i	−0.999383	+	0.0351321i	$0.988815\pi$
$558$		0			0
$559$		−	24.4790i		−	0.0437906i
$560$	−28.3640			−0.0506500
$561$		0			0
$562$		−	581.466i		−	1.03464i
$563$		−	420.979i		−	0.747743i	−0.927481	−	0.373871i	$-0.878030\pi$
$563$		−	420.979i		−	0.747743i	0.927481	−	0.373871i	$-0.121970\pi$
$564$		0			0
$565$	205.918			0.364457
$566$		−	236.412i		−	0.417688i
$567$		0			0
$568$	131.719			0.231900
$569$			519.422i			0.912868i	0.889757	+	0.456434i	$0.150874\pi$
$569$			519.422i			0.912868i	−0.889757	+	0.456434i	$0.849126\pi$
$570$		0			0
$571$		−	824.370i		−	1.44373i	−0.692034	−	0.721865i	$-0.743285\pi$
$571$		−	824.370i		−	1.44373i	0.692034	−	0.721865i	$-0.256715\pi$
$572$			472.555i			0.826145i
$573$		0			0
$574$			251.107i			0.437469i
$575$	−98.8937	−	499.232i	−0.171989	−	0.868230i
$576$		0			0
$577$	326.554			0.565951			0.282975	−	0.959127i	$-0.408679\pi$
$577$	326.554			0.565951			0.282975	+	0.959127i	$0.408679\pi$
$578$	186.401			0.322493
$579$		0			0
$580$		−	172.666i		−	0.297700i
$581$	−446.862			−0.769126
$582$		0			0
$583$	1007.35			1.72787
$584$	−10.5213			−0.0180160
$585$		0			0
$586$		−	508.867i		−	0.868374i
$587$	142.994			0.243602			0.121801	−	0.992555i	$-0.461133\pi$
$587$	142.994			0.243602			0.121801	+	0.992555i	$0.461133\pi$
$588$		0			0
$589$		−	314.756i		−	0.534391i
$590$		−	95.7112i		−	0.162222i
$591$		0			0
$592$		−	210.928i		−	0.356296i
$593$	−340.100			−0.573525			−0.286763	−	0.958002i	$-0.592579\pi$
$593$	−340.100			−0.573525			−0.286763	+	0.958002i	$0.592579\pi$
$594$		0			0
$595$			88.9051i			0.149420i
$596$			17.4506i			0.0292796i
$597$		0			0
$598$	73.5768	+	371.428i	0.123038	+	0.621118i
$599$	−810.734			−1.35348			−0.676739	−	0.736223i	$-0.736608\pi$
$599$	−810.734			−1.35348			−0.676739	+	0.736223i	$0.736608\pi$
$600$		0			0
$601$	−917.012			−1.52581			−0.762906	−	0.646510i	$-0.776228\pi$
$601$	−917.012			−1.52581			−0.762906	+	0.646510i	$0.776228\pi$
$602$	12.4422			0.0206681
$603$		0			0
$604$	−172.348			−0.285345
$605$			493.144i			0.815114i
$606$		0			0
$607$	−891.833			−1.46925			−0.734624	−	0.678475i	$-0.762641\pi$
$607$	−891.833			−1.46925			−0.734624	+	0.678475i	$0.762641\pi$
$608$			153.850i			0.253043i
$609$		0			0
$610$	170.090			0.278836
$611$	233.123			0.381543
$612$		0			0
$613$		−	451.763i		−	0.736970i	−0.929634	−	0.368485i	$-0.879877\pi$
$613$		−	451.763i		−	0.736970i	0.929634	−	0.368485i	$-0.120123\pi$
$614$	225.397			0.367096
$615$		0			0
$616$	−240.191			−0.389920
$617$			123.296i			0.199832i	0.994996	+	0.0999161i	$0.0318574\pi$
$617$			123.296i			0.199832i	−0.994996	+	0.0999161i	$0.968143\pi$
$618$		0			0
$619$			417.170i			0.673941i	0.941515	+	0.336971i	$0.109402\pi$
$619$			417.170i			0.673941i	−0.941515	+	0.336971i	$0.890598\pi$
$620$			39.2293i			0.0632730i
$621$		0			0
$622$	300.666			0.483385
$623$	573.715			0.920891
$624$		0			0
$625$	417.815			0.668505
$626$		−	9.55091i		−	0.0152570i
$627$		0			0
$628$		−	355.714i		−	0.566424i
$629$	−661.138			−1.05109
$630$		0			0
$631$			553.468i			0.877129i	0.898700	+	0.438564i	$0.144513\pi$
$631$			553.468i			0.877129i	−0.898700	+	0.438564i	$0.855487\pi$
$632$			74.8239i			0.118392i
$633$		0			0
$634$	−371.766			−0.586381
$635$		−	379.231i		−	0.597215i
$636$		0			0
$637$	−366.635			−0.575565
$638$		−	1462.16i		−	2.29179i
$639$		0			0
$640$		−	19.1749i		−	0.0299608i
$641$			524.194i			0.817776i	0.912585	+	0.408888i	$0.134083\pi$
$641$			524.194i			0.817776i	−0.912585	+	0.408888i	$0.865917\pi$
$642$		0			0
$643$			946.150i			1.47146i	0.677274	+	0.735731i	$0.263161\pi$
$643$			946.150i			1.47146i	−0.677274	+	0.735731i	$0.736839\pi$
$644$	−188.790	+	37.3977i	−0.293152	+	0.0580710i
$645$		0			0
$646$	482.232			0.746490
$647$	157.004			0.242665			0.121332	−	0.992612i	$-0.461283\pi$
$647$	157.004			0.242665			0.121332	+	0.992612i	$0.461283\pi$
$648$		0			0
$649$		−	810.497i		−	1.24884i
$650$	−364.282			−0.560434
$651$		0			0
$652$	−232.207			−0.356146
$653$	111.109			0.170152			0.0850761	−	0.996374i	$-0.472887\pi$
$653$	111.109			0.170152			0.0850761	+	0.996374i	$0.472887\pi$
$654$		0			0
$655$			83.5282i			0.127524i
$656$	−169.756			−0.258774
$657$		0			0
$658$			118.492i			0.180079i
$659$		−	397.086i		−	0.602559i	−0.953536	−	0.301279i	$-0.902586\pi$
$659$		−	397.086i		−	0.602559i	0.953536	−	0.301279i	$-0.0974137\pi$
$660$		0			0
$661$		−	755.142i		−	1.14242i	−0.820803	−	0.571212i	$-0.806474\pi$
$661$		−	755.142i		−	1.14242i	0.820803	−	0.571212i	$-0.193526\pi$
$662$	499.317			0.754256
$663$		0			0
$664$		−	302.092i		−	0.454958i
$665$		−	192.855i		−	0.290007i
$666$		0			0
$667$	−227.659	−	1149.26i	−0.341318	−	1.72303i
$668$	563.918			0.844189
$669$		0			0
$670$	10.5203			0.0157019
$671$	1440.35			2.14657
$672$		0			0
$673$	394.477			0.586147			0.293073	−	0.956090i	$-0.405322\pi$
$673$	394.477			0.586147			0.293073	+	0.956090i	$0.405322\pi$
$674$		−	134.880i		−	0.200119i
$675$		0			0
$676$	−66.9743			−0.0990745
$677$		−	776.614i		−	1.14714i	−0.819156	−	0.573571i	$-0.805558\pi$
$677$		−	776.614i		−	1.14714i	0.819156	−	0.573571i	$-0.194442\pi$
$678$		0			0
$679$	500.034			0.736427
$680$	−60.1025			−0.0883860
$681$		0			0
$682$			332.200i			0.487096i
$683$	−809.349			−1.18499			−0.592496	−	0.805574i	$-0.701857\pi$
$683$	−809.349			−1.18499			−0.592496	+	0.805574i	$0.701857\pi$
$684$		0			0
$685$	165.784			0.242021
$686$		−	476.282i		−	0.694288i
$687$		0			0
$688$			8.41129i			0.0122257i
$689$		−	577.748i		−	0.838531i
$690$		0			0
$691$	−118.726			−0.171818			−0.0859088	−	0.996303i	$-0.527379\pi$
$691$	−118.726			−0.171818			−0.0859088	+	0.996303i	$0.527379\pi$
$692$	−231.461			−0.334481
$693$		0			0
$694$	492.900			0.710231
$695$		−	295.637i		−	0.425376i
$696$		0			0
$697$			532.089i			0.763398i
$698$	649.072			0.929903
$699$		0			0
$700$		−	185.158i		−	0.264511i
$701$			751.269i			1.07171i	0.844310	+	0.535856i	$0.180011\pi$
$701$			751.269i			1.07171i	−0.844310	+	0.535856i	$0.819989\pi$
$702$		0			0
$703$	1434.15			2.04005
$704$		−	162.376i		−	0.230648i
$705$		0			0
$706$	−318.227			−0.450746
$707$			102.720i			0.145289i
$708$		0			0
$709$		−	80.3638i		−	0.113348i	−0.998393	−	0.0566740i	$-0.981950\pi$
$709$		−	80.3638i		−	0.113348i	0.998393	−	0.0566740i	$-0.0180496\pi$
$710$		−	111.621i		−	0.157213i
$711$		0			0
$712$			387.848i			0.544731i
$713$	51.7235	+	261.109i	0.0725435	+	0.366212i
$714$		0			0
$715$	400.452			0.560073
$716$	−369.809			−0.516493
$717$		0			0
$718$			451.095i			0.628266i
$719$	211.128			0.293641			0.146821	−	0.989163i	$-0.453096\pi$
$719$	211.128			0.293641			0.146821	+	0.989163i	$0.453096\pi$
$720$		0			0
$721$	−32.5505			−0.0451463
$722$	−535.536			−0.741739
$723$		0			0
$724$			698.314i			0.964522i
$725$	1127.15			1.55469
$726$		0			0
$727$		−	870.940i		−	1.19799i	−0.800752	−	0.598996i	$-0.795567\pi$
$727$		−	870.940i		−	1.19799i	0.800752	−	0.598996i	$-0.204433\pi$
$728$			137.757i			0.189227i
$729$		0			0
$730$			8.91598i			0.0122137i
$731$	26.3646			0.0360665
$732$		0			0
$733$		−	1194.53i		−	1.62964i	−0.579713	−	0.814821i	$-0.696835\pi$
$733$		−	1194.53i		−	1.62964i	0.579713	−	0.814821i	$-0.303165\pi$
$734$			15.3359i			0.0208935i
$735$		0			0
$736$	−25.2820	−	127.628i	−0.0343505	−	0.173407i
$737$	89.0872			0.120878
$738$		0			0
$739$	−1344.03			−1.81872			−0.909360	−	0.416011i	$-0.863428\pi$
$739$	−1344.03			−1.81872			−0.909360	+	0.416011i	$0.863428\pi$
$740$	−178.744			−0.241546
$741$		0			0
$742$	293.659			0.395766
$743$		−	1070.72i		−	1.44107i	−0.693417	−	0.720537i	$-0.743896\pi$
$743$		−	1070.72i		−	1.44107i	0.693417	−	0.720537i	$-0.256104\pi$
$744$		0			0
$745$	14.7880			0.0198497
$746$		−	356.859i		−	0.478363i
$747$		0			0
$748$	−508.957			−0.680424
$749$	179.304			0.239391
$750$		0			0
$751$			812.180i			1.08146i	0.841195	+	0.540732i	$0.181853\pi$
$751$			812.180i			1.08146i	−0.841195	+	0.540732i	$0.818147\pi$
$752$	−80.1040			−0.106521
$753$		0			0
$754$	−838.599			−1.11220
$755$			146.051i			0.193445i
$756$		0			0
$757$		−	624.896i		−	0.825491i	−0.910846	−	0.412745i	$-0.864570\pi$
$757$		−	624.896i		−	0.825491i	0.910846	−	0.412745i	$-0.135430\pi$
$758$		−	955.779i		−	1.26092i
$759$		0			0
$760$	130.375			0.171546
$761$	−1119.42			−1.47098			−0.735491	−	0.677534i	$-0.763049\pi$
$761$	−1119.42			−1.47098			−0.735491	+	0.677534i	$0.763049\pi$
$762$		0			0
$763$	−5.72570			−0.00750419
$764$		−	638.326i		−	0.835505i
$765$		0			0
$766$		−	403.804i		−	0.527160i
$767$	−464.847			−0.606058
$768$		0			0
$769$			483.747i			0.629059i	0.949248	+	0.314530i	$0.101847\pi$
$769$			483.747i			0.629059i	−0.949248	+	0.314530i	$0.898153\pi$
$770$			203.542i			0.264341i
$771$		0			0
$772$	742.992			0.962425
$773$		−	1101.62i		−	1.42512i	−0.701609	−	0.712562i	$-0.747535\pi$
$773$		−	1101.62i		−	1.42512i	0.701609	−	0.712562i	$-0.252465\pi$
$774$		0			0
$775$	−256.086			−0.330433
$776$			338.038i			0.435615i
$777$		0			0
$778$			168.862i			0.217046i
$779$		−	1154.22i		−	1.48166i
$780$		0			0
$781$		−	945.227i		−	1.21028i
$782$	−400.041	+	79.2446i	−0.511561	+	0.101336i
$783$		0			0
$784$	125.981			0.160689
$785$	−301.439			−0.383999
$786$		0			0
$787$			1186.68i			1.50785i	0.656961	+	0.753925i	$0.271842\pi$
$787$			1186.68i			1.50785i	−0.656961	+	0.753925i	$0.728158\pi$
$788$	381.609			0.484276
$789$		0			0
$790$	63.4072			0.0802623
$791$	508.330			0.642642
$792$		0			0
$793$		−	826.086i		−	1.04172i
$794$	−191.693			−0.241427
$795$		0			0
$796$		−	330.408i		−	0.415085i
$797$		−	89.5551i		−	0.112365i	−0.998421	−	0.0561826i	$-0.982107\pi$
$797$		−	89.5551i		−	0.112365i	0.998421	−	0.0561826i	$-0.0178929\pi$
$798$		0			0
$799$			251.081i			0.314244i
$800$	125.172			0.156465
$801$		0			0
$802$			191.595i			0.238896i
$803$			75.5019i			0.0940247i
$804$		0			0
$805$	31.6915	+	159.984i	0.0393684	+	0.198738i
$806$	190.527			0.236386
$807$		0			0
$808$	−69.4414			−0.0859423
$809$	1189.39			1.47020			0.735099	−	0.677960i	$-0.237136\pi$
$809$	1189.39			1.47020			0.735099	+	0.677960i	$0.237136\pi$
$810$		0			0
$811$	−1569.97			−1.93585			−0.967923	−	0.251247i	$-0.919159\pi$
$811$	−1569.97			−1.93585			−0.967923	+	0.251247i	$0.919159\pi$
$812$		−	426.244i		−	0.524931i
$813$		0			0
$814$	−1513.63			−1.85950
$815$			196.777i			0.241444i
$816$		0			0
$817$	−57.1906			−0.0700008
$818$	−383.885			−0.469297
$819$		0			0
$820$			143.854i			0.175432i
$821$	−49.4062			−0.0601780			−0.0300890	−	0.999547i	$-0.509579\pi$
$821$	−49.4062			−0.0601780			−0.0300890	+	0.999547i	$0.509579\pi$
$822$		0			0
$823$	−41.3668			−0.0502635			−0.0251317	−	0.999684i	$-0.508001\pi$
$823$	−41.3668			−0.0502635			−0.0251317	+	0.999684i	$0.508001\pi$
$824$		−	22.0051i		−	0.0267052i
$825$		0			0
$826$		−	236.273i		−	0.286045i
$827$			101.706i			0.122982i	0.998108	+	0.0614912i	$0.0195856\pi$
$827$			101.706i			0.122982i	−0.998108	+	0.0614912i	$0.980414\pi$
$828$		0			0
$829$	1215.98			1.46680			0.733401	−	0.679796i	$-0.237932\pi$
$829$	1215.98			1.46680			0.733401	+	0.679796i	$0.237932\pi$
$830$	−255.998			−0.308432
$831$		0			0
$832$	−93.1280			−0.111933
$833$		−	394.878i		−	0.474043i
$834$		0			0
$835$		−	477.875i		−	0.572305i
$836$	1104.04			1.32062
$837$		0			0
$838$			303.189i			0.361801i
$839$		−	593.795i		−	0.707741i	−0.935294	−	0.353871i	$-0.884865\pi$
$839$		−	593.795i		−	0.707741i	0.935294	−	0.353871i	$-0.115135\pi$
$840$		0			0
$841$	1753.77			2.08533
$842$			805.225i			0.956325i
$843$		0			0
$844$	435.772			0.516317
$845$			56.7553i			0.0671661i
$846$		0			0
$847$			1217.38i			1.43728i
$848$			198.522i			0.234106i
$849$		0			0
$850$		−	392.344i		−	0.461581i
$851$	−1189.72	+	235.673i	−1.39802	+	0.276936i
$852$		0			0
$853$	−708.023			−0.830038			−0.415019	−	0.909813i	$-0.636225\pi$
$853$	−708.023			−0.830038			−0.415019	+	0.909813i	$0.636225\pi$
$854$	419.884			0.491668
$855$		0			0
$856$			121.215i			0.141606i
$857$	−1406.05			−1.64067			−0.820335	−	0.571884i	$-0.806213\pi$
$857$	−1406.05			−1.64067			−0.820335	+	0.571884i	$0.806213\pi$
$858$		0			0
$859$	848.615			0.987911			0.493955	−	0.869487i	$-0.335551\pi$
$859$	848.615			0.987911			0.493955	+	0.869487i	$0.335551\pi$
$860$	7.12789			0.00828824
$861$		0			0
$862$			259.088i			0.300566i
$863$	869.000			1.00695			0.503476	−	0.864009i	$-0.332054\pi$
$863$	869.000			1.00695			0.503476	+	0.864009i	$0.332054\pi$
$864$		0			0
$865$			196.144i			0.226756i
$866$			1059.69i			1.22366i
$867$		0			0
$868$			96.8415i			0.111569i
$869$	536.942			0.617885
$870$		0			0
$871$		−	51.0944i		−	0.0586618i
$872$		−	3.87074i		−	0.00443892i
$873$		0			0
$874$	867.775	−	171.899i	0.992877	−	0.196681i
$875$	−334.181			−0.381922
$876$		0			0
$877$	223.104			0.254394			0.127197	−	0.991877i	$-0.459402\pi$
$877$	223.104			0.254394			0.127197	+	0.991877i	$0.459402\pi$
$878$	129.398			0.147378
$879$		0			0
$880$	−137.601			−0.156364
$881$		−	151.919i		−	0.172439i	−0.996276	−	0.0862194i	$-0.972521\pi$
$881$		−	151.919i		−	0.172439i	0.996276	−	0.0862194i	$-0.0274786\pi$
$882$		0			0
$883$	−658.614			−0.745882			−0.372941	−	0.927855i	$-0.621651\pi$
$883$	−658.614			−0.745882			−0.372941	+	0.927855i	$0.621651\pi$
$884$			291.904i			0.330208i
$885$		0			0
$886$	518.552			0.585273
$887$	623.832			0.703306			0.351653	−	0.936130i	$-0.385620\pi$
$887$	623.832			0.703306			0.351653	+	0.936130i	$0.385620\pi$
$888$		0			0
$889$		−	936.172i		−	1.05306i
$890$	328.670			0.369292
$891$		0			0
$892$	−458.168			−0.513641
$893$		−	544.649i		−	0.609909i
$894$		0			0
$895$			313.383i			0.350149i
$896$		−	47.3352i		−	0.0528295i
$897$		0			0
$898$	580.595			0.646542
$899$	−589.524			−0.655755
$900$		0			0
$901$	622.253			0.690625
$902$			1218.18i			1.35053i
$903$		0			0
$904$			343.646i			0.380139i
$905$	591.765			0.653884
$906$		0			0
$907$			111.606i			0.123050i	0.998106	+	0.0615249i	$0.0195964\pi$
$907$			111.606i			0.123050i	−0.998106	+	0.0615249i	$0.980404\pi$
$908$		−	545.180i		−	0.600418i
$909$		0			0
$910$	116.738			0.128284
$911$			1401.85i			1.53880i	0.638767	+	0.769400i	$0.279445\pi$
$911$			1401.85i			1.53880i	−0.638767	+	0.769400i	$0.720555\pi$
$912$		0			0
$913$	−2167.83			−2.37441
$914$			1188.06i			1.29985i
$915$		0			0
$916$		−	310.796i		−	0.339297i
$917$			206.198i			0.224861i
$918$		0			0
$919$		−	952.820i		−	1.03680i	−0.855138	−	0.518401i	$-0.826528\pi$
$919$		−	952.820i		−	1.03680i	0.855138	−	0.518401i	$-0.173472\pi$
$920$	−108.154	+	21.4244i	−0.117559	+	0.0232874i
$921$		0			0
$922$	−50.8557			−0.0551580
$923$	−542.118			−0.587344
$924$		0			0
$925$		−	1166.83i		−	1.26143i
$926$	−869.995			−0.939520
$927$		0			0
$928$	288.154			0.310510
$929$	977.171			1.05185			0.525926	−	0.850530i	$-0.323719\pi$
$929$	977.171			1.05185			0.525926	+	0.850530i	$0.323719\pi$
$930$		0			0
$931$			856.575i			0.920059i
$932$	−400.544			−0.429768
$933$		0			0
$934$			713.724i			0.764159i
$935$			431.300i			0.461283i
$936$		0			0
$937$			1381.30i			1.47418i	0.675796	+	0.737089i	$0.263800\pi$
$937$			1381.30i			1.47418i	−0.675796	+	0.737089i	$0.736200\pi$
$938$	25.9703			0.0276869
$939$		0			0
$940$			67.8817i			0.0722145i
$941$		−	918.677i		−	0.976277i	−0.872766	−	0.488138i	$-0.837676\pi$
$941$		−	918.677i		−	0.976277i	0.872766	−	0.488138i	$-0.162324\pi$
$942$		0			0
$943$	189.671	+	957.491i	0.201136	+	1.01537i
$944$	159.727			0.169203
$945$		0			0
$946$	60.3601			0.0638056
$947$	−346.985			−0.366404			−0.183202	−	0.983075i	$-0.558646\pi$
$947$	−346.985			−0.366404			−0.183202	+	0.983075i	$0.558646\pi$
$948$		0			0
$949$	43.3028			0.0456299
$950$			851.079i			0.895873i
$951$		0			0
$952$	−148.369			−0.155850
$953$			1718.04i			1.80277i	0.433023	+	0.901383i	$0.357447\pi$
$953$			1718.04i			1.80277i	−0.433023	+	0.901383i	$0.642553\pi$
$954$		0			0
$955$	−540.929			−0.566418
$956$	−881.198			−0.921755
$957$		0			0
$958$			70.8724i			0.0739795i
$959$	409.256			0.426753
$960$		0			0
$961$	−827.062			−0.860626
$962$			868.117i			0.902409i
$963$		0			0
$964$			662.371i			0.687107i
$965$		−	629.626i		−	0.652462i
$966$		0			0
$967$	−1277.14			−1.32073			−0.660363	−	0.750946i	$-0.729598\pi$
$967$	−1277.14			−1.32073			−0.660363	+	0.750946i	$0.729598\pi$
$968$	−822.983			−0.850189
$969$		0			0
$970$	286.459			0.295319
$971$			1001.34i			1.03125i	0.856814	+	0.515626i	$0.172440\pi$
$971$			1001.34i			1.03125i	−0.856814	+	0.515626i	$0.827560\pi$
$972$		0			0
$973$		−	729.809i		−	0.750061i
$974$	508.461			0.522034
$975$		0			0
$976$			283.854i			0.290834i
$977$			980.126i			1.00320i	0.865100	+	0.501600i	$0.167255\pi$
$977$			980.126i			1.00320i	−0.865100	+	0.501600i	$0.832745\pi$
$978$		0			0
$979$	2783.23			2.84293
$980$		−	106.758i		−	0.108937i
$981$		0			0
$982$	−730.298			−0.743684
$983$		−	1017.15i		−	1.03474i	−0.855761	−	0.517371i	$-0.826911\pi$
$983$		−	1017.15i		−	1.03474i	0.855761	−	0.517371i	$-0.173089\pi$
$984$		0			0
$985$		−	323.383i		−	0.328308i
$986$		−	903.199i		−	0.916023i
$987$		0			0
$988$		−	633.202i		−	0.640893i
$989$	47.4430	−	9.39807i	0.0479707	−	0.00950259i
$990$		0			0
$991$	1094.77			1.10471			0.552356	−	0.833608i	$-0.313729\pi$
$991$	1094.77			1.10471			0.552356	+	0.833608i	$0.313729\pi$
$992$	−65.4677			−0.0659957
$993$		0			0
$994$		−	275.549i		−	0.277212i
$995$	−279.994			−0.281401
$996$		0			0
$997$	385.578			0.386738			0.193369	−	0.981126i	$-0.438059\pi$
$997$	385.578			0.386738			0.193369	+	0.981126i	$0.438059\pi$
$998$	−86.3020			−0.0864749
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	414.3.b.c.91.5		8
3.2	odd	2		138.3.b.a.91.2	yes	8
12.11	even	2		1104.3.c.c.1057.7		8
23.22	odd	2	inner	414.3.b.c.91.8		8
69.68	even	2		138.3.b.a.91.1	✓	8
276.275	odd	2		1104.3.c.c.1057.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.3.b.a.91.1	✓	8	69.68	even	2
138.3.b.a.91.2	yes	8	3.2	odd	2
414.3.b.c.91.5		8	1.1	even	1	trivial
414.3.b.c.91.8		8	23.22	odd	2	inner
1104.3.c.c.1057.6		8	276.275	odd	2
1104.3.c.c.1057.7		8	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	414.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} +2.00000 q^{4} -1.69484i q^{5} -4.18388i q^{7} +2.82843 q^{8} +O(q^{10})\)	\(q+1.41421 q^{2} +2.00000 q^{4} -1.69484i q^{5} -4.18388i q^{7} +2.82843 q^{8} -2.39686i q^{10} -20.2970i q^{11} -11.6410 q^{13} -5.91690i q^{14} +4.00000 q^{16} -12.5377i q^{17} +27.1971i q^{19} -3.38968i q^{20} -28.7043i q^{22} +(-4.46926 - 22.5616i) q^{23} +22.1275 q^{25} -16.4629 q^{26} -8.36776i q^{28} +50.9389 q^{29} -11.5732 q^{31} +5.65685 q^{32} -17.7310i q^{34} -7.09100 q^{35} -52.7319i q^{37} +38.4625i q^{38} -4.79373i q^{40} -42.4390 q^{41} +2.10282i q^{43} -40.5940i q^{44} +(-6.32049 - 31.9069i) q^{46} -20.0260 q^{47} +31.4951 q^{49} +31.2930 q^{50} -23.2820 q^{52} +49.6304i q^{53} -34.4001 q^{55} -11.8338i q^{56} +72.0384 q^{58} +39.9319 q^{59} +70.9635i q^{61} -16.3669 q^{62} +8.00000 q^{64} +19.7296i q^{65} +4.38918i q^{67} -25.0755i q^{68} -10.0282 q^{70} +46.5698 q^{71} -3.71985 q^{73} -74.5741i q^{74} +54.3941i q^{76} -84.9203 q^{77} +26.4542i q^{79} -6.77935i q^{80} -60.0178 q^{82} -106.806i q^{83} -21.2494 q^{85} +2.97384i q^{86} -57.4086i q^{88} +137.125i q^{89} +48.7046i q^{91} +(-8.93852 - 45.1232i) q^{92} -28.3210 q^{94} +46.0946 q^{95} +119.514i q^{97} +44.5408 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{4}+O(q^{10}) \) 8 * q + 16 * q^4	\( 8 q + 16 q^{4} + 16 q^{13} + 32 q^{16} - 16 q^{23} + 72 q^{25} - 32 q^{26} + 144 q^{29} - 128 q^{31} + 112 q^{35} + 16 q^{41} - 80 q^{46} + 112 q^{47} + 40 q^{49} + 160 q^{50} + 32 q^{52} - 64 q^{55} + 128 q^{58} - 80 q^{59} + 96 q^{62} + 64 q^{64} - 144 q^{70} - 32 q^{71} + 64 q^{73} - 224 q^{77} + 48 q^{85} - 32 q^{92} - 16 q^{94} - 112 q^{95} - 224 q^{98}+O(q^{100}) \) 8 * q + 16 * q^4 + 16 * q^13 + 32 * q^16 - 16 * q^23 + 72 * q^25 - 32 * q^26 + 144 * q^29 - 128 * q^31 + 112 * q^35 + 16 * q^41 - 80 * q^46 + 112 * q^47 + 40 * q^49 + 160 * q^50 + 32 * q^52 - 64 * q^55 + 128 * q^58 - 80 * q^59 + 96 * q^62 + 64 * q^64 - 144 * q^70 - 32 * q^71 + 64 * q^73 - 224 * q^77 + 48 * q^85 - 32 * q^92 - 16 * q^94 - 112 * q^95 - 224 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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