LMFDB - Embedded newform 672.2.i.d.209.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [672,2,Mod(209,672)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(672, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("672.209");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 672 = 2^{5} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	672.i (of order $2$, degree $1$, not 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:	$5.36594701583$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.3317760000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 8x^{6} + 13x^{4} + 12x^{2} + 36 $ x^8 - 8x^6 + 13x^4 + 12*x^2 + 36
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 168)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			209.3
Root			$0.578737 + 0.965926i$ of defining polynomial
Character	$\chi$	$=$	672.209
Dual form			672.2.i.d.209.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.58114 + 0.707107i) q^{3} +1.41421i q^{5} +(-1.00000 - 2.44949i) q^{7} +(2.00000 - 2.23607i) q^{9} +O(q^{10})$	$q+(-1.58114 + 0.707107i) q^{3} +1.41421i q^{5} +(-1.00000 - 2.44949i) q^{7} +(2.00000 - 2.23607i) q^{9} +3.46410 q^{11} -3.16228 q^{13} +(-1.00000 - 2.23607i) q^{15} +3.16228 q^{19} +(3.31319 + 3.16588i) q^{21} -4.47214i q^{23} +3.00000 q^{25} +(-1.58114 + 4.94975i) q^{27} +6.92820 q^{29} +4.89898i q^{31} +(-5.47723 + 2.44949i) q^{33} +(3.46410 - 1.41421i) q^{35} +(5.00000 - 2.23607i) q^{39} +10.9545 q^{41} +7.74597i q^{43} +(3.16228 + 2.82843i) q^{45} +10.9545 q^{47} +(-5.00000 + 4.89898i) q^{49} +4.89898i q^{55} +(-5.00000 + 2.23607i) q^{57} -9.89949i q^{59} -3.16228 q^{61} +(-7.47723 - 2.66291i) q^{63} -4.47214i q^{65} -7.74597i q^{67} +(3.16228 + 7.07107i) q^{69} +8.94427i q^{71} -14.6969i q^{73} +(-4.74342 + 2.12132i) q^{75} +(-3.46410 - 8.48528i) q^{77} +10.0000 q^{79} +(-1.00000 - 8.94427i) q^{81} +7.07107i q^{83} +(-10.9545 + 4.89898i) q^{87} +(3.16228 + 7.74597i) q^{91} +(-3.46410 - 7.74597i) q^{93} +4.47214i q^{95} +4.89898i q^{97} +(6.92820 - 7.74597i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{7} + 16 q^{9}+O(q^{10}) $ 8 * q - 8 * q^7 + 16 * q^9	$ 8 q - 8 q^{7} + 16 q^{9} - 8 q^{15} + 24 q^{25} + 40 q^{39} - 40 q^{49} - 40 q^{57} - 16 q^{63} + 80 q^{79} - 8 q^{81}+O(q^{100}) $ 8 * q - 8 * q^7 + 16 * q^9 - 8 * q^15 + 24 * q^25 + 40 * q^39 - 40 * q^49 - 40 * q^57 - 16 * q^63 + 80 * q^79 - 8 * q^81

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/672\mathbb{Z}\right)^\times$.

$n$	$127$	$421$	$449$	$577$
$\chi(n)$	$1$	$-1$	$-1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−1.58114	+	0.707107i	−0.912871	+	0.408248i
$3$	−1.58114	+	0.707107i	−0.912871	+	0.408248i
$4$		0			0
$5$			1.41421i			0.632456i	0.948683	+	0.316228i	$0.102416\pi$
$5$			1.41421i			0.632456i	−0.948683	+	0.316228i	$0.897584\pi$
$6$		0			0
$7$	−1.00000	−	2.44949i	−0.377964	−	0.925820i
$7$	−1.00000	−	2.44949i	−0.377964	−	0.925820i
$8$		0			0
$9$	2.00000	−	2.23607i	0.666667	−	0.745356i
$10$		0			0
$11$	3.46410			1.04447			0.522233	−	0.852803i	$-0.325099\pi$
$11$	3.46410			1.04447			0.522233	+	0.852803i	$0.325099\pi$
$12$		0			0
$13$	−3.16228			−0.877058			−0.438529	−	0.898717i	$-0.644500\pi$
$13$	−3.16228			−0.877058			−0.438529	+	0.898717i	$0.644500\pi$
$14$		0			0
$15$	−1.00000	−	2.23607i	−0.258199	−	0.577350i
$16$		0			0
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		0			0
$19$	3.16228			0.725476			0.362738	−	0.931891i	$-0.381842\pi$
$19$	3.16228			0.725476			0.362738	+	0.931891i	$0.381842\pi$
$20$		0			0
$21$	3.31319	+	3.16588i	0.722997	+	0.690851i
$22$		0			0
$23$		−	4.47214i		−	0.932505i	−0.884652	−	0.466252i	$-0.845604\pi$
$23$		−	4.47214i		−	0.932505i	0.884652	−	0.466252i	$-0.154396\pi$
$24$		0			0
$25$	3.00000			0.600000
$26$		0			0
$27$	−1.58114	+	4.94975i	−0.304290	+	0.952579i
$28$		0			0
$29$	6.92820			1.28654			0.643268	−	0.765641i	$-0.277578\pi$
$29$	6.92820			1.28654			0.643268	+	0.765641i	$0.277578\pi$
$30$		0			0
$31$			4.89898i			0.879883i	0.898027	+	0.439941i	$0.145001\pi$
$31$			4.89898i			0.879883i	−0.898027	+	0.439941i	$0.854999\pi$
$32$		0			0
$33$	−5.47723	+	2.44949i	−0.953463	+	0.426401i
$34$		0			0
$35$	3.46410	−	1.41421i	0.585540	−	0.239046i
$36$		0			0
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$	5.00000	−	2.23607i	0.800641	−	0.358057i
$40$		0			0
$41$	10.9545			1.71080			0.855399	−	0.517970i	$-0.173312\pi$
$41$	10.9545			1.71080			0.855399	+	0.517970i	$0.173312\pi$
$42$		0			0
$43$			7.74597i			1.18125i	0.806947	+	0.590624i	$0.201119\pi$
$43$			7.74597i			1.18125i	−0.806947	+	0.590624i	$0.798881\pi$
$44$		0			0
$45$	3.16228	+	2.82843i	0.471405	+	0.421637i
$46$		0			0
$47$	10.9545			1.59787			0.798935	−	0.601417i	$-0.205397\pi$
$47$	10.9545			1.59787			0.798935	+	0.601417i	$0.205397\pi$
$48$		0			0
$49$	−5.00000	+	4.89898i	−0.714286	+	0.699854i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$			4.89898i			0.660578i
$56$		0			0
$57$	−5.00000	+	2.23607i	−0.662266	+	0.296174i
$58$		0			0
$59$		−	9.89949i		−	1.28880i	−0.764687	−	0.644402i	$-0.777106\pi$
$59$		−	9.89949i		−	1.28880i	0.764687	−	0.644402i	$-0.222894\pi$
$60$		0			0
$61$	−3.16228			−0.404888			−0.202444	−	0.979294i	$-0.564888\pi$
$61$	−3.16228			−0.404888			−0.202444	+	0.979294i	$0.564888\pi$
$62$		0			0
$63$	−7.47723	−	2.66291i	−0.942042	−	0.335495i
$64$		0			0
$65$		−	4.47214i		−	0.554700i
$66$		0			0
$67$		−	7.74597i		−	0.946320i	−0.880976	−	0.473160i	$-0.843113\pi$
$67$		−	7.74597i		−	0.946320i	0.880976	−	0.473160i	$-0.156887\pi$
$68$		0			0
$69$	3.16228	+	7.07107i	0.380693	+	0.851257i
$70$		0			0
$71$			8.94427i			1.06149i	0.847532	+	0.530745i	$0.178088\pi$
$71$			8.94427i			1.06149i	−0.847532	+	0.530745i	$0.821912\pi$
$72$		0			0
$73$		−	14.6969i		−	1.72015i	−0.510171	−	0.860073i	$-0.670418\pi$
$73$		−	14.6969i		−	1.72015i	0.510171	−	0.860073i	$-0.329582\pi$
$74$		0			0
$75$	−4.74342	+	2.12132i	−0.547723	+	0.244949i
$76$		0			0
$77$	−3.46410	−	8.48528i	−0.394771	−	0.966988i
$78$		0			0
$79$	10.0000			1.12509			0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000			1.12509			0.562544	+	0.826767i	$0.309823\pi$
$80$		0			0
$81$	−1.00000	−	8.94427i	−0.111111	−	0.993808i
$82$		0			0
$83$			7.07107i			0.776151i	0.921628	+	0.388075i	$0.126860\pi$
$83$			7.07107i			0.776151i	−0.921628	+	0.388075i	$0.873140\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−10.9545	+	4.89898i	−1.17444	+	0.525226i
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$	3.16228	+	7.74597i	0.331497	+	0.811998i
$92$		0			0
$93$	−3.46410	−	7.74597i	−0.359211	−	0.803219i
$94$		0			0
$95$			4.47214i			0.458831i
$96$		0			0
$97$			4.89898i			0.497416i	0.968579	+	0.248708i	$0.0800060\pi$
$97$			4.89898i			0.497416i	−0.968579	+	0.248708i	$0.919994\pi$
$98$		0			0
$99$	6.92820	−	7.74597i	0.696311	−	0.778499i
$100$		0			0
$101$		−	15.5563i		−	1.54791i	−0.633238	−	0.773957i	$-0.718274\pi$
$101$		−	15.5563i		−	1.54791i	0.633238	−	0.773957i	$-0.281726\pi$
$102$		0			0
$103$		−	9.79796i		−	0.965422i	−0.875780	−	0.482711i	$-0.839652\pi$
$103$		−	9.79796i		−	0.965422i	0.875780	−	0.482711i	$-0.160348\pi$
$104$		0			0
$105$	−4.47723	+	4.68556i	−0.436932	+	0.457264i
$106$		0			0
$107$	−10.3923			−1.00466			−0.502331	−	0.864675i	$-0.667524\pi$
$107$	−10.3923			−1.00466			−0.502331	+	0.864675i	$0.667524\pi$
$108$		0			0
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			4.47214i			0.420703i	0.977626	+	0.210352i	$0.0674609\pi$
$113$			4.47214i			0.420703i	−0.977626	+	0.210352i	$0.932539\pi$
$114$		0			0
$115$	6.32456			0.589768
$116$		0			0
$117$	−6.32456	+	7.07107i	−0.584705	+	0.653720i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	1.00000			0.0909091
$122$		0			0
$123$	−17.3205	+	7.74597i	−1.56174	+	0.698430i
$124$		0			0
$125$			11.3137i			1.01193i
$126$		0			0
$127$	−8.00000			−0.709885			−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000			−0.709885			−0.354943	+	0.934888i	$0.615500\pi$
$128$		0			0
$129$	−5.47723	−	12.2474i	−0.482243	−	1.07833i
$130$		0			0
$131$		−	1.41421i		−	0.123560i	−0.998090	−	0.0617802i	$-0.980322\pi$
$131$		−	1.41421i		−	0.123560i	0.998090	−	0.0617802i	$-0.0196778\pi$
$132$		0			0
$133$	−3.16228	−	7.74597i	−0.274204	−	0.671660i
$134$		0			0
$135$	−7.00000	−	2.23607i	−0.602464	−	0.192450i
$136$		0			0
$137$			17.8885i			1.52832i	0.645026	+	0.764161i	$0.276847\pi$
$137$			17.8885i			1.52832i	−0.645026	+	0.764161i	$0.723153\pi$
$138$		0			0
$139$	−15.8114			−1.34110			−0.670552	−	0.741862i	$-0.733943\pi$
$139$	−15.8114			−1.34110			−0.670552	+	0.741862i	$0.733943\pi$
$140$		0			0
$141$	−17.3205	+	7.74597i	−1.45865	+	0.652328i
$142$		0			0
$143$	−10.9545			−0.916057
$144$		0			0
$145$			9.79796i			0.813676i
$146$		0			0
$147$	4.44159	−	11.2815i	0.366336	−	0.930482i
$148$		0			0
$149$	−6.92820			−0.567581			−0.283790	−	0.958886i	$-0.591592\pi$
$149$	−6.92820			−0.567581			−0.283790	+	0.958886i	$0.591592\pi$
$150$		0			0
$151$	−8.00000			−0.651031			−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000			−0.651031			−0.325515	+	0.945537i	$0.605538\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−6.92820			−0.556487
$156$		0			0
$157$	15.8114			1.26189			0.630943	−	0.775829i	$-0.282668\pi$
$157$	15.8114			1.26189			0.630943	+	0.775829i	$0.282668\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−10.9545	+	4.47214i	−0.863332	+	0.352454i
$162$		0			0
$163$		−	23.2379i		−	1.82013i	−0.414462	−	0.910066i	$-0.636030\pi$
$163$		−	23.2379i		−	1.82013i	0.414462	−	0.910066i	$-0.363970\pi$
$164$		0			0
$165$	−3.46410	−	7.74597i	−0.269680	−	0.603023i
$166$		0			0
$167$	21.9089			1.69536			0.847681	−	0.530506i	$-0.177998\pi$
$167$	21.9089			1.69536			0.847681	+	0.530506i	$0.177998\pi$
$168$		0			0
$169$	−3.00000			−0.230769
$170$		0			0
$171$	6.32456	−	7.07107i	0.483651	−	0.540738i
$172$		0			0
$173$		−	7.07107i		−	0.537603i	−0.963196	−	0.268802i	$-0.913372\pi$
$173$		−	7.07107i		−	0.537603i	0.963196	−	0.268802i	$-0.0866276\pi$
$174$		0			0
$175$	−3.00000	−	7.34847i	−0.226779	−	0.555492i
$176$		0			0
$177$	7.00000	+	15.6525i	0.526152	+	1.17651i
$178$		0			0
$179$	10.3923			0.776757			0.388379	−	0.921500i	$-0.373035\pi$
$179$	10.3923			0.776757			0.388379	+	0.921500i	$0.373035\pi$
$180$		0			0
$181$	−3.16228			−0.235050			−0.117525	−	0.993070i	$-0.537496\pi$
$181$	−3.16228			−0.235050			−0.117525	+	0.993070i	$0.537496\pi$
$182$		0			0
$183$	5.00000	−	2.23607i	0.369611	−	0.165295i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	13.7055	−	1.07676i	0.996928	−	0.0783231i
$190$		0			0
$191$			8.94427i			0.647185i	0.946197	+	0.323592i	$0.104891\pi$
$191$			8.94427i			0.647185i	−0.946197	+	0.323592i	$0.895109\pi$
$192$		0			0
$193$	−4.00000			−0.287926			−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000			−0.287926			−0.143963	+	0.989583i	$0.545985\pi$
$194$		0			0
$195$	3.16228	+	7.07107i	0.226455	+	0.506370i
$196$		0			0
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$		0			0
$201$	5.47723	+	12.2474i	0.386334	+	0.863868i
$202$		0			0
$203$	−6.92820	−	16.9706i	−0.486265	−	1.19110i
$204$		0			0
$205$			15.4919i			1.08200i
$206$		0			0
$207$	−10.0000	−	8.94427i	−0.695048	−	0.621670i
$208$		0			0
$209$	10.9545			0.757735
$210$		0			0
$211$		−	7.74597i		−	0.533254i	−0.963800	−	0.266627i	$-0.914091\pi$
$211$		−	7.74597i		−	0.533254i	0.963800	−	0.266627i	$-0.0859092\pi$
$212$		0			0
$213$	−6.32456	−	14.1421i	−0.433351	−	0.969003i
$214$		0			0
$215$	−10.9545			−0.747087
$216$		0			0
$217$	12.0000	−	4.89898i	0.814613	−	0.332564i
$218$		0			0
$219$	10.3923	+	23.2379i	0.702247	+	1.57027i
$220$		0			0
$221$		0			0
$222$		0			0
$223$			9.79796i			0.656120i	0.944657	+	0.328060i	$0.106395\pi$
$223$			9.79796i			0.656120i	−0.944657	+	0.328060i	$0.893605\pi$
$224$		0			0
$225$	6.00000	−	6.70820i	0.400000	−	0.447214i
$226$		0			0
$227$		−	9.89949i		−	0.657053i	−0.944495	−	0.328526i	$-0.893448\pi$
$227$		−	9.89949i		−	0.657053i	0.944495	−	0.328526i	$-0.106552\pi$
$228$		0			0
$229$	15.8114			1.04485			0.522423	−	0.852686i	$-0.325028\pi$
$229$	15.8114			1.04485			0.522423	+	0.852686i	$0.325028\pi$
$230$		0			0
$231$	11.4772	+	10.9669i	0.755146	+	0.721570i
$232$		0			0
$233$		−	8.94427i		−	0.585959i	−0.956119	−	0.292979i	$-0.905353\pi$
$233$		−	8.94427i		−	0.585959i	0.956119	−	0.292979i	$-0.0946467\pi$
$234$		0			0
$235$			15.4919i			1.01058i
$236$		0			0
$237$	−15.8114	+	7.07107i	−1.02706	+	0.459315i
$238$		0			0
$239$		−	4.47214i		−	0.289278i	−0.989484	−	0.144639i	$-0.953798\pi$
$239$		−	4.47214i		−	0.289278i	0.989484	−	0.144639i	$-0.0462022\pi$
$240$		0			0
$241$			4.89898i			0.315571i	0.987473	+	0.157786i	$0.0504355\pi$
$241$			4.89898i			0.315571i	−0.987473	+	0.157786i	$0.949565\pi$
$242$		0			0
$243$	7.90569	+	13.4350i	0.507151	+	0.861858i
$244$		0			0
$245$	−6.92820	−	7.07107i	−0.442627	−	0.451754i
$246$		0			0
$247$	−10.0000			−0.636285
$248$		0			0
$249$	−5.00000	−	11.1803i	−0.316862	−	0.708525i
$250$		0			0
$251$			15.5563i			0.981908i	0.871185	+	0.490954i	$0.163352\pi$
$251$			15.5563i			0.981908i	−0.871185	+	0.490954i	$0.836648\pi$
$252$		0			0
$253$		−	15.4919i		−	0.973970i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	10.9545			0.683320			0.341660	−	0.939824i	$-0.389011\pi$
$257$	10.9545			0.683320			0.341660	+	0.939824i	$0.389011\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	13.8564	−	15.4919i	0.857690	−	0.958927i
$262$		0			0
$263$			8.94427i			0.551527i	0.961225	+	0.275764i	$0.0889307\pi$
$263$			8.94427i			0.551527i	−0.961225	+	0.275764i	$0.911069\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$			9.89949i			0.603583i	0.953374	+	0.301791i	$0.0975846\pi$
$269$			9.89949i			0.603583i	−0.953374	+	0.301791i	$0.902415\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$	−10.4772	−	10.0114i	−0.634111	−	0.605916i
$274$		0			0
$275$	10.3923			0.626680
$276$		0			0
$277$			15.4919i			0.930820i	0.885095	+	0.465410i	$0.154093\pi$
$277$			15.4919i			0.930820i	−0.885095	+	0.465410i	$0.845907\pi$
$278$		0			0
$279$	10.9545	+	9.79796i	0.655826	+	0.586588i
$280$		0			0
$281$		−	8.94427i		−	0.533571i	−0.963756	−	0.266785i	$-0.914039\pi$
$281$		−	8.94427i		−	0.533571i	0.963756	−	0.266785i	$-0.0859614\pi$
$282$		0			0
$283$	−15.8114			−0.939889			−0.469945	−	0.882696i	$-0.655726\pi$
$283$	−15.8114			−0.939889			−0.469945	+	0.882696i	$0.655726\pi$
$284$		0			0
$285$	−3.16228	−	7.07107i	−0.187317	−	0.418854i
$286$		0			0
$287$	−10.9545	−	26.8328i	−0.646621	−	1.58389i
$288$		0			0
$289$	−17.0000			−1.00000
$290$		0			0
$291$	−3.46410	−	7.74597i	−0.203069	−	0.454077i
$292$		0			0
$293$			26.8701i			1.56977i	0.619644	+	0.784883i	$0.287277\pi$
$293$			26.8701i			1.56977i	−0.619644	+	0.784883i	$0.712723\pi$
$294$		0			0
$295$	14.0000			0.815112
$296$		0			0
$297$	−5.47723	+	17.1464i	−0.317821	+	0.994937i
$298$		0			0
$299$			14.1421i			0.817861i
$300$		0			0
$301$	18.9737	−	7.74597i	1.09362	−	0.446470i
$302$		0			0
$303$	11.0000	+	24.5967i	0.631933	+	1.41305i
$304$		0			0
$305$		−	4.47214i		−	0.256074i
$306$		0			0
$307$	3.16228			0.180481			0.0902404	−	0.995920i	$-0.471236\pi$
$307$	3.16228			0.180481			0.0902404	+	0.995920i	$0.471236\pi$
$308$		0			0
$309$	6.92820	+	15.4919i	0.394132	+	0.881305i
$310$		0			0
$311$	−10.9545			−0.621170			−0.310585	−	0.950546i	$-0.600525\pi$
$311$	−10.9545			−0.621170			−0.310585	+	0.950546i	$0.600525\pi$
$312$		0			0
$313$		−	9.79796i		−	0.553813i	−0.960897	−	0.276907i	$-0.910691\pi$
$313$		−	9.79796i		−	0.553813i	0.960897	−	0.276907i	$-0.0893093\pi$
$314$		0			0
$315$	3.76593	−	10.5744i	0.212186	−	0.595800i
$316$		0			0
$317$	6.92820			0.389127			0.194563	−	0.980890i	$-0.437671\pi$
$317$	6.92820			0.389127			0.194563	+	0.980890i	$0.437671\pi$
$318$		0			0
$319$	24.0000			1.34374
$320$		0			0
$321$	16.4317	−	7.34847i	0.917127	−	0.410152i
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−9.48683			−0.526235
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−10.9545	−	26.8328i	−0.603938	−	1.47934i
$330$		0			0
$331$			7.74597i			0.425757i	0.977079	+	0.212878i	$0.0682838\pi$
$331$			7.74597i			0.425757i	−0.977079	+	0.212878i	$0.931716\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	10.9545			0.598506
$336$		0			0
$337$	8.00000			0.435788			0.217894	−	0.975972i	$-0.430081\pi$
$337$	8.00000			0.435788			0.217894	+	0.975972i	$0.430081\pi$
$338$		0			0
$339$	−3.16228	−	7.07107i	−0.171751	−	0.384048i
$340$		0			0
$341$			16.9706i			0.919007i
$342$		0			0
$343$	17.0000	+	7.34847i	0.917914	+	0.396780i
$344$		0			0
$345$	−10.0000	+	4.47214i	−0.538382	+	0.240772i
$346$		0			0
$347$	17.3205			0.929814			0.464907	−	0.885360i	$-0.346088\pi$
$347$	17.3205			0.929814			0.464907	+	0.885360i	$0.346088\pi$
$348$		0			0
$349$	−22.1359			−1.18491			−0.592455	−	0.805604i	$-0.701841\pi$
$349$	−22.1359			−1.18491			−0.592455	+	0.805604i	$0.701841\pi$
$350$		0			0
$351$	5.00000	−	15.6525i	0.266880	−	0.835467i
$352$		0			0
$353$	−10.9545			−0.583047			−0.291523	−	0.956564i	$-0.594162\pi$
$353$	−10.9545			−0.583047			−0.291523	+	0.956564i	$0.594162\pi$
$354$		0			0
$355$	−12.6491			−0.671345
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	31.3050i		−	1.65221i	−0.563515	−	0.826106i	$-0.690551\pi$
$359$		−	31.3050i		−	1.65221i	0.563515	−	0.826106i	$-0.309449\pi$
$360$		0			0
$361$	−9.00000			−0.473684
$362$		0			0
$363$	−1.58114	+	0.707107i	−0.0829883	+	0.0371135i
$364$		0			0
$365$	20.7846			1.08792
$366$		0			0
$367$		−	19.5959i		−	1.02290i	−0.859313	−	0.511449i	$-0.829109\pi$
$367$		−	19.5959i		−	1.02290i	0.859313	−	0.511449i	$-0.170891\pi$
$368$		0			0
$369$	21.9089	−	24.4949i	1.14053	−	1.27515i
$370$		0			0
$371$		0			0
$372$		0			0
$373$			30.9839i			1.60428i	0.597133	+	0.802142i	$0.296306\pi$
$373$			30.9839i			1.60428i	−0.597133	+	0.802142i	$0.703694\pi$
$374$		0			0
$375$	−8.00000	−	17.8885i	−0.413118	−	0.923760i
$376$		0			0
$377$	−21.9089			−1.12837
$378$		0			0
$379$			23.2379i			1.19365i	0.802371	+	0.596825i	$0.203571\pi$
$379$			23.2379i			1.19365i	−0.802371	+	0.596825i	$0.796429\pi$
$380$		0			0
$381$	12.6491	−	5.65685i	0.648034	−	0.289809i
$382$		0			0
$383$	−10.9545			−0.559746			−0.279873	−	0.960037i	$-0.590292\pi$
$383$	−10.9545			−0.559746			−0.279873	+	0.960037i	$0.590292\pi$
$384$		0			0
$385$	12.0000	−	4.89898i	0.611577	−	0.249675i
$386$		0			0
$387$	17.3205	+	15.4919i	0.880451	+	0.787499i
$388$		0			0
$389$	27.7128			1.40510			0.702548	−	0.711637i	$-0.252046\pi$
$389$	27.7128			1.40510			0.702548	+	0.711637i	$0.252046\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	1.00000	+	2.23607i	0.0504433	+	0.112795i
$394$		0			0
$395$			14.1421i			0.711568i
$396$		0			0
$397$	−22.1359			−1.11097			−0.555486	−	0.831526i	$-0.687468\pi$
$397$	−22.1359			−1.11097			−0.555486	+	0.831526i	$0.687468\pi$
$398$		0			0
$399$	10.4772	+	10.0114i	0.524517	+	0.501196i
$400$		0			0
$401$			4.47214i			0.223328i	0.993746	+	0.111664i	$0.0356180\pi$
$401$			4.47214i			0.223328i	−0.993746	+	0.111664i	$0.964382\pi$
$402$		0			0
$403$		−	15.4919i		−	0.771708i
$404$		0			0
$405$	12.6491	−	1.41421i	0.628539	−	0.0702728i
$406$		0			0
$407$		0			0
$408$		0			0
$409$			9.79796i			0.484478i	0.970217	+	0.242239i	$0.0778818\pi$
$409$			9.79796i			0.484478i	−0.970217	+	0.242239i	$0.922118\pi$
$410$		0			0
$411$	−12.6491	−	28.2843i	−0.623935	−	1.39516i
$412$		0			0
$413$	−24.2487	+	9.89949i	−1.19320	+	0.487122i
$414$		0			0
$415$	−10.0000			−0.490881
$416$		0			0
$417$	25.0000	−	11.1803i	1.22426	−	0.547504i
$418$		0			0
$419$			24.0416i			1.17451i	0.809402	+	0.587255i	$0.199792\pi$
$419$			24.0416i			1.17451i	−0.809402	+	0.587255i	$0.800208\pi$
$420$		0			0
$421$		−	30.9839i		−	1.51006i	−0.655690	−	0.755031i	$-0.727622\pi$
$421$		−	30.9839i		−	1.51006i	0.655690	−	0.755031i	$-0.272378\pi$
$422$		0			0
$423$	21.9089	−	24.4949i	1.06525	−	1.19098i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	3.16228	+	7.74597i	0.153033	+	0.374854i
$428$		0			0
$429$	17.3205	−	7.74597i	0.836242	−	0.373979i
$430$		0			0
$431$			22.3607i			1.07708i	0.842601	+	0.538538i	$0.181023\pi$
$431$			22.3607i			1.07708i	−0.842601	+	0.538538i	$0.818977\pi$
$432$		0			0
$433$			14.6969i			0.706290i	0.935569	+	0.353145i	$0.114888\pi$
$433$			14.6969i			0.706290i	−0.935569	+	0.353145i	$0.885112\pi$
$434$		0			0
$435$	−6.92820	−	15.4919i	−0.332182	−	0.742781i
$436$		0			0
$437$		−	14.1421i		−	0.676510i
$438$		0			0
$439$			24.4949i			1.16908i	0.811366	+	0.584539i	$0.198725\pi$
$439$			24.4949i			1.16908i	−0.811366	+	0.584539i	$0.801275\pi$
$440$		0			0
$441$	0.954451	+	20.9783i	0.0454501	+	0.998967i
$442$		0			0
$443$	−17.3205			−0.822922			−0.411461	−	0.911427i	$-0.634981\pi$
$443$	−17.3205			−0.822922			−0.411461	+	0.911427i	$0.634981\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	10.9545	−	4.89898i	0.518128	−	0.231714i
$448$		0			0
$449$		−	8.94427i		−	0.422106i	−0.977475	−	0.211053i	$-0.932311\pi$
$449$		−	8.94427i		−	0.422106i	0.977475	−	0.211053i	$-0.0676893\pi$
$450$		0			0
$451$	37.9473			1.78687
$452$		0			0
$453$	12.6491	−	5.65685i	0.594307	−	0.265782i
$454$		0			0
$455$	−10.9545	+	4.47214i	−0.513553	+	0.209657i
$456$		0			0
$457$	8.00000			0.374224			0.187112	−	0.982339i	$-0.440087\pi$
$457$	8.00000			0.374224			0.187112	+	0.982339i	$0.440087\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	15.5563i		−	0.724531i	−0.932075	−	0.362266i	$-0.882003\pi$
$461$		−	15.5563i		−	0.724531i	0.932075	−	0.362266i	$-0.117997\pi$
$462$		0			0
$463$	−14.0000			−0.650635			−0.325318	−	0.945605i	$-0.605471\pi$
$463$	−14.0000			−0.650635			−0.325318	+	0.945605i	$0.605471\pi$
$464$		0			0
$465$	10.9545	−	4.89898i	0.508001	−	0.227185i
$466$		0			0
$467$			7.07107i			0.327210i	0.986526	+	0.163605i	$0.0523123\pi$
$467$			7.07107i			0.327210i	−0.986526	+	0.163605i	$0.947688\pi$
$468$		0			0
$469$	−18.9737	+	7.74597i	−0.876122	+	0.357676i
$470$		0			0
$471$	−25.0000	+	11.1803i	−1.15194	+	0.515163i
$472$		0			0
$473$			26.8328i			1.23377i
$474$		0			0
$475$	9.48683			0.435286
$476$		0			0
$477$		0			0
$478$		0			0
$479$	10.9545			0.500522			0.250261	−	0.968178i	$-0.419484\pi$
$479$	10.9545			0.500522			0.250261	+	0.968178i	$0.419484\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$	14.1582	−	14.8170i	0.644222	−	0.674198i
$484$		0			0
$485$	−6.92820			−0.314594
$486$		0			0
$487$	−8.00000			−0.362515			−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000			−0.362515			−0.181257	+	0.983436i	$0.558017\pi$
$488$		0			0
$489$	16.4317	+	36.7423i	0.743066	+	1.66155i
$490$		0			0
$491$	−3.46410			−0.156333			−0.0781664	−	0.996940i	$-0.524907\pi$
$491$	−3.46410			−0.156333			−0.0781664	+	0.996940i	$0.524907\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	10.9545	+	9.79796i	0.492366	+	0.440386i
$496$		0			0
$497$	21.9089	−	8.94427i	0.982749	−	0.401205i
$498$		0			0
$499$		−	7.74597i		−	0.346757i	−0.984855	−	0.173379i	$-0.944532\pi$
$499$		−	7.74597i		−	0.346757i	0.984855	−	0.173379i	$-0.0554684\pi$
$500$		0			0
$501$	−34.6410	+	15.4919i	−1.54765	+	0.692129i
$502$		0			0
$503$	−32.8634			−1.46530			−0.732652	−	0.680603i	$-0.761718\pi$
$503$	−32.8634			−1.46530			−0.732652	+	0.680603i	$0.761718\pi$
$504$		0			0
$505$	22.0000			0.978987
$506$		0			0
$507$	4.74342	−	2.12132i	0.210663	−	0.0942111i
$508$		0			0
$509$			9.89949i			0.438787i	0.975636	+	0.219394i	$0.0704079\pi$
$509$			9.89949i			0.438787i	−0.975636	+	0.219394i	$0.929592\pi$
$510$		0			0
$511$	−36.0000	+	14.6969i	−1.59255	+	0.650154i
$512$		0			0
$513$	−5.00000	+	15.6525i	−0.220755	+	0.691074i
$514$		0			0
$515$	13.8564			0.610586
$516$		0			0
$517$	37.9473			1.66892
$518$		0			0
$519$	5.00000	+	11.1803i	0.219476	+	0.490762i
$520$		0			0
$521$	−32.8634			−1.43977			−0.719885	−	0.694094i	$-0.755805\pi$
$521$	−32.8634			−1.43977			−0.719885	+	0.694094i	$0.755805\pi$
$522$		0			0
$523$	3.16228			0.138277			0.0691384	−	0.997607i	$-0.477975\pi$
$523$	3.16228			0.138277			0.0691384	+	0.997607i	$0.477975\pi$
$524$		0			0
$525$	9.93957	+	9.49763i	0.433798	+	0.414511i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	3.00000			0.130435
$530$		0			0
$531$	−22.1359	−	19.7990i	−0.960618	−	0.859203i
$532$		0			0
$533$	−34.6410			−1.50047
$534$		0			0
$535$		−	14.6969i		−	0.635404i
$536$		0			0
$537$	−16.4317	+	7.34847i	−0.709079	+	0.317110i
$538$		0			0
$539$	−17.3205	+	16.9706i	−0.746047	+	0.730974i
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$	5.00000	−	2.23607i	0.214571	−	0.0959589i
$544$		0			0
$545$		0			0
$546$		0			0
$547$			7.74597i			0.331194i	0.986194	+	0.165597i	$0.0529550\pi$
$547$			7.74597i			0.331194i	−0.986194	+	0.165597i	$0.947045\pi$
$548$		0			0
$549$	−6.32456	+	7.07107i	−0.269925	+	0.301786i
$550$		0			0
$551$	21.9089			0.933351
$552$		0			0
$553$	−10.0000	−	24.4949i	−0.425243	−	1.04163i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$		0			0
$559$		−	24.4949i		−	1.03602i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	1.41421i		−	0.0596020i	−0.999556	−	0.0298010i	$-0.990513\pi$
$563$		−	1.41421i		−	0.0596020i	0.999556	−	0.0298010i	$-0.00948736\pi$
$564$		0			0
$565$	−6.32456			−0.266076
$566$		0			0
$567$	−20.9089	+	11.3938i	−0.878091	+	0.478493i
$568$		0			0
$569$		−	22.3607i		−	0.937408i	−0.883355	−	0.468704i	$-0.844721\pi$
$569$		−	22.3607i		−	0.937408i	0.883355	−	0.468704i	$-0.155279\pi$
$570$		0			0
$571$		−	7.74597i		−	0.324159i	−0.986778	−	0.162079i	$-0.948180\pi$
$571$		−	7.74597i		−	0.324159i	0.986778	−	0.162079i	$-0.0518200\pi$
$572$		0			0
$573$	−6.32456	−	14.1421i	−0.264212	−	0.590796i
$574$		0			0
$575$		−	13.4164i		−	0.559503i
$576$		0			0
$577$			29.3939i			1.22368i	0.790980	+	0.611842i	$0.209571\pi$
$577$			29.3939i			1.22368i	−0.790980	+	0.611842i	$0.790429\pi$
$578$		0			0
$579$	6.32456	−	2.82843i	0.262840	−	0.117545i
$580$		0			0
$581$	17.3205	−	7.07107i	0.718576	−	0.293357i
$582$		0			0
$583$		0			0
$584$		0			0
$585$	−10.0000	−	8.94427i	−0.413449	−	0.369800i
$586$		0			0
$587$		−	35.3553i		−	1.45927i	−0.683836	−	0.729636i	$-0.739690\pi$
$587$		−	35.3553i		−	1.45927i	0.683836	−	0.729636i	$-0.260310\pi$
$588$		0			0
$589$			15.4919i			0.638334i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−32.8634			−1.34954			−0.674768	−	0.738030i	$-0.735756\pi$
$593$	−32.8634			−1.34954			−0.674768	+	0.738030i	$0.735756\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$			8.94427i			0.365453i	0.983164	+	0.182727i	$0.0584923\pi$
$599$			8.94427i			0.365453i	−0.983164	+	0.182727i	$0.941508\pi$
$600$		0			0
$601$		−	24.4949i		−	0.999168i	−0.866266	−	0.499584i	$-0.833486\pi$
$601$		−	24.4949i		−	0.999168i	0.866266	−	0.499584i	$-0.166514\pi$
$602$		0			0
$603$	−17.3205	−	15.4919i	−0.705346	−	0.630880i
$604$		0			0
$605$			1.41421i			0.0574960i
$606$		0			0
$607$			19.5959i			0.795374i	0.917521	+	0.397687i	$0.130187\pi$
$607$			19.5959i			0.795374i	−0.917521	+	0.397687i	$0.869813\pi$
$608$		0			0
$609$	22.9545	+	21.9338i	0.930161	+	0.888804i
$610$		0			0
$611$	−34.6410			−1.40143
$612$		0			0
$613$		−	46.4758i		−	1.87714i	−0.345089	−	0.938570i	$-0.612151\pi$
$613$		−	46.4758i		−	1.87714i	0.345089	−	0.938570i	$-0.387849\pi$
$614$		0			0
$615$	−10.9545	−	24.4949i	−0.441726	−	0.987730i
$616$		0			0
$617$			31.3050i			1.26029i	0.776478	+	0.630145i	$0.217005\pi$
$617$			31.3050i			1.26029i	−0.776478	+	0.630145i	$0.782995\pi$
$618$		0			0
$619$	3.16228			0.127103			0.0635513	−	0.997979i	$-0.479757\pi$
$619$	3.16228			0.127103			0.0635513	+	0.997979i	$0.479757\pi$
$620$		0			0
$621$	22.1359	+	7.07107i	0.888285	+	0.283752i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$	−17.3205	+	7.74597i	−0.691714	+	0.309344i
$628$		0			0
$629$		0			0
$630$		0			0
$631$	10.0000			0.398094			0.199047	−	0.979990i	$-0.436215\pi$
$631$	10.0000			0.398094			0.199047	+	0.979990i	$0.436215\pi$
$632$		0			0
$633$	5.47723	+	12.2474i	0.217700	+	0.486792i
$634$		0			0
$635$		−	11.3137i		−	0.448971i
$636$		0			0
$637$	15.8114	−	15.4919i	0.626470	−	0.613813i
$638$		0			0
$639$	20.0000	+	17.8885i	0.791188	+	0.707660i
$640$		0			0
$641$			31.3050i			1.23647i	0.785993	+	0.618236i	$0.212152\pi$
$641$			31.3050i			1.23647i	−0.785993	+	0.618236i	$0.787848\pi$
$642$		0			0
$643$	3.16228			0.124708			0.0623540	−	0.998054i	$-0.480139\pi$
$643$	3.16228			0.124708			0.0623540	+	0.998054i	$0.480139\pi$
$644$		0			0
$645$	17.3205	−	7.74597i	0.681994	−	0.304997i
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$		0			0
$649$		−	34.2929i		−	1.34611i
$650$		0			0
$651$	−15.5096	+	16.2312i	−0.607868	+	0.636153i
$652$		0			0
$653$	−48.4974			−1.89785			−0.948925	−	0.315501i	$-0.897828\pi$
$653$	−48.4974			−1.89785			−0.948925	+	0.315501i	$0.897828\pi$
$654$		0			0
$655$	2.00000			0.0781465
$656$		0			0
$657$	−32.8634	−	29.3939i	−1.28212	−	1.14676i
$658$		0			0
$659$	24.2487			0.944596			0.472298	−	0.881439i	$-0.343425\pi$
$659$	24.2487			0.944596			0.472298	+	0.881439i	$0.343425\pi$
$660$		0			0
$661$	−41.1096			−1.59898			−0.799489	−	0.600680i	$-0.794896\pi$
$661$	−41.1096			−1.59898			−0.799489	+	0.600680i	$0.794896\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	10.9545	−	4.47214i	0.424795	−	0.173422i
$666$		0			0
$667$		−	30.9839i		−	1.19970i
$668$		0			0
$669$	−6.92820	−	15.4919i	−0.267860	−	0.598953i
$670$		0			0
$671$	−10.9545			−0.422892
$672$		0			0
$673$	−46.0000			−1.77317			−0.886585	−	0.462566i	$-0.846929\pi$
$673$	−46.0000			−1.77317			−0.886585	+	0.462566i	$0.846929\pi$
$674$		0			0
$675$	−4.74342	+	14.8492i	−0.182574	+	0.571548i
$676$		0			0
$677$		−	7.07107i		−	0.271763i	−0.990725	−	0.135882i	$-0.956613\pi$
$677$		−	7.07107i		−	0.271763i	0.990725	−	0.135882i	$-0.0433867\pi$
$678$		0			0
$679$	12.0000	−	4.89898i	0.460518	−	0.188006i
$680$		0			0
$681$	7.00000	+	15.6525i	0.268241	+	0.599804i
$682$		0			0
$683$	−31.1769			−1.19295			−0.596476	−	0.802631i	$-0.703433\pi$
$683$	−31.1769			−1.19295			−0.596476	+	0.802631i	$0.703433\pi$
$684$		0			0
$685$	−25.2982			−0.966595
$686$		0			0
$687$	−25.0000	+	11.1803i	−0.953809	+	0.426557i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	3.16228			0.120299			0.0601494	−	0.998189i	$-0.480842\pi$
$691$	3.16228			0.120299			0.0601494	+	0.998189i	$0.480842\pi$
$692$		0			0
$693$	−25.9019	−	9.22460i	−0.983931	−	0.350413i
$694$		0			0
$695$		−	22.3607i		−	0.848189i
$696$		0			0
$697$		0			0
$698$		0			0
$699$	6.32456	+	14.1421i	0.239217	+	0.534905i
$700$		0			0
$701$	−20.7846			−0.785024			−0.392512	−	0.919747i	$-0.628394\pi$
$701$	−20.7846			−0.785024			−0.392512	+	0.919747i	$0.628394\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	−10.9545	−	24.4949i	−0.412568	−	0.922531i
$706$		0			0
$707$	−38.1051	+	15.5563i	−1.43309	+	0.585057i
$708$		0			0
$709$		−	30.9839i		−	1.16362i	−0.813323	−	0.581812i	$-0.802344\pi$
$709$		−	30.9839i		−	1.16362i	0.813323	−	0.581812i	$-0.197656\pi$
$710$		0			0
$711$	20.0000	−	22.3607i	0.750059	−	0.838591i
$712$		0			0
$713$	21.9089			0.820495
$714$		0			0
$715$		−	15.4919i		−	0.579365i
$716$		0			0
$717$	3.16228	+	7.07107i	0.118097	+	0.264074i
$718$		0			0
$719$	−32.8634			−1.22560			−0.612798	−	0.790239i	$-0.709956\pi$
$719$	−32.8634			−1.22560			−0.612798	+	0.790239i	$0.709956\pi$
$720$		0			0
$721$	−24.0000	+	9.79796i	−0.893807	+	0.364895i
$722$		0			0
$723$	−3.46410	−	7.74597i	−0.128831	−	0.288076i
$724$		0			0
$725$	20.7846			0.771921
$726$		0			0
$727$		−	19.5959i		−	0.726772i	−0.931639	−	0.363386i	$-0.881621\pi$
$727$		−	19.5959i		−	0.726772i	0.931639	−	0.363386i	$-0.118379\pi$
$728$		0			0
$729$	−22.0000	−	15.6525i	−0.814815	−	0.579721i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	15.8114			0.584007			0.292003	−	0.956417i	$-0.405678\pi$
$733$	15.8114			0.584007			0.292003	+	0.956417i	$0.405678\pi$
$734$		0			0
$735$	15.9545	+	6.28136i	0.588489	+	0.231691i
$736$		0			0
$737$		−	26.8328i		−	0.988399i
$738$		0			0
$739$			23.2379i			0.854820i	0.904058	+	0.427410i	$0.140574\pi$
$739$			23.2379i			0.854820i	−0.904058	+	0.427410i	$0.859426\pi$
$740$		0			0
$741$	15.8114	−	7.07107i	0.580846	−	0.259762i
$742$		0			0
$743$			49.1935i			1.80473i	0.430968	+	0.902367i	$0.358172\pi$
$743$			49.1935i			1.80473i	−0.430968	+	0.902367i	$0.641828\pi$
$744$		0			0
$745$		−	9.79796i		−	0.358969i
$746$		0			0
$747$	15.8114	+	14.1421i	0.578508	+	0.517434i
$748$		0			0
$749$	10.3923	+	25.4558i	0.379727	+	0.930136i
$750$		0			0
$751$	40.0000			1.45962			0.729810	−	0.683650i	$-0.239608\pi$
$751$	40.0000			1.45962			0.729810	+	0.683650i	$0.239608\pi$
$752$		0			0
$753$	−11.0000	−	24.5967i	−0.400862	−	0.896355i
$754$		0			0
$755$		−	11.3137i		−	0.411748i
$756$		0			0
$757$			46.4758i			1.68919i	0.535404	+	0.844596i	$0.320159\pi$
$757$			46.4758i			1.68919i	−0.535404	+	0.844596i	$0.679841\pi$
$758$		0			0
$759$	10.9545	+	24.4949i	0.397621	+	0.889108i
$760$		0			0
$761$	10.9545			0.397099			0.198549	−	0.980091i	$-0.436377\pi$
$761$	10.9545			0.397099			0.198549	+	0.980091i	$0.436377\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$			31.3050i			1.13036i
$768$		0			0
$769$			24.4949i			0.883309i	0.897185	+	0.441654i	$0.145608\pi$
$769$			24.4949i			0.883309i	−0.897185	+	0.441654i	$0.854392\pi$
$770$		0			0
$771$	−17.3205	+	7.74597i	−0.623783	+	0.278964i
$772$		0			0
$773$			35.3553i			1.27164i	0.771836	+	0.635822i	$0.219339\pi$
$773$			35.3553i			1.27164i	−0.771836	+	0.635822i	$0.780661\pi$
$774$		0			0
$775$			14.6969i			0.527930i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	34.6410			1.24114
$780$		0			0
$781$			30.9839i			1.10869i
$782$		0			0
$783$	−10.9545	+	34.2929i	−0.391480	+	1.22553i
$784$		0			0
$785$			22.3607i			0.798087i
$786$		0			0
$787$	−53.7587			−1.91629			−0.958146	−	0.286281i	$-0.907581\pi$
$787$	−53.7587			−1.91629			−0.958146	+	0.286281i	$0.907581\pi$
$788$		0			0
$789$	−6.32456	−	14.1421i	−0.225160	−	0.503473i
$790$		0			0
$791$	10.9545	−	4.47214i	0.389495	−	0.159011i
$792$		0			0
$793$	10.0000			0.355110
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	49.4975i		−	1.75329i	−0.481137	−	0.876645i	$-0.659776\pi$
$797$		−	49.4975i		−	1.75329i	0.481137	−	0.876645i	$-0.340224\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	50.9117i		−	1.79663i
$804$		0			0
$805$	−6.32456	−	15.4919i	−0.222911	−	0.546019i
$806$		0			0
$807$	−7.00000	−	15.6525i	−0.246412	−	0.550993i
$808$		0			0
$809$		−	22.3607i		−	0.786160i	−0.919504	−	0.393080i	$-0.871410\pi$
$809$		−	22.3607i		−	0.786160i	0.919504	−	0.393080i	$-0.128590\pi$
$810$		0			0
$811$	22.1359			0.777298			0.388649	−	0.921386i	$-0.372942\pi$
$811$	22.1359			0.777298			0.388649	+	0.921386i	$0.372942\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	32.8634			1.15115
$816$		0			0
$817$			24.4949i			0.856968i
$818$		0			0
$819$	23.6451	+	8.42087i	0.826225	+	0.294249i
$820$		0			0
$821$	−13.8564			−0.483592			−0.241796	−	0.970327i	$-0.577736\pi$
$821$	−13.8564			−0.483592			−0.241796	+	0.970327i	$0.577736\pi$
$822$		0			0
$823$	−26.0000			−0.906303			−0.453152	−	0.891434i	$-0.649700\pi$
$823$	−26.0000			−0.906303			−0.453152	+	0.891434i	$0.649700\pi$
$824$		0			0
$825$	−16.4317	+	7.34847i	−0.572078	+	0.255841i
$826$		0			0
$827$	51.9615			1.80688			0.903440	−	0.428715i	$-0.141034\pi$
$827$	51.9615			1.80688			0.903440	+	0.428715i	$0.141034\pi$
$828$		0			0
$829$	34.7851			1.20813			0.604067	−	0.796933i	$-0.293546\pi$
$829$	34.7851			1.20813			0.604067	+	0.796933i	$0.293546\pi$
$830$		0			0
$831$	−10.9545	−	24.4949i	−0.380006	−	0.849719i
$832$		0			0
$833$		0			0
$834$		0			0
$835$			30.9839i			1.07224i
$836$		0			0
$837$	−24.2487	−	7.74597i	−0.838158	−	0.267740i
$838$		0			0
$839$	−21.9089			−0.756379			−0.378190	−	0.925728i	$-0.623453\pi$
$839$	−21.9089			−0.756379			−0.378190	+	0.925728i	$0.623453\pi$
$840$		0			0
$841$	19.0000			0.655172
$842$		0			0
$843$	6.32456	+	14.1421i	0.217829	+	0.487081i
$844$		0			0
$845$		−	4.24264i		−	0.145951i
$846$		0			0
$847$	−1.00000	−	2.44949i	−0.0343604	−	0.0841655i
$848$		0			0
$849$	25.0000	−	11.1803i	0.857998	−	0.383708i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	53.7587			1.84066			0.920332	−	0.391139i	$-0.127919\pi$
$853$	53.7587			1.84066			0.920332	+	0.391139i	$0.127919\pi$
$854$		0			0
$855$	10.0000	+	8.94427i	0.341993	+	0.305888i
$856$		0			0
$857$	54.7723			1.87098			0.935492	−	0.353347i	$-0.114957\pi$
$857$	54.7723			1.87098			0.935492	+	0.353347i	$0.114957\pi$
$858$		0			0
$859$	3.16228			0.107896			0.0539478	−	0.998544i	$-0.482820\pi$
$859$	3.16228			0.107896			0.0539478	+	0.998544i	$0.482820\pi$
$860$		0			0
$861$	36.2942	+	34.6804i	1.23690	+	1.18191i
$862$		0			0
$863$			8.94427i			0.304467i	0.988345	+	0.152233i	$0.0486465\pi$
$863$			8.94427i			0.304467i	−0.988345	+	0.152233i	$0.951353\pi$
$864$		0			0
$865$	10.0000			0.340010
$866$		0			0
$867$	26.8794	−	12.0208i	0.912871	−	0.408248i
$868$		0			0
$869$	34.6410			1.17512
$870$		0			0
$871$			24.4949i			0.829978i
$872$		0			0
$873$	10.9545	+	9.79796i	0.370752	+	0.331611i
$874$		0			0
$875$	27.7128	−	11.3137i	0.936864	−	0.382473i
$876$		0			0
$877$		−	15.4919i		−	0.523125i	−0.965186	−	0.261563i	$-0.915762\pi$
$877$		−	15.4919i		−	0.523125i	0.965186	−	0.261563i	$-0.0842378\pi$
$878$		0			0
$879$	−19.0000	−	42.4853i	−0.640854	−	1.43299i
$880$		0			0
$881$	−32.8634			−1.10719			−0.553597	−	0.832785i	$-0.686745\pi$
$881$	−32.8634			−1.10719			−0.553597	+	0.832785i	$0.686745\pi$
$882$		0			0
$883$			23.2379i			0.782018i	0.920387	+	0.391009i	$0.127874\pi$
$883$			23.2379i			0.782018i	−0.920387	+	0.391009i	$0.872126\pi$
$884$		0			0
$885$	−22.1359	+	9.89949i	−0.744092	+	0.332768i
$886$		0			0
$887$	21.9089			0.735629			0.367814	−	0.929899i	$-0.380106\pi$
$887$	21.9089			0.735629			0.367814	+	0.929899i	$0.380106\pi$
$888$		0			0
$889$	8.00000	+	19.5959i	0.268311	+	0.657226i
$890$		0			0
$891$	−3.46410	−	30.9839i	−0.116052	−	1.03800i
$892$		0			0
$893$	34.6410			1.15922
$894$		0			0
$895$			14.6969i			0.491264i
$896$		0			0
$897$	−10.0000	−	22.3607i	−0.333890	−	0.746601i
$898$		0			0
$899$			33.9411i			1.13200i
$900$		0			0
$901$		0			0
$902$		0			0
$903$	−24.5228	+	25.6639i	−0.816067	+	0.854040i
$904$		0			0
$905$		−	4.47214i		−	0.148659i
$906$		0			0
$907$		−	38.7298i		−	1.28600i	−0.765865	−	0.643002i	$-0.777689\pi$
$907$		−	38.7298i		−	1.28600i	0.765865	−	0.643002i	$-0.222311\pi$
$908$		0			0
$909$	−34.7851	−	31.1127i	−1.15375	−	1.03194i
$910$		0			0
$911$		−	31.3050i		−	1.03718i	−0.855023	−	0.518590i	$-0.826457\pi$
$911$		−	31.3050i		−	1.03718i	0.855023	−	0.518590i	$-0.173543\pi$
$912$		0			0
$913$			24.4949i			0.810663i
$914$		0			0
$915$	3.16228	+	7.07107i	0.104542	+	0.233762i
$916$		0			0
$917$	−3.46410	+	1.41421i	−0.114395	+	0.0467014i
$918$		0			0
$919$	10.0000			0.329870			0.164935	−	0.986304i	$-0.447259\pi$
$919$	10.0000			0.329870			0.164935	+	0.986304i	$0.447259\pi$
$920$		0			0
$921$	−5.00000	+	2.23607i	−0.164756	+	0.0736809i
$922$		0			0
$923$		−	28.2843i		−	0.930988i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−21.9089	−	19.5959i	−0.719583	−	0.643614i
$928$		0			0
$929$	21.9089			0.718808			0.359404	−	0.933182i	$-0.382980\pi$
$929$	21.9089			0.718808			0.359404	+	0.933182i	$0.382980\pi$
$930$		0			0
$931$	−15.8114	+	15.4919i	−0.518197	+	0.507728i
$932$		0			0
$933$	17.3205	−	7.74597i	0.567048	−	0.253592i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	44.0908i		−	1.44038i	−0.693775	−	0.720192i	$-0.744054\pi$
$937$		−	44.0908i		−	1.44038i	0.693775	−	0.720192i	$-0.255946\pi$
$938$		0			0
$939$	6.92820	+	15.4919i	0.226093	+	0.505560i
$940$		0			0
$941$			1.41421i			0.0461020i	0.999734	+	0.0230510i	$0.00733802\pi$
$941$			1.41421i			0.0461020i	−0.999734	+	0.0230510i	$0.992662\pi$
$942$		0			0
$943$		−	48.9898i		−	1.59533i
$944$		0			0
$945$	1.52277	+	19.3825i	0.0495359	+	0.630513i
$946$		0			0
$947$	−17.3205			−0.562841			−0.281420	−	0.959585i	$-0.590806\pi$
$947$	−17.3205			−0.562841			−0.281420	+	0.959585i	$0.590806\pi$
$948$		0			0
$949$			46.4758i			1.50867i
$950$		0			0
$951$	−10.9545	+	4.89898i	−0.355222	+	0.158860i
$952$		0			0
$953$			17.8885i			0.579467i	0.957107	+	0.289733i	$0.0935666\pi$
$953$			17.8885i			0.579467i	−0.957107	+	0.289733i	$0.906433\pi$
$954$		0			0
$955$	−12.6491			−0.409316
$956$		0			0
$957$	−37.9473	+	16.9706i	−1.22666	+	0.548580i
$958$		0			0
$959$	43.8178	−	17.8885i	1.41495	−	0.577651i
$960$		0			0
$961$	7.00000			0.225806
$962$		0			0
$963$	−20.7846	+	23.2379i	−0.669775	+	0.748831i
$964$		0			0
$965$		−	5.65685i		−	0.182101i
$966$		0			0
$967$	−32.0000			−1.02905			−0.514525	−	0.857475i	$-0.672032\pi$
$967$	−32.0000			−1.02905			−0.514525	+	0.857475i	$0.672032\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			41.0122i			1.31614i	0.752955	+	0.658072i	$0.228628\pi$
$971$			41.0122i			1.31614i	−0.752955	+	0.658072i	$0.771372\pi$
$972$		0			0
$973$	15.8114	+	38.7298i	0.506890	+	1.24162i
$974$		0			0
$975$	15.0000	−	6.70820i	0.480384	−	0.214834i
$976$		0			0
$977$		−	8.94427i		−	0.286153i	−0.989712	−	0.143076i	$-0.954301\pi$
$977$		−	8.94427i		−	0.286153i	0.989712	−	0.143076i	$-0.0456994\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$	43.8178			1.39757			0.698785	−	0.715331i	$-0.253724\pi$
$983$	43.8178			1.39757			0.698785	+	0.715331i	$0.253724\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	36.2942	+	34.6804i	1.15526	+	1.10389i
$988$		0			0
$989$	34.6410			1.10152
$990$		0			0
$991$	−2.00000			−0.0635321			−0.0317660	−	0.999495i	$-0.510113\pi$
$991$	−2.00000			−0.0635321			−0.0317660	+	0.999495i	$0.510113\pi$
$992$		0			0
$993$	−5.47723	−	12.2474i	−0.173814	−	0.388661i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−3.16228			−0.100150			−0.0500752	−	0.998745i	$-0.515946\pi$
$997$	−3.16228			−0.100150			−0.0500752	+	0.998745i	$0.515946\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.2.i.d.209.3		8
3.2	odd	2	inner	672.2.i.d.209.1		8
4.3	odd	2		168.2.i.e.125.6	yes	8
7.6	odd	2	inner	672.2.i.d.209.6		8
8.3	odd	2		168.2.i.e.125.1	✓	8
8.5	even	2	inner	672.2.i.d.209.5		8
12.11	even	2		168.2.i.e.125.4	yes	8
21.20	even	2	inner	672.2.i.d.209.8		8
24.5	odd	2	inner	672.2.i.d.209.7		8
24.11	even	2		168.2.i.e.125.7	yes	8
28.27	even	2		168.2.i.e.125.5	yes	8
56.13	odd	2	inner	672.2.i.d.209.4		8
56.27	even	2		168.2.i.e.125.2	yes	8
84.83	odd	2		168.2.i.e.125.3	yes	8
168.83	odd	2		168.2.i.e.125.8	yes	8
168.125	even	2	inner	672.2.i.d.209.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.2.i.e.125.1	✓	8	8.3	odd	2
168.2.i.e.125.2	yes	8	56.27	even	2
168.2.i.e.125.3	yes	8	84.83	odd	2
168.2.i.e.125.4	yes	8	12.11	even	2
168.2.i.e.125.5	yes	8	28.27	even	2
168.2.i.e.125.6	yes	8	4.3	odd	2
168.2.i.e.125.7	yes	8	24.11	even	2
168.2.i.e.125.8	yes	8	168.83	odd	2
672.2.i.d.209.1		8	3.2	odd	2	inner
672.2.i.d.209.2		8	168.125	even	2	inner
672.2.i.d.209.3		8	1.1	even	1	trivial
672.2.i.d.209.4		8	56.13	odd	2	inner
672.2.i.d.209.5		8	8.5	even	2	inner
672.2.i.d.209.6		8	7.6	odd	2	inner
672.2.i.d.209.7		8	24.5	odd	2	inner
672.2.i.d.209.8		8	21.20	even	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:	\( N \)	\(=\)	\( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	672.i (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.58114 + 0.707107i) q^{3} +1.41421i q^{5} +(-1.00000 - 2.44949i) q^{7} +(2.00000 - 2.23607i) q^{9} +O(q^{10})\)	\(q+(-1.58114 + 0.707107i) q^{3} +1.41421i q^{5} +(-1.00000 - 2.44949i) q^{7} +(2.00000 - 2.23607i) q^{9} +3.46410 q^{11} -3.16228 q^{13} +(-1.00000 - 2.23607i) q^{15} +3.16228 q^{19} +(3.31319 + 3.16588i) q^{21} -4.47214i q^{23} +3.00000 q^{25} +(-1.58114 + 4.94975i) q^{27} +6.92820 q^{29} +4.89898i q^{31} +(-5.47723 + 2.44949i) q^{33} +(3.46410 - 1.41421i) q^{35} +(5.00000 - 2.23607i) q^{39} +10.9545 q^{41} +7.74597i q^{43} +(3.16228 + 2.82843i) q^{45} +10.9545 q^{47} +(-5.00000 + 4.89898i) q^{49} +4.89898i q^{55} +(-5.00000 + 2.23607i) q^{57} -9.89949i q^{59} -3.16228 q^{61} +(-7.47723 - 2.66291i) q^{63} -4.47214i q^{65} -7.74597i q^{67} +(3.16228 + 7.07107i) q^{69} +8.94427i q^{71} -14.6969i q^{73} +(-4.74342 + 2.12132i) q^{75} +(-3.46410 - 8.48528i) q^{77} +10.0000 q^{79} +(-1.00000 - 8.94427i) q^{81} +7.07107i q^{83} +(-10.9545 + 4.89898i) q^{87} +(3.16228 + 7.74597i) q^{91} +(-3.46410 - 7.74597i) q^{93} +4.47214i q^{95} +4.89898i q^{97} +(6.92820 - 7.74597i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{7} + 16 q^{9}+O(q^{10}) \) 8 * q - 8 * q^7 + 16 * q^9	\( 8 q - 8 q^{7} + 16 q^{9} - 8 q^{15} + 24 q^{25} + 40 q^{39} - 40 q^{49} - 40 q^{57} - 16 q^{63} + 80 q^{79} - 8 q^{81}+O(q^{100}) \) 8 * q - 8 * q^7 + 16 * q^9 - 8 * q^15 + 24 * q^25 + 40 * q^39 - 40 * q^49 - 40 * q^57 - 16 * q^63 + 80 * q^79 - 8 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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