LMFDB - Embedded newform 700.2.c.h.699.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [700,2,Mod(699,700)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(700, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("700.699");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 700 = 2^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	700.c (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:	$5.58952814149$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.49787136.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 $ x^8 + 3x^6 + 5x^4 + 12*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			699.1
Root			$-1.09445 - 0.895644i$ of defining polynomial
Character	$\chi$	$=$	700.699
Dual form			700.2.c.h.699.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.09445 - 0.895644i) q^{2} +(0.395644 + 1.96048i) q^{4} -2.64575 q^{7} +(1.32288 - 2.50000i) q^{8} +3.00000 q^{9} +O(q^{10})$	$q+(-1.09445 - 0.895644i) q^{2} +(0.395644 + 1.96048i) q^{4} -2.64575 q^{7} +(1.32288 - 2.50000i) q^{8} +3.00000 q^{9} -0.818350i q^{11} +(2.89564 + 2.36965i) q^{14} +(-3.68693 + 1.55130i) q^{16} +(-3.28335 - 2.68693i) q^{18} +(-0.732950 + 0.895644i) q^{22} +4.28245 q^{23} +(-1.04678 - 5.18693i) q^{28} +8.16515 q^{29} +(5.42458 + 1.60436i) q^{32} +(1.18693 + 5.88143i) q^{36} -12.1652i q^{37} +13.0381 q^{43} +(1.60436 - 0.323775i) q^{44} +(-4.68693 - 3.83555i) q^{46} +7.00000 q^{49} +10.0000i q^{53} +(-3.50000 + 6.61438i) q^{56} +(-8.93635 - 7.31307i) q^{58} -7.93725 q^{63} +(-4.50000 - 6.61438i) q^{64} +4.47315 q^{67} -16.5022i q^{71} +(3.96863 - 7.50000i) q^{72} +(-10.8956 + 13.3142i) q^{74} +2.16515i q^{77} -14.8655i q^{79} +9.00000 q^{81} +(-14.2695 - 11.6775i) q^{86} +(-2.04588 - 1.08258i) q^{88} +(1.69433 + 8.39564i) q^{92} +(-7.66115 - 6.26951i) q^{98} -2.45505i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 6 q^{4} + 24 q^{9}+O(q^{10}) $ 8 * q - 6 * q^4 + 24 * q^9	$ 8 q - 6 q^{4} + 24 q^{9} + 14 q^{14} - 2 q^{16} - 8 q^{29} - 18 q^{36} + 22 q^{44} - 10 q^{46} + 56 q^{49} - 28 q^{56} - 36 q^{64} - 78 q^{74} + 72 q^{81} - 50 q^{86}+O(q^{100}) $ 8 * q - 6 * q^4 + 24 * q^9 + 14 * q^14 - 2 * q^16 - 8 * q^29 - 18 * q^36 + 22 * q^44 - 10 * q^46 + 56 * q^49 - 28 * q^56 - 36 * q^64 - 78 * q^74 + 72 * q^81 - 50 * q^86

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/700\mathbb{Z}\right)^\times$.

$n$	$101$	$351$	$477$
$\chi(n)$	$-1$	$-1$	$-1$

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 $

$2$	−1.09445	−	0.895644i	−0.773893	−	0.633316i
$2$	−1.09445	−	0.895644i	−0.773893	−	0.633316i
$3$		0			0		1.00000			$0$
$3$		0			0		−1.00000			$\pi$
$4$	0.395644	+	1.96048i	0.197822	+	0.980238i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−2.64575			−1.00000
$7$	−2.64575			−1.00000
$8$	1.32288	−	2.50000i	0.467707	−	0.883883i
$9$	3.00000			1.00000
$10$		0			0
$11$		−	0.818350i		−	0.246742i	−0.992361	−	0.123371i	$-0.960630\pi$
$11$		−	0.818350i		−	0.246742i	0.992361	−	0.123371i	$-0.0393705\pi$
$12$		0			0
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$	2.89564	+	2.36965i	0.773893	+	0.633316i
$15$		0			0
$16$	−3.68693	+	1.55130i	−0.921733	+	0.387825i
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$	−3.28335	−	2.68693i	−0.773893	−	0.633316i
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		0			0
$21$		0			0
$22$	−0.732950	+	0.895644i	−0.156266	+	0.190952i
$23$	4.28245			0.892953			0.446476	−	0.894795i	$-0.352679\pi$
$23$	4.28245			0.892953			0.446476	+	0.894795i	$0.352679\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$	−1.04678	−	5.18693i	−0.197822	−	0.980238i
$29$	8.16515			1.51623			0.758115	−	0.652121i	$-0.226120\pi$
$29$	8.16515			1.51623			0.758115	+	0.652121i	$0.226120\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$	5.42458	+	1.60436i	0.958939	+	0.283613i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	1.18693	+	5.88143i	0.197822	+	0.980238i
$37$		−	12.1652i		−	1.99994i	−0.00783774	−	0.999969i	$-0.502495\pi$
$37$		−	12.1652i		−	1.99994i	0.00783774	−	0.999969i	$-0.497505\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$	13.0381			1.98828			0.994142	−	0.108078i	$-0.0344695\pi$
$43$	13.0381			1.98828			0.994142	+	0.108078i	$0.0344695\pi$
$44$	1.60436	−	0.323775i	0.241866	−	0.0488110i
$45$		0			0
$46$	−4.68693	−	3.83555i	−0.691050	−	0.565521i
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$		0			0
$51$		0			0
$52$		0			0
$53$			10.0000i			1.37361i	0.726844	+	0.686803i	$0.240986\pi$
$53$			10.0000i			1.37361i	−0.726844	+	0.686803i	$0.759014\pi$
$54$		0			0
$55$		0			0
$56$	−3.50000	+	6.61438i	−0.467707	+	0.883883i
$57$		0			0
$58$	−8.93635	−	7.31307i	−1.17340	−	0.960253i
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		0			0
$63$	−7.93725			−1.00000
$64$	−4.50000	−	6.61438i	−0.562500	−	0.826797i
$65$		0			0
$66$		0			0
$67$	4.47315			0.546483			0.273241	−	0.961946i	$-0.411904\pi$
$67$	4.47315			0.546483			0.273241	+	0.961946i	$0.411904\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	16.5022i		−	1.95845i	−0.202787	−	0.979223i	$-0.565000\pi$
$71$		−	16.5022i		−	1.95845i	0.202787	−	0.979223i	$-0.435000\pi$
$72$	3.96863	−	7.50000i	0.467707	−	0.883883i
$73$		0			0			−	1.00000i	$-0.5\pi$
$73$		0			0				1.00000i	$0.5\pi$
$74$	−10.8956	+	13.3142i	−1.26659	+	1.54774i
$75$		0			0
$76$		0			0
$77$			2.16515i			0.246742i
$78$		0			0
$79$		−	14.8655i		−	1.67249i	−0.548352	−	0.836247i	$-0.684745\pi$
$79$		−	14.8655i		−	1.67249i	0.548352	−	0.836247i	$-0.315255\pi$
$80$		0			0
$81$	9.00000			1.00000
$82$		0			0
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$		0			0
$85$		0			0
$86$	−14.2695	−	11.6775i	−1.53872	−	1.25921i
$87$		0			0
$88$	−2.04588	−	1.08258i	−0.218091	−	0.115403i
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$		0			0
$92$	1.69433	+	8.39564i	0.176646	+	0.875306i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0			−	1.00000i	$-0.5\pi$
$97$		0			0				1.00000i	$0.5\pi$
$98$	−7.66115	−	6.26951i	−0.773893	−	0.633316i
$99$		−	2.45505i		−	0.246742i
$100$		0			0
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$	8.95644	−	10.9445i	0.869926	−	1.06302i
$107$	5.29150			0.511549			0.255774	−	0.966736i	$-0.417670\pi$
$107$	5.29150			0.511549			0.255774	+	0.966736i	$0.417670\pi$
$108$		0			0
$109$	−18.1652			−1.73991			−0.869953	−	0.493135i	$-0.835851\pi$
$109$	−18.1652			−1.73991			−0.869953	+	0.493135i	$0.835851\pi$
$110$		0			0
$111$		0			0
$112$	9.75470	−	4.10436i	0.921733	−	0.387825i
$113$			19.3303i			1.81844i	0.416314	+	0.909221i	$0.363322\pi$
$113$			19.3303i			1.81844i	−0.416314	+	0.909221i	$0.636678\pi$
$114$		0			0
$115$		0			0
$116$	3.23049	+	16.0076i	0.299944	+	1.48627i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	10.3303			0.939118
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$	8.68693	+	7.10895i	0.773893	+	0.633316i
$127$	−21.7937			−1.93387			−0.966937	−	0.255014i	$-0.917920\pi$
$127$	−21.7937			−1.93387			−0.966937	+	0.255014i	$0.917920\pi$
$128$	−0.999100	+	11.2695i	−0.0883088	+	0.996093i
$129$		0			0
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$		0			0
$134$	−4.89564	−	4.00635i	−0.422919	−	0.346096i
$135$		0			0
$136$		0			0
$137$			10.0000i			0.854358i	0.904167	+	0.427179i	$0.140493\pi$
$137$			10.0000i			0.854358i	−0.904167	+	0.427179i	$0.859507\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$		0			0
$142$	−14.7801	+	18.0608i	−1.24031	+	1.51563i
$143$		0			0
$144$	−11.0608	+	4.65390i	−0.921733	+	0.387825i
$145$		0			0
$146$		0			0
$147$		0			0
$148$	23.8495	−	4.81307i	1.96042	−	0.395632i
$149$	1.83485			0.150317			0.0751583	−	0.997172i	$-0.476054\pi$
$149$	1.83485			0.150317			0.0751583	+	0.997172i	$0.476054\pi$
$150$		0			0
$151$			18.1389i			1.47612i	0.674735	+	0.738060i	$0.264258\pi$
$151$			18.1389i			1.47612i	−0.674735	+	0.738060i	$0.735742\pi$
$152$		0			0
$153$		0			0
$154$	1.93920	−	2.36965i	0.156266	−	0.190952i
$155$		0			0
$156$		0			0
$157$		0			0			−	1.00000i	$-0.5\pi$
$157$		0			0				1.00000i	$0.5\pi$
$158$	−13.3142	+	16.2695i	−1.05922	+	1.29433i
$159$		0			0
$160$		0			0
$161$	−11.3303			−0.892953
$162$	−9.85005	−	8.06080i	−0.773893	−	0.633316i
$163$	15.8745			1.24339			0.621694	−	0.783260i	$-0.286445\pi$
$163$	15.8745			1.24339			0.621694	+	0.783260i	$0.286445\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	−13.0000			−1.00000
$170$		0			0
$171$		0			0
$172$	5.15843	+	25.5608i	0.393326	+	1.94899i
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$		0			0
$176$	1.26951	+	3.01720i	0.0956927	+	0.227430i
$177$		0			0
$178$		0			0
$179$			26.4575i			1.97753i	0.149487	+	0.988764i	$0.452238\pi$
$179$			26.4575i			1.97753i	−0.149487	+	0.988764i	$0.547762\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$	5.66515	−	10.7061i	0.417641	−	0.789266i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$			26.4575i			1.91440i	0.289430	+	0.957199i	$0.406534\pi$
$191$			26.4575i			1.91440i	−0.289430	+	0.957199i	$0.593466\pi$
$192$		0			0
$193$		−	9.33030i		−	0.671610i	−0.941932	−	0.335805i	$-0.890992\pi$
$193$		−	9.33030i		−	0.671610i	0.941932	−	0.335805i	$-0.109008\pi$
$194$		0			0
$195$		0			0
$196$	2.76951	+	13.7233i	0.197822	+	0.980238i
$197$			22.1652i			1.57920i	0.613621	+	0.789601i	$0.289712\pi$
$197$			22.1652i			1.57920i	−0.613621	+	0.789601i	$0.710288\pi$
$198$	−2.19885	+	2.68693i	−0.156266	+	0.190952i
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−21.6030			−1.51623
$204$		0			0
$205$		0			0
$206$		0			0
$207$	12.8474			0.892953
$208$		0			0
$209$		0			0
$210$		0			0
$211$		−	26.4575i		−	1.82141i	−0.413057	−	0.910705i	$-0.635539\pi$
$211$		−	26.4575i		−	1.82141i	0.413057	−	0.910705i	$-0.364461\pi$
$212$	−19.6048	+	3.95644i	−1.34646	+	0.271729i
$213$		0			0
$214$	−5.79129	−	4.73930i	−0.395884	−	0.323972i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	19.8809	+	16.2695i	1.34650	+	1.10191i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$	−14.3521	−	4.24473i	−0.958939	−	0.283613i
$225$		0			0
$226$	17.3131	−	21.1561i	1.15165	−	1.40728i
$227$		0			0		1.00000			$0$
$227$		0			0		−1.00000			$\pi$
$228$		0			0
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$		0			0
$232$	10.8015	−	20.4129i	0.709152	−	1.34017i
$233$		−	29.3303i		−	1.92149i	−0.277429	−	0.960746i	$-0.589482\pi$
$233$		−	29.3303i		−	1.92149i	0.277429	−	0.960746i	$-0.410518\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	26.4575i		−	1.71139i	−0.517477	−	0.855697i	$-0.673129\pi$
$239$		−	26.4575i		−	1.71139i	0.517477	−	0.855697i	$-0.326871\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$	−11.3060	−	9.25227i	−0.726778	−	0.594759i
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$	−3.14033	−	15.5608i	−0.197822	−	0.980238i
$253$		−	3.50455i		−	0.220329i
$254$	23.8521	+	19.5194i	1.49661	+	1.22475i
$255$		0			0
$256$	11.1869	−	11.4391i	0.699183	−	0.714943i
$257$		0			0			−	1.00000i	$-0.5\pi$
$257$		0			0				1.00000i	$0.5\pi$
$258$		0			0
$259$			32.1860i			1.99994i
$260$		0			0
$261$	24.4955			1.51623
$262$		0			0
$263$	−30.3586			−1.87199			−0.935995	−	0.352014i	$-0.885497\pi$
$263$	−30.3586			−1.87199			−0.935995	+	0.352014i	$0.885497\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	1.76978	+	8.76951i	0.108106	+	0.535683i
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$		0			0
$273$		0			0
$274$	8.95644	−	10.9445i	0.541078	−	0.661182i
$275$		0			0
$276$		0			0
$277$		−	10.0000i		−	0.600842i	−0.953807	−	0.300421i	$-0.902873\pi$
$277$		−	10.0000i		−	0.600842i	0.953807	−	0.300421i	$-0.0971271\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−31.3303			−1.86901			−0.934505	−	0.355951i	$-0.884157\pi$
$281$	−31.3303			−1.86901			−0.934505	+	0.355951i	$0.884157\pi$
$282$		0			0
$283$		0			0		1.00000			$0$
$283$		0			0		−1.00000			$\pi$
$284$	32.3521	−	6.52898i	1.91974	−	0.387424i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	16.2737	+	4.81307i	0.958939	+	0.283613i
$289$	−17.0000			−1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$		0			0
$296$	−30.4129	−	16.0930i	−1.76771	−	0.935386i
$297$		0			0
$298$	−2.00815	−	1.64337i	−0.116329	−	0.0951979i
$299$		0			0
$300$		0			0
$301$	−34.4955			−1.98828
$302$	16.2460	−	19.8521i	0.934850	−	1.14236i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$	−4.24473	+	0.856629i	−0.241866	+	0.0488110i
$309$		0			0
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$		0			0			−	1.00000i	$-0.5\pi$
$313$		0			0				1.00000i	$0.5\pi$
$314$		0			0
$315$		0			0
$316$	29.1434	−	5.88143i	1.63944	−	0.330856i
$317$			7.83485i			0.440049i	0.975494	+	0.220024i	$0.0706137\pi$
$317$			7.83485i			0.440049i	−0.975494	+	0.220024i	$0.929386\pi$
$318$		0			0
$319$		−	6.68195i		−	0.374118i
$320$		0			0
$321$		0			0
$322$	12.4005	+	10.1479i	0.691050	+	0.565521i
$323$		0			0
$324$	3.56080	+	17.6443i	0.197822	+	0.980238i
$325$		0			0
$326$	−17.3739	−	14.2179i	−0.962249	−	0.787457i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$			33.8227i			1.85906i	0.368744	+	0.929531i	$0.379788\pi$
$331$			33.8227i			1.85906i	−0.368744	+	0.929531i	$0.620212\pi$
$332$		0			0
$333$		−	36.4955i		−	1.99994i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		−	30.0000i		−	1.63420i	−0.576493	−	0.817102i	$-0.695579\pi$
$337$		−	30.0000i		−	1.63420i	0.576493	−	0.817102i	$-0.304421\pi$
$338$	14.2279	+	11.6434i	0.773893	+	0.633316i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−18.5203			−1.00000
$344$	17.2477	−	32.5951i	0.929935	−	1.75741i
$345$		0			0
$346$		0			0
$347$	21.9844			1.18018			0.590091	−	0.807337i	$-0.299092\pi$
$347$	21.9844			1.18018			0.590091	+	0.807337i	$0.299092\pi$
$348$		0			0
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$		0			0
$351$		0			0
$352$	1.31293	−	4.43920i	0.0699792	−	0.236610i
$353$		0			0			−	1.00000i	$-0.5\pi$
$353$		0			0				1.00000i	$0.5\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	23.6965	−	28.9564i	1.25240	−	1.53040i
$359$		−	11.5921i		−	0.611805i	−0.952063	−	0.305903i	$-0.901042\pi$
$359$		−	11.5921i		−	0.611805i	0.952063	−	0.305903i	$-0.0989582\pi$
$360$		0			0
$361$	−19.0000			−1.00000
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$	−15.7891	+	6.64337i	−0.823064	+	0.346310i
$369$		0			0
$370$		0			0
$371$		−	26.4575i		−	1.37361i
$372$		0			0
$373$		−	16.4955i		−	0.854102i	−0.904227	−	0.427051i	$-0.859552\pi$
$373$		−	16.4955i		−	0.854102i	0.904227	−	0.427051i	$-0.140448\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$			28.9126i			1.48514i	0.669769	+	0.742569i	$0.266393\pi$
$379$			28.9126i			1.48514i	−0.669769	+	0.742569i	$0.733607\pi$
$380$		0			0
$381$		0			0
$382$	23.6965	−	28.9564i	1.21242	−	1.48154i
$383$		0			0		1.00000			$0$
$383$		0			0		−1.00000			$\pi$
$384$		0			0
$385$		0			0
$386$	−8.35663	+	10.2116i	−0.425341	+	0.519754i
$387$	39.1142			1.98828
$388$		0			0
$389$	28.1652			1.42803			0.714015	−	0.700130i	$-0.246875\pi$
$389$	28.1652			1.42803			0.714015	+	0.700130i	$0.246875\pi$
$390$		0			0
$391$		0			0
$392$	9.26013	−	17.5000i	0.467707	−	0.883883i
$393$		0			0
$394$	19.8521	−	24.2587i	1.00013	−	1.22213i
$395$		0			0
$396$	4.81307	−	0.971326i	0.241866	−	0.0488110i
$397$		0			0			−	1.00000i	$-0.5\pi$
$397$		0			0				1.00000i	$0.5\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	1.33030			0.0664322			0.0332161	−	0.999448i	$-0.489425\pi$
$401$	1.33030			0.0664322			0.0332161	+	0.999448i	$0.489425\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$	23.6434	+	19.3486i	1.17340	+	0.960253i
$407$	−9.95536			−0.493469
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	−14.0608	−	11.5067i	−0.691050	−	0.565521i
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	−14.4955			−0.706465			−0.353233	−	0.935536i	$-0.614918\pi$
$421$	−14.4955			−0.706465			−0.353233	+	0.935536i	$0.614918\pi$
$422$	−23.6965	+	28.9564i	−1.15353	+	1.40958i
$423$		0			0
$424$	25.0000	+	13.2288i	1.21411	+	0.642445i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	2.09355	+	10.3739i	0.101196	+	0.501440i
$429$		0			0
$430$		0			0
$431$			26.4575i			1.27441i	0.770693	+	0.637207i	$0.219910\pi$
$431$			26.4575i			1.27441i	−0.770693	+	0.637207i	$0.780090\pi$
$432$		0			0
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		0			0
$435$		0			0
$436$	−7.18693	−	35.6123i	−0.344192	−	1.70552i
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	21.0000			1.00000
$442$		0			0
$443$	37.0405			1.75985			0.879924	−	0.475114i	$-0.157593\pi$
$443$	37.0405			1.75985			0.879924	+	0.475114i	$0.157593\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	11.9059	+	17.5000i	0.562500	+	0.826797i
$449$	37.6606			1.77731			0.888657	−	0.458573i	$-0.151639\pi$
$449$	37.6606			1.77731			0.888657	+	0.458573i	$0.151639\pi$
$450$		0			0
$451$		0			0
$452$	−37.8966	+	7.64792i	−1.78251	+	0.359728i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	33.6606i		−	1.57458i	−0.616585	−	0.787288i	$-0.711484\pi$
$457$		−	33.6606i		−	1.57458i	0.616585	−	0.787288i	$-0.288516\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	−15.8745			−0.737751			−0.368875	−	0.929479i	$-0.620257\pi$
$463$	−15.8745			−0.737751			−0.368875	+	0.929479i	$0.620257\pi$
$464$	−30.1044	+	12.6666i	−1.39756	+	0.588032i
$465$		0			0
$466$	−26.2695	+	32.1006i	−1.21691	+	1.48703i
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$		0			0
$469$	−11.8348			−0.546483
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	10.6697i		−	0.490593i
$474$		0			0
$475$		0			0
$476$		0			0
$477$			30.0000i			1.37361i
$478$	−23.6965	+	28.9564i	−1.08385	+	1.32444i
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	4.08712	+	20.2523i	0.185778	+	0.920560i
$485$		0			0
$486$		0			0
$487$	−39.3049			−1.78107			−0.890537	−	0.454911i	$-0.849671\pi$
$487$	−39.3049			−1.78107			−0.890537	+	0.454911i	$0.849671\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	35.4594i		−	1.60026i	−0.599827	−	0.800129i	$-0.704764\pi$
$491$		−	35.4594i		−	1.60026i	0.599827	−	0.800129i	$-0.295236\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$			43.6606i			1.95845i
$498$		0			0
$499$			26.4575i			1.18440i	0.805791	+	0.592200i	$0.201741\pi$
$499$			26.4575i			1.18440i	−0.805791	+	0.592200i	$0.798259\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$	−10.5000	+	19.8431i	−0.467707	+	0.883883i
$505$		0			0
$506$	−3.13883	+	3.83555i	−0.139538	+	0.170511i
$507$		0			0
$508$	−8.62253	−	42.7259i	−0.382563	−	1.89566i
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$		0			0
$512$	−22.4889	+	2.50000i	−0.993878	+	0.110485i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$	28.8272	−	35.2259i	1.26659	−	1.54774i
$519$		0			0
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$	−26.8091	−	21.9392i	−1.17340	−	0.960253i
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$		0			0
$525$		0			0
$526$	33.2259	+	27.1905i	1.44872	+	1.18556i
$527$		0			0
$528$		0			0
$529$	−4.66061			−0.202635
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	5.91742	−	11.1829i	0.255594	−	0.483027i
$537$		0			0
$538$		0			0
$539$		−	5.72845i		−	0.246742i
$540$		0			0
$541$	44.4955			1.91301			0.956504	−	0.291718i	$-0.0942267\pi$
$541$	44.4955			1.91301			0.956504	+	0.291718i	$0.0942267\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−30.1679			−1.28988			−0.644942	−	0.764231i	$-0.723119\pi$
$547$	−30.1679			−1.28988			−0.644942	+	0.764231i	$0.723119\pi$
$548$	−19.6048	+	3.95644i	−0.837474	+	0.169011i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$			39.3303i			1.67249i
$554$	−8.95644	+	10.9445i	−0.380523	+	0.464987i
$555$		0			0
$556$		0			0
$557$		−	32.1652i		−	1.36288i	−0.731873	−	0.681441i	$-0.761354\pi$
$557$		−	32.1652i		−	1.36288i	0.731873	−	0.681441i	$-0.238646\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	34.2895	+	28.0608i	1.44641	+	1.18367i
$563$		0			0		1.00000			$0$
$563$		0			0		−1.00000			$\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−23.8118			−1.00000
$568$	−41.2554	−	21.8303i	−1.73104	−	0.915979i
$569$	−47.6606			−1.99804			−0.999018	−	0.0443003i	$-0.985894\pi$
$569$	−47.6606			−1.99804			−0.999018	+	0.0443003i	$0.985894\pi$
$570$		0			0
$571$			27.2759i			1.14146i	0.821138	+	0.570730i	$0.193340\pi$
$571$			27.2759i			1.14146i	−0.821138	+	0.570730i	$0.806660\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−13.5000	−	19.8431i	−0.562500	−	0.826797i
$577$		0			0			−	1.00000i	$-0.5\pi$
$577$		0			0				1.00000i	$0.5\pi$
$578$	18.6057	+	15.2259i	0.773893	+	0.633316i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	8.18350			0.338926
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		1.00000			$0$
$587$		0			0		−1.00000			$\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	18.8718	+	44.8521i	0.775627	+	1.84341i
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$	0.725947	+	3.59718i	0.0297359	+	0.147346i
$597$		0			0
$598$		0			0
$599$		−	46.2331i		−	1.88903i	−0.328465	−	0.944516i	$-0.606531\pi$
$599$		−	46.2331i		−	1.88903i	0.328465	−	0.944516i	$-0.393469\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$	37.7536	+	30.8956i	1.53872	+	1.25921i
$603$	13.4195			0.546483
$604$	−35.5608	+	7.17653i	−1.44695	+	0.292009i
$605$		0			0
$606$		0			0
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$			46.4955i			1.87793i	0.344008	+	0.938967i	$0.388215\pi$
$613$			46.4955i			1.87793i	−0.344008	+	0.938967i	$0.611785\pi$
$614$		0			0
$615$		0			0
$616$	5.41288	+	2.86423i	0.218091	+	0.115403i
$617$			23.6606i			0.952540i	0.879299	+	0.476270i	$0.158012\pi$
$617$			23.6606i			0.952540i	−0.879299	+	0.476270i	$0.841988\pi$
$618$		0			0
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		−	9.95536i		−	0.396316i	−0.980170	−	0.198158i	$-0.936504\pi$
$631$		−	9.95536i		−	0.396316i	0.980170	−	0.198158i	$-0.0634960\pi$
$632$	−37.1636	−	19.6652i	−1.47829	−	0.782238i
$633$		0			0
$634$	7.01723	−	8.57485i	0.278690	−	0.340551i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	−5.98465	+	7.31307i	−0.236935	+	0.289527i
$639$		−	49.5065i		−	1.95845i
$640$		0			0
$641$	41.3303			1.63245			0.816224	−	0.577735i	$-0.196063\pi$
$641$	41.3303			1.63245			0.816224	+	0.577735i	$0.196063\pi$
$642$		0			0
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$	−4.48277	−	22.2128i	−0.176646	−	0.875306i
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$	11.9059	−	22.5000i	0.467707	−	0.883883i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	6.28065	+	31.1216i	0.245969	+	1.21882i
$653$			50.0000i			1.95665i	0.207072	+	0.978326i	$0.433606\pi$
$653$			50.0000i			1.95665i	−0.207072	+	0.978326i	$0.566394\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$			26.4575i			1.03064i	0.856998	+	0.515319i	$0.172327\pi$
$659$			26.4575i			1.03064i	−0.856998	+	0.515319i	$0.827673\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$	30.2931	−	37.0172i	1.17737	−	1.43872i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−32.6869	+	39.9425i	−1.26659	+	1.54774i
$667$	34.9669			1.35392
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$			30.0000i			1.15642i	0.815890	+	0.578208i	$0.196248\pi$
$673$			30.0000i			1.15642i	−0.815890	+	0.578208i	$0.803752\pi$
$674$	−26.8693	+	32.8335i	−1.03497	+	1.26470i
$675$		0			0
$676$	−5.14337	−	25.4862i	−0.197822	−	0.980238i
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	47.6791			1.82439			0.912194	−	0.409757i	$-0.134387\pi$
$683$	47.6791			1.82439			0.912194	+	0.409757i	$0.134387\pi$
$684$		0			0
$685$		0			0
$686$	20.2695	+	16.5876i	0.773893	+	0.633316i
$687$		0			0
$688$	−48.0704	+	20.2259i	−1.83267	+	0.771107i
$689$		0			0
$690$		0			0
$691$		0			0			−	1.00000i	$-0.5\pi$
$691$		0			0				1.00000i	$0.5\pi$
$692$		0			0
$693$			6.49545i			0.246742i
$694$	−24.0608	−	19.6902i	−0.913335	−	0.747428i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−2.00000			−0.0755390			−0.0377695	−	0.999286i	$-0.512025\pi$
$701$	−2.00000			−0.0755390			−0.0377695	+	0.999286i	$0.512025\pi$
$702$		0			0
$703$		0			0
$704$	−5.41288	+	3.68258i	−0.204006	+	0.138792i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−6.00000			−0.225335			−0.112667	−	0.993633i	$-0.535939\pi$
$709$	−6.00000			−0.225335			−0.112667	+	0.993633i	$0.535939\pi$
$710$		0			0
$711$		−	44.5964i		−	1.67249i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−51.8693	+	10.4678i	−1.93845	+	0.391198i
$717$		0			0
$718$	−10.3824	+	12.6869i	−0.387466	+	0.473472i
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$	20.7946	+	17.0172i	0.773893	+	0.633316i
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	27.0000			1.00000
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0			−	1.00000i	$-0.5\pi$
$733$		0			0				1.00000i	$0.5\pi$
$734$		0			0
$735$		0			0
$736$	23.2305	+	6.87058i	0.856287	+	0.253253i
$737$		−	3.66061i		−	0.134840i
$738$		0			0
$739$		−	37.0961i		−	1.36460i	−0.731072	−	0.682300i	$-0.760980\pi$
$739$		−	37.0961i		−	1.36460i	0.731072	−	0.682300i	$-0.239020\pi$
$740$		0			0
$741$		0			0
$742$	−23.6965	+	28.9564i	−0.869926	+	1.06302i
$743$	−37.0405			−1.35888			−0.679442	−	0.733729i	$-0.737778\pi$
$743$	−37.0405			−1.35888			−0.679442	+	0.733729i	$0.737778\pi$
$744$		0			0
$745$		0			0
$746$	−14.7741	+	18.0535i	−0.540917	+	0.660984i
$747$		0			0
$748$		0			0
$749$	−14.0000			−0.511549
$750$		0			0
$751$			26.4575i			0.965448i	0.875772	+	0.482724i	$0.160353\pi$
$751$			26.4575i			0.965448i	−0.875772	+	0.482724i	$0.839647\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	17.8348i		−	0.648219i	−0.946020	−	0.324109i	$-0.894935\pi$
$757$		−	17.8348i		−	0.648219i	0.946020	−	0.324109i	$-0.105065\pi$
$758$	25.8954	−	31.6434i	0.940562	−	1.14934i
$759$		0			0
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$	48.0605			1.73991
$764$	−51.8693	+	10.4678i	−1.87657	+	0.378710i
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$		0			0
$772$	18.2918	−	3.69148i	0.658338	−	0.132859i
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$	−42.8085	−	35.0324i	−1.53872	−	1.25921i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	−30.8254	−	25.2259i	−1.10514	−	0.904394i
$779$		0			0
$780$		0			0
$781$	−13.5045			−0.483231
$782$		0			0
$783$		0			0
$784$	−25.8085	+	10.8591i	−0.921733	+	0.387825i
$785$		0			0
$786$		0			0
$787$		0			0		1.00000			$0$
$787$		0			0		−1.00000			$\pi$
$788$	−43.4542	+	8.76951i	−1.54799	+	0.312401i
$789$		0			0
$790$		0			0
$791$		−	51.1432i		−	1.81844i
$792$	−6.13763	−	3.24773i	−0.218091	−	0.115403i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$	−1.45595	−	1.19148i	−0.0514114	−	0.0420725i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	17.6606			0.620914			0.310457	−	0.950587i	$-0.399518\pi$
$809$	17.6606			0.620914			0.310457	+	0.950587i	$0.399518\pi$
$810$		0			0
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$	−8.54708	−	42.3521i	−0.299944	−	1.48627i
$813$		0			0
$814$	10.8956	+	8.91645i	0.381892	+	0.312522i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	22.0000			0.767805			0.383903	−	0.923374i	$-0.374580\pi$
$821$	22.0000			0.767805			0.383903	+	0.923374i	$0.374580\pi$
$822$		0			0
$823$	3.90105			0.135982			0.0679910	−	0.997686i	$-0.478341\pi$
$823$	3.90105			0.135982			0.0679910	+	0.997686i	$0.478341\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	56.6254			1.96906			0.984529	−	0.175224i	$-0.0560649\pi$
$827$	56.6254			1.96906			0.984529	+	0.175224i	$0.0560649\pi$
$828$	5.08298	+	25.1869i	0.176646	+	0.875306i
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	37.6697			1.29896
$842$	15.8646	+	12.9828i	0.546729	+	0.447416i
$843$		0			0
$844$	51.8693	−	10.4678i	1.78542	−	0.360315i
$845$		0			0
$846$		0			0
$847$	−27.3314			−0.939118
$848$	−15.5130	−	36.8693i	−0.532719	−	1.26610i
$849$		0			0
$850$		0			0
$851$		−	52.0967i		−	1.78585i
$852$		0			0
$853$		0			0			−	1.00000i	$-0.5\pi$
$853$		0			0				1.00000i	$0.5\pi$
$854$		0			0
$855$		0			0
$856$	7.00000	−	13.2288i	0.239255	−	0.452150i
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		0			0
$859$		0			0			−	1.00000i	$-0.5\pi$
$859$		0			0				1.00000i	$0.5\pi$
$860$		0			0
$861$		0			0
$862$	23.6965	−	28.9564i	0.807106	−	0.986260i
$863$	22.1751			0.754848			0.377424	−	0.926041i	$-0.376810\pi$
$863$	22.1751			0.754848			0.377424	+	0.926041i	$0.376810\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−12.1652			−0.412674
$870$		0			0
$871$		0			0
$872$	−24.0302	+	45.4129i	−0.813766	+	1.53787i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$		−	50.0000i		−	1.68838i	−0.536044	−	0.844190i	$-0.680082\pi$
$877$		−	50.0000i		−	1.68838i	0.536044	−	0.844190i	$-0.319918\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	−22.9835	−	18.8085i	−0.773893	−	0.633316i
$883$	−39.4956			−1.32913			−0.664566	−	0.747230i	$-0.731383\pi$
$883$	−39.4956			−1.32913			−0.664566	+	0.747230i	$0.731383\pi$
$884$		0			0
$885$		0			0
$886$	−40.5390	−	33.1751i	−1.36193	−	1.11454i
$887$		0			0		1.00000			$0$
$887$		0			0		−1.00000			$\pi$
$888$		0			0
$889$	57.6606			1.93387
$890$		0			0
$891$		−	7.36515i		−	0.246742i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$	2.64337	−	29.8163i	0.0883088	−	0.996093i
$897$		0			0
$898$	−41.2177	−	33.7305i	−1.37545	−	1.12560i
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$	48.3258	+	25.5716i	1.60729	+	0.850498i
$905$		0			0
$906$		0			0
$907$	5.29150			0.175701			0.0878507	−	0.996134i	$-0.472000\pi$
$907$	5.29150			0.175701			0.0878507	+	0.996134i	$0.472000\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			42.9597i			1.42332i	0.702525	+	0.711659i	$0.252056\pi$
$911$			42.9597i			1.42332i	−0.702525	+	0.711659i	$0.747944\pi$
$912$		0			0
$913$		0			0
$914$	−30.1479	+	36.8399i	−0.997204	+	1.21855i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$			23.0490i			0.760315i	0.924922	+	0.380158i	$0.124130\pi$
$919$			23.0490i			0.760315i	−0.924922	+	0.380158i	$0.875870\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$	17.3739	+	14.2179i	0.570941	+	0.467229i
$927$		0			0
$928$	44.2925	+	13.0998i	1.45397	+	0.430022i
$929$		0			0		1.00000			$0$
$929$		0			0		−1.00000			$\pi$
$930$		0			0
$931$		0			0
$932$	57.5014	−	11.6044i	1.88352	−	0.380113i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0			−	1.00000i	$-0.5\pi$
$937$		0			0				1.00000i	$0.5\pi$
$938$	12.9527	+	10.5998i	0.422919	+	0.346096i
$939$		0			0
$940$		0			0
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	−9.55625	+	11.6775i	−0.310700	+	0.379667i
$947$	−58.2065			−1.89146			−0.945729	−	0.324956i	$-0.894650\pi$
$947$	−58.2065			−1.89146			−0.945729	+	0.324956i	$0.894650\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$			10.6697i			0.345625i	0.984955	+	0.172813i	$0.0552855\pi$
$953$			10.6697i			0.345625i	−0.984955	+	0.172813i	$0.944714\pi$
$954$	26.8693	−	32.8335i	0.869926	−	1.06302i
$955$		0			0
$956$	51.8693	−	10.4678i	1.67757	−	0.338551i
$957$		0			0
$958$		0			0
$959$		−	26.4575i		−	0.854358i
$960$		0			0
$961$	−31.0000			−1.00000
$962$		0			0
$963$	15.8745			0.511549
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−47.6235			−1.53147			−0.765735	−	0.643157i	$-0.777624\pi$
$967$	−47.6235			−1.53147			−0.765735	+	0.643157i	$0.777624\pi$
$968$	13.6657	−	25.8258i	0.439232	−	0.830071i
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$		0			0
$973$		0			0
$974$	43.0172	+	35.2032i	1.37836	+	1.12798i
$975$		0			0
$976$		0			0
$977$		−	13.6606i		−	0.437041i	−0.975832	−	0.218521i	$-0.929877\pi$
$977$		−	13.6606i		−	0.437041i	0.975832	−	0.218521i	$-0.0701231\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−54.4955			−1.73991
$982$	−31.7590	+	38.8085i	−1.01347	+	1.23843i
$983$		0			0		1.00000			$0$
$983$		0			0		−1.00000			$\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	55.8348			1.77544
$990$		0			0
$991$			8.31865i			0.264251i	0.991233	+	0.132125i	$0.0421802\pi$
$991$			8.31865i			0.264251i	−0.991233	+	0.132125i	$0.957820\pi$
$992$		0			0
$993$		0			0
$994$	39.1044	−	47.7844i	1.24031	−	1.51563i
$995$		0			0
$996$		0			0
$997$		0			0			−	1.00000i	$-0.5\pi$
$997$		0			0				1.00000i	$0.5\pi$
$998$	23.6965	−	28.9564i	0.750100	−	0.916600i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	700.2.c.h.699.1		8
4.3	odd	2	inner	700.2.c.h.699.7		8
5.2	odd	4		700.2.g.c.251.3	✓	4
5.3	odd	4		700.2.g.e.251.2	yes	4
5.4	even	2	inner	700.2.c.h.699.8		8
7.6	odd	2	CM	700.2.c.h.699.1		8
20.3	even	4		700.2.g.e.251.1	yes	4
20.7	even	4		700.2.g.c.251.4	yes	4
20.19	odd	2	inner	700.2.c.h.699.2		8
28.27	even	2	inner	700.2.c.h.699.7		8
35.13	even	4		700.2.g.e.251.2	yes	4
35.27	even	4		700.2.g.c.251.3	✓	4
35.34	odd	2	inner	700.2.c.h.699.8		8
140.27	odd	4		700.2.g.c.251.4	yes	4
140.83	odd	4		700.2.g.e.251.1	yes	4
140.139	even	2	inner	700.2.c.h.699.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
700.2.c.h.699.1		8	1.1	even	1	trivial
700.2.c.h.699.1		8	7.6	odd	2	CM
700.2.c.h.699.2		8	20.19	odd	2	inner
700.2.c.h.699.2		8	140.139	even	2	inner
700.2.c.h.699.7		8	4.3	odd	2	inner
700.2.c.h.699.7		8	28.27	even	2	inner
700.2.c.h.699.8		8	5.4	even	2	inner
700.2.c.h.699.8		8	35.34	odd	2	inner
700.2.g.c.251.3	✓	4	5.2	odd	4
700.2.g.c.251.3	✓	4	35.27	even	4
700.2.g.c.251.4	yes	4	20.7	even	4
700.2.g.c.251.4	yes	4	140.27	odd	4
700.2.g.e.251.1	yes	4	20.3	even	4
700.2.g.e.251.1	yes	4	140.83	odd	4
700.2.g.e.251.2	yes	4	5.3	odd	4
700.2.g.e.251.2	yes	4	35.13	even	4

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:	\( N \)	\(=\)	\( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	700.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.09445 - 0.895644i) q^{2} +(0.395644 + 1.96048i) q^{4} -2.64575 q^{7} +(1.32288 - 2.50000i) q^{8} +3.00000 q^{9} +O(q^{10})\)	\(q+(-1.09445 - 0.895644i) q^{2} +(0.395644 + 1.96048i) q^{4} -2.64575 q^{7} +(1.32288 - 2.50000i) q^{8} +3.00000 q^{9} -0.818350i q^{11} +(2.89564 + 2.36965i) q^{14} +(-3.68693 + 1.55130i) q^{16} +(-3.28335 - 2.68693i) q^{18} +(-0.732950 + 0.895644i) q^{22} +4.28245 q^{23} +(-1.04678 - 5.18693i) q^{28} +8.16515 q^{29} +(5.42458 + 1.60436i) q^{32} +(1.18693 + 5.88143i) q^{36} -12.1652i q^{37} +13.0381 q^{43} +(1.60436 - 0.323775i) q^{44} +(-4.68693 - 3.83555i) q^{46} +7.00000 q^{49} +10.0000i q^{53} +(-3.50000 + 6.61438i) q^{56} +(-8.93635 - 7.31307i) q^{58} -7.93725 q^{63} +(-4.50000 - 6.61438i) q^{64} +4.47315 q^{67} -16.5022i q^{71} +(3.96863 - 7.50000i) q^{72} +(-10.8956 + 13.3142i) q^{74} +2.16515i q^{77} -14.8655i q^{79} +9.00000 q^{81} +(-14.2695 - 11.6775i) q^{86} +(-2.04588 - 1.08258i) q^{88} +(1.69433 + 8.39564i) q^{92} +(-7.66115 - 6.26951i) q^{98} -2.45505i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 6 q^{4} + 24 q^{9}+O(q^{10}) \) 8 * q - 6 * q^4 + 24 * q^9	\( 8 q - 6 q^{4} + 24 q^{9} + 14 q^{14} - 2 q^{16} - 8 q^{29} - 18 q^{36} + 22 q^{44} - 10 q^{46} + 56 q^{49} - 28 q^{56} - 36 q^{64} - 78 q^{74} + 72 q^{81} - 50 q^{86}+O(q^{100}) \) 8 * q - 6 * q^4 + 24 * q^9 + 14 * q^14 - 2 * q^16 - 8 * q^29 - 18 * q^36 + 22 * q^44 - 10 * q^46 + 56 * q^49 - 28 * q^56 - 36 * q^64 - 78 * q^74 + 72 * q^81 - 50 * q^86

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more