LMFDB - Embedded newform 176.9.h.e.65.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [176,9,Mod(65,176)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(176, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("176.65");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 176 = 2^{4} \cdot 11 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	176.h (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:	$71.6986353708$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 7944x^{6} + 15215349x^{4} + 1757611988x^{2} + 38177252100 $ x^8 + 7944x^6 + 15215349x^4 + 1757611988*x^2 + 38177252100
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{20}\cdot 3^{3}\cdot 11^{2} $
Twist minimal:	no (minimal twist has level 22)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			65.4
Root			$70.2090i$ of defining polynomial
Character	$\chi$	$=$	176.65
Dual form			176.9.h.e.65.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-107.356 q^{3} -1065.90 q^{5} +3358.46i q^{7} +4964.40 q^{9} +O(q^{10})$	$q-107.356 q^{3} -1065.90 q^{5} +3358.46i q^{7} +4964.40 q^{9} +(5944.35 + 13380.0i) q^{11} -18360.6i q^{13} +114431. q^{15} -3045.27i q^{17} -180007. i q^{19} -360553. i q^{21} -128952. q^{23} +745509. q^{25} +171406. q^{27} -803829. i q^{29} -790758. q^{31} +(-638164. - 1.43643e6i) q^{33} -3.57977e6i q^{35} +1.85903e6 q^{37} +1.97112e6i q^{39} +1.91113e6i q^{41} +5.87142e6i q^{43} -5.29153e6 q^{45} +6.46157e6 q^{47} -5.51448e6 q^{49} +326929. i q^{51} +1.03543e7 q^{53} +(-6.33606e6 - 1.42617e7i) q^{55} +1.93249e7i q^{57} +2.46965e6 q^{59} -1.97635e7i q^{61} +1.66727e7i q^{63} +1.95705e7i q^{65} -1.24402e7 q^{67} +1.38438e7 q^{69} +3.54223e7 q^{71} +1.52941e7i q^{73} -8.00352e7 q^{75} +(-4.49361e7 + 1.99639e7i) q^{77} +3.11795e7i q^{79} -5.09729e7 q^{81} -3.73727e7i q^{83} +3.24594e6i q^{85} +8.62962e7i q^{87} -4.94298e7 q^{89} +6.16633e7 q^{91} +8.48929e7 q^{93} +1.91869e8i q^{95} +4.09642e7 q^{97} +(2.95101e7 + 6.64235e7i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 182 q^{3} - 1410 q^{5} + 44582 q^{9}+O(q^{10}) $ 8 * q - 182 * q^3 - 1410 * q^5 + 44582 * q^9	$ 8 q - 182 q^{3} - 1410 q^{5} + 44582 q^{9} - 5808 q^{11} - 87958 q^{15} + 802026 q^{23} + 999558 q^{25} + 1561354 q^{27} - 196726 q^{31} - 4286722 q^{33} + 8627998 q^{37} + 1146988 q^{45} + 14335392 q^{47} - 6714712 q^{49} + 55946352 q^{53} + 10078442 q^{55} - 21793110 q^{59} + 113809034 q^{67} - 171636914 q^{69} - 16741974 q^{71} - 346496844 q^{75} - 137074080 q^{77} + 85282724 q^{81} + 42055422 q^{89} + 146801952 q^{91} + 253251118 q^{93} + 100034782 q^{97} + 333541978 q^{99}+O(q^{100}) $ 8 * q - 182 * q^3 - 1410 * q^5 + 44582 * q^9 - 5808 * q^11 - 87958 * q^15 + 802026 * q^23 + 999558 * q^25 + 1561354 * q^27 - 196726 * q^31 - 4286722 * q^33 + 8627998 * q^37 + 1146988 * q^45 + 14335392 * q^47 - 6714712 * q^49 + 55946352 * q^53 + 10078442 * q^55 - 21793110 * q^59 + 113809034 * q^67 - 171636914 * q^69 - 16741974 * q^71 - 346496844 * q^75 - 137074080 * q^77 + 85282724 * q^81 + 42055422 * q^89 + 146801952 * q^91 + 253251118 * q^93 + 100034782 * q^97 + 333541978 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/176\mathbb{Z}\right)^\times$.

$n$	$111$	$133$	$145$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$	−107.356			−1.32539			−0.662694	−	0.748890i	$-0.730587\pi$
$3$	−107.356			−1.32539			−0.662694	+	0.748890i	$0.730587\pi$
$4$		0			0
$5$	−1065.90			−1.70543			−0.852717	−	0.522374i	$-0.825047\pi$
$5$	−1065.90			−1.70543			−0.852717	+	0.522374i	$0.825047\pi$
$6$		0			0
$7$			3358.46i			1.39878i	0.714742	+	0.699388i	$0.246544\pi$
$7$			3358.46i			1.39878i	−0.714742	+	0.699388i	$0.753456\pi$
$8$		0			0
$9$	4964.40			0.756652
$10$		0			0
$11$	5944.35	+	13380.0i	0.406007	+	0.913870i
$11$	5944.35	+	13380.0i	0.406007	+	0.913870i
$12$		0			0
$13$		−	18360.6i		−	0.642855i	−0.946934	−	0.321427i	$-0.895837\pi$
$13$		−	18360.6i		−	0.642855i	0.946934	−	0.321427i	$-0.104163\pi$
$14$		0			0
$15$	114431.			2.26036
$16$		0			0
$17$		−	3045.27i		−	0.0364611i	−0.999834	−	0.0182306i	$-0.994197\pi$
$17$		−	3045.27i		−	0.0364611i	0.999834	−	0.0182306i	$-0.00580329\pi$
$18$		0			0
$19$		−	180007.i		−	1.38126i	−0.723209	−	0.690629i	$-0.757334\pi$
$19$		−	180007.i		−	1.38126i	0.723209	−	0.690629i	$-0.242666\pi$
$20$		0			0
$21$		−	360553.i		−	1.85392i
$22$		0			0
$23$	−128952.			−0.460804			−0.230402	−	0.973096i	$-0.574004\pi$
$23$	−128952.			−0.460804			−0.230402	+	0.973096i	$0.574004\pi$
$24$		0			0
$25$	745509.			1.90850
$26$		0			0
$27$	171406.			0.322530
$28$		0			0
$29$		−	803829.i		−	1.13651i	−0.822854	−	0.568253i	$-0.807620\pi$
$29$		−	803829.i		−	1.13651i	0.822854	−	0.568253i	$-0.192380\pi$
$30$		0			0
$31$	−790758.			−0.856242			−0.428121	−	0.903721i	$-0.640824\pi$
$31$	−790758.			−0.856242			−0.428121	+	0.903721i	$0.640824\pi$
$32$		0			0
$33$	−638164.	−	1.43643e6i	−0.538117	−	1.21123i
$34$		0			0
$35$		−	3.57977e6i		−	2.38552i
$36$		0			0
$37$	1.85903e6			0.991929			0.495964	−	0.868343i	$-0.334815\pi$
$37$	1.85903e6			0.991929			0.495964	+	0.868343i	$0.334815\pi$
$38$		0			0
$39$			1.97112e6i			0.852032i
$40$		0			0
$41$			1.91113e6i			0.676325i	0.941088	+	0.338163i	$0.109805\pi$
$41$			1.91113e6i			0.676325i	−0.941088	+	0.338163i	$0.890195\pi$
$42$		0			0
$43$			5.87142e6i			1.71739i	0.512486	+	0.858696i	$0.328725\pi$
$43$			5.87142e6i			1.71739i	−0.512486	+	0.858696i	$0.671275\pi$
$44$		0			0
$45$	−5.29153e6			−1.29042
$46$		0			0
$47$	6.46157e6			1.32418			0.662090	−	0.749425i	$-0.269670\pi$
$47$	6.46157e6			1.32418			0.662090	+	0.749425i	$0.269670\pi$
$48$		0			0
$49$	−5.51448e6			−0.956577
$50$		0			0
$51$			326929.i			0.0483251i
$52$		0			0
$53$	1.03543e7			1.31225			0.656125	−	0.754652i	$-0.272194\pi$
$53$	1.03543e7			1.31225			0.656125	+	0.754652i	$0.272194\pi$
$54$		0			0
$55$	−6.33606e6	−	1.42617e7i	−0.692419	−	1.55854i
$56$		0			0
$57$			1.93249e7i			1.83070i
$58$		0			0
$59$	2.46965e6			0.203810			0.101905	−	0.994794i	$-0.467506\pi$
$59$	2.46965e6			0.203810			0.101905	+	0.994794i	$0.467506\pi$
$60$		0			0
$61$		−	1.97635e7i		−	1.42740i	−0.700452	−	0.713700i	$-0.747018\pi$
$61$		−	1.97635e7i		−	1.42740i	0.700452	−	0.713700i	$-0.252982\pi$
$62$		0			0
$63$			1.66727e7i			1.05839i
$64$		0			0
$65$			1.95705e7i			1.09635i
$66$		0			0
$67$	−1.24402e7			−0.617348			−0.308674	−	0.951168i	$-0.599885\pi$
$67$	−1.24402e7			−0.617348			−0.308674	+	0.951168i	$0.599885\pi$
$68$		0			0
$69$	1.38438e7			0.610744
$70$		0			0
$71$	3.54223e7			1.39394			0.696969	−	0.717102i	$-0.254532\pi$
$71$	3.54223e7			1.39394			0.696969	+	0.717102i	$0.254532\pi$
$72$		0			0
$73$			1.52941e7i			0.538557i	0.963062	+	0.269278i	$0.0867852\pi$
$73$			1.52941e7i			0.538557i	−0.963062	+	0.269278i	$0.913215\pi$
$74$		0			0
$75$	−8.00352e7			−2.52951
$76$		0			0
$77$	−4.49361e7	+	1.99639e7i	−1.27830	+	0.567914i
$78$		0			0
$79$			3.11795e7i			0.800499i	0.916406	+	0.400250i	$0.131077\pi$
$79$			3.11795e7i			0.800499i	−0.916406	+	0.400250i	$0.868923\pi$
$80$		0			0
$81$	−5.09729e7			−1.18413
$82$		0			0
$83$		−	3.73727e7i		−	0.787485i	−0.919221	−	0.393743i	$-0.871180\pi$
$83$		−	3.73727e7i		−	0.787485i	0.919221	−	0.393743i	$-0.128820\pi$
$84$		0			0
$85$			3.24594e6i			0.0621821i
$86$		0			0
$87$			8.62962e7i			1.50631i
$88$		0			0
$89$	−4.94298e7			−0.787823			−0.393911	−	0.919148i	$-0.628878\pi$
$89$	−4.94298e7			−0.787823			−0.393911	+	0.919148i	$0.628878\pi$
$90$		0			0
$91$	6.16633e7			0.899210
$92$		0			0
$93$	8.48929e7			1.13485
$94$		0			0
$95$			1.91869e8i			2.35564i
$96$		0			0
$97$	4.09642e7			0.462719			0.231360	−	0.972868i	$-0.425683\pi$
$97$	4.09642e7			0.462719			0.231360	+	0.972868i	$0.425683\pi$
$98$		0			0
$99$	2.95101e7	+	6.64235e7i	0.307206	+	0.691482i
$100$		0			0
$101$		−	1.18153e8i		−	1.13542i	−0.823227	−	0.567712i	$-0.807829\pi$
$101$		−	1.18153e8i		−	1.13542i	0.823227	−	0.567712i	$-0.192171\pi$
$102$		0			0
$103$	−1.55085e8			−1.37791			−0.688954	−	0.724805i	$-0.741930\pi$
$103$	−1.55085e8			−1.37791			−0.688954	+	0.724805i	$0.741930\pi$
$104$		0			0
$105$			3.84311e8i			3.16174i
$106$		0			0
$107$		−	1.58572e7i		−	0.120974i	−0.998169	−	0.0604869i	$-0.980735\pi$
$107$		−	1.58572e7i		−	0.120974i	0.998169	−	0.0604869i	$-0.0192653\pi$
$108$		0			0
$109$			1.63146e8i			1.15577i	0.816118	+	0.577885i	$0.196122\pi$
$109$			1.63146e8i			1.15577i	−0.816118	+	0.577885i	$0.803878\pi$
$110$		0			0
$111$	−1.99579e8			−1.31469
$112$		0			0
$113$	9.10485e6			0.0558417			0.0279209	−	0.999610i	$-0.491111\pi$
$113$	9.10485e6			0.0558417			0.0279209	+	0.999610i	$0.491111\pi$
$114$		0			0
$115$	1.37449e8			0.785871
$116$		0			0
$117$		−	9.11491e7i		−	0.486417i
$118$		0			0
$119$	1.02274e7			0.0510010
$120$		0			0
$121$	−1.43688e8	+	1.59071e8i	−0.670316	+	0.742076i
$122$		0			0
$123$		−	2.05172e8i		−	0.896393i
$124$		0			0
$125$	−3.78270e8			−1.54939
$126$		0			0
$127$		−	1.11942e8i		−	0.430306i	−0.976580	−	0.215153i	$-0.930975\pi$
$127$		−	1.11942e8i		−	0.430306i	0.976580	−	0.215153i	$-0.0690251\pi$
$128$		0			0
$129$		−	6.30335e8i		−	2.27621i
$130$		0			0
$131$		−	9.91228e7i		−	0.336580i	−0.985738	−	0.168290i	$-0.946175\pi$
$131$		−	9.91228e7i		−	0.336580i	0.985738	−	0.168290i	$-0.0538245\pi$
$132$		0			0
$133$	6.04547e8			1.93207
$134$		0			0
$135$	−1.82701e8			−0.550054
$136$		0			0
$137$	4.10441e7			0.116511			0.0582557	−	0.998302i	$-0.481446\pi$
$137$	4.10441e7			0.116511			0.0582557	+	0.998302i	$0.481446\pi$
$138$		0			0
$139$			3.47447e8i			0.930742i	0.885116	+	0.465371i	$0.154079\pi$
$139$			3.47447e8i			0.930742i	−0.885116	+	0.465371i	$0.845921\pi$
$140$		0			0
$141$	−6.93691e8			−1.75505
$142$		0			0
$143$	2.45664e8	−	1.09142e8i	0.587485	−	0.261004i
$144$		0			0
$145$			8.56798e8i			1.93824i
$146$		0			0
$147$	5.92014e8			1.26784
$148$		0			0
$149$		−	1.07915e8i		−	0.218945i	−0.993990	−	0.109473i	$-0.965084\pi$
$149$		−	1.07915e8i		−	0.218945i	0.993990	−	0.109473i	$-0.0349162\pi$
$150$		0			0
$151$			5.27667e8i			1.01497i	0.861661	+	0.507484i	$0.169424\pi$
$151$			5.27667e8i			1.01497i	−0.861661	+	0.507484i	$0.830576\pi$
$152$		0			0
$153$		−	1.51179e7i		−	0.0275884i
$154$		0			0
$155$	8.42866e8			1.46026
$156$		0			0
$157$	2.06743e8			0.340277			0.170138	−	0.985420i	$-0.445579\pi$
$157$	2.06743e8			0.340277			0.170138	+	0.985420i	$0.445579\pi$
$158$		0			0
$159$	−1.11160e9			−1.73924
$160$		0			0
$161$		−	4.33080e8i		−	0.644562i
$162$		0			0
$163$	−1.02174e9			−1.44741			−0.723705	−	0.690110i	$-0.757562\pi$
$163$	−1.02174e9			−1.44741			−0.723705	+	0.690110i	$0.757562\pi$
$164$		0			0
$165$	6.80217e8	+	1.53108e9i	0.917723	+	2.06568i
$166$		0			0
$167$		−	1.22957e8i		−	0.158084i	−0.996871	−	0.0790422i	$-0.974814\pi$
$167$		−	1.22957e8i		−	0.158084i	0.996871	−	0.0790422i	$-0.0251862\pi$
$168$		0			0
$169$	4.78620e8			0.586738
$170$		0			0
$171$		−	8.93626e8i		−	1.04513i
$172$		0			0
$173$		−	3.45610e8i		−	0.385835i	−0.981215	−	0.192918i	$-0.938205\pi$
$173$		−	3.45610e8i		−	0.385835i	0.981215	−	0.192918i	$-0.0617950\pi$
$174$		0			0
$175$			2.50377e9i			2.66957i
$176$		0			0
$177$	−2.65132e8			−0.270128
$178$		0			0
$179$	−1.26831e9			−1.23541			−0.617707	−	0.786409i	$-0.711938\pi$
$179$	−1.26831e9			−1.23541			−0.617707	+	0.786409i	$0.711938\pi$
$180$		0			0
$181$	−5.10123e8			−0.475292			−0.237646	−	0.971352i	$-0.576376\pi$
$181$	−5.10123e8			−0.475292			−0.237646	+	0.971352i	$0.576376\pi$
$182$		0			0
$183$			2.12174e9i			1.89186i
$184$		0			0
$185$	−1.98154e9			−1.69167
$186$		0			0
$187$	4.07456e7	−	1.81022e7i	0.0333207	−	0.0148035i
$188$		0			0
$189$			5.75660e8i			0.451148i
$190$		0			0
$191$	−1.27397e9			−0.957253			−0.478626	−	0.878019i	$-0.658865\pi$
$191$	−1.27397e9			−0.957253			−0.478626	+	0.878019i	$0.658865\pi$
$192$		0			0
$193$		−	5.01886e8i		−	0.361722i	−0.983509	−	0.180861i	$-0.942112\pi$
$193$		−	5.01886e8i		−	0.361722i	0.983509	−	0.180861i	$-0.0578885\pi$
$194$		0			0
$195$		−	2.10101e9i		−	1.45308i
$196$		0			0
$197$			5.13021e8i			0.340620i	0.985391	+	0.170310i	$0.0544769\pi$
$197$			5.13021e8i			0.340620i	−0.985391	+	0.170310i	$0.945523\pi$
$198$		0			0
$199$	2.60295e9			1.65979			0.829895	−	0.557920i	$-0.188400\pi$
$199$	2.60295e9			1.65979			0.829895	+	0.557920i	$0.188400\pi$
$200$		0			0
$201$	1.33554e9			0.818225
$202$		0			0
$203$	2.69963e9			1.58972
$204$		0			0
$205$		−	2.03707e9i		−	1.15343i
$206$		0			0
$207$	−6.40168e8			−0.348668
$208$		0			0
$209$	2.40849e9	−	1.07003e9i	1.26229	−	0.560801i
$210$		0			0
$211$		−	9.08887e8i		−	0.458543i	−0.973363	−	0.229271i	$-0.926366\pi$
$211$		−	9.08887e8i		−	0.458543i	0.973363	−	0.229271i	$-0.0736344\pi$
$212$		0			0
$213$	−3.80281e9			−1.84751
$214$		0			0
$215$		−	6.25832e9i		−	2.92890i
$216$		0			0
$217$		−	2.65573e9i		−	1.19769i
$218$		0			0
$219$		−	1.64192e9i		−	0.713796i
$220$		0			0
$221$	−5.59129e7			−0.0234392
$222$		0			0
$223$	2.95821e9			1.19622			0.598108	−	0.801415i	$-0.295919\pi$
$223$	2.95821e9			1.19622			0.598108	+	0.801415i	$0.295919\pi$
$224$		0			0
$225$	3.70100e9			1.44407
$226$		0			0
$227$		−	2.51073e9i		−	0.945576i	−0.881176	−	0.472788i	$-0.843248\pi$
$227$		−	2.51073e9i		−	0.945576i	0.881176	−	0.472788i	$-0.156752\pi$
$228$		0			0
$229$	3.15387e9			1.14684			0.573419	−	0.819262i	$-0.305617\pi$
$229$	3.15387e9			1.14684			0.573419	+	0.819262i	$0.305617\pi$
$230$		0			0
$231$	4.82418e9	−	2.14325e9i	1.69424	−	0.752706i
$232$		0			0
$233$		−	5.76200e9i		−	1.95501i	−0.210905	−	0.977506i	$-0.567641\pi$
$233$		−	5.76200e9i		−	1.95501i	0.210905	−	0.977506i	$-0.432359\pi$
$234$		0			0
$235$	−6.88736e9			−2.25830
$236$		0			0
$237$		−	3.34732e9i		−	1.06097i
$238$		0			0
$239$			9.46263e8i			0.290015i	0.989431	+	0.145007i	$0.0463206\pi$
$239$			9.46263e8i			0.290015i	−0.989431	+	0.145007i	$0.953679\pi$
$240$		0			0
$241$			4.91602e9i			1.45729i	0.684892	+	0.728644i	$0.259849\pi$
$241$			4.91602e9i			1.45729i	−0.684892	+	0.728644i	$0.740151\pi$
$242$		0			0
$243$	4.34767e9			1.24690
$244$		0			0
$245$	5.87786e9			1.63138
$246$		0			0
$247$	−3.30503e9			−0.887948
$248$		0			0
$249$			4.01220e9i			1.04372i
$250$		0			0
$251$	−1.92828e9			−0.485820			−0.242910	−	0.970049i	$-0.578102\pi$
$251$	−1.92828e9			−0.485820			−0.242910	+	0.970049i	$0.578102\pi$
$252$		0			0
$253$	−7.66535e8	−	1.72537e9i	−0.187090	−	0.421115i
$254$		0			0
$255$		−	3.48473e8i		−	0.0824153i
$256$		0			0
$257$	−1.82805e9			−0.419039			−0.209520	−	0.977804i	$-0.567190\pi$
$257$	−1.82805e9			−0.419039			−0.209520	+	0.977804i	$0.567190\pi$
$258$		0			0
$259$			6.24350e9i			1.38749i
$260$		0			0
$261$		−	3.99052e9i		−	0.859940i
$262$		0			0
$263$		−	8.47338e9i		−	1.77106i	−0.464581	−	0.885531i	$-0.653795\pi$
$263$		−	8.47338e9i		−	1.77106i	0.464581	−	0.885531i	$-0.346205\pi$
$264$		0			0
$265$	−1.10366e10			−2.23796
$266$		0			0
$267$	5.30660e9			1.04417
$268$		0			0
$269$	5.95203e9			1.13673			0.568363	−	0.822778i	$-0.307577\pi$
$269$	5.95203e9			1.13673			0.568363	+	0.822778i	$0.307577\pi$
$270$		0			0
$271$			6.96805e8i			0.129192i	0.997912	+	0.0645958i	$0.0205758\pi$
$271$			6.96805e8i			0.129192i	−0.997912	+	0.0645958i	$0.979424\pi$
$272$		0			0
$273$	−6.61995e9			−1.19180
$274$		0			0
$275$	4.43157e9	+	9.97489e9i	0.774866	+	1.74412i
$276$		0			0
$277$			3.75478e9i			0.637772i	0.947793	+	0.318886i	$0.103309\pi$
$277$			3.75478e9i			0.637772i	−0.947793	+	0.318886i	$0.896691\pi$
$278$		0			0
$279$	−3.92564e9			−0.647878
$280$		0			0
$281$			5.93778e9i			0.952355i	0.879349	+	0.476178i	$0.157978\pi$
$281$			5.93778e9i			0.952355i	−0.879349	+	0.476178i	$0.842022\pi$
$282$		0			0
$283$		−	1.82263e9i		−	0.284154i	−0.989856	−	0.142077i	$-0.954622\pi$
$283$		−	1.82263e9i		−	0.284154i	0.989856	−	0.142077i	$-0.0453781\pi$
$284$		0			0
$285$		−	2.05983e10i		−	3.12214i
$286$		0			0
$287$	−6.41847e9			−0.946028
$288$		0			0
$289$	6.96648e9			0.998671
$290$		0			0
$291$	−4.39777e9			−0.613282
$292$		0			0
$293$			4.48121e9i			0.608029i	0.952667	+	0.304015i	$0.0983272\pi$
$293$			4.48121e9i			0.608029i	−0.952667	+	0.304015i	$0.901673\pi$
$294$		0			0
$295$	−2.63238e9			−0.347585
$296$		0			0
$297$	1.01890e9	+	2.29340e9i	0.130950	+	0.294750i
$298$		0			0
$299$			2.36763e9i			0.296230i
$300$		0			0
$301$	−1.97190e10			−2.40225
$302$		0			0
$303$			1.26845e10i			1.50488i
$304$		0			0
$305$			2.10659e10i			2.43434i
$306$		0			0
$307$			1.48030e10i			1.66646i	0.552926	+	0.833230i	$0.313511\pi$
$307$			1.48030e10i			1.66646i	−0.552926	+	0.833230i	$0.686489\pi$
$308$		0			0
$309$	1.66493e10			1.82626
$310$		0			0
$311$	−2.33763e9			−0.249882			−0.124941	−	0.992164i	$-0.539874\pi$
$311$	−2.33763e9			−0.249882			−0.124941	+	0.992164i	$0.539874\pi$
$312$		0			0
$313$	3.98216e9			0.414898			0.207449	−	0.978246i	$-0.433484\pi$
$313$	3.98216e9			0.414898			0.207449	+	0.978246i	$0.433484\pi$
$314$		0			0
$315$		−	1.77714e10i		−	1.80501i
$316$		0			0
$317$	6.72620e9			0.666090			0.333045	−	0.942911i	$-0.391924\pi$
$317$	6.72620e9			0.666090			0.333045	+	0.942911i	$0.391924\pi$
$318$		0			0
$319$	1.07552e10	−	4.77824e9i	1.03862	−	0.461430i
$320$		0			0
$321$			1.70237e9i			0.160337i
$322$		0			0
$323$	−5.48170e8			−0.0503623
$324$		0			0
$325$		−	1.36880e10i		−	1.22689i
$326$		0			0
$327$		−	1.75148e10i		−	1.53184i
$328$		0			0
$329$			2.17010e10i			1.85223i
$330$		0			0
$331$	−1.67921e9			−0.139892			−0.0699460	−	0.997551i	$-0.522283\pi$
$331$	−1.67921e9			−0.139892			−0.0699460	+	0.997551i	$0.522283\pi$
$332$		0			0
$333$	9.22898e9			0.750545
$334$		0			0
$335$	1.32600e10			1.05285
$336$		0			0
$337$			5.65480e8i			0.0438427i	0.999760	+	0.0219214i	$0.00697835\pi$
$337$			5.65480e8i			0.0438427i	−0.999760	+	0.0219214i	$0.993022\pi$
$338$		0			0
$339$	−9.77464e8			−0.0740119
$340$		0			0
$341$	−4.70054e9	−	1.05803e10i	−0.347641	−	0.782494i
$342$		0			0
$343$			8.40706e8i			0.0607390i
$344$		0			0
$345$	−1.47561e10			−1.04158
$346$		0			0
$347$			2.55153e10i			1.75988i	0.475083	+	0.879941i	$0.342418\pi$
$347$			2.55153e10i			1.75988i	−0.475083	+	0.879941i	$0.657582\pi$
$348$		0			0
$349$			8.37613e9i			0.564601i	0.959326	+	0.282300i	$0.0910975\pi$
$349$			8.37613e9i			0.564601i	−0.959326	+	0.282300i	$0.908903\pi$
$350$		0			0
$351$		−	3.14711e9i		−	0.207340i
$352$		0			0
$353$	−2.09179e10			−1.34716			−0.673580	−	0.739114i	$-0.735245\pi$
$353$	−2.09179e10			−1.34716			−0.673580	+	0.739114i	$0.735245\pi$
$354$		0			0
$355$	−3.77565e10			−2.37727
$356$		0			0
$357$	−1.09798e9			−0.0675961
$358$		0			0
$359$		−	5.13471e9i		−	0.309128i	−0.987983	−	0.154564i	$-0.950603\pi$
$359$		−	5.13471e9i		−	0.309128i	0.987983	−	0.154564i	$-0.0493972\pi$
$360$		0			0
$361$	−1.54190e10			−0.907875
$362$		0			0
$363$	1.54258e10	−	1.70772e10i	0.888429	−	0.983538i
$364$		0			0
$365$		−	1.63019e10i		−	0.918473i
$366$		0			0
$367$	−8.84680e9			−0.487666			−0.243833	−	0.969817i	$-0.578405\pi$
$367$	−8.84680e9			−0.487666			−0.243833	+	0.969817i	$0.578405\pi$
$368$		0			0
$369$			9.48762e9i			0.511743i
$370$		0			0
$371$			3.47745e10i			1.83555i
$372$		0			0
$373$			1.42801e10i			0.737729i	0.929483	+	0.368865i	$0.120253\pi$
$373$			1.42801e10i			0.737729i	−0.929483	+	0.368865i	$0.879747\pi$
$374$		0			0
$375$	4.06097e10			2.05355
$376$		0			0
$377$	−1.47588e10			−0.730608
$378$		0			0
$379$	−5.57885e9			−0.270388			−0.135194	−	0.990819i	$-0.543166\pi$
$379$	−5.57885e9			−0.270388			−0.135194	+	0.990819i	$0.543166\pi$
$380$		0			0
$381$			1.20177e10i			0.570323i
$382$		0			0
$383$	−1.03414e10			−0.480600			−0.240300	−	0.970699i	$-0.577246\pi$
$383$	−1.03414e10			−0.480600			−0.240300	+	0.970699i	$0.577246\pi$
$384$		0			0
$385$	4.78972e10	−	2.12794e10i	2.18006	−	0.968539i
$386$		0			0
$387$			2.91481e10i			1.29947i
$388$		0			0
$389$	−4.04072e10			−1.76466			−0.882329	−	0.470633i	$-0.844026\pi$
$389$	−4.04072e10			−1.76466			−0.882329	+	0.470633i	$0.844026\pi$
$390$		0			0
$391$			3.92693e8i			0.0168014i
$392$		0			0
$393$			1.06415e10i			0.446099i
$394$		0			0
$395$		−	3.32341e10i		−	1.36520i
$396$		0			0
$397$	1.92978e10			0.776867			0.388433	−	0.921477i	$-0.373016\pi$
$397$	1.92978e10			0.776867			0.388433	+	0.921477i	$0.373016\pi$
$398$		0			0
$399$	−6.49020e10			−2.56075
$400$		0			0
$401$	1.71839e10			0.664576			0.332288	−	0.943178i	$-0.392179\pi$
$401$	1.71839e10			0.664576			0.332288	+	0.943178i	$0.392179\pi$
$402$		0			0
$403$			1.45188e10i			0.550439i
$404$		0			0
$405$	5.43318e10			2.01945
$406$		0			0
$407$	1.10508e10	+	2.48738e10i	0.402730	+	0.906494i
$408$		0			0
$409$			4.43496e10i			1.58488i	0.609950	+	0.792440i	$0.291190\pi$
$409$			4.43496e10i			1.58488i	−0.609950	+	0.792440i	$0.708810\pi$
$410$		0			0
$411$	−4.40635e9			−0.154423
$412$		0			0
$413$			8.29421e9i			0.285085i
$414$		0			0
$415$			3.98354e10i			1.34300i
$416$		0			0
$417$		−	3.73007e10i		−	1.23359i
$418$		0			0
$419$	−3.48472e10			−1.13061			−0.565304	−	0.824882i	$-0.691241\pi$
$419$	−3.48472e10			−1.13061			−0.565304	+	0.824882i	$0.691241\pi$
$420$		0			0
$421$	−4.46769e10			−1.42218			−0.711090	−	0.703101i	$-0.751798\pi$
$421$	−4.46769e10			−1.42218			−0.711090	+	0.703101i	$0.751798\pi$
$422$		0			0
$423$	3.20778e10			1.00194
$424$		0			0
$425$		−	2.27028e9i		−	0.0695862i
$426$		0			0
$427$	6.63752e10			1.99661
$428$		0			0
$429$	−2.63736e10	+	1.17171e10i	−0.778646	+	0.345931i
$430$		0			0
$431$			2.19637e10i			0.636497i	0.948007	+	0.318248i	$0.103095\pi$
$431$			2.19637e10i			0.636497i	−0.948007	+	0.318248i	$0.896905\pi$
$432$		0			0
$433$	2.09111e9			0.0594874			0.0297437	−	0.999558i	$-0.490531\pi$
$433$	2.09111e9			0.0594874			0.0297437	+	0.999558i	$0.490531\pi$
$434$		0			0
$435$		−	9.19827e10i		−	2.56891i
$436$		0			0
$437$			2.32122e10i			0.636489i
$438$		0			0
$439$		−	6.24447e9i		−	0.168127i	−0.996460	−	0.0840635i	$-0.973210\pi$
$439$		−	6.24447e9i		−	0.168127i	0.996460	−	0.0840635i	$-0.0267899\pi$
$440$		0			0
$441$	−2.73760e10			−0.723796
$442$		0			0
$443$	3.60739e8			0.00936652			0.00468326	−	0.999989i	$-0.498509\pi$
$443$	3.60739e8			0.00936652			0.00468326	+	0.999989i	$0.498509\pi$
$444$		0			0
$445$	5.26870e10			1.34358
$446$		0			0
$447$			1.15853e10i			0.290187i
$448$		0			0
$449$	1.40883e10			0.346635			0.173317	−	0.984866i	$-0.444551\pi$
$449$	1.40883e10			0.346635			0.173317	+	0.984866i	$0.444551\pi$
$450$		0			0
$451$	−2.55709e10	+	1.13605e10i	−0.618073	+	0.274593i
$452$		0			0
$453$		−	5.66484e10i		−	1.34523i
$454$		0			0
$455$	−6.57267e10			−1.53354
$456$		0			0
$457$			4.48437e10i			1.02810i	0.857760	+	0.514051i	$0.171856\pi$
$457$			4.48437e10i			1.02810i	−0.857760	+	0.514051i	$0.828144\pi$
$458$		0			0
$459$		−	5.21977e8i		−	0.0117598i
$460$		0			0
$461$		−	7.44959e10i		−	1.64941i	−0.565562	−	0.824706i	$-0.691341\pi$
$461$		−	7.44959e10i		−	1.64941i	0.565562	−	0.824706i	$-0.308659\pi$
$462$		0			0
$463$	6.03013e10			1.31221			0.656105	−	0.754670i	$-0.272203\pi$
$463$	6.03013e10			1.31221			0.656105	+	0.754670i	$0.272203\pi$
$464$		0			0
$465$	−9.04870e10			−1.93542
$466$		0			0
$467$	5.22205e10			1.09793			0.548963	−	0.835846i	$-0.315023\pi$
$467$	5.22205e10			1.09793			0.548963	+	0.835846i	$0.315023\pi$
$468$		0			0
$469$		−	4.17801e10i		−	0.863532i
$470$		0			0
$471$	−2.21952e10			−0.450998
$472$		0			0
$473$	−7.85594e10	+	3.49018e10i	−1.56947	+	0.697274i
$474$		0			0
$475$		−	1.34197e11i		−	2.63614i
$476$		0			0
$477$	5.14028e10			0.992917
$478$		0			0
$479$			6.42483e10i			1.22045i	0.792228	+	0.610225i	$0.208921\pi$
$479$			6.42483e10i			1.22045i	−0.792228	+	0.610225i	$0.791079\pi$
$480$		0			0
$481$		−	3.41329e10i		−	0.637666i
$482$		0			0
$483$			4.64939e10i			0.854295i
$484$		0			0
$485$	−4.36636e10			−0.789137
$486$		0			0
$487$	3.07857e10			0.547309			0.273655	−	0.961828i	$-0.411767\pi$
$487$	3.07857e10			0.547309			0.273655	+	0.961828i	$0.411767\pi$
$488$		0			0
$489$	1.09691e11			1.91838
$490$		0			0
$491$			4.47915e10i			0.770671i	0.922776	+	0.385336i	$0.125914\pi$
$491$			4.47915e10i			0.770671i	−0.922776	+	0.385336i	$0.874086\pi$
$492$		0			0
$493$	−2.44788e9			−0.0414383
$494$		0			0
$495$	−3.14547e10	−	7.08005e10i	−0.523920	−	1.17928i
$496$		0			0
$497$			1.18964e11i			1.94981i
$498$		0			0
$499$	−3.39394e10			−0.547396			−0.273698	−	0.961816i	$-0.588247\pi$
$499$	−3.39394e10			−0.547396			−0.273698	+	0.961816i	$0.588247\pi$
$500$		0			0
$501$			1.32003e10i			0.209523i
$502$		0			0
$503$		−	7.83988e10i		−	1.22472i	−0.790578	−	0.612361i	$-0.790220\pi$
$503$		−	7.83988e10i		−	1.22472i	0.790578	−	0.612361i	$-0.209780\pi$
$504$		0			0
$505$			1.25939e11i			1.93639i
$506$		0			0
$507$	−5.13829e10			−0.777655
$508$		0			0
$509$	7.95844e10			1.18565			0.592825	−	0.805331i	$-0.298013\pi$
$509$	7.95844e10			1.18565			0.592825	+	0.805331i	$0.298013\pi$
$510$		0			0
$511$	−5.13645e10			−0.753321
$512$		0			0
$513$		−	3.08542e10i		−	0.445497i
$514$		0			0
$515$	1.65304e11			2.34993
$516$		0			0
$517$	3.84099e10	+	8.64556e10i	0.537626	+	1.21013i
$518$		0			0
$519$			3.71035e10i			0.511382i
$520$		0			0
$521$	1.61542e10			0.219247			0.109623	−	0.993973i	$-0.465036\pi$
$521$	1.61542e10			0.219247			0.109623	+	0.993973i	$0.465036\pi$
$522$		0			0
$523$		−	1.90690e10i		−	0.254872i	−0.991847	−	0.127436i	$-0.959325\pi$
$523$		−	1.90690e10i		−	0.254872i	0.991847	−	0.127436i	$-0.0406747\pi$
$524$		0			0
$525$		−	2.68795e11i		−	3.53822i
$526$		0			0
$527$			2.40807e9i			0.0312196i
$528$		0			0
$529$	−6.16824e10			−0.787660
$530$		0			0
$531$	1.22603e10			0.154214
$532$		0			0
$533$	3.50895e10			0.434779
$534$		0			0
$535$			1.69021e10i			0.206313i
$536$		0			0
$537$	1.36161e11			1.63740
$538$		0			0
$539$	−3.27800e10	−	7.37835e10i	−0.388377	−	0.874187i
$540$		0			0
$541$		−	1.58088e11i		−	1.84548i	−0.385427	−	0.922739i	$-0.625946\pi$
$541$		−	1.58088e11i		−	1.84548i	0.385427	−	0.922739i	$-0.374054\pi$
$542$		0			0
$543$	5.47650e10			0.629947
$544$		0			0
$545$		−	1.73897e11i		−	1.97109i
$546$		0			0
$547$		−	3.30999e10i		−	0.369724i	−0.982764	−	0.184862i	$-0.940816\pi$
$547$		−	3.30999e10i		−	0.369724i	0.982764	−	0.184862i	$-0.0591839\pi$
$548$		0			0
$549$		−	9.81141e10i		−	1.08005i
$550$		0			0
$551$	−1.44695e11			−1.56981
$552$		0			0
$553$	−1.04715e11			−1.11972
$554$		0			0
$555$	2.12731e11			2.24212
$556$		0			0
$557$		−	1.79805e11i		−	1.86802i	−0.357250	−	0.934009i	$-0.616285\pi$
$557$		−	1.79805e11i		−	1.86802i	0.357250	−	0.934009i	$-0.383715\pi$
$558$		0			0
$559$	1.07803e11			1.10403
$560$		0			0
$561$	−4.37430e9	+	1.94338e9i	−0.0441629	+	0.0196204i
$562$		0			0
$563$		−	8.00717e10i		−	0.796977i	−0.917173	−	0.398488i	$-0.869535\pi$
$563$		−	8.00717e10i		−	0.796977i	0.917173	−	0.398488i	$-0.130465\pi$
$564$		0			0
$565$	−9.70482e9			−0.0952344
$566$		0			0
$567$		−	1.71191e11i		−	1.65633i
$568$		0			0
$569$			9.97788e10i			0.951896i	0.879474	+	0.475948i	$0.157895\pi$
$569$			9.97788e10i			0.951896i	−0.879474	+	0.475948i	$0.842105\pi$
$570$		0			0
$571$		−	3.65344e10i		−	0.343683i	−0.985125	−	0.171841i	$-0.945028\pi$
$571$		−	3.65344e10i		−	0.343683i	0.985125	−	0.171841i	$-0.0549717\pi$
$572$		0			0
$573$	1.36769e11			1.26873
$574$		0			0
$575$	−9.61348e10			−0.879446
$576$		0			0
$577$	−1.45081e11			−1.30890			−0.654452	−	0.756103i	$-0.727101\pi$
$577$	−1.45081e11			−1.30890			−0.654452	+	0.756103i	$0.727101\pi$
$578$		0			0
$579$			5.38806e10i			0.479422i
$580$		0			0
$581$	1.25515e11			1.10152
$582$		0			0
$583$	6.15495e10	+	1.38540e11i	0.532783	+	1.19923i
$584$		0			0
$585$			9.71555e10i			0.829553i
$586$		0			0
$587$	−5.23812e10			−0.441187			−0.220594	−	0.975366i	$-0.570799\pi$
$587$	−5.23812e10			−0.441187			−0.220594	+	0.975366i	$0.570799\pi$
$588$		0			0
$589$			1.42342e11i			1.18269i
$590$		0			0
$591$		−	5.50761e10i		−	0.451453i
$592$		0			0
$593$			1.47372e11i			1.19178i	0.803066	+	0.595890i	$0.203201\pi$
$593$			1.47372e11i			1.19178i	−0.803066	+	0.595890i	$0.796799\pi$
$594$		0			0
$595$	−1.09014e10			−0.0869788
$596$		0			0
$597$	−2.79443e11			−2.19986
$598$		0			0
$599$	2.85450e10			0.221729			0.110865	−	0.993836i	$-0.464638\pi$
$599$	2.85450e10			0.221729			0.110865	+	0.993836i	$0.464638\pi$
$600$		0			0
$601$			1.04044e9i			0.00797480i	0.999992	+	0.00398740i	$0.00126923\pi$
$601$			1.04044e9i			0.00797480i	−0.999992	+	0.00398740i	$0.998731\pi$
$602$		0			0
$603$	−6.17583e10			−0.467118
$604$		0			0
$605$	1.53157e11	−	1.69553e11i	1.14318	−	1.26556i
$606$		0			0
$607$			6.76881e10i			0.498606i	0.968426	+	0.249303i	$0.0802015\pi$
$607$			6.76881e10i			0.498606i	−0.968426	+	0.249303i	$0.919798\pi$
$608$		0			0
$609$	−2.89823e11			−2.10699
$610$		0			0
$611$		−	1.18638e11i		−	0.851255i
$612$		0			0
$613$			4.18024e10i			0.296046i	0.988984	+	0.148023i	$0.0472910\pi$
$613$			4.18024e10i			0.296046i	−0.988984	+	0.148023i	$0.952709\pi$
$614$		0			0
$615$			2.18692e11i			1.52874i
$616$		0			0
$617$	2.06804e11			1.42698			0.713492	−	0.700663i	$-0.247112\pi$
$617$	2.06804e11			1.42698			0.713492	+	0.700663i	$0.247112\pi$
$618$		0			0
$619$	−1.05226e11			−0.716741			−0.358370	−	0.933579i	$-0.616668\pi$
$619$	−1.05226e11			−0.716741			−0.358370	+	0.933579i	$0.616668\pi$
$620$		0			0
$621$	−2.21031e10			−0.148623
$622$		0			0
$623$		−	1.66008e11i		−	1.10199i
$624$		0			0
$625$	1.11982e11			0.733882
$626$		0			0
$627$	−2.58567e11	+	1.14874e11i	−1.67302	+	0.743279i
$628$		0			0
$629$		−	5.66126e9i		−	0.0361669i
$630$		0			0
$631$	−6.14446e10			−0.387584			−0.193792	−	0.981043i	$-0.562079\pi$
$631$	−6.14446e10			−0.387584			−0.193792	+	0.981043i	$0.562079\pi$
$632$		0			0
$633$			9.75748e10i			0.607747i
$634$		0			0
$635$			1.19318e11i			0.733859i
$636$		0			0
$637$			1.01249e11i			0.614940i
$638$		0			0
$639$	1.75850e11			1.05473
$640$		0			0
$641$	2.29437e11			1.35904			0.679519	−	0.733658i	$-0.262189\pi$
$641$	2.29437e11			1.35904			0.679519	+	0.733658i	$0.262189\pi$
$642$		0			0
$643$	−9.30078e10			−0.544096			−0.272048	−	0.962284i	$-0.587701\pi$
$643$	−9.30078e10			−0.544096			−0.272048	+	0.962284i	$0.587701\pi$
$644$		0			0
$645$			6.71871e11i			3.88192i
$646$		0			0
$647$	1.65018e11			0.941702			0.470851	−	0.882213i	$-0.343947\pi$
$647$	1.65018e11			0.941702			0.470851	+	0.882213i	$0.343947\pi$
$648$		0			0
$649$	1.46804e10	+	3.30438e10i	0.0827486	+	0.186256i
$650$		0			0
$651$			2.85110e11i			1.58741i
$652$		0			0
$653$	1.43934e11			0.791610			0.395805	−	0.918335i	$-0.370466\pi$
$653$	1.43934e11			0.791610			0.395805	+	0.918335i	$0.370466\pi$
$654$		0			0
$655$			1.05655e11i			0.574015i
$656$		0			0
$657$			7.59258e10i			0.407500i
$658$		0			0
$659$			1.10006e11i			0.583279i	0.956528	+	0.291640i	$0.0942008\pi$
$659$			1.10006e11i			0.583279i	−0.956528	+	0.291640i	$0.905799\pi$
$660$		0			0
$661$	−1.14673e11			−0.600695			−0.300348	−	0.953830i	$-0.597103\pi$
$661$	−1.14673e11			−0.600695			−0.300348	+	0.953830i	$0.597103\pi$
$662$		0			0
$663$	6.00261e9			0.0310660
$664$		0			0
$665$	−6.44384e11			−3.29502
$666$		0			0
$667$			1.03655e11i			0.523706i
$668$		0			0
$669$	−3.17583e11			−1.58545
$670$		0			0
$671$	2.64436e11	−	1.17482e11i	1.30446	−	0.579535i
$672$		0			0
$673$		−	1.40462e11i		−	0.684698i	−0.939573	−	0.342349i	$-0.888777\pi$
$673$		−	1.40462e11i		−	0.684698i	0.939573	−	0.342349i	$-0.111223\pi$
$674$		0			0
$675$	1.27785e11			0.615550
$676$		0			0
$677$		−	1.42844e11i		−	0.679998i	−0.940426	−	0.339999i	$-0.889573\pi$
$677$		−	1.42844e11i		−	0.679998i	0.940426	−	0.339999i	$-0.110427\pi$
$678$		0			0
$679$			1.37577e11i			0.647241i
$680$		0			0
$681$			2.69543e11i			1.25325i
$682$		0			0
$683$	−2.64521e11			−1.21556			−0.607782	−	0.794104i	$-0.707941\pi$
$683$	−2.64521e11			−1.21556			−0.607782	+	0.794104i	$0.707941\pi$
$684$		0			0
$685$	−4.37487e10			−0.198702
$686$		0			0
$687$	−3.38588e11			−1.52000
$688$		0			0
$689$		−	1.90111e11i		−	0.843586i
$690$		0			0
$691$	−1.00992e11			−0.442969			−0.221484	−	0.975164i	$-0.571090\pi$
$691$	−1.00992e11			−0.442969			−0.221484	+	0.975164i	$0.571090\pi$
$692$		0			0
$693$	−2.23081e11	+	9.91087e10i	−0.967229	+	0.429713i
$694$		0			0
$695$		−	3.70342e11i		−	1.58732i
$696$		0			0
$697$	5.81992e9			0.0246596
$698$		0			0
$699$			6.18588e11i			2.59115i
$700$		0			0
$701$			2.91296e10i			0.120632i	0.998179	+	0.0603160i	$0.0192108\pi$
$701$			2.91296e10i			0.120632i	−0.998179	+	0.0603160i	$0.980789\pi$
$702$		0			0
$703$		−	3.34639e11i		−	1.37011i
$704$		0			0
$705$	7.39402e11			2.99312
$706$		0			0
$707$	3.96812e11			1.58821
$708$		0			0
$709$	2.06523e10			0.0817305			0.0408652	−	0.999165i	$-0.486989\pi$
$709$	2.06523e10			0.0817305			0.0408652	+	0.999165i	$0.486989\pi$
$710$		0			0
$711$			1.54787e11i			0.605700i
$712$		0			0
$713$	1.01970e11			0.394560
$714$		0			0
$715$	−2.61852e11	+	1.16334e11i	−1.00192	+	0.445124i
$716$		0			0
$717$		−	1.01587e11i		−	0.384382i
$718$		0			0
$719$	6.12833e10			0.229312			0.114656	−	0.993405i	$-0.463423\pi$
$719$	6.12833e10			0.229312			0.114656	+	0.993405i	$0.463423\pi$
$720$		0			0
$721$		−	5.20846e11i		−	1.92739i
$722$		0			0
$723$		−	5.27766e11i		−	1.93147i
$724$		0			0
$725$		−	5.99262e11i		−	2.16903i
$726$		0			0
$727$	1.89951e11			0.679994			0.339997	−	0.940427i	$-0.389574\pi$
$727$	1.89951e11			0.679994			0.339997	+	0.940427i	$0.389574\pi$
$728$		0			0
$729$	−1.32317e11			−0.468497
$730$		0			0
$731$	1.78801e10			0.0626181
$732$		0			0
$733$			5.03570e11i			1.74439i	0.489158	+	0.872195i	$0.337304\pi$
$733$			5.03570e11i			1.74439i	−0.489158	+	0.872195i	$0.662696\pi$
$734$		0			0
$735$	−6.31026e11			−2.16221
$736$		0			0
$737$	−7.39492e10	−	1.66450e11i	−0.250648	−	0.564175i
$738$		0			0
$739$			3.10991e11i			1.04273i	0.853335	+	0.521363i	$0.174576\pi$
$739$			3.10991e11i			1.04273i	−0.853335	+	0.521363i	$0.825424\pi$
$740$		0			0
$741$	3.54816e11			1.17688
$742$		0			0
$743$			6.45324e10i			0.211750i	0.994379	+	0.105875i	$0.0337643\pi$
$743$			6.45324e10i			0.211750i	−0.994379	+	0.105875i	$0.966236\pi$
$744$		0			0
$745$			1.15026e11i			0.373396i
$746$		0			0
$747$		−	1.85533e11i		−	0.595853i
$748$		0			0
$749$	5.32558e10			0.169215
$750$		0			0
$751$	5.37663e11			1.69025			0.845124	−	0.534570i	$-0.179526\pi$
$751$	5.37663e11			1.69025			0.845124	+	0.534570i	$0.179526\pi$
$752$		0			0
$753$	2.07013e11			0.643900
$754$		0			0
$755$		−	5.62438e11i		−	1.73096i
$756$		0			0
$757$	−3.02429e11			−0.920957			−0.460478	−	0.887671i	$-0.652322\pi$
$757$	−3.02429e11			−0.920957			−0.460478	+	0.887671i	$0.652322\pi$
$758$		0			0
$759$	8.22925e10	+	1.85230e11i	0.247967	+	0.558140i
$760$		0			0
$761$			1.76863e11i			0.527349i	0.964612	+	0.263675i	$0.0849345\pi$
$761$			1.76863e11i			0.527349i	−0.964612	+	0.263675i	$0.915065\pi$
$762$		0			0
$763$	−5.47921e11			−1.61666
$764$		0			0
$765$			1.61141e10i			0.0470502i
$766$		0			0
$767$		−	4.53441e10i		−	0.131021i
$768$		0			0
$769$		−	1.46197e11i		−	0.418056i	−0.977910	−	0.209028i	$-0.932970\pi$
$769$		−	1.46197e11i		−	0.418056i	0.977910	−	0.209028i	$-0.0670300\pi$
$770$		0			0
$771$	1.96252e11			0.555389
$772$		0			0
$773$	2.18958e11			0.613258			0.306629	−	0.951829i	$-0.400799\pi$
$773$	2.18958e11			0.613258			0.306629	+	0.951829i	$0.400799\pi$
$774$		0			0
$775$	−5.89517e11			−1.63414
$776$		0			0
$777$		−	6.70279e11i		−	1.83896i
$778$		0			0
$779$	3.44017e11			0.934180
$780$		0			0
$781$	2.10563e11	+	4.73949e11i	0.565949	+	1.27388i
$782$		0			0
$783$		−	1.37781e11i		−	0.366557i
$784$		0			0
$785$	−2.20366e11			−0.580319
$786$		0			0
$787$			3.82192e11i			0.996282i	0.867096	+	0.498141i	$0.165984\pi$
$787$			3.82192e11i			0.996282i	−0.867096	+	0.498141i	$0.834016\pi$
$788$		0			0
$789$			9.09671e11i			2.34734i
$790$		0			0
$791$			3.05783e10i			0.0781101i
$792$		0			0
$793$	−3.62870e11			−0.917610
$794$		0			0
$795$	1.18485e12			2.96616
$796$		0			0
$797$	−1.99828e11			−0.495248			−0.247624	−	0.968856i	$-0.579650\pi$
$797$	−1.99828e11			−0.495248			−0.247624	+	0.968856i	$0.579650\pi$
$798$		0			0
$799$		−	1.96772e10i		−	0.0482811i
$800$		0			0
$801$	−2.45389e11			−0.596108
$802$		0			0
$803$	−2.04634e11	+	9.09133e10i	−0.492171	+	0.218658i
$804$		0			0
$805$			4.61618e11i			1.09926i
$806$		0			0
$807$	−6.38989e11			−1.50660
$808$		0			0
$809$			7.04635e11i			1.64502i	0.568754	+	0.822508i	$0.307426\pi$
$809$			7.04635e11i			1.64502i	−0.568754	+	0.822508i	$0.692574\pi$
$810$		0			0
$811$			2.30141e11i			0.531999i	0.963973	+	0.265999i	$0.0857019\pi$
$811$			2.30141e11i			0.531999i	−0.963973	+	0.265999i	$0.914298\pi$
$812$		0			0
$813$		−	7.48065e10i		−	0.171229i
$814$		0			0
$815$	1.08907e12			2.46846
$816$		0			0
$817$	1.05690e12			2.37216
$818$		0			0
$819$	3.06121e11			0.680390
$820$		0			0
$821$		−	4.32504e11i		−	0.951957i	−0.879457	−	0.475979i	$-0.842094\pi$
$821$		−	4.32504e11i		−	0.951957i	0.879457	−	0.475979i	$-0.157906\pi$
$822$		0			0
$823$	5.60310e8			0.00122132			0.000610659	−	1.00000i	$-0.499806\pi$
$823$	5.60310e8			0.00122132			0.000610659		1.00000i	$0.499806\pi$
$824$		0			0
$825$	−4.75757e11	−	1.07087e12i	−1.02700	−	2.31164i
$826$		0			0
$827$			7.69936e11i			1.64601i	0.568033	+	0.823006i	$0.307704\pi$
$827$			7.69936e11i			1.64601i	−0.568033	+	0.823006i	$0.692296\pi$
$828$		0			0
$829$	−6.88427e11			−1.45760			−0.728802	−	0.684724i	$-0.759923\pi$
$829$	−6.88427e11			−1.45760			−0.728802	+	0.684724i	$0.759923\pi$
$830$		0			0
$831$		−	4.03100e11i		−	0.845295i
$832$		0			0
$833$			1.67931e10i			0.0348779i
$834$		0			0
$835$			1.31060e11i			0.269602i
$836$		0			0
$837$	−1.35540e11			−0.276164
$838$		0			0
$839$	−7.23133e11			−1.45939			−0.729693	−	0.683775i	$-0.760337\pi$
$839$	−7.23133e11			−1.45939			−0.729693	+	0.683775i	$0.760337\pi$
$840$		0			0
$841$	−1.45895e11			−0.291645
$842$		0			0
$843$		−	6.37459e11i		−	1.26224i
$844$		0			0
$845$	−5.10159e11			−1.00064
$846$		0			0
$847$	−5.34233e11	−	4.82572e11i	−1.03800	−	0.937623i
$848$		0			0
$849$			1.95671e11i			0.376614i
$850$		0			0
$851$	−2.39726e11			−0.457085
$852$		0			0
$853$			1.84780e11i			0.349027i	0.984655	+	0.174514i	$0.0558353\pi$
$853$			1.84780e11i			0.349027i	−0.984655	+	0.174514i	$0.944165\pi$
$854$		0			0
$855$			9.52512e11i			1.78240i
$856$		0			0
$857$			3.73581e11i			0.692566i	0.938130	+	0.346283i	$0.112556\pi$
$857$			3.73581e11i			0.692566i	−0.938130	+	0.346283i	$0.887444\pi$
$858$		0			0
$859$	−7.68889e11			−1.41218			−0.706091	−	0.708121i	$-0.749543\pi$
$859$	−7.68889e11			−1.41218			−0.706091	+	0.708121i	$0.749543\pi$
$860$		0			0
$861$	6.89064e11			1.25385
$862$		0			0
$863$	−3.07507e11			−0.554386			−0.277193	−	0.960814i	$-0.589404\pi$
$863$	−3.07507e11			−0.554386			−0.277193	+	0.960814i	$0.589404\pi$
$864$		0			0
$865$			3.68384e11i			0.658017i
$866$		0			0
$867$	−7.47897e11			−1.32363
$868$		0			0
$869$	−4.17181e11	+	1.85342e11i	−0.731552	+	0.325009i
$870$		0			0
$871$			2.28410e11i			0.396865i
$872$		0			0
$873$	2.03362e11			0.350117
$874$		0			0
$875$		−	1.27040e12i		−	2.16725i
$876$		0			0
$877$		−	8.14445e11i		−	1.37678i	−0.725342	−	0.688389i	$-0.758318\pi$
$877$		−	8.14445e11i		−	1.37678i	0.725342	−	0.688389i	$-0.241682\pi$
$878$		0			0
$879$		−	4.81086e11i		−	0.805874i
$880$		0			0
$881$	6.15951e11			1.02245			0.511226	−	0.859447i	$-0.329192\pi$
$881$	6.15951e11			1.02245			0.511226	+	0.859447i	$0.329192\pi$
$882$		0			0
$883$	7.84739e11			1.29087			0.645435	−	0.763815i	$-0.276676\pi$
$883$	7.84739e11			1.29087			0.645435	+	0.763815i	$0.276676\pi$
$884$		0			0
$885$	2.82603e11			0.460685
$886$		0			0
$887$			4.96777e10i			0.0802541i	0.999195	+	0.0401270i	$0.0127763\pi$
$887$			4.96777e10i			0.0802541i	−0.999195	+	0.0401270i	$0.987224\pi$
$888$		0			0
$889$	3.75953e11			0.601902
$890$		0			0
$891$	−3.03001e11	−	6.82016e11i	−0.480765	−	1.08214i
$892$		0			0
$893$		−	1.16313e12i		−	1.82903i
$894$		0			0
$895$	1.35188e12			2.10692
$896$		0			0
$897$		−	2.54180e11i		−	0.392620i
$898$		0			0
$899$			6.35634e11i			0.973124i
$900$		0			0
$901$		−	3.15316e10i		−	0.0478461i
$902$		0			0
$903$	2.11696e12			3.18391
$904$		0			0
$905$	5.43738e11			0.810580
$906$		0			0
$907$	2.72646e11			0.402875			0.201438	−	0.979501i	$-0.435439\pi$
$907$	2.72646e11			0.402875			0.201438	+	0.979501i	$0.435439\pi$
$908$		0			0
$909$		−	5.86557e11i		−	0.859122i
$910$		0			0
$911$	−3.22636e11			−0.468424			−0.234212	−	0.972186i	$-0.575251\pi$
$911$	−3.22636e11			−0.468424			−0.234212	+	0.972186i	$0.575251\pi$
$912$		0			0
$913$	5.00046e11	−	2.22157e11i	0.719659	−	0.319725i
$914$		0			0
$915$		−	2.26156e12i		−	3.22644i
$916$		0			0
$917$	3.32900e11			0.470800
$918$		0			0
$919$			8.15014e11i			1.14262i	0.820733	+	0.571311i	$0.193565\pi$
$919$			8.15014e11i			1.14262i	−0.820733	+	0.571311i	$0.806435\pi$
$920$		0			0
$921$		−	1.58919e12i		−	2.20871i
$922$		0			0
$923$		−	6.50373e11i		−	0.896099i
$924$		0			0
$925$	1.38593e12			1.89310
$926$		0			0
$927$	−7.69902e11			−1.04260
$928$		0			0
$929$	−1.16871e12			−1.56908			−0.784541	−	0.620077i	$-0.787101\pi$
$929$	−1.16871e12			−1.56908			−0.784541	+	0.620077i	$0.787101\pi$
$930$		0			0
$931$			9.92644e11i			1.32128i
$932$		0			0
$933$	2.50960e11			0.331190
$934$		0			0
$935$	−4.34306e10	+	1.92950e10i	−0.0568263	+	0.0252464i
$936$		0			0
$937$		−	6.97255e11i		−	0.904552i	−0.891878	−	0.452276i	$-0.850612\pi$
$937$		−	6.97255e11i		−	0.904552i	0.891878	−	0.452276i	$-0.149388\pi$
$938$		0			0
$939$	−4.27511e11			−0.549901
$940$		0			0
$941$			1.00973e12i			1.28779i	0.765113	+	0.643896i	$0.222683\pi$
$941$			1.00973e12i			1.28779i	−0.765113	+	0.643896i	$0.777317\pi$
$942$		0			0
$943$		−	2.46444e11i		−	0.311653i
$944$		0			0
$945$		−	6.13593e11i		−	0.769402i
$946$		0			0
$947$	6.74858e11			0.839097			0.419549	−	0.907733i	$-0.362188\pi$
$947$	6.74858e11			0.839097			0.419549	+	0.907733i	$0.362188\pi$
$948$		0			0
$949$	2.80808e11			0.346214
$950$		0			0
$951$	−7.22101e11			−0.882827
$952$		0			0
$953$		−	4.22219e11i		−	0.511878i	−0.966693	−	0.255939i	$-0.917615\pi$
$953$		−	4.22219e11i		−	0.511878i	0.966693	−	0.255939i	$-0.0823846\pi$
$954$		0			0
$955$	1.35792e12			1.63253
$956$		0			0
$957$	−1.15464e12	+	5.12975e11i	−1.37657	+	0.611573i
$958$		0			0
$959$			1.37845e11i			0.162973i
$960$		0			0
$961$	−2.27593e11			−0.266849
$962$		0			0
$963$		−	7.87214e10i		−	0.0915351i
$964$		0			0
$965$			5.34958e11i			0.616894i
$966$		0			0
$967$			1.16711e12i			1.33477i	0.744712	+	0.667386i	$0.232587\pi$
$967$			1.16711e12i			1.33477i	−0.744712	+	0.667386i	$0.767413\pi$
$968$		0			0
$969$	5.88496e10			0.0667495
$970$		0			0
$971$	3.25447e11			0.366103			0.183051	−	0.983103i	$-0.441403\pi$
$971$	3.25447e11			0.366103			0.183051	+	0.983103i	$0.441403\pi$
$972$		0			0
$973$	−1.16689e12			−1.30190
$974$		0			0
$975$			1.46949e12i			1.62611i
$976$		0			0
$977$	−2.82408e11			−0.309955			−0.154978	−	0.987918i	$-0.549531\pi$
$977$	−2.82408e11			−0.309955			−0.154978	+	0.987918i	$0.549531\pi$
$978$		0			0
$979$	−2.93828e11	−	6.61369e11i	−0.319862	−	0.719967i
$980$		0			0
$981$			8.09923e11i			0.874516i
$982$		0			0
$983$	1.04658e12			1.12088			0.560439	−	0.828196i	$-0.310633\pi$
$983$	1.04658e12			1.12088			0.560439	+	0.828196i	$0.310633\pi$
$984$		0			0
$985$		−	5.46827e11i		−	0.580905i
$986$		0			0
$987$		−	2.32974e12i		−	2.45492i
$988$		0			0
$989$		−	7.57131e11i		−	0.791381i
$990$		0			0
$991$	4.32249e11			0.448167			0.224083	−	0.974570i	$-0.428061\pi$
$991$	4.32249e11			0.448167			0.224083	+	0.974570i	$0.428061\pi$
$992$		0			0
$993$	1.80274e11			0.185411
$994$		0			0
$995$	−2.77447e12			−2.83066
$996$		0			0
$997$			6.50471e11i			0.658336i	0.944271	+	0.329168i	$0.106768\pi$
$997$			6.50471e11i			0.658336i	−0.944271	+	0.329168i	$0.893232\pi$
$998$		0			0
$999$	3.18649e11			0.319927

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	176.9.h.e.65.4		8
4.3	odd	2		22.9.b.a.21.3	✓	8
11.10	odd	2	inner	176.9.h.e.65.3		8
12.11	even	2		198.9.d.a.109.8		8
44.43	even	2		22.9.b.a.21.7	yes	8
132.131	odd	2		198.9.d.a.109.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.9.b.a.21.3	✓	8	4.3	odd	2
22.9.b.a.21.7	yes	8	44.43	even	2
176.9.h.e.65.3		8	11.10	odd	2	inner
176.9.h.e.65.4		8	1.1	even	1	trivial
198.9.d.a.109.4		8	132.131	odd	2
198.9.d.a.109.8		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:	\( N \)	\(=\)	\( 176 = 2^{4} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	176.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-107.356 q^{3} -1065.90 q^{5} +3358.46i q^{7} +4964.40 q^{9} +O(q^{10})\)	\(q-107.356 q^{3} -1065.90 q^{5} +3358.46i q^{7} +4964.40 q^{9} +(5944.35 + 13380.0i) q^{11} -18360.6i q^{13} +114431. q^{15} -3045.27i q^{17} -180007. i q^{19} -360553. i q^{21} -128952. q^{23} +745509. q^{25} +171406. q^{27} -803829. i q^{29} -790758. q^{31} +(-638164. - 1.43643e6i) q^{33} -3.57977e6i q^{35} +1.85903e6 q^{37} +1.97112e6i q^{39} +1.91113e6i q^{41} +5.87142e6i q^{43} -5.29153e6 q^{45} +6.46157e6 q^{47} -5.51448e6 q^{49} +326929. i q^{51} +1.03543e7 q^{53} +(-6.33606e6 - 1.42617e7i) q^{55} +1.93249e7i q^{57} +2.46965e6 q^{59} -1.97635e7i q^{61} +1.66727e7i q^{63} +1.95705e7i q^{65} -1.24402e7 q^{67} +1.38438e7 q^{69} +3.54223e7 q^{71} +1.52941e7i q^{73} -8.00352e7 q^{75} +(-4.49361e7 + 1.99639e7i) q^{77} +3.11795e7i q^{79} -5.09729e7 q^{81} -3.73727e7i q^{83} +3.24594e6i q^{85} +8.62962e7i q^{87} -4.94298e7 q^{89} +6.16633e7 q^{91} +8.48929e7 q^{93} +1.91869e8i q^{95} +4.09642e7 q^{97} +(2.95101e7 + 6.64235e7i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 182 q^{3} - 1410 q^{5} + 44582 q^{9}+O(q^{10}) \) 8 * q - 182 * q^3 - 1410 * q^5 + 44582 * q^9	\( 8 q - 182 q^{3} - 1410 q^{5} + 44582 q^{9} - 5808 q^{11} - 87958 q^{15} + 802026 q^{23} + 999558 q^{25} + 1561354 q^{27} - 196726 q^{31} - 4286722 q^{33} + 8627998 q^{37} + 1146988 q^{45} + 14335392 q^{47} - 6714712 q^{49} + 55946352 q^{53} + 10078442 q^{55} - 21793110 q^{59} + 113809034 q^{67} - 171636914 q^{69} - 16741974 q^{71} - 346496844 q^{75} - 137074080 q^{77} + 85282724 q^{81} + 42055422 q^{89} + 146801952 q^{91} + 253251118 q^{93} + 100034782 q^{97} + 333541978 q^{99}+O(q^{100}) \) 8 * q - 182 * q^3 - 1410 * q^5 + 44582 * q^9 - 5808 * q^11 - 87958 * q^15 + 802026 * q^23 + 999558 * q^25 + 1561354 * q^27 - 196726 * q^31 - 4286722 * q^33 + 8627998 * q^37 + 1146988 * q^45 + 14335392 * q^47 - 6714712 * q^49 + 55946352 * q^53 + 10078442 * q^55 - 21793110 * q^59 + 113809034 * q^67 - 171636914 * q^69 - 16741974 * q^71 - 346496844 * q^75 - 137074080 * q^77 + 85282724 * q^81 + 42055422 * q^89 + 146801952 * q^91 + 253251118 * q^93 + 100034782 * q^97 + 333541978 * q^99

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