LMFDB - Embedded newform 297.3.c.b.109.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [297,3,Mod(109,297)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(297, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("297.109");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 297 = 3^{3} \cdot 11 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	297.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$8.09266385150$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 24x^{14} + 420x^{12} + 1908x^{10} + 20196x^{8} - 91800x^{6} + 597348x^{4} - 4428432x^{2} + 8714304 $ x^16 + 24x^14 + 420x^12 + 1908x^10 + 20196x^8 - 91800x^6 + 597348x^4 - 4428432*x^2 + 8714304
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			109.14
Root			$-1.73205 + 2.58810i$ of defining polynomial
Character	$\chi$	$=$	297.109
Dual form			297.3.c.b.109.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.58810i q^{2} -2.69827 q^{4} +2.26558 q^{5} -2.77010i q^{7} +3.36900i q^{8} +O(q^{10})$	\(q+2.58810i q^{2} -2.69827 q^{4} +2.26558 q^{5} -2.77010i q^{7} +3.36900i q^{8} +5.86356i q^{10} +(8.52496 + 6.95162i) q^{11} +4.77781i q^{13} +7.16930 q^{14} -19.5124 q^{16} +15.6020i q^{17} +17.6274i q^{19} -6.11316 q^{20} +(-17.9915 + 22.0635i) q^{22} +28.6517 q^{23} -19.8671 q^{25} -12.3655 q^{26} +7.47449i q^{28} +6.13893i q^{29} -6.74010 q^{31} -37.0241i q^{32} -40.3795 q^{34} -6.27589i q^{35} +9.37576 q^{37} -45.6215 q^{38} +7.63275i q^{40} +4.10503i q^{41} -41.0702i q^{43} +(-23.0027 - 18.7574i) q^{44} +74.1535i q^{46} -62.5137 q^{47} +41.3266 q^{49} -51.4182i q^{50} -12.8918i q^{52} +17.3897 q^{53} +(19.3140 + 15.7495i) q^{55} +9.33246 q^{56} -15.8882 q^{58} +29.4623 q^{59} -90.5313i q^{61} -17.4441i q^{62} +17.7726 q^{64} +10.8245i q^{65} +36.1196 q^{67} -42.0984i q^{68} +16.2426 q^{70} -21.7376 q^{71} -65.0074i q^{73} +24.2654i q^{74} -47.5635i q^{76} +(19.2567 - 23.6150i) q^{77} +67.7956i q^{79} -44.2070 q^{80} -10.6242 q^{82} +100.991i q^{83} +35.3476i q^{85} +106.294 q^{86} +(-23.4200 + 28.7206i) q^{88} +146.161 q^{89} +13.2350 q^{91} -77.3101 q^{92} -161.792i q^{94} +39.9363i q^{95} +153.935 q^{97} +106.957i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 32 q^{4}+O(q^{10}) $ 16 * q - 32 * q^4	$ 16 q - 32 q^{4} + 160 q^{16} + 102 q^{22} + 12 q^{25} + 68 q^{31} + 156 q^{34} - 124 q^{37} - 272 q^{49} + 182 q^{55} + 492 q^{58} - 680 q^{64} - 400 q^{67} + 324 q^{70} - 444 q^{82} - 510 q^{88} + 120 q^{91} + 152 q^{97}+O(q^{100}) $ 16 * q - 32 * q^4 + 160 * q^16 + 102 * q^22 + 12 * q^25 + 68 * q^31 + 156 * q^34 - 124 * q^37 - 272 * q^49 + 182 * q^55 + 492 * q^58 - 680 * q^64 - 400 * q^67 + 324 * q^70 - 444 * q^82 - 510 * q^88 + 120 * q^91 + 152 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/297\mathbb{Z}\right)^\times$.

$n$	$56$	$244$
$\chi(n)$	$1$	$-1$

Coefficient data

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

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

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

$2$			2.58810i			1.29405i	0.762468	+	0.647026i	$0.223987\pi$
$2$			2.58810i			1.29405i	−0.762468	+	0.647026i	$0.776013\pi$
$3$		0			0
$3$		0			0
$4$	−2.69827			−0.674569
$5$	2.26558			0.453117			0.226558	−	0.973998i	$-0.427253\pi$
$5$	2.26558			0.453117			0.226558	+	0.973998i	$0.427253\pi$
$6$		0			0
$7$		−	2.77010i		−	0.395728i	−0.980229	−	0.197864i	$-0.936599\pi$
$7$		−	2.77010i		−	0.395728i	0.980229	−	0.197864i	$-0.0634005\pi$
$8$			3.36900i			0.421125i
$9$		0			0
$10$			5.86356i			0.586356i
$11$	8.52496	+	6.95162i	0.774997	+	0.631965i
$11$	8.52496	+	6.95162i	0.774997	+	0.631965i
$12$		0			0
$13$			4.77781i			0.367524i	0.982971	+	0.183762i	$0.0588275\pi$
$13$			4.77781i			0.367524i	−0.982971	+	0.183762i	$0.941173\pi$
$14$	7.16930			0.512093
$15$		0			0
$16$	−19.5124			−1.21953
$17$			15.6020i			0.917764i	0.888497	+	0.458882i	$0.151750\pi$
$17$			15.6020i			0.917764i	−0.888497	+	0.458882i	$0.848250\pi$
$18$		0			0
$19$			17.6274i			0.927757i	0.885899	+	0.463879i	$0.153543\pi$
$19$			17.6274i			0.927757i	−0.885899	+	0.463879i	$0.846457\pi$
$20$	−6.11316			−0.305658
$21$		0			0
$22$	−17.9915	+	22.0635i	−0.817795	+	1.00289i
$23$	28.6517			1.24573			0.622863	−	0.782331i	$-0.285970\pi$
$23$	28.6517			1.24573			0.622863	+	0.782331i	$0.285970\pi$
$24$		0			0
$25$	−19.8671			−0.794685
$26$	−12.3655			−0.475594
$27$		0			0
$28$			7.47449i			0.266946i
$29$			6.13893i			0.211687i	0.994383	+	0.105844i	$0.0337543\pi$
$29$			6.13893i			0.211687i	−0.994383	+	0.105844i	$0.966246\pi$
$30$		0			0
$31$	−6.74010			−0.217423			−0.108711	−	0.994073i	$-0.534672\pi$
$31$	−6.74010			−0.217423			−0.108711	+	0.994073i	$0.534672\pi$
$32$		−	37.0241i		−	1.15700i
$33$		0			0
$34$	−40.3795			−1.18763
$35$		−	6.27589i		−	0.179311i
$36$		0			0
$37$	9.37576			0.253399			0.126700	−	0.991941i	$-0.459562\pi$
$37$	9.37576			0.253399			0.126700	+	0.991941i	$0.459562\pi$
$38$	−45.6215			−1.20057
$39$		0			0
$40$			7.63275i			0.190819i
$41$			4.10503i			0.100123i	0.998746	+	0.0500613i	$0.0159417\pi$
$41$			4.10503i			0.100123i	−0.998746	+	0.0500613i	$0.984058\pi$
$42$		0			0
$43$		−	41.0702i		−	0.955121i	−0.878599	−	0.477561i	$-0.841521\pi$
$43$		−	41.0702i		−	0.955121i	0.878599	−	0.477561i	$-0.158479\pi$
$44$	−23.0027	−	18.7574i	−0.522788	−	0.426304i
$45$		0			0
$46$			74.1535i			1.61203i
$47$	−62.5137			−1.33008			−0.665039	−	0.746808i	$-0.731585\pi$
$47$	−62.5137			−1.33008			−0.665039	+	0.746808i	$0.731585\pi$
$48$		0			0
$49$	41.3266			0.843399
$50$		−	51.4182i		−	1.02836i
$51$		0			0
$52$		−	12.8918i		−	0.247920i
$53$	17.3897			0.328107			0.164054	−	0.986451i	$-0.447543\pi$
$53$	17.3897			0.328107			0.164054	+	0.986451i	$0.447543\pi$
$54$		0			0
$55$	19.3140	+	15.7495i	0.351164	+	0.286354i
$56$	9.33246			0.166651
$57$		0			0
$58$	−15.8882			−0.273934
$59$	29.4623			0.499362			0.249681	−	0.968328i	$-0.419674\pi$
$59$	29.4623			0.499362			0.249681	+	0.968328i	$0.419674\pi$
$60$		0			0
$61$		−	90.5313i		−	1.48412i	−0.670334	−	0.742060i	$-0.733849\pi$
$61$		−	90.5313i		−	1.48412i	0.670334	−	0.742060i	$-0.266151\pi$
$62$		−	17.4441i		−	0.281356i
$63$		0			0
$64$	17.7726			0.277696
$65$			10.8245i			0.166531i
$66$		0			0
$67$	36.1196			0.539099			0.269550	−	0.962987i	$-0.413125\pi$
$67$	36.1196			0.539099			0.269550	+	0.962987i	$0.413125\pi$
$68$		−	42.0984i		−	0.619095i
$69$		0			0
$70$	16.2426			0.232038
$71$	−21.7376			−0.306164			−0.153082	−	0.988214i	$-0.548920\pi$
$71$	−21.7376			−0.306164			−0.153082	+	0.988214i	$0.548920\pi$
$72$		0			0
$73$		−	65.0074i		−	0.890512i	−0.895403	−	0.445256i	$-0.853113\pi$
$73$		−	65.0074i		−	0.890512i	0.895403	−	0.445256i	$-0.146887\pi$
$74$			24.2654i			0.327911i
$75$		0			0
$76$		−	47.5635i		−	0.625836i
$77$	19.2567	−	23.6150i	0.250087	−	0.306688i
$78$		0			0
$79$			67.7956i			0.858172i	0.903264	+	0.429086i	$0.141164\pi$
$79$			67.7956i			0.858172i	−0.903264	+	0.429086i	$0.858836\pi$
$80$	−44.2070			−0.552587
$81$		0			0
$82$	−10.6242			−0.129564
$83$			100.991i			1.21675i	0.793648	+	0.608377i	$0.208179\pi$
$83$			100.991i			1.21675i	−0.793648	+	0.608377i	$0.791821\pi$
$84$		0			0
$85$			35.3476i			0.415854i
$86$	106.294			1.23598
$87$		0			0
$88$	−23.4200	+	28.7206i	−0.266136	+	0.326370i
$89$	146.161			1.64225			0.821127	−	0.570746i	$-0.193346\pi$
$89$	146.161			1.64225			0.821127	+	0.570746i	$0.193346\pi$
$90$		0			0
$91$	13.2350			0.145440
$92$	−77.3101			−0.840327
$93$		0			0
$94$		−	161.792i		−	1.72119i
$95$			39.9363i			0.420382i
$96$		0			0
$97$	153.935			1.58696			0.793481	−	0.608595i	$-0.208267\pi$
$97$	153.935			1.58696			0.793481	+	0.608595i	$0.208267\pi$
$98$			106.957i			1.09140i
$99$		0			0
$100$	53.6070			0.536070
$101$			93.9088i			0.929790i	0.885366	+	0.464895i	$0.153908\pi$
$101$			93.9088i			0.929790i	−0.885366	+	0.464895i	$0.846092\pi$
$102$		0			0
$103$	33.1519			0.321863			0.160931	−	0.986966i	$-0.448550\pi$
$103$	33.1519			0.321863			0.160931	+	0.986966i	$0.448550\pi$
$104$	−16.0964			−0.154773
$105$		0			0
$106$			45.0063i			0.424588i
$107$		−	35.2805i		−	0.329724i	−0.986317	−	0.164862i	$-0.947282\pi$
$107$		−	35.2805i		−	0.329724i	0.986317	−	0.164862i	$-0.0527179\pi$
$108$		0			0
$109$		−	203.649i		−	1.86834i	−0.356834	−	0.934168i	$-0.616144\pi$
$109$		−	203.649i		−	1.86834i	0.356834	−	0.934168i	$-0.383856\pi$
$110$	−40.7612	+	49.9866i	−0.370557	+	0.454424i
$111$		0			0
$112$			54.0513i			0.482601i
$113$	22.9552			0.203143			0.101572	−	0.994828i	$-0.467613\pi$
$113$	22.9552			0.203143			0.101572	+	0.994828i	$0.467613\pi$
$114$		0			0
$115$	64.9127			0.564459
$116$		−	16.5645i		−	0.142798i
$117$		0			0
$118$			76.2516i			0.646200i
$119$	43.2191			0.363185
$120$		0			0
$121$	24.3500	+	118.525i	0.201240	+	0.979542i
$122$	234.304			1.92053
$123$		0			0
$124$	18.1866			0.146666
$125$	−101.650			−0.813202
$126$		0			0
$127$		−	232.445i		−	1.83028i	−0.403142	−	0.915138i	$-0.632082\pi$
$127$		−	232.445i		−	1.83028i	0.403142	−	0.915138i	$-0.367918\pi$
$128$		−	102.099i		−	0.797650i
$129$		0			0
$130$	−28.0150			−0.215500
$131$		−	0.969092i		−	0.00739765i	−0.999993	−	0.00369883i	$-0.998823\pi$
$131$		−	0.969092i		−	0.00739765i	0.999993	−	0.00369883i	$-0.00117738\pi$
$132$		0			0
$133$	48.8296			0.367140
$134$			93.4814i			0.697622i
$135$		0			0
$136$	−52.5631			−0.386493
$137$	−139.204			−1.01609			−0.508044	−	0.861331i	$-0.669631\pi$
$137$	−139.204			−1.01609			−0.508044	+	0.861331i	$0.669631\pi$
$138$		0			0
$139$		−	123.856i		−	0.891051i	−0.895269	−	0.445525i	$-0.853017\pi$
$139$		−	123.856i		−	0.891051i	0.895269	−	0.445525i	$-0.146983\pi$
$140$			16.9341i			0.120958i
$141$		0			0
$142$		−	56.2592i		−	0.396191i
$143$	−33.2135	+	40.7306i	−0.232262	+	0.284830i
$144$		0			0
$145$			13.9083i			0.0959190i
$146$	168.246			1.15237
$147$		0			0
$148$	−25.2984			−0.170935
$149$		−	155.857i		−	1.04602i	−0.852327	−	0.523009i	$-0.824809\pi$
$149$		−	155.857i		−	1.04602i	0.852327	−	0.523009i	$-0.175191\pi$
$150$		0			0
$151$			183.313i			1.21399i	0.794704	+	0.606997i	$0.207626\pi$
$151$			183.313i			1.21399i	−0.794704	+	0.606997i	$0.792374\pi$
$152$	−59.3867			−0.390702
$153$		0			0
$154$	61.1180	+	49.8382i	0.396870	+	0.323625i
$155$	−15.2703			−0.0985178
$156$		0			0
$157$	155.404			0.989837			0.494918	−	0.868939i	$-0.335198\pi$
$157$	155.404			0.989837			0.494918	+	0.868939i	$0.335198\pi$
$158$	−175.462			−1.11052
$159$		0			0
$160$		−	83.8812i		−	0.524258i
$161$		−	79.3680i		−	0.492969i
$162$		0			0
$163$	−293.730			−1.80203			−0.901013	−	0.433791i	$-0.857176\pi$
$163$	−293.730			−1.80203			−0.901013	+	0.433791i	$0.857176\pi$
$164$		−	11.0765i		−	0.0675396i
$165$		0			0
$166$	−261.374			−1.57454
$167$		−	212.738i		−	1.27388i	−0.770914	−	0.636940i	$-0.780200\pi$
$167$		−	212.738i		−	1.27388i	0.770914	−	0.636940i	$-0.219800\pi$
$168$		0			0
$169$	146.173			0.864926
$170$	−91.4832			−0.538137
$171$		0			0
$172$			110.819i			0.644295i
$173$		−	217.448i		−	1.25693i	−0.777839	−	0.628464i	$-0.783684\pi$
$173$		−	217.448i		−	1.25693i	0.777839	−	0.628464i	$-0.216316\pi$
$174$		0			0
$175$			55.0339i			0.314480i
$176$	−166.343	−	135.643i	−0.945128	−	0.770698i
$177$		0			0
$178$			378.279i			2.12516i
$179$	−15.6152			−0.0872356			−0.0436178	−	0.999048i	$-0.513888\pi$
$179$	−15.6152			−0.0872356			−0.0436178	+	0.999048i	$0.513888\pi$
$180$		0			0
$181$	−197.378			−1.09048			−0.545242	−	0.838278i	$-0.683562\pi$
$181$	−197.378			−1.09048			−0.545242	+	0.838278i	$0.683562\pi$
$182$			34.2535i			0.188206i
$183$		0			0
$184$			96.5275i			0.524606i
$185$	21.2416			0.114819
$186$		0			0
$187$	−108.459	+	133.006i	−0.579995	+	0.711264i
$188$	168.679			0.897229
$189$		0			0
$190$	−103.359			−0.543996
$191$	−249.438			−1.30596			−0.652978	−	0.757377i	$-0.726481\pi$
$191$	−249.438			−1.30596			−0.652978	+	0.757377i	$0.726481\pi$
$192$		0			0
$193$		−	159.688i		−	0.827398i	−0.910414	−	0.413699i	$-0.864237\pi$
$193$		−	159.688i		−	0.827398i	0.910414	−	0.413699i	$-0.135763\pi$
$194$			398.400i			2.05361i
$195$		0			0
$196$	−111.510			−0.568930
$197$		−	221.284i		−	1.12327i	−0.827385	−	0.561635i	$-0.810172\pi$
$197$		−	221.284i		−	1.12327i	0.827385	−	0.561635i	$-0.189828\pi$
$198$		0			0
$199$	145.598			0.731646			0.365823	−	0.930684i	$-0.380787\pi$
$199$	145.598			0.731646			0.365823	+	0.930684i	$0.380787\pi$
$200$		−	66.9324i		−	0.334662i
$201$		0			0
$202$	−243.046			−1.20320
$203$	17.0054			0.0837706
$204$		0			0
$205$			9.30028i			0.0453672i
$206$			85.8004i			0.416507i
$207$		0			0
$208$		−	93.2265i		−	0.448204i
$209$	−122.539	+	150.273i	−0.586310	+	0.719009i
$210$		0			0
$211$			197.252i			0.934841i	0.884035	+	0.467421i	$0.154817\pi$
$211$			197.252i			0.934841i	−0.884035	+	0.467421i	$0.845183\pi$
$212$	−46.9222			−0.221331
$213$		0			0
$214$	91.3096			0.426680
$215$		−	93.0479i		−	0.432781i
$216$		0			0
$217$			18.6707i			0.0860403i
$218$	527.063			2.41772
$219$		0			0
$220$	−52.1145	−	42.4964i	−0.236884	−	0.193165i
$221$	−74.5433			−0.337300
$222$		0			0
$223$	−315.239			−1.41363			−0.706813	−	0.707400i	$-0.749868\pi$
$223$	−315.239			−1.41363			−0.706813	+	0.707400i	$0.749868\pi$
$224$	−102.560			−0.457859
$225$		0			0
$226$			59.4104i			0.262878i
$227$		−	200.524i		−	0.883364i	−0.897172	−	0.441682i	$-0.854382\pi$
$227$		−	200.524i		−	0.883364i	0.897172	−	0.441682i	$-0.145618\pi$
$228$		0			0
$229$	118.287			0.516538			0.258269	−	0.966073i	$-0.416848\pi$
$229$	118.287			0.516538			0.258269	+	0.966073i	$0.416848\pi$
$230$			168.001i			0.730438i
$231$		0			0
$232$	−20.6821			−0.0891468
$233$			55.0349i			0.236201i	0.993002	+	0.118101i	$0.0376806\pi$
$233$			55.0349i			0.236201i	−0.993002	+	0.118101i	$0.962319\pi$
$234$		0			0
$235$	−141.630			−0.602681
$236$	−79.4975			−0.336854
$237$		0			0
$238$			111.855i			0.469980i
$239$			416.587i			1.74304i	0.490359	+	0.871520i	$0.336866\pi$
$239$			416.587i			1.74304i	−0.490359	+	0.871520i	$0.663134\pi$
$240$		0			0
$241$		−	100.066i		−	0.415210i	−0.978213	−	0.207605i	$-0.933433\pi$
$241$		−	100.066i		−	0.415210i	0.978213	−	0.207605i	$-0.0665668\pi$
$242$	−306.754	+	63.0203i	−1.26758	+	0.260414i
$243$		0			0
$244$			244.278i			1.00114i
$245$	93.6287			0.382158
$246$		0			0
$247$	−84.2203			−0.340973
$248$		−	22.7074i		−	0.0915621i
$249$		0			0
$250$		−	263.081i		−	1.05232i
$251$	415.369			1.65485			0.827427	−	0.561573i	$-0.189804\pi$
$251$	415.369			1.65485			0.827427	+	0.561573i	$0.189804\pi$
$252$		0			0
$253$	244.255	+	199.176i	0.965433	+	0.787255i
$254$	601.591			2.36847
$255$		0			0
$256$	335.334			1.30990
$257$	−349.553			−1.36013			−0.680064	−	0.733152i	$-0.738048\pi$
$257$	−349.553			−1.36013			−0.680064	+	0.733152i	$0.738048\pi$
$258$		0			0
$259$		−	25.9718i		−	0.100277i
$260$		−	29.2075i		−	0.112337i
$261$		0			0
$262$	2.50811			0.00957294
$263$		−	61.8274i		−	0.235085i	−0.993068	−	0.117543i	$-0.962498\pi$
$263$		−	61.8274i		−	0.235085i	0.993068	−	0.117543i	$-0.0375017\pi$
$264$		0			0
$265$	39.3978			0.148671
$266$			126.376i			0.475098i
$267$		0			0
$268$	−97.4607			−0.363659
$269$	404.745			1.50463			0.752314	−	0.658805i	$-0.228938\pi$
$269$	404.745			1.50463			0.752314	+	0.658805i	$0.228938\pi$
$270$		0			0
$271$		−	36.4953i		−	0.134669i	−0.997730	−	0.0673346i	$-0.978551\pi$
$271$		−	36.4953i		−	0.134669i	0.997730	−	0.0673346i	$-0.0214495\pi$
$272$		−	304.432i		−	1.11924i
$273$		0			0
$274$		−	360.274i		−	1.31487i
$275$	−169.367	−	138.109i	−0.615879	−	0.502214i
$276$		0			0
$277$		−	373.711i		−	1.34914i	−0.738212	−	0.674569i	$-0.764329\pi$
$277$		−	373.711i		−	1.34914i	0.738212	−	0.674569i	$-0.235671\pi$
$278$	320.552			1.15307
$279$		0			0
$280$	21.1435			0.0755124
$281$			467.385i			1.66329i	0.555307	+	0.831645i	$0.312601\pi$
$281$			467.385i			1.66329i	−0.555307	+	0.831645i	$0.687399\pi$
$282$		0			0
$283$			250.833i			0.886334i	0.896439	+	0.443167i	$0.146145\pi$
$283$			250.833i			0.886334i	−0.896439	+	0.443167i	$0.853855\pi$
$284$	58.6541			0.206528
$285$		0			0
$286$	−105.415	−	85.9599i	−0.368584	−	0.300559i
$287$	11.3713			0.0396214
$288$		0			0
$289$	45.5779			0.157709
$290$	−35.9960			−0.124124
$291$		0			0
$292$			175.408i			0.600711i
$293$			480.875i			1.64121i	0.571494	+	0.820606i	$0.306364\pi$
$293$			480.875i			1.64121i	−0.571494	+	0.820606i	$0.693636\pi$
$294$		0			0
$295$	66.7494			0.226269
$296$			31.5870i			0.106713i
$297$		0			0
$298$	403.373			1.35360
$299$			136.892i			0.457833i
$300$		0			0
$301$	−113.769			−0.377969
$302$	−474.433			−1.57097
$303$		0			0
$304$		−	343.953i		−	1.13142i
$305$		−	205.106i		−	0.672479i
$306$		0			0
$307$			524.467i			1.70836i	0.519977	+	0.854180i	$0.325940\pi$
$307$			524.467i			1.70836i	−0.519977	+	0.854180i	$0.674060\pi$
$308$	−51.9598	+	63.7197i	−0.168701	+	0.206882i
$309$		0			0
$310$		−	39.5210i		−	0.127487i
$311$	171.208			0.550509			0.275255	−	0.961371i	$-0.411238\pi$
$311$	171.208			0.550509			0.275255	+	0.961371i	$0.411238\pi$
$312$		0			0
$313$	−18.1732			−0.0580612			−0.0290306	−	0.999579i	$-0.509242\pi$
$313$	−18.1732			−0.0580612			−0.0290306	+	0.999579i	$0.509242\pi$
$314$			402.202i			1.28090i
$315$		0			0
$316$		−	182.931i		−	0.578896i
$317$	−35.9037			−0.113261			−0.0566304	−	0.998395i	$-0.518036\pi$
$317$	−35.9037			−0.113261			−0.0566304	+	0.998395i	$0.518036\pi$
$318$		0			0
$319$	−42.6755	+	52.3341i	−0.133779	+	0.164057i
$320$	40.2652			0.125829
$321$		0			0
$322$	205.412			0.637927
$323$	−275.022			−0.851462
$324$		0			0
$325$		−	94.9213i		−	0.292066i
$326$		−	760.204i		−	2.33191i
$327$		0			0
$328$	−13.8298			−0.0421642
$329$			173.169i			0.526350i
$330$		0			0
$331$	−558.496			−1.68730			−0.843650	−	0.536894i	$-0.819598\pi$
$331$	−558.496			−1.68730			−0.843650	+	0.536894i	$0.819598\pi$
$332$		−	272.500i		−	0.820785i
$333$		0			0
$334$	550.587			1.64846
$335$	81.8321			0.244275
$336$		0			0
$337$		−	192.148i		−	0.570171i	−0.958502	−	0.285086i	$-0.907978\pi$
$337$		−	192.148i		−	0.570171i	0.958502	−	0.285086i	$-0.0920221\pi$
$338$			378.310i			1.11926i
$339$		0			0
$340$		−	95.3775i		−	0.280522i
$341$	−57.4591	−	46.8546i	−0.168502	−	0.137404i
$342$		0			0
$343$		−	250.213i		−	0.729485i
$344$	138.366			0.402225
$345$		0			0
$346$	562.779			1.62653
$347$		−	614.420i		−	1.77066i	−0.464960	−	0.885332i	$-0.653931\pi$
$347$		−	614.420i		−	1.77066i	0.464960	−	0.885332i	$-0.346069\pi$
$348$		0			0
$349$			78.1287i			0.223865i	0.993716	+	0.111932i	$0.0357040\pi$
$349$			78.1287i			0.223865i	−0.993716	+	0.111932i	$0.964296\pi$
$350$	−142.433			−0.406953
$351$		0			0
$352$	257.378	−	315.629i	0.731186	−	0.896674i
$353$	119.346			0.338091			0.169046	−	0.985608i	$-0.445931\pi$
$353$	119.346			0.338091			0.169046	+	0.985608i	$0.445931\pi$
$354$		0			0
$355$	−49.2484			−0.138728
$356$	−394.381			−1.10781
$357$		0			0
$358$		−	40.4137i		−	0.112887i
$359$		−	105.084i		−	0.292713i	−0.989232	−	0.146357i	$-0.953245\pi$
$359$		−	105.084i		−	0.292713i	0.989232	−	0.146357i	$-0.0467547\pi$
$360$		0			0
$361$	50.2751			0.139266
$362$		−	510.834i		−	1.41114i
$363$		0			0
$364$	−35.7116			−0.0981089
$365$		−	147.280i		−	0.403506i
$366$		0			0
$367$	122.400			0.333515			0.166758	−	0.985998i	$-0.446670\pi$
$367$	122.400			0.333515			0.166758	+	0.985998i	$0.446670\pi$
$368$	−559.063			−1.51919
$369$		0			0
$370$			54.9754i			0.148582i
$371$		−	48.1712i		−	0.129841i
$372$		0			0
$373$			210.545i			0.564464i	0.959346	+	0.282232i	$0.0910748\pi$
$373$			210.545i			0.564464i	−0.959346	+	0.282232i	$0.908925\pi$
$374$	−344.234	−	280.703i	−0.920412	−	0.750543i
$375$		0			0
$376$		−	210.609i		−	0.560129i
$377$	−29.3306			−0.0778000
$378$		0			0
$379$	−139.703			−0.368611			−0.184305	−	0.982869i	$-0.559004\pi$
$379$	−139.703			−0.368611			−0.184305	+	0.982869i	$0.559004\pi$
$380$		−	107.759i		−	0.283577i
$381$		0			0
$382$		−	645.570i		−	1.68997i
$383$	118.047			0.308217			0.154108	−	0.988054i	$-0.450749\pi$
$383$	118.047			0.308217			0.154108	+	0.988054i	$0.450749\pi$
$384$		0			0
$385$	43.6276	−	53.5017i	0.113318	−	0.138965i
$386$	413.289			1.07070
$387$		0			0
$388$	−415.360			−1.07051
$389$	−321.392			−0.826200			−0.413100	−	0.910686i	$-0.635554\pi$
$389$	−321.392			−0.826200			−0.413100	+	0.910686i	$0.635554\pi$
$390$		0			0
$391$			447.023i			1.14328i
$392$			139.229i			0.355176i
$393$		0			0
$394$	572.706			1.45357
$395$			153.596i			0.388852i
$396$		0			0
$397$	373.565			0.940969			0.470485	−	0.882408i	$-0.344079\pi$
$397$	373.565			0.940969			0.470485	+	0.882408i	$0.344079\pi$
$398$			376.822i			0.946788i
$399$		0			0
$400$	387.656			0.969139
$401$	33.2906			0.0830190			0.0415095	−	0.999138i	$-0.486783\pi$
$401$	33.2906			0.0830190			0.0415095	+	0.999138i	$0.486783\pi$
$402$		0			0
$403$		−	32.2029i		−	0.0799079i
$404$		−	253.392i		−	0.627207i
$405$		0			0
$406$			44.0118i			0.108404i
$407$	79.9281	+	65.1767i	0.196383	+	0.160139i
$408$		0			0
$409$		−	634.765i		−	1.55199i	−0.630737	−	0.775997i	$-0.717247\pi$
$409$		−	634.765i		−	1.55199i	0.630737	−	0.775997i	$-0.282753\pi$
$410$	−24.0701			−0.0587075
$411$		0			0
$412$	−89.4528			−0.217118
$413$		−	81.6136i		−	0.197612i
$414$		0			0
$415$			228.803i			0.551332i
$416$	176.894			0.425226
$417$		0			0
$418$	−388.922	−	317.143i	−0.930434	−	0.758716i
$419$	619.928			1.47954			0.739771	−	0.672858i	$-0.234934\pi$
$419$	619.928			1.47954			0.739771	+	0.672858i	$0.234934\pi$
$420$		0			0
$421$	99.4160			0.236143			0.118071	−	0.993005i	$-0.462329\pi$
$421$	99.4160			0.236143			0.118071	+	0.993005i	$0.462329\pi$
$422$	−510.507			−1.20973
$423$		0			0
$424$			58.5859i			0.138174i
$425$		−	309.967i		−	0.729334i
$426$		0			0
$427$	−250.781			−0.587308
$428$			95.1965i			0.222422i
$429$		0			0
$430$	240.818			0.560041
$431$			48.0676i			0.111526i	0.998444	+	0.0557629i	$0.0177591\pi$
$431$			48.0676i			0.111526i	−0.998444	+	0.0557629i	$0.982241\pi$
$432$		0			0
$433$	−47.7826			−0.110352			−0.0551762	−	0.998477i	$-0.517572\pi$
$433$	−47.7826			−0.110352			−0.0551762	+	0.998477i	$0.517572\pi$
$434$	−48.3218			−0.111341
$435$		0			0
$436$			549.500i			1.26032i
$437$			505.054i			1.15573i
$438$		0			0
$439$			307.563i			0.700599i	0.936638	+	0.350299i	$0.113920\pi$
$439$			307.563i			0.700599i	−0.936638	+	0.350299i	$0.886080\pi$
$440$	−53.0599	+	65.0689i	−0.120591	+	0.147884i
$441$		0			0
$442$		−	192.926i		−	0.436483i
$443$	−162.321			−0.366413			−0.183207	−	0.983074i	$-0.558648\pi$
$443$	−162.321			−0.366413			−0.183207	+	0.983074i	$0.558648\pi$
$444$		0			0
$445$	331.139			0.744132
$446$		−	815.870i		−	1.82930i
$447$		0			0
$448$		−	49.2318i		−	0.109892i
$449$	315.559			0.702805			0.351402	−	0.936224i	$-0.385705\pi$
$449$	315.559			0.702805			0.351402	+	0.936224i	$0.385705\pi$
$450$		0			0
$451$	−28.5366	+	34.9952i	−0.0632740	+	0.0775947i
$452$	−61.9394			−0.137034
$453$		0			0
$454$	518.976			1.14312
$455$	29.9850			0.0659010
$456$		0			0
$457$			707.374i			1.54787i	0.633268	+	0.773933i	$0.281713\pi$
$457$			707.374i			1.54787i	−0.633268	+	0.773933i	$0.718287\pi$
$458$			306.139i			0.668427i
$459$		0			0
$460$	−175.152			−0.380766
$461$			548.281i			1.18933i	0.803974	+	0.594665i	$0.202715\pi$
$461$			548.281i			1.18933i	−0.803974	+	0.594665i	$0.797285\pi$
$462$		0			0
$463$	−508.909			−1.09916			−0.549578	−	0.835442i	$-0.685212\pi$
$463$	−508.909			−1.09916			−0.549578	+	0.835442i	$0.685212\pi$
$464$		−	119.785i		−	0.258158i
$465$		0			0
$466$	−142.436			−0.305657
$467$	−83.0067			−0.177745			−0.0888723	−	0.996043i	$-0.528326\pi$
$467$	−83.0067			−0.177745			−0.0888723	+	0.996043i	$0.528326\pi$
$468$		0			0
$469$		−	100.055i		−	0.213337i
$470$		−	366.553i		−	0.779900i
$471$		0			0
$472$			99.2586i			0.210294i
$473$	285.504	−	350.122i	0.603603	−	0.740216i
$474$		0			0
$475$		−	350.206i		−	0.737275i
$476$	−116.617			−0.244993
$477$		0			0
$478$	−1078.17			−2.25558
$479$			924.326i			1.92970i	0.262802	+	0.964850i	$0.415353\pi$
$479$			924.326i			1.92970i	−0.262802	+	0.964850i	$0.584647\pi$
$480$		0			0
$481$			44.7956i			0.0931301i
$482$	258.980			0.537303
$483$		0			0
$484$	−65.7030	−	319.812i	−0.135750	−	0.660768i
$485$	348.753			0.719078
$486$		0			0
$487$	−842.889			−1.73078			−0.865389	−	0.501101i	$-0.832929\pi$
$487$	−842.889			−1.73078			−0.865389	+	0.501101i	$0.832929\pi$
$488$	305.000			0.625000
$489$		0			0
$490$			242.321i			0.494532i
$491$			809.414i			1.64850i	0.566226	+	0.824250i	$0.308403\pi$
$491$			809.414i			1.64850i	−0.566226	+	0.824250i	$0.691597\pi$
$492$		0			0
$493$	−95.7795			−0.194279
$494$		−	217.971i		−	0.441236i
$495$		0			0
$496$	131.516			0.265152
$497$			60.2154i			0.121158i
$498$		0			0
$499$	−616.064			−1.23460			−0.617298	−	0.786729i	$-0.711773\pi$
$499$	−616.064			−1.23460			−0.617298	+	0.786729i	$0.711773\pi$
$500$	274.280			0.548560
$501$		0			0
$502$			1075.02i			2.14147i
$503$			326.862i			0.649825i	0.945744	+	0.324912i	$0.105335\pi$
$503$			326.862i			0.649825i	−0.945744	+	0.324912i	$0.894665\pi$
$504$		0			0
$505$			212.758i			0.421303i
$506$	−515.487	+	632.156i	−1.01875	+	1.24932i
$507$		0			0
$508$			627.200i			1.23465i
$509$	−525.516			−1.03245			−0.516224	−	0.856454i	$-0.672663\pi$
$509$	−525.516			−1.03245			−0.516224	+	0.856454i	$0.672663\pi$
$510$		0			0
$511$	−180.077			−0.352401
$512$			459.481i			0.897423i
$513$		0			0
$514$		−	904.679i		−	1.76008i
$515$	75.1083			0.145841
$516$		0			0
$517$	−532.927	−	434.571i	−1.03081	−	0.840564i
$518$	67.2177			0.129764
$519$		0			0
$520$	−36.4678			−0.0701304
$521$	536.860			1.03044			0.515221	−	0.857058i	$-0.327710\pi$
$521$	536.860			1.03044			0.515221	+	0.857058i	$0.327710\pi$
$522$		0			0
$523$			955.587i			1.82713i	0.406697	+	0.913563i	$0.366681\pi$
$523$			955.587i			1.82713i	−0.406697	+	0.913563i	$0.633319\pi$
$524$			2.61488i			0.00499022i
$525$		0			0
$526$	160.016			0.304212
$527$		−	105.159i		−	0.199543i
$528$		0			0
$529$	291.919			0.551831
$530$			101.966i			0.192388i
$531$		0			0
$532$	−131.756			−0.247661
$533$	−19.6130			−0.0367974
$534$		0			0
$535$		−	79.9309i		−	0.149404i
$536$			121.687i			0.227028i
$537$		0			0
$538$			1047.52i			1.94706i
$539$	352.307	+	287.286i	0.653631	+	0.532999i
$540$		0			0
$541$		−	673.434i		−	1.24480i	−0.782701	−	0.622398i	$-0.786159\pi$
$541$		−	673.434i		−	1.24480i	0.782701	−	0.622398i	$-0.213841\pi$
$542$	94.4537			0.174269
$543$		0			0
$544$	577.650			1.06186
$545$		−	461.383i		−	0.846574i
$546$		0			0
$547$			580.591i			1.06141i	0.847557	+	0.530705i	$0.178073\pi$
$547$			580.591i			1.06141i	−0.847557	+	0.530705i	$0.821927\pi$
$548$	375.611			0.685421
$549$		0			0
$550$	357.440	−	438.338i	0.649890	−	0.796978i
$551$	−108.213			−0.196394
$552$		0			0
$553$	187.800			0.339603
$554$	967.203			1.74585
$555$		0			0
$556$			334.198i			0.601075i
$557$		−	482.932i		−	0.867023i	−0.901148	−	0.433511i	$-0.857274\pi$
$557$		−	482.932i		−	0.867023i	0.901148	−	0.433511i	$-0.142726\pi$
$558$		0			0
$559$	196.225			0.351029
$560$			122.458i			0.218674i
$561$		0			0
$562$	−1209.64			−2.15238
$563$		−	318.429i		−	0.565593i	−0.959180	−	0.282796i	$-0.908738\pi$
$563$		−	318.429i		−	0.565593i	0.959180	−	0.282796i	$-0.0912621\pi$
$564$		0			0
$565$	52.0069			0.0920476
$566$	−649.181			−1.14696
$567$		0			0
$568$		−	73.2340i		−	0.128933i
$569$		−	706.234i		−	1.24119i	−0.784133	−	0.620593i	$-0.786892\pi$
$569$		−	706.234i		−	1.24119i	0.784133	−	0.620593i	$-0.213108\pi$
$570$		0			0
$571$			721.508i			1.26359i	0.775137	+	0.631794i	$0.217681\pi$
$571$			721.508i			1.26359i	−0.775137	+	0.631794i	$0.782319\pi$
$572$	89.6191	−	109.902i	0.156677	−	0.192137i
$573$		0			0
$574$			29.4302i			0.0512721i
$575$	−569.227			−0.989960
$576$		0			0
$577$	40.7749			0.0706670			0.0353335	−	0.999376i	$-0.488751\pi$
$577$	40.7749			0.0706670			0.0353335	+	0.999376i	$0.488751\pi$
$578$			117.960i			0.204084i
$579$		0			0
$580$		−	37.5283i		−	0.0647039i
$581$	279.754			0.481504
$582$		0			0
$583$	148.247	+	120.887i	0.254282	+	0.207353i
$584$	219.010			0.375017
$585$		0			0
$586$	−1244.55			−2.12381
$587$	−612.103			−1.04276			−0.521382	−	0.853323i	$-0.674583\pi$
$587$	−612.103			−1.04276			−0.521382	+	0.853323i	$0.674583\pi$
$588$		0			0
$589$		−	118.810i		−	0.201715i
$590$			172.754i			0.292804i
$591$		0			0
$592$	−182.944			−0.309027
$593$		−	397.874i		−	0.670950i	−0.942049	−	0.335475i	$-0.891103\pi$
$593$		−	397.874i		−	0.670950i	0.942049	−	0.335475i	$-0.108897\pi$
$594$		0			0
$595$	97.9163			0.164565
$596$			420.544i			0.705611i
$597$		0			0
$598$	−354.291			−0.592460
$599$	−177.464			−0.296267			−0.148134	−	0.988967i	$-0.547327\pi$
$599$	−177.464			−0.296267			−0.148134	+	0.988967i	$0.547327\pi$
$600$		0			0
$601$		−	528.583i		−	0.879506i	−0.898119	−	0.439753i	$-0.855066\pi$
$601$		−	528.583i		−	0.879506i	0.898119	−	0.439753i	$-0.144934\pi$
$602$		−	294.445i		−	0.489111i
$603$		0			0
$604$		−	494.629i		−	0.818922i
$605$	55.1670	+	268.527i	0.0911850	+	0.443847i
$606$		0			0
$607$		−	976.318i		−	1.60843i	−0.594338	−	0.804216i	$-0.702586\pi$
$607$		−	976.318i		−	1.60843i	0.594338	−	0.804216i	$-0.297414\pi$
$608$	652.639			1.07342
$609$		0			0
$610$	530.836			0.870223
$611$		−	298.678i		−	0.488835i
$612$		0			0
$613$			300.244i			0.489794i	0.969549	+	0.244897i	$0.0787541\pi$
$613$			300.244i			0.489794i	−0.969549	+	0.244897i	$0.921246\pi$
$614$	−1357.37			−2.21071
$615$		0			0
$616$	79.5589	+	64.8757i	0.129154	+	0.105318i
$617$	170.760			0.276758			0.138379	−	0.990379i	$-0.455811\pi$
$617$	170.760			0.276758			0.138379	+	0.990379i	$0.455811\pi$
$618$		0			0
$619$	706.831			1.14189			0.570946	−	0.820988i	$-0.306577\pi$
$619$	706.831			1.14189			0.570946	+	0.820988i	$0.306577\pi$
$620$	41.2033			0.0664570
$621$		0			0
$622$			443.105i			0.712387i
$623$		−	404.879i		−	0.649886i
$624$		0			0
$625$	266.381			0.426210
$626$		−	47.0340i		−	0.0751342i
$627$		0			0
$628$	−419.324			−0.667713
$629$			146.281i			0.232561i
$630$		0			0
$631$	1029.67			1.63180			0.815901	−	0.578192i	$-0.196242\pi$
$631$	1029.67			1.63180			0.815901	+	0.578192i	$0.196242\pi$
$632$	−228.403			−0.361398
$633$		0			0
$634$		−	92.9224i		−	0.146565i
$635$		−	526.623i		−	0.829328i
$636$		0			0
$637$			197.450i			0.309969i
$638$	−135.446	−	110.449i	−0.212298	−	0.173117i
$639$		0			0
$640$		−	231.314i		−	0.361429i
$641$	−1087.18			−1.69606			−0.848031	−	0.529947i	$-0.822212\pi$
$641$	−1087.18			−1.69606			−0.848031	+	0.529947i	$0.822212\pi$
$642$		0			0
$643$	−407.038			−0.633030			−0.316515	−	0.948588i	$-0.602513\pi$
$643$	−407.038			−0.633030			−0.316515	+	0.948588i	$0.602513\pi$
$644$			214.157i			0.332541i
$645$		0			0
$646$		−	711.786i		−	1.10184i
$647$	−339.932			−0.525397			−0.262698	−	0.964878i	$-0.584612\pi$
$647$	−339.932			−0.525397			−0.262698	+	0.964878i	$0.584612\pi$
$648$		0			0
$649$	251.165	+	204.811i	0.387004	+	0.315579i
$650$	245.666			0.377948
$651$		0			0
$652$	792.565			1.21559
$653$	−1010.70			−1.54778			−0.773892	−	0.633318i	$-0.781693\pi$
$653$	−1010.70			−1.54778			−0.773892	+	0.633318i	$0.781693\pi$
$654$		0			0
$655$		−	2.19556i		−	0.00335200i
$656$		−	80.0990i		−	0.122102i
$657$		0			0
$658$	−448.179			−0.681124
$659$		−	143.154i		−	0.217229i	−0.994084	−	0.108614i	$-0.965359\pi$
$659$		−	143.154i		−	0.217229i	0.994084	−	0.108614i	$-0.0346413\pi$
$660$		0			0
$661$	211.821			0.320455			0.160227	−	0.987080i	$-0.448777\pi$
$661$	211.821			0.320455			0.160227	+	0.987080i	$0.448777\pi$
$662$		−	1445.45i		−	2.18345i
$663$		0			0
$664$	−340.238			−0.512406
$665$	110.628			0.166357
$666$		0			0
$667$			175.891i			0.263704i
$668$			574.025i			0.859319i
$669$		0			0
$670$			211.790i			0.316104i
$671$	629.339	−	771.776i	0.937912	−	1.15019i
$672$		0			0
$673$		−	781.979i		−	1.16193i	−0.813928	−	0.580965i	$-0.802675\pi$
$673$		−	781.979i		−	1.16193i	0.813928	−	0.580965i	$-0.197325\pi$
$674$	497.298			0.737831
$675$		0			0
$676$	−394.414			−0.583452
$677$			372.112i			0.549649i	0.961494	+	0.274824i	$0.0886197\pi$
$677$			372.112i			0.549649i	−0.961494	+	0.274824i	$0.911380\pi$
$678$		0			0
$679$		−	426.416i		−	0.628006i
$680$	−119.086			−0.175127
$681$		0			0
$682$	121.265	−	148.710i	0.177807	−	0.218050i
$683$	−988.869			−1.44783			−0.723915	−	0.689889i	$-0.757659\pi$
$683$	−988.869			−1.44783			−0.723915	+	0.689889i	$0.757659\pi$
$684$		0			0
$685$	−315.378			−0.460406
$686$	647.578			0.943991
$687$		0			0
$688$			801.379i			1.16479i
$689$			83.0846i			0.120587i
$690$		0			0
$691$	−1155.92			−1.67283			−0.836414	−	0.548098i	$-0.815352\pi$
$691$	−1155.92			−1.67283			−0.836414	+	0.548098i	$0.815352\pi$
$692$			586.736i			0.847884i
$693$		0			0
$694$	1590.18			2.29133
$695$		−	280.606i		−	0.403750i
$696$		0			0
$697$	−64.0466			−0.0918890
$698$	−202.205			−0.289692
$699$		0			0
$700$		−	148.497i		−	0.212138i
$701$		−	96.2667i		−	0.137328i	−0.997640	−	0.0686638i	$-0.978126\pi$
$701$		−	96.2667i		−	0.137328i	0.997640	−	0.0686638i	$-0.0218736\pi$
$702$		0			0
$703$			165.270i			0.235093i
$704$	151.511	+	123.548i	0.215214	+	0.175495i
$705$		0			0
$706$			308.880i			0.437508i
$707$	260.137			0.367944
$708$		0			0
$709$	537.434			0.758017			0.379008	−	0.925393i	$-0.376265\pi$
$709$	537.434			0.758017			0.379008	+	0.925393i	$0.376265\pi$
$710$		−	127.460i		−	0.179521i
$711$		0			0
$712$			492.415i			0.691594i
$713$	−193.115			−0.270849
$714$		0			0
$715$	−75.2479	+	92.2786i	−0.105242	+	0.129061i
$716$	42.1340			0.0588464
$717$		0			0
$718$	271.969			0.378786
$719$	−621.354			−0.864192			−0.432096	−	0.901828i	$-0.642226\pi$
$719$	−621.354			−0.864192			−0.432096	+	0.901828i	$0.642226\pi$
$720$		0			0
$721$		−	91.8339i		−	0.127370i
$722$			130.117i			0.180218i
$723$		0			0
$724$	532.579			0.735607
$725$		−	121.963i		−	0.168225i
$726$		0			0
$727$	164.513			0.226290			0.113145	−	0.993578i	$-0.463907\pi$
$727$	164.513			0.226290			0.113145	+	0.993578i	$0.463907\pi$
$728$			44.5887i			0.0612482i
$729$		0			0
$730$	381.175			0.522157
$731$	640.777			0.876576
$732$		0			0
$733$		−	368.801i		−	0.503139i	−0.967839	−	0.251569i	$-0.919053\pi$
$733$		−	368.801i		−	0.503139i	0.967839	−	0.251569i	$-0.0809467\pi$
$734$			316.784i			0.431586i
$735$		0			0
$736$		−	1060.80i		−	1.44131i
$737$	307.919	+	251.090i	0.417800	+	0.340692i
$738$		0			0
$739$		−	353.880i		−	0.478863i	−0.970913	−	0.239432i	$-0.923039\pi$
$739$		−	353.880i		−	0.478863i	0.970913	−	0.239432i	$-0.0769611\pi$
$740$	−57.3156			−0.0774535
$741$		0			0
$742$	124.672			0.168021
$743$		−	771.832i		−	1.03880i	−0.854530	−	0.519402i	$-0.826155\pi$
$743$		−	771.832i		−	1.03880i	0.854530	−	0.519402i	$-0.173845\pi$
$744$		0			0
$745$		−	353.106i		−	0.473968i
$746$	−544.912			−0.730445
$747$		0			0
$748$	292.652	−	358.888i	0.391246	−	0.479796i
$749$	−97.7305			−0.130481
$750$		0			0
$751$	1091.78			1.45377			0.726885	−	0.686759i	$-0.240967\pi$
$751$	1091.78			1.45377			0.726885	+	0.686759i	$0.240967\pi$
$752$	1219.79			1.62207
$753$		0			0
$754$		−	75.9106i		−	0.100677i
$755$			415.311i			0.550081i
$756$		0			0
$757$	874.040			1.15461			0.577305	−	0.816529i	$-0.304104\pi$
$757$	874.040			1.15461			0.577305	+	0.816529i	$0.304104\pi$
$758$		−	361.567i		−	0.477001i
$759$		0			0
$760$	−134.545			−0.177033
$761$			604.496i			0.794344i	0.917744	+	0.397172i	$0.130008\pi$
$761$			604.496i			0.794344i	−0.917744	+	0.397172i	$0.869992\pi$
$762$		0			0
$763$	−564.127			−0.739353
$764$	673.051			0.880957
$765$		0			0
$766$			305.518i			0.398848i
$767$			140.765i			0.183527i
$768$		0			0
$769$		−	27.4337i		−	0.0356745i	−0.999841	−	0.0178372i	$-0.994322\pi$
$769$		−	27.4337i		−	0.0356745i	0.999841	−	0.0178372i	$-0.00567807\pi$
$770$	138.468	+	112.913i	0.179828	+	0.146640i
$771$		0			0
$772$			430.882i			0.558137i
$773$	−12.3376			−0.0159607			−0.00798033	−	0.999968i	$-0.502540\pi$
$773$	−12.3376			−0.0159607			−0.00798033	+	0.999968i	$0.502540\pi$
$774$		0			0
$775$	133.906			0.172783
$776$			518.608i			0.668309i
$777$		0			0
$778$		−	831.795i		−	1.06914i
$779$	−72.3609			−0.0928895
$780$		0			0
$781$	−185.312	−	151.112i	−0.237276	−	0.193485i
$782$	−1156.94			−1.47947
$783$		0			0
$784$	−806.381			−1.02855
$785$	352.081			0.448511
$786$		0			0
$787$		−	773.569i		−	0.982934i	−0.870896	−	0.491467i	$-0.836461\pi$
$787$		−	773.569i		−	0.982934i	0.870896	−	0.491467i	$-0.163539\pi$
$788$			597.086i			0.757723i
$789$		0			0
$790$	−397.523			−0.503194
$791$		−	63.5881i		−	0.0803895i
$792$		0			0
$793$	432.541			0.545449
$794$			966.824i			1.21766i
$795$		0			0
$796$	−392.862			−0.493546
$797$	1300.61			1.63189			0.815943	−	0.578132i	$-0.196218\pi$
$797$	1300.61			1.63189			0.815943	+	0.578132i	$0.196218\pi$
$798$		0			0
$799$		−	975.338i		−	1.22070i
$800$			735.563i			0.919454i
$801$		0			0
$802$			86.1595i			0.107431i
$803$	451.906	−	554.185i	0.562773	−	0.690144i
$804$		0			0
$805$		−	179.815i		−	0.223372i
$806$	83.3444			0.103405
$807$		0			0
$808$	−316.379			−0.391558
$809$		−	601.736i		−	0.743802i	−0.928272	−	0.371901i	$-0.878706\pi$
$809$		−	601.736i		−	0.743802i	0.928272	−	0.371901i	$-0.121294\pi$
$810$		0			0
$811$			270.354i			0.333358i	0.986011	+	0.166679i	$0.0533044\pi$
$811$			270.354i			0.333358i	−0.986011	+	0.166679i	$0.946696\pi$
$812$	−45.8853			−0.0565090
$813$		0			0
$814$	−168.684	+	206.862i	−0.207229	+	0.254130i
$815$	−665.470			−0.816528
$816$		0			0
$817$	723.961			0.886121
$818$	1642.84			2.00836
$819$		0			0
$820$		−	25.0947i		−	0.0306033i
$821$		−	634.169i		−	0.772435i	−0.922408	−	0.386218i	$-0.873781\pi$
$821$		−	634.169i		−	0.772435i	0.922408	−	0.386218i	$-0.126219\pi$
$822$		0			0
$823$	986.267			1.19838			0.599190	−	0.800607i	$-0.295489\pi$
$823$	986.267			1.19838			0.599190	+	0.800607i	$0.295489\pi$
$824$			111.689i			0.135544i
$825$		0			0
$826$	211.224			0.255720
$827$			1325.17i			1.60239i	0.598406	+	0.801193i	$0.295801\pi$
$827$			1325.17i			1.60239i	−0.598406	+	0.801193i	$0.704199\pi$
$828$		0			0
$829$	994.324			1.19943			0.599713	−	0.800215i	$-0.295281\pi$
$829$	994.324			1.19943			0.599713	+	0.800215i	$0.295281\pi$
$830$	−592.165			−0.713452
$831$		0			0
$832$			84.9139i			0.102060i
$833$			644.776i			0.774041i
$834$		0			0
$835$		−	481.975i		−	0.577216i
$836$	330.643	−	405.477i	0.395507	−	0.485021i
$837$		0			0
$838$			1604.44i			1.91460i
$839$	200.336			0.238779			0.119390	−	0.992847i	$-0.461906\pi$
$839$	200.336			0.238779			0.119390	+	0.992847i	$0.461906\pi$
$840$		0			0
$841$	803.314			0.955189
$842$			257.299i			0.305581i
$843$		0			0
$844$		−	532.239i		−	0.630615i
$845$	331.166			0.391912
$846$		0			0
$847$	328.325	−	67.4519i	0.387633	−	0.0796363i
$848$	−339.315			−0.400136
$849$		0			0
$850$	802.226			0.943795
$851$	268.631			0.315666
$852$		0			0
$853$		−	649.483i		−	0.761411i	−0.924696	−	0.380705i	$-0.875681\pi$
$853$		−	649.483i		−	0.761411i	0.924696	−	0.380705i	$-0.124319\pi$
$854$		−	649.046i		−	0.760007i
$855$		0			0
$856$	118.860			0.138855
$857$			567.728i			0.662459i	0.943550	+	0.331230i	$0.107463\pi$
$857$			567.728i			0.662459i	−0.943550	+	0.331230i	$0.892537\pi$
$858$		0			0
$859$	−792.266			−0.922312			−0.461156	−	0.887319i	$-0.652565\pi$
$859$	−792.266			−0.922312			−0.461156	+	0.887319i	$0.652565\pi$
$860$			251.069i			0.291941i
$861$		0			0
$862$	−124.404			−0.144320
$863$	311.026			0.360401			0.180200	−	0.983630i	$-0.442325\pi$
$863$	311.026			0.360401			0.180200	+	0.983630i	$0.442325\pi$
$864$		0			0
$865$		−	492.648i		−	0.569535i
$866$		−	123.666i		−	0.142802i
$867$		0			0
$868$		−	50.3788i		−	0.0580401i
$869$	−471.289	+	577.955i	−0.542335	+	0.665080i
$870$		0			0
$871$			172.573i			0.198132i
$872$	686.092			0.786803
$873$		0			0
$874$	−1307.13			−1.49557
$875$			281.581i			0.321807i
$876$		0			0
$877$			1006.14i			1.14725i	0.819117	+	0.573627i	$0.194464\pi$
$877$			1006.14i			1.14725i	−0.819117	+	0.573627i	$0.805536\pi$
$878$	−796.004			−0.906611
$879$		0			0
$880$	−376.863	−	307.310i	−0.428253	−	0.349216i
$881$	−529.675			−0.601221			−0.300610	−	0.953747i	$-0.597190\pi$
$881$	−529.675			−0.601221			−0.300610	+	0.953747i	$0.597190\pi$
$882$		0			0
$883$	−365.789			−0.414257			−0.207128	−	0.978314i	$-0.566412\pi$
$883$	−365.789			−0.414257			−0.207128	+	0.978314i	$0.566412\pi$
$884$	201.138			0.227532
$885$		0			0
$886$		−	420.103i		−	0.474157i
$887$		−	335.515i		−	0.378259i	−0.981952	−	0.189129i	$-0.939433\pi$
$887$		−	335.515i		−	0.378259i	0.981952	−	0.189129i	$-0.0605665\pi$
$888$		0			0
$889$	−643.895			−0.724292
$890$			857.021i			0.962945i
$891$		0			0
$892$	850.600			0.953588
$893$		−	1101.95i		−	1.23399i
$894$		0			0
$895$	−35.3775			−0.0395279
$896$	−282.825			−0.315653
$897$		0			0
$898$			816.700i			0.909466i
$899$		−	41.3770i		−	0.0460256i
$900$		0			0
$901$			271.314i			0.301125i
$902$	−90.5712	−	73.8556i	−0.100412	−	0.0818799i
$903$		0			0
$904$			77.3360i			0.0855487i
$905$	−447.176			−0.494117
$906$		0			0
$907$	81.1929			0.0895181			0.0447590	−	0.998998i	$-0.485748\pi$
$907$	81.1929			0.0895181			0.0447590	+	0.998998i	$0.485748\pi$
$908$			541.068i			0.595890i
$909$		0			0
$910$			77.6042i			0.0852793i
$911$	997.277			1.09471			0.547353	−	0.836902i	$-0.315636\pi$
$911$	997.277			1.09471			0.547353	+	0.836902i	$0.315636\pi$
$912$		0			0
$913$	−702.048	+	860.942i	−0.768947	+	0.942981i
$914$	−1830.76			−2.00302
$915$		0			0
$916$	−319.171			−0.348440
$917$	−2.68448			−0.00292746
$918$		0			0
$919$		−	493.638i		−	0.537147i	−0.963259	−	0.268573i	$-0.913448\pi$
$919$		−	493.638i		−	0.537147i	0.963259	−	0.268573i	$-0.0865522\pi$
$920$			218.691i			0.237708i
$921$		0			0
$922$	−1419.01			−1.53905
$923$		−	103.858i		−	0.112522i
$924$		0			0
$925$	−186.270			−0.201373
$926$		−	1317.11i		−	1.42236i
$927$		0			0
$928$	227.288			0.244923
$929$	1166.86			1.25604			0.628020	−	0.778197i	$-0.283866\pi$
$929$	1166.86			1.25604			0.628020	+	0.778197i	$0.283866\pi$
$930$		0			0
$931$			728.479i			0.782470i
$932$		−	148.499i		−	0.159334i
$933$		0			0
$934$		−	214.830i		−	0.230011i
$935$	−245.723	+	301.337i	−0.262805	+	0.322286i
$936$		0			0
$937$			651.648i			0.695462i	0.937594	+	0.347731i	$0.113048\pi$
$937$			651.648i			0.695462i	−0.937594	+	0.347731i	$0.886952\pi$
$938$	258.953			0.276069
$939$		0			0
$940$	382.156			0.406549
$941$			1254.08i			1.33271i	0.745633	+	0.666357i	$0.232147\pi$
$941$			1254.08i			1.33271i	−0.745633	+	0.666357i	$0.767853\pi$
$942$		0			0
$943$			117.616i			0.124725i
$944$	−574.881			−0.608985
$945$		0			0
$946$	906.152	+	738.915i	0.957877	+	0.781094i
$947$	−313.733			−0.331291			−0.165646	−	0.986185i	$-0.552971\pi$
$947$	−313.733			−0.331291			−0.165646	+	0.986185i	$0.552971\pi$
$948$		0			0
$949$	310.593			0.327284
$950$	906.368			0.954072
$951$		0			0
$952$			145.605i			0.152946i
$953$		−	686.321i		−	0.720169i	−0.932920	−	0.360085i	$-0.882748\pi$
$953$		−	686.321i		−	0.720169i	0.932920	−	0.360085i	$-0.117252\pi$
$954$		0			0
$955$	−565.122			−0.591750
$956$		−	1124.07i		−	1.17580i
$957$		0			0
$958$	−2392.25			−2.49713
$959$			385.609i			0.402095i
$960$		0			0
$961$	−915.571			−0.952727
$962$	−115.936			−0.120515
$963$		0			0
$964$			270.004i			0.280087i
$965$		−	361.786i		−	0.374908i
$966$		0			0
$967$		−	1514.43i		−	1.56611i	−0.621954	−	0.783054i	$-0.713661\pi$
$967$		−	1514.43i		−	1.56611i	0.621954	−	0.783054i	$-0.286339\pi$
$968$	−399.309	+	82.0352i	−0.412510	+	0.0847471i
$969$		0			0
$970$			902.609i			0.930524i
$971$	−560.702			−0.577448			−0.288724	−	0.957412i	$-0.593231\pi$
$971$	−560.702			−0.577448			−0.288724	+	0.957412i	$0.593231\pi$
$972$		0			0
$973$	−343.093			−0.352614
$974$		−	2181.48i		−	2.23972i
$975$		0			0
$976$			1766.48i			1.80992i
$977$	−1134.55			−1.16126			−0.580631	−	0.814167i	$-0.697194\pi$
$977$	−1134.55			−1.16126			−0.580631	+	0.814167i	$0.697194\pi$
$978$		0			0
$979$	1246.01	+	1016.05i	1.27274	+	1.03785i
$980$	−252.636			−0.257792
$981$		0			0
$982$	−2094.85			−2.13324
$983$	−55.6429			−0.0566052			−0.0283026	−	0.999599i	$-0.509010\pi$
$983$	−55.6429			−0.0566052			−0.0283026	+	0.999599i	$0.509010\pi$
$984$		0			0
$985$		−	501.338i		−	0.508972i
$986$		−	247.887i		−	0.251407i
$987$		0			0
$988$	227.249			0.230009
$989$		−	1176.73i		−	1.18982i
$990$		0			0
$991$	−292.719			−0.295377			−0.147689	−	0.989034i	$-0.547183\pi$
$991$	−292.719			−0.295377			−0.147689	+	0.989034i	$0.547183\pi$
$992$			249.546i			0.251559i
$993$		0			0
$994$	−155.843			−0.156784
$995$	329.863			0.331521
$996$		0			0
$997$			578.641i			0.580382i	0.956969	+	0.290191i	$0.0937188\pi$
$997$			578.641i			0.580382i	−0.956969	+	0.290191i	$0.906281\pi$
$998$		−	1594.44i		−	1.59763i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	297.3.c.b.109.14	yes	16
3.2	odd	2	inner	297.3.c.b.109.3	✓	16
11.10	odd	2	inner	297.3.c.b.109.4	yes	16
33.32	even	2	inner	297.3.c.b.109.13	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
297.3.c.b.109.3	✓	16	3.2	odd	2	inner
297.3.c.b.109.4	yes	16	11.10	odd	2	inner
297.3.c.b.109.13	yes	16	33.32	even	2	inner
297.3.c.b.109.14	yes	16	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 297 = 3^{3} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	297.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.58810i q^{2} -2.69827 q^{4} +2.26558 q^{5} -2.77010i q^{7} +3.36900i q^{8} +O(q^{10})\)	\(q+2.58810i q^{2} -2.69827 q^{4} +2.26558 q^{5} -2.77010i q^{7} +3.36900i q^{8} +5.86356i q^{10} +(8.52496 + 6.95162i) q^{11} +4.77781i q^{13} +7.16930 q^{14} -19.5124 q^{16} +15.6020i q^{17} +17.6274i q^{19} -6.11316 q^{20} +(-17.9915 + 22.0635i) q^{22} +28.6517 q^{23} -19.8671 q^{25} -12.3655 q^{26} +7.47449i q^{28} +6.13893i q^{29} -6.74010 q^{31} -37.0241i q^{32} -40.3795 q^{34} -6.27589i q^{35} +9.37576 q^{37} -45.6215 q^{38} +7.63275i q^{40} +4.10503i q^{41} -41.0702i q^{43} +(-23.0027 - 18.7574i) q^{44} +74.1535i q^{46} -62.5137 q^{47} +41.3266 q^{49} -51.4182i q^{50} -12.8918i q^{52} +17.3897 q^{53} +(19.3140 + 15.7495i) q^{55} +9.33246 q^{56} -15.8882 q^{58} +29.4623 q^{59} -90.5313i q^{61} -17.4441i q^{62} +17.7726 q^{64} +10.8245i q^{65} +36.1196 q^{67} -42.0984i q^{68} +16.2426 q^{70} -21.7376 q^{71} -65.0074i q^{73} +24.2654i q^{74} -47.5635i q^{76} +(19.2567 - 23.6150i) q^{77} +67.7956i q^{79} -44.2070 q^{80} -10.6242 q^{82} +100.991i q^{83} +35.3476i q^{85} +106.294 q^{86} +(-23.4200 + 28.7206i) q^{88} +146.161 q^{89} +13.2350 q^{91} -77.3101 q^{92} -161.792i q^{94} +39.9363i q^{95} +153.935 q^{97} +106.957i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 32 q^{4}+O(q^{10}) \) 16 * q - 32 * q^4	\( 16 q - 32 q^{4} + 160 q^{16} + 102 q^{22} + 12 q^{25} + 68 q^{31} + 156 q^{34} - 124 q^{37} - 272 q^{49} + 182 q^{55} + 492 q^{58} - 680 q^{64} - 400 q^{67} + 324 q^{70} - 444 q^{82} - 510 q^{88} + 120 q^{91} + 152 q^{97}+O(q^{100}) \) 16 * q - 32 * q^4 + 160 * q^16 + 102 * q^22 + 12 * q^25 + 68 * q^31 + 156 * q^34 - 124 * q^37 - 272 * q^49 + 182 * q^55 + 492 * q^58 - 680 * q^64 - 400 * q^67 + 324 * q^70 - 444 * q^82 - 510 * q^88 + 120 * q^91 + 152 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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