LMFDB - Embedded newform 99.3.c.c.10.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [99,3,Mod(10,99)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(99, 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("99.10");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 99 = 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	99.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:	$2.69755461717$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{46})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 46x^{2} + 2116 $ x^4 + 46*x^2 + 2116
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			10.4
Root			$-3.39116 + 5.87367i$ of defining polynomial
Character	$\chi$	$=$	99.10
Dual form			99.3.c.c.10.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205i q^{2} +1.00000 q^{4} +6.78233 q^{5} -11.7473i q^{7} +8.66025i q^{8} +O(q^{10})$	\(q+1.73205i q^{2} +1.00000 q^{4} +6.78233 q^{5} -11.7473i q^{7} +8.66025i q^{8} +11.7473i q^{10} +(-6.78233 + 8.66025i) q^{11} +11.7473i q^{13} +20.3470 q^{14} -11.0000 q^{16} -10.3923i q^{17} +6.78233 q^{20} +(-15.0000 - 11.7473i) q^{22} -33.9116 q^{23} +21.0000 q^{25} -20.3470 q^{26} -11.7473i q^{28} -34.6410i q^{29} -10.0000 q^{31} +15.5885i q^{32} +18.0000 q^{34} -79.6743i q^{35} +50.0000 q^{37} +58.7367i q^{40} +34.6410i q^{41} -46.9894i q^{43} +(-6.78233 + 8.66025i) q^{44} -58.7367i q^{46} -33.9116 q^{47} -89.0000 q^{49} +36.3731i q^{50} +11.7473i q^{52} -33.9116 q^{53} +(-46.0000 + 58.7367i) q^{55} +101.735 q^{56} +60.0000 q^{58} +67.8233 q^{59} +58.7367i q^{61} -17.3205i q^{62} -71.0000 q^{64} +79.6743i q^{65} -10.0000 q^{67} -10.3923i q^{68} +138.000 q^{70} -33.9116 q^{71} -70.4840i q^{73} +86.6025i q^{74} +(101.735 + 79.6743i) q^{77} +58.7367i q^{79} -74.6056 q^{80} -60.0000 q^{82} -76.2102i q^{83} -70.4840i q^{85} +81.3880 q^{86} +(-75.0000 - 58.7367i) q^{88} -13.5647 q^{89} +138.000 q^{91} -33.9116 q^{92} -58.7367i q^{94} -40.0000 q^{97} -154.153i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{4}+O(q^{10}) $ 4 * q + 4 * q^4	$ 4 q + 4 q^{4} - 44 q^{16} - 60 q^{22} + 84 q^{25} - 40 q^{31} + 72 q^{34} + 200 q^{37} - 356 q^{49} - 184 q^{55} + 240 q^{58} - 284 q^{64} - 40 q^{67} + 552 q^{70} - 240 q^{82} - 300 q^{88} + 552 q^{91} - 160 q^{97}+O(q^{100}) $ 4 * q + 4 * q^4 - 44 * q^16 - 60 * q^22 + 84 * q^25 - 40 * q^31 + 72 * q^34 + 200 * q^37 - 356 * q^49 - 184 * q^55 + 240 * q^58 - 284 * q^64 - 40 * q^67 + 552 * q^70 - 240 * q^82 - 300 * q^88 + 552 * q^91 - 160 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/99\mathbb{Z}\right)^\times$.

$n$	$46$	$56$
$\chi(n)$	$-1$	$1$

Coefficient data

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

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

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

$2$			1.73205i			0.866025i	0.901388	+	0.433013i	$0.142549\pi$
$2$			1.73205i			0.866025i	−0.901388	+	0.433013i	$0.857451\pi$
$3$		0			0
$3$		0			0
$4$	1.00000			0.250000
$5$	6.78233			1.35647			0.678233	−	0.734847i	$-0.262746\pi$
$5$	6.78233			1.35647			0.678233	+	0.734847i	$0.262746\pi$
$6$		0			0
$7$		−	11.7473i		−	1.67819i	−0.543984	−	0.839096i	$-0.683085\pi$
$7$		−	11.7473i		−	1.67819i	0.543984	−	0.839096i	$-0.316915\pi$
$8$			8.66025i			1.08253i
$9$		0			0
$10$			11.7473i			1.17473i
$11$	−6.78233	+	8.66025i	−0.616575	+	0.787296i
$11$	−6.78233	+	8.66025i	−0.616575	+	0.787296i
$12$		0			0
$13$			11.7473i			0.903642i	0.892109	+	0.451821i	$0.149225\pi$
$13$			11.7473i			0.903642i	−0.892109	+	0.451821i	$0.850775\pi$
$14$	20.3470			1.45336
$15$		0			0
$16$	−11.0000			−0.687500
$17$		−	10.3923i		−	0.611312i	−0.952142	−	0.305656i	$-0.901124\pi$
$17$		−	10.3923i		−	0.611312i	0.952142	−	0.305656i	$-0.0988758\pi$
$18$		0			0
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$	6.78233			0.339116
$21$		0			0
$22$	−15.0000	−	11.7473i	−0.681818	−	0.533970i
$23$	−33.9116			−1.47442			−0.737210	−	0.675664i	$-0.763857\pi$
$23$	−33.9116			−1.47442			−0.737210	+	0.675664i	$0.763857\pi$
$24$		0			0
$25$	21.0000			0.840000
$26$	−20.3470			−0.782577
$27$		0			0
$28$		−	11.7473i		−	0.419548i
$29$		−	34.6410i		−	1.19452i	−0.802049	−	0.597259i	$-0.796256\pi$
$29$		−	34.6410i		−	1.19452i	0.802049	−	0.597259i	$-0.203744\pi$
$30$		0			0
$31$	−10.0000			−0.322581			−0.161290	−	0.986907i	$-0.551566\pi$
$31$	−10.0000			−0.322581			−0.161290	+	0.986907i	$0.551566\pi$
$32$			15.5885i			0.487139i
$33$		0			0
$34$	18.0000			0.529412
$35$		−	79.6743i		−	2.27641i
$36$		0			0
$37$	50.0000			1.35135			0.675676	−	0.737199i	$-0.263852\pi$
$37$	50.0000			1.35135			0.675676	+	0.737199i	$0.263852\pi$
$38$		0			0
$39$		0			0
$40$			58.7367i			1.46842i
$41$			34.6410i			0.844903i	0.906386	+	0.422451i	$0.138830\pi$
$41$			34.6410i			0.844903i	−0.906386	+	0.422451i	$0.861170\pi$
$42$		0			0
$43$		−	46.9894i		−	1.09278i	−0.837532	−	0.546388i	$-0.816002\pi$
$43$		−	46.9894i		−	1.09278i	0.837532	−	0.546388i	$-0.183998\pi$
$44$	−6.78233	+	8.66025i	−0.154144	+	0.196824i
$45$		0			0
$46$		−	58.7367i		−	1.27688i
$47$	−33.9116			−0.721524			−0.360762	−	0.932658i	$-0.617483\pi$
$47$	−33.9116			−0.721524			−0.360762	+	0.932658i	$0.617483\pi$
$48$		0			0
$49$	−89.0000			−1.81633
$50$			36.3731i			0.727461i
$51$		0			0
$52$			11.7473i			0.225910i
$53$	−33.9116			−0.639842			−0.319921	−	0.947444i	$-0.603656\pi$
$53$	−33.9116			−0.639842			−0.319921	+	0.947444i	$0.603656\pi$
$54$		0			0
$55$	−46.0000	+	58.7367i	−0.836364	+	1.06794i
$56$	101.735			1.81670
$57$		0			0
$58$	60.0000			1.03448
$59$	67.8233			1.14955			0.574774	−	0.818312i	$-0.305090\pi$
$59$	67.8233			1.14955			0.574774	+	0.818312i	$0.305090\pi$
$60$		0			0
$61$			58.7367i			0.962897i	0.876474	+	0.481448i	$0.159889\pi$
$61$			58.7367i			0.962897i	−0.876474	+	0.481448i	$0.840111\pi$
$62$		−	17.3205i		−	0.279363i
$63$		0			0
$64$	−71.0000			−1.10938
$65$			79.6743i			1.22576i
$66$		0			0
$67$	−10.0000			−0.149254			−0.0746269	−	0.997212i	$-0.523777\pi$
$67$	−10.0000			−0.149254			−0.0746269	+	0.997212i	$0.523777\pi$
$68$		−	10.3923i		−	0.152828i
$69$		0			0
$70$	138.000			1.97143
$71$	−33.9116			−0.477629			−0.238814	−	0.971065i	$-0.576759\pi$
$71$	−33.9116			−0.477629			−0.238814	+	0.971065i	$0.576759\pi$
$72$		0			0
$73$		−	70.4840i		−	0.965535i	−0.875749	−	0.482767i	$-0.839632\pi$
$73$		−	70.4840i		−	0.965535i	0.875749	−	0.482767i	$-0.160368\pi$
$74$			86.6025i			1.17030i
$75$		0			0
$76$		0			0
$77$	101.735	+	79.6743i	1.32123	+	1.03473i
$78$		0			0
$79$			58.7367i			0.743503i	0.928332	+	0.371751i	$0.121243\pi$
$79$			58.7367i			0.743503i	−0.928332	+	0.371751i	$0.878757\pi$
$80$	−74.6056			−0.932570
$81$		0			0
$82$	−60.0000			−0.731707
$83$		−	76.2102i		−	0.918196i	−0.888386	−	0.459098i	$-0.848173\pi$
$83$		−	76.2102i		−	0.918196i	0.888386	−	0.459098i	$-0.151827\pi$
$84$		0			0
$85$		−	70.4840i		−	0.829224i
$86$	81.3880			0.946372
$87$		0			0
$88$	−75.0000	−	58.7367i	−0.852273	−	0.667463i
$89$	−13.5647			−0.152412			−0.0762060	−	0.997092i	$-0.524281\pi$
$89$	−13.5647			−0.152412			−0.0762060	+	0.997092i	$0.524281\pi$
$90$		0			0
$91$	138.000			1.51648
$92$	−33.9116			−0.368605
$93$		0			0
$94$		−	58.7367i		−	0.624859i
$95$		0			0
$96$		0			0
$97$	−40.0000			−0.412371			−0.206186	−	0.978513i	$-0.566105\pi$
$97$	−40.0000			−0.412371			−0.206186	+	0.978513i	$0.566105\pi$
$98$		−	154.153i		−	1.57298i
$99$		0			0
$100$	21.0000			0.210000
$101$			69.2820i			0.685961i	0.939343	+	0.342980i	$0.111436\pi$
$101$			69.2820i			0.685961i	−0.939343	+	0.342980i	$0.888564\pi$
$102$		0			0
$103$	170.000			1.65049			0.825243	−	0.564778i	$-0.191038\pi$
$103$	170.000			1.65049			0.825243	+	0.564778i	$0.191038\pi$
$104$	−101.735			−0.978221
$105$		0			0
$106$		−	58.7367i		−	0.554120i
$107$		−	93.5307i		−	0.874119i	−0.899433	−	0.437060i	$-0.856020\pi$
$107$		−	93.5307i		−	0.874119i	0.899433	−	0.437060i	$-0.143980\pi$
$108$		0			0
$109$			176.210i			1.61661i	0.588766	+	0.808303i	$0.299614\pi$
$109$			176.210i			1.61661i	−0.588766	+	0.808303i	$0.700386\pi$
$110$	−101.735	−	79.6743i	−0.924863	−	0.724312i
$111$		0			0
$112$			129.221i			1.15376i
$113$	67.8233			0.600206			0.300103	−	0.953907i	$-0.402979\pi$
$113$	67.8233			0.600206			0.300103	+	0.953907i	$0.402979\pi$
$114$		0			0
$115$	−230.000			−2.00000
$116$		−	34.6410i		−	0.298629i
$117$		0			0
$118$			117.473i			0.995537i
$119$	−122.082			−1.02590
$120$		0			0
$121$	−29.0000	−	117.473i	−0.239669	−	0.970855i
$122$	−101.735			−0.833893
$123$		0			0
$124$	−10.0000			−0.0806452
$125$	−27.1293			−0.217035
$126$		0			0
$127$		−	105.726i		−	0.832489i	−0.909253	−	0.416244i	$-0.863346\pi$
$127$		−	105.726i		−	0.832489i	0.909253	−	0.416244i	$-0.136654\pi$
$128$		−	60.6218i		−	0.473608i
$129$		0			0
$130$	−138.000			−1.06154
$131$			34.6410i			0.264435i	0.991221	+	0.132218i	$0.0422098\pi$
$131$			34.6410i			0.264435i	−0.991221	+	0.132218i	$0.957790\pi$
$132$		0			0
$133$		0			0
$134$		−	17.3205i		−	0.129258i
$135$		0			0
$136$	90.0000			0.661765
$137$	271.293			1.98024			0.990121	−	0.140214i	$-0.0447792\pi$
$137$	271.293			1.98024			0.990121	+	0.140214i	$0.0447792\pi$
$138$		0			0
$139$			117.473i			0.845132i	0.906332	+	0.422566i	$0.138871\pi$
$139$			117.473i			0.845132i	−0.906332	+	0.422566i	$0.861129\pi$
$140$		−	79.6743i		−	0.569102i
$141$		0			0
$142$		−	58.7367i		−	0.413639i
$143$	−101.735	−	79.6743i	−0.711433	−	0.557163i
$144$		0			0
$145$		−	234.947i		−	1.62032i
$146$	122.082			0.836178
$147$		0			0
$148$	50.0000			0.337838
$149$			138.564i			0.929960i	0.885321	+	0.464980i	$0.153939\pi$
$149$			138.564i			0.929960i	−0.885321	+	0.464980i	$0.846061\pi$
$150$		0			0
$151$			58.7367i			0.388985i	0.980904	+	0.194492i	$0.0623060\pi$
$151$			58.7367i			0.388985i	−0.980904	+	0.194492i	$0.937694\pi$
$152$		0			0
$153$		0			0
$154$	−138.000	+	176.210i	−0.896104	+	1.14422i
$155$	−67.8233			−0.437570
$156$		0			0
$157$	170.000			1.08280			0.541401	−	0.840764i	$-0.317894\pi$
$157$	170.000			1.08280			0.541401	+	0.840764i	$0.317894\pi$
$158$	−101.735			−0.643892
$159$		0			0
$160$			105.726i			0.660788i
$161$			398.372i			2.47436i
$162$		0			0
$163$	140.000			0.858896			0.429448	−	0.903092i	$-0.358708\pi$
$163$	140.000			0.858896			0.429448	+	0.903092i	$0.358708\pi$
$164$			34.6410i			0.211226i
$165$		0			0
$166$	132.000			0.795181
$167$			200.918i			1.20310i	0.798835	+	0.601551i	$0.205450\pi$
$167$			200.918i			1.20310i	−0.798835	+	0.601551i	$0.794550\pi$
$168$		0			0
$169$	31.0000			0.183432
$170$	122.082			0.718129
$171$		0			0
$172$		−	46.9894i		−	0.273194i
$173$		−	304.841i		−	1.76209i	−0.473036	−	0.881043i	$-0.656842\pi$
$173$		−	304.841i		−	1.76209i	0.473036	−	0.881043i	$-0.343158\pi$
$174$		0			0
$175$		−	246.694i		−	1.40968i
$176$	74.6056	−	95.2628i	0.423896	−	0.541266i
$177$		0			0
$178$		−	23.4947i		−	0.131993i
$179$	149.211			0.833582			0.416791	−	0.909002i	$-0.363155\pi$
$179$	149.211			0.833582			0.416791	+	0.909002i	$0.363155\pi$
$180$		0			0
$181$	−130.000			−0.718232			−0.359116	−	0.933293i	$-0.616922\pi$
$181$	−130.000			−0.718232			−0.359116	+	0.933293i	$0.616922\pi$
$182$			239.023i			1.31331i
$183$		0			0
$184$		−	293.684i		−	1.59611i
$185$	339.116			1.83306
$186$		0			0
$187$	90.0000	+	70.4840i	0.481283	+	0.376920i
$188$	−33.9116			−0.180381
$189$		0			0
$190$		0			0
$191$	88.1703			0.461625			0.230812	−	0.972998i	$-0.425862\pi$
$191$	88.1703			0.461625			0.230812	+	0.972998i	$0.425862\pi$
$192$		0			0
$193$			187.957i			0.973873i	0.873437	+	0.486936i	$0.161886\pi$
$193$			187.957i			0.973873i	−0.873437	+	0.486936i	$0.838114\pi$
$194$		−	69.2820i		−	0.357124i
$195$		0			0
$196$	−89.0000			−0.454082
$197$			41.5692i			0.211011i	0.994419	+	0.105506i	$0.0336461\pi$
$197$			41.5692i			0.211011i	−0.994419	+	0.105506i	$0.966354\pi$
$198$		0			0
$199$	−202.000			−1.01508			−0.507538	−	0.861630i	$-0.669444\pi$
$199$	−202.000			−1.01508			−0.507538	+	0.861630i	$0.669444\pi$
$200$			181.865i			0.909327i
$201$		0			0
$202$	−120.000			−0.594059
$203$	−406.940			−2.00463
$204$		0			0
$205$			234.947i			1.14608i
$206$			294.449i			1.42936i
$207$		0			0
$208$		−	129.221i		−	0.621254i
$209$		0			0
$210$		0			0
$211$			117.473i			0.556746i	0.960473	+	0.278373i	$0.0897951\pi$
$211$			117.473i			0.556746i	−0.960473	+	0.278373i	$0.910205\pi$
$212$	−33.9116			−0.159961
$213$		0			0
$214$	162.000			0.757009
$215$		−	318.697i		−	1.48231i
$216$		0			0
$217$			117.473i			0.541352i
$218$	−305.205			−1.40002
$219$		0			0
$220$	−46.0000	+	58.7367i	−0.209091	+	0.266985i
$221$	122.082			0.552407
$222$		0			0
$223$	−250.000			−1.12108			−0.560538	−	0.828129i	$-0.689406\pi$
$223$	−250.000			−1.12108			−0.560538	+	0.828129i	$0.689406\pi$
$224$	183.123			0.817513
$225$		0			0
$226$			117.473i			0.519794i
$227$		−	45.0333i		−	0.198385i	−0.995068	−	0.0991923i	$-0.968374\pi$
$227$		−	45.0333i		−	0.198385i	0.995068	−	0.0991923i	$-0.0316259\pi$
$228$		0			0
$229$	158.000			0.689956			0.344978	−	0.938611i	$-0.387886\pi$
$229$	158.000			0.689956			0.344978	+	0.938611i	$0.387886\pi$
$230$		−	398.372i		−	1.73205i
$231$		0			0
$232$	300.000			1.29310
$233$			62.3538i			0.267613i	0.991007	+	0.133807i	$0.0427201\pi$
$233$			62.3538i			0.267613i	−0.991007	+	0.133807i	$0.957280\pi$
$234$		0			0
$235$	−230.000			−0.978723
$236$	67.8233			0.287387
$237$		0			0
$238$		−	211.452i		−	0.888454i
$239$		−	381.051i		−	1.59436i	−0.603744	−	0.797178i	$-0.706325\pi$
$239$		−	381.051i		−	1.59436i	0.603744	−	0.797178i	$-0.293675\pi$
$240$		0			0
$241$			234.947i			0.974883i	0.873156	+	0.487441i	$0.162070\pi$
$241$			234.947i			0.974883i	−0.873156	+	0.487441i	$0.837930\pi$
$242$	203.470	−	50.2295i	0.840785	−	0.207560i
$243$		0			0
$244$			58.7367i			0.240724i
$245$	−603.627			−2.46379
$246$		0			0
$247$		0			0
$248$		−	86.6025i		−	0.349204i
$249$		0			0
$250$		−	46.9894i		−	0.187957i
$251$	−135.647			−0.540425			−0.270212	−	0.962801i	$-0.587094\pi$
$251$	−135.647			−0.540425			−0.270212	+	0.962801i	$0.587094\pi$
$252$		0			0
$253$	230.000	−	293.684i	0.909091	−	1.16080i
$254$	183.123			0.720956
$255$		0			0
$256$	−179.000			−0.699219
$257$	−339.116			−1.31952			−0.659760	−	0.751477i	$-0.729342\pi$
$257$	−339.116			−1.31952			−0.659760	+	0.751477i	$0.729342\pi$
$258$		0			0
$259$		−	587.367i		−	2.26783i
$260$			79.6743i			0.306440i
$261$		0			0
$262$	−60.0000			−0.229008
$263$		−	96.9948i		−	0.368802i	−0.982851	−	0.184401i	$-0.940966\pi$
$263$		−	96.9948i		−	0.368802i	0.982851	−	0.184401i	$-0.0590345\pi$
$264$		0			0
$265$	−230.000			−0.867925
$266$		0			0
$267$		0			0
$268$	−10.0000			−0.0373134
$269$	−33.9116			−0.126066			−0.0630328	−	0.998011i	$-0.520077\pi$
$269$	−33.9116			−0.126066			−0.0630328	+	0.998011i	$0.520077\pi$
$270$		0			0
$271$		−	176.210i		−	0.650222i	−0.945676	−	0.325111i	$-0.894598\pi$
$271$		−	176.210i		−	0.650222i	0.945676	−	0.325111i	$-0.105402\pi$
$272$			114.315i			0.420277i
$273$		0			0
$274$			469.894i			1.71494i
$275$	−142.429	+	181.865i	−0.517923	+	0.661328i
$276$		0			0
$277$		−	11.7473i		−	0.0424092i	−0.999775	−	0.0212046i	$-0.993250\pi$
$277$		−	11.7473i		−	0.0424092i	0.999775	−	0.0212046i	$-0.00675013\pi$
$278$	−203.470			−0.731906
$279$		0			0
$280$	690.000			2.46429
$281$		−	190.526i		−	0.678027i	−0.940782	−	0.339014i	$-0.889907\pi$
$281$		−	190.526i		−	0.678027i	0.940782	−	0.339014i	$-0.110093\pi$
$282$		0			0
$283$			399.410i			1.41134i	0.708540	+	0.705671i	$0.249354\pi$
$283$			399.410i			1.41134i	−0.708540	+	0.705671i	$0.750646\pi$
$284$	−33.9116			−0.119407
$285$		0			0
$286$	138.000	−	176.210i	0.482517	−	0.616119i
$287$	406.940			1.41791
$288$		0			0
$289$	181.000			0.626298
$290$	406.940			1.40324
$291$		0			0
$292$		−	70.4840i		−	0.241384i
$293$		−	422.620i		−	1.44239i	−0.692732	−	0.721195i	$-0.743593\pi$
$293$		−	422.620i		−	1.44239i	0.692732	−	0.721195i	$-0.256407\pi$
$294$		0			0
$295$	460.000			1.55932
$296$			433.013i			1.46288i
$297$		0			0
$298$	−240.000			−0.805369
$299$		−	398.372i		−	1.33235i
$300$		0			0
$301$	−552.000			−1.83389
$302$	−101.735			−0.336871
$303$		0			0
$304$		0			0
$305$			398.372i			1.30614i
$306$		0			0
$307$		−	70.4840i		−	0.229590i	−0.993389	−	0.114795i	$-0.963379\pi$
$307$		−	70.4840i		−	0.229590i	0.993389	−	0.114795i	$-0.0366211\pi$
$308$	101.735	+	79.6743i	0.330308	+	0.258683i
$309$		0			0
$310$		−	117.473i		−	0.378946i
$311$	373.028			1.19945			0.599724	−	0.800207i	$-0.295277\pi$
$311$	373.028			1.19945			0.599724	+	0.800207i	$0.295277\pi$
$312$		0			0
$313$	−70.0000			−0.223642			−0.111821	−	0.993728i	$-0.535668\pi$
$313$	−70.0000			−0.223642			−0.111821	+	0.993728i	$0.535668\pi$
$314$			294.449i			0.937735i
$315$		0			0
$316$			58.7367i			0.185876i
$317$	169.558			0.534884			0.267442	−	0.963574i	$-0.413822\pi$
$317$	169.558			0.534884			0.267442	+	0.963574i	$0.413822\pi$
$318$		0			0
$319$	300.000	+	234.947i	0.940439	+	0.736510i
$320$	−481.545			−1.50483
$321$		0			0
$322$	−690.000			−2.14286
$323$		0			0
$324$		0			0
$325$			246.694i			0.759059i
$326$			242.487i			0.743826i
$327$		0			0
$328$	−300.000			−0.914634
$329$			398.372i			1.21086i
$330$		0			0
$331$	−40.0000			−0.120846			−0.0604230	−	0.998173i	$-0.519245\pi$
$331$	−40.0000			−0.120846			−0.0604230	+	0.998173i	$0.519245\pi$
$332$		−	76.2102i		−	0.229549i
$333$		0			0
$334$	−348.000			−1.04192
$335$	−67.8233			−0.202458
$336$		0			0
$337$		−	164.463i		−	0.488020i	−0.969773	−	0.244010i	$-0.921537\pi$
$337$		−	164.463i		−	0.488020i	0.969773	−	0.244010i	$-0.0784630\pi$
$338$			53.6936i			0.158857i
$339$		0			0
$340$		−	70.4840i		−	0.207306i
$341$	67.8233	−	86.6025i	0.198895	−	0.253966i
$342$		0			0
$343$			469.894i			1.36995i
$344$	406.940			1.18296
$345$		0			0
$346$	528.000			1.52601
$347$			491.902i			1.41759i	0.705416	+	0.708793i	$0.250760\pi$
$347$			491.902i			1.41759i	−0.705416	+	0.708793i	$0.749240\pi$
$348$		0			0
$349$		−	646.104i		−	1.85130i	−0.378381	−	0.925650i	$-0.623519\pi$
$349$		−	646.104i		−	1.85130i	0.378381	−	0.925650i	$-0.376481\pi$
$350$	427.287			1.22082
$351$		0			0
$352$	−135.000	−	105.726i	−0.383523	−	0.300358i
$353$	−135.647			−0.384268			−0.192134	−	0.981369i	$-0.561541\pi$
$353$	−135.647			−0.384268			−0.192134	+	0.981369i	$0.561541\pi$
$354$		0			0
$355$	−230.000			−0.647887
$356$	−13.5647			−0.0381030
$357$		0			0
$358$			258.441i			0.721904i
$359$			103.923i			0.289479i	0.989470	+	0.144740i	$0.0462345\pi$
$359$			103.923i			0.289479i	−0.989470	+	0.144740i	$0.953766\pi$
$360$		0			0
$361$	361.000			1.00000
$362$		−	225.167i		−	0.622007i
$363$		0			0
$364$	138.000			0.379121
$365$		−	478.046i		−	1.30972i
$366$		0			0
$367$	−130.000			−0.354223			−0.177112	−	0.984191i	$-0.556675\pi$
$367$	−130.000			−0.354223			−0.177112	+	0.984191i	$0.556675\pi$
$368$	373.028			1.01366
$369$		0			0
$370$			587.367i			1.58748i
$371$			398.372i			1.07378i
$372$		0			0
$373$		−	481.641i		−	1.29126i	−0.763649	−	0.645631i	$-0.776594\pi$
$373$		−	481.641i		−	1.29126i	0.763649	−	0.645631i	$-0.223406\pi$
$374$	−122.082	+	155.885i	−0.326422	+	0.416804i
$375$		0			0
$376$		−	293.684i		−	0.781073i
$377$	406.940			1.07942
$378$		0			0
$379$	308.000			0.812665			0.406332	−	0.913725i	$-0.366807\pi$
$379$	308.000			0.812665			0.406332	+	0.913725i	$0.366807\pi$
$380$		0			0
$381$		0			0
$382$			152.715i			0.399779i
$383$	−237.382			−0.619795			−0.309898	−	0.950770i	$-0.600295\pi$
$383$	−237.382			−0.619795			−0.309898	+	0.950770i	$0.600295\pi$
$384$		0			0
$385$	690.000	+	540.378i	1.79221	+	1.40358i
$386$	−325.552			−0.843399
$387$		0			0
$388$	−40.0000			−0.103093
$389$	−644.321			−1.65635			−0.828177	−	0.560467i	$-0.810622\pi$
$389$	−644.321			−1.65635			−0.828177	+	0.560467i	$0.810622\pi$
$390$		0			0
$391$			352.420i			0.901330i
$392$		−	770.763i		−	1.96623i
$393$		0			0
$394$	−72.0000			−0.182741
$395$			398.372i			1.00854i
$396$		0			0
$397$	−310.000			−0.780856			−0.390428	−	0.920633i	$-0.627673\pi$
$397$	−310.000			−0.780856			−0.390428	+	0.920633i	$0.627673\pi$
$398$		−	349.874i		−	0.879081i
$399$		0			0
$400$	−231.000			−0.577500
$401$	556.151			1.38691			0.693455	−	0.720500i	$-0.256088\pi$
$401$	556.151			1.38691			0.693455	+	0.720500i	$0.256088\pi$
$402$		0			0
$403$		−	117.473i		−	0.291497i
$404$			69.2820i			0.171490i
$405$		0			0
$406$		−	704.840i		−	1.73606i
$407$	−339.116	+	433.013i	−0.833210	+	1.06391i
$408$		0			0
$409$		−	587.367i		−	1.43611i	−0.695989	−	0.718053i	$-0.745034\pi$
$409$		−	587.367i		−	1.43611i	0.695989	−	0.718053i	$-0.254966\pi$
$410$	−406.940			−0.992536
$411$		0			0
$412$	170.000			0.412621
$413$		−	796.743i		−	1.92916i
$414$		0			0
$415$		−	516.883i		−	1.24550i
$416$	−183.123			−0.440199
$417$		0			0
$418$		0			0
$419$	−54.2586			−0.129496			−0.0647478	−	0.997902i	$-0.520624\pi$
$419$	−54.2586			−0.129496			−0.0647478	+	0.997902i	$0.520624\pi$
$420$		0			0
$421$	−310.000			−0.736342			−0.368171	−	0.929758i	$-0.620016\pi$
$421$	−310.000			−0.736342			−0.368171	+	0.929758i	$0.620016\pi$
$422$	−203.470			−0.482156
$423$		0			0
$424$		−	293.684i		−	0.692650i
$425$		−	218.238i		−	0.513502i
$426$		0			0
$427$	690.000			1.61593
$428$		−	93.5307i		−	0.218530i
$429$		0			0
$430$	552.000			1.28372
$431$		−	103.923i		−	0.241121i	−0.992706	−	0.120560i	$-0.961531\pi$
$431$		−	103.923i		−	0.241121i	0.992706	−	0.120560i	$-0.0384691\pi$
$432$		0			0
$433$	−610.000			−1.40878			−0.704388	−	0.709815i	$-0.748778\pi$
$433$	−610.000			−1.40878			−0.704388	+	0.709815i	$0.748778\pi$
$434$	−203.470			−0.468825
$435$		0			0
$436$			176.210i			0.404152i
$437$		0			0
$438$		0			0
$439$			411.157i			0.936576i	0.883576	+	0.468288i	$0.155129\pi$
$439$			411.157i			0.936576i	−0.883576	+	0.468288i	$0.844871\pi$
$440$	−508.675	−	398.372i	−1.15608	−	0.905390i
$441$		0			0
$442$			211.452i			0.478398i
$443$	271.293			0.612400			0.306200	−	0.951967i	$-0.400942\pi$
$443$	271.293			0.612400			0.306200	+	0.951967i	$0.400942\pi$
$444$		0			0
$445$	−92.0000			−0.206742
$446$		−	433.013i		−	0.970880i
$447$		0			0
$448$			834.061i			1.86174i
$449$	−217.035			−0.483373			−0.241687	−	0.970354i	$-0.577701\pi$
$449$	−217.035			−0.483373			−0.241687	+	0.970354i	$0.577701\pi$
$450$		0			0
$451$	−300.000	−	234.947i	−0.665188	−	0.520946i
$452$	67.8233			0.150052
$453$		0			0
$454$	78.0000			0.171806
$455$	935.962			2.05706
$456$		0			0
$457$			164.463i			0.359875i	0.983678	+	0.179937i	$0.0575895\pi$
$457$			164.463i			0.359875i	−0.983678	+	0.179937i	$0.942410\pi$
$458$			273.664i			0.597520i
$459$		0			0
$460$	−230.000			−0.500000
$461$		−	658.179i		−	1.42772i	−0.700288	−	0.713860i	$-0.746945\pi$
$461$		−	658.179i		−	1.42772i	0.700288	−	0.713860i	$-0.253055\pi$
$462$		0			0
$463$	−670.000			−1.44708			−0.723542	−	0.690280i	$-0.757487\pi$
$463$	−670.000			−1.44708			−0.723542	+	0.690280i	$0.757487\pi$
$464$			381.051i			0.821231i
$465$		0			0
$466$	−108.000			−0.231760
$467$	−339.116			−0.726160			−0.363080	−	0.931758i	$-0.618275\pi$
$467$	−339.116			−0.726160			−0.363080	+	0.931758i	$0.618275\pi$
$468$		0			0
$469$			117.473i			0.250476i
$470$		−	398.372i		−	0.847599i
$471$		0			0
$472$			587.367i			1.24442i
$473$	406.940	+	318.697i	0.860338	+	0.673779i
$474$		0			0
$475$		0			0
$476$	−122.082			−0.256475
$477$		0			0
$478$	660.000			1.38075
$479$			484.974i			1.01247i	0.862395	+	0.506236i	$0.168964\pi$
$479$			484.974i			1.01247i	−0.862395	+	0.506236i	$0.831036\pi$
$480$		0			0
$481$			587.367i			1.22114i
$482$	−406.940			−0.844273
$483$		0			0
$484$	−29.0000	−	117.473i	−0.0599174	−	0.242714i
$485$	−271.293			−0.559367
$486$		0			0
$487$	230.000			0.472279			0.236140	−	0.971719i	$-0.424118\pi$
$487$	230.000			0.472279			0.236140	+	0.971719i	$0.424118\pi$
$488$	−508.675			−1.04237
$489$		0			0
$490$		−	1045.51i		−	2.13370i
$491$		−	17.3205i		−	0.0352760i	−0.999844	−	0.0176380i	$-0.994385\pi$
$491$		−	17.3205i		−	0.0352760i	0.999844	−	0.0176380i	$-0.00561464\pi$
$492$		0			0
$493$	−360.000			−0.730223
$494$		0			0
$495$		0			0
$496$	110.000			0.221774
$497$			398.372i			0.801553i
$498$		0			0
$499$	152.000			0.304609			0.152305	−	0.988334i	$-0.451331\pi$
$499$	152.000			0.304609			0.152305	+	0.988334i	$0.451331\pi$
$500$	−27.1293			−0.0542586
$501$		0			0
$502$		−	234.947i		−	0.468022i
$503$			769.031i			1.52889i	0.644690	+	0.764444i	$0.276986\pi$
$503$			769.031i			1.52889i	−0.644690	+	0.764444i	$0.723014\pi$
$504$		0			0
$505$			469.894i			0.930482i
$506$	508.675	+	398.372i	1.00529	+	0.787296i
$507$		0			0
$508$		−	105.726i		−	0.208122i
$509$	−644.321			−1.26586			−0.632929	−	0.774210i	$-0.718147\pi$
$509$	−644.321			−1.26586			−0.632929	+	0.774210i	$0.718147\pi$
$510$		0			0
$511$	−828.000			−1.62035
$512$		−	552.524i		−	1.07915i
$513$		0			0
$514$		−	587.367i		−	1.14274i
$515$	1153.00			2.23883
$516$		0			0
$517$	230.000	−	293.684i	0.444874	−	0.568053i
$518$	1017.35			1.96400
$519$		0			0
$520$	−690.000			−1.32692
$521$	149.211			0.286394			0.143197	−	0.989694i	$-0.454262\pi$
$521$	149.211			0.286394			0.143197	+	0.989694i	$0.454262\pi$
$522$		0			0
$523$			281.936i			0.539075i	0.962990	+	0.269537i	$0.0868708\pi$
$523$			281.936i			0.539075i	−0.962990	+	0.269537i	$0.913129\pi$
$524$			34.6410i			0.0661088i
$525$		0			0
$526$	168.000			0.319392
$527$			103.923i			0.197197i
$528$		0			0
$529$	621.000			1.17391
$530$		−	398.372i		−	0.751645i
$531$		0			0
$532$		0			0
$533$	−406.940			−0.763489
$534$		0			0
$535$		−	634.356i		−	1.18571i
$536$		−	86.6025i		−	0.161572i
$537$		0			0
$538$		−	58.7367i		−	0.109176i
$539$	603.627	−	770.763i	1.11990	−	1.42999i
$540$		0			0
$541$			881.051i			1.62856i	0.580473	+	0.814280i	$0.302868\pi$
$541$			881.051i			1.62856i	−0.580473	+	0.814280i	$0.697132\pi$
$542$	305.205			0.563109
$543$		0			0
$544$	162.000			0.297794
$545$			1195.12i			2.19287i
$546$		0			0
$547$		−	187.957i		−	0.343615i	−0.985131	−	0.171808i	$-0.945039\pi$
$547$		−	187.957i		−	0.343615i	0.985131	−	0.171808i	$-0.0549607\pi$
$548$	271.293			0.495061
$549$		0			0
$550$	−315.000	−	246.694i	−0.572727	−	0.448535i
$551$		0			0
$552$		0			0
$553$	690.000			1.24774
$554$	20.3470			0.0367274
$555$		0			0
$556$			117.473i			0.211283i
$557$			478.046i			0.858251i	0.903245	+	0.429126i	$0.141178\pi$
$557$			478.046i			0.858251i	−0.903245	+	0.429126i	$0.858822\pi$
$558$		0			0
$559$	552.000			0.987478
$560$			876.418i			1.56503i
$561$		0			0
$562$	330.000			0.587189
$563$			1084.26i			1.92587i	0.269737	+	0.962934i	$0.413063\pi$
$563$			1084.26i			1.92587i	−0.269737	+	0.962934i	$0.586937\pi$
$564$		0			0
$565$	460.000			0.814159
$566$	−691.798			−1.22226
$567$		0			0
$568$		−	293.684i		−	0.517048i
$569$		−	606.218i		−	1.06541i	−0.846301	−	0.532705i	$-0.821176\pi$
$569$		−	606.218i		−	1.06541i	0.846301	−	0.532705i	$-0.178824\pi$
$570$		0			0
$571$			117.473i			0.205733i	0.994695	+	0.102866i	$0.0328014\pi$
$571$			117.473i			0.205733i	−0.994695	+	0.102866i	$0.967199\pi$
$572$	−101.735	−	79.6743i	−0.177858	−	0.139291i
$573$		0			0
$574$			704.840i			1.22794i
$575$	−712.145			−1.23851
$576$		0			0
$577$	−400.000			−0.693241			−0.346620	−	0.938005i	$-0.612671\pi$
$577$	−400.000			−0.693241			−0.346620	+	0.938005i	$0.612671\pi$
$578$			313.501i			0.542390i
$579$		0			0
$580$		−	234.947i		−	0.405081i
$581$	−895.268			−1.54091
$582$		0			0
$583$	230.000	−	293.684i	0.394511	−	0.503745i
$584$	610.410			1.04522
$585$		0			0
$586$	732.000			1.24915
$587$	881.703			1.50205			0.751025	−	0.660274i	$-0.229560\pi$
$587$	881.703			1.50205			0.751025	+	0.660274i	$0.229560\pi$
$588$		0			0
$589$		0			0
$590$			796.743i			1.35041i
$591$		0			0
$592$	−550.000			−0.929054
$593$		−	613.146i		−	1.03397i	−0.855994	−	0.516986i	$-0.827054\pi$
$593$		−	613.146i		−	1.03397i	0.855994	−	0.516986i	$-0.172946\pi$
$594$		0			0
$595$	−828.000			−1.39160
$596$			138.564i			0.232490i
$597$		0			0
$598$	690.000			1.15385
$599$	1064.83			1.77767			0.888836	−	0.458225i	$-0.151515\pi$
$599$	1064.83			1.77767			0.888836	+	0.458225i	$0.151515\pi$
$600$		0			0
$601$		−	117.473i		−	0.195463i	−0.995213	−	0.0977316i	$-0.968841\pi$
$601$		−	117.473i		−	0.195463i	0.995213	−	0.0977316i	$-0.0311587\pi$
$602$		−	956.092i		−	1.58819i
$603$		0			0
$604$			58.7367i			0.0972462i
$605$	−196.688	−	796.743i	−0.325103	−	1.31693i
$606$		0			0
$607$		−	693.093i		−	1.14183i	−0.821008	−	0.570917i	$-0.806588\pi$
$607$		−	693.093i		−	1.14183i	0.821008	−	0.570917i	$-0.193412\pi$
$608$		0			0
$609$		0			0
$610$	−690.000			−1.13115
$611$		−	398.372i		−	0.651999i
$612$		0			0
$613$		−	105.726i		−	0.172473i	−0.996275	−	0.0862366i	$-0.972516\pi$
$613$		−	105.726i		−	0.172473i	0.996275	−	0.0862366i	$-0.0274841\pi$
$614$	122.082			0.198831
$615$		0			0
$616$	−690.000	+	881.051i	−1.12013	+	1.43028i
$617$	67.8233			0.109924			0.0549622	−	0.998488i	$-0.482496\pi$
$617$	67.8233			0.109924			0.0549622	+	0.998488i	$0.482496\pi$
$618$		0			0
$619$	−490.000			−0.791599			−0.395800	−	0.918337i	$-0.629533\pi$
$619$	−490.000			−0.791599			−0.395800	+	0.918337i	$0.629533\pi$
$620$	−67.8233			−0.109392
$621$		0			0
$622$			646.104i			1.03875i
$623$			159.349i			0.255776i
$624$		0			0
$625$	−709.000			−1.13440
$626$		−	121.244i		−	0.193680i
$627$		0			0
$628$	170.000			0.270701
$629$		−	519.615i		−	0.826097i
$630$		0			0
$631$	338.000			0.535658			0.267829	−	0.963467i	$-0.413694\pi$
$631$	338.000			0.535658			0.267829	+	0.963467i	$0.413694\pi$
$632$	−508.675			−0.804865
$633$		0			0
$634$			293.684i			0.463223i
$635$		−	717.069i		−	1.12924i
$636$		0			0
$637$		−	1045.51i		−	1.64131i
$638$	−406.940	+	519.615i	−0.637837	+	0.814444i
$639$		0			0
$640$		−	411.157i		−	0.642433i
$641$	−257.729			−0.402073			−0.201036	−	0.979584i	$-0.564431\pi$
$641$	−257.729			−0.402073			−0.201036	+	0.979584i	$0.564431\pi$
$642$		0			0
$643$	−130.000			−0.202177			−0.101089	−	0.994877i	$-0.532233\pi$
$643$	−130.000			−0.202177			−0.101089	+	0.994877i	$0.532233\pi$
$644$			398.372i			0.618590i
$645$		0			0
$646$		0			0
$647$	−1051.26			−1.62482			−0.812412	−	0.583084i	$-0.801846\pi$
$647$	−1051.26			−1.62482			−0.812412	+	0.583084i	$0.801846\pi$
$648$		0			0
$649$	−460.000	+	587.367i	−0.708783	+	0.905034i
$650$	−427.287			−0.657364
$651$		0			0
$652$	140.000			0.214724
$653$	−237.382			−0.363525			−0.181762	−	0.983342i	$-0.558180\pi$
$653$	−237.382			−0.363525			−0.181762	+	0.983342i	$0.558180\pi$
$654$		0			0
$655$			234.947i			0.358697i
$656$		−	381.051i		−	0.580871i
$657$		0			0
$658$	−690.000			−1.04863
$659$		−	398.372i		−	0.604509i	−0.953227	−	0.302255i	$-0.902261\pi$
$659$		−	398.372i		−	0.604509i	0.953227	−	0.302255i	$-0.0977393\pi$
$660$		0			0
$661$	278.000			0.420575			0.210287	−	0.977640i	$-0.432560\pi$
$661$	278.000			0.420575			0.210287	+	0.977640i	$0.432560\pi$
$662$		−	69.2820i		−	0.104656i
$663$		0			0
$664$	660.000			0.993976
$665$		0			0
$666$		0			0
$667$			1174.73i			1.76122i
$668$			200.918i			0.300775i
$669$		0			0
$670$		−	117.473i		−	0.175333i
$671$	−508.675	−	398.372i	−0.758085	−	0.593698i
$672$		0			0
$673$		−	540.378i		−	0.802939i	−0.915872	−	0.401469i	$-0.868500\pi$
$673$		−	540.378i		−	0.802939i	0.915872	−	0.401469i	$-0.131500\pi$
$674$	284.858			0.422638
$675$		0			0
$676$	31.0000			0.0458580
$677$		−	491.902i		−	0.726591i	−0.931674	−	0.363296i	$-0.881651\pi$
$677$		−	491.902i		−	0.726591i	0.931674	−	0.363296i	$-0.118349\pi$
$678$		0			0
$679$			469.894i			0.692038i
$680$	610.410			0.897661
$681$		0			0
$682$	150.000	+	117.473i	0.219941	+	0.172248i
$683$	−949.526			−1.39023			−0.695114	−	0.718899i	$-0.744646\pi$
$683$	−949.526			−1.39023			−0.695114	+	0.718899i	$0.744646\pi$
$684$		0			0
$685$	1840.00			2.68613
$686$	−813.880			−1.18641
$687$		0			0
$688$			516.883i			0.751283i
$689$		−	398.372i		−	0.578188i
$690$		0			0
$691$	482.000			0.697540			0.348770	−	0.937208i	$-0.386599\pi$
$691$	482.000			0.697540			0.348770	+	0.937208i	$0.386599\pi$
$692$		−	304.841i		−	0.440522i
$693$		0			0
$694$	−852.000			−1.22767
$695$			796.743i			1.14639i
$696$		0			0
$697$	360.000			0.516499
$698$	1119.08			1.60327
$699$		0			0
$700$		−	246.694i		−	0.352420i
$701$		−	207.846i		−	0.296499i	−0.988950	−	0.148250i	$-0.952636\pi$
$701$		−	207.846i		−	0.296499i	0.988950	−	0.148250i	$-0.0473639\pi$
$702$		0			0
$703$		0			0
$704$	481.545	−	614.878i	0.684013	−	0.873406i
$705$		0			0
$706$		−	234.947i		−	0.332786i
$707$	813.880			1.15117
$708$		0			0
$709$	−1210.00			−1.70663			−0.853315	−	0.521397i	$-0.825411\pi$
$709$	−1210.00			−1.70663			−0.853315	+	0.521397i	$0.825411\pi$
$710$		−	398.372i		−	0.561087i
$711$		0			0
$712$		−	117.473i		−	0.164991i
$713$	339.116			0.475619
$714$		0			0
$715$	−690.000	−	540.378i	−0.965035	−	0.755773i
$716$	149.211			0.208396
$717$		0			0
$718$	−180.000			−0.250696
$719$	−1132.65			−1.57531			−0.787656	−	0.616115i	$-0.788705\pi$
$719$	−1132.65			−1.57531			−0.787656	+	0.616115i	$0.788705\pi$
$720$		0			0
$721$		−	1997.05i		−	2.76983i
$722$			625.270i			0.866025i
$723$		0			0
$724$	−130.000			−0.179558
$725$		−	727.461i		−	1.00339i
$726$		0			0
$727$	1130.00			1.55433			0.777166	−	0.629295i	$-0.216656\pi$
$727$	1130.00			1.55433			0.777166	+	0.629295i	$0.216656\pi$
$728$			1195.12i			1.64164i
$729$		0			0
$730$	828.000			1.13425
$731$	−488.328			−0.668027
$732$		0			0
$733$			928.040i			1.26608i	0.774117	+	0.633042i	$0.218194\pi$
$733$			928.040i			1.26608i	−0.774117	+	0.633042i	$0.781806\pi$
$734$		−	225.167i		−	0.306766i
$735$		0			0
$736$		−	528.630i		−	0.718248i
$737$	67.8233	−	86.6025i	0.0920262	−	0.117507i
$738$		0			0
$739$		−	1057.26i		−	1.43066i	−0.698785	−	0.715332i	$-0.746276\pi$
$739$		−	1057.26i		−	1.43066i	0.698785	−	0.715332i	$-0.253724\pi$
$740$	339.116			0.458266
$741$		0			0
$742$	−690.000			−0.929919
$743$		−	131.636i		−	0.177168i	−0.996069	−	0.0885840i	$-0.971766\pi$
$743$		−	131.636i		−	0.177168i	0.996069	−	0.0885840i	$-0.0282342\pi$
$744$		0			0
$745$			939.787i			1.26146i
$746$	834.227			1.11827
$747$		0			0
$748$	90.0000	+	70.4840i	0.120321	+	0.0942300i
$749$	−1098.74			−1.46694
$750$		0			0
$751$	−298.000			−0.396804			−0.198402	−	0.980121i	$-0.563575\pi$
$751$	−298.000			−0.396804			−0.198402	+	0.980121i	$0.563575\pi$
$752$	373.028			0.496048
$753$		0			0
$754$			704.840i			0.934802i
$755$			398.372i			0.527645i
$756$		0			0
$757$	−490.000			−0.647292			−0.323646	−	0.946178i	$-0.604909\pi$
$757$	−490.000			−0.647292			−0.323646	+	0.946178i	$0.604909\pi$
$758$			533.472i			0.703788i
$759$		0			0
$760$		0			0
$761$			502.295i			0.660046i	0.943973	+	0.330023i	$0.107056\pi$
$761$			502.295i			0.660046i	−0.943973	+	0.330023i	$0.892944\pi$
$762$		0			0
$763$	2070.00			2.71298
$764$	88.1703			0.115406
$765$		0			0
$766$		−	411.157i		−	0.536758i
$767$			796.743i			1.03878i
$768$		0			0
$769$		−	234.947i		−	0.305522i	−0.988263	−	0.152761i	$-0.951183\pi$
$769$		−	234.947i		−	0.305522i	0.988263	−	0.152761i	$-0.0488166\pi$
$770$	−935.962	+	1195.12i	−1.21553	+	1.55210i
$771$		0			0
$772$			187.957i			0.243468i
$773$	1390.38			1.79868			0.899339	−	0.437253i	$-0.144048\pi$
$773$	1390.38			1.79868			0.899339	+	0.437253i	$0.144048\pi$
$774$		0			0
$775$	−210.000			−0.270968
$776$		−	346.410i		−	0.446405i
$777$		0			0
$778$		−	1116.00i		−	1.43444i
$779$		0			0
$780$		0			0
$781$	230.000	−	293.684i	0.294494	−	0.376035i
$782$	−610.410			−0.780575
$783$		0			0
$784$	979.000			1.24872
$785$	1153.00			1.46878
$786$		0			0
$787$			187.957i			0.238828i	0.992845	+	0.119414i	$0.0381015\pi$
$787$			187.957i			0.238828i	−0.992845	+	0.119414i	$0.961898\pi$
$788$			41.5692i			0.0527528i
$789$		0			0
$790$	−690.000			−0.873418
$791$		−	796.743i		−	1.00726i
$792$		0			0
$793$	−690.000			−0.870113
$794$		−	536.936i		−	0.676241i
$795$		0			0
$796$	−202.000			−0.253769
$797$	−237.382			−0.297844			−0.148922	−	0.988849i	$-0.547580\pi$
$797$	−237.382			−0.297844			−0.148922	+	0.988849i	$0.547580\pi$
$798$		0			0
$799$			352.420i			0.441077i
$800$			327.358i			0.409197i
$801$		0			0
$802$			963.282i			1.20110i
$803$	610.410	+	478.046i	0.760162	+	0.595325i
$804$		0			0
$805$			2701.89i			3.35638i
$806$	203.470			0.252444
$807$		0			0
$808$	−600.000			−0.742574
$809$			1143.15i			1.41305i	0.707691	+	0.706523i	$0.249737\pi$
$809$			1143.15i			1.41305i	−0.707691	+	0.706523i	$0.750263\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$	−406.940			−0.501157
$813$		0			0
$814$	−750.000	−	587.367i	−0.921376	−	0.721581i
$815$	949.526			1.16506
$816$		0			0
$817$		0			0
$818$	1017.35			1.24370
$819$		0			0
$820$			234.947i			0.286520i
$821$		−	969.948i		−	1.18142i	−0.806883	−	0.590712i	$-0.798847\pi$
$821$		−	969.948i		−	1.18142i	0.806883	−	0.590712i	$-0.201153\pi$
$822$		0			0
$823$	−310.000			−0.376671			−0.188335	−	0.982105i	$-0.560309\pi$
$823$	−310.000			−0.376671			−0.188335	+	0.982105i	$0.560309\pi$
$824$			1472.24i			1.78670i
$825$		0			0
$826$	1380.00			1.67070
$827$		−	893.738i		−	1.08070i	−0.841441	−	0.540350i	$-0.818292\pi$
$827$		−	893.738i		−	1.08070i	0.841441	−	0.540350i	$-0.181708\pi$
$828$		0			0
$829$	−850.000			−1.02533			−0.512666	−	0.858588i	$-0.671342\pi$
$829$	−850.000			−1.02533			−0.512666	+	0.858588i	$0.671342\pi$
$830$	895.268			1.07864
$831$		0			0
$832$		−	834.061i		−	1.00248i
$833$			924.915i			1.11034i
$834$		0			0
$835$			1362.69i			1.63197i
$836$		0			0
$837$		0			0
$838$		−	93.9787i		−	0.112146i
$839$	−440.851			−0.525449			−0.262724	−	0.964871i	$-0.584621\pi$
$839$	−440.851			−0.525449			−0.262724	+	0.964871i	$0.584621\pi$
$840$		0			0
$841$	−359.000			−0.426873
$842$		−	536.936i		−	0.637691i
$843$		0			0
$844$			117.473i			0.139186i
$845$	210.252			0.248819
$846$		0			0
$847$	−1380.00	+	340.673i	−1.62928	+	0.402211i
$848$	373.028			0.439892
$849$		0			0
$850$	378.000			0.444706
$851$	−1695.58			−1.99246
$852$		0			0
$853$		−	1069.01i		−	1.25323i	−0.779328	−	0.626617i	$-0.784439\pi$
$853$		−	1069.01i		−	1.25323i	0.779328	−	0.626617i	$-0.215561\pi$
$854$			1195.12i			1.39943i
$855$		0			0
$856$	810.000			0.946262
$857$			963.020i			1.12371i	0.827235	+	0.561855i	$0.189912\pi$
$857$			963.020i			1.12371i	−0.827235	+	0.561855i	$0.810088\pi$
$858$		0			0
$859$	−820.000			−0.954598			−0.477299	−	0.878741i	$-0.658384\pi$
$859$	−820.000			−0.954598			−0.477299	+	0.878741i	$0.658384\pi$
$860$		−	318.697i		−	0.370578i
$861$		0			0
$862$	180.000			0.208817
$863$	779.968			0.903787			0.451893	−	0.892072i	$-0.350749\pi$
$863$	779.968			0.903787			0.451893	+	0.892072i	$0.350749\pi$
$864$		0			0
$865$		−	2067.53i		−	2.39021i
$866$		−	1056.55i		−	1.22004i
$867$		0			0
$868$			117.473i			0.135338i
$869$	−508.675	−	398.372i	−0.585356	−	0.458425i
$870$		0			0
$871$		−	117.473i		−	0.134872i
$872$	−1526.02			−1.75003
$873$		0			0
$874$		0			0
$875$			318.697i			0.364226i
$876$		0			0
$877$		−	693.093i		−	0.790300i	−0.918617	−	0.395150i	$-0.870693\pi$
$877$		−	693.093i		−	0.790300i	0.918617	−	0.395150i	$-0.129307\pi$
$878$	−712.145			−0.811099
$879$		0			0
$880$	506.000	−	646.104i	0.575000	−	0.734209i
$881$	1492.11			1.69366			0.846829	−	0.531865i	$-0.178509\pi$
$881$	1492.11			1.69366			0.846829	+	0.531865i	$0.178509\pi$
$882$		0			0
$883$	230.000			0.260476			0.130238	−	0.991483i	$-0.458426\pi$
$883$	230.000			0.260476			0.130238	+	0.991483i	$0.458426\pi$
$884$	122.082			0.138102
$885$		0			0
$886$			469.894i			0.530354i
$887$		−	547.328i		−	0.617055i	−0.951215	−	0.308528i	$-0.900164\pi$
$887$		−	547.328i		−	0.617055i	0.951215	−	0.308528i	$-0.0998362\pi$
$888$		0			0
$889$	−1242.00			−1.39708
$890$		−	159.349i		−	0.179043i
$891$		0			0
$892$	−250.000			−0.280269
$893$		0			0
$894$		0			0
$895$	1012.00			1.13073
$896$	−712.145			−0.794804
$897$		0			0
$898$		−	375.915i		−	0.418613i
$899$			346.410i			0.385328i
$900$		0			0
$901$			352.420i			0.391143i
$902$	406.940	−	519.615i	0.451153	−	0.576070i
$903$		0			0
$904$			587.367i			0.649742i
$905$	−881.703			−0.974257
$906$		0			0
$907$	−430.000			−0.474090			−0.237045	−	0.971499i	$-0.576179\pi$
$907$	−430.000			−0.474090			−0.237045	+	0.971499i	$0.576179\pi$
$908$		−	45.0333i		−	0.0495962i
$909$		0			0
$910$			1621.13i			1.78146i
$911$	−155.994			−0.171233			−0.0856167	−	0.996328i	$-0.527286\pi$
$911$	−155.994			−0.171233			−0.0856167	+	0.996328i	$0.527286\pi$
$912$		0			0
$913$	660.000	+	516.883i	0.722892	+	0.566137i
$914$	−284.858			−0.311661
$915$		0			0
$916$	158.000			0.172489
$917$	406.940			0.443773
$918$		0			0
$919$			1233.47i			1.34219i	0.741372	+	0.671094i	$0.234175\pi$
$919$			1233.47i			1.34219i	−0.741372	+	0.671094i	$0.765825\pi$
$920$		−	1991.86i		−	2.16506i
$921$		0			0
$922$	1140.00			1.23644
$923$		−	398.372i		−	0.431605i
$924$		0			0
$925$	1050.00			1.13514
$926$		−	1160.47i		−	1.25321i
$927$		0			0
$928$	540.000			0.581897
$929$	−949.526			−1.02209			−0.511047	−	0.859552i	$-0.670742\pi$
$929$	−949.526			−1.02209			−0.511047	+	0.859552i	$0.670742\pi$
$930$		0			0
$931$		0			0
$932$			62.3538i			0.0669033i
$933$		0			0
$934$		−	587.367i		−	0.628873i
$935$	610.410	+	478.046i	0.652845	+	0.511279i
$936$		0			0
$937$		−	986.777i		−	1.05312i	−0.850137	−	0.526562i	$-0.823481\pi$
$937$		−	986.777i		−	1.05312i	0.850137	−	0.526562i	$-0.176519\pi$
$938$	−203.470			−0.216919
$939$		0			0
$940$	−230.000			−0.244681
$941$			34.6410i			0.0368130i	0.999831	+	0.0184065i	$0.00585930\pi$
$941$			34.6410i			0.0368130i	−0.999831	+	0.0184065i	$0.994141\pi$
$942$		0			0
$943$		−	1174.73i		−	1.24574i
$944$	−746.056			−0.790314
$945$		0			0
$946$	−552.000	+	704.840i	−0.583510	+	0.745074i
$947$	271.293			0.286476			0.143238	−	0.989688i	$-0.454248\pi$
$947$	271.293			0.286476			0.143238	+	0.989688i	$0.454248\pi$
$948$		0			0
$949$	828.000			0.872497
$950$		0			0
$951$		0			0
$952$		−	1057.26i		−	1.11057i
$953$			145.492i			0.152668i	0.997082	+	0.0763338i	$0.0243215\pi$
$953$			145.492i			0.152668i	−0.997082	+	0.0763338i	$0.975679\pi$
$954$		0			0
$955$	598.000			0.626178
$956$		−	381.051i		−	0.398589i
$957$		0			0
$958$	−840.000			−0.876827
$959$		−	3186.97i		−	3.32323i
$960$		0			0
$961$	−861.000			−0.895942
$962$	−1017.35			−1.05754
$963$		0			0
$964$			234.947i			0.243721i
$965$			1274.79i			1.32103i
$966$		0			0
$967$		−	693.093i		−	0.716746i	−0.933579	−	0.358373i	$-0.883332\pi$
$967$		−	693.093i		−	0.716746i	0.933579	−	0.358373i	$-0.116668\pi$
$968$	1017.35	−	251.147i	1.05098	−	0.259450i
$969$		0			0
$970$		−	469.894i		−	0.484426i
$971$	−868.138			−0.894066			−0.447033	−	0.894517i	$-0.647519\pi$
$971$	−868.138			−0.894066			−0.447033	+	0.894517i	$0.647519\pi$
$972$		0			0
$973$	1380.00			1.41829
$974$			398.372i			0.409006i
$975$		0			0
$976$		−	646.104i		−	0.661992i
$977$	881.703			0.902459			0.451230	−	0.892408i	$-0.350986\pi$
$977$	881.703			0.902459			0.451230	+	0.892408i	$0.350986\pi$
$978$		0			0
$979$	92.0000	−	117.473i	0.0939734	−	0.119993i
$980$	−603.627			−0.615946
$981$		0			0
$982$	30.0000			0.0305499
$983$	−33.9116			−0.0344981			−0.0172491	−	0.999851i	$-0.505491\pi$
$983$	−33.9116			−0.0344981			−0.0172491	+	0.999851i	$0.505491\pi$
$984$		0			0
$985$			281.936i			0.286230i
$986$		−	623.538i		−	0.632392i
$987$		0			0
$988$		0			0
$989$			1593.49i			1.61121i
$990$		0			0
$991$	830.000			0.837538			0.418769	−	0.908093i	$-0.362462\pi$
$991$	830.000			0.837538			0.418769	+	0.908093i	$0.362462\pi$
$992$		−	155.885i		−	0.157142i
$993$		0			0
$994$	−690.000			−0.694165
$995$	−1370.03			−1.37692
$996$		0			0
$997$			1186.48i			1.19005i	0.803707	+	0.595026i	$0.202858\pi$
$997$			1186.48i			1.19005i	−0.803707	+	0.595026i	$0.797142\pi$
$998$			263.272i			0.263799i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	99.3.c.c.10.4	yes	4
3.2	odd	2	inner	99.3.c.c.10.1	✓	4
4.3	odd	2		1584.3.j.i.1297.4		4
11.10	odd	2	inner	99.3.c.c.10.2	yes	4
12.11	even	2		1584.3.j.i.1297.2		4
33.32	even	2	inner	99.3.c.c.10.3	yes	4
44.43	even	2		1584.3.j.i.1297.3		4
132.131	odd	2		1584.3.j.i.1297.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.3.c.c.10.1	✓	4	3.2	odd	2	inner
99.3.c.c.10.2	yes	4	11.10	odd	2	inner
99.3.c.c.10.3	yes	4	33.32	even	2	inner
99.3.c.c.10.4	yes	4	1.1	even	1	trivial
1584.3.j.i.1297.1		4	132.131	odd	2
1584.3.j.i.1297.2		4	12.11	even	2
1584.3.j.i.1297.3		4	44.43	even	2
1584.3.j.i.1297.4		4	4.3	odd	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:	\( N \)	\(=\)	\( 99 = 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	99.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.73205i q^{2} +1.00000 q^{4} +6.78233 q^{5} -11.7473i q^{7} +8.66025i q^{8} +O(q^{10})\)	\(q+1.73205i q^{2} +1.00000 q^{4} +6.78233 q^{5} -11.7473i q^{7} +8.66025i q^{8} +11.7473i q^{10} +(-6.78233 + 8.66025i) q^{11} +11.7473i q^{13} +20.3470 q^{14} -11.0000 q^{16} -10.3923i q^{17} +6.78233 q^{20} +(-15.0000 - 11.7473i) q^{22} -33.9116 q^{23} +21.0000 q^{25} -20.3470 q^{26} -11.7473i q^{28} -34.6410i q^{29} -10.0000 q^{31} +15.5885i q^{32} +18.0000 q^{34} -79.6743i q^{35} +50.0000 q^{37} +58.7367i q^{40} +34.6410i q^{41} -46.9894i q^{43} +(-6.78233 + 8.66025i) q^{44} -58.7367i q^{46} -33.9116 q^{47} -89.0000 q^{49} +36.3731i q^{50} +11.7473i q^{52} -33.9116 q^{53} +(-46.0000 + 58.7367i) q^{55} +101.735 q^{56} +60.0000 q^{58} +67.8233 q^{59} +58.7367i q^{61} -17.3205i q^{62} -71.0000 q^{64} +79.6743i q^{65} -10.0000 q^{67} -10.3923i q^{68} +138.000 q^{70} -33.9116 q^{71} -70.4840i q^{73} +86.6025i q^{74} +(101.735 + 79.6743i) q^{77} +58.7367i q^{79} -74.6056 q^{80} -60.0000 q^{82} -76.2102i q^{83} -70.4840i q^{85} +81.3880 q^{86} +(-75.0000 - 58.7367i) q^{88} -13.5647 q^{89} +138.000 q^{91} -33.9116 q^{92} -58.7367i q^{94} -40.0000 q^{97} -154.153i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{4}+O(q^{10}) \) 4 * q + 4 * q^4	\( 4 q + 4 q^{4} - 44 q^{16} - 60 q^{22} + 84 q^{25} - 40 q^{31} + 72 q^{34} + 200 q^{37} - 356 q^{49} - 184 q^{55} + 240 q^{58} - 284 q^{64} - 40 q^{67} + 552 q^{70} - 240 q^{82} - 300 q^{88} + 552 q^{91} - 160 q^{97}+O(q^{100}) \) 4 * q + 4 * q^4 - 44 * q^16 - 60 * q^22 + 84 * q^25 - 40 * q^31 + 72 * q^34 + 200 * q^37 - 356 * q^49 - 184 * q^55 + 240 * q^58 - 284 * q^64 - 40 * q^67 + 552 * q^70 - 240 * q^82 - 300 * q^88 + 552 * q^91 - 160 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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