LMFDB - Embedded newform 124.3.c.a.61.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [124,3,Mod(61,124)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(124, 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("124.61");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 124 = 2^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	124.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:	$3.37875527807$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			61.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	124.61
Dual form			124.3.c.a.61.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.46410i q^{3} -6.00000 q^{5} -10.0000 q^{7} -3.00000 q^{9} +O(q^{10})$	$q-3.46410i q^{3} -6.00000 q^{5} -10.0000 q^{7} -3.00000 q^{9} +17.3205i q^{11} -17.3205i q^{13} +20.7846i q^{15} -13.8564i q^{17} -14.0000 q^{19} +34.6410i q^{21} -34.6410i q^{23} +11.0000 q^{25} -20.7846i q^{27} -17.3205i q^{29} -31.0000 q^{31} +60.0000 q^{33} +60.0000 q^{35} +3.46410i q^{37} -60.0000 q^{39} +54.0000 q^{41} +72.7461i q^{43} +18.0000 q^{45} +30.0000 q^{47} +51.0000 q^{49} -48.0000 q^{51} +31.1769i q^{53} -103.923i q^{55} +48.4974i q^{57} -6.00000 q^{59} -17.3205i q^{61} +30.0000 q^{63} +103.923i q^{65} -110.000 q^{67} -120.000 q^{69} +66.0000 q^{71} -90.0666i q^{73} -38.1051i q^{75} -173.205i q^{77} -69.2820i q^{79} -99.0000 q^{81} +38.1051i q^{83} +83.1384i q^{85} -60.0000 q^{87} +34.6410i q^{89} +173.205i q^{91} +107.387i q^{93} +84.0000 q^{95} +110.000 q^{97} -51.9615i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 12 q^{5} - 20 q^{7} - 6 q^{9}+O(q^{10}) $ 2 * q - 12 * q^5 - 20 * q^7 - 6 * q^9	$ 2 q - 12 q^{5} - 20 q^{7} - 6 q^{9} - 28 q^{19} + 22 q^{25} - 62 q^{31} + 120 q^{33} + 120 q^{35} - 120 q^{39} + 108 q^{41} + 36 q^{45} + 60 q^{47} + 102 q^{49} - 96 q^{51} - 12 q^{59} + 60 q^{63} - 220 q^{67} - 240 q^{69} + 132 q^{71} - 198 q^{81} - 120 q^{87} + 168 q^{95} + 220 q^{97}+O(q^{100}) $ 2 * q - 12 * q^5 - 20 * q^7 - 6 * q^9 - 28 * q^19 + 22 * q^25 - 62 * q^31 + 120 * q^33 + 120 * q^35 - 120 * q^39 + 108 * q^41 + 36 * q^45 + 60 * q^47 + 102 * q^49 - 96 * q^51 - 12 * q^59 + 60 * q^63 - 220 * q^67 - 240 * q^69 + 132 * q^71 - 198 * q^81 - 120 * q^87 + 168 * q^95 + 220 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/124\mathbb{Z}\right)^\times$.

$n$	$63$	$65$
$\chi(n)$	$1$	$-1$

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 $

$2$	0	0
$2$	0	0
$3$	− 3.46410i	− 1.15470i	−0.816497	−	0.577350i	$-0.804087\pi$
$3$	− 3.46410i	− 1.15470i	0.816497	−	0.577350i	$-0.195913\pi$
$4$	0	0
$5$	−6.00000	−1.20000	−0.600000	−	0.800000i	$-0.704833\pi$
$5$	−6.00000	−1.20000	−0.600000	+	0.800000i	$0.704833\pi$
$6$	0	0
$7$	−10.0000	−1.42857	−0.714286	−	0.699854i	$-0.753248\pi$
$7$	−10.0000	−1.42857	−0.714286	+	0.699854i	$0.753248\pi$
$8$	0	0
$9$	−3.00000	−0.333333
$10$	0	0
$11$	17.3205i	1.57459i	0.616575	+	0.787296i	$0.288520\pi$
$11$	17.3205i	1.57459i	−0.616575	+	0.787296i	$0.711480\pi$
$12$	0	0
$13$	− 17.3205i	− 1.33235i	−0.745797	−	0.666173i	$-0.767931\pi$
$13$	− 17.3205i	− 1.33235i	0.745797	−	0.666173i	$-0.232069\pi$
$14$	0	0
$15$	20.7846i	1.38564i
$16$	0	0
$17$	− 13.8564i	− 0.815083i	−0.913187	−	0.407541i	$-0.866386\pi$
$17$	− 13.8564i	− 0.815083i	0.913187	−	0.407541i	$-0.133614\pi$
$18$	0	0
$19$	−14.0000	−0.736842	−0.368421	−	0.929659i	$-0.620102\pi$
$19$	−14.0000	−0.736842	−0.368421	+	0.929659i	$0.620102\pi$
$20$	0	0
$21$	34.6410i	1.64957i
$22$	0	0
$23$	− 34.6410i	− 1.50613i	−0.657945	−	0.753066i	$-0.728574\pi$
$23$	− 34.6410i	− 1.50613i	0.657945	−	0.753066i	$-0.271426\pi$
$24$	0	0
$25$	11.0000	0.440000
$26$	0	0
$27$	− 20.7846i	− 0.769800i
$28$	0	0
$29$	− 17.3205i	− 0.597259i	−0.954369	−	0.298629i	$-0.903471\pi$
$29$	− 17.3205i	− 0.597259i	0.954369	−	0.298629i	$-0.0965295\pi$
$30$	0	0
$31$	−31.0000	−1.00000
$31$	−31.0000	−1.00000
$32$	0	0
$33$	60.0000	1.81818
$34$	0	0
$35$	60.0000	1.71429
$36$	0	0
$37$	3.46410i	0.0936244i	0.998904	+	0.0468122i	$0.0149062\pi$
$37$	3.46410i	0.0936244i	−0.998904	+	0.0468122i	$0.985094\pi$
$38$	0	0
$39$	−60.0000	−1.53846
$40$	0	0
$41$	54.0000	1.31707	0.658537	−	0.752549i	$-0.271176\pi$
$41$	54.0000	1.31707	0.658537	+	0.752549i	$0.271176\pi$
$42$	0	0
$43$	72.7461i	1.69177i	0.533365	+	0.845885i	$0.320927\pi$
$43$	72.7461i	1.69177i	−0.533365	+	0.845885i	$0.679073\pi$
$44$	0	0
$45$	18.0000	0.400000
$46$	0	0
$47$	30.0000	0.638298	0.319149	−	0.947705i	$-0.396603\pi$
$47$	30.0000	0.638298	0.319149	+	0.947705i	$0.396603\pi$
$48$	0	0
$49$	51.0000	1.04082
$50$	0	0
$51$	−48.0000	−0.941176
$52$	0	0
$53$	31.1769i	0.588244i	0.955768	+	0.294122i	$0.0950271\pi$
$53$	31.1769i	0.588244i	−0.955768	+	0.294122i	$0.904973\pi$
$54$	0	0
$55$	− 103.923i	− 1.88951i
$56$	0	0
$57$	48.4974i	0.850832i
$58$	0	0
$59$	−6.00000	−0.101695	−0.0508475	−	0.998706i	$-0.516192\pi$
$59$	−6.00000	−0.101695	−0.0508475	+	0.998706i	$0.516192\pi$
$60$	0	0
$61$	− 17.3205i	− 0.283943i	−0.989871	−	0.141971i	$-0.954656\pi$
$61$	− 17.3205i	− 0.283943i	0.989871	−	0.141971i	$-0.0453441\pi$
$62$	0	0
$63$	30.0000	0.476190
$64$	0	0
$65$	103.923i	1.59882i
$66$	0	0
$67$	−110.000	−1.64179	−0.820896	−	0.571078i	$-0.806525\pi$
$67$	−110.000	−1.64179	−0.820896	+	0.571078i	$0.806525\pi$
$68$	0	0
$69$	−120.000	−1.73913
$70$	0	0
$71$	66.0000	0.929577	0.464789	−	0.885422i	$-0.346130\pi$
$71$	66.0000	0.929577	0.464789	+	0.885422i	$0.346130\pi$
$72$	0	0
$73$	− 90.0666i	− 1.23379i	−0.787046	−	0.616895i	$-0.788390\pi$
$73$	− 90.0666i	− 1.23379i	0.787046	−	0.616895i	$-0.211610\pi$
$74$	0	0
$75$	− 38.1051i	− 0.508068i
$76$	0	0
$77$	− 173.205i	− 2.24942i
$78$	0	0
$79$	− 69.2820i	− 0.876988i	−0.898734	−	0.438494i	$-0.855512\pi$
$79$	− 69.2820i	− 0.876988i	0.898734	−	0.438494i	$-0.144488\pi$
$80$	0	0
$81$	−99.0000	−1.22222
$82$	0	0
$83$	38.1051i	0.459098i	0.973297	+	0.229549i	$0.0737251\pi$
$83$	38.1051i	0.459098i	−0.973297	+	0.229549i	$0.926275\pi$
$84$	0	0
$85$	83.1384i	0.978099i
$86$	0	0
$87$	−60.0000	−0.689655
$88$	0	0
$89$	34.6410i	0.389225i	0.980880	+	0.194612i	$0.0623449\pi$
$89$	34.6410i	0.389225i	−0.980880	+	0.194612i	$0.937655\pi$
$90$	0	0
$91$	173.205i	1.90335i
$92$	0	0
$93$	107.387i	1.15470i
$94$	0	0
$95$	84.0000	0.884211
$96$	0	0
$97$	110.000	1.13402	0.567010	−	0.823711i	$-0.308100\pi$
$97$	110.000	1.13402	0.567010	+	0.823711i	$0.308100\pi$
$98$	0	0
$99$	− 51.9615i	− 0.524864i
$100$	0	0
$101$	−174.000	−1.72277	−0.861386	−	0.507951i	$-0.830403\pi$
$101$	−174.000	−1.72277	−0.861386	+	0.507951i	$0.830403\pi$
$102$	0	0
$103$	−110.000	−1.06796	−0.533981	−	0.845497i	$-0.679304\pi$
$103$	−110.000	−1.06796	−0.533981	+	0.845497i	$0.679304\pi$
$104$	0	0
$105$	− 207.846i	− 1.97949i
$106$	0	0
$107$	−30.0000	−0.280374	−0.140187	−	0.990125i	$-0.544770\pi$
$107$	−30.0000	−0.280374	−0.140187	+	0.990125i	$0.544770\pi$
$108$	0	0
$109$	26.0000	0.238532	0.119266	−	0.992862i	$-0.461946\pi$
$109$	26.0000	0.238532	0.119266	+	0.992862i	$0.461946\pi$
$110$	0	0
$111$	12.0000	0.108108
$112$	0	0
$113$	30.0000	0.265487	0.132743	−	0.991150i	$-0.457621\pi$
$113$	30.0000	0.265487	0.132743	+	0.991150i	$0.457621\pi$
$114$	0	0
$115$	207.846i	1.80736i
$116$	0	0
$117$	51.9615i	0.444116i
$118$	0	0
$119$	138.564i	1.16440i
$120$	0	0
$121$	−179.000	−1.47934
$122$	0	0
$123$	− 187.061i	− 1.52083i
$124$	0	0
$125$	84.0000	0.672000
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	0	0
$129$	252.000	1.95349
$130$	0	0
$131$	66.0000	0.503817	0.251908	−	0.967751i	$-0.418942\pi$
$131$	66.0000	0.503817	0.251908	+	0.967751i	$0.418942\pi$
$132$	0	0
$133$	140.000	1.05263
$134$	0	0
$135$	124.708i	0.923760i
$136$	0	0
$137$	− 228.631i	− 1.66884i	−0.551131	−	0.834419i	$-0.685804\pi$
$137$	− 228.631i	− 1.66884i	0.551131	−	0.834419i	$-0.314196\pi$
$138$	0	0
$139$	− 190.526i	− 1.37069i	−0.728220	−	0.685344i	$-0.759652\pi$
$139$	− 190.526i	− 1.37069i	0.728220	−	0.685344i	$-0.240348\pi$
$140$	0	0
$141$	− 103.923i	− 0.737043i
$142$	0	0
$143$	300.000	2.09790
$144$	0	0
$145$	103.923i	0.716711i
$146$	0	0
$147$	− 176.669i	− 1.20183i
$148$	0	0
$149$	114.000	0.765101	0.382550	−	0.923935i	$-0.375046\pi$
$149$	114.000	0.765101	0.382550	+	0.923935i	$0.375046\pi$
$150$	0	0
$151$	− 34.6410i	− 0.229411i	−0.993400	−	0.114705i	$-0.963408\pi$
$151$	− 34.6410i	− 0.229411i	0.993400	−	0.114705i	$-0.0365924\pi$
$152$	0	0
$153$	41.5692i	0.271694i
$154$	0	0
$155$	186.000	1.20000
$156$	0	0
$157$	−230.000	−1.46497	−0.732484	−	0.680784i	$-0.761639\pi$
$157$	−230.000	−1.46497	−0.732484	+	0.680784i	$0.761639\pi$
$158$	0	0
$159$	108.000	0.679245
$160$	0	0
$161$	346.410i	2.15162i
$162$	0	0
$163$	50.0000	0.306748	0.153374	−	0.988168i	$-0.450986\pi$
$163$	50.0000	0.306748	0.153374	+	0.988168i	$0.450986\pi$
$164$	0	0
$165$	−360.000	−2.18182
$166$	0	0
$167$	228.631i	1.36905i	0.728991	+	0.684523i	$0.239990\pi$
$167$	228.631i	1.36905i	−0.728991	+	0.684523i	$0.760010\pi$
$168$	0	0
$169$	−131.000	−0.775148
$170$	0	0
$171$	42.0000	0.245614
$172$	0	0
$173$	−30.0000	−0.173410	−0.0867052	−	0.996234i	$-0.527634\pi$
$173$	−30.0000	−0.173410	−0.0867052	+	0.996234i	$0.527634\pi$
$174$	0	0
$175$	−110.000	−0.628571
$176$	0	0
$177$	20.7846i	0.117427i
$178$	0	0
$179$	− 155.885i	− 0.870864i	−0.900222	−	0.435432i	$-0.856596\pi$
$179$	− 155.885i	− 0.870864i	0.900222	−	0.435432i	$-0.143404\pi$
$180$	0	0
$181$	− 190.526i	− 1.05263i	−0.850290	−	0.526314i	$-0.823574\pi$
$181$	− 190.526i	− 1.05263i	0.850290	−	0.526314i	$-0.176426\pi$
$182$	0	0
$183$	−60.0000	−0.327869
$184$	0	0
$185$	− 20.7846i	− 0.112349i
$186$	0	0
$187$	240.000	1.28342
$188$	0	0
$189$	207.846i	1.09971i
$190$	0	0
$191$	−18.0000	−0.0942408	−0.0471204	−	0.998889i	$-0.515004\pi$
$191$	−18.0000	−0.0942408	−0.0471204	+	0.998889i	$0.515004\pi$
$192$	0	0
$193$	−110.000	−0.569948	−0.284974	−	0.958535i	$-0.591985\pi$
$193$	−110.000	−0.569948	−0.284974	+	0.958535i	$0.591985\pi$
$194$	0	0
$195$	360.000	1.84615
$196$	0	0
$197$	− 51.9615i	− 0.263764i	−0.991265	−	0.131882i	$-0.957898\pi$
$197$	− 51.9615i	− 0.263764i	0.991265	−	0.131882i	$-0.0421020\pi$
$198$	0	0
$199$	34.6410i	0.174075i	0.996205	+	0.0870377i	$0.0277401\pi$
$199$	34.6410i	0.174075i	−0.996205	+	0.0870377i	$0.972260\pi$
$200$	0	0
$201$	381.051i	1.89578i
$202$	0	0
$203$	173.205i	0.853227i
$204$	0	0
$205$	−324.000	−1.58049
$206$	0	0
$207$	103.923i	0.502044i
$208$	0	0
$209$	− 242.487i	− 1.16023i
$210$	0	0
$211$	−22.0000	−0.104265	−0.0521327	−	0.998640i	$-0.516602\pi$
$211$	−22.0000	−0.104265	−0.0521327	+	0.998640i	$0.516602\pi$
$212$	0	0
$213$	− 228.631i	− 1.07338i
$214$	0	0
$215$	− 436.477i	− 2.03012i
$216$	0	0
$217$	310.000	1.42857
$218$	0	0
$219$	−312.000	−1.42466
$220$	0	0
$221$	−240.000	−1.08597
$222$	0	0
$223$	− 332.554i	− 1.49127i	−0.666353	−	0.745636i	$-0.732146\pi$
$223$	− 332.554i	− 1.49127i	0.666353	−	0.745636i	$-0.267854\pi$
$224$	0	0
$225$	−33.0000	−0.146667
$226$	0	0
$227$	−390.000	−1.71806	−0.859031	−	0.511924i	$-0.828933\pi$
$227$	−390.000	−1.71806	−0.859031	+	0.511924i	$0.828933\pi$
$228$	0	0
$229$	363.731i	1.58834i	0.607693	+	0.794172i	$0.292095\pi$
$229$	363.731i	1.58834i	−0.607693	+	0.794172i	$0.707905\pi$
$230$	0	0
$231$	−600.000	−2.59740
$232$	0	0
$233$	390.000	1.67382	0.836910	−	0.547341i	$-0.184360\pi$
$233$	390.000	1.67382	0.836910	+	0.547341i	$0.184360\pi$
$234$	0	0
$235$	−180.000	−0.765957
$236$	0	0
$237$	−240.000	−1.01266
$238$	0	0
$239$	− 69.2820i	− 0.289883i	−0.989440	−	0.144941i	$-0.953701\pi$
$239$	− 69.2820i	− 0.289883i	0.989440	−	0.144941i	$-0.0462994\pi$
$240$	0	0
$241$	− 346.410i	− 1.43739i	−0.695327	−	0.718693i	$-0.744741\pi$
$241$	− 346.410i	− 1.43739i	0.695327	−	0.718693i	$-0.255259\pi$
$242$	0	0
$243$	155.885i	0.641500i
$244$	0	0
$245$	−306.000	−1.24898
$246$	0	0
$247$	242.487i	0.981729i
$248$	0	0
$249$	132.000	0.530120
$250$	0	0
$251$	− 190.526i	− 0.759066i	−0.925178	−	0.379533i	$-0.876085\pi$
$251$	− 190.526i	− 0.759066i	0.925178	−	0.379533i	$-0.123915\pi$
$252$	0	0
$253$	600.000	2.37154
$254$	0	0
$255$	288.000	1.12941
$256$	0	0
$257$	−270.000	−1.05058	−0.525292	−	0.850922i	$-0.676044\pi$
$257$	−270.000	−1.05058	−0.525292	+	0.850922i	$0.676044\pi$
$258$	0	0
$259$	− 34.6410i	− 0.133749i
$260$	0	0
$261$	51.9615i	0.199086i
$262$	0	0
$263$	117.779i	0.447831i	0.974609	+	0.223915i	$0.0718839\pi$
$263$	117.779i	0.447831i	−0.974609	+	0.223915i	$0.928116\pi$
$264$	0	0
$265$	− 187.061i	− 0.705892i
$266$	0	0
$267$	120.000	0.449438
$268$	0	0
$269$	121.244i	0.450720i	0.974276	+	0.225360i	$0.0723557\pi$
$269$	121.244i	0.450720i	−0.974276	+	0.225360i	$0.927644\pi$
$270$	0	0
$271$	484.974i	1.78957i	0.446494	+	0.894786i	$0.352672\pi$
$271$	484.974i	1.78957i	−0.446494	+	0.894786i	$0.647328\pi$
$272$	0	0
$273$	600.000	2.19780
$274$	0	0
$275$	190.526i	0.692820i
$276$	0	0
$277$	446.869i	1.61325i	0.591066	+	0.806623i	$0.298707\pi$
$277$	446.869i	1.61325i	−0.591066	+	0.806623i	$0.701293\pi$
$278$	0	0
$279$	93.0000	0.333333
$280$	0	0
$281$	66.0000	0.234875	0.117438	−	0.993080i	$-0.462532\pi$
$281$	66.0000	0.234875	0.117438	+	0.993080i	$0.462532\pi$
$282$	0	0
$283$	410.000	1.44876	0.724382	−	0.689399i	$-0.242125\pi$
$283$	410.000	1.44876	0.724382	+	0.689399i	$0.242125\pi$
$284$	0	0
$285$	− 290.985i	− 1.02100i
$286$	0	0
$287$	−540.000	−1.88153
$288$	0	0
$289$	97.0000	0.335640
$290$	0	0
$291$	− 381.051i	− 1.30945i
$292$	0	0
$293$	−150.000	−0.511945	−0.255973	−	0.966684i	$-0.582396\pi$
$293$	−150.000	−0.511945	−0.255973	+	0.966684i	$0.582396\pi$
$294$	0	0
$295$	36.0000	0.122034
$296$	0	0
$297$	360.000	1.21212
$298$	0	0
$299$	−600.000	−2.00669
$300$	0	0
$301$	− 727.461i	− 2.41682i
$302$	0	0
$303$	602.754i	1.98929i
$304$	0	0
$305$	103.923i	0.340731i
$306$	0	0
$307$	370.000	1.20521	0.602606	−	0.798039i	$-0.294129\pi$
$307$	370.000	1.20521	0.602606	+	0.798039i	$0.294129\pi$
$308$	0	0
$309$	381.051i	1.23318i
$310$	0	0
$311$	594.000	1.90997	0.954984	−	0.296658i	$-0.0958720\pi$
$311$	594.000	1.90997	0.954984	+	0.296658i	$0.0958720\pi$
$312$	0	0
$313$	− 297.913i	− 0.951798i	−0.879500	−	0.475899i	$-0.842123\pi$
$313$	− 297.913i	− 0.951798i	0.879500	−	0.475899i	$-0.157877\pi$
$314$	0	0
$315$	−180.000	−0.571429
$316$	0	0
$317$	−270.000	−0.851735	−0.425868	−	0.904786i	$-0.640031\pi$
$317$	−270.000	−0.851735	−0.425868	+	0.904786i	$0.640031\pi$
$318$	0	0
$319$	300.000	0.940439
$320$	0	0
$321$	103.923i	0.323748i
$322$	0	0
$323$	193.990i	0.600587i
$324$	0	0
$325$	− 190.526i	− 0.586233i
$326$	0	0
$327$	− 90.0666i	− 0.275433i
$328$	0	0
$329$	−300.000	−0.911854
$330$	0	0
$331$	− 606.218i	− 1.83147i	−0.401779	−	0.915737i	$-0.631608\pi$
$331$	− 606.218i	− 1.83147i	0.401779	−	0.915737i	$-0.368392\pi$
$332$	0	0
$333$	− 10.3923i	− 0.0312081i
$334$	0	0
$335$	660.000	1.97015
$336$	0	0
$337$	401.836i	1.19239i	0.802839	+	0.596196i	$0.203322\pi$
$337$	401.836i	1.19239i	−0.802839	+	0.596196i	$0.796678\pi$
$338$	0	0
$339$	− 103.923i	− 0.306558i
$340$	0	0
$341$	− 536.936i	− 1.57459i
$342$	0	0
$343$	−20.0000	−0.0583090
$344$	0	0
$345$	720.000	2.08696
$346$	0	0
$347$	17.3205i	0.0499150i	0.999689	+	0.0249575i	$0.00794505\pi$
$347$	17.3205i	0.0499150i	−0.999689	+	0.0249575i	$0.992055\pi$
$348$	0	0
$349$	154.000	0.441261	0.220630	−	0.975357i	$-0.429189\pi$
$349$	154.000	0.441261	0.220630	+	0.975357i	$0.429189\pi$
$350$	0	0
$351$	−360.000	−1.02564
$352$	0	0
$353$	− 415.692i	− 1.17760i	−0.808279	−	0.588799i	$-0.799601\pi$
$353$	− 415.692i	− 1.17760i	0.808279	−	0.588799i	$-0.200399\pi$
$354$	0	0
$355$	−396.000	−1.11549
$356$	0	0
$357$	480.000	1.34454
$358$	0	0
$359$	−282.000	−0.785515	−0.392758	−	0.919642i	$-0.628479\pi$
$359$	−282.000	−0.785515	−0.392758	+	0.919642i	$0.628479\pi$
$360$	0	0
$361$	−165.000	−0.457064
$362$	0	0
$363$	620.074i	1.70819i
$364$	0	0
$365$	540.400i	1.48055i
$366$	0	0
$367$	− 152.420i	− 0.415315i	−0.978202	−	0.207657i	$-0.933416\pi$
$367$	− 152.420i	− 0.415315i	0.978202	−	0.207657i	$-0.0665839\pi$
$368$	0	0
$369$	−162.000	−0.439024
$370$	0	0
$371$	− 311.769i	− 0.840348i
$372$	0	0
$373$	410.000	1.09920	0.549598	−	0.835429i	$-0.314781\pi$
$373$	410.000	1.09920	0.549598	+	0.835429i	$0.314781\pi$
$374$	0	0
$375$	− 290.985i	− 0.775959i
$376$	0	0
$377$	−300.000	−0.795756
$378$	0	0
$379$	−526.000	−1.38786	−0.693931	−	0.720041i	$-0.744123\pi$
$379$	−526.000	−1.38786	−0.693931	+	0.720041i	$0.744123\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 55.4256i	− 0.144714i	−0.997379	−	0.0723572i	$-0.976948\pi$
$383$	− 55.4256i	− 0.144714i	0.997379	−	0.0723572i	$-0.0230522\pi$
$384$	0	0
$385$	1039.23i	2.69930i
$386$	0	0
$387$	− 218.238i	− 0.563924i
$388$	0	0
$389$	433.013i	1.11314i	0.830800	+	0.556572i	$0.187884\pi$
$389$	433.013i	1.11314i	−0.830800	+	0.556572i	$0.812116\pi$
$390$	0	0
$391$	−480.000	−1.22762
$392$	0	0
$393$	− 228.631i	− 0.581758i
$394$	0	0
$395$	415.692i	1.05239i
$396$	0	0
$397$	−190.000	−0.478589	−0.239295	−	0.970947i	$-0.576916\pi$
$397$	−190.000	−0.478589	−0.239295	+	0.970947i	$0.576916\pi$
$398$	0	0
$399$	− 484.974i	− 1.21547i
$400$	0	0
$401$	− 207.846i	− 0.518319i	−0.965834	−	0.259160i	$-0.916554\pi$
$401$	− 207.846i	− 0.518319i	0.965834	−	0.259160i	$-0.0834456\pi$
$402$	0	0
$403$	536.936i	1.33235i
$404$	0	0
$405$	594.000	1.46667
$406$	0	0
$407$	−60.0000	−0.147420
$408$	0	0
$409$	450.333i	1.10106i	0.834816	+	0.550530i	$0.185574\pi$
$409$	450.333i	1.10106i	−0.834816	+	0.550530i	$0.814426\pi$
$410$	0	0
$411$	−792.000	−1.92701
$412$	0	0
$413$	60.0000	0.145278
$414$	0	0
$415$	− 228.631i	− 0.550917i
$416$	0	0
$417$	−660.000	−1.58273
$418$	0	0
$419$	234.000	0.558473	0.279236	−	0.960222i	$-0.409919\pi$
$419$	234.000	0.558473	0.279236	+	0.960222i	$0.409919\pi$
$420$	0	0
$421$	−206.000	−0.489311	−0.244656	−	0.969610i	$-0.578675\pi$
$421$	−206.000	−0.489311	−0.244656	+	0.969610i	$0.578675\pi$
$422$	0	0
$423$	−90.0000	−0.212766
$424$	0	0
$425$	− 152.420i	− 0.358636i
$426$	0	0
$427$	173.205i	0.405633i
$428$	0	0
$429$	− 1039.23i	− 2.42245i
$430$	0	0
$431$	−354.000	−0.821346	−0.410673	−	0.911783i	$-0.634706\pi$
$431$	−354.000	−0.821346	−0.410673	+	0.911783i	$0.634706\pi$
$432$	0	0
$433$	− 207.846i	− 0.480014i	−0.970771	−	0.240007i	$-0.922850\pi$
$433$	− 207.846i	− 0.480014i	0.970771	−	0.240007i	$-0.0771498\pi$
$434$	0	0
$435$	360.000	0.827586
$436$	0	0
$437$	484.974i	1.10978i
$438$	0	0
$439$	−506.000	−1.15262	−0.576310	−	0.817231i	$-0.695508\pi$
$439$	−506.000	−1.15262	−0.576310	+	0.817231i	$0.695508\pi$
$440$	0	0
$441$	−153.000	−0.346939
$442$	0	0
$443$	−30.0000	−0.0677201	−0.0338600	−	0.999427i	$-0.510780\pi$
$443$	−30.0000	−0.0677201	−0.0338600	+	0.999427i	$0.510780\pi$
$444$	0	0
$445$	− 207.846i	− 0.467070i
$446$	0	0
$447$	− 394.908i	− 0.883462i
$448$	0	0
$449$	554.256i	1.23442i	0.786797	+	0.617212i	$0.211738\pi$
$449$	554.256i	1.23442i	−0.786797	+	0.617212i	$0.788262\pi$
$450$	0	0
$451$	935.307i	2.07385i
$452$	0	0
$453$	−120.000	−0.264901
$454$	0	0
$455$	− 1039.23i	− 2.28402i
$456$	0	0
$457$	− 311.769i	− 0.682208i	−0.940026	−	0.341104i	$-0.889199\pi$
$457$	− 311.769i	− 0.682208i	0.940026	−	0.341104i	$-0.110801\pi$
$458$	0	0
$459$	−288.000	−0.627451
$460$	0	0
$461$	814.064i	1.76587i	0.469500	+	0.882933i	$0.344434\pi$
$461$	814.064i	1.76587i	−0.469500	+	0.882933i	$0.655566\pi$
$462$	0	0
$463$	− 762.102i	− 1.64601i	−0.568035	−	0.823005i	$-0.692296\pi$
$463$	− 762.102i	− 1.64601i	0.568035	−	0.823005i	$-0.307704\pi$
$464$	0	0
$465$	− 644.323i	− 1.38564i
$466$	0	0
$467$	570.000	1.22056	0.610278	−	0.792187i	$-0.291058\pi$
$467$	570.000	1.22056	0.610278	+	0.792187i	$0.291058\pi$
$468$	0	0
$469$	1100.00	2.34542
$470$	0	0
$471$	796.743i	1.69160i
$472$	0	0
$473$	−1260.00	−2.66385
$474$	0	0
$475$	−154.000	−0.324211
$476$	0	0
$477$	− 93.5307i	− 0.196081i
$478$	0	0
$479$	642.000	1.34029	0.670146	−	0.742229i	$-0.266231\pi$
$479$	642.000	1.34029	0.670146	+	0.742229i	$0.266231\pi$
$480$	0	0
$481$	60.0000	0.124740
$482$	0	0
$483$	1200.00	2.48447
$484$	0	0
$485$	−660.000	−1.36082
$486$	0	0
$487$	− 713.605i	− 1.46531i	−0.680601	−	0.732654i	$-0.738281\pi$
$487$	− 713.605i	− 1.46531i	0.680601	−	0.732654i	$-0.261719\pi$
$488$	0	0
$489$	− 173.205i	− 0.354203i
$490$	0	0
$491$	− 329.090i	− 0.670244i	−0.942175	−	0.335122i	$-0.891223\pi$
$491$	− 329.090i	− 0.670244i	0.942175	−	0.335122i	$-0.108777\pi$
$492$	0	0
$493$	−240.000	−0.486815
$494$	0	0
$495$	311.769i	0.629837i
$496$	0	0
$497$	−660.000	−1.32797
$498$	0	0
$499$	− 363.731i	− 0.728919i	−0.931219	−	0.364460i	$-0.881254\pi$
$499$	− 363.731i	− 0.728919i	0.931219	−	0.364460i	$-0.118746\pi$
$500$	0	0
$501$	792.000	1.58084
$502$	0	0
$503$	−30.0000	−0.0596421	−0.0298211	−	0.999555i	$-0.509494\pi$
$503$	−30.0000	−0.0596421	−0.0298211	+	0.999555i	$0.509494\pi$
$504$	0	0
$505$	1044.00	2.06733
$506$	0	0
$507$	453.797i	0.895064i
$508$	0	0
$509$	− 225.167i	− 0.442371i	−0.975232	−	0.221185i	$-0.929007\pi$
$509$	− 225.167i	− 0.442371i	0.975232	−	0.221185i	$-0.0709926\pi$
$510$	0	0
$511$	900.666i	1.76256i
$512$	0	0
$513$	290.985i	0.567221i
$514$	0	0
$515$	660.000	1.28155
$516$	0	0
$517$	519.615i	1.00506i
$518$	0	0
$519$	103.923i	0.200237i
$520$	0	0
$521$	66.0000	0.126679	0.0633397	−	0.997992i	$-0.479825\pi$
$521$	66.0000	0.126679	0.0633397	+	0.997992i	$0.479825\pi$
$522$	0	0
$523$	− 481.510i	− 0.920669i	−0.887745	−	0.460335i	$-0.847729\pi$
$523$	− 481.510i	− 0.920669i	0.887745	−	0.460335i	$-0.152271\pi$
$524$	0	0
$525$	381.051i	0.725812i
$526$	0	0
$527$	429.549i	0.815083i
$528$	0	0
$529$	−671.000	−1.26843
$530$	0	0
$531$	18.0000	0.0338983
$532$	0	0
$533$	− 935.307i	− 1.75480i
$534$	0	0
$535$	180.000	0.336449
$536$	0	0
$537$	−540.000	−1.00559
$538$	0	0
$539$	883.346i	1.63886i
$540$	0	0
$541$	418.000	0.772643	0.386322	−	0.922364i	$-0.373745\pi$
$541$	418.000	0.772643	0.386322	+	0.922364i	$0.373745\pi$
$542$	0	0
$543$	−660.000	−1.21547
$544$	0	0
$545$	−156.000	−0.286239
$546$	0	0
$547$	610.000	1.11517	0.557587	−	0.830119i	$-0.311727\pi$
$547$	610.000	1.11517	0.557587	+	0.830119i	$0.311727\pi$
$548$	0	0
$549$	51.9615i	0.0946476i
$550$	0	0
$551$	242.487i	0.440086i
$552$	0	0
$553$	692.820i	1.25284i
$554$	0	0
$555$	−72.0000	−0.129730
$556$	0	0
$557$	190.526i	0.342057i	0.985266	+	0.171028i	$0.0547090\pi$
$557$	190.526i	0.342057i	−0.985266	+	0.171028i	$0.945291\pi$
$558$	0	0
$559$	1260.00	2.25403
$560$	0	0
$561$	− 831.384i	− 1.48197i
$562$	0	0
$563$	90.0000	0.159858	0.0799290	−	0.996801i	$-0.474531\pi$
$563$	90.0000	0.159858	0.0799290	+	0.996801i	$0.474531\pi$
$564$	0	0
$565$	−180.000	−0.318584
$566$	0	0
$567$	990.000	1.74603
$568$	0	0
$569$	− 381.051i	− 0.669686i	−0.942274	−	0.334843i	$-0.891317\pi$
$569$	− 381.051i	− 0.669686i	0.942274	−	0.334843i	$-0.108683\pi$
$570$	0	0
$571$	− 190.526i	− 0.333670i	−0.985985	−	0.166835i	$-0.946645\pi$
$571$	− 190.526i	− 0.333670i	0.985985	−	0.166835i	$-0.0533547\pi$
$572$	0	0
$573$	62.3538i	0.108820i
$574$	0	0
$575$	− 381.051i	− 0.662698i
$576$	0	0
$577$	−50.0000	−0.0866551	−0.0433276	−	0.999061i	$-0.513796\pi$
$577$	−50.0000	−0.0866551	−0.0433276	+	0.999061i	$0.513796\pi$
$578$	0	0
$579$	381.051i	0.658119i
$580$	0	0
$581$	− 381.051i	− 0.655854i
$582$	0	0
$583$	−540.000	−0.926244
$584$	0	0
$585$	− 311.769i	− 0.532939i
$586$	0	0
$587$	433.013i	0.737671i	0.929495	+	0.368835i	$0.120243\pi$
$587$	433.013i	0.737671i	−0.929495	+	0.368835i	$0.879757\pi$
$588$	0	0
$589$	434.000	0.736842
$590$	0	0
$591$	−180.000	−0.304569
$592$	0	0
$593$	30.0000	0.0505902	0.0252951	−	0.999680i	$-0.491947\pi$
$593$	30.0000	0.0505902	0.0252951	+	0.999680i	$0.491947\pi$
$594$	0	0
$595$	− 831.384i	− 1.39728i
$596$	0	0
$597$	120.000	0.201005
$598$	0	0
$599$	198.000	0.330551	0.165275	−	0.986247i	$-0.447149\pi$
$599$	198.000	0.330551	0.165275	+	0.986247i	$0.447149\pi$
$600$	0	0
$601$	− 519.615i	− 0.864584i	−0.901734	−	0.432292i	$-0.857705\pi$
$601$	− 519.615i	− 0.864584i	0.901734	−	0.432292i	$-0.142295\pi$
$602$	0	0
$603$	330.000	0.547264
$604$	0	0
$605$	1074.00	1.77521
$606$	0	0
$607$	130.000	0.214168	0.107084	−	0.994250i	$-0.465849\pi$
$607$	130.000	0.214168	0.107084	+	0.994250i	$0.465849\pi$
$608$	0	0
$609$	600.000	0.985222
$610$	0	0
$611$	− 519.615i	− 0.850434i
$612$	0	0
$613$	142.028i	0.231694i	0.993267	+	0.115847i	$0.0369582\pi$
$613$	142.028i	0.231694i	−0.993267	+	0.115847i	$0.963042\pi$
$614$	0	0
$615$	1122.37i	1.82499i
$616$	0	0
$617$	−990.000	−1.60454	−0.802269	−	0.596963i	$-0.796374\pi$
$617$	−990.000	−1.60454	−0.802269	+	0.596963i	$0.796374\pi$
$618$	0	0
$619$	294.449i	0.475684i	0.971304	+	0.237842i	$0.0764401\pi$
$619$	294.449i	0.475684i	−0.971304	+	0.237842i	$0.923560\pi$
$620$	0	0
$621$	−720.000	−1.15942
$622$	0	0
$623$	− 346.410i	− 0.556036i
$624$	0	0
$625$	−779.000	−1.24640
$626$	0	0
$627$	−840.000	−1.33971
$628$	0	0
$629$	48.0000	0.0763116
$630$	0	0
$631$	− 34.6410i	− 0.0548986i	−0.999623	−	0.0274493i	$-0.991262\pi$
$631$	− 34.6410i	− 0.0548986i	0.999623	−	0.0274493i	$-0.00873848\pi$
$632$	0	0
$633$	76.2102i	0.120395i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 883.346i	− 1.38673i
$638$	0	0
$639$	−198.000	−0.309859
$640$	0	0
$641$	831.384i	1.29701i	0.761210	+	0.648506i	$0.224606\pi$
$641$	831.384i	1.29701i	−0.761210	+	0.648506i	$0.775394\pi$
$642$	0	0
$643$	− 31.1769i	− 0.0484866i	−0.999706	−	0.0242433i	$-0.992282\pi$
$643$	− 31.1769i	− 0.0484866i	0.999706	−	0.0242433i	$-0.00771765\pi$
$644$	0	0
$645$	−1512.00	−2.34419
$646$	0	0
$647$	− 464.190i	− 0.717449i	−0.933443	−	0.358725i	$-0.883212\pi$
$647$	− 464.190i	− 0.717449i	0.933443	−	0.358725i	$-0.116788\pi$
$648$	0	0
$649$	− 103.923i	− 0.160128i
$650$	0	0
$651$	− 1073.87i	− 1.64957i
$652$	0	0
$653$	1170.00	1.79173	0.895865	−	0.444326i	$-0.146557\pi$
$653$	1170.00	1.79173	0.895865	+	0.444326i	$0.146557\pi$
$654$	0	0
$655$	−396.000	−0.604580
$656$	0	0
$657$	270.200i	0.411263i
$658$	0	0
$659$	66.0000	0.100152	0.0500759	−	0.998745i	$-0.484054\pi$
$659$	66.0000	0.100152	0.0500759	+	0.998745i	$0.484054\pi$
$660$	0	0
$661$	−814.000	−1.23147	−0.615734	−	0.787954i	$-0.711140\pi$
$661$	−814.000	−1.23147	−0.615734	+	0.787954i	$0.711140\pi$
$662$	0	0
$663$	831.384i	1.25397i
$664$	0	0
$665$	−840.000	−1.26316
$666$	0	0
$667$	−600.000	−0.899550
$668$	0	0
$669$	−1152.00	−1.72197
$670$	0	0
$671$	300.000	0.447094
$672$	0	0
$673$	55.4256i	0.0823561i	0.999152	+	0.0411780i	$0.0131111\pi$
$673$	55.4256i	0.0823561i	−0.999152	+	0.0411780i	$0.986889\pi$
$674$	0	0
$675$	− 228.631i	− 0.338712i
$676$	0	0
$677$	− 897.202i	− 1.32526i	−0.748946	−	0.662631i	$-0.769440\pi$
$677$	− 897.202i	− 1.32526i	0.748946	−	0.662631i	$-0.230560\pi$
$678$	0	0
$679$	−1100.00	−1.62003
$680$	0	0
$681$	1351.00i	1.98385i
$682$	0	0
$683$	−630.000	−0.922401	−0.461201	−	0.887296i	$-0.652581\pi$
$683$	−630.000	−0.922401	−0.461201	+	0.887296i	$0.652581\pi$
$684$	0	0
$685$	1371.78i	2.00260i
$686$	0	0
$687$	1260.00	1.83406
$688$	0	0
$689$	540.000	0.783745
$690$	0	0
$691$	74.0000	0.107091	0.0535456	−	0.998565i	$-0.482948\pi$
$691$	74.0000	0.107091	0.0535456	+	0.998565i	$0.482948\pi$
$692$	0	0
$693$	519.615i	0.749806i
$694$	0	0
$695$	1143.15i	1.64483i
$696$	0	0
$697$	− 748.246i	− 1.07352i
$698$	0	0
$699$	− 1351.00i	− 1.93276i
$700$	0	0
$701$	1002.00	1.42939	0.714693	−	0.699438i	$-0.246566\pi$
$701$	1002.00	1.42939	0.714693	+	0.699438i	$0.246566\pi$
$702$	0	0
$703$	− 48.4974i	− 0.0689864i
$704$	0	0
$705$	623.538i	0.884451i
$706$	0	0
$707$	1740.00	2.46110
$708$	0	0
$709$	433.013i	0.610737i	0.952234	+	0.305369i	$0.0987797\pi$
$709$	433.013i	0.610737i	−0.952234	+	0.305369i	$0.901220\pi$
$710$	0	0
$711$	207.846i	0.292329i
$712$	0	0
$713$	1073.87i	1.50613i
$714$	0	0
$715$	−1800.00	−2.51748
$716$	0	0
$717$	−240.000	−0.334728
$718$	0	0
$719$	− 346.410i	− 0.481794i	−0.970551	−	0.240897i	$-0.922558\pi$
$719$	− 346.410i	− 0.481794i	0.970551	−	0.240897i	$-0.0774417\pi$
$720$	0	0
$721$	1100.00	1.52566
$722$	0	0
$723$	−1200.00	−1.65975
$724$	0	0
$725$	− 190.526i	− 0.262794i
$726$	0	0
$727$	290.000	0.398900	0.199450	−	0.979908i	$-0.436085\pi$
$727$	290.000	0.398900	0.199450	+	0.979908i	$0.436085\pi$
$728$	0	0
$729$	−351.000	−0.481481
$730$	0	0
$731$	1008.00	1.37893
$732$	0	0
$733$	−230.000	−0.313779	−0.156889	−	0.987616i	$-0.550147\pi$
$733$	−230.000	−0.313779	−0.156889	+	0.987616i	$0.550147\pi$
$734$	0	0
$735$	1060.02i	1.44220i
$736$	0	0
$737$	− 1905.26i	− 2.58515i
$738$	0	0
$739$	467.654i	0.632820i	0.948623	+	0.316410i	$0.102477\pi$
$739$	467.654i	0.632820i	−0.948623	+	0.316410i	$0.897523\pi$
$740$	0	0
$741$	840.000	1.13360
$742$	0	0
$743$	− 1295.57i	− 1.74371i	−0.489767	−	0.871853i	$-0.662918\pi$
$743$	− 1295.57i	− 1.74371i	0.489767	−	0.871853i	$-0.337082\pi$
$744$	0	0
$745$	−684.000	−0.918121
$746$	0	0
$747$	− 114.315i	− 0.153033i
$748$	0	0
$749$	300.000	0.400534
$750$	0	0
$751$	−2.00000	−0.00266312	−0.00133156	−	0.999999i	$-0.500424\pi$
$751$	−2.00000	−0.00266312	−0.00133156	+	0.999999i	$0.500424\pi$
$752$	0	0
$753$	−660.000	−0.876494
$754$	0	0
$755$	207.846i	0.275293i
$756$	0	0
$757$	− 869.490i	− 1.14860i	−0.818645	−	0.574300i	$-0.805274\pi$
$757$	− 869.490i	− 1.14860i	0.818645	−	0.574300i	$-0.194726\pi$
$758$	0	0
$759$	− 2078.46i	− 2.73842i
$760$	0	0
$761$	311.769i	0.409684i	0.978795	+	0.204842i	$0.0656680\pi$
$761$	311.769i	0.409684i	−0.978795	+	0.204842i	$0.934332\pi$
$762$	0	0
$763$	−260.000	−0.340760
$764$	0	0
$765$	− 249.415i	− 0.326033i
$766$	0	0
$767$	103.923i	0.135493i
$768$	0	0
$769$	−626.000	−0.814044	−0.407022	−	0.913418i	$-0.633433\pi$
$769$	−626.000	−0.814044	−0.407022	+	0.913418i	$0.633433\pi$
$770$	0	0
$771$	935.307i	1.21311i
$772$	0	0
$773$	− 259.808i	− 0.336103i	−0.985778	−	0.168052i	$-0.946253\pi$
$773$	− 259.808i	− 0.336103i	0.985778	−	0.168052i	$-0.0537475\pi$
$774$	0	0
$775$	−341.000	−0.440000
$776$	0	0
$777$	−120.000	−0.154440
$778$	0	0
$779$	−756.000	−0.970475
$780$	0	0
$781$	1143.15i	1.46370i
$782$	0	0
$783$	−360.000	−0.459770
$784$	0	0
$785$	1380.00	1.75796
$786$	0	0
$787$	398.372i	0.506190i	0.967441	+	0.253095i	$0.0814486\pi$
$787$	398.372i	0.506190i	−0.967441	+	0.253095i	$0.918551\pi$
$788$	0	0
$789$	408.000	0.517110
$790$	0	0
$791$	−300.000	−0.379267
$792$	0	0
$793$	−300.000	−0.378310
$794$	0	0
$795$	−648.000	−0.815094
$796$	0	0
$797$	952.628i	1.19527i	0.801769	+	0.597634i	$0.203892\pi$
$797$	952.628i	1.19527i	−0.801769	+	0.597634i	$0.796108\pi$
$798$	0	0
$799$	− 415.692i	− 0.520266i
$800$	0	0
$801$	− 103.923i	− 0.129742i
$802$	0	0
$803$	1560.00	1.94271
$804$	0	0
$805$	− 2078.46i	− 2.58194i
$806$	0	0
$807$	420.000	0.520446
$808$	0	0
$809$	1351.00i	1.66996i	0.550278	+	0.834981i	$0.314522\pi$
$809$	1351.00i	1.66996i	−0.550278	+	0.834981i	$0.685478\pi$
$810$	0	0
$811$	154.000	0.189889	0.0949445	−	0.995483i	$-0.469733\pi$
$811$	154.000	0.189889	0.0949445	+	0.995483i	$0.469733\pi$
$812$	0	0
$813$	1680.00	2.06642
$814$	0	0
$815$	−300.000	−0.368098
$816$	0	0
$817$	− 1018.45i	− 1.24657i
$818$	0	0
$819$	− 519.615i	− 0.634451i
$820$	0	0
$821$	571.577i	0.696196i	0.937458	+	0.348098i	$0.113172\pi$
$821$	571.577i	0.696196i	−0.937458	+	0.348098i	$0.886828\pi$
$822$	0	0
$823$	741.318i	0.900751i	0.892839	+	0.450375i	$0.148710\pi$
$823$	741.318i	0.900751i	−0.892839	+	0.450375i	$0.851290\pi$
$824$	0	0
$825$	660.000	0.800000
$826$	0	0
$827$	446.869i	0.540350i	0.962811	+	0.270175i	$0.0870815\pi$
$827$	446.869i	0.540350i	−0.962811	+	0.270175i	$0.912919\pi$
$828$	0	0
$829$	− 225.167i	− 0.271612i	−0.990735	−	0.135806i	$-0.956638\pi$
$829$	− 225.167i	− 0.271612i	0.990735	−	0.135806i	$-0.0433624\pi$
$830$	0	0
$831$	1548.00	1.86282
$832$	0	0
$833$	− 706.677i	− 0.848351i
$834$	0	0
$835$	− 1371.78i	− 1.64286i
$836$	0	0
$837$	644.323i	0.769800i
$838$	0	0
$839$	−234.000	−0.278903	−0.139452	−	0.990229i	$-0.544534\pi$
$839$	−234.000	−0.278903	−0.139452	+	0.990229i	$0.544534\pi$
$840$	0	0
$841$	541.000	0.643282
$842$	0	0
$843$	− 228.631i	− 0.271211i
$844$	0	0
$845$	786.000	0.930178
$846$	0	0
$847$	1790.00	2.11334
$848$	0	0
$849$	− 1420.28i	− 1.67289i
$850$	0	0
$851$	120.000	0.141011
$852$	0	0
$853$	370.000	0.433763	0.216882	−	0.976198i	$-0.430411\pi$
$853$	370.000	0.433763	0.216882	+	0.976198i	$0.430411\pi$
$854$	0	0
$855$	−252.000	−0.294737
$856$	0	0
$857$	450.000	0.525088	0.262544	−	0.964920i	$-0.415439\pi$
$857$	450.000	0.525088	0.262544	+	0.964920i	$0.415439\pi$
$858$	0	0
$859$	1333.68i	1.55260i	0.630367	+	0.776298i	$0.282905\pi$
$859$	1333.68i	1.55260i	−0.630367	+	0.776298i	$0.717095\pi$
$860$	0	0
$861$	1870.61i	2.17261i
$862$	0	0
$863$	471.118i	0.545907i	0.962027	+	0.272954i	$0.0880005\pi$
$863$	471.118i	0.545907i	−0.962027	+	0.272954i	$0.911999\pi$
$864$	0	0
$865$	180.000	0.208092
$866$	0	0
$867$	− 336.018i	− 0.387564i
$868$	0	0
$869$	1200.00	1.38090
$870$	0	0
$871$	1905.26i	2.18744i
$872$	0	0
$873$	−330.000	−0.378007
$874$	0	0
$875$	−840.000	−0.960000
$876$	0	0
$877$	−1310.00	−1.49373	−0.746864	−	0.664976i	$-0.768442\pi$
$877$	−1310.00	−1.49373	−0.746864	+	0.664976i	$0.768442\pi$
$878$	0	0
$879$	519.615i	0.591144i
$880$	0	0
$881$	− 762.102i	− 0.865042i	−0.901624	−	0.432521i	$-0.857624\pi$
$881$	− 762.102i	− 0.865042i	0.901624	−	0.432521i	$-0.142376\pi$
$882$	0	0
$883$	620.074i	0.702236i	0.936331	+	0.351118i	$0.114198\pi$
$883$	620.074i	0.702236i	−0.936331	+	0.351118i	$0.885802\pi$
$884$	0	0
$885$	− 124.708i	− 0.140913i
$886$	0	0
$887$	−1530.00	−1.72492	−0.862458	−	0.506129i	$-0.831076\pi$
$887$	−1530.00	−1.72492	−0.862458	+	0.506129i	$0.831076\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	− 1714.73i	− 1.92450i
$892$	0	0
$893$	−420.000	−0.470325
$894$	0	0
$895$	935.307i	1.04504i
$896$	0	0
$897$	2078.46i	2.31712i
$898$	0	0
$899$	536.936i	0.597259i
$900$	0	0
$901$	432.000	0.479467
$902$	0	0
$903$	−2520.00	−2.79070
$904$	0	0
$905$	1143.15i	1.26315i
$906$	0	0
$907$	1450.00	1.59868	0.799338	−	0.600881i	$-0.205183\pi$
$907$	1450.00	1.59868	0.799338	+	0.600881i	$0.205183\pi$
$908$	0	0
$909$	522.000	0.574257
$910$	0	0
$911$	− 762.102i	− 0.836556i	−0.908319	−	0.418278i	$-0.862634\pi$
$911$	− 762.102i	− 0.836556i	0.908319	−	0.418278i	$-0.137366\pi$
$912$	0	0
$913$	−660.000	−0.722892
$914$	0	0
$915$	360.000	0.393443
$916$	0	0
$917$	−660.000	−0.719738
$918$	0	0
$919$	226.000	0.245919	0.122960	−	0.992412i	$-0.460761\pi$
$919$	226.000	0.245919	0.122960	+	0.992412i	$0.460761\pi$
$920$	0	0
$921$	− 1281.72i	− 1.39166i
$922$	0	0
$923$	− 1143.15i	− 1.23852i
$924$	0	0
$925$	38.1051i	0.0411947i
$926$	0	0
$927$	330.000	0.355987
$928$	0	0
$929$	− 554.256i	− 0.596616i	−0.954470	−	0.298308i	$-0.903578\pi$
$929$	− 554.256i	− 0.596616i	0.954470	−	0.298308i	$-0.0964223\pi$
$930$	0	0
$931$	−714.000	−0.766917
$932$	0	0
$933$	− 2057.68i	− 2.20544i
$934$	0	0
$935$	−1440.00	−1.54011
$936$	0	0
$937$	1570.00	1.67556	0.837780	−	0.546008i	$-0.183853\pi$
$937$	1570.00	1.67556	0.837780	+	0.546008i	$0.183853\pi$
$938$	0	0
$939$	−1032.00	−1.09904
$940$	0	0
$941$	− 571.577i	− 0.607414i	−0.952765	−	0.303707i	$-0.901776\pi$
$941$	− 571.577i	− 0.607414i	0.952765	−	0.303707i	$-0.0982244\pi$
$942$	0	0
$943$	− 1870.61i	− 1.98368i
$944$	0	0
$945$	− 1247.08i	− 1.31966i
$946$	0	0
$947$	− 72.7461i	− 0.0768175i	−0.999262	−	0.0384087i	$-0.987771\pi$
$947$	− 72.7461i	− 0.0768175i	0.999262	−	0.0384087i	$-0.0122289\pi$
$948$	0	0
$949$	−1560.00	−1.64384
$950$	0	0
$951$	935.307i	0.983499i
$952$	0	0
$953$	394.908i	0.414384i	0.978300	+	0.207192i	$0.0664324\pi$
$953$	394.908i	0.414384i	−0.978300	+	0.207192i	$0.933568\pi$
$954$	0	0
$955$	108.000	0.113089
$956$	0	0
$957$	− 1039.23i	− 1.08593i
$958$	0	0
$959$	2286.31i	2.38405i
$960$	0	0
$961$	961.000	1.00000
$962$	0	0
$963$	90.0000	0.0934579
$964$	0	0
$965$	660.000	0.683938
$966$	0	0
$967$	34.6410i	0.0358232i	0.999840	+	0.0179116i	$0.00570174\pi$
$967$	34.6410i	0.0358232i	−0.999840	+	0.0179116i	$0.994298\pi$
$968$	0	0
$969$	672.000	0.693498
$970$	0	0
$971$	354.000	0.364573	0.182286	−	0.983245i	$-0.441650\pi$
$971$	354.000	0.364573	0.182286	+	0.983245i	$0.441650\pi$
$972$	0	0
$973$	1905.26i	1.95813i
$974$	0	0
$975$	−660.000	−0.676923
$976$	0	0
$977$	−270.000	−0.276356	−0.138178	−	0.990407i	$-0.544125\pi$
$977$	−270.000	−0.276356	−0.138178	+	0.990407i	$0.544125\pi$
$978$	0	0
$979$	−600.000	−0.612870
$980$	0	0
$981$	−78.0000	−0.0795107
$982$	0	0
$983$	796.743i	0.810522i	0.914201	+	0.405261i	$0.132819\pi$
$983$	796.743i	0.810522i	−0.914201	+	0.405261i	$0.867181\pi$
$984$	0	0
$985$	311.769i	0.316517i
$986$	0	0
$987$	1039.23i	1.05292i
$988$	0	0
$989$	2520.00	2.54803
$990$	0	0
$991$	− 277.128i	− 0.279645i	−0.990177	−	0.139822i	$-0.955347\pi$
$991$	− 277.128i	− 0.279645i	0.990177	−	0.139822i	$-0.0446532\pi$
$992$	0	0
$993$	−2100.00	−2.11480
$994$	0	0
$995$	− 207.846i	− 0.208891i
$996$	0	0
$997$	−70.0000	−0.0702106	−0.0351053	−	0.999384i	$-0.511177\pi$
$997$	−70.0000	−0.0702106	−0.0351053	+	0.999384i	$0.511177\pi$
$998$	0	0
$999$	72.0000	0.0720721

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	124.3.c.a.61.1	✓	2
3.2	odd	2		1116.3.h.c.433.1		2
4.3	odd	2		496.3.e.a.433.2		2
5.2	odd	4		3100.3.f.a.1549.1		4
5.3	odd	4		3100.3.f.a.1549.4		4
5.4	even	2		3100.3.d.a.1301.2		2
31.30	odd	2	inner	124.3.c.a.61.2	yes	2
93.92	even	2		1116.3.h.c.433.2		2
124.123	even	2		496.3.e.a.433.1		2
155.92	even	4		3100.3.f.a.1549.3		4
155.123	even	4		3100.3.f.a.1549.2		4
155.154	odd	2		3100.3.d.a.1301.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
124.3.c.a.61.1	✓	2	1.1	even	1	trivial
124.3.c.a.61.2	yes	2	31.30	odd	2	inner
496.3.e.a.433.1		2	124.123	even	2
496.3.e.a.433.2		2	4.3	odd	2
1116.3.h.c.433.1		2	3.2	odd	2
1116.3.h.c.433.2		2	93.92	even	2
3100.3.d.a.1301.1		2	155.154	odd	2
3100.3.d.a.1301.2		2	5.4	even	2
3100.3.f.a.1549.1		4	5.2	odd	4
3100.3.f.a.1549.2		4	155.123	even	4
3100.3.f.a.1549.3		4	155.92	even	4
3100.3.f.a.1549.4		4	5.3	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 124 = 2^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	124.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.46410i q^{3} -6.00000 q^{5} -10.0000 q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q-3.46410i q^{3} -6.00000 q^{5} -10.0000 q^{7} -3.00000 q^{9} +17.3205i q^{11} -17.3205i q^{13} +20.7846i q^{15} -13.8564i q^{17} -14.0000 q^{19} +34.6410i q^{21} -34.6410i q^{23} +11.0000 q^{25} -20.7846i q^{27} -17.3205i q^{29} -31.0000 q^{31} +60.0000 q^{33} +60.0000 q^{35} +3.46410i q^{37} -60.0000 q^{39} +54.0000 q^{41} +72.7461i q^{43} +18.0000 q^{45} +30.0000 q^{47} +51.0000 q^{49} -48.0000 q^{51} +31.1769i q^{53} -103.923i q^{55} +48.4974i q^{57} -6.00000 q^{59} -17.3205i q^{61} +30.0000 q^{63} +103.923i q^{65} -110.000 q^{67} -120.000 q^{69} +66.0000 q^{71} -90.0666i q^{73} -38.1051i q^{75} -173.205i q^{77} -69.2820i q^{79} -99.0000 q^{81} +38.1051i q^{83} +83.1384i q^{85} -60.0000 q^{87} +34.6410i q^{89} +173.205i q^{91} +107.387i q^{93} +84.0000 q^{95} +110.000 q^{97} -51.9615i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 12 q^{5} - 20 q^{7} - 6 q^{9}+O(q^{10}) \) 2 * q - 12 * q^5 - 20 * q^7 - 6 * q^9	\( 2 q - 12 q^{5} - 20 q^{7} - 6 q^{9} - 28 q^{19} + 22 q^{25} - 62 q^{31} + 120 q^{33} + 120 q^{35} - 120 q^{39} + 108 q^{41} + 36 q^{45} + 60 q^{47} + 102 q^{49} - 96 q^{51} - 12 q^{59} + 60 q^{63} - 220 q^{67} - 240 q^{69} + 132 q^{71} - 198 q^{81} - 120 q^{87} + 168 q^{95} + 220 q^{97}+O(q^{100}) \) 2 * q - 12 * q^5 - 20 * q^7 - 6 * q^9 - 28 * q^19 + 22 * q^25 - 62 * q^31 + 120 * q^33 + 120 * q^35 - 120 * q^39 + 108 * q^41 + 36 * q^45 + 60 * q^47 + 102 * q^49 - 96 * q^51 - 12 * q^59 + 60 * q^63 - 220 * q^67 - 240 * q^69 + 132 * q^71 - 198 * q^81 - 120 * q^87 + 168 * q^95 + 220 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

Properties

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