LMFDB - Embedded newform 3100.3.f.a.1549.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3100,3,Mod(1549,3100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3100, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3100.1549");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 3100 = 2^{2} \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	3100.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$84.4688819517$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 124)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1549.3
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	3100.1549
Dual form			3100.3.f.a.1549.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.46410 q^{3} -10.0000i q^{7} +3.00000 q^{9} +O(q^{10})$	$q+3.46410 q^{3} -10.0000i q^{7} +3.00000 q^{9} -17.3205i q^{11} +17.3205 q^{13} -13.8564 q^{17} +14.0000 q^{19} -34.6410i q^{21} +34.6410 q^{23} -20.7846 q^{27} -17.3205i q^{29} -31.0000 q^{31} -60.0000i q^{33} +3.46410 q^{37} +60.0000 q^{39} +54.0000 q^{41} -72.7461 q^{43} +30.0000i q^{47} -51.0000 q^{49} -48.0000 q^{51} -31.1769 q^{53} +48.4974 q^{57} +6.00000 q^{59} +17.3205i q^{61} -30.0000i q^{63} -110.000i q^{67} +120.000 q^{69} +66.0000 q^{71} +90.0666 q^{73} -173.205 q^{77} -69.2820i q^{79} -99.0000 q^{81} -38.1051 q^{83} -60.0000i q^{87} +34.6410i q^{89} -173.205i q^{91} -107.387 q^{93} +110.000i q^{97} -51.9615i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{9}+O(q^{10}) $ 4 * q + 12 * q^9	$ 4 q + 12 q^{9} + 56 q^{19} - 124 q^{31} + 240 q^{39} + 216 q^{41} - 204 q^{49} - 192 q^{51} + 24 q^{59} + 480 q^{69} + 264 q^{71} - 396 q^{81}+O(q^{100}) $ 4 * q + 12 * q^9 + 56 * q^19 - 124 * q^31 + 240 * q^39 + 216 * q^41 - 204 * q^49 - 192 * q^51 + 24 * q^59 + 480 * q^69 + 264 * q^71 - 396 * q^81

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3100\mathbb{Z}\right)^\times$.

$n$	$1551$	$1801$	$2977$
$\chi(n)$	$1$	$-1$	$-1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	3.46410	1.15470	0.577350	−	0.816497i	$-0.304087\pi$
$3$	3.46410	1.15470	0.577350	+	0.816497i	$0.304087\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 10.0000i	− 1.42857i	−0.699854	−	0.714286i	$-0.746752\pi$
$7$	− 10.0000i	− 1.42857i	0.699854	−	0.714286i	$-0.253248\pi$
$8$	0	0
$9$	3.00000	0.333333
$10$	0	0
$11$	− 17.3205i	− 1.57459i	−0.616575	−	0.787296i	$-0.711480\pi$
$11$	− 17.3205i	− 1.57459i	0.616575	−	0.787296i	$-0.288520\pi$
$12$	0	0
$13$	17.3205	1.33235	0.666173	−	0.745797i	$-0.267931\pi$
$13$	17.3205	1.33235	0.666173	+	0.745797i	$0.267931\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−13.8564	−0.815083	−0.407541	−	0.913187i	$-0.633614\pi$
$17$	−13.8564	−0.815083	−0.407541	+	0.913187i	$0.633614\pi$
$18$	0	0
$19$	14.0000	0.736842	0.368421	−	0.929659i	$-0.379898\pi$
$19$	14.0000	0.736842	0.368421	+	0.929659i	$0.379898\pi$
$20$	0	0
$21$	− 34.6410i	− 1.64957i
$22$	0	0
$23$	34.6410	1.50613	0.753066	−	0.657945i	$-0.228574\pi$
$23$	34.6410	1.50613	0.753066	+	0.657945i	$0.228574\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−20.7846	−0.769800
$28$	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
$33$	− 60.0000i	− 1.81818i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	3.46410	0.0936244	0.0468122	−	0.998904i	$-0.485094\pi$
$37$	3.46410	0.0936244	0.0468122	+	0.998904i	$0.485094\pi$
$38$	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
$43$	−72.7461	−1.69177	−0.845885	−	0.533365i	$-0.820927\pi$
$43$	−72.7461	−1.69177	−0.845885	+	0.533365i	$0.820927\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	30.0000i	0.638298i	0.947705	+	0.319149i	$0.103397\pi$
$47$	30.0000i	0.638298i	−0.947705	+	0.319149i	$0.896603\pi$
$48$	0	0
$49$	−51.0000	−1.04082
$50$	0	0
$51$	−48.0000	−0.941176
$52$	0	0
$53$	−31.1769	−0.588244	−0.294122	−	0.955768i	$-0.595027\pi$
$53$	−31.1769	−0.588244	−0.294122	+	0.955768i	$0.595027\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	48.4974	0.850832
$58$	0	0
$59$	6.00000	0.101695	0.0508475	−	0.998706i	$-0.483808\pi$
$59$	6.00000	0.101695	0.0508475	+	0.998706i	$0.483808\pi$
$60$	0	0
$61$	17.3205i	0.283943i	0.989871	+	0.141971i	$0.0453441\pi$
$61$	17.3205i	0.283943i	−0.989871	+	0.141971i	$0.954656\pi$
$62$	0	0
$63$	− 30.0000i	− 0.476190i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 110.000i	− 1.64179i	−0.571078	−	0.820896i	$-0.693475\pi$
$67$	− 110.000i	− 1.64179i	0.571078	−	0.820896i	$-0.306525\pi$
$68$	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
$73$	90.0666	1.23379	0.616895	−	0.787046i	$-0.288390\pi$
$73$	90.0666	1.23379	0.616895	+	0.787046i	$0.288390\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−173.205	−2.24942
$78$	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
$83$	−38.1051	−0.459098	−0.229549	−	0.973297i	$-0.573725\pi$
$83$	−38.1051	−0.459098	−0.229549	+	0.973297i	$0.573725\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 60.0000i	− 0.689655i
$88$	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
$93$	−107.387	−1.15470
$94$	0	0
$95$	0	0
$96$	0	0
$97$	110.000i	1.13402i	0.823711	+	0.567010i	$0.191900\pi$
$97$	110.000i	1.13402i	−0.823711	+	0.567010i	$0.808100\pi$
$98$	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
$103$	110.000i	1.06796i	0.845497	+	0.533981i	$0.179304\pi$
$103$	110.000i	1.06796i	−0.845497	+	0.533981i	$0.820696\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 30.0000i	− 0.280374i	−0.990125	−	0.140187i	$-0.955230\pi$
$107$	− 30.0000i	− 0.280374i	0.990125	−	0.140187i	$-0.0447703\pi$
$108$	0	0
$109$	−26.0000	−0.238532	−0.119266	−	0.992862i	$-0.538054\pi$
$109$	−26.0000	−0.238532	−0.119266	+	0.992862i	$0.538054\pi$
$110$	0	0
$111$	12.0000	0.108108
$112$	0	0
$113$	− 30.0000i	− 0.265487i	−0.991150	−	0.132743i	$-0.957621\pi$
$113$	− 30.0000i	− 0.265487i	0.991150	−	0.132743i	$-0.0423786\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	51.9615	0.444116
$118$	0	0
$119$	138.564i	1.16440i
$120$	0	0
$121$	−179.000	−1.47934
$122$	0	0
$123$	187.061	1.52083
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	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
$133$	− 140.000i	− 1.05263i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−228.631	−1.66884	−0.834419	−	0.551131i	$-0.814196\pi$
$137$	−228.631	−1.66884	−0.834419	+	0.551131i	$0.814196\pi$
$138$	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
$143$	− 300.000i	− 2.09790i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−176.669	−1.20183
$148$	0	0
$149$	−114.000	−0.765101	−0.382550	−	0.923935i	$-0.624954\pi$
$149$	−114.000	−0.765101	−0.382550	+	0.923935i	$0.624954\pi$
$150$	0	0
$151$	34.6410i	0.229411i	0.993400	+	0.114705i	$0.0365924\pi$
$151$	34.6410i	0.229411i	−0.993400	+	0.114705i	$0.963408\pi$
$152$	0	0
$153$	−41.5692	−0.271694
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 230.000i	− 1.46497i	−0.680784	−	0.732484i	$-0.738361\pi$
$157$	− 230.000i	− 1.46497i	0.680784	−	0.732484i	$-0.261639\pi$
$158$	0	0
$159$	−108.000	−0.679245
$160$	0	0
$161$	− 346.410i	− 2.15162i
$162$	0	0
$163$	− 50.0000i	− 0.306748i	−0.988168	−	0.153374i	$-0.950986\pi$
$163$	− 50.0000i	− 0.306748i	0.988168	−	0.153374i	$-0.0490140\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	228.631	1.36905	0.684523	−	0.728991i	$-0.260010\pi$
$167$	228.631	1.36905	0.684523	+	0.728991i	$0.260010\pi$
$168$	0	0
$169$	131.000	0.775148
$170$	0	0
$171$	42.0000	0.245614
$172$	0	0
$173$	30.0000i	0.173410i	0.996234	+	0.0867052i	$0.0276338\pi$
$173$	30.0000i	0.173410i	−0.996234	+	0.0867052i	$0.972366\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	20.7846	0.117427
$178$	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	$0.176426\pi$
$181$	190.526i	1.05263i	−0.850290	+	0.526314i	$0.823574\pi$
$182$	0	0
$183$	60.0000i	0.327869i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	240.000i	1.28342i
$188$	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
$193$	110.000i	0.569948i	0.958535	+	0.284974i	$0.0919850\pi$
$193$	110.000i	0.569948i	−0.958535	+	0.284974i	$0.908015\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−51.9615	−0.263764	−0.131882	−	0.991265i	$-0.542102\pi$
$197$	−51.9615	−0.263764	−0.131882	+	0.991265i	$0.542102\pi$
$198$	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
$203$	−173.205	−0.853227
$204$	0	0
$205$	0	0
$206$	0	0
$207$	103.923	0.502044
$208$	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
$213$	228.631	1.07338
$214$	0	0
$215$	0	0
$216$	0	0
$217$	310.000i	1.42857i
$218$	0	0
$219$	312.000	1.42466
$220$	0	0
$221$	−240.000	−1.08597
$222$	0	0
$223$	332.554	1.49127	0.745636	−	0.666353i	$-0.232146\pi$
$223$	332.554	1.49127	0.745636	+	0.666353i	$0.232146\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 390.000i	− 1.71806i	−0.511924	−	0.859031i	$-0.671067\pi$
$227$	− 390.000i	− 1.71806i	0.511924	−	0.859031i	$-0.328933\pi$
$228$	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
$233$	− 390.000i	− 1.67382i	−0.547341	−	0.836910i	$-0.684360\pi$
$233$	− 390.000i	− 1.67382i	0.547341	−	0.836910i	$-0.315640\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 240.000i	− 1.01266i
$238$	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	$0.255259\pi$
$241$	346.410i	1.43739i	−0.695327	+	0.718693i	$0.744741\pi$
$242$	0	0
$243$	−155.885	−0.641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	242.487	0.981729
$248$	0	0
$249$	−132.000	−0.530120
$250$	0	0
$251$	190.526i	0.759066i	0.925178	+	0.379533i	$0.123915\pi$
$251$	190.526i	0.759066i	−0.925178	+	0.379533i	$0.876085\pi$
$252$	0	0
$253$	− 600.000i	− 2.37154i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 270.000i	− 1.05058i	−0.850922	−	0.525292i	$-0.823956\pi$
$257$	− 270.000i	− 1.05058i	0.850922	−	0.525292i	$-0.176044\pi$
$258$	0	0
$259$	− 34.6410i	− 0.133749i
$260$	0	0
$261$	− 51.9615i	− 0.199086i
$262$	0	0
$263$	−117.779	−0.447831	−0.223915	−	0.974609i	$-0.571884\pi$
$263$	−117.779	−0.447831	−0.223915	+	0.974609i	$0.571884\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	120.000i	0.449438i
$268$	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	$-0.647328\pi$
$271$	− 484.974i	− 1.78957i	0.446494	−	0.894786i	$-0.352672\pi$
$272$	0	0
$273$	− 600.000i	− 2.19780i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	446.869	1.61325	0.806623	−	0.591066i	$-0.201293\pi$
$277$	446.869	1.61325	0.806623	+	0.591066i	$0.201293\pi$
$278$	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
$283$	− 410.000i	− 1.44876i	−0.689399	−	0.724382i	$-0.742125\pi$
$283$	− 410.000i	− 1.44876i	0.689399	−	0.724382i	$-0.257875\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 540.000i	− 1.88153i
$288$	0	0
$289$	−97.0000	−0.335640
$290$	0	0
$291$	381.051i	1.30945i
$292$	0	0
$293$	150.000i	0.511945i	0.966684	+	0.255973i	$0.0823957\pi$
$293$	150.000i	0.511945i	−0.966684	+	0.255973i	$0.917604\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	360.000i	1.21212i
$298$	0	0
$299$	600.000	2.00669
$300$	0	0
$301$	727.461i	2.41682i
$302$	0	0
$303$	−602.754	−1.98929
$304$	0	0
$305$	0	0
$306$	0	0
$307$	370.000i	1.20521i	0.798039	+	0.602606i	$0.205871\pi$
$307$	370.000i	1.20521i	−0.798039	+	0.602606i	$0.794129\pi$
$308$	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
$313$	297.913	0.951798	0.475899	−	0.879500i	$-0.342123\pi$
$313$	297.913	0.951798	0.475899	+	0.879500i	$0.342123\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 270.000i	− 0.851735i	−0.904786	−	0.425868i	$-0.859969\pi$
$317$	− 270.000i	− 0.851735i	0.904786	−	0.425868i	$-0.140031\pi$
$318$	0	0
$319$	−300.000	−0.940439
$320$	0	0
$321$	− 103.923i	− 0.323748i
$322$	0	0
$323$	−193.990	−0.600587
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−90.0666	−0.275433
$328$	0	0
$329$	300.000	0.911854
$330$	0	0
$331$	606.218i	1.83147i	0.401779	+	0.915737i	$0.368392\pi$
$331$	606.218i	1.83147i	−0.401779	+	0.915737i	$0.631608\pi$
$332$	0	0
$333$	10.3923	0.0312081
$334$	0	0
$335$	0	0
$336$	0	0
$337$	401.836	1.19239	0.596196	−	0.802839i	$-0.296678\pi$
$337$	401.836	1.19239	0.596196	+	0.802839i	$0.296678\pi$
$338$	0	0
$339$	− 103.923i	− 0.306558i
$340$	0	0
$341$	536.936i	1.57459i
$342$	0	0
$343$	20.0000i	0.0583090i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	17.3205	0.0499150	0.0249575	−	0.999689i	$-0.492055\pi$
$347$	17.3205	0.0499150	0.0249575	+	0.999689i	$0.492055\pi$
$348$	0	0
$349$	−154.000	−0.441261	−0.220630	−	0.975357i	$-0.570811\pi$
$349$	−154.000	−0.441261	−0.220630	+	0.975357i	$0.570811\pi$
$350$	0	0
$351$	−360.000	−1.02564
$352$	0	0
$353$	415.692	1.17760	0.588799	−	0.808279i	$-0.299601\pi$
$353$	415.692	1.17760	0.588799	+	0.808279i	$0.299601\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	480.000i	1.34454i
$358$	0	0
$359$	282.000	0.785515	0.392758	−	0.919642i	$-0.371521\pi$
$359$	282.000	0.785515	0.392758	+	0.919642i	$0.371521\pi$
$360$	0	0
$361$	−165.000	−0.457064
$362$	0	0
$363$	−620.074	−1.70819
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−152.420	−0.415315	−0.207657	−	0.978202i	$-0.566584\pi$
$367$	−152.420	−0.415315	−0.207657	+	0.978202i	$0.566584\pi$
$368$	0	0
$369$	162.000	0.439024
$370$	0	0
$371$	311.769i	0.840348i
$372$	0	0
$373$	− 410.000i	− 1.09920i	−0.835429	−	0.549598i	$-0.814781\pi$
$373$	− 410.000i	− 1.09920i	0.835429	−	0.549598i	$-0.185219\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 300.000i	− 0.795756i
$378$	0	0
$379$	526.000	1.38786	0.693931	−	0.720041i	$-0.255877\pi$
$379$	526.000	1.38786	0.693931	+	0.720041i	$0.255877\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	55.4256	0.144714	0.0723572	−	0.997379i	$-0.476948\pi$
$383$	55.4256	0.144714	0.0723572	+	0.997379i	$0.476948\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−218.238	−0.563924
$388$	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
$393$	228.631	0.581758
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 190.000i	− 0.478589i	−0.970947	−	0.239295i	$-0.923084\pi$
$397$	− 190.000i	− 0.478589i	0.970947	−	0.239295i	$-0.0769162\pi$
$398$	0	0
$399$	− 484.974i	− 1.21547i
$400$	0	0
$401$	207.846i	0.518319i	0.965834	+	0.259160i	$0.0834456\pi$
$401$	207.846i	0.518319i	−0.965834	+	0.259160i	$0.916554\pi$
$402$	0	0
$403$	−536.936	−1.33235
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 60.0000i	− 0.147420i
$408$	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
$413$	− 60.0000i	− 0.145278i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 660.000i	− 1.58273i
$418$	0	0
$419$	−234.000	−0.558473	−0.279236	−	0.960222i	$-0.590081\pi$
$419$	−234.000	−0.558473	−0.279236	+	0.960222i	$0.590081\pi$
$420$	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
$423$	90.0000i	0.212766i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	173.205	0.405633
$428$	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
$433$	207.846	0.480014	0.240007	−	0.970771i	$-0.422850\pi$
$433$	207.846	0.480014	0.240007	+	0.970771i	$0.422850\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	484.974	1.10978
$438$	0	0
$439$	506.000	1.15262	0.576310	−	0.817231i	$-0.304492\pi$
$439$	506.000	1.15262	0.576310	+	0.817231i	$0.304492\pi$
$440$	0	0
$441$	−153.000	−0.346939
$442$	0	0
$443$	30.0000i	0.0677201i	0.999427	+	0.0338600i	$0.0107800\pi$
$443$	30.0000i	0.0677201i	−0.999427	+	0.0338600i	$0.989220\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−394.908	−0.883462
$448$	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
$453$	120.000i	0.264901i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−311.769	−0.682208	−0.341104	−	0.940026i	$-0.610801\pi$
$457$	−311.769	−0.682208	−0.341104	+	0.940026i	$0.610801\pi$
$458$	0	0
$459$	288.000	0.627451
$460$	0	0
$461$	− 814.064i	− 1.76587i	−0.469500	−	0.882933i	$-0.655566\pi$
$461$	− 814.064i	− 1.76587i	0.469500	−	0.882933i	$-0.344434\pi$
$462$	0	0
$463$	762.102	1.64601	0.823005	−	0.568035i	$-0.192296\pi$
$463$	762.102	1.64601	0.823005	+	0.568035i	$0.192296\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	570.000i	1.22056i	0.792187	+	0.610278i	$0.208942\pi$
$467$	570.000i	1.22056i	−0.792187	+	0.610278i	$0.791058\pi$
$468$	0	0
$469$	−1100.00	−2.34542
$470$	0	0
$471$	− 796.743i	− 1.69160i
$472$	0	0
$473$	1260.00i	2.66385i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−93.5307	−0.196081
$478$	0	0
$479$	−642.000	−1.34029	−0.670146	−	0.742229i	$-0.733769\pi$
$479$	−642.000	−1.34029	−0.670146	+	0.742229i	$0.733769\pi$
$480$	0	0
$481$	60.0000	0.124740
$482$	0	0
$483$	− 1200.00i	− 2.48447i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−713.605	−1.46531	−0.732654	−	0.680601i	$-0.761719\pi$
$487$	−713.605	−1.46531	−0.732654	+	0.680601i	$0.761719\pi$
$488$	0	0
$489$	− 173.205i	− 0.354203i
$490$	0	0
$491$	329.090i	0.670244i	0.942175	+	0.335122i	$0.108777\pi$
$491$	329.090i	0.670244i	−0.942175	+	0.335122i	$0.891223\pi$
$492$	0	0
$493$	240.000i	0.486815i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 660.000i	− 1.32797i
$498$	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
$503$	30.0000i	0.0596421i	0.999555	+	0.0298211i	$0.00949375\pi$
$503$	30.0000i	0.0596421i	−0.999555	+	0.0298211i	$0.990506\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	453.797	0.895064
$508$	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
$513$	−290.985	−0.567221
$514$	0	0
$515$	0	0
$516$	0	0
$517$	519.615	1.00506
$518$	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
$523$	481.510	0.920669	0.460335	−	0.887745i	$-0.347729\pi$
$523$	481.510	0.920669	0.460335	+	0.887745i	$0.347729\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	429.549	0.815083
$528$	0	0
$529$	671.000	1.26843
$530$	0	0
$531$	18.0000	0.0338983
$532$	0	0
$533$	935.307	1.75480
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 540.000i	− 1.00559i
$538$	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
$543$	660.000i	1.21547i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	610.000i	1.11517i	0.830119	+	0.557587i	$0.188273\pi$
$547$	610.000i	1.11517i	−0.830119	+	0.557587i	$0.811727\pi$
$548$	0	0
$549$	51.9615i	0.0946476i
$550$	0	0
$551$	− 242.487i	− 0.440086i
$552$	0	0
$553$	−692.820	−1.25284
$554$	0	0
$555$	0	0
$556$	0	0
$557$	190.526	0.342057	0.171028	−	0.985266i	$-0.445291\pi$
$557$	190.526	0.342057	0.171028	+	0.985266i	$0.445291\pi$
$558$	0	0
$559$	−1260.00	−2.25403
$560$	0	0
$561$	831.384i	1.48197i
$562$	0	0
$563$	− 90.0000i	− 0.159858i	−0.996801	−	0.0799290i	$-0.974531\pi$
$563$	− 90.0000i	− 0.159858i	0.996801	−	0.0799290i	$-0.0254693\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	990.000i	1.74603i
$568$	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	$0.0533547\pi$
$571$	190.526i	0.333670i	−0.985985	+	0.166835i	$0.946645\pi$
$572$	0	0
$573$	−62.3538	−0.108820
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 50.0000i	− 0.0866551i	−0.999061	−	0.0433276i	$-0.986204\pi$
$577$	− 50.0000i	− 0.0866551i	0.999061	−	0.0433276i	$-0.0137959\pi$
$578$	0	0
$579$	381.051i	0.658119i
$580$	0	0
$581$	381.051i	0.655854i
$582$	0	0
$583$	540.000i	0.926244i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	433.013	0.737671	0.368835	−	0.929495i	$-0.379757\pi$
$587$	433.013	0.737671	0.368835	+	0.929495i	$0.379757\pi$
$588$	0	0
$589$	−434.000	−0.736842
$590$	0	0
$591$	−180.000	−0.304569
$592$	0	0
$593$	− 30.0000i	− 0.0505902i	−0.999680	−	0.0252951i	$-0.991947\pi$
$593$	− 30.0000i	− 0.0505902i	0.999680	−	0.0252951i	$-0.00805254\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	120.000i	0.201005i
$598$	0	0
$599$	−198.000	−0.330551	−0.165275	−	0.986247i	$-0.552851\pi$
$599$	−198.000	−0.330551	−0.165275	+	0.986247i	$0.552851\pi$
$600$	0	0
$601$	519.615i	0.864584i	0.901734	+	0.432292i	$0.142295\pi$
$601$	519.615i	0.864584i	−0.901734	+	0.432292i	$0.857705\pi$
$602$	0	0
$603$	− 330.000i	− 0.547264i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	130.000i	0.214168i	0.994250	+	0.107084i	$0.0341514\pi$
$607$	130.000i	0.214168i	−0.994250	+	0.107084i	$0.965849\pi$
$608$	0	0
$609$	−600.000	−0.985222
$610$	0	0
$611$	519.615i	0.850434i
$612$	0	0
$613$	−142.028	−0.231694	−0.115847	−	0.993267i	$-0.536958\pi$
$613$	−142.028	−0.231694	−0.115847	+	0.993267i	$0.536958\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 990.000i	− 1.60454i	−0.596963	−	0.802269i	$-0.703626\pi$
$617$	− 990.000i	− 1.60454i	0.596963	−	0.802269i	$-0.296374\pi$
$618$	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
$623$	346.410	0.556036
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 840.000i	− 1.33971i
$628$	0	0
$629$	−48.0000	−0.0763116
$630$	0	0
$631$	34.6410i	0.0548986i	0.999623	+	0.0274493i	$0.00873848\pi$
$631$	34.6410i	0.0548986i	−0.999623	+	0.0274493i	$0.991262\pi$
$632$	0	0
$633$	−76.2102	−0.120395
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−883.346	−1.38673
$638$	0	0
$639$	198.000	0.309859
$640$	0	0
$641$	− 831.384i	− 1.29701i	−0.761210	−	0.648506i	$-0.775394\pi$
$641$	− 831.384i	− 1.29701i	0.761210	−	0.648506i	$-0.224606\pi$
$642$	0	0
$643$	31.1769	0.0484866	0.0242433	−	0.999706i	$-0.492282\pi$
$643$	31.1769	0.0484866	0.0242433	+	0.999706i	$0.492282\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−464.190	−0.717449	−0.358725	−	0.933443i	$-0.616788\pi$
$647$	−464.190	−0.717449	−0.358725	+	0.933443i	$0.616788\pi$
$648$	0	0
$649$	− 103.923i	− 0.160128i
$650$	0	0
$651$	1073.87i	1.64957i
$652$	0	0
$653$	− 1170.00i	− 1.79173i	−0.444326	−	0.895865i	$-0.646557\pi$
$653$	− 1170.00i	− 1.79173i	0.444326	−	0.895865i	$-0.353443\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	270.200	0.411263
$658$	0	0
$659$	−66.0000	−0.100152	−0.0500759	−	0.998745i	$-0.515946\pi$
$659$	−66.0000	−0.100152	−0.0500759	+	0.998745i	$0.515946\pi$
$660$	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
$663$	−831.384	−1.25397
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 600.000i	− 0.899550i
$668$	0	0
$669$	1152.00	1.72197
$670$	0	0
$671$	300.000	0.447094
$672$	0	0
$673$	−55.4256	−0.0823561	−0.0411780	−	0.999152i	$-0.513111\pi$
$673$	−55.4256	−0.0823561	−0.0411780	+	0.999152i	$0.513111\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−897.202	−1.32526	−0.662631	−	0.748946i	$-0.730560\pi$
$677$	−897.202	−1.32526	−0.662631	+	0.748946i	$0.730560\pi$
$678$	0	0
$679$	1100.00	1.62003
$680$	0	0
$681$	− 1351.00i	− 1.98385i
$682$	0	0
$683$	630.000i	0.922401i	0.887296	+	0.461201i	$0.152581\pi$
$683$	630.000i	0.922401i	−0.887296	+	0.461201i	$0.847419\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	1260.00i	1.83406i
$688$	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
$693$	−519.615	−0.749806
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−748.246	−1.07352
$698$	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
$703$	48.4974	0.0689864
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1740.00i	2.46110i
$708$	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
$713$	−1073.87	−1.50613
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 240.000i	− 0.334728i
$718$	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
$723$	1200.00i	1.65975i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	290.000i	0.398900i	0.979908	+	0.199450i	$0.0639155\pi$
$727$	290.000i	0.398900i	−0.979908	+	0.199450i	$0.936085\pi$
$728$	0	0
$729$	351.000	0.481481
$730$	0	0
$731$	1008.00	1.37893
$732$	0	0
$733$	230.000i	0.313779i	0.987616	+	0.156889i	$0.0501467\pi$
$733$	230.000i	0.313779i	−0.987616	+	0.156889i	$0.949853\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1905.26	−2.58515
$738$	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
$743$	1295.57	1.74371	0.871853	−	0.489767i	$-0.162918\pi$
$743$	1295.57	1.74371	0.871853	+	0.489767i	$0.162918\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−114.315	−0.153033
$748$	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
$753$	660.000i	0.876494i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−869.490	−1.14860	−0.574300	−	0.818645i	$-0.694726\pi$
$757$	−869.490	−1.14860	−0.574300	+	0.818645i	$0.694726\pi$
$758$	0	0
$759$	− 2078.46i	− 2.73842i
$760$	0	0
$761$	− 311.769i	− 0.409684i	−0.978795	−	0.204842i	$-0.934332\pi$
$761$	− 311.769i	− 0.409684i	0.978795	−	0.204842i	$-0.0656680\pi$
$762$	0	0
$763$	260.000i	0.340760i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	103.923	0.135493
$768$	0	0
$769$	626.000	0.814044	0.407022	−	0.913418i	$-0.366567\pi$
$769$	626.000	0.814044	0.407022	+	0.913418i	$0.366567\pi$
$770$	0	0
$771$	− 935.307i	− 1.21311i
$772$	0	0
$773$	259.808	0.336103	0.168052	−	0.985778i	$-0.446253\pi$
$773$	259.808	0.336103	0.168052	+	0.985778i	$0.446253\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 120.000i	− 0.154440i
$778$	0	0
$779$	756.000	0.970475
$780$	0	0
$781$	− 1143.15i	− 1.46370i
$782$	0	0
$783$	360.000i	0.459770i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	398.372	0.506190	0.253095	−	0.967441i	$-0.418551\pi$
$787$	398.372	0.506190	0.253095	+	0.967441i	$0.418551\pi$
$788$	0	0
$789$	−408.000	−0.517110
$790$	0	0
$791$	−300.000	−0.379267
$792$	0	0
$793$	300.000i	0.378310i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	952.628	1.19527	0.597634	−	0.801769i	$-0.296108\pi$
$797$	952.628	1.19527	0.597634	+	0.801769i	$0.296108\pi$
$798$	0	0
$799$	− 415.692i	− 0.520266i
$800$	0	0
$801$	103.923i	0.129742i
$802$	0	0
$803$	− 1560.00i	− 1.94271i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	420.000i	0.520446i
$808$	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
$813$	− 1680.00i	− 2.06642i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−1018.45	−1.24657
$818$	0	0
$819$	− 519.615i	− 0.634451i
$820$	0	0
$821$	− 571.577i	− 0.696196i	−0.937458	−	0.348098i	$-0.886828\pi$
$821$	− 571.577i	− 0.696196i	0.937458	−	0.348098i	$-0.113172\pi$
$822$	0	0
$823$	−741.318	−0.900751	−0.450375	−	0.892839i	$-0.648710\pi$
$823$	−741.318	−0.900751	−0.450375	+	0.892839i	$0.648710\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	446.869	0.540350	0.270175	−	0.962811i	$-0.412919\pi$
$827$	446.869	0.540350	0.270175	+	0.962811i	$0.412919\pi$
$828$	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
$833$	706.677	0.848351
$834$	0	0
$835$	0	0
$836$	0	0
$837$	644.323	0.769800
$838$	0	0
$839$	234.000	0.278903	0.139452	−	0.990229i	$-0.455466\pi$
$839$	234.000	0.278903	0.139452	+	0.990229i	$0.455466\pi$
$840$	0	0
$841$	541.000	0.643282
$842$	0	0
$843$	228.631	0.271211
$844$	0	0
$845$	0	0
$846$	0	0
$847$	1790.00i	2.11334i
$848$	0	0
$849$	− 1420.28i	− 1.67289i
$850$	0	0
$851$	120.000	0.141011
$852$	0	0
$853$	− 370.000i	− 0.433763i	−0.976198	−	0.216882i	$-0.930411\pi$
$853$	− 370.000i	− 0.433763i	0.976198	−	0.216882i	$-0.0695886\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	450.000i	0.525088i	0.964920	+	0.262544i	$0.0845614\pi$
$857$	450.000i	0.525088i	−0.964920	+	0.262544i	$0.915439\pi$
$858$	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
$863$	−471.118	−0.545907	−0.272954	−	0.962027i	$-0.588001\pi$
$863$	−471.118	−0.545907	−0.272954	+	0.962027i	$0.588001\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−336.018	−0.387564
$868$	0	0
$869$	−1200.00	−1.38090
$870$	0	0
$871$	− 1905.26i	− 2.18744i
$872$	0	0
$873$	330.000i	0.378007i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 1310.00i	− 1.49373i	−0.664976	−	0.746864i	$-0.731558\pi$
$877$	− 1310.00i	− 1.49373i	0.664976	−	0.746864i	$-0.268442\pi$
$878$	0	0
$879$	519.615i	0.591144i
$880$	0	0
$881$	762.102i	0.865042i	0.901624	+	0.432521i	$0.142376\pi$
$881$	762.102i	0.865042i	−0.901624	+	0.432521i	$0.857624\pi$
$882$	0	0
$883$	−620.074	−0.702236	−0.351118	−	0.936331i	$-0.614198\pi$
$883$	−620.074	−0.702236	−0.351118	+	0.936331i	$0.614198\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1530.00i	− 1.72492i	−0.506129	−	0.862458i	$-0.668924\pi$
$887$	− 1530.00i	− 1.72492i	0.506129	−	0.862458i	$-0.331076\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	1714.73i	1.92450i
$892$	0	0
$893$	420.000i	0.470325i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	2078.46	2.31712
$898$	0	0
$899$	536.936i	0.597259i
$900$	0	0
$901$	432.000	0.479467
$902$	0	0
$903$	2520.00i	2.79070i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1450.00i	1.59868i	0.600881	+	0.799338i	$0.294817\pi$
$907$	1450.00i	1.59868i	−0.600881	+	0.799338i	$0.705183\pi$
$908$	0	0
$909$	−522.000	−0.574257
$910$	0	0
$911$	762.102i	0.836556i	0.908319	+	0.418278i	$0.137366\pi$
$911$	762.102i	0.836556i	−0.908319	+	0.418278i	$0.862634\pi$
$912$	0	0
$913$	660.000i	0.722892i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 660.000i	− 0.719738i
$918$	0	0
$919$	−226.000	−0.245919	−0.122960	−	0.992412i	$-0.539239\pi$
$919$	−226.000	−0.245919	−0.122960	+	0.992412i	$0.539239\pi$
$920$	0	0
$921$	1281.72i	1.39166i
$922$	0	0
$923$	1143.15	1.23852
$924$	0	0
$925$	0	0
$926$	0	0
$927$	330.000i	0.355987i
$928$	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
$933$	2057.68	2.20544
$934$	0	0
$935$	0	0
$936$	0	0
$937$	1570.00i	1.67556i	0.546008	+	0.837780i	$0.316147\pi$
$937$	1570.00i	1.67556i	−0.546008	+	0.837780i	$0.683853\pi$
$938$	0	0
$939$	1032.00	1.09904
$940$	0	0
$941$	571.577i	0.607414i	0.952765	+	0.303707i	$0.0982244\pi$
$941$	571.577i	0.607414i	−0.952765	+	0.303707i	$0.901776\pi$
$942$	0	0
$943$	1870.61	1.98368
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−72.7461	−0.0768175	−0.0384087	−	0.999262i	$-0.512229\pi$
$947$	−72.7461	−0.0768175	−0.0384087	+	0.999262i	$0.512229\pi$
$948$	0	0
$949$	1560.00	1.64384
$950$	0	0
$951$	− 935.307i	− 0.983499i
$952$	0	0
$953$	−394.908	−0.414384	−0.207192	−	0.978300i	$-0.566432\pi$
$953$	−394.908	−0.414384	−0.207192	+	0.978300i	$0.566432\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−1039.23	−1.08593
$958$	0	0
$959$	2286.31i	2.38405i
$960$	0	0
$961$	961.000	1.00000
$962$	0	0
$963$	− 90.0000i	− 0.0934579i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	34.6410	0.0358232	0.0179116	−	0.999840i	$-0.494298\pi$
$967$	34.6410	0.0358232	0.0179116	+	0.999840i	$0.494298\pi$
$968$	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
$973$	−1905.26	−1.95813
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 270.000i	− 0.276356i	−0.990407	−	0.138178i	$-0.955875\pi$
$977$	− 270.000i	− 0.276356i	0.990407	−	0.138178i	$-0.0441246\pi$
$978$	0	0
$979$	600.000	0.612870
$980$	0	0
$981$	−78.0000	−0.0795107
$982$	0	0
$983$	−796.743	−0.810522	−0.405261	−	0.914201i	$-0.632819\pi$
$983$	−796.743	−0.810522	−0.405261	+	0.914201i	$0.632819\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	1039.23	1.05292
$988$	0	0
$989$	−2520.00	−2.54803
$990$	0	0
$991$	277.128i	0.279645i	0.990177	+	0.139822i	$0.0446532\pi$
$991$	277.128i	0.279645i	−0.990177	+	0.139822i	$0.955347\pi$
$992$	0	0
$993$	2100.00i	2.11480i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 70.0000i	− 0.0702106i	−0.999384	−	0.0351053i	$-0.988823\pi$
$997$	− 70.0000i	− 0.0702106i	0.999384	−	0.0351053i	$-0.0111767\pi$
$998$	0	0
$999$	−72.0000	−0.0720721

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
124.3.c.a.61.1	✓	2	155.123	even	4
124.3.c.a.61.2	yes	2	5.3	odd	4
496.3.e.a.433.1		2	20.3	even	4
496.3.e.a.433.2		2	620.123	odd	4
1116.3.h.c.433.1		2	465.278	odd	4
1116.3.h.c.433.2		2	15.8	even	4
3100.3.d.a.1301.1		2	5.2	odd	4
3100.3.d.a.1301.2		2	155.92	even	4
3100.3.f.a.1549.1		4	31.30	odd	2	inner
3100.3.f.a.1549.2		4	5.4	even	2	inner
3100.3.f.a.1549.3		4	1.1	even	1	trivial
3100.3.f.a.1549.4		4	155.154	odd	2	inner

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:	\( N \)	\(=\)	\( 3100 = 2^{2} \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	3100.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.46410 q^{3} -10.0000i q^{7} +3.00000 q^{9} +O(q^{10})\)	\(q+3.46410 q^{3} -10.0000i q^{7} +3.00000 q^{9} -17.3205i q^{11} +17.3205 q^{13} -13.8564 q^{17} +14.0000 q^{19} -34.6410i q^{21} +34.6410 q^{23} -20.7846 q^{27} -17.3205i q^{29} -31.0000 q^{31} -60.0000i q^{33} +3.46410 q^{37} +60.0000 q^{39} +54.0000 q^{41} -72.7461 q^{43} +30.0000i q^{47} -51.0000 q^{49} -48.0000 q^{51} -31.1769 q^{53} +48.4974 q^{57} +6.00000 q^{59} +17.3205i q^{61} -30.0000i q^{63} -110.000i q^{67} +120.000 q^{69} +66.0000 q^{71} +90.0666 q^{73} -173.205 q^{77} -69.2820i q^{79} -99.0000 q^{81} -38.1051 q^{83} -60.0000i q^{87} +34.6410i q^{89} -173.205i q^{91} -107.387 q^{93} +110.000i q^{97} -51.9615i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{9}+O(q^{10}) \) 4 * q + 12 * q^9	\( 4 q + 12 q^{9} + 56 q^{19} - 124 q^{31} + 240 q^{39} + 216 q^{41} - 204 q^{49} - 192 q^{51} + 24 q^{59} + 480 q^{69} + 264 q^{71} - 396 q^{81}+O(q^{100}) \) 4 * q + 12 * q^9 + 56 * q^19 - 124 * q^31 + 240 * q^39 + 216 * q^41 - 204 * q^49 - 192 * q^51 + 24 * q^59 + 480 * q^69 + 264 * q^71 - 396 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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