LMFDB - Embedded newform 2352.3.f.b.97.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2352,3,Mod(97,2352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2352, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("2352.97");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2352 = 2^{4} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2352.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:	$64.0873581775$
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 $
Twist minimal:	no (minimal twist has level 21)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			97.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	2352.97
Dual form			2352.3.f.b.97.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{3} -3.46410i q^{5} -3.00000 q^{9} +O(q^{10})$	$q-1.73205i q^{3} -3.46410i q^{5} -3.00000 q^{9} -10.0000 q^{11} +12.1244i q^{13} -6.00000 q^{15} +6.92820i q^{17} -32.9090i q^{19} -40.0000 q^{23} +13.0000 q^{25} +5.19615i q^{27} +16.0000 q^{29} +5.19615i q^{31} +17.3205i q^{33} +5.00000 q^{37} +21.0000 q^{39} +24.2487i q^{41} +19.0000 q^{43} +10.3923i q^{45} +51.9615i q^{47} +12.0000 q^{51} -32.0000 q^{53} +34.6410i q^{55} -57.0000 q^{57} +41.5692i q^{59} +20.7846i q^{61} +42.0000 q^{65} -59.0000 q^{67} +69.2820i q^{69} +26.0000 q^{71} +19.0526i q^{73} -22.5167i q^{75} -47.0000 q^{79} +9.00000 q^{81} +24.2487i q^{83} +24.0000 q^{85} -27.7128i q^{87} +117.779i q^{89} +9.00000 q^{93} -114.000 q^{95} -48.4974i q^{97} +30.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{9}+O(q^{10}) $ 2 * q - 6 * q^9	$ 2 q - 6 q^{9} - 20 q^{11} - 12 q^{15} - 80 q^{23} + 26 q^{25} + 32 q^{29} + 10 q^{37} + 42 q^{39} + 38 q^{43} + 24 q^{51} - 64 q^{53} - 114 q^{57} + 84 q^{65} - 118 q^{67} + 52 q^{71} - 94 q^{79} + 18 q^{81} + 48 q^{85} + 18 q^{93} - 228 q^{95} + 60 q^{99}+O(q^{100}) $ 2 * q - 6 * q^9 - 20 * q^11 - 12 * q^15 - 80 * q^23 + 26 * q^25 + 32 * q^29 + 10 * q^37 + 42 * q^39 + 38 * q^43 + 24 * q^51 - 64 * q^53 - 114 * q^57 + 84 * q^65 - 118 * q^67 + 52 * q^71 - 94 * q^79 + 18 * q^81 + 48 * q^85 + 18 * q^93 - 228 * q^95 + 60 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times$.

$n$	$785$	$1471$	$1765$	$2257$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	− 1.73205i	− 0.577350i
$3$	− 1.73205i	− 0.577350i
$4$	0	0
$5$	− 3.46410i	− 0.692820i	−0.938083	−	0.346410i	$-0.887401\pi$
$5$	− 3.46410i	− 0.692820i	0.938083	−	0.346410i	$-0.112599\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−3.00000	−0.333333
$10$	0	0
$11$	−10.0000	−0.909091	−0.454545	−	0.890724i	$-0.650198\pi$
$11$	−10.0000	−0.909091	−0.454545	+	0.890724i	$0.650198\pi$
$12$	0	0
$13$	12.1244i	0.932643i	0.884615	+	0.466321i	$0.154421\pi$
$13$	12.1244i	0.932643i	−0.884615	+	0.466321i	$0.845579\pi$
$14$	0	0
$15$	−6.00000	−0.400000
$16$	0	0
$17$	6.92820i	0.407541i	0.979019	+	0.203771i	$0.0653197\pi$
$17$	6.92820i	0.407541i	−0.979019	+	0.203771i	$0.934680\pi$
$18$	0	0
$19$	− 32.9090i	− 1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$19$	− 32.9090i	− 1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−40.0000	−1.73913	−0.869565	−	0.493818i	$-0.835601\pi$
$23$	−40.0000	−1.73913	−0.869565	+	0.493818i	$0.835601\pi$
$24$	0	0
$25$	13.0000	0.520000
$26$	0	0
$27$	5.19615i	0.192450i
$28$	0	0
$29$	16.0000	0.551724	0.275862	−	0.961197i	$-0.411037\pi$
$29$	16.0000	0.551724	0.275862	+	0.961197i	$0.411037\pi$
$30$	0	0
$31$	5.19615i	0.167618i	0.996482	+	0.0838089i	$0.0267085\pi$
$31$	5.19615i	0.167618i	−0.996482	+	0.0838089i	$0.973291\pi$
$32$	0	0
$33$	17.3205i	0.524864i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	5.00000	0.135135	0.0675676	−	0.997715i	$-0.478476\pi$
$37$	5.00000	0.135135	0.0675676	+	0.997715i	$0.478476\pi$
$38$	0	0
$39$	21.0000	0.538462
$40$	0	0
$41$	24.2487i	0.591432i	0.955276	+	0.295716i	$0.0955582\pi$
$41$	24.2487i	0.591432i	−0.955276	+	0.295716i	$0.904442\pi$
$42$	0	0
$43$	19.0000	0.441860	0.220930	−	0.975290i	$-0.429091\pi$
$43$	19.0000	0.441860	0.220930	+	0.975290i	$0.429091\pi$
$44$	0	0
$45$	10.3923i	0.230940i
$46$	0	0
$47$	51.9615i	1.10556i	0.833326	+	0.552782i	$0.186434\pi$
$47$	51.9615i	1.10556i	−0.833326	+	0.552782i	$0.813566\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	12.0000	0.235294
$52$	0	0
$53$	−32.0000	−0.603774	−0.301887	−	0.953344i	$-0.597616\pi$
$53$	−32.0000	−0.603774	−0.301887	+	0.953344i	$0.597616\pi$
$54$	0	0
$55$	34.6410i	0.629837i
$56$	0	0
$57$	−57.0000	−1.00000
$58$	0	0
$59$	41.5692i	0.704563i	0.935894	+	0.352282i	$0.114594\pi$
$59$	41.5692i	0.704563i	−0.935894	+	0.352282i	$0.885406\pi$
$60$	0	0
$61$	20.7846i	0.340731i	0.985381	+	0.170366i	$0.0544949\pi$
$61$	20.7846i	0.340731i	−0.985381	+	0.170366i	$0.945505\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	42.0000	0.646154
$66$	0	0
$67$	−59.0000	−0.880597	−0.440299	−	0.897851i	$-0.645127\pi$
$67$	−59.0000	−0.880597	−0.440299	+	0.897851i	$0.645127\pi$
$68$	0	0
$69$	69.2820i	1.00409i
$70$	0	0
$71$	26.0000	0.366197	0.183099	−	0.983095i	$-0.441387\pi$
$71$	26.0000	0.366197	0.183099	+	0.983095i	$0.441387\pi$
$72$	0	0
$73$	19.0526i	0.260994i	0.991449	+	0.130497i	$0.0416573\pi$
$73$	19.0526i	0.260994i	−0.991449	+	0.130497i	$0.958343\pi$
$74$	0	0
$75$	− 22.5167i	− 0.300222i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−47.0000	−0.594937	−0.297468	−	0.954732i	$-0.596142\pi$
$79$	−47.0000	−0.594937	−0.297468	+	0.954732i	$0.596142\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	24.2487i	0.292153i	0.989273	+	0.146077i	$0.0466646\pi$
$83$	24.2487i	0.292153i	−0.989273	+	0.146077i	$0.953335\pi$
$84$	0	0
$85$	24.0000	0.282353
$86$	0	0
$87$	− 27.7128i	− 0.318538i
$88$	0	0
$89$	117.779i	1.32336i	0.749784	+	0.661682i	$0.230157\pi$
$89$	117.779i	1.32336i	−0.749784	+	0.661682i	$0.769843\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	9.00000	0.0967742
$94$	0	0
$95$	−114.000	−1.20000
$96$	0	0
$97$	− 48.4974i	− 0.499973i	−0.968249	−	0.249987i	$-0.919574\pi$
$97$	− 48.4974i	− 0.499973i	0.968249	−	0.249987i	$-0.0804263\pi$
$98$	0	0
$99$	30.0000	0.303030
$100$	0	0
$101$	128.172i	1.26903i	0.772912	+	0.634514i	$0.218800\pi$
$101$	128.172i	1.26903i	−0.772912	+	0.634514i	$0.781200\pi$
$102$	0	0
$103$	− 8.66025i	− 0.0840801i	−0.999116	−	0.0420401i	$-0.986614\pi$
$103$	− 8.66025i	− 0.0840801i	0.999116	−	0.0420401i	$-0.0133857\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	212.000	1.98131	0.990654	−	0.136397i	$-0.0435524\pi$
$107$	212.000	1.98131	0.990654	+	0.136397i	$0.0435524\pi$
$108$	0	0
$109$	17.0000	0.155963	0.0779817	−	0.996955i	$-0.475152\pi$
$109$	17.0000	0.155963	0.0779817	+	0.996955i	$0.475152\pi$
$110$	0	0
$111$	− 8.66025i	− 0.0780203i
$112$	0	0
$113$	142.000	1.25664	0.628319	−	0.777956i	$-0.283743\pi$
$113$	142.000	1.25664	0.628319	+	0.777956i	$0.283743\pi$
$114$	0	0
$115$	138.564i	1.20490i
$116$	0	0
$117$	− 36.3731i	− 0.310881i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−21.0000	−0.173554
$122$	0	0
$123$	42.0000	0.341463
$124$	0	0
$125$	− 131.636i	− 1.05309i
$126$	0	0
$127$	145.000	1.14173	0.570866	−	0.821043i	$-0.306608\pi$
$127$	145.000	1.14173	0.570866	+	0.821043i	$0.306608\pi$
$128$	0	0
$129$	− 32.9090i	− 0.255108i
$130$	0	0
$131$	148.956i	1.13707i	0.822659	+	0.568536i	$0.192490\pi$
$131$	148.956i	1.13707i	−0.822659	+	0.568536i	$0.807510\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	18.0000	0.133333
$136$	0	0
$137$	−116.000	−0.846715	−0.423358	−	0.905963i	$-0.639149\pi$
$137$	−116.000	−0.846715	−0.423358	+	0.905963i	$0.639149\pi$
$138$	0	0
$139$	− 84.8705i	− 0.610579i	−0.952260	−	0.305290i	$-0.901247\pi$
$139$	− 84.8705i	− 0.610579i	0.952260	−	0.305290i	$-0.0987532\pi$
$140$	0	0
$141$	90.0000	0.638298
$142$	0	0
$143$	− 121.244i	− 0.847857i
$144$	0	0
$145$	− 55.4256i	− 0.382246i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	124.000	0.832215	0.416107	−	0.909315i	$-0.363394\pi$
$149$	124.000	0.832215	0.416107	+	0.909315i	$0.363394\pi$
$150$	0	0
$151$	46.0000	0.304636	0.152318	−	0.988332i	$-0.451326\pi$
$151$	46.0000	0.304636	0.152318	+	0.988332i	$0.451326\pi$
$152$	0	0
$153$	− 20.7846i	− 0.135847i
$154$	0	0
$155$	18.0000	0.116129
$156$	0	0
$157$	− 187.061i	− 1.19147i	−0.803179	−	0.595737i	$-0.796860\pi$
$157$	− 187.061i	− 1.19147i	0.803179	−	0.595737i	$-0.203140\pi$
$158$	0	0
$159$	55.4256i	0.348589i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	58.0000	0.355828	0.177914	−	0.984046i	$-0.443065\pi$
$163$	58.0000	0.355828	0.177914	+	0.984046i	$0.443065\pi$
$164$	0	0
$165$	60.0000	0.363636
$166$	0	0
$167$	266.736i	1.59722i	0.601849	+	0.798610i	$0.294431\pi$
$167$	266.736i	1.59722i	−0.601849	+	0.798610i	$0.705569\pi$
$168$	0	0
$169$	22.0000	0.130178
$170$	0	0
$171$	98.7269i	0.577350i
$172$	0	0
$173$	− 124.708i	− 0.720854i	−0.932788	−	0.360427i	$-0.882631\pi$
$173$	− 124.708i	− 0.720854i	0.932788	−	0.360427i	$-0.117369\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	72.0000	0.406780
$178$	0	0
$179$	−10.0000	−0.0558659	−0.0279330	−	0.999610i	$-0.508892\pi$
$179$	−10.0000	−0.0558659	−0.0279330	+	0.999610i	$0.508892\pi$
$180$	0	0
$181$	327.358i	1.80861i	0.426892	+	0.904303i	$0.359609\pi$
$181$	327.358i	1.80861i	−0.426892	+	0.904303i	$0.640391\pi$
$182$	0	0
$183$	36.0000	0.196721
$184$	0	0
$185$	− 17.3205i	− 0.0936244i
$186$	0	0
$187$	− 69.2820i	− 0.370492i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2.00000	0.0104712	0.00523560	−	0.999986i	$-0.498333\pi$
$191$	2.00000	0.0104712	0.00523560	+	0.999986i	$0.498333\pi$
$192$	0	0
$193$	−235.000	−1.21762	−0.608808	−	0.793317i	$-0.708352\pi$
$193$	−235.000	−1.21762	−0.608808	+	0.793317i	$0.708352\pi$
$194$	0	0
$195$	− 72.7461i	− 0.373057i
$196$	0	0
$197$	100.000	0.507614	0.253807	−	0.967255i	$-0.418317\pi$
$197$	100.000	0.507614	0.253807	+	0.967255i	$0.418317\pi$
$198$	0	0
$199$	− 200.918i	− 1.00964i	−0.863225	−	0.504819i	$-0.831559\pi$
$199$	− 200.918i	− 1.00964i	0.863225	−	0.504819i	$-0.168441\pi$
$200$	0	0
$201$	102.191i	0.508413i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	84.0000	0.409756
$206$	0	0
$207$	120.000	0.579710
$208$	0	0
$209$	329.090i	1.57459i
$210$	0	0
$211$	−2.00000	−0.00947867	−0.00473934	−	0.999989i	$-0.501509\pi$
$211$	−2.00000	−0.00947867	−0.00473934	+	0.999989i	$0.501509\pi$
$212$	0	0
$213$	− 45.0333i	− 0.211424i
$214$	0	0
$215$	− 65.8179i	− 0.306130i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	33.0000	0.150685
$220$	0	0
$221$	−84.0000	−0.380090
$222$	0	0
$223$	339.482i	1.52234i	0.648552	+	0.761170i	$0.275375\pi$
$223$	339.482i	1.52234i	−0.648552	+	0.761170i	$0.724625\pi$
$224$	0	0
$225$	−39.0000	−0.173333
$226$	0	0
$227$	162.813i	0.717237i	0.933484	+	0.358618i	$0.116752\pi$
$227$	162.813i	0.717237i	−0.933484	+	0.358618i	$0.883248\pi$
$228$	0	0
$229$	8.66025i	0.0378177i	0.999821	+	0.0189089i	$0.00601923\pi$
$229$	8.66025i	0.0378177i	−0.999821	+	0.0189089i	$0.993981\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−170.000	−0.729614	−0.364807	−	0.931083i	$-0.618865\pi$
$233$	−170.000	−0.729614	−0.364807	+	0.931083i	$0.618865\pi$
$234$	0	0
$235$	180.000	0.765957
$236$	0	0
$237$	81.4064i	0.343487i
$238$	0	0
$239$	−142.000	−0.594142	−0.297071	−	0.954855i	$-0.596010\pi$
$239$	−142.000	−0.594142	−0.297071	+	0.954855i	$0.596010\pi$
$240$	0	0
$241$	152.420i	0.632450i	0.948684	+	0.316225i	$0.102415\pi$
$241$	152.420i	0.632450i	−0.948684	+	0.316225i	$0.897585\pi$
$242$	0	0
$243$	− 15.5885i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	399.000	1.61538
$248$	0	0
$249$	42.0000	0.168675
$250$	0	0
$251$	290.985i	1.15930i	0.814865	+	0.579650i	$0.196811\pi$
$251$	290.985i	1.15930i	−0.814865	+	0.579650i	$0.803189\pi$
$252$	0	0
$253$	400.000	1.58103
$254$	0	0
$255$	− 41.5692i	− 0.163017i
$256$	0	0
$257$	− 439.941i	− 1.71183i	−0.517115	−	0.855916i	$-0.672994\pi$
$257$	− 439.941i	− 1.71183i	0.517115	−	0.855916i	$-0.327006\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−48.0000	−0.183908
$262$	0	0
$263$	−136.000	−0.517110	−0.258555	−	0.965997i	$-0.583246\pi$
$263$	−136.000	−0.517110	−0.258555	+	0.965997i	$0.583246\pi$
$264$	0	0
$265$	110.851i	0.418307i
$266$	0	0
$267$	204.000	0.764045
$268$	0	0
$269$	225.167i	0.837051i	0.908205	+	0.418525i	$0.137453\pi$
$269$	225.167i	0.837051i	−0.908205	+	0.418525i	$0.862547\pi$
$270$	0	0
$271$	367.195i	1.35496i	0.735540	+	0.677481i	$0.236928\pi$
$271$	367.195i	1.35496i	−0.735540	+	0.677481i	$0.763072\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−130.000	−0.472727
$276$	0	0
$277$	395.000	1.42599	0.712996	−	0.701168i	$-0.247338\pi$
$277$	395.000	1.42599	0.712996	+	0.701168i	$0.247338\pi$
$278$	0	0
$279$	− 15.5885i	− 0.0558726i
$280$	0	0
$281$	100.000	0.355872	0.177936	−	0.984042i	$-0.443058\pi$
$281$	100.000	0.355872	0.177936	+	0.984042i	$0.443058\pi$
$282$	0	0
$283$	− 358.535i	− 1.26691i	−0.773781	−	0.633453i	$-0.781637\pi$
$283$	− 358.535i	− 1.26691i	0.773781	−	0.633453i	$-0.218363\pi$
$284$	0	0
$285$	197.454i	0.692820i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	241.000	0.833910
$290$	0	0
$291$	−84.0000	−0.288660
$292$	0	0
$293$	242.487i	0.827601i	0.910368	+	0.413801i	$0.135799\pi$
$293$	242.487i	0.827601i	−0.910368	+	0.413801i	$0.864201\pi$
$294$	0	0
$295$	144.000	0.488136
$296$	0	0
$297$	− 51.9615i	− 0.174955i
$298$	0	0
$299$	− 484.974i	− 1.62199i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	222.000	0.732673
$304$	0	0
$305$	72.0000	0.236066
$306$	0	0
$307$	− 181.865i	− 0.592395i	−0.955127	−	0.296198i	$-0.904281\pi$
$307$	− 181.865i	− 0.592395i	0.955127	−	0.296198i	$-0.0957187\pi$
$308$	0	0
$309$	−15.0000	−0.0485437
$310$	0	0
$311$	550.792i	1.77104i	0.464605	+	0.885518i	$0.346196\pi$
$311$	550.792i	1.77104i	−0.464605	+	0.885518i	$0.653804\pi$
$312$	0	0
$313$	202.650i	0.647444i	0.946152	+	0.323722i	$0.104934\pi$
$313$	202.650i	0.647444i	−0.946152	+	0.323722i	$0.895066\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	292.000	0.921136	0.460568	−	0.887625i	$-0.347646\pi$
$317$	292.000	0.921136	0.460568	+	0.887625i	$0.347646\pi$
$318$	0	0
$319$	−160.000	−0.501567
$320$	0	0
$321$	− 367.195i	− 1.14391i
$322$	0	0
$323$	228.000	0.705882
$324$	0	0
$325$	157.617i	0.484974i
$326$	0	0
$327$	− 29.4449i	− 0.0900455i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−5.00000	−0.0151057	−0.00755287	−	0.999971i	$-0.502404\pi$
$331$	−5.00000	−0.0151057	−0.00755287	+	0.999971i	$0.502404\pi$
$332$	0	0
$333$	−15.0000	−0.0450450
$334$	0	0
$335$	204.382i	0.610096i
$336$	0	0
$337$	−439.000	−1.30267	−0.651335	−	0.758790i	$-0.725791\pi$
$337$	−439.000	−1.30267	−0.651335	+	0.758790i	$0.725791\pi$
$338$	0	0
$339$	− 245.951i	− 0.725520i
$340$	0	0
$341$	− 51.9615i	− 0.152380i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	240.000	0.695652
$346$	0	0
$347$	−220.000	−0.634006	−0.317003	−	0.948425i	$-0.602676\pi$
$347$	−220.000	−0.634006	−0.317003	+	0.948425i	$0.602676\pi$
$348$	0	0
$349$	− 339.482i	− 0.972728i	−0.873756	−	0.486364i	$-0.838323\pi$
$349$	− 339.482i	− 0.972728i	0.873756	−	0.486364i	$-0.161677\pi$
$350$	0	0
$351$	−63.0000	−0.179487
$352$	0	0
$353$	− 308.305i	− 0.873385i	−0.899611	−	0.436693i	$-0.856150\pi$
$353$	− 308.305i	− 0.873385i	0.899611	−	0.436693i	$-0.143850\pi$
$354$	0	0
$355$	− 90.0666i	− 0.253709i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−292.000	−0.813370	−0.406685	−	0.913568i	$-0.633315\pi$
$359$	−292.000	−0.813370	−0.406685	+	0.913568i	$0.633315\pi$
$360$	0	0
$361$	−722.000	−2.00000
$362$	0	0
$363$	36.3731i	0.100201i
$364$	0	0
$365$	66.0000	0.180822
$366$	0	0
$367$	538.668i	1.46776i	0.679279	+	0.733880i	$0.262292\pi$
$367$	538.668i	1.46776i	−0.679279	+	0.733880i	$0.737708\pi$
$368$	0	0
$369$	− 72.7461i	− 0.197144i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−205.000	−0.549598	−0.274799	−	0.961502i	$-0.588611\pi$
$373$	−205.000	−0.549598	−0.274799	+	0.961502i	$0.588611\pi$
$374$	0	0
$375$	−228.000	−0.608000
$376$	0	0
$377$	193.990i	0.514562i
$378$	0	0
$379$	523.000	1.37995	0.689974	−	0.723835i	$-0.257622\pi$
$379$	523.000	1.37995	0.689974	+	0.723835i	$0.257622\pi$
$380$	0	0
$381$	− 251.147i	− 0.659179i
$382$	0	0
$383$	76.2102i	0.198982i	0.995038	+	0.0994912i	$0.0317215\pi$
$383$	76.2102i	0.198982i	−0.995038	+	0.0994912i	$0.968278\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−57.0000	−0.147287
$388$	0	0
$389$	−74.0000	−0.190231	−0.0951157	−	0.995466i	$-0.530322\pi$
$389$	−74.0000	−0.190231	−0.0951157	+	0.995466i	$0.530322\pi$
$390$	0	0
$391$	− 277.128i	− 0.708768i
$392$	0	0
$393$	258.000	0.656489
$394$	0	0
$395$	162.813i	0.412184i
$396$	0	0
$397$	323.894i	0.815853i	0.913015	+	0.407926i	$0.133748\pi$
$397$	323.894i	0.815853i	−0.913015	+	0.407926i	$0.866252\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−128.000	−0.319202	−0.159601	−	0.987182i	$-0.551021\pi$
$401$	−128.000	−0.319202	−0.159601	+	0.987182i	$0.551021\pi$
$402$	0	0
$403$	−63.0000	−0.156328
$404$	0	0
$405$	− 31.1769i	− 0.0769800i
$406$	0	0
$407$	−50.0000	−0.122850
$408$	0	0
$409$	− 296.181i	− 0.724158i	−0.932147	−	0.362079i	$-0.882067\pi$
$409$	− 296.181i	− 0.724158i	0.932147	−	0.362079i	$-0.117933\pi$
$410$	0	0
$411$	200.918i	0.488851i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	84.0000	0.202410
$416$	0	0
$417$	−147.000	−0.352518
$418$	0	0
$419$	412.228i	0.983838i	0.870641	+	0.491919i	$0.163704\pi$
$419$	412.228i	0.983838i	−0.870641	+	0.491919i	$0.836296\pi$
$420$	0	0
$421$	107.000	0.254157	0.127078	−	0.991893i	$-0.459440\pi$
$421$	107.000	0.254157	0.127078	+	0.991893i	$0.459440\pi$
$422$	0	0
$423$	− 155.885i	− 0.368521i
$424$	0	0
$425$	90.0666i	0.211922i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−210.000	−0.489510
$430$	0	0
$431$	−262.000	−0.607889	−0.303944	−	0.952690i	$-0.598304\pi$
$431$	−262.000	−0.607889	−0.303944	+	0.952690i	$0.598304\pi$
$432$	0	0
$433$	− 36.3731i	− 0.0840025i	−0.999118	−	0.0420012i	$-0.986627\pi$
$433$	− 36.3731i	− 0.0840025i	0.999118	−	0.0420012i	$-0.0133733\pi$
$434$	0	0
$435$	−96.0000	−0.220690
$436$	0	0
$437$	1316.36i	3.01226i
$438$	0	0
$439$	− 311.769i	− 0.710180i	−0.934832	−	0.355090i	$-0.884450\pi$
$439$	− 311.769i	− 0.710180i	0.934832	−	0.355090i	$-0.115550\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	212.000	0.478555	0.239278	−	0.970951i	$-0.423089\pi$
$443$	212.000	0.478555	0.239278	+	0.970951i	$0.423089\pi$
$444$	0	0
$445$	408.000	0.916854
$446$	0	0
$447$	− 214.774i	− 0.480479i
$448$	0	0
$449$	−782.000	−1.74165	−0.870824	−	0.491595i	$-0.836414\pi$
$449$	−782.000	−1.74165	−0.870824	+	0.491595i	$0.836414\pi$
$450$	0	0
$451$	− 242.487i	− 0.537665i
$452$	0	0
$453$	− 79.6743i	− 0.175882i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	677.000	1.48140	0.740700	−	0.671836i	$-0.234494\pi$
$457$	677.000	1.48140	0.740700	+	0.671836i	$0.234494\pi$
$458$	0	0
$459$	−36.0000	−0.0784314
$460$	0	0
$461$	484.974i	1.05200i	0.850483	+	0.526002i	$0.176310\pi$
$461$	484.974i	1.05200i	−0.850483	+	0.526002i	$0.823690\pi$
$462$	0	0
$463$	−443.000	−0.956803	−0.478402	−	0.878141i	$-0.658784\pi$
$463$	−443.000	−0.956803	−0.478402	+	0.878141i	$0.658784\pi$
$464$	0	0
$465$	− 31.1769i	− 0.0670471i
$466$	0	0
$467$	− 45.0333i	− 0.0964311i	−0.998837	−	0.0482155i	$-0.984647\pi$
$467$	− 45.0333i	− 0.0964311i	0.998837	−	0.0482155i	$-0.0153534\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−324.000	−0.687898
$472$	0	0
$473$	−190.000	−0.401691
$474$	0	0
$475$	− 427.817i	− 0.900666i
$476$	0	0
$477$	96.0000	0.201258
$478$	0	0
$479$	− 55.4256i	− 0.115711i	−0.998325	−	0.0578556i	$-0.981574\pi$
$479$	− 55.4256i	− 0.115711i	0.998325	−	0.0578556i	$-0.0184263\pi$
$480$	0	0
$481$	60.6218i	0.126033i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−168.000	−0.346392
$486$	0	0
$487$	67.0000	0.137577	0.0687885	−	0.997631i	$-0.478087\pi$
$487$	67.0000	0.137577	0.0687885	+	0.997631i	$0.478087\pi$
$488$	0	0
$489$	− 100.459i	− 0.205438i
$490$	0	0
$491$	68.0000	0.138493	0.0692464	−	0.997600i	$-0.477941\pi$
$491$	68.0000	0.138493	0.0692464	+	0.997600i	$0.477941\pi$
$492$	0	0
$493$	110.851i	0.224850i
$494$	0	0
$495$	− 103.923i	− 0.209946i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−509.000	−1.02004	−0.510020	−	0.860163i	$-0.670362\pi$
$499$	−509.000	−1.02004	−0.510020	+	0.860163i	$0.670362\pi$
$500$	0	0
$501$	462.000	0.922156
$502$	0	0
$503$	− 654.715i	− 1.30162i	−0.759240	−	0.650810i	$-0.774429\pi$
$503$	− 654.715i	− 1.30162i	0.759240	−	0.650810i	$-0.225571\pi$
$504$	0	0
$505$	444.000	0.879208
$506$	0	0
$507$	− 38.1051i	− 0.0751580i
$508$	0	0
$509$	869.490i	1.70823i	0.520083	+	0.854115i	$0.325901\pi$
$509$	869.490i	1.70823i	−0.520083	+	0.854115i	$0.674099\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	171.000	0.333333
$514$	0	0
$515$	−30.0000	−0.0582524
$516$	0	0
$517$	− 519.615i	− 1.00506i
$518$	0	0
$519$	−216.000	−0.416185
$520$	0	0
$521$	− 429.549i	− 0.824469i	−0.911078	−	0.412235i	$-0.864748\pi$
$521$	− 429.549i	− 0.824469i	0.911078	−	0.412235i	$-0.135252\pi$
$522$	0	0
$523$	985.537i	1.88439i	0.335063	+	0.942196i	$0.391243\pi$
$523$	985.537i	1.88439i	−0.335063	+	0.942196i	$0.608757\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−36.0000	−0.0683112
$528$	0	0
$529$	1071.00	2.02457
$530$	0	0
$531$	− 124.708i	− 0.234854i
$532$	0	0
$533$	−294.000	−0.551595
$534$	0	0
$535$	− 734.390i	− 1.37269i
$536$	0	0
$537$	17.3205i	0.0322542i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−121.000	−0.223660	−0.111830	−	0.993727i	$-0.535671\pi$
$541$	−121.000	−0.223660	−0.111830	+	0.993727i	$0.535671\pi$
$542$	0	0
$543$	567.000	1.04420
$544$	0	0
$545$	− 58.8897i	− 0.108055i
$546$	0	0
$547$	−926.000	−1.69287	−0.846435	−	0.532492i	$-0.821256\pi$
$547$	−926.000	−1.69287	−0.846435	+	0.532492i	$0.821256\pi$
$548$	0	0
$549$	− 62.3538i	− 0.113577i
$550$	0	0
$551$	− 526.543i	− 0.955614i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−30.0000	−0.0540541
$556$	0	0
$557$	−662.000	−1.18851	−0.594255	−	0.804277i	$-0.702553\pi$
$557$	−662.000	−1.18851	−0.594255	+	0.804277i	$0.702553\pi$
$558$	0	0
$559$	230.363i	0.412098i
$560$	0	0
$561$	−120.000	−0.213904
$562$	0	0
$563$	− 322.161i	− 0.572223i	−0.958196	−	0.286111i	$-0.907637\pi$
$563$	− 322.161i	− 0.572223i	0.958196	−	0.286111i	$-0.0923627\pi$
$564$	0	0
$565$	− 491.902i	− 0.870624i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−758.000	−1.33216	−0.666081	−	0.745880i	$-0.732029\pi$
$569$	−758.000	−1.33216	−0.666081	+	0.745880i	$0.732029\pi$
$570$	0	0
$571$	865.000	1.51489	0.757443	−	0.652901i	$-0.226448\pi$
$571$	865.000	1.51489	0.757443	+	0.652901i	$0.226448\pi$
$572$	0	0
$573$	− 3.46410i	− 0.00604555i
$574$	0	0
$575$	−520.000	−0.904348
$576$	0	0
$577$	− 1072.14i	− 1.85813i	−0.369920	−	0.929064i	$-0.620615\pi$
$577$	− 1072.14i	− 1.85813i	0.369920	−	0.929064i	$-0.379385\pi$
$578$	0	0
$579$	407.032i	0.702991i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	320.000	0.548885
$584$	0	0
$585$	−126.000	−0.215385
$586$	0	0
$587$	− 339.482i	− 0.578334i	−0.957279	−	0.289167i	$-0.906622\pi$
$587$	− 339.482i	− 0.578334i	0.957279	−	0.289167i	$-0.0933783\pi$
$588$	0	0
$589$	171.000	0.290323
$590$	0	0
$591$	− 173.205i	− 0.293071i
$592$	0	0
$593$	− 245.951i	− 0.414758i	−0.978261	−	0.207379i	$-0.933507\pi$
$593$	− 245.951i	− 0.414758i	0.978261	−	0.207379i	$-0.0664933\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−348.000	−0.582915
$598$	0	0
$599$	284.000	0.474124	0.237062	−	0.971495i	$-0.423816\pi$
$599$	284.000	0.474124	0.237062	+	0.971495i	$0.423816\pi$
$600$	0	0
$601$	− 594.093i	− 0.988508i	−0.869317	−	0.494254i	$-0.835441\pi$
$601$	− 594.093i	− 0.988508i	0.869317	−	0.494254i	$-0.164559\pi$
$602$	0	0
$603$	177.000	0.293532
$604$	0	0
$605$	72.7461i	0.120242i
$606$	0	0
$607$	− 8.66025i	− 0.0142673i	−0.999975	−	0.00713365i	$-0.997729\pi$
$607$	− 8.66025i	− 0.0142673i	0.999975	−	0.00713365i	$-0.00227073\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−630.000	−1.03110
$612$	0	0
$613$	878.000	1.43230	0.716150	−	0.697946i	$-0.245903\pi$
$613$	878.000	1.43230	0.716150	+	0.697946i	$0.245903\pi$
$614$	0	0
$615$	− 145.492i	− 0.236573i
$616$	0	0
$617$	−194.000	−0.314425	−0.157212	−	0.987565i	$-0.550251\pi$
$617$	−194.000	−0.314425	−0.157212	+	0.987565i	$0.550251\pi$
$618$	0	0
$619$	611.414i	0.987745i	0.869534	+	0.493872i	$0.164419\pi$
$619$	611.414i	0.987745i	−0.869534	+	0.493872i	$0.835581\pi$
$620$	0	0
$621$	− 207.846i	− 0.334696i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−131.000	−0.209600
$626$	0	0
$627$	570.000	0.909091
$628$	0	0
$629$	34.6410i	0.0550732i
$630$	0	0
$631$	250.000	0.396197	0.198098	−	0.980182i	$-0.436523\pi$
$631$	250.000	0.396197	0.198098	+	0.980182i	$0.436523\pi$
$632$	0	0
$633$	3.46410i	0.00547251i
$634$	0	0
$635$	− 502.295i	− 0.791015i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−78.0000	−0.122066
$640$	0	0
$641$	−1124.00	−1.75351	−0.876755	−	0.480937i	$-0.840297\pi$
$641$	−1124.00	−1.75351	−0.876755	+	0.480937i	$0.840297\pi$
$642$	0	0
$643$	569.845i	0.886228i	0.896465	+	0.443114i	$0.146126\pi$
$643$	569.845i	0.886228i	−0.896465	+	0.443114i	$0.853874\pi$
$644$	0	0
$645$	−114.000	−0.176744
$646$	0	0
$647$	1084.26i	1.67583i	0.545799	+	0.837916i	$0.316226\pi$
$647$	1084.26i	1.67583i	−0.545799	+	0.837916i	$0.683774\pi$
$648$	0	0
$649$	− 415.692i	− 0.640512i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−1010.00	−1.54671	−0.773354	−	0.633975i	$-0.781422\pi$
$653$	−1010.00	−1.54671	−0.773354	+	0.633975i	$0.781422\pi$
$654$	0	0
$655$	516.000	0.787786
$656$	0	0
$657$	− 57.1577i	− 0.0869980i
$658$	0	0
$659$	908.000	1.37785	0.688923	−	0.724835i	$-0.258084\pi$
$659$	908.000	1.37785	0.688923	+	0.724835i	$0.258084\pi$
$660$	0	0
$661$	722.265i	1.09269i	0.837562	+	0.546343i	$0.183980\pi$
$661$	722.265i	1.09269i	−0.837562	+	0.546343i	$0.816020\pi$
$662$	0	0
$663$	145.492i	0.219445i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−640.000	−0.959520
$668$	0	0
$669$	588.000	0.878924
$670$	0	0
$671$	− 207.846i	− 0.309756i
$672$	0	0
$673$	−1027.00	−1.52600	−0.763001	−	0.646397i	$-0.776275\pi$
$673$	−1027.00	−1.52600	−0.763001	+	0.646397i	$0.776275\pi$
$674$	0	0
$675$	67.5500i	0.100074i
$676$	0	0
$677$	− 561.184i	− 0.828928i	−0.910066	−	0.414464i	$-0.863969\pi$
$677$	− 561.184i	− 0.828928i	0.910066	−	0.414464i	$-0.136031\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	282.000	0.414097
$682$	0	0
$683$	−976.000	−1.42899	−0.714495	−	0.699641i	$-0.753343\pi$
$683$	−976.000	−1.42899	−0.714495	+	0.699641i	$0.753343\pi$
$684$	0	0
$685$	401.836i	0.586622i
$686$	0	0
$687$	15.0000	0.0218341
$688$	0	0
$689$	− 387.979i	− 0.563105i
$690$	0	0
$691$	− 566.381i	− 0.819654i	−0.912163	−	0.409827i	$-0.865589\pi$
$691$	− 566.381i	− 0.819654i	0.912163	−	0.409827i	$-0.134411\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−294.000	−0.423022
$696$	0	0
$697$	−168.000	−0.241033
$698$	0	0
$699$	294.449i	0.421243i
$700$	0	0
$701$	352.000	0.502140	0.251070	−	0.967969i	$-0.419218\pi$
$701$	352.000	0.502140	0.251070	+	0.967969i	$0.419218\pi$
$702$	0	0
$703$	− 164.545i	− 0.234061i
$704$	0	0
$705$	− 311.769i	− 0.442226i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−1150.00	−1.62200	−0.811001	−	0.585044i	$-0.801077\pi$
$709$	−1150.00	−1.62200	−0.811001	+	0.585044i	$0.801077\pi$
$710$	0	0
$711$	141.000	0.198312
$712$	0	0
$713$	− 207.846i	− 0.291509i
$714$	0	0
$715$	−420.000	−0.587413
$716$	0	0
$717$	245.951i	0.343028i
$718$	0	0
$719$	973.413i	1.35384i	0.736056	+	0.676921i	$0.236686\pi$
$719$	973.413i	1.35384i	−0.736056	+	0.676921i	$0.763314\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	264.000	0.365145
$724$	0	0
$725$	208.000	0.286897
$726$	0	0
$727$	− 206.114i	− 0.283513i	−0.989902	−	0.141757i	$-0.954725\pi$
$727$	− 206.114i	− 0.283513i	0.989902	−	0.141757i	$-0.0452750\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	0	0
$731$	131.636i	0.180076i
$732$	0	0
$733$	1245.34i	1.69897i	0.527613	+	0.849485i	$0.323087\pi$
$733$	1245.34i	1.69897i	−0.527613	+	0.849485i	$0.676913\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	590.000	0.800543
$738$	0	0
$739$	−311.000	−0.420839	−0.210419	−	0.977611i	$-0.567483\pi$
$739$	−311.000	−0.420839	−0.210419	+	0.977611i	$0.567483\pi$
$740$	0	0
$741$	− 691.088i	− 0.932643i
$742$	0	0
$743$	−394.000	−0.530283	−0.265141	−	0.964210i	$-0.585419\pi$
$743$	−394.000	−0.530283	−0.265141	+	0.964210i	$0.585419\pi$
$744$	0	0
$745$	− 429.549i	− 0.576575i
$746$	0	0
$747$	− 72.7461i	− 0.0973844i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	79.0000	0.105193	0.0525965	−	0.998616i	$-0.483250\pi$
$751$	79.0000	0.105193	0.0525965	+	0.998616i	$0.483250\pi$
$752$	0	0
$753$	504.000	0.669323
$754$	0	0
$755$	− 159.349i	− 0.211058i
$756$	0	0
$757$	−250.000	−0.330251	−0.165125	−	0.986273i	$-0.552803\pi$
$757$	−250.000	−0.330251	−0.165125	+	0.986273i	$0.552803\pi$
$758$	0	0
$759$	− 692.820i	− 0.912807i
$760$	0	0
$761$	− 949.164i	− 1.24726i	−0.781720	−	0.623629i	$-0.785657\pi$
$761$	− 949.164i	− 1.24726i	0.781720	−	0.623629i	$-0.214343\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−72.0000	−0.0941176
$766$	0	0
$767$	−504.000	−0.657106
$768$	0	0
$769$	− 860.829i	− 1.11941i	−0.828691	−	0.559707i	$-0.810914\pi$
$769$	− 860.829i	− 1.11941i	0.828691	−	0.559707i	$-0.189086\pi$
$770$	0	0
$771$	−762.000	−0.988327
$772$	0	0
$773$	225.167i	0.291289i	0.989337	+	0.145645i	$0.0465256\pi$
$773$	225.167i	0.291289i	−0.989337	+	0.145645i	$0.953474\pi$
$774$	0	0
$775$	67.5500i	0.0871613i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	798.000	1.02439
$780$	0	0
$781$	−260.000	−0.332907
$782$	0	0
$783$	83.1384i	0.106179i
$784$	0	0
$785$	−648.000	−0.825478
$786$	0	0
$787$	− 249.415i	− 0.316919i	−0.987365	−	0.158460i	$-0.949347\pi$
$787$	− 249.415i	− 0.316919i	0.987365	−	0.158460i	$-0.0506527\pi$
$788$	0	0
$789$	235.559i	0.298554i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−252.000	−0.317781
$794$	0	0
$795$	192.000	0.241509
$796$	0	0
$797$	− 1357.93i	− 1.70380i	−0.523705	−	0.851900i	$-0.675451\pi$
$797$	− 1357.93i	− 1.70380i	0.523705	−	0.851900i	$-0.324549\pi$
$798$	0	0
$799$	−360.000	−0.450563
$800$	0	0
$801$	− 353.338i	− 0.441122i
$802$	0	0
$803$	− 190.526i	− 0.237267i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	390.000	0.483271
$808$	0	0
$809$	−1418.00	−1.75278	−0.876391	−	0.481601i	$-0.840055\pi$
$809$	−1418.00	−1.75278	−0.876391	+	0.481601i	$0.840055\pi$
$810$	0	0
$811$	− 872.954i	− 1.07639i	−0.842820	−	0.538196i	$-0.819106\pi$
$811$	− 872.954i	− 1.07639i	0.842820	−	0.538196i	$-0.180894\pi$
$812$	0	0
$813$	636.000	0.782288
$814$	0	0
$815$	− 200.918i	− 0.246525i
$816$	0	0
$817$	− 625.270i	− 0.765325i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	250.000	0.304507	0.152253	−	0.988341i	$-0.451347\pi$
$821$	250.000	0.304507	0.152253	+	0.988341i	$0.451347\pi$
$822$	0	0
$823$	−206.000	−0.250304	−0.125152	−	0.992138i	$-0.539942\pi$
$823$	−206.000	−0.250304	−0.125152	+	0.992138i	$0.539942\pi$
$824$	0	0
$825$	225.167i	0.272929i
$826$	0	0
$827$	−1234.00	−1.49214	−0.746070	−	0.665867i	$-0.768062\pi$
$827$	−1234.00	−1.49214	−0.746070	+	0.665867i	$0.768062\pi$
$828$	0	0
$829$	− 344.678i	− 0.415776i	−0.978153	−	0.207888i	$-0.933341\pi$
$829$	− 344.678i	− 0.415776i	0.978153	−	0.207888i	$-0.0666589\pi$
$830$	0	0
$831$	− 684.160i	− 0.823297i
$832$	0	0
$833$	0	0
$834$	0	0
$835$	924.000	1.10659
$836$	0	0
$837$	−27.0000	−0.0322581
$838$	0	0
$839$	− 484.974i	− 0.578038i	−0.957323	−	0.289019i	$-0.906671\pi$
$839$	− 484.974i	− 0.578038i	0.957323	−	0.289019i	$-0.0933291\pi$
$840$	0	0
$841$	−585.000	−0.695600
$842$	0	0
$843$	− 173.205i	− 0.205463i
$844$	0	0
$845$	− 76.2102i	− 0.0901896i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−621.000	−0.731449
$850$	0	0
$851$	−200.000	−0.235018
$852$	0	0
$853$	278.860i	0.326917i	0.986550	+	0.163458i	$0.0522650\pi$
$853$	278.860i	0.326917i	−0.986550	+	0.163458i	$0.947735\pi$
$854$	0	0
$855$	342.000	0.400000
$856$	0	0
$857$	637.395i	0.743751i	0.928283	+	0.371876i	$0.121285\pi$
$857$	637.395i	0.743751i	−0.928283	+	0.371876i	$0.878715\pi$
$858$	0	0
$859$	609.682i	0.709758i	0.934912	+	0.354879i	$0.115478\pi$
$859$	609.682i	0.709758i	−0.934912	+	0.354879i	$0.884522\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−670.000	−0.776362	−0.388181	−	0.921583i	$-0.626896\pi$
$863$	−670.000	−0.776362	−0.388181	+	0.921583i	$0.626896\pi$
$864$	0	0
$865$	−432.000	−0.499422
$866$	0	0
$867$	− 417.424i	− 0.481458i
$868$	0	0
$869$	470.000	0.540852
$870$	0	0
$871$	− 715.337i	− 0.821282i
$872$	0	0
$873$	145.492i	0.166658i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−394.000	−0.449259	−0.224629	−	0.974444i	$-0.572117\pi$
$877$	−394.000	−0.449259	−0.224629	+	0.974444i	$0.572117\pi$
$878$	0	0
$879$	420.000	0.477816
$880$	0	0
$881$	1163.94i	1.32116i	0.750758	+	0.660578i	$0.229689\pi$
$881$	1163.94i	1.32116i	−0.750758	+	0.660578i	$0.770311\pi$
$882$	0	0
$883$	−737.000	−0.834655	−0.417327	−	0.908756i	$-0.637033\pi$
$883$	−737.000	−0.834655	−0.417327	+	0.908756i	$0.637033\pi$
$884$	0	0
$885$	− 249.415i	− 0.281825i
$886$	0	0
$887$	730.925i	0.824042i	0.911174	+	0.412021i	$0.135177\pi$
$887$	730.925i	0.824042i	−0.911174	+	0.412021i	$0.864823\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−90.0000	−0.101010
$892$	0	0
$893$	1710.00	1.91489
$894$	0	0
$895$	34.6410i	0.0387050i
$896$	0	0
$897$	−840.000	−0.936455
$898$	0	0
$899$	83.1384i	0.0924788i
$900$	0	0
$901$	− 221.703i	− 0.246063i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1134.00	1.25304
$906$	0	0
$907$	235.000	0.259096	0.129548	−	0.991573i	$-0.458647\pi$
$907$	235.000	0.259096	0.129548	+	0.991573i	$0.458647\pi$
$908$	0	0
$909$	− 384.515i	− 0.423009i
$910$	0	0
$911$	740.000	0.812294	0.406147	−	0.913808i	$-0.366872\pi$
$911$	740.000	0.812294	0.406147	+	0.913808i	$0.366872\pi$
$912$	0	0
$913$	− 242.487i	− 0.265594i
$914$	0	0
$915$	− 124.708i	− 0.136293i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−1517.00	−1.65071	−0.825354	−	0.564616i	$-0.809024\pi$
$919$	−1517.00	−1.65071	−0.825354	+	0.564616i	$0.809024\pi$
$920$	0	0
$921$	−315.000	−0.342020
$922$	0	0
$923$	315.233i	0.341531i
$924$	0	0
$925$	65.0000	0.0702703
$926$	0	0
$927$	25.9808i	0.0280267i
$928$	0	0
$929$	1111.98i	1.19696i	0.801137	+	0.598480i	$0.204229\pi$
$929$	1111.98i	1.19696i	−0.801137	+	0.598480i	$0.795771\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	954.000	1.02251
$934$	0	0
$935$	−240.000	−0.256684
$936$	0	0
$937$	− 836.581i	− 0.892829i	−0.894826	−	0.446414i	$-0.852701\pi$
$937$	− 836.581i	− 0.892829i	0.894826	−	0.446414i	$-0.147299\pi$
$938$	0	0
$939$	351.000	0.373802
$940$	0	0
$941$	394.908i	0.419668i	0.977737	+	0.209834i	$0.0672924\pi$
$941$	394.908i	0.419668i	−0.977737	+	0.209834i	$0.932708\pi$
$942$	0	0
$943$	− 969.948i	− 1.02858i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	338.000	0.356917	0.178458	−	0.983947i	$-0.442889\pi$
$947$	338.000	0.356917	0.178458	+	0.983947i	$0.442889\pi$
$948$	0	0
$949$	−231.000	−0.243414
$950$	0	0
$951$	− 505.759i	− 0.531818i
$952$	0	0
$953$	−1244.00	−1.30535	−0.652676	−	0.757637i	$-0.726354\pi$
$953$	−1244.00	−1.30535	−0.652676	+	0.757637i	$0.726354\pi$
$954$	0	0
$955$	− 6.92820i	− 0.00725466i
$956$	0	0
$957$	277.128i	0.289580i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	934.000	0.971904
$962$	0	0
$963$	−636.000	−0.660436
$964$	0	0
$965$	814.064i	0.843590i
$966$	0	0
$967$	1741.00	1.80041	0.900207	−	0.435463i	$-0.143415\pi$
$967$	1741.00	1.80041	0.900207	+	0.435463i	$0.143415\pi$
$968$	0	0
$969$	− 394.908i	− 0.407541i
$970$	0	0
$971$	− 1281.72i	− 1.32000i	−0.751267	−	0.659999i	$-0.770557\pi$
$971$	− 1281.72i	− 1.32000i	0.751267	−	0.659999i	$-0.229443\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	273.000	0.280000
$976$	0	0
$977$	262.000	0.268168	0.134084	−	0.990970i	$-0.457191\pi$
$977$	262.000	0.268168	0.134084	+	0.990970i	$0.457191\pi$
$978$	0	0
$979$	− 1177.79i	− 1.20306i
$980$	0	0
$981$	−51.0000	−0.0519878
$982$	0	0
$983$	1108.51i	1.12768i	0.825883	+	0.563842i	$0.190677\pi$
$983$	1108.51i	1.12768i	−0.825883	+	0.563842i	$0.809323\pi$
$984$	0	0
$985$	− 346.410i	− 0.351685i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−760.000	−0.768453
$990$	0	0
$991$	67.0000	0.0676085	0.0338042	−	0.999428i	$-0.489238\pi$
$991$	67.0000	0.0676085	0.0338042	+	0.999428i	$0.489238\pi$
$992$	0	0
$993$	8.66025i	0.00872130i
$994$	0	0
$995$	−696.000	−0.699497
$996$	0	0
$997$	989.001i	0.991977i	0.868329	+	0.495988i	$0.165194\pi$
$997$	989.001i	0.991977i	−0.868329	+	0.495988i	$0.834806\pi$
$998$	0	0
$999$	25.9808i	0.0260068i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.3.f.b.97.1		2
4.3	odd	2		147.3.d.a.97.2		2
7.2	even	3		336.3.bh.c.241.1		2
7.3	odd	6		336.3.bh.c.145.1		2
7.6	odd	2	inner	2352.3.f.b.97.2		2
12.11	even	2		441.3.d.d.244.2		2
21.2	odd	6		1008.3.cg.f.577.1		2
21.17	even	6		1008.3.cg.f.145.1		2
28.3	even	6		21.3.f.c.19.1	yes	2
28.11	odd	6		147.3.f.e.19.1		2
28.19	even	6		147.3.f.e.31.1		2
28.23	odd	6		21.3.f.c.10.1	✓	2
28.27	even	2		147.3.d.a.97.1		2
84.11	even	6		441.3.m.b.19.1		2
84.23	even	6		63.3.m.a.10.1		2
84.47	odd	6		441.3.m.b.325.1		2
84.59	odd	6		63.3.m.a.19.1		2
84.83	odd	2		441.3.d.d.244.1		2
140.3	odd	12		525.3.s.d.124.1		4
140.23	even	12		525.3.s.d.199.2		4
140.59	even	6		525.3.o.b.376.1		2
140.79	odd	6		525.3.o.b.451.1		2
140.87	odd	12		525.3.s.d.124.2		4
140.107	even	12		525.3.s.d.199.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.3.f.c.10.1	✓	2	28.23	odd	6
21.3.f.c.19.1	yes	2	28.3	even	6
63.3.m.a.10.1		2	84.23	even	6
63.3.m.a.19.1		2	84.59	odd	6
147.3.d.a.97.1		2	28.27	even	2
147.3.d.a.97.2		2	4.3	odd	2
147.3.f.e.19.1		2	28.11	odd	6
147.3.f.e.31.1		2	28.19	even	6
336.3.bh.c.145.1		2	7.3	odd	6
336.3.bh.c.241.1		2	7.2	even	3
441.3.d.d.244.1		2	84.83	odd	2
441.3.d.d.244.2		2	12.11	even	2
441.3.m.b.19.1		2	84.11	even	6
441.3.m.b.325.1		2	84.47	odd	6
525.3.o.b.376.1		2	140.59	even	6
525.3.o.b.451.1		2	140.79	odd	6
525.3.s.d.124.1		4	140.3	odd	12
525.3.s.d.124.2		4	140.87	odd	12
525.3.s.d.199.1		4	140.107	even	12
525.3.s.d.199.2		4	140.23	even	12
1008.3.cg.f.145.1		2	21.17	even	6
1008.3.cg.f.577.1		2	21.2	odd	6
2352.3.f.b.97.1		2	1.1	even	1	trivial
2352.3.f.b.97.2		2	7.6	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:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2352.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.73205i q^{3} -3.46410i q^{5} -3.00000 q^{9} +O(q^{10})\)	\(q-1.73205i q^{3} -3.46410i q^{5} -3.00000 q^{9} -10.0000 q^{11} +12.1244i q^{13} -6.00000 q^{15} +6.92820i q^{17} -32.9090i q^{19} -40.0000 q^{23} +13.0000 q^{25} +5.19615i q^{27} +16.0000 q^{29} +5.19615i q^{31} +17.3205i q^{33} +5.00000 q^{37} +21.0000 q^{39} +24.2487i q^{41} +19.0000 q^{43} +10.3923i q^{45} +51.9615i q^{47} +12.0000 q^{51} -32.0000 q^{53} +34.6410i q^{55} -57.0000 q^{57} +41.5692i q^{59} +20.7846i q^{61} +42.0000 q^{65} -59.0000 q^{67} +69.2820i q^{69} +26.0000 q^{71} +19.0526i q^{73} -22.5167i q^{75} -47.0000 q^{79} +9.00000 q^{81} +24.2487i q^{83} +24.0000 q^{85} -27.7128i q^{87} +117.779i q^{89} +9.00000 q^{93} -114.000 q^{95} -48.4974i q^{97} +30.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{9}+O(q^{10}) \) 2 * q - 6 * q^9	\( 2 q - 6 q^{9} - 20 q^{11} - 12 q^{15} - 80 q^{23} + 26 q^{25} + 32 q^{29} + 10 q^{37} + 42 q^{39} + 38 q^{43} + 24 q^{51} - 64 q^{53} - 114 q^{57} + 84 q^{65} - 118 q^{67} + 52 q^{71} - 94 q^{79} + 18 q^{81} + 48 q^{85} + 18 q^{93} - 228 q^{95} + 60 q^{99}+O(q^{100}) \) 2 * q - 6 * q^9 - 20 * q^11 - 12 * q^15 - 80 * q^23 + 26 * q^25 + 32 * q^29 + 10 * q^37 + 42 * q^39 + 38 * q^43 + 24 * q^51 - 64 * q^53 - 114 * q^57 + 84 * q^65 - 118 * q^67 + 52 * q^71 - 94 * q^79 + 18 * q^81 + 48 * q^85 + 18 * q^93 - 228 * q^95 + 60 * q^99

Introduction

L-functions

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Database

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