LMFDB - Embedded newform 378.3.b.c.323.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,3,Mod(323,378)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(378, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("378.323");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	378.b (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:	$10.2997539928$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.157351936.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + x^{4} + 16 $ x^8 + x^4 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			323.4
Root			$1.28897 - 0.581861i$ of defining polynomial
Character	$\chi$	$=$	378.323
Dual form			378.3.b.c.323.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{2} -2.00000 q^{4} +8.15587i q^{5} -2.64575 q^{7} +2.82843i q^{8} +O(q^{10})$	\(q-1.41421i q^{2} -2.00000 q^{4} +8.15587i q^{5} -2.64575 q^{7} +2.82843i q^{8} +11.5341 q^{10} -10.0931i q^{11} -23.1134 q^{13} +3.74166i q^{14} +4.00000 q^{16} -24.0304i q^{17} -2.53036 q^{19} -16.3117i q^{20} -14.2738 q^{22} -26.0480i q^{23} -41.5182 q^{25} +32.6872i q^{26} +5.29150 q^{28} -25.7460i q^{29} -28.5927 q^{31} -5.65685i q^{32} -33.9841 q^{34} -21.5784i q^{35} -16.0607 q^{37} +3.57847i q^{38} -23.0683 q^{40} +43.4028i q^{41} +40.4441 q^{43} +20.1862i q^{44} -36.8375 q^{46} +73.7549i q^{47} +7.00000 q^{49} +58.7156i q^{50} +46.2267 q^{52} -16.7549i q^{53} +82.3182 q^{55} -7.48331i q^{56} -36.4104 q^{58} +61.5881i q^{59} -111.774 q^{61} +40.4362i q^{62} -8.00000 q^{64} -188.510i q^{65} +67.4163 q^{67} +48.0608i q^{68} -30.5165 q^{70} +32.0254i q^{71} -78.5199 q^{73} +22.7133i q^{74} +5.06072 q^{76} +26.7039i q^{77} -114.902 q^{79} +32.6235i q^{80} +61.3809 q^{82} -116.363i q^{83} +195.989 q^{85} -57.1966i q^{86} +28.5477 q^{88} +59.2528i q^{89} +61.1522 q^{91} +52.0961i q^{92} +104.305 q^{94} -20.6373i q^{95} +116.942 q^{97} -9.89949i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 16 q^{4}+O(q^{10}) $ 8 * q - 16 * q^4	$ 8 q - 16 q^{4} + 16 q^{10} - 40 q^{13} + 32 q^{16} + 40 q^{19} - 16 q^{22} - 40 q^{31} - 16 q^{34} - 8 q^{37} - 32 q^{40} + 40 q^{43} - 64 q^{46} + 56 q^{49} + 80 q^{52} + 56 q^{55} + 112 q^{58} - 88 q^{61} - 64 q^{64} + 240 q^{67} - 112 q^{70} - 288 q^{73} - 80 q^{76} - 424 q^{79} + 144 q^{82} + 200 q^{85} + 32 q^{88} + 56 q^{91} + 288 q^{94} + 600 q^{97}+O(q^{100}) $ 8 * q - 16 * q^4 + 16 * q^10 - 40 * q^13 + 32 * q^16 + 40 * q^19 - 16 * q^22 - 40 * q^31 - 16 * q^34 - 8 * q^37 - 32 * q^40 + 40 * q^43 - 64 * q^46 + 56 * q^49 + 80 * q^52 + 56 * q^55 + 112 * q^58 - 88 * q^61 - 64 * q^64 + 240 * q^67 - 112 * q^70 - 288 * q^73 - 80 * q^76 - 424 * q^79 + 144 * q^82 + 200 * q^85 + 32 * q^88 + 56 * q^91 + 288 * q^94 + 600 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/378\mathbb{Z}\right)^\times$.

$n$	$29$	$325$
$\chi(n)$	$-1$	$1$

Coefficient data

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

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

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

$2$	− 1.41421i	− 0.707107i
$2$	− 1.41421i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−2.00000	−0.500000
$5$	8.15587i	1.63117i	0.578634	+	0.815587i	$0.303586\pi$
$5$	8.15587i	1.63117i	−0.578634	+	0.815587i	$0.696414\pi$
$6$	0	0
$7$	−2.64575	−0.377964
$7$	−2.64575	−0.377964
$8$	2.82843i	0.353553i
$9$	0	0
$10$	11.5341	1.15341
$11$	− 10.0931i	− 0.917557i	−0.888551	−	0.458778i	$-0.848287\pi$
$11$	− 10.0931i	− 0.917557i	0.888551	−	0.458778i	$-0.151713\pi$
$12$	0	0
$13$	−23.1134	−1.77795	−0.888976	−	0.457955i	$-0.848582\pi$
$13$	−23.1134	−1.77795	−0.888976	+	0.457955i	$0.848582\pi$
$14$	3.74166i	0.267261i
$15$	0	0
$16$	4.00000	0.250000
$17$	− 24.0304i	− 1.41355i	−0.707438	−	0.706776i	$-0.750149\pi$
$17$	− 24.0304i	− 1.41355i	0.707438	−	0.706776i	$-0.249851\pi$
$18$	0	0
$19$	−2.53036	−0.133177	−0.0665884	−	0.997781i	$-0.521211\pi$
$19$	−2.53036	−0.133177	−0.0665884	+	0.997781i	$0.521211\pi$
$20$	− 16.3117i	− 0.815587i
$21$	0	0
$22$	−14.2738	−0.648811
$23$	− 26.0480i	− 1.13252i	−0.824225	−	0.566262i	$-0.808389\pi$
$23$	− 26.0480i	− 1.13252i	0.824225	−	0.566262i	$-0.191611\pi$
$24$	0	0
$25$	−41.5182	−1.66073
$26$	32.6872i	1.25720i
$27$	0	0
$28$	5.29150	0.188982
$29$	− 25.7460i	− 0.887795i	−0.896078	−	0.443897i	$-0.853595\pi$
$29$	− 25.7460i	− 0.887795i	0.896078	−	0.443897i	$-0.146405\pi$
$30$	0	0
$31$	−28.5927	−0.922347	−0.461173	−	0.887310i	$-0.652571\pi$
$31$	−28.5927	−0.922347	−0.461173	+	0.887310i	$0.652571\pi$
$32$	− 5.65685i	− 0.176777i
$33$	0	0
$34$	−33.9841	−0.999532
$35$	− 21.5784i	− 0.616526i
$36$	0	0
$37$	−16.0607	−0.434073	−0.217037	−	0.976163i	$-0.569639\pi$
$37$	−16.0607	−0.434073	−0.217037	+	0.976163i	$0.569639\pi$
$38$	3.57847i	0.0941702i
$39$	0	0
$40$	−23.0683	−0.576707
$41$	43.4028i	1.05861i	0.848433	+	0.529303i	$0.177546\pi$
$41$	43.4028i	1.05861i	−0.848433	+	0.529303i	$0.822454\pi$
$42$	0	0
$43$	40.4441	0.940560	0.470280	−	0.882517i	$-0.344153\pi$
$43$	40.4441	0.940560	0.470280	+	0.882517i	$0.344153\pi$
$44$	20.1862i	0.458778i
$45$	0	0
$46$	−36.8375	−0.800815
$47$	73.7549i	1.56925i	0.619968	+	0.784627i	$0.287145\pi$
$47$	73.7549i	1.56925i	−0.619968	+	0.784627i	$0.712855\pi$
$48$	0	0
$49$	7.00000	0.142857
$50$	58.7156i	1.17431i
$51$	0	0
$52$	46.2267	0.888976
$53$	− 16.7549i	− 0.316130i	−0.987429	−	0.158065i	$-0.949474\pi$
$53$	− 16.7549i	− 0.316130i	0.987429	−	0.158065i	$-0.0505256\pi$
$54$	0	0
$55$	82.3182	1.49669
$56$	− 7.48331i	− 0.133631i
$57$	0	0
$58$	−36.4104	−0.627766
$59$	61.5881i	1.04387i	0.852986	+	0.521933i	$0.174789\pi$
$59$	61.5881i	1.04387i	−0.852986	+	0.521933i	$0.825211\pi$
$60$	0	0
$61$	−111.774	−1.83236	−0.916181	−	0.400765i	$-0.868744\pi$
$61$	−111.774	−1.83236	−0.916181	+	0.400765i	$0.868744\pi$
$62$	40.4362i	0.652198i
$63$	0	0
$64$	−8.00000	−0.125000
$65$	− 188.510i	− 2.90015i
$66$	0	0
$67$	67.4163	1.00621	0.503107	−	0.864224i	$-0.332190\pi$
$67$	67.4163	1.00621	0.503107	+	0.864224i	$0.332190\pi$
$68$	48.0608i	0.706776i
$69$	0	0
$70$	−30.5165	−0.435950
$71$	32.0254i	0.451062i	0.974236	+	0.225531i	$0.0724117\pi$
$71$	32.0254i	0.451062i	−0.974236	+	0.225531i	$0.927588\pi$
$72$	0	0
$73$	−78.5199	−1.07562	−0.537808	−	0.843068i	$-0.680747\pi$
$73$	−78.5199	−1.07562	−0.537808	+	0.843068i	$0.680747\pi$
$74$	22.7133i	0.306936i
$75$	0	0
$76$	5.06072	0.0665884
$77$	26.7039i	0.346804i
$78$	0	0
$79$	−114.902	−1.45445	−0.727225	−	0.686399i	$-0.759190\pi$
$79$	−114.902	−1.45445	−0.727225	+	0.686399i	$0.759190\pi$
$80$	32.6235i	0.407794i
$81$	0	0
$82$	61.3809	0.748547
$83$	− 116.363i	− 1.40196i	−0.713182	−	0.700979i	$-0.752747\pi$
$83$	− 116.363i	− 1.40196i	0.713182	−	0.700979i	$-0.247253\pi$
$84$	0	0
$85$	195.989	2.30575
$86$	− 57.1966i	− 0.665076i
$87$	0	0
$88$	28.5477	0.324405
$89$	59.2528i	0.665762i	0.942969	+	0.332881i	$0.108021\pi$
$89$	59.2528i	0.665762i	−0.942969	+	0.332881i	$0.891979\pi$
$90$	0	0
$91$	61.1522	0.672002
$92$	52.0961i	0.566262i
$93$	0	0
$94$	104.305	1.10963
$95$	− 20.6373i	− 0.217235i
$96$	0	0
$97$	116.942	1.20558	0.602792	−	0.797898i	$-0.294055\pi$
$97$	116.942	1.20558	0.602792	+	0.797898i	$0.294055\pi$
$98$	− 9.89949i	− 0.101015i
$99$	0	0
$100$	83.0365	0.830365
$101$	− 125.712i	− 1.24467i	−0.782752	−	0.622334i	$-0.786184\pi$
$101$	− 125.712i	− 1.24467i	0.782752	−	0.622334i	$-0.213816\pi$
$102$	0	0
$103$	−100.493	−0.975660	−0.487830	−	0.872939i	$-0.662211\pi$
$103$	−100.493	−0.975660	−0.487830	+	0.872939i	$0.662211\pi$
$104$	− 65.3745i	− 0.628601i
$105$	0	0
$106$	−23.6950	−0.223538
$107$	57.1655i	0.534257i	0.963661	+	0.267128i	$0.0860748\pi$
$107$	57.1655i	0.534257i	−0.963661	+	0.267128i	$0.913925\pi$
$108$	0	0
$109$	−123.970	−1.13734	−0.568669	−	0.822566i	$-0.692542\pi$
$109$	−123.970	−1.13734	−0.568669	+	0.822566i	$0.692542\pi$
$110$	− 116.416i	− 1.05832i
$111$	0	0
$112$	−10.5830	−0.0944911
$113$	− 68.2894i	− 0.604331i	−0.953256	−	0.302165i	$-0.902291\pi$
$113$	− 68.2894i	− 0.604331i	0.953256	−	0.302165i	$-0.0977095\pi$
$114$	0	0
$115$	212.445	1.84734
$116$	51.4921i	0.443897i
$117$	0	0
$118$	87.0988	0.738125
$119$	63.5784i	0.534272i
$120$	0	0
$121$	19.1288	0.158090
$122$	158.072i	1.29568i
$123$	0	0
$124$	57.1855	0.461173
$125$	− 134.721i	− 1.07776i
$126$	0	0
$127$	−206.711	−1.62765	−0.813825	−	0.581111i	$-0.802618\pi$
$127$	−206.711	−1.62765	−0.813825	+	0.581111i	$0.802618\pi$
$128$	11.3137i	0.0883883i
$129$	0	0
$130$	−266.593	−2.05071
$131$	− 109.861i	− 0.838631i	−0.907841	−	0.419315i	$-0.862270\pi$
$131$	− 109.861i	− 0.838631i	0.907841	−	0.419315i	$-0.137730\pi$
$132$	0	0
$133$	6.69470	0.0503361
$134$	− 95.3410i	− 0.711500i
$135$	0	0
$136$	67.9682	0.499766
$137$	− 1.06460i	− 0.00777079i	−0.999992	−	0.00388540i	$-0.998763\pi$
$137$	− 1.06460i	− 0.00777079i	0.999992	−	0.00388540i	$-0.00123676\pi$
$138$	0	0
$139$	95.1004	0.684175	0.342088	−	0.939668i	$-0.388866\pi$
$139$	95.1004	0.684175	0.342088	+	0.939668i	$0.388866\pi$
$140$	43.1568i	0.308263i
$141$	0	0
$142$	45.2908	0.318949
$143$	233.286i	1.63137i
$144$	0	0
$145$	209.981	1.44815
$146$	111.044i	0.760575i
$147$	0	0
$148$	32.1214	0.217037
$149$	172.608i	1.15844i	0.815171	+	0.579220i	$0.196643\pi$
$149$	172.608i	1.15844i	−0.815171	+	0.579220i	$0.803357\pi$
$150$	0	0
$151$	−49.7864	−0.329711	−0.164856	−	0.986318i	$-0.552716\pi$
$151$	−49.7864	−0.329711	−0.164856	+	0.986318i	$0.552716\pi$
$152$	− 7.15694i	− 0.0470851i
$153$	0	0
$154$	37.7650	0.245227
$155$	− 233.199i	− 1.50451i
$156$	0	0
$157$	190.704	1.21467	0.607337	−	0.794445i	$-0.292238\pi$
$157$	190.704	1.21467	0.607337	+	0.794445i	$0.292238\pi$
$158$	162.495i	1.02845i
$159$	0	0
$160$	46.1366	0.288354
$161$	68.9167i	0.428054i
$162$	0	0
$163$	134.589	0.825700	0.412850	−	0.910799i	$-0.364533\pi$
$163$	134.589	0.825700	0.412850	+	0.910799i	$0.364533\pi$
$164$	− 86.8057i	− 0.529303i
$165$	0	0
$166$	−164.561	−0.991334
$167$	142.698i	0.854478i	0.904139	+	0.427239i	$0.140514\pi$
$167$	142.698i	0.854478i	−0.904139	+	0.427239i	$0.859486\pi$
$168$	0	0
$169$	365.228	2.16111
$170$	− 277.170i	− 1.63041i
$171$	0	0
$172$	−80.8882	−0.470280
$173$	− 79.2843i	− 0.458291i	−0.973392	−	0.229145i	$-0.926407\pi$
$173$	− 79.2843i	− 0.458291i	0.973392	−	0.229145i	$-0.0735931\pi$
$174$	0	0
$175$	109.847	0.627697
$176$	− 40.3725i	− 0.229389i
$177$	0	0
$178$	83.7962	0.470765
$179$	42.0080i	0.234682i	0.993092	+	0.117341i	$0.0374370\pi$
$179$	42.0080i	0.234682i	−0.993092	+	0.117341i	$0.962563\pi$
$180$	0	0
$181$	−15.6586	−0.0865117	−0.0432559	−	0.999064i	$-0.513773\pi$
$181$	−15.6586	−0.0865117	−0.0432559	+	0.999064i	$0.513773\pi$
$182$	− 86.4823i	− 0.475177i
$183$	0	0
$184$	73.6750	0.400408
$185$	− 130.989i	− 0.708049i
$186$	0	0
$187$	−242.542	−1.29701
$188$	− 147.510i	− 0.784627i
$189$	0	0
$190$	−29.1855	−0.153608
$191$	39.5316i	0.206972i	0.994631	+	0.103486i	$0.0329997\pi$
$191$	39.5316i	0.206972i	−0.994631	+	0.103486i	$0.967000\pi$
$192$	0	0
$193$	−301.266	−1.56096	−0.780482	−	0.625178i	$-0.785026\pi$
$193$	−301.266	−1.56096	−0.780482	+	0.625178i	$0.785026\pi$
$194$	− 165.380i	− 0.852476i
$195$	0	0
$196$	−14.0000	−0.0714286
$197$	− 211.873i	− 1.07550i	−0.843104	−	0.537750i	$-0.819275\pi$
$197$	− 211.873i	− 1.07550i	0.843104	−	0.537750i	$-0.180725\pi$
$198$	0	0
$199$	−154.099	−0.774367	−0.387183	−	0.922003i	$-0.626552\pi$
$199$	−154.099	−0.774367	−0.387183	+	0.922003i	$0.626552\pi$
$200$	− 117.431i	− 0.587156i
$201$	0	0
$202$	−177.783	−0.880113
$203$	68.1176i	0.335555i
$204$	0	0
$205$	−353.988	−1.72677
$206$	142.119i	0.689896i
$207$	0	0
$208$	−92.4535	−0.444488
$209$	25.5392i	0.122197i
$210$	0	0
$211$	47.5886	0.225538	0.112769	−	0.993621i	$-0.464028\pi$
$211$	47.5886	0.225538	0.112769	+	0.993621i	$0.464028\pi$
$212$	33.5098i	0.158065i
$213$	0	0
$214$	80.8442	0.377777
$215$	329.857i	1.53422i
$216$	0	0
$217$	75.6493	0.348614
$218$	175.320i	0.804220i
$219$	0	0
$220$	−164.636	−0.748347
$221$	555.423i	2.51323i
$222$	0	0
$223$	229.673	1.02992	0.514961	−	0.857213i	$-0.327806\pi$
$223$	229.673	1.02992	0.514961	+	0.857213i	$0.327806\pi$
$224$	14.9666i	0.0668153i
$225$	0	0
$226$	−96.5758	−0.427326
$227$	335.272i	1.47697i	0.674270	+	0.738485i	$0.264458\pi$
$227$	335.272i	1.47697i	−0.674270	+	0.738485i	$0.735542\pi$
$228$	0	0
$229$	250.272	1.09289	0.546446	−	0.837494i	$-0.315980\pi$
$229$	250.272	1.09289	0.546446	+	0.837494i	$0.315980\pi$
$230$	− 300.442i	− 1.30627i
$231$	0	0
$232$	72.8208	0.313883
$233$	− 106.984i	− 0.459159i	−0.973290	−	0.229580i	$-0.926265\pi$
$233$	− 106.984i	− 0.459159i	0.973290	−	0.229580i	$-0.0737351\pi$
$234$	0	0
$235$	−601.536	−2.55973
$236$	− 123.176i	− 0.521933i
$237$	0	0
$238$	89.9134	0.377788
$239$	− 132.247i	− 0.553333i	−0.960966	−	0.276667i	$-0.910770\pi$
$239$	− 132.247i	− 0.553333i	0.960966	−	0.276667i	$-0.0892298\pi$
$240$	0	0
$241$	−244.171	−1.01316	−0.506579	−	0.862193i	$-0.669090\pi$
$241$	−244.171	−1.01316	−0.506579	+	0.862193i	$0.669090\pi$
$242$	− 27.0523i	− 0.111786i
$243$	0	0
$244$	223.548	0.916181
$245$	57.0911i	0.233025i
$246$	0	0
$247$	58.4851	0.236782
$248$	− 80.8725i	− 0.326099i
$249$	0	0
$250$	−190.524	−0.762095
$251$	359.596i	1.43265i	0.697765	+	0.716327i	$0.254178\pi$
$251$	359.596i	1.43265i	−0.697765	+	0.716327i	$0.745822\pi$
$252$	0	0
$253$	−262.906	−1.03915
$254$	292.334i	1.15092i
$255$	0	0
$256$	16.0000	0.0625000
$257$	− 291.293i	− 1.13344i	−0.823911	−	0.566719i	$-0.808213\pi$
$257$	− 291.293i	− 1.13344i	0.823911	−	0.566719i	$-0.191787\pi$
$258$	0	0
$259$	42.4927	0.164064
$260$	377.019i	1.45007i
$261$	0	0
$262$	−155.366	−0.593001
$263$	142.164i	0.540546i	0.962784	+	0.270273i	$0.0871139\pi$
$263$	142.164i	0.540546i	−0.962784	+	0.270273i	$0.912886\pi$
$264$	0	0
$265$	136.651	0.515664
$266$	− 9.46774i	− 0.0355930i
$267$	0	0
$268$	−134.833	−0.503107
$269$	− 241.624i	− 0.898232i	−0.893473	−	0.449116i	$-0.851739\pi$
$269$	− 241.624i	− 0.898232i	0.893473	−	0.449116i	$-0.148261\pi$
$270$	0	0
$271$	−513.653	−1.89540	−0.947699	−	0.319167i	$-0.896597\pi$
$271$	−513.653	−1.89540	−0.947699	+	0.319167i	$0.896597\pi$
$272$	− 96.1215i	− 0.353388i
$273$	0	0
$274$	−1.50557	−0.00549478
$275$	419.049i	1.52381i
$276$	0	0
$277$	−281.988	−1.01801	−0.509003	−	0.860765i	$-0.669986\pi$
$277$	−281.988	−1.01801	−0.509003	+	0.860765i	$0.669986\pi$
$278$	− 134.492i	− 0.483785i
$279$	0	0
$280$	61.0329	0.217975
$281$	344.327i	1.22536i	0.790329	+	0.612682i	$0.209910\pi$
$281$	344.327i	1.22536i	−0.790329	+	0.612682i	$0.790090\pi$
$282$	0	0
$283$	290.012	1.02478	0.512389	−	0.858754i	$-0.328761\pi$
$283$	290.012	1.02478	0.512389	+	0.858754i	$0.328761\pi$
$284$	− 64.0508i	− 0.225531i
$285$	0	0
$286$	329.916	1.15355
$287$	− 114.833i	− 0.400115i
$288$	0	0
$289$	−288.459	−0.998128
$290$	− 296.959i	− 1.02400i
$291$	0	0
$292$	157.040	0.537808
$293$	− 218.973i	− 0.747348i	−0.927560	−	0.373674i	$-0.878098\pi$
$293$	− 218.973i	− 0.747348i	0.927560	−	0.373674i	$-0.121902\pi$
$294$	0	0
$295$	−502.305	−1.70273
$296$	− 45.4266i	− 0.153468i
$297$	0	0
$298$	244.104	0.819141
$299$	602.058i	2.01357i
$300$	0	0
$301$	−107.005	−0.355498
$302$	70.4086i	0.233141i
$303$	0	0
$304$	−10.1214	−0.0332942
$305$	− 911.615i	− 2.98890i
$306$	0	0
$307$	77.7286	0.253188	0.126594	−	0.991955i	$-0.459596\pi$
$307$	77.7286	0.253188	0.126594	+	0.991955i	$0.459596\pi$
$308$	− 53.4078i	− 0.173402i
$309$	0	0
$310$	−329.793	−1.06385
$311$	− 255.899i	− 0.822828i	−0.911449	−	0.411414i	$-0.865035\pi$
$311$	− 255.899i	− 0.822828i	0.911449	−	0.411414i	$-0.134965\pi$
$312$	0	0
$313$	−296.845	−0.948388	−0.474194	−	0.880420i	$-0.657260\pi$
$313$	−296.845	−0.948388	−0.474194	+	0.880420i	$0.657260\pi$
$314$	− 269.696i	− 0.858904i
$315$	0	0
$316$	229.803	0.727225
$317$	− 372.238i	− 1.17425i	−0.809496	−	0.587126i	$-0.800259\pi$
$317$	− 372.238i	− 1.17425i	0.809496	−	0.587126i	$-0.199741\pi$
$318$	0	0
$319$	−259.858	−0.814602
$320$	− 65.2470i	− 0.203897i
$321$	0	0
$322$	97.4629	0.302680
$323$	60.8055i	0.188252i
$324$	0	0
$325$	959.626	2.95270
$326$	− 190.338i	− 0.583858i
$327$	0	0
$328$	−122.762	−0.374274
$329$	− 195.137i	− 0.593122i
$330$	0	0
$331$	189.269	0.571811	0.285906	−	0.958258i	$-0.407706\pi$
$331$	189.269	0.571811	0.285906	+	0.958258i	$0.407706\pi$
$332$	232.725i	0.700979i
$333$	0	0
$334$	201.805	0.604208
$335$	549.838i	1.64131i
$336$	0	0
$337$	280.154	0.831317	0.415659	−	0.909521i	$-0.363551\pi$
$337$	280.154	0.831317	0.415659	+	0.909521i	$0.363551\pi$
$338$	− 516.510i	− 1.52814i
$339$	0	0
$340$	−391.977	−1.15287
$341$	288.590i	0.846305i
$342$	0	0
$343$	−18.5203	−0.0539949
$344$	114.393i	0.332538i
$345$	0	0
$346$	−112.125	−0.324060
$347$	− 462.187i	− 1.33195i	−0.745974	−	0.665976i	$-0.768016\pi$
$347$	− 462.187i	− 1.33195i	0.745974	−	0.665976i	$-0.231984\pi$
$348$	0	0
$349$	541.980	1.55295	0.776476	−	0.630146i	$-0.217005\pi$
$349$	541.980	1.55295	0.776476	+	0.630146i	$0.217005\pi$
$350$	− 155.347i	− 0.443849i
$351$	0	0
$352$	−57.0953	−0.162203
$353$	− 123.276i	− 0.349224i	−0.984637	−	0.174612i	$-0.944133\pi$
$353$	− 123.276i	− 0.349224i	0.984637	−	0.174612i	$-0.0558671\pi$
$354$	0	0
$355$	−261.195	−0.735761
$356$	− 118.506i	− 0.332881i
$357$	0	0
$358$	59.4083	0.165945
$359$	90.2319i	0.251342i	0.992072	+	0.125671i	$0.0401084\pi$
$359$	90.2319i	0.251342i	−0.992072	+	0.125671i	$0.959892\pi$
$360$	0	0
$361$	−354.597	−0.982264
$362$	22.1446i	0.0611730i
$363$	0	0
$364$	−122.304	−0.336001
$365$	− 640.398i	− 1.75452i
$366$	0	0
$367$	539.070	1.46885	0.734427	−	0.678687i	$-0.237451\pi$
$367$	539.070	1.46885	0.734427	+	0.678687i	$0.237451\pi$
$368$	− 104.192i	− 0.283131i
$369$	0	0
$370$	−185.247	−0.500667
$371$	44.3293i	0.119486i
$372$	0	0
$373$	263.822	0.707298	0.353649	−	0.935378i	$-0.384941\pi$
$373$	263.822	0.707298	0.353649	+	0.935378i	$0.384941\pi$
$374$	343.006i	0.917127i
$375$	0	0
$376$	−208.610	−0.554815
$377$	595.078i	1.57846i
$378$	0	0
$379$	667.087	1.76012	0.880062	−	0.474859i	$-0.157501\pi$
$379$	667.087	1.76012	0.880062	+	0.474859i	$0.157501\pi$
$380$	41.2746i	0.108617i
$381$	0	0
$382$	55.9062	0.146351
$383$	− 311.448i	− 0.813181i	−0.913611	−	0.406590i	$-0.866718\pi$
$383$	− 311.448i	− 0.813181i	0.913611	−	0.406590i	$-0.133282\pi$
$384$	0	0
$385$	−217.794	−0.565698
$386$	426.055i	1.10377i
$387$	0	0
$388$	−233.883	−0.602792
$389$	70.9463i	0.182381i	0.995833	+	0.0911906i	$0.0290672\pi$
$389$	70.9463i	0.182381i	−0.995833	+	0.0911906i	$0.970933\pi$
$390$	0	0
$391$	−625.944	−1.60088
$392$	19.7990i	0.0505076i
$393$	0	0
$394$	−299.634	−0.760493
$395$	− 937.123i	− 2.37246i
$396$	0	0
$397$	324.719	0.817933	0.408966	−	0.912549i	$-0.365889\pi$
$397$	324.719	0.817933	0.408966	+	0.912549i	$0.365889\pi$
$398$	217.929i	0.547560i
$399$	0	0
$400$	−166.073	−0.415182
$401$	443.824i	1.10679i	0.832918	+	0.553397i	$0.186669\pi$
$401$	443.824i	1.10679i	−0.832918	+	0.553397i	$0.813331\pi$
$402$	0	0
$403$	660.875	1.63989
$404$	251.423i	0.622334i
$405$	0	0
$406$	96.3329	0.237273
$407$	162.103i	0.398287i
$408$	0	0
$409$	−223.255	−0.545856	−0.272928	−	0.962034i	$-0.587992\pi$
$409$	−223.255	−0.545856	−0.272928	+	0.962034i	$0.587992\pi$
$410$	500.615i	1.22101i
$411$	0	0
$412$	200.986	0.487830
$413$	− 162.947i	− 0.394545i
$414$	0	0
$415$	949.038	2.28684
$416$	130.749i	0.314300i
$417$	0	0
$418$	36.1179	0.0864065
$419$	289.036i	0.689823i	0.938635	+	0.344911i	$0.112091\pi$
$419$	289.036i	0.689823i	−0.938635	+	0.344911i	$0.887909\pi$
$420$	0	0
$421$	−703.671	−1.67143	−0.835714	−	0.549165i	$-0.814946\pi$
$421$	−703.671	−1.67143	−0.835714	+	0.549165i	$0.814946\pi$
$422$	− 67.3005i	− 0.159480i
$423$	0	0
$424$	47.3901	0.111769
$425$	997.699i	2.34753i
$426$	0	0
$427$	295.726	0.692568
$428$	− 114.331i	− 0.267128i
$429$	0	0
$430$	466.488	1.08486
$431$	− 53.1941i	− 0.123420i	−0.998094	−	0.0617101i	$-0.980345\pi$
$431$	− 53.1941i	− 0.123420i	0.998094	−	0.0617101i	$-0.0196554\pi$
$432$	0	0
$433$	−378.031	−0.873051	−0.436525	−	0.899692i	$-0.643791\pi$
$433$	−378.031	−0.873051	−0.436525	+	0.899692i	$0.643791\pi$
$434$	− 106.984i	− 0.246508i
$435$	0	0
$436$	247.940	0.568669
$437$	65.9109i	0.150826i
$438$	0	0
$439$	477.045	1.08666	0.543332	−	0.839518i	$-0.317163\pi$
$439$	477.045	1.08666	0.543332	+	0.839518i	$0.317163\pi$
$440$	232.831i	0.529162i
$441$	0	0
$442$	785.487	1.77712
$443$	− 67.0689i	− 0.151397i	−0.997131	−	0.0756986i	$-0.975881\pi$
$443$	− 67.0689i	− 0.151397i	0.997131	−	0.0756986i	$-0.0241187\pi$
$444$	0	0
$445$	−483.258	−1.08597
$446$	− 324.806i	− 0.728265i
$447$	0	0
$448$	21.1660	0.0472456
$449$	− 378.496i	− 0.842974i	−0.906834	−	0.421487i	$-0.861508\pi$
$449$	− 378.496i	− 0.842974i	0.906834	−	0.421487i	$-0.138492\pi$
$450$	0	0
$451$	438.070	0.971331
$452$	136.579i	0.302165i
$453$	0	0
$454$	474.146	1.04438
$455$	498.750i	1.09615i
$456$	0	0
$457$	−290.251	−0.635123	−0.317561	−	0.948238i	$-0.602864\pi$
$457$	−290.251	−0.635123	−0.317561	+	0.948238i	$0.602864\pi$
$458$	− 353.938i	− 0.772791i
$459$	0	0
$460$	−424.889	−0.923672
$461$	− 168.959i	− 0.366506i	−0.983066	−	0.183253i	$-0.941337\pi$
$461$	− 168.959i	− 0.366506i	0.983066	−	0.183253i	$-0.0586628\pi$
$462$	0	0
$463$	146.852	0.317175	0.158588	−	0.987345i	$-0.449306\pi$
$463$	146.852	0.317175	0.158588	+	0.987345i	$0.449306\pi$
$464$	− 102.984i	− 0.221949i
$465$	0	0
$466$	−151.298	−0.324675
$467$	− 171.667i	− 0.367595i	−0.982964	−	0.183798i	$-0.941161\pi$
$467$	− 171.667i	− 0.367595i	0.982964	−	0.183798i	$-0.0588391\pi$
$468$	0	0
$469$	−178.367	−0.380313
$470$	850.700i	1.81000i
$471$	0	0
$472$	−174.198	−0.369063
$473$	− 408.207i	− 0.863017i
$474$	0	0
$475$	105.056	0.221171
$476$	− 127.157i	− 0.267136i
$477$	0	0
$478$	−187.025	−0.391266
$479$	− 233.833i	− 0.488169i	−0.969754	−	0.244085i	$-0.921513\pi$
$479$	− 233.833i	− 0.488169i	0.969754	−	0.244085i	$-0.0784875\pi$
$480$	0	0
$481$	371.217	0.771761
$482$	345.310i	0.716411i
$483$	0	0
$484$	−38.2577	−0.0790448
$485$	953.761i	1.96652i
$486$	0	0
$487$	609.989	1.25254	0.626272	−	0.779605i	$-0.284580\pi$
$487$	609.989	1.25254	0.626272	+	0.779605i	$0.284580\pi$
$488$	− 316.145i	− 0.647838i
$489$	0	0
$490$	80.7390	0.164773
$491$	22.0433i	0.0448947i	0.999748	+	0.0224474i	$0.00714582\pi$
$491$	22.0433i	0.0448947i	−0.999748	+	0.0224474i	$0.992854\pi$
$492$	0	0
$493$	−618.687	−1.25494
$494$	− 82.7104i	− 0.167430i
$495$	0	0
$496$	−114.371	−0.230587
$497$	− 84.7313i	− 0.170486i
$498$	0	0
$499$	−154.587	−0.309793	−0.154897	−	0.987931i	$-0.549504\pi$
$499$	−154.587	−0.309793	−0.154897	+	0.987931i	$0.549504\pi$
$500$	269.441i	0.538882i
$501$	0	0
$502$	508.546	1.01304
$503$	− 163.201i	− 0.324456i	−0.986753	−	0.162228i	$-0.948132\pi$
$503$	− 163.201i	− 0.324456i	0.986753	−	0.162228i	$-0.0518680\pi$
$504$	0	0
$505$	1025.29	2.03027
$506$	371.806i	0.734793i
$507$	0	0
$508$	413.423	0.813825
$509$	− 960.560i	− 1.88715i	−0.331156	−	0.943576i	$-0.607439\pi$
$509$	− 960.560i	− 1.88715i	0.331156	−	0.943576i	$-0.392561\pi$
$510$	0	0
$511$	207.744	0.406544
$512$	− 22.6274i	− 0.0441942i
$513$	0	0
$514$	−411.951	−0.801461
$515$	− 819.608i	− 1.59147i
$516$	0	0
$517$	744.418	1.43988
$518$	− 60.0937i	− 0.116011i
$519$	0	0
$520$	533.186	1.02536
$521$	− 606.242i	− 1.16361i	−0.813328	−	0.581806i	$-0.802346\pi$
$521$	− 606.242i	− 1.16361i	0.813328	−	0.581806i	$-0.197654\pi$
$522$	0	0
$523$	−815.032	−1.55838	−0.779189	−	0.626789i	$-0.784369\pi$
$523$	−815.032	−1.55838	−0.779189	+	0.626789i	$0.784369\pi$
$524$	219.721i	0.419315i
$525$	0	0
$526$	201.050	0.382224
$527$	687.095i	1.30378i
$528$	0	0
$529$	−149.501	−0.282610
$530$	− 193.254i	− 0.364629i
$531$	0	0
$532$	−13.3894	−0.0251680
$533$	− 1003.19i	− 1.88215i
$534$	0	0
$535$	−466.234	−0.871466
$536$	190.682i	0.355750i
$537$	0	0
$538$	−341.709	−0.635146
$539$	− 70.6519i	− 0.131080i
$540$	0	0
$541$	−1043.95	−1.92967	−0.964835	−	0.262856i	$-0.915336\pi$
$541$	−1043.95	−1.92967	−0.964835	+	0.262856i	$0.915336\pi$
$542$	726.415i	1.34025i
$543$	0	0
$544$	−135.936	−0.249883
$545$	− 1011.08i	− 1.85520i
$546$	0	0
$547$	226.177	0.413486	0.206743	−	0.978395i	$-0.433714\pi$
$547$	226.177	0.413486	0.206743	+	0.978395i	$0.433714\pi$
$548$	2.12920i	0.00388540i
$549$	0	0
$550$	592.624	1.07750
$551$	65.1467i	0.118234i
$552$	0	0
$553$	304.001	0.549731
$554$	398.791i	0.719839i
$555$	0	0
$556$	−190.201	−0.342088
$557$	394.275i	0.707854i	0.935273	+	0.353927i	$0.115154\pi$
$557$	394.275i	0.707854i	−0.935273	+	0.353927i	$0.884846\pi$
$558$	0	0
$559$	−934.799	−1.67227
$560$	− 86.3136i	− 0.154131i
$561$	0	0
$562$	486.953	0.866464
$563$	267.789i	0.475647i	0.971308	+	0.237824i	$0.0764340\pi$
$563$	267.789i	0.475647i	−0.971308	+	0.237824i	$0.923566\pi$
$564$	0	0
$565$	556.959	0.985769
$566$	− 410.139i	− 0.724627i
$567$	0	0
$568$	−90.5816	−0.159475
$569$	146.709i	0.257837i	0.991655	+	0.128919i	$0.0411506\pi$
$569$	146.709i	0.257837i	−0.991655	+	0.128919i	$0.958849\pi$
$570$	0	0
$571$	31.1222	0.0545047	0.0272524	−	0.999629i	$-0.491324\pi$
$571$	31.1222	0.0545047	0.0272524	+	0.999629i	$0.491324\pi$
$572$	− 466.572i	− 0.815686i
$573$	0	0
$574$	−162.399	−0.282924
$575$	1081.47i	1.88082i
$576$	0	0
$577$	−88.2056	−0.152869	−0.0764347	−	0.997075i	$-0.524354\pi$
$577$	−88.2056	−0.152869	−0.0764347	+	0.997075i	$0.524354\pi$
$578$	407.943i	0.705783i
$579$	0	0
$580$	−419.963	−0.724074
$581$	307.866i	0.529890i
$582$	0	0
$583$	−169.109	−0.290068
$584$	− 222.088i	− 0.380287i
$585$	0	0
$586$	−309.675	−0.528455
$587$	− 338.925i	− 0.577385i	−0.957422	−	0.288692i	$-0.906780\pi$
$587$	− 338.925i	− 0.577385i	0.957422	−	0.288692i	$-0.0932205\pi$
$588$	0	0
$589$	72.3499	0.122835
$590$	710.367i	1.20401i
$591$	0	0
$592$	−64.2429	−0.108518
$593$	1085.36i	1.83029i	0.403122	+	0.915146i	$0.367925\pi$
$593$	1085.36i	1.83029i	−0.403122	+	0.915146i	$0.632075\pi$
$594$	0	0
$595$	−518.537	−0.871491
$596$	− 345.215i	− 0.579220i
$597$	0	0
$598$	851.439	1.42381
$599$	986.458i	1.64684i	0.567431	+	0.823421i	$0.307937\pi$
$599$	986.458i	1.64684i	−0.567431	+	0.823421i	$0.692063\pi$
$600$	0	0
$601$	27.3803	0.0455580	0.0227790	−	0.999741i	$-0.492749\pi$
$601$	27.3803	0.0455580	0.0227790	+	0.999741i	$0.492749\pi$
$602$	151.328i	0.251375i
$603$	0	0
$604$	99.5728	0.164856
$605$	156.012i	0.257872i
$606$	0	0
$607$	−223.363	−0.367978	−0.183989	−	0.982928i	$-0.558901\pi$
$607$	−223.363	−0.367978	−0.183989	+	0.982928i	$0.558901\pi$
$608$	14.3139i	0.0235426i
$609$	0	0
$610$	−1289.22	−2.11347
$611$	− 1704.72i	− 2.79006i
$612$	0	0
$613$	555.018	0.905413	0.452706	−	0.891660i	$-0.350459\pi$
$613$	555.018	0.905413	0.452706	+	0.891660i	$0.350459\pi$
$614$	− 109.925i	− 0.179031i
$615$	0	0
$616$	−75.5300	−0.122614
$617$	− 1042.15i	− 1.68906i	−0.535507	−	0.844531i	$-0.679879\pi$
$617$	− 1042.15i	− 1.68906i	0.535507	−	0.844531i	$-0.320121\pi$
$618$	0	0
$619$	−139.825	−0.225889	−0.112944	−	0.993601i	$-0.536028\pi$
$619$	−139.825	−0.225889	−0.112944	+	0.993601i	$0.536028\pi$
$620$	466.397i	0.752254i
$621$	0	0
$622$	−361.896	−0.581827
$623$	− 156.768i	− 0.251634i
$624$	0	0
$625$	60.8077	0.0972924
$626$	419.803i	0.670612i
$627$	0	0
$628$	−381.407	−0.607337
$629$	385.945i	0.613585i
$630$	0	0
$631$	−655.037	−1.03809	−0.519047	−	0.854746i	$-0.673713\pi$
$631$	−655.037	−1.03809	−0.519047	+	0.854746i	$0.673713\pi$
$632$	− 324.991i	− 0.514226i
$633$	0	0
$634$	−526.424	−0.830321
$635$	− 1685.91i	− 2.65498i
$636$	0	0
$637$	−161.794	−0.253993
$638$	367.495i	0.576011i
$639$	0	0
$640$	−92.2731	−0.144177
$641$	448.897i	0.700307i	0.936692	+	0.350153i	$0.113871\pi$
$641$	448.897i	0.700307i	−0.936692	+	0.350153i	$0.886129\pi$
$642$	0	0
$643$	−388.947	−0.604894	−0.302447	−	0.953166i	$-0.597803\pi$
$643$	−388.947	−0.604894	−0.302447	+	0.953166i	$0.597803\pi$
$644$	− 137.833i	− 0.214027i
$645$	0	0
$646$	85.9919	0.133114
$647$	458.666i	0.708912i	0.935073	+	0.354456i	$0.115334\pi$
$647$	458.666i	0.708912i	−0.935073	+	0.354456i	$0.884666\pi$
$648$	0	0
$649$	621.617	0.957807
$650$	− 1357.12i	− 2.08787i
$651$	0	0
$652$	−269.178	−0.412850
$653$	821.317i	1.25776i	0.777502	+	0.628880i	$0.216486\pi$
$653$	821.317i	1.25776i	−0.777502	+	0.628880i	$0.783514\pi$
$654$	0	0
$655$	896.009	1.36795
$656$	173.611i	0.264651i
$657$	0	0
$658$	−275.966	−0.419401
$659$	− 1091.86i	− 1.65685i	−0.560099	−	0.828426i	$-0.689237\pi$
$659$	− 1091.86i	− 1.65685i	0.560099	−	0.828426i	$-0.310763\pi$
$660$	0	0
$661$	−755.123	−1.14239	−0.571197	−	0.820813i	$-0.693521\pi$
$661$	−755.123	−1.14239	−0.571197	+	0.820813i	$0.693521\pi$
$662$	− 267.667i	− 0.404332i
$663$	0	0
$664$	329.123	0.495667
$665$	54.6011i	0.0821069i
$666$	0	0
$667$	−670.634	−1.00545
$668$	− 285.396i	− 0.427239i
$669$	0	0
$670$	777.589	1.16058
$671$	1128.15i	1.68130i
$672$	0	0
$673$	−624.087	−0.927322	−0.463661	−	0.886013i	$-0.653464\pi$
$673$	−624.087	−0.927322	−0.463661	+	0.886013i	$0.653464\pi$
$674$	− 396.197i	− 0.587830i
$675$	0	0
$676$	−730.455	−1.08056
$677$	− 1316.10i	− 1.94401i	−0.234951	−	0.972007i	$-0.575493\pi$
$677$	− 1316.10i	− 1.94401i	0.234951	−	0.972007i	$-0.424507\pi$
$678$	0	0
$679$	−309.398	−0.455668
$680$	554.340i	0.815205i
$681$	0	0
$682$	408.128	0.598428
$683$	− 156.335i	− 0.228895i	−0.993429	−	0.114448i	$-0.963490\pi$
$683$	− 156.335i	− 0.228895i	0.993429	−	0.114448i	$-0.0365098\pi$
$684$	0	0
$685$	8.68273	0.0126755
$686$	26.1916i	0.0381802i
$687$	0	0
$688$	161.776	0.235140
$689$	387.262i	0.562065i
$690$	0	0
$691$	−551.411	−0.797990	−0.398995	−	0.916953i	$-0.630641\pi$
$691$	−551.411	−0.797990	−0.398995	+	0.916953i	$0.630641\pi$
$692$	158.569i	0.229145i
$693$	0	0
$694$	−653.631	−0.941832
$695$	775.626i	1.11601i
$696$	0	0
$697$	1042.99	1.49639
$698$	− 766.476i	− 1.09810i
$699$	0	0
$700$	−219.694	−0.313848
$701$	− 857.762i	− 1.22363i	−0.791002	−	0.611813i	$-0.790441\pi$
$701$	− 857.762i	− 1.22363i	0.791002	−	0.611813i	$-0.209559\pi$
$702$	0	0
$703$	40.6394	0.0578085
$704$	80.7450i	0.114695i
$705$	0	0
$706$	−174.339	−0.246939
$707$	332.601i	0.470440i
$708$	0	0
$709$	943.333	1.33051	0.665256	−	0.746615i	$-0.268322\pi$
$709$	943.333	1.33051	0.665256	+	0.746615i	$0.268322\pi$
$710$	369.386i	0.520262i
$711$	0	0
$712$	−167.592	−0.235382
$713$	744.785i	1.04458i
$714$	0	0
$715$	−1902.65	−2.66105
$716$	− 84.0160i	− 0.117341i
$717$	0	0
$718$	127.607	0.177726
$719$	− 175.405i	− 0.243957i	−0.992533	−	0.121978i	$-0.961076\pi$
$719$	− 175.405i	− 0.243957i	0.992533	−	0.121978i	$-0.0389238\pi$
$720$	0	0
$721$	265.880	0.368765
$722$	501.476i	0.694565i
$723$	0	0
$724$	31.3172	0.0432559
$725$	1068.93i	1.47439i
$726$	0	0
$727$	657.028	0.903753	0.451877	−	0.892080i	$-0.350755\pi$
$727$	657.028	0.903753	0.451877	+	0.892080i	$0.350755\pi$
$728$	172.965i	0.237589i
$729$	0	0
$730$	−905.660	−1.24063
$731$	− 971.886i	− 1.32953i
$732$	0	0
$733$	−42.0489	−0.0573655	−0.0286828	−	0.999589i	$-0.509131\pi$
$733$	−42.0489	−0.0573655	−0.0286828	+	0.999589i	$0.509131\pi$
$734$	− 762.360i	− 1.03864i
$735$	0	0
$736$	−147.350	−0.200204
$737$	− 680.441i	− 0.923258i
$738$	0	0
$739$	−415.509	−0.562258	−0.281129	−	0.959670i	$-0.590709\pi$
$739$	−415.509	−0.562258	−0.281129	+	0.959670i	$0.590709\pi$
$740$	261.978i	0.354025i
$741$	0	0
$742$	62.6912	0.0844894
$743$	− 1107.62i	− 1.49074i	−0.666648	−	0.745372i	$-0.732272\pi$
$743$	− 1107.62i	− 1.49074i	0.666648	−	0.745372i	$-0.267728\pi$
$744$	0	0
$745$	−1407.76	−1.88962
$746$	− 373.101i	− 0.500135i
$747$	0	0
$748$	485.083	0.648507
$749$	− 151.246i	− 0.201930i
$750$	0	0
$751$	−546.081	−0.727138	−0.363569	−	0.931567i	$-0.618442\pi$
$751$	−546.081	−0.727138	−0.363569	+	0.931567i	$0.618442\pi$
$752$	295.020i	0.392313i
$753$	0	0
$754$	841.567	1.11614
$755$	− 406.051i	− 0.537817i
$756$	0	0
$757$	−738.177	−0.975135	−0.487567	−	0.873085i	$-0.662116\pi$
$757$	−738.177	−0.975135	−0.487567	+	0.873085i	$0.662116\pi$
$758$	− 943.404i	− 1.24460i
$759$	0	0
$760$	58.3710	0.0768040
$761$	828.998i	1.08935i	0.838646	+	0.544677i	$0.183348\pi$
$761$	828.998i	1.08935i	−0.838646	+	0.544677i	$0.816652\pi$
$762$	0	0
$763$	327.994	0.429874
$764$	− 79.0633i	− 0.103486i
$765$	0	0
$766$	−440.454	−0.575006
$767$	− 1423.51i	− 1.85594i
$768$	0	0
$769$	−1124.68	−1.46252	−0.731259	−	0.682100i	$-0.761067\pi$
$769$	−1124.68	−1.46252	−0.731259	+	0.682100i	$0.761067\pi$
$770$	308.007i	0.400009i
$771$	0	0
$772$	602.532	0.780482
$773$	93.3214i	0.120726i	0.998176	+	0.0603631i	$0.0192259\pi$
$773$	93.3214i	0.120726i	−0.998176	+	0.0603631i	$0.980774\pi$
$774$	0	0
$775$	1187.12	1.53177
$776$	330.761i	0.426238i
$777$	0	0
$778$	100.333	0.128963
$779$	− 109.825i	− 0.140982i
$780$	0	0
$781$	323.237	0.413875
$782$	885.219i	1.13199i
$783$	0	0
$784$	28.0000	0.0357143
$785$	1555.35i	1.98134i
$786$	0	0
$787$	−876.284	−1.11345	−0.556724	−	0.830697i	$-0.687942\pi$
$787$	−876.284	−1.11345	−0.556724	+	0.830697i	$0.687942\pi$
$788$	423.747i	0.537750i
$789$	0	0
$790$	−1325.29	−1.67758
$791$	180.677i	0.228416i
$792$	0	0
$793$	2583.47	3.25785
$794$	− 459.222i	− 0.578366i
$795$	0	0
$796$	308.198	0.387183
$797$	− 233.264i	− 0.292678i	−0.989235	−	0.146339i	$-0.953251\pi$
$797$	− 233.264i	− 0.292678i	0.989235	−	0.146339i	$-0.0467490\pi$
$798$	0	0
$799$	1772.36	2.21822
$800$	234.863i	0.293578i
$801$	0	0
$802$	627.662	0.782621
$803$	792.511i	0.986938i
$804$	0	0
$805$	−562.075	−0.698230
$806$	− 934.618i	− 1.15958i
$807$	0	0
$808$	355.566	0.440057
$809$	493.652i	0.610200i	0.952320	+	0.305100i	$0.0986898\pi$
$809$	493.652i	0.610200i	−0.952320	+	0.305100i	$0.901310\pi$
$810$	0	0
$811$	−172.045	−0.212139	−0.106069	−	0.994359i	$-0.533827\pi$
$811$	−172.045	−0.212139	−0.106069	+	0.994359i	$0.533827\pi$
$812$	− 136.235i	− 0.167777i
$813$	0	0
$814$	229.248	0.281631
$815$	1097.69i	1.34686i
$816$	0	0
$817$	−102.338	−0.125261
$818$	315.730i	0.385979i
$819$	0	0
$820$	707.976	0.863385
$821$	165.561i	0.201658i	0.994904	+	0.100829i	$0.0321494\pi$
$821$	165.561i	0.201658i	−0.994904	+	0.100829i	$0.967851\pi$
$822$	0	0
$823$	1597.55	1.94113	0.970567	−	0.240829i	$-0.0774194\pi$
$823$	1597.55	1.94113	0.970567	+	0.240829i	$0.0774194\pi$
$824$	− 284.237i	− 0.344948i
$825$	0	0
$826$	−230.442	−0.278985
$827$	987.791i	1.19443i	0.802082	+	0.597214i	$0.203725\pi$
$827$	987.791i	1.19443i	−0.802082	+	0.597214i	$0.796275\pi$
$828$	0	0
$829$	−56.3597	−0.0679851	−0.0339926	−	0.999422i	$-0.510822\pi$
$829$	−56.3597	−0.0679851	−0.0339926	+	0.999422i	$0.510822\pi$
$830$	− 1342.14i	− 1.61704i
$831$	0	0
$832$	184.907	0.222244
$833$	− 168.213i	− 0.201936i
$834$	0	0
$835$	−1163.83	−1.39380
$836$	− 51.0785i	− 0.0610986i
$837$	0	0
$838$	408.758	0.487778
$839$	1178.15i	1.40423i	0.712064	+	0.702114i	$0.247760\pi$
$839$	1178.15i	1.40423i	−0.712064	+	0.702114i	$0.752240\pi$
$840$	0	0
$841$	178.141	0.211821
$842$	995.141i	1.18188i
$843$	0	0
$844$	−95.1772	−0.112769
$845$	2978.75i	3.52515i
$846$	0	0
$847$	−50.6101	−0.0597522
$848$	− 67.0197i	− 0.0790326i
$849$	0	0
$850$	1410.96	1.65995
$851$	418.350i	0.491599i
$852$	0	0
$853$	−413.635	−0.484918	−0.242459	−	0.970162i	$-0.577954\pi$
$853$	−413.635	−0.484918	−0.242459	+	0.970162i	$0.577954\pi$
$854$	− 418.220i	− 0.489719i
$855$	0	0
$856$	−161.688	−0.188888
$857$	− 798.554i	− 0.931802i	−0.884837	−	0.465901i	$-0.845730\pi$
$857$	− 798.554i	− 0.931802i	0.884837	−	0.465901i	$-0.154270\pi$
$858$	0	0
$859$	−515.234	−0.599807	−0.299903	−	0.953970i	$-0.596954\pi$
$859$	−515.234	−0.599807	−0.299903	+	0.953970i	$0.596954\pi$
$860$	− 659.713i	− 0.767109i
$861$	0	0
$862$	−75.2279	−0.0872713
$863$	− 751.986i	− 0.871363i	−0.900101	−	0.435682i	$-0.856507\pi$
$863$	− 751.986i	− 0.871363i	0.900101	−	0.435682i	$-0.143493\pi$
$864$	0	0
$865$	646.632	0.747552
$866$	534.617i	0.617340i
$867$	0	0
$868$	−151.299	−0.174307
$869$	1159.72i	1.33454i
$870$	0	0
$871$	−1558.22	−1.78900
$872$	− 350.640i	− 0.402110i
$873$	0	0
$874$	93.2121	0.106650
$875$	356.437i	0.407357i
$876$	0	0
$877$	−462.092	−0.526901	−0.263451	−	0.964673i	$-0.584861\pi$
$877$	−462.092	−0.526901	−0.263451	+	0.964673i	$0.584861\pi$
$878$	− 674.644i	− 0.768387i
$879$	0	0
$880$	329.273	0.374174
$881$	462.516i	0.524990i	0.964933	+	0.262495i	$0.0845453\pi$
$881$	462.516i	0.524990i	−0.964933	+	0.262495i	$0.915455\pi$
$882$	0	0
$883$	−355.670	−0.402797	−0.201399	−	0.979509i	$-0.564549\pi$
$883$	−355.670	−0.402797	−0.201399	+	0.979509i	$0.564549\pi$
$884$	− 1110.85i	− 1.25661i
$885$	0	0
$886$	−94.8498	−0.107054
$887$	1248.15i	1.40716i	0.710617	+	0.703579i	$0.248416\pi$
$887$	1248.15i	1.40716i	−0.710617	+	0.703579i	$0.751584\pi$
$888$	0	0
$889$	546.907	0.615194
$890$	683.431i	0.767900i
$891$	0	0
$892$	−459.345	−0.514961
$893$	− 186.626i	− 0.208988i
$894$	0	0
$895$	−342.612	−0.382807
$896$	− 29.9333i	− 0.0334077i
$897$	0	0
$898$	−535.273	−0.596073
$899$	736.150i	0.818854i
$900$	0	0
$901$	−402.627	−0.446867
$902$	− 619.525i	− 0.686835i
$903$	0	0
$904$	193.152	0.213663
$905$	− 127.710i	− 0.141116i
$906$	0	0
$907$	760.252	0.838205	0.419103	−	0.907939i	$-0.362345\pi$
$907$	760.252	0.838205	0.419103	+	0.907939i	$0.362345\pi$
$908$	− 670.544i	− 0.738485i
$909$	0	0
$910$	705.338	0.775097
$911$	− 1171.62i	− 1.28608i	−0.765833	−	0.643040i	$-0.777673\pi$
$911$	− 1171.62i	− 1.28608i	0.765833	−	0.643040i	$-0.222327\pi$
$912$	0	0
$913$	−1174.46	−1.28638
$914$	410.477i	0.449099i
$915$	0	0
$916$	−500.545	−0.546446
$917$	290.664i	0.316973i
$918$	0	0
$919$	1022.10	1.11219	0.556096	−	0.831118i	$-0.312299\pi$
$919$	1022.10	1.11219	0.556096	+	0.831118i	$0.312299\pi$
$920$	600.884i	0.653135i
$921$	0	0
$922$	−238.945	−0.259159
$923$	− 740.215i	− 0.801967i
$924$	0	0
$925$	666.813	0.720878
$926$	− 207.680i	− 0.224277i
$927$	0	0
$928$	−145.642	−0.156941
$929$	891.630i	0.959774i	0.877330	+	0.479887i	$0.159322\pi$
$929$	891.630i	0.959774i	−0.877330	+	0.479887i	$0.840678\pi$
$930$	0	0
$931$	−17.7125	−0.0190253
$932$	213.968i	0.229580i
$933$	0	0
$934$	−242.774	−0.259929
$935$	− 1978.14i	− 2.11566i
$936$	0	0
$937$	−561.744	−0.599513	−0.299757	−	0.954016i	$-0.596906\pi$
$937$	−561.744	−0.599513	−0.299757	+	0.954016i	$0.596906\pi$
$938$	252.249i	0.268922i
$939$	0	0
$940$	1203.07	1.27986
$941$	1447.20i	1.53794i	0.639287	+	0.768968i	$0.279230\pi$
$941$	1447.20i	1.53794i	−0.639287	+	0.768968i	$0.720770\pi$
$942$	0	0
$943$	1130.56	1.19890
$944$	246.353i	0.260967i
$945$	0	0
$946$	−577.292	−0.610245
$947$	− 1655.42i	− 1.74807i	−0.485864	−	0.874034i	$-0.661495\pi$
$947$	− 1655.42i	− 1.74807i	0.485864	−	0.874034i	$-0.338505\pi$
$948$	0	0
$949$	1814.86	1.91239
$950$	− 148.572i	− 0.156391i
$951$	0	0
$952$	−179.827	−0.188894
$953$	− 367.955i	− 0.386102i	−0.981189	−	0.193051i	$-0.938162\pi$
$953$	− 367.955i	− 0.386102i	0.981189	−	0.193051i	$-0.0618383\pi$
$954$	0	0
$955$	−322.415	−0.337607
$956$	264.493i	0.276667i
$957$	0	0
$958$	−330.690	−0.345188
$959$	2.81666i	0.00293708i
$960$	0	0
$961$	−143.455	−0.149277
$962$	− 524.980i	− 0.545718i
$963$	0	0
$964$	488.342	0.506579
$965$	− 2457.09i	− 2.54620i
$966$	0	0
$967$	1139.36	1.17825	0.589123	−	0.808043i	$-0.299473\pi$
$967$	1139.36	1.17825	0.589123	+	0.808043i	$0.299473\pi$
$968$	54.1045i	0.0558931i
$969$	0	0
$970$	1348.82	1.39054
$971$	− 79.5809i	− 0.0819577i	−0.999160	−	0.0409788i	$-0.986952\pi$
$971$	− 79.5809i	− 0.0819577i	0.999160	−	0.0409788i	$-0.0130476\pi$
$972$	0	0
$973$	−251.612	−0.258594
$974$	− 862.655i	− 0.885682i
$975$	0	0
$976$	−447.096	−0.458090
$977$	1598.58i	1.63622i	0.575063	+	0.818109i	$0.304977\pi$
$977$	1598.58i	1.63622i	−0.575063	+	0.818109i	$0.695023\pi$
$978$	0	0
$979$	598.046	0.610875
$980$	− 114.182i	− 0.116512i
$981$	0	0
$982$	31.1740	0.0317454
$983$	850.160i	0.864863i	0.901667	+	0.432431i	$0.142344\pi$
$983$	850.160i	0.864863i	−0.901667	+	0.432431i	$0.857656\pi$
$984$	0	0
$985$	1728.01	1.75433
$986$	874.956i	0.887379i
$987$	0	0
$988$	−116.970	−0.118391
$989$	− 1053.49i	− 1.06521i
$990$	0	0
$991$	319.206	0.322105	0.161053	−	0.986946i	$-0.448511\pi$
$991$	319.206	0.322105	0.161053	+	0.986946i	$0.448511\pi$
$992$	161.745i	0.163049i
$993$	0	0
$994$	−119.828	−0.120551
$995$	− 1256.81i	− 1.26313i
$996$	0	0
$997$	1372.68	1.37681	0.688404	−	0.725328i	$-0.258312\pi$
$997$	1372.68	1.37681	0.688404	+	0.725328i	$0.258312\pi$
$998$	218.619i	0.219057i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.3.b.c.323.4	✓	8
3.2	odd	2	inner	378.3.b.c.323.5	yes	8
4.3	odd	2		3024.3.d.h.1457.8		8
9.2	odd	6		1134.3.q.f.701.8		16
9.4	even	3		1134.3.q.f.1079.8		16
9.5	odd	6		1134.3.q.f.1079.1		16
9.7	even	3		1134.3.q.f.701.1		16
12.11	even	2		3024.3.d.h.1457.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.3.b.c.323.4	✓	8	1.1	even	1	trivial
378.3.b.c.323.5	yes	8	3.2	odd	2	inner
1134.3.q.f.701.1		16	9.7	even	3
1134.3.q.f.701.8		16	9.2	odd	6
1134.3.q.f.1079.1		16	9.5	odd	6
1134.3.q.f.1079.8		16	9.4	even	3
3024.3.d.h.1457.1		8	12.11	even	2
3024.3.d.h.1457.8		8	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	378.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} -2.00000 q^{4} +8.15587i q^{5} -2.64575 q^{7} +2.82843i q^{8} +O(q^{10})\)	\(q-1.41421i q^{2} -2.00000 q^{4} +8.15587i q^{5} -2.64575 q^{7} +2.82843i q^{8} +11.5341 q^{10} -10.0931i q^{11} -23.1134 q^{13} +3.74166i q^{14} +4.00000 q^{16} -24.0304i q^{17} -2.53036 q^{19} -16.3117i q^{20} -14.2738 q^{22} -26.0480i q^{23} -41.5182 q^{25} +32.6872i q^{26} +5.29150 q^{28} -25.7460i q^{29} -28.5927 q^{31} -5.65685i q^{32} -33.9841 q^{34} -21.5784i q^{35} -16.0607 q^{37} +3.57847i q^{38} -23.0683 q^{40} +43.4028i q^{41} +40.4441 q^{43} +20.1862i q^{44} -36.8375 q^{46} +73.7549i q^{47} +7.00000 q^{49} +58.7156i q^{50} +46.2267 q^{52} -16.7549i q^{53} +82.3182 q^{55} -7.48331i q^{56} -36.4104 q^{58} +61.5881i q^{59} -111.774 q^{61} +40.4362i q^{62} -8.00000 q^{64} -188.510i q^{65} +67.4163 q^{67} +48.0608i q^{68} -30.5165 q^{70} +32.0254i q^{71} -78.5199 q^{73} +22.7133i q^{74} +5.06072 q^{76} +26.7039i q^{77} -114.902 q^{79} +32.6235i q^{80} +61.3809 q^{82} -116.363i q^{83} +195.989 q^{85} -57.1966i q^{86} +28.5477 q^{88} +59.2528i q^{89} +61.1522 q^{91} +52.0961i q^{92} +104.305 q^{94} -20.6373i q^{95} +116.942 q^{97} -9.89949i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{4}+O(q^{10}) \) 8 * q - 16 * q^4	\( 8 q - 16 q^{4} + 16 q^{10} - 40 q^{13} + 32 q^{16} + 40 q^{19} - 16 q^{22} - 40 q^{31} - 16 q^{34} - 8 q^{37} - 32 q^{40} + 40 q^{43} - 64 q^{46} + 56 q^{49} + 80 q^{52} + 56 q^{55} + 112 q^{58} - 88 q^{61} - 64 q^{64} + 240 q^{67} - 112 q^{70} - 288 q^{73} - 80 q^{76} - 424 q^{79} + 144 q^{82} + 200 q^{85} + 32 q^{88} + 56 q^{91} + 288 q^{94} + 600 q^{97}+O(q^{100}) \) 8 * q - 16 * q^4 + 16 * q^10 - 40 * q^13 + 32 * q^16 + 40 * q^19 - 16 * q^22 - 40 * q^31 - 16 * q^34 - 8 * q^37 - 32 * q^40 + 40 * q^43 - 64 * q^46 + 56 * q^49 + 80 * q^52 + 56 * q^55 + 112 * q^58 - 88 * q^61 - 64 * q^64 + 240 * q^67 - 112 * q^70 - 288 * q^73 - 80 * q^76 - 424 * q^79 + 144 * q^82 + 200 * q^85 + 32 * q^88 + 56 * q^91 + 288 * q^94 + 600 * q^97

Introduction

L-functions

Modular forms

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Database

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