LMFDB - Embedded newform 378.3.b.c.323.7

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			323.7
Root			$-0.581861 - 1.28897i$ of defining polynomial
Character	$\chi$	$=$	378.323
Dual form			378.3.b.c.323.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{2} -2.00000 q^{4} -0.672556i q^{5} +2.64575 q^{7} -2.82843i q^{8} +O(q^{10})$	\(q+1.41421i q^{2} -2.00000 q^{4} -0.672556i q^{5} +2.64575 q^{7} -2.82843i q^{8} +0.951138 q^{10} -13.2647i q^{11} +4.62808 q^{13} +3.74166i q^{14} +4.00000 q^{16} -15.2020i q^{17} +4.04508 q^{19} +1.34511i q^{20} +18.7591 q^{22} +9.26566i q^{23} +24.5477 q^{25} +6.54510i q^{26} -5.29150 q^{28} -27.5450i q^{29} +44.0486 q^{31} +5.65685i q^{32} +21.4988 q^{34} -1.77942i q^{35} -2.90984 q^{37} +5.72060i q^{38} -1.90228 q^{40} +23.9470i q^{41} +62.8940 q^{43} +26.5294i q^{44} -13.1036 q^{46} -55.1568i q^{47} +7.00000 q^{49} +34.7156i q^{50} -9.25617 q^{52} -1.84323i q^{53} -8.92125 q^{55} -7.48331i q^{56} +38.9546 q^{58} +13.2450i q^{59} -20.5346 q^{61} +62.2941i q^{62} -8.00000 q^{64} -3.11265i q^{65} -75.2985 q^{67} +30.4039i q^{68} +2.51647 q^{70} +91.3391i q^{71} +40.4610 q^{73} -4.11514i q^{74} -8.09016 q^{76} -35.0951i q^{77} -84.4365 q^{79} -2.69022i q^{80} -33.8662 q^{82} -80.0194i q^{83} -10.2242 q^{85} +88.9456i q^{86} -37.5182 q^{88} -109.600i q^{89} +12.2448 q^{91} -18.5313i q^{92} +78.0035 q^{94} -2.72054i q^{95} +126.396 q^{97} +9.89949i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	− 0.672556i	− 0.134511i	−0.997736	−	0.0672556i	$-0.978576\pi$
$5$	− 0.672556i	− 0.134511i	0.997736	−	0.0672556i	$-0.0214243\pi$
$6$	0	0
$7$	2.64575	0.377964
$7$	2.64575	0.377964
$8$	− 2.82843i	− 0.353553i
$9$	0	0
$10$	0.951138	0.0951138
$11$	− 13.2647i	− 1.20588i	−0.797786	−	0.602941i	$-0.793996\pi$
$11$	− 13.2647i	− 1.20588i	0.797786	−	0.602941i	$-0.206004\pi$
$12$	0	0
$13$	4.62808	0.356006	0.178003	−	0.984030i	$-0.443036\pi$
$13$	4.62808	0.356006	0.178003	+	0.984030i	$0.443036\pi$
$14$	3.74166i	0.267261i
$15$	0	0
$16$	4.00000	0.250000
$17$	− 15.2020i	− 0.894232i	−0.894476	−	0.447116i	$-0.852451\pi$
$17$	− 15.2020i	− 0.894232i	0.894476	−	0.447116i	$-0.147549\pi$
$18$	0	0
$19$	4.04508	0.212899	0.106449	−	0.994318i	$-0.466052\pi$
$19$	4.04508	0.212899	0.106449	+	0.994318i	$0.466052\pi$
$20$	1.34511i	0.0672556i
$21$	0	0
$22$	18.7591	0.852687
$23$	9.26566i	0.402855i	0.979503	+	0.201427i	$0.0645580\pi$
$23$	9.26566i	0.402855i	−0.979503	+	0.201427i	$0.935442\pi$
$24$	0	0
$25$	24.5477	0.981907
$26$	6.54510i	0.251735i
$27$	0	0
$28$	−5.29150	−0.188982
$29$	− 27.5450i	− 0.949829i	−0.880032	−	0.474914i	$-0.842479\pi$
$29$	− 27.5450i	− 0.949829i	0.880032	−	0.474914i	$-0.157521\pi$
$30$	0	0
$31$	44.0486	1.42092	0.710461	−	0.703736i	$-0.248486\pi$
$31$	44.0486	1.42092	0.710461	+	0.703736i	$0.248486\pi$
$32$	5.65685i	0.176777i
$33$	0	0
$34$	21.4988	0.632318
$35$	− 1.77942i	− 0.0508405i
$36$	0	0
$37$	−2.90984	−0.0786445	−0.0393222	−	0.999227i	$-0.512520\pi$
$37$	−2.90984	−0.0786445	−0.0393222	+	0.999227i	$0.512520\pi$
$38$	5.72060i	0.150542i
$39$	0	0
$40$	−1.90228	−0.0475569
$41$	23.9470i	0.584073i	0.956407	+	0.292037i	$0.0943329\pi$
$41$	23.9470i	0.584073i	−0.956407	+	0.292037i	$0.905667\pi$
$42$	0	0
$43$	62.8940	1.46265	0.731326	−	0.682028i	$-0.238902\pi$
$43$	62.8940	1.46265	0.731326	+	0.682028i	$0.238902\pi$
$44$	26.5294i	0.602941i
$45$	0	0
$46$	−13.1036	−0.284861
$47$	− 55.1568i	− 1.17355i	−0.809751	−	0.586774i	$-0.800398\pi$
$47$	− 55.1568i	− 1.17355i	0.809751	−	0.586774i	$-0.199602\pi$
$48$	0	0
$49$	7.00000	0.142857
$50$	34.7156i	0.694313i
$51$	0	0
$52$	−9.25617	−0.178003
$53$	− 1.84323i	− 0.0347779i	−0.999849	−	0.0173889i	$-0.994465\pi$
$53$	− 1.84323i	− 0.0347779i	0.999849	−	0.0173889i	$-0.00553535\pi$
$54$	0	0
$55$	−8.92125	−0.162205
$56$	− 7.48331i	− 0.133631i
$57$	0	0
$58$	38.9546	0.671630
$59$	13.2450i	0.224492i	0.993680	+	0.112246i	$0.0358044\pi$
$59$	13.2450i	0.224492i	−0.993680	+	0.112246i	$0.964196\pi$
$60$	0	0
$61$	−20.5346	−0.336633	−0.168316	−	0.985733i	$-0.553833\pi$
$61$	−20.5346	−0.336633	−0.168316	+	0.985733i	$0.553833\pi$
$62$	62.2941i	1.00474i
$63$	0	0
$64$	−8.00000	−0.125000
$65$	− 3.11265i	− 0.0478869i
$66$	0	0
$67$	−75.2985	−1.12386	−0.561929	−	0.827185i	$-0.689941\pi$
$67$	−75.2985	−1.12386	−0.561929	+	0.827185i	$0.689941\pi$
$68$	30.4039i	0.447116i
$69$	0	0
$70$	2.51647	0.0359496
$71$	91.3391i	1.28647i	0.765670	+	0.643233i	$0.222407\pi$
$71$	91.3391i	1.28647i	−0.765670	+	0.643233i	$0.777593\pi$
$72$	0	0
$73$	40.4610	0.554261	0.277130	−	0.960832i	$-0.410617\pi$
$73$	40.4610	0.554261	0.277130	+	0.960832i	$0.410617\pi$
$74$	− 4.11514i	− 0.0556100i
$75$	0	0
$76$	−8.09016	−0.106449
$77$	− 35.0951i	− 0.455780i
$78$	0	0
$79$	−84.4365	−1.06882	−0.534408	−	0.845226i	$-0.679466\pi$
$79$	−84.4365	−1.06882	−0.534408	+	0.845226i	$0.679466\pi$
$80$	− 2.69022i	− 0.0336278i
$81$	0	0
$82$	−33.8662	−0.413002
$83$	− 80.0194i	− 0.964089i	−0.876147	−	0.482045i	$-0.839894\pi$
$83$	− 80.0194i	− 0.964089i	0.876147	−	0.482045i	$-0.160106\pi$
$84$	0	0
$85$	−10.2242	−0.120284
$86$	88.9456i	1.03425i
$87$	0	0
$88$	−37.5182	−0.426344
$89$	− 109.600i	− 1.23146i	−0.787957	−	0.615730i	$-0.788861\pi$
$89$	− 109.600i	− 1.23146i	0.787957	−	0.615730i	$-0.211139\pi$
$90$	0	0
$91$	12.2448	0.134558
$92$	− 18.5313i	− 0.201427i
$93$	0	0
$94$	78.0035	0.829824
$95$	− 2.72054i	− 0.0286373i
$96$	0	0
$97$	126.396	1.30306	0.651528	−	0.758624i	$-0.274128\pi$
$97$	126.396	1.30306	0.651528	+	0.758624i	$0.274128\pi$
$98$	9.89949i	0.101015i
$99$	0	0
$100$	−49.0953	−0.490953
$101$	− 119.202i	− 1.18022i	−0.807324	−	0.590108i	$-0.799085\pi$
$101$	− 119.202i	− 1.18022i	0.807324	−	0.590108i	$-0.200915\pi$
$102$	0	0
$103$	−132.242	−1.28390	−0.641952	−	0.766745i	$-0.721875\pi$
$103$	−132.242	−1.28390	−0.641952	+	0.766745i	$0.721875\pi$
$104$	− 13.0902i	− 0.125867i
$105$	0	0
$106$	2.60672	0.0245917
$107$	109.283i	1.02134i	0.859777	+	0.510669i	$0.170602\pi$
$107$	109.283i	1.02134i	−0.859777	+	0.510669i	$0.829398\pi$
$108$	0	0
$109$	−184.589	−1.69347	−0.846736	−	0.532013i	$-0.821436\pi$
$109$	−184.589	−1.69347	−0.846736	+	0.532013i	$0.821436\pi$
$110$	− 12.6166i	− 0.114696i
$111$	0	0
$112$	10.5830	0.0944911
$113$	35.8528i	0.317281i	0.987336	+	0.158641i	$0.0507111\pi$
$113$	35.8528i	0.317281i	−0.987336	+	0.158641i	$0.949289\pi$
$114$	0	0
$115$	6.23168	0.0541885
$116$	55.0901i	0.474914i
$117$	0	0
$118$	−18.7313	−0.158739
$119$	− 40.2206i	− 0.337988i
$120$	0	0
$121$	−54.9522	−0.454150
$122$	− 29.0403i	− 0.238035i
$123$	0	0
$124$	−88.0972	−0.710461
$125$	− 33.3236i	− 0.266589i
$126$	0	0
$127$	96.3439	0.758613	0.379307	−	0.925271i	$-0.376163\pi$
$127$	96.3439	0.758613	0.379307	+	0.925271i	$0.376163\pi$
$128$	− 11.3137i	− 0.0883883i
$129$	0	0
$130$	4.40195	0.0338611
$131$	76.2958i	0.582411i	0.956661	+	0.291205i	$0.0940563\pi$
$131$	76.2958i	0.582411i	−0.956661	+	0.291205i	$0.905944\pi$
$132$	0	0
$133$	10.7023	0.0804682
$134$	− 106.488i	− 0.794688i
$135$	0	0
$136$	−42.9976	−0.316159
$137$	− 173.775i	− 1.26843i	−0.773156	−	0.634216i	$-0.781323\pi$
$137$	− 173.775i	− 1.26843i	0.773156	−	0.634216i	$-0.218677\pi$
$138$	0	0
$139$	−85.9389	−0.618265	−0.309133	−	0.951019i	$-0.600039\pi$
$139$	−85.9389	−0.618265	−0.309133	+	0.951019i	$0.600039\pi$
$140$	3.55883i	0.0254202i
$141$	0	0
$142$	−129.173	−0.909669
$143$	− 61.3901i	− 0.429302i
$144$	0	0
$145$	−18.5256	−0.127763
$146$	57.2205i	0.391921i
$147$	0	0
$148$	5.81969	0.0393222
$149$	0.416605i	0.00279600i	0.999999	+	0.00139800i	$0.000444998\pi$
$149$	0.416605i	0.00279600i	−0.999999	+	0.00139800i	$0.999555\pi$
$150$	0	0
$151$	−170.051	−1.12617	−0.563084	−	0.826400i	$-0.690385\pi$
$151$	−170.051	−1.12617	−0.563084	+	0.826400i	$0.690385\pi$
$152$	− 11.4412i	− 0.0752711i
$153$	0	0
$154$	49.6320	0.322285
$155$	− 29.6252i	− 0.191130i
$156$	0	0
$157$	−146.513	−0.933202	−0.466601	−	0.884468i	$-0.654522\pi$
$157$	−146.513	−0.933202	−0.466601	+	0.884468i	$0.654522\pi$
$158$	− 119.411i	− 0.755767i
$159$	0	0
$160$	3.80455	0.0237785
$161$	24.5146i	0.152265i
$162$	0	0
$163$	85.8374	0.526610	0.263305	−	0.964713i	$-0.415188\pi$
$163$	85.8374	0.526610	0.263305	+	0.964713i	$0.415188\pi$
$164$	− 47.8940i	− 0.292037i
$165$	0	0
$166$	113.165	0.681714
$167$	244.178i	1.46214i	0.682300	+	0.731072i	$0.260980\pi$
$167$	244.178i	1.46214i	−0.682300	+	0.731072i	$0.739020\pi$
$168$	0	0
$169$	−147.581	−0.873259
$170$	− 14.4592i	− 0.0850539i
$171$	0	0
$172$	−125.788	−0.731326
$173$	− 144.775i	− 0.836847i	−0.908252	−	0.418424i	$-0.862583\pi$
$173$	− 144.775i	− 0.836847i	0.908252	−	0.418424i	$-0.137417\pi$
$174$	0	0
$175$	64.9470	0.371126
$176$	− 53.0588i	− 0.301470i
$177$	0	0
$178$	154.998	0.870774
$179$	192.258i	1.07407i	0.843561	+	0.537033i	$0.180455\pi$
$179$	192.258i	1.07407i	−0.843561	+	0.537033i	$0.819545\pi$
$180$	0	0
$181$	298.952	1.65167	0.825835	−	0.563912i	$-0.190704\pi$
$181$	298.952	1.65167	0.825835	+	0.563912i	$0.190704\pi$
$182$	17.3167i	0.0951467i
$183$	0	0
$184$	26.2072	0.142431
$185$	1.95703i	0.0105786i
$186$	0	0
$187$	−201.649	−1.07834
$188$	110.314i	0.586774i
$189$	0	0
$190$	3.84743	0.0202496
$191$	225.355i	1.17987i	0.807451	+	0.589935i	$0.200846\pi$
$191$	225.355i	1.17987i	−0.807451	+	0.589935i	$0.799154\pi$
$192$	0	0
$193$	−343.910	−1.78192	−0.890958	−	0.454086i	$-0.849966\pi$
$193$	−343.910	−1.78192	−0.890958	+	0.454086i	$0.849966\pi$
$194$	178.752i	0.921400i
$195$	0	0
$196$	−14.0000	−0.0714286
$197$	111.867i	0.567851i	0.958846	+	0.283926i	$0.0916369\pi$
$197$	111.867i	0.567851i	−0.958846	+	0.283926i	$0.908363\pi$
$198$	0	0
$199$	240.040	1.20623	0.603116	−	0.797654i	$-0.293926\pi$
$199$	240.040	1.20623	0.603116	+	0.797654i	$0.293926\pi$
$200$	− 69.4313i	− 0.347156i
$201$	0	0
$202$	168.577	0.834539
$203$	− 72.8773i	− 0.359002i
$204$	0	0
$205$	16.1057	0.0785644
$206$	− 187.018i	− 0.907857i
$207$	0	0
$208$	18.5123	0.0890016
$209$	− 53.6567i	− 0.256731i
$210$	0	0
$211$	216.294	1.02509	0.512544	−	0.858661i	$-0.328703\pi$
$211$	216.294	1.02509	0.512544	+	0.858661i	$0.328703\pi$
$212$	3.68645i	0.0173889i
$213$	0	0
$214$	−154.550	−0.722195
$215$	− 42.2998i	− 0.196743i
$216$	0	0
$217$	116.542	0.537058
$218$	− 261.048i	− 1.19747i
$219$	0	0
$220$	17.8425	0.0811023
$221$	− 70.3559i	− 0.318352i
$222$	0	0
$223$	410.400	1.84036	0.920180	−	0.391495i	$-0.128042\pi$
$223$	410.400	1.84036	0.920180	+	0.391495i	$0.128042\pi$
$224$	14.9666i	0.0668153i
$225$	0	0
$226$	−50.7035	−0.224352
$227$	418.066i	1.84170i	0.389916	+	0.920850i	$0.372504\pi$
$227$	418.066i	1.84170i	−0.389916	+	0.920850i	$0.627496\pi$
$228$	0	0
$229$	−38.0368	−0.166099	−0.0830497	−	0.996545i	$-0.526466\pi$
$229$	−38.0368	−0.166099	−0.0830497	+	0.996545i	$0.526466\pi$
$230$	8.81292i	0.0383171i
$231$	0	0
$232$	−77.9091	−0.335815
$233$	− 101.421i	− 0.435281i	−0.976029	−	0.217641i	$-0.930164\pi$
$233$	− 101.421i	− 0.435281i	0.976029	−	0.217641i	$-0.0698361\pi$
$234$	0	0
$235$	−37.0960	−0.157855
$236$	− 26.4900i	− 0.112246i
$237$	0	0
$238$	56.8805	0.238994
$239$	403.709i	1.68916i	0.535431	+	0.844579i	$0.320149\pi$
$239$	403.709i	1.68916i	−0.535431	+	0.844579i	$0.679851\pi$
$240$	0	0
$241$	452.244	1.87653	0.938266	−	0.345914i	$-0.112431\pi$
$241$	452.244	1.87653	0.938266	+	0.345914i	$0.112431\pi$
$242$	− 77.7142i	− 0.321133i
$243$	0	0
$244$	41.0692	0.168316
$245$	− 4.70789i	− 0.0192159i
$246$	0	0
$247$	18.7210	0.0757933
$248$	− 124.588i	− 0.502372i
$249$	0	0
$250$	47.1267	0.188507
$251$	− 55.2568i	− 0.220146i	−0.993923	−	0.110073i	$-0.964891\pi$
$251$	− 55.2568i	− 0.220146i	0.993923	−	0.110073i	$-0.0351085\pi$
$252$	0	0
$253$	122.906	0.485795
$254$	136.251i	0.536421i
$255$	0	0
$256$	16.0000	0.0625000
$257$	66.7940i	0.259899i	0.991521	+	0.129949i	$0.0414815\pi$
$257$	66.7940i	0.259899i	−0.991521	+	0.129949i	$0.958519\pi$
$258$	0	0
$259$	−7.69873	−0.0297248
$260$	6.22529i	0.0239434i
$261$	0	0
$262$	−107.899	−0.411827
$263$	361.502i	1.37453i	0.726406	+	0.687265i	$0.241189\pi$
$263$	361.502i	1.37453i	−0.726406	+	0.687265i	$0.758811\pi$
$264$	0	0
$265$	−1.23967	−0.00467801
$266$	15.1353i	0.0568996i
$267$	0	0
$268$	150.597	0.561929
$269$	172.459i	0.641111i	0.947230	+	0.320555i	$0.103870\pi$
$269$	172.459i	0.641111i	−0.947230	+	0.320555i	$0.896130\pi$
$270$	0	0
$271$	91.2119	0.336575	0.168288	−	0.985738i	$-0.446176\pi$
$271$	91.2119	0.336575	0.168288	+	0.985738i	$0.446176\pi$
$272$	− 60.8078i	− 0.223558i
$273$	0	0
$274$	245.755	0.896917
$275$	− 325.617i	− 1.18406i
$276$	0	0
$277$	−52.0409	−0.187873	−0.0939367	−	0.995578i	$-0.529945\pi$
$277$	−52.0409	−0.187873	−0.0939367	+	0.995578i	$0.529945\pi$
$278$	− 121.536i	− 0.437180i
$279$	0	0
$280$	−5.03295	−0.0179748
$281$	413.083i	1.47005i	0.678042	+	0.735023i	$0.262829\pi$
$281$	413.083i	1.47005i	−0.678042	+	0.735023i	$0.737171\pi$
$282$	0	0
$283$	108.973	0.385063	0.192531	−	0.981291i	$-0.438330\pi$
$283$	108.973	0.385063	0.192531	+	0.981291i	$0.438330\pi$
$284$	− 182.678i	− 0.643233i
$285$	0	0
$286$	86.8187	0.303562
$287$	63.3578i	0.220759i
$288$	0	0
$289$	57.9007	0.200348
$290$	− 26.1991i	− 0.0903418i
$291$	0	0
$292$	−80.9221	−0.277130
$293$	− 403.517i	− 1.37719i	−0.725145	−	0.688596i	$-0.758227\pi$
$293$	− 403.517i	− 1.37719i	0.725145	−	0.688596i	$-0.241773\pi$
$294$	0	0
$295$	8.90801	0.0301966
$296$	8.23028i	0.0278050i
$297$	0	0
$298$	−0.589168	−0.00197707
$299$	42.8822i	0.143419i
$300$	0	0
$301$	166.402	0.552830
$302$	− 240.489i	− 0.796321i
$303$	0	0
$304$	16.1803	0.0532247
$305$	13.8107i	0.0452809i
$306$	0	0
$307$	−91.2879	−0.297355	−0.148677	−	0.988886i	$-0.547502\pi$
$307$	−91.2879	−0.297355	−0.148677	+	0.988886i	$0.547502\pi$
$308$	70.1902i	0.227890i
$309$	0	0
$310$	41.8963	0.135149
$311$	− 12.1248i	− 0.0389864i	−0.999810	−	0.0194932i	$-0.993795\pi$
$311$	− 12.1248i	− 0.0389864i	0.999810	−	0.0194932i	$-0.00620527\pi$
$312$	0	0
$313$	87.0221	0.278026	0.139013	−	0.990291i	$-0.455607\pi$
$313$	87.0221	0.278026	0.139013	+	0.990291i	$0.455607\pi$
$314$	− 207.200i	− 0.659874i
$315$	0	0
$316$	168.873	0.534408
$317$	− 193.797i	− 0.611348i	−0.952136	−	0.305674i	$-0.901118\pi$
$317$	− 193.797i	− 0.611348i	0.952136	−	0.305674i	$-0.0988817\pi$
$318$	0	0
$319$	−365.377	−1.14538
$320$	5.38045i	0.0168139i
$321$	0	0
$322$	−34.6689	−0.107667
$323$	− 61.4931i	− 0.190381i
$324$	0	0
$325$	113.609	0.349565
$326$	121.392i	0.372369i
$327$	0	0
$328$	67.7323	0.206501
$329$	− 145.931i	− 0.443560i
$330$	0	0
$331$	−417.814	−1.26228	−0.631138	−	0.775670i	$-0.717412\pi$
$331$	−417.814	−1.26228	−0.631138	+	0.775670i	$0.717412\pi$
$332$	160.039i	0.482045i
$333$	0	0
$334$	−345.320	−1.03389
$335$	50.6425i	0.151172i
$336$	0	0
$337$	63.6695	0.188930	0.0944652	−	0.995528i	$-0.469886\pi$
$337$	63.6695	0.188930	0.0944652	+	0.995528i	$0.469886\pi$
$338$	− 208.711i	− 0.617488i
$339$	0	0
$340$	20.4483	0.0601422
$341$	− 584.291i	− 1.71346i
$342$	0	0
$343$	18.5203	0.0539949
$344$	− 177.891i	− 0.517125i
$345$	0	0
$346$	204.742	0.591740
$347$	− 310.877i	− 0.895901i	−0.894058	−	0.447950i	$-0.852154\pi$
$347$	− 310.877i	− 0.895901i	0.894058	−	0.447950i	$-0.147846\pi$
$348$	0	0
$349$	415.607	1.19085	0.595426	−	0.803410i	$-0.296983\pi$
$349$	415.607	1.19085	0.595426	+	0.803410i	$0.296983\pi$
$350$	91.8490i	0.262426i
$351$	0	0
$352$	75.0365	0.213172
$353$	150.239i	0.425605i	0.977095	+	0.212803i	$0.0682591\pi$
$353$	150.239i	0.425605i	−0.977095	+	0.212803i	$0.931741\pi$
$354$	0	0
$355$	61.4307	0.173044
$356$	219.200i	0.615730i
$357$	0	0
$358$	−271.894	−0.759479
$359$	14.0939i	0.0392589i	0.999807	+	0.0196294i	$0.00624865\pi$
$359$	14.0939i	0.0392589i	−0.999807	+	0.0196294i	$0.993751\pi$
$360$	0	0
$361$	−344.637	−0.954674
$362$	422.782i	1.16791i
$363$	0	0
$364$	−24.4895	−0.0672789
$365$	− 27.2123i	− 0.0745543i
$366$	0	0
$367$	−44.6289	−0.121605	−0.0608024	−	0.998150i	$-0.519366\pi$
$367$	−44.6289	−0.121605	−0.0608024	+	0.998150i	$0.519366\pi$
$368$	37.0626i	0.100714i
$369$	0	0
$370$	−2.76766	−0.00748017
$371$	− 4.87672i	− 0.0131448i
$372$	0	0
$373$	−259.881	−0.696732	−0.348366	−	0.937359i	$-0.613263\pi$
$373$	−259.881	−0.696732	−0.348366	+	0.937359i	$0.613263\pi$
$374$	− 285.175i	− 0.762500i
$375$	0	0
$376$	−156.007	−0.414912
$377$	− 127.481i	− 0.338145i
$378$	0	0
$379$	−185.662	−0.489872	−0.244936	−	0.969539i	$-0.578767\pi$
$379$	−185.662	−0.489872	−0.244936	+	0.969539i	$0.578767\pi$
$380$	5.44108i	0.0143186i
$381$	0	0
$382$	−318.700	−0.834294
$383$	604.452i	1.57820i	0.614262	+	0.789102i	$0.289454\pi$
$383$	604.452i	1.57820i	−0.614262	+	0.789102i	$0.710546\pi$
$384$	0	0
$385$	−23.6034	−0.0613076
$386$	− 486.362i	− 1.26000i
$387$	0	0
$388$	−252.793	−0.651528
$389$	506.617i	1.30236i	0.758924	+	0.651179i	$0.225725\pi$
$389$	506.617i	1.30236i	−0.758924	+	0.651179i	$0.774275\pi$
$390$	0	0
$391$	140.856	0.360246
$392$	− 19.7990i	− 0.0505076i
$393$	0	0
$394$	−158.203	−0.401531
$395$	56.7883i	0.143768i
$396$	0	0
$397$	−232.366	−0.585305	−0.292652	−	0.956219i	$-0.594538\pi$
$397$	−232.366	−0.585305	−0.292652	+	0.956219i	$0.594538\pi$
$398$	339.468i	0.852935i
$399$	0	0
$400$	98.1907	0.245477
$401$	− 647.469i	− 1.61464i	−0.590116	−	0.807318i	$-0.700918\pi$
$401$	− 647.469i	− 1.61464i	0.590116	−	0.807318i	$-0.299082\pi$
$402$	0	0
$403$	203.861	0.505857
$404$	238.404i	0.590108i
$405$	0	0
$406$	103.064	0.253852
$407$	38.5982i	0.0948359i
$408$	0	0
$409$	−148.863	−0.363967	−0.181984	−	0.983302i	$-0.558252\pi$
$409$	−148.863	−0.363967	−0.181984	+	0.983302i	$0.558252\pi$
$410$	22.7769i	0.0555534i
$411$	0	0
$412$	264.484	0.641952
$413$	35.0430i	0.0848498i
$414$	0	0
$415$	−53.8175	−0.129681
$416$	26.1804i	0.0629336i
$417$	0	0
$418$	75.8821	0.181536
$419$	− 418.288i	− 0.998301i	−0.866515	−	0.499150i	$-0.833645\pi$
$419$	− 418.288i	− 0.998301i	0.866515	−	0.499150i	$-0.166355\pi$
$420$	0	0
$421$	494.465	1.17450	0.587251	−	0.809405i	$-0.300210\pi$
$421$	494.465	1.17450	0.587251	+	0.809405i	$0.300210\pi$
$422$	305.885i	0.724847i
$423$	0	0
$424$	−5.21343	−0.0122958
$425$	− 373.172i	− 0.878053i
$426$	0	0
$427$	−54.3294	−0.127235
$428$	− 218.566i	− 0.510669i
$429$	0	0
$430$	59.8209	0.139118
$431$	804.276i	1.86607i	0.359786	+	0.933035i	$0.382850\pi$
$431$	804.276i	1.86607i	−0.359786	+	0.933035i	$0.617150\pi$
$432$	0	0
$433$	−855.394	−1.97551	−0.987753	−	0.156023i	$-0.950133\pi$
$433$	−855.394	−1.97551	−0.987753	+	0.156023i	$0.950133\pi$
$434$	164.815i	0.379757i
$435$	0	0
$436$	369.177	0.846736
$437$	37.4803i	0.0857673i
$438$	0	0
$439$	−736.810	−1.67838	−0.839191	−	0.543837i	$-0.816971\pi$
$439$	−736.810	−1.67838	−0.839191	+	0.543837i	$0.816971\pi$
$440$	25.2331i	0.0573480i
$441$	0	0
$442$	99.4983	0.225109
$443$	138.024i	0.311566i	0.987791	+	0.155783i	$0.0497900\pi$
$443$	138.024i	0.311566i	−0.987791	+	0.155783i	$0.950210\pi$
$444$	0	0
$445$	−73.7121	−0.165645
$446$	580.394i	1.30133i
$447$	0	0
$448$	−21.1660	−0.0472456
$449$	− 488.441i	− 1.08784i	−0.839137	−	0.543921i	$-0.816939\pi$
$449$	− 488.441i	− 1.08784i	0.839137	−	0.543921i	$-0.183061\pi$
$450$	0	0
$451$	317.650	0.704323
$452$	− 71.7055i	− 0.158641i
$453$	0	0
$454$	−591.235	−1.30228
$455$	− 8.23529i	− 0.0180995i
$456$	0	0
$457$	−331.337	−0.725026	−0.362513	−	0.931979i	$-0.618081\pi$
$457$	−331.337	−0.725026	−0.362513	+	0.931979i	$0.618081\pi$
$458$	− 53.7921i	− 0.117450i
$459$	0	0
$460$	−12.4634	−0.0270943
$461$	20.8885i	0.0453112i	0.999743	+	0.0226556i	$0.00721212\pi$
$461$	20.8885i	0.0453112i	−0.999743	+	0.0226556i	$0.992788\pi$
$462$	0	0
$463$	106.427	0.229864	0.114932	−	0.993373i	$-0.463335\pi$
$463$	106.427	0.229864	0.114932	+	0.993373i	$0.463335\pi$
$464$	− 110.180i	− 0.237457i
$465$	0	0
$466$	143.430	0.307790
$467$	− 472.833i	− 1.01249i	−0.862390	−	0.506245i	$-0.831033\pi$
$467$	− 472.833i	− 1.01249i	0.862390	−	0.506245i	$-0.168967\pi$
$468$	0	0
$469$	−199.221	−0.424779
$470$	− 52.4617i	− 0.111621i
$471$	0	0
$472$	37.4625	0.0793697
$473$	− 834.270i	− 1.76378i
$474$	0	0
$475$	99.2972	0.209047
$476$	80.4412i	0.168994i
$477$	0	0
$478$	−570.930	−1.19442
$479$	373.539i	0.779832i	0.920850	+	0.389916i	$0.127496\pi$
$479$	373.539i	0.779832i	−0.920850	+	0.389916i	$0.872504\pi$
$480$	0	0
$481$	−13.4670	−0.0279979
$482$	639.570i	1.32691i
$483$	0	0
$484$	109.904	0.227075
$485$	− 85.0087i	− 0.175276i
$486$	0	0
$487$	−642.695	−1.31970	−0.659851	−	0.751396i	$-0.729381\pi$
$487$	−642.695	−1.31970	−0.659851	+	0.751396i	$0.729381\pi$
$488$	58.0806i	0.119018i
$489$	0	0
$490$	6.65797	0.0135877
$491$	805.881i	1.64131i	0.571427	+	0.820653i	$0.306390\pi$
$491$	805.881i	1.64131i	−0.571427	+	0.820653i	$0.693610\pi$
$492$	0	0
$493$	−418.738	−0.849368
$494$	26.4754i	0.0535940i
$495$	0	0
$496$	176.194	0.355231
$497$	241.661i	0.486239i
$498$	0	0
$499$	408.219	0.818075	0.409037	−	0.912518i	$-0.365865\pi$
$499$	408.219	0.818075	0.409037	+	0.912518i	$0.365865\pi$
$500$	66.6472i	0.133294i
$501$	0	0
$502$	78.1449	0.155667
$503$	− 708.495i	− 1.40854i	−0.709933	−	0.704269i	$-0.751275\pi$
$503$	− 708.495i	− 1.40854i	0.709933	−	0.704269i	$-0.248725\pi$
$504$	0	0
$505$	−80.1699	−0.158752
$506$	173.816i	0.343509i
$507$	0	0
$508$	−192.688	−0.379307
$509$	842.863i	1.65592i	0.560787	+	0.827960i	$0.310499\pi$
$509$	842.863i	1.65592i	−0.560787	+	0.827960i	$0.689501\pi$
$510$	0	0
$511$	107.050	0.209491
$512$	22.6274i	0.0441942i
$513$	0	0
$514$	−94.4610	−0.183776
$515$	88.9402i	0.172699i
$516$	0	0
$517$	−731.638	−1.41516
$518$	− 10.8876i	− 0.0210186i
$519$	0	0
$520$	−8.80389	−0.0169306
$521$	798.331i	1.53231i	0.642659	+	0.766153i	$0.277831\pi$
$521$	798.331i	1.53231i	−0.642659	+	0.766153i	$0.722169\pi$
$522$	0	0
$523$	101.371	0.193825	0.0969127	−	0.995293i	$-0.469103\pi$
$523$	101.371	0.193825	0.0969127	+	0.995293i	$0.469103\pi$
$524$	− 152.592i	− 0.291205i
$525$	0	0
$526$	−511.240	−0.971940
$527$	− 669.625i	− 1.27063i
$528$	0	0
$529$	443.148	0.837708
$530$	− 1.75316i	− 0.00330786i
$531$	0	0
$532$	−21.4045	−0.0402341
$533$	110.829i	0.207934i
$534$	0	0
$535$	73.4991	0.137381
$536$	212.976i	0.397344i
$537$	0	0
$538$	−243.894	−0.453334
$539$	− 92.8529i	− 0.172269i
$540$	0	0
$541$	469.805	0.868402	0.434201	−	0.900816i	$-0.357031\pi$
$541$	469.805	0.868402	0.434201	+	0.900816i	$0.357031\pi$
$542$	128.993i	0.237995i
$543$	0	0
$544$	85.9952	0.158079
$545$	124.146i	0.227791i
$546$	0	0
$547$	−742.985	−1.35829	−0.679145	−	0.734004i	$-0.737649\pi$
$547$	−742.985	−1.35829	−0.679145	+	0.734004i	$0.737649\pi$
$548$	347.551i	0.634216i
$549$	0	0
$550$	460.493	0.837259
$551$	− 111.422i	− 0.202217i
$552$	0	0
$553$	−223.398	−0.403975
$554$	− 73.5970i	− 0.132847i
$555$	0	0
$556$	171.878	0.309133
$557$	− 506.745i	− 0.909775i	−0.890549	−	0.454887i	$-0.849679\pi$
$557$	− 506.745i	− 0.909775i	0.890549	−	0.454887i	$-0.150321\pi$
$558$	0	0
$559$	291.079	0.520713
$560$	− 7.11767i	− 0.0127101i
$561$	0	0
$562$	−584.187	−1.03948
$563$	− 900.954i	− 1.60027i	−0.599818	−	0.800137i	$-0.704760\pi$
$563$	− 900.954i	− 1.60027i	0.599818	−	0.800137i	$-0.295240\pi$
$564$	0	0
$565$	24.1130	0.0426779
$566$	154.111i	0.272281i
$567$	0	0
$568$	258.346	0.454835
$569$	− 590.261i	− 1.03737i	−0.854967	−	0.518683i	$-0.826423\pi$
$569$	− 590.261i	− 1.03737i	0.854967	−	0.518683i	$-0.173577\pi$
$570$	0	0
$571$	941.922	1.64960	0.824800	−	0.565425i	$-0.191288\pi$
$571$	941.922	1.64960	0.824800	+	0.565425i	$0.191288\pi$
$572$	122.780i	0.214651i
$573$	0	0
$574$	−89.6014	−0.156100
$575$	227.450i	0.395566i
$576$	0	0
$577$	236.677	0.410185	0.205092	−	0.978743i	$-0.434251\pi$
$577$	236.677	0.410185	0.205092	+	0.978743i	$0.434251\pi$
$578$	81.8839i	0.141668i
$579$	0	0
$580$	37.0512	0.0638813
$581$	− 211.711i	− 0.364391i
$582$	0	0
$583$	−24.4499	−0.0419380
$584$	− 114.441i	− 0.195961i
$585$	0	0
$586$	570.660	0.973822
$587$	− 920.582i	− 1.56828i	−0.620582	−	0.784141i	$-0.713104\pi$
$587$	− 920.582i	− 1.56828i	0.620582	−	0.784141i	$-0.286896\pi$
$588$	0	0
$589$	178.180	0.302513
$590$	12.5978i	0.0213522i
$591$	0	0
$592$	−11.6394	−0.0196611
$593$	− 760.444i	− 1.28237i	−0.767388	−	0.641184i	$-0.778444\pi$
$593$	− 760.444i	− 1.28237i	0.767388	−	0.641184i	$-0.221556\pi$
$594$	0	0
$595$	−27.0506	−0.0454632
$596$	− 0.833209i	− 0.00139800i
$597$	0	0
$598$	−60.6447	−0.101412
$599$	399.798i	0.667442i	0.942672	+	0.333721i	$0.108304\pi$
$599$	399.798i	0.667442i	−0.942672	+	0.333721i	$0.891696\pi$
$600$	0	0
$601$	664.811	1.10617	0.553087	−	0.833124i	$-0.313450\pi$
$601$	664.811	1.10617	0.553087	+	0.833124i	$0.313450\pi$
$602$	235.328i	0.390910i
$603$	0	0
$604$	340.103	0.563084
$605$	36.9584i	0.0610883i
$606$	0	0
$607$	372.670	0.613954	0.306977	−	0.951717i	$-0.400682\pi$
$607$	372.670	0.613954	0.306977	+	0.951717i	$0.400682\pi$
$608$	22.8824i	0.0376355i
$609$	0	0
$610$	−19.5312	−0.0320184
$611$	− 255.270i	− 0.417791i
$612$	0	0
$613$	512.686	0.836356	0.418178	−	0.908365i	$-0.362669\pi$
$613$	512.686	0.836356	0.418178	+	0.908365i	$0.362669\pi$
$614$	− 129.101i	− 0.210262i
$615$	0	0
$616$	−99.2639	−0.161143
$617$	921.484i	1.49349i	0.665110	+	0.746745i	$0.268385\pi$
$617$	921.484i	1.49349i	−0.665110	+	0.746745i	$0.731615\pi$
$618$	0	0
$619$	−1104.63	−1.78454	−0.892270	−	0.451503i	$-0.850888\pi$
$619$	−1104.63	−1.78454	−0.892270	+	0.451503i	$0.850888\pi$
$620$	59.2503i	0.0955650i
$621$	0	0
$622$	17.1470	0.0275676
$623$	− 289.974i	− 0.465448i
$624$	0	0
$625$	591.280	0.946048
$626$	123.068i	0.196594i
$627$	0	0
$628$	293.026	0.466601
$629$	44.2353i	0.0703264i
$630$	0	0
$631$	119.817	0.189884	0.0949418	−	0.995483i	$-0.469733\pi$
$631$	119.817	0.189884	0.0949418	+	0.995483i	$0.469733\pi$
$632$	238.823i	0.377884i
$633$	0	0
$634$	274.071	0.432288
$635$	− 64.7967i	− 0.102042i
$636$	0	0
$637$	32.3966	0.0508581
$638$	− 516.721i	− 0.809907i
$639$	0	0
$640$	−7.60910	−0.0118892
$641$	− 82.6281i	− 0.128905i	−0.997921	−	0.0644525i	$-0.979470\pi$
$641$	− 82.6281i	− 0.128905i	0.997921	−	0.0644525i	$-0.0205301\pi$
$642$	0	0
$643$	126.741	0.197109	0.0985547	−	0.995132i	$-0.468578\pi$
$643$	126.741	0.197109	0.0985547	+	0.995132i	$0.468578\pi$
$644$	− 49.0293i	− 0.0761324i
$645$	0	0
$646$	86.9643	0.134620
$647$	738.224i	1.14099i	0.821299	+	0.570497i	$0.193250\pi$
$647$	738.224i	1.14099i	−0.821299	+	0.570497i	$0.806750\pi$
$648$	0	0
$649$	175.691	0.270710
$650$	160.667i	0.247180i
$651$	0	0
$652$	−171.675	−0.263305
$653$	− 69.1073i	− 0.105831i	−0.998599	−	0.0529153i	$-0.983149\pi$
$653$	− 69.1073i	− 0.105831i	0.998599	−	0.0529153i	$-0.0168513\pi$
$654$	0	0
$655$	51.3132	0.0783408
$656$	95.7880i	0.146018i
$657$	0	0
$658$	206.378	0.313644
$659$	− 427.081i	− 0.648075i	−0.946044	−	0.324037i	$-0.894960\pi$
$659$	− 427.081i	− 0.648075i	0.946044	−	0.324037i	$-0.105040\pi$
$660$	0	0
$661$	93.3442	0.141217	0.0706083	−	0.997504i	$-0.477506\pi$
$661$	93.3442	0.141217	0.0706083	+	0.997504i	$0.477506\pi$
$662$	− 590.878i	− 0.892565i
$663$	0	0
$664$	−226.329	−0.340857
$665$	− 7.19788i	− 0.0108239i
$666$	0	0
$667$	255.223	0.382643
$668$	− 488.356i	− 0.731072i
$669$	0	0
$670$	−71.6193	−0.106894
$671$	272.385i	0.405939i
$672$	0	0
$673$	468.530	0.696181	0.348091	−	0.937461i	$-0.386830\pi$
$673$	468.530	0.696181	0.348091	+	0.937461i	$0.386830\pi$
$674$	90.0423i	0.133594i
$675$	0	0
$676$	295.162	0.436630
$677$	− 1286.77i	− 1.90070i	−0.311183	−	0.950350i	$-0.600725\pi$
$677$	− 1286.77i	− 1.90070i	0.311183	−	0.950350i	$-0.399275\pi$
$678$	0	0
$679$	334.414	0.492509
$680$	28.9183i	0.0425269i
$681$	0	0
$682$	826.313	1.21160
$683$	478.532i	0.700632i	0.936632	+	0.350316i	$0.113926\pi$
$683$	478.532i	0.700632i	−0.936632	+	0.350316i	$0.886074\pi$
$684$	0	0
$685$	−116.874	−0.170618
$686$	26.1916i	0.0381802i
$687$	0	0
$688$	251.576	0.365663
$689$	− 8.53061i	− 0.0123811i
$690$	0	0
$691$	886.514	1.28294	0.641471	−	0.767147i	$-0.278324\pi$
$691$	886.514	1.28294	0.641471	+	0.767147i	$0.278324\pi$
$692$	289.549i	0.418424i
$693$	0	0
$694$	439.647	0.633497
$695$	57.7987i	0.0831636i
$696$	0	0
$697$	364.041	0.522297
$698$	587.758i	0.842060i
$699$	0	0
$700$	−129.894	−0.185563
$701$	503.322i	0.718006i	0.933336	+	0.359003i	$0.116883\pi$
$701$	503.322i	0.718006i	−0.933336	+	0.359003i	$0.883117\pi$
$702$	0	0
$703$	−11.7705	−0.0167433
$704$	106.118i	0.150735i
$705$	0	0
$706$	−212.470	−0.300948
$707$	− 315.378i	− 0.446080i
$708$	0	0
$709$	−102.128	−0.144046	−0.0720229	−	0.997403i	$-0.522945\pi$
$709$	−102.128	−0.144046	−0.0720229	+	0.997403i	$0.522945\pi$
$710$	86.8761i	0.122361i
$711$	0	0
$712$	−309.996	−0.435387
$713$	408.139i	0.572425i
$714$	0	0
$715$	−41.2883	−0.0577459
$716$	− 384.516i	− 0.537033i
$717$	0	0
$718$	−19.9318	−0.0277602
$719$	− 375.610i	− 0.522406i	−0.965284	−	0.261203i	$-0.915881\pi$
$719$	− 375.610i	− 0.522406i	0.965284	−	0.261203i	$-0.0841192\pi$
$720$	0	0
$721$	−349.880	−0.485270
$722$	− 487.391i	− 0.675057i
$723$	0	0
$724$	−597.904	−0.825835
$725$	− 676.166i	− 0.932643i
$726$	0	0
$727$	−967.189	−1.33038	−0.665192	−	0.746672i	$-0.731650\pi$
$727$	−967.189	−1.33038	−0.665192	+	0.746672i	$0.731650\pi$
$728$	− 34.6334i	− 0.0475734i
$729$	0	0
$730$	38.4840	0.0527178
$731$	− 956.112i	− 1.30795i
$732$	0	0
$733$	44.8714	0.0612162	0.0306081	−	0.999531i	$-0.490256\pi$
$733$	44.8714	0.0612162	0.0306081	+	0.999531i	$0.490256\pi$
$734$	− 63.1148i	− 0.0859875i
$735$	0	0
$736$	−52.4145	−0.0712153
$737$	998.812i	1.35524i
$738$	0	0
$739$	790.331	1.06946	0.534730	−	0.845023i	$-0.320413\pi$
$739$	790.331	1.06946	0.534730	+	0.845023i	$0.320413\pi$
$740$	− 3.91407i	− 0.00528928i
$741$	0	0
$742$	6.89673	0.00929478
$743$	198.537i	0.267210i	0.991035	+	0.133605i	$0.0426554\pi$
$743$	198.537i	0.267210i	−0.991035	+	0.133605i	$0.957345\pi$
$744$	0	0
$745$	0.280190	0.000376094 0
$746$	− 367.527i	− 0.492664i
$747$	0	0
$748$	403.299	0.539169
$749$	289.136i	0.386030i
$750$	0	0
$751$	527.315	0.702151	0.351076	−	0.936347i	$-0.385816\pi$
$751$	527.315	0.702151	0.351076	+	0.936347i	$0.385816\pi$
$752$	− 220.627i	− 0.293387i
$753$	0	0
$754$	180.285	0.239105
$755$	114.369i	0.151482i
$756$	0	0
$757$	−986.410	−1.30305	−0.651526	−	0.758626i	$-0.725871\pi$
$757$	−986.410	−1.30305	−0.651526	+	0.758626i	$0.725871\pi$
$758$	− 262.565i	− 0.346392i
$759$	0	0
$760$	−7.69485	−0.0101248
$761$	1431.24i	1.88073i	0.340166	+	0.940366i	$0.389517\pi$
$761$	1431.24i	1.88073i	−0.340166	+	0.940366i	$0.610483\pi$
$762$	0	0
$763$	−488.375	−0.640073
$764$	− 450.710i	− 0.589935i
$765$	0	0
$766$	−854.825	−1.11596
$767$	61.2990i	0.0799204i
$768$	0	0
$769$	321.382	0.417922	0.208961	−	0.977924i	$-0.432992\pi$
$769$	321.382	0.417922	0.208961	+	0.977924i	$0.432992\pi$
$770$	− 33.3803i	− 0.0433510i
$771$	0	0
$772$	687.819	0.890958
$773$	731.933i	0.946873i	0.880828	+	0.473436i	$0.156987\pi$
$773$	731.933i	0.946873i	−0.880828	+	0.473436i	$0.843013\pi$
$774$	0	0
$775$	1081.29	1.39521
$776$	− 357.503i	− 0.460700i
$777$	0	0
$778$	−716.465	−0.920906
$779$	96.8674i	0.124348i
$780$	0	0
$781$	1211.59	1.55133
$782$	199.201i	0.254732i
$783$	0	0
$784$	28.0000	0.0357143
$785$	98.5381i	0.125526i
$786$	0	0
$787$	−790.685	−1.00468	−0.502341	−	0.864669i	$-0.667528\pi$
$787$	−790.685	−1.00468	−0.502341	+	0.864669i	$0.667528\pi$
$788$	− 223.733i	− 0.283926i
$789$	0	0
$790$	−80.3108	−0.101659
$791$	94.8575i	0.119921i
$792$	0	0
$793$	−95.0358	−0.119843
$794$	− 328.615i	− 0.413873i
$795$	0	0
$796$	−480.080	−0.603116
$797$	1157.18i	1.45192i	0.687739	+	0.725958i	$0.258603\pi$
$797$	1157.18i	1.45192i	−0.687739	+	0.725958i	$0.741397\pi$
$798$	0	0
$799$	−838.491	−1.04943
$800$	138.863i	0.173578i
$801$	0	0
$802$	915.660	1.14172
$803$	− 536.703i	− 0.668373i
$804$	0	0
$805$	16.4875	0.0204813
$806$	288.302i	0.357695i
$807$	0	0
$808$	−337.154	−0.417269
$809$	951.837i	1.17656i	0.808658	+	0.588280i	$0.200195\pi$
$809$	951.837i	1.17656i	−0.808658	+	0.588280i	$0.799805\pi$
$810$	0	0
$811$	−737.263	−0.909079	−0.454540	−	0.890727i	$-0.650196\pi$
$811$	−737.263	−0.909079	−0.454540	+	0.890727i	$0.650196\pi$
$812$	145.755i	0.179501i
$813$	0	0
$814$	−54.5861	−0.0670591
$815$	− 57.7304i	− 0.0708349i
$816$	0	0
$817$	254.411	0.311397
$818$	− 210.523i	− 0.257364i
$819$	0	0
$820$	−32.2114	−0.0392822
$821$	1089.87i	1.32750i	0.747957	+	0.663748i	$0.231035\pi$
$821$	1089.87i	1.32750i	−0.747957	+	0.663748i	$0.768965\pi$
$822$	0	0
$823$	1121.28	1.36243	0.681216	−	0.732083i	$-0.261452\pi$
$823$	1121.28	1.36243	0.681216	+	0.732083i	$0.261452\pi$
$824$	374.037i	0.453928i
$825$	0	0
$826$	−49.5583	−0.0599979
$827$	− 1322.75i	− 1.59946i	−0.600362	−	0.799729i	$-0.704977\pi$
$827$	− 1322.75i	− 1.59946i	0.600362	−	0.799729i	$-0.295023\pi$
$828$	0	0
$829$	−303.153	−0.365686	−0.182843	−	0.983142i	$-0.558530\pi$
$829$	−303.153	−0.365686	−0.182843	+	0.983142i	$0.558530\pi$
$830$	− 76.1095i	− 0.0916982i
$831$	0	0
$832$	−37.0247	−0.0445008
$833$	− 106.414i	− 0.127747i
$834$	0	0
$835$	164.224	0.196675
$836$	107.313i	0.128365i
$837$	0	0
$838$	591.549	0.705905
$839$	− 48.7743i	− 0.0581339i	−0.999577	−	0.0290669i	$-0.990746\pi$
$839$	− 48.7743i	− 0.0581339i	0.999577	−	0.0290669i	$-0.00925360\pi$
$840$	0	0
$841$	82.2710	0.0978252
$842$	699.279i	0.830498i
$843$	0	0
$844$	−432.587	−0.512544
$845$	99.2564i	0.117463i
$846$	0	0
$847$	−145.390	−0.171653
$848$	− 7.37291i	− 0.00869447i
$849$	0	0
$850$	527.746	0.620877
$851$	− 26.9616i	− 0.0316823i
$852$	0	0
$853$	−795.011	−0.932017	−0.466009	−	0.884780i	$-0.654308\pi$
$853$	−795.011	−0.932017	−0.466009	+	0.884780i	$0.654308\pi$
$854$	− 76.8334i	− 0.0899689i
$855$	0	0
$856$	309.100	0.361098
$857$	− 515.098i	− 0.601048i	−0.953774	−	0.300524i	$-0.902838\pi$
$857$	− 515.098i	− 0.601048i	0.953774	−	0.300524i	$-0.0971616\pi$
$858$	0	0
$859$	731.616	0.851706	0.425853	−	0.904792i	$-0.359974\pi$
$859$	731.616	0.851706	0.425853	+	0.904792i	$0.359974\pi$
$860$	84.5995i	0.0983715i
$861$	0	0
$862$	−1137.42	−1.31951
$863$	1675.40i	1.94137i	0.240350	+	0.970686i	$0.422738\pi$
$863$	1675.40i	1.94137i	−0.240350	+	0.970686i	$0.577262\pi$
$864$	0	0
$865$	−97.3691	−0.112565
$866$	− 1209.71i	− 1.39689i
$867$	0	0
$868$	−233.083	−0.268529
$869$	1120.02i	1.28887i
$870$	0	0
$871$	−348.488	−0.400101
$872$	522.095i	0.598733i
$873$	0	0
$874$	−53.0052	−0.0606467
$875$	− 88.1659i	− 0.100761i
$876$	0	0
$877$	−1563.11	−1.78234	−0.891170	−	0.453670i	$-0.850115\pi$
$877$	−1563.11	−1.78234	−0.891170	+	0.453670i	$0.850115\pi$
$878$	− 1042.01i	− 1.18680i
$879$	0	0
$880$	−35.6850	−0.0405512
$881$	− 751.835i	− 0.853389i	−0.904396	−	0.426694i	$-0.859678\pi$
$881$	− 751.835i	− 0.853389i	0.904396	−	0.426694i	$-0.140322\pi$
$882$	0	0
$883$	−1005.78	−1.13905	−0.569527	−	0.821973i	$-0.692873\pi$
$883$	−1005.78	−1.13905	−0.569527	+	0.821973i	$0.692873\pi$
$884$	140.712i	0.159176i
$885$	0	0
$886$	−195.195	−0.220310
$887$	1413.71i	1.59381i	0.604106	+	0.796904i	$0.293530\pi$
$887$	1413.71i	1.59381i	−0.604106	+	0.796904i	$0.706470\pi$
$888$	0	0
$889$	254.902	0.286729
$890$	− 104.245i	− 0.117129i
$891$	0	0
$892$	−820.801	−0.920180
$893$	− 223.113i	− 0.249847i
$894$	0	0
$895$	129.304	0.144474
$896$	− 29.9333i	− 0.0334077i
$897$	0	0
$898$	690.760	0.769220
$899$	− 1213.32i	− 1.34963i
$900$	0	0
$901$	−28.0207	−0.0310995
$902$	449.224i	0.498032i
$903$	0	0
$904$	101.407	0.112176
$905$	− 201.062i	− 0.222168i
$906$	0	0
$907$	−158.562	−0.174821	−0.0874104	−	0.996172i	$-0.527859\pi$
$907$	−158.562	−0.174821	−0.0874104	+	0.996172i	$0.527859\pi$
$908$	− 836.132i	− 0.920850i
$909$	0	0
$910$	11.6465	0.0127983
$911$	85.1896i	0.0935121i	0.998906	+	0.0467561i	$0.0148883\pi$
$911$	85.1896i	0.0935121i	−0.998906	+	0.0467561i	$0.985112\pi$
$912$	0	0
$913$	−1061.43	−1.16258
$914$	− 468.581i	− 0.512671i
$915$	0	0
$916$	76.0735	0.0830497
$917$	201.860i	0.220131i
$918$	0	0
$919$	−89.1494	−0.0970070	−0.0485035	−	0.998823i	$-0.515445\pi$
$919$	−89.1494	−0.0970070	−0.0485035	+	0.998823i	$0.515445\pi$
$920$	− 17.6258i	− 0.0191585i
$921$	0	0
$922$	−29.5408	−0.0320399
$923$	422.725i	0.457990i
$924$	0	0
$925$	−71.4299	−0.0772215
$926$	150.511i	0.162538i
$927$	0	0
$928$	155.818	0.167908
$929$	271.528i	0.292280i	0.989264	+	0.146140i	$0.0466850\pi$
$929$	271.528i	0.292280i	−0.989264	+	0.146140i	$0.953315\pi$
$930$	0	0
$931$	28.3155	0.0304141
$932$	202.841i	0.217641i
$933$	0	0
$934$	668.686	0.715938
$935$	135.620i	0.145049i
$936$	0	0
$937$	−1238.24	−1.32149	−0.660747	−	0.750609i	$-0.729760\pi$
$937$	−1238.24	−1.32149	−0.660747	+	0.750609i	$0.729760\pi$
$938$	− 281.741i	− 0.300364i
$939$	0	0
$940$	74.1921	0.0789277
$941$	796.254i	0.846179i	0.906088	+	0.423089i	$0.139054\pi$
$941$	796.254i	0.846179i	−0.906088	+	0.423089i	$0.860946\pi$
$942$	0	0
$943$	−221.885	−0.235297
$944$	52.9800i	0.0561229i
$945$	0	0
$946$	1179.84	1.24718
$947$	− 716.676i	− 0.756786i	−0.925645	−	0.378393i	$-0.876477\pi$
$947$	− 716.676i	− 0.756786i	0.925645	−	0.378393i	$-0.123523\pi$
$948$	0	0
$949$	187.257	0.197320
$950$	140.427i	0.147818i
$951$	0	0
$952$	−113.761	−0.119497
$953$	1105.45i	1.15996i	0.814629	+	0.579982i	$0.196940\pi$
$953$	1105.45i	1.15996i	−0.814629	+	0.579982i	$0.803060\pi$
$954$	0	0
$955$	151.564	0.158706
$956$	− 807.418i	− 0.844579i
$957$	0	0
$958$	−528.265	−0.551424
$959$	− 459.766i	− 0.479422i
$960$	0	0
$961$	979.278	1.01902
$962$	− 19.0452i	− 0.0197975i
$963$	0	0
$964$	−904.489	−0.938266
$965$	231.299i	0.239688i
$966$	0	0
$967$	176.547	0.182572	0.0912858	−	0.995825i	$-0.470902\pi$
$967$	176.547	0.182572	0.0912858	+	0.995825i	$0.470902\pi$
$968$	155.428i	0.160566i
$969$	0	0
$970$	120.220	0.123939
$971$	− 1493.78i	− 1.53840i	−0.639009	−	0.769199i	$-0.720655\pi$
$971$	− 1493.78i	− 1.53840i	0.639009	−	0.769199i	$-0.279345\pi$
$972$	0	0
$973$	−227.373	−0.233682
$974$	− 908.908i	− 0.933171i
$975$	0	0
$976$	−82.1384	−0.0841582
$977$	− 720.120i	− 0.737073i	−0.929613	−	0.368536i	$-0.879859\pi$
$977$	− 720.120i	− 0.737073i	0.929613	−	0.368536i	$-0.120141\pi$
$978$	0	0
$979$	−1453.81	−1.48500
$980$	9.41579i	0.00960795i
$981$	0	0
$982$	−1139.69	−1.16058
$983$	727.718i	0.740303i	0.928971	+	0.370151i	$0.120694\pi$
$983$	727.718i	0.740303i	−0.928971	+	0.370151i	$0.879306\pi$
$984$	0	0
$985$	75.2366	0.0763824
$986$	− 592.185i	− 0.600594i
$987$	0	0
$988$	−37.4419	−0.0378967
$989$	582.755i	0.589236i
$990$	0	0
$991$	−682.016	−0.688210	−0.344105	−	0.938931i	$-0.611818\pi$
$991$	−682.016	−0.688210	−0.344105	+	0.938931i	$0.611818\pi$
$992$	249.176i	0.251186i
$993$	0	0
$994$	−341.760	−0.343823
$995$	− 161.440i	− 0.162252i
$996$	0	0
$997$	−210.841	−0.211476	−0.105738	−	0.994394i	$-0.533720\pi$
$997$	−210.841	−0.211476	−0.105738	+	0.994394i	$0.533720\pi$
$998$	577.309i	0.578466i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.3.b.c.323.7	yes	8
3.2	odd	2	inner	378.3.b.c.323.2	✓	8
4.3	odd	2		3024.3.d.h.1457.4		8
9.2	odd	6		1134.3.q.f.701.3		16
9.4	even	3		1134.3.q.f.1079.3		16
9.5	odd	6		1134.3.q.f.1079.6		16
9.7	even	3		1134.3.q.f.701.6		16
12.11	even	2		3024.3.d.h.1457.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.3.b.c.323.2	✓	8	3.2	odd	2	inner
378.3.b.c.323.7	yes	8	1.1	even	1	trivial
1134.3.q.f.701.3		16	9.2	odd	6
1134.3.q.f.701.6		16	9.7	even	3
1134.3.q.f.1079.3		16	9.4	even	3
1134.3.q.f.1079.6		16	9.5	odd	6
3024.3.d.h.1457.4		8	4.3	odd	2
3024.3.d.h.1457.5		8	12.11	even	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)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} -2.00000 q^{4} -0.672556i q^{5} +2.64575 q^{7} -2.82843i q^{8} +O(q^{10})\)	\(q+1.41421i q^{2} -2.00000 q^{4} -0.672556i q^{5} +2.64575 q^{7} -2.82843i q^{8} +0.951138 q^{10} -13.2647i q^{11} +4.62808 q^{13} +3.74166i q^{14} +4.00000 q^{16} -15.2020i q^{17} +4.04508 q^{19} +1.34511i q^{20} +18.7591 q^{22} +9.26566i q^{23} +24.5477 q^{25} +6.54510i q^{26} -5.29150 q^{28} -27.5450i q^{29} +44.0486 q^{31} +5.65685i q^{32} +21.4988 q^{34} -1.77942i q^{35} -2.90984 q^{37} +5.72060i q^{38} -1.90228 q^{40} +23.9470i q^{41} +62.8940 q^{43} +26.5294i q^{44} -13.1036 q^{46} -55.1568i q^{47} +7.00000 q^{49} +34.7156i q^{50} -9.25617 q^{52} -1.84323i q^{53} -8.92125 q^{55} -7.48331i q^{56} +38.9546 q^{58} +13.2450i q^{59} -20.5346 q^{61} +62.2941i q^{62} -8.00000 q^{64} -3.11265i q^{65} -75.2985 q^{67} +30.4039i q^{68} +2.51647 q^{70} +91.3391i q^{71} +40.4610 q^{73} -4.11514i q^{74} -8.09016 q^{76} -35.0951i q^{77} -84.4365 q^{79} -2.69022i q^{80} -33.8662 q^{82} -80.0194i q^{83} -10.2242 q^{85} +88.9456i q^{86} -37.5182 q^{88} -109.600i q^{89} +12.2448 q^{91} -18.5313i q^{92} +78.0035 q^{94} -2.72054i q^{95} +126.396 q^{97} +9.89949i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{4}+O(q^{10}) \) 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

Varieties

Fields

Representations

Groups

Database

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