LMFDB - Embedded newform 360.4.f.e.289.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [360,4,Mod(289,360)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(360, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("360.289");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 360 = 2^{3} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	360.f (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:	$21.2406876021$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			289.4
Root			$-1.22474 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	360.289
Dual form			360.4.f.e.289.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(10.7980 + 2.89898i) q^{5} -12.6969i q^{7} +O(q^{10})$	$q+(10.7980 + 2.89898i) q^{5} -12.6969i q^{7} -59.1918 q^{11} -42.2020i q^{13} -126.384i q^{17} +19.1918 q^{19} +78.3133i q^{23} +(108.192 + 62.6061i) q^{25} -148.384 q^{29} -139.151 q^{31} +(36.8082 - 137.101i) q^{35} -66.5653i q^{37} +203.212 q^{41} -288.879i q^{43} -360.434i q^{47} +181.788 q^{49} -686.888i q^{53} +(-639.151 - 171.596i) q^{55} -83.1102 q^{59} -208.829 q^{61} +(122.343 - 455.696i) q^{65} +192.293i q^{67} -500.767 q^{71} -122.706i q^{73} +751.555i q^{77} +289.616 q^{79} -573.950i q^{83} +(366.384 - 1364.69i) q^{85} -565.151 q^{89} -535.837 q^{91} +(207.233 + 55.6367i) q^{95} +643.959i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5}+O(q^{10}) $ 4 * q + 4 * q^5	$ 4 q + 4 q^{5} - 80 q^{11} - 80 q^{19} + 276 q^{25} - 280 q^{29} + 384 q^{31} + 304 q^{35} + 1048 q^{41} + 492 q^{49} - 1616 q^{55} + 1392 q^{59} - 1384 q^{61} - 608 q^{65} - 1376 q^{71} + 1472 q^{79} + 1152 q^{85} - 1320 q^{89} + 992 q^{91} + 1456 q^{95}+O(q^{100}) $ 4 * q + 4 * q^5 - 80 * q^11 - 80 * q^19 + 276 * q^25 - 280 * q^29 + 384 * q^31 + 304 * q^35 + 1048 * q^41 + 492 * q^49 - 1616 * q^55 + 1392 * q^59 - 1384 * q^61 - 608 * q^65 - 1376 * q^71 + 1472 * q^79 + 1152 * q^85 - 1320 * q^89 + 992 * q^91 + 1456 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/360\mathbb{Z}\right)^\times$.

$n$	$181$	$217$	$271$	$281$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	10.7980	+	2.89898i	0.965799	+	0.259293i
$5$	10.7980	+	2.89898i	0.965799	+	0.259293i
$6$		0			0
$7$		−	12.6969i		−	0.685570i	−0.939414	−	0.342785i	$-0.888630\pi$
$7$		−	12.6969i		−	0.685570i	0.939414	−	0.342785i	$-0.111370\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−59.1918			−1.62246			−0.811228	−	0.584730i	$-0.801200\pi$
$11$	−59.1918			−1.62246			−0.811228	+	0.584730i	$0.801200\pi$
$12$		0			0
$13$		−	42.2020i		−	0.900365i	−0.892937	−	0.450182i	$-0.851359\pi$
$13$		−	42.2020i		−	0.900365i	0.892937	−	0.450182i	$-0.148641\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	126.384i		−	1.80309i	−0.432685	−	0.901545i	$-0.642434\pi$
$17$		−	126.384i		−	1.80309i	0.432685	−	0.901545i	$-0.357566\pi$
$18$		0			0
$19$	19.1918			0.231732			0.115866	−	0.993265i	$-0.463036\pi$
$19$	19.1918			0.231732			0.115866	+	0.993265i	$0.463036\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			78.3133i			0.709976i	0.934871	+	0.354988i	$0.115515\pi$
$23$			78.3133i			0.709976i	−0.934871	+	0.354988i	$0.884485\pi$
$24$		0			0
$25$	108.192	+	62.6061i	0.865535	+	0.500849i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−148.384			−0.950143			−0.475072	−	0.879947i	$-0.657578\pi$
$29$	−148.384			−0.950143			−0.475072	+	0.879947i	$0.657578\pi$
$30$		0			0
$31$	−139.151			−0.806202			−0.403101	−	0.915156i	$-0.632068\pi$
$31$	−139.151			−0.806202			−0.403101	+	0.915156i	$0.632068\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	36.8082	−	137.101i	0.177763	−	0.662123i
$36$		0			0
$37$		−	66.5653i		−	0.295764i	−0.989005	−	0.147882i	$-0.952754\pi$
$37$		−	66.5653i		−	0.295764i	0.989005	−	0.147882i	$-0.0472456\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	203.212			0.774059			0.387030	−	0.922067i	$-0.373501\pi$
$41$	203.212			0.774059			0.387030	+	0.922067i	$0.373501\pi$
$42$		0			0
$43$		−	288.879i		−	1.02450i	−0.858836	−	0.512251i	$-0.828812\pi$
$43$		−	288.879i		−	1.02450i	0.858836	−	0.512251i	$-0.171188\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	360.434i		−	1.11861i	−0.828962	−	0.559305i	$-0.811068\pi$
$47$		−	360.434i		−	1.11861i	0.828962	−	0.559305i	$-0.188932\pi$
$48$		0			0
$49$	181.788			0.529993
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	686.888i		−	1.78021i	−0.455753	−	0.890106i	$-0.650630\pi$
$53$		−	686.888i		−	1.78021i	0.455753	−	0.890106i	$-0.349370\pi$
$54$		0			0
$55$	−639.151	−	171.596i	−1.56697	−	0.420691i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−83.1102			−0.183390			−0.0916951	−	0.995787i	$-0.529229\pi$
$59$	−83.1102			−0.183390			−0.0916951	+	0.995787i	$0.529229\pi$
$60$		0			0
$61$	−208.829			−0.438324			−0.219162	−	0.975688i	$-0.570332\pi$
$61$	−208.829			−0.438324			−0.219162	+	0.975688i	$0.570332\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	122.343	−	455.696i	0.233458	−	0.869571i
$66$		0			0
$67$			192.293i			0.350632i	0.984512	+	0.175316i	$0.0560946\pi$
$67$			192.293i			0.350632i	−0.984512	+	0.175316i	$0.943905\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−500.767			−0.837044			−0.418522	−	0.908207i	$-0.637452\pi$
$71$	−500.767			−0.837044			−0.418522	+	0.908207i	$0.637452\pi$
$72$		0			0
$73$		−	122.706i		−	0.196735i	−0.995150	−	0.0983676i	$-0.968638\pi$
$73$		−	122.706i		−	0.196735i	0.995150	−	0.0983676i	$-0.0313621\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			751.555i			1.11231i
$78$		0			0
$79$	289.616			0.412461			0.206230	−	0.978503i	$-0.433880\pi$
$79$	289.616			0.412461			0.206230	+	0.978503i	$0.433880\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	573.950i		−	0.759026i	−0.925186	−	0.379513i	$-0.876091\pi$
$83$		−	573.950i		−	0.759026i	0.925186	−	0.379513i	$-0.123909\pi$
$84$		0			0
$85$	366.384	−	1364.69i	0.467528	−	1.74142i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−565.151			−0.673100			−0.336550	−	0.941666i	$-0.609260\pi$
$89$	−565.151			−0.673100			−0.336550	+	0.941666i	$0.609260\pi$
$90$		0			0
$91$	−535.837			−0.617263
$92$		0			0
$93$		0			0
$94$		0			0
$95$	207.233	+	55.6367i	0.223807	+	0.0600864i
$96$		0			0
$97$			643.959i			0.674063i	0.941493	+	0.337032i	$0.109423\pi$
$97$			643.959i			0.674063i	−0.941493	+	0.337032i	$0.890577\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	745.918			0.734868			0.367434	−	0.930050i	$-0.380236\pi$
$101$	745.918			0.734868			0.367434	+	0.930050i	$0.380236\pi$
$102$		0			0
$103$			738.413i			0.706389i	0.935550	+	0.353194i	$0.114905\pi$
$103$			738.413i			0.706389i	−0.935550	+	0.353194i	$0.885095\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			558.536i			0.504633i	0.967645	+	0.252316i	$0.0811923\pi$
$107$			558.536i			0.504633i	−0.967645	+	0.252316i	$0.918808\pi$
$108$		0			0
$109$	1523.69			1.33893			0.669465	−	0.742843i	$-0.266523\pi$
$109$	1523.69			1.33893			0.669465	+	0.742843i	$0.266523\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	58.4245i		−	0.0486382i	−0.999704	−	0.0243191i	$-0.992258\pi$
$113$		−	58.4245i		−	0.0486382i	0.999704	−	0.0243191i	$-0.00774177\pi$
$114$		0			0
$115$	−227.029	+	845.623i	−0.184091	+	0.685694i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1604.69			−1.23615
$120$		0			0
$121$	2172.67			1.63236
$122$		0			0
$123$		0			0
$124$		0			0
$125$	986.757	+	989.664i	0.706066	+	0.708146i
$126$		0			0
$127$			941.283i			0.657680i	0.944386	+	0.328840i	$0.106658\pi$
$127$			941.283i			0.657680i	−0.944386	+	0.328840i	$0.893342\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	214.343			0.142956			0.0714779	−	0.997442i	$-0.477228\pi$
$131$	214.343			0.142956			0.0714779	+	0.997442i	$0.477228\pi$
$132$		0			0
$133$		−	243.678i		−	0.158869i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			499.514i			0.311506i	0.987796	+	0.155753i	$0.0497805\pi$
$137$			499.514i			0.311506i	−0.987796	+	0.155753i	$0.950220\pi$
$138$		0			0
$139$	−2660.48			−1.62345			−0.811723	−	0.584042i	$-0.801470\pi$
$139$	−2660.48			−1.62345			−0.811723	+	0.584042i	$0.801470\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			2498.02i			1.46080i
$144$		0			0
$145$	−1602.24	−	430.161i	−0.917647	−	0.246365i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1334.89			0.733947			0.366973	−	0.930231i	$-0.380394\pi$
$149$	1334.89			0.733947			0.366973	+	0.930231i	$0.380394\pi$
$150$		0			0
$151$	−259.478			−0.139841			−0.0699205	−	0.997553i	$-0.522275\pi$
$151$	−259.478			−0.139841			−0.0699205	+	0.997553i	$0.522275\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−1502.55	−	403.396i	−0.778629	−	0.209042i
$156$		0			0
$157$			1063.58i			0.540654i	0.962769	+	0.270327i	$0.0871319\pi$
$157$			1063.58i			0.540654i	−0.962769	+	0.270327i	$0.912868\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	994.339			0.486738
$162$		0			0
$163$		−	3139.67i		−	1.50870i	−0.656474	−	0.754349i	$-0.727953\pi$
$163$		−	3139.67i		−	1.50870i	0.656474	−	0.754349i	$-0.272047\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			936.411i			0.433902i	0.976182	+	0.216951i	$0.0696112\pi$
$167$			936.411i			0.433902i	−0.976182	+	0.216951i	$0.930389\pi$
$168$		0			0
$169$	415.988			0.189344
$170$		0			0
$171$		0			0
$172$		0			0
$173$			1439.78i			0.632741i	0.948636	+	0.316371i	$0.102464\pi$
$173$			1439.78i			0.632741i	−0.948636	+	0.316371i	$0.897536\pi$
$174$		0			0
$175$	794.906	−	1373.71i	0.343367	−	0.593385i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	2494.42			1.04158			0.520788	−	0.853686i	$-0.325638\pi$
$179$	2494.42			1.04158			0.520788	+	0.853686i	$0.325638\pi$
$180$		0			0
$181$	−1590.36			−0.653096			−0.326548	−	0.945181i	$-0.605886\pi$
$181$	−1590.36			−0.653096			−0.326548	+	0.945181i	$0.605886\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	192.971	−	718.769i	0.0766894	−	0.285649i
$186$		0			0
$187$			7480.88i			2.92543i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	2652.77			1.00496			0.502480	−	0.864589i	$-0.332421\pi$
$191$	2652.77			1.00496			0.502480	+	0.864589i	$0.332421\pi$
$192$		0			0
$193$			3001.61i			1.11948i	0.828667	+	0.559742i	$0.189100\pi$
$193$			3001.61i			1.11948i	−0.828667	+	0.559742i	$0.810900\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			2176.06i			0.786994i	0.919326	+	0.393497i	$0.128735\pi$
$197$			2176.06i			0.786994i	−0.919326	+	0.393497i	$0.871265\pi$
$198$		0			0
$199$	4270.30			1.52117			0.760587	−	0.649236i	$-0.224911\pi$
$199$	4270.30			1.52117			0.760587	+	0.649236i	$0.224911\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			1884.02i			0.651390i
$204$		0			0
$205$	2194.28	+	589.108i	0.747585	+	0.200708i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−1136.00			−0.375975
$210$		0			0
$211$	−2978.78			−0.971886			−0.485943	−	0.873991i	$-0.661524\pi$
$211$	−2978.78			−0.971886			−0.485943	+	0.873991i	$0.661524\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	837.453	−	3119.30i	0.265646	−	0.989462i
$216$		0			0
$217$			1766.79i			0.552708i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−5333.65			−1.62344
$222$		0			0
$223$		−	3183.16i		−	0.955875i	−0.878394	−	0.477937i	$-0.841385\pi$
$223$		−	3183.16i		−	0.955875i	0.878394	−	0.477937i	$-0.158615\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			5910.41i			1.72814i	0.503372	+	0.864070i	$0.332092\pi$
$227$			5910.41i			1.72814i	−0.503372	+	0.864070i	$0.667908\pi$
$228$		0			0
$229$	−3465.59			−1.00006			−0.500028	−	0.866009i	$-0.666677\pi$
$229$	−3465.59			−1.00006			−0.500028	+	0.866009i	$0.666677\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	4611.51i		−	1.29661i	−0.761380	−	0.648305i	$-0.775478\pi$
$233$		−	4611.51i		−	1.29661i	0.761380	−	0.648305i	$-0.224522\pi$
$234$		0			0
$235$	1044.89	−	3891.95i	0.290047	−	1.08035i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	4816.47			1.30356			0.651782	−	0.758406i	$-0.274022\pi$
$239$	4816.47			1.30356			0.651782	+	0.758406i	$0.274022\pi$
$240$		0			0
$241$	3057.47			0.817217			0.408608	−	0.912710i	$-0.366014\pi$
$241$	3057.47			0.817217			0.408608	+	0.912710i	$0.366014\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1962.94	+	526.999i	0.511867	+	0.137423i
$246$		0			0
$247$		−	809.935i		−	0.208643i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−7498.54			−1.88567			−0.942836	−	0.333258i	$-0.891852\pi$
$251$	−7498.54			−1.88567			−0.942836	+	0.333258i	$0.891852\pi$
$252$		0			0
$253$		−	4635.51i		−	1.15190i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			2272.33i			0.551532i	0.961225	+	0.275766i	$0.0889316\pi$
$257$			2272.33i			0.551532i	−0.961225	+	0.275766i	$0.911068\pi$
$258$		0			0
$259$	−845.176			−0.202767
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	5271.34i		−	1.23591i	−0.786213	−	0.617955i	$-0.787961\pi$
$263$		−	5271.34i		−	1.23591i	0.786213	−	0.617955i	$-0.212039\pi$
$264$		0			0
$265$	1991.27	−	7416.99i	0.461596	−	1.71933i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−4315.61			−0.978169			−0.489085	−	0.872236i	$-0.662669\pi$
$269$	−4315.61			−0.978169			−0.489085	+	0.872236i	$0.662669\pi$
$270$		0			0
$271$	4471.27			1.00225			0.501125	−	0.865375i	$-0.332919\pi$
$271$	4471.27			1.00225			0.501125	+	0.865375i	$0.332919\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−6404.07	−	3705.77i	−1.40429	−	0.812605i
$276$		0			0
$277$			8112.09i			1.75960i	0.475348	+	0.879798i	$0.342322\pi$
$277$			8112.09i			1.75960i	−0.475348	+	0.879798i	$0.657678\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	3811.46			0.809155			0.404577	−	0.914504i	$-0.367419\pi$
$281$	3811.46			0.809155			0.404577	+	0.914504i	$0.367419\pi$
$282$		0			0
$283$			2362.60i			0.496261i	0.968727	+	0.248131i	$0.0798162\pi$
$283$			2362.60i			0.496261i	−0.968727	+	0.248131i	$0.920184\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	2580.17i		−	0.530672i
$288$		0			0
$289$	−11059.8			−2.25114
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	6337.79i		−	1.26368i	−0.775099	−	0.631839i	$-0.782300\pi$
$293$		−	6337.79i		−	1.26368i	0.775099	−	0.631839i	$-0.217700\pi$
$294$		0			0
$295$	−897.420	−	240.935i	−0.177118	−	0.0475517i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	3304.98			0.639237
$300$		0			0
$301$	−3667.87			−0.702368
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−2254.92	−	605.390i	−0.423333	−	0.113654i
$306$		0			0
$307$		−	4170.44i		−	0.775308i	−0.921805	−	0.387654i	$-0.873285\pi$
$307$		−	4170.44i		−	0.775308i	0.921805	−	0.387654i	$-0.126715\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	5083.71			0.926915			0.463457	−	0.886119i	$-0.346609\pi$
$311$	5083.71			0.926915			0.463457	+	0.886119i	$0.346609\pi$
$312$		0			0
$313$			1522.86i			0.275007i	0.990501	+	0.137504i	$0.0439078\pi$
$313$			1522.86i			0.275007i	−0.990501	+	0.137504i	$0.956092\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	4996.09i		−	0.885200i	−0.896719	−	0.442600i	$-0.854056\pi$
$317$		−	4996.09i		−	0.885200i	0.896719	−	0.442600i	$-0.145944\pi$
$318$		0			0
$319$	8783.10			1.54157
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	2425.53i		−	0.417834i
$324$		0			0
$325$	2642.11	−	4565.92i	0.450947	−	0.779297i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−4576.40			−0.766885
$330$		0			0
$331$	2832.04			0.470281			0.235141	−	0.971961i	$-0.424445\pi$
$331$	2832.04			0.470281			0.235141	+	0.971961i	$0.424445\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−557.453	+	2076.37i	−0.0909162	+	0.338639i
$336$		0			0
$337$		−	1684.43i		−	0.272276i	−0.990690	−	0.136138i	$-0.956531\pi$
$337$		−	1684.43i		−	0.272276i	0.990690	−	0.136138i	$-0.0434690\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	8236.60			1.30803
$342$		0			0
$343$		−	6663.20i		−	1.04892i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	9350.30i		−	1.44654i	−0.690564	−	0.723271i	$-0.742638\pi$
$347$		−	9350.30i		−	1.44654i	0.690564	−	0.723271i	$-0.257362\pi$
$348$		0			0
$349$	−4174.20			−0.640228			−0.320114	−	0.947379i	$-0.603721\pi$
$349$	−4174.20			−0.640228			−0.320114	+	0.947379i	$0.603721\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			6217.04i			0.937394i	0.883359	+	0.468697i	$0.155276\pi$
$353$			6217.04i			0.937394i	−0.883359	+	0.468697i	$0.844724\pi$
$354$		0			0
$355$	−5407.27	−	1451.71i	−0.808416	−	0.217039i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	8149.73			1.19812			0.599062	−	0.800703i	$-0.295540\pi$
$359$	8149.73			1.19812			0.599062	+	0.800703i	$0.295540\pi$
$360$		0			0
$361$	−6490.67			−0.946300
$362$		0			0
$363$		0			0
$364$		0			0
$365$	355.723	−	1324.98i	0.0510120	−	0.190007i
$366$		0			0
$367$			11868.2i			1.68806i	0.536299	+	0.844028i	$0.319822\pi$
$367$			11868.2i			1.68806i	−0.536299	+	0.844028i	$0.680178\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−8721.37			−1.22046
$372$		0			0
$373$		−	3286.85i		−	0.456265i	−0.973630	−	0.228132i	$-0.926738\pi$
$373$		−	3286.85i		−	0.456265i	0.973630	−	0.228132i	$-0.0732619\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			6262.09i			0.855475i
$378$		0			0
$379$	−8326.24			−1.12847			−0.564235	−	0.825614i	$-0.690829\pi$
$379$	−8326.24			−1.12847			−0.564235	+	0.825614i	$0.690829\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	4249.23i		−	0.566908i	−0.958986	−	0.283454i	$-0.908520\pi$
$383$		−	4249.23i		−	0.566908i	0.958986	−	0.283454i	$-0.0914803\pi$
$384$		0			0
$385$	−2178.74	+	8115.26i	−0.288413	+	1.07426i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−7841.83			−1.02210			−0.511050	−	0.859551i	$-0.670743\pi$
$389$	−7841.83			−1.02210			−0.511050	+	0.859551i	$0.670743\pi$
$390$		0			0
$391$	9897.52			1.28015
$392$		0			0
$393$		0			0
$394$		0			0
$395$	3127.27	+	839.592i	0.398354	+	0.106948i
$396$		0			0
$397$		−	401.794i		−	0.0507946i	−0.999677	−	0.0253973i	$-0.991915\pi$
$397$		−	401.794i		−	0.0507946i	0.999677	−	0.0253973i	$-0.00808508\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−9732.89			−1.21206			−0.606032	−	0.795441i	$-0.707239\pi$
$401$	−9732.89			−1.21206			−0.606032	+	0.795441i	$0.707239\pi$
$402$		0			0
$403$			5872.46i			0.725876i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			3940.12i			0.479864i
$408$		0			0
$409$	5629.67			0.680609			0.340305	−	0.940315i	$-0.389470\pi$
$409$	5629.67			0.680609			0.340305	+	0.940315i	$0.389470\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			1055.25i			0.125727i
$414$		0			0
$415$	1663.87	−	6197.49i	0.196810	−	0.733067i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	9773.77			1.13957			0.569785	−	0.821794i	$-0.307026\pi$
$419$	9773.77			1.13957			0.569785	+	0.821794i	$0.307026\pi$
$420$		0			0
$421$	3037.95			0.351688			0.175844	−	0.984418i	$-0.443735\pi$
$421$	3037.95			0.351688			0.175844	+	0.984418i	$0.443735\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	7912.39	−	13673.7i	0.903076	−	1.56064i
$426$		0			0
$427$			2651.48i			0.300502i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−1873.45			−0.209376			−0.104688	−	0.994505i	$-0.533384\pi$
$431$	−1873.45			−0.209376			−0.104688	+	0.994505i	$0.533384\pi$
$432$		0			0
$433$		−	11855.8i		−	1.31583i	−0.753092	−	0.657915i	$-0.771439\pi$
$433$		−	11855.8i		−	1.31583i	0.753092	−	0.657915i	$-0.228561\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			1502.98i			0.164524i
$438$		0			0
$439$	16243.8			1.76600			0.882999	−	0.469375i	$-0.155521\pi$
$439$	16243.8			1.76600			0.882999	+	0.469375i	$0.155521\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			3905.52i			0.418864i	0.977823	+	0.209432i	$0.0671614\pi$
$443$			3905.52i			0.418864i	−0.977823	+	0.209432i	$0.932839\pi$
$444$		0			0
$445$	−6102.48	−	1638.36i	−0.650079	−	0.174530i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−6852.71			−0.720265			−0.360133	−	0.932901i	$-0.617269\pi$
$449$	−6852.71			−0.720265			−0.360133	+	0.932901i	$0.617269\pi$
$450$		0			0
$451$	−12028.5			−1.25588
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−5785.94	−	1553.38i	−0.596152	−	0.160052i
$456$		0			0
$457$			5800.75i			0.593758i	0.954915	+	0.296879i	$0.0959458\pi$
$457$			5800.75i			0.593758i	−0.954915	+	0.296879i	$0.904054\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	5263.63			0.531783			0.265891	−	0.964003i	$-0.414334\pi$
$461$	5263.63			0.531783			0.265891	+	0.964003i	$0.414334\pi$
$462$		0			0
$463$		−	13636.6i		−	1.36879i	−0.729113	−	0.684394i	$-0.760067\pi$
$463$		−	13636.6i		−	1.36879i	0.729113	−	0.684394i	$-0.239933\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			13158.3i			1.30384i	0.758287	+	0.651920i	$0.226036\pi$
$467$			13158.3i			1.30384i	−0.758287	+	0.651920i	$0.773964\pi$
$468$		0			0
$469$	2441.53			0.240383
$470$		0			0
$471$		0			0
$472$		0			0
$473$			17099.3i			1.66221i
$474$		0			0
$475$	2076.40	+	1201.53i	0.200572	+	0.116063i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−6775.90			−0.646344			−0.323172	−	0.946340i	$-0.604749\pi$
$479$	−6775.90			−0.646344			−0.323172	+	0.946340i	$0.604749\pi$
$480$		0			0
$481$	−2809.19			−0.266295
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−1866.82	+	6953.44i	−0.174780	+	0.651010i
$486$		0			0
$487$		−	5933.64i		−	0.552113i	−0.961141	−	0.276057i	$-0.910972\pi$
$487$		−	5933.64i		−	0.552113i	0.961141	−	0.276057i	$-0.0890277\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	11934.5			1.09694			0.548468	−	0.836171i	$-0.315211\pi$
$491$	11934.5			1.09694			0.548468	+	0.836171i	$0.315211\pi$
$492$		0			0
$493$			18753.3i			1.71319i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			6358.21i			0.573853i
$498$		0			0
$499$	1567.65			0.140636			0.0703182	−	0.997525i	$-0.477599\pi$
$499$	1567.65			0.140636			0.0703182	+	0.997525i	$0.477599\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			450.509i			0.0399348i	0.999801	+	0.0199674i	$0.00635624\pi$
$503$			450.509i			0.0399348i	−0.999801	+	0.0199674i	$0.993644\pi$
$504$		0			0
$505$	8054.40	+	2162.40i	0.709734	+	0.190546i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−7016.11			−0.610969			−0.305485	−	0.952197i	$-0.598818\pi$
$509$	−7016.11			−0.610969			−0.305485	+	0.952197i	$0.598818\pi$
$510$		0			0
$511$	−1557.99			−0.134876
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−2140.64	+	7973.36i	−0.183161	+	0.682229i
$516$		0			0
$517$			21334.7i			1.81489i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	16356.0			1.37538			0.687688	−	0.726006i	$-0.258626\pi$
$521$	16356.0			1.37538			0.687688	+	0.726006i	$0.258626\pi$
$522$		0			0
$523$			13190.5i			1.10283i	0.834232	+	0.551414i	$0.185911\pi$
$523$			13190.5i			1.10283i	−0.834232	+	0.551414i	$0.814089\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			17586.4i			1.45366i
$528$		0			0
$529$	6034.03			0.495934
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	8575.97i		−	0.696935i
$534$		0			0
$535$	−1619.18	+	6031.05i	−0.130847	+	0.487374i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−10760.4			−0.859891
$540$		0			0
$541$	21997.6			1.74815			0.874076	−	0.485789i	$-0.161468\pi$
$541$	21997.6			1.74815			0.874076	+	0.485789i	$0.161468\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	16452.8	+	4417.16i	1.29314	+	0.347175i
$546$		0			0
$547$		−	212.663i		−	0.0166230i	−0.999965	−	0.00831151i	$-0.997354\pi$
$547$		−	212.663i		−	0.0166230i	0.999965	−	0.00831151i	$-0.00264567\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−2847.76			−0.220179
$552$		0			0
$553$		−	3677.24i		−	0.282771i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	6641.66i		−	0.505235i	−0.967566	−	0.252618i	$-0.918709\pi$
$557$		−	6641.66i		−	0.505235i	0.967566	−	0.252618i	$-0.0812915\pi$
$558$		0			0
$559$	−12191.3			−0.922425
$560$		0			0
$561$		0			0
$562$		0			0
$563$			7122.94i			0.533208i	0.963806	+	0.266604i	$0.0859016\pi$
$563$			7122.94i			0.533208i	−0.963806	+	0.266604i	$0.914098\pi$
$564$		0			0
$565$	169.371	−	630.865i	0.0126115	−	0.0469747i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	5951.39			0.438480			0.219240	−	0.975671i	$-0.429642\pi$
$569$	5951.39			0.438480			0.219240	+	0.975671i	$0.429642\pi$
$570$		0			0
$571$	−23612.2			−1.73054			−0.865272	−	0.501303i	$-0.832854\pi$
$571$	−23612.2			−1.73054			−0.865272	+	0.501303i	$0.832854\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−4902.89	+	8472.86i	−0.355591	+	0.614509i
$576$		0			0
$577$		−	25398.1i		−	1.83247i	−0.400636	−	0.916237i	$-0.631211\pi$
$577$		−	25398.1i		−	1.83247i	0.400636	−	0.916237i	$-0.368789\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−7287.41			−0.520366
$582$		0			0
$583$			40658.1i			2.88832i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	17994.6i		−	1.26528i	−0.774448	−	0.632638i	$-0.781972\pi$
$587$		−	17994.6i		−	1.26528i	0.774448	−	0.632638i	$-0.218028\pi$
$588$		0			0
$589$	−2670.56			−0.186823
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	14842.0i		−	1.02780i	−0.857850	−	0.513900i	$-0.828200\pi$
$593$		−	14842.0i		−	1.02780i	0.857850	−	0.513900i	$-0.171800\pi$
$594$		0			0
$595$	−17327.3	−	4651.95i	−1.19387	−	0.320523i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	25071.6			1.71018			0.855089	−	0.518481i	$-0.173502\pi$
$599$	25071.6			1.71018			0.855089	+	0.518481i	$0.173502\pi$
$600$		0			0
$601$	−2772.27			−0.188159			−0.0940794	−	0.995565i	$-0.529991\pi$
$601$	−2772.27			−0.188159			−0.0940794	+	0.995565i	$0.529991\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	23460.4	+	6298.54i	1.57653	+	0.423259i
$606$		0			0
$607$		−	19585.8i		−	1.30966i	−0.755776	−	0.654830i	$-0.772740\pi$
$607$		−	19585.8i		−	1.30966i	0.755776	−	0.654830i	$-0.227260\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−15211.0			−1.00716
$612$		0			0
$613$		−	24247.5i		−	1.59763i	−0.601579	−	0.798814i	$-0.705461\pi$
$613$		−	24247.5i		−	1.59763i	0.601579	−	0.798814i	$-0.294539\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	1225.83i		−	0.0799841i	−0.999200	−	0.0399920i	$-0.987267\pi$
$617$		−	1225.83i		−	0.0799841i	0.999200	−	0.0399920i	$-0.0127333\pi$
$618$		0			0
$619$	12430.1			0.807122			0.403561	−	0.914953i	$-0.367772\pi$
$619$	12430.1			0.807122			0.403561	+	0.914953i	$0.367772\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			7175.69i			0.461457i
$624$		0			0
$625$	7785.95	+	13546.9i	0.498301	+	0.867004i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−8412.77			−0.533289
$630$		0			0
$631$	−6215.00			−0.392100			−0.196050	−	0.980594i	$-0.562812\pi$
$631$	−6215.00			−0.392100			−0.196050	+	0.980594i	$0.562812\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−2728.76	+	10163.9i	−0.170531	+	0.635186i
$636$		0			0
$637$		−	7671.81i		−	0.477187i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−5059.76			−0.311776			−0.155888	−	0.987775i	$-0.549824\pi$
$641$	−5059.76			−0.311776			−0.155888	+	0.987775i	$0.549824\pi$
$642$		0			0
$643$		−	8676.69i		−	0.532154i	−0.963952	−	0.266077i	$-0.914272\pi$
$643$		−	8676.69i		−	0.532154i	0.963952	−	0.266077i	$-0.0857276\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	22787.4i		−	1.38465i	−0.721587	−	0.692323i	$-0.756587\pi$
$647$		−	22787.4i		−	1.38465i	0.721587	−	0.692323i	$-0.243413\pi$
$648$		0			0
$649$	4919.45			0.297543
$650$		0			0
$651$		0			0
$652$		0			0
$653$			26813.1i			1.60686i	0.595402	+	0.803428i	$0.296993\pi$
$653$			26813.1i			1.60686i	−0.595402	+	0.803428i	$0.703007\pi$
$654$		0			0
$655$	2314.47	+	621.376i	0.138067	+	0.0370674i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−1935.78			−0.114427			−0.0572134	−	0.998362i	$-0.518222\pi$
$659$	−1935.78			−0.114427			−0.0572134	+	0.998362i	$0.518222\pi$
$660$		0			0
$661$	3371.84			0.198411			0.0992053	−	0.995067i	$-0.468370\pi$
$661$	3371.84			0.198411			0.0992053	+	0.995067i	$0.468370\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	706.416	−	2631.22i	0.0411935	−	0.153435i
$666$		0			0
$667$		−	11620.4i		−	0.674579i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	12360.9			0.711161
$672$		0			0
$673$			22061.3i			1.26360i	0.775133	+	0.631798i	$0.217683\pi$
$673$			22061.3i			1.26360i	−0.775133	+	0.631798i	$0.782317\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	12747.5i		−	0.723674i	−0.932241	−	0.361837i	$-0.882150\pi$
$677$		−	12747.5i		−	0.723674i	0.932241	−	0.361837i	$-0.117850\pi$
$678$		0			0
$679$	8176.31			0.462118
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	24562.0i		−	1.37605i	−0.725689	−	0.688023i	$-0.758479\pi$
$683$		−	24562.0i		−	1.37605i	0.725689	−	0.688023i	$-0.241521\pi$
$684$		0			0
$685$	−1448.08	+	5393.73i	−0.0807713	+	0.300853i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−28988.1			−1.60284
$690$		0			0
$691$	−15853.1			−0.872767			−0.436383	−	0.899761i	$-0.643741\pi$
$691$	−15853.1			−0.872767			−0.436383	+	0.899761i	$0.643741\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−28727.8	−	7712.68i	−1.56792	−	0.420948i
$696$		0			0
$697$		−	25682.7i		−	1.39570i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	18288.0			0.985347			0.492673	−	0.870214i	$-0.336020\pi$
$701$	18288.0			0.985347			0.492673	+	0.870214i	$0.336020\pi$
$702$		0			0
$703$		−	1277.51i		−	0.0685380i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	9470.88i		−	0.503804i
$708$		0			0
$709$	−11985.3			−0.634864			−0.317432	−	0.948281i	$-0.602821\pi$
$709$	−11985.3			−0.634864			−0.317432	+	0.948281i	$0.602821\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	10897.4i		−	0.572384i
$714$		0			0
$715$	−7241.70	+	26973.5i	−0.378775	+	1.41084i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	17036.3			0.883653			0.441827	−	0.897101i	$-0.354331\pi$
$719$	17036.3			0.883653			0.441827	+	0.897101i	$0.354331\pi$
$720$		0			0
$721$	9375.59			0.484279
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−16053.9	−	9289.73i	−0.822382	−	0.475878i
$726$		0			0
$727$			1219.85i			0.0622307i	0.999516	+	0.0311154i	$0.00990592\pi$
$727$			1219.85i			0.0622307i	−0.999516	+	0.0311154i	$0.990094\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−36509.5			−1.84727
$732$		0			0
$733$		−	17750.6i		−	0.894453i	−0.894421	−	0.447226i	$-0.852412\pi$
$733$		−	17750.6i		−	0.894453i	0.894421	−	0.447226i	$-0.147588\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	11382.2i		−	0.568884i
$738$		0			0
$739$	25825.4			1.28553			0.642763	−	0.766065i	$-0.277788\pi$
$739$	25825.4			1.28553			0.642763	+	0.766065i	$0.277788\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	11549.8i		−	0.570282i	−0.958486	−	0.285141i	$-0.907959\pi$
$743$		−	11549.8i		−	0.570282i	0.958486	−	0.285141i	$-0.0920405\pi$
$744$		0			0
$745$	14414.0	+	3869.81i	0.708845	+	0.190307i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	7091.69			0.345961
$750$		0			0
$751$	29468.3			1.43184			0.715921	−	0.698182i	$-0.246007\pi$
$751$	29468.3			1.43184			0.715921	+	0.698182i	$0.246007\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−2801.83	−	752.220i	−0.135058	−	0.0362597i
$756$		0			0
$757$		−	13820.2i		−	0.663545i	−0.943359	−	0.331772i	$-0.892353\pi$
$757$		−	13820.2i		−	0.663545i	0.943359	−	0.331772i	$-0.107647\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1188.09			0.0565943			0.0282971	−	0.999600i	$-0.490992\pi$
$761$	1188.09			0.0565943			0.0282971	+	0.999600i	$0.490992\pi$
$762$		0			0
$763$		−	19346.2i		−	0.917931i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			3507.42i			0.165118i
$768$		0			0
$769$	−29907.3			−1.40245			−0.701225	−	0.712940i	$-0.747363\pi$
$769$	−29907.3			−1.40245			−0.701225	+	0.712940i	$0.747363\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			27528.3i			1.28088i	0.768007	+	0.640442i	$0.221249\pi$
$773$			27528.3i			1.28088i	−0.768007	+	0.640442i	$0.778751\pi$
$774$		0			0
$775$	−15055.0	−	8711.71i	−0.697796	−	0.403785i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	3900.02			0.179374
$780$		0			0
$781$	29641.3			1.35807
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−3083.29	+	11484.5i	−0.140188	+	0.522163i
$786$		0			0
$787$			10038.2i			0.454666i	0.973817	+	0.227333i	$0.0730006\pi$
$787$			10038.2i			0.454666i	−0.973817	+	0.227333i	$0.926999\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−741.812			−0.0333449
$792$		0			0
$793$			8812.99i			0.394651i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	41075.3i		−	1.82555i	−0.408466	−	0.912773i	$-0.633936\pi$
$797$		−	41075.3i		−	1.82555i	0.408466	−	0.912773i	$-0.366064\pi$
$798$		0			0
$799$	−45552.9			−2.01695
$800$		0			0
$801$		0			0
$802$		0			0
$803$			7263.20i			0.319194i
$804$		0			0
$805$	10736.8	+	2882.57i	0.470091	+	0.126208i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−2134.47			−0.0927616			−0.0463808	−	0.998924i	$-0.514769\pi$
$809$	−2134.47			−0.0927616			−0.0463808	+	0.998924i	$0.514769\pi$
$810$		0			0
$811$	5866.38			0.254003			0.127001	−	0.991903i	$-0.459465\pi$
$811$	5866.38			0.254003			0.127001	+	0.991903i	$0.459465\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	9101.83	−	33902.0i	0.391194	−	1.45710i
$816$		0			0
$817$		−	5544.11i		−	0.237410i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	28572.6			1.21460			0.607301	−	0.794472i	$-0.292252\pi$
$821$	28572.6			1.21460			0.607301	+	0.794472i	$0.292252\pi$
$822$		0			0
$823$		−	15900.5i		−	0.673460i	−0.941601	−	0.336730i	$-0.890679\pi$
$823$		−	15900.5i		−	0.673460i	0.941601	−	0.336730i	$-0.109321\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			34717.6i			1.45979i	0.683558	+	0.729896i	$0.260431\pi$
$827$			34717.6i			1.45979i	−0.683558	+	0.729896i	$0.739569\pi$
$828$		0			0
$829$	812.559			0.0340426			0.0170213	−	0.999855i	$-0.494582\pi$
$829$	812.559			0.0340426			0.0170213	+	0.999855i	$0.494582\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	22975.0i		−	0.955626i
$834$		0			0
$835$	−2714.64	+	10111.3i	−0.112508	+	0.419062i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−852.588			−0.0350829			−0.0175415	−	0.999846i	$-0.505584\pi$
$839$	−852.588			−0.0350829			−0.0175415	+	0.999846i	$0.505584\pi$
$840$		0			0
$841$	−2371.29			−0.0972277
$842$		0			0
$843$		0			0
$844$		0			0
$845$	4491.82	+	1205.94i	0.182868	+	0.0490954i
$846$		0			0
$847$		−	27586.3i		−	1.11910i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	5212.95			0.209985
$852$		0			0
$853$			8851.54i			0.355300i	0.984094	+	0.177650i	$0.0568495\pi$
$853$			8851.54i			0.355300i	−0.984094	+	0.177650i	$0.943150\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	1712.28i		−	0.0682502i	−0.999418	−	0.0341251i	$-0.989136\pi$
$857$		−	1712.28i		−	0.0682502i	0.999418	−	0.0341251i	$-0.0108645\pi$
$858$		0			0
$859$	−33276.7			−1.32175			−0.660877	−	0.750495i	$-0.729815\pi$
$859$	−33276.7			−1.32175			−0.660877	+	0.750495i	$0.729815\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	5913.49i		−	0.233253i	−0.993176	−	0.116627i	$-0.962792\pi$
$863$		−	5913.49i		−	0.233253i	0.993176	−	0.116627i	$-0.0372081\pi$
$864$		0			0
$865$	−4173.89	+	15546.7i	−0.164065	+	0.611101i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−17142.9			−0.669199
$870$		0			0
$871$	8115.15			0.315696
$872$		0			0
$873$		0			0
$874$		0			0
$875$	12565.7	−	12528.8i	0.485484	−	0.484058i
$876$		0			0
$877$		−	38661.1i		−	1.48859i	−0.667851	−	0.744295i	$-0.732786\pi$
$877$		−	38661.1i		−	1.48859i	0.667851	−	0.744295i	$-0.267214\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−20876.4			−0.798348			−0.399174	−	0.916875i	$-0.630703\pi$
$881$	−20876.4			−0.798348			−0.399174	+	0.916875i	$0.630703\pi$
$882$		0			0
$883$			14713.8i			0.560770i	0.959888	+	0.280385i	$0.0904622\pi$
$883$			14713.8i			0.560770i	−0.959888	+	0.280385i	$0.909538\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	38213.1i		−	1.44653i	−0.690571	−	0.723264i	$-0.742641\pi$
$887$		−	38213.1i		−	1.44653i	0.690571	−	0.723264i	$-0.257359\pi$
$888$		0			0
$889$	11951.4			0.450886
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	6917.38i		−	0.259218i
$894$		0			0
$895$	26934.7	+	7231.29i	1.00595	+	0.270073i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	20647.7			0.766007
$900$		0			0
$901$	−86811.4			−3.20989
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−17172.6	−	4610.42i	−0.630760	−	0.169343i
$906$		0			0
$907$		−	30912.4i		−	1.13168i	−0.824516	−	0.565838i	$-0.808553\pi$
$907$		−	30912.4i		−	1.13168i	0.824516	−	0.565838i	$-0.191447\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	1390.58			0.0505730			0.0252865	−	0.999680i	$-0.491950\pi$
$911$	1390.58			0.0505730			0.0252865	+	0.999680i	$0.491950\pi$
$912$		0			0
$913$			33973.2i			1.23149i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	2721.50i		−	0.0980063i
$918$		0			0
$919$	14064.7			0.504843			0.252422	−	0.967617i	$-0.418773\pi$
$919$	14064.7			0.504843			0.252422	+	0.967617i	$0.418773\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			21133.4i			0.753645i
$924$		0			0
$925$	4167.40	−	7201.82i	0.148133	−	0.255994i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	51692.7			1.82560			0.912799	−	0.408408i	$-0.133916\pi$
$929$	51692.7			1.82560			0.912799	+	0.408408i	$0.133916\pi$
$930$		0			0
$931$	3488.84			0.122816
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−21686.9	+	80778.3i	−0.758544	+	2.82538i
$936$		0			0
$937$			12479.3i			0.435092i	0.976050	+	0.217546i	$0.0698052\pi$
$937$			12479.3i			0.435092i	−0.976050	+	0.217546i	$0.930195\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−41800.9			−1.44811			−0.724054	−	0.689743i	$-0.757723\pi$
$941$	−41800.9			−1.44811			−0.724054	+	0.689743i	$0.757723\pi$
$942$		0			0
$943$			15914.2i			0.549563i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	6482.77i		−	0.222452i	−0.993795	−	0.111226i	$-0.964522\pi$
$947$		−	6482.77i		−	0.222452i	0.993795	−	0.111226i	$-0.0354777\pi$
$948$		0			0
$949$	−5178.45			−0.177133
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	28635.8i		−	0.973351i	−0.873583	−	0.486675i	$-0.838209\pi$
$953$		−	28635.8i		−	0.973351i	0.873583	−	0.486675i	$-0.161791\pi$
$954$		0			0
$955$	28644.5	+	7690.32i	0.970590	+	0.260579i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	6342.30			0.213560
$960$		0			0
$961$	−10428.0			−0.350038
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−8701.60	+	32411.2i	−0.290274	+	1.08120i
$966$		0			0
$967$			3086.04i			0.102627i	0.998683	+	0.0513135i	$0.0163408\pi$
$967$			3086.04i			0.102627i	−0.998683	+	0.0513135i	$0.983659\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−11625.0			−0.384205			−0.192102	−	0.981375i	$-0.561531\pi$
$971$	−11625.0			−0.384205			−0.192102	+	0.981375i	$0.561531\pi$
$972$		0			0
$973$			33780.0i			1.11299i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			38232.0i			1.25194i	0.779845	+	0.625972i	$0.215298\pi$
$977$			38232.0i			1.25194i	−0.779845	+	0.625972i	$0.784702\pi$
$978$		0			0
$979$	33452.3			1.09207
$980$		0			0
$981$		0			0
$982$		0			0
$983$			54290.3i			1.76154i	0.473546	+	0.880769i	$0.342974\pi$
$983$			54290.3i			1.76154i	−0.473546	+	0.880769i	$0.657026\pi$
$984$		0			0
$985$	−6308.35	+	23497.0i	−0.204062	+	0.760078i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	22623.0			0.727371
$990$		0			0
$991$	13673.7			0.438303			0.219152	−	0.975691i	$-0.429671\pi$
$991$	13673.7			0.438303			0.219152	+	0.975691i	$0.429671\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	46110.5	+	12379.5i	1.46915	+	0.394429i
$996$		0			0
$997$			3392.56i			0.107767i	0.998547	+	0.0538834i	$0.0171599\pi$
$997$			3392.56i			0.107767i	−0.998547	+	0.0538834i	$0.982840\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	360.4.f.e.289.4		4
3.2	odd	2		40.4.c.a.9.1	✓	4
4.3	odd	2		720.4.f.m.289.4		4
5.2	odd	4		1800.4.a.bk.1.2		2
5.3	odd	4		1800.4.a.bp.1.1		2
5.4	even	2	inner	360.4.f.e.289.3		4
12.11	even	2		80.4.c.c.49.4		4
15.2	even	4		200.4.a.k.1.1		2
15.8	even	4		200.4.a.l.1.2		2
15.14	odd	2		40.4.c.a.9.4	yes	4
20.19	odd	2		720.4.f.m.289.3		4
24.5	odd	2		320.4.c.g.129.4		4
24.11	even	2		320.4.c.h.129.1		4
60.23	odd	4		400.4.a.v.1.1		2
60.47	odd	4		400.4.a.x.1.2		2
60.59	even	2		80.4.c.c.49.1		4
120.29	odd	2		320.4.c.g.129.1		4
120.53	even	4		1600.4.a.ce.1.1		2
120.59	even	2		320.4.c.h.129.4		4
120.77	even	4		1600.4.a.cl.1.2		2
120.83	odd	4		1600.4.a.cm.1.2		2
120.107	odd	4		1600.4.a.cf.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.4.c.a.9.1	✓	4	3.2	odd	2
40.4.c.a.9.4	yes	4	15.14	odd	2
80.4.c.c.49.1		4	60.59	even	2
80.4.c.c.49.4		4	12.11	even	2
200.4.a.k.1.1		2	15.2	even	4
200.4.a.l.1.2		2	15.8	even	4
320.4.c.g.129.1		4	120.29	odd	2
320.4.c.g.129.4		4	24.5	odd	2
320.4.c.h.129.1		4	24.11	even	2
320.4.c.h.129.4		4	120.59	even	2
360.4.f.e.289.3		4	5.4	even	2	inner
360.4.f.e.289.4		4	1.1	even	1	trivial
400.4.a.v.1.1		2	60.23	odd	4
400.4.a.x.1.2		2	60.47	odd	4
720.4.f.m.289.3		4	20.19	odd	2
720.4.f.m.289.4		4	4.3	odd	2
1600.4.a.ce.1.1		2	120.53	even	4
1600.4.a.cf.1.1		2	120.107	odd	4
1600.4.a.cl.1.2		2	120.77	even	4
1600.4.a.cm.1.2		2	120.83	odd	4
1800.4.a.bk.1.2		2	5.2	odd	4
1800.4.a.bp.1.1		2	5.3	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	360.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(10.7980 + 2.89898i) q^{5} -12.6969i q^{7} +O(q^{10})\)	\(q+(10.7980 + 2.89898i) q^{5} -12.6969i q^{7} -59.1918 q^{11} -42.2020i q^{13} -126.384i q^{17} +19.1918 q^{19} +78.3133i q^{23} +(108.192 + 62.6061i) q^{25} -148.384 q^{29} -139.151 q^{31} +(36.8082 - 137.101i) q^{35} -66.5653i q^{37} +203.212 q^{41} -288.879i q^{43} -360.434i q^{47} +181.788 q^{49} -686.888i q^{53} +(-639.151 - 171.596i) q^{55} -83.1102 q^{59} -208.829 q^{61} +(122.343 - 455.696i) q^{65} +192.293i q^{67} -500.767 q^{71} -122.706i q^{73} +751.555i q^{77} +289.616 q^{79} -573.950i q^{83} +(366.384 - 1364.69i) q^{85} -565.151 q^{89} -535.837 q^{91} +(207.233 + 55.6367i) q^{95} +643.959i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5}+O(q^{10}) \) 4 * q + 4 * q^5	\( 4 q + 4 q^{5} - 80 q^{11} - 80 q^{19} + 276 q^{25} - 280 q^{29} + 384 q^{31} + 304 q^{35} + 1048 q^{41} + 492 q^{49} - 1616 q^{55} + 1392 q^{59} - 1384 q^{61} - 608 q^{65} - 1376 q^{71} + 1472 q^{79} + 1152 q^{85} - 1320 q^{89} + 992 q^{91} + 1456 q^{95}+O(q^{100}) \) 4 * q + 4 * q^5 - 80 * q^11 - 80 * q^19 + 276 * q^25 - 280 * q^29 + 384 * q^31 + 304 * q^35 + 1048 * q^41 + 492 * q^49 - 1616 * q^55 + 1392 * q^59 - 1384 * q^61 - 608 * q^65 - 1376 * q^71 + 1472 * q^79 + 1152 * q^85 - 1320 * q^89 + 992 * q^91 + 1456 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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