LMFDB - Embedded newform 1089.2.d.g.1088.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1089,2,Mod(1088,1089)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1089, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1089.1088");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1089 = 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1089.d (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:	$8.69570878012$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 2x^{14} - 16x^{12} - 72x^{10} + 26x^{8} + 360x^{6} + 725x^{4} + 1000x^{2} + 625 $ x^16 + 2x^14 - 16x^12 - 72x^10 + 26x^8 + 360x^6 + 725x^4 + 1000*x^2 + 625
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 99)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1088.2
Root			$0.0783900 + 1.17295i$ of defining polynomial
Character	$\chi$	$=$	1089.1088
Dual form			1089.2.d.g.1088.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.43632 q^{2} +3.93565 q^{4} +3.79576i q^{5} +0.367791i q^{7} -4.71586 q^{8} -9.24768i q^{10} -0.948263i q^{13} -0.896057i q^{14} +3.61803 q^{16} -3.43470 q^{17} +4.26229i q^{19} +14.9388i q^{20} +4.96800i q^{23} -9.40778 q^{25} +2.31027i q^{26} +1.44750i q^{28} -2.48342 q^{29} +3.51391 q^{31} +0.617031 q^{32} +8.36801 q^{34} -1.39605 q^{35} -7.18234 q^{37} -10.3843i q^{38} -17.9003i q^{40} -2.71910 q^{41} -1.88749i q^{43} -12.1036i q^{46} +0.0206077i q^{47} +6.86473 q^{49} +22.9204 q^{50} -3.73203i q^{52} -5.54524i q^{53} -1.73445i q^{56} +6.05040 q^{58} -6.62419i q^{59} +9.75250i q^{61} -8.56101 q^{62} -8.73935 q^{64} +3.59938 q^{65} +4.46351 q^{67} -13.5178 q^{68} +3.40122 q^{70} -10.3266i q^{71} +4.39587i q^{73} +17.4985 q^{74} +16.7749i q^{76} +10.9371i q^{79} +13.7332i q^{80} +6.62460 q^{82} -9.18325 q^{83} -13.0373i q^{85} +4.59852i q^{86} -3.04837i q^{89} +0.348763 q^{91} +19.5523i q^{92} -0.0502070i q^{94} -16.1786 q^{95} -15.0868 q^{97} -16.7247 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{4} + 40 q^{16} - 32 q^{25} + 16 q^{31} + 40 q^{34} + 8 q^{37} + 16 q^{49} + 32 q^{58} - 104 q^{64} + 96 q^{67} - 64 q^{70} + 88 q^{82} + 48 q^{91}+O(q^{100}) $ 16 * q + 16 * q^4 + 40 * q^16 - 32 * q^25 + 16 * q^31 + 40 * q^34 + 8 * q^37 + 16 * q^49 + 32 * q^58 - 104 * q^64 + 96 * q^67 - 64 * q^70 + 88 * q^82 + 48 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times$.

$n$	$244$	$848$
$\chi(n)$	$-1$	$-1$

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 $

$2$	−2.43632	−1.72274	−0.861369	−	0.507980i	$-0.830392\pi$
$2$	−2.43632	−1.72274	−0.861369	+	0.507980i	$0.830392\pi$
$3$	0	0
$3$	0	0
$4$	3.93565	1.96782
$5$	3.79576i	1.69751i	0.528782	+	0.848757i	$0.322649\pi$
$5$	3.79576i	1.69751i	−0.528782	+	0.848757i	$0.677351\pi$
$6$	0	0
$7$	0.367791i	0.139012i	0.997582	+	0.0695060i	$0.0221423\pi$
$7$	0.367791i	0.139012i	−0.997582	+	0.0695060i	$0.977858\pi$
$8$	−4.71586	−1.66731
$9$	0	0
$10$	− 9.24768i	− 2.92437i
$11$	0	0
$11$	0	0
$12$	0	0
$13$	− 0.948263i	− 0.263001i	−0.991316	−	0.131500i	$-0.958021\pi$
$13$	− 0.948263i	− 0.263001i	0.991316	−	0.131500i	$-0.0419795\pi$
$14$	− 0.896057i	− 0.239481i
$15$	0	0
$16$	3.61803	0.904508
$17$	−3.43470	−0.833036	−0.416518	−	0.909127i	$-0.636750\pi$
$17$	−3.43470	−0.833036	−0.416518	+	0.909127i	$0.636750\pi$
$18$	0	0
$19$	4.26229i	0.977837i	0.872329	+	0.488919i	$0.162609\pi$
$19$	4.26229i	0.977837i	−0.872329	+	0.488919i	$0.837391\pi$
$20$	14.9388i	3.34041i
$21$	0	0
$22$	0	0
$23$	4.96800i	1.03590i	0.855411	+	0.517950i	$0.173305\pi$
$23$	4.96800i	1.03590i	−0.855411	+	0.517950i	$0.826695\pi$
$24$	0	0
$25$	−9.40778	−1.88156
$26$	2.31027i	0.453081i
$27$	0	0
$28$	1.44750i	0.273551i
$29$	−2.48342	−0.461159	−0.230580	−	0.973053i	$-0.574062\pi$
$29$	−2.48342	−0.461159	−0.230580	+	0.973053i	$0.574062\pi$
$30$	0	0
$31$	3.51391	0.631117	0.315559	−	0.948906i	$-0.397808\pi$
$31$	3.51391	0.631117	0.315559	+	0.948906i	$0.397808\pi$
$32$	0.617031	0.109077
$33$	0	0
$34$	8.36801	1.43510
$35$	−1.39605	−0.235975
$36$	0	0
$37$	−7.18234	−1.18077	−0.590385	−	0.807122i	$-0.701024\pi$
$37$	−7.18234	−1.18077	−0.590385	+	0.807122i	$0.701024\pi$
$38$	− 10.3843i	− 1.68456i
$39$	0	0
$40$	− 17.9003i	− 2.83028i
$41$	−2.71910	−0.424653	−0.212326	−	0.977199i	$-0.568104\pi$
$41$	−2.71910	−0.424653	−0.212326	+	0.977199i	$0.568104\pi$
$42$	0	0
$43$	− 1.88749i	− 0.287839i	−0.989589	−	0.143919i	$-0.954029\pi$
$43$	− 1.88749i	− 0.287839i	0.989589	−	0.143919i	$-0.0459706\pi$
$44$	0	0
$45$	0	0
$46$	− 12.1036i	− 1.78458i
$47$	0.0206077i	0.00300594i	0.999999	+	0.00150297i	$0.000478411\pi$
$47$	0.0206077i	0.00300594i	−0.999999	+	0.00150297i	$0.999522\pi$
$48$	0	0
$49$	6.86473	0.980676
$50$	22.9204	3.24143
$51$	0	0
$52$	− 3.73203i	− 0.517539i
$53$	− 5.54524i	− 0.761697i	−0.924637	−	0.380849i	$-0.875632\pi$
$53$	− 5.54524i	− 0.761697i	0.924637	−	0.380849i	$-0.124368\pi$
$54$	0	0
$55$	0	0
$56$	− 1.73445i	− 0.231776i
$57$	0	0
$58$	6.05040	0.794456
$59$	− 6.62419i	− 0.862396i	−0.902257	−	0.431198i	$-0.858091\pi$
$59$	− 6.62419i	− 0.862396i	0.902257	−	0.431198i	$-0.141909\pi$
$60$	0	0
$61$	9.75250i	1.24868i	0.781153	+	0.624340i	$0.214632\pi$
$61$	9.75250i	1.24868i	−0.781153	+	0.624340i	$0.785368\pi$
$62$	−8.56101	−1.08725
$63$	0	0
$64$	−8.73935	−1.09242
$65$	3.59938	0.446448
$66$	0	0
$67$	4.46351	0.545305	0.272652	−	0.962113i	$-0.412099\pi$
$67$	4.46351	0.545305	0.272652	+	0.962113i	$0.412099\pi$
$68$	−13.5178	−1.63927
$69$	0	0
$70$	3.40122	0.406523
$71$	− 10.3266i	− 1.22555i	−0.790258	−	0.612774i	$-0.790054\pi$
$71$	− 10.3266i	− 1.22555i	0.790258	−	0.612774i	$-0.209946\pi$
$72$	0	0
$73$	4.39587i	0.514498i	0.966345	+	0.257249i	$0.0828160\pi$
$73$	4.39587i	0.514498i	−0.966345	+	0.257249i	$0.917184\pi$
$74$	17.4985	2.03416
$75$	0	0
$76$	16.7749i	1.92421i
$77$	0	0
$78$	0	0
$79$	10.9371i	1.23052i	0.788324	+	0.615261i	$0.210949\pi$
$79$	10.9371i	1.23052i	−0.788324	+	0.615261i	$0.789051\pi$
$80$	13.7332i	1.53542i
$81$	0	0
$82$	6.62460	0.731565
$83$	−9.18325	−1.00799	−0.503996	−	0.863706i	$-0.668137\pi$
$83$	−9.18325	−1.00799	−0.503996	+	0.863706i	$0.668137\pi$
$84$	0	0
$85$	− 13.0373i	− 1.41409i
$86$	4.59852i	0.495871i
$87$	0	0
$88$	0	0
$89$	− 3.04837i	− 0.323127i	−0.986862	−	0.161563i	$-0.948346\pi$
$89$	− 3.04837i	− 0.323127i	0.986862	−	0.161563i	$-0.0516536\pi$
$90$	0	0
$91$	0.348763	0.0365603
$92$	19.5523i	2.03847i
$93$	0	0
$94$	− 0.0502070i	− 0.00517845i
$95$	−16.1786	−1.65989
$96$	0	0
$97$	−15.0868	−1.53184	−0.765919	−	0.642938i	$-0.777715\pi$
$97$	−15.0868	−1.53184	−0.765919	+	0.642938i	$0.777715\pi$
$98$	−16.7247	−1.68945
$99$	0	0
$100$	−37.0257	−3.70257
$101$	17.4598	1.73732	0.868658	−	0.495413i	$-0.164983\pi$
$101$	17.4598	1.73732	0.868658	+	0.495413i	$0.164983\pi$
$102$	0	0
$103$	−9.31561	−0.917894	−0.458947	−	0.888464i	$-0.651773\pi$
$103$	−9.31561	−0.917894	−0.458947	+	0.888464i	$0.651773\pi$
$104$	4.47187i	0.438503i
$105$	0	0
$106$	13.5100i	1.31220i
$107$	−15.4255	−1.49124	−0.745620	−	0.666371i	$-0.767847\pi$
$107$	−15.4255	−1.49124	−0.745620	+	0.666371i	$0.767847\pi$
$108$	0	0
$109$	− 10.6286i	− 1.01803i	−0.860757	−	0.509016i	$-0.830009\pi$
$109$	− 10.6286i	− 1.01803i	0.860757	−	0.509016i	$-0.169991\pi$
$110$	0	0
$111$	0	0
$112$	1.33068i	0.125738i
$113$	− 1.04702i	− 0.0984953i	−0.998787	−	0.0492476i	$-0.984318\pi$
$113$	− 1.04702i	− 0.0984953i	0.998787	−	0.0492476i	$-0.0156824\pi$
$114$	0	0
$115$	−18.8573	−1.75846
$116$	−9.77386	−0.907480
$117$	0	0
$118$	16.1386i	1.48568i
$119$	− 1.26325i	− 0.115802i
$120$	0	0
$121$	0	0
$122$	− 23.7602i	− 2.15115i
$123$	0	0
$124$	13.8295	1.24193
$125$	− 16.7309i	− 1.49646i
$126$	0	0
$127$	− 17.5450i	− 1.55686i	−0.627729	−	0.778432i	$-0.716016\pi$
$127$	− 17.5450i	− 1.55686i	0.627729	−	0.778432i	$-0.283984\pi$
$128$	20.0578	1.77287
$129$	0	0
$130$	−8.76923	−0.769112
$131$	−13.8890	−1.21349	−0.606744	−	0.794897i	$-0.707525\pi$
$131$	−13.8890	−1.21349	−0.606744	+	0.794897i	$0.707525\pi$
$132$	0	0
$133$	−1.56764	−0.135931
$134$	−10.8745	−0.939417
$135$	0	0
$136$	16.1975	1.38893
$137$	1.03773i	0.0886593i	0.999017	+	0.0443296i	$0.0141152\pi$
$137$	1.03773i	0.0886593i	−0.999017	+	0.0443296i	$0.985885\pi$
$138$	0	0
$139$	19.8351i	1.68239i	0.540732	+	0.841195i	$0.318147\pi$
$139$	19.8351i	1.68239i	−0.540732	+	0.841195i	$0.681853\pi$
$140$	−5.49435	−0.464358
$141$	0	0
$142$	25.1590i	2.11130i
$143$	0	0
$144$	0	0
$145$	− 9.42646i	− 0.782825i
$146$	− 10.7097i	− 0.886345i
$147$	0	0
$148$	−28.2672	−2.32355
$149$	−8.03599	−0.658334	−0.329167	−	0.944272i	$-0.606768\pi$
$149$	−8.03599	−0.658334	−0.329167	+	0.944272i	$0.606768\pi$
$150$	0	0
$151$	− 10.0735i	− 0.819773i	−0.912137	−	0.409886i	$-0.865568\pi$
$151$	− 10.0735i	− 0.819773i	0.912137	−	0.409886i	$-0.134432\pi$
$152$	− 20.1004i	− 1.63036i
$153$	0	0
$154$	0	0
$155$	13.3380i	1.07133i
$156$	0	0
$157$	−16.1823	−1.29149	−0.645746	−	0.763552i	$-0.723453\pi$
$157$	−16.1823	−1.29149	−0.645746	+	0.763552i	$0.723453\pi$
$158$	− 26.6463i	− 2.11986i
$159$	0	0
$160$	2.34210i	0.185159i
$161$	−1.82719	−0.144003
$162$	0	0
$163$	4.27584	0.334910	0.167455	−	0.985880i	$-0.446445\pi$
$163$	4.27584	0.334910	0.167455	+	0.985880i	$0.446445\pi$
$164$	−10.7014	−0.835642
$165$	0	0
$166$	22.3733	1.73651
$167$	−18.3529	−1.42019	−0.710094	−	0.704107i	$-0.751347\pi$
$167$	−18.3529	−1.42019	−0.710094	+	0.704107i	$0.751347\pi$
$168$	0	0
$169$	12.1008	0.930831
$170$	31.7630i	2.43611i
$171$	0	0
$172$	− 7.42848i	− 0.566416i
$173$	6.18239	0.470038	0.235019	−	0.971991i	$-0.424485\pi$
$173$	6.18239	0.470038	0.235019	+	0.971991i	$0.424485\pi$
$174$	0	0
$175$	− 3.46010i	− 0.261559i
$176$	0	0
$177$	0	0
$178$	7.42680i	0.556662i
$179$	9.22567i	0.689559i	0.938684	+	0.344780i	$0.112046\pi$
$179$	9.22567i	0.689559i	−0.938684	+	0.344780i	$0.887954\pi$
$180$	0	0
$181$	−4.18767	−0.311267	−0.155634	−	0.987815i	$-0.549742\pi$
$181$	−4.18767	−0.311267	−0.155634	+	0.987815i	$0.549742\pi$
$182$	−0.849698	−0.0629838
$183$	0	0
$184$	− 23.4284i	− 1.72716i
$185$	− 27.2624i	− 2.00437i
$186$	0	0
$187$	0	0
$188$	0.0811047i	0.00591517i
$189$	0	0
$190$	39.4163	2.85956
$191$	20.1830i	1.46039i	0.683240	+	0.730194i	$0.260570\pi$
$191$	20.1830i	1.46039i	−0.683240	+	0.730194i	$0.739430\pi$
$192$	0	0
$193$	− 8.01396i	− 0.576857i	−0.957501	−	0.288429i	$-0.906867\pi$
$193$	− 8.01396i	− 0.576857i	0.957501	−	0.288429i	$-0.0931328\pi$
$194$	36.7564	2.63895
$195$	0	0
$196$	27.0172	1.92980
$197$	21.1710	1.50837	0.754187	−	0.656659i	$-0.228031\pi$
$197$	21.1710	1.50837	0.754187	+	0.656659i	$0.228031\pi$
$198$	0	0
$199$	−10.5160	−0.745457	−0.372729	−	0.927940i	$-0.621578\pi$
$199$	−10.5160	−0.745457	−0.372729	+	0.927940i	$0.621578\pi$
$200$	44.3658	3.13713
$201$	0	0
$202$	−42.5376	−2.99294
$203$	− 0.913380i	− 0.0641067i
$204$	0	0
$205$	− 10.3211i	− 0.720854i
$206$	22.6958	1.58129
$207$	0	0
$208$	− 3.43085i	− 0.237886i
$209$	0	0
$210$	0	0
$211$	− 3.69822i	− 0.254596i	−0.991864	−	0.127298i	$-0.959369\pi$
$211$	− 3.69822i	− 0.254596i	0.991864	−	0.127298i	$-0.0406305\pi$
$212$	− 21.8241i	− 1.49889i
$213$	0	0
$214$	37.5815	2.56902
$215$	7.16444	0.488611
$216$	0	0
$217$	1.29239i	0.0877329i
$218$	25.8946i	1.75380i
$219$	0	0
$220$	0	0
$221$	3.25699i	0.219089i
$222$	0	0
$223$	−17.8971	−1.19848	−0.599240	−	0.800570i	$-0.704530\pi$
$223$	−17.8971	−1.19848	−0.599240	+	0.800570i	$0.704530\pi$
$224$	0.226939i	0.0151630i
$225$	0	0
$226$	2.55087i	0.169681i
$227$	−17.3784	−1.15344	−0.576722	−	0.816940i	$-0.695668\pi$
$227$	−17.3784	−1.15344	−0.576722	+	0.816940i	$0.695668\pi$
$228$	0	0
$229$	4.27666	0.282609	0.141305	−	0.989966i	$-0.454870\pi$
$229$	4.27666	0.282609	0.141305	+	0.989966i	$0.454870\pi$
$230$	45.9425	3.02936
$231$	0	0
$232$	11.7114	0.768894
$233$	23.3828	1.53186	0.765929	−	0.642925i	$-0.222279\pi$
$233$	23.3828	1.53186	0.765929	+	0.642925i	$0.222279\pi$
$234$	0	0
$235$	−0.0782219	−0.00510263
$236$	− 26.0705i	− 1.69704i
$237$	0	0
$238$	3.07768i	0.199497i
$239$	19.9957	1.29342	0.646708	−	0.762738i	$-0.276145\pi$
$239$	19.9957	1.29342	0.646708	+	0.762738i	$0.276145\pi$
$240$	0	0
$241$	− 2.32570i	− 0.149811i	−0.997191	−	0.0749057i	$-0.976134\pi$
$241$	− 2.32570i	− 0.149811i	0.997191	−	0.0749057i	$-0.0238656\pi$
$242$	0	0
$243$	0	0
$244$	38.3824i	2.45718i
$245$	26.0569i	1.66471i
$246$	0	0
$247$	4.04178	0.257172
$248$	−16.5711	−1.05227
$249$	0	0
$250$	40.7618i	2.57800i
$251$	3.21004i	0.202616i	0.994855	+	0.101308i	$0.0323027\pi$
$251$	3.21004i	0.202616i	−0.994855	+	0.101308i	$0.967697\pi$
$252$	0	0
$253$	0	0
$254$	42.7451i	2.68207i
$255$	0	0
$256$	−31.3885	−1.96178
$257$	18.0428i	1.12548i	0.826635	+	0.562739i	$0.190252\pi$
$257$	18.0428i	1.12548i	−0.826635	+	0.562739i	$0.809748\pi$
$258$	0	0
$259$	− 2.64160i	− 0.164141i
$260$	14.1659	0.878531
$261$	0	0
$262$	33.8381	2.09052
$263$	2.05966	0.127004	0.0635019	−	0.997982i	$-0.479773\pi$
$263$	2.05966	0.127004	0.0635019	+	0.997982i	$0.479773\pi$
$264$	0	0
$265$	21.0484	1.29299
$266$	3.81926	0.234174
$267$	0	0
$268$	17.5668	1.07306
$269$	− 2.87997i	− 0.175595i	−0.996138	−	0.0877974i	$-0.972017\pi$
$269$	− 2.87997i	− 0.175595i	0.996138	−	0.0877974i	$-0.0279828\pi$
$270$	0	0
$271$	− 10.1705i	− 0.617814i	−0.951092	−	0.308907i	$-0.900037\pi$
$271$	− 10.1705i	− 0.617814i	0.951092	−	0.308907i	$-0.0999632\pi$
$272$	−12.4268	−0.753488
$273$	0	0
$274$	− 2.52824i	− 0.152737i
$275$	0	0
$276$	0	0
$277$	7.31072i	0.439259i	0.975583	+	0.219629i	$0.0704848\pi$
$277$	7.31072i	0.439259i	−0.975583	+	0.219629i	$0.929515\pi$
$278$	− 48.3246i	− 2.89832i
$279$	0	0
$280$	6.58356	0.393443
$281$	2.30543	0.137530	0.0687652	−	0.997633i	$-0.478094\pi$
$281$	2.30543	0.137530	0.0687652	+	0.997633i	$0.478094\pi$
$282$	0	0
$283$	24.4328i	1.45238i	0.687494	+	0.726190i	$0.258711\pi$
$283$	24.4328i	1.45238i	−0.687494	+	0.726190i	$0.741289\pi$
$284$	− 40.6420i	− 2.41166i
$285$	0	0
$286$	0	0
$287$	− 1.00006i	− 0.0590318i
$288$	0	0
$289$	−5.20286	−0.306051
$290$	22.9659i	1.34860i
$291$	0	0
$292$	17.3006i	1.01244i
$293$	10.6614	0.622846	0.311423	−	0.950271i	$-0.399194\pi$
$293$	10.6614	0.622846	0.311423	+	0.950271i	$0.399194\pi$
$294$	0	0
$295$	25.1438	1.46393
$296$	33.8709	1.96871
$297$	0	0
$298$	19.5782	1.13414
$299$	4.71097	0.272443
$300$	0	0
$301$	0.694201	0.0400131
$302$	24.5423i	1.41225i
$303$	0	0
$304$	15.4211i	0.884462i
$305$	−37.0181	−2.11965
$306$	0	0
$307$	− 4.56848i	− 0.260737i	−0.991466	−	0.130369i	$-0.958384\pi$
$307$	− 4.56848i	− 0.260737i	0.991466	−	0.130369i	$-0.0416160\pi$
$308$	0	0
$309$	0	0
$310$	− 32.4955i	− 1.84562i
$311$	− 9.38221i	− 0.532017i	−0.963971	−	0.266008i	$-0.914295\pi$
$311$	− 9.38221i	− 0.532017i	0.963971	−	0.266008i	$-0.0857049\pi$
$312$	0	0
$313$	−5.01601	−0.283522	−0.141761	−	0.989901i	$-0.545276\pi$
$313$	−5.01601	−0.283522	−0.141761	+	0.989901i	$0.545276\pi$
$314$	39.4253	2.22490
$315$	0	0
$316$	43.0446i	2.42145i
$317$	11.6762i	0.655801i	0.944712	+	0.327900i	$0.106341\pi$
$317$	11.6762i	0.655801i	−0.944712	+	0.327900i	$0.893659\pi$
$318$	0	0
$319$	0	0
$320$	− 33.1725i	− 1.85440i
$321$	0	0
$322$	4.45162	0.248079
$323$	− 14.6397i	− 0.814574i
$324$	0	0
$325$	8.92105i	0.494851i
$326$	−10.4173	−0.576961
$327$	0	0
$328$	12.8229	0.708026
$329$	−0.00757934	−0.000417863 0
$330$	0	0
$331$	−29.0516	−1.59682	−0.798411	−	0.602113i	$-0.794326\pi$
$331$	−29.0516	−1.59682	−0.798411	+	0.602113i	$0.794326\pi$
$332$	−36.1420	−1.98355
$333$	0	0
$334$	44.7134	2.44661
$335$	16.9424i	0.925663i
$336$	0	0
$337$	− 28.8803i	− 1.57321i	−0.617456	−	0.786605i	$-0.711837\pi$
$337$	− 28.8803i	− 1.57321i	0.617456	−	0.786605i	$-0.288163\pi$
$338$	−29.4814	−1.60358
$339$	0	0
$340$	− 51.3101i	− 2.78268i
$341$	0	0
$342$	0	0
$343$	5.09933i	0.275338i
$344$	8.90111i	0.479916i
$345$	0	0
$346$	−15.0623	−0.809752
$347$	−22.1430	−1.18870	−0.594350	−	0.804207i	$-0.702590\pi$
$347$	−22.1430	−1.18870	−0.594350	+	0.804207i	$0.702590\pi$
$348$	0	0
$349$	− 15.6625i	− 0.838393i	−0.907895	−	0.419197i	$-0.862312\pi$
$349$	− 15.6625i	− 0.838393i	0.907895	−	0.419197i	$-0.137688\pi$
$350$	8.42991i	0.450598i
$351$	0	0
$352$	0	0
$353$	− 11.1636i	− 0.594180i	−0.954850	−	0.297090i	$-0.903984\pi$
$353$	− 11.1636i	− 0.594180i	0.954850	−	0.297090i	$-0.0960161\pi$
$354$	0	0
$355$	39.1975	2.08038
$356$	− 11.9973i	− 0.635856i
$357$	0	0
$358$	− 22.4767i	− 1.18793i
$359$	16.5643	0.874233	0.437116	−	0.899405i	$-0.356000\pi$
$359$	16.5643	0.874233	0.437116	+	0.899405i	$0.356000\pi$
$360$	0	0
$361$	0.832847	0.0438340
$362$	10.2025	0.536232
$363$	0	0
$364$	1.37261	0.0719442
$365$	−16.6857	−0.873368
$366$	0	0
$367$	9.81027	0.512092	0.256046	−	0.966665i	$-0.417580\pi$
$367$	9.81027	0.512092	0.256046	+	0.966665i	$0.417580\pi$
$368$	17.9744i	0.936981i
$369$	0	0
$370$	66.4200i	3.45301i
$371$	2.03949	0.105885
$372$	0	0
$373$	12.6350i	0.654216i	0.944987	+	0.327108i	$0.106074\pi$
$373$	12.6350i	0.654216i	−0.944987	+	0.327108i	$0.893926\pi$
$374$	0	0
$375$	0	0
$376$	− 0.0971830i	− 0.00501183i
$377$	2.35493i	0.121285i
$378$	0	0
$379$	35.1020	1.80307	0.901535	−	0.432706i	$-0.142441\pi$
$379$	35.1020	1.80307	0.901535	+	0.432706i	$0.142441\pi$
$380$	−63.6735	−3.26638
$381$	0	0
$382$	− 49.1721i	− 2.51587i
$383$	22.5865i	1.15412i	0.816702	+	0.577059i	$0.195800\pi$
$383$	22.5865i	1.15412i	−0.816702	+	0.577059i	$0.804200\pi$
$384$	0	0
$385$	0	0
$386$	19.5246i	0.993774i
$387$	0	0
$388$	−59.3765	−3.01439
$389$	− 7.35884i	− 0.373108i	−0.982445	−	0.186554i	$-0.940268\pi$
$389$	− 7.35884i	− 0.373108i	0.982445	−	0.186554i	$-0.0597319\pi$
$390$	0	0
$391$	− 17.0636i	− 0.862942i
$392$	−32.3731	−1.63509
$393$	0	0
$394$	−51.5794	−2.59853
$395$	−41.5146	−2.08883
$396$	0	0
$397$	16.7327	0.839790	0.419895	−	0.907573i	$-0.362067\pi$
$397$	16.7327	0.839790	0.419895	+	0.907573i	$0.362067\pi$
$398$	25.6202	1.28423
$399$	0	0
$400$	−34.0377	−1.70188
$401$	5.07343i	0.253355i	0.991944	+	0.126677i	$0.0404313\pi$
$401$	5.07343i	0.253355i	−0.991944	+	0.126677i	$0.959569\pi$
$402$	0	0
$403$	− 3.33211i	− 0.165984i
$404$	68.7157	3.41873
$405$	0	0
$406$	2.22528i	0.110439i
$407$	0	0
$408$	0	0
$409$	32.3477i	1.59949i	0.600340	+	0.799745i	$0.295032\pi$
$409$	32.3477i	1.59949i	−0.600340	+	0.799745i	$0.704968\pi$
$410$	25.1454i	1.24184i
$411$	0	0
$412$	−36.6630	−1.80625
$413$	2.43632	0.119883
$414$	0	0
$415$	− 34.8574i	− 1.71108i
$416$	− 0.585108i	− 0.0286873i
$417$	0	0
$418$	0	0
$419$	27.9075i	1.36337i	0.731645	+	0.681686i	$0.238753\pi$
$419$	27.9075i	1.36337i	−0.731645	+	0.681686i	$0.761247\pi$
$420$	0	0
$421$	26.7286	1.30267	0.651337	−	0.758788i	$-0.274208\pi$
$421$	26.7286	1.30267	0.651337	+	0.758788i	$0.274208\pi$
$422$	9.01005i	0.438603i
$423$	0	0
$424$	26.1506i	1.26998i
$425$	32.3129	1.56740
$426$	0	0
$427$	−3.58689	−0.173582
$428$	−60.7094	−2.93450
$429$	0	0
$430$	−17.4549	−0.841748
$431$	0.716622	0.0345185	0.0172592	−	0.999851i	$-0.494506\pi$
$431$	0.716622	0.0345185	0.0172592	+	0.999851i	$0.494506\pi$
$432$	0	0
$433$	11.4324	0.549407	0.274703	−	0.961529i	$-0.411420\pi$
$433$	11.4324	0.549407	0.274703	+	0.961529i	$0.411420\pi$
$434$	− 3.14867i	− 0.151141i
$435$	0	0
$436$	− 41.8303i	− 2.00331i
$437$	−21.1751	−1.01294
$438$	0	0
$439$	− 8.57525i	− 0.409274i	−0.978838	−	0.204637i	$-0.934399\pi$
$439$	− 8.57525i	− 0.409274i	0.978838	−	0.204637i	$-0.0656015\pi$
$440$	0	0
$441$	0	0
$442$	− 7.93508i	− 0.377433i
$443$	13.9769i	0.664063i	0.943268	+	0.332031i	$0.107734\pi$
$443$	13.9769i	0.664063i	−0.943268	+	0.332031i	$0.892266\pi$
$444$	0	0
$445$	11.5709	0.548512
$446$	43.6031	2.06467
$447$	0	0
$448$	− 3.21426i	− 0.151859i
$449$	21.6079i	1.01974i	0.860252	+	0.509870i	$0.170306\pi$
$449$	21.6079i	1.01974i	−0.860252	+	0.509870i	$0.829694\pi$
$450$	0	0
$451$	0	0
$452$	− 4.12070i	− 0.193821i
$453$	0	0
$454$	42.3393	1.98708
$455$	1.32382i	0.0620616i
$456$	0	0
$457$	− 9.80506i	− 0.458662i	−0.973349	−	0.229331i	$-0.926346\pi$
$457$	− 9.80506i	− 0.458662i	0.973349	−	0.229331i	$-0.0736537\pi$
$458$	−10.4193	−0.486862
$459$	0	0
$460$	−74.2159	−3.46033
$461$	−12.3585	−0.575595	−0.287797	−	0.957691i	$-0.592923\pi$
$461$	−12.3585	−0.575595	−0.287797	+	0.957691i	$0.592923\pi$
$462$	0	0
$463$	−33.8717	−1.57415	−0.787076	−	0.616856i	$-0.788406\pi$
$463$	−33.8717	−1.57415	−0.787076	+	0.616856i	$0.788406\pi$
$464$	−8.98509	−0.417122
$465$	0	0
$466$	−56.9680	−2.63899
$467$	− 15.7369i	− 0.728219i	−0.931356	−	0.364109i	$-0.881373\pi$
$467$	− 15.7369i	− 0.728219i	0.931356	−	0.364109i	$-0.118627\pi$
$468$	0	0
$469$	1.64164i	0.0758040i
$470$	0.190573	0.00879050
$471$	0	0
$472$	31.2387i	1.43788i
$473$	0	0
$474$	0	0
$475$	− 40.0987i	− 1.83986i
$476$	− 4.97171i	− 0.227878i
$477$	0	0
$478$	−48.7159	−2.22821
$479$	−12.4691	−0.569726	−0.284863	−	0.958568i	$-0.591948\pi$
$479$	−12.4691	−0.569726	−0.284863	+	0.958568i	$0.591948\pi$
$480$	0	0
$481$	6.81075i	0.310544i
$482$	5.66614i	0.258086i
$483$	0	0
$484$	0	0
$485$	− 57.2660i	− 2.60032i
$486$	0	0
$487$	−6.95609	−0.315210	−0.157605	−	0.987502i	$-0.550377\pi$
$487$	−6.95609	−0.315210	−0.157605	+	0.987502i	$0.550377\pi$
$488$	− 45.9914i	− 2.08193i
$489$	0	0
$490$	− 63.4828i	− 2.86786i
$491$	0.301920	0.0136255	0.00681273	−	0.999977i	$-0.497831\pi$
$491$	0.301920	0.0136255	0.00681273	+	0.999977i	$0.497831\pi$
$492$	0	0
$493$	8.52979	0.384162
$494$	−9.84705	−0.443040
$495$	0	0
$496$	12.7135	0.570851
$497$	3.79805	0.170366
$498$	0	0
$499$	6.61671	0.296205	0.148102	−	0.988972i	$-0.452683\pi$
$499$	6.61671	0.296205	0.148102	+	0.988972i	$0.452683\pi$
$500$	− 65.8469i	− 2.94476i
$501$	0	0
$502$	− 7.82068i	− 0.349054i
$503$	−10.3292	−0.460554	−0.230277	−	0.973125i	$-0.573963\pi$
$503$	−10.3292	−0.460554	−0.230277	+	0.973125i	$0.573963\pi$
$504$	0	0
$505$	66.2732i	2.94912i
$506$	0	0
$507$	0	0
$508$	− 69.0508i	− 3.06363i
$509$	20.3250i	0.900891i	0.892804	+	0.450445i	$0.148735\pi$
$509$	20.3250i	0.900891i	−0.892804	+	0.450445i	$0.851265\pi$
$510$	0	0
$511$	−1.61676	−0.0715214
$512$	36.3567	1.60675
$513$	0	0
$514$	− 43.9580i	− 1.93890i
$515$	− 35.3598i	− 1.55814i
$516$	0	0
$517$	0	0
$518$	6.43579i	0.282772i
$519$	0	0
$520$	−16.9742	−0.744366
$521$	− 11.9982i	− 0.525650i	−0.964843	−	0.262825i	$-0.915346\pi$
$521$	− 11.9982i	− 0.525650i	0.964843	−	0.262825i	$-0.0846542\pi$
$522$	0	0
$523$	37.8361i	1.65446i	0.561865	+	0.827229i	$0.310084\pi$
$523$	37.8361i	1.65446i	−0.561865	+	0.827229i	$0.689916\pi$
$524$	−54.6623	−2.38793
$525$	0	0
$526$	−5.01798	−0.218794
$527$	−12.0692	−0.525743
$528$	0	0
$529$	−1.68106	−0.0730898
$530$	−51.2806	−2.22749
$531$	0	0
$532$	−6.16966	−0.267489
$533$	2.57842i	0.111684i
$534$	0	0
$535$	− 58.5515i	− 2.53140i
$536$	−21.0493	−0.909191
$537$	0	0
$538$	7.01652i	0.302504i
$539$	0	0
$540$	0	0
$541$	36.8317i	1.58352i	0.610834	+	0.791759i	$0.290834\pi$
$541$	36.8317i	1.58352i	−0.610834	+	0.791759i	$0.709166\pi$
$542$	24.7786i	1.06433i
$543$	0	0
$544$	−2.11931	−0.0908648
$545$	40.3435	1.72812
$546$	0	0
$547$	36.6744i	1.56809i	0.620706	+	0.784043i	$0.286846\pi$
$547$	36.6744i	1.56809i	−0.620706	+	0.784043i	$0.713154\pi$
$548$	4.08414i	0.174466i
$549$	0	0
$550$	0	0
$551$	− 10.5851i	− 0.450939i
$552$	0	0
$553$	−4.02258	−0.171057
$554$	− 17.8113i	− 0.756728i
$555$	0	0
$556$	78.0639i	3.31065i
$557$	−7.46772	−0.316418	−0.158209	−	0.987406i	$-0.550572\pi$
$557$	−7.46772	−0.316418	−0.158209	+	0.987406i	$0.550572\pi$
$558$	0	0
$559$	−1.78983	−0.0757018
$560$	−5.05095	−0.213441
$561$	0	0
$562$	−5.61676	−0.236929
$563$	−3.91481	−0.164990	−0.0824948	−	0.996591i	$-0.526289\pi$
$563$	−3.91481	−0.164990	−0.0824948	+	0.996591i	$0.526289\pi$
$564$	0	0
$565$	3.97423	0.167197
$566$	− 59.5261i	− 2.50207i
$567$	0	0
$568$	48.6990i	2.04336i
$569$	37.6712	1.57926	0.789628	−	0.613585i	$-0.210273\pi$
$569$	37.6712	1.57926	0.789628	+	0.613585i	$0.210273\pi$
$570$	0	0
$571$	30.3411i	1.26974i	0.772621	+	0.634868i	$0.218946\pi$
$571$	30.3411i	1.26974i	−0.772621	+	0.634868i	$0.781054\pi$
$572$	0	0
$573$	0	0
$574$	2.43647i	0.101696i
$575$	− 46.7379i	− 1.94911i
$576$	0	0
$577$	−3.31962	−0.138198	−0.0690988	−	0.997610i	$-0.522012\pi$
$577$	−3.31962	−0.138198	−0.0690988	+	0.997610i	$0.522012\pi$
$578$	12.6758	0.527245
$579$	0	0
$580$	− 37.0992i	− 1.54046i
$581$	− 3.37752i	− 0.140123i
$582$	0	0
$583$	0	0
$584$	− 20.7303i	− 0.857826i
$585$	0	0
$586$	−25.9746	−1.07300
$587$	40.9075i	1.68843i	0.536003	+	0.844216i	$0.319933\pi$
$587$	40.9075i	1.68843i	−0.536003	+	0.844216i	$0.680067\pi$
$588$	0	0
$589$	14.9773i	0.617130i
$590$	−61.2583	−2.52197
$591$	0	0
$592$	−25.9860	−1.06802
$593$	4.62924	0.190100	0.0950500	−	0.995472i	$-0.469699\pi$
$593$	4.62924	0.190100	0.0950500	+	0.995472i	$0.469699\pi$
$594$	0	0
$595$	4.79500	0.196576
$596$	−31.6268	−1.29549
$597$	0	0
$598$	−11.4774	−0.469347
$599$	30.5789i	1.24942i	0.780858	+	0.624709i	$0.214782\pi$
$599$	30.5789i	1.24942i	−0.780858	+	0.624709i	$0.785218\pi$
$600$	0	0
$601$	31.2300i	1.27390i	0.770905	+	0.636950i	$0.219804\pi$
$601$	31.2300i	1.27390i	−0.770905	+	0.636950i	$0.780196\pi$
$602$	−1.69129	−0.0689320
$603$	0	0
$604$	− 39.6459i	− 1.61317i
$605$	0	0
$606$	0	0
$607$	− 30.7667i	− 1.24878i	−0.781112	−	0.624391i	$-0.785347\pi$
$607$	− 30.7667i	− 1.24878i	0.781112	−	0.624391i	$-0.214653\pi$
$608$	2.62997i	0.106659i
$609$	0	0
$610$	90.1880	3.65160
$611$	0.0195415	0.000790566 0
$612$	0	0
$613$	25.4710i	1.02876i	0.857561	+	0.514382i	$0.171979\pi$
$613$	25.4710i	1.02876i	−0.857561	+	0.514382i	$0.828021\pi$
$614$	11.1303i	0.449181i
$615$	0	0
$616$	0	0
$617$	4.20930i	0.169460i	0.996404	+	0.0847300i	$0.0270028\pi$
$617$	4.20930i	0.169460i	−0.996404	+	0.0847300i	$0.972997\pi$
$618$	0	0
$619$	38.5032	1.54757	0.773787	−	0.633446i	$-0.218360\pi$
$619$	38.5032	1.54757	0.773787	+	0.633446i	$0.218360\pi$
$620$	52.4935i	2.10819i
$621$	0	0
$622$	22.8581i	0.916525i
$623$	1.12116	0.0449185
$624$	0	0
$625$	16.4675	0.658700
$626$	12.2206	0.488433
$627$	0	0
$628$	−63.6880	−2.54143
$629$	24.6692	0.983624
$630$	0	0
$631$	−11.8328	−0.471058	−0.235529	−	0.971867i	$-0.575682\pi$
$631$	−11.8328	−0.471058	−0.235529	+	0.971867i	$0.575682\pi$
$632$	− 51.5779i	− 2.05166i
$633$	0	0
$634$	− 28.4469i	− 1.12977i
$635$	66.5964	2.64280
$636$	0	0
$637$	− 6.50957i	− 0.257918i
$638$	0	0
$639$	0	0
$640$	76.1345i	3.00948i
$641$	35.9401i	1.41955i	0.704430	+	0.709774i	$0.251203\pi$
$641$	35.9401i	1.41955i	−0.704430	+	0.709774i	$0.748797\pi$
$642$	0	0
$643$	3.25527	0.128375	0.0641876	−	0.997938i	$-0.479554\pi$
$643$	3.25527	0.128375	0.0641876	+	0.997938i	$0.479554\pi$
$644$	−7.19118	−0.283372
$645$	0	0
$646$	35.6669i	1.40330i
$647$	12.1550i	0.477863i	0.971036	+	0.238931i	$0.0767971\pi$
$647$	12.1550i	0.477863i	−0.971036	+	0.238931i	$0.923203\pi$
$648$	0	0
$649$	0	0
$650$	− 21.7345i	− 0.852498i
$651$	0	0
$652$	16.8282	0.659043
$653$	25.5204i	0.998690i	0.866403	+	0.499345i	$0.166426\pi$
$653$	25.5204i	0.998690i	−0.866403	+	0.499345i	$0.833574\pi$
$654$	0	0
$655$	− 52.7193i	− 2.05992i
$656$	−9.83781	−0.384102
$657$	0	0
$658$	0.0184657	0.000719867 0
$659$	27.9441	1.08855	0.544273	−	0.838908i	$-0.316806\pi$
$659$	27.9441	1.08855	0.544273	+	0.838908i	$0.316806\pi$
$660$	0	0
$661$	−27.3798	−1.06495	−0.532475	−	0.846446i	$-0.678738\pi$
$661$	−27.3798	−1.06495	−0.532475	+	0.846446i	$0.678738\pi$
$662$	70.7790	2.75091
$663$	0	0
$664$	43.3069	1.68063
$665$	− 5.95037i	− 0.230745i
$666$	0	0
$667$	− 12.3376i	− 0.477715i
$668$	−72.2304	−2.79468
$669$	0	0
$670$	− 41.2771i	− 1.59467i
$671$	0	0
$672$	0	0
$673$	− 13.5488i	− 0.522267i	−0.965303	−	0.261134i	$-0.915904\pi$
$673$	− 13.5488i	− 0.522267i	0.965303	−	0.261134i	$-0.0840963\pi$
$674$	70.3616i	2.71023i
$675$	0	0
$676$	47.6245	1.83171
$677$	36.6970	1.41038	0.705190	−	0.709018i	$-0.250862\pi$
$677$	36.6970	1.41038	0.705190	+	0.709018i	$0.250862\pi$
$678$	0	0
$679$	− 5.54881i	− 0.212944i
$680$	61.4819i	2.35772i
$681$	0	0
$682$	0	0
$683$	30.8347i	1.17986i	0.807455	+	0.589929i	$0.200844\pi$
$683$	30.8347i	1.17986i	−0.807455	+	0.589929i	$0.799156\pi$
$684$	0	0
$685$	−3.93897	−0.150500
$686$	− 12.4236i	− 0.474335i
$687$	0	0
$688$	− 6.82899i	− 0.260353i
$689$	−5.25835	−0.200327
$690$	0	0
$691$	−47.7199	−1.81535	−0.907676	−	0.419672i	$-0.862145\pi$
$691$	−47.7199	−1.81535	−0.907676	+	0.419672i	$0.862145\pi$
$692$	24.3317	0.924953
$693$	0	0
$694$	53.9474	2.04782
$695$	−75.2892	−2.85588
$696$	0	0
$697$	9.33929	0.353751
$698$	38.1588i	1.44433i
$699$	0	0
$700$	− 13.6177i	− 0.514703i
$701$	−14.0665	−0.531284	−0.265642	−	0.964072i	$-0.585584\pi$
$701$	−14.0665	−0.531284	−0.265642	+	0.964072i	$0.585584\pi$
$702$	0	0
$703$	− 30.6133i	− 1.15460i
$704$	0	0
$705$	0	0
$706$	27.1981i	1.02362i
$707$	6.42157i	0.241508i
$708$	0	0
$709$	−9.07416	−0.340787	−0.170394	−	0.985376i	$-0.554504\pi$
$709$	−9.07416	−0.340787	−0.170394	+	0.985376i	$0.554504\pi$
$710$	−95.4975	−3.58396
$711$	0	0
$712$	14.3757i	0.538751i
$713$	17.4571i	0.653775i
$714$	0	0
$715$	0	0
$716$	36.3090i	1.35693i
$717$	0	0
$718$	−40.3560	−1.50607
$719$	33.9415i	1.26580i	0.774232	+	0.632901i	$0.218136\pi$
$719$	33.9415i	1.26580i	−0.774232	+	0.632901i	$0.781864\pi$
$720$	0	0
$721$	− 3.42620i	− 0.127598i
$722$	−2.02908	−0.0755146
$723$	0	0
$724$	−16.4812	−0.612519
$725$	23.3635	0.867697
$726$	0	0
$727$	30.8992	1.14599	0.572993	−	0.819560i	$-0.305782\pi$
$727$	30.8992	1.14599	0.572993	+	0.819560i	$0.305782\pi$
$728$	−1.64472	−0.0609572
$729$	0	0
$730$	40.6516	1.50458
$731$	6.48294i	0.239780i
$732$	0	0
$733$	41.6009i	1.53657i	0.640110	+	0.768283i	$0.278889\pi$
$733$	41.6009i	1.53657i	−0.640110	+	0.768283i	$0.721111\pi$
$734$	−23.9009	−0.882200
$735$	0	0
$736$	3.06541i	0.112993i
$737$	0	0
$738$	0	0
$739$	− 14.3707i	− 0.528634i	−0.964436	−	0.264317i	$-0.914854\pi$
$739$	− 14.3707i	− 0.528634i	0.964436	−	0.264317i	$-0.0851465\pi$
$740$	− 107.295i	− 3.94426i
$741$	0	0
$742$	−4.96885	−0.182412
$743$	−13.4298	−0.492692	−0.246346	−	0.969182i	$-0.579230\pi$
$743$	−13.4298	−0.492692	−0.246346	+	0.969182i	$0.579230\pi$
$744$	0	0
$745$	− 30.5027i	− 1.11753i
$746$	− 30.7829i	− 1.12704i
$747$	0	0
$748$	0	0
$749$	− 5.67337i	− 0.207301i
$750$	0	0
$751$	−36.3646	−1.32696	−0.663481	−	0.748193i	$-0.730922\pi$
$751$	−36.3646	−1.32696	−0.663481	+	0.748193i	$0.730922\pi$
$752$	0.0745594i	0.00271890i
$753$	0	0
$754$	− 5.73737i	− 0.208943i
$755$	38.2367	1.39158
$756$	0	0
$757$	39.9016	1.45025	0.725124	−	0.688618i	$-0.241782\pi$
$757$	39.9016	1.45025	0.725124	+	0.688618i	$0.241782\pi$
$758$	−85.5197	−3.10622
$759$	0	0
$760$	76.2962	2.76755
$761$	−3.41015	−0.123618	−0.0618089	−	0.998088i	$-0.519687\pi$
$761$	−3.41015	−0.123618	−0.0618089	+	0.998088i	$0.519687\pi$
$762$	0	0
$763$	3.90910	0.141519
$764$	79.4331i	2.87379i
$765$	0	0
$766$	− 55.0280i	− 1.98824i
$767$	−6.28147	−0.226811
$768$	0	0
$769$	− 21.4154i	− 0.772260i	−0.922444	−	0.386130i	$-0.873812\pi$
$769$	− 21.4154i	− 0.772260i	0.922444	−	0.386130i	$-0.126188\pi$
$770$	0	0
$771$	0	0
$772$	− 31.5401i	− 1.13515i
$773$	− 35.8866i	− 1.29075i	−0.763866	−	0.645375i	$-0.776701\pi$
$773$	− 35.8866i	− 1.29075i	0.763866	−	0.645375i	$-0.223299\pi$
$774$	0	0
$775$	−33.0581	−1.18748
$776$	71.1474	2.55404
$777$	0	0
$778$	17.9285i	0.642767i
$779$	− 11.5896i	− 0.415241i
$780$	0	0
$781$	0	0
$782$	41.5723i	1.48662i
$783$	0	0
$784$	24.8368	0.887029
$785$	− 61.4243i	− 2.19233i
$786$	0	0
$787$	48.0635i	1.71328i	0.515917	+	0.856639i	$0.327451\pi$
$787$	48.0635i	1.71328i	−0.515917	+	0.856639i	$0.672549\pi$
$788$	83.3218	2.96822
$789$	0	0
$790$	101.143	3.59850
$791$	0.385085	0.0136920
$792$	0	0
$793$	9.24793	0.328404
$794$	−40.7662	−1.44674
$795$	0	0
$796$	−41.3872	−1.46693
$797$	− 6.04580i	− 0.214153i	−0.994251	−	0.107077i	$-0.965851\pi$
$797$	− 6.04580i	− 0.214153i	0.994251	−	0.107077i	$-0.0341490\pi$
$798$	0	0
$799$	− 0.0707812i	− 0.00250406i
$800$	−5.80489	−0.205234
$801$	0	0
$802$	− 12.3605i	− 0.436464i
$803$	0	0
$804$	0	0
$805$	− 6.93557i	− 0.244447i
$806$	8.11809i	0.285947i
$807$	0	0
$808$	−82.3379	−2.89664
$809$	7.43425	0.261374	0.130687	−	0.991424i	$-0.458282\pi$
$809$	7.43425	0.261374	0.130687	+	0.991424i	$0.458282\pi$
$810$	0	0
$811$	29.1631i	1.02405i	0.858970	+	0.512027i	$0.171105\pi$
$811$	29.1631i	1.02405i	−0.858970	+	0.512027i	$0.828895\pi$
$812$	− 3.59474i	− 0.126151i
$813$	0	0
$814$	0	0
$815$	16.2301i	0.568514i
$816$	0	0
$817$	8.04502	0.281460
$818$	− 78.8093i	− 2.75550i
$819$	0	0
$820$	− 40.6201i	− 1.41851i
$821$	−1.07240	−0.0374270	−0.0187135	−	0.999825i	$-0.505957\pi$
$821$	−1.07240	−0.0374270	−0.0187135	+	0.999825i	$0.505957\pi$
$822$	0	0
$823$	−8.94619	−0.311845	−0.155922	−	0.987769i	$-0.549835\pi$
$823$	−8.94619	−0.311845	−0.155922	+	0.987769i	$0.549835\pi$
$824$	43.9311	1.53041
$825$	0	0
$826$	−5.93565	−0.206528
$827$	35.3031	1.22761	0.613805	−	0.789458i	$-0.289638\pi$
$827$	35.3031	1.22761	0.613805	+	0.789458i	$0.289638\pi$
$828$	0	0
$829$	27.0734	0.940298	0.470149	−	0.882587i	$-0.344200\pi$
$829$	27.0734	0.940298	0.470149	+	0.882587i	$0.344200\pi$
$830$	84.9237i	2.94774i
$831$	0	0
$832$	8.28720i	0.287307i
$833$	−23.5783	−0.816938
$834$	0	0
$835$	− 69.6630i	− 2.41079i
$836$	0	0
$837$	0	0
$838$	− 67.9916i	− 2.34873i
$839$	− 37.2329i	− 1.28542i	−0.766108	−	0.642712i	$-0.777809\pi$
$839$	− 37.2329i	− 1.28542i	0.766108	−	0.642712i	$-0.222191\pi$
$840$	0	0
$841$	−22.8326	−0.787332
$842$	−65.1195	−2.24417
$843$	0	0
$844$	− 14.5549i	− 0.501001i
$845$	45.9317i	1.58010i
$846$	0	0
$847$	0	0
$848$	− 20.0629i	− 0.688962i
$849$	0	0
$850$	−78.7245	−2.70023
$851$	− 35.6819i	− 1.22316i
$852$	0	0
$853$	− 35.1190i	− 1.20245i	−0.799080	−	0.601225i	$-0.794680\pi$
$853$	− 35.1190i	− 1.20245i	0.799080	−	0.601225i	$-0.205320\pi$
$854$	8.73880	0.299035
$855$	0	0
$856$	72.7445	2.48636
$857$	29.9637	1.02354	0.511770	−	0.859122i	$-0.328990\pi$
$857$	29.9637	1.02354	0.511770	+	0.859122i	$0.328990\pi$
$858$	0	0
$859$	17.2684	0.589189	0.294595	−	0.955622i	$-0.404815\pi$
$859$	17.2684	0.589189	0.294595	+	0.955622i	$0.404815\pi$
$860$	28.1967	0.961500
$861$	0	0
$862$	−1.74592	−0.0594663
$863$	− 34.8150i	− 1.18512i	−0.805528	−	0.592558i	$-0.798118\pi$
$863$	− 34.8150i	− 1.18512i	0.805528	−	0.592558i	$-0.201882\pi$
$864$	0	0
$865$	23.4668i	0.797897i
$866$	−27.8530	−0.946484
$867$	0	0
$868$	5.08638i	0.172643i
$869$	0	0
$870$	0	0
$871$	− 4.23258i	− 0.143416i
$872$	50.1228i	1.69737i
$873$	0	0
$874$	51.5893	1.74503
$875$	6.15348	0.208025
$876$	0	0
$877$	− 1.98289i	− 0.0669576i	−0.999439	−	0.0334788i	$-0.989341\pi$
$877$	− 1.98289i	− 0.0669576i	0.999439	−	0.0334788i	$-0.0106586\pi$
$878$	20.8920i	0.705072i
$879$	0	0
$880$	0	0
$881$	− 48.9571i	− 1.64941i	−0.565566	−	0.824703i	$-0.691342\pi$
$881$	− 48.9571i	− 1.64941i	0.565566	−	0.824703i	$-0.308658\pi$
$882$	0	0
$883$	34.7553	1.16961	0.584804	−	0.811175i	$-0.301171\pi$
$883$	34.7553	1.16961	0.584804	+	0.811175i	$0.301171\pi$
$884$	12.8184i	0.431129i
$885$	0	0
$886$	− 34.0522i	− 1.14401i
$887$	14.4931	0.486629	0.243315	−	0.969947i	$-0.421765\pi$
$887$	14.4931	0.486629	0.243315	+	0.969947i	$0.421765\pi$
$888$	0	0
$889$	6.45289	0.216423
$890$	−28.1903	−0.944943
$891$	0	0
$892$	−70.4368	−2.35840
$893$	−0.0878361	−0.00293932
$894$	0	0
$895$	−35.0184	−1.17054
$896$	7.37708i	0.246451i
$897$	0	0
$898$	− 52.6437i	− 1.75674i
$899$	−8.72651	−0.291045
$900$	0	0
$901$	19.0462i	0.634521i
$902$	0	0
$903$	0	0
$904$	4.93759i	0.164222i
$905$	− 15.8954i	− 0.528381i
$906$	0	0
$907$	22.7691	0.756037	0.378019	−	0.925798i	$-0.376606\pi$
$907$	22.7691	0.756037	0.378019	+	0.925798i	$0.376606\pi$
$908$	−68.3952	−2.26978
$909$	0	0
$910$	− 3.22525i	− 0.106916i
$911$	− 22.6944i	− 0.751900i	−0.926640	−	0.375950i	$-0.877316\pi$
$911$	− 22.6944i	− 0.751900i	0.926640	−	0.375950i	$-0.122684\pi$
$912$	0	0
$913$	0	0
$914$	23.8883i	0.790153i
$915$	0	0
$916$	16.8314	0.556126
$917$	− 5.10826i	− 0.168690i
$918$	0	0
$919$	1.25513i	0.0414029i	0.999786	+	0.0207014i	$0.00658994\pi$
$919$	1.25513i	0.0414029i	−0.999786	+	0.0207014i	$0.993410\pi$
$920$	88.9286	2.93189
$921$	0	0
$922$	30.1093	0.991598
$923$	−9.79237	−0.322320
$924$	0	0
$925$	67.5699	2.22169
$926$	82.5223	2.71185
$927$	0	0
$928$	−1.53235	−0.0503017
$929$	2.87512i	0.0943296i	0.998887	+	0.0471648i	$0.0150186\pi$
$929$	2.87512i	0.0943296i	−0.998887	+	0.0471648i	$0.984981\pi$
$930$	0	0
$931$	29.2595i	0.958941i
$932$	92.0265	3.01443
$933$	0	0
$934$	38.3402i	1.25453i
$935$	0	0
$936$	0	0
$937$	30.8451i	1.00767i	0.863801	+	0.503833i	$0.168077\pi$
$937$	30.8451i	1.00767i	−0.863801	+	0.503833i	$0.831923\pi$
$938$	− 3.99956i	− 0.130590i
$939$	0	0
$940$	−0.307854	−0.0100411
$941$	−6.54086	−0.213226	−0.106613	−	0.994301i	$-0.534001\pi$
$941$	−6.54086	−0.213226	−0.106613	+	0.994301i	$0.534001\pi$
$942$	0	0
$943$	− 13.5085i	− 0.439898i
$944$	− 23.9665i	− 0.780044i
$945$	0	0
$946$	0	0
$947$	33.2579i	1.08074i	0.841429	+	0.540368i	$0.181715\pi$
$947$	33.2579i	1.08074i	−0.841429	+	0.540368i	$0.818285\pi$
$948$	0	0
$949$	4.16844	0.135313
$950$	97.6933i	3.16959i
$951$	0	0
$952$	5.95731i	0.193078i
$953$	−54.9548	−1.78016	−0.890081	−	0.455803i	$-0.849352\pi$
$953$	−54.9548	−1.78016	−0.890081	+	0.455803i	$0.849352\pi$
$954$	0	0
$955$	−76.6097	−2.47903
$956$	78.6961	2.54521
$957$	0	0
$958$	30.3786	0.981488
$959$	−0.381668	−0.0123247
$960$	0	0
$961$	−18.6524	−0.601691
$962$	− 16.5932i	− 0.534985i
$963$	0	0
$964$	− 9.15313i	− 0.294803i
$965$	30.4190	0.979224
$966$	0	0
$967$	− 34.0395i	− 1.09464i	−0.836925	−	0.547318i	$-0.815649\pi$
$967$	− 34.0395i	− 1.09464i	0.836925	−	0.547318i	$-0.184351\pi$
$968$	0	0
$969$	0	0
$970$	139.518i	4.47966i
$971$	− 38.0638i	− 1.22153i	−0.791814	−	0.610763i	$-0.790863\pi$
$971$	− 38.0638i	− 1.22153i	0.791814	−	0.610763i	$-0.209137\pi$
$972$	0	0
$973$	−7.29517	−0.233872
$974$	16.9472	0.543025
$975$	0	0
$976$	35.2849i	1.12944i
$977$	32.1005i	1.02699i	0.858094	+	0.513493i	$0.171649\pi$
$977$	32.1005i	1.02699i	−0.858094	+	0.513493i	$0.828351\pi$
$978$	0	0
$979$	0	0
$980$	102.551i	3.27586i
$981$	0	0
$982$	−0.735573	−0.0234731
$983$	21.4305i	0.683526i	0.939786	+	0.341763i	$0.111024\pi$
$983$	21.4305i	0.683526i	−0.939786	+	0.341763i	$0.888976\pi$
$984$	0	0
$985$	80.3602i	2.56049i
$986$	−20.7813	−0.661811
$987$	0	0
$988$	15.9070	0.506069
$989$	9.37704	0.298172
$990$	0	0
$991$	−29.3068	−0.930962	−0.465481	−	0.885058i	$-0.654119\pi$
$991$	−29.3068	−0.930962	−0.465481	+	0.885058i	$0.654119\pi$
$992$	2.16819	0.0688402
$993$	0	0
$994$	−9.25326	−0.293496
$995$	− 39.9161i	− 1.26542i
$996$	0	0
$997$	32.7622i	1.03759i	0.854899	+	0.518795i	$0.173619\pi$
$997$	32.7622i	1.03759i	−0.854899	+	0.518795i	$0.826381\pi$
$998$	−16.1204	−0.510283
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.2.d.g.1088.2		16
3.2	odd	2	inner	1089.2.d.g.1088.15		16
11.6	odd	10		99.2.j.a.8.1	✓	16
11.9	even	5		99.2.j.a.62.4	yes	16
11.10	odd	2	inner	1089.2.d.g.1088.16		16
33.17	even	10		99.2.j.a.8.4	yes	16
33.20	odd	10		99.2.j.a.62.1	yes	16
33.32	even	2	inner	1089.2.d.g.1088.1		16
44.31	odd	10		1584.2.cd.c.161.1		16
44.39	even	10		1584.2.cd.c.305.4		16
99.20	odd	30		891.2.u.c.458.1		32
99.31	even	15		891.2.u.c.755.1		32
99.50	even	30		891.2.u.c.107.4		32
99.61	odd	30		891.2.u.c.701.4		32
99.83	even	30		891.2.u.c.701.1		32
99.86	odd	30		891.2.u.c.755.4		32
99.94	odd	30		891.2.u.c.107.1		32
99.97	even	15		891.2.u.c.458.4		32
132.83	odd	10		1584.2.cd.c.305.1		16
132.119	even	10		1584.2.cd.c.161.4		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.2.j.a.8.1	✓	16	11.6	odd	10
99.2.j.a.8.4	yes	16	33.17	even	10
99.2.j.a.62.1	yes	16	33.20	odd	10
99.2.j.a.62.4	yes	16	11.9	even	5
891.2.u.c.107.1		32	99.94	odd	30
891.2.u.c.107.4		32	99.50	even	30
891.2.u.c.458.1		32	99.20	odd	30
891.2.u.c.458.4		32	99.97	even	15
891.2.u.c.701.1		32	99.83	even	30
891.2.u.c.701.4		32	99.61	odd	30
891.2.u.c.755.1		32	99.31	even	15
891.2.u.c.755.4		32	99.86	odd	30
1089.2.d.g.1088.1		16	33.32	even	2	inner
1089.2.d.g.1088.2		16	1.1	even	1	trivial
1089.2.d.g.1088.15		16	3.2	odd	2	inner
1089.2.d.g.1088.16		16	11.10	odd	2	inner
1584.2.cd.c.161.1		16	44.31	odd	10
1584.2.cd.c.161.4		16	132.119	even	10
1584.2.cd.c.305.1		16	132.83	odd	10
1584.2.cd.c.305.4		16	44.39	even	10

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:	\( N \)	\(=\)	\( 1089 = 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1089.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.43632 q^{2} +3.93565 q^{4} +3.79576i q^{5} +0.367791i q^{7} -4.71586 q^{8} -9.24768i q^{10} -0.948263i q^{13} -0.896057i q^{14} +3.61803 q^{16} -3.43470 q^{17} +4.26229i q^{19} +14.9388i q^{20} +4.96800i q^{23} -9.40778 q^{25} +2.31027i q^{26} +1.44750i q^{28} -2.48342 q^{29} +3.51391 q^{31} +0.617031 q^{32} +8.36801 q^{34} -1.39605 q^{35} -7.18234 q^{37} -10.3843i q^{38} -17.9003i q^{40} -2.71910 q^{41} -1.88749i q^{43} -12.1036i q^{46} +0.0206077i q^{47} +6.86473 q^{49} +22.9204 q^{50} -3.73203i q^{52} -5.54524i q^{53} -1.73445i q^{56} +6.05040 q^{58} -6.62419i q^{59} +9.75250i q^{61} -8.56101 q^{62} -8.73935 q^{64} +3.59938 q^{65} +4.46351 q^{67} -13.5178 q^{68} +3.40122 q^{70} -10.3266i q^{71} +4.39587i q^{73} +17.4985 q^{74} +16.7749i q^{76} +10.9371i q^{79} +13.7332i q^{80} +6.62460 q^{82} -9.18325 q^{83} -13.0373i q^{85} +4.59852i q^{86} -3.04837i q^{89} +0.348763 q^{91} +19.5523i q^{92} -0.0502070i q^{94} -16.1786 q^{95} -15.0868 q^{97} -16.7247 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{4} + 40 q^{16} - 32 q^{25} + 16 q^{31} + 40 q^{34} + 8 q^{37} + 16 q^{49} + 32 q^{58} - 104 q^{64} + 96 q^{67} - 64 q^{70} + 88 q^{82} + 48 q^{91}+O(q^{100}) \) 16 * q + 16 * q^4 + 40 * q^16 - 32 * q^25 + 16 * q^31 + 40 * q^34 + 8 * q^37 + 16 * q^49 + 32 * q^58 - 104 * q^64 + 96 * q^67 - 64 * q^70 + 88 * q^82 + 48 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more