LMFDB - Embedded newform 1089.3.c.k.604.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1089,3,Mod(604,1089)]

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([0, 1]))

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

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

chi := DirichletCharacter("1089.604");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1089 = 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1089.c (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:	$29.6731007888$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.523388583936.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 28x^{6} + 262x^{4} + 948x^{2} + 1089 $ x^8 + 28x^6 + 262x^4 + 948*x^2 + 1089
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2\cdot 11 $
Twist minimal:	no (minimal twist has level 121)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			604.8
Root			$3.57626i$ of defining polynomial
Character	$\chi$	$=$	1089.604
Dual form			1089.3.c.k.604.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.57626i q^{2} -8.78965 q^{4} +3.44471 q^{5} -3.88348i q^{7} -17.1290i q^{8} +O(q^{10})$	$q+3.57626i q^{2} -8.78965 q^{4} +3.44471 q^{5} -3.88348i q^{7} -17.1290i q^{8} +12.3192i q^{10} +9.80240i q^{13} +13.8883 q^{14} +26.0993 q^{16} +20.9255i q^{17} -26.0425i q^{19} -30.2778 q^{20} -26.4380 q^{23} -13.1340 q^{25} -35.0559 q^{26} +34.1344i q^{28} +27.3115i q^{29} -34.3712 q^{31} +24.8219i q^{32} -74.8349 q^{34} -13.3774i q^{35} -20.9105 q^{37} +93.1349 q^{38} -59.0045i q^{40} -13.1594i q^{41} -38.9159i q^{43} -94.5492i q^{46} -30.9376 q^{47} +33.9186 q^{49} -46.9706i q^{50} -86.1596i q^{52} -31.3996 q^{53} -66.5202 q^{56} -97.6732 q^{58} +9.12891 q^{59} -21.5050i q^{61} -122.921i q^{62} +15.6278 q^{64} +33.7664i q^{65} -78.8374 q^{67} -183.927i q^{68} +47.8412 q^{70} -109.362 q^{71} -33.8711i q^{73} -74.7815i q^{74} +228.905i q^{76} +40.8466i q^{79} +89.9045 q^{80} +47.0614 q^{82} -119.555i q^{83} +72.0820i q^{85} +139.174 q^{86} -22.2767 q^{89} +38.0674 q^{91} +232.381 q^{92} -110.641i q^{94} -89.7089i q^{95} +134.669 q^{97} +121.302i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 24 q^{4} + 4 q^{5}+O(q^{10}) $ 8 * q - 24 * q^4 + 4 * q^5	$ 8 q - 24 q^{4} + 4 q^{5} + 4 q^{14} - 24 q^{16} - 52 q^{20} + 12 q^{23} - 16 q^{25} - 168 q^{26} - 116 q^{31} - 180 q^{34} - 4 q^{37} + 132 q^{38} + 244 q^{47} + 88 q^{49} - 268 q^{53} + 12 q^{56} + 88 q^{58} + 56 q^{59} - 40 q^{64} + 284 q^{67} - 188 q^{70} - 272 q^{71} + 356 q^{80} - 180 q^{82} - 336 q^{86} + 24 q^{89} + 140 q^{91} + 156 q^{92} + 152 q^{97}+O(q^{100}) $ 8 * q - 24 * q^4 + 4 * q^5 + 4 * q^14 - 24 * q^16 - 52 * q^20 + 12 * q^23 - 16 * q^25 - 168 * q^26 - 116 * q^31 - 180 * q^34 - 4 * q^37 + 132 * q^38 + 244 * q^47 + 88 * q^49 - 268 * q^53 + 12 * q^56 + 88 * q^58 + 56 * q^59 - 40 * q^64 + 284 * q^67 - 188 * q^70 - 272 * q^71 + 356 * q^80 - 180 * q^82 - 336 * q^86 + 24 * q^89 + 140 * q^91 + 156 * q^92 + 152 * q^97

Display 100 coefficients 10 coefficients

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 $

$2$	3.57626i	1.78813i	0.447936	+	0.894065i	$0.352159\pi$
$2$	3.57626i	1.78813i	−0.447936	+	0.894065i	$0.647841\pi$
$3$	0	0
$3$	0	0
$4$	−8.78965	−2.19741
$5$	3.44471	0.688941	0.344471	−	0.938797i	$-0.388058\pi$
$5$	3.44471	0.688941	0.344471	+	0.938797i	$0.388058\pi$
$6$	0	0
$7$	− 3.88348i	− 0.554783i	−0.960757	−	0.277391i	$-0.910530\pi$
$7$	− 3.88348i	− 0.554783i	0.960757	−	0.277391i	$-0.0894698\pi$
$8$	− 17.1290i	− 2.14113i
$9$	0	0
$10$	12.3192i	1.23192i
$11$	0	0
$11$	0	0
$12$	0	0
$13$	9.80240i	0.754030i	0.926207	+	0.377015i	$0.123050\pi$
$13$	9.80240i	0.754030i	−0.926207	+	0.377015i	$0.876950\pi$
$14$	13.8883	0.992024
$15$	0	0
$16$	26.0993	1.63121
$17$	20.9255i	1.23091i	0.788172	+	0.615455i	$0.211028\pi$
$17$	20.9255i	1.23091i	−0.788172	+	0.615455i	$0.788972\pi$
$18$	0	0
$19$	− 26.0425i	− 1.37066i	−0.728233	−	0.685330i	$-0.759658\pi$
$19$	− 26.0425i	− 1.37066i	0.728233	−	0.685330i	$-0.240342\pi$
$20$	−30.2778	−1.51389
$21$	0	0
$22$	0	0
$23$	−26.4380	−1.14948	−0.574739	−	0.818337i	$-0.694896\pi$
$23$	−26.4380	−1.14948	−0.574739	+	0.818337i	$0.694896\pi$
$24$	0	0
$25$	−13.1340	−0.525360
$26$	−35.0559	−1.34831
$27$	0	0
$28$	34.1344i	1.21909i
$29$	27.3115i	0.941777i	0.882193	+	0.470889i	$0.156067\pi$
$29$	27.3115i	0.941777i	−0.882193	+	0.470889i	$0.843933\pi$
$30$	0	0
$31$	−34.3712	−1.10875	−0.554375	−	0.832267i	$-0.687043\pi$
$31$	−34.3712	−1.10875	−0.554375	+	0.832267i	$0.687043\pi$
$32$	24.8219i	0.775684i
$33$	0	0
$34$	−74.8349	−2.20103
$35$	− 13.3774i	− 0.382213i
$36$	0	0
$37$	−20.9105	−0.565149	−0.282575	−	0.959245i	$-0.591189\pi$
$37$	−20.9105	−0.565149	−0.282575	+	0.959245i	$0.591189\pi$
$38$	93.1349	2.45092
$39$	0	0
$40$	− 59.0045i	− 1.47511i
$41$	− 13.1594i	− 0.320961i	−0.987039	−	0.160480i	$-0.948696\pi$
$41$	− 13.1594i	− 0.320961i	0.987039	−	0.160480i	$-0.0513043\pi$
$42$	0	0
$43$	− 38.9159i	− 0.905022i	−0.891759	−	0.452511i	$-0.850528\pi$
$43$	− 38.9159i	− 0.905022i	0.891759	−	0.452511i	$-0.149472\pi$
$44$	0	0
$45$	0	0
$46$	− 94.5492i	− 2.05542i
$47$	−30.9376	−0.658246	−0.329123	−	0.944287i	$-0.606753\pi$
$47$	−30.9376	−0.658246	−0.329123	+	0.944287i	$0.606753\pi$
$48$	0	0
$49$	33.9186	0.692216
$50$	− 46.9706i	− 0.939413i
$51$	0	0
$52$	− 86.1596i	− 1.65692i
$53$	−31.3996	−0.592446	−0.296223	−	0.955119i	$-0.595727\pi$
$53$	−31.3996	−0.592446	−0.296223	+	0.955119i	$0.595727\pi$
$54$	0	0
$55$	0	0
$56$	−66.5202	−1.18786
$57$	0	0
$58$	−97.6732	−1.68402
$59$	9.12891	0.154727	0.0773636	−	0.997003i	$-0.475350\pi$
$59$	9.12891	0.154727	0.0773636	+	0.997003i	$0.475350\pi$
$60$	0	0
$61$	− 21.5050i	− 0.352541i	−0.984342	−	0.176271i	$-0.943597\pi$
$61$	− 21.5050i	− 0.352541i	0.984342	−	0.176271i	$-0.0564034\pi$
$62$	− 122.921i	− 1.98259i
$63$	0	0
$64$	15.6278	0.244184
$65$	33.7664i	0.519483i
$66$	0	0
$67$	−78.8374	−1.17668	−0.588339	−	0.808614i	$-0.700218\pi$
$67$	−78.8374	−1.17668	−0.588339	+	0.808614i	$0.700218\pi$
$68$	− 183.927i	− 2.70481i
$69$	0	0
$70$	47.8412	0.683446
$71$	−109.362	−1.54031	−0.770157	−	0.637854i	$-0.779822\pi$
$71$	−109.362	−1.54031	−0.770157	+	0.637854i	$0.779822\pi$
$72$	0	0
$73$	− 33.8711i	− 0.463988i	−0.972717	−	0.231994i	$-0.925475\pi$
$73$	− 33.8711i	− 0.463988i	0.972717	−	0.231994i	$-0.0745250\pi$
$74$	− 74.7815i	− 1.01056i
$75$	0	0
$76$	228.905i	3.01191i
$77$	0	0
$78$	0	0
$79$	40.8466i	0.517046i	0.966005	+	0.258523i	$0.0832357\pi$
$79$	40.8466i	0.517046i	−0.966005	+	0.258523i	$0.916764\pi$
$80$	89.9045	1.12381
$81$	0	0
$82$	47.0614	0.573920
$83$	− 119.555i	− 1.44042i	−0.693757	−	0.720209i	$-0.744046\pi$
$83$	− 119.555i	− 1.44042i	0.693757	−	0.720209i	$-0.255954\pi$
$84$	0	0
$85$	72.0820i	0.848024i
$86$	139.174	1.61830
$87$	0	0
$88$	0	0
$89$	−22.2767	−0.250299	−0.125150	−	0.992138i	$-0.539941\pi$
$89$	−22.2767	−0.250299	−0.125150	+	0.992138i	$0.539941\pi$
$90$	0	0
$91$	38.0674	0.418323
$92$	232.381	2.52588
$93$	0	0
$94$	− 110.641i	− 1.17703i
$95$	− 89.7089i	− 0.944304i
$96$	0	0
$97$	134.669	1.38834	0.694172	−	0.719809i	$-0.255771\pi$
$97$	134.669	1.38834	0.694172	+	0.719809i	$0.255771\pi$
$98$	121.302i	1.23777i
$99$	0	0
$100$	115.443	1.15443
$101$	133.428i	1.32106i	0.750798	+	0.660532i	$0.229669\pi$
$101$	133.428i	1.32106i	−0.750798	+	0.660532i	$0.770331\pi$
$102$	0	0
$103$	−148.863	−1.44527	−0.722637	−	0.691228i	$-0.757070\pi$
$103$	−148.863	−1.44527	−0.722637	+	0.691228i	$0.757070\pi$
$104$	167.906	1.61448
$105$	0	0
$106$	− 112.293i	− 1.05937i
$107$	33.9604i	0.317387i	0.987328	+	0.158694i	$0.0507282\pi$
$107$	33.9604i	0.317387i	−0.987328	+	0.158694i	$0.949272\pi$
$108$	0	0
$109$	− 28.8970i	− 0.265110i	−0.991176	−	0.132555i	$-0.957682\pi$
$109$	− 28.8970i	− 0.265110i	0.991176	−	0.132555i	$-0.0423181\pi$
$110$	0	0
$111$	0	0
$112$	− 101.356i	− 0.904966i
$113$	47.7721	0.422762	0.211381	−	0.977404i	$-0.432204\pi$
$113$	47.7721	0.422762	0.211381	+	0.977404i	$0.432204\pi$
$114$	0	0
$115$	−91.0711	−0.791923
$116$	− 240.059i	− 2.06947i
$117$	0	0
$118$	32.6474i	0.276673i
$119$	81.2636	0.682887
$120$	0	0
$121$	0	0
$122$	76.9076	0.630390
$123$	0	0
$124$	302.111	2.43638
$125$	−131.360	−1.05088
$126$	0	0
$127$	− 76.0605i	− 0.598902i	−0.954112	−	0.299451i	$-0.903197\pi$
$127$	− 76.0605i	− 0.598902i	0.954112	−	0.299451i	$-0.0968034\pi$
$128$	155.176i	1.21232i
$129$	0	0
$130$	−120.757	−0.928903
$131$	− 138.150i	− 1.05458i	−0.849686	−	0.527289i	$-0.823208\pi$
$131$	− 138.150i	− 1.05458i	0.849686	−	0.527289i	$-0.176792\pi$
$132$	0	0
$133$	−101.136	−0.760418
$134$	− 281.943i	− 2.10405i
$135$	0	0
$136$	358.433	2.63554
$137$	150.107	1.09567	0.547835	−	0.836586i	$-0.315452\pi$
$137$	150.107	1.09567	0.547835	+	0.836586i	$0.315452\pi$
$138$	0	0
$139$	254.426i	1.83041i	0.402994	+	0.915203i	$0.367969\pi$
$139$	254.426i	1.83041i	−0.402994	+	0.915203i	$0.632031\pi$
$140$	117.583i	0.839879i
$141$	0	0
$142$	− 391.108i	− 2.75428i
$143$	0	0
$144$	0	0
$145$	94.0802i	0.648829i
$146$	121.132	0.829671
$147$	0	0
$148$	183.796	1.24187
$149$	112.116i	0.752458i	0.926527	+	0.376229i	$0.122779\pi$
$149$	112.116i	0.752458i	−0.926527	+	0.376229i	$0.877221\pi$
$150$	0	0
$151$	− 156.588i	− 1.03701i	−0.855076	−	0.518503i	$-0.826489\pi$
$151$	− 156.588i	− 1.03701i	0.855076	−	0.518503i	$-0.173511\pi$
$152$	−446.084	−2.93476
$153$	0	0
$154$	0	0
$155$	−118.399	−0.763863
$156$	0	0
$157$	−77.5695	−0.494074	−0.247037	−	0.969006i	$-0.579457\pi$
$157$	−77.5695	−0.494074	−0.247037	+	0.969006i	$0.579457\pi$
$158$	−146.078	−0.924546
$159$	0	0
$160$	85.5041i	0.534401i
$161$	102.671i	0.637711i
$162$	0	0
$163$	−131.119	−0.804410	−0.402205	−	0.915550i	$-0.631756\pi$
$163$	−131.119	−0.804410	−0.402205	+	0.915550i	$0.631756\pi$
$164$	115.666i	0.705283i
$165$	0	0
$166$	427.559	2.57566
$167$	71.3060i	0.426982i	0.976945	+	0.213491i	$0.0684834\pi$
$167$	71.3060i	0.426982i	−0.976945	+	0.213491i	$0.931517\pi$
$168$	0	0
$169$	72.9130	0.431438
$170$	−257.784	−1.51638
$171$	0	0
$172$	342.057i	1.98871i
$173$	65.0980i	0.376289i	0.982141	+	0.188145i	$0.0602474\pi$
$173$	65.0980i	0.376289i	−0.982141	+	0.188145i	$0.939753\pi$
$174$	0	0
$175$	51.0056i	0.291461i
$176$	0	0
$177$	0	0
$178$	− 79.6671i	− 0.447568i
$179$	138.569	0.774128	0.387064	−	0.922053i	$-0.373489\pi$
$179$	138.569	0.774128	0.387064	+	0.922053i	$0.373489\pi$
$180$	0	0
$181$	−310.372	−1.71476	−0.857381	−	0.514682i	$-0.827910\pi$
$181$	−310.372	−1.71476	−0.857381	+	0.514682i	$0.827910\pi$
$182$	136.139i	0.748016i
$183$	0	0
$184$	452.858i	2.46118i
$185$	−72.0306	−0.389355
$186$	0	0
$187$	0	0
$188$	271.930	1.44644
$189$	0	0
$190$	320.822	1.68854
$191$	55.3125	0.289594	0.144797	−	0.989461i	$-0.453747\pi$
$191$	55.3125	0.289594	0.144797	+	0.989461i	$0.453747\pi$
$192$	0	0
$193$	− 285.835i	− 1.48101i	−0.672051	−	0.740505i	$-0.734587\pi$
$193$	− 285.835i	− 1.48101i	0.672051	−	0.740505i	$-0.265413\pi$
$194$	481.613i	2.48254i
$195$	0	0
$196$	−298.133	−1.52108
$197$	− 40.4364i	− 0.205261i	−0.994720	−	0.102630i	$-0.967274\pi$
$197$	− 40.4364i	− 0.205261i	0.994720	−	0.102630i	$-0.0327259\pi$
$198$	0	0
$199$	−243.226	−1.22224	−0.611120	−	0.791538i	$-0.709281\pi$
$199$	−243.226	−1.22224	−0.611120	+	0.791538i	$0.709281\pi$
$200$	224.973i	1.12486i
$201$	0	0
$202$	−477.172	−2.36224
$203$	106.064	0.522482
$204$	0	0
$205$	− 45.3302i	− 0.221123i
$206$	− 532.374i	− 2.58434i
$207$	0	0
$208$	255.836i	1.22998i
$209$	0	0
$210$	0	0
$211$	155.386i	0.736428i	0.929741	+	0.368214i	$0.120031\pi$
$211$	155.386i	0.736428i	−0.929741	+	0.368214i	$0.879969\pi$
$212$	275.992	1.30185
$213$	0	0
$214$	−121.451	−0.567530
$215$	− 134.054i	− 0.623507i
$216$	0	0
$217$	133.480i	0.615115i
$218$	103.343	0.474051
$219$	0	0
$220$	0	0
$221$	−205.120	−0.928143
$222$	0	0
$223$	321.673	1.44248	0.721240	−	0.692685i	$-0.243572\pi$
$223$	321.673	1.44248	0.721240	+	0.692685i	$0.243572\pi$
$224$	96.3953	0.430336
$225$	0	0
$226$	170.845i	0.755953i
$227$	− 165.544i	− 0.729269i	−0.931151	−	0.364635i	$-0.881194\pi$
$227$	− 165.544i	− 0.729269i	0.931151	−	0.364635i	$-0.118806\pi$
$228$	0	0
$229$	305.728	1.33505	0.667527	−	0.744585i	$-0.267353\pi$
$229$	305.728	1.33505	0.667527	+	0.744585i	$0.267353\pi$
$230$	− 325.694i	− 1.41606i
$231$	0	0
$232$	467.820	2.01647
$233$	340.396i	1.46093i	0.682951	+	0.730464i	$0.260696\pi$
$233$	340.396i	1.46093i	−0.682951	+	0.730464i	$0.739304\pi$
$234$	0	0
$235$	−106.571	−0.453493
$236$	−80.2399	−0.340000
$237$	0	0
$238$	290.620i	1.22109i
$239$	223.562i	0.935405i	0.883886	+	0.467703i	$0.154918\pi$
$239$	223.562i	0.935405i	−0.883886	+	0.467703i	$0.845082\pi$
$240$	0	0
$241$	− 215.440i	− 0.893944i	−0.894548	−	0.446972i	$-0.852502\pi$
$241$	− 215.440i	− 0.893944i	0.894548	−	0.446972i	$-0.147498\pi$
$242$	0	0
$243$	0	0
$244$	189.022i	0.774679i
$245$	116.840	0.476896
$246$	0	0
$247$	255.279	1.03352
$248$	588.746i	2.37398i
$249$	0	0
$250$	− 469.779i	− 1.87912i
$251$	−132.573	−0.528178	−0.264089	−	0.964498i	$-0.585071\pi$
$251$	−132.573	−0.528178	−0.264089	+	0.964498i	$0.585071\pi$
$252$	0	0
$253$	0	0
$254$	272.012	1.07091
$255$	0	0
$256$	−492.441	−1.92360
$257$	−22.0414	−0.0857642	−0.0428821	−	0.999080i	$-0.513654\pi$
$257$	−22.0414	−0.0857642	−0.0428821	+	0.999080i	$0.513654\pi$
$258$	0	0
$259$	81.2056i	0.313535i
$260$	− 296.795i	− 1.14152i
$261$	0	0
$262$	494.060	1.88573
$263$	88.6951i	0.337244i	0.985681	+	0.168622i	$0.0539317\pi$
$263$	88.6951i	0.337244i	−0.985681	+	0.168622i	$0.946068\pi$
$264$	0	0
$265$	−108.163	−0.408160
$266$	− 361.688i	− 1.35973i
$267$	0	0
$268$	692.953	2.58565
$269$	−482.027	−1.79192	−0.895960	−	0.444134i	$-0.853511\pi$
$269$	−482.027	−1.79192	−0.895960	+	0.444134i	$0.853511\pi$
$270$	0	0
$271$	− 101.261i	− 0.373657i	−0.982393	−	0.186829i	$-0.940179\pi$
$271$	− 101.261i	− 0.373657i	0.982393	−	0.186829i	$-0.0598209\pi$
$272$	546.140i	2.00787i
$273$	0	0
$274$	536.821i	1.95920i
$275$	0	0
$276$	0	0
$277$	108.610i	0.392093i	0.980595	+	0.196046i	$0.0628103\pi$
$277$	108.610i	0.392093i	−0.980595	+	0.196046i	$0.937190\pi$
$278$	−909.895	−3.27300
$279$	0	0
$280$	−229.143	−0.818367
$281$	− 180.797i	− 0.643405i	−0.946841	−	0.321703i	$-0.895745\pi$
$281$	− 180.797i	− 0.643405i	0.946841	−	0.321703i	$-0.104255\pi$
$282$	0	0
$283$	334.701i	1.18269i	0.806419	+	0.591345i	$0.201403\pi$
$283$	334.701i	1.18269i	−0.806419	+	0.591345i	$0.798597\pi$
$284$	961.256	3.38471
$285$	0	0
$286$	0	0
$287$	−51.1042	−0.178063
$288$	0	0
$289$	−148.875	−0.515138
$290$	−336.455	−1.16019
$291$	0	0
$292$	297.715i	1.01957i
$293$	83.8747i	0.286262i	0.989704	+	0.143131i	$0.0457170\pi$
$293$	83.8747i	0.286262i	−0.989704	+	0.143131i	$0.954283\pi$
$294$	0	0
$295$	31.4464	0.106598
$296$	358.177i	1.21006i
$297$	0	0
$298$	−400.957	−1.34549
$299$	− 259.156i	− 0.866742i
$300$	0	0
$301$	−151.129	−0.502090
$302$	560.000	1.85430
$303$	0	0
$304$	− 679.693i	− 2.23583i
$305$	− 74.0785i	− 0.242880i
$306$	0	0
$307$	− 402.102i	− 1.30978i	−0.755724	−	0.654890i	$-0.772715\pi$
$307$	− 402.102i	− 1.30978i	0.755724	−	0.654890i	$-0.227285\pi$
$308$	0	0
$309$	0	0
$310$	− 423.425i	− 1.36589i
$311$	−360.643	−1.15962	−0.579812	−	0.814751i	$-0.696874\pi$
$311$	−360.643	−1.15962	−0.579812	+	0.814751i	$0.696874\pi$
$312$	0	0
$313$	16.5121	0.0527543	0.0263771	−	0.999652i	$-0.491603\pi$
$313$	16.5121	0.0527543	0.0263771	+	0.999652i	$0.491603\pi$
$314$	− 277.409i	− 0.883468i
$315$	0	0
$316$	− 359.027i	− 1.13616i
$317$	−10.1481	−0.0320129	−0.0160064	−	0.999872i	$-0.505095\pi$
$317$	−10.1481	−0.0320129	−0.0160064	+	0.999872i	$0.505095\pi$
$318$	0	0
$319$	0	0
$320$	53.8330	0.168228
$321$	0	0
$322$	−367.180	−1.14031
$323$	544.952	1.68716
$324$	0	0
$325$	− 128.745i	− 0.396137i
$326$	− 468.915i	− 1.43839i
$327$	0	0
$328$	−225.408	−0.687219
$329$	120.145i	0.365184i
$330$	0	0
$331$	−175.740	−0.530937	−0.265469	−	0.964119i	$-0.585527\pi$
$331$	−175.740	−0.530937	−0.265469	+	0.964119i	$0.585527\pi$
$332$	1050.84i	3.16519i
$333$	0	0
$334$	−255.009	−0.763500
$335$	−271.572	−0.810662
$336$	0	0
$337$	− 343.792i	− 1.02015i	−0.860129	−	0.510077i	$-0.829617\pi$
$337$	− 343.792i	− 1.02015i	0.860129	−	0.510077i	$-0.170383\pi$
$338$	260.756i	0.771468i
$339$	0	0
$340$	− 633.576i	− 1.86346i
$341$	0	0
$342$	0	0
$343$	− 322.013i	− 0.938812i
$344$	−666.593	−1.93777
$345$	0	0
$346$	−232.808	−0.672854
$347$	− 117.905i	− 0.339784i	−0.985463	−	0.169892i	$-0.945658\pi$
$347$	− 117.905i	− 0.339784i	0.985463	−	0.169892i	$-0.0543419\pi$
$348$	0	0
$349$	− 43.8220i	− 0.125565i	−0.998027	−	0.0627823i	$-0.980003\pi$
$349$	− 43.8220i	− 0.125565i	0.998027	−	0.0627823i	$-0.0199974\pi$
$350$	−182.409	−0.521170
$351$	0	0
$352$	0	0
$353$	−345.503	−0.978761	−0.489381	−	0.872070i	$-0.662777\pi$
$353$	−345.503	−0.978761	−0.489381	+	0.872070i	$0.662777\pi$
$354$	0	0
$355$	−376.721	−1.06119
$356$	195.804	0.550011
$357$	0	0
$358$	495.559i	1.38424i
$359$	− 170.429i	− 0.474733i	−0.971420	−	0.237366i	$-0.923716\pi$
$359$	− 170.429i	− 0.474733i	0.971420	−	0.237366i	$-0.0762842\pi$
$360$	0	0
$361$	−317.214	−0.878709
$362$	− 1109.97i	− 3.06622i
$363$	0	0
$364$	−334.599	−0.919228
$365$	− 116.676i	− 0.319660i
$366$	0	0
$367$	198.656	0.541298	0.270649	−	0.962678i	$-0.412762\pi$
$367$	198.656	0.541298	0.270649	+	0.962678i	$0.412762\pi$
$368$	−690.014	−1.87504
$369$	0	0
$370$	− 257.600i	− 0.696217i
$371$	121.940i	0.328679i
$372$	0	0
$373$	159.274i	0.427009i	0.976942	+	0.213505i	$0.0684878\pi$
$373$	159.274i	0.427009i	−0.976942	+	0.213505i	$0.931512\pi$
$374$	0	0
$375$	0	0
$376$	529.931i	1.40939i
$377$	−267.718	−0.710129
$378$	0	0
$379$	47.8228	0.126181	0.0630907	−	0.998008i	$-0.479904\pi$
$379$	47.8228	0.126181	0.0630907	+	0.998008i	$0.479904\pi$
$380$	788.510i	2.07503i
$381$	0	0
$382$	197.812i	0.517833i
$383$	71.2421	0.186011	0.0930053	−	0.995666i	$-0.470353\pi$
$383$	71.2421	0.186011	0.0930053	+	0.995666i	$0.470353\pi$
$384$	0	0
$385$	0	0
$386$	1022.22	2.64824
$387$	0	0
$388$	−1183.70	−3.05076
$389$	−445.646	−1.14562	−0.572810	−	0.819688i	$-0.694147\pi$
$389$	−445.646	−1.14562	−0.572810	+	0.819688i	$0.694147\pi$
$390$	0	0
$391$	− 553.227i	− 1.41490i
$392$	− 580.993i	− 1.48212i
$393$	0	0
$394$	144.611	0.367033
$395$	140.705i	0.356214i
$396$	0	0
$397$	83.3626	0.209981	0.104991	−	0.994473i	$-0.466519\pi$
$397$	83.3626	0.209981	0.104991	+	0.994473i	$0.466519\pi$
$398$	− 869.838i	− 2.18552i
$399$	0	0
$400$	−342.789	−0.856972
$401$	289.781	0.722647	0.361324	−	0.932441i	$-0.382325\pi$
$401$	289.781	0.722647	0.361324	+	0.932441i	$0.382325\pi$
$402$	0	0
$403$	− 336.921i	− 0.836031i
$404$	− 1172.78i	− 2.90292i
$405$	0	0
$406$	379.312i	0.934266i
$407$	0	0
$408$	0	0
$409$	540.792i	1.32223i	0.750285	+	0.661114i	$0.229916\pi$
$409$	540.792i	1.32223i	−0.750285	+	0.661114i	$0.770084\pi$
$410$	162.113	0.395397
$411$	0	0
$412$	1308.45	3.17586
$413$	− 35.4519i	− 0.0858400i
$414$	0	0
$415$	− 411.831i	− 0.992364i
$416$	−243.314	−0.584889
$417$	0	0
$418$	0	0
$419$	−824.379	−1.96749	−0.983746	−	0.179563i	$-0.942531\pi$
$419$	−824.379	−1.96749	−0.983746	+	0.179563i	$0.942531\pi$
$420$	0	0
$421$	−47.7281	−0.113368	−0.0566842	−	0.998392i	$-0.518053\pi$
$421$	−47.7281	−0.113368	−0.0566842	+	0.998392i	$0.518053\pi$
$422$	−555.702	−1.31683
$423$	0	0
$424$	537.846i	1.26850i
$425$	− 274.835i	− 0.646671i
$426$	0	0
$427$	−83.5143	−0.195584
$428$	− 298.500i	− 0.697431i
$429$	0	0
$430$	479.412	1.11491
$431$	648.502i	1.50464i	0.658795	+	0.752322i	$0.271066\pi$
$431$	648.502i	1.50464i	−0.658795	+	0.752322i	$0.728934\pi$
$432$	0	0
$433$	−418.276	−0.965996	−0.482998	−	0.875621i	$-0.660452\pi$
$433$	−418.276	−0.965996	−0.482998	+	0.875621i	$0.660452\pi$
$434$	−477.359	−1.09991
$435$	0	0
$436$	253.994i	0.582556i
$437$	688.513i	1.57554i
$438$	0	0
$439$	678.114i	1.54468i	0.635210	+	0.772340i	$0.280914\pi$
$439$	678.114i	1.54468i	−0.635210	+	0.772340i	$0.719086\pi$
$440$	0	0
$441$	0	0
$442$	− 733.561i	− 1.65964i
$443$	824.561	1.86131	0.930655	−	0.365897i	$-0.119238\pi$
$443$	824.561	1.86131	0.930655	+	0.365897i	$0.119238\pi$
$444$	0	0
$445$	−76.7365	−0.172442
$446$	1150.39i	2.57934i
$447$	0	0
$448$	− 60.6900i	− 0.135469i
$449$	−156.027	−0.347499	−0.173750	−	0.984790i	$-0.555588\pi$
$449$	−156.027	−0.347499	−0.173750	+	0.984790i	$0.555588\pi$
$450$	0	0
$451$	0	0
$452$	−419.900	−0.928981
$453$	0	0
$454$	592.029	1.30403
$455$	131.131	0.288200
$456$	0	0
$457$	459.120i	1.00464i	0.864682	+	0.502319i	$0.167520\pi$
$457$	459.120i	1.00464i	−0.864682	+	0.502319i	$0.832480\pi$
$458$	1093.36i	2.38725i
$459$	0	0
$460$	800.483	1.74018
$461$	536.047i	1.16279i	0.813621	+	0.581396i	$0.197493\pi$
$461$	536.047i	1.16279i	−0.813621	+	0.581396i	$0.802507\pi$
$462$	0	0
$463$	−133.000	−0.287258	−0.143629	−	0.989632i	$-0.545877\pi$
$463$	−133.000	−0.287258	−0.143629	+	0.989632i	$0.545877\pi$
$464$	712.813i	1.53623i
$465$	0	0
$466$	−1217.35	−2.61233
$467$	484.649	1.03779	0.518896	−	0.854837i	$-0.326343\pi$
$467$	484.649	1.03779	0.518896	+	0.854837i	$0.326343\pi$
$468$	0	0
$469$	306.163i	0.652801i
$470$	− 381.125i	− 0.810905i
$471$	0	0
$472$	− 156.369i	− 0.331291i
$473$	0	0
$474$	0	0
$475$	342.043i	0.720090i
$476$	−714.278	−1.50058
$477$	0	0
$478$	−799.516	−1.67263
$479$	866.399i	1.80877i	0.426722	+	0.904383i	$0.359668\pi$
$479$	866.399i	1.80877i	−0.426722	+	0.904383i	$0.640332\pi$
$480$	0	0
$481$	− 204.973i	− 0.426140i
$482$	770.471	1.59849
$483$	0	0
$484$	0	0
$485$	463.896	0.956487
$486$	0	0
$487$	580.104	1.19118	0.595589	−	0.803289i	$-0.296919\pi$
$487$	580.104	1.19118	0.595589	+	0.803289i	$0.296919\pi$
$488$	−368.360	−0.754837
$489$	0	0
$490$	417.849i	0.852753i
$491$	417.674i	0.850660i	0.905038	+	0.425330i	$0.139842\pi$
$491$	417.674i	0.850660i	−0.905038	+	0.425330i	$0.860158\pi$
$492$	0	0
$493$	−571.506	−1.15924
$494$	912.946i	1.84807i
$495$	0	0
$496$	−897.066	−1.80860
$497$	424.706i	0.854540i
$498$	0	0
$499$	802.655	1.60853	0.804264	−	0.594272i	$-0.202560\pi$
$499$	802.655	1.60853	0.804264	+	0.594272i	$0.202560\pi$
$500$	1154.61	2.30922
$501$	0	0
$502$	− 474.114i	− 0.944451i
$503$	− 886.237i	− 1.76190i	−0.473207	−	0.880951i	$-0.656904\pi$
$503$	− 886.237i	− 1.76190i	0.473207	−	0.880951i	$-0.343096\pi$
$504$	0	0
$505$	459.618i	0.910136i
$506$	0	0
$507$	0	0
$508$	668.545i	1.31603i
$509$	70.6698	0.138840	0.0694202	−	0.997588i	$-0.477885\pi$
$509$	70.6698	0.138840	0.0694202	+	0.997588i	$0.477885\pi$
$510$	0	0
$511$	−131.538	−0.257412
$512$	− 1140.39i	− 2.22733i
$513$	0	0
$514$	− 78.8259i	− 0.153358i
$515$	−512.790	−0.995708
$516$	0	0
$517$	0	0
$518$	−290.412	−0.560642
$519$	0	0
$520$	578.385	1.11228
$521$	101.366	0.194561	0.0972806	−	0.995257i	$-0.468986\pi$
$521$	101.366	0.194561	0.0972806	+	0.995257i	$0.468986\pi$
$522$	0	0
$523$	774.697i	1.48126i	0.671916	+	0.740628i	$0.265472\pi$
$523$	774.697i	1.48126i	−0.671916	+	0.740628i	$0.734528\pi$
$524$	1214.29i	2.31734i
$525$	0	0
$526$	−317.197	−0.603036
$527$	− 719.234i	− 1.36477i
$528$	0	0
$529$	169.968	0.321300
$530$	− 386.817i	− 0.729844i
$531$	0	0
$532$	888.947	1.67095
$533$	128.994	0.242014
$534$	0	0
$535$	116.984i	0.218661i
$536$	1350.41i	2.51942i
$537$	0	0
$538$	− 1723.85i	− 3.20419i
$539$	0	0
$540$	0	0
$541$	− 252.507i	− 0.466741i	−0.972388	−	0.233370i	$-0.925025\pi$
$541$	− 252.507i	− 0.466741i	0.972388	−	0.233370i	$-0.0749755\pi$
$542$	362.136	0.668148
$543$	0	0
$544$	−519.409	−0.954797
$545$	− 99.5416i	− 0.182645i
$546$	0	0
$547$	475.673i	0.869604i	0.900526	+	0.434802i	$0.143182\pi$
$547$	475.673i	0.869604i	−0.900526	+	0.434802i	$0.856818\pi$
$548$	−1319.39	−2.40764
$549$	0	0
$550$	0	0
$551$	711.262	1.29086
$552$	0	0
$553$	158.627	0.286848
$554$	−388.417	−0.701113
$555$	0	0
$556$	− 2236.32i	− 4.02215i
$557$	− 762.642i	− 1.36920i	−0.728921	−	0.684598i	$-0.759978\pi$
$557$	− 762.642i	− 1.36920i	0.728921	−	0.684598i	$-0.240022\pi$
$558$	0	0
$559$	381.469	0.682414
$560$	− 349.142i	− 0.623468i
$561$	0	0
$562$	646.577	1.15049
$563$	50.9757i	0.0905430i	0.998975	+	0.0452715i	$0.0144153\pi$
$563$	50.9757i	0.0905430i	−0.998975	+	0.0452715i	$0.985585\pi$
$564$	0	0
$565$	164.561	0.291258
$566$	−1196.98	−2.11480
$567$	0	0
$568$	1873.27i	3.29801i
$569$	619.186i	1.08820i	0.839020	+	0.544101i	$0.183129\pi$
$569$	619.186i	1.08820i	−0.839020	+	0.544101i	$0.816871\pi$
$570$	0	0
$571$	− 501.853i	− 0.878902i	−0.898266	−	0.439451i	$-0.855173\pi$
$571$	− 501.853i	− 0.878902i	0.898266	−	0.439451i	$-0.144827\pi$
$572$	0	0
$573$	0	0
$574$	− 182.762i	− 0.318401i
$575$	347.237	0.603890
$576$	0	0
$577$	360.865	0.625416	0.312708	−	0.949849i	$-0.398764\pi$
$577$	360.865	0.625416	0.312708	+	0.949849i	$0.398764\pi$
$578$	− 532.415i	− 0.921133i
$579$	0	0
$580$	− 826.932i	− 1.42574i
$581$	−464.288	−0.799119
$582$	0	0
$583$	0	0
$584$	−580.180	−0.993458
$585$	0	0
$586$	−299.958	−0.511874
$587$	−188.412	−0.320974	−0.160487	−	0.987038i	$-0.551307\pi$
$587$	−188.412	−0.320974	−0.160487	+	0.987038i	$0.551307\pi$
$588$	0	0
$589$	895.115i	1.51972i
$590$	112.461i	0.190611i
$591$	0	0
$592$	−545.751	−0.921876
$593$	231.751i	0.390811i	0.980723	+	0.195406i	$0.0626023\pi$
$593$	231.751i	0.390811i	−0.980723	+	0.195406i	$0.937398\pi$
$594$	0	0
$595$	279.929	0.470469
$596$	− 985.463i	− 1.65346i
$597$	0	0
$598$	926.809	1.54985
$599$	−688.162	−1.14885	−0.574426	−	0.818557i	$-0.694775\pi$
$599$	−688.162	−1.14885	−0.574426	+	0.818557i	$0.694775\pi$
$600$	0	0
$601$	825.672i	1.37383i	0.726737	+	0.686916i	$0.241036\pi$
$601$	825.672i	1.37383i	−0.726737	+	0.686916i	$0.758964\pi$
$602$	− 540.478i	− 0.897803i
$603$	0	0
$604$	1376.35i	2.27873i
$605$	0	0
$606$	0	0
$607$	− 795.811i	− 1.31106i	−0.755171	−	0.655528i	$-0.772446\pi$
$607$	− 795.811i	− 1.31106i	0.755171	−	0.655528i	$-0.227554\pi$
$608$	646.425	1.06320
$609$	0	0
$610$	264.924	0.434302
$611$	− 303.262i	− 0.496338i
$612$	0	0
$613$	1080.32i	1.76236i	0.472785	+	0.881178i	$0.343249\pi$
$613$	1080.32i	1.76236i	−0.472785	+	0.881178i	$0.656751\pi$
$614$	1438.02	2.34206
$615$	0	0
$616$	0	0
$617$	592.426	0.960172	0.480086	−	0.877222i	$-0.340606\pi$
$617$	592.426	0.960172	0.480086	+	0.877222i	$0.340606\pi$
$618$	0	0
$619$	−236.638	−0.382291	−0.191145	−	0.981562i	$-0.561220\pi$
$619$	−236.638	−0.382291	−0.191145	+	0.981562i	$0.561220\pi$
$620$	1040.68	1.67852
$621$	0	0
$622$	− 1289.75i	− 2.07356i
$623$	86.5109i	0.138862i
$624$	0	0
$625$	−124.148	−0.198637
$626$	59.0516i	0.0943316i
$627$	0	0
$628$	681.809	1.08568
$629$	− 437.562i	− 0.695648i
$630$	0	0
$631$	648.168	1.02721	0.513603	−	0.858028i	$-0.328310\pi$
$631$	648.168	1.02721	0.513603	+	0.858028i	$0.328310\pi$
$632$	699.663	1.10706
$633$	0	0
$634$	− 36.2922i	− 0.0572432i
$635$	− 262.006i	− 0.412608i
$636$	0	0
$637$	332.483i	0.521952i
$638$	0	0
$639$	0	0
$640$	534.537i	0.835215i
$641$	−625.207	−0.975362	−0.487681	−	0.873022i	$-0.662157\pi$
$641$	−625.207	−0.975362	−0.487681	+	0.873022i	$0.662157\pi$
$642$	0	0
$643$	−824.383	−1.28209	−0.641044	−	0.767504i	$-0.721499\pi$
$643$	−824.383	−1.28209	−0.641044	+	0.767504i	$0.721499\pi$
$644$	− 902.446i	− 1.40131i
$645$	0	0
$646$	1948.89i	3.01686i
$647$	−427.317	−0.660460	−0.330230	−	0.943901i	$-0.607126\pi$
$647$	−427.317	−0.660460	−0.330230	+	0.943901i	$0.607126\pi$
$648$	0	0
$649$	0	0
$650$	460.425	0.708346
$651$	0	0
$652$	1152.49	1.76762
$653$	1280.69	1.96124	0.980622	−	0.195908i	$-0.0627655\pi$
$653$	1280.69	1.96124	0.980622	+	0.195908i	$0.0627655\pi$
$654$	0	0
$655$	− 475.886i	− 0.726543i
$656$	− 343.451i	− 0.523554i
$657$	0	0
$658$	−429.671	−0.652996
$659$	602.881i	0.914842i	0.889250	+	0.457421i	$0.151227\pi$
$659$	602.881i	0.914842i	−0.889250	+	0.457421i	$0.848773\pi$
$660$	0	0
$661$	263.343	0.398401	0.199200	−	0.979959i	$-0.436166\pi$
$661$	263.343	0.398401	0.199200	+	0.979959i	$0.436166\pi$
$662$	− 628.493i	− 0.949385i
$663$	0	0
$664$	−2047.86	−3.08412
$665$	−348.383	−0.523884
$666$	0	0
$667$	− 722.062i	− 1.08255i
$668$	− 626.755i	− 0.938255i
$669$	0	0
$670$	− 971.212i	− 1.44957i
$671$	0	0
$672$	0	0
$673$	− 684.366i	− 1.01689i	−0.861095	−	0.508444i	$-0.830221\pi$
$673$	− 684.366i	− 1.01689i	0.861095	−	0.508444i	$-0.169779\pi$
$674$	1229.49	1.82417
$675$	0	0
$676$	−640.880	−0.948047
$677$	− 607.633i	− 0.897538i	−0.893648	−	0.448769i	$-0.851863\pi$
$677$	− 607.633i	− 0.897538i	0.893648	−	0.448769i	$-0.148137\pi$
$678$	0	0
$679$	− 522.986i	− 0.770229i
$680$	1234.70	1.81573
$681$	0	0
$682$	0	0
$683$	−419.287	−0.613890	−0.306945	−	0.951727i	$-0.599307\pi$
$683$	−419.287	−0.613890	−0.306945	+	0.951727i	$0.599307\pi$
$684$	0	0
$685$	517.074	0.754852
$686$	1151.60	1.67872
$687$	0	0
$688$	− 1015.68i	− 1.47628i
$689$	− 307.792i	− 0.446722i
$690$	0	0
$691$	220.369	0.318913	0.159457	−	0.987205i	$-0.449026\pi$
$691$	220.369	0.318913	0.159457	+	0.987205i	$0.449026\pi$
$692$	− 572.189i	− 0.826863i
$693$	0	0
$694$	421.660	0.607579
$695$	876.424i	1.26104i
$696$	0	0
$697$	275.366	0.395074
$698$	156.719	0.224526
$699$	0	0
$700$	− 448.321i	− 0.640459i
$701$	897.679i	1.28057i	0.768138	+	0.640284i	$0.221183\pi$
$701$	897.679i	1.28057i	−0.768138	+	0.640284i	$0.778817\pi$
$702$	0	0
$703$	544.563i	0.774628i
$704$	0	0
$705$	0	0
$706$	− 1235.61i	− 1.75015i
$707$	518.163	0.732904
$708$	0	0
$709$	834.168	1.17654	0.588271	−	0.808664i	$-0.299809\pi$
$709$	834.168	1.17654	0.588271	+	0.808664i	$0.299809\pi$
$710$	− 1347.25i	− 1.89754i
$711$	0	0
$712$	381.578i	0.535924i
$713$	908.707	1.27448
$714$	0	0
$715$	0	0
$716$	−1217.97	−1.70108
$717$	0	0
$718$	609.499	0.848884
$719$	−182.321	−0.253575	−0.126788	−	0.991930i	$-0.540467\pi$
$719$	−182.321	−0.253575	−0.126788	+	0.991930i	$0.540467\pi$
$720$	0	0
$721$	578.107i	0.801812i
$722$	− 1134.44i	− 1.57125i
$723$	0	0
$724$	2728.06	3.76804
$725$	− 358.710i	− 0.494772i
$726$	0	0
$727$	861.755	1.18536	0.592679	−	0.805439i	$-0.298070\pi$
$727$	861.755	1.18536	0.592679	+	0.805439i	$0.298070\pi$
$728$	− 652.058i	− 0.895684i
$729$	0	0
$730$	417.264	0.571595
$731$	814.334	1.11400
$732$	0	0
$733$	− 76.6339i	− 0.104548i	−0.998633	−	0.0522741i	$-0.983353\pi$
$733$	− 76.6339i	− 0.104548i	0.998633	−	0.0522741i	$-0.0166470\pi$
$734$	710.447i	0.967912i
$735$	0	0
$736$	− 656.241i	− 0.891632i
$737$	0	0
$738$	0	0
$739$	970.927i	1.31384i	0.753961	+	0.656919i	$0.228141\pi$
$739$	970.927i	1.31384i	−0.753961	+	0.656919i	$0.771859\pi$
$740$	633.124	0.855573
$741$	0	0
$742$	−436.089	−0.587721
$743$	1232.91i	1.65937i	0.558236	+	0.829683i	$0.311479\pi$
$743$	1232.91i	1.65937i	−0.558236	+	0.829683i	$0.688521\pi$
$744$	0	0
$745$	386.208i	0.518400i
$746$	−569.607	−0.763549
$747$	0	0
$748$	0	0
$749$	131.885	0.176081
$750$	0	0
$751$	511.106	0.680568	0.340284	−	0.940323i	$-0.389477\pi$
$751$	511.106	0.680568	0.340284	+	0.940323i	$0.389477\pi$
$752$	−807.450	−1.07374
$753$	0	0
$754$	− 957.431i	− 1.26980i
$755$	− 539.400i	− 0.714437i
$756$	0	0
$757$	−1085.07	−1.43338	−0.716690	−	0.697391i	$-0.754344\pi$
$757$	−1085.07	−1.43338	−0.716690	+	0.697391i	$0.754344\pi$
$758$	171.027i	0.225629i
$759$	0	0
$760$	−1536.63	−2.02188
$761$	− 1283.94i	− 1.68718i	−0.536990	−	0.843589i	$-0.680439\pi$
$761$	− 1283.94i	− 1.68718i	0.536990	−	0.843589i	$-0.319561\pi$
$762$	0	0
$763$	−112.221	−0.147078
$764$	−486.178	−0.636358
$765$	0	0
$766$	254.780i	0.332611i
$767$	89.4852i	0.116669i
$768$	0	0
$769$	261.621i	0.340209i	0.985426	+	0.170105i	$0.0544105\pi$
$769$	261.621i	0.340209i	−0.985426	+	0.170105i	$0.945589\pi$
$770$	0	0
$771$	0	0
$772$	2512.39i	3.25439i
$773$	−428.491	−0.554322	−0.277161	−	0.960824i	$-0.589393\pi$
$773$	−428.491	−0.554322	−0.277161	+	0.960824i	$0.589393\pi$
$774$	0	0
$775$	451.432	0.582493
$776$	− 2306.76i	− 2.97262i
$777$	0	0
$778$	− 1593.75i	− 2.04852i
$779$	−342.704	−0.439928
$780$	0	0
$781$	0	0
$782$	1978.49	2.53003
$783$	0	0
$784$	885.253	1.12915
$785$	−267.204	−0.340388
$786$	0	0
$787$	226.672i	0.288020i	0.989576	+	0.144010i	$0.0459998\pi$
$787$	226.672i	0.288020i	−0.989576	+	0.144010i	$0.954000\pi$
$788$	355.422i	0.451043i
$789$	0	0
$790$	−503.196	−0.636958
$791$	− 185.522i	− 0.234541i
$792$	0	0
$793$	210.801	0.265827
$794$	298.127i	0.375474i
$795$	0	0
$796$	2137.87	2.68576
$797$	1008.01	1.26475	0.632376	−	0.774661i	$-0.282080\pi$
$797$	1008.01	1.26475	0.632376	+	0.774661i	$0.282080\pi$
$798$	0	0
$799$	− 647.383i	− 0.810241i
$800$	− 326.011i	− 0.407513i
$801$	0	0
$802$	1036.33i	1.29219i
$803$	0	0
$804$	0	0
$805$	353.673i	0.439345i
$806$	1204.92	1.49493
$807$	0	0
$808$	2285.48	2.82857
$809$	590.222i	0.729569i	0.931092	+	0.364785i	$0.118857\pi$
$809$	590.222i	0.729569i	−0.931092	+	0.364785i	$0.881143\pi$
$810$	0	0
$811$	51.3911i	0.0633676i	0.999498	+	0.0316838i	$0.0100870\pi$
$811$	51.3911i	0.0633676i	−0.999498	+	0.0316838i	$0.989913\pi$
$812$	−932.263	−1.14811
$813$	0	0
$814$	0	0
$815$	−451.666	−0.554191
$816$	0	0
$817$	−1013.47	−1.24048
$818$	−1934.01	−2.36432
$819$	0	0
$820$	398.437i	0.485899i
$821$	1265.51i	1.54143i	0.637181	+	0.770714i	$0.280100\pi$
$821$	1265.51i	1.54143i	−0.637181	+	0.770714i	$0.719900\pi$
$822$	0	0
$823$	−645.137	−0.783885	−0.391943	−	0.919990i	$-0.628197\pi$
$823$	−645.137	−0.783885	−0.391943	+	0.919990i	$0.628197\pi$
$824$	2549.88i	3.09452i
$825$	0	0
$826$	126.785	0.153493
$827$	− 823.126i	− 0.995315i	−0.867374	−	0.497658i	$-0.834194\pi$
$827$	− 823.126i	− 0.995315i	0.867374	−	0.497658i	$-0.165806\pi$
$828$	0	0
$829$	−129.174	−0.155820	−0.0779098	−	0.996960i	$-0.524825\pi$
$829$	−129.174	−0.155820	−0.0779098	+	0.996960i	$0.524825\pi$
$830$	1472.82	1.77448
$831$	0	0
$832$	153.189i	0.184122i
$833$	709.762i	0.852055i
$834$	0	0
$835$	245.628i	0.294165i
$836$	0	0
$837$	0	0
$838$	− 2948.20i	− 3.51813i
$839$	−263.217	−0.313728	−0.156864	−	0.987620i	$-0.550138\pi$
$839$	−263.217	−0.313728	−0.156864	+	0.987620i	$0.550138\pi$
$840$	0	0
$841$	95.0799	0.113056
$842$	− 170.688i	− 0.202717i
$843$	0	0
$844$	− 1365.79i	− 1.61824i
$845$	251.164	0.297236
$846$	0	0
$847$	0	0
$848$	−819.510	−0.966403
$849$	0	0
$850$	982.882	1.15633
$851$	552.833	0.649627
$852$	0	0
$853$	− 668.377i	− 0.783560i	−0.920059	−	0.391780i	$-0.871859\pi$
$853$	− 668.377i	− 0.783560i	0.920059	−	0.391780i	$-0.128141\pi$
$854$	− 298.669i	− 0.349729i
$855$	0	0
$856$	581.710	0.679567
$857$	930.246i	1.08547i	0.839905	+	0.542734i	$0.182611\pi$
$857$	930.246i	1.08547i	−0.839905	+	0.542734i	$0.817389\pi$
$858$	0	0
$859$	492.266	0.573068	0.286534	−	0.958070i	$-0.407497\pi$
$859$	492.266	0.573068	0.286534	+	0.958070i	$0.407497\pi$
$860$	1178.29i	1.37010i
$861$	0	0
$862$	−2319.21	−2.69050
$863$	−927.386	−1.07461	−0.537304	−	0.843389i	$-0.680557\pi$
$863$	−927.386	−1.07461	−0.537304	+	0.843389i	$0.680557\pi$
$864$	0	0
$865$	224.244i	0.259241i
$866$	− 1495.87i	− 1.72733i
$867$	0	0
$868$	− 1173.24i	− 1.35166i
$869$	0	0
$870$	0	0
$871$	− 772.796i	− 0.887251i
$872$	−494.978	−0.567635
$873$	0	0
$874$	−2462.30	−2.81728
$875$	510.135i	0.583012i
$876$	0	0
$877$	− 1024.32i	− 1.16798i	−0.811759	−	0.583992i	$-0.801490\pi$
$877$	− 1024.32i	− 1.16798i	0.811759	−	0.583992i	$-0.198510\pi$
$878$	−2425.11	−2.76209
$879$	0	0
$880$	0	0
$881$	−882.106	−1.00126	−0.500628	−	0.865663i	$-0.666897\pi$
$881$	−882.106	−1.00126	−0.500628	+	0.865663i	$0.666897\pi$
$882$	0	0
$883$	−1251.63	−1.41748	−0.708738	−	0.705471i	$-0.750735\pi$
$883$	−1251.63	−1.41748	−0.708738	+	0.705471i	$0.750735\pi$
$884$	1802.93	2.03951
$885$	0	0
$886$	2948.85i	3.32827i
$887$	903.715i	1.01884i	0.860517	+	0.509422i	$0.170141\pi$
$887$	903.715i	1.01884i	−0.860517	+	0.509422i	$0.829859\pi$
$888$	0	0
$889$	−295.379	−0.332260
$890$	− 274.430i	− 0.308348i
$891$	0	0
$892$	−2827.39	−3.16972
$893$	805.693i	0.902232i
$894$	0	0
$895$	477.329	0.533329
$896$	602.624	0.672572
$897$	0	0
$898$	− 557.994i	− 0.621374i
$899$	− 938.732i	− 1.04420i
$900$	0	0
$901$	− 657.052i	− 0.729247i
$902$	0	0
$903$	0	0
$904$	− 818.289i	− 0.905187i
$905$	−1069.14	−1.18137
$906$	0	0
$907$	−197.589	−0.217849	−0.108924	−	0.994050i	$-0.534741\pi$
$907$	−197.589	−0.217849	−0.108924	+	0.994050i	$0.534741\pi$
$908$	1455.07i	1.60251i
$909$	0	0
$910$	468.959i	0.515339i
$911$	1169.51	1.28376	0.641881	−	0.766805i	$-0.278154\pi$
$911$	1169.51	1.28376	0.641881	+	0.766805i	$0.278154\pi$
$912$	0	0
$913$	0	0
$914$	−1641.93	−1.79643
$915$	0	0
$916$	−2687.24	−2.93367
$917$	−536.502	−0.585062
$918$	0	0
$919$	321.126i	0.349429i	0.984619	+	0.174715i	$0.0559003\pi$
$919$	321.126i	0.349429i	−0.984619	+	0.174715i	$0.944100\pi$
$920$	1559.96i	1.69561i
$921$	0	0
$922$	−1917.04	−2.07922
$923$	− 1072.01i	− 1.16144i
$924$	0	0
$925$	274.639	0.296907
$926$	− 475.644i	− 0.513655i
$927$	0	0
$928$	−677.924	−0.730522
$929$	20.3301	0.0218838	0.0109419	−	0.999940i	$-0.496517\pi$
$929$	20.3301	0.0218838	0.0109419	+	0.999940i	$0.496517\pi$
$930$	0	0
$931$	− 883.326i	− 0.948793i
$932$	− 2991.96i	− 3.21026i
$933$	0	0
$934$	1733.23i	1.85571i
$935$	0	0
$936$	0	0
$937$	− 10.7489i	− 0.0114716i	−0.999984	−	0.00573581i	$-0.998174\pi$
$937$	− 10.7489i	− 0.0114716i	0.999984	−	0.00573581i	$-0.00182578\pi$
$938$	−1094.92	−1.16729
$939$	0	0
$940$	936.720	0.996511
$941$	− 817.514i	− 0.868771i	−0.900727	−	0.434386i	$-0.856966\pi$
$941$	− 817.514i	− 0.868771i	0.900727	−	0.434386i	$-0.143034\pi$
$942$	0	0
$943$	347.908i	0.368937i
$944$	238.258	0.252392
$945$	0	0
$946$	0	0
$947$	615.594	0.650046	0.325023	−	0.945706i	$-0.394628\pi$
$947$	615.594	0.650046	0.325023	+	0.945706i	$0.394628\pi$
$948$	0	0
$949$	332.018	0.349861
$950$	−1223.23	−1.28762
$951$	0	0
$952$	− 1391.97i	− 1.46215i
$953$	681.521i	0.715132i	0.933888	+	0.357566i	$0.116393\pi$
$953$	681.521i	0.715132i	−0.933888	+	0.357566i	$0.883607\pi$
$954$	0	0
$955$	190.535	0.199514
$956$	− 1965.03i	− 2.05547i
$957$	0	0
$958$	−3098.47	−3.23431
$959$	− 582.936i	− 0.607859i
$960$	0	0
$961$	220.382	0.229326
$962$	733.038	0.761994
$963$	0	0
$964$	1893.65i	1.96436i
$965$	− 984.617i	− 1.02033i
$966$	0	0
$967$	− 947.157i	− 0.979479i	−0.871869	−	0.489740i	$-0.837092\pi$
$967$	− 947.157i	− 0.979479i	0.871869	−	0.489740i	$-0.162908\pi$
$968$	0	0
$969$	0	0
$970$	1659.01i	1.71032i
$971$	1296.64	1.33536	0.667682	−	0.744446i	$-0.267287\pi$
$971$	1296.64	1.33536	0.667682	+	0.744446i	$0.267287\pi$
$972$	0	0
$973$	988.059	1.01548
$974$	2074.60i	2.12998i
$975$	0	0
$976$	− 561.267i	− 0.575068i
$977$	−1923.23	−1.96851	−0.984255	−	0.176754i	$-0.943440\pi$
$977$	−1923.23	−1.96851	−0.984255	+	0.176754i	$0.943440\pi$
$978$	0	0
$979$	0	0
$980$	−1026.98	−1.04794
$981$	0	0
$982$	−1493.71	−1.52109
$983$	489.785	0.498256	0.249128	−	0.968471i	$-0.419856\pi$
$983$	489.785	0.498256	0.249128	+	0.968471i	$0.419856\pi$
$984$	0	0
$985$	− 139.291i	− 0.141413i
$986$	− 2043.86i	− 2.07288i
$987$	0	0
$988$	−2243.82	−2.27107
$989$	1028.86i	1.04030i
$990$	0	0
$991$	−159.005	−0.160450	−0.0802248	−	0.996777i	$-0.525564\pi$
$991$	−159.005	−0.160450	−0.0802248	+	0.996777i	$0.525564\pi$
$992$	− 853.159i	− 0.860039i
$993$	0	0
$994$	−1518.86	−1.52803
$995$	−837.841	−0.842051
$996$	0	0
$997$	− 309.823i	− 0.310755i	−0.987855	−	0.155378i	$-0.950341\pi$
$997$	− 309.823i	− 0.310755i	0.987855	−	0.155378i	$-0.0496594\pi$
$998$	2870.51i	2.87626i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.3.c.k.604.8		8
3.2	odd	2		121.3.b.c.120.1	✓	8
11.10	odd	2	inner	1089.3.c.k.604.1		8
33.2	even	10		121.3.d.f.40.8		32
33.5	odd	10		121.3.d.f.118.8		32
33.8	even	10		121.3.d.f.112.8		32
33.14	odd	10		121.3.d.f.112.1		32
33.17	even	10		121.3.d.f.118.1		32
33.20	odd	10		121.3.d.f.40.1		32
33.26	odd	10		121.3.d.f.94.8		32
33.29	even	10		121.3.d.f.94.1		32
33.32	even	2		121.3.b.c.120.8	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
121.3.b.c.120.1	✓	8	3.2	odd	2
121.3.b.c.120.8	yes	8	33.32	even	2
121.3.d.f.40.1		32	33.20	odd	10
121.3.d.f.40.8		32	33.2	even	10
121.3.d.f.94.1		32	33.29	even	10
121.3.d.f.94.8		32	33.26	odd	10
121.3.d.f.112.1		32	33.14	odd	10
121.3.d.f.112.8		32	33.8	even	10
121.3.d.f.118.1		32	33.17	even	10
121.3.d.f.118.8		32	33.5	odd	10
1089.3.c.k.604.1		8	11.10	odd	2	inner
1089.3.c.k.604.8		8	1.1	even	1	trivial

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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1089.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.57626i q^{2} -8.78965 q^{4} +3.44471 q^{5} -3.88348i q^{7} -17.1290i q^{8} +O(q^{10})\)	\(q+3.57626i q^{2} -8.78965 q^{4} +3.44471 q^{5} -3.88348i q^{7} -17.1290i q^{8} +12.3192i q^{10} +9.80240i q^{13} +13.8883 q^{14} +26.0993 q^{16} +20.9255i q^{17} -26.0425i q^{19} -30.2778 q^{20} -26.4380 q^{23} -13.1340 q^{25} -35.0559 q^{26} +34.1344i q^{28} +27.3115i q^{29} -34.3712 q^{31} +24.8219i q^{32} -74.8349 q^{34} -13.3774i q^{35} -20.9105 q^{37} +93.1349 q^{38} -59.0045i q^{40} -13.1594i q^{41} -38.9159i q^{43} -94.5492i q^{46} -30.9376 q^{47} +33.9186 q^{49} -46.9706i q^{50} -86.1596i q^{52} -31.3996 q^{53} -66.5202 q^{56} -97.6732 q^{58} +9.12891 q^{59} -21.5050i q^{61} -122.921i q^{62} +15.6278 q^{64} +33.7664i q^{65} -78.8374 q^{67} -183.927i q^{68} +47.8412 q^{70} -109.362 q^{71} -33.8711i q^{73} -74.7815i q^{74} +228.905i q^{76} +40.8466i q^{79} +89.9045 q^{80} +47.0614 q^{82} -119.555i q^{83} +72.0820i q^{85} +139.174 q^{86} -22.2767 q^{89} +38.0674 q^{91} +232.381 q^{92} -110.641i q^{94} -89.7089i q^{95} +134.669 q^{97} +121.302i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 24 q^{4} + 4 q^{5}+O(q^{10}) \) 8 * q - 24 * q^4 + 4 * q^5	\( 8 q - 24 q^{4} + 4 q^{5} + 4 q^{14} - 24 q^{16} - 52 q^{20} + 12 q^{23} - 16 q^{25} - 168 q^{26} - 116 q^{31} - 180 q^{34} - 4 q^{37} + 132 q^{38} + 244 q^{47} + 88 q^{49} - 268 q^{53} + 12 q^{56} + 88 q^{58} + 56 q^{59} - 40 q^{64} + 284 q^{67} - 188 q^{70} - 272 q^{71} + 356 q^{80} - 180 q^{82} - 336 q^{86} + 24 q^{89} + 140 q^{91} + 156 q^{92} + 152 q^{97}+O(q^{100}) \) 8 * q - 24 * q^4 + 4 * q^5 + 4 * q^14 - 24 * q^16 - 52 * q^20 + 12 * q^23 - 16 * q^25 - 168 * q^26 - 116 * q^31 - 180 * q^34 - 4 * q^37 + 132 * q^38 + 244 * q^47 + 88 * q^49 - 268 * q^53 + 12 * q^56 + 88 * q^58 + 56 * q^59 - 40 * q^64 + 284 * q^67 - 188 * q^70 - 272 * q^71 + 356 * q^80 - 180 * q^82 - 336 * q^86 + 24 * q^89 + 140 * q^91 + 156 * q^92 + 152 * q^97

Introduction

L-functions

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Database

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