LMFDB - Embedded newform 1089.3.c.l.604.16

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:	$16$
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} - 21x^{14} + 227x^{12} - 1488x^{10} + 24225x^{8} - 62832x^{6} + 64372x^{4} + 7986x^{2} + 14641 $ x^16 - 21x^14 + 227x^12 - 1488x^10 + 24225x^8 - 62832x^6 + 64372x^4 + 7986*x^2 + 14641
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{12}\cdot 11^{4} $
Twist minimal:	no (minimal twist has level 99)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			604.16
Root			$3.67414 + 1.19380i$ of defining polynomial
Character	$\chi$	$=$	1089.604
Dual form			1089.3.c.l.604.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.86322i q^{2} -10.9245 q^{4} +4.15040 q^{5} -7.44465i q^{7} -26.7509i q^{8} +O(q^{10})$	$q+3.86322i q^{2} -10.9245 q^{4} +4.15040 q^{5} -7.44465i q^{7} -26.7509i q^{8} +16.0339i q^{10} +7.32428i q^{13} +28.7603 q^{14} +59.6467 q^{16} -3.57908i q^{17} +29.9894i q^{19} -45.3410 q^{20} -20.6531 q^{23} -7.77420 q^{25} -28.2953 q^{26} +81.3291i q^{28} +17.1852i q^{29} +15.7747 q^{31} +123.425i q^{32} +13.8268 q^{34} -30.8983i q^{35} -32.2745 q^{37} -115.856 q^{38} -111.027i q^{40} +16.2215i q^{41} +33.7299i q^{43} -79.7874i q^{46} -38.9036 q^{47} -6.42281 q^{49} -30.0335i q^{50} -80.0141i q^{52} -36.6137 q^{53} -199.151 q^{56} -66.3902 q^{58} -69.9524 q^{59} -22.7511i q^{61} +60.9413i q^{62} -238.231 q^{64} +30.3987i q^{65} -63.0682 q^{67} +39.0996i q^{68} +119.367 q^{70} +31.5933 q^{71} +103.388i q^{73} -124.683i q^{74} -327.620i q^{76} +18.2635i q^{79} +247.557 q^{80} -62.6675 q^{82} +118.991i q^{83} -14.8546i q^{85} -130.306 q^{86} +154.739 q^{89} +54.5267 q^{91} +225.624 q^{92} -150.293i q^{94} +124.468i q^{95} -69.2168 q^{97} -24.8128i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 44 q^{4}+O(q^{10}) $ 16 * q - 44 * q^4	$ 16 q - 44 q^{4} + 244 q^{16} + 16 q^{25} - 80 q^{31} + 328 q^{34} - 280 q^{37} + 436 q^{49} + 140 q^{58} - 656 q^{64} + 300 q^{67} + 308 q^{70} + 580 q^{82} + 768 q^{91} - 720 q^{97}+O(q^{100}) $ 16 * q - 44 * q^4 + 244 * q^16 + 16 * q^25 - 80 * q^31 + 328 * q^34 - 280 * q^37 + 436 * q^49 + 140 * q^58 - 656 * q^64 + 300 * q^67 + 308 * q^70 + 580 * q^82 + 768 * q^91 - 720 * 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 $

$2$	3.86322i	1.93161i	0.259266	+	0.965806i	$0.416519\pi$
$2$	3.86322i	1.93161i	−0.259266	+	0.965806i	$0.583481\pi$
$3$	0	0
$3$	0	0
$4$	−10.9245	−2.73112
$5$	4.15040	0.830079	0.415040	−	0.909803i	$-0.363768\pi$
$5$	4.15040	0.830079	0.415040	+	0.909803i	$0.363768\pi$
$6$	0	0
$7$	− 7.44465i	− 1.06352i	−0.846895	−	0.531761i	$-0.821531\pi$
$7$	− 7.44465i	− 1.06352i	0.846895	−	0.531761i	$-0.178469\pi$
$8$	− 26.7509i	− 3.34386i
$9$	0	0
$10$	16.0339i	1.60339i
$11$	0	0
$11$	0	0
$12$	0	0
$13$	7.32428i	0.563406i	0.959502	+	0.281703i	$0.0908993\pi$
$13$	7.32428i	0.563406i	−0.959502	+	0.281703i	$0.909101\pi$
$14$	28.7603	2.05431
$15$	0	0
$16$	59.6467	3.72792
$17$	− 3.57908i	− 0.210534i	−0.994444	−	0.105267i	$-0.966430\pi$
$17$	− 3.57908i	− 0.210534i	0.994444	−	0.105267i	$-0.0335697\pi$
$18$	0	0
$19$	29.9894i	1.57839i	0.614142	+	0.789196i	$0.289502\pi$
$19$	29.9894i	1.57839i	−0.614142	+	0.789196i	$0.710498\pi$
$20$	−45.3410	−2.26705
$21$	0	0
$22$	0	0
$23$	−20.6531	−0.897959	−0.448980	−	0.893542i	$-0.648212\pi$
$23$	−20.6531	−0.897959	−0.448980	+	0.893542i	$0.648212\pi$
$24$	0	0
$25$	−7.77420	−0.310968
$26$	−28.2953	−1.08828
$27$	0	0
$28$	81.3291i	2.90461i
$29$	17.1852i	0.592592i	0.955096	+	0.296296i	$0.0957515\pi$
$29$	17.1852i	0.592592i	−0.955096	+	0.296296i	$0.904248\pi$
$30$	0	0
$31$	15.7747	0.508862	0.254431	−	0.967091i	$-0.418112\pi$
$31$	15.7747	0.508862	0.254431	+	0.967091i	$0.418112\pi$
$32$	123.425i	3.85703i
$33$	0	0
$34$	13.8268	0.406670
$35$	− 30.8983i	− 0.882807i
$36$	0	0
$37$	−32.2745	−0.872283	−0.436141	−	0.899878i	$-0.643655\pi$
$37$	−32.2745	−0.872283	−0.436141	+	0.899878i	$0.643655\pi$
$38$	−115.856	−3.04884
$39$	0	0
$40$	− 111.027i	− 2.77567i
$41$	16.2215i	0.395647i	0.980238	+	0.197824i	$0.0633874\pi$
$41$	16.2215i	0.395647i	−0.980238	+	0.197824i	$0.936613\pi$
$42$	0	0
$43$	33.7299i	0.784415i	0.919877	+	0.392208i	$0.128289\pi$
$43$	33.7299i	0.784415i	−0.919877	+	0.392208i	$0.871711\pi$
$44$	0	0
$45$	0	0
$46$	− 79.7874i	− 1.73451i
$47$	−38.9036	−0.827737	−0.413868	−	0.910337i	$-0.635823\pi$
$47$	−38.9036	−0.827737	−0.413868	+	0.910337i	$0.635823\pi$
$48$	0	0
$49$	−6.42281	−0.131078
$50$	− 30.0335i	− 0.600670i
$51$	0	0
$52$	− 80.0141i	− 1.53873i
$53$	−36.6137	−0.690824	−0.345412	−	0.938451i	$-0.612261\pi$
$53$	−36.6137	−0.690824	−0.345412	+	0.938451i	$0.612261\pi$
$54$	0	0
$55$	0	0
$56$	−199.151	−3.55627
$57$	0	0
$58$	−66.3902	−1.14466
$59$	−69.9524	−1.18563	−0.592817	−	0.805338i	$-0.701984\pi$
$59$	−69.9524	−1.18563	−0.592817	+	0.805338i	$0.701984\pi$
$60$	0	0
$61$	− 22.7511i	− 0.372968i	−0.982458	−	0.186484i	$-0.940291\pi$
$61$	− 22.7511i	− 0.372968i	0.982458	−	0.186484i	$-0.0597093\pi$
$62$	60.9413i	0.982925i
$63$	0	0
$64$	−238.231	−3.72236
$65$	30.3987i	0.467672i
$66$	0	0
$67$	−63.0682	−0.941317	−0.470658	−	0.882315i	$-0.655984\pi$
$67$	−63.0682	−0.941317	−0.470658	+	0.882315i	$0.655984\pi$
$68$	39.0996i	0.574994i
$69$	0	0
$70$	119.367	1.70524
$71$	31.5933	0.444976	0.222488	−	0.974935i	$-0.428582\pi$
$71$	31.5933	0.444976	0.222488	+	0.974935i	$0.428582\pi$
$72$	0	0
$73$	103.388i	1.41628i	0.706075	+	0.708138i	$0.250464\pi$
$73$	103.388i	1.41628i	−0.706075	+	0.708138i	$0.749536\pi$
$74$	− 124.683i	− 1.68491i
$75$	0	0
$76$	− 327.620i	− 4.31078i
$77$	0	0
$78$	0	0
$79$	18.2635i	0.231183i	0.993297	+	0.115592i	$0.0368764\pi$
$79$	18.2635i	0.231183i	−0.993297	+	0.115592i	$0.963124\pi$
$80$	247.557	3.09447
$81$	0	0
$82$	−62.6675	−0.764237
$83$	118.991i	1.43362i	0.697268	+	0.716811i	$0.254399\pi$
$83$	118.991i	1.43362i	−0.697268	+	0.716811i	$0.745601\pi$
$84$	0	0
$85$	− 14.8546i	− 0.174760i
$86$	−130.306	−1.51519
$87$	0	0
$88$	0	0
$89$	154.739	1.73864	0.869320	−	0.494249i	$-0.164557\pi$
$89$	154.739	1.73864	0.869320	+	0.494249i	$0.164557\pi$
$90$	0	0
$91$	54.5267	0.599195
$92$	225.624	2.45244
$93$	0	0
$94$	− 150.293i	− 1.59887i
$95$	124.468i	1.31019i
$96$	0	0
$97$	−69.2168	−0.713575	−0.356788	−	0.934186i	$-0.616128\pi$
$97$	−69.2168	−0.713575	−0.356788	+	0.934186i	$0.616128\pi$
$98$	− 24.8128i	− 0.253191i
$99$	0	0
$100$	84.9293	0.849293
$101$	137.148i	1.35790i	0.734186	+	0.678948i	$0.237564\pi$
$101$	137.148i	1.35790i	−0.734186	+	0.678948i	$0.762436\pi$
$102$	0	0
$103$	20.5216	0.199239	0.0996193	−	0.995026i	$-0.468238\pi$
$103$	20.5216	0.199239	0.0996193	+	0.995026i	$0.468238\pi$
$104$	195.931	1.88395
$105$	0	0
$106$	− 141.447i	− 1.33440i
$107$	− 59.3985i	− 0.555127i	−0.960707	−	0.277563i	$-0.910473\pi$
$107$	− 59.3985i	− 0.555127i	0.960707	−	0.277563i	$-0.0895268\pi$
$108$	0	0
$109$	− 37.5085i	− 0.344115i	−0.985087	−	0.172058i	$-0.944958\pi$
$109$	− 37.5085i	− 0.344115i	0.985087	−	0.172058i	$-0.0550415\pi$
$110$	0	0
$111$	0	0
$112$	− 444.049i	− 3.96472i
$113$	−117.154	−1.03676	−0.518380	−	0.855151i	$-0.673465\pi$
$113$	−117.154	−1.03676	−0.518380	+	0.855151i	$0.673465\pi$
$114$	0	0
$115$	−85.7184	−0.745377
$116$	− 187.739i	− 1.61844i
$117$	0	0
$118$	− 270.242i	− 2.29018i
$119$	−26.6450	−0.223907
$120$	0	0
$121$	0	0
$122$	87.8925	0.720430
$123$	0	0
$124$	−172.331	−1.38977
$125$	−136.026	−1.08821
$126$	0	0
$127$	30.4604i	0.239846i	0.992783	+	0.119923i	$0.0382648\pi$
$127$	30.4604i	0.239846i	−0.992783	+	0.119923i	$0.961735\pi$
$128$	− 426.641i	− 3.33313i
$129$	0	0
$130$	−117.437	−0.903361
$131$	− 179.208i	− 1.36800i	−0.729482	−	0.684000i	$-0.760239\pi$
$131$	− 179.208i	− 1.36800i	0.729482	−	0.684000i	$-0.239761\pi$
$132$	0	0
$133$	223.261	1.67865
$134$	− 243.647i	− 1.81826i
$135$	0	0
$136$	−95.7435	−0.703996
$137$	−192.096	−1.40216	−0.701080	−	0.713082i	$-0.747299\pi$
$137$	−192.096	−1.40216	−0.701080	+	0.713082i	$0.747299\pi$
$138$	0	0
$139$	− 24.5728i	− 0.176783i	−0.996086	−	0.0883914i	$-0.971827\pi$
$139$	− 24.5728i	− 0.176783i	0.996086	−	0.0883914i	$-0.0281726\pi$
$140$	337.548i	2.41106i
$141$	0	0
$142$	122.052i	0.859520i
$143$	0	0
$144$	0	0
$145$	71.3253i	0.491899i
$146$	−399.411	−2.73569
$147$	0	0
$148$	352.582	2.38231
$149$	28.8346i	0.193521i	0.995308	+	0.0967603i	$0.0308480\pi$
$149$	28.8346i	0.193521i	−0.995308	+	0.0967603i	$0.969152\pi$
$150$	0	0
$151$	140.463i	0.930218i	0.885254	+	0.465109i	$0.153985\pi$
$151$	140.463i	0.930218i	−0.885254	+	0.465109i	$0.846015\pi$
$152$	802.244	5.27792
$153$	0	0
$154$	0	0
$155$	65.4714	0.422396
$156$	0	0
$157$	264.782	1.68651	0.843255	−	0.537514i	$-0.180636\pi$
$157$	264.782	1.68651	0.843255	+	0.537514i	$0.180636\pi$
$158$	−70.5559	−0.446556
$159$	0	0
$160$	512.262i	3.20164i
$161$	153.755i	0.954999i
$162$	0	0
$163$	64.2445	0.394138	0.197069	−	0.980390i	$-0.436858\pi$
$163$	64.2445	0.394138	0.197069	+	0.980390i	$0.436858\pi$
$164$	− 177.212i	− 1.08056i
$165$	0	0
$166$	−459.687	−2.76920
$167$	− 114.868i	− 0.687830i	−0.939001	−	0.343915i	$-0.888247\pi$
$167$	− 114.868i	− 0.687830i	0.939001	−	0.343915i	$-0.111753\pi$
$168$	0	0
$169$	115.355	0.682573
$170$	57.3866	0.337568
$171$	0	0
$172$	− 368.482i	− 2.14234i
$173$	233.750i	1.35116i	0.737288	+	0.675578i	$0.236106\pi$
$173$	233.750i	1.35116i	−0.737288	+	0.675578i	$0.763894\pi$
$174$	0	0
$175$	57.8762i	0.330721i
$176$	0	0
$177$	0	0
$178$	597.791i	3.35838i
$179$	−222.362	−1.24225	−0.621123	−	0.783713i	$-0.713323\pi$
$179$	−222.362	−1.24225	−0.621123	+	0.783713i	$0.713323\pi$
$180$	0	0
$181$	269.413	1.48847	0.744234	−	0.667920i	$-0.232815\pi$
$181$	269.413	1.48847	0.744234	+	0.667920i	$0.232815\pi$
$182$	210.649i	1.15741i
$183$	0	0
$184$	552.488i	3.00265i
$185$	−133.952	−0.724064
$186$	0	0
$187$	0	0
$188$	425.003	2.26065
$189$	0	0
$190$	−480.848	−2.53078
$191$	−244.952	−1.28247	−0.641236	−	0.767344i	$-0.721578\pi$
$191$	−244.952	−1.28247	−0.641236	+	0.767344i	$0.721578\pi$
$192$	0	0
$193$	263.453i	1.36504i	0.730867	+	0.682520i	$0.239116\pi$
$193$	263.453i	1.36504i	−0.730867	+	0.682520i	$0.760884\pi$
$194$	− 267.400i	− 1.37835i
$195$	0	0
$196$	70.1660	0.357990
$197$	− 256.209i	− 1.30055i	−0.759697	−	0.650277i	$-0.774653\pi$
$197$	− 256.209i	− 1.30055i	0.759697	−	0.650277i	$-0.225347\pi$
$198$	0	0
$199$	−7.63062	−0.0383448	−0.0191724	−	0.999816i	$-0.506103\pi$
$199$	−7.63062	−0.0383448	−0.0191724	+	0.999816i	$0.506103\pi$
$200$	207.967i	1.03983i
$201$	0	0
$202$	−529.832	−2.62293
$203$	127.938	0.630235
$204$	0	0
$205$	67.3259i	0.328419i
$206$	79.2794i	0.384852i
$207$	0	0
$208$	436.869i	2.10033i
$209$	0	0
$210$	0	0
$211$	− 375.591i	− 1.78005i	−0.455911	−	0.890025i	$-0.650687\pi$
$211$	− 375.591i	− 1.78005i	0.455911	−	0.890025i	$-0.349313\pi$
$212$	399.986	1.88673
$213$	0	0
$214$	229.470	1.07229
$215$	139.992i	0.651127i
$216$	0	0
$217$	− 117.437i	− 0.541186i
$218$	144.904	0.664697
$219$	0	0
$220$	0	0
$221$	26.2142	0.118616
$222$	0	0
$223$	−28.1852	−0.126391	−0.0631955	−	0.998001i	$-0.520129\pi$
$223$	−28.1852	−0.126391	−0.0631955	+	0.998001i	$0.520129\pi$
$224$	918.855	4.10203
$225$	0	0
$226$	− 452.591i	− 2.00262i
$227$	272.560i	1.20071i	0.799735	+	0.600353i	$0.204973\pi$
$227$	272.560i	1.20071i	−0.799735	+	0.600353i	$0.795027\pi$
$228$	0	0
$229$	0.590239	0.00257746	0.00128873	−	0.999999i	$-0.499590\pi$
$229$	0.590239	0.00257746	0.00128873	+	0.999999i	$0.499590\pi$
$230$	− 331.149i	− 1.43978i
$231$	0	0
$232$	459.719	1.98155
$233$	− 185.559i	− 0.796391i	−0.917301	−	0.398196i	$-0.869637\pi$
$233$	− 185.559i	− 0.796391i	0.917301	−	0.398196i	$-0.130363\pi$
$234$	0	0
$235$	−161.465	−0.687087
$236$	764.194	3.23811
$237$	0	0
$238$	− 102.935i	− 0.432502i
$239$	− 269.209i	− 1.12640i	−0.826322	−	0.563198i	$-0.809571\pi$
$239$	− 269.209i	− 1.12640i	0.826322	−	0.563198i	$-0.190429\pi$
$240$	0	0
$241$	249.403i	1.03487i	0.855723	+	0.517434i	$0.173113\pi$
$241$	249.403i	1.03487i	−0.855723	+	0.517434i	$0.826887\pi$
$242$	0	0
$243$	0	0
$244$	248.544i	1.01862i
$245$	−26.6572	−0.108805
$246$	0	0
$247$	−219.651	−0.889276
$248$	− 421.988i	− 1.70156i
$249$	0	0
$250$	− 525.499i	− 2.10199i
$251$	−321.026	−1.27899	−0.639493	−	0.768797i	$-0.720856\pi$
$251$	−321.026	−1.27899	−0.639493	+	0.768797i	$0.720856\pi$
$252$	0	0
$253$	0	0
$254$	−117.675	−0.463289
$255$	0	0
$256$	695.285	2.71596
$257$	11.5105	0.0447878	0.0223939	−	0.999749i	$-0.492871\pi$
$257$	11.5105	0.0447878	0.0223939	+	0.999749i	$0.492871\pi$
$258$	0	0
$259$	240.272i	0.927692i
$260$	− 332.090i	− 1.27727i
$261$	0	0
$262$	692.321	2.64245
$263$	− 184.999i	− 0.703417i	−0.936110	−	0.351708i	$-0.885601\pi$
$263$	− 184.999i	− 0.703417i	0.936110	−	0.351708i	$-0.114399\pi$
$264$	0	0
$265$	−151.961	−0.573439
$266$	862.507i	3.24251i
$267$	0	0
$268$	688.989	2.57085
$269$	−242.153	−0.900198	−0.450099	−	0.892979i	$-0.648611\pi$
$269$	−242.153	−0.900198	−0.450099	+	0.892979i	$0.648611\pi$
$270$	0	0
$271$	243.989i	0.900329i	0.892946	+	0.450164i	$0.148635\pi$
$271$	243.989i	0.900329i	−0.892946	+	0.450164i	$0.851365\pi$
$272$	− 213.480i	− 0.784853i
$273$	0	0
$274$	− 742.110i	− 2.70843i
$275$	0	0
$276$	0	0
$277$	− 144.134i	− 0.520340i	−0.965563	−	0.260170i	$-0.916221\pi$
$277$	− 144.134i	− 0.520340i	0.965563	−	0.260170i	$-0.0837785\pi$
$278$	94.9303	0.341476
$279$	0	0
$280$	−826.556	−2.95198
$281$	− 359.273i	− 1.27855i	−0.768977	−	0.639277i	$-0.779234\pi$
$281$	− 359.273i	− 1.27855i	0.768977	−	0.639277i	$-0.220766\pi$
$282$	0	0
$283$	443.556i	1.56734i	0.621180	+	0.783668i	$0.286654\pi$
$283$	443.556i	1.56734i	−0.621180	+	0.783668i	$0.713346\pi$
$284$	−345.141	−1.21528
$285$	0	0
$286$	0	0
$287$	120.764	0.420780
$288$	0	0
$289$	276.190	0.955675
$290$	−275.546	−0.950157
$291$	0	0
$292$	− 1129.46i	− 3.86802i
$293$	514.650i	1.75648i	0.478217	+	0.878242i	$0.341283\pi$
$293$	514.650i	1.75648i	−0.478217	+	0.878242i	$0.658717\pi$
$294$	0	0
$295$	−290.330	−0.984170
$296$	863.371i	2.91679i
$297$	0	0
$298$	−111.394	−0.373807
$299$	− 151.269i	− 0.505916i
$300$	0	0
$301$	251.107	0.834242
$302$	−542.639	−1.79682
$303$	0	0
$304$	1788.77i	5.88411i
$305$	− 94.4260i	− 0.309593i
$306$	0	0
$307$	− 145.625i	− 0.474349i	−0.971467	−	0.237175i	$-0.923779\pi$
$307$	− 145.625i	− 0.474349i	0.971467	−	0.237175i	$-0.0762214\pi$
$308$	0	0
$309$	0	0
$310$	252.931i	0.815906i
$311$	−396.344	−1.27442	−0.637208	−	0.770692i	$-0.719911\pi$
$311$	−396.344	−1.27442	−0.637208	+	0.770692i	$0.719911\pi$
$312$	0	0
$313$	−300.680	−0.960638	−0.480319	−	0.877094i	$-0.659479\pi$
$313$	−300.680	−0.960638	−0.480319	+	0.877094i	$0.659479\pi$
$314$	1022.91i	3.25768i
$315$	0	0
$316$	− 199.519i	− 0.631390i
$317$	270.066	0.851943	0.425972	−	0.904737i	$-0.359932\pi$
$317$	270.066	0.851943	0.425972	+	0.904737i	$0.359932\pi$
$318$	0	0
$319$	0	0
$320$	−988.754	−3.08986
$321$	0	0
$322$	−593.989	−1.84469
$323$	107.334	0.332305
$324$	0	0
$325$	− 56.9405i	− 0.175201i
$326$	248.191i	0.761321i
$327$	0	0
$328$	433.941	1.32299
$329$	289.624i	0.880316i
$330$	0	0
$331$	172.536	0.521257	0.260629	−	0.965439i	$-0.416070\pi$
$331$	172.536	0.521257	0.260629	+	0.965439i	$0.416070\pi$
$332$	− 1299.91i	− 3.91540i
$333$	0	0
$334$	443.760	1.32862
$335$	−261.758	−0.781368
$336$	0	0
$337$	− 276.335i	− 0.819986i	−0.912089	−	0.409993i	$-0.865531\pi$
$337$	− 276.335i	− 0.819986i	0.912089	−	0.409993i	$-0.134469\pi$
$338$	445.642i	1.31847i
$339$	0	0
$340$	162.279i	0.477291i
$341$	0	0
$342$	0	0
$343$	− 316.972i	− 0.924117i
$344$	902.303	2.62298
$345$	0	0
$346$	−903.029	−2.60991
$347$	144.475i	0.416354i	0.978091	+	0.208177i	$0.0667530\pi$
$347$	144.475i	0.416354i	−0.978091	+	0.208177i	$0.933247\pi$
$348$	0	0
$349$	147.891i	0.423755i	0.977296	+	0.211878i	$0.0679578\pi$
$349$	147.891i	0.423755i	−0.977296	+	0.211878i	$0.932042\pi$
$350$	−223.589	−0.638825
$351$	0	0
$352$	0	0
$353$	24.8683	0.0704485	0.0352243	−	0.999379i	$-0.488785\pi$
$353$	24.8683	0.0704485	0.0352243	+	0.999379i	$0.488785\pi$
$354$	0	0
$355$	131.125	0.369365
$356$	−1690.45	−4.74844
$357$	0	0
$358$	− 859.034i	− 2.39954i
$359$	− 233.674i	− 0.650902i	−0.945559	−	0.325451i	$-0.894484\pi$
$359$	− 233.674i	− 0.650902i	0.945559	−	0.325451i	$-0.105516\pi$
$360$	0	0
$361$	−538.366	−1.49132
$362$	1040.80i	2.87514i
$363$	0	0
$364$	−595.677	−1.63648
$365$	429.102i	1.17562i
$366$	0	0
$367$	−548.153	−1.49361	−0.746803	−	0.665046i	$-0.768412\pi$
$367$	−548.153	−1.49361	−0.746803	+	0.665046i	$0.768412\pi$
$368$	−1231.89	−3.34752
$369$	0	0
$370$	− 517.486i	− 1.39861i
$371$	272.576i	0.734706i
$372$	0	0
$373$	− 84.3957i	− 0.226262i	−0.993580	−	0.113131i	$-0.963912\pi$
$373$	− 84.3957i	− 0.226262i	0.993580	−	0.113131i	$-0.0360879\pi$
$374$	0	0
$375$	0	0
$376$	1040.71i	2.76784i
$377$	−125.869	−0.333870
$378$	0	0
$379$	134.386	0.354580	0.177290	−	0.984159i	$-0.443267\pi$
$379$	134.386	0.354580	0.177290	+	0.984159i	$0.443267\pi$
$380$	− 1359.75i	− 3.57829i
$381$	0	0
$382$	− 946.305i	− 2.47724i
$383$	−413.818	−1.08046	−0.540232	−	0.841516i	$-0.681664\pi$
$383$	−413.818	−1.08046	−0.540232	+	0.841516i	$0.681664\pi$
$384$	0	0
$385$	0	0
$386$	−1017.78	−2.63673
$387$	0	0
$388$	756.159	1.94886
$389$	−214.844	−0.552297	−0.276148	−	0.961115i	$-0.589058\pi$
$389$	−214.844	−0.552297	−0.276148	+	0.961115i	$0.589058\pi$
$390$	0	0
$391$	73.9189i	0.189051i
$392$	171.816i	0.438306i
$393$	0	0
$394$	989.793	2.51216
$395$	75.8007i	0.191900i
$396$	0	0
$397$	633.921	1.59678	0.798389	−	0.602142i	$-0.205686\pi$
$397$	633.921	1.59678	0.798389	+	0.602142i	$0.205686\pi$
$398$	− 29.4788i	− 0.0740673i
$399$	0	0
$400$	−463.705	−1.15926
$401$	316.692	0.789756	0.394878	−	0.918733i	$-0.370787\pi$
$401$	316.692	0.789756	0.394878	+	0.918733i	$0.370787\pi$
$402$	0	0
$403$	115.539i	0.286696i
$404$	− 1498.27i	− 3.70858i
$405$	0	0
$406$	494.252i	1.21737i
$407$	0	0
$408$	0	0
$409$	378.419i	0.925231i	0.886559	+	0.462615i	$0.153089\pi$
$409$	378.419i	0.925231i	−0.886559	+	0.462615i	$0.846911\pi$
$410$	−260.095	−0.634378
$411$	0	0
$412$	−224.188	−0.544145
$413$	520.771i	1.26095i
$414$	0	0
$415$	493.858i	1.19002i
$416$	−903.998	−2.17307
$417$	0	0
$418$	0	0
$419$	187.826	0.448273	0.224137	−	0.974558i	$-0.428044\pi$
$419$	187.826	0.448273	0.224137	+	0.974558i	$0.428044\pi$
$420$	0	0
$421$	−153.479	−0.364558	−0.182279	−	0.983247i	$-0.558347\pi$
$421$	−153.479	−0.364558	−0.182279	+	0.983247i	$0.558347\pi$
$422$	1450.99	3.43837
$423$	0	0
$424$	979.449i	2.31002i
$425$	27.8245i	0.0654693i
$426$	0	0
$427$	−169.374	−0.396660
$428$	648.899i	1.51612i
$429$	0	0
$430$	−540.822	−1.25772
$431$	198.527i	0.460619i	0.973117	+	0.230310i	$0.0739739\pi$
$431$	198.527i	0.460619i	−0.973117	+	0.230310i	$0.926026\pi$
$432$	0	0
$433$	82.6034	0.190770	0.0953849	−	0.995440i	$-0.469592\pi$
$433$	82.6034	0.190770	0.0953849	+	0.995440i	$0.469592\pi$
$434$	453.687	1.04536
$435$	0	0
$436$	409.762i	0.939821i
$437$	− 619.374i	− 1.41733i
$438$	0	0
$439$	− 231.295i	− 0.526868i	−0.964677	−	0.263434i	$-0.915145\pi$
$439$	− 231.295i	− 0.526868i	0.964677	−	0.263434i	$-0.0848550\pi$
$440$	0	0
$441$	0	0
$442$	101.271i	0.229120i
$443$	602.563	1.36019	0.680094	−	0.733125i	$-0.261939\pi$
$443$	602.563	1.36019	0.680094	+	0.733125i	$0.261939\pi$
$444$	0	0
$445$	642.228	1.44321
$446$	− 108.886i	− 0.244138i
$447$	0	0
$448$	1773.55i	3.95881i
$449$	266.735	0.594065	0.297032	−	0.954867i	$-0.404003\pi$
$449$	266.735	0.594065	0.297032	+	0.954867i	$0.404003\pi$
$450$	0	0
$451$	0	0
$452$	1279.85	2.83152
$453$	0	0
$454$	−1052.96	−2.31930
$455$	226.308	0.497379
$456$	0	0
$457$	816.782i	1.78727i	0.448796	+	0.893634i	$0.351853\pi$
$457$	816.782i	1.78727i	−0.448796	+	0.893634i	$0.648147\pi$
$458$	2.28023i	0.00497866i
$459$	0	0
$460$	936.431	2.03572
$461$	− 193.244i	− 0.419185i	−0.977789	−	0.209592i	$-0.932786\pi$
$461$	− 193.244i	− 0.419185i	0.977789	−	0.209592i	$-0.0672137\pi$
$462$	0	0
$463$	−530.341	−1.14545	−0.572723	−	0.819749i	$-0.694113\pi$
$463$	−530.341	−1.14545	−0.572723	+	0.819749i	$0.694113\pi$
$464$	1025.04i	2.20913i
$465$	0	0
$466$	716.857	1.53832
$467$	−101.521	−0.217390	−0.108695	−	0.994075i	$-0.534667\pi$
$467$	−101.521	−0.217390	−0.108695	+	0.994075i	$0.534667\pi$
$468$	0	0
$469$	469.521i	1.00111i
$470$	− 623.777i	− 1.32719i
$471$	0	0
$472$	1871.29i	3.96459i
$473$	0	0
$474$	0	0
$475$	− 233.144i	− 0.490829i
$476$	291.083	0.611519
$477$	0	0
$478$	1040.01	2.17576
$479$	− 471.107i	− 0.983522i	−0.870730	−	0.491761i	$-0.836353\pi$
$479$	− 471.107i	− 0.983522i	0.870730	−	0.491761i	$-0.163647\pi$
$480$	0	0
$481$	− 236.387i	− 0.491450i
$482$	−963.501	−1.99896
$483$	0	0
$484$	0	0
$485$	−287.277	−0.592324
$486$	0	0
$487$	716.246	1.47073	0.735366	−	0.677670i	$-0.237010\pi$
$487$	716.246	1.47073	0.735366	+	0.677670i	$0.237010\pi$
$488$	−608.611	−1.24715
$489$	0	0
$490$	− 102.983i	− 0.210169i
$491$	− 802.375i	− 1.63417i	−0.576520	−	0.817083i	$-0.695590\pi$
$491$	− 802.375i	− 1.63417i	0.576520	−	0.817083i	$-0.304410\pi$
$492$	0	0
$493$	61.5071	0.124761
$494$	− 848.561i	− 1.71774i
$495$	0	0
$496$	940.910	1.89700
$497$	− 235.201i	− 0.473241i
$498$	0	0
$499$	178.361	0.357437	0.178719	−	0.983900i	$-0.442805\pi$
$499$	178.361	0.357437	0.178719	+	0.983900i	$0.442805\pi$
$500$	1486.02	2.97203
$501$	0	0
$502$	− 1240.19i	− 2.47051i
$503$	452.616i	0.899833i	0.893071	+	0.449916i	$0.148546\pi$
$503$	452.616i	0.899833i	−0.893071	+	0.449916i	$0.851454\pi$
$504$	0	0
$505$	569.217i	1.12716i
$506$	0	0
$507$	0	0
$508$	− 332.765i	− 0.655049i
$509$	908.282	1.78444	0.892222	−	0.451597i	$-0.149146\pi$
$509$	908.282	1.78444	0.892222	+	0.451597i	$0.149146\pi$
$510$	0	0
$511$	769.688	1.50624
$512$	979.477i	1.91304i
$513$	0	0
$514$	44.4675i	0.0865127i
$515$	85.1727	0.165384
$516$	0	0
$517$	0	0
$518$	−928.225	−1.79194
$519$	0	0
$520$	813.192	1.56383
$521$	257.259	0.493779	0.246889	−	0.969044i	$-0.420592\pi$
$521$	257.259	0.493779	0.246889	+	0.969044i	$0.420592\pi$
$522$	0	0
$523$	− 212.744i	− 0.406776i	−0.979098	−	0.203388i	$-0.934805\pi$
$523$	− 212.744i	− 0.406776i	0.979098	−	0.203388i	$-0.0651953\pi$
$524$	1957.76i	3.73618i
$525$	0	0
$526$	714.691	1.35873
$527$	− 56.4590i	− 0.107133i
$528$	0	0
$529$	−102.451	−0.193669
$530$	− 587.061i	− 1.10766i
$531$	0	0
$532$	−2439.01	−4.58461
$533$	−118.811	−0.222910
$534$	0	0
$535$	− 246.528i	− 0.460799i
$536$	1687.13i	3.14763i
$537$	0	0
$538$	− 935.492i	− 1.73883i
$539$	0	0
$540$	0	0
$541$	59.5988i	0.110164i	0.998482	+	0.0550821i	$0.0175420\pi$
$541$	59.5988i	0.110164i	−0.998482	+	0.0550821i	$0.982458\pi$
$542$	−942.585	−1.73909
$543$	0	0
$544$	441.747	0.812035
$545$	− 155.675i	− 0.285643i
$546$	0	0
$547$	− 535.776i	− 0.979481i	−0.871868	−	0.489740i	$-0.837092\pi$
$547$	− 535.776i	− 0.979481i	0.871868	−	0.489740i	$-0.162908\pi$
$548$	2098.55	3.82947
$549$	0	0
$550$	0	0
$551$	−515.374	−0.935343
$552$	0	0
$553$	135.965	0.245868
$554$	556.822	1.00509
$555$	0	0
$556$	268.446i	0.482816i
$557$	− 431.831i	− 0.775281i	−0.921811	−	0.387640i	$-0.873290\pi$
$557$	− 431.831i	− 0.775281i	0.921811	−	0.387640i	$-0.126710\pi$
$558$	0	0
$559$	−247.047	−0.441945
$560$	− 1842.98i	− 3.29103i
$561$	0	0
$562$	1387.95	2.46967
$563$	− 155.811i	− 0.276751i	−0.990380	−	0.138376i	$-0.955812\pi$
$563$	− 155.811i	− 0.276751i	0.990380	−	0.138376i	$-0.0441882\pi$
$564$	0	0
$565$	−486.235	−0.860593
$566$	−1713.56	−3.02749
$567$	0	0
$568$	− 845.148i	− 1.48794i
$569$	622.591i	1.09419i	0.837072	+	0.547093i	$0.184265\pi$
$569$	622.591i	1.09419i	−0.837072	+	0.547093i	$0.815735\pi$
$570$	0	0
$571$	327.913i	0.574279i	0.957889	+	0.287139i	$0.0927043\pi$
$571$	327.913i	0.574279i	−0.957889	+	0.287139i	$0.907296\pi$
$572$	0	0
$573$	0	0
$574$	466.537i	0.812783i
$575$	160.561	0.279237
$576$	0	0
$577$	−731.904	−1.26846	−0.634232	−	0.773143i	$-0.718684\pi$
$577$	−731.904	−1.26846	−0.634232	+	0.773143i	$0.718684\pi$
$578$	1066.98i	1.84599i
$579$	0	0
$580$	− 779.193i	− 1.34344i
$581$	885.843	1.52469
$582$	0	0
$583$	0	0
$584$	2765.72	4.73583
$585$	0	0
$586$	−1988.21	−3.39284
$587$	−109.316	−0.186228	−0.0931142	−	0.995655i	$-0.529682\pi$
$587$	−109.316	−0.186228	−0.0931142	+	0.995655i	$0.529682\pi$
$588$	0	0
$589$	473.075i	0.803184i
$590$	− 1121.61i	− 1.90103i
$591$	0	0
$592$	−1925.06	−3.25180
$593$	903.281i	1.52324i	0.648024	+	0.761620i	$0.275596\pi$
$593$	903.281i	1.52324i	−0.648024	+	0.761620i	$0.724404\pi$
$594$	0	0
$595$	−110.587	−0.185861
$596$	− 315.003i	− 0.528529i
$597$	0	0
$598$	584.385	0.977233
$599$	779.269	1.30095	0.650475	−	0.759528i	$-0.274570\pi$
$599$	779.269	1.30095	0.650475	+	0.759528i	$0.274570\pi$
$600$	0	0
$601$	564.553i	0.939356i	0.882838	+	0.469678i	$0.155630\pi$
$601$	564.553i	0.939356i	−0.882838	+	0.469678i	$0.844370\pi$
$602$	970.082i	1.61143i
$603$	0	0
$604$	− 1534.49i	− 2.54054i
$605$	0	0
$606$	0	0
$607$	134.510i	0.221597i	0.993843	+	0.110799i	$0.0353409\pi$
$607$	134.510i	0.221597i	−0.993843	+	0.110799i	$0.964659\pi$
$608$	−3701.44	−6.08790
$609$	0	0
$610$	364.789	0.598014
$611$	− 284.941i	− 0.466352i
$612$	0	0
$613$	540.234i	0.881295i	0.897680	+	0.440648i	$0.145251\pi$
$613$	540.234i	0.881295i	−0.897680	+	0.440648i	$0.854749\pi$
$614$	562.583	0.916259
$615$	0	0
$616$	0	0
$617$	−684.941	−1.11011	−0.555057	−	0.831812i	$-0.687304\pi$
$617$	−684.941	−1.11011	−0.555057	+	0.831812i	$0.687304\pi$
$618$	0	0
$619$	1089.66	1.76035	0.880177	−	0.474646i	$-0.157424\pi$
$619$	1089.66	1.76035	0.880177	+	0.474646i	$0.157424\pi$
$620$	−715.242	−1.15362
$621$	0	0
$622$	− 1531.16i	− 2.46168i
$623$	− 1151.98i	− 1.84908i
$624$	0	0
$625$	−370.207	−0.592331
$626$	− 1161.59i	− 1.85558i
$627$	0	0
$628$	−2892.61	−4.60607
$629$	115.513i	0.183645i
$630$	0	0
$631$	858.142	1.35997	0.679986	−	0.733225i	$-0.261986\pi$
$631$	858.142	1.35997	0.679986	+	0.733225i	$0.261986\pi$
$632$	488.564	0.773044
$633$	0	0
$634$	1043.33i	1.64562i
$635$	126.423i	0.199091i
$636$	0	0
$637$	− 47.0425i	− 0.0738501i
$638$	0	0
$639$	0	0
$640$	− 1770.73i	− 2.76677i
$641$	−703.848	−1.09805	−0.549024	−	0.835807i	$-0.685000\pi$
$641$	−703.848	−1.09805	−0.549024	+	0.835807i	$0.685000\pi$
$642$	0	0
$643$	−484.714	−0.753833	−0.376916	−	0.926247i	$-0.623016\pi$
$643$	−484.714	−0.753833	−0.376916	+	0.926247i	$0.623016\pi$
$644$	− 1679.69i	− 2.60822i
$645$	0	0
$646$	414.657i	0.641884i
$647$	762.533	1.17857	0.589283	−	0.807926i	$-0.299410\pi$
$647$	762.533	1.17857	0.589283	+	0.807926i	$0.299410\pi$
$648$	0	0
$649$	0	0
$650$	219.974	0.338421
$651$	0	0
$652$	−701.839	−1.07644
$653$	297.917	0.456229	0.228114	−	0.973634i	$-0.426744\pi$
$653$	297.917	0.456229	0.228114	+	0.973634i	$0.426744\pi$
$654$	0	0
$655$	− 743.784i	− 1.13555i
$656$	967.561i	1.47494i
$657$	0	0
$658$	−1118.88	−1.70043
$659$	622.572i	0.944722i	0.881405	+	0.472361i	$0.156598\pi$
$659$	622.572i	0.944722i	−0.881405	+	0.472361i	$0.843402\pi$
$660$	0	0
$661$	1201.96	1.81840	0.909199	−	0.416361i	$-0.136695\pi$
$661$	1201.96	1.81840	0.909199	+	0.416361i	$0.136695\pi$
$662$	666.546i	1.00687i
$663$	0	0
$664$	3183.10	4.79383
$665$	926.621	1.39342
$666$	0	0
$667$	− 354.927i	− 0.532124i
$668$	1254.87i	1.87855i
$669$	0	0
$670$	− 1011.23i	− 1.50930i
$671$	0	0
$672$	0	0
$673$	− 293.533i	− 0.436156i	−0.975931	−	0.218078i	$-0.930021\pi$
$673$	− 293.533i	− 0.436156i	0.975931	−	0.218078i	$-0.0699788\pi$
$674$	1067.55	1.58390
$675$	0	0
$676$	−1260.19	−1.86419
$677$	171.767i	0.253718i	0.991921	+	0.126859i	$0.0404896\pi$
$677$	171.767i	0.253718i	−0.991921	+	0.126859i	$0.959510\pi$
$678$	0	0
$679$	515.295i	0.758903i
$680$	−397.373	−0.584373
$681$	0	0
$682$	0	0
$683$	371.395	0.543770	0.271885	−	0.962330i	$-0.412353\pi$
$683$	371.395	0.543770	0.271885	+	0.962330i	$0.412353\pi$
$684$	0	0
$685$	−797.275	−1.16390
$686$	1224.53	1.78504
$687$	0	0
$688$	2011.87i	2.92423i
$689$	− 268.169i	− 0.389215i
$690$	0	0
$691$	473.877	0.685785	0.342892	−	0.939375i	$-0.388593\pi$
$691$	473.877	0.685785	0.342892	+	0.939375i	$0.388593\pi$
$692$	− 2553.60i	− 3.69018i
$693$	0	0
$694$	−558.138	−0.804234
$695$	− 101.987i	− 0.146744i
$696$	0	0
$697$	58.0582	0.0832972
$698$	−571.334	−0.818531
$699$	0	0
$700$	− 632.269i	− 0.903241i
$701$	− 528.656i	− 0.754145i	−0.926184	−	0.377073i	$-0.876931\pi$
$701$	− 528.656i	− 0.754145i	0.926184	−	0.377073i	$-0.123069\pi$
$702$	0	0
$703$	− 967.893i	− 1.37680i
$704$	0	0
$705$	0	0
$706$	96.0719i	0.136079i
$707$	1021.02	1.44415
$708$	0	0
$709$	−1077.52	−1.51978	−0.759888	−	0.650054i	$-0.774746\pi$
$709$	−1077.52	−1.51978	−0.759888	+	0.650054i	$0.774746\pi$
$710$	506.564i	0.713470i
$711$	0	0
$712$	− 4139.41i	− 5.81377i
$713$	−325.797	−0.456938
$714$	0	0
$715$	0	0
$716$	2429.19	3.39273
$717$	0	0
$718$	902.734	1.25729
$719$	1230.24	1.71104	0.855522	−	0.517766i	$-0.173236\pi$
$719$	1230.24	1.71104	0.855522	+	0.517766i	$0.173236\pi$
$720$	0	0
$721$	− 152.776i	− 0.211895i
$722$	− 2079.83i	− 2.88065i
$723$	0	0
$724$	−2943.20	−4.06519
$725$	− 133.601i	− 0.184277i
$726$	0	0
$727$	−389.945	−0.536376	−0.268188	−	0.963367i	$-0.586425\pi$
$727$	−389.945	−0.536376	−0.268188	+	0.963367i	$0.586425\pi$
$728$	− 1458.64i	− 2.00362i
$729$	0	0
$730$	−1657.72	−2.27084
$731$	120.722	0.165146
$732$	0	0
$733$	187.861i	0.256290i	0.991755	+	0.128145i	$0.0409024\pi$
$733$	187.861i	0.256290i	−0.991755	+	0.128145i	$0.959098\pi$
$734$	− 2117.64i	− 2.88507i
$735$	0	0
$736$	− 2549.10i	− 3.46345i
$737$	0	0
$738$	0	0
$739$	− 735.434i	− 0.995175i	−0.867414	−	0.497588i	$-0.834219\pi$
$739$	− 735.434i	− 0.995175i	0.867414	−	0.497588i	$-0.165781\pi$
$740$	1463.36	1.97751
$741$	0	0
$742$	−1053.02	−1.41917
$743$	− 335.749i	− 0.451884i	−0.974141	−	0.225942i	$-0.927454\pi$
$743$	− 335.749i	− 0.451884i	0.974141	−	0.225942i	$-0.0725459\pi$
$744$	0	0
$745$	119.675i	0.160637i
$746$	326.039	0.437050
$747$	0	0
$748$	0	0
$749$	−442.201	−0.590389
$750$	0	0
$751$	373.497	0.497333	0.248666	−	0.968589i	$-0.420008\pi$
$751$	373.497	0.497333	0.248666	+	0.968589i	$0.420008\pi$
$752$	−2320.47	−3.08573
$753$	0	0
$754$	− 486.260i	− 0.644908i
$755$	582.977i	0.772155i
$756$	0	0
$757$	1248.69	1.64952	0.824762	−	0.565480i	$-0.191309\pi$
$757$	1248.69	1.64952	0.824762	+	0.565480i	$0.191309\pi$
$758$	519.163i	0.684912i
$759$	0	0
$760$	3329.63	4.38109
$761$	274.084i	0.360162i	0.983652	+	0.180081i	$0.0576360\pi$
$761$	274.084i	0.360162i	−0.983652	+	0.180081i	$0.942364\pi$
$762$	0	0
$763$	−279.238	−0.365974
$764$	2675.98	3.50259
$765$	0	0
$766$	− 1598.67i	− 2.08704i
$767$	− 512.351i	− 0.667993i
$768$	0	0
$769$	1313.03i	1.70745i	0.520722	+	0.853726i	$0.325663\pi$
$769$	1313.03i	1.70745i	−0.520722	+	0.853726i	$0.674337\pi$
$770$	0	0
$771$	0	0
$772$	− 2878.09i	− 3.72809i
$773$	573.729	0.742211	0.371105	−	0.928591i	$-0.378979\pi$
$773$	573.729	0.742211	0.371105	+	0.928591i	$0.378979\pi$
$774$	0	0
$775$	−122.636	−0.158240
$776$	1851.61i	2.38610i
$777$	0	0
$778$	− 829.989i	− 1.06682i
$779$	−486.475	−0.624487
$780$	0	0
$781$	0	0
$782$	−285.565	−0.365173
$783$	0	0
$784$	−383.099	−0.488647
$785$	1098.95	1.39994
$786$	0	0
$787$	274.155i	0.348354i	0.984714	+	0.174177i	$0.0557266\pi$
$787$	274.155i	0.348354i	−0.984714	+	0.174177i	$0.944273\pi$
$788$	2798.95i	3.55197i
$789$	0	0
$790$	−292.835	−0.370677
$791$	872.169i	1.10262i
$792$	0	0
$793$	166.635	0.210133
$794$	2448.98i	3.08436i
$795$	0	0
$796$	83.3607	0.104724
$797$	343.282	0.430718	0.215359	−	0.976535i	$-0.430908\pi$
$797$	343.282	0.430718	0.215359	+	0.976535i	$0.430908\pi$
$798$	0	0
$799$	139.239i	0.174267i
$800$	− 959.530i	− 1.19941i
$801$	0	0
$802$	1223.45i	1.52550i
$803$	0	0
$804$	0	0
$805$	638.144i	0.792725i
$806$	−446.351	−0.553786
$807$	0	0
$808$	3668.82	4.54062
$809$	− 565.999i	− 0.699628i	−0.936819	−	0.349814i	$-0.886245\pi$
$809$	− 565.999i	− 0.699628i	0.936819	−	0.349814i	$-0.113755\pi$
$810$	0	0
$811$	15.6724i	0.0193248i	0.999953	+	0.00966241i	$0.00307569\pi$
$811$	15.6724i	0.0193248i	−0.999953	+	0.00966241i	$0.996924\pi$
$812$	−1397.65	−1.72125
$813$	0	0
$814$	0	0
$815$	266.640	0.327166
$816$	0	0
$817$	−1011.54	−1.23811
$818$	−1461.92	−1.78719
$819$	0	0
$820$	− 735.501i	− 0.896953i
$821$	− 1400.47i	− 1.70581i	−0.522067	−	0.852904i	$-0.674839\pi$
$821$	− 1400.47i	− 1.70581i	0.522067	−	0.852904i	$-0.325161\pi$
$822$	0	0
$823$	593.743	0.721437	0.360719	−	0.932675i	$-0.382531\pi$
$823$	593.743	0.721437	0.360719	+	0.932675i	$0.382531\pi$
$824$	− 548.970i	− 0.666226i
$825$	0	0
$826$	−2011.85	−2.43566
$827$	544.028i	0.657833i	0.944359	+	0.328916i	$0.106683\pi$
$827$	544.028i	0.657833i	−0.944359	+	0.328916i	$0.893317\pi$
$828$	0	0
$829$	586.695	0.707714	0.353857	−	0.935300i	$-0.384870\pi$
$829$	586.695	0.707714	0.353857	+	0.935300i	$0.384870\pi$
$830$	−1907.88	−2.29866
$831$	0	0
$832$	− 1744.87i	− 2.09720i
$833$	22.9877i	0.0275963i
$834$	0	0
$835$	− 476.747i	− 0.570954i
$836$	0	0
$837$	0	0
$838$	725.615i	0.865889i
$839$	−661.038	−0.787888	−0.393944	−	0.919134i	$-0.628890\pi$
$839$	−661.038	−0.787888	−0.393944	+	0.919134i	$0.628890\pi$
$840$	0	0
$841$	545.670	0.648834
$842$	− 592.923i	− 0.704184i
$843$	0	0
$844$	4103.14i	4.86154i
$845$	478.769	0.566590
$846$	0	0
$847$	0	0
$848$	−2183.88	−2.57534
$849$	0	0
$850$	−107.492	−0.126461
$851$	666.567	0.783275
$852$	0	0
$853$	473.383i	0.554962i	0.960731	+	0.277481i	$0.0894996\pi$
$853$	473.383i	0.554962i	−0.960731	+	0.277481i	$0.910500\pi$
$854$	− 654.329i	− 0.766193i
$855$	0	0
$856$	−1588.96	−1.85627
$857$	461.983i	0.539070i	0.962991	+	0.269535i	$0.0868701\pi$
$857$	461.983i	0.539070i	−0.962991	+	0.269535i	$0.913130\pi$
$858$	0	0
$859$	−787.819	−0.917135	−0.458567	−	0.888660i	$-0.651637\pi$
$859$	−787.819	−0.917135	−0.458567	+	0.888660i	$0.651637\pi$
$860$	− 1529.35i	− 1.77831i
$861$	0	0
$862$	−766.954	−0.889738
$863$	−864.133	−1.00131	−0.500657	−	0.865646i	$-0.666908\pi$
$863$	−864.133	−1.00131	−0.500657	+	0.865646i	$0.666908\pi$
$864$	0	0
$865$	970.156i	1.12157i
$866$	319.115i	0.368493i
$867$	0	0
$868$	1282.94i	1.47805i
$869$	0	0
$870$	0	0
$871$	− 461.930i	− 0.530344i
$872$	−1003.39	−1.15067
$873$	0	0
$874$	2392.78	2.73773
$875$	1012.67i	1.15733i
$876$	0	0
$877$	75.2272i	0.0857779i	0.999080	+	0.0428889i	$0.0136562\pi$
$877$	75.2272i	0.0857779i	−0.999080	+	0.0428889i	$0.986344\pi$
$878$	893.544	1.01770
$879$	0	0
$880$	0	0
$881$	405.703	0.460503	0.230252	−	0.973131i	$-0.426045\pi$
$881$	405.703	0.460503	0.230252	+	0.973131i	$0.426045\pi$
$882$	0	0
$883$	−136.243	−0.154296	−0.0771480	−	0.997020i	$-0.524581\pi$
$883$	−136.243	−0.154296	−0.0771480	+	0.997020i	$0.524581\pi$
$884$	−286.377	−0.323955
$885$	0	0
$886$	2327.84i	2.62735i
$887$	71.2823i	0.0803634i	0.999192	+	0.0401817i	$0.0127937\pi$
$887$	71.2823i	0.0803634i	−0.999192	+	0.0401817i	$0.987206\pi$
$888$	0	0
$889$	226.767	0.255081
$890$	2481.07i	2.78772i
$891$	0	0
$892$	307.909	0.345189
$893$	− 1166.70i	− 1.30649i
$894$	0	0
$895$	−922.891	−1.03116
$896$	−3176.19	−3.54486
$897$	0	0
$898$	1030.46i	1.14750i
$899$	271.092i	0.301548i
$900$	0	0
$901$	131.043i	0.145442i
$902$	0	0
$903$	0	0
$904$	3133.97i	3.46678i
$905$	1118.17	1.23555
$906$	0	0
$907$	597.914	0.659221	0.329611	−	0.944117i	$-0.393083\pi$
$907$	597.914	0.659221	0.329611	+	0.944117i	$0.393083\pi$
$908$	− 2977.58i	− 3.27928i
$909$	0	0
$910$	874.277i	0.960744i
$911$	1753.01	1.92427	0.962136	−	0.272571i	$-0.0878740\pi$
$911$	1753.01	1.92427	0.962136	+	0.272571i	$0.0878740\pi$
$912$	0	0
$913$	0	0
$914$	−3155.41	−3.45231
$915$	0	0
$916$	−6.44806	−0.00703937
$917$	−1334.14	−1.45490
$918$	0	0
$919$	556.541i	0.605595i	0.953055	+	0.302797i	$0.0979205\pi$
$919$	556.541i	0.605595i	−0.953055	+	0.302797i	$0.902079\pi$
$920$	2293.04i	2.49244i
$921$	0	0
$922$	746.545	0.809702
$923$	231.398i	0.250702i
$924$	0	0
$925$	250.908	0.271252
$926$	− 2048.83i	− 2.21256i
$927$	0	0
$928$	−2121.08	−2.28564
$929$	−703.792	−0.757580	−0.378790	−	0.925483i	$-0.623660\pi$
$929$	−703.792	−0.757580	−0.378790	+	0.925483i	$0.623660\pi$
$930$	0	0
$931$	− 192.617i	− 0.206892i
$932$	2027.14i	2.17504i
$933$	0	0
$934$	− 392.199i	− 0.419913i
$935$	0	0
$936$	0	0
$937$	407.451i	0.434846i	0.976077	+	0.217423i	$0.0697651\pi$
$937$	407.451i	0.434846i	−0.976077	+	0.217423i	$0.930235\pi$
$938$	−1813.86	−1.93376
$939$	0	0
$940$	1763.93	1.87652
$941$	− 467.484i	− 0.496795i	−0.968658	−	0.248398i	$-0.920096\pi$
$941$	− 467.484i	− 0.496795i	0.968658	−	0.248398i	$-0.0799040\pi$
$942$	0	0
$943$	− 335.025i	− 0.355275i
$944$	−4172.42	−4.41994
$945$	0	0
$946$	0	0
$947$	−1795.45	−1.89594	−0.947969	−	0.318363i	$-0.896867\pi$
$947$	−1795.45	−1.89594	−0.947969	+	0.318363i	$0.896867\pi$
$948$	0	0
$949$	−757.243	−0.797938
$950$	900.687	0.948092
$951$	0	0
$952$	712.777i	0.748715i
$953$	− 456.624i	− 0.479144i	−0.970879	−	0.239572i	$-0.922993\pi$
$953$	− 456.624i	− 0.479144i	0.970879	−	0.239572i	$-0.0770071\pi$
$954$	0	0
$955$	−1016.65	−1.06455
$956$	2940.97i	3.07633i
$957$	0	0
$958$	1819.99	1.89978
$959$	1430.09i	1.49123i
$960$	0	0
$961$	−712.158	−0.741059
$962$	913.217	0.949290
$963$	0	0
$964$	− 2724.61i	− 2.82636i
$965$	1093.43i	1.13309i
$966$	0	0
$967$	42.6322i	0.0440871i	0.999757	+	0.0220436i	$0.00701725\pi$
$967$	42.6322i	0.0440871i	−0.999757	+	0.0220436i	$0.992983\pi$
$968$	0	0
$969$	0	0
$970$	− 1109.82i	− 1.14414i
$971$	570.592	0.587633	0.293816	−	0.955862i	$-0.405075\pi$
$971$	570.592	0.587633	0.293816	+	0.955862i	$0.405075\pi$
$972$	0	0
$973$	−182.936	−0.188012
$974$	2767.02i	2.84088i
$975$	0	0
$976$	− 1357.03i	− 1.39040i
$977$	−1604.15	−1.64191	−0.820956	−	0.570992i	$-0.806559\pi$
$977$	−1604.15	−1.64191	−0.820956	+	0.570992i	$0.806559\pi$
$978$	0	0
$979$	0	0
$980$	291.217	0.297160
$981$	0	0
$982$	3099.76	3.15657
$983$	−1274.76	−1.29680	−0.648401	−	0.761299i	$-0.724562\pi$
$983$	−1274.76	−1.29680	−0.648401	+	0.761299i	$0.724562\pi$
$984$	0	0
$985$	− 1063.37i	− 1.07956i
$986$	237.616i	0.240989i
$987$	0	0
$988$	2399.58	2.42872
$989$	− 696.625i	− 0.704373i
$990$	0	0
$991$	−1336.90	−1.34904	−0.674519	−	0.738258i	$-0.735649\pi$
$991$	−1336.90	−1.34904	−0.674519	+	0.738258i	$0.735649\pi$
$992$	1946.99i	1.96270i
$993$	0	0
$994$	908.634	0.914118
$995$	−31.6701	−0.0318292
$996$	0	0
$997$	− 1800.83i	− 1.80625i	−0.429383	−	0.903123i	$-0.641269\pi$
$997$	− 1800.83i	− 1.80625i	0.429383	−	0.903123i	$-0.358731\pi$
$998$	689.049i	0.690430i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.3.c.l.604.16		16
3.2	odd	2	inner	1089.3.c.l.604.1		16
11.6	odd	10		99.3.k.b.19.4	yes	16
11.9	even	5		99.3.k.b.73.4	yes	16
11.10	odd	2	inner	1089.3.c.l.604.2		16
33.17	even	10		99.3.k.b.19.1	✓	16
33.20	odd	10		99.3.k.b.73.1	yes	16
33.32	even	2	inner	1089.3.c.l.604.15		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.3.k.b.19.1	✓	16	33.17	even	10
99.3.k.b.19.4	yes	16	11.6	odd	10
99.3.k.b.73.1	yes	16	33.20	odd	10
99.3.k.b.73.4	yes	16	11.9	even	5
1089.3.c.l.604.1		16	3.2	odd	2	inner
1089.3.c.l.604.2		16	11.10	odd	2	inner
1089.3.c.l.604.15		16	33.32	even	2	inner
1089.3.c.l.604.16		16	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)

\(f(q)\)	\(=\)	\(q+3.86322i q^{2} -10.9245 q^{4} +4.15040 q^{5} -7.44465i q^{7} -26.7509i q^{8} +O(q^{10})\)	\(q+3.86322i q^{2} -10.9245 q^{4} +4.15040 q^{5} -7.44465i q^{7} -26.7509i q^{8} +16.0339i q^{10} +7.32428i q^{13} +28.7603 q^{14} +59.6467 q^{16} -3.57908i q^{17} +29.9894i q^{19} -45.3410 q^{20} -20.6531 q^{23} -7.77420 q^{25} -28.2953 q^{26} +81.3291i q^{28} +17.1852i q^{29} +15.7747 q^{31} +123.425i q^{32} +13.8268 q^{34} -30.8983i q^{35} -32.2745 q^{37} -115.856 q^{38} -111.027i q^{40} +16.2215i q^{41} +33.7299i q^{43} -79.7874i q^{46} -38.9036 q^{47} -6.42281 q^{49} -30.0335i q^{50} -80.0141i q^{52} -36.6137 q^{53} -199.151 q^{56} -66.3902 q^{58} -69.9524 q^{59} -22.7511i q^{61} +60.9413i q^{62} -238.231 q^{64} +30.3987i q^{65} -63.0682 q^{67} +39.0996i q^{68} +119.367 q^{70} +31.5933 q^{71} +103.388i q^{73} -124.683i q^{74} -327.620i q^{76} +18.2635i q^{79} +247.557 q^{80} -62.6675 q^{82} +118.991i q^{83} -14.8546i q^{85} -130.306 q^{86} +154.739 q^{89} +54.5267 q^{91} +225.624 q^{92} -150.293i q^{94} +124.468i q^{95} -69.2168 q^{97} -24.8128i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 44 q^{4}+O(q^{10}) \) 16 * q - 44 * q^4	\( 16 q - 44 q^{4} + 244 q^{16} + 16 q^{25} - 80 q^{31} + 328 q^{34} - 280 q^{37} + 436 q^{49} + 140 q^{58} - 656 q^{64} + 300 q^{67} + 308 q^{70} + 580 q^{82} + 768 q^{91} - 720 q^{97}+O(q^{100}) \) 16 * q - 44 * q^4 + 244 * q^16 + 16 * q^25 - 80 * q^31 + 328 * q^34 - 280 * q^37 + 436 * q^49 + 140 * q^58 - 656 * q^64 + 300 * q^67 + 308 * q^70 + 580 * q^82 + 768 * q^91 - 720 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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