LMFDB - Embedded newform 1089.3.c.l.604.11

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			604.11
Root			$1.32111 + 0.429256i$ of defining polynomial
Character	$\chi$	$=$	1089.604
Dual form			1089.3.c.l.604.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.38910i q^{2} +2.07040 q^{4} -4.37284 q^{5} -0.612830i q^{7} +8.43240i q^{8} +O(q^{10})$	$q+1.38910i q^{2} +2.07040 q^{4} -4.37284 q^{5} -0.612830i q^{7} +8.43240i q^{8} -6.07432i q^{10} -8.94888i q^{13} +0.851284 q^{14} -3.43188 q^{16} +2.97439i q^{17} -10.0047i q^{19} -9.05351 q^{20} -28.6467 q^{23} -5.87827 q^{25} +12.4309 q^{26} -1.26880i q^{28} +35.3868i q^{29} -50.3715 q^{31} +28.9624i q^{32} -4.13173 q^{34} +2.67981i q^{35} +1.74660 q^{37} +13.8976 q^{38} -36.8735i q^{40} -59.4772i q^{41} -43.6490i q^{43} -39.7932i q^{46} +41.5674 q^{47} +48.6244 q^{49} -8.16552i q^{50} -18.5277i q^{52} -65.6252 q^{53} +5.16763 q^{56} -49.1559 q^{58} -56.3923 q^{59} +85.5613i q^{61} -69.9711i q^{62} -53.9592 q^{64} +39.1320i q^{65} +27.8960 q^{67} +6.15816i q^{68} -3.72253 q^{70} -117.876 q^{71} -118.075i q^{73} +2.42621i q^{74} -20.7138i q^{76} -125.059i q^{79} +15.0071 q^{80} +82.6199 q^{82} -32.6748i q^{83} -13.0065i q^{85} +60.6329 q^{86} +19.9195 q^{89} -5.48414 q^{91} -59.3101 q^{92} +57.7413i q^{94} +43.7491i q^{95} -175.072 q^{97} +67.5443i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.38910i	0.694551i	0.937763	+	0.347275i	$0.112893\pi$
$2$	1.38910i	0.694551i	−0.937763	+	0.347275i	$0.887107\pi$
$3$	0	0
$3$	0	0
$4$	2.07040	0.517599
$5$	−4.37284	−0.874568	−0.437284	−	0.899323i	$-0.644060\pi$
$5$	−4.37284	−0.874568	−0.437284	+	0.899323i	$0.644060\pi$
$6$	0	0
$7$	− 0.612830i	− 0.0875472i	−0.999041	−	0.0437736i	$-0.986062\pi$
$7$	− 0.612830i	− 0.0875472i	0.999041	−	0.0437736i	$-0.0139380\pi$
$8$	8.43240i	1.05405i
$9$	0	0
$10$	− 6.07432i	− 0.607432i
$11$	0	0
$11$	0	0
$12$	0	0
$13$	− 8.94888i	− 0.688375i	−0.938901	−	0.344188i	$-0.888154\pi$
$13$	− 8.94888i	− 0.688375i	0.938901	−	0.344188i	$-0.111846\pi$
$14$	0.851284	0.0608060
$15$	0	0
$16$	−3.43188	−0.214492
$17$	2.97439i	0.174964i	0.996166	+	0.0874820i	$0.0278820\pi$
$17$	2.97439i	0.174964i	−0.996166	+	0.0874820i	$0.972118\pi$
$18$	0	0
$19$	− 10.0047i	− 0.526565i	−0.964719	−	0.263283i	$-0.915195\pi$
$19$	− 10.0047i	− 0.526565i	0.964719	−	0.263283i	$-0.0848051\pi$
$20$	−9.05351	−0.452675
$21$	0	0
$22$	0	0
$23$	−28.6467	−1.24551	−0.622755	−	0.782417i	$-0.713987\pi$
$23$	−28.6467	−1.24551	−0.622755	+	0.782417i	$0.713987\pi$
$24$	0	0
$25$	−5.87827	−0.235131
$26$	12.4309	0.478112
$27$	0	0
$28$	− 1.26880i	− 0.0453143i
$29$	35.3868i	1.22023i	0.792311	+	0.610117i	$0.208878\pi$
$29$	35.3868i	1.22023i	−0.792311	+	0.610117i	$0.791122\pi$
$30$	0	0
$31$	−50.3715	−1.62489	−0.812443	−	0.583040i	$-0.801863\pi$
$31$	−50.3715	−1.62489	−0.812443	+	0.583040i	$0.801863\pi$
$32$	28.9624i	0.905074i
$33$	0	0
$34$	−4.13173	−0.121521
$35$	2.67981i	0.0765659i
$36$	0	0
$37$	1.74660	0.0472055	0.0236028	−	0.999721i	$-0.492486\pi$
$37$	1.74660	0.0472055	0.0236028	+	0.999721i	$0.492486\pi$
$38$	13.8976	0.365726
$39$	0	0
$40$	− 36.8735i	− 0.921838i
$41$	− 59.4772i	− 1.45066i	−0.688400	−	0.725331i	$-0.741687\pi$
$41$	− 59.4772i	− 1.45066i	0.688400	−	0.725331i	$-0.258313\pi$
$42$	0	0
$43$	− 43.6490i	− 1.01509i	−0.861625	−	0.507546i	$-0.830553\pi$
$43$	− 43.6490i	− 1.01509i	0.861625	−	0.507546i	$-0.169447\pi$
$44$	0	0
$45$	0	0
$46$	− 39.7932i	− 0.865070i
$47$	41.5674	0.884412	0.442206	−	0.896913i	$-0.354196\pi$
$47$	41.5674	0.884412	0.442206	+	0.896913i	$0.354196\pi$
$48$	0	0
$49$	48.6244	0.992335
$50$	− 8.16552i	− 0.163310i
$51$	0	0
$52$	− 18.5277i	− 0.356302i
$53$	−65.6252	−1.23821	−0.619105	−	0.785308i	$-0.712505\pi$
$53$	−65.6252	−1.23821	−0.619105	+	0.785308i	$0.712505\pi$
$54$	0	0
$55$	0	0
$56$	5.16763	0.0922791
$57$	0	0
$58$	−49.1559	−0.847515
$59$	−56.3923	−0.955801	−0.477901	−	0.878414i	$-0.658602\pi$
$59$	−56.3923	−0.955801	−0.477901	+	0.878414i	$0.658602\pi$
$60$	0	0
$61$	85.5613i	1.40264i	0.712845	+	0.701322i	$0.247407\pi$
$61$	85.5613i	1.40264i	−0.712845	+	0.701322i	$0.752593\pi$
$62$	− 69.9711i	− 1.12857i
$63$	0	0
$64$	−53.9592	−0.843112
$65$	39.1320i	0.602031i
$66$	0	0
$67$	27.8960	0.416359	0.208179	−	0.978091i	$-0.433246\pi$
$67$	27.8960	0.416359	0.208179	+	0.978091i	$0.433246\pi$
$68$	6.15816i	0.0905611i
$69$	0	0
$70$	−3.72253	−0.0531789
$71$	−117.876	−1.66023	−0.830116	−	0.557591i	$-0.811726\pi$
$71$	−117.876	−1.66023	−0.830116	+	0.557591i	$0.811726\pi$
$72$	0	0
$73$	− 118.075i	− 1.61747i	−0.588176	−	0.808733i	$-0.700154\pi$
$73$	− 118.075i	− 1.61747i	0.588176	−	0.808733i	$-0.299846\pi$
$74$	2.42621i	0.0327866i
$75$	0	0
$76$	− 20.7138i	− 0.272550i
$77$	0	0
$78$	0	0
$79$	− 125.059i	− 1.58303i	−0.611150	−	0.791515i	$-0.709293\pi$
$79$	− 125.059i	− 1.58303i	0.611150	−	0.791515i	$-0.290707\pi$
$80$	15.0071	0.187588
$81$	0	0
$82$	82.6199	1.00756
$83$	− 32.6748i	− 0.393672i	−0.980436	−	0.196836i	$-0.936933\pi$
$83$	− 32.6748i	− 0.393672i	0.980436	−	0.196836i	$-0.0630666\pi$
$84$	0	0
$85$	− 13.0065i	− 0.153018i
$86$	60.6329	0.705033
$87$	0	0
$88$	0	0
$89$	19.9195	0.223815	0.111908	−	0.993719i	$-0.464304\pi$
$89$	19.9195	0.223815	0.111908	+	0.993719i	$0.464304\pi$
$90$	0	0
$91$	−5.48414	−0.0602653
$92$	−59.3101	−0.644675
$93$	0	0
$94$	57.7413i	0.614269i
$95$	43.7491i	0.460517i
$96$	0	0
$97$	−175.072	−1.80486	−0.902432	−	0.430831i	$-0.858220\pi$
$97$	−175.072	−1.80486	−0.902432	+	0.430831i	$0.858220\pi$
$98$	67.5443i	0.689228i
$99$	0	0
$100$	−12.1704	−0.121704
$101$	− 84.5611i	− 0.837238i	−0.908162	−	0.418619i	$-0.862514\pi$
$101$	− 84.5611i	− 0.837238i	0.908162	−	0.418619i	$-0.137486\pi$
$102$	0	0
$103$	−32.1822	−0.312448	−0.156224	−	0.987722i	$-0.549932\pi$
$103$	−32.1822	−0.312448	−0.156224	+	0.987722i	$0.549932\pi$
$104$	75.4605	0.725582
$105$	0	0
$106$	− 91.1600i	− 0.860000i
$107$	42.9460i	0.401365i	0.979656	+	0.200682i	$0.0643159\pi$
$107$	42.9460i	0.401365i	−0.979656	+	0.200682i	$0.935684\pi$
$108$	0	0
$109$	− 18.6259i	− 0.170879i	−0.996343	−	0.0854397i	$-0.972770\pi$
$109$	− 18.6259i	− 0.170879i	0.996343	−	0.0854397i	$-0.0272295\pi$
$110$	0	0
$111$	0	0
$112$	2.10316i	0.0187782i
$113$	−168.265	−1.48908	−0.744538	−	0.667581i	$-0.767330\pi$
$113$	−168.265	−1.48908	−0.744538	+	0.667581i	$0.767330\pi$
$114$	0	0
$115$	125.268	1.08928
$116$	73.2647i	0.631592i
$117$	0	0
$118$	− 78.3346i	− 0.663853i
$119$	1.82279	0.0153176
$120$	0	0
$121$	0	0
$122$	−118.853	−0.974208
$123$	0	0
$124$	−104.289	−0.841040
$125$	135.026	1.08021
$126$	0	0
$127$	95.5531i	0.752386i	0.926541	+	0.376193i	$0.122767\pi$
$127$	95.5531i	0.752386i	−0.926541	+	0.376193i	$0.877233\pi$
$128$	40.8946i	0.319489i
$129$	0	0
$130$	−54.3584	−0.418141
$131$	− 201.569i	− 1.53869i	−0.638833	−	0.769346i	$-0.720582\pi$
$131$	− 201.569i	− 1.53869i	0.638833	−	0.769346i	$-0.279418\pi$
$132$	0	0
$133$	−6.13120	−0.0460993
$134$	38.7504i	0.289182i
$135$	0	0
$136$	−25.0812	−0.184421
$137$	131.931	0.963003	0.481501	−	0.876445i	$-0.340092\pi$
$137$	131.931	0.963003	0.481501	+	0.876445i	$0.340092\pi$
$138$	0	0
$139$	− 55.9770i	− 0.402712i	−0.979518	−	0.201356i	$-0.935465\pi$
$139$	− 55.9770i	− 0.402712i	0.979518	−	0.201356i	$-0.0645348\pi$
$140$	5.54826i	0.0396305i
$141$	0	0
$142$	− 163.742i	− 1.15312i
$143$	0	0
$144$	0	0
$145$	− 154.741i	− 1.06718i
$146$	164.018	1.12341
$147$	0	0
$148$	3.61616	0.0244335
$149$	− 30.3800i	− 0.203892i	−0.994790	−	0.101946i	$-0.967493\pi$
$149$	− 30.3800i	− 0.203892i	0.994790	−	0.101946i	$-0.0325070\pi$
$150$	0	0
$151$	113.516i	0.751764i	0.926668	+	0.375882i	$0.122660\pi$
$151$	113.516i	0.751764i	−0.926668	+	0.375882i	$0.877340\pi$
$152$	84.3639	0.555026
$153$	0	0
$154$	0	0
$155$	220.266	1.42107
$156$	0	0
$157$	22.3951	0.142644	0.0713221	−	0.997453i	$-0.477278\pi$
$157$	22.3951	0.142644	0.0713221	+	0.997453i	$0.477278\pi$
$158$	173.720	1.09949
$159$	0	0
$160$	− 126.648i	− 0.791549i
$161$	17.5556i	0.109041i
$162$	0	0
$163$	−127.335	−0.781194	−0.390597	−	0.920562i	$-0.627731\pi$
$163$	−127.335	−0.781194	−0.390597	+	0.920562i	$0.627731\pi$
$164$	− 123.141i	− 0.750862i
$165$	0	0
$166$	45.3886	0.273425
$167$	248.138i	1.48586i	0.669370	+	0.742929i	$0.266564\pi$
$167$	248.138i	1.48586i	−0.669370	+	0.742929i	$0.733436\pi$
$168$	0	0
$169$	88.9175	0.526139
$170$	18.0674	0.106279
$171$	0	0
$172$	− 90.3706i	− 0.525411i
$173$	187.474i	1.08367i	0.840486	+	0.541833i	$0.182269\pi$
$173$	187.474i	1.08367i	−0.840486	+	0.541833i	$0.817731\pi$
$174$	0	0
$175$	3.60238i	0.0205850i
$176$	0	0
$177$	0	0
$178$	27.6703i	0.155451i
$179$	−23.0143	−0.128572	−0.0642858	−	0.997932i	$-0.520477\pi$
$179$	−23.0143	−0.128572	−0.0642858	+	0.997932i	$0.520477\pi$
$180$	0	0
$181$	−139.288	−0.769546	−0.384773	−	0.923011i	$-0.625720\pi$
$181$	−139.288	−0.769546	−0.384773	+	0.923011i	$0.625720\pi$
$182$	− 7.61803i	− 0.0418573i
$183$	0	0
$184$	− 241.561i	− 1.31283i
$185$	−7.63762	−0.0412844
$186$	0	0
$187$	0	0
$188$	86.0609	0.457771
$189$	0	0
$190$	−60.7720	−0.319852
$191$	−87.5488	−0.458370	−0.229185	−	0.973383i	$-0.573606\pi$
$191$	−87.5488	−0.458370	−0.229185	+	0.973383i	$0.573606\pi$
$192$	0	0
$193$	141.096i	0.731067i	0.930798	+	0.365534i	$0.119113\pi$
$193$	141.096i	0.731067i	−0.930798	+	0.365534i	$0.880887\pi$
$194$	− 243.193i	− 1.25357i
$195$	0	0
$196$	100.672	0.513632
$197$	− 180.754i	− 0.917532i	−0.888557	−	0.458766i	$-0.848292\pi$
$197$	− 180.754i	− 0.917532i	0.888557	−	0.458766i	$-0.151708\pi$
$198$	0	0
$199$	57.3438	0.288160	0.144080	−	0.989566i	$-0.453978\pi$
$199$	57.3438	0.288160	0.144080	+	0.989566i	$0.453978\pi$
$200$	− 49.5679i	− 0.247840i
$201$	0	0
$202$	117.464	0.581505
$203$	21.6861	0.106828
$204$	0	0
$205$	260.084i	1.26870i
$206$	− 44.7043i	− 0.217011i
$207$	0	0
$208$	30.7115i	0.147651i
$209$	0	0
$210$	0	0
$211$	− 54.7860i	− 0.259649i	−0.991537	−	0.129825i	$-0.958559\pi$
$211$	− 54.7860i	− 0.259649i	0.991537	−	0.129825i	$-0.0414414\pi$
$212$	−135.870	−0.640896
$213$	0	0
$214$	−59.6564	−0.278768
$215$	190.870i	0.887767i
$216$	0	0
$217$	30.8692i	0.142254i
$218$	25.8732	0.118684
$219$	0	0
$220$	0	0
$221$	26.6174	0.120441
$222$	0	0
$223$	292.895	1.31343	0.656716	−	0.754138i	$-0.271945\pi$
$223$	292.895	1.31343	0.656716	+	0.754138i	$0.271945\pi$
$224$	17.7490	0.0792367
$225$	0	0
$226$	− 233.738i	− 1.03424i
$227$	86.2266i	0.379853i	0.981798	+	0.189927i	$0.0608250\pi$
$227$	86.2266i	0.379853i	−0.981798	+	0.189927i	$0.939175\pi$
$228$	0	0
$229$	50.2263	0.219329	0.109664	−	0.993969i	$-0.465022\pi$
$229$	50.2263	0.219329	0.109664	+	0.993969i	$0.465022\pi$
$230$	174.009i	0.756563i
$231$	0	0
$232$	−298.396	−1.28619
$233$	− 427.125i	− 1.83315i	−0.399860	−	0.916576i	$-0.630941\pi$
$233$	− 427.125i	− 1.83315i	0.399860	−	0.916576i	$-0.369059\pi$
$234$	0	0
$235$	−181.767	−0.773479
$236$	−116.754	−0.494722
$237$	0	0
$238$	2.53205i	0.0106389i
$239$	− 19.0899i	− 0.0798741i	−0.999202	−	0.0399371i	$-0.987284\pi$
$239$	− 19.0899i	− 0.0798741i	0.999202	−	0.0399371i	$-0.0127157\pi$
$240$	0	0
$241$	53.8901i	0.223611i	0.993730	+	0.111805i	$0.0356633\pi$
$241$	53.8901i	0.223611i	−0.993730	+	0.111805i	$0.964337\pi$
$242$	0	0
$243$	0	0
$244$	177.146i	0.726007i
$245$	−212.627	−0.867865
$246$	0	0
$247$	−89.5312	−0.362474
$248$	− 424.752i	− 1.71271i
$249$	0	0
$250$	187.565i	0.750258i
$251$	399.601	1.59204	0.796018	−	0.605273i	$-0.206936\pi$
$251$	399.601	1.59204	0.796018	+	0.605273i	$0.206936\pi$
$252$	0	0
$253$	0	0
$254$	−132.733	−0.522571
$255$	0	0
$256$	−272.644	−1.06501
$257$	−458.433	−1.78379	−0.891894	−	0.452245i	$-0.850623\pi$
$257$	−458.433	−1.78379	−0.891894	+	0.452245i	$0.850623\pi$
$258$	0	0
$259$	− 1.07037i	− 0.00413271i
$260$	81.0188i	0.311611i
$261$	0	0
$262$	279.999	1.06870
$263$	262.279i	0.997259i	0.866815	+	0.498630i	$0.166163\pi$
$263$	262.279i	0.997259i	−0.866815	+	0.498630i	$0.833837\pi$
$264$	0	0
$265$	286.968	1.08290
$266$	− 8.51687i	− 0.0320183i
$267$	0	0
$268$	57.7558	0.215507
$269$	72.6738	0.270163	0.135081	−	0.990834i	$-0.456870\pi$
$269$	72.6738	0.270163	0.135081	+	0.990834i	$0.456870\pi$
$270$	0	0
$271$	201.240i	0.742583i	0.928516	+	0.371292i	$0.121085\pi$
$271$	201.240i	0.742583i	−0.928516	+	0.371292i	$0.878915\pi$
$272$	− 10.2077i	− 0.0375284i
$273$	0	0
$274$	183.266i	0.668854i
$275$	0	0
$276$	0	0
$277$	357.053i	1.28900i	0.764604	+	0.644500i	$0.222934\pi$
$277$	357.053i	1.28900i	−0.764604	+	0.644500i	$0.777066\pi$
$278$	77.7577	0.279704
$279$	0	0
$280$	−22.5972	−0.0807043
$281$	276.466i	0.983863i	0.870634	+	0.491932i	$0.163709\pi$
$281$	276.466i	0.983863i	−0.870634	+	0.491932i	$0.836291\pi$
$282$	0	0
$283$	286.805i	1.01345i	0.862109	+	0.506723i	$0.169143\pi$
$283$	286.805i	1.01345i	−0.862109	+	0.506723i	$0.830857\pi$
$284$	−244.051	−0.859334
$285$	0	0
$286$	0	0
$287$	−36.4494	−0.127001
$288$	0	0
$289$	280.153	0.969388
$290$	214.951	0.741209
$291$	0	0
$292$	− 244.462i	− 0.837198i
$293$	− 149.014i	− 0.508579i	−0.967128	−	0.254289i	$-0.918158\pi$
$293$	− 149.014i	− 0.508579i	0.967128	−	0.254289i	$-0.0818416\pi$
$294$	0	0
$295$	246.594	0.835913
$296$	14.7281i	0.0497570i
$297$	0	0
$298$	42.2009	0.141614
$299$	256.356i	0.857379i
$300$	0	0
$301$	−26.7494	−0.0888684
$302$	−157.686	−0.522138
$303$	0	0
$304$	34.3350i	0.112944i
$305$	− 374.146i	− 1.22671i
$306$	0	0
$307$	277.276i	0.903179i	0.892226	+	0.451590i	$0.149143\pi$
$307$	277.276i	0.903179i	−0.892226	+	0.451590i	$0.850857\pi$
$308$	0	0
$309$	0	0
$310$	305.973i	0.987008i
$311$	209.678	0.674205	0.337102	−	0.941468i	$-0.390553\pi$
$311$	209.678	0.674205	0.337102	+	0.941468i	$0.390553\pi$
$312$	0	0
$313$	106.849	0.341370	0.170685	−	0.985326i	$-0.445402\pi$
$313$	106.849	0.341370	0.170685	+	0.985326i	$0.445402\pi$
$314$	31.1091i	0.0990737i
$315$	0	0
$316$	− 258.922i	− 0.819374i
$317$	−489.691	−1.54477	−0.772384	−	0.635156i	$-0.780936\pi$
$317$	−489.691	−1.54477	−0.772384	+	0.635156i	$0.780936\pi$
$318$	0	0
$319$	0	0
$320$	235.955	0.737359
$321$	0	0
$322$	−24.3865	−0.0757344
$323$	29.7580	0.0921299
$324$	0	0
$325$	52.6040i	0.161858i
$326$	− 176.881i	− 0.542579i
$327$	0	0
$328$	501.535	1.52907
$329$	− 25.4737i	− 0.0774278i
$330$	0	0
$331$	−333.988	−1.00903	−0.504514	−	0.863404i	$-0.668328\pi$
$331$	−333.988	−1.00903	−0.504514	+	0.863404i	$0.668328\pi$
$332$	− 67.6497i	− 0.203764i
$333$	0	0
$334$	−344.690	−1.03200
$335$	−121.985	−0.364134
$336$	0	0
$337$	501.440i	1.48795i	0.668206	+	0.743976i	$0.267062\pi$
$337$	501.440i	1.48795i	−0.668206	+	0.743976i	$0.732938\pi$
$338$	123.516i	0.365431i
$339$	0	0
$340$	− 26.9286i	− 0.0792019i
$341$	0	0
$342$	0	0
$343$	− 59.8272i	− 0.174423i
$344$	368.065	1.06996
$345$	0	0
$346$	−260.421	−0.752661
$347$	− 203.786i	− 0.587280i	−0.955916	−	0.293640i	$-0.905133\pi$
$347$	− 203.786i	− 0.587280i	0.955916	−	0.293640i	$-0.0948667\pi$
$348$	0	0
$349$	422.541i	1.21072i	0.795952	+	0.605359i	$0.206971\pi$
$349$	422.541i	1.21072i	−0.795952	+	0.605359i	$0.793029\pi$
$350$	−5.00408	−0.0142974
$351$	0	0
$352$	0	0
$353$	−314.375	−0.890581	−0.445290	−	0.895386i	$-0.646900\pi$
$353$	−314.375	−0.890581	−0.445290	+	0.895386i	$0.646900\pi$
$354$	0	0
$355$	515.455	1.45198
$356$	41.2413	0.115846
$357$	0	0
$358$	− 31.9692i	− 0.0892995i
$359$	414.902i	1.15571i	0.816138	+	0.577857i	$0.196111\pi$
$359$	414.902i	1.15571i	−0.816138	+	0.577857i	$0.803889\pi$
$360$	0	0
$361$	260.905	0.722729
$362$	− 193.485i	− 0.534489i
$363$	0	0
$364$	−11.3543	−0.0311933
$365$	516.323i	1.41458i
$366$	0	0
$367$	462.999	1.26158	0.630789	−	0.775954i	$-0.282731\pi$
$367$	462.999	1.26158	0.630789	+	0.775954i	$0.282731\pi$
$368$	98.3121	0.267152
$369$	0	0
$370$	− 10.6094i	− 0.0286741i
$371$	40.2171i	0.108402i
$372$	0	0
$373$	656.230i	1.75933i	0.475594	+	0.879665i	$0.342233\pi$
$373$	656.230i	1.75933i	−0.475594	+	0.879665i	$0.657767\pi$
$374$	0	0
$375$	0	0
$376$	350.513i	0.932215i
$377$	316.672	0.839979
$378$	0	0
$379$	−451.003	−1.18998	−0.594990	−	0.803733i	$-0.702844\pi$
$379$	−451.003	−1.18998	−0.594990	+	0.803733i	$0.702844\pi$
$380$	90.5780i	0.238363i
$381$	0	0
$382$	− 121.614i	− 0.318362i
$383$	−315.916	−0.824847	−0.412424	−	0.910992i	$-0.635318\pi$
$383$	−315.916	−0.824847	−0.412424	+	0.910992i	$0.635318\pi$
$384$	0	0
$385$	0	0
$386$	−195.997	−0.507763
$387$	0	0
$388$	−362.468	−0.934196
$389$	214.374	0.551091	0.275545	−	0.961288i	$-0.411142\pi$
$389$	214.374	0.551091	0.275545	+	0.961288i	$0.411142\pi$
$390$	0	0
$391$	− 85.2065i	− 0.217919i
$392$	410.021i	1.04597i
$393$	0	0
$394$	251.085	0.637272
$395$	546.864i	1.38447i
$396$	0	0
$397$	369.231	0.930052	0.465026	−	0.885297i	$-0.346045\pi$
$397$	369.231	0.930052	0.465026	+	0.885297i	$0.346045\pi$
$398$	79.6565i	0.200142i
$399$	0	0
$400$	20.1735	0.0504338
$401$	111.431	0.277883	0.138941	−	0.990301i	$-0.455630\pi$
$401$	111.431	0.277883	0.138941	+	0.990301i	$0.455630\pi$
$402$	0	0
$403$	450.768i	1.11853i
$404$	− 175.075i	− 0.433354i
$405$	0	0
$406$	30.1242i	0.0741975i
$407$	0	0
$408$	0	0
$409$	203.166i	0.496739i	0.968665	+	0.248370i	$0.0798947\pi$
$409$	203.166i	0.496739i	−0.968665	+	0.248370i	$0.920105\pi$
$410$	−361.283	−0.881179
$411$	0	0
$412$	−66.6299	−0.161723
$413$	34.5589i	0.0836777i
$414$	0	0
$415$	142.881i	0.344293i
$416$	259.181	0.623031
$417$	0	0
$418$	0	0
$419$	−519.060	−1.23881	−0.619404	−	0.785073i	$-0.712626\pi$
$419$	−519.060	−1.23881	−0.619404	+	0.785073i	$0.712626\pi$
$420$	0	0
$421$	333.468	0.792085	0.396042	−	0.918232i	$-0.370383\pi$
$421$	333.468	0.792085	0.396042	+	0.918232i	$0.370383\pi$
$422$	76.1033	0.180340
$423$	0	0
$424$	− 553.378i	− 1.30514i
$425$	− 17.4843i	− 0.0411394i
$426$	0	0
$427$	52.4345	0.122798
$428$	88.9153i	0.207746i
$429$	0	0
$430$	−265.138	−0.616599
$431$	− 731.630i	− 1.69752i	−0.528781	−	0.848759i	$-0.677351\pi$
$431$	− 731.630i	− 1.69752i	0.528781	−	0.848759i	$-0.322649\pi$
$432$	0	0
$433$	−373.113	−0.861693	−0.430846	−	0.902425i	$-0.641785\pi$
$433$	−373.113	−0.861693	−0.430846	+	0.902425i	$0.641785\pi$
$434$	−42.8804	−0.0988028
$435$	0	0
$436$	− 38.5629i	− 0.0884470i
$437$	286.603i	0.655842i
$438$	0	0
$439$	− 294.954i	− 0.671878i	−0.941884	−	0.335939i	$-0.890946\pi$
$439$	− 294.954i	− 0.671878i	0.941884	−	0.335939i	$-0.109054\pi$
$440$	0	0
$441$	0	0
$442$	36.9743i	0.0836523i
$443$	−505.564	−1.14123	−0.570614	−	0.821219i	$-0.693295\pi$
$443$	−505.564	−1.14123	−0.570614	+	0.821219i	$0.693295\pi$
$444$	0	0
$445$	−87.1050	−0.195741
$446$	406.861i	0.912245i
$447$	0	0
$448$	33.0678i	0.0738121i
$449$	26.5854	0.0592103	0.0296051	−	0.999562i	$-0.490575\pi$
$449$	26.5854	0.0592103	0.0296051	+	0.999562i	$0.490575\pi$
$450$	0	0
$451$	0	0
$452$	−348.376	−0.770744
$453$	0	0
$454$	−119.778	−0.263827
$455$	23.9813	0.0527061
$456$	0	0
$457$	176.695i	0.386642i	0.981136	+	0.193321i	$0.0619259\pi$
$457$	176.695i	0.386642i	−0.981136	+	0.193321i	$0.938074\pi$
$458$	69.7695i	0.152335i
$459$	0	0
$460$	259.353	0.563812
$461$	155.031i	0.336292i	0.985762	+	0.168146i	$0.0537781\pi$
$461$	155.031i	0.336292i	−0.985762	+	0.168146i	$0.946222\pi$
$462$	0	0
$463$	−192.026	−0.414744	−0.207372	−	0.978262i	$-0.566491\pi$
$463$	−192.026	−0.414744	−0.207372	+	0.978262i	$0.566491\pi$
$464$	− 121.443i	− 0.261731i
$465$	0	0
$466$	593.320	1.27322
$467$	−323.120	−0.691906	−0.345953	−	0.938252i	$-0.612444\pi$
$467$	−323.120	−0.691906	−0.345953	+	0.938252i	$0.612444\pi$
$468$	0	0
$469$	− 17.0955i	− 0.0364510i
$470$	− 252.494i	− 0.537220i
$471$	0	0
$472$	− 475.522i	− 1.00746i
$473$	0	0
$474$	0	0
$475$	58.8106i	0.123812i
$476$	3.77390	0.00792837
$477$	0	0
$478$	26.5178	0.0554766
$479$	− 458.454i	− 0.957105i	−0.878059	−	0.478553i	$-0.841162\pi$
$479$	− 458.454i	− 0.957105i	0.878059	−	0.478553i	$-0.158838\pi$
$480$	0	0
$481$	− 15.6302i	− 0.0324951i
$482$	−74.8589	−0.155309
$483$	0	0
$484$	0	0
$485$	765.561	1.57848
$486$	0	0
$487$	246.917	0.507015	0.253508	−	0.967333i	$-0.418416\pi$
$487$	246.917	0.507015	0.253508	+	0.967333i	$0.418416\pi$
$488$	−721.487	−1.47846
$489$	0	0
$490$	− 295.360i	− 0.602776i
$491$	− 43.3100i	− 0.0882078i	−0.999027	−	0.0441039i	$-0.985957\pi$
$491$	− 43.3100i	− 0.0882078i	0.999027	−	0.0441039i	$-0.0140433\pi$
$492$	0	0
$493$	−105.254	−0.213497
$494$	− 124.368i	− 0.251757i
$495$	0	0
$496$	172.869	0.348526
$497$	72.2382i	0.145349i
$498$	0	0
$499$	−493.199	−0.988375	−0.494187	−	0.869355i	$-0.664534\pi$
$499$	−493.199	−0.988375	−0.494187	+	0.869355i	$0.664534\pi$
$500$	279.557	0.559113
$501$	0	0
$502$	555.086i	1.10575i
$503$	299.110i	0.594651i	0.954776	+	0.297326i	$0.0960947\pi$
$503$	299.110i	0.594651i	−0.954776	+	0.297326i	$0.903905\pi$
$504$	0	0
$505$	369.772i	0.732222i
$506$	0	0
$507$	0	0
$508$	197.833i	0.389434i
$509$	541.162	1.06319	0.531593	−	0.847000i	$-0.321594\pi$
$509$	541.162	1.06319	0.531593	+	0.847000i	$0.321594\pi$
$510$	0	0
$511$	−72.3599	−0.141605
$512$	− 215.151i	− 0.420217i
$513$	0	0
$514$	− 636.811i	− 1.23893i
$515$	140.728	0.273257
$516$	0	0
$517$	0	0
$518$	1.48686	0.00287038
$519$	0	0
$520$	−329.977	−0.634571
$521$	206.634	0.396609	0.198305	−	0.980140i	$-0.436456\pi$
$521$	206.634	0.396609	0.198305	+	0.980140i	$0.436456\pi$
$522$	0	0
$523$	56.3408i	0.107726i	0.998548	+	0.0538631i	$0.0171535\pi$
$523$	56.3408i	0.107726i	−0.998548	+	0.0538631i	$0.982847\pi$
$524$	− 417.327i	− 0.796425i
$525$	0	0
$526$	−364.332	−0.692647
$527$	− 149.824i	− 0.284297i
$528$	0	0
$529$	291.635	0.551295
$530$	398.628i	0.752129i
$531$	0	0
$532$	−12.6940	−0.0238609
$533$	−532.254	−0.998601
$534$	0	0
$535$	− 187.796i	− 0.351021i
$536$	235.230i	0.438863i
$537$	0	0
$538$	100.951i	0.187642i
$539$	0	0
$540$	0	0
$541$	− 448.655i	− 0.829307i	−0.909979	−	0.414654i	$-0.863903\pi$
$541$	− 448.655i	− 0.829307i	0.909979	−	0.414654i	$-0.136097\pi$
$542$	−279.543	−0.515762
$543$	0	0
$544$	−86.1453	−0.158355
$545$	81.4479i	0.149446i
$546$	0	0
$547$	829.805i	1.51701i	0.651666	+	0.758506i	$0.274070\pi$
$547$	829.805i	1.51701i	−0.651666	+	0.758506i	$0.725930\pi$
$548$	273.150	0.498449
$549$	0	0
$550$	0	0
$551$	354.036	0.642533
$552$	0	0
$553$	−76.6401	−0.138590
$554$	−495.983	−0.895276
$555$	0	0
$556$	− 115.894i	− 0.208443i
$557$	− 1010.81i	− 1.81475i	−0.420325	−	0.907374i	$-0.638084\pi$
$557$	− 1010.81i	− 1.81475i	0.420325	−	0.907374i	$-0.361916\pi$
$558$	0	0
$559$	−390.609	−0.698764
$560$	− 9.19677i	− 0.0164228i
$561$	0	0
$562$	−384.039	−0.683343
$563$	8.04144i	0.0142832i	0.999974	+	0.00714159i	$0.00227326\pi$
$563$	8.04144i	0.0142832i	−0.999974	+	0.00714159i	$0.997727\pi$
$564$	0	0
$565$	735.798	1.30230
$566$	−398.402	−0.703890
$567$	0	0
$568$	− 993.981i	− 1.74997i
$569$	− 660.882i	− 1.16148i	−0.814089	−	0.580740i	$-0.802763\pi$
$569$	− 660.882i	− 1.16148i	0.814089	−	0.580740i	$-0.197237\pi$
$570$	0	0
$571$	− 472.496i	− 0.827489i	−0.910393	−	0.413745i	$-0.864221\pi$
$571$	− 472.496i	− 0.827489i	0.910393	−	0.413745i	$-0.135779\pi$
$572$	0	0
$573$	0	0
$574$	− 50.6319i	− 0.0882090i
$575$	168.393	0.292858
$576$	0	0
$577$	−193.531	−0.335409	−0.167705	−	0.985837i	$-0.553636\pi$
$577$	−193.531	−0.335409	−0.167705	+	0.985837i	$0.553636\pi$
$578$	389.161i	0.673289i
$579$	0	0
$580$	− 320.375i	− 0.552370i
$581$	−20.0241	−0.0344648
$582$	0	0
$583$	0	0
$584$	995.655	1.70489
$585$	0	0
$586$	206.995	0.353234
$587$	436.101	0.742932	0.371466	−	0.928446i	$-0.378855\pi$
$587$	436.101	0.742932	0.371466	+	0.928446i	$0.378855\pi$
$588$	0	0
$589$	503.953i	0.855608i
$590$	342.545i	0.580584i
$591$	0	0
$592$	−5.99413	−0.0101252
$593$	− 848.153i	− 1.43027i	−0.698984	−	0.715137i	$-0.746364\pi$
$593$	− 848.153i	− 1.43027i	0.698984	−	0.715137i	$-0.253636\pi$
$594$	0	0
$595$	−7.97079	−0.0133963
$596$	− 62.8985i	− 0.105534i
$597$	0	0
$598$	−356.105	−0.595493
$599$	1018.56	1.70044	0.850220	−	0.526427i	$-0.176469\pi$
$599$	1018.56	1.70044	0.850220	+	0.526427i	$0.176469\pi$
$600$	0	0
$601$	585.339i	0.973942i	0.873418	+	0.486971i	$0.161898\pi$
$601$	585.339i	0.973942i	−0.873418	+	0.486971i	$0.838102\pi$
$602$	− 37.1576i	− 0.0617237i
$603$	0	0
$604$	235.024i	0.389112i
$605$	0	0
$606$	0	0
$607$	− 994.914i	− 1.63907i	−0.573031	−	0.819534i	$-0.694233\pi$
$607$	− 994.914i	− 1.63907i	0.573031	−	0.819534i	$-0.305767\pi$
$608$	289.761	0.476580
$609$	0	0
$610$	519.727	0.852011
$611$	− 371.982i	− 0.608808i
$612$	0	0
$613$	− 661.114i	− 1.07849i	−0.842149	−	0.539245i	$-0.818710\pi$
$613$	− 661.114i	− 1.07849i	0.842149	−	0.539245i	$-0.181290\pi$
$614$	−385.165	−0.627304
$615$	0	0
$616$	0	0
$617$	−118.682	−0.192354	−0.0961770	−	0.995364i	$-0.530661\pi$
$617$	−118.682	−0.192354	−0.0961770	+	0.995364i	$0.530661\pi$
$618$	0	0
$619$	−107.767	−0.174099	−0.0870495	−	0.996204i	$-0.527744\pi$
$619$	−107.767	−0.174099	−0.0870495	+	0.996204i	$0.527744\pi$
$620$	456.039	0.735546
$621$	0	0
$622$	291.264i	0.468270i
$623$	− 12.2073i	− 0.0195944i
$624$	0	0
$625$	−443.489	−0.709583
$626$	148.424i	0.237099i
$627$	0	0
$628$	46.3668	0.0738325
$629$	5.19508i	0.00825926i
$630$	0	0
$631$	−472.024	−0.748057	−0.374029	−	0.927417i	$-0.622024\pi$
$631$	−472.024	−0.748057	−0.374029	+	0.927417i	$0.622024\pi$
$632$	1054.55	1.66859
$633$	0	0
$634$	− 680.231i	− 1.07292i
$635$	− 417.838i	− 0.658013i
$636$	0	0
$637$	− 435.134i	− 0.683099i
$638$	0	0
$639$	0	0
$640$	− 178.826i	− 0.279415i
$641$	−589.141	−0.919096	−0.459548	−	0.888153i	$-0.651989\pi$
$641$	−589.141	−0.919096	−0.459548	+	0.888153i	$0.651989\pi$
$642$	0	0
$643$	986.500	1.53421	0.767107	−	0.641519i	$-0.221696\pi$
$643$	986.500	1.53421	0.767107	+	0.641519i	$0.221696\pi$
$644$	36.3470i	0.0564394i
$645$	0	0
$646$	41.3368i	0.0639889i
$647$	731.747	1.13098	0.565492	−	0.824754i	$-0.308686\pi$
$647$	731.747	1.13098	0.565492	+	0.824754i	$0.308686\pi$
$648$	0	0
$649$	0	0
$650$	−73.0723	−0.112419
$651$	0	0
$652$	−263.633	−0.404345
$653$	−105.534	−0.161615	−0.0808073	−	0.996730i	$-0.525750\pi$
$653$	−105.534	−0.161615	−0.0808073	+	0.996730i	$0.525750\pi$
$654$	0	0
$655$	881.427i	1.34569i
$656$	204.118i	0.311156i
$657$	0	0
$658$	35.3856	0.0537775
$659$	− 592.799i	− 0.899543i	−0.893144	−	0.449772i	$-0.851505\pi$
$659$	− 592.799i	− 0.899543i	0.893144	−	0.449772i	$-0.148495\pi$
$660$	0	0
$661$	531.402	0.803936	0.401968	−	0.915654i	$-0.368326\pi$
$661$	531.402	0.803936	0.401968	+	0.915654i	$0.368326\pi$
$662$	− 463.944i	− 0.700821i
$663$	0	0
$664$	275.527	0.414950
$665$	26.8108	0.0403169
$666$	0	0
$667$	− 1013.72i	− 1.51981i
$668$	513.745i	0.769079i
$669$	0	0
$670$	− 169.449i	− 0.252910i
$671$	0	0
$672$	0	0
$673$	475.091i	0.705931i	0.935636	+	0.352965i	$0.114827\pi$
$673$	475.091i	0.705931i	−0.935636	+	0.352965i	$0.885173\pi$
$674$	−696.551	−1.03346
$675$	0	0
$676$	184.094	0.272329
$677$	− 209.774i	− 0.309859i	−0.987926	−	0.154929i	$-0.950485\pi$
$677$	− 209.774i	− 0.309859i	0.987926	−	0.154929i	$-0.0495150\pi$
$678$	0	0
$679$	107.289i	0.158011i
$680$	109.676	0.161288
$681$	0	0
$682$	0	0
$683$	−661.462	−0.968466	−0.484233	−	0.874939i	$-0.660901\pi$
$683$	−661.462	−0.968466	−0.484233	+	0.874939i	$0.660901\pi$
$684$	0	0
$685$	−576.915	−0.842211
$686$	83.1061	0.121146
$687$	0	0
$688$	149.798i	0.217730i
$689$	587.272i	0.852354i
$690$	0	0
$691$	−1052.83	−1.52363	−0.761817	−	0.647792i	$-0.775693\pi$
$691$	−1052.83	−1.52363	−0.761817	+	0.647792i	$0.775693\pi$
$692$	388.145i	0.560904i
$693$	0	0
$694$	283.080	0.407896
$695$	244.778i	0.352199i
$696$	0	0
$697$	176.908	0.253814
$698$	−586.952	−0.840906
$699$	0	0
$700$	7.45836i	0.0106548i
$701$	879.048i	1.25399i	0.779023	+	0.626996i	$0.215716\pi$
$701$	879.048i	1.25399i	−0.779023	+	0.626996i	$0.784284\pi$
$702$	0	0
$703$	− 17.4743i	− 0.0248568i
$704$	0	0
$705$	0	0
$706$	− 436.699i	− 0.618554i
$707$	−51.8216	−0.0732978
$708$	0	0
$709$	724.871	1.02239	0.511193	−	0.859466i	$-0.329204\pi$
$709$	724.871	1.02239	0.511193	+	0.859466i	$0.329204\pi$
$710$	716.019i	1.00848i
$711$	0	0
$712$	167.970i	0.235912i
$713$	1442.98	2.02381
$714$	0	0
$715$	0	0
$716$	−47.6487	−0.0665485
$717$	0	0
$718$	−576.341	−0.802703
$719$	737.609	1.02588	0.512941	−	0.858424i	$-0.328556\pi$
$719$	737.609	1.02588	0.512941	+	0.858424i	$0.328556\pi$
$720$	0	0
$721$	19.7222i	0.0273540i
$722$	362.424i	0.501972i
$723$	0	0
$724$	−288.381	−0.398316
$725$	− 208.013i	− 0.286915i
$726$	0	0
$727$	−764.559	−1.05166	−0.525832	−	0.850589i	$-0.676246\pi$
$727$	−764.559	−1.05166	−0.525832	+	0.850589i	$0.676246\pi$
$728$	− 46.2445i	− 0.0635226i
$729$	0	0
$730$	−717.225	−0.982500
$731$	129.829	0.177605
$732$	0	0
$733$	− 1064.74i	− 1.45257i	−0.687392	−	0.726287i	$-0.741244\pi$
$733$	− 1064.74i	− 1.45257i	0.687392	−	0.726287i	$-0.258756\pi$
$734$	643.153i	0.876230i
$735$	0	0
$736$	− 829.677i	− 1.12728i
$737$	0	0
$738$	0	0
$739$	− 1170.71i	− 1.58418i	−0.610402	−	0.792092i	$-0.708992\pi$
$739$	− 1170.71i	− 1.58418i	0.610402	−	0.792092i	$-0.291008\pi$
$740$	−15.8129	−0.0213688
$741$	0	0
$742$	−55.8656	−0.0752906
$743$	696.700i	0.937685i	0.883282	+	0.468843i	$0.155329\pi$
$743$	696.700i	0.937685i	−0.883282	+	0.468843i	$0.844671\pi$
$744$	0	0
$745$	132.847i	0.178318i
$746$	−911.571	−1.22194
$747$	0	0
$748$	0	0
$749$	26.3186	0.0351384
$750$	0	0
$751$	426.754	0.568247	0.284124	−	0.958788i	$-0.408297\pi$
$751$	426.754	0.568247	0.284124	+	0.958788i	$0.408297\pi$
$752$	−142.654	−0.189700
$753$	0	0
$754$	439.890i	0.583408i
$755$	− 496.389i	− 0.657468i
$756$	0	0
$757$	−928.212	−1.22617	−0.613086	−	0.790016i	$-0.710072\pi$
$757$	−928.212	−1.22617	−0.613086	+	0.790016i	$0.710072\pi$
$758$	− 626.489i	− 0.826502i
$759$	0	0
$760$	−368.910	−0.485408
$761$	48.2758i	0.0634373i	0.999497	+	0.0317186i	$0.0100981\pi$
$761$	48.2758i	0.0634373i	−0.999497	+	0.0317186i	$0.989902\pi$
$762$	0	0
$763$	−11.4145	−0.0149600
$764$	−181.261	−0.237252
$765$	0	0
$766$	− 438.840i	− 0.572898i
$767$	504.648i	0.657950i
$768$	0	0
$769$	435.589i	0.566435i	0.959056	+	0.283218i	$0.0914019\pi$
$769$	435.589i	0.566435i	−0.959056	+	0.283218i	$0.908598\pi$
$770$	0	0
$771$	0	0
$772$	292.124i	0.378400i
$773$	351.292	0.454453	0.227226	−	0.973842i	$-0.427034\pi$
$773$	351.292	0.454453	0.227226	+	0.973842i	$0.427034\pi$
$774$	0	0
$775$	296.097	0.382061
$776$	− 1476.28i	− 1.90242i
$777$	0	0
$778$	297.788i	0.382761i
$779$	−595.053	−0.763868
$780$	0	0
$781$	0	0
$782$	118.360	0.151356
$783$	0	0
$784$	−166.873	−0.212848
$785$	−97.9304	−0.124752
$786$	0	0
$787$	− 430.733i	− 0.547310i	−0.961828	−	0.273655i	$-0.911767\pi$
$787$	− 430.733i	− 0.547310i	0.961828	−	0.273655i	$-0.0882327\pi$
$788$	− 374.232i	− 0.474913i
$789$	0	0
$790$	−759.650	−0.961583
$791$	103.118i	0.130364i
$792$	0	0
$793$	765.678	0.965546
$794$	512.899i	0.645968i
$795$	0	0
$796$	118.724	0.149151
$797$	−391.319	−0.490990	−0.245495	−	0.969398i	$-0.578950\pi$
$797$	−391.319	−0.490990	−0.245495	+	0.969398i	$0.578950\pi$
$798$	0	0
$799$	123.637i	0.154740i
$800$	− 170.249i	− 0.212811i
$801$	0	0
$802$	154.789i	0.193004i
$803$	0	0
$804$	0	0
$805$	− 76.7677i	− 0.0953636i
$806$	−626.163	−0.776877
$807$	0	0
$808$	713.053	0.882491
$809$	337.874i	0.417643i	0.977954	+	0.208822i	$0.0669628\pi$
$809$	337.874i	0.417643i	−0.977954	+	0.208822i	$0.933037\pi$
$810$	0	0
$811$	− 1478.21i	− 1.82270i	−0.411630	−	0.911351i	$-0.635040\pi$
$811$	− 1478.21i	− 1.82270i	0.411630	−	0.911351i	$-0.364960\pi$
$812$	44.8988	0.0552941
$813$	0	0
$814$	0	0
$815$	556.814	0.683207
$816$	0	0
$817$	−436.696	−0.534512
$818$	−282.219	−0.345011
$819$	0	0
$820$	538.477i	0.656679i
$821$	13.0232i	0.0158626i	0.999969	+	0.00793131i	$0.00252464\pi$
$821$	13.0232i	0.0158626i	−0.999969	+	0.00793131i	$0.997475\pi$
$822$	0	0
$823$	10.0793	0.0122470	0.00612350	−	0.999981i	$-0.498051\pi$
$823$	10.0793	0.0122470	0.00612350	+	0.999981i	$0.498051\pi$
$824$	− 271.373i	− 0.329336i
$825$	0	0
$826$	−48.0058	−0.0581184
$827$	12.5612i	0.0151888i	0.999971	+	0.00759442i	$0.00241740\pi$
$827$	12.5612i	0.0151888i	−0.999971	+	0.00759442i	$0.997583\pi$
$828$	0	0
$829$	−205.335	−0.247690	−0.123845	−	0.992302i	$-0.539523\pi$
$829$	−205.335	−0.247690	−0.123845	+	0.992302i	$0.539523\pi$
$830$	−198.477	−0.239129
$831$	0	0
$832$	482.874i	0.580378i
$833$	144.628i	0.173623i
$834$	0	0
$835$	− 1085.07i	− 1.29948i
$836$	0	0
$837$	0	0
$838$	− 721.028i	− 0.860415i
$839$	1108.32	1.32100	0.660499	−	0.750827i	$-0.270345\pi$
$839$	1108.32	1.32100	0.660499	+	0.750827i	$0.270345\pi$
$840$	0	0
$841$	−411.225	−0.488972
$842$	463.220i	0.550143i
$843$	0	0
$844$	− 113.429i	− 0.134394i
$845$	−388.822	−0.460145
$846$	0	0
$847$	0	0
$848$	225.218	0.265587
$849$	0	0
$850$	24.2874	0.0285734
$851$	−50.0345	−0.0587950
$852$	0	0
$853$	1392.87i	1.63290i	0.577413	+	0.816452i	$0.304062\pi$
$853$	1392.87i	1.63290i	−0.577413	+	0.816452i	$0.695938\pi$
$854$	72.8369i	0.0852891i
$855$	0	0
$856$	−362.138	−0.423059
$857$	1297.45i	1.51395i	0.653444	+	0.756974i	$0.273323\pi$
$857$	1297.45i	1.51395i	−0.653444	+	0.756974i	$0.726677\pi$
$858$	0	0
$859$	−406.557	−0.473290	−0.236645	−	0.971596i	$-0.576048\pi$
$859$	−406.557	−0.473290	−0.236645	+	0.971596i	$0.576048\pi$
$860$	395.176i	0.459507i
$861$	0	0
$862$	1016.31	1.17901
$863$	−323.659	−0.375040	−0.187520	−	0.982261i	$-0.560045\pi$
$863$	−323.659	−0.375040	−0.187520	+	0.982261i	$0.560045\pi$
$864$	0	0
$865$	− 819.794i	− 0.947739i
$866$	− 518.292i	− 0.598490i
$867$	0	0
$868$	63.9114i	0.0736306i
$869$	0	0
$870$	0	0
$871$	− 249.638i	− 0.286611i
$872$	157.061	0.180115
$873$	0	0
$874$	−398.121	−0.455516
$875$	− 82.7478i	− 0.0945690i
$876$	0	0
$877$	− 621.473i	− 0.708635i	−0.935125	−	0.354318i	$-0.884713\pi$
$877$	− 621.473i	− 0.708635i	0.935125	−	0.354318i	$-0.115287\pi$
$878$	409.722	0.466654
$879$	0	0
$880$	0	0
$881$	−1346.48	−1.52835	−0.764176	−	0.645007i	$-0.776854\pi$
$881$	−1346.48	−1.52835	−0.764176	+	0.645007i	$0.776854\pi$
$882$	0	0
$883$	537.489	0.608708	0.304354	−	0.952559i	$-0.401559\pi$
$883$	537.489	0.608708	0.304354	+	0.952559i	$0.401559\pi$
$884$	55.1086	0.0623401
$885$	0	0
$886$	− 702.280i	− 0.792641i
$887$	− 467.641i	− 0.527217i	−0.964630	−	0.263608i	$-0.915087\pi$
$887$	− 467.641i	− 0.527217i	0.964630	−	0.263608i	$-0.0849127\pi$
$888$	0	0
$889$	58.5578	0.0658693
$890$	− 120.998i	− 0.135952i
$891$	0	0
$892$	606.409	0.679831
$893$	− 415.871i	− 0.465701i
$894$	0	0
$895$	100.638	0.112445
$896$	25.0615	0.0279704
$897$	0	0
$898$	36.9299i	0.0411246i
$899$	− 1782.49i	− 1.98274i
$900$	0	0
$901$	− 195.195i	− 0.216642i
$902$	0	0
$903$	0	0
$904$	− 1418.88i	− 1.56956i
$905$	609.084	0.673021
$906$	0	0
$907$	163.842	0.180642	0.0903209	−	0.995913i	$-0.471211\pi$
$907$	163.842	0.180642	0.0903209	+	0.995913i	$0.471211\pi$
$908$	178.523i	0.196612i
$909$	0	0
$910$	33.3124i	0.0366071i
$911$	−964.379	−1.05859	−0.529297	−	0.848437i	$-0.677544\pi$
$911$	−964.379	−1.05859	−0.529297	+	0.848437i	$0.677544\pi$
$912$	0	0
$913$	0	0
$914$	−245.448	−0.268542
$915$	0	0
$916$	103.988	0.113524
$917$	−123.527	−0.134708
$918$	0	0
$919$	− 1592.00i	− 1.73232i	−0.499769	−	0.866159i	$-0.666582\pi$
$919$	− 1592.00i	− 1.73232i	0.499769	−	0.866159i	$-0.333418\pi$
$920$	1056.31i	1.14816i
$921$	0	0
$922$	−215.354	−0.233572
$923$	1054.86i	1.14286i
$924$	0	0
$925$	−10.2670	−0.0110995
$926$	− 266.744i	− 0.288061i
$927$	0	0
$928$	−1024.89	−1.10440
$929$	1071.44	1.15333	0.576665	−	0.816980i	$-0.304354\pi$
$929$	1071.44	1.15333	0.576665	+	0.816980i	$0.304354\pi$
$930$	0	0
$931$	− 486.475i	− 0.522529i
$932$	− 884.317i	− 0.948838i
$933$	0	0
$934$	− 448.847i	− 0.480564i
$935$	0	0
$936$	0	0
$937$	535.151i	0.571132i	0.958359	+	0.285566i	$0.0921816\pi$
$937$	535.151i	0.571132i	−0.958359	+	0.285566i	$0.907818\pi$
$938$	23.7474	0.0253171
$939$	0	0
$940$	−376.331	−0.400352
$941$	− 1477.02i	− 1.56963i	−0.619733	−	0.784813i	$-0.712759\pi$
$941$	− 1477.02i	− 1.56963i	0.619733	−	0.784813i	$-0.287241\pi$
$942$	0	0
$943$	1703.83i	1.80682i
$944$	193.531	0.205012
$945$	0	0
$946$	0	0
$947$	425.172	0.448968	0.224484	−	0.974478i	$-0.427930\pi$
$947$	425.172	0.448968	0.224484	+	0.974478i	$0.427930\pi$
$948$	0	0
$949$	−1056.64	−1.11342
$950$	−81.6939	−0.0859936
$951$	0	0
$952$	15.3705i	0.0161455i
$953$	544.839i	0.571710i	0.958273	+	0.285855i	$0.0922775\pi$
$953$	544.839i	0.571710i	−0.958273	+	0.285855i	$0.907723\pi$
$954$	0	0
$955$	382.837	0.400876
$956$	− 39.5237i	− 0.0413428i
$957$	0	0
$958$	636.839	0.664759
$959$	− 80.8515i	− 0.0843081i
$960$	0	0
$961$	1576.29	1.64026
$962$	21.7119	0.0225695
$963$	0	0
$964$	111.574i	0.115741i
$965$	− 616.990i	− 0.639368i
$966$	0	0
$967$	− 1196.81i	− 1.23765i	−0.785529	−	0.618824i	$-0.787609\pi$
$967$	− 1196.81i	− 1.23765i	0.785529	−	0.618824i	$-0.212391\pi$
$968$	0	0
$969$	0	0
$970$	1063.44i	1.09633i
$971$	−813.320	−0.837611	−0.418806	−	0.908076i	$-0.637551\pi$
$971$	−813.320	−0.837611	−0.418806	+	0.908076i	$0.637551\pi$
$972$	0	0
$973$	−34.3044	−0.0352563
$974$	342.992i	0.352148i
$975$	0	0
$976$	− 293.636i	− 0.300856i
$977$	−682.907	−0.698983	−0.349492	−	0.936939i	$-0.613646\pi$
$977$	−682.907	−0.698983	−0.349492	+	0.936939i	$0.613646\pi$
$978$	0	0
$979$	0	0
$980$	−440.222	−0.449206
$981$	0	0
$982$	60.1620	0.0612648
$983$	−1471.66	−1.49711	−0.748556	−	0.663072i	$-0.769252\pi$
$983$	−1471.66	−1.49711	−0.748556	+	0.663072i	$0.769252\pi$
$984$	0	0
$985$	790.407i	0.802444i
$986$	− 146.209i	− 0.148285i
$987$	0	0
$988$	−185.365	−0.187616
$989$	1250.40i	1.26431i
$990$	0	0
$991$	−399.921	−0.403553	−0.201777	−	0.979432i	$-0.564671\pi$
$991$	−399.921	−0.403553	−0.201777	+	0.979432i	$0.564671\pi$
$992$	− 1458.88i	− 1.47064i
$993$	0	0
$994$	−100.346	−0.100952
$995$	−250.755	−0.252016
$996$	0	0
$997$	1250.46i	1.25422i	0.778931	+	0.627109i	$0.215762\pi$
$997$	1250.46i	1.25422i	−0.778931	+	0.627109i	$0.784238\pi$
$998$	− 685.104i	− 0.686477i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.3.c.l.604.11		16
3.2	odd	2	inner	1089.3.c.l.604.6		16
11.6	odd	10		99.3.k.b.19.3	yes	16
11.9	even	5		99.3.k.b.73.3	yes	16
11.10	odd	2	inner	1089.3.c.l.604.5		16
33.17	even	10		99.3.k.b.19.2	✓	16
33.20	odd	10		99.3.k.b.73.2	yes	16
33.32	even	2	inner	1089.3.c.l.604.12		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.3.k.b.19.2	✓	16	33.17	even	10
99.3.k.b.19.3	yes	16	11.6	odd	10
99.3.k.b.73.2	yes	16	33.20	odd	10
99.3.k.b.73.3	yes	16	11.9	even	5
1089.3.c.l.604.5		16	11.10	odd	2	inner
1089.3.c.l.604.6		16	3.2	odd	2	inner
1089.3.c.l.604.11		16	1.1	even	1	trivial
1089.3.c.l.604.12		16	33.32	even	2	inner

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)

\(f(q)\)	\(=\)	\(q+1.38910i q^{2} +2.07040 q^{4} -4.37284 q^{5} -0.612830i q^{7} +8.43240i q^{8} +O(q^{10})\)	\(q+1.38910i q^{2} +2.07040 q^{4} -4.37284 q^{5} -0.612830i q^{7} +8.43240i q^{8} -6.07432i q^{10} -8.94888i q^{13} +0.851284 q^{14} -3.43188 q^{16} +2.97439i q^{17} -10.0047i q^{19} -9.05351 q^{20} -28.6467 q^{23} -5.87827 q^{25} +12.4309 q^{26} -1.26880i q^{28} +35.3868i q^{29} -50.3715 q^{31} +28.9624i q^{32} -4.13173 q^{34} +2.67981i q^{35} +1.74660 q^{37} +13.8976 q^{38} -36.8735i q^{40} -59.4772i q^{41} -43.6490i q^{43} -39.7932i q^{46} +41.5674 q^{47} +48.6244 q^{49} -8.16552i q^{50} -18.5277i q^{52} -65.6252 q^{53} +5.16763 q^{56} -49.1559 q^{58} -56.3923 q^{59} +85.5613i q^{61} -69.9711i q^{62} -53.9592 q^{64} +39.1320i q^{65} +27.8960 q^{67} +6.15816i q^{68} -3.72253 q^{70} -117.876 q^{71} -118.075i q^{73} +2.42621i q^{74} -20.7138i q^{76} -125.059i q^{79} +15.0071 q^{80} +82.6199 q^{82} -32.6748i q^{83} -13.0065i q^{85} +60.6329 q^{86} +19.9195 q^{89} -5.48414 q^{91} -59.3101 q^{92} +57.7413i q^{94} +43.7491i q^{95} -175.072 q^{97} +67.5443i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 44 q^{4}+O(q^{10}) \) 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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