LMFDB - Embedded newform 3584.2.b.g.1793.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3584,2,Mod(1793,3584)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3584, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3584.1793");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3584 = 2^{9} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3584.b (of order $2$, degree $1$, not 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:	$28.6183840844$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 4x^{9} + 12x^{8} - 12x^{7} + 3x^{6} - 48x^{5} + 158x^{4} - 80x^{3} - 66x^{2} + 72x + 36 $ x^10 - 4x^9 + 12x^8 - 12x^7 + 3x^6 - 48x^5 + 158x^4 - 80x^3 - 66x^2 + 72*x + 36
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{9} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1793.9
Root			$1.68909 - 0.861941i$ of defining polynomial
Character	$\chi$	$=$	3584.1793
Dual form			3584.2.b.g.1793.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.99347i q^{3} +0.369776i q^{5} -1.00000 q^{7} -5.96088 q^{9} +O(q^{10})$	$q+2.99347i q^{3} +0.369776i q^{5} -1.00000 q^{7} -5.96088 q^{9} -3.93465i q^{11} -6.07457i q^{13} -1.10691 q^{15} +6.75635 q^{17} +0.117109i q^{19} -2.99347i q^{21} -8.67135 q^{23} +4.86327 q^{25} -8.86331i q^{27} +6.27619i q^{29} -0.808080 q^{31} +11.7783 q^{33} -0.369776i q^{35} +10.1743i q^{37} +18.1840 q^{39} +7.27490 q^{41} +0.0365776i q^{43} -2.20419i q^{45} +4.06779 q^{47} +1.00000 q^{49} +20.2250i q^{51} +2.53918i q^{53} +1.45494 q^{55} -0.350562 q^{57} +5.53975i q^{59} +6.02663i q^{61} +5.96088 q^{63} +2.24623 q^{65} +4.88072i q^{67} -25.9574i q^{69} -2.35733 q^{71} +6.82414 q^{73} +14.5581i q^{75} +3.93465i q^{77} +6.35733 q^{79} +8.64944 q^{81} +8.65033i q^{83} +2.49834i q^{85} -18.7876 q^{87} +3.56443 q^{89} +6.07457i q^{91} -2.41897i q^{93} -0.0433041 q^{95} -0.211290 q^{97} +23.4540i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 10 q^{7} - 22 q^{9}+O(q^{10}) $ 10 * q - 10 * q^7 - 22 * q^9	$ 10 q - 10 q^{7} - 22 q^{9} + 16 q^{15} + 16 q^{17} - 8 q^{23} - 30 q^{25} - 8 q^{31} + 12 q^{33} - 20 q^{41} - 24 q^{47} + 10 q^{49} + 32 q^{55} - 28 q^{57} + 22 q^{63} + 32 q^{65} - 48 q^{73} + 40 q^{79} + 62 q^{81} + 8 q^{87} - 16 q^{89} - 32 q^{95} + 32 q^{97}+O(q^{100}) $ 10 * q - 10 * q^7 - 22 * q^9 + 16 * q^15 + 16 * q^17 - 8 * q^23 - 30 * q^25 - 8 * q^31 + 12 * q^33 - 20 * q^41 - 24 * q^47 + 10 * q^49 + 32 * q^55 - 28 * q^57 + 22 * q^63 + 32 * q^65 - 48 * q^73 + 40 * q^79 + 62 * q^81 + 8 * q^87 - 16 * q^89 - 32 * q^95 + 32 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3584\mathbb{Z}\right)^\times$.

$n$	$1023$	$1025$	$1541$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	2.99347i	1.72828i	0.503250	+	0.864141i	$0.332138\pi$
$3$	2.99347i	1.72828i	−0.503250	+	0.864141i	$0.667862\pi$
$4$	0	0
$5$	0.369776i	0.165369i	0.996576	+	0.0826844i	$0.0263493\pi$
$5$	0.369776i	0.165369i	−0.996576	+	0.0826844i	$0.973651\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−5.96088	−1.98696
$10$	0	0
$11$	− 3.93465i	− 1.18634i	−0.805077	−	0.593170i	$-0.797876\pi$
$11$	− 3.93465i	− 1.18634i	0.805077	−	0.593170i	$-0.202124\pi$
$12$	0	0
$13$	− 6.07457i	− 1.68478i	−0.538867	−	0.842391i	$-0.681147\pi$
$13$	− 6.07457i	− 1.68478i	0.538867	−	0.842391i	$-0.318853\pi$
$14$	0	0
$15$	−1.10691	−0.285804
$16$	0	0
$17$	6.75635	1.63866	0.819328	−	0.573325i	$-0.194347\pi$
$17$	6.75635	1.63866	0.819328	+	0.573325i	$0.194347\pi$
$18$	0	0
$19$	0.117109i	0.0268666i	0.999910	+	0.0134333i	$0.00427609\pi$
$19$	0.117109i	0.0268666i	−0.999910	+	0.0134333i	$0.995724\pi$
$20$	0	0
$21$	− 2.99347i	− 0.653229i
$22$	0	0
$23$	−8.67135	−1.80810	−0.904050	−	0.427426i	$-0.859420\pi$
$23$	−8.67135	−1.80810	−0.904050	+	0.427426i	$0.859420\pi$
$24$	0	0
$25$	4.86327	0.972653
$26$	0	0
$27$	− 8.86331i	− 1.70574i
$28$	0	0
$29$	6.27619i	1.16546i	0.812666	+	0.582730i	$0.198015\pi$
$29$	6.27619i	1.16546i	−0.812666	+	0.582730i	$0.801985\pi$
$30$	0	0
$31$	−0.808080	−0.145136	−0.0725678	−	0.997363i	$-0.523119\pi$
$31$	−0.808080	−0.145136	−0.0725678	+	0.997363i	$0.523119\pi$
$32$	0	0
$33$	11.7783	2.05033
$34$	0	0
$35$	− 0.369776i	− 0.0625035i
$36$	0	0
$37$	10.1743i	1.67264i	0.548243	+	0.836319i	$0.315297\pi$
$37$	10.1743i	1.67264i	−0.548243	+	0.836319i	$0.684703\pi$
$38$	0	0
$39$	18.1840	2.91178
$40$	0	0
$41$	7.27490	1.13615	0.568074	−	0.822977i	$-0.307689\pi$
$41$	7.27490	1.13615	0.568074	+	0.822977i	$0.307689\pi$
$42$	0	0
$43$	0.0365776i	0.00557804i	0.999996	+	0.00278902i	$0.000887774\pi$
$43$	0.0365776i	0.00557804i	−0.999996	+	0.00278902i	$0.999112\pi$
$44$	0	0
$45$	− 2.20419i	− 0.328581i
$46$	0	0
$47$	4.06779	0.593349	0.296674	−	0.954979i	$-0.404122\pi$
$47$	4.06779	0.593349	0.296674	+	0.954979i	$0.404122\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	20.2250i	2.83206i
$52$	0	0
$53$	2.53918i	0.348784i	0.984676	+	0.174392i	$0.0557959\pi$
$53$	2.53918i	0.348784i	−0.984676	+	0.174392i	$0.944204\pi$
$54$	0	0
$55$	1.45494	0.196184
$56$	0	0
$57$	−0.350562	−0.0464331
$58$	0	0
$59$	5.53975i	0.721213i	0.932718	+	0.360607i	$0.117430\pi$
$59$	5.53975i	0.721213i	−0.932718	+	0.360607i	$0.882570\pi$
$60$	0	0
$61$	6.02663i	0.771631i	0.922576	+	0.385815i	$0.126080\pi$
$61$	6.02663i	0.771631i	−0.922576	+	0.385815i	$0.873920\pi$
$62$	0	0
$63$	5.96088	0.751000
$64$	0	0
$65$	2.24623	0.278610
$66$	0	0
$67$	4.88072i	0.596275i	0.954523	+	0.298138i	$0.0963655\pi$
$67$	4.88072i	0.596275i	−0.954523	+	0.298138i	$0.903635\pi$
$68$	0	0
$69$	− 25.9574i	− 3.12491i
$70$	0	0
$71$	−2.35733	−0.279763	−0.139882	−	0.990168i	$-0.544672\pi$
$71$	−2.35733	−0.279763	−0.139882	+	0.990168i	$0.544672\pi$
$72$	0	0
$73$	6.82414	0.798706	0.399353	−	0.916797i	$-0.369235\pi$
$73$	6.82414	0.798706	0.399353	+	0.916797i	$0.369235\pi$
$74$	0	0
$75$	14.5581i	1.68102i
$76$	0	0
$77$	3.93465i	0.448395i
$78$	0	0
$79$	6.35733	0.715255	0.357627	−	0.933864i	$-0.383586\pi$
$79$	6.35733	0.715255	0.357627	+	0.933864i	$0.383586\pi$
$80$	0	0
$81$	8.64944	0.961049
$82$	0	0
$83$	8.65033i	0.949497i	0.880122	+	0.474748i	$0.157461\pi$
$83$	8.65033i	0.949497i	−0.880122	+	0.474748i	$0.842539\pi$
$84$	0	0
$85$	2.49834i	0.270983i
$86$	0	0
$87$	−18.7876	−2.01424
$88$	0	0
$89$	3.56443	0.377829	0.188915	−	0.981994i	$-0.439503\pi$
$89$	3.56443	0.377829	0.188915	+	0.981994i	$0.439503\pi$
$90$	0	0
$91$	6.07457i	0.636788i
$92$	0	0
$93$	− 2.41897i	− 0.250835i
$94$	0	0
$95$	−0.0433041	−0.00444290
$96$	0	0
$97$	−0.211290	−0.0214532	−0.0107266	−	0.999942i	$-0.503414\pi$
$97$	−0.211290	−0.0214532	−0.0107266	+	0.999942i	$0.503414\pi$
$98$	0	0
$99$	23.4540i	2.35721i
$100$	0	0
$101$	− 9.84907i	− 0.980019i	−0.871717	−	0.490010i	$-0.836993\pi$
$101$	− 9.84907i	− 0.980019i	0.871717	−	0.490010i	$-0.163007\pi$
$102$	0	0
$103$	11.6587	1.14877	0.574385	−	0.818585i	$-0.305241\pi$
$103$	11.6587	1.14877	0.574385	+	0.818585i	$0.305241\pi$
$104$	0	0
$105$	1.10691	0.108024
$106$	0	0
$107$	− 10.4269i	− 1.00801i	−0.863701	−	0.504005i	$-0.831859\pi$
$107$	− 10.4269i	− 1.00801i	0.863701	−	0.504005i	$-0.168141\pi$
$108$	0	0
$109$	− 13.2092i	− 1.26521i	−0.774473	−	0.632607i	$-0.781985\pi$
$109$	− 13.2092i	− 1.26521i	0.774473	−	0.632607i	$-0.218015\pi$
$110$	0	0
$111$	−30.4564	−2.89079
$112$	0	0
$113$	19.3760	1.82274	0.911369	−	0.411590i	$-0.135027\pi$
$113$	19.3760	1.82274	0.911369	+	0.411590i	$0.135027\pi$
$114$	0	0
$115$	− 3.20645i	− 0.299003i
$116$	0	0
$117$	36.2098i	3.34759i
$118$	0	0
$119$	−6.75635	−0.619354
$120$	0	0
$121$	−4.48145	−0.407405
$122$	0	0
$123$	21.7772i	1.96359i
$124$	0	0
$125$	3.64720i	0.326215i
$126$	0	0
$127$	18.5931	1.64987	0.824936	−	0.565227i	$-0.191211\pi$
$127$	18.5931	1.64987	0.824936	+	0.565227i	$0.191211\pi$
$128$	0	0
$129$	−0.109494	−0.00964043
$130$	0	0
$131$	3.06663i	0.267933i	0.990986	+	0.133966i	$0.0427714\pi$
$131$	3.06663i	0.267933i	−0.990986	+	0.133966i	$0.957229\pi$
$132$	0	0
$133$	− 0.117109i	− 0.0101546i
$134$	0	0
$135$	3.27744	0.282077
$136$	0	0
$137$	−6.90239	−0.589711	−0.294855	−	0.955542i	$-0.595271\pi$
$137$	−6.90239	−0.589711	−0.294855	+	0.955542i	$0.595271\pi$
$138$	0	0
$139$	18.9927i	1.61094i	0.592634	+	0.805472i	$0.298088\pi$
$139$	18.9927i	1.61094i	−0.592634	+	0.805472i	$0.701912\pi$
$140$	0	0
$141$	12.1768i	1.02547i
$142$	0	0
$143$	−23.9013	−1.99873
$144$	0	0
$145$	−2.32078	−0.192731
$146$	0	0
$147$	2.99347i	0.246897i
$148$	0	0
$149$	12.6726i	1.03818i	0.854720	+	0.519090i	$0.173729\pi$
$149$	12.6726i	1.03818i	−0.854720	+	0.519090i	$0.826271\pi$
$150$	0	0
$151$	9.87845	0.803897	0.401949	−	0.915662i	$-0.368333\pi$
$151$	9.87845	0.803897	0.401949	+	0.915662i	$0.368333\pi$
$152$	0	0
$153$	−40.2738	−3.25594
$154$	0	0
$155$	− 0.298809i	− 0.0240009i
$156$	0	0
$157$	− 11.0877i	− 0.884898i	−0.896794	−	0.442449i	$-0.854110\pi$
$157$	− 11.0877i	− 0.884898i	0.896794	−	0.442449i	$-0.145890\pi$
$158$	0	0
$159$	−7.60097	−0.602796
$160$	0	0
$161$	8.67135	0.683398
$162$	0	0
$163$	− 10.6611i	− 0.835045i	−0.908667	−	0.417523i	$-0.862898\pi$
$163$	− 10.6611i	− 0.835045i	0.908667	−	0.417523i	$-0.137102\pi$
$164$	0	0
$165$	4.35532i	0.339061i
$166$	0	0
$167$	13.6840	1.05890	0.529448	−	0.848342i	$-0.322399\pi$
$167$	13.6840	1.05890	0.529448	+	0.848342i	$0.322399\pi$
$168$	0	0
$169$	−23.9004	−1.83849
$170$	0	0
$171$	− 0.698072i	− 0.0533829i
$172$	0	0
$173$	− 0.242470i	− 0.0184346i	−0.999958	−	0.00921731i	$-0.997066\pi$
$173$	− 0.242470i	− 0.0184346i	0.999958	−	0.00921731i	$-0.00293400\pi$
$174$	0	0
$175$	−4.86327	−0.367628
$176$	0	0
$177$	−16.5831	−1.24646
$178$	0	0
$179$	0.776129i	0.0580106i	0.999579	+	0.0290053i	$0.00923397\pi$
$179$	0.776129i	0.0580106i	−0.999579	+	0.0290053i	$0.990766\pi$
$180$	0	0
$181$	− 15.0614i	− 1.11951i	−0.828659	−	0.559754i	$-0.810896\pi$
$181$	− 15.0614i	− 1.11951i	0.828659	−	0.559754i	$-0.189104\pi$
$182$	0	0
$183$	−18.0406	−1.33360
$184$	0	0
$185$	−3.76220	−0.276602
$186$	0	0
$187$	− 26.5839i	− 1.94400i
$188$	0	0
$189$	8.86331i	0.644711i
$190$	0	0
$191$	15.4209	1.11582	0.557910	−	0.829902i	$-0.311603\pi$
$191$	15.4209	1.11582	0.557910	+	0.829902i	$0.311603\pi$
$192$	0	0
$193$	9.75377	0.702092	0.351046	−	0.936358i	$-0.385826\pi$
$193$	9.75377	0.702092	0.351046	+	0.936358i	$0.385826\pi$
$194$	0	0
$195$	6.72402i	0.481517i
$196$	0	0
$197$	− 21.3127i	− 1.51847i	−0.650817	−	0.759234i	$-0.725574\pi$
$197$	− 21.3127i	− 1.51847i	0.650817	−	0.759234i	$-0.274426\pi$
$198$	0	0
$199$	−12.2290	−0.866892	−0.433446	−	0.901180i	$-0.642702\pi$
$199$	−12.2290	−0.866892	−0.433446	+	0.901180i	$0.642702\pi$
$200$	0	0
$201$	−14.6103	−1.03053
$202$	0	0
$203$	− 6.27619i	− 0.440502i
$204$	0	0
$205$	2.69008i	0.187884i
$206$	0	0
$207$	51.6888	3.59262
$208$	0	0
$209$	0.460782	0.0318730
$210$	0	0
$211$	− 23.6543i	− 1.62843i	−0.580566	−	0.814213i	$-0.697169\pi$
$211$	− 23.6543i	− 1.62843i	0.580566	−	0.814213i	$-0.302831\pi$
$212$	0	0
$213$	− 7.05659i	− 0.483509i
$214$	0	0
$215$	−0.0135255	−0.000922434 0
$216$	0	0
$217$	0.808080	0.0548561
$218$	0	0
$219$	20.4279i	1.38039i
$220$	0	0
$221$	− 41.0419i	− 2.76078i
$222$	0	0
$223$	−10.2530	−0.686591	−0.343296	−	0.939227i	$-0.611543\pi$
$223$	−10.2530	−0.686591	−0.343296	+	0.939227i	$0.611543\pi$
$224$	0	0
$225$	−28.9893	−1.93262
$226$	0	0
$227$	− 11.9299i	− 0.791818i	−0.918290	−	0.395909i	$-0.870430\pi$
$227$	− 11.9299i	− 0.791818i	0.918290	−	0.395909i	$-0.129570\pi$
$228$	0	0
$229$	5.89932i	0.389838i	0.980819	+	0.194919i	$0.0624444\pi$
$229$	5.89932i	0.389838i	−0.980819	+	0.194919i	$0.937556\pi$
$230$	0	0
$231$	−11.7783	−0.774953
$232$	0	0
$233$	−8.61032	−0.564081	−0.282040	−	0.959403i	$-0.591011\pi$
$233$	−8.61032	−0.564081	−0.282040	+	0.959403i	$0.591011\pi$
$234$	0	0
$235$	1.50417i	0.0981213i
$236$	0	0
$237$	19.0305i	1.23616i
$238$	0	0
$239$	−18.0485	−1.16746	−0.583729	−	0.811949i	$-0.698407\pi$
$239$	−18.0485	−1.16746	−0.583729	+	0.811949i	$0.698407\pi$
$240$	0	0
$241$	15.9635	1.02830	0.514148	−	0.857701i	$-0.328108\pi$
$241$	15.9635	1.02830	0.514148	+	0.857701i	$0.328108\pi$
$242$	0	0
$243$	− 0.698072i	− 0.0447813i
$244$	0	0
$245$	0.369776i	0.0236241i
$246$	0	0
$247$	0.711386	0.0452644
$248$	0	0
$249$	−25.8945	−1.64100
$250$	0	0
$251$	20.6242i	1.30179i	0.759168	+	0.650895i	$0.225606\pi$
$251$	20.6242i	1.30179i	−0.759168	+	0.650895i	$0.774394\pi$
$252$	0	0
$253$	34.1187i	2.14502i
$254$	0	0
$255$	−7.47870	−0.468334
$256$	0	0
$257$	−5.66877	−0.353608	−0.176804	−	0.984246i	$-0.556576\pi$
$257$	−5.66877	−0.353608	−0.176804	+	0.984246i	$0.556576\pi$
$258$	0	0
$259$	− 10.1743i	− 0.632198i
$260$	0	0
$261$	− 37.4116i	− 2.31572i
$262$	0	0
$263$	11.2910	0.696231	0.348115	−	0.937452i	$-0.386822\pi$
$263$	11.2910	0.696231	0.348115	+	0.937452i	$0.386822\pi$
$264$	0	0
$265$	−0.938928	−0.0576779
$266$	0	0
$267$	10.6700i	0.652995i
$268$	0	0
$269$	0.242470i	0.0147836i	0.999973	+	0.00739182i	$0.00235291\pi$
$269$	0.242470i	0.0147836i	−0.999973	+	0.00739182i	$0.997647\pi$
$270$	0	0
$271$	2.25299	0.136859	0.0684297	−	0.997656i	$-0.478201\pi$
$271$	2.25299	0.136859	0.0684297	+	0.997656i	$0.478201\pi$
$272$	0	0
$273$	−18.1840	−1.10055
$274$	0	0
$275$	− 19.1352i	− 1.15390i
$276$	0	0
$277$	17.5105i	1.05211i	0.850452	+	0.526053i	$0.176329\pi$
$277$	17.5105i	1.05211i	−0.850452	+	0.526053i	$0.823671\pi$
$278$	0	0
$279$	4.81687	0.288378
$280$	0	0
$281$	32.7772	1.95532	0.977661	−	0.210190i	$-0.0674083\pi$
$281$	32.7772	1.95532	0.977661	+	0.210190i	$0.0674083\pi$
$282$	0	0
$283$	24.7190i	1.46939i	0.678396	+	0.734696i	$0.262675\pi$
$283$	24.7190i	1.46939i	−0.678396	+	0.734696i	$0.737325\pi$
$284$	0	0
$285$	− 0.129629i	− 0.00767859i
$286$	0	0
$287$	−7.27490	−0.429424
$288$	0	0
$289$	28.6483	1.68519
$290$	0	0
$291$	− 0.632490i	− 0.0370772i
$292$	0	0
$293$	15.7531i	0.920303i	0.887840	+	0.460152i	$0.152205\pi$
$293$	15.7531i	0.920303i	−0.887840	+	0.460152i	$0.847795\pi$
$294$	0	0
$295$	−2.04846	−0.119266
$296$	0	0
$297$	−34.8740	−2.02359
$298$	0	0
$299$	52.6747i	3.04625i
$300$	0	0
$301$	− 0.0365776i	− 0.00210830i
$302$	0	0
$303$	29.4829	1.69375
$304$	0	0
$305$	−2.22850	−0.127604
$306$	0	0
$307$	− 15.7863i	− 0.900971i	−0.892784	−	0.450485i	$-0.851251\pi$
$307$	− 15.7863i	− 0.900971i	0.892784	−	0.450485i	$-0.148749\pi$
$308$	0	0
$309$	34.9001i	1.98540i
$310$	0	0
$311$	14.2186	0.806261	0.403130	−	0.915143i	$-0.367922\pi$
$311$	14.2186	0.806261	0.403130	+	0.915143i	$0.367922\pi$
$312$	0	0
$313$	−26.8794	−1.51931	−0.759656	−	0.650325i	$-0.774633\pi$
$313$	−26.8794	−1.51931	−0.759656	+	0.650325i	$0.774633\pi$
$314$	0	0
$315$	2.20419i	0.124192i
$316$	0	0
$317$	− 0.832075i	− 0.0467340i	−0.999727	−	0.0233670i	$-0.992561\pi$
$317$	− 0.832075i	− 0.0467340i	0.999727	−	0.0233670i	$-0.00743863\pi$
$318$	0	0
$319$	24.6946	1.38263
$320$	0	0
$321$	31.2127	1.74212
$322$	0	0
$323$	0.791229i	0.0440252i
$324$	0	0
$325$	− 29.5422i	− 1.63871i
$326$	0	0
$327$	39.5414	2.18665
$328$	0	0
$329$	−4.06779	−0.224265
$330$	0	0
$331$	13.4164i	0.737433i	0.929542	+	0.368716i	$0.120203\pi$
$331$	13.4164i	0.737433i	−0.929542	+	0.368716i	$0.879797\pi$
$332$	0	0
$333$	− 60.6475i	− 3.32346i
$334$	0	0
$335$	−1.80477	−0.0986054
$336$	0	0
$337$	14.6613	0.798653	0.399326	−	0.916809i	$-0.369244\pi$
$337$	14.6613	0.798653	0.399326	+	0.916809i	$0.369244\pi$
$338$	0	0
$339$	58.0014i	3.15021i
$340$	0	0
$341$	3.17951i	0.172180i
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	9.59843	0.516762
$346$	0	0
$347$	6.02974i	0.323693i	0.986816	+	0.161847i	$0.0517450\pi$
$347$	6.02974i	0.323693i	−0.986816	+	0.161847i	$0.948255\pi$
$348$	0	0
$349$	24.9754i	1.33690i	0.743756	+	0.668452i	$0.233043\pi$
$349$	24.9754i	1.33690i	−0.743756	+	0.668452i	$0.766957\pi$
$350$	0	0
$351$	−53.8408	−2.87381
$352$	0	0
$353$	−22.2968	−1.18674	−0.593370	−	0.804930i	$-0.702203\pi$
$353$	−22.2968	−1.18674	−0.593370	+	0.804930i	$0.702203\pi$
$354$	0	0
$355$	− 0.871682i	− 0.0462641i
$356$	0	0
$357$	− 20.2250i	− 1.07042i
$358$	0	0
$359$	−26.4896	−1.39807	−0.699035	−	0.715087i	$-0.746387\pi$
$359$	−26.4896	−1.39807	−0.699035	+	0.715087i	$0.746387\pi$
$360$	0	0
$361$	18.9863	0.999278
$362$	0	0
$363$	− 13.4151i	− 0.704110i
$364$	0	0
$365$	2.52340i	0.132081i
$366$	0	0
$367$	−10.2138	−0.533157	−0.266579	−	0.963813i	$-0.585893\pi$
$367$	−10.2138	−0.533157	−0.266579	+	0.963813i	$0.585893\pi$
$368$	0	0
$369$	−43.3648	−2.25748
$370$	0	0
$371$	− 2.53918i	− 0.131828i
$372$	0	0
$373$	16.9320i	0.876708i	0.898802	+	0.438354i	$0.144438\pi$
$373$	16.9320i	0.876708i	−0.898802	+	0.438354i	$0.855562\pi$
$374$	0	0
$375$	−10.9178	−0.563792
$376$	0	0
$377$	38.1251	1.96354
$378$	0	0
$379$	− 1.34665i	− 0.0691729i	−0.999402	−	0.0345865i	$-0.988989\pi$
$379$	− 1.34665i	− 0.0691729i	0.999402	−	0.0345865i	$-0.0110114\pi$
$380$	0	0
$381$	55.6579i	2.85144i
$382$	0	0
$383$	14.0473	0.717784	0.358892	−	0.933379i	$-0.383155\pi$
$383$	14.0473	0.717784	0.358892	+	0.933379i	$0.383155\pi$
$384$	0	0
$385$	−1.45494	−0.0741505
$386$	0	0
$387$	− 0.218035i	− 0.0110833i
$388$	0	0
$389$	− 4.17440i	− 0.211651i	−0.994385	−	0.105825i	$-0.966252\pi$
$389$	− 4.17440i	− 0.211651i	0.994385	−	0.105825i	$-0.0337484\pi$
$390$	0	0
$391$	−58.5867	−2.96285
$392$	0	0
$393$	−9.17987	−0.463063
$394$	0	0
$395$	2.35079i	0.118281i
$396$	0	0
$397$	− 21.4578i	− 1.07694i	−0.842645	−	0.538469i	$-0.819003\pi$
$397$	− 21.4578i	− 1.07694i	0.842645	−	0.538469i	$-0.180997\pi$
$398$	0	0
$399$	0.350562	0.0175501
$400$	0	0
$401$	−0.747050	−0.0373059	−0.0186530	−	0.999826i	$-0.505938\pi$
$401$	−0.747050	−0.0373059	−0.0186530	+	0.999826i	$0.505938\pi$
$402$	0	0
$403$	4.90874i	0.244522i
$404$	0	0
$405$	3.19835i	0.158927i
$406$	0	0
$407$	40.0321	1.98432
$408$	0	0
$409$	−7.69274	−0.380382	−0.190191	−	0.981747i	$-0.560911\pi$
$409$	−7.69274	−0.380382	−0.190191	+	0.981747i	$0.560911\pi$
$410$	0	0
$411$	− 20.6621i	− 1.01919i
$412$	0	0
$413$	− 5.53975i	− 0.272593i
$414$	0	0
$415$	−3.19868	−0.157017
$416$	0	0
$417$	−56.8542	−2.78416
$418$	0	0
$419$	16.5196i	0.807036i	0.914972	+	0.403518i	$0.132213\pi$
$419$	16.5196i	0.807036i	−0.914972	+	0.403518i	$0.867787\pi$
$420$	0	0
$421$	17.5105i	0.853411i	0.904391	+	0.426705i	$0.140326\pi$
$421$	17.5105i	0.853411i	−0.904391	+	0.426705i	$0.859674\pi$
$422$	0	0
$423$	−24.2476	−1.17896
$424$	0	0
$425$	32.8579	1.59384
$426$	0	0
$427$	− 6.02663i	− 0.291649i
$428$	0	0
$429$	− 71.5478i	− 3.45436i
$430$	0	0
$431$	27.3912	1.31939	0.659693	−	0.751535i	$-0.270686\pi$
$431$	27.3912	1.31939	0.659693	+	0.751535i	$0.270686\pi$
$432$	0	0
$433$	−28.9749	−1.39245	−0.696223	−	0.717826i	$-0.745137\pi$
$433$	−28.9749	−1.39245	−0.696223	+	0.717826i	$0.745137\pi$
$434$	0	0
$435$	− 6.94720i	− 0.333093i
$436$	0	0
$437$	− 1.01549i	− 0.0485776i
$438$	0	0
$439$	34.1407	1.62945	0.814724	−	0.579848i	$-0.196888\pi$
$439$	34.1407	1.62945	0.814724	+	0.579848i	$0.196888\pi$
$440$	0	0
$441$	−5.96088	−0.283851
$442$	0	0
$443$	18.9122i	0.898546i	0.893394	+	0.449273i	$0.148317\pi$
$443$	18.9122i	0.898546i	−0.893394	+	0.449273i	$0.851683\pi$
$444$	0	0
$445$	1.31804i	0.0624811i
$446$	0	0
$447$	−37.9351	−1.79427
$448$	0	0
$449$	−28.8293	−1.36054	−0.680270	−	0.732962i	$-0.738137\pi$
$449$	−28.8293	−1.36054	−0.680270	+	0.732962i	$0.738137\pi$
$450$	0	0
$451$	− 28.6242i	− 1.34786i
$452$	0	0
$453$	29.5709i	1.38936i
$454$	0	0
$455$	−2.24623	−0.105305
$456$	0	0
$457$	24.0906	1.12691	0.563456	−	0.826146i	$-0.309472\pi$
$457$	24.0906	1.12691	0.563456	+	0.826146i	$0.309472\pi$
$458$	0	0
$459$	− 59.8836i	− 2.79513i
$460$	0	0
$461$	21.2107i	0.987882i	0.869496	+	0.493941i	$0.164444\pi$
$461$	21.2107i	0.987882i	−0.869496	+	0.493941i	$0.835556\pi$
$462$	0	0
$463$	−15.5565	−0.722973	−0.361486	−	0.932377i	$-0.617731\pi$
$463$	−15.5565	−0.722973	−0.361486	+	0.932377i	$0.617731\pi$
$464$	0	0
$465$	0.894475	0.0414803
$466$	0	0
$467$	2.50838i	0.116074i	0.998314	+	0.0580371i	$0.0184842\pi$
$467$	2.50838i	0.116074i	−0.998314	+	0.0580371i	$0.981516\pi$
$468$	0	0
$469$	− 4.88072i	− 0.225371i
$470$	0	0
$471$	33.1909	1.52935
$472$	0	0
$473$	0.143920	0.00661746
$474$	0	0
$475$	0.569532i	0.0261319i
$476$	0	0
$477$	− 15.1358i	− 0.693019i
$478$	0	0
$479$	25.1184	1.14769	0.573845	−	0.818964i	$-0.305451\pi$
$479$	25.1184	1.14769	0.573845	+	0.818964i	$0.305451\pi$
$480$	0	0
$481$	61.8042	2.81803
$482$	0	0
$483$	25.9574i	1.18110i
$484$	0	0
$485$	− 0.0781298i	− 0.00354769i
$486$	0	0
$487$	−17.0804	−0.773987	−0.386993	−	0.922082i	$-0.626486\pi$
$487$	−17.0804	−0.773987	−0.386993	+	0.922082i	$0.626486\pi$
$488$	0	0
$489$	31.9138	1.44319
$490$	0	0
$491$	12.8459i	0.579727i	0.957068	+	0.289864i	$0.0936099\pi$
$491$	12.8459i	0.579727i	−0.957068	+	0.289864i	$0.906390\pi$
$492$	0	0
$493$	42.4041i	1.90979i
$494$	0	0
$495$	−8.67271	−0.389809
$496$	0	0
$497$	2.35733	0.105740
$498$	0	0
$499$	− 3.94086i	− 0.176417i	−0.996102	−	0.0882086i	$-0.971886\pi$
$499$	− 3.94086i	− 0.176417i	0.996102	−	0.0882086i	$-0.0281142\pi$
$500$	0	0
$501$	40.9625i	1.83007i
$502$	0	0
$503$	−38.0573	−1.69689	−0.848447	−	0.529281i	$-0.822462\pi$
$503$	−38.0573	−1.69689	−0.848447	+	0.529281i	$0.822462\pi$
$504$	0	0
$505$	3.64195	0.162065
$506$	0	0
$507$	− 71.5451i	− 3.17743i
$508$	0	0
$509$	16.0223i	0.710176i	0.934833	+	0.355088i	$0.115549\pi$
$509$	16.0223i	0.710176i	−0.934833	+	0.355088i	$0.884451\pi$
$510$	0	0
$511$	−6.82414	−0.301882
$512$	0	0
$513$	1.03797	0.0458276
$514$	0	0
$515$	4.31112i	0.189971i
$516$	0	0
$517$	− 16.0053i	− 0.703914i
$518$	0	0
$519$	0.725826	0.0318602
$520$	0	0
$521$	17.2497	0.755722	0.377861	−	0.925862i	$-0.376660\pi$
$521$	17.2497	0.755722	0.377861	+	0.925862i	$0.376660\pi$
$522$	0	0
$523$	− 36.6235i	− 1.60143i	−0.599043	−	0.800717i	$-0.704452\pi$
$523$	− 36.6235i	− 1.60143i	0.599043	−	0.800717i	$-0.295548\pi$
$524$	0	0
$525$	− 14.5581i	− 0.635366i
$526$	0	0
$527$	−5.45967	−0.237827
$528$	0	0
$529$	52.1922	2.26923
$530$	0	0
$531$	− 33.0217i	− 1.43302i
$532$	0	0
$533$	− 44.1919i	− 1.91416i
$534$	0	0
$535$	3.85563	0.166693
$536$	0	0
$537$	−2.32332	−0.100259
$538$	0	0
$539$	− 3.93465i	− 0.169477i
$540$	0	0
$541$	− 26.2577i	− 1.12891i	−0.825465	−	0.564454i	$-0.809087\pi$
$541$	− 26.2577i	− 1.12891i	0.825465	−	0.564454i	$-0.190913\pi$
$542$	0	0
$543$	45.0860	1.93483
$544$	0	0
$545$	4.88445	0.209227
$546$	0	0
$547$	3.75319i	0.160475i	0.996776	+	0.0802374i	$0.0255678\pi$
$547$	3.75319i	0.160475i	−0.996776	+	0.0802374i	$0.974432\pi$
$548$	0	0
$549$	− 35.9240i	− 1.53320i
$550$	0	0
$551$	−0.734998	−0.0313120
$552$	0	0
$553$	−6.35733	−0.270341
$554$	0	0
$555$	− 11.2620i	− 0.478047i
$556$	0	0
$557$	− 4.15042i	− 0.175859i	−0.996127	−	0.0879294i	$-0.971975\pi$
$557$	− 4.15042i	− 0.175859i	0.996127	−	0.0879294i	$-0.0280250\pi$
$558$	0	0
$559$	0.222193	0.00939778
$560$	0	0
$561$	79.5781	3.35979
$562$	0	0
$563$	3.13556i	0.132148i	0.997815	+	0.0660740i	$0.0210473\pi$
$563$	3.13556i	0.132148i	−0.997815	+	0.0660740i	$0.978953\pi$
$564$	0	0
$565$	7.16477i	0.301424i
$566$	0	0
$567$	−8.64944	−0.363242
$568$	0	0
$569$	−15.9020	−0.666647	−0.333323	−	0.942813i	$-0.608170\pi$
$569$	−15.9020	−0.666647	−0.333323	+	0.942813i	$0.608170\pi$
$570$	0	0
$571$	− 8.50038i	− 0.355730i	−0.984055	−	0.177865i	$-0.943081\pi$
$571$	− 8.50038i	− 0.355730i	0.984055	−	0.177865i	$-0.0569190\pi$
$572$	0	0
$573$	46.1621i	1.92845i
$574$	0	0
$575$	−42.1711	−1.75865
$576$	0	0
$577$	−32.4207	−1.34969	−0.674846	−	0.737959i	$-0.735790\pi$
$577$	−32.4207	−1.34969	−0.674846	+	0.737959i	$0.735790\pi$
$578$	0	0
$579$	29.1976i	1.21341i
$580$	0	0
$581$	− 8.65033i	− 0.358876i
$582$	0	0
$583$	9.99079	0.413776
$584$	0	0
$585$	−13.3895	−0.553587
$586$	0	0
$587$	− 18.3326i	− 0.756666i	−0.925670	−	0.378333i	$-0.876497\pi$
$587$	− 18.3326i	− 0.756666i	0.925670	−	0.378333i	$-0.123503\pi$
$588$	0	0
$589$	− 0.0946334i	− 0.00389930i
$590$	0	0
$591$	63.7991	2.62434
$592$	0	0
$593$	−24.3751	−1.00096	−0.500482	−	0.865747i	$-0.666844\pi$
$593$	−24.3751	−1.00096	−0.500482	+	0.865747i	$0.666844\pi$
$594$	0	0
$595$	− 2.49834i	− 0.102422i
$596$	0	0
$597$	− 36.6072i	− 1.49823i
$598$	0	0
$599$	41.8673	1.71065	0.855325	−	0.518091i	$-0.173357\pi$
$599$	41.8673	1.71065	0.855325	+	0.518091i	$0.173357\pi$
$600$	0	0
$601$	−28.9937	−1.18268	−0.591340	−	0.806423i	$-0.701401\pi$
$601$	−28.9937	−1.18268	−0.591340	+	0.806423i	$0.701401\pi$
$602$	0	0
$603$	− 29.0934i	− 1.18478i
$604$	0	0
$605$	− 1.65713i	− 0.0673720i
$606$	0	0
$607$	−32.7589	−1.32964	−0.664821	−	0.747003i	$-0.731492\pi$
$607$	−32.7589	−1.32964	−0.664821	+	0.747003i	$0.731492\pi$
$608$	0	0
$609$	18.7876	0.761312
$610$	0	0
$611$	− 24.7101i	− 0.999663i
$612$	0	0
$613$	− 1.94718i	− 0.0786459i	−0.999227	−	0.0393230i	$-0.987480\pi$
$613$	− 1.94718i	− 0.0786459i	0.999227	−	0.0393230i	$-0.0125201\pi$
$614$	0	0
$615$	−8.05269	−0.324716
$616$	0	0
$617$	1.14258	0.0459985	0.0229992	−	0.999735i	$-0.492678\pi$
$617$	1.14258	0.0459985	0.0229992	+	0.999735i	$0.492678\pi$
$618$	0	0
$619$	− 34.4805i	− 1.38589i	−0.720992	−	0.692943i	$-0.756314\pi$
$619$	− 34.4805i	− 1.38589i	0.720992	−	0.692943i	$-0.243686\pi$
$620$	0	0
$621$	76.8568i	3.08416i
$622$	0	0
$623$	−3.56443	−0.142806
$624$	0	0
$625$	22.9677	0.918707
$626$	0	0
$627$	1.37934i	0.0550855i
$628$	0	0
$629$	68.7409i	2.74088i
$630$	0	0
$631$	−7.20195	−0.286705	−0.143352	−	0.989672i	$-0.545788\pi$
$631$	−7.20195	−0.286705	−0.143352	+	0.989672i	$0.545788\pi$
$632$	0	0
$633$	70.8084	2.81438
$634$	0	0
$635$	6.87528i	0.272837i
$636$	0	0
$637$	− 6.07457i	− 0.240683i
$638$	0	0
$639$	14.0517	0.555878
$640$	0	0
$641$	17.6807	0.698345	0.349173	−	0.937058i	$-0.386463\pi$
$641$	17.6807	0.698345	0.349173	+	0.937058i	$0.386463\pi$
$642$	0	0
$643$	− 36.9615i	− 1.45762i	−0.684716	−	0.728810i	$-0.740074\pi$
$643$	− 36.9615i	− 1.45762i	0.684716	−	0.728810i	$-0.259926\pi$
$644$	0	0
$645$	− 0.0404883i	− 0.00159423i
$646$	0	0
$647$	−18.6129	−0.731747	−0.365873	−	0.930665i	$-0.619230\pi$
$647$	−18.6129	−0.731747	−0.365873	+	0.930665i	$0.619230\pi$
$648$	0	0
$649$	21.7969	0.855605
$650$	0	0
$651$	2.41897i	0.0948068i
$652$	0	0
$653$	28.0079i	1.09604i	0.836467	+	0.548018i	$0.184617\pi$
$653$	28.0079i	1.09604i	−0.836467	+	0.548018i	$0.815383\pi$
$654$	0	0
$655$	−1.13397	−0.0443077
$656$	0	0
$657$	−40.6779	−1.58700
$658$	0	0
$659$	40.9629i	1.59569i	0.602865	+	0.797843i	$0.294026\pi$
$659$	40.9629i	1.59569i	−0.602865	+	0.797843i	$0.705974\pi$
$660$	0	0
$661$	− 16.8098i	− 0.653825i	−0.945055	−	0.326912i	$-0.893992\pi$
$661$	− 16.8098i	− 0.653825i	0.945055	−	0.326912i	$-0.106008\pi$
$662$	0	0
$663$	122.858	4.77140
$664$	0	0
$665$	0.0433041	0.00167926
$666$	0	0
$667$	− 54.4230i	− 2.10727i
$668$	0	0
$669$	− 30.6920i	− 1.18662i
$670$	0	0
$671$	23.7127	0.915417
$672$	0	0
$673$	−8.69460	−0.335152	−0.167576	−	0.985859i	$-0.553594\pi$
$673$	−8.69460	−0.335152	−0.167576	+	0.985859i	$0.553594\pi$
$674$	0	0
$675$	− 43.1046i	− 1.65910i
$676$	0	0
$677$	39.2305i	1.50775i	0.657017	+	0.753876i	$0.271818\pi$
$677$	39.2305i	1.50775i	−0.657017	+	0.753876i	$0.728182\pi$
$678$	0	0
$679$	0.211290	0.00810855
$680$	0	0
$681$	35.7119	1.36848
$682$	0	0
$683$	41.6065i	1.59203i	0.605277	+	0.796015i	$0.293062\pi$
$683$	41.6065i	1.59203i	−0.605277	+	0.796015i	$0.706938\pi$
$684$	0	0
$685$	− 2.55234i	− 0.0975198i
$686$	0	0
$687$	−17.6595	−0.673751
$688$	0	0
$689$	15.4244	0.587624
$690$	0	0
$691$	− 15.0503i	− 0.572541i	−0.958149	−	0.286271i	$-0.907584\pi$
$691$	− 15.0503i	− 0.572541i	0.958149	−	0.286271i	$-0.0924156\pi$
$692$	0	0
$693$	− 23.4540i	− 0.890942i
$694$	0	0
$695$	−7.02306	−0.266400
$696$	0	0
$697$	49.1518	1.86176
$698$	0	0
$699$	− 25.7747i	− 0.974890i
$700$	0	0
$701$	− 32.4666i	− 1.22625i	−0.789987	−	0.613123i	$-0.789913\pi$
$701$	− 32.4666i	− 1.22625i	0.789987	−	0.613123i	$-0.210087\pi$
$702$	0	0
$703$	−1.19150	−0.0449381
$704$	0	0
$705$	−4.50270	−0.169581
$706$	0	0
$707$	9.84907i	0.370412i
$708$	0	0
$709$	− 40.9483i	− 1.53785i	−0.639342	−	0.768923i	$-0.720793\pi$
$709$	− 40.9483i	− 1.53785i	0.639342	−	0.768923i	$-0.279207\pi$
$710$	0	0
$711$	−37.8952	−1.42118
$712$	0	0
$713$	7.00714	0.262420
$714$	0	0
$715$	− 8.83812i	− 0.330527i
$716$	0	0
$717$	− 54.0276i	− 2.01770i
$718$	0	0
$719$	−42.4750	−1.58405	−0.792025	−	0.610489i	$-0.790973\pi$
$719$	−42.4750	−1.58405	−0.792025	+	0.610489i	$0.790973\pi$
$720$	0	0
$721$	−11.6587	−0.434194
$722$	0	0
$723$	47.7862i	1.77719i
$724$	0	0
$725$	30.5228i	1.13359i
$726$	0	0
$727$	2.84716	0.105595	0.0527977	−	0.998605i	$-0.483186\pi$
$727$	2.84716	0.105595	0.0527977	+	0.998605i	$0.483186\pi$
$728$	0	0
$729$	28.0380	1.03844
$730$	0	0
$731$	0.247131i	0.00914049i
$732$	0	0
$733$	− 16.3001i	− 0.602058i	−0.953615	−	0.301029i	$-0.902670\pi$
$733$	− 16.3001i	− 0.602058i	0.953615	−	0.301029i	$-0.0973302\pi$
$734$	0	0
$735$	−1.10691	−0.0408291
$736$	0	0
$737$	19.2039	0.707386
$738$	0	0
$739$	− 1.70803i	− 0.0628308i	−0.999506	−	0.0314154i	$-0.989999\pi$
$739$	− 1.70803i	− 0.0628308i	0.999506	−	0.0314154i	$-0.0100015\pi$
$740$	0	0
$741$	2.12951i	0.0782297i
$742$	0	0
$743$	−20.6504	−0.757591	−0.378796	−	0.925480i	$-0.623662\pi$
$743$	−20.6504	−0.757591	−0.378796	+	0.925480i	$0.623662\pi$
$744$	0	0
$745$	−4.68602	−0.171683
$746$	0	0
$747$	− 51.5635i	− 1.88661i
$748$	0	0
$749$	10.4269i	0.380992i
$750$	0	0
$751$	−32.5679	−1.18842	−0.594210	−	0.804310i	$-0.702535\pi$
$751$	−32.5679	−1.18842	−0.594210	+	0.804310i	$0.702535\pi$
$752$	0	0
$753$	−61.7380	−2.24986
$754$	0	0
$755$	3.65281i	0.132940i
$756$	0	0
$757$	− 17.4453i	− 0.634062i	−0.948415	−	0.317031i	$-0.897314\pi$
$757$	− 17.4453i	− 0.634062i	0.948415	−	0.317031i	$-0.102686\pi$
$758$	0	0
$759$	−102.133	−3.70721
$760$	0	0
$761$	3.14810	0.114119	0.0570593	−	0.998371i	$-0.481828\pi$
$761$	3.14810	0.114119	0.0570593	+	0.998371i	$0.481828\pi$
$762$	0	0
$763$	13.2092i	0.478206i
$764$	0	0
$765$	− 14.8923i	− 0.538431i
$766$	0	0
$767$	33.6516	1.21509
$768$	0	0
$769$	46.3568	1.67167	0.835834	−	0.548982i	$-0.184985\pi$
$769$	46.3568	1.67167	0.835834	+	0.548982i	$0.184985\pi$
$770$	0	0
$771$	− 16.9693i	− 0.611134i
$772$	0	0
$773$	− 3.39055i	− 0.121950i	−0.998139	−	0.0609748i	$-0.980579\pi$
$773$	− 3.39055i	− 0.121950i	0.998139	−	0.0609748i	$-0.0194209\pi$
$774$	0	0
$775$	−3.92991	−0.141167
$776$	0	0
$777$	30.4564	1.09262
$778$	0	0
$779$	0.851956i	0.0305245i
$780$	0	0
$781$	9.27524i	0.331894i
$782$	0	0
$783$	55.6278	1.98798
$784$	0	0
$785$	4.09998	0.146335
$786$	0	0
$787$	− 0.966714i	− 0.0344596i	−0.999852	−	0.0172298i	$-0.994515\pi$
$787$	− 0.966714i	− 0.0344596i	0.999852	−	0.0172298i	$-0.00548469\pi$
$788$	0	0
$789$	33.7992i	1.20328i
$790$	0	0
$791$	−19.3760	−0.688930
$792$	0	0
$793$	36.6092	1.30003
$794$	0	0
$795$	− 2.81066i	− 0.0996837i
$796$	0	0
$797$	− 13.8418i	− 0.490301i	−0.969485	−	0.245150i	$-0.921163\pi$
$797$	− 13.8418i	− 0.490301i	0.969485	−	0.245150i	$-0.0788373\pi$
$798$	0	0
$799$	27.4834	0.972294
$800$	0	0
$801$	−21.2471	−0.750731
$802$	0	0
$803$	− 26.8506i	− 0.947537i
$804$	0	0
$805$	3.20645i	0.113013i
$806$	0	0
$807$	−0.725826	−0.0255503
$808$	0	0
$809$	−0.856545	−0.0301145	−0.0150573	−	0.999887i	$-0.504793\pi$
$809$	−0.856545	−0.0301145	−0.0150573	+	0.999887i	$0.504793\pi$
$810$	0	0
$811$	− 25.6209i	− 0.899671i	−0.893111	−	0.449836i	$-0.851483\pi$
$811$	− 25.6209i	− 0.899671i	0.893111	−	0.449836i	$-0.148517\pi$
$812$	0	0
$813$	6.74427i	0.236532i
$814$	0	0
$815$	3.94223	0.138090
$816$	0	0
$817$	−0.00428357	−0.000149863 0
$818$	0	0
$819$	− 36.2098i	− 1.26527i
$820$	0	0
$821$	− 33.7705i	− 1.17860i	−0.807915	−	0.589299i	$-0.799404\pi$
$821$	− 33.7705i	− 1.17860i	0.807915	−	0.589299i	$-0.200596\pi$
$822$	0	0
$823$	−8.12997	−0.283393	−0.141696	−	0.989910i	$-0.545256\pi$
$823$	−8.12997	−0.283393	−0.141696	+	0.989910i	$0.545256\pi$
$824$	0	0
$825$	57.2808	1.99426
$826$	0	0
$827$	19.2110i	0.668033i	0.942567	+	0.334016i	$0.108404\pi$
$827$	19.2110i	0.668033i	−0.942567	+	0.334016i	$0.891596\pi$
$828$	0	0
$829$	− 33.0789i	− 1.14888i	−0.818547	−	0.574439i	$-0.805220\pi$
$829$	− 33.0789i	− 1.14888i	0.818547	−	0.574439i	$-0.194780\pi$
$830$	0	0
$831$	−52.4173	−1.81834
$832$	0	0
$833$	6.75635	0.234094
$834$	0	0
$835$	5.06000i	0.175108i
$836$	0	0
$837$	7.16227i	0.247564i
$838$	0	0
$839$	7.13889	0.246462	0.123231	−	0.992378i	$-0.460674\pi$
$839$	7.13889	0.246462	0.123231	+	0.992378i	$0.460674\pi$
$840$	0	0
$841$	−10.3906	−0.358295
$842$	0	0
$843$	98.1175i	3.37935i
$844$	0	0
$845$	− 8.83778i	− 0.304029i
$846$	0	0
$847$	4.48145	0.153985
$848$	0	0
$849$	−73.9957	−2.53953
$850$	0	0
$851$	− 88.2245i	− 3.02430i
$852$	0	0
$853$	− 3.34699i	− 0.114599i	−0.998357	−	0.0572993i	$-0.981751\pi$
$853$	− 3.34699i	− 0.114599i	0.998357	−	0.0572993i	$-0.0182489\pi$
$854$	0	0
$855$	0.258130	0.00882787
$856$	0	0
$857$	−13.9947	−0.478050	−0.239025	−	0.971013i	$-0.576828\pi$
$857$	−13.9947	−0.478050	−0.239025	+	0.971013i	$0.576828\pi$
$858$	0	0
$859$	35.0781i	1.19685i	0.801179	+	0.598424i	$0.204206\pi$
$859$	35.0781i	1.19685i	−0.801179	+	0.598424i	$0.795794\pi$
$860$	0	0
$861$	− 21.7772i	− 0.742165i
$862$	0	0
$863$	4.18861	0.142582	0.0712910	−	0.997456i	$-0.477288\pi$
$863$	4.18861	0.142582	0.0712910	+	0.997456i	$0.477288\pi$
$864$	0	0
$865$	0.0896594	0.00304851
$866$	0	0
$867$	85.7579i	2.91249i
$868$	0	0
$869$	− 25.0138i	− 0.848536i
$870$	0	0
$871$	29.6483	1.00459
$872$	0	0
$873$	1.25947	0.0426267
$874$	0	0
$875$	− 3.64720i	− 0.123298i
$876$	0	0
$877$	0.663532i	0.0224059i	0.999937	+	0.0112029i	$0.00356608\pi$
$877$	0.663532i	0.0224059i	−0.999937	+	0.0112029i	$0.996434\pi$
$878$	0	0
$879$	−47.1563	−1.59054
$880$	0	0
$881$	−26.5838	−0.895631	−0.447816	−	0.894126i	$-0.647798\pi$
$881$	−26.5838	−0.895631	−0.447816	+	0.894126i	$0.647798\pi$
$882$	0	0
$883$	− 38.1327i	− 1.28327i	−0.767011	−	0.641634i	$-0.778257\pi$
$883$	− 38.1327i	− 1.28327i	0.767011	−	0.641634i	$-0.221743\pi$
$884$	0	0
$885$	− 6.13202i	− 0.206126i
$886$	0	0
$887$	−50.6497	−1.70065	−0.850326	−	0.526257i	$-0.823595\pi$
$887$	−50.6497	−1.70065	−0.850326	+	0.526257i	$0.823595\pi$
$888$	0	0
$889$	−18.5931	−0.623593
$890$	0	0
$891$	− 34.0325i	− 1.14013i
$892$	0	0
$893$	0.476375i	0.0159413i
$894$	0	0
$895$	−0.286994	−0.00959315
$896$	0	0
$897$	−157.680	−5.26479
$898$	0	0
$899$	− 5.07167i	− 0.169150i
$900$	0	0
$901$	17.1556i	0.571536i
$902$	0	0
$903$	0.109494	0.00364374
$904$	0	0
$905$	5.56936	0.185132
$906$	0	0
$907$	− 36.3370i	− 1.20655i	−0.797533	−	0.603276i	$-0.793862\pi$
$907$	− 36.3370i	− 1.20655i	0.797533	−	0.603276i	$-0.206138\pi$
$908$	0	0
$909$	58.7091i	1.94726i
$910$	0	0
$911$	25.1125	0.832015	0.416008	−	0.909361i	$-0.363429\pi$
$911$	25.1125	0.832015	0.416008	+	0.909361i	$0.363429\pi$
$912$	0	0
$913$	34.0360	1.12643
$914$	0	0
$915$	− 6.67096i	− 0.220535i
$916$	0	0
$917$	− 3.06663i	− 0.101269i
$918$	0	0
$919$	13.4558	0.443866	0.221933	−	0.975062i	$-0.428763\pi$
$919$	13.4558	0.443866	0.221933	+	0.975062i	$0.428763\pi$
$920$	0	0
$921$	47.2558	1.55713
$922$	0	0
$923$	14.3197i	0.471340i
$924$	0	0
$925$	49.4801i	1.62690i
$926$	0	0
$927$	−69.4963	−2.28256
$928$	0	0
$929$	−12.3637	−0.405641	−0.202820	−	0.979216i	$-0.565011\pi$
$929$	−12.3637	−0.405641	−0.202820	+	0.979216i	$0.565011\pi$
$930$	0	0
$931$	0.117109i	0.00383809i
$932$	0	0
$933$	42.5629i	1.39345i
$934$	0	0
$935$	9.83007	0.321478
$936$	0	0
$937$	−41.9442	−1.37026	−0.685129	−	0.728422i	$-0.740254\pi$
$937$	−41.9442	−1.37026	−0.685129	+	0.728422i	$0.740254\pi$
$938$	0	0
$939$	− 80.4627i	− 2.62580i
$940$	0	0
$941$	− 15.5943i	− 0.508360i	−0.967157	−	0.254180i	$-0.918194\pi$
$941$	− 15.5943i	− 0.508360i	0.967157	−	0.254180i	$-0.0818056\pi$
$942$	0	0
$943$	−63.0832	−2.05427
$944$	0	0
$945$	−3.27744	−0.106615
$946$	0	0
$947$	13.8044i	0.448583i	0.974522	+	0.224291i	$0.0720067\pi$
$947$	13.8044i	0.448583i	−0.974522	+	0.224291i	$0.927993\pi$
$948$	0	0
$949$	− 41.4537i	− 1.34564i
$950$	0	0
$951$	2.49080	0.0807696
$952$	0	0
$953$	4.06472	0.131669	0.0658346	−	0.997831i	$-0.479029\pi$
$953$	4.06472	0.131669	0.0658346	+	0.997831i	$0.479029\pi$
$954$	0	0
$955$	5.70229i	0.184522i
$956$	0	0
$957$	73.9226i	2.38958i
$958$	0	0
$959$	6.90239	0.222890
$960$	0	0
$961$	−30.3470	−0.978936
$962$	0	0
$963$	62.1536i	2.00287i
$964$	0	0
$965$	3.60671i	0.116104i
$966$	0	0
$967$	54.3821	1.74881	0.874406	−	0.485195i	$-0.161251\pi$
$967$	54.3821	1.74881	0.874406	+	0.485195i	$0.161251\pi$
$968$	0	0
$969$	−2.36852	−0.0760879
$970$	0	0
$971$	− 18.5975i	− 0.596821i	−0.954438	−	0.298410i	$-0.903544\pi$
$971$	− 18.5975i	− 0.596821i	0.954438	−	0.298410i	$-0.0964564\pi$
$972$	0	0
$973$	− 18.9927i	− 0.608879i
$974$	0	0
$975$	88.4339	2.83215
$976$	0	0
$977$	−2.81062	−0.0899197	−0.0449598	−	0.998989i	$-0.514316\pi$
$977$	−2.81062	−0.0899197	−0.0449598	+	0.998989i	$0.514316\pi$
$978$	0	0
$979$	− 14.0248i	− 0.448234i
$980$	0	0
$981$	78.7385i	2.51393i
$982$	0	0
$983$	44.4447	1.41757	0.708783	−	0.705427i	$-0.249245\pi$
$983$	44.4447	1.41757	0.708783	+	0.705427i	$0.249245\pi$
$984$	0	0
$985$	7.88093	0.251107
$986$	0	0
$987$	− 12.1768i	− 0.387593i
$988$	0	0
$989$	− 0.317177i	− 0.0100857i
$990$	0	0
$991$	43.6101	1.38532	0.692660	−	0.721264i	$-0.256439\pi$
$991$	43.6101	1.38532	0.692660	+	0.721264i	$0.256439\pi$
$992$	0	0
$993$	−40.1617	−1.27449
$994$	0	0
$995$	− 4.52200i	− 0.143357i
$996$	0	0
$997$	8.23286i	0.260737i	0.991466	+	0.130369i	$0.0416161\pi$
$997$	8.23286i	0.260737i	−0.991466	+	0.130369i	$0.958384\pi$
$998$	0	0
$999$	90.1776	2.85309

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3584.2.b.g.1793.9		10
4.3	odd	2		3584.2.b.h.1793.2		10
8.3	odd	2		3584.2.b.h.1793.9		10
8.5	even	2	inner	3584.2.b.g.1793.2		10
16.3	odd	4		3584.2.a.o.1.9	yes	10
16.5	even	4		3584.2.a.p.1.9	yes	10
16.11	odd	4		3584.2.a.o.1.2	✓	10
16.13	even	4		3584.2.a.p.1.2	yes	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3584.2.a.o.1.2	✓	10	16.11	odd	4
3584.2.a.o.1.9	yes	10	16.3	odd	4
3584.2.a.p.1.2	yes	10	16.13	even	4
3584.2.a.p.1.9	yes	10	16.5	even	4
3584.2.b.g.1793.2		10	8.5	even	2	inner
3584.2.b.g.1793.9		10	1.1	even	1	trivial
3584.2.b.h.1793.2		10	4.3	odd	2
3584.2.b.h.1793.9		10	8.3	odd	2

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:	\( N \)	\(=\)	\( 3584 = 2^{9} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3584.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.99347i q^{3} +0.369776i q^{5} -1.00000 q^{7} -5.96088 q^{9} +O(q^{10})\)	\(q+2.99347i q^{3} +0.369776i q^{5} -1.00000 q^{7} -5.96088 q^{9} -3.93465i q^{11} -6.07457i q^{13} -1.10691 q^{15} +6.75635 q^{17} +0.117109i q^{19} -2.99347i q^{21} -8.67135 q^{23} +4.86327 q^{25} -8.86331i q^{27} +6.27619i q^{29} -0.808080 q^{31} +11.7783 q^{33} -0.369776i q^{35} +10.1743i q^{37} +18.1840 q^{39} +7.27490 q^{41} +0.0365776i q^{43} -2.20419i q^{45} +4.06779 q^{47} +1.00000 q^{49} +20.2250i q^{51} +2.53918i q^{53} +1.45494 q^{55} -0.350562 q^{57} +5.53975i q^{59} +6.02663i q^{61} +5.96088 q^{63} +2.24623 q^{65} +4.88072i q^{67} -25.9574i q^{69} -2.35733 q^{71} +6.82414 q^{73} +14.5581i q^{75} +3.93465i q^{77} +6.35733 q^{79} +8.64944 q^{81} +8.65033i q^{83} +2.49834i q^{85} -18.7876 q^{87} +3.56443 q^{89} +6.07457i q^{91} -2.41897i q^{93} -0.0433041 q^{95} -0.211290 q^{97} +23.4540i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 10 q^{7} - 22 q^{9}+O(q^{10}) \) 10 * q - 10 * q^7 - 22 * q^9	\( 10 q - 10 q^{7} - 22 q^{9} + 16 q^{15} + 16 q^{17} - 8 q^{23} - 30 q^{25} - 8 q^{31} + 12 q^{33} - 20 q^{41} - 24 q^{47} + 10 q^{49} + 32 q^{55} - 28 q^{57} + 22 q^{63} + 32 q^{65} - 48 q^{73} + 40 q^{79} + 62 q^{81} + 8 q^{87} - 16 q^{89} - 32 q^{95} + 32 q^{97}+O(q^{100}) \) 10 * q - 10 * q^7 - 22 * q^9 + 16 * q^15 + 16 * q^17 - 8 * q^23 - 30 * q^25 - 8 * q^31 + 12 * q^33 - 20 * q^41 - 24 * q^47 + 10 * q^49 + 32 * q^55 - 28 * q^57 + 22 * q^63 + 32 * q^65 - 48 * q^73 + 40 * q^79 + 62 * q^81 + 8 * q^87 - 16 * q^89 - 32 * q^95 + 32 * q^97

Introduction

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Modular forms

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