LMFDB - Embedded newform 5850.2.e.f.5149.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5850,2,Mod(5149,5850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5850, 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("5850.5149");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5850 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5850.e (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:	$46.7124851824$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1950)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			5149.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	5850.5149
Dual form			5850.2.e.f.5149.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{8} +O(q^{10})$	$q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{8} -4.00000 q^{11} -1.00000i q^{13} +1.00000 q^{16} -1.00000 q^{19} +4.00000i q^{22} -4.00000i q^{23} -1.00000 q^{26} -3.00000 q^{29} +4.00000 q^{31} -1.00000i q^{32} +5.00000i q^{37} +1.00000i q^{38} -9.00000 q^{41} +2.00000i q^{43} +4.00000 q^{44} -4.00000 q^{46} -3.00000i q^{47} +7.00000 q^{49} +1.00000i q^{52} -1.00000i q^{53} +3.00000i q^{58} +10.0000 q^{59} +4.00000 q^{61} -4.00000i q^{62} -1.00000 q^{64} +9.00000i q^{67} -7.00000 q^{71} +4.00000i q^{73} +5.00000 q^{74} +1.00000 q^{76} -11.0000 q^{79} +9.00000i q^{82} -6.00000i q^{83} +2.00000 q^{86} -4.00000i q^{88} +10.0000 q^{89} +4.00000i q^{92} -3.00000 q^{94} +12.0000i q^{97} -7.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4}+O(q^{10}) $ 2 * q - 2 * q^4	$ 2 q - 2 q^{4} - 8 q^{11} + 2 q^{16} - 2 q^{19} - 2 q^{26} - 6 q^{29} + 8 q^{31} - 18 q^{41} + 8 q^{44} - 8 q^{46} + 14 q^{49} + 20 q^{59} + 8 q^{61} - 2 q^{64} - 14 q^{71} + 10 q^{74} + 2 q^{76} - 22 q^{79} + 4 q^{86} + 20 q^{89} - 6 q^{94}+O(q^{100}) $ 2 * q - 2 * q^4 - 8 * q^11 + 2 * q^16 - 2 * q^19 - 2 * q^26 - 6 * q^29 + 8 * q^31 - 18 * q^41 + 8 * q^44 - 8 * q^46 + 14 * q^49 + 20 * q^59 + 8 * q^61 - 2 * q^64 - 14 * q^71 + 10 * q^74 + 2 * q^76 - 22 * q^79 + 4 * q^86 + 20 * q^89 - 6 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5850\mathbb{Z}\right)^\times$.

$n$	$2251$	$3251$	$3277$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	1.00000i	0.353553i
$9$	0	0
$10$	0	0
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	− 1.00000i	− 0.277350i
$13$	− 1.00000i	− 0.277350i
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0	0
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	0	0
$21$	0	0
$22$	4.00000i	0.852803i
$23$	− 4.00000i	− 0.834058i	−0.908893	−	0.417029i	$-0.863071\pi$
$23$	− 4.00000i	− 0.834058i	0.908893	−	0.417029i	$-0.136929\pi$
$24$	0	0
$25$	0	0
$26$	−1.00000	−0.196116
$27$	0	0
$28$	0	0
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	5.00000i	0.821995i	0.911636	+	0.410997i	$0.134819\pi$
$37$	5.00000i	0.821995i	−0.911636	+	0.410997i	$0.865181\pi$
$38$	1.00000i	0.162221i
$39$	0	0
$40$	0	0
$41$	−9.00000	−1.40556	−0.702782	−	0.711405i	$-0.748059\pi$
$41$	−9.00000	−1.40556	−0.702782	+	0.711405i	$0.748059\pi$
$42$	0	0
$43$	2.00000i	0.304997i	0.988304	+	0.152499i	$0.0487319\pi$
$43$	2.00000i	0.304997i	−0.988304	+	0.152499i	$0.951268\pi$
$44$	4.00000	0.603023
$45$	0	0
$46$	−4.00000	−0.589768
$47$	− 3.00000i	− 0.437595i	−0.975770	−	0.218797i	$-0.929787\pi$
$47$	− 3.00000i	− 0.437595i	0.975770	−	0.218797i	$-0.0702134\pi$
$48$	0	0
$49$	7.00000	1.00000
$50$	0	0
$51$	0	0
$52$	1.00000i	0.138675i
$53$	− 1.00000i	− 0.137361i	−0.997639	−	0.0686803i	$-0.978121\pi$
$53$	− 1.00000i	− 0.137361i	0.997639	−	0.0686803i	$-0.0218788\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	3.00000i	0.393919i
$59$	10.0000	1.30189	0.650945	−	0.759125i	$-0.274373\pi$
$59$	10.0000	1.30189	0.650945	+	0.759125i	$0.274373\pi$
$60$	0	0
$61$	4.00000	0.512148	0.256074	−	0.966657i	$-0.417571\pi$
$61$	4.00000	0.512148	0.256074	+	0.966657i	$0.417571\pi$
$62$	− 4.00000i	− 0.508001i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	9.00000i	1.09952i	0.835321	+	0.549762i	$0.185282\pi$
$67$	9.00000i	1.09952i	−0.835321	+	0.549762i	$0.814718\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−7.00000	−0.830747	−0.415374	−	0.909651i	$-0.636349\pi$
$71$	−7.00000	−0.830747	−0.415374	+	0.909651i	$0.636349\pi$
$72$	0	0
$73$	4.00000i	0.468165i	0.972217	+	0.234082i	$0.0752085\pi$
$73$	4.00000i	0.468165i	−0.972217	+	0.234082i	$0.924791\pi$
$74$	5.00000	0.581238
$75$	0	0
$76$	1.00000	0.114708
$77$	0	0
$78$	0	0
$79$	−11.0000	−1.23760	−0.618798	−	0.785550i	$-0.712380\pi$
$79$	−11.0000	−1.23760	−0.618798	+	0.785550i	$0.712380\pi$
$80$	0	0
$81$	0	0
$82$	9.00000i	0.993884i
$83$	− 6.00000i	− 0.658586i	−0.944228	−	0.329293i	$-0.893190\pi$
$83$	− 6.00000i	− 0.658586i	0.944228	−	0.329293i	$-0.106810\pi$
$84$	0	0
$85$	0	0
$86$	2.00000	0.215666
$87$	0	0
$88$	− 4.00000i	− 0.426401i
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	0	0
$91$	0	0
$92$	4.00000i	0.417029i
$93$	0	0
$94$	−3.00000	−0.309426
$95$	0	0
$96$	0	0
$97$	12.0000i	1.21842i	0.793011	+	0.609208i	$0.208512\pi$
$97$	12.0000i	1.21842i	−0.793011	+	0.609208i	$0.791488\pi$
$98$	− 7.00000i	− 0.707107i
$99$	0	0
$100$	0	0
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	−1.00000	−0.0971286
$107$	9.00000i	0.870063i	0.900415	+	0.435031i	$0.143263\pi$
$107$	9.00000i	0.870063i	−0.900415	+	0.435031i	$0.856737\pi$
$108$	0	0
$109$	1.00000	0.0957826	0.0478913	−	0.998853i	$-0.484750\pi$
$109$	1.00000	0.0957826	0.0478913	+	0.998853i	$0.484750\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 2.00000i	− 0.188144i	−0.995565	−	0.0940721i	$-0.970012\pi$
$113$	− 2.00000i	− 0.188144i	0.995565	−	0.0940721i	$-0.0299884\pi$
$114$	0	0
$115$	0	0
$116$	3.00000	0.278543
$117$	0	0
$118$	− 10.0000i	− 0.920575i
$119$	0	0
$120$	0	0
$121$	5.00000	0.454545
$122$	− 4.00000i	− 0.362143i
$123$	0	0
$124$	−4.00000	−0.359211
$125$	0	0
$126$	0	0
$127$	17.0000i	1.50851i	0.656584	+	0.754253i	$0.272001\pi$
$127$	17.0000i	1.50851i	−0.656584	+	0.754253i	$0.727999\pi$
$128$	1.00000i	0.0883883i
$129$	0	0
$130$	0	0
$131$	3.00000	0.262111	0.131056	−	0.991375i	$-0.458163\pi$
$131$	3.00000	0.262111	0.131056	+	0.991375i	$0.458163\pi$
$132$	0	0
$133$	0	0
$134$	9.00000	0.777482
$135$	0	0
$136$	0	0
$137$	23.0000i	1.96502i	0.186203	+	0.982511i	$0.440382\pi$
$137$	23.0000i	1.96502i	−0.186203	+	0.982511i	$0.559618\pi$
$138$	0	0
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	0	0
$141$	0	0
$142$	7.00000i	0.587427i
$143$	4.00000i	0.334497i
$144$	0	0
$145$	0	0
$146$	4.00000	0.331042
$147$	0	0
$148$	− 5.00000i	− 0.410997i
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	12.0000	0.976546	0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000	0.976546	0.488273	+	0.872691i	$0.337627\pi$
$152$	− 1.00000i	− 0.0811107i
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 2.00000i	− 0.159617i	−0.996810	−	0.0798087i	$-0.974569\pi$
$157$	− 2.00000i	− 0.159617i	0.996810	−	0.0798087i	$-0.0254309\pi$
$158$	11.0000i	0.875113i
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	8.00000i	0.626608i	0.949653	+	0.313304i	$0.101436\pi$
$163$	8.00000i	0.626608i	−0.949653	+	0.313304i	$0.898564\pi$
$164$	9.00000	0.702782
$165$	0	0
$166$	−6.00000	−0.465690
$167$	7.00000i	0.541676i	0.962625	+	0.270838i	$0.0873008\pi$
$167$	7.00000i	0.541676i	−0.962625	+	0.270838i	$0.912699\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	0	0
$172$	− 2.00000i	− 0.152499i
$173$	3.00000i	0.228086i	0.993476	+	0.114043i	$0.0363801\pi$
$173$	3.00000i	0.228086i	−0.993476	+	0.114043i	$0.963620\pi$
$174$	0	0
$175$	0	0
$176$	−4.00000	−0.301511
$177$	0	0
$178$	− 10.0000i	− 0.749532i
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	−12.0000	−0.891953	−0.445976	−	0.895045i	$-0.647144\pi$
$181$	−12.0000	−0.891953	−0.445976	+	0.895045i	$0.647144\pi$
$182$	0	0
$183$	0	0
$184$	4.00000	0.294884
$185$	0	0
$186$	0	0
$187$	0	0
$188$	3.00000i	0.218797i
$189$	0	0
$190$	0	0
$191$	18.0000	1.30243	0.651217	−	0.758891i	$-0.274259\pi$
$191$	18.0000	1.30243	0.651217	+	0.758891i	$0.274259\pi$
$192$	0	0
$193$	20.0000i	1.43963i	0.694165	+	0.719816i	$0.255774\pi$
$193$	20.0000i	1.43963i	−0.694165	+	0.719816i	$0.744226\pi$
$194$	12.0000	0.861550
$195$	0	0
$196$	−7.00000	−0.500000
$197$	12.0000i	0.854965i	0.904024	+	0.427482i	$0.140599\pi$
$197$	12.0000i	0.854965i	−0.904024	+	0.427482i	$0.859401\pi$
$198$	0	0
$199$	−1.00000	−0.0708881	−0.0354441	−	0.999372i	$-0.511285\pi$
$199$	−1.00000	−0.0708881	−0.0354441	+	0.999372i	$0.511285\pi$
$200$	0	0
$201$	0	0
$202$	− 2.00000i	− 0.140720i
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	− 1.00000i	− 0.0693375i
$209$	4.00000	0.276686
$210$	0	0
$211$	24.0000	1.65223	0.826114	−	0.563503i	$-0.190547\pi$
$211$	24.0000	1.65223	0.826114	+	0.563503i	$0.190547\pi$
$212$	1.00000i	0.0686803i
$213$	0	0
$214$	9.00000	0.615227
$215$	0	0
$216$	0	0
$217$	0	0
$218$	− 1.00000i	− 0.0677285i
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 14.0000i	− 0.937509i	−0.883328	−	0.468755i	$-0.844703\pi$
$223$	− 14.0000i	− 0.937509i	0.883328	−	0.468755i	$-0.155297\pi$
$224$	0	0
$225$	0	0
$226$	−2.00000	−0.133038
$227$	− 10.0000i	− 0.663723i	−0.943328	−	0.331862i	$-0.892323\pi$
$227$	− 10.0000i	− 0.663723i	0.943328	−	0.331862i	$-0.107677\pi$
$228$	0	0
$229$	−5.00000	−0.330409	−0.165205	−	0.986259i	$-0.552828\pi$
$229$	−5.00000	−0.330409	−0.165205	+	0.986259i	$0.552828\pi$
$230$	0	0
$231$	0	0
$232$	− 3.00000i	− 0.196960i
$233$	− 8.00000i	− 0.524097i	−0.965055	−	0.262049i	$-0.915602\pi$
$233$	− 8.00000i	− 0.524097i	0.965055	−	0.262049i	$-0.0843981\pi$
$234$	0	0
$235$	0	0
$236$	−10.0000	−0.650945
$237$	0	0
$238$	0	0
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	0	0
$241$	−12.0000	−0.772988	−0.386494	−	0.922292i	$-0.626314\pi$
$241$	−12.0000	−0.772988	−0.386494	+	0.922292i	$0.626314\pi$
$242$	− 5.00000i	− 0.321412i
$243$	0	0
$244$	−4.00000	−0.256074
$245$	0	0
$246$	0	0
$247$	1.00000i	0.0636285i
$248$	4.00000i	0.254000i
$249$	0	0
$250$	0	0
$251$	−23.0000	−1.45175	−0.725874	−	0.687828i	$-0.758564\pi$
$251$	−23.0000	−1.45175	−0.725874	+	0.687828i	$0.758564\pi$
$252$	0	0
$253$	16.0000i	1.00591i
$254$	17.0000	1.06667
$255$	0	0
$256$	1.00000	0.0625000
$257$	24.0000i	1.49708i	0.663090	+	0.748539i	$0.269245\pi$
$257$	24.0000i	1.49708i	−0.663090	+	0.748539i	$0.730755\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	− 3.00000i	− 0.185341i
$263$	− 6.00000i	− 0.369976i	−0.982741	−	0.184988i	$-0.940775\pi$
$263$	− 6.00000i	− 0.369976i	0.982741	−	0.184988i	$-0.0592246\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	− 9.00000i	− 0.549762i
$269$	−11.0000	−0.670682	−0.335341	−	0.942097i	$-0.608852\pi$
$269$	−11.0000	−0.670682	−0.335341	+	0.942097i	$0.608852\pi$
$270$	0	0
$271$	24.0000	1.45790	0.728948	−	0.684569i	$-0.240010\pi$
$271$	24.0000	1.45790	0.728948	+	0.684569i	$0.240010\pi$
$272$	0	0
$273$	0	0
$274$	23.0000	1.38948
$275$	0	0
$276$	0	0
$277$	26.0000i	1.56219i	0.624413	+	0.781094i	$0.285338\pi$
$277$	26.0000i	1.56219i	−0.624413	+	0.781094i	$0.714662\pi$
$278$	− 12.0000i	− 0.719712i
$279$	0	0
$280$	0	0
$281$	5.00000	0.298275	0.149137	−	0.988816i	$-0.452350\pi$
$281$	5.00000	0.298275	0.149137	+	0.988816i	$0.452350\pi$
$282$	0	0
$283$	4.00000i	0.237775i	0.992908	+	0.118888i	$0.0379328\pi$
$283$	4.00000i	0.237775i	−0.992908	+	0.118888i	$0.962067\pi$
$284$	7.00000	0.415374
$285$	0	0
$286$	4.00000	0.236525
$287$	0	0
$288$	0	0
$289$	17.0000	1.00000
$290$	0	0
$291$	0	0
$292$	− 4.00000i	− 0.234082i
$293$	16.0000i	0.934730i	0.884064	+	0.467365i	$0.154797\pi$
$293$	16.0000i	0.934730i	−0.884064	+	0.467365i	$0.845203\pi$
$294$	0	0
$295$	0	0
$296$	−5.00000	−0.290619
$297$	0	0
$298$	0	0
$299$	−4.00000	−0.231326
$300$	0	0
$301$	0	0
$302$	− 12.0000i	− 0.690522i
$303$	0	0
$304$	−1.00000	−0.0573539
$305$	0	0
$306$	0	0
$307$	− 3.00000i	− 0.171219i	−0.996329	−	0.0856095i	$-0.972716\pi$
$307$	− 3.00000i	− 0.171219i	0.996329	−	0.0856095i	$-0.0272838\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	30.0000	1.70114	0.850572	−	0.525859i	$-0.176256\pi$
$311$	30.0000	1.70114	0.850572	+	0.525859i	$0.176256\pi$
$312$	0	0
$313$	1.00000i	0.0565233i	0.999601	+	0.0282617i	$0.00899717\pi$
$313$	1.00000i	0.0565233i	−0.999601	+	0.0282617i	$0.991003\pi$
$314$	−2.00000	−0.112867
$315$	0	0
$316$	11.0000	0.618798
$317$	32.0000i	1.79730i	0.438667	+	0.898650i	$0.355451\pi$
$317$	32.0000i	1.79730i	−0.438667	+	0.898650i	$0.644549\pi$
$318$	0	0
$319$	12.0000	0.671871
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	8.00000	0.443079
$327$	0	0
$328$	− 9.00000i	− 0.496942i
$329$	0	0
$330$	0	0
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	6.00000i	0.329293i
$333$	0	0
$334$	7.00000	0.383023
$335$	0	0
$336$	0	0
$337$	− 22.0000i	− 1.19842i	−0.800593	−	0.599208i	$-0.795482\pi$
$337$	− 22.0000i	− 1.19842i	0.800593	−	0.599208i	$-0.204518\pi$
$338$	1.00000i	0.0543928i
$339$	0	0
$340$	0	0
$341$	−16.0000	−0.866449
$342$	0	0
$343$	0	0
$344$	−2.00000	−0.107833
$345$	0	0
$346$	3.00000	0.161281
$347$	− 17.0000i	− 0.912608i	−0.889824	−	0.456304i	$-0.849173\pi$
$347$	− 17.0000i	− 0.912608i	0.889824	−	0.456304i	$-0.150827\pi$
$348$	0	0
$349$	−18.0000	−0.963518	−0.481759	−	0.876304i	$-0.660002\pi$
$349$	−18.0000	−0.963518	−0.481759	+	0.876304i	$0.660002\pi$
$350$	0	0
$351$	0	0
$352$	4.00000i	0.213201i
$353$	15.0000i	0.798369i	0.916871	+	0.399185i	$0.130707\pi$
$353$	15.0000i	0.798369i	−0.916871	+	0.399185i	$0.869293\pi$
$354$	0	0
$355$	0	0
$356$	−10.0000	−0.529999
$357$	0	0
$358$	12.0000i	0.634220i
$359$	−17.0000	−0.897226	−0.448613	−	0.893726i	$-0.648082\pi$
$359$	−17.0000	−0.897226	−0.448613	+	0.893726i	$0.648082\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	12.0000i	0.630706i
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	21.0000i	1.09619i	0.836416	+	0.548096i	$0.184647\pi$
$367$	21.0000i	1.09619i	−0.836416	+	0.548096i	$0.815353\pi$
$368$	− 4.00000i	− 0.208514i
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 10.0000i	− 0.517780i	−0.965907	−	0.258890i	$-0.916643\pi$
$373$	− 10.0000i	− 0.517780i	0.965907	−	0.258890i	$-0.0833568\pi$
$374$	0	0
$375$	0	0
$376$	3.00000	0.154713
$377$	3.00000i	0.154508i
$378$	0	0
$379$	28.0000	1.43826	0.719132	−	0.694874i	$-0.244540\pi$
$379$	28.0000	1.43826	0.719132	+	0.694874i	$0.244540\pi$
$380$	0	0
$381$	0	0
$382$	− 18.0000i	− 0.920960i
$383$	5.00000i	0.255488i	0.991807	+	0.127744i	$0.0407736\pi$
$383$	5.00000i	0.255488i	−0.991807	+	0.127744i	$0.959226\pi$
$384$	0	0
$385$	0	0
$386$	20.0000	1.01797
$387$	0	0
$388$	− 12.0000i	− 0.609208i
$389$	−19.0000	−0.963338	−0.481669	−	0.876353i	$-0.659969\pi$
$389$	−19.0000	−0.963338	−0.481669	+	0.876353i	$0.659969\pi$
$390$	0	0
$391$	0	0
$392$	7.00000i	0.353553i
$393$	0	0
$394$	12.0000	0.604551
$395$	0	0
$396$	0	0
$397$	− 3.00000i	− 0.150566i	−0.997162	−	0.0752828i	$-0.976014\pi$
$397$	− 3.00000i	− 0.150566i	0.997162	−	0.0752828i	$-0.0239860\pi$
$398$	1.00000i	0.0501255i
$399$	0	0
$400$	0	0
$401$	26.0000	1.29838	0.649189	−	0.760627i	$-0.275108\pi$
$401$	26.0000	1.29838	0.649189	+	0.760627i	$0.275108\pi$
$402$	0	0
$403$	− 4.00000i	− 0.199254i
$404$	−2.00000	−0.0995037
$405$	0	0
$406$	0	0
$407$	− 20.0000i	− 0.991363i
$408$	0	0
$409$	40.0000	1.97787	0.988936	−	0.148340i	$-0.0473931\pi$
$409$	40.0000	1.97787	0.988936	+	0.148340i	$0.0473931\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	−1.00000	−0.0490290
$417$	0	0
$418$	− 4.00000i	− 0.195646i
$419$	−37.0000	−1.80757	−0.903784	−	0.427989i	$-0.859222\pi$
$419$	−37.0000	−1.80757	−0.903784	+	0.427989i	$0.859222\pi$
$420$	0	0
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	− 24.0000i	− 1.16830i
$423$	0	0
$424$	1.00000	0.0485643
$425$	0	0
$426$	0	0
$427$	0	0
$428$	− 9.00000i	− 0.435031i
$429$	0	0
$430$	0	0
$431$	3.00000	0.144505	0.0722525	−	0.997386i	$-0.476981\pi$
$431$	3.00000	0.144505	0.0722525	+	0.997386i	$0.476981\pi$
$432$	0	0
$433$	− 19.0000i	− 0.913082i	−0.889702	−	0.456541i	$-0.849088\pi$
$433$	− 19.0000i	− 0.913082i	0.889702	−	0.456541i	$-0.150912\pi$
$434$	0	0
$435$	0	0
$436$	−1.00000	−0.0478913
$437$	4.00000i	0.191346i
$438$	0	0
$439$	19.0000	0.906821	0.453410	−	0.891302i	$-0.350207\pi$
$439$	19.0000	0.906821	0.453410	+	0.891302i	$0.350207\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	3.00000i	0.142534i	0.997457	+	0.0712672i	$0.0227043\pi$
$443$	3.00000i	0.142534i	−0.997457	+	0.0712672i	$0.977296\pi$
$444$	0	0
$445$	0	0
$446$	−14.0000	−0.662919
$447$	0	0
$448$	0	0
$449$	−9.00000	−0.424736	−0.212368	−	0.977190i	$-0.568118\pi$
$449$	−9.00000	−0.424736	−0.212368	+	0.977190i	$0.568118\pi$
$450$	0	0
$451$	36.0000	1.69517
$452$	2.00000i	0.0940721i
$453$	0	0
$454$	−10.0000	−0.469323
$455$	0	0
$456$	0	0
$457$	− 32.0000i	− 1.49690i	−0.663193	−	0.748448i	$-0.730799\pi$
$457$	− 32.0000i	− 1.49690i	0.663193	−	0.748448i	$-0.269201\pi$
$458$	5.00000i	0.233635i
$459$	0	0
$460$	0	0
$461$	24.0000	1.11779	0.558896	−	0.829238i	$-0.311225\pi$
$461$	24.0000	1.11779	0.558896	+	0.829238i	$0.311225\pi$
$462$	0	0
$463$	− 14.0000i	− 0.650635i	−0.945605	−	0.325318i	$-0.894529\pi$
$463$	− 14.0000i	− 0.650635i	0.945605	−	0.325318i	$-0.105471\pi$
$464$	−3.00000	−0.139272
$465$	0	0
$466$	−8.00000	−0.370593
$467$	15.0000i	0.694117i	0.937843	+	0.347059i	$0.112820\pi$
$467$	15.0000i	0.694117i	−0.937843	+	0.347059i	$0.887180\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	10.0000i	0.460287i
$473$	− 8.00000i	− 0.367840i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	8.00000i	0.365911i
$479$	25.0000	1.14228	0.571140	−	0.820853i	$-0.306501\pi$
$479$	25.0000	1.14228	0.571140	+	0.820853i	$0.306501\pi$
$480$	0	0
$481$	5.00000	0.227980
$482$	12.0000i	0.546585i
$483$	0	0
$484$	−5.00000	−0.227273
$485$	0	0
$486$	0	0
$487$	38.0000i	1.72194i	0.508652	+	0.860972i	$0.330144\pi$
$487$	38.0000i	1.72194i	−0.508652	+	0.860972i	$0.669856\pi$
$488$	4.00000i	0.181071i
$489$	0	0
$490$	0	0
$491$	32.0000	1.44414	0.722070	−	0.691820i	$-0.243191\pi$
$491$	32.0000	1.44414	0.722070	+	0.691820i	$0.243191\pi$
$492$	0	0
$493$	0	0
$494$	1.00000	0.0449921
$495$	0	0
$496$	4.00000	0.179605
$497$	0	0
$498$	0	0
$499$	1.00000	0.0447661	0.0223831	−	0.999749i	$-0.492875\pi$
$499$	1.00000	0.0447661	0.0223831	+	0.999749i	$0.492875\pi$
$500$	0	0
$501$	0	0
$502$	23.0000i	1.02654i
$503$	− 16.0000i	− 0.713405i	−0.934218	−	0.356702i	$-0.883901\pi$
$503$	− 16.0000i	− 0.713405i	0.934218	−	0.356702i	$-0.116099\pi$
$504$	0	0
$505$	0	0
$506$	16.0000	0.711287
$507$	0	0
$508$	− 17.0000i	− 0.754253i
$509$	24.0000	1.06378	0.531891	−	0.846813i	$-0.321482\pi$
$509$	24.0000	1.06378	0.531891	+	0.846813i	$0.321482\pi$
$510$	0	0
$511$	0	0
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	24.0000	1.05859
$515$	0	0
$516$	0	0
$517$	12.0000i	0.527759i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	0	0
$523$	− 30.0000i	− 1.31181i	−0.754844	−	0.655904i	$-0.772288\pi$
$523$	− 30.0000i	− 1.31181i	0.754844	−	0.655904i	$-0.227712\pi$
$524$	−3.00000	−0.131056
$525$	0	0
$526$	−6.00000	−0.261612
$527$	0	0
$528$	0	0
$529$	7.00000	0.304348
$530$	0	0
$531$	0	0
$532$	0	0
$533$	9.00000i	0.389833i
$534$	0	0
$535$	0	0
$536$	−9.00000	−0.388741
$537$	0	0
$538$	11.0000i	0.474244i
$539$	−28.0000	−1.20605
$540$	0	0
$541$	−26.0000	−1.11783	−0.558914	−	0.829226i	$-0.688782\pi$
$541$	−26.0000	−1.11783	−0.558914	+	0.829226i	$0.688782\pi$
$542$	− 24.0000i	− 1.03089i
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	22.0000i	0.940652i	0.882493	+	0.470326i	$0.155864\pi$
$547$	22.0000i	0.940652i	−0.882493	+	0.470326i	$0.844136\pi$
$548$	− 23.0000i	− 0.982511i
$549$	0	0
$550$	0	0
$551$	3.00000	0.127804
$552$	0	0
$553$	0	0
$554$	26.0000	1.10463
$555$	0	0
$556$	−12.0000	−0.508913
$557$	42.0000i	1.77960i	0.456354	+	0.889799i	$0.349155\pi$
$557$	42.0000i	1.77960i	−0.456354	+	0.889799i	$0.650845\pi$
$558$	0	0
$559$	2.00000	0.0845910
$560$	0	0
$561$	0	0
$562$	− 5.00000i	− 0.210912i
$563$	21.0000i	0.885044i	0.896758	+	0.442522i	$0.145916\pi$
$563$	21.0000i	0.885044i	−0.896758	+	0.442522i	$0.854084\pi$
$564$	0	0
$565$	0	0
$566$	4.00000	0.168133
$567$	0	0
$568$	− 7.00000i	− 0.293713i
$569$	−8.00000	−0.335377	−0.167689	−	0.985840i	$-0.553630\pi$
$569$	−8.00000	−0.335377	−0.167689	+	0.985840i	$0.553630\pi$
$570$	0	0
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	− 4.00000i	− 0.167248i
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 12.0000i	− 0.499567i	−0.968302	−	0.249783i	$-0.919641\pi$
$577$	− 12.0000i	− 0.499567i	0.968302	−	0.249783i	$-0.0803594\pi$
$578$	− 17.0000i	− 0.707107i
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	4.00000i	0.165663i
$584$	−4.00000	−0.165521
$585$	0	0
$586$	16.0000	0.660954
$587$	12.0000i	0.495293i	0.968850	+	0.247647i	$0.0796572\pi$
$587$	12.0000i	0.495293i	−0.968850	+	0.247647i	$0.920343\pi$
$588$	0	0
$589$	−4.00000	−0.164817
$590$	0	0
$591$	0	0
$592$	5.00000i	0.205499i
$593$	1.00000i	0.0410651i	0.999789	+	0.0205325i	$0.00653617\pi$
$593$	1.00000i	0.0410651i	−0.999789	+	0.0205325i	$0.993464\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	4.00000i	0.163572i
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	45.0000	1.83559	0.917794	−	0.397057i	$-0.129968\pi$
$601$	45.0000	1.83559	0.917794	+	0.397057i	$0.129968\pi$
$602$	0	0
$603$	0	0
$604$	−12.0000	−0.488273
$605$	0	0
$606$	0	0
$607$	7.00000i	0.284121i	0.989858	+	0.142061i	$0.0453728\pi$
$607$	7.00000i	0.284121i	−0.989858	+	0.142061i	$0.954627\pi$
$608$	1.00000i	0.0405554i
$609$	0	0
$610$	0	0
$611$	−3.00000	−0.121367
$612$	0	0
$613$	− 2.00000i	− 0.0807792i	−0.999184	−	0.0403896i	$-0.987140\pi$
$613$	− 2.00000i	− 0.0807792i	0.999184	−	0.0403896i	$-0.0128599\pi$
$614$	−3.00000	−0.121070
$615$	0	0
$616$	0	0
$617$	− 27.0000i	− 1.08698i	−0.839416	−	0.543490i	$-0.817103\pi$
$617$	− 27.0000i	− 1.08698i	0.839416	−	0.543490i	$-0.182897\pi$
$618$	0	0
$619$	−8.00000	−0.321547	−0.160774	−	0.986991i	$-0.551399\pi$
$619$	−8.00000	−0.321547	−0.160774	+	0.986991i	$0.551399\pi$
$620$	0	0
$621$	0	0
$622$	− 30.0000i	− 1.20289i
$623$	0	0
$624$	0	0
$625$	0	0
$626$	1.00000	0.0399680
$627$	0	0
$628$	2.00000i	0.0798087i
$629$	0	0
$630$	0	0
$631$	−28.0000	−1.11466	−0.557331	−	0.830290i	$-0.688175\pi$
$631$	−28.0000	−1.11466	−0.557331	+	0.830290i	$0.688175\pi$
$632$	− 11.0000i	− 0.437557i
$633$	0	0
$634$	32.0000	1.27088
$635$	0	0
$636$	0	0
$637$	− 7.00000i	− 0.277350i
$638$	− 12.0000i	− 0.475085i
$639$	0	0
$640$	0	0
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	0	0
$643$	− 19.0000i	− 0.749287i	−0.927169	−	0.374643i	$-0.877765\pi$
$643$	− 19.0000i	− 0.749287i	0.927169	−	0.374643i	$-0.122235\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 24.0000i	− 0.943537i	−0.881722	−	0.471769i	$-0.843616\pi$
$647$	− 24.0000i	− 0.943537i	0.881722	−	0.471769i	$-0.156384\pi$
$648$	0	0
$649$	−40.0000	−1.57014
$650$	0	0
$651$	0	0
$652$	− 8.00000i	− 0.313304i
$653$	26.0000i	1.01746i	0.860927	+	0.508729i	$0.169885\pi$
$653$	26.0000i	1.01746i	−0.860927	+	0.508729i	$0.830115\pi$
$654$	0	0
$655$	0	0
$656$	−9.00000	−0.351391
$657$	0	0
$658$	0	0
$659$	13.0000	0.506408	0.253204	−	0.967413i	$-0.418516\pi$
$659$	13.0000	0.506408	0.253204	+	0.967413i	$0.418516\pi$
$660$	0	0
$661$	−13.0000	−0.505641	−0.252821	−	0.967513i	$-0.581358\pi$
$661$	−13.0000	−0.505641	−0.252821	+	0.967513i	$0.581358\pi$
$662$	− 4.00000i	− 0.155464i
$663$	0	0
$664$	6.00000	0.232845
$665$	0	0
$666$	0	0
$667$	12.0000i	0.464642i
$668$	− 7.00000i	− 0.270838i
$669$	0	0
$670$	0	0
$671$	−16.0000	−0.617673
$672$	0	0
$673$	11.0000i	0.424019i	0.977268	+	0.212009i	$0.0680008\pi$
$673$	11.0000i	0.424019i	−0.977268	+	0.212009i	$0.931999\pi$
$674$	−22.0000	−0.847408
$675$	0	0
$676$	1.00000	0.0384615
$677$	2.00000i	0.0768662i	0.999261	+	0.0384331i	$0.0122367\pi$
$677$	2.00000i	0.0768662i	−0.999261	+	0.0384331i	$0.987763\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	16.0000i	0.612672i
$683$	28.0000i	1.07139i	0.844411	+	0.535695i	$0.179950\pi$
$683$	28.0000i	1.07139i	−0.844411	+	0.535695i	$0.820050\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	2.00000i	0.0762493i
$689$	−1.00000	−0.0380970
$690$	0	0
$691$	−3.00000	−0.114125	−0.0570627	−	0.998371i	$-0.518173\pi$
$691$	−3.00000	−0.114125	−0.0570627	+	0.998371i	$0.518173\pi$
$692$	− 3.00000i	− 0.114043i
$693$	0	0
$694$	−17.0000	−0.645311
$695$	0	0
$696$	0	0
$697$	0	0
$698$	18.0000i	0.681310i
$699$	0	0
$700$	0	0
$701$	38.0000	1.43524	0.717620	−	0.696435i	$-0.245231\pi$
$701$	38.0000	1.43524	0.717620	+	0.696435i	$0.245231\pi$
$702$	0	0
$703$	− 5.00000i	− 0.188579i
$704$	4.00000	0.150756
$705$	0	0
$706$	15.0000	0.564532
$707$	0	0
$708$	0	0
$709$	−46.0000	−1.72757	−0.863783	−	0.503864i	$-0.831911\pi$
$709$	−46.0000	−1.72757	−0.863783	+	0.503864i	$0.831911\pi$
$710$	0	0
$711$	0	0
$712$	10.0000i	0.374766i
$713$	− 16.0000i	− 0.599205i
$714$	0	0
$715$	0	0
$716$	12.0000	0.448461
$717$	0	0
$718$	17.0000i	0.634434i
$719$	−36.0000	−1.34257	−0.671287	−	0.741198i	$-0.734258\pi$
$719$	−36.0000	−1.34257	−0.671287	+	0.741198i	$0.734258\pi$
$720$	0	0
$721$	0	0
$722$	18.0000i	0.669891i
$723$	0	0
$724$	12.0000	0.445976
$725$	0	0
$726$	0	0
$727$	− 52.0000i	− 1.92857i	−0.264861	−	0.964287i	$-0.585326\pi$
$727$	− 52.0000i	− 1.92857i	0.264861	−	0.964287i	$-0.414674\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	− 49.0000i	− 1.80986i	−0.425564	−	0.904928i	$-0.639924\pi$
$733$	− 49.0000i	− 1.80986i	0.425564	−	0.904928i	$-0.360076\pi$
$734$	21.0000	0.775124
$735$	0	0
$736$	−4.00000	−0.147442
$737$	− 36.0000i	− 1.32608i
$738$	0	0
$739$	−53.0000	−1.94964	−0.974818	−	0.223001i	$-0.928415\pi$
$739$	−53.0000	−1.94964	−0.974818	+	0.223001i	$0.928415\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	27.0000i	0.990534i	0.868741	+	0.495267i	$0.164930\pi$
$743$	27.0000i	0.990534i	−0.868741	+	0.495267i	$0.835070\pi$
$744$	0	0
$745$	0	0
$746$	−10.0000	−0.366126
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−43.0000	−1.56909	−0.784546	−	0.620070i	$-0.787104\pi$
$751$	−43.0000	−1.56909	−0.784546	+	0.620070i	$0.787104\pi$
$752$	− 3.00000i	− 0.109399i
$753$	0	0
$754$	3.00000	0.109254
$755$	0	0
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	− 28.0000i	− 1.01701i
$759$	0	0
$760$	0	0
$761$	15.0000	0.543750	0.271875	−	0.962333i	$-0.412356\pi$
$761$	15.0000	0.543750	0.271875	+	0.962333i	$0.412356\pi$
$762$	0	0
$763$	0	0
$764$	−18.0000	−0.651217
$765$	0	0
$766$	5.00000	0.180657
$767$	− 10.0000i	− 0.361079i
$768$	0	0
$769$	−16.0000	−0.576975	−0.288487	−	0.957484i	$-0.593152\pi$
$769$	−16.0000	−0.576975	−0.288487	+	0.957484i	$0.593152\pi$
$770$	0	0
$771$	0	0
$772$	− 20.0000i	− 0.719816i
$773$	32.0000i	1.15096i	0.817816	+	0.575480i	$0.195185\pi$
$773$	32.0000i	1.15096i	−0.817816	+	0.575480i	$0.804815\pi$
$774$	0	0
$775$	0	0
$776$	−12.0000	−0.430775
$777$	0	0
$778$	19.0000i	0.681183i
$779$	9.00000	0.322458
$780$	0	0
$781$	28.0000	1.00192
$782$	0	0
$783$	0	0
$784$	7.00000	0.250000
$785$	0	0
$786$	0	0
$787$	− 32.0000i	− 1.14068i	−0.821410	−	0.570338i	$-0.806812\pi$
$787$	− 32.0000i	− 1.14068i	0.821410	−	0.570338i	$-0.193188\pi$
$788$	− 12.0000i	− 0.427482i
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	− 4.00000i	− 0.142044i
$794$	−3.00000	−0.106466
$795$	0	0
$796$	1.00000	0.0354441
$797$	− 22.0000i	− 0.779280i	−0.920967	−	0.389640i	$-0.872599\pi$
$797$	− 22.0000i	− 0.779280i	0.920967	−	0.389640i	$-0.127401\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	− 26.0000i	− 0.918092i
$803$	− 16.0000i	− 0.564628i
$804$	0	0
$805$	0	0
$806$	−4.00000	−0.140894
$807$	0	0
$808$	2.00000i	0.0703598i
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	0	0
$811$	12.0000	0.421377	0.210688	−	0.977553i	$-0.432429\pi$
$811$	12.0000	0.421377	0.210688	+	0.977553i	$0.432429\pi$
$812$	0	0
$813$	0	0
$814$	−20.0000	−0.701000
$815$	0	0
$816$	0	0
$817$	− 2.00000i	− 0.0699711i
$818$	− 40.0000i	− 1.39857i
$819$	0	0
$820$	0	0
$821$	6.00000	0.209401	0.104701	−	0.994504i	$-0.466612\pi$
$821$	6.00000	0.209401	0.104701	+	0.994504i	$0.466612\pi$
$822$	0	0
$823$	− 31.0000i	− 1.08059i	−0.841475	−	0.540296i	$-0.818312\pi$
$823$	− 31.0000i	− 1.08059i	0.841475	−	0.540296i	$-0.181688\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	−46.0000	−1.59765	−0.798823	−	0.601566i	$-0.794544\pi$
$829$	−46.0000	−1.59765	−0.798823	+	0.601566i	$0.794544\pi$
$830$	0	0
$831$	0	0
$832$	1.00000i	0.0346688i
$833$	0	0
$834$	0	0
$835$	0	0
$836$	−4.00000	−0.138343
$837$	0	0
$838$	37.0000i	1.27814i
$839$	24.0000	0.828572	0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000	0.828572	0.414286	+	0.910147i	$0.364031\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	− 2.00000i	− 0.0689246i
$843$	0	0
$844$	−24.0000	−0.826114
$845$	0	0
$846$	0	0
$847$	0	0
$848$	− 1.00000i	− 0.0343401i
$849$	0	0
$850$	0	0
$851$	20.0000	0.685591
$852$	0	0
$853$	41.0000i	1.40381i	0.712269	+	0.701907i	$0.247668\pi$
$853$	41.0000i	1.40381i	−0.712269	+	0.701907i	$0.752332\pi$
$854$	0	0
$855$	0	0
$856$	−9.00000	−0.307614
$857$	42.0000i	1.43469i	0.696717	+	0.717346i	$0.254643\pi$
$857$	42.0000i	1.43469i	−0.696717	+	0.717346i	$0.745357\pi$
$858$	0	0
$859$	−22.0000	−0.750630	−0.375315	−	0.926897i	$-0.622466\pi$
$859$	−22.0000	−0.750630	−0.375315	+	0.926897i	$0.622466\pi$
$860$	0	0
$861$	0	0
$862$	− 3.00000i	− 0.102180i
$863$	− 57.0000i	− 1.94030i	−0.242500	−	0.970151i	$-0.577968\pi$
$863$	− 57.0000i	− 1.94030i	0.242500	−	0.970151i	$-0.422032\pi$
$864$	0	0
$865$	0	0
$866$	−19.0000	−0.645646
$867$	0	0
$868$	0	0
$869$	44.0000	1.49260
$870$	0	0
$871$	9.00000	0.304953
$872$	1.00000i	0.0338643i
$873$	0	0
$874$	4.00000	0.135302
$875$	0	0
$876$	0	0
$877$	23.0000i	0.776655i	0.921521	+	0.388327i	$0.126947\pi$
$877$	23.0000i	0.776655i	−0.921521	+	0.388327i	$0.873053\pi$
$878$	− 19.0000i	− 0.641219i
$879$	0	0
$880$	0	0
$881$	−56.0000	−1.88669	−0.943344	−	0.331816i	$-0.892339\pi$
$881$	−56.0000	−1.88669	−0.943344	+	0.331816i	$0.892339\pi$
$882$	0	0
$883$	36.0000i	1.21150i	0.795656	+	0.605748i	$0.207126\pi$
$883$	36.0000i	1.21150i	−0.795656	+	0.605748i	$0.792874\pi$
$884$	0	0
$885$	0	0
$886$	3.00000	0.100787
$887$	− 44.0000i	− 1.47738i	−0.674048	−	0.738688i	$-0.735446\pi$
$887$	− 44.0000i	− 1.47738i	0.674048	−	0.738688i	$-0.264554\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	14.0000i	0.468755i
$893$	3.00000i	0.100391i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	9.00000i	0.300334i
$899$	−12.0000	−0.400222
$900$	0	0
$901$	0	0
$902$	− 36.0000i	− 1.19867i
$903$	0	0
$904$	2.00000	0.0665190
$905$	0	0
$906$	0	0
$907$	30.0000i	0.996134i	0.867139	+	0.498067i	$0.165957\pi$
$907$	30.0000i	0.996134i	−0.867139	+	0.498067i	$0.834043\pi$
$908$	10.0000i	0.331862i
$909$	0	0
$910$	0	0
$911$	6.00000	0.198789	0.0993944	−	0.995048i	$-0.468309\pi$
$911$	6.00000	0.198789	0.0993944	+	0.995048i	$0.468309\pi$
$912$	0	0
$913$	24.0000i	0.794284i
$914$	−32.0000	−1.05847
$915$	0	0
$916$	5.00000	0.165205
$917$	0	0
$918$	0	0
$919$	−1.00000	−0.0329870	−0.0164935	−	0.999864i	$-0.505250\pi$
$919$	−1.00000	−0.0329870	−0.0164935	+	0.999864i	$0.505250\pi$
$920$	0	0
$921$	0	0
$922$	− 24.0000i	− 0.790398i
$923$	7.00000i	0.230408i
$924$	0	0
$925$	0	0
$926$	−14.0000	−0.460069
$927$	0	0
$928$	3.00000i	0.0984798i
$929$	1.00000	0.0328089	0.0164045	−	0.999865i	$-0.494778\pi$
$929$	1.00000	0.0328089	0.0164045	+	0.999865i	$0.494778\pi$
$930$	0	0
$931$	−7.00000	−0.229416
$932$	8.00000i	0.262049i
$933$	0	0
$934$	15.0000	0.490815
$935$	0	0
$936$	0	0
$937$	− 34.0000i	− 1.11073i	−0.831606	−	0.555366i	$-0.812578\pi$
$937$	− 34.0000i	− 1.11073i	0.831606	−	0.555366i	$-0.187422\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	24.0000	0.782378	0.391189	−	0.920310i	$-0.372064\pi$
$941$	24.0000	0.782378	0.391189	+	0.920310i	$0.372064\pi$
$942$	0	0
$943$	36.0000i	1.17232i
$944$	10.0000	0.325472
$945$	0	0
$946$	−8.00000	−0.260102
$947$	− 4.00000i	− 0.129983i	−0.997886	−	0.0649913i	$-0.979298\pi$
$947$	− 4.00000i	− 0.129983i	0.997886	−	0.0649913i	$-0.0207020\pi$
$948$	0	0
$949$	4.00000	0.129845
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 56.0000i	− 1.81402i	−0.421111	−	0.907009i	$-0.638360\pi$
$953$	− 56.0000i	− 1.81402i	0.421111	−	0.907009i	$-0.361640\pi$
$954$	0	0
$955$	0	0
$956$	8.00000	0.258738
$957$	0	0
$958$	− 25.0000i	− 0.807713i
$959$	0	0
$960$	0	0
$961$	−15.0000	−0.483871
$962$	− 5.00000i	− 0.161206i
$963$	0	0
$964$	12.0000	0.386494
$965$	0	0
$966$	0	0
$967$	− 8.00000i	− 0.257263i	−0.991692	−	0.128631i	$-0.958942\pi$
$967$	− 8.00000i	− 0.257263i	0.991692	−	0.128631i	$-0.0410584\pi$
$968$	5.00000i	0.160706i
$969$	0	0
$970$	0	0
$971$	−15.0000	−0.481373	−0.240686	−	0.970603i	$-0.577373\pi$
$971$	−15.0000	−0.481373	−0.240686	+	0.970603i	$0.577373\pi$
$972$	0	0
$973$	0	0
$974$	38.0000	1.21760
$975$	0	0
$976$	4.00000	0.128037
$977$	30.0000i	0.959785i	0.877327	+	0.479893i	$0.159324\pi$
$977$	30.0000i	0.959785i	−0.877327	+	0.479893i	$0.840676\pi$
$978$	0	0
$979$	−40.0000	−1.27841
$980$	0	0
$981$	0	0
$982$	− 32.0000i	− 1.02116i
$983$	32.0000i	1.02064i	0.859984	+	0.510321i	$0.170473\pi$
$983$	32.0000i	1.02064i	−0.859984	+	0.510321i	$0.829527\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	− 1.00000i	− 0.0318142i
$989$	8.00000	0.254385
$990$	0	0
$991$	−7.00000	−0.222362	−0.111181	−	0.993800i	$-0.535463\pi$
$991$	−7.00000	−0.222362	−0.111181	+	0.993800i	$0.535463\pi$
$992$	− 4.00000i	− 0.127000i
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	14.0000i	0.443384i	0.975117	+	0.221692i	$0.0711580\pi$
$997$	14.0000i	0.443384i	−0.975117	+	0.221692i	$0.928842\pi$
$998$	− 1.00000i	− 0.0316544i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5850.2.e.f.5149.1		2
3.2	odd	2		1950.2.e.h.1249.2		2
5.2	odd	4		5850.2.a.bp.1.1		1
5.3	odd	4		5850.2.a.l.1.1		1
5.4	even	2	inner	5850.2.e.f.5149.2		2
15.2	even	4		1950.2.a.j.1.1	✓	1
15.8	even	4		1950.2.a.s.1.1	yes	1
15.14	odd	2		1950.2.e.h.1249.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1950.2.a.j.1.1	✓	1	15.2	even	4
1950.2.a.s.1.1	yes	1	15.8	even	4
1950.2.e.h.1249.1		2	15.14	odd	2
1950.2.e.h.1249.2		2	3.2	odd	2
5850.2.a.l.1.1		1	5.3	odd	4
5850.2.a.bp.1.1		1	5.2	odd	4
5850.2.e.f.5149.1		2	1.1	even	1	trivial
5850.2.e.f.5149.2		2	5.4	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:	\( N \)	\(=\)	\( 5850 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5850.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{8} -4.00000 q^{11} -1.00000i q^{13} +1.00000 q^{16} -1.00000 q^{19} +4.00000i q^{22} -4.00000i q^{23} -1.00000 q^{26} -3.00000 q^{29} +4.00000 q^{31} -1.00000i q^{32} +5.00000i q^{37} +1.00000i q^{38} -9.00000 q^{41} +2.00000i q^{43} +4.00000 q^{44} -4.00000 q^{46} -3.00000i q^{47} +7.00000 q^{49} +1.00000i q^{52} -1.00000i q^{53} +3.00000i q^{58} +10.0000 q^{59} +4.00000 q^{61} -4.00000i q^{62} -1.00000 q^{64} +9.00000i q^{67} -7.00000 q^{71} +4.00000i q^{73} +5.00000 q^{74} +1.00000 q^{76} -11.0000 q^{79} +9.00000i q^{82} -6.00000i q^{83} +2.00000 q^{86} -4.00000i q^{88} +10.0000 q^{89} +4.00000i q^{92} -3.00000 q^{94} +12.0000i q^{97} -7.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4}+O(q^{10}) \) 2 * q - 2 * q^4	\( 2 q - 2 q^{4} - 8 q^{11} + 2 q^{16} - 2 q^{19} - 2 q^{26} - 6 q^{29} + 8 q^{31} - 18 q^{41} + 8 q^{44} - 8 q^{46} + 14 q^{49} + 20 q^{59} + 8 q^{61} - 2 q^{64} - 14 q^{71} + 10 q^{74} + 2 q^{76} - 22 q^{79} + 4 q^{86} + 20 q^{89} - 6 q^{94}+O(q^{100}) \) 2 * q - 2 * q^4 - 8 * q^11 + 2 * q^16 - 2 * q^19 - 2 * q^26 - 6 * q^29 + 8 * q^31 - 18 * q^41 + 8 * q^44 - 8 * q^46 + 14 * q^49 + 20 * q^59 + 8 * q^61 - 2 * q^64 - 14 * q^71 + 10 * q^74 + 2 * q^76 - 22 * q^79 + 4 * q^86 + 20 * q^89 - 6 * q^94

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