LMFDB - Embedded newform 1600.3.g.k.351.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1600,3,Mod(351,1600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1600, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1600.351");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1600 = 2^{6} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1600.g (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:	$43.5968422976$
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} + 19x^{14} + 301x^{12} + 1102x^{10} + 3238x^{8} + 1102x^{6} + 301x^{4} + 19x^{2} + 1 $ x^16 + 19x^14 + 301x^12 + 1102x^10 + 3238x^8 + 1102x^6 + 301x^4 + 19*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{37} $
Twist minimal:	no (minimal twist has level 320)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			351.11
Root			$1.94376 - 3.36668i$ of defining polynomial
Character	$\chi$	$=$	1600.351
Dual form			1600.3.g.k.351.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.90845 q^{3} +3.95502i q^{7} -5.35782 q^{9} +O(q^{10})$	$q+1.90845 q^{3} +3.95502i q^{7} -5.35782 q^{9} -6.18667 q^{11} -4.94185i q^{13} +22.4311 q^{17} +10.3923 q^{19} +7.54796i q^{21} -39.6083i q^{23} -27.4012 q^{27} -30.4354i q^{29} +24.7156i q^{31} -11.8070 q^{33} +24.0264i q^{37} -9.43127i q^{39} +31.0735 q^{41} +52.2110 q^{43} +13.1640i q^{47} +33.3578 q^{49} +42.8087 q^{51} +17.9596i q^{53} +19.8332 q^{57} +104.421 q^{59} -57.2849i q^{61} -21.1903i q^{63} +99.3796 q^{67} -75.5905i q^{69} -16.7156i q^{71} +96.3357 q^{73} -24.4684i q^{77} +139.578i q^{79} -4.07345 q^{81} -91.7458 q^{83} -58.0844i q^{87} +94.7156 q^{89} +19.5451 q^{91} +47.1686i q^{93} -143.796 q^{97} +33.1471 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 96 q^{9}+O(q^{10}) $ 16 * q + 96 * q^9	$ 16 q + 96 q^{9} - 48 q^{41} + 352 q^{49} + 480 q^{81} + 1152 q^{89}+O(q^{100}) $ 16 * q + 96 * q^9 - 48 * q^41 + 352 * q^49 + 480 * q^81 + 1152 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1151$
$\chi(n)$	$1$	$-1$	$-1$

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
$3$	1.90845	0.636150	0.318075	−	0.948066i	$-0.396964\pi$
$3$	1.90845	0.636150	0.318075	+	0.948066i	$0.396964\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.95502i	0.565003i	0.959267	+	0.282501i	$0.0911642\pi$
$7$	3.95502i	0.565003i	−0.959267	+	0.282501i	$0.908836\pi$
$8$	0	0
$9$	−5.35782	−0.595313
$10$	0	0
$11$	−6.18667	−0.562425	−0.281212	−	0.959646i	$-0.590737\pi$
$11$	−6.18667	−0.562425	−0.281212	+	0.959646i	$0.590737\pi$
$12$	0	0
$13$	− 4.94185i	− 0.380142i	−0.981770	−	0.190071i	$-0.939128\pi$
$13$	− 4.94185i	− 0.380142i	0.981770	−	0.190071i	$-0.0608718\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	22.4311	1.31948	0.659740	−	0.751494i	$-0.270667\pi$
$17$	22.4311	1.31948	0.659740	+	0.751494i	$0.270667\pi$
$18$	0	0
$19$	10.3923	0.546963	0.273482	−	0.961877i	$-0.411825\pi$
$19$	10.3923	0.546963	0.273482	+	0.961877i	$0.411825\pi$
$20$	0	0
$21$	7.54796i	0.359427i
$22$	0	0
$23$	− 39.6083i	− 1.72210i	−0.508520	−	0.861050i	$-0.669807\pi$
$23$	− 39.6083i	− 1.72210i	0.508520	−	0.861050i	$-0.330193\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−27.4012	−1.01486
$28$	0	0
$29$	− 30.4354i	− 1.04950i	−0.851258	−	0.524748i	$-0.824160\pi$
$29$	− 30.4354i	− 1.04950i	0.851258	−	0.524748i	$-0.175840\pi$
$30$	0	0
$31$	24.7156i	0.797278i	0.917108	+	0.398639i	$0.130517\pi$
$31$	24.7156i	0.797278i	−0.917108	+	0.398639i	$0.869483\pi$
$32$	0	0
$33$	−11.8070	−0.357787
$34$	0	0
$35$	0	0
$36$	0	0
$37$	24.0264i	0.649361i	0.945824	+	0.324680i	$0.105257\pi$
$37$	24.0264i	0.649361i	−0.945824	+	0.324680i	$0.894743\pi$
$38$	0	0
$39$	− 9.43127i	− 0.241827i
$40$	0	0
$41$	31.0735	0.757889	0.378945	−	0.925419i	$-0.376287\pi$
$41$	31.0735	0.757889	0.378945	+	0.925419i	$0.376287\pi$
$42$	0	0
$43$	52.2110	1.21421	0.607105	−	0.794622i	$-0.292331\pi$
$43$	52.2110	1.21421	0.607105	+	0.794622i	$0.292331\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	13.1640i	0.280086i	0.990145	+	0.140043i	$0.0447241\pi$
$47$	13.1640i	0.280086i	−0.990145	+	0.140043i	$0.955276\pi$
$48$	0	0
$49$	33.3578	0.680772
$50$	0	0
$51$	42.8087	0.839387
$52$	0	0
$53$	17.9596i	0.338860i	0.985542	+	0.169430i	$0.0541926\pi$
$53$	17.9596i	0.338860i	−0.985542	+	0.169430i	$0.945807\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	19.8332	0.347951
$58$	0	0
$59$	104.421	1.76985	0.884924	−	0.465735i	$-0.154210\pi$
$59$	104.421	1.76985	0.884924	+	0.465735i	$0.154210\pi$
$60$	0	0
$61$	− 57.2849i	− 0.939097i	−0.882907	−	0.469548i	$-0.844417\pi$
$61$	− 57.2849i	− 0.939097i	0.882907	−	0.469548i	$-0.155583\pi$
$62$	0	0
$63$	− 21.1903i	− 0.336354i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	99.3796	1.48328	0.741639	−	0.670799i	$-0.234049\pi$
$67$	99.3796	1.48328	0.741639	+	0.670799i	$0.234049\pi$
$68$	0	0
$69$	− 75.5905i	− 1.09551i
$70$	0	0
$71$	− 16.7156i	− 0.235431i	−0.993047	−	0.117716i	$-0.962443\pi$
$71$	− 16.7156i	− 0.235431i	0.993047	−	0.117716i	$-0.0375572\pi$
$72$	0	0
$73$	96.3357	1.31967	0.659833	−	0.751412i	$-0.270627\pi$
$73$	96.3357	1.31967	0.659833	+	0.751412i	$0.270627\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 24.4684i	− 0.317772i
$78$	0	0
$79$	139.578i	1.76681i	0.468608	+	0.883406i	$0.344756\pi$
$79$	139.578i	1.76681i	−0.468608	+	0.883406i	$0.655244\pi$
$80$	0	0
$81$	−4.07345	−0.0502895
$82$	0	0
$83$	−91.7458	−1.10537	−0.552686	−	0.833390i	$-0.686397\pi$
$83$	−91.7458	−1.10537	−0.552686	+	0.833390i	$0.686397\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 58.0844i	− 0.667637i
$88$	0	0
$89$	94.7156	1.06422	0.532110	−	0.846675i	$-0.321399\pi$
$89$	94.7156	1.06422	0.532110	+	0.846675i	$0.321399\pi$
$90$	0	0
$91$	19.5451	0.214781
$92$	0	0
$93$	47.1686i	0.507189i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−143.796	−1.48243	−0.741216	−	0.671267i	$-0.765751\pi$
$97$	−143.796	−1.48243	−0.741216	+	0.671267i	$0.765751\pi$
$98$	0	0
$99$	33.1471	0.334819
$100$	0	0
$101$	− 66.5595i	− 0.659005i	−0.944155	−	0.329502i	$-0.893119\pi$
$101$	− 66.5595i	− 0.659005i	0.944155	−	0.329502i	$-0.106881\pi$
$102$	0	0
$103$	153.063i	1.48605i	0.669264	+	0.743024i	$0.266609\pi$
$103$	153.063i	1.48605i	−0.669264	+	0.743024i	$0.733391\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−75.7953	−0.708367	−0.354184	−	0.935176i	$-0.615241\pi$
$107$	−75.7953	−0.708367	−0.354184	+	0.935176i	$0.615241\pi$
$108$	0	0
$109$	− 124.088i	− 1.13842i	−0.822192	−	0.569211i	$-0.807249\pi$
$109$	− 124.088i	− 1.13842i	0.822192	−	0.569211i	$-0.192751\pi$
$110$	0	0
$111$	45.8531i	0.413091i
$112$	0	0
$113$	5.19589	0.0459814	0.0229907	−	0.999736i	$-0.492681\pi$
$113$	5.19589	0.0459814	0.0229907	+	0.999736i	$0.492681\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	26.4775i	0.226303i
$118$	0	0
$119$	88.7156i	0.745510i
$120$	0	0
$121$	−82.7251	−0.683678
$122$	0	0
$123$	59.3021	0.482131
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 109.383i	− 0.861287i	−0.902522	−	0.430644i	$-0.858287\pi$
$127$	− 109.383i	− 0.861287i	0.902522	−	0.430644i	$-0.141713\pi$
$128$	0	0
$129$	99.6422	0.772420
$130$	0	0
$131$	139.062	1.06154	0.530771	−	0.847515i	$-0.321902\pi$
$131$	139.062	1.06154	0.530771	+	0.847515i	$0.321902\pi$
$132$	0	0
$133$	41.1018i	0.309036i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−5.42829	−0.0396226	−0.0198113	−	0.999804i	$-0.506307\pi$
$137$	−5.42829	−0.0396226	−0.0198113	+	0.999804i	$0.506307\pi$
$138$	0	0
$139$	191.765	1.37961	0.689803	−	0.723998i	$-0.257697\pi$
$139$	191.765	1.37961	0.689803	+	0.723998i	$0.257697\pi$
$140$	0	0
$141$	25.1229i	0.178177i
$142$	0	0
$143$	30.5736i	0.213801i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	63.6617	0.433073
$148$	0	0
$149$	− 184.959i	− 1.24133i	−0.784075	−	0.620667i	$-0.786862\pi$
$149$	− 184.959i	− 1.24133i	0.784075	−	0.620667i	$-0.213138\pi$
$150$	0	0
$151$	− 121.431i	− 0.804181i	−0.915600	−	0.402090i	$-0.868284\pi$
$151$	− 121.431i	− 0.804181i	0.915600	−	0.402090i	$-0.131716\pi$
$152$	0	0
$153$	−120.182	−0.785503
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 261.156i	− 1.66341i	−0.555216	−	0.831706i	$-0.687364\pi$
$157$	− 261.156i	− 1.66341i	0.555216	−	0.831706i	$-0.312636\pi$
$158$	0	0
$159$	34.2749i	0.215566i
$160$	0	0
$161$	156.652	0.972991
$162$	0	0
$163$	−199.302	−1.22271	−0.611356	−	0.791356i	$-0.709375\pi$
$163$	−199.302	−1.22271	−0.611356	+	0.791356i	$0.709375\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	168.999i	1.01197i	0.862542	+	0.505986i	$0.168871\pi$
$167$	168.999i	1.01197i	−0.862542	+	0.505986i	$0.831129\pi$
$168$	0	0
$169$	144.578	0.855492
$170$	0	0
$171$	−55.6801	−0.325614
$172$	0	0
$173$	− 295.307i	− 1.70697i	−0.521113	−	0.853487i	$-0.674483\pi$
$173$	− 295.307i	− 1.70697i	0.521113	−	0.853487i	$-0.325517\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	199.282	1.12589
$178$	0	0
$179$	241.258	1.34781	0.673906	−	0.738817i	$-0.264615\pi$
$179$	241.258	1.34781	0.673906	+	0.738817i	$0.264615\pi$
$180$	0	0
$181$	− 137.833i	− 0.761511i	−0.924676	−	0.380755i	$-0.875664\pi$
$181$	− 137.833i	− 0.761511i	0.924676	−	0.380755i	$-0.124336\pi$
$182$	0	0
$183$	− 109.325i	− 0.597406i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−138.774	−0.742108
$188$	0	0
$189$	− 108.372i	− 0.573398i
$190$	0	0
$191$	340.294i	1.78164i	0.454354	+	0.890821i	$0.349870\pi$
$191$	340.294i	1.78164i	−0.454354	+	0.890821i	$0.650130\pi$
$192$	0	0
$193$	−96.5680	−0.500353	−0.250176	−	0.968200i	$-0.580489\pi$
$193$	−96.5680	−0.500353	−0.250176	+	0.968200i	$0.580489\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	229.054i	1.16271i	0.813651	+	0.581354i	$0.197477\pi$
$197$	229.054i	1.16271i	−0.813651	+	0.581354i	$0.802523\pi$
$198$	0	0
$199$	51.5782i	0.259187i	0.991567	+	0.129593i	$0.0413672\pi$
$199$	51.5782i	0.259187i	−0.991567	+	0.129593i	$0.958633\pi$
$200$	0	0
$201$	189.661	0.943587
$202$	0	0
$203$	120.373	0.592968
$204$	0	0
$205$	0	0
$206$	0	0
$207$	212.214i	1.02519i
$208$	0	0
$209$	−64.2938	−0.307626
$210$	0	0
$211$	77.7042	0.368266	0.184133	−	0.982901i	$-0.441052\pi$
$211$	77.7042	0.368266	0.184133	+	0.982901i	$0.441052\pi$
$212$	0	0
$213$	− 31.9010i	− 0.149770i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−97.7508	−0.450465
$218$	0	0
$219$	183.852	0.839506
$220$	0	0
$221$	− 110.851i	− 0.501589i
$222$	0	0
$223$	98.9917i	0.443909i	0.975057	+	0.221954i	$0.0712436\pi$
$223$	98.9917i	0.443909i	−0.975057	+	0.221954i	$0.928756\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−172.584	−0.760280	−0.380140	−	0.924929i	$-0.624124\pi$
$227$	−172.584	−0.760280	−0.380140	+	0.924929i	$0.624124\pi$
$228$	0	0
$229$	− 151.690i	− 0.662401i	−0.943560	−	0.331201i	$-0.892546\pi$
$229$	− 151.690i	− 0.662401i	0.943560	−	0.331201i	$-0.107454\pi$
$230$	0	0
$231$	− 46.6968i	− 0.202151i
$232$	0	0
$233$	−289.007	−1.24037	−0.620187	−	0.784454i	$-0.712943\pi$
$233$	−289.007	−1.24037	−0.620187	+	0.784454i	$0.712943\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	266.378i	1.12396i
$238$	0	0
$239$	17.5593i	0.0734699i	0.999325	+	0.0367349i	$0.0116957\pi$
$239$	17.5593i	0.0734699i	−0.999325	+	0.0367349i	$0.988304\pi$
$240$	0	0
$241$	−101.936	−0.422971	−0.211485	−	0.977381i	$-0.567830\pi$
$241$	−101.936	−0.422971	−0.211485	+	0.977381i	$0.567830\pi$
$242$	0	0
$243$	238.837	0.982867
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 51.3572i	− 0.207924i
$248$	0	0
$249$	−175.092	−0.703182
$250$	0	0
$251$	123.966	0.493889	0.246944	−	0.969030i	$-0.420573\pi$
$251$	123.966	0.493889	0.246944	+	0.969030i	$0.420573\pi$
$252$	0	0
$253$	245.044i	0.968552i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	118.534	0.461223	0.230612	−	0.973046i	$-0.425927\pi$
$257$	118.534	0.461223	0.230612	+	0.973046i	$0.425927\pi$
$258$	0	0
$259$	−95.0247	−0.366891
$260$	0	0
$261$	163.067i	0.624779i
$262$	0	0
$263$	− 223.071i	− 0.848177i	−0.905621	−	0.424089i	$-0.860595\pi$
$263$	− 223.071i	− 0.848177i	0.905621	−	0.424089i	$-0.139405\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	180.760	0.677004
$268$	0	0
$269$	− 249.061i	− 0.925877i	−0.886390	−	0.462938i	$-0.846795\pi$
$269$	− 249.061i	− 0.925877i	0.886390	−	0.462938i	$-0.153205\pi$
$270$	0	0
$271$	130.991i	0.483360i	0.970356	+	0.241680i	$0.0776984\pi$
$271$	130.991i	0.483360i	−0.970356	+	0.241680i	$0.922302\pi$
$272$	0	0
$273$	37.3008	0.136633
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 39.7756i	− 0.143594i	−0.997419	−	0.0717971i	$-0.977127\pi$
$277$	− 39.7756i	− 0.143594i	0.997419	−	0.0717971i	$-0.0228734\pi$
$278$	0	0
$279$	− 132.422i	− 0.474630i
$280$	0	0
$281$	58.0640	0.206634	0.103317	−	0.994649i	$-0.467054\pi$
$281$	58.0640	0.206634	0.103317	+	0.994649i	$0.467054\pi$
$282$	0	0
$283$	−241.288	−0.852607	−0.426304	−	0.904580i	$-0.640184\pi$
$283$	−241.288	−0.852607	−0.426304	+	0.904580i	$0.640184\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	122.896i	0.428209i
$288$	0	0
$289$	214.156	0.741025
$290$	0	0
$291$	−274.427	−0.943049
$292$	0	0
$293$	− 87.8283i	− 0.299755i	−0.988705	−	0.149878i	$-0.952112\pi$
$293$	− 87.8283i	− 0.299755i	0.988705	−	0.149878i	$-0.0478880\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	169.522	0.570782
$298$	0	0
$299$	−195.738	−0.654642
$300$	0	0
$301$	206.496i	0.686032i
$302$	0	0
$303$	− 127.025i	− 0.419226i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	138.914	0.452490	0.226245	−	0.974070i	$-0.427355\pi$
$307$	138.914	0.452490	0.226245	+	0.974070i	$0.427355\pi$
$308$	0	0
$309$	292.113i	0.945350i
$310$	0	0
$311$	− 330.863i	− 1.06387i	−0.846786	−	0.531933i	$-0.821466\pi$
$311$	− 330.863i	− 1.06387i	0.846786	−	0.531933i	$-0.178534\pi$
$312$	0	0
$313$	385.575	1.23187	0.615935	−	0.787797i	$-0.288779\pi$
$313$	385.575	1.23187	0.615935	+	0.787797i	$0.288779\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 414.112i	− 1.30635i	−0.757208	−	0.653174i	$-0.773437\pi$
$317$	− 414.112i	− 1.30635i	0.757208	−	0.653174i	$-0.226563\pi$
$318$	0	0
$319$	188.294i	0.590263i
$320$	0	0
$321$	−144.652	−0.450628
$322$	0	0
$323$	233.111	0.721707
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 236.816i	− 0.724207i
$328$	0	0
$329$	−52.0640	−0.158249
$330$	0	0
$331$	52.4704	0.158521	0.0792604	−	0.996854i	$-0.474744\pi$
$331$	52.4704	0.158521	0.0792604	+	0.996854i	$0.474744\pi$
$332$	0	0
$333$	− 128.729i	− 0.386573i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−469.871	−1.39428	−0.697138	−	0.716937i	$-0.745544\pi$
$337$	−469.871	−1.39428	−0.697138	+	0.716937i	$0.745544\pi$
$338$	0	0
$339$	9.91611	0.0292511
$340$	0	0
$341$	− 152.908i	− 0.448409i
$342$	0	0
$343$	325.727i	0.949641i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	267.604	0.771192	0.385596	−	0.922668i	$-0.373996\pi$
$347$	267.604	0.771192	0.385596	+	0.922668i	$0.373996\pi$
$348$	0	0
$349$	662.916i	1.89947i	0.313053	+	0.949736i	$0.398648\pi$
$349$	662.916i	1.89947i	−0.313053	+	0.949736i	$0.601352\pi$
$350$	0	0
$351$	135.412i	0.385790i
$352$	0	0
$353$	366.925	1.03945	0.519723	−	0.854335i	$-0.326035\pi$
$353$	366.925	1.03945	0.519723	+	0.854335i	$0.326035\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	169.309i	0.474256i
$358$	0	0
$359$	− 234.863i	− 0.654213i	−0.944987	−	0.327107i	$-0.893926\pi$
$359$	− 234.863i	− 0.654213i	0.944987	−	0.327107i	$-0.106074\pi$
$360$	0	0
$361$	−253.000	−0.700831
$362$	0	0
$363$	−157.877	−0.434922
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 455.060i	− 1.23994i	−0.784624	−	0.619972i	$-0.787144\pi$
$367$	− 455.060i	− 1.23994i	0.784624	−	0.619972i	$-0.212856\pi$
$368$	0	0
$369$	−166.486	−0.451181
$370$	0	0
$371$	−71.0304	−0.191457
$372$	0	0
$373$	518.534i	1.39017i	0.718926	+	0.695086i	$0.244634\pi$
$373$	518.534i	1.39017i	−0.718926	+	0.695086i	$0.755366\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−150.407	−0.398957
$378$	0	0
$379$	−563.918	−1.48791	−0.743955	−	0.668230i	$-0.767052\pi$
$379$	−563.918	−1.48791	−0.743955	+	0.668230i	$0.767052\pi$
$380$	0	0
$381$	− 208.753i	− 0.547908i
$382$	0	0
$383$	− 56.8436i	− 0.148417i	−0.997243	−	0.0742083i	$-0.976357\pi$
$383$	− 56.8436i	− 0.148417i	0.997243	−	0.0742083i	$-0.0236430\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−279.737	−0.722835
$388$	0	0
$389$	188.677i	0.485031i	0.970147	+	0.242516i	$0.0779726\pi$
$389$	188.677i	0.485031i	−0.970147	+	0.242516i	$0.922027\pi$
$390$	0	0
$391$	− 888.460i	− 2.27228i
$392$	0	0
$393$	265.393	0.675300
$394$	0	0
$395$	0	0
$396$	0	0
$397$	143.716i	0.362005i	0.983483	+	0.181003i	$0.0579342\pi$
$397$	143.716i	0.362005i	−0.983483	+	0.181003i	$0.942066\pi$
$398$	0	0
$399$	78.4407i	0.196593i
$400$	0	0
$401$	1.28437	0.00320291	0.00160145	−	0.999999i	$-0.499490\pi$
$401$	1.28437	0.00320291	0.00160145	+	0.999999i	$0.499490\pi$
$402$	0	0
$403$	122.141	0.303079
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 148.643i	− 0.365217i
$408$	0	0
$409$	76.3767	0.186740	0.0933700	−	0.995631i	$-0.470236\pi$
$409$	76.3767	0.186740	0.0933700	+	0.995631i	$0.470236\pi$
$410$	0	0
$411$	−10.3596	−0.0252059
$412$	0	0
$413$	412.987i	0.999969i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	365.974	0.877636
$418$	0	0
$419$	−33.3906	−0.0796912	−0.0398456	−	0.999206i	$-0.512687\pi$
$419$	−33.3906	−0.0796912	−0.0398456	+	0.999206i	$0.512687\pi$
$420$	0	0
$421$	392.318i	0.931871i	0.884819	+	0.465935i	$0.154282\pi$
$421$	392.318i	0.931871i	−0.884819	+	0.465935i	$0.845718\pi$
$422$	0	0
$423$	− 70.5305i	− 0.166739i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	226.563	0.530592
$428$	0	0
$429$	58.3482i	0.136010i
$430$	0	0
$431$	840.460i	1.95002i	0.222156	+	0.975011i	$0.428691\pi$
$431$	840.460i	1.95002i	−0.222156	+	0.975011i	$0.571309\pi$
$432$	0	0
$433$	275.552	0.636380	0.318190	−	0.948027i	$-0.396925\pi$
$433$	275.552	0.636380	0.318190	+	0.948027i	$0.396925\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 411.622i	− 0.941926i
$438$	0	0
$439$	− 477.725i	− 1.08821i	−0.839016	−	0.544106i	$-0.816869\pi$
$439$	− 477.725i	− 1.08821i	0.839016	−	0.544106i	$-0.183131\pi$
$440$	0	0
$441$	−178.725	−0.405272
$442$	0	0
$443$	−625.709	−1.41244	−0.706218	−	0.707994i	$-0.749600\pi$
$443$	−625.709	−1.41244	−0.706218	+	0.707994i	$0.749600\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 352.984i	− 0.789674i
$448$	0	0
$449$	−104.377	−0.232465	−0.116232	−	0.993222i	$-0.537082\pi$
$449$	−104.377	−0.232465	−0.116232	+	0.993222i	$0.537082\pi$
$450$	0	0
$451$	−192.241	−0.426256
$452$	0	0
$453$	− 231.746i	− 0.511580i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−198.564	−0.434495	−0.217248	−	0.976117i	$-0.569708\pi$
$457$	−198.564	−0.434495	−0.217248	+	0.976117i	$0.569708\pi$
$458$	0	0
$459$	−614.640	−1.33908
$460$	0	0
$461$	11.1338i	0.0241515i	0.999927	+	0.0120757i	$0.00384392\pi$
$461$	11.1338i	0.0241515i	−0.999927	+	0.0120757i	$0.996156\pi$
$462$	0	0
$463$	− 15.9944i	− 0.0345451i	−0.999851	−	0.0172725i	$-0.994502\pi$
$463$	− 15.9944i	− 0.0345451i	0.999851	−	0.0172725i	$-0.00549830\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−539.567	−1.15539	−0.577695	−	0.816253i	$-0.696048\pi$
$467$	−539.567	−1.15539	−0.577695	+	0.816253i	$0.696048\pi$
$468$	0	0
$469$	393.048i	0.838056i
$470$	0	0
$471$	− 498.403i	− 1.05818i
$472$	0	0
$473$	−323.013	−0.682902
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 96.2240i	− 0.201728i
$478$	0	0
$479$	127.744i	0.266689i	0.991070	+	0.133344i	$0.0425716\pi$
$479$	127.744i	0.266689i	−0.991070	+	0.133344i	$0.957428\pi$
$480$	0	0
$481$	118.735	0.246849
$482$	0	0
$483$	298.962	0.618969
$484$	0	0
$485$	0	0
$486$	0	0
$487$	624.466i	1.28227i	0.767428	+	0.641135i	$0.221536\pi$
$487$	624.466i	1.28227i	−0.767428	+	0.641135i	$0.778464\pi$
$488$	0	0
$489$	−380.358	−0.777828
$490$	0	0
$491$	424.601	0.864769	0.432384	−	0.901689i	$-0.357672\pi$
$491$	424.601	0.864769	0.432384	+	0.901689i	$0.357672\pi$
$492$	0	0
$493$	− 682.701i	− 1.38479i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	66.1107	0.133019
$498$	0	0
$499$	−591.631	−1.18563	−0.592816	−	0.805338i	$-0.701984\pi$
$499$	−591.631	−1.18563	−0.592816	+	0.805338i	$0.701984\pi$
$500$	0	0
$501$	322.527i	0.643766i
$502$	0	0
$503$	5.71879i	0.0113694i	0.999984	+	0.00568468i	$0.00180950\pi$
$503$	5.71879i	0.0113694i	−0.999984	+	0.00568468i	$0.998191\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	275.920	0.544221
$508$	0	0
$509$	439.708i	0.863867i	0.901905	+	0.431933i	$0.142168\pi$
$509$	439.708i	0.863867i	−0.901905	+	0.431933i	$0.857832\pi$
$510$	0	0
$511$	381.009i	0.745615i
$512$	0	0
$513$	−284.761	−0.555091
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 81.4416i	− 0.157527i
$518$	0	0
$519$	− 563.578i	− 1.08589i
$520$	0	0
$521$	80.5687	0.154642	0.0773212	−	0.997006i	$-0.475363\pi$
$521$	80.5687	0.154642	0.0773212	+	0.997006i	$0.475363\pi$
$522$	0	0
$523$	585.089	1.11872	0.559359	−	0.828926i	$-0.311047\pi$
$523$	585.089	1.11872	0.559359	+	0.828926i	$0.311047\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	554.400i	1.05199i
$528$	0	0
$529$	−1039.82	−1.96563
$530$	0	0
$531$	−559.469	−1.05361
$532$	0	0
$533$	− 153.560i	− 0.288105i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	460.430	0.857411
$538$	0	0
$539$	−206.374	−0.382883
$540$	0	0
$541$	248.685i	0.459676i	0.973229	+	0.229838i	$0.0738196\pi$
$541$	248.685i	0.459676i	−0.973229	+	0.229838i	$0.926180\pi$
$542$	0	0
$543$	− 263.048i	− 0.484435i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	47.9916	0.0877360	0.0438680	−	0.999037i	$-0.486032\pi$
$547$	47.9916	0.0877360	0.0438680	+	0.999037i	$0.486032\pi$
$548$	0	0
$549$	306.922i	0.559056i
$550$	0	0
$551$	− 316.294i	− 0.574036i
$552$	0	0
$553$	−552.034	−0.998254
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1091.71i	1.95998i	0.199045	+	0.979990i	$0.436216\pi$
$557$	1091.71i	1.95998i	−0.199045	+	0.979990i	$0.563784\pi$
$558$	0	0
$559$	− 258.019i	− 0.461572i
$560$	0	0
$561$	−264.844	−0.472092
$562$	0	0
$563$	427.231	0.758846	0.379423	−	0.925223i	$-0.376122\pi$
$563$	427.231	0.758846	0.379423	+	0.925223i	$0.376122\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 16.1106i	− 0.0284137i
$568$	0	0
$569$	−54.2298	−0.0953072	−0.0476536	−	0.998864i	$-0.515174\pi$
$569$	−54.2298	−0.0953072	−0.0476536	+	0.998864i	$0.515174\pi$
$570$	0	0
$571$	−1050.42	−1.83961	−0.919806	−	0.392373i	$-0.871654\pi$
$571$	−1050.42	−1.83961	−0.919806	+	0.392373i	$0.871654\pi$
$572$	0	0
$573$	649.434i	1.13339i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	161.031	0.279083	0.139542	−	0.990216i	$-0.455437\pi$
$577$	161.031	0.279083	0.139542	+	0.990216i	$0.455437\pi$
$578$	0	0
$579$	−184.295	−0.318299
$580$	0	0
$581$	− 362.856i	− 0.624538i
$582$	0	0
$583$	− 111.110i	− 0.190583i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	186.363	0.317484	0.158742	−	0.987320i	$-0.449256\pi$
$587$	186.363	0.317484	0.158742	+	0.987320i	$0.449256\pi$
$588$	0	0
$589$	256.852i	0.436082i
$590$	0	0
$591$	437.137i	0.739657i
$592$	0	0
$593$	−1011.51	−1.70576	−0.852879	−	0.522109i	$-0.825145\pi$
$593$	−1011.51	−1.70576	−0.852879	+	0.522109i	$0.825145\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	98.4344i	0.164882i
$598$	0	0
$599$	− 647.616i	− 1.08116i	−0.841292	−	0.540581i	$-0.818204\pi$
$599$	− 647.616i	− 1.08116i	0.841292	−	0.540581i	$-0.181796\pi$
$600$	0	0
$601$	40.3767	0.0671825	0.0335913	−	0.999436i	$-0.489306\pi$
$601$	40.3767	0.0671825	0.0335913	+	0.999436i	$0.489306\pi$
$602$	0	0
$603$	−532.458	−0.883014
$604$	0	0
$605$	0	0
$606$	0	0
$607$	799.902i	1.31780i	0.752233	+	0.658898i	$0.228977\pi$
$607$	799.902i	1.31780i	−0.752233	+	0.658898i	$0.771023\pi$
$608$	0	0
$609$	229.725	0.377217
$610$	0	0
$611$	65.0546	0.106472
$612$	0	0
$613$	− 900.584i	− 1.46914i	−0.678532	−	0.734571i	$-0.737384\pi$
$613$	− 900.584i	− 1.46914i	0.678532	−	0.734571i	$-0.262616\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−255.001	−0.413292	−0.206646	−	0.978416i	$-0.566255\pi$
$617$	−255.001	−0.413292	−0.206646	+	0.978416i	$0.566255\pi$
$618$	0	0
$619$	168.789	0.272679	0.136340	−	0.990662i	$-0.456466\pi$
$619$	168.789	0.272679	0.136340	+	0.990662i	$0.456466\pi$
$620$	0	0
$621$	1085.31i	1.74769i
$622$	0	0
$623$	374.602i	0.601288i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−122.702	−0.195696
$628$	0	0
$629$	538.939i	0.856818i
$630$	0	0
$631$	− 665.431i	− 1.05457i	−0.849690	−	0.527283i	$-0.823211\pi$
$631$	− 665.431i	− 1.05457i	0.849690	−	0.527283i	$-0.176789\pi$
$632$	0	0
$633$	148.295	0.234273
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 164.849i	− 0.258790i
$638$	0	0
$639$	89.5593i	0.140155i
$640$	0	0
$641$	552.633	0.862142	0.431071	−	0.902318i	$-0.358136\pi$
$641$	552.633	0.862142	0.431071	+	0.902318i	$0.358136\pi$
$642$	0	0
$643$	−901.490	−1.40201	−0.701003	−	0.713159i	$-0.747264\pi$
$643$	−901.490	−1.40201	−0.701003	+	0.713159i	$0.747264\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 67.8163i	− 0.104817i	−0.998626	−	0.0524083i	$-0.983310\pi$
$647$	− 67.8163i	− 0.104817i	0.998626	−	0.0524083i	$-0.0166897\pi$
$648$	0	0
$649$	−646.019	−0.995407
$650$	0	0
$651$	−186.553	−0.286563
$652$	0	0
$653$	− 124.351i	− 0.190431i	−0.995457	−	0.0952153i	$-0.969646\pi$
$653$	− 124.351i	− 0.190431i	0.995457	−	0.0952153i	$-0.0303540\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−516.149	−0.785615
$658$	0	0
$659$	−260.582	−0.395420	−0.197710	−	0.980261i	$-0.563350\pi$
$659$	−260.582	−0.395420	−0.197710	+	0.980261i	$0.563350\pi$
$660$	0	0
$661$	447.965i	0.677708i	0.940839	+	0.338854i	$0.110039\pi$
$661$	447.965i	0.677708i	−0.940839	+	0.338854i	$0.889961\pi$
$662$	0	0
$663$	− 211.554i	− 0.319086i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1205.49	−1.80734
$668$	0	0
$669$	188.921i	0.282393i
$670$	0	0
$671$	354.403i	0.528171i
$672$	0	0
$673$	−288.775	−0.429086	−0.214543	−	0.976715i	$-0.568826\pi$
$673$	−288.775	−0.429086	−0.214543	+	0.976715i	$0.568826\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 248.339i	− 0.366823i	−0.983036	−	0.183412i	$-0.941286\pi$
$677$	− 248.339i	− 0.366823i	0.983036	−	0.183412i	$-0.0587141\pi$
$678$	0	0
$679$	− 568.716i	− 0.837578i
$680$	0	0
$681$	−329.367	−0.483652
$682$	0	0
$683$	451.900	0.661640	0.330820	−	0.943694i	$-0.392675\pi$
$683$	451.900	0.661640	0.330820	+	0.943694i	$0.392675\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 289.493i	− 0.421387i
$688$	0	0
$689$	88.7534	0.128815
$690$	0	0
$691$	193.513	0.280048	0.140024	−	0.990148i	$-0.455282\pi$
$691$	193.513	0.280048	0.140024	+	0.990148i	$0.455282\pi$
$692$	0	0
$693$	131.097i	0.189174i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	697.013	1.00002
$698$	0	0
$699$	−551.555	−0.789064
$700$	0	0
$701$	− 928.025i	− 1.32386i	−0.749566	−	0.661929i	$-0.769738\pi$
$701$	− 928.025i	− 1.32386i	0.749566	−	0.661929i	$-0.230262\pi$
$702$	0	0
$703$	249.689i	0.355177i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	263.244	0.372339
$708$	0	0
$709$	− 75.9885i	− 0.107177i	−0.998563	−	0.0535885i	$-0.982934\pi$
$709$	− 75.9885i	− 0.107177i	0.998563	−	0.0535885i	$-0.0170659\pi$
$710$	0	0
$711$	− 747.834i	− 1.05181i
$712$	0	0
$713$	978.944	1.37299
$714$	0	0
$715$	0	0
$716$	0	0
$717$	33.5111i	0.0467379i
$718$	0	0
$719$	395.578i	0.550178i	0.961419	+	0.275089i	$0.0887074\pi$
$719$	395.578i	0.550178i	−0.961419	+	0.275089i	$0.911293\pi$
$720$	0	0
$721$	−605.367	−0.839622
$722$	0	0
$723$	−194.540	−0.269073
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 352.113i	− 0.484337i	−0.970234	−	0.242168i	$-0.922141\pi$
$727$	− 352.113i	− 0.484337i	0.970234	−	0.242168i	$-0.0778587\pi$
$728$	0	0
$729$	492.469	0.675540
$730$	0	0
$731$	1171.15	1.60213
$732$	0	0
$733$	− 156.532i	− 0.213550i	−0.994283	−	0.106775i	$-0.965947\pi$
$733$	− 156.532i	− 0.213550i	0.994283	−	0.106775i	$-0.0340525\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−614.829	−0.834232
$738$	0	0
$739$	−183.376	−0.248140	−0.124070	−	0.992273i	$-0.539595\pi$
$739$	−183.376	−0.248140	−0.124070	+	0.992273i	$0.539595\pi$
$740$	0	0
$741$	− 98.0126i	− 0.132271i
$742$	0	0
$743$	1028.34i	1.38404i	0.721878	+	0.692021i	$0.243279\pi$
$743$	1028.34i	1.38404i	−0.721878	+	0.692021i	$0.756721\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	491.557	0.658042
$748$	0	0
$749$	− 299.772i	− 0.400230i
$750$	0	0
$751$	740.038i	0.985403i	0.870198	+	0.492702i	$0.163991\pi$
$751$	740.038i	0.985403i	−0.870198	+	0.492702i	$0.836009\pi$
$752$	0	0
$753$	236.583	0.314188
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 934.857i	− 1.23495i	−0.786591	−	0.617475i	$-0.788156\pi$
$757$	− 934.857i	− 1.23495i	0.786591	−	0.617475i	$-0.211844\pi$
$758$	0	0
$759$	467.654i	0.616144i
$760$	0	0
$761$	436.735	0.573896	0.286948	−	0.957946i	$-0.407359\pi$
$761$	436.735	0.573896	0.286948	+	0.957946i	$0.407359\pi$
$762$	0	0
$763$	490.770	0.643211
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 516.033i	− 0.672793i
$768$	0	0
$769$	768.275	0.999057	0.499529	−	0.866297i	$-0.333507\pi$
$769$	768.275	0.999057	0.499529	+	0.866297i	$0.333507\pi$
$770$	0	0
$771$	226.217	0.293407
$772$	0	0
$773$	792.266i	1.02492i	0.858710	+	0.512462i	$0.171266\pi$
$773$	792.266i	1.02492i	−0.858710	+	0.512462i	$0.828734\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−181.350	−0.233398
$778$	0	0
$779$	322.925	0.414538
$780$	0	0
$781$	103.414i	0.132413i
$782$	0	0
$783$	833.966i	1.06509i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	965.852	1.22726	0.613629	−	0.789594i	$-0.289709\pi$
$787$	965.852	1.22726	0.613629	+	0.789594i	$0.289709\pi$
$788$	0	0
$789$	− 425.719i	− 0.539568i
$790$	0	0
$791$	20.5499i	0.0259796i
$792$	0	0
$793$	−283.093	−0.356990
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 358.505i	− 0.449818i	−0.974380	−	0.224909i	$-0.927792\pi$
$797$	− 358.505i	− 0.449818i	0.974380	−	0.224909i	$-0.0722085\pi$
$798$	0	0
$799$	295.284i	0.369567i
$800$	0	0
$801$	−507.469	−0.633544
$802$	0	0
$803$	−595.997	−0.742213
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 475.320i	− 0.588997i
$808$	0	0
$809$	−222.881	−0.275502	−0.137751	−	0.990467i	$-0.543987\pi$
$809$	−222.881	−0.275502	−0.137751	+	0.990467i	$0.543987\pi$
$810$	0	0
$811$	−1292.40	−1.59358	−0.796792	−	0.604253i	$-0.793471\pi$
$811$	−1292.40	−1.59358	−0.796792	+	0.604253i	$0.793471\pi$
$812$	0	0
$813$	249.989i	0.307490i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	542.593	0.664129
$818$	0	0
$819$	−104.719	−0.127862
$820$	0	0
$821$	− 1276.69i	− 1.55505i	−0.628855	−	0.777523i	$-0.716476\pi$
$821$	− 1276.69i	− 1.55505i	0.628855	−	0.777523i	$-0.283524\pi$
$822$	0	0
$823$	− 1177.68i	− 1.43096i	−0.698631	−	0.715482i	$-0.746207\pi$
$823$	− 1177.68i	− 1.43096i	0.698631	−	0.715482i	$-0.253793\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−154.182	−0.186435	−0.0932176	−	0.995646i	$-0.529715\pi$
$827$	−154.182	−0.186435	−0.0932176	+	0.995646i	$0.529715\pi$
$828$	0	0
$829$	243.838i	0.294135i	0.989126	+	0.147067i	$0.0469834\pi$
$829$	243.838i	0.294135i	−0.989126	+	0.147067i	$0.953017\pi$
$830$	0	0
$831$	− 75.9097i	− 0.0913474i
$832$	0	0
$833$	748.254	0.898264
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 677.238i	− 0.809125i
$838$	0	0
$839$	1521.51i	1.81348i	0.421694	+	0.906738i	$0.361436\pi$
$839$	1521.51i	1.81348i	−0.421694	+	0.906738i	$0.638564\pi$
$840$	0	0
$841$	−85.3127	−0.101442
$842$	0	0
$843$	110.812	0.131450
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 327.179i	− 0.386280i
$848$	0	0
$849$	−460.486	−0.542386
$850$	0	0
$851$	951.643	1.11826
$852$	0	0
$853$	59.6651i	0.0699474i	0.999388	+	0.0349737i	$0.0111347\pi$
$853$	59.6651i	0.0699474i	−0.999388	+	0.0349737i	$0.988865\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1263.47	−1.47430	−0.737149	−	0.675730i	$-0.763828\pi$
$857$	−1263.47	−1.47430	−0.737149	+	0.675730i	$0.763828\pi$
$858$	0	0
$859$	1067.99	1.24330	0.621649	−	0.783296i	$-0.286463\pi$
$859$	1067.99	1.24330	0.621649	+	0.783296i	$0.286463\pi$
$860$	0	0
$861$	234.541i	0.272406i
$862$	0	0
$863$	− 439.103i	− 0.508810i	−0.967098	−	0.254405i	$-0.918120\pi$
$863$	− 439.103i	− 0.508810i	0.967098	−	0.254405i	$-0.0818795\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	408.707	0.471403
$868$	0	0
$869$	− 863.525i	− 0.993699i
$870$	0	0
$871$	− 491.119i	− 0.563856i
$872$	0	0
$873$	770.432	0.882511
$874$	0	0
$875$	0	0
$876$	0	0
$877$	932.204i	1.06295i	0.847075	+	0.531473i	$0.178361\pi$
$877$	932.204i	1.06295i	−0.847075	+	0.531473i	$0.821639\pi$
$878$	0	0
$879$	− 167.616i	− 0.190689i
$880$	0	0
$881$	223.073	0.253205	0.126602	−	0.991954i	$-0.459593\pi$
$881$	223.073	0.253205	0.126602	+	0.991954i	$0.459593\pi$
$882$	0	0
$883$	−790.958	−0.895762	−0.447881	−	0.894093i	$-0.647821\pi$
$883$	−790.958	−0.895762	−0.447881	+	0.894093i	$0.647821\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	233.462i	0.263204i	0.991303	+	0.131602i	$0.0420122\pi$
$887$	233.462i	0.263204i	−0.991303	+	0.131602i	$0.957988\pi$
$888$	0	0
$889$	432.614	0.486630
$890$	0	0
$891$	25.2011	0.0282841
$892$	0	0
$893$	136.805i	0.153197i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−373.556	−0.416451
$898$	0	0
$899$	752.230	0.836741
$900$	0	0
$901$	402.854i	0.447118i
$902$	0	0
$903$	394.087i	0.436420i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−1679.77	−1.85201	−0.926004	−	0.377514i	$-0.876779\pi$
$907$	−1679.77	−1.85201	−0.926004	+	0.377514i	$0.876779\pi$
$908$	0	0
$909$	356.613i	0.392314i
$910$	0	0
$911$	44.0903i	0.0483977i	0.999707	+	0.0241988i	$0.00770348\pi$
$911$	44.0903i	0.0483977i	−0.999707	+	0.0241988i	$0.992297\pi$
$912$	0	0
$913$	567.601	0.621688
$914$	0	0
$915$	0	0
$916$	0	0
$917$	549.993i	0.599774i
$918$	0	0
$919$	− 153.303i	− 0.166815i	−0.996516	−	0.0834076i	$-0.973420\pi$
$919$	− 153.303i	− 0.166815i	0.996516	−	0.0834076i	$-0.0265804\pi$
$920$	0	0
$921$	265.111	0.287851
$922$	0	0
$923$	−82.6061	−0.0894974
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 820.084i	− 0.884664i
$928$	0	0
$929$	1413.68	1.52172	0.760861	−	0.648915i	$-0.224777\pi$
$929$	1413.68	1.52172	0.760861	+	0.648915i	$0.224777\pi$
$930$	0	0
$931$	346.665	0.372357
$932$	0	0
$933$	− 631.435i	− 0.676779i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	90.2102	0.0962755	0.0481378	−	0.998841i	$-0.484671\pi$
$937$	90.2102	0.0962755	0.0481378	+	0.998841i	$0.484671\pi$
$938$	0	0
$939$	735.851	0.783654
$940$	0	0
$941$	− 1347.77i	− 1.43227i	−0.697961	−	0.716136i	$-0.745909\pi$
$941$	− 1347.77i	− 1.43227i	0.697961	−	0.716136i	$-0.254091\pi$
$942$	0	0
$943$	− 1230.77i	− 1.30516i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	910.489	0.961446	0.480723	−	0.876873i	$-0.340374\pi$
$947$	910.489	0.961446	0.480723	+	0.876873i	$0.340374\pi$
$948$	0	0
$949$	− 476.076i	− 0.501661i
$950$	0	0
$951$	− 790.313i	− 0.831033i
$952$	0	0
$953$	−86.1971	−0.0904481	−0.0452241	−	0.998977i	$-0.514400\pi$
$953$	−86.1971	−0.0904481	−0.0452241	+	0.998977i	$0.514400\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	359.349i	0.375496i
$958$	0	0
$959$	− 21.4690i	− 0.0223869i
$960$	0	0
$961$	350.137	0.364347
$962$	0	0
$963$	406.097	0.421700
$964$	0	0
$965$	0	0
$966$	0	0
$967$	419.290i	0.433599i	0.976216	+	0.216799i	$0.0695618\pi$
$967$	419.290i	0.433599i	−0.976216	+	0.216799i	$0.930438\pi$
$968$	0	0
$969$	444.881	0.459114
$970$	0	0
$971$	829.403	0.854174	0.427087	−	0.904210i	$-0.359540\pi$
$971$	829.403	0.854174	0.427087	+	0.904210i	$0.359540\pi$
$972$	0	0
$973$	758.435i	0.779481i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1667.21	1.70646	0.853231	−	0.521533i	$-0.174640\pi$
$977$	1667.21	1.70646	0.853231	+	0.521533i	$0.174640\pi$
$978$	0	0
$979$	−585.975	−0.598544
$980$	0	0
$981$	664.840i	0.677717i
$982$	0	0
$983$	776.192i	0.789616i	0.918764	+	0.394808i	$0.129189\pi$
$983$	776.192i	0.789616i	−0.918764	+	0.394808i	$0.870811\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−99.3616	−0.100670
$988$	0	0
$989$	− 2067.99i	− 2.09099i
$990$	0	0
$991$	1392.66i	1.40531i	0.711530	+	0.702655i	$0.248002\pi$
$991$	1392.66i	1.40531i	−0.711530	+	0.702655i	$0.751998\pi$
$992$	0	0
$993$	100.137	0.100843
$994$	0	0
$995$	0	0
$996$	0	0
$997$	1400.32i	1.40453i	0.711915	+	0.702266i	$0.247828\pi$
$997$	1400.32i	1.40453i	−0.711915	+	0.702266i	$0.752172\pi$
$998$	0	0
$999$	− 658.350i	− 0.659009i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1600.3.g.k.351.11		16
4.3	odd	2	inner	1600.3.g.k.351.6		16
5.2	odd	4		320.3.e.b.159.5	✓	16
5.3	odd	4		320.3.e.b.159.11	yes	16
5.4	even	2	inner	1600.3.g.k.351.5		16
8.3	odd	2	inner	1600.3.g.k.351.9		16
8.5	even	2	inner	1600.3.g.k.351.8		16
20.3	even	4		320.3.e.b.159.7	yes	16
20.7	even	4		320.3.e.b.159.9	yes	16
20.19	odd	2	inner	1600.3.g.k.351.12		16
40.3	even	4		320.3.e.b.159.10	yes	16
40.13	odd	4		320.3.e.b.159.6	yes	16
40.19	odd	2	inner	1600.3.g.k.351.7		16
40.27	even	4		320.3.e.b.159.8	yes	16
40.29	even	2	inner	1600.3.g.k.351.10		16
40.37	odd	4		320.3.e.b.159.12	yes	16
80.3	even	4		1280.3.h.n.1279.9		16
80.13	odd	4		1280.3.h.n.1279.5		16
80.27	even	4		1280.3.h.n.1279.11		16
80.37	odd	4		1280.3.h.n.1279.7		16
80.43	even	4		1280.3.h.n.1279.8		16
80.53	odd	4		1280.3.h.n.1279.12		16
80.67	even	4		1280.3.h.n.1279.6		16
80.77	odd	4		1280.3.h.n.1279.10		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.3.e.b.159.5	✓	16	5.2	odd	4
320.3.e.b.159.6	yes	16	40.13	odd	4
320.3.e.b.159.7	yes	16	20.3	even	4
320.3.e.b.159.8	yes	16	40.27	even	4
320.3.e.b.159.9	yes	16	20.7	even	4
320.3.e.b.159.10	yes	16	40.3	even	4
320.3.e.b.159.11	yes	16	5.3	odd	4
320.3.e.b.159.12	yes	16	40.37	odd	4
1280.3.h.n.1279.5		16	80.13	odd	4
1280.3.h.n.1279.6		16	80.67	even	4
1280.3.h.n.1279.7		16	80.37	odd	4
1280.3.h.n.1279.8		16	80.43	even	4
1280.3.h.n.1279.9		16	80.3	even	4
1280.3.h.n.1279.10		16	80.77	odd	4
1280.3.h.n.1279.11		16	80.27	even	4
1280.3.h.n.1279.12		16	80.53	odd	4
1600.3.g.k.351.5		16	5.4	even	2	inner
1600.3.g.k.351.6		16	4.3	odd	2	inner
1600.3.g.k.351.7		16	40.19	odd	2	inner
1600.3.g.k.351.8		16	8.5	even	2	inner
1600.3.g.k.351.9		16	8.3	odd	2	inner
1600.3.g.k.351.10		16	40.29	even	2	inner
1600.3.g.k.351.11		16	1.1	even	1	trivial
1600.3.g.k.351.12		16	20.19	odd	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:	\( N \)	\(=\)	\( 1600 = 2^{6} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1600.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.90845 q^{3} +3.95502i q^{7} -5.35782 q^{9} +O(q^{10})\)	\(q+1.90845 q^{3} +3.95502i q^{7} -5.35782 q^{9} -6.18667 q^{11} -4.94185i q^{13} +22.4311 q^{17} +10.3923 q^{19} +7.54796i q^{21} -39.6083i q^{23} -27.4012 q^{27} -30.4354i q^{29} +24.7156i q^{31} -11.8070 q^{33} +24.0264i q^{37} -9.43127i q^{39} +31.0735 q^{41} +52.2110 q^{43} +13.1640i q^{47} +33.3578 q^{49} +42.8087 q^{51} +17.9596i q^{53} +19.8332 q^{57} +104.421 q^{59} -57.2849i q^{61} -21.1903i q^{63} +99.3796 q^{67} -75.5905i q^{69} -16.7156i q^{71} +96.3357 q^{73} -24.4684i q^{77} +139.578i q^{79} -4.07345 q^{81} -91.7458 q^{83} -58.0844i q^{87} +94.7156 q^{89} +19.5451 q^{91} +47.1686i q^{93} -143.796 q^{97} +33.1471 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 96 q^{9}+O(q^{10}) \) 16 * q + 96 * q^9	\( 16 q + 96 q^{9} - 48 q^{41} + 352 q^{49} + 480 q^{81} + 1152 q^{89}+O(q^{100}) \) 16 * q + 96 * q^9 - 48 * q^41 + 352 * q^49 + 480 * q^81 + 1152 * q^89

Introduction

L-functions

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