LMFDB - Embedded newform 4896.2.f.c.2449.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4896,2,Mod(2449,4896)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4896, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4896.2449");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4896 = 2^{5} \cdot 3^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4896.f (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:	$39.0947568296$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.1649659456.5
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 2x^{5} + 4x^{4} - 4x^{3} - 8x + 16 $ x^8 - x^7 - 2x^5 + 4x^4 - 4x^3 - 8x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 136)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2449.6
Root			$0.814732 - 1.15595i$ of defining polynomial
Character	$\chi$	$=$	4896.2449
Dual form			4896.2.f.c.2449.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.77580i q^{5} +0.423267 q^{7} +O(q^{10})$	$q+1.77580i q^{5} +0.423267 q^{7} -1.66235i q^{11} +6.39959i q^{13} +1.00000 q^{17} +7.19684i q^{19} +5.68219 q^{23} +1.84653 q^{25} +3.37031i q^{29} -0.266431 q^{31} +0.751637i q^{35} -3.50598i q^{37} -8.79516 q^{41} -7.53429i q^{43} +2.10546 q^{47} -6.82085 q^{49} -3.32469i q^{53} +2.95199 q^{55} -3.98269i q^{59} +9.74118i q^{61} -11.3644 q^{65} -4.82556i q^{67} -12.8897 q^{71} +13.3280 q^{73} -0.703615i q^{77} +9.68219 q^{79} +8.81648i q^{83} +1.77580i q^{85} -6.39007 q^{89} +2.70873i q^{91} -12.7801 q^{95} +3.41576 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{7}+O(q^{10}) $ 8 * q - 8 * q^7	$ 8 q - 8 q^{7} + 8 q^{17} + 12 q^{23} - 8 q^{25} + 12 q^{31} - 28 q^{47} - 8 q^{49} - 44 q^{55} - 24 q^{65} + 8 q^{73} + 44 q^{79} - 8 q^{89} - 16 q^{95} + 8 q^{97}+O(q^{100}) $ 8 * q - 8 * q^7 + 8 * q^17 + 12 * q^23 - 8 * q^25 + 12 * q^31 - 28 * q^47 - 8 * q^49 - 44 * q^55 - 24 * q^65 + 8 * q^73 + 44 * q^79 - 8 * q^89 - 16 * q^95 + 8 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4896\mathbb{Z}\right)^\times$.

$n$	$613$	$2143$	$3809$	$4321$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.77580i	0.794162i	0.917784	+	0.397081i	$0.129977\pi$
$5$	1.77580i	0.794162i	−0.917784	+	0.397081i	$0.870023\pi$
$6$	0	0
$7$	0.423267	0.159980	0.0799899	−	0.996796i	$-0.474511\pi$
$7$	0.423267	0.159980	0.0799899	+	0.996796i	$0.474511\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 1.66235i	− 0.501216i	−0.968089	−	0.250608i	$-0.919369\pi$
$11$	− 1.66235i	− 0.501216i	0.968089	−	0.250608i	$-0.0806305\pi$
$12$	0	0
$13$	6.39959i	1.77493i	0.460879	+	0.887463i	$0.347534\pi$
$13$	6.39959i	1.77493i	−0.460879	+	0.887463i	$0.652466\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	7.19684i	1.65107i	0.564352	+	0.825534i	$0.309126\pi$
$19$	7.19684i	1.65107i	−0.564352	+	0.825534i	$0.690874\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.68219	1.18482	0.592410	−	0.805637i	$-0.298177\pi$
$23$	5.68219	1.18482	0.592410	+	0.805637i	$0.298177\pi$
$24$	0	0
$25$	1.84653	0.369307
$26$	0	0
$27$	0	0
$28$	0	0
$29$	3.37031i	0.625850i	0.949778	+	0.312925i	$0.101309\pi$
$29$	3.37031i	0.625850i	−0.949778	+	0.312925i	$0.898691\pi$
$30$	0	0
$31$	−0.266431	−0.0478524	−0.0239262	−	0.999714i	$-0.507617\pi$
$31$	−0.266431	−0.0478524	−0.0239262	+	0.999714i	$0.507617\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.751637i	0.127050i
$36$	0	0
$37$	− 3.50598i	− 0.576380i	−0.957573	−	0.288190i	$-0.906946\pi$
$37$	− 3.50598i	− 0.576380i	0.957573	−	0.288190i	$-0.0930535\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−8.79516	−1.37357	−0.686786	−	0.726859i	$-0.740979\pi$
$41$	−8.79516	−1.37357	−0.686786	+	0.726859i	$0.740979\pi$
$42$	0	0
$43$	− 7.53429i	− 1.14897i	−0.818516	−	0.574484i	$-0.805203\pi$
$43$	− 7.53429i	− 1.14897i	0.818516	−	0.574484i	$-0.194797\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.10546	0.307113	0.153556	−	0.988140i	$-0.450927\pi$
$47$	2.10546	0.307113	0.153556	+	0.988140i	$0.450927\pi$
$48$	0	0
$49$	−6.82085	−0.974406
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 3.32469i	− 0.456681i	−0.973581	−	0.228341i	$-0.926670\pi$
$53$	− 3.32469i	− 0.456681i	0.973581	−	0.228341i	$-0.0733300\pi$
$54$	0	0
$55$	2.95199	0.398047
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 3.98269i	− 0.518502i	−0.965810	−	0.259251i	$-0.916524\pi$
$59$	− 3.98269i	− 0.518502i	0.965810	−	0.259251i	$-0.0834757\pi$
$60$	0	0
$61$	9.74118i	1.24723i	0.781732	+	0.623615i	$0.214337\pi$
$61$	9.74118i	1.24723i	−0.781732	+	0.623615i	$0.785663\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−11.3644	−1.40958
$66$	0	0
$67$	− 4.82556i	− 0.589536i	−0.955569	−	0.294768i	$-0.904758\pi$
$67$	− 4.82556i	− 0.589536i	0.955569	−	0.294768i	$-0.0952423\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−12.8897	−1.52973	−0.764866	−	0.644190i	$-0.777195\pi$
$71$	−12.8897	−1.52973	−0.764866	+	0.644190i	$0.777195\pi$
$72$	0	0
$73$	13.3280	1.55993	0.779963	−	0.625825i	$-0.215238\pi$
$73$	13.3280	1.55993	0.779963	+	0.625825i	$0.215238\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 0.703615i	− 0.0801844i
$78$	0	0
$79$	9.68219	1.08933	0.544666	−	0.838653i	$-0.316656\pi$
$79$	9.68219	1.08933	0.544666	+	0.838653i	$0.316656\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	8.81648i	0.967735i	0.875141	+	0.483867i	$0.160768\pi$
$83$	8.81648i	0.967735i	−0.875141	+	0.483867i	$0.839232\pi$
$84$	0	0
$85$	1.77580i	0.192613i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−6.39007	−0.677347	−0.338673	−	0.940904i	$-0.609978\pi$
$89$	−6.39007	−0.677347	−0.338673	+	0.940904i	$0.609978\pi$
$90$	0	0
$91$	2.70873i	0.283952i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−12.7801	−1.31122
$96$	0	0
$97$	3.41576	0.346818	0.173409	−	0.984850i	$-0.444522\pi$
$97$	3.41576	0.346818	0.173409	+	0.984850i	$0.444522\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1.25348i	0.124726i	0.998054	+	0.0623629i	$0.0198636\pi$
$101$	1.25348i	0.124726i	−0.998054	+	0.0623629i	$0.980136\pi$
$102$	0	0
$103$	−15.6417	−1.54122	−0.770611	−	0.637306i	$-0.780049\pi$
$103$	−15.6417	−1.54122	−0.770611	+	0.637306i	$0.780049\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	13.6186i	1.31656i	0.752771	+	0.658282i	$0.228717\pi$
$107$	13.6186i	1.31656i	−0.752771	+	0.658282i	$0.771283\pi$
$108$	0	0
$109$	17.8997i	1.71448i	0.514920	+	0.857238i	$0.327822\pi$
$109$	17.8997i	1.71448i	−0.514920	+	0.857238i	$0.672178\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−5.36439	−0.504639	−0.252320	−	0.967644i	$-0.581193\pi$
$113$	−5.36439	−0.504639	−0.252320	+	0.967644i	$0.581193\pi$
$114$	0	0
$115$	10.0904i	0.940938i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0.423267	0.0388008
$120$	0	0
$121$	8.23661	0.748783
$122$	0	0
$123$	0	0
$124$	0	0
$125$	12.1581i	1.08745i
$126$	0	0
$127$	2.58424	0.229314	0.114657	−	0.993405i	$-0.463423\pi$
$127$	2.58424	0.229314	0.114657	+	0.993405i	$0.463423\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 12.4277i	− 1.08581i	−0.839793	−	0.542906i	$-0.817324\pi$
$131$	− 12.4277i	− 1.08581i	0.839793	−	0.542906i	$-0.182676\pi$
$132$	0	0
$133$	3.04618i	0.264137i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	12.3604	1.05602	0.528012	−	0.849237i	$-0.322938\pi$
$137$	12.3604	1.05602	0.528012	+	0.849237i	$0.322938\pi$
$138$	0	0
$139$	− 9.95651i	− 0.844500i	−0.906479	−	0.422250i	$-0.861240\pi$
$139$	− 9.95651i	− 0.844500i	0.906479	−	0.422250i	$-0.138760\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	10.6383	0.889621
$144$	0	0
$145$	−5.98499	−0.497027
$146$	0	0
$147$	0	0
$148$	0	0
$149$	13.1115i	1.07413i	0.843539	+	0.537067i	$0.180468\pi$
$149$	13.1115i	1.07413i	−0.843539	+	0.537067i	$0.819532\pi$
$150$	0	0
$151$	−2.89791	−0.235828	−0.117914	−	0.993024i	$-0.537621\pi$
$151$	−2.89791	−0.235828	−0.117914	+	0.993024i	$0.537621\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 0.473128i	− 0.0380026i
$156$	0	0
$157$	− 8.83339i	− 0.704981i	−0.935815	−	0.352490i	$-0.885335\pi$
$157$	− 8.83339i	− 0.704981i	0.935815	−	0.352490i	$-0.114665\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	2.40508	0.189547
$162$	0	0
$163$	− 1.24816i	− 0.0977634i	−0.998805	−	0.0488817i	$-0.984434\pi$
$163$	− 1.24816i	− 0.0977634i	0.998805	−	0.0488817i	$-0.0155657\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−8.04321	−0.622402	−0.311201	−	0.950344i	$-0.600731\pi$
$167$	−8.04321	−0.622402	−0.311201	+	0.950344i	$0.600731\pi$
$168$	0	0
$169$	−27.9547	−2.15036
$170$	0	0
$171$	0	0
$172$	0	0
$173$	14.3481i	1.09086i	0.838155	+	0.545431i	$0.183634\pi$
$173$	14.3481i	1.09086i	−0.838155	+	0.545431i	$0.816366\pi$
$174$	0	0
$175$	0.781576	0.0590816
$176$	0	0
$177$	0	0
$178$	0	0
$179$	6.55574i	0.489999i	0.969523	+	0.245000i	$0.0787878\pi$
$179$	6.55574i	0.489999i	−0.969523	+	0.245000i	$0.921212\pi$
$180$	0	0
$181$	0.907794i	0.0674758i	0.999431	+	0.0337379i	$0.0107411\pi$
$181$	0.907794i	0.0674758i	−0.999431	+	0.0337379i	$0.989259\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	6.22593	0.457739
$186$	0	0
$187$	− 1.66235i	− 0.121563i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−25.4335	−1.84030	−0.920151	−	0.391564i	$-0.871934\pi$
$191$	−25.4335	−1.84030	−0.920151	+	0.391564i	$0.871934\pi$
$192$	0	0
$193$	−1.08708	−0.0782500	−0.0391250	−	0.999234i	$-0.512457\pi$
$193$	−1.08708	−0.0782500	−0.0391250	+	0.999234i	$0.512457\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 2.31502i	− 0.164939i	−0.996594	−	0.0824693i	$-0.973719\pi$
$197$	− 2.31502i	− 0.164939i	0.996594	−	0.0824693i	$-0.0262807\pi$
$198$	0	0
$199$	1.11296	0.0788959	0.0394480	−	0.999222i	$-0.487440\pi$
$199$	1.11296	0.0788959	0.0394480	+	0.999222i	$0.487440\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1.42654i	0.100123i
$204$	0	0
$205$	− 15.6184i	− 1.09084i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	11.9636	0.827542
$210$	0	0
$211$	− 7.37281i	− 0.507565i	−0.967261	−	0.253783i	$-0.918325\pi$
$211$	− 7.37281i	− 0.507565i	0.967261	−	0.253783i	$-0.0816748\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	13.3794	0.912467
$216$	0	0
$217$	−0.112771	−0.00765541
$218$	0	0
$219$	0	0
$220$	0	0
$221$	6.39959i	0.430483i
$222$	0	0
$223$	13.0034	0.870770	0.435385	−	0.900244i	$-0.356612\pi$
$223$	13.0034	0.870770	0.435385	+	0.900244i	$0.356612\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2.75143i	0.182619i	0.995823	+	0.0913095i	$0.0291053\pi$
$227$	2.75143i	0.182619i	−0.995823	+	0.0913095i	$0.970895\pi$
$228$	0	0
$229$	23.0217i	1.52132i	0.649152	+	0.760659i	$0.275124\pi$
$229$	23.0217i	1.52132i	−0.649152	+	0.760659i	$0.724876\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2.27730	−0.149191	−0.0745956	−	0.997214i	$-0.523767\pi$
$233$	−2.27730	−0.149191	−0.0745956	+	0.997214i	$0.523767\pi$
$234$	0	0
$235$	3.73888i	0.243897i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−15.7404	−1.01816	−0.509081	−	0.860718i	$-0.670015\pi$
$239$	−15.7404	−1.01816	−0.509081	+	0.860718i	$0.670015\pi$
$240$	0	0
$241$	10.1892	0.656342	0.328171	−	0.944618i	$-0.393568\pi$
$241$	10.1892	0.656342	0.328171	+	0.944618i	$0.393568\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 12.1125i	− 0.773837i
$246$	0	0
$247$	−46.0568	−2.93052
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 1.82382i	− 0.115119i	−0.998342	−	0.0575593i	$-0.981668\pi$
$251$	− 1.82382i	− 0.115119i	0.998342	−	0.0575593i	$-0.0183318\pi$
$252$	0	0
$253$	− 9.44577i	− 0.593850i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−0.763392	−0.0476191	−0.0238095	−	0.999717i	$-0.507580\pi$
$257$	−0.763392	−0.0476191	−0.0238095	+	0.999717i	$0.507580\pi$
$258$	0	0
$259$	− 1.48397i	− 0.0922092i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−5.97037	−0.368149	−0.184074	−	0.982912i	$-0.558929\pi$
$263$	−5.97037	−0.368149	−0.184074	+	0.982912i	$0.558929\pi$
$264$	0	0
$265$	5.90399	0.362679
$266$	0	0
$267$	0	0
$268$	0	0
$269$	9.79145i	0.596995i	0.954410	+	0.298498i	$0.0964855\pi$
$269$	9.79145i	0.596995i	−0.954410	+	0.298498i	$0.903514\pi$
$270$	0	0
$271$	−25.9000	−1.57331	−0.786655	−	0.617393i	$-0.788189\pi$
$271$	−25.9000	−1.57331	−0.786655	+	0.617393i	$0.788189\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 3.06958i	− 0.185102i
$276$	0	0
$277$	15.7553i	0.946644i	0.880890	+	0.473322i	$0.156945\pi$
$277$	15.7553i	0.946644i	−0.880890	+	0.473322i	$0.843055\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−4.83980	−0.288718	−0.144359	−	0.989525i	$-0.546112\pi$
$281$	−4.83980	−0.288718	−0.144359	+	0.989525i	$0.546112\pi$
$282$	0	0
$283$	20.6113i	1.22522i	0.790387	+	0.612608i	$0.209879\pi$
$283$	20.6113i	1.22522i	−0.790387	+	0.612608i	$0.790121\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3.72270	−0.219744
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 21.7682i	− 1.27171i	−0.771807	−	0.635857i	$-0.780647\pi$
$293$	− 21.7682i	− 1.27171i	0.771807	−	0.635857i	$-0.219353\pi$
$294$	0	0
$295$	7.07246	0.411775
$296$	0	0
$297$	0	0
$298$	0	0
$299$	36.3637i	2.10297i
$300$	0	0
$301$	− 3.18901i	− 0.183812i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−17.2984	−0.990503
$306$	0	0
$307$	5.16301i	0.294669i	0.989087	+	0.147334i	$0.0470693\pi$
$307$	5.16301i	0.294669i	−0.989087	+	0.147334i	$0.952931\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−21.9472	−1.24451	−0.622256	−	0.782814i	$-0.713784\pi$
$311$	−21.9472	−1.24451	−0.622256	+	0.782814i	$0.713784\pi$
$312$	0	0
$313$	−11.0357	−0.623775	−0.311888	−	0.950119i	$-0.600961\pi$
$313$	−11.0357	−0.623775	−0.311888	+	0.950119i	$0.600961\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	10.3823i	0.583127i	0.956552	+	0.291563i	$0.0941754\pi$
$317$	10.3823i	0.583127i	−0.956552	+	0.291563i	$0.905825\pi$
$318$	0	0
$319$	5.60261	0.313686
$320$	0	0
$321$	0	0
$322$	0	0
$323$	7.19684i	0.400443i
$324$	0	0
$325$	11.8170i	0.655492i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0.891171	0.0491318
$330$	0	0
$331$	− 8.08415i	− 0.444345i	−0.975007	−	0.222173i	$-0.928685\pi$
$331$	− 8.08415i	− 0.444345i	0.975007	−	0.222173i	$-0.0713149\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	8.56923	0.468187
$336$	0	0
$337$	−22.4882	−1.22501	−0.612506	−	0.790466i	$-0.709838\pi$
$337$	−22.4882	−1.22501	−0.612506	+	0.790466i	$0.709838\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0.442900i	0.0239844i
$342$	0	0
$343$	−5.84990	−0.315865
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 20.1059i	− 1.07934i	−0.841876	−	0.539670i	$-0.818549\pi$
$347$	− 20.1059i	− 1.07934i	0.841876	−	0.539670i	$-0.181451\pi$
$348$	0	0
$349$	− 28.0947i	− 1.50387i	−0.659236	−	0.751936i	$-0.729120\pi$
$349$	− 28.0947i	− 1.50387i	0.659236	−	0.751936i	$-0.270880\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	29.4726	1.56867	0.784333	−	0.620340i	$-0.213005\pi$
$353$	29.4726	1.56867	0.784333	+	0.620340i	$0.213005\pi$
$354$	0	0
$355$	− 22.8896i	− 1.21485i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	23.1629	1.22249	0.611246	−	0.791441i	$-0.290669\pi$
$359$	23.1629	1.22249	0.611246	+	0.791441i	$0.290669\pi$
$360$	0	0
$361$	−32.7945	−1.72603
$362$	0	0
$363$	0	0
$364$	0	0
$365$	23.6679i	1.23883i
$366$	0	0
$367$	14.0582	0.733833	0.366917	−	0.930254i	$-0.380413\pi$
$367$	14.0582	0.733833	0.366917	+	0.930254i	$0.380413\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 1.40723i	− 0.0730598i
$372$	0	0
$373$	2.53567i	0.131292i	0.997843	+	0.0656460i	$0.0209108\pi$
$373$	2.53567i	0.131292i	−0.997843	+	0.0656460i	$0.979089\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−21.5686	−1.11084
$378$	0	0
$379$	6.32817i	0.325056i	0.986704	+	0.162528i	$0.0519648\pi$
$379$	6.32817i	0.325056i	−0.986704	+	0.162528i	$0.948035\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	9.31301	0.475873	0.237936	−	0.971281i	$-0.423529\pi$
$383$	9.31301	0.475873	0.237936	+	0.971281i	$0.423529\pi$
$384$	0	0
$385$	1.24948	0.0636794
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 1.34471i	− 0.0681796i	−0.999419	−	0.0340898i	$-0.989147\pi$
$389$	− 1.34471i	− 0.0681796i	0.999419	−	0.0340898i	$-0.0108532\pi$
$390$	0	0
$391$	5.68219	0.287361
$392$	0	0
$393$	0	0
$394$	0	0
$395$	17.1936i	0.865106i
$396$	0	0
$397$	− 2.58638i	− 0.129807i	−0.997892	−	0.0649033i	$-0.979326\pi$
$397$	− 2.58638i	− 0.129807i	0.997892	−	0.0649033i	$-0.0206739\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	0.642351	0.0320775	0.0160387	−	0.999871i	$-0.494894\pi$
$401$	0.642351	0.0320775	0.0160387	+	0.999871i	$0.494894\pi$
$402$	0	0
$403$	− 1.70505i	− 0.0849345i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−5.82816	−0.288891
$408$	0	0
$409$	31.2466	1.54505	0.772523	−	0.634987i	$-0.218994\pi$
$409$	31.2466	1.54505	0.772523	+	0.634987i	$0.218994\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 1.68574i	− 0.0829498i
$414$	0	0
$415$	−15.6563	−0.768538
$416$	0	0
$417$	0	0
$418$	0	0
$419$	1.59625i	0.0779819i	0.999240	+	0.0389909i	$0.0124143\pi$
$419$	1.59625i	0.0779819i	−0.999240	+	0.0389909i	$0.987586\pi$
$420$	0	0
$421$	0.982124i	0.0478658i	0.999714	+	0.0239329i	$0.00761881\pi$
$421$	0.982124i	0.0478658i	−0.999714	+	0.0239329i	$0.992381\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1.84653	0.0895700
$426$	0	0
$427$	4.12312i	0.199532i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	17.9014	0.862279	0.431140	−	0.902285i	$-0.358112\pi$
$431$	17.9014	0.862279	0.431140	+	0.902285i	$0.358112\pi$
$432$	0	0
$433$	−11.7695	−0.565605	−0.282802	−	0.959178i	$-0.591264\pi$
$433$	−11.7695	−0.565605	−0.282802	+	0.959178i	$0.591264\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	40.8938i	1.95622i
$438$	0	0
$439$	32.4287	1.54774	0.773868	−	0.633347i	$-0.218319\pi$
$439$	32.4287	1.54774	0.773868	+	0.633347i	$0.218319\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	11.9790i	0.569141i	0.958655	+	0.284571i	$0.0918510\pi$
$443$	11.9790i	0.569141i	−0.958655	+	0.284571i	$0.908149\pi$
$444$	0	0
$445$	− 11.3475i	− 0.537923i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−1.78908	−0.0844319	−0.0422159	−	0.999109i	$-0.513442\pi$
$449$	−1.78908	−0.0844319	−0.0422159	+	0.999109i	$0.513442\pi$
$450$	0	0
$451$	14.6206i	0.688457i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−4.81017	−0.225504
$456$	0	0
$457$	−0.785137	−0.0367272	−0.0183636	−	0.999831i	$-0.505846\pi$
$457$	−0.785137	−0.0367272	−0.0183636	+	0.999831i	$0.505846\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	16.8562i	0.785071i	0.919737	+	0.392536i	$0.128402\pi$
$461$	16.8562i	0.785071i	−0.919737	+	0.392536i	$0.871598\pi$
$462$	0	0
$463$	6.30693	0.293108	0.146554	−	0.989203i	$-0.453182\pi$
$463$	6.30693	0.293108	0.146554	+	0.989203i	$0.453182\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 10.4361i	− 0.482926i	−0.970410	−	0.241463i	$-0.922373\pi$
$467$	− 10.4361i	− 0.482926i	0.970410	−	0.241463i	$-0.0776273\pi$
$468$	0	0
$469$	− 2.04250i	− 0.0943138i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−12.5246	−0.575881
$474$	0	0
$475$	13.2892i	0.609750i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	43.2579	1.97650	0.988252	−	0.152836i	$-0.0488405\pi$
$479$	43.2579	1.97650	0.988252	+	0.152836i	$0.0488405\pi$
$480$	0	0
$481$	22.4368	1.02303
$482$	0	0
$483$	0	0
$484$	0	0
$485$	6.06571i	0.275430i
$486$	0	0
$487$	−16.6788	−0.755790	−0.377895	−	0.925849i	$-0.623352\pi$
$487$	−16.6788	−0.755790	−0.377895	+	0.925849i	$0.623352\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	12.5231i	0.565158i	0.959244	+	0.282579i	$0.0911900\pi$
$491$	12.5231i	0.565158i	−0.959244	+	0.282579i	$0.908810\pi$
$492$	0	0
$493$	3.37031i	0.151791i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−5.45580	−0.244726
$498$	0	0
$499$	38.7525i	1.73480i	0.497610	+	0.867401i	$0.334211\pi$
$499$	38.7525i	1.73480i	−0.497610	+	0.867401i	$0.665789\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	11.7066	0.521974	0.260987	−	0.965342i	$-0.415952\pi$
$503$	11.7066	0.521974	0.260987	+	0.965342i	$0.415952\pi$
$504$	0	0
$505$	−2.22593	−0.0990525
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 33.5637i	− 1.48769i	−0.668354	−	0.743843i	$-0.733001\pi$
$509$	− 33.5637i	− 1.48769i	0.668354	−	0.743843i	$-0.266999\pi$
$510$	0	0
$511$	5.64131	0.249557
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 27.7765i	− 1.22398i
$516$	0	0
$517$	− 3.50000i	− 0.153930i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	10.5910	0.463999	0.232000	−	0.972716i	$-0.425473\pi$
$521$	10.5910	0.463999	0.232000	+	0.972716i	$0.425473\pi$
$522$	0	0
$523$	4.33078i	0.189372i	0.995507	+	0.0946859i	$0.0301847\pi$
$523$	4.33078i	0.189372i	−0.995507	+	0.0946859i	$0.969815\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−0.266431	−0.0116059
$528$	0	0
$529$	9.28732	0.403797
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 56.2854i	− 2.43799i
$534$	0	0
$535$	−24.1840	−1.04557
$536$	0	0
$537$	0	0
$538$	0	0
$539$	11.3386i	0.488388i
$540$	0	0
$541$	− 31.9236i	− 1.37250i	−0.727364	−	0.686251i	$-0.759255\pi$
$541$	− 31.9236i	− 1.37250i	0.727364	−	0.686251i	$-0.240745\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−31.7862	−1.36157
$546$	0	0
$547$	− 16.6310i	− 0.711091i	−0.934659	−	0.355546i	$-0.884295\pi$
$547$	− 16.6310i	− 0.711091i	0.934659	−	0.355546i	$-0.115705\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−24.2556	−1.03332
$552$	0	0
$553$	4.09815	0.174271
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 18.1879i	− 0.770647i	−0.922782	−	0.385323i	$-0.874090\pi$
$557$	− 18.1879i	− 0.770647i	0.922782	−	0.385323i	$-0.125910\pi$
$558$	0	0
$559$	48.2163	2.03933
$560$	0	0
$561$	0	0
$562$	0	0
$563$	28.6686i	1.20824i	0.796895	+	0.604118i	$0.206475\pi$
$563$	28.6686i	1.20824i	−0.796895	+	0.604118i	$0.793525\pi$
$564$	0	0
$565$	− 9.52608i	− 0.400765i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	31.7795	1.33227	0.666133	−	0.745833i	$-0.267948\pi$
$569$	31.7795	1.33227	0.666133	+	0.745833i	$0.267948\pi$
$570$	0	0
$571$	20.3148i	0.850149i	0.905158	+	0.425074i	$0.139752\pi$
$571$	20.3148i	0.850149i	−0.905158	+	0.425074i	$0.860248\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	10.4924	0.437562
$576$	0	0
$577$	13.2433	0.551328	0.275664	−	0.961254i	$-0.411102\pi$
$577$	13.2433	0.551328	0.275664	+	0.961254i	$0.411102\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	3.73172i	0.154818i
$582$	0	0
$583$	−5.52678	−0.228896
$584$	0	0
$585$	0	0
$586$	0	0
$587$	18.6800i	0.771007i	0.922706	+	0.385503i	$0.125972\pi$
$587$	18.6800i	0.771007i	−0.922706	+	0.385503i	$0.874028\pi$
$588$	0	0
$589$	− 1.91746i	− 0.0790076i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	11.7741	0.483503	0.241752	−	0.970338i	$-0.422278\pi$
$593$	11.7741	0.483503	0.241752	+	0.970338i	$0.422278\pi$
$594$	0	0
$595$	0.751637i	0.0308141i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0.511116	0.0208836	0.0104418	−	0.999945i	$-0.496676\pi$
$599$	0.511116	0.0208836	0.0104418	+	0.999945i	$0.496676\pi$
$600$	0	0
$601$	35.5686	1.45087	0.725436	−	0.688290i	$-0.241638\pi$
$601$	35.5686	1.45087	0.725436	+	0.688290i	$0.241638\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	14.6266i	0.594655i
$606$	0	0
$607$	10.2068	0.414281	0.207140	−	0.978311i	$-0.433584\pi$
$607$	10.2068	0.414281	0.207140	+	0.978311i	$0.433584\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	13.4741i	0.545103i
$612$	0	0
$613$	19.3573i	0.781835i	0.920426	+	0.390917i	$0.127842\pi$
$613$	19.3573i	0.781835i	−0.920426	+	0.390917i	$0.872158\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−36.6624	−1.47597	−0.737986	−	0.674816i	$-0.764223\pi$
$617$	−36.6624	−1.47597	−0.737986	+	0.674816i	$0.764223\pi$
$618$	0	0
$619$	10.7742i	0.433053i	0.976277	+	0.216527i	$0.0694728\pi$
$619$	10.7742i	0.433053i	−0.976277	+	0.216527i	$0.930527\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−2.70471	−0.108362
$624$	0	0
$625$	−12.3576	−0.494306
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 3.50598i	− 0.139793i
$630$	0	0
$631$	−23.5212	−0.936365	−0.468183	−	0.883632i	$-0.655091\pi$
$631$	−23.5212	−0.936365	−0.468183	+	0.883632i	$0.655091\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	4.58909i	0.182112i
$636$	0	0
$637$	− 43.6506i	− 1.72950i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−16.5842	−0.655038	−0.327519	−	0.944845i	$-0.606213\pi$
$641$	−16.5842	−0.655038	−0.327519	+	0.944845i	$0.606213\pi$
$642$	0	0
$643$	1.13762i	0.0448633i	0.999748	+	0.0224316i	$0.00714081\pi$
$643$	1.13762i	0.0448633i	−0.999748	+	0.0224316i	$0.992859\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−9.03974	−0.355389	−0.177694	−	0.984086i	$-0.556864\pi$
$647$	−9.03974	−0.355389	−0.177694	+	0.984086i	$0.556864\pi$
$648$	0	0
$649$	−6.62061	−0.259881
$650$	0	0
$651$	0	0
$652$	0	0
$653$	12.1018i	0.473581i	0.971561	+	0.236790i	$0.0760955\pi$
$653$	12.1018i	0.473581i	−0.971561	+	0.236790i	$0.923905\pi$
$654$	0	0
$655$	22.0691	0.862311
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 25.6152i	− 0.997828i	−0.866651	−	0.498914i	$-0.833732\pi$
$659$	− 25.6152i	− 0.997828i	0.866651	−	0.498914i	$-0.166268\pi$
$660$	0	0
$661$	− 22.9592i	− 0.893009i	−0.894781	−	0.446505i	$-0.852669\pi$
$661$	− 22.9592i	− 0.893009i	0.894781	−	0.446505i	$-0.147331\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−5.40941	−0.209768
$666$	0	0
$667$	19.1507i	0.741519i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	16.1932	0.625132
$672$	0	0
$673$	−39.6985	−1.53026	−0.765132	−	0.643873i	$-0.777326\pi$
$673$	−39.6985	−1.53026	−0.765132	+	0.643873i	$0.777326\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	12.1196i	0.465795i	0.972501	+	0.232897i	$0.0748206\pi$
$677$	12.1196i	0.465795i	−0.972501	+	0.232897i	$0.925179\pi$
$678$	0	0
$679$	1.44578	0.0554839
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 3.47509i	− 0.132971i	−0.997787	−	0.0664854i	$-0.978821\pi$
$683$	− 3.47509i	− 0.132971i	0.997787	−	0.0664854i	$-0.0211786\pi$
$684$	0	0
$685$	21.9497i	0.838654i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	21.2766	0.810576
$690$	0	0
$691$	− 20.1984i	− 0.768385i	−0.923253	−	0.384193i	$-0.874480\pi$
$691$	− 20.1984i	− 0.768385i	0.923253	−	0.384193i	$-0.125520\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	17.6808	0.670670
$696$	0	0
$697$	−8.79516	−0.333140
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 49.3933i	− 1.86556i	−0.360445	−	0.932780i	$-0.617375\pi$
$701$	− 49.3933i	− 1.86556i	0.360445	−	0.932780i	$-0.382625\pi$
$702$	0	0
$703$	25.2320	0.951643
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0.530556i	0.0199536i
$708$	0	0
$709$	− 18.3067i	− 0.687522i	−0.939057	−	0.343761i	$-0.888299\pi$
$709$	− 18.3067i	− 0.687522i	0.939057	−	0.343761i	$-0.111701\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−1.51391	−0.0566964
$714$	0	0
$715$	18.8915i	0.706503i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−36.4504	−1.35937	−0.679686	−	0.733503i	$-0.737884\pi$
$719$	−36.4504	−1.35937	−0.679686	+	0.733503i	$0.737884\pi$
$720$	0	0
$721$	−6.62061	−0.246564
$722$	0	0
$723$	0	0
$724$	0	0
$725$	6.22338i	0.231131i
$726$	0	0
$727$	−10.7047	−0.397016	−0.198508	−	0.980099i	$-0.563610\pi$
$727$	−10.7047	−0.397016	−0.198508	+	0.980099i	$0.563610\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 7.53429i	− 0.278666i
$732$	0	0
$733$	− 1.89967i	− 0.0701658i	−0.999384	−	0.0350829i	$-0.988830\pi$
$733$	− 1.89967i	− 0.0701658i	0.999384	−	0.0350829i	$-0.0111695\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−8.02175	−0.295485
$738$	0	0
$739$	− 44.1249i	− 1.62316i	−0.584241	−	0.811580i	$-0.698608\pi$
$739$	− 44.1249i	− 1.62316i	0.584241	−	0.811580i	$-0.301392\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	15.7390	0.577407	0.288704	−	0.957418i	$-0.406776\pi$
$743$	15.7390	0.577407	0.288704	+	0.957418i	$0.406776\pi$
$744$	0	0
$745$	−23.2834	−0.853037
$746$	0	0
$747$	0	0
$748$	0	0
$749$	5.76432i	0.210624i
$750$	0	0
$751$	11.5186	0.420320	0.210160	−	0.977667i	$-0.432601\pi$
$751$	11.5186	0.420320	0.210160	+	0.977667i	$0.432601\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 5.14611i	− 0.187286i
$756$	0	0
$757$	29.7285i	1.08050i	0.841504	+	0.540251i	$0.181671\pi$
$757$	29.7285i	1.08050i	−0.841504	+	0.540251i	$0.818329\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	54.0455	1.95915	0.979573	−	0.201088i	$-0.0644479\pi$
$761$	54.0455	1.95915	0.979573	+	0.201088i	$0.0644479\pi$
$762$	0	0
$763$	7.57633i	0.274282i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	25.4876	0.920303
$768$	0	0
$769$	34.3237	1.23774	0.618872	−	0.785492i	$-0.287590\pi$
$769$	34.3237	1.23774	0.618872	+	0.785492i	$0.287590\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 51.2521i	− 1.84341i	−0.387894	−	0.921704i	$-0.626797\pi$
$773$	− 51.2521i	− 1.84341i	0.387894	−	0.921704i	$-0.373203\pi$
$774$	0	0
$775$	−0.491973	−0.0176722
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 63.2973i	− 2.26786i
$780$	0	0
$781$	21.4272i	0.766726i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	15.6863	0.559869
$786$	0	0
$787$	− 10.6372i	− 0.379177i	−0.981864	−	0.189588i	$-0.939285\pi$
$787$	− 10.6372i	− 0.379177i	0.981864	−	0.189588i	$-0.0607153\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−2.27057	−0.0807320
$792$	0	0
$793$	−62.3395	−2.21374
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 32.6441i	− 1.15631i	−0.815926	−	0.578157i	$-0.803772\pi$
$797$	− 32.6441i	− 1.15631i	0.815926	−	0.578157i	$-0.196228\pi$
$798$	0	0
$799$	2.10546	0.0744858
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 22.1558i	− 0.781860i
$804$	0	0
$805$	4.27095i	0.150531i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	28.0568	0.986424	0.493212	−	0.869909i	$-0.335823\pi$
$809$	28.0568	0.986424	0.493212	+	0.869909i	$0.335823\pi$
$810$	0	0
$811$	− 9.17973i	− 0.322344i	−0.986926	−	0.161172i	$-0.948473\pi$
$811$	− 9.17973i	− 0.322344i	0.986926	−	0.161172i	$-0.0515274\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	2.21648	0.0776400
$816$	0	0
$817$	54.2231	1.89703
$818$	0	0
$819$	0	0
$820$	0	0
$821$	19.3407i	0.674995i	0.941326	+	0.337498i	$0.109580\pi$
$821$	19.3407i	0.674995i	−0.941326	+	0.337498i	$0.890420\pi$
$822$	0	0
$823$	18.8278	0.656295	0.328148	−	0.944626i	$-0.393576\pi$
$823$	18.8278	0.656295	0.328148	+	0.944626i	$0.393576\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	40.6687i	1.41419i	0.707119	+	0.707094i	$0.249994\pi$
$827$	40.6687i	1.41419i	−0.707119	+	0.707094i	$0.750006\pi$
$828$	0	0
$829$	14.7047i	0.510714i	0.966847	+	0.255357i	$0.0821930\pi$
$829$	14.7047i	0.510714i	−0.966847	+	0.255357i	$0.917807\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−6.82085	−0.236328
$834$	0	0
$835$	− 14.2831i	− 0.494288i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−18.7883	−0.648645	−0.324322	−	0.945947i	$-0.605136\pi$
$839$	−18.7883	−0.648645	−0.324322	+	0.945947i	$0.605136\pi$
$840$	0	0
$841$	17.6410	0.608311
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 49.6420i	− 1.70774i
$846$	0	0
$847$	3.48628	0.119790
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 19.9217i	− 0.682906i
$852$	0	0
$853$	− 54.4758i	− 1.86521i	−0.360893	−	0.932607i	$-0.617528\pi$
$853$	− 54.4758i	− 1.86521i	0.360893	−	0.932607i	$-0.382472\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−40.3258	−1.37750	−0.688752	−	0.724997i	$-0.741841\pi$
$857$	−40.3258	−1.37750	−0.688752	+	0.724997i	$0.741841\pi$
$858$	0	0
$859$	26.8916i	0.917530i	0.888558	+	0.458765i	$0.151708\pi$
$859$	26.8916i	0.917530i	−0.888558	+	0.458765i	$0.848292\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−37.8313	−1.28779	−0.643895	−	0.765114i	$-0.722683\pi$
$863$	−37.8313	−1.28779	−0.643895	+	0.765114i	$0.722683\pi$
$864$	0	0
$865$	−25.4793	−0.866322
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 16.0951i	− 0.545991i
$870$	0	0
$871$	30.8816	1.04638
$872$	0	0
$873$	0	0
$874$	0	0
$875$	5.14611i	0.173970i
$876$	0	0
$877$	18.7783i	0.634098i	0.948409	+	0.317049i	$0.102692\pi$
$877$	18.7783i	0.634098i	−0.948409	+	0.317049i	$0.897308\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−10.1746	−0.342789	−0.171395	−	0.985202i	$-0.554827\pi$
$881$	−10.1746	−0.342789	−0.171395	+	0.985202i	$0.554827\pi$
$882$	0	0
$883$	27.3719i	0.921138i	0.887624	+	0.460569i	$0.152355\pi$
$883$	27.3719i	0.921138i	−0.887624	+	0.460569i	$0.847645\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−42.4031	−1.42376	−0.711878	−	0.702303i	$-0.752155\pi$
$887$	−42.4031	−1.42376	−0.711878	+	0.702303i	$0.752155\pi$
$888$	0	0
$889$	1.09382	0.0366856
$890$	0	0
$891$	0	0
$892$	0	0
$893$	15.1527i	0.507064i
$894$	0	0
$895$	−11.6417	−0.389139
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 0.897954i	− 0.0299484i
$900$	0	0
$901$	− 3.32469i	− 0.110762i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1.61206	−0.0535867
$906$	0	0
$907$	− 19.2006i	− 0.637545i	−0.947831	−	0.318773i	$-0.896729\pi$
$907$	− 19.2006i	− 0.637545i	0.947831	−	0.318773i	$-0.103271\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	11.4022	0.377771	0.188886	−	0.981999i	$-0.439512\pi$
$911$	11.4022	0.377771	0.188886	+	0.981999i	$0.439512\pi$
$912$	0	0
$913$	14.6560	0.485044
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 5.26023i	− 0.173708i
$918$	0	0
$919$	30.6301	1.01039	0.505196	−	0.863005i	$-0.331420\pi$
$919$	30.6301	1.01039	0.505196	+	0.863005i	$0.331420\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 82.4890i	− 2.71516i
$924$	0	0
$925$	− 6.47392i	− 0.212861i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−58.0272	−1.90381	−0.951905	−	0.306394i	$-0.900877\pi$
$929$	−58.0272	−1.90381	−0.951905	+	0.306394i	$0.900877\pi$
$930$	0	0
$931$	− 49.0885i	− 1.60881i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	2.95199	0.0965405
$936$	0	0
$937$	42.3191	1.38250	0.691252	−	0.722614i	$-0.257059\pi$
$937$	42.3191	1.38250	0.691252	+	0.722614i	$0.257059\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	38.8057i	1.26503i	0.774548	+	0.632515i	$0.217977\pi$
$941$	38.8057i	1.26503i	−0.774548	+	0.632515i	$0.782023\pi$
$942$	0	0
$943$	−49.9758	−1.62744
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 36.5421i	− 1.18746i	−0.804665	−	0.593729i	$-0.797655\pi$
$947$	− 36.5421i	− 1.18746i	0.804665	−	0.593729i	$-0.202345\pi$
$948$	0	0
$949$	85.2938i	2.76875i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−8.36871	−0.271089	−0.135545	−	0.990771i	$-0.543278\pi$
$953$	−8.36871	−0.271089	−0.135545	+	0.990771i	$0.543278\pi$
$954$	0	0
$955$	− 45.1648i	− 1.46150i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	5.23176	0.168942
$960$	0	0
$961$	−30.9290	−0.997710
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 1.93044i	− 0.0621432i
$966$	0	0
$967$	24.7679	0.796480	0.398240	−	0.917281i	$-0.369621\pi$
$967$	24.7679	0.796480	0.398240	+	0.917281i	$0.369621\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 20.8495i	− 0.669093i	−0.942379	−	0.334546i	$-0.891417\pi$
$971$	− 20.8495i	− 0.669093i	0.942379	−	0.334546i	$-0.108583\pi$
$972$	0	0
$973$	− 4.21426i	− 0.135103i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	44.3041	1.41741	0.708707	−	0.705503i	$-0.249279\pi$
$977$	44.3041	1.41741	0.708707	+	0.705503i	$0.249279\pi$
$978$	0	0
$979$	10.6225i	0.339497i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	17.4807	0.557548	0.278774	−	0.960357i	$-0.410072\pi$
$983$	17.4807	0.557548	0.278774	+	0.960357i	$0.410072\pi$
$984$	0	0
$985$	4.11102	0.130988
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 42.8113i	− 1.36132i
$990$	0	0
$991$	−46.2299	−1.46854	−0.734271	−	0.678857i	$-0.762476\pi$
$991$	−46.2299	−1.46854	−0.734271	+	0.678857i	$0.762476\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	1.97640i	0.0626562i
$996$	0	0
$997$	− 29.1043i	− 0.921743i	−0.887467	−	0.460872i	$-0.847537\pi$
$997$	− 29.1043i	− 0.921743i	0.887467	−	0.460872i	$-0.152463\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4896.2.f.c.2449.6		8
3.2	odd	2		544.2.c.a.273.8		8
4.3	odd	2		1224.2.f.d.613.6		8
8.3	odd	2		1224.2.f.d.613.5		8
8.5	even	2	inner	4896.2.f.c.2449.3		8
12.11	even	2		136.2.c.a.69.3	✓	8
24.5	odd	2		544.2.c.a.273.1		8
24.11	even	2		136.2.c.a.69.4	yes	8
48.5	odd	4		4352.2.a.be.1.8		8
48.11	even	4		4352.2.a.bc.1.1		8
48.29	odd	4		4352.2.a.be.1.1		8
48.35	even	4		4352.2.a.bc.1.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.2.c.a.69.3	✓	8	12.11	even	2
136.2.c.a.69.4	yes	8	24.11	even	2
544.2.c.a.273.1		8	24.5	odd	2
544.2.c.a.273.8		8	3.2	odd	2
1224.2.f.d.613.5		8	8.3	odd	2
1224.2.f.d.613.6		8	4.3	odd	2
4352.2.a.bc.1.1		8	48.11	even	4
4352.2.a.bc.1.8		8	48.35	even	4
4352.2.a.be.1.1		8	48.29	odd	4
4352.2.a.be.1.8		8	48.5	odd	4
4896.2.f.c.2449.3		8	8.5	even	2	inner
4896.2.f.c.2449.6		8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4896 = 2^{5} \cdot 3^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4896.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.77580i q^{5} +0.423267 q^{7} +O(q^{10})\)	\(q+1.77580i q^{5} +0.423267 q^{7} -1.66235i q^{11} +6.39959i q^{13} +1.00000 q^{17} +7.19684i q^{19} +5.68219 q^{23} +1.84653 q^{25} +3.37031i q^{29} -0.266431 q^{31} +0.751637i q^{35} -3.50598i q^{37} -8.79516 q^{41} -7.53429i q^{43} +2.10546 q^{47} -6.82085 q^{49} -3.32469i q^{53} +2.95199 q^{55} -3.98269i q^{59} +9.74118i q^{61} -11.3644 q^{65} -4.82556i q^{67} -12.8897 q^{71} +13.3280 q^{73} -0.703615i q^{77} +9.68219 q^{79} +8.81648i q^{83} +1.77580i q^{85} -6.39007 q^{89} +2.70873i q^{91} -12.7801 q^{95} +3.41576 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{7}+O(q^{10}) \) 8 * q - 8 * q^7	\( 8 q - 8 q^{7} + 8 q^{17} + 12 q^{23} - 8 q^{25} + 12 q^{31} - 28 q^{47} - 8 q^{49} - 44 q^{55} - 24 q^{65} + 8 q^{73} + 44 q^{79} - 8 q^{89} - 16 q^{95} + 8 q^{97}+O(q^{100}) \) 8 * q - 8 * q^7 + 8 * q^17 + 12 * q^23 - 8 * q^25 + 12 * q^31 - 28 * q^47 - 8 * q^49 - 44 * q^55 - 24 * q^65 + 8 * q^73 + 44 * q^79 - 8 * q^89 - 16 * q^95 + 8 * q^97

Introduction

L-functions

Modular forms

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Database

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