LMFDB - Embedded newform 3168.2.f.g.1585.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3168,2,Mod(1585,3168)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3168, 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("3168.1585");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3168 = 2^{5} \cdot 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3168.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:	$25.2966073603$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	10.0.578281160704.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 2x^{8} - 2x^{7} - 3x^{6} - 6x^{5} - 6x^{4} - 8x^{3} + 16x^{2} + 32 $ x^10 + 2x^8 - 2x^7 - 3x^6 - 6x^5 - 6x^4 - 8x^3 + 16*x^2 + 32
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 88)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1585.10
Root			$0.437403 - 1.34487i$ of defining polynomial
Character	$\chi$	$=$	3168.1585
Dual form			3168.2.f.g.1585.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.16794i q^{5} -0.933222 q^{7} -1.00000i q^{11} +2.93322i q^{13} -2.44626 q^{17} +2.68283i q^{19} -3.47129 q^{23} -12.3717 q^{25} +4.57245i q^{29} +3.65781 q^{31} -3.88961i q^{35} -4.53806i q^{37} -4.12618 q^{41} -11.4650i q^{43} +3.26265 q^{47} -6.12910 q^{49} -0.650132i q^{53} +4.16794 q^{55} +2.90965i q^{59} +10.7060i q^{61} -12.2255 q^{65} +5.42623i q^{67} -7.75129 q^{71} +13.0650 q^{73} +0.933222i q^{77} -4.83666 q^{79} +0.659664i q^{83} -10.1959i q^{85} +2.74629 q^{89} -2.73735i q^{91} -11.1819 q^{95} +1.82905 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 4 q^{17} - 12 q^{23} - 6 q^{25} + 4 q^{31} - 4 q^{41} - 4 q^{47} - 6 q^{49} + 8 q^{55} - 16 q^{65} - 12 q^{71} - 4 q^{73} - 16 q^{79} + 4 q^{89} + 24 q^{95} - 20 q^{97}+O(q^{100}) $ 10 * q + 4 * q^17 - 12 * q^23 - 6 * q^25 + 4 * q^31 - 4 * q^41 - 4 * q^47 - 6 * q^49 + 8 * q^55 - 16 * q^65 - 12 * q^71 - 4 * q^73 - 16 * q^79 + 4 * q^89 + 24 * q^95 - 20 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3168\mathbb{Z}\right)^\times$.

$n$	$353$	$991$	$1189$	$1729$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$5$	4.16794i	1.86396i	0.362511	+	0.931979i	$0.381919\pi$
$5$	4.16794i	1.86396i	−0.362511	+	0.931979i	$0.618081\pi$
$6$	0	0
$7$	−0.933222	−0.352725	−0.176362	−	0.984325i	$-0.556433\pi$
$7$	−0.933222	−0.352725	−0.176362	+	0.984325i	$0.556433\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 1.00000i	− 0.301511i
$11$	− 1.00000i	− 0.301511i
$12$	0	0
$13$	2.93322i	0.813530i	0.913533	+	0.406765i	$0.133343\pi$
$13$	2.93322i	0.813530i	−0.913533	+	0.406765i	$0.866657\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.44626	−0.593306	−0.296653	−	0.954985i	$-0.595870\pi$
$17$	−2.44626	−0.593306	−0.296653	+	0.954985i	$0.595870\pi$
$18$	0	0
$19$	2.68283i	0.615484i	0.951470	+	0.307742i	$0.0995734\pi$
$19$	2.68283i	0.615484i	−0.951470	+	0.307742i	$0.900427\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3.47129	−0.723813	−0.361907	−	0.932214i	$-0.617874\pi$
$23$	−3.47129	−0.723813	−0.361907	+	0.932214i	$0.617874\pi$
$24$	0	0
$25$	−12.3717	−2.47434
$26$	0	0
$27$	0	0
$28$	0	0
$29$	4.57245i	0.849082i	0.905409	+	0.424541i	$0.139565\pi$
$29$	4.57245i	0.849082i	−0.905409	+	0.424541i	$0.860435\pi$
$30$	0	0
$31$	3.65781	0.656962	0.328481	−	0.944511i	$-0.393463\pi$
$31$	3.65781	0.656962	0.328481	+	0.944511i	$0.393463\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 3.88961i	− 0.657465i
$36$	0	0
$37$	− 4.53806i	− 0.746053i	−0.927821	−	0.373027i	$-0.878320\pi$
$37$	− 4.53806i	− 0.746053i	0.927821	−	0.373027i	$-0.121680\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−4.12618	−0.644402	−0.322201	−	0.946671i	$-0.604423\pi$
$41$	−4.12618	−0.644402	−0.322201	+	0.946671i	$0.604423\pi$
$42$	0	0
$43$	− 11.4650i	− 1.74839i	−0.485573	−	0.874196i	$-0.661389\pi$
$43$	− 11.4650i	− 1.74839i	0.485573	−	0.874196i	$-0.338611\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3.26265	0.475907	0.237953	−	0.971277i	$-0.423523\pi$
$47$	3.26265	0.475907	0.237953	+	0.971277i	$0.423523\pi$
$48$	0	0
$49$	−6.12910	−0.875585
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 0.650132i	− 0.0893025i	−0.999003	−	0.0446512i	$-0.985782\pi$
$53$	− 0.650132i	− 0.0893025i	0.999003	−	0.0446512i	$-0.0142177\pi$
$54$	0	0
$55$	4.16794	0.562005
$56$	0	0
$57$	0	0
$58$	0	0
$59$	2.90965i	0.378804i	0.981900	+	0.189402i	$0.0606550\pi$
$59$	2.90965i	0.378804i	−0.981900	+	0.189402i	$0.939345\pi$
$60$	0	0
$61$	10.7060i	1.37076i	0.728184	+	0.685382i	$0.240365\pi$
$61$	10.7060i	1.37076i	−0.728184	+	0.685382i	$0.759635\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−12.2255	−1.51639
$66$	0	0
$67$	5.42623i	0.662919i	0.943469	+	0.331460i	$0.107541\pi$
$67$	5.42623i	0.662919i	−0.943469	+	0.331460i	$0.892459\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−7.75129	−0.919909	−0.459955	−	0.887942i	$-0.652134\pi$
$71$	−7.75129	−0.919909	−0.459955	+	0.887942i	$0.652134\pi$
$72$	0	0
$73$	13.0650	1.52915	0.764574	−	0.644536i	$-0.222949\pi$
$73$	13.0650	1.52915	0.764574	+	0.644536i	$0.222949\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.933222i	0.106351i
$78$	0	0
$79$	−4.83666	−0.544166	−0.272083	−	0.962274i	$-0.587713\pi$
$79$	−4.83666	−0.544166	−0.272083	+	0.962274i	$0.587713\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	0.659664i	0.0724075i	0.999344	+	0.0362038i	$0.0115265\pi$
$83$	0.659664i	0.0724075i	−0.999344	+	0.0362038i	$0.988473\pi$
$84$	0	0
$85$	− 10.1959i	− 1.10590i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	2.74629	0.291107	0.145553	−	0.989350i	$-0.453504\pi$
$89$	2.74629	0.291107	0.145553	+	0.989350i	$0.453504\pi$
$90$	0	0
$91$	− 2.73735i	− 0.286952i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−11.1819	−1.14724
$96$	0	0
$97$	1.82905	0.185712	0.0928561	−	0.995680i	$-0.470400\pi$
$97$	1.82905	0.185712	0.0928561	+	0.995680i	$0.470400\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 12.6811i	− 1.26182i	−0.775857	−	0.630908i	$-0.782682\pi$
$101$	− 12.6811i	− 1.26182i	0.775857	−	0.630908i	$-0.217318\pi$
$102$	0	0
$103$	3.86644	0.380972	0.190486	−	0.981690i	$-0.438994\pi$
$103$	3.86644	0.380972	0.190486	+	0.981690i	$0.438994\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 12.7821i	− 1.23570i	−0.786298	−	0.617848i	$-0.788005\pi$
$107$	− 12.7821i	− 1.23570i	0.786298	−	0.617848i	$-0.211995\pi$
$108$	0	0
$109$	1.85065i	0.177260i	0.996065	+	0.0886299i	$0.0282489\pi$
$109$	1.85065i	0.177260i	−0.996065	+	0.0886299i	$0.971751\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−16.6815	−1.56926	−0.784632	−	0.619962i	$-0.787148\pi$
$113$	−16.6815	−1.56926	−0.784632	+	0.619962i	$0.787148\pi$
$114$	0	0
$115$	− 14.4681i	− 1.34916i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2.28291	0.209274
$120$	0	0
$121$	−1.00000	−0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 30.7248i	− 2.74811i
$126$	0	0
$127$	6.51013	0.577680	0.288840	−	0.957377i	$-0.406730\pi$
$127$	6.51013	0.577680	0.288840	+	0.957377i	$0.406730\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 6.91569i	− 0.604227i	−0.953272	−	0.302114i	$-0.902308\pi$
$131$	− 6.91569i	− 0.604227i	0.953272	−	0.302114i	$-0.0976922\pi$
$132$	0	0
$133$	− 2.50368i	− 0.217097i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−5.81952	−0.497195	−0.248598	−	0.968607i	$-0.579970\pi$
$137$	−5.81952	−0.497195	−0.248598	+	0.968607i	$0.579970\pi$
$138$	0	0
$139$	− 9.53529i	− 0.808772i	−0.914588	−	0.404386i	$-0.867485\pi$
$139$	− 9.53529i	− 0.808772i	0.914588	−	0.404386i	$-0.132515\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	2.93322	0.245288
$144$	0	0
$145$	−19.0577	−1.58265
$146$	0	0
$147$	0	0
$148$	0	0
$149$	13.9221i	1.14055i	0.821455	+	0.570273i	$0.193162\pi$
$149$	13.9221i	1.14055i	−0.821455	+	0.570273i	$0.806838\pi$
$150$	0	0
$151$	−15.2226	−1.23880	−0.619398	−	0.785077i	$-0.712623\pi$
$151$	−15.2226	−1.23880	−0.619398	+	0.785077i	$0.712623\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	15.2455i	1.22455i
$156$	0	0
$157$	− 0.873940i	− 0.0697480i	−0.999392	−	0.0348740i	$-0.988897\pi$
$157$	− 0.873940i	− 0.0697480i	0.999392	−	0.0348740i	$-0.0111030\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	3.23948	0.255307
$162$	0	0
$163$	− 7.52686i	− 0.589549i	−0.955567	−	0.294775i	$-0.904755\pi$
$163$	− 7.52686i	− 0.589549i	0.955567	−	0.294775i	$-0.0952446\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0.142904	0.0110583	0.00552913	−	0.999985i	$-0.498240\pi$
$167$	0.142904	0.0110583	0.00552913	+	0.999985i	$0.498240\pi$
$168$	0	0
$169$	4.39621	0.338170
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1.12910i	0.0858436i	0.999078	+	0.0429218i	$0.0136666\pi$
$173$	1.12910i	0.0858436i	−0.999078	+	0.0429218i	$0.986333\pi$
$174$	0	0
$175$	11.5456	0.872762
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 15.1620i	− 1.13326i	−0.823972	−	0.566631i	$-0.808246\pi$
$179$	− 15.1620i	− 1.13326i	0.823972	−	0.566631i	$-0.191754\pi$
$180$	0	0
$181$	− 16.9211i	− 1.25773i	−0.777513	−	0.628867i	$-0.783519\pi$
$181$	− 16.9211i	− 1.25773i	0.777513	−	0.628867i	$-0.216481\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	18.9144	1.39061
$186$	0	0
$187$	2.44626i	0.178888i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−16.6720	−1.20635	−0.603174	−	0.797610i	$-0.706097\pi$
$191$	−16.6720	−1.20635	−0.603174	+	0.797610i	$0.706097\pi$
$192$	0	0
$193$	0.180700	0.0130071	0.00650354	−	0.999979i	$-0.497930\pi$
$193$	0.180700	0.0130071	0.00650354	+	0.999979i	$0.497930\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	8.38885i	0.597681i	0.954303	+	0.298840i	$0.0965998\pi$
$197$	8.38885i	0.597681i	−0.954303	+	0.298840i	$0.903400\pi$
$198$	0	0
$199$	−22.0996	−1.56660	−0.783298	−	0.621646i	$-0.786464\pi$
$199$	−22.0996	−1.56660	−0.783298	+	0.621646i	$0.786464\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 4.26711i	− 0.299492i
$204$	0	0
$205$	− 17.1977i	− 1.20114i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2.68283	0.185575
$210$	0	0
$211$	15.0390i	1.03533i	0.855585	+	0.517663i	$0.173198\pi$
$211$	15.0390i	1.03533i	−0.855585	+	0.517663i	$0.826802\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	47.7853	3.25893
$216$	0	0
$217$	−3.41355	−0.231727
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 7.17543i	− 0.482672i
$222$	0	0
$223$	10.2766	0.688172	0.344086	−	0.938938i	$-0.388189\pi$
$223$	10.2766	0.688172	0.344086	+	0.938938i	$0.388189\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	15.6688i	1.03998i	0.854173	+	0.519989i	$0.174064\pi$
$227$	15.6688i	1.03998i	−0.854173	+	0.519989i	$0.825936\pi$
$228$	0	0
$229$	14.0747i	0.930083i	0.885289	+	0.465041i	$0.153961\pi$
$229$	14.0747i	0.930083i	−0.885289	+	0.465041i	$0.846039\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	12.0993	0.792652	0.396326	−	0.918110i	$-0.370285\pi$
$233$	12.0993	0.792652	0.396326	+	0.918110i	$0.370285\pi$
$234$	0	0
$235$	13.5985i	0.887070i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−21.7479	−1.40675	−0.703377	−	0.710817i	$-0.748325\pi$
$239$	−21.7479	−1.40675	−0.703377	+	0.710817i	$0.748325\pi$
$240$	0	0
$241$	−6.93967	−0.447024	−0.223512	−	0.974701i	$-0.571752\pi$
$241$	−6.93967	−0.447024	−0.223512	+	0.974701i	$0.571752\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 25.5457i	− 1.63205i
$246$	0	0
$247$	−7.86935	−0.500714
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 17.7279i	− 1.11897i	−0.828840	−	0.559486i	$-0.810999\pi$
$251$	− 17.7279i	− 1.11897i	0.828840	−	0.559486i	$-0.189001\pi$
$252$	0	0
$253$	3.47129i	0.218238i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−17.7545	−1.10750	−0.553748	−	0.832684i	$-0.686803\pi$
$257$	−17.7545	−1.10750	−0.553748	+	0.832684i	$0.686803\pi$
$258$	0	0
$259$	4.23502i	0.263152i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−27.6424	−1.70450	−0.852252	−	0.523132i	$-0.824763\pi$
$263$	−27.6424	−1.70450	−0.852252	+	0.523132i	$0.824763\pi$
$264$	0	0
$265$	2.70971	0.166456
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 10.6283i	− 0.648020i	−0.946054	−	0.324010i	$-0.894969\pi$
$269$	− 10.6283i	− 0.648020i	0.946054	−	0.324010i	$-0.105031\pi$
$270$	0	0
$271$	−19.7481	−1.19961	−0.599805	−	0.800146i	$-0.704755\pi$
$271$	−19.7481	−1.19961	−0.599805	+	0.800146i	$0.704755\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	12.3717i	0.746042i
$276$	0	0
$277$	12.5353i	0.753172i	0.926382	+	0.376586i	$0.122902\pi$
$277$	12.5353i	0.753172i	−0.926382	+	0.376586i	$0.877098\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	23.8037	1.42001	0.710006	−	0.704195i	$-0.248692\pi$
$281$	23.8037	1.42001	0.710006	+	0.704195i	$0.248692\pi$
$282$	0	0
$283$	27.7294i	1.64834i	0.566340	+	0.824172i	$0.308359\pi$
$283$	27.7294i	1.64834i	−0.566340	+	0.824172i	$0.691641\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3.85065	0.227297
$288$	0	0
$289$	−11.0158	−0.647988
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 5.61389i	− 0.327967i	−0.986463	−	0.163984i	$-0.947566\pi$
$293$	− 5.61389i	− 0.327967i	0.986463	−	0.163984i	$-0.0524344\pi$
$294$	0	0
$295$	−12.1272	−0.706075
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 10.1821i	− 0.588843i
$300$	0	0
$301$	10.6994i	0.616701i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−44.6220	−2.55505
$306$	0	0
$307$	27.8554i	1.58979i	0.606747	+	0.794895i	$0.292474\pi$
$307$	27.8554i	1.58979i	−0.606747	+	0.794895i	$0.707526\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−24.4075	−1.38402	−0.692012	−	0.721886i	$-0.743276\pi$
$311$	−24.4075	−1.38402	−0.692012	+	0.721886i	$0.743276\pi$
$312$	0	0
$313$	9.13514	0.516349	0.258174	−	0.966098i	$-0.416879\pi$
$313$	9.13514	0.516349	0.258174	+	0.966098i	$0.416879\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	17.9501i	1.00818i	0.863652	+	0.504088i	$0.168171\pi$
$317$	17.9501i	1.00818i	−0.863652	+	0.504088i	$0.831829\pi$
$318$	0	0
$319$	4.57245	0.256008
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 6.56291i	− 0.365170i
$324$	0	0
$325$	− 36.2890i	− 2.01295i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−3.04478	−0.167864
$330$	0	0
$331$	13.1934i	0.725173i	0.931950	+	0.362586i	$0.118106\pi$
$331$	13.1934i	0.725173i	−0.931950	+	0.362586i	$0.881894\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−22.6162	−1.23565
$336$	0	0
$337$	23.8712	1.30035	0.650173	−	0.759787i	$-0.274697\pi$
$337$	23.8712	1.30035	0.650173	+	0.759787i	$0.274697\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 3.65781i	− 0.198082i
$342$	0	0
$343$	12.2524	0.661566
$344$	0	0
$345$	0	0
$346$	0	0
$347$	37.2210i	1.99813i	0.0432373	+	0.999065i	$0.486233\pi$
$347$	37.2210i	1.99813i	−0.0432373	+	0.999065i	$0.513767\pi$
$348$	0	0
$349$	28.4140i	1.52097i	0.649358	+	0.760483i	$0.275038\pi$
$349$	28.4140i	1.52097i	−0.649358	+	0.760483i	$0.724962\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	6.72158	0.357754	0.178877	−	0.983871i	$-0.442754\pi$
$353$	6.72158	0.357754	0.178877	+	0.983871i	$0.442754\pi$
$354$	0	0
$355$	− 32.3069i	− 1.71467i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	19.8260	1.04638	0.523188	−	0.852217i	$-0.324743\pi$
$359$	19.8260	1.04638	0.523188	+	0.852217i	$0.324743\pi$
$360$	0	0
$361$	11.8024	0.621179
$362$	0	0
$363$	0	0
$364$	0	0
$365$	54.4543i	2.85027i
$366$	0	0
$367$	−4.54822	−0.237415	−0.118708	−	0.992929i	$-0.537875\pi$
$367$	−4.54822	−0.237415	−0.118708	+	0.992929i	$0.537875\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0.606718i	0.0314992i
$372$	0	0
$373$	33.3992i	1.72934i	0.502336	+	0.864672i	$0.332474\pi$
$373$	33.3992i	1.72934i	−0.502336	+	0.864672i	$0.667526\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−13.4120	−0.690753
$378$	0	0
$379$	− 9.24843i	− 0.475060i	−0.971380	−	0.237530i	$-0.923662\pi$
$379$	− 9.24843i	− 0.475060i	0.971380	−	0.237530i	$-0.0763378\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	31.6080	1.61509	0.807547	−	0.589803i	$-0.200795\pi$
$383$	31.6080	1.61509	0.807547	+	0.589803i	$0.200795\pi$
$384$	0	0
$385$	−3.88961	−0.198233
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 4.99543i	− 0.253278i	−0.991949	−	0.126639i	$-0.959581\pi$
$389$	− 4.99543i	− 0.253278i	0.991949	−	0.126639i	$-0.0404190\pi$
$390$	0	0
$391$	8.49168	0.429443
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 20.1589i	− 1.01430i
$396$	0	0
$397$	17.7950i	0.893107i	0.894757	+	0.446553i	$0.147349\pi$
$397$	17.7950i	0.893107i	−0.894757	+	0.446553i	$0.852651\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	12.1739	0.607934	0.303967	−	0.952683i	$-0.401689\pi$
$401$	12.1739	0.607934	0.303967	+	0.952683i	$0.401689\pi$
$402$	0	0
$403$	10.7292i	0.534458i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−4.53806	−0.224943
$408$	0	0
$409$	0.587997	0.0290746	0.0145373	−	0.999894i	$-0.495372\pi$
$409$	0.587997	0.0290746	0.0145373	+	0.999894i	$0.495372\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 2.71535i	− 0.133614i
$414$	0	0
$415$	−2.74944	−0.134965
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 11.1507i	− 0.544748i	−0.962191	−	0.272374i	$-0.912191\pi$
$419$	− 11.1507i	− 0.544748i	0.962191	−	0.272374i	$-0.0878088\pi$
$420$	0	0
$421$	− 34.8398i	− 1.69799i	−0.528401	−	0.848995i	$-0.677208\pi$
$421$	− 34.8398i	− 1.69799i	0.528401	−	0.848995i	$-0.322792\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	30.2645	1.46804
$426$	0	0
$427$	− 9.99108i	− 0.483502i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−24.8647	−1.19769	−0.598846	−	0.800864i	$-0.704374\pi$
$431$	−24.8647	−1.19769	−0.598846	+	0.800864i	$0.704374\pi$
$432$	0	0
$433$	−0.547704	−0.0263210	−0.0131605	−	0.999913i	$-0.504189\pi$
$433$	−0.547704	−0.0263210	−0.0131605	+	0.999913i	$0.504189\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 9.31288i	− 0.445495i
$438$	0	0
$439$	−27.5308	−1.31397	−0.656987	−	0.753902i	$-0.728169\pi$
$439$	−27.5308	−1.31397	−0.656987	+	0.753902i	$0.728169\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 8.05454i	− 0.382683i	−0.981524	−	0.191341i	$-0.938716\pi$
$443$	− 8.05454i	− 0.382683i	0.981524	−	0.191341i	$-0.0612838\pi$
$444$	0	0
$445$	11.4464i	0.542611i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	33.7705	1.59373	0.796865	−	0.604158i	$-0.206490\pi$
$449$	33.7705	1.59373	0.796865	+	0.604158i	$0.206490\pi$
$450$	0	0
$451$	4.12618i	0.194294i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	11.4091	0.534867
$456$	0	0
$457$	−3.83220	−0.179263	−0.0896313	−	0.995975i	$-0.528569\pi$
$457$	−3.83220	−0.179263	−0.0896313	+	0.995975i	$0.528569\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	15.8380i	0.737649i	0.929499	+	0.368825i	$0.120240\pi$
$461$	15.8380i	0.737649i	−0.929499	+	0.368825i	$0.879760\pi$
$462$	0	0
$463$	−30.4131	−1.41341	−0.706707	−	0.707506i	$-0.749820\pi$
$463$	−30.4131	−1.41341	−0.706707	+	0.707506i	$0.749820\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	20.1815i	0.933887i	0.884287	+	0.466943i	$0.154645\pi$
$467$	20.1815i	0.933887i	−0.884287	+	0.466943i	$0.845355\pi$
$468$	0	0
$469$	− 5.06388i	− 0.233828i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−11.4650	−0.527160
$474$	0	0
$475$	− 33.1912i	− 1.52292i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	12.5604	0.573901	0.286951	−	0.957945i	$-0.407358\pi$
$479$	12.5604	0.573901	0.286951	+	0.957945i	$0.407358\pi$
$480$	0	0
$481$	13.3111	0.606936
$482$	0	0
$483$	0	0
$484$	0	0
$485$	7.62338i	0.346160i
$486$	0	0
$487$	−14.2519	−0.645814	−0.322907	−	0.946431i	$-0.604660\pi$
$487$	−14.2519	−0.645814	−0.322907	+	0.946431i	$0.604660\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 20.4046i	− 0.920849i	−0.887699	−	0.460424i	$-0.847697\pi$
$491$	− 20.4046i	− 0.920849i	0.887699	−	0.460424i	$-0.152303\pi$
$492$	0	0
$493$	− 11.1854i	− 0.503765i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	7.23368	0.324475
$498$	0	0
$499$	− 33.3751i	− 1.49407i	−0.664782	−	0.747037i	$-0.731476\pi$
$499$	− 33.3751i	− 1.49407i	0.664782	−	0.747037i	$-0.268524\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	14.8148	0.660561	0.330281	−	0.943883i	$-0.392857\pi$
$503$	14.8148	0.660561	0.330281	+	0.943883i	$0.392857\pi$
$504$	0	0
$505$	52.8541	2.35197
$506$	0	0
$507$	0	0
$508$	0	0
$509$	23.9464i	1.06140i	0.847559	+	0.530702i	$0.178072\pi$
$509$	23.9464i	1.06140i	−0.847559	+	0.530702i	$0.821928\pi$
$510$	0	0
$511$	−12.1926	−0.539369
$512$	0	0
$513$	0	0
$514$	0	0
$515$	16.1151i	0.710116i
$516$	0	0
$517$	− 3.26265i	− 0.143491i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−20.6157	−0.903189	−0.451594	−	0.892223i	$-0.649145\pi$
$521$	−20.6157	−0.903189	−0.451594	+	0.892223i	$0.649145\pi$
$522$	0	0
$523$	22.6014i	0.988292i	0.869379	+	0.494146i	$0.164519\pi$
$523$	22.6014i	0.988292i	−0.869379	+	0.494146i	$0.835481\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−8.94797	−0.389779
$528$	0	0
$529$	−10.9502	−0.476095
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 12.1030i	− 0.524240i
$534$	0	0
$535$	53.2752	2.30329
$536$	0	0
$537$	0	0
$538$	0	0
$539$	6.12910i	0.263999i
$540$	0	0
$541$	5.25411i	0.225892i	0.993601	+	0.112946i	$0.0360287\pi$
$541$	5.25411i	0.225892i	−0.993601	+	0.112946i	$0.963971\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−7.71338	−0.330405
$546$	0	0
$547$	− 31.4365i	− 1.34413i	−0.740493	−	0.672064i	$-0.765408\pi$
$547$	− 31.4365i	− 1.34413i	0.740493	−	0.672064i	$-0.234592\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−12.2671	−0.522596
$552$	0	0
$553$	4.51368	0.191941
$554$	0	0
$555$	0	0
$556$	0	0
$557$	13.8905i	0.588561i	0.955719	+	0.294280i	$0.0950800\pi$
$557$	13.8905i	0.588561i	−0.955719	+	0.294280i	$0.904920\pi$
$558$	0	0
$559$	33.6293	1.42237
$560$	0	0
$561$	0	0
$562$	0	0
$563$	20.5160i	0.864647i	0.901719	+	0.432324i	$0.142306\pi$
$563$	20.5160i	0.864647i	−0.901719	+	0.432324i	$0.857694\pi$
$564$	0	0
$565$	− 69.5275i	− 2.92504i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	20.9308	0.877463	0.438731	−	0.898618i	$-0.355428\pi$
$569$	20.9308	0.877463	0.438731	+	0.898618i	$0.355428\pi$
$570$	0	0
$571$	− 6.00582i	− 0.251336i	−0.992072	−	0.125668i	$-0.959893\pi$
$571$	− 6.00582i	− 0.251336i	0.992072	−	0.125668i	$-0.0401074\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	42.9457	1.79096
$576$	0	0
$577$	−45.4212	−1.89091	−0.945455	−	0.325753i	$-0.894382\pi$
$577$	−45.4212	−1.89091	−0.945455	+	0.325753i	$0.894382\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 0.615613i	− 0.0255399i
$582$	0	0
$583$	−0.650132	−0.0269257
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 4.95230i	− 0.204403i	−0.994764	−	0.102202i	$-0.967411\pi$
$587$	− 4.95230i	− 0.204403i	0.994764	−	0.102202i	$-0.0325887\pi$
$588$	0	0
$589$	9.81329i	0.404350i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	17.6332	0.724110	0.362055	−	0.932157i	$-0.382075\pi$
$593$	17.6332	0.724110	0.362055	+	0.932157i	$0.382075\pi$
$594$	0	0
$595$	9.51502i	0.390078i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	45.1533	1.84491	0.922456	−	0.386101i	$-0.126178\pi$
$599$	45.1533	1.84491	0.922456	+	0.386101i	$0.126178\pi$
$600$	0	0
$601$	−15.6375	−0.637867	−0.318933	−	0.947777i	$-0.603325\pi$
$601$	−15.6375	−0.637867	−0.318933	+	0.947777i	$0.603325\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 4.16794i	− 0.169451i
$606$	0	0
$607$	41.9318	1.70196	0.850979	−	0.525200i	$-0.176010\pi$
$607$	41.9318	1.70196	0.850979	+	0.525200i	$0.176010\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	9.57008i	0.387164i
$612$	0	0
$613$	27.8914i	1.12652i	0.826278	+	0.563262i	$0.190454\pi$
$613$	27.8914i	1.12652i	−0.826278	+	0.563262i	$0.809546\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	14.7471	0.593697	0.296849	−	0.954925i	$-0.404064\pi$
$617$	14.7471	0.593697	0.296849	+	0.954925i	$0.404064\pi$
$618$	0	0
$619$	5.12150i	0.205851i	0.994689	+	0.102925i	$0.0328203\pi$
$619$	5.12150i	0.205851i	−0.994689	+	0.102925i	$0.967180\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−2.56290	−0.102681
$624$	0	0
$625$	66.2007	2.64803
$626$	0	0
$627$	0	0
$628$	0	0
$629$	11.1013i	0.442638i
$630$	0	0
$631$	48.5097	1.93114	0.965570	−	0.260142i	$-0.0837693\pi$
$631$	48.5097	1.93114	0.965570	+	0.260142i	$0.0837693\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	27.1338i	1.07677i
$636$	0	0
$637$	− 17.9780i	− 0.712314i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	7.22155	0.285234	0.142617	−	0.989778i	$-0.454448\pi$
$641$	7.22155	0.285234	0.142617	+	0.989778i	$0.454448\pi$
$642$	0	0
$643$	11.9607i	0.471684i	0.971791	+	0.235842i	$0.0757847\pi$
$643$	11.9607i	0.471684i	−0.971791	+	0.235842i	$0.924215\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−26.1952	−1.02984	−0.514920	−	0.857238i	$-0.672178\pi$
$647$	−26.1952	−1.02984	−0.514920	+	0.857238i	$0.672178\pi$
$648$	0	0
$649$	2.90965	0.114214
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 34.2180i	− 1.33906i	−0.742787	−	0.669528i	$-0.766497\pi$
$653$	− 34.2180i	− 1.33906i	0.742787	−	0.669528i	$-0.233503\pi$
$654$	0	0
$655$	28.8242	1.12625
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 2.73052i	− 0.106366i	−0.998585	−	0.0531831i	$-0.983063\pi$
$659$	− 2.73052i	− 0.106366i	0.998585	−	0.0531831i	$-0.0169367\pi$
$660$	0	0
$661$	− 11.2478i	− 0.437488i	−0.975782	−	0.218744i	$-0.929804\pi$
$661$	− 11.2478i	− 0.437488i	0.975782	−	0.218744i	$-0.0701959\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	10.4352	0.404659
$666$	0	0
$667$	− 15.8723i	− 0.614577i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	10.7060	0.413301
$672$	0	0
$673$	−20.0432	−0.772609	−0.386305	−	0.922371i	$-0.626249\pi$
$673$	−20.0432	−0.772609	−0.386305	+	0.922371i	$0.626249\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 27.1414i	− 1.04313i	−0.853212	−	0.521564i	$-0.825349\pi$
$677$	− 27.1414i	− 1.04313i	0.853212	−	0.521564i	$-0.174651\pi$
$678$	0	0
$679$	−1.70691	−0.0655053
$680$	0	0
$681$	0	0
$682$	0	0
$683$	41.8019i	1.59951i	0.600330	+	0.799753i	$0.295036\pi$
$683$	41.8019i	1.59951i	−0.600330	+	0.799753i	$0.704964\pi$
$684$	0	0
$685$	− 24.2554i	− 0.926751i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1.90698	0.0726502
$690$	0	0
$691$	28.7753i	1.09466i	0.836916	+	0.547332i	$0.184356\pi$
$691$	28.7753i	1.09466i	−0.836916	+	0.547332i	$0.815644\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	39.7425	1.50752
$696$	0	0
$697$	10.0937	0.382327
$698$	0	0
$699$	0	0
$700$	0	0
$701$	22.5826i	0.852933i	0.904503	+	0.426467i	$0.140242\pi$
$701$	22.5826i	0.852933i	−0.904503	+	0.426467i	$0.859758\pi$
$702$	0	0
$703$	12.1749	0.459184
$704$	0	0
$705$	0	0
$706$	0	0
$707$	11.8343i	0.445074i
$708$	0	0
$709$	14.5420i	0.546138i	0.961994	+	0.273069i	$0.0880387\pi$
$709$	14.5420i	0.546138i	−0.961994	+	0.273069i	$0.911961\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−12.6973	−0.475518
$714$	0	0
$715$	12.2255i	0.457207i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−5.98413	−0.223170	−0.111585	−	0.993755i	$-0.535593\pi$
$719$	−5.98413	−0.223170	−0.111585	+	0.993755i	$0.535593\pi$
$720$	0	0
$721$	−3.60825	−0.134378
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 56.5690i	− 2.10092i
$726$	0	0
$727$	13.7234	0.508973	0.254486	−	0.967076i	$-0.418094\pi$
$727$	13.7234	0.508973	0.254486	+	0.967076i	$0.418094\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	28.0463i	1.03733i
$732$	0	0
$733$	46.4631i	1.71615i	0.513522	+	0.858076i	$0.328340\pi$
$733$	46.4631i	1.71615i	−0.513522	+	0.858076i	$0.671660\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	5.42623	0.199878
$738$	0	0
$739$	12.5779i	0.462686i	0.972872	+	0.231343i	$0.0743119\pi$
$739$	12.5779i	0.462686i	−0.972872	+	0.231343i	$0.925688\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−9.36221	−0.343466	−0.171733	−	0.985144i	$-0.554937\pi$
$743$	−9.36221	−0.343466	−0.171733	+	0.985144i	$0.554937\pi$
$744$	0	0
$745$	−58.0266	−2.12593
$746$	0	0
$747$	0	0
$748$	0	0
$749$	11.9286i	0.435861i
$750$	0	0
$751$	−3.96209	−0.144579	−0.0722894	−	0.997384i	$-0.523031\pi$
$751$	−3.96209	−0.144579	−0.0722894	+	0.997384i	$0.523031\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 63.4468i	− 2.30906i
$756$	0	0
$757$	− 3.56879i	− 0.129710i	−0.997895	−	0.0648550i	$-0.979342\pi$
$757$	− 3.56879i	− 0.129710i	0.997895	−	0.0648550i	$-0.0206585\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−49.8018	−1.80532	−0.902658	−	0.430359i	$-0.858387\pi$
$761$	−49.8018	−1.80532	−0.902658	+	0.430359i	$0.858387\pi$
$762$	0	0
$763$	− 1.72707i	− 0.0625240i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−8.53465	−0.308168
$768$	0	0
$769$	30.5772	1.10264	0.551321	−	0.834293i	$-0.314124\pi$
$769$	30.5772	1.10264	0.551321	+	0.834293i	$0.314124\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	23.9315i	0.860756i	0.902649	+	0.430378i	$0.141620\pi$
$773$	23.9315i	0.860756i	−0.902649	+	0.430378i	$0.858380\pi$
$774$	0	0
$775$	−45.2534	−1.62555
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 11.0699i	− 0.396619i
$780$	0	0
$781$	7.75129i	0.277363i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	3.64253	0.130007
$786$	0	0
$787$	− 13.4992i	− 0.481196i	−0.970625	−	0.240598i	$-0.922656\pi$
$787$	− 13.4992i	− 0.481196i	0.970625	−	0.240598i	$-0.0773436\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	15.5676	0.553519
$792$	0	0
$793$	−31.4031	−1.11516
$794$	0	0
$795$	0	0
$796$	0	0
$797$	23.8879i	0.846154i	0.906094	+	0.423077i	$0.139050\pi$
$797$	23.8879i	0.846154i	−0.906094	+	0.423077i	$0.860950\pi$
$798$	0	0
$799$	−7.98130	−0.282358
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 13.0650i	− 0.461056i
$804$	0	0
$805$	13.5020i	0.475882i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−20.1858	−0.709694	−0.354847	−	0.934924i	$-0.615467\pi$
$809$	−20.1858	−0.709694	−0.354847	+	0.934924i	$0.615467\pi$
$810$	0	0
$811$	3.78003i	0.132735i	0.997795	+	0.0663675i	$0.0211410\pi$
$811$	3.78003i	0.132735i	−0.997795	+	0.0663675i	$0.978859\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	31.3715	1.09890
$816$	0	0
$817$	30.7586	1.07611
$818$	0	0
$819$	0	0
$820$	0	0
$821$	32.5119i	1.13467i	0.823486	+	0.567336i	$0.192026\pi$
$821$	32.5119i	1.13467i	−0.823486	+	0.567336i	$0.807974\pi$
$822$	0	0
$823$	−20.7651	−0.723826	−0.361913	−	0.932212i	$-0.617876\pi$
$823$	−20.7651	−0.723826	−0.361913	+	0.932212i	$0.617876\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 11.2140i	− 0.389948i	−0.980808	−	0.194974i	$-0.937538\pi$
$827$	− 11.2140i	− 0.389948i	0.980808	−	0.194974i	$-0.0624622\pi$
$828$	0	0
$829$	− 6.82205i	− 0.236940i	−0.992958	−	0.118470i	$-0.962201\pi$
$829$	− 6.82205i	− 0.236940i	0.992958	−	0.118470i	$-0.0377989\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	14.9934	0.519490
$834$	0	0
$835$	0.595616i	0.0206121i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−19.3720	−0.668796	−0.334398	−	0.942432i	$-0.608533\pi$
$839$	−19.3720	−0.668796	−0.334398	+	0.942432i	$0.608533\pi$
$840$	0	0
$841$	8.09273	0.279060
$842$	0	0
$843$	0	0
$844$	0	0
$845$	18.3231i	0.630334i
$846$	0	0
$847$	0.933222	0.0320659
$848$	0	0
$849$	0	0
$850$	0	0
$851$	15.7529i	0.540003i
$852$	0	0
$853$	− 11.1509i	− 0.381800i	−0.981609	−	0.190900i	$-0.938859\pi$
$853$	− 11.1509i	− 0.381800i	0.981609	−	0.190900i	$-0.0611406\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	30.4078	1.03871	0.519355	−	0.854558i	$-0.326172\pi$
$857$	30.4078	1.03871	0.519355	+	0.854558i	$0.326172\pi$
$858$	0	0
$859$	27.7908i	0.948210i	0.880468	+	0.474105i	$0.157228\pi$
$859$	27.7908i	0.948210i	−0.880468	+	0.474105i	$0.842772\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0.221827	0.00755109	0.00377554	−	0.999993i	$-0.498798\pi$
$863$	0.221827	0.00755109	0.00377554	+	0.999993i	$0.498798\pi$
$864$	0	0
$865$	−4.70600	−0.160009
$866$	0	0
$867$	0	0
$868$	0	0
$869$	4.83666i	0.164072i
$870$	0	0
$871$	−15.9163	−0.539304
$872$	0	0
$873$	0	0
$874$	0	0
$875$	28.6731i	0.969328i
$876$	0	0
$877$	15.5470i	0.524985i	0.964934	+	0.262492i	$0.0845444\pi$
$877$	15.5470i	0.524985i	−0.964934	+	0.262492i	$0.915456\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−21.4338	−0.722125	−0.361062	−	0.932542i	$-0.617586\pi$
$881$	−21.4338	−0.722125	−0.361062	+	0.932542i	$0.617586\pi$
$882$	0	0
$883$	14.2700i	0.480224i	0.970745	+	0.240112i	$0.0771842\pi$
$883$	14.2700i	0.480224i	−0.970745	+	0.240112i	$0.922816\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	23.6109	0.792775	0.396388	−	0.918083i	$-0.370264\pi$
$887$	23.6109	0.792775	0.396388	+	0.918083i	$0.370264\pi$
$888$	0	0
$889$	−6.07540	−0.203762
$890$	0	0
$891$	0	0
$892$	0	0
$893$	8.75315i	0.292913i
$894$	0	0
$895$	63.1944	2.11235
$896$	0	0
$897$	0	0
$898$	0	0
$899$	16.7251i	0.557815i
$900$	0	0
$901$	1.59039i	0.0529837i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	70.5260	2.34436
$906$	0	0
$907$	45.2882i	1.50377i	0.659294	+	0.751885i	$0.270855\pi$
$907$	45.2882i	1.50377i	−0.659294	+	0.751885i	$0.729145\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−22.4539	−0.743931	−0.371966	−	0.928247i	$-0.621316\pi$
$911$	−22.4539	−0.743931	−0.371966	+	0.928247i	$0.621316\pi$
$912$	0	0
$913$	0.659664	0.0218317
$914$	0	0
$915$	0	0
$916$	0	0
$917$	6.45388i	0.213126i
$918$	0	0
$919$	−41.5329	−1.37004	−0.685022	−	0.728522i	$-0.740207\pi$
$919$	−41.5329	−1.37004	−0.685022	+	0.728522i	$0.740207\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 22.7363i	− 0.748373i
$924$	0	0
$925$	56.1436i	1.84599i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	37.2997	1.22376	0.611882	−	0.790949i	$-0.290413\pi$
$929$	37.2997	1.22376	0.611882	+	0.790949i	$0.290413\pi$
$930$	0	0
$931$	− 16.4433i	− 0.538909i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−10.1959	−0.333441
$936$	0	0
$937$	13.1897	0.430888	0.215444	−	0.976516i	$-0.430880\pi$
$937$	13.1897	0.430888	0.215444	+	0.976516i	$0.430880\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 24.6716i	− 0.804270i	−0.915580	−	0.402135i	$-0.868268\pi$
$941$	− 24.6716i	− 0.804270i	0.915580	−	0.402135i	$-0.131732\pi$
$942$	0	0
$943$	14.3232	0.466426
$944$	0	0
$945$	0	0
$946$	0	0
$947$	38.8667i	1.26300i	0.775376	+	0.631500i	$0.217560\pi$
$947$	38.8667i	1.26300i	−0.775376	+	0.631500i	$0.782440\pi$
$948$	0	0
$949$	38.3227i	1.24401i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−52.5807	−1.70326	−0.851628	−	0.524147i	$-0.824384\pi$
$953$	−52.5807	−1.70326	−0.851628	+	0.524147i	$0.824384\pi$
$954$	0	0
$955$	− 69.4881i	− 2.24858i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	5.43091	0.175373
$960$	0	0
$961$	−17.6204	−0.568401
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0.753147i	0.0242447i
$966$	0	0
$967$	2.98370	0.0959493	0.0479746	−	0.998849i	$-0.484723\pi$
$967$	2.98370	0.0959493	0.0479746	+	0.998849i	$0.484723\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	15.0443i	0.482793i	0.970427	+	0.241397i	$0.0776055\pi$
$971$	15.0443i	0.482793i	−0.970427	+	0.241397i	$0.922395\pi$
$972$	0	0
$973$	8.89854i	0.285274i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	11.4313	0.365720	0.182860	−	0.983139i	$-0.441464\pi$
$977$	11.4313	0.365720	0.182860	+	0.983139i	$0.441464\pi$
$978$	0	0
$979$	− 2.74629i	− 0.0877720i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	58.1032	1.85321	0.926603	−	0.376042i	$-0.122715\pi$
$983$	58.1032	1.85321	0.926603	+	0.376042i	$0.122715\pi$
$984$	0	0
$985$	−34.9642	−1.11405
$986$	0	0
$987$	0	0
$988$	0	0
$989$	39.7982i	1.26551i
$990$	0	0
$991$	−30.2307	−0.960311	−0.480156	−	0.877183i	$-0.659420\pi$
$991$	−30.2307	−0.960311	−0.480156	+	0.877183i	$0.659420\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 92.1096i	− 2.92007i
$996$	0	0
$997$	13.3996i	0.424368i	0.977230	+	0.212184i	$0.0680577\pi$
$997$	13.3996i	0.424368i	−0.977230	+	0.212184i	$0.931942\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3168.2.f.g.1585.10		10
3.2	odd	2		352.2.c.a.177.2		10
4.3	odd	2		792.2.f.g.397.3		10
8.3	odd	2		792.2.f.g.397.4		10
8.5	even	2	inner	3168.2.f.g.1585.1		10
12.11	even	2		88.2.c.a.45.8	yes	10
24.5	odd	2		352.2.c.a.177.9		10
24.11	even	2		88.2.c.a.45.7	✓	10
33.32	even	2		3872.2.c.f.1937.2		10
48.5	odd	4		2816.2.a.q.1.1		5
48.11	even	4		2816.2.a.r.1.5		5
48.29	odd	4		2816.2.a.p.1.5		5
48.35	even	4		2816.2.a.o.1.1		5
132.35	odd	10		968.2.o.h.565.5		40
132.47	even	10		968.2.o.g.493.10		40
132.59	even	10		968.2.o.g.269.4		40
132.71	even	10		968.2.o.g.245.2		40
132.83	odd	10		968.2.o.h.245.9		40
132.95	odd	10		968.2.o.h.269.7		40
132.107	odd	10		968.2.o.h.493.1		40
132.119	even	10		968.2.o.g.565.6		40
132.131	odd	2		968.2.c.d.485.3		10
264.35	odd	10		968.2.o.h.565.9		40
264.59	even	10		968.2.o.g.269.10		40
264.83	odd	10		968.2.o.h.245.5		40
264.107	odd	10		968.2.o.h.493.7		40
264.131	odd	2		968.2.c.d.485.4		10
264.179	even	10		968.2.o.g.493.4		40
264.197	even	2		3872.2.c.f.1937.9		10
264.203	even	10		968.2.o.g.245.6		40
264.227	odd	10		968.2.o.h.269.1		40
264.251	even	10		968.2.o.g.565.2		40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
88.2.c.a.45.7	✓	10	24.11	even	2
88.2.c.a.45.8	yes	10	12.11	even	2
352.2.c.a.177.2		10	3.2	odd	2
352.2.c.a.177.9		10	24.5	odd	2
792.2.f.g.397.3		10	4.3	odd	2
792.2.f.g.397.4		10	8.3	odd	2
968.2.c.d.485.3		10	132.131	odd	2
968.2.c.d.485.4		10	264.131	odd	2
968.2.o.g.245.2		40	132.71	even	10
968.2.o.g.245.6		40	264.203	even	10
968.2.o.g.269.4		40	132.59	even	10
968.2.o.g.269.10		40	264.59	even	10
968.2.o.g.493.4		40	264.179	even	10
968.2.o.g.493.10		40	132.47	even	10
968.2.o.g.565.2		40	264.251	even	10
968.2.o.g.565.6		40	132.119	even	10
968.2.o.h.245.5		40	264.83	odd	10
968.2.o.h.245.9		40	132.83	odd	10
968.2.o.h.269.1		40	264.227	odd	10
968.2.o.h.269.7		40	132.95	odd	10
968.2.o.h.493.1		40	132.107	odd	10
968.2.o.h.493.7		40	264.107	odd	10
968.2.o.h.565.5		40	132.35	odd	10
968.2.o.h.565.9		40	264.35	odd	10
2816.2.a.o.1.1		5	48.35	even	4
2816.2.a.p.1.5		5	48.29	odd	4
2816.2.a.q.1.1		5	48.5	odd	4
2816.2.a.r.1.5		5	48.11	even	4
3168.2.f.g.1585.1		10	8.5	even	2	inner
3168.2.f.g.1585.10		10	1.1	even	1	trivial
3872.2.c.f.1937.2		10	33.32	even	2
3872.2.c.f.1937.9		10	264.197	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3168 = 2^{5} \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3168.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.16794i q^{5} -0.933222 q^{7} -1.00000i q^{11} +2.93322i q^{13} -2.44626 q^{17} +2.68283i q^{19} -3.47129 q^{23} -12.3717 q^{25} +4.57245i q^{29} +3.65781 q^{31} -3.88961i q^{35} -4.53806i q^{37} -4.12618 q^{41} -11.4650i q^{43} +3.26265 q^{47} -6.12910 q^{49} -0.650132i q^{53} +4.16794 q^{55} +2.90965i q^{59} +10.7060i q^{61} -12.2255 q^{65} +5.42623i q^{67} -7.75129 q^{71} +13.0650 q^{73} +0.933222i q^{77} -4.83666 q^{79} +0.659664i q^{83} -10.1959i q^{85} +2.74629 q^{89} -2.73735i q^{91} -11.1819 q^{95} +1.82905 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 4 q^{17} - 12 q^{23} - 6 q^{25} + 4 q^{31} - 4 q^{41} - 4 q^{47} - 6 q^{49} + 8 q^{55} - 16 q^{65} - 12 q^{71} - 4 q^{73} - 16 q^{79} + 4 q^{89} + 24 q^{95} - 20 q^{97}+O(q^{100}) \) 10 * q + 4 * q^17 - 12 * q^23 - 6 * q^25 + 4 * q^31 - 4 * q^41 - 4 * q^47 - 6 * q^49 + 8 * q^55 - 16 * q^65 - 12 * q^71 - 4 * q^73 - 16 * q^79 + 4 * q^89 + 24 * q^95 - 20 * q^97

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