LMFDB - Embedded newform 4176.2.a.bt.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4176,2,Mod(1,4176)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4176, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("4176.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4176 = 2^{4} \cdot 3^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4176.a (trivial)

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:	yes
Analytic conductor:	$33.3455278841$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 261)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	4176.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{5} -2.23607 q^{7} +O(q^{10})$	$q+2.00000 q^{5} -2.23607 q^{7} -1.76393 q^{11} -5.47214 q^{13} +5.47214 q^{17} +0.472136 q^{23} -1.00000 q^{25} +1.00000 q^{29} +8.94427 q^{31} -4.47214 q^{35} -4.00000 q^{37} -2.00000 q^{41} -0.472136 q^{43} +8.70820 q^{47} -2.00000 q^{49} -8.00000 q^{53} -3.52786 q^{55} -12.4721 q^{59} +6.94427 q^{61} -10.9443 q^{65} -8.23607 q^{67} -8.94427 q^{71} -2.00000 q^{73} +3.94427 q^{77} -12.4721 q^{79} -16.9443 q^{83} +10.9443 q^{85} -14.4164 q^{89} +12.2361 q^{91} +8.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{5}+O(q^{10}) $ 2 * q + 4 * q^5	$ 2 q + 4 q^{5} - 8 q^{11} - 2 q^{13} + 2 q^{17} - 8 q^{23} - 2 q^{25} + 2 q^{29} - 8 q^{37} - 4 q^{41} + 8 q^{43} + 4 q^{47} - 4 q^{49} - 16 q^{53} - 16 q^{55} - 16 q^{59} - 4 q^{61} - 4 q^{65} - 12 q^{67} - 4 q^{73} - 10 q^{77} - 16 q^{79} - 16 q^{83} + 4 q^{85} - 2 q^{89} + 20 q^{91} + 16 q^{97}+O(q^{100}) $ 2 * q + 4 * q^5 - 8 * q^11 - 2 * q^13 + 2 * q^17 - 8 * q^23 - 2 * q^25 + 2 * q^29 - 8 * q^37 - 4 * q^41 + 8 * q^43 + 4 * q^47 - 4 * q^49 - 16 * q^53 - 16 * q^55 - 16 * q^59 - 4 * q^61 - 4 * q^65 - 12 * q^67 - 4 * q^73 - 10 * q^77 - 16 * q^79 - 16 * q^83 + 4 * q^85 - 2 * q^89 + 20 * q^91 + 16 * q^97

Display 100 coefficients 10 coefficients

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
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	0	0
$7$	−2.23607	−0.845154	−0.422577	−	0.906327i	$-0.638874\pi$
$7$	−2.23607	−0.845154	−0.422577	+	0.906327i	$0.638874\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.76393	−0.531846	−0.265923	−	0.963994i	$-0.585677\pi$
$11$	−1.76393	−0.531846	−0.265923	+	0.963994i	$0.585677\pi$
$12$	0	0
$13$	−5.47214	−1.51770	−0.758849	−	0.651267i	$-0.774238\pi$
$13$	−5.47214	−1.51770	−0.758849	+	0.651267i	$0.774238\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.47214	1.32719	0.663594	−	0.748093i	$-0.269030\pi$
$17$	5.47214	1.32719	0.663594	+	0.748093i	$0.269030\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0.472136	0.0984472	0.0492236	−	0.998788i	$-0.484325\pi$
$23$	0.472136	0.0984472	0.0492236	+	0.998788i	$0.484325\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	8.94427	1.60644	0.803219	−	0.595683i	$-0.203119\pi$
$31$	8.94427	1.60644	0.803219	+	0.595683i	$0.203119\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−4.47214	−0.755929
$36$	0	0
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	−0.472136	−0.0720001	−0.0360000	−	0.999352i	$-0.511462\pi$
$43$	−0.472136	−0.0720001	−0.0360000	+	0.999352i	$0.511462\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.70820	1.27022	0.635111	−	0.772421i	$-0.280954\pi$
$47$	8.70820	1.27022	0.635111	+	0.772421i	$0.280954\pi$
$48$	0	0
$49$	−2.00000	−0.285714
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−8.00000	−1.09888	−0.549442	−	0.835532i	$-0.685160\pi$
$53$	−8.00000	−1.09888	−0.549442	+	0.835532i	$0.685160\pi$
$54$	0	0
$55$	−3.52786	−0.475697
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−12.4721	−1.62373	−0.811867	−	0.583843i	$-0.801549\pi$
$59$	−12.4721	−1.62373	−0.811867	+	0.583843i	$0.801549\pi$
$60$	0	0
$61$	6.94427	0.889123	0.444561	−	0.895748i	$-0.353360\pi$
$61$	6.94427	0.889123	0.444561	+	0.895748i	$0.353360\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−10.9443	−1.35747
$66$	0	0
$67$	−8.23607	−1.00620	−0.503098	−	0.864229i	$-0.667806\pi$
$67$	−8.23607	−1.00620	−0.503098	+	0.864229i	$0.667806\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−8.94427	−1.06149	−0.530745	−	0.847532i	$-0.678088\pi$
$71$	−8.94427	−1.06149	−0.530745	+	0.847532i	$0.678088\pi$
$72$	0	0
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.94427	0.449492
$78$	0	0
$79$	−12.4721	−1.40322	−0.701612	−	0.712559i	$-0.747536\pi$
$79$	−12.4721	−1.40322	−0.701612	+	0.712559i	$0.747536\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−16.9443	−1.85988	−0.929938	−	0.367717i	$-0.880140\pi$
$83$	−16.9443	−1.85988	−0.929938	+	0.367717i	$0.880140\pi$
$84$	0	0
$85$	10.9443	1.18707
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−14.4164	−1.52814	−0.764068	−	0.645136i	$-0.776801\pi$
$89$	−14.4164	−1.52814	−0.764068	+	0.645136i	$0.776801\pi$
$90$	0	0
$91$	12.2361	1.28269
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	8.00000	0.812277	0.406138	−	0.913812i	$-0.366875\pi$
$97$	8.00000	0.812277	0.406138	+	0.913812i	$0.366875\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	15.9443	1.58651	0.793257	−	0.608887i	$-0.208384\pi$
$101$	15.9443	1.58651	0.793257	+	0.608887i	$0.208384\pi$
$102$	0	0
$103$	−0.944272	−0.0930419	−0.0465209	−	0.998917i	$-0.514813\pi$
$103$	−0.944272	−0.0930419	−0.0465209	+	0.998917i	$0.514813\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.47214	0.432338	0.216169	−	0.976356i	$-0.430644\pi$
$107$	4.47214	0.432338	0.216169	+	0.976356i	$0.430644\pi$
$108$	0	0
$109$	−15.4721	−1.48196	−0.740981	−	0.671526i	$-0.765639\pi$
$109$	−15.4721	−1.48196	−0.740981	+	0.671526i	$0.765639\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−3.47214	−0.326631	−0.163316	−	0.986574i	$-0.552219\pi$
$113$	−3.47214	−0.326631	−0.163316	+	0.986574i	$0.552219\pi$
$114$	0	0
$115$	0.944272	0.0880538
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−12.2361	−1.12168
$120$	0	0
$121$	−7.88854	−0.717140
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−12.0000	−1.07331
$126$	0	0
$127$	−16.9443	−1.50356	−0.751780	−	0.659413i	$-0.770805\pi$
$127$	−16.9443	−1.50356	−0.751780	+	0.659413i	$0.770805\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−13.7639	−1.20256	−0.601280	−	0.799038i	$-0.705342\pi$
$131$	−13.7639	−1.20256	−0.601280	+	0.799038i	$0.705342\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	3.29180	0.279206	0.139603	−	0.990208i	$-0.455417\pi$
$139$	3.29180	0.279206	0.139603	+	0.990208i	$0.455417\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	9.65248	0.807181
$144$	0	0
$145$	2.00000	0.166091
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−13.8885	−1.13779	−0.568897	−	0.822409i	$-0.692630\pi$
$149$	−13.8885	−1.13779	−0.568897	+	0.822409i	$0.692630\pi$
$150$	0	0
$151$	8.94427	0.727875	0.363937	−	0.931423i	$-0.381432\pi$
$151$	8.94427	0.727875	0.363937	+	0.931423i	$0.381432\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	17.8885	1.43684
$156$	0	0
$157$	12.0000	0.957704	0.478852	−	0.877896i	$-0.341053\pi$
$157$	12.0000	0.957704	0.478852	+	0.877896i	$0.341053\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1.05573	−0.0832030
$162$	0	0
$163$	8.47214	0.663589	0.331794	−	0.943352i	$-0.392346\pi$
$163$	8.47214	0.663589	0.331794	+	0.943352i	$0.392346\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	7.52786	0.582524	0.291262	−	0.956643i	$-0.405925\pi$
$167$	7.52786	0.582524	0.291262	+	0.956643i	$0.405925\pi$
$168$	0	0
$169$	16.9443	1.30341
$170$	0	0
$171$	0	0
$172$	0	0
$173$	23.8885	1.81621	0.908106	−	0.418740i	$-0.137528\pi$
$173$	23.8885	1.81621	0.908106	+	0.418740i	$0.137528\pi$
$174$	0	0
$175$	2.23607	0.169031
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−21.4164	−1.60074	−0.800369	−	0.599508i	$-0.795363\pi$
$179$	−21.4164	−1.60074	−0.800369	+	0.599508i	$0.795363\pi$
$180$	0	0
$181$	−4.52786	−0.336553	−0.168277	−	0.985740i	$-0.553820\pi$
$181$	−4.52786	−0.336553	−0.168277	+	0.985740i	$0.553820\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−8.00000	−0.588172
$186$	0	0
$187$	−9.65248	−0.705859
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−13.8885	−1.00494	−0.502470	−	0.864595i	$-0.667575\pi$
$191$	−13.8885	−1.00494	−0.502470	+	0.864595i	$0.667575\pi$
$192$	0	0
$193$	21.8885	1.57557	0.787786	−	0.615949i	$-0.211227\pi$
$193$	21.8885	1.57557	0.787786	+	0.615949i	$0.211227\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−16.0000	−1.13995	−0.569976	−	0.821661i	$-0.693048\pi$
$197$	−16.0000	−1.13995	−0.569976	+	0.821661i	$0.693048\pi$
$198$	0	0
$199$	−2.70820	−0.191979	−0.0959897	−	0.995382i	$-0.530602\pi$
$199$	−2.70820	−0.191979	−0.0959897	+	0.995382i	$0.530602\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−2.23607	−0.156941
$204$	0	0
$205$	−4.00000	−0.279372
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	24.9443	1.71723	0.858617	−	0.512617i	$-0.171324\pi$
$211$	24.9443	1.71723	0.858617	+	0.512617i	$0.171324\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−0.944272	−0.0643988
$216$	0	0
$217$	−20.0000	−1.35769
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−29.9443	−2.01427
$222$	0	0
$223$	14.2361	0.953318	0.476659	−	0.879088i	$-0.341848\pi$
$223$	14.2361	0.953318	0.476659	+	0.879088i	$0.341848\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	8.94427	0.593652	0.296826	−	0.954932i	$-0.404072\pi$
$227$	8.94427	0.593652	0.296826	+	0.954932i	$0.404072\pi$
$228$	0	0
$229$	24.0000	1.58596	0.792982	−	0.609245i	$-0.208527\pi$
$229$	24.0000	1.58596	0.792982	+	0.609245i	$0.208527\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−22.9443	−1.50313	−0.751565	−	0.659659i	$-0.770701\pi$
$233$	−22.9443	−1.50313	−0.751565	+	0.659659i	$0.770701\pi$
$234$	0	0
$235$	17.4164	1.13612
$236$	0	0
$237$	0	0
$238$	0	0
$239$	16.9443	1.09603	0.548017	−	0.836467i	$-0.315383\pi$
$239$	16.9443	1.09603	0.548017	+	0.836467i	$0.315383\pi$
$240$	0	0
$241$	−6.88854	−0.443730	−0.221865	−	0.975077i	$-0.571214\pi$
$241$	−6.88854	−0.443730	−0.221865	+	0.975077i	$0.571214\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−4.00000	−0.255551
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−13.7639	−0.868772	−0.434386	−	0.900727i	$-0.643035\pi$
$251$	−13.7639	−0.868772	−0.434386	+	0.900727i	$0.643035\pi$
$252$	0	0
$253$	−0.832816	−0.0523587
$254$	0	0
$255$	0	0
$256$	0	0
$257$	5.88854	0.367317	0.183659	−	0.982990i	$-0.441206\pi$
$257$	5.88854	0.367317	0.183659	+	0.982990i	$0.441206\pi$
$258$	0	0
$259$	8.94427	0.555770
$260$	0	0
$261$	0	0
$262$	0	0
$263$	20.0000	1.23325	0.616626	−	0.787256i	$-0.288499\pi$
$263$	20.0000	1.23325	0.616626	+	0.787256i	$0.288499\pi$
$264$	0	0
$265$	−16.0000	−0.982872
$266$	0	0
$267$	0	0
$268$	0	0
$269$	3.00000	0.182913	0.0914566	−	0.995809i	$-0.470848\pi$
$269$	3.00000	0.182913	0.0914566	+	0.995809i	$0.470848\pi$
$270$	0	0
$271$	−20.4721	−1.24359	−0.621797	−	0.783179i	$-0.713597\pi$
$271$	−20.4721	−1.24359	−0.621797	+	0.783179i	$0.713597\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.76393	0.106369
$276$	0	0
$277$	−31.4721	−1.89098	−0.945489	−	0.325655i	$-0.894415\pi$
$277$	−31.4721	−1.89098	−0.945489	+	0.325655i	$0.894415\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	3.05573	0.181644	0.0908221	−	0.995867i	$-0.471051\pi$
$283$	3.05573	0.181644	0.0908221	+	0.995867i	$0.471051\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.47214	0.263982
$288$	0	0
$289$	12.9443	0.761428
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−10.0557	−0.587462	−0.293731	−	0.955888i	$-0.594897\pi$
$293$	−10.0557	−0.587462	−0.293731	+	0.955888i	$0.594897\pi$
$294$	0	0
$295$	−24.9443	−1.45231
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−2.58359	−0.149413
$300$	0	0
$301$	1.05573	0.0608512
$302$	0	0
$303$	0	0
$304$	0	0
$305$	13.8885	0.795256
$306$	0	0
$307$	9.88854	0.564369	0.282185	−	0.959360i	$-0.408941\pi$
$307$	9.88854	0.564369	0.282185	+	0.959360i	$0.408941\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−4.23607	−0.240205	−0.120103	−	0.992761i	$-0.538322\pi$
$311$	−4.23607	−0.240205	−0.120103	+	0.992761i	$0.538322\pi$
$312$	0	0
$313$	−19.9443	−1.12732	−0.563658	−	0.826008i	$-0.690607\pi$
$313$	−19.9443	−1.12732	−0.563658	+	0.826008i	$0.690607\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−3.00000	−0.168497	−0.0842484	−	0.996445i	$-0.526849\pi$
$317$	−3.00000	−0.168497	−0.0842484	+	0.996445i	$0.526849\pi$
$318$	0	0
$319$	−1.76393	−0.0987612
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	5.47214	0.303539
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−19.4721	−1.07353
$330$	0	0
$331$	0.944272	0.0519019	0.0259509	−	0.999663i	$-0.491739\pi$
$331$	0.944272	0.0519019	0.0259509	+	0.999663i	$0.491739\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−16.4721	−0.899969
$336$	0	0
$337$	7.88854	0.429716	0.214858	−	0.976645i	$-0.431071\pi$
$337$	7.88854	0.429716	0.214858	+	0.976645i	$0.431071\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−15.7771	−0.854377
$342$	0	0
$343$	20.1246	1.08663
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	23.8885	1.27872	0.639362	−	0.768906i	$-0.279198\pi$
$349$	23.8885	1.27872	0.639362	+	0.768906i	$0.279198\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	28.0000	1.49029	0.745145	−	0.666903i	$-0.232380\pi$
$353$	28.0000	1.49029	0.745145	+	0.666903i	$0.232380\pi$
$354$	0	0
$355$	−17.8885	−0.949425
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−28.9443	−1.52762	−0.763810	−	0.645441i	$-0.776674\pi$
$359$	−28.9443	−1.52762	−0.763810	+	0.645441i	$0.776674\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−4.00000	−0.209370
$366$	0	0
$367$	13.4164	0.700331	0.350165	−	0.936688i	$-0.386125\pi$
$367$	13.4164	0.700331	0.350165	+	0.936688i	$0.386125\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	17.8885	0.928727
$372$	0	0
$373$	−23.8885	−1.23690	−0.618451	−	0.785823i	$-0.712239\pi$
$373$	−23.8885	−1.23690	−0.618451	+	0.785823i	$0.712239\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−5.47214	−0.281829
$378$	0	0
$379$	−17.4164	−0.894621	−0.447310	−	0.894379i	$-0.647618\pi$
$379$	−17.4164	−0.894621	−0.447310	+	0.894379i	$0.647618\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−1.41641	−0.0723751	−0.0361875	−	0.999345i	$-0.511521\pi$
$383$	−1.41641	−0.0723751	−0.0361875	+	0.999345i	$0.511521\pi$
$384$	0	0
$385$	7.88854	0.402037
$386$	0	0
$387$	0	0
$388$	0	0
$389$	5.00000	0.253510	0.126755	−	0.991934i	$-0.459544\pi$
$389$	5.00000	0.253510	0.126755	+	0.991934i	$0.459544\pi$
$390$	0	0
$391$	2.58359	0.130658
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−24.9443	−1.25508
$396$	0	0
$397$	27.8885	1.39969	0.699843	−	0.714297i	$-0.253253\pi$
$397$	27.8885	1.39969	0.699843	+	0.714297i	$0.253253\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	17.0557	0.851722	0.425861	−	0.904789i	$-0.359971\pi$
$401$	17.0557	0.851722	0.425861	+	0.904789i	$0.359971\pi$
$402$	0	0
$403$	−48.9443	−2.43809
$404$	0	0
$405$	0	0
$406$	0	0
$407$	7.05573	0.349739
$408$	0	0
$409$	−8.00000	−0.395575	−0.197787	−	0.980245i	$-0.563376\pi$
$409$	−8.00000	−0.395575	−0.197787	+	0.980245i	$0.563376\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	27.8885	1.37231
$414$	0	0
$415$	−33.8885	−1.66352
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−10.5836	−0.517042	−0.258521	−	0.966006i	$-0.583235\pi$
$419$	−10.5836	−0.517042	−0.258521	+	0.966006i	$0.583235\pi$
$420$	0	0
$421$	−9.05573	−0.441349	−0.220675	−	0.975347i	$-0.570826\pi$
$421$	−9.05573	−0.441349	−0.220675	+	0.975347i	$0.570826\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−5.47214	−0.265438
$426$	0	0
$427$	−15.5279	−0.751446
$428$	0	0
$429$	0	0
$430$	0	0
$431$	35.3050	1.70058	0.850290	−	0.526315i	$-0.176427\pi$
$431$	35.3050	1.70058	0.850290	+	0.526315i	$0.176427\pi$
$432$	0	0
$433$	−15.8885	−0.763555	−0.381777	−	0.924254i	$-0.624688\pi$
$433$	−15.8885	−0.763555	−0.381777	+	0.924254i	$0.624688\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−3.65248	−0.174323	−0.0871616	−	0.996194i	$-0.527780\pi$
$439$	−3.65248	−0.174323	−0.0871616	+	0.996194i	$0.527780\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−26.2361	−1.24651	−0.623257	−	0.782017i	$-0.714191\pi$
$443$	−26.2361	−1.24651	−0.623257	+	0.782017i	$0.714191\pi$
$444$	0	0
$445$	−28.8328	−1.36681
$446$	0	0
$447$	0	0
$448$	0	0
$449$	22.4164	1.05790	0.528948	−	0.848654i	$-0.322587\pi$
$449$	22.4164	1.05790	0.528948	+	0.848654i	$0.322587\pi$
$450$	0	0
$451$	3.52786	0.166121
$452$	0	0
$453$	0	0
$454$	0	0
$455$	24.4721	1.14727
$456$	0	0
$457$	−15.0000	−0.701670	−0.350835	−	0.936437i	$-0.614102\pi$
$457$	−15.0000	−0.701670	−0.350835	+	0.936437i	$0.614102\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−10.0000	−0.465746	−0.232873	−	0.972507i	$-0.574813\pi$
$461$	−10.0000	−0.465746	−0.232873	+	0.972507i	$0.574813\pi$
$462$	0	0
$463$	16.1246	0.749374	0.374687	−	0.927151i	$-0.377750\pi$
$463$	16.1246	0.749374	0.374687	+	0.927151i	$0.377750\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−41.8885	−1.93837	−0.969185	−	0.246333i	$-0.920774\pi$
$467$	−41.8885	−1.93837	−0.969185	+	0.246333i	$0.920774\pi$
$468$	0	0
$469$	18.4164	0.850391
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0.832816	0.0382929
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−20.9443	−0.956968	−0.478484	−	0.878096i	$-0.658814\pi$
$479$	−20.9443	−0.956968	−0.478484	+	0.878096i	$0.658814\pi$
$480$	0	0
$481$	21.8885	0.998032
$482$	0	0
$483$	0	0
$484$	0	0
$485$	16.0000	0.726523
$486$	0	0
$487$	34.8328	1.57843	0.789213	−	0.614120i	$-0.210489\pi$
$487$	34.8328	1.57843	0.789213	+	0.614120i	$0.210489\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	8.00000	0.361035	0.180517	−	0.983572i	$-0.442223\pi$
$491$	8.00000	0.361035	0.180517	+	0.983572i	$0.442223\pi$
$492$	0	0
$493$	5.47214	0.246453
$494$	0	0
$495$	0	0
$496$	0	0
$497$	20.0000	0.897123
$498$	0	0
$499$	3.29180	0.147361	0.0736805	−	0.997282i	$-0.476525\pi$
$499$	3.29180	0.147361	0.0736805	+	0.997282i	$0.476525\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	8.70820	0.388280	0.194140	−	0.980974i	$-0.437808\pi$
$503$	8.70820	0.388280	0.194140	+	0.980974i	$0.437808\pi$
$504$	0	0
$505$	31.8885	1.41902
$506$	0	0
$507$	0	0
$508$	0	0
$509$	2.00000	0.0886484	0.0443242	−	0.999017i	$-0.485887\pi$
$509$	2.00000	0.0886484	0.0443242	+	0.999017i	$0.485887\pi$
$510$	0	0
$511$	4.47214	0.197836
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−1.88854	−0.0832192
$516$	0	0
$517$	−15.3607	−0.675562
$518$	0	0
$519$	0	0
$520$	0	0
$521$	9.05573	0.396739	0.198369	−	0.980127i	$-0.436435\pi$
$521$	9.05573	0.396739	0.198369	+	0.980127i	$0.436435\pi$
$522$	0	0
$523$	25.1803	1.10106	0.550530	−	0.834816i	$-0.314426\pi$
$523$	25.1803	1.10106	0.550530	+	0.834816i	$0.314426\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	48.9443	2.13205
$528$	0	0
$529$	−22.7771	−0.990308
$530$	0	0
$531$	0	0
$532$	0	0
$533$	10.9443	0.474049
$534$	0	0
$535$	8.94427	0.386695
$536$	0	0
$537$	0	0
$538$	0	0
$539$	3.52786	0.151956
$540$	0	0
$541$	10.9443	0.470531	0.235266	−	0.971931i	$-0.424404\pi$
$541$	10.9443	0.470531	0.235266	+	0.971931i	$0.424404\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−30.9443	−1.32551
$546$	0	0
$547$	−22.5967	−0.966167	−0.483084	−	0.875574i	$-0.660483\pi$
$547$	−22.5967	−0.966167	−0.483084	+	0.875574i	$0.660483\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	27.8885	1.18594
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−5.88854	−0.249506	−0.124753	−	0.992188i	$-0.539814\pi$
$557$	−5.88854	−0.249506	−0.124753	+	0.992188i	$0.539814\pi$
$558$	0	0
$559$	2.58359	0.109274
$560$	0	0
$561$	0	0
$562$	0	0
$563$	29.5410	1.24501	0.622503	−	0.782618i	$-0.286116\pi$
$563$	29.5410	1.24501	0.622503	+	0.782618i	$0.286116\pi$
$564$	0	0
$565$	−6.94427	−0.292148
$566$	0	0
$567$	0	0
$568$	0	0
$569$	3.47214	0.145560	0.0727798	−	0.997348i	$-0.476813\pi$
$569$	3.47214	0.145560	0.0727798	+	0.997348i	$0.476813\pi$
$570$	0	0
$571$	−12.9443	−0.541701	−0.270850	−	0.962621i	$-0.587305\pi$
$571$	−12.9443	−0.541701	−0.270850	+	0.962621i	$0.587305\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−0.472136	−0.0196894
$576$	0	0
$577$	−3.88854	−0.161882	−0.0809411	−	0.996719i	$-0.525793\pi$
$577$	−3.88854	−0.161882	−0.0809411	+	0.996719i	$0.525793\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	37.8885	1.57188
$582$	0	0
$583$	14.1115	0.584437
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−6.36068	−0.262533	−0.131267	−	0.991347i	$-0.541904\pi$
$587$	−6.36068	−0.262533	−0.131267	+	0.991347i	$0.541904\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−40.8328	−1.67680	−0.838401	−	0.545053i	$-0.816509\pi$
$593$	−40.8328	−1.67680	−0.838401	+	0.545053i	$0.816509\pi$
$594$	0	0
$595$	−24.4721	−1.00326
$596$	0	0
$597$	0	0
$598$	0	0
$599$	38.5967	1.57702	0.788510	−	0.615022i	$-0.210853\pi$
$599$	38.5967	1.57702	0.788510	+	0.615022i	$0.210853\pi$
$600$	0	0
$601$	33.8885	1.38234	0.691171	−	0.722691i	$-0.257095\pi$
$601$	33.8885	1.38234	0.691171	+	0.722691i	$0.257095\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−15.7771	−0.641430
$606$	0	0
$607$	42.8328	1.73853	0.869265	−	0.494346i	$-0.164592\pi$
$607$	42.8328	1.73853	0.869265	+	0.494346i	$0.164592\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−47.6525	−1.92781
$612$	0	0
$613$	4.41641	0.178377	0.0891885	−	0.996015i	$-0.471573\pi$
$613$	4.41641	0.178377	0.0891885	+	0.996015i	$0.471573\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	41.7771	1.68188	0.840941	−	0.541127i	$-0.182002\pi$
$617$	41.7771	1.68188	0.840941	+	0.541127i	$0.182002\pi$
$618$	0	0
$619$	−24.4721	−0.983618	−0.491809	−	0.870703i	$-0.663664\pi$
$619$	−24.4721	−0.983618	−0.491809	+	0.870703i	$0.663664\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	32.2361	1.29151
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−21.8885	−0.872753
$630$	0	0
$631$	−11.6525	−0.463878	−0.231939	−	0.972730i	$-0.574507\pi$
$631$	−11.6525	−0.463878	−0.231939	+	0.972730i	$0.574507\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−33.8885	−1.34483
$636$	0	0
$637$	10.9443	0.433628
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−15.3607	−0.606710	−0.303355	−	0.952878i	$-0.598107\pi$
$641$	−15.3607	−0.606710	−0.303355	+	0.952878i	$0.598107\pi$
$642$	0	0
$643$	12.7082	0.501163	0.250581	−	0.968096i	$-0.419378\pi$
$643$	12.7082	0.501163	0.250581	+	0.968096i	$0.419378\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0.472136	0.0185616	0.00928079	−	0.999957i	$-0.497046\pi$
$647$	0.472136	0.0185616	0.00928079	+	0.999957i	$0.497046\pi$
$648$	0	0
$649$	22.0000	0.863576
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−27.9443	−1.09354	−0.546772	−	0.837282i	$-0.684144\pi$
$653$	−27.9443	−1.09354	−0.546772	+	0.837282i	$0.684144\pi$
$654$	0	0
$655$	−27.5279	−1.07560
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−33.0689	−1.28818	−0.644090	−	0.764949i	$-0.722764\pi$
$659$	−33.0689	−1.28818	−0.644090	+	0.764949i	$0.722764\pi$
$660$	0	0
$661$	26.3050	1.02314	0.511572	−	0.859240i	$-0.329063\pi$
$661$	26.3050	1.02314	0.511572	+	0.859240i	$0.329063\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0.472136	0.0182812
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−12.2492	−0.472876
$672$	0	0
$673$	4.05573	0.156337	0.0781684	−	0.996940i	$-0.475093\pi$
$673$	4.05573	0.156337	0.0781684	+	0.996940i	$0.475093\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	35.0000	1.34516	0.672580	−	0.740025i	$-0.265186\pi$
$677$	35.0000	1.34516	0.672580	+	0.740025i	$0.265186\pi$
$678$	0	0
$679$	−17.8885	−0.686499
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−16.0000	−0.612223	−0.306111	−	0.951996i	$-0.599028\pi$
$683$	−16.0000	−0.612223	−0.306111	+	0.951996i	$0.599028\pi$
$684$	0	0
$685$	−12.0000	−0.458496
$686$	0	0
$687$	0	0
$688$	0	0
$689$	43.7771	1.66777
$690$	0	0
$691$	−26.1246	−0.993827	−0.496914	−	0.867800i	$-0.665533\pi$
$691$	−26.1246	−0.993827	−0.496914	+	0.867800i	$0.665533\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	6.58359	0.249730
$696$	0	0
$697$	−10.9443	−0.414544
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−6.11146	−0.230827	−0.115413	−	0.993318i	$-0.536819\pi$
$701$	−6.11146	−0.230827	−0.115413	+	0.993318i	$0.536819\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−35.6525	−1.34085
$708$	0	0
$709$	−35.8885	−1.34782	−0.673911	−	0.738812i	$-0.735387\pi$
$709$	−35.8885	−1.34782	−0.673911	+	0.738812i	$0.735387\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	4.22291	0.158149
$714$	0	0
$715$	19.3050	0.721964
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−16.4721	−0.614307	−0.307154	−	0.951660i	$-0.599377\pi$
$719$	−16.4721	−0.614307	−0.307154	+	0.951660i	$0.599377\pi$
$720$	0	0
$721$	2.11146	0.0786347
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−1.00000	−0.0371391
$726$	0	0
$727$	11.5279	0.427545	0.213772	−	0.976883i	$-0.431425\pi$
$727$	11.5279	0.427545	0.213772	+	0.976883i	$0.431425\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−2.58359	−0.0955576
$732$	0	0
$733$	18.9443	0.699723	0.349861	−	0.936802i	$-0.386229\pi$
$733$	18.9443	0.699723	0.349861	+	0.936802i	$0.386229\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	14.5279	0.535141
$738$	0	0
$739$	32.9443	1.21187	0.605937	−	0.795512i	$-0.292798\pi$
$739$	32.9443	1.21187	0.605937	+	0.795512i	$0.292798\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	30.5967	1.12249	0.561243	−	0.827651i	$-0.310323\pi$
$743$	30.5967	1.12249	0.561243	+	0.827651i	$0.310323\pi$
$744$	0	0
$745$	−27.7771	−1.01767
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−10.0000	−0.365392
$750$	0	0
$751$	0.944272	0.0344570	0.0172285	−	0.999852i	$-0.494516\pi$
$751$	0.944272	0.0344570	0.0172285	+	0.999852i	$0.494516\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	17.8885	0.651031
$756$	0	0
$757$	−28.8328	−1.04795	−0.523973	−	0.851735i	$-0.675551\pi$
$757$	−28.8328	−1.04795	−0.523973	+	0.851735i	$0.675551\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	38.9443	1.41173	0.705864	−	0.708347i	$-0.250559\pi$
$761$	38.9443	1.41173	0.705864	+	0.708347i	$0.250559\pi$
$762$	0	0
$763$	34.5967	1.25249
$764$	0	0
$765$	0	0
$766$	0	0
$767$	68.2492	2.46434
$768$	0	0
$769$	−35.7771	−1.29015	−0.645077	−	0.764117i	$-0.723175\pi$
$769$	−35.7771	−1.29015	−0.645077	+	0.764117i	$0.723175\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−10.0000	−0.359675	−0.179838	−	0.983696i	$-0.557557\pi$
$773$	−10.0000	−0.359675	−0.179838	+	0.983696i	$0.557557\pi$
$774$	0	0
$775$	−8.94427	−0.321288
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	15.7771	0.564549
$782$	0	0
$783$	0	0
$784$	0	0
$785$	24.0000	0.856597
$786$	0	0
$787$	−3.05573	−0.108925	−0.0544625	−	0.998516i	$-0.517345\pi$
$787$	−3.05573	−0.108925	−0.0544625	+	0.998516i	$0.517345\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	7.76393	0.276054
$792$	0	0
$793$	−38.0000	−1.34942
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−25.7771	−0.913071	−0.456536	−	0.889705i	$-0.650910\pi$
$797$	−25.7771	−0.913071	−0.456536	+	0.889705i	$0.650910\pi$
$798$	0	0
$799$	47.6525	1.68582
$800$	0	0
$801$	0	0
$802$	0	0
$803$	3.52786	0.124496
$804$	0	0
$805$	−2.11146	−0.0744191
$806$	0	0
$807$	0	0
$808$	0	0
$809$	12.3050	0.432619	0.216310	−	0.976325i	$-0.430598\pi$
$809$	12.3050	0.432619	0.216310	+	0.976325i	$0.430598\pi$
$810$	0	0
$811$	−1.18034	−0.0414473	−0.0207237	−	0.999785i	$-0.506597\pi$
$811$	−1.18034	−0.0414473	−0.0207237	+	0.999785i	$0.506597\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	16.9443	0.593532
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	28.0000	0.977207	0.488603	−	0.872506i	$-0.337507\pi$
$821$	28.0000	0.977207	0.488603	+	0.872506i	$0.337507\pi$
$822$	0	0
$823$	15.3050	0.533497	0.266749	−	0.963766i	$-0.414051\pi$
$823$	15.3050	0.533497	0.266749	+	0.963766i	$0.414051\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	9.88854	0.343858	0.171929	−	0.985109i	$-0.445000\pi$
$827$	9.88854	0.343858	0.171929	+	0.985109i	$0.445000\pi$
$828$	0	0
$829$	10.9443	0.380110	0.190055	−	0.981773i	$-0.439133\pi$
$829$	10.9443	0.380110	0.190055	+	0.981773i	$0.439133\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−10.9443	−0.379197
$834$	0	0
$835$	15.0557	0.521025
$836$	0	0
$837$	0	0
$838$	0	0
$839$	6.81966	0.235441	0.117720	−	0.993047i	$-0.462441\pi$
$839$	6.81966	0.235441	0.117720	+	0.993047i	$0.462441\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	0	0
$844$	0	0
$845$	33.8885	1.16580
$846$	0	0
$847$	17.6393	0.606094
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−1.88854	−0.0647384
$852$	0	0
$853$	−32.8328	−1.12417	−0.562087	−	0.827078i	$-0.690001\pi$
$853$	−32.8328	−1.12417	−0.562087	+	0.827078i	$0.690001\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	5.88854	0.201149	0.100574	−	0.994930i	$-0.467932\pi$
$857$	5.88854	0.201149	0.100574	+	0.994930i	$0.467932\pi$
$858$	0	0
$859$	36.2492	1.23681	0.618404	−	0.785861i	$-0.287780\pi$
$859$	36.2492	1.23681	0.618404	+	0.785861i	$0.287780\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	49.8885	1.69823	0.849113	−	0.528211i	$-0.177137\pi$
$863$	49.8885	1.69823	0.849113	+	0.528211i	$0.177137\pi$
$864$	0	0
$865$	47.7771	1.62447
$866$	0	0
$867$	0	0
$868$	0	0
$869$	22.0000	0.746299
$870$	0	0
$871$	45.0689	1.52710
$872$	0	0
$873$	0	0
$874$	0	0
$875$	26.8328	0.907115
$876$	0	0
$877$	15.8885	0.536518	0.268259	−	0.963347i	$-0.413552\pi$
$877$	15.8885	0.536518	0.268259	+	0.963347i	$0.413552\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	36.4164	1.22690	0.613450	−	0.789734i	$-0.289781\pi$
$881$	36.4164	1.22690	0.613450	+	0.789734i	$0.289781\pi$
$882$	0	0
$883$	−46.8328	−1.57605	−0.788025	−	0.615643i	$-0.788896\pi$
$883$	−46.8328	−1.57605	−0.788025	+	0.615643i	$0.788896\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	47.0689	1.58042	0.790209	−	0.612837i	$-0.209972\pi$
$887$	47.0689	1.58042	0.790209	+	0.612837i	$0.209972\pi$
$888$	0	0
$889$	37.8885	1.27074
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−42.8328	−1.43174
$896$	0	0
$897$	0	0
$898$	0	0
$899$	8.94427	0.298308
$900$	0	0
$901$	−43.7771	−1.45843
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−9.05573	−0.301023
$906$	0	0
$907$	10.3607	0.344021	0.172010	−	0.985095i	$-0.444974\pi$
$907$	10.3607	0.344021	0.172010	+	0.985095i	$0.444974\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−1.18034	−0.0391064	−0.0195532	−	0.999809i	$-0.506224\pi$
$911$	−1.18034	−0.0391064	−0.0195532	+	0.999809i	$0.506224\pi$
$912$	0	0
$913$	29.8885	0.989166
$914$	0	0
$915$	0	0
$916$	0	0
$917$	30.7771	1.01635
$918$	0	0
$919$	−51.6525	−1.70386	−0.851929	−	0.523657i	$-0.824567\pi$
$919$	−51.6525	−1.70386	−0.851929	+	0.523657i	$0.824567\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	48.9443	1.61102
$924$	0	0
$925$	4.00000	0.131519
$926$	0	0
$927$	0	0
$928$	0	0
$929$	14.9443	0.490306	0.245153	−	0.969484i	$-0.421162\pi$
$929$	14.9443	0.490306	0.245153	+	0.969484i	$0.421162\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−19.3050	−0.631339
$936$	0	0
$937$	−5.94427	−0.194191	−0.0970954	−	0.995275i	$-0.530955\pi$
$937$	−5.94427	−0.194191	−0.0970954	+	0.995275i	$0.530955\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−37.7771	−1.23150	−0.615749	−	0.787942i	$-0.711146\pi$
$941$	−37.7771	−1.23150	−0.615749	+	0.787942i	$0.711146\pi$
$942$	0	0
$943$	−0.944272	−0.0307497
$944$	0	0
$945$	0	0
$946$	0	0
$947$	2.23607	0.0726624	0.0363312	−	0.999340i	$-0.488433\pi$
$947$	2.23607	0.0726624	0.0363312	+	0.999340i	$0.488433\pi$
$948$	0	0
$949$	10.9443	0.355266
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−54.9443	−1.77982	−0.889910	−	0.456137i	$-0.849233\pi$
$953$	−54.9443	−1.77982	−0.889910	+	0.456137i	$0.849233\pi$
$954$	0	0
$955$	−27.7771	−0.898845
$956$	0	0
$957$	0	0
$958$	0	0
$959$	13.4164	0.433238
$960$	0	0
$961$	49.0000	1.58065
$962$	0	0
$963$	0	0
$964$	0	0
$965$	43.7771	1.40923
$966$	0	0
$967$	−34.8328	−1.12015	−0.560074	−	0.828443i	$-0.689227\pi$
$967$	−34.8328	−1.12015	−0.560074	+	0.828443i	$0.689227\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0.944272	0.0303031	0.0151516	−	0.999885i	$-0.495177\pi$
$971$	0.944272	0.0303031	0.0151516	+	0.999885i	$0.495177\pi$
$972$	0	0
$973$	−7.36068	−0.235973
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−17.8885	−0.572305	−0.286153	−	0.958184i	$-0.592376\pi$
$977$	−17.8885	−0.572305	−0.286153	+	0.958184i	$0.592376\pi$
$978$	0	0
$979$	25.4296	0.812732
$980$	0	0
$981$	0	0
$982$	0	0
$983$	10.1115	0.322505	0.161253	−	0.986913i	$-0.448447\pi$
$983$	10.1115	0.322505	0.161253	+	0.986913i	$0.448447\pi$
$984$	0	0
$985$	−32.0000	−1.01960
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−0.222912	−0.00708820
$990$	0	0
$991$	37.7639	1.19961	0.599805	−	0.800146i	$-0.295245\pi$
$991$	37.7639	1.19961	0.599805	+	0.800146i	$0.295245\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−5.41641	−0.171712
$996$	0	0
$997$	17.8885	0.566536	0.283268	−	0.959041i	$-0.408581\pi$
$997$	17.8885	0.566536	0.283268	+	0.959041i	$0.408581\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4176.2.a.bt.1.1		2
3.2	odd	2		4176.2.a.bm.1.1		2
4.3	odd	2		261.2.a.c.1.2	yes	2
12.11	even	2		261.2.a.a.1.1	✓	2
20.19	odd	2		6525.2.a.q.1.1		2
60.59	even	2		6525.2.a.z.1.2		2
116.115	odd	2		7569.2.a.h.1.1		2
348.347	even	2		7569.2.a.j.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
261.2.a.a.1.1	✓	2	12.11	even	2
261.2.a.c.1.2	yes	2	4.3	odd	2
4176.2.a.bm.1.1		2	3.2	odd	2
4176.2.a.bt.1.1		2	1.1	even	1	trivial
6525.2.a.q.1.1		2	20.19	odd	2
6525.2.a.z.1.2		2	60.59	even	2
7569.2.a.h.1.1		2	116.115	odd	2
7569.2.a.j.1.2		2	348.347	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:	\( N \)	\(=\)	\( 4176 = 2^{4} \cdot 3^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4176.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{5} -2.23607 q^{7} +O(q^{10})\)	\(q+2.00000 q^{5} -2.23607 q^{7} -1.76393 q^{11} -5.47214 q^{13} +5.47214 q^{17} +0.472136 q^{23} -1.00000 q^{25} +1.00000 q^{29} +8.94427 q^{31} -4.47214 q^{35} -4.00000 q^{37} -2.00000 q^{41} -0.472136 q^{43} +8.70820 q^{47} -2.00000 q^{49} -8.00000 q^{53} -3.52786 q^{55} -12.4721 q^{59} +6.94427 q^{61} -10.9443 q^{65} -8.23607 q^{67} -8.94427 q^{71} -2.00000 q^{73} +3.94427 q^{77} -12.4721 q^{79} -16.9443 q^{83} +10.9443 q^{85} -14.4164 q^{89} +12.2361 q^{91} +8.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{5}+O(q^{10}) \) 2 * q + 4 * q^5	\( 2 q + 4 q^{5} - 8 q^{11} - 2 q^{13} + 2 q^{17} - 8 q^{23} - 2 q^{25} + 2 q^{29} - 8 q^{37} - 4 q^{41} + 8 q^{43} + 4 q^{47} - 4 q^{49} - 16 q^{53} - 16 q^{55} - 16 q^{59} - 4 q^{61} - 4 q^{65} - 12 q^{67} - 4 q^{73} - 10 q^{77} - 16 q^{79} - 16 q^{83} + 4 q^{85} - 2 q^{89} + 20 q^{91} + 16 q^{97}+O(q^{100}) \) 2 * q + 4 * q^5 - 8 * q^11 - 2 * q^13 + 2 * q^17 - 8 * q^23 - 2 * q^25 + 2 * q^29 - 8 * q^37 - 4 * q^41 + 8 * q^43 + 4 * q^47 - 4 * q^49 - 16 * q^53 - 16 * q^55 - 16 * q^59 - 4 * q^61 - 4 * q^65 - 12 * q^67 - 4 * q^73 - 10 * q^77 - 16 * q^79 - 16 * q^83 + 4 * q^85 - 2 * q^89 + 20 * q^91 + 16 * q^97

Introduction

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