LMFDB - Embedded newform 1440.4.k.c.721.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1440,4,Mod(721,1440)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1440, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1440.721");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1440 = 2^{5} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1440.k (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:	$84.9627504083$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4x^{11} + 7x^{10} - 12x^{9} + 21x^{8} - 68x^{6} + 336x^{4} - 768x^{3} + 1792x^{2} - 4096x + 4096 $ x^12 - 4x^11 + 7x^10 - 12x^9 + 21x^8 - 68x^6 + 336x^4 - 768x^3 + 1792x^2 - 4096*x + 4096
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{30}\cdot 5^{4} $
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			721.12
Root			$-1.86176 - 0.730647i$ of defining polynomial
Character	$\chi$	$=$	1440.721
Dual form			1440.4.k.c.721.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.00000i q^{5} +34.6280 q^{7} +O(q^{10})$	$q+5.00000i q^{5} +34.6280 q^{7} -7.91595i q^{11} -11.0346i q^{13} -57.6152 q^{17} +141.133i q^{19} +129.328 q^{23} -25.0000 q^{25} +89.1664i q^{29} +78.3307 q^{31} +173.140i q^{35} +249.332i q^{37} -91.8705 q^{41} -194.184i q^{43} -72.6149 q^{47} +856.096 q^{49} +456.782i q^{53} +39.5797 q^{55} +341.098i q^{59} -217.067i q^{61} +55.1731 q^{65} -529.237i q^{67} +381.540 q^{71} -876.902 q^{73} -274.113i q^{77} +203.950 q^{79} -996.654i q^{83} -288.076i q^{85} -172.456 q^{89} -382.107i q^{91} -705.664 q^{95} +1058.49 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 28 q^{7}+O(q^{10}) $ 12 * q - 28 * q^7	$ 12 q - 28 q^{7} + 604 q^{23} - 300 q^{25} + 264 q^{31} - 40 q^{41} - 940 q^{47} + 1308 q^{49} - 440 q^{55} - 1592 q^{71} + 432 q^{73} - 2016 q^{79} + 424 q^{89} - 1520 q^{95} - 1584 q^{97}+O(q^{100}) $ 12 * q - 28 * q^7 + 604 * q^23 - 300 * q^25 + 264 * q^31 - 40 * q^41 - 940 * q^47 + 1308 * q^49 - 440 * q^55 - 1592 * q^71 + 432 * q^73 - 2016 * q^79 + 424 * q^89 - 1520 * q^95 - 1584 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1440\mathbb{Z}\right)^\times$.

$n$	$577$	$641$	$901$	$991$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$5$	5.00000i	0.447214i
$5$	5.00000i	0.447214i
$6$	0	0
$7$	34.6280	1.86973	0.934867	−	0.354998i	$-0.115518\pi$
$7$	34.6280	1.86973	0.934867	+	0.354998i	$0.115518\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 7.91595i	− 0.216977i	−0.994098	−	0.108489i	$-0.965399\pi$
$11$	− 7.91595i	− 0.216977i	0.994098	−	0.108489i	$-0.0346011\pi$
$12$	0	0
$13$	− 11.0346i	− 0.235420i	−0.993048	−	0.117710i	$-0.962445\pi$
$13$	− 11.0346i	− 0.235420i	0.993048	−	0.117710i	$-0.0375553\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−57.6152	−0.821985	−0.410992	−	0.911639i	$-0.634818\pi$
$17$	−57.6152	−0.821985	−0.410992	+	0.911639i	$0.634818\pi$
$18$	0	0
$19$	141.133i	1.70411i	0.523453	+	0.852055i	$0.324644\pi$
$19$	141.133i	1.70411i	−0.523453	+	0.852055i	$0.675356\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	129.328	1.17247	0.586233	−	0.810143i	$-0.300610\pi$
$23$	129.328	1.17247	0.586233	+	0.810143i	$0.300610\pi$
$24$	0	0
$25$	−25.0000	−0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	89.1664i	0.570958i	0.958385	+	0.285479i	$0.0921527\pi$
$29$	89.1664i	0.570958i	−0.958385	+	0.285479i	$0.907847\pi$
$30$	0	0
$31$	78.3307	0.453826	0.226913	−	0.973915i	$-0.427137\pi$
$31$	78.3307	0.453826	0.226913	+	0.973915i	$0.427137\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	173.140i	0.836171i
$36$	0	0
$37$	249.332i	1.10783i	0.832572	+	0.553917i	$0.186868\pi$
$37$	249.332i	1.10783i	−0.832572	+	0.553917i	$0.813132\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−91.8705	−0.349946	−0.174973	−	0.984573i	$-0.555984\pi$
$41$	−91.8705	−0.349946	−0.174973	+	0.984573i	$0.555984\pi$
$42$	0	0
$43$	− 194.184i	− 0.688668i	−0.938847	−	0.344334i	$-0.888105\pi$
$43$	− 194.184i	− 0.688668i	0.938847	−	0.344334i	$-0.111895\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−72.6149	−0.225361	−0.112680	−	0.993631i	$-0.535944\pi$
$47$	−72.6149	−0.225361	−0.112680	+	0.993631i	$0.535944\pi$
$48$	0	0
$49$	856.096	2.49591
$50$	0	0
$51$	0	0
$52$	0	0
$53$	456.782i	1.18385i	0.805995	+	0.591923i	$0.201631\pi$
$53$	456.782i	1.18385i	−0.805995	+	0.591923i	$0.798369\pi$
$54$	0	0
$55$	39.5797	0.0970351
$56$	0	0
$57$	0	0
$58$	0	0
$59$	341.098i	0.752664i	0.926485	+	0.376332i	$0.122815\pi$
$59$	341.098i	0.752664i	−0.926485	+	0.376332i	$0.877185\pi$
$60$	0	0
$61$	− 217.067i	− 0.455616i	−0.973706	−	0.227808i	$-0.926844\pi$
$61$	− 217.067i	− 0.455616i	0.973706	−	0.227808i	$-0.0731559\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	55.1731	0.105283
$66$	0	0
$67$	− 529.237i	− 0.965024i	−0.875889	−	0.482512i	$-0.839724\pi$
$67$	− 529.237i	− 0.965024i	0.875889	−	0.482512i	$-0.160276\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	381.540	0.637754	0.318877	−	0.947796i	$-0.396694\pi$
$71$	381.540	0.637754	0.318877	+	0.947796i	$0.396694\pi$
$72$	0	0
$73$	−876.902	−1.40594	−0.702970	−	0.711220i	$-0.748143\pi$
$73$	−876.902	−1.40594	−0.702970	+	0.711220i	$0.748143\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 274.113i	− 0.405690i
$78$	0	0
$79$	203.950	0.290458	0.145229	−	0.989398i	$-0.453608\pi$
$79$	203.950	0.290458	0.145229	+	0.989398i	$0.453608\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 996.654i	− 1.31804i	−0.752127	−	0.659018i	$-0.770972\pi$
$83$	− 996.654i	− 1.31804i	0.752127	−	0.659018i	$-0.229028\pi$
$84$	0	0
$85$	− 288.076i	− 0.367603i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−172.456	−0.205396	−0.102698	−	0.994713i	$-0.532748\pi$
$89$	−172.456	−0.205396	−0.102698	+	0.994713i	$0.532748\pi$
$90$	0	0
$91$	− 382.107i	− 0.440172i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−705.664	−0.762101
$96$	0	0
$97$	1058.49	1.10797	0.553984	−	0.832527i	$-0.313107\pi$
$97$	1058.49	1.10797	0.553984	+	0.832527i	$0.313107\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	689.992i	0.679770i	0.940467	+	0.339885i	$0.110388\pi$
$101$	689.992i	0.679770i	−0.940467	+	0.339885i	$0.889612\pi$
$102$	0	0
$103$	−1289.96	−1.23402	−0.617009	−	0.786956i	$-0.711656\pi$
$103$	−1289.96	−1.23402	−0.617009	+	0.786956i	$0.711656\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1843.74i	1.66580i	0.553420	+	0.832902i	$0.313322\pi$
$107$	1843.74i	1.66580i	−0.553420	+	0.832902i	$0.686678\pi$
$108$	0	0
$109$	340.598i	0.299297i	0.988739	+	0.149648i	$0.0478142\pi$
$109$	340.598i	0.299297i	−0.988739	+	0.149648i	$0.952186\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−157.831	−0.131393	−0.0656967	−	0.997840i	$-0.520927\pi$
$113$	−157.831	−0.131393	−0.0656967	+	0.997840i	$0.520927\pi$
$114$	0	0
$115$	646.639i	0.524343i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1995.10	−1.53689
$120$	0	0
$121$	1268.34	0.952921
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 125.000i	− 0.0894427i
$126$	0	0
$127$	−494.704	−0.345652	−0.172826	−	0.984952i	$-0.555290\pi$
$127$	−494.704	−0.345652	−0.172826	+	0.984952i	$0.555290\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 488.783i	− 0.325993i	−0.986627	−	0.162997i	$-0.947884\pi$
$131$	− 488.783i	− 0.325993i	0.986627	−	0.162997i	$-0.0521160\pi$
$132$	0	0
$133$	4887.14i	3.18623i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1532.13	0.955464	0.477732	−	0.878506i	$-0.341459\pi$
$137$	1532.13	0.955464	0.477732	+	0.878506i	$0.341459\pi$
$138$	0	0
$139$	− 755.095i	− 0.460765i	−0.973100	−	0.230382i	$-0.926002\pi$
$139$	− 755.095i	− 0.460765i	0.973100	−	0.230382i	$-0.0739977\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−87.3495	−0.0510807
$144$	0	0
$145$	−445.832	−0.255340
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1446.67i	0.795410i	0.917513	+	0.397705i	$0.130193\pi$
$149$	1446.67i	0.795410i	−0.917513	+	0.397705i	$0.869807\pi$
$150$	0	0
$151$	1230.20	0.662998	0.331499	−	0.943456i	$-0.392446\pi$
$151$	1230.20	0.662998	0.331499	+	0.943456i	$0.392446\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	391.654i	0.202957i
$156$	0	0
$157$	− 1773.74i	− 0.901657i	−0.892611	−	0.450828i	$-0.851129\pi$
$157$	− 1773.74i	− 0.901657i	0.892611	−	0.450828i	$-0.148871\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	4478.36	2.19220
$162$	0	0
$163$	1711.28i	0.822319i	0.911563	+	0.411159i	$0.134876\pi$
$163$	1711.28i	0.822319i	−0.911563	+	0.411159i	$0.865124\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3913.26	1.81328	0.906638	−	0.421909i	$-0.138640\pi$
$167$	3913.26	1.81328	0.906638	+	0.421909i	$0.138640\pi$
$168$	0	0
$169$	2075.24	0.944578
$170$	0	0
$171$	0	0
$172$	0	0
$173$	3985.34i	1.75144i	0.482815	+	0.875722i	$0.339614\pi$
$173$	3985.34i	1.75144i	−0.482815	+	0.875722i	$0.660386\pi$
$174$	0	0
$175$	−865.699	−0.373947
$176$	0	0
$177$	0	0
$178$	0	0
$179$	884.734i	0.369431i	0.982792	+	0.184715i	$0.0591363\pi$
$179$	884.734i	0.369431i	−0.982792	+	0.184715i	$0.940864\pi$
$180$	0	0
$181$	2810.77i	1.15427i	0.816649	+	0.577134i	$0.195829\pi$
$181$	2810.77i	1.15427i	−0.816649	+	0.577134i	$0.804171\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1246.66	−0.495439
$186$	0	0
$187$	456.079i	0.178352i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−749.321	−0.283869	−0.141935	−	0.989876i	$-0.545332\pi$
$191$	−749.321	−0.283869	−0.141935	+	0.989876i	$0.545332\pi$
$192$	0	0
$193$	−3969.83	−1.48060	−0.740298	−	0.672279i	$-0.765315\pi$
$193$	−3969.83	−1.48060	−0.740298	+	0.672279i	$0.765315\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2243.20i	0.811275i	0.914034	+	0.405638i	$0.132951\pi$
$197$	2243.20i	0.811275i	−0.914034	+	0.405638i	$0.867049\pi$
$198$	0	0
$199$	672.030	0.239392	0.119696	−	0.992811i	$-0.461808\pi$
$199$	672.030	0.239392	0.119696	+	0.992811i	$0.461808\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	3087.65i	1.06754i
$204$	0	0
$205$	− 459.353i	− 0.156500i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1117.20	0.369753
$210$	0	0
$211$	1935.07i	0.631356i	0.948866	+	0.315678i	$0.102232\pi$
$211$	1935.07i	0.631356i	−0.948866	+	0.315678i	$0.897768\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	970.918	0.307982
$216$	0	0
$217$	2712.43	0.848535
$218$	0	0
$219$	0	0
$220$	0	0
$221$	635.762i	0.193511i
$222$	0	0
$223$	−2492.56	−0.748494	−0.374247	−	0.927329i	$-0.622099\pi$
$223$	−2492.56	−0.748494	−0.374247	+	0.927329i	$0.622099\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3155.27i	0.922567i	0.887253	+	0.461283i	$0.152611\pi$
$227$	3155.27i	0.922567i	−0.887253	+	0.461283i	$0.847389\pi$
$228$	0	0
$229$	− 2299.43i	− 0.663539i	−0.943360	−	0.331770i	$-0.892354\pi$
$229$	− 2299.43i	− 0.663539i	0.943360	−	0.331770i	$-0.107646\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−741.991	−0.208624	−0.104312	−	0.994545i	$-0.533264\pi$
$233$	−741.991	−0.208624	−0.104312	+	0.994545i	$0.533264\pi$
$234$	0	0
$235$	− 363.074i	− 0.100785i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	5786.01	1.56597	0.782984	−	0.622042i	$-0.213697\pi$
$239$	5786.01	1.56597	0.782984	+	0.622042i	$0.213697\pi$
$240$	0	0
$241$	265.054	0.0708449	0.0354224	−	0.999372i	$-0.488722\pi$
$241$	265.054	0.0708449	0.0354224	+	0.999372i	$0.488722\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	4280.48i	1.11620i
$246$	0	0
$247$	1557.35	0.401181
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1762.02i	0.443098i	0.975149	+	0.221549i	$0.0711113\pi$
$251$	1762.02i	0.443098i	−0.975149	+	0.221549i	$0.928889\pi$
$252$	0	0
$253$	− 1023.75i	− 0.254398i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1507.84	0.365980	0.182990	−	0.983115i	$-0.441422\pi$
$257$	1507.84	0.365980	0.182990	+	0.983115i	$0.441422\pi$
$258$	0	0
$259$	8633.85i	2.07136i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	2772.54	0.650046	0.325023	−	0.945706i	$-0.394628\pi$
$263$	2772.54	0.650046	0.325023	+	0.945706i	$0.394628\pi$
$264$	0	0
$265$	−2283.91	−0.529432
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 5166.61i	− 1.17106i	−0.810652	−	0.585528i	$-0.800887\pi$
$269$	− 5166.61i	− 1.17106i	0.810652	−	0.585528i	$-0.199113\pi$
$270$	0	0
$271$	1458.79	0.326994	0.163497	−	0.986544i	$-0.447723\pi$
$271$	1458.79	0.326994	0.163497	+	0.986544i	$0.447723\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	197.899i	0.0433954i
$276$	0	0
$277$	1994.60i	0.432650i	0.976321	+	0.216325i	$0.0694070\pi$
$277$	1994.60i	0.432650i	−0.976321	+	0.216325i	$0.930593\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	311.583	0.0661477	0.0330739	−	0.999453i	$-0.489470\pi$
$281$	311.583	0.0661477	0.0330739	+	0.999453i	$0.489470\pi$
$282$	0	0
$283$	− 6072.33i	− 1.27549i	−0.770249	−	0.637743i	$-0.779868\pi$
$283$	− 6072.33i	− 1.27549i	0.770249	−	0.637743i	$-0.220132\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3181.29	−0.654305
$288$	0	0
$289$	−1593.49	−0.324341
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 2321.06i	− 0.462791i	−0.972860	−	0.231395i	$-0.925671\pi$
$293$	− 2321.06i	− 0.462791i	0.972860	−	0.231395i	$-0.0743291\pi$
$294$	0	0
$295$	−1705.49	−0.336602
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 1427.08i	− 0.276021i
$300$	0	0
$301$	− 6724.19i	− 1.28763i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	1085.34	0.203758
$306$	0	0
$307$	− 2499.43i	− 0.464658i	−0.972637	−	0.232329i	$-0.925365\pi$
$307$	− 2499.43i	− 0.464658i	0.972637	−	0.232329i	$-0.0746347\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−3052.83	−0.556624	−0.278312	−	0.960491i	$-0.589775\pi$
$311$	−3052.83	−0.556624	−0.278312	+	0.960491i	$0.589775\pi$
$312$	0	0
$313$	6179.23	1.11588	0.557941	−	0.829881i	$-0.311592\pi$
$313$	6179.23	1.11588	0.557941	+	0.829881i	$0.311592\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 3116.20i	− 0.552124i	−0.961140	−	0.276062i	$-0.910971\pi$
$317$	− 3116.20i	− 0.552124i	0.961140	−	0.276062i	$-0.0890295\pi$
$318$	0	0
$319$	705.836	0.123885
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 8131.39i	− 1.40075i
$324$	0	0
$325$	275.866i	0.0470839i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−2514.51	−0.421365
$330$	0	0
$331$	4157.26i	0.690343i	0.938540	+	0.345171i	$0.112179\pi$
$331$	4157.26i	0.690343i	−0.938540	+	0.345171i	$0.887821\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	2646.19	0.431572
$336$	0	0
$337$	−8123.42	−1.31309	−0.656544	−	0.754287i	$-0.727983\pi$
$337$	−8123.42	−1.31309	−0.656544	+	0.754287i	$0.727983\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 620.062i	− 0.0984699i
$342$	0	0
$343$	17767.5	2.79695
$344$	0	0
$345$	0	0
$346$	0	0
$347$	4620.55i	0.714824i	0.933947	+	0.357412i	$0.116341\pi$
$347$	4620.55i	0.714824i	−0.933947	+	0.357412i	$0.883659\pi$
$348$	0	0
$349$	5560.98i	0.852929i	0.904504	+	0.426465i	$0.140241\pi$
$349$	5560.98i	0.852929i	−0.904504	+	0.426465i	$0.859759\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−12031.8	−1.81413	−0.907066	−	0.420989i	$-0.861683\pi$
$353$	−12031.8	−1.81413	−0.907066	+	0.420989i	$0.861683\pi$
$354$	0	0
$355$	1907.70i	0.285212i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1533.24	0.225408	0.112704	−	0.993629i	$-0.464049\pi$
$359$	1533.24	0.225408	0.112704	+	0.993629i	$0.464049\pi$
$360$	0	0
$361$	−13059.5	−1.90399
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 4384.51i	− 0.628755i
$366$	0	0
$367$	5480.04	0.779444	0.389722	−	0.920933i	$-0.372571\pi$
$367$	5480.04	0.779444	0.389722	+	0.920933i	$0.372571\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	15817.4i	2.21348i
$372$	0	0
$373$	6225.70i	0.864221i	0.901821	+	0.432111i	$0.142231\pi$
$373$	6225.70i	0.864221i	−0.901821	+	0.432111i	$0.857769\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	983.917	0.134415
$378$	0	0
$379$	11172.0i	1.51416i	0.653325	+	0.757078i	$0.273374\pi$
$379$	11172.0i	1.51416i	−0.653325	+	0.757078i	$0.726626\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	7621.03	1.01675	0.508376	−	0.861135i	$-0.330246\pi$
$383$	7621.03	1.01675	0.508376	+	0.861135i	$0.330246\pi$
$384$	0	0
$385$	1370.57	0.181430
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 5546.31i	− 0.722903i	−0.932391	−	0.361451i	$-0.882281\pi$
$389$	− 5546.31i	− 0.722903i	0.932391	−	0.361451i	$-0.117719\pi$
$390$	0	0
$391$	−7451.25	−0.963749
$392$	0	0
$393$	0	0
$394$	0	0
$395$	1019.75i	0.129897i
$396$	0	0
$397$	− 11025.2i	− 1.39380i	−0.717169	−	0.696900i	$-0.754562\pi$
$397$	− 11025.2i	− 1.39380i	0.717169	−	0.696900i	$-0.245438\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−10522.3	−1.31037	−0.655186	−	0.755467i	$-0.727410\pi$
$401$	−10522.3	−1.31037	−0.655186	+	0.755467i	$0.727410\pi$
$402$	0	0
$403$	− 864.350i	− 0.106840i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1973.70	0.240375
$408$	0	0
$409$	2320.17	0.280502	0.140251	−	0.990116i	$-0.455209\pi$
$409$	2320.17	0.280502	0.140251	+	0.990116i	$0.455209\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	11811.5i	1.40728i
$414$	0	0
$415$	4983.27	0.589444
$416$	0	0
$417$	0	0
$418$	0	0
$419$	9212.17i	1.07409i	0.843554	+	0.537045i	$0.180459\pi$
$419$	9212.17i	1.07409i	−0.843554	+	0.537045i	$0.819541\pi$
$420$	0	0
$421$	6967.70i	0.806615i	0.915064	+	0.403308i	$0.132140\pi$
$421$	6967.70i	0.806615i	−0.915064	+	0.403308i	$0.867860\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1440.38	0.164397
$426$	0	0
$427$	− 7516.59i	− 0.851882i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−11247.1	−1.25697	−0.628486	−	0.777821i	$-0.716325\pi$
$431$	−11247.1	−1.25697	−0.628486	+	0.777821i	$0.716325\pi$
$432$	0	0
$433$	2589.27	0.287372	0.143686	−	0.989623i	$-0.454104\pi$
$433$	2589.27	0.287372	0.143686	+	0.989623i	$0.454104\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	18252.4i	1.99801i
$438$	0	0
$439$	−4220.01	−0.458793	−0.229396	−	0.973333i	$-0.573675\pi$
$439$	−4220.01	−0.458793	−0.229396	+	0.973333i	$0.573675\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 9764.04i	− 1.04719i	−0.851969	−	0.523593i	$-0.824591\pi$
$443$	− 9764.04i	− 1.04719i	0.851969	−	0.523593i	$-0.175409\pi$
$444$	0	0
$445$	− 862.279i	− 0.0918560i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	17159.3	1.80355	0.901777	−	0.432202i	$-0.142263\pi$
$449$	17159.3	1.80355	0.901777	+	0.432202i	$0.142263\pi$
$450$	0	0
$451$	727.242i	0.0759302i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1910.53	0.196851
$456$	0	0
$457$	13027.3	1.33346	0.666730	−	0.745299i	$-0.267693\pi$
$457$	13027.3	1.33346	0.666730	+	0.745299i	$0.267693\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	313.396i	0.0316623i	0.999875	+	0.0158312i	$0.00503943\pi$
$461$	313.396i	0.0316623i	−0.999875	+	0.0158312i	$0.994961\pi$
$462$	0	0
$463$	12166.5	1.22123	0.610613	−	0.791930i	$-0.290923\pi$
$463$	12166.5	1.22123	0.610613	+	0.791930i	$0.290923\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 1844.42i	− 0.182761i	−0.995816	−	0.0913806i	$-0.970872\pi$
$467$	− 1844.42i	− 0.182761i	0.995816	−	0.0913806i	$-0.0291280\pi$
$468$	0	0
$469$	− 18326.4i	− 1.80434i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−1537.15	−0.149425
$474$	0	0
$475$	− 3528.32i	− 0.340822i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−16355.9	−1.56017	−0.780084	−	0.625675i	$-0.784824\pi$
$479$	−16355.9	−1.56017	−0.780084	+	0.625675i	$0.784824\pi$
$480$	0	0
$481$	2751.28	0.260806
$482$	0	0
$483$	0	0
$484$	0	0
$485$	5292.43i	0.495499i
$486$	0	0
$487$	−11824.6	−1.10025	−0.550126	−	0.835082i	$-0.685420\pi$
$487$	−11824.6	−1.10025	−0.550126	+	0.835082i	$0.685420\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 7931.40i	− 0.729000i	−0.931203	−	0.364500i	$-0.881240\pi$
$491$	− 7931.40i	− 0.729000i	0.931203	−	0.364500i	$-0.118760\pi$
$492$	0	0
$493$	− 5137.34i	− 0.469319i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	13212.0	1.19243
$498$	0	0
$499$	− 2734.69i	− 0.245333i	−0.992448	−	0.122667i	$-0.960855\pi$
$499$	− 2734.69i	− 0.245333i	0.992448	−	0.122667i	$-0.0391446\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	7297.64	0.646890	0.323445	−	0.946247i	$-0.395159\pi$
$503$	7297.64	0.646890	0.323445	+	0.946247i	$0.395159\pi$
$504$	0	0
$505$	−3449.96	−0.304002
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 10972.4i	− 0.955486i	−0.878500	−	0.477743i	$-0.841455\pi$
$509$	− 10972.4i	− 0.955486i	0.878500	−	0.477743i	$-0.158545\pi$
$510$	0	0
$511$	−30365.3	−2.62873
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 6449.81i	− 0.551869i
$516$	0	0
$517$	574.815i	0.0488982i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−7693.43	−0.646939	−0.323470	−	0.946239i	$-0.604849\pi$
$521$	−7693.43	−0.646939	−0.323470	+	0.946239i	$0.604849\pi$
$522$	0	0
$523$	− 14535.3i	− 1.21527i	−0.794217	−	0.607634i	$-0.792119\pi$
$523$	− 14535.3i	− 1.21527i	0.794217	−	0.607634i	$-0.207881\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−4513.04	−0.373038
$528$	0	0
$529$	4558.69	0.374676
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1013.76i	0.0823840i
$534$	0	0
$535$	−9218.70	−0.744970
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 6776.81i	− 0.541555i
$540$	0	0
$541$	19131.8i	1.52040i	0.649686	+	0.760202i	$0.274900\pi$
$541$	19131.8i	1.52040i	−0.649686	+	0.760202i	$0.725100\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−1702.99	−0.133850
$546$	0	0
$547$	− 7142.78i	− 0.558324i	−0.960244	−	0.279162i	$-0.909943\pi$
$547$	− 7142.78i	− 0.558324i	0.960244	−	0.279162i	$-0.0900566\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−12584.3	−0.972974
$552$	0	0
$553$	7062.39	0.543080
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 11347.8i	− 0.863236i	−0.902057	−	0.431618i	$-0.857943\pi$
$557$	− 11347.8i	− 0.863236i	0.902057	−	0.431618i	$-0.142057\pi$
$558$	0	0
$559$	−2142.74	−0.162126
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 8802.19i	− 0.658913i	−0.944171	−	0.329456i	$-0.893135\pi$
$563$	− 8802.19i	− 0.658913i	0.944171	−	0.329456i	$-0.106865\pi$
$564$	0	0
$565$	− 789.153i	− 0.0587609i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−16714.7	−1.23149	−0.615744	−	0.787946i	$-0.711145\pi$
$569$	−16714.7	−1.23149	−0.615744	+	0.787946i	$0.711145\pi$
$570$	0	0
$571$	17386.8i	1.27428i	0.770746	+	0.637142i	$0.219884\pi$
$571$	17386.8i	1.27428i	−0.770746	+	0.637142i	$0.780116\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−3233.20	−0.234493
$576$	0	0
$577$	2260.52	0.163097	0.0815483	−	0.996669i	$-0.474014\pi$
$577$	2260.52	0.163097	0.0815483	+	0.996669i	$0.474014\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 34512.1i	− 2.46438i
$582$	0	0
$583$	3615.86	0.256867
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 25591.0i	− 1.79941i	−0.436498	−	0.899705i	$-0.643781\pi$
$587$	− 25591.0i	− 1.79941i	0.436498	−	0.899705i	$-0.356219\pi$
$588$	0	0
$589$	11055.0i	0.773369i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	10053.7	0.696218	0.348109	−	0.937454i	$-0.386824\pi$
$593$	10053.7	0.696218	0.348109	+	0.937454i	$0.386824\pi$
$594$	0	0
$595$	− 9975.49i	− 0.687320i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−7086.68	−0.483395	−0.241698	−	0.970352i	$-0.577704\pi$
$599$	−7086.68	−0.483395	−0.241698	+	0.970352i	$0.577704\pi$
$600$	0	0
$601$	6673.11	0.452915	0.226458	−	0.974021i	$-0.427286\pi$
$601$	6673.11	0.452915	0.226458	+	0.974021i	$0.427286\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	6341.69i	0.426159i
$606$	0	0
$607$	15205.6	1.01677	0.508383	−	0.861131i	$-0.330243\pi$
$607$	15205.6	1.01677	0.508383	+	0.861131i	$0.330243\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	801.278i	0.0530544i
$612$	0	0
$613$	− 8147.64i	− 0.536836i	−0.963303	−	0.268418i	$-0.913499\pi$
$613$	− 8147.64i	− 0.536836i	0.963303	−	0.268418i	$-0.0865008\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−449.110	−0.0293039	−0.0146519	−	0.999893i	$-0.504664\pi$
$617$	−449.110	−0.0293039	−0.0146519	+	0.999893i	$0.504664\pi$
$618$	0	0
$619$	− 27800.4i	− 1.80516i	−0.430524	−	0.902579i	$-0.641671\pi$
$619$	− 27800.4i	− 1.80516i	0.430524	−	0.902579i	$-0.358329\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−5971.79	−0.384037
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 14365.3i	− 0.910623i
$630$	0	0
$631$	−14793.5	−0.933315	−0.466657	−	0.884438i	$-0.654542\pi$
$631$	−14793.5	−0.933315	−0.466657	+	0.884438i	$0.654542\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 2473.52i	− 0.154580i
$636$	0	0
$637$	− 9446.70i	− 0.587585i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	13853.8	0.853655	0.426828	−	0.904333i	$-0.359631\pi$
$641$	13853.8	0.853655	0.426828	+	0.904333i	$0.359631\pi$
$642$	0	0
$643$	4978.60i	0.305345i	0.988277	+	0.152672i	$0.0487880\pi$
$643$	4978.60i	0.305345i	−0.988277	+	0.152672i	$0.951212\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−1903.39	−0.115657	−0.0578284	−	0.998327i	$-0.518418\pi$
$647$	−1903.39	−0.115657	−0.0578284	+	0.998327i	$0.518418\pi$
$648$	0	0
$649$	2700.11	0.163311
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 10877.0i	− 0.651839i	−0.945397	−	0.325920i	$-0.894326\pi$
$653$	− 10877.0i	− 0.651839i	0.945397	−	0.325920i	$-0.105674\pi$
$654$	0	0
$655$	2443.91	0.145789
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 30611.3i	− 1.80948i	−0.425965	−	0.904739i	$-0.640065\pi$
$659$	− 30611.3i	− 1.80948i	0.425965	−	0.904739i	$-0.359935\pi$
$660$	0	0
$661$	− 16098.8i	− 0.947310i	−0.880710	−	0.473655i	$-0.842934\pi$
$661$	− 16098.8i	− 0.947310i	0.880710	−	0.473655i	$-0.157066\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−24435.7	−1.42493
$666$	0	0
$667$	11531.7i	0.669429i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−1718.29	−0.0988583
$672$	0	0
$673$	−23837.8	−1.36535	−0.682673	−	0.730724i	$-0.739183\pi$
$673$	−23837.8	−1.36535	−0.682673	+	0.730724i	$0.739183\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 25235.3i	− 1.43260i	−0.697791	−	0.716302i	$-0.745833\pi$
$677$	− 25235.3i	− 1.43260i	0.697791	−	0.716302i	$-0.254167\pi$
$678$	0	0
$679$	36653.2	2.07161
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 19975.3i	− 1.11909i	−0.828802	−	0.559543i	$-0.810977\pi$
$683$	− 19975.3i	− 1.11909i	0.828802	−	0.559543i	$-0.189023\pi$
$684$	0	0
$685$	7660.64i	0.427297i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	5040.42	0.278700
$690$	0	0
$691$	− 13772.3i	− 0.758209i	−0.925354	−	0.379104i	$-0.876232\pi$
$691$	− 13772.3i	− 0.758209i	0.925354	−	0.379104i	$-0.123768\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	3775.48	0.206060
$696$	0	0
$697$	5293.14	0.287650
$698$	0	0
$699$	0	0
$700$	0	0
$701$	6347.90i	0.342021i	0.985269	+	0.171011i	$0.0547032\pi$
$701$	6347.90i	0.342021i	−0.985269	+	0.171011i	$0.945297\pi$
$702$	0	0
$703$	−35188.9	−1.88787
$704$	0	0
$705$	0	0
$706$	0	0
$707$	23893.0i	1.27099i
$708$	0	0
$709$	17910.5i	0.948719i	0.880331	+	0.474360i	$0.157320\pi$
$709$	17910.5i	0.948719i	−0.880331	+	0.474360i	$0.842680\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	10130.3	0.532096
$714$	0	0
$715$	− 436.748i	− 0.0228440i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	36601.6	1.89849	0.949243	−	0.314545i	$-0.101852\pi$
$719$	36601.6	1.89849	0.949243	+	0.314545i	$0.101852\pi$
$720$	0	0
$721$	−44668.8	−2.30728
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 2229.16i	− 0.114192i
$726$	0	0
$727$	−2644.18	−0.134893	−0.0674466	−	0.997723i	$-0.521485\pi$
$727$	−2644.18	−0.134893	−0.0674466	+	0.997723i	$0.521485\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	11187.9i	0.566075i
$732$	0	0
$733$	33452.1i	1.68565i	0.538188	+	0.842825i	$0.319109\pi$
$733$	33452.1i	1.68565i	−0.538188	+	0.842825i	$0.680891\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−4189.42	−0.209388
$738$	0	0
$739$	− 25834.9i	− 1.28600i	−0.765867	−	0.642999i	$-0.777690\pi$
$739$	− 25834.9i	− 1.28600i	0.765867	−	0.642999i	$-0.222310\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	7625.09	0.376497	0.188249	−	0.982121i	$-0.439719\pi$
$743$	7625.09	0.376497	0.188249	+	0.982121i	$0.439719\pi$
$744$	0	0
$745$	−7233.37	−0.355718
$746$	0	0
$747$	0	0
$748$	0	0
$749$	63845.0i	3.11461i
$750$	0	0
$751$	−25362.7	−1.23235	−0.616177	−	0.787607i	$-0.711320\pi$
$751$	−25362.7	−1.23235	−0.616177	+	0.787607i	$0.711320\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	6151.02i	0.296502i
$756$	0	0
$757$	− 38948.0i	− 1.87000i	−0.354648	−	0.935000i	$-0.615399\pi$
$757$	− 38948.0i	− 1.87000i	0.354648	−	0.935000i	$-0.384601\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	13722.2	0.653654	0.326827	−	0.945084i	$-0.394021\pi$
$761$	13722.2	0.653654	0.326827	+	0.945084i	$0.394021\pi$
$762$	0	0
$763$	11794.2i	0.559606i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	3763.89	0.177192
$768$	0	0
$769$	18689.5	0.876414	0.438207	−	0.898874i	$-0.355614\pi$
$769$	18689.5	0.876414	0.438207	+	0.898874i	$0.355614\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 6386.49i	− 0.297162i	−0.988900	−	0.148581i	$-0.952529\pi$
$773$	− 6386.49i	− 0.297162i	0.988900	−	0.148581i	$-0.0474705\pi$
$774$	0	0
$775$	−1958.27	−0.0907653
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 12965.9i	− 0.596345i
$780$	0	0
$781$	− 3020.25i	− 0.138378i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	8868.71	0.403233
$786$	0	0
$787$	19410.3i	0.879167i	0.898202	+	0.439583i	$0.144874\pi$
$787$	19410.3i	0.879167i	−0.898202	+	0.439583i	$0.855126\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−5465.35	−0.245671
$792$	0	0
$793$	−2395.25	−0.107261
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 30740.8i	− 1.36624i	−0.730305	−	0.683121i	$-0.760622\pi$
$797$	− 30740.8i	− 1.36624i	0.730305	−	0.683121i	$-0.239378\pi$
$798$	0	0
$799$	4183.72	0.185243
$800$	0	0
$801$	0	0
$802$	0	0
$803$	6941.51i	0.305057i
$804$	0	0
$805$	22391.8i	0.980382i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−31805.5	−1.38223	−0.691114	−	0.722746i	$-0.742880\pi$
$809$	−31805.5	−1.38223	−0.691114	+	0.722746i	$0.742880\pi$
$810$	0	0
$811$	− 31014.7i	− 1.34288i	−0.741060	−	0.671439i	$-0.765677\pi$
$811$	− 31014.7i	− 1.34288i	0.741060	−	0.671439i	$-0.234323\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−8556.41	−0.367752
$816$	0	0
$817$	27405.7	1.17357
$818$	0	0
$819$	0	0
$820$	0	0
$821$	12021.4i	0.511021i	0.966806	+	0.255511i	$0.0822436\pi$
$821$	12021.4i	0.511021i	−0.966806	+	0.255511i	$0.917756\pi$
$822$	0	0
$823$	−15489.0	−0.656032	−0.328016	−	0.944672i	$-0.606380\pi$
$823$	−15489.0	−0.656032	−0.328016	+	0.944672i	$0.606380\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	17097.2i	0.718898i	0.933165	+	0.359449i	$0.117035\pi$
$827$	17097.2i	0.718898i	−0.933165	+	0.359449i	$0.882965\pi$
$828$	0	0
$829$	− 9885.97i	− 0.414178i	−0.978322	−	0.207089i	$-0.933601\pi$
$829$	− 9885.97i	− 0.414178i	0.978322	−	0.207089i	$-0.0663990\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−49324.2	−2.05160
$834$	0	0
$835$	19566.3i	0.810922i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−39204.3	−1.61321	−0.806606	−	0.591090i	$-0.798698\pi$
$839$	−39204.3	−1.61321	−0.806606	+	0.591090i	$0.798698\pi$
$840$	0	0
$841$	16438.4	0.674007
$842$	0	0
$843$	0	0
$844$	0	0
$845$	10376.2i	0.422428i
$846$	0	0
$847$	43920.0	1.78171
$848$	0	0
$849$	0	0
$850$	0	0
$851$	32245.5i	1.29890i
$852$	0	0
$853$	− 45054.1i	− 1.80847i	−0.427038	−	0.904234i	$-0.640443\pi$
$853$	− 45054.1i	− 1.80847i	0.427038	−	0.904234i	$-0.359557\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	48464.2	1.93174	0.965872	−	0.259021i	$-0.0834000\pi$
$857$	48464.2	1.93174	0.965872	+	0.259021i	$0.0834000\pi$
$858$	0	0
$859$	− 1000.76i	− 0.0397503i	−0.999802	−	0.0198751i	$-0.993673\pi$
$859$	− 1000.76i	− 0.0397503i	0.999802	−	0.0198751i	$-0.00632687\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	22485.5	0.886922	0.443461	−	0.896294i	$-0.353750\pi$
$863$	22485.5	0.886922	0.443461	+	0.896294i	$0.353750\pi$
$864$	0	0
$865$	−19926.7	−0.783270
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 1614.46i	− 0.0630228i
$870$	0	0
$871$	−5839.94	−0.227186
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 4328.50i	− 0.167234i
$876$	0	0
$877$	43726.2i	1.68361i	0.539780	+	0.841806i	$0.318507\pi$
$877$	43726.2i	1.68361i	−0.539780	+	0.841806i	$0.681493\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−20290.0	−0.775923	−0.387962	−	0.921675i	$-0.626821\pi$
$881$	−20290.0	−0.775923	−0.387962	+	0.921675i	$0.626821\pi$
$882$	0	0
$883$	14887.3i	0.567382i	0.958916	+	0.283691i	$0.0915590\pi$
$883$	14887.3i	0.567382i	−0.958916	+	0.283691i	$0.908441\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	34359.3	1.30065	0.650323	−	0.759658i	$-0.274633\pi$
$887$	34359.3	1.30065	0.650323	+	0.759658i	$0.274633\pi$
$888$	0	0
$889$	−17130.6	−0.646278
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 10248.3i	− 0.384040i
$894$	0	0
$895$	−4423.67	−0.165214
$896$	0	0
$897$	0	0
$898$	0	0
$899$	6984.47i	0.259116i
$900$	0	0
$901$	− 26317.6i	− 0.973103i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−14053.8	−0.516205
$906$	0	0
$907$	− 53363.9i	− 1.95360i	−0.214144	−	0.976802i	$-0.568696\pi$
$907$	− 53363.9i	− 1.95360i	0.214144	−	0.976802i	$-0.431304\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−25213.9	−0.916984	−0.458492	−	0.888699i	$-0.651610\pi$
$911$	−25213.9	−0.916984	−0.458492	+	0.888699i	$0.651610\pi$
$912$	0	0
$913$	−7889.46	−0.285984
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 16925.5i	− 0.609521i
$918$	0	0
$919$	−3747.02	−0.134497	−0.0672485	−	0.997736i	$-0.521422\pi$
$919$	−3747.02	−0.134497	−0.0672485	+	0.997736i	$0.521422\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 4210.15i	− 0.150140i
$924$	0	0
$925$	− 6233.29i	− 0.221567i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	27726.7	0.979209	0.489604	−	0.871945i	$-0.337141\pi$
$929$	27726.7	0.979209	0.489604	+	0.871945i	$0.337141\pi$
$930$	0	0
$931$	120823.i	4.25330i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−2280.40	−0.0797614
$936$	0	0
$937$	29333.2	1.02270	0.511352	−	0.859371i	$-0.329145\pi$
$937$	29333.2	1.02270	0.511352	+	0.859371i	$0.329145\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 24218.4i	− 0.838998i	−0.907756	−	0.419499i	$-0.862206\pi$
$941$	− 24218.4i	− 0.838998i	0.907756	−	0.419499i	$-0.137794\pi$
$942$	0	0
$943$	−11881.4	−0.410299
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 50824.6i	− 1.74401i	−0.489497	−	0.872005i	$-0.662820\pi$
$947$	− 50824.6i	− 1.74401i	0.489497	−	0.872005i	$-0.337180\pi$
$948$	0	0
$949$	9676.28i	0.330986i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	10155.1	0.345180	0.172590	−	0.984994i	$-0.444786\pi$
$953$	10155.1	0.345180	0.172590	+	0.984994i	$0.444786\pi$
$954$	0	0
$955$	− 3746.61i	− 0.126950i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	53054.5	1.78646
$960$	0	0
$961$	−23655.3	−0.794042
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 19849.2i	− 0.662142i
$966$	0	0
$967$	28225.7	0.938652	0.469326	−	0.883025i	$-0.344497\pi$
$967$	28225.7	0.938652	0.469326	+	0.883025i	$0.344497\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 47631.4i	− 1.57422i	−0.616814	−	0.787109i	$-0.711577\pi$
$971$	− 47631.4i	− 1.57422i	0.616814	−	0.787109i	$-0.288423\pi$
$972$	0	0
$973$	− 26147.4i	− 0.861508i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−13542.4	−0.443460	−0.221730	−	0.975108i	$-0.571170\pi$
$977$	−13542.4	−0.443460	−0.221730	+	0.975108i	$0.571170\pi$
$978$	0	0
$979$	1365.15i	0.0445663i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−35108.5	−1.13915	−0.569576	−	0.821938i	$-0.692893\pi$
$983$	−35108.5	−1.13915	−0.569576	+	0.821938i	$0.692893\pi$
$984$	0	0
$985$	−11216.0	−0.362813
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 25113.4i	− 0.807440i
$990$	0	0
$991$	42816.1	1.37245	0.686225	−	0.727390i	$-0.259267\pi$
$991$	42816.1	1.37245	0.686225	+	0.727390i	$0.259267\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	3360.15i	0.107059i
$996$	0	0
$997$	− 13029.0i	− 0.413873i	−0.978354	−	0.206936i	$-0.933651\pi$
$997$	− 13029.0i	− 0.413873i	0.978354	−	0.206936i	$-0.0663493\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1440.4.k.c.721.12		12
3.2	odd	2		160.4.d.a.81.10		12
4.3	odd	2		360.4.k.c.181.12		12
8.3	odd	2		360.4.k.c.181.11		12
8.5	even	2	inner	1440.4.k.c.721.6		12
12.11	even	2		40.4.d.a.21.1	✓	12
15.2	even	4		800.4.f.c.49.10		12
15.8	even	4		800.4.f.b.49.3		12
15.14	odd	2		800.4.d.d.401.3		12
24.5	odd	2		160.4.d.a.81.3		12
24.11	even	2		40.4.d.a.21.2	yes	12
48.5	odd	4		1280.4.a.bd.1.5		6
48.11	even	4		1280.4.a.bb.1.2		6
48.29	odd	4		1280.4.a.ba.1.2		6
48.35	even	4		1280.4.a.bc.1.5		6
60.23	odd	4		200.4.f.c.149.4		12
60.47	odd	4		200.4.f.b.149.9		12
60.59	even	2		200.4.d.b.101.12		12
120.29	odd	2		800.4.d.d.401.10		12
120.53	even	4		800.4.f.c.49.9		12
120.59	even	2		200.4.d.b.101.11		12
120.77	even	4		800.4.f.b.49.4		12
120.83	odd	4		200.4.f.b.149.10		12
120.107	odd	4		200.4.f.c.149.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.4.d.a.21.1	✓	12	12.11	even	2
40.4.d.a.21.2	yes	12	24.11	even	2
160.4.d.a.81.3		12	24.5	odd	2
160.4.d.a.81.10		12	3.2	odd	2
200.4.d.b.101.11		12	120.59	even	2
200.4.d.b.101.12		12	60.59	even	2
200.4.f.b.149.9		12	60.47	odd	4
200.4.f.b.149.10		12	120.83	odd	4
200.4.f.c.149.3		12	120.107	odd	4
200.4.f.c.149.4		12	60.23	odd	4
360.4.k.c.181.11		12	8.3	odd	2
360.4.k.c.181.12		12	4.3	odd	2
800.4.d.d.401.3		12	15.14	odd	2
800.4.d.d.401.10		12	120.29	odd	2
800.4.f.b.49.3		12	15.8	even	4
800.4.f.b.49.4		12	120.77	even	4
800.4.f.c.49.9		12	120.53	even	4
800.4.f.c.49.10		12	15.2	even	4
1280.4.a.ba.1.2		6	48.29	odd	4
1280.4.a.bb.1.2		6	48.11	even	4
1280.4.a.bc.1.5		6	48.35	even	4
1280.4.a.bd.1.5		6	48.5	odd	4
1440.4.k.c.721.6		12	8.5	even	2	inner
1440.4.k.c.721.12		12	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:	\( N \)	\(=\)	\( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1440.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+5.00000i q^{5} +34.6280 q^{7} +O(q^{10})\)	\(q+5.00000i q^{5} +34.6280 q^{7} -7.91595i q^{11} -11.0346i q^{13} -57.6152 q^{17} +141.133i q^{19} +129.328 q^{23} -25.0000 q^{25} +89.1664i q^{29} +78.3307 q^{31} +173.140i q^{35} +249.332i q^{37} -91.8705 q^{41} -194.184i q^{43} -72.6149 q^{47} +856.096 q^{49} +456.782i q^{53} +39.5797 q^{55} +341.098i q^{59} -217.067i q^{61} +55.1731 q^{65} -529.237i q^{67} +381.540 q^{71} -876.902 q^{73} -274.113i q^{77} +203.950 q^{79} -996.654i q^{83} -288.076i q^{85} -172.456 q^{89} -382.107i q^{91} -705.664 q^{95} +1058.49 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 28 q^{7}+O(q^{10}) \) 12 * q - 28 * q^7	\( 12 q - 28 q^{7} + 604 q^{23} - 300 q^{25} + 264 q^{31} - 40 q^{41} - 940 q^{47} + 1308 q^{49} - 440 q^{55} - 1592 q^{71} + 432 q^{73} - 2016 q^{79} + 424 q^{89} - 1520 q^{95} - 1584 q^{97}+O(q^{100}) \) 12 * q - 28 * q^7 + 604 * q^23 - 300 * q^25 + 264 * q^31 - 40 * q^41 - 940 * q^47 + 1308 * q^49 - 440 * q^55 - 1592 * q^71 + 432 * q^73 - 2016 * q^79 + 424 * q^89 - 1520 * q^95 - 1584 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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