LMFDB - Embedded newform 1344.4.a.ba.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1344,4,Mod(1,1344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1344, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("1344.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1344 = 2^{6} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1344.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:	$79.2985670477$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 21)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1344.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+3.00000 q^{3} +18.0000 q^{5} +7.00000 q^{7} +9.00000 q^{9} +O(q^{10})$

$q+3.00000 q^{3} +18.0000 q^{5} +7.00000 q^{7} +9.00000 q^{9} +36.0000 q^{11} +34.0000 q^{13} +54.0000 q^{15} +42.0000 q^{17} +124.000 q^{19} +21.0000 q^{21} +199.000 q^{25} +27.0000 q^{27} -102.000 q^{29} -160.000 q^{31} +108.000 q^{33} +126.000 q^{35} -398.000 q^{37} +102.000 q^{39} -318.000 q^{41} +268.000 q^{43} +162.000 q^{45} +240.000 q^{47} +49.0000 q^{49} +126.000 q^{51} +498.000 q^{53} +648.000 q^{55} +372.000 q^{57} +132.000 q^{59} -398.000 q^{61} +63.0000 q^{63} +612.000 q^{65} -92.0000 q^{67} -720.000 q^{71} -502.000 q^{73} +597.000 q^{75} +252.000 q^{77} -1024.00 q^{79} +81.0000 q^{81} +204.000 q^{83} +756.000 q^{85} -306.000 q^{87} +354.000 q^{89} +238.000 q^{91} -480.000 q^{93} +2232.00 q^{95} -286.000 q^{97} +324.000 q^{99} +O(q^{100})$

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
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	18.0000	1.60997	0.804984	−	0.593296i	$-0.202174\pi$
$5$	18.0000	1.60997	0.804984	+	0.593296i	$0.202174\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	36.0000	0.986764	0.493382	−	0.869813i	$-0.335760\pi$
$11$	36.0000	0.986764	0.493382	+	0.869813i	$0.335760\pi$
$12$	0	0
$13$	34.0000	0.725377	0.362689	−	0.931910i	$-0.381859\pi$
$13$	34.0000	0.725377	0.362689	+	0.931910i	$0.381859\pi$
$14$	0	0
$15$	54.0000	0.929516
$16$	0	0
$17$	42.0000	0.599206	0.299603	−	0.954064i	$-0.403146\pi$
$17$	42.0000	0.599206	0.299603	+	0.954064i	$0.403146\pi$
$18$	0	0
$19$	124.000	1.49724	0.748620	−	0.663000i	$-0.230717\pi$
$19$	124.000	1.49724	0.748620	+	0.663000i	$0.230717\pi$
$20$	0	0
$21$	21.0000	0.218218
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	199.000	1.59200
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−102.000	−0.653135	−0.326568	−	0.945174i	$-0.605892\pi$
$29$	−102.000	−0.653135	−0.326568	+	0.945174i	$0.605892\pi$
$30$	0	0
$31$	−160.000	−0.926995	−0.463498	−	0.886098i	$-0.653406\pi$
$31$	−160.000	−0.926995	−0.463498	+	0.886098i	$0.653406\pi$
$32$	0	0
$33$	108.000	0.569709
$34$	0	0
$35$	126.000	0.608511
$36$	0	0
$37$	−398.000	−1.76840	−0.884200	−	0.467109i	$-0.845296\pi$
$37$	−398.000	−1.76840	−0.884200	+	0.467109i	$0.845296\pi$
$38$	0	0
$39$	102.000	0.418797
$40$	0	0
$41$	−318.000	−1.21130	−0.605649	−	0.795732i	$-0.707087\pi$
$41$	−318.000	−1.21130	−0.605649	+	0.795732i	$0.707087\pi$
$42$	0	0
$43$	268.000	0.950456	0.475228	−	0.879863i	$-0.342366\pi$
$43$	268.000	0.950456	0.475228	+	0.879863i	$0.342366\pi$
$44$	0	0
$45$	162.000	0.536656
$46$	0	0
$47$	240.000	0.744843	0.372421	−	0.928064i	$-0.378528\pi$
$47$	240.000	0.744843	0.372421	+	0.928064i	$0.378528\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	126.000	0.345952
$52$	0	0
$53$	498.000	1.29067	0.645335	−	0.763899i	$-0.276718\pi$
$53$	498.000	1.29067	0.645335	+	0.763899i	$0.276718\pi$
$54$	0	0
$55$	648.000	1.58866
$56$	0	0
$57$	372.000	0.864432
$58$	0	0
$59$	132.000	0.291270	0.145635	−	0.989338i	$-0.453477\pi$
$59$	132.000	0.291270	0.145635	+	0.989338i	$0.453477\pi$
$60$	0	0
$61$	−398.000	−0.835388	−0.417694	−	0.908588i	$-0.637162\pi$
$61$	−398.000	−0.835388	−0.417694	+	0.908588i	$0.637162\pi$
$62$	0	0
$63$	63.0000	0.125988
$64$	0	0
$65$	612.000	1.16783
$66$	0	0
$67$	−92.0000	−0.167755	−0.0838775	−	0.996476i	$-0.526730\pi$
$67$	−92.0000	−0.167755	−0.0838775	+	0.996476i	$0.526730\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−720.000	−1.20350	−0.601748	−	0.798686i	$-0.705529\pi$
$71$	−720.000	−1.20350	−0.601748	+	0.798686i	$0.705529\pi$
$72$	0	0
$73$	−502.000	−0.804858	−0.402429	−	0.915451i	$-0.631834\pi$
$73$	−502.000	−0.804858	−0.402429	+	0.915451i	$0.631834\pi$
$74$	0	0
$75$	597.000	0.919142
$76$	0	0
$77$	252.000	0.372962
$78$	0	0
$79$	−1024.00	−1.45834	−0.729171	−	0.684332i	$-0.760094\pi$
$79$	−1024.00	−1.45834	−0.729171	+	0.684332i	$0.760094\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	204.000	0.269782	0.134891	−	0.990860i	$-0.456932\pi$
$83$	204.000	0.269782	0.134891	+	0.990860i	$0.456932\pi$
$84$	0	0
$85$	756.000	0.964703
$86$	0	0
$87$	−306.000	−0.377088
$88$	0	0
$89$	354.000	0.421617	0.210809	−	0.977527i	$-0.432390\pi$
$89$	354.000	0.421617	0.210809	+	0.977527i	$0.432390\pi$
$90$	0	0
$91$	238.000	0.274167
$92$	0	0
$93$	−480.000	−0.535201
$94$	0	0
$95$	2232.00	2.41051
$96$	0	0
$97$	−286.000	−0.299370	−0.149685	−	0.988734i	$-0.547826\pi$
$97$	−286.000	−0.299370	−0.149685	+	0.988734i	$0.547826\pi$
$98$	0	0
$99$	324.000	0.328921
$100$	0	0
$101$	−414.000	−0.407867	−0.203933	−	0.978985i	$-0.565373\pi$
$101$	−414.000	−0.407867	−0.203933	+	0.978985i	$0.565373\pi$
$102$	0	0
$103$	56.0000	0.0535713	0.0267857	−	0.999641i	$-0.491473\pi$
$103$	56.0000	0.0535713	0.0267857	+	0.999641i	$0.491473\pi$
$104$	0	0
$105$	378.000	0.351324
$106$	0	0
$107$	−12.0000	−0.0108419	−0.00542095	−	0.999985i	$-0.501726\pi$
$107$	−12.0000	−0.0108419	−0.00542095	+	0.999985i	$0.501726\pi$
$108$	0	0
$109$	−1478.00	−1.29878	−0.649389	−	0.760457i	$-0.724975\pi$
$109$	−1478.00	−1.29878	−0.649389	+	0.760457i	$0.724975\pi$
$110$	0	0
$111$	−1194.00	−1.02099
$112$	0	0
$113$	402.000	0.334664	0.167332	−	0.985901i	$-0.446485\pi$
$113$	402.000	0.334664	0.167332	+	0.985901i	$0.446485\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	306.000	0.241792
$118$	0	0
$119$	294.000	0.226478
$120$	0	0
$121$	−35.0000	−0.0262960
$122$	0	0
$123$	−954.000	−0.699344
$124$	0	0
$125$	1332.00	0.953102
$126$	0	0
$127$	1280.00	0.894344	0.447172	−	0.894448i	$-0.352431\pi$
$127$	1280.00	0.894344	0.447172	+	0.894448i	$0.352431\pi$
$128$	0	0
$129$	804.000	0.548746
$130$	0	0
$131$	−1764.00	−1.17650	−0.588250	−	0.808679i	$-0.700183\pi$
$131$	−1764.00	−1.17650	−0.588250	+	0.808679i	$0.700183\pi$
$132$	0	0
$133$	868.000	0.565903
$134$	0	0
$135$	486.000	0.309839
$136$	0	0
$137$	−2358.00	−1.47049	−0.735246	−	0.677800i	$-0.762934\pi$
$137$	−2358.00	−1.47049	−0.735246	+	0.677800i	$0.762934\pi$
$138$	0	0
$139$	52.0000	0.0317308	0.0158654	−	0.999874i	$-0.494950\pi$
$139$	52.0000	0.0317308	0.0158654	+	0.999874i	$0.494950\pi$
$140$	0	0
$141$	720.000	0.430035
$142$	0	0
$143$	1224.00	0.715776
$144$	0	0
$145$	−1836.00	−1.05153
$146$	0	0
$147$	147.000	0.0824786
$148$	0	0
$149$	1746.00	0.959986	0.479993	−	0.877272i	$-0.340639\pi$
$149$	1746.00	0.959986	0.479993	+	0.877272i	$0.340639\pi$
$150$	0	0
$151$	−232.000	−0.125032	−0.0625162	−	0.998044i	$-0.519913\pi$
$151$	−232.000	−0.125032	−0.0625162	+	0.998044i	$0.519913\pi$
$152$	0	0
$153$	378.000	0.199735
$154$	0	0
$155$	−2880.00	−1.49243
$156$	0	0
$157$	−1694.00	−0.861120	−0.430560	−	0.902562i	$-0.641684\pi$
$157$	−1694.00	−0.861120	−0.430560	+	0.902562i	$0.641684\pi$
$158$	0	0
$159$	1494.00	0.745169
$160$	0	0
$161$	0	0
$162$	0	0
$163$	2932.00	1.40891	0.704454	−	0.709750i	$-0.251192\pi$
$163$	2932.00	1.40891	0.704454	+	0.709750i	$0.251192\pi$
$164$	0	0
$165$	1944.00	0.917213
$166$	0	0
$167$	1176.00	0.544920	0.272460	−	0.962167i	$-0.412163\pi$
$167$	1176.00	0.544920	0.272460	+	0.962167i	$0.412163\pi$
$168$	0	0
$169$	−1041.00	−0.473828
$170$	0	0
$171$	1116.00	0.499080
$172$	0	0
$173$	−870.000	−0.382340	−0.191170	−	0.981557i	$-0.561228\pi$
$173$	−870.000	−0.382340	−0.191170	+	0.981557i	$0.561228\pi$
$174$	0	0
$175$	1393.00	0.601719
$176$	0	0
$177$	396.000	0.168165
$178$	0	0
$179$	2316.00	0.967072	0.483536	−	0.875324i	$-0.339352\pi$
$179$	2316.00	0.967072	0.483536	+	0.875324i	$0.339352\pi$
$180$	0	0
$181$	106.000	0.0435299	0.0217650	−	0.999763i	$-0.493071\pi$
$181$	106.000	0.0435299	0.0217650	+	0.999763i	$0.493071\pi$
$182$	0	0
$183$	−1194.00	−0.482312
$184$	0	0
$185$	−7164.00	−2.84707
$186$	0	0
$187$	1512.00	0.591275
$188$	0	0
$189$	189.000	0.0727393
$190$	0	0
$191$	−1128.00	−0.427326	−0.213663	−	0.976907i	$-0.568539\pi$
$191$	−1128.00	−0.427326	−0.213663	+	0.976907i	$0.568539\pi$
$192$	0	0
$193$	4034.00	1.50453	0.752263	−	0.658862i	$-0.228962\pi$
$193$	4034.00	1.50453	0.752263	+	0.658862i	$0.228962\pi$
$194$	0	0
$195$	1836.00	0.674250
$196$	0	0
$197$	1314.00	0.475221	0.237611	−	0.971360i	$-0.423636\pi$
$197$	1314.00	0.475221	0.237611	+	0.971360i	$0.423636\pi$
$198$	0	0
$199$	5096.00	1.81531	0.907653	−	0.419722i	$-0.137872\pi$
$199$	5096.00	1.81531	0.907653	+	0.419722i	$0.137872\pi$
$200$	0	0
$201$	−276.000	−0.0968534
$202$	0	0
$203$	−714.000	−0.246862
$204$	0	0
$205$	−5724.00	−1.95015
$206$	0	0
$207$	0	0
$208$	0	0
$209$	4464.00	1.47742
$210$	0	0
$211$	3076.00	1.00360	0.501802	−	0.864982i	$-0.332670\pi$
$211$	3076.00	1.00360	0.501802	+	0.864982i	$0.332670\pi$
$212$	0	0
$213$	−2160.00	−0.694839
$214$	0	0
$215$	4824.00	1.53020
$216$	0	0
$217$	−1120.00	−0.350371
$218$	0	0
$219$	−1506.00	−0.464685
$220$	0	0
$221$	1428.00	0.434650
$222$	0	0
$223$	−1888.00	−0.566950	−0.283475	−	0.958980i	$-0.591487\pi$
$223$	−1888.00	−0.566950	−0.283475	+	0.958980i	$0.591487\pi$
$224$	0	0
$225$	1791.00	0.530667
$226$	0	0
$227$	4716.00	1.37891	0.689454	−	0.724330i	$-0.257851\pi$
$227$	4716.00	1.37891	0.689454	+	0.724330i	$0.257851\pi$
$228$	0	0
$229$	1690.00	0.487678	0.243839	−	0.969816i	$-0.421593\pi$
$229$	1690.00	0.487678	0.243839	+	0.969816i	$0.421593\pi$
$230$	0	0
$231$	756.000	0.215330
$232$	0	0
$233$	138.000	0.0388012	0.0194006	−	0.999812i	$-0.493824\pi$
$233$	138.000	0.0388012	0.0194006	+	0.999812i	$0.493824\pi$
$234$	0	0
$235$	4320.00	1.19917
$236$	0	0
$237$	−3072.00	−0.841974
$238$	0	0
$239$	1896.00	0.513147	0.256573	−	0.966525i	$-0.417406\pi$
$239$	1896.00	0.513147	0.256573	+	0.966525i	$0.417406\pi$
$240$	0	0
$241$	−3598.00	−0.961691	−0.480846	−	0.876805i	$-0.659670\pi$
$241$	−3598.00	−0.961691	−0.480846	+	0.876805i	$0.659670\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	882.000	0.229996
$246$	0	0
$247$	4216.00	1.08606
$248$	0	0
$249$	612.000	0.155759
$250$	0	0
$251$	3060.00	0.769504	0.384752	−	0.923020i	$-0.374287\pi$
$251$	3060.00	0.769504	0.384752	+	0.923020i	$0.374287\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	2268.00	0.556971
$256$	0	0
$257$	−6822.00	−1.65582	−0.827908	−	0.560864i	$-0.810469\pi$
$257$	−6822.00	−1.65582	−0.827908	+	0.560864i	$0.810469\pi$
$258$	0	0
$259$	−2786.00	−0.668392
$260$	0	0
$261$	−918.000	−0.217712
$262$	0	0
$263$	2592.00	0.607717	0.303858	−	0.952717i	$-0.401725\pi$
$263$	2592.00	0.607717	0.303858	+	0.952717i	$0.401725\pi$
$264$	0	0
$265$	8964.00	2.07794
$266$	0	0
$267$	1062.00	0.243421
$268$	0	0
$269$	−8214.00	−1.86177	−0.930886	−	0.365311i	$-0.880963\pi$
$269$	−8214.00	−1.86177	−0.930886	+	0.365311i	$0.880963\pi$
$270$	0	0
$271$	−5344.00	−1.19788	−0.598939	−	0.800795i	$-0.704411\pi$
$271$	−5344.00	−1.19788	−0.598939	+	0.800795i	$0.704411\pi$
$272$	0	0
$273$	714.000	0.158290
$274$	0	0
$275$	7164.00	1.57093
$276$	0	0
$277$	6514.00	1.41295	0.706477	−	0.707736i	$-0.250283\pi$
$277$	6514.00	1.41295	0.706477	+	0.707736i	$0.250283\pi$
$278$	0	0
$279$	−1440.00	−0.308998
$280$	0	0
$281$	6618.00	1.40497	0.702485	−	0.711698i	$-0.252074\pi$
$281$	6618.00	1.40497	0.702485	+	0.711698i	$0.252074\pi$
$282$	0	0
$283$	−3260.00	−0.684759	−0.342380	−	0.939562i	$-0.611233\pi$
$283$	−3260.00	−0.684759	−0.342380	+	0.939562i	$0.611233\pi$
$284$	0	0
$285$	6696.00	1.39171
$286$	0	0
$287$	−2226.00	−0.457828
$288$	0	0
$289$	−3149.00	−0.640953
$290$	0	0
$291$	−858.000	−0.172841
$292$	0	0
$293$	−5118.00	−1.02047	−0.510233	−	0.860036i	$-0.670441\pi$
$293$	−5118.00	−1.02047	−0.510233	+	0.860036i	$0.670441\pi$
$294$	0	0
$295$	2376.00	0.468936
$296$	0	0
$297$	972.000	0.189903
$298$	0	0
$299$	0	0
$300$	0	0
$301$	1876.00	0.359239
$302$	0	0
$303$	−1242.00	−0.235482
$304$	0	0
$305$	−7164.00	−1.34495
$306$	0	0
$307$	−452.000	−0.0840293	−0.0420147	−	0.999117i	$-0.513378\pi$
$307$	−452.000	−0.0840293	−0.0420147	+	0.999117i	$0.513378\pi$
$308$	0	0
$309$	168.000	0.0309294
$310$	0	0
$311$	5016.00	0.914570	0.457285	−	0.889320i	$-0.348822\pi$
$311$	5016.00	0.914570	0.457285	+	0.889320i	$0.348822\pi$
$312$	0	0
$313$	5402.00	0.975524	0.487762	−	0.872977i	$-0.337813\pi$
$313$	5402.00	0.975524	0.487762	+	0.872977i	$0.337813\pi$
$314$	0	0
$315$	1134.00	0.202837
$316$	0	0
$317$	−10086.0	−1.78702	−0.893511	−	0.449041i	$-0.851766\pi$
$317$	−10086.0	−1.78702	−0.893511	+	0.449041i	$0.851766\pi$
$318$	0	0
$319$	−3672.00	−0.644491
$320$	0	0
$321$	−36.0000	−0.00625958
$322$	0	0
$323$	5208.00	0.897154
$324$	0	0
$325$	6766.00	1.15480
$326$	0	0
$327$	−4434.00	−0.749849
$328$	0	0
$329$	1680.00	0.281524
$330$	0	0
$331$	8044.00	1.33577	0.667883	−	0.744267i	$-0.267201\pi$
$331$	8044.00	1.33577	0.667883	+	0.744267i	$0.267201\pi$
$332$	0	0
$333$	−3582.00	−0.589467
$334$	0	0
$335$	−1656.00	−0.270080
$336$	0	0
$337$	4178.00	0.675342	0.337671	−	0.941264i	$-0.390361\pi$
$337$	4178.00	0.675342	0.337671	+	0.941264i	$0.390361\pi$
$338$	0	0
$339$	1206.00	0.193218
$340$	0	0
$341$	−5760.00	−0.914726
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−156.000	−0.0241341	−0.0120670	−	0.999927i	$-0.503841\pi$
$347$	−156.000	−0.0241341	−0.0120670	+	0.999927i	$0.503841\pi$
$348$	0	0
$349$	12418.0	1.90464	0.952321	−	0.305097i	$-0.0986888\pi$
$349$	12418.0	1.90464	0.952321	+	0.305097i	$0.0986888\pi$
$350$	0	0
$351$	918.000	0.139599
$352$	0	0
$353$	−7830.00	−1.18059	−0.590296	−	0.807187i	$-0.700989\pi$
$353$	−7830.00	−1.18059	−0.590296	+	0.807187i	$0.700989\pi$
$354$	0	0
$355$	−12960.0	−1.93759
$356$	0	0
$357$	882.000	0.130757
$358$	0	0
$359$	−9312.00	−1.36899	−0.684497	−	0.729016i	$-0.739978\pi$
$359$	−9312.00	−1.36899	−0.684497	+	0.729016i	$0.739978\pi$
$360$	0	0
$361$	8517.00	1.24173
$362$	0	0
$363$	−105.000	−0.0151820
$364$	0	0
$365$	−9036.00	−1.29580
$366$	0	0
$367$	−3760.00	−0.534797	−0.267398	−	0.963586i	$-0.586164\pi$
$367$	−3760.00	−0.534797	−0.267398	+	0.963586i	$0.586164\pi$
$368$	0	0
$369$	−2862.00	−0.403766
$370$	0	0
$371$	3486.00	0.487828
$372$	0	0
$373$	−5870.00	−0.814845	−0.407422	−	0.913240i	$-0.633572\pi$
$373$	−5870.00	−0.814845	−0.407422	+	0.913240i	$0.633572\pi$
$374$	0	0
$375$	3996.00	0.550273
$376$	0	0
$377$	−3468.00	−0.473769
$378$	0	0
$379$	1852.00	0.251005	0.125502	−	0.992093i	$-0.459946\pi$
$379$	1852.00	0.251005	0.125502	+	0.992093i	$0.459946\pi$
$380$	0	0
$381$	3840.00	0.516350
$382$	0	0
$383$	2160.00	0.288175	0.144087	−	0.989565i	$-0.453975\pi$
$383$	2160.00	0.288175	0.144087	+	0.989565i	$0.453975\pi$
$384$	0	0
$385$	4536.00	0.600457
$386$	0	0
$387$	2412.00	0.316819
$388$	0	0
$389$	6786.00	0.884483	0.442241	−	0.896896i	$-0.354183\pi$
$389$	6786.00	0.884483	0.442241	+	0.896896i	$0.354183\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−5292.00	−0.679252
$394$	0	0
$395$	−18432.0	−2.34788
$396$	0	0
$397$	6514.00	0.823497	0.411748	−	0.911298i	$-0.364918\pi$
$397$	6514.00	0.823497	0.411748	+	0.911298i	$0.364918\pi$
$398$	0	0
$399$	2604.00	0.326724
$400$	0	0
$401$	3330.00	0.414694	0.207347	−	0.978267i	$-0.433517\pi$
$401$	3330.00	0.414694	0.207347	+	0.978267i	$0.433517\pi$
$402$	0	0
$403$	−5440.00	−0.672421
$404$	0	0
$405$	1458.00	0.178885
$406$	0	0
$407$	−14328.0	−1.74499
$408$	0	0
$409$	−5398.00	−0.652601	−0.326301	−	0.945266i	$-0.605802\pi$
$409$	−5398.00	−0.652601	−0.326301	+	0.945266i	$0.605802\pi$
$410$	0	0
$411$	−7074.00	−0.848990
$412$	0	0
$413$	924.000	0.110090
$414$	0	0
$415$	3672.00	0.434341
$416$	0	0
$417$	156.000	0.0183198
$418$	0	0
$419$	−13092.0	−1.52646	−0.763229	−	0.646128i	$-0.776387\pi$
$419$	−13092.0	−1.52646	−0.763229	+	0.646128i	$0.776387\pi$
$420$	0	0
$421$	322.000	0.0372763	0.0186381	−	0.999826i	$-0.494067\pi$
$421$	322.000	0.0372763	0.0186381	+	0.999826i	$0.494067\pi$
$422$	0	0
$423$	2160.00	0.248281
$424$	0	0
$425$	8358.00	0.953935
$426$	0	0
$427$	−2786.00	−0.315747
$428$	0	0
$429$	3672.00	0.413254
$430$	0	0
$431$	2616.00	0.292363	0.146181	−	0.989258i	$-0.453302\pi$
$431$	2616.00	0.292363	0.146181	+	0.989258i	$0.453302\pi$
$432$	0	0
$433$	4322.00	0.479681	0.239841	−	0.970812i	$-0.422905\pi$
$433$	4322.00	0.479681	0.239841	+	0.970812i	$0.422905\pi$
$434$	0	0
$435$	−5508.00	−0.607100
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−9016.00	−0.980205	−0.490103	−	0.871665i	$-0.663041\pi$
$439$	−9016.00	−0.980205	−0.490103	+	0.871665i	$0.663041\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	0	0
$443$	5268.00	0.564989	0.282495	−	0.959269i	$-0.408838\pi$
$443$	5268.00	0.564989	0.282495	+	0.959269i	$0.408838\pi$
$444$	0	0
$445$	6372.00	0.678790
$446$	0	0
$447$	5238.00	0.554248
$448$	0	0
$449$	−5310.00	−0.558117	−0.279058	−	0.960274i	$-0.590022\pi$
$449$	−5310.00	−0.558117	−0.279058	+	0.960274i	$0.590022\pi$
$450$	0	0
$451$	−11448.0	−1.19527
$452$	0	0
$453$	−696.000	−0.0721875
$454$	0	0
$455$	4284.00	0.441400
$456$	0	0
$457$	15770.0	1.61420	0.807100	−	0.590415i	$-0.201036\pi$
$457$	15770.0	1.61420	0.807100	+	0.590415i	$0.201036\pi$
$458$	0	0
$459$	1134.00	0.115317
$460$	0	0
$461$	5370.00	0.542529	0.271264	−	0.962505i	$-0.412558\pi$
$461$	5370.00	0.542529	0.271264	+	0.962505i	$0.412558\pi$
$462$	0	0
$463$	−3328.00	−0.334050	−0.167025	−	0.985953i	$-0.553416\pi$
$463$	−3328.00	−0.334050	−0.167025	+	0.985953i	$0.553416\pi$
$464$	0	0
$465$	−8640.00	−0.861657
$466$	0	0
$467$	−4548.00	−0.450656	−0.225328	−	0.974283i	$-0.572345\pi$
$467$	−4548.00	−0.450656	−0.225328	+	0.974283i	$0.572345\pi$
$468$	0	0
$469$	−644.000	−0.0634055
$470$	0	0
$471$	−5082.00	−0.497168
$472$	0	0
$473$	9648.00	0.937876
$474$	0	0
$475$	24676.0	2.38361
$476$	0	0
$477$	4482.00	0.430224
$478$	0	0
$479$	−8064.00	−0.769214	−0.384607	−	0.923080i	$-0.625663\pi$
$479$	−8064.00	−0.769214	−0.384607	+	0.923080i	$0.625663\pi$
$480$	0	0
$481$	−13532.0	−1.28276
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−5148.00	−0.481977
$486$	0	0
$487$	16616.0	1.54608	0.773042	−	0.634355i	$-0.218734\pi$
$487$	16616.0	1.54608	0.773042	+	0.634355i	$0.218734\pi$
$488$	0	0
$489$	8796.00	0.813433
$490$	0	0
$491$	7140.00	0.656260	0.328130	−	0.944633i	$-0.393582\pi$
$491$	7140.00	0.656260	0.328130	+	0.944633i	$0.393582\pi$
$492$	0	0
$493$	−4284.00	−0.391362
$494$	0	0
$495$	5832.00	0.529553
$496$	0	0
$497$	−5040.00	−0.454879
$498$	0	0
$499$	9124.00	0.818530	0.409265	−	0.912416i	$-0.365785\pi$
$499$	9124.00	0.818530	0.409265	+	0.912416i	$0.365785\pi$
$500$	0	0
$501$	3528.00	0.314610
$502$	0	0
$503$	−6552.00	−0.580794	−0.290397	−	0.956906i	$-0.593787\pi$
$503$	−6552.00	−0.580794	−0.290397	+	0.956906i	$0.593787\pi$
$504$	0	0
$505$	−7452.00	−0.656653
$506$	0	0
$507$	−3123.00	−0.273565
$508$	0	0
$509$	−2790.00	−0.242956	−0.121478	−	0.992594i	$-0.538763\pi$
$509$	−2790.00	−0.242956	−0.121478	+	0.992594i	$0.538763\pi$
$510$	0	0
$511$	−3514.00	−0.304208
$512$	0	0
$513$	3348.00	0.288144
$514$	0	0
$515$	1008.00	0.0862481
$516$	0	0
$517$	8640.00	0.734984
$518$	0	0
$519$	−2610.00	−0.220744
$520$	0	0
$521$	−14862.0	−1.24974	−0.624871	−	0.780728i	$-0.714849\pi$
$521$	−14862.0	−1.24974	−0.624871	+	0.780728i	$0.714849\pi$
$522$	0	0
$523$	−17660.0	−1.47652	−0.738258	−	0.674518i	$-0.764351\pi$
$523$	−17660.0	−1.47652	−0.738258	+	0.674518i	$0.764351\pi$
$524$	0	0
$525$	4179.00	0.347403
$526$	0	0
$527$	−6720.00	−0.555461
$528$	0	0
$529$	−12167.0	−1.00000
$530$	0	0
$531$	1188.00	0.0970900
$532$	0	0
$533$	−10812.0	−0.878649
$534$	0	0
$535$	−216.000	−0.0174551
$536$	0	0
$537$	6948.00	0.558340
$538$	0	0
$539$	1764.00	0.140966
$540$	0	0
$541$	19834.0	1.57621	0.788106	−	0.615540i	$-0.211062\pi$
$541$	19834.0	1.57621	0.788106	+	0.615540i	$0.211062\pi$
$542$	0	0
$543$	318.000	0.0251320
$544$	0	0
$545$	−26604.0	−2.09099
$546$	0	0
$547$	−20972.0	−1.63930	−0.819651	−	0.572863i	$-0.805833\pi$
$547$	−20972.0	−1.63930	−0.819651	+	0.572863i	$0.805833\pi$
$548$	0	0
$549$	−3582.00	−0.278463
$550$	0	0
$551$	−12648.0	−0.977900
$552$	0	0
$553$	−7168.00	−0.551201
$554$	0	0
$555$	−21492.0	−1.64376
$556$	0	0
$557$	−21174.0	−1.61072	−0.805360	−	0.592786i	$-0.798028\pi$
$557$	−21174.0	−1.61072	−0.805360	+	0.592786i	$0.798028\pi$
$558$	0	0
$559$	9112.00	0.689439
$560$	0	0
$561$	4536.00	0.341373
$562$	0	0
$563$	17772.0	1.33037	0.665187	−	0.746677i	$-0.268352\pi$
$563$	17772.0	1.33037	0.665187	+	0.746677i	$0.268352\pi$
$564$	0	0
$565$	7236.00	0.538798
$566$	0	0
$567$	567.000	0.0419961
$568$	0	0
$569$	8250.00	0.607835	0.303917	−	0.952698i	$-0.401705\pi$
$569$	8250.00	0.607835	0.303917	+	0.952698i	$0.401705\pi$
$570$	0	0
$571$	−20756.0	−1.52121	−0.760606	−	0.649214i	$-0.775098\pi$
$571$	−20756.0	−1.52121	−0.760606	+	0.649214i	$0.775098\pi$
$572$	0	0
$573$	−3384.00	−0.246717
$574$	0	0
$575$	0	0
$576$	0	0
$577$	2.00000	0.000144300 0	7.21500e−5	−	1.00000i	$-0.499977\pi$
$577$	2.00000	0.000144300 0	7.21500e−5		1.00000i	$0.499977\pi$
$578$	0	0
$579$	12102.0	0.868639
$580$	0	0
$581$	1428.00	0.101968
$582$	0	0
$583$	17928.0	1.27359
$584$	0	0
$585$	5508.00	0.389278
$586$	0	0
$587$	−26364.0	−1.85376	−0.926881	−	0.375354i	$-0.877521\pi$
$587$	−26364.0	−1.85376	−0.926881	+	0.375354i	$0.877521\pi$
$588$	0	0
$589$	−19840.0	−1.38793
$590$	0	0
$591$	3942.00	0.274369
$592$	0	0
$593$	2298.00	0.159136	0.0795679	−	0.996829i	$-0.474646\pi$
$593$	2298.00	0.159136	0.0795679	+	0.996829i	$0.474646\pi$
$594$	0	0
$595$	5292.00	0.364623
$596$	0	0
$597$	15288.0	1.04807
$598$	0	0
$599$	3072.00	0.209547	0.104773	−	0.994496i	$-0.466588\pi$
$599$	3072.00	0.209547	0.104773	+	0.994496i	$0.466588\pi$
$600$	0	0
$601$	24554.0	1.66652	0.833260	−	0.552881i	$-0.186472\pi$
$601$	24554.0	1.66652	0.833260	+	0.552881i	$0.186472\pi$
$602$	0	0
$603$	−828.000	−0.0559184
$604$	0	0
$605$	−630.000	−0.0423358
$606$	0	0
$607$	16832.0	1.12552	0.562759	−	0.826621i	$-0.309740\pi$
$607$	16832.0	1.12552	0.562759	+	0.826621i	$0.309740\pi$
$608$	0	0
$609$	−2142.00	−0.142526
$610$	0	0
$611$	8160.00	0.540292
$612$	0	0
$613$	2482.00	0.163535	0.0817676	−	0.996651i	$-0.473943\pi$
$613$	2482.00	0.163535	0.0817676	+	0.996651i	$0.473943\pi$
$614$	0	0
$615$	−17172.0	−1.12592
$616$	0	0
$617$	−15798.0	−1.03080	−0.515400	−	0.856950i	$-0.672357\pi$
$617$	−15798.0	−1.03080	−0.515400	+	0.856950i	$0.672357\pi$
$618$	0	0
$619$	15460.0	1.00386	0.501930	−	0.864908i	$-0.332623\pi$
$619$	15460.0	1.00386	0.501930	+	0.864908i	$0.332623\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	2478.00	0.159356
$624$	0	0
$625$	−899.000	−0.0575360
$626$	0	0
$627$	13392.0	0.852990
$628$	0	0
$629$	−16716.0	−1.05964
$630$	0	0
$631$	−7720.00	−0.487050	−0.243525	−	0.969895i	$-0.578304\pi$
$631$	−7720.00	−0.487050	−0.243525	+	0.969895i	$0.578304\pi$
$632$	0	0
$633$	9228.00	0.579431
$634$	0	0
$635$	23040.0	1.43987
$636$	0	0
$637$	1666.00	0.103625
$638$	0	0
$639$	−6480.00	−0.401166
$640$	0	0
$641$	−17262.0	−1.06366	−0.531832	−	0.846850i	$-0.678496\pi$
$641$	−17262.0	−1.06366	−0.531832	+	0.846850i	$0.678496\pi$
$642$	0	0
$643$	12220.0	0.749471	0.374735	−	0.927132i	$-0.377734\pi$
$643$	12220.0	0.749471	0.374735	+	0.927132i	$0.377734\pi$
$644$	0	0
$645$	14472.0	0.883464
$646$	0	0
$647$	13560.0	0.823955	0.411977	−	0.911194i	$-0.364838\pi$
$647$	13560.0	0.823955	0.411977	+	0.911194i	$0.364838\pi$
$648$	0	0
$649$	4752.00	0.287415
$650$	0	0
$651$	−3360.00	−0.202287
$652$	0	0
$653$	−23094.0	−1.38398	−0.691989	−	0.721908i	$-0.743265\pi$
$653$	−23094.0	−1.38398	−0.691989	+	0.721908i	$0.743265\pi$
$654$	0	0
$655$	−31752.0	−1.89413
$656$	0	0
$657$	−4518.00	−0.268286
$658$	0	0
$659$	−22548.0	−1.33285	−0.666423	−	0.745574i	$-0.732175\pi$
$659$	−22548.0	−1.33285	−0.666423	+	0.745574i	$0.732175\pi$
$660$	0	0
$661$	−17462.0	−1.02752	−0.513762	−	0.857933i	$-0.671748\pi$
$661$	−17462.0	−1.02752	−0.513762	+	0.857933i	$0.671748\pi$
$662$	0	0
$663$	4284.00	0.250945
$664$	0	0
$665$	15624.0	0.911087
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−5664.00	−0.327329
$670$	0	0
$671$	−14328.0	−0.824331
$672$	0	0
$673$	−22462.0	−1.28655	−0.643274	−	0.765636i	$-0.722424\pi$
$673$	−22462.0	−1.28655	−0.643274	+	0.765636i	$0.722424\pi$
$674$	0	0
$675$	5373.00	0.306381
$676$	0	0
$677$	25554.0	1.45069	0.725347	−	0.688383i	$-0.241679\pi$
$677$	25554.0	1.45069	0.725347	+	0.688383i	$0.241679\pi$
$678$	0	0
$679$	−2002.00	−0.113151
$680$	0	0
$681$	14148.0	0.796112
$682$	0	0
$683$	−9276.00	−0.519672	−0.259836	−	0.965653i	$-0.583669\pi$
$683$	−9276.00	−0.519672	−0.259836	+	0.965653i	$0.583669\pi$
$684$	0	0
$685$	−42444.0	−2.36745
$686$	0	0
$687$	5070.00	0.281561
$688$	0	0
$689$	16932.0	0.936223
$690$	0	0
$691$	−27380.0	−1.50736	−0.753679	−	0.657243i	$-0.771723\pi$
$691$	−27380.0	−1.50736	−0.753679	+	0.657243i	$0.771723\pi$
$692$	0	0
$693$	2268.00	0.124321
$694$	0	0
$695$	936.000	0.0510856
$696$	0	0
$697$	−13356.0	−0.725817
$698$	0	0
$699$	414.000	0.0224019
$700$	0	0
$701$	−25830.0	−1.39171	−0.695853	−	0.718184i	$-0.744973\pi$
$701$	−25830.0	−1.39171	−0.695853	+	0.718184i	$0.744973\pi$
$702$	0	0
$703$	−49352.0	−2.64772
$704$	0	0
$705$	12960.0	0.692343
$706$	0	0
$707$	−2898.00	−0.154159
$708$	0	0
$709$	6226.00	0.329792	0.164896	−	0.986311i	$-0.447271\pi$
$709$	6226.00	0.329792	0.164896	+	0.986311i	$0.447271\pi$
$710$	0	0
$711$	−9216.00	−0.486114
$712$	0	0
$713$	0	0
$714$	0	0
$715$	22032.0	1.15238
$716$	0	0
$717$	5688.00	0.296265
$718$	0	0
$719$	−15072.0	−0.781767	−0.390884	−	0.920440i	$-0.627831\pi$
$719$	−15072.0	−0.781767	−0.390884	+	0.920440i	$0.627831\pi$
$720$	0	0
$721$	392.000	0.0202480
$722$	0	0
$723$	−10794.0	−0.555233
$724$	0	0
$725$	−20298.0	−1.03979
$726$	0	0
$727$	−32920.0	−1.67942	−0.839708	−	0.543038i	$-0.817274\pi$
$727$	−32920.0	−1.67942	−0.839708	+	0.543038i	$0.817274\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	11256.0	0.569519
$732$	0	0
$733$	6946.00	0.350009	0.175004	−	0.984568i	$-0.444006\pi$
$733$	6946.00	0.350009	0.175004	+	0.984568i	$0.444006\pi$
$734$	0	0
$735$	2646.00	0.132788
$736$	0	0
$737$	−3312.00	−0.165535
$738$	0	0
$739$	2356.00	0.117276	0.0586379	−	0.998279i	$-0.481324\pi$
$739$	2356.00	0.117276	0.0586379	+	0.998279i	$0.481324\pi$
$740$	0	0
$741$	12648.0	0.627039
$742$	0	0
$743$	−23520.0	−1.16133	−0.580663	−	0.814144i	$-0.697207\pi$
$743$	−23520.0	−1.16133	−0.580663	+	0.814144i	$0.697207\pi$
$744$	0	0
$745$	31428.0	1.54555
$746$	0	0
$747$	1836.00	0.0899273
$748$	0	0
$749$	−84.0000	−0.00409785
$750$	0	0
$751$	3008.00	0.146156	0.0730782	−	0.997326i	$-0.476718\pi$
$751$	3008.00	0.146156	0.0730782	+	0.997326i	$0.476718\pi$
$752$	0	0
$753$	9180.00	0.444273
$754$	0	0
$755$	−4176.00	−0.201298
$756$	0	0
$757$	20770.0	0.997224	0.498612	−	0.866825i	$-0.333843\pi$
$757$	20770.0	0.997224	0.498612	+	0.866825i	$0.333843\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	11538.0	0.549609	0.274804	−	0.961500i	$-0.411387\pi$
$761$	11538.0	0.549609	0.274804	+	0.961500i	$0.411387\pi$
$762$	0	0
$763$	−10346.0	−0.490892
$764$	0	0
$765$	6804.00	0.321568
$766$	0	0
$767$	4488.00	0.211281
$768$	0	0
$769$	8498.00	0.398499	0.199249	−	0.979949i	$-0.436150\pi$
$769$	8498.00	0.398499	0.199249	+	0.979949i	$0.436150\pi$
$770$	0	0
$771$	−20466.0	−0.955986
$772$	0	0
$773$	32322.0	1.50393	0.751967	−	0.659200i	$-0.229105\pi$
$773$	32322.0	1.50393	0.751967	+	0.659200i	$0.229105\pi$
$774$	0	0
$775$	−31840.0	−1.47578
$776$	0	0
$777$	−8358.00	−0.385896
$778$	0	0
$779$	−39432.0	−1.81360
$780$	0	0
$781$	−25920.0	−1.18757
$782$	0	0
$783$	−2754.00	−0.125696
$784$	0	0
$785$	−30492.0	−1.38638
$786$	0	0
$787$	−26228.0	−1.18796	−0.593982	−	0.804479i	$-0.702445\pi$
$787$	−26228.0	−1.18796	−0.593982	+	0.804479i	$0.702445\pi$
$788$	0	0
$789$	7776.00	0.350866
$790$	0	0
$791$	2814.00	0.126491
$792$	0	0
$793$	−13532.0	−0.605972
$794$	0	0
$795$	26892.0	1.19970
$796$	0	0
$797$	43338.0	1.92611	0.963056	−	0.269302i	$-0.0867931\pi$
$797$	43338.0	1.92611	0.963056	+	0.269302i	$0.0867931\pi$
$798$	0	0
$799$	10080.0	0.446314
$800$	0	0
$801$	3186.00	0.140539
$802$	0	0
$803$	−18072.0	−0.794206
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−24642.0	−1.07489
$808$	0	0
$809$	−28902.0	−1.25604	−0.628022	−	0.778195i	$-0.716135\pi$
$809$	−28902.0	−1.25604	−0.628022	+	0.778195i	$0.716135\pi$
$810$	0	0
$811$	−27164.0	−1.17615	−0.588075	−	0.808807i	$-0.700114\pi$
$811$	−27164.0	−1.17615	−0.588075	+	0.808807i	$0.700114\pi$
$812$	0	0
$813$	−16032.0	−0.691595
$814$	0	0
$815$	52776.0	2.26830
$816$	0	0
$817$	33232.0	1.42306
$818$	0	0
$819$	2142.00	0.0913889
$820$	0	0
$821$	17202.0	0.731247	0.365624	−	0.930763i	$-0.380856\pi$
$821$	17202.0	0.731247	0.365624	+	0.930763i	$0.380856\pi$
$822$	0	0
$823$	−5992.00	−0.253789	−0.126894	−	0.991916i	$-0.540501\pi$
$823$	−5992.00	−0.253789	−0.126894	+	0.991916i	$0.540501\pi$
$824$	0	0
$825$	21492.0	0.906976
$826$	0	0
$827$	−25884.0	−1.08836	−0.544181	−	0.838968i	$-0.683159\pi$
$827$	−25884.0	−1.08836	−0.544181	+	0.838968i	$0.683159\pi$
$828$	0	0
$829$	1474.00	0.0617541	0.0308770	−	0.999523i	$-0.490170\pi$
$829$	1474.00	0.0617541	0.0308770	+	0.999523i	$0.490170\pi$
$830$	0	0
$831$	19542.0	0.815770
$832$	0	0
$833$	2058.00	0.0856008
$834$	0	0
$835$	21168.0	0.877304
$836$	0	0
$837$	−4320.00	−0.178400
$838$	0	0
$839$	33528.0	1.37964	0.689818	−	0.723983i	$-0.257690\pi$
$839$	33528.0	1.37964	0.689818	+	0.723983i	$0.257690\pi$
$840$	0	0
$841$	−13985.0	−0.573414
$842$	0	0
$843$	19854.0	0.811160
$844$	0	0
$845$	−18738.0	−0.762848
$846$	0	0
$847$	−245.000	−0.00993896
$848$	0	0
$849$	−9780.00	−0.395346
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−1190.00	−0.0477665	−0.0238832	−	0.999715i	$-0.507603\pi$
$853$	−1190.00	−0.0477665	−0.0238832	+	0.999715i	$0.507603\pi$
$854$	0	0
$855$	20088.0	0.803503
$856$	0	0
$857$	34578.0	1.37825	0.689126	−	0.724642i	$-0.257995\pi$
$857$	34578.0	1.37825	0.689126	+	0.724642i	$0.257995\pi$
$858$	0	0
$859$	44404.0	1.76373	0.881865	−	0.471501i	$-0.156288\pi$
$859$	44404.0	1.76373	0.881865	+	0.471501i	$0.156288\pi$
$860$	0	0
$861$	−6678.00	−0.264327
$862$	0	0
$863$	−38328.0	−1.51182	−0.755910	−	0.654676i	$-0.772805\pi$
$863$	−38328.0	−1.51182	−0.755910	+	0.654676i	$0.772805\pi$
$864$	0	0
$865$	−15660.0	−0.615556
$866$	0	0
$867$	−9447.00	−0.370054
$868$	0	0
$869$	−36864.0	−1.43904
$870$	0	0
$871$	−3128.00	−0.121686
$872$	0	0
$873$	−2574.00	−0.0997900
$874$	0	0
$875$	9324.00	0.360239
$876$	0	0
$877$	38842.0	1.49555	0.747777	−	0.663950i	$-0.231121\pi$
$877$	38842.0	1.49555	0.747777	+	0.663950i	$0.231121\pi$
$878$	0	0
$879$	−15354.0	−0.589167
$880$	0	0
$881$	−35046.0	−1.34022	−0.670108	−	0.742264i	$-0.733752\pi$
$881$	−35046.0	−1.34022	−0.670108	+	0.742264i	$0.733752\pi$
$882$	0	0
$883$	−14204.0	−0.541339	−0.270670	−	0.962672i	$-0.587245\pi$
$883$	−14204.0	−0.541339	−0.270670	+	0.962672i	$0.587245\pi$
$884$	0	0
$885$	7128.00	0.270740
$886$	0	0
$887$	−26136.0	−0.989359	−0.494679	−	0.869076i	$-0.664714\pi$
$887$	−26136.0	−0.989359	−0.494679	+	0.869076i	$0.664714\pi$
$888$	0	0
$889$	8960.00	0.338030
$890$	0	0
$891$	2916.00	0.109640
$892$	0	0
$893$	29760.0	1.11521
$894$	0	0
$895$	41688.0	1.55696
$896$	0	0
$897$	0	0
$898$	0	0
$899$	16320.0	0.605453
$900$	0	0
$901$	20916.0	0.773377
$902$	0	0
$903$	5628.00	0.207407
$904$	0	0
$905$	1908.00	0.0700818
$906$	0	0
$907$	9052.00	0.331386	0.165693	−	0.986177i	$-0.447014\pi$
$907$	9052.00	0.331386	0.165693	+	0.986177i	$0.447014\pi$
$908$	0	0
$909$	−3726.00	−0.135956
$910$	0	0
$911$	5016.00	0.182423	0.0912116	−	0.995832i	$-0.470926\pi$
$911$	5016.00	0.182423	0.0912116	+	0.995832i	$0.470926\pi$
$912$	0	0
$913$	7344.00	0.266211
$914$	0	0
$915$	−21492.0	−0.776507
$916$	0	0
$917$	−12348.0	−0.444675
$918$	0	0
$919$	44552.0	1.59917	0.799584	−	0.600555i	$-0.205054\pi$
$919$	44552.0	1.59917	0.799584	+	0.600555i	$0.205054\pi$
$920$	0	0
$921$	−1356.00	−0.0485144
$922$	0	0
$923$	−24480.0	−0.872989
$924$	0	0
$925$	−79202.0	−2.81529
$926$	0	0
$927$	504.000	0.0178571
$928$	0	0
$929$	24234.0	0.855858	0.427929	−	0.903812i	$-0.359243\pi$
$929$	24234.0	0.855858	0.427929	+	0.903812i	$0.359243\pi$
$930$	0	0
$931$	6076.00	0.213891
$932$	0	0
$933$	15048.0	0.528027
$934$	0	0
$935$	27216.0	0.951934
$936$	0	0
$937$	−13894.0	−0.484415	−0.242208	−	0.970224i	$-0.577872\pi$
$937$	−13894.0	−0.484415	−0.242208	+	0.970224i	$0.577872\pi$
$938$	0	0
$939$	16206.0	0.563219
$940$	0	0
$941$	−46758.0	−1.61984	−0.809919	−	0.586542i	$-0.800489\pi$
$941$	−46758.0	−1.61984	−0.809919	+	0.586542i	$0.800489\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	3402.00	0.117108
$946$	0	0
$947$	−13812.0	−0.473949	−0.236974	−	0.971516i	$-0.576156\pi$
$947$	−13812.0	−0.473949	−0.236974	+	0.971516i	$0.576156\pi$
$948$	0	0
$949$	−17068.0	−0.583826
$950$	0	0
$951$	−30258.0	−1.03174
$952$	0	0
$953$	−58518.0	−1.98907	−0.994535	−	0.104402i	$-0.966707\pi$
$953$	−58518.0	−1.98907	−0.994535	+	0.104402i	$0.966707\pi$
$954$	0	0
$955$	−20304.0	−0.687981
$956$	0	0
$957$	−11016.0	−0.372097
$958$	0	0
$959$	−16506.0	−0.555794
$960$	0	0
$961$	−4191.00	−0.140680
$962$	0	0
$963$	−108.000	−0.00361397
$964$	0	0
$965$	72612.0	2.42224
$966$	0	0
$967$	19640.0	0.653133	0.326567	−	0.945174i	$-0.394108\pi$
$967$	19640.0	0.653133	0.326567	+	0.945174i	$0.394108\pi$
$968$	0	0
$969$	15624.0	0.517972
$970$	0	0
$971$	58308.0	1.92708	0.963539	−	0.267568i	$-0.0862200\pi$
$971$	58308.0	1.92708	0.963539	+	0.267568i	$0.0862200\pi$
$972$	0	0
$973$	364.000	0.0119931
$974$	0	0
$975$	20298.0	0.666724
$976$	0	0
$977$	−23550.0	−0.771168	−0.385584	−	0.922673i	$-0.626000\pi$
$977$	−23550.0	−0.771168	−0.385584	+	0.922673i	$0.626000\pi$
$978$	0	0
$979$	12744.0	0.416037
$980$	0	0
$981$	−13302.0	−0.432926
$982$	0	0
$983$	15768.0	0.511619	0.255809	−	0.966727i	$-0.417658\pi$
$983$	15768.0	0.511619	0.255809	+	0.966727i	$0.417658\pi$
$984$	0	0
$985$	23652.0	0.765092
$986$	0	0
$987$	5040.00	0.162538
$988$	0	0
$989$	0	0
$990$	0	0
$991$	35264.0	1.13037	0.565186	−	0.824964i	$-0.308805\pi$
$991$	35264.0	1.13037	0.565186	+	0.824964i	$0.308805\pi$
$992$	0	0
$993$	24132.0	0.771204
$994$	0	0
$995$	91728.0	2.92259
$996$	0	0
$997$	29338.0	0.931940	0.465970	−	0.884801i	$-0.345706\pi$
$997$	29338.0	0.931940	0.465970	+	0.884801i	$0.345706\pi$
$998$	0	0
$999$	−10746.0	−0.340329

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.4.a.ba.1.1		1
4.3	odd	2		1344.4.a.n.1.1		1
8.3	odd	2		336.4.a.f.1.1		1
8.5	even	2		21.4.a.a.1.1	✓	1
24.5	odd	2		63.4.a.c.1.1		1
24.11	even	2		1008.4.a.v.1.1		1
40.13	odd	4		525.4.d.c.274.2		2
40.29	even	2		525.4.a.g.1.1		1
40.37	odd	4		525.4.d.c.274.1		2
56.5	odd	6		147.4.e.g.67.1		2
56.13	odd	2		147.4.a.c.1.1		1
56.27	even	2		2352.4.a.r.1.1		1
56.37	even	6		147.4.e.i.67.1		2
56.45	odd	6		147.4.e.g.79.1		2
56.53	even	6		147.4.e.i.79.1		2
120.29	odd	2		1575.4.a.b.1.1		1
168.5	even	6		441.4.e.d.361.1		2
168.53	odd	6		441.4.e.b.226.1		2
168.101	even	6		441.4.e.d.226.1		2
168.125	even	2		441.4.a.j.1.1		1
168.149	odd	6		441.4.e.b.361.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.4.a.a.1.1	✓	1	8.5	even	2
63.4.a.c.1.1		1	24.5	odd	2
147.4.a.c.1.1		1	56.13	odd	2
147.4.e.g.67.1		2	56.5	odd	6
147.4.e.g.79.1		2	56.45	odd	6
147.4.e.i.67.1		2	56.37	even	6
147.4.e.i.79.1		2	56.53	even	6
336.4.a.f.1.1		1	8.3	odd	2
441.4.a.j.1.1		1	168.125	even	2
441.4.e.b.226.1		2	168.53	odd	6
441.4.e.b.361.1		2	168.149	odd	6
441.4.e.d.226.1		2	168.101	even	6
441.4.e.d.361.1		2	168.5	even	6
525.4.a.g.1.1		1	40.29	even	2
525.4.d.c.274.1		2	40.37	odd	4
525.4.d.c.274.2		2	40.13	odd	4
1008.4.a.v.1.1		1	24.11	even	2
1344.4.a.n.1.1		1	4.3	odd	2
1344.4.a.ba.1.1		1	1.1	even	1	trivial
1575.4.a.b.1.1		1	120.29	odd	2
2352.4.a.r.1.1		1	56.27	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:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1344.a (trivial)

Introduction

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