LMFDB - Embedded newform 784.4.a.z.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [784,4,Mod(1,784)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(784, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("784.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	784.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:	$46.2574974445$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{22}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 22 $ x^2 - 22
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 98)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$4.69042$ of defining polynomial
Character	$\chi$	$=$	784.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.38083 q^{3} +9.38083 q^{5} +61.0000 q^{9} +O(q^{10})$	$q+9.38083 q^{3} +9.38083 q^{5} +61.0000 q^{9} -20.0000 q^{11} +65.6658 q^{13} +88.0000 q^{15} +56.2850 q^{17} -9.38083 q^{19} -48.0000 q^{23} -37.0000 q^{25} +318.948 q^{27} -166.000 q^{29} +206.378 q^{31} -187.617 q^{33} -78.0000 q^{37} +616.000 q^{39} +393.995 q^{41} -436.000 q^{43} +572.231 q^{45} -206.378 q^{47} +528.000 q^{51} +62.0000 q^{53} -187.617 q^{55} -88.0000 q^{57} +666.039 q^{59} +272.044 q^{61} +616.000 q^{65} -580.000 q^{67} -450.280 q^{69} +544.000 q^{71} -600.373 q^{73} -347.091 q^{75} +680.000 q^{79} +1345.00 q^{81} -196.997 q^{83} +528.000 q^{85} -1557.22 q^{87} -1500.93 q^{89} +1936.00 q^{93} -88.0000 q^{95} -656.658 q^{97} -1220.00 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 122 q^{9}+O(q^{10}) $ 2 * q + 122 * q^9	$ 2 q + 122 q^{9} - 40 q^{11} + 176 q^{15} - 96 q^{23} - 74 q^{25} - 332 q^{29} - 156 q^{37} + 1232 q^{39} - 872 q^{43} + 1056 q^{51} + 124 q^{53} - 176 q^{57} + 1232 q^{65} - 1160 q^{67} + 1088 q^{71} + 1360 q^{79} + 2690 q^{81} + 1056 q^{85} + 3872 q^{93} - 176 q^{95} - 2440 q^{99}+O(q^{100}) $ 2 * q + 122 * q^9 - 40 * q^11 + 176 * q^15 - 96 * q^23 - 74 * q^25 - 332 * q^29 - 156 * q^37 + 1232 * q^39 - 872 * q^43 + 1056 * q^51 + 124 * q^53 - 176 * q^57 + 1232 * q^65 - 1160 * q^67 + 1088 * q^71 + 1360 * q^79 + 2690 * q^81 + 1056 * q^85 + 3872 * q^93 - 176 * q^95 - 2440 * q^99

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
$3$	9.38083	1.80534	0.902671	−	0.430331i	$-0.141603\pi$
$3$	9.38083	1.80534	0.902671	+	0.430331i	$0.141603\pi$
$4$	0	0
$5$	9.38083	0.839047	0.419524	−	0.907744i	$-0.362197\pi$
$5$	9.38083	0.839047	0.419524	+	0.907744i	$0.362197\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	61.0000	2.25926
$10$	0	0
$11$	−20.0000	−0.548202	−0.274101	−	0.961701i	$-0.588380\pi$
$11$	−20.0000	−0.548202	−0.274101	+	0.961701i	$0.588380\pi$
$12$	0	0
$13$	65.6658	1.40096	0.700478	−	0.713674i	$-0.252970\pi$
$13$	65.6658	1.40096	0.700478	+	0.713674i	$0.252970\pi$
$14$	0	0
$15$	88.0000	1.51477
$16$	0	0
$17$	56.2850	0.803007	0.401503	−	0.915858i	$-0.368488\pi$
$17$	56.2850	0.803007	0.401503	+	0.915858i	$0.368488\pi$
$18$	0	0
$19$	−9.38083	−0.113269	−0.0566345	−	0.998395i	$-0.518037\pi$
$19$	−9.38083	−0.113269	−0.0566345	+	0.998395i	$0.518037\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−48.0000	−0.435161	−0.217580	−	0.976042i	$-0.569816\pi$
$23$	−48.0000	−0.435161	−0.217580	+	0.976042i	$0.569816\pi$
$24$	0	0
$25$	−37.0000	−0.296000
$26$	0	0
$27$	318.948	2.27339
$28$	0	0
$29$	−166.000	−1.06295	−0.531473	−	0.847075i	$-0.678361\pi$
$29$	−166.000	−1.06295	−0.531473	+	0.847075i	$0.678361\pi$
$30$	0	0
$31$	206.378	1.19570	0.597849	−	0.801609i	$-0.296022\pi$
$31$	206.378	1.19570	0.597849	+	0.801609i	$0.296022\pi$
$32$	0	0
$33$	−187.617	−0.989693
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−78.0000	−0.346571	−0.173285	−	0.984872i	$-0.555438\pi$
$37$	−78.0000	−0.346571	−0.173285	+	0.984872i	$0.555438\pi$
$38$	0	0
$39$	616.000	2.52920
$40$	0	0
$41$	393.995	1.50077	0.750386	−	0.661000i	$-0.229868\pi$
$41$	393.995	1.50077	0.750386	+	0.661000i	$0.229868\pi$
$42$	0	0
$43$	−436.000	−1.54626	−0.773132	−	0.634245i	$-0.781311\pi$
$43$	−436.000	−1.54626	−0.773132	+	0.634245i	$0.781311\pi$
$44$	0	0
$45$	572.231	1.89562
$46$	0	0
$47$	−206.378	−0.640497	−0.320249	−	0.947334i	$-0.603766\pi$
$47$	−206.378	−0.640497	−0.320249	+	0.947334i	$0.603766\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	528.000	1.44970
$52$	0	0
$53$	62.0000	0.160686	0.0803430	−	0.996767i	$-0.474398\pi$
$53$	62.0000	0.160686	0.0803430	+	0.996767i	$0.474398\pi$
$54$	0	0
$55$	−187.617	−0.459968
$56$	0	0
$57$	−88.0000	−0.204489
$58$	0	0
$59$	666.039	1.46968	0.734838	−	0.678243i	$-0.237258\pi$
$59$	666.039	1.46968	0.734838	+	0.678243i	$0.237258\pi$
$60$	0	0
$61$	272.044	0.571011	0.285506	−	0.958377i	$-0.407838\pi$
$61$	272.044	0.571011	0.285506	+	0.958377i	$0.407838\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	616.000	1.17547
$66$	0	0
$67$	−580.000	−1.05759	−0.528793	−	0.848751i	$-0.677355\pi$
$67$	−580.000	−1.05759	−0.528793	+	0.848751i	$0.677355\pi$
$68$	0	0
$69$	−450.280	−0.785613
$70$	0	0
$71$	544.000	0.909309	0.454654	−	0.890668i	$-0.349763\pi$
$71$	544.000	0.909309	0.454654	+	0.890668i	$0.349763\pi$
$72$	0	0
$73$	−600.373	−0.962580	−0.481290	−	0.876561i	$-0.659832\pi$
$73$	−600.373	−0.962580	−0.481290	+	0.876561i	$0.659832\pi$
$74$	0	0
$75$	−347.091	−0.534381
$76$	0	0
$77$	0	0
$78$	0	0
$79$	680.000	0.968430	0.484215	−	0.874949i	$-0.339105\pi$
$79$	680.000	0.968430	0.484215	+	0.874949i	$0.339105\pi$
$80$	0	0
$81$	1345.00	1.84499
$82$	0	0
$83$	−196.997	−0.260521	−0.130261	−	0.991480i	$-0.541581\pi$
$83$	−196.997	−0.260521	−0.130261	+	0.991480i	$0.541581\pi$
$84$	0	0
$85$	528.000	0.673760
$86$	0	0
$87$	−1557.22	−1.91898
$88$	0	0
$89$	−1500.93	−1.78762	−0.893812	−	0.448441i	$-0.851979\pi$
$89$	−1500.93	−1.78762	−0.893812	+	0.448441i	$0.851979\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	1936.00	2.15864
$94$	0	0
$95$	−88.0000	−0.0950380
$96$	0	0
$97$	−656.658	−0.687356	−0.343678	−	0.939088i	$-0.611673\pi$
$97$	−656.658	−0.687356	−0.343678	+	0.939088i	$0.611673\pi$
$98$	0	0
$99$	−1220.00	−1.23853
$100$	0	0
$101$	121.951	0.120144	0.0600721	−	0.998194i	$-0.480867\pi$
$101$	121.951	0.120144	0.0600721	+	0.998194i	$0.480867\pi$
$102$	0	0
$103$	1369.60	1.31020	0.655101	−	0.755541i	$-0.272626\pi$
$103$	1369.60	1.31020	0.655101	+	0.755541i	$0.272626\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	260.000	0.234908	0.117454	−	0.993078i	$-0.462527\pi$
$107$	260.000	0.234908	0.117454	+	0.993078i	$0.462527\pi$
$108$	0	0
$109$	1882.00	1.65379	0.826894	−	0.562358i	$-0.190106\pi$
$109$	1882.00	1.65379	0.826894	+	0.562358i	$0.190106\pi$
$110$	0	0
$111$	−731.705	−0.625679
$112$	0	0
$113$	−1286.00	−1.07059	−0.535295	−	0.844665i	$-0.679800\pi$
$113$	−1286.00	−1.07059	−0.535295	+	0.844665i	$0.679800\pi$
$114$	0	0
$115$	−450.280	−0.365120
$116$	0	0
$117$	4005.62	3.16512
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−931.000	−0.699474
$122$	0	0
$123$	3696.00	2.70941
$124$	0	0
$125$	−1519.69	−1.08741
$126$	0	0
$127$	−2312.00	−1.61541	−0.807704	−	0.589588i	$-0.799290\pi$
$127$	−2312.00	−1.61541	−0.807704	+	0.589588i	$0.799290\pi$
$128$	0	0
$129$	−4090.04	−2.79154
$130$	0	0
$131$	253.282	0.168927	0.0844633	−	0.996427i	$-0.473082\pi$
$131$	253.282	0.168927	0.0844633	+	0.996427i	$0.473082\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	2992.00	1.90748
$136$	0	0
$137$	−1114.00	−0.694711	−0.347356	−	0.937733i	$-0.612920\pi$
$137$	−1114.00	−0.694711	−0.347356	+	0.937733i	$0.612920\pi$
$138$	0	0
$139$	−1378.98	−0.841466	−0.420733	−	0.907185i	$-0.638227\pi$
$139$	−1378.98	−0.841466	−0.420733	+	0.907185i	$0.638227\pi$
$140$	0	0
$141$	−1936.00	−1.15632
$142$	0	0
$143$	−1313.32	−0.768007
$144$	0	0
$145$	−1557.22	−0.891862
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−946.000	−0.520130	−0.260065	−	0.965591i	$-0.583744\pi$
$149$	−946.000	−0.520130	−0.260065	+	0.965591i	$0.583744\pi$
$150$	0	0
$151$	−832.000	−0.448392	−0.224196	−	0.974544i	$-0.571976\pi$
$151$	−832.000	−0.448392	−0.224196	+	0.974544i	$0.571976\pi$
$152$	0	0
$153$	3433.38	1.81420
$154$	0	0
$155$	1936.00	1.00325
$156$	0	0
$157$	2879.92	1.46396	0.731982	−	0.681324i	$-0.238596\pi$
$157$	2879.92	1.46396	0.731982	+	0.681324i	$0.238596\pi$
$158$	0	0
$159$	581.612	0.290093
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−636.000	−0.305616	−0.152808	−	0.988256i	$-0.548832\pi$
$163$	−636.000	−0.305616	−0.152808	+	0.988256i	$0.548832\pi$
$164$	0	0
$165$	−1760.00	−0.830399
$166$	0	0
$167$	656.658	0.304274	0.152137	−	0.988359i	$-0.451385\pi$
$167$	656.658	0.304274	0.152137	+	0.988359i	$0.451385\pi$
$168$	0	0
$169$	2115.00	0.962676
$170$	0	0
$171$	−572.231	−0.255904
$172$	0	0
$173$	666.039	0.292705	0.146353	−	0.989232i	$-0.453247\pi$
$173$	666.039	0.292705	0.146353	+	0.989232i	$0.453247\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	6248.00	2.65327
$178$	0	0
$179$	3228.00	1.34789	0.673944	−	0.738782i	$-0.264599\pi$
$179$	3228.00	1.34789	0.673944	+	0.738782i	$0.264599\pi$
$180$	0	0
$181$	−2823.63	−1.15955	−0.579776	−	0.814776i	$-0.696860\pi$
$181$	−2823.63	−1.15955	−0.579776	+	0.814776i	$0.696860\pi$
$182$	0	0
$183$	2552.00	1.03087
$184$	0	0
$185$	−731.705	−0.290789
$186$	0	0
$187$	−1125.70	−0.440210
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2136.00	0.809191	0.404596	−	0.914496i	$-0.367412\pi$
$191$	2136.00	0.809191	0.404596	+	0.914496i	$0.367412\pi$
$192$	0	0
$193$	1658.00	0.618370	0.309185	−	0.951002i	$-0.399944\pi$
$193$	1658.00	0.618370	0.309185	+	0.951002i	$0.399944\pi$
$194$	0	0
$195$	5778.59	2.12212
$196$	0	0
$197$	−978.000	−0.353704	−0.176852	−	0.984237i	$-0.556591\pi$
$197$	−978.000	−0.353704	−0.176852	+	0.984237i	$0.556591\pi$
$198$	0	0
$199$	4934.32	1.75771	0.878855	−	0.477088i	$-0.158308\pi$
$199$	4934.32	1.75771	0.878855	+	0.477088i	$0.158308\pi$
$200$	0	0
$201$	−5440.88	−1.90930
$202$	0	0
$203$	0	0
$204$	0	0
$205$	3696.00	1.25922
$206$	0	0
$207$	−2928.00	−0.983140
$208$	0	0
$209$	187.617	0.0620943
$210$	0	0
$211$	−1556.00	−0.507675	−0.253838	−	0.967247i	$-0.581693\pi$
$211$	−1556.00	−0.507675	−0.253838	+	0.967247i	$0.581693\pi$
$212$	0	0
$213$	5103.17	1.64161
$214$	0	0
$215$	−4090.04	−1.29739
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−5632.00	−1.73779
$220$	0	0
$221$	3696.00	1.12498
$222$	0	0
$223$	−2889.30	−0.867630	−0.433815	−	0.901002i	$-0.642833\pi$
$223$	−2889.30	−0.867630	−0.433815	+	0.901002i	$0.642833\pi$
$224$	0	0
$225$	−2257.00	−0.668741
$226$	0	0
$227$	1979.36	0.578742	0.289371	−	0.957217i	$-0.406554\pi$
$227$	1979.36	0.578742	0.289371	+	0.957217i	$0.406554\pi$
$228$	0	0
$229$	2767.35	0.798565	0.399282	−	0.916828i	$-0.369259\pi$
$229$	2767.35	0.798565	0.399282	+	0.916828i	$0.369259\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−6490.00	−1.82478	−0.912391	−	0.409321i	$-0.865766\pi$
$233$	−6490.00	−1.82478	−0.912391	+	0.409321i	$0.865766\pi$
$234$	0	0
$235$	−1936.00	−0.537407
$236$	0	0
$237$	6378.97	1.74835
$238$	0	0
$239$	4296.00	1.16270	0.581350	−	0.813654i	$-0.302525\pi$
$239$	4296.00	1.16270	0.581350	+	0.813654i	$0.302525\pi$
$240$	0	0
$241$	4521.56	1.20854	0.604272	−	0.796778i	$-0.293464\pi$
$241$	4521.56	1.20854	0.604272	+	0.796778i	$0.293464\pi$
$242$	0	0
$243$	4005.62	1.05745
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−616.000	−0.158685
$248$	0	0
$249$	−1848.00	−0.470330
$250$	0	0
$251$	−5581.59	−1.40361	−0.701807	−	0.712367i	$-0.747623\pi$
$251$	−5581.59	−1.40361	−0.701807	+	0.712367i	$0.747623\pi$
$252$	0	0
$253$	960.000	0.238556
$254$	0	0
$255$	4953.08	1.21637
$256$	0	0
$257$	−1500.93	−0.364302	−0.182151	−	0.983271i	$-0.558306\pi$
$257$	−1500.93	−0.364302	−0.182151	+	0.983271i	$0.558306\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−10126.0	−2.40147
$262$	0	0
$263$	400.000	0.0937835	0.0468917	−	0.998900i	$-0.485068\pi$
$263$	400.000	0.0937835	0.0468917	+	0.998900i	$0.485068\pi$
$264$	0	0
$265$	581.612	0.134823
$266$	0	0
$267$	−14080.0	−3.22727
$268$	0	0
$269$	−272.044	−0.0616610	−0.0308305	−	0.999525i	$-0.509815\pi$
$269$	−272.044	−0.0616610	−0.0308305	+	0.999525i	$0.509815\pi$
$270$	0	0
$271$	−6904.29	−1.54762	−0.773812	−	0.633416i	$-0.781652\pi$
$271$	−6904.29	−1.54762	−0.773812	+	0.633416i	$0.781652\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	740.000	0.162268
$276$	0	0
$277$	−6770.00	−1.46848	−0.734242	−	0.678888i	$-0.762462\pi$
$277$	−6770.00	−1.46848	−0.734242	+	0.678888i	$0.762462\pi$
$278$	0	0
$279$	12589.1	2.70139
$280$	0	0
$281$	1878.00	0.398691	0.199345	−	0.979929i	$-0.436118\pi$
$281$	1878.00	0.398691	0.199345	+	0.979929i	$0.436118\pi$
$282$	0	0
$283$	−384.614	−0.0807878	−0.0403939	−	0.999184i	$-0.512861\pi$
$283$	−384.614	−0.0807878	−0.0403939	+	0.999184i	$0.512861\pi$
$284$	0	0
$285$	−825.513	−0.171576
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−1745.00	−0.355180
$290$	0	0
$291$	−6160.00	−1.24091
$292$	0	0
$293$	−3742.95	−0.746299	−0.373149	−	0.927771i	$-0.621722\pi$
$293$	−3742.95	−0.746299	−0.373149	+	0.927771i	$0.621722\pi$
$294$	0	0
$295$	6248.00	1.23313
$296$	0	0
$297$	−6378.97	−1.24628
$298$	0	0
$299$	−3151.96	−0.609641
$300$	0	0
$301$	0	0
$302$	0	0
$303$	1144.00	0.216901
$304$	0	0
$305$	2552.00	0.479105
$306$	0	0
$307$	−722.324	−0.134284	−0.0671420	−	0.997743i	$-0.521388\pi$
$307$	−722.324	−0.134284	−0.0671420	+	0.997743i	$0.521388\pi$
$308$	0	0
$309$	12848.0	2.36536
$310$	0	0
$311$	−7279.53	−1.32728	−0.663640	−	0.748052i	$-0.730989\pi$
$311$	−7279.53	−1.32728	−0.663640	+	0.748052i	$0.730989\pi$
$312$	0	0
$313$	1519.69	0.274435	0.137218	−	0.990541i	$-0.456184\pi$
$313$	1519.69	0.274435	0.137218	+	0.990541i	$0.456184\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2358.00	0.417787	0.208893	−	0.977938i	$-0.433014\pi$
$317$	2358.00	0.417787	0.208893	+	0.977938i	$0.433014\pi$
$318$	0	0
$319$	3320.00	0.582709
$320$	0	0
$321$	2439.02	0.424089
$322$	0	0
$323$	−528.000	−0.0909557
$324$	0	0
$325$	−2429.64	−0.414683
$326$	0	0
$327$	17654.7	2.98565
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−2372.00	−0.393888	−0.196944	−	0.980415i	$-0.563102\pi$
$331$	−2372.00	−0.393888	−0.196944	+	0.980415i	$0.563102\pi$
$332$	0	0
$333$	−4758.00	−0.782993
$334$	0	0
$335$	−5440.88	−0.887365
$336$	0	0
$337$	−250.000	−0.0404106	−0.0202053	−	0.999796i	$-0.506432\pi$
$337$	−250.000	−0.0404106	−0.0202053	+	0.999796i	$0.506432\pi$
$338$	0	0
$339$	−12063.7	−1.93278
$340$	0	0
$341$	−4127.57	−0.655485
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−4224.00	−0.659167
$346$	0	0
$347$	−9540.00	−1.47589	−0.737945	−	0.674861i	$-0.764204\pi$
$347$	−9540.00	−1.47589	−0.737945	+	0.674861i	$0.764204\pi$
$348$	0	0
$349$	−5712.93	−0.876235	−0.438117	−	0.898918i	$-0.644355\pi$
$349$	−5712.93	−0.876235	−0.438117	+	0.898918i	$0.644355\pi$
$350$	0	0
$351$	20944.0	3.18492
$352$	0	0
$353$	4390.23	0.661950	0.330975	−	0.943640i	$-0.392622\pi$
$353$	4390.23	0.661950	0.330975	+	0.943640i	$0.392622\pi$
$354$	0	0
$355$	5103.17	0.762953
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−1840.00	−0.270506	−0.135253	−	0.990811i	$-0.543185\pi$
$359$	−1840.00	−0.270506	−0.135253	+	0.990811i	$0.543185\pi$
$360$	0	0
$361$	−6771.00	−0.987170
$362$	0	0
$363$	−8733.55	−1.26279
$364$	0	0
$365$	−5632.00	−0.807650
$366$	0	0
$367$	2964.34	0.421628	0.210814	−	0.977526i	$-0.432389\pi$
$367$	2964.34	0.421628	0.210814	+	0.977526i	$0.432389\pi$
$368$	0	0
$369$	24033.7	3.39063
$370$	0	0
$371$	0	0
$372$	0	0
$373$	3982.00	0.552762	0.276381	−	0.961048i	$-0.410865\pi$
$373$	3982.00	0.552762	0.276381	+	0.961048i	$0.410865\pi$
$374$	0	0
$375$	−14256.0	−1.96314
$376$	0	0
$377$	−10900.5	−1.48914
$378$	0	0
$379$	−2676.00	−0.362683	−0.181342	−	0.983420i	$-0.558044\pi$
$379$	−2676.00	−0.362683	−0.181342	+	0.983420i	$0.558044\pi$
$380$	0	0
$381$	−21688.5	−2.91636
$382$	0	0
$383$	−7035.62	−0.938652	−0.469326	−	0.883025i	$-0.655503\pi$
$383$	−7035.62	−0.938652	−0.469326	+	0.883025i	$0.655503\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−26596.0	−3.49341
$388$	0	0
$389$	8658.00	1.12848	0.564239	−	0.825611i	$-0.309170\pi$
$389$	8658.00	1.12848	0.564239	+	0.825611i	$0.309170\pi$
$390$	0	0
$391$	−2701.68	−0.349437
$392$	0	0
$393$	2376.00	0.304970
$394$	0	0
$395$	6378.97	0.812558
$396$	0	0
$397$	−9052.50	−1.14441	−0.572207	−	0.820109i	$-0.693912\pi$
$397$	−9052.50	−1.14441	−0.572207	+	0.820109i	$0.693912\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−5706.00	−0.710584	−0.355292	−	0.934755i	$-0.615619\pi$
$401$	−5706.00	−0.710584	−0.355292	+	0.934755i	$0.615619\pi$
$402$	0	0
$403$	13552.0	1.67512
$404$	0	0
$405$	12617.2	1.54804
$406$	0	0
$407$	1560.00	0.189991
$408$	0	0
$409$	2420.25	0.292601	0.146301	−	0.989240i	$-0.453263\pi$
$409$	2420.25	0.292601	0.146301	+	0.989240i	$0.453263\pi$
$410$	0	0
$411$	−10450.2	−1.25419
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−1848.00	−0.218590
$416$	0	0
$417$	−12936.0	−1.51913
$418$	0	0
$419$	1510.31	0.176095	0.0880473	−	0.996116i	$-0.471937\pi$
$419$	1510.31	0.176095	0.0880473	+	0.996116i	$0.471937\pi$
$420$	0	0
$421$	−16770.0	−1.94138	−0.970689	−	0.240341i	$-0.922741\pi$
$421$	−16770.0	−1.94138	−0.970689	+	0.240341i	$0.922741\pi$
$422$	0	0
$423$	−12589.1	−1.44705
$424$	0	0
$425$	−2082.54	−0.237690
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−12320.0	−1.38652
$430$	0	0
$431$	−1336.00	−0.149311	−0.0746553	−	0.997209i	$-0.523786\pi$
$431$	−1336.00	−0.149311	−0.0746553	+	0.997209i	$0.523786\pi$
$432$	0	0
$433$	−11163.2	−1.23896	−0.619479	−	0.785013i	$-0.712656\pi$
$433$	−11163.2	−1.23896	−0.619479	+	0.785013i	$0.712656\pi$
$434$	0	0
$435$	−14608.0	−1.61011
$436$	0	0
$437$	450.280	0.0492902
$438$	0	0
$439$	3602.24	0.391630	0.195815	−	0.980641i	$-0.437265\pi$
$439$	3602.24	0.391630	0.195815	+	0.980641i	$0.437265\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−6348.00	−0.680818	−0.340409	−	0.940277i	$-0.610566\pi$
$443$	−6348.00	−0.680818	−0.340409	+	0.940277i	$0.610566\pi$
$444$	0	0
$445$	−14080.0	−1.49990
$446$	0	0
$447$	−8874.27	−0.939012
$448$	0	0
$449$	7170.00	0.753615	0.376808	−	0.926292i	$-0.377022\pi$
$449$	7170.00	0.753615	0.376808	+	0.926292i	$0.377022\pi$
$450$	0	0
$451$	−7879.90	−0.822727
$452$	0	0
$453$	−7804.85	−0.809501
$454$	0	0
$455$	0	0
$456$	0	0
$457$	6866.00	0.702796	0.351398	−	0.936226i	$-0.385706\pi$
$457$	6866.00	0.702796	0.351398	+	0.936226i	$0.385706\pi$
$458$	0	0
$459$	17952.0	1.82555
$460$	0	0
$461$	1378.98	0.139318	0.0696590	−	0.997571i	$-0.477809\pi$
$461$	1378.98	0.139318	0.0696590	+	0.997571i	$0.477809\pi$
$462$	0	0
$463$	−2648.00	−0.265795	−0.132897	−	0.991130i	$-0.542428\pi$
$463$	−2648.00	−0.265795	−0.132897	+	0.991130i	$0.542428\pi$
$464$	0	0
$465$	18161.3	1.81120
$466$	0	0
$467$	12335.8	1.22234	0.611170	−	0.791500i	$-0.290699\pi$
$467$	12335.8	1.22234	0.611170	+	0.791500i	$0.290699\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	27016.0	2.64295
$472$	0	0
$473$	8720.00	0.847666
$474$	0	0
$475$	347.091	0.0335276
$476$	0	0
$477$	3782.00	0.363031
$478$	0	0
$479$	13339.5	1.27244	0.636221	−	0.771507i	$-0.280497\pi$
$479$	13339.5	1.27244	0.636221	+	0.771507i	$0.280497\pi$
$480$	0	0
$481$	−5121.93	−0.485530
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−6160.00	−0.576724
$486$	0	0
$487$	−13936.0	−1.29672	−0.648358	−	0.761336i	$-0.724544\pi$
$487$	−13936.0	−1.29672	−0.648358	+	0.761336i	$0.724544\pi$
$488$	0	0
$489$	−5966.21	−0.551741
$490$	0	0
$491$	12276.0	1.12833	0.564163	−	0.825663i	$-0.309199\pi$
$491$	12276.0	1.12833	0.564163	+	0.825663i	$0.309199\pi$
$492$	0	0
$493$	−9343.31	−0.853553
$494$	0	0
$495$	−11444.6	−1.03919
$496$	0	0
$497$	0	0
$498$	0	0
$499$	2220.00	0.199160	0.0995800	−	0.995030i	$-0.468250\pi$
$499$	2220.00	0.199160	0.0995800	+	0.995030i	$0.468250\pi$
$500$	0	0
$501$	6160.00	0.549318
$502$	0	0
$503$	11294.5	1.00119	0.500594	−	0.865682i	$-0.333115\pi$
$503$	11294.5	1.00119	0.500594	+	0.865682i	$0.333115\pi$
$504$	0	0
$505$	1144.00	0.100807
$506$	0	0
$507$	19840.5	1.73796
$508$	0	0
$509$	−15881.7	−1.38300	−0.691499	−	0.722377i	$-0.743049\pi$
$509$	−15881.7	−1.38300	−0.691499	+	0.722377i	$0.743049\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−2992.00	−0.257505
$514$	0	0
$515$	12848.0	1.09932
$516$	0	0
$517$	4127.57	0.351122
$518$	0	0
$519$	6248.00	0.528433
$520$	0	0
$521$	11613.5	0.976575	0.488287	−	0.872683i	$-0.337622\pi$
$521$	11613.5	0.976575	0.488287	+	0.872683i	$0.337622\pi$
$522$	0	0
$523$	−12617.2	−1.05490	−0.527450	−	0.849586i	$-0.676852\pi$
$523$	−12617.2	−1.05490	−0.527450	+	0.849586i	$0.676852\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	11616.0	0.960154
$528$	0	0
$529$	−9863.00	−0.810635
$530$	0	0
$531$	40628.4	3.32038
$532$	0	0
$533$	25872.0	2.10252
$534$	0	0
$535$	2439.02	0.197099
$536$	0	0
$537$	30281.3	2.43340
$538$	0	0
$539$	0	0
$540$	0	0
$541$	1798.00	0.142887	0.0714437	−	0.997445i	$-0.477239\pi$
$541$	1798.00	0.142887	0.0714437	+	0.997445i	$0.477239\pi$
$542$	0	0
$543$	−26488.0	−2.09339
$544$	0	0
$545$	17654.7	1.38761
$546$	0	0
$547$	−1276.00	−0.0997401	−0.0498700	−	0.998756i	$-0.515881\pi$
$547$	−1276.00	−0.0997401	−0.0498700	+	0.998756i	$0.515881\pi$
$548$	0	0
$549$	16594.7	1.29006
$550$	0	0
$551$	1557.22	0.120399
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−6864.00	−0.524974
$556$	0	0
$557$	2694.00	0.204934	0.102467	−	0.994736i	$-0.467326\pi$
$557$	2694.00	0.204934	0.102467	+	0.994736i	$0.467326\pi$
$558$	0	0
$559$	−28630.3	−2.16625
$560$	0	0
$561$	−10560.0	−0.794730
$562$	0	0
$563$	15769.2	1.18045	0.590223	−	0.807240i	$-0.299040\pi$
$563$	15769.2	1.18045	0.590223	+	0.807240i	$0.299040\pi$
$564$	0	0
$565$	−12063.7	−0.898276
$566$	0	0
$567$	0	0
$568$	0	0
$569$	12606.0	0.928772	0.464386	−	0.885633i	$-0.346275\pi$
$569$	12606.0	0.928772	0.464386	+	0.885633i	$0.346275\pi$
$570$	0	0
$571$	−6852.00	−0.502184	−0.251092	−	0.967963i	$-0.580790\pi$
$571$	−6852.00	−0.502184	−0.251092	+	0.967963i	$0.580790\pi$
$572$	0	0
$573$	20037.5	1.46087
$574$	0	0
$575$	1776.00	0.128808
$576$	0	0
$577$	14371.4	1.03690	0.518449	−	0.855108i	$-0.326509\pi$
$577$	14371.4	1.03690	0.518449	+	0.855108i	$0.326509\pi$
$578$	0	0
$579$	15553.4	1.11637
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−1240.00	−0.0880884
$584$	0	0
$585$	37576.0	2.65569
$586$	0	0
$587$	−18977.4	−1.33438	−0.667191	−	0.744887i	$-0.732503\pi$
$587$	−18977.4	−1.33438	−0.667191	+	0.744887i	$0.732503\pi$
$588$	0	0
$589$	−1936.00	−0.135435
$590$	0	0
$591$	−9174.45	−0.638556
$592$	0	0
$593$	8217.61	0.569067	0.284534	−	0.958666i	$-0.408161\pi$
$593$	8217.61	0.569067	0.284534	+	0.958666i	$0.408161\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	46288.0	3.17327
$598$	0	0
$599$	19104.0	1.30312	0.651559	−	0.758598i	$-0.274115\pi$
$599$	19104.0	1.30312	0.651559	+	0.758598i	$0.274115\pi$
$600$	0	0
$601$	21538.4	1.46185	0.730923	−	0.682460i	$-0.239090\pi$
$601$	21538.4	1.46185	0.730923	+	0.682460i	$0.239090\pi$
$602$	0	0
$603$	−35380.0	−2.38936
$604$	0	0
$605$	−8733.55	−0.586892
$606$	0	0
$607$	−13733.5	−0.918331	−0.459166	−	0.888351i	$-0.651852\pi$
$607$	−13733.5	−0.918331	−0.459166	+	0.888351i	$0.651852\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−13552.0	−0.897308
$612$	0	0
$613$	28034.0	1.84712	0.923558	−	0.383458i	$-0.125267\pi$
$613$	28034.0	1.84712	0.923558	+	0.383458i	$0.125267\pi$
$614$	0	0
$615$	34671.6	2.27332
$616$	0	0
$617$	−8258.00	−0.538824	−0.269412	−	0.963025i	$-0.586829\pi$
$617$	−8258.00	−0.538824	−0.269412	+	0.963025i	$0.586829\pi$
$618$	0	0
$619$	5131.31	0.333191	0.166595	−	0.986025i	$-0.446723\pi$
$619$	5131.31	0.333191	0.166595	+	0.986025i	$0.446723\pi$
$620$	0	0
$621$	−15309.5	−0.989291
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−9631.00	−0.616384
$626$	0	0
$627$	1760.00	0.112101
$628$	0	0
$629$	−4390.23	−0.278299
$630$	0	0
$631$	−912.000	−0.0575375	−0.0287687	−	0.999586i	$-0.509159\pi$
$631$	−912.000	−0.0575375	−0.0287687	+	0.999586i	$0.509159\pi$
$632$	0	0
$633$	−14596.6	−0.916527
$634$	0	0
$635$	−21688.5	−1.35540
$636$	0	0
$637$	0	0
$638$	0	0
$639$	33184.0	2.05436
$640$	0	0
$641$	−890.000	−0.0548407	−0.0274203	−	0.999624i	$-0.508729\pi$
$641$	−890.000	−0.0548407	−0.0274203	+	0.999624i	$0.508729\pi$
$642$	0	0
$643$	29352.6	1.80024	0.900120	−	0.435642i	$-0.143479\pi$
$643$	29352.6	1.80024	0.900120	+	0.435642i	$0.143479\pi$
$644$	0	0
$645$	−38368.0	−2.34223
$646$	0	0
$647$	−11876.1	−0.721637	−0.360818	−	0.932636i	$-0.617503\pi$
$647$	−11876.1	−0.721637	−0.360818	+	0.932636i	$0.617503\pi$
$648$	0	0
$649$	−13320.8	−0.805680
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−21526.0	−1.29001	−0.645006	−	0.764178i	$-0.723145\pi$
$653$	−21526.0	−1.29001	−0.645006	+	0.764178i	$0.723145\pi$
$654$	0	0
$655$	2376.00	0.141737
$656$	0	0
$657$	−36622.8	−2.17472
$658$	0	0
$659$	−23452.0	−1.38628	−0.693141	−	0.720802i	$-0.743774\pi$
$659$	−23452.0	−1.38628	−0.693141	+	0.720802i	$0.743774\pi$
$660$	0	0
$661$	−26669.7	−1.56934	−0.784668	−	0.619916i	$-0.787167\pi$
$661$	−26669.7	−1.56934	−0.784668	+	0.619916i	$0.787167\pi$
$662$	0	0
$663$	34671.6	2.03097
$664$	0	0
$665$	0	0
$666$	0	0
$667$	7968.00	0.462552
$668$	0	0
$669$	−27104.0	−1.56637
$670$	0	0
$671$	−5440.88	−0.313030
$672$	0	0
$673$	−13858.0	−0.793739	−0.396870	−	0.917875i	$-0.629904\pi$
$673$	−13858.0	−0.793739	−0.396870	+	0.917875i	$0.629904\pi$
$674$	0	0
$675$	−11801.1	−0.672924
$676$	0	0
$677$	32448.3	1.84208	0.921041	−	0.389466i	$-0.127340\pi$
$677$	32448.3	1.84208	0.921041	+	0.389466i	$0.127340\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	18568.0	1.04483
$682$	0	0
$683$	27812.0	1.55812	0.779060	−	0.626949i	$-0.215696\pi$
$683$	27812.0	1.55812	0.779060	+	0.626949i	$0.215696\pi$
$684$	0	0
$685$	−10450.2	−0.582895
$686$	0	0
$687$	25960.0	1.44168
$688$	0	0
$689$	4071.28	0.225114
$690$	0	0
$691$	1303.94	0.0717859	0.0358929	−	0.999356i	$-0.488572\pi$
$691$	1303.94	0.0717859	0.0358929	+	0.999356i	$0.488572\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−12936.0	−0.706029
$696$	0	0
$697$	22176.0	1.20513
$698$	0	0
$699$	−60881.6	−3.29435
$700$	0	0
$701$	22906.0	1.23416	0.617081	−	0.786900i	$-0.288315\pi$
$701$	22906.0	1.23416	0.617081	+	0.786900i	$0.288315\pi$
$702$	0	0
$703$	731.705	0.0392557
$704$	0	0
$705$	−18161.3	−0.970204
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−15086.0	−0.799107	−0.399553	−	0.916710i	$-0.630835\pi$
$709$	−15086.0	−0.799107	−0.399553	+	0.916710i	$0.630835\pi$
$710$	0	0
$711$	41480.0	2.18793
$712$	0	0
$713$	−9906.16	−0.520321
$714$	0	0
$715$	−12320.0	−0.644394
$716$	0	0
$717$	40300.1	2.09907
$718$	0	0
$719$	20544.0	1.06559	0.532797	−	0.846243i	$-0.321141\pi$
$719$	20544.0	1.06559	0.532797	+	0.846243i	$0.321141\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	42416.0	2.18184
$724$	0	0
$725$	6142.00	0.314632
$726$	0	0
$727$	7223.24	0.368494	0.184247	−	0.982880i	$-0.441015\pi$
$727$	7223.24	0.368494	0.184247	+	0.982880i	$0.441015\pi$
$728$	0	0
$729$	1261.00	0.0640654
$730$	0	0
$731$	−24540.3	−1.24166
$732$	0	0
$733$	29427.7	1.48286	0.741430	−	0.671031i	$-0.234148\pi$
$733$	29427.7	1.48286	0.741430	+	0.671031i	$0.234148\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	11600.0	0.579771
$738$	0	0
$739$	−32668.0	−1.62613	−0.813066	−	0.582171i	$-0.802203\pi$
$739$	−32668.0	−1.62613	−0.813066	+	0.582171i	$0.802203\pi$
$740$	0	0
$741$	−5778.59	−0.286480
$742$	0	0
$743$	37056.0	1.82968	0.914840	−	0.403816i	$-0.132316\pi$
$743$	37056.0	1.82968	0.914840	+	0.403816i	$0.132316\pi$
$744$	0	0
$745$	−8874.27	−0.436413
$746$	0	0
$747$	−12016.8	−0.588586
$748$	0	0
$749$	0	0
$750$	0	0
$751$	19608.0	0.952738	0.476369	−	0.879246i	$-0.341953\pi$
$751$	19608.0	0.952738	0.476369	+	0.879246i	$0.341953\pi$
$752$	0	0
$753$	−52360.0	−2.53400
$754$	0	0
$755$	−7804.85	−0.376222
$756$	0	0
$757$	19378.0	0.930390	0.465195	−	0.885208i	$-0.345984\pi$
$757$	19378.0	0.930390	0.465195	+	0.885208i	$0.345984\pi$
$758$	0	0
$759$	9005.60	0.430675
$760$	0	0
$761$	−13977.4	−0.665810	−0.332905	−	0.942960i	$-0.608029\pi$
$761$	−13977.4	−0.665810	−0.332905	+	0.942960i	$0.608029\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	32208.0	1.52220
$766$	0	0
$767$	43736.0	2.05895
$768$	0	0
$769$	8536.56	0.400307	0.200154	−	0.979765i	$-0.435856\pi$
$769$	8536.56	0.400307	0.200154	+	0.979765i	$0.435856\pi$
$770$	0	0
$771$	−14080.0	−0.657690
$772$	0	0
$773$	−29296.3	−1.36315	−0.681576	−	0.731748i	$-0.738705\pi$
$773$	−29296.3	−1.36315	−0.681576	+	0.731748i	$0.738705\pi$
$774$	0	0
$775$	−7636.00	−0.353927
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−3696.00	−0.169991
$780$	0	0
$781$	−10880.0	−0.498485
$782$	0	0
$783$	−52945.4	−2.41649
$784$	0	0
$785$	27016.0	1.22833
$786$	0	0
$787$	−13780.4	−0.624167	−0.312084	−	0.950055i	$-0.601027\pi$
$787$	−13780.4	−0.624167	−0.312084	+	0.950055i	$0.601027\pi$
$788$	0	0
$789$	3752.33	0.169311
$790$	0	0
$791$	0	0
$792$	0	0
$793$	17864.0	0.799961
$794$	0	0
$795$	5456.00	0.243402
$796$	0	0
$797$	−34868.6	−1.54970	−0.774848	−	0.632148i	$-0.782174\pi$
$797$	−34868.6	−1.54970	−0.774848	+	0.632148i	$0.782174\pi$
$798$	0	0
$799$	−11616.0	−0.514324
$800$	0	0
$801$	−91556.9	−4.03871
$802$	0	0
$803$	12007.5	0.527689
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−2552.00	−0.111319
$808$	0	0
$809$	14034.0	0.609900	0.304950	−	0.952368i	$-0.401360\pi$
$809$	14034.0	0.609900	0.304950	+	0.952368i	$0.401360\pi$
$810$	0	0
$811$	6632.25	0.287164	0.143582	−	0.989638i	$-0.454138\pi$
$811$	6632.25	0.287164	0.143582	+	0.989638i	$0.454138\pi$
$812$	0	0
$813$	−64768.0	−2.79399
$814$	0	0
$815$	−5966.21	−0.256426
$816$	0	0
$817$	4090.04	0.175144
$818$	0	0
$819$	0	0
$820$	0	0
$821$	28622.0	1.21670	0.608352	−	0.793667i	$-0.291831\pi$
$821$	28622.0	1.21670	0.608352	+	0.793667i	$0.291831\pi$
$822$	0	0
$823$	−24688.0	−1.04565	−0.522825	−	0.852440i	$-0.675122\pi$
$823$	−24688.0	−1.04565	−0.522825	+	0.852440i	$0.675122\pi$
$824$	0	0
$825$	6941.82	0.292949
$826$	0	0
$827$	30756.0	1.29322	0.646609	−	0.762822i	$-0.276187\pi$
$827$	30756.0	1.29322	0.646609	+	0.762822i	$0.276187\pi$
$828$	0	0
$829$	23236.3	0.973499	0.486750	−	0.873542i	$-0.338182\pi$
$829$	23236.3	0.973499	0.486750	+	0.873542i	$0.338182\pi$
$830$	0	0
$831$	−63508.2	−2.65111
$832$	0	0
$833$	0	0
$834$	0	0
$835$	6160.00	0.255300
$836$	0	0
$837$	65824.0	2.71829
$838$	0	0
$839$	24033.7	0.988957	0.494479	−	0.869190i	$-0.335359\pi$
$839$	24033.7	0.988957	0.494479	+	0.869190i	$0.335359\pi$
$840$	0	0
$841$	3167.00	0.129854
$842$	0	0
$843$	17617.2	0.719773
$844$	0	0
$845$	19840.5	0.807731
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−3608.00	−0.145850
$850$	0	0
$851$	3744.00	0.150814
$852$	0	0
$853$	−23574.0	−0.946260	−0.473130	−	0.880993i	$-0.656876\pi$
$853$	−23574.0	−0.946260	−0.473130	+	0.880993i	$0.656876\pi$
$854$	0	0
$855$	−5368.00	−0.214715
$856$	0	0
$857$	24484.0	0.975912	0.487956	−	0.872868i	$-0.337743\pi$
$857$	24484.0	0.975912	0.487956	+	0.872868i	$0.337743\pi$
$858$	0	0
$859$	32954.9	1.30897	0.654485	−	0.756075i	$-0.272885\pi$
$859$	32954.9	1.30897	0.654485	+	0.756075i	$0.272885\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−40872.0	−1.61217	−0.806083	−	0.591803i	$-0.798416\pi$
$863$	−40872.0	−1.61217	−0.806083	+	0.591803i	$0.798416\pi$
$864$	0	0
$865$	6248.00	0.245593
$866$	0	0
$867$	−16369.6	−0.641222
$868$	0	0
$869$	−13600.0	−0.530896
$870$	0	0
$871$	−38086.2	−1.48163
$872$	0	0
$873$	−40056.2	−1.55292
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−12006.0	−0.462273	−0.231137	−	0.972921i	$-0.574244\pi$
$877$	−12006.0	−0.462273	−0.231137	+	0.972921i	$0.574244\pi$
$878$	0	0
$879$	−35112.0	−1.34732
$880$	0	0
$881$	35722.2	1.36607	0.683037	−	0.730383i	$-0.260659\pi$
$881$	35722.2	1.36607	0.683037	+	0.730383i	$0.260659\pi$
$882$	0	0
$883$	−19588.0	−0.746533	−0.373267	−	0.927724i	$-0.621762\pi$
$883$	−19588.0	−0.746533	−0.373267	+	0.927724i	$0.621762\pi$
$884$	0	0
$885$	58611.4	2.22622
$886$	0	0
$887$	40243.8	1.52340	0.761699	−	0.647931i	$-0.224366\pi$
$887$	40243.8	1.52340	0.761699	+	0.647931i	$0.224366\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−26900.0	−1.01143
$892$	0	0
$893$	1936.00	0.0725485
$894$	0	0
$895$	30281.3	1.13094
$896$	0	0
$897$	−29568.0	−1.10061
$898$	0	0
$899$	−34258.8	−1.27096
$900$	0	0
$901$	3489.67	0.129032
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−26488.0	−0.972918
$906$	0	0
$907$	−15868.0	−0.580913	−0.290457	−	0.956888i	$-0.593807\pi$
$907$	−15868.0	−0.580913	−0.290457	+	0.956888i	$0.593807\pi$
$908$	0	0
$909$	7439.00	0.271437
$910$	0	0
$911$	−39832.0	−1.44862	−0.724310	−	0.689474i	$-0.757842\pi$
$911$	−39832.0	−1.44862	−0.724310	+	0.689474i	$0.757842\pi$
$912$	0	0
$913$	3939.95	0.142818
$914$	0	0
$915$	23939.9	0.864949
$916$	0	0
$917$	0	0
$918$	0	0
$919$	30528.0	1.09578	0.547892	−	0.836549i	$-0.315430\pi$
$919$	30528.0	1.09578	0.547892	+	0.836549i	$0.315430\pi$
$920$	0	0
$921$	−6776.00	−0.242429
$922$	0	0
$923$	35722.2	1.27390
$924$	0	0
$925$	2886.00	0.102585
$926$	0	0
$927$	83545.7	2.96009
$928$	0	0
$929$	−16604.1	−0.586396	−0.293198	−	0.956052i	$-0.594720\pi$
$929$	−16604.1	−0.586396	−0.293198	+	0.956052i	$0.594720\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−68288.0	−2.39619
$934$	0	0
$935$	−10560.0	−0.369357
$936$	0	0
$937$	29943.6	1.04399	0.521993	−	0.852950i	$-0.325189\pi$
$937$	29943.6	1.04399	0.521993	+	0.852950i	$0.325189\pi$
$938$	0	0
$939$	14256.0	0.495449
$940$	0	0
$941$	5375.22	0.186214	0.0931068	−	0.995656i	$-0.470320\pi$
$941$	5375.22	0.186214	0.0931068	+	0.995656i	$0.470320\pi$
$942$	0	0
$943$	−18911.8	−0.653077
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−45212.0	−1.55142	−0.775709	−	0.631091i	$-0.782607\pi$
$947$	−45212.0	−1.55142	−0.775709	+	0.631091i	$0.782607\pi$
$948$	0	0
$949$	−39424.0	−1.34853
$950$	0	0
$951$	22120.0	0.754248
$952$	0	0
$953$	34218.0	1.16310	0.581548	−	0.813512i	$-0.302447\pi$
$953$	34218.0	1.16310	0.581548	+	0.813512i	$0.302447\pi$
$954$	0	0
$955$	20037.5	0.678950
$956$	0	0
$957$	31144.4	1.05199
$958$	0	0
$959$	0	0
$960$	0	0
$961$	12801.0	0.429694
$962$	0	0
$963$	15860.0	0.530718
$964$	0	0
$965$	15553.4	0.518842
$966$	0	0
$967$	−14464.0	−0.481004	−0.240502	−	0.970649i	$-0.577312\pi$
$967$	−14464.0	−0.481004	−0.240502	+	0.970649i	$0.577312\pi$
$968$	0	0
$969$	−4953.08	−0.164206
$970$	0	0
$971$	−37832.9	−1.25038	−0.625188	−	0.780474i	$-0.714978\pi$
$971$	−37832.9	−1.25038	−0.625188	+	0.780474i	$0.714978\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−22792.0	−0.748644
$976$	0	0
$977$	42062.0	1.37736	0.688681	−	0.725065i	$-0.258190\pi$
$977$	42062.0	1.37736	0.688681	+	0.725065i	$0.258190\pi$
$978$	0	0
$979$	30018.7	0.979980
$980$	0	0
$981$	114802.	3.73634
$982$	0	0
$983$	43020.5	1.39587	0.697935	−	0.716161i	$-0.254102\pi$
$983$	43020.5	1.39587	0.697935	+	0.716161i	$0.254102\pi$
$984$	0	0
$985$	−9174.45	−0.296774
$986$	0	0
$987$	0	0
$988$	0	0
$989$	20928.0	0.672873
$990$	0	0
$991$	−21272.0	−0.681864	−0.340932	−	0.940088i	$-0.610743\pi$
$991$	−21272.0	−0.681864	−0.340932	+	0.940088i	$0.610743\pi$
$992$	0	0
$993$	−22251.3	−0.711102
$994$	0	0
$995$	46288.0	1.47480
$996$	0	0
$997$	121.951	0.00387384	0.00193692	−	0.999998i	$-0.499383\pi$
$997$	121.951	0.00387384	0.00193692	+	0.999998i	$0.499383\pi$
$998$	0	0
$999$	−24878.0	−0.787892

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.4.a.z.1.2		2
4.3	odd	2		98.4.a.h.1.1	✓	2
7.6	odd	2	inner	784.4.a.z.1.1		2
12.11	even	2		882.4.a.w.1.1		2
20.19	odd	2		2450.4.a.bs.1.2		2
28.3	even	6		98.4.c.g.79.1		4
28.11	odd	6		98.4.c.g.79.2		4
28.19	even	6		98.4.c.g.67.1		4
28.23	odd	6		98.4.c.g.67.2		4
28.27	even	2		98.4.a.h.1.2	yes	2
84.11	even	6		882.4.g.bi.667.2		4
84.23	even	6		882.4.g.bi.361.2		4
84.47	odd	6		882.4.g.bi.361.1		4
84.59	odd	6		882.4.g.bi.667.1		4
84.83	odd	2		882.4.a.w.1.2		2
140.139	even	2		2450.4.a.bs.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
98.4.a.h.1.1	✓	2	4.3	odd	2
98.4.a.h.1.2	yes	2	28.27	even	2
98.4.c.g.67.1		4	28.19	even	6
98.4.c.g.67.2		4	28.23	odd	6
98.4.c.g.79.1		4	28.3	even	6
98.4.c.g.79.2		4	28.11	odd	6
784.4.a.z.1.1		2	7.6	odd	2	inner
784.4.a.z.1.2		2	1.1	even	1	trivial
882.4.a.w.1.1		2	12.11	even	2
882.4.a.w.1.2		2	84.83	odd	2
882.4.g.bi.361.1		4	84.47	odd	6
882.4.g.bi.361.2		4	84.23	even	6
882.4.g.bi.667.1		4	84.59	odd	6
882.4.g.bi.667.2		4	84.11	even	6
2450.4.a.bs.1.1		2	140.139	even	2
2450.4.a.bs.1.2		2	20.19	odd	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:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	784.a (trivial)

\(f(q)\)	\(=\)	\(q+9.38083 q^{3} +9.38083 q^{5} +61.0000 q^{9} +O(q^{10})\)	\(q+9.38083 q^{3} +9.38083 q^{5} +61.0000 q^{9} -20.0000 q^{11} +65.6658 q^{13} +88.0000 q^{15} +56.2850 q^{17} -9.38083 q^{19} -48.0000 q^{23} -37.0000 q^{25} +318.948 q^{27} -166.000 q^{29} +206.378 q^{31} -187.617 q^{33} -78.0000 q^{37} +616.000 q^{39} +393.995 q^{41} -436.000 q^{43} +572.231 q^{45} -206.378 q^{47} +528.000 q^{51} +62.0000 q^{53} -187.617 q^{55} -88.0000 q^{57} +666.039 q^{59} +272.044 q^{61} +616.000 q^{65} -580.000 q^{67} -450.280 q^{69} +544.000 q^{71} -600.373 q^{73} -347.091 q^{75} +680.000 q^{79} +1345.00 q^{81} -196.997 q^{83} +528.000 q^{85} -1557.22 q^{87} -1500.93 q^{89} +1936.00 q^{93} -88.0000 q^{95} -656.658 q^{97} -1220.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 122 q^{9}+O(q^{10}) \) 2 * q + 122 * q^9	\( 2 q + 122 q^{9} - 40 q^{11} + 176 q^{15} - 96 q^{23} - 74 q^{25} - 332 q^{29} - 156 q^{37} + 1232 q^{39} - 872 q^{43} + 1056 q^{51} + 124 q^{53} - 176 q^{57} + 1232 q^{65} - 1160 q^{67} + 1088 q^{71} + 1360 q^{79} + 2690 q^{81} + 1056 q^{85} + 3872 q^{93} - 176 q^{95} - 2440 q^{99}+O(q^{100}) \) 2 * q + 122 * q^9 - 40 * q^11 + 176 * q^15 - 96 * q^23 - 74 * q^25 - 332 * q^29 - 156 * q^37 + 1232 * q^39 - 872 * q^43 + 1056 * q^51 + 124 * q^53 - 176 * q^57 + 1232 * q^65 - 1160 * q^67 + 1088 * q^71 + 1360 * q^79 + 2690 * q^81 + 1056 * q^85 + 3872 * q^93 - 176 * q^95 - 2440 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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