LMFDB - Embedded newform 1936.4.a.bm.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1936,4,Mod(1,1936)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1936, 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("1936.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1936 = 2^{4} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1936.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:	$114.227697771$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.978025.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 99x^{2} + 100x + 2420 $ x^4 - 2x^3 - 99x^2 + 100*x + 2420
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 11 $
Twist minimal:	no (minimal twist has level 22)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$8.19378$ of defining polynomial
Character	$\chi$	$=$	1936.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+7.81182 q^{3} +14.9181 q^{5} +21.7679 q^{7} +34.0245 q^{9} +O(q^{10})$	$q+7.81182 q^{3} +14.9181 q^{5} +21.7679 q^{7} +34.0245 q^{9} +44.0286 q^{13} +116.538 q^{15} +24.9145 q^{17} +21.9573 q^{19} +170.046 q^{21} -177.749 q^{23} +97.5508 q^{25} +54.8738 q^{27} +149.396 q^{29} -75.1436 q^{31} +324.736 q^{35} +222.336 q^{37} +343.943 q^{39} +253.121 q^{41} +130.623 q^{43} +507.582 q^{45} -499.093 q^{47} +130.839 q^{49} +194.627 q^{51} +12.9421 q^{53} +171.526 q^{57} -35.5614 q^{59} -538.343 q^{61} +740.639 q^{63} +656.824 q^{65} +519.621 q^{67} -1388.54 q^{69} -78.4486 q^{71} -1144.07 q^{73} +762.049 q^{75} +772.546 q^{79} -489.997 q^{81} +537.242 q^{83} +371.678 q^{85} +1167.06 q^{87} +667.089 q^{89} +958.407 q^{91} -587.008 q^{93} +327.561 q^{95} -179.654 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} + 25 q^{5} - 3 q^{7} + 102 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 + 25 * q^5 - 3 * q^7 + 102 * q^9	$ 4 q - 4 q^{3} + 25 q^{5} - 3 q^{7} + 102 q^{9} - 41 q^{13} - 68 q^{15} + 52 q^{17} - 16 q^{19} + 25 q^{21} - 314 q^{23} - 21 q^{25} - 286 q^{27} + 561 q^{29} - 199 q^{31} + 714 q^{35} + 357 q^{37} + 1038 q^{39} + 32 q^{41} + 721 q^{43} + 1326 q^{45} - 403 q^{47} + 823 q^{49} + 174 q^{51} - 133 q^{53} - 1031 q^{57} - 1016 q^{59} - 919 q^{61} + 1367 q^{63} + 69 q^{65} - 289 q^{67} - 1620 q^{69} + 1205 q^{71} - 1234 q^{73} + 911 q^{75} + 603 q^{79} - 1400 q^{81} - 1514 q^{83} + 717 q^{85} + 1061 q^{87} - 1101 q^{89} + 2306 q^{91} - 2298 q^{93} + 1766 q^{95} + 2116 q^{97}+O(q^{100}) $ 4 * q - 4 * q^3 + 25 * q^5 - 3 * q^7 + 102 * q^9 - 41 * q^13 - 68 * q^15 + 52 * q^17 - 16 * q^19 + 25 * q^21 - 314 * q^23 - 21 * q^25 - 286 * q^27 + 561 * q^29 - 199 * q^31 + 714 * q^35 + 357 * q^37 + 1038 * q^39 + 32 * q^41 + 721 * q^43 + 1326 * q^45 - 403 * q^47 + 823 * q^49 + 174 * q^51 - 133 * q^53 - 1031 * q^57 - 1016 * q^59 - 919 * q^61 + 1367 * q^63 + 69 * q^65 - 289 * q^67 - 1620 * q^69 + 1205 * q^71 - 1234 * q^73 + 911 * q^75 + 603 * q^79 - 1400 * q^81 - 1514 * q^83 + 717 * q^85 + 1061 * q^87 - 1101 * q^89 + 2306 * q^91 - 2298 * q^93 + 1766 * q^95 + 2116 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	7.81182	1.50338	0.751692	−	0.659514i	$-0.229238\pi$
$3$	7.81182	1.50338	0.751692	+	0.659514i	$0.229238\pi$
$4$	0	0
$5$	14.9181	1.33432	0.667159	−	0.744915i	$-0.267510\pi$
$5$	14.9181	1.33432	0.667159	+	0.744915i	$0.267510\pi$
$6$	0	0
$7$	21.7679	1.17535	0.587677	−	0.809096i	$-0.300043\pi$
$7$	21.7679	1.17535	0.587677	+	0.809096i	$0.300043\pi$
$8$	0	0
$9$	34.0245	1.26017
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	44.0286	0.939333	0.469666	−	0.882844i	$-0.344374\pi$
$13$	44.0286	0.939333	0.469666	+	0.882844i	$0.344374\pi$
$14$	0	0
$15$	116.538	2.00599
$16$	0	0
$17$	24.9145	0.355450	0.177725	−	0.984080i	$-0.443126\pi$
$17$	24.9145	0.355450	0.177725	+	0.984080i	$0.443126\pi$
$18$	0	0
$19$	21.9573	0.265123	0.132562	−	0.991175i	$-0.457680\pi$
$19$	21.9573	0.265123	0.132562	+	0.991175i	$0.457680\pi$
$20$	0	0
$21$	170.046	1.76701
$22$	0	0
$23$	−177.749	−1.61145	−0.805723	−	0.592293i	$-0.798223\pi$
$23$	−177.749	−1.61145	−0.805723	+	0.592293i	$0.798223\pi$
$24$	0	0
$25$	97.5508	0.780407
$26$	0	0
$27$	54.8738	0.391128
$28$	0	0
$29$	149.396	0.956628	0.478314	−	0.878189i	$-0.341248\pi$
$29$	149.396	0.956628	0.478314	+	0.878189i	$0.341248\pi$
$30$	0	0
$31$	−75.1436	−0.435361	−0.217681	−	0.976020i	$-0.569849\pi$
$31$	−75.1436	−0.435361	−0.217681	+	0.976020i	$0.569849\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	324.736	1.56830
$36$	0	0
$37$	222.336	0.987889	0.493944	−	0.869493i	$-0.335555\pi$
$37$	222.336	0.987889	0.493944	+	0.869493i	$0.335555\pi$
$38$	0	0
$39$	343.943	1.41218
$40$	0	0
$41$	253.121	0.964168	0.482084	−	0.876125i	$-0.339880\pi$
$41$	253.121	0.964168	0.482084	+	0.876125i	$0.339880\pi$
$42$	0	0
$43$	130.623	0.463253	0.231626	−	0.972805i	$-0.425595\pi$
$43$	130.623	0.463253	0.231626	+	0.972805i	$0.425595\pi$
$44$	0	0
$45$	507.582	1.68146
$46$	0	0
$47$	−499.093	−1.54894	−0.774471	−	0.632610i	$-0.781984\pi$
$47$	−499.093	−1.54894	−0.774471	+	0.632610i	$0.781984\pi$
$48$	0	0
$49$	130.839	0.381456
$50$	0	0
$51$	194.627	0.534378
$52$	0	0
$53$	12.9421	0.0335422	0.0167711	−	0.999859i	$-0.494661\pi$
$53$	12.9421	0.0335422	0.0167711	+	0.999859i	$0.494661\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	171.526	0.398582
$58$	0	0
$59$	−35.5614	−0.0784694	−0.0392347	−	0.999230i	$-0.512492\pi$
$59$	−35.5614	−0.0784694	−0.0392347	+	0.999230i	$0.512492\pi$
$60$	0	0
$61$	−538.343	−1.12996	−0.564982	−	0.825103i	$-0.691117\pi$
$61$	−538.343	−1.12996	−0.564982	+	0.825103i	$0.691117\pi$
$62$	0	0
$63$	740.639	1.48114
$64$	0	0
$65$	656.824	1.25337
$66$	0	0
$67$	519.621	0.947491	0.473745	−	0.880662i	$-0.342902\pi$
$67$	519.621	0.947491	0.473745	+	0.880662i	$0.342902\pi$
$68$	0	0
$69$	−1388.54	−2.42262
$70$	0	0
$71$	−78.4486	−0.131129	−0.0655644	−	0.997848i	$-0.520885\pi$
$71$	−78.4486	−0.131129	−0.0655644	+	0.997848i	$0.520885\pi$
$72$	0	0
$73$	−1144.07	−1.83429	−0.917145	−	0.398554i	$-0.869512\pi$
$73$	−1144.07	−1.83429	−0.917145	+	0.398554i	$0.869512\pi$
$74$	0	0
$75$	762.049	1.17325
$76$	0	0
$77$	0	0
$78$	0	0
$79$	772.546	1.10023	0.550115	−	0.835089i	$-0.314584\pi$
$79$	772.546	1.10023	0.550115	+	0.835089i	$0.314584\pi$
$80$	0	0
$81$	−489.997	−0.672149
$82$	0	0
$83$	537.242	0.710481	0.355241	−	0.934775i	$-0.384399\pi$
$83$	537.242	0.710481	0.355241	+	0.934775i	$0.384399\pi$
$84$	0	0
$85$	371.678	0.474284
$86$	0	0
$87$	1167.06	1.43818
$88$	0	0
$89$	667.089	0.794509	0.397255	−	0.917708i	$-0.369963\pi$
$89$	667.089	0.794509	0.397255	+	0.917708i	$0.369963\pi$
$90$	0	0
$91$	958.407	1.10405
$92$	0	0
$93$	−587.008	−0.654515
$94$	0	0
$95$	327.561	0.353759
$96$	0	0
$97$	−179.654	−0.188052	−0.0940262	−	0.995570i	$-0.529974\pi$
$97$	−179.654	−0.188052	−0.0940262	+	0.995570i	$0.529974\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	410.736	0.404651	0.202325	−	0.979318i	$-0.435150\pi$
$101$	410.736	0.404651	0.202325	+	0.979318i	$0.435150\pi$
$102$	0	0
$103$	−1367.66	−1.30834	−0.654172	−	0.756345i	$-0.726983\pi$
$103$	−1367.66	−1.30834	−0.654172	+	0.756345i	$0.726983\pi$
$104$	0	0
$105$	2536.78	2.35775
$106$	0	0
$107$	−395.241	−0.357097	−0.178548	−	0.983931i	$-0.557140\pi$
$107$	−395.241	−0.357097	−0.178548	+	0.983931i	$0.557140\pi$
$108$	0	0
$109$	−505.826	−0.444490	−0.222245	−	0.974991i	$-0.571338\pi$
$109$	−505.826	−0.444490	−0.222245	+	0.974991i	$0.571338\pi$
$110$	0	0
$111$	1736.85	1.48518
$112$	0	0
$113$	1537.62	1.28007	0.640033	−	0.768347i	$-0.278921\pi$
$113$	1537.62	1.28007	0.640033	+	0.768347i	$0.278921\pi$
$114$	0	0
$115$	−2651.68	−2.15018
$116$	0	0
$117$	1498.05	1.18371
$118$	0	0
$119$	542.335	0.417780
$120$	0	0
$121$	0	0
$122$	0	0
$123$	1977.34	1.44952
$124$	0	0
$125$	−409.491	−0.293008
$126$	0	0
$127$	−697.332	−0.487230	−0.243615	−	0.969872i	$-0.578333\pi$
$127$	−697.332	−0.487230	−0.243615	+	0.969872i	$0.578333\pi$
$128$	0	0
$129$	1020.40	0.696447
$130$	0	0
$131$	−259.910	−0.173347	−0.0866735	−	0.996237i	$-0.527624\pi$
$131$	−259.910	−0.173347	−0.0866735	+	0.996237i	$0.527624\pi$
$132$	0	0
$133$	477.962	0.311613
$134$	0	0
$135$	818.615	0.521890
$136$	0	0
$137$	−2083.56	−1.29935	−0.649674	−	0.760213i	$-0.725094\pi$
$137$	−2083.56	−1.29935	−0.649674	+	0.760213i	$0.725094\pi$
$138$	0	0
$139$	173.429	0.105828	0.0529138	−	0.998599i	$-0.483149\pi$
$139$	173.429	0.105828	0.0529138	+	0.998599i	$0.483149\pi$
$140$	0	0
$141$	−3898.83	−2.32866
$142$	0	0
$143$	0	0
$144$	0	0
$145$	2228.71	1.27645
$146$	0	0
$147$	1022.09	0.573475
$148$	0	0
$149$	449.055	0.246899	0.123450	−	0.992351i	$-0.460604\pi$
$149$	449.055	0.246899	0.123450	+	0.992351i	$0.460604\pi$
$150$	0	0
$151$	28.5358	0.0153789	0.00768944	−	0.999970i	$-0.497552\pi$
$151$	28.5358	0.0153789	0.00768944	+	0.999970i	$0.497552\pi$
$152$	0	0
$153$	847.702	0.447926
$154$	0	0
$155$	−1121.00	−0.580910
$156$	0	0
$157$	1723.86	0.876300	0.438150	−	0.898902i	$-0.355634\pi$
$157$	1723.86	0.876300	0.438150	+	0.898902i	$0.355634\pi$
$158$	0	0
$159$	101.101	0.0504268
$160$	0	0
$161$	−3869.22	−1.89402
$162$	0	0
$163$	−3318.22	−1.59450	−0.797249	−	0.603651i	$-0.793712\pi$
$163$	−3318.22	−1.59450	−0.797249	+	0.603651i	$0.793712\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3015.60	−1.39733	−0.698665	−	0.715449i	$-0.746222\pi$
$167$	−3015.60	−1.39733	−0.698665	+	0.715449i	$0.746222\pi$
$168$	0	0
$169$	−258.486	−0.117654
$170$	0	0
$171$	747.084	0.334099
$172$	0	0
$173$	2209.41	0.970973	0.485486	−	0.874244i	$-0.338643\pi$
$173$	2209.41	0.970973	0.485486	+	0.874244i	$0.338643\pi$
$174$	0	0
$175$	2123.47	0.917254
$176$	0	0
$177$	−277.799	−0.117970
$178$	0	0
$179$	2271.83	0.948629	0.474315	−	0.880355i	$-0.342696\pi$
$179$	2271.83	0.948629	0.474315	+	0.880355i	$0.342696\pi$
$180$	0	0
$181$	−624.435	−0.256430	−0.128215	−	0.991746i	$-0.540925\pi$
$181$	−624.435	−0.256430	−0.128215	+	0.991746i	$0.540925\pi$
$182$	0	0
$183$	−4205.44	−1.69877
$184$	0	0
$185$	3316.85	1.31816
$186$	0	0
$187$	0	0
$188$	0	0
$189$	1194.48	0.459714
$190$	0	0
$191$	1344.20	0.509229	0.254615	−	0.967043i	$-0.418051\pi$
$191$	1344.20	0.509229	0.254615	+	0.967043i	$0.418051\pi$
$192$	0	0
$193$	−4623.76	−1.72449	−0.862243	−	0.506496i	$-0.830941\pi$
$193$	−4623.76	−1.72449	−0.862243	+	0.506496i	$0.830941\pi$
$194$	0	0
$195$	5130.99	1.88430
$196$	0	0
$197$	−664.691	−0.240392	−0.120196	−	0.992750i	$-0.538352\pi$
$197$	−664.691	−0.240392	−0.120196	+	0.992750i	$0.538352\pi$
$198$	0	0
$199$	3042.82	1.08392	0.541959	−	0.840405i	$-0.317683\pi$
$199$	3042.82	1.08392	0.541959	+	0.840405i	$0.317683\pi$
$200$	0	0
$201$	4059.19	1.42444
$202$	0	0
$203$	3252.04	1.12438
$204$	0	0
$205$	3776.10	1.28651
$206$	0	0
$207$	−6047.82	−2.03069
$208$	0	0
$209$	0	0
$210$	0	0
$211$	2591.61	0.845563	0.422781	−	0.906232i	$-0.361054\pi$
$211$	2591.61	0.845563	0.422781	+	0.906232i	$0.361054\pi$
$212$	0	0
$213$	−612.826	−0.197137
$214$	0	0
$215$	1948.66	0.618127
$216$	0	0
$217$	−1635.72	−0.511703
$218$	0	0
$219$	−8937.26	−2.75764
$220$	0	0
$221$	1096.95	0.333886
$222$	0	0
$223$	−2637.20	−0.791929	−0.395965	−	0.918266i	$-0.629590\pi$
$223$	−2637.20	−0.791929	−0.395965	+	0.918266i	$0.629590\pi$
$224$	0	0
$225$	3319.11	0.983441
$226$	0	0
$227$	250.670	0.0732932	0.0366466	−	0.999328i	$-0.488332\pi$
$227$	250.670	0.0732932	0.0366466	+	0.999328i	$0.488332\pi$
$228$	0	0
$229$	1799.31	0.519222	0.259611	−	0.965713i	$-0.416406\pi$
$229$	1799.31	0.519222	0.259611	+	0.965713i	$0.416406\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−3180.48	−0.894248	−0.447124	−	0.894472i	$-0.647552\pi$
$233$	−3180.48	−0.894248	−0.447124	+	0.894472i	$0.647552\pi$
$234$	0	0
$235$	−7445.54	−2.06678
$236$	0	0
$237$	6034.98	1.65407
$238$	0	0
$239$	5013.80	1.35697	0.678485	−	0.734614i	$-0.262637\pi$
$239$	5013.80	1.35697	0.678485	+	0.734614i	$0.262637\pi$
$240$	0	0
$241$	−6074.13	−1.62352	−0.811761	−	0.583990i	$-0.801491\pi$
$241$	−6074.13	−1.62352	−0.811761	+	0.583990i	$0.801491\pi$
$242$	0	0
$243$	−5309.35	−1.40163
$244$	0	0
$245$	1951.88	0.508984
$246$	0	0
$247$	966.746	0.249039
$248$	0	0
$249$	4196.83	1.06813
$250$	0	0
$251$	5569.69	1.40062	0.700310	−	0.713838i	$-0.253045\pi$
$251$	5569.69	1.40062	0.700310	+	0.713838i	$0.253045\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	2903.48	0.713031
$256$	0	0
$257$	−5907.05	−1.43374	−0.716871	−	0.697206i	$-0.754426\pi$
$257$	−5907.05	−1.43374	−0.716871	+	0.697206i	$0.754426\pi$
$258$	0	0
$259$	4839.79	1.16112
$260$	0	0
$261$	5083.13	1.20551
$262$	0	0
$263$	−6853.74	−1.60692	−0.803459	−	0.595360i	$-0.797009\pi$
$263$	−6853.74	−1.60692	−0.803459	+	0.595360i	$0.797009\pi$
$264$	0	0
$265$	193.072	0.0447559
$266$	0	0
$267$	5211.18	1.19445
$268$	0	0
$269$	1101.36	0.249633	0.124817	−	0.992180i	$-0.460166\pi$
$269$	1101.36	0.249633	0.124817	+	0.992180i	$0.460166\pi$
$270$	0	0
$271$	−2571.69	−0.576455	−0.288227	−	0.957562i	$-0.593066\pi$
$271$	−2571.69	−0.576455	−0.288227	+	0.957562i	$0.593066\pi$
$272$	0	0
$273$	7486.90	1.65981
$274$	0	0
$275$	0	0
$276$	0	0
$277$	7340.10	1.59214	0.796072	−	0.605202i	$-0.206908\pi$
$277$	7340.10	1.59214	0.796072	+	0.605202i	$0.206908\pi$
$278$	0	0
$279$	−2556.72	−0.548627
$280$	0	0
$281$	7492.94	1.59072	0.795359	−	0.606139i	$-0.207282\pi$
$281$	7492.94	1.59072	0.795359	+	0.606139i	$0.207282\pi$
$282$	0	0
$283$	−6494.10	−1.36408	−0.682039	−	0.731316i	$-0.738907\pi$
$283$	−6494.10	−1.36408	−0.682039	+	0.731316i	$0.738907\pi$
$284$	0	0
$285$	2558.85	0.531835
$286$	0	0
$287$	5509.91	1.13324
$288$	0	0
$289$	−4292.27	−0.873655
$290$	0	0
$291$	−1403.42	−0.282715
$292$	0	0
$293$	2684.87	0.535331	0.267666	−	0.963512i	$-0.413748\pi$
$293$	2684.87	0.535331	0.267666	+	0.963512i	$0.413748\pi$
$294$	0	0
$295$	−530.509	−0.104703
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−7826.03	−1.51368
$300$	0	0
$301$	2843.39	0.544486
$302$	0	0
$303$	3208.59	0.608345
$304$	0	0
$305$	−8031.08	−1.50773
$306$	0	0
$307$	8331.66	1.54890	0.774451	−	0.632633i	$-0.218026\pi$
$307$	8331.66	1.54890	0.774451	+	0.632633i	$0.218026\pi$
$308$	0	0
$309$	−10683.9	−1.96695
$310$	0	0
$311$	5020.35	0.915363	0.457682	−	0.889116i	$-0.348680\pi$
$311$	5020.35	0.915363	0.457682	+	0.889116i	$0.348680\pi$
$312$	0	0
$313$	3022.08	0.545744	0.272872	−	0.962050i	$-0.412026\pi$
$313$	3022.08	0.545744	0.272872	+	0.962050i	$0.412026\pi$
$314$	0	0
$315$	11049.0	1.97631
$316$	0	0
$317$	−10540.1	−1.86749	−0.933744	−	0.357942i	$-0.883478\pi$
$317$	−10540.1	−1.86749	−0.933744	+	0.357942i	$0.883478\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−3087.55	−0.536854
$322$	0	0
$323$	547.054	0.0942381
$324$	0	0
$325$	4295.02	0.733061
$326$	0	0
$327$	−3951.42	−0.668239
$328$	0	0
$329$	−10864.2	−1.82055
$330$	0	0
$331$	−309.871	−0.0514563	−0.0257281	−	0.999669i	$-0.508190\pi$
$331$	−309.871	−0.0514563	−0.0257281	+	0.999669i	$0.508190\pi$
$332$	0	0
$333$	7564.88	1.24490
$334$	0	0
$335$	7751.78	1.26425
$336$	0	0
$337$	−11164.7	−1.80468	−0.902340	−	0.431024i	$-0.858152\pi$
$337$	−11164.7	−1.80468	−0.902340	+	0.431024i	$0.858152\pi$
$338$	0	0
$339$	12011.6	1.92443
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−4618.28	−0.727008
$344$	0	0
$345$	−20714.5	−3.23255
$346$	0	0
$347$	2515.48	0.389159	0.194580	−	0.980887i	$-0.437666\pi$
$347$	2515.48	0.389159	0.194580	+	0.980887i	$0.437666\pi$
$348$	0	0
$349$	11376.6	1.74491	0.872457	−	0.488692i	$-0.162526\pi$
$349$	11376.6	1.74491	0.872457	+	0.488692i	$0.162526\pi$
$350$	0	0
$351$	2416.01	0.367400
$352$	0	0
$353$	6210.08	0.936343	0.468172	−	0.883638i	$-0.344913\pi$
$353$	6210.08	0.936343	0.468172	+	0.883638i	$0.344913\pi$
$354$	0	0
$355$	−1170.31	−0.174968
$356$	0	0
$357$	4236.62	0.628084
$358$	0	0
$359$	−2487.51	−0.365698	−0.182849	−	0.983141i	$-0.558532\pi$
$359$	−2487.51	−0.365698	−0.182849	+	0.983141i	$0.558532\pi$
$360$	0	0
$361$	−6376.88	−0.929710
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−17067.4	−2.44753
$366$	0	0
$367$	−6719.53	−0.955739	−0.477870	−	0.878431i	$-0.658591\pi$
$367$	−6719.53	−0.955739	−0.477870	+	0.878431i	$0.658591\pi$
$368$	0	0
$369$	8612.31	1.21501
$370$	0	0
$371$	281.722	0.0394239
$372$	0	0
$373$	8614.37	1.19580	0.597902	−	0.801569i	$-0.296001\pi$
$373$	8614.37	1.19580	0.597902	+	0.801569i	$0.296001\pi$
$374$	0	0
$375$	−3198.87	−0.440503
$376$	0	0
$377$	6577.70	0.898592
$378$	0	0
$379$	8009.10	1.08549	0.542744	−	0.839898i	$-0.317386\pi$
$379$	8009.10	1.08549	0.542744	+	0.839898i	$0.317386\pi$
$380$	0	0
$381$	−5447.43	−0.732494
$382$	0	0
$383$	−8011.95	−1.06891	−0.534454	−	0.845198i	$-0.679483\pi$
$383$	−8011.95	−1.06891	−0.534454	+	0.845198i	$0.679483\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	4444.39	0.583775
$388$	0	0
$389$	8029.60	1.04657	0.523287	−	0.852157i	$-0.324706\pi$
$389$	8029.60	1.04657	0.523287	+	0.852157i	$0.324706\pi$
$390$	0	0
$391$	−4428.53	−0.572789
$392$	0	0
$393$	−2030.37	−0.260607
$394$	0	0
$395$	11524.9	1.46806
$396$	0	0
$397$	4435.00	0.560670	0.280335	−	0.959902i	$-0.409554\pi$
$397$	4435.00	0.560670	0.280335	+	0.959902i	$0.409554\pi$
$398$	0	0
$399$	3733.75	0.468475
$400$	0	0
$401$	−3929.21	−0.489315	−0.244657	−	0.969610i	$-0.578675\pi$
$401$	−3929.21	−0.489315	−0.244657	+	0.969610i	$0.578675\pi$
$402$	0	0
$403$	−3308.46	−0.408949
$404$	0	0
$405$	−7309.84	−0.896861
$406$	0	0
$407$	0	0
$408$	0	0
$409$	14564.2	1.76077	0.880384	−	0.474261i	$-0.157285\pi$
$409$	14564.2	1.76077	0.880384	+	0.474261i	$0.157285\pi$
$410$	0	0
$411$	−16276.4	−1.95342
$412$	0	0
$413$	−774.094	−0.0922293
$414$	0	0
$415$	8014.65	0.948008
$416$	0	0
$417$	1354.79	0.159099
$418$	0	0
$419$	−4028.77	−0.469734	−0.234867	−	0.972028i	$-0.575465\pi$
$419$	−4028.77	−0.469734	−0.234867	+	0.972028i	$0.575465\pi$
$420$	0	0
$421$	−8527.60	−0.987197	−0.493599	−	0.869690i	$-0.664319\pi$
$421$	−8527.60	−0.987197	−0.493599	+	0.869690i	$0.664319\pi$
$422$	0	0
$423$	−16981.4	−1.95192
$424$	0	0
$425$	2430.43	0.277396
$426$	0	0
$427$	−11718.6	−1.32811
$428$	0	0
$429$	0	0
$430$	0	0
$431$	13415.2	1.49928	0.749639	−	0.661847i	$-0.230227\pi$
$431$	13415.2	1.49928	0.749639	+	0.661847i	$0.230227\pi$
$432$	0	0
$433$	4132.31	0.458628	0.229314	−	0.973352i	$-0.426352\pi$
$433$	4132.31	0.458628	0.229314	+	0.973352i	$0.426352\pi$
$434$	0	0
$435$	17410.3	1.91899
$436$	0	0
$437$	−3902.88	−0.427231
$438$	0	0
$439$	3358.46	0.365126	0.182563	−	0.983194i	$-0.441561\pi$
$439$	3358.46	0.365126	0.182563	+	0.983194i	$0.441561\pi$
$440$	0	0
$441$	4451.74	0.480698
$442$	0	0
$443$	441.749	0.0473773	0.0236886	−	0.999719i	$-0.492459\pi$
$443$	441.749	0.0473773	0.0236886	+	0.999719i	$0.492459\pi$
$444$	0	0
$445$	9951.73	1.06013
$446$	0	0
$447$	3507.93	0.371185
$448$	0	0
$449$	−408.478	−0.0429338	−0.0214669	−	0.999770i	$-0.506834\pi$
$449$	−408.478	−0.0429338	−0.0214669	+	0.999770i	$0.506834\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	222.916	0.0231204
$454$	0	0
$455$	14297.6	1.47315
$456$	0	0
$457$	−1522.85	−0.155877	−0.0779384	−	0.996958i	$-0.524834\pi$
$457$	−1522.85	−0.155877	−0.0779384	+	0.996958i	$0.524834\pi$
$458$	0	0
$459$	1367.15	0.139027
$460$	0	0
$461$	13861.4	1.40041	0.700203	−	0.713944i	$-0.253093\pi$
$461$	13861.4	1.40041	0.700203	+	0.713944i	$0.253093\pi$
$462$	0	0
$463$	6502.26	0.652669	0.326334	−	0.945254i	$-0.394186\pi$
$463$	6502.26	0.652669	0.326334	+	0.945254i	$0.394186\pi$
$464$	0	0
$465$	−8757.07	−0.873332
$466$	0	0
$467$	−423.973	−0.0420110	−0.0210055	−	0.999779i	$-0.506687\pi$
$467$	−423.973	−0.0420110	−0.0210055	+	0.999779i	$0.506687\pi$
$468$	0	0
$469$	11311.0	1.11364
$470$	0	0
$471$	13466.5	1.31742
$472$	0	0
$473$	0	0
$474$	0	0
$475$	2141.95	0.206904
$476$	0	0
$477$	440.348	0.0422687
$478$	0	0
$479$	4557.34	0.434718	0.217359	−	0.976092i	$-0.430256\pi$
$479$	4557.34	0.434718	0.217359	+	0.976092i	$0.430256\pi$
$480$	0	0
$481$	9789.15	0.927956
$482$	0	0
$483$	−30225.6	−2.84744
$484$	0	0
$485$	−2680.10	−0.250922
$486$	0	0
$487$	6708.92	0.624251	0.312126	−	0.950041i	$-0.398959\pi$
$487$	6708.92	0.624251	0.312126	+	0.950041i	$0.398959\pi$
$488$	0	0
$489$	−25921.3	−2.39714
$490$	0	0
$491$	14527.7	1.33528	0.667641	−	0.744483i	$-0.267304\pi$
$491$	14527.7	1.33528	0.667641	+	0.744483i	$0.267304\pi$
$492$	0	0
$493$	3722.13	0.340034
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1707.66	−0.154123
$498$	0	0
$499$	9699.53	0.870162	0.435081	−	0.900391i	$-0.356720\pi$
$499$	9699.53	0.870162	0.435081	+	0.900391i	$0.356720\pi$
$500$	0	0
$501$	−23557.3	−2.10072
$502$	0	0
$503$	15707.1	1.39234	0.696170	−	0.717877i	$-0.254886\pi$
$503$	15707.1	1.39234	0.696170	+	0.717877i	$0.254886\pi$
$504$	0	0
$505$	6127.41	0.539933
$506$	0	0
$507$	−2019.25	−0.176879
$508$	0	0
$509$	−1474.08	−0.128364	−0.0641822	−	0.997938i	$-0.520444\pi$
$509$	−1474.08	−0.128364	−0.0641822	+	0.997938i	$0.520444\pi$
$510$	0	0
$511$	−24903.9	−2.15594
$512$	0	0
$513$	1204.88	0.103697
$514$	0	0
$515$	−20402.9	−1.74575
$516$	0	0
$517$	0	0
$518$	0	0
$519$	17259.5	1.45975
$520$	0	0
$521$	−9272.84	−0.779752	−0.389876	−	0.920867i	$-0.627482\pi$
$521$	−9272.84	−0.779752	−0.389876	+	0.920867i	$0.627482\pi$
$522$	0	0
$523$	−4242.50	−0.354707	−0.177353	−	0.984147i	$-0.556754\pi$
$523$	−4242.50	−0.354707	−0.177353	+	0.984147i	$0.556754\pi$
$524$	0	0
$525$	16588.2	1.37899
$526$	0	0
$527$	−1872.17	−0.154749
$528$	0	0
$529$	19427.7	1.59676
$530$	0	0
$531$	−1209.96	−0.0988844
$532$	0	0
$533$	11144.6	0.905675
$534$	0	0
$535$	−5896.25	−0.476481
$536$	0	0
$537$	17747.1	1.42615
$538$	0	0
$539$	0	0
$540$	0	0
$541$	6520.25	0.518166	0.259083	−	0.965855i	$-0.416580\pi$
$541$	6520.25	0.518166	0.259083	+	0.965855i	$0.416580\pi$
$542$	0	0
$543$	−4877.97	−0.385513
$544$	0	0
$545$	−7545.98	−0.593091
$546$	0	0
$547$	6370.78	0.497980	0.248990	−	0.968506i	$-0.419901\pi$
$547$	6370.78	0.497980	0.248990	+	0.968506i	$0.419901\pi$
$548$	0	0
$549$	−18316.8	−1.42394
$550$	0	0
$551$	3280.33	0.253624
$552$	0	0
$553$	16816.7	1.29316
$554$	0	0
$555$	25910.6	1.98170
$556$	0	0
$557$	−6103.99	−0.464335	−0.232167	−	0.972676i	$-0.574582\pi$
$557$	−6103.99	−0.464335	−0.232167	+	0.972676i	$0.574582\pi$
$558$	0	0
$559$	5751.15	0.435148
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−13592.6	−1.01751	−0.508756	−	0.860911i	$-0.669894\pi$
$563$	−13592.6	−1.01751	−0.508756	+	0.860911i	$0.669894\pi$
$564$	0	0
$565$	22938.5	1.70802
$566$	0	0
$567$	−10666.2	−0.790013
$568$	0	0
$569$	−17197.6	−1.26706	−0.633532	−	0.773717i	$-0.718395\pi$
$569$	−17197.6	−1.26706	−0.633532	+	0.773717i	$0.718395\pi$
$570$	0	0
$571$	2475.65	0.181441	0.0907203	−	0.995876i	$-0.471083\pi$
$571$	2475.65	0.181441	0.0907203	+	0.995876i	$0.471083\pi$
$572$	0	0
$573$	10500.6	0.765567
$574$	0	0
$575$	−17339.6	−1.25758
$576$	0	0
$577$	−20339.7	−1.46751	−0.733755	−	0.679414i	$-0.762234\pi$
$577$	−20339.7	−1.46751	−0.733755	+	0.679414i	$0.762234\pi$
$578$	0	0
$579$	−36120.0	−2.59256
$580$	0	0
$581$	11694.6	0.835067
$582$	0	0
$583$	0	0
$584$	0	0
$585$	22348.1	1.57945
$586$	0	0
$587$	14818.4	1.04195	0.520973	−	0.853573i	$-0.325569\pi$
$587$	14818.4	1.04195	0.520973	+	0.853573i	$0.325569\pi$
$588$	0	0
$589$	−1649.95	−0.115424
$590$	0	0
$591$	−5192.44	−0.361402
$592$	0	0
$593$	−11123.2	−0.770281	−0.385140	−	0.922858i	$-0.625847\pi$
$593$	−11123.2	−0.770281	−0.385140	+	0.922858i	$0.625847\pi$
$594$	0	0
$595$	8090.63	0.557451
$596$	0	0
$597$	23770.0	1.62955
$598$	0	0
$599$	−8285.18	−0.565147	−0.282574	−	0.959246i	$-0.591188\pi$
$599$	−8285.18	−0.565147	−0.282574	+	0.959246i	$0.591188\pi$
$600$	0	0
$601$	−27318.3	−1.85414	−0.927069	−	0.374890i	$-0.877680\pi$
$601$	−27318.3	−1.85414	−0.927069	+	0.374890i	$0.877680\pi$
$602$	0	0
$603$	17679.8	1.19399
$604$	0	0
$605$	0	0
$606$	0	0
$607$	16327.7	1.09180	0.545898	−	0.837851i	$-0.316188\pi$
$607$	16327.7	1.09180	0.545898	+	0.837851i	$0.316188\pi$
$608$	0	0
$609$	25404.3	1.69037
$610$	0	0
$611$	−21974.4	−1.45497
$612$	0	0
$613$	21025.3	1.38533	0.692663	−	0.721262i	$-0.256437\pi$
$613$	21025.3	1.38533	0.692663	+	0.721262i	$0.256437\pi$
$614$	0	0
$615$	29498.2	1.93412
$616$	0	0
$617$	871.824	0.0568854	0.0284427	−	0.999595i	$-0.490945\pi$
$617$	871.824	0.0568854	0.0284427	+	0.999595i	$0.490945\pi$
$618$	0	0
$619$	−9397.52	−0.610207	−0.305104	−	0.952319i	$-0.598691\pi$
$619$	−9397.52	−0.610207	−0.305104	+	0.952319i	$0.598691\pi$
$620$	0	0
$621$	−9753.76	−0.630282
$622$	0	0
$623$	14521.1	0.933829
$624$	0	0
$625$	−18302.7	−1.17137
$626$	0	0
$627$	0	0
$628$	0	0
$629$	5539.40	0.351145
$630$	0	0
$631$	14535.9	0.917062	0.458531	−	0.888678i	$-0.348376\pi$
$631$	14535.9	0.917062	0.458531	+	0.888678i	$0.348376\pi$
$632$	0	0
$633$	20245.2	1.27121
$634$	0	0
$635$	−10402.9	−0.650120
$636$	0	0
$637$	5760.67	0.358314
$638$	0	0
$639$	−2669.17	−0.165244
$640$	0	0
$641$	−11419.7	−0.703666	−0.351833	−	0.936063i	$-0.614441\pi$
$641$	−11419.7	−0.703666	−0.351833	+	0.936063i	$0.614441\pi$
$642$	0	0
$643$	−23969.9	−1.47011	−0.735055	−	0.678007i	$-0.762844\pi$
$643$	−23969.9	−1.47011	−0.735055	+	0.678007i	$0.762844\pi$
$644$	0	0
$645$	15222.5	0.929282
$646$	0	0
$647$	−8314.25	−0.505204	−0.252602	−	0.967570i	$-0.581286\pi$
$647$	−8314.25	−0.505204	−0.252602	+	0.967570i	$0.581286\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−12777.9	−0.769287
$652$	0	0
$653$	−30079.3	−1.80260	−0.901298	−	0.433200i	$-0.857384\pi$
$653$	−30079.3	−1.80260	−0.901298	+	0.433200i	$0.857384\pi$
$654$	0	0
$655$	−3877.38	−0.231300
$656$	0	0
$657$	−38926.3	−2.31151
$658$	0	0
$659$	−10041.6	−0.593572	−0.296786	−	0.954944i	$-0.595915\pi$
$659$	−10041.6	−0.593572	−0.296786	+	0.954944i	$0.595915\pi$
$660$	0	0
$661$	1402.50	0.0825281	0.0412640	−	0.999148i	$-0.486862\pi$
$661$	1402.50	0.0825281	0.0412640	+	0.999148i	$0.486862\pi$
$662$	0	0
$663$	8569.17	0.501959
$664$	0	0
$665$	7130.31	0.415792
$666$	0	0
$667$	−26555.1	−1.54155
$668$	0	0
$669$	−20601.4	−1.19057
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−20385.4	−1.16761	−0.583803	−	0.811895i	$-0.698436\pi$
$673$	−20385.4	−1.16761	−0.583803	+	0.811895i	$0.698436\pi$
$674$	0	0
$675$	5352.98	0.305239
$676$	0	0
$677$	7385.00	0.419245	0.209622	−	0.977782i	$-0.432777\pi$
$677$	7385.00	0.419245	0.209622	+	0.977782i	$0.432777\pi$
$678$	0	0
$679$	−3910.68	−0.221028
$680$	0	0
$681$	1958.19	0.110188
$682$	0	0
$683$	25844.0	1.44787	0.723935	−	0.689868i	$-0.242332\pi$
$683$	25844.0	1.44787	0.723935	+	0.689868i	$0.242332\pi$
$684$	0	0
$685$	−31082.8	−1.73374
$686$	0	0
$687$	14055.9	0.780591
$688$	0	0
$689$	569.822	0.0315073
$690$	0	0
$691$	−8438.80	−0.464583	−0.232292	−	0.972646i	$-0.574622\pi$
$691$	−8438.80	−0.464583	−0.232292	+	0.972646i	$0.574622\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	2587.23	0.141208
$696$	0	0
$697$	6306.39	0.342714
$698$	0	0
$699$	−24845.3	−1.34440
$700$	0	0
$701$	12983.4	0.699536	0.349768	−	0.936836i	$-0.386260\pi$
$701$	12983.4	0.699536	0.349768	+	0.936836i	$0.386260\pi$
$702$	0	0
$703$	4881.90	0.261912
$704$	0	0
$705$	−58163.2	−3.10717
$706$	0	0
$707$	8940.83	0.475608
$708$	0	0
$709$	−17918.6	−0.949150	−0.474575	−	0.880215i	$-0.657398\pi$
$709$	−17918.6	−0.949150	−0.474575	+	0.880215i	$0.657398\pi$
$710$	0	0
$711$	26285.4	1.38647
$712$	0	0
$713$	13356.7	0.701560
$714$	0	0
$715$	0	0
$716$	0	0
$717$	39166.9	2.04005
$718$	0	0
$719$	−3517.35	−0.182441	−0.0912204	−	0.995831i	$-0.529077\pi$
$719$	−3517.35	−0.182441	−0.0912204	+	0.995831i	$0.529077\pi$
$720$	0	0
$721$	−29771.0	−1.53777
$722$	0	0
$723$	−47449.9	−2.44078
$724$	0	0
$725$	14573.7	0.746559
$726$	0	0
$727$	29438.9	1.50183	0.750913	−	0.660401i	$-0.229614\pi$
$727$	29438.9	1.50183	0.750913	+	0.660401i	$0.229614\pi$
$728$	0	0
$729$	−28245.8	−1.43503
$730$	0	0
$731$	3254.41	0.164663
$732$	0	0
$733$	−3435.90	−0.173135	−0.0865673	−	0.996246i	$-0.527590\pi$
$733$	−3435.90	−0.173135	−0.0865673	+	0.996246i	$0.527590\pi$
$734$	0	0
$735$	15247.7	0.765199
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−33601.8	−1.67261	−0.836307	−	0.548262i	$-0.815290\pi$
$739$	−33601.8	−1.67261	−0.836307	+	0.548262i	$0.815290\pi$
$740$	0	0
$741$	7552.04	0.374401
$742$	0	0
$743$	−1193.70	−0.0589404	−0.0294702	−	0.999566i	$-0.509382\pi$
$743$	−1193.70	−0.0589404	−0.0294702	+	0.999566i	$0.509382\pi$
$744$	0	0
$745$	6699.06	0.329443
$746$	0	0
$747$	18279.4	0.895324
$748$	0	0
$749$	−8603.54	−0.419715
$750$	0	0
$751$	−24509.4	−1.19089	−0.595446	−	0.803396i	$-0.703024\pi$
$751$	−24509.4	−1.19089	−0.595446	+	0.803396i	$0.703024\pi$
$752$	0	0
$753$	43509.4	2.10567
$754$	0	0
$755$	425.701	0.0205203
$756$	0	0
$757$	10803.8	0.518721	0.259360	−	0.965781i	$-0.416488\pi$
$757$	10803.8	0.518721	0.259360	+	0.965781i	$0.416488\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−8175.91	−0.389457	−0.194728	−	0.980857i	$-0.562383\pi$
$761$	−8175.91	−0.389457	−0.194728	+	0.980857i	$0.562383\pi$
$762$	0	0
$763$	−11010.8	−0.522432
$764$	0	0
$765$	12646.1	0.597676
$766$	0	0
$767$	−1565.72	−0.0737089
$768$	0	0
$769$	28895.9	1.35502	0.677511	−	0.735513i	$-0.263059\pi$
$769$	28895.9	1.35502	0.677511	+	0.735513i	$0.263059\pi$
$770$	0	0
$771$	−46144.8	−2.15547
$772$	0	0
$773$	−17996.8	−0.837387	−0.418693	−	0.908128i	$-0.637512\pi$
$773$	−17996.8	−0.837387	−0.418693	+	0.908128i	$0.637512\pi$
$774$	0	0
$775$	−7330.32	−0.339759
$776$	0	0
$777$	37807.5	1.74561
$778$	0	0
$779$	5557.85	0.255623
$780$	0	0
$781$	0	0
$782$	0	0
$783$	8197.94	0.374164
$784$	0	0
$785$	25716.8	1.16926
$786$	0	0
$787$	−15876.7	−0.719114	−0.359557	−	0.933123i	$-0.617072\pi$
$787$	−15876.7	−0.719114	−0.359557	+	0.933123i	$0.617072\pi$
$788$	0	0
$789$	−53540.1	−2.41582
$790$	0	0
$791$	33470.8	1.50453
$792$	0	0
$793$	−23702.5	−1.06141
$794$	0	0
$795$	1508.24	0.0672854
$796$	0	0
$797$	16497.7	0.733223	0.366611	−	0.930374i	$-0.380518\pi$
$797$	16497.7	0.733223	0.366611	+	0.930374i	$0.380518\pi$
$798$	0	0
$799$	−12434.7	−0.550572
$800$	0	0
$801$	22697.3	1.00121
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−57721.5	−2.52722
$806$	0	0
$807$	8603.65	0.375295
$808$	0	0
$809$	9102.95	0.395603	0.197801	−	0.980242i	$-0.436620\pi$
$809$	9102.95	0.395603	0.197801	+	0.980242i	$0.436620\pi$
$810$	0	0
$811$	36576.2	1.58368	0.791841	−	0.610728i	$-0.209123\pi$
$811$	36576.2	1.58368	0.791841	+	0.610728i	$0.209123\pi$
$812$	0	0
$813$	−20089.6	−0.866633
$814$	0	0
$815$	−49501.7	−2.12757
$816$	0	0
$817$	2868.13	0.122819
$818$	0	0
$819$	32609.3	1.39128
$820$	0	0
$821$	3644.21	0.154913	0.0774567	−	0.996996i	$-0.475320\pi$
$821$	3644.21	0.154913	0.0774567	+	0.996996i	$0.475320\pi$
$822$	0	0
$823$	16895.9	0.715618	0.357809	−	0.933795i	$-0.383524\pi$
$823$	16895.9	0.715618	0.357809	+	0.933795i	$0.383524\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−38778.1	−1.63053	−0.815263	−	0.579090i	$-0.803408\pi$
$827$	−38778.1	−1.63053	−0.815263	+	0.579090i	$0.803408\pi$
$828$	0	0
$829$	−12394.3	−0.519267	−0.259634	−	0.965707i	$-0.583602\pi$
$829$	−12394.3	−0.519267	−0.259634	+	0.965707i	$0.583602\pi$
$830$	0	0
$831$	57339.5	2.39361
$832$	0	0
$833$	3259.80	0.135589
$834$	0	0
$835$	−44987.1	−1.86448
$836$	0	0
$837$	−4123.41	−0.170282
$838$	0	0
$839$	−24563.8	−1.01077	−0.505386	−	0.862893i	$-0.668650\pi$
$839$	−24563.8	−1.01077	−0.505386	+	0.862893i	$0.668650\pi$
$840$	0	0
$841$	−2069.74	−0.0848635
$842$	0	0
$843$	58533.5	2.39146
$844$	0	0
$845$	−3856.13	−0.156988
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−50730.7	−2.05073
$850$	0	0
$851$	−39520.1	−1.59193
$852$	0	0
$853$	12207.6	0.490013	0.245007	−	0.969521i	$-0.421210\pi$
$853$	12207.6	0.490013	0.245007	+	0.969521i	$0.421210\pi$
$854$	0	0
$855$	11145.1	0.445794
$856$	0	0
$857$	−7281.72	−0.290244	−0.145122	−	0.989414i	$-0.546357\pi$
$857$	−7281.72	−0.290244	−0.145122	+	0.989414i	$0.546357\pi$
$858$	0	0
$859$	−5927.39	−0.235437	−0.117718	−	0.993047i	$-0.537558\pi$
$859$	−5927.39	−0.235437	−0.117718	+	0.993047i	$0.537558\pi$
$860$	0	0
$861$	43042.4	1.70369
$862$	0	0
$863$	38561.9	1.52104	0.760522	−	0.649312i	$-0.224943\pi$
$863$	38561.9	1.52104	0.760522	+	0.649312i	$0.224943\pi$
$864$	0	0
$865$	32960.3	1.29559
$866$	0	0
$867$	−33530.4	−1.31344
$868$	0	0
$869$	0	0
$870$	0	0
$871$	22878.2	0.890009
$872$	0	0
$873$	−6112.63	−0.236977
$874$	0	0
$875$	−8913.73	−0.344388
$876$	0	0
$877$	−28512.7	−1.09784	−0.548919	−	0.835875i	$-0.684961\pi$
$877$	−28512.7	−1.09784	−0.548919	+	0.835875i	$0.684961\pi$
$878$	0	0
$879$	20973.7	0.804809
$880$	0	0
$881$	40747.6	1.55826	0.779128	−	0.626865i	$-0.215662\pi$
$881$	40747.6	1.55826	0.779128	+	0.626865i	$0.215662\pi$
$882$	0	0
$883$	3595.59	0.137034	0.0685171	−	0.997650i	$-0.478173\pi$
$883$	3595.59	0.137034	0.0685171	+	0.997650i	$0.478173\pi$
$884$	0	0
$885$	−4144.24	−0.157409
$886$	0	0
$887$	−7715.30	−0.292057	−0.146028	−	0.989280i	$-0.546649\pi$
$887$	−7715.30	−0.292057	−0.146028	+	0.989280i	$0.546649\pi$
$888$	0	0
$889$	−15179.4	−0.572667
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−10958.7	−0.410660
$894$	0	0
$895$	33891.5	1.26577
$896$	0	0
$897$	−61135.5	−2.27565
$898$	0	0
$899$	−11226.2	−0.416478
$900$	0	0
$901$	322.446	0.0119226
$902$	0	0
$903$	22212.0	0.818571
$904$	0	0
$905$	−9315.41	−0.342160
$906$	0	0
$907$	33713.7	1.23423	0.617114	−	0.786874i	$-0.288302\pi$
$907$	33713.7	1.23423	0.617114	+	0.786874i	$0.288302\pi$
$908$	0	0
$909$	13975.1	0.509927
$910$	0	0
$911$	−15030.7	−0.546642	−0.273321	−	0.961923i	$-0.588122\pi$
$911$	−15030.7	−0.546642	−0.273321	+	0.961923i	$0.588122\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−62737.3	−2.26670
$916$	0	0
$917$	−5657.69	−0.203744
$918$	0	0
$919$	−17090.7	−0.613460	−0.306730	−	0.951797i	$-0.599235\pi$
$919$	−17090.7	−0.613460	−0.306730	+	0.951797i	$0.599235\pi$
$920$	0	0
$921$	65085.4	2.32860
$922$	0	0
$923$	−3453.98	−0.123173
$924$	0	0
$925$	21689.1	0.770955
$926$	0	0
$927$	−46533.9	−1.64873
$928$	0	0
$929$	13160.3	0.464773	0.232386	−	0.972624i	$-0.425347\pi$
$929$	13160.3	0.464773	0.232386	+	0.972624i	$0.425347\pi$
$930$	0	0
$931$	2872.87	0.101133
$932$	0	0
$933$	39218.1	1.37614
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−4056.29	−0.141423	−0.0707114	−	0.997497i	$-0.522527\pi$
$937$	−4056.29	−0.141423	−0.0707114	+	0.997497i	$0.522527\pi$
$938$	0	0
$939$	23607.9	0.820463
$940$	0	0
$941$	−33397.6	−1.15699	−0.578497	−	0.815684i	$-0.696361\pi$
$941$	−33397.6	−1.15699	−0.578497	+	0.815684i	$0.696361\pi$
$942$	0	0
$943$	−44992.1	−1.55370
$944$	0	0
$945$	17819.5	0.613405
$946$	0	0
$947$	−25660.8	−0.880533	−0.440267	−	0.897867i	$-0.645116\pi$
$947$	−25660.8	−0.880533	−0.440267	+	0.897867i	$0.645116\pi$
$948$	0	0
$949$	−50371.7	−1.72301
$950$	0	0
$951$	−82337.7	−2.80755
$952$	0	0
$953$	−48852.8	−1.66054	−0.830271	−	0.557359i	$-0.811815\pi$
$953$	−48852.8	−1.66054	−0.830271	+	0.557359i	$0.811815\pi$
$954$	0	0
$955$	20052.9	0.679474
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−45354.6	−1.52719
$960$	0	0
$961$	−24144.4	−0.810461
$962$	0	0
$963$	−13447.8	−0.450001
$964$	0	0
$965$	−68977.9	−2.30101
$966$	0	0
$967$	−48861.8	−1.62491	−0.812456	−	0.583023i	$-0.801870\pi$
$967$	−48861.8	−1.62491	−0.812456	+	0.583023i	$0.801870\pi$
$968$	0	0
$969$	4273.48	0.141676
$970$	0	0
$971$	−34887.5	−1.15303	−0.576516	−	0.817086i	$-0.695588\pi$
$971$	−34887.5	−1.15303	−0.576516	+	0.817086i	$0.695588\pi$
$972$	0	0
$973$	3775.17	0.124385
$974$	0	0
$975$	33551.9	1.10207
$976$	0	0
$977$	−19012.4	−0.622580	−0.311290	−	0.950315i	$-0.600761\pi$
$977$	−19012.4	−0.622580	−0.311290	+	0.950315i	$0.600761\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−17210.5	−0.560130
$982$	0	0
$983$	7447.89	0.241659	0.120829	−	0.992673i	$-0.461445\pi$
$983$	7447.89	0.241659	0.120829	+	0.992673i	$0.461445\pi$
$984$	0	0
$985$	−9915.95	−0.320760
$986$	0	0
$987$	−84869.1	−2.73699
$988$	0	0
$989$	−23218.2	−0.746506
$990$	0	0
$991$	−5313.37	−0.170318	−0.0851588	−	0.996367i	$-0.527140\pi$
$991$	−5313.37	−0.170318	−0.0851588	+	0.996367i	$0.527140\pi$
$992$	0	0
$993$	−2420.65	−0.0773586
$994$	0	0
$995$	45393.2	1.44629
$996$	0	0
$997$	−53523.9	−1.70022	−0.850110	−	0.526604i	$-0.823465\pi$
$997$	−53523.9	−1.70022	−0.850110	+	0.526604i	$0.823465\pi$
$998$	0	0
$999$	12200.4	0.386391

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1936.4.a.bm.1.4		4
4.3	odd	2		242.4.a.o.1.1		4
11.7	odd	10		176.4.m.b.49.1		8
11.8	odd	10		176.4.m.b.97.1		8
11.10	odd	2		1936.4.a.bn.1.4		4
12.11	even	2		2178.4.a.bt.1.1		4
44.3	odd	10		242.4.c.q.9.2		8
44.7	even	10		22.4.c.b.5.2	✓	8
44.15	odd	10		242.4.c.q.27.2		8
44.19	even	10		22.4.c.b.9.2	yes	8
44.27	odd	10		242.4.c.n.3.1		8
44.31	odd	10		242.4.c.n.81.1		8
44.35	even	10		242.4.c.r.81.1		8
44.39	even	10		242.4.c.r.3.1		8
44.43	even	2		242.4.a.n.1.1		4
132.95	odd	10		198.4.f.d.181.1		8
132.107	odd	10		198.4.f.d.163.1		8
132.131	odd	2		2178.4.a.by.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.4.c.b.5.2	✓	8	44.7	even	10
22.4.c.b.9.2	yes	8	44.19	even	10
176.4.m.b.49.1		8	11.7	odd	10
176.4.m.b.97.1		8	11.8	odd	10
198.4.f.d.163.1		8	132.107	odd	10
198.4.f.d.181.1		8	132.95	odd	10
242.4.a.n.1.1		4	44.43	even	2
242.4.a.o.1.1		4	4.3	odd	2
242.4.c.n.3.1		8	44.27	odd	10
242.4.c.n.81.1		8	44.31	odd	10
242.4.c.q.9.2		8	44.3	odd	10
242.4.c.q.27.2		8	44.15	odd	10
242.4.c.r.3.1		8	44.39	even	10
242.4.c.r.81.1		8	44.35	even	10
1936.4.a.bm.1.4		4	1.1	even	1	trivial
1936.4.a.bn.1.4		4	11.10	odd	2
2178.4.a.bt.1.1		4	12.11	even	2
2178.4.a.by.1.1		4	132.131	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:	\( N \)	\(=\)	\( 1936 = 2^{4} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1936.a (trivial)

\(f(q)\)	\(=\)	\(q+7.81182 q^{3} +14.9181 q^{5} +21.7679 q^{7} +34.0245 q^{9} +O(q^{10})\)	\(q+7.81182 q^{3} +14.9181 q^{5} +21.7679 q^{7} +34.0245 q^{9} +44.0286 q^{13} +116.538 q^{15} +24.9145 q^{17} +21.9573 q^{19} +170.046 q^{21} -177.749 q^{23} +97.5508 q^{25} +54.8738 q^{27} +149.396 q^{29} -75.1436 q^{31} +324.736 q^{35} +222.336 q^{37} +343.943 q^{39} +253.121 q^{41} +130.623 q^{43} +507.582 q^{45} -499.093 q^{47} +130.839 q^{49} +194.627 q^{51} +12.9421 q^{53} +171.526 q^{57} -35.5614 q^{59} -538.343 q^{61} +740.639 q^{63} +656.824 q^{65} +519.621 q^{67} -1388.54 q^{69} -78.4486 q^{71} -1144.07 q^{73} +762.049 q^{75} +772.546 q^{79} -489.997 q^{81} +537.242 q^{83} +371.678 q^{85} +1167.06 q^{87} +667.089 q^{89} +958.407 q^{91} -587.008 q^{93} +327.561 q^{95} -179.654 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 25 q^{5} - 3 q^{7} + 102 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 + 25 * q^5 - 3 * q^7 + 102 * q^9	\( 4 q - 4 q^{3} + 25 q^{5} - 3 q^{7} + 102 q^{9} - 41 q^{13} - 68 q^{15} + 52 q^{17} - 16 q^{19} + 25 q^{21} - 314 q^{23} - 21 q^{25} - 286 q^{27} + 561 q^{29} - 199 q^{31} + 714 q^{35} + 357 q^{37} + 1038 q^{39} + 32 q^{41} + 721 q^{43} + 1326 q^{45} - 403 q^{47} + 823 q^{49} + 174 q^{51} - 133 q^{53} - 1031 q^{57} - 1016 q^{59} - 919 q^{61} + 1367 q^{63} + 69 q^{65} - 289 q^{67} - 1620 q^{69} + 1205 q^{71} - 1234 q^{73} + 911 q^{75} + 603 q^{79} - 1400 q^{81} - 1514 q^{83} + 717 q^{85} + 1061 q^{87} - 1101 q^{89} + 2306 q^{91} - 2298 q^{93} + 1766 q^{95} + 2116 q^{97}+O(q^{100}) \) 4 * q - 4 * q^3 + 25 * q^5 - 3 * q^7 + 102 * q^9 - 41 * q^13 - 68 * q^15 + 52 * q^17 - 16 * q^19 + 25 * q^21 - 314 * q^23 - 21 * q^25 - 286 * q^27 + 561 * q^29 - 199 * q^31 + 714 * q^35 + 357 * q^37 + 1038 * q^39 + 32 * q^41 + 721 * q^43 + 1326 * q^45 - 403 * q^47 + 823 * q^49 + 174 * q^51 - 133 * q^53 - 1031 * q^57 - 1016 * q^59 - 919 * q^61 + 1367 * q^63 + 69 * q^65 - 289 * q^67 - 1620 * q^69 + 1205 * q^71 - 1234 * q^73 + 911 * q^75 + 603 * q^79 - 1400 * q^81 - 1514 * q^83 + 717 * q^85 + 1061 * q^87 - 1101 * q^89 + 2306 * q^91 - 2298 * q^93 + 1766 * q^95 + 2116 * q^97

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