LMFDB - Embedded newform 7600.2.a.ce.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7600,2,Mod(1,7600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7600, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("7600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7600 = 2^{4} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7600.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:	$60.6863055362$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} + 4 $ x^4 - 6*x^2 + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 380)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.28825$ of defining polynomial
Character	$\chi$	$=$	7600.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.28825 q^{3} -2.82843 q^{7} +2.23607 q^{9} +O(q^{10})$	$q+2.28825 q^{3} -2.82843 q^{7} +2.23607 q^{9} +5.23607 q^{11} +4.03631 q^{13} -1.08036 q^{17} +1.00000 q^{19} -6.47214 q^{21} +7.40492 q^{23} -1.74806 q^{27} +4.47214 q^{29} +4.00000 q^{31} +11.9814 q^{33} +6.86474 q^{37} +9.23607 q^{39} -6.00000 q^{41} -8.48528 q^{43} -8.48528 q^{47} +1.00000 q^{49} -2.47214 q^{51} -6.86474 q^{53} +2.28825 q^{57} +10.4721 q^{59} +1.70820 q^{61} -6.32456 q^{63} -1.62054 q^{67} +16.9443 q^{69} +1.52786 q^{71} -13.7295 q^{73} -14.8098 q^{77} +11.4164 q^{79} -10.7082 q^{81} +5.24419 q^{83} +10.2333 q^{87} -14.9443 q^{89} -11.4164 q^{91} +9.15298 q^{93} +4.44897 q^{97} +11.7082 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 12 q^{11} + 4 q^{19} - 8 q^{21} + 16 q^{31} + 28 q^{39} - 24 q^{41} + 4 q^{49} + 8 q^{51} + 24 q^{59} - 20 q^{61} + 32 q^{69} + 24 q^{71} - 8 q^{79} - 16 q^{81} - 24 q^{89} + 8 q^{91} + 20 q^{99}+O(q^{100}) $ 4 * q + 12 * q^11 + 4 * q^19 - 8 * q^21 + 16 * q^31 + 28 * q^39 - 24 * q^41 + 4 * q^49 + 8 * q^51 + 24 * q^59 - 20 * q^61 + 32 * q^69 + 24 * q^71 - 8 * q^79 - 16 * q^81 - 24 * q^89 + 8 * q^91 + 20 * 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
$3$	2.28825	1.32112	0.660560	−	0.750774i	$-0.270319\pi$
$3$	2.28825	1.32112	0.660560	+	0.750774i	$0.270319\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.82843	−1.06904	−0.534522	−	0.845154i	$-0.679509\pi$
$7$	−2.82843	−1.06904	−0.534522	+	0.845154i	$0.679509\pi$
$8$	0	0
$9$	2.23607	0.745356
$10$	0	0
$11$	5.23607	1.57873	0.789367	−	0.613922i	$-0.210409\pi$
$11$	5.23607	1.57873	0.789367	+	0.613922i	$0.210409\pi$
$12$	0	0
$13$	4.03631	1.11947	0.559735	−	0.828671i	$-0.310903\pi$
$13$	4.03631	1.11947	0.559735	+	0.828671i	$0.310903\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.08036	−0.262027	−0.131013	−	0.991381i	$-0.541823\pi$
$17$	−1.08036	−0.262027	−0.131013	+	0.991381i	$0.541823\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−6.47214	−1.41234
$22$	0	0
$23$	7.40492	1.54403	0.772016	−	0.635603i	$-0.219248\pi$
$23$	7.40492	1.54403	0.772016	+	0.635603i	$0.219248\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.74806	−0.336415
$28$	0	0
$29$	4.47214	0.830455	0.415227	−	0.909718i	$-0.363702\pi$
$29$	4.47214	0.830455	0.415227	+	0.909718i	$0.363702\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	0	0
$33$	11.9814	2.08570
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6.86474	1.12856	0.564278	−	0.825585i	$-0.309155\pi$
$37$	6.86474	1.12856	0.564278	+	0.825585i	$0.309155\pi$
$38$	0	0
$39$	9.23607	1.47895
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	−8.48528	−1.29399	−0.646997	−	0.762493i	$-0.723975\pi$
$43$	−8.48528	−1.29399	−0.646997	+	0.762493i	$0.723975\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−8.48528	−1.23771	−0.618853	−	0.785507i	$-0.712402\pi$
$47$	−8.48528	−1.23771	−0.618853	+	0.785507i	$0.712402\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−2.47214	−0.346168
$52$	0	0
$53$	−6.86474	−0.942944	−0.471472	−	0.881881i	$-0.656277\pi$
$53$	−6.86474	−0.942944	−0.471472	+	0.881881i	$0.656277\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.28825	0.303086
$58$	0	0
$59$	10.4721	1.36336	0.681678	−	0.731652i	$-0.261251\pi$
$59$	10.4721	1.36336	0.681678	+	0.731652i	$0.261251\pi$
$60$	0	0
$61$	1.70820	0.218713	0.109357	−	0.994003i	$-0.465121\pi$
$61$	1.70820	0.218713	0.109357	+	0.994003i	$0.465121\pi$
$62$	0	0
$63$	−6.32456	−0.796819
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−1.62054	−0.197981	−0.0989905	−	0.995088i	$-0.531561\pi$
$67$	−1.62054	−0.197981	−0.0989905	+	0.995088i	$0.531561\pi$
$68$	0	0
$69$	16.9443	2.03985
$70$	0	0
$71$	1.52786	0.181324	0.0906621	−	0.995882i	$-0.471102\pi$
$71$	1.52786	0.181324	0.0906621	+	0.995882i	$0.471102\pi$
$72$	0	0
$73$	−13.7295	−1.60691	−0.803457	−	0.595363i	$-0.797008\pi$
$73$	−13.7295	−1.60691	−0.803457	+	0.595363i	$0.797008\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−14.8098	−1.68774
$78$	0	0
$79$	11.4164	1.28445	0.642223	−	0.766518i	$-0.278012\pi$
$79$	11.4164	1.28445	0.642223	+	0.766518i	$0.278012\pi$
$80$	0	0
$81$	−10.7082	−1.18980
$82$	0	0
$83$	5.24419	0.575625	0.287812	−	0.957687i	$-0.407072\pi$
$83$	5.24419	0.575625	0.287812	+	0.957687i	$0.407072\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	10.2333	1.09713
$88$	0	0
$89$	−14.9443	−1.58409	−0.792045	−	0.610463i	$-0.790983\pi$
$89$	−14.9443	−1.58409	−0.792045	+	0.610463i	$0.790983\pi$
$90$	0	0
$91$	−11.4164	−1.19676
$92$	0	0
$93$	9.15298	0.949120
$94$	0	0
$95$	0	0
$96$	0	0
$97$	4.44897	0.451725	0.225862	−	0.974159i	$-0.427480\pi$
$97$	4.44897	0.451725	0.225862	+	0.974159i	$0.427480\pi$
$98$	0	0
$99$	11.7082	1.17672
$100$	0	0
$101$	0.763932	0.0760141	0.0380070	−	0.999277i	$-0.487899\pi$
$101$	0.763932	0.0760141	0.0380070	+	0.999277i	$0.487899\pi$
$102$	0	0
$103$	15.3500	1.51248	0.756241	−	0.654293i	$-0.227034\pi$
$103$	15.3500	1.51248	0.756241	+	0.654293i	$0.227034\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	16.4304	1.58838	0.794192	−	0.607666i	$-0.207894\pi$
$107$	16.4304	1.58838	0.794192	+	0.607666i	$0.207894\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	15.7082	1.49096
$112$	0	0
$113$	12.1089	1.13911	0.569556	−	0.821952i	$-0.307115\pi$
$113$	12.1089	1.13911	0.569556	+	0.821952i	$0.307115\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	9.02546	0.834404
$118$	0	0
$119$	3.05573	0.280118
$120$	0	0
$121$	16.4164	1.49240
$122$	0	0
$123$	−13.7295	−1.23794
$124$	0	0
$125$	0	0
$126$	0	0
$127$	10.1058	0.896747	0.448374	−	0.893846i	$-0.352003\pi$
$127$	10.1058	0.896747	0.448374	+	0.893846i	$0.352003\pi$
$128$	0	0
$129$	−19.4164	−1.70952
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	−2.82843	−0.245256
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1.08036	−0.0923016	−0.0461508	−	0.998934i	$-0.514695\pi$
$137$	−1.08036	−0.0923016	−0.0461508	+	0.998934i	$0.514695\pi$
$138$	0	0
$139$	−15.7082	−1.33235	−0.666176	−	0.745794i	$-0.732070\pi$
$139$	−15.7082	−1.33235	−0.666176	+	0.745794i	$0.732070\pi$
$140$	0	0
$141$	−19.4164	−1.63516
$142$	0	0
$143$	21.1344	1.76735
$144$	0	0
$145$	0	0
$146$	0	0
$147$	2.28825	0.188731
$148$	0	0
$149$	18.6525	1.52807	0.764035	−	0.645175i	$-0.223215\pi$
$149$	18.6525	1.52807	0.764035	+	0.645175i	$0.223215\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	0	0
$153$	−2.41577	−0.195303
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−19.3863	−1.54720	−0.773599	−	0.633676i	$-0.781545\pi$
$157$	−19.3863	−1.54720	−0.773599	+	0.633676i	$0.781545\pi$
$158$	0	0
$159$	−15.7082	−1.24574
$160$	0	0
$161$	−20.9443	−1.65064
$162$	0	0
$163$	8.48528	0.664619	0.332309	−	0.943170i	$-0.392172\pi$
$163$	8.48528	0.664619	0.332309	+	0.943170i	$0.392172\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	4.70401	0.364007	0.182004	−	0.983298i	$-0.441742\pi$
$167$	4.70401	0.364007	0.182004	+	0.983298i	$0.441742\pi$
$168$	0	0
$169$	3.29180	0.253215
$170$	0	0
$171$	2.23607	0.170996
$172$	0	0
$173$	−13.1893	−1.00276	−0.501382	−	0.865226i	$-0.667175\pi$
$173$	−13.1893	−1.00276	−0.501382	+	0.865226i	$0.667175\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	23.9628	1.80116
$178$	0	0
$179$	13.5279	1.01112	0.505560	−	0.862791i	$-0.331286\pi$
$179$	13.5279	1.01112	0.505560	+	0.862791i	$0.331286\pi$
$180$	0	0
$181$	6.00000	0.445976	0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000	0.445976	0.222988	+	0.974821i	$0.428419\pi$
$182$	0	0
$183$	3.90879	0.288946
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−5.65685	−0.413670
$188$	0	0
$189$	4.94427	0.359643
$190$	0	0
$191$	20.9443	1.51547	0.757737	−	0.652560i	$-0.226305\pi$
$191$	20.9443	1.51547	0.757737	+	0.652560i	$0.226305\pi$
$192$	0	0
$193$	4.44897	0.320244	0.160122	−	0.987097i	$-0.448811\pi$
$193$	4.44897	0.320244	0.160122	+	0.987097i	$0.448811\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−12.6491	−0.901212	−0.450606	−	0.892723i	$-0.648792\pi$
$197$	−12.6491	−0.901212	−0.450606	+	0.892723i	$0.648792\pi$
$198$	0	0
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	−3.70820	−0.261557
$202$	0	0
$203$	−12.6491	−0.887794
$204$	0	0
$205$	0	0
$206$	0	0
$207$	16.5579	1.15085
$208$	0	0
$209$	5.23607	0.362186
$210$	0	0
$211$	11.4164	0.785938	0.392969	−	0.919552i	$-0.371448\pi$
$211$	11.4164	0.785938	0.392969	+	0.919552i	$0.371448\pi$
$212$	0	0
$213$	3.49613	0.239551
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−11.3137	−0.768025
$218$	0	0
$219$	−31.4164	−2.12292
$220$	0	0
$221$	−4.36068	−0.293331
$222$	0	0
$223$	28.6668	1.91967	0.959836	−	0.280560i	$-0.0905202\pi$
$223$	28.6668	1.91967	0.959836	+	0.280560i	$0.0905202\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−15.3500	−1.01882	−0.509408	−	0.860525i	$-0.670136\pi$
$227$	−15.3500	−1.01882	−0.509408	+	0.860525i	$0.670136\pi$
$228$	0	0
$229$	−5.70820	−0.377209	−0.188604	−	0.982053i	$-0.560396\pi$
$229$	−5.70820	−0.377209	−0.188604	+	0.982053i	$0.560396\pi$
$230$	0	0
$231$	−33.8885	−2.22970
$232$	0	0
$233$	12.6491	0.828671	0.414335	−	0.910124i	$-0.364014\pi$
$233$	12.6491	0.828671	0.414335	+	0.910124i	$0.364014\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	26.1235	1.69691
$238$	0	0
$239$	−5.88854	−0.380898	−0.190449	−	0.981697i	$-0.560994\pi$
$239$	−5.88854	−0.380898	−0.190449	+	0.981697i	$0.560994\pi$
$240$	0	0
$241$	24.8328	1.59962	0.799811	−	0.600252i	$-0.204933\pi$
$241$	24.8328	1.59962	0.799811	+	0.600252i	$0.204933\pi$
$242$	0	0
$243$	−19.2588	−1.23545
$244$	0	0
$245$	0	0
$246$	0	0
$247$	4.03631	0.256824
$248$	0	0
$249$	12.0000	0.760469
$250$	0	0
$251$	−29.8885	−1.88655	−0.943274	−	0.332015i	$-0.892272\pi$
$251$	−29.8885	−1.88655	−0.943274	+	0.332015i	$0.892272\pi$
$252$	0	0
$253$	38.7727	2.43762
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4.70401	0.293428	0.146714	−	0.989179i	$-0.453130\pi$
$257$	4.70401	0.293428	0.146714	+	0.989179i	$0.453130\pi$
$258$	0	0
$259$	−19.4164	−1.20648
$260$	0	0
$261$	10.0000	0.618984
$262$	0	0
$263$	−9.56564	−0.589843	−0.294921	−	0.955522i	$-0.595293\pi$
$263$	−9.56564	−0.589843	−0.294921	+	0.955522i	$0.595293\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−34.1962	−2.09277
$268$	0	0
$269$	−2.94427	−0.179515	−0.0897577	−	0.995964i	$-0.528609\pi$
$269$	−2.94427	−0.179515	−0.0897577	+	0.995964i	$0.528609\pi$
$270$	0	0
$271$	19.1246	1.16174	0.580869	−	0.813997i	$-0.302713\pi$
$271$	19.1246	1.16174	0.580869	+	0.813997i	$0.302713\pi$
$272$	0	0
$273$	−26.1235	−1.58107
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−3.24109	−0.194738	−0.0973691	−	0.995248i	$-0.531043\pi$
$277$	−3.24109	−0.194738	−0.0973691	+	0.995248i	$0.531043\pi$
$278$	0	0
$279$	8.94427	0.535480
$280$	0	0
$281$	1.41641	0.0844958	0.0422479	−	0.999107i	$-0.486548\pi$
$281$	1.41641	0.0844958	0.0422479	+	0.999107i	$0.486548\pi$
$282$	0	0
$283$	−16.5579	−0.984265	−0.492133	−	0.870520i	$-0.663782\pi$
$283$	−16.5579	−0.984265	−0.492133	+	0.870520i	$0.663782\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	16.9706	1.00174
$288$	0	0
$289$	−15.8328	−0.931342
$290$	0	0
$291$	10.1803	0.596782
$292$	0	0
$293$	−6.86474	−0.401042	−0.200521	−	0.979689i	$-0.564264\pi$
$293$	−6.86474	−0.401042	−0.200521	+	0.979689i	$0.564264\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−9.15298	−0.531110
$298$	0	0
$299$	29.8885	1.72850
$300$	0	0
$301$	24.0000	1.38334
$302$	0	0
$303$	1.74806	0.100424
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−4.03631	−0.230364	−0.115182	−	0.993344i	$-0.536745\pi$
$307$	−4.03631	−0.230364	−0.115182	+	0.993344i	$0.536745\pi$
$308$	0	0
$309$	35.1246	1.99817
$310$	0	0
$311$	−3.70820	−0.210273	−0.105136	−	0.994458i	$-0.533528\pi$
$311$	−3.70820	−0.210273	−0.105136	+	0.994458i	$0.533528\pi$
$312$	0	0
$313$	27.4589	1.55207	0.776036	−	0.630689i	$-0.217228\pi$
$313$	27.4589	1.55207	0.776036	+	0.630689i	$0.217228\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	13.1893	0.740784	0.370392	−	0.928875i	$-0.379223\pi$
$317$	13.1893	0.740784	0.370392	+	0.928875i	$0.379223\pi$
$318$	0	0
$319$	23.4164	1.31107
$320$	0	0
$321$	37.5967	2.09845
$322$	0	0
$323$	−1.08036	−0.0601130
$324$	0	0
$325$	0	0
$326$	0	0
$327$	4.57649	0.253081
$328$	0	0
$329$	24.0000	1.32316
$330$	0	0
$331$	19.4164	1.06722	0.533611	−	0.845730i	$-0.320835\pi$
$331$	19.4164	1.06722	0.533611	+	0.845730i	$0.320835\pi$
$332$	0	0
$333$	15.3500	0.841176
$334$	0	0
$335$	0	0
$336$	0	0
$337$	7.27740	0.396425	0.198213	−	0.980159i	$-0.436486\pi$
$337$	7.27740	0.396425	0.198213	+	0.980159i	$0.436486\pi$
$338$	0	0
$339$	27.7082	1.50490
$340$	0	0
$341$	20.9443	1.13420
$342$	0	0
$343$	16.9706	0.916324
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−34.8639	−1.87159	−0.935795	−	0.352544i	$-0.885317\pi$
$347$	−34.8639	−1.87159	−0.935795	+	0.352544i	$0.885317\pi$
$348$	0	0
$349$	−1.41641	−0.0758186	−0.0379093	−	0.999281i	$-0.512070\pi$
$349$	−1.41641	−0.0758186	−0.0379093	+	0.999281i	$0.512070\pi$
$350$	0	0
$351$	−7.05573	−0.376607
$352$	0	0
$353$	15.8902	0.845750	0.422875	−	0.906188i	$-0.361021\pi$
$353$	15.8902	0.845750	0.422875	+	0.906188i	$0.361021\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	6.99226	0.370069
$358$	0	0
$359$	11.1246	0.587135	0.293567	−	0.955938i	$-0.405158\pi$
$359$	11.1246	0.587135	0.293567	+	0.955938i	$0.405158\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	37.5648	1.97164
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−8.48528	−0.442928	−0.221464	−	0.975169i	$-0.571084\pi$
$367$	−8.48528	−0.442928	−0.221464	+	0.975169i	$0.571084\pi$
$368$	0	0
$369$	−13.4164	−0.698430
$370$	0	0
$371$	19.4164	1.00805
$372$	0	0
$373$	−34.3237	−1.77721	−0.888606	−	0.458670i	$-0.848326\pi$
$373$	−34.3237	−1.77721	−0.888606	+	0.458670i	$0.848326\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	18.0509	0.929670
$378$	0	0
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	0	0
$381$	23.1246	1.18471
$382$	0	0
$383$	−23.6777	−1.20987	−0.604936	−	0.796274i	$-0.706801\pi$
$383$	−23.6777	−1.20987	−0.604936	+	0.796274i	$0.706801\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−18.9737	−0.964486
$388$	0	0
$389$	2.94427	0.149281	0.0746403	−	0.997211i	$-0.476219\pi$
$389$	2.94427	0.149281	0.0746403	+	0.997211i	$0.476219\pi$
$390$	0	0
$391$	−8.00000	−0.404577
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	30.7000	1.54079	0.770395	−	0.637566i	$-0.220059\pi$
$397$	30.7000	1.54079	0.770395	+	0.637566i	$0.220059\pi$
$398$	0	0
$399$	−6.47214	−0.324012
$400$	0	0
$401$	4.47214	0.223328	0.111664	−	0.993746i	$-0.464382\pi$
$401$	4.47214	0.223328	0.111664	+	0.993746i	$0.464382\pi$
$402$	0	0
$403$	16.1452	0.804252
$404$	0	0
$405$	0	0
$406$	0	0
$407$	35.9442	1.78169
$408$	0	0
$409$	−9.41641	−0.465611	−0.232806	−	0.972523i	$-0.574791\pi$
$409$	−9.41641	−0.465611	−0.232806	+	0.972523i	$0.574791\pi$
$410$	0	0
$411$	−2.47214	−0.121941
$412$	0	0
$413$	−29.6197	−1.45749
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−35.9442	−1.76020
$418$	0	0
$419$	3.05573	0.149282	0.0746410	−	0.997210i	$-0.476219\pi$
$419$	3.05573	0.149282	0.0746410	+	0.997210i	$0.476219\pi$
$420$	0	0
$421$	14.0000	0.682318	0.341159	−	0.940006i	$-0.389181\pi$
$421$	14.0000	0.682318	0.341159	+	0.940006i	$0.389181\pi$
$422$	0	0
$423$	−18.9737	−0.922531
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−4.83153	−0.233814
$428$	0	0
$429$	48.3607	2.33488
$430$	0	0
$431$	−10.4721	−0.504425	−0.252213	−	0.967672i	$-0.581158\pi$
$431$	−10.4721	−0.504425	−0.252213	+	0.967672i	$0.581158\pi$
$432$	0	0
$433$	−37.9774	−1.82508	−0.912540	−	0.408989i	$-0.865882\pi$
$433$	−37.9774	−1.82508	−0.912540	+	0.408989i	$0.865882\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	7.40492	0.354225
$438$	0	0
$439$	−34.8328	−1.66248	−0.831240	−	0.555914i	$-0.812368\pi$
$439$	−34.8328	−1.66248	−0.831240	+	0.555914i	$0.812368\pi$
$440$	0	0
$441$	2.23607	0.106479
$442$	0	0
$443$	−5.24419	−0.249159	−0.124580	−	0.992210i	$-0.539758\pi$
$443$	−5.24419	−0.249159	−0.124580	+	0.992210i	$0.539758\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	42.6814	2.01876
$448$	0	0
$449$	−7.52786	−0.355262	−0.177631	−	0.984097i	$-0.556843\pi$
$449$	−7.52786	−0.355262	−0.177631	+	0.984097i	$0.556843\pi$
$450$	0	0
$451$	−31.4164	−1.47934
$452$	0	0
$453$	18.3060	0.860089
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−37.9473	−1.77510	−0.887551	−	0.460710i	$-0.847595\pi$
$457$	−37.9473	−1.77510	−0.887551	+	0.460710i	$0.847595\pi$
$458$	0	0
$459$	1.88854	0.0881497
$460$	0	0
$461$	−11.8885	−0.553705	−0.276852	−	0.960912i	$-0.589291\pi$
$461$	−11.8885	−0.553705	−0.276852	+	0.960912i	$0.589291\pi$
$462$	0	0
$463$	−33.5285	−1.55820	−0.779100	−	0.626900i	$-0.784324\pi$
$463$	−33.5285	−1.55820	−0.779100	+	0.626900i	$0.784324\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−11.7264	−0.542632	−0.271316	−	0.962490i	$-0.587459\pi$
$467$	−11.7264	−0.542632	−0.271316	+	0.962490i	$0.587459\pi$
$468$	0	0
$469$	4.58359	0.211651
$470$	0	0
$471$	−44.3607	−2.04403
$472$	0	0
$473$	−44.4295	−2.04287
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−15.3500	−0.702829
$478$	0	0
$479$	−6.76393	−0.309052	−0.154526	−	0.987989i	$-0.549385\pi$
$479$	−6.76393	−0.309052	−0.154526	+	0.987989i	$0.549385\pi$
$480$	0	0
$481$	27.7082	1.26339
$482$	0	0
$483$	−47.9256	−2.18069
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−17.7658	−0.805044	−0.402522	−	0.915410i	$-0.631866\pi$
$487$	−17.7658	−0.805044	−0.402522	+	0.915410i	$0.631866\pi$
$488$	0	0
$489$	19.4164	0.878040
$490$	0	0
$491$	29.8885	1.34885	0.674426	−	0.738343i	$-0.264391\pi$
$491$	29.8885	1.34885	0.674426	+	0.738343i	$0.264391\pi$
$492$	0	0
$493$	−4.83153	−0.217601
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−4.32145	−0.193844
$498$	0	0
$499$	−23.1246	−1.03520	−0.517600	−	0.855623i	$-0.673174\pi$
$499$	−23.1246	−1.03520	−0.517600	+	0.855623i	$0.673174\pi$
$500$	0	0
$501$	10.7639	0.480897
$502$	0	0
$503$	−39.1853	−1.74719	−0.873593	−	0.486656i	$-0.838216\pi$
$503$	−39.1853	−1.74719	−0.873593	+	0.486656i	$0.838216\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	7.53244	0.334527
$508$	0	0
$509$	−37.4164	−1.65845	−0.829227	−	0.558913i	$-0.811219\pi$
$509$	−37.4164	−1.65845	−0.829227	+	0.558913i	$0.811219\pi$
$510$	0	0
$511$	38.8328	1.71786
$512$	0	0
$513$	−1.74806	−0.0771789
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−44.4295	−1.95401
$518$	0	0
$519$	−30.1803	−1.32477
$520$	0	0
$521$	35.8885	1.57231	0.786153	−	0.618032i	$-0.212070\pi$
$521$	35.8885	1.57231	0.786153	+	0.618032i	$0.212070\pi$
$522$	0	0
$523$	−3.62365	−0.158451	−0.0792255	−	0.996857i	$-0.525245\pi$
$523$	−3.62365	−0.158451	−0.0792255	+	0.996857i	$0.525245\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−4.32145	−0.188245
$528$	0	0
$529$	31.8328	1.38404
$530$	0	0
$531$	23.4164	1.01619
$532$	0	0
$533$	−24.2179	−1.04899
$534$	0	0
$535$	0	0
$536$	0	0
$537$	30.9551	1.33581
$538$	0	0
$539$	5.23607	0.225533
$540$	0	0
$541$	−1.70820	−0.0734414	−0.0367207	−	0.999326i	$-0.511691\pi$
$541$	−1.70820	−0.0734414	−0.0367207	+	0.999326i	$0.511691\pi$
$542$	0	0
$543$	13.7295	0.589188
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−1.20788	−0.0516453	−0.0258227	−	0.999667i	$-0.508221\pi$
$547$	−1.20788	−0.0516453	−0.0258227	+	0.999667i	$0.508221\pi$
$548$	0	0
$549$	3.81966	0.163019
$550$	0	0
$551$	4.47214	0.190519
$552$	0	0
$553$	−32.2905	−1.37313
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−11.5687	−0.490184	−0.245092	−	0.969500i	$-0.578818\pi$
$557$	−11.5687	−0.490184	−0.245092	+	0.969500i	$0.578818\pi$
$558$	0	0
$559$	−34.2492	−1.44859
$560$	0	0
$561$	−12.9443	−0.546508
$562$	0	0
$563$	17.3531	0.731347	0.365673	−	0.930743i	$-0.380839\pi$
$563$	17.3531	0.731347	0.365673	+	0.930743i	$0.380839\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	30.2874	1.27195
$568$	0	0
$569$	−21.0557	−0.882702	−0.441351	−	0.897335i	$-0.645501\pi$
$569$	−21.0557	−0.882702	−0.441351	+	0.897335i	$0.645501\pi$
$570$	0	0
$571$	15.1246	0.632945	0.316473	−	0.948602i	$-0.397501\pi$
$571$	15.1246	0.632945	0.316473	+	0.948602i	$0.397501\pi$
$572$	0	0
$573$	47.9256	2.00212
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−5.65685	−0.235498	−0.117749	−	0.993043i	$-0.537568\pi$
$577$	−5.65685	−0.235498	−0.117749	+	0.993043i	$0.537568\pi$
$578$	0	0
$579$	10.1803	0.423080
$580$	0	0
$581$	−14.8328	−0.615369
$582$	0	0
$583$	−35.9442	−1.48866
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−37.0246	−1.52817	−0.764084	−	0.645117i	$-0.776809\pi$
$587$	−37.0246	−1.52817	−0.764084	+	0.645117i	$0.776809\pi$
$588$	0	0
$589$	4.00000	0.164817
$590$	0	0
$591$	−28.9443	−1.19061
$592$	0	0
$593$	−14.8098	−0.608167	−0.304084	−	0.952645i	$-0.598350\pi$
$593$	−14.8098	−0.608167	−0.304084	+	0.952645i	$0.598350\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	36.6119	1.49843
$598$	0	0
$599$	36.0000	1.47092	0.735460	−	0.677568i	$-0.236966\pi$
$599$	36.0000	1.47092	0.735460	+	0.677568i	$0.236966\pi$
$600$	0	0
$601$	−36.8328	−1.50244	−0.751221	−	0.660051i	$-0.770535\pi$
$601$	−36.8328	−1.50244	−0.751221	+	0.660051i	$0.770535\pi$
$602$	0	0
$603$	−3.62365	−0.147566
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−12.9343	−0.524985	−0.262493	−	0.964934i	$-0.584545\pi$
$607$	−12.9343	−0.524985	−0.262493	+	0.964934i	$0.584545\pi$
$608$	0	0
$609$	−28.9443	−1.17288
$610$	0	0
$611$	−34.2492	−1.38558
$612$	0	0
$613$	−20.2117	−0.816341	−0.408170	−	0.912906i	$-0.633833\pi$
$613$	−20.2117	−0.816341	−0.408170	+	0.912906i	$0.633833\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−28.5393	−1.14895	−0.574475	−	0.818522i	$-0.694794\pi$
$617$	−28.5393	−1.14895	−0.574475	+	0.818522i	$0.694794\pi$
$618$	0	0
$619$	−0.875388	−0.0351848	−0.0175924	−	0.999845i	$-0.505600\pi$
$619$	−0.875388	−0.0351848	−0.0175924	+	0.999845i	$0.505600\pi$
$620$	0	0
$621$	−12.9443	−0.519436
$622$	0	0
$623$	42.2688	1.69346
$624$	0	0
$625$	0	0
$626$	0	0
$627$	11.9814	0.478491
$628$	0	0
$629$	−7.41641	−0.295712
$630$	0	0
$631$	−11.7082	−0.466096	−0.233048	−	0.972465i	$-0.574870\pi$
$631$	−11.7082	−0.466096	−0.233048	+	0.972465i	$0.574870\pi$
$632$	0	0
$633$	26.1235	1.03832
$634$	0	0
$635$	0	0
$636$	0	0
$637$	4.03631	0.159924
$638$	0	0
$639$	3.41641	0.135151
$640$	0	0
$641$	−47.8885	−1.89148	−0.945742	−	0.324919i	$-0.894663\pi$
$641$	−47.8885	−1.89148	−0.945742	+	0.324919i	$0.894663\pi$
$642$	0	0
$643$	11.7264	0.462443	0.231221	−	0.972901i	$-0.425728\pi$
$643$	11.7264	0.462443	0.231221	+	0.972901i	$0.425728\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−34.8639	−1.37064	−0.685320	−	0.728242i	$-0.740338\pi$
$647$	−34.8639	−1.37064	−0.685320	+	0.728242i	$0.740338\pi$
$648$	0	0
$649$	54.8328	2.15238
$650$	0	0
$651$	−25.8885	−1.01465
$652$	0	0
$653$	−3.24109	−0.126834	−0.0634168	−	0.997987i	$-0.520200\pi$
$653$	−3.24109	−0.126834	−0.0634168	+	0.997987i	$0.520200\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−30.7000	−1.19772
$658$	0	0
$659$	27.0557	1.05394	0.526971	−	0.849883i	$-0.323328\pi$
$659$	27.0557	1.05394	0.526971	+	0.849883i	$0.323328\pi$
$660$	0	0
$661$	−26.0000	−1.01128	−0.505641	−	0.862744i	$-0.668744\pi$
$661$	−26.0000	−1.01128	−0.505641	+	0.862744i	$0.668744\pi$
$662$	0	0
$663$	−9.97831	−0.387525
$664$	0	0
$665$	0	0
$666$	0	0
$667$	33.1158	1.28225
$668$	0	0
$669$	65.5967	2.53612
$670$	0	0
$671$	8.94427	0.345290
$672$	0	0
$673$	42.8090	1.65016	0.825082	−	0.565013i	$-0.191129\pi$
$673$	42.8090	1.65016	0.825082	+	0.565013i	$0.191129\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−30.0022	−1.15308	−0.576540	−	0.817069i	$-0.695597\pi$
$677$	−30.0022	−1.15308	−0.576540	+	0.817069i	$0.695597\pi$
$678$	0	0
$679$	−12.5836	−0.482914
$680$	0	0
$681$	−35.1246	−1.34598
$682$	0	0
$683$	−10.1058	−0.386689	−0.193344	−	0.981131i	$-0.561933\pi$
$683$	−10.1058	−0.386689	−0.193344	+	0.981131i	$0.561933\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−13.0618	−0.498338
$688$	0	0
$689$	−27.7082	−1.05560
$690$	0	0
$691$	15.1246	0.575367	0.287684	−	0.957725i	$-0.407115\pi$
$691$	15.1246	0.575367	0.287684	+	0.957725i	$0.407115\pi$
$692$	0	0
$693$	−33.1158	−1.25797
$694$	0	0
$695$	0	0
$696$	0	0
$697$	6.48218	0.245530
$698$	0	0
$699$	28.9443	1.09477
$700$	0	0
$701$	41.1246	1.55326	0.776628	−	0.629960i	$-0.216929\pi$
$701$	41.1246	1.55326	0.776628	+	0.629960i	$0.216929\pi$
$702$	0	0
$703$	6.86474	0.258908
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−2.16073	−0.0812625
$708$	0	0
$709$	6.58359	0.247252	0.123626	−	0.992329i	$-0.460548\pi$
$709$	6.58359	0.247252	0.123626	+	0.992329i	$0.460548\pi$
$710$	0	0
$711$	25.5279	0.957370
$712$	0	0
$713$	29.6197	1.10927
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−13.4744	−0.503212
$718$	0	0
$719$	21.5967	0.805423	0.402711	−	0.915327i	$-0.368068\pi$
$719$	21.5967	0.805423	0.402711	+	0.915327i	$0.368068\pi$
$720$	0	0
$721$	−43.4164	−1.61691
$722$	0	0
$723$	56.8236	2.11329
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−8.48528	−0.314702	−0.157351	−	0.987543i	$-0.550295\pi$
$727$	−8.48528	−0.314702	−0.157351	+	0.987543i	$0.550295\pi$
$728$	0	0
$729$	−11.9443	−0.442380
$730$	0	0
$731$	9.16718	0.339061
$732$	0	0
$733$	−33.9411	−1.25364	−0.626822	−	0.779162i	$-0.715645\pi$
$733$	−33.9411	−1.25364	−0.626822	+	0.779162i	$0.715645\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−8.48528	−0.312559
$738$	0	0
$739$	−34.8328	−1.28135	−0.640673	−	0.767814i	$-0.721345\pi$
$739$	−34.8328	−1.28135	−0.640673	+	0.767814i	$0.721345\pi$
$740$	0	0
$741$	9.23607	0.339295
$742$	0	0
$743$	−6.86474	−0.251843	−0.125921	−	0.992040i	$-0.540189\pi$
$743$	−6.86474	−0.251843	−0.125921	+	0.992040i	$0.540189\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	11.7264	0.429045
$748$	0	0
$749$	−46.4721	−1.69805
$750$	0	0
$751$	−47.4164	−1.73025	−0.865125	−	0.501557i	$-0.832761\pi$
$751$	−47.4164	−1.73025	−0.865125	+	0.501557i	$0.832761\pi$
$752$	0	0
$753$	−68.3923	−2.49236
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−0.825324	−0.0299969	−0.0149985	−	0.999888i	$-0.504774\pi$
$757$	−0.825324	−0.0299969	−0.0149985	+	0.999888i	$0.504774\pi$
$758$	0	0
$759$	88.7214	3.22038
$760$	0	0
$761$	24.5410	0.889611	0.444806	−	0.895627i	$-0.353273\pi$
$761$	24.5410	0.889611	0.444806	+	0.895627i	$0.353273\pi$
$762$	0	0
$763$	−5.65685	−0.204792
$764$	0	0
$765$	0	0
$766$	0	0
$767$	42.2688	1.52624
$768$	0	0
$769$	28.5410	1.02922	0.514608	−	0.857426i	$-0.327938\pi$
$769$	28.5410	1.02922	0.514608	+	0.857426i	$0.327938\pi$
$770$	0	0
$771$	10.7639	0.387654
$772$	0	0
$773$	15.3500	0.552102	0.276051	−	0.961143i	$-0.410974\pi$
$773$	15.3500	0.552102	0.276051	+	0.961143i	$0.410974\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−44.4295	−1.59390
$778$	0	0
$779$	−6.00000	−0.214972
$780$	0	0
$781$	8.00000	0.286263
$782$	0	0
$783$	−7.81758	−0.279378
$784$	0	0
$785$	0	0
$786$	0	0
$787$	3.62365	0.129169	0.0645845	−	0.997912i	$-0.479428\pi$
$787$	3.62365	0.129169	0.0645845	+	0.997912i	$0.479428\pi$
$788$	0	0
$789$	−21.8885	−0.779253
$790$	0	0
$791$	−34.2492	−1.21776
$792$	0	0
$793$	6.89484	0.244843
$794$	0	0
$795$	0	0
$796$	0	0
$797$	14.1120	0.499874	0.249937	−	0.968262i	$-0.419590\pi$
$797$	14.1120	0.499874	0.249937	+	0.968262i	$0.419590\pi$
$798$	0	0
$799$	9.16718	0.324312
$800$	0	0
$801$	−33.4164	−1.18071
$802$	0	0
$803$	−71.8885	−2.53689
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−6.73722	−0.237161
$808$	0	0
$809$	−16.4721	−0.579129	−0.289565	−	0.957158i	$-0.593511\pi$
$809$	−16.4721	−0.579129	−0.289565	+	0.957158i	$0.593511\pi$
$810$	0	0
$811$	−8.00000	−0.280918	−0.140459	−	0.990086i	$-0.544858\pi$
$811$	−8.00000	−0.280918	−0.140459	+	0.990086i	$0.544858\pi$
$812$	0	0
$813$	43.7618	1.53479
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−8.48528	−0.296862
$818$	0	0
$819$	−25.5279	−0.892016
$820$	0	0
$821$	21.0557	0.734850	0.367425	−	0.930053i	$-0.380239\pi$
$821$	21.0557	0.734850	0.367425	+	0.930053i	$0.380239\pi$
$822$	0	0
$823$	−2.00310	−0.0698238	−0.0349119	−	0.999390i	$-0.511115\pi$
$823$	−2.00310	−0.0698238	−0.0349119	+	0.999390i	$0.511115\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−20.5942	−0.716131	−0.358065	−	0.933696i	$-0.616564\pi$
$827$	−20.5942	−0.716131	−0.358065	+	0.933696i	$0.616564\pi$
$828$	0	0
$829$	18.0000	0.625166	0.312583	−	0.949890i	$-0.398806\pi$
$829$	18.0000	0.625166	0.312583	+	0.949890i	$0.398806\pi$
$830$	0	0
$831$	−7.41641	−0.257272
$832$	0	0
$833$	−1.08036	−0.0374324
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−6.99226	−0.241688
$838$	0	0
$839$	13.3050	0.459338	0.229669	−	0.973269i	$-0.426236\pi$
$839$	13.3050	0.459338	0.229669	+	0.973269i	$0.426236\pi$
$840$	0	0
$841$	−9.00000	−0.310345
$842$	0	0
$843$	3.24109	0.111629
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−46.4326	−1.59544
$848$	0	0
$849$	−37.8885	−1.30033
$850$	0	0
$851$	50.8328	1.74253
$852$	0	0
$853$	10.4884	0.359115	0.179558	−	0.983747i	$-0.442533\pi$
$853$	10.4884	0.359115	0.179558	+	0.983747i	$0.442533\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	40.8059	1.39390	0.696951	−	0.717119i	$-0.254540\pi$
$857$	40.8059	1.39390	0.696951	+	0.717119i	$0.254540\pi$
$858$	0	0
$859$	−40.0000	−1.36478	−0.682391	−	0.730987i	$-0.739060\pi$
$859$	−40.0000	−1.36478	−0.682391	+	0.730987i	$0.739060\pi$
$860$	0	0
$861$	38.8328	1.32342
$862$	0	0
$863$	20.5942	0.701035	0.350518	−	0.936556i	$-0.386006\pi$
$863$	20.5942	0.701035	0.350518	+	0.936556i	$0.386006\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−36.2294	−1.23041
$868$	0	0
$869$	59.7771	2.02780
$870$	0	0
$871$	−6.54102	−0.221634
$872$	0	0
$873$	9.94820	0.336696
$874$	0	0
$875$	0	0
$876$	0	0
$877$	4.44897	0.150231	0.0751155	−	0.997175i	$-0.476067\pi$
$877$	4.44897	0.150231	0.0751155	+	0.997175i	$0.476067\pi$
$878$	0	0
$879$	−15.7082	−0.529825
$880$	0	0
$881$	−30.6525	−1.03271	−0.516354	−	0.856375i	$-0.672711\pi$
$881$	−30.6525	−1.03271	−0.516354	+	0.856375i	$0.672711\pi$
$882$	0	0
$883$	14.9675	0.503695	0.251848	−	0.967767i	$-0.418962\pi$
$883$	14.9675	0.503695	0.251848	+	0.967767i	$0.418962\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−36.3268	−1.21973	−0.609867	−	0.792504i	$-0.708777\pi$
$887$	−36.3268	−1.21973	−0.609867	+	0.792504i	$0.708777\pi$
$888$	0	0
$889$	−28.5836	−0.958663
$890$	0	0
$891$	−56.0689	−1.87838
$892$	0	0
$893$	−8.48528	−0.283949
$894$	0	0
$895$	0	0
$896$	0	0
$897$	68.3923	2.28355
$898$	0	0
$899$	17.8885	0.596616
$900$	0	0
$901$	7.41641	0.247076
$902$	0	0
$903$	54.9179	1.82755
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−8.86784	−0.294452	−0.147226	−	0.989103i	$-0.547034\pi$
$907$	−8.86784	−0.294452	−0.147226	+	0.989103i	$0.547034\pi$
$908$	0	0
$909$	1.70820	0.0566575
$910$	0	0
$911$	37.5279	1.24335	0.621677	−	0.783274i	$-0.286452\pi$
$911$	37.5279	1.24335	0.621677	+	0.783274i	$0.286452\pi$
$912$	0	0
$913$	27.4589	0.908759
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	26.8328	0.885133	0.442566	−	0.896736i	$-0.354068\pi$
$919$	26.8328	0.885133	0.442566	+	0.896736i	$0.354068\pi$
$920$	0	0
$921$	−9.23607	−0.304339
$922$	0	0
$923$	6.16693	0.202987
$924$	0	0
$925$	0	0
$926$	0	0
$927$	34.3237	1.12734
$928$	0	0
$929$	16.4721	0.540433	0.270217	−	0.962800i	$-0.412905\pi$
$929$	16.4721	0.540433	0.270217	+	0.962800i	$0.412905\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	−8.48528	−0.277796
$934$	0	0
$935$	0	0
$936$	0	0
$937$	51.6768	1.68821	0.844104	−	0.536180i	$-0.180133\pi$
$937$	51.6768	1.68821	0.844104	+	0.536180i	$0.180133\pi$
$938$	0	0
$939$	62.8328	2.05047
$940$	0	0
$941$	−21.0557	−0.686397	−0.343199	−	0.939263i	$-0.611510\pi$
$941$	−21.0557	−0.686397	−0.343199	+	0.939263i	$0.611510\pi$
$942$	0	0
$943$	−44.4295	−1.44682
$944$	0	0
$945$	0	0
$946$	0	0
$947$	28.6969	0.932525	0.466263	−	0.884646i	$-0.345600\pi$
$947$	28.6969	0.932525	0.466263	+	0.884646i	$0.345600\pi$
$948$	0	0
$949$	−55.4164	−1.79889
$950$	0	0
$951$	30.1803	0.978665
$952$	0	0
$953$	−27.0764	−0.877090	−0.438545	−	0.898709i	$-0.644506\pi$
$953$	−27.0764	−0.877090	−0.438545	+	0.898709i	$0.644506\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	53.5825	1.73208
$958$	0	0
$959$	3.05573	0.0986746
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	36.7394	1.18391
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−16.5579	−0.532466	−0.266233	−	0.963909i	$-0.585779\pi$
$967$	−16.5579	−0.532466	−0.266233	+	0.963909i	$0.585779\pi$
$968$	0	0
$969$	−2.47214	−0.0794164
$970$	0	0
$971$	−4.36068	−0.139941	−0.0699704	−	0.997549i	$-0.522290\pi$
$971$	−4.36068	−0.139941	−0.0699704	+	0.997549i	$0.522290\pi$
$972$	0	0
$973$	44.4295	1.42434
$974$	0	0
$975$	0	0
$976$	0	0
$977$	47.1304	1.50784	0.753918	−	0.656969i	$-0.228162\pi$
$977$	47.1304	1.50784	0.753918	+	0.656969i	$0.228162\pi$
$978$	0	0
$979$	−78.2492	−2.50086
$980$	0	0
$981$	4.47214	0.142784
$982$	0	0
$983$	33.4009	1.06532	0.532662	−	0.846328i	$-0.321192\pi$
$983$	33.4009	1.06532	0.532662	+	0.846328i	$0.321192\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	54.9179	1.74806
$988$	0	0
$989$	−62.8328	−1.99797
$990$	0	0
$991$	14.8328	0.471180	0.235590	−	0.971853i	$-0.424298\pi$
$991$	14.8328	0.471180	0.235590	+	0.971853i	$0.424298\pi$
$992$	0	0
$993$	44.4295	1.40993
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−30.7000	−0.972280	−0.486140	−	0.873881i	$-0.661595\pi$
$997$	−30.7000	−0.972280	−0.486140	+	0.873881i	$0.661595\pi$
$998$	0	0
$999$	−12.0000	−0.379663

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7600.2.a.ce.1.4		4
4.3	odd	2		1900.2.a.j.1.1		4
5.2	odd	4		1520.2.d.f.609.1		4
5.3	odd	4		1520.2.d.f.609.4		4
5.4	even	2	inner	7600.2.a.ce.1.1		4
20.3	even	4		380.2.c.a.229.1	✓	4
20.7	even	4		380.2.c.a.229.4	yes	4
20.19	odd	2		1900.2.a.j.1.4		4
60.23	odd	4		3420.2.f.a.1369.3		4
60.47	odd	4		3420.2.f.a.1369.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.2.c.a.229.1	✓	4	20.3	even	4
380.2.c.a.229.4	yes	4	20.7	even	4
1520.2.d.f.609.1		4	5.2	odd	4
1520.2.d.f.609.4		4	5.3	odd	4
1900.2.a.j.1.1		4	4.3	odd	2
1900.2.a.j.1.4		4	20.19	odd	2
3420.2.f.a.1369.3		4	60.23	odd	4
3420.2.f.a.1369.4		4	60.47	odd	4
7600.2.a.ce.1.1		4	5.4	even	2	inner
7600.2.a.ce.1.4		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 7600 = 2^{4} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7600.a (trivial)

\(f(q)\)	\(=\)	\(q+2.28825 q^{3} -2.82843 q^{7} +2.23607 q^{9} +O(q^{10})\)	\(q+2.28825 q^{3} -2.82843 q^{7} +2.23607 q^{9} +5.23607 q^{11} +4.03631 q^{13} -1.08036 q^{17} +1.00000 q^{19} -6.47214 q^{21} +7.40492 q^{23} -1.74806 q^{27} +4.47214 q^{29} +4.00000 q^{31} +11.9814 q^{33} +6.86474 q^{37} +9.23607 q^{39} -6.00000 q^{41} -8.48528 q^{43} -8.48528 q^{47} +1.00000 q^{49} -2.47214 q^{51} -6.86474 q^{53} +2.28825 q^{57} +10.4721 q^{59} +1.70820 q^{61} -6.32456 q^{63} -1.62054 q^{67} +16.9443 q^{69} +1.52786 q^{71} -13.7295 q^{73} -14.8098 q^{77} +11.4164 q^{79} -10.7082 q^{81} +5.24419 q^{83} +10.2333 q^{87} -14.9443 q^{89} -11.4164 q^{91} +9.15298 q^{93} +4.44897 q^{97} +11.7082 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 12 q^{11} + 4 q^{19} - 8 q^{21} + 16 q^{31} + 28 q^{39} - 24 q^{41} + 4 q^{49} + 8 q^{51} + 24 q^{59} - 20 q^{61} + 32 q^{69} + 24 q^{71} - 8 q^{79} - 16 q^{81} - 24 q^{89} + 8 q^{91} + 20 q^{99}+O(q^{100}) \) 4 * q + 12 * q^11 + 4 * q^19 - 8 * q^21 + 16 * q^31 + 28 * q^39 - 24 * q^41 + 4 * q^49 + 8 * q^51 + 24 * q^59 - 20 * q^61 + 32 * q^69 + 24 * q^71 - 8 * q^79 - 16 * q^81 - 24 * q^89 + 8 * q^91 + 20 * q^99

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