LMFDB - Embedded newform 7600.2.a.w.1.2

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:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 760)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ 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)

$f(q)$	$=$	$q+1.23607 q^{3} -4.47214 q^{7} -1.47214 q^{9} +O(q^{10})$	$q+1.23607 q^{3} -4.47214 q^{7} -1.47214 q^{9} -3.23607 q^{13} -6.47214 q^{17} -1.00000 q^{19} -5.52786 q^{21} -2.00000 q^{23} -5.52786 q^{27} +2.00000 q^{29} +1.52786 q^{31} +4.76393 q^{37} -4.00000 q^{39} +3.52786 q^{41} +0.472136 q^{43} -12.4721 q^{47} +13.0000 q^{49} -8.00000 q^{51} +11.2361 q^{53} -1.23607 q^{57} +10.4721 q^{59} -4.47214 q^{61} +6.58359 q^{63} -1.23607 q^{67} -2.47214 q^{69} +1.52786 q^{71} +6.47214 q^{73} +6.47214 q^{79} -2.41641 q^{81} +14.9443 q^{83} +2.47214 q^{87} -6.94427 q^{89} +14.4721 q^{91} +1.88854 q^{93} +4.18034 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 6 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 6 * q^9	$ 2 q - 2 q^{3} + 6 q^{9} - 2 q^{13} - 4 q^{17} - 2 q^{19} - 20 q^{21} - 4 q^{23} - 20 q^{27} + 4 q^{29} + 12 q^{31} + 14 q^{37} - 8 q^{39} + 16 q^{41} - 8 q^{43} - 16 q^{47} + 26 q^{49} - 16 q^{51} + 18 q^{53} + 2 q^{57} + 12 q^{59} + 40 q^{63} + 2 q^{67} + 4 q^{69} + 12 q^{71} + 4 q^{73} + 4 q^{79} + 22 q^{81} + 12 q^{83} - 4 q^{87} + 4 q^{89} + 20 q^{91} - 32 q^{93} - 14 q^{97}+O(q^{100}) $ 2 * q - 2 * q^3 + 6 * q^9 - 2 * q^13 - 4 * q^17 - 2 * q^19 - 20 * q^21 - 4 * q^23 - 20 * q^27 + 4 * q^29 + 12 * q^31 + 14 * q^37 - 8 * q^39 + 16 * q^41 - 8 * q^43 - 16 * q^47 + 26 * q^49 - 16 * q^51 + 18 * q^53 + 2 * q^57 + 12 * q^59 + 40 * q^63 + 2 * q^67 + 4 * q^69 + 12 * q^71 + 4 * q^73 + 4 * q^79 + 22 * q^81 + 12 * q^83 - 4 * q^87 + 4 * q^89 + 20 * q^91 - 32 * q^93 - 14 * 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
$3$	1.23607	0.713644	0.356822	−	0.934172i	$-0.383860\pi$
$3$	1.23607	0.713644	0.356822	+	0.934172i	$0.383860\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4.47214	−1.69031	−0.845154	−	0.534522i	$-0.820491\pi$
$7$	−4.47214	−1.69031	−0.845154	+	0.534522i	$0.820491\pi$
$8$	0	0
$9$	−1.47214	−0.490712
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	−3.23607	−0.897524	−0.448762	−	0.893651i	$-0.648135\pi$
$13$	−3.23607	−0.897524	−0.448762	+	0.893651i	$0.648135\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−6.47214	−1.56972	−0.784862	−	0.619671i	$-0.787266\pi$
$17$	−6.47214	−1.56972	−0.784862	+	0.619671i	$0.787266\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−5.52786	−1.20628
$22$	0	0
$23$	−2.00000	−0.417029	−0.208514	−	0.978019i	$-0.566863\pi$
$23$	−2.00000	−0.417029	−0.208514	+	0.978019i	$0.566863\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−5.52786	−1.06384
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	1.52786	0.274412	0.137206	−	0.990543i	$-0.456188\pi$
$31$	1.52786	0.274412	0.137206	+	0.990543i	$0.456188\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	4.76393	0.783186	0.391593	−	0.920139i	$-0.371924\pi$
$37$	4.76393	0.783186	0.391593	+	0.920139i	$0.371924\pi$
$38$	0	0
$39$	−4.00000	−0.640513
$40$	0	0
$41$	3.52786	0.550960	0.275480	−	0.961307i	$-0.411163\pi$
$41$	3.52786	0.550960	0.275480	+	0.961307i	$0.411163\pi$
$42$	0	0
$43$	0.472136	0.0720001	0.0360000	−	0.999352i	$-0.488538\pi$
$43$	0.472136	0.0720001	0.0360000	+	0.999352i	$0.488538\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−12.4721	−1.81925	−0.909624	−	0.415433i	$-0.863630\pi$
$47$	−12.4721	−1.81925	−0.909624	+	0.415433i	$0.863630\pi$
$48$	0	0
$49$	13.0000	1.85714
$50$	0	0
$51$	−8.00000	−1.12022
$52$	0	0
$53$	11.2361	1.54339	0.771696	−	0.635991i	$-0.219409\pi$
$53$	11.2361	1.54339	0.771696	+	0.635991i	$0.219409\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−1.23607	−0.163721
$58$	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
$61$	−4.47214	−0.572598	−0.286299	−	0.958140i	$-0.592425\pi$
$61$	−4.47214	−0.572598	−0.286299	+	0.958140i	$0.592425\pi$
$62$	0	0
$63$	6.58359	0.829455
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−1.23607	−0.151010	−0.0755049	−	0.997145i	$-0.524057\pi$
$67$	−1.23607	−0.151010	−0.0755049	+	0.997145i	$0.524057\pi$
$68$	0	0
$69$	−2.47214	−0.297610
$70$	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
$73$	6.47214	0.757506	0.378753	−	0.925498i	$-0.376353\pi$
$73$	6.47214	0.757506	0.378753	+	0.925498i	$0.376353\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	6.47214	0.728172	0.364086	−	0.931365i	$-0.381381\pi$
$79$	6.47214	0.728172	0.364086	+	0.931365i	$0.381381\pi$
$80$	0	0
$81$	−2.41641	−0.268490
$82$	0	0
$83$	14.9443	1.64035	0.820173	−	0.572115i	$-0.193877\pi$
$83$	14.9443	1.64035	0.820173	+	0.572115i	$0.193877\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	2.47214	0.265041
$88$	0	0
$89$	−6.94427	−0.736091	−0.368046	−	0.929808i	$-0.619973\pi$
$89$	−6.94427	−0.736091	−0.368046	+	0.929808i	$0.619973\pi$
$90$	0	0
$91$	14.4721	1.51709
$92$	0	0
$93$	1.88854	0.195833
$94$	0	0
$95$	0	0
$96$	0	0
$97$	4.18034	0.424449	0.212225	−	0.977221i	$-0.431929\pi$
$97$	4.18034	0.424449	0.212225	+	0.977221i	$0.431929\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	8.47214	0.843009	0.421505	−	0.906826i	$-0.361502\pi$
$101$	8.47214	0.843009	0.421505	+	0.906826i	$0.361502\pi$
$102$	0	0
$103$	−17.2361	−1.69832	−0.849160	−	0.528135i	$-0.822891\pi$
$103$	−17.2361	−1.69832	−0.849160	+	0.528135i	$0.822891\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−7.70820	−0.745180	−0.372590	−	0.927996i	$-0.621530\pi$
$107$	−7.70820	−0.745180	−0.372590	+	0.927996i	$0.621530\pi$
$108$	0	0
$109$	13.4164	1.28506	0.642529	−	0.766261i	$-0.277885\pi$
$109$	13.4164	1.28506	0.642529	+	0.766261i	$0.277885\pi$
$110$	0	0
$111$	5.88854	0.558916
$112$	0	0
$113$	15.2361	1.43329	0.716644	−	0.697439i	$-0.245677\pi$
$113$	15.2361	1.43329	0.716644	+	0.697439i	$0.245677\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	4.76393	0.440426
$118$	0	0
$119$	28.9443	2.65332
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	4.36068	0.393189
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−6.76393	−0.600202	−0.300101	−	0.953907i	$-0.597020\pi$
$127$	−6.76393	−0.600202	−0.300101	+	0.953907i	$0.597020\pi$
$128$	0	0
$129$	0.583592	0.0513824
$130$	0	0
$131$	8.94427	0.781465	0.390732	−	0.920504i	$-0.372222\pi$
$131$	8.94427	0.781465	0.390732	+	0.920504i	$0.372222\pi$
$132$	0	0
$133$	4.47214	0.387783
$134$	0	0
$135$	0	0
$136$	0	0
$137$	11.4164	0.975370	0.487685	−	0.873020i	$-0.337842\pi$
$137$	11.4164	0.975370	0.487685	+	0.873020i	$0.337842\pi$
$138$	0	0
$139$	−4.94427	−0.419368	−0.209684	−	0.977769i	$-0.567243\pi$
$139$	−4.94427	−0.419368	−0.209684	+	0.977769i	$0.567243\pi$
$140$	0	0
$141$	−15.4164	−1.29830
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	16.0689	1.32534
$148$	0	0
$149$	−21.4164	−1.75450	−0.877250	−	0.480033i	$-0.840625\pi$
$149$	−21.4164	−1.75450	−0.877250	+	0.480033i	$0.840625\pi$
$150$	0	0
$151$	−17.8885	−1.45575	−0.727875	−	0.685710i	$-0.759492\pi$
$151$	−17.8885	−1.45575	−0.727875	+	0.685710i	$0.759492\pi$
$152$	0	0
$153$	9.52786	0.770282
$154$	0	0
$155$	0	0
$156$	0	0
$157$	15.4164	1.23036	0.615182	−	0.788385i	$-0.289083\pi$
$157$	15.4164	1.23036	0.615182	+	0.788385i	$0.289083\pi$
$158$	0	0
$159$	13.8885	1.10143
$160$	0	0
$161$	8.94427	0.704907
$162$	0	0
$163$	8.47214	0.663589	0.331794	−	0.943352i	$-0.392346\pi$
$163$	8.47214	0.663589	0.331794	+	0.943352i	$0.392346\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−10.1803	−0.787778	−0.393889	−	0.919158i	$-0.628871\pi$
$167$	−10.1803	−0.787778	−0.393889	+	0.919158i	$0.628871\pi$
$168$	0	0
$169$	−2.52786	−0.194451
$170$	0	0
$171$	1.47214	0.112577
$172$	0	0
$173$	−14.6525	−1.11401	−0.557004	−	0.830510i	$-0.688049\pi$
$173$	−14.6525	−1.11401	−0.557004	+	0.830510i	$0.688049\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	12.9443	0.972951
$178$	0	0
$179$	5.52786	0.413172	0.206586	−	0.978428i	$-0.433765\pi$
$179$	5.52786	0.413172	0.206586	+	0.978428i	$0.433765\pi$
$180$	0	0
$181$	23.8885	1.77562	0.887811	−	0.460209i	$-0.152225\pi$
$181$	23.8885	1.77562	0.887811	+	0.460209i	$0.152225\pi$
$182$	0	0
$183$	−5.52786	−0.408631
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	24.7214	1.79821
$190$	0	0
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	0	0
$193$	−5.70820	−0.410886	−0.205443	−	0.978669i	$-0.565863\pi$
$193$	−5.70820	−0.410886	−0.205443	+	0.978669i	$0.565863\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−0.944272	−0.0672766	−0.0336383	−	0.999434i	$-0.510709\pi$
$197$	−0.944272	−0.0672766	−0.0336383	+	0.999434i	$0.510709\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	−1.52786	−0.107767
$202$	0	0
$203$	−8.94427	−0.627765
$204$	0	0
$205$	0	0
$206$	0	0
$207$	2.94427	0.204641
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−7.05573	−0.485736	−0.242868	−	0.970059i	$-0.578088\pi$
$211$	−7.05573	−0.485736	−0.242868	+	0.970059i	$0.578088\pi$
$212$	0	0
$213$	1.88854	0.129401
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−6.83282	−0.463842
$218$	0	0
$219$	8.00000	0.540590
$220$	0	0
$221$	20.9443	1.40886
$222$	0	0
$223$	−25.2361	−1.68993	−0.844966	−	0.534820i	$-0.820379\pi$
$223$	−25.2361	−1.68993	−0.844966	+	0.534820i	$0.820379\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	10.1803	0.675693	0.337846	−	0.941201i	$-0.390302\pi$
$227$	10.1803	0.675693	0.337846	+	0.941201i	$0.390302\pi$
$228$	0	0
$229$	17.4164	1.15091	0.575454	−	0.817834i	$-0.304825\pi$
$229$	17.4164	1.15091	0.575454	+	0.817834i	$0.304825\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	12.9443	0.848007	0.424004	−	0.905660i	$-0.360624\pi$
$233$	12.9443	0.848007	0.424004	+	0.905660i	$0.360624\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	8.00000	0.519656
$238$	0	0
$239$	−28.9443	−1.87225	−0.936125	−	0.351668i	$-0.885614\pi$
$239$	−28.9443	−1.87225	−0.936125	+	0.351668i	$0.885614\pi$
$240$	0	0
$241$	21.4164	1.37955	0.689776	−	0.724023i	$-0.257709\pi$
$241$	21.4164	1.37955	0.689776	+	0.724023i	$0.257709\pi$
$242$	0	0
$243$	13.5967	0.872232
$244$	0	0
$245$	0	0
$246$	0	0
$247$	3.23607	0.205906
$248$	0	0
$249$	18.4721	1.17062
$250$	0	0
$251$	−4.00000	−0.252478	−0.126239	−	0.992000i	$-0.540291\pi$
$251$	−4.00000	−0.252478	−0.126239	+	0.992000i	$0.540291\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−10.2918	−0.641985	−0.320992	−	0.947082i	$-0.604016\pi$
$257$	−10.2918	−0.641985	−0.320992	+	0.947082i	$0.604016\pi$
$258$	0	0
$259$	−21.3050	−1.32383
$260$	0	0
$261$	−2.94427	−0.182246
$262$	0	0
$263$	−11.8885	−0.733079	−0.366540	−	0.930402i	$-0.619458\pi$
$263$	−11.8885	−0.733079	−0.366540	+	0.930402i	$0.619458\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−8.58359	−0.525307
$268$	0	0
$269$	−25.4164	−1.54967	−0.774833	−	0.632166i	$-0.782166\pi$
$269$	−25.4164	−1.54967	−0.774833	+	0.632166i	$0.782166\pi$
$270$	0	0
$271$	−0.944272	−0.0573604	−0.0286802	−	0.999589i	$-0.509130\pi$
$271$	−0.944272	−0.0573604	−0.0286802	+	0.999589i	$0.509130\pi$
$272$	0	0
$273$	17.8885	1.08266
$274$	0	0
$275$	0	0
$276$	0	0
$277$	26.4721	1.59056	0.795278	−	0.606245i	$-0.207325\pi$
$277$	26.4721	1.59056	0.795278	+	0.606245i	$0.207325\pi$
$278$	0	0
$279$	−2.24922	−0.134657
$280$	0	0
$281$	21.4164	1.27760	0.638798	−	0.769375i	$-0.279432\pi$
$281$	21.4164	1.27760	0.638798	+	0.769375i	$0.279432\pi$
$282$	0	0
$283$	19.8885	1.18225	0.591126	−	0.806580i	$-0.298684\pi$
$283$	19.8885	1.18225	0.591126	+	0.806580i	$0.298684\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−15.7771	−0.931292
$288$	0	0
$289$	24.8885	1.46403
$290$	0	0
$291$	5.16718	0.302906
$292$	0	0
$293$	−17.7082	−1.03452	−0.517262	−	0.855827i	$-0.673049\pi$
$293$	−17.7082	−1.03452	−0.517262	+	0.855827i	$0.673049\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	6.47214	0.374293
$300$	0	0
$301$	−2.11146	−0.121702
$302$	0	0
$303$	10.4721	0.601608
$304$	0	0
$305$	0	0
$306$	0	0
$307$	13.2361	0.755422	0.377711	−	0.925923i	$-0.376711\pi$
$307$	13.2361	0.755422	0.377711	+	0.925923i	$0.376711\pi$
$308$	0	0
$309$	−21.3050	−1.21200
$310$	0	0
$311$	20.0000	1.13410	0.567048	−	0.823685i	$-0.308085\pi$
$311$	20.0000	1.13410	0.567048	+	0.823685i	$0.308085\pi$
$312$	0	0
$313$	−16.0000	−0.904373	−0.452187	−	0.891923i	$-0.649356\pi$
$313$	−16.0000	−0.904373	−0.452187	+	0.891923i	$0.649356\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−21.1246	−1.18648	−0.593238	−	0.805027i	$-0.702151\pi$
$317$	−21.1246	−1.18648	−0.593238	+	0.805027i	$0.702151\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−9.52786	−0.531794
$322$	0	0
$323$	6.47214	0.360119
$324$	0	0
$325$	0	0
$326$	0	0
$327$	16.5836	0.917075
$328$	0	0
$329$	55.7771	3.07509
$330$	0	0
$331$	32.9443	1.81078	0.905390	−	0.424580i	$-0.139578\pi$
$331$	32.9443	1.81078	0.905390	+	0.424580i	$0.139578\pi$
$332$	0	0
$333$	−7.01316	−0.384319
$334$	0	0
$335$	0	0
$336$	0	0
$337$	31.2361	1.70154	0.850769	−	0.525541i	$-0.176137\pi$
$337$	31.2361	1.70154	0.850769	+	0.525541i	$0.176137\pi$
$338$	0	0
$339$	18.8328	1.02286
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−26.8328	−1.44884
$344$	0	0
$345$	0	0
$346$	0	0
$347$	10.9443	0.587519	0.293760	−	0.955879i	$-0.405093\pi$
$347$	10.9443	0.587519	0.293760	+	0.955879i	$0.405093\pi$
$348$	0	0
$349$	−7.88854	−0.422264	−0.211132	−	0.977458i	$-0.567715\pi$
$349$	−7.88854	−0.422264	−0.211132	+	0.977458i	$0.567715\pi$
$350$	0	0
$351$	17.8885	0.954820
$352$	0	0
$353$	−22.4721	−1.19607	−0.598036	−	0.801470i	$-0.704052\pi$
$353$	−22.4721	−1.19607	−0.598036	+	0.801470i	$0.704052\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	35.7771	1.89352
$358$	0	0
$359$	−5.88854	−0.310785	−0.155393	−	0.987853i	$-0.549664\pi$
$359$	−5.88854	−0.310785	−0.155393	+	0.987853i	$0.549664\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−13.5967	−0.713644
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−30.3607	−1.58481	−0.792407	−	0.609992i	$-0.791172\pi$
$367$	−30.3607	−1.58481	−0.792407	+	0.609992i	$0.791172\pi$
$368$	0	0
$369$	−5.19350	−0.270363
$370$	0	0
$371$	−50.2492	−2.60881
$372$	0	0
$373$	21.1246	1.09379	0.546895	−	0.837201i	$-0.315810\pi$
$373$	21.1246	1.09379	0.546895	+	0.837201i	$0.315810\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−6.47214	−0.333332
$378$	0	0
$379$	−21.8885	−1.12434	−0.562169	−	0.827022i	$-0.690033\pi$
$379$	−21.8885	−1.12434	−0.562169	+	0.827022i	$0.690033\pi$
$380$	0	0
$381$	−8.36068	−0.428331
$382$	0	0
$383$	−20.6525	−1.05529	−0.527646	−	0.849464i	$-0.676925\pi$
$383$	−20.6525	−1.05529	−0.527646	+	0.849464i	$0.676925\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−0.695048	−0.0353313
$388$	0	0
$389$	37.7771	1.91537	0.957687	−	0.287811i	$-0.0929275\pi$
$389$	37.7771	1.91537	0.957687	+	0.287811i	$0.0929275\pi$
$390$	0	0
$391$	12.9443	0.654620
$392$	0	0
$393$	11.0557	0.557688
$394$	0	0
$395$	0	0
$396$	0	0
$397$	23.4164	1.17524	0.587618	−	0.809139i	$-0.300066\pi$
$397$	23.4164	1.17524	0.587618	+	0.809139i	$0.300066\pi$
$398$	0	0
$399$	5.52786	0.276739
$400$	0	0
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	0	0
$403$	−4.94427	−0.246292
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−24.4721	−1.21007	−0.605035	−	0.796199i	$-0.706841\pi$
$409$	−24.4721	−1.21007	−0.605035	+	0.796199i	$0.706841\pi$
$410$	0	0
$411$	14.1115	0.696067
$412$	0	0
$413$	−46.8328	−2.30449
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−6.11146	−0.299279
$418$	0	0
$419$	29.8885	1.46015	0.730075	−	0.683367i	$-0.239485\pi$
$419$	29.8885	1.46015	0.730075	+	0.683367i	$0.239485\pi$
$420$	0	0
$421$	20.4721	0.997751	0.498875	−	0.866674i	$-0.333747\pi$
$421$	20.4721	0.997751	0.498875	+	0.866674i	$0.333747\pi$
$422$	0	0
$423$	18.3607	0.892727
$424$	0	0
$425$	0	0
$426$	0	0
$427$	20.0000	0.967868
$428$	0	0
$429$	0	0
$430$	0	0
$431$	17.8885	0.861661	0.430830	−	0.902433i	$-0.358221\pi$
$431$	17.8885	0.861661	0.430830	+	0.902433i	$0.358221\pi$
$432$	0	0
$433$	12.1803	0.585350	0.292675	−	0.956212i	$-0.405455\pi$
$433$	12.1803	0.585350	0.292675	+	0.956212i	$0.405455\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.00000	0.0956730
$438$	0	0
$439$	−22.4721	−1.07254	−0.536268	−	0.844048i	$-0.680166\pi$
$439$	−22.4721	−1.07254	−0.536268	+	0.844048i	$0.680166\pi$
$440$	0	0
$441$	−19.1378	−0.911322
$442$	0	0
$443$	−23.8885	−1.13498	−0.567489	−	0.823381i	$-0.692085\pi$
$443$	−23.8885	−1.13498	−0.567489	+	0.823381i	$0.692085\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−26.4721	−1.25209
$448$	0	0
$449$	9.41641	0.444388	0.222194	−	0.975003i	$-0.428678\pi$
$449$	9.41641	0.444388	0.222194	+	0.975003i	$0.428678\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−22.1115	−1.03889
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−33.8885	−1.58524	−0.792620	−	0.609716i	$-0.791283\pi$
$457$	−33.8885	−1.58524	−0.792620	+	0.609716i	$0.791283\pi$
$458$	0	0
$459$	35.7771	1.66993
$460$	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
$463$	31.8885	1.48199	0.740993	−	0.671513i	$-0.234355\pi$
$463$	31.8885	1.48199	0.740993	+	0.671513i	$0.234355\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	6.00000	0.277647	0.138823	−	0.990317i	$-0.455668\pi$
$467$	6.00000	0.277647	0.138823	+	0.990317i	$0.455668\pi$
$468$	0	0
$469$	5.52786	0.255253
$470$	0	0
$471$	19.0557	0.878042
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−16.5410	−0.757361
$478$	0	0
$479$	24.9443	1.13973	0.569866	−	0.821737i	$-0.306995\pi$
$479$	24.9443	1.13973	0.569866	+	0.821737i	$0.306995\pi$
$480$	0	0
$481$	−15.4164	−0.702928
$482$	0	0
$483$	11.0557	0.503053
$484$	0	0
$485$	0	0
$486$	0	0
$487$	12.6525	0.573338	0.286669	−	0.958030i	$-0.407452\pi$
$487$	12.6525	0.573338	0.286669	+	0.958030i	$0.407452\pi$
$488$	0	0
$489$	10.4721	0.473566
$490$	0	0
$491$	−7.05573	−0.318421	−0.159210	−	0.987245i	$-0.550895\pi$
$491$	−7.05573	−0.318421	−0.159210	+	0.987245i	$0.550895\pi$
$492$	0	0
$493$	−12.9443	−0.582981
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−6.83282	−0.306494
$498$	0	0
$499$	14.8328	0.664008	0.332004	−	0.943278i	$-0.392275\pi$
$499$	14.8328	0.664008	0.332004	+	0.943278i	$0.392275\pi$
$500$	0	0
$501$	−12.5836	−0.562193
$502$	0	0
$503$	−37.7771	−1.68440	−0.842199	−	0.539168i	$-0.818739\pi$
$503$	−37.7771	−1.68440	−0.842199	+	0.539168i	$0.818739\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−3.12461	−0.138769
$508$	0	0
$509$	2.00000	0.0886484	0.0443242	−	0.999017i	$-0.485887\pi$
$509$	2.00000	0.0886484	0.0443242	+	0.999017i	$0.485887\pi$
$510$	0	0
$511$	−28.9443	−1.28042
$512$	0	0
$513$	5.52786	0.244061
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−18.1115	−0.795005
$520$	0	0
$521$	10.0000	0.438108	0.219054	−	0.975713i	$-0.429703\pi$
$521$	10.0000	0.438108	0.219054	+	0.975713i	$0.429703\pi$
$522$	0	0
$523$	−21.2361	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$523$	−21.2361	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−9.88854	−0.430752
$528$	0	0
$529$	−19.0000	−0.826087
$530$	0	0
$531$	−15.4164	−0.669015
$532$	0	0
$533$	−11.4164	−0.494500
$534$	0	0
$535$	0	0
$536$	0	0
$537$	6.83282	0.294858
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−30.3607	−1.30531	−0.652654	−	0.757656i	$-0.726344\pi$
$541$	−30.3607	−1.30531	−0.652654	+	0.757656i	$0.726344\pi$
$542$	0	0
$543$	29.5279	1.26716
$544$	0	0
$545$	0	0
$546$	0	0
$547$	0.291796	0.0124763	0.00623815	−	0.999981i	$-0.498014\pi$
$547$	0.291796	0.0124763	0.00623815	+	0.999981i	$0.498014\pi$
$548$	0	0
$549$	6.58359	0.280981
$550$	0	0
$551$	−2.00000	−0.0852029
$552$	0	0
$553$	−28.9443	−1.23084
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−39.4164	−1.67013	−0.835063	−	0.550154i	$-0.814569\pi$
$557$	−39.4164	−1.67013	−0.835063	+	0.550154i	$0.814569\pi$
$558$	0	0
$559$	−1.52786	−0.0646218
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−29.2361	−1.23215	−0.616077	−	0.787686i	$-0.711279\pi$
$563$	−29.2361	−1.23215	−0.616077	+	0.787686i	$0.711279\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	10.8065	0.453831
$568$	0	0
$569$	12.4721	0.522859	0.261430	−	0.965223i	$-0.415806\pi$
$569$	12.4721	0.522859	0.261430	+	0.965223i	$0.415806\pi$
$570$	0	0
$571$	3.05573	0.127878	0.0639391	−	0.997954i	$-0.479634\pi$
$571$	3.05573	0.127878	0.0639391	+	0.997954i	$0.479634\pi$
$572$	0	0
$573$	−9.88854	−0.413100
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−20.9443	−0.871921	−0.435961	−	0.899966i	$-0.643591\pi$
$577$	−20.9443	−0.871921	−0.435961	+	0.899966i	$0.643591\pi$
$578$	0	0
$579$	−7.05573	−0.293226
$580$	0	0
$581$	−66.8328	−2.77269
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	25.7771	1.06393	0.531967	−	0.846765i	$-0.321453\pi$
$587$	25.7771	1.06393	0.531967	+	0.846765i	$0.321453\pi$
$588$	0	0
$589$	−1.52786	−0.0629545
$590$	0	0
$591$	−1.16718	−0.0480115
$592$	0	0
$593$	19.0557	0.782525	0.391262	−	0.920279i	$-0.372038\pi$
$593$	19.0557	0.782525	0.391262	+	0.920279i	$0.372038\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	38.4721	1.57193	0.785964	−	0.618272i	$-0.212167\pi$
$599$	38.4721	1.57193	0.785964	+	0.618272i	$0.212167\pi$
$600$	0	0
$601$	42.3607	1.72793	0.863964	−	0.503553i	$-0.167974\pi$
$601$	42.3607	1.72793	0.863964	+	0.503553i	$0.167974\pi$
$602$	0	0
$603$	1.81966	0.0741023
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−15.1246	−0.613889	−0.306945	−	0.951727i	$-0.599307\pi$
$607$	−15.1246	−0.613889	−0.306945	+	0.951727i	$0.599307\pi$
$608$	0	0
$609$	−11.0557	−0.448001
$610$	0	0
$611$	40.3607	1.63282
$612$	0	0
$613$	39.4164	1.59201	0.796007	−	0.605288i	$-0.206942\pi$
$613$	39.4164	1.59201	0.796007	+	0.605288i	$0.206942\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−42.2492	−1.70089	−0.850445	−	0.526064i	$-0.823667\pi$
$617$	−42.2492	−1.70089	−0.850445	+	0.526064i	$0.823667\pi$
$618$	0	0
$619$	11.0557	0.444367	0.222184	−	0.975005i	$-0.428682\pi$
$619$	11.0557	0.444367	0.222184	+	0.975005i	$0.428682\pi$
$620$	0	0
$621$	11.0557	0.443651
$622$	0	0
$623$	31.0557	1.24422
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−30.8328	−1.22938
$630$	0	0
$631$	20.0000	0.796187	0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000	0.796187	0.398094	+	0.917345i	$0.369672\pi$
$632$	0	0
$633$	−8.72136	−0.346643
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−42.0689	−1.66683
$638$	0	0
$639$	−2.24922	−0.0889779
$640$	0	0
$641$	34.3607	1.35717	0.678583	−	0.734524i	$-0.262595\pi$
$641$	34.3607	1.35717	0.678583	+	0.734524i	$0.262595\pi$
$642$	0	0
$643$	−6.00000	−0.236617	−0.118308	−	0.992977i	$-0.537747\pi$
$643$	−6.00000	−0.236617	−0.118308	+	0.992977i	$0.537747\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−9.05573	−0.356017	−0.178009	−	0.984029i	$-0.556966\pi$
$647$	−9.05573	−0.356017	−0.178009	+	0.984029i	$0.556966\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−8.44582	−0.331018
$652$	0	0
$653$	21.5279	0.842450	0.421225	−	0.906956i	$-0.361600\pi$
$653$	21.5279	0.842450	0.421225	+	0.906956i	$0.361600\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−9.52786	−0.371717
$658$	0	0
$659$	−28.3607	−1.10478	−0.552388	−	0.833587i	$-0.686283\pi$
$659$	−28.3607	−1.10478	−0.552388	+	0.833587i	$0.686283\pi$
$660$	0	0
$661$	−27.5279	−1.07071	−0.535355	−	0.844627i	$-0.679822\pi$
$661$	−27.5279	−1.07071	−0.535355	+	0.844627i	$0.679822\pi$
$662$	0	0
$663$	25.8885	1.00543
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−4.00000	−0.154881
$668$	0	0
$669$	−31.1935	−1.20601
$670$	0	0
$671$	0	0
$672$	0	0
$673$	19.8197	0.763992	0.381996	−	0.924164i	$-0.375237\pi$
$673$	19.8197	0.763992	0.381996	+	0.924164i	$0.375237\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	13.1246	0.504420	0.252210	−	0.967673i	$-0.418843\pi$
$677$	13.1246	0.504420	0.252210	+	0.967673i	$0.418843\pi$
$678$	0	0
$679$	−18.6950	−0.717450
$680$	0	0
$681$	12.5836	0.482204
$682$	0	0
$683$	13.8197	0.528795	0.264397	−	0.964414i	$-0.414827\pi$
$683$	13.8197	0.528795	0.264397	+	0.964414i	$0.414827\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	21.5279	0.821339
$688$	0	0
$689$	−36.3607	−1.38523
$690$	0	0
$691$	17.8885	0.680512	0.340256	−	0.940333i	$-0.389486\pi$
$691$	17.8885	0.680512	0.340256	+	0.940333i	$0.389486\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−22.8328	−0.864855
$698$	0	0
$699$	16.0000	0.605176
$700$	0	0
$701$	23.3050	0.880216	0.440108	−	0.897945i	$-0.354940\pi$
$701$	23.3050	0.880216	0.440108	+	0.897945i	$0.354940\pi$
$702$	0	0
$703$	−4.76393	−0.179675
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−37.8885	−1.42495
$708$	0	0
$709$	0.111456	0.00418582	0.00209291	−	0.999998i	$-0.499334\pi$
$709$	0.111456	0.00418582	0.00209291	+	0.999998i	$0.499334\pi$
$710$	0	0
$711$	−9.52786	−0.357323
$712$	0	0
$713$	−3.05573	−0.114438
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−35.7771	−1.33612
$718$	0	0
$719$	45.8885	1.71135	0.855677	−	0.517510i	$-0.173141\pi$
$719$	45.8885	1.71135	0.855677	+	0.517510i	$0.173141\pi$
$720$	0	0
$721$	77.0820	2.87069
$722$	0	0
$723$	26.4721	0.984509
$724$	0	0
$725$	0	0
$726$	0	0
$727$	45.4164	1.68440	0.842201	−	0.539164i	$-0.181260\pi$
$727$	45.4164	1.68440	0.842201	+	0.539164i	$0.181260\pi$
$728$	0	0
$729$	24.0557	0.890953
$730$	0	0
$731$	−3.05573	−0.113020
$732$	0	0
$733$	−16.9443	−0.625851	−0.312925	−	0.949778i	$-0.601309\pi$
$733$	−16.9443	−0.625851	−0.312925	+	0.949778i	$0.601309\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−4.00000	−0.147142	−0.0735712	−	0.997290i	$-0.523440\pi$
$739$	−4.00000	−0.147142	−0.0735712	+	0.997290i	$0.523440\pi$
$740$	0	0
$741$	4.00000	0.146944
$742$	0	0
$743$	39.1246	1.43534	0.717671	−	0.696382i	$-0.245208\pi$
$743$	39.1246	1.43534	0.717671	+	0.696382i	$0.245208\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−22.0000	−0.804938
$748$	0	0
$749$	34.4721	1.25958
$750$	0	0
$751$	−20.9443	−0.764267	−0.382134	−	0.924107i	$-0.624811\pi$
$751$	−20.9443	−0.764267	−0.382134	+	0.924107i	$0.624811\pi$
$752$	0	0
$753$	−4.94427	−0.180179
$754$	0	0
$755$	0	0
$756$	0	0
$757$	12.0000	0.436147	0.218074	−	0.975932i	$-0.430023\pi$
$757$	12.0000	0.436147	0.218074	+	0.975932i	$0.430023\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−0.472136	−0.0171149	−0.00855746	−	0.999963i	$-0.502724\pi$
$761$	−0.472136	−0.0171149	−0.00855746	+	0.999963i	$0.502724\pi$
$762$	0	0
$763$	−60.0000	−2.17215
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−33.8885	−1.22364
$768$	0	0
$769$	−2.58359	−0.0931667	−0.0465834	−	0.998914i	$-0.514833\pi$
$769$	−2.58359	−0.0931667	−0.0465834	+	0.998914i	$0.514833\pi$
$770$	0	0
$771$	−12.7214	−0.458149
$772$	0	0
$773$	−14.2918	−0.514040	−0.257020	−	0.966406i	$-0.582741\pi$
$773$	−14.2918	−0.514040	−0.257020	+	0.966406i	$0.582741\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−26.3344	−0.944740
$778$	0	0
$779$	−3.52786	−0.126399
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−11.0557	−0.395099
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−54.5410	−1.94418	−0.972089	−	0.234614i	$-0.924617\pi$
$787$	−54.5410	−1.94418	−0.972089	+	0.234614i	$0.924617\pi$
$788$	0	0
$789$	−14.6950	−0.523158
$790$	0	0
$791$	−68.1378	−2.42270
$792$	0	0
$793$	14.4721	0.513921
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−14.6525	−0.519017	−0.259509	−	0.965741i	$-0.583561\pi$
$797$	−14.6525	−0.519017	−0.259509	+	0.965741i	$0.583561\pi$
$798$	0	0
$799$	80.7214	2.85572
$800$	0	0
$801$	10.2229	0.361209
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−31.4164	−1.10591
$808$	0	0
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	0	0
$811$	−41.3050	−1.45041	−0.725207	−	0.688531i	$-0.758256\pi$
$811$	−41.3050	−1.45041	−0.725207	+	0.688531i	$0.758256\pi$
$812$	0	0
$813$	−1.16718	−0.0409349
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−0.472136	−0.0165179
$818$	0	0
$819$	−21.3050	−0.744455
$820$	0	0
$821$	42.9443	1.49877	0.749383	−	0.662137i	$-0.230350\pi$
$821$	42.9443	1.49877	0.749383	+	0.662137i	$0.230350\pi$
$822$	0	0
$823$	−22.5836	−0.787215	−0.393607	−	0.919279i	$-0.628773\pi$
$823$	−22.5836	−0.787215	−0.393607	+	0.919279i	$0.628773\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	38.7639	1.34795	0.673977	−	0.738752i	$-0.264585\pi$
$827$	38.7639	1.34795	0.673977	+	0.738752i	$0.264585\pi$
$828$	0	0
$829$	−9.41641	−0.327045	−0.163523	−	0.986540i	$-0.552286\pi$
$829$	−9.41641	−0.327045	−0.163523	+	0.986540i	$0.552286\pi$
$830$	0	0
$831$	32.7214	1.13509
$832$	0	0
$833$	−84.1378	−2.91520
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−8.44582	−0.291930
$838$	0	0
$839$	17.5279	0.605129	0.302565	−	0.953129i	$-0.402157\pi$
$839$	17.5279	0.605129	0.302565	+	0.953129i	$0.402157\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	26.4721	0.911749
$844$	0	0
$845$	0	0
$846$	0	0
$847$	49.1935	1.69031
$848$	0	0
$849$	24.5836	0.843707
$850$	0	0
$851$	−9.52786	−0.326611
$852$	0	0
$853$	40.9443	1.40191	0.700953	−	0.713208i	$-0.252758\pi$
$853$	40.9443	1.40191	0.700953	+	0.713208i	$0.252758\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−21.3475	−0.729218	−0.364609	−	0.931161i	$-0.618797\pi$
$857$	−21.3475	−0.729218	−0.364609	+	0.931161i	$0.618797\pi$
$858$	0	0
$859$	37.8885	1.29274	0.646370	−	0.763024i	$-0.276286\pi$
$859$	37.8885	1.29274	0.646370	+	0.763024i	$0.276286\pi$
$860$	0	0
$861$	−19.5016	−0.664611
$862$	0	0
$863$	21.2361	0.722884	0.361442	−	0.932395i	$-0.382285\pi$
$863$	21.2361	0.722884	0.361442	+	0.932395i	$0.382285\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	30.7639	1.04480
$868$	0	0
$869$	0	0
$870$	0	0
$871$	4.00000	0.135535
$872$	0	0
$873$	−6.15403	−0.208282
$874$	0	0
$875$	0	0
$876$	0	0
$877$	8.18034	0.276230	0.138115	−	0.990416i	$-0.455896\pi$
$877$	8.18034	0.276230	0.138115	+	0.990416i	$0.455896\pi$
$878$	0	0
$879$	−21.8885	−0.738282
$880$	0	0
$881$	40.2492	1.35603	0.678015	−	0.735048i	$-0.262840\pi$
$881$	40.2492	1.35603	0.678015	+	0.735048i	$0.262840\pi$
$882$	0	0
$883$	35.5279	1.19561	0.597804	−	0.801642i	$-0.296040\pi$
$883$	35.5279	1.19561	0.597804	+	0.801642i	$0.296040\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	36.2918	1.21856	0.609280	−	0.792955i	$-0.291459\pi$
$887$	36.2918	1.21856	0.609280	+	0.792955i	$0.291459\pi$
$888$	0	0
$889$	30.2492	1.01453
$890$	0	0
$891$	0	0
$892$	0	0
$893$	12.4721	0.417364
$894$	0	0
$895$	0	0
$896$	0	0
$897$	8.00000	0.267112
$898$	0	0
$899$	3.05573	0.101914
$900$	0	0
$901$	−72.7214	−2.42270
$902$	0	0
$903$	−2.60990	−0.0868521
$904$	0	0
$905$	0	0
$906$	0	0
$907$	34.1803	1.13494	0.567470	−	0.823394i	$-0.307922\pi$
$907$	34.1803	1.13494	0.567470	+	0.823394i	$0.307922\pi$
$908$	0	0
$909$	−12.4721	−0.413675
$910$	0	0
$911$	1.16718	0.0386705	0.0193353	−	0.999813i	$-0.493845\pi$
$911$	1.16718	0.0386705	0.0193353	+	0.999813i	$0.493845\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−40.0000	−1.32092
$918$	0	0
$919$	−33.8885	−1.11788	−0.558940	−	0.829208i	$-0.688792\pi$
$919$	−33.8885	−1.11788	−0.558940	+	0.829208i	$0.688792\pi$
$920$	0	0
$921$	16.3607	0.539103
$922$	0	0
$923$	−4.94427	−0.162743
$924$	0	0
$925$	0	0
$926$	0	0
$927$	25.3738	0.833386
$928$	0	0
$929$	41.7771	1.37066	0.685331	−	0.728232i	$-0.259658\pi$
$929$	41.7771	1.37066	0.685331	+	0.728232i	$0.259658\pi$
$930$	0	0
$931$	−13.0000	−0.426058
$932$	0	0
$933$	24.7214	0.809341
$934$	0	0
$935$	0	0
$936$	0	0
$937$	35.4164	1.15700	0.578502	−	0.815681i	$-0.303638\pi$
$937$	35.4164	1.15700	0.578502	+	0.815681i	$0.303638\pi$
$938$	0	0
$939$	−19.7771	−0.645401
$940$	0	0
$941$	−16.8328	−0.548734	−0.274367	−	0.961625i	$-0.588468\pi$
$941$	−16.8328	−0.548734	−0.274367	+	0.961625i	$0.588468\pi$
$942$	0	0
$943$	−7.05573	−0.229766
$944$	0	0
$945$	0	0
$946$	0	0
$947$	9.05573	0.294272	0.147136	−	0.989116i	$-0.452995\pi$
$947$	9.05573	0.294272	0.147136	+	0.989116i	$0.452995\pi$
$948$	0	0
$949$	−20.9443	−0.679880
$950$	0	0
$951$	−26.1115	−0.846722
$952$	0	0
$953$	−2.29180	−0.0742386	−0.0371193	−	0.999311i	$-0.511818\pi$
$953$	−2.29180	−0.0742386	−0.0371193	+	0.999311i	$0.511818\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−51.0557	−1.64868
$960$	0	0
$961$	−28.6656	−0.924698
$962$	0	0
$963$	11.3475	0.365669
$964$	0	0
$965$	0	0
$966$	0	0
$967$	32.8328	1.05583	0.527916	−	0.849297i	$-0.322974\pi$
$967$	32.8328	1.05583	0.527916	+	0.849297i	$0.322974\pi$
$968$	0	0
$969$	8.00000	0.256997
$970$	0	0
$971$	44.3607	1.42360	0.711801	−	0.702381i	$-0.247880\pi$
$971$	44.3607	1.42360	0.711801	+	0.702381i	$0.247880\pi$
$972$	0	0
$973$	22.1115	0.708861
$974$	0	0
$975$	0	0
$976$	0	0
$977$	3.81966	0.122202	0.0611009	−	0.998132i	$-0.480539\pi$
$977$	3.81966	0.122202	0.0611009	+	0.998132i	$0.480539\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−19.7508	−0.630594
$982$	0	0
$983$	−34.7639	−1.10880	−0.554399	−	0.832251i	$-0.687052\pi$
$983$	−34.7639	−1.10880	−0.554399	+	0.832251i	$0.687052\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	68.9443	2.19452
$988$	0	0
$989$	−0.944272	−0.0300261
$990$	0	0
$991$	−9.88854	−0.314120	−0.157060	−	0.987589i	$-0.550202\pi$
$991$	−9.88854	−0.314120	−0.157060	+	0.987589i	$0.550202\pi$
$992$	0	0
$993$	40.7214	1.29225
$994$	0	0
$995$	0	0
$996$	0	0
$997$	33.3050	1.05478	0.527389	−	0.849624i	$-0.323171\pi$
$997$	33.3050	1.05478	0.527389	+	0.849624i	$0.323171\pi$
$998$	0	0
$999$	−26.3344	−0.833183

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7600.2.a.w.1.2		2
4.3	odd	2		3800.2.a.q.1.1		2
5.2	odd	4		1520.2.d.c.609.2		4
5.3	odd	4		1520.2.d.c.609.3		4
5.4	even	2		7600.2.a.be.1.1		2
20.3	even	4		760.2.d.b.609.2	✓	4
20.7	even	4		760.2.d.b.609.3	yes	4
20.19	odd	2		3800.2.a.k.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
760.2.d.b.609.2	✓	4	20.3	even	4
760.2.d.b.609.3	yes	4	20.7	even	4
1520.2.d.c.609.2		4	5.2	odd	4
1520.2.d.c.609.3		4	5.3	odd	4
3800.2.a.k.1.2		2	20.19	odd	2
3800.2.a.q.1.1		2	4.3	odd	2
7600.2.a.w.1.2		2	1.1	even	1	trivial
7600.2.a.be.1.1		2	5.4	even	2

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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)

\(f(q)\)	\(=\)	\(q+1.23607 q^{3} -4.47214 q^{7} -1.47214 q^{9} +O(q^{10})\)	\(q+1.23607 q^{3} -4.47214 q^{7} -1.47214 q^{9} -3.23607 q^{13} -6.47214 q^{17} -1.00000 q^{19} -5.52786 q^{21} -2.00000 q^{23} -5.52786 q^{27} +2.00000 q^{29} +1.52786 q^{31} +4.76393 q^{37} -4.00000 q^{39} +3.52786 q^{41} +0.472136 q^{43} -12.4721 q^{47} +13.0000 q^{49} -8.00000 q^{51} +11.2361 q^{53} -1.23607 q^{57} +10.4721 q^{59} -4.47214 q^{61} +6.58359 q^{63} -1.23607 q^{67} -2.47214 q^{69} +1.52786 q^{71} +6.47214 q^{73} +6.47214 q^{79} -2.41641 q^{81} +14.9443 q^{83} +2.47214 q^{87} -6.94427 q^{89} +14.4721 q^{91} +1.88854 q^{93} +4.18034 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 6 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 6 * q^9	\( 2 q - 2 q^{3} + 6 q^{9} - 2 q^{13} - 4 q^{17} - 2 q^{19} - 20 q^{21} - 4 q^{23} - 20 q^{27} + 4 q^{29} + 12 q^{31} + 14 q^{37} - 8 q^{39} + 16 q^{41} - 8 q^{43} - 16 q^{47} + 26 q^{49} - 16 q^{51} + 18 q^{53} + 2 q^{57} + 12 q^{59} + 40 q^{63} + 2 q^{67} + 4 q^{69} + 12 q^{71} + 4 q^{73} + 4 q^{79} + 22 q^{81} + 12 q^{83} - 4 q^{87} + 4 q^{89} + 20 q^{91} - 32 q^{93} - 14 q^{97}+O(q^{100}) \) 2 * q - 2 * q^3 + 6 * q^9 - 2 * q^13 - 4 * q^17 - 2 * q^19 - 20 * q^21 - 4 * q^23 - 20 * q^27 + 4 * q^29 + 12 * q^31 + 14 * q^37 - 8 * q^39 + 16 * q^41 - 8 * q^43 - 16 * q^47 + 26 * q^49 - 16 * q^51 + 18 * q^53 + 2 * q^57 + 12 * q^59 + 40 * q^63 + 2 * q^67 + 4 * q^69 + 12 * q^71 + 4 * q^73 + 4 * q^79 + 22 * q^81 + 12 * q^83 - 4 * q^87 + 4 * q^89 + 20 * q^91 - 32 * q^93 - 14 * q^97

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