LMFDB - Embedded newform 7600.2.a.ck.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:	$6$
Coefficient field:	6.6.66064384.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 9x^{4} + 13x^{2} - 1 $ x^6 - 9x^4 + 13x^2 - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 95)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.30397$ 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)

$f(q)$	$=$	$q+0.537080 q^{3} -3.18676 q^{7} -2.71155 q^{9} +O(q^{10})$	$q+0.537080 q^{3} -3.18676 q^{7} -2.71155 q^{9} -4.15544 q^{11} -2.07086 q^{13} -5.79470 q^{17} +1.00000 q^{19} -1.71155 q^{21} -2.60794 q^{23} -3.06756 q^{27} +6.00000 q^{29} -2.59933 q^{31} -2.23180 q^{33} -4.30266 q^{37} -1.11222 q^{39} -0.599328 q^{41} +3.18676 q^{43} +11.7086 q^{47} +3.15544 q^{49} -3.11222 q^{51} -11.7503 q^{53} +0.537080 q^{57} +1.71155 q^{59} -8.75476 q^{61} +8.64104 q^{63} +4.76228 q^{67} -1.40067 q^{69} -13.7115 q^{71} +2.72714 q^{73} +13.2424 q^{77} +1.40067 q^{79} +6.48711 q^{81} -7.07154 q^{83} +3.22248 q^{87} +16.5353 q^{89} +6.59933 q^{91} -1.39605 q^{93} -2.07086 q^{97} +11.2677 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 14 q^{9}+O(q^{10}) $ 6 * q + 14 * q^9	$ 6 q + 14 q^{9} - 2 q^{11} + 6 q^{19} + 20 q^{21} + 36 q^{29} + 8 q^{39} + 12 q^{41} - 4 q^{49} - 4 q^{51} - 20 q^{59} - 14 q^{61} - 24 q^{69} - 52 q^{71} + 24 q^{79} + 38 q^{81} + 24 q^{89} + 24 q^{91} + 30 q^{99}+O(q^{100}) $ 6 * q + 14 * q^9 - 2 * q^11 + 6 * q^19 + 20 * q^21 + 36 * q^29 + 8 * q^39 + 12 * q^41 - 4 * q^49 - 4 * q^51 - 20 * q^59 - 14 * q^61 - 24 * q^69 - 52 * q^71 + 24 * q^79 + 38 * q^81 + 24 * q^89 + 24 * q^91 + 30 * 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
$3$	0.537080	0.310083	0.155042	−	0.987908i	$-0.450449\pi$
$3$	0.537080	0.310083	0.155042	+	0.987908i	$0.450449\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.18676	−1.20448	−0.602241	−	0.798314i	$-0.705725\pi$
$7$	−3.18676	−1.20448	−0.602241	+	0.798314i	$0.705725\pi$
$8$	0	0
$9$	−2.71155	−0.903848
$10$	0	0
$11$	−4.15544	−1.25291	−0.626456	−	0.779457i	$-0.715495\pi$
$11$	−4.15544	−1.25291	−0.626456	+	0.779457i	$0.715495\pi$
$12$	0	0
$13$	−2.07086	−0.574353	−0.287176	−	0.957878i	$-0.592717\pi$
$13$	−2.07086	−0.574353	−0.287176	+	0.957878i	$0.592717\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−5.79470	−1.40542	−0.702710	−	0.711476i	$-0.748027\pi$
$17$	−5.79470	−1.40542	−0.702710	+	0.711476i	$0.748027\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−1.71155	−0.373490
$22$	0	0
$23$	−2.60794	−0.543793	−0.271896	−	0.962327i	$-0.587651\pi$
$23$	−2.60794	−0.543793	−0.271896	+	0.962327i	$0.587651\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−3.06756	−0.590352
$28$	0	0
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	−2.59933	−0.466853	−0.233427	−	0.972374i	$-0.574994\pi$
$31$	−2.59933	−0.466853	−0.233427	+	0.972374i	$0.574994\pi$
$32$	0	0
$33$	−2.23180	−0.388507
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−4.30266	−0.707353	−0.353677	−	0.935368i	$-0.615069\pi$
$37$	−4.30266	−0.707353	−0.353677	+	0.935368i	$0.615069\pi$
$38$	0	0
$39$	−1.11222	−0.178097
$40$	0	0
$41$	−0.599328	−0.0935993	−0.0467997	−	0.998904i	$-0.514902\pi$
$41$	−0.599328	−0.0935993	−0.0467997	+	0.998904i	$0.514902\pi$
$42$	0	0
$43$	3.18676	0.485976	0.242988	−	0.970029i	$-0.421872\pi$
$43$	3.18676	0.485976	0.242988	+	0.970029i	$0.421872\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	11.7086	1.70787	0.853937	−	0.520376i	$-0.174208\pi$
$47$	11.7086	1.70787	0.853937	+	0.520376i	$0.174208\pi$
$48$	0	0
$49$	3.15544	0.450777
$50$	0	0
$51$	−3.11222	−0.435798
$52$	0	0
$53$	−11.7503	−1.61403	−0.807017	−	0.590529i	$-0.798919\pi$
$53$	−11.7503	−1.61403	−0.807017	+	0.590529i	$0.798919\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0.537080	0.0711380
$58$	0	0
$59$	1.71155	0.222824	0.111412	−	0.993774i	$-0.464463\pi$
$59$	1.71155	0.222824	0.111412	+	0.993774i	$0.464463\pi$
$60$	0	0
$61$	−8.75476	−1.12093	−0.560466	−	0.828177i	$-0.689378\pi$
$61$	−8.75476	−1.12093	−0.560466	+	0.828177i	$0.689378\pi$
$62$	0	0
$63$	8.64104	1.08867
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.76228	0.581805	0.290902	−	0.956753i	$-0.406044\pi$
$67$	4.76228	0.581805	0.290902	+	0.956753i	$0.406044\pi$
$68$	0	0
$69$	−1.40067	−0.168621
$70$	0	0
$71$	−13.7115	−1.62726	−0.813631	−	0.581382i	$-0.802512\pi$
$71$	−13.7115	−1.62726	−0.813631	+	0.581382i	$0.802512\pi$
$72$	0	0
$73$	2.72714	0.319188	0.159594	−	0.987183i	$-0.448982\pi$
$73$	2.72714	0.319188	0.159594	+	0.987183i	$0.448982\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	13.2424	1.50911
$78$	0	0
$79$	1.40067	0.157588	0.0787939	−	0.996891i	$-0.474893\pi$
$79$	1.40067	0.157588	0.0787939	+	0.996891i	$0.474893\pi$
$80$	0	0
$81$	6.48711	0.720790
$82$	0	0
$83$	−7.07154	−0.776203	−0.388101	−	0.921617i	$-0.626869\pi$
$83$	−7.07154	−0.776203	−0.388101	+	0.921617i	$0.626869\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	3.22248	0.345486
$88$	0	0
$89$	16.5353	1.75274	0.876370	−	0.481639i	$-0.159958\pi$
$89$	16.5353	1.75274	0.876370	+	0.481639i	$0.159958\pi$
$90$	0	0
$91$	6.59933	0.691798
$92$	0	0
$93$	−1.39605	−0.144763
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−2.07086	−0.210264	−0.105132	−	0.994458i	$-0.533526\pi$
$97$	−2.07086	−0.210264	−0.105132	+	0.994458i	$0.533526\pi$
$98$	0	0
$99$	11.2677	1.13244
$100$	0	0
$101$	−1.71155	−0.170305	−0.0851525	−	0.996368i	$-0.527138\pi$
$101$	−1.71155	−0.170305	−0.0851525	+	0.996368i	$0.527138\pi$
$102$	0	0
$103$	5.75296	0.566856	0.283428	−	0.958994i	$-0.408528\pi$
$103$	5.75296	0.566856	0.283428	+	0.958994i	$0.408528\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−15.4324	−1.49191	−0.745955	−	0.665996i	$-0.768007\pi$
$107$	−15.4324	−1.49191	−0.745955	+	0.665996i	$0.768007\pi$
$108$	0	0
$109$	11.7115	1.12176	0.560881	−	0.827896i	$-0.310462\pi$
$109$	11.7115	1.12176	0.560881	+	0.827896i	$0.310462\pi$
$110$	0	0
$111$	−2.31087	−0.219338
$112$	0	0
$113$	10.5927	0.996477	0.498239	−	0.867040i	$-0.333980\pi$
$113$	10.5927	0.996477	0.498239	+	0.867040i	$0.333980\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	5.61523	0.519128
$118$	0	0
$119$	18.4663	1.69280
$120$	0	0
$121$	6.26765	0.569787
$122$	0	0
$123$	−0.321887	−0.0290236
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−6.07484	−0.539055	−0.269528	−	0.962993i	$-0.586868\pi$
$127$	−6.07484	−0.539055	−0.269528	+	0.962993i	$0.586868\pi$
$128$	0	0
$129$	1.71155	0.150693
$130$	0	0
$131$	13.5785	1.18636	0.593181	−	0.805069i	$-0.297872\pi$
$131$	13.5785	1.18636	0.593181	+	0.805069i	$0.297872\pi$
$132$	0	0
$133$	−3.18676	−0.276327
$134$	0	0
$135$	0	0
$136$	0	0
$137$	7.94302	0.678618	0.339309	−	0.940675i	$-0.389807\pi$
$137$	7.94302	0.678618	0.339309	+	0.940675i	$0.389807\pi$
$138$	0	0
$139$	−3.26765	−0.277159	−0.138579	−	0.990351i	$-0.544254\pi$
$139$	−3.26765	−0.277159	−0.138579	+	0.990351i	$0.544254\pi$
$140$	0	0
$141$	6.28845	0.529583
$142$	0	0
$143$	8.60532	0.719613
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1.69472	0.139778
$148$	0	0
$149$	8.44389	0.691751	0.345875	−	0.938280i	$-0.387582\pi$
$149$	8.44389	0.691751	0.345875	+	0.938280i	$0.387582\pi$
$150$	0	0
$151$	−0.887783	−0.0722468	−0.0361234	−	0.999347i	$-0.511501\pi$
$151$	−0.887783	−0.0722468	−0.0361234	+	0.999347i	$0.511501\pi$
$152$	0	0
$153$	15.7126	1.27029
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−4.14172	−0.330545	−0.165273	−	0.986248i	$-0.552850\pi$
$157$	−4.14172	−0.330545	−0.165273	+	0.986248i	$0.552850\pi$
$158$	0	0
$159$	−6.31087	−0.500485
$160$	0	0
$161$	8.31087	0.654989
$162$	0	0
$163$	−24.7126	−1.93564	−0.967819	−	0.251647i	$-0.919028\pi$
$163$	−24.7126	−1.93564	−0.967819	+	0.251647i	$0.919028\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3.60464	−0.278935	−0.139468	−	0.990227i	$-0.544539\pi$
$167$	−3.60464	−0.278935	−0.139468	+	0.990227i	$0.544539\pi$
$168$	0	0
$169$	−8.71155	−0.670119
$170$	0	0
$171$	−2.71155	−0.207357
$172$	0	0
$173$	22.4205	1.70460	0.852300	−	0.523054i	$-0.175207\pi$
$173$	22.4205	1.70460	0.852300	+	0.523054i	$0.175207\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0.919237	0.0690941
$178$	0	0
$179$	5.13464	0.383781	0.191890	−	0.981416i	$-0.438538\pi$
$179$	5.13464	0.383781	0.191890	+	0.981416i	$0.438538\pi$
$180$	0	0
$181$	20.8462	1.54948	0.774742	−	0.632277i	$-0.217880\pi$
$181$	20.8462	1.54948	0.774742	+	0.632277i	$0.217880\pi$
$182$	0	0
$183$	−4.70201	−0.347583
$184$	0	0
$185$	0	0
$186$	0	0
$187$	24.0795	1.76087
$188$	0	0
$189$	9.77557	0.711068
$190$	0	0
$191$	5.26765	0.381154	0.190577	−	0.981672i	$-0.438964\pi$
$191$	5.26765	0.381154	0.190577	+	0.981672i	$0.438964\pi$
$192$	0	0
$193$	−2.07086	−0.149064	−0.0745318	−	0.997219i	$-0.523746\pi$
$193$	−2.07086	−0.149064	−0.0745318	+	0.997219i	$0.523746\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−10.4318	−0.743232	−0.371616	−	0.928387i	$-0.621196\pi$
$197$	−10.4318	−0.743232	−0.371616	+	0.928387i	$0.621196\pi$
$198$	0	0
$199$	2.73235	0.193691	0.0968455	−	0.995299i	$-0.469125\pi$
$199$	2.73235	0.193691	0.0968455	+	0.995299i	$0.469125\pi$
$200$	0	0
$201$	2.55773	0.180408
$202$	0	0
$203$	−19.1206	−1.34200
$204$	0	0
$205$	0	0
$206$	0	0
$207$	7.07154	0.491506
$208$	0	0
$209$	−4.15544	−0.287438
$210$	0	0
$211$	15.7340	1.08317	0.541585	−	0.840646i	$-0.317824\pi$
$211$	15.7340	1.08317	0.541585	+	0.840646i	$0.317824\pi$
$212$	0	0
$213$	−7.36420	−0.504586
$214$	0	0
$215$	0	0
$216$	0	0
$217$	8.28343	0.562316
$218$	0	0
$219$	1.46469	0.0989748
$220$	0	0
$221$	12.0000	0.807207
$222$	0	0
$223$	18.8219	1.26041	0.630203	−	0.776430i	$-0.282972\pi$
$223$	18.8219	1.26041	0.630203	+	0.776430i	$0.282972\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−14.4418	−0.958533	−0.479267	−	0.877669i	$-0.659097\pi$
$227$	−14.4418	−0.958533	−0.479267	+	0.877669i	$0.659097\pi$
$228$	0	0
$229$	−4.17785	−0.276080	−0.138040	−	0.990427i	$-0.544080\pi$
$229$	−4.17785	−0.276080	−0.138040	+	0.990427i	$0.544080\pi$
$230$	0	0
$231$	7.11222	0.467950
$232$	0	0
$233$	−12.0847	−0.791697	−0.395849	−	0.918316i	$-0.629550\pi$
$233$	−12.0847	−0.791697	−0.395849	+	0.918316i	$0.629550\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0.752273	0.0488654
$238$	0	0
$239$	11.3541	0.734435	0.367218	−	0.930135i	$-0.380310\pi$
$239$	11.3541	0.734435	0.367218	+	0.930135i	$0.380310\pi$
$240$	0	0
$241$	−3.40067	−0.219057	−0.109528	−	0.993984i	$-0.534934\pi$
$241$	−3.40067	−0.219057	−0.109528	+	0.993984i	$0.534934\pi$
$242$	0	0
$243$	12.6868	0.813857
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−2.07086	−0.131766
$248$	0	0
$249$	−3.79798	−0.240687
$250$	0	0
$251$	−3.04322	−0.192086	−0.0960432	−	0.995377i	$-0.530619\pi$
$251$	−3.04322	−0.192086	−0.0960432	+	0.995377i	$0.530619\pi$
$252$	0	0
$253$	10.8371	0.681324
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−17.2881	−1.07840	−0.539201	−	0.842177i	$-0.681274\pi$
$257$	−17.2881	−1.07840	−0.539201	+	0.842177i	$0.681274\pi$
$258$	0	0
$259$	13.7115	0.851994
$260$	0	0
$261$	−16.2693	−1.00704
$262$	0	0
$263$	−1.19336	−0.0735859	−0.0367930	−	0.999323i	$-0.511714\pi$
$263$	−1.19336	−0.0735859	−0.0367930	+	0.999323i	$0.511714\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	8.88078	0.543495
$268$	0	0
$269$	22.1089	1.34800	0.674000	−	0.738731i	$-0.264575\pi$
$269$	22.1089	1.34800	0.674000	+	0.738731i	$0.264575\pi$
$270$	0	0
$271$	4.08644	0.248234	0.124117	−	0.992268i	$-0.460390\pi$
$271$	4.08644	0.248234	0.124117	+	0.992268i	$0.460390\pi$
$272$	0	0
$273$	3.54437	0.214515
$274$	0	0
$275$	0	0
$276$	0	0
$277$	10.0199	0.602037	0.301019	−	0.953618i	$-0.402673\pi$
$277$	10.0199	0.602037	0.301019	+	0.953618i	$0.402673\pi$
$278$	0	0
$279$	7.04820	0.421964
$280$	0	0
$281$	0.599328	0.0357529	0.0178765	−	0.999840i	$-0.494309\pi$
$281$	0.599328	0.0357529	0.0178765	+	0.999840i	$0.494309\pi$
$282$	0	0
$283$	5.41856	0.322100	0.161050	−	0.986946i	$-0.448512\pi$
$283$	5.41856	0.322100	0.161050	+	0.986946i	$0.448512\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.90991	0.112739
$288$	0	0
$289$	16.5785	0.975207
$290$	0	0
$291$	−1.11222	−0.0651993
$292$	0	0
$293$	−3.46691	−0.202539	−0.101269	−	0.994859i	$-0.532290\pi$
$293$	−3.46691	−0.202539	−0.101269	+	0.994859i	$0.532290\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	12.7470	0.739658
$298$	0	0
$299$	5.40067	0.312329
$300$	0	0
$301$	−10.1554	−0.585350
$302$	0	0
$303$	−0.919237	−0.0528088
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−16.5901	−0.946846	−0.473423	−	0.880835i	$-0.656982\pi$
$307$	−16.5901	−0.946846	−0.473423	+	0.880835i	$0.656982\pi$
$308$	0	0
$309$	3.08980	0.175772
$310$	0	0
$311$	4.15544	0.235633	0.117817	−	0.993035i	$-0.462411\pi$
$311$	4.15544	0.235633	0.117817	+	0.993035i	$0.462411\pi$
$312$	0	0
$313$	0.919237	0.0519583	0.0259792	−	0.999662i	$-0.491730\pi$
$313$	0.919237	0.0519583	0.0259792	+	0.999662i	$0.491730\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	26.7292	1.50126	0.750630	−	0.660723i	$-0.229750\pi$
$317$	26.7292	1.50126	0.750630	+	0.660723i	$0.229750\pi$
$318$	0	0
$319$	−24.9326	−1.39596
$320$	0	0
$321$	−8.28845	−0.462616
$322$	0	0
$323$	−5.79470	−0.322426
$324$	0	0
$325$	0	0
$326$	0	0
$327$	6.29004	0.347840
$328$	0	0
$329$	−37.3125	−2.05710
$330$	0	0
$331$	−8.00000	−0.439720	−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000	−0.439720	−0.219860	+	0.975531i	$0.570560\pi$
$332$	0	0
$333$	11.6669	0.639340
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−22.5040	−1.22587	−0.612935	−	0.790133i	$-0.710011\pi$
$337$	−22.5040	−1.22587	−0.612935	+	0.790133i	$0.710011\pi$
$338$	0	0
$339$	5.68913	0.308991
$340$	0	0
$341$	10.8013	0.584926
$342$	0	0
$343$	12.2517	0.661530
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−14.4543	−0.775946	−0.387973	−	0.921671i	$-0.626825\pi$
$347$	−14.4543	−0.775946	−0.387973	+	0.921671i	$0.626825\pi$
$348$	0	0
$349$	−13.3541	−0.714828	−0.357414	−	0.933946i	$-0.616342\pi$
$349$	−13.3541	−0.714828	−0.357414	+	0.933946i	$0.616342\pi$
$350$	0	0
$351$	6.35248	0.339070
$352$	0	0
$353$	−17.6410	−0.938937	−0.469469	−	0.882949i	$-0.655554\pi$
$353$	−17.6410	−0.938937	−0.469469	+	0.882949i	$0.655554\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	9.91789	0.524910
$358$	0	0
$359$	−12.4663	−0.657947	−0.328973	−	0.944339i	$-0.606703\pi$
$359$	−12.4663	−0.657947	−0.328973	+	0.944339i	$0.606703\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	3.36623	0.176681
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−16.4291	−0.857594	−0.428797	−	0.903401i	$-0.641062\pi$
$367$	−16.4291	−0.857594	−0.428797	+	0.903401i	$0.641062\pi$
$368$	0	0
$369$	1.62511	0.0845996
$370$	0	0
$371$	37.4455	1.94407
$372$	0	0
$373$	5.29334	0.274079	0.137039	−	0.990566i	$-0.456241\pi$
$373$	5.29334	0.274079	0.137039	+	0.990566i	$0.456241\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−12.4252	−0.639928
$378$	0	0
$379$	−14.5353	−0.746629	−0.373314	−	0.927705i	$-0.621779\pi$
$379$	−14.5353	−0.746629	−0.373314	+	0.927705i	$0.621779\pi$
$380$	0	0
$381$	−3.26268	−0.167152
$382$	0	0
$383$	0.453598	0.0231778	0.0115889	−	0.999933i	$-0.496311\pi$
$383$	0.453598	0.0231778	0.0115889	+	0.999933i	$0.496311\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−8.64104	−0.439249
$388$	0	0
$389$	16.1554	0.819113	0.409557	−	0.912285i	$-0.365683\pi$
$389$	16.1554	0.819113	0.409557	+	0.912285i	$0.365683\pi$
$390$	0	0
$391$	15.1122	0.764258
$392$	0	0
$393$	7.29276	0.367871
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−32.7563	−1.64399	−0.821995	−	0.569495i	$-0.807139\pi$
$397$	−32.7563	−1.64399	−0.821995	+	0.569495i	$0.807139\pi$
$398$	0	0
$399$	−1.71155	−0.0856844
$400$	0	0
$401$	12.0864	0.603568	0.301784	−	0.953376i	$-0.402418\pi$
$401$	12.0864	0.603568	0.301784	+	0.953376i	$0.402418\pi$
$402$	0	0
$403$	5.38284	0.268138
$404$	0	0
$405$	0	0
$406$	0	0
$407$	17.8794	0.886251
$408$	0	0
$409$	−19.1346	−0.946147	−0.473073	−	0.881023i	$-0.656855\pi$
$409$	−19.1346	−0.946147	−0.473073	+	0.881023i	$0.656855\pi$
$410$	0	0
$411$	4.26604	0.210428
$412$	0	0
$413$	−5.45428	−0.268388
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−1.75499	−0.0859423
$418$	0	0
$419$	−8.04484	−0.393016	−0.196508	−	0.980502i	$-0.562960\pi$
$419$	−8.04484	−0.393016	−0.196508	+	0.980502i	$0.562960\pi$
$420$	0	0
$421$	−29.3591	−1.43087	−0.715437	−	0.698678i	$-0.753772\pi$
$421$	−29.3591	−1.43087	−0.715437	+	0.698678i	$0.753772\pi$
$422$	0	0
$423$	−31.7484	−1.54366
$424$	0	0
$425$	0	0
$426$	0	0
$427$	27.8993	1.35014
$428$	0	0
$429$	4.62175	0.223140
$430$	0	0
$431$	32.0448	1.54355	0.771773	−	0.635898i	$-0.219370\pi$
$431$	32.0448	1.54355	0.771773	+	0.635898i	$0.219370\pi$
$432$	0	0
$433$	−0.482831	−0.0232034	−0.0116017	−	0.999933i	$-0.503693\pi$
$433$	−0.482831	−0.0232034	−0.0116017	+	0.999933i	$0.503693\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2.60794	−0.124755
$438$	0	0
$439$	27.3591	1.30578	0.652889	−	0.757454i	$-0.273557\pi$
$439$	27.3591	1.30578	0.652889	+	0.757454i	$0.273557\pi$
$440$	0	0
$441$	−8.55611	−0.407434
$442$	0	0
$443$	23.3815	1.11089	0.555444	−	0.831554i	$-0.312548\pi$
$443$	23.3815	1.11089	0.555444	+	0.831554i	$0.312548\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	4.53505	0.214500
$448$	0	0
$449$	23.1346	1.09179	0.545895	−	0.837853i	$-0.316190\pi$
$449$	23.1346	1.09179	0.545895	+	0.837853i	$0.316190\pi$
$450$	0	0
$451$	2.49047	0.117272
$452$	0	0
$453$	−0.476811	−0.0224025
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−21.2503	−0.994049	−0.497025	−	0.867736i	$-0.665574\pi$
$457$	−21.2503	−0.994049	−0.497025	+	0.867736i	$0.665574\pi$
$458$	0	0
$459$	17.7756	0.829692
$460$	0	0
$461$	31.5785	1.47076	0.735379	−	0.677656i	$-0.237004\pi$
$461$	31.5785	1.47076	0.735379	+	0.677656i	$0.237004\pi$
$462$	0	0
$463$	−15.6119	−0.725547	−0.362774	−	0.931877i	$-0.618170\pi$
$463$	−15.6119	−0.725547	−0.362774	+	0.931877i	$0.618170\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−29.8264	−1.38020	−0.690101	−	0.723713i	$-0.742434\pi$
$467$	−29.8264	−1.38020	−0.690101	+	0.723713i	$0.742434\pi$
$468$	0	0
$469$	−15.1762	−0.700774
$470$	0	0
$471$	−2.22443	−0.102496
$472$	0	0
$473$	−13.2424	−0.608885
$474$	0	0
$475$	0	0
$476$	0	0
$477$	31.8616	1.45884
$478$	0	0
$479$	−4.53531	−0.207223	−0.103612	−	0.994618i	$-0.533040\pi$
$479$	−4.53531	−0.207223	−0.103612	+	0.994618i	$0.533040\pi$
$480$	0	0
$481$	8.91020	0.406270
$482$	0	0
$483$	4.46360	0.203101
$484$	0	0
$485$	0	0
$486$	0	0
$487$	39.2550	1.77881	0.889407	−	0.457116i	$-0.151118\pi$
$487$	39.2550	1.77881	0.889407	+	0.457116i	$0.151118\pi$
$488$	0	0
$489$	−13.2726	−0.600209
$490$	0	0
$491$	−38.8910	−1.75513	−0.877564	−	0.479460i	$-0.840832\pi$
$491$	−38.8910	−1.75513	−0.877564	+	0.479460i	$0.840832\pi$
$492$	0	0
$493$	−34.7682	−1.56588
$494$	0	0
$495$	0	0
$496$	0	0
$497$	43.6954	1.96001
$498$	0	0
$499$	−4.73235	−0.211849	−0.105924	−	0.994374i	$-0.533780\pi$
$499$	−4.73235	−0.211849	−0.105924	+	0.994374i	$0.533780\pi$
$500$	0	0
$501$	−1.93598	−0.0864931
$502$	0	0
$503$	1.85567	0.0827400	0.0413700	−	0.999144i	$-0.486828\pi$
$503$	1.85567	0.0827400	0.0413700	+	0.999144i	$0.486828\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−4.67880	−0.207793
$508$	0	0
$509$	22.8878	1.01448	0.507242	−	0.861804i	$-0.330665\pi$
$509$	22.8878	1.01448	0.507242	+	0.861804i	$0.330665\pi$
$510$	0	0
$511$	−8.69074	−0.384456
$512$	0	0
$513$	−3.06756	−0.135436
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−48.6543	−2.13982
$518$	0	0
$519$	12.0416	0.528568
$520$	0	0
$521$	−29.7340	−1.30267	−0.651334	−	0.758791i	$-0.725790\pi$
$521$	−29.7340	−1.30267	−0.651334	+	0.758791i	$0.725790\pi$
$522$	0	0
$523$	30.6497	1.34022	0.670109	−	0.742263i	$-0.266248\pi$
$523$	30.6497	1.34022	0.670109	+	0.742263i	$0.266248\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	15.0623	0.656125
$528$	0	0
$529$	−16.1987	−0.704289
$530$	0	0
$531$	−4.64093	−0.201399
$532$	0	0
$533$	1.24112	0.0537590
$534$	0	0
$535$	0	0
$536$	0	0
$537$	2.75771	0.119004
$538$	0	0
$539$	−13.1122	−0.564783
$540$	0	0
$541$	2.21946	0.0954220	0.0477110	−	0.998861i	$-0.484807\pi$
$541$	2.21946	0.0954220	0.0477110	+	0.998861i	$0.484807\pi$
$542$	0	0
$543$	11.1961	0.480469
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−14.4297	−0.616970	−0.308485	−	0.951229i	$-0.599822\pi$
$547$	−14.4297	−0.616970	−0.308485	+	0.951229i	$0.599822\pi$
$548$	0	0
$549$	23.7389	1.01315
$550$	0	0
$551$	6.00000	0.255609
$552$	0	0
$553$	−4.46360	−0.189812
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−19.4610	−0.824588	−0.412294	−	0.911051i	$-0.635272\pi$
$557$	−19.4610	−0.824588	−0.412294	+	0.911051i	$0.635272\pi$
$558$	0	0
$559$	−6.59933	−0.279122
$560$	0	0
$561$	12.9326	0.546016
$562$	0	0
$563$	−12.3649	−0.521118	−0.260559	−	0.965458i	$-0.583907\pi$
$563$	−12.3649	−0.521118	−0.260559	+	0.965458i	$0.583907\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−20.6729	−0.868179
$568$	0	0
$569$	15.0898	0.632597	0.316299	−	0.948660i	$-0.397560\pi$
$569$	15.0898	0.632597	0.316299	+	0.948660i	$0.397560\pi$
$570$	0	0
$571$	−17.1571	−0.718000	−0.359000	−	0.933337i	$-0.616882\pi$
$571$	−17.1571	−0.718000	−0.359000	+	0.933337i	$0.616882\pi$
$572$	0	0
$573$	2.82915	0.118190
$574$	0	0
$575$	0	0
$576$	0	0
$577$	22.5165	0.937374	0.468687	−	0.883364i	$-0.344727\pi$
$577$	22.5165	0.937374	0.468687	+	0.883364i	$0.344727\pi$
$578$	0	0
$579$	−1.11222	−0.0462222
$580$	0	0
$581$	22.5353	0.934922
$582$	0	0
$583$	48.8278	2.02224
$584$	0	0
$585$	0	0
$586$	0	0
$587$	7.49544	0.309370	0.154685	−	0.987964i	$-0.450564\pi$
$587$	7.49544	0.309370	0.154685	+	0.987964i	$0.450564\pi$
$588$	0	0
$589$	−2.59933	−0.107103
$590$	0	0
$591$	−5.60269	−0.230464
$592$	0	0
$593$	−27.8094	−1.14199	−0.570997	−	0.820952i	$-0.693443\pi$
$593$	−27.8094	−1.14199	−0.570997	+	0.820952i	$0.693443\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1.46749	0.0600603
$598$	0	0
$599$	45.4903	1.85869	0.929343	−	0.369219i	$-0.120375\pi$
$599$	45.4903	1.85869	0.929343	+	0.369219i	$0.120375\pi$
$600$	0	0
$601$	−16.5993	−0.677101	−0.338550	−	0.940948i	$-0.609937\pi$
$601$	−16.5993	−0.677101	−0.338550	+	0.940948i	$0.609937\pi$
$602$	0	0
$603$	−12.9131	−0.525863
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−5.08417	−0.206360	−0.103180	−	0.994663i	$-0.532902\pi$
$607$	−5.08417	−0.206360	−0.103180	+	0.994663i	$0.532902\pi$
$608$	0	0
$609$	−10.2693	−0.416132
$610$	0	0
$611$	−24.2469	−0.980923
$612$	0	0
$613$	−4.63706	−0.187289	−0.0936445	−	0.995606i	$-0.529852\pi$
$613$	−4.63706	−0.187289	−0.0936445	+	0.995606i	$0.529852\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−40.2874	−1.62191	−0.810955	−	0.585108i	$-0.801052\pi$
$617$	−40.2874	−1.62191	−0.810955	+	0.585108i	$0.801052\pi$
$618$	0	0
$619$	43.3815	1.74365	0.871825	−	0.489818i	$-0.162937\pi$
$619$	43.3815	1.74365	0.871825	+	0.489818i	$0.162937\pi$
$620$	0	0
$621$	8.00000	0.321029
$622$	0	0
$623$	−52.6940	−2.11114
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−2.23180	−0.0891296
$628$	0	0
$629$	24.9326	0.994129
$630$	0	0
$631$	−7.53369	−0.299911	−0.149956	−	0.988693i	$-0.547913\pi$
$631$	−7.53369	−0.299911	−0.149956	+	0.988693i	$0.547913\pi$
$632$	0	0
$633$	8.45040	0.335873
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−6.53446	−0.258905
$638$	0	0
$639$	37.1795	1.47080
$640$	0	0
$641$	−23.3075	−0.920591	−0.460296	−	0.887766i	$-0.652257\pi$
$641$	−23.3075	−0.920591	−0.460296	+	0.887766i	$0.652257\pi$
$642$	0	0
$643$	1.87419	0.0739110	0.0369555	−	0.999317i	$-0.488234\pi$
$643$	1.87419	0.0739110	0.0369555	+	0.999317i	$0.488234\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−47.0371	−1.84922	−0.924609	−	0.380917i	$-0.875608\pi$
$647$	−47.0371	−1.84922	−0.924609	+	0.380917i	$0.875608\pi$
$648$	0	0
$649$	−7.11222	−0.279179
$650$	0	0
$651$	4.44887	0.174365
$652$	0	0
$653$	−24.1630	−0.945571	−0.472785	−	0.881178i	$-0.656751\pi$
$653$	−24.1630	−0.945571	−0.472785	+	0.881178i	$0.656751\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−7.39477	−0.288497
$658$	0	0
$659$	−10.2885	−0.400781	−0.200391	−	0.979716i	$-0.564221\pi$
$659$	−10.2885	−0.400781	−0.200391	+	0.979716i	$0.564221\pi$
$660$	0	0
$661$	5.93598	0.230883	0.115441	−	0.993314i	$-0.463172\pi$
$661$	5.93598	0.230883	0.115441	+	0.993314i	$0.463172\pi$
$662$	0	0
$663$	6.44496	0.250302
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−15.6476	−0.605879
$668$	0	0
$669$	10.1089	0.390831
$670$	0	0
$671$	36.3799	1.40443
$672$	0	0
$673$	40.7053	1.56907	0.784537	−	0.620082i	$-0.212901\pi$
$673$	40.7053	1.56907	0.784537	+	0.620082i	$0.212901\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−18.8761	−0.725469	−0.362734	−	0.931893i	$-0.618157\pi$
$677$	−18.8761	−0.725469	−0.362734	+	0.931893i	$0.618157\pi$
$678$	0	0
$679$	6.59933	0.253259
$680$	0	0
$681$	−7.75638	−0.297225
$682$	0	0
$683$	−19.6576	−0.752179	−0.376089	−	0.926583i	$-0.622731\pi$
$683$	−19.6576	−0.752179	−0.376089	+	0.926583i	$0.622731\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−2.24384	−0.0856079
$688$	0	0
$689$	24.3333	0.927025
$690$	0	0
$691$	48.2451	1.83533	0.917665	−	0.397354i	$-0.130072\pi$
$691$	48.2451	1.83533	0.917665	+	0.397354i	$0.130072\pi$
$692$	0	0
$693$	−35.9073	−1.36401
$694$	0	0
$695$	0	0
$696$	0	0
$697$	3.47293	0.131546
$698$	0	0
$699$	−6.49047	−0.245492
$700$	0	0
$701$	0.512889	0.0193715	0.00968577	−	0.999953i	$-0.496917\pi$
$701$	0.512889	0.0193715	0.00968577	+	0.999953i	$0.496917\pi$
$702$	0	0
$703$	−4.30266	−0.162278
$704$	0	0
$705$	0	0
$706$	0	0
$707$	5.45428	0.205129
$708$	0	0
$709$	−8.84618	−0.332225	−0.166113	−	0.986107i	$-0.553122\pi$
$709$	−8.84618	−0.332225	−0.166113	+	0.986107i	$0.553122\pi$
$710$	0	0
$711$	−3.79798	−0.142436
$712$	0	0
$713$	6.77889	0.253871
$714$	0	0
$715$	0	0
$716$	0	0
$717$	6.09806	0.227736
$718$	0	0
$719$	−7.84456	−0.292553	−0.146276	−	0.989244i	$-0.546729\pi$
$719$	−7.84456	−0.292553	−0.146276	+	0.989244i	$0.546729\pi$
$720$	0	0
$721$	−18.3333	−0.682767
$722$	0	0
$723$	−1.82643	−0.0679258
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−2.91130	−0.107974	−0.0539870	−	0.998542i	$-0.517193\pi$
$727$	−2.91130	−0.107974	−0.0539870	+	0.998542i	$0.517193\pi$
$728$	0	0
$729$	−12.6475	−0.468427
$730$	0	0
$731$	−18.4663	−0.683001
$732$	0	0
$733$	33.7775	1.24760	0.623800	−	0.781584i	$-0.285588\pi$
$733$	33.7775	1.24760	0.623800	+	0.781584i	$0.285588\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−19.7893	−0.728950
$738$	0	0
$739$	35.8030	1.31703	0.658517	−	0.752566i	$-0.271184\pi$
$739$	35.8030	1.31703	0.658517	+	0.752566i	$0.271184\pi$
$740$	0	0
$741$	−1.11222	−0.0408583
$742$	0	0
$743$	−5.66948	−0.207993	−0.103996	−	0.994578i	$-0.533163\pi$
$743$	−5.66948	−0.207993	−0.103996	+	0.994578i	$0.533163\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	19.1748	0.701569
$748$	0	0
$749$	49.1795	1.79698
$750$	0	0
$751$	27.4679	1.00232	0.501159	−	0.865355i	$-0.332907\pi$
$751$	27.4679	1.00232	0.501159	+	0.865355i	$0.332907\pi$
$752$	0	0
$753$	−1.63445	−0.0595628
$754$	0	0
$755$	0	0
$756$	0	0
$757$	41.6370	1.51332	0.756662	−	0.653806i	$-0.226829\pi$
$757$	41.6370	1.51332	0.756662	+	0.653806i	$0.226829\pi$
$758$	0	0
$759$	5.82040	0.211267
$760$	0	0
$761$	−9.46967	−0.343275	−0.171638	−	0.985160i	$-0.554906\pi$
$761$	−9.46967	−0.343275	−0.171638	+	0.985160i	$0.554906\pi$
$762$	0	0
$763$	−37.3219	−1.35114
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−3.54437	−0.127980
$768$	0	0
$769$	1.90858	0.0688253	0.0344127	−	0.999408i	$-0.489044\pi$
$769$	1.90858	0.0688253	0.0344127	+	0.999408i	$0.489044\pi$
$770$	0	0
$771$	−9.28510	−0.334395
$772$	0	0
$773$	28.4007	1.02150	0.510751	−	0.859729i	$-0.329367\pi$
$773$	28.4007	1.02150	0.510751	+	0.859729i	$0.329367\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	7.36420	0.264189
$778$	0	0
$779$	−0.599328	−0.0214732
$780$	0	0
$781$	56.9775	2.03881
$782$	0	0
$783$	−18.4053	−0.657753
$784$	0	0
$785$	0	0
$786$	0	0
$787$	15.6708	0.558605	0.279303	−	0.960203i	$-0.409897\pi$
$787$	15.6708	0.558605	0.279303	+	0.960203i	$0.409897\pi$
$788$	0	0
$789$	−0.640931	−0.0228178
$790$	0	0
$791$	−33.7564	−1.20024
$792$	0	0
$793$	18.1299	0.643811
$794$	0	0
$795$	0	0
$796$	0	0
$797$	36.9225	1.30786	0.653932	−	0.756554i	$-0.273118\pi$
$797$	36.9225	1.30786	0.653932	+	0.756554i	$0.273118\pi$
$798$	0	0
$799$	−67.8478	−2.40028
$800$	0	0
$801$	−44.8362	−1.58421
$802$	0	0
$803$	−11.3325	−0.399914
$804$	0	0
$805$	0	0
$806$	0	0
$807$	11.8742	0.417993
$808$	0	0
$809$	43.0465	1.51343	0.756716	−	0.653743i	$-0.226802\pi$
$809$	43.0465	1.51343	0.756716	+	0.653743i	$0.226802\pi$
$810$	0	0
$811$	−32.2469	−1.13234	−0.566170	−	0.824288i	$-0.691575\pi$
$811$	−32.2469	−1.13234	−0.566170	+	0.824288i	$0.691575\pi$
$812$	0	0
$813$	2.19475	0.0769731
$814$	0	0
$815$	0	0
$816$	0	0
$817$	3.18676	0.111491
$818$	0	0
$819$	−17.8944	−0.625280
$820$	0	0
$821$	5.18121	0.180826	0.0904128	−	0.995904i	$-0.471181\pi$
$821$	5.18121	0.180826	0.0904128	+	0.995904i	$0.471181\pi$
$822$	0	0
$823$	−25.8517	−0.901133	−0.450567	−	0.892743i	$-0.648778\pi$
$823$	−25.8517	−0.901133	−0.450567	+	0.892743i	$0.648778\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−9.97816	−0.346974	−0.173487	−	0.984836i	$-0.555504\pi$
$827$	−9.97816	−0.346974	−0.173487	+	0.984836i	$0.555504\pi$
$828$	0	0
$829$	21.9808	0.763425	0.381713	−	0.924281i	$-0.375334\pi$
$829$	21.9808	0.763425	0.381713	+	0.924281i	$0.375334\pi$
$830$	0	0
$831$	5.38149	0.186682
$832$	0	0
$833$	−18.2848	−0.633531
$834$	0	0
$835$	0	0
$836$	0	0
$837$	7.97359	0.275607
$838$	0	0
$839$	16.1089	0.556140	0.278070	−	0.960561i	$-0.410305\pi$
$839$	16.1089	0.556140	0.278070	+	0.960561i	$0.410305\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	0.321887	0.0110864
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−19.9735	−0.686298
$848$	0	0
$849$	2.91020	0.0998779
$850$	0	0
$851$	11.2211	0.384653
$852$	0	0
$853$	−44.9615	−1.53945	−0.769727	−	0.638373i	$-0.779608\pi$
$853$	−44.9615	−1.53945	−0.769727	+	0.638373i	$0.779608\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	46.2431	1.57963	0.789817	−	0.613343i	$-0.210176\pi$
$857$	46.2431	1.57963	0.789817	+	0.613343i	$0.210176\pi$
$858$	0	0
$859$	10.7772	0.367713	0.183856	−	0.982953i	$-0.441142\pi$
$859$	10.7772	0.367713	0.183856	+	0.982953i	$0.441142\pi$
$860$	0	0
$861$	1.02578	0.0349584
$862$	0	0
$863$	22.7966	0.776007	0.388003	−	0.921658i	$-0.373165\pi$
$863$	22.7966	0.776007	0.388003	+	0.921658i	$0.373165\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	8.90400	0.302396
$868$	0	0
$869$	−5.82040	−0.197444
$870$	0	0
$871$	−9.86201	−0.334161
$872$	0	0
$873$	5.61523	0.190047
$874$	0	0
$875$	0	0
$876$	0	0
$877$	4.94644	0.167029	0.0835146	−	0.996507i	$-0.473385\pi$
$877$	4.94644	0.167029	0.0835146	+	0.996507i	$0.473385\pi$
$878$	0	0
$879$	−1.86201	−0.0628039
$880$	0	0
$881$	2.53033	0.0852490	0.0426245	−	0.999091i	$-0.486428\pi$
$881$	2.53033	0.0852490	0.0426245	+	0.999091i	$0.486428\pi$
$882$	0	0
$883$	29.7430	1.00093	0.500465	−	0.865757i	$-0.333162\pi$
$883$	29.7430	1.00093	0.500465	+	0.865757i	$0.333162\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	45.5450	1.52925	0.764626	−	0.644474i	$-0.222924\pi$
$887$	45.5450	1.52925	0.764626	+	0.644474i	$0.222924\pi$
$888$	0	0
$889$	19.3591	0.649282
$890$	0	0
$891$	−26.9568	−0.903086
$892$	0	0
$893$	11.7086	0.391813
$894$	0	0
$895$	0	0
$896$	0	0
$897$	2.90059	0.0968480
$898$	0	0
$899$	−15.5960	−0.520155
$900$	0	0
$901$	68.0897	2.26840
$902$	0	0
$903$	−5.45428	−0.181507
$904$	0	0
$905$	0	0
$906$	0	0
$907$	33.2034	1.10250	0.551250	−	0.834340i	$-0.314151\pi$
$907$	33.2034	1.10250	0.551250	+	0.834340i	$0.314151\pi$
$908$	0	0
$909$	4.64093	0.153930
$910$	0	0
$911$	−55.7788	−1.84803	−0.924017	−	0.382351i	$-0.875114\pi$
$911$	−55.7788	−1.84803	−0.924017	+	0.382351i	$0.875114\pi$
$912$	0	0
$913$	29.3853	0.972513
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−43.2715	−1.42895
$918$	0	0
$919$	28.5769	0.942665	0.471333	−	0.881956i	$-0.343773\pi$
$919$	28.5769	0.942665	0.471333	+	0.881956i	$0.343773\pi$
$920$	0	0
$921$	−8.91020	−0.293601
$922$	0	0
$923$	28.3947	0.934622
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−15.5994	−0.512352
$928$	0	0
$929$	−54.4937	−1.78788	−0.893940	−	0.448186i	$-0.852070\pi$
$929$	−54.4937	−1.78788	−0.893940	+	0.448186i	$0.852070\pi$
$930$	0	0
$931$	3.15544	0.103415
$932$	0	0
$933$	2.23180	0.0730659
$934$	0	0
$935$	0	0
$936$	0	0
$937$	37.1484	1.21359	0.606793	−	0.794860i	$-0.292456\pi$
$937$	37.1484	1.21359	0.606793	+	0.794860i	$0.292456\pi$
$938$	0	0
$939$	0.493704	0.0161114
$940$	0	0
$941$	32.3973	1.05612	0.528061	−	0.849206i	$-0.322919\pi$
$941$	32.3973	1.05612	0.528061	+	0.849206i	$0.322919\pi$
$942$	0	0
$943$	1.56301	0.0508986
$944$	0	0
$945$	0	0
$946$	0	0
$947$	35.4662	1.15250	0.576249	−	0.817275i	$-0.304516\pi$
$947$	35.4662	1.15250	0.576249	+	0.817275i	$0.304516\pi$
$948$	0	0
$949$	−5.64752	−0.183326
$950$	0	0
$951$	14.3557	0.465516
$952$	0	0
$953$	14.3411	0.464553	0.232277	−	0.972650i	$-0.425383\pi$
$953$	14.3411	0.464553	0.232277	+	0.972650i	$0.425383\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−13.3908	−0.432864
$958$	0	0
$959$	−25.3125	−0.817383
$960$	0	0
$961$	−24.2435	−0.782048
$962$	0	0
$963$	41.8458	1.34846
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−27.9351	−0.898331	−0.449165	−	0.893449i	$-0.648279\pi$
$967$	−27.9351	−0.898331	−0.449165	+	0.893449i	$0.648279\pi$
$968$	0	0
$969$	−3.11222	−0.0999788
$970$	0	0
$971$	42.5993	1.36708	0.683539	−	0.729914i	$-0.260440\pi$
$971$	42.5993	1.36708	0.683539	+	0.729914i	$0.260440\pi$
$972$	0	0
$973$	10.4132	0.333833
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−39.5597	−1.26563	−0.632813	−	0.774304i	$-0.718100\pi$
$977$	−39.5597	−1.26563	−0.632813	+	0.774304i	$0.718100\pi$
$978$	0	0
$979$	−68.7114	−2.19603
$980$	0	0
$981$	−31.7564	−1.01390
$982$	0	0
$983$	−39.7689	−1.26843	−0.634215	−	0.773157i	$-0.718677\pi$
$983$	−39.7689	−1.26843	−0.634215	+	0.773157i	$0.718677\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−20.0398	−0.637874
$988$	0	0
$989$	−8.31087	−0.264270
$990$	0	0
$991$	−55.2019	−1.75355	−0.876773	−	0.480905i	$-0.840308\pi$
$991$	−55.2019	−1.75355	−0.876773	+	0.480905i	$0.840308\pi$
$992$	0	0
$993$	−4.29664	−0.136350
$994$	0	0
$995$	0	0
$996$	0	0
$997$	11.6543	0.369097	0.184548	−	0.982823i	$-0.440918\pi$
$997$	11.6543	0.369097	0.184548	+	0.982823i	$0.440918\pi$
$998$	0	0
$999$	13.1987	0.417587

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7600.2.a.ck.1.4		6
4.3	odd	2		475.2.a.j.1.6		6
5.2	odd	4		1520.2.d.h.609.3		6
5.3	odd	4		1520.2.d.h.609.4		6
5.4	even	2	inner	7600.2.a.ck.1.3		6
12.11	even	2		4275.2.a.br.1.1		6
20.3	even	4		95.2.b.b.39.1	✓	6
20.7	even	4		95.2.b.b.39.6	yes	6
20.19	odd	2		475.2.a.j.1.1		6
60.23	odd	4		855.2.c.d.514.6		6
60.47	odd	4		855.2.c.d.514.1		6
60.59	even	2		4275.2.a.br.1.6		6
76.75	even	2		9025.2.a.bx.1.1		6
380.227	odd	4		1805.2.b.e.1084.1		6
380.303	odd	4		1805.2.b.e.1084.6		6
380.379	even	2		9025.2.a.bx.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.2.b.b.39.1	✓	6	20.3	even	4
95.2.b.b.39.6	yes	6	20.7	even	4
475.2.a.j.1.1		6	20.19	odd	2
475.2.a.j.1.6		6	4.3	odd	2
855.2.c.d.514.1		6	60.47	odd	4
855.2.c.d.514.6		6	60.23	odd	4
1520.2.d.h.609.3		6	5.2	odd	4
1520.2.d.h.609.4		6	5.3	odd	4
1805.2.b.e.1084.1		6	380.227	odd	4
1805.2.b.e.1084.6		6	380.303	odd	4
4275.2.a.br.1.1		6	12.11	even	2
4275.2.a.br.1.6		6	60.59	even	2
7600.2.a.ck.1.3		6	5.4	even	2	inner
7600.2.a.ck.1.4		6	1.1	even	1	trivial
9025.2.a.bx.1.1		6	76.75	even	2
9025.2.a.bx.1.6		6	380.379	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+0.537080 q^{3} -3.18676 q^{7} -2.71155 q^{9} +O(q^{10})\)	\(q+0.537080 q^{3} -3.18676 q^{7} -2.71155 q^{9} -4.15544 q^{11} -2.07086 q^{13} -5.79470 q^{17} +1.00000 q^{19} -1.71155 q^{21} -2.60794 q^{23} -3.06756 q^{27} +6.00000 q^{29} -2.59933 q^{31} -2.23180 q^{33} -4.30266 q^{37} -1.11222 q^{39} -0.599328 q^{41} +3.18676 q^{43} +11.7086 q^{47} +3.15544 q^{49} -3.11222 q^{51} -11.7503 q^{53} +0.537080 q^{57} +1.71155 q^{59} -8.75476 q^{61} +8.64104 q^{63} +4.76228 q^{67} -1.40067 q^{69} -13.7115 q^{71} +2.72714 q^{73} +13.2424 q^{77} +1.40067 q^{79} +6.48711 q^{81} -7.07154 q^{83} +3.22248 q^{87} +16.5353 q^{89} +6.59933 q^{91} -1.39605 q^{93} -2.07086 q^{97} +11.2677 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 14 q^{9}+O(q^{10}) \) 6 * q + 14 * q^9	\( 6 q + 14 q^{9} - 2 q^{11} + 6 q^{19} + 20 q^{21} + 36 q^{29} + 8 q^{39} + 12 q^{41} - 4 q^{49} - 4 q^{51} - 20 q^{59} - 14 q^{61} - 24 q^{69} - 52 q^{71} + 24 q^{79} + 38 q^{81} + 24 q^{89} + 24 q^{91} + 30 q^{99}+O(q^{100}) \) 6 * q + 14 * q^9 - 2 * q^11 + 6 * q^19 + 20 * q^21 + 36 * q^29 + 8 * q^39 + 12 * q^41 - 4 * q^49 - 4 * q^51 - 20 * q^59 - 14 * q^61 - 24 * q^69 - 52 * q^71 + 24 * q^79 + 38 * q^81 + 24 * q^89 + 24 * q^91 + 30 * q^99

Introduction

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