LMFDB - Embedded newform 7600.2.a.bz.1.3

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:	$3$
Coefficient field:	3.3.993.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x + 3 $ x^3 - x^2 - 6*x + 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 950)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.25342$ 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)

$f(q)$	$=$	$q+3.25342 q^{3} -0.0778929 q^{7} +7.58473 q^{9} +O(q^{10})$	$q+3.25342 q^{3} -0.0778929 q^{7} +7.58473 q^{9} +4.50684 q^{11} +5.33131 q^{13} -7.33131 q^{17} -1.00000 q^{19} -0.253418 q^{21} +3.40920 q^{23} +14.9160 q^{27} -1.33131 q^{29} +2.50684 q^{31} +14.6626 q^{33} +5.50684 q^{37} +17.3450 q^{39} +0.506836 q^{43} -5.66262 q^{47} -6.99393 q^{49} -23.8518 q^{51} +12.9358 q^{53} -3.25342 q^{57} -7.56499 q^{59} -2.15579 q^{61} -0.590796 q^{63} -4.58473 q^{67} +11.0916 q^{69} +10.8579 q^{71} +5.09763 q^{73} -0.351050 q^{77} -17.0137 q^{79} +25.7739 q^{81} -13.1695 q^{83} -4.33131 q^{87} +15.0137 q^{89} -0.415271 q^{91} +8.15579 q^{93} -7.67629 q^{97} +34.1831 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{3} + 2 q^{7} + 5 q^{9}+O(q^{10}) $ 3 * q + 2 * q^3 + 2 * q^7 + 5 * q^9	$ 3 q + 2 q^{3} + 2 q^{7} + 5 q^{9} - 2 q^{11} + 6 q^{13} - 12 q^{17} - 3 q^{19} + 7 q^{21} - 2 q^{23} + 17 q^{27} + 6 q^{29} - 8 q^{31} + 24 q^{33} + q^{37} + 11 q^{39} - 14 q^{43} + 3 q^{47} + 9 q^{49} - 15 q^{51} + 10 q^{53} - 2 q^{57} - 6 q^{59} - 2 q^{61} - 14 q^{63} + 4 q^{67} + 6 q^{71} + 12 q^{73} + 10 q^{77} - 20 q^{79} + 23 q^{81} - 4 q^{83} - 3 q^{87} + 14 q^{89} - 19 q^{91} + 20 q^{93} + 28 q^{97} + 36 q^{99}+O(q^{100}) $ 3 * q + 2 * q^3 + 2 * q^7 + 5 * q^9 - 2 * q^11 + 6 * q^13 - 12 * q^17 - 3 * q^19 + 7 * q^21 - 2 * q^23 + 17 * q^27 + 6 * q^29 - 8 * q^31 + 24 * q^33 + q^37 + 11 * q^39 - 14 * q^43 + 3 * q^47 + 9 * q^49 - 15 * q^51 + 10 * q^53 - 2 * q^57 - 6 * q^59 - 2 * q^61 - 14 * q^63 + 4 * q^67 + 6 * q^71 + 12 * q^73 + 10 * q^77 - 20 * q^79 + 23 * q^81 - 4 * q^83 - 3 * q^87 + 14 * q^89 - 19 * q^91 + 20 * q^93 + 28 * q^97 + 36 * 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
$3$	3.25342	1.87836	0.939181	−	0.343423i	$-0.111586\pi$
$3$	3.25342	1.87836	0.939181	+	0.343423i	$0.111586\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.0778929	−0.0294407	−0.0147204	−	0.999892i	$-0.504686\pi$
$7$	−0.0778929	−0.0294407	−0.0147204	+	0.999892i	$0.504686\pi$
$8$	0	0
$9$	7.58473	2.52824
$10$	0	0
$11$	4.50684	1.35886	0.679431	−	0.733739i	$-0.262227\pi$
$11$	4.50684	1.35886	0.679431	+	0.733739i	$0.262227\pi$
$12$	0	0
$13$	5.33131	1.47864	0.739320	−	0.673354i	$-0.235147\pi$
$13$	5.33131	1.47864	0.739320	+	0.673354i	$0.235147\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−7.33131	−1.77810	−0.889052	−	0.457806i	$-0.848635\pi$
$17$	−7.33131	−1.77810	−0.889052	+	0.457806i	$0.848635\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−0.253418	−0.0553003
$22$	0	0
$23$	3.40920	0.710868	0.355434	−	0.934701i	$-0.384333\pi$
$23$	3.40920	0.710868	0.355434	+	0.934701i	$0.384333\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	14.9160	2.87059
$28$	0	0
$29$	−1.33131	−0.247218	−0.123609	−	0.992331i	$-0.539447\pi$
$29$	−1.33131	−0.247218	−0.123609	+	0.992331i	$0.539447\pi$
$30$	0	0
$31$	2.50684	0.450241	0.225121	−	0.974331i	$-0.427722\pi$
$31$	2.50684	0.450241	0.225121	+	0.974331i	$0.427722\pi$
$32$	0	0
$33$	14.6626	2.55243
$34$	0	0
$35$	0	0
$36$	0	0
$37$	5.50684	0.905318	0.452659	−	0.891684i	$-0.350475\pi$
$37$	5.50684	0.905318	0.452659	+	0.891684i	$0.350475\pi$
$38$	0	0
$39$	17.3450	2.77742
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	0.506836	0.0772918	0.0386459	−	0.999253i	$-0.487696\pi$
$43$	0.506836	0.0772918	0.0386459	+	0.999253i	$0.487696\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−5.66262	−0.825978	−0.412989	−	0.910736i	$-0.635515\pi$
$47$	−5.66262	−0.825978	−0.412989	+	0.910736i	$0.635515\pi$
$48$	0	0
$49$	−6.99393	−0.999133
$50$	0	0
$51$	−23.8518	−3.33992
$52$	0	0
$53$	12.9358	1.77687	0.888433	−	0.459006i	$-0.151794\pi$
$53$	12.9358	1.77687	0.888433	+	0.459006i	$0.151794\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−3.25342	−0.430926
$58$	0	0
$59$	−7.56499	−0.984878	−0.492439	−	0.870347i	$-0.663894\pi$
$59$	−7.56499	−0.984878	−0.492439	+	0.870347i	$0.663894\pi$
$60$	0	0
$61$	−2.15579	−0.276020	−0.138010	−	0.990431i	$-0.544071\pi$
$61$	−2.15579	−0.276020	−0.138010	+	0.990431i	$0.544071\pi$
$62$	0	0
$63$	−0.590796	−0.0744333
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−4.58473	−0.560114	−0.280057	−	0.959983i	$-0.590353\pi$
$67$	−4.58473	−0.560114	−0.280057	+	0.959983i	$0.590353\pi$
$68$	0	0
$69$	11.0916	1.33527
$70$	0	0
$71$	10.8579	1.28859	0.644297	−	0.764775i	$-0.277150\pi$
$71$	10.8579	1.28859	0.644297	+	0.764775i	$0.277150\pi$
$72$	0	0
$73$	5.09763	0.596633	0.298316	−	0.954467i	$-0.403575\pi$
$73$	5.09763	0.596633	0.298316	+	0.954467i	$0.403575\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−0.351050	−0.0400059
$78$	0	0
$79$	−17.0137	−1.91419	−0.957094	−	0.289778i	$-0.906418\pi$
$79$	−17.0137	−1.91419	−0.957094	+	0.289778i	$0.906418\pi$
$80$	0	0
$81$	25.7739	2.86377
$82$	0	0
$83$	−13.1695	−1.44554	−0.722768	−	0.691091i	$-0.757130\pi$
$83$	−13.1695	−1.44554	−0.722768	+	0.691091i	$0.757130\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−4.33131	−0.464365
$88$	0	0
$89$	15.0137	1.59145	0.795723	−	0.605661i	$-0.207091\pi$
$89$	15.0137	1.59145	0.795723	+	0.605661i	$0.207091\pi$
$90$	0	0
$91$	−0.415271	−0.0435322
$92$	0	0
$93$	8.15579	0.845716
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−7.67629	−0.779410	−0.389705	−	0.920940i	$-0.627423\pi$
$97$	−7.67629	−0.779410	−0.389705	+	0.920940i	$0.627423\pi$
$98$	0	0
$99$	34.1831	3.43553
$100$	0	0
$101$	−4.15579	−0.413516	−0.206758	−	0.978392i	$-0.566291\pi$
$101$	−4.15579	−0.413516	−0.206758	+	0.978392i	$0.566291\pi$
$102$	0	0
$103$	2.35105	0.231656	0.115828	−	0.993269i	$-0.463048\pi$
$103$	2.35105	0.231656	0.115828	+	0.993269i	$0.463048\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−14.0334	−1.35666	−0.678331	−	0.734757i	$-0.737296\pi$
$107$	−14.0334	−1.35666	−0.678331	+	0.734757i	$0.737296\pi$
$108$	0	0
$109$	−0.0778929	−0.00746078	−0.00373039	−	0.999993i	$-0.501187\pi$
$109$	−0.0778929	−0.00746078	−0.00373039	+	0.999993i	$0.501187\pi$
$110$	0	0
$111$	17.9160	1.70052
$112$	0	0
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	40.4365	3.73836
$118$	0	0
$119$	0.571057	0.0523487
$120$	0	0
$121$	9.31157	0.846506
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	17.8321	1.58234	0.791171	−	0.611596i	$-0.209472\pi$
$127$	17.8321	1.58234	0.791171	+	0.611596i	$0.209472\pi$
$128$	0	0
$129$	1.64895	0.145182
$130$	0	0
$131$	1.49316	0.130458	0.0652292	−	0.997870i	$-0.479222\pi$
$131$	1.49316	0.130458	0.0652292	+	0.997870i	$0.479222\pi$
$132$	0	0
$133$	0.0778929	0.00675417
$134$	0	0
$135$	0	0
$136$	0	0
$137$	8.42894	0.720133	0.360067	−	0.932927i	$-0.382754\pi$
$137$	8.42894	0.720133	0.360067	+	0.932927i	$0.382754\pi$
$138$	0	0
$139$	−8.81841	−0.747968	−0.373984	−	0.927435i	$-0.622008\pi$
$139$	−8.81841	−0.747968	−0.373984	+	0.927435i	$0.622008\pi$
$140$	0	0
$141$	−18.4229	−1.55149
$142$	0	0
$143$	24.0273	2.00927
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−22.7542	−1.87673
$148$	0	0
$149$	17.6763	1.44810	0.724049	−	0.689748i	$-0.242279\pi$
$149$	17.6763	1.44810	0.724049	+	0.689748i	$0.242279\pi$
$150$	0	0
$151$	−20.0000	−1.62758	−0.813788	−	0.581161i	$-0.802599\pi$
$151$	−20.0000	−1.62758	−0.813788	+	0.581161i	$0.802599\pi$
$152$	0	0
$153$	−55.6060	−4.49548
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−0.506836	−0.0404499	−0.0202250	−	0.999795i	$-0.506438\pi$
$157$	−0.506836	−0.0404499	−0.0202250	+	0.999795i	$0.506438\pi$
$158$	0	0
$159$	42.0855	3.33760
$160$	0	0
$161$	−0.265553	−0.0209285
$162$	0	0
$163$	0.830542	0.0650531	0.0325265	−	0.999471i	$-0.489645\pi$
$163$	0.830542	0.0650531	0.0325265	+	0.999471i	$0.489645\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	16.5068	1.27734	0.638669	−	0.769482i	$-0.279485\pi$
$167$	16.5068	1.27734	0.638669	+	0.769482i	$0.279485\pi$
$168$	0	0
$169$	15.4229	1.18638
$170$	0	0
$171$	−7.58473	−0.580019
$172$	0	0
$173$	−18.8321	−1.43178	−0.715888	−	0.698215i	$-0.753978\pi$
$173$	−18.8321	−1.43178	−0.715888	+	0.698215i	$0.753978\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−24.6121	−1.84996
$178$	0	0
$179$	7.15579	0.534849	0.267424	−	0.963579i	$-0.413827\pi$
$179$	7.15579	0.534849	0.267424	+	0.963579i	$0.413827\pi$
$180$	0	0
$181$	12.5205	0.930642	0.465321	−	0.885142i	$-0.345939\pi$
$181$	12.5205	0.930642	0.465321	+	0.885142i	$0.345939\pi$
$182$	0	0
$183$	−7.01367	−0.518466
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−33.0410	−2.41620
$188$	0	0
$189$	−1.16185	−0.0845124
$190$	0	0
$191$	4.90237	0.354723	0.177361	−	0.984146i	$-0.443244\pi$
$191$	4.90237	0.354723	0.177361	+	0.984146i	$0.443244\pi$
$192$	0	0
$193$	−18.1558	−1.30688	−0.653441	−	0.756977i	$-0.726675\pi$
$193$	−18.1558	−1.30688	−0.653441	+	0.756977i	$0.726675\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2.98633	0.212767	0.106384	−	0.994325i	$-0.466073\pi$
$197$	2.98633	0.212767	0.106384	+	0.994325i	$0.466073\pi$
$198$	0	0
$199$	3.06422	0.217217	0.108608	−	0.994085i	$-0.465361\pi$
$199$	3.06422	0.217217	0.108608	+	0.994085i	$0.465361\pi$
$200$	0	0
$201$	−14.9160	−1.05210
$202$	0	0
$203$	0.103700	0.00727829
$204$	0	0
$205$	0	0
$206$	0	0
$207$	25.8579	1.79725
$208$	0	0
$209$	−4.50684	−0.311744
$210$	0	0
$211$	−19.2534	−1.32546	−0.662730	−	0.748858i	$-0.730602\pi$
$211$	−19.2534	−1.32546	−0.662730	+	0.748858i	$0.730602\pi$
$212$	0	0
$213$	35.3252	2.42045
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−0.195265	−0.0132554
$218$	0	0
$219$	16.5847	1.12069
$220$	0	0
$221$	−39.0855	−2.62918
$222$	0	0
$223$	−14.5068	−0.971450	−0.485725	−	0.874112i	$-0.661444\pi$
$223$	−14.5068	−0.971450	−0.485725	+	0.874112i	$0.661444\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−21.5984	−1.43354	−0.716768	−	0.697312i	$-0.754379\pi$
$227$	−21.5984	−1.43354	−0.716768	+	0.697312i	$0.754379\pi$
$228$	0	0
$229$	−19.0137	−1.25646	−0.628229	−	0.778028i	$-0.716220\pi$
$229$	−19.0137	−1.25646	−0.628229	+	0.778028i	$0.716220\pi$
$230$	0	0
$231$	−1.14211	−0.0751456
$232$	0	0
$233$	6.01367	0.393969	0.196984	−	0.980407i	$-0.436885\pi$
$233$	6.01367	0.393969	0.196984	+	0.980407i	$0.436885\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−55.3526	−3.59554
$238$	0	0
$239$	−15.4092	−0.996739	−0.498369	−	0.866965i	$-0.666068\pi$
$239$	−15.4092	−0.996739	−0.498369	+	0.866965i	$0.666068\pi$
$240$	0	0
$241$	−4.81841	−0.310381	−0.155190	−	0.987885i	$-0.549599\pi$
$241$	−4.81841	−0.310381	−0.155190	+	0.987885i	$0.549599\pi$
$242$	0	0
$243$	39.1052	2.50860
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−5.33131	−0.339223
$248$	0	0
$249$	−42.8458	−2.71524
$250$	0	0
$251$	−1.52051	−0.0959736	−0.0479868	−	0.998848i	$-0.515281\pi$
$251$	−1.52051	−0.0959736	−0.0479868	+	0.998848i	$0.515281\pi$
$252$	0	0
$253$	15.3647	0.965972
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−16.5068	−1.02967	−0.514834	−	0.857290i	$-0.672146\pi$
$257$	−16.5068	−1.02967	−0.514834	+	0.857290i	$0.672146\pi$
$258$	0	0
$259$	−0.428943	−0.0266532
$260$	0	0
$261$	−10.0976	−0.625028
$262$	0	0
$263$	−18.0273	−1.11161	−0.555807	−	0.831311i	$-0.687591\pi$
$263$	−18.0273	−1.11161	−0.555807	+	0.831311i	$0.687591\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	48.8458	2.98931
$268$	0	0
$269$	20.2089	1.23216	0.616080	−	0.787683i	$-0.288720\pi$
$269$	20.2089	1.23216	0.616080	+	0.787683i	$0.288720\pi$
$270$	0	0
$271$	−6.08396	−0.369574	−0.184787	−	0.982779i	$-0.559160\pi$
$271$	−6.08396	−0.369574	−0.184787	+	0.982779i	$0.559160\pi$
$272$	0	0
$273$	−1.35105	−0.0817693
$274$	0	0
$275$	0	0
$276$	0	0
$277$	31.3647	1.88452	0.942262	−	0.334877i	$-0.108695\pi$
$277$	31.3647	1.88452	0.942262	+	0.334877i	$0.108695\pi$
$278$	0	0
$279$	19.0137	1.13832
$280$	0	0
$281$	11.3252	0.675607	0.337804	−	0.941217i	$-0.390316\pi$
$281$	11.3252	0.675607	0.337804	+	0.941217i	$0.390316\pi$
$282$	0	0
$283$	26.1437	1.55408	0.777039	−	0.629452i	$-0.216721\pi$
$283$	26.1437	1.55408	0.777039	+	0.629452i	$0.216721\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	36.7481	2.16165
$290$	0	0
$291$	−24.9742	−1.46401
$292$	0	0
$293$	−1.33131	−0.0777760	−0.0388880	−	0.999244i	$-0.512382\pi$
$293$	−1.33131	−0.0777760	−0.0388880	+	0.999244i	$0.512382\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	67.2241	3.90074
$298$	0	0
$299$	18.1755	1.05112
$300$	0	0
$301$	−0.0394789	−0.00227553
$302$	0	0
$303$	−13.5205	−0.776733
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−2.84421	−0.162328	−0.0811639	−	0.996701i	$-0.525864\pi$
$307$	−2.84421	−0.162328	−0.0811639	+	0.996701i	$0.525864\pi$
$308$	0	0
$309$	7.64895	0.435134
$310$	0	0
$311$	16.3895	0.929361	0.464681	−	0.885478i	$-0.346169\pi$
$311$	16.3895	0.929361	0.464681	+	0.885478i	$0.346169\pi$
$312$	0	0
$313$	21.0471	1.18965	0.594826	−	0.803855i	$-0.297221\pi$
$313$	21.0471	1.18965	0.594826	+	0.803855i	$0.297221\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	30.8902	1.73497	0.867484	−	0.497465i	$-0.165736\pi$
$317$	30.8902	1.73497	0.867484	+	0.497465i	$0.165736\pi$
$318$	0	0
$319$	−6.00000	−0.335936
$320$	0	0
$321$	−45.6566	−2.54830
$322$	0	0
$323$	7.33131	0.407925
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−0.253418	−0.0140140
$328$	0	0
$329$	0.441078	0.0243174
$330$	0	0
$331$	−28.3845	−1.56015	−0.780076	−	0.625685i	$-0.784819\pi$
$331$	−28.3845	−1.56015	−0.780076	+	0.625685i	$0.784819\pi$
$332$	0	0
$333$	41.7679	2.28886
$334$	0	0
$335$	0	0
$336$	0	0
$337$	17.8716	0.973526	0.486763	−	0.873534i	$-0.338178\pi$
$337$	17.8716	0.973526	0.486763	+	0.873534i	$0.338178\pi$
$338$	0	0
$339$	19.5205	1.06021
$340$	0	0
$341$	11.2979	0.611816
$342$	0	0
$343$	1.09003	0.0588560
$344$	0	0
$345$	0	0
$346$	0	0
$347$	33.0410	1.77373	0.886867	−	0.462024i	$-0.152877\pi$
$347$	33.0410	1.77373	0.886867	+	0.462024i	$0.152877\pi$
$348$	0	0
$349$	−19.7158	−1.05536	−0.527681	−	0.849443i	$-0.676938\pi$
$349$	−19.7158	−1.05536	−0.527681	+	0.849443i	$0.676938\pi$
$350$	0	0
$351$	79.5220	4.24457
$352$	0	0
$353$	5.90997	0.314556	0.157278	−	0.987554i	$-0.449728\pi$
$353$	5.90997	0.314556	0.157278	+	0.987554i	$0.449728\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	1.85789	0.0983298
$358$	0	0
$359$	7.00760	0.369847	0.184924	−	0.982753i	$-0.440796\pi$
$359$	7.00760	0.369847	0.184924	+	0.982753i	$0.440796\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	30.2944	1.59005
$364$	0	0
$365$	0	0
$366$	0	0
$367$	17.7158	0.924756	0.462378	−	0.886683i	$-0.346996\pi$
$367$	17.7158	0.924756	0.462378	+	0.886683i	$0.346996\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−1.00760	−0.0523122
$372$	0	0
$373$	−15.4487	−0.799902	−0.399951	−	0.916536i	$-0.630973\pi$
$373$	−15.4487	−0.799902	−0.399951	+	0.916536i	$0.630973\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−7.09763	−0.365547
$378$	0	0
$379$	−34.4563	−1.76990	−0.884950	−	0.465685i	$-0.845808\pi$
$379$	−34.4563	−1.76990	−0.884950	+	0.465685i	$0.845808\pi$
$380$	0	0
$381$	58.0152	2.97221
$382$	0	0
$383$	−12.0273	−0.614569	−0.307284	−	0.951618i	$-0.599420\pi$
$383$	−12.0273	−0.614569	−0.307284	+	0.951618i	$0.599420\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3.84421	0.195412
$388$	0	0
$389$	−15.3647	−0.779022	−0.389511	−	0.921022i	$-0.627356\pi$
$389$	−15.3647	−0.779022	−0.389511	+	0.921022i	$0.627356\pi$
$390$	0	0
$391$	−24.9939	−1.26400
$392$	0	0
$393$	4.85789	0.245048
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−1.32524	−0.0665121	−0.0332560	−	0.999447i	$-0.510588\pi$
$397$	−1.32524	−0.0665121	−0.0332560	+	0.999447i	$0.510588\pi$
$398$	0	0
$399$	0.253418	0.0126868
$400$	0	0
$401$	−6.46736	−0.322964	−0.161482	−	0.986876i	$-0.551627\pi$
$401$	−6.46736	−0.322964	−0.161482	+	0.986876i	$0.551627\pi$
$402$	0	0
$403$	13.3647	0.665744
$404$	0	0
$405$	0	0
$406$	0	0
$407$	24.8184	1.23020
$408$	0	0
$409$	−1.36472	−0.0674812	−0.0337406	−	0.999431i	$-0.510742\pi$
$409$	−1.36472	−0.0674812	−0.0337406	+	0.999431i	$0.510742\pi$
$410$	0	0
$411$	27.4229	1.35267
$412$	0	0
$413$	0.589259	0.0289955
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−28.6900	−1.40495
$418$	0	0
$419$	−16.6231	−0.812094	−0.406047	−	0.913852i	$-0.633093\pi$
$419$	−16.6231	−0.812094	−0.406047	+	0.913852i	$0.633093\pi$
$420$	0	0
$421$	−31.1128	−1.51635	−0.758174	−	0.652053i	$-0.773908\pi$
$421$	−31.1128	−1.51635	−0.758174	+	0.652053i	$0.773908\pi$
$422$	0	0
$423$	−42.9495	−2.08827
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0.167920	0.00812623
$428$	0	0
$429$	78.1710	3.77413
$430$	0	0
$431$	10.1831	0.490504	0.245252	−	0.969459i	$-0.421129\pi$
$431$	10.1831	0.490504	0.245252	+	0.969459i	$0.421129\pi$
$432$	0	0
$433$	−22.1953	−1.06664	−0.533318	−	0.845915i	$-0.679055\pi$
$433$	−22.1953	−1.06664	−0.533318	+	0.845915i	$0.679055\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−3.40920	−0.163084
$438$	0	0
$439$	9.32524	0.445070	0.222535	−	0.974925i	$-0.428567\pi$
$439$	9.32524	0.445070	0.222535	+	0.974925i	$0.428567\pi$
$440$	0	0
$441$	−53.0471	−2.52605
$442$	0	0
$443$	−13.9879	−0.664584	−0.332292	−	0.943177i	$-0.607822\pi$
$443$	−13.9879	−0.664584	−0.332292	+	0.943177i	$0.607822\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	57.5084	2.72005
$448$	0	0
$449$	13.4932	0.636782	0.318391	−	0.947960i	$-0.396858\pi$
$449$	13.4932	0.636782	0.318391	+	0.947960i	$0.396858\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−65.0684	−3.05718
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−9.68236	−0.452922	−0.226461	−	0.974020i	$-0.572716\pi$
$457$	−9.68236	−0.452922	−0.226461	+	0.974020i	$0.572716\pi$
$458$	0	0
$459$	−109.354	−5.10421
$460$	0	0
$461$	8.66262	0.403459	0.201729	−	0.979441i	$-0.435344\pi$
$461$	8.66262	0.403459	0.201729	+	0.979441i	$0.435344\pi$
$462$	0	0
$463$	28.0015	1.30134	0.650671	−	0.759360i	$-0.274488\pi$
$463$	28.0015	1.30134	0.650671	+	0.759360i	$0.274488\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	16.9742	0.785472	0.392736	−	0.919651i	$-0.371529\pi$
$467$	16.9742	0.785472	0.392736	+	0.919651i	$0.371529\pi$
$468$	0	0
$469$	0.357118	0.0164902
$470$	0	0
$471$	−1.64895	−0.0759796
$472$	0	0
$473$	2.28423	0.105029
$474$	0	0
$475$	0	0
$476$	0	0
$477$	98.1144	4.49235
$478$	0	0
$479$	−10.0532	−0.459340	−0.229670	−	0.973269i	$-0.573765\pi$
$479$	−10.0532	−0.459340	−0.229670	+	0.973269i	$0.573765\pi$
$480$	0	0
$481$	29.3587	1.33864
$482$	0	0
$483$	−0.863954	−0.0393113
$484$	0	0
$485$	0	0
$486$	0	0
$487$	19.2089	0.870440	0.435220	−	0.900324i	$-0.356671\pi$
$487$	19.2089	0.870440	0.435220	+	0.900324i	$0.356671\pi$
$488$	0	0
$489$	2.70210	0.122193
$490$	0	0
$491$	−5.32524	−0.240325	−0.120162	−	0.992754i	$-0.538342\pi$
$491$	−5.32524	−0.240325	−0.120162	+	0.992754i	$0.538342\pi$
$492$	0	0
$493$	9.76025	0.439580
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−0.845752	−0.0379372
$498$	0	0
$499$	11.9605	0.535426	0.267713	−	0.963499i	$-0.413732\pi$
$499$	11.9605	0.535426	0.267713	+	0.963499i	$0.413732\pi$
$500$	0	0
$501$	53.7036	2.39930
$502$	0	0
$503$	−19.3313	−0.861941	−0.430970	−	0.902366i	$-0.641829\pi$
$503$	−19.3313	−0.861941	−0.430970	+	0.902366i	$0.641829\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	50.1771	2.22844
$508$	0	0
$509$	−36.1573	−1.60265	−0.801323	−	0.598232i	$-0.795870\pi$
$509$	−36.1573	−1.60265	−0.801323	+	0.598232i	$0.795870\pi$
$510$	0	0
$511$	−0.397069	−0.0175653
$512$	0	0
$513$	−14.9160	−0.658559
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−25.5205	−1.12239
$518$	0	0
$519$	−61.2686	−2.68939
$520$	0	0
$521$	−31.5205	−1.38094	−0.690469	−	0.723362i	$-0.742596\pi$
$521$	−31.5205	−1.38094	−0.690469	+	0.723362i	$0.742596\pi$
$522$	0	0
$523$	5.59840	0.244801	0.122400	−	0.992481i	$-0.460941\pi$
$523$	5.59840	0.244801	0.122400	+	0.992481i	$0.460941\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−18.3784	−0.800575
$528$	0	0
$529$	−11.3773	−0.494667
$530$	0	0
$531$	−57.3784	−2.49001
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	23.2808	1.00464
$538$	0	0
$539$	−31.5205	−1.35768
$540$	0	0
$541$	15.1968	0.653362	0.326681	−	0.945135i	$-0.394070\pi$
$541$	15.1968	0.653362	0.326681	+	0.945135i	$0.394070\pi$
$542$	0	0
$543$	40.7344	1.74808
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−26.6505	−1.13949	−0.569746	−	0.821821i	$-0.692959\pi$
$547$	−26.6505	−1.13949	−0.569746	+	0.821821i	$0.692959\pi$
$548$	0	0
$549$	−16.3511	−0.697846
$550$	0	0
$551$	1.33131	0.0567158
$552$	0	0
$553$	1.32524	0.0563551
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−12.8458	−0.544292	−0.272146	−	0.962256i	$-0.587733\pi$
$557$	−12.8458	−0.544292	−0.272146	+	0.962256i	$0.587733\pi$
$558$	0	0
$559$	2.70210	0.114287
$560$	0	0
$561$	−107.496	−4.53849
$562$	0	0
$563$	10.1968	0.429744	0.214872	−	0.976642i	$-0.431067\pi$
$563$	10.1968	0.429744	0.214872	+	0.976642i	$0.431067\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−2.00760	−0.0843115
$568$	0	0
$569$	−24.3784	−1.02200	−0.510998	−	0.859582i	$-0.670724\pi$
$569$	−24.3784	−1.02200	−0.510998	+	0.859582i	$0.670724\pi$
$570$	0	0
$571$	−8.32371	−0.348336	−0.174168	−	0.984716i	$-0.555724\pi$
$571$	−8.32371	−0.348336	−0.174168	+	0.984716i	$0.555724\pi$
$572$	0	0
$573$	15.9495	0.666298
$574$	0	0
$575$	0	0
$576$	0	0
$577$	7.29290	0.303607	0.151804	−	0.988411i	$-0.451492\pi$
$577$	7.29290	0.303607	0.151804	+	0.988411i	$0.451492\pi$
$578$	0	0
$579$	−59.0684	−2.45480
$580$	0	0
$581$	1.02581	0.0425576
$582$	0	0
$583$	58.2994	2.41452
$584$	0	0
$585$	0	0
$586$	0	0
$587$	5.53264	0.228357	0.114178	−	0.993460i	$-0.463576\pi$
$587$	5.53264	0.228357	0.114178	+	0.993460i	$0.463576\pi$
$588$	0	0
$589$	−2.50684	−0.103292
$590$	0	0
$591$	9.71577	0.399653
$592$	0	0
$593$	26.3252	1.08105	0.540524	−	0.841329i	$-0.318226\pi$
$593$	26.3252	1.08105	0.540524	+	0.841329i	$0.318226\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	9.96919	0.408012
$598$	0	0
$599$	−22.2994	−0.911130	−0.455565	−	0.890202i	$-0.650563\pi$
$599$	−22.2994	−0.911130	−0.455565	+	0.890202i	$0.650563\pi$
$600$	0	0
$601$	32.4947	1.32549	0.662743	−	0.748847i	$-0.269392\pi$
$601$	32.4947	1.32549	0.662743	+	0.748847i	$0.269392\pi$
$602$	0	0
$603$	−34.7739	−1.41610
$604$	0	0
$605$	0	0
$606$	0	0
$607$	8.35105	0.338959	0.169479	−	0.985534i	$-0.445791\pi$
$607$	8.35105	0.338959	0.169479	+	0.985534i	$0.445791\pi$
$608$	0	0
$609$	0.337378	0.0136713
$610$	0	0
$611$	−30.1892	−1.22132
$612$	0	0
$613$	38.2994	1.54690	0.773450	−	0.633858i	$-0.218529\pi$
$613$	38.2994	1.54690	0.773450	+	0.633858i	$0.218529\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	14.3526	0.577813	0.288907	−	0.957357i	$-0.406708\pi$
$617$	14.3526	0.577813	0.288907	+	0.957357i	$0.406708\pi$
$618$	0	0
$619$	8.62314	0.346593	0.173297	−	0.984870i	$-0.444558\pi$
$619$	8.62314	0.346593	0.173297	+	0.984870i	$0.444558\pi$
$620$	0	0
$621$	50.8518	2.04061
$622$	0	0
$623$	−1.16946	−0.0468533
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−14.6626	−0.585569
$628$	0	0
$629$	−40.3723	−1.60975
$630$	0	0
$631$	10.3374	0.411525	0.205762	−	0.978602i	$-0.434033\pi$
$631$	10.3374	0.411525	0.205762	+	0.978602i	$0.434033\pi$
$632$	0	0
$633$	−62.6394	−2.48969
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−37.2868	−1.47736
$638$	0	0
$639$	82.3541	3.25788
$640$	0	0
$641$	5.88369	0.232392	0.116196	−	0.993226i	$-0.462930\pi$
$641$	5.88369	0.232392	0.116196	+	0.993226i	$0.462930\pi$
$642$	0	0
$643$	−25.5084	−1.00595	−0.502976	−	0.864300i	$-0.667762\pi$
$643$	−25.5084	−1.00595	−0.502976	+	0.864300i	$0.667762\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−5.09157	−0.200170	−0.100085	−	0.994979i	$-0.531911\pi$
$647$	−5.09157	−0.200170	−0.100085	+	0.994979i	$0.531911\pi$
$648$	0	0
$649$	−34.0942	−1.33831
$650$	0	0
$651$	−0.635277	−0.0248985
$652$	0	0
$653$	11.1816	0.437570	0.218785	−	0.975773i	$-0.429791\pi$
$653$	11.1816	0.437570	0.218785	+	0.975773i	$0.429791\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	38.6642	1.50843
$658$	0	0
$659$	13.7542	0.535787	0.267894	−	0.963449i	$-0.413672\pi$
$659$	13.7542	0.535787	0.267894	+	0.963449i	$0.413672\pi$
$660$	0	0
$661$	−12.6171	−0.490747	−0.245374	−	0.969429i	$-0.578911\pi$
$661$	−12.6171	−0.490747	−0.245374	+	0.969429i	$0.578911\pi$
$662$	0	0
$663$	−127.161	−4.93854
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−4.53871	−0.175740
$668$	0	0
$669$	−47.1968	−1.82473
$670$	0	0
$671$	−9.71577	−0.375073
$672$	0	0
$673$	−2.35105	−0.0906263	−0.0453132	−	0.998973i	$-0.514429\pi$
$673$	−2.35105	−0.0906263	−0.0453132	+	0.998973i	$0.514429\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	23.7663	0.913414	0.456707	−	0.889617i	$-0.349029\pi$
$677$	23.7663	0.913414	0.456707	+	0.889617i	$0.349029\pi$
$678$	0	0
$679$	0.597928	0.0229464
$680$	0	0
$681$	−70.2686	−2.69270
$682$	0	0
$683$	28.1695	1.07787	0.538937	−	0.842346i	$-0.318826\pi$
$683$	28.1695	1.07787	0.538937	+	0.842346i	$0.318826\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−61.8594	−2.36008
$688$	0	0
$689$	68.9647	2.62734
$690$	0	0
$691$	−2.32371	−0.0883979	−0.0441990	−	0.999023i	$-0.514074\pi$
$691$	−2.32371	−0.0883979	−0.0441990	+	0.999023i	$0.514074\pi$
$692$	0	0
$693$	−2.66262	−0.101145
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	19.5650	0.740016
$700$	0	0
$701$	23.7036	0.895274	0.447637	−	0.894215i	$-0.352266\pi$
$701$	23.7036	0.895274	0.447637	+	0.894215i	$0.352266\pi$
$702$	0	0
$703$	−5.50684	−0.207694
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0.323706	0.0121742
$708$	0	0
$709$	9.05315	0.339998	0.169999	−	0.985444i	$-0.445624\pi$
$709$	9.05315	0.339998	0.169999	+	0.985444i	$0.445624\pi$
$710$	0	0
$711$	−129.044	−4.83953
$712$	0	0
$713$	8.54631	0.320062
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−50.1326	−1.87224
$718$	0	0
$719$	−35.0734	−1.30802	−0.654008	−	0.756488i	$-0.726914\pi$
$719$	−35.0734	−1.30802	−0.654008	+	0.756488i	$0.726914\pi$
$720$	0	0
$721$	−0.183130	−0.00682012
$722$	0	0
$723$	−15.6763	−0.583008
$724$	0	0
$725$	0	0
$726$	0	0
$727$	30.1386	1.11778	0.558890	−	0.829242i	$-0.311227\pi$
$727$	30.1386	1.11778	0.558890	+	0.829242i	$0.311227\pi$
$728$	0	0
$729$	49.9039	1.84829
$730$	0	0
$731$	−3.71577	−0.137433
$732$	0	0
$733$	−47.8321	−1.76672	−0.883359	−	0.468697i	$-0.844724\pi$
$733$	−47.8321	−1.76672	−0.883359	+	0.468697i	$0.844724\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−20.6626	−0.761117
$738$	0	0
$739$	22.4674	0.826475	0.413238	−	0.910623i	$-0.364398\pi$
$739$	22.4674	0.826475	0.413238	+	0.910623i	$0.364398\pi$
$740$	0	0
$741$	−17.3450	−0.637184
$742$	0	0
$743$	−11.7926	−0.432629	−0.216314	−	0.976324i	$-0.569404\pi$
$743$	−11.7926	−0.432629	−0.216314	+	0.976324i	$0.569404\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−99.8868	−3.65467
$748$	0	0
$749$	1.09310	0.0399411
$750$	0	0
$751$	2.03948	0.0744216	0.0372108	−	0.999307i	$-0.488153\pi$
$751$	2.03948	0.0744216	0.0372108	+	0.999307i	$0.488153\pi$
$752$	0	0
$753$	−4.94685	−0.180273
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−43.3526	−1.57568	−0.787838	−	0.615882i	$-0.788800\pi$
$757$	−43.3526	−1.57568	−0.787838	+	0.615882i	$0.788800\pi$
$758$	0	0
$759$	49.9879	1.81444
$760$	0	0
$761$	6.62921	0.240309	0.120154	−	0.992755i	$-0.461661\pi$
$761$	6.62921	0.240309	0.120154	+	0.992755i	$0.461661\pi$
$762$	0	0
$763$	0.00606730	0.000219651 0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−40.3313	−1.45628
$768$	0	0
$769$	−19.6429	−0.708340	−0.354170	−	0.935181i	$-0.615237\pi$
$769$	−19.6429	−0.708340	−0.354170	+	0.935181i	$0.615237\pi$
$770$	0	0
$771$	−53.7036	−1.93409
$772$	0	0
$773$	−14.4107	−0.518318	−0.259159	−	0.965835i	$-0.583445\pi$
$773$	−14.4107	−0.518318	−0.259159	+	0.965835i	$0.583445\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−1.39553	−0.0500644
$778$	0	0
$779$	0	0
$780$	0	0
$781$	48.9347	1.75102
$782$	0	0
$783$	−19.8579	−0.709663
$784$	0	0
$785$	0	0
$786$	0	0
$787$	24.3176	0.866830	0.433415	−	0.901194i	$-0.357308\pi$
$787$	24.3176	0.866830	0.433415	+	0.901194i	$0.357308\pi$
$788$	0	0
$789$	−58.6505	−2.08801
$790$	0	0
$791$	−0.467357	−0.0166173
$792$	0	0
$793$	−11.4932	−0.408134
$794$	0	0
$795$	0	0
$796$	0	0
$797$	7.30397	0.258720	0.129360	−	0.991598i	$-0.458708\pi$
$797$	7.30397	0.258720	0.129360	+	0.991598i	$0.458708\pi$
$798$	0	0
$799$	41.5144	1.46868
$800$	0	0
$801$	113.875	4.02356
$802$	0	0
$803$	22.9742	0.810742
$804$	0	0
$805$	0	0
$806$	0	0
$807$	65.7481	2.31444
$808$	0	0
$809$	−0.584729	−0.0205580	−0.0102790	−	0.999947i	$-0.503272\pi$
$809$	−0.584729	−0.0205580	−0.0102790	+	0.999947i	$0.503272\pi$
$810$	0	0
$811$	−1.90997	−0.0670682	−0.0335341	−	0.999438i	$-0.510676\pi$
$811$	−1.90997	−0.0670682	−0.0335341	+	0.999438i	$0.510676\pi$
$812$	0	0
$813$	−19.7937	−0.694194
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−0.506836	−0.0177319
$818$	0	0
$819$	−3.14972	−0.110060
$820$	0	0
$821$	−38.2226	−1.33398	−0.666989	−	0.745067i	$-0.732417\pi$
$821$	−38.2226	−1.33398	−0.666989	+	0.745067i	$0.732417\pi$
$822$	0	0
$823$	−45.7481	−1.59468	−0.797340	−	0.603531i	$-0.793760\pi$
$823$	−45.7481	−1.59468	−0.797340	+	0.603531i	$0.793760\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−22.6687	−0.788268	−0.394134	−	0.919053i	$-0.628955\pi$
$827$	−22.6687	−0.788268	−0.394134	+	0.919053i	$0.628955\pi$
$828$	0	0
$829$	−4.63028	−0.160816	−0.0804081	−	0.996762i	$-0.525622\pi$
$829$	−4.63028	−0.160816	−0.0804081	+	0.996762i	$0.525622\pi$
$830$	0	0
$831$	102.043	3.53982
$832$	0	0
$833$	51.2747	1.77656
$834$	0	0
$835$	0	0
$836$	0	0
$837$	37.3921	1.29246
$838$	0	0
$839$	50.4826	1.74285	0.871426	−	0.490527i	$-0.163196\pi$
$839$	50.4826	1.74285	0.871426	+	0.490527i	$0.163196\pi$
$840$	0	0
$841$	−27.2276	−0.938883
$842$	0	0
$843$	36.8458	1.26904
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−0.725305	−0.0249218
$848$	0	0
$849$	85.0562	2.91912
$850$	0	0
$851$	18.7739	0.643562
$852$	0	0
$853$	−36.2105	−1.23982	−0.619912	−	0.784672i	$-0.712832\pi$
$853$	−36.2105	−1.23982	−0.619912	+	0.784672i	$0.712832\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−25.1968	−0.860706	−0.430353	−	0.902661i	$-0.641611\pi$
$857$	−25.1968	−0.860706	−0.430353	+	0.902661i	$0.641611\pi$
$858$	0	0
$859$	−18.8579	−0.643423	−0.321711	−	0.946838i	$-0.604258\pi$
$859$	−18.8579	−0.643423	−0.321711	+	0.946838i	$0.604258\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	15.0137	0.511071	0.255536	−	0.966800i	$-0.417748\pi$
$863$	15.0137	0.511071	0.255536	+	0.966800i	$0.417748\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	119.557	4.06037
$868$	0	0
$869$	−76.6778	−2.60112
$870$	0	0
$871$	−24.4426	−0.828206
$872$	0	0
$873$	−58.2226	−1.97054
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−35.1128	−1.18568	−0.592838	−	0.805322i	$-0.701993\pi$
$877$	−35.1128	−1.18568	−0.592838	+	0.805322i	$0.701993\pi$
$878$	0	0
$879$	−4.33131	−0.146091
$880$	0	0
$881$	−41.3389	−1.39274	−0.696372	−	0.717681i	$-0.745203\pi$
$881$	−41.3389	−1.39274	−0.696372	+	0.717681i	$0.745203\pi$
$882$	0	0
$883$	−12.6353	−0.425211	−0.212605	−	0.977138i	$-0.568195\pi$
$883$	−12.6353	−0.425211	−0.212605	+	0.977138i	$0.568195\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	4.15579	0.139538	0.0697688	−	0.997563i	$-0.477774\pi$
$887$	4.15579	0.139538	0.0697688	+	0.997563i	$0.477774\pi$
$888$	0	0
$889$	−1.38899	−0.0465853
$890$	0	0
$891$	116.159	3.89147
$892$	0	0
$893$	5.66262	0.189492
$894$	0	0
$895$	0	0
$896$	0	0
$897$	59.1326	1.97438
$898$	0	0
$899$	−3.33738	−0.111308
$900$	0	0
$901$	−94.8362	−3.15945
$902$	0	0
$903$	−0.128441	−0.00427426
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−41.7542	−1.38643	−0.693213	−	0.720733i	$-0.743805\pi$
$907$	−41.7542	−1.38643	−0.693213	+	0.720733i	$0.743805\pi$
$908$	0	0
$909$	−31.5205	−1.04547
$910$	0	0
$911$	9.12998	0.302490	0.151245	−	0.988496i	$-0.451672\pi$
$911$	9.12998	0.302490	0.151245	+	0.988496i	$0.451672\pi$
$912$	0	0
$913$	−59.3526	−1.96428
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−0.116307	−0.00384079
$918$	0	0
$919$	−19.0197	−0.627403	−0.313702	−	0.949522i	$-0.601569\pi$
$919$	−19.0197	−0.627403	−0.313702	+	0.949522i	$0.601569\pi$
$920$	0	0
$921$	−9.25342	−0.304910
$922$	0	0
$923$	57.8868	1.90537
$924$	0	0
$925$	0	0
$926$	0	0
$927$	17.8321	0.585682
$928$	0	0
$929$	−7.77239	−0.255004	−0.127502	−	0.991838i	$-0.540696\pi$
$929$	−7.77239	−0.255004	−0.127502	+	0.991838i	$0.540696\pi$
$930$	0	0
$931$	6.99393	0.229217
$932$	0	0
$933$	53.3218	1.74568
$934$	0	0
$935$	0	0
$936$	0	0
$937$	45.9818	1.50216	0.751080	−	0.660211i	$-0.229533\pi$
$937$	45.9818	1.50216	0.751080	+	0.660211i	$0.229533\pi$
$938$	0	0
$939$	68.4750	2.23460
$940$	0	0
$941$	−40.6242	−1.32431	−0.662156	−	0.749366i	$-0.730358\pi$
$941$	−40.6242	−1.32431	−0.662156	+	0.749366i	$0.730358\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−2.28423	−0.0742274	−0.0371137	−	0.999311i	$-0.511816\pi$
$947$	−2.28423	−0.0742274	−0.0371137	+	0.999311i	$0.511816\pi$
$948$	0	0
$949$	27.1771	0.882205
$950$	0	0
$951$	100.499	3.25890
$952$	0	0
$953$	57.5084	1.86288	0.931439	−	0.363896i	$-0.118554\pi$
$953$	57.5084	1.86288	0.931439	+	0.363896i	$0.118554\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−19.5205	−0.631008
$958$	0	0
$959$	−0.656555	−0.0212013
$960$	0	0
$961$	−24.7158	−0.797283
$962$	0	0
$963$	−106.440	−3.42997
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−4.70210	−0.151209	−0.0756047	−	0.997138i	$-0.524089\pi$
$967$	−4.70210	−0.151209	−0.0756047	+	0.997138i	$0.524089\pi$
$968$	0	0
$969$	23.8518	0.766231
$970$	0	0
$971$	−14.9863	−0.480934	−0.240467	−	0.970657i	$-0.577301\pi$
$971$	−14.9863	−0.480934	−0.240467	+	0.970657i	$0.577301\pi$
$972$	0	0
$973$	0.686891	0.0220207
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−8.89737	−0.284652	−0.142326	−	0.989820i	$-0.545458\pi$
$977$	−8.89737	−0.284652	−0.142326	+	0.989820i	$0.545458\pi$
$978$	0	0
$979$	67.6642	2.16256
$980$	0	0
$981$	−0.590796	−0.0188627
$982$	0	0
$983$	−1.60947	−0.0513341	−0.0256671	−	0.999671i	$-0.508171\pi$
$983$	−1.60947	−0.0513341	−0.0256671	+	0.999671i	$0.508171\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	1.43501	0.0456769
$988$	0	0
$989$	1.72791	0.0549443
$990$	0	0
$991$	−16.8974	−0.536763	−0.268381	−	0.963313i	$-0.586489\pi$
$991$	−16.8974	−0.536763	−0.268381	+	0.963313i	$0.586489\pi$
$992$	0	0
$993$	−92.3465	−2.93053
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−24.1558	−0.765021	−0.382511	−	0.923951i	$-0.624940\pi$
$997$	−24.1558	−0.765021	−0.382511	+	0.923951i	$0.624940\pi$
$998$	0	0
$999$	82.1402	2.59880

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7600.2.a.bz.1.3		3
4.3	odd	2		950.2.a.j.1.1	✓	3
5.4	even	2		7600.2.a.bk.1.1		3
12.11	even	2		8550.2.a.cp.1.2		3
20.3	even	4		950.2.b.h.799.4		6
20.7	even	4		950.2.b.h.799.3		6
20.19	odd	2		950.2.a.l.1.3	yes	3
60.59	even	2		8550.2.a.ci.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.a.j.1.1	✓	3	4.3	odd	2
950.2.a.l.1.3	yes	3	20.19	odd	2
950.2.b.h.799.3		6	20.7	even	4
950.2.b.h.799.4		6	20.3	even	4
7600.2.a.bk.1.1		3	5.4	even	2
7600.2.a.bz.1.3		3	1.1	even	1	trivial
8550.2.a.ci.1.2		3	60.59	even	2
8550.2.a.cp.1.2		3	12.11	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)

\(f(q)\)	\(=\)	\(q+3.25342 q^{3} -0.0778929 q^{7} +7.58473 q^{9} +O(q^{10})\)	\(q+3.25342 q^{3} -0.0778929 q^{7} +7.58473 q^{9} +4.50684 q^{11} +5.33131 q^{13} -7.33131 q^{17} -1.00000 q^{19} -0.253418 q^{21} +3.40920 q^{23} +14.9160 q^{27} -1.33131 q^{29} +2.50684 q^{31} +14.6626 q^{33} +5.50684 q^{37} +17.3450 q^{39} +0.506836 q^{43} -5.66262 q^{47} -6.99393 q^{49} -23.8518 q^{51} +12.9358 q^{53} -3.25342 q^{57} -7.56499 q^{59} -2.15579 q^{61} -0.590796 q^{63} -4.58473 q^{67} +11.0916 q^{69} +10.8579 q^{71} +5.09763 q^{73} -0.351050 q^{77} -17.0137 q^{79} +25.7739 q^{81} -13.1695 q^{83} -4.33131 q^{87} +15.0137 q^{89} -0.415271 q^{91} +8.15579 q^{93} -7.67629 q^{97} +34.1831 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{3} + 2 q^{7} + 5 q^{9}+O(q^{10}) \) 3 * q + 2 * q^3 + 2 * q^7 + 5 * q^9	\( 3 q + 2 q^{3} + 2 q^{7} + 5 q^{9} - 2 q^{11} + 6 q^{13} - 12 q^{17} - 3 q^{19} + 7 q^{21} - 2 q^{23} + 17 q^{27} + 6 q^{29} - 8 q^{31} + 24 q^{33} + q^{37} + 11 q^{39} - 14 q^{43} + 3 q^{47} + 9 q^{49} - 15 q^{51} + 10 q^{53} - 2 q^{57} - 6 q^{59} - 2 q^{61} - 14 q^{63} + 4 q^{67} + 6 q^{71} + 12 q^{73} + 10 q^{77} - 20 q^{79} + 23 q^{81} - 4 q^{83} - 3 q^{87} + 14 q^{89} - 19 q^{91} + 20 q^{93} + 28 q^{97} + 36 q^{99}+O(q^{100}) \) 3 * q + 2 * q^3 + 2 * q^7 + 5 * q^9 - 2 * q^11 + 6 * q^13 - 12 * q^17 - 3 * q^19 + 7 * q^21 - 2 * q^23 + 17 * q^27 + 6 * q^29 - 8 * q^31 + 24 * q^33 + q^37 + 11 * q^39 - 14 * q^43 + 3 * q^47 + 9 * q^49 - 15 * q^51 + 10 * q^53 - 2 * q^57 - 6 * q^59 - 2 * q^61 - 14 * q^63 + 4 * q^67 + 6 * q^71 + 12 * q^73 + 10 * q^77 - 20 * q^79 + 23 * q^81 - 4 * q^83 - 3 * q^87 + 14 * q^89 - 19 * q^91 + 20 * q^93 + 28 * q^97 + 36 * q^99

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