LMFDB - Embedded newform 7600.2.a.cf.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("7600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$60.6863055362$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.11344.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 4x + 3 $ x^4 - 2x^3 - 4x^2 + 4*x + 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
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			1.1
Root			$-0.552409$ of defining polynomial
Character	$\chi$	$=$	7600.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.87834 q^{3} +3.10482 q^{7} +5.28487 q^{9} +O(q^{10})$	$q-2.87834 q^{3} +3.10482 q^{7} +5.28487 q^{9} +1.10482 q^{11} -1.77353 q^{13} -7.75669 q^{17} -1.00000 q^{19} -8.93674 q^{21} -6.65187 q^{23} -6.57664 q^{27} +7.75669 q^{29} -6.57664 q^{31} -3.18005 q^{33} +1.40652 q^{37} +5.10482 q^{39} +2.81995 q^{41} +3.10482 q^{43} +1.46492 q^{47} +2.63990 q^{49} +22.3264 q^{51} +2.59348 q^{53} +2.87834 q^{57} +5.38969 q^{59} +4.07523 q^{61} +16.4086 q^{63} -15.8151 q^{67} +19.1464 q^{69} -7.14638 q^{71} -0.243310 q^{73} +3.43026 q^{77} +9.38969 q^{79} +3.07523 q^{81} +8.86151 q^{83} -22.3264 q^{87} +0.813048 q^{89} -5.50648 q^{91} +18.9298 q^{93} -3.19689 q^{97} +5.83882 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} + 4 q^{7} + 8 q^{9}+O(q^{10}) $ 4 * q + 2 * q^3 + 4 * q^7 + 8 * q^9	$ 4 q + 2 q^{3} + 4 q^{7} + 8 q^{9} - 4 q^{11} - 2 q^{13} - 4 q^{17} - 4 q^{19} - 4 q^{21} - 8 q^{23} - 4 q^{27} + 4 q^{29} - 4 q^{31} - 8 q^{33} + 6 q^{37} + 12 q^{39} + 16 q^{41} + 4 q^{43} - 12 q^{47} + 20 q^{49} + 36 q^{51} + 10 q^{53} - 2 q^{57} + 20 q^{61} + 20 q^{63} - 18 q^{67} + 28 q^{69} + 20 q^{71} - 28 q^{73} + 40 q^{77} + 16 q^{79} + 16 q^{81} - 36 q^{87} + 4 q^{89} + 36 q^{91} + 40 q^{93} - 30 q^{97} + 4 q^{99}+O(q^{100}) $ 4 * q + 2 * q^3 + 4 * q^7 + 8 * q^9 - 4 * q^11 - 2 * q^13 - 4 * q^17 - 4 * q^19 - 4 * q^21 - 8 * q^23 - 4 * q^27 + 4 * q^29 - 4 * q^31 - 8 * q^33 + 6 * q^37 + 12 * q^39 + 16 * q^41 + 4 * q^43 - 12 * q^47 + 20 * q^49 + 36 * q^51 + 10 * q^53 - 2 * q^57 + 20 * q^61 + 20 * q^63 - 18 * q^67 + 28 * q^69 + 20 * q^71 - 28 * q^73 + 40 * q^77 + 16 * q^79 + 16 * q^81 - 36 * q^87 + 4 * q^89 + 36 * q^91 + 40 * q^93 - 30 * q^97 + 4 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−2.87834	−1.66181	−0.830907	−	0.556412i	$-0.812178\pi$
$3$	−2.87834	−1.66181	−0.830907	+	0.556412i	$0.812178\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.10482	1.17351	0.586756	−	0.809764i	$-0.300405\pi$
$7$	3.10482	1.17351	0.586756	+	0.809764i	$0.300405\pi$
$8$	0	0
$9$	5.28487	1.76162
$10$	0	0
$11$	1.10482	0.333115	0.166558	−	0.986032i	$-0.446735\pi$
$11$	1.10482	0.333115	0.166558	+	0.986032i	$0.446735\pi$
$12$	0	0
$13$	−1.77353	−0.491888	−0.245944	−	0.969284i	$-0.579098\pi$
$13$	−1.77353	−0.491888	−0.245944	+	0.969284i	$0.579098\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−7.75669	−1.88127	−0.940637	−	0.339415i	$-0.889771\pi$
$17$	−7.75669	−1.88127	−0.940637	+	0.339415i	$0.889771\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−8.93674	−1.95016
$22$	0	0
$23$	−6.65187	−1.38701	−0.693505	−	0.720451i	$-0.743935\pi$
$23$	−6.65187	−1.38701	−0.693505	+	0.720451i	$0.743935\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−6.57664	−1.26567
$28$	0	0
$29$	7.75669	1.44038	0.720191	−	0.693776i	$-0.244054\pi$
$29$	7.75669	1.44038	0.720191	+	0.693776i	$0.244054\pi$
$30$	0	0
$31$	−6.57664	−1.18120	−0.590600	−	0.806965i	$-0.701109\pi$
$31$	−6.57664	−1.18120	−0.590600	+	0.806965i	$0.701109\pi$
$32$	0	0
$33$	−3.18005	−0.553576
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.40652	0.231231	0.115616	−	0.993294i	$-0.463116\pi$
$37$	1.40652	0.231231	0.115616	+	0.993294i	$0.463116\pi$
$38$	0	0
$39$	5.10482	0.817425
$40$	0	0
$41$	2.81995	0.440402	0.220201	−	0.975454i	$-0.429329\pi$
$41$	2.81995	0.440402	0.220201	+	0.975454i	$0.429329\pi$
$42$	0	0
$43$	3.10482	0.473480	0.236740	−	0.971573i	$-0.423921\pi$
$43$	3.10482	0.473480	0.236740	+	0.971573i	$0.423921\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.46492	0.213680	0.106840	−	0.994276i	$-0.465927\pi$
$47$	1.46492	0.213680	0.106840	+	0.994276i	$0.465927\pi$
$48$	0	0
$49$	2.63990	0.377129
$50$	0	0
$51$	22.3264	3.12633
$52$	0	0
$53$	2.59348	0.356241	0.178121	−	0.984009i	$-0.442998\pi$
$53$	2.59348	0.356241	0.178121	+	0.984009i	$0.442998\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.87834	0.381246
$58$	0	0
$59$	5.38969	0.701678	0.350839	−	0.936436i	$-0.385897\pi$
$59$	5.38969	0.701678	0.350839	+	0.936436i	$0.385897\pi$
$60$	0	0
$61$	4.07523	0.521780	0.260890	−	0.965369i	$-0.415984\pi$
$61$	4.07523	0.521780	0.260890	+	0.965369i	$0.415984\pi$
$62$	0	0
$63$	16.4086	2.06728
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−15.8151	−1.93212	−0.966060	−	0.258318i	$-0.916832\pi$
$67$	−15.8151	−1.93212	−0.966060	+	0.258318i	$0.916832\pi$
$68$	0	0
$69$	19.1464	2.30495
$70$	0	0
$71$	−7.14638	−0.848119	−0.424059	−	0.905634i	$-0.639395\pi$
$71$	−7.14638	−0.848119	−0.424059	+	0.905634i	$0.639395\pi$
$72$	0	0
$73$	−0.243310	−0.0284773	−0.0142387	−	0.999899i	$-0.504532\pi$
$73$	−0.243310	−0.0284773	−0.0142387	+	0.999899i	$0.504532\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.43026	0.390915
$78$	0	0
$79$	9.38969	1.05642	0.528211	−	0.849113i	$-0.322863\pi$
$79$	9.38969	1.05642	0.528211	+	0.849113i	$0.322863\pi$
$80$	0	0
$81$	3.07523	0.341692
$82$	0	0
$83$	8.86151	0.972677	0.486338	−	0.873771i	$-0.338332\pi$
$83$	8.86151	0.972677	0.486338	+	0.873771i	$0.338332\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−22.3264	−2.39364
$88$	0	0
$89$	0.813048	0.0861829	0.0430914	−	0.999071i	$-0.486279\pi$
$89$	0.813048	0.0861829	0.0430914	+	0.999071i	$0.486279\pi$
$90$	0	0
$91$	−5.50648	−0.577236
$92$	0	0
$93$	18.9298	1.96293
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−3.19689	−0.324595	−0.162297	−	0.986742i	$-0.551890\pi$
$97$	−3.19689	−0.324595	−0.162297	+	0.986742i	$0.551890\pi$
$98$	0	0
$99$	5.83882	0.586824
$100$	0	0
$101$	−3.01887	−0.300389	−0.150195	−	0.988656i	$-0.547990\pi$
$101$	−3.01887	−0.300389	−0.150195	+	0.988656i	$0.547990\pi$
$102$	0	0
$103$	6.05839	0.596951	0.298476	−	0.954417i	$-0.403522\pi$
$103$	6.05839	0.596951	0.298476	+	0.954417i	$0.403522\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	20.3848	1.97068	0.985338	−	0.170616i	$-0.0545759\pi$
$107$	20.3848	1.97068	0.985338	+	0.170616i	$0.0545759\pi$
$108$	0	0
$109$	4.72710	0.452774	0.226387	−	0.974037i	$-0.427309\pi$
$109$	4.72710	0.452774	0.226387	+	0.974037i	$0.427309\pi$
$110$	0	0
$111$	−4.04846	−0.384263
$112$	0	0
$113$	7.98316	0.750993	0.375496	−	0.926824i	$-0.377472\pi$
$113$	7.98316	0.750993	0.375496	+	0.926824i	$0.377472\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−9.37285	−0.866520
$118$	0	0
$119$	−24.0831	−2.20770
$120$	0	0
$121$	−9.77938	−0.889034
$122$	0	0
$123$	−8.11679	−0.731866
$124$	0	0
$125$	0	0
$126$	0	0
$127$	4.63503	0.411293	0.205646	−	0.978626i	$-0.434070\pi$
$127$	4.63503	0.411293	0.205646	+	0.978626i	$0.434070\pi$
$128$	0	0
$129$	−8.93674	−0.786836
$130$	0	0
$131$	−15.5134	−1.35541	−0.677705	−	0.735334i	$-0.737025\pi$
$131$	−15.5134	−1.35541	−0.677705	+	0.735334i	$0.737025\pi$
$132$	0	0
$133$	−3.10482	−0.269222
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0.813048	0.0694634	0.0347317	−	0.999397i	$-0.488942\pi$
$137$	0.813048	0.0694634	0.0347317	+	0.999397i	$0.488942\pi$
$138$	0	0
$139$	12.7687	1.08302	0.541512	−	0.840693i	$-0.317852\pi$
$139$	12.7687	1.08302	0.541512	+	0.840693i	$0.317852\pi$
$140$	0	0
$141$	−4.21654	−0.355097
$142$	0	0
$143$	−1.95942	−0.163855
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−7.59854	−0.626717
$148$	0	0
$149$	−13.7982	−1.13040	−0.565198	−	0.824955i	$-0.691200\pi$
$149$	−13.7982	−1.13040	−0.565198	+	0.824955i	$0.691200\pi$
$150$	0	0
$151$	−5.02269	−0.408740	−0.204370	−	0.978894i	$-0.565515\pi$
$151$	−5.02269	−0.408740	−0.204370	+	0.978894i	$0.565515\pi$
$152$	0	0
$153$	−40.9931	−3.31409
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−7.18695	−0.573581	−0.286791	−	0.957993i	$-0.592588\pi$
$157$	−7.18695	−0.573581	−0.286791	+	0.957993i	$0.592588\pi$
$158$	0	0
$159$	−7.46492	−0.592007
$160$	0	0
$161$	−20.6529	−1.62767
$162$	0	0
$163$	7.25528	0.568277	0.284139	−	0.958783i	$-0.408292\pi$
$163$	7.25528	0.568277	0.284139	+	0.958783i	$0.408292\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−7.33129	−0.567312	−0.283656	−	0.958926i	$-0.591547\pi$
$167$	−7.33129	−0.567312	−0.283656	+	0.958926i	$0.591547\pi$
$168$	0	0
$169$	−9.85461	−0.758047
$170$	0	0
$171$	−5.28487	−0.404144
$172$	0	0
$173$	5.77353	0.438953	0.219477	−	0.975618i	$-0.429565\pi$
$173$	5.77353	0.438953	0.219477	+	0.975618i	$0.429565\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−15.5134	−1.16606
$178$	0	0
$179$	4.48379	0.335134	0.167567	−	0.985861i	$-0.446409\pi$
$179$	4.48379	0.335134	0.167567	+	0.985861i	$0.446409\pi$
$180$	0	0
$181$	−2.73400	−0.203217	−0.101608	−	0.994824i	$-0.532399\pi$
$181$	−2.73400	−0.203217	−0.101608	+	0.994824i	$0.532399\pi$
$182$	0	0
$183$	−11.7299	−0.867101
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−8.56974	−0.626681
$188$	0	0
$189$	−20.4193	−1.48528
$190$	0	0
$191$	17.7230	1.28239	0.641196	−	0.767377i	$-0.278438\pi$
$191$	17.7230	1.28239	0.641196	+	0.767377i	$0.278438\pi$
$192$	0	0
$193$	13.5371	0.974423	0.487212	−	0.873284i	$-0.338014\pi$
$193$	13.5371	0.974423	0.487212	+	0.873284i	$0.338014\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	21.5134	1.53276	0.766382	−	0.642385i	$-0.222055\pi$
$197$	21.5134	1.53276	0.766382	+	0.642385i	$0.222055\pi$
$198$	0	0
$199$	−7.09410	−0.502888	−0.251444	−	0.967872i	$-0.580905\pi$
$199$	−7.09410	−0.502888	−0.251444	+	0.967872i	$0.580905\pi$
$200$	0	0
$201$	45.5213	3.21082
$202$	0	0
$203$	24.0831	1.69030
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−35.1543	−2.44339
$208$	0	0
$209$	−1.10482	−0.0764219
$210$	0	0
$211$	−11.0604	−0.761431	−0.380716	−	0.924692i	$-0.624322\pi$
$211$	−11.0604	−0.761431	−0.380716	+	0.924692i	$0.624322\pi$
$212$	0	0
$213$	20.5697	1.40942
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−20.4193	−1.38615
$218$	0	0
$219$	0.700331	0.0473240
$220$	0	0
$221$	13.7567	0.925375
$222$	0	0
$223$	−12.6350	−0.846104	−0.423052	−	0.906105i	$-0.639041\pi$
$223$	−12.6350	−0.846104	−0.423052	+	0.906105i	$0.639041\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−6.62813	−0.439925	−0.219962	−	0.975508i	$-0.570593\pi$
$227$	−6.62813	−0.439925	−0.219962	+	0.975508i	$0.570593\pi$
$228$	0	0
$229$	−0.0752308	−0.00497139	−0.00248570	−	0.999997i	$-0.500791\pi$
$229$	−0.0752308	−0.00497139	−0.00248570	+	0.999997i	$0.500791\pi$
$230$	0	0
$231$	−9.87348	−0.649627
$232$	0	0
$233$	−9.51338	−0.623242	−0.311621	−	0.950206i	$-0.600872\pi$
$233$	−9.51338	−0.623242	−0.311621	+	0.950206i	$0.600872\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−27.0268	−1.75558
$238$	0	0
$239$	2.92984	0.189515	0.0947577	−	0.995500i	$-0.469792\pi$
$239$	2.92984	0.189515	0.0947577	+	0.995500i	$0.469792\pi$
$240$	0	0
$241$	−9.42743	−0.607274	−0.303637	−	0.952788i	$-0.598201\pi$
$241$	−9.42743	−0.607274	−0.303637	+	0.952788i	$0.598201\pi$
$242$	0	0
$243$	10.8783	0.697846
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.77353	0.112847
$248$	0	0
$249$	−25.5065	−1.61641
$250$	0	0
$251$	14.9298	0.942363	0.471181	−	0.882036i	$-0.343828\pi$
$251$	14.9298	0.942363	0.471181	+	0.882036i	$0.343828\pi$
$252$	0	0
$253$	−7.34911	−0.462035
$254$	0	0
$255$	0	0
$256$	0	0
$257$	15.1295	0.943755	0.471877	−	0.881664i	$-0.343576\pi$
$257$	15.1295	0.943755	0.471877	+	0.881664i	$0.343576\pi$
$258$	0	0
$259$	4.36700	0.271352
$260$	0	0
$261$	40.9931	2.53741
$262$	0	0
$263$	−15.2216	−0.938605	−0.469302	−	0.883038i	$-0.655495\pi$
$263$	−15.2216	−0.938605	−0.469302	+	0.883038i	$0.655495\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−2.34023	−0.143220
$268$	0	0
$269$	11.5203	0.702404	0.351202	−	0.936300i	$-0.385773\pi$
$269$	11.5203	0.702404	0.351202	+	0.936300i	$0.385773\pi$
$270$	0	0
$271$	−2.16118	−0.131282	−0.0656411	−	0.997843i	$-0.520909\pi$
$271$	−2.16118	−0.131282	−0.0656411	+	0.997843i	$0.520909\pi$
$272$	0	0
$273$	15.8495	0.959258
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−23.4205	−1.40720	−0.703602	−	0.710595i	$-0.748426\pi$
$277$	−23.4205	−1.40720	−0.703602	+	0.710595i	$0.748426\pi$
$278$	0	0
$279$	−34.7567	−2.08083
$280$	0	0
$281$	25.6824	1.53209	0.766043	−	0.642789i	$-0.222223\pi$
$281$	25.6824	1.53209	0.766043	+	0.642789i	$0.222223\pi$
$282$	0	0
$283$	7.38587	0.439045	0.219522	−	0.975607i	$-0.429550\pi$
$283$	7.38587	0.439045	0.219522	+	0.975607i	$0.429550\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	8.75543	0.516817
$288$	0	0
$289$	43.1662	2.53919
$290$	0	0
$291$	9.20174	0.539416
$292$	0	0
$293$	24.1522	1.41099	0.705494	−	0.708716i	$-0.250725\pi$
$293$	24.1522	1.41099	0.705494	+	0.708716i	$0.250725\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−7.26600	−0.421616
$298$	0	0
$299$	11.7973	0.682253
$300$	0	0
$301$	9.63990	0.555635
$302$	0	0
$303$	8.68936	0.499190
$304$	0	0
$305$	0	0
$306$	0	0
$307$	4.38482	0.250255	0.125127	−	0.992141i	$-0.460066\pi$
$307$	4.38482	0.250255	0.125127	+	0.992141i	$0.460066\pi$
$308$	0	0
$309$	−17.4381	−0.992022
$310$	0	0
$311$	15.7123	0.890963	0.445481	−	0.895291i	$-0.353033\pi$
$311$	15.7123	0.890963	0.445481	+	0.895291i	$0.353033\pi$
$312$	0	0
$313$	−24.7794	−1.40061	−0.700307	−	0.713842i	$-0.746953\pi$
$313$	−24.7794	−1.40061	−0.700307	+	0.713842i	$0.746953\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	27.3798	1.53780	0.768900	−	0.639369i	$-0.220804\pi$
$317$	27.3798	1.53780	0.768900	+	0.639369i	$0.220804\pi$
$318$	0	0
$319$	8.56974	0.479813
$320$	0	0
$321$	−58.6745	−3.27489
$322$	0	0
$323$	7.75669	0.431594
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−13.6062	−0.752426
$328$	0	0
$329$	4.54831	0.250756
$330$	0	0
$331$	−4.70033	−0.258354	−0.129177	−	0.991622i	$-0.541233\pi$
$331$	−4.70033	−0.258354	−0.129177	+	0.991622i	$0.541233\pi$
$332$	0	0
$333$	7.43329	0.407342
$334$	0	0
$335$	0	0
$336$	0	0
$337$	20.6360	1.12412	0.562058	−	0.827098i	$-0.310010\pi$
$337$	20.6360	1.12412	0.562058	+	0.827098i	$0.310010\pi$
$338$	0	0
$339$	−22.9783	−1.24801
$340$	0	0
$341$	−7.26600	−0.393476
$342$	0	0
$343$	−13.5373	−0.730947
$344$	0	0
$345$	0	0
$346$	0	0
$347$	30.0148	1.61128	0.805639	−	0.592407i	$-0.201822\pi$
$347$	30.0148	1.61128	0.805639	+	0.592407i	$0.201822\pi$
$348$	0	0
$349$	−14.7340	−0.788693	−0.394347	−	0.918962i	$-0.629029\pi$
$349$	−14.7340	−0.788693	−0.394347	+	0.918962i	$0.629029\pi$
$350$	0	0
$351$	11.6638	0.622570
$352$	0	0
$353$	29.6302	1.57705	0.788527	−	0.615000i	$-0.210844\pi$
$353$	29.6302	1.57705	0.788527	+	0.615000i	$0.210844\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	69.3195	3.66878
$358$	0	0
$359$	−21.7577	−1.14833	−0.574163	−	0.818741i	$-0.694672\pi$
$359$	−21.7577	−1.14833	−0.574163	+	0.818741i	$0.694672\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	28.1484	1.47741
$364$	0	0
$365$	0	0
$366$	0	0
$367$	30.0347	1.56780	0.783898	−	0.620889i	$-0.213228\pi$
$367$	30.0347	1.56780	0.783898	+	0.620889i	$0.213228\pi$
$368$	0	0
$369$	14.9031	0.775823
$370$	0	0
$371$	8.05227	0.418053
$372$	0	0
$373$	37.3055	1.93161	0.965803	−	0.259277i	$-0.0834843\pi$
$373$	37.3055	1.93161	0.965803	+	0.259277i	$0.0834843\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−13.7567	−0.708506
$378$	0	0
$379$	14.7794	0.759166	0.379583	−	0.925158i	$-0.376068\pi$
$379$	14.7794	0.759166	0.379583	+	0.925158i	$0.376068\pi$
$380$	0	0
$381$	−13.3412	−0.683492
$382$	0	0
$383$	−29.0020	−1.48193	−0.740967	−	0.671541i	$-0.765633\pi$
$383$	−29.0020	−1.48193	−0.740967	+	0.671541i	$0.765633\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	16.4086	0.834094
$388$	0	0
$389$	21.1987	1.07481	0.537407	−	0.843323i	$-0.319404\pi$
$389$	21.1987	1.07481	0.537407	+	0.843323i	$0.319404\pi$
$390$	0	0
$391$	51.5965	2.60935
$392$	0	0
$393$	44.6529	2.25244
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−10.3856	−0.521238	−0.260619	−	0.965442i	$-0.583927\pi$
$397$	−10.3856	−0.521238	−0.260619	+	0.965442i	$0.583927\pi$
$398$	0	0
$399$	8.93674	0.447397
$400$	0	0
$401$	25.6638	1.28159	0.640796	−	0.767712i	$-0.278605\pi$
$401$	25.6638	1.28159	0.640796	+	0.767712i	$0.278605\pi$
$402$	0	0
$403$	11.6638	0.581017
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.55395	0.0770266
$408$	0	0
$409$	5.71611	0.282644	0.141322	−	0.989964i	$-0.454865\pi$
$409$	5.71611	0.282644	0.141322	+	0.989964i	$0.454865\pi$
$410$	0	0
$411$	−2.34023	−0.115435
$412$	0	0
$413$	16.7340	0.823427
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−36.7526	−1.79978
$418$	0	0
$419$	−15.4303	−0.753818	−0.376909	−	0.926250i	$-0.623013\pi$
$419$	−15.4303	−0.753818	−0.376909	+	0.926250i	$0.623013\pi$
$420$	0	0
$421$	−1.18005	−0.0575121	−0.0287561	−	0.999586i	$-0.509155\pi$
$421$	−1.18005	−0.0575121	−0.0287561	+	0.999586i	$0.509155\pi$
$422$	0	0
$423$	7.74190	0.376424
$424$	0	0
$425$	0	0
$426$	0	0
$427$	12.6529	0.612315
$428$	0	0
$429$	5.63990	0.272297
$430$	0	0
$431$	14.0255	0.675585	0.337792	−	0.941221i	$-0.390320\pi$
$431$	14.0255	0.675585	0.337792	+	0.941221i	$0.390320\pi$
$432$	0	0
$433$	−25.5894	−1.22975	−0.614874	−	0.788625i	$-0.710793\pi$
$433$	−25.5894	−1.22975	−0.614874	+	0.788625i	$0.710793\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	6.65187	0.318202
$438$	0	0
$439$	21.9732	1.04873	0.524363	−	0.851495i	$-0.324304\pi$
$439$	21.9732	1.04873	0.524363	+	0.851495i	$0.324304\pi$
$440$	0	0
$441$	13.9515	0.664358
$442$	0	0
$443$	−19.3721	−0.920395	−0.460197	−	0.887817i	$-0.652221\pi$
$443$	−19.3721	−0.920395	−0.460197	+	0.887817i	$0.652221\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	39.7161	1.87851
$448$	0	0
$449$	−19.1841	−0.905355	−0.452677	−	0.891674i	$-0.649531\pi$
$449$	−19.1841	−0.905355	−0.452677	+	0.891674i	$0.649531\pi$
$450$	0	0
$451$	3.11553	0.146705
$452$	0	0
$453$	14.4570	0.679250
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−8.36010	−0.391069	−0.195534	−	0.980697i	$-0.562644\pi$
$457$	−8.36010	−0.391069	−0.195534	+	0.980697i	$0.562644\pi$
$458$	0	0
$459$	51.0130	2.38108
$460$	0	0
$461$	−25.0268	−1.16561	−0.582806	−	0.812611i	$-0.698045\pi$
$461$	−25.0268	−1.16561	−0.582806	+	0.812611i	$0.698045\pi$
$462$	0	0
$463$	32.3749	1.50459	0.752294	−	0.658827i	$-0.228947\pi$
$463$	32.3749	1.50459	0.752294	+	0.658827i	$0.228947\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	15.2216	0.704372	0.352186	−	0.935930i	$-0.385438\pi$
$467$	15.2216	0.704372	0.352186	+	0.935930i	$0.385438\pi$
$468$	0	0
$469$	−49.1030	−2.26736
$470$	0	0
$471$	20.6865	0.953185
$472$	0	0
$473$	3.43026	0.157724
$474$	0	0
$475$	0	0
$476$	0	0
$477$	13.7062	0.627563
$478$	0	0
$479$	20.0485	0.916038	0.458019	−	0.888943i	$-0.348559\pi$
$479$	20.0485	0.916038	0.458019	+	0.888943i	$0.348559\pi$
$480$	0	0
$481$	−2.49451	−0.113740
$482$	0	0
$483$	59.4460	2.70489
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−31.6508	−1.43424	−0.717118	−	0.696952i	$-0.754539\pi$
$487$	−31.6508	−1.43424	−0.717118	+	0.696952i	$0.754539\pi$
$488$	0	0
$489$	−20.8832	−0.944371
$490$	0	0
$491$	−24.8033	−1.11936	−0.559679	−	0.828710i	$-0.689076\pi$
$491$	−24.8033	−1.11936	−0.559679	+	0.828710i	$0.689076\pi$
$492$	0	0
$493$	−60.1662	−2.70975
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−22.1882	−0.995277
$498$	0	0
$499$	33.0237	1.47834	0.739171	−	0.673518i	$-0.235217\pi$
$499$	33.0237	1.47834	0.739171	+	0.673518i	$0.235217\pi$
$500$	0	0
$501$	21.1020	0.942767
$502$	0	0
$503$	−3.54396	−0.158017	−0.0790087	−	0.996874i	$-0.525175\pi$
$503$	−3.54396	−0.158017	−0.0790087	+	0.996874i	$0.525175\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	28.3650	1.25973
$508$	0	0
$509$	−2.11679	−0.0938250	−0.0469125	−	0.998899i	$-0.514938\pi$
$509$	−2.11679	−0.0938250	−0.0469125	+	0.998899i	$0.514938\pi$
$510$	0	0
$511$	−0.755435	−0.0334185
$512$	0	0
$513$	6.57664	0.290366
$514$	0	0
$515$	0	0
$516$	0	0
$517$	1.61847	0.0711802
$518$	0	0
$519$	−16.6182	−0.729458
$520$	0	0
$521$	−8.60748	−0.377101	−0.188550	−	0.982064i	$-0.560379\pi$
$521$	−8.60748	−0.377101	−0.188550	+	0.982064i	$0.560379\pi$
$522$	0	0
$523$	36.8349	1.61068	0.805340	−	0.592814i	$-0.201983\pi$
$523$	36.8349	1.61068	0.805340	+	0.592814i	$0.201983\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	51.0130	2.22216
$528$	0	0
$529$	21.2474	0.923799
$530$	0	0
$531$	28.4838	1.23609
$532$	0	0
$533$	−5.00125	−0.216628
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−12.9059	−0.556931
$538$	0	0
$539$	2.91661	0.125627
$540$	0	0
$541$	−32.0079	−1.37613	−0.688063	−	0.725651i	$-0.741539\pi$
$541$	−32.0079	−1.37613	−0.688063	+	0.725651i	$0.741539\pi$
$542$	0	0
$543$	7.86941	0.337709
$544$	0	0
$545$	0	0
$546$	0	0
$547$	17.6240	0.753550	0.376775	−	0.926305i	$-0.377033\pi$
$547$	17.6240	0.753550	0.376775	+	0.926305i	$0.377033\pi$
$548$	0	0
$549$	21.5371	0.919179
$550$	0	0
$551$	−7.75669	−0.330446
$552$	0	0
$553$	29.1533	1.23972
$554$	0	0
$555$	0	0
$556$	0	0
$557$	34.6865	1.46972	0.734858	−	0.678221i	$-0.237249\pi$
$557$	34.6865	1.46972	0.734858	+	0.678221i	$0.237249\pi$
$558$	0	0
$559$	−5.50648	−0.232899
$560$	0	0
$561$	24.6667	1.04143
$562$	0	0
$563$	13.5073	0.569263	0.284632	−	0.958637i	$-0.408129\pi$
$563$	13.5073	0.569263	0.284632	+	0.958637i	$0.408129\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	9.54803	0.400980
$568$	0	0
$569$	32.3588	1.35655	0.678276	−	0.734807i	$-0.262727\pi$
$569$	32.3588	1.35655	0.678276	+	0.734807i	$0.262727\pi$
$570$	0	0
$571$	8.93293	0.373831	0.186916	−	0.982376i	$-0.440151\pi$
$571$	8.93293	0.373831	0.186916	+	0.982376i	$0.440151\pi$
$572$	0	0
$573$	−51.0130	−2.13110
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−14.9059	−0.620541	−0.310270	−	0.950648i	$-0.600420\pi$
$577$	−14.9059	−0.620541	−0.310270	+	0.950648i	$0.600420\pi$
$578$	0	0
$579$	−38.9645	−1.61931
$580$	0	0
$581$	27.5134	1.14145
$582$	0	0
$583$	2.86532	0.118669
$584$	0	0
$585$	0	0
$586$	0	0
$587$	7.37207	0.304278	0.152139	−	0.988359i	$-0.451384\pi$
$587$	7.37207	0.304278	0.152139	+	0.988359i	$0.451384\pi$
$588$	0	0
$589$	6.57664	0.270986
$590$	0	0
$591$	−61.9229	−2.54717
$592$	0	0
$593$	17.6853	0.726247	0.363124	−	0.931741i	$-0.381710\pi$
$593$	17.6853	0.726247	0.363124	+	0.931741i	$0.381710\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	20.4193	0.835705
$598$	0	0
$599$	−5.30657	−0.216821	−0.108410	−	0.994106i	$-0.534576\pi$
$599$	−5.30657	−0.216821	−0.108410	+	0.994106i	$0.534576\pi$
$600$	0	0
$601$	43.7875	1.78613	0.893065	−	0.449927i	$-0.148550\pi$
$601$	43.7875	1.78613	0.893065	+	0.449927i	$0.148550\pi$
$602$	0	0
$603$	−83.5806	−3.40367
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−7.24535	−0.294080	−0.147040	−	0.989131i	$-0.546975\pi$
$607$	−7.24535	−0.294080	−0.147040	+	0.989131i	$0.546975\pi$
$608$	0	0
$609$	−69.3195	−2.80897
$610$	0	0
$611$	−2.59807	−0.105107
$612$	0	0
$613$	−20.8962	−0.843988	−0.421994	−	0.906599i	$-0.638670\pi$
$613$	−20.8962	−0.843988	−0.421994	+	0.906599i	$0.638670\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	22.0494	0.887677	0.443839	−	0.896107i	$-0.353616\pi$
$617$	22.0494	0.887677	0.443839	+	0.896107i	$0.353616\pi$
$618$	0	0
$619$	0.0484607	0.00194780	0.000973900	−	1.00000i	$-0.499690\pi$
$619$	0.0484607	0.00194780	0.000973900		1.00000i	$0.499690\pi$
$620$	0	0
$621$	43.7470	1.75550
$622$	0	0
$623$	2.52437	0.101137
$624$	0	0
$625$	0	0
$626$	0	0
$627$	3.18005	0.126999
$628$	0	0
$629$	−10.9100	−0.435009
$630$	0	0
$631$	32.6182	1.29851	0.649255	−	0.760571i	$-0.275081\pi$
$631$	32.6182	1.29851	0.649255	+	0.760571i	$0.275081\pi$
$632$	0	0
$633$	31.8357	1.26536
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−4.68193	−0.185505
$638$	0	0
$639$	−37.7677	−1.49407
$640$	0	0
$641$	−13.4136	−0.529806	−0.264903	−	0.964275i	$-0.585340\pi$
$641$	−13.4136	−0.529806	−0.264903	+	0.964275i	$0.585340\pi$
$642$	0	0
$643$	23.8052	0.938783	0.469392	−	0.882990i	$-0.344473\pi$
$643$	23.8052	0.938783	0.469392	+	0.882990i	$0.344473\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−48.4580	−1.90508	−0.952540	−	0.304412i	$-0.901540\pi$
$647$	−48.4580	−1.90508	−0.952540	+	0.304412i	$0.901540\pi$
$648$	0	0
$649$	5.95463	0.233740
$650$	0	0
$651$	58.7737	2.30352
$652$	0	0
$653$	−22.2351	−0.870128	−0.435064	−	0.900399i	$-0.643274\pi$
$653$	−22.2351	−0.870128	−0.435064	+	0.900399i	$0.643274\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−1.28586	−0.0501663
$658$	0	0
$659$	10.7072	0.417095	0.208547	−	0.978012i	$-0.433126\pi$
$659$	10.7072	0.417095	0.208547	+	0.978012i	$0.433126\pi$
$660$	0	0
$661$	−44.7980	−1.74244	−0.871220	−	0.490893i	$-0.836670\pi$
$661$	−44.7980	−1.74244	−0.871220	+	0.490893i	$0.836670\pi$
$662$	0	0
$663$	−39.5965	−1.53780
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−51.5965	−1.99782
$668$	0	0
$669$	36.3680	1.40607
$670$	0	0
$671$	4.50239	0.173813
$672$	0	0
$673$	43.2772	1.66821	0.834106	−	0.551604i	$-0.185984\pi$
$673$	43.2772	1.66821	0.834106	+	0.551604i	$0.185984\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−11.5330	−0.443251	−0.221625	−	0.975132i	$-0.571136\pi$
$677$	−11.5330	−0.443251	−0.221625	+	0.975132i	$0.571136\pi$
$678$	0	0
$679$	−9.92575	−0.380915
$680$	0	0
$681$	19.0780	0.731072
$682$	0	0
$683$	0.916090	0.0350532	0.0175266	−	0.999846i	$-0.494421\pi$
$683$	0.916090	0.0350532	0.0175266	+	0.999846i	$0.494421\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0.216540	0.00826153
$688$	0	0
$689$	−4.59960	−0.175231
$690$	0	0
$691$	−14.8278	−0.564077	−0.282039	−	0.959403i	$-0.591011\pi$
$691$	−14.8278	−0.564077	−0.282039	+	0.959403i	$0.591011\pi$
$692$	0	0
$693$	18.1285	0.688644
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−21.8735	−0.828517
$698$	0	0
$699$	27.3828	1.03571
$700$	0	0
$701$	36.1722	1.36620	0.683102	−	0.730323i	$-0.260631\pi$
$701$	36.1722	1.36620	0.683102	+	0.730323i	$0.260631\pi$
$702$	0	0
$703$	−1.40652	−0.0530481
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−9.37305	−0.352510
$708$	0	0
$709$	10.4331	0.391823	0.195911	−	0.980622i	$-0.437234\pi$
$709$	10.4331	0.391823	0.195911	+	0.980622i	$0.437234\pi$
$710$	0	0
$711$	49.6233	1.86102
$712$	0	0
$713$	43.7470	1.63834
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−8.43308	−0.314939
$718$	0	0
$719$	−7.73373	−0.288420	−0.144210	−	0.989547i	$-0.546064\pi$
$719$	−7.73373	−0.288420	−0.144210	+	0.989547i	$0.546064\pi$
$720$	0	0
$721$	18.8102	0.700529
$722$	0	0
$723$	27.1354	1.00918
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−39.5241	−1.46587	−0.732934	−	0.680300i	$-0.761849\pi$
$727$	−39.5241	−1.46587	−0.732934	+	0.680300i	$0.761849\pi$
$728$	0	0
$729$	−40.5373	−1.50138
$730$	0	0
$731$	−24.0831	−0.890746
$732$	0	0
$733$	−50.7498	−1.87449	−0.937243	−	0.348677i	$-0.886631\pi$
$733$	−50.7498	−1.87449	−0.937243	+	0.348677i	$0.886631\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−17.4728	−0.643619
$738$	0	0
$739$	−0.419276	−0.0154233	−0.00771165	−	0.999970i	$-0.502455\pi$
$739$	−0.419276	−0.0154233	−0.00771165	+	0.999970i	$0.502455\pi$
$740$	0	0
$741$	−5.10482	−0.187530
$742$	0	0
$743$	19.5511	0.717259	0.358630	−	0.933480i	$-0.383244\pi$
$743$	19.5511	0.717259	0.358630	+	0.933480i	$0.383244\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	46.8319	1.71349
$748$	0	0
$749$	63.2912	2.31261
$750$	0	0
$751$	−8.76768	−0.319937	−0.159969	−	0.987122i	$-0.551139\pi$
$751$	−8.76768	−0.319937	−0.159969	+	0.987122i	$0.551139\pi$
$752$	0	0
$753$	−42.9732	−1.56603
$754$	0	0
$755$	0	0
$756$	0	0
$757$	48.0535	1.74653	0.873267	−	0.487241i	$-0.161997\pi$
$757$	48.0535	1.74653	0.873267	+	0.487241i	$0.161997\pi$
$758$	0	0
$759$	21.1533	0.767815
$760$	0	0
$761$	29.3947	1.06556	0.532779	−	0.846254i	$-0.321148\pi$
$761$	29.3947	1.06556	0.532779	+	0.846254i	$0.321148\pi$
$762$	0	0
$763$	14.6768	0.531336
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−9.55875	−0.345146
$768$	0	0
$769$	−13.3790	−0.482458	−0.241229	−	0.970468i	$-0.577551\pi$
$769$	−13.3790	−0.482458	−0.241229	+	0.970468i	$0.577551\pi$
$770$	0	0
$771$	−43.5480	−1.56834
$772$	0	0
$773$	−0.133625	−0.00480617	−0.00240309	−	0.999997i	$-0.500765\pi$
$773$	−0.133625	−0.00480617	−0.00240309	+	0.999997i	$0.500765\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−12.5697	−0.450937
$778$	0	0
$779$	−2.81995	−0.101035
$780$	0	0
$781$	−7.89545	−0.282522
$782$	0	0
$783$	−51.0130	−1.82305
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−4.84060	−0.172549	−0.0862744	−	0.996271i	$-0.527496\pi$
$787$	−4.84060	−0.172549	−0.0862744	+	0.996271i	$0.527496\pi$
$788$	0	0
$789$	43.8130	1.55979
$790$	0	0
$791$	24.7863	0.881299
$792$	0	0
$793$	−7.22753	−0.256657
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−11.6993	−0.414410	−0.207205	−	0.978298i	$-0.566437\pi$
$797$	−11.6993	−0.414410	−0.207205	+	0.978298i	$0.566437\pi$
$798$	0	0
$799$	−11.3629	−0.401991
$800$	0	0
$801$	4.29685	0.151822
$802$	0	0
$803$	−0.268814	−0.00948624
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−33.1593	−1.16726
$808$	0	0
$809$	−33.1987	−1.16720	−0.583601	−	0.812040i	$-0.698357\pi$
$809$	−33.1987	−1.16720	−0.583601	+	0.812040i	$0.698357\pi$
$810$	0	0
$811$	26.5766	0.933232	0.466616	−	0.884460i	$-0.345473\pi$
$811$	26.5766	0.933232	0.466616	+	0.884460i	$0.345473\pi$
$812$	0	0
$813$	6.22061	0.218166
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−3.10482	−0.108624
$818$	0	0
$819$	−29.1010	−1.01687
$820$	0	0
$821$	25.5807	0.892773	0.446387	−	0.894840i	$-0.352711\pi$
$821$	25.5807	0.892773	0.446387	+	0.894840i	$0.352711\pi$
$822$	0	0
$823$	12.9783	0.452395	0.226198	−	0.974081i	$-0.427371\pi$
$823$	12.9783	0.452395	0.226198	+	0.974081i	$0.427371\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−12.5167	−0.435248	−0.217624	−	0.976033i	$-0.569831\pi$
$827$	−12.5167	−0.435248	−0.217624	+	0.976033i	$0.569831\pi$
$828$	0	0
$829$	−19.4391	−0.675149	−0.337574	−	0.941299i	$-0.609606\pi$
$829$	−19.4391	−0.675149	−0.337574	+	0.941299i	$0.609606\pi$
$830$	0	0
$831$	67.4124	2.33851
$832$	0	0
$833$	−20.4769	−0.709482
$834$	0	0
$835$	0	0
$836$	0	0
$837$	43.2522	1.49501
$838$	0	0
$839$	43.9972	1.51895	0.759475	−	0.650536i	$-0.225456\pi$
$839$	43.9972	1.51895	0.759475	+	0.650536i	$0.225456\pi$
$840$	0	0
$841$	31.1662	1.07470
$842$	0	0
$843$	−73.9229	−2.54604
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−30.3632	−1.04329
$848$	0	0
$849$	−21.2591	−0.729610
$850$	0	0
$851$	−9.35601	−0.320720
$852$	0	0
$853$	−3.30374	−0.113118	−0.0565590	−	0.998399i	$-0.518013\pi$
$853$	−3.30374	−0.113118	−0.0565590	+	0.998399i	$0.518013\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−6.02374	−0.205767	−0.102883	−	0.994693i	$-0.532807\pi$
$857$	−6.02374	−0.205767	−0.102883	+	0.994693i	$0.532807\pi$
$858$	0	0
$859$	36.0158	1.22884	0.614421	−	0.788978i	$-0.289390\pi$
$859$	36.0158	1.22884	0.614421	+	0.788978i	$0.289390\pi$
$860$	0	0
$861$	−25.2012	−0.858853
$862$	0	0
$863$	29.9250	1.01866	0.509329	−	0.860572i	$-0.329894\pi$
$863$	29.9250	1.01866	0.509329	+	0.860572i	$0.329894\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−124.247	−4.21966
$868$	0	0
$869$	10.3739	0.351911
$870$	0	0
$871$	28.0485	0.950386
$872$	0	0
$873$	−16.8951	−0.571813
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−0.618980	−0.0209015	−0.0104507	−	0.999945i	$-0.503327\pi$
$877$	−0.618980	−0.0209015	−0.0104507	+	0.999945i	$0.503327\pi$
$878$	0	0
$879$	−69.5184	−2.34480
$880$	0	0
$881$	−21.2423	−0.715672	−0.357836	−	0.933784i	$-0.616485\pi$
$881$	−21.2423	−0.715672	−0.357836	+	0.933784i	$0.616485\pi$
$882$	0	0
$883$	−48.8971	−1.64552	−0.822760	−	0.568389i	$-0.807567\pi$
$883$	−48.8971	−1.64552	−0.822760	+	0.568389i	$0.807567\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	58.2797	1.95684	0.978421	−	0.206622i	$-0.0662469\pi$
$887$	58.2797	1.95684	0.978421	+	0.206622i	$0.0662469\pi$
$888$	0	0
$889$	14.3909	0.482657
$890$	0	0
$891$	3.39757	0.113823
$892$	0	0
$893$	−1.46492	−0.0490216
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−33.9566	−1.13378
$898$	0	0
$899$	−51.0130	−1.70138
$900$	0	0
$901$	−20.1168	−0.670187
$902$	0	0
$903$	−27.7470	−0.923361
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−36.5154	−1.21247	−0.606237	−	0.795284i	$-0.707322\pi$
$907$	−36.5154	−1.21247	−0.606237	+	0.795284i	$0.707322\pi$
$908$	0	0
$909$	−15.9543	−0.529172
$910$	0	0
$911$	−6.62735	−0.219574	−0.109787	−	0.993955i	$-0.535017\pi$
$911$	−6.62735	−0.219574	−0.109787	+	0.993955i	$0.535017\pi$
$912$	0	0
$913$	9.79036	0.324014
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−48.1662	−1.59059
$918$	0	0
$919$	−7.91688	−0.261154	−0.130577	−	0.991438i	$-0.541683\pi$
$919$	−7.91688	−0.261154	−0.130577	+	0.991438i	$0.541683\pi$
$920$	0	0
$921$	−12.6210	−0.415877
$922$	0	0
$923$	12.6743	0.417179
$924$	0	0
$925$	0	0
$926$	0	0
$927$	32.0178	1.05160
$928$	0	0
$929$	35.8597	1.17652	0.588259	−	0.808673i	$-0.299814\pi$
$929$	35.8597	1.17652	0.588259	+	0.808673i	$0.299814\pi$
$930$	0	0
$931$	−2.63990	−0.0865192
$932$	0	0
$933$	−45.2254	−1.48061
$934$	0	0
$935$	0	0
$936$	0	0
$937$	11.4420	0.373793	0.186896	−	0.982380i	$-0.440157\pi$
$937$	11.4420	0.373793	0.186896	+	0.982380i	$0.440157\pi$
$938$	0	0
$939$	71.3236	2.32756
$940$	0	0
$941$	−0.360100	−0.0117389	−0.00586945	−	0.999983i	$-0.501868\pi$
$941$	−0.360100	−0.0117389	−0.00586945	+	0.999983i	$0.501868\pi$
$942$	0	0
$943$	−18.7579	−0.610843
$944$	0	0
$945$	0	0
$946$	0	0
$947$	2.63807	0.0857256	0.0428628	−	0.999081i	$-0.486352\pi$
$947$	2.63807	0.0857256	0.0428628	+	0.999081i	$0.486352\pi$
$948$	0	0
$949$	0.431517	0.0140076
$950$	0	0
$951$	−78.8084	−2.55554
$952$	0	0
$953$	−10.4430	−0.338282	−0.169141	−	0.985592i	$-0.554099\pi$
$953$	−10.4430	−0.338282	−0.169141	+	0.985592i	$0.554099\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−24.6667	−0.797360
$958$	0	0
$959$	2.52437	0.0815160
$960$	0	0
$961$	12.2522	0.395232
$962$	0	0
$963$	107.731	3.47159
$964$	0	0
$965$	0	0
$966$	0	0
$967$	33.2732	1.06999	0.534996	−	0.844854i	$-0.320313\pi$
$967$	33.2732	1.06999	0.534996	+	0.844854i	$0.320313\pi$
$968$	0	0
$969$	−22.3264	−0.717228
$970$	0	0
$971$	−23.5657	−0.756258	−0.378129	−	0.925753i	$-0.623432\pi$
$971$	−23.5657	−0.756258	−0.378129	+	0.925753i	$0.623432\pi$
$972$	0	0
$973$	39.6444	1.27094
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−0.402439	−0.0128752	−0.00643759	−	0.999979i	$-0.502049\pi$
$977$	−0.402439	−0.0128752	−0.00643759	+	0.999979i	$0.502049\pi$
$978$	0	0
$979$	0.898271	0.0287089
$980$	0	0
$981$	24.9821	0.797617
$982$	0	0
$983$	11.4927	0.366561	0.183281	−	0.983061i	$-0.441328\pi$
$983$	11.4927	0.366561	0.183281	+	0.983061i	$0.441328\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−13.0916	−0.416710
$988$	0	0
$989$	−20.6529	−0.656723
$990$	0	0
$991$	−6.05761	−0.192426	−0.0962132	−	0.995361i	$-0.530673\pi$
$991$	−6.05761	−0.192426	−0.0962132	+	0.995361i	$0.530673\pi$
$992$	0	0
$993$	13.5292	0.429335
$994$	0	0
$995$	0	0
$996$	0	0
$997$	48.1086	1.52362	0.761808	−	0.647803i	$-0.224312\pi$
$997$	48.1086	1.52362	0.761808	+	0.647803i	$0.224312\pi$
$998$	0	0
$999$	−9.25020	−0.292663

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7600.2.a.cf.1.1		4
4.3	odd	2		475.2.a.i.1.3		4
5.4	even	2		1520.2.a.t.1.4		4
12.11	even	2		4275.2.a.bo.1.2		4
20.3	even	4		475.2.b.e.324.2		8
20.7	even	4		475.2.b.e.324.7		8
20.19	odd	2		95.2.a.b.1.2	✓	4
40.19	odd	2		6080.2.a.cc.1.4		4
40.29	even	2		6080.2.a.ch.1.1		4
60.59	even	2		855.2.a.m.1.3		4
76.75	even	2		9025.2.a.bf.1.2		4
140.139	even	2		4655.2.a.y.1.2		4
380.379	even	2		1805.2.a.p.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.2.a.b.1.2	✓	4	20.19	odd	2
475.2.a.i.1.3		4	4.3	odd	2
475.2.b.e.324.2		8	20.3	even	4
475.2.b.e.324.7		8	20.7	even	4
855.2.a.m.1.3		4	60.59	even	2
1520.2.a.t.1.4		4	5.4	even	2
1805.2.a.p.1.3		4	380.379	even	2
4275.2.a.bo.1.2		4	12.11	even	2
4655.2.a.y.1.2		4	140.139	even	2
6080.2.a.cc.1.4		4	40.19	odd	2
6080.2.a.ch.1.1		4	40.29	even	2
7600.2.a.cf.1.1		4	1.1	even	1	trivial
9025.2.a.bf.1.2		4	76.75	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)

\(f(q)\)	\(=\)	\(q-2.87834 q^{3} +3.10482 q^{7} +5.28487 q^{9} +O(q^{10})\)	\(q-2.87834 q^{3} +3.10482 q^{7} +5.28487 q^{9} +1.10482 q^{11} -1.77353 q^{13} -7.75669 q^{17} -1.00000 q^{19} -8.93674 q^{21} -6.65187 q^{23} -6.57664 q^{27} +7.75669 q^{29} -6.57664 q^{31} -3.18005 q^{33} +1.40652 q^{37} +5.10482 q^{39} +2.81995 q^{41} +3.10482 q^{43} +1.46492 q^{47} +2.63990 q^{49} +22.3264 q^{51} +2.59348 q^{53} +2.87834 q^{57} +5.38969 q^{59} +4.07523 q^{61} +16.4086 q^{63} -15.8151 q^{67} +19.1464 q^{69} -7.14638 q^{71} -0.243310 q^{73} +3.43026 q^{77} +9.38969 q^{79} +3.07523 q^{81} +8.86151 q^{83} -22.3264 q^{87} +0.813048 q^{89} -5.50648 q^{91} +18.9298 q^{93} -3.19689 q^{97} +5.83882 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} + 4 q^{7} + 8 q^{9}+O(q^{10}) \) 4 * q + 2 * q^3 + 4 * q^7 + 8 * q^9	\( 4 q + 2 q^{3} + 4 q^{7} + 8 q^{9} - 4 q^{11} - 2 q^{13} - 4 q^{17} - 4 q^{19} - 4 q^{21} - 8 q^{23} - 4 q^{27} + 4 q^{29} - 4 q^{31} - 8 q^{33} + 6 q^{37} + 12 q^{39} + 16 q^{41} + 4 q^{43} - 12 q^{47} + 20 q^{49} + 36 q^{51} + 10 q^{53} - 2 q^{57} + 20 q^{61} + 20 q^{63} - 18 q^{67} + 28 q^{69} + 20 q^{71} - 28 q^{73} + 40 q^{77} + 16 q^{79} + 16 q^{81} - 36 q^{87} + 4 q^{89} + 36 q^{91} + 40 q^{93} - 30 q^{97} + 4 q^{99}+O(q^{100}) \) 4 * q + 2 * q^3 + 4 * q^7 + 8 * q^9 - 4 * q^11 - 2 * q^13 - 4 * q^17 - 4 * q^19 - 4 * q^21 - 8 * q^23 - 4 * q^27 + 4 * q^29 - 4 * q^31 - 8 * q^33 + 6 * q^37 + 12 * q^39 + 16 * q^41 + 4 * q^43 - 12 * q^47 + 20 * q^49 + 36 * q^51 + 10 * q^53 - 2 * q^57 + 20 * q^61 + 20 * q^63 - 18 * q^67 + 28 * q^69 + 20 * q^71 - 28 * q^73 + 40 * q^77 + 16 * q^79 + 16 * q^81 - 36 * q^87 + 4 * q^89 + 36 * q^91 + 40 * q^93 - 30 * q^97 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more