LMFDB - Embedded newform 7600.2.a.cn.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:	$6$
Coefficient field:	6.6.253565184.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 11x^{4} + 20x^{3} + 22x^{2} - 32x + 8 $ x^6 - 2x^5 - 11x^4 + 20x^3 + 22x^2 - 32*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 760)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.366738$ 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)

$f(q)$	$=$	$q+0.366738 q^{3} -3.08675 q^{7} -2.86550 q^{9} +4.94567 q^{11} -3.79016 q^{13} +3.97713 q^{17} -1.00000 q^{19} -1.13203 q^{21} -6.56426 q^{23} -2.15110 q^{27} +5.14084 q^{29} -4.25626 q^{31} +1.81376 q^{33} -7.75503 q^{37} -1.39000 q^{39} -2.25626 q^{41} +10.2276 q^{43} -0.935652 q^{47} +2.52804 q^{49} +1.45856 q^{51} -7.52117 q^{53} -0.366738 q^{57} +10.4356 q^{59} +10.9163 q^{61} +8.84510 q^{63} +6.80924 q^{67} -2.40736 q^{69} +2.98093 q^{71} -2.19638 q^{73} -15.2661 q^{77} +3.26652 q^{79} +7.80762 q^{81} +4.09936 q^{83} +1.88534 q^{87} -12.4134 q^{89} +11.6993 q^{91} -1.56093 q^{93} -10.6172 q^{97} -14.1718 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{3} + 6 q^{7} + 8 q^{9} + 2 q^{11} - 14 q^{13} - 10 q^{17} - 6 q^{19} + 18 q^{21} + 2 q^{23} + 2 q^{27} - 2 q^{29} - 8 q^{31} - 8 q^{33} - 4 q^{37} + 18 q^{39} + 4 q^{41} + 4 q^{43} + 4 q^{47}+ \cdots + 22 q^{99}+O(q^{100}) $ 6 * q + 2 * q^3 + 6 * q^7 + 8 * q^9 + 2 * q^11 - 14 * q^13 - 10 * q^17 - 6 * q^19 + 18 * q^21 + 2 * q^23 + 2 * q^27 - 2 * q^29 - 8 * q^31 - 8 * q^33 - 4 * q^37 + 18 * q^39 + 4 * q^41 + 4 * q^43 + 4 * q^47 + 2 * q^49 + 34 * q^51 + 14 * q^53 - 2 * q^57 + 2 * q^59 + 10 * q^61 + 44 * q^63 + 42 * q^67 + 18 * q^69 + 8 * q^71 + 2 * q^73 - 24 * q^77 + 20 * q^79 + 6 * q^81 + 16 * q^83 + 30 * q^87 + 32 * q^89 - 10 * q^91 - 12 * q^93 - 40 * q^97 + 22 * q^99

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
$3$	0.366738	0.211736	0.105868	−	0.994380i	$-0.466238\pi$
$3$	0.366738	0.211736	0.105868	+	0.994380i	$0.466238\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.08675	−1.16668	−0.583341	−	0.812227i	$-0.698255\pi$
$7$	−3.08675	−1.16668	−0.583341	+	0.812227i	$0.698255\pi$
$8$	0	0
$9$	−2.86550	−0.955168
$10$	0	0
$11$	4.94567	1.49118	0.745588	−	0.666407i	$-0.232169\pi$
$11$	4.94567	1.49118	0.745588	+	0.666407i	$0.232169\pi$
$12$	0	0
$13$	−3.79016	−1.05120	−0.525601	−	0.850731i	$-0.676160\pi$
$13$	−3.79016	−1.05120	−0.525601	+	0.850731i	$0.676160\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.97713	0.964596	0.482298	−	0.876007i	$-0.339802\pi$
$17$	3.97713	0.964596	0.482298	+	0.876007i	$0.339802\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−1.13203	−0.247029
$22$	0	0
$23$	−6.56426	−1.36874	−0.684372	−	0.729133i	$-0.739923\pi$
$23$	−6.56426	−1.36874	−0.684372	+	0.729133i	$0.739923\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−2.15110	−0.413980
$28$	0	0
$29$	5.14084	0.954630	0.477315	−	0.878732i	$-0.341610\pi$
$29$	5.14084	0.954630	0.477315	+	0.878732i	$0.341610\pi$
$30$	0	0
$31$	−4.25626	−0.764447	−0.382224	−	0.924070i	$-0.624842\pi$
$31$	−4.25626	−0.764447	−0.382224	+	0.924070i	$0.624842\pi$
$32$	0	0
$33$	1.81376	0.315736
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−7.75503	−1.27492	−0.637459	−	0.770484i	$-0.720015\pi$
$37$	−7.75503	−1.27492	−0.637459	+	0.770484i	$0.720015\pi$
$38$	0	0
$39$	−1.39000	−0.222577
$40$	0	0
$41$	−2.25626	−0.352369	−0.176184	−	0.984357i	$-0.556376\pi$
$41$	−2.25626	−0.352369	−0.176184	+	0.984357i	$0.556376\pi$
$42$	0	0
$43$	10.2276	1.55969	0.779846	−	0.625971i	$-0.215297\pi$
$43$	10.2276	1.55969	0.779846	+	0.625971i	$0.215297\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−0.935652	−0.136479	−0.0682395	−	0.997669i	$-0.521738\pi$
$47$	−0.935652	−0.136479	−0.0682395	+	0.997669i	$0.521738\pi$
$48$	0	0
$49$	2.52804	0.361149
$50$	0	0
$51$	1.45856	0.204240
$52$	0	0
$53$	−7.52117	−1.03311	−0.516556	−	0.856253i	$-0.672786\pi$
$53$	−7.52117	−1.03311	−0.516556	+	0.856253i	$0.672786\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−0.366738	−0.0485756
$58$	0	0
$59$	10.4356	1.35860	0.679298	−	0.733863i	$-0.262285\pi$
$59$	10.4356	1.35860	0.679298	+	0.733863i	$0.262285\pi$
$60$	0	0
$61$	10.9163	1.39769	0.698847	−	0.715271i	$-0.253697\pi$
$61$	10.9163	1.39769	0.698847	+	0.715271i	$0.253697\pi$
$62$	0	0
$63$	8.84510	1.11438
$64$	0	0
$65$	0	0
$66$	0	0
$67$	6.80924	0.831881	0.415940	−	0.909392i	$-0.363453\pi$
$67$	6.80924	0.831881	0.415940	+	0.909392i	$0.363453\pi$
$68$	0	0
$69$	−2.40736	−0.289812
$70$	0	0
$71$	2.98093	0.353771	0.176886	−	0.984231i	$-0.443398\pi$
$71$	2.98093	0.353771	0.176886	+	0.984231i	$0.443398\pi$
$72$	0	0
$73$	−2.19638	−0.257066	−0.128533	−	0.991705i	$-0.541027\pi$
$73$	−2.19638	−0.257066	−0.128533	+	0.991705i	$0.541027\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−15.2661	−1.73973
$78$	0	0
$79$	3.26652	0.367513	0.183756	−	0.982972i	$-0.441174\pi$
$79$	3.26652	0.367513	0.183756	+	0.982972i	$0.441174\pi$
$80$	0	0
$81$	7.80762	0.867513
$82$	0	0
$83$	4.09936	0.449963	0.224982	−	0.974363i	$-0.427768\pi$
$83$	4.09936	0.449963	0.224982	+	0.974363i	$0.427768\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1.88534	0.202129
$88$	0	0
$89$	−12.4134	−1.31581	−0.657907	−	0.753099i	$-0.728558\pi$
$89$	−12.4134	−1.31581	−0.657907	+	0.753099i	$0.728558\pi$
$90$	0	0
$91$	11.6993	1.22642
$92$	0	0
$93$	−1.56093	−0.161861
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−10.6172	−1.07801	−0.539007	−	0.842302i	$-0.681200\pi$
$97$	−10.6172	−1.07801	−0.539007	+	0.842302i	$0.681200\pi$
$98$	0	0
$99$	−14.1718	−1.42432
$100$	0	0
$101$	−13.1119	−1.30469	−0.652343	−	0.757924i	$-0.726214\pi$
$101$	−13.1119	−1.30469	−0.652343	+	0.757924i	$0.726214\pi$
$102$	0	0
$103$	4.95691	0.488419	0.244209	−	0.969723i	$-0.421472\pi$
$103$	4.95691	0.488419	0.244209	+	0.969723i	$0.421472\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	18.9142	1.82850	0.914251	−	0.405148i	$-0.132780\pi$
$107$	18.9142	1.82850	0.914251	+	0.405148i	$0.132780\pi$
$108$	0	0
$109$	8.90118	0.852578	0.426289	−	0.904587i	$-0.359821\pi$
$109$	8.90118	0.852578	0.426289	+	0.904587i	$0.359821\pi$
$110$	0	0
$111$	−2.84406	−0.269946
$112$	0	0
$113$	9.85071	0.926677	0.463339	−	0.886181i	$-0.346651\pi$
$113$	9.85071	0.926677	0.463339	+	0.886181i	$0.346651\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	10.8607	1.00407
$118$	0	0
$119$	−12.2764	−1.12538
$120$	0	0
$121$	13.4596	1.22360
$122$	0	0
$123$	−0.827456	−0.0746092
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−3.76384	−0.333987	−0.166993	−	0.985958i	$-0.553406\pi$
$127$	−3.76384	−0.333987	−0.166993	+	0.985958i	$0.553406\pi$
$128$	0	0
$129$	3.75084	0.330243
$130$	0	0
$131$	−10.0488	−0.877966	−0.438983	−	0.898495i	$-0.644661\pi$
$131$	−10.0488	−0.877966	−0.438983	+	0.898495i	$0.644661\pi$
$132$	0	0
$133$	3.08675	0.267655
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−14.6244	−1.24945	−0.624724	−	0.780845i	$-0.714789\pi$
$137$	−14.6244	−1.24945	−0.624724	+	0.780845i	$0.714789\pi$
$138$	0	0
$139$	12.0773	1.02438	0.512191	−	0.858872i	$-0.328834\pi$
$139$	12.0773	1.02438	0.512191	+	0.858872i	$0.328834\pi$
$140$	0	0
$141$	−0.343139	−0.0288975
$142$	0	0
$143$	−18.7449	−1.56753
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0.927129	0.0764683
$148$	0	0
$149$	12.5739	1.03009	0.515046	−	0.857162i	$-0.327775\pi$
$149$	12.5739	1.03009	0.515046	+	0.857162i	$0.327775\pi$
$150$	0	0
$151$	0.390336	0.0317651	0.0158826	−	0.999874i	$-0.494944\pi$
$151$	0.390336	0.0317651	0.0158826	+	0.999874i	$0.494944\pi$
$152$	0	0
$153$	−11.3965	−0.921351
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−17.6517	−1.40876	−0.704379	−	0.709824i	$-0.748774\pi$
$157$	−17.6517	−1.40876	−0.704379	+	0.709824i	$0.748774\pi$
$158$	0	0
$159$	−2.75830	−0.218747
$160$	0	0
$161$	20.2623	1.59689
$162$	0	0
$163$	5.72733	0.448599	0.224300	−	0.974520i	$-0.427991\pi$
$163$	5.72733	0.448599	0.224300	+	0.974520i	$0.427991\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	8.22343	0.636348	0.318174	−	0.948032i	$-0.396930\pi$
$167$	8.22343	0.636348	0.318174	+	0.948032i	$0.396930\pi$
$168$	0	0
$169$	1.36534	0.105027
$170$	0	0
$171$	2.86550	0.219131
$172$	0	0
$173$	15.1427	1.15127	0.575637	−	0.817705i	$-0.304754\pi$
$173$	15.1427	1.15127	0.575637	+	0.817705i	$0.304754\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	3.82712	0.287664
$178$	0	0
$179$	−4.07415	−0.304516	−0.152258	−	0.988341i	$-0.548654\pi$
$179$	−4.07415	−0.304516	−0.152258	+	0.988341i	$0.548654\pi$
$180$	0	0
$181$	2.92981	0.217771	0.108885	−	0.994054i	$-0.465272\pi$
$181$	2.92981	0.217771	0.108885	+	0.994054i	$0.465272\pi$
$182$	0	0
$183$	4.00343	0.295942
$184$	0	0
$185$	0	0
$186$	0	0
$187$	19.6696	1.43838
$188$	0	0
$189$	6.63992	0.482983
$190$	0	0
$191$	5.41644	0.391920	0.195960	−	0.980612i	$-0.437218\pi$
$191$	5.41644	0.391920	0.195960	+	0.980612i	$0.437218\pi$
$192$	0	0
$193$	24.6275	1.77273	0.886363	−	0.462991i	$-0.153224\pi$
$193$	24.6275	1.77273	0.886363	+	0.462991i	$0.153224\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3.93923	0.280658	0.140329	−	0.990105i	$-0.455184\pi$
$197$	3.93923	0.280658	0.140329	+	0.990105i	$0.455184\pi$
$198$	0	0
$199$	20.8079	1.47504	0.737518	−	0.675328i	$-0.235998\pi$
$199$	20.8079	1.47504	0.737518	+	0.675328i	$0.235998\pi$
$200$	0	0
$201$	2.49720	0.176139
$202$	0	0
$203$	−15.8685	−1.11375
$204$	0	0
$205$	0	0
$206$	0	0
$207$	18.8099	1.30738
$208$	0	0
$209$	−4.94567	−0.342099
$210$	0	0
$211$	22.9069	1.57698	0.788488	−	0.615050i	$-0.210864\pi$
$211$	22.9069	1.57698	0.788488	+	0.615050i	$0.210864\pi$
$212$	0	0
$213$	1.09322	0.0749061
$214$	0	0
$215$	0	0
$216$	0	0
$217$	13.1380	0.891868
$218$	0	0
$219$	−0.805494	−0.0544302
$220$	0	0
$221$	−15.0740	−1.01399
$222$	0	0
$223$	16.6488	1.11489	0.557444	−	0.830214i	$-0.311782\pi$
$223$	16.6488	1.11489	0.557444	+	0.830214i	$0.311782\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−11.8343	−0.785469	−0.392735	−	0.919652i	$-0.628471\pi$
$227$	−11.8343	−0.785469	−0.392735	+	0.919652i	$0.628471\pi$
$228$	0	0
$229$	4.84288	0.320026	0.160013	−	0.987115i	$-0.448846\pi$
$229$	4.84288	0.320026	0.160013	+	0.987115i	$0.448846\pi$
$230$	0	0
$231$	−5.59864	−0.368363
$232$	0	0
$233$	−9.60851	−0.629474	−0.314737	−	0.949179i	$-0.601916\pi$
$233$	−9.60851	−0.629474	−0.314737	+	0.949179i	$0.601916\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1.19796	0.0778157
$238$	0	0
$239$	9.49984	0.614493	0.307247	−	0.951630i	$-0.400592\pi$
$239$	9.49984	0.614493	0.307247	+	0.951630i	$0.400592\pi$
$240$	0	0
$241$	−21.8687	−1.40868	−0.704342	−	0.709860i	$-0.748758\pi$
$241$	−21.8687	−1.40868	−0.704342	+	0.709860i	$0.748758\pi$
$242$	0	0
$243$	9.31665	0.597663
$244$	0	0
$245$	0	0
$246$	0	0
$247$	3.79016	0.241162
$248$	0	0
$249$	1.50339	0.0952734
$250$	0	0
$251$	−2.03109	−0.128201	−0.0641007	−	0.997943i	$-0.520418\pi$
$251$	−2.03109	−0.128201	−0.0641007	+	0.997943i	$0.520418\pi$
$252$	0	0
$253$	−32.4647	−2.04104
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−7.62917	−0.475894	−0.237947	−	0.971278i	$-0.576475\pi$
$257$	−7.62917	−0.475894	−0.237947	+	0.971278i	$0.576475\pi$
$258$	0	0
$259$	23.9379	1.48743
$260$	0	0
$261$	−14.7311	−0.911831
$262$	0	0
$263$	−30.3299	−1.87022	−0.935112	−	0.354352i	$-0.884702\pi$
$263$	−30.3299	−1.87022	−0.935112	+	0.354352i	$0.884702\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−4.55245	−0.278605
$268$	0	0
$269$	24.3382	1.48393	0.741963	−	0.670441i	$-0.233895\pi$
$269$	24.3382	1.48393	0.741963	+	0.670441i	$0.233895\pi$
$270$	0	0
$271$	−3.62084	−0.219951	−0.109975	−	0.993934i	$-0.535077\pi$
$271$	−3.62084	−0.219951	−0.109975	+	0.993934i	$0.535077\pi$
$272$	0	0
$273$	4.29057	0.259677
$274$	0	0
$275$	0	0
$276$	0	0
$277$	29.6842	1.78355	0.891776	−	0.452477i	$-0.149459\pi$
$277$	29.6842	1.78355	0.891776	+	0.452477i	$0.149459\pi$
$278$	0	0
$279$	12.1963	0.730175
$280$	0	0
$281$	29.1363	1.73813	0.869064	−	0.494700i	$-0.164722\pi$
$281$	29.1363	1.73813	0.869064	+	0.494700i	$0.164722\pi$
$282$	0	0
$283$	24.2081	1.43902	0.719511	−	0.694481i	$-0.244366\pi$
$283$	24.2081	1.43902	0.719511	+	0.694481i	$0.244366\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	6.96452	0.411103
$288$	0	0
$289$	−1.18244	−0.0695551
$290$	0	0
$291$	−3.89373	−0.228254
$292$	0	0
$293$	15.9338	0.930865	0.465433	−	0.885083i	$-0.345899\pi$
$293$	15.9338	0.930865	0.465433	+	0.885083i	$0.345899\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−10.6386	−0.617316
$298$	0	0
$299$	24.8796	1.43883
$300$	0	0
$301$	−31.5700	−1.81967
$302$	0	0
$303$	−4.80864	−0.276249
$304$	0	0
$305$	0	0
$306$	0	0
$307$	25.9373	1.48032	0.740161	−	0.672429i	$-0.234749\pi$
$307$	25.9373	1.48032	0.740161	+	0.672429i	$0.234749\pi$
$308$	0	0
$309$	1.81788	0.103416
$310$	0	0
$311$	2.58526	0.146597	0.0732983	−	0.997310i	$-0.476647\pi$
$311$	2.58526	0.146597	0.0732983	+	0.997310i	$0.476647\pi$
$312$	0	0
$313$	−17.1027	−0.966701	−0.483351	−	0.875427i	$-0.660580\pi$
$313$	−17.1027	−0.966701	−0.483351	+	0.875427i	$0.660580\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−24.5591	−1.37938	−0.689690	−	0.724105i	$-0.742253\pi$
$317$	−24.5591	−1.37938	−0.689690	+	0.724105i	$0.742253\pi$
$318$	0	0
$319$	25.4249	1.42352
$320$	0	0
$321$	6.93654	0.387160
$322$	0	0
$323$	−3.97713	−0.221293
$324$	0	0
$325$	0	0
$326$	0	0
$327$	3.26440	0.180522
$328$	0	0
$329$	2.88813	0.159228
$330$	0	0
$331$	−16.4328	−0.903227	−0.451614	−	0.892214i	$-0.649151\pi$
$331$	−16.4328	−0.903227	−0.451614	+	0.892214i	$0.649151\pi$
$332$	0	0
$333$	22.2221	1.21776
$334$	0	0
$335$	0	0
$336$	0	0
$337$	12.0174	0.654627	0.327314	−	0.944916i	$-0.393857\pi$
$337$	12.0174	0.654627	0.327314	+	0.944916i	$0.393857\pi$
$338$	0	0
$339$	3.61263	0.196211
$340$	0	0
$341$	−21.0501	−1.13992
$342$	0	0
$343$	13.8038	0.745336
$344$	0	0
$345$	0	0
$346$	0	0
$347$	28.3681	1.52288	0.761439	−	0.648236i	$-0.224493\pi$
$347$	28.3681	1.52288	0.761439	+	0.648236i	$0.224493\pi$
$348$	0	0
$349$	33.9733	1.81855	0.909275	−	0.416195i	$-0.136637\pi$
$349$	33.9733	1.81855	0.909275	+	0.416195i	$0.136637\pi$
$350$	0	0
$351$	8.15302	0.435176
$352$	0	0
$353$	−26.6181	−1.41674	−0.708369	−	0.705842i	$-0.750569\pi$
$353$	−26.6181	−1.41674	−0.708369	+	0.705842i	$0.750569\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−4.50222	−0.238283
$358$	0	0
$359$	−5.84964	−0.308732	−0.154366	−	0.988014i	$-0.549334\pi$
$359$	−5.84964	−0.308732	−0.154366	+	0.988014i	$0.549334\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	4.93616	0.259081
$364$	0	0
$365$	0	0
$366$	0	0
$367$	2.41146	0.125877	0.0629385	−	0.998017i	$-0.479953\pi$
$367$	2.41146	0.125877	0.0629385	+	0.998017i	$0.479953\pi$
$368$	0	0
$369$	6.46533	0.336571
$370$	0	0
$371$	23.2160	1.20531
$372$	0	0
$373$	−27.2074	−1.40874	−0.704372	−	0.709831i	$-0.748771\pi$
$373$	−27.2074	−1.40874	−0.704372	+	0.709831i	$0.748771\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−19.4846	−1.00351
$378$	0	0
$379$	37.0749	1.90441	0.952204	−	0.305461i	$-0.0988107\pi$
$379$	37.0749	1.90441	0.952204	+	0.305461i	$0.0988107\pi$
$380$	0	0
$381$	−1.38034	−0.0707170
$382$	0	0
$383$	−0.238749	−0.0121995	−0.00609975	−	0.999981i	$-0.501942\pi$
$383$	−0.238749	−0.0121995	−0.00609975	+	0.999981i	$0.501942\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−29.3072	−1.48977
$388$	0	0
$389$	21.3399	1.08197	0.540987	−	0.841031i	$-0.318051\pi$
$389$	21.3399	1.08197	0.540987	+	0.841031i	$0.318051\pi$
$390$	0	0
$391$	−26.1069	−1.32028
$392$	0	0
$393$	−3.68527	−0.185897
$394$	0	0
$395$	0	0
$396$	0	0
$397$	17.7416	0.890424	0.445212	−	0.895425i	$-0.353128\pi$
$397$	17.7416	0.890424	0.445212	+	0.895425i	$0.353128\pi$
$398$	0	0
$399$	1.13203	0.0566723
$400$	0	0
$401$	36.4310	1.81928	0.909638	−	0.415401i	$-0.136359\pi$
$401$	36.4310	1.81928	0.909638	+	0.415401i	$0.136359\pi$
$402$	0	0
$403$	16.1319	0.803589
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−38.3538	−1.90113
$408$	0	0
$409$	−0.530833	−0.0262480	−0.0131240	−	0.999914i	$-0.504178\pi$
$409$	−0.530833	−0.0262480	−0.0131240	+	0.999914i	$0.504178\pi$
$410$	0	0
$411$	−5.36332	−0.264553
$412$	0	0
$413$	−32.2120	−1.58505
$414$	0	0
$415$	0	0
$416$	0	0
$417$	4.42919	0.216898
$418$	0	0
$419$	−29.3816	−1.43538	−0.717692	−	0.696361i	$-0.754801\pi$
$419$	−29.3816	−1.43538	−0.717692	+	0.696361i	$0.754801\pi$
$420$	0	0
$421$	2.95912	0.144219	0.0721093	−	0.997397i	$-0.477027\pi$
$421$	2.95912	0.144219	0.0721093	+	0.997397i	$0.477027\pi$
$422$	0	0
$423$	2.68112	0.130360
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−33.6960	−1.63067
$428$	0	0
$429$	−6.87446	−0.331902
$430$	0	0
$431$	32.4518	1.56315	0.781573	−	0.623814i	$-0.214418\pi$
$431$	32.4518	1.56315	0.781573	+	0.623814i	$0.214418\pi$
$432$	0	0
$433$	−27.7821	−1.33512	−0.667562	−	0.744554i	$-0.732662\pi$
$433$	−27.7821	−1.33512	−0.667562	+	0.744554i	$0.732662\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	6.56426	0.314011
$438$	0	0
$439$	0.0388357	0.00185352	0.000926762	−	1.00000i	$-0.499705\pi$
$439$	0.0388357	0.00185352	0.000926762		1.00000i	$0.499705\pi$
$440$	0	0
$441$	−7.24412	−0.344958
$442$	0	0
$443$	3.98842	0.189496	0.0947479	−	0.995501i	$-0.469796\pi$
$443$	3.98842	0.189496	0.0947479	+	0.995501i	$0.469796\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	4.61132	0.218108
$448$	0	0
$449$	23.1695	1.09343	0.546717	−	0.837317i	$-0.315877\pi$
$449$	23.1695	1.09343	0.546717	+	0.837317i	$0.315877\pi$
$450$	0	0
$451$	−11.1587	−0.525444
$452$	0	0
$453$	0.143151	0.00672582
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−11.5398	−0.539811	−0.269905	−	0.962887i	$-0.586992\pi$
$457$	−11.5398	−0.539811	−0.269905	+	0.962887i	$0.586992\pi$
$458$	0	0
$459$	−8.55521	−0.399323
$460$	0	0
$461$	8.96090	0.417351	0.208675	−	0.977985i	$-0.433085\pi$
$461$	8.96090	0.417351	0.208675	+	0.977985i	$0.433085\pi$
$462$	0	0
$463$	−4.43012	−0.205885	−0.102943	−	0.994687i	$-0.532826\pi$
$463$	−4.43012	−0.205885	−0.102943	+	0.994687i	$0.532826\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−30.6874	−1.42004	−0.710021	−	0.704180i	$-0.751315\pi$
$467$	−30.6874	−1.42004	−0.710021	+	0.704180i	$0.751315\pi$
$468$	0	0
$469$	−21.0184	−0.970541
$470$	0	0
$471$	−6.47354	−0.298285
$472$	0	0
$473$	50.5823	2.32578
$474$	0	0
$475$	0	0
$476$	0	0
$477$	21.5519	0.986796
$478$	0	0
$479$	18.0047	0.822657	0.411328	−	0.911487i	$-0.365065\pi$
$479$	18.0047	0.822657	0.411328	+	0.911487i	$0.365065\pi$
$480$	0	0
$481$	29.3928	1.34020
$482$	0	0
$483$	7.43093	0.338119
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−14.7434	−0.668087	−0.334043	−	0.942558i	$-0.608413\pi$
$487$	−14.7434	−0.668087	−0.334043	+	0.942558i	$0.608413\pi$
$488$	0	0
$489$	2.10043	0.0949847
$490$	0	0
$491$	−40.0805	−1.80881	−0.904404	−	0.426677i	$-0.859684\pi$
$491$	−40.0805	−1.80881	−0.904404	+	0.426677i	$0.859684\pi$
$492$	0	0
$493$	20.4458	0.920832
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−9.20139	−0.412739
$498$	0	0
$499$	−12.7838	−0.572282	−0.286141	−	0.958188i	$-0.592373\pi$
$499$	−12.7838	−0.572282	−0.286141	+	0.958188i	$0.592373\pi$
$500$	0	0
$501$	3.01584	0.134738
$502$	0	0
$503$	−11.9101	−0.531046	−0.265523	−	0.964105i	$-0.585545\pi$
$503$	−11.9101	−0.531046	−0.265523	+	0.964105i	$0.585545\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0.500723	0.0222379
$508$	0	0
$509$	15.7374	0.697549	0.348775	−	0.937207i	$-0.386598\pi$
$509$	15.7374	0.697549	0.348775	+	0.937207i	$0.386598\pi$
$510$	0	0
$511$	6.77967	0.299915
$512$	0	0
$513$	2.15110	0.0949734
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−4.62743	−0.203514
$518$	0	0
$519$	5.55338	0.243766
$520$	0	0
$521$	5.67909	0.248806	0.124403	−	0.992232i	$-0.460299\pi$
$521$	5.67909	0.248806	0.124403	+	0.992232i	$0.460299\pi$
$522$	0	0
$523$	−36.8094	−1.60956	−0.804781	−	0.593572i	$-0.797717\pi$
$523$	−36.8094	−1.60956	−0.804781	+	0.593572i	$0.797717\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−16.9277	−0.737382
$528$	0	0
$529$	20.0896	0.873459
$530$	0	0
$531$	−29.9032	−1.29769
$532$	0	0
$533$	8.55160	0.370411
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−1.49414	−0.0644770
$538$	0	0
$539$	12.5029	0.538537
$540$	0	0
$541$	−24.9051	−1.07076	−0.535378	−	0.844613i	$-0.679831\pi$
$541$	−24.9051	−1.07076	−0.535378	+	0.844613i	$0.679831\pi$
$542$	0	0
$543$	1.07447	0.0461100
$544$	0	0
$545$	0	0
$546$	0	0
$547$	9.17066	0.392109	0.196055	−	0.980593i	$-0.437187\pi$
$547$	9.17066	0.392109	0.196055	+	0.980593i	$0.437187\pi$
$548$	0	0
$549$	−31.2808	−1.33503
$550$	0	0
$551$	−5.14084	−0.219007
$552$	0	0
$553$	−10.0830	−0.428771
$554$	0	0
$555$	0	0
$556$	0	0
$557$	16.3168	0.691364	0.345682	−	0.938352i	$-0.387648\pi$
$557$	16.3168	0.691364	0.345682	+	0.938352i	$0.387648\pi$
$558$	0	0
$559$	−38.7642	−1.63955
$560$	0	0
$561$	7.21357	0.304557
$562$	0	0
$563$	23.4254	0.987264	0.493632	−	0.869671i	$-0.335669\pi$
$563$	23.4254	0.987264	0.493632	+	0.869671i	$0.335669\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−24.1002	−1.01211
$568$	0	0
$569$	−21.5994	−0.905495	−0.452748	−	0.891639i	$-0.649556\pi$
$569$	−21.5994	−0.905495	−0.452748	+	0.891639i	$0.649556\pi$
$570$	0	0
$571$	13.9099	0.582111	0.291055	−	0.956706i	$-0.405994\pi$
$571$	13.9099	0.582111	0.291055	+	0.956706i	$0.405994\pi$
$572$	0	0
$573$	1.98641	0.0829836
$574$	0	0
$575$	0	0
$576$	0	0
$577$	27.0208	1.12489	0.562445	−	0.826834i	$-0.309861\pi$
$577$	27.0208	1.12489	0.562445	+	0.826834i	$0.309861\pi$
$578$	0	0
$579$	9.03183	0.375350
$580$	0	0
$581$	−12.6537	−0.524964
$582$	0	0
$583$	−37.1972	−1.54055
$584$	0	0
$585$	0	0
$586$	0	0
$587$	2.29463	0.0947096	0.0473548	−	0.998878i	$-0.484921\pi$
$587$	2.29463	0.0947096	0.0473548	+	0.998878i	$0.484921\pi$
$588$	0	0
$589$	4.25626	0.175376
$590$	0	0
$591$	1.44466	0.0594255
$592$	0	0
$593$	22.6929	0.931888	0.465944	−	0.884814i	$-0.345715\pi$
$593$	22.6929	0.931888	0.465944	+	0.884814i	$0.345715\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	7.63105	0.312318
$598$	0	0
$599$	4.95675	0.202527	0.101264	−	0.994860i	$-0.467711\pi$
$599$	4.95675	0.202527	0.101264	+	0.994860i	$0.467711\pi$
$600$	0	0
$601$	−33.5372	−1.36801	−0.684005	−	0.729477i	$-0.739763\pi$
$601$	−33.5372	−1.36801	−0.684005	+	0.729477i	$0.739763\pi$
$602$	0	0
$603$	−19.5119	−0.794586
$604$	0	0
$605$	0	0
$606$	0	0
$607$	32.9103	1.33579	0.667893	−	0.744257i	$-0.267196\pi$
$607$	32.9103	1.33579	0.667893	+	0.744257i	$0.267196\pi$
$608$	0	0
$609$	−5.81957	−0.235821
$610$	0	0
$611$	3.54628	0.143467
$612$	0	0
$613$	44.4090	1.79366	0.896831	−	0.442374i	$-0.145863\pi$
$613$	44.4090	1.79366	0.896831	+	0.442374i	$0.145863\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−18.2962	−0.736579	−0.368290	−	0.929711i	$-0.620056\pi$
$617$	−18.2962	−0.736579	−0.368290	+	0.929711i	$0.620056\pi$
$618$	0	0
$619$	−2.76770	−0.111243	−0.0556216	−	0.998452i	$-0.517714\pi$
$619$	−2.76770	−0.111243	−0.0556216	+	0.998452i	$0.517714\pi$
$620$	0	0
$621$	14.1204	0.566632
$622$	0	0
$623$	38.3170	1.53514
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−1.81376	−0.0724347
$628$	0	0
$629$	−30.8428	−1.22978
$630$	0	0
$631$	5.74488	0.228700	0.114350	−	0.993441i	$-0.463521\pi$
$631$	5.74488	0.228700	0.114350	+	0.993441i	$0.463521\pi$
$632$	0	0
$633$	8.40083	0.333903
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−9.58170	−0.379641
$638$	0	0
$639$	−8.54186	−0.337911
$640$	0	0
$641$	19.3436	0.764028	0.382014	−	0.924157i	$-0.375231\pi$
$641$	19.3436	0.764028	0.382014	+	0.924157i	$0.375231\pi$
$642$	0	0
$643$	−11.0400	−0.435375	−0.217687	−	0.976019i	$-0.569851\pi$
$643$	−11.0400	−0.435375	−0.217687	+	0.976019i	$0.569851\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	3.35938	0.132071	0.0660353	−	0.997817i	$-0.478965\pi$
$647$	3.35938	0.132071	0.0660353	+	0.997817i	$0.478965\pi$
$648$	0	0
$649$	51.6109	2.02590
$650$	0	0
$651$	4.81821	0.188841
$652$	0	0
$653$	−44.1434	−1.72747	−0.863733	−	0.503950i	$-0.831879\pi$
$653$	−44.1434	−1.72747	−0.863733	+	0.503950i	$0.831879\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	6.29372	0.245542
$658$	0	0
$659$	−35.1874	−1.37071	−0.685354	−	0.728210i	$-0.740352\pi$
$659$	−35.1874	−1.37071	−0.685354	+	0.728210i	$0.740352\pi$
$660$	0	0
$661$	10.3574	0.402855	0.201428	−	0.979503i	$-0.435442\pi$
$661$	10.3574	0.402855	0.201428	+	0.979503i	$0.435442\pi$
$662$	0	0
$663$	−5.52819	−0.214697
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−33.7458	−1.30664
$668$	0	0
$669$	6.10575	0.236062
$670$	0	0
$671$	53.9886	2.08421
$672$	0	0
$673$	−36.0384	−1.38918	−0.694589	−	0.719407i	$-0.744414\pi$
$673$	−36.0384	−1.38918	−0.694589	+	0.719407i	$0.744414\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−11.4531	−0.440177	−0.220089	−	0.975480i	$-0.570635\pi$
$677$	−11.4531	−0.440177	−0.220089	+	0.975480i	$0.570635\pi$
$678$	0	0
$679$	32.7727	1.25770
$680$	0	0
$681$	−4.34008	−0.166312
$682$	0	0
$683$	−42.9554	−1.64364	−0.821821	−	0.569745i	$-0.807042\pi$
$683$	−42.9554	−1.64364	−0.821821	+	0.569745i	$0.807042\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	1.77607	0.0677611
$688$	0	0
$689$	28.5065	1.08601
$690$	0	0
$691$	−5.23235	−0.199048	−0.0995241	−	0.995035i	$-0.531732\pi$
$691$	−5.23235	−0.199048	−0.0995241	+	0.995035i	$0.531732\pi$
$692$	0	0
$693$	43.7449	1.66173
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−8.97345	−0.339894
$698$	0	0
$699$	−3.52380	−0.133282
$700$	0	0
$701$	−17.3930	−0.656925	−0.328463	−	0.944517i	$-0.606531\pi$
$701$	−17.3930	−0.656925	−0.328463	+	0.944517i	$0.606531\pi$
$702$	0	0
$703$	7.75503	0.292486
$704$	0	0
$705$	0	0
$706$	0	0
$707$	40.4733	1.52215
$708$	0	0
$709$	−21.5383	−0.808888	−0.404444	−	0.914563i	$-0.632535\pi$
$709$	−21.5383	−0.808888	−0.404444	+	0.914563i	$0.632535\pi$
$710$	0	0
$711$	−9.36024	−0.351036
$712$	0	0
$713$	27.9392	1.04633
$714$	0	0
$715$	0	0
$716$	0	0
$717$	3.48395	0.130110
$718$	0	0
$719$	−30.8923	−1.15209	−0.576044	−	0.817418i	$-0.695404\pi$
$719$	−30.8923	−1.15209	−0.576044	+	0.817418i	$0.695404\pi$
$720$	0	0
$721$	−15.3007	−0.569830
$722$	0	0
$723$	−8.02006	−0.298269
$724$	0	0
$725$	0	0
$726$	0	0
$727$	20.9947	0.778649	0.389325	−	0.921101i	$-0.372708\pi$
$727$	20.9947	0.778649	0.389325	+	0.921101i	$0.372708\pi$
$728$	0	0
$729$	−20.0061	−0.740967
$730$	0	0
$731$	40.6765	1.50447
$732$	0	0
$733$	22.0911	0.815955	0.407978	−	0.912992i	$-0.366234\pi$
$733$	22.0911	0.815955	0.407978	+	0.912992i	$0.366234\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	33.6762	1.24048
$738$	0	0
$739$	7.81209	0.287372	0.143686	−	0.989623i	$-0.454104\pi$
$739$	7.81209	0.287372	0.143686	+	0.989623i	$0.454104\pi$
$740$	0	0
$741$	1.39000	0.0510628
$742$	0	0
$743$	−17.9012	−0.656731	−0.328365	−	0.944551i	$-0.606498\pi$
$743$	−17.9012	−0.656731	−0.328365	+	0.944551i	$0.606498\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−11.7467	−0.429790
$748$	0	0
$749$	−58.3834	−2.13328
$750$	0	0
$751$	48.8802	1.78366	0.891832	−	0.452367i	$-0.149420\pi$
$751$	48.8802	1.78366	0.891832	+	0.452367i	$0.149420\pi$
$752$	0	0
$753$	−0.744878	−0.0271449
$754$	0	0
$755$	0	0
$756$	0	0
$757$	6.75011	0.245337	0.122668	−	0.992448i	$-0.460855\pi$
$757$	6.75011	0.245337	0.122668	+	0.992448i	$0.460855\pi$
$758$	0	0
$759$	−11.9060	−0.432161
$760$	0	0
$761$	47.5563	1.72392	0.861958	−	0.506980i	$-0.169238\pi$
$761$	47.5563	1.72392	0.861958	+	0.506980i	$0.169238\pi$
$762$	0	0
$763$	−27.4757	−0.994689
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−39.5525	−1.42816
$768$	0	0
$769$	−0.849714	−0.0306415	−0.0153207	−	0.999883i	$-0.504877\pi$
$769$	−0.849714	−0.0306415	−0.0153207	+	0.999883i	$0.504877\pi$
$770$	0	0
$771$	−2.79790	−0.100764
$772$	0	0
$773$	−4.74109	−0.170525	−0.0852627	−	0.996359i	$-0.527173\pi$
$773$	−4.74109	−0.170525	−0.0852627	+	0.996359i	$0.527173\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	8.77891	0.314942
$778$	0	0
$779$	2.25626	0.0808390
$780$	0	0
$781$	14.7427	0.527535
$782$	0	0
$783$	−11.0585	−0.395197
$784$	0	0
$785$	0	0
$786$	0	0
$787$	19.0208	0.678018	0.339009	−	0.940783i	$-0.389908\pi$
$787$	19.0208	0.678018	0.339009	+	0.940783i	$0.389908\pi$
$788$	0	0
$789$	−11.1231	−0.395994
$790$	0	0
$791$	−30.4067	−1.08114
$792$	0	0
$793$	−41.3747	−1.46926
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−18.8784	−0.668706	−0.334353	−	0.942448i	$-0.608518\pi$
$797$	−18.8784	−0.668706	−0.334353	+	0.942448i	$0.608518\pi$
$798$	0	0
$799$	−3.72121	−0.131647
$800$	0	0
$801$	35.5705	1.25682
$802$	0	0
$803$	−10.8625	−0.383331
$804$	0	0
$805$	0	0
$806$	0	0
$807$	8.92573	0.314201
$808$	0	0
$809$	−1.68753	−0.0593305	−0.0296652	−	0.999560i	$-0.509444\pi$
$809$	−1.68753	−0.0593305	−0.0296652	+	0.999560i	$0.509444\pi$
$810$	0	0
$811$	−38.3960	−1.34827	−0.674133	−	0.738610i	$-0.735483\pi$
$811$	−38.3960	−1.34827	−0.674133	+	0.738610i	$0.735483\pi$
$812$	0	0
$813$	−1.32790	−0.0465715
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−10.2276	−0.357818
$818$	0	0
$819$	−33.5244	−1.17144
$820$	0	0
$821$	−11.4671	−0.400204	−0.200102	−	0.979775i	$-0.564127\pi$
$821$	−11.4671	−0.400204	−0.200102	+	0.979775i	$0.564127\pi$
$822$	0	0
$823$	46.0927	1.60669	0.803345	−	0.595515i	$-0.203052\pi$
$823$	46.0927	1.60669	0.803345	+	0.595515i	$0.203052\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.44674	0.0503081	0.0251540	−	0.999684i	$-0.491992\pi$
$827$	1.44674	0.0503081	0.0251540	+	0.999684i	$0.491992\pi$
$828$	0	0
$829$	−43.8553	−1.52316	−0.761578	−	0.648073i	$-0.775575\pi$
$829$	−43.8553	−1.52316	−0.761578	+	0.648073i	$0.775575\pi$
$830$	0	0
$831$	10.8863	0.377642
$832$	0	0
$833$	10.0544	0.348363
$834$	0	0
$835$	0	0
$836$	0	0
$837$	9.15565	0.316465
$838$	0	0
$839$	45.7264	1.57865	0.789325	−	0.613976i	$-0.210431\pi$
$839$	45.7264	1.57865	0.789325	+	0.613976i	$0.210431\pi$
$840$	0	0
$841$	−2.57179	−0.0886824
$842$	0	0
$843$	10.6854	0.368024
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−41.5466	−1.42756
$848$	0	0
$849$	8.87802	0.304693
$850$	0	0
$851$	50.9061	1.74504
$852$	0	0
$853$	28.3388	0.970303	0.485151	−	0.874430i	$-0.338764\pi$
$853$	28.3388	0.970303	0.485151	+	0.874430i	$0.338764\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	18.7526	0.640575	0.320288	−	0.947320i	$-0.396220\pi$
$857$	18.7526	0.640575	0.320288	+	0.947320i	$0.396220\pi$
$858$	0	0
$859$	14.2675	0.486801	0.243401	−	0.969926i	$-0.421737\pi$
$859$	14.2675	0.486801	0.243401	+	0.969926i	$0.421737\pi$
$860$	0	0
$861$	2.55415	0.0870453
$862$	0	0
$863$	32.9573	1.12188	0.560940	−	0.827857i	$-0.310440\pi$
$863$	32.9573	1.12188	0.560940	+	0.827857i	$0.310440\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−0.433644	−0.0147273
$868$	0	0
$869$	16.1552	0.548026
$870$	0	0
$871$	−25.8081	−0.874475
$872$	0	0
$873$	30.4236	1.02968
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−41.9346	−1.41603	−0.708016	−	0.706197i	$-0.750409\pi$
$877$	−41.9346	−1.41603	−0.708016	+	0.706197i	$0.750409\pi$
$878$	0	0
$879$	5.84354	0.197098
$880$	0	0
$881$	7.69223	0.259158	0.129579	−	0.991569i	$-0.458637\pi$
$881$	7.69223	0.259158	0.129579	+	0.991569i	$0.458637\pi$
$882$	0	0
$883$	32.8546	1.10565	0.552823	−	0.833299i	$-0.313551\pi$
$883$	32.8546	1.10565	0.552823	+	0.833299i	$0.313551\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1.97358	0.0662664	0.0331332	−	0.999451i	$-0.489451\pi$
$887$	1.97358	0.0662664	0.0331332	+	0.999451i	$0.489451\pi$
$888$	0	0
$889$	11.6180	0.389656
$890$	0	0
$891$	38.6139	1.29361
$892$	0	0
$893$	0.935652	0.0313104
$894$	0	0
$895$	0	0
$896$	0	0
$897$	9.12430	0.304651
$898$	0	0
$899$	−21.8808	−0.729764
$900$	0	0
$901$	−29.9127	−0.996536
$902$	0	0
$903$	−11.5779	−0.385289
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−6.21342	−0.206313	−0.103157	−	0.994665i	$-0.532894\pi$
$907$	−6.21342	−0.206313	−0.103157	+	0.994665i	$0.532894\pi$
$908$	0	0
$909$	37.5723	1.24619
$910$	0	0
$911$	40.4200	1.33917	0.669586	−	0.742734i	$-0.266471\pi$
$911$	40.4200	1.33917	0.669586	+	0.742734i	$0.266471\pi$
$912$	0	0
$913$	20.2741	0.670974
$914$	0	0
$915$	0	0
$916$	0	0
$917$	31.0181	1.02431
$918$	0	0
$919$	−53.0177	−1.74889	−0.874446	−	0.485122i	$-0.838775\pi$
$919$	−53.0177	−1.74889	−0.874446	+	0.485122i	$0.838775\pi$
$920$	0	0
$921$	9.51220	0.313438
$922$	0	0
$923$	−11.2982	−0.371885
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−14.2040	−0.466522
$928$	0	0
$929$	38.0520	1.24845	0.624224	−	0.781246i	$-0.285415\pi$
$929$	38.0520	1.24845	0.624224	+	0.781246i	$0.285415\pi$
$930$	0	0
$931$	−2.52804	−0.0828533
$932$	0	0
$933$	0.948112	0.0310398
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−12.8035	−0.418271	−0.209135	−	0.977887i	$-0.567065\pi$
$937$	−12.8035	−0.418271	−0.209135	+	0.977887i	$0.567065\pi$
$938$	0	0
$939$	−6.27220	−0.204686
$940$	0	0
$941$	40.1622	1.30925	0.654625	−	0.755954i	$-0.272827\pi$
$941$	40.1622	1.30925	0.654625	+	0.755954i	$0.272827\pi$
$942$	0	0
$943$	14.8107	0.482303
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−25.3713	−0.824458	−0.412229	−	0.911080i	$-0.635250\pi$
$947$	−25.3713	−0.824458	−0.412229	+	0.911080i	$0.635250\pi$
$948$	0	0
$949$	8.32463	0.270229
$950$	0	0
$951$	−9.00676	−0.292064
$952$	0	0
$953$	−6.25884	−0.202744	−0.101372	−	0.994849i	$-0.532323\pi$
$953$	−6.25884	−0.202744	−0.101372	+	0.994849i	$0.532323\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	9.32426	0.301411
$958$	0	0
$959$	45.1420	1.45771
$960$	0	0
$961$	−12.8842	−0.415620
$962$	0	0
$963$	−54.1986	−1.74653
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−16.6425	−0.535186	−0.267593	−	0.963532i	$-0.586228\pi$
$967$	−16.6425	−0.535186	−0.267593	+	0.963532i	$0.586228\pi$
$968$	0	0
$969$	−1.45856	−0.0468558
$970$	0	0
$971$	20.5468	0.659377	0.329688	−	0.944090i	$-0.393056\pi$
$971$	20.5468	0.659377	0.329688	+	0.944090i	$0.393056\pi$
$972$	0	0
$973$	−37.2796	−1.19513
$974$	0	0
$975$	0	0
$976$	0	0
$977$	48.7487	1.55961	0.779804	−	0.626024i	$-0.215319\pi$
$977$	48.7487	1.55961	0.779804	+	0.626024i	$0.215319\pi$
$978$	0	0
$979$	−61.3924	−1.96211
$980$	0	0
$981$	−25.5064	−0.814355
$982$	0	0
$983$	−34.9881	−1.11595	−0.557973	−	0.829859i	$-0.688421\pi$
$983$	−34.9881	−1.11595	−0.557973	+	0.829859i	$0.688421\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	1.05919	0.0337142
$988$	0	0
$989$	−67.1366	−2.13482
$990$	0	0
$991$	−12.4515	−0.395535	−0.197768	−	0.980249i	$-0.563369\pi$
$991$	−12.4515	−0.395535	−0.197768	+	0.980249i	$0.563369\pi$
$992$	0	0
$993$	−6.02652	−0.191246
$994$	0	0
$995$	0	0
$996$	0	0
$997$	13.4889	0.427199	0.213599	−	0.976921i	$-0.431481\pi$
$997$	13.4889	0.427199	0.213599	+	0.976921i	$0.431481\pi$
$998$	0	0
$999$	16.6818	0.527790

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7600.2.a.cn.1.3		6
4.3	odd	2		3800.2.a.z.1.4		6
5.2	odd	4		1520.2.d.k.609.6		12
5.3	odd	4		1520.2.d.k.609.7		12
5.4	even	2		7600.2.a.cg.1.4		6
20.3	even	4		760.2.d.e.609.6	✓	12
20.7	even	4		760.2.d.e.609.7	yes	12
20.19	odd	2		3800.2.a.be.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
760.2.d.e.609.6	✓	12	20.3	even	4
760.2.d.e.609.7	yes	12	20.7	even	4
1520.2.d.k.609.6		12	5.2	odd	4
1520.2.d.k.609.7		12	5.3	odd	4
3800.2.a.z.1.4		6	4.3	odd	2
3800.2.a.be.1.3		6	20.19	odd	2
7600.2.a.cg.1.4		6	5.4	even	2
7600.2.a.cn.1.3		6	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+0.366738 q^{3} -3.08675 q^{7} -2.86550 q^{9} +4.94567 q^{11} -3.79016 q^{13} +3.97713 q^{17} -1.00000 q^{19} -1.13203 q^{21} -6.56426 q^{23} -2.15110 q^{27} +5.14084 q^{29} -4.25626 q^{31} +1.81376 q^{33} -7.75503 q^{37} -1.39000 q^{39} -2.25626 q^{41} +10.2276 q^{43} -0.935652 q^{47} +2.52804 q^{49} +1.45856 q^{51} -7.52117 q^{53} -0.366738 q^{57} +10.4356 q^{59} +10.9163 q^{61} +8.84510 q^{63} +6.80924 q^{67} -2.40736 q^{69} +2.98093 q^{71} -2.19638 q^{73} -15.2661 q^{77} +3.26652 q^{79} +7.80762 q^{81} +4.09936 q^{83} +1.88534 q^{87} -12.4134 q^{89} +11.6993 q^{91} -1.56093 q^{93} -10.6172 q^{97} -14.1718 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{3} + 6 q^{7} + 8 q^{9} + 2 q^{11} - 14 q^{13} - 10 q^{17} - 6 q^{19} + 18 q^{21} + 2 q^{23} + 2 q^{27} - 2 q^{29} - 8 q^{31} - 8 q^{33} - 4 q^{37} + 18 q^{39} + 4 q^{41} + 4 q^{43} + 4 q^{47}+ \cdots + 22 q^{99}+O(q^{100}) \) 6 * q + 2 * q^3 + 6 * q^7 + 8 * q^9 + 2 * q^11 - 14 * q^13 - 10 * q^17 - 6 * q^19 + 18 * q^21 + 2 * q^23 + 2 * q^27 - 2 * q^29 - 8 * q^31 - 8 * q^33 - 4 * q^37 + 18 * q^39 + 4 * q^41 + 4 * q^43 + 4 * q^47 + 2 * q^49 + 34 * q^51 + 14 * q^53 - 2 * q^57 + 2 * q^59 + 10 * q^61 + 44 * q^63 + 42 * q^67 + 18 * q^69 + 8 * q^71 + 2 * q^73 - 24 * q^77 + 20 * q^79 + 6 * q^81 + 16 * q^83 + 30 * q^87 + 32 * q^89 - 10 * q^91 - 12 * q^93 - 40 * q^97 + 22 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more