LMFDB - Embedded newform 3040.2.f.b.1521.25

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3040,2,Mod(1521,3040)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3040.1521"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3040, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3040 = 2^{5} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3040.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [44] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$24.2745222145$
Analytic rank:	$0$
Dimension:	$44$
Twist minimal:	no (minimal twist has level 760)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1521.25
Character	$\chi$	$=$	3040.1521
Dual form			3040.2.f.b.1521.20

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.530865i q^{3} +1.00000i q^{5} -1.59876 q^{7} +2.71818 q^{9} -4.08419i q^{11} +4.29317i q^{13} -0.530865 q^{15} -7.51422 q^{17} -1.00000i q^{19} -0.848726i q^{21} +7.97383 q^{23} -1.00000 q^{25} +3.03558i q^{27} +5.77844i q^{29} -3.57200 q^{31} +2.16815 q^{33} -1.59876i q^{35} +1.10308i q^{37} -2.27909 q^{39} +2.90996 q^{41} +4.31616i q^{43} +2.71818i q^{45} -6.15476 q^{47} -4.44396 q^{49} -3.98904i q^{51} +11.5004i q^{53} +4.08419 q^{55} +0.530865 q^{57} -6.91548i q^{59} +4.61636i q^{61} -4.34572 q^{63} -4.29317 q^{65} +10.2207i q^{67} +4.23303i q^{69} -11.3535 q^{71} +15.1543 q^{73} -0.530865i q^{75} +6.52964i q^{77} +3.08138 q^{79} +6.54306 q^{81} +3.40976i q^{83} -7.51422i q^{85} -3.06757 q^{87} -13.5566 q^{89} -6.86375i q^{91} -1.89625i q^{93} +1.00000 q^{95} +3.07007 q^{97} -11.1016i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 44 q - 4 q^{7} - 60 q^{9} + 24 q^{17} - 4 q^{23} - 44 q^{25} + 40 q^{33} + 24 q^{39} - 32 q^{41} + 20 q^{47} + 108 q^{49} - 8 q^{55} - 20 q^{63} + 12 q^{65} + 8 q^{71} - 88 q^{73} - 40 q^{79} + 116 q^{81}+ \cdots + 116 q^{97}+O(q^{100}) $ 44 * q - 4 * q^7 - 60 * q^9 + 24 * q^17 - 4 * q^23 - 44 * q^25 + 40 * q^33 + 24 * q^39 - 32 * q^41 + 20 * q^47 + 108 * q^49 - 8 * q^55 - 20 * q^63 + 12 * q^65 + 8 * q^71 - 88 * q^73 - 40 * q^79 + 116 * q^81 - 48 * q^87 - 64 * q^89 + 44 * q^95 + 116 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3040\mathbb{Z}\right)^\times$.

$n$	$191$	$1217$	$1921$	$2661$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	0.530865i	0.306495i	0.988188	+	0.153248i	$0.0489732\pi$
$3$	0.530865i	0.306495i	−0.988188	+	0.153248i	$0.951027\pi$
$4$	0	0
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	0	0
$7$	−1.59876	−0.604275	−0.302137	−	0.953264i	$-0.597700\pi$
$7$	−1.59876	−0.604275	−0.302137	+	0.953264i	$0.597700\pi$
$8$	0	0
$9$	2.71818	0.906061
$10$	0	0
$11$	− 4.08419i	− 1.23143i	−0.787970	−	0.615714i	$-0.788868\pi$
$11$	− 4.08419i	− 1.23143i	0.787970	−	0.615714i	$-0.211132\pi$
$12$	0	0
$13$	4.29317i	1.19071i	0.803463	+	0.595355i	$0.202989\pi$
$13$	4.29317i	1.19071i	−0.803463	+	0.595355i	$0.797011\pi$
$14$	0	0
$15$	−0.530865	−0.137069
$16$	0	0
$17$	−7.51422	−1.82247	−0.911233	−	0.411891i	$-0.864869\pi$
$17$	−7.51422	−1.82247	−0.911233	+	0.411891i	$0.864869\pi$
$18$	0	0
$19$	− 1.00000i	− 0.229416i
$19$	− 1.00000i	− 0.229416i
$20$	0	0
$21$	− 0.848726i	− 0.185207i
$22$	0	0
$23$	7.97383	1.66266	0.831329	−	0.555781i	$-0.187581\pi$
$23$	7.97383	1.66266	0.831329	+	0.555781i	$0.187581\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	3.03558i	0.584198i
$28$	0	0
$29$	5.77844i	1.07303i	0.843891	+	0.536515i	$0.180260\pi$
$29$	5.77844i	1.07303i	−0.843891	+	0.536515i	$0.819740\pi$
$30$	0	0
$31$	−3.57200	−0.641551	−0.320775	−	0.947155i	$-0.603943\pi$
$31$	−3.57200	−0.641551	−0.320775	+	0.947155i	$0.603943\pi$
$32$	0	0
$33$	2.16815	0.377427
$34$	0	0
$35$	− 1.59876i	− 0.270240i
$36$	0	0
$37$	1.10308i	0.181346i	0.995881	+	0.0906728i	$0.0289017\pi$
$37$	1.10308i	0.181346i	−0.995881	+	0.0906728i	$0.971098\pi$
$38$	0	0
$39$	−2.27909	−0.364947
$40$	0	0
$41$	2.90996	0.454459	0.227230	−	0.973841i	$-0.427033\pi$
$41$	2.90996	0.454459	0.227230	+	0.973841i	$0.427033\pi$
$42$	0	0
$43$	4.31616i	0.658208i	0.944294	+	0.329104i	$0.106747\pi$
$43$	4.31616i	0.658208i	−0.944294	+	0.329104i	$0.893253\pi$
$44$	0	0
$45$	2.71818i	0.405203i
$46$	0	0
$47$	−6.15476	−0.897764	−0.448882	−	0.893591i	$-0.648178\pi$
$47$	−6.15476	−0.897764	−0.448882	+	0.893591i	$0.648178\pi$
$48$	0	0
$49$	−4.44396	−0.634852
$50$	0	0
$51$	− 3.98904i	− 0.558577i
$52$	0	0
$53$	11.5004i	1.57971i	0.613296	+	0.789853i	$0.289843\pi$
$53$	11.5004i	1.57971i	−0.613296	+	0.789853i	$0.710157\pi$
$54$	0	0
$55$	4.08419	0.550712
$56$	0	0
$57$	0.530865	0.0703148
$58$	0	0
$59$	− 6.91548i	− 0.900319i	−0.892948	−	0.450159i	$-0.851367\pi$
$59$	− 6.91548i	− 0.900319i	0.892948	−	0.450159i	$-0.148633\pi$
$60$	0	0
$61$	4.61636i	0.591064i	0.955333	+	0.295532i	$0.0954969\pi$
$61$	4.61636i	0.591064i	−0.955333	+	0.295532i	$0.904503\pi$
$62$	0	0
$63$	−4.34572	−0.547510
$64$	0	0
$65$	−4.29317	−0.532502
$66$	0	0
$67$	10.2207i	1.24866i	0.781162	+	0.624328i	$0.214627\pi$
$67$	10.2207i	1.24866i	−0.781162	+	0.624328i	$0.785373\pi$
$68$	0	0
$69$	4.23303i	0.509596i
$70$	0	0
$71$	−11.3535	−1.34741	−0.673704	−	0.739001i	$-0.735298\pi$
$71$	−11.3535	−1.34741	−0.673704	+	0.739001i	$0.735298\pi$
$72$	0	0
$73$	15.1543	1.77368	0.886840	−	0.462078i	$-0.152896\pi$
$73$	15.1543	1.77368	0.886840	+	0.462078i	$0.152896\pi$
$74$	0	0
$75$	− 0.530865i	− 0.0612990i
$76$	0	0
$77$	6.52964i	0.744121i
$78$	0	0
$79$	3.08138	0.346683	0.173341	−	0.984862i	$-0.444544\pi$
$79$	3.08138	0.346683	0.173341	+	0.984862i	$0.444544\pi$
$80$	0	0
$81$	6.54306	0.727007
$82$	0	0
$83$	3.40976i	0.374270i	0.982334	+	0.187135i	$0.0599202\pi$
$83$	3.40976i	0.374270i	−0.982334	+	0.187135i	$0.940080\pi$
$84$	0	0
$85$	− 7.51422i	− 0.815032i
$86$	0	0
$87$	−3.06757	−0.328878
$88$	0	0
$89$	−13.5566	−1.43699	−0.718497	−	0.695530i	$-0.755170\pi$
$89$	−13.5566	−1.43699	−0.718497	+	0.695530i	$0.755170\pi$
$90$	0	0
$91$	− 6.86375i	− 0.719516i
$92$	0	0
$93$	− 1.89625i	− 0.196632i
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	3.07007	0.311718	0.155859	−	0.987779i	$-0.450185\pi$
$97$	3.07007	0.311718	0.155859	+	0.987779i	$0.450185\pi$
$98$	0	0
$99$	− 11.1016i	− 1.11575i
$100$	0	0
$101$	− 10.5727i	− 1.05202i	−0.850478	−	0.526010i	$-0.823687\pi$
$101$	− 10.5727i	− 1.05202i	0.850478	−	0.526010i	$-0.176313\pi$
$102$	0	0
$103$	−10.5955	−1.04401	−0.522004	−	0.852943i	$-0.674815\pi$
$103$	−10.5955	−1.04401	−0.522004	+	0.852943i	$0.674815\pi$
$104$	0	0
$105$	0.848726	0.0828272
$106$	0	0
$107$	2.02637i	0.195897i	0.995191	+	0.0979484i	$0.0312280\pi$
$107$	2.02637i	0.195897i	−0.995191	+	0.0979484i	$0.968772\pi$
$108$	0	0
$109$	9.89833i	0.948088i	0.880501	+	0.474044i	$0.157206\pi$
$109$	9.89833i	0.948088i	−0.880501	+	0.474044i	$0.842794\pi$
$110$	0	0
$111$	−0.585588	−0.0555815
$112$	0	0
$113$	−10.9289	−1.02811	−0.514054	−	0.857758i	$-0.671857\pi$
$113$	−10.9289	−1.02811	−0.514054	+	0.857758i	$0.671857\pi$
$114$	0	0
$115$	7.97383i	0.743563i
$116$	0	0
$117$	11.6696i	1.07886i
$118$	0	0
$119$	12.0134	1.10127
$120$	0	0
$121$	−5.68058	−0.516416
$122$	0	0
$123$	1.54480i	0.139290i
$124$	0	0
$125$	− 1.00000i	− 0.0894427i
$126$	0	0
$127$	−4.07012	−0.361165	−0.180583	−	0.983560i	$-0.557798\pi$
$127$	−4.07012	−0.361165	−0.180583	+	0.983560i	$0.557798\pi$
$128$	0	0
$129$	−2.29130	−0.201737
$130$	0	0
$131$	17.8997i	1.56390i	0.623338	+	0.781952i	$0.285776\pi$
$131$	17.8997i	1.56390i	−0.623338	+	0.781952i	$0.714224\pi$
$132$	0	0
$133$	1.59876i	0.138630i
$134$	0	0
$135$	−3.03558	−0.261261
$136$	0	0
$137$	2.57962	0.220392	0.110196	−	0.993910i	$-0.464852\pi$
$137$	2.57962	0.220392	0.110196	+	0.993910i	$0.464852\pi$
$138$	0	0
$139$	− 13.5951i	− 1.15312i	−0.817055	−	0.576560i	$-0.804395\pi$
$139$	− 13.5951i	− 1.15312i	0.817055	−	0.576560i	$-0.195605\pi$
$140$	0	0
$141$	− 3.26735i	− 0.275160i
$142$	0	0
$143$	17.5341	1.46627
$144$	0	0
$145$	−5.77844	−0.479874
$146$	0	0
$147$	− 2.35915i	− 0.194579i
$148$	0	0
$149$	− 1.38602i	− 0.113547i	−0.998387	−	0.0567735i	$-0.981919\pi$
$149$	− 1.38602i	− 0.113547i	0.998387	−	0.0567735i	$-0.0180813\pi$
$150$	0	0
$151$	−17.6513	−1.43644	−0.718220	−	0.695816i	$-0.755043\pi$
$151$	−17.6513	−1.43644	−0.718220	+	0.695816i	$0.755043\pi$
$152$	0	0
$153$	−20.4250	−1.65127
$154$	0	0
$155$	− 3.57200i	− 0.286910i
$156$	0	0
$157$	12.4964i	0.997325i	0.866796	+	0.498662i	$0.166175\pi$
$157$	12.4964i	0.997325i	−0.866796	+	0.498662i	$0.833825\pi$
$158$	0	0
$159$	−6.10518	−0.484172
$160$	0	0
$161$	−12.7482	−1.00470
$162$	0	0
$163$	10.6418i	0.833533i	0.909013	+	0.416767i	$0.136837\pi$
$163$	10.6418i	0.833533i	−0.909013	+	0.416767i	$0.863163\pi$
$164$	0	0
$165$	2.16815i	0.168790i
$166$	0	0
$167$	−9.32577	−0.721650	−0.360825	−	0.932634i	$-0.617505\pi$
$167$	−9.32577	−0.721650	−0.360825	+	0.932634i	$0.617505\pi$
$168$	0	0
$169$	−5.43128	−0.417791
$170$	0	0
$171$	− 2.71818i	− 0.207865i
$172$	0	0
$173$	− 14.2559i	− 1.08386i	−0.840424	−	0.541929i	$-0.817694\pi$
$173$	− 14.2559i	− 1.08386i	0.840424	−	0.541929i	$-0.182306\pi$
$174$	0	0
$175$	1.59876	0.120855
$176$	0	0
$177$	3.67119	0.275943
$178$	0	0
$179$	15.8824i	1.18711i	0.804794	+	0.593554i	$0.202276\pi$
$179$	15.8824i	1.18711i	−0.804794	+	0.593554i	$0.797724\pi$
$180$	0	0
$181$	9.70924i	0.721682i	0.932627	+	0.360841i	$0.117510\pi$
$181$	9.70924i	0.721682i	−0.932627	+	0.360841i	$0.882490\pi$
$182$	0	0
$183$	−2.45066	−0.181158
$184$	0	0
$185$	−1.10308	−0.0811002
$186$	0	0
$187$	30.6895i	2.24424i
$188$	0	0
$189$	− 4.85317i	− 0.353016i
$190$	0	0
$191$	−5.34269	−0.386584	−0.193292	−	0.981141i	$-0.561916\pi$
$191$	−5.34269	−0.386584	−0.193292	+	0.981141i	$0.561916\pi$
$192$	0	0
$193$	−6.68316	−0.481065	−0.240532	−	0.970641i	$-0.577322\pi$
$193$	−6.68316	−0.481065	−0.240532	+	0.970641i	$0.577322\pi$
$194$	0	0
$195$	− 2.27909i	− 0.163209i
$196$	0	0
$197$	− 1.87200i	− 0.133374i	−0.997774	−	0.0666872i	$-0.978757\pi$
$197$	− 1.87200i	− 0.133374i	0.997774	−	0.0666872i	$-0.0212429\pi$
$198$	0	0
$199$	−11.7535	−0.833184	−0.416592	−	0.909094i	$-0.636776\pi$
$199$	−11.7535	−0.833184	−0.416592	+	0.909094i	$0.636776\pi$
$200$	0	0
$201$	−5.42580	−0.382707
$202$	0	0
$203$	− 9.23835i	− 0.648405i
$204$	0	0
$205$	2.90996i	0.203240i
$206$	0	0
$207$	21.6743	1.50647
$208$	0	0
$209$	−4.08419	−0.282509
$210$	0	0
$211$	16.3391i	1.12483i	0.826856	+	0.562413i	$0.190127\pi$
$211$	16.3391i	1.12483i	−0.826856	+	0.562413i	$0.809873\pi$
$212$	0	0
$213$	− 6.02715i	− 0.412974i
$214$	0	0
$215$	−4.31616	−0.294360
$216$	0	0
$217$	5.71078	0.387673
$218$	0	0
$219$	8.04490i	0.543624i
$220$	0	0
$221$	− 32.2598i	− 2.17003i
$222$	0	0
$223$	−14.1309	−0.946278	−0.473139	−	0.880988i	$-0.656879\pi$
$223$	−14.1309	−0.946278	−0.473139	+	0.880988i	$0.656879\pi$
$224$	0	0
$225$	−2.71818	−0.181212
$226$	0	0
$227$	− 14.1373i	− 0.938324i	−0.883112	−	0.469162i	$-0.844556\pi$
$227$	− 14.1373i	− 0.938324i	0.883112	−	0.469162i	$-0.155444\pi$
$228$	0	0
$229$	13.1301i	0.867665i	0.900994	+	0.433832i	$0.142839\pi$
$229$	13.1301i	0.867665i	−0.900994	+	0.433832i	$0.857161\pi$
$230$	0	0
$231$	−3.46636	−0.228069
$232$	0	0
$233$	−16.8001	−1.10061	−0.550306	−	0.834963i	$-0.685489\pi$
$233$	−16.8001	−1.10061	−0.550306	+	0.834963i	$0.685489\pi$
$234$	0	0
$235$	− 6.15476i	− 0.401492i
$236$	0	0
$237$	1.63580i	0.106257i
$238$	0	0
$239$	12.7091	0.822083	0.411042	−	0.911617i	$-0.365165\pi$
$239$	12.7091	0.822083	0.411042	+	0.911617i	$0.365165\pi$
$240$	0	0
$241$	−2.78627	−0.179479	−0.0897396	−	0.995965i	$-0.528603\pi$
$241$	−2.78627	−0.179479	−0.0897396	+	0.995965i	$0.528603\pi$
$242$	0	0
$243$	12.5802i	0.807022i
$244$	0	0
$245$	− 4.44396i	− 0.283915i
$246$	0	0
$247$	4.29317	0.273168
$248$	0	0
$249$	−1.81012	−0.114712
$250$	0	0
$251$	− 19.1306i	− 1.20752i	−0.797168	−	0.603758i	$-0.793669\pi$
$251$	− 19.1306i	− 1.20752i	0.797168	−	0.603758i	$-0.206331\pi$
$252$	0	0
$253$	− 32.5666i	− 2.04744i
$254$	0	0
$255$	3.98904	0.249803
$256$	0	0
$257$	15.3571	0.957952	0.478976	−	0.877828i	$-0.341008\pi$
$257$	15.3571	0.957952	0.478976	+	0.877828i	$0.341008\pi$
$258$	0	0
$259$	− 1.76356i	− 0.109583i
$260$	0	0
$261$	15.7069i	0.972231i
$262$	0	0
$263$	30.6398	1.88933	0.944665	−	0.328037i	$-0.106387\pi$
$263$	30.6398	1.88933	0.944665	+	0.328037i	$0.106387\pi$
$264$	0	0
$265$	−11.5004	−0.706466
$266$	0	0
$267$	− 7.19671i	− 0.440432i
$268$	0	0
$269$	21.9765i	1.33993i	0.742394	+	0.669964i	$0.233690\pi$
$269$	21.9765i	1.33993i	−0.742394	+	0.669964i	$0.766310\pi$
$270$	0	0
$271$	−2.44649	−0.148614	−0.0743069	−	0.997235i	$-0.523674\pi$
$271$	−2.44649	−0.148614	−0.0743069	+	0.997235i	$0.523674\pi$
$272$	0	0
$273$	3.64372	0.220528
$274$	0	0
$275$	4.08419i	0.246286i
$276$	0	0
$277$	− 27.0414i	− 1.62476i	−0.583127	−	0.812381i	$-0.698171\pi$
$277$	− 27.0414i	− 1.62476i	0.583127	−	0.812381i	$-0.301829\pi$
$278$	0	0
$279$	−9.70936	−0.581284
$280$	0	0
$281$	15.9317	0.950404	0.475202	−	0.879877i	$-0.342375\pi$
$281$	15.9317	0.950404	0.475202	+	0.879877i	$0.342375\pi$
$282$	0	0
$283$	− 4.15406i	− 0.246933i	−0.992349	−	0.123467i	$-0.960599\pi$
$283$	− 4.15406i	− 0.246933i	0.992349	−	0.123467i	$-0.0394012\pi$
$284$	0	0
$285$	0.530865i	0.0314457i
$286$	0	0
$287$	−4.65233	−0.274618
$288$	0	0
$289$	39.4635	2.32138
$290$	0	0
$291$	1.62979i	0.0955401i
$292$	0	0
$293$	21.0272i	1.22842i	0.789141	+	0.614212i	$0.210526\pi$
$293$	21.0272i	1.22842i	−0.789141	+	0.614212i	$0.789474\pi$
$294$	0	0
$295$	6.91548	0.402635
$296$	0	0
$297$	12.3979	0.719398
$298$	0	0
$299$	34.2330i	1.97974i
$300$	0	0
$301$	− 6.90050i	− 0.397738i
$302$	0	0
$303$	5.61266	0.322439
$304$	0	0
$305$	−4.61636	−0.264332
$306$	0	0
$307$	− 16.6865i	− 0.952348i	−0.879351	−	0.476174i	$-0.842023\pi$
$307$	− 16.6865i	− 0.952348i	0.879351	−	0.476174i	$-0.157977\pi$
$308$	0	0
$309$	− 5.62479i	− 0.319983i
$310$	0	0
$311$	9.12451	0.517403	0.258702	−	0.965957i	$-0.416705\pi$
$311$	9.12451	0.517403	0.258702	+	0.965957i	$0.416705\pi$
$312$	0	0
$313$	−23.5688	−1.33219	−0.666093	−	0.745869i	$-0.732035\pi$
$313$	−23.5688	−1.33219	−0.666093	+	0.745869i	$0.732035\pi$
$314$	0	0
$315$	− 4.34572i	− 0.244854i
$316$	0	0
$317$	− 13.6522i	− 0.766786i	−0.923585	−	0.383393i	$-0.874756\pi$
$317$	− 13.6522i	− 0.766786i	0.923585	−	0.383393i	$-0.125244\pi$
$318$	0	0
$319$	23.6002	1.32136
$320$	0	0
$321$	−1.07573	−0.0600414
$322$	0	0
$323$	7.51422i	0.418102i
$324$	0	0
$325$	− 4.29317i	− 0.238142i
$326$	0	0
$327$	−5.25468	−0.290584
$328$	0	0
$329$	9.83999	0.542496
$330$	0	0
$331$	22.6976i	1.24757i	0.781595	+	0.623786i	$0.214406\pi$
$331$	22.6976i	1.24757i	−0.781595	+	0.623786i	$0.785594\pi$
$332$	0	0
$333$	2.99838i	0.164310i
$334$	0	0
$335$	−10.2207	−0.558416
$336$	0	0
$337$	35.6697	1.94305	0.971527	−	0.236930i	$-0.0761413\pi$
$337$	35.6697	1.94305	0.971527	+	0.236930i	$0.0761413\pi$
$338$	0	0
$339$	− 5.80179i	− 0.315110i
$340$	0	0
$341$	14.5887i	0.790024i
$342$	0	0
$343$	18.2962	0.987900
$344$	0	0
$345$	−4.23303	−0.227898
$346$	0	0
$347$	6.33393i	0.340023i	0.985442	+	0.170012i	$0.0543805\pi$
$347$	6.33393i	0.340023i	−0.985442	+	0.170012i	$0.945619\pi$
$348$	0	0
$349$	− 17.8252i	− 0.954159i	−0.878860	−	0.477079i	$-0.841695\pi$
$349$	− 17.8252i	− 0.954159i	0.878860	−	0.477079i	$-0.158305\pi$
$350$	0	0
$351$	−13.0323	−0.695611
$352$	0	0
$353$	27.7834	1.47876	0.739381	−	0.673287i	$-0.235118\pi$
$353$	27.7834	1.47876	0.739381	+	0.673287i	$0.235118\pi$
$354$	0	0
$355$	− 11.3535i	− 0.602579i
$356$	0	0
$357$	6.37752i	0.337534i
$358$	0	0
$359$	−9.93716	−0.524463	−0.262232	−	0.965005i	$-0.584458\pi$
$359$	−9.93716	−0.524463	−0.262232	+	0.965005i	$0.584458\pi$
$360$	0	0
$361$	−1.00000	−0.0526316
$362$	0	0
$363$	− 3.01562i	− 0.158279i
$364$	0	0
$365$	15.1543i	0.793213i
$366$	0	0
$367$	6.50515	0.339566	0.169783	−	0.985481i	$-0.445693\pi$
$367$	6.50515	0.339566	0.169783	+	0.985481i	$0.445693\pi$
$368$	0	0
$369$	7.90980	0.411768
$370$	0	0
$371$	− 18.3864i	− 0.954576i
$372$	0	0
$373$	18.1569i	0.940129i	0.882632	+	0.470065i	$0.155769\pi$
$373$	18.1569i	0.940129i	−0.882632	+	0.470065i	$0.844231\pi$
$374$	0	0
$375$	0.530865	0.0274138
$376$	0	0
$377$	−24.8078	−1.27767
$378$	0	0
$379$	30.5842i	1.57101i	0.618858	+	0.785503i	$0.287596\pi$
$379$	30.5842i	1.57101i	−0.618858	+	0.785503i	$0.712404\pi$
$380$	0	0
$381$	− 2.16069i	− 0.110695i
$382$	0	0
$383$	−32.9375	−1.68303	−0.841514	−	0.540236i	$-0.818335\pi$
$383$	−32.9375	−1.68303	−0.841514	+	0.540236i	$0.818335\pi$
$384$	0	0
$385$	−6.52964	−0.332781
$386$	0	0
$387$	11.7321i	0.596376i
$388$	0	0
$389$	− 13.4993i	− 0.684444i	−0.939619	−	0.342222i	$-0.888821\pi$
$389$	− 13.4993i	− 0.684444i	0.939619	−	0.342222i	$-0.111179\pi$
$390$	0	0
$391$	−59.9171	−3.03014
$392$	0	0
$393$	−9.50233	−0.479329
$394$	0	0
$395$	3.08138i	0.155041i
$396$	0	0
$397$	29.6811i	1.48965i	0.667259	+	0.744826i	$0.267467\pi$
$397$	29.6811i	1.48965i	−0.667259	+	0.744826i	$0.732533\pi$
$398$	0	0
$399$	−0.848726	−0.0424894
$400$	0	0
$401$	5.56949	0.278127	0.139064	−	0.990283i	$-0.455591\pi$
$401$	5.56949	0.278127	0.139064	+	0.990283i	$0.455591\pi$
$402$	0	0
$403$	− 15.3352i	− 0.763901i
$404$	0	0
$405$	6.54306i	0.325127i
$406$	0	0
$407$	4.50519	0.223314
$408$	0	0
$409$	−29.9040	−1.47866	−0.739330	−	0.673343i	$-0.764858\pi$
$409$	−29.9040	−1.47866	−0.739330	+	0.673343i	$0.764858\pi$
$410$	0	0
$411$	1.36943i	0.0675491i
$412$	0	0
$413$	11.0562i	0.544040i
$414$	0	0
$415$	−3.40976	−0.167379
$416$	0	0
$417$	7.21715	0.353425
$418$	0	0
$419$	− 0.927860i	− 0.0453289i	−0.999743	−	0.0226645i	$-0.992785\pi$
$419$	− 0.927860i	− 0.0453289i	0.999743	−	0.0226645i	$-0.00721494\pi$
$420$	0	0
$421$	− 36.2436i	− 1.76641i	−0.468990	−	0.883204i	$-0.655382\pi$
$421$	− 36.2436i	− 1.76641i	0.468990	−	0.883204i	$-0.344618\pi$
$422$	0	0
$423$	−16.7298	−0.813429
$424$	0	0
$425$	7.51422	0.364493
$426$	0	0
$427$	− 7.38045i	− 0.357165i
$428$	0	0
$429$	9.30824i	0.449406i
$430$	0	0
$431$	−14.2580	−0.686785	−0.343392	−	0.939192i	$-0.611576\pi$
$431$	−14.2580	−0.686785	−0.343392	+	0.939192i	$0.611576\pi$
$432$	0	0
$433$	8.82519	0.424112	0.212056	−	0.977258i	$-0.431984\pi$
$433$	8.82519	0.424112	0.212056	+	0.977258i	$0.431984\pi$
$434$	0	0
$435$	− 3.06757i	− 0.147079i
$436$	0	0
$437$	− 7.97383i	− 0.381440i
$438$	0	0
$439$	20.6663	0.986351	0.493175	−	0.869930i	$-0.335836\pi$
$439$	20.6663	0.986351	0.493175	+	0.869930i	$0.335836\pi$
$440$	0	0
$441$	−12.0795	−0.575215
$442$	0	0
$443$	11.4417i	0.543612i	0.962352	+	0.271806i	$0.0876209\pi$
$443$	11.4417i	0.543612i	−0.962352	+	0.271806i	$0.912379\pi$
$444$	0	0
$445$	− 13.5566i	− 0.642643i
$446$	0	0
$447$	0.735789	0.0348016
$448$	0	0
$449$	28.4103	1.34077	0.670383	−	0.742015i	$-0.266130\pi$
$449$	28.4103	1.34077	0.670383	+	0.742015i	$0.266130\pi$
$450$	0	0
$451$	− 11.8848i	− 0.559634i
$452$	0	0
$453$	− 9.37045i	− 0.440262i
$454$	0	0
$455$	6.86375	0.321777
$456$	0	0
$457$	−35.1259	−1.64312	−0.821561	−	0.570121i	$-0.806896\pi$
$457$	−35.1259	−1.64312	−0.821561	+	0.570121i	$0.806896\pi$
$458$	0	0
$459$	− 22.8100i	− 1.06468i
$460$	0	0
$461$	− 30.0315i	− 1.39871i	−0.714777	−	0.699353i	$-0.753472\pi$
$461$	− 30.0315i	− 1.39871i	0.714777	−	0.699353i	$-0.246528\pi$
$462$	0	0
$463$	2.07091	0.0962432	0.0481216	−	0.998841i	$-0.484677\pi$
$463$	2.07091	0.0962432	0.0481216	+	0.998841i	$0.484677\pi$
$464$	0	0
$465$	1.89625	0.0879366
$466$	0	0
$467$	40.1423i	1.85756i	0.370628	+	0.928781i	$0.379142\pi$
$467$	40.1423i	1.85756i	−0.370628	+	0.928781i	$0.620858\pi$
$468$	0	0
$469$	− 16.3404i	− 0.754531i
$470$	0	0
$471$	−6.63393	−0.305675
$472$	0	0
$473$	17.6280	0.810536
$474$	0	0
$475$	1.00000i	0.0458831i
$476$	0	0
$477$	31.2603i	1.43131i
$478$	0	0
$479$	26.0532	1.19040	0.595200	−	0.803577i	$-0.297073\pi$
$479$	26.0532	1.19040	0.595200	+	0.803577i	$0.297073\pi$
$480$	0	0
$481$	−4.73571	−0.215930
$482$	0	0
$483$	− 6.76759i	− 0.307936i
$484$	0	0
$485$	3.07007i	0.139405i
$486$	0	0
$487$	7.11676	0.322491	0.161245	−	0.986914i	$-0.448449\pi$
$487$	7.11676	0.322491	0.161245	+	0.986914i	$0.448449\pi$
$488$	0	0
$489$	−5.64938	−0.255474
$490$	0	0
$491$	− 37.1832i	− 1.67805i	−0.544090	−	0.839027i	$-0.683125\pi$
$491$	− 37.1832i	− 1.67805i	0.544090	−	0.839027i	$-0.316875\pi$
$492$	0	0
$493$	− 43.4205i	− 1.95556i
$494$	0	0
$495$	11.1016	0.498978
$496$	0	0
$497$	18.1515	0.814204
$498$	0	0
$499$	− 10.4518i	− 0.467888i	−0.972250	−	0.233944i	$-0.924837\pi$
$499$	− 10.4518i	− 0.467888i	0.972250	−	0.233944i	$-0.0751633\pi$
$500$	0	0
$501$	− 4.95073i	− 0.221182i
$502$	0	0
$503$	30.2318	1.34797	0.673985	−	0.738745i	$-0.264581\pi$
$503$	30.2318	1.34797	0.673985	+	0.738745i	$0.264581\pi$
$504$	0	0
$505$	10.5727	0.470478
$506$	0	0
$507$	− 2.88328i	− 0.128051i
$508$	0	0
$509$	− 12.5208i	− 0.554975i	−0.960729	−	0.277488i	$-0.910498\pi$
$509$	− 12.5208i	− 0.554975i	0.960729	−	0.277488i	$-0.0895017\pi$
$510$	0	0
$511$	−24.2281	−1.07179
$512$	0	0
$513$	3.03558	0.134024
$514$	0	0
$515$	− 10.5955i	− 0.466894i
$516$	0	0
$517$	25.1372i	1.10553i
$518$	0	0
$519$	7.56797	0.332197
$520$	0	0
$521$	15.0001	0.657168	0.328584	−	0.944475i	$-0.393429\pi$
$521$	15.0001	0.657168	0.328584	+	0.944475i	$0.393429\pi$
$522$	0	0
$523$	− 23.4506i	− 1.02543i	−0.858560	−	0.512713i	$-0.828641\pi$
$523$	− 23.4506i	− 1.02543i	0.858560	−	0.512713i	$-0.171359\pi$
$524$	0	0
$525$	0.848726i	0.0370414i
$526$	0	0
$527$	26.8408	1.16920
$528$	0	0
$529$	40.5819	1.76443
$530$	0	0
$531$	− 18.7975i	− 0.815743i
$532$	0	0
$533$	12.4929i	0.541129i
$534$	0	0
$535$	−2.02637	−0.0876077
$536$	0	0
$537$	−8.43142	−0.363843
$538$	0	0
$539$	18.1500i	0.781775i
$540$	0	0
$541$	6.30059i	0.270884i	0.990785	+	0.135442i	$0.0432454\pi$
$541$	6.30059i	0.270884i	−0.990785	+	0.135442i	$0.956755\pi$
$542$	0	0
$543$	−5.15429	−0.221192
$544$	0	0
$545$	−9.89833	−0.423998
$546$	0	0
$547$	2.94592i	0.125958i	0.998015	+	0.0629792i	$0.0200602\pi$
$547$	2.94592i	0.125958i	−0.998015	+	0.0629792i	$0.979940\pi$
$548$	0	0
$549$	12.5481i	0.535540i
$550$	0	0
$551$	5.77844	0.246170
$552$	0	0
$553$	−4.92639	−0.209492
$554$	0	0
$555$	− 0.585588i	− 0.0248568i
$556$	0	0
$557$	− 1.01592i	− 0.0430460i	−0.999768	−	0.0215230i	$-0.993148\pi$
$557$	− 1.01592i	− 0.0430460i	0.999768	−	0.0215230i	$-0.00685151\pi$
$558$	0	0
$559$	−18.5300	−0.783735
$560$	0	0
$561$	−16.2920	−0.687848
$562$	0	0
$563$	20.8276i	0.877781i	0.898541	+	0.438890i	$0.144628\pi$
$563$	20.8276i	0.877781i	−0.898541	+	0.438890i	$0.855372\pi$
$564$	0	0
$565$	− 10.9289i	− 0.459784i
$566$	0	0
$567$	−10.4608	−0.439312
$568$	0	0
$569$	21.6656	0.908269	0.454135	−	0.890933i	$-0.349948\pi$
$569$	21.6656	0.908269	0.454135	+	0.890933i	$0.349948\pi$
$570$	0	0
$571$	− 11.1870i	− 0.468160i	−0.972217	−	0.234080i	$-0.924792\pi$
$571$	− 11.1870i	− 0.468160i	0.972217	−	0.234080i	$-0.0752078\pi$
$572$	0	0
$573$	− 2.83625i	− 0.118486i
$574$	0	0
$575$	−7.97383	−0.332532
$576$	0	0
$577$	−7.17954	−0.298888	−0.149444	−	0.988770i	$-0.547748\pi$
$577$	−7.17954	−0.298888	−0.149444	+	0.988770i	$0.547748\pi$
$578$	0	0
$579$	− 3.54786i	− 0.147444i
$580$	0	0
$581$	− 5.45139i	− 0.226162i
$582$	0	0
$583$	46.9699	1.94530
$584$	0	0
$585$	−11.6696	−0.482479
$586$	0	0
$587$	− 21.8913i	− 0.903550i	−0.892132	−	0.451775i	$-0.850791\pi$
$587$	− 21.8913i	− 0.903550i	0.892132	−	0.451775i	$-0.149209\pi$
$588$	0	0
$589$	3.57200i	0.147182i
$590$	0	0
$591$	0.993778	0.0408786
$592$	0	0
$593$	−15.5372	−0.638036	−0.319018	−	0.947749i	$-0.603353\pi$
$593$	−15.5372	−0.638036	−0.319018	+	0.947749i	$0.603353\pi$
$594$	0	0
$595$	12.0134i	0.492503i
$596$	0	0
$597$	− 6.23953i	− 0.255367i
$598$	0	0
$599$	33.3360	1.36207	0.681037	−	0.732249i	$-0.261529\pi$
$599$	33.3360	1.36207	0.681037	+	0.732249i	$0.261529\pi$
$600$	0	0
$601$	0.180219	0.00735130	0.00367565	−	0.999993i	$-0.498830\pi$
$601$	0.180219	0.00735130	0.00367565	+	0.999993i	$0.498830\pi$
$602$	0	0
$603$	27.7817i	1.13136i
$604$	0	0
$605$	− 5.68058i	− 0.230948i
$606$	0	0
$607$	21.7657	0.883441	0.441720	−	0.897153i	$-0.354368\pi$
$607$	21.7657	0.883441	0.441720	+	0.897153i	$0.354368\pi$
$608$	0	0
$609$	4.90432	0.198733
$610$	0	0
$611$	− 26.4234i	− 1.06898i
$612$	0	0
$613$	− 0.0959402i	− 0.00387499i	−0.999998	−	0.00193749i	$-0.999383\pi$
$613$	− 0.0959402i	− 0.00387499i	0.999998	−	0.00193749i	$-0.000616724\pi$
$614$	0	0
$615$	−1.54480	−0.0622922
$616$	0	0
$617$	46.4973	1.87191	0.935956	−	0.352117i	$-0.114538\pi$
$617$	46.4973	1.87191	0.935956	+	0.352117i	$0.114538\pi$
$618$	0	0
$619$	6.37072i	0.256061i	0.991770	+	0.128030i	$0.0408655\pi$
$619$	6.37072i	0.256061i	−0.991770	+	0.128030i	$0.959135\pi$
$620$	0	0
$621$	24.2052i	0.971322i
$622$	0	0
$623$	21.6737	0.868339
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	− 2.16815i	− 0.0865876i
$628$	0	0
$629$	− 8.28880i	− 0.330496i
$630$	0	0
$631$	28.9645	1.15306	0.576530	−	0.817076i	$-0.304406\pi$
$631$	28.9645	1.15306	0.576530	+	0.817076i	$0.304406\pi$
$632$	0	0
$633$	−8.67383	−0.344754
$634$	0	0
$635$	− 4.07012i	− 0.161518i
$636$	0	0
$637$	− 19.0787i	− 0.755925i
$638$	0	0
$639$	−30.8608	−1.22083
$640$	0	0
$641$	23.1989	0.916301	0.458151	−	0.888875i	$-0.348512\pi$
$641$	23.1989	0.916301	0.458151	+	0.888875i	$0.348512\pi$
$642$	0	0
$643$	− 2.62001i	− 0.103323i	−0.998665	−	0.0516616i	$-0.983548\pi$
$643$	− 2.62001i	− 0.103323i	0.998665	−	0.0516616i	$-0.0164517\pi$
$644$	0	0
$645$	− 2.29130i	− 0.0902197i
$646$	0	0
$647$	6.48788	0.255065	0.127532	−	0.991834i	$-0.459294\pi$
$647$	6.48788	0.255065	0.127532	+	0.991834i	$0.459294\pi$
$648$	0	0
$649$	−28.2441	−1.10868
$650$	0	0
$651$	3.03165i	0.118820i
$652$	0	0
$653$	− 26.3459i	− 1.03099i	−0.856891	−	0.515497i	$-0.827607\pi$
$653$	− 26.3459i	− 1.03099i	0.856891	−	0.515497i	$-0.172393\pi$
$654$	0	0
$655$	−17.8997	−0.699399
$656$	0	0
$657$	41.1922	1.60706
$658$	0	0
$659$	− 17.9787i	− 0.700350i	−0.936684	−	0.350175i	$-0.886122\pi$
$659$	− 17.9787i	− 0.700350i	0.936684	−	0.350175i	$-0.113878\pi$
$660$	0	0
$661$	− 20.4574i	− 0.795700i	−0.917450	−	0.397850i	$-0.869756\pi$
$661$	− 20.4574i	− 0.795700i	0.917450	−	0.397850i	$-0.130244\pi$
$662$	0	0
$663$	17.1256	0.665103
$664$	0	0
$665$	−1.59876	−0.0619973
$666$	0	0
$667$	46.0763i	1.78408i
$668$	0	0
$669$	− 7.50162i	− 0.290029i
$670$	0	0
$671$	18.8541	0.727853
$672$	0	0
$673$	−26.5170	−1.02215	−0.511077	−	0.859535i	$-0.670753\pi$
$673$	−26.5170	−1.02215	−0.511077	+	0.859535i	$0.670753\pi$
$674$	0	0
$675$	− 3.03558i	− 0.116840i
$676$	0	0
$677$	24.0962i	0.926092i	0.886334	+	0.463046i	$0.153244\pi$
$677$	24.0962i	0.926092i	−0.886334	+	0.463046i	$0.846756\pi$
$678$	0	0
$679$	−4.90831	−0.188363
$680$	0	0
$681$	7.50498	0.287592
$682$	0	0
$683$	− 36.1606i	− 1.38365i	−0.722066	−	0.691824i	$-0.756807\pi$
$683$	− 36.1606i	− 1.38365i	0.722066	−	0.691824i	$-0.243193\pi$
$684$	0	0
$685$	2.57962i	0.0985623i
$686$	0	0
$687$	−6.97034	−0.265935
$688$	0	0
$689$	−49.3733	−1.88097
$690$	0	0
$691$	− 21.7649i	− 0.827976i	−0.910282	−	0.413988i	$-0.864136\pi$
$691$	− 21.7649i	− 0.827976i	0.910282	−	0.413988i	$-0.135864\pi$
$692$	0	0
$693$	17.7487i	0.674219i
$694$	0	0
$695$	13.5951	0.515691
$696$	0	0
$697$	−21.8661	−0.828237
$698$	0	0
$699$	− 8.91860i	− 0.337332i
$700$	0	0
$701$	9.06451i	0.342362i	0.985240	+	0.171181i	$0.0547583\pi$
$701$	9.06451i	0.342362i	−0.985240	+	0.171181i	$0.945242\pi$
$702$	0	0
$703$	1.10308	0.0416035
$704$	0	0
$705$	3.26735	0.123055
$706$	0	0
$707$	16.9032i	0.635709i
$708$	0	0
$709$	− 21.2734i	− 0.798938i	−0.916747	−	0.399469i	$-0.869195\pi$
$709$	− 21.2734i	− 0.798938i	0.916747	−	0.399469i	$-0.130805\pi$
$710$	0	0
$711$	8.37576	0.314116
$712$	0	0
$713$	−28.4825	−1.06668
$714$	0	0
$715$	17.5341i	0.655738i
$716$	0	0
$717$	6.74682i	0.251965i
$718$	0	0
$719$	−38.8544	−1.44902	−0.724512	−	0.689262i	$-0.757935\pi$
$719$	−38.8544	−1.44902	−0.724512	+	0.689262i	$0.757935\pi$
$720$	0	0
$721$	16.9397	0.630867
$722$	0	0
$723$	− 1.47913i	− 0.0550095i
$724$	0	0
$725$	− 5.77844i	− 0.214606i
$726$	0	0
$727$	−22.3400	−0.828546	−0.414273	−	0.910153i	$-0.635964\pi$
$727$	−22.3400	−0.828546	−0.414273	+	0.910153i	$0.635964\pi$
$728$	0	0
$729$	12.9508	0.479659
$730$	0	0
$731$	− 32.4326i	− 1.19956i
$732$	0	0
$733$	7.54019i	0.278503i	0.990257	+	0.139252i	$0.0444697\pi$
$733$	7.54019i	0.278503i	−0.990257	+	0.139252i	$0.955530\pi$
$734$	0	0
$735$	2.35915	0.0870184
$736$	0	0
$737$	41.7432	1.53763
$738$	0	0
$739$	20.2897i	0.746369i	0.927757	+	0.373184i	$0.121734\pi$
$739$	20.2897i	0.746369i	−0.927757	+	0.373184i	$0.878266\pi$
$740$	0	0
$741$	2.27909i	0.0837245i
$742$	0	0
$743$	40.5952	1.48929	0.744647	−	0.667458i	$-0.232618\pi$
$743$	40.5952	1.48929	0.744647	+	0.667458i	$0.232618\pi$
$744$	0	0
$745$	1.38602	0.0507798
$746$	0	0
$747$	9.26835i	0.339111i
$748$	0	0
$749$	− 3.23968i	− 0.118376i
$750$	0	0
$751$	41.1087	1.50008	0.750038	−	0.661394i	$-0.230035\pi$
$751$	41.1087	1.50008	0.750038	+	0.661394i	$0.230035\pi$
$752$	0	0
$753$	10.1558	0.370098
$754$	0	0
$755$	− 17.6513i	− 0.642396i
$756$	0	0
$757$	23.1182i	0.840245i	0.907467	+	0.420123i	$0.138013\pi$
$757$	23.1182i	0.840245i	−0.907467	+	0.420123i	$0.861987\pi$
$758$	0	0
$759$	17.2885	0.627532
$760$	0	0
$761$	−19.5773	−0.709676	−0.354838	−	0.934928i	$-0.615464\pi$
$761$	−19.5773	−0.709676	−0.354838	+	0.934928i	$0.615464\pi$
$762$	0	0
$763$	− 15.8251i	− 0.572906i
$764$	0	0
$765$	− 20.4250i	− 0.738468i
$766$	0	0
$767$	29.6893	1.07202
$768$	0	0
$769$	37.2201	1.34219	0.671095	−	0.741371i	$-0.265824\pi$
$769$	37.2201	1.34219	0.671095	+	0.741371i	$0.265824\pi$
$770$	0	0
$771$	8.15257i	0.293608i
$772$	0	0
$773$	− 7.99430i	− 0.287535i	−0.989611	−	0.143768i	$-0.954078\pi$
$773$	− 7.99430i	− 0.287535i	0.989611	−	0.143768i	$-0.0459218\pi$
$774$	0	0
$775$	3.57200	0.128310
$776$	0	0
$777$	0.936214	0.0335865
$778$	0	0
$779$	− 2.90996i	− 0.104260i
$780$	0	0
$781$	46.3696i	1.65924i
$782$	0	0
$783$	−17.5409	−0.626862
$784$	0	0
$785$	−12.4964	−0.446017
$786$	0	0
$787$	− 34.8314i	− 1.24160i	−0.783967	−	0.620802i	$-0.786807\pi$
$787$	− 34.8314i	− 1.24160i	0.783967	−	0.620802i	$-0.213193\pi$
$788$	0	0
$789$	16.2656i	0.579070i
$790$	0	0
$791$	17.4728	0.621260
$792$	0	0
$793$	−19.8188	−0.703786
$794$	0	0
$795$	− 6.10518i	− 0.216528i
$796$	0	0
$797$	10.7193i	0.379697i	0.981813	+	0.189849i	$0.0607998\pi$
$797$	10.7193i	0.379697i	−0.981813	+	0.189849i	$0.939200\pi$
$798$	0	0
$799$	46.2482	1.63615
$800$	0	0
$801$	−36.8492	−1.30200
$802$	0	0
$803$	− 61.8931i	− 2.18416i
$804$	0	0
$805$	− 12.7482i	− 0.449316i
$806$	0	0
$807$	−11.6665	−0.410681
$808$	0	0
$809$	28.7727	1.01159	0.505797	−	0.862652i	$-0.331198\pi$
$809$	28.7727	1.01159	0.505797	+	0.862652i	$0.331198\pi$
$810$	0	0
$811$	− 17.7551i	− 0.623465i	−0.950170	−	0.311733i	$-0.899091\pi$
$811$	− 17.7551i	− 0.623465i	0.950170	−	0.311733i	$-0.100909\pi$
$812$	0	0
$813$	− 1.29876i	− 0.0455494i
$814$	0	0
$815$	−10.6418	−0.372767
$816$	0	0
$817$	4.31616	0.151003
$818$	0	0
$819$	− 18.6569i	− 0.651925i
$820$	0	0
$821$	− 44.3665i	− 1.54840i	−0.632939	−	0.774202i	$-0.718152\pi$
$821$	− 44.3665i	− 1.54840i	0.632939	−	0.774202i	$-0.281848\pi$
$822$	0	0
$823$	−52.3287	−1.82406	−0.912032	−	0.410118i	$-0.865487\pi$
$823$	−52.3287	−1.82406	−0.912032	+	0.410118i	$0.865487\pi$
$824$	0	0
$825$	−2.16815	−0.0754854
$826$	0	0
$827$	− 36.8109i	− 1.28004i	−0.768358	−	0.640021i	$-0.778926\pi$
$827$	− 36.8109i	− 1.28004i	0.768358	−	0.640021i	$-0.221074\pi$
$828$	0	0
$829$	− 6.63815i	− 0.230553i	−0.993333	−	0.115276i	$-0.963225\pi$
$829$	− 6.63815i	− 0.230553i	0.993333	−	0.115276i	$-0.0367753\pi$
$830$	0	0
$831$	14.3553	0.497981
$832$	0	0
$833$	33.3929	1.15700
$834$	0	0
$835$	− 9.32577i	− 0.322732i
$836$	0	0
$837$	− 10.8431i	− 0.374793i
$838$	0	0
$839$	−34.1930	−1.18047	−0.590236	−	0.807231i	$-0.700965\pi$
$839$	−34.1930	−1.18047	−0.590236	+	0.807231i	$0.700965\pi$
$840$	0	0
$841$	−4.39042	−0.151394
$842$	0	0
$843$	8.45757i	0.291294i
$844$	0	0
$845$	− 5.43128i	− 0.186842i
$846$	0	0
$847$	9.08188	0.312057
$848$	0	0
$849$	2.20524	0.0756838
$850$	0	0
$851$	8.79578i	0.301516i
$852$	0	0
$853$	27.3027i	0.934826i	0.884039	+	0.467413i	$0.154814\pi$
$853$	27.3027i	0.934826i	−0.884039	+	0.467413i	$0.845186\pi$
$854$	0	0
$855$	2.71818	0.0929599
$856$	0	0
$857$	8.87790	0.303263	0.151632	−	0.988437i	$-0.451547\pi$
$857$	8.87790	0.303263	0.151632	+	0.988437i	$0.451547\pi$
$858$	0	0
$859$	36.8942i	1.25881i	0.777076	+	0.629407i	$0.216702\pi$
$859$	36.8942i	1.25881i	−0.777076	+	0.629407i	$0.783298\pi$
$860$	0	0
$861$	− 2.46976i	− 0.0841691i
$862$	0	0
$863$	25.5457	0.869587	0.434793	−	0.900530i	$-0.356821\pi$
$863$	25.5457	0.869587	0.434793	+	0.900530i	$0.356821\pi$
$864$	0	0
$865$	14.2559	0.484716
$866$	0	0
$867$	20.9498i	0.711493i
$868$	0	0
$869$	− 12.5849i	− 0.426915i
$870$	0	0
$871$	−43.8791	−1.48679
$872$	0	0
$873$	8.34501	0.282436
$874$	0	0
$875$	1.59876i	0.0540480i
$876$	0	0
$877$	22.6731i	0.765615i	0.923828	+	0.382808i	$0.125043\pi$
$877$	22.6731i	0.765615i	−0.923828	+	0.382808i	$0.874957\pi$
$878$	0	0
$879$	−11.1626	−0.376506
$880$	0	0
$881$	32.3070	1.08845	0.544225	−	0.838939i	$-0.316824\pi$
$881$	32.3070	1.08845	0.544225	+	0.838939i	$0.316824\pi$
$882$	0	0
$883$	23.7637i	0.799711i	0.916578	+	0.399856i	$0.130940\pi$
$883$	23.7637i	0.799711i	−0.916578	+	0.399856i	$0.869060\pi$
$884$	0	0
$885$	3.67119i	0.123406i
$886$	0	0
$887$	40.8668	1.37217	0.686087	−	0.727520i	$-0.259327\pi$
$887$	40.8668	1.37217	0.686087	+	0.727520i	$0.259327\pi$
$888$	0	0
$889$	6.50715	0.218243
$890$	0	0
$891$	− 26.7231i	− 0.895257i
$892$	0	0
$893$	6.15476i	0.205961i
$894$	0	0
$895$	−15.8824	−0.530891
$896$	0	0
$897$	−18.1731	−0.606782
$898$	0	0
$899$	− 20.6406i	− 0.688403i
$900$	0	0
$901$	− 86.4168i	− 2.87896i
$902$	0	0
$903$	3.66324	0.121905
$904$	0	0
$905$	−9.70924	−0.322746
$906$	0	0
$907$	− 6.07897i	− 0.201849i	−0.994894	−	0.100925i	$-0.967820\pi$
$907$	− 6.07897i	− 0.201849i	0.994894	−	0.100925i	$-0.0321801\pi$
$908$	0	0
$909$	− 28.7385i	− 0.953195i
$910$	0	0
$911$	−36.8291	−1.22020	−0.610102	−	0.792323i	$-0.708871\pi$
$911$	−36.8291	−1.22020	−0.610102	+	0.792323i	$0.708871\pi$
$912$	0	0
$913$	13.9261	0.460887
$914$	0	0
$915$	− 2.45066i	− 0.0810164i
$916$	0	0
$917$	− 28.6173i	− 0.945028i
$918$	0	0
$919$	−21.5592	−0.711171	−0.355586	−	0.934644i	$-0.615719\pi$
$919$	−21.5592	−0.711171	−0.355586	+	0.934644i	$0.615719\pi$
$920$	0	0
$921$	8.85827	0.291890
$922$	0	0
$923$	− 48.7423i	− 1.60437i
$924$	0	0
$925$	− 1.10308i	− 0.0362691i
$926$	0	0
$927$	−28.8006	−0.945934
$928$	0	0
$929$	−36.9410	−1.21200	−0.605998	−	0.795466i	$-0.707226\pi$
$929$	−36.9410	−1.21200	−0.605998	+	0.795466i	$0.707226\pi$
$930$	0	0
$931$	4.44396i	0.145645i
$932$	0	0
$933$	4.84388i	0.158582i
$934$	0	0
$935$	−30.6895	−1.00365
$936$	0	0
$937$	−55.5587	−1.81502	−0.907512	−	0.420026i	$-0.862021\pi$
$937$	−55.5587	−1.81502	−0.907512	+	0.420026i	$0.862021\pi$
$938$	0	0
$939$	− 12.5118i	− 0.408308i
$940$	0	0
$941$	− 3.59643i	− 0.117240i	−0.998280	−	0.0586201i	$-0.981330\pi$
$941$	− 3.59643i	− 0.117240i	0.998280	−	0.0586201i	$-0.0186701\pi$
$942$	0	0
$943$	23.2035	0.755610
$944$	0	0
$945$	4.85317	0.157874
$946$	0	0
$947$	29.4296i	0.956334i	0.878269	+	0.478167i	$0.158699\pi$
$947$	29.4296i	0.956334i	−0.878269	+	0.478167i	$0.841301\pi$
$948$	0	0
$949$	65.0600i	2.11194i
$950$	0	0
$951$	7.24749	0.235016
$952$	0	0
$953$	54.0253	1.75005	0.875026	−	0.484077i	$-0.160844\pi$
$953$	54.0253	1.75005	0.875026	+	0.484077i	$0.160844\pi$
$954$	0	0
$955$	− 5.34269i	− 0.172885i
$956$	0	0
$957$	12.5285i	0.404990i
$958$	0	0
$959$	−4.12420	−0.133177
$960$	0	0
$961$	−18.2408	−0.588413
$962$	0	0
$963$	5.50805i	0.177494i
$964$	0	0
$965$	− 6.68316i	− 0.215139i
$966$	0	0
$967$	4.11964	0.132479	0.0662394	−	0.997804i	$-0.478900\pi$
$967$	4.11964	0.132479	0.0662394	+	0.997804i	$0.478900\pi$
$968$	0	0
$969$	−3.98904	−0.128146
$970$	0	0
$971$	22.6082i	0.725531i	0.931881	+	0.362765i	$0.118167\pi$
$971$	22.6082i	0.725531i	−0.931881	+	0.362765i	$0.881833\pi$
$972$	0	0
$973$	21.7353i	0.696801i
$974$	0	0
$975$	2.27909	0.0729894
$976$	0	0
$977$	33.7930	1.08113	0.540567	−	0.841301i	$-0.318210\pi$
$977$	33.7930	1.08113	0.540567	+	0.841301i	$0.318210\pi$
$978$	0	0
$979$	55.3676i	1.76956i
$980$	0	0
$981$	26.9055i	0.859026i
$982$	0	0
$983$	−54.5710	−1.74054	−0.870272	−	0.492572i	$-0.836057\pi$
$983$	−54.5710	−1.74054	−0.870272	+	0.492572i	$0.836057\pi$
$984$	0	0
$985$	1.87200	0.0596468
$986$	0	0
$987$	5.22371i	0.166272i
$988$	0	0
$989$	34.4163i	1.09437i
$990$	0	0
$991$	−33.8783	−1.07618	−0.538090	−	0.842888i	$-0.680854\pi$
$991$	−33.8783	−1.07618	−0.538090	+	0.842888i	$0.680854\pi$
$992$	0	0
$993$	−12.0493	−0.382374
$994$	0	0
$995$	− 11.7535i	− 0.372611i
$996$	0	0
$997$	− 41.4264i	− 1.31199i	−0.754767	−	0.655993i	$-0.772250\pi$
$997$	− 41.4264i	− 1.31199i	0.754767	−	0.655993i	$-0.227750\pi$
$998$	0	0
$999$	−3.34850	−0.105942

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3040.2.f.b.1521.25		44
4.3	odd	2		760.2.f.b.381.9	✓	44
8.3	odd	2		760.2.f.b.381.10	yes	44
8.5	even	2	inner	3040.2.f.b.1521.20		44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
760.2.f.b.381.9	✓	44	4.3	odd	2
760.2.f.b.381.10	yes	44	8.3	odd	2
3040.2.f.b.1521.20		44	8.5	even	2	inner
3040.2.f.b.1521.25		44	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3040 = 2^{5} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3040.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.530865i q^{3} +1.00000i q^{5} -1.59876 q^{7} +2.71818 q^{9} -4.08419i q^{11} +4.29317i q^{13} -0.530865 q^{15} -7.51422 q^{17} -1.00000i q^{19} -0.848726i q^{21} +7.97383 q^{23} -1.00000 q^{25} +3.03558i q^{27} +5.77844i q^{29} -3.57200 q^{31} +2.16815 q^{33} -1.59876i q^{35} +1.10308i q^{37} -2.27909 q^{39} +2.90996 q^{41} +4.31616i q^{43} +2.71818i q^{45} -6.15476 q^{47} -4.44396 q^{49} -3.98904i q^{51} +11.5004i q^{53} +4.08419 q^{55} +0.530865 q^{57} -6.91548i q^{59} +4.61636i q^{61} -4.34572 q^{63} -4.29317 q^{65} +10.2207i q^{67} +4.23303i q^{69} -11.3535 q^{71} +15.1543 q^{73} -0.530865i q^{75} +6.52964i q^{77} +3.08138 q^{79} +6.54306 q^{81} +3.40976i q^{83} -7.51422i q^{85} -3.06757 q^{87} -13.5566 q^{89} -6.86375i q^{91} -1.89625i q^{93} +1.00000 q^{95} +3.07007 q^{97} -11.1016i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 4 q^{7} - 60 q^{9} + 24 q^{17} - 4 q^{23} - 44 q^{25} + 40 q^{33} + 24 q^{39} - 32 q^{41} + 20 q^{47} + 108 q^{49} - 8 q^{55} - 20 q^{63} + 12 q^{65} + 8 q^{71} - 88 q^{73} - 40 q^{79} + 116 q^{81}+ \cdots + 116 q^{97}+O(q^{100}) \) 44 * q - 4 * q^7 - 60 * q^9 + 24 * q^17 - 4 * q^23 - 44 * q^25 + 40 * q^33 + 24 * q^39 - 32 * q^41 + 20 * q^47 + 108 * q^49 - 8 * q^55 - 20 * q^63 + 12 * q^65 + 8 * q^71 - 88 * q^73 - 40 * q^79 + 116 * q^81 - 48 * q^87 - 64 * q^89 + 44 * q^95 + 116 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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