LMFDB - Embedded newform 7605.2.a.cf.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7605,2,Mod(1,7605)] 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("7605.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7605, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 7605 = 3^{2} \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7605.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,-2,0,2,4,0,10,-6,0,-2,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$60.7262307372$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.4752.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 3x^{2} + 4x + 1 $ x^4 - 2x^3 - 3x^2 + 4*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 65)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.49551$ of defining polynomial
Character	$\chi$	$=$	7605.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.49551 q^{2} +0.236543 q^{4} +1.00000 q^{5} +4.82684 q^{7} -2.63726 q^{8} +1.49551 q^{10} +1.06939 q^{11} +7.21857 q^{14} -4.41713 q^{16} -3.55889 q^{17} +5.73205 q^{19} +0.236543 q^{20} +1.59928 q^{22} +7.08580 q^{23} +1.00000 q^{25} +1.14176 q^{28} -1.47309 q^{29} -1.46410 q^{31} -1.33133 q^{32} -5.32235 q^{34} +4.82684 q^{35} -0.0253983 q^{37} +8.57233 q^{38} -2.63726 q^{40} -0.267949 q^{41} +3.55889 q^{43} +0.252957 q^{44} +10.5969 q^{46} -6.51793 q^{47} +16.2984 q^{49} +1.49551 q^{50} -0.991015 q^{53} +1.06939 q^{55} -12.7296 q^{56} -2.20301 q^{58} -8.72307 q^{59} +6.33734 q^{61} -2.18958 q^{62} +6.84325 q^{64} +5.17316 q^{67} -0.841831 q^{68} +7.21857 q^{70} +7.76488 q^{71} +10.1088 q^{73} -0.0379833 q^{74} +1.35588 q^{76} +5.16177 q^{77} +8.78347 q^{79} -4.41713 q^{80} -0.400720 q^{82} +0.725474 q^{83} -3.55889 q^{85} +5.32235 q^{86} -2.82026 q^{88} -13.5065 q^{89} +1.67610 q^{92} -9.74761 q^{94} +5.73205 q^{95} -3.43870 q^{97} +24.3743 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} + 2 q^{4} + 4 q^{5} + 10 q^{7} - 6 q^{8} - 2 q^{10} - 2 q^{14} + 2 q^{16} + 2 q^{17} + 16 q^{19} + 2 q^{20} + 12 q^{22} + 10 q^{23} + 4 q^{25} + 8 q^{28} - 8 q^{29} + 8 q^{31} - 4 q^{32} - 4 q^{34}+ \cdots + 24 q^{98}+O(q^{100}) $ 4 * q - 2 * q^2 + 2 * q^4 + 4 * q^5 + 10 * q^7 - 6 * q^8 - 2 * q^10 - 2 * q^14 + 2 * q^16 + 2 * q^17 + 16 * q^19 + 2 * q^20 + 12 * q^22 + 10 * q^23 + 4 * q^25 + 8 * q^28 - 8 * q^29 + 8 * q^31 - 4 * q^32 - 4 * q^34 + 10 * q^35 - 2 * q^37 - 8 * q^38 - 6 * q^40 - 8 * q^41 - 2 * q^43 - 12 * q^44 + 16 * q^46 - 8 * q^47 + 12 * q^49 - 2 * q^50 + 12 * q^53 - 12 * q^56 + 22 * q^58 - 12 * q^59 + 28 * q^61 - 4 * q^62 + 4 * q^64 + 30 * q^67 - 14 * q^68 - 2 * q^70 - 4 * q^71 - 8 * q^73 + 10 * q^74 + 20 * q^76 - 18 * q^77 - 8 * q^79 + 2 * q^80 + 4 * q^82 + 12 * q^83 + 2 * q^85 + 4 * q^86 - 18 * q^88 + 12 * q^89 - 22 * q^92 - 32 * q^94 + 16 * q^95 + 2 * q^97 + 24 * q^98

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 $

$2$	1.49551	1.05748	0.528742	−	0.848783i	$-0.322664\pi$
$2$	1.49551	1.05748	0.528742	+	0.848783i	$0.322664\pi$
$3$	0	0
$3$	0	0
$4$	0.236543	0.118272
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	4.82684	1.82437	0.912187	−	0.409775i	$-0.134393\pi$
$7$	4.82684	1.82437	0.912187	+	0.409775i	$0.134393\pi$
$8$	−2.63726	−0.932413
$9$	0	0
$10$	1.49551	0.472921
$11$	1.06939	0.322433	0.161217	−	0.986919i	$-0.448458\pi$
$11$	1.06939	0.322433	0.161217	+	0.986919i	$0.448458\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	7.21857	1.92924
$15$	0	0
$16$	−4.41713	−1.10428
$17$	−3.55889	−0.863157	−0.431579	−	0.902075i	$-0.642043\pi$
$17$	−3.55889	−0.863157	−0.431579	+	0.902075i	$0.642043\pi$
$18$	0	0
$19$	5.73205	1.31502	0.657511	−	0.753445i	$-0.271609\pi$
$19$	5.73205	1.31502	0.657511	+	0.753445i	$0.271609\pi$
$20$	0.236543	0.0528927
$21$	0	0
$22$	1.59928	0.340968
$23$	7.08580	1.47749	0.738746	−	0.673984i	$-0.235418\pi$
$23$	7.08580	1.47749	0.738746	+	0.673984i	$0.235418\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	1.14176	0.215772
$29$	−1.47309	−0.273545	−0.136773	−	0.990602i	$-0.543673\pi$
$29$	−1.47309	−0.273545	−0.136773	+	0.990602i	$0.543673\pi$
$30$	0	0
$31$	−1.46410	−0.262960	−0.131480	−	0.991319i	$-0.541973\pi$
$31$	−1.46410	−0.262960	−0.131480	+	0.991319i	$0.541973\pi$
$32$	−1.33133	−0.235348
$33$	0	0
$34$	−5.32235	−0.912775
$35$	4.82684	0.815885
$36$	0	0
$37$	−0.0253983	−0.00417545	−0.00208772	−	0.999998i	$-0.500665\pi$
$37$	−0.0253983	−0.00417545	−0.00208772	+	0.999998i	$0.500665\pi$
$38$	8.57233	1.39061
$39$	0	0
$40$	−2.63726	−0.416988
$41$	−0.267949	−0.0418466	−0.0209233	−	0.999781i	$-0.506661\pi$
$41$	−0.267949	−0.0418466	−0.0209233	+	0.999781i	$0.506661\pi$
$42$	0	0
$43$	3.55889	0.542726	0.271363	−	0.962477i	$-0.412526\pi$
$43$	3.55889	0.542726	0.271363	+	0.962477i	$0.412526\pi$
$44$	0.252957	0.0381347
$45$	0	0
$46$	10.5969	1.56242
$47$	−6.51793	−0.950738	−0.475369	−	0.879787i	$-0.657685\pi$
$47$	−6.51793	−0.950738	−0.475369	+	0.879787i	$0.657685\pi$
$48$	0	0
$49$	16.2984	2.32834
$50$	1.49551	0.211497
$51$	0	0
$52$	0	0
$53$	−0.991015	−0.136126	−0.0680632	−	0.997681i	$-0.521682\pi$
$53$	−0.991015	−0.136126	−0.0680632	+	0.997681i	$0.521682\pi$
$54$	0	0
$55$	1.06939	0.144196
$56$	−12.7296	−1.70107
$57$	0	0
$58$	−2.20301	−0.289270
$59$	−8.72307	−1.13565	−0.567823	−	0.823151i	$-0.692214\pi$
$59$	−8.72307	−1.13565	−0.567823	+	0.823151i	$0.692214\pi$
$60$	0	0
$61$	6.33734	0.811413	0.405707	−	0.914003i	$-0.367026\pi$
$61$	6.33734	0.811413	0.405707	+	0.914003i	$0.367026\pi$
$62$	−2.18958	−0.278076
$63$	0	0
$64$	6.84325	0.855406
$65$	0	0
$66$	0	0
$67$	5.17316	0.632002	0.316001	−	0.948759i	$-0.397660\pi$
$67$	5.17316	0.632002	0.316001	+	0.948759i	$0.397660\pi$
$68$	−0.841831	−0.102087
$69$	0	0
$70$	7.21857	0.862785
$71$	7.76488	0.921521	0.460761	−	0.887524i	$-0.347577\pi$
$71$	7.76488	0.921521	0.460761	+	0.887524i	$0.347577\pi$
$72$	0	0
$73$	10.1088	1.18314	0.591572	−	0.806252i	$-0.298507\pi$
$73$	10.1088	1.18314	0.591572	+	0.806252i	$0.298507\pi$
$74$	−0.0379833	−0.00441547
$75$	0	0
$76$	1.35588	0.155530
$77$	5.16177	0.588238
$78$	0	0
$79$	8.78347	0.988218	0.494109	−	0.869400i	$-0.335494\pi$
$79$	8.78347	0.988218	0.494109	+	0.869400i	$0.335494\pi$
$80$	−4.41713	−0.493851
$81$	0	0
$82$	−0.400720	−0.0442521
$83$	0.725474	0.0796311	0.0398155	−	0.999207i	$-0.487323\pi$
$83$	0.725474	0.0796311	0.0398155	+	0.999207i	$0.487323\pi$
$84$	0	0
$85$	−3.55889	−0.386016
$86$	5.32235	0.573923
$87$	0	0
$88$	−2.82026	−0.300641
$89$	−13.5065	−1.43169	−0.715845	−	0.698259i	$-0.753958\pi$
$89$	−13.5065	−1.43169	−0.715845	+	0.698259i	$0.753958\pi$
$90$	0	0
$91$	0	0
$92$	1.67610	0.174745
$93$	0	0
$94$	−9.74761	−1.00539
$95$	5.73205	0.588096
$96$	0	0
$97$	−3.43870	−0.349147	−0.174574	−	0.984644i	$-0.555855\pi$
$97$	−3.43870	−0.349147	−0.174574	+	0.984644i	$0.555855\pi$
$98$	24.3743	2.46218
$99$	0	0
$100$	0.236543	0.0236543
$101$	−2.85527	−0.284110	−0.142055	−	0.989859i	$-0.545371\pi$
$101$	−2.85527	−0.284110	−0.142055	+	0.989859i	$0.545371\pi$
$102$	0	0
$103$	−5.54488	−0.546354	−0.273177	−	0.961964i	$-0.588074\pi$
$103$	−5.54488	−0.546354	−0.273177	+	0.961964i	$0.588074\pi$
$104$	0	0
$105$	0	0
$106$	−1.48207	−0.143951
$107$	4.44111	0.429338	0.214669	−	0.976687i	$-0.431133\pi$
$107$	4.44111	0.429338	0.214669	+	0.976687i	$0.431133\pi$
$108$	0	0
$109$	13.7804	1.31993	0.659963	−	0.751298i	$-0.270572\pi$
$109$	13.7804	1.31993	0.659963	+	0.751298i	$0.270572\pi$
$110$	1.59928	0.152485
$111$	0	0
$112$	−21.3208	−2.01463
$113$	8.04399	0.756715	0.378358	−	0.925660i	$-0.376489\pi$
$113$	8.04399	0.756715	0.378358	+	0.925660i	$0.376489\pi$
$114$	0	0
$115$	7.08580	0.660755
$116$	−0.348448	−0.0323526
$117$	0	0
$118$	−13.0454	−1.20093
$119$	−17.1782	−1.57472
$120$	0	0
$121$	−9.85641	−0.896037
$122$	9.47754	0.858056
$123$	0	0
$124$	−0.346323	−0.0311007
$125$	1.00000	0.0894427
$126$	0	0
$127$	0.706653	0.0627053	0.0313526	−	0.999508i	$-0.490019\pi$
$127$	0.706653	0.0627053	0.0313526	+	0.999508i	$0.490019\pi$
$128$	12.8968	1.13993
$129$	0	0
$130$	0	0
$131$	−6.26554	−0.547423	−0.273711	−	0.961812i	$-0.588251\pi$
$131$	−6.26554	−0.547423	−0.273711	+	0.961812i	$0.588251\pi$
$132$	0	0
$133$	27.6677	2.39909
$134$	7.73650	0.668332
$135$	0	0
$136$	9.38573	0.804820
$137$	−16.3058	−1.39310	−0.696549	−	0.717509i	$-0.745282\pi$
$137$	−16.3058	−1.39310	−0.696549	+	0.717509i	$0.745282\pi$
$138$	0	0
$139$	−6.82528	−0.578913	−0.289456	−	0.957191i	$-0.593475\pi$
$139$	−6.82528	−0.578913	−0.289456	+	0.957191i	$0.593475\pi$
$140$	1.14176	0.0964960
$141$	0	0
$142$	11.6124	0.974494
$143$	0	0
$144$	0	0
$145$	−1.47309	−0.122333
$146$	15.1178	1.25116
$147$	0	0
$148$	−0.00600778	−0.000493837 0
$149$	8.43955	0.691395	0.345698	−	0.938346i	$-0.387642\pi$
$149$	8.43955	0.691395	0.345698	+	0.938346i	$0.387642\pi$
$150$	0	0
$151$	−1.37017	−0.111503	−0.0557513	−	0.998445i	$-0.517755\pi$
$151$	−1.37017	−0.111503	−0.0557513	+	0.998445i	$0.517755\pi$
$152$	−15.1169	−1.22614
$153$	0	0
$154$	7.71947	0.622052
$155$	−1.46410	−0.117599
$156$	0	0
$157$	11.9700	0.955311	0.477656	−	0.878547i	$-0.341487\pi$
$157$	11.9700	0.955311	0.477656	+	0.878547i	$0.341487\pi$
$158$	13.1357	1.04502
$159$	0	0
$160$	−1.33133	−0.105251
$161$	34.2020	2.69550
$162$	0	0
$163$	−22.5713	−1.76792	−0.883962	−	0.467559i	$-0.845134\pi$
$163$	−22.5713	−1.76792	−0.883962	+	0.467559i	$0.845134\pi$
$164$	−0.0633815	−0.00494927
$165$	0	0
$166$	1.08495	0.0842085
$167$	−8.19700	−0.634303	−0.317152	−	0.948375i	$-0.602726\pi$
$167$	−8.19700	−0.634303	−0.317152	+	0.948375i	$0.602726\pi$
$168$	0	0
$169$	0	0
$170$	−5.32235	−0.408205
$171$	0	0
$172$	0.841831	0.0641890
$173$	9.16772	0.697009	0.348505	−	0.937307i	$-0.386690\pi$
$173$	9.16772	0.697009	0.348505	+	0.937307i	$0.386690\pi$
$174$	0	0
$175$	4.82684	0.364875
$176$	−4.72364	−0.356057
$177$	0	0
$178$	−20.1991	−1.51399
$179$	10.0370	0.750200	0.375100	−	0.926984i	$-0.377608\pi$
$179$	10.0370	0.750200	0.375100	+	0.926984i	$0.377608\pi$
$180$	0	0
$181$	17.0238	1.26537	0.632686	−	0.774408i	$-0.281952\pi$
$181$	17.0238	1.26537	0.632686	+	0.774408i	$0.281952\pi$
$182$	0	0
$183$	0	0
$184$	−18.6871	−1.37763
$185$	−0.0253983	−0.00186732
$186$	0	0
$187$	−3.80584	−0.278310
$188$	−1.54177	−0.112445
$189$	0	0
$190$	8.57233	0.621902
$191$	3.87741	0.280559	0.140280	−	0.990112i	$-0.455200\pi$
$191$	3.87741	0.280559	0.140280	+	0.990112i	$0.455200\pi$
$192$	0	0
$193$	1.25394	0.0902608	0.0451304	−	0.998981i	$-0.485630\pi$
$193$	1.25394	0.0902608	0.0451304	+	0.998981i	$0.485630\pi$
$194$	−5.14261	−0.369218
$195$	0	0
$196$	3.85527	0.275376
$197$	15.2820	1.08879	0.544397	−	0.838828i	$-0.316758\pi$
$197$	15.2820	1.08879	0.544397	+	0.838828i	$0.316758\pi$
$198$	0	0
$199$	−13.2296	−0.937822	−0.468911	−	0.883246i	$-0.655353\pi$
$199$	−13.2296	−0.937822	−0.468911	+	0.883246i	$0.655353\pi$
$200$	−2.63726	−0.186483
$201$	0	0
$202$	−4.27007	−0.300441
$203$	−7.11035	−0.499049
$204$	0	0
$205$	−0.267949	−0.0187144
$206$	−8.29242	−0.577760
$207$	0	0
$208$	0	0
$209$	6.12979	0.424007
$210$	0	0
$211$	−4.81042	−0.331163	−0.165582	−	0.986196i	$-0.552950\pi$
$211$	−4.81042	−0.331163	−0.165582	+	0.986196i	$0.552950\pi$
$212$	−0.234418	−0.0160999
$213$	0	0
$214$	6.64172	0.454018
$215$	3.55889	0.242714
$216$	0	0
$217$	−7.06698	−0.479738
$218$	20.6088	1.39580
$219$	0	0
$220$	0.252957	0.0170543
$221$	0	0
$222$	0	0
$223$	14.7132	0.985271	0.492635	−	0.870236i	$-0.336034\pi$
$223$	14.7132	0.985271	0.492635	+	0.870236i	$0.336034\pi$
$224$	−6.42612	−0.429363
$225$	0	0
$226$	12.0299	0.800214
$227$	−14.9028	−0.989134	−0.494567	−	0.869140i	$-0.664673\pi$
$227$	−14.9028	−0.989134	−0.494567	+	0.869140i	$0.664673\pi$
$228$	0	0
$229$	−19.3074	−1.27587	−0.637933	−	0.770092i	$-0.720210\pi$
$229$	−19.3074	−1.27587	−0.637933	+	0.770092i	$0.720210\pi$
$230$	10.5969	0.698737
$231$	0	0
$232$	3.88492	0.255057
$233$	21.1937	1.38845	0.694224	−	0.719759i	$-0.255748\pi$
$233$	21.1937	1.38845	0.694224	+	0.719759i	$0.255748\pi$
$234$	0	0
$235$	−6.51793	−0.425183
$236$	−2.06338	−0.134315
$237$	0	0
$238$	−25.6901	−1.66524
$239$	14.8971	0.963612	0.481806	−	0.876278i	$-0.339981\pi$
$239$	14.8971	0.963612	0.481806	+	0.876278i	$0.339981\pi$
$240$	0	0
$241$	9.39168	0.604971	0.302486	−	0.953154i	$-0.402184\pi$
$241$	9.39168	0.604971	0.302486	+	0.953154i	$0.402184\pi$
$242$	−14.7403	−0.947544
$243$	0	0
$244$	1.49905	0.0959671
$245$	16.2984	1.04126
$246$	0	0
$247$	0	0
$248$	3.86122	0.245188
$249$	0	0
$250$	1.49551	0.0945842
$251$	−11.3163	−0.714281	−0.357140	−	0.934051i	$-0.616248\pi$
$251$	−11.3163	−0.714281	−0.357140	+	0.934051i	$0.616248\pi$
$252$	0	0
$253$	7.57748	0.476392
$254$	1.05680	0.0663098
$255$	0	0
$256$	5.60076	0.350047
$257$	26.5319	1.65502	0.827508	−	0.561453i	$-0.189758\pi$
$257$	26.5319	1.65502	0.827508	+	0.561453i	$0.189758\pi$
$258$	0	0
$259$	−0.122593	−0.00761758
$260$	0	0
$261$	0	0
$262$	−9.37017	−0.578891
$263$	14.1408	0.871957	0.435979	−	0.899957i	$-0.356402\pi$
$263$	14.1408	0.871957	0.435979	+	0.899957i	$0.356402\pi$
$264$	0	0
$265$	−0.991015	−0.0608776
$266$	41.3772	2.53700
$267$	0	0
$268$	1.22368	0.0747479
$269$	−24.7745	−1.51053	−0.755264	−	0.655421i	$-0.772491\pi$
$269$	−24.7745	−1.51053	−0.755264	+	0.655421i	$0.772491\pi$
$270$	0	0
$271$	18.7171	1.13698	0.568492	−	0.822689i	$-0.307527\pi$
$271$	18.7171	1.13698	0.568492	+	0.822689i	$0.307527\pi$
$272$	15.7201	0.953170
$273$	0	0
$274$	−24.3854	−1.47318
$275$	1.06939	0.0644866
$276$	0	0
$277$	−22.6647	−1.36179	−0.680893	−	0.732382i	$-0.738408\pi$
$277$	−22.6647	−1.36179	−0.680893	+	0.732382i	$0.738408\pi$
$278$	−10.2073	−0.612191
$279$	0	0
$280$	−12.7296	−0.760742
$281$	27.8384	1.66070	0.830351	−	0.557241i	$-0.188140\pi$
$281$	27.8384	1.66070	0.830351	+	0.557241i	$0.188140\pi$
$282$	0	0
$283$	7.92007	0.470799	0.235400	−	0.971899i	$-0.424360\pi$
$283$	7.92007	0.470799	0.235400	+	0.971899i	$0.424360\pi$
$284$	1.83673	0.108990
$285$	0	0
$286$	0	0
$287$	−1.29335	−0.0763439
$288$	0	0
$289$	−4.33431	−0.254959
$290$	−2.20301	−0.129365
$291$	0	0
$292$	2.39117	0.139932
$293$	0.272971	0.0159471	0.00797356	−	0.999968i	$-0.497462\pi$
$293$	0.272971	0.0159471	0.00797356	+	0.999968i	$0.497462\pi$
$294$	0	0
$295$	−8.72307	−0.507877
$296$	0.0669819	0.00389324
$297$	0	0
$298$	12.6214	0.731139
$299$	0	0
$300$	0	0
$301$	17.1782	0.990134
$302$	−2.04909	−0.117912
$303$	0	0
$304$	−25.3192	−1.45216
$305$	6.33734	0.362875
$306$	0	0
$307$	6.85224	0.391078	0.195539	−	0.980696i	$-0.437354\pi$
$307$	6.85224	0.391078	0.195539	+	0.980696i	$0.437354\pi$
$308$	1.22098	0.0695719
$309$	0	0
$310$	−2.18958	−0.124360
$311$	−10.6447	−0.603605	−0.301803	−	0.953370i	$-0.597588\pi$
$311$	−10.6447	−0.603605	−0.301803	+	0.953370i	$0.597588\pi$
$312$	0	0
$313$	17.8236	1.00745	0.503724	−	0.863865i	$-0.331963\pi$
$313$	17.8236	1.00745	0.503724	+	0.863865i	$0.331963\pi$
$314$	17.9012	1.01023
$315$	0	0
$316$	2.07767	0.116878
$317$	−8.17161	−0.458963	−0.229482	−	0.973313i	$-0.573703\pi$
$317$	−8.17161	−0.458963	−0.229482	+	0.973313i	$0.573703\pi$
$318$	0	0
$319$	−1.57530	−0.0882000
$320$	6.84325	0.382549
$321$	0	0
$322$	51.1494	2.85044
$323$	−20.3997	−1.13507
$324$	0	0
$325$	0	0
$326$	−33.7556	−1.86955
$327$	0	0
$328$	0.706653	0.0390184
$329$	−31.4610	−1.73450
$330$	0	0
$331$	24.9395	1.37080	0.685400	−	0.728167i	$-0.259627\pi$
$331$	24.9395	1.37080	0.685400	+	0.728167i	$0.259627\pi$
$332$	0.171606	0.00941809
$333$	0	0
$334$	−12.2587	−0.670765
$335$	5.17316	0.282640
$336$	0	0
$337$	−19.6057	−1.06799	−0.533996	−	0.845487i	$-0.679310\pi$
$337$	−19.6057	−1.06799	−0.533996	+	0.845487i	$0.679310\pi$
$338$	0	0
$339$	0	0
$340$	−0.841831	−0.0456547
$341$	−1.56569	−0.0847871
$342$	0	0
$343$	44.8817	2.42339
$344$	−9.38573	−0.506045
$345$	0	0
$346$	13.7104	0.737076
$347$	−17.0810	−0.916955	−0.458478	−	0.888706i	$-0.651605\pi$
$347$	−17.0810	−0.916955	−0.458478	+	0.888706i	$0.651605\pi$
$348$	0	0
$349$	−28.3719	−1.51871	−0.759357	−	0.650674i	$-0.774486\pi$
$349$	−28.3719	−1.51871	−0.759357	+	0.650674i	$0.774486\pi$
$350$	7.21857	0.385849
$351$	0	0
$352$	−1.42371	−0.0758840
$353$	−21.2520	−1.13113	−0.565564	−	0.824704i	$-0.691342\pi$
$353$	−21.2520	−1.13113	−0.565564	+	0.824704i	$0.691342\pi$
$354$	0	0
$355$	7.76488	0.412117
$356$	−3.19488	−0.169328
$357$	0	0
$358$	15.0104	0.793325
$359$	−32.6519	−1.72330	−0.861650	−	0.507502i	$-0.830569\pi$
$359$	−32.6519	−1.72330	−0.861650	+	0.507502i	$0.830569\pi$
$360$	0	0
$361$	13.8564	0.729285
$362$	25.4593	1.33811
$363$	0	0
$364$	0	0
$365$	10.1088	0.529118
$366$	0	0
$367$	−5.91837	−0.308936	−0.154468	−	0.987998i	$-0.549366\pi$
$367$	−5.91837	−0.308936	−0.154468	+	0.987998i	$0.549366\pi$
$368$	−31.2989	−1.63157
$369$	0	0
$370$	−0.0379833	−0.00197466
$371$	−4.78347	−0.248345
$372$	0	0
$373$	−13.3185	−0.689607	−0.344803	−	0.938675i	$-0.612054\pi$
$373$	−13.3185	−0.689607	−0.344803	+	0.938675i	$0.612054\pi$
$374$	−5.69166	−0.294309
$375$	0	0
$376$	17.1895	0.886480
$377$	0	0
$378$	0	0
$379$	25.4186	1.30566	0.652832	−	0.757503i	$-0.273581\pi$
$379$	25.4186	1.30566	0.652832	+	0.757503i	$0.273581\pi$
$380$	1.35588	0.0695550
$381$	0	0
$382$	5.79869	0.296687
$383$	10.8268	0.553226	0.276613	−	0.960981i	$-0.410788\pi$
$383$	10.8268	0.553226	0.276613	+	0.960981i	$0.410788\pi$
$384$	0	0
$385$	5.16177	0.263068
$386$	1.87528	0.0954493
$387$	0	0
$388$	−0.813402	−0.0412942
$389$	23.0370	1.16802	0.584011	−	0.811746i	$-0.301482\pi$
$389$	23.0370	1.16802	0.584011	+	0.811746i	$0.301482\pi$
$390$	0	0
$391$	−25.2176	−1.27531
$392$	−42.9831	−2.17097
$393$	0	0
$394$	22.8543	1.15138
$395$	8.78347	0.441944
$396$	0	0
$397$	21.0864	1.05830	0.529149	−	0.848529i	$-0.322511\pi$
$397$	21.0864	1.05830	0.529149	+	0.848529i	$0.322511\pi$
$398$	−19.7850	−0.991731
$399$	0	0
$400$	−4.41713	−0.220857
$401$	19.7769	0.987611	0.493805	−	0.869572i	$-0.335605\pi$
$401$	19.7769	0.987611	0.493805	+	0.869572i	$0.335605\pi$
$402$	0	0
$403$	0	0
$404$	−0.675394	−0.0336021
$405$	0	0
$406$	−10.6336	−0.527736
$407$	−0.0271606	−0.00134630
$408$	0	0
$409$	−31.8809	−1.57641	−0.788204	−	0.615414i	$-0.788989\pi$
$409$	−31.8809	−1.57641	−0.788204	+	0.615414i	$0.788989\pi$
$410$	−0.400720	−0.0197902
$411$	0	0
$412$	−1.31160	−0.0646181
$413$	−42.1048	−2.07184
$414$	0	0
$415$	0.725474	0.0356121
$416$	0	0
$417$	0	0
$418$	9.16715	0.448380
$419$	−30.7296	−1.50124	−0.750621	−	0.660733i	$-0.770245\pi$
$419$	−30.7296	−1.50124	−0.750621	+	0.660733i	$0.770245\pi$
$420$	0	0
$421$	17.9820	0.876391	0.438195	−	0.898880i	$-0.355618\pi$
$421$	17.9820	0.876391	0.438195	+	0.898880i	$0.355618\pi$
$422$	−7.19403	−0.350200
$423$	0	0
$424$	2.61357	0.126926
$425$	−3.55889	−0.172631
$426$	0	0
$427$	30.5893	1.48032
$428$	1.05051	0.0507785
$429$	0	0
$430$	5.32235	0.256666
$431$	−4.89949	−0.236000	−0.118000	−	0.993014i	$-0.537648\pi$
$431$	−4.89949	−0.236000	−0.118000	+	0.993014i	$0.537648\pi$
$432$	0	0
$433$	−19.2394	−0.924588	−0.462294	−	0.886727i	$-0.652974\pi$
$433$	−19.2394	−0.924588	−0.462294	+	0.886727i	$0.652974\pi$
$434$	−10.5687	−0.507315
$435$	0	0
$436$	3.25967	0.156110
$437$	40.6162	1.94294
$438$	0	0
$439$	−8.55974	−0.408534	−0.204267	−	0.978915i	$-0.565481\pi$
$439$	−8.55974	−0.408534	−0.204267	+	0.978915i	$0.565481\pi$
$440$	−2.82026	−0.134451
$441$	0	0
$442$	0	0
$443$	37.9652	1.80378	0.901891	−	0.431965i	$-0.142179\pi$
$443$	37.9652	1.80378	0.901891	+	0.431965i	$0.142179\pi$
$444$	0	0
$445$	−13.5065	−0.640271
$446$	22.0037	1.04191
$447$	0	0
$448$	33.0313	1.56058
$449$	26.7062	1.26035	0.630173	−	0.776455i	$-0.282984\pi$
$449$	26.7062	1.26035	0.630173	+	0.776455i	$0.282984\pi$
$450$	0	0
$451$	−0.286542	−0.0134927
$452$	1.90275	0.0894979
$453$	0	0
$454$	−22.2873	−1.04599
$455$	0	0
$456$	0	0
$457$	−4.27127	−0.199801	−0.0999007	−	0.994997i	$-0.531853\pi$
$457$	−4.27127	−0.199801	−0.0999007	+	0.994997i	$0.531853\pi$
$458$	−28.8743	−1.34921
$459$	0	0
$460$	1.67610	0.0781485
$461$	−20.6423	−0.961407	−0.480704	−	0.876883i	$-0.659619\pi$
$461$	−20.6423	−0.961407	−0.480704	+	0.876883i	$0.659619\pi$
$462$	0	0
$463$	−32.1040	−1.49200	−0.745999	−	0.665947i	$-0.768028\pi$
$463$	−32.1040	−1.49200	−0.745999	+	0.665947i	$0.768028\pi$
$464$	6.50682	0.302071
$465$	0	0
$466$	31.6954	1.46826
$467$	−23.3774	−1.08178	−0.540888	−	0.841095i	$-0.681912\pi$
$467$	−23.3774	−1.08178	−0.540888	+	0.841095i	$0.681912\pi$
$468$	0	0
$469$	24.9700	1.15301
$470$	−9.74761	−0.449624
$471$	0	0
$472$	23.0050	1.05889
$473$	3.80584	0.174993
$474$	0	0
$475$	5.73205	0.263005
$476$	−4.06338	−0.186245
$477$	0	0
$478$	22.2787	1.01900
$479$	5.17534	0.236467	0.118234	−	0.992986i	$-0.462277\pi$
$479$	5.17534	0.236467	0.118234	+	0.992986i	$0.462277\pi$
$480$	0	0
$481$	0	0
$482$	14.0453	0.639747
$483$	0	0
$484$	−2.33147	−0.105976
$485$	−3.43870	−0.156143
$486$	0	0
$487$	−30.7729	−1.39445	−0.697227	−	0.716850i	$-0.745583\pi$
$487$	−30.7729	−1.39445	−0.697227	+	0.716850i	$0.745583\pi$
$488$	−16.7132	−0.756572
$489$	0	0
$490$	24.3743	1.10112
$491$	−35.7983	−1.61556	−0.807778	−	0.589487i	$-0.799330\pi$
$491$	−35.7983	−1.61556	−0.807778	+	0.589487i	$0.799330\pi$
$492$	0	0
$493$	5.24255	0.236113
$494$	0	0
$495$	0	0
$496$	6.46713	0.290383
$497$	37.4798	1.68120
$498$	0	0
$499$	28.8971	1.29361	0.646805	−	0.762655i	$-0.276105\pi$
$499$	28.8971	1.29361	0.646805	+	0.762655i	$0.276105\pi$
$500$	0.236543	0.0105785
$501$	0	0
$502$	−16.9237	−0.755340
$503$	7.86321	0.350603	0.175302	−	0.984515i	$-0.443910\pi$
$503$	7.86321	0.350603	0.175302	+	0.984515i	$0.443910\pi$
$504$	0	0
$505$	−2.85527	−0.127058
$506$	11.3322	0.503777
$507$	0	0
$508$	0.167154	0.00741625
$509$	28.0113	1.24158	0.620790	−	0.783977i	$-0.286812\pi$
$509$	28.0113	1.24158	0.620790	+	0.783977i	$0.286812\pi$
$510$	0	0
$511$	48.7935	2.15850
$512$	−17.4176	−0.769757
$513$	0	0
$514$	39.6787	1.75015
$515$	−5.54488	−0.244337
$516$	0	0
$517$	−6.97020	−0.306549
$518$	−0.183339	−0.00805546
$519$	0	0
$520$	0	0
$521$	37.5609	1.64557	0.822786	−	0.568351i	$-0.192419\pi$
$521$	37.5609	1.64557	0.822786	+	0.568351i	$0.192419\pi$
$522$	0	0
$523$	−45.3106	−1.98129	−0.990647	−	0.136450i	$-0.956431\pi$
$523$	−45.3106	−1.98129	−0.990647	+	0.136450i	$0.956431\pi$
$524$	−1.48207	−0.0647446
$525$	0	0
$526$	21.1476	0.922080
$527$	5.21058	0.226976
$528$	0	0
$529$	27.2086	1.18298
$530$	−1.48207	−0.0643770
$531$	0	0
$532$	6.54460	0.283744
$533$	0	0
$534$	0	0
$535$	4.44111	0.192006
$536$	−13.6430	−0.589287
$537$	0	0
$538$	−37.0504	−1.59736
$539$	17.4293	0.750733
$540$	0	0
$541$	19.7445	0.848882	0.424441	−	0.905456i	$-0.360471\pi$
$541$	19.7445	0.848882	0.424441	+	0.905456i	$0.360471\pi$
$542$	27.9916	1.20234
$543$	0	0
$544$	4.73806	0.203143
$545$	13.7804	0.590289
$546$	0	0
$547$	−11.8312	−0.505867	−0.252934	−	0.967484i	$-0.581395\pi$
$547$	−11.8312	−0.505867	−0.252934	+	0.967484i	$0.581395\pi$
$548$	−3.85702	−0.164764
$549$	0	0
$550$	1.59928	0.0681935
$551$	−8.44381	−0.359718
$552$	0	0
$553$	42.3964	1.80288
$554$	−33.8952	−1.44007
$555$	0	0
$556$	−1.61447	−0.0684689
$557$	4.04621	0.171443	0.0857217	−	0.996319i	$-0.472680\pi$
$557$	4.04621	0.171443	0.0857217	+	0.996319i	$0.472680\pi$
$558$	0	0
$559$	0	0
$560$	−21.3208	−0.900968
$561$	0	0
$562$	41.6326	1.75617
$563$	−3.89926	−0.164334	−0.0821671	−	0.996619i	$-0.526184\pi$
$563$	−3.89926	−0.164334	−0.0821671	+	0.996619i	$0.526184\pi$
$564$	0	0
$565$	8.04399	0.338413
$566$	11.8445	0.497863
$567$	0	0
$568$	−20.4780	−0.859239
$569$	17.3356	0.726744	0.363372	−	0.931644i	$-0.381625\pi$
$569$	17.3356	0.726744	0.363372	+	0.931644i	$0.381625\pi$
$570$	0	0
$571$	29.5118	1.23503	0.617515	−	0.786559i	$-0.288140\pi$
$571$	29.5118	1.23503	0.617515	+	0.786559i	$0.288140\pi$
$572$	0	0
$573$	0	0
$574$	−1.93421	−0.0807324
$575$	7.08580	0.295498
$576$	0	0
$577$	−28.3684	−1.18099	−0.590496	−	0.807041i	$-0.701068\pi$
$577$	−28.3684	−1.18099	−0.590496	+	0.807041i	$0.701068\pi$
$578$	−6.48199	−0.269615
$579$	0	0
$580$	−0.348448	−0.0144685
$581$	3.50174	0.145277
$582$	0	0
$583$	−1.05978	−0.0438917
$584$	−26.6595	−1.10318
$585$	0	0
$586$	0.408230	0.0168638
$587$	34.3877	1.41933	0.709667	−	0.704538i	$-0.248846\pi$
$587$	34.3877	1.41933	0.709667	+	0.704538i	$0.248846\pi$
$588$	0	0
$589$	−8.39230	−0.345799
$590$	−13.0454	−0.537071
$591$	0	0
$592$	0.112187	0.00461088
$593$	5.47612	0.224877	0.112439	−	0.993659i	$-0.464134\pi$
$593$	5.47612	0.224877	0.112439	+	0.993659i	$0.464134\pi$
$594$	0	0
$595$	−17.1782	−0.704237
$596$	1.99632	0.0817724
$597$	0	0
$598$	0	0
$599$	−38.6039	−1.57731	−0.788657	−	0.614833i	$-0.789223\pi$
$599$	−38.6039	−1.57731	−0.788657	+	0.614833i	$0.789223\pi$
$600$	0	0
$601$	6.57896	0.268361	0.134181	−	0.990957i	$-0.457160\pi$
$601$	6.57896	0.268361	0.134181	+	0.990957i	$0.457160\pi$
$602$	25.6901	1.04705
$603$	0	0
$604$	−0.324103	−0.0131876
$605$	−9.85641	−0.400720
$606$	0	0
$607$	−16.7664	−0.680525	−0.340263	−	0.940330i	$-0.610516\pi$
$607$	−16.7664	−0.680525	−0.340263	+	0.940330i	$0.610516\pi$
$608$	−7.63126	−0.309488
$609$	0	0
$610$	9.47754	0.383734
$611$	0	0
$612$	0	0
$613$	−28.7789	−1.16237	−0.581184	−	0.813772i	$-0.697410\pi$
$613$	−28.7789	−1.16237	−0.581184	+	0.813772i	$0.697410\pi$
$614$	10.2476	0.413558
$615$	0	0
$616$	−13.6129	−0.548481
$617$	−37.3291	−1.50281	−0.751406	−	0.659840i	$-0.770624\pi$
$617$	−37.3291	−1.50281	−0.751406	+	0.659840i	$0.770624\pi$
$618$	0	0
$619$	−12.7535	−0.512606	−0.256303	−	0.966597i	$-0.582504\pi$
$619$	−12.7535	−0.512606	−0.256303	+	0.966597i	$0.582504\pi$
$620$	−0.346323	−0.0139087
$621$	0	0
$622$	−15.9192	−0.638303
$623$	−65.1939	−2.61194
$624$	0	0
$625$	1.00000	0.0400000
$626$	26.6553	1.06536
$627$	0	0
$628$	2.83143	0.112986
$629$	0.0903896	0.00360407
$630$	0	0
$631$	−28.7242	−1.14349	−0.571746	−	0.820430i	$-0.693734\pi$
$631$	−28.7242	−1.14349	−0.571746	+	0.820430i	$0.693734\pi$
$632$	−23.1643	−0.921427
$633$	0	0
$634$	−12.2207	−0.485346
$635$	0.706653	0.0280427
$636$	0	0
$637$	0	0
$638$	−2.35588	−0.0932701
$639$	0	0
$640$	12.8968	0.509791
$641$	22.3970	0.884630	0.442315	−	0.896860i	$-0.354157\pi$
$641$	22.3970	0.884630	0.442315	+	0.896860i	$0.354157\pi$
$642$	0	0
$643$	14.1642	0.558581	0.279290	−	0.960207i	$-0.409901\pi$
$643$	14.1642	0.558581	0.279290	+	0.960207i	$0.409901\pi$
$644$	8.09025	0.318801
$645$	0	0
$646$	−30.5080	−1.20032
$647$	23.6097	0.928193	0.464096	−	0.885785i	$-0.346379\pi$
$647$	23.6097	0.928193	0.464096	+	0.885785i	$0.346379\pi$
$648$	0	0
$649$	−9.32835	−0.366170
$650$	0	0
$651$	0	0
$652$	−5.33910	−0.209095
$653$	−33.2765	−1.30221	−0.651105	−	0.758987i	$-0.725694\pi$
$653$	−33.2765	−1.30221	−0.651105	+	0.758987i	$0.725694\pi$
$654$	0	0
$655$	−6.26554	−0.244815
$656$	1.18357	0.0462105
$657$	0	0
$658$	−47.0502	−1.83421
$659$	−23.0908	−0.899491	−0.449745	−	0.893157i	$-0.648485\pi$
$659$	−23.0908	−0.899491	−0.449745	+	0.893157i	$0.648485\pi$
$660$	0	0
$661$	13.4365	0.522620	0.261310	−	0.965255i	$-0.415845\pi$
$661$	13.4365	0.522620	0.261310	+	0.965255i	$0.415845\pi$
$662$	37.2972	1.44960
$663$	0	0
$664$	−1.91326	−0.0742491
$665$	27.6677	1.07291
$666$	0	0
$667$	−10.4380	−0.404161
$668$	−1.93895	−0.0750200
$669$	0	0
$670$	7.73650	0.298887
$671$	6.77708	0.261626
$672$	0	0
$673$	−1.94524	−0.0749835	−0.0374918	−	0.999297i	$-0.511937\pi$
$673$	−1.94524	−0.0749835	−0.0374918	+	0.999297i	$0.511937\pi$
$674$	−29.3205	−1.12938
$675$	0	0
$676$	0	0
$677$	−24.8683	−0.955768	−0.477884	−	0.878423i	$-0.658596\pi$
$677$	−24.8683	−0.955768	−0.477884	+	0.878423i	$0.658596\pi$
$678$	0	0
$679$	−16.5981	−0.636975
$680$	9.38573	0.359926
$681$	0	0
$682$	−2.34151	−0.0896610
$683$	−14.6221	−0.559500	−0.279750	−	0.960073i	$-0.590252\pi$
$683$	−14.6221	−0.559500	−0.279750	+	0.960073i	$0.590252\pi$
$684$	0	0
$685$	−16.3058	−0.623013
$686$	67.1210	2.56269
$687$	0	0
$688$	−15.7201	−0.599323
$689$	0	0
$690$	0	0
$691$	3.52451	0.134079	0.0670393	−	0.997750i	$-0.478645\pi$
$691$	3.52451	0.134079	0.0670393	+	0.997750i	$0.478645\pi$
$692$	2.16856	0.0824364
$693$	0	0
$694$	−25.5447	−0.969665
$695$	−6.82528	−0.258898
$696$	0	0
$697$	0.953601	0.0361202
$698$	−42.4304	−1.60602
$699$	0	0
$700$	1.14176	0.0431543
$701$	1.53457	0.0579599	0.0289800	−	0.999580i	$-0.490774\pi$
$701$	1.53457	0.0579599	0.0289800	+	0.999580i	$0.490774\pi$
$702$	0	0
$703$	−0.145584	−0.00549081
$704$	7.31810	0.275811
$705$	0	0
$706$	−31.7825	−1.19615
$707$	−13.7819	−0.518322
$708$	0	0
$709$	−14.0052	−0.525978	−0.262989	−	0.964799i	$-0.584708\pi$
$709$	−14.0052	−0.525978	−0.262989	+	0.964799i	$0.584708\pi$
$710$	11.6124	0.435807
$711$	0	0
$712$	35.6203	1.33493
$713$	−10.3743	−0.388522
$714$	0	0
$715$	0	0
$716$	2.37418	0.0887274
$717$	0	0
$718$	−48.8312	−1.82236
$719$	−22.4761	−0.838218	−0.419109	−	0.907936i	$-0.637657\pi$
$719$	−22.4761	−0.838218	−0.419109	+	0.907936i	$0.637657\pi$
$720$	0	0
$721$	−26.7643	−0.996753
$722$	20.7224	0.771206
$723$	0	0
$724$	4.02687	0.149658
$725$	−1.47309	−0.0547091
$726$	0	0
$727$	−10.3421	−0.383566	−0.191783	−	0.981437i	$-0.561427\pi$
$727$	−10.3421	−0.383566	−0.191783	+	0.981437i	$0.561427\pi$
$728$	0	0
$729$	0	0
$730$	15.1178	0.559534
$731$	−12.6657	−0.468458
$732$	0	0
$733$	−27.3533	−1.01032	−0.505159	−	0.863026i	$-0.668566\pi$
$733$	−27.3533	−1.01032	−0.505159	+	0.863026i	$0.668566\pi$
$734$	−8.85096	−0.326695
$735$	0	0
$736$	−9.43355	−0.347725
$737$	5.53212	0.203778
$738$	0	0
$739$	13.4517	0.494827	0.247413	−	0.968910i	$-0.420419\pi$
$739$	13.4517	0.494827	0.247413	+	0.968910i	$0.420419\pi$
$740$	−0.00600778	−0.000220851 0
$741$	0	0
$742$	−7.15372	−0.262621
$743$	16.3926	0.601388	0.300694	−	0.953721i	$-0.402782\pi$
$743$	16.3926	0.601388	0.300694	+	0.953721i	$0.402782\pi$
$744$	0	0
$745$	8.43955	0.309201
$746$	−19.9179	−0.729248
$747$	0	0
$748$	−0.900245	−0.0329162
$749$	21.4365	0.783274
$750$	0	0
$751$	−27.6655	−1.00953	−0.504764	−	0.863257i	$-0.668421\pi$
$751$	−27.6655	−1.00953	−0.504764	+	0.863257i	$0.668421\pi$
$752$	28.7906	1.04988
$753$	0	0
$754$	0	0
$755$	−1.37017	−0.0498654
$756$	0	0
$757$	22.9978	0.835870	0.417935	−	0.908477i	$-0.362754\pi$
$757$	22.9978	0.835870	0.417935	+	0.908477i	$0.362754\pi$
$758$	38.0136	1.38072
$759$	0	0
$760$	−15.1169	−0.548349
$761$	7.66442	0.277835	0.138918	−	0.990304i	$-0.455638\pi$
$761$	7.66442	0.277835	0.138918	+	0.990304i	$0.455638\pi$
$762$	0	0
$763$	66.5160	2.40804
$764$	0.917174	0.0331822
$765$	0	0
$766$	16.1916	0.585027
$767$	0	0
$768$	0	0
$769$	−7.23095	−0.260755	−0.130377	−	0.991464i	$-0.541619\pi$
$769$	−7.23095	−0.260755	−0.130377	+	0.991464i	$0.541619\pi$
$770$	7.71947	0.278190
$771$	0	0
$772$	0.296612	0.0106753
$773$	33.5995	1.20849	0.604246	−	0.796798i	$-0.293475\pi$
$773$	33.5995	1.20849	0.604246	+	0.796798i	$0.293475\pi$
$774$	0	0
$775$	−1.46410	−0.0525921
$776$	9.06877	0.325550
$777$	0	0
$778$	34.4520	1.23516
$779$	−1.53590	−0.0550293
$780$	0	0
$781$	8.30368	0.297129
$782$	−37.7131	−1.34862
$783$	0	0
$784$	−71.9921	−2.57115
$785$	11.9700	0.427228
$786$	0	0
$787$	−2.97168	−0.105929	−0.0529645	−	0.998596i	$-0.516867\pi$
$787$	−2.97168	−0.105929	−0.0529645	+	0.998596i	$0.516867\pi$
$788$	3.61484	0.128773
$789$	0	0
$790$	13.1357	0.467349
$791$	38.8270	1.38053
$792$	0	0
$793$	0	0
$794$	31.5349	1.11913
$795$	0	0
$796$	−3.12937	−0.110918
$797$	22.5751	0.799650	0.399825	−	0.916591i	$-0.369071\pi$
$797$	22.5751	0.799650	0.399825	+	0.916591i	$0.369071\pi$
$798$	0	0
$799$	23.1966	0.820636
$800$	−1.33133	−0.0470696
$801$	0	0
$802$	29.5765	1.04438
$803$	10.8102	0.381485
$804$	0	0
$805$	34.2020	1.20546
$806$	0	0
$807$	0	0
$808$	7.53009	0.264908
$809$	−13.6584	−0.480204	−0.240102	−	0.970748i	$-0.577181\pi$
$809$	−13.6584	−0.480204	−0.240102	+	0.970748i	$0.577181\pi$
$810$	0	0
$811$	14.1147	0.495636	0.247818	−	0.968807i	$-0.420287\pi$
$811$	14.1147	0.495636	0.247818	+	0.968807i	$0.420287\pi$
$812$	−1.68190	−0.0590233
$813$	0	0
$814$	−0.0406189	−0.00142369
$815$	−22.5713	−0.790640
$816$	0	0
$817$	20.3997	0.713696
$818$	−47.6781	−1.66703
$819$	0	0
$820$	−0.0633815	−0.00221338
$821$	−1.58562	−0.0553383	−0.0276692	−	0.999617i	$-0.508808\pi$
$821$	−1.58562	−0.0553383	−0.0276692	+	0.999617i	$0.508808\pi$
$822$	0	0
$823$	18.5648	0.647127	0.323563	−	0.946206i	$-0.395119\pi$
$823$	18.5648	0.647127	0.323563	+	0.946206i	$0.395119\pi$
$824$	14.6233	0.509427
$825$	0	0
$826$	−62.9681	−2.19094
$827$	−9.01023	−0.313316	−0.156658	−	0.987653i	$-0.550072\pi$
$827$	−9.01023	−0.313316	−0.156658	+	0.987653i	$0.550072\pi$
$828$	0	0
$829$	47.1177	1.63647	0.818233	−	0.574887i	$-0.194954\pi$
$829$	47.1177	1.63647	0.818233	+	0.574887i	$0.194954\pi$
$830$	1.08495	0.0376592
$831$	0	0
$832$	0	0
$833$	−58.0041	−2.00972
$834$	0	0
$835$	−8.19700	−0.283669
$836$	1.44996	0.0501479
$837$	0	0
$838$	−45.9564	−1.58754
$839$	53.4766	1.84622	0.923108	−	0.384541i	$-0.125640\pi$
$839$	53.4766	1.84622	0.923108	+	0.384541i	$0.125640\pi$
$840$	0	0
$841$	−26.8300	−0.925173
$842$	26.8923	0.926769
$843$	0	0
$844$	−1.13787	−0.0391672
$845$	0	0
$846$	0	0
$847$	−47.5753	−1.63471
$848$	4.37745	0.150322
$849$	0	0
$850$	−5.32235	−0.182555
$851$	−0.179967	−0.00616919
$852$	0	0
$853$	27.7756	0.951019	0.475510	−	0.879711i	$-0.342264\pi$
$853$	27.7756	0.951019	0.475510	+	0.879711i	$0.342264\pi$
$854$	45.7465	1.56541
$855$	0	0
$856$	−11.7124	−0.400321
$857$	−53.6917	−1.83407	−0.917037	−	0.398801i	$-0.869426\pi$
$857$	−53.6917	−1.83407	−0.917037	+	0.398801i	$0.869426\pi$
$858$	0	0
$859$	2.08958	0.0712955	0.0356477	−	0.999364i	$-0.488651\pi$
$859$	2.08958	0.0712955	0.0356477	+	0.999364i	$0.488651\pi$
$860$	0.841831	0.0287062
$861$	0	0
$862$	−7.32722	−0.249566
$863$	−1.75413	−0.0597113	−0.0298557	−	0.999554i	$-0.509505\pi$
$863$	−1.75413	−0.0597113	−0.0298557	+	0.999554i	$0.509505\pi$
$864$	0	0
$865$	9.16772	0.311712
$866$	−28.7727	−0.977737
$867$	0	0
$868$	−1.67165	−0.0567394
$869$	9.39295	0.318634
$870$	0	0
$871$	0	0
$872$	−36.3426	−1.23072
$873$	0	0
$874$	60.7418	2.05462
$875$	4.82684	0.163177
$876$	0	0
$877$	−21.5672	−0.728272	−0.364136	−	0.931346i	$-0.618636\pi$
$877$	−21.5672	−0.728272	−0.364136	+	0.931346i	$0.618636\pi$
$878$	−12.8012	−0.432018
$879$	0	0
$880$	−4.72364	−0.159234
$881$	−25.0263	−0.843158	−0.421579	−	0.906792i	$-0.638524\pi$
$881$	−25.0263	−0.843158	−0.421579	+	0.906792i	$0.638524\pi$
$882$	0	0
$883$	−48.7832	−1.64169	−0.820843	−	0.571154i	$-0.806496\pi$
$883$	−48.7832	−1.64169	−0.820843	+	0.571154i	$0.806496\pi$
$884$	0	0
$885$	0	0
$886$	56.7772	1.90747
$887$	33.7933	1.13467	0.567334	−	0.823488i	$-0.307975\pi$
$887$	33.7933	1.13467	0.567334	+	0.823488i	$0.307975\pi$
$888$	0	0
$889$	3.41090	0.114398
$890$	−20.1991	−0.677076
$891$	0	0
$892$	3.48031	0.116530
$893$	−37.3611	−1.25024
$894$	0	0
$895$	10.0370	0.335500
$896$	62.2508	2.07965
$897$	0	0
$898$	39.9394	1.33279
$899$	2.15675	0.0719316
$900$	0	0
$901$	3.52691	0.117499
$902$	−0.428526	−0.0142683
$903$	0	0
$904$	−21.2141	−0.705571
$905$	17.0238	0.565892
$906$	0	0
$907$	−34.6270	−1.14977	−0.574885	−	0.818234i	$-0.694953\pi$
$907$	−34.6270	−1.14977	−0.574885	+	0.818234i	$0.694953\pi$
$908$	−3.52516	−0.116986
$909$	0	0
$910$	0	0
$911$	−31.1865	−1.03326	−0.516628	−	0.856210i	$-0.672813\pi$
$911$	−31.1865	−1.03326	−0.516628	+	0.856210i	$0.672813\pi$
$912$	0	0
$913$	0.775814	0.0256757
$914$	−6.38771	−0.211287
$915$	0	0
$916$	−4.56702	−0.150899
$917$	−30.2428	−0.998704
$918$	0	0
$919$	−51.9220	−1.71275	−0.856374	−	0.516356i	$-0.827288\pi$
$919$	−51.9220	−1.71275	−0.856374	+	0.516356i	$0.827288\pi$
$920$	−18.6871	−0.616096
$921$	0	0
$922$	−30.8707	−1.01667
$923$	0	0
$924$	0	0
$925$	−0.0253983	−0.000835090 0
$926$	−48.0117	−1.57776
$927$	0	0
$928$	1.96117	0.0643784
$929$	20.4915	0.672304	0.336152	−	0.941808i	$-0.390874\pi$
$929$	20.4915	0.672304	0.336152	+	0.941808i	$0.390874\pi$
$930$	0	0
$931$	93.4231	3.06182
$932$	5.01324	0.164214
$933$	0	0
$934$	−34.9610	−1.14396
$935$	−3.80584	−0.124464
$936$	0	0
$937$	−39.6806	−1.29631	−0.648154	−	0.761510i	$-0.724459\pi$
$937$	−39.6806	−1.29631	−0.648154	+	0.761510i	$0.724459\pi$
$938$	37.3428	1.21929
$939$	0	0
$940$	−1.54177	−0.0502870
$941$	−19.6189	−0.639557	−0.319779	−	0.947492i	$-0.603609\pi$
$941$	−19.6189	−0.639557	−0.319779	+	0.947492i	$0.603609\pi$
$942$	0	0
$943$	−1.89864	−0.0618281
$944$	38.5309	1.25408
$945$	0	0
$946$	5.69166	0.185052
$947$	57.2124	1.85915	0.929576	−	0.368631i	$-0.120173\pi$
$947$	57.2124	1.85915	0.929576	+	0.368631i	$0.120173\pi$
$948$	0	0
$949$	0	0
$950$	8.57233	0.278123
$951$	0	0
$952$	45.3034	1.46829
$953$	−27.4770	−0.890066	−0.445033	−	0.895514i	$-0.646808\pi$
$953$	−27.4770	−0.890066	−0.445033	+	0.895514i	$0.646808\pi$
$954$	0	0
$955$	3.87741	0.125470
$956$	3.52380	0.113968
$957$	0	0
$958$	7.73976	0.250060
$959$	−78.7055	−2.54153
$960$	0	0
$961$	−28.8564	−0.930852
$962$	0	0
$963$	0	0
$964$	2.22154	0.0715509
$965$	1.25394	0.0403659
$966$	0	0
$967$	10.3643	0.333293	0.166647	−	0.986017i	$-0.446706\pi$
$967$	10.3643	0.333293	0.166647	+	0.986017i	$0.446706\pi$
$968$	25.9939	0.835477
$969$	0	0
$970$	−5.14261	−0.165119
$971$	−41.7515	−1.33987	−0.669935	−	0.742420i	$-0.733678\pi$
$971$	−41.7515	−1.33987	−0.669935	+	0.742420i	$0.733678\pi$
$972$	0	0
$973$	−32.9445	−1.05615
$974$	−46.0212	−1.47461
$975$	0	0
$976$	−27.9929	−0.896030
$977$	−13.7938	−0.441303	−0.220652	−	0.975353i	$-0.570818\pi$
$977$	−13.7938	−0.441303	−0.220652	+	0.975353i	$0.570818\pi$
$978$	0	0
$979$	−14.4437	−0.461624
$980$	3.85527	0.123152
$981$	0	0
$982$	−53.5367	−1.70842
$983$	37.9997	1.21200	0.606002	−	0.795463i	$-0.292773\pi$
$983$	37.9997	1.21200	0.606002	+	0.795463i	$0.292773\pi$
$984$	0	0
$985$	15.2820	0.486924
$986$	7.84028	0.249685
$987$	0	0
$988$	0	0
$989$	25.2176	0.801873
$990$	0	0
$991$	−52.5530	−1.66940	−0.834700	−	0.550705i	$-0.814359\pi$
$991$	−52.5530	−1.66940	−0.834700	+	0.550705i	$0.814359\pi$
$992$	1.94920	0.0618873
$993$	0	0
$994$	56.0513	1.77784
$995$	−13.2296	−0.419407
$996$	0	0
$997$	18.5899	0.588749	0.294375	−	0.955690i	$-0.404889\pi$
$997$	18.5899	0.588749	0.294375	+	0.955690i	$0.404889\pi$
$998$	43.2158	1.36797
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7605.2.a.cf.1.4		4
3.2	odd	2		845.2.a.m.1.1		4
13.6	odd	12		585.2.bu.c.361.1		8
13.11	odd	12		585.2.bu.c.316.1		8
13.12	even	2		7605.2.a.cj.1.1		4
15.14	odd	2		4225.2.a.bi.1.4		4
39.2	even	12		845.2.m.g.316.1		8
39.5	even	4		845.2.c.g.506.7		8
39.8	even	4		845.2.c.g.506.2		8
39.11	even	12		65.2.m.a.56.4	yes	8
39.17	odd	6		845.2.e.n.146.1		8
39.20	even	12		845.2.m.g.361.1		8
39.23	odd	6		845.2.e.n.191.1		8
39.29	odd	6		845.2.e.m.191.4		8
39.32	even	12		65.2.m.a.36.4	✓	8
39.35	odd	6		845.2.e.m.146.4		8
39.38	odd	2		845.2.a.l.1.4		4
156.11	odd	12		1040.2.da.b.641.3		8
156.71	odd	12		1040.2.da.b.881.3		8
195.32	odd	12		325.2.m.b.49.4		8
195.89	even	12		325.2.n.d.251.1		8
195.128	odd	12		325.2.m.b.199.4		8
195.149	even	12		325.2.n.d.101.1		8
195.167	odd	12		325.2.m.c.199.1		8
195.188	odd	12		325.2.m.c.49.1		8
195.194	odd	2		4225.2.a.bl.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.2.m.a.36.4	✓	8	39.32	even	12
65.2.m.a.56.4	yes	8	39.11	even	12
325.2.m.b.49.4		8	195.32	odd	12
325.2.m.b.199.4		8	195.128	odd	12
325.2.m.c.49.1		8	195.188	odd	12
325.2.m.c.199.1		8	195.167	odd	12
325.2.n.d.101.1		8	195.149	even	12
325.2.n.d.251.1		8	195.89	even	12
585.2.bu.c.316.1		8	13.11	odd	12
585.2.bu.c.361.1		8	13.6	odd	12
845.2.a.l.1.4		4	39.38	odd	2
845.2.a.m.1.1		4	3.2	odd	2
845.2.c.g.506.2		8	39.8	even	4
845.2.c.g.506.7		8	39.5	even	4
845.2.e.m.146.4		8	39.35	odd	6
845.2.e.m.191.4		8	39.29	odd	6
845.2.e.n.146.1		8	39.17	odd	6
845.2.e.n.191.1		8	39.23	odd	6
845.2.m.g.316.1		8	39.2	even	12
845.2.m.g.361.1		8	39.20	even	12
1040.2.da.b.641.3		8	156.11	odd	12
1040.2.da.b.881.3		8	156.71	odd	12
4225.2.a.bi.1.4		4	15.14	odd	2
4225.2.a.bl.1.1		4	195.194	odd	2
7605.2.a.cf.1.4		4	1.1	even	1	trivial
7605.2.a.cj.1.1		4	13.12	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 7605 = 3^{2} \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7605.a (trivial)

\(f(q)\)	\(=\)	\(q+1.49551 q^{2} +0.236543 q^{4} +1.00000 q^{5} +4.82684 q^{7} -2.63726 q^{8} +1.49551 q^{10} +1.06939 q^{11} +7.21857 q^{14} -4.41713 q^{16} -3.55889 q^{17} +5.73205 q^{19} +0.236543 q^{20} +1.59928 q^{22} +7.08580 q^{23} +1.00000 q^{25} +1.14176 q^{28} -1.47309 q^{29} -1.46410 q^{31} -1.33133 q^{32} -5.32235 q^{34} +4.82684 q^{35} -0.0253983 q^{37} +8.57233 q^{38} -2.63726 q^{40} -0.267949 q^{41} +3.55889 q^{43} +0.252957 q^{44} +10.5969 q^{46} -6.51793 q^{47} +16.2984 q^{49} +1.49551 q^{50} -0.991015 q^{53} +1.06939 q^{55} -12.7296 q^{56} -2.20301 q^{58} -8.72307 q^{59} +6.33734 q^{61} -2.18958 q^{62} +6.84325 q^{64} +5.17316 q^{67} -0.841831 q^{68} +7.21857 q^{70} +7.76488 q^{71} +10.1088 q^{73} -0.0379833 q^{74} +1.35588 q^{76} +5.16177 q^{77} +8.78347 q^{79} -4.41713 q^{80} -0.400720 q^{82} +0.725474 q^{83} -3.55889 q^{85} +5.32235 q^{86} -2.82026 q^{88} -13.5065 q^{89} +1.67610 q^{92} -9.74761 q^{94} +5.73205 q^{95} -3.43870 q^{97} +24.3743 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} + 2 q^{4} + 4 q^{5} + 10 q^{7} - 6 q^{8} - 2 q^{10} - 2 q^{14} + 2 q^{16} + 2 q^{17} + 16 q^{19} + 2 q^{20} + 12 q^{22} + 10 q^{23} + 4 q^{25} + 8 q^{28} - 8 q^{29} + 8 q^{31} - 4 q^{32} - 4 q^{34}+ \cdots + 24 q^{98}+O(q^{100}) \) 4 * q - 2 * q^2 + 2 * q^4 + 4 * q^5 + 10 * q^7 - 6 * q^8 - 2 * q^10 - 2 * q^14 + 2 * q^16 + 2 * q^17 + 16 * q^19 + 2 * q^20 + 12 * q^22 + 10 * q^23 + 4 * q^25 + 8 * q^28 - 8 * q^29 + 8 * q^31 - 4 * q^32 - 4 * q^34 + 10 * q^35 - 2 * q^37 - 8 * q^38 - 6 * q^40 - 8 * q^41 - 2 * q^43 - 12 * q^44 + 16 * q^46 - 8 * q^47 + 12 * q^49 - 2 * q^50 + 12 * q^53 - 12 * q^56 + 22 * q^58 - 12 * q^59 + 28 * q^61 - 4 * q^62 + 4 * q^64 + 30 * q^67 - 14 * q^68 - 2 * q^70 - 4 * q^71 - 8 * q^73 + 10 * q^74 + 20 * q^76 - 18 * q^77 - 8 * q^79 + 2 * q^80 + 4 * q^82 + 12 * q^83 + 2 * q^85 + 4 * q^86 - 18 * q^88 + 12 * q^89 - 22 * q^92 - 32 * q^94 + 16 * q^95 + 2 * q^97 + 24 * q^98

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