LMFDB - Embedded newform 7605.2.a.cn.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7605,2,Mod(1,7605)]

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

lf = mfeigenbasis(mf)

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")

//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);

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

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

Self dual:	yes
Analytic conductor:	$60.7262307372$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.3352656.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 6x^{3} + 10x^{2} + 6x - 6 $ x^5 - 2x^4 - 6x^3 + 10x^2 + 6x - 6
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 585)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.12283$ 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)

$f(q)$	$=$	$q-2.12283 q^{2} +2.50640 q^{4} +1.00000 q^{5} -1.46707 q^{7} -1.07500 q^{8} +O(q^{10})$	$q-2.12283 q^{2} +2.50640 q^{4} +1.00000 q^{5} -1.46707 q^{7} -1.07500 q^{8} -2.12283 q^{10} +0.344239 q^{11} +3.11433 q^{14} -2.73076 q^{16} -5.13563 q^{17} -0.506400 q^{19} +2.50640 q^{20} -0.730760 q^{22} -1.07500 q^{23} +1.00000 q^{25} -3.67706 q^{28} -8.04847 q^{29} -9.17129 q^{31} +7.94693 q^{32} +10.9021 q^{34} -1.46707 q^{35} -2.73076 q^{37} +1.07500 q^{38} -1.07500 q^{40} +6.12713 q^{41} -5.87488 q^{43} +0.862801 q^{44} +2.28204 q^{46} +4.26369 q^{47} -4.84771 q^{49} -2.12283 q^{50} +4.07866 q^{53} +0.344239 q^{55} +1.57710 q^{56} +17.0855 q^{58} -5.06127 q^{59} -5.17129 q^{61} +19.4691 q^{62} -11.4085 q^{64} -1.19719 q^{67} -12.8719 q^{68} +3.11433 q^{70} +0.445772 q^{71} +7.52634 q^{73} +5.79693 q^{74} -1.26924 q^{76} -0.505022 q^{77} +0.834915 q^{79} -2.73076 q^{80} -13.0068 q^{82} +15.2506 q^{83} -5.13563 q^{85} +12.4714 q^{86} -0.370057 q^{88} +11.9499 q^{89} -2.69438 q^{92} -9.05109 q^{94} -0.506400 q^{95} +1.40056 q^{97} +10.2909 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 2 q^{2} + 6 q^{4} + 5 q^{5} - q^{7} + 6 q^{8}+O(q^{10}) $ 5 * q + 2 * q^2 + 6 * q^4 + 5 * q^5 - q^7 + 6 * q^8	$ 5 q + 2 q^{2} + 6 q^{4} + 5 q^{5} - q^{7} + 6 q^{8} + 2 q^{10} + 8 q^{11} + 4 q^{14} + 4 q^{16} + 4 q^{19} + 6 q^{20} + 14 q^{22} + 6 q^{23} + 5 q^{25} + 2 q^{28} - 16 q^{29} - 9 q^{31} + 14 q^{32} - q^{35} + 4 q^{37} - 6 q^{38} + 6 q^{40} + 6 q^{41} + 15 q^{43} + 14 q^{44} + 16 q^{46} + 10 q^{47} + 10 q^{49} + 2 q^{50} + 20 q^{53} + 8 q^{55} + 2 q^{56} + 4 q^{58} + 12 q^{59} + 11 q^{61} + 22 q^{62} + 4 q^{64} - 5 q^{67} - 50 q^{68} + 4 q^{70} + 10 q^{71} - q^{73} + 26 q^{74} - 24 q^{76} - 42 q^{77} - 17 q^{79} + 4 q^{80} + 16 q^{82} + 16 q^{83} + 44 q^{86} + 20 q^{88} + 4 q^{89} + 34 q^{92} - 16 q^{94} + 4 q^{95} + 11 q^{97} + 30 q^{98}+O(q^{100}) $ 5 * q + 2 * q^2 + 6 * q^4 + 5 * q^5 - q^7 + 6 * q^8 + 2 * q^10 + 8 * q^11 + 4 * q^14 + 4 * q^16 + 4 * q^19 + 6 * q^20 + 14 * q^22 + 6 * q^23 + 5 * q^25 + 2 * q^28 - 16 * q^29 - 9 * q^31 + 14 * q^32 - q^35 + 4 * q^37 - 6 * q^38 + 6 * q^40 + 6 * q^41 + 15 * q^43 + 14 * q^44 + 16 * q^46 + 10 * q^47 + 10 * q^49 + 2 * q^50 + 20 * q^53 + 8 * q^55 + 2 * q^56 + 4 * q^58 + 12 * q^59 + 11 * q^61 + 22 * q^62 + 4 * q^64 - 5 * q^67 - 50 * q^68 + 4 * q^70 + 10 * q^71 - q^73 + 26 * q^74 - 24 * q^76 - 42 * q^77 - 17 * q^79 + 4 * q^80 + 16 * q^82 + 16 * q^83 + 44 * q^86 + 20 * q^88 + 4 * q^89 + 34 * q^92 - 16 * q^94 + 4 * q^95 + 11 * q^97 + 30 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.12283	−1.50107	−0.750533	−	0.660833i	$-0.770203\pi$
$2$	−2.12283	−1.50107	−0.750533	+	0.660833i	$0.770203\pi$
$3$	0	0
$3$	0	0
$4$	2.50640	1.25320
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.46707	−0.554499	−0.277250	−	0.960798i	$-0.589423\pi$
$7$	−1.46707	−0.554499	−0.277250	+	0.960798i	$0.589423\pi$
$8$	−1.07500	−0.380070
$9$	0	0
$10$	−2.12283	−0.671297
$11$	0.344239	0.103792	0.0518960	−	0.998652i	$-0.483474\pi$
$11$	0.344239	0.103792	0.0518960	+	0.998652i	$0.483474\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	3.11433	0.832340
$15$	0	0
$16$	−2.73076	−0.682690
$17$	−5.13563	−1.24557	−0.622786	−	0.782392i	$-0.713999\pi$
$17$	−5.13563	−1.24557	−0.622786	+	0.782392i	$0.713999\pi$
$18$	0	0
$19$	−0.506400	−0.116176	−0.0580880	−	0.998311i	$-0.518500\pi$
$19$	−0.506400	−0.116176	−0.0580880	+	0.998311i	$0.518500\pi$
$20$	2.50640	0.560448
$21$	0	0
$22$	−0.730760	−0.155799
$23$	−1.07500	−0.224153	−0.112076	−	0.993700i	$-0.535750\pi$
$23$	−1.07500	−0.224153	−0.112076	+	0.993700i	$0.535750\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	−3.67706	−0.694898
$29$	−8.04847	−1.49456	−0.747281	−	0.664508i	$-0.768641\pi$
$29$	−8.04847	−1.49456	−0.747281	+	0.664508i	$0.768641\pi$
$30$	0	0
$31$	−9.17129	−1.64721	−0.823607	−	0.567162i	$-0.808041\pi$
$31$	−9.17129	−1.64721	−0.823607	+	0.567162i	$0.808041\pi$
$32$	7.94693	1.40483
$33$	0	0
$34$	10.9021	1.86969
$35$	−1.46707	−0.247980
$36$	0	0
$37$	−2.73076	−0.448934	−0.224467	−	0.974482i	$-0.572064\pi$
$37$	−2.73076	−0.448934	−0.224467	+	0.974482i	$0.572064\pi$
$38$	1.07500	0.174388
$39$	0	0
$40$	−1.07500	−0.169972
$41$	6.12713	0.956897	0.478449	−	0.878116i	$-0.341199\pi$
$41$	6.12713	0.956897	0.478449	+	0.878116i	$0.341199\pi$
$42$	0	0
$43$	−5.87488	−0.895911	−0.447956	−	0.894056i	$-0.647848\pi$
$43$	−5.87488	−0.895911	−0.447956	+	0.894056i	$0.647848\pi$
$44$	0.862801	0.130072
$45$	0	0
$46$	2.28204	0.336468
$47$	4.26369	0.621924	0.310962	−	0.950422i	$-0.399349\pi$
$47$	4.26369	0.621924	0.310962	+	0.950422i	$0.399349\pi$
$48$	0	0
$49$	−4.84771	−0.692531
$50$	−2.12283	−0.300213
$51$	0	0
$52$	0	0
$53$	4.07866	0.560248	0.280124	−	0.959964i	$-0.409624\pi$
$53$	4.07866	0.560248	0.280124	+	0.959964i	$0.409624\pi$
$54$	0	0
$55$	0.344239	0.0464172
$56$	1.57710	0.210748
$57$	0	0
$58$	17.0855	2.24344
$59$	−5.06127	−0.658921	−0.329460	−	0.944169i	$-0.606867\pi$
$59$	−5.06127	−0.658921	−0.329460	+	0.944169i	$0.606867\pi$
$60$	0	0
$61$	−5.17129	−0.662116	−0.331058	−	0.943610i	$-0.607406\pi$
$61$	−5.17129	−0.662116	−0.331058	+	0.943610i	$0.607406\pi$
$62$	19.4691	2.47258
$63$	0	0
$64$	−11.4085	−1.42606
$65$	0	0
$66$	0	0
$67$	−1.19719	−0.146260	−0.0731300	−	0.997322i	$-0.523299\pi$
$67$	−1.19719	−0.146260	−0.0731300	+	0.997322i	$0.523299\pi$
$68$	−12.8719	−1.56095
$69$	0	0
$70$	3.11433	0.372234
$71$	0.445772	0.0529033	0.0264517	−	0.999650i	$-0.491579\pi$
$71$	0.445772	0.0529033	0.0264517	+	0.999650i	$0.491579\pi$
$72$	0	0
$73$	7.52634	0.880892	0.440446	−	0.897779i	$-0.354820\pi$
$73$	7.52634	0.880892	0.440446	+	0.897779i	$0.354820\pi$
$74$	5.79693	0.673880
$75$	0	0
$76$	−1.26924	−0.145592
$77$	−0.505022	−0.0575526
$78$	0	0
$79$	0.834915	0.0939352	0.0469676	−	0.998896i	$-0.485044\pi$
$79$	0.834915	0.0939352	0.0469676	+	0.998896i	$0.485044\pi$
$80$	−2.73076	−0.305308
$81$	0	0
$82$	−13.0068	−1.43637
$83$	15.2506	1.67397	0.836985	−	0.547226i	$-0.184316\pi$
$83$	15.2506	1.67397	0.836985	+	0.547226i	$0.184316\pi$
$84$	0	0
$85$	−5.13563	−0.557037
$86$	12.4714	1.34482
$87$	0	0
$88$	−0.370057	−0.0394482
$89$	11.9499	1.26669	0.633343	−	0.773872i	$-0.281682\pi$
$89$	11.9499	1.26669	0.633343	+	0.773872i	$0.281682\pi$
$90$	0	0
$91$	0	0
$92$	−2.69438	−0.280908
$93$	0	0
$94$	−9.05109	−0.933548
$95$	−0.506400	−0.0519555
$96$	0	0
$97$	1.40056	0.142206	0.0711029	−	0.997469i	$-0.477348\pi$
$97$	1.40056	0.142206	0.0711029	+	0.997469i	$0.477348\pi$
$98$	10.2909	1.03953
$99$	0	0
$100$	2.50640	0.250640
$101$	−10.0940	−1.00439	−0.502196	−	0.864754i	$-0.667474\pi$
$101$	−10.0940	−1.00439	−0.502196	+	0.864754i	$0.667474\pi$
$102$	0	0
$103$	15.5592	1.53309	0.766545	−	0.642190i	$-0.221974\pi$
$103$	15.5592	1.53309	0.766545	+	0.642190i	$0.221974\pi$
$104$	0	0
$105$	0	0
$106$	−8.65830	−0.840969
$107$	−2.11369	−0.204339	−0.102169	−	0.994767i	$-0.532578\pi$
$107$	−2.11369	−0.204339	−0.102169	+	0.994767i	$0.532578\pi$
$108$	0	0
$109$	−4.12514	−0.395117	−0.197558	−	0.980291i	$-0.563301\pi$
$109$	−4.12514	−0.395117	−0.197558	+	0.980291i	$0.563301\pi$
$110$	−0.730760	−0.0696753
$111$	0	0
$112$	4.00621	0.378551
$113$	11.3915	1.07162	0.535809	−	0.844339i	$-0.320007\pi$
$113$	11.3915	1.07162	0.535809	+	0.844339i	$0.320007\pi$
$114$	0	0
$115$	−1.07500	−0.100244
$116$	−20.1727	−1.87299
$117$	0	0
$118$	10.7442	0.989083
$119$	7.53431	0.690669
$120$	0	0
$121$	−10.8815	−0.989227
$122$	10.9778	0.993881
$123$	0	0
$124$	−22.9869	−2.06429
$125$	1.00000	0.0894427
$126$	0	0
$127$	−21.0135	−1.86464	−0.932322	−	0.361630i	$-0.882220\pi$
$127$	−21.0135	−1.86464	−0.932322	+	0.361630i	$0.882220\pi$
$128$	8.32432	0.735773
$129$	0	0
$130$	0	0
$131$	13.4726	1.17711	0.588555	−	0.808457i	$-0.299697\pi$
$131$	13.4726	1.17711	0.588555	+	0.808457i	$0.299697\pi$
$132$	0	0
$133$	0.742922	0.0644195
$134$	2.54143	0.219546
$135$	0	0
$136$	5.52080	0.473404
$137$	10.2020	0.871618	0.435809	−	0.900039i	$-0.356462\pi$
$137$	10.2020	0.871618	0.435809	+	0.900039i	$0.356462\pi$
$138$	0	0
$139$	12.1520	1.03072	0.515360	−	0.856974i	$-0.327658\pi$
$139$	12.1520	1.03072	0.515360	+	0.856974i	$0.327658\pi$
$140$	−3.67706	−0.310768
$141$	0	0
$142$	−0.946296	−0.0794114
$143$	0	0
$144$	0	0
$145$	−8.04847	−0.668389
$146$	−15.9771	−1.32228
$147$	0	0
$148$	−6.84438	−0.562604
$149$	0.667286	0.0546662	0.0273331	−	0.999626i	$-0.491299\pi$
$149$	0.667286	0.0546662	0.0273331	+	0.999626i	$0.491299\pi$
$150$	0	0
$151$	15.9198	1.29553	0.647767	−	0.761839i	$-0.275703\pi$
$151$	15.9198	1.29553	0.647767	+	0.761839i	$0.275703\pi$
$152$	0.544379	0.0441550
$153$	0	0
$154$	1.07207	0.0863902
$155$	−9.17129	−0.736656
$156$	0	0
$157$	17.4611	1.39355	0.696774	−	0.717290i	$-0.254618\pi$
$157$	17.4611	1.39355	0.696774	+	0.717290i	$0.254618\pi$
$158$	−1.77238	−0.141003
$159$	0	0
$160$	7.94693	0.628260
$161$	1.57710	0.124293
$162$	0	0
$163$	−14.5585	−1.14031	−0.570156	−	0.821536i	$-0.693117\pi$
$163$	−14.5585	−1.14031	−0.570156	+	0.821536i	$0.693117\pi$
$164$	15.3570	1.19918
$165$	0	0
$166$	−32.3744	−2.51274
$167$	15.5211	1.20106	0.600530	−	0.799603i	$-0.294956\pi$
$167$	15.5211	1.20106	0.600530	+	0.799603i	$0.294956\pi$
$168$	0	0
$169$	0	0
$170$	10.9021	0.836149
$171$	0	0
$172$	−14.7248	−1.12276
$173$	8.32589	0.633006	0.316503	−	0.948592i	$-0.397491\pi$
$173$	8.32589	0.633006	0.316503	+	0.948592i	$0.397491\pi$
$174$	0	0
$175$	−1.46707	−0.110900
$176$	−0.940034	−0.0708577
$177$	0	0
$178$	−25.3675	−1.90138
$179$	25.2627	1.88822	0.944111	−	0.329629i	$-0.106924\pi$
$179$	25.2627	1.88822	0.944111	+	0.329629i	$0.106924\pi$
$180$	0	0
$181$	−21.6041	−1.60582	−0.802909	−	0.596102i	$-0.796716\pi$
$181$	−21.6041	−1.60582	−0.802909	+	0.596102i	$0.796716\pi$
$182$	0	0
$183$	0	0
$184$	1.15562	0.0851937
$185$	−2.73076	−0.200769
$186$	0	0
$187$	−1.76788	−0.129280
$188$	10.6865	0.779394
$189$	0	0
$190$	1.07500	0.0779886
$191$	−9.31447	−0.673972	−0.336986	−	0.941510i	$-0.609407\pi$
$191$	−9.31447	−0.673972	−0.336986	+	0.941510i	$0.609407\pi$
$192$	0	0
$193$	7.60007	0.547065	0.273533	−	0.961863i	$-0.411808\pi$
$193$	7.60007	0.547065	0.273533	+	0.961863i	$0.411808\pi$
$194$	−2.97316	−0.213460
$195$	0	0
$196$	−12.1503	−0.867879
$197$	10.9433	0.779676	0.389838	−	0.920884i	$-0.372531\pi$
$197$	10.9433	0.779676	0.389838	+	0.920884i	$0.372531\pi$
$198$	0	0
$199$	−14.1968	−1.00638	−0.503192	−	0.864175i	$-0.667841\pi$
$199$	−14.1968	−1.00638	−0.503192	+	0.864175i	$0.667841\pi$
$200$	−1.07500	−0.0760139
$201$	0	0
$202$	21.4278	1.50766
$203$	11.8076	0.828734
$204$	0	0
$205$	6.12713	0.427937
$206$	−33.0294	−2.30127
$207$	0	0
$208$	0	0
$209$	−0.174322	−0.0120581
$210$	0	0
$211$	10.3298	0.711132	0.355566	−	0.934651i	$-0.384288\pi$
$211$	10.3298	0.711132	0.355566	+	0.934651i	$0.384288\pi$
$212$	10.2228	0.702102
$213$	0	0
$214$	4.48701	0.306726
$215$	−5.87488	−0.400664
$216$	0	0
$217$	13.4549	0.913378
$218$	8.75697	0.593097
$219$	0	0
$220$	0.862801	0.0581700
$221$	0	0
$222$	0	0
$223$	−24.5061	−1.64105	−0.820526	−	0.571609i	$-0.806319\pi$
$223$	−24.5061	−1.64105	−0.820526	+	0.571609i	$0.806319\pi$
$224$	−11.6587	−0.778979
$225$	0	0
$226$	−24.1821	−1.60857
$227$	3.00524	0.199465	0.0997323	−	0.995014i	$-0.468201\pi$
$227$	3.00524	0.199465	0.0997323	+	0.995014i	$0.468201\pi$
$228$	0	0
$229$	21.9741	1.45209	0.726046	−	0.687646i	$-0.241356\pi$
$229$	21.9741	1.45209	0.726046	+	0.687646i	$0.241356\pi$
$230$	2.28204	0.150473
$231$	0	0
$232$	8.65210	0.568038
$233$	−7.22372	−0.473242	−0.236621	−	0.971602i	$-0.576040\pi$
$233$	−7.22372	−0.473242	−0.236621	+	0.971602i	$0.576040\pi$
$234$	0	0
$235$	4.26369	0.278133
$236$	−12.6856	−0.825759
$237$	0	0
$238$	−15.9940	−1.03674
$239$	0.492963	0.0318871	0.0159436	−	0.999873i	$-0.494925\pi$
$239$	0.492963	0.0318871	0.0159436	+	0.999873i	$0.494925\pi$
$240$	0	0
$241$	26.2768	1.69264	0.846320	−	0.532675i	$-0.178813\pi$
$241$	26.2768	1.69264	0.846320	+	0.532675i	$0.178813\pi$
$242$	23.0996	1.48490
$243$	0	0
$244$	−12.9613	−0.829764
$245$	−4.84771	−0.309709
$246$	0	0
$247$	0	0
$248$	9.85914	0.626056
$249$	0	0
$250$	−2.12283	−0.134259
$251$	21.5543	1.36049	0.680246	−	0.732984i	$-0.261873\pi$
$251$	21.5543	1.36049	0.680246	+	0.732984i	$0.261873\pi$
$252$	0	0
$253$	−0.370057	−0.0232653
$254$	44.6080	2.79895
$255$	0	0
$256$	5.14580	0.321613
$257$	−12.8121	−0.799194	−0.399597	−	0.916691i	$-0.630850\pi$
$257$	−12.8121	−0.799194	−0.399597	+	0.916691i	$0.630850\pi$
$258$	0	0
$259$	4.00621	0.248934
$260$	0	0
$261$	0	0
$262$	−28.6001	−1.76692
$263$	24.4379	1.50691	0.753454	−	0.657500i	$-0.228386\pi$
$263$	24.4379	1.50691	0.753454	+	0.657500i	$0.228386\pi$
$264$	0	0
$265$	4.07866	0.250550
$266$	−1.57710	−0.0966980
$267$	0	0
$268$	−3.00064	−0.183293
$269$	−23.8268	−1.45274	−0.726372	−	0.687302i	$-0.758795\pi$
$269$	−23.8268	−1.45274	−0.726372	+	0.687302i	$0.758795\pi$
$270$	0	0
$271$	−20.7754	−1.26201	−0.631007	−	0.775777i	$-0.717358\pi$
$271$	−20.7754	−1.26201	−0.631007	+	0.775777i	$0.717358\pi$
$272$	14.0242	0.850340
$273$	0	0
$274$	−21.6571	−1.30836
$275$	0.344239	0.0207584
$276$	0	0
$277$	0.468111	0.0281261	0.0140630	−	0.999901i	$-0.495523\pi$
$277$	0.468111	0.0281261	0.0140630	+	0.999901i	$0.495523\pi$
$278$	−25.7966	−1.54718
$279$	0	0
$280$	1.57710	0.0942495
$281$	−24.5123	−1.46228	−0.731141	−	0.682226i	$-0.761012\pi$
$281$	−24.5123	−1.46228	−0.731141	+	0.682226i	$0.761012\pi$
$282$	0	0
$283$	22.8949	1.36096	0.680481	−	0.732766i	$-0.261771\pi$
$283$	22.8949	1.36096	0.680481	+	0.732766i	$0.261771\pi$
$284$	1.11728	0.0662985
$285$	0	0
$286$	0	0
$287$	−8.98891	−0.530599
$288$	0	0
$289$	9.37467	0.551451
$290$	17.0855	1.00330
$291$	0	0
$292$	18.8640	1.10393
$293$	19.2404	1.12404	0.562019	−	0.827125i	$-0.310025\pi$
$293$	19.2404	1.12404	0.562019	+	0.827125i	$0.310025\pi$
$294$	0	0
$295$	−5.06127	−0.294678
$296$	2.93557	0.170626
$297$	0	0
$298$	−1.41653	−0.0820575
$299$	0	0
$300$	0	0
$301$	8.61885	0.496782
$302$	−33.7950	−1.94468
$303$	0	0
$304$	1.38286	0.0793122
$305$	−5.17129	−0.296107
$306$	0	0
$307$	16.4739	0.940216	0.470108	−	0.882609i	$-0.344215\pi$
$307$	16.4739	0.940216	0.470108	+	0.882609i	$0.344215\pi$
$308$	−1.26579	−0.0721249
$309$	0	0
$310$	19.4691	1.10577
$311$	−9.54562	−0.541283	−0.270641	−	0.962680i	$-0.587236\pi$
$311$	−9.54562	−0.541283	−0.270641	+	0.962680i	$0.587236\pi$
$312$	0	0
$313$	21.9067	1.23824	0.619120	−	0.785297i	$-0.287490\pi$
$313$	21.9067	1.23824	0.619120	+	0.785297i	$0.287490\pi$
$314$	−37.0669	−2.09181
$315$	0	0
$316$	2.09263	0.117720
$317$	−27.1321	−1.52389	−0.761945	−	0.647642i	$-0.775755\pi$
$317$	−27.1321	−1.52389	−0.761945	+	0.647642i	$0.775755\pi$
$318$	0	0
$319$	−2.77060	−0.155124
$320$	−11.4085	−0.637752
$321$	0	0
$322$	−3.34790	−0.186571
$323$	2.60068	0.144706
$324$	0	0
$325$	0	0
$326$	30.9053	1.71168
$327$	0	0
$328$	−6.58666	−0.363688
$329$	−6.25512	−0.344856
$330$	0	0
$331$	−1.13794	−0.0625468	−0.0312734	−	0.999511i	$-0.509956\pi$
$331$	−1.13794	−0.0625468	−0.0312734	+	0.999511i	$0.509956\pi$
$332$	38.2241	2.09782
$333$	0	0
$334$	−32.9486	−1.80287
$335$	−1.19719	−0.0654095
$336$	0	0
$337$	−26.9676	−1.46902	−0.734510	−	0.678598i	$-0.762588\pi$
$337$	−26.9676	−1.46902	−0.734510	+	0.678598i	$0.762588\pi$
$338$	0	0
$339$	0	0
$340$	−12.8719	−0.698079
$341$	−3.15712	−0.170967
$342$	0	0
$343$	17.3814	0.938507
$344$	6.31550	0.340509
$345$	0	0
$346$	−17.6744	−0.950184
$347$	6.53390	0.350758	0.175379	−	0.984501i	$-0.443885\pi$
$347$	6.53390	0.350758	0.175379	+	0.984501i	$0.443885\pi$
$348$	0	0
$349$	1.05768	0.0566162	0.0283081	−	0.999599i	$-0.490988\pi$
$349$	1.05768	0.0566162	0.0283081	+	0.999599i	$0.490988\pi$
$350$	3.11433	0.166468
$351$	0	0
$352$	2.73564	0.145810
$353$	−24.5686	−1.30766	−0.653828	−	0.756643i	$-0.726838\pi$
$353$	−24.5686	−1.30766	−0.653828	+	0.756643i	$0.726838\pi$
$354$	0	0
$355$	0.445772	0.0236591
$356$	29.9512	1.58741
$357$	0	0
$358$	−53.6283	−2.83435
$359$	7.44704	0.393040	0.196520	−	0.980500i	$-0.437036\pi$
$359$	7.44704	0.393040	0.196520	+	0.980500i	$0.437036\pi$
$360$	0	0
$361$	−18.7436	−0.986503
$362$	45.8617	2.41044
$363$	0	0
$364$	0	0
$365$	7.52634	0.393947
$366$	0	0
$367$	−0.0445814	−0.00232713	−0.00116356	−	0.999999i	$-0.500370\pi$
$367$	−0.0445814	−0.00232713	−0.00116356	+	0.999999i	$0.500370\pi$
$368$	2.93557	0.153027
$369$	0	0
$370$	5.79693	0.301368
$371$	−5.98368	−0.310657
$372$	0	0
$373$	−3.34236	−0.173061	−0.0865304	−	0.996249i	$-0.527578\pi$
$373$	−3.34236	−0.173061	−0.0865304	+	0.996249i	$0.527578\pi$
$374$	3.75291	0.194058
$375$	0	0
$376$	−4.58347	−0.236374
$377$	0	0
$378$	0	0
$379$	26.2190	1.34678	0.673391	−	0.739287i	$-0.264837\pi$
$379$	26.2190	1.34678	0.673391	+	0.739287i	$0.264837\pi$
$380$	−1.26924	−0.0651106
$381$	0	0
$382$	19.7730	1.01168
$383$	12.8991	0.659113	0.329557	−	0.944136i	$-0.393101\pi$
$383$	12.8991	0.659113	0.329557	+	0.944136i	$0.393101\pi$
$384$	0	0
$385$	−0.505022	−0.0257383
$386$	−16.1336	−0.821181
$387$	0	0
$388$	3.51037	0.178212
$389$	−5.08578	−0.257859	−0.128930	−	0.991654i	$-0.541154\pi$
$389$	−5.08578	−0.257859	−0.128930	+	0.991654i	$0.541154\pi$
$390$	0	0
$391$	5.52080	0.279199
$392$	5.21129	0.263210
$393$	0	0
$394$	−23.2307	−1.17034
$395$	0.834915	0.0420091
$396$	0	0
$397$	11.0642	0.555296	0.277648	−	0.960683i	$-0.410445\pi$
$397$	11.0642	0.555296	0.277648	+	0.960683i	$0.410445\pi$
$398$	30.1373	1.51065
$399$	0	0
$400$	−2.73076	−0.136538
$401$	−23.0953	−1.15332	−0.576662	−	0.816983i	$-0.695645\pi$
$401$	−23.0953	−1.15332	−0.576662	+	0.816983i	$0.695645\pi$
$402$	0	0
$403$	0	0
$404$	−25.2996	−1.25870
$405$	0	0
$406$	−25.0656	−1.24398
$407$	−0.940034	−0.0465958
$408$	0	0
$409$	−14.4211	−0.713080	−0.356540	−	0.934280i	$-0.616044\pi$
$409$	−14.4211	−0.713080	−0.356540	+	0.934280i	$0.616044\pi$
$410$	−13.0068	−0.642362
$411$	0	0
$412$	38.9975	1.92127
$413$	7.42522	0.365371
$414$	0	0
$415$	15.2506	0.748622
$416$	0	0
$417$	0	0
$418$	0.370057	0.0181001
$419$	15.1685	0.741027	0.370514	−	0.928827i	$-0.379182\pi$
$419$	15.1685	0.741027	0.370514	+	0.928827i	$0.379182\pi$
$420$	0	0
$421$	−10.7707	−0.524932	−0.262466	−	0.964941i	$-0.584536\pi$
$421$	−10.7707	−0.524932	−0.262466	+	0.964941i	$0.584536\pi$
$422$	−21.9284	−1.06746
$423$	0	0
$424$	−4.38456	−0.212933
$425$	−5.13563	−0.249115
$426$	0	0
$427$	7.58664	0.367143
$428$	−5.29776	−0.256077
$429$	0	0
$430$	12.4714	0.601423
$431$	−41.3139	−1.99002	−0.995010	−	0.0997722i	$-0.968189\pi$
$431$	−41.3139	−1.99002	−0.995010	+	0.0997722i	$0.968189\pi$
$432$	0	0
$433$	18.4524	0.886766	0.443383	−	0.896332i	$-0.353778\pi$
$433$	18.4524	0.886766	0.443383	+	0.896332i	$0.353778\pi$
$434$	−28.5625	−1.37104
$435$	0	0
$436$	−10.3393	−0.495160
$437$	0.544379	0.0260412
$438$	0	0
$439$	7.88139	0.376158	0.188079	−	0.982154i	$-0.439774\pi$
$439$	7.88139	0.376158	0.188079	+	0.982154i	$0.439774\pi$
$440$	−0.370057	−0.0176418
$441$	0	0
$442$	0	0
$443$	−38.2769	−1.81859	−0.909294	−	0.416154i	$-0.863378\pi$
$443$	−38.2769	−1.81859	−0.909294	+	0.416154i	$0.863378\pi$
$444$	0	0
$445$	11.9499	0.566479
$446$	52.0223	2.46333
$447$	0	0
$448$	16.7370	0.790747
$449$	12.5444	0.592006	0.296003	−	0.955187i	$-0.404346\pi$
$449$	12.5444	0.592006	0.296003	+	0.955187i	$0.404346\pi$
$450$	0	0
$451$	2.10920	0.0993182
$452$	28.5516	1.34295
$453$	0	0
$454$	−6.37960	−0.299410
$455$	0	0
$456$	0	0
$457$	20.2687	0.948128	0.474064	−	0.880490i	$-0.342786\pi$
$457$	20.2687	0.948128	0.474064	+	0.880490i	$0.342786\pi$
$458$	−46.6473	−2.17969
$459$	0	0
$460$	−2.69438	−0.125626
$461$	12.5001	0.582188	0.291094	−	0.956694i	$-0.405981\pi$
$461$	12.5001	0.582188	0.291094	+	0.956694i	$0.405981\pi$
$462$	0	0
$463$	15.0923	0.701400	0.350700	−	0.936488i	$-0.385944\pi$
$463$	15.0923	0.701400	0.350700	+	0.936488i	$0.385944\pi$
$464$	21.9784	1.02032
$465$	0	0
$466$	15.3347	0.710367
$467$	−14.8821	−0.688662	−0.344331	−	0.938848i	$-0.611894\pi$
$467$	−14.8821	−0.688662	−0.344331	+	0.938848i	$0.611894\pi$
$468$	0	0
$469$	1.75636	0.0811011
$470$	−9.05109	−0.417496
$471$	0	0
$472$	5.44086	0.250436
$473$	−2.02236	−0.0929884
$474$	0	0
$475$	−0.506400	−0.0232352
$476$	18.8840	0.865546
$477$	0	0
$478$	−1.04648	−0.0478647
$479$	0.304161	0.0138975	0.00694873	−	0.999976i	$-0.497788\pi$
$479$	0.304161	0.0138975	0.00694873	+	0.999976i	$0.497788\pi$
$480$	0	0
$481$	0	0
$482$	−55.7812	−2.54076
$483$	0	0
$484$	−27.2734	−1.23970
$485$	1.40056	0.0635964
$486$	0	0
$487$	−41.7163	−1.89035	−0.945174	−	0.326567i	$-0.894108\pi$
$487$	−41.7163	−1.89035	−0.945174	+	0.326567i	$0.894108\pi$
$488$	5.55914	0.251650
$489$	0	0
$490$	10.2909	0.464894
$491$	−40.5595	−1.83043	−0.915213	−	0.402971i	$-0.867978\pi$
$491$	−40.5595	−1.83043	−0.915213	+	0.402971i	$0.867978\pi$
$492$	0	0
$493$	41.3339	1.86159
$494$	0	0
$495$	0	0
$496$	25.0446	1.12454
$497$	−0.653977	−0.0293349
$498$	0	0
$499$	−21.7238	−0.972493	−0.486246	−	0.873822i	$-0.661634\pi$
$499$	−21.7238	−0.972493	−0.486246	+	0.873822i	$0.661634\pi$
$500$	2.50640	0.112090
$501$	0	0
$502$	−45.7560	−2.04219
$503$	−24.4271	−1.08915	−0.544575	−	0.838712i	$-0.683309\pi$
$503$	−24.4271	−1.08915	−0.544575	+	0.838712i	$0.683309\pi$
$504$	0	0
$505$	−10.0940	−0.449177
$506$	0.785567	0.0349227
$507$	0	0
$508$	−52.6681	−2.33677
$509$	31.5025	1.39632	0.698161	−	0.715941i	$-0.254002\pi$
$509$	31.5025	1.39632	0.698161	+	0.715941i	$0.254002\pi$
$510$	0	0
$511$	−11.0417	−0.488454
$512$	−27.5723	−1.21854
$513$	0	0
$514$	27.1978	1.19964
$515$	15.5592	0.685619
$516$	0	0
$517$	1.46773	0.0645507
$518$	−8.50449	−0.373666
$519$	0	0
$520$	0	0
$521$	38.2684	1.67657	0.838284	−	0.545235i	$-0.183559\pi$
$521$	38.2684	1.67657	0.838284	+	0.545235i	$0.183559\pi$
$522$	0	0
$523$	15.0017	0.655979	0.327989	−	0.944681i	$-0.393629\pi$
$523$	15.0017	0.655979	0.327989	+	0.944681i	$0.393629\pi$
$524$	33.7678	1.47515
$525$	0	0
$526$	−51.8776	−2.26197
$527$	47.1004	2.05172
$528$	0	0
$529$	−21.8444	−0.949755
$530$	−8.65830	−0.376093
$531$	0	0
$532$	1.86206	0.0807305
$533$	0	0
$534$	0	0
$535$	−2.11369	−0.0913830
$536$	1.28698	0.0555890
$537$	0	0
$538$	50.5801	2.18066
$539$	−1.66877	−0.0718791
$540$	0	0
$541$	11.9341	0.513089	0.256544	−	0.966532i	$-0.417416\pi$
$541$	11.9341	0.513089	0.256544	+	0.966532i	$0.417416\pi$
$542$	44.1025	1.89437
$543$	0	0
$544$	−40.8125	−1.74982
$545$	−4.12514	−0.176702
$546$	0	0
$547$	23.5978	1.00897	0.504484	−	0.863421i	$-0.331683\pi$
$547$	23.5978	1.00897	0.504484	+	0.863421i	$0.331683\pi$
$548$	25.5703	1.09231
$549$	0	0
$550$	−0.730760	−0.0311597
$551$	4.07574	0.173632
$552$	0	0
$553$	−1.22488	−0.0520870
$554$	−0.993719	−0.0422191
$555$	0	0
$556$	30.4578	1.29170
$557$	14.6794	0.621986	0.310993	−	0.950412i	$-0.399338\pi$
$557$	14.6794	0.621986	0.310993	+	0.950412i	$0.399338\pi$
$558$	0	0
$559$	0	0
$560$	4.00621	0.169293
$561$	0	0
$562$	52.0354	2.19498
$563$	30.5339	1.28685	0.643425	−	0.765509i	$-0.277513\pi$
$563$	30.5339	1.28685	0.643425	+	0.765509i	$0.277513\pi$
$564$	0	0
$565$	11.3915	0.479242
$566$	−48.6020	−2.04289
$567$	0	0
$568$	−0.479204	−0.0201070
$569$	19.1284	0.801904	0.400952	−	0.916099i	$-0.368679\pi$
$569$	19.1284	0.801904	0.400952	+	0.916099i	$0.368679\pi$
$570$	0	0
$571$	−25.5461	−1.06907	−0.534535	−	0.845146i	$-0.679513\pi$
$571$	−25.5461	−1.06907	−0.534535	+	0.845146i	$0.679513\pi$
$572$	0	0
$573$	0	0
$574$	19.0819	0.796464
$575$	−1.07500	−0.0448306
$576$	0	0
$577$	−28.2497	−1.17605	−0.588025	−	0.808843i	$-0.700094\pi$
$577$	−28.2497	−1.17605	−0.588025	+	0.808843i	$0.700094\pi$
$578$	−19.9008	−0.827765
$579$	0	0
$580$	−20.1727	−0.837625
$581$	−22.3736	−0.928215
$582$	0	0
$583$	1.40404	0.0581492
$584$	−8.09081	−0.334800
$585$	0	0
$586$	−40.8441	−1.68725
$587$	0.795048	0.0328152	0.0164076	−	0.999865i	$-0.494777\pi$
$587$	0.795048	0.0328152	0.0164076	+	0.999865i	$0.494777\pi$
$588$	0	0
$589$	4.64434	0.191367
$590$	10.7442	0.442332
$591$	0	0
$592$	7.45705	0.306483
$593$	−36.0924	−1.48214	−0.741068	−	0.671430i	$-0.765680\pi$
$593$	−36.0924	−1.48214	−0.741068	+	0.671430i	$0.765680\pi$
$594$	0	0
$595$	7.53431	0.308877
$596$	1.67248	0.0685076
$597$	0	0
$598$	0	0
$599$	2.37758	0.0971454	0.0485727	−	0.998820i	$-0.484533\pi$
$599$	2.37758	0.0971454	0.0485727	+	0.998820i	$0.484533\pi$
$600$	0	0
$601$	28.1377	1.14776	0.573880	−	0.818940i	$-0.305438\pi$
$601$	28.1377	1.14776	0.573880	+	0.818940i	$0.305438\pi$
$602$	−18.2963	−0.745703
$603$	0	0
$604$	39.9014	1.62356
$605$	−10.8815	−0.442396
$606$	0	0
$607$	9.14045	0.370999	0.185500	−	0.982644i	$-0.440610\pi$
$607$	9.14045	0.370999	0.185500	+	0.982644i	$0.440610\pi$
$608$	−4.02432	−0.163208
$609$	0	0
$610$	10.9778	0.444477
$611$	0	0
$612$	0	0
$613$	38.7304	1.56431	0.782153	−	0.623086i	$-0.214121\pi$
$613$	38.7304	1.56431	0.782153	+	0.623086i	$0.214121\pi$
$614$	−34.9713	−1.41133
$615$	0	0
$616$	0.542898	0.0218740
$617$	37.6345	1.51511	0.757554	−	0.652773i	$-0.226394\pi$
$617$	37.6345	1.51511	0.757554	+	0.652773i	$0.226394\pi$
$618$	0	0
$619$	−33.3469	−1.34033	−0.670163	−	0.742214i	$-0.733776\pi$
$619$	−33.3469	−1.34033	−0.670163	+	0.742214i	$0.733776\pi$
$620$	−22.9869	−0.923177
$621$	0	0
$622$	20.2637	0.812501
$623$	−17.5313	−0.702376
$624$	0	0
$625$	1.00000	0.0400000
$626$	−46.5041	−1.85868
$627$	0	0
$628$	43.7645	1.74639
$629$	14.0242	0.559180
$630$	0	0
$631$	37.0603	1.47535	0.737673	−	0.675158i	$-0.235925\pi$
$631$	37.0603	1.47535	0.737673	+	0.675158i	$0.235925\pi$
$632$	−0.897533	−0.0357019
$633$	0	0
$634$	57.5968	2.28746
$635$	−21.0135	−0.833894
$636$	0	0
$637$	0	0
$638$	5.88150	0.232851
$639$	0	0
$640$	8.32432	0.329048
$641$	34.1064	1.34712	0.673561	−	0.739131i	$-0.264764\pi$
$641$	34.1064	1.34712	0.673561	+	0.739131i	$0.264764\pi$
$642$	0	0
$643$	−7.27484	−0.286892	−0.143446	−	0.989658i	$-0.545818\pi$
$643$	−7.27484	−0.286892	−0.143446	+	0.989658i	$0.545818\pi$
$644$	3.95283	0.155763
$645$	0	0
$646$	−5.52080	−0.217213
$647$	46.7782	1.83904	0.919521	−	0.393041i	$-0.128577\pi$
$647$	46.7782	1.83904	0.919521	+	0.393041i	$0.128577\pi$
$648$	0	0
$649$	−1.74228	−0.0683907
$650$	0	0
$651$	0	0
$652$	−36.4895	−1.42904
$653$	−1.79978	−0.0704309	−0.0352154	−	0.999380i	$-0.511212\pi$
$653$	−1.79978	−0.0704309	−0.0352154	+	0.999380i	$0.511212\pi$
$654$	0	0
$655$	13.4726	0.526420
$656$	−16.7317	−0.653264
$657$	0	0
$658$	13.2786	0.517652
$659$	27.2721	1.06237	0.531186	−	0.847255i	$-0.321747\pi$
$659$	27.2721	1.06237	0.531186	+	0.847255i	$0.321747\pi$
$660$	0	0
$661$	−24.1264	−0.938409	−0.469204	−	0.883090i	$-0.655459\pi$
$661$	−24.1264	−0.938409	−0.469204	+	0.883090i	$0.655459\pi$
$662$	2.41565	0.0938870
$663$	0	0
$664$	−16.3944	−0.636225
$665$	0.742922	0.0288093
$666$	0	0
$667$	8.65210	0.335010
$668$	38.9021	1.50517
$669$	0	0
$670$	2.54143	0.0981840
$671$	−1.78016	−0.0687224
$672$	0	0
$673$	38.8867	1.49897	0.749485	−	0.662021i	$-0.230301\pi$
$673$	38.8867	1.49897	0.749485	+	0.662021i	$0.230301\pi$
$674$	57.2476	2.20510
$675$	0	0
$676$	0	0
$677$	−23.0788	−0.886991	−0.443495	−	0.896277i	$-0.646262\pi$
$677$	−23.0788	−0.886991	−0.443495	+	0.896277i	$0.646262\pi$
$678$	0	0
$679$	−2.05472	−0.0788530
$680$	5.52080	0.211713
$681$	0	0
$682$	6.70202	0.256634
$683$	−0.915569	−0.0350333	−0.0175166	−	0.999847i	$-0.505576\pi$
$683$	−0.915569	−0.0350333	−0.0175166	+	0.999847i	$0.505576\pi$
$684$	0	0
$685$	10.2020	0.389799
$686$	−36.8977	−1.40876
$687$	0	0
$688$	16.0429	0.611630
$689$	0	0
$690$	0	0
$691$	−7.17959	−0.273124	−0.136562	−	0.990631i	$-0.543605\pi$
$691$	−7.17959	−0.273124	−0.136562	+	0.990631i	$0.543605\pi$
$692$	20.8680	0.793283
$693$	0	0
$694$	−13.8704	−0.526512
$695$	12.1520	0.460952
$696$	0	0
$697$	−31.4667	−1.19188
$698$	−2.24527	−0.0849847
$699$	0	0
$700$	−3.67706	−0.138980
$701$	−32.6040	−1.23143	−0.615717	−	0.787967i	$-0.711134\pi$
$701$	−32.6040	−1.23143	−0.615717	+	0.787967i	$0.711134\pi$
$702$	0	0
$703$	1.38286	0.0521554
$704$	−3.92724	−0.148013
$705$	0	0
$706$	52.1550	1.96288
$707$	14.8086	0.556934
$708$	0	0
$709$	−3.51909	−0.132162	−0.0660811	−	0.997814i	$-0.521050\pi$
$709$	−3.51909	−0.132162	−0.0660811	+	0.997814i	$0.521050\pi$
$710$	−0.946296	−0.0355139
$711$	0	0
$712$	−12.8461	−0.481429
$713$	9.85914	0.369228
$714$	0	0
$715$	0	0
$716$	63.3184	2.36632
$717$	0	0
$718$	−15.8088	−0.589979
$719$	−21.5297	−0.802924	−0.401462	−	0.915876i	$-0.631498\pi$
$719$	−21.5297	−0.802924	−0.401462	+	0.915876i	$0.631498\pi$
$720$	0	0
$721$	−22.8263	−0.850098
$722$	39.7894	1.48081
$723$	0	0
$724$	−54.1484	−2.01241
$725$	−8.04847	−0.298913
$726$	0	0
$727$	51.8935	1.92462	0.962312	−	0.271948i	$-0.0876679\pi$
$727$	51.8935	1.92462	0.962312	+	0.271948i	$0.0876679\pi$
$728$	0	0
$729$	0	0
$730$	−15.9771	−0.591340
$731$	30.1712	1.11592
$732$	0	0
$733$	22.8890	0.845423	0.422711	−	0.906264i	$-0.361078\pi$
$733$	22.8890	0.845423	0.422711	+	0.906264i	$0.361078\pi$
$734$	0.0946386	0.00349317
$735$	0	0
$736$	−8.54295	−0.314897
$737$	−0.412120	−0.0151806
$738$	0	0
$739$	−25.5974	−0.941615	−0.470807	−	0.882236i	$-0.656037\pi$
$739$	−25.5974	−0.941615	−0.470807	+	0.882236i	$0.656037\pi$
$740$	−6.84438	−0.251604
$741$	0	0
$742$	12.7023	0.466317
$743$	28.1012	1.03093	0.515467	−	0.856910i	$-0.327619\pi$
$743$	28.1012	1.03093	0.515467	+	0.856910i	$0.327619\pi$
$744$	0	0
$745$	0.667286	0.0244475
$746$	7.09525	0.259776
$747$	0	0
$748$	−4.43102	−0.162014
$749$	3.10093	0.113306
$750$	0	0
$751$	9.86987	0.360157	0.180078	−	0.983652i	$-0.442365\pi$
$751$	9.86987	0.360157	0.180078	+	0.983652i	$0.442365\pi$
$752$	−11.6431	−0.424581
$753$	0	0
$754$	0	0
$755$	15.9198	0.579380
$756$	0	0
$757$	1.86104	0.0676407	0.0338204	−	0.999428i	$-0.489233\pi$
$757$	1.86104	0.0676407	0.0338204	+	0.999428i	$0.489233\pi$
$758$	−55.6585	−2.02161
$759$	0	0
$760$	0.544379	0.0197467
$761$	−31.7926	−1.15248	−0.576240	−	0.817280i	$-0.695481\pi$
$761$	−31.7926	−1.15248	−0.576240	+	0.817280i	$0.695481\pi$
$762$	0	0
$763$	6.05186	0.219092
$764$	−23.3458	−0.844621
$765$	0	0
$766$	−27.3826	−0.989373
$767$	0	0
$768$	0	0
$769$	19.3350	0.697238	0.348619	−	0.937265i	$-0.386651\pi$
$769$	19.3350	0.697238	0.348619	+	0.937265i	$0.386651\pi$
$770$	1.07207	0.0386349
$771$	0	0
$772$	19.0488	0.685582
$773$	−1.64035	−0.0589993	−0.0294997	−	0.999565i	$-0.509391\pi$
$773$	−1.64035	−0.0589993	−0.0294997	+	0.999565i	$0.509391\pi$
$774$	0	0
$775$	−9.17129	−0.329443
$776$	−1.50561	−0.0540481
$777$	0	0
$778$	10.7962	0.387064
$779$	−3.10278	−0.111169
$780$	0	0
$781$	0.153452	0.00549094
$782$	−11.7197	−0.419096
$783$	0	0
$784$	13.2379	0.472784
$785$	17.4611	0.623214
$786$	0	0
$787$	32.2921	1.15109	0.575544	−	0.817771i	$-0.304790\pi$
$787$	32.2921	1.15109	0.575544	+	0.817771i	$0.304790\pi$
$788$	27.4282	0.977089
$789$	0	0
$790$	−1.77238	−0.0630585
$791$	−16.7120	−0.594212
$792$	0	0
$793$	0	0
$794$	−23.4874	−0.833535
$795$	0	0
$796$	−35.5828	−1.26120
$797$	−41.0414	−1.45376	−0.726880	−	0.686764i	$-0.759030\pi$
$797$	−41.0414	−1.45376	−0.726880	+	0.686764i	$0.759030\pi$
$798$	0	0
$799$	−21.8967	−0.774651
$800$	7.94693	0.280967
$801$	0	0
$802$	49.0273	1.73121
$803$	2.59086	0.0914295
$804$	0	0
$805$	1.57710	0.0555853
$806$	0	0
$807$	0	0
$808$	10.8511	0.381739
$809$	−22.6382	−0.795916	−0.397958	−	0.917404i	$-0.630281\pi$
$809$	−22.6382	−0.795916	−0.397958	+	0.917404i	$0.630281\pi$
$810$	0	0
$811$	6.81520	0.239314	0.119657	−	0.992815i	$-0.461820\pi$
$811$	6.81520	0.239314	0.119657	+	0.992815i	$0.461820\pi$
$812$	29.5947	1.03857
$813$	0	0
$814$	1.99553	0.0699433
$815$	−14.5585	−0.509963
$816$	0	0
$817$	2.97504	0.104083
$818$	30.6136	1.07038
$819$	0	0
$820$	15.3570	0.536291
$821$	42.0112	1.46620	0.733100	−	0.680121i	$-0.238073\pi$
$821$	42.0112	1.46620	0.733100	+	0.680121i	$0.238073\pi$
$822$	0	0
$823$	33.9531	1.18353	0.591766	−	0.806110i	$-0.298431\pi$
$823$	33.9531	1.18353	0.591766	+	0.806110i	$0.298431\pi$
$824$	−16.7261	−0.582681
$825$	0	0
$826$	−15.7625	−0.548446
$827$	24.6063	0.855644	0.427822	−	0.903863i	$-0.359281\pi$
$827$	24.6063	0.855644	0.427822	+	0.903863i	$0.359281\pi$
$828$	0	0
$829$	−14.8486	−0.515712	−0.257856	−	0.966183i	$-0.583016\pi$
$829$	−14.8486	−0.515712	−0.257856	+	0.966183i	$0.583016\pi$
$830$	−32.3744	−1.12373
$831$	0	0
$832$	0	0
$833$	24.8961	0.862597
$834$	0	0
$835$	15.5211	0.537130
$836$	−0.436922	−0.0151113
$837$	0	0
$838$	−32.2000	−1.11233
$839$	7.23578	0.249807	0.124903	−	0.992169i	$-0.460138\pi$
$839$	7.23578	0.249807	0.124903	+	0.992169i	$0.460138\pi$
$840$	0	0
$841$	35.7778	1.23372
$842$	22.8644	0.787958
$843$	0	0
$844$	25.8906	0.891191
$845$	0	0
$846$	0	0
$847$	15.9639	0.548526
$848$	−11.1379	−0.382476
$849$	0	0
$850$	10.9021	0.373937
$851$	2.93557	0.100630
$852$	0	0
$853$	−36.9206	−1.26414	−0.632068	−	0.774913i	$-0.717794\pi$
$853$	−36.9206	−1.26414	−0.632068	+	0.774913i	$0.717794\pi$
$854$	−16.1051	−0.551106
$855$	0	0
$856$	2.27222	0.0776629
$857$	16.2642	0.555573	0.277786	−	0.960643i	$-0.410399\pi$
$857$	16.2642	0.555573	0.277786	+	0.960643i	$0.410399\pi$
$858$	0	0
$859$	25.2975	0.863141	0.431571	−	0.902079i	$-0.357960\pi$
$859$	25.2975	0.863141	0.431571	+	0.902079i	$0.357960\pi$
$860$	−14.7248	−0.502112
$861$	0	0
$862$	87.7023	2.98715
$863$	−55.7647	−1.89825	−0.949127	−	0.314894i	$-0.898031\pi$
$863$	−55.7647	−1.89825	−0.949127	+	0.314894i	$0.898031\pi$
$864$	0	0
$865$	8.32589	0.283089
$866$	−39.1713	−1.33109
$867$	0	0
$868$	33.7234	1.14465
$869$	0.287410	0.00974972
$870$	0	0
$871$	0	0
$872$	4.43452	0.150172
$873$	0	0
$874$	−1.15562	−0.0390895
$875$	−1.46707	−0.0495959
$876$	0	0
$877$	−31.4341	−1.06145	−0.530727	−	0.847543i	$-0.678081\pi$
$877$	−31.4341	−1.06145	−0.530727	+	0.847543i	$0.678081\pi$
$878$	−16.7308	−0.564638
$879$	0	0
$880$	−0.940034	−0.0316885
$881$	31.0181	1.04502	0.522512	−	0.852632i	$-0.324995\pi$
$881$	31.0181	1.04502	0.522512	+	0.852632i	$0.324995\pi$
$882$	0	0
$883$	−43.7232	−1.47140	−0.735701	−	0.677307i	$-0.763147\pi$
$883$	−43.7232	−1.47140	−0.735701	+	0.677307i	$0.763147\pi$
$884$	0	0
$885$	0	0
$886$	81.2552	2.72982
$887$	18.2525	0.612858	0.306429	−	0.951894i	$-0.400866\pi$
$887$	18.2525	0.612858	0.306429	+	0.951894i	$0.400866\pi$
$888$	0	0
$889$	30.8282	1.03394
$890$	−25.3675	−0.850322
$891$	0	0
$892$	−61.4221	−2.05657
$893$	−2.15913	−0.0722526
$894$	0	0
$895$	25.2627	0.844438
$896$	−12.2123	−0.407986
$897$	0	0
$898$	−26.6296	−0.888640
$899$	73.8149	2.46186
$900$	0	0
$901$	−20.9465	−0.697829
$902$	−4.47746	−0.149083
$903$	0	0
$904$	−12.2458	−0.407290
$905$	−21.6041	−0.718144
$906$	0	0
$907$	18.1625	0.603077	0.301538	−	0.953454i	$-0.402500\pi$
$907$	18.1625	0.603077	0.301538	+	0.953454i	$0.402500\pi$
$908$	7.53233	0.249969
$909$	0	0
$910$	0	0
$911$	43.3147	1.43508	0.717541	−	0.696517i	$-0.245268\pi$
$911$	43.3147	1.43508	0.717541	+	0.696517i	$0.245268\pi$
$912$	0	0
$913$	5.24985	0.173745
$914$	−43.0269	−1.42320
$915$	0	0
$916$	55.0759	1.81976
$917$	−19.7653	−0.652707
$918$	0	0
$919$	4.11950	0.135890	0.0679449	−	0.997689i	$-0.478356\pi$
$919$	4.11950	0.135890	0.0679449	+	0.997689i	$0.478356\pi$
$920$	1.15562	0.0380998
$921$	0	0
$922$	−26.5356	−0.873903
$923$	0	0
$924$	0	0
$925$	−2.73076	−0.0897868
$926$	−32.0384	−1.05285
$927$	0	0
$928$	−63.9606	−2.09961
$929$	48.3466	1.58620	0.793101	−	0.609091i	$-0.208465\pi$
$929$	48.3466	1.58620	0.793101	+	0.609091i	$0.208465\pi$
$930$	0	0
$931$	2.45488	0.0804554
$932$	−18.1055	−0.593067
$933$	0	0
$934$	31.5922	1.03373
$935$	−1.76788	−0.0578160
$936$	0	0
$937$	−17.7174	−0.578801	−0.289401	−	0.957208i	$-0.593456\pi$
$937$	−17.7174	−0.578801	−0.289401	+	0.957208i	$0.593456\pi$
$938$	−3.72845	−0.121738
$939$	0	0
$940$	10.6865	0.348556
$941$	25.7342	0.838912	0.419456	−	0.907776i	$-0.362221\pi$
$941$	25.7342	0.838912	0.419456	+	0.907776i	$0.362221\pi$
$942$	0	0
$943$	−6.58666	−0.214491
$944$	13.8211	0.449839
$945$	0	0
$946$	4.29313	0.139582
$947$	48.3882	1.57241	0.786203	−	0.617968i	$-0.212044\pi$
$947$	48.3882	1.57241	0.786203	+	0.617968i	$0.212044\pi$
$948$	0	0
$949$	0	0
$950$	1.07500	0.0348776
$951$	0	0
$952$	−8.09938	−0.262502
$953$	13.8256	0.447855	0.223928	−	0.974606i	$-0.428112\pi$
$953$	13.8256	0.447855	0.223928	+	0.974606i	$0.428112\pi$
$954$	0	0
$955$	−9.31447	−0.301409
$956$	1.23556	0.0399610
$957$	0	0
$958$	−0.645681	−0.0208610
$959$	−14.9671	−0.483311
$960$	0	0
$961$	53.1126	1.71331
$962$	0	0
$963$	0	0
$964$	65.8602	2.12122
$965$	7.60007	0.244655
$966$	0	0
$967$	34.8289	1.12002	0.560010	−	0.828486i	$-0.310797\pi$
$967$	34.8289	1.12002	0.560010	+	0.828486i	$0.310797\pi$
$968$	11.6976	0.375975
$969$	0	0
$970$	−2.97316	−0.0954623
$971$	7.81940	0.250937	0.125468	−	0.992098i	$-0.459957\pi$
$971$	7.81940	0.250937	0.125468	+	0.992098i	$0.459957\pi$
$972$	0	0
$973$	−17.8278	−0.571534
$974$	88.5566	2.83754
$975$	0	0
$976$	14.1216	0.452020
$977$	4.91587	0.157273	0.0786363	−	0.996903i	$-0.474943\pi$
$977$	4.91587	0.157273	0.0786363	+	0.996903i	$0.474943\pi$
$978$	0	0
$979$	4.11362	0.131472
$980$	−12.1503	−0.388127
$981$	0	0
$982$	86.1009	2.74759
$983$	−5.23823	−0.167074	−0.0835368	−	0.996505i	$-0.526622\pi$
$983$	−5.23823	−0.167074	−0.0835368	+	0.996505i	$0.526622\pi$
$984$	0	0
$985$	10.9433	0.348682
$986$	−87.7448	−2.79436
$987$	0	0
$988$	0	0
$989$	6.31550	0.200821
$990$	0	0
$991$	−1.77878	−0.0565049	−0.0282525	−	0.999601i	$-0.508994\pi$
$991$	−1.77878	−0.0565049	−0.0282525	+	0.999601i	$0.508994\pi$
$992$	−72.8837	−2.31406
$993$	0	0
$994$	1.38828	0.0440336
$995$	−14.1968	−0.450068
$996$	0	0
$997$	43.9171	1.39087	0.695434	−	0.718590i	$-0.255212\pi$
$997$	43.9171	1.39087	0.695434	+	0.718590i	$0.255212\pi$
$998$	46.1160	1.45978
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7605.2.a.cn.1.1		5
3.2	odd	2		7605.2.a.cl.1.5		5
13.4	even	6		585.2.j.i.406.1	yes	10
13.10	even	6		585.2.j.i.451.1	yes	10
13.12	even	2		7605.2.a.cm.1.5		5
39.17	odd	6		585.2.j.h.406.5	✓	10
39.23	odd	6		585.2.j.h.451.5	yes	10
39.38	odd	2		7605.2.a.co.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
585.2.j.h.406.5	✓	10	39.17	odd	6
585.2.j.h.451.5	yes	10	39.23	odd	6
585.2.j.i.406.1	yes	10	13.4	even	6
585.2.j.i.451.1	yes	10	13.10	even	6
7605.2.a.cl.1.5		5	3.2	odd	2
7605.2.a.cm.1.5		5	13.12	even	2
7605.2.a.cn.1.1		5	1.1	even	1	trivial
7605.2.a.co.1.1		5	39.38	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-2.12283 q^{2} +2.50640 q^{4} +1.00000 q^{5} -1.46707 q^{7} -1.07500 q^{8} +O(q^{10})\)	\(q-2.12283 q^{2} +2.50640 q^{4} +1.00000 q^{5} -1.46707 q^{7} -1.07500 q^{8} -2.12283 q^{10} +0.344239 q^{11} +3.11433 q^{14} -2.73076 q^{16} -5.13563 q^{17} -0.506400 q^{19} +2.50640 q^{20} -0.730760 q^{22} -1.07500 q^{23} +1.00000 q^{25} -3.67706 q^{28} -8.04847 q^{29} -9.17129 q^{31} +7.94693 q^{32} +10.9021 q^{34} -1.46707 q^{35} -2.73076 q^{37} +1.07500 q^{38} -1.07500 q^{40} +6.12713 q^{41} -5.87488 q^{43} +0.862801 q^{44} +2.28204 q^{46} +4.26369 q^{47} -4.84771 q^{49} -2.12283 q^{50} +4.07866 q^{53} +0.344239 q^{55} +1.57710 q^{56} +17.0855 q^{58} -5.06127 q^{59} -5.17129 q^{61} +19.4691 q^{62} -11.4085 q^{64} -1.19719 q^{67} -12.8719 q^{68} +3.11433 q^{70} +0.445772 q^{71} +7.52634 q^{73} +5.79693 q^{74} -1.26924 q^{76} -0.505022 q^{77} +0.834915 q^{79} -2.73076 q^{80} -13.0068 q^{82} +15.2506 q^{83} -5.13563 q^{85} +12.4714 q^{86} -0.370057 q^{88} +11.9499 q^{89} -2.69438 q^{92} -9.05109 q^{94} -0.506400 q^{95} +1.40056 q^{97} +10.2909 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 2 q^{2} + 6 q^{4} + 5 q^{5} - q^{7} + 6 q^{8}+O(q^{10}) \) 5 * q + 2 * q^2 + 6 * q^4 + 5 * q^5 - q^7 + 6 * q^8	\( 5 q + 2 q^{2} + 6 q^{4} + 5 q^{5} - q^{7} + 6 q^{8} + 2 q^{10} + 8 q^{11} + 4 q^{14} + 4 q^{16} + 4 q^{19} + 6 q^{20} + 14 q^{22} + 6 q^{23} + 5 q^{25} + 2 q^{28} - 16 q^{29} - 9 q^{31} + 14 q^{32} - q^{35} + 4 q^{37} - 6 q^{38} + 6 q^{40} + 6 q^{41} + 15 q^{43} + 14 q^{44} + 16 q^{46} + 10 q^{47} + 10 q^{49} + 2 q^{50} + 20 q^{53} + 8 q^{55} + 2 q^{56} + 4 q^{58} + 12 q^{59} + 11 q^{61} + 22 q^{62} + 4 q^{64} - 5 q^{67} - 50 q^{68} + 4 q^{70} + 10 q^{71} - q^{73} + 26 q^{74} - 24 q^{76} - 42 q^{77} - 17 q^{79} + 4 q^{80} + 16 q^{82} + 16 q^{83} + 44 q^{86} + 20 q^{88} + 4 q^{89} + 34 q^{92} - 16 q^{94} + 4 q^{95} + 11 q^{97} + 30 q^{98}+O(q^{100}) \) 5 * q + 2 * q^2 + 6 * q^4 + 5 * q^5 - q^7 + 6 * q^8 + 2 * q^10 + 8 * q^11 + 4 * q^14 + 4 * q^16 + 4 * q^19 + 6 * q^20 + 14 * q^22 + 6 * q^23 + 5 * q^25 + 2 * q^28 - 16 * q^29 - 9 * q^31 + 14 * q^32 - q^35 + 4 * q^37 - 6 * q^38 + 6 * q^40 + 6 * q^41 + 15 * q^43 + 14 * q^44 + 16 * q^46 + 10 * q^47 + 10 * q^49 + 2 * q^50 + 20 * q^53 + 8 * q^55 + 2 * q^56 + 4 * q^58 + 12 * q^59 + 11 * q^61 + 22 * q^62 + 4 * q^64 - 5 * q^67 - 50 * q^68 + 4 * q^70 + 10 * q^71 - q^73 + 26 * q^74 - 24 * q^76 - 42 * q^77 - 17 * q^79 + 4 * q^80 + 16 * q^82 + 16 * q^83 + 44 * q^86 + 20 * q^88 + 4 * q^89 + 34 * q^92 - 16 * q^94 + 4 * q^95 + 11 * q^97 + 30 * q^98

Introduction

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