LMFDB - Embedded newform 7605.2.a.bx.1.2

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:	$3$
Coefficient field:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 195)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.81361$ 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)

$f(q)$	$=$	$q-0.289169 q^{2} -1.91638 q^{4} -1.00000 q^{5} -4.91638 q^{7} +1.13249 q^{8} +O(q^{10})$	$q-0.289169 q^{2} -1.91638 q^{4} -1.00000 q^{5} -4.91638 q^{7} +1.13249 q^{8} +0.289169 q^{10} +4.91638 q^{11} +1.42166 q^{14} +3.50528 q^{16} +4.33804 q^{17} -2.57834 q^{19} +1.91638 q^{20} -1.42166 q^{22} +6.33804 q^{23} +1.00000 q^{25} +9.42166 q^{28} -6.00000 q^{29} -1.42166 q^{31} -3.27861 q^{32} -1.25443 q^{34} +4.91638 q^{35} -9.49472 q^{37} +0.745574 q^{38} -1.13249 q^{40} +4.33804 q^{41} -1.15667 q^{43} -9.42166 q^{44} -1.83276 q^{46} -5.42166 q^{47} +17.1708 q^{49} -0.289169 q^{50} +0.338044 q^{53} -4.91638 q^{55} -5.56777 q^{56} +1.73501 q^{58} -11.2544 q^{59} -10.1708 q^{61} +0.411100 q^{62} -6.06249 q^{64} +7.25443 q^{67} -8.31335 q^{68} -1.42166 q^{70} +0.916382 q^{71} +3.15667 q^{73} +2.74557 q^{74} +4.94108 q^{76} -24.1708 q^{77} -3.49472 q^{79} -3.50528 q^{80} -1.25443 q^{82} +11.2544 q^{83} -4.33804 q^{85} +0.334474 q^{86} +5.56777 q^{88} -0.338044 q^{89} -12.1461 q^{92} +1.56777 q^{94} +2.57834 q^{95} +12.3380 q^{97} -4.96526 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 8 q^{4} - 3 q^{5} - q^{7} + 6 q^{8}+O(q^{10}) $ 3 * q + 8 * q^4 - 3 * q^5 - q^7 + 6 * q^8	$ 3 q + 8 q^{4} - 3 q^{5} - q^{7} + 6 q^{8} + q^{11} + 6 q^{14} + 26 q^{16} + q^{17} - 6 q^{19} - 8 q^{20} - 6 q^{22} + 7 q^{23} + 3 q^{25} + 30 q^{28} - 18 q^{29} - 6 q^{31} + 22 q^{32} + 22 q^{34} + q^{35} - 13 q^{37} + 28 q^{38} - 6 q^{40} + q^{41} - 30 q^{44} + 22 q^{46} - 18 q^{47} + 12 q^{49} - 11 q^{53} - q^{55} + 16 q^{56} - 8 q^{59} + 9 q^{61} - 28 q^{62} + 30 q^{64} - 4 q^{67} - 18 q^{68} - 6 q^{70} - 11 q^{71} + 6 q^{73} + 34 q^{74} - 4 q^{76} - 33 q^{77} + 5 q^{79} - 26 q^{80} + 22 q^{82} + 8 q^{83} - q^{85} + 56 q^{86} - 16 q^{88} + 11 q^{89} - 2 q^{92} - 28 q^{94} + 6 q^{95} + 25 q^{97} + 10 q^{98}+O(q^{100}) $ 3 * q + 8 * q^4 - 3 * q^5 - q^7 + 6 * q^8 + q^11 + 6 * q^14 + 26 * q^16 + q^17 - 6 * q^19 - 8 * q^20 - 6 * q^22 + 7 * q^23 + 3 * q^25 + 30 * q^28 - 18 * q^29 - 6 * q^31 + 22 * q^32 + 22 * q^34 + q^35 - 13 * q^37 + 28 * q^38 - 6 * q^40 + q^41 - 30 * q^44 + 22 * q^46 - 18 * q^47 + 12 * q^49 - 11 * q^53 - q^55 + 16 * q^56 - 8 * q^59 + 9 * q^61 - 28 * q^62 + 30 * q^64 - 4 * q^67 - 18 * q^68 - 6 * q^70 - 11 * q^71 + 6 * q^73 + 34 * q^74 - 4 * q^76 - 33 * q^77 + 5 * q^79 - 26 * q^80 + 22 * q^82 + 8 * q^83 - q^85 + 56 * q^86 - 16 * q^88 + 11 * q^89 - 2 * q^92 - 28 * q^94 + 6 * q^95 + 25 * q^97 + 10 * 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 $

$2$	−0.289169	−0.204473	−0.102237	−	0.994760i	$-0.532600\pi$
$2$	−0.289169	−0.204473	−0.102237	+	0.994760i	$0.532600\pi$
$3$	0	0
$3$	0	0
$4$	−1.91638	−0.958191
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−4.91638	−1.85822	−0.929109	−	0.369807i	$-0.879424\pi$
$7$	−4.91638	−1.85822	−0.929109	+	0.369807i	$0.879424\pi$
$8$	1.13249	0.400397
$9$	0	0
$10$	0.289169	0.0914431
$11$	4.91638	1.48234	0.741172	−	0.671315i	$-0.234270\pi$
$11$	4.91638	1.48234	0.741172	+	0.671315i	$0.234270\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	1.42166	0.379955
$15$	0	0
$16$	3.50528	0.876320
$17$	4.33804	1.05213	0.526065	−	0.850444i	$-0.323667\pi$
$17$	4.33804	1.05213	0.526065	+	0.850444i	$0.323667\pi$
$18$	0	0
$19$	−2.57834	−0.591511	−0.295756	−	0.955264i	$-0.595571\pi$
$19$	−2.57834	−0.591511	−0.295756	+	0.955264i	$0.595571\pi$
$20$	1.91638	0.428516
$21$	0	0
$22$	−1.42166	−0.303100
$23$	6.33804	1.32157	0.660787	−	0.750574i	$-0.270223\pi$
$23$	6.33804	1.32157	0.660787	+	0.750574i	$0.270223\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	9.42166	1.78053
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	−1.42166	−0.255338	−0.127669	−	0.991817i	$-0.540750\pi$
$31$	−1.42166	−0.255338	−0.127669	+	0.991817i	$0.540750\pi$
$32$	−3.27861	−0.579581
$33$	0	0
$34$	−1.25443	−0.215132
$35$	4.91638	0.831020
$36$	0	0
$37$	−9.49472	−1.56092	−0.780461	−	0.625204i	$-0.785016\pi$
$37$	−9.49472	−1.56092	−0.780461	+	0.625204i	$0.785016\pi$
$38$	0.745574	0.120948
$39$	0	0
$40$	−1.13249	−0.179063
$41$	4.33804	0.677489	0.338744	−	0.940878i	$-0.389998\pi$
$41$	4.33804	0.677489	0.338744	+	0.940878i	$0.389998\pi$
$42$	0	0
$43$	−1.15667	−0.176391	−0.0881956	−	0.996103i	$-0.528110\pi$
$43$	−1.15667	−0.176391	−0.0881956	+	0.996103i	$0.528110\pi$
$44$	−9.42166	−1.42037
$45$	0	0
$46$	−1.83276	−0.270226
$47$	−5.42166	−0.790831	−0.395415	−	0.918502i	$-0.629399\pi$
$47$	−5.42166	−0.790831	−0.395415	+	0.918502i	$0.629399\pi$
$48$	0	0
$49$	17.1708	2.45297
$50$	−0.289169	−0.0408946
$51$	0	0
$52$	0	0
$53$	0.338044	0.0464340	0.0232170	−	0.999730i	$-0.492609\pi$
$53$	0.338044	0.0464340	0.0232170	+	0.999730i	$0.492609\pi$
$54$	0	0
$55$	−4.91638	−0.662925
$56$	−5.56777	−0.744025
$57$	0	0
$58$	1.73501	0.227818
$59$	−11.2544	−1.46520	−0.732601	−	0.680659i	$-0.761694\pi$
$59$	−11.2544	−1.46520	−0.732601	+	0.680659i	$0.761694\pi$
$60$	0	0
$61$	−10.1708	−1.30224	−0.651119	−	0.758975i	$-0.725700\pi$
$61$	−10.1708	−1.30224	−0.651119	+	0.758975i	$0.725700\pi$
$62$	0.411100	0.0522098
$63$	0	0
$64$	−6.06249	−0.757812
$65$	0	0
$66$	0	0
$67$	7.25443	0.886269	0.443135	−	0.896455i	$-0.353866\pi$
$67$	7.25443	0.886269	0.443135	+	0.896455i	$0.353866\pi$
$68$	−8.31335	−1.00814
$69$	0	0
$70$	−1.42166	−0.169921
$71$	0.916382	0.108754	0.0543772	−	0.998520i	$-0.482683\pi$
$71$	0.916382	0.108754	0.0543772	+	0.998520i	$0.482683\pi$
$72$	0	0
$73$	3.15667	0.369461	0.184730	−	0.982789i	$-0.440859\pi$
$73$	3.15667	0.369461	0.184730	+	0.982789i	$0.440859\pi$
$74$	2.74557	0.319166
$75$	0	0
$76$	4.94108	0.566780
$77$	−24.1708	−2.75452
$78$	0	0
$79$	−3.49472	−0.393187	−0.196593	−	0.980485i	$-0.562988\pi$
$79$	−3.49472	−0.393187	−0.196593	+	0.980485i	$0.562988\pi$
$80$	−3.50528	−0.391902
$81$	0	0
$82$	−1.25443	−0.138528
$83$	11.2544	1.23533	0.617667	−	0.786440i	$-0.288078\pi$
$83$	11.2544	1.23533	0.617667	+	0.786440i	$0.288078\pi$
$84$	0	0
$85$	−4.33804	−0.470527
$86$	0.334474	0.0360672
$87$	0	0
$88$	5.56777	0.593527
$89$	−0.338044	−0.0358326	−0.0179163	−	0.999839i	$-0.505703\pi$
$89$	−0.338044	−0.0358326	−0.0179163	+	0.999839i	$0.505703\pi$
$90$	0	0
$91$	0	0
$92$	−12.1461	−1.26632
$93$	0	0
$94$	1.56777	0.161704
$95$	2.57834	0.264532
$96$	0	0
$97$	12.3380	1.25274	0.626369	−	0.779526i	$-0.284540\pi$
$97$	12.3380	1.25274	0.626369	+	0.779526i	$0.284540\pi$
$98$	−4.96526	−0.501567
$99$	0	0
$100$	−1.91638	−0.191638
$101$	−10.6761	−1.06231	−0.531155	−	0.847274i	$-0.678242\pi$
$101$	−10.6761	−1.06231	−0.531155	+	0.847274i	$0.678242\pi$
$102$	0	0
$103$	−14.5089	−1.42960	−0.714800	−	0.699329i	$-0.753482\pi$
$103$	−14.5089	−1.42960	−0.714800	+	0.699329i	$0.753482\pi$
$104$	0	0
$105$	0	0
$106$	−0.0977518	−0.00949450
$107$	−4.17081	−0.403207	−0.201604	−	0.979467i	$-0.564615\pi$
$107$	−4.17081	−0.403207	−0.201604	+	0.979467i	$0.564615\pi$
$108$	0	0
$109$	3.83276	0.367112	0.183556	−	0.983009i	$-0.441239\pi$
$109$	3.83276	0.367112	0.183556	+	0.983009i	$0.441239\pi$
$110$	1.42166	0.135550
$111$	0	0
$112$	−17.2333	−1.62839
$113$	0.843326	0.0793334	0.0396667	−	0.999213i	$-0.487370\pi$
$113$	0.843326	0.0793334	0.0396667	+	0.999213i	$0.487370\pi$
$114$	0	0
$115$	−6.33804	−0.591026
$116$	11.4983	1.06759
$117$	0	0
$118$	3.25443	0.299594
$119$	−21.3275	−1.95509
$120$	0	0
$121$	13.1708	1.19735
$122$	2.94108	0.266273
$123$	0	0
$124$	2.72445	0.244663
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	1.83276	0.162631	0.0813157	−	0.996688i	$-0.474088\pi$
$127$	1.83276	0.162631	0.0813157	+	0.996688i	$0.474088\pi$
$128$	8.31029	0.734533
$129$	0	0
$130$	0	0
$131$	−5.83276	−0.509611	−0.254805	−	0.966992i	$-0.582011\pi$
$131$	−5.83276	−0.509611	−0.254805	+	0.966992i	$0.582011\pi$
$132$	0	0
$133$	12.6761	1.09916
$134$	−2.09775	−0.181218
$135$	0	0
$136$	4.91281	0.421270
$137$	−16.5089	−1.41045	−0.705223	−	0.708985i	$-0.749153\pi$
$137$	−16.5089	−1.41045	−0.705223	+	0.708985i	$0.749153\pi$
$138$	0	0
$139$	7.49472	0.635694	0.317847	−	0.948142i	$-0.397040\pi$
$139$	7.49472	0.635694	0.317847	+	0.948142i	$0.397040\pi$
$140$	−9.42166	−0.796276
$141$	0	0
$142$	−0.264989	−0.0222374
$143$	0	0
$144$	0	0
$145$	6.00000	0.498273
$146$	−0.912811	−0.0755448
$147$	0	0
$148$	18.1955	1.49566
$149$	20.4842	1.67813	0.839064	−	0.544033i	$-0.183103\pi$
$149$	20.4842	1.67813	0.839064	+	0.544033i	$0.183103\pi$
$150$	0	0
$151$	16.4111	1.33552	0.667758	−	0.744378i	$-0.267254\pi$
$151$	16.4111	1.33552	0.667758	+	0.744378i	$0.267254\pi$
$152$	−2.91995	−0.236839
$153$	0	0
$154$	6.98944	0.563225
$155$	1.42166	0.114191
$156$	0	0
$157$	−21.6655	−1.72910	−0.864549	−	0.502549i	$-0.832396\pi$
$157$	−21.6655	−1.72910	−0.864549	+	0.502549i	$0.832396\pi$
$158$	1.01056	0.0803961
$159$	0	0
$160$	3.27861	0.259197
$161$	−31.1602	−2.45577
$162$	0	0
$163$	−6.07306	−0.475678	−0.237839	−	0.971305i	$-0.576439\pi$
$163$	−6.07306	−0.475678	−0.237839	+	0.971305i	$0.576439\pi$
$164$	−8.31335	−0.649163
$165$	0	0
$166$	−3.25443	−0.252592
$167$	−0.745574	−0.0576942	−0.0288471	−	0.999584i	$-0.509184\pi$
$167$	−0.745574	−0.0576942	−0.0288471	+	0.999584i	$0.509184\pi$
$168$	0	0
$169$	0	0
$170$	1.25443	0.0962101
$171$	0	0
$172$	2.21663	0.169016
$173$	−0.843326	−0.0641169	−0.0320584	−	0.999486i	$-0.510206\pi$
$173$	−0.843326	−0.0641169	−0.0320584	+	0.999486i	$0.510206\pi$
$174$	0	0
$175$	−4.91638	−0.371644
$176$	17.2333	1.29901
$177$	0	0
$178$	0.0977518	0.00732681
$179$	18.9894	1.41934	0.709669	−	0.704536i	$-0.248845\pi$
$179$	18.9894	1.41934	0.709669	+	0.704536i	$0.248845\pi$
$180$	0	0
$181$	17.4947	1.30037	0.650186	−	0.759775i	$-0.274691\pi$
$181$	17.4947	1.30037	0.650186	+	0.759775i	$0.274691\pi$
$182$	0	0
$183$	0	0
$184$	7.17780	0.529154
$185$	9.49472	0.698066
$186$	0	0
$187$	21.3275	1.55962
$188$	10.3900	0.757767
$189$	0	0
$190$	−0.745574	−0.0540896
$191$	22.5089	1.62868	0.814342	−	0.580386i	$-0.197098\pi$
$191$	22.5089	1.62868	0.814342	+	0.580386i	$0.197098\pi$
$192$	0	0
$193$	−2.65139	−0.190851	−0.0954257	−	0.995437i	$-0.530421\pi$
$193$	−2.65139	−0.190851	−0.0954257	+	0.995437i	$0.530421\pi$
$194$	−3.56777	−0.256151
$195$	0	0
$196$	−32.9058	−2.35042
$197$	−12.9894	−0.925459	−0.462730	−	0.886500i	$-0.653130\pi$
$197$	−12.9894	−0.925459	−0.462730	+	0.886500i	$0.653130\pi$
$198$	0	0
$199$	−2.84333	−0.201558	−0.100779	−	0.994909i	$-0.532134\pi$
$199$	−2.84333	−0.201558	−0.100779	+	0.994909i	$0.532134\pi$
$200$	1.13249	0.0800794
$201$	0	0
$202$	3.08719	0.217214
$203$	29.4983	2.07037
$204$	0	0
$205$	−4.33804	−0.302982
$206$	4.19550	0.292315
$207$	0	0
$208$	0	0
$209$	−12.6761	−0.876823
$210$	0	0
$211$	6.31335	0.434629	0.217314	−	0.976102i	$-0.430270\pi$
$211$	6.31335	0.434629	0.217314	+	0.976102i	$0.430270\pi$
$212$	−0.647822	−0.0444926
$213$	0	0
$214$	1.20607	0.0824450
$215$	1.15667	0.0788845
$216$	0	0
$217$	6.98944	0.474474
$218$	−1.10831	−0.0750645
$219$	0	0
$220$	9.42166	0.635208
$221$	0	0
$222$	0	0
$223$	−19.2544	−1.28937	−0.644686	−	0.764448i	$-0.723012\pi$
$223$	−19.2544	−1.28937	−0.644686	+	0.764448i	$0.723012\pi$
$224$	16.1189	1.07699
$225$	0	0
$226$	−0.243863	−0.0162215
$227$	−13.0872	−0.868627	−0.434314	−	0.900762i	$-0.643009\pi$
$227$	−13.0872	−0.868627	−0.434314	+	0.900762i	$0.643009\pi$
$228$	0	0
$229$	24.5089	1.61959	0.809795	−	0.586713i	$-0.199578\pi$
$229$	24.5089	1.61959	0.809795	+	0.586713i	$0.199578\pi$
$230$	1.83276	0.120849
$231$	0	0
$232$	−6.79497	−0.446111
$233$	−8.33804	−0.546243	−0.273122	−	0.961979i	$-0.588056\pi$
$233$	−8.33804	−0.546243	−0.273122	+	0.961979i	$0.588056\pi$
$234$	0	0
$235$	5.42166	0.353670
$236$	21.5678	1.40394
$237$	0	0
$238$	6.16724	0.399763
$239$	8.91638	0.576753	0.288376	−	0.957517i	$-0.406885\pi$
$239$	8.91638	0.576753	0.288376	+	0.957517i	$0.406885\pi$
$240$	0	0
$241$	6.00000	0.386494	0.193247	−	0.981150i	$-0.438098\pi$
$241$	6.00000	0.386494	0.193247	+	0.981150i	$0.438098\pi$
$242$	−3.80858	−0.244825
$243$	0	0
$244$	19.4911	1.24779
$245$	−17.1708	−1.09700
$246$	0	0
$247$	0	0
$248$	−1.61003	−0.102237
$249$	0	0
$250$	0.289169	0.0182886
$251$	6.31335	0.398495	0.199248	−	0.979949i	$-0.436150\pi$
$251$	6.31335	0.398495	0.199248	+	0.979949i	$0.436150\pi$
$252$	0	0
$253$	31.1602	1.95903
$254$	−0.529977	−0.0332537
$255$	0	0
$256$	9.72191	0.607619
$257$	11.1567	0.695934	0.347967	−	0.937507i	$-0.386872\pi$
$257$	11.1567	0.695934	0.347967	+	0.937507i	$0.386872\pi$
$258$	0	0
$259$	46.6797	2.90053
$260$	0	0
$261$	0	0
$262$	1.68665	0.104202
$263$	−8.00000	−0.493301	−0.246651	−	0.969104i	$-0.579330\pi$
$263$	−8.00000	−0.493301	−0.246651	+	0.969104i	$0.579330\pi$
$264$	0	0
$265$	−0.338044	−0.0207659
$266$	−3.66553	−0.224748
$267$	0	0
$268$	−13.9022	−0.849215
$269$	−18.6761	−1.13870	−0.569351	−	0.822095i	$-0.692805\pi$
$269$	−18.6761	−1.13870	−0.569351	+	0.822095i	$0.692805\pi$
$270$	0	0
$271$	−6.57834	−0.399606	−0.199803	−	0.979836i	$-0.564030\pi$
$271$	−6.57834	−0.399606	−0.199803	+	0.979836i	$0.564030\pi$
$272$	15.2061	0.922003
$273$	0	0
$274$	4.77384	0.288398
$275$	4.91638	0.296469
$276$	0	0
$277$	25.6655	1.54209	0.771046	−	0.636779i	$-0.219734\pi$
$277$	25.6655	1.54209	0.771046	+	0.636779i	$0.219734\pi$
$278$	−2.16724	−0.129982
$279$	0	0
$280$	5.56777	0.332738
$281$	3.15667	0.188311	0.0941557	−	0.995557i	$-0.469985\pi$
$281$	3.15667	0.188311	0.0941557	+	0.995557i	$0.469985\pi$
$282$	0	0
$283$	−3.47002	−0.206271	−0.103136	−	0.994667i	$-0.532888\pi$
$283$	−3.47002	−0.206271	−0.103136	+	0.994667i	$0.532888\pi$
$284$	−1.75614	−0.104208
$285$	0	0
$286$	0	0
$287$	−21.3275	−1.25892
$288$	0	0
$289$	1.81863	0.106978
$290$	−1.73501	−0.101883
$291$	0	0
$292$	−6.04939	−0.354014
$293$	28.6550	1.67404	0.837020	−	0.547172i	$-0.184296\pi$
$293$	28.6550	1.67404	0.837020	+	0.547172i	$0.184296\pi$
$294$	0	0
$295$	11.2544	0.655258
$296$	−10.7527	−0.624989
$297$	0	0
$298$	−5.92337	−0.343132
$299$	0	0
$300$	0	0
$301$	5.68665	0.327773
$302$	−4.74557	−0.273077
$303$	0	0
$304$	−9.03780	−0.518353
$305$	10.1708	0.582379
$306$	0	0
$307$	1.92694	0.109977	0.0549883	−	0.998487i	$-0.482488\pi$
$307$	1.92694	0.109977	0.0549883	+	0.998487i	$0.482488\pi$
$308$	46.3205	2.63935
$309$	0	0
$310$	−0.411100	−0.0233489
$311$	29.9789	1.69995	0.849973	−	0.526826i	$-0.176618\pi$
$311$	29.9789	1.69995	0.849973	+	0.526826i	$0.176618\pi$
$312$	0	0
$313$	−16.3133	−0.922085	−0.461042	−	0.887378i	$-0.652524\pi$
$313$	−16.3133	−0.922085	−0.461042	+	0.887378i	$0.652524\pi$
$314$	6.26499	0.353554
$315$	0	0
$316$	6.69721	0.376748
$317$	30.6761	1.72294	0.861470	−	0.507808i	$-0.169544\pi$
$317$	30.6761	1.72294	0.861470	+	0.507808i	$0.169544\pi$
$318$	0	0
$319$	−29.4983	−1.65159
$320$	6.06249	0.338904
$321$	0	0
$322$	9.01056	0.502139
$323$	−11.1849	−0.622347
$324$	0	0
$325$	0	0
$326$	1.75614	0.0972634
$327$	0	0
$328$	4.91281	0.271265
$329$	26.6550	1.46954
$330$	0	0
$331$	10.0978	0.555023	0.277511	−	0.960722i	$-0.410490\pi$
$331$	10.0978	0.555023	0.277511	+	0.960722i	$0.410490\pi$
$332$	−21.5678	−1.18369
$333$	0	0
$334$	0.215597	0.0117969
$335$	−7.25443	−0.396352
$336$	0	0
$337$	−1.32391	−0.0721180	−0.0360590	−	0.999350i	$-0.511480\pi$
$337$	−1.32391	−0.0721180	−0.0360590	+	0.999350i	$0.511480\pi$
$338$	0	0
$339$	0	0
$340$	8.31335	0.450855
$341$	−6.98944	−0.378499
$342$	0	0
$343$	−50.0036	−2.69994
$344$	−1.30993	−0.0706265
$345$	0	0
$346$	0.243863	0.0131102
$347$	7.49472	0.402338	0.201169	−	0.979557i	$-0.435526\pi$
$347$	7.49472	0.402338	0.201169	+	0.979557i	$0.435526\pi$
$348$	0	0
$349$	22.1461	1.18545	0.592727	−	0.805403i	$-0.298051\pi$
$349$	22.1461	1.18545	0.592727	+	0.805403i	$0.298051\pi$
$350$	1.42166	0.0759911
$351$	0	0
$352$	−16.1189	−0.859139
$353$	4.50885	0.239982	0.119991	−	0.992775i	$-0.461713\pi$
$353$	4.50885	0.239982	0.119991	+	0.992775i	$0.461713\pi$
$354$	0	0
$355$	−0.916382	−0.0486365
$356$	0.647822	0.0343345
$357$	0	0
$358$	−5.49115	−0.290216
$359$	−20.4111	−1.07726	−0.538628	−	0.842543i	$-0.681057\pi$
$359$	−20.4111	−1.07726	−0.538628	+	0.842543i	$0.681057\pi$
$360$	0	0
$361$	−12.3522	−0.650115
$362$	−5.05892	−0.265891
$363$	0	0
$364$	0	0
$365$	−3.15667	−0.165228
$366$	0	0
$367$	−10.3133	−0.538352	−0.269176	−	0.963091i	$-0.586751\pi$
$367$	−10.3133	−0.538352	−0.269176	+	0.963091i	$0.586751\pi$
$368$	22.2166	1.15812
$369$	0	0
$370$	−2.74557	−0.142736
$371$	−1.66196	−0.0862844
$372$	0	0
$373$	18.6761	0.967011	0.483506	−	0.875341i	$-0.339363\pi$
$373$	18.6761	0.967011	0.483506	+	0.875341i	$0.339363\pi$
$374$	−6.16724	−0.318900
$375$	0	0
$376$	−6.14000	−0.316646
$377$	0	0
$378$	0	0
$379$	28.7527	1.47693	0.738464	−	0.674293i	$-0.235552\pi$
$379$	28.7527	1.47693	0.738464	+	0.674293i	$0.235552\pi$
$380$	−4.94108	−0.253472
$381$	0	0
$382$	−6.50885	−0.333022
$383$	−14.2439	−0.727827	−0.363914	−	0.931433i	$-0.618560\pi$
$383$	−14.2439	−0.727827	−0.363914	+	0.931433i	$0.618560\pi$
$384$	0	0
$385$	24.1708	1.23186
$386$	0.766699	0.0390240
$387$	0	0
$388$	−23.6444	−1.20036
$389$	−34.6761	−1.75815	−0.879074	−	0.476686i	$-0.841838\pi$
$389$	−34.6761	−1.75815	−0.879074	+	0.476686i	$0.841838\pi$
$390$	0	0
$391$	27.4947	1.39047
$392$	19.4458	0.982163
$393$	0	0
$394$	3.75614	0.189231
$395$	3.49472	0.175838
$396$	0	0
$397$	−7.18137	−0.360423	−0.180211	−	0.983628i	$-0.557678\pi$
$397$	−7.18137	−0.360423	−0.180211	+	0.983628i	$0.557678\pi$
$398$	0.822200	0.0412132
$399$	0	0
$400$	3.50528	0.175264
$401$	37.8610	1.89069	0.945345	−	0.326072i	$-0.105725\pi$
$401$	37.8610	1.89069	0.945345	+	0.326072i	$0.105725\pi$
$402$	0	0
$403$	0	0
$404$	20.4595	1.01790
$405$	0	0
$406$	−8.52998	−0.423336
$407$	−46.6797	−2.31382
$408$	0	0
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	1.25443	0.0619517
$411$	0	0
$412$	27.8045	1.36983
$413$	55.3311	2.72266
$414$	0	0
$415$	−11.2544	−0.552458
$416$	0	0
$417$	0	0
$418$	3.66553	0.179287
$419$	33.4983	1.63650	0.818249	−	0.574864i	$-0.194945\pi$
$419$	33.4983	1.63650	0.818249	+	0.574864i	$0.194945\pi$
$420$	0	0
$421$	−13.5194	−0.658896	−0.329448	−	0.944174i	$-0.606863\pi$
$421$	−13.5194	−0.658896	−0.329448	+	0.944174i	$0.606863\pi$
$422$	−1.82562	−0.0888699
$423$	0	0
$424$	0.382833	0.0185920
$425$	4.33804	0.210426
$426$	0	0
$427$	50.0036	2.41984
$428$	7.99286	0.386349
$429$	0	0
$430$	−0.334474	−0.0161298
$431$	12.4111	0.597822	0.298911	−	0.954281i	$-0.403377\pi$
$431$	12.4111	0.597822	0.298911	+	0.954281i	$0.403377\pi$
$432$	0	0
$433$	−17.3239	−0.832534	−0.416267	−	0.909242i	$-0.636662\pi$
$433$	−17.3239	−0.832534	−0.416267	+	0.909242i	$0.636662\pi$
$434$	−2.02113	−0.0970171
$435$	0	0
$436$	−7.34504	−0.351763
$437$	−16.3416	−0.781725
$438$	0	0
$439$	0.651393	0.0310893	0.0155446	−	0.999879i	$-0.495052\pi$
$439$	0.651393	0.0310893	0.0155446	+	0.999879i	$0.495052\pi$
$440$	−5.56777	−0.265433
$441$	0	0
$442$	0	0
$443$	8.84690	0.420329	0.210164	−	0.977666i	$-0.432600\pi$
$443$	8.84690	0.420329	0.210164	+	0.977666i	$0.432600\pi$
$444$	0	0
$445$	0.338044	0.0160248
$446$	5.56777	0.263642
$447$	0	0
$448$	29.8055	1.40818
$449$	4.33804	0.204725	0.102362	−	0.994747i	$-0.467360\pi$
$449$	4.33804	0.204725	0.102362	+	0.994747i	$0.467360\pi$
$450$	0	0
$451$	21.3275	1.00427
$452$	−1.61613	−0.0760166
$453$	0	0
$454$	3.78440	0.177611
$455$	0	0
$456$	0	0
$457$	−15.3275	−0.716989	−0.358495	−	0.933532i	$-0.616710\pi$
$457$	−15.3275	−0.716989	−0.358495	+	0.933532i	$0.616710\pi$
$458$	−7.08719	−0.331163
$459$	0	0
$460$	12.1461	0.566315
$461$	−11.8575	−0.552257	−0.276128	−	0.961121i	$-0.589052\pi$
$461$	−11.8575	−0.552257	−0.276128	+	0.961121i	$0.589052\pi$
$462$	0	0
$463$	26.4147	1.22759	0.613797	−	0.789464i	$-0.289641\pi$
$463$	26.4147	1.22759	0.613797	+	0.789464i	$0.289641\pi$
$464$	−21.0317	−0.976372
$465$	0	0
$466$	2.41110	0.111692
$467$	33.6691	1.55802	0.779010	−	0.627012i	$-0.215722\pi$
$467$	33.6691	1.55802	0.779010	+	0.627012i	$0.215722\pi$
$468$	0	0
$469$	−35.6655	−1.64688
$470$	−1.56777	−0.0723160
$471$	0	0
$472$	−12.7456	−0.586663
$473$	−5.68665	−0.261473
$474$	0	0
$475$	−2.57834	−0.118302
$476$	40.8716	1.87335
$477$	0	0
$478$	−2.57834	−0.117930
$479$	10.7491	0.491141	0.245570	−	0.969379i	$-0.421025\pi$
$479$	10.7491	0.491141	0.245570	+	0.969379i	$0.421025\pi$
$480$	0	0
$481$	0	0
$482$	−1.73501	−0.0790276
$483$	0	0
$484$	−25.2403	−1.14729
$485$	−12.3380	−0.560242
$486$	0	0
$487$	−22.7491	−1.03086	−0.515431	−	0.856931i	$-0.672368\pi$
$487$	−22.7491	−1.03086	−0.515431	+	0.856931i	$0.672368\pi$
$488$	−11.5184	−0.521413
$489$	0	0
$490$	4.96526	0.224307
$491$	17.6867	0.798187	0.399094	−	0.916910i	$-0.369325\pi$
$491$	17.6867	0.798187	0.399094	+	0.916910i	$0.369325\pi$
$492$	0	0
$493$	−26.0283	−1.17225
$494$	0	0
$495$	0	0
$496$	−4.98333	−0.223758
$497$	−4.50528	−0.202089
$498$	0	0
$499$	−19.9305	−0.892212	−0.446106	−	0.894980i	$-0.647190\pi$
$499$	−19.9305	−0.892212	−0.446106	+	0.894980i	$0.647190\pi$
$500$	1.91638	0.0857032
$501$	0	0
$502$	−1.82562	−0.0814815
$503$	−33.3522	−1.48710	−0.743550	−	0.668680i	$-0.766860\pi$
$503$	−33.3522	−1.48710	−0.743550	+	0.668680i	$0.766860\pi$
$504$	0	0
$505$	10.6761	0.475080
$506$	−9.01056	−0.400568
$507$	0	0
$508$	−3.51227	−0.155832
$509$	13.8363	0.613285	0.306642	−	0.951825i	$-0.400794\pi$
$509$	13.8363	0.613285	0.306642	+	0.951825i	$0.400794\pi$
$510$	0	0
$511$	−15.5194	−0.686538
$512$	−19.4319	−0.858775
$513$	0	0
$514$	−3.22616	−0.142300
$515$	14.5089	0.639336
$516$	0	0
$517$	−26.6550	−1.17228
$518$	−13.4983	−0.593081
$519$	0	0
$520$	0	0
$521$	23.3522	1.02308	0.511539	−	0.859260i	$-0.329076\pi$
$521$	23.3522	1.02308	0.511539	+	0.859260i	$0.329076\pi$
$522$	0	0
$523$	−4.00000	−0.174908	−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000	−0.174908	−0.0874539	+	0.996169i	$0.527873\pi$
$524$	11.1778	0.488304
$525$	0	0
$526$	2.31335	0.100867
$527$	−6.16724	−0.268649
$528$	0	0
$529$	17.1708	0.746557
$530$	0.0977518	0.00424607
$531$	0	0
$532$	−24.2922	−1.05320
$533$	0	0
$534$	0	0
$535$	4.17081	0.180320
$536$	8.21560	0.354860
$537$	0	0
$538$	5.40054	0.232834
$539$	84.4182	3.63615
$540$	0	0
$541$	−16.1744	−0.695391	−0.347695	−	0.937608i	$-0.613036\pi$
$541$	−16.1744	−0.695391	−0.347695	+	0.937608i	$0.613036\pi$
$542$	1.90225	0.0817086
$543$	0	0
$544$	−14.2227	−0.609795
$545$	−3.83276	−0.164178
$546$	0	0
$547$	9.68665	0.414171	0.207086	−	0.978323i	$-0.433602\pi$
$547$	9.68665	0.414171	0.207086	+	0.978323i	$0.433602\pi$
$548$	31.6373	1.35148
$549$	0	0
$550$	−1.42166	−0.0606199
$551$	15.4700	0.659045
$552$	0	0
$553$	17.1814	0.730626
$554$	−7.42166	−0.315316
$555$	0	0
$556$	−14.3627	−0.609116
$557$	0.647822	0.0274491	0.0137246	−	0.999906i	$-0.495631\pi$
$557$	0.647822	0.0274491	0.0137246	+	0.999906i	$0.495631\pi$
$558$	0	0
$559$	0	0
$560$	17.2333	0.728240
$561$	0	0
$562$	−0.912811	−0.0385046
$563$	16.3169	0.687676	0.343838	−	0.939029i	$-0.388273\pi$
$563$	16.3169	0.687676	0.343838	+	0.939029i	$0.388273\pi$
$564$	0	0
$565$	−0.843326	−0.0354790
$566$	1.00342	0.0421769
$567$	0	0
$568$	1.03780	0.0435450
$569$	−44.6550	−1.87203	−0.936017	−	0.351956i	$-0.885517\pi$
$569$	−44.6550	−1.87203	−0.936017	+	0.351956i	$0.885517\pi$
$570$	0	0
$571$	−6.67252	−0.279236	−0.139618	−	0.990205i	$-0.544587\pi$
$571$	−6.67252	−0.279236	−0.139618	+	0.990205i	$0.544587\pi$
$572$	0	0
$573$	0	0
$574$	6.16724	0.257415
$575$	6.33804	0.264315
$576$	0	0
$577$	−15.3275	−0.638091	−0.319046	−	0.947739i	$-0.603362\pi$
$577$	−15.3275	−0.638091	−0.319046	+	0.947739i	$0.603362\pi$
$578$	−0.525891	−0.0218742
$579$	0	0
$580$	−11.4983	−0.477440
$581$	−55.3311	−2.29552
$582$	0	0
$583$	1.66196	0.0688312
$584$	3.57492	0.147931
$585$	0	0
$586$	−8.28611	−0.342296
$587$	12.2650	0.506230	0.253115	−	0.967436i	$-0.418545\pi$
$587$	12.2650	0.506230	0.253115	+	0.967436i	$0.418545\pi$
$588$	0	0
$589$	3.66553	0.151035
$590$	−3.25443	−0.133983
$591$	0	0
$592$	−33.2817	−1.36787
$593$	−5.85389	−0.240390	−0.120195	−	0.992750i	$-0.538352\pi$
$593$	−5.85389	−0.240390	−0.120195	+	0.992750i	$0.538352\pi$
$594$	0	0
$595$	21.3275	0.874342
$596$	−39.2555	−1.60797
$597$	0	0
$598$	0	0
$599$	−27.1355	−1.10873	−0.554364	−	0.832274i	$-0.687039\pi$
$599$	−27.1355	−1.10873	−0.554364	+	0.832274i	$0.687039\pi$
$600$	0	0
$601$	34.1708	1.39386	0.696928	−	0.717141i	$-0.254550\pi$
$601$	34.1708	1.39386	0.696928	+	0.717141i	$0.254550\pi$
$602$	−1.64440	−0.0670208
$603$	0	0
$604$	−31.4499	−1.27968
$605$	−13.1708	−0.535469
$606$	0	0
$607$	10.3133	0.418606	0.209303	−	0.977851i	$-0.432881\pi$
$607$	10.3133	0.418606	0.209303	+	0.977851i	$0.432881\pi$
$608$	8.45335	0.342829
$609$	0	0
$610$	−2.94108	−0.119081
$611$	0	0
$612$	0	0
$613$	−0.484156	−0.0195549	−0.00977744	−	0.999952i	$-0.503112\pi$
$613$	−0.484156	−0.0195549	−0.00977744	+	0.999952i	$0.503112\pi$
$614$	−0.557212	−0.0224872
$615$	0	0
$616$	−27.3733	−1.10290
$617$	−7.15667	−0.288117	−0.144058	−	0.989569i	$-0.546015\pi$
$617$	−7.15667	−0.288117	−0.144058	+	0.989569i	$0.546015\pi$
$618$	0	0
$619$	−5.42166	−0.217915	−0.108958	−	0.994046i	$-0.534751\pi$
$619$	−5.42166	−0.217915	−0.108958	+	0.994046i	$0.534751\pi$
$620$	−2.72445	−0.109416
$621$	0	0
$622$	−8.66895	−0.347593
$623$	1.66196	0.0665848
$624$	0	0
$625$	1.00000	0.0400000
$626$	4.71731	0.188542
$627$	0	0
$628$	41.5194	1.65681
$629$	−41.1885	−1.64229
$630$	0	0
$631$	10.7244	0.426934	0.213467	−	0.976950i	$-0.431524\pi$
$631$	10.7244	0.426934	0.213467	+	0.976950i	$0.431524\pi$
$632$	−3.95775	−0.157431
$633$	0	0
$634$	−8.87056	−0.352295
$635$	−1.83276	−0.0727310
$636$	0	0
$637$	0	0
$638$	8.52998	0.337705
$639$	0	0
$640$	−8.31029	−0.328493
$641$	0.362741	0.0143274	0.00716370	−	0.999974i	$-0.497720\pi$
$641$	0.362741	0.0143274	0.00716370	+	0.999974i	$0.497720\pi$
$642$	0	0
$643$	9.39697	0.370580	0.185290	−	0.982684i	$-0.440678\pi$
$643$	9.39697	0.370580	0.185290	+	0.982684i	$0.440678\pi$
$644$	59.7149	2.35310
$645$	0	0
$646$	3.23433	0.127253
$647$	−18.0036	−0.707793	−0.353897	−	0.935285i	$-0.615144\pi$
$647$	−18.0036	−0.707793	−0.353897	+	0.935285i	$0.615144\pi$
$648$	0	0
$649$	−55.3311	−2.17193
$650$	0	0
$651$	0	0
$652$	11.6383	0.455791
$653$	34.8222	1.36270	0.681349	−	0.731959i	$-0.261394\pi$
$653$	34.8222	1.36270	0.681349	+	0.731959i	$0.261394\pi$
$654$	0	0
$655$	5.83276	0.227905
$656$	15.2061	0.593697
$657$	0	0
$658$	−7.70778	−0.300480
$659$	−11.4700	−0.446809	−0.223404	−	0.974726i	$-0.571717\pi$
$659$	−11.4700	−0.446809	−0.223404	+	0.974726i	$0.571717\pi$
$660$	0	0
$661$	−12.1672	−0.473251	−0.236625	−	0.971601i	$-0.576041\pi$
$661$	−12.1672	−0.473251	−0.236625	+	0.971601i	$0.576041\pi$
$662$	−2.91995	−0.113487
$663$	0	0
$664$	12.7456	0.494624
$665$	−12.6761	−0.491558
$666$	0	0
$667$	−38.0283	−1.47246
$668$	1.42880	0.0552821
$669$	0	0
$670$	2.09775	0.0810432
$671$	−50.0036	−1.93037
$672$	0	0
$673$	−27.9789	−1.07851	−0.539253	−	0.842144i	$-0.681293\pi$
$673$	−27.9789	−1.07851	−0.539253	+	0.842144i	$0.681293\pi$
$674$	0.382833	0.0147462
$675$	0	0
$676$	0	0
$677$	−22.9930	−0.883693	−0.441847	−	0.897091i	$-0.645676\pi$
$677$	−22.9930	−0.883693	−0.441847	+	0.897091i	$0.645676\pi$
$678$	0	0
$679$	−60.6585	−2.32786
$680$	−4.91281	−0.188398
$681$	0	0
$682$	2.02113	0.0773929
$683$	−28.6066	−1.09460	−0.547301	−	0.836936i	$-0.684345\pi$
$683$	−28.6066	−1.09460	−0.547301	+	0.836936i	$0.684345\pi$
$684$	0	0
$685$	16.5089	0.630771
$686$	14.4595	0.552065
$687$	0	0
$688$	−4.05447	−0.154575
$689$	0	0
$690$	0	0
$691$	19.4005	0.738031	0.369016	−	0.929423i	$-0.379695\pi$
$691$	19.4005	0.738031	0.369016	+	0.929423i	$0.379695\pi$
$692$	1.61613	0.0614362
$693$	0	0
$694$	−2.16724	−0.0822672
$695$	−7.49472	−0.284291
$696$	0	0
$697$	18.8186	0.712806
$698$	−6.40396	−0.242393
$699$	0	0
$700$	9.42166	0.356105
$701$	38.9683	1.47181	0.735906	−	0.677083i	$-0.236756\pi$
$701$	38.9683	1.47181	0.735906	+	0.677083i	$0.236756\pi$
$702$	0	0
$703$	24.4806	0.923303
$704$	−29.8055	−1.12334
$705$	0	0
$706$	−1.30382	−0.0490698
$707$	52.4877	1.97400
$708$	0	0
$709$	17.5194	0.657955	0.328978	−	0.944338i	$-0.393296\pi$
$709$	17.5194	0.657955	0.328978	+	0.944338i	$0.393296\pi$
$710$	0.264989	0.00994485
$711$	0	0
$712$	−0.382833	−0.0143473
$713$	−9.01056	−0.337448
$714$	0	0
$715$	0	0
$716$	−36.3910	−1.36000
$717$	0	0
$718$	5.90225	0.220270
$719$	4.33447	0.161649	0.0808243	−	0.996728i	$-0.474245\pi$
$719$	4.33447	0.161649	0.0808243	+	0.996728i	$0.474245\pi$
$720$	0	0
$721$	71.3311	2.65651
$722$	3.57186	0.132931
$723$	0	0
$724$	−33.5266	−1.24600
$725$	−6.00000	−0.222834
$726$	0	0
$727$	22.1672	0.822137	0.411069	−	0.911604i	$-0.365156\pi$
$727$	22.1672	0.822137	0.411069	+	0.911604i	$0.365156\pi$
$728$	0	0
$729$	0	0
$730$	0.912811	0.0337846
$731$	−5.01770	−0.185586
$732$	0	0
$733$	2.83976	0.104889	0.0524444	−	0.998624i	$-0.483299\pi$
$733$	2.83976	0.104889	0.0524444	+	0.998624i	$0.483299\pi$
$734$	2.98230	0.110079
$735$	0	0
$736$	−20.7799	−0.765959
$737$	35.6655	1.31376
$738$	0	0
$739$	43.9305	1.61601	0.808005	−	0.589176i	$-0.200547\pi$
$739$	43.9305	1.61601	0.808005	+	0.589176i	$0.200547\pi$
$740$	−18.1955	−0.668880
$741$	0	0
$742$	0.480585	0.0176428
$743$	41.0872	1.50734	0.753671	−	0.657251i	$-0.228281\pi$
$743$	41.0872	1.50734	0.753671	+	0.657251i	$0.228281\pi$
$744$	0	0
$745$	−20.4842	−0.750481
$746$	−5.40054	−0.197728
$747$	0	0
$748$	−40.8716	−1.49441
$749$	20.5053	0.749247
$750$	0	0
$751$	23.6902	0.864468	0.432234	−	0.901761i	$-0.357725\pi$
$751$	23.6902	0.864468	0.432234	+	0.901761i	$0.357725\pi$
$752$	−19.0045	−0.693021
$753$	0	0
$754$	0	0
$755$	−16.4111	−0.597261
$756$	0	0
$757$	9.32391	0.338883	0.169442	−	0.985540i	$-0.445804\pi$
$757$	9.32391	0.338883	0.169442	+	0.985540i	$0.445804\pi$
$758$	−8.31438	−0.301992
$759$	0	0
$760$	2.91995	0.105918
$761$	−42.8222	−1.55230	−0.776152	−	0.630546i	$-0.782831\pi$
$761$	−42.8222	−1.55230	−0.776152	+	0.630546i	$0.782831\pi$
$762$	0	0
$763$	−18.8433	−0.682174
$764$	−43.1355	−1.56059
$765$	0	0
$766$	4.11888	0.148821
$767$	0	0
$768$	0	0
$769$	17.3239	0.624716	0.312358	−	0.949964i	$-0.398881\pi$
$769$	17.3239	0.624716	0.312358	+	0.949964i	$0.398881\pi$
$770$	−6.98944	−0.251882
$771$	0	0
$772$	5.08108	0.182872
$773$	−11.6373	−0.418563	−0.209282	−	0.977855i	$-0.567113\pi$
$773$	−11.6373	−0.418563	−0.209282	+	0.977855i	$0.567113\pi$
$774$	0	0
$775$	−1.42166	−0.0510676
$776$	13.9728	0.501593
$777$	0	0
$778$	10.0272	0.359494
$779$	−11.1849	−0.400742
$780$	0	0
$781$	4.50528	0.161212
$782$	−7.95061	−0.284313
$783$	0	0
$784$	60.1885	2.14959
$785$	21.6655	0.773276
$786$	0	0
$787$	−20.9411	−0.746469	−0.373234	−	0.927737i	$-0.621751\pi$
$787$	−20.9411	−0.746469	−0.373234	+	0.927737i	$0.621751\pi$
$788$	24.8927	0.886766
$789$	0	0
$790$	−1.01056	−0.0359542
$791$	−4.14611	−0.147419
$792$	0	0
$793$	0	0
$794$	2.07663	0.0736967
$795$	0	0
$796$	5.44890	0.193131
$797$	−20.3380	−0.720410	−0.360205	−	0.932873i	$-0.617293\pi$
$797$	−20.3380	−0.720410	−0.360205	+	0.932873i	$0.617293\pi$
$798$	0	0
$799$	−23.5194	−0.832057
$800$	−3.27861	−0.115916
$801$	0	0
$802$	−10.9482	−0.386595
$803$	15.5194	0.547668
$804$	0	0
$805$	31.1602	1.09825
$806$	0	0
$807$	0	0
$808$	−12.0906	−0.425346
$809$	−7.68665	−0.270248	−0.135124	−	0.990829i	$-0.543143\pi$
$809$	−7.68665	−0.270248	−0.135124	+	0.990829i	$0.543143\pi$
$810$	0	0
$811$	44.4111	1.55948	0.779742	−	0.626101i	$-0.215350\pi$
$811$	44.4111	1.55948	0.779742	+	0.626101i	$0.215350\pi$
$812$	−56.5300	−1.98381
$813$	0	0
$814$	13.4983	0.473115
$815$	6.07306	0.212730
$816$	0	0
$817$	2.98230	0.104337
$818$	−4.04836	−0.141548
$819$	0	0
$820$	8.31335	0.290315
$821$	−46.4630	−1.62157	−0.810785	−	0.585343i	$-0.800960\pi$
$821$	−46.4630	−1.62157	−0.810785	+	0.585343i	$0.800960\pi$
$822$	0	0
$823$	46.5089	1.62120	0.810598	−	0.585603i	$-0.199142\pi$
$823$	46.5089	1.62120	0.810598	+	0.585603i	$0.199142\pi$
$824$	−16.4312	−0.572408
$825$	0	0
$826$	−16.0000	−0.556711
$827$	39.4005	1.37009	0.685045	−	0.728500i	$-0.259782\pi$
$827$	39.4005	1.37009	0.685045	+	0.728500i	$0.259782\pi$
$828$	0	0
$829$	47.6444	1.65476	0.827379	−	0.561644i	$-0.189831\pi$
$829$	47.6444	1.65476	0.827379	+	0.561644i	$0.189831\pi$
$830$	3.25443	0.112963
$831$	0	0
$832$	0	0
$833$	74.4877	2.58085
$834$	0	0
$835$	0.745574	0.0258017
$836$	24.2922	0.840164
$837$	0	0
$838$	−9.68665	−0.334620
$839$	−39.9058	−1.37770	−0.688851	−	0.724903i	$-0.741885\pi$
$839$	−39.9058	−1.37770	−0.688851	+	0.724903i	$0.741885\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	3.90939	0.134726
$843$	0	0
$844$	−12.0988	−0.416457
$845$	0	0
$846$	0	0
$847$	−64.7527	−2.22493
$848$	1.18494	0.0406910
$849$	0	0
$850$	−1.25443	−0.0430265
$851$	−60.1779	−2.06287
$852$	0	0
$853$	−29.5019	−1.01012	−0.505062	−	0.863083i	$-0.668530\pi$
$853$	−29.5019	−1.01012	−0.505062	+	0.863083i	$0.668530\pi$
$854$	−14.4595	−0.494793
$855$	0	0
$856$	−4.72342	−0.161443
$857$	−8.33804	−0.284822	−0.142411	−	0.989808i	$-0.545485\pi$
$857$	−8.33804	−0.284822	−0.142411	+	0.989808i	$0.545485\pi$
$858$	0	0
$859$	−4.17081	−0.142306	−0.0711531	−	0.997465i	$-0.522668\pi$
$859$	−4.17081	−0.142306	−0.0711531	+	0.997465i	$0.522668\pi$
$860$	−2.21663	−0.0755864
$861$	0	0
$862$	−3.58890	−0.122238
$863$	−3.93051	−0.133796	−0.0668981	−	0.997760i	$-0.521310\pi$
$863$	−3.93051	−0.133796	−0.0668981	+	0.997760i	$0.521310\pi$
$864$	0	0
$865$	0.843326	0.0286739
$866$	5.00953	0.170231
$867$	0	0
$868$	−13.3944	−0.454637
$869$	−17.1814	−0.582838
$870$	0	0
$871$	0	0
$872$	4.34058	0.146991
$873$	0	0
$874$	4.72548	0.159842
$875$	4.91638	0.166204
$876$	0	0
$877$	23.0177	0.777253	0.388626	−	0.921395i	$-0.372950\pi$
$877$	23.0177	0.777253	0.388626	+	0.921395i	$0.372950\pi$
$878$	−0.188362	−0.00635692
$879$	0	0
$880$	−17.2333	−0.580934
$881$	15.3522	0.517228	0.258614	−	0.965981i	$-0.416734\pi$
$881$	15.3522	0.517228	0.258614	+	0.965981i	$0.416734\pi$
$882$	0	0
$883$	42.8011	1.44037	0.720185	−	0.693782i	$-0.244057\pi$
$883$	42.8011	1.44037	0.720185	+	0.693782i	$0.244057\pi$
$884$	0	0
$885$	0	0
$886$	−2.55824	−0.0859459
$887$	53.1885	1.78590	0.892948	−	0.450160i	$-0.148633\pi$
$887$	53.1885	1.78590	0.892948	+	0.450160i	$0.148633\pi$
$888$	0	0
$889$	−9.01056	−0.302205
$890$	−0.0977518	−0.00327665
$891$	0	0
$892$	36.8988	1.23546
$893$	13.9789	0.467785
$894$	0	0
$895$	−18.9894	−0.634747
$896$	−40.8566	−1.36492
$897$	0	0
$898$	−1.25443	−0.0418607
$899$	8.52998	0.284491
$900$	0	0
$901$	1.46645	0.0488546
$902$	−6.16724	−0.205347
$903$	0	0
$904$	0.955062	0.0317649
$905$	−17.4947	−0.581544
$906$	0	0
$907$	11.8116	0.392199	0.196099	−	0.980584i	$-0.437172\pi$
$907$	11.8116	0.392199	0.196099	+	0.980584i	$0.437172\pi$
$908$	25.0800	0.832311
$909$	0	0
$910$	0	0
$911$	−44.1955	−1.46426	−0.732131	−	0.681164i	$-0.761474\pi$
$911$	−44.1955	−1.46426	−0.732131	+	0.681164i	$0.761474\pi$
$912$	0	0
$913$	55.3311	1.83119
$914$	4.43223	0.146605
$915$	0	0
$916$	−46.9683	−1.55188
$917$	28.6761	0.946968
$918$	0	0
$919$	55.2096	1.82120	0.910599	−	0.413291i	$-0.135621\pi$
$919$	55.2096	1.82120	0.910599	+	0.413291i	$0.135621\pi$
$920$	−7.17780	−0.236645
$921$	0	0
$922$	3.42880	0.112922
$923$	0	0
$924$	0	0
$925$	−9.49472	−0.312184
$926$	−7.63829	−0.251010
$927$	0	0
$928$	19.6716	0.645753
$929$	22.9930	0.754376	0.377188	−	0.926137i	$-0.376891\pi$
$929$	22.9930	0.754376	0.377188	+	0.926137i	$0.376891\pi$
$930$	0	0
$931$	−44.2721	−1.45096
$932$	15.9789	0.523405
$933$	0	0
$934$	−9.73604	−0.318573
$935$	−21.3275	−0.697483
$936$	0	0
$937$	7.97887	0.260658	0.130329	−	0.991471i	$-0.458397\pi$
$937$	7.97887	0.260658	0.130329	+	0.991471i	$0.458397\pi$
$938$	10.3133	0.336743
$939$	0	0
$940$	−10.3900	−0.338884
$941$	41.5019	1.35292	0.676461	−	0.736478i	$-0.263513\pi$
$941$	41.5019	1.35292	0.676461	+	0.736478i	$0.263513\pi$
$942$	0	0
$943$	27.4947	0.895351
$944$	−39.4499	−1.28399
$945$	0	0
$946$	1.64440	0.0534641
$947$	47.4499	1.54192	0.770958	−	0.636886i	$-0.219778\pi$
$947$	47.4499	1.54192	0.770958	+	0.636886i	$0.219778\pi$
$948$	0	0
$949$	0	0
$950$	0.745574	0.0241896
$951$	0	0
$952$	−24.1533	−0.782811
$953$	−30.3663	−0.983661	−0.491831	−	0.870691i	$-0.663672\pi$
$953$	−30.3663	−0.983661	−0.491831	+	0.870691i	$0.663672\pi$
$954$	0	0
$955$	−22.5089	−0.728369
$956$	−17.0872	−0.552639
$957$	0	0
$958$	−3.10831	−0.100425
$959$	81.1638	2.62092
$960$	0	0
$961$	−28.9789	−0.934802
$962$	0	0
$963$	0	0
$964$	−11.4983	−0.370335
$965$	2.65139	0.0853514
$966$	0	0
$967$	41.0943	1.32150	0.660752	−	0.750604i	$-0.270237\pi$
$967$	41.0943	1.32150	0.660752	+	0.750604i	$0.270237\pi$
$968$	14.9159	0.479414
$969$	0	0
$970$	3.56777	0.114554
$971$	−38.1744	−1.22507	−0.612537	−	0.790442i	$-0.709851\pi$
$971$	−38.1744	−1.22507	−0.612537	+	0.790442i	$0.709851\pi$
$972$	0	0
$973$	−36.8469	−1.18126
$974$	6.57834	0.210784
$975$	0	0
$976$	−35.6515	−1.14118
$977$	−10.4806	−0.335304	−0.167652	−	0.985846i	$-0.553618\pi$
$977$	−10.4806	−0.335304	−0.167652	+	0.985846i	$0.553618\pi$
$978$	0	0
$979$	−1.66196	−0.0531163
$980$	32.9058	1.05114
$981$	0	0
$982$	−5.11442	−0.163208
$983$	−0.0766264	−0.00244400	−0.00122200	−	0.999999i	$-0.500389\pi$
$983$	−0.0766264	−0.00244400	−0.00122200	+	0.999999i	$0.500389\pi$
$984$	0	0
$985$	12.9894	0.413878
$986$	7.52656	0.239694
$987$	0	0
$988$	0	0
$989$	−7.33105	−0.233114
$990$	0	0
$991$	−13.8575	−0.440197	−0.220098	−	0.975478i	$-0.570638\pi$
$991$	−13.8575	−0.440197	−0.220098	+	0.975478i	$0.570638\pi$
$992$	4.66107	0.147989
$993$	0	0
$994$	1.30279	0.0413219
$995$	2.84333	0.0901395
$996$	0	0
$997$	−10.3416	−0.327522	−0.163761	−	0.986500i	$-0.552363\pi$
$997$	−10.3416	−0.327522	−0.163761	+	0.986500i	$0.552363\pi$
$998$	5.76328	0.182433
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7605.2.a.bx.1.2		3
3.2	odd	2		2535.2.a.bc.1.2		3
13.12	even	2		585.2.a.n.1.2		3
39.38	odd	2		195.2.a.e.1.2	✓	3
52.51	odd	2		9360.2.a.dd.1.1		3
65.12	odd	4		2925.2.c.w.2224.4		6
65.38	odd	4		2925.2.c.w.2224.3		6
65.64	even	2		2925.2.a.bh.1.2		3
156.155	even	2		3120.2.a.bj.1.1		3
195.38	even	4		975.2.c.i.274.4		6
195.77	even	4		975.2.c.i.274.3		6
195.194	odd	2		975.2.a.o.1.2		3
273.272	even	2		9555.2.a.bq.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
195.2.a.e.1.2	✓	3	39.38	odd	2
585.2.a.n.1.2		3	13.12	even	2
975.2.a.o.1.2		3	195.194	odd	2
975.2.c.i.274.3		6	195.77	even	4
975.2.c.i.274.4		6	195.38	even	4
2535.2.a.bc.1.2		3	3.2	odd	2
2925.2.a.bh.1.2		3	65.64	even	2
2925.2.c.w.2224.3		6	65.38	odd	4
2925.2.c.w.2224.4		6	65.12	odd	4
3120.2.a.bj.1.1		3	156.155	even	2
7605.2.a.bx.1.2		3	1.1	even	1	trivial
9360.2.a.dd.1.1		3	52.51	odd	2
9555.2.a.bq.1.2		3	273.272	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-0.289169 q^{2} -1.91638 q^{4} -1.00000 q^{5} -4.91638 q^{7} +1.13249 q^{8} +O(q^{10})\)	\(q-0.289169 q^{2} -1.91638 q^{4} -1.00000 q^{5} -4.91638 q^{7} +1.13249 q^{8} +0.289169 q^{10} +4.91638 q^{11} +1.42166 q^{14} +3.50528 q^{16} +4.33804 q^{17} -2.57834 q^{19} +1.91638 q^{20} -1.42166 q^{22} +6.33804 q^{23} +1.00000 q^{25} +9.42166 q^{28} -6.00000 q^{29} -1.42166 q^{31} -3.27861 q^{32} -1.25443 q^{34} +4.91638 q^{35} -9.49472 q^{37} +0.745574 q^{38} -1.13249 q^{40} +4.33804 q^{41} -1.15667 q^{43} -9.42166 q^{44} -1.83276 q^{46} -5.42166 q^{47} +17.1708 q^{49} -0.289169 q^{50} +0.338044 q^{53} -4.91638 q^{55} -5.56777 q^{56} +1.73501 q^{58} -11.2544 q^{59} -10.1708 q^{61} +0.411100 q^{62} -6.06249 q^{64} +7.25443 q^{67} -8.31335 q^{68} -1.42166 q^{70} +0.916382 q^{71} +3.15667 q^{73} +2.74557 q^{74} +4.94108 q^{76} -24.1708 q^{77} -3.49472 q^{79} -3.50528 q^{80} -1.25443 q^{82} +11.2544 q^{83} -4.33804 q^{85} +0.334474 q^{86} +5.56777 q^{88} -0.338044 q^{89} -12.1461 q^{92} +1.56777 q^{94} +2.57834 q^{95} +12.3380 q^{97} -4.96526 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 8 q^{4} - 3 q^{5} - q^{7} + 6 q^{8}+O(q^{10}) \) 3 * q + 8 * q^4 - 3 * q^5 - q^7 + 6 * q^8	\( 3 q + 8 q^{4} - 3 q^{5} - q^{7} + 6 q^{8} + q^{11} + 6 q^{14} + 26 q^{16} + q^{17} - 6 q^{19} - 8 q^{20} - 6 q^{22} + 7 q^{23} + 3 q^{25} + 30 q^{28} - 18 q^{29} - 6 q^{31} + 22 q^{32} + 22 q^{34} + q^{35} - 13 q^{37} + 28 q^{38} - 6 q^{40} + q^{41} - 30 q^{44} + 22 q^{46} - 18 q^{47} + 12 q^{49} - 11 q^{53} - q^{55} + 16 q^{56} - 8 q^{59} + 9 q^{61} - 28 q^{62} + 30 q^{64} - 4 q^{67} - 18 q^{68} - 6 q^{70} - 11 q^{71} + 6 q^{73} + 34 q^{74} - 4 q^{76} - 33 q^{77} + 5 q^{79} - 26 q^{80} + 22 q^{82} + 8 q^{83} - q^{85} + 56 q^{86} - 16 q^{88} + 11 q^{89} - 2 q^{92} - 28 q^{94} + 6 q^{95} + 25 q^{97} + 10 q^{98}+O(q^{100}) \) 3 * q + 8 * q^4 - 3 * q^5 - q^7 + 6 * q^8 + q^11 + 6 * q^14 + 26 * q^16 + q^17 - 6 * q^19 - 8 * q^20 - 6 * q^22 + 7 * q^23 + 3 * q^25 + 30 * q^28 - 18 * q^29 - 6 * q^31 + 22 * q^32 + 22 * q^34 + q^35 - 13 * q^37 + 28 * q^38 - 6 * q^40 + q^41 - 30 * q^44 + 22 * q^46 - 18 * q^47 + 12 * q^49 - 11 * q^53 - q^55 + 16 * q^56 - 8 * q^59 + 9 * q^61 - 28 * q^62 + 30 * q^64 - 4 * q^67 - 18 * q^68 - 6 * q^70 - 11 * q^71 + 6 * q^73 + 34 * q^74 - 4 * q^76 - 33 * q^77 + 5 * q^79 - 26 * q^80 + 22 * q^82 + 8 * q^83 - q^85 + 56 * q^86 - 16 * q^88 + 11 * q^89 - 2 * q^92 - 28 * q^94 + 6 * q^95 + 25 * q^97 + 10 * q^98

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