LMFDB - Embedded newform 7605.2.a.ch.1.1

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 = [4,0,0,4,-4,0,0,0,0,0,0] 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:	$1$
Dimension:	$4$
Coefficient field:	4.4.13824.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} + 6 $ x^4 - 6*x^2 + 6
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 195)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.17533$ 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)

$f(q)$	$=$	$q-2.17533 q^{2} +2.73205 q^{4} -1.00000 q^{5} -3.90738 q^{7} -1.59245 q^{8} +2.17533 q^{10} -1.59245 q^{11} +8.49983 q^{14} -2.00000 q^{16} +3.76778 q^{17} -7.91695 q^{19} -2.73205 q^{20} +3.46410 q^{22} -6.22231 q^{23} +1.00000 q^{25} -10.6752 q^{28} +5.03573 q^{29} +0.184901 q^{31} +7.53556 q^{32} -8.19615 q^{34} +3.90738 q^{35} -1.64934 q^{37} +17.2220 q^{38} +1.59245 q^{40} +10.3715 q^{41} +6.74162 q^{43} -4.35066 q^{44} +13.5356 q^{46} +6.58288 q^{47} +8.26761 q^{49} -2.17533 q^{50} +5.51641 q^{53} +1.59245 q^{55} +6.22231 q^{56} -10.9544 q^{58} -4.32983 q^{59} -9.73205 q^{61} -0.402220 q^{62} -12.3923 q^{64} +6.16043 q^{67} +10.2938 q^{68} -8.49983 q^{70} -1.24347 q^{71} +2.25803 q^{73} +3.58786 q^{74} -21.6295 q^{76} +6.22231 q^{77} -4.33940 q^{79} +2.00000 q^{80} -22.5614 q^{82} +9.12801 q^{83} -3.76778 q^{85} -14.6652 q^{86} +2.53590 q^{88} -9.14883 q^{89} -16.9997 q^{92} -14.3199 q^{94} +7.91695 q^{95} -9.58288 q^{97} -17.9848 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{4} - 4 q^{5} + 12 q^{14} - 8 q^{16} - 12 q^{19} - 4 q^{20} + 4 q^{25} - 12 q^{28} + 12 q^{29} - 12 q^{31} - 12 q^{34} - 24 q^{37} + 12 q^{41} + 16 q^{43} + 24 q^{46} + 24 q^{47} - 4 q^{49} - 12 q^{58}+ \cdots - 24 q^{98}+O(q^{100}) $ 4 * q + 4 * q^4 - 4 * q^5 + 12 * q^14 - 8 * q^16 - 12 * q^19 - 4 * q^20 + 4 * q^25 - 12 * q^28 + 12 * q^29 - 12 * q^31 - 12 * q^34 - 24 * q^37 + 12 * q^41 + 16 * q^43 + 24 * q^46 + 24 * q^47 - 4 * q^49 - 12 * q^58 - 12 * q^59 - 32 * q^61 - 8 * q^64 + 12 * q^67 - 12 * q^70 - 12 * q^71 - 24 * q^73 - 24 * q^74 - 24 * q^76 - 8 * q^79 + 8 * q^80 - 12 * q^82 + 12 * q^86 + 24 * q^88 + 12 * q^89 - 24 * q^92 - 12 * q^94 + 12 * q^95 - 36 * 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 $

$2$	−2.17533	−1.53819	−0.769095	−	0.639135i	$-0.779292\pi$
$2$	−2.17533	−1.53819	−0.769095	+	0.639135i	$0.779292\pi$
$3$	0	0
$3$	0	0
$4$	2.73205	1.36603
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−3.90738	−1.47685	−0.738425	−	0.674335i	$-0.764430\pi$
$7$	−3.90738	−1.47685	−0.738425	+	0.674335i	$0.764430\pi$
$8$	−1.59245	−0.563016
$9$	0	0
$10$	2.17533	0.687899
$11$	−1.59245	−0.480142	−0.240071	−	0.970755i	$-0.577171\pi$
$11$	−1.59245	−0.480142	−0.240071	+	0.970755i	$0.577171\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	8.49983	2.27167
$15$	0	0
$16$	−2.00000	−0.500000
$17$	3.76778	0.913820	0.456910	−	0.889513i	$-0.348956\pi$
$17$	3.76778	0.913820	0.456910	+	0.889513i	$0.348956\pi$
$18$	0	0
$19$	−7.91695	−1.81627	−0.908137	−	0.418674i	$-0.862495\pi$
$19$	−7.91695	−1.81627	−0.908137	+	0.418674i	$0.862495\pi$
$20$	−2.73205	−0.610905
$21$	0	0
$22$	3.46410	0.738549
$23$	−6.22231	−1.29744	−0.648720	−	0.761027i	$-0.724695\pi$
$23$	−6.22231	−1.29744	−0.648720	+	0.761027i	$0.724695\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	−10.6752	−2.01741
$29$	5.03573	0.935111	0.467556	−	0.883964i	$-0.345135\pi$
$29$	5.03573	0.935111	0.467556	+	0.883964i	$0.345135\pi$
$30$	0	0
$31$	0.184901	0.0332092	0.0166046	−	0.999862i	$-0.494714\pi$
$31$	0.184901	0.0332092	0.0166046	+	0.999862i	$0.494714\pi$
$32$	7.53556	1.33211
$33$	0	0
$34$	−8.19615	−1.40563
$35$	3.90738	0.660468
$36$	0	0
$37$	−1.64934	−0.271151	−0.135575	−	0.990767i	$-0.543288\pi$
$37$	−1.64934	−0.271151	−0.135575	+	0.990767i	$0.543288\pi$
$38$	17.2220	2.79377
$39$	0	0
$40$	1.59245	0.251789
$41$	10.3715	1.61975	0.809877	−	0.586600i	$-0.199534\pi$
$41$	10.3715	1.61975	0.809877	+	0.586600i	$0.199534\pi$
$42$	0	0
$43$	6.74162	1.02809	0.514044	−	0.857764i	$-0.328147\pi$
$43$	6.74162	1.02809	0.514044	+	0.857764i	$0.328147\pi$
$44$	−4.35066	−0.655886
$45$	0	0
$46$	13.5356	1.99571
$47$	6.58288	0.960211	0.480106	−	0.877211i	$-0.340598\pi$
$47$	6.58288	0.960211	0.480106	+	0.877211i	$0.340598\pi$
$48$	0	0
$49$	8.26761	1.18109
$50$	−2.17533	−0.307638
$51$	0	0
$52$	0	0
$53$	5.51641	0.757737	0.378869	−	0.925450i	$-0.376313\pi$
$53$	5.51641	0.757737	0.378869	+	0.925450i	$0.376313\pi$
$54$	0	0
$55$	1.59245	0.214726
$56$	6.22231	0.831491
$57$	0	0
$58$	−10.9544	−1.43838
$59$	−4.32983	−0.563696	−0.281848	−	0.959459i	$-0.590947\pi$
$59$	−4.32983	−0.563696	−0.281848	+	0.959459i	$0.590947\pi$
$60$	0	0
$61$	−9.73205	−1.24606	−0.623031	−	0.782197i	$-0.714099\pi$
$61$	−9.73205	−1.24606	−0.623031	+	0.782197i	$0.714099\pi$
$62$	−0.402220	−0.0510820
$63$	0	0
$64$	−12.3923	−1.54904
$65$	0	0
$66$	0	0
$67$	6.16043	0.752616	0.376308	−	0.926495i	$-0.377194\pi$
$67$	6.16043	0.752616	0.376308	+	0.926495i	$0.377194\pi$
$68$	10.2938	1.24830
$69$	0	0
$70$	−8.49983	−1.01592
$71$	−1.24347	−0.147573	−0.0737866	−	0.997274i	$-0.523508\pi$
$71$	−1.24347	−0.147573	−0.0737866	+	0.997274i	$0.523508\pi$
$72$	0	0
$73$	2.25803	0.264283	0.132141	−	0.991231i	$-0.457815\pi$
$73$	2.25803	0.264283	0.132141	+	0.991231i	$0.457815\pi$
$74$	3.58786	0.417081
$75$	0	0
$76$	−21.6295	−2.48108
$77$	6.22231	0.709098
$78$	0	0
$79$	−4.33940	−0.488221	−0.244111	−	0.969747i	$-0.578496\pi$
$79$	−4.33940	−0.488221	−0.244111	+	0.969747i	$0.578496\pi$
$80$	2.00000	0.223607
$81$	0	0
$82$	−22.5614	−2.49149
$83$	9.12801	1.00193	0.500964	−	0.865468i	$-0.332979\pi$
$83$	9.12801	1.00193	0.500964	+	0.865468i	$0.332979\pi$
$84$	0	0
$85$	−3.76778	−0.408673
$86$	−14.6652	−1.58139
$87$	0	0
$88$	2.53590	0.270328
$89$	−9.14883	−0.969774	−0.484887	−	0.874577i	$-0.661139\pi$
$89$	−9.14883	−0.969774	−0.484887	+	0.874577i	$0.661139\pi$
$90$	0	0
$91$	0	0
$92$	−16.9997	−1.77234
$93$	0	0
$94$	−14.3199	−1.47699
$95$	7.91695	0.812262
$96$	0	0
$97$	−9.58288	−0.972994	−0.486497	−	0.873682i	$-0.661726\pi$
$97$	−9.58288	−0.972994	−0.486497	+	0.873682i	$0.661726\pi$
$98$	−17.9848	−1.81673
$99$	0	0
$100$	2.73205	0.273205
$101$	12.2223	1.21616	0.608082	−	0.793874i	$-0.291939\pi$
$101$	12.2223	1.21616	0.608082	+	0.793874i	$0.291939\pi$
$102$	0	0
$103$	14.7939	1.45769	0.728845	−	0.684679i	$-0.240058\pi$
$103$	14.7939	1.45769	0.728845	+	0.684679i	$0.240058\pi$
$104$	0	0
$105$	0	0
$106$	−12.0000	−1.16554
$107$	−13.9378	−1.34742	−0.673708	−	0.738998i	$-0.735299\pi$
$107$	−13.9378	−1.34742	−0.673708	+	0.738998i	$0.735299\pi$
$108$	0	0
$109$	15.4330	1.47822	0.739108	−	0.673587i	$-0.235247\pi$
$109$	15.4330	1.47822	0.739108	+	0.673587i	$0.235247\pi$
$110$	−3.46410	−0.330289
$111$	0	0
$112$	7.81476	0.738425
$113$	3.46410	0.325875	0.162938	−	0.986636i	$-0.447903\pi$
$113$	3.46410	0.325875	0.162938	+	0.986636i	$0.447903\pi$
$114$	0	0
$115$	6.22231	0.580233
$116$	13.7579	1.27739
$117$	0	0
$118$	9.41880	0.867071
$119$	−14.7221	−1.34958
$120$	0	0
$121$	−8.46410	−0.769464
$122$	21.1704	1.91668
$123$	0	0
$124$	0.505159	0.0453646
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	2.67718	0.237561	0.118781	−	0.992921i	$-0.462102\pi$
$127$	2.67718	0.237561	0.118781	+	0.992921i	$0.462102\pi$
$128$	11.8862	1.05060
$129$	0	0
$130$	0	0
$131$	12.5713	1.09836	0.549179	−	0.835705i	$-0.314940\pi$
$131$	12.5713	1.09836	0.549179	+	0.835705i	$0.314940\pi$
$132$	0	0
$133$	30.9345	2.68236
$134$	−13.4009	−1.15767
$135$	0	0
$136$	−6.00000	−0.514496
$137$	9.28419	0.793202	0.396601	−	0.917991i	$-0.370190\pi$
$137$	9.28419	0.793202	0.396601	+	0.917991i	$0.370190\pi$
$138$	0	0
$139$	4.51641	0.383077	0.191538	−	0.981485i	$-0.438652\pi$
$139$	4.51641	0.383077	0.191538	+	0.981485i	$0.438652\pi$
$140$	10.6752	0.902215
$141$	0	0
$142$	2.70496	0.226995
$143$	0	0
$144$	0	0
$145$	−5.03573	−0.418194
$146$	−4.91196	−0.406517
$147$	0	0
$148$	−4.50609	−0.370399
$149$	11.9431	0.978417	0.489209	−	0.872167i	$-0.337286\pi$
$149$	11.9431	0.978417	0.489209	+	0.872167i	$0.337286\pi$
$150$	0	0
$151$	−19.8363	−1.61426	−0.807129	−	0.590376i	$-0.798980\pi$
$151$	−19.8363	−1.61426	−0.807129	+	0.590376i	$0.798980\pi$
$152$	12.6074	1.02259
$153$	0	0
$154$	−13.5356	−1.09073
$155$	−0.184901	−0.0148516
$156$	0	0
$157$	15.7413	1.25629	0.628146	−	0.778096i	$-0.283814\pi$
$157$	15.7413	1.25629	0.628146	+	0.778096i	$0.283814\pi$
$158$	9.43963	0.750976
$159$	0	0
$160$	−7.53556	−0.595738
$161$	24.3129	1.91613
$162$	0	0
$163$	1.37148	0.107423	0.0537113	−	0.998557i	$-0.482895\pi$
$163$	1.37148	0.107423	0.0537113	+	0.998557i	$0.482895\pi$
$164$	28.3354	2.21262
$165$	0	0
$166$	−19.8564	−1.54116
$167$	−15.0295	−1.16301	−0.581507	−	0.813541i	$-0.697537\pi$
$167$	−15.0295	−1.16301	−0.581507	+	0.813541i	$0.697537\pi$
$168$	0	0
$169$	0	0
$170$	8.19615	0.628616
$171$	0	0
$172$	18.4185	1.40439
$173$	−8.96334	−0.681470	−0.340735	−	0.940159i	$-0.610676\pi$
$173$	−8.96334	−0.681470	−0.340735	+	0.940159i	$0.610676\pi$
$174$	0	0
$175$	−3.90738	−0.295370
$176$	3.18490	0.240071
$177$	0	0
$178$	19.9017	1.49170
$179$	−14.2054	−1.06176	−0.530880	−	0.847447i	$-0.678139\pi$
$179$	−14.2054	−1.06176	−0.530880	+	0.847447i	$0.678139\pi$
$180$	0	0
$181$	−22.7314	−1.68961	−0.844805	−	0.535075i	$-0.820283\pi$
$181$	−22.7314	−1.68961	−0.844805	+	0.535075i	$0.820283\pi$
$182$	0	0
$183$	0	0
$184$	9.90871	0.730480
$185$	1.64934	0.121262
$186$	0	0
$187$	−6.00000	−0.438763
$188$	17.9848	1.31167
$189$	0	0
$190$	−17.2220	−1.24941
$191$	−11.8654	−0.858549	−0.429275	−	0.903174i	$-0.641231\pi$
$191$	−11.8654	−0.858549	−0.429275	+	0.903174i	$0.641231\pi$
$192$	0	0
$193$	−9.25838	−0.666432	−0.333216	−	0.942850i	$-0.608134\pi$
$193$	−9.25838	−0.666432	−0.333216	+	0.942850i	$0.608134\pi$
$194$	20.8459	1.49665
$195$	0	0
$196$	22.5875	1.61339
$197$	−8.11769	−0.578361	−0.289181	−	0.957275i	$-0.593383\pi$
$197$	−8.11769	−0.578361	−0.289181	+	0.957275i	$0.593383\pi$
$198$	0	0
$199$	8.07111	0.572146	0.286073	−	0.958208i	$-0.407650\pi$
$199$	8.07111	0.572146	0.286073	+	0.958208i	$0.407650\pi$
$200$	−1.59245	−0.112603
$201$	0	0
$202$	−26.5875	−1.87069
$203$	−19.6765	−1.38102
$204$	0	0
$205$	−10.3715	−0.724376
$206$	−32.1817	−2.24220
$207$	0	0
$208$	0	0
$209$	12.6074	0.872069
$210$	0	0
$211$	4.58786	0.315842	0.157921	−	0.987452i	$-0.449521\pi$
$211$	4.58786	0.315842	0.157921	+	0.987452i	$0.449521\pi$
$212$	15.0711	1.03509
$213$	0	0
$214$	30.3192	2.07258
$215$	−6.74162	−0.459775
$216$	0	0
$217$	−0.722478	−0.0490450
$218$	−33.5719	−2.27377
$219$	0	0
$220$	4.35066	0.293321
$221$	0	0
$222$	0	0
$223$	26.3083	1.76174	0.880868	−	0.473362i	$-0.156960\pi$
$223$	26.3083	1.76174	0.880868	+	0.473362i	$0.156960\pi$
$224$	−29.4443	−1.96733
$225$	0	0
$226$	−7.53556	−0.501258
$227$	10.4499	0.693587	0.346794	−	0.937941i	$-0.387270\pi$
$227$	10.4499	0.693587	0.346794	+	0.937941i	$0.387270\pi$
$228$	0	0
$229$	22.0930	1.45995	0.729974	−	0.683475i	$-0.239532\pi$
$229$	22.0930	1.45995	0.729974	+	0.683475i	$0.239532\pi$
$230$	−13.5356	−0.892508
$231$	0	0
$232$	−8.01915	−0.526483
$233$	−8.75821	−0.573769	−0.286885	−	0.957965i	$-0.592620\pi$
$233$	−8.75821	−0.573769	−0.286885	+	0.957965i	$0.592620\pi$
$234$	0	0
$235$	−6.58288	−0.429420
$236$	−11.8293	−0.770023
$237$	0	0
$238$	32.0255	2.07590
$239$	6.31291	0.408348	0.204174	−	0.978935i	$-0.434549\pi$
$239$	6.31291	0.408348	0.204174	+	0.978935i	$0.434549\pi$
$240$	0	0
$241$	−21.1733	−1.36389	−0.681946	−	0.731402i	$-0.738866\pi$
$241$	−21.1733	−1.36389	−0.681946	+	0.731402i	$0.738866\pi$
$242$	18.4122	1.18358
$243$	0	0
$244$	−26.5885	−1.70215
$245$	−8.26761	−0.528198
$246$	0	0
$247$	0	0
$248$	−0.294445	−0.0186973
$249$	0	0
$250$	2.17533	0.137580
$251$	−26.8098	−1.69222	−0.846111	−	0.533007i	$-0.821062\pi$
$251$	−26.8098	−1.69222	−0.846111	+	0.533007i	$0.821062\pi$
$252$	0	0
$253$	9.90871	0.622956
$254$	−5.82374	−0.365414
$255$	0	0
$256$	−1.07180	−0.0669873
$257$	1.00957	0.0629754	0.0314877	−	0.999504i	$-0.489975\pi$
$257$	1.00957	0.0629754	0.0314877	+	0.999504i	$0.489975\pi$
$258$	0	0
$259$	6.44461	0.400449
$260$	0	0
$261$	0	0
$262$	−27.3467	−1.68948
$263$	25.0691	1.54583	0.772915	−	0.634510i	$-0.218798\pi$
$263$	25.0691	1.54583	0.772915	+	0.634510i	$0.218798\pi$
$264$	0	0
$265$	−5.51641	−0.338870
$266$	−67.2927	−4.12598
$267$	0	0
$268$	16.8306	1.02809
$269$	6.66983	0.406667	0.203333	−	0.979110i	$-0.434823\pi$
$269$	6.66983	0.406667	0.203333	+	0.979110i	$0.434823\pi$
$270$	0	0
$271$	−20.7840	−1.26254	−0.631270	−	0.775563i	$-0.717466\pi$
$271$	−20.7840	−1.26254	−0.631270	+	0.775563i	$0.717466\pi$
$272$	−7.53556	−0.456910
$273$	0	0
$274$	−20.1962	−1.22009
$275$	−1.59245	−0.0960284
$276$	0	0
$277$	31.4251	1.88815	0.944076	−	0.329727i	$-0.106957\pi$
$277$	31.4251	1.88815	0.944076	+	0.329727i	$0.106957\pi$
$278$	−9.82467	−0.589245
$279$	0	0
$280$	−6.22231	−0.371854
$281$	6.46277	0.385536	0.192768	−	0.981244i	$-0.438253\pi$
$281$	6.46277	0.385536	0.192768	+	0.981244i	$0.438253\pi$
$282$	0	0
$283$	−7.14094	−0.424485	−0.212242	−	0.977217i	$-0.568077\pi$
$283$	−7.14094	−0.424485	−0.212242	+	0.977217i	$0.568077\pi$
$284$	−3.39723	−0.201589
$285$	0	0
$286$	0	0
$287$	−40.5253	−2.39213
$288$	0	0
$289$	−2.80385	−0.164932
$290$	10.9544	0.643262
$291$	0	0
$292$	6.16906	0.361017
$293$	−7.63484	−0.446032	−0.223016	−	0.974815i	$-0.571590\pi$
$293$	−7.63484	−0.446032	−0.223016	+	0.974815i	$0.571590\pi$
$294$	0	0
$295$	4.32983	0.252092
$296$	2.62650	0.152662
$297$	0	0
$298$	−25.9802	−1.50499
$299$	0	0
$300$	0	0
$301$	−26.3421	−1.51833
$302$	43.1505	2.48303
$303$	0	0
$304$	15.8339	0.908137
$305$	9.73205	0.557256
$306$	0	0
$307$	−1.25137	−0.0714193	−0.0357097	−	0.999362i	$-0.511369\pi$
$307$	−1.25137	−0.0714193	−0.0357097	+	0.999362i	$0.511369\pi$
$308$	16.9997	0.968645
$309$	0	0
$310$	0.402220	0.0228446
$311$	−33.5782	−1.90405	−0.952023	−	0.306028i	$-0.901000\pi$
$311$	−33.5782	−1.90405	−0.952023	+	0.306028i	$0.901000\pi$
$312$	0	0
$313$	16.8462	0.952206	0.476103	−	0.879390i	$-0.342049\pi$
$313$	16.8462	0.952206	0.476103	+	0.879390i	$0.342049\pi$
$314$	−34.2424	−1.93241
$315$	0	0
$316$	−11.8555	−0.666922
$317$	21.0142	1.18028	0.590138	−	0.807302i	$-0.299073\pi$
$317$	21.0142	1.18028	0.590138	+	0.807302i	$0.299073\pi$
$318$	0	0
$319$	−8.01915	−0.448986
$320$	12.3923	0.692751
$321$	0	0
$322$	−52.8885	−2.94736
$323$	−29.8293	−1.65975
$324$	0	0
$325$	0	0
$326$	−2.98342	−0.165236
$327$	0	0
$328$	−16.5161	−0.911947
$329$	−25.7218	−1.41809
$330$	0	0
$331$	−13.8069	−0.758894	−0.379447	−	0.925214i	$-0.623886\pi$
$331$	−13.8069	−0.758894	−0.379447	+	0.925214i	$0.623886\pi$
$332$	24.9382	1.36866
$333$	0	0
$334$	32.6940	1.78894
$335$	−6.16043	−0.336580
$336$	0	0
$337$	14.4201	0.785512	0.392756	−	0.919643i	$-0.371522\pi$
$337$	14.4201	0.785512	0.392756	+	0.919643i	$0.371522\pi$
$338$	0	0
$339$	0	0
$340$	−10.2938	−0.558258
$341$	−0.294445	−0.0159451
$342$	0	0
$343$	−4.95302	−0.267438
$344$	−10.7357	−0.578830
$345$	0	0
$346$	19.4982	1.04823
$347$	−10.6788	−0.573268	−0.286634	−	0.958040i	$-0.592536\pi$
$347$	−10.6788	−0.573268	−0.286634	+	0.958040i	$0.592536\pi$
$348$	0	0
$349$	−18.4225	−0.986131	−0.493066	−	0.869992i	$-0.664124\pi$
$349$	−18.4225	−0.986131	−0.493066	+	0.869992i	$0.664124\pi$
$350$	8.49983	0.454335
$351$	0	0
$352$	−12.0000	−0.639602
$353$	−2.23297	−0.118849	−0.0594244	−	0.998233i	$-0.518927\pi$
$353$	−2.23297	−0.118849	−0.0594244	+	0.998233i	$0.518927\pi$
$354$	0	0
$355$	1.24347	0.0659967
$356$	−24.9951	−1.32474
$357$	0	0
$358$	30.9014	1.63319
$359$	14.7638	0.779203	0.389601	−	0.920984i	$-0.372613\pi$
$359$	14.7638	0.779203	0.389601	+	0.920984i	$0.372613\pi$
$360$	0	0
$361$	43.6781	2.29885
$362$	49.4482	2.59894
$363$	0	0
$364$	0	0
$365$	−2.25803	−0.118191
$366$	0	0
$367$	−9.68897	−0.505760	−0.252880	−	0.967498i	$-0.581378\pi$
$367$	−9.68897	−0.505760	−0.252880	+	0.967498i	$0.581378\pi$
$368$	12.4446	0.648720
$369$	0	0
$370$	−3.58786	−0.186524
$371$	−21.5547	−1.11906
$372$	0	0
$373$	−17.8389	−0.923663	−0.461831	−	0.886968i	$-0.652807\pi$
$373$	−17.8389	−0.923663	−0.461831	+	0.886968i	$0.652807\pi$
$374$	13.0520	0.674901
$375$	0	0
$376$	−10.4829	−0.540615
$377$	0	0
$378$	0	0
$379$	−5.90570	−0.303355	−0.151678	−	0.988430i	$-0.548468\pi$
$379$	−5.90570	−0.303355	−0.151678	+	0.988430i	$0.548468\pi$
$380$	21.6295	1.10957
$381$	0	0
$382$	25.8111	1.32061
$383$	26.6853	1.36356	0.681778	−	0.731559i	$-0.261207\pi$
$383$	26.6853	1.36356	0.681778	+	0.731559i	$0.261207\pi$
$384$	0	0
$385$	−6.22231	−0.317118
$386$	20.1400	1.02510
$387$	0	0
$388$	−26.1809	−1.32913
$389$	15.7770	0.799926	0.399963	−	0.916531i	$-0.369023\pi$
$389$	15.7770	0.799926	0.399963	+	0.916531i	$0.369023\pi$
$390$	0	0
$391$	−23.4443	−1.18563
$392$	−13.1658	−0.664971
$393$	0	0
$394$	17.6586	0.889629
$395$	4.33940	0.218339
$396$	0	0
$397$	−29.9660	−1.50395	−0.751975	−	0.659191i	$-0.770899\pi$
$397$	−29.9660	−1.50395	−0.751975	+	0.659191i	$0.770899\pi$
$398$	−17.5573	−0.880069
$399$	0	0
$400$	−2.00000	−0.100000
$401$	25.0936	1.25312	0.626558	−	0.779375i	$-0.284463\pi$
$401$	25.0936	1.25312	0.626558	+	0.779375i	$0.284463\pi$
$402$	0	0
$403$	0	0
$404$	33.3920	1.66131
$405$	0	0
$406$	42.8028	2.12427
$407$	2.62650	0.130191
$408$	0	0
$409$	26.4859	1.30964	0.654822	−	0.755783i	$-0.272743\pi$
$409$	26.4859	1.30964	0.654822	+	0.755783i	$0.272743\pi$
$410$	22.5614	1.11423
$411$	0	0
$412$	40.4178	1.99124
$413$	16.9183	0.832495
$414$	0	0
$415$	−9.12801	−0.448076
$416$	0	0
$417$	0	0
$418$	−27.4251	−1.34141
$419$	26.0426	1.27227	0.636133	−	0.771579i	$-0.280533\pi$
$419$	26.0426	1.27227	0.636133	+	0.771579i	$0.280533\pi$
$420$	0	0
$421$	−18.9577	−0.923940	−0.461970	−	0.886896i	$-0.652857\pi$
$421$	−18.9577	−0.923940	−0.461970	+	0.886896i	$0.652857\pi$
$422$	−9.98011	−0.485824
$423$	0	0
$424$	−8.78461	−0.426618
$425$	3.76778	0.182764
$426$	0	0
$427$	38.0268	1.84025
$428$	−38.0787	−1.84060
$429$	0	0
$430$	14.6652	0.707221
$431$	−20.8720	−1.00537	−0.502684	−	0.864470i	$-0.667654\pi$
$431$	−20.8720	−1.00537	−0.502684	+	0.864470i	$0.667654\pi$
$432$	0	0
$433$	−10.5707	−0.507995	−0.253998	−	0.967205i	$-0.581746\pi$
$433$	−10.5707	−0.507995	−0.253998	+	0.967205i	$0.581746\pi$
$434$	1.57163	0.0754405
$435$	0	0
$436$	42.1638	2.01928
$437$	49.2617	2.35651
$438$	0	0
$439$	−12.3586	−0.589841	−0.294921	−	0.955522i	$-0.595293\pi$
$439$	−12.3586	−0.589841	−0.294921	+	0.955522i	$0.595293\pi$
$440$	−2.53590	−0.120894
$441$	0	0
$442$	0	0
$443$	−17.3365	−0.823682	−0.411841	−	0.911256i	$-0.635114\pi$
$443$	−17.3365	−0.823682	−0.411841	+	0.911256i	$0.635114\pi$
$444$	0	0
$445$	9.14883	0.433696
$446$	−57.2292	−2.70988
$447$	0	0
$448$	48.4214	2.28770
$449$	−11.4026	−0.538123	−0.269062	−	0.963123i	$-0.586713\pi$
$449$	−11.4026	−0.538123	−0.269062	+	0.963123i	$0.586713\pi$
$450$	0	0
$451$	−16.5161	−0.777711
$452$	9.46410	0.445154
$453$	0	0
$454$	−22.7321	−1.06687
$455$	0	0
$456$	0	0
$457$	−2.29336	−0.107279	−0.0536394	−	0.998560i	$-0.517082\pi$
$457$	−2.29336	−0.107279	−0.0536394	+	0.998560i	$0.517082\pi$
$458$	−48.0596	−2.24568
$459$	0	0
$460$	16.9997	0.792613
$461$	18.8173	0.876410	0.438205	−	0.898875i	$-0.355614\pi$
$461$	18.8173	0.876410	0.438205	+	0.898875i	$0.355614\pi$
$462$	0	0
$463$	8.95234	0.416050	0.208025	−	0.978123i	$-0.433296\pi$
$463$	8.95234	0.416050	0.208025	+	0.978123i	$0.433296\pi$
$464$	−10.0715	−0.467556
$465$	0	0
$466$	19.0520	0.882565
$467$	−13.7750	−0.637433	−0.318716	−	0.947850i	$-0.603252\pi$
$467$	−13.7750	−0.637433	−0.318716	+	0.947850i	$0.603252\pi$
$468$	0	0
$469$	−24.0711	−1.11150
$470$	14.3199	0.660528
$471$	0	0
$472$	6.89504	0.317370
$473$	−10.7357	−0.493628
$474$	0	0
$475$	−7.91695	−0.363255
$476$	−40.2216	−1.84356
$477$	0	0
$478$	−13.7326	−0.628116
$479$	−19.2130	−0.877864	−0.438932	−	0.898520i	$-0.644643\pi$
$479$	−19.2130	−0.877864	−0.438932	+	0.898520i	$0.644643\pi$
$480$	0	0
$481$	0	0
$482$	46.0589	2.09792
$483$	0	0
$484$	−23.1244	−1.05111
$485$	9.58288	0.435136
$486$	0	0
$487$	−5.00331	−0.226722	−0.113361	−	0.993554i	$-0.536162\pi$
$487$	−5.00331	−0.226722	−0.113361	+	0.993554i	$0.536162\pi$
$488$	15.4978	0.701553
$489$	0	0
$490$	17.9848	0.812468
$491$	−17.0357	−0.768812	−0.384406	−	0.923164i	$-0.625594\pi$
$491$	−17.0357	−0.768812	−0.384406	+	0.923164i	$0.625594\pi$
$492$	0	0
$493$	18.9735	0.854524
$494$	0	0
$495$	0	0
$496$	−0.369802	−0.0166046
$497$	4.85872	0.217943
$498$	0	0
$499$	−12.1438	−0.543633	−0.271817	−	0.962349i	$-0.587624\pi$
$499$	−12.1438	−0.543633	−0.271817	+	0.962349i	$0.587624\pi$
$500$	−2.73205	−0.122181
$501$	0	0
$502$	58.3202	2.60296
$503$	18.2877	0.815408	0.407704	−	0.913114i	$-0.366330\pi$
$503$	18.2877	0.815408	0.407704	+	0.913114i	$0.366330\pi$
$504$	0	0
$505$	−12.2223	−0.543886
$506$	−21.5547	−0.958223
$507$	0	0
$508$	7.31418	0.324514
$509$	10.2938	0.456263	0.228131	−	0.973630i	$-0.426738\pi$
$509$	10.2938	0.456263	0.228131	+	0.973630i	$0.426738\pi$
$510$	0	0
$511$	−8.82299	−0.390306
$512$	−21.4409	−0.947564
$513$	0	0
$514$	−2.19615	−0.0968681
$515$	−14.7939	−0.651899
$516$	0	0
$517$	−10.4829	−0.461038
$518$	−14.0191	−0.615966
$519$	0	0
$520$	0	0
$521$	−15.5987	−0.683392	−0.341696	−	0.939811i	$-0.611001\pi$
$521$	−15.5987	−0.683392	−0.341696	+	0.939811i	$0.611001\pi$
$522$	0	0
$523$	−18.9084	−0.826805	−0.413403	−	0.910548i	$-0.635660\pi$
$523$	−18.9084	−0.826805	−0.413403	+	0.910548i	$0.635660\pi$
$524$	34.3454	1.50039
$525$	0	0
$526$	−54.5336	−2.37778
$527$	0.696665	0.0303472
$528$	0	0
$529$	15.7171	0.683352
$530$	12.0000	0.521247
$531$	0	0
$532$	84.5147	3.66418
$533$	0	0
$534$	0	0
$535$	13.9378	0.602583
$536$	−9.81017	−0.423735
$537$	0	0
$538$	−14.5091	−0.625530
$539$	−13.1658	−0.567089
$540$	0	0
$541$	30.4610	1.30962	0.654810	−	0.755793i	$-0.272749\pi$
$541$	30.4610	1.30962	0.654810	+	0.755793i	$0.272749\pi$
$542$	45.2120	1.94202
$543$	0	0
$544$	28.3923	1.21731
$545$	−15.4330	−0.661078
$546$	0	0
$547$	−39.1332	−1.67322	−0.836608	−	0.547801i	$-0.815465\pi$
$547$	−39.1332	−1.67322	−0.836608	+	0.547801i	$0.815465\pi$
$548$	25.3649	1.08353
$549$	0	0
$550$	3.46410	0.147710
$551$	−39.8676	−1.69842
$552$	0	0
$553$	16.9557	0.721029
$554$	−68.3599	−2.90434
$555$	0	0
$556$	12.3391	0.523293
$557$	15.8935	0.673428	0.336714	−	0.941607i	$-0.390684\pi$
$557$	15.8935	0.673428	0.336714	+	0.941607i	$0.390684\pi$
$558$	0	0
$559$	0	0
$560$	−7.81476	−0.330234
$561$	0	0
$562$	−14.0586	−0.593028
$563$	8.98051	0.378483	0.189242	−	0.981931i	$-0.439397\pi$
$563$	8.98051	0.378483	0.189242	+	0.981931i	$0.439397\pi$
$564$	0	0
$565$	−3.46410	−0.145736
$566$	15.5339	0.652938
$567$	0	0
$568$	1.98017	0.0830861
$569$	6.10145	0.255786	0.127893	−	0.991788i	$-0.459179\pi$
$569$	6.10145	0.255786	0.127893	+	0.991788i	$0.459179\pi$
$570$	0	0
$571$	−25.5346	−1.06859	−0.534295	−	0.845298i	$-0.679423\pi$
$571$	−25.5346	−1.06859	−0.534295	+	0.845298i	$0.679423\pi$
$572$	0	0
$573$	0	0
$574$	88.1558	3.67955
$575$	−6.22231	−0.259488
$576$	0	0
$577$	−25.7658	−1.07264	−0.536322	−	0.844013i	$-0.680187\pi$
$577$	−25.7658	−1.07264	−0.536322	+	0.844013i	$0.680187\pi$
$578$	6.09929	0.253697
$579$	0	0
$580$	−13.7579	−0.571264
$581$	−35.6666	−1.47970
$582$	0	0
$583$	−8.78461	−0.363821
$584$	−3.59581	−0.148796
$585$	0	0
$586$	16.6083	0.686082
$587$	35.8717	1.48058	0.740292	−	0.672285i	$-0.234687\pi$
$587$	35.8717	1.48058	0.740292	+	0.672285i	$0.234687\pi$
$588$	0	0
$589$	−1.46385	−0.0603169
$590$	−9.41880	−0.387766
$591$	0	0
$592$	3.29869	0.135575
$593$	−35.5276	−1.45894	−0.729471	−	0.684012i	$-0.760234\pi$
$593$	−35.5276	−1.45894	−0.729471	+	0.684012i	$0.760234\pi$
$594$	0	0
$595$	14.7221	0.603549
$596$	32.6292	1.33654
$597$	0	0
$598$	0	0
$599$	−20.7406	−0.847438	−0.423719	−	0.905794i	$-0.639276\pi$
$599$	−20.7406	−0.847438	−0.423719	+	0.905794i	$0.639276\pi$
$600$	0	0
$601$	1.16333	0.0474533	0.0237267	−	0.999718i	$-0.492447\pi$
$601$	1.16333	0.0474533	0.0237267	+	0.999718i	$0.492447\pi$
$602$	57.3027	2.33548
$603$	0	0
$604$	−54.1938	−2.20512
$605$	8.46410	0.344115
$606$	0	0
$607$	1.02650	0.0416642	0.0208321	−	0.999783i	$-0.493368\pi$
$607$	1.02650	0.0416642	0.0208321	+	0.999783i	$0.493368\pi$
$608$	−59.6586	−2.41948
$609$	0	0
$610$	−21.1704	−0.857164
$611$	0	0
$612$	0	0
$613$	19.2456	0.777321	0.388660	−	0.921381i	$-0.372938\pi$
$613$	19.2456	0.777321	0.388660	+	0.921381i	$0.372938\pi$
$614$	2.72214	0.109856
$615$	0	0
$616$	−9.90871	−0.399234
$617$	−20.8415	−0.839047	−0.419524	−	0.907744i	$-0.637803\pi$
$617$	−20.8415	−0.839047	−0.419524	+	0.907744i	$0.637803\pi$
$618$	0	0
$619$	−29.0909	−1.16926	−0.584630	−	0.811300i	$-0.698760\pi$
$619$	−29.0909	−1.16926	−0.584630	+	0.811300i	$0.698760\pi$
$620$	−0.505159	−0.0202877
$621$	0	0
$622$	73.0436	2.92878
$623$	35.7479	1.43221
$624$	0	0
$625$	1.00000	0.0400000
$626$	−36.6461	−1.46467
$627$	0	0
$628$	43.0060	1.71613
$629$	−6.21436	−0.247783
$630$	0	0
$631$	8.20672	0.326704	0.163352	−	0.986568i	$-0.447769\pi$
$631$	8.20672	0.326704	0.163352	+	0.986568i	$0.447769\pi$
$632$	6.91029	0.274876
$633$	0	0
$634$	−45.7128	−1.81549
$635$	−2.67718	−0.106241
$636$	0	0
$637$	0	0
$638$	17.4443	0.690625
$639$	0	0
$640$	−11.8862	−0.469844
$641$	13.7910	0.544713	0.272356	−	0.962196i	$-0.412197\pi$
$641$	13.7910	0.544713	0.272356	+	0.962196i	$0.412197\pi$
$642$	0	0
$643$	25.6507	1.01156	0.505782	−	0.862661i	$-0.331204\pi$
$643$	25.6507	1.01156	0.505782	+	0.862661i	$0.331204\pi$
$644$	66.4241	2.61748
$645$	0	0
$646$	64.8885	2.55301
$647$	0.170682	0.00671021	0.00335511	−	0.999994i	$-0.498932\pi$
$647$	0.170682	0.00671021	0.00335511	+	0.999994i	$0.498932\pi$
$648$	0	0
$649$	6.89504	0.270654
$650$	0	0
$651$	0	0
$652$	3.74695	0.146742
$653$	−22.5895	−0.883995	−0.441998	−	0.897016i	$-0.645730\pi$
$653$	−22.5895	−0.883995	−0.441998	+	0.897016i	$0.645730\pi$
$654$	0	0
$655$	−12.5713	−0.491201
$656$	−20.7430	−0.809877
$657$	0	0
$658$	55.9533	2.18129
$659$	6.41145	0.249755	0.124877	−	0.992172i	$-0.460146\pi$
$659$	6.41145	0.249755	0.124877	+	0.992172i	$0.460146\pi$
$660$	0	0
$661$	−12.9276	−0.502826	−0.251413	−	0.967880i	$-0.580895\pi$
$661$	−12.9276	−0.502826	−0.251413	+	0.967880i	$0.580895\pi$
$662$	30.0345	1.16732
$663$	0	0
$664$	−14.5359	−0.564102
$665$	−30.9345	−1.19959
$666$	0	0
$667$	−31.3338	−1.21325
$668$	−41.0613	−1.58871
$669$	0	0
$670$	13.4009	0.517724
$671$	15.4978	0.598286
$672$	0	0
$673$	1.21308	0.0467606	0.0233803	−	0.999727i	$-0.492557\pi$
$673$	1.21308	0.0467606	0.0233803	+	0.999727i	$0.492557\pi$
$674$	−31.3684	−1.20827
$675$	0	0
$676$	0	0
$677$	−40.7052	−1.56443	−0.782214	−	0.623010i	$-0.785910\pi$
$677$	−40.7052	−1.56443	−0.782214	+	0.623010i	$0.785910\pi$
$678$	0	0
$679$	37.4439	1.43697
$680$	6.00000	0.230089
$681$	0	0
$682$	0.640515	0.0245266
$683$	−38.4730	−1.47213	−0.736064	−	0.676912i	$-0.763318\pi$
$683$	−38.4730	−1.47213	−0.736064	+	0.676912i	$0.763318\pi$
$684$	0	0
$685$	−9.28419	−0.354731
$686$	10.7744	0.411370
$687$	0	0
$688$	−13.4832	−0.514044
$689$	0	0
$690$	0	0
$691$	−28.8579	−1.09781	−0.548903	−	0.835886i	$-0.684954\pi$
$691$	−28.8579	−1.09781	−0.548903	+	0.835886i	$0.684954\pi$
$692$	−24.4883	−0.930905
$693$	0	0
$694$	23.2299	0.881795
$695$	−4.51641	−0.171317
$696$	0	0
$697$	39.0774	1.48016
$698$	40.0749	1.51686
$699$	0	0
$700$	−10.6752	−0.403483
$701$	−19.2663	−0.727679	−0.363840	−	0.931462i	$-0.618534\pi$
$701$	−19.2663	−0.727679	−0.363840	+	0.931462i	$0.618534\pi$
$702$	0	0
$703$	13.0578	0.492484
$704$	19.7341	0.743758
$705$	0	0
$706$	4.85743	0.182812
$707$	−47.7572	−1.79609
$708$	0	0
$709$	9.47200	0.355728	0.177864	−	0.984055i	$-0.443081\pi$
$709$	9.47200	0.355728	0.177864	+	0.984055i	$0.443081\pi$
$710$	−2.70496	−0.101515
$711$	0	0
$712$	14.5691	0.545999
$713$	−1.15051	−0.0430869
$714$	0	0
$715$	0	0
$716$	−38.8098	−1.45039
$717$	0	0
$718$	−32.1161	−1.19856
$719$	−1.53846	−0.0573750	−0.0286875	−	0.999588i	$-0.509133\pi$
$719$	−1.53846	−0.0573750	−0.0286875	+	0.999588i	$0.509133\pi$
$720$	0	0
$721$	−57.8055	−2.15279
$722$	−95.0142	−3.53606
$723$	0	0
$724$	−62.1032	−2.30805
$725$	5.03573	0.187022
$726$	0	0
$727$	21.5454	0.799074	0.399537	−	0.916717i	$-0.369171\pi$
$727$	21.5454	0.799074	0.399537	+	0.916717i	$0.369171\pi$
$728$	0	0
$729$	0	0
$730$	4.91196	0.181800
$731$	25.4009	0.939488
$732$	0	0
$733$	4.51108	0.166621	0.0833103	−	0.996524i	$-0.473451\pi$
$733$	4.51108	0.166621	0.0833103	+	0.996524i	$0.473451\pi$
$734$	21.0767	0.777955
$735$	0	0
$736$	−46.8885	−1.72833
$737$	−9.81017	−0.361362
$738$	0	0
$739$	4.30901	0.158509	0.0792547	−	0.996854i	$-0.474746\pi$
$739$	4.30901	0.158509	0.0792547	+	0.996854i	$0.474746\pi$
$740$	4.50609	0.165647
$741$	0	0
$742$	46.8885	1.72133
$743$	24.4955	0.898653	0.449326	−	0.893368i	$-0.351664\pi$
$743$	24.4955	0.898653	0.449326	+	0.893368i	$0.351664\pi$
$744$	0	0
$745$	−11.9431	−0.437562
$746$	38.8054	1.42077
$747$	0	0
$748$	−16.3923	−0.599362
$749$	54.4602	1.98993
$750$	0	0
$751$	38.1432	1.39186	0.695932	−	0.718108i	$-0.254992\pi$
$751$	38.1432	1.39186	0.695932	+	0.718108i	$0.254992\pi$
$752$	−13.1658	−0.480106
$753$	0	0
$754$	0	0
$755$	19.8363	0.721918
$756$	0	0
$757$	−43.0576	−1.56496	−0.782478	−	0.622679i	$-0.786044\pi$
$757$	−43.0576	−1.56496	−0.782478	+	0.622679i	$0.786044\pi$
$758$	12.8468	0.466618
$759$	0	0
$760$	−12.6074	−0.457317
$761$	22.0377	0.798868	0.399434	−	0.916762i	$-0.369207\pi$
$761$	22.0377	0.798868	0.399434	+	0.916762i	$0.369207\pi$
$762$	0	0
$763$	−60.3027	−2.18310
$764$	−32.4168	−1.17280
$765$	0	0
$766$	−58.0493	−2.09741
$767$	0	0
$768$	0	0
$769$	−44.5991	−1.60829	−0.804143	−	0.594436i	$-0.797375\pi$
$769$	−44.5991	−1.60829	−0.804143	+	0.594436i	$0.797375\pi$
$770$	13.5356	0.487788
$771$	0	0
$772$	−25.2944	−0.910364
$773$	39.4820	1.42007	0.710035	−	0.704167i	$-0.248679\pi$
$773$	39.4820	1.42007	0.710035	+	0.704167i	$0.248679\pi$
$774$	0	0
$775$	0.184901	0.00664184
$776$	15.2603	0.547811
$777$	0	0
$778$	−34.3202	−1.23044
$779$	−82.1105	−2.94191
$780$	0	0
$781$	1.98017	0.0708560
$782$	50.9990	1.82372
$783$	0	0
$784$	−16.5352	−0.590543
$785$	−15.7413	−0.561830
$786$	0	0
$787$	−23.7221	−0.845603	−0.422801	−	0.906222i	$-0.638953\pi$
$787$	−23.7221	−0.845603	−0.422801	+	0.906222i	$0.638953\pi$
$788$	−22.1779	−0.790056
$789$	0	0
$790$	−9.43963	−0.335847
$791$	−13.5356	−0.481269
$792$	0	0
$793$	0	0
$794$	65.1859	2.31336
$795$	0	0
$796$	22.0507	0.781566
$797$	−13.9046	−0.492527	−0.246263	−	0.969203i	$-0.579203\pi$
$797$	−13.9046	−0.492527	−0.246263	+	0.969203i	$0.579203\pi$
$798$	0	0
$799$	24.8028	0.877461
$800$	7.53556	0.266422
$801$	0	0
$802$	−54.5868	−1.92753
$803$	−3.59581	−0.126893
$804$	0	0
$805$	−24.3129	−0.856917
$806$	0	0
$807$	0	0
$808$	−19.4634	−0.684721
$809$	19.6203	0.689815	0.344907	−	0.938637i	$-0.387910\pi$
$809$	19.6203	0.689815	0.344907	+	0.938637i	$0.387910\pi$
$810$	0	0
$811$	39.5653	1.38932	0.694662	−	0.719336i	$-0.255554\pi$
$811$	39.5653	1.38932	0.694662	+	0.719336i	$0.255554\pi$
$812$	−53.7572	−1.88651
$813$	0	0
$814$	−5.71350	−0.200258
$815$	−1.37148	−0.0480409
$816$	0	0
$817$	−53.3731	−1.86729
$818$	−57.6156	−2.01448
$819$	0	0
$820$	−28.3354	−0.989515
$821$	13.5772	0.473848	0.236924	−	0.971528i	$-0.423861\pi$
$821$	13.5772	0.473848	0.236924	+	0.971528i	$0.423861\pi$
$822$	0	0
$823$	−31.3721	−1.09356	−0.546782	−	0.837275i	$-0.684147\pi$
$823$	−31.3721	−1.09356	−0.546782	+	0.837275i	$0.684147\pi$
$824$	−23.5586	−0.820703
$825$	0	0
$826$	−36.8028	−1.28053
$827$	−28.8614	−1.00361	−0.501805	−	0.864981i	$-0.667330\pi$
$827$	−28.8614	−1.00361	−0.501805	+	0.864981i	$0.667330\pi$
$828$	0	0
$829$	−52.8165	−1.83439	−0.917196	−	0.398435i	$-0.869553\pi$
$829$	−52.8165	−1.83439	−0.917196	+	0.398435i	$0.869553\pi$
$830$	19.8564	0.689226
$831$	0	0
$832$	0	0
$833$	31.1505	1.07930
$834$	0	0
$835$	15.0295	0.520116
$836$	34.4439	1.19127
$837$	0	0
$838$	−56.6513	−1.95699
$839$	19.0367	0.657221	0.328610	−	0.944466i	$-0.393420\pi$
$839$	19.0367	0.657221	0.328610	+	0.944466i	$0.393420\pi$
$840$	0	0
$841$	−3.64145	−0.125567
$842$	41.2391	1.42119
$843$	0	0
$844$	12.5343	0.431448
$845$	0	0
$846$	0	0
$847$	33.0724	1.13638
$848$	−11.0328	−0.378869
$849$	0	0
$850$	−8.19615	−0.281126
$851$	10.2627	0.351802
$852$	0	0
$853$	−39.9748	−1.36871	−0.684355	−	0.729149i	$-0.739916\pi$
$853$	−39.9748	−1.36871	−0.684355	+	0.729149i	$0.739916\pi$
$854$	−82.7208	−2.83065
$855$	0	0
$856$	22.1952	0.758617
$857$	−21.0950	−0.720593	−0.360296	−	0.932838i	$-0.617324\pi$
$857$	−21.0950	−0.720593	−0.360296	+	0.932838i	$0.617324\pi$
$858$	0	0
$859$	−15.5693	−0.531218	−0.265609	−	0.964081i	$-0.585573\pi$
$859$	−15.5693	−0.531218	−0.265609	+	0.964081i	$0.585573\pi$
$860$	−18.4185	−0.628064
$861$	0	0
$862$	45.4034	1.54645
$863$	22.4771	0.765129	0.382565	−	0.923929i	$-0.375041\pi$
$863$	22.4771	0.765129	0.382565	+	0.923929i	$0.375041\pi$
$864$	0	0
$865$	8.96334	0.304763
$866$	22.9947	0.781392
$867$	0	0
$868$	−1.97385	−0.0669967
$869$	6.91029	0.234415
$870$	0	0
$871$	0	0
$872$	−24.5763	−0.832259
$873$	0	0
$874$	−107.160	−3.62475
$875$	3.90738	0.132094
$876$	0	0
$877$	−19.0115	−0.641973	−0.320986	−	0.947084i	$-0.604014\pi$
$877$	−19.0115	−0.641973	−0.320986	+	0.947084i	$0.604014\pi$
$878$	26.8839	0.907288
$879$	0	0
$880$	−3.18490	−0.107363
$881$	−7.96171	−0.268237	−0.134118	−	0.990965i	$-0.542820\pi$
$881$	−7.96171	−0.268237	−0.134118	+	0.990965i	$0.542820\pi$
$882$	0	0
$883$	40.5067	1.36316	0.681580	−	0.731743i	$-0.261293\pi$
$883$	40.5067	1.36316	0.681580	+	0.731743i	$0.261293\pi$
$884$	0	0
$885$	0	0
$886$	37.7126	1.26698
$887$	35.4614	1.19068	0.595339	−	0.803474i	$-0.297018\pi$
$887$	35.4614	1.19068	0.595339	+	0.803474i	$0.297018\pi$
$888$	0	0
$889$	−10.4607	−0.350842
$890$	−19.9017	−0.667107
$891$	0	0
$892$	71.8757	2.40658
$893$	−52.1163	−1.74401
$894$	0	0
$895$	14.2054	0.474834
$896$	−46.4439	−1.55158
$897$	0	0
$898$	24.8044	0.827735
$899$	0.931110	0.0310543
$900$	0	0
$901$	20.7846	0.692436
$902$	35.9279	1.19627
$903$	0	0
$904$	−5.51641	−0.183473
$905$	22.7314	0.755616
$906$	0	0
$907$	−44.0720	−1.46339	−0.731693	−	0.681635i	$-0.761269\pi$
$907$	−44.0720	−1.46339	−0.731693	+	0.681635i	$0.761269\pi$
$908$	28.5498	0.947458
$909$	0	0
$910$	0	0
$911$	11.9337	0.395380	0.197690	−	0.980265i	$-0.436656\pi$
$911$	11.9337	0.395380	0.197690	+	0.980265i	$0.436656\pi$
$912$	0	0
$913$	−14.5359	−0.481068
$914$	4.98881	0.165015
$915$	0	0
$916$	60.3593	1.99433
$917$	−49.1208	−1.62211
$918$	0	0
$919$	−19.0139	−0.627211	−0.313606	−	0.949553i	$-0.601537\pi$
$919$	−19.0139	−0.627211	−0.313606	+	0.949553i	$0.601537\pi$
$920$	−9.90871	−0.326681
$921$	0	0
$922$	−40.9338	−1.34808
$923$	0	0
$924$	0	0
$925$	−1.64934	−0.0542301
$926$	−19.4743	−0.639964
$927$	0	0
$928$	37.9470	1.24567
$929$	−38.9268	−1.27715	−0.638574	−	0.769560i	$-0.720475\pi$
$929$	−38.9268	−1.27715	−0.638574	+	0.769560i	$0.720475\pi$
$930$	0	0
$931$	−65.4542	−2.14518
$932$	−23.9279	−0.783783
$933$	0	0
$934$	29.9652	0.980492
$935$	6.00000	0.196221
$936$	0	0
$937$	−25.3721	−0.828871	−0.414436	−	0.910079i	$-0.636021\pi$
$937$	−25.3721	−0.828871	−0.414436	+	0.910079i	$0.636021\pi$
$938$	52.3626	1.70970
$939$	0	0
$940$	−17.9848	−0.586598
$941$	−43.6828	−1.42402	−0.712010	−	0.702170i	$-0.752215\pi$
$941$	−43.6828	−1.42402	−0.712010	+	0.702170i	$0.752215\pi$
$942$	0	0
$943$	−64.5345	−2.10153
$944$	8.65966	0.281848
$945$	0	0
$946$	23.3537	0.759293
$947$	45.4528	1.47702	0.738509	−	0.674244i	$-0.235530\pi$
$947$	45.4528	1.47702	0.738509	+	0.674244i	$0.235530\pi$
$948$	0	0
$949$	0	0
$950$	17.2220	0.558754
$951$	0	0
$952$	23.4443	0.759833
$953$	26.0297	0.843184	0.421592	−	0.906786i	$-0.361472\pi$
$953$	26.0297	0.843184	0.421592	+	0.906786i	$0.361472\pi$
$954$	0	0
$955$	11.8654	0.383955
$956$	17.2472	0.557814
$957$	0	0
$958$	41.7945	1.35032
$959$	−36.2768	−1.17144
$960$	0	0
$961$	−30.9658	−0.998897
$962$	0	0
$963$	0	0
$964$	−57.8466	−1.86311
$965$	9.25838	0.298038
$966$	0	0
$967$	0.640515	0.0205976	0.0102988	−	0.999947i	$-0.496722\pi$
$967$	0.640515	0.0205976	0.0102988	+	0.999947i	$0.496722\pi$
$968$	13.4787	0.433221
$969$	0	0
$970$	−20.8459	−0.669321
$971$	19.1837	0.615633	0.307817	−	0.951446i	$-0.400402\pi$
$971$	19.1837	0.615633	0.307817	+	0.951446i	$0.400402\pi$
$972$	0	0
$973$	−17.6473	−0.565747
$974$	10.8838	0.348741
$975$	0	0
$976$	19.4641	0.623031
$977$	−18.6100	−0.595387	−0.297694	−	0.954661i	$-0.596217\pi$
$977$	−18.6100	−0.595387	−0.297694	+	0.954661i	$0.596217\pi$
$978$	0	0
$979$	14.5691	0.465629
$980$	−22.5875	−0.721532
$981$	0	0
$982$	37.0583	1.18258
$983$	18.3850	0.586392	0.293196	−	0.956052i	$-0.405281\pi$
$983$	18.3850	0.586392	0.293196	+	0.956052i	$0.405281\pi$
$984$	0	0
$985$	8.11769	0.258651
$986$	−41.2736	−1.31442
$987$	0	0
$988$	0	0
$989$	−41.9485	−1.33388
$990$	0	0
$991$	−46.8691	−1.48885	−0.744424	−	0.667707i	$-0.767276\pi$
$991$	−46.8691	−1.48885	−0.744424	+	0.667707i	$0.767276\pi$
$992$	1.39333	0.0442383
$993$	0	0
$994$	−10.5693	−0.335238
$995$	−8.07111	−0.255871
$996$	0	0
$997$	−28.6625	−0.907750	−0.453875	−	0.891065i	$-0.649959\pi$
$997$	−28.6625	−0.907750	−0.453875	+	0.891065i	$0.649959\pi$
$998$	26.4168	0.836210
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7605.2.a.ch.1.1		4
3.2	odd	2		2535.2.a.bk.1.4		4
13.6	odd	12		585.2.bu.d.361.4		8
13.11	odd	12		585.2.bu.d.316.4		8
13.12	even	2		7605.2.a.ci.1.4		4
39.11	even	12		195.2.bb.b.121.1	✓	8
39.32	even	12		195.2.bb.b.166.1	yes	8
39.38	odd	2		2535.2.a.bj.1.1		4
195.32	odd	12		975.2.w.i.49.1		8
195.89	even	12		975.2.bc.j.901.4		8
195.128	odd	12		975.2.w.i.199.1		8
195.149	even	12		975.2.bc.j.751.4		8
195.167	odd	12		975.2.w.h.199.4		8
195.188	odd	12		975.2.w.h.49.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
195.2.bb.b.121.1	✓	8	39.11	even	12
195.2.bb.b.166.1	yes	8	39.32	even	12
585.2.bu.d.316.4		8	13.11	odd	12
585.2.bu.d.361.4		8	13.6	odd	12
975.2.w.h.49.4		8	195.188	odd	12
975.2.w.h.199.4		8	195.167	odd	12
975.2.w.i.49.1		8	195.32	odd	12
975.2.w.i.199.1		8	195.128	odd	12
975.2.bc.j.751.4		8	195.149	even	12
975.2.bc.j.901.4		8	195.89	even	12
2535.2.a.bj.1.1		4	39.38	odd	2
2535.2.a.bk.1.4		4	3.2	odd	2
7605.2.a.ch.1.1		4	1.1	even	1	trivial
7605.2.a.ci.1.4		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)

\(f(q)\)	\(=\)	\(q-2.17533 q^{2} +2.73205 q^{4} -1.00000 q^{5} -3.90738 q^{7} -1.59245 q^{8} +2.17533 q^{10} -1.59245 q^{11} +8.49983 q^{14} -2.00000 q^{16} +3.76778 q^{17} -7.91695 q^{19} -2.73205 q^{20} +3.46410 q^{22} -6.22231 q^{23} +1.00000 q^{25} -10.6752 q^{28} +5.03573 q^{29} +0.184901 q^{31} +7.53556 q^{32} -8.19615 q^{34} +3.90738 q^{35} -1.64934 q^{37} +17.2220 q^{38} +1.59245 q^{40} +10.3715 q^{41} +6.74162 q^{43} -4.35066 q^{44} +13.5356 q^{46} +6.58288 q^{47} +8.26761 q^{49} -2.17533 q^{50} +5.51641 q^{53} +1.59245 q^{55} +6.22231 q^{56} -10.9544 q^{58} -4.32983 q^{59} -9.73205 q^{61} -0.402220 q^{62} -12.3923 q^{64} +6.16043 q^{67} +10.2938 q^{68} -8.49983 q^{70} -1.24347 q^{71} +2.25803 q^{73} +3.58786 q^{74} -21.6295 q^{76} +6.22231 q^{77} -4.33940 q^{79} +2.00000 q^{80} -22.5614 q^{82} +9.12801 q^{83} -3.76778 q^{85} -14.6652 q^{86} +2.53590 q^{88} -9.14883 q^{89} -16.9997 q^{92} -14.3199 q^{94} +7.91695 q^{95} -9.58288 q^{97} -17.9848 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{4} - 4 q^{5} + 12 q^{14} - 8 q^{16} - 12 q^{19} - 4 q^{20} + 4 q^{25} - 12 q^{28} + 12 q^{29} - 12 q^{31} - 12 q^{34} - 24 q^{37} + 12 q^{41} + 16 q^{43} + 24 q^{46} + 24 q^{47} - 4 q^{49} - 12 q^{58}+ \cdots - 24 q^{98}+O(q^{100}) \) 4 * q + 4 * q^4 - 4 * q^5 + 12 * q^14 - 8 * q^16 - 12 * q^19 - 4 * q^20 + 4 * q^25 - 12 * q^28 + 12 * q^29 - 12 * q^31 - 12 * q^34 - 24 * q^37 + 12 * q^41 + 16 * q^43 + 24 * q^46 + 24 * q^47 - 4 * q^49 - 12 * q^58 - 12 * q^59 - 32 * q^61 - 8 * q^64 + 12 * q^67 - 12 * q^70 - 12 * q^71 - 24 * q^73 - 24 * q^74 - 24 * q^76 - 8 * q^79 + 8 * q^80 - 12 * q^82 + 12 * q^86 + 24 * q^88 + 12 * q^89 - 24 * q^92 - 12 * q^94 + 12 * q^95 - 36 * q^97 - 24 * q^98

Introduction

L-functions

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